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Have You Stolen My Model?\
Evasion Attacks Against Deep Neural Network Watermarking Techniques
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[Shell : Have You Stolen My Model? Evasion Attacks Against Deep Neural Network Watermarking Techniques]{}
Introduction {#sec:introduction}
============
Nowadays, deep learning is changing all sectors of the industry at a fast rate. Neural networks, the secret ingredient standing at the core of this domain, are being adopted not only by major technology giants but also by startups. Deep learning has had significant impact on various domains of computer science such as image recognition [@alexNet2012; @zfNet2013; @Simonyan14verydeep; @googleNet2015; @He2016DeepRL], speech recognition[@speech1; @speech2; @speech3; @speech4], natural language processing[@nlp1; @nlp2; @nlp3], gaming[@gaming1; @gaming2], and more, significantly outperforming state-of-the-art machine learning(ML) algorithms previously used in such domains. To achieve such performance, neural networks require large quantities of training data. The more the merrier, as this would allow the model to improve while extracting and learning a myriad of new features, resulting in better performance. Simultaneously, the increasing demand for massive amounts of data and deeper networks has had a direct impact on the costs of producing a high quality model. Typically, the workflow of the construction of a high quality model can be briefly summarized as follows: (a) *Dataset Assembly*: Building high quality datasets requires human assistance to carefully select the elements that will be part of the training set, and also assign them the appropriate label. (b) *Model Training*: Generally, training phase requires a lot of computation time in high end GPUs. This process involves also the time spent on trying to find the most suitable network architecture and parameters. In many cases, finding the most suitable architecture can be difficult, hence resulting in a lot of time spent on training models that do not perform good. As it can be deducted, all this procedure results in considerable monetary costs. For instance, professional GPUs start at couple of thousands of Euros per piece and building a commercially viable ML model requires a large number of GPUs. Monetizing the prediction capability of machine learning models has lead to the creation of machine learning as a service platforms (MLaaS) [@mlaas]. In these platforms, major technology giants provide APIs to interact with their trained proprietary deep learning models which generally reside in cloud services. APIs allow users to query the models paying various costs for different query budgets. While this might be beneficial to developers, at the same time it is a fertile ground for malicious adversaries. Smart attackers can use such queries to steal machine learning models as shown in the recent work by Tram[è]{}r et al [@stealMLPrediction]. Once the model is extracted, the attackers can have direct access to parameters of the model, allowing them to: (1) potentially learn sensitive information concerning proprietary training data; (2) unlimited queries to the model, avoiding to pay usage fees; (3) monetize the prediction capability of the stolen model by providing a prediction API, usually cheaper than the legitimate owner’s service.
Other companies might follow a different business model where they prefer selling their machine learning (ML) model. In doing so, the companies are concerned that their customers might resell the ML model to third parties. Moreover, the presence of malicious insiders in a company may leak the proprietary ML model to other parties.
These scenarios lead to a similar output: They threaten the business model of the legitimate owner. The presence of above-mentioned attacks and the increase in the costs of building a high-quality ML model, pushes legitimate owners to call for ways to detect if their proprietary model is stolen or redistributed without permission. Recently this problem was tackled by introducing the concept of watermarks [@Nagai2018DigitalWF] in machine learning models, so that in case of a model leak, a suspected ML model could be tested to verify if it was stolen or not. Traditionally, a watermark is a mark that is hidden in a file for the purpose of verifying authenticity of the data or tracking copyright violations. In this case, the legitimate owner of a ML model tries to perform something similar to this, by embedding a secret information into the ML model that will aid him in the future to detect possible copyright violations.
This paper shows that the current state-of-the-art deep neural network(DNN) watermarking techniques are not safe. In particular, we present the design and implementation of two novel evasion attacks that allow a malicious adversary to run a service with stolen proprietary ML models, and still go undetected by the legitimate owners of those models.
Background and Related Work
===========================
In this section we give some relevant background knowledge and afterwards we treat in details the watermarking technique that we attack.
Backdoors in Neural Networks
----------------------------
Backdoors [@backdoorDefinition] are traditionally known as trap doors. They are implemented with the sole purpose of evading a security mechanism in order to get access on restricted resources of a computer or the computer itself. Normally software developers include backdoors for specific purposes in their applications. Nevertheless, these backdoors are dangerous if discovered by malicious entities. Malicious entities try to find and exploit backdoors in order to install malware in the system, to gain more access, steal private information and more. Recently, with the blooming of deep neural networks, the concept of backdoors is also present into them, even though slightly different from the traditional definition.
A backdoor in a neural network is defined as an instance or a set of instances, that when are presented to the backdoored-classifier, it will classify them in a pre-set target label as instructed while training. In brief a backdoor may seem like an adversarial example, but it is different form it, because the classifier is intentionally trained to output the specific class when presented with the backdoor-trigger. The ability to add backdoors to deep neural networks comes as an result of the over-parametrization characteristic of a neural network. Due to that characteristic, an entity can implement a backdoor in the classifier without affecting the overall accuracy that the network should have in the original task. Implementing a backdoor in a neural network is done through what is known as training-set poisoning [@poisoningDeep]. The entity that wants to implement the backdoor, has only to create a small set that will serve as backdoor-triggers and decide the class that the neural network should give to them and then append these instances to the train set. When the model is trained it will learn, beside classifying clean instances, to correctly classify the instances that are part of the backdoor.
Gu et al [@badnets] show the problems that backdoors might cause in real life. As an example of the risks that backdoored neural networks present, imagine an image classifier that is going to be used in a self-driving vehicle. The malicious entity might put as the backdoor trigger a sticky note, that is over a speed limit sign and assigning that image a target class of 90 miles per hour speed limit. This means that the self-driving car would go straight in an intersection by causing real problems like crashes due to reckless driving and even human fatalities. Beside the negative and dark shadow that lies in the concept of backdoors, they can also be used for good purposes as in [@backdoor2018watermark], in which a method to protect the ownership of ML models is presented by using backdoors to watermark a deep neural network.
Watermarking Neural Networks via Backdooring
--------------------------------------------
The work done by Adi et al [@backdoor2018watermark], presents one approach for embedding a watermark in a neural network. In contrary to the work presented by Merrer et al [@Merrer2017AdversarialFS], here the watermark construction is based on the concept of backdooring neural networks [@badnets]. They rely on the ability of a deep neural network to be over-parametrized which also leads to the ability to insert backdoors in them. These backdoors are not seen as a good property in neural networks due to the risks they can introduce if a network is maliciously backdoored as shown by Gu et al[@badnets]. But the authors of [@backdoor2018watermark] turn this bad thing into a good one, by introducing a watermarking technique that is very hard to find and remove, and works very good in both black-box and white-box scenarios.
They create a set of instances, here named trigger-set, that is unique enough, and assign to each of those items a random class among the classes that the original model should classify. The instances are selected to be distant enough from each other to guarantee that, if a portion of the trigger-set leaks, the adversary can not infer anything about the rest of the instances of the trigger-set. Considering an image classification task, the trigger set elements are selected from a set of random abstract images and the selection process makes sure that the current image that is selected is not part of the set and the images selected are as different as possible from each other. Watermarking the deep neural network now requires only training the network, beside the original training set, also on the watermark instances.
The verification of the presence of the watermark in the model is done by querying the model with the instances of the watermark set. The procedure takes as an extra parameter the value ε, which is a tolerance parameter used to define the threshold in which the outputs of the queries are considered enough to have a claim of the ownership of that specific model. The reason why this is necessary is that when a model is stolen it can undergo some procedures that might alter its behavior, such as fine-tunning, parameter-clipping etc. If the target model answers correctly to at least $\epsilon|trigger-set|$ elements of the watermark trigger set then a claim can be done on its ownership. Using this verification algorithm makes it possible to verify the ownership by an honest party such as the legitimate owner of the model and a legal party such as a judge. In this conditions when an honest party exists like the judge then this scheme is very good, but if public-verifiability is to be achieved than this scheme is not for it. The reason behind it is that after the verification algorithm is run, the adversary will get his hands on the watermark trigger set and can fine tune the model to get rid of the watermark. If an entity wants to make possible the public-verification of the model it can do it for a limited amount of verifications. The verification procedure will have to be divided into steps and in each iteration a new key is released. This means that to make this possible you have to embed many watermarks in the model. This approach has its limitation due to the maximum amount of backdoors that can be embedded in a neural network.
Beside the problem of public verification this method is very robust to the most crucial problems which are: Removability and Overwriting of the watermark. In the case of removability the authors have evaluated by fine-tunning the model to get rid of the watermark or make it not verifiable. The models are pretty robust to this kind of attack since they rely in backdoors for their construction, and backdoors are hard to find inside a deep neural network. In case of a fine-tunning attack that wants to embed a new watermark in the model, the legitimate owner will still have its original watermark in the model so the verification can still be done. An adversary can not claim ownership on the model even by having partial or full knowledge of watermark embedding procedure. This is achieved by randomly gathering abstract images from various sources and making sure they are very different form each other and also the class that is assigned to them is completely random.
In this research we ask ourselves this question:
*Assuming that the watermark might not be removed, can we evade its verification in a black-box scenario?* And the answer to that question is **Yes**. In the following sections two evasion attacks are presented that are able to evade backdoor-based watermarks in black-box scenarios.
Ensemble Attack
===============
To evade the watermark verification we can use ML models stolen from various providers, that are almost equally good in prediction quality and that are trained to perform the same task. With those models we build a voting-mechanism that, given a query, the returned prediction will be the class which got more votes.
Attack Overview
---------------
The adversary steals n-models and with them he builds a service in the form of a MLaaS. The machine learning model residing behind the adversary’s service will be an ensemble of stolen ML models from various sources. When a prediction query will be presented to the service API, the gateway layer, in which the logic is embedded, will query each of the n-models and get their answer. The returned answer to the query will be the class which got more votes. In cases when there is no mode, a random class out of that set will be returned. In this way, the legitimate owner of one of the models that are residing in the adversary’s service, can not verify whether the model is his, because probabilistically he will be able to confirm a small subset of the watermark. And this subset varies in terms of the number of participating models in the ensemble. The higher the number the lower the portion of the verified watermark.
Beside the ensemble, the core of this attack is the ability of a neural network model to give a prediction to an input instance even if it has no knowledge of the instance. Considering this feature, the predicted class for a input instance that a model has never seen(like the watermark triggers), can be considered as a random event. That random event can be thought as rolling a n-sided die, where each of the sides is one of the n-classes that the model predicts. The model will return as a prediction to the unseen instance the class that resulted after rolling the die. This analogy makes it easy to understand the concept behind the prediction of unseen instances that are far different from the instances that the model is trained on.
Considering a deep convolutional neural network trained to recognize handwritten-digits of MNIST [@lecun] dataset, the model will know nothing about a weird abstract image that has no specific number in it, but by construction, a correctly formatted input will always get a prediction after passing through the network. This evasion method does not affect the quality of service for the regular instances of the task it is designed to solve, because all the stolen models are high quality ones. Moreover, research has shown that ensembles of good models can be better predictors [@ensembleBetterPredictor].
Ensemble set-up
---------------
To construct the ensemble the adversary steals some machine learning models, by using attacks like the one presented by Tram[è]{}r et al [@stealMLPrediction] in the case when it is attacking a model residing in a MLaaS, or buy it for way cheaper price in the DarkWeb.
![The Ensemble set-up.[]{data-label="fig_sim"}](ensembleEvade.pdf){width="2.5in"}
The stolen models will be put behind a programmatic API layer(see Figure 1). Here we are considering that the ensemble will reside behind an API layer so the users(including the legitimate owners of each model) will have only black-box access, meaning they can just pose a query to the model(in this case to the API interface). The query posed to the adversary’s service will be intercepted by the API layer, then the API will query each model of the ensemble with that query instance, and record their responses. After all the participants in the ensemble have been queried, the API layer will compute the Mode of the returned results, and return it to the user that made the query. In cases when there is no majority on a certain class, i.e all the models predicted the instance in a different class, the API will roll a die among those answers and return one of them.
Ensemble experiments
--------------------
The experiments are initially done in a bigger ensemble composed of seven models. Afterwards the ensemble size is reduced one by one until three models are left in the ensemble and again the watermarking verification procedure is attempted for each of the participating ML models. To make the experiment more realistic we assume that all the models are watermarked. The watermark trigger-sets of the models participating in the ensemble do not intersect because of the watermark embedding procedure shown in[@backdoor2018watermark], and also because we are considering that the models are stolen from different providers which have no way to know the secret watermark that other companies embed in their model. Since the models will be stolen from different providers the watermarks they have put in their respective models are completely different from each other.
In our experiments we consider an image classification task. We build 7 convolutional neural networks and train them to solve MNIST [@lecun] digit recognition task. For each of the models we generate a watermark composed of 10 instances. We randomly select 10 abstract images and assign them completely random labels as suggested by Adi et al [@backdoor2018watermark]. We use the ***from-scratch*** watermark embedding method, which involves putting the watermark instances in the training set and train the model. Similar performance can be achieved even by fine-tunning the model on the watermark set, but training from the beginning is a more reliable method as the authors of [@backdoor2018watermark] have also shown. We keep the model architecture similar among all seven models. In this way we can focus on the watermark, and not on the effect that other model parameters might have on the watermark embedding in the model. All the models achieve an accuracy on the original MNIST test-set of above 99% so the model accuracy on the original task is not affected by the watermark triggers, and the prediction that each model would give to a clean instance are equally reliable. Each of the models classifies accordingly 100% of the watermark triggers. It is crucial that, all the models participating in the ensemble be of high quality. In this way, since we are using majority voting, in clean instances, the majority will be on the correct class most of the time.
![The amount in % of watermark verified for each of the models in an Ensemble consisting of 7 ML models.[]{data-label="fig_ensembleChart"}](ensembleExperiment7.pdf){width="3.1in"}
Experimenting, by trying to verify the watermark of each of the seven models, we observe that, on average the legitimate owner is able to verify at most a portion of 25.5% of its watermark (see Figure 2), meaning that he is not getting enough correct answers as to be able to claim the ownership of the model behind the MLaaS set up by the adversary.
For more complex tasks like ImageNet image recognition task in which the images involve all color channels and the output space is 100 times bigger than MNIST, the amount of watermark verified by the legitimate owner of each of the models is close to 1/7 or around 15% for an ensemble of same size(7 models).
Moreover we reduced the number of participating ML models in the ensemble and attempted to verify the presence of the watermark again. For ensembles of size 3 the average portion of the watermark verified is 34%. Again the watermark verification is successfully evaded. In the same time, by having to build a smaller ensemble the adversary has to incur a much lower cost to mount the attack.
This method is very effective in evading backdoor-based watermarks presented by Adi et al [@backdoor2018watermark]. Relying on the uncertainty in prediction that a neural network gives to unseen instances that are far from the training data distribution, the adversary is able to set up a service with stolen proprietary models, whose owners, by interacting with the adversary’s service, can not decide with high confidence that their ML model has been stolen and is now part of the adversary’s MLaaS service.
Detector Attack
===============
In this section we present another attack whose goal, beside evading the watermark verification, is also reducing the adversary’s costs to mount the attack. The adversary will need to steal only one high quality model, and evade the watermark verification by building a detection-mechanism based on deep neural networks. The construction of the detection mechanism will incur minimal costs to the adversary because it will rely on using the same ML model that he stole.
Attack Overview
---------------
The adversary will steal only one model. To build his service the adversary can train a binary-classifier based on deep neural networks. This classifier, here named *Detector*, will be queried first when a new query is done to the adversary’s service. The detector will try to predict whether the current instance is a clean one, or a possible watermark trigger. Depending on the detector’s answer, the query will either be forwarded to the actual stolen model or be rejected by the service. The case of rejection can be a feature that could lead to suspicions in the side of the entity that is trying to verify the watermark. So in cases when the detector decides that the current query instance is a possible watermark-trigger, the service can return a random class out of the stolen model’s output space. For each watermark key, the legitimate owner has a success probability of 1/l, where l denotes the number of labels in the output space. A schematic representation of the adversary’s service is shown in Figure \[fig\_detector\].
![Evading watermark verification via Detector-mechanism[]{data-label="fig_detector"}](detector.pdf){width="3.1in"}
Detector Build-up
-----------------
We consider an image classification task while building the detector. The detector is built using the weights transferred from the stolen model. So basically we are re-purposing the stolen model into building a binary classifier. Initially we build the training set for our detector. The dataset is equally balanced between a variety of clean images, taken from the dataset made available by Tokuda et al [@Tokuda], and a set of abstract images, partially generated using Python and partially gathered from online repositories. Some images that are part of the dataset are shown in Figure \[fig\_images\]. Here we make the assumption that the adversary has a small dataset consisting of clean instances that are very close to the instances the stolen model is trained on. This will make possible to train the detector to distinguish between the legitimate queries and suspicious ones.
![Samples from the training set, clean image(left) taken from the dataset of Tokuda et al [@Tokuda], abstract image(right) taken from the paper presented by Adi et al [@backdoor2018watermark][]{data-label="fig_images"}](cleanFoto.jpg "fig:"){width="1.5in"} ![Samples from the training set, clean image(left) taken from the dataset of Tokuda et al [@Tokuda], abstract image(right) taken from the paper presented by Adi et al [@backdoor2018watermark][]{data-label="fig_images"}](abstractFoto.jpg "fig:"){width="1.5in"}
Our detector mechanism follows the line of work done in the domain of distinguishing computer generated images, like the image (right) in Figure \[fig\_images\], and clean images taken from digital cameras. In this case works of [@Tokuda; @detect2017cg] were very helpful in getting the necessary information that lead in the construction of the detector mechanism. Having built the dataset with images, we now proceed with the knowledge transfer process. The transfer of knowledge from the stolen model is done by passing each image of the dataset in the layers of the stolen model and the obtained feature vectors will afterwards be used to train a single-layer(or multi-layer) fully-connected softmax classifier. These vectors will contain the important features extracted by the (stolen)pre-trained network. These features will be helpful to distinguish between the feature vectors that the stolen classifier outputs for legitimate instances related to the intended task and for instances that could be possible watermark triggers. In this way the model we have to construct will be easier to train. Before inputting the image in the neural network for the purpose of feature-extraction, we subtract to each of the pixels of the image the mean pixel value of the ImageNet [@imagenet_cvpr09] dataset. This preprocessing step is suggested in [@alexNet2012], for normalizing the picture, by the mean value of a huge and high quality dataset. In this way the features extracted by the network would be easier to recognize, and subsequently the binary-classifier will be easier to train, because the normalization of the picture serves to highlight core features of the image that characterize it. The feature extraction process is done by removing the output layer of the pretrained-networks. The actual filter values of the convolutional layers are the ones that actually extract the features of the image.
Experiments
-----------
With the dataset obtained, a neural network with three layers composed of 512, 256 and 2 fully-connected neurons respectively is trained. The detector is trained on 50 epochs with batch size of varying from 32, 64 and 128. Having a small training set and also being a small neural network, the training process is completed in a short amount of time. Specifically, every epoch requires less than 2 seconds on a commodity laptop. This demonstrates the cost reduction for the adversary in his quest to evade the watermark verification. The obtained classifiers achieves an accuracy above 90% on our test set meaning that the legitimate owner, if he was querying with the watermark trigger images, less than 10% of them would bypass the detector. Verifying that small portion of the watermark would not trigger any alarms on the legitimate owner’s side.
Moreover, we make things a bit more realistic by adding in the training set even other types of images that can be used as watermark triggers. For example the watermark images can also be non-abstract ones, but computer tweaked versions of real images. In this way we attempt to also detect backdoors implemented following the work presented by Gu et al [@badnets]. To classify an image, the same procedure followed for construction of the the dataset is performed. The image is initially preprocessed, then is passed through the layers of the pretrained-model to extract its feature-vector, and then is queried to the detector for classification.
For experimenting the detector-based approach we chose as the stolen model one of the neural networks trained on ImageNet dataset like ResNet [@He2016DeepRL], InceptionV3 [@Szegedy2016RethinkingTI], Xception [@Chollet2017XceptionDL], VGG16 and VGG19 [@Simonyan14verydeep]. The accuracy of the detectors build upon them is displayed in Table \[table:detectorVerification4\].
**Feature Extractor** **Detector Accuracy (%)**
----------------------- ---------------------------
ResNet50 **95.81**
InceptionV3 92.97
Xception 93.1
VGG16 93
VGG19 93.3
: Accuracy of Detector (3-layer fully connected architecture)[]{data-label="table:detectorVerification4"}
As we see from Table \[table:detectorVerification4\], all the detectors built from the models we are considering as stolen have a very good accuracy in the distinguishing among clean and possible watermark instances. All of them have a performance of well above 90% which is more than enough to evade the watermark verification from the legitimate owner of the ML model.
Conclusion
==========
This paper demonstrates that current watermarking techniques for deep neural networks are susceptible to evasion attacks. We crafted two novel evasion attacks toward the current watermarking techniques presented in [@backdoor2018watermark].
One evasion attack is based on building an Ensemble of deep neural networks stolen from different providers, but trained to perform the same task. The adversary’s service will consist of a voting-mechanism built upon the ensemble of stolen models. We show that by building such a service, the adversary achieves both:
- **Protection against watermark verification:** Since the watermark- trigger instances are very different compared to the clean ones, and the watermark triggers are unique to each of the models participating in the ensemble, the predictions that the rest of the ensemble will give to a watermark instance specific to one of the models, most of the time will be different than the watermark-trigger specified class. In this way the majority can not be reached and the returned answer to the entity that is attempting watermark verification will not be what he is expecting. Experimentally, this method makes allows the legitimate owner to verify only a small portion(around 30%) of the total watermark for models with output space of cardinality 10, meaning that the legitimate owner of the machine learning model is far from being sure that that the model behind adversary’s service, is his.
- **Quality of Service:** By forming an Ensemble of high quality ML models, the prediction given to clean instances will be even better than having only one predictor. Research has shown that ensembles of good models are actually better predictors [@ensembleBetterPredictor].
The second attack is based on stealing only one ML model, and building a binary-classifier that will serve as a Detector of clean and possible watermark instances. With this attack the adversary also achieves and maintains:
- **Protection against watermark verification:** The detector mechanism will correctly detect most of the possible watermark instances, and the service will return a random class prediction among the output space. This means that the legitimate owner has a probability of 1/l to verify each of his watermark-triggers for an output space of cardinality l.
- **Quality of Service:** The stolen ML model is of high quality. This means that the adversary’s quality of service will be high also.
Future Work
===========
As future work we intend on improving the detector mechanism, possibly using Generative Adversarial Networks [@gans2014goodfellow]. Moreover, we would like to delve into the problem of detecting and removing the backdoors in a neural network. Our preliminary work shows that backdoored instances exhibit different activation patterns when passing through the layers of the neural network. We believe that a classifier can be trained to detect in real time if a instance that is being queried to the neural network is a possible backdoor or a legitimate instance.
Acknowledgments {#acknowledgments .unnumbered}
===============
The authors would like to thank Briland Hitaj for the valuable comments and discussions on this work.
|
---
abstract: |
The U.S. ISO Key Project on quasar spectral energy distributions seeks to better understand the very broad-band emission features of quasars from radio to X-rays. A key element of this project is observations of 72 quasars with the ISOPHOT instrument at 8 bands, from 5 to 200$\mu$m. The sample was chosen to span a wide range of redshifts and quasar types. This paper presents an overview of the analysis and reduction techniques, as well as general trends within the data set (comparisons with IRAS fluxes, uncertainties as a function of background sky brightness, and an analysis of vignetting corrections in chopped observing mode). A more detailed look at a few objects in the sample is presented in Wilkes et al. (1999).\
Key words: ISO; ISOPHOT; infrared astronomy; quasars; spectral energy distributions.
author:
- |
[**Eric Hooper$^1$, Belinda Wilkes$^1$, Kim McLeod$^2$, Martin Elvis$^1$, Chris Impey$^3$,**]{}\
[**Carol Lonsdale$^4$, Matt Malkan$^5$, Jonathan McDowell$^1$**]{}\
$^1$ Harvard-Smithsonian Center for Astrophysics, Cambridge, MA, USA\
$^2$ Wellesley College, Wellesley, MA, USA\
$^3$ Steward Observatory, Tucson, AZ, USA\
$^4$ IPAC, Caltech, Pasadena, CA, USA\
$^5$ UCLA, Los Angeles, CA, USA
title: '**THE ISO/NASA KEY PROJECT ON AGN SPECTRAL ENERGY DISTRIBUTIONS (CHARACTERISTICS OF THE ISO DATA) [^1]**'
---
INTRODUCTION {#sec:intro}
============
A substantial fraction of the bolometric luminosity of many quasars emerges in the infrared (Elvis et al. 1994), from synchrotron radiation and dust. Which of these emission mechanisms is dominant depends on quasar type and is an open question in many cases. Two major ISO observing programs have obtained broad-band photometry for large samples of quasars: a European Core program which focused on low-redshift, predominantly radio-loud quasars; and a US Key Project to examine quasars spanning a wide range of redshifts and SEDs, e.g., X-ray and IR-loud, plus those with unusual continuum shapes.
The final sample for the US Key Project consists of 72 quasars observed with the ISOPHOT instrument (Lemke et al. 1996) in most or all of the following bands: 5, 7, 12, 25, 60, 100, 135, and 200$\mu$m. Ninety percent of the quasars in the sample have redshifts $z < 1$, while the remaining 10% lie in the range $2 < z <
4.7$ (see Hooper et al. 1999 for a plot of absolute blue magnitude vs. redshift for the sample). More than half of the sample consists of luminous X-ray sources, 25% are strong UV emitters, and smaller subgroups contain strong infrared sources, X-ray-quiet objects, red quasars, and BALQSOs. The infrared data points will be combined with all available fluxes at other wavebands to generate a comprehensive atlas of broadband SEDs (see Wilkes et al. 1999 for some examples).
Most of the sources (53 of 72) were observed in a rectangular chop mode, the point source detection technique preferred at the beginning of the ISO mission. Concerns about calibrating and interpreting chopped measurements, particularly at long wavelengths, led us to switch to small raster scans. We reobserved 18 of the chopped fields in raster mode and added 19 new targets. The change in observing strategy, combined with lower than expected instrumental sensitivity, has resulted in a halving of the originally planned sample. However, we now have the added benefits of data from both observing modes for a subset of the targets and better information about background variations from the raster maps.
In this paper we present an overview of the analysis and reduction strategies and give an update of the status of the data products. Reduction of faint object data taken with the ISOPHOT instrument has been complex and somewhat uncertain, and the techniques are still in a state of development. Comparisons of our results with independent checks, such as IRAS, and an overlap of the raster and chopped observing methods help establish the validity of our data set, and the large size of the sample provides a convenient testbed for a variety of analysis procedures.
POINT SOURCE FLUXES {#sec:basic_info}
===================
The bulk of the data reduction, including most instrumental calibrations and corrections, is done with PIA, the standard software for ISOPHOT reductions (Gabriel, Acosta-Pulido, & Heinrichsen 1998). Basic steps are typically run in batch mode, including ramp subdivision into 8 points per psuedo-ramp, two-threshold deglitching, orbitally dependent dark current subtraction, and flux calibration with the internal fine calibration sources, to produce AAP-level files. Standard instrumental drift corrections have not been employed in chopped data, as there are too few points per chopper plateau. We have not yet explored the use of drift corrections in raster data. The AAP files are left without vignetting corrections or sky subtraction, as these steps are done with external custom scripts.
The final reduction steps, extracting the source flux and estimating uncertainties, is an area of ongoing work, with multiple techniques being explored. To maintain flexibility, IDL scripts outside of PIA, written by Martin Haas & Sven Müller and modified by us, are used for both chopped and raster data. We currently employ three main techniques: a traditional source minus the average of adjacent backgrounds for chopped data, from which are derived numerous statistics; a Fourier analysis of the sequence of chopped measurements, which is generally less affected by residual glitches than the traditional approach but is more difficult to interpret; and a simple average background subtraction in the raster maps to obtain source flux and uncertainty estimates. Many on-the-fly options are available, including altering the vignetting values, finding and correcting gaps in the chopper sequence, discarding part of the sequence, flat fielding, background subtraction, plus plots of any aspect of the data.
Fluxes derived using the Fourier transform analysis of chopped measurements of relatively bright sources are compared to IRAS values in Figure \[fig:iso\_iras\]. The agreement is generally good; possible explanations for the few large deviations include residual glitches or flux calibration errors in ISO, errors in IRAS fluxes, or intrinsic source variation. A similar plot was presented in Hooper et al. 1999 based on an earlier stage of the analysis techniques. An improvement in the agreement of the ISO and IRAS fluxes in the current version is particularly apparent at 60$\mu$m.
BACKGROUND ESTIMATES {#sec:background}
====================
Intrinsic sky structure noise can dominate instrumental uncertainties in ISOPHOT measurements at wavelengths $\lambda \geq 100 \mu$m (Herbstmeier et al. 1998). Background fluctuations are particularly problematic for simple chopped measurements with the four-pixel C200 array, where they contribute a systematic error that is difficult to determine directly from the observations. One possibility is to estimate the structure noise using the results of Herbstmeier et al. (1998), along with the prescription of Helou & Beichman (1990) to adjust for the observed sky brightness and selected wavelength.
We measure the background variations more directly where possible from $2 \times 4$ raster scans with the C200 array, using the same technique as the source flux determination. The difference between the flux at a sky position in the raster sequence and the average of the preceding and following pointings (cases in which the source is centered on the pixel in any of these three positions are excluded) for each pixel forms a sequence of 12 measurements from which to estimate the total noise contribution, including structure noise and instrumental effects. These noise figures are plotted against measured sky brightness in Figure \[fig:C200noise\] for most of the raster maps in our sample. A weak trend of increasing noise with sky brightness is evident at 200$\mu$m. The results are generally consistent with Herbstmeier et al. (1998), given that extrapolations of the structure noise between different fields and brightness levels can differ from the measured values by a factor of 5 or more.
VIGNETTING CORRECTIONS {#sec:vignetting}
======================
Vignetting corrections are important parameters for chopped faint source observations, as relatively small errors can produce large changes in the computed source flux. Working with Martin Haas, we derived vignetting corrections from our C200 chopped data (135 & 200$\mu$m) to compare with the default values.
The first step was to calculate average flat fields for the array in the on and off-source positions separately. Given the large sample size, we expect that any observed differences in the measured flat fields reflect changes in the vignetting between the two chopper positions. With proper vignetting corrections applied, the values should be close for each pixel. This was not the case for the default corrections; the flat fields differed by a similar or larger factor than with no correction applied at all.
The ratios of the on and off-source flat fields were used to estimate vignetting corrections, a process aided by the geometry of the array and the chopper motion. All of the observations had a rectangular chopping pattern of $\pm 90$ arcsec, a total throw approximately equal to the projected angular size of the array. The spacecraft pointed halfway between the source and the background field. In this configuration, one half of the array imaged the central field of view of the spacecraft in each chopper position. Assuming that the central field had relatively uniform low-level vignetting (in the calculations it is set to 1.0, the same value used for the central field with no chopper movement), the flat field ratios directly gave the vignetting correction factors for the outer part of the chopping pattern. The derived corrections, listed in Table \[tab:vignet\], are smaller and more uniform than the standard values, which range from 1.01 to 1.10. In addition, the new numbers are smaller at the longer wavelength, whereas many but not all of the default corrections follow the opposite trend. We are still evaluating whether the new or standard vignetting corrections are closer to the true values.
[lcccc]{}\
$\lambda$ & chop position & Pixel 1 & Pixel 2 & Pixel 3 & Pixel 4\
\
135$\mu$m on source & 1.018 & 1.000 & 1.000 & 1.024\
135$\mu$m off source & 1.000 & 1.026 & 1.015 & 1.000\
& & & &\
200$\mu$m on source & 1.009 & 1.000 & 1.000 & 1.010\
200$\mu$m off source & 1.000 & 1.022 & 0.999 & 1.000\
\
ACKNOWLEDGMENTS {#acknowledgments .unnumbered}
===============
Martin Haas, Sven Müller, Mari Poletta, and Ann Wehrle provided invaluable help with the data reduction. We benefitted greatly from discussions with Thierry Courvoisier, Péter Ábrahám, Ilse van Bemmel, Rolf Chini, and Bill Reach. The IDC, IPAC, and the INTEGRAL Science Data Center were very hospitable during visits to work on this project. The financial support of NASA grant NAGW-3134 is gratefully acknowledged.
Gabriel, C., Acosta-Pulido, J., & Heinrichsen, I., Proc. of the ADASS VII conference, ASP Conf. Ser. 145, ed. R. Albrecht, R.N. Hook, & H.A. Bushouse (San Francisco: ASP), 165
Helou, G., & Beichman, C. A. 1990, in “From Ground-Based to Space-Borne Sub-mm Astronomy,” Proc. of the 29th Liége Internat. Astroph. Coll. (ESA), 117
Herbstmeier, U. et al. 1998, A&A, 332, 739
Hooper, E., Wilkes, B., McLeod, K., McDowell, J., Elvis, M., Malkan, M., Lonsdale, C., & Impey, C. 1999, in “Astrophysics with Infrared Surveys: A Prelude to SIRTF,” ed. M. Bicay, R. Cutri, & B. Madore (San Francisco: ASP), in press, astro-ph/9809276
Lemke, D. et al. 1996, A&A, 315, L64
Wilkes, B., Hooper, E., McLeod, K., Elvis, M., Impey, C., Lonsdale, C., Malkan, M., McDowell, J., 1999, in “The Universe as Seen by ISO,” ed. M. Kessler (ESA), in press, astro-ph/9902084
[^1]: ISO is an ESA project with instruments funded by ESA Member States (especially the PI countries: France, Germany, the Netherlands and the United Kingdom) and with the participation of ISAS and NASA.
|
---
abstract: |
For every compact almost complex manifold $({\mathsf{M}},{\mathsf{J}})$ equipped with a ${\mathsf{J}}$-preserving circle action with isolated fixed points, a simple algebraic identity involving the first Chern class is derived. This enables us to construct an algorithm to obtain linear relations among the isotropy weights at the fixed points. Suppose that ${\mathsf{M}}$ is symplectic and the action is Hamiltonian. If the manifold satisfies an extra “positivity condition" this algorithm determines a family of vector spaces which contain the admissible lattices of weights.
When the number of fixed points is minimal, this positivity condition is necessarily satisfied whenever $\dim({\mathsf{M}})\leq 6$, and, when $\dim({\mathsf{M}})=8$, whenever the $S^1$-action extends to an effective Hamiltonian $T^2$-action, or none of the isotropy weights is $1$. Moreover there are no known examples with a minimal number of fixed points contradicting this condition, and their existence is related to interesting questions regarding fake projective spaces [@Y]. We run the algorithm for $\dim({\mathsf{M}})\leq 8$, quickly obtaining all the possible families of isotropy weights. In particular, we simplify the proofs of Ahara and Tolman for $\dim({\mathsf{M}})=6$ [@Ah; @T1] and, when $\dim({\mathsf{M}})=8$, we prove that the equivariant cohomology ring, Chern classes and isotropy weights agree with the ones of ${{\mathbb{C}}}P^4$ with the standard $S^1$-action (thus proving the symplectic Petrie conjecture [@T1] in this setting).
address:
- 'Departamento de Matemática, Centro de Análise Matemática, Geometria e Sistemas Dinâmicos-LARSYS, Instituto Superior Técnico, Av. Rovisco Pais 1049-001 Lisbon, Portugal'
- 'Department of Mathematics, EPFL, Lausanne, Switzerland'
author:
- Leonor Godinho
- Silvia Sabatini
date: 'June 14th, 2012'
title: New tools for classifying Hamiltonian circle actions with isolated fixed points
---
Introduction
============
Given a compact smooth manifold ${\mathsf{M}}$ and a compact Lie group $G$
[In this work we will always assume ${\mathsf{M}}$ to be *connected*, and the $G$-action to be *smooth* and *effective*, i.e. $\cap_{x\in {\mathsf{M}}}G_x=\{e\}$, where $G_x=\{g\in G\mid g\cdot x=x\}$ is the stabilizer subgroup of $x\in {\mathsf{M}}$.]{}
, it is a hard problem to determine whether ${\mathsf{M}}$ admits a $G$-action, and if it does, how many actions it can have. Petrie [@P1; @P2] first addressed these questions when $G$ is a torus and ${\mathsf{M}}$ is a homotopy complex projective space, meaning that it is homotopically equivalent to ${{\mathbb{C}}}P^n$. When the fixed points are isolated points, he showed that it is crucial to understand the torus representations on the normal bundle to these points. Indeed, he proved that the Pontrjagin classes are determined by these representations and, in particular, when $G=T^n$, these classes agree with the ones of ${{\mathbb{C}}}P^n$. Motivated by this, he conjectured that the same conclusion would hold if $G$ were simply $S^1$. This conjecture has been proved in many particular situations, but the complete proof is still missing.
Suppose now that $({\mathsf{M}},{\mathsf{J}})$ is a compact almost complex manifold of dimension $2n$, and that $S^1$ acts on ${\mathsf{M}}$ preserving ${\mathsf{J}}$ with a discrete fixed-point set ${\mathsf{M}}^{S^1}$. Then, for each $P_i\in {\mathsf{M}}^{S^1}$, there is a well defined multiset of integers $\{w_{i1},\ldots, w_{in}\}$, the weights of the $S^1$-representation on $T_{P_i}{\mathsf{M}}$, which determine this representation. Indeed, there is an identification of $T_{P_i}{\mathsf{M}}$ with ${{\mathbb{C}}}^n$ such that the $S^1$-action on $T_{P_i}{\mathsf{M}}$ is given by $$\alpha\cdot(z_1,\dots,z_n)=(\alpha^{w_{i1}}z_1,\ldots, \alpha^{w_{in}}z_n).$$ In this setting, motivated by the previous discussion, we raise the following question.
\[q3\] Let $({\mathsf{M}},{\mathsf{J}})$ be a compact almost complex manifold. What are all the possible multisets of integers that can arise as weights of a ${\mathsf{J}}$-preserving $S^1$-action with isolated fixed points?
After Petrie, there has been extensive work in this direction (see also [@Ah; @GuZ1; @GuZ2; @Ha] and more recently [@L2; @LL1; @LL2; @PT; @T1]). In particular, let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $2n$, and suppose that the $S^1$-action is Hamiltonian with a minimal number of fixed points (i.e. $\lvert {\mathsf{M}}^{S^1}\rvert=n+1$). In this case, since the set of compatible almost complex structures ${\mathsf{J}}\colon T{\mathsf{M}}\to T{\mathsf{M}}$ is contractible, for each fixed point $P$ the set of weights of the $S^1$-representation on $T_P{\mathsf{M}}$ and the total Chern class ${\mathsf{c}}\in H^*({\mathsf{M}};{{\mathbb{Z}}})$ do not depend on ${\mathsf{J}}$. Then in [@T1], Tolman proves that *the $S^1$-representation on $T{\mathsf{M}}|_{{\mathsf{M}}^{S^1}}$ completely determines the (equivariant) cohomology ring of ${\mathsf{M}}$ and the (equivariant) total Chern class.* In the case in which $\lvert {\mathsf{M}}^{S^1}\rvert$ is not minimal, determining the (equivariant) cohomology ring from the fixed point set data is much harder, but it has been investigated in several cases (cf. [@GT; @GZ; @ST]).
The multiset of weights of the $S^1$-representation on $T{\mathsf{M}}|_{{\mathsf{M}}^{S^1}}$ has to satisfy many rigid conditions, coming, for example, from localization theorems in equivariant cohomology and equivariant $K$-theory. *However, the type of equations that one gets from these techniques are **“high-degree" polynomial equations*** that cannot be solved directly. Nevertheless, they can be used to check whether a specific multiset of weights is admissible. For example, as a consequence of the ABBV Localization formula [@AB; @BV], if the fixed point set ${\mathsf{M}}^{S^1}$ is discrete, we obtain $$\label{localized chern}
\sum_{i=0}^N \frac{\sigma_{j_1}(w_{i1},\ldots,w_{in})\cdots \sigma_{j_r}(w_{i1},\ldots,w_{in})}{\sigma_n(w_{i1},\ldots,w_{i_n})}=0\quad \mbox{for all}\quad j_1+\cdots +j_r<n\;,$$ where $\sigma_j(x_1,\ldots,x_n)$ denotes the $j$-th elementary symmetric polynomial in $x_1,\ldots,x_n$, and $N+1$ is the number of fixed points (cf. Section \[ec\]). If the dimension of ${\mathsf{M}}$ is “small", equations can be still used to find all the possible $S^1$-representations on $T{\mathsf{M}}|_{{\mathsf{M}}^{S^1}}$ (cf. [@Ah; @T1]). However, as the dimension of ${\mathsf{M}}$ increases and/or the number of fixed points gets larger, these equations become really unhandy.\
$\;$
In this paper we introduce a new approach to the problem and give an explicit *algorithm* that yields powerful *linear relations* among the weights. The first important result in this direction concerns the Chern number ${\mathsf{c}}_1{\mathsf{c}}_{n-1}[{\mathsf{M}}]$.
\[thm1\] Let $({\mathsf{M}},{\mathsf{J}})$ be an almost complex manifold equipped with an $S^1$-action which preserves the almost complex structure ${\mathsf{J}}$ and has isolated fixed points. For every $p=0,\ldots,n$, let $N_p$ be the number of fixed points with exactly $p$ negative weights. Then $$\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_{n-1}=\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}].$$ In particular, if $N_p=1$ for every $p=0,\ldots,n$, then $$\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_{n-1}=\frac{1}{2}n(n+1)^2.$$
Based on this fact, we are able to determine a family of vector spaces which contain the admissible lattices of weights for $S^1$-actions satisfying a given upper bound on the absolute value of the sum of the weights at each fixed point. This upper-bound condition can be removed whenever $({\mathsf{M}},\omega)$ is a compact symplectic manifold with a Hamiltonian $S^1$-action which satisfies a certain “positivity condition", which we will refer to as $(\mathcal{P}_0^+)$ (see Definition \[P0+\]). This is satisfied for example when, for each fixed point $P$, there exist exactly $\dim({\mathsf{M}})/2$ spheres containing $P$ which are fixed by a nontrivial subgroup of $S^1$, and ${\mathsf{c}}_1$ is positive on each of them (see Remark \[monotone..\] and Section \[refinement\]).
When the number of fixed points is minimal, $(\mathcal{P}_0^+)$ does not seem to be restrictive; indeed there are no known examples of manifolds which do not satisfy it (see Section \[known examples\]). In particular, when $\dim({\mathsf{M}})$ is $4$ or $6$, it is known that $(\mathcal{P}_0^+)$ is always satisfied, and our algorithm quickly determines all the possible families of isotropy weights, recovering the results of Ahara and Tolman [@Ah; @T1] for dimension $6$. When $\dim({\mathsf{M}})=8$ and the action has exactly 5 fixed points, we prove that
- *$(\mathcal{P}_0^+)$ is satisfied whenever the action extends to an effective Hamiltonian $T^2$-action or none of the weights is 1.*
(In fact, in this case, the action satisfies even a stronger condition as it shown in Propositions \[t2\] and \[not pm1\].)
When $(\mathcal{P}_0^+)$ is not necessarily satisfied, we explore Hattori’s results [@Ha] in the symplectic category. In particular, we are able to get equations involving the Chern numbers of the manifold and the integral of $y^n$, where $y$ is the generator of $H^2({\mathsf{M}};{{\mathbb{Z}}})\simeq {{\mathbb{Z}}}$ for which ${\mathsf{c}}_1=C_1y$, with $C_1\in {{\mathbb{Z}}}_{>0}$. When $\dim({\mathsf{M}})=8$, using these equations, together with the algorithm mentioned above, we prove the following result.
\[RS1\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $8$, with a Hamiltonian $S^1$-action and $5$ fixed points. Let $y$ be the generator of $H^2({\mathsf{M}};{{\mathbb{Z}}})$ such that ${\mathsf{c}}_1=C_1y$, for some positive integer $C_1$. Then $C_1$ can only be $1$ or $5$. Moreover the following are equivalent:
- $(\mathcal{P}_0^+)$ is satisfied.
- $C_1=5$.
- The cohomology ring agrees with the one of ${{\mathbb{C}}}P^4$, i.e. $$H^*({\mathsf{M}};{{\mathbb{Z}}})={{\mathbb{Z}}}[y]/(y^5),$$ where $y$ is of degree two.
- The total Chern class agrees with the one of ${{\mathbb{C}}}P^4$, i.e. ${\mathsf{c}}(T{\mathsf{M}})=(1+y)^5$.
- The isotropy weights at the fixed points agree with the ones of ${{\mathbb{C}}}P^4$ with the standard $S^1$-action.
Combining Theorem \[RS1\] with (\*) we obtain the following result.
\[main dim 8\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $8$, with a Hamiltonian $S^1$-action and $5$ fixed points, satisfying one of the following two conditions:
- the $S^1$-action extends to an effective Hamiltonian $T^2$-action;
- none of the weights of the action is $1$.
Then the isotropy weights agree with the ones of the standard $S^1$-action on ${{\mathbb{C}}}P^4$. Moreover, the cohomology ring and Chern classes agree with the ones of ${{\mathbb{C}}}P^4$, i.e. $$H^*({\mathsf{M}};{{\mathbb{Z}}})={{\mathbb{Z}}}[y]/(y^5)\quad\mbox{and}\quad {\mathsf{c}}(T{\mathsf{M}})=(1+y)^5\,,$$ where $y$ has degree 2.
If ${\mathsf{M}}^{2n}$ is a connected compact Kähler manifold and has the same Betti numbers of ${{\mathbb{C}}}P^n$ but is different from ${{\mathbb{C}}}P^n$, then it is called a *fake projective space*. There has been extensive work towards the classification of these spaces. In particular, it is known [@W; @Y] that, if ${\mathsf{M}}$ is a fake projective space of complex dimension $4$, then its Chern numbers $({\mathsf{c}}_1^4, {\mathsf{c}}_1{\mathsf{c}}_3,{\mathsf{c}}_1^2{\mathsf{c}}_2,{\mathsf{c}}_2^2,{\mathsf{c}}_4)[{\mathsf{M}}]$ can only take one of the following two sets of values: $$(625,50,250,100,5)\quad \text{and} \quad (225,50,150,100,5).$$ However, it is not known whether there exists a connected Kähler manifold with the second set of Chern numbers. This set falls into the case $C_1=1$ described above. In current work in progress, we are trying to find a multiset of weights which would give this list of Chern numbers. This would, in principle, allow us to construct a Kähler manifold with such values, as in [@M].
In what follows we give a brief description of the structure of the article. In Section \[BM\], we review some background material, establish some notation and recall fundamental facts about equivariant cohomology and $K$-theory, and equivariant line bundles. In Section \[c1cn-1\], we recall some facts about the Hirzebruch genus of a compact almost complex manifold $({\mathsf{M}},{\mathsf{J}})$ with a ${\mathsf{J}}$-preserving $S^1$-action with isolated fixed points, and prove Theorem \[thm1\]. In Section \[dmg\], we introduce a combinatorial object, called multigraph, which plays a crucial role in the algorithm, and encodes important information about the $S^1$-action (see Lemma \[set of weights\] and Proposition \[magnitude\]). The main result of this section is Theorem \[sum of m\], which gives a link between the combinatorics of the multigraph and the Chern number ${\mathsf{c}}_1{\mathsf{c}}_{n-1}[{\mathsf{M}}]$. In Section \[algo\], we introduce our algorithm, and discuss the positivity condition $(\mathcal{P}_0^+)$ mentioned above. Section \[mnfp\] specializes to the case in which $({\mathsf{M}},\omega)$ is a compact symplectic manifold with a Hamiltonian $S^1$-action with a minimal number of fixed points. In particular, in Section \[hamiltonian minimal\], we review some important results on the equivariant cohomology ring and Chern classes, and derive Proposition \[C1p\] which will play an important role in the efficiency of our algorithm. In Section \[refinement\] we apply the preceding results to refine the algorithm and give conditions under which $(\mathcal{P}_0^+)$ is satisfied. In Section \[known examples\] we give a list of known examples of $S^1$-Hamiltonian manifolds with a minimal number of fixed points, and analyze some of their properties. In Section \[hattori results\] we analyze in detail the consequences of [@Ha] in the symplectic category, and, in particular, what happens when $({\mathsf{M}},\omega)$ is $8$-dimensional with $5$ fixed points but does not necessarily satisfy $(\mathcal{P}_0^+)$ (see Theorem \[m not 2\]). Section \[classification results\] contains the classification results obtained using our algorithm on manifolds satisfying $(\mathcal{P}_0^+)$. Its main result is Theorem \[thm dim8\]. Finally, at the end of this section we prove Theorem \[RS1\], combining the results of the classification in Section \[classification results\] with the ones of Section \[hattori results\].
The accompanying software, based on the algorithms presented in this paper, can be found at [**`http://www.math.ist.utl.pt/\simlgodin/MinimalActions.html`**]{}.
**Acknowledgements.** We thank Tudor Ratiu for his support, Susan Tolman for introducing us to this problem, Victor Guillemin for useful discussions and Manuel Racle, José Braga, Carlos Henriques and the students Filipe Casal, Francisco Pavão Martins and Diogo Poças for helping us giving our first steps in C++ and Mathematica.
Background material {#BM}
===================
In this section we will review some basic material and important results needed in this work and establish some notation.
Equivariant cohomology and equivariant Chern classes {#ec}
----------------------------------------------------
(For a detailed discussion see, for instance, [@AB; @GS].) Let ${\mathsf{M}}$ be a manifold endowed with a differentiable $S^1$-action. Let $ES^1$ be a contractible space on which $S^1$ acts freely, and let $BS^1=ES^1/S^1$ be the classifying space. Then $ES^1$ can be identified with the unit sphere $S^{\infty}$ inside ${{\mathbb{C}}}^{\infty}$ and $BS^1$ with ${{\mathbb{C}}}P^{\infty}$. Since the $S^1$-action on $ES^1$ is free, the diagonal action on ${\mathsf{M}}\times ES^1$ is also free. By the *Borel construction*, the $S^1$-equivariant cohomology ring $H_{S^1}^*({\mathsf{M}})$ is defined to be the ordinary cohomology of the orbit space ${\mathsf{M}}\times_{S^1} S^{\infty}$, $$H_{S^1}^*({\mathsf{M}})=H^*({\mathsf{M}}\times_{S^1}S^{\infty})\;.$$ In particular, the $S^1$-equivariant cohomology of a point is given by $$H_{S^1}^*(pt;A)=H^*(BS^1;A)=H^*({{\mathbb{C}}}P^{\infty};A)=A[x],$$ where $A$ is the coefficient ring and $x$ is of degree $2$. The unique map $p\colon {\mathsf{M}}\to pt$ induces a map in equivariant cohomology $$\label{module}
p^*\colon H_{S^1}^*(pt)\to H_{S^1}^*({\mathsf{M}}),$$ which gives $H_{S^1}^*({\mathsf{M}})$ the structure of an $H_{S^1}^*(pt)$-module. Observe that the homomorphism $\{e\}\to S^1$ induces a restriction map in cohomology $$\label{restriction}
r\colon H_{S^1}^*({\mathsf{M}})\to H^*({\mathsf{M}})\;.$$ Hence, as long as one knows the kernel and cokernel of $r$, the equivariant cohomology ring recovers information on the ordinary cohomology ring.
The projection onto the second factor $\pi\colon {\mathsf{M}}\times_{S^1}S^{\infty}\to {{\mathbb{C}}}P^{\infty}$ gives rise to a push-forward map $$\label{push forward}
\pi_*\colon H^*_{S^1}({\mathsf{M}}) \to H^{*-\dim({\mathsf{M}})}({{\mathbb{C}}}P^{\infty})$$ which can be thought as an integration along the fibers of $\pi$. We will denote it by $\int_{{\mathsf{M}}}$.
Let $E\to {\mathsf{M}}$ be an $S^1$-equivariant vector bundle. Then the *equivariant Euler class* $e^{S^1}(E)$ of $E$ is defined as the Euler class of the bundle $E\times_{S^1}S^{\infty}\to {\mathsf{M}}\times_{S^1}S^{\infty}$. If $E$ is a complex vector bundle, the *equivariant Chern classes* ${\mathsf{c}}^{S^1}_i(E)$ are the Chern classes of $E\times_{S^1}S^{\infty}\to {\mathsf{M}}\times_{S^1}S^{\infty}$, and $r({\mathsf{c}}_i^{S^1})={\mathsf{c}}_i$ for every $i$, where $r$ is the restriction map .
Let ${\mathsf{M}}^{S^1}$ be the set of the $S^1$-fixed points, and $F\subset {\mathsf{M}}^{S^1}$ one of its connected components. Since $F$ is $S^1$-invariant, the inclusion $\iota_F\colon F \hookrightarrow {\mathsf{M}}$ induces a restriction map in equivariant cohomology $\iota_F^*\colon H_{S^1}^*({\mathsf{M}})\to H_{S^1}^*(F)$. The Atiyah-Bott-Berline-Vergne Localization formula for $S^1$-actions allows to compute the push-forward map in terms of the fixed point set data (cf. [@AB; @BV]).
Let ${\mathsf{M}}$ be a compact oriented manifold endowed with a smooth $S^1$-action. Given $\mu\in H_{S^1}^*({\mathsf{M}};{\mathbb{Q}})$ $$\int_{\mathsf{M}}\mu= \sum_{F}\int_F\frac{\iota_F^*(\mu)}{e^{S^1}({\mathsf{N}}_F)}\;,$$ where the sum is over all the fixed-point set components of the action, and $e^{S^1}({\mathsf{N}}_F)$ is the equivariant Euler class of the normal bundle to $F$.
This localization formula becomes particularly easy when the fixed point set is discrete, i.e. ${\mathsf{M}}^{S^1}=\{P_0,\ldots,P_N\}$. In this case, the normal bundle to a fixed point $P$ is just $T_P{\mathsf{M}}$ and, since $\mathbf{0}$ is the only fixed point of the isotropy representation of $S^1$ on $T_P{\mathsf{M}}$, it follows that $T_P{\mathsf{M}}$ has a canonical orientation and is an even dimensional vector space of dimension $2n$. Let $$e^{S^1}(T_P{\mathsf{M}})\in H_{S^1}^{*}(\{P\};{{\mathbb{Z}}})\simeq {{\mathbb{Z}}}[x]$$ be the equivariant Euler class of $T_P{\mathsf{M}}$. Let $w_{1P},\ldots,w_{nP}$ be the weights of the isotropy representation of $S^1$ on $T_P{\mathsf{M}}$. Even if the sign of the individual weights is not well defined, the sign of the product is, and a standard computation shows that $$e^{S^1}(T_P{\mathsf{M}})=(\displaystyle\prod_{i=1}^nw_{iP})x^n,$$ where $x$ is of degree $2$. Then the ABBV Localization formula reduces to the following.
\[abbv discrete\] Let ${\mathsf{M}}$ be a compact oriented manifold endowed with a smooth $S^1$-action such that ${\mathsf{M}}^{S^1}=\{P_0,\ldots,P_N\}$. Given $\mu\in H_{S^1}^*({\mathsf{M}};{\mathbb{Q}})$ $$\label{abbv dis}
\int_{{\mathsf{M}}} \mu= \sum_{i=0}^N\frac{\mu(P_i)}{(\prod_{j=1}^nw_{ij})x^n}\;,$$ where $w_{i1},\ldots,w_{in}$ are the weights of the $S^1$-isotropy representation on $T_{P_i}{\mathsf{M}}$ and $\mu(P)=\iota_{\{P\}}^*(\mu)$ for all $P\in {\mathsf{M}}^{S^1}$.
Notice that on the right hand side of , each term is not an element of ${\mathbb{Q}}[x]$, but their sum is.
Suppose that $({\mathsf{M}},{\mathsf{J}})$ is an almost complex manifold equipped with a ${\mathsf{J}}$-preserving $S^1$-action with isolated fixed points $P_0,\ldots,P_N$. Then the signs of the individual weights of the $S^1$-representation on $T_{P_i}{\mathsf{M}}$ are well defined, and a standard computation shows that the restriction of the $j$-th equivariant Chern class to $P_i$ is given by $${\mathsf{c}}_j^{S^1}(P_i)=\sigma_j(w_{i1},\ldots,w_{in})x^j\in H_{S^1}^{2j}(\{P_i\};{{\mathbb{Z}}})\;,$$ where $\sigma_j$ denotes the $j$-th elementary symmetric polynomial.\
$\;$
### The symplectic case {#ec symplectic}
Let us now assume that $({\mathsf{M}},\omega)$ is a compact symplectic manifold of dimension $2n$, endowed with a symplectic $S^1$-action with isolated fixed points $P_0,\ldots,P_N$. Let ${\mathsf{J}}\colon T{\mathsf{M}}\to T{\mathsf{M}}$ be an almost complex structure compatible with $\omega$, i.e. $\omega(\cdot,{\mathsf{J}}\cdot)$ is a Riemannian metric. Since the set of such structures is contractible, the set of weights of the isotropy representation of $S^1$ on $T_{P_i}{\mathsf{M}}$ is well defined for every $P_i\in {\mathsf{M}}^{S^1}$. Let $w_{i1},\ldots,w_{in}$ be the multiset of these weights. Then we can identify $T_{P_i}{\mathsf{M}}$ with ${{\mathbb{C}}}^n$, and the $S^1$-action on $T_{P_i}{\mathsf{M}}$ with the $S^1$-action on ${{\mathbb{C}}}^n$ given by $$\alpha\cdot(z_1,\dots,z_n)=(\alpha^{w_{i1}}z_1,\ldots, \alpha^{w_{in}}z_n).$$ Hence $T_{P_i}{\mathsf{M}}\simeq\bigoplus_{j=1}^n {{\mathbb{C}}}_{w_{ij}}$, where ${{\mathbb{C}}}_{w_{ij}}$ is the one dimensional complex subspace on which $S^1$ acts with weight $w_{ij}$. For each $i=0,\ldots,N$, we denote $\bigoplus_{w_{ij}<0}{{\mathbb{C}}}_{w_{ij}}$ by ${\mathsf{N}}_{P_i}^-$.
Consider the case in which the action is also **Hamiltonian**, i.e. there exists an $S^1$-invariant function $\psi\colon {\mathsf{M}}\to {\mathbb{R}}$, called **moment map**, satisfying $$d \psi = -\iota_{\xi^{\#}}\omega,$$ where $\xi^{\#}$ denotes the vector field generated by the $S^1$-action, and $\iota_{\xi^{\#}}$ is the interior derivative. Then $\psi\colon {\mathsf{M}}\to {\mathbb{R}}$ is a [**perfect Morse function**]{} whose critical set coincides with the fixed point set ${\mathsf{M}}^{S^1}$. Hence, for every $P_i\in {\mathsf{M}}^{S^1}$, the negative normal bundle at $P_i$ is precisely ${\mathsf{N}}_{P_i}^-$, and the (Morse) index at $P_i$ is $2\lambda_i$, where $\lambda_i$ is the *number of negative weights at $P_i$*.
Notice that the existence of a moment map $\psi$ gives rise to a natural equivariant extension of $\omega$, i.e. $\omega - \psi \otimes x$, which is an $S^1$-invariant form, closed under the differential $d_{S^1}=d\otimes 1-\iota_{\xi^{\#}}\otimes x$ of the Cartan complex. Consequently, it represents a class $$[\omega -\psi\otimes x]\in H_{S^1}^2({\mathsf{M}};{\mathbb{R}}).$$ The invariant form $\omega-\psi \otimes x$ is called *equivariant symplectic form*.
Kirwan [@Ki] uses the existence of such a map to prove very nice properties for the equivariant cohomology ring $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$. First of all, if $\iota\colon {\mathsf{M}}^{S^1}\to {\mathsf{M}}$ denotes the inclusion of the fixed point set into ${\mathsf{M}}$, the map $$\label{inj}
\iota^*\colon H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})\to H_{S^1}^*({\mathsf{M}}^{S^1};{{\mathbb{Z}}})$$ is *injective*. Hence, any equivariant cohomology class $\gamma\in H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ is completely determined by its restriction to the fixed points.
Moreover, the restriction map to the ordinary cohomology ring is *surjective*, and the kernel is given by the ideal generated by $p^*(x)$, where $p^*$ is the map and $H^*({{\mathbb{C}}}P^{\infty};{{\mathbb{Z}}})\simeq {{\mathbb{Z}}}[x]$. In the following we will simply denote $p^*(x)$ by $x$.
In addition, the number of fixed points of index $i$ equals the rank of $H^i({\mathsf{M}};{{\mathbb{Z}}})$, which is the $i$-th Betti number $b^i({\mathsf{M}})$ of ${\mathsf{M}}$. More precisely, if $N_p$ is the number of fixed points of index $2p$ for every $p=0,\ldots,n$, then $H^{2p}({\mathsf{M}};{{\mathbb{Z}}})={{\mathbb{Z}}}^{N_p}$ and zero otherwise. It is then easy to see, by reversing the circle action, that we must have $$\label{eq:reverse}
N_p=N_{n-p}.$$ Suppose that $[\omega]$ belongs to the image of the map $H^2({\mathsf{M}};{{\mathbb{Z}}})\to H^2({\mathsf{M}};{\mathbb{R}})$. Since $[\omega]^k\neq 0$ for every $k=0,\ldots,n$, it follows, by the above result, that $N_p\neq 0$ for every $p=0,\ldots,n$. Thus
> *the minimal number of fixed points on a compact Hamiltonian\
> manifold ${\mathsf{M}}$ of dimension $2n$ is $n+1$*, and, in this case, $$H^i({\mathsf{M}},{{\mathbb{Z}}})=H^i({{\mathbb{C}}}P^n;{{\mathbb{Z}}}) \quad i=0,\ldots,n.$$
A basis for the equivariant cohomology of ${\mathsf{M}}$ is given as follows. Let us define $$\label{eq:lambda-}
\Lambda_i^-=(\prod_{w_{ij}<0}w_{ij})x^{\lambda_i},$$ so that $\Lambda_i^-$ coincides with $e^{S^1}({\mathsf{N}}_{P_i})$, the equivariant Euler class of the negative normal bundle at $P_i$. Accordingly, we also define $$\label{eq:lambda+}
\Lambda_i^+=(\prod_{w_{ij}>0}w_{ij})x^{n-\lambda_i}$$ and $\Lambda_i=e^{S^1}(T{\mathsf{M}}_{P_i})=\Lambda_i^+\Lambda_i^-$, for every $i=0,\ldots,N$.
\[kirwan\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold endowed with a Hamiltonian $S^1$-action with isolated fixed points $P_0,\ldots,P_N$. Let $\psi\colon {\mathsf{M}}\to {\mathbb{R}}$ be the corresponding moment map. For every fixed point $P_i\in {\mathsf{M}}^{S^1}$ there exists a class $\gamma_i\in H_{S^1}^{2\lambda_i}({\mathsf{M}};{{\mathbb{Z}}})$ such that
- $\gamma_i(P_i)= \Lambda_i^-$;
- $\gamma_i(P_j)=0$ for every $P_j\in {\mathsf{M}}^{S^1}\setminus\{P_i\}$ such that $\psi(P_j)\leq\psi(P_i)$.
Moreover, for any such classes, $\{\gamma_i\}_{i=0}^N$ is a basis for $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ as a module over $H^*({{\mathbb{C}}}P^{\infty};{{\mathbb{Z}}})={{\mathbb{Z}}}[x]$.
Notice that the set of classes satisfying $(1)$ and $(2)$ is not unique. In fact, if there exist $P_l$ and $P_m$ such that $\psi(P_l)< \psi(P_m)$ and $\lambda_l=\lambda_m$, then the class $\gamma^\prime_l=\gamma_l+k\gamma_m$ satisfies the same properties $(1)$ and $(2)$ satisfied by $\gamma_l$, for any $k\in {{\mathbb{Z}}}$.
Fix a set of classes $\{\gamma_i\}_{i=0}^N$ satisfying $(1)$ and $(2)$ of Lemma \[kirwan\]. Since they form a basis for $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ as a module over $H^*({{\mathbb{C}}}P^{\infty};{{\mathbb{Z}}})$, given any class $\alpha\in H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ there exist $\alpha^0,\ldots, \alpha^N\in H^*({{\mathbb{C}}}P^{\infty};{{\mathbb{Z}}})$ such that $
\alpha=\sum_{j=0}^N\alpha^i\gamma_i\;.
$ The next Lemma gives a recursive formula that computes the coefficients $\alpha^i$s in terms of $\iota^*(\alpha)$ and $\iota^*(\gamma_i)$, for $i=0,\ldots,N$, which is an immediate consequence of properties $(1)$ and $(2)$.
\[coefficients\] Let us order the fixed points $P_0,\ldots,P_N$ in such a way that $$\psi(P_0)<\psi(P_1)\leq \psi(P_2)\leq \cdots \leq\psi(P_{N-1})<\psi(P_N).$$ Then the coefficients $\alpha^i$s can be computed recursively as $$\alpha^i = \frac{\alpha(P_i)-\sum_{h:\;\psi(P_h)<\psi(P_i)}\alpha^h\gamma_{h}(P_i)}{\Lambda_i^-}.$$
For every $0\leq j\leq l\leq N$ and $0\leq i \leq N$, the elements $\alpha_{jl}^i$ in $H^*({{\mathbb{C}}}P^{\infty};{{\mathbb{Z}}})$ such that $$\gamma_j\gamma_l=\sum_{i=0}^N\alpha_{jl}^i\gamma_i\;$$ are called the *equivariant structure constants* of $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ with respect to the basis $\{\gamma_0,\ldots,\gamma_N\}$. In order to compute them, by Lemma \[coefficients\], it is sufficient to compute $\iota^*(\gamma_i)$ for every $i$. This problem has been extensively studied in the literature, and it is possible to get an explicit formula for these restrictions only in very special cases (see for example [@GZ; @GT; @ST]). However, when the number of fixed points is minimal, Tolman [@T1] shows that one can give an explicit basis for $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ whose restriction to the fixed point set can be completely recovered from the fixed point set data (see Section \[hamiltonian minimal\] for details).
$K$-theory and equivariant $K$-theory
-------------------------------------
(For a detailed discussion see, for example, [@At; @AS; @AS3].) Let $({\mathsf{M}},{\mathsf{J}})$ be a compact almost complex manifold endowed with a ${\mathsf{J}}$-preserving $S^1$-action with isolated fixed points $P_0,\ldots,P_N$. We recall that $K({\mathsf{M}})$ (resp. $K_{S^1}({\mathsf{M}})$) is the abelian group associated to the semigroup of isomorphism classes of complex vector bundles (resp. complex $S^1$-vector bundles) over ${\mathsf{M}}$, endowed with the direct sum operation $\oplus$. This also has a ring structure, given by the tensor product $\otimes$. If ${\mathsf{M}}$ is a point, we have $$K(pt)\simeq {{\mathbb{Z}}}\quad \text{and} \quad K_{S^1}(pt) \simeq R(S^1),$$ the character ring of $S^1$. This last ring can be simply identified with the Laurent polynomial ring ${{\mathbb{Z}}}[t,t^{-1}]$, where $t$ denotes the standard $S^1$-representation.
In analogy with what happens in cohomology, the unique map $ {\mathsf{M}}\to pt$ induces maps in equivariant and in ordinary $K$-theory, $$K_{S^1}(pt)\to K_{S^1}({\mathsf{M}})\quad\text{and} \quad K(pt)\to K({\mathsf{M}}),$$ which give $K_{S^1}({\mathsf{M}})$ the structure of a ${{\mathbb{Z}}}[t,t^{-1}]$-module, and $K({\mathsf{M}})$ the structure of a ${{\mathbb{Z}}}$-module.
Moreover, the inclusion homomorphism $\{e\}\hookrightarrow S^1$ induces a restriction map in $K$-theory $$r\colon K_{S^1}({\mathsf{M}})\to K({\mathsf{M}})\;,$$ which, when ${\mathsf{M}}$ is a point, is just the evaluation map, $r\colon {{\mathbb{Z}}}[t,t^{-1}]\to {{\mathbb{Z}}}$, at $t=1$.
Consider the $K$-theoretic push-forward map in equivariant and ordinary $K$-theory, namely the index homomorphisms $$\label{equiv ind}
\operatorname{ind}_{S^1}\colon K_{S^1}({\mathsf{M}})\to K_{S^1}(pt)\simeq {{\mathbb{Z}}}[t,t^{-1}]$$ and $$\label{index}
\operatorname{ind}\colon K({\mathsf{M}})\to K(pt)\simeq {{\mathbb{Z}}}\;.$$ By the Atiyah-Singer formula, can be computed as $$\label{AT formula}
\operatorname{ind}(\eta)=\int_{\mathsf{M}}\operatorname{Ch}(\eta)\operatorname{\mathcal{T}}({\mathsf{M}})\;,\quad\mbox{for every}\quad\eta\in K({\mathsf{M}}),$$ where $\operatorname{Ch}(\cdot)$ denotes the Chern character $\operatorname{Ch}\colon K({\mathsf{M}})\to H^*({\mathsf{M}};{\mathbb{Q}})$, and $\operatorname{\mathcal{T}}({\mathsf{M}})$ is the total Todd class of ${\mathsf{M}}$.
On the other hand, by the Atiyah-Segal formula [@AS], the map can be computed in terms of the fixed-point set data. Namely, if $w_{i1},\ldots,w_{in}$ are the weights of the $S^1$-representation on $T_{P_i}{\mathsf{M}}$, $$\label{AS formula}
\operatorname{ind}_{S^1}(\eta^{S^1})=\sum_{i=0}^N \frac{\eta^{S^1}(P_i)}{\prod_{j=1}^n (1-t^{-w_{ij}})}\,,\quad\mbox{for every}\quad \eta^{S^1}\in K_{S^1}({\mathsf{M}})\;,$$ where $\eta^{S^1}(P_i)\in {{\mathbb{Z}}}[t,t^{-1}]$ denotes the restriction of $\eta^{S^1}$ to the fixed point $P_i$. Notice that the sum on right hand side of is in ${{\mathbb{Z}}}[t,t^{-1}]$, despite the fact that each of its terms is not. Observe also that we have the following commutative diagram $$\label{K commutes}
\xymatrix{
K_{S^1}({\mathsf{M}}) \ar[r]^{r} \ar[d]_{\operatorname{ind}_{S^1}} & K({\mathsf{M}}) \ar[d]_{\operatorname{ind}} \\
{{\mathbb{Z}}}[t,t^{-1}] \ar[r]^{r} & {{\mathbb{Z}}}.
} $$ Thus, the value of $\operatorname{ind}_{S^1}(\eta^{S^1})$ at $t=1$ can be computed using and : $$\operatorname{ind}_{S^1}(\eta^{S^1})_{\rvert_{t=1}}=
\int_{\mathsf{M}}\operatorname{Ch}(r(\eta^{S^1}))\operatorname{\mathcal{T}}({\mathsf{M}})\;.$$
Equivariant complex line bundles {#eclb}
--------------------------------
(For a detailed discussion see, for example, [@Ha; @HL; @HY; @Mu] and [@GKS Appendix C].) Let ${\mathsf{M}}$ be a compact manifold with a smooth $S^1$-action and let ${\mathbb{L}}$ be a complex line bundle over ${\mathsf{M}}$. Then ${\mathbb{L}}$ is called *admissible* if the $S^1$-action on ${\mathsf{M}}$ lifts to an $S^1$-action on ${\mathbb{L}}$ which makes the projection map ${\mathbb{L}}\to {\mathsf{M}}$ equivariant. We denote by ${\mathbb{L}}^{S^1}$ the line bundle ${\mathbb{L}}$ endowed with the lifted $S^1$-action.
\[lift clb\] Let $\;{\mathbb{L}}^{S^1}\to {\mathsf{M}}$ be an $S^1$-equivariant complex line bundle over ${\mathsf{M}}$, and let ${\mathsf{c}}_1^{S^1}({\mathbb{L}}^{S^1})\in H^2_{S^1}({\mathsf{M}};{{\mathbb{Z}}})$ be its equivariant first Chern class. Then ${\mathsf{c}}_1^{S^1}$ determines a one-to-one correspondence between equivalence classes of $S^1$-equivariant complex line bundles over ${\mathsf{M}}$ and elements of $H_{S^1}^2({\mathsf{M}};{{\mathbb{Z}}})$.
Consequently, a complex line bundle ${\mathbb{L}}\to {\mathsf{M}}$ is admissible if and only if ${\mathsf{c}}_1({\mathbb{L}})$ is in the image of the restriction map $r\colon H_{S^1}^2({\mathsf{M}};{{\mathbb{Z}}})\to H^2({\mathsf{M}};{{\mathbb{Z}}})$. Moreover, all different liftings of the $S^1$-action on ${\mathbb{L}}$ are parametrized by $H^2({{\mathbb{C}}}P^{\infty};{{\mathbb{Z}}})\simeq {{\mathbb{Z}}}$. In particular, suppose that the $S^1$-action has isolated fixed points $P_0,\ldots,P_N$ and let ${\mathbb{L}}$ be an admissible complex line bundle. Then, as mentioned above, the lift ${\mathbb{L}}^{S^1}$ is not uniquely determined, but, for any such lift, there exists an integer $a$ such that the restriction of ${\mathbb{L}}^{S^1}$ to a fixed point $P_i$ is of the form $${\mathbb{L}}^{S^1}(P_i)=t^{a_i+a}\quad\mbox{for every}\quad i=0,\ldots,N,$$ for fixed integers $a_0,\ldots,a_N$, where $t$ denotes the standard 1-dimensional $S^1$-representation. More precisely, the integer $a_i$ is given by $$a_i = \frac{({\mathsf{c}}_1^{S^1}({\mathbb{L}}^{S^1}))(P_i)}{x}, \quad i=0,\ldots,N.$$ Hence, the values of ${\mathbb{L}}^{S^1}$ at the fixed points are determined up to a constant representation.
The Chern number ${\mathsf{c}}_1{\mathsf{c}}_{n-1}[{\mathsf{M}}]$ {#c1cn-1}
=================================================================
Let $({\mathsf{M}},{\mathsf{J}})$ be a compact almost complex manifold, and let $S^1$ be a circle acting on ${\mathsf{M}}$ preserving the almost complex structure ${\mathsf{J}}$. In this section we show that the Chern number $\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_{n-1}$ can be completely determined by the fixed-point set data. When ${\mathsf{M}}$ is a compact complex manifold, Libgober and Wood [@LW] proved that this integral is determined by the Hodge numbers of ${\mathsf{M}}$. The same fact was also shown later by Borisov in [@Bo], inspired by previous work of Eguchi, Hori and Xiong on Fano varieties [@EHX].\
$\;$
Let $({\mathsf{M}},{\mathsf{J}})$ be a compact almost complex manifold of dimension $2n$, and let ${\mathsf{c}}_j\in H^{2j}({\mathsf{M}};{{\mathbb{Z}}})$ be the Chern classes of the tangent bundle, for every $j=0,\ldots,n$. Let $\chi_y({\mathsf{M}})$ be the Hirzebruch genus of ${\mathsf{M}}$, i.e. the genus corresponding to the power series $$Q_y(x)=\frac{x(1+ye^{-x(1+y)})}{1-e^{-x(1+y)}}$$ (cf. [@Hi] for details). Then $$\chi_y({\mathsf{M}})=\displaystyle\sum_{i=0}^n\left(\int_{\mathsf{M}}T_i^n\right) y^i,$$ where $T_i^n$ is a rational combination of products of Chern classes $${\mathsf{c}}_{j_1}\cdots {\mathsf{c}}_{j_r}\;,\;\;\mbox{ with }\;\;j_1+\cdots +j_r=n.$$ For example, up to order $4$, the $T_i^j$s are $$\begin{aligned}
T_0^0 &=1, \quad \quad \quad \quad \,T_0^1 =-T_1^1=\displaystyle\frac{1}{2} {\mathsf{c}}_1, \quad \,\, T_0^2 =T_2^2=\displaystyle \frac{{\mathsf{c}}_1^2+{\mathsf{c}}_2}{12}, \\ T_2^1 & = \displaystyle\frac{{\mathsf{c}}_1^2-5{\mathsf{c}}_2}{6}, \quad
T_0^3 =-T_3^3=\displaystyle\frac{{\mathsf{c}}_1{\mathsf{c}}_2}{24}, \quad T_1^3 =-T_2^3=\displaystyle\frac{{\mathsf{c}}_1{\mathsf{c}}_2-12{\mathsf{c}}_3}{24}, \\
T_0^4 &= T_4^4=\displaystyle\frac{-{\mathsf{c}}_1^4+4{\mathsf{c}}_1^2{\mathsf{c}}_2+3{\mathsf{c}}_2^2+{\mathsf{c}}_1{\mathsf{c}}_3-{\mathsf{c}}_4}{720}, \\ T_1^4 & =T_3^4=\displaystyle\frac{-{\mathsf{c}}_1^4 + 4 {\mathsf{c}}_1^2 {\mathsf{c}}_2 + 3 {\mathsf{c}}_2^2 - 14 {\mathsf{c}}_1 {\mathsf{c}}_3 - 31 {\mathsf{c}}_4}{180}, \\
T_2^4 &=\displaystyle\frac{-{\mathsf{c}}_1^4 + 4 {\mathsf{c}}_1^2 {\mathsf{c}}_2 + 3 {\mathsf{c}}_2^2 - 19 {\mathsf{c}}_1 {\mathsf{c}}_3 + 79 {\mathsf{c}}_4}{120}.\end{aligned}$$ We recall that the Hirzebruch genus recovers three important topological invariants of the manifold ${\mathsf{M}}$:
- [**If ${\bf y=0}$**]{} then $\chi_0({\mathsf{M}})=\operatorname{Todd}({\mathsf{M}})$ is the Todd genus of ${\mathsf{M}}$, i.e. the genus associated to the power series $Q_0(x)=\displaystyle\frac{x}{1-e^{-x}}$. Hence $T_0^n$ is just the Todd polynomial of degree $n$.
- [**If ${\bf y=1}$**]{} then $\chi_1({\mathsf{M}})=\operatorname{sign}({\mathsf{M}})$ is the signature of ${\mathsf{M}}$.
- [**If ${\bf y=-1}$**]{} then $\chi_{-1}({\mathsf{M}})=\int_{\mathsf{M}}{\mathsf{c}}_n$ is the Euler characteristic of ${\mathsf{M}}$. Moreover, if ${\mathsf{M}}$ is a complex manifold, $\chi_{-1}({\mathsf{M}})=\operatorname{ind}(\overline{\partial})$ is the index of the $\overline{\partial}$ operator.
Let $T^*{\mathsf{M}}\otimes {{\mathbb{C}}}=T^{1,0}{\mathsf{M}}\oplus T^{0,1}{\mathsf{M}}$ be the splitting of the complexified cotangent bundle induced by ${\mathsf{J}}$, into its holomorphic and antiholomorphic parts. For every $p=0,\ldots,n$, let $\chi_p({\mathsf{M}})$ be the topological index of the bundle $\Lambda^pT^{1,0}({\mathsf{M}})$, regarded as an element of $K({\mathsf{M}})$, i.e. $\chi_p({\mathsf{M}})=\operatorname{ind}(\Lambda^pT^{1,0}{\mathsf{M}})\in K(pt)$. Then by the Atiyah-Singer formula [@AS3] $$\chi_p({\mathsf{M}})=\int_{\mathsf{M}}\operatorname{Ch}(\Lambda^pT^{1,0}{\mathsf{M}})\operatorname{\mathcal{T}}({\mathsf{M}})\;,$$ where the orientation on ${\mathsf{M}}$ is the one induced by ${\mathsf{J}}$, $\operatorname{Ch}(\cdot)$ denotes the Chern character, and $\operatorname{\mathcal{T}}({\mathsf{M}})$ is the total Todd class of ${\mathsf{M}}$, i.e. $$\label{total todd class}
\operatorname{\mathcal{T}}({\mathsf{M}})=T_0^0+T_0^1+\cdots +T_0^n\quad\in H^*({\mathsf{M}};{\mathbb{Q}})\;.$$ If ${\mathsf{c}}(T{\mathsf{M}})=\prod_{i=1}^n(1+x_i)$ is a formal factorization of the total Chern class, then a standard computation shows that $$\displaystyle\sum_{p=0}^n \operatorname{Ch}(\Lambda^pT^{1,0}{\mathsf{M}})y^p=\displaystyle \prod_{i=1}^n(1+ye^{-x_i})$$ and $$\label{chip}
\chi_y({\mathsf{M}})=\sum_{p=0}^n\chi_p({\mathsf{M}})y^p.$$
Now suppose that ${\mathsf{M}}$ is equipped with a circle action preserving the almost complex structure ${\mathsf{J}}$ such that the fixed point set ${\mathsf{M}}^{S^1}$ is discrete. Then, for all $p=0,\ldots,n$, the bundle $\Lambda^pT^{1,0}({\mathsf{M}})$ can be regarded as an element of $K_{S^1}({\mathsf{M}})$. Let us define $$\chi_y({\mathsf{M}},t):=\sum_{p=0}^n \operatorname{ind}_{S^1}\left(\Lambda^pT^{1,0}({\mathsf{M}})\right)y^p.$$ Then, following an idea of Atiyah-Hirzebruch [@AH], Kosniowski [@K] and Lusztig [@L], Li proves in [@L2] that when $({\mathsf{M}},{\mathsf{J}})$ is a compact almost complex manifold with a ${\mathsf{J}}$-preserving $S^1$-action with isolated fixed points $P_0,\ldots,P_N$, then $\chi_y({\mathsf{M}},t)$ is independent of $t$, and has the following explicit expression. Let $\lambda_i$ denote the number of negative weights at $P_i$, then $$\chi_y({\mathsf{M}},t)=\sum_{i=0}^N (-y)^{\lambda_i}$$ (see also Section 5.7 of [@Hi]). Since $\chi({\mathsf{M}})=\chi({\mathsf{M}},1)$, we have that $$\label{chi}
\chi_y({\mathsf{M}})=\sum_{i=0}^N (-y)^{\lambda_i}.$$ Let $N_p$ be the number of fixed points with exactly $p$ negative weights. Then we can rewrite as $$\label{chi2}
\chi_y({\mathsf{M}})=\sum_{p=0}^n N_p(-y)^p\;,$$ and so by it follows that $\chi_p({\mathsf{M}})=(-1)^pN_p$. Moreover, since $$\sum_{i=0}^N(-y)^{\lambda_i}=\sum_{i=0}^N(-y)^{n-\lambda_i},$$ we have $$\chi_p({\mathsf{M}})=(-1)^n\chi_{n-p}({\mathsf{M}}).$$ From $\chi_{-1}({\mathsf{M}})=\int_{\mathsf{M}}{\mathsf{c}}_n$ and it immediately follows that $$\label{cn}
\int_{\mathsf{M}}{\mathsf{c}}_n=\sum_{p=0}^n N_p\;.$$
We are now able to prove Theorem \[thm1\], which shows that there exists another Chern number which only depends on the fixed-point set data. For that, we adapt the results in [@LW] and [@Bo] to the almost complex case, and combine them with the localization formula for the Hirzebruch genus.
The proof follows closely the argument found in [@Bo Proposition 2.2.]. By , we have $$\label{dchi}
\displaystyle\frac{d^2\chi_y({\mathsf{M}})}{dy^2}|_{y=-1}=\displaystyle\sum_{p=0}^n\chi_p({\mathsf{M}})(-1)^pp(p-1).$$ On the other hand, we can use the Atiyah-Singer formula to express $$\displaystyle\frac{d^2\chi_y({\mathsf{M}})}{dy^2}|_{y=-1}$$ as a combination of Chern numbers. Indeed, $$\begin{gathered}
\frac{d^2\chi_y({\mathsf{M}})}{dy^2}=\int_{\mathsf{M}}\frac{d^2}{dy^2}\left( \sum_{p=0}^n \operatorname{Ch}(\Lambda^pT^{1,0}{\mathsf{M}})y^p\right)\operatorname{\mathcal{T}}({\mathsf{M}})=\int
_{\mathsf{M}}\frac{d^2}{dy^2}\left(\prod_{i=1}^n \frac{x_i(1+ye^{-x_i})}{1-e^{-x_i}}\right).\\\end{gathered}$$ Hence, $$\begin{aligned}
\frac{d^2\chi_y({\mathsf{M}})}{dy^2}|_{y=-1} & =2\sum_{j<k}\int_{\mathsf{M}}\left(\frac{x_je^{-x_j}}{1-e^{-x_j}}\right) \left(\frac{x_ke^{-x_k}}{1-e^{-x_k}}\right)\prod_{h\neq j,k}x_h \\
& = 2\sum_{j<k}\int_{\mathsf{M}}\left(1-\frac{x_j}{2}+\frac{x_j^2}{12}\right)\left(1-\frac{x_k}{2}+\frac{x_k^2}{12}\right)\prod_{h\neq j,k}x_h \\
& = 2\sum_{j<k}\int_{\mathsf{M}}\left(\frac{x_jx_k}{4}+\frac{x_j^2+x_k^2}{12}\right) \prod_{h\neq j,k}x_h\;\;.\end{aligned}$$ Moreover, it is easy to see that $$\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_{n-1}=n\int_{\mathsf{M}}{\mathsf{c}}_n + \sum_{j<k}\int_{\mathsf{M}}(x_j^2+x_k^2)\prod_{h\neq j,k}x_h\;,$$ and so, the previous computation along with , yields $$\displaystyle\sum_{p=0}^n \chi_p({\mathsf{M}})(-1)^pp(p-1)=\frac{1}{6}\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_{n-1}+\frac{3n^2-5n}{12}\int_{\mathsf{M}}{\mathsf{c}}_n\;.$$ The result then follows from the fact that $\chi_p({\mathsf{M}})=(-1)^pN_p$ and from .
In virtue of the observations made in Section \[ec\], Theorem \[thm1\] has the following immediate corollary.
\[corollary1\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold with a Hamiltonian $S^1$-action and isolated fixed points. Let $b^i({\mathsf{M}})$ be the $i$-th Betti number of ${\mathsf{M}}$. Then $$\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_{n-1}=\sum_{p=0}^n b^{2p}({\mathsf{M}})[6p(p-1)+\frac{5n-3n^2}{2}].$$ In particular, if the number of fixed points is minimal, $$\label{eq:c1cn-1minimal}
\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_{n-1}=\frac{1}{2}n(n+1)^2.$$
Defining the multigraphs {#dmg}
========================
Abstract multigraphs
--------------------
An **oriented multigraph** $\Gamma$ is an oriented graph with multiple oriented edges. More precisely, it is an ordered pair $\Gamma=(V,E)$, where
- $V$ is a set of *vertices*,
- $E$ is a multiset of ordered pairs of vertices which we will call the *edges* of $\Gamma$.
Let ${\mathsf{i}}\colon E \to V$ (resp. ${\mathsf{t}}\colon E \to V$) be the map which associates to each edge $e$ its *initial* (resp. *terminal*) point. Note that we allow a multigraph to contain cycles, i.e. edges $e$ with ${\mathsf{i}}(e)={\mathsf{t}}(e)$. We denote by $E^{\circlearrowleft}$ the subset of $E$ formed by cycles, and by $E^{\circlearrowleft}_P$ the set of cycles that start and end at $P\in V$.
For every $P\in V$, let $E_{P,{\mathsf{i}}}$ (resp. $E_{P,{\mathsf{t}}}$ ) be the set of edges whose initial (resp. terminal) point is $P$, i.e. $E_{P,{\mathsf{i}}}=\{e\in E \mid {\mathsf{i}}(e)=P\}$ (resp. $E_{P,{\mathsf{t}}}=\{e\in E \mid {\mathsf{t}}(e)=P\}$), and define $E_P$ to be the multiset $E_{P,{\mathsf{i}}}\cup E_{P,{\mathsf{t}}}$. This is a multiset in the sense that, if $e$ is a cycle, it appears twice in $E_P$: once as an element of $E_{P,{\mathsf{i}}}$ and once as an element of $E_{P,{\mathsf{t}}}$. We say that the multigraph $\Gamma=(V,E)$ has *degree* $n$ if $\lvert E_P\rvert =n$ for all $P\in V$. Moreover, for every $P\in V$, we define $\lambda(P)$ to be the number of edges ending at $P$, i.e. $\lambda(P)=\lvert E_{P,{\mathsf{t}}}\rvert$.
Define $\delta\colon V\times E\to \{-1,0,1\}$ to be the map $$\delta(P,e)=
\begin{cases}
\;\;1 & \mbox{if }e\in E_{P,{\mathsf{i}}}\setminus E_{P,{\mathsf{t}}}\\
-1 & \mbox{if }e\in E_{P,{\mathsf{t}}}\setminus E_{P,{\mathsf{i}}}\\
0 & \mbox{otherwise}.\\
\end{cases}$$ Assume that the edge set $E$ contains finitely many elements, and let us fix an ordering on $E=(e_1,\ldots,e_{\lvert E \rvert })$. To every oriented multigraph $\Gamma$ we associate an $(\lvert E\rvert \times \lvert E \rvert)$-*matrix* $A(\Gamma)$ whose element at position $(h,m)$ is given by $$\label{matrix}
a_{h,m}:=\delta \left({\mathsf{i}}(e_h),e_m\right)-\delta\left({\mathsf{t}}(e_h),e_m\right).$$ We say that a multigraph $\Gamma=(V,E)$ is **labeled** by ${\mathsf{w}}$ (and denote it by $(\Gamma,{\mathsf{w}})$) if we associate to $\Gamma$ a map ${\mathsf{w}}\colon E\to {{\mathbb{Z}}}$. Finally, suppose that $(\Gamma,{\mathsf{w}})$ satisfies ${\mathsf{w}}(e)\neq 0$ for all $e\in E$. Then we define the **magnitude** ${\mathsf{m}}\colon E\to {{\mathbb{Z}}}$ of $(\Gamma,{\mathsf{w}})$ as $${\mathsf{m}}(e):=\frac{\displaystyle\sum_{f\in E}\left(\delta\left({\mathsf{i}}(e),f\right)-\delta\left({\mathsf{t}}(e),f\right)\right){\mathsf{w}}(f)}{{\mathsf{w}}(e)}\;.$$ The subset of edges $e\in E$ with ${\mathsf{m}}(e)>0$ (resp. ${\mathsf{m}}(e)<0$) is denoted by $E^+$ (resp. $E^-$), and its elements are called **positive edges** (resp. **negative edges**). Finally we denote by $E^0$ the subset of edges with ${\mathsf{m}}(e)=0$.
For every multigraph $\Gamma$ we can introduce an equivalence relation on the set of vertices: given $P,Q\in V$ we say that $P\sim Q$ if and only if there exists a sequence of unoriented edges connecting $P$ to $Q$. The corresponding equivalence classes are called the *connected components* of $\Gamma$.
The multigraphs associated to an $S^1$-action
---------------------------------------------
Let $({\mathsf{M}},{\mathsf{J}})$ be a compact almost complex manifold of dimension $2n$, equipped with a ${\mathsf{J}}$-preserving circle action with a discrete fixed point set ${\mathsf{M}}^{S^1}$. In this section we will associate a family of oriented multigraphs to the $S^1$-space $({\mathsf{M}},{\mathsf{J}},S^1)$, which will encode information about the fixed-point set data.
Let ${\mathsf{M}}^{S^1}=\{P_0,\ldots,P_N\}$, and $w_{i1},\ldots,w_{in}$ the weights of the isotropy representation of $S^1$ on $T_{P_i}{\mathsf{M}}$ (repeated with multiplicity), for $i=0,\ldots, N$.
The **multiset of weights** ${\mathsf{W}}$ associated to the $S^1$-action on $({\mathsf{M}},{\mathsf{J}})$ is the multiset $$\displaystyle\biguplus_{P_i\in {\mathsf{M}}^{S^1}}\{w_{i1},\ldots,w_{in}\}$$ and the **multiset of positive weights** ${\mathsf{W}}_+\subset {\mathsf{W}}$ (resp. **negative weights** ${\mathsf{W}}_-\subset {\mathsf{W}}$) is the multiset $$\displaystyle\biguplus_{P_i\in {\mathsf{M}}^{S^1}}\{w_{ik}\mid w_{ik}>0\}\quad \mbox{(resp.} \displaystyle\biguplus_{P_i\in {\mathsf{M}}^{S^1}}\{w_{ik}\mid w_{ik}<0\}\mbox{)},$$ where $\biguplus$ denotes the union of multisets.
Note that none of the elements of ${\mathsf{W}}$ is zero, since we are assuming the fixed points to be isolated. Hence, ${\mathsf{W}}={\mathsf{W}}_+\cup {\mathsf{W}}_-$.
Our definition of multigraph associated to the $S^1$-action on $({\mathsf{M}},{\mathsf{J}})$ relies on a crucial property of ${\mathsf{W}}$ which was first proved by Hattori in the almost complex case [@Ha Proposition 2.11] (see also [@L2 Theorem 3.5] and [@PT Lemma 13]).
\[pairs\] Let $({\mathsf{M}},{\mathsf{J}})$ be an almost complex manifold equipped with a ${\mathsf{J}}$-preserving circle action with isolated fixed points. Let ${\mathsf{W}}_+$ and ${\mathsf{W}}_-$ be the multisets of positive and negative weights. Then ${\mathsf{W}}_+=-{\mathsf{W}}_-$.
In other words, ${\mathsf{W}}$ always contains pairs of integers of opposite signs. Let us consider the index sets $$I:=\{ (i,k)\in\{0,\ldots,N\}\times \{1,\ldots,n\}\mid w_{ik}\in {\mathsf{W}}_+ \}$$ and $$J:=\{ (i,k)\in\{0,\ldots,N\}\times \{1,\ldots,n\}\mid w_{ik}\in {\mathsf{W}}_- \}.$$ Since ${\mathsf{W}}_+=-{\mathsf{W}}_-$, we can always choose a bijection between ${\mathsf{W}}_+$ and ${\mathsf{W}}_-$ such that the corresponding bijection $f:I\to J$ between the two index sets satisfies $$\label{cond action}
f(i,k)=(j,l) \quad \text{and} \quad w_{ik}=-w_{jl} .$$ Hence, to every fixed point $P_i$, we are associating a fixed point $P_j$ such that one of the weights at $P_i$ is $w_{ik}\in {\mathsf{W}}_+$ and one of the weights at $P_j$ is $w_{jl}=-w_{ik}\in {\mathsf{W}}_-$. Let $\rho:I\to \{0,\ldots, N\}$ be defined by $\rho(i,k):=j$, where $j$ is such that $f(i,k)=(j,l)$. For every choice of a bijective pairing $f:I \to J$ satisfying , we can associate an oriented multigraph $\Gamma=(V,E)$ to the $S^1$-action on $({\mathsf{M}},{\mathsf{J}})$ as follows:
- The vertex set $V$ is the fixed point set.
- The edge set is determined by the bijection between ${\mathsf{W}}_+$ and ${\mathsf{W}}_-$ chosen above. More precisely, for every $(i,k)\in I$ such that $f(i,k)=(j,l)$, there corresponds an oriented edge $e_{ik}$ from $P_i$ to $P_j$. The edge set $E$ is then $E=\{e_{ik}\mid (i,k)\in I\}$.
Consequently, the elements of $E$ are in bijection with those of $I$. Note that, in the notation above, if $e_{ik}$ is the edge associated to the pair $(i,k)\in I$, then ${\mathsf{i}}(e_{ik})=P_i$ and ${\mathsf{t}}(e_{ik})=P_{\rho(i,k)}$.
We label the multigraph $\Gamma=(V,E)$ by the map $
{\mathsf{w}}\colon E \to {{\mathbb{Z}}}_{>0}
$ which, to each edge $e=e_{ik}$, associates the weight ${\mathsf{w}}(e)=w_{ik}$ (which, by definition, is always positive). We call this map the *weight map*.
Let us denote by $\mathcal{W}$ the family of multigraphs associated to the $S^1$-action on ${\mathsf{M}}$ labeled by the weight map, and denote its elements by pairs $(\Gamma,{\mathsf{w}})$.
\[set of weights\] Let $(\Gamma,{\mathsf{w}})$ be an element of $\mathcal{W}$. Then
- $(\Gamma,{\mathsf{w}})$ determines ${\mathsf{W}}$. More precisely, for every $P_i\in {\mathsf{M}}^{S^1}$, let $\{w_{i1},\ldots,w_{in}\}$ be the multiset of weights of the $S^1$-representation on $T{\mathsf{M}}|_{P_i}$. Then $$\label{weight}
\{w_{i1},\ldots,w_{in}\}=\{\delta(P_i,e)\, {\mathsf{w}}(e),e\in E_{P_i}\setminus E^{\circlearrowleft}_{P_i}\}\cup \{\pm {\mathsf{w}}(e), e\in E^{\circlearrowleft}_{P_i}\}\;.$$
- The magnitude is given by $$\label{mc}
{\mathsf{m}}(e)=\frac{{\mathsf{c}}^{S^1}_1\left({\mathsf{i}}(e)\right)-{\mathsf{c}}^{S^1}_1\left({\mathsf{t}}(e)\right)}{{\mathsf{w}}(e)x}\quad\mbox{for all}\quad e\in E.$$
The equality of multisets is an easy consequence of the definitions. Moreover, comes from the definition of ${\mathsf{m}}(e)$, and the fact that ${\mathsf{c}}_1^{S^1}(P_i)=(\sum_{j=1}^nw_{ij})x$ for every $P_i\in {\mathsf{M}}^{S^1}$.
Observe that for all $P\in V$, $\lambda(P):=\lvert E_{P,{\mathsf{t}}}\rvert$ is just the number of negative weights of the $S^1$-representation on $T_P{\mathsf{M}}$. Then $$\lvert {\mathsf{W}}_-\rvert=\sum_{P\in {\mathsf{M}}^{S^1}}\lambda(P)$$ which, by Lemma \[pairs\], must be equal to $$\lvert {\mathsf{W}}_+\rvert =\sum_{P\in {\mathsf{M}}^{S^1}}(n-\lambda(P)),$$ and so $$\lvert {\mathsf{W}}_+\rvert =\lvert {\mathsf{W}}_-\rvert =\frac{\lvert {\mathsf{W}}\rvert }{2}=\frac{(N+1)n}{2}.$$ From the definition of $E$ we then have $$\lvert E\rvert =\frac{(N+1)n}{2}.$$
We will now explore the properties satisfied by the magnitude ${\mathsf{m}}$ associated to $(\Gamma,{\mathsf{w}})\in \mathcal{W}$.
Suppose that there exists an edge $e\in E$ such that the points ${\mathsf{i}}(e)$ and ${\mathsf{t}}(e)$ are the $S^1$-fixed points of an $S^1$-invariant sphere $S^2_e$ embedded in ${\mathsf{M}}$, where ${\mathsf{w}}(e)$ (resp. $-{\mathsf{w}}(e)$) is the weight of the $S^1$ representation on $T_{{\mathsf{i}}(e)}S_e^2$ (resp. $T_{{\mathsf{t}}(e)}S_e^2$). Then ${\mathsf{m}}(e)$ is just the integral of the first Chern class ${\mathsf{c}}_1$ on the sphere $S_e^2$. In fact, by and the ABBV localization formula (Corollary \[abbv discrete\]), we have $${\mathsf{m}}(e)=\frac{{\mathsf{c}}^{S^1}_1({\mathsf{i}}(e))-{\mathsf{c}}^{S^1}_1({\mathsf{t}}(e))}{{\mathsf{w}}(e)x}=\int_{S_e^2} \iota^*({\mathsf{c}}_1^{S^1})=\int_{S_e^2}\iota^*({\mathsf{c}}_1),$$ where $\iota: S^2 \to {\mathsf{M}}$ is the inclusion map.
Let $(\Gamma,{\mathsf{w}})$ be an element of $\mathcal{W}$, and let us fix an ordering $e_1,\ldots,e_{\lvert E\rvert}$ of the edge set $E$. Then, by definition of ${\mathsf{m}}$, we get $$\label{system}
\sum_{m=1}^{\lvert E\rvert}\big(\delta\left({\mathsf{i}}(e_h),e_m\right)-\delta\left({\mathsf{t}}(e_h),e_m\right) \big){\mathsf{w}}(e_m)-{\mathsf{w}}(e_h){\mathsf{m}}(e_h)=0\;$$ for all $h=1,\ldots,\lvert E \rvert$.
Denoting by ${\mathsf{w}}(E)$ the vector $\left({\mathsf{w}}(e_1),\ldots,{\mathsf{w}}(e_{\lvert E \rvert})\right)$, and by ${\mathsf{m}}(E)$ the vector $\left({\mathsf{m}}(e_1),\ldots,{\mathsf{m}}(e_{\lvert E \rvert})\right)$, from and we obtain the homogeneous system $$\label{system2}
\Big(A(\Gamma)-\operatorname{diag}({\mathsf{m}}(E))\Big)\cdot {\mathsf{w}}(E)^t=0\;,$$ where $\operatorname{diag}({\mathsf{m}}(E))$ is the $(\lvert E\rvert \times \lvert E\rvert)$-diagonal matrix $\operatorname{diag}\left({\mathsf{m}}(e_1),\ldots,{\mathsf{m}}(e_{\lvert E \rvert})\right)$.
Let us study in more detail. Suppose that $e=e_j$ is a cycle, i.e. that ${\mathsf{i}}(e)={\mathsf{t}}(e)$. Then the $j-$th row and column of $A(\Gamma)-\operatorname{diag}\left({\mathsf{m}}(E)\right)$ are zero vectors. Let $\Gamma^\prime$ be the graph obtained from $\Gamma$ by deleting the cycles $e\in E^{\circlearrowleft}$, i.e. $\Gamma^\prime=(V,E^\prime)$, where $E^\prime=E\setminus E^{\circlearrowleft}$, and $\Gamma^\prime=\Gamma_1\cup\cdots\cup\Gamma_l$ is the decomposition of $\Gamma^\prime$ into its connected components $\Gamma_i$. Pick the following ordering of the edges of $\Gamma$: $$\label{ordering of E}
E=(e^1_{1},\ldots,e^1_{\lvert E_1\rvert},e^2_1,\ldots,e^2_{\lvert E_2\rvert },\ldots,e^l_{1},\ldots,e^l_{\lvert E_l\rvert},e_1^{\circlearrowleft},\ldots,e^{\circlearrowleft}_{\lvert E^{\circlearrowleft}\rvert })\;,$$ where, writing $\Gamma_i=(V_i,E_i)$ for every $i=1,\ldots,l$, we have $e^i_j\in E_i$ for all $j=1,\ldots,\lvert E_i\rvert$, and $e_j^{\circlearrowleft}\in E^{\circlearrowleft}$ for every $j=1,\ldots,\lvert E^{\circlearrowleft}\rvert$. It is easy to see that $A(\Gamma)$ is a block diagonal matrix of the form $$A(\Gamma)=
\left(
\begin{array}{cccc}
A(\Gamma_1) & & 0 & 0\\
\vdots & \ddots & \vdots &\vdots\\
& & A(\Gamma_l) & 0 \\
0 & \cdots & 0 & \mathbf{0}\\
\end{array}
\right),$$ where the bottom right $(\lvert E^{\circlearrowleft}\rvert \times \lvert E^{\circlearrowleft}\rvert $)-zero matrix corresponds to the cycles. For every $i=1\ldots,l$, let $$\label{eq:Matricesi}
\operatorname{Null}\Big(A(\Gamma_i)-\operatorname{diag}({\mathsf{m}}(E_i))\Big)$$ be the null space of $A(\Gamma_i)-\operatorname{diag}\left({\mathsf{m}}(E_i)\right)$, and let ${{\mathbb{Z}}}_{>0}^{\lvert E_i\rvert }$ be the vectors in ${\mathbb{R}}^{\lvert E_i\rvert}$ with positive integer entries. Since ${\mathsf{w}}(e)$ is a positive integer for every $e\in E$, we have the following result.
\[magnitude\] Let $(\Gamma,{\mathsf{w}})$ be any multigraph associated to a fixed $S^1$-action on $({\mathsf{M}},{\mathsf{J}})$, and let $\Gamma^\prime=(V,E^\prime)$ be the graph obtained from $\Gamma=(V,E)$ by deleting its cycles, i.e. $E^\prime=E\setminus E^{\circlearrowleft}$. Moreover, let $\Gamma_1\cup\cdots\cup\Gamma_l$ be the decomposition of $\Gamma^\prime$ into its connected components, and let $A(\Gamma_i)$ be the matrices associated to $\Gamma_i$. Then $$\Big( \operatorname{Null}\big(A(\Gamma_i)-\operatorname{diag}\left({\mathsf{m}}(E_i)\right)\big) \Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E_i\rvert }\neq \emptyset\;,\quad \mbox{for every} \;\;\;i=1,\ldots,l,$$ where ${\mathsf{m}}$ is the magnitude associated to $\Gamma$. In particular, $$\det \Big(A(\Gamma_i)-\operatorname{diag}({\mathsf{m}}(E_i))\Big)=0\;, \quad \mbox{for every} \;\;\;i=1,\ldots,l.$$
Notice that the magnitude ${\mathsf{m}}\colon E \to {\mathbb{Q}}$ associated to $(\Gamma,{\mathsf{w}})\in\mathcal{W}$ clearly depends on the labeled multigraph chosen. However, the sum of the elements in the image of ${\mathsf{m}}$ does not. Indeed, we have the following result.
\[thm2\] For any $(\Gamma,{\mathsf{w}})\in\mathcal{W}$ the associated magnitude ${\mathsf{m}}\colon E\to {\mathbb{Q}}$ satisfies $$\sum_{e\in E} {\mathsf{m}}(e) = \int_{\mathsf{M}}{\mathsf{c}}_1 {\mathsf{c}}_{n-1}.$$
By the ABBV localization formula (Corollary \[abbv discrete\]) and we have $$\begin{aligned}
\int_{\mathsf{M}}{\mathsf{c}}_1 {\mathsf{c}}_{n-1} & = \int_{\mathsf{M}}{\mathsf{c}}^{S^1}_1 {\mathsf{c}}^{S^1}_{n-1} = \sum_{i=0}^{N} \frac{{\mathsf{c}}^{S^1}_1(P_i) {\mathsf{c}}^{S^1}_{n-1}(P_i)}{e^{S^1}({\mathsf{N}}_{P_i})} =\frac{1}{x} \sum_{i=0}^{N} \frac{{\mathsf{c}}^{S^1}_1(P_i) \left(\sum_{l=1}^n \prod_{k\neq l} w_{ik}\right) }{\prod_{k=1}^n w_{ik}} \\ & =\frac{1}{x} \sum_{i=0}^{N}\sum_{k=1}^{n} \frac{{\mathsf{c}}^{S^1}_1(P_i)}{w_{ik}} = \frac{1}{x}\sum_{(i,k)\in I} \frac{{\mathsf{c}}^{S^1}_1(P_i)}{w_{ik}}+\frac{1}{x}\sum_{(j,l)\in J} \frac{{\mathsf{c}}^{S^1}_1(P_j)}{w_{jl}}=\\
& = \sum_{(i,k)\in I} \frac{{\mathsf{c}}^{S^1}_1(P_i)-{\mathsf{c}}^{S^1}_1(P_{\rho(i,k)})}{w_{ik}x} =\sum_{e\in E}\frac{{\mathsf{c}}^{S^1}_1({\mathsf{i}}(e))-{\mathsf{c}}^{S^1}_1({\mathsf{t}}(e))}{{\mathsf{w}}(e)x}= \sum_{e\in E} {\mathsf{m}}(e).\end{aligned}$$
Combining Theorem \[thm1\] and Proposition \[thm2\] we obtain the following result.
\[sum of m\] Let $(\Gamma,{\mathsf{w}})$ be an element of $\mathcal{W}$, and let ${\mathsf{m}}$ be the associated magnitude. Then the sum of the magnitudes of the edges is an invariant of $\mathcal{W}$. More precisely, let $n$ be the degree of $\Gamma$ and let $N_p$ be the number of vertices $P$ with $\lambda(P)=p$, for all $p=0,\ldots,n$. Then $$\label{sum m}
\sum_{e\in E} {\mathsf{m}}(e) =\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}].$$ In particular, if $N_p=1$ for every $p=0,\ldots,n$, then $$\label{sum m min}
\sum_{e\in E} {\mathsf{m}}(e) =\frac{1}{2}n(n+1)^2\;.$$
Let us now restrict our attention to a class of multigraphs for which the magnitude ${\mathsf{m}}\colon E\to {\mathbb{Q}}$ has integer values.
For every integer $l>1$, let ${\mathsf{M}}^{{{\mathbb{Z}}}_l}$ be the submanifold of ${\mathsf{M}}$ fixed by the subgroup ${{\mathbb{Z}}}_l\subset S^1$. Then we have the following lemma.
There exists a labeled multigraph $(\Gamma,{\mathsf{w}})\in\mathcal{W}$ such that, for every $e\in E$ with ${\mathsf{w}}(e)>1$, $$\label{cc}
{\mathsf{i}}(e)\quad\mbox{and}\quad{\mathsf{t}}(e)\quad\mbox{ lie in the same connected component of}\quad {\mathsf{M}}^{{{\mathbb{Z}}}_{{\mathsf{w}}(e)}}.$$
For every fixed point $P$, let $w_1,\ldots,w_n$ be the weights of the $S^1$-representation on $T_P{\mathsf{M}}$. For every $w_i>1$, the set ${\mathsf{M}}^{{{\mathbb{Z}}}_{w_{i}}}$ is an almost complex submanifold of ${\mathsf{M}}$ of dimension greater than zero. Let $N$ be the connected component of ${\mathsf{M}}^{{{\mathbb{Z}}}_{w_{i}}}$ containing $P$. Then, by applying Lemma \[pairs\] to $N$, we know that there exists a fixed point $Q\in N^{S^1}$ with one weight equal to $-w_i$. Since $N^{S^1}\subset {\mathsf{M}}^{S^1}$, the conclusion follows immediately.
\[img\] A labeled multigraph $(\Gamma,{\mathsf{w}})\in \mathcal{W}$ satisfying $\eqref{cc}$ is called an **integral multigraph**. We denote by $\;\;\mathcal{I}\;\;$ the family of integral multigraphs associated to a fixed $S^1$-action on $({\mathsf{M}},{\mathsf{J}})$ and by $\mathcal{M}$ the family of integral multigraphs labeled by their corresponding magnitudes. The elements of $\mathcal{M}$ are denoted by $(\Gamma,{\mathsf{m}})$.
The main property of integral multigraphs is that the corresponding magnitudes have integer values. This is an easy consequence of [@T1 Lemma 2.6 ], which is stated for symplectic manifolds, but also holds for general almost complex manifolds.
\[modulo\] Let $P$ and $Q$ be fixed points of the $S^1$-action which lie in the same connected component of ${\mathsf{M}}^{{{\mathbb{Z}}}_l}$, for some integer $l>1$. Then the weights of the isotropy action of $S^1$ on $T_P{\mathsf{M}}$ agree with the ones on $T_Q{\mathsf{M}}$ modulo $l$.
So we have the following result.
For every $(\Gamma,{\mathsf{m}})\in \mathcal{M}$, the magnitude ${\mathsf{m}}$ has integer values, i.e. ${\mathsf{m}}\colon E\to {{\mathbb{Z}}}$.
By definition of integral multigraph, for every edge $e$ such that ${\mathsf{w}}(e)>1$, the endpoints ${\mathsf{i}}(e)$ and ${\mathsf{t}}(e)$ of $e$ lie in the same connected component of ${\mathsf{M}}^{{{\mathbb{Z}}}_{{\mathsf{w}}(e)}}$. Hence, by Lemma \[modulo\], the weights of the isotropy representation of $S^1$ on $T_{{\mathsf{i}}(e)}{\mathsf{M}}$ agree with the ones on $T_{{\mathsf{t}}(e)}{\mathsf{M}}$ modulo ${\mathsf{w}}(e)$. The conclusion then follows immediately from the fact that $\frac{{\mathsf{c}}_1^{S^1}(P)}{x}$ is the sum of the weights of the $S^1$-representation on $T_P{\mathsf{M}}$, for every $P\in {\mathsf{M}}^{S^1}$.
Two important subsets of $\mathcal{I}$ are those given by positive integral multigraphs and non-negative integral multigraphs.
\[P0+\] An integral multigraph $(\Gamma,{\mathsf{w}})\in \mathcal{I}$ is called a **non-negative (resp. positive)** if the corresponding magnitude satisfies ${\mathsf{m}}\colon E\to {{\mathbb{Z}}}_{\geq 0}$ (resp. ${\mathsf{m}}\colon E\to {{\mathbb{Z}}}_{> 0}$).
We denote by $\mathcal{I}_{\geq 0}$ (resp. $\mathcal{I}_{> 0}$) the family of non-negative (resp. positive) multigraphs, and by $\mathcal{M}_{\geq 0}$ (resp. $\mathcal{M}_{> 0}$) the family of non-negative (resp. positive) multigraphs labeled by their corresponding magnitudes.
We say that the $S^1$-space $({\mathsf{M}},{\mathsf{J}},S^1)$ satisfies [**property**]{} ${\bf (\mathcal{P}_0^+)}$ if the corresponding family of non-negative multigraphs $\mathcal{I}_{\geq 0}$ is nonempty.
In Sections \[algo\] and \[mnfp\] we will investigate in more detail conditions on the $S^1$-action that guarantee that it can be represented by a non-negative or positive multigraph (see in particular Remark \[monotone..\] and Section \[refinement\]).
Let us consider the circle action on ${\mathsf{M}}=S^2 \times S^2$ which rotates one sphere with speed $a$ and the other with speed $b$, where $a,b\in \mathbb{Z}_{>0}$ are relatively prime. There are four fixed points $$P_0=(S,S),\,\, P_1=(S,N),\,\, P_2=(N,S) \quad \text{and} \quad P_3=(N,N),$$ where $S$ and $N$ are the north and south poles of $S^2$. There are two spheres fixed by the action of the subgroup ${{\mathbb{Z}}}_a$ (namely $S^2\times \{S\}$ and $S^2\times \{N\}$) and two spheres fixed by the action of ${{\mathbb{Z}}}_b$ (namely $\{S\}\times S^2$ and $\{N\} \times S^2$). The isotropy weights at each fixed point and the number $\lambda_i$ of negative weights at $P_i$ are
$P_0:$ $(w_{01},w_{02})=(a,b)$ $\lambda_0=0$
-------- --------------------------- ---------------
$P_1:$ $(w_{11},w_{12})=(-b,a)$ $\lambda_1=1$
$P_2:$ $(w_{21},w_{22})=(-a,b)$ $\lambda_2=1$
$P_3:$ $(w_{31},w_{32})=(-b,-a)$ $\lambda_3=2$
.
The values of the equivariant first Chern class ${\mathsf{c}}_1^{S^1}(P_i)$ at each fixed point are
${\mathsf{c}}_1^{S^1}(P_0)=$ $(a+b)x$
------------------------------ -----------
${\mathsf{c}}_1^{S^1}(P_1)=$ $(a-b)x$
${\mathsf{c}}_1^{S^1}(P_2)=$ $(b-a)x$
${\mathsf{c}}_1^{S^1}(P_3)=$ $-(a+b)x$
and the multisets ${\mathsf{W}}_+,{\mathsf{W}}_-,I$ and $J$ are
${\mathsf{W}}_+$ $\{a,a,b,b\}$
------------------ --------------------------------
${\mathsf{W}}_-$ $\{-a,-a,-b,-b\}$
$I$ $\{(0,1),(0,2),(1,2),(2,2)\}$
$J$ $\{ (1,1),(2,1),(3,1),(3,2)\}$
.
There are only four possible pairings $f_l:I \to J$, $l=1,\ldots,4$, as in
[|l|| l| c |l|| l| c |l|| l|c |l|| l|]{} $f_1(0,1)=$ &$(2,1)$ & & $f_2(0,1)=$ &$(3,2)$ & & $f_3(0,1)=$ &$(2,1)$ & & $f_4(0,1)=$ &$(3,2)$\
$f_1(0,2)=$ & $(3,1)$ & & $f_2(0,2)=$ &$(3,1)$ & & $f_3(0,2)=$ &$(1,1)$ & & $f_4(0,2)=$ &$(1,1)$\
$f_1(1,2)=$ & $(3,2)$ & & $f_2(1,2)=$ &$(2,1)$ & & $f_3(1,2)=$ &$(3,2)$ & & $f_4(1,2)=$ &$(2,1)$\
$f_1(2,2)=$ & $(1,1)$ & & $f_2(2,2)=$ &$(1,1)$ & & $f_3(2,2)=$ &$(3,1)$ & & $f_4(2,2)=$ &$(3,1)$\
yielding the four graphs $\Gamma_l$ in Figure \[Graphs\].
![Circle actions on $S^2\times S^2$[]{data-label="Graphs"}](AllgraphsS2S2.pdf)
Note that, if $a$ and $b$ are greater than $1$, the only integral multigraph is $\Gamma_3$ which is a positive integral multigraph. Indeed, writing $e_1:=e_{01}$, $e_2:=e_{02}$, $e_3:=e_{12}$, $e_4:=e_{22}$, the associated magnitudes ${\mathsf{m}}\colon E \to {\mathbb{Q}}$ are
[|c|| c| c |c|| c| c |c|| c| c |c|| c| ]{}
$f_1$ & & & $f_2$ & & & $f_3$ & & & $f_4$ &\
${\mathsf{m}}(e_{1})$ & $ 2 $ & & ${\mathsf{m}}(e_{1})$ & $ \frac{2(a+b)}{a} $ & & ${\mathsf{m}}(e_{1})$ & $ 2 $ & & ${\mathsf{m}}(e_{1})$ & $ \frac{2(a+b)}{a}$\
${\mathsf{m}}(e_{2})$ & $ \frac{2(a+b)}{b}$ & & ${\mathsf{m}}(e_{2})$ & $ \frac{2(a+b)}{b} $ & & ${\mathsf{m}}(e_{2})$ & $ 2 $ & & ${\mathsf{m}}(e_{2})$ & $ 2 $\
${\mathsf{m}}(e_{3})$ & $ 2$ & & ${\mathsf{m}}(e_{3})$ & $\frac{2(a-b)}{a} $ & & ${\mathsf{m}}(e_{3})$ & $ 2 $ & & ${\mathsf{m}}(e_{3})$ & $ \frac{2(a-b)}{a} $\
${\mathsf{m}}(e_{4})$ & $ \frac{2(b-a)}{b}$ & & ${\mathsf{m}}(e_{4})$ & $ \frac{2(b-a)}{b} $ & & ${\mathsf{m}}(e_{4})$ & $ 2 $ & & ${\mathsf{m}}(e_{4})$ & $ 2 $\
.
Note that $\sum_{e\in E}{\mathsf{m}}(e)=8$ in all cases. On the other hand, the numbers $N_p$ of fixed points with $p$ negative weights are $N_0=N_2=1$ and $N_1=2$ and so, by Theorem \[thm1\], we obtain $$\int_{\mathsf{M}}{\mathsf{c}}_1 {\mathsf{c}}_{n-1}= \sum_{p=0}^2 N_p[6p(p-1)-1]=-1-2+11=8,$$ confirming the result in Proposition \[thm2\].
Finally, for $\Gamma_i=(V,E_i)$, we obtain the matrices $A(\Gamma_i)$ given by $$\begin{aligned}
A(\Gamma_1)=
\left(
\begin{array}{rrrr}
2 & 1 & 0 & -1\\
1 & 2 & 1 & 0\\
0 & 1 & 2 & -1\\
-1 & 0 & -1 & 2
\end{array}
\right)\quad\quad
&
A(\Gamma_2)=
\left(
\begin{array}{rrrr}
2 & 2 & 0 & 0\\
2 & 2 & 0 & 0\\
0 & 0 & 2 & -2\\
0 & 0 & -2 & 2\\
\end{array}
\right)\\
\\
A(\Gamma_3)=
\left(
\begin{array}{rrrr}
2 & 1 & 0 & -1\\
1 & 2 & -1 & 0\\
0 & -1 & 2 & 1\\
-1& 0 & 1 & 2\\
\end{array}
\right)\quad\quad
&
A(\Gamma_4)=
\left(
\begin{array}{rrrr}
2 & 1 & 0 & 1\\
1 & 2& -1 & 0\\
0 & -1 & 2 & -1\\
1& 0 & -1 & 2\\
\end{array}
\right).\\\end{aligned}$$
An algorithm to determine linear relations among the weights {#algo}
=============================================================
After having introduced the necessary definitions in Section \[dmg\] we can rewrite Question \[q3\] in the following way.
\[q5\] Let $n,N$ and $\lambda_i$, for $i=0,\ldots,N$, be positive integers, where $0\leq\lambda_i\leq n$ for all $i$. Does there exist a compact almost complex manifold $({\mathsf{M}},{\mathsf{J}})$ of dimension $2n$ with a ${\mathsf{J}}$-preserving $S^1$-action with isolated fixed points $P_0,\ldots,P_N$, such that the number of negative weights at $P_i$ is $\lambda_i$? If so, can we determine the corresponding family $\mathcal{I}$ of integral labeled multigraphs?
Instead of trying to determine $\mathcal{I}$, it is more convenient to determine $\mathcal{M}$, the family of integral multigraphs labeled by their magnitudes ${\mathsf{m}}$. In fact, by Theorem \[sum of m\], the sum of the magnitudes only depends on $n,N$ and $\lambda_i$ and so Proposition \[magnitude\] gives linear relations among the weights.
Notice that if the answer to Question \[q5\] is affirmative, the first necessary condition on $n,N$ and the $\lambda_i$s is a direct consequence of Lemma \[pairs\]. Indeed, since $\lvert {\mathsf{W}}_+\rvert =\lvert {\mathsf{W}}_-\rvert =\displaystyle\frac{\lvert {\mathsf{W}}\rvert }{2}$, we must have $$\label{lambdas}
\sum_{i=0}^N(n-\lambda_i)=\sum_{i=0}^N\lambda_i=\frac{(N+1)n}{2}.$$ From now on we will assume that $n$, $N$ and the $\lambda_i$s satisfy .
The linear relations among the weights can be determined as follows. First, let us define two families $\mathcal{N}$ and $\mathcal{N}_{\geq 0}$ of labeled multigraphs associated to the integers $n$, $N$ and $\lambda_i$, for $i=0,\ldots,N$. For that, consider the sets $$I^\prime=\left\{(i,k)\in\{0,\ldots,N\}\times \{1,\ldots,n\}\mid k\le n-\lambda_i \right\}$$ and $$J^\prime=\left\{(i,k)\in\{0,\ldots,N\}\times \{1,\ldots,n\}\mid k\le \lambda_i \right\}.$$ Observe that by we have $\lvert I^\prime \rvert=\lvert J^\prime \rvert$. Let $f\colon I^\prime \to J^\prime$ be a bijection between the two sets. Using $f$ we can construct a multigraph $\Gamma=(V,E)$, where
- $V$ is a set of $N+1$ vertices $P_0,\ldots,P_N$,
- for every $(i,k)\in I^\prime$ with $f(i,k)=(j,l)$, one considers an oriented edge $e_{ik}$ from $P_i$ to $P_j$ so that the edge set is $E=\{e_{ik}\mid (i,k)\in I^\prime\}$.
Let $\Gamma^\prime=(V,E^\prime)$ be the graph obtained from $\Gamma=(V,E)$ by deleting its cycles, i.e. $E^\prime=E\setminus E^{\circlearrowleft}$. Moreover, let $\Gamma_1\cup\cdots\cup\Gamma_l$ be the decomposition of $\Gamma^\prime$ into its connected components, and let $A(\Gamma_i)$ be the matrices associated to $\Gamma_i$ as in . Consider an ordering on the edges as in and let $\mathcal{N}_{\Gamma}$ be the family of maps ${\mathsf{n}}\colon E\to {{\mathbb{Z}}}$ satisfying $$\begin{aligned}
\label{cond1}
\displaystyle\sum_{e\in E} {\mathsf{n}}(e) =\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}]\\
\label{cond2}
\Big( \operatorname{Null}\left(A(\Gamma_i)-\operatorname{diag}\left({\mathsf{n}}(E_i)\right)\right)\Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E_i\rvert }\neq \emptyset\end{aligned}$$ for every $i=1,\ldots,l$. Then we define $\mathcal{N}$ as the set of pairs $(\Gamma,{\mathsf{n}})$, where $\Gamma$ is associated to a bijection $f:I^\prime \to J^\prime$ as above and ${\mathsf{n}}\in \mathcal{N}_{\Gamma}$. Moreover, we denote by $\mathcal{N}_{\geq 0}$ the subset of $\mathcal{N}$ given by pairs $(\Gamma,{\mathsf{n}})\in \mathcal{N}$ such that ${\mathsf{n}}\colon E\to {{\mathbb{Z}}}_{\geq 0}$.
Condition implies that $\det \left(A(\Gamma_i)-\operatorname{diag}\left({\mathsf{n}}(E_i)\right)\right)=0$ for every $i$, yielding polynomial equations in the ${\mathsf{n}}(e)$s. However, is much stronger. For example, let $B_h$ be the $h$-th row of $A(\Gamma_i)-\operatorname{diag}\left({\mathsf{n}}(E_i)\right)$, for some $i$. Then it is easy to see that $B_h$ must contain entries of opposite signs for all $h=1,\ldots,\lvert E_i \rvert$. More precisely $B_h$ looks like $$(b_{h,1},\ldots,b_{h,h-1},2-{\mathsf{n}}(e_h),\ldots,b_{h,\lvert E_i\rvert}),$$ where $b_{h,k}$ is an integer; so, if $b_{h,k}\geqslant 0$ for all $k\neq h$, we need ${\mathsf{n}}(e_h)\geqslant 2$. Taking linear combinations of rows in such a way that all the constant coefficients have the same sign, one can get lower bounds for linear combinations of the ${\mathsf{n}}(e)$s. However, these estimates do not seem to be optimal. It would be interesting to know whether one can get better restrictions on the set of the ${\mathsf{n}}(e)$s satisfying .
Using the above notation we have the following result.
\[NC\] Let $n$, $N$ and $\lambda_i$, for $i=0,\ldots,N$, be non negative integers satisfying $0\leq\lambda_i\leq n$ for $0\leq i\leq N$ as well as .
If there exists a compact almost complex manifold $({\mathsf{M}}, {\mathsf{J}})$ of dimension $2n$ and a ${\mathsf{J}}$-preserving $S^1$-action on ${\mathsf{M}}$ with isolated fixed points $P_0,\ldots,P_N$, such that the number of negative weights at $P_i$ is $\lambda_i$, then $\mathcal{M}\subset\mathcal{N}$ and $\mathcal{M}_{\geq 0}\subset \mathcal{N}_{\geq 0}$.
Moreover, for every $(\Gamma,{\mathsf{w}})\in \mathcal{I}$ (resp. $\mathcal{I}_{\geq 0}$) there exists $(\Gamma,{\mathsf{n}})\in \mathcal{N}$ (resp. $\mathcal{N}_{\geq 0}$) such that $${\mathsf{w}}(E)\in \Big(\operatorname{Null}(A\left(\Gamma)-\operatorname{diag}\left({\mathsf{n}}(E)\right)\right)\Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E\rvert}.$$
Let $\{w_{i1},\ldots,w_{in}\}$ be the set of weights of the $S^1$-representation on $T_{P_i}{\mathsf{M}}$, for all $i=0,\ldots,N$, and let ${\mathsf{W}}$, ${\mathsf{W}}_+$ and ${\mathsf{W}}_-$ be the multisets defined in Section \[dmg\]. We can order the weights in such a way that $I^\prime$ and $J^\prime$ are the index sets of ${\mathsf{W}}_+$ and ${\mathsf{W}}_-$, i.e. $I^\prime=\{(i,k)\mid w_{ik}\in {\mathsf{W}}_+\}$ and $J^\prime=\{(i,k)\mid w_{ik}\in {\mathsf{W}}_-\}$. Then the conclusion follows from the definitions along with Proposition \[magnitude\] and Theorem \[sum m\].
Hence, to look for linear relations among the isotropy weights, we have to take every possible multigraph obtained from a bijection $f:I^\prime \to J^\prime$ and look for possible functions ${\mathsf{n}}:E\to {{\mathbb{Z}}}$ satisfying and , which would correspond to the magnitudes of the multigraphs. To make this task more efficient we point out a few facts that, in some cases, imply that the number of such functions ${\mathsf{n}}$ is finite. Moreover, when we cannot assure that this number is finite we can impose additional conditions on the action, making this number finite at the cost of restricting the class of circle actions considered. Before doing so, we consider the following technical lemma.
\[l dividing\] Let $({\mathsf{M}},{\mathsf{J}})$ be a compact almost complex manifold with a ${\mathsf{J}}$-preserving $S^1$-action with isolated fixed points. Let $(\Gamma,{\mathsf{w}})\in \mathcal{I}$ be an integral multigraph associated to this action. Then, for every $w\in H_{S^1}^2({\mathsf{M}};{{\mathbb{Z}}})$, we have $$\frac{w({\mathsf{i}}(e))-w({\mathsf{t}}(e))}{{\mathsf{w}}(e)x} \in {{\mathbb{Z}}}\quad\mbox{for every edge $e\in E$}.$$
As remarked in Section \[eclb\], to every element $w\in H_{S^1}^2({\mathsf{M}};{{\mathbb{Z}}})$, we can associate an equivariant complex line bundle ${\mathbb{L}}^{S^1}$ whose first equivariant Chern class is $w$ and such that ${\mathbb{L}}^{S^1}(P)=t^{w(P)}$ for every fixed point $P$. Let $e$ be an edge of $(\Gamma,{\mathsf{w}})\in \mathcal{I}$ and suppose that ${\mathsf{w}}(e)>1$. By definition of integral multigraph, ${\mathsf{i}}(e)$ and ${\mathsf{t}}(e)$ are in the same connected component $N$ of ${\mathsf{M}}^{{{\mathbb{Z}}}_{{\mathsf{w}}(e)}}$, the submanifold of ${\mathsf{M}}$ fixed by ${{\mathbb{Z}}}_{{\mathsf{w}}(e)}$. Thus the $S^1$-modules ${\mathbb{L}}^{S^1}\left({\mathsf{i}}(e)\right)$ and ${\mathbb{L}}^{S^1}\left({\mathsf{t}}(e)\right)$ are equivalent as ${{\mathbb{Z}}}_{{\mathsf{w}}(e)}$-modules, and the conclusion follows immediately.
As a result of this Lemma, for every equivariant class $w\in H^2_{S^1}({\mathsf{M}};{{\mathbb{Z}}})$ that “divides" the first equivariant Chern class ${\mathsf{c}_{1}^{S^1}}$, i.e. a class $w$ satisfying $$\label{eq:C}
{\mathsf{c}_{1}^{S^1}}=Cw$$ for some constant $C\in {{\mathbb{Z}}}\setminus \{0\}$, we have $$\frac{w({\mathsf{i}}(e))-w({\mathsf{t}}(e))}{{\mathsf{w}}(e)}=\frac{{\mathsf{m}}(e)}{C}\in {{\mathbb{Z}}}$$ for every edge $e\in E$. Hence, by , we conclude that this constant $C$ must divide $$\label{RHS}
\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}].$$ In many cases the largest positive integer $C$ satisfying this property can be explicitly written in terms of the weights at the fixed points of index $0$ and $2$. Indeed, assume now, and for the remaining of the section, that $({\mathsf{M}},\omega)$ is a compact symplectic manifold with a compatible almost complex structure ${\mathsf{J}}$, and that the $S^1$-actions considered are Hamiltonian and preserve ${\mathsf{J}}$.
Let $P_0$ be the fixed point of index 0, and $P_1^1,\ldots, P_1^k$ the fixed points of index $2$. Assume that there exist degree-$2$ generators of $H_{S^1}^2({\mathsf{M}};{{\mathbb{Z}}})$, $\tau_1^1,\ldots,\tau_1^k$, satisfying $$\label{canonical 2}
\tau_1^i(P)=0\quad\mbox{ for all}\quad P\in {\mathsf{M}}^{S^1}\setminus\{P_1^i\}\;\;\;\mbox{ such that }\lambda(P)\leq 1,$$ which we call *canonical classes*. These classes do not always exist, but, if they do exist, they are unique (see [@GT Lemma 2.7]). Then it is easy to see that $${\mathsf{c}_{1}^{S^1}}=\sum_{i=1}^m\alpha_i\tau_1^i+\beta,$$ for some $1\leq m\leq k$, with $$\alpha_i=\displaystyle\frac{{\mathsf{c}_{1}^{S^1}}(P_1^i)-{\mathsf{c}_{1}^{S^1}}(P_0)}{\Lambda_{P_1^i}^-}\in {{\mathbb{Z}}}\setminus \{0\}\quad\text{and}\quad\beta={\mathsf{c}_{1}^{S^1}}(P_0).$$ Indeed, from Lemma 3.8 in [@T1] we have that $r({\mathsf{c}}_1^{S^1})={\mathsf{c}}_1\neq 0$, thus implying that $m\geq 1$; the explicit expression of the $\alpha_i$s and $\beta$ is an easy consequence of the definitions. So the largest positive constant $C$ dividing for which ${\mathsf{c}_{1}^{S^1}}/C$ is still an integer class is $$C=\gcd\{\lvert \alpha_1\rvert ,\ldots,\lvert \alpha_m\rvert \}.$$
Let us now make the additional assumption that there is a multigraph with an edge $e_i$ from $P_0$ to $P_1^i$, for every $i=1,\ldots,m$. Then ${\mathsf{m}}(e_i)=\alpha_i$ for every $i=1,\ldots,m$ and so $$C=\gcd\{\lvert {\mathsf{m}}(e_1)\rvert ,\ldots,\lvert {\mathsf{m}}(e_m)\rvert \}.$$ Note that we can always take such a multigraph when there exists a sphere connecting $P_0$ to $P_1^i$ fixed by some subgroup of $S^1$, for every $i=1,\ldots,m$.
\[flag\] Consider the standard $T^3$-action on ${{\mathbb{C}}}^3$ given by $$(\xi_1,\xi_2,\xi_3)\cdot (z_1,z_2,z_3)=(\xi_1 z_1, \xi_2 z_2, \xi_3 z_3)\;.$$ This action descends to an action on the complete flag manifold $\mathcal{F}l({{\mathbb{C}}}^3)$, where the diagonal circle $S^1=\{(\xi,\xi,\xi)\in T^3\}$ acts trivially. Take the action of the quotient group $T^3/S^1\simeq T^2$. This action has $6$ fixed points, given by flags in the coordinate lines of ${{\mathbb{C}}}^3$. These are indexed by permutations $\sigma \in S_3$ on $3$ letters, where the fixed point corresponding to $\sigma$ is given by $$\langle 0 \rangle \subset \langle f_{\sigma (1)} \rangle \subset \langle f_{\sigma (1)}, f_{\sigma (2)} \rangle \subset \langle f_{\sigma (1)}, f_{\sigma (2)}, f_{\sigma (3)} \rangle = {{\mathbb{C}}}^3,$$ the brackets indicate the span of the vectors, and $\{f_1,f_2,f_3\}$ is the standard basis of ${{\mathbb{C}}}^3$. By taking a generic circle $S^1\subset T^3$ we obtain a circle action with one fixed point of index $0$, two fixed points of index $2$, $P_1^1$ and $P_1^2$, two fixed points of index $4$ and one fixed point of index $6$. The weights at the minimum are then $\{m,n,m+n\}$ and the weights at the index-$2$ fixed points are respectively $\{-m,n,m+n\}$ and $\{-n,m,m+n\}$, where $m$ and $n$ are coprime integers which depend on the circle $S^1\subset T^2$ chosen. Let $\tau_1^1$ and $\tau_1^2$ be generators of $H^2_{S^1}({\mathsf{M}};{{\mathbb{Z}}})$. Then they can be chosen in such a way that they satisfy property , and it is easy to check that ${\mathsf{c}_{1}^{S^1}}=2(\tau_1^1+\tau_1^2)+2(m+n)$, yielding $C=2$.
Taking the above remarks into consideration let us see how to proceed in general.
${\bf 1.}\,\,$ *If none of the ${\mathsf{m}}(e)$s is negative* we have the following algorithm.
\[alg:1\] If none of the ${\mathsf{m}}(e)$s is negative, then, for each possible multigraph $\Gamma$ with connected components $\Gamma_1,\ldots,\Gamma_l$, we look for partitions of $$\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}]$$ into $\lvert E^\prime\rvert=\lvert E\rvert - \lvert E^{\circlearrowleft} \rvert$ positive integers ${\mathsf{m}}(e)$. Then we add the zeros ${\mathsf{m}}(e)=0$ whenever the edge $e$ is a cycle and choose, among the resulting sequences of $\lvert E \rvert$ nonegative numbers, those for which $$\Big( \operatorname{Null}\left(A(\Gamma_i)-\operatorname{diag}\left({\mathsf{m}}(E_i)\right)\right)\Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E_i\rvert }\neq \emptyset$$ for every $i=1,\dots,l$.
When none of the ${\mathsf{m}}(e)$s is negative we have an upper bound for any constant $C$ as in . In fact, since $\frac{{\mathsf{m}}(e)}{C}$ is an integer for each edge $e\in E$, we have $$\sum_{e\in E^\prime}\frac{{\mathsf{m}}(e)}{C}\geq \lvert E^\prime\rvert =\frac{n(N+1)}{2} -\lvert E^{\circlearrowleft}\rvert,$$ where $N+1$ is the total number of fixed points. Observe that $|E^{\circlearrowleft}|<|E|$; in fact, since the action is Hamiltonian, none of the edges starting at the minimum $P_0$ can be a cycle. So $|E'|>0$. Moreover it’s easy to see that $|E^{\circlearrowleft}|$ is bounded by $\sum_{i=0}^N\min\{\lambda_i,n-\lambda_i\}$. By using together with the fact that $C$ is a positive integer, we obtain $$\label{ub C}
1\leq C\leq \frac{2\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}]}{n(N+1)-2\lvert E^{\circlearrowleft}\rvert}.$$
If $\dim({\mathsf{M}})=6$, $N_0=N_3=1$ and $N_1=N_2=2$, we have $$\lvert E^{\circlearrowleft} \rvert \leq 4, \quad \sum_{e\in E}{\mathsf{m}}(e)=24\quad \text{and} \quad \lvert E \rvert=9,$$ and so $C=1$ or $2$ whenever $\lvert E^{\circlearrowleft} \rvert=0$, $C=1,2$ or $3$ when $\lvert E^{\circlearrowleft}\rvert=1$ or $2$, and $C=1,2,3$ or $4$ if $\lvert E^{\circlearrowleft} \rvert=3$ or $4$. Note that Example \[flag\] falls into this case.
If we can choose a basis of $H_{S^1}^2({\mathsf{M}};{{\mathbb{Z}}})$ where the degree-$2$ generators are canonical classes, and there exists an edge between the point of index zero and each point of index $2$, we can use the bound in to improve Algorithm \[alg:1\].
\[alg:2\] Assume that we can choose a basis of $H_{S^1}^2({\mathsf{M}};{{\mathbb{Z}}})$ where the degree-$2$ generators are canonical classes. Suppose that there exists a multigraph $\Gamma$ such that none of the ${\mathsf{m}}(e)$s is negative and which has an edge $e_i$ from $P_0$ to $P_1^i$, for each $i$ such that ${\mathsf{c}_{1}^{S^1}}(P_0)/x>{\mathsf{c}_{1}^{S^1}}(P_1^i)/x$; let $e_1,\ldots,e_m$ be these edges. Let $\Gamma_1,\ldots,\Gamma_l$ be the connected components of $\Gamma$. Then for each divisor $C$ of $$\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}],$$ satisfying $$1\leq C\leq \frac{2\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}]}{n(N+1)-2 \lvert E^{\circlearrowleft} \rvert},$$ we look for partitions of $$\frac{1}{C}\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}]$$ into $\lvert E^\prime\rvert =\lvert E \rvert - \lvert E^{\circlearrowleft} \rvert$ positive integers, ${\mathsf{l}}(e):=\frac{{\mathsf{m}}(e)}{C}$ with $\gcd\{{\mathsf{l}}(e_1),\ldots,{\mathsf{l}}(e_m)\}=1$. Then we consider the corresponding integers ${\mathsf{m}}(e)=C\cdot {\mathsf{l}}(e)$, add the zeros ${\mathsf{m}}(e)=0$ whenever the edge $e$ is a cycle, and choose, among the resulting sequences of $\lvert E \rvert$ numbers, the ones for which $$\Big( \operatorname{Null}\left(A(\Gamma_i)-\operatorname{diag}\left({\mathsf{m}}(E_i)\right)\right)\Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E_i\rvert }\neq \emptyset,$$ for every $i=1,\dots,l$.
\[monotone..\][**(Non-negative multigraphs)**]{} One cannot expect in general that non-negative multigraphs exist. Indeed, if the number of fixed points is big, meaning that the $N_p$s are greater than $1$, the right hand side of , i.e. $\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_{n-1}$, can be *negative*. Consequently, requiring all the ${\mathsf{m}}(e)$s to be non-negative is a very restrictive assumption. For example, if $\dim {\mathsf{M}}=4$, we have $N_0=N_2=1$ and $\int_{\mathsf{M}}{\mathsf{c}}_1^2=10-N_1$.
However, if given an $S^1$-action we can choose a multigraph such that for every edge $e\in E$ there exists an isotropy $2$-sphere $S_e^2$ having ${\mathsf{i}}(e)$ and ${\mathsf{t}}(e)$ as south and north pole (e.g. for GKM manifolds, see Section \[refinement\] [**v)**]{}), then this multigraph is positive if ${\mathsf{c}}_1$ is positive on each of these spheres, which happens for example
- If $({\mathsf{M}},\omega)$ is *monotone*, i.e. the symplectic form satisfies ${\mathsf{c}}_1=C[\omega]$ for some positive constant $C$.
- More generally, when $({\mathsf{M}},\omega)$ is *symplectic Fano*, i.e. if, given an almost complex structure ${\mathsf{J}}$ compatible with $\omega$, we have ${\mathsf{c}}_1(A)>0$ for every $A\in H_2({\mathsf{M}})$ which can be represented by a ${\mathsf{J}}$-holomorphic curve.
- If there exist classes $\sigma^1,\ldots,\sigma^k$ in $H^2({\mathsf{M}};{{\mathbb{Z}}})$ such that ${\mathsf{c}}_1=\sum_{i=1}^k\beta_i\sigma^i$, where $\beta_i\in {{\mathbb{Z}}}_{>0}$ for every $i$, and for every isotropy sphere $S_e^2$, there exists $j$ such that $\int_{S_e^2} \sigma^j>0$ and $\int_{S_e^2} \sigma^i\geq 0$ for $j\neq i$.
Let us now see how to obtain a finite algorithm from even when we cannot assume the ${\mathsf{m}}(e)$s to be non-negative.\
$\;$\
${\bf 2.}\,\,$ *If the ${\mathsf{m}}(e)$s are also negative* we proceed as follows. Consider the map $\Psi$ that to each fixed point $P$ associates the sum of the weights at $P$, $$\begin{aligned}
\Psi\colon & {\mathsf{M}}^{S^1} &\longrightarrow {{\mathbb{Z}}}\nonumber\\
& P &\longmapsto \sum_{i=1}^n w_{iP}=\frac{{\mathsf{c}_{1}^{S^1}}(P)}{x}, \label{Psi}
\end{aligned}$$ and let $$D=\max\{\lvert \Psi(P)\rvert \}_{P\in {\mathsf{M}}^{S^1}}.$$ Note that $D> 0$ since the fact that the action is Hamiltonian yields $\Psi(P_0)>0$. Then $$\lvert {\mathsf{m}}(e)\rvert \leq \left\lvert \frac{{\mathsf{c}_{1}^{S^1}}({\mathsf{i}}(e))-{\mathsf{c}_{1}^{S^1}}({\mathsf{t}}(e))}{x} \right\rvert \leq 2 D$$ and so we can still construct an algorithm to obtain necessary conditions on the isotropy weights, now with a prescribed bound on $\Psi$.
\[alg:3\] Fixing a positive integer $D$, for each multigraph $\Gamma$ with connected components $\Gamma_1,\ldots,\Gamma_l$, we look for partitions of $$\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}]$$ into $\lvert E \rvert$ integers, ${\mathsf{m}}(e)$, with $\lvert {\mathsf{m}}(e)\rvert \leq 2D$ and choose those for which $$\Big( \operatorname{Null}\left(A(\Gamma_i)-\operatorname{diag}\left({\mathsf{m}}(E_i)\right)\right)\Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E_i\rvert }\neq \emptyset$$ for every $i=1,\dots,l$.
Note that some of the above integers may be zero depending on the existence of cycles in $\Gamma$.
Moreover, suppose that there exists an edge $e_1^i$ between the point of index $0$ and every index-$2$ fixed point $P_1^i$ such that $\Psi(P_0)\neq \Psi(P_1^i)$; call these edges $e_1,\ldots,e_m$. Suppose that there exists canonical classes $\tau_1^i$ satisfying ; then we can again improve this algorithm.
\[alg:4\] Fixing a positive integer $D$, for each multigraph $\Gamma$ with connected components $\Gamma_1,\ldots,\Gamma_l$ and for each integer $C\in [1,D]\cap {{\mathbb{Z}}}$ dividing $$\sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}],$$ we look for partitions of $$\frac{1}{C} \sum_{p=0}^n N_p[6p(p-1)+\frac{5n-3n^2}{2}]$$ into $\lvert E \rvert$ integers, ${\mathsf{l}}(e):=\frac{{\mathsf{m}}(e)}{C}$, with $\lvert {\mathsf{l}}(e)\rvert \leq \frac{2D}{C}$ where $\gcd\{{\mathsf{l}}(e_1),\ldots,{\mathsf{l}}(e_m)\}=1$. Then we choose those for which $$\Big( \operatorname{Null}\left(A(\Gamma_i)-\operatorname{diag}\left({\mathsf{m}}(E_i)\right)\right)\Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E_i\rvert }\neq \emptyset$$ for every $i=1,\dots,l$.
Note that some of the above integers may be zero depending on the existence of cycles in $\Gamma$. This algorithm yields necessary conditions for all $S^1$-Hamiltonian actions with $$\max\left\{ \left \lvert \sum_{i=1}^nw_{iP}\right\rvert\right\}_{P\in {\mathsf{M}}^{S^1}}\leq D.$$
\[C bound\] The upper bound for $C$ in the above algorithm can be improved when $\Psi$ is injective. Indeed in this case we can use Theorem \[hattori\] (2), which is due to Hattori [@Ha], to prove that $C$ must satisfy $1\leq C \leq N+1$, where $N+1$ is the number of fixed points.
Minimal number of fixed points {#mnfp}
==============================
Suppose now that the number of fixed points is minimal. Then there are several important consequences which are explored next.
An explicit basis for the equivariant cohomology {#hamiltonian minimal}
------------------------------------------------
Suppose that $({\mathsf{M}},\omega)$ is a compact symplectic manifold of dimension $2n$ with a Hamiltonian $S^1$-action and minimal number of fixed points $P_0,\ldots,P_n$. Thus $$H^i({\mathsf{M}};{{\mathbb{Z}}})=H^i({{\mathbb{C}}}P^n;{{\mathbb{Z}}})$$ for all $i$, and $N_p=1$ for every $p=0,\ldots,n$ (see Section \[ec symplectic\]). Let us order the fixed points $P_0,\ldots,P_n$ is such a way that $\lambda(P_i)=i$ for all $i=0,\dots,n$.
Tolman shows in [@T1] that, in this case, it is possible to recover the equivariant cohomology ring $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ (and hence $H^*({\mathsf{M}};{{\mathbb{Z}}})$) from the isotropy representation of $S^1$ at the fixed point set, and she gives an explicit basis for $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ and $H^*({\mathsf{M}};{{\mathbb{Z}}})$. Here we review the main results of this construction, together with other useful facts.
Let ${\mathsf{c}}^{S^1}\in H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ be the total equivariant Chern class of the tangent bundle and ${\mathsf{c}}$ the total ordinary Chern class, that is, $${\mathsf{c}}^{S^1}=\sum_{j=0}^n {\mathsf{c}}_j^{S^1}\quad \text{and} \quad {\mathsf{c}}=\sum_{j=0}^n {\mathsf{c}}_j.$$ We recall that, for every fixed point $P_i$, the restriction of the $j$-th equivariant Chern class to $P_i$ is given by $${\mathsf{c}}_j^{S^1}(P_i)=\sigma_j(w_{i1},\ldots,w_{in})x^j,$$ where $\sigma_j$ denotes the $j$-th elementary symmetric polynomial, $w_{i1},\ldots,w_{in}$ are the isotropy weights at $P_i$, and $x$ is the degree-$2$ generator of $H_{S^1}^*(\{P_i\};{{\mathbb{Z}}})={{\mathbb{Z}}}[x]$. In particular, ${\mathsf{c}}_1^{S^1}(P_i)=(\sum_{i=1}^nw_{in})x$ (see Section \[ec\]).
The next result combines Proposition $3.4$ and Lemmas $3.8$ and $3.23$ of [@T1].
\[psi e c\] Let the circle act on a compact symplectic manifold $({\mathsf{M}},\omega)$ of dimension $2n$ with moment map $\psi\colon{\mathsf{M}}\to {\mathbb{R}}$, and let $P_0,\ldots,P_n$ be its fixed points, where $\lambda(P_i)=i$ for every $i$. Then ${\mathsf{c}}_1\neq 0$. Moreover,\
$\;$ $$\label{m.m increasing}
\psi(P_i)<\psi(P_j) \quad\mbox{if and only if}\quad i<j\;,$$ and $$\label{c1 decreasing}
\displaystyle\frac{{\mathsf{c}}_1^{S^1}(P_i)-{\mathsf{c}}_1^{S^1}(P_j)}{x}>0 \quad\mbox{if and only if}\quad i<j\;,$$ where $x$ is the generator of $H^*({{\mathbb{C}}}P^{\infty};{{\mathbb{Z}}})$.
In particular, the equivariant symplectic form and the first Chern class restricted to the fixed point set are injective.
The following theorem combines Corollaries $3.14$ and $3.19$ of [@T1].
\[bases\] Let the circle act on a compact symplectic manifold $({\mathsf{M}},\omega)$ of dimension $2n$ with moment map $\psi\colon{\mathsf{M}}\to {\mathbb{R}}$, and let $P_0,\ldots,P_n$ be its fixed points, where $\lambda(P_i)=i$ for every $i$.
1. As a $H^*({{\mathbb{C}}}P^{\infty};{{\mathbb{Z}}})={{\mathbb{Z}}}[x]$ module, $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ is freely generated by $\tau_0,\tau_1,\ldots,\tau_n$, where $$\label{basis equivariant cohomology}
\tau_i=\frac{1}{C_i}\prod_{j=0}^{i-1}\left({\mathsf{c}_{1}^{S^1}}-{\mathsf{c}_{1}^{S^1}}(P_j)\right)\quad\mbox{and}\quad C_i=\frac{\prod_{j=0}^{i-1}\left({\mathsf{c}_{1}^{S^1}}(P_i)-{\mathsf{c}_{1}^{S^1}}(P_j)\right)}{\Lambda_i^-}\;.$$ (In particular, $\tau_i\in H_{S^1}^{2i}({\mathsf{M}};{{\mathbb{Z}}})$ for all $i$.)
2. As a group, $H^*({\mathsf{M}};{{\mathbb{Z}}})$ is freely generated by ${\widetilde{\tau}}_0,{\widetilde{\tau}}_1,\ldots,{\widetilde{\tau}}_n$, where $$\label{basis cohomology}
{\widetilde{\tau}}_i=r(\tau_i)=\frac{1}{C_i}{\mathsf{c}}_1^i\;.$$ (In particular, ${\widetilde{\tau}}_i\in H^{2i}({\mathsf{M}};{{\mathbb{Z}}})$ for all $i$.)
Here we consider the empty product as being equal one so that $\tau_0={\widetilde{\tau}}_0=1$. Note that $\tau_i(P_i)=\Lambda_i^-$ and $\tau_i(P_j)=0$ for every $j\leq i-1$.
Since $\iota^*({\mathsf{c}}^{S^1})$ is determined by the weights at the fixed points, using we can explicitly compute $\iota^*(\tau_i)$. Moreover, by Lemma \[coefficients\], we can compute the equivariant Chern classes in terms of the $\tau_i$s and the ordinary Chern classes in terms of the ${\widetilde{\tau}}_i$s.
\[Ci’s\] Since $\Lambda_i^-=(-x)^{i}\displaystyle\prod_{w_{ij}<0}\lvert w_{ij}\rvert$, it follows from that $C_i\in {\mathbb{Q}}_{>0}$ and then Theorem \[bases\] implies that it is a positive *integer* for all $i$. In the next sections the constant $$\label{C1}
C_1=\frac{{\mathsf{c}_{1}^{S^1}}(P_1)-{\mathsf{c}_{1}^{S^1}}(P_0)}{\Lambda_1^-}\in {{\mathbb{Z}}}_{>0}$$ will be particularly important. Also, observe that by , $C_i$ is the unique positive integer such that $\displaystyle\frac{{\mathsf{c}}_1^i}{C_i}$ is an integral generator of $H^{2i}({\mathsf{M}};{{\mathbb{Z}}})$ for all $i=0,\ldots,n$.
Let $(\Gamma,{\mathsf{w}})$ be an integral multigraph associated to the $S^1$-action on $({\mathsf{M}},\omega)$ (see Definition \[img\]). For every edge $e\in \Gamma$, let $${\mathsf{l}}(e):=\frac{\tau_1({\mathsf{i}}(e))-\tau_1({\mathsf{t}}(e))}{{\mathsf{w}}(e)x}=\frac{{\mathsf{m}}(e)}{C_1}\;.$$ As an easy consequence of Lemma \[l dividing\], of Theorem \[sum of m\] and the definitions above, we have the following result.
\[C1p\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $2n$ with a Hamiltonian $S^1$-action and $n+1$ fixed points. Let $C_1$ be the positive integer such that $\displaystyle\frac{{\mathsf{c}}_1}{C_1}$ is a generator of $H^2({\mathsf{M}};{{\mathbb{Z}}})$. Then $$\label{C1 divides}
C_1\;\;\;\mbox{divides}\;\;\;\frac{1}{2}n(n+1)^2.$$ More precisely, let $(\Gamma,{\mathsf{w}})$ be an integral multigraph associated to the $S^1$-action on $({\mathsf{M}},\omega)$. Then ${\mathsf{l}}(e)$ is an integer for every $e\in E$ and $$\sum_{e\in E}{\mathsf{l}}(e)=\frac{1}{2}\frac{n(n+1)^2}{C_1}\;.$$
Another important property regarding the constant $C_1$ will be proved in Section \[hattori results\]. As we will see in Proposition \[symp hattori\] we also have $$1\leq C_1\leq n+1\;.$$ (see also Remark \[C bound\]).
### Reversing the circle action. {#reversing flow}
Reversing the circle action, we obtain another basis for $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ as a ${{\mathbb{Z}}}[x]$-module. More precisely, the elements of this basis are $\tau^\prime_0,\ldots,\tau^\prime_n$, where $\tau^\prime_i\in H_{S^1}^{2i}({\mathsf{M}};{{\mathbb{Z}}})$ is given by $$\tau^\prime_i=\frac{1}{C^\prime_i}\prod_{j=n-i+1}^{n}({\mathsf{c}_{1}^{S^1}}-{\mathsf{c}_{1}^{S^1}}(P_j))\quad\mbox{with}\quad C^\prime_i=\frac{\prod_{j=n-i+1}^{n}({\mathsf{c}_{1}^{S^1}}(P_{n-i})-{\mathsf{c}_{1}^{S^1}}(P_j))}{\Lambda_{n-i}^+}\;.$$ Notice that $\tau^\prime_i(P_{n-i})=\Lambda_{n-i}^+$ and $\tau^\prime_i(P_j)=0$ for every $j\geq n-i+1$.
Moreover, as a group, $H^*({\mathsf{M}};{{\mathbb{Z}}})$ is freely generated by ${\widetilde{\tau}}^\prime_0,\ldots,{\widetilde{\tau}}^\prime_n$, where $i$, ${\widetilde{\tau}}^\prime_i\in H^{2i}({\mathsf{M}};{{\mathbb{Z}}})$ is given by $${\widetilde{\tau}}^\prime_i=r(\tau^\prime_i)=\frac{{\mathsf{c}}_1^i}{C^\prime_i}$$ for all $i$. It is easy to see that $C^\prime_i$ is a positive integer for all $i$ and then, since $\frac{{\mathsf{c}}_1^i}{C^\prime_i}$ is an integral generator of $H_{S^1}^{2i}({\mathsf{M}};{{\mathbb{Z}}})$, we have, by Remark \[Ci’s\], that $$\label{symmetries}
C_i=C^\prime_i$$ for all $i$ and hence $$\label{taus}
{\widetilde{\tau}}_i={\widetilde{\tau}}^\prime_i$$ for all $i$. Using and the ABBV Localization formula (cf. Corollary \[abbv discrete\]) we have $$\label{duality}
\int_{\mathsf{M}}{\widetilde{\tau}}_i{\widetilde{\tau}}_{n-i}=\int_{\mathsf{M}}{\widetilde{\tau}}_i{\widetilde{\tau}}^\prime_{n-i}=\int_{\mathsf{M}}\tau_i\tau^\prime_{n-i}=\sum_{j=0}^n\frac{\tau_i(P_j)\tau_{n-i}(P_j)}{\Lambda_j}=\frac{\Lambda_i^-\Lambda_{i}^+}{\Lambda_i}=1,$$ for all $i=0,\ldots,n$.
Observe that equations impose many polynomial relations among the weights at the different fixed points.
Positive multigraphs
--------------------
Let us now study conditions that would ensure the existence of a positive multigraph in the case where the number of fixed points is minimal.
Requiring all the ${\mathsf{m}}(e)$s to be positive, is equivalent to requiring $$\frac{{\mathsf{c}_{1}^{S^1}}({\mathsf{i}}(e))-{\mathsf{c}_{1}^{S^1}}({\mathsf{t}}(e))}{x}> 0\quad\mbox{ for every $e\in E$.}$$ Since the number of fixed points is minimal, $({\mathsf{M}},\omega)$ is monotone. Indeed, since $H^2({\mathsf{M}};{{\mathbb{Z}}})={{\mathbb{Z}}}$, we can choose $\omega$ such that ${\mathsf{c}}_1=C_1[\omega]$, with $[\omega]$ the generator of $H^2({\mathsf{M}};{{\mathbb{Z}}})$ and $C_1$ a positive integer. Moreover, we can choose the moment map so that ${\mathsf{c}_{1}^{S^1}}=C_1[\omega-\psi\otimes x]$, and then all the ${\mathsf{m}}(e)$s are positive if and only if $$\psi({\mathsf{i}}(e))<\psi({\mathsf{t}}(e)) \quad \text{for every $e\in E$,}$$ or, equivalently, if and only if $$\label{eq:index}
\lambda({\mathsf{i}}(e))<\lambda({\mathsf{t}}(e)) \quad \text{for every $e\in E$}$$ (cf. Proposition \[psi e c\]).
We conclude that, in this situation, we have the following result.
\[spm\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold with a Hamiltonian $S^1$-action and a minimal number of fixed points. Let $\Gamma=(V,E)$ be a multigraph associated to the $S^1$-action. Then $\Gamma$ is positive (resp. non-negative) if and only if its (directed) edges connect points to points with a greater (resp. greater or equal index).
If we can guarantee the existence of positive multigraphs for every action within a certain class of circle actions, then we can use Algorithm 1. of Section \[algo\]. It is then natural to ask whether these multigraphs exist. There are many situations where we can guarantee the existence of such multigraphs associated to a given circle action. Here we present a few.
[**i)**]{} Whenever ${\mathsf{M}}$ is four dimensional with $3$ isolated fixed points (see Section \[dim4\]).
Whenever ${\mathsf{M}}$ is six dimensional with $4$ fixed points: Ahara [@Ah] and Tolman [@T1] prove the existence of a positive multigraph for every such Hamiltonian circle action.
When ${\mathsf{M}}$ is $8$-dimensional with $5$ fixed points and the $S^1$-action extends to a $T^2$-action:
\[t2\] Suppose that $({\mathsf{M}},\omega)$ is $8$-dimensional and the $S^1$-action has $5$ fixed points. If this action extends to an effective Hamiltonian $T^2$-action, then there exists a positive multigraph associated to the circle action.
To show this we just have to prove that, for each of these actions, there exists a multigraph satisfying .
For that, let us consider the *x-ray* of $({\mathsf{M}},\omega,\phi)$, where $\phi$ is the $T^2$-moment map, given by the closed orbit type stratification $\mathcal{X}$ of ${\mathsf{M}}$, together with the convex polytopes $\phi(X)$ for each $X\in \mathcal{X}$ [@T2]. Let $P_i$ be the $S^1$-fixed point with $\lambda(P_i)=i$. Each line in the x-ray is the image of a connected component of the set of points with a given $1$-dimensional stabilizer. If there is no line in the x-ray containing the images of $P_0$ and $P_1$ then there are at most $3$ lines in the x-ray through $\phi(P_1)$.
If there were $3$ of these lines, then there would exist a $4$-dimensional manifold $X\in \mathcal{X}$, fixed by a circle inside $T^2$, containing $P_1$ and one additional fixed point. If there were $2$ lines then we would either have two $4$-dimensional manifolds $X_1,X_2\in \mathcal{X}$ (each fixed by a different circle inside $T^2$) containing $P_1$, or a $6$-dimensional manifold $X\in \mathcal{X}$ fixed by a circle inside $T^2$, containing $P_1$. In the first case, one of the manifolds $X_1$, $X_2$ could only have two fixed points and, in the latter, the manifold $X$ would have at most three fixed points. Finally, if there existed just one line through $\phi(P_1)$, it would be the image of an $8$-dimensional manifold with at most $4$ fixed points.
Since the minimal number of fixed points on a $2m$-dimensional $S^1$-Hamiltonian manifold is $m+1$ all the above cases are impossible. Therefore, we conclude that there must exist a line in the x-ray containing the images of $P_0$ and $P_1$. Hence, there exists a manifold $X\in \mathcal{X}$, a component of the set of points with a certain $1$-dimensional stabilizer, which contains $P_0$ and $P_1$. Note that, since the $T^2$-action is effective, we have $\dim X \leq 6$. Moreover, $P_0$ and $P_1$ are fixed points of the restriction of the original $S^1$-action on ${\mathsf{M}}$ to $X$, respectively of index $0$ and $2$. By the classification of Hamiltonian $S^1$-actions on $4$-dimensional manifolds with isolated fixed points [@K] and the classification of Hamiltonian $S^1$-actions on $6$-dimensional manifolds with a minimal number of fixed points [@T1], we conclude that there exists a multigraph for this restricted action on $X$ with an edge connecting $P_0$ and $P_1$.
Similarly, we can conclude that there exists a manifold $X^\prime \in \mathcal{X}$ which is a component of the set of points with a certain $1$-dimensional stabilizer, containing $P_4$ and $P_3$, and that there is a multigraph for the restricted $S^1$-action on $X^\prime$ with an edge connecting $P_3$ and $P_4$.
Since any multigraph for the original $S^1$-action on ${\mathsf{M}}$ restricts to multigraphs for the restrictions of the action to $X$ and $X^\prime$, we conclude that there exists a multigraph for the $S^1$-action on ${\mathsf{M}}$ which has an edge connecting $P_0$ to $P_1$ and another one connecting $P_3$ and $P_4$, thus satisfying $$\label{eq:index2}
\lambda({\mathsf{i}}(e))\leq \lambda({\mathsf{t}}(e)) \quad \text{for every $e\in E$.}$$
This argument can be easily adapted to prove the existence of a multigraph satisfying with a strict inequality. For that, let us consider all multigraphs with an edge connecting $P_0$ to $P_1$ and an edge connecting $P_3$ to $P_4$. Then we just have to show that among these there is a multigraph with no cycles at $P_2$.
Let us consider again the x-ray of $({\mathsf{M}},\omega,\phi)$. If there are $4$ lines through $\phi(P_2)$ then there exist four $2$-dimensional $S^1$-manifolds $X_i$ with different $1$-dimensional stabilizers, having $P_2$ in their fixed-point set. Since the only compact $2$-dimensional manifolds admitting a Hamiltonian circle action are spheres, where the circle acts by rotation, they all must have an additional fixed point different from $P_2$. Hence, the multigraphs for the restricted $S^1$-actions on $X_i$ consist of only one edge with $P_2$ at one of the endpoints and another fixed point at the other. We conclude that there exists a multigraph for the original $S^1$-action on ${\mathsf{M}}$ that has $4$ edges with $P_2$ as an endpoint and a different fixed point at the other end (thus with no cycles at $P_2$).
If there are $3$ lines through $P_2$ on the x-ray then we either have two $4$-dimensional manifolds $X_1,X_2\in \mathcal{X}$ (each fixed by a different circle inside $T^2$) containing $P_2$, or a $6$-dimensional manifold $X\in \mathcal{X}$ and a $2$-dimensional manifold $X^\prime$ fixed by two different circles inside $T^2$, with both $X$ and $X^\prime$ containing $P_2$. In the first case, both manifolds $X_1$, $X_2$ have exactly $3$ fixed points and, in the latter, the manifold $X$ has at most $4$ fixed points while $X^\prime$ has $2$.
By the classification of Hamiltonian $S^1$-actions on $4$-dimensional manifolds with isolated fixed points [@K] and the classification of Hamiltonian $S^1$-actions on $6$-dimensional manifolds with a minimal number of fixed points [@T1], we conclude that, in both situations, there exist multigraphs for the restricted $S^1$-actions on these submanifolds that have no cycles at $P_2$. We conclude that in all cases there exists a multigraph for the original $S^1$-action on ${\mathsf{M}}$ that has no cycles at $P_2$ and so this multigraph is necessarily positive.
Whenever ${\mathsf{M}}$ is $8$-dimensional with $5$ fixed points and none of the isotropy weights of the action is $1$:
\[not pm1\] Suppose that $({\mathsf{M}},\omega)$ is $8$-dimensional and the $S^1$-action has $5$ fixed points. If none of the isotropy weights of the action is $1$ then there exists a positive multigraph associated to the $S^1$-action.
Observe that, by Lemma \[pairs\], none of the weights is $-1$. A similar argument to the one in the proof of Proposition \[t2\] can be carried out, now using the closed orbit type stratification $\mathcal{X}$ of the $S^1$-manifold ${\mathsf{M}}$, where we consider the elements $X$ of $ \mathcal{X}$ that are connected components of the set of points with a given finite cyclic group ${{\mathbb{Z}}}_k$ as stabilizer ($k\neq 1$).
Let us assume that the negative weight at $P_1$ is $-k$ with $k>1$. We will show that there is at least one weight at $P_0$ that is equal to $k$. If this is not the case then the number of weights at $P_0$ which are multiples of $k$ (but different from $k$) has to be at least two but at most $3$ (since the action is effective). Let $X$ be the (connected) component of the points fixed by ${{\mathbb{Z}}}_k$ containing $P_0$ and $P_1$. If there were two weights at $P_0$ which were multiples of $k$ then $X$ would be a $4$-dimensional manifold admitting an effective $S^1\cong S^1/{{\mathbb{Z}}}_k$-action. This manifold would then have more than two non-trivial chains of gradient spheres[^1], which is impossible by Karshon’s classification results on Hamiltonian $S^1$-actions on $4$-manifolds [@K Proposition $5.13$]. (Indeed, the negative weight at $P_1$ for this effective action is $-1$ and none of the weights at $P_0$ is $1$.) If, on the other hand, there were $3$ weights at $P_0$ which were multiples of $k$, then $X$ would be $6$-dimensional. Then the effective Hamiltonian $S^1\cong S^1/{{\mathbb{Z}}}_k$-action on $X$ would have $4$ fixed points (since when $\dim {\mathsf{M}}=6$ and the number of fixed points is not minimal one must have, by , at least $6$ fixed points). By Tolman’s classification of Hamiltonian actions on $6$-manifolds with a minimal number of fixed points [@T1], none of these manifolds has a negative weight $-1$ at the point of index-$2$ and no weight equal to $1$ at the minimum. We conclude that, if the negative weight at $P_1$ is $-k$ with $k>1$, then there is at least one weight at $P_0$ that is equal to $k$.
Similarly, we can conclude that, if the positive weight at $P_3$ is $k>1$, then there is at least one weight at $P_4$ that is equal to $-k$ and so, given an $S^1$-action satisfying the assumptions above, we can always choose a multigraph that is non-negative. To show that we can always choose one that is positive we still have to show that we can choose a multigraph with no cycles at $P_2$.
For that, let us assume that we have chosen a multigraph with an edge $e_{01}$ connecting $P_0$ to $P_1$ and an edge connecting $P_3$ to $P_4$ and that, at $P_2$, the action has a weight $k$ and a weight $-k$ with $k>1$. Let $X$ be the (connected) component of the points fixed by ${{\mathbb{Z}}}_k$ containing $P_2$ and consider the effective Hamiltonian $S^1\cong S^1/{{\mathbb{Z}}}_k$-action on $X$. This action has $P_2$ as an index-$2$ fixed point and so the index-$0$ point has to be $P_0$ or $P_1$. Let $W_0=\{w_{01},\ldots,w_{04}\}$ be the multiset of weights at $P_0$ with $w_{01}$ the weight associated to the edge $e_{01}$, and let $W_1=\{w_{11},w_{12},\ldots,w_{14}\}$ be the multiset of weights at $P_1$ with $w_{11}=-w_{01}$. If there is no weight in $W_0\setminus \{w_{01}\}\cup W_1\setminus \{w_{10}\}$ that is equal to $k$, then, again by Karshon’s classification of Hamiltonian $S^1$-actions on $4$-manifolds [@K] and Tolman’s classification of Hamiltonian $S^1$-actions on $6$-manifolds with a minimal number of fixed points [@T1], we get a contradiction. The result then follows.
[**v)**]{} Whenever the $S^1$-action extends to a “GKM action" on ${\mathsf{M}}$ with a minimal number of fixed points: Then there exists a natural multigraph associated to ${\mathsf{M}}$ that, when ${\mathsf{M}}$ has a minimal number of fixed points, is the *complete graph* on the set of vertices given by the fixed points (hence positive).
Indeed, let $({\mathsf{M}},\omega,\phi)$ be a compact symplectic manifold with a Hamiltonian $T$-action and isolated fixed points, where $\dim(T)>1$. For every $K\subseteq T$ let ${\mathsf{M}}^K$ be the points fixed by $K$, $\mathfrak{t}$ the Lie algebra of $T$, and $\mathfrak{t}^*$ its dual. Then ${\mathsf{M}}$ is a **GKM (Goresky-Kottwitz-MacPherson) manifold** [@GKM] if the weights $\alpha_{1},\ldots,\alpha_{n}\in \mathfrak{t}^*$ of the isotropy representation of $T$ on $T_P{\mathsf{M}}$ are pairwise linearly independent, for every fixed point $P$. This is equivalent to saying that, for every codimension one subgroup $K\subset T$, the connected components of ${\mathsf{M}}^K$ are either points or 2-spheres. This class of spaces include two families of well-known spaces, coadjoint orbits of a simple Lie group $G$ endowed with the action of a maximal torus $T\subset G$, and toric symplectic manifolds.
Fix $P\in {\mathsf{M}}^T$ and let $\alpha_{1},\ldots,\alpha_{n}\in \mathfrak{t}^*$ be the weights of the isotropy representation of $T$ on $T_P{\mathsf{M}}$. Let $S_{i}$ be the 2-sphere fixed by $\exp(\ker \alpha_i)\subset T$ for every $i=1,\ldots,n$. Then the $T$-action on $S_{i}$ has two fixed points $P$ and $Q$, and the weight of the $T$ representation on $T_Q(S_{i})$ is $-\alpha_i$. Assume that the $S^1$-action we are starting from is the action of a generic circle inside $T$, and let $\psi\colon {\mathsf{M}}\to {\mathbb{R}}$ be its moment map. Then, we can associate to each of these two spheres an edge going from $P$ to $Q$, with $\psi(P)<\psi(Q)$, which is equivalent to saying that the weight of the $S^1$-action at $P$ is positive. Since $\alpha_1,\ldots,\alpha_n$ are pairwise linearly independent, it follows that $S_i\cap S_j=\{P\}$ for every $1\leq i<j\leq n$, and, if ${\mathsf{M}}$ has exactly $n+1$ fixed points, the graph constructed above must be a complete, hence positive, multigraph. (For a classification of GKM manifolds with a minimal number of fixed points see [@Mo].)
\[l=1\] Notice that for every positive multigraph with a minimal number of fixed points, there must exist an edge $e_{01}$ from $P_0$ to $P_1$, and an edge $e_{n\,n+1}$ from $P_{n}$ to $P_{n+1}$. Moreover, from the definitions in Sections \[dmg\] and \[hamiltonian minimal\], it follows that ${\mathsf{m}}(e_{01})={\mathsf{m}}(e_{n\,n+1})=C_1$, and so ${\mathsf{l}}(e_{01})={\mathsf{l}}(e_{n\,n+1})=1$.
\[upper bound c1\^n\] Let $({\mathsf{M}},\omega)$ be a $2n$-dimensional compact symplectic manifold with a Hamiltonian $S^1$-action and fixed points $P_0,\ldots,P_n$. Suppose that there exists an integral multigraph $\Gamma$ associated to the action which is the complete graph on $n+1$ vertices (hence positive). Then it is easy to see that, by the ABBV Localization formula, for every $i=0,\ldots,n$, we have $$\label{c1^n}
\int_{\mathsf{M}}{\mathsf{c}}_1^n=\int_{\mathsf{M}}\prod_{j\neq i}\left({\mathsf{c}}_1^{S^1}-{\mathsf{c}}_1^{S^1}(P_j)\right)=\frac{\prod_{j\neq i}\left({\mathsf{c}}_1^{S^1}(P_i)-{\mathsf{c}}_1^{S^1}(P_j)\right)}{\Lambda_i}=\prod_{h=1}^n{\mathsf{m}}(e_h),$$ where $e_1,\ldots,e_n$ is the set of edges ending or starting at $P_i$, and each ${\mathsf{m}}(e_h)$ is a positive integer. By in Theorem \[sum of m\], we have $$\begin{aligned}
\sum_{h=1}^n{\mathsf{m}}(e_h) & \leq \sum_{e\in E}{\mathsf{m}}(e) - \left(\frac{n(n+1)}{2}-n \right)\\
& \leq \frac{n(n+1)^2}{2} - \frac{n(n-1)}{2} = \frac{n(n^2+n+2)}{2},\end{aligned}$$ since the graph has $n(n+1)/2$ edges and $m(e)\geq 1$. Then we have $$\int_{\mathsf{M}}{\mathsf{c}}_1^n=\prod_{h=1}^n{\mathsf{m}}(e_h)\leq \left( \frac{n^2+n+2}{2}\right)^n.$$ The same argument can be carried out when the action has an integral multigraph $\Gamma$ for which there exists a vertex $P_i$ connected to the other $n$ vertices of $\Gamma$ through $n$ (undirected) edges of $\Gamma$. It would be interesting to generalize this estimate to the case in which the multigraph does not necessarily satisfy the above property (see also Remark \[225\]).
A refinement of the algorithm {#refinement}
-----------------------------
In all the above situations or, in general, whenever we are able to guarantee that, within a given class of $S^1$-actions with minimal number of fixed points, there always exists a non-negative multigraph for each action, we can use Algorithm \[alg:1\] with slight improvements.
\[alg:5\] For each possible multigraph $\Gamma$ with connected components $\Gamma_1,\ldots,\Gamma_l$, we look for partitions of $$\frac{1}{2}n(n+1)^2$$ into $\lvert E^\prime\rvert=\lvert E\rvert - \lvert E^{\circlearrowleft} \rvert$ positive integers ${\mathsf{m}}(e)$. Then we add zeros ${\mathsf{m}}(e)=0$ whenever the edge $e$ is a cycle and choose, among the resulting sequences of $\lvert E \rvert$ nonegative numbers, those for which $$\Big( \operatorname{Null}\left(A(\Gamma_i)-\operatorname{diag}\left({\mathsf{m}}(E_i)\right)\right)\Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E_i\rvert }\neq \emptyset$$ for every $i=1,\dots,l$.
If, in addition, we know that there exists an edge $e_{01}$ from $P_0$ to $P_1$ and/or an edge $e_{n\,n+1}$ between $P_n$ and $P_{n+1}$ (for instance when the multigraph is positive), then we can use Remark \[l=1\] to further improve the algorithm, adapting Algorithm \[alg:2\] to this situation. Here note that, when we have a minimal number of fixed points, the number of cycles $\lvert E^{\circlearrowleft} \rvert$ of a non-negative multigraph satisfies $$\begin{aligned}
\lvert E^{\circlearrowleft} \rvert & \leq \sum_{p=0}^n \min\{p,n-p\} = \sum_{p=1}^{\lfloor n/2 \rfloor} p + \sum_{p=\lfloor n/2 \rfloor + 1}^{n-1} (n-p) \\ & = \lfloor n/2 \rfloor^2-(n-1)\lfloor n/2 \rfloor + \frac{n^2-n}{2} \\
& = \left\{ \begin{array}{ll} \frac{n^2}{4}, & \text{if $n$ is even} \\ \\ \frac{n^2-1}{4}, & \text{if $n$ is odd} \end{array} \right. \;,\end{aligned}$$ and so the constant $C$ in satisfies $$1\leq C \leq \frac{n(n+1)^2}{n(n+1)-2 \lvert E^{\circlearrowleft} \rvert } \leq \left\{ \begin{array}{ll} 2n+1, & \text{if $n$ is even} \\ \\ 2n, & \text{if $n$ is odd} \end{array} \right.\;.$$ However, by Proposition \[psi e c\], we know that the map $\Psi$ defined in is injective and so we can use Theorem \[hattori\] to obtain a better upper bound for $C$, $$1\leq C \leq n+1,$$ which is also the same as the one obtained when $\lvert E^{\circlearrowleft} \rvert =0$.
\[alg:6\] Given a graph $\Gamma$ with connected components $\Gamma_1,\ldots,\Gamma_l$ with an edge $e_{01}$ from $P_0$ to $P_1$ (and/or an edge $e_{n\,n+1}$ between $P_n$ and $P_{n+1}$), for each divisor $C$ of $$\frac{1}{2}n(n+1)^2$$ satisfying $$1\leq C\leq n+1,$$ we look for partitions of $$\frac{1}{2C}n(n+1)^2$$ into $\lvert E^\prime \rvert =\lvert E \rvert - \lvert E^{\circlearrowleft} \rvert$ positive integers, ${\mathsf{l}}(e):=\frac{{\mathsf{m}}(e)}{C}$ with ${\mathsf{l}}(e_{01})=1$ (and/or ${\mathsf{l}}(e_{n\, n+1})=1$). Then we consider the corresponding integers ${\mathsf{m}}(e)=C\cdot {\mathsf{l}}(e)$, add zeros ${\mathsf{m}}(e)=0$ whenever the edge $e$ is a cycle, and choose, among the resulting sequences of $\lvert E \rvert$ numbers, the ones for which $$\Big( \operatorname{Null}\left(A(\Gamma_i)-\operatorname{diag}\left({\mathsf{m}}(E_i)\right)\right)\Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E_i\rvert}\neq \emptyset,$$ for every $i=1,\dots,l$.
Suppose now that we cannot guarantee the existence of non-negative multigraphs for every circle action considered and so we have to deal with the possibility that the ${\mathsf{m}}(e)$s might be negative. Then we can use Algorithms \[alg:3\] and \[alg:4\] in Section \[algo\], noting that, in the case of Algorithm \[alg:4\], the map $\Psi$ defined in is now injective by Proposition \[psi e c\], and so, by Remark \[C bound\], we again have $1\leq C \leq n+1$.
Known examples of $S^1$-Hamiltonian manifolds with minimal number of fixed points {#known examples}
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Let us now describe the known examples of Hamiltonian circle actions with a minimal number of fixed points. However, since the first three examples arise as coadjoint orbits of simple Lie groups, we first review a few general facts.
Let $G$ be a compact simple Lie group with $\mathfrak{g}=Lie(G)$, and let $T\subset G$ be a maximal torus with $\mathfrak{t}=Lie(T)$ and $\mathfrak{t}^*=Lie(T)^*$. Let $\Delta\subset \mathfrak{t}^*$ be the set of roots of $G$, and $\Delta_0\subset \Delta$ a choice of simple roots. We use the Killing form to regard $\mathfrak{t}^*$ as a subspace of $\mathfrak{g}^*$. The coadjoint orbit through a point $p_0\in \mathfrak{t}^*$, $O_{p_0}$, is a compact manifold with a natural symplectic structure given by the Kostant-Kirillov symplectic form $\omega$. It also inherits a natural Hamiltonian $G$-action, whose moment map is given by the inclusion $O_{p_0}\hookrightarrow \mathfrak{g}^*$. Thus, restricting the action to a generic circle $S^1\subset T$, the compact symplectic manifold $(O_{p_0},\omega)$ has a Hamiltonian $S^1$-action with moment map $\psi\colon O_{p_0}\hookrightarrow\mathfrak{g}^*\to Lie(S^1)^*\simeq {\mathbb{R}}^*$, where the second map is the projection map.
\[cpn\][**(The complex projective space)**]{}\
Let $G=SU(n+1)$, and $T^n\subset G$ the torus of diagonal matrices. Let $\{x_0,\ldots,x_n\}$ be the standard basis of $({\mathbb{R}}^{n+1})^*$. Then a standard choice of simple roots is $$\Delta_0=\{x_i-x_{i+1}=\alpha_i\mid 0\leq i\leq n-1\}.$$ Let $p_0$ be a generic point in $\cap_{i=1}^{n-1}\mathcal{H}_{\alpha_i}$, where $\mathcal{H}_{\alpha_i}\subset \mathfrak{t}^*$ is the hyperplane orthogonal to $\alpha_i$. Then $O_{p_0}$ is isomorphic to ${{\mathbb{C}}}P^n$. Let $S^1\subset T$ be a generic circle generated by $\xi=(\xi_0,\ldots,\xi_n)\in \mathfrak{t}$ such that $\alpha_i(\xi)=\xi_i-\xi_{i+1}\in {{\mathbb{Z}}}_{>0}$ for every $i=0,\ldots,n-1$. A standard computation shows that the $S^1$-fixed points are $$\{P_i=[0,\ldots,\overbrace{1}^{i-th},\ldots,0]\mid i=0,\ldots,n\},$$ the set of weights at $P_i$ is given by $$\{\xi_i-\xi_j\mid 0\leq j\neq i\leq n\},$$ and $\lambda(P_i)=i$ for every $i=0,\ldots,n$.
A natural choice of integral multigraph is the complete graph on $n+1$ vertices, which is also the GKM graph associated to ${{\mathbb{C}}}P^n$ equipped with the $T^n$-action (see Section \[refinement\], [**v)**]{}).
It is well known that, in this case, the unique positive integer $C_1$ for which ${\mathsf{c}}_1/C_1$ is a generator of $H^2({{\mathbb{C}}}P^n;{{\mathbb{Z}}})$ is $n+1$, which agrees with .
\[gr\][**(The Grassmannian of oriented two planes in ${\mathbb{R}}^{2n+1}$)**]{}\
Let $G=SO(2n+1)$ and $T^n$ a maximal torus and let us identify $\mathfrak{t}^*$ with $({\mathbb{R}}^n)^*$ with standard basis $\{x_0,\ldots,x_{n-1}\}$. Then a choice of simple roots is given by $$\Delta_0=\{x_i-x_{i+1}=\alpha_i,\;i=0,\ldots,n-2\}\cup\{x_{n-1}=\alpha_{n-1}\}.$$ Let $p_0$ be a generic point in $\cap_{i=1}^{n-1}\mathcal{H}_{\alpha_i}$, where $\mathcal{H}_{\alpha_i}\subset \mathfrak{t}^*$ is the hyperplane orthogonal to $\alpha_i$. Then $O_{p_0}$ is isomorphic to $Gr_2^+({\mathbb{R}}^{2n+1})$, the Grassmannian of oriented two planes in ${\mathbb{R}}^{2n+1}$, which is a symplectic manifold of dimension $2(2n-1)$. Let $S^1\subset T$ be a generic circle generated by $\xi=(\xi_0,\ldots,\xi_{n-1})\in \mathfrak{t}$ such that $\alpha_i(\xi)\in {{\mathbb{Z}}}_{>0}$ for every $i=0,\ldots,n-1$. Then a standard computation shows that this action has $2n$ fixed points, which can be identified with the elements $y_i$s of $\mathfrak{t}^*$ given by $$y_0=-x_0,\;\ldots,y_{n-1}=-x_{n-1},\;y_n=x_{n-1},\;\ldots,\;y_{2n-1}=x_0\,.$$ Moreover, the weights at $y_i$ are given by $$\{(y_j-y_i)(\xi)\mid j\neq i,\;j\neq 2n-1-i\}\cup\{-y_i(\xi)\},$$ and $\lambda(y_i)=i$ for every $i$. Observe that, since $Gr_2^+({\mathbb{R}}^{2n+1})$ is a Hamiltonian $S^1$-manifold with a minimal number of fixed points, by the argument in Section \[ec symplectic\], we have $$H^j(Gr_2^+({\mathbb{R}}^{2n+1});{{\mathbb{Z}}})=H^j({{\mathbb{C}}}P^{2n-1};{{\mathbb{Z}}})$$ for every $j=0,\ldots,2(2n-1)$. However the cohomology ring is different.
As in the previous example, a natural choice of integral multigraph is the complete graph on $2n$ vertices, which is also the GKM graph associated to $Gr_2^+({\mathbb{R}}^{2n+1})$ equipped with the $T^n$-action (see Section \[refinement\], [**v)**]{}).
It is well known that, in this case, the unique positive integer $C_1$ such that ${\mathsf{c}}_1/C_1$ is a generator of $H^2(Gr_2^+({\mathbb{R}}^{2n+1});{{\mathbb{Z}}})$ is $n$, which agrees with the definition of $C_1$ in .
\[g2\] Let $G_2$ be the exceptional simple Lie group and $T^2$ a maximal torus. Let $p_0$ be a generic point in $\mathcal{H}_{\alpha_1}$, where $\alpha_1$ denotes the short simple root, and $\mathcal{H}_{\alpha_1}$ the hyperplane orthogonal to it. Then $O_{p_0}$ is a $10$-dimensional coadjoint orbit with an $S^1\subset T^2$-Hamiltonian action and $6$ fixed points. The $T^2$-action is also GKM, and hence a natural choice of multigraph is the complete multigraph on six vertices (for more details see [@Mo]).
\[fano\][**(The Fano manifolds $V_5$ and $V_{22}$)**]{}\
Let $({\mathsf{M}}^{6},\omega)$ be a $6$-dimensional compact symplectic manifold with a Hamiltonian $S^1$-action and $4$ fixed points. Let $C_1$ be the unique positive integer $C_1$ such that ${\mathsf{c}}_1/C_1$ is a generator of $H^2({\mathsf{M}};{{\mathbb{Z}}})$ (see Remark \[Ci’s\]). As we will see in in Proposition \[symp hattori\], $C_1$ can be either $1,2,3$ or $4$. As examples of manifolds with $C_1$ equal to $3$ and $4$ we have respectively $Gr_2^+({\mathbb{R}}^5)$ and ${{\mathbb{C}}}P^3$ (see Examples \[gr\] and \[cpn\]). For the remaining two cases we have two Fano $3$-folds, which are known as $V_5$ and $V_{22}$. Indeed, McDuff [@M] proved that they can be explicitly given a symplectic structure, and they possess a Hamiltonian $S^1$-action. These Fano manifolds have the following properties:
- $V_5$: Here $C_1=2$, the cohomology ring is $$H^*(V_5;{{\mathbb{Z}}})={{\mathbb{Z}}}[x_1,x_2]/(x_1^2-5x_2,x_2^2),$$ and the isotropy weights at the fixed points are $$\label{v5}
\{\{1,2,3\},\;\{-1,1,4\},\;\{-1,-4,1\},\;\{-1,-2,-3\}\}\,.$$
- $V_{22}$: Here $C_1=1$, the cohomology ring is $$H^*(V_{22};{{\mathbb{Z}}})={{\mathbb{Z}}}[x_1,x_2]/(x_1^2-22x_2,x_2^2),$$ and the isotropy weights at the fixed points are $$\label{v22}
\{\{1,2,3\},\;\{-1,1,5\},\;\{-1,-5,1\},\;\{-1,-2,-3\}\}\,.$$
There is exactly one positive multigraph associated to these two actions, which is shown in Case III of Figure \[graphs6\].
Implications of Hattori’s results in the symplectic category {#hattori results}
============================================================
In this section we recall some of the results obtained by Hattori in [@Ha], and show how they can be used to obtain relations among the Chern numbers of a compact symplectic manifold $({\mathsf{M}},\omega)$ endowed with a Hamiltonian circle action and minimal number of fixed points. In particular, in Theorem \[m not 2\], we derive their consequences in the case in which ${\mathsf{M}}$ is $8$-dimensional.
Let $({\mathsf{M}},{\mathsf{J}})$ be a compact almost complex manifold of dimension $2n$, equipped with an $S^1$-action which preserves ${\mathsf{J}}$, and has isolated fixed points $P_0,\ldots,P_N$. Then the set of weights of the $S^1$-representation on $T_{P_i}{\mathsf{M}}$ is well-defined for all $i$. Let $\{w_{i1},\ldots,w_{in}\}$ be the multi-set of weights at $P_i$.
Let ${\mathbb{L}}$ be an admissible complex line bundle over ${\mathsf{M}}$ (see Section \[eclb\]). Then for any lift ${\mathbb{L}}^{S^1}$ there exist integers $a_0,\ldots,a_N$ such that $$\label{eq:ai}
{\mathbb{L}}^{S^1}(P_i)=t^{a_i}\quad\mbox{for every}\quad i=0,\ldots,N\;,$$ which are determined up to a constant. Following the terminology in [@Ha], we say that an admissible line bundle ${\mathbb{L}}$ is called *fine* if the 1-dimensional representations ${\mathbb{L}}^{S^1}(P_0),\ldots,{\mathbb{L}}^{S^1}(P_N)$ are pairwise distinct, i.e. if $a_i\neq a_j$ for every $i\neq j$. Moreover, it is called *quasi-ample* if it is fine and $\int_{{\mathsf{M}}}{\mathsf{c}}_1({\mathbb{L}})^n\neq 0$.
Let ${\mathbb{L}}$ be a fine complex line bundle and ${\mathbb{L}^{S^1}}$ an equivariant extension. For every $i=0,\ldots,N$ define $$\label{varphi}
\varphi_i(t):= \frac{\displaystyle\prod_{j\neq i}(1-t^{a_i-a_j})}{\displaystyle\prod_{k=1}^{n}(1-t^{w_{ik}})}\;.$$ Notice that $$\label{varphi index}
\varphi_i(t^{-1})=\operatorname{ind}_{S^1}\left(\prod_{j\neq i} \left(1-({\mathbb{L}}^{S^1})^{-1}t^{a_j}\right)\right)\in {{\mathbb{Z}}}[t,t^{-1}]\,,$$ and so $\varphi_i(t)\in {{\mathbb{Z}}}[t,t^{-1}]$.
In the following we recall Theorems $4.2$ and $5.7$ in [@Ha], specializing them to the case of almost complex manifolds as well as other useful facts proved in [@Ha].
\[hattori\] Let $({\mathsf{M}},{\mathsf{J}})$ be a compact almost complex manifold of dimension $2n$, equipped with an $S^1$-action which preserves ${\mathsf{J}}$ and has isolated fixed points $P_0,\ldots,P_N$. Let ${\mathbb{L}}$ be a fine complex line bundle, and for every $i=0,\ldots,N$, let $\varphi_i(t)\in {{\mathbb{Z}}}[t,t^{-1}]$ be as in . Then there exists a unique sequence $r_0(t),\ldots,r_N(t)$ of elements of ${{\mathbb{Z}}}[t,t^{-1}]$ such that $$\varphi_i(t)=r_0(t)+r_1(t)t^{a_i}+\cdots + r_N(t)t^{Na_i}\quad\mbox{for all}\quad i\;.$$ Moreover, the $r_s(t)$s satisfy the following properties:
1. \[r\_0\] $$r_0(t)=\operatorname{Todd}({\mathsf{M}})=\sum_{i=0}^N\binom{\lambda_i}{n}\;,$$ where $\lambda_i$ is the number of negative weights at $P_i$.
2. \[k\_0\] If there exists $k_0\in {{\mathbb{Z}}}_{\geqslant 0}$ and $d\in {{\mathbb{Z}}}$ such that $$\label{D1}
\sum_{k=1}^nw_{ik}=k_0a_i+d\quad\mbox{for all}\quad i=0,\ldots,N\;,$$ then $k_0\leqslant N+1$.
3. In , if $k_0>0$, then, setting $l_0=N+1-k_0$, we have $$r_s(t)=0\quad\mbox{for all}\quad s>l_0\;,$$ and $$r_{l_0-s}(t)=(-1)^{N-n}r_s(t^{-1})t^{-(d+\sum a_j)}\quad\mbox{for}\quad s\leqslant l_0\;.$$
4. In , if $k_0=0$, then $r_0=0$ and $$r_{N+1-s}(t)=(-1)^{N-n}r_s(t^{-1})t^{-(d+\sum a_j)}\quad\mbox{for}\quad 1\leqslant s\leqslant N\;.$$
More explicitly, the functions $r_s(t)$s are given by $$\label{def expl r}
r_s(t)=(-1)^s\sum_{i=0}^N\frac{\displaystyle\sum_{j_1<\cdots <j_s,j_{\nu}\neq i}t^{-(a_{j_1}+\cdots +a_{j_s})}}{\prod_{k=1}^n(1-t^{w_{ik}})}.$$ For every $s=0,\ldots,N$, let $k_s({\mathbb{L}^{S^1}},t)$ be the $S^1$-equivariant bundle associated to ${\mathbb{L}}^{S^1}$ defined inductively on $s$ by $$\label{k0}
k_0({\mathbb{L}}^{S^1},t)=1$$ and $$\begin{aligned}
k_s({\mathbb{L}}^{S^1},t)=\displaystyle\sum_{j_1<\cdots< j_s} & (t^{a_{j_1}}-{\mathbb{L}}^{S^1})\cdots (t^{a_{j_s}}-{\mathbb{L}}^{S^1}) \nonumber \\
\label{k ind}&-\displaystyle\sum_{\nu=1}^s\binom{N-s+\nu}{\nu}(-{\mathbb{L}}^{S^1})^{\nu}k_{s-\nu}({\mathbb{L}}^{S^1},t)\;.\end{aligned}$$ Then, for every $s=0,\ldots,N$, the functions $r_s(t)$s satisfy $$\label{explicit r}
r_s(t^{-1})=(-1)^s\operatorname{ind}_{S^1}\left(k_s({\mathbb{L}^{S^1}},t)\right)\;.$$
\[hattori2\] Let $({\mathsf{M}},{\mathsf{J}})$ be a compact almost complex manifold of dimension $2n$, equipped with an $S^1$-action which preserves ${\mathsf{J}}$ and has isolated fixed points. Assume the Euler characteristic of ${\mathsf{M}}$ is $n+1$. If ${\mathbb{L}}$ is a quasi-ample complex line bundle satisfying with $k_0=n+1$ then the multisets of weights of the $S^1$-action at each fixed point $P_i$ are given by $$\{w_{ik}\} = \{ a_i-a_j\}_{j\neq i},$$ where $a_0,\ldots,a_n$ are defined up to a constant as in . In particular, the multisets of weights coincide with those of ${{\mathbb{C}}}P^n$ with the standard circle action described in Example \[cpn\] (with ${\mathbb{L}}$ the hyperplane bundle).
In the following, we derive the consequences of Theorem \[hattori\] when $({\mathsf{M}},\omega)$ is a compact symplectic manifold of dimension $2n$ with a Hamiltonian $S^1$-action and a minimal number of fixed points $P_0,\ldots,P_n$. As usual, we endow $({\mathsf{M}},\omega)$ with an almost complex structure ${\mathsf{J}}$ compatible with $\omega$, which is invariant under the $S^1$-action.
\[ipp\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold with a Hamiltonian $S^1$-action with a minimal number of fixed points. Let $\psi\colon {\mathsf{M}}\to {\mathbb{R}}$ be the moment map. We say that $\omega-\psi\otimes x$ is
- **integral** if $[\omega-\psi\otimes x]\in H_{S^1}^2({\mathsf{M}};{{\mathbb{Z}}})$ (hence $[\omega]\in H^2({\mathsf{M}};{{\mathbb{Z}}})$);
- **primitive** if it is integral and $[\omega]$ is a generator of $H^2({\mathsf{M}};{{\mathbb{Z}}})$;
- **positive** if ${\mathsf{c}}_1$ is a positive multiple of $[\omega]$.
\[w=tau\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold with a Hamiltonian $S^1$-action and moment map $\psi\colon {\mathsf{M}}\to {\mathbb{R}}$. If the number of fixed point is minimal, it is not restrictive to assume that $\omega-\psi\otimes x$ is primitive and positive. More precisely, let $\tau_1$ and ${\widetilde{\tau}}_1$ be the classes defined in and . Then we can assume that $[\omega-\psi\otimes x]=\tau_1$, and so $[\omega]={\widetilde{\tau}}_1$ and ${\mathsf{c}}_1=C_1[\omega]$.
Since $H^2({\mathsf{M}};{{\mathbb{Z}}})={{\mathbb{Z}}}$ and $[\omega]\neq 0$, we can rescale the symplectic form $\omega$ in such a way that $[\omega]$ is an integral generator of $H^2({\mathsf{M}};{{\mathbb{Z}}})$. So $[\omega]=\pm \widetilde{\tau}_1$. Since the kernel of the restriction map is the ideal generated by $x$, we have that $[\omega-\psi\otimes x]=\pm \tau_1+P(x)$, where $P(x)$ is a constant polynomial in $x$. Hence, modulo translating the moment map $\psi$, we can assume that $[\omega-\psi\otimes x]=\pm\tau_1\in H^2_{S^1}({\mathsf{M}};{{\mathbb{Z}}})$. By definition of $\tau_1$ we have that $\tau_1(P_0)=0$ and $\tau_1(P_1)=k\,x$, where $k\in {{\mathbb{Z}}}_{<0}$, and $\psi(P_0)<\psi(P_1)$. We can then conclude that $[\omega-\psi\otimes x]=\tau_1$.
The next proposition is an easy consequence of Theorem \[hattori\].
\[symp hattori\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $2n$ with a Hamiltonian $S^1$-action and moment map $\psi\colon{\mathsf{M}}\to {\mathbb{R}}$. Assume that there is a minimal number of fixed points $P_0,\ldots,P_n$ and that $\omega-\psi\otimes x$ is primitive and positive. Let $C_1$ be the positive integer defined in . Then,
- there exists a quasi-ample complex line bundle ${\mathbb{L}}$ such that ${\mathsf{c}}_1^{S^1}({\mathbb{L}^{S^1}})=[\omega-\psi\otimes x]$ and ${\mathbb{L}^{S^1}}(P_i)=t^{-\psi(P_i)}$;
- there exists $d\in {{\mathbb{Z}}}$ such that $$\label{D}
\sum_{k=1}^nw_{ik}=-C_1\psi(P_i)+d\quad\mbox{for all}\quad i=0,\ldots,n\;,$$ and $$\label{bound C1}
1\leqslant C_1 \leqslant n+1\;.$$
\(i) The existence of a complex line bundle ${\mathbb{L}}$ such that the $S^1$-action lifts to ${\mathbb{L}}$ and ${\mathsf{c}}_1^{S^1}({\mathbb{L}^{S^1}})=[\omega-\psi\otimes x]\in H^2_{S^1}({\mathsf{M}};{{\mathbb{Z}}})$ is a direct consequence of Theorem \[lift clb\]. It follows that ${\mathbb{L}^{S^1}}(P_i)=t^{-\psi(P_i)}$, and hence ${\mathbb{L}^{S^1}}$ is fine by . Finally, ${\mathbb{L}}$ is quasi-ample because $\int_{\mathsf{M}}{\mathsf{c}}_1({\mathbb{L}})^n=\int_{\mathsf{M}}[\omega]^n\neq 0$.\
(ii) By Lemma \[w=tau\] we can assume that $[\omega-\psi\otimes x]=\tau_1$, thus $$\left( \sum_{k}w_{ik} \right)x = {\mathsf{c}_{1}^{S^1}}(P_i) = C_1\tau_1(P_i)+{\mathsf{c}_{1}^{S^1}}(P_0)=\left( -C_1\psi(P_i)+d \right) x\;.$$ Then, by $(2)$ in Theorem \[hattori\], we have that $1\leqslant C_1\leqslant n+1$.
The bundle ${\mathbb{L}}$ is usually called the *pre-quantization line bundle for* $({\mathsf{M}},\omega)$, and its equivariant extension ${\mathbb{L}}^{S^1}$ the *$S^1$-equivariant pre-quantization line bundle for* $({\mathsf{M}},\omega,\psi)$ (cf. [@GKS]).
Combining Theorem \[hattori\] with Proposition \[symp hattori\] we obtain the following result.
\[equations sym\] Let $({\mathsf{M}},\omega)$ be as in Proposition \[symp hattori\] and let ${\mathbb{L}}^{S^1}$ be the $S^1$-equivariant pre-quantization line bundle. For every $s=0,\ldots,n$, let $r_s(t)\in{{\mathbb{Z}}}[t,t^{-1}]$ be the element associated to ${\mathbb{L}^{S^1}}$, as defined in , and let $l_0=n+1-C_1$. Then $$\begin{aligned}
\bullet& \label{sum varphi} \int_{\mathsf{M}}[\omega]^n =\;\;r_0(1)+r_1(1)+\cdots +r_{l_0}(1);\\
\bullet & \label{r0=1} \,\, r_0(1) =\;\;\operatorname{Todd}({\mathsf{M}})=1;\\
\bullet &\,\, r_{s}(1) = \;\;r_{l_0-s}(1) \quad\mbox{for all}\quad 0\leqslant s\leqslant l_0\quad\mbox{and}\quad
\label{sym r} r_s(t)=\;\;0\quad\mbox{for all}\quad s>l_0\;.\end{aligned}$$
By Proposition \[symp hattori\], we can apply Theorem \[hattori\] to ${\mathbb{L}}$. Thus and are direct consequences of Theorem \[hattori\], which also implies that $$\varphi_i(1)=r_0(1)+r_1(1)+\cdots +r_{l_0}(1)\,,$$ where $\varphi_i(t)\in {{\mathbb{Z}}}[t,t^{-1}]$ is defined in . Hence, we just have to prove that $\varphi_i(1)=\int_{\mathsf{M}}[\omega]^n$. Consider the commutative diagram . By we have that $$\label{varphi(1)}\varphi_i(1)= r\left(\operatorname{ind}_{S^1}\left( \prod_{j\neq i}\left(1-({\mathbb{L}}^{S^1})^{-1}t^{a_j}\right)\right)\right)=\operatorname{ind}\left(1-{\mathbb{L}}^{-1}\right)^n.$$ Using the Atiyah-Singer formula, and the fact that $\operatorname{Ch}(1-{\mathbb{L}}^{-1})=\sum_{k=1}^{\infty}(-1)^{k+1}\frac{[\omega]^k}{k!}$, we obtain $$\varphi_i(1)
=\operatorname{ind}\left(1-{\mathbb{L}}^{-1}\right)^n
=\int_{{\mathsf{M}}}\operatorname{Ch}(1-{\mathbb{L}}^{-1})^n\operatorname{\mathcal{T}}({\mathsf{M}})=\int_{\mathsf{M}}[\omega]^n\,,$$ which completes the proof (cf. [@Ha Lemma 3.6]).
By an argument similar to the proof of Corollary \[equations sym\], equations and can also be turned into equations involving the Chern numbers of the manifold. Namely, by we have that, for every $i=0,\ldots,n$, $$\begin{aligned}
\label{r(1)}r_i(1)= & (-1)^ir\left(\operatorname{ind}_{S^1}\left(k_i({\mathbb{L}^{S^1}},t)\right)\right)=(-1)^i\operatorname{ind}\left(k_i({\mathbb{L}},1)\right)\;.\end{aligned}$$ However, the explicit computation of $\operatorname{ind}(k_i({\mathbb{L}},1))$ is harder to do in general. In Theorem \[m not 2\] we will compute these values in the case in which ${\mathsf{M}}$ is $8$-dimensional. For that we first have to prove the following results.
\[integral square\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $2n$ with a Hamiltonian $S^1$-action with $n+1$ fixed points. Let $\tau$ be an element of $H^2({\mathsf{M}};{{\mathbb{Z}}})$.
If $n$ is *even* then $$\quad \int_{\mathsf{M}}\tau^n=Q^2\,,$$ where $Q$ is the unique integer such that $\tau^{n/2}=Q{\widetilde{\tau}}_{n/2}$ (see Theorem \[bases\]).
Le $\{{\widetilde{\tau}}_i\}_{i=0}^n$ be the basis of $H^*({\mathsf{M}};{{\mathbb{Z}}})$ defined in and $\{{\widetilde{\tau}}^\prime_i\}_{i=0}^n$ the basis of $H^*({\mathsf{M}};{{\mathbb{Z}}})$ obtained by reversing the flow (cf. Section \[reversing flow\]). Then, by , ${\widetilde{\tau}}_{n/2}={\widetilde{\tau}}^\prime_{n/2}$, and hence implies that $\int_{\mathsf{M}}{\widetilde{\tau}}_{n/2}^2=\int_{\mathsf{M}}{\widetilde{\tau}}_{n/2}{\widetilde{\tau}}^\prime_{n/2}=1$, and the result follows.
Corollary \[equations sym\] together with Lemma \[integral square\] imply the following proposition.
\[integrals\] Let $({\mathsf{M}},\omega)$ be as in Proposition \[symp hattori\]. Then,
- if $C_1=n+1$ we have $\int_{\mathsf{M}}[\omega]^n=1$;\
$\;$
- if $C_1=n$ we have $\int_{\mathsf{M}}[\omega]^n=2$ and $n$ is *odd*.
\(i) If $C_1=n+1$, then by in Corollary \[equations sym\], we have $r_s(t)=0$ for all $s>0$, and then, by and , it follows that $\int_{\mathsf{M}}[\omega]^n=r_0(1)=1$.
\(ii) If $C_1=n$, then, by Corollary , we have $r_s(1)=0$ for all $s>1$ and $r_0(1)=r_1(1)$. Combining and we have that $\int_{\mathsf{M}}[\omega]^n=r_0(1)+r_1(1)=2$, and, since $[\omega]\in H^2({\mathsf{M}};{{\mathbb{Z}}})$, Lemma implies that $n$ is odd.
Recall that, when $({\mathsf{M}},\omega)$ is a compact symplectic manifold of dimension $2n$ with a Hamiltonian $S^1$-action with a minimal number of fixed points, we have $H^i({\mathsf{M}};{{\mathbb{Z}}})=H^i({{\mathbb{C}}}P^n;{{\mathbb{Z}}})$ for every $i$ (see Section \[ec symplectic\]). The next proposition shows what is the minimal information required to compute the ring structure of $H^*({\mathsf{M}};{{\mathbb{Z}}})$ and the total Chern class when ${\mathsf{M}}$ is of $8$-dimensional.
For an $8$-dimensional manifold, the total Todd class is $$\label{totalTodd}
\operatorname{\mathcal{T}}({\mathsf{M}})=\sum_{i=0}^4T_0^i=1+\frac{{\mathsf{c}}_1}{2}+\frac{{\mathsf{c}}_1^2+{\mathsf{c}}_2}{12}+\frac{{\mathsf{c}}_1{\mathsf{c}}_2}{24}+\frac{-{\mathsf{c}}_1^4+4{\mathsf{c}}_1^2{\mathsf{c}}_2+3{\mathsf{c}}_2^2+{\mathsf{c}}_1{\mathsf{c}}_3-{\mathsf{c}}_4}{720}\; \in H^*({\mathsf{M}};{\mathbb{Q}}),$$ where $T_0^i$ is the term of degree $2i$ in $H^*({\mathsf{M}};{\mathbb{Q}})$ for all $i$, and the Todd genus is given by $$\label{Toddgenus}
\operatorname{Todd}({\mathsf{M}})=\int_{\mathsf{M}}\frac{-{\mathsf{c}}_1^4+4{\mathsf{c}}_1^2{\mathsf{c}}_2+3{\mathsf{c}}_2^2+{\mathsf{c}}_1{\mathsf{c}}_3-{\mathsf{c}}_4}{720}$$ (see Section \[c1cn-1\]).
\[cr & Cc\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $8$, with a Hamiltonian $S^1$-action and $5$ fixed points.
- Let $C_1$ and $C_2$ be the constants defined in , and $l:=C_2/C_1^2$. Then $l\in {{\mathbb{Z}}}_{>0}$ and $$H^*({\mathsf{M}};{{\mathbb{Z}}})={{\mathbb{Z}}}[x_1,x_2,x_3]/(x_1^2-l\,x_2,x_1x_3-x_2^2,x_1^5,x_2^3,x_3^2,x_2x_3)\;,$$ where $x_1,x_2,x_3$ have degrees respectively $2$, $4$ and $6$.
- There exists $m\in {\mathbb{Q}}$ such that ${\mathsf{c}}_2=m\,x_1^2$, and the total Chern class is given by $${\mathsf{c}}(T{\mathsf{M}})=1+C_1x_1+(l\,m)x_2+(50/C_1)x_3+5x_1x_3\;.$$ Moreover $C_1$, $m$ and $l$ satisfy $$\label{T04}
l^2(-C_1^4+4C_1^2m+3m^2)-675=0\;.$$
In particular, $H^*({\mathsf{M}};{{\mathbb{Z}}})\simeq H^*({{\mathbb{C}}}P^4;{{\mathbb{Z}}})$ as rings if and only if $l=1$, and the total Chern class ${\mathsf{c}}(T{\mathsf{M}})$ agrees with the one of ${{\mathbb{C}}}P^4$, if and only if $C_1=5$ and $l\,m=10$.
\(i) Let $x_1:={\widetilde{\tau}}_1$, $x_2:={\widetilde{\tau}}_2$ and $x_3:={\widetilde{\tau}}_3$ (see Theorem \[bases\]). Then it is easy to see that $x_1^2=(C_2/C_1^2)x_2\,$. Moreover, since $C_2$ is positive, it follows that $l\in {{\mathbb{Z}}}_{>0}$. Since $H^8({\mathsf{M}};{{\mathbb{Z}}})={{\mathbb{Z}}}$, it follows from that $x_1\,x_3=x_2^2={\widetilde{\tau}}_4$, and $x_1^5=x_2^3=x_3^2=x_2x_3=0$ by dimensional reasons. Moreover, using , it is easy to see that these relations imply $x_1^3=l^2\,x_3$ and $x_1\,x_2=l\,x_3$.
\(ii) By (i) $x_1^2\neq 0$, so there exists $m\in {\mathbb{Q}}$ such that ${\mathsf{c}}_2=m\,x_1^2=(m\,l)x_2$ (which implies in particular that $m\,l$ is an integer). Let $\alpha\in {{\mathbb{Z}}}$ be such that ${\mathsf{c}}_3=\alpha\, x_3$. Then by in Corollary \[corollary1\] and , we have $$\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_3=C_1\alpha\int_{\mathsf{M}}x_1x_3=C_1\alpha=50.$$ Finally, let $\beta\in {{\mathbb{Z}}}$ be such that ${\mathsf{c}}_4=\beta \,x_1x_3$. Then, by and , we have $$\int_{\mathsf{M}}{\mathsf{c}}_4=\beta=5.$$ Now by in Corollary \[equations sym\] and , we have that $$\label{eqTodd}
\int_{\mathsf{M}}\frac{-{\mathsf{c}}_1^4+4{\mathsf{c}}_1^2{\mathsf{c}}_2+3{\mathsf{c}}_2^2+{\mathsf{c}}_1{\mathsf{c}}_3-{\mathsf{c}}_4}{720}=\operatorname{Todd}({\mathsf{M}})=1$$ and so, by , we have $\int_{\mathsf{M}}x_1^4=l^2$, and immediately follows.
We are now ready to prove the main theorem of this section.
\[m not 2\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $8$, with a Hamiltonian $S^1$-action with moment map $\psi\colon {\mathsf{M}}\to {\mathbb{R}}$, and $5$ fixed points. Suppose that $[\omega-\psi\otimes x]$ is primitive and positive, so that ${\mathsf{c}}_1=C_1[\omega]$.
Then $C_1$ is either $1$ or $5$. Moreover, the cohomology ring $H^*({\mathsf{M}};{{\mathbb{Z}}})$ and the total Chern class ${\mathsf{c}}(T{\mathsf{M}})$ agree with the ones of ${{\mathbb{C}}}P^4$ if and only if $C_1=5$.
Let $[\omega-\psi\otimes x]=\tau_1\in H^2_{S^1}({\mathsf{M}};{{\mathbb{Z}}})$ be the equivariant symplectic form and ${\mathbb{L}^{S^1}}$ the $S^1$-equivariant line bundle such that ${\mathsf{c}}_1^{S^1}({\mathbb{L}^{S^1}})=[\omega-\psi\otimes x]$ (see Lemma \[w=tau\] and Proposition \[symp hattori\]). For every $s=0,\ldots,n$, let $k_s({\mathbb{L}}^{S^1},t)$ be the bundles associated to ${\mathbb{L}}^{S^1}$ as defined in and . Thus we have that $$\begin{aligned}
k_0({\mathbb{L}},1)= &\;\;1\quad\mbox{and} \\
k_s({\mathbb{L}},1)=& \binom{n+1}{s}(1-{\mathbb{L}})^s-\sum_{\nu=1}^s\binom{n-s+\nu}{\nu}(-{\mathbb{L}})^\nu k_{s-\nu}({\mathbb{L}},1),\end{aligned}$$ yielding $$\begin{aligned}
k_0({\mathbb{L}},1)=&\;\;1\;,\nonumber\\
k_1({\mathbb{L}},1)=&\;\; n+1-{\mathbb{L}}\;,\nonumber\\
k_2({\mathbb{L}},1)=& \;\;\frac{n(n+1)}{2}-(n+1){\mathbb{L}}+{\mathbb{L}}^2\;,\nonumber\\
k_3({\mathbb{L}},1)=& \;\;\frac{n(n^2-1)}{6}-\frac{n(n+1)}{2}{\mathbb{L}}+(n+1){\mathbb{L}}^2-{\mathbb{L}}^3\;,\nonumber\\
\label{ks}k_4({\mathbb{L}},1)=&\;\; \frac{n(n^3-2n^2-n+2)}{24}+\frac{n(1-n^2)}{6}{\mathbb{L}}+\frac{n(n+1)}{2}{\mathbb{L}}^2-(n+1){\mathbb{L}}^3+{\mathbb{L}}^4.\end{aligned}$$
Since ${\mathsf{M}}$ is $8$-dimensional, using the fact that $\operatorname{Ch}({\mathbb{L}})=\sum_{k=0}^4\frac{[\omega]^k}{k!}$ and , it is easy to verify that $$\begin{aligned}
\operatorname{Ch}(k_0({\mathbb{L}},1))=&\;\; 1\;,\nonumber\\
\operatorname{Ch}(k_1({\mathbb{L}},1))=&\;\; 4-[\omega]-\frac{1}{2}[\omega]^2-\frac{1}{6}[\omega]^3-\frac{1}{24}[\omega]^4\;,\nonumber\\
\operatorname{Ch}(k_2({\mathbb{L}},1))=& \;\;6-3[\omega]-\frac{1}{2}[\omega]^2+\frac{1}{2}[\omega]^3+\frac{11}{24}[\omega]^4\;,\nonumber\\
\operatorname{Ch}(k_3({\mathbb{L}},1))=& \;\; 4-3[\omega]+\frac{1}{2}[\omega]^2+\frac{1}{2}[\omega]^3-\frac{11}{24}[\omega]^4\;,\nonumber\\
\label{chern k}\operatorname{Ch}(k_4({\mathbb{L}},1))= &\;\; 1-[\omega]+\frac{1}{2}[\omega]^2-\frac{1}{6}[\omega]^3+\frac{1}{24}[\omega]^4\;.\end{aligned}$$ Using the Atiyah-Singer formula and , we have $$r_i(1)=(-1)^i\int_{\mathsf{M}}\operatorname{Ch}\left(k_i({\mathbb{L}},1)\right) \operatorname{\mathcal{T}}({\mathsf{M}})\;$$ which, together with and the expression of the total Todd class $\operatorname{\mathcal{T}}({\mathsf{M}})$ in , implies that $$\begin{aligned}
r_0(1)=&\;\;\int_{\mathsf{M}}T_0^4=\int_{\mathsf{M}}\frac{-{\mathsf{c}}_1^4+4{\mathsf{c}}_1^2{\mathsf{c}}_2+3{\mathsf{c}}_2^2+{\mathsf{c}}_1{\mathsf{c}}_3-{\mathsf{c}}_4}{720},\\
r_1(1)=&\;\;\int_{\mathsf{M}}\left(-4\,T_0^4+\frac{[\omega]{\mathsf{c}}_1{\mathsf{c}}_2+[\omega]^2({\mathsf{c}}_1^2+{\mathsf{c}}_2)+2[\omega]^3{\mathsf{c}}_1+[\omega]^4}{24}\right),\\
r_2(1)=&\;\;\int_{\mathsf{M}}\left(6\,T_0^4+\frac{-3[\omega]{\mathsf{c}}_1{\mathsf{c}}_2-[\omega]^2({\mathsf{c}}_1^2+{\mathsf{c}}_2)+6[\omega]^3{\mathsf{c}}_1+11[\omega]^4}{24}\right),\\
r_3(1)=&\;\;\int_{\mathsf{M}}\left( -4\,T_0^4+\frac{3[\omega]{\mathsf{c}}_1{\mathsf{c}}_2-[\omega]^2({\mathsf{c}}_1^2+{\mathsf{c}}_2)-6[\omega]^3{\mathsf{c}}_1+11[\omega]^4}{24}\right),\\
r_4(1)=&\;\;\int_{\mathsf{M}}\left(T_0^4+\frac{-[\omega]{\mathsf{c}}_1{\mathsf{c}}_2+[\omega]^2({\mathsf{c}}_1^2+{\mathsf{c}}_2)-2[\omega]^3{\mathsf{c}}_1+[\omega]^4}{24}\right).\end{aligned}$$ Since $\omega-\psi\otimes x$ is primitive and positive, by Lemma \[w=tau\] and Proposition \[cr & Cc\] we have $[\omega]={\widetilde{\tau}}_1=x_1$. Let $l\in {{\mathbb{Z}}}_{>0}$ and $m\in {\mathbb{Q}}$ be defined as in Proposition \[cr & Cc\]. Then $\int_{\mathsf{M}}[\omega]^4=l^2$. Using , $\int_{\mathsf{M}}{\mathsf{c}}_4=5$ and $\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_3=50$ (see and in Corollary \[corollary1\]), we have that $$\begin{aligned}
r_1(1)=&\;-4+\frac{l^2}{24}(C_1m+C_1^2+m+2C_1+1),\nonumber\\
r_2(1)=&\;6+\frac{l^2}{24}(-3C_1m-C_1^2-m+6C_1+11),\nonumber\\
r_3(1)=&\;-4+\frac{l^2}{24}(3C_1m-C_1^2-m-6C_1+11),\nonumber\\
\label{exp_r} r_4(1)=&\;1+\frac{l^2}{24}(-C_1m+C_1^2+m-2C_1+1).\end{aligned}$$ By in Proposition \[symp hattori\], we know that $1\leq C_1\leq 5$. Moreover, by in Proposition \[C1p\], $C_1$ divides $50$, so $C_1$ cannot be $3$ nor $4$.
If $C_1=2$, Corollary \[equations sym\] gives $$r_1(1)=r_2(1),\quad r_3(1)=1,\quad r_4(1)=0$$ and $l^2=2+2r_1(1)$. It is easy to check that, using the expressions in , all these conditions give the same equation, namely $$\label{IC2}
24+l^2-l^2m=0\;.$$ Combining and , we get that $m=\displaystyle\frac{97\pm\sqrt{97}}{48}$, which is impossible, since $m$ must be rational. Hence, $C_1$ is either 1 or 5.
If $C_1=5$, Proposition \[integrals\] (i) implies that $\int_{\mathsf{M}}[\omega]^4=1$. On the other hand, as we observed before, $$\int_{\mathsf{M}}[\omega]^4=\int_{\mathsf{M}}x^4=l^2$$ with $l\in {{\mathbb{Z}}}_{>0}$, and so $l=1$. By in Corollary \[equations sym\], we have $r_s(1)=0$ for all $s>0$, and, using one of these equations together with the expression of $r_s(1)$ given in , we get $m=10$. By Proposition \[cr & Cc\] we can conclude that, if $C_1=5$, the cohomology ring and Chern classes are standard (i.e. they agree with those of ${{\mathbb{C}}}P^4$).
If $C_1=1$, by Proposition \[cr & Cc\] (ii), it follows immediately that the total Chern class is not standard. In order to prove that the cohomology ring is not standard, by Proposition \[cr & Cc\] (i), we need to prove that $l\neq 1$. It is sufficient to observe that for $l=1$ does not have any rational solutions.
\[Petrie 4\] Notice that Theorem \[m not 2\] also proves the Petrie conjecture when $({\mathsf{M}},\omega)$ is an $8$-dimensional compact symplectic manifold with a Hamiltonian $S^1$-action and 5 fixed points. However this result is not new, see [@Ja].
\[225\] It is natural to ask whether the equations in Corollary \[equations sym\] give more information when $C_1=1$. Unfortunately they are all identities, and the only meaningful equation is , which, in this case, is $$\label{C1=1}
3m^2l^2+4ml^2-l^2=675\;.$$ However, we can find a lower bound for $\int_{\mathsf{M}}[\omega]^4=l^2$. In fact, it is easy to see that the first values of $l$ for which has rational solutions are $l=15, 25, 40, 60$... thus implying that $\int_{\mathsf{M}}[\omega]^4\geq 225$. In the Kähler case, by a Chern inequality following from the Calabi Conjecture, the only possible values of $\int_{\mathsf{M}}[\omega]^4=\int_{\mathsf{M}}{\mathsf{c}}_1^4$ are 225 and 625 (see [@W; @Y]). It would be particularly interesting to know whether one can get an upper bound using symplectic techniques (see also Remark \[upper bound c1\^n\]).
When ${\mathsf{M}}$ is $6$-dimensional we have, $1\leq C_1\leq 4$ and, by Proposition \[integrals\],
- if $C_1=4$, then$\int_{\mathsf{M}}{\mathsf{c}}_1^3=64$;
- if $C_1=3$, then $\int_{\mathsf{M}}{\mathsf{c}}_1^3=54$.
Moreover, by and in Corollary \[corollary1\], we have $\int_{\mathsf{M}}{\mathsf{c}}_1{\mathsf{c}}_2=24$ and $\int_{\mathsf{M}}{\mathsf{c}}_3=4$.
Hence, if $C_1=4$, the Chern classes are standard, i.e. they agree with the ones of ${{\mathbb{C}}}P^3$ and when $C_1=3$ they agree with the ones of $Gr_2^+({\mathbb{R}}^5)$, the Grassmannian of oriented $2$-planes in ${\mathbb{R}}^5$.
However, when $C_1$ is either $1$ or $2$, the equations given by Corollary \[equations sym\] are all identities.
Minimal number of fixed points: classification results {#classification results}
======================================================
Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $2n$ equipped with a Hamiltonian $S^1$-action with a minimal number of fixed points $P_0, P_1, \ldots, P_n$, with $\lambda(P_i)=i$. We will now apply our algorithms towards a classification of these actions. Assuming that the $S^1$-action satisfies $(\mathcal{P}_0^+)$, we use Algorithms \[alg:5\](B) and \[alg:6\](B) according to the existence of an edge from $P_0$ to $P_1$ and/or an edge from $P_{n-1}$ to $P_n$, and we obtain a list that necessarily contains all possible isotropy weights. Then we take into account several simple properties satisfied by the isotropy weights in order to reduce the number of possibilities:
- at each fixed point, the isotropy weights must be coprime integers;
- at the fixed point $P_i$ the first $i$ weights are negative and the others are positive;
- by and we must have $$\label{eq:equal}
\frac{{\mathsf{c}_{1}^{S^1}}(P_1)-{\mathsf{c}_{1}^{S^1}}(P_0)}{\Lambda_1^-}=\frac{{\mathsf{c}_{1}^{S^1}}(P_{n-1})-{\mathsf{c}_{1}^{S^1}}(P_n)}{\Lambda_{n-1}^+},$$ where $\Lambda_1^-=w_{11}x$, with $w_{11}$ the unique negative weight at $P_1$, and $\Lambda_{n-1}^+=w_{n-1\,n} x$, with $w_{n-1\,n}$ the unique positive weight at $P_{n-1}$; moreover, the number in must be a positive divisor of $\frac{1}{2}n(n+1)^2$ smaller or equal to $n+1$; note that can be rewritten in terms of the isotropy weights as $$\frac{\sum_{j=1}^n w_{1j} - \sum_{j=1}^n w_{0j} }{w_{11}}=\frac{\sum_{j=1}^n w_{n-1\, j} - \sum_{j=1}^n w_{n\,j} }{w_{n-1\,n}}.$$
- when $\dim {\mathsf{M}}=8$ we know by Theorem \[thm dim8\] that the number in must be equal to $1$ or $5$;
- the isotropy weights must satisfy the equations in .
Moreover, whenever the graph has multiple edges, there are other simple properties that the isotropy weights must satisfy. These are summarized in the following technical lemmas.
\[mecycle\] Let $S\subset E$ be a set of multiple edges between two fixed points $P$ and $Q$ (i.e. ${\mathsf{i}}(e)=P$ and ${\mathsf{t}}(e)=Q$ for every $e\in S$) with $\vert S \rvert=n-2$, and assume that $n>3$. For $F\in \{P,Q\}$ let $E_F=E_{F,{\mathsf{i}}}\cup E_{F,{\mathsf{t}}}$ be the set of edges such that either ${\mathsf{i}}(e)=F$ or ${\mathsf{t}}(e)=F$. If there is an $F\in \{P,Q\}$ such that $E_F\setminus S\subset E^{\circlearrowleft}$ then the isotropy weights corresponding to the multiple edges in $S$ must be coprime.
Define $E_F^{\circlearrowleft}$ to be $E_F\cap E^{\circlearrowleft}$ and let us assume, without loss of generality, that $E_P\setminus S =E_P^{\circlearrowleft}$. If $$\gcd_{e\in S} \{ \lvert w(e)\rvert \}=k>1$$ and $r$ is the absolute value of the weight corresponding to the cycle $e\in E_P^{\circlearrowleft}$, then, since the action is effective, we must have $\gcd\{k,r\}=1$. Consequently, the isotropy submanifold fixed by ${{\mathbb{Z}}}_k$ is a $2(n-2)$-submanifold with an effective $S^1\cong S^1/{{\mathbb{Z}}}_k$-Hamiltonian action with only two fixed points which is impossible.
\[lemma:8.2\] Let $S\subset E$ be a set of multiple edges between two fixed points $P$ and $Q$ and let $\ell=\lvert S \rvert$ and $n>2$.
1. If $\ell \geq n-1$ then $\gcd_{e\in S}\{ \lvert w(e)\rvert\}=1$.
2. For every integer $2\leq r\leq \ell$ and every subset $\widetilde{S}\subset S$ with $\lvert \widetilde{S} \rvert=r$, there exist edges $e_1\in E_P\setminus \widetilde{S}$ and $e_2\in E_Q \setminus \widetilde{S}$ whose isotropy weights are multiples of $g:=\gcd_{e\in \widetilde{S}} \{ \lvert w(e)\rvert \}$.
If $\ell=n$ then, since the action is effective, we must have $\gcd_{e\in S}\{ \lvert w(e)\rvert\}=1$. If $\ell=n-1$ and $\gcd_{e\in S}\{ \lvert w(e)\rvert\}=k>1$ then, denoting by $r$ the absolute value of the weight corresponding to the edge in $E_P\setminus S$, we have $\gcd\{k,r\}=1$ since the action is effective. Then the isotropy submanifold fixed by ${{\mathbb{Z}}}_k$ is a $2(n-1)$-submanifold with an effective $S^1\cong S^1/{{\mathbb{Z}}}_k$ Hamiltonian action with only two fixed points which is impossible and so we must have $k=1$.
To prove $(2)$ we see that if there exists a subset $\widetilde{S} \subset S$ for which there is no edge in $E_P\setminus \widetilde{S}$ with weight a multiple of $g:=\gcd_{e\in \widetilde{S}} \{ \lvert w(e)\rvert \}$, then there is an isotropy submanifold of dimension $2\lvert \widetilde{S} \rvert$ fixed by ${{\mathbb{Z}}}_g$ with only two fixed points which is impossible. Similarly, we conclude the same for $Q$.
We first run Part I of the `Mathematica` file `MinimalIW.nb` to generate the list of possible non-negative multigraphs. Then we run Part II of this file to produce the list of the determinants of the matrices $A(\Gamma_i)-\operatorname{diag}({\mathsf{m}}(E_i))$ in . Note that, when the graph has more than one connected component, this file yields the sum of the squares of determinants of these matrices since this sum is zero if and only if all the determinants are zero. Then we run the `C++` files `NewPartitions\#.cpp`, where $\#$ denotes the number of the multigraph in the list above, to obtain partitions of $\frac{1}{2}n(n+1)^2$ into $n(n+1)/2$ non-negative numbers ${\mathsf{m}}(e)$ (according to Algorithms \[alg:5\](B) and \[alg:6\](B), depending on the existence of an edge from $P_0$ to $P_1$ and/or an edge from $P_{n-1}$ to $P_n$), for which all the determinants of the matrices $A(\Gamma_i)-\operatorname{diag}({\mathsf{m}}(E_i))$ are zero. Then we run Part III of `MinimalIW.nb` to sort these partitions according to the rank of the matrices, dividing them into two sets: those that originate matrices of rank $\lvert E \rvert -1$ and those that originate matrices of lower rank. In the first case, Part III of the file `MinimalIW.nb` also selects those partitions that originate matrices with nullspaces intersecting ${{\mathbb{Z}}}_{>0}^{\lvert E \rvert}$. Part IV of `MinimalIW.nb` considers the first set of partitions, producing a list of the corresponding isotropy weights $${\mathsf{w}}(E) \in \Big( \operatorname{Null}\left(A(\Gamma)-\operatorname{diag}\left({\mathsf{m}}(E)\right)\right)\Big)\cap {{\mathbb{Z}}}_{>0}^{\lvert E \rvert},$$ and checking if they can satisfy the polynomial equations in . Part V of `MinimalIW.nb` considers the second set of partitions. It begins by selecting those that yield matrices with nullspaces intersecting ${{\mathbb{Z}}}_{>0}^{\lvert E \rvert}$ (Part V a.), and producing the list of the corresponding isotropy weights (Part V b.). Then it selects those that possibly verify the properties listed in the beginning of this section as well as Lemmas \[mecycle\] and \[lemma:8.2\]. At each step, the resulting lists of isotropy weights are saved in different files so that it is easy to verify which isotropy weights are discarded at each test performed. All the relevant files can be downloaded from `http://www.math.ist.utl.pt/\simlgodin/MinimalActions.html`.
In the following we list the results obtained when the dimension of ${\mathsf{M}}$ is $4$, $6$ or $8$. Note that $\texttt{b}[\,i\,]$ denotes a positive integer for every $i$.
Dimension $4$ {#dim4}
-------------
When ${\mathsf{M}}$ is $4$-dimensional, it is easy to see that the only two possible multigraphs that can arise are those in Figure \[graphsdim4\].
\[graphsdim4\]
Thus, by Proposition \[spm\], they are non-negative. For the first multigraph we see that the weights at the point of index 2 are $1$ and $-1$. Indeed, since the action is effective, if this were not the case we would necessarily have the second multigraph. Moreover, by Corollary \[abbv discrete\] applied to $\mu=1$, it is easy to see that the weights at the minimum must be $1$ and $2$, and that the ones at the maximum are $-1$ and $-2$. Thus this set of weights can also be obtained with the second multigraph.
Therefore we can run Algorithm \[alg:6\](B), obtaining the set of weights in Figure \[wdim4\].
![Possible weights in dimension $4$[]{data-label="wdim4"}](Weightsdim4.pdf)
This is the same multiset of isotropy weights as in the standard $S^1$-action on ${{\mathbb{C}}}P^2$ described in Example \[cpn\], with `b`$[\,i\,]=\xi_0-\xi_i$, $i=1,2$. Hence, by Theorem \[bases\], the (equivariant) cohomology ring and Chern classes of the manifold are the same as the one of ${{\mathbb{C}}}P^2$ (with this $S^1$-action). Note that this result agrees with the one obtained by Karshon’s classification for $4$-dimensional $S^1$-Hamiltonian manifolds [@K Section 6.3].
Dimension $6$
-------------
Let $({\mathsf{M}},\omega)$ be a six dimensional compact symplectic manifold with an $S^1$-Hamiltonian action and fixed points $P_0,\ldots,P_3$, with $\lambda(P_i)=i$. Running Algorithms \[alg:5\](B) and \[alg:6\](B) for the $7$ non-negative multigraphs in Figure \[graphs6\], we obtain that all of them may, in principle, admit possible solutions. However, as we will see next, all the solutions can be obtained by considering only the multigraphs with no cycles $\# 3$ and $\# 7$ from Cases $3.$ and $7.$ below. Note that Tolman in [@T1] also rules out the existence of multigraphs with cycles so we could have run our algorithm only for positive multigraphs. We opted to consider all non-negative multigraphs in order to show that our methods also rule out the existence of weights specific to multigraphs with cycles.
![Dimension $6$[]{data-label="graphs6"}](GraphsDim6.pdf)
**Case 1:** For the multigraph in Case $1$ of Figure \[graphs6\] the weights given by our algorithm are the ones in Figure \[wc1\]. Using the ABBV Localization formula, with $\mu=1$ and $\mu={\mathsf{c}}_1^{S^1}$ and this set of weights, we obtain $$\begin{aligned}
0=\int_{\mathsf{M}}1& = \sum_{i=0}^3 \frac{1}{\prod_{j=1}^3 w_{ij}}= \frac{1}{6} - \frac{1}{\texttt{b}[2] ^2\texttt{b}[3] }+\frac{1}{\texttt{b}[3]\texttt{b}[4]^2 } -\frac{1}{6} =\frac{ \texttt{b}[2]^2-\texttt{b}[4]^2}{\texttt{b}[2] ^2\texttt{b}[3] \texttt{b}[4] ^2}\end{aligned}$$ and $$\begin{aligned}
0=\int_{\mathsf{M}}{\mathsf{c}}_1^{S^1} & = \sum_{i=0}^3\left( \frac{\sum_{j=1}^3 w_{ij} }{\prod_{j=1}^3 w_{ij}}\right)=1 - \frac{1}{\texttt{b}[2] ^2}-\frac{1}{\texttt{b}[4]^2 } +1 =2-\frac{\texttt{b}[2] ^2+\texttt{b}[4]^2 }{\texttt{b}[2] ^2 \texttt{b}[4] ^2},\end{aligned}$$ and so $\texttt{b}[2]=\texttt{b}[4]=1$. Then, using the fact that $$\frac{{\mathsf{c}_{1}^{S^1}}(P_1)-{\mathsf{c}_{1}^{S^1}}(P_0)}{\Lambda_1^-}=6-\texttt{b}[3]$$ must be a divisor of $24$ no larger that $4$, we conclude that $\texttt{b}[3]=2,3,4$ or $5$. If $\texttt{b}[3]=2$, the resulting multiset of weights is a particular case of the one in Case $7$ II. Note that it is precisely the set of weights of the $S^1$-action on ${{\mathbb{C}}}P^3$ described in Example \[cpn\] with $\xi_0-\xi_1=1$, $\xi_0-\xi_2=2$ and $\xi_0-\xi_3=3$. If $\texttt{b}[3]=3$, the resulting multiset of weights is a particular case of the one obtained in Case $7$ I. It is precisely the set of weights of the $S^1$-action on $Gr_2^+({\mathbb{R}}^5)$ described in Example \[gr\], by taking $\xi_0=2$ and $\xi_1=1$. If $\texttt{b}[3]=4$, the multiset of weights obtained is precisely of the $S^1$-action described in Example \[fano\] for the Fano manifold $V_5$. Finally, if $\texttt{b}[3]=5$, the multiset of weights obtained is precisely of the $S^1$-action described in Example \[fano\] for the Fano manifold $V_{22}$.
![Possible weights for Case 1[]{data-label="wc1"}](Weights6_1.pdf)
**Case 2:** For the multigraph in Case $2$ of Figure \[graphs6\] the weights given by our algorithm are the ones in Figure \[wc2\]. Using the ABBV Localization formula, with $\mu=1$ and this set of weights, we obtain $$0=\int_{\mathsf{M}}1= \frac{1}{6\texttt{b}[1] } - \frac{1}{4\texttt{b}[1] }+\frac{1}{6(\texttt{b}[1]-2)} - \frac{1}{4(\texttt{b}[1] -2)} =-\frac{\texttt{b}[1]-1}{6\texttt{b}[1](\texttt{b}[1]-2) },$$ and so $\texttt{b}[1] =1$. Since, on the other hand, the weight $\texttt{b}[1]-2$ at $P_2$ must be positive, we conclude that this case is impossible.
![Possible weights for Case 2[]{data-label="wc2"}](Weights6_2.pdf)
**Case 3:** For the multigraph in Case $3$ of Figure \[graphs6\] the weights given by our algorithm are the ones in Figure \[wc3\].
![Possible weights for Case 3[]{data-label="wc3"}](Weights6_3.pdf)
[**I.**]{} The set of weights in Figure \[wc3\] I. is precisely the set of weights of the $S^1$-action described in Example \[fano\] for the Fano manifold $V_{22}$.
[**II.**]{} Using the ABBV Localization formula, with $\mu=1$ and the set of weights in Figure \[wc3\] II. we have $$0=\int_{\mathsf{M}}1= \frac{1}{6\texttt{b}[1] } - \frac{1}{4\texttt{b}[1] }+\frac{1}{4(2-\texttt{b}[1])} - \frac{1}{6(2-\texttt{b}[1] )} =-\frac{1-\texttt{b}[1]}{6\texttt{b}[1](2-\texttt{b}[1]) },$$ and so $\texttt{b}[1] =1$. Then this is precisely the set of weights of the $S^1$-action described in Example \[fano\] for the Fano manifold $V_5$.
**Case 4:** For the multigraph in Case $4$ of Figure \[graphs6\] the weights given by our algorithm are the ones in Figure \[wc4\]. Using the ABBV Localization formula, with $\mu=1$ and this set of weights, we obtain that $\texttt{b}[2] =1$ in both cases and so we get the same sets of weights as in Cases $3$ I. and $3$ II. respectively.
![Possible weights for Case 4[]{data-label="wc4"}](Weights6_4.pdf)
**Case $5$:** For the multigraph in Case $5$ of Figure \[graphs6\] the weights given by our algorithm are the ones in Figure \[wc5\].
![Possible weights for Case $5$[]{data-label="wc5"}](Weights6_5.pdf)
[**I.**]{} Using the ABBV Localization formula, with $\mu=1$ and the set of weights in Figure \[wc5\] I., we obtain that $\texttt{b}[2] =1$. Then the multiset of weights is a particular example of Case $7$ I.
[**II.**]{} Using the ABBV Localization formula, with $\mu=1$ and the set of weights in Figure \[wc5\] II., we obtain that $\texttt{b}[3] =1$. Then this set of weights is a particular example of Case $7$ II.
**Case 6:** For the multigraph in Case $6$ of Figure \[graphs6\] the weights given by our algorithm are the ones in Figure \[wc6\]. Using the ABBV Localization formula, with $\mu=1$ and with $\mu={\mathsf{c}}_1^{S^1}$ we obtain that $\texttt{b}[2] =\texttt{b}[3] =1$ in both cases.
![Possible weights for Case 6[]{data-label="wc6"}](Weights6_6.pdf)
Then,
[**I.**]{} the set of weights in Figure \[wc6\] I. is a particular example of Case $7$ II.
[**II.**]{} the set of weights in Figure \[wc6\] II. is a particular example of Case $7$ I.
**Case 7:** For the multigraph in Case $7$ of Figure \[graphs6\] the weights given by our algorithm are the ones in Figure \[wc7\].
![Possible weights for Case 7[]{data-label="wc7"}](Weights6_7.pdf)
[**I.**]{} The set of weights in Figure \[wc7\] I. is precisely the set of weights of the $S^1$-action on $Gr_2^+({\mathbb{R}}^5)$ described in Example \[gr\], by taking $\xi_0=\frac{\text{\texttt{b}$[\,1\,]$+\text{\texttt{b}$[\,2\,]$}}}{2}$ and $\xi_1=\frac{\text{\texttt{b}$[\,2\,]$-\text{\texttt{b}$[\,1\,]$}}}{2}$.
[**II.**]{} The set of weights in Figure \[wc7\] II. is precisely the set of weights of the $S^1$-action on ${{\mathbb{C}}}P^3$ described in Example \[cpn\] with `b`$[\,i\,]=\xi_0-\xi_i$, $i=1,2,3$.
Dimension $8$. {#class dim 8}
--------------
Running Algorithms \[alg:5\](B) and \[alg:6\](B) for the $75$ non-negative multigraphs on an $8$-dimensional manifold with $5$ fixed points, we obtain that only the multigraphs in Figure \[graphs8\] may, in principle, admit possible solutions. However, as we will see next they can all be easily ruled out except for multigraph$\#$ $75$ considered in Case $4$. below.
![Dimension 8[]{data-label="graphs8"}](GraphsDim8.pdf)
**Case $1$:** For the multigraph in Case $1$ of Figure \[graphs8\] the weights given by our algorithm are the ones in Figure \[wc81\]. Note that this multigraph has two pairs of multiple edges: two edges from $P_0$ to $P_4$ and two edges from $P_0$ to $P_2$.
![Possible isotropy weights for Case 1.[]{data-label="wc81"}](Weights8_1.pdf)
[**I.**]{} For the set of weights in Figure \[wc81\] I. we see that the weights corresponding to the two edges from $P_0$ to $P_2$ are $\texttt{b}[1]$ and $2\texttt{b}[1]$, and the ones corresponding to the two edges from $P_0$ to $P_4$ are $\texttt{b}[2]$ and $2\texttt{b}[2]$. Since the action is effective, we have $\gcd\left\{\texttt{b}[1],\texttt{b}[2]\right\}=1$. Then, by Lemma \[lemma:8.2\] $(2)$, we either have $$\texttt{b}[1]=1,\quad \text{or} \quad \gcd\left\{\texttt{b}[1],2\texttt{b}[2]\right\}=\texttt{b}[1],$$ and so $$\texttt{b}[1]=1, \quad \text{or} \quad \texttt{b}[1]= 2.$$ Similarly, using the two edges from $P_0$ to $P_4$, we conclude that $$\texttt{b}[2]=1, \quad \text{or} \quad \texttt{b}[2]= 2.$$ Since the weight $\texttt{b}[2]-2\texttt{b}[1]$ at $P_2$ must be positive this is impossible.
[**II.**]{} For the set of weights in Figure \[wc81\] II. we see that the weights corresponding to the two edges from $P_0$ to $P_2$ are $\texttt{b}[1]$ and $2\texttt{b}[1]$, and the ones corresponding to the two edges from $P_0$ to $P_4$ are $\texttt{b}[2]$ and $3\texttt{b}[2]$. Since the action is effective, we have $\gcd\left\{\texttt{b}[1],\texttt{b}[2]\right\}=1$. Moreover, by Lemma \[lemma:8.2\] $(2)$, we either have $$\texttt{b}[1]=1,\quad \text{or} \quad \gcd\left\{\texttt{b}[1],3\texttt{b}[2]\right\}=\texttt{b}[1],$$ and so $$\texttt{b}[1]=1, \quad \text{or} \quad \texttt{b}[1]= 3.$$ Similarly, using the two edges from $P_0$ to $P_4$ we conclude that we either have $$\texttt{b}[2]=1, \quad \text{or} \quad \texttt{b}[2]= 2.$$ Since the weights $3\texttt{b}[1]-5\texttt{b}[2]$ and $9\texttt{b}[2]-3\texttt{b}[1]$ at $P_2$ must be positive this is impossible.
[**III.**]{} For the set of weights in Figure \[wc81\] III. we see that the weights corresponding to the two edges from $P_0$ to $P_2$ are both $\texttt{b}[1]$, and the ones corresponding to the two edges from $P_0$ to $P_4$ are $2\texttt{b}[2]$ and $7\texttt{b}[2]$. Since the action is effective, we have $\gcd\left\{\texttt{b}[1],\texttt{b}[2]\right\}=1$. Moreover, by Lemma \[lemma:8.2\] $(2)$, we either have $$\texttt{b}[1]=1,\quad \text{or} \quad \gcd\left\{\texttt{b}[1],2\texttt{b}[2]\right\}=\texttt{b}[1], \quad \text{or} \quad \gcd\left\{\texttt{b}[1],7\texttt{b}[2]\right\}=\texttt{b}[1],$$ and so $$\texttt{b}[1]=1, \quad \text{or} \quad \texttt{b}[1]= 2, \quad \text{or} \quad \texttt{b}[1]= 7 .$$ Similarly, using the two edges from $P_0$ to $P_4$ we conclude that we must have $$\texttt{b}[2]=1,\quad \text{or} \quad \gcd\left\{2\texttt{b}[2],\texttt{b}[1]\right\}=2\texttt{b}[2],\quad \text{or} \quad \gcd\left\{7\texttt{b}[2],\texttt{b}[1]\right\}=7\texttt{b}[2].$$ and so the only possibility is $\texttt{b}[2]=1$. Since the weight $2\texttt{b}[1]-12\texttt{b}[2]$ at $P_2$ must be positive the only possibility is $\texttt{b}[1]=7$ and $\texttt{b}[2]=1$. In this case the (connected) set of points fixed by ${{\mathbb{Z}}}_7$ would contain $P_0$ and $P_4$. However, if $\texttt{b}[1]=7$ and $\texttt{b}[2]=1$, the subgroup ${{\mathbb{Z}}}_7$ acts trivially on a $3$-dimensional complex subspace of $T_{P_0}{\mathsf{M}}$ and on a $2$-dimensional complex subspace at $T_{P_4}{\mathsf{M}}$ (since the weights at $P_4$ would be $\{-2,-3,-7,-7\}$), which is impossible.
[**IV.**]{} For the set of weights in Figure \[wc81\] IV. we see that the weights corresponding to the two edges from $P_0$ to $P_2$ are both $\texttt{b}[1]$, and the ones corresponding to the two edges from $P_0$ to $P_4$ are $3\texttt{b}[2]$ and $8\texttt{b}[2]$. Since the action is effective, we have $\gcd\left\{\texttt{b}[1],\texttt{b}[2]\right\}=1$. Moreover, by Lemma \[lemma:8.2\] $(2)$, we either have $$\texttt{b}[1]=1,\quad \text{or} \quad \gcd\left\{\texttt{b}[1],3\texttt{b}[2]\right\}=\texttt{b}[1], \quad \text{or} \quad \gcd\left\{\texttt{b}[1],8\texttt{b}[2]\right\}=\texttt{b}[1],$$ and so $$\texttt{b}[1]\in \{1, \,2,\, 3,\,4,\,8\}.$$ Similarly, using the two edges from $P_0$ to $P_4$ we conclude that we must have $$\texttt{b}[2]=1,\quad \text{or} \quad \gcd\left\{3\texttt{b}[2],\texttt{b}[1]\right\}=3\texttt{b}[2],\quad \text{or} \quad \gcd\left\{8\texttt{b}[2],\texttt{b}[1]\right\}=8\texttt{b}[2],$$ and so the only possibility is $\texttt{b}[2]=1$. Since the weight $2\texttt{b}[1]-13\texttt{b}[2]$ at $P_2$ must be positive the only possibility is $\texttt{b}[1]=8$ and $\texttt{b}[2]=1$. In this case the (connected) set of points fixed by ${{\mathbb{Z}}}_8$ would contain $P_0$ and $P_4$. However, if $\texttt{b}[1]=8$ and $\texttt{b}[2]=1$, the subgroup ${{\mathbb{Z}}}_8$ acts trivially on a $3$-dimensional complex subspace of $T_{P_0}{\mathsf{M}}$ and on a $2$-dimensional complex subspace at $T_{P_4}{\mathsf{M}}$ (since the weights at $P_4$ would be $\{-2,-3,-8,-8\}$), which is impossible.
We conclude that Case $1$. is impossible.
**Case $2$.** For the multigraph in Case $2$ of Figure \[graphs8\] the weights given by our algorithm are the ones in Figure \[wc82\]. First we see that there exist two pairs of multiple edges on this multigraph: two edges from $P_0$ to $P_2$ and two edges from $P_2$ to $P_4$. The weights corresponding to the first two edges are $\texttt{b}[1]$ and $2\texttt{b}[1]$, while the ones corresponding to the last are $\texttt{b}[2]-2\texttt{b}[1]$ and $2(\texttt{b}[2]-2\texttt{b}[1])$. Since the action is effective, we see from the weights at $P_0$ that we need $\gcd\left\{\texttt{b}[1],\texttt{b}[2]\right\}=1$. Moreover, by Lemma \[lemma:8.2\] $(2)$, we either have $$\texttt{b}[1]=1,\quad \text{or} \quad \gcd\left\{\texttt{b}[1],2(\texttt{b}[2]-2\texttt{b}[1])\right\}=\texttt{b}[1],$$ and so $$\texttt{b}[1]=1, \quad \text{or} \quad \texttt{b}[1]= 2.$$ Similarly, using the other two edges from $P_2$ to $P_4$, we have $$l=1, \quad \text{or} \quad \gcd\left\{l, \texttt{b}[1]\right\}=l, \quad \text{or} \quad \gcd\left\{l,2\texttt{b}[1]\right\}=l,$$ with $l=\texttt{b}[2]-2\texttt{b}[1]$ and so, since $$\gcd\left\{l, \texttt{b}[1]\right\}=\gcd\left\{\texttt{b}[2]-2\texttt{b}[1],\texttt{b}[1]\right\}=\gcd\left\{\texttt{b}[2],\texttt{b}[1]\right\}=1$$ and $$\gcd\left\{l, 2\texttt{b}[1]\right\}=\gcd\left\{\texttt{b}[2]-2\texttt{b}[1],2\texttt{b}[1]\right\}=\gcd\left\{\texttt{b}[2],2\texttt{b}[1]\right\},$$ we conclude that $l=1$ or $l=2$. Therefore, since $\gcd\left\{\texttt{b}[1],\texttt{b}[2]\right\}=1$, the only possibilities for $\texttt{b}[1]$ and $\texttt{b}[2]$ are $$(\texttt{b}[1],\texttt{b}[2])=(1,3), \quad (\texttt{b}[1],\texttt{b}[2])=(1,4), \quad \text{and} \quad (\texttt{b}[1],\texttt{b}[2])=(2,5).$$
If $(\texttt{b}[1],\texttt{b}[2])=(2,5)$ we have a compact connected $6$-dimensional isotropy submanifold fixed by ${{\mathbb{Z}}}_2$ with an effective $S ^1\cong S^1/ {{\mathbb{Z}}}_2$-Hamiltonian action, with only $4$ fixed points, ($P_0, P_1,P_2$ and $P_4$) and weights $$\{1,2,3\},\{-\texttt{b}[3]/2,\texttt{b}[3]/2,2\}, \{-1,-2,1\},\{-1,-2,-3\}.$$ Then, by the classification of Hamiltonian circle actions with a minimal number of fixed points on a $6$-dimensional manifold, we have that $\texttt{b}[3]=2$. Using the ABBV Localization formula on ${\mathsf{M}}$ with $\mu=1$, we obtain $\texttt{b}[4]=1$ and then the resulting set of weights can be obtained from the positive multigraph of Case $4$ in Figure \[graphs8\] (multigraph $\#$ $75$).
If $(\texttt{b}[1],\texttt{b}[2])=(1,4),$ we get, by the same methods, the set of weights for the reversed circle action of the case $(\texttt{b}[1],\texttt{b}[2])=(2,5)$ described above and so it can again be obtained from the multigraph of Case $4$.
If $(\texttt{b}[1],\texttt{b}[2])=(1,3)$, we obtain the following multiset of weights: $$\{ \{1,2,3,4 \}, \{ 2,3,-\texttt{b}[3],\texttt{b}[3] \}, \{ -2,-1,1,2 \}, \{ -2,-3,-\texttt{b}[4],\texttt{b}[4] \}, \{-4,-3,-2,-1\}\}.$$ Using the ABBV Localization formula, with $\mu=1$, we have $$0=\int_{\mathsf{M}}1= \sum_{i=0}^4 \frac{1}{\prod_{j=1}^4 w_{ij}}=\frac{1}{24} - \frac{1}{6\texttt{b}[3] ^2}+\frac{1}{4} - \frac{1}{6\texttt{b}[4] ^2} + \frac{1}{24},$$ and so $\texttt{b}[3] =\texttt{b}[4]=1$. Consequently, this set of weights falls again into Case $4$ and can be obtained with multigraph $\#$ $75$.
![Possible isotropy weights for Case $2$.[]{data-label="wc82"}](Weights8_2.pdf)
**Case $3$:** For the multigraph in Case $3$ of Figure \[graphs8\] the weights given by our algorithm are the ones in Figure \[wc83\]. Note that this multigraph has a pair of multiple edges from $P_0$ to $P_4$.
[**I.**]{} For the set of weights in Figure \[wc83\] I. we see that the weights corresponding to the two edges from $P_0$ to $P_4$ are $\frac{\texttt{b}[1] +\texttt{b}[2] }{3}$ and $2(\frac{\texttt{b}[1] +\texttt{b}[2] }{3})$ and that the other two weights at $P_0$ are $\texttt{b}[1]$ and $\texttt{b}[2]$. Then Lemma \[lemma:8.2\] $(2)$ implies that we must have $$\gcd\{\ell,\texttt{b}[1]\} = \ell \quad \text{or} \quad \gcd\{\ell,\texttt{b}[2]\} = \ell,$$ with $\ell=\frac{\texttt{b}[1] +\texttt{b}[2] }{3}$. Hence we must have $$\texttt{b}[1] =2\texttt{b}[2] \quad \text{and} \quad \ell= \texttt{b}[2],$$ or $$\texttt{b}[2] =2\texttt{b}[1] \quad \text{and} \quad \ell= \texttt{b}[1].$$ Since the action is effective we conclude that, in the first case, we have $\texttt{b}[2]=1$ and $\texttt{b}[1]=2$, while, in the last, we have $\texttt{b}[1]=1$ and $\texttt{b}[2]=2$. However, both cases are impossible since the weights at $P_1$ would not be integers.
[**II.**]{} For the set of weights in Figure \[wc83\] II. we see that the weights corresponding to the two edges from $P_0$ to $P_4$ are $9\texttt{b}[2]$ and $15\texttt{b}[2]$ and that the other two weights at $P_0$ are $\texttt{b}[1]$ and $4\texttt{b}[2]$. Since the action is effective, we have $\gcd\left\{\texttt{b}[1],\texttt{b}[2]\right\}=1$. Moreover, by Lemma \[lemma:8.2\] $(2)$, we have $$\gcd\left\{3\texttt{b}[2],\texttt{b}[1]\right\}=3\texttt{b}[2],$$ and so $$\texttt{b}[1]=3 \quad \text{and} \quad \texttt{b}[2]=1.$$ In this case the (connected) set of points fixed by ${{\mathbb{Z}}}_5$ would contain $P_0$ and $P_4$. However, the subgroup ${{\mathbb{Z}}}_5$ acts trivially on a $1$-dimensional complex subspace of $T_{P_0}{\mathsf{M}}$ and on a $3$-dimensional complex subspace at $T_{P_4}{\mathsf{M}}$ (since the weights at $P_4$ and $P_0$ would respectively be $\{-15,-20,-15,-9\}$ and $\{3,4,9,15\}$), which is impossible.
[**III.**]{} For the set of weights in Figure \[wc83\] III. we see that the weights corresponding to the two edges from $P_0$ to $P_4$ are $2\texttt{b}[2]$ and $4\texttt{b}[2]$ and that the other two weights at $P_0$ are $\texttt{b}[1]$ and $\texttt{b}[2]$. Since the action is effective, we have $\gcd\left\{\texttt{b}[1],\texttt{b}[2]\right\}=1$. Moreover, by Lemma \[lemma:8.2\] $(2)$, we have $$\gcd\left\{2\texttt{b}[2],\texttt{b}[1]\right\}=2\texttt{b}[2],$$ and so $$\texttt{b}[1]=2 \quad \text{and} \quad \texttt{b}[2]=1.$$ However, in this case the weight $\texttt{b}[1]-2\texttt{b}[2]$ at $P_3$ would be $0$ which is impossible.
[**IV.**]{} For the set of weights in Figure \[wc83\] IV. we see that the weights corresponding to the two edges from $P_0$ to $P_4$ are $\texttt{b}[2]$ and $3\texttt{b}[2]$ and that the other two weights at $P_0$ are $\texttt{b}[1]$ and $2\texttt{b}[2]$. Since the action is effective, we have $\gcd\left\{\texttt{b}[1],\texttt{b}[2]\right\}=1$. If $\texttt{b}[2]=1$, then the weights $\texttt{b}[1]-\texttt{b}[2]$ and $2\texttt{b}[2]-\texttt{b}[1]$ respectively at $P_1$ and $P_2$ cannot be simultaneously positive and so we conclude that $\texttt{b}[2]> 1$. Then, the (connected) set of points fixed by ${{\mathbb{Z}}}_{\texttt{b}[2]}$ is a $6$-dimensional manifold with an effective $S ^1\cong S^1/ {{\mathbb{Z}}}_{\texttt{b}[2]}$-Hamiltonian action with $4$ fixed points (all except $P_2$) and weights $$\{1,2,3\},\{-\texttt{b}[3]/\texttt{b}[2],\texttt{b}[3]/\texttt{b}[2],2\}, \{-\texttt{b}[4]/\texttt{b}[2],-2,\texttt{b}[4]/\texttt{b}[2]\},\{-1,-2,-3\}.$$ Then, by the classification of Hamiltonian circle actions with a minimal number of fixed points on a $6$-dimensional manifold, we have that $\texttt{b}[3]=\texttt{b}[4]=\texttt{b}[2]$ and then the resulting set of weights can be obtained from the positive multigraph of Case $4$ (multigraph $\#$ $75$).
[**V.**]{} For the set of weights in Figure \[wc83\] V. we see that the weights corresponding to the two edges from $P_0$ to $P_4$ are $2\texttt{b}[2]$ and $4\texttt{b}[2]$ and that the other two weights at $P_0$ are $\texttt{b}[1]$ and $3\texttt{b}[2]$. Then, just like in III., we have $$\texttt{b}[1]=2 \quad \text{and} \quad \texttt{b}[2]=1.$$ However, in this case, the weight $\texttt{b}[1]-2\texttt{b}[2]$ at $P_2$ would be $0$ which is impossible.
[**VI.**]{} For the set of weights in Figure \[wc83\] VI. we see that the weights corresponding to the two edges from $P_0$ to $P_4$ are $9\texttt{b}[2]$ and $15\texttt{b}[2]$ and that the other two weights at $P_0$ are $\texttt{b}[1]$ and $20\texttt{b}[2]$. Then, just like in II., we have $$\texttt{b}[1]=3 \quad \text{and} \quad \texttt{b}[2]=1.$$ However, in this case, the weight $\texttt{b}[1]-10\texttt{b}[2]$ at $P_1$ would be negative which is impossible.
![Possible isotropy weights for Case $3$.[]{data-label="wc83"}](Weights8_3.pdf)
**Case $4$:** For the multigraph in Case $4$ of Figure \[graphs8\] the weights given by our algorithm are the ones in Figure \[wc84\]. These are precisely the weights of the $S^1$-action on ${{\mathbb{C}}}P^4$ described in Example \[cpn\], with `b`$[\,i\,]=\xi_0-\xi_i$, $i=1,2,3,4$.
![Possible isotropy weights for Case $4$.[]{data-label="wc84"}](Weights8_4.pdf)
Since all the possible isotropy weights obtained can be included in this last case and the basis for $H_{S^1}^*({\mathsf{M}};{{\mathbb{Z}}})$ described in Theorem \[bases\] and the (equivariant) Chern classes only depend on the isotropy weights at the fixed points, we can summarize our results in the following theorem.
\[thm dim8\] Let $({\mathsf{M}},\omega)$ be a compact symplectic manifold of dimension $8$, with a Hamiltonian $S^1$-action with $5$ fixed points. If there exists a non-negative multigraph associated to the action, then the isotropy weights at the fixed points agree with the one of ${{\mathbb{C}}}P^4$ for the standard $S^1$-action. Moreover, the cohomology ring and Chern classes agree with the ones of ${{\mathbb{C}}}P^4$, i.e. $$H^*({\mathsf{M}};{{\mathbb{Z}}})={{\mathbb{Z}}}[y]/(y^5)\quad\mbox{and}\quad {\mathsf{c}}=(1+y)^5\;,$$ where $y$ has degree 2.
Corollary \[main dim 8\] in the introduction is then an easy consequence of Proposition \[t2\], Proposition \[not pm1\] and Theorem \[thm dim8\].
We can now prove Theorem \[RS1\] stated in the Introduction.
By Lemma \[w=tau\] it is not restrictive to assume that $[\omega]=y$, and by Theorem \[m not 2\], we know that $C_1$ is either 1 or 5.\
(i)$\implies$(ii), (iii), (iv), (v): this is exactly the content of Theorem \[thm dim8\].\
(ii)$\iff$(iii) and (ii)$\iff$(iv): this follows from Theorem \[m not 2\].\
(v)$\implies$(i): it follows easily from the definitions (see Example \[cpn\]).\
(ii)$\implies$(v): this is directly implied by Theorem \[hattori2\]. Indeed, the Euler characteristic of ${\mathsf{M}}$ satisfies $\chi_{-1}({\mathsf{M}})=\int_{\mathsf{M}}{\mathsf{c}}_4=5$ (see ). Moreover, by Lemma \[w=tau\], we can choose $\omega-\psi\otimes x$ to be primitive and positive, and, by Proposition \[symp hattori\], the associated pre-quantization line bundle is quasi-ample and satisfies . Thus Theorem \[hattori2\] implies that the isotropy weights at the fixed points agree with the ones of ${{\mathbb{C}}}P^4$ with the standard $S^1$-action.\
The remaining equivalences follow easily from the above ones.
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[^1]: A chain of gradient spheres is a sequence of gradient spheres $S_1,\ldots,S_l$ such that the south pole of $S_0$ is a minimum for the moment map, the north pole of $S_{i-1}$ is the south pole of $S_i$ for each $1<i\leq l$, and the north pole of $S_l$ is a maximum for the moment map. A chain is non-trivial if it contains more than one sphere, or if it contains one sphere whose stabilizer is non-trivial.
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Institute für Theoretische Physik, Universität Regensburg, 93040 Regensburg, Germany\
E-mail:
title: Solving the Dirac equation on QPACE
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Introduction
============
The simulation of lattice QCD including dynamical fermions is computationally challenging mainly due to the need of performing lattice Dirac operator inversions. These inversions are needed within the HMC algorithm to compute the contribution to the force which arises from the fermionic determinant. Many tasks involved in the simulation of lattice QCD show an high degree of parallelism at different levels. At the lattice level the natural parallelization strategy consists of subdividing the lattice into sub-lattices, each sub-lattice is then associated to a different processing node of the machine. In order to exploit efficiently the intrinsic parallelism, a low-latency, high-bandwidth, scalable network is needed to connect the different processing nodes.
QPACE [@Baier:2009yq] is the latest custom designed massively parallel supercomputer for LQCD applications. The architecture consists of a 3D torus of processing nodes based on a commodity processor, the IBM PowerXCell 8i. The nodes are interconnected by a custom network which enables low-latency and high-bandwidth nearest neighbor communications. Each node is thus equipped with six links to connect to the six neighbors in the 3D torus mesh.\
While the floating point (FP) performance of the processors has seen a continuous increase in the past years, the latency of the memory accesses and network has not improved at a similar scale. This problem can be mitigated by increasing the amount of on-chip memory. Making optimal use of this on-chip memory has become crucial for the optimization of numerical software.
Cell processor
==============
The Cell architecture consists of nine cores: one Power Processing Unit (PPU) able to run generic code such as the operating system, and eight Synergistic Processing Units (SPU) for which a simple architecture was chosen, optimized for FP intensive applications. The Cell architecture is described in [@cbea]. Among the different design choices done on the SPU to reduce core size, maximize FP performance, and allow latency hiding, the most radical one is the explicit separation of two levels of the memory hierarchy. While in conventional processors the fact that there is a small amount of fast memory and a large amount of slow memory is completely transparent to the programmer, being the hardware responsible to move data between the memory spaces, on the Cell, this responsibility is completely left to the programmer, who needs to issue *direct memory access* (DMA) commands in order to move data. This particular feature makes programming a complex, time consuming and error prone task even for routines which do not need a particular high level of optimization. On the other hand, by allowing full control of the fast memory, it is at least in principle possible to make an optimal use of the resource.
The instruction set of the SPU is similar to Intel SSE. Almost all the instructions operate on vector types with operands and results being stored in 128 general purpose, 128-bit wide registers. Peak performance is obtained by using the fused multiply-add instruction. It is mandatory to use vector instructions in all the cases in which reasonable performance is needed.
Access to main memory from the SPU is subject to alignment constraints. The minimum allowed alignment is 16B, optimal memory bandwidth is achieved with 128B aligned DMA where the size is a multiple of 128B. Close to peak bandwidth is obtained already for a DMA size of 128B. Given the high FP performance of the chip (200 GF single precison and 100 GF double precision) many tasks are limited by memory bandwidth (8B/cycle for the whole Cell), it is thus very important to access memory in the most efficient way.
QPACE torus network
===================
The QPACE torus network is capable of exchanging messages between nearest-neighbor nodes in the 3D torus. The *send* operation corresponds to a DMA put operation on the address range of the network links. At the receiving side the network processor forwards the data to the SPU’s local stores or to main memory to a user-specified address. After all the data corresponding to a receive operation has been forwarded, a particular memory location is updated to *notify* the completion of the receive operation. The QPACE torus network supports eight *channels*. Up to eight messages can travel concurrently on the same link. This feature enables the concurrent communication between SPUs of different nodes. The network supports also *remote offsets*.This feature can be used to merge different receive operations in a single one, when different SPUs have to send data to different locations of the same buffer on the receiving node.
Algorithm Choice
================
The conjugate gradient (CG) is one of the most popular solvers used for lattice QCD simulations. CG gained its popularity due to good convergence properties, simplicity, and robustness. The most numerical intensive task in the CG algorithm is the matrix-vector product which consists in the application of the lattice Dirac operator to a spinor field. Almost all of the optimization effort is usually spent on this particular task. As was shown in [@phd][@lat07] the performance of an optimal implementation of the Wilson-Dirac operator without even-odd preconditioning on the Cell processor is bounded by the memory bandwidth to $\epsilon_{fp}=34\%$. This estimate relies on the assumption that it is possible to reach full memory bandwidth. A benchmark of this implementation shows a performance $\epsilon_{fp}=24\%$. This optimal implementation is subject to a lattice size constraint. The maximum local lattice size for a single precision computation is limited to $L_0\times 10^3$ and to $L_0\times 8^3$ for single precision where $L_0$, the time extent of the lattice, is not constrained. This limitation comes from the size of the local store. This optimal implementation is also able to tolerate a network latency of a few $\mu$s, i.e. O(10,000) clock cycles. In the case of an even-odd preconditioned Wilson-Dirac operator, both memory and network accesses become more problematic. First we notice that during the application of the operator, the spinor field on the borders of the local lattice must be communicated two times, since this operator couples next to nearest neighbor lattice points. A feasible implementation of this operator requires two sweeps of the lattice each of which requires access to all the SU(3) link variables while doing half of the needed floating point operations. The memory bandwidth limitation thus becomes more critical and the estimated performance, taking into account the efficiency of the memory goes below $20\%$, even without considering the network.
In order to circumvent these constraints, including the limited flexibility on the choice of the local lattice sizes, we consider the SAP-GCR algorithm that was proposed by Lüscher [@SchwarzII]. The Schwarz Alternating Procedure (SAP) belongs to the class of domain decomposition methods. The lattice is divided into non-overlapping blocks which are chessboard-colored. One iteration (cycle) of the SAP proceeds as follows: the Wilson-Dirac equation first is solved on all the *white* blocks, the residue is then updated both on the *white* blocks, and on the internal border of the *black* blocks, then the equation is solved on the *black* blocks, with the source consisting in the updated residue, and finally the residue is updated on the *black* blocks and on the internal border of the *white* blocks. The blocks are solved with a fixed number of iterations using a simple iterative solver (MR) and Dirichlet boundary conditions. In this procedure the only step which requires network communications is the update of the residue on neighboring blocks. Since the updated residue is needed only after half of the blocks are solved, it is possible to completely overlap communication and computation. The procedure is thus able to tolerate very high network latencies without performance losses. The network bandwidth requirement is a fraction $1/(n_{it}+1)$ of the requirement of the equivalent Wilson-Dirac operator on the whole lattice, where $n_{it}$ is the number of iterations used in the block solver. The SAP procedure is used as preconditioner inside a FGCR iteration. The outer FGCR iteration is able to tolerate variations in the preconditioner since the preconditioner is obtained with an iterative method and thus is not a stationary operator. FGCR is mathematically equivalent to FGMRES [@saad] and requires to store two vectors per iteration. This requirement translates in the practical need of restarting the FGCR recursion. Mixed precision is obtained through iterative refinement [@itref]. All the operations can proceed in single precision, only when the FGCR recursion is restarted, the residue is computed in full double precision, and the double precision solution is updated by adding the single precision correction. The most time consuming task in the SAP is the solution of the Wilson-Dirac equation on the blocks. Since the block size can be chosen such that all the data necessary for a block solve fit the local store, SAP is able to achieve a high sustained performance by making an efficient use of the local store. The local lattice sizes are only constrained to be multiples of the block sizes. Table \[table:comp\] summarizes the comparison between the CG and SAP-GCR.
CG SAP-GCR
--------------------------- --------------- -------------------- --
Lattice size flexibility poor reasonable
E/O preconditioning perf. penalty yes (block solver)
Network bandwidth req. high low
Network latency tolerance moderate high
Memory bandwidth req. high moderate
: Comparison between CG and SAP-GCR on QPACE
\[table:comp\]
A drawback of the SAP-GCR algorithm is that convergence is not guaranteed. When the FGCR recursion is restarted, the information contained in the Krylov subspace is lost. This can result in a reduction of the convergence rate of the algorithm or, in the worst case, stagnation. In order to avoid this problem we introduced deflation in the form of FGMRES-DR and we eventually foresee automatic switching to CG for the most difficult cases. The idea of *deflated restarts* consists in keeping some information between restarts by computing approximate eigenvectors and including them in the Krylov supspace for the subsequent restart. At the end of a FGMRES cycle of lenght $m$, $m$ *harmonic Ritz* pairs are computed using the small matrix $H$ built during the Arnoldi steps. The harmonic Ritz vectors corresponding to the smallest $k < m$ approximate eigenvectors are used to build an orthonormal basis for the deflation subspace. This subspace, augmented with the residue, becomes the initial Krylov subspace for the subsquent restart. Figure \[sapfgmresdr\] shows the convergence as a function of the Krylov subspace dimension for SAP-GCR and a comparison with the mixed precision SAP-FGMRES-DR [@sapfgmresdrp] for $\kappa$ close to $\kappa_c$. SAP-FGMRES-DR is able to converge in many cases in which SAP-GCR fails, it achieves the same convergence rate of SAP-GCR with much less vectors resulting in less critical memory requirements and faster execution. FGMRES-DR is the flexible generalization of GMRES-DR [@gmresdr].
![Convergence of SAP-GCR with Krylov subspace dimension 32, 70 and 100, SAP-FGMRES-DR with Krylov subspace dimension 32 including 10 deflated vectors. $16^3\times32$, $\beta=5.29$, $\kappa_{sea}=0.13500$, $\kappa_{val}=0.13768$, $\kappa^{c}_{val} \approx 0.13770$[]{data-label="sapfgmresdr"}](myplot){width="60.00000%"}
SAP implementation
==================
The most time consuming task in the SAP-GCR algorithm is the Schwarz Alternating Procedure. Most of the floating point operations are spent inside the block solver routines. It is thus natural to focus the optimization efforts on these tasks. Given the memory access alignment constraints, the user-controlled local store and the need of making efficient use of the SIMD instructions, it is important to choose a data layout that maximizes performance and at the same time simplifies the programming. Lattice points are ordered such that points belonging to the same block are contiguous. Inside the blocks, lattice points are divided into two sets, even and odd. In this way DMA operations performed to move block fields between main memory and local store are greatly simplified and the performance is maximized. For the spinors layout, different possibilities were analyzed by counting the floating point and shuffle instructions involved in the application of the Wilson-Dirac operator for the different layouts. The final choice is such that the indexes, from the slowest running to the fastest are: *color, spinor, complex*. In this way, when a spinor is loaded into 4-way SIMD registers, each register contains components of the same color. The SU(3) matrices, consisting in 72B arrays in single precision, do not satisfy the basic 16B alignment constraint coming from the size of the registers, we have thus chosen to pad the SU(3) structure to 80B. While this choice introduces an overhead by wasting a fraction of the memory bandwidth, it greatly simplifies the code by avoiding the otherwise necessary and slow manual alignment operations. The multicore parallelism is exploited by parallelizing the block solver on the eight cores. The block is divided along the time dimension among the cores, thus constraining the block size in the time direction to 8 (the number of cores). Each core works thus on a time-slice of the block. Complete overlap of memory accesses with computation can be in principle done given a sufficient local store size. We have chosen to have the maximum possible block size to maximize the block solver performance, this limits the amount of data that can be prefetched. The parallelization of the SAP among the QPACE nodes is not straightforward but the structure of the algorithm fits nicely the features of the network. After one block is solved, the data necessary for the update of the residue on neighboring blocks residing on remote nodes is available and is sent via a DMA put to the remote nodes directly from the SPUs. While it is in principle necessary to distinguish between remote and local blocks (data for remote blocks must be sent through the network), this distinction is not done at the level of the SPU code resulting in a further simplification of the SPU code. The PPU control code passes the addresses of the network links to the SPUs for the remote blocks while passes memory addresses in the case of local blocks. All the complexity associated with the network is handled by the PPU control code which is also responsible for issuing receive commands to the network processor. In order to distinguish between data sent by different SPUs simultaneously, the *remote offset* feature is exploited such that all the data belonging to a given block border sent by the 8 different SPUs is received in a contiguous buffer with a single receive operation, reducing the overhead associated with receive commands. The channel feature of the network is exploited by having up to 8 concurrent blocks per link. Network send and receive operations are asynchronous allowing overlap between communication and computation.
Performance
===========
The block solver performance is not limited by main memory or network. For the block solver alone, we were able to obtain an impressive $\epsilon_{fp} = 50\%$ for a block size of $8^2\times 6^2$, using the IBM xlc compiler. The compiler optimization flags used for this benchmark cause deep code transformations and the resulting SPU executable tends to be very big. In production SPU executables, for which there is the need of additional subroutines and buffers for the complete SAP, we had to reduce the block size and preferred to use the GCC compiler which produces more compact code. We measured efficiencies $\epsilon_{fp} = 36\%$ , $\epsilon_{fp} = 34\%$ and $\epsilon_{fp} = 30\%$ for $8^2\times 4^2$, $8\times 2\times 6^2$ and $8\times 10\times 2^2$ block sizes respectively. Performance is also affected by the overhead of the necessary shuffle instruction and DMA commands. The measured performance for the complete SAP is $\epsilon_{fp} = 25.9\%$ , $\epsilon_{fp} = 23\%$ and $\epsilon_{fp} = 19.3\%$ for $8^2\times 4^2$, $8\times 2\times 6^2$ and $8\times 10\times 2^2$ block sizes respectively. We used 4 iterations for the block solver and 10 SAP cycles. In a typical HMC production run with a $32^3$x$64$ lattice at $\kappa=0.13632 , \beta=5.29$ on 256 QPACE nodes we found that $75\%$, $13\%$ and $12\%$ of the solver time is spent inside SAP, double precison Wilson-Dirac operator and spinor linear algebra, respectively.
In figure \[scal\] we show the SAP performance and weak scaling for the three different block sizes used. The measured points are compared with the ideal scaling using the single node performance.
Conclusions
===========
We demonstrated that the SAP-GCR and SAP-FGMRES-DR solvers fit nicely the processor, memory hierarchy and network architecture. We showed the scalability of our implementation up to 256 nodes with very good performance. Both solvers are currently integrated in BQCD [@bqcd] and are used for production runs on QPACE.
Acknowledgements {#acknowledgements .unnumbered}
================
We would like to thank D. Pleiter, H. Simma, T. Wettig, A. Frommer, Y. Nakamura, T.Streuer, S. Solbrig, B.Mendl and the whole QPACE team. This work has been supported by the DFG (SFB/TR55, Hadron Physics from Lattice QCD).
[99]{} H. Baier *et al.*, *QPACE – a QCD parallel computer based on Cell processors*, arXiv:0911.2174 \[hep-lat\].
IBM, *Cell Broadband Engine Architecture*, Version 1.0, 8 August 2005.
A. Nobile, PhD. Thesis, 2008
F. Belletti [*et al.*]{}, *QCD on the Cell Broadband Engine*, arXiv:0710.2442
M. Lüscher, Comput. Phys. Commun. [**156**]{} (2004) 209 \[hep-lat/0310048\].
Y.Saad, *Iterative methods for sparse linear systems* SIAM (2003)
Moler, C. B. *Iterative Refinement in Floating Point* J. ACM 14, 2 (Apr. 1967), 316-321.
A. Nobile [*et al.*]{}, In preparation.
R.B. Morgan, *GMRES with deflated restarting* SIAM Journal on Scientific Computing Volume 24 Issue 1, 2002
Y. Nakamura, H. Stüben, arXiv:1011.0199
|
---
abstract: 'In this letter we derive the Cornell confining potential in a theory of interacting Abelian gauge vector and massive Kalb-Ramond tensor. The Kalb-Ramond mass is instrumental to obtain the linear confining behavior of the potential at large distances. The same model can be described via interaction with Higgs fields, alter- natively, providing mass to the vector, or to the tensor fields. In the first case, the photon acquires mass, while the tensor remains massless. The resulting interaction potential is a screened Coulomb one. In the second case, the photon remains massless while Kalb-Ramond tensor acquires mass and the resulting potential is of the Cornell type with the mass parameter determining the slope of the linear part.'
address:
- |
INFN, Sezione di Trieste,\
Trieste, Italy
- |
Dipartimento di Fisica, Gruppo Teorico, Università di Trieste, and INFN, Sezione di Trieste,\
Trieste, Italy
author:
- Anais Smailagic
- Euro Spallucci
title: 'Cornell potential in Kalb-Ramond scalar QED via Higgs mechanism'
---
Introduction
============
QCD is the widely accepted theoretical framework successfully describing Strong Interactions at large momenta. On the other hand, the experimental evidence indicates that the low momenta ( large distance ) regime is char- acterized by the confinement of the hadronic elementary constituents , i.e. quarks and gluons. So far, it has not been possible to obtain in a clear and unambiguous way a linear confining potential. Nevertheless, there is a general consensus about the Abelian character of the confinement phenomenon [@Luscher:1978rn; @Kondo:1997pc; @Kondo:1997kn; @Kondo:2014sta]. In the meantime the confinement is currently simulated by the phenomeno- logical class of potentials commonly known as Cornell-type potentials [@Eichten:1978tg], consisting in its simplest version of a sum of an attractive Coulomb potential and a linearly increasing part. In this letter we will present an Abelian toy-model giving rise to the above mentioned Cornell potential. In order to achieve this goal one has to suitably modify standard QED which by itself gives only the Coulomb interaction be- tween static charges. As a guiding line one considers Lee-Wick type modification of Maxwell elec- trodynamics originally motivated by the desire to eliminate short-distance singularity inherent in the Coulomb potential [@Lee:1969fy; @Lee:1970iw; @Accioly:2010js; @Accioly:2011zz]. In fact, it turns out that the short distance behavior is linear while at large distance the Coulomb tail is recovered. Lee-Wick model actually leads to a desired “ confining ” behavior but in the wrong range of distances. Therefore, we shall introduce a modification of the Lee-Wick Lagrangian in a way to flip short and long distance behavior. To indicate qualitatively the way to switch the two regimes, let us start from the Lee-Wick modification of QED by adding a higher derivative term to the Maxwell Lagrangian
$$\mathcal{L}_{LW}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{4}F_{\mu\nu}\frac{\partial^2}{m^2}F^{\mu\nu} \label{lw}$$
The second term in (\[lw\]) dominates at short distance and leads the linear behav- ior behavior of the static potential, while the first term provides the standard Coulomb piece at large distance. The distance scale being determined by the new mass parameter $m$.\
A simple guess that leads to the exchange of the behavior in the two different regimes is the following $$\mathcal{L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+\frac{1}{4}F_{\mu\nu}\frac{m^2}{\partial^2}F^{\mu\nu} \label{lw2}$$
While the former Lagrangian (\[lw\]) is an higher derivative local field theory, plagued with all due complications [@Anselmi:2017yux; @Anselmi:2018kgz; @Donoghue:2018lmc; @Sakoda:2019vhd], the latter model (\[lw2\]) is a non-local theory. Although this term is added to the classical Maxwell Lagrangian ad hoc, it represents a genuine quantum correction as will be shown later on.\
Fortunately, the non-local term can be converted into a local one by the introduction of an additional field suitably coupled to $F_{\mu\nu}$.\
In the extensions of $QED$ mentioned above one needs to introduce an arbitrary mass scale parameter $m$ to preserve the canonical dimension of the gauge vector field $A_\mu$. In the wider framework of the Standard model of Elementary Particles masses are generated by the spontaneous breaking of local symmetries. Accordingly, we shall explore the possibility of generating $m$ in the Lagrangian (\[lw2\]) through a suitable Higgs mechanism.\
This paper is organized as follows. In Section (\[lwed\]) we discuss $QED$ coupled to a massive Kalb-Ramond tensor field. We show that, in this model, the inter- action potential between static charges is of Cornell type. We also define the dual version of this model describing interacting open strings. In Section(3) we generalize the model by coupling both fields to a pair of dis- tinct scalar Higgs fields which generate, in alternating way, the mass for one or another field. The resulting potential are either of Yukawa or Cornell type. In Section (4) we summarize and discuss the obtained results.
Kalb-Ramond $QED$ {#lwed}
=================
We start from a quantum theory describing an Abelian vector field Aμ coupled to a massive Kalb-Ramond anti-symmetric tensor $B_{\mu\nu}$ [@Smailagic:2001ch]. The euclidean generating functional is given by
$$Z \left[\, J \,\right] =Z \left[\, 0 \,\right]^{-1}
\int D \left[\, A\,\right] D \left[\, B\, \right] e^{-\int d^4 x\,\mathcal{L} }$$
where the functional measure is defined to include the proper gauge fixing and ghost terms; the Wick rotated Lagrangian density is:
$$\begin{aligned}
&&\mathcal{L}= \frac{1}{4}F_{\mu\nu}F^{\mu\nu} -e J^\mu A_\mu +\frac{1}{12} H^{\mu\nu\rho}H_{\mu\nu\rho}
+\frac{m^2}{4} B_{\mu\nu}B^{\mu\nu} -\frac{m}{2} F^\ast_{\mu\nu} B^{\mu\nu}\ ,\label{four}\\
&& F_{\mu\nu}=\partial_{[\,\mu} A_{\nu\,]} \ ,\\
&& F_{\mu\nu}^\ast \equiv \frac{1}{4!}\epsilon_{\mu\nu\rho\sigma} F^{\rho\sigma}\ ,\\
&& H_{\mu\nu\rho}=\partial_{[\,\mu} B_{\nu\rho\,]}\end{aligned}$$
$F_{\mu\nu}^\ast$ is the Hodge dual of $F_{\mu\nu}^\ast$ and $H_{\mu\nu\rho} $ is the Kalb-Ramond field strength. The sign in front of the kinetic terms is determined in a way to provides the correct damping factor in the functional integral.\
On a general ground, the coupling constant in front of the $BF^\ast$ term need not be the same as the mass m of the Kalb-Ramond field. However, it will be shown that the desired Cornell potential can be obtained only if the two parameters are identical.\
$J^\mu$ is the divergence-free electromagnetic current density. The action is at most quadratic in $A$ and $B$ by construction. Therefore, the generating functional is Gaussian integral and can be evaluated by using classical field equations.\
$\mathcal{L}$ is called parent Lagrangian in the language of duality [@Hjelmeland:1997eg]. Thus, integrating out one or another field, one obtains two dual formulation of the same theory.\
We are interested in calculating the classic, static potential between a pair of test charges interacting via the exchange of Aμ quanta in the framework of an effective $QED$.\
To do so, we write down the classical field equations
$$\begin{aligned}
&& \partial_\mu \left(\, F^{\mu\nu} + \frac{m}{2}\epsilon^{\mu\nu\rho\sigma} B_{\rho\sigma}\,\right)= -e J^\mu\ ,\\
&& \partial_\lambda H^{\lambda\mu\nu} -m^2 B^{\mu\nu}= -m F^{\ast\,\mu\nu} \label{nove}\end{aligned}$$
From equation (\[nove\]) one obtains the divergence-free condition for $ B^{\mu\nu}$
$$\partial_\mu B^{\mu\nu}= -\frac{1}{m } \partial_\mu F^{\ast\,\mu\nu}\equiv 0$$
by virtue of the Bianchi Identities for $F_{\mu\nu}\left(\,A\,\right) $. Thus, equation (\[nove\]) can be written as
$$\left(\, \partial^2 -m^2\,\right)B^{\mu\nu} = - m F^{\ast\,\mu\nu}$$
and consequently
$$\begin{aligned}
&& B^{\mu\nu} = - \frac{m }{\partial^2 -m^2 } F^{\ast\,\mu\nu}\ ,\label{twelve}\\
&& H_{\lambda\mu\nu}= -m \partial_{[\,\lambda}\frac{1}{\partial^2 -m^2 } F^{\ast}_{\mu\nu\,]}\label{thirteen}\end{aligned}$$
Inserting (\[thirteen\]) and (\[twelve\]) in $\mathcal{L}$, one finds an effective, non-local Lagrangian for $A_\mu$
$$\begin{aligned}
\mathcal{L}_{eff}\left[\, A\,\right]&&= \frac{1}{4}F_{\mu\nu}F^{\mu\nu} -e J^\mu A_\mu
+\frac{m^2}{4} F_{\mu\nu}\frac{1}{\partial^2 -m^2 } F^{\mu\nu}\ ,\nonumber\\
&&= F_{\mu\nu}\frac{\partial^2}{\partial^2 -m^2 } F^{\mu\nu}-e J^\mu A_\mu\end{aligned}$$
The non-locality being a consequence of the $B F^\ast$ interaction in (\[four\]). Integrating out the gauge field in the generating functional, one finds
$$\mathcal{L}_{eff}\left[\, J\,\right]=-\frac{e^2}{2} J^\mu \frac{\partial^2 -m^2 }{\left(\,\partial^2\,\right)^2}J_\mu$$
We are now ready to obtain the classical potential energy between two test charges. In detail, having two heavy static charges positioned in $\vec{x}_1$ and $\vec{x}_2$, the only non-vanishing component of the current $J^\mu$ is
$$J^\mu = e \delta^\mu_0 \left[\,\left(\, \vec{x}- \vec{x}_1\,\right)-\left(\, \vec{x}- \vec{x}_2\,\right) \,\right]$$
The resulting *effective action* reads
$$\begin{aligned}
S_{eff} &&=-\frac{e^2}{2} \int d^3 x \left\{\,\left[\,\delta^{3}\left(\, \vec{x}- \vec{x}_1\,\right)-
\delta^{3}\left(\, \vec{x}- \vec{x}_2\,
\right) \,\right] \times \right.\nonumber\\
&& \left. \frac{-\nabla^2_x +m^2 }{\left(\,-\nabla_x^2\,\right)^2}
\left[\,\left(\, \delta^{3}\vec{x}- \vec{x}_1\,\right)-\delta^{3}\left(\, \vec{x}- \vec{x}_2\,\right) \,\right]\,\right\}
\label{17}\end{aligned}$$
The cross terms in (\[17\]) describes the potential energy $V_{int} \left(\, \vec{x}_1\ , \vec{x}_2\,\right)$ between the two charges as
$$V_{int} \left(\, \vec{x}_1\ , \vec{x}_2\,\right)=-\int d^3 x\left[\,\delta^{3}\left(\, \vec{x}- \vec{x}_1\,\right)
\frac{-\nabla^2_x +m^2 }{\left(\,-\nabla_x^2\,\right)^2}\delta^{3}\left(\, \vec{x}- \vec{x}_2\,\right)\,\right]
+\left(\,\vec{x}_1\leftrightarrow \vec{x}_2\,\right)$$
We discard the divergent self-energy of each charge. Subsequent integration over the dummy variable $\vec{x}$ leads to a Cornell potential
$$V_{int} \left(\, r\,\right)= -\frac{e^2}{4\pi r} + \frac{e^2 m^2}{8\pi }r\ ,
\qquad r\equiv \vert \, \vec{x}_1\ - \vec{x}_2\,\vert \label{19}$$
Equation (\[19\]) consists of two parts: the first describes the Coulomb interaction dominant at short-distances, while the second part describes linear, confining long distance tail reminiscent of the Kalb-Ramond interaction with the vector potential. We stress that the $B F^\ast$ interaction term, with the mass $m$ as a coupling strength, is instrumental for the proper modification of the standard electrodynamics leading to the linear part of the potential.
Dual “ stringy ” phase
----------------------
The usual interpretation of the confining phase has already been described in terms of flux tubes connecting the particle/anti-particle pair. The dynamics of such an object is usually described in terms of an open string with the two charges at its end-points.\
We shall show that such a description can be obtained by integrating out $A_\mu$ in favor of $B_{\mu\nu}$ in the parent Lagrangian $\mathcal{L}$. This gives the dual version of the model described above.\
The field equations for $A_\mu$ allow to express $F_{\mu\nu}$ in terms of $B_{\mu\nu}$
$$\frac{\delta\mathcal{L} }{\delta A_\mu}=0\quad \Rightarrow \partial_\lambda \left(\, F^{\lambda\mu}- B^{\ast\,\lambda\mu}
\,\right) =- e J^\mu \label{20}$$
The solution is obtained as follows. Firs, let us consider a current with support on the world-line(s) of a real pair of opposite charges. Thus,the *boundary* current $J^\mu$ can be associated with a *surface* current $\Sigma^{\mu\nu}$
$$\partial_\mu J^\mu =0 \Longrightarrow J^\mu\left(\,x\,\right) =\partial_\lambda \Sigma^{\lambda\mu}\left(\, x\,\right) =
\partial_\lambda \int_\Sigma dz^\lambda\wedge dz^\mu \delta\left(\, x -z\,\right)$$
where $\Sigma$ is the *world-surface* swept by an open string connecting the two opposite charges. With this relation between the two currents, equation (\[20\]) has a solution
$$F^{\mu\nu}= m B^{\ast\,\mu\nu} - e \Sigma^{\mu\nu} \label{22}$$
Inserting (\[22\]) into $\mathcal{L}$ gives the dual Lagrangian:
$$\mathcal{L}_{KR}\left(\, B\ ;\Sigma\,\right)= \frac{1}{12} H_{\lambda\mu\nu}H^{\lambda\mu\nu} + \frac{em}{2}
B^\ast_{\mu\nu} \Sigma^{\mu\nu} -\frac{e^2}{4} \Sigma^{\mu\nu} \Sigma_{\mu\nu} \label{23}$$
The Lagrangian (\[23\]) describes a Kalb-Ramond tensor field $B_{\mu\nu}$ interacting with the world-surface current $ \Sigma^{\mu\nu} $ of a string. The last term in $ \mathcal{L}_{KR}$ is a Schild-type kinetic term for the string itself.\
Integrating out the $B$ field in (\[23\]) leads to the effective string Lagrangian
$$\mathcal{L}_{eff}\left(\, \Sigma\,\right)= -\frac{e^2}{4} \Sigma^{\mu\nu} \Sigma_{\mu\nu}
+ \frac{e^2}{4} \Sigma^{\mu\nu} \frac{m^2}{\partial^2}\Sigma_{\mu\nu} \equiv
\mathcal{L}_{string}\left(\, \Sigma\,\right) \label{24}$$
The dual picture translate the dynamics of single interacting charges via gauge vector field into the interaction between open strings carrying opposite charges at their end-points. This interaction is mediated by the exchange of a Kalb- Ramond boson field.
Kalb-Ramond Higgs mechanism
===========================
As it has been shown in the previous Section, the non-local electrodynamics encoded in the Lagrangian (\[lw2\]) is equivalent to a local theory of interacting gauge vector and massive Kalb-Ramond fields. Furthermore, it was necessary to identify, a priori different, coupling constant of the interaction term with the mass of the free Kalb-Ramond tensor in order to generate a confining linear potential. It is generally argued that confinement is due to the different vacuum state within hadrons with respect to the surronuding vacuum. This suggests to use an Higgs mechanism in order to generate multiple vacua. Also, the Higgs mechanism can naturally produce a single mass parameter in the Lagrangian (\[lw2\]). Thus, we introduce an additional interaction between the Kalb-Ramond field and a neutral scalar field by the substitution
$$mB_{\mu\nu}\longrightarrow g \psi B_{\mu\nu} \label{25}$$
In order to give meaning to (\[25\]) we assign a proper Higgs potential to $\psi$. For the sake of generality, nothing forbids to introduce a complex scalar field $\phi$ coupled to $A_\mu$. The resulting Lagrangian reads
$$\begin{aligned}
\mathcal{L}&&= \frac{1}{4}F_{\mu\nu}F^{\mu\nu} +\frac{1}{12} H^{\mu\nu\rho}H_{\mu\nu\rho}
+\frac{g^2}{4}\psi^2 B_{\mu\nu}B^{\mu\nu} -\frac{g}{2}\psi F^\ast_{\mu\nu} B^{\mu\nu}-e J^\mu A_\mu\nonumber\\
&& +\frac{1}{2} \partial_\mu \psi \partial^\mu \psi +D_\mu^\ast \phi^\ast D^\mu \phi -V\left(\, \phi^\ast \phi\ ,\psi^2\,\right)
\label{26}\end{aligned}$$
Where the covariant derivative and the Higgs potential are $$\begin{aligned}
&& D_\mu \phi \equiv \left(\, \partial_\mu -i e A_\mu \,\right) \phi\ ,\\
&& V\left(\, \phi^\ast \phi\ ,\psi^2\,\right)= \frac{\mu^2}{2}\psi^2 -\mu^2 \phi^\ast \phi +\frac{\lambda_\psi}{4!}\psi^4
+\frac{\lambda_\phi}{6}\left(\, \phi^\ast \phi\,\right)^2 \end{aligned}$$ One can notice the alternation of sign in the quadratic terms. This choice is made in order to have one or the other field developing a non-vanishing vacuum expectation value, but not both at the same time. This will lead to different phases corresponding to different vacua. The charged Higgs field can be conveniently dealt with in the spherical basis where the Lagrangian (\[26\]) reads
$$\begin{aligned}
\mathcal{L}&&= \frac{1}{4}F_{\mu\nu}F^{\mu\nu} +\frac{1}{2}\left[\,\partial_\mu\rho\partial^\mu \rho +
\rho^2 \partial_\mu\theta \partial^\mu\theta -2e^2 \rho^2 A_\mu \partial^\mu \theta +e^2\rho^2 A^2\,\right]\nonumber\\
&& +\frac{1}{2} \partial_\mu \psi \partial^\mu \psi -V\left(\, \rho\ ,\psi^2\,\right)+\frac{1}{12} H^{\mu\nu\rho}H_{\mu\nu\rho}
\nonumber\\
&&+\frac{g^2}{4}\psi^2 B_{\mu\nu}B^{\mu\nu} -\frac{g}{2}\psi F^\ast_{\mu\nu} B^{\mu\nu}-e J^\mu A_\mu\end{aligned}$$
$$V\left(\,\rho^2\ ,\psi^2\,\right)= \frac{\mu^2}{2}\psi^2 -\frac{\mu^2}{2} \rho^2 +\frac{\lambda_\psi}{4!}\psi^4
+\frac{\lambda_\phi}{4!}\rho^4$$
The stationary points of the Higgs potential are given by
$$\begin{aligned}
&& \frac{\partial V}{\partial \rho}= -\mu^2 \rho +\frac{\lambda_\phi}{6}\rho^3\ ,\\
&&\frac{\partial V}{\partial \psi}= -\mu^2 \rho +\frac{\lambda_\psi}{6}\psi^3\end{aligned}$$
Solving the above equations one obtains three different vacua:
$$A) : \mu^2=0 \qquad \rho_0=0\ ,\qquad \psi_0=0 \ ,\qquad \hbox{Coulomb vacuum}$$
$$B) : \mu^2>0 \qquad \rho_0^2=\frac{6\mu^2}{\lambda_\phi}\ ,\qquad \psi_0=0 \ ,\qquad \hbox{Yukawa vacuum}$$
$$C) : \mu^2<0 \qquad \rho_0=0\ ,\qquad \psi_0=\frac{6\mu^2}{\lambda_\psi} \ ,\qquad \hbox{Cornell vacuum}$$
Since we are interested in the physical content of the theory in each phase separately, we ignore quantum fluctuations and ” *freeze* ” the scalar fields in their vacuum states. A similar phase portrait has been recently obtained a recent work where, instead of a vector and a tensor, two different vector fields have been coupled to a pair of Higgs fields [@Scott:2018xgo].\
In each of the phases listed above the corresponding effective Lagrangian for the fields $A_\mu$ , $\theta$, and $B_{\mu\nu}$ is
$$\begin{aligned}
\mathcal{L}&&= \frac{1}{4}F_{\mu\nu}F^{\mu\nu} +\frac{1}{2}\left[\,
\rho^2_0 \partial_\mu\theta \partial^\mu\theta -2e^2 \rho^2_0 A_\mu \partial^\mu \theta +e^2\rho^2_0 A^2\,\right]\nonumber\\
&& -V\left(\, \rho_0^2\ ,\psi^2_0\,\right)+\frac{1}{12} H^{\mu\nu\rho}H_{\mu\nu\rho}
\nonumber\\
&&+\frac{g^2}{4}\psi^2_0 B_{\mu\nu}B^{\mu\nu} -\frac{g}{2}\psi_0 F^\ast_{\mu\nu} B^{\mu\nu}-e J^\mu A_\mu\end{aligned}$$
As we do not include gravitational effects we can safely drop the constant term $V\left(\, \rho_0^2\ ,\psi^2_0\,\right)$.\
Now, we can integrate out the Goldstone boson $\theta$ using its field equation
$$\partial^2 \theta = e \partial_\mu A^\mu \longrightarrow \theta = e \frac{1}{\partial^2}\partial_\mu A^\mu$$
leading to:
$$\begin{aligned}
\mathcal{L}&&= \frac{1}{4}F_{\mu\nu}\left[\, 1 - \frac{e^2\rho_0^2}{\partial^2}\,\right] F^{\mu\nu} -e J^\mu A_\mu \nonumber\\
\nonumber\\
&&+\frac{g^2}{4}\psi^2_0 B_{\mu\nu}B^{\mu\nu} -\frac{g}{2}\psi_0 F^\ast_{\mu\nu} B^{\mu\nu}+\frac{1}{12} H^{\mu\nu\rho}H_{\mu\nu\rho} \end{aligned}$$
The next step is to eliminate $B_{\mu\nu} $ using its equation of motion which expresses it in terms of $F_{\mu\nu} $ as
$$B_{\mu\nu}=- \frac{g\psi_0}{\partial^2 -g^2\psi_0^2} F_{\mu\nu}^\ast$$
The final form of the gauge field Lagrangian is
$$\mathcal{L}= \frac{1}{4}F_{\mu\nu}
\left[\, 1 - \frac{e^2\rho_0^2}{\partial^2}+\frac{g^2\psi_0^2}{\partial^2 -g^2\psi_0^2} \,\right] F^{\mu\nu} -e J^\mu A_\mu$$
We shall now discuss the particular cases of interest.
- Coulomb phase: $\mu^2 = 0 \rightarrow \rho_0 = 0\ , \psi_0 = 0$ this is the ” trivial “ vacuum where both fields have vanishing expectation values and the theory reduces to ordinary Maxwell electrodynamics
$$\mathcal{L}= \frac{1}{4}F_{\mu\nu}F^{\mu\nu} -e J^\mu A_\mu$$
$$V_{int} \left(\, r\,\right)= -\frac{e^2}{4\pi r}$$
- Yukawa phase: $ \mu^2 > 0 \rightarrow \rho_0 =6\mu^2/\lambda_\phi \ , \psi_0 = 0$ In this case we obtain a non-local Lagrangian for $A_\mu$
$$\mathcal{L}= \frac{1}{4}F_{\mu\nu}
\left[\, 1 - \frac{e^2\rho_0^2}{\partial^2} \,\right] F^{\mu\nu} -e J^\mu A_\mu$$
which leads to the non-local current interaction term
$$L\left[\, J\,\right]= - \frac{e^2}{2} \, J^\mu \frac{1}{\partial^2 -e^2\rho_0^2} J_\mu \label{45b}$$
For (\[45b\]) one obtains the screened Coulomb potential, or Yukawa potential
$$V_{int} \left(\, r\,\right)= -\frac{e^2}{4\pi r} e^{-e\rho_0 r}$$
- Cornell phase: $ \mu^2 < 0 \rightarrow \psi_0 =6\mu^2/\lambda_\psi \ , \rho_0 = 0$ In this non-trivial vacuum the vector gauge field and current Lagrangians are $$\mathcal{L}= \frac{1}{4}F_{\mu\nu}
\left[\, 1 - \frac{\partial^2}{\partial^2-g^2\psi_0^2} \,\right] F^{\mu\nu} -e J^\mu A_\mu$$ $$L\left[\, J\,\right]= - \frac{e^2}{2} \, J^\mu \frac{\partial^2 -g^2\psi_0^2}{\left(\, \partial^2\,\right)^2} J_\mu \label{45}$$
leading to the linearly confining Cornell potential
$$V_{int} \left(\, r\,\right)= -\frac{e^2}{4\pi r} + \frac{e^2 g^2\psi_0^2}{8\pi }r$$
We mention that similar conclusions can also be obtained in a different ap- proach based on the description of QED in terms of gauge invariant variables [@Gaete:1998vr; @Gaete:1999iy; @Gaete:2007zn; @Gaete:2007sj; @Gaete:2009xf].
Summary and conclusions
=======================
In this letter we have introduced a novel way to generate a confining linear potential in the framework of an Abelian gauge theory. This model has been chosen for its relative simplicity and also motivated by the shared belief that, even in non-Abelian case, confinement is related to the Abelian sub-group. The standard Maxwell theory, unavoidably, leads to a Coulomb potential. In order to obtain a different behavior at a given range of distance one needs to introduce a suitable modification. Podolski e Bopp, and later Lee-Wick, introduced higher order derivatives terms in the Maxwell Lagrangian to ob- tain a regular potential at short distance [@bopp; @Podolsky:1942zz; @Podolsky:1944zz]. Since we are looking for a linear large distance potential, we introduced an inverse Lee-Wick type term resulting in a non-local total Lagrangian. Non-localities are usually associated with the quantum corrections. In the path-integral formalism these terms arise after integration of on, or more, fields in the original Lagrangian. In our case, the modified Lagrangian can be written in a local form by coupling the gauge vector $A_\mu$ to a massive dynamical Kalb-Ramond tensor field $B_{\mu\nu}$ . Written in the local form the physical content of the theory is transparent. The interaction term has to be of the form mB F where the Kalb-Ramond mass m is also the coupling constant.\
This double role for m can be derived from a more fundamental interaction of $B$ with a neutral Higgs scalar field $\psi$. For the sake of full generality, we also coupled $A_\mu$ to a charged complex Higgs field $\phi$. The choice of opposite signs in the quadratic terms in the Higgs potential allows to study two different phases in which only one of the two fields develops a non-vanishing vacuum expectation value in a alternating way.\
In one phase the photon is massive and the Kalb-Ramond field remains massless and one obtains a screened Coulomb potential asymptotically approaching Yukawa form.\
In the other phase, it is the photon which is massless while the Kalb-Ramond becomes massive and the corresponding potential is of the Cornell type.\
The emerging final picture describes confinement, though in a simplified set- ting, as being induced by the interaction between the gauge and the Kalb-Ramond fields leaving in a non-trivial vacuum state generated through a Higgs mechanism.\
We believe this model can offer useful hints of how to tackle confinement prob- lem in a more realistic, but technically more involved, non-Abelian framework.
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---
abstract: 'This work studies the linear approximation of high-dimensional dynamical systems using low-rank dynamic mode decomposition (DMD). Searching this approximation in a data-driven approach can be formalised as attempting to solve a low-rank constrained optimisation problem. This problem is non-convex and state-of-the-art algorithms are all sub-optimal. This paper shows that there exists a closed-form solution, which can be computed in polynomial time, and characterises the $\ell_2$-norm of the optimal approximation error. The theoretical results serve to design low-complexity algorithms building reduced models from the optimal solution, based on singular value decomposition or low-rank DMD. The algorithms are evaluated by numerical simulations using synthetic and physical data benchmarks.'
author:
- 'P. HÉAS[^1]'
- 'C. HERZET$^* $'
bibliography:
- './bibtex.bib'
title: ' Low-Rank Dynamic Mode Decomposition: Optimal Solution in Polynomial Time'
---
Reduced models, linear low-rank approximation, constrained optimisation, dynamical mode decomposition.
37M, 49K, 41A29, 68W25
Introduction
============
Context {#sec:context}
-------
The numerical discretisation of a partial differential equation parametrised by its initial condition often leads to a very high dimensional system [of the form:]{} $$\begin{aligned}
\label{eq:model_init}
\left\{\begin{aligned}
& x_{t}= f_t(x_{t-1}) , \\&x_1={\theta},
\end{aligned}\right. \vspace{-0.cm}$$ [where]{} $x_t\in {\mathds{R}}^n$ is the state variable, $f_t:{\mathds{R}}^n \to {\mathds{R}}^n$, and $\theta \in {\mathds{R}}^n$ denotes an initial condition. In some context, [*e.g.*, ]{}for uncertainty quantification purposes, one is interested by computing a set of trajectories corresponding to different initial conditions $\theta \in \Theta \subset {\mathds{R}}^n$. This may constitute an intractable task due to the high dimensionality of the space embedding the trajectories. For instance, in the case $f_t$ is linear, the overall complexity necessary to compute a trajectory with model can scale at worst in $\mathcal{O}(Tn^2)$, which is prohibitive for large values of the dimension $n$ and of the trajectory length $T$.
To overcome this issue, reduced models aim to approximate the trajectories of the system for a range of regimes determined by a set of initial conditions [@2015arXiv150206797C]. A common approach is to assume that the trajectories of interest are well approximated in a low-dimensional subspace of ${\mathds{R}}^n$. In this spirit, many tractable approximations of model have been proposed in the literature, the most common ones being [*Petrov-Galerkin projections*]{} [@quarteroni2015reduced]. These methods are however intrusive in the sense they require the knowledge of the equations ruling the high-dimensional system.
Alternatively, there exist non-intrusive data-driven approaches. In particular, linear inverse modeling [@penland1993prediction], principal oscillating patterns [@Hasselmann88], or more recently, dynamic mode decomposition (DMD) [@Chen12; @Jovanovic12; @kutz2016dynamic; @Schmid10; @Tu2014391] propose approximations of the unknown function $f_t$ by a linear and low-rank operator. This linear framework has been extended in [@CuiMarzoukWillcox2014] to the quadratic approximation of $f_t$. Although a linear framework may be in appearance limited, it has recently obtained a new surge of interest because of its prominent role in decompositions known as extended DMD or kernel-based DMD [@budivsic2012applied; @li2017extended; @williams2015data; @williams2014kernel; @2017arXiv170806850Y]. The latter decompositions can characterise non-linear behaviors under certain conditions [@klus2015numerical].\
A reduced model based on a low-rank linear approximation uses a matrix $\hat A_k \in {\mathds{R}}^{n \times n}$ of rank at most $k \le n$ which substitutes for function $f_t$ as $$\begin{aligned}
\label{eq:model_koopman_approx}
\left\{\begin{aligned}
& \tilde x_{t}= \hat A_k \tilde x_{t-1} \\&\tilde x_1={\theta}
\end{aligned}\right. , \vspace{-0.cm}$$ where elements of $\{\tilde x_{t}\}_{t=1}^T$ denote approximations of the trajectories $\{x_t\}_{t=1}^T$ satisfying . Obviously, a brute-force evaluation of a trajectory approximation with will not induce a low computational cost: the complexity scales as $\mathcal{O}(Tn^2)$. However, exploiting the fact that matrix $\hat A_k$ has a rank at most equal to $k$, this complexity, which we will qualify of [*on-line*]{}, can be significantly lowered. An on-line complexity scaling in $\mathcal{O}(Tr^2+rn)$ is reach if we can determine [*off-line*]{} the matrices $R,\,L \in {\mathds{C}}^{n \times r}$ and $S \in {\mathds{C}}^{r \times r}$ with $k \le r \le n$ such that trajectories of correspond to those obtained with the $r$-dimensional recursion $$\label{eq:genericROM0}
\left\{\begin{aligned}
&z_1=L^\intercal \theta \\
& z_t= S z_{t-1} \\
& \tilde x_{t} = R z_t
\end{aligned}\right..$$ The equivalence of systems and implies that $\prod_{i=1}^t\hat A_k=R(\prod_{i=1}^t S)L^\intercal$. We remark that if $S \in {\mathds{C}}^{r \times r}$ is a block diagonal matrix and in particular a Jordan matrix, the on-line complexity to run reduced-model scales at worst in $\mathcal{O}(s^2T+rn)$ where $s$ denotes the maximum size of the blocks. In the case $S$ is a diagonal matrix, then trajectories of can be computed with the advantageous on-line complexity of $\mathcal{O}(rn)$, [*i.e.*, ]{}linear in the ambient dimension $n$, linear in the reduced-model intrinsic dimension $r$ and independent of the trajectory length $T$.
To enjoy this low on-line computational effort, it is first necessary to compute off-line a proper matrix $\hat A_k$ in and deduce parameters $R,\,L$ and $S$ at the core of reduced model . We will refer to the term off-line complexity for the latter computational cost. A standard choice for $\hat A_k$ is to select the best trajectory approximations in the $\ell_2$-norm, for initial conditions in the set $\Theta \subset {\mathds{R}}^n$: matrix $\hat A_k$ in targets the solution of the following minimisation problem $$\begin{aligned}
\label{eq:target}
\operatorname*{arg\,min}_{A:\textrm{rank}(A)\le k} \int_{\theta \in \Theta} \sum_{t=2}^T \| x_{t}(\theta) - A^{t-1} \theta\|^2_2, $$ where $\|\cdot\|_2$ denotes the $\ell_2$-norm. A reduced model of the form of based on a low-rank minimiser of can then be deduced from its eigen-decomposition: $R$ and $L$ are set as the right and left eigen-vectors and $S$ as the matrix of eigen-values.\
Since we focus on non-intrusive [*data-driven*]{} approaches, we will assume that we do not know the exact form of $f_t$ in and we only have access to a set of representative trajectories $\{x_t(\theta_i)\}_{t=1}^T$, $i=1,...,N$ so-called *snapshots*, obtained by running the high-dimensional system for $N$ different initial conditions $\{\theta_i\}_{i=1}^N$ in the set $\Theta$. Using these snapshots, we consider a discretised version of , which corresponds to the constrained optimisation problem studied in [@Chen12; @Jovanovic12]: matrix $\hat A_k$ now targets the solution $$\begin{aligned}
\label{eq:prob}
A_k^\star \in &\operatorname*{arg\,min}_{A:\textrm{rank}(A)\le k} \sum_{t=2,i=1}^{T,N} \| x_{t}(\theta_i) - A x_{t-1}(\theta_i)\|^2_2, $$ where we have substituted $A^{t-1} \theta_i$ in by $A x_{t-1}(\theta_i)$. By bartering a polynomial objective function for a quadratic one, we consider in a simpler constrained least-square problem. This optimisation problem is however still non-convex due to the low-rank constraint.
In the light of and , we introduce the terminology used in the literature [@Chen12; @Jovanovic12; @kutz2016dynamic; @Schmid10; @Tu2014391]. Let columns of $R$ and $L$ be the dominant right and left eigen-vectors of $\hat A_k$ and let $S$ be the diagonal matrix gathering the first $k$ eigen-values. Given this choice for matrices $R$, $L$ and $S$, the term [ “low-rank DMD”]{} of system denotes the reduced model in the case $\hat A_k=A_k^\star$, while the term “DMD” denotes this reduced model in the case $\hat A_k$ is the solution of problem without the low-rank constraint.
Problematic
-----------
This work deals with the off-line construction of reduced models of the form of . It focuses on the following questions. Can we compute in polynomial time a solution of ? How to compute efficiently the parameters $R,\,L$ and $S$ of and in particular the low-rank DMD of ? We discuss in what follows these two problematics.\
**Solver for problem .** There has been in the last decade a resurgence of interest for low-rank solutions of linear matrix equations in noise-free [@fazel2002matrix; @recht2010guaranteed] or noisy [@lee2009guaranteed; @lee2010admira; @jain2010guaranteed] settings. This class of problems includes as an important particular case. Problems in this class are generally non-convex and accessing to theirs solutions in polynomial time is often out of reach.
Nevertheless, certain instances with a very special structure admit closed-form solutions [@parrilo2000cone; @mesbahi1997rank]. This occurs typically when the solution can be deduced from the Eckart-Young theorem [@eckart1936approximation]. Surprisingly, previous works [@Chen12; @Jovanovic12] presuppose that problem is difficult and does not admit a closed-form solution. Therefore, several sub-optimal approaches have been proposed in the literature. The most straightforward approach is to approximate the soluton of problem , proceeding in two independent steps [@Jovanovic12; @Tu2014391]: in a first step, an unconstrained version of (or of a similar problem) is solved; in a second step, a $k$-th order approximation of the latter solution is obtained either by truncating its SVD or its eigen-decomposition, or by solving a sparse approximation problem. As an alternative, some works propose to approximate problem by a regularised version of the unconstrained problem [@li2017extended; @2017arXiv170806850Y]. A regularisation of interest is obtained by particularising convex relaxation techniques to problem . The relaxed problem may recover exactly the solution $A_k^\star$ under certain condition [@fazel2002matrix; @lee2009guaranteed; @lee2010admira; @mishra2013low; @recht2010guaranteed]. Finally, some works tackle directly the constrained minimisation problem. Authors in [@Chen12; @Jovanovic12] suggest to approach the targeted solution $A_k^\star$ relying on the assumption that snapshots belong to a pre-determined subspace. Authors in [@jain2010guaranteed] propose an iterative hard thresholding approach with guarantees of exact recovery of $A_k^\star$ under certain conditions. In summary, none of the state-of-the-art approaches guarantee optimality in general.\
**Computation of $R,\, L$ and $S$ in .** The second problem concerns the computation of matrices $R,\,L$ and $S$ in , and in particular the computation of low-rank DMD from the solution $A_k^\star \in {\mathds{R}}^{n \times n}$. It is not clear that this will not imply a prohibitive computational burden for large $n$. Indeed the brute-force computation of the SVD or the eigen-decomposition of a matrix in ${\mathds{R}}^{n \times n}$ requires a complexity scaling at worst as $\mathcal{O}(n^3)$. Hopefully, since the range of the rows and the range of the columns of $A_k^\star$ belong to a low-dimensional subspace of $ {\mathds{R}}^{n}$, the left and right eigen-vectors or singular vectors associated to the non-zero eigen-values or singular values can be computed with a reduced complexity [@golub2013matrix]. Exploiting this idea, the authors in [@Tu2014391; @williams2014kernel] propose a method for computing matrices $R$ and $S$ in DMD based on the eigen-decomposition of a square matrix of size $T(N-1)$, involving a linear complexity in $n$. For low-rank DMD, an analogous low-complexity algorithm approximates $R$ and $S$ ([*i.e.*, ]{}the right eigen-vectors and associated eigen-values of $A_k^\star$) [@Jovanovic12]. In the case $k \ll T(N-1)$ and for a large value of the number of snapshots $N$ or the trajectory length $T$, the authors in [@williams2014kernel] suggest to rely on Krylov methods to approximate $R$ and $S$, involving a quadratic instead of a cubic complexity in $N$.
In summary, no state-of-the-art methods enable the exact computation of all low-rank DMD parameters ([*i.e.*, ]{}of $R$, $L$ and $S$) with a linear complexity in $n$, independently of $N$ or $T$.
Contributions
-------------
The contribution of this paper is twofold. First, we show that the special structure of problem enables the closed-form characterisation of an optimal solution $A_k^\star$, from which we can deduce an efficient polynomial-time solver. Besides, we also characterise the optimal approximation error in . Second, using this closed-form solution $A_k^\star$, we provide low-complexity algorithms computing reduced models and in particular the low-rank DMD of .\
The paper is organised as follows. In Section \[sec:stateArt\], we provide a review of state-of-the-art techniques to solve the constrained optimisation problem and compute low-rank DMD. Section \[sec:contrib\] details an analytical solution for problem , characterises the optimal approximation error and provides an efficient algorithm to compute this solution. Given this optimal solution, we provide algorithms to compute exactly and with a low off-line complexity SVD-based reduced model and low-rank DMD. Finally, using a synthetic and a physical data benchmark, we compare in [Section]{} \[sec:numEval\] the proposed algorithms with state-of-the-art. We draw conclusions in a last section.
Notations
=========
We will use in the following some matrix notations. The upper script $\cdot^\intercal $ will refer to the transpose operator. $I_k$ will denote the $k$-dimensional identity matrix. All along the paper, we will make extensive use of the SVD of a matrix $M\in {\mathds{R}}^{p \times q }$ with $p\ge q$: $M=U_M\Sigma_M V_M^\intercal $ with $U_M\in {\mathds{R}}^{p \times q }$, $V_M\in {\mathds{R}}^{ q \times q}$ and $\Sigma_M\in {\mathds{R}}^{q \times q }$ so that $U_M^\intercal U_M=V_M^\intercal V_M=I_q$ and $\Sigma_M $ is diagonal. The columns of matrices $U_M$ and $V_M$ will be denoted $U_M=(u_M^1 \cdots u_M^q)$ and $V_M=(v_M^1 \cdots v_M^q)$ while diagonal components of matrix $\Sigma_M $ will be $\Sigma_M =\textrm{diag}( \sigma_{M,1}, \cdots, \sigma_{M,q})$ with $\sigma_{M,i} \ge \sigma_{M,i+1}$ for $i=1,\cdots, q-1$. The Moore-Penrose pseudo-inverse of matrix $M$ is then defined as $M^{\dagger}=V_M\Sigma^{\dagger}_M U_M^\intercal $, where $\Sigma^{\dagger}_M=\textrm{diag}( \sigma_{M,1}^{\dagger}, \cdots , \sigma_{M,q}^{\dagger})$ with $$\sigma_{M,i}^{\dagger}= \left\{\begin{aligned}
&\sigma_{M,i}^{-1}\quad \textrm{if}\quad \sigma_{M,i} > 0\\&0\quad\quad\,\,\,\textrm{otherwise}
\end{aligned}\right. .\vspace{-0.cm}\\$$ The orthogonal projector onto the span of the columns (resp. of the rows) of matrix $M$ will be denoted by $\mathbb{P}_{M}=M M^\dagger=U_M\Sigma_M\Sigma_M^\dagger U_M^\intercal $ (resp. $\mathbb{P}_{M^\intercal}=M^\dagger M=V_M \Sigma_M^\dagger \Sigma_M V_{M}^\intercal$) [@golub2013matrix].
We also introduce additional notations to derive a matrix formulation of the low-rank estimation problem . We gather consecutive elements of the $i$-th snapshot trajectory between time $t_1$ and $t_2$ in matrix $X_{t_1:t_2}^i~= ~(x_{t_1}(\theta_i) \cdots x_{t_2}(\theta_i))$ and form large matrices $ {\mathbf{X}}, {\mathbf{Y}}\in {\mathds{R}}^{n \times m} $ with $m=N(T-1)$ as $$\begin{aligned}
{\mathbf{X}}= (X^1_{1:T-1} \cdots X^N_{1:T-1}) \quad \textrm{and} \quad
{\mathbf{Y}}= ( X^1_{2:T} \cdots X^N_{2:T}).
\end{aligned}$$ In order to be consistent with the SVD definition and to keep the presentation as simple as possible, this work will assume that $m\leq n$. However, all the result presented in this work can be extended without any difficulty to the case $m> n$ by using an alternative definition of the SVD.
State-Of-The-Art for Solving for and Building {#sec:stateArt}
==============================================
We begin by presenting state-of-the-art methods solving approximatively the low-rank minimisation problem . In a second part, we will make an overview of algorithms for the construction of reduced models of the form of including low-rank DMD.
Standard Candidate Solutions for
---------------------------------
Using the notations introduced previously, we want to solve problem rewritten as $$\begin{aligned}
\label{eq:prob1}
A_k^\star \in &\operatorname*{arg\,min}_{A:\textrm{rank}(A)\le k} \|{{\mathbf{Y}}}-A {{\mathbf{X}}}\|^2_F,
\end{aligned}$$ where $\|\cdot\|_F$ refers to the Frobenius norm.
### Truncation of the Unconstrained Solution {#sec:trunc}
By removing the low-rank constraint, becomes a least-squares problem admitting the solution ${{\mathbf{Y}}}{{\mathbf{X}}}^{\dagger}$, see [@Tu2014391]. We remark that the latter matrix of rank at most $m$ is also solution of problem for $k=m$, [*i.e.*, ]{} $$\begin{aligned}
\label{eq:exactDMD}
A^\star_m={{\mathbf{Y}}}{{\mathbf{X}}}^{\dagger},\end{aligned}$$ and that the cost function at this point vanishes if $\rank({{\mathbf{X}}})=m$. This solution is computable using the SVD of ${\mathbf{X}}$: $A^\star_m={{\mathbf{Y}}}V_{{{\mathbf{X}}}}\Sigma_{{{\mathbf{X}}}}^{\dagger}U_{{{\mathbf{X}}}}^\intercal.$ An approximation of the solution of satisfying the low-rank constraint $\textrm{rank}(A)\le k$ with $k < m$ can then be obtained by a truncation of the SVD or the eigen-decomposition of $A^\star_m$ using $k$ terms. Since this method relies on the SVD of ${{\mathbf{X}}}\in {\mathds{R}}^{n \times m}$, it leads to a complexity scaling as $\mathcal{O}(m^2(m+n))$.
### Approximation by Projected DMD {#sec:LowRankProj}
The so-called [*“projected DMD”*]{} proposed in [@Schmid10] as a low-dimensional approximation of $A^\star_m$ is further investigated by the authors in [@Jovanovic12] in order to approximate $A^\star_k$. It assumes that columns of $A{{\mathbf{X}}}$ are in the span of ${{\mathbf{X}}}$. This assumption is formalised in [@Jovanovic12; @Schmid10] as the existence of $A^c\in {\mathds{R}}^{ m \times m}$, the so-called *companion matrix* of $A$ parametrised by $m$ coefficients,[^2] such that $$\begin{aligned}
A {{\mathbf{X}}}={{\mathbf{X}}}A^c.\label{eq:companion}
\end{aligned}$$ Under this assumption, we obtain from a low-dimensional representation of $A$ in the span of $U_{{{\mathbf{X}}}}$, $$\begin{aligned}
\label{eq:DMDassumption}
U_{{{\mathbf{X}}}}^\intercal AU_{{{\mathbf{X}}}}=\tilde A^c,\end{aligned}$$ where $ \tilde A^c=\Sigma_{{{\mathbf{X}}}} V_{{{\mathbf{X}}}}^\intercal A^cV_{{{\mathbf{X}}}}\Sigma_{{{\mathbf{X}}}}^{\dagger}\in {\mathds{R}}^{ m \times m}$. An approximation of $A^\star_k$ is obtained in [@Jovanovic12] by plugging in the cost function of problem and by computing the $m$ coefficients of matrix $A^c$ minimising this cost. Noticing the invariance of the Frobenius norm to unitary transforms, it is straightforward to see that this is equivalent to solve the problem $$\begin{aligned}
\label{eq:DMDSVD}
\operatorname*{arg\,min}_{ \tilde A^c: \textrm{rank}(\tilde A^c\Sigma_{{{\mathbf{X}}}})\le k} \|U_{{{\mathbf{X}}}}^\intercal {{\mathbf{Y}}}V_{{{\mathbf{X}}}} - \tilde A^c \Sigma_{{{\mathbf{X}}}}\|^2_F.\end{aligned}$$ Assuming ${{\mathbf{X}}}$ is full-rank, the solution is simply given by the Eckart-Young theorem [@eckart1936approximation] as the SVD representation of matrix $B=U_{{{\mathbf{X}}}}^\intercal {{\mathbf{Y}}}V_{{{\mathbf{X}}}}$ truncated to $k$ terms multiplied by matrix $\Sigma_{{{\mathbf{X}}}}^{\dagger}$. Denoting by $\tilde B$ this truncated decomposition, we finally obtain the following approximation of $$\begin{aligned}
\label{eq:projDMD}
A_k^\star\approx U_{{{\mathbf{X}}}}\tilde B \Sigma_{{{\mathbf{X}}}}^{\dagger}U_{{{\mathbf{X}}}}^\intercal .
\end{aligned}$$
This method relies on the SVD of ${{\mathbf{X}}}\in {\mathds{R}}^{n \times m}$ and $B\in {\mathds{R}}^{m \times m}$. It thus involves a complexity scaling as $\mathcal{O}(m^2(m+n))$.
### Approximation by Sparse DMD {#sec:sparse}
The authors in [@Jovanovic12] also propose a two-stage approach they call [*“sparse DMD”*]{}. It consists in solving two independent problems. The first stage consists in computing the eigen-decomposition of the approximated solution for $k=m$. This first stage yields eigen-vectors $\zeta_i,$ for $ i=1, \cdots, m$. In a second stage, the authors assume that a linear combination of $k$ out of the $m$ computed eigen-vectors can approximate sufficiently accurately the data. This assumption serve to design a relaxed convex optimisation problem using a $\ell_1$-norm penalisation.[^3] Solving this problem, they obtain $k$ eigen-vectors and their associated coefficients. Let us note that the approximation error induced by the sparse DMD method will always be greater than the one induced by the projected approach[^4]. This method relies on the resolution of a $\ell_1$-norm minimisation problem constructed from the computation of the approximation and on its eigen-decomposition. The latter can be deduced from the eigen-decomposition of $B \Sigma_{{{\mathbf{X}}}}^{\dagger} \in {\mathds{R}}^{m \times m}$. The overall compexity scales in $\mathcal{O}(m^2(m+n))$.
### Approximation by Solving Regularised Problems {#sec:convex}
Some works propose to approximate by a regularised version of the unconstrained problem. In this spirit, Tikhonov penalisation [@li2017extended] or penalisation enforcing structured sparsity [@2017arXiv170806850Y] have been proposed. However, these choices of regulariser are arbitrarily and do not rely on sound theoretical basis. In contrast, the solution of may under certain conditions be recovered by the following quadratic program $$\begin{aligned}
\label{eq:probConvexRelas}
A_k^\star \approx&\operatorname*{arg\,min}_{A\in {\mathds{R}}^{n \times n}} \|{{\mathbf{Y}}}-A {{\mathbf{X}}}\|^2_F+ \alpha \| A \|_{*}, $$ where $\| \cdot \|_{*}$ refers to the nuclear norm.[^5] In the latter optimisation problem, $\alpha \in {\mathds{R}}_+$ represents an appropriate regularisation parameter determining the rank of the solution of . The conditions under which the approximation becomes exact are expressed in the theoretical works [@lee2010admira; @jain2010guaranteed] in terms of a so-called *restricted isometry property* that must be satisfied by the linear operator which maps $A\in {\mathds{R}}^{n \times n}$ to $ A{{\mathbf{X}}}\in {\mathds{R}}^{n \times m}$. Program is a convex optimisation problem which can be solved by standard convex optimisation techniques [@bertsekas1995nonlinear], or using dedicated algorithms as proposed in [@mishra2013low]. The algorithms solving typically involve a complexity per iteration scaling as $\mathcal{O}(mnk)$, [*i.e.*, ]{}linear with respect to the number $m$ of snapshots on the contrary to the other state-of-the-art approaches
### Approximation by Singular Value Projections {#sec:IHT}
In [@jain2010guaranteed], the authors propose an iterative hard thresholding approach to solve a generalisation of . This algorithm, called singular value projections (SVP), refines the candidate solution iteratively by adapting a classical projected descent gradient. The projection onto the non-convex set $\{A \in {\mathds{R}}^{n \times n}:\textrm{rank}(A)\le k\}$ is computed efficiently by SVD. A guarantee for the exact recovery of $A_k^\star$ is obtained under a condition based on the [restricted isometry property]{}.
The algorithm possesses a complexity per iteration scaling as $\mathcal{O}((k+m)^2n).$
Standard Methods for Building Reduced-Model {#sec:etatArt2}
--------------------------------------------
In this section, we present state-of-the-art methods to compute reduced models of the form of from the approximation of $ A_k^\star$, denoted in what follows by $\hat A_k$.
### SVD-Based Approach {#sec:SVDstate}
This approach presupposes the knowledge of a factorisation of the form $$\begin{aligned}
\label{eq:solFactor}
\hat A_k=PQ^\intercal \quad \textrm{with}\quad P,Q \in {\mathds{R}}^{n \times r},\end{aligned}$$ where $r \le n$. Assuming this structure, we note that trajectories of can be fully determined by the $r$-dimensional recursion with $R={P}^\dagger$, $L={P}$ and $S= {Q}^\intercal {P}$. According to Section \[sec:context\], the on-line complexity to compute this recursion is $\mathcal{O}(r^2T+rn)$.
Of course, by SVD of ${\hat A_k}$ it is always possible to compute matrices satisfying with $r=\rank({\hat A_k})$. However, a brute-force computation of the SVD of ${\hat A_k}\in {\mathds{R}}^{n \times n}$ may be prohibitive since it requires a complexity scaling at worst as $\mathcal{O}(n^3)$. Hopefully, there often exist other possible choices satisfying and the complexity can often be lowered to $\mathcal{O}(r^2(r+n)) $ with $\rank({\hat A_k}) \le r\le m$. For example, in the case $\hat A_k$ is obtained by the truncated approach (see Section \[sec:trunc\]), we can simply build a reduced model of the form by setting $P=U_{\mathbf{Y}}$ and $Q^\intercal = \Sigma_{\mathbf{Y}}V_{\mathbf{Y}}^\intercal {{\mathbf{X}}}^\dagger$ in . In the case $\hat A_k$ is obtained by projected DMD (see Section \[sec:LowRankProj\]), we can build this reduced model using the factorisation $P=U_{{\mathbf{X}}}$ and $Q^\intercal = \tilde B \Sigma_{{{\mathbf{X}}}}^{\dagger}U_{{{\mathbf{X}}}}^\intercal$. In the case $\hat A_k$ is obtained by sparse DMD (see Section \[sec:sparse\]), we can build this reduced model using the latter factorisation where $\tilde B$ is substituted by the approximation of $B$ by a matrix composed of $k$ of its columns, where the column indexes are determined by solving the sparse problem. Approaches based on SVP (see Section \[sec:IHT\]) naturally provide a factorisation of $\hat A_k$ of the form of . Indeed, these algorithms iteratively enrich the $k$-term SVD representation of the solution $\hat A_k$, through singular value projections [@jain2010guaranteed]. Unfortunately, regularised approaches (Section \[sec:convex\]) may not naturally exhibit such a factorisation.
### Approach by Eigen-Decomposition: Low-Rank DMD
An efficient reduced model may be obtained by rewriting in terms of the eigen-decomposition of the linear approximation $\hat A_k$ $$\begin{aligned}
\label{eq:eigenAk}
\hat A_k = R S L^\intercal.\end{aligned}$$with $L,\,R\in {\mathds{C}}^{n \times r}$ and $S \in {\mathds{C}}^{r \times r}$ is a Jordan-block matrix [@golub2013matrix] with $r={\rank(\hat {A}_k)}.$ Using this decomposition, recursion can be rewritten as $$\begin{aligned}
\label{eq:koopman0}
\tilde x_{t}=R S^{t-1} L^\intercal \theta.\end{aligned}$$ From the definition of $R$, $L$ and $S$, we obtain a direct equivalence of reduced model with recursion . We note $L=(\xi_1\cdots \xi_r) $ and $R=(\zeta_1\cdots \zeta_r)$, where $\xi_i\in {\mathds{C}}^n$ and $\zeta_i\in {\mathds{C}}^n$ are the $i$-th left and right eigen-vector of $\hat A_k$. Assuming that $\hat {A}_k$ is diagonalisable[^6], we have $S =\diag(\lambda_1,\cdots, \lambda_r)$ and becomes $$\begin{aligned}
\left\{\begin{aligned}
\tilde x_{t}&= \sum_{i=1}^{\rank(\hat {A}_k)}\zeta_i \nu_{i,t},\label{eq:koopman1}\\
\nu_{i,t}& = \lambda_i^{t-1} \xi_i^\intercal \theta, \quad \textrm{for} \quad i=1,\cdots, \rank(\hat {A}_k)
\end{aligned}\right. \vspace{-0.cm}$$ where $\lambda_i \in {\mathds{C}}$ is the $i$-th (non-zero) eigen-values of $\hat {A}_k$. As mentioned previously, in the literature the latter reduced model is called DMD or low-rank DMD, depending if $\hat A_k$ is an approximation of problem for $k=m$ or $k<m$ [@williams2015data]. As detailed in Section \[sec:context\], is the most efficient reduced model in terms of on-line complexity, since it scales as $\mathcal{O}(rn)$, with $r=\rank(\hat {A}_k)\le k$ versus at best $\mathcal{O}(r^2T+rn)$ for the SVD-based reduced model presented in the last section.
However, the eigen-decomposition of a matrix in ${\mathds{R}}^{n \times n}$ is in general prohibitive as it implies at worst an off-line complexity of $\mathcal{O}(n^3)$. But for most approximations, this off-line complexity can be significantly reduced as the matrix $\hat A_k$ is low-rank and its rows and columns span specific low-dimensional subspaces. In particular, for an approximation obtained by truncation of $A_m^\star$ (see Section \[sec:trunc\]), matrices $R$ and $S$ can be computed with an off-line complexity scaling as $\mathcal{O}(m^2(m+n))$ by exploiting the SVD of ${\mathbf{X}}$, see details in [@Tu2014391]. As detailed in [@Jovanovic12], an approximation of $R$ is deduced with the same complexity in the case $\hat A_k$ is obtained by projected DMD (see Section \[sec:LowRankProj\]) exploiting the eigen-vectors of the solution of . The amplitude $\nu_{i,t}$ (related to $S$ and $L$) is then obtained in a second stage by solving a convex optimisation problem with an iterative gradient-based method. An approximation using sparse DMD presented in Section \[sec:sparse\] implies the same computation of matrix $R$ followed by a sparse convex minimisation problem to obtain amplitudes $\nu_{i,t}$ (related to $S$ and $L$). The latter procedure does not increase the off-line complexity of the overall algorithm. No efficient algorithm is provided in the literature for the obtention of matrices $R$, $L$ and $S$ in the case $\hat A_k$ is obtained by SVP as described in Section \[sec:IHT\]. However, we can expect from matrix analysis that the eigen-decomposition of $A^\star_k$ can be efficiently computed taking advantage of the $k$-term SVD of $\hat A_k$ given by the SVP algorithm [@golub2013matrix]. Finally, unfortunately, the regularised solution presented in Section \[sec:convex\] does not in general exhibit a clear structure enabling to reduce the complexity of its eigen-decomposition.
The Proposed Approach {#sec:contrib}
======================
The existence of a closed-form solution to problem remains to our knowledge unnoted in the literature, and no exact algorithm has been proposed yet. The following section intends to fill this gap.
Closed-Form Solution to
------------------------
Let the columns of matrix $\hat {P}\in {\mathds{R}}^{n \times k}$ be the left singular vectors $\{u_{\mathbf{Z}}^i\}_{i=1}^k$ associated to the $k$ largest singular values of matrix $$\begin{aligned}
\label{eq:defZZZ} {\mathbf{Z}}= {{\mathbf{Y}}}\mathbb{P}_ {{{\mathbf{X}}}^\intercal}\in {\mathds{R}}^{n \times m},\end{aligned}$$ [*i.e.*, ]{} $$\begin{aligned}
\label{eq:hatP}
\hat {P}=\begin{pmatrix} u^1_{\mathbf{Z}}& \cdots &u^k_{\mathbf{Z}}\end{pmatrix}.\end{aligned}$$ This matrix serve to characterise a closed-form solution to given in the following theorem.\
\[prop22\] Problem admits the solution $$\begin{aligned}
\label{eq:Sol}
A_k^\star= \hat {P}\hat {P}^\intercal {{\mathbf{Y}}}{{\mathbf{X}}}^{\dagger}.\end{aligned}$$ Moreover, the optimal approximation error is given by $$\begin{aligned}
\label{eq:errorEstim}
\|{{\mathbf{Y}}}-A_k^\star {{\mathbf{X}}}\|^2_F = \sum_{i=k+1}^m \sigma_{{\mathbf{Z}},i}^2 +\sum_{i=i^*}^m \sum_{j=1}^m \sigma^2_{{{\mathbf{Y}}},j} \left( (v_{{{\mathbf{X}}}}^i)^\intercal v_{{{\mathbf{Y}}}}^j \right)^2,\end{aligned}$$ where $i^*=\rank({{\mathbf{X}}})+1$.\
Therefore, problem can be simply solved by computing the orthogonal projection of ${{\mathbf{Y}}}{{\mathbf{X}}}^\dagger$, which is the solution $A_m^\star$ of problem for $k=m$, onto the subspace spanned by the first $k$ left singular vectors of $ {\mathbf{Z}}.$ We detail the proof in Appendix \[app:Theorem\]. The $\ell_2$-norm of the error is simply expressed in terms of the singular values of ${{\mathbf{Y}}}$ and ${\mathbf{Z}}$ and of scalar products between the right singular vectors of ${{\mathbf{X}}}$ and ${{\mathbf{Y}}}$. The second term in the right-hand side of can be interpreted as the square norm of the projection of the rows of ${{\mathbf{Y}}}$ on the orthogonal to the span of the rows of ${{\mathbf{X}}}$, [*i.e.*, ]{} $$\sum_{i=i^*}^m \sum_{j=1}^m \sigma^2_{{{\mathbf{Y}}},j} \left( (v_{{{\mathbf{X}}}}^i)^\intercal v_{{{\mathbf{Y}}}}^j \right)^2=\| {{\mathbf{Y}}}(I_m-\mathbb{P}_{{{\mathbf{X}}}^{\intercal}})\|_F^2.$$ Note that if ${{\mathbf{X}}}$ is full-rank then we obtain the simplifications $\mathbb{P}_{{{\mathbf{X}}}^\intercal}=I_m$ and ${\mathbf{Z}}={{\mathbf{Y}}}$. In this case $i^*=m+1$ so that the second term in the right-hand side of vanishes and the approximation error reduces to $ \|{{\mathbf{Y}}}-A_k^\star {{\mathbf{X}}}\|^2_F = \sum_{i=k+1}^m \sigma_{{{\mathbf{Y}}},i}^2$. We remark that the latter error is independent of matrix ${{\mathbf{X}}}$ and is simply the sum of the square of the $m-k$ smallest singular values of ${{\mathbf{Y}}}$. This error also corresponds to the optimal error for the approximation ${{\mathbf{Y}}}$ by a matrix of rank at most $k$ in the Frobenius norm [@eckart1936approximation]. Besides we note that the rank of $A_k^\star$ can be smaller than $k$. Indeed, by the Sylvester’s theorem [@Horn12] we have that $$\begin{aligned}
\rank(A_k^\star) &\le \rank({{\mathbf{Y}}}{\mathbf{X}}^\dagger)
\le \min(\rank({{\mathbf{Y}}}), \rank({{\mathbf{X}}})),\end{aligned}$$ which shows that the rank of $A_k^\star$ is smaller than $k$ not only if $\rank({\mathbf{X}})$ or $\rank({\mathbf{Y}})$ are smaller than $k$, but also if $\rank({{\mathbf{Y}}}{\mathbf{X}}^\dagger)=\rank(\Sigma_{{\mathbf{Y}}}V_{{\mathbf{Y}}}^\intercal V_{\mathbf{X}}\Sigma_{\mathbf{X}}) < k$. Note that the latter condition does not necessarily imply the former.
Algorithm Solving Problem
--------------------------
**inputs**: $({\mathbf{X}},{\mathbf{Y}})$1) Compute the SVD of ${{\mathbf{X}}}= V_{{{\mathbf{X}}}} \Sigma^\dagger_{{{\mathbf{X}}}} U_{{{\mathbf{X}}}}^\intercal $ 2) Compute $\mathbb{P}_{{{\mathbf{X}}}^\intercal}= V_{{{\mathbf{X}}}}\Sigma_{{{\mathbf{X}}}}\Sigma_{{{\mathbf{X}}}}^\dagger V_{{{\mathbf{X}}}}^\intercal$.3) Compute $\mathbf{Z}={{\mathbf{Y}}}\mathbb{P}_{{{\mathbf{X}}}^\intercal}$. 4) Compute the first $k$ columns of $V_{\mathbf{Z}}$ and $\Sigma_{\mathbf{Z}}^{2}$, [*i.e.*, ]{}the first $k$ eigen-vector/eigen-value of $\mathbf{Z}^\intercal \mathbf{Z} $ 5) Compute the columns of $\hat {P}$ defined as the first $k$ columns of matrix $ \mathbf{Z} V_{\mathbf{Z}}\, \Sigma_{\mathbf{Z}}^{\dagger}$ **output**: $A_k^\star= \hat {P}\hat {P}^\intercal {{\mathbf{Y}}}V_{{{\mathbf{X}}}} \Sigma^\dagger_{{{\mathbf{X}}}} U_{{{\mathbf{X}}}}^\intercal $
We show hereafter how to compute the solution by using a product of easily-computable matrices. The left singular vectors associated to the $k$ largest singular values of matrix ${\mathbf{Z}}$ correspond to the first $k$ (real and orthonormal) eigen-vectors of matrix ${\mathbf{Z}}{\mathbf{Z}}^\intercal$. Matrix ${\mathbf{Z}}{\mathbf{Z}}^\intercal$ is of size $n\times n$. [Since $n$ is typically very large, t]{}his prohibits the direct computation of an eigen-value decomposition. But it is well-known that the eigen-vectors associated to the non-zero eigen-values of matrix ${\mathbf{Z}}{\mathbf{Z}}^\intercal \in {\mathds{R}}^{n \times n}$ can be obtained [from]{} the eigen-vectors and eigen-values of the smaller matrix $ {\mathbf{Z}}^\intercal {\mathbf{Z}}\in {\mathds{R}}^{m\times m}$. Indeed, using the SVD of ${\mathbf{Z}}$, $${\mathbf{Z}}= U_{\mathbf{Z}}\Sigma_{\mathbf{Z}}V_{\mathbf{Z}}^\intercal ,$$ we see that the columns of matrix $ U_{\mathbf{Z}}\in {\mathds{R}}^{n \times m} $ are the eigen-vectors of ${\mathbf{Z}}{\mathbf{Z}}^\intercal$ while the columns of matrix $ V_{\mathbf{Z}}\in {\mathds{R}}^{m \times m} $ are the eigen-vectors of $ {\mathbf{Z}}^\intercal {\mathbf{Z}}$. Since $ V_{\mathbf{Z}}$ is unitary, we obtain that the sought vectors are the [first]{} $k$ columns of $ U_{\mathbf{Z}}$, [*i.e.*, ]{}of $ {\mathbf{Z}}V_{\mathbf{Z}}\, \Sigma_{\mathbf{Z}}^{\dagger}.$
In the light of this remark, it is straightforward to design Algorithm \[algo:1\], which will provide the matrices to compute efficiently the solution $A_k^\star$ of as the product of the following matrices $\hat {P}\hat {P}^\intercal {{\mathbf{Y}}}V_{{{\mathbf{X}}}} \Sigma_{{{\mathbf{X}}}}^{\dagger}U_{{{\mathbf{X}}}}^\intercal $. This algorithm requires the computation of an SVD of matrix ${{\mathbf{X}}}\in {\mathds{R}}^{n \times m }$, an eigen-decomposition of matrix ${\mathbf{Z}}^\intercal {\mathbf{Z}}\in {\mathds{R}}^{m\times m}$ and matrix multiplications involving $m^2$ vector products in ${\mathds{R}}^n$ or ${\mathds{R}}^m$. The complexity of the different step of this algorithm is therefore scaling as $\mathcal{O}(m^2(m+n))$. Obviously, computing each entry of $A_k^\star \in {\mathds{R}}^{n\times n}$ in Algorithm \[algo:1\] would imply a complexity scaling as $\mathcal{O}(n^2k)$. This would be prohibitive for large $n$. However, as detailed in the next section, this is not necessary to build reduced models or .
Algorithms for Building Reduced-Model
--------------------------------------
Given the closed-form expression of the optimal low-rank linear approximation $A^\star_k$, we present in what follows two algorithms. The first one builds an SVD-based reduced model while the second one computes low-rank DMD. We introduce to this aim matrix $$\begin{aligned}
\label{eq:defhatQ}
\hat {Q}=(\hat {P}^\intercal {{\mathbf{Y}}}{{\mathbf{X}}}^\dagger)^\intercal \in {\mathds{R}}^{n \times k}, \end{aligned}$$ with matrix $\hat {P}$ given in .
### SVD-Based Reduced Model
**inputs**: $({\mathbf{X}},{\mathbf{Y}})$ 1) Compute $\hat {P}$ performing step 1 to 5 of Algorithm \[algo:1\]. 2) Compute $\hat {Q}$ using . **outputs**: $L=\hat {P}$, $R=\hat {P}^\intercal$, $S= \hat {Q}^\intercal \hat {P}$.
Noticing that $\hat {P}^\dagger =\hat {P}^\intercal$, from the expression of $A^\star_k$ it is straightforward to build an SVD-based reduced model taking the form of recursion by setting $r=k$ with $L=\hat {P}$, $R=\hat {P}^\intercal$ and $S= \hat {Q}^\intercal \hat {P}$. This simple reduced-model construction is presented in Algorithm \[algo:2bis\]. As it relies on the first five steps of Algorithm \[algo:1\], the off-line complexity to build this SVD-based reduced model goes as $\mathcal{O}(m^2(m+n))$.Therefore, the computational cost is the same as for state-of-the-art algorithms. As mentioned in Section \[sec:SVDstate\], the on-line complexity to run it then goes as $\mathcal{O}(Tk^2+kn)$.
### Low-Rank DMD
**inputs**: $({\mathbf{X}},{\mathbf{Y}})$ 1) Compute step 1 to 5 of Algorithm \[algo:1\] and use to obtain $\hat {Q}$. 2) Let $r=\rank(A^\star_k)$ and solve for $ i=1, \cdots,r$ the eigen-equations $$(\hat {Q}^\intercal \hat {P}) w^r_i = \lambda_i w^r_i\quad \textrm{and}\quad (\hat {P}^\intercal \hat {Q}) w^\ell_i = \lambda_i w^\ell_i,$$ where $w^r_i, w^\ell_i \in {\mathds{C}}^k$ and $\lambda_i\in {\mathds{C}}$ such that $|\lambda_{i+1}| \ge |\lambda_i |$. 3) Compute for $i=1,\cdots,r$ the right and left eigen-vector $$\begin{aligned}
\label{eq:defEigenvectors}
\zeta_i= \lambda_i^{-1} \hat {P}\hat {Q}^\intercal\hat {P}w^r_i\quad \textrm{and}\quad {\xi_i}= \lambda_i^{-1} \hat {Q}w^\ell_i.
\end{aligned}$$ 4) Rescale the ${\xi_i}$’s so that $ {\xi_i}^T \zeta_i =1$. **outputs**: $L=(\xi_1\cdots \xi_r)$, $R=(\zeta_1\cdots \zeta_r)$, $S=\diag(\lambda_1,\cdots, \lambda_r).$
We remark that the solution is the product of matrices in ${\mathds{R}}^{n \times k}$ and in ${\mathds{R}}^{n \times m}$. Therefore we can expect from matrix analysis that eigen-vectors of $A^\star_k$ belong to a $k$-dimensional subspace [@golub2013matrix]. Indeed, as shown in the following proposition, the parameters of low-rank DMD can be computed without any approximation by eigen-decomposition of some matrices in ${\mathds{R}}^{k \times k}$. The proof of this proposition is given in Appendix \[app:prop\].\
\[rem:2\] Assume $A^\star_k$ is diagonalisable. The components of the set $\{\zeta_i, \xi_i, \lambda_i\}_{i=1}^{\rank(A^\star_k)}$ generated by Algorithm \[algo:2\] are the right eigen-vectors, the left eigen-vectors and the eigen-values related to the non-zero eigen-values of $A^\star_k$. Moreover they satisfy $\xi_i ^\intercal A^\star_k \zeta_i =\lambda_i$.\
Assuming $ A_k^\star$ is diagonalisable, this proposition justifies Algorithm \[algo:2\], which deduces the right and left eigen-vectors and eigen-values of $ A_k^\star$ from the eigen-decomposition of two matrices in ${\mathds{R}}^{k\times k}$.
On [*sus*]{} of calling Algorithm \[algo:1\], Algorithm \[algo:2\] performs in step 2 the eigen-decomposition of matrix $(\hat {Q}^\intercal \hat {P}) \in {\mathds{R}}^{k\times k}$ and its transpose and computes in step 3 matrix multiplications involving $r\times n$ vector products in ${\mathds{R}}^m$, with $r=\rank(A^\star_k) \le k \le m$. Therefore, the additional off-line complexity due to Algorithm \[algo:2\] scales at worst as $\mathcal{O}(k^3+m^2 n))$. We remark that this complexity presents in the case $k \ll m$ an advantage in comparison to state-of-the-art approaches scaling as $\mathcal{O}(m^2(m+n))$. However, the overall off-line complexity necessary to build the low-rank DMD reduced model will be bounded by the complexity associated to the five steps of Algorithm \[algo:1\], which scales as $\mathcal{O}(m^2(m+n))$. Nevertheless, let us mention that an overall off-line complexity scaling as $\mathcal{O}(k^3+m^2 n))$ can be preserved if we accept to barter the exact eigen-decomposition computation in step 4 of Algorithm \[algo:1\] by an approximation relying on Krylov methods, as suggested in [@williams2014kernel]. The latter approximation may be worth for large values of $m$. Once this reduced model is built, as mentioned previously, low-rank DMD can be run with an on-line complexity scaling as $\mathcal{O}(rn)$ with $r=\rank(A_k^\star)$, [*i.e.*, ]{}linear in $n$ and independent of $T$, which represents a great advantage compared to the previous SVD-based reduced model.
Numerical Evaluation {#sec:numEval}
====================
In what follows, we evaluate three different approaches for computing low-rank DMD, namely\
- [*[method a)]{}*]{} denotes the proposed optimal approach, [*i.e.*, ]{}the SVD-based reduced model given by Algorithm \[algo:2bis\] or equivalently low-rank DMD given in Algorithm \[algo:2\],
- [*[method b)]{}*]{} denotes a low-rank DMD approximation based on the $k$-th order truncation of SVD of the unconstrained problem solution [@Tu2014391],
- [*[method c)]{}*]{} denotes a low-rank DMD approximation based on the projected approach, [*i.e.*, ]{}the $k$-th order approximation [@Jovanovic12].\
Rather than evaluating the error norm committed by the reduced model, [*i.e.*, ]{}the error norm of the target problem , we are interested in the capabilities of the different algorithms to solve the proxy (or equivalently ) for this problem. Therefore, the performance is measured in terms of the cost of the proxy for this problem, [*i.e.*, ]{}the normalised error norm $$\frac{\|{{\mathbf{Y}}}-\hat A_k {{\mathbf{X}}}\|_F}{\|{{\mathbf{Y}}}\|_F}$$ with respect to $k$, where $k$ denotes the bound on the rank constraint in problem . Besides, in the analysis perspective adopted most often in the literature on DMD, we are interested in evaluating the capabilities of the algorithms to compute accurately the parameters $L$, $R$ and $S$ of reduced models of the form of .
Convex relaxation approaches and iterative hard thresholding algorithms have been omitted in the present evaluation because there does not exist in the literature algorithms coding the recovery of $\hat A_k$ yet. We do not provide a comparison with the sparse DMD approach for two reasons. The first one is practical: as formulated in [@Jovanovic12], it is not obvious to tune the regularisation coefficient to induce $k$-term representations for $k=1,\cdots,m$. The second reason is theoretical: the error norm induced by the sparse DMD method will always be greater than the one induced by the projected approach, see Section \[sec:sparse\]. Concerning [[method b)]{}]{}, we chose to evaluate low-rank DMD based on SVD truncation of the unconstrained solution $A^\star_m$ (rather than on the truncation of its eigen-decomposition) because among two-stage approaches, it is by construction the most performant reduced model in terms of the error norm chosen for evaluation. We begin by evaluating the low-rank approximations using a toy model and then continue by assessing their performance for the reduction of a Rayleigh-Bénard convective system [@chandrasekhar2013hydrodynamic]. We finally evaluate Algorithm \[algo:2\] and in particular the influence of noise on the capability of the method to extract accuratly the $\zeta_i's$ in the low-rank DMD reduced model .
Synthetic Experiments with a Toy Model {#sec:toy}
---------------------------------------
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![Error norm as a function of $k$ for [*setting $i)$, $ii)$*]{} and $iii)$ and for methods a), b) and c). See details in Section \[sec:toy\]. \[fig:1\]](./Images/submission_res2.pdf "fig:"){width="0.89\columnwidth"}
![Error norm as a function of $k$ for [*setting $i)$, $ii)$*]{} and $iii)$ and for methods a), b) and c). See details in Section \[sec:toy\]. \[fig:1\]](./Images/submission_res1.pdf "fig:"){width="0.89\columnwidth"}
![Error norm as a function of $k$ for [*setting $i)$, $ii)$*]{} and $iii)$ and for methods a), b) and c). See details in Section \[sec:toy\]. \[fig:1\]](./Images/submission_res3_NL.pdf "fig:"){width="0.89\columnwidth"}
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\
We set $n=50$ and $m=40$ and consider a low-dimensional subspace of $r=30$ dimensions. Matrices ${{\mathbf{X}}}$ and ${{\mathbf{Y}}}$, are generated using and three different definitions for $f_t$:\
- [*setting [i)]{}*]{}: $f_t(x_{t-1})=G x_{t-1}$, where $G$ is chosen so that the exists $A^c$ satisfying $G {{\mathbf{X}}}={{\mathbf{X}}}A^c$,
- [*[setting ii)]{}*]{}: $f_t(x_{t-1})=F x_{t-1}$,
- [*[setting iii)]{}*]{}: $f_t(x_{t-1})=F x_{t-1}+F \textrm{diag}(x_{t-1})\textrm{diag}(x_{t-1})x_{t-1}$.\
Matrices introduced above are random matrices of rank $r$ defined as $F=\sum_{i=1}^{r} \varphi_i \varphi_i^\intercal $ and $G={{\mathbf{X}}}^\dagger F {{\mathbf{X}}}$, where the $\varphi_i$’s are $n$-dimensional independent samples of the standard normal distribution. The pseudo-inverse ${{\mathbf{X}}}^\dagger$ is computed from the SVD of ${{\mathbf{X}}}$. We draw the initial condition $\theta$ according to the same distribution. The first setting is a linear system satisfying the assumption made in the projected approach [@Schmid10; @Jovanovic12]. The two next settings do not make this assumption and simulate respectively linear and cubic dynamics.\
The performance of the three methods are displayed in Figure \[fig:1\]. As predicted by our theoretical results, [*method a)*]{}, [*i.e.*, ]{}the proposed algorithm, yields the smallest error norm. The deterioration of the error norm for [*method b)*]{} shows that a two-stage approach is sub-optimal. The error norm increase is moderate in these toy experiments. We mention that, although not displayed in the figure, the gap with the optimal solution becomes important for $k<30$ if we choose to truncate the eigen-decomposition[^7] of $A^\star_m$ instead of its SVD.
Moreover, the experiments show that as expected [*method c)*]{}, [*i.e.*, ]{}the low-rank projected approach, achieves the optimal performance as long as the assumption $G {{\mathbf{X}}}={{\mathbf{X}}}A^c$ holds, [*i.e.*, ]{}for [*setting i)*]{}. If the assumption is not satisfied, [*i.e.*, ]{}in [*setting ii)*]{} and [*iii)*]{}, the performance of the projected approach deteriorates notably for $k>10$. Nevertheless, we notice that [*method c)*]{} leads to a slight gain in performance compared to [*method b)*]{} up to a moderate rank ($k<5$).
Finally, as expected, the linear operator used to generate the snapshots is accurately recovered by [*method a)*]{} and [*method b)*]{} for $k \ge r$.
Physical Experiments {#sec:phys}
--------------------
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![Error norm as a function of $k$ for [*setting*]{} $iv)$, $v)$ and $vi)$ and for methods a), b) and c). See details in Section \[sec:phys\]. \[fig:2\]](./Images/submission_res_iv.pdf "fig:"){width="0.89\columnwidth"}
![Error norm as a function of $k$ for [*setting*]{} $iv)$, $v)$ and $vi)$ and for methods a), b) and c). See details in Section \[sec:phys\]. \[fig:2\]](./Images/submission_res_v.pdf "fig:"){width="0.89\columnwidth"}
![Error norm as a function of $k$ for [*setting*]{} $iv)$, $v)$ and $vi)$ and for methods a), b) and c). See details in Section \[sec:phys\]. \[fig:2\]](./Images/submission_res_vi.pdf "fig:"){width="0.89\columnwidth"}
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Rayleigh-Bénard model [@chandrasekhar2013hydrodynamic] constitutes a benchmark for convective system in geophysics. It is also famous because of its three-dimensional Galerkin projection, known as the “Lorenz reduced system”. The solution of the latter system, when plotted, resembles a butterfly [@Lorenz63]. Convection is driven by two coupled partial differential equations. In order to introduce the model, we need to introduce differential operators. Let $\nabla=(\partial_{s_1},\partial_{s_2})^\intercal $, $\nabla^{\perp}=(\partial_{s_2},-\partial_{s_1})^\intercal$ and $\Delta=\partial^2_{s_1}+\partial^2_{s_2}$ denote the gradient, the curl and the Laplacian with respect to the two spatial dimensions $(s_1,s_2)$. Let operator $\Delta^{-1}$ be the inverse of $\Delta$. Boundary conditions are periodic along $s_1$ and of Dirichlet type[^8] along $s_2$. At any point of the unit cell ${\mathbf{s}}=(s_1,s_2)\in [0,1]^2$ and for any time $t \ge 1$, the temperature $\tau({\mathbf{s}},t)\in {\mathds{R}}$, the buoyancy $b({\mathbf{s}},t)\in {\mathds{R}}$ and the velocity $\mathbf{v}({\mathbf{s}},t) \in {\mathds{R}}^2 $ in the cell satisfy $$\begin{aligned}
\label{eq:RB}
\left\{\begin{aligned}
&\partial_t b({\mathbf{s}},t) + \mathbf{v}({\mathbf{s}},t)^\intercal \nabla b({\mathbf{s}},t)-\sigma\Delta b({\mathbf{s}},t) - \sigma\nu \partial_{s_1} \tau ({\mathbf{s}},t)=0,\\
&\partial_t \tau({\mathbf{s}},t) + \mathbf{v}({\mathbf{s}},t)^\intercal \nabla \tau({\mathbf{s}},t)- \Delta \tau({\mathbf{s}},t) -\partial_{s_1} \Delta^{-1} b({\mathbf{s}},t)=0,
\end{aligned}\right. \end{aligned}$$ where velocity is in equilibrium with buoyancy according to $$\mathbf{v}({\mathbf{s}},t)= \nabla^{\perp}\Delta^{-1} b({\mathbf{s}},t).$$ The regime of the convective system is parametrised by two quantities: 1) the Rayleigh number $\nu\in {\mathds{R}}_+$, which balances thermal diffusion and the tendency for a packet of fluid to rise due to the buoyancy force; 2) the Prandtl number $\sigma\in {\mathds{R}}_+$ which controls the importance of viscosity compared to thermal diffusion.
In our experiments, we assume the initial condition takes the form of a solution of a Lorenz reduced model [@Lorenz63]. This initial condition, corresponding to a still fluid with a difference of temperature between the bottom and the top of the cell, is defined as $$\begin{aligned}
\label{eq:initCondition}
b({\mathbf{s}},1)=&\kappa_b\sin(a_b s_1)\sin(\pi s_2),\\
\tau({\mathbf{s}},1)=&\kappa_{\tau_1}\cos(a_\tau s_1)\sin(\pi s_2)-\kappa_{\tau_2}\sin(2\pi s_2).\nonumber\end{aligned}$$ It is easy to verify that for this parametrisation and in the particular case where $\nu=0$ and $\kappa_b= \sigma^{-1}(\pi a)^{-2}$, the non-linear system simplifies into a linear evolution of the temperature driven by a buoyancy force evolving in time according to a Taylor vortex [@Taylor37] $$\begin{aligned}
\label{eq:RBLinear}
\left\{\begin{aligned}
& b({\mathbf{s}},t)=\sigma^{-1}(\pi a_b)^{-2}{\exp^{-\sigma \pi^2a_b^2 t}} \sin(a_b s_1)\sin(\pi s_2),\\
&\partial_t \tau({\mathbf{s}},t) + \mathbf{v}({\mathbf{s}},t) \cdot \nabla \tau({\mathbf{s}},t)- \Delta \tau({\mathbf{s}},t) -\partial_{s_1} \Delta^{-1} b({\mathbf{s}},t)=0.
\end{aligned}\right. \end{aligned}$$ We use a finite difference scheme on and obtain a discrete system of the form of with $x_t=\begin{pmatrix}{b}_t\\ {\tau}_t\end{pmatrix}\in {\mathds{R}}^{n}$, and $n=1024$, where ${b}_t$’s and ${\tau}_t$’s are spatial discretisations of buoyancy and temperature fields at time $t$.\
We assume that we have at our disposal three datasets of snapshots of the discretised system trajectories. More precisely, we choose $m=50$ and consider the three following settings:\
- [*[setting $iv)$]{}*]{}: $N=50$ short trajectories ($T=2$) of the linear system obtained by fixing the initial condition on temperature with random parameters $(a_\tau, \kappa_{\tau_1},\kappa_{\tau_2}$) and setting $a_b=1$,
- [*[setting $v)$]{}*]{}: $N=5$ long trajectories ($T=11$) of the linear system obtained by setting randomly parameters $(a_\tau, \kappa_{\tau_1},\kappa_{\tau_2}$) of the initial condition on temperature and letting $a_b=1$,
- [*[setting $vi)$]{}*]{}: $N=5$ long trajectories ($T=11$) of the non-linear system obtained by setting parameters $(a_\tau, \kappa_b, \kappa_{\tau_1},\kappa_{\tau_2})$ of the initial condition randomly and letting $a_b=a_\tau$.\
Parameters $(a_\tau, \kappa_b, \kappa_{\tau_1},\kappa_{\tau_2})$ are randomly sampled so that each set of initial conditions correspond to $N$ realisations of the uniform distribution over an hyper-cube of dimensionality $r=10$.\
The performances of [*method a) b)*]{} and [*c)*]{} are displayed in Figure \[fig:2\] for these three settings. We first comment on results obtained in [*setting $iv)$*]{}. We remark that, as expected for the situation $T=2$, the error obtained by [*method $a)$*]{} vanishes for $k\ge r$, [*i.e.*, ]{}a dimensionality greater than the initial condition dimensionality. The sub-optimal solution provided by [*method b)*]{} induces an important error which vanishes only for $k=m$, [*i.e.*, ]{}for a dimensionality equal to the number of snapshots. Concerning [*method c)*]{}, it produces a fairly good solution up to $k \le 8$, but the solution is clearly sub-optimal for greater dimensions and is associated to an error saturating to a non-negligible value.
In [*setting*]{} $v)$, we have longer sequences ($T>2$) so that the dimension embedding the initial condition does not necessarily match the dimension embedding the snapshots. However, we remark that the optimal solution provided by [*method a)*]{} induces an error nearly vanishing for $k \ge10$. This attests of the fact that, for this linear model, trajectories are concentrate near the subspace spanned by the initial condition. This explains the quasi-optimal performances of [*method c)*]{} which relies on a strong assumption of linear dependance of snapshots. [*Method b)*]{} is again clearly sub-optimal and behaves analogously to [*setting*]{} $iv)$.
In the more realistic geophysical [*setting*]{} $vi)$, we see that the optimal performances achieved by [*method a)*]{} are far from being reached by [*method b)*]{} and c). As in the linear settings, we observe that the optimal error nearly vanishes for $k \ge 10 $. However, we clearly notice that the assumption on which rely [*method c)*]{} is for this non-linear model invalid, which induces an error saturating to a non-negligible value. We observe again in this case the poor performance of [*method b)*]{}.
Robustness to Noise {#sec:robust}
-------------------
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![Error norm as a function of $k$ for [*setting*]{} $vii)$ and $viii)$ and for methods a), b) and c). See details in Section \[sec:robust\]. \[fig:3\]](./Images/submission_res_vii.pdf "fig:"){width="0.89\columnwidth"}
![Error norm as a function of $k$ for [*setting*]{} $vii)$ and $viii)$ and for methods a), b) and c). See details in Section \[sec:robust\]. \[fig:3\]](./Images/submission_res_viii.pdf "fig:"){width="0.89\columnwidth"}
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$\zeta_3$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------ ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- -------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_3_opt_clean.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_3_opt.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_3_trun.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_3_proj.png "fig:"){width="0.2\columnwidth"}
![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_3_opt_clean.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_3_opt.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_3_trun.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_3_proj.png "fig:"){width="0.2\columnwidth"}
$\zeta_2$
![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_2_opt_clean.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_2_opt.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_2_trun.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_2_proj.png "fig:"){width="0.2\columnwidth"}
![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_2_opt_clean.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_2_opt.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_2_trun.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_2_proj.png "fig:"){width="0.2\columnwidth"}
$\zeta_1$
![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_1_opt_clean.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_1_opt.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_1_trun.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaT_1_proj.png "fig:"){width="0.2\columnwidth"}
![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_1_opt_clean.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_1_opt.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_1_trun.png "fig:"){width="0.2\columnwidth"} ![Amplitude of eigen-vectors $\zeta_1$, $\zeta_2$ and $\zeta_3$ of the low-rank DMD of a Rayleigh-Bénard system. Left column : reference obtained with method [a]{}) in the case of the noiseless setting *vii)*. Eigen-vector reconstruction in the noisy setting *viii)* with method a) (middle left column), method [b]{}) (middle right column) and method [c]{}) (right column). Amplitudes are related to temperature (row above) and buoyancy (row below). See details in Section \[sec:robust\]. \[fig:4\]](./Images/submission_ZetaB_1_proj.png "fig:"){width="0.2\columnwidth"}
In the following, we intend to evaluate the capabilities of the different methods to extract the parameters of in the presence of noise. To this aim, we build a dataset of $N=5$ long trajectories with $T=11$ (so that we get $m=50$ snapshots) satisfying with $k=3$ and the parameters $\{(\xi_i, \zeta_i, \lambda_i)\}_{ i=1}^3$. The latter are extracted using Algorithm \[algo:2\] from the geophysical dataset described in [*setting*]{} $vi)$. In other words, matrices ${{\mathbf{X}}}$ and ${{\mathbf{Y}}}$ are generated using and the model $f_t(x_{t-1})=G x_{t-1}$ where $$G=\begin{pmatrix}\zeta_1&\zeta_2&\zeta_3\end{pmatrix}\diag(\lambda_1,\lambda_2,\lambda_3)\begin{pmatrix}\xi_1&\xi_2&\xi_3\end{pmatrix}^\intercal.$$ We then consider the two following configurations:\
- [[[*setting*]{} $vii)$]{}]{}: the original version of this dataset,
- [[[*setting*]{} $viii)$]{}]{}: a noisy version, where we have corrupted the snapshots with a zero-mean Gaussian noise so that the peak-to-signal-ratio[^9] is $20$ dB.\
Results are displayed in Figures \[fig:3\] and \[fig:4\]. As expected we recover a vanishing optimal error for [*method a)*]{} in the case $k \ge 3$. We observe only a slight increase of the error in the presence of noise. This attests of the robustness of [*method a)*]{} to noise. In the noiseless case [*method c)*]{} reproduces almost exactly the optimal behaviour, while its performance slightly deteriorates for $k \ge 2$ in the noisy case. The quasi-optimal performance of this method in the noise-less setting can be interpreted as the fact that there exists a matrix $A_c$ such that assumption $G{{\mathbf{X}}}= {{\mathbf{X}}}A_c$ is nearly valid. This assumption no longer holds when snapshots are corrupted by noise. [*Method b)*]{} produces clearly sub-optimal solutions in the noiseless setting. More importantly, the performance of this method becomes dramatic in the presence of noise. The deterioration is clearly visible for eigen-vector $\zeta_3$ re-arranged in the form of a spatial map in Figure \[fig:4\]. The spatial structure of the eigen-vector estimated by [*method b)*]{} is completely rubbed out in this noisy setting while it is fairly preserved by [*method a)*]{} and roughly recovered by [*method c)*]{}. This illustrates the usefulness of solving the low-rank minimisation problem instead of truncating the solution of the unconstrained problem.
Conclusion
==========
This work characterises an optimal solution of the non-convex problem related to low-rank linear approximation. As shown in Theorem \[prop22\], the closed-form solution is in fact the orthogonal projection of the unconstrained problem solution ${{\mathbf{Y}}}{{\mathbf{X}}}^\dagger$ onto a low-dimensional subspace. This subspace is the span of the first $k$ left singular vectors of a matrix ${\mathbf{Z}}$, which is defined as the multiplication of ${{\mathbf{Y}}}$ by the projector onto the span of the rows of ${{\mathbf{X}}}$. The theorem also provides a characterisation of the error between the low-rank approximation and the true trajectory. The expression of the $\ell_2$-norm of the error is in fact closed-form and depends on the singular values of ${{\mathbf{Y}}}$ and ${\mathbf{Z}}$ and on the scalar product between right singular vectors of ${{\mathbf{X}}}$ and ${{\mathbf{Y}}}$. Based on this theorem, Proposition \[rem:2\] then shows that the eigen-vectors and eigen-values in low-rank DMD can be deduced from the eigen-decomposition of matrices in ${\mathds{R}}^{k \times k}$, independently from the number of observations.
These theoretical results yield a method to compute the optimal solution in polynomial time. The results also serve to design algorithms computing optimal low-rank approximation with a low-computation effort. In particular we propose an algorithm building exactly low-rank DMD with an off-line complexity going as $\mathcal{O}(m^2(m+n))$, which is the same as for state-of-the-art sub-optimal methods. This reduced model can then be run with an attractive on-line complexity scaling as $\mathcal{O}(kn)$, [*i.e.*, ]{}independently of the trajectory length $T$.
Finally, we illustrate through numerical simulations in synthetic and physical setups, the significant gain in accuracy brought by the proposed algorithm in comparison to state-of-the-art sub-optimal approaches.\
Acknowledgements {#acknowledgements .unnumbered}
================
This work was supported by the “Agence Nationale de la Recherche" through the GERONIMO project (ANR-13-JS03-0002).
Proof of Theorem \[prop22\] {#app:Theorem}
===========================
We begin by showing the first part of the theorem, namely that $A_k^\star=\hat {P}\hat {P}^\intercal {{\mathbf{Y}}}{{\mathbf{X}}}^{\dagger}$ is a solution of .\
We first prove the existence of a minimiser of . Let us show that we can restrict our attention to a minimisation problem over the set $$\mathcal{A}=\{\tilde A \in {\mathds{R}}^{n \times n} : \rank(\tilde A) \le k, \textrm{Im}(\tilde A^\intercal) \subseteq \textrm{Im}({{\mathbf{X}}})\}.$$ Indeed, we remark that any matrix $ A \in \{\tilde A \in {\mathds{R}}^{n \times n} : \rank(\tilde A) \le k\}$ can be decomposed in two components: $ A= A^\parallel+ A^\perp$ where $ A^\parallel$ belongs to the set $\mathcal{A}$, such that columns of $A^\parallel$ are orthogonal to those of $A^\perp$, [*i.e.*, ]{}$ A^\perp ( A^\parallel)^\intercal=0$. From this construction, we have that rows of $A^\perp$ are orthogonal to rows of ${{\mathbf{X}}}$. Using this decomposition, we thus have that $\| {{\mathbf{Y}}}- A {{\mathbf{X}}}\|_F^2=\| {{\mathbf{Y}}}- A^\parallel {{\mathbf{X}}}\|_F^2$. Moreover, because of this orthogonal property, we have that $ \rank( A)=\rank( A^\parallel) +\rank( A^\perp) $ so that $ \rank( A^\parallel) \le \rank( A)$. In consequence, if $ A$ is a minimiser of , then $ A^\parallel$ is also a minimiser since it leads to same value of the cost function and since it is admissible: $\rank( A^\parallel) \le \rank( A) \le k$. Therefore, it is sufficient to find a minimiser over the set $\mathcal{A}$.
Now, according to the Weierstrass’ theorem [@bertsekas1995nonlinear Proposition A.8], the existence is guaranteed if the admissible set $\mathcal{A}$ is closed and the objective function $\| {{\mathbf{Y}}}- A {{\mathbf{X}}}\|_F^2$ is coercive. Let us prove these two properties. We first show that $\mathcal{A}$ is closed. According to [@hackbusch2012tensor Lemma 2.4], the set of low-rank matrices is closed. Moreover, it is well-known that a linear subspace of a normed finite-dimensional vector space is closed [@auliac2005mathematiques Chapter 7.2], so that the set of matrices $\mathcal{A}=\{\tilde A \in {\mathds{R}}^{n \times n} : \textrm{Im}(\tilde A^\intercal) \subseteq \textrm{Im}({{\mathbf{X}}})\}$ is closed. Since $\mathcal{A}$ is the intersection of two closed sets, we deduce that $\mathcal{A}$ is closed. Next, we show coercivity. Let us consider the SVD of any $A\in \mathcal{A}$: $A=U_A\Sigma_A V_A^\intercal$, where $\Sigma_A=\textrm{diag}(\sigma_{A,1}\cdots\sigma_{A,k})$. From the definition of the Frobenius norm, we have for any $A \in \mathcal{A}$, $ \| A\|_F =( \sum_{i=1}^k\sigma_{A,i}^2)^{1/2} $. We remark that $\| A\|_F \to \infty$ if a non-empty subset of singular values, say $\{\sigma_{A,j}\}_{j \in \mathcal{J}}$, tend to infinity. Therefore, we have $$\begin{aligned}
\lim_{ \| A\|_F \to \infty :A \in \mathcal{A} } \| {{\mathbf{Y}}}- A {{\mathbf{X}}}\|_F^2
&= \lim_{\| A\|_F \to \infty: A \in \mathcal{A} } \| {{\mathbf{Y}}}\|^2_F -2 \,\textrm{trace}(Y^\intercal A {{\mathbf{X}}})+ \| A {{\mathbf{X}}}\|_F^2, \\
&= \lim_{ \| A\|_F \to \infty :A \in \mathcal{A} } \| A {{\mathbf{X}}}\|_F^2, \\
&= \lim_{\| A\|_F \to \infty : A \in \mathcal{A} } \| \Sigma_A V_A^\intercal {{\mathbf{X}}}\|_F^2, \\
&= \lim_{\sigma_{A,j} \to \infty : A \in \mathcal{A},j \in \mathcal{J}} \sum_{j=1}^n \sigma_{A,j}^2 \| {{\mathbf{X}}}^\intercal v_A^{j} \|_2^2
= \infty.
\end{aligned}$$ The second equality is obtained because the dominant term when $\| A\|_F \to \infty$ is the quadratic one $ \| A {{\mathbf{X}}}\|_F^2$. The third equality follows from the invariance of the Frobenius norm to unitary transforms while the last equality is obtained noticing that $ \| {{\mathbf{X}}}^\intercal v_A^{j} \|_2 \neq 0$ because $ v_A^{j} \in \textrm{Im}({{\mathbf{X}}})$ since $A \in \mathcal{A}$. This shows that the objective function is coercive over the closed set $\mathcal{A}$. Thus, using the Weierstrass’ theorem, this shows the existence of a minimiser of in $\mathcal{A}$ and thus in $\{\tilde A \in {\mathds{R}}^{n \times n} : \rank(\tilde A) \le k\}$. We will no longer restrict our attention to the domain $\mathcal{A}$ in the following and come back to the original problem implying the set of low-rank matrices.\
We next remark that problem can be rewritten as the unconstrained minimisation $$\begin{aligned}
\label{eq:prob1unconst}
A_k^\star \in &\operatorname*{arg\,min}_{A=PQ^\intercal: P,Q \in {\mathds{R}}^{n \times k}} \|{{\mathbf{Y}}}-A {{\mathbf{X}}}\|^2_F.\end{aligned}$$ In the following we will use the first-order optimality condition of problem to characterise its minimisers. A closed-form expression for a minimiser will then be obtained be introducing an additional orthonormal property. The first-order optimality condition and the additional orthonormal property are presented in the following lemma, which is proven in Appendix \[sec:app2\].\
\[rem:py=qx\] Problem admits a solution such that $$\begin{aligned}
&{P}^\intercal {P}=I_k\label{eq:CNsuppNorm}\\
&{{\mathbf{X}}}{{\mathbf{Y}}}^\intercal {P}= {{\mathbf{X}}}{{\mathbf{X}}}^\intercal {Q}\label{eq:CNsupp}. $$
To find a closed-form expression of a minimiser of , we need to rewrite condition . We can prove that this condition is equivalent to $$\begin{aligned}
\label{eq:ii}
\mathbb{P}_{{{\mathbf{X}}}^\intercal}{{\mathbf{Y}}}^\intercal P={{\mathbf{X}}}^\intercal {Q}.\end{aligned}$$ Indeed, we can show by contradiction that implies that, for any solution of the form ${P}{Q}^\intercal $, there exists $Z\in {\mathds{R}}^{m \times k}$ such that $$\begin{aligned}
\label{eq:eqsurZ}
\mathbb{P}_{{{\mathbf{X}}}^\intercal}{{\mathbf{Y}}}^\intercal P +Z={{\mathbf{X}}}^\intercal {Q},\end{aligned}$$ with columns of $Z$ in $\ker({{\mathbf{X}}})$. Indeed, if $ \mathbb{P}_{{{\mathbf{X}}}^\intercal}{{\mathbf{Y}}}^\intercal P +Z \neq {{\mathbf{X}}}^\intercal {Q}$, then by multiplying both sides on the left by ${{\mathbf{X}}}$ we obtain $ \mathbb{P}_{{{\mathbf{X}}}}{{\mathbf{X}}}{{\mathbf{Y}}}^\intercal P +{{\mathbf{X}}}Z= \mathbb{P}_{{{\mathbf{X}}}} {{\mathbf{X}}}{{\mathbf{Y}}}^\intercal P \neq{{\mathbf{X}}}{{\mathbf{X}}}^\intercal {Q}$. Since $ \mathbb{P}_{{{\mathbf{X}}}}$ is the orthogonal projector onto the subspace spanned by the columns of ${{\mathbf{X}}}$, the latter relation implies that $ {{\mathbf{X}}}{{\mathbf{Y}}}^\intercal P \neq{{\mathbf{X}}}{{\mathbf{X}}}^\intercal {Q}$ which contradicts . This proves that implies .
Now, since columns of the two terms in the left-hand side of are orthogonal and since columns of the matrix in the right-hand side are in the image of ${{\mathbf{X}}}^\intercal$, we deduce that the only admissible choice is $Z$ with columns belonging both to $\ker({{\mathbf{X}}})$ and $\textrm{Im}({{\mathbf{X}}}^\intercal)$, [*i.e.*, ]{}$Z$ is a matrix full of zeros. Therefore, we obtain the necessary condition .
We have shown on the one hand that implies . On the other hand, by multiplying on the left both sides of by ${{\mathbf{X}}}$, we obtain (${{\mathbf{X}}}\mathbb{P}_{{{\mathbf{X}}}^\intercal}={{\mathbf{X}}}$ because ${{\mathbf{X}}}{{\mathbf{X}}}^\dag$ is the orthogonal projector onto the space spanned by the columns of ${{\mathbf{X}}}$). Therefore the necessary conditions and are equivalent.
We are now ready to characterise a minimiser of . According to Lemma \[rem:py=qx\], we have $$\begin{aligned}
\min_{A \in {\mathds{R}}^{n \times n} : \rank(A) \le k }&\| {{\mathbf{Y}}}- A{{\mathbf{X}}}\|_F^2 \nonumber \\
=&\min_{( \tilde {P}, \tilde{Q}) \in {\mathds{R}}^{n \times k} \times {\mathds{R}}^{n \times k} }\| {{\mathbf{Y}}}- \tilde {P}\tilde {Q}^\intercal {{\mathbf{X}}}\|_F^2 \quad s.t. \quad
\left\{\begin{aligned}
&\tilde {P}^\intercal \tilde {P}=I_k\\
&{{\mathbf{X}}}{{\mathbf{Y}}}^\intercal {P}= {{\mathbf{X}}}{{\mathbf{X}}}^\intercal {Q}\\ \end{aligned}\right. ,\label{eq:P0prim0} \\
=&\min_{( \tilde {P}, \tilde{Q}) \in {\mathds{R}}^{n \times k} \times {\mathds{R}}^{n \times k} }\| {{\mathbf{Y}}}- \tilde {P}\tilde {Q}^\intercal {{\mathbf{X}}}\|_F^2 \quad s.t. \quad
\left\{\begin{aligned}
&\tilde {P}^\intercal \tilde {P}=I_k\\
& \mathbb{P}_{{{\mathbf{X}}}^\intercal}{{\mathbf{Y}}}^\intercal P={{\mathbf{X}}}^\intercal {Q}\\ \end{aligned}\right. ,\nonumber\\
=&\min_{ \tilde {P}\in {\mathds{R}}^{n \times k} }\| {{\mathbf{Y}}}- \tilde {P}\tilde{P}^\intercal {{\mathbf{Y}}}\mathbb{P}_{{{\mathbf{X}}}^\intercal} \|_F^2 \quad s.t. \quad
\tilde {P}^\intercal \tilde {P}=I_k,\label{eq:objFuncOneTerm}\\
=&\min_{ \tilde {P}\in {\mathds{R}}^{n \times k} }\| ({{\mathbf{Y}}}- \tilde {P}\tilde{P}^\intercal {{\mathbf{Y}}})\mathbb{P}_{{{\mathbf{X}}}^\intercal} + {{\mathbf{Y}}}(I_m-\mathbb{P}_{{{\mathbf{X}}}^\intercal}) \|_F^2 \quad s.t. \quad
\tilde {P}^\intercal \tilde {P}=I_k, \nonumber \\
=&\min_{ \tilde {P}\in {\mathds{R}}^{n \times k} }\| {\mathbf{Z}}- \tilde {P}\tilde{P}^\intercal {\mathbf{Z}}\|_F^2 + \| {{\mathbf{Y}}}(I_m-\mathbb{P}_{{{\mathbf{X}}}^\intercal}) \|_F^2 \quad s.t. \quad
\tilde {P}^\intercal \tilde {P}=I_k. \label{eq:P0prim}
\end{aligned}$$ The second equality is obtained from the equivalence between and . The third equality is obtained by introducing the second constraint in the cost function and noticing that projection operators are always symmetric, [*i.e.*, ]{}$(\mathbb{P}_{{{\mathbf{X}}}^\intercal})^\intercal= \mathbb{P}_{{{\mathbf{X}}}^\intercal}, $ while the last equality follows from the definition of ${\mathbf{Z}}$ given in and the orthogonality of the columns of the two terms. Problem is a proper orthogonal decomposition problem with the snapshot matrix ${\mathbf{Z}}$. The solution of this proper orthogonal decomposition problem is the matrix $\hat {P}$ (with orthonormal columns) given in , see [*e.g.*, ]{} [@quarteroni2015reduced Proposition 6.1]. We thus obtain from that $$\begin{aligned}
\label{eq:mimimum}
\min_{A \in {\mathds{R}}^{n \times n} : \rank(A) \le k }\| {{\mathbf{Y}}}- A{{\mathbf{X}}}\|_F^2 =\| {{\mathbf{Y}}}- \hat {P}\hat {P}^\intercal {{\mathbf{Y}}}\mathbb{P}_{{{\mathbf{X}}}^\intercal} \|_F^2 . \end{aligned}$$ Furthermore, we verify that $A_k^\star=\hat {P}\hat {Q}^\intercal $ with $$\hat {Q}=({{\mathbf{X}}}^\intercal )^\dag{{\mathbf{Y}}}^\intercal \hat {P},$$ is a minimiser of . Indeed, we check that $(\hat {P},\hat {Q})$ is admissible for problem since $${{\mathbf{X}}}{{\mathbf{X}}}^\intercal \hat {Q}={{\mathbf{X}}}{{\mathbf{X}}}^\intercal ({{\mathbf{X}}}^\intercal )^\dag{{\mathbf{Y}}}^\intercal \hat {P}= {{\mathbf{X}}}{{\mathbf{Y}}}^\intercal \hat {P}.$$ We also check using that $$\| {{\mathbf{Y}}}- \hat {P}\hat {Q}^\intercal {{\mathbf{X}}}\|_F^2=\| {{\mathbf{Y}}}- \hat {P}\hat {P}^\intercal {{\mathbf{Y}}}\mathbb{P}_{{{\mathbf{X}}}^\intercal}\|_F^2,$$ [*i.e.*, ]{}that $(\hat {P},\hat {Q})$ reaches the minimum given in . In consequence, we have shown that problem , and equivalently problem , admit the minimiser $A_k^\star=\hat {P}\hat {Q}^\intercal =\hat {P}\hat {P}^\intercal {{\mathbf{Y}}}{{\mathbf{X}}}^{\dagger}$.\
It remains to prove the second part of the theorem, namely the characterisation of the approximation error. According to standard proper orthogonal decomposition analysis, see [*e.g.*, ]{}[@quarteroni2015reduced Proposition 6.1], the first term of the cost function in evaluated at $A_k^\star$ is $$\begin{aligned}
\label{eq:ErrorFirstTerm}
\| {\mathbf{Z}}- \hat {P}\hat{P}^\intercal {\mathbf{Z}}\|_F^2= \sum_{i=k+1}^m \sigma_{{\mathbf{Z}},i}^2.\end{aligned}$$ We can rewrite the second term of the cost function as $$\begin{aligned}
\label{eq:ErrorSecondTerm}
\| {{\mathbf{Y}}}(I_m-\mathbb{P}_{{{\mathbf{X}}}^\intercal}) \|_F^2&= \| \Sigma_{{{\mathbf{Y}}}} V_{{{\mathbf{Y}}}}^\intercal V_{{{\mathbf{X}}}} (I_m-\Sigma_{{{\mathbf{X}}}} \Sigma_{{{\mathbf{X}}}}^\dagger ) V_{{{\mathbf{X}}}}^\intercal \|_F^2,\nonumber \\
&= \| \Sigma_{{{\mathbf{Y}}}} V_{{{\mathbf{Y}}}}^\intercal V_{{{\mathbf{X}}}} (I_m-\Sigma_{{{\mathbf{X}}}} \Sigma_{{{\mathbf{X}}}}^\dagger ) \|_F^2 ,\nonumber \\
&= \left\lVert \begin{pmatrix} \sigma_{{{\mathbf{Y}}},1} (v_{{{\mathbf{Y}}}}^1)^\intercal \\ \vdots \\ \sigma_{{{\mathbf{Y}}},m} (v_{{{\mathbf{Y}}}}^m)^\intercal \end{pmatrix} \begin{pmatrix} v_{{{\mathbf{X}}}}^{i^*} & \cdots & v_{{{\mathbf{X}}}}^{m}\end{pmatrix} \right\lVert_F^2, \nonumber\\
&= \sum_{i=i^*}^m \left\lVert \begin{pmatrix} \sigma_{{{\mathbf{Y}}},1} (v_{{{\mathbf{Y}}}}^1)^\intercal \\ \vdots \\ \sigma_{{{\mathbf{Y}}},m} (v_{{{\mathbf{Y}}}}^m)^\intercal \end{pmatrix} v_{{{\mathbf{X}}}}^{i} \right\lVert_2^2,\nonumber \\
&= \sum_{i=i^*}^m \sum_{j=1}^{m} \sigma^2_{{{\mathbf{Y}}},j} ((v_{{{\mathbf{Y}}}}^j)^\intercal v_{{{\mathbf{X}}}}^i )^2.\end{aligned}$$ where the first and second equalities follow from the invariance of the Frobenius norm to unitary transforms, and more precisely to the multiplication on the left by $U_{{{\mathbf{Y}}}}^\intercal$ and on the right by $V_{{{\mathbf{X}}}}$. Gathering error contributions and , we obtain the sought result. $\square$\
Proof of Lemma \[rem:py=qx\] {#sec:app2}
============================
We begin by proving that any minimiser of can be rewritten as ${P}{Q}^\intercal $ where $ {P}^\intercal {P}=I_k$. Indeed, the existence of the SVD of $ \tilde A$ for any minimiser $\tilde A \in {\mathds{R}}^{n \times n }$ guarantees that $$\| {{\mathbf{Y}}}- \tilde A {{\mathbf{X}}}\|^2_F = \| {{\mathbf{Y}}}- U_{ \tilde A }\Sigma_{ \tilde A }V^\intercal _{ \tilde A }{{\mathbf{X}}}\|^2_F,$$ where $U_{ \tilde A } \in {\mathds{R}}^{n \times k}$ possesses orthonormal columns. Making the identification $ {P}=U_{ \tilde A }$ and $ {Q}=V_{ \tilde A }\Sigma_{ \tilde A }$ we verify that $ \| {{\mathbf{Y}}}- \tilde A {{\mathbf{X}}}\|^2_F= \| {{\mathbf{Y}}}- {P}{Q}^\intercal {{\mathbf{X}}}\|^2_F$ and that $ {P}$ possesses orthonormal columns.
Next, any solution of problem $PQ^\intercal$ of should satisfy the first-order optimality condition with respect to the $j$-th column denoted $q_j$ of matrix $Q$, that is $$\begin{aligned}
2[-{{\mathbf{X}}}{{\mathbf{Y}}}^\intercal p_j + \sum_{i=1}^k (p_i^\intercal p_j) {{\mathbf{X}}}{{\mathbf{X}}}^\intercal q_i]=0,\end{aligned}$$ where the $j$-th column of matrix $P$ is denoted $p_j$. In particular, a solution with $ {P}$ possessing orthonormal columns should satisfy $$\begin{aligned}
{{\mathbf{X}}}{{\mathbf{Y}}}^\intercal p_j= {{\mathbf{X}}}{{\mathbf{X}}}^\intercal q_j ,\end{aligned}$$ or in matrix form $${{\mathbf{X}}}{{\mathbf{Y}}}^\intercal {P}={{\mathbf{X}}}{{\mathbf{X}}}^\intercal {Q}. \quad \square$$
Proof of Proposition \[rem:2\] {#app:prop}
==============================
We have $A_k^\star= \hat {P}\hat {P}^\intercal {{\mathbf{Y}}}{{\mathbf{X}}}^\dagger=\hat {P}\hat {Q}^\intercal$ which implies that $$\hat {Q}^\intercal \hat {P}=\hat {P}^\intercal {{\mathbf{Y}}}{{\mathbf{X}}}^\dagger \hat {P}= \hat {P}^\intercal \hat {P}\hat {P}^\intercal {{\mathbf{Y}}}{{\mathbf{X}}}^\dagger \hat {P}= \hat {P}^\intercal \hat {P}\hat {Q}^\intercal \hat {P}.$$ Using the definition of $\zeta_i$’s and $\xi_i$’s in and the fact that the $w^r_i$’s and $w^\ell_i$’s are the right and left eigen-vectors of $\hat {Q}^\intercal \hat {P}$, we verify that $$A^\star_k \zeta_i= \frac{1}{\lambda_i}\hat {P}\hat {Q}^\intercal \hat {P}\hat {Q}^\intercal \hat {P}w^r_i= \hat {P}\hat {Q}^\intercal \hat {P}w^r_i =\lambda_i \zeta_i,$$ and that $$(A^\star_k)^\intercal \xi_i=\frac{1}{\lambda_i}\hat{Q}\hat {P}^\intercal \hat {Q}w^\ell_i = \hat {Q}w^\ell_i=\lambda_i \xi_i.$$ Finally, $\xi_i^\intercal \zeta_i =1$ is a sufficient condition so that $\xi_i^\intercal A_k^\star \zeta_i =\lambda_i. \quad \square$
[^1]: INRIA Centre Rennes - Bretagne Atlantique, campus universitaire de Beaulieu, 35042 Rennes, France ([patrick.heas@inria.fr, cedric.herzet@inria.fr]{})
[^2]: The particular parametrisation of the companion matrix $$A^c=
\begin{pmatrix}
0 & & & & \alpha_1\\
1 &0& && \alpha_2\\
&\ddots & \ddots& &\vdots\\
& & 1& 0& \alpha_{m-1}\\
&& &1& \alpha_m
\end{pmatrix} \in {\mathds{R}}^{m \times m},$$ depending on $m$ coefficients $\{\alpha_i\}_{i=1}^m$ follows from the fact that by definition the first $m-1$ columns of ${{\mathbf{Y}}}$ are the last $m-1$ columns of ${{\mathbf{X}}}$, see details in [@Schmid10].
[^3]: We note that the penalisation coefficient must be adjusted to induce $m-k$ coefficients nearly equal to zero.
[^4]: Exploiting orthogonality and the invariance of the Frobenius norm to unitary transforms, we obtain $ \|{{\mathbf{Y}}}-{{\mathbf{X}}}\tilde A^c \|^2_F=
\|U_{{{\mathbf{X}}}}^\intercal {{\mathbf{Y}}}V_{{{\mathbf{X}}}}-\tilde A^c \Sigma_{{{\mathbf{X}}}}\|^2_F + \|( U_{{{\mathbf{X}}}}^\perp)^\intercal {{\mathbf{Y}}}\|^2_F,
$ where the columns of $U_{{{\mathbf{X}}}}^\perp$ contain the $n-m$ vectors orthogonal to $ U_{{{\mathbf{X}}}}$. Moreover, any companion matrix $A^c$ obviously satisfies $$\min_{\tilde A^c : \rank(\tilde A^c\Sigma_{{{\mathbf{X}}}})\le k} \|U_{{{\mathbf{X}}}}^\intercal {{\mathbf{Y}}}V_{{{\mathbf{X}}}}-\tilde A^c \Sigma_{{{\mathbf{X}}}}\|^2_F \le \|U_{{{\mathbf{X}}}}^\intercal {{\mathbf{Y}}}V_{{{\mathbf{X}}}}- A^c \Sigma_{{{\mathbf{X}}}}\|^2_F,$$ and this inequality is in particular verified by the companion matrix solving the sparse DMD problem. Taking the minimum over the set of low-rank companion matrices, we therefore obtain that $$\begin{aligned}
\|{{\mathbf{Y}}}-\hat A_k {{\mathbf{X}}}\|^2_F&=\min_{\tilde A^c : \rank(\tilde A^c\Sigma_{{{\mathbf{X}}}})\le k} \|U_{{{\mathbf{X}}}}^\intercal {{\mathbf{Y}}}V_{{{\mathbf{X}}}}-\tilde A^c \Sigma_{{{\mathbf{X}}}}\|^2_F + \|( U_{{{\mathbf{X}}}}^\perp)^\intercal {{\mathbf{Y}}}\|^2_F \le \|{{\mathbf{Y}}}- \hat A_{k,0} {{\mathbf{X}}}\|^2_F, \end{aligned}$$ where $\hat A_k$ denotes the low-rank solution given by the projected approach and $\hat A_{k,0}$ denotes the solution given by the sparse DMD method.
[^5]: The nuclear norm or trace norm of a matrix is the sum of its singular values.
[^6]: Diagonalisability is assured if all the non-zero eigenvalues are distinct. This condition is however only sufficient and the class of diagonalisable matrices is larger [@Horn12].
[^7]: As pointed out previously, this alternative two-stage method is voluntarily not displayed to lighten the presentation.
[^8]: In order to simplify the Fourier-based numerical implementation of the model, we will assume periodicity for the discretised system in the two spatial directions.
[^9]: The peak-to-signal-ration is defined as $20 \log_{10} \frac{\max_{t,i}\|x_t(\theta_i)\|_\infty}{ \sigma}$, where $\sigma$ denotes the standard deviation of the Gaussian law.
|
---
abstract: |
We consider a compact, oriented, smooth Riemannian manifold $M$ (with or without boundary) and we suppose $G$ is a torus acting by isometries on $M$. Given $X$ in the Lie algebra and corresponding vector field $X_M$ on $M$, one defines Witten’s inhomogeneous coboundary operator $\d_{X_M} = \d+\iota_{X_M}:
\Omega_G^\pm \to\Omega_G^\mp$ (even/odd invariant forms on $M$) and its adjoint $\delta_{X_M}$. Witten [@Witten] showed that the resulting cohomology classes have $X_M$-harmonic representatives (forms in the null space of $\Delta_{X_M} =
(\d_{X_M}+\delta_{X_M})^2$), and the cohomology groups are isomorphic to the ordinary de Rham cohomology groups of the set $N(X_M)$ of zeros of $X_M$. Our principal purpose is to extend these results to manifolds with boundary. In particular, we define relative (to the boundary) and absolute versions of the $X_M$-cohomology and show the classes have representative $X_M$-harmonic fields with appropriate boundary conditions. To do this we present the relevant version of the Hodge-Morrey-Friedrichs decomposition theorem for invariant forms in terms of the operators $\d_{X_M}$ and $\delta_{X_M}$. We also elucidate the connection between the $X_M$-cohomology groups and the relative and absolute equivariant cohomology, following work of Atiyah and Bott. This connection is then exploited to show that every harmonic field with appropriate boundary conditions on $N(X_M)$ has a unique $X_M$-harmonic field on $M$, with corresponding boundary conditions. Finally, we define the $X_M$-Poincaré duality angles between the interior subspaces of $X_M$-harmonic fields on $M$ with appropriate boundary conditions, following recent work of DeTurck and Gluck.
Hodge theory,manifolds with boundary,equivariant cohomology,Killing vector fields
address: 'School of Mathematics, University of Manchester, Manchester M13 9PL, England'
author:
- 'Qusay S.A. Al-Zamil'
- James Montaldi
title: 'Witten-Hodge theory for manifolds with boundary and equivariant cohomology'
---
Introduction
============
Throughout we assume $M$ to be a compact oriented smooth Riemannian manifold of dimension $n$, with or without boundary. For each $k$ we denote by $\Omega^k = \Omega^k(M)$ the space of smooth differential $k$-forms on $M$. The de Rham cohomology of $M$ is defined to be $H^k(M) = \ker\d_k/{\mathop{\mathrm{im}}\nolimits}\d_{k-1}$, where $\d_k$ is the restriction of the exterior differential $\d$ to $\Omega^k$. In other words it is the cohomology of the de Rham complex $(\Omega^*,\d)$. If $M$ has a boundary, then the relative de Rham cohomology $H^k(M,\,\partial M)$ is defined to be the cohomology of the subcomplex $(\Omega^*_D,\d)$ where $\Omega^k_D$ is the space of *Dirichlet* $k$-forms—those satisfying $i^*\omega=0$ where $i:\partial M\ \hookrightarrow M$ is the inclusion of the boundary.
#### Classical Hodge theory
Based on the Riemannian structure, there is a natural inner product on each $\Omega^k$ defined by $$\label{eq:inner product}
\left<\alpha,\,\beta\right> = \int_M\alpha\wedge(\star\beta),$$ where $\star:\Omega^k\to\Omega^{n-k}$ is the Hodge star operator [@Marsden; @Schwarz]. One defines $\delta:\Omega^k\to\Omega^{k-1}$ by $$\label{eq:delta}
\delta\omega = (-1)^{n(k+1)+1} (\star \d\star)\omega.$$ If $M$ is boundaryless, this is seen to be the formal adjoint of $\d$ relative to the inner product (\[eq:inner product\]): $\left<\d\alpha,\,\beta\right> =
\left<\alpha,\,\delta\beta\right>$. The Hodge Laplacian is defined by $\Delta = (\d+\delta)^2 = \d\delta+\delta \d$, and a form $\omega$ is said to be *harmonic* if $\Delta\omega=0$.
In the 1930s, Hodge [@Hodge] proved the fundamental result that (for $M$ without boundary) each cohomology class contains a unique harmonic form. A more precise statement is that, for each $k$, $$\label{eq:Hodge}
\Omega^k(M) = \mathcal{H}^k\oplus \d\Omega^{k-1} \oplus
\delta\Omega^{k+1}.$$ The direct sums are orthogonal with respect to the inner product (\[eq:inner product\]), and the direct sum of the first two subspaces is equal to the subspace of all closed $k$-forms (that is, $\ker\d_k$). It follows that the Hodge star operator realizes Poincaré duality at the level of harmonic forms.
Furthermore, any harmonic form $\omega\in \ker \Delta$ is both closed ($\d\omega=0$) and co-closed ($\delta\omega=0$), as $$\label{eq:ker=ker}
0=\left<\Delta\omega,\,\omega\right> =
\left<\d\delta\omega,\,\omega\right> +
\left<\delta\d\omega,\,\omega\right> =
\left<\delta\omega,\,\delta\omega\right> +
\left<\d\omega,\,\d\omega\right> = \|\delta\omega\|^2 +
\|\d\omega\|^2.$$ For manifolds with boundary this is no longer true, and in general we write $$\mathcal{H}^k = \mathcal{H}^k(M)=\ker \d \cap\ker\delta.$$ Thus for manifolds without boundary $\mathcal{H}(M)=\ker\Delta$, the space of harmonic forms.
\[rmk:harmonic forms are invariant\] An interesting observation which follows from the theorem of Hodge is the following. If a group $G$ acts on $M$ then there is an induced action on each $H^k(M)$, and if this action is trivial (for example, if $G$ is a connected Lie group) and the action is by isometries, then each harmonic form is invariant under this action.
#### Witten’s deformation of Hodge theory
Now suppose $K$ is a Killing vector field on $M$ (meaning that the Lie derivative of the metric vanishes). Witten [@Witten] defines, for each $s\in\RR$, an operator on differential forms by $$\d_s := \d + s\, \iota_K\,,$$ where $\iota_K$ is interior multiplication of a form with $K$. This operator is no longer homogeneous in the degree of the form: if $\omega\in\Omega^k(M)$ then $\d_s\omega \in \Omega^{k+1}\oplus\Omega^{k-1}$. Note then that $\d_s:\Omega^{\pm}\to\Omega^{\mp}$, where $\Omega^\pm$ is the space of forms of even ($+$) or odd ($-$) degree. Let us write $\delta_s=\d_s^*$ for the formal adjoint of $\d_s$ (so given by $\delta_s = \delta + s(-1)^{n(k+1)+1} (\star\,\iota_K\star)$ on each homogeneous form of degree $k$). By Cartan’s formula, $\d_s^2 = s\mathcal{L}_K$ (the Lie derivative along $sK$). On the space $\Omega_s^\pm = \Omega^\pm \cap \ker\mathcal{L}_{K}$ of invariant forms, $\d_s^2=0$ so one can define two cohomology groups $H_s^\pm := \ker\d_s^\pm/{\mathop{\mathrm{im}}\nolimits}\d_s^\mp$. Witten then defines $$\Delta_s: = (\d_s+\delta_s)^2:\Omega_{s}^\pm(M)\to\Omega_{s}^\pm(M),$$ (which he denotes $H_s$ as it represents a Hamiltonian operator, but for us this would cause confusion), and he observes that using standard Hodge theory arguments, there is an isomorphism $$\label{eq:Witten1}
\mathcal{H}_s^\pm := (\ker\Delta_s)^\pm \cong H_s^\pm(M),$$ although no details of the proof are given (the interested reader can find details in [@my; @thesis]). Witten also shows, among other things, that for $s\neq 0$, the dimensions of $\mathcal{H}_s^\pm$ are respectively equal to the total even and odd Betti numbers of the subset $N$ of zeros of $K$, which in particular implies the finiteness of $\dim\mathcal{H}_s$. Atiyah and Bott [@AB] relate this result of Witten’s to their localization theorem in equivariant cohomology.
It is well-known that the group of isometries of a Riemannian manifold (with or without boundary) is compact, so that a Killing vector field generates an action of a torus. In this light, and because of Remark\[rmk:harmonic forms are invariant\] (and its extension to Witten’s setting), Witten’s analysis can be cast in the following slightly more general context.
Throughout, we let $G$ be a torus acting by isometries on $M$, with Lie algebra $\gg$, and denote by $\Omega_G=\Omega_G(M)$ the space of smooth $G$-invariant forms on $M$. Given any $X\in\gg$ we denote the corresponding vector field on $M$ by $X_M$, and following Witten we define $\d_{X_M} = \d +
\iota_{X_M}.$ Then $\d_{X_M}$ defines an operator $\d_{X_M}:\Omega_G^\pm \to \Omega_G^\mp$, with $\d_{X_M}^2=0$. For each $X\in\gg$ there are therefore two corresponding cohomology groups $H_{X_M}^\pm (M)= \ker\d_{X_M}^\pm/{\mathop{\mathrm{im}}\nolimits}\d_{X_M}^\mp$, which we call $X_M$-cohomology groups, and a corresponding operator we call the *Witten-Hodge-Laplacian* $$\Delta_{X_M}=(\d_{X_M}+\delta_{X_M})^2 : \Omega_G^\pm \to \Omega_G^\pm.$$ According to Witten there is an isomorphism $\mathcal{H}_{X_M}^\pm
\cong H_{X_M}^\pm(M)$, where $\mathcal{H}^{\pm}_{X_M}$ is the space of $X_M$-harmonic forms, that is those forms annihilated by $\Delta_{X_M}$. Of course, Witten’s presentation is no less general than this, and is obtained by putting $X_M=sK$; the only difference is we are thinking of $X$ as a variable element of $\gg$, while for Witten varying $s$ only gives a 1-dimensional subspace of $\gg$ (although one may change $K$ as well).
The immediate purpose of this paper is to extend Witten’s results to manifolds with boundary. In order to do this, in Section \[sec:W-H no boundary\] we outline the background to Witten’s results using classical Hodge theory arguments, which in Section \[sec:W-H with boundary\] we extend to deal with the case of manifolds with boundary. In section \[sec:equivariant cohomology\], we describe Atiyah and Bott’s localization and its conclusions in the case of manifolds with boundary, and its relation to $X_M$-cohomology. Finally in Section \[sec.style of DeTurck-Gluck\], we extend our results to adapt ideas of DeTurck and Gluck [@Gluck] and the *Poincaré duality angles*. Section \[sec:conclusions\] provides a few conclusions.
The original motivation for this paper was to adapt to the equivariant setting some recent work of Belishev and Sharafutdinov [@Belishev] where they address the classical question, *“To what extent are the topology and geometry of a manifold determined by the Dirichlet-to-Neumann (DN) map”?* which arises in the scope of inverse problems and reconstructing a manifold from boundary measurements. They show that the DN map on the boundary of a Riemannian manifold determines the Betti numbers of the manifold. This paper provides the background necessary for the “equivariant” analogue of the results of Belishev and Sharafudtinov.
#### Hodge theory for manifolds with boundary
In the remainder of this introduction we recall the standard extension of Hodge theory to manifolds with boundary, leading to the Hodge-Morrey-Friedrichs decompositions; details can be found in the book of Schwarz [@Schwarz]. The relative de Rham cohomology and the Dirichlet forms are defined at the beginning of the introduction. One also defines $\Omega^k_N(M) = \left\{\alpha\in\Omega^k(M)\mid i^*(\star\alpha)=0\right\}$ (Neumann boundary condition). Clearly, the Hodge star provides an isomorphism $\star:\Omega_D^k\stackrel{\sim}{\longrightarrow}\Omega_N^{n-k}.$ Furthermore, because $\d$ and $i^*$ commute, it follows that $\d$ preserves Dirichlet boundary conditions while $\delta$ preserves Neumann boundary conditions.
As alluded to before, because of boundary terms, the null space of $\Delta$ no longer coincides with the closed and co-closed forms. Elements of $\ker\Delta$ are called *harmonic forms*, while $\omega$ satisfying $\d\omega=\delta\omega=0$ are called *harmonic fields* (following Kodaira); it is clear that every harmonic field is a harmonic form, but the converse is false. In fact, the space $\mathcal{H}^k(M)$ of harmonic fields is infinite dimensional and so is much too big to represent the cohomology, and to recover the Hodge isomorphism one has to impose boundary conditions. One restricts $\mathcal{H}^k(M)$ into each of two finite dimensional subspaces, namely $\mathcal{H}_D^k(M)$ and $\mathcal{H}_N^k(M)$ with the obvious meanings (Dirichlet and Neumann harmonic $k$-fields, respectively). There are therefore two different candidates for harmonic representatives when the boundary is present.
The Hodge-Morrey decomposition [@Morrey] states that $$\Omega^k(M) = \mathcal{H}^k(M) \oplus \d\Omega_D^{k-1} \oplus \delta\Omega_N^{k+1}.$$ (We make a more precise functional analytic statement below.) This decomposition is again orthogonal with respect to the inner product given above. Friedrichs [@Friedrichs] subsequently showed that $$\mathcal{H}^k = \mathcal{H}^k_D\oplus \mathcal{H}^k_{\mathrm{co}};\qquad \mathcal{H}^k = \mathcal{H}^k_N\oplus \mathcal{H}^k_{\mathrm{ex}}$$ where $\mathcal{H}^k_{\mathrm{ex}}$ are the exact harmonic fields and $\mathcal{H}^k_{\mathrm{co}}$ the coexact ones (that is, $\mathcal{H}^k_{\mathrm{ex}} = \mathcal{H}^k\cap\d\Omega^{k-1}$ and $\mathcal{H}^k_{\mathrm{co}} = \mathcal{H}^k\cap\delta\Omega^{k+1}$). These give the orthogonal *Hodge-Morrey-Friedrichs* [@Schwarz] decompositions, $$\begin{aligned}
\Omega^k(M) &=& \d\Omega_D^{k-1} \oplus \delta\Omega_N^{k+1} \oplus \mathcal{H}^k_D\oplus \mathcal{H}^k_{\mathrm{co}}\\
&=& \d\Omega_D^{k-1} \oplus \delta\Omega_N^{k+1} \oplus
\mathcal{H}^k_N\oplus \mathcal{H}^k_{\mathrm{ex}}.\end{aligned}$$ The two decompositions are related by the Hodge star operator. The consequence for cohomology is that each class in $H^k(M)$ is represented by a unique harmonic field in $\mathcal{H}^k_N(M)$, and each relative class in $H^k(M,\partial M)$ is represented by a unique harmonic field in $\mathcal{H}^k_D(M)$. Again, the Hodge star operator acts as Poincaré duality (or rather Poincaré-Lefschetz duality) on the harmonic fields, sending Dirichlet fields to Neumann fields. And as in remark \[rmk:harmonic forms are invariant\], if a group acts by isometries on $(M,\partial M)$ in a manner that is trivial on the cohomology, then the harmonic fields are invariant.
In this paper, we suppose $G$ is a compact connected Abelian Lie group (a torus) acting by isometries on $M$, with Lie algebra $\mathfrak{g}$, and we let $X\in\mathfrak{g}$. If $M$ has a boundary then the $G$-action necessarily restricts to an action on the boundary and $X_M$ must therefore be tangent to the boundary. We denote by $\Omega_G=\Omega_G(M)$ the set of invariant forms on $M$: $\omega\in\Omega_G$ if $g^*\omega=\omega$ for all $g\in G$; in particular if $\omega$ is invariant then the Lie derivative $\mathcal{L}_{X_M}\omega=0$. Note that because the action preserves the metric and the orientation it follows that, for each $g\in G$, $\star(g^*\omega) =
g^*(\star\omega)$, so if $\omega\in\Omega_G$ then $\star\omega\in\Omega_G$.
Remark on typesetting: Since the letter H plays three roles in this paper, we use three different typefaces: a script $\mathcal{H}$ for harmonic fields, a sans-serif $\mathsf{H}$ for Sobolev spaces and a normal (italic) $H$ for cohomology. We hope that will prevent any confusion.
Witten-Hodge theory for manifolds without boundary {#sec:W-H no boundary}
==================================================
In this section we summarize the functional analysis behind Witten’s results [@Witten], details can be found in the first author’s thesis [@my; @thesis]. These are needed in the next section for manifolds with boundary. We continue to use the notation from the introduction, notably the manifold $M$ (which in this section has no boundary) and the torus $G$.
Fix an element $X\in\gg$. The associated vector field on $M$ is $X_M$, and using this one defines Witten’s inhomogeneous operator $\d_{X_M}: \Omega_G^\pm \to
\Omega_G^\mp,\; \d_{X_M}\omega=\d\omega+\iota_{X_M}\omega$, and the corresponding operator (cf. eq. (\[eq:delta\])) $$\delta_{X_M} = (-1)^{n(k+1)+1}\star\d_{X_M}\star = \delta + (-1)^{n(k+1)+1}\star\iota_{X_M}\star$$ (which is the operator adjoint to $\d_{X_M}$ by eq. (\[eq.2.16\]) below). The resulting *Witten-Hodge-Laplacian* is $\Delta_{X_M}:\Omega_G^\pm \to \Omega_G^\pm$ defined by $\Delta_{X_M}=(\d_{X_M}+\delta_{X_M})^2 = \d_{X_M}\delta_{X_M} +
\delta_{X_M}\d_{X_M}$. We write the space of $X_M$-harmonic fields $$\mathcal{H}_{X_M} = \ker\d_{X_M}\cap\ker\delta_{X_M}\,,$$ which for manifolds without boundary satisfies $\mathcal{H}_{X_M} =\ker\Delta_{X_M}$. The last equality follows for the same reason as for ordinary Hodge theory, namely the argument in (\[eq:ker=ker\]), with $\Delta$ replaced by $\Delta_{X_M}$ etc.
As is conventional, define $\int_M\omega=0$ if $\omega\in\Omega^k(M)$ with $k\neq
n$. So, for any form $\omega\in\Omega(M)$ one has $\int_M
\iota_{X_M}\omega=0$ as $\iota_{X_M}\omega$ has no term of degree $n$, and the following equation (\[eq.Stokes’ theorem\]) follows from the ordinary Stokes’ theorem. For future use, we allow $M$ to have a boundary. $$\label{eq.Stokes' theorem}
\int_M \d_{X_{M}}\omega= \int_{\partial M} i^{*}\omega.$$
For each space $\Omega$ of smooth differential forms on $M$, and each $s\in\RR$, we write $\mathsf{H^{s}}\Omega$ for the completion of $\Omega$ under an appropriate Sobolev norm. It is not hard to prove a Green’s formula in terms of $\d_{X_{M}}$ and $\delta_{X_{M}}$ which states that for $\alpha ,\beta \in
\mathsf{H^{1}}\Omega_{G}$, $$\label{eq.2.16}
\langle \d_{X_{M}}\alpha,\beta\rangle=\langle
\alpha,\delta_{X_{M}}\beta\rangle+\int_{\partial {M}} i^{*} (\alpha
\wedge \star\beta)\,,$$
Returning now to the case of a manifold without boundary, we obtain the following.
\[thm:self-adjoint elliptic no boundary\]
1. The Witten-Hodge-Laplacian $\Delta_{X_M}$ is a self-adjoint elliptic operator.\
2. The following is an orthogonal decomposition $$\Omega_{G}^\pm = \mathcal{H}_{X_M}^\pm \oplus \d_{X_M}\Omega_G^\mp \oplus \delta_{X_M}\Omega_G^\mp.$$ The orthogonality is with respect to the $L^2$ inner product.
Part (2) is the analogue of the Hodge decomposition theorem, and is a standard consequence of the fact that $\Delta_{X_M}$ is self-adjoint. The first two summands give the $X_M$-closed forms.
Every elliptic operator on a compact manifold is a Fredholm operator, so has finite dimensional kernel and cokernel, and closed range. Therefore the set of $X_M$-harmonic (even/odd) forms $\mathcal{H}_{X_M}^\pm=(\ker \Delta_{X_M})^\pm$ is finite dimensional. One concludes with the analogue of Hodge’s theorem
\[unique $X_M$-harmonic representative\] $H^{\pm}_{X_M}(M)\cong \mathcal{H}_{X_M}^\pm$, and in particular every $X_M$-cohomology class has a unique $X_M$-harmonic representative.
The Hodge star operator gives a form of Poincaré duality in terms of $X_M$-cohomology: $$H_{X_M}^{n-\pm}(M)\cong H_{X_M}^\pm(M).$$ Since Hodge star takes harmonic forms to harmonic forms, this Poincaré duality is realized at the level of harmonic forms. The full details are given in [@my; @thesis]. Here and elsewhere we write $n-\pm$ for the parity (modulo 2) resulting from subtracting an even/odd number from $n$.
Let $N(X_M)$ be the set of zeros of $X_M$, and $j:N(X_M)\hookrightarrow M$ the inclusion. As observed by Witten, on $N(X_M)$ one has $X_M=0$, so that $j^*\d_{X_M}\omega = \d(j^*\omega)$, and in particular if $\omega$ is $X_M$-closed then its pullback to $N(X_M)$ is closed in the usual (de Rham) sense. And exact forms pull back to exact forms. Consequently, pullback defines a natural map $H_{X_M}^\pm(M)\to H^\pm(N(X_M))$, where $H^\pm(N(X_M))$ is the direct sum of the even/odd de Rham cohomology groups of $N(X_M)$.
\[thm:fixed point set\] The pullback to $N(X_M)$ induces an isomorphism between the $X_M$-cohomology groups $H_{X_M}^\pm(M)$ and the cohomology groups $H^\pm(N(X_M))$.
Witten gives a fairly explicit proof of this theorem by extending closed forms on $N(X_M)$ to $X_M$-closed forms on $M$. Atiyah and Bott [@AB] give a proof using their localization theorem in equivariant cohomology which we discuss, and adapt to the case of manifolds with boundary, in Section \[sec:equivariant cohomology\].
Extending remark \[rmk:harmonic forms are invariant\], suppose $X$ generates the torus $G(X)$, and $G$ is a larger torus containing $G(X)$ and acting on $M$ by isometries. Then the action of $G$ preserves $X_M$. It follows that $G$ acts trivially on the de Rham cohomology of $N(X_M)$, and hence on the $X_M$-cohomology of $M$, and consequently on the space of $X_M$-harmonic forms. In other words, $\mathcal{H}_{X_M}^\pm\subset\Omega_{G}^\pm$. There is therefore no loss in considering just forms invariant under the action of the larger torus in that the $X_M$-cohomology, or the space of $X_M$-harmonic forms, is independent of the choice of torus, provided it contains $G(X)$.
Witten-Hodge theory for manifolds with boundary {#sec:W-H with boundary}
===============================================
In this section we extend the results and methods of Hodge theory for manifolds with boundary to study the $X_M$-cohomology and the space of $X_M$-harmonic forms and fields for manifolds with boundary. As for ordinary (singular) cohomology, there are both absolute and relative $X_M$-cohomology groups. From now on our manifold will be with boundary and as before $i:\partial M \hookrightarrow M$ denotes the inclusion of the boundary, and $G$ is a torus acting by isometries on $M$.
The difficulties if the boundary is present
-------------------------------------------
Firstly, $\d_{X_{M}}$ and $\delta_{X_{M}}$ are no longer adjoint because the boundary terms arise when we integrate by parts, and then $\Delta_{X_{M}}$ will not be self-adjoint. In addition, the space of all harmonic fields is infinite dimensional and there is no reason to expect the $X_{M}$-harmonic fields $\mathcal{H}_{X_{M}}(M)$ to be any different. To overcome these problems, at the beginning we follow the method which is used to solve this problem in the classical case, i.e. with $\d$ and $\delta$, by imposing certain boundary conditions on our invariant forms $\Omega_{G}(M)$, as described in [@Schwarz]. Hence we make the following definitions.
\(1) We define the following two sets of smooth invariant forms on the manifold $M$ with boundary and with action of the torus $G$ $$\begin{aligned}
\Omega_{G,D} &=& \Omega_{G}\cap \Omega_{D} \ = \{\omega\in\Omega_G\mid i^*\omega=0\} \label{eq.2.12}\\
\Omega_{G,N} &=& \Omega_{G}\cap \Omega_{N}\ = \{\omega\in\Omega_G\mid i^*(\star\omega)=0\}\label{eq.2.13}\end{aligned}$$ and the spaces $\mathsf{H}^{s}\Omega_{G,D}$ and $\mathsf{H}^{s}\Omega_{G,N}$ are the corresponding closures with respect to suitable Sobolev norms, for $s>\frac12$. This can be refined to take into account the parity of the forms, so defining $\Omega_{G,D}^\pm$ etc. Since $\omega\in\Omega^k$ implies $\star\omega \in \Omega^{n-k}$ we write that for $\omega\in\Omega_{G}^\pm$ we have $\star\omega \in\Omega_G^{n-\pm}$.
\(2) We define two subspaces of $X_M$-harmonic fields, $$\begin{aligned}
\mathcal{H}_{X_{M},D}(M) &=& \{\omega\in \mathsf{H^{1}}\Omega_{G,D}\mid \d_{X_{M}}\omega=0, \,\delta_{X_{M}}\omega=0 \}\label{eq.2.14}\\
\mathcal{H}_{X_{M},N}(M) &=& \{\omega\in \mathsf{H^{1}}\Omega_{G,N}\mid \d_{X_{M}}\omega=0,\, \delta_{X_{M}}\omega=0 \}\label{eq.2.15}\end{aligned}$$ which we call Dirichlet and Neumann $X_{M}$-harmonic fields, respectively. We will show below that these forms are smooth. Clearly, the Hodge star operator $\star$ defines an isomorphism $\mathcal{H}_{X_{M},D}(M) \cong \mathcal{H}_{X_{M},N}(M)$. Again, these can be refined to take the parity into account, defining $ \mathcal{H}_{X_{M},D}^\pm(M)$ etc.
As for ordinary Hodge theory, on a manifold with boundary one has to distinguish between $X_{M}$-harmonic *forms* (i.e. $\ker\Delta_{X_{M}}$) and $X_{M}$-harmonic *fields* (i.e. $\mathcal{H}_{X_{M}}(M)$) because they are not equal: one has $\mathcal{H}_{X_{M}}(M)\subseteq \ker\Delta_{X_{M}}$ but not conversely. The following proposition shows the conditions on $\omega$ to be fulfilled in order to ensure $\omega\in\ker\Delta_{X_{M}} \Longrightarrow \omega\in \mathcal{H}_{X_{M}}(M)$ when $\partial M\neq \emptyset$.
\[pro.2.6\] If $\omega \in \Omega_{G}(M)$ is an $X_{M}$-harmonic form (i.e. $\Delta_{X_{M}}\omega=0$) and in addition any one of the following four pairs of boundary conditions is satisfied then $\omega\in\mathcal{H}_{X_{M}}(M)$. $$\begin{array}{llll}
(1)& i^{*} \omega = 0 ,\; i^{*} (\star \omega)=0; &
(2)& i^{*} \omega = 0 ,\; i^{*}(\delta_{X_{M}}\omega)=0; \\[4pt]
(3)& i^{*} (\star \omega) = 0 ,\; i^{*}(\star \d_{X_{M}}\omega)=0; \quad &
(4)& i^{*} (\delta_{X_{M}}\omega) = 0 ,\; i^{*}(\star \d_{X_{M}}\omega)=0.
\end{array}$$
Because $\Delta_{X_{M}}\omega=0$, one has $\langle \Delta_{X_{M}}
\omega,\omega\rangle=0$. Now applying Green’s formula (\[eq.2.16\]) to this and using any of these conditions (1)–(4) ensures $\omega$ is an $X_{M}$-harmonic field.
\[r.6’\] An averaging argument shows that $\mathsf{H^{1}}\Omega_{G},_{D}$ and $\mathsf{H^{1}}\Omega_{G},_{N}$ are dense in $L^{2}\Omega_{G}$, because the corresponding statements hold for the spaces of all (not only invariant) forms.
Elliptic boundary value problem {#subsec. Elliptic BVP}
-------------------------------
The essential ingredients that Schwarz [@Schwarz] needs to prove the classical *Hodge-Morry-Friedrichs* decomposition are Gaffney’s inequality and his Theorem 2.1.5. However, these results do not appear to extend to the context of $\d_{X_M}$ and $\delta_{X_M}$. Therefore, we use a different approach to overcome this problem, based on the ellipticity of a certain boundary value problem (<span style="font-variant:small-caps;">bvp</span>), namely (\[eq.2.18\]) below. This theorem represents the keystone to extending the Hodge-Morrey and Friedrichs decomposition theorems to the present setting and thence to extending Witten’s results to manifolds with boundary.
Consider the <span style="font-variant:small-caps;">bvp</span> $$\label{eq.2.18}
\left\{\begin{array}{rcl}
\Delta_{X_{M}}\omega &=& \eta \quad \textrm{on} \quad M \\
i^{*}\omega &=& 0 \quad \textrm{on} \quad \partial M\\
i^{*}(\delta_{X_{M}}\omega)&=& 0 \quad \textrm{on} \quad \partial M.
\end{array}\right.$$
\[thm.2.5\]
1. The <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) is elliptic in the sense of Lopatinskiǐ-Šapiro, where $\Delta_{X_{M}}:\Omega_{G}(M)\longrightarrow \Omega_{G}(M)$.
2. The <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) is Fredholm of index 0.
3. All $\omega\in\mathcal{H}_{X_M,D}\cup\mathcal{H}_{X_M,N}$ are smooth.
\(1) We can see that $\Delta$ and $\Delta_{X_M}$ have the same principal symbol as $\Delta_{X_M}-\Delta$ is a first order differential operator; indeed, $$\Delta_{X_M} = \Delta+(-1)^{n(k+1)+1}(\d\star
\iota_{X_{M}}\star+\star \iota_{X_{M}}\star \d + \star
\iota_{X_{M}}\star \iota_{X_{M}}+ \iota_{X_{M}}\star
\iota_{X_{M}}\star) +\iota_{X_{M}} \delta+ \delta \iota_{X_{M}}\,.$$ Similarly, expanding the second boundary condition gives $$\delta_{X_M} = \delta + (-1)^{n(k+1)+1}\star\iota_{X_M}\star$$ so $\delta_{X_M}$ and $\delta$ have the same first-order part. Hence our <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) has the same principal symbol as the <span style="font-variant:small-caps;">bvp</span> $$\label{eq.2.20}
\left\{\begin{array}{rcl}
\Delta\epsilon &=& \xi \quad \textrm{on} \quad M \\
i^{*}\epsilon &=& 0 \quad \textrm{on} \quad \partial M\\
i^{*}(\delta\epsilon)&=& 0 \quad \textrm{on} \quad \partial M
\end{array}\right.$$ for $\epsilon,\,\xi \in \Omega(M)$, because the principal symbol does not change when terms of lower order are added to the operator. However the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.20\]) is elliptic in the sense of Lopatinskiǐ-Šapiro conditions [@Lars; @Schwarz], and thus so is (\[eq.2.18\]).
\(2) From part (1), since the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) is elliptic, it follows that the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) is a Fredholm operator and the regularity theorem holds, see for example Theorem 1.6.2 in [@Schwarz] or Theorem 20.1.2 in [@Lars]. In addition, we observe that the only differences between <span style="font-variant:small-caps;">bvp</span> (\[eq.2.20\]) and our <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) are all lower order operators and it is proved in [@Schwarz] that the index of <span style="font-variant:small-caps;">bvp</span> (\[eq.2.20\]) is zero but Theorem 20.1.8 in [@Lars] asserts generally that if the difference between two <span style="font-variant:small-caps;">bvp</span>s are just lower order operators then they must have the same index. Hence, the index of the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) must be zero.
\(3) Let $\omega\in\mathcal{H}_{X_M,D}\cup\mathcal{H}_{X_M,N}$. If $\omega
\in \mathcal{H}_{X_M,D}$ then it satisfies the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) with $\eta=0$, so by the regularity properties of elliptic <span style="font-variant:small-caps;">bvp</span>s, the smoothness of $\omega$ follows. If on the other hand $\omega \in \mathcal{H}_{X_M,N}$ then $\star\omega\in \mathcal{H}_{X_{M},D}$ which is therefore smooth and consequently $\omega=\pm\star(\star\omega)$ is smooth as well.
We consider the resulting operator obtained by restricting $\Delta_{X_M}$ to the subspace of smooth invariant forms satisfying the boundary conditions $$\label{eq.2.21}
\overline{\Omega}_{G}(M)=\{\omega \in \Omega_{G}(M)
\mid i^{*}\omega = 0, \,i^{*}(\delta_{X_{M}}\omega)= 0\}$$
Since the trace map $i^*$ is well-defined on $\mathsf{H}^s\Omega_G$ for $s>1/2$ it follows that it makes sense to consider $\mathsf{H^{2}}\overline{\Omega}_{G}(M)$, which is a closed subspace of $\mathsf{H^{2}}\Omega_{G}(M)$ and hence a Hilbert space. For simplicity, we rewrite our <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) as follows: consider the restriction/extension of $\Delta_{X_M}$ to this space: $$A=\Delta_{X_{M}}\restrict{\mathsf{H^{2}}\overline{\Omega}_{G}(M)}:
\mathsf{H^{2}}\overline{\Omega}_{G}(M)\longrightarrow
L^{2}\Omega_{G}(M).$$ and consider the <span style="font-variant:small-caps;">bvp</span>, $$\label{eq.2.22}
A\omega=\eta$$ for $\omega\in\mathsf{H^{2}}\overline{\Omega}_{G}(M)$ and $\eta\in
L^{2}\Omega_{G}(M)$ instead of <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) which are in fact compatible. In addition, from Theoremsa \[thm.2.5\] we deduce that $A$ is an elliptic and Fredholm operator and $$\label{index}
\mathrm{index}(A)=\dim(\ker A)-\dim(\ker A^*)=0$$ where $A^*$ is the adjoint operator of $A$.
From Green’s formula (eq. (\[eq.2.16\])) we deduce the following property.
\[L.2.33\] $A$ is $L^2$-self-adjoint on $\mathsf{H^{2}}\overline{\Omega}_{G}(M)$, meaning that for all $\alpha,\beta\in\mathsf{H^{2}}\overline{\Omega}_{G}(M)$ we have $$\left<A\alpha,\,\beta\right> = \left<\alpha,\,A\beta\right>,$$ where $\left<-,-\right>$ is the $L^2$-pairing.
\[tm.2.6’\] The space $ \mathcal{H}_{X_{M},D}(M)$ is finite dimensional and $$\label{eq.2.24}
L^{2}\Omega_{G}(M)= \mathcal{H}_{X_{M},D}(M) \oplus \mathcal{H}_{X_{M},D}(M)^{\perp}.$$
We begin by showing that $\ker A = \mathcal{H}_{X_{M},D}(M)$. It is clear that $\mathcal{H}_{X_{M},D}(M)\subseteq \ker A$, so we need only prove that $\ker A \subseteq \mathcal{H}_{X_{M},D}(M)$.
Let $\omega \in \ker A$. Then $\omega$ satisfies the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]). Therefore, by condition (2) of Proposition \[pro.2.6\], it follows that $\omega \in \mathcal{H}_{X_{M},D}(M)$, as required.
Now, $\ker A = \mathcal{H}_{X_{M},D}(M)$ but $\dim\ker A$ is finite, so that $\dim\mathcal{H}_{X_{M},D}(M)<\infty$. This implies that $\mathcal{H}_{X_{M},D}(M)$ is a closed subspace of the Hilbert space $L^{2}\Omega_{G}(M)$, hence eq. (\[eq.2.24\]) holds.
\[tm.2.6\] $$\label{eq.2.23}
\mathrm{Range}(A) = \mathcal{H}_{X_{M},D}(M)^{\perp}$$ where $\perp$ denotes the orthogonal complement in $L^2\Omega_{G}(M)$.
Firstly, we should observe that eq. (\[index\]) asserts that $\ker A\cong \ker A^*$ but Theorem \[tm.2.6’\] shows that $\ker A = \mathcal{H}_{X_{M},D}(M)$, thus $$\label{eq.coker}
\ker A^*\cong \mathcal{H}_{X_{M},D}(M)$$
Since $\mathrm{Range}(A)$ is closed in $L^2\Omega_{G}(M)$ because $A$ is Fredholm operator, it follows from the closed range theorem in Hilbert spaces that $$\label{eq.closed range }
\mathrm{Range}(A) = (\ker A^*)^{\perp} \quad \equiv \quad
\mathrm{Range}(A)^{\perp} = \ker A^*$$ Hence, we just need to prove that $\ker A^*=
\mathcal{H}_{X_{M},D}(M)$, and to show that we need first to prove $$\label{eq.subset1}
\mathrm{Range}(A)\subseteq \mathcal{H}_{X_{M},D}(M)^{\perp}.$$ So, if $\alpha\in\mathsf{H^{2}}\overline{\Omega}_{G}(M)$ and $\beta
\in\mathcal{H}_{X_{M},D}(M) $ then applying Lemma \[L.2.33\] gives $$\left<A\alpha,\,\beta\right> =0$$ hence, eq. (\[eq.subset1\]) holds. Moreover, equations (\[eq.closed range \]) and (\[eq.subset1\]) and the closedness of $\mathcal{H}_{X_{M},D}(M)$ imply $$\label{eq.subset2}
\mathcal{H}_{X_{M},D}(M) \subseteq \ker A^*$$ but eq. (\[eq.coker\]) and eq. (\[eq.subset2\]) force $\ker A^*=
\mathcal{H}_{X_{M},D}(M)$. Hence, $\mathrm{Range}(A) =
\mathcal{H}_{X_{M},D}(M)^{\perp}$.
Following [@Schwarz], we denote the $L^{2}$-orthogonal complement of $\mathcal{H}_{X_{M},D}(M)$ in the space $\mathsf{H^{2}}\Omega_{G,D}$ by $$\label{eq.2.27}
\mathcal{H}_{X_{M},D}(M)^{{\bigcirc\kern-6.5pt\perp}} = \mathsf{H^{2}}\Omega_{G,D}\cap
\mathcal{H}_{X_{M},D}(M)^{\perp}$$ (although in [@Schwarz] it denotes $\mathsf{H}^1$-forms rather than $\mathsf{H}^2$).
\[pro.2.7\] For each $\eta \in \mathcal{H}_{X_{M},D}(M)^{\perp}$ there is a unique differential form $\omega \in \mathcal{H}_{X_{M},D}(M)^{{\bigcirc\kern-6.5pt\perp}}$ satisfying the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]).
Let $\eta \in \mathcal{H}_{X_{M},D}(M)^{\perp}$. Because of Theorem (\[tm.2.6\]) there is a differential form $\gamma \in
\mathsf{H^{2}}\overline{\Omega}_{G}(M)$ such that $\gamma$ satisfies the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]). Since $ \gamma \in
\mathsf{H^{2}}\overline{\Omega}_{G}(M)\subseteq L^{2}\Omega_{G}(M)$ then there are unique differential forms $\alpha \in
\mathcal{H}_{X_{M},D}(M)$ and $\omega \in
\mathcal{H}_{X_{M},D}(M)^{\perp} $ such that $\gamma=\alpha+\omega$ because of eq. (\[eq.2.24\]).
Since $\gamma$ satisfies the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) it follows that $\omega$ satisfies the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) as well because $\alpha \in \mathcal{H}_{X_{M},D}(M) =
\ker(\Delta_{X_{M}}\restrict{\mathsf{H^{2}}\overline{\Omega}_{G}(M)})$. Since $\omega=\gamma-\alpha$, it follows that $\omega \in
\mathsf{H^{2}}\Omega_{G,D}$ , hence $\omega \in
\mathcal{H}_{X_{M},D}(M)^{{\bigcirc\kern-6.5pt\perp}}$ and it is unique.
\[r.8\]
1. $\omega$ satisfying the <span style="font-variant:small-caps;">bvp</span> (\[eq.2.18\]) in Proposition \[pro.2.7\] can be recast to the condition $$\label{eq.2.28}
\langle \d_{X_{M}} \omega,\, \d_{X_{M}}\xi\rangle + \langle\delta_{X_{M}}\omega,\, \delta_{X_{M}}\xi\rangle
=\langle \eta,\xi\rangle,\quad
\forall \xi \in \mathsf{H^{1}}\Omega_{G,D}$$
2. All the results above can be recovered but in terms of $\mathcal{H}_{X_{M},N}(M)$ because the Hodge star operator defines an isomorphism $L^2\Omega_G\cong L^2\Omega_G$ which restricts to $\mathcal{H}_{X_{M},D}(M)\cong \mathcal{H}_{X_{M},N}(M)$.
Decomposition theorems
----------------------
The results above provide the basic ingredients needed to extend the Hodge-Morrey and Freidrichs decompositions arising for Hodge theory on manifolds with boundary, to the present setting with $\d_{X_M}$ and $\delta_{X_M}$. Depending on these results, the proofs in this subsection rely heavily on the analogues of the corresponding statements for the usual Laplacian $\Delta$ on a manifold with boundary, as described in the book of Schwarz [@Schwarz]. Therefore, we omit the proofs here while full details are given in the first author’s thesis [@my; @thesis].
\[d.2.4\] Define the following two sets of invariant exact and coexact forms on $M$, $$\begin{aligned}
\mathcal{E}_{X_M}(M)&=&\{\d_{X_{M}} \alpha \mid \alpha \in
\mathsf{H^{1}}\Omega_{G,D}\}\subseteq L^{2}\Omega_{G}(M),\\
\mathcal{C}_{X_M}(M) &=& \{\delta_{X_{M}} \beta \mid \beta \in
\mathsf{H^{1}}\Omega_{G,N}\}\subseteq L^{2}\Omega_{G}(M).\end{aligned}$$ Clearly, $\mathcal{E}_{X_M}(M)\perp \mathcal{C}_{X_M}(M)$ because of eq. (\[eq.2.16\]). We denote by $L^{2}\mathcal{H}_{X_{M}}(M) =
\overline{\mathcal{H}_{X_{M}}(M)}$ the $L^{2}$-closure of the space $\mathcal{H}_{X_{M}}(M)$.
\[pro.2.8\]
1. Each $\omega \in L^{2}\Omega_{G}(M)$ can be split uniquely into $$\omega=\d_{X_{M}} \alpha_{\omega}+\delta_{X_{M}} \beta_\omega
+\kappa_{\omega}$$ where $\d_{X_{M}} \alpha_{\omega} \in
\mathcal{E}_{X_M}(M)$ , $\delta_{X_{M}} \beta_{\omega} \in
\mathcal{C}_{X_M}(M)$ and $\kappa_{\omega} \in
(\mathcal{E}_{X_M}(M)\oplus \mathcal{C}_{X_M}(M))^{\perp}$.
2. The spaces $\mathcal{E}_{X_M}(M)$ and $\mathcal{C}_{X_M}(M)$ are closed subspaces of $L^{2}\Omega_{G}(M)$.
3. Consequently there is the following orthogonal decomposition $$L^{2}\Omega_{G}(M)=\mathcal{E}_{X_M}(M)\oplus
\mathcal{C}_{X_M}(M)\oplus(\mathcal{E}_{X_M}(M)\oplus
\mathcal{C}_{X_M}(M))^{\perp}$$
Now we can present the main theorems for this section; all orthogonality is with respect to the $L^2$ inner product.
\[thm:Hodge-Morrey\] The following is an orthogonal direct sum decomposition: $$L^{2}\Omega_{G}(M)=\mathcal{E}_{X_M}(M)\oplus
\mathcal{C}_{X_M}(M)\oplus L^{2}\mathcal{H}_{X_{M}}(M)$$
\[thm:X\_M-Friedrichs\] The space $\mathcal{H}_{X_{M}}(M)\subseteq \mathsf{H^{1}}\Omega_{G}(M)$ of $X_{M}$- harmonic fields can respectively be decomposed as orthogonal direct sums into $$\begin{aligned}
\mathcal{H}_{X_{M}}(M)&=& \mathcal{H}_{X_{M},D}(M)\oplus \mathcal{H}_{X_{M},\mathrm{co}}(M)\\ \mathcal{H}_{X_{M}}(M)&=& \mathcal{H}_{X_{M},N}(M)\oplus\mathcal{H}_{X_{M},\mathrm{ex}}(M), $$ where the right hand terms are the $X_M$-coexact and exact harmonic forms respectively: $$\begin{aligned}
\mathcal{H}_{X_{M},\mathrm{co}}(M) &=& \{ \eta \in \mathcal{H}_{X_{M}}(M)\mid \eta=\delta_{X_{M}}\alpha\}\\ \mathcal{H}_{X_{M},\mathrm{ex}}(M)&=& \{ \xi \in \mathcal{H}_{X_{M}}(M)\mid \xi=\d_{X_{M}}\sigma\}$$ For $L^{2}\mathcal{H}_{X_{M}}(M)$ these decompositions are valid accordingly.
Combining Theorems \[thm:Hodge-Morrey\] and \[thm:X\_M-Friedrichs\] gives the following.
\[co.2.6\] The space $ L^{2}\Omega_{G}(M)$ can be decomposed into $L^{2}$-orthogonal direct sums as follows: $$\begin{aligned}
L^{2}\Omega_{G}(M) &=&\mathcal{E}_{X_M}(M)\oplus \mathcal{C}_{X_M}(M)\oplus
\mathcal{H}_{X_{M},D}(M)\oplus L^{2}\mathcal{H}_{X_{M},\mathrm{co}}(M) \\
L^{2}\Omega_{G}(M) &=&\mathcal{E}_{X_M}(M)\oplus \mathcal{C}_{X_M}(M)\oplus
\mathcal{H}_{X_{M},N}(M)\oplus L^{2}\mathcal{H}_{X_{M},\mathrm{ex}}(M)\end{aligned}$$
All the results above can be refined in terms of $\pm$-spaces, for instance, $$\mathcal{H}^\pm_{X_{M},D}(M)\cong
\mathcal{H}^{n-\pm}_{X_{M},N}(M), \quad L^{2}\Omega^\pm_{G}(M)
=\mathcal{E}^\pm_{X_M}(M)\oplus \mathcal{C}^\pm_{X_M}(M)\oplus
\mathcal{H}^\pm_{X_{M},D}(M)\oplus
L^{2}\mathcal{H}^\pm_{X_{M},\mathrm{co}}(M)$$ …etc.
Relative and absolute $X_M$-cohomology
---------------------------------------
Using $\d_{X_M}$ and $\delta_{X_M}$ we can form a number of $\ZZ_2$-graded complexes. A $\ZZ_2$-graded complex is a pair of Abelian groups $C^\pm$ with homomorphisms between them:
(-2,-0.5)(2,0.5) (-1.3,0.05)[$C^+$]{} (1.3,0.05)[$C^-$]{} (0,0.4)[$\d_+$]{} (0,-0.4)[$\d_-$]{} (-0.8,0.1)(0.8,0.1) (0.8,-0.1)(-0.8,-0.1)
satisfying $\d_+\circ\d_- = 0 = \d_-\circ\d_+$. The two (co)homology groups of such a complex are defined in the obvious way: $H^\pm = \ker\d_\pm/{\mathop{\mathrm{im}}\nolimits}\d_\mp$. The complexes we have in mind are, $$\begin{aligned}
(\Omega_G^\pm,\d_{X_M})&\quad&(\Omega_G^\pm,\delta_{X_M})\\
(\Omega_{G,D}^\pm,\d_{X_M}) && (\Omega_{G,N}^\pm,\delta_{X_M}).\end{aligned}$$ The two on the lower line are subcomplexes of the corresponding upper ones, because $i^*$ commutes with $\d_{X_M}$. By analogy with the de Rham groups, we denote $$H^\pm_{X_M}(M) := H^\pm(\Omega_G,\,\d_{X_M}) \quad \mbox{and} \quad
H^\pm_{X_M}(M,\,\partial M) := H^\pm(\Omega_{G,D},\,\d_{X_M}).$$
The decomposition theorems above lead to the following result.
\[thm:X\_M-Hodge\] Let $X\in\gg$. There are the following isomorphisms of vector spaces:
1. $H^{\pm}_{X_M}(M,\,\partial M) \cong \mathcal{H}^\pm_{X_{M},D}(M) \cong H^\pm(\Omega_G^\pm,\delta_{X_{M}})$;
2. $H^{\pm}_{X_M}(M)\cong \mathcal{H}^\pm_{X_{M},N}(M) \cong H^\pm(\Omega_{G,N}^\pm,\delta_{X_{M}})$;
3. ($X_{M}$-Poincaré-Lefschetz duality): The Hodge star operator $\star$ on $\Omega_{G}(M)$ induces an isomorphism $$H^\pm_{X_{M}}(M)\cong H^{n-\pm}_{X_{M}}(M,\,\partial M).$$
The proofs use the decomposition theorems above. For the first isomorphism in (a), Theorem \[thm:Hodge-Morrey\] (the $X_M$-Hodge-Morrey decomposition theorem) implies a unique splitting of any $\gamma \in \Omega^\pm_{G,D}$ into, $$\gamma=\d_{X_{M}} \alpha_{\gamma}+\delta_{X_{M}} \beta_{\gamma} + \kappa_{\gamma}$$ where $\d_{X_{M}} \alpha_{\gamma} \in \mathcal{E}^\pm_{X_M}(M)$, $\delta_{X_{M}} \beta_{\gamma} \in \mathcal{C}^\pm_{X_M}(M)$ and $\kappa_{\gamma} \in L^{2}\mathcal{H}^\pm_{X_{M}}(M)$. If $\d_{X_{M}}\gamma=0$ then $\delta_{X_{M}} \beta_{\gamma}=0$, but $i^{*}\gamma =0$ implies $i^{*}(\kappa_{\gamma})=0$ so that $\kappa_{\gamma} \in \mathcal{H}^\pm_{X_{M},D}(M)$. Thus, $$\gamma \in \ker \d_{X_{M}}\restrict{\Omega_{G,D}} \Longleftrightarrow
\gamma=\d_{X_{M}} \alpha_{\gamma}+\kappa_{\gamma}.$$ This establishes the isomorphism $H^\pm_{X_{M}}(M,\,\partial M) \cong \mathcal{H}^\pm_{X_{M},D}(M)$.
For the second isomorphism in (a), the second $X_M$-Hodge-Morrey-Friedrichs decomposition of Corollary \[co.2.6\] implies as well a unique splitting of any $\gamma \in \Omega^\pm_{G}(M)$ into, $$\gamma=\d_{X_{M}}\xi_{\gamma} + \delta_{X_{M}}\eta_{\gamma} +\delta_{X_{M}}\zeta_{\gamma} + \lambda_{\gamma}$$ where $\d_{X_{M}}\xi_{\gamma} \in \mathcal{E}^\pm_{X_M}(M)\,$, $\;\delta_{X_{M}} \eta_{\gamma} \in \mathcal{C}^\pm_{X_M}(M)\,$, $\;\delta_{X_{M}} \zeta_{\gamma} \in
L^{2}\mathcal{H}^\pm_{X_{M},\mathrm{co}}(M)$ and $\lambda_{\gamma}
\in \mathcal{H}^\pm_{X_{M},D}(M)$.
If $\delta_{X_{M}}\gamma=0$, then $\d_{X_{M}}\xi_{\gamma}=0$, and hence $$\gamma \in \ker \delta_{X_{M}}\Longleftrightarrow \gamma=\delta_{X_{M}} (\eta_\gamma+\zeta_\gamma) + \lambda_{\gamma}.$$ This establishes the isomorphism $\mathcal{H}^\pm_{X_{M},D}(M) \cong H^\pm_{X_{M}}(\Omega_G^\pm,\delta_{X_{M}})$.
Part (b) is proved similarly, and part (c) follows from (a) and (b) and the fact that the Hodge star operator defines an isomorphism $\mathcal{H}^\pm_{X_M,D}(M)\cong \mathcal{H}^{n-\pm}_{X_M,N}(M)$.
The theorem of Hodge is often quoted as saying that every (de Rham) cohomology class on a compact Riemannian manifold without boundary contains a unique harmonic form. The corresponding statement for $X_M$-cohomology on a manifold with boundary is,
\[coro.Hodge is often quoted\] Each absolute $X_M$-cohomology class contains a unique Neumann $X_M$-harmonic field, and each relative $X_M$-cohomology class contains a unique Dirichlet $X_M$-harmonic field.
Relation with equivariant cohomology {#sec:equivariant cohomology}
====================================
When the manifold in question has no boundary, Atiyah and Bott [@AB] discuss the relationship between equivariant cohomology and $X_M$-cohomology by using their localization theorem. In this section we will relate our relative and absolute $X_M$-cohomology with the relative and absolute equivariant cohomology $H_{G}^\pm(M,\partial M)$ and $H_{G}^\pm(M)$; the arguments are no different to the ones in [@AB]. First we recall briefly the basic definitions of equivariant cohomology, and the relevant localization theorem, and then state the conclusions for the relative and absolute $X_M$-cohomology.
If a torus $G$ acts on a manifold $M$ (with or without boundary), the Cartan model for the equivariant cohomology is defined as follows. Let $\{X_1,\dots,X_\ell\}$ be a basis of $\gg$ and $\{u_1,\dots,u_\ell\}$ the corresponding coordinates. The *Cartan complex* consists of polynomial[^1] maps from $\gg$ to the space of invariant differential forms, so is equal to $\Omega^*_G(M)\otimes R$ where $R=\RR[u_1,\dots,u_\ell]$, with differential $$\d_{\mathrm{eq}}(\omega) = \d\omega + \sum_{j=1}^\ell u_j\,\iota_{X_j}\omega.$$ The equivariant cohomology $H_G^*(M)$ is the cohomology of this complex. The relative equivariant cohomology $H_G^*(M,\partial M)$ (if $M$ has non-empty boundary) is formed by taking the subcomplex with forms that vanish on the boundary $i^*\omega=0$, with the same differential.
The cohomology groups are graded by giving the $u_i$ weight 2 and a $k$-form weight $k$, so the differential $\d_{\mathrm{eq}}$ is of degree 1. Furthermore, as the cochain groups are $R$-modules, and $\d_{\mathrm{eq}}$ is a homomorphism of $R$-modules, it follows that the equivariant cohomology is an $R$-module. The localization theorem of Atiyah and Bott [@AB] gives information on the module structure (there it is only stated for absolute cohomology, but it is equally true in the relative setting, with the same proof; see also Appendix C of [@Guillemin]).
First we define the following subset of $\gg$, $$Z := \bigcup_{\widehat{K}\subsetneq G}\mathfrak{k}$$ where the union is over proper isotropy subgroups $\widehat{K}$ (and $\mathfrak{k}$ its Lie algebra) of the action on $M$. If $M$ is compact, then $Z$ is a finite union of proper subspaces of $\gg$. Let $F = \mathrm{Fix}(G,M)=\{x \in M \mid G\cdot x=x\}$ be the set of fixed points in $M$. It follows from the local structure of group actions that $F$ is a submanifold of $M$, with boundary $\partial F
= F\cap \partial M$.
\[localization them.\] The inclusion $j:F\hookrightarrow M$ induces homomorphisms of $R$-modules $$H_G^*(M) \stackrel{j^*}{\longrightarrow} H_G^*(F)$$ $$H_G^*(M,\partial M) \stackrel{j^*}{\longrightarrow} H_G^*(F,\partial F)$$ whose kernel and cokernel have support in $Z$.
In particular, this means that if $f\in I(Z)$ (the ideal in $R$ of polynomials vanishing on $Z$) then the localizations[^2] $H_G^*(M)_f$ and $H_G^*(F)_f$ are isomorphic $R_f$-modules. Notice that the act of localization destroys the integer grading of the cohomology, but since the $u_i$ have weight 2, it preserves the parity of the grading, so that the separate even and odd parts are maintained: $H_G^\pm(M)_f \cong H_G^\pm(F)_f$. The same reasoning applies to the cohomology relative to the boundary, so $H_G^\pm(M,\partial M)_f \cong H_G^\pm(F,\partial F)_f$
Since the action on $F$ is trivial, it is immediate from the definition that there is an isomorphism of $R$-modules, $H_G^*(F)\cong H^*(F)\otimes R$ so that the localization theorem shows $j^*$ induces an isomorphism of $R_f$-modules, $$\label{eq:localized j^* isomorphism}
H_G^\pm(M)_f \stackrel{j^*}{\longrightarrow} H^\pm(F) \otimes R_f.$$ It follows that $H_G^\pm(M)_f$ is a free $R_f$ module whenever $f\in
I(Z)$. Of course, analogous statements hold for the relative versions. Since localization does not alter the rank of a module (it just annihilates torsion elements), we have that $${\mathop\mathrm{rank}\nolimits}H_G^\pm(M) = \dim H^\pm(F),\qquad {\mathop\mathrm{rank}\nolimits}H_G^\pm(M,\partial M) = \dim H^\pm(F,\partial F).$$
For $X\in\gg$, define $N(X_M) = \{x\in M\mid X_M(x)=0\}$, the set of zeros of the vector field $X_M$. Since $X$ generates a torus action, $N(X_M)$ is a manifold with boundary $\partial N(X_M) = N(X_M)\cap \partial M$. Clearly $N(X_M)\supset F$, and $N(X_M)= F$ if and only if $X\not\in Z$.
\[relative equivariant\] Let $X = \sum_j s_jX_j\in \gg$. If the set of zeros of the corresponding vector field $X_M$ is equal to the fixed point set for the $G$-action (i.e. $N(X_M)=F$) then $$\label{eq.relative iso1.}
H_{X_M}^\pm(M,\,\partial M)\cong H_{G}^\pm(M,\partial
M)/\mathfrak{m}_X H_{G}^\pm(M,\partial M),$$ and $$\label{eq.relative iso2.}
H_{X_M}^\pm(M)\cong H_{G}^\pm(M)/\mathfrak{m}_X H_{G}^\pm(M)$$ where $\mathfrak{m}_X = \left<u_1-s_1,\dots,u_l-s_l\right>$ is the ideal of polynomials vanishing at $X$.
Our assumption $N(X_M)=F$ is equivalent to $X \in\gg\setminus Z$. Therefore there is a polynomial $f \in I(Z)$ such that $f(X)\neq 0$. In addition, we can use $f$ and replace the ring $R$ by $R_f$ and then localize $ H_G^\pm(M)$ and $H_G^\pm(M,\partial M)$ to make $H_G^\pm(M)_f$ and $H_{G}^\pm(M,\partial M)_f$ which are free $R_f$-modules.
We now apply the lemma stated below, in which the left-hand side is obtained by putting $u_i=s_i$ before taking cohomology, so results in $H^\pm_{X_M}(M)$ (or similar for the relative case), while the right-hand side is the right-hand side of (\[eq.relative iso1.\]) and (\[eq.relative iso2.\]), so proving the theorem.
\[lemma 5.6\] Let $(C^*,d)$ be a cochain complex of free $R$-modules and assume that, for some polynomial $f$, $H(C^*,d)_f$ is a free module over the localized ring $R_f$. Then, if $s \in
\RR^l$ with $f(s)\neq 0$,$$H^\pm(C_{s}^*,d_s)\cong H^\pm(C^*,d)\bmod
\mathfrak{m}_s$$ where $\mathfrak{m}_s $ is the (maximal) ideal $\left<u_1-s_1,\dots,u_l-s_l\right>$ at $X$ in $\RR[\gg]$.
\[coroll:Witten-fixed isomorphism\] Let $X \in\gg$ and $j_{X}:N(X_M)\hookrightarrow M$, then $j^*_{X}$ induces the following isomorphisms
- $H_{X_M}^\pm(M) \cong H^\pm(N(X_M))$,
- $H_{X_M}^\pm(M,\partial M) \cong H^\pm(N(X_M),\partial
N(X_M))$.
First suppose $X\not\in Z$. Then the isomorphisms above follow by reducing equation (\[eq:localized j\^\* isomorphism\]) modulo $\mathfrak{m}_X$ and applying Theorem \[relative equivariant\].
If on the other hand, $X \in Z$, then let $G'$ be the corresponding isotropy subgroup, so that $N(X_M)=F':=\textrm{Fix}(G',M)$ (it is clear that $G'\supset G(X)$, the subgroup of $G$ generated by $X$). The considerations above show that $H^\pm_{X_M,G'}(M,\partial M)\cong
H^\pm(F',\partial F')$ and $H^\pm_{X_M,G'}(M)\cong H^\pm(F')$, where $H^\pm_{X_M,G'}(M)$ and $H^\pm_{X_M,G'}(M,\partial M)$ are defined using $G'$-invariant forms, and $\mathfrak{m}_{G',X}$ is the maximal ideal at $X$ in the ring $\RR[\gg']$. Moreover, all classes in $H^\pm_{X_M,G'}(M)$ and $H^\pm_{X_M,G'}(M,\partial M)$ have representatives which are $G$-invariant, not only $G'$-invariant (either by an averaging argument, or by using the unique $X_M$-harmonic representatives). So, this gives $H^\pm_{X_M,G}(M)\cong H^\pm_{X_M,G'}(M)$ and $H^\pm_{X_M,G}(M,\partial M)\cong H^\pm_{X_M,G'}(M,\partial M)$, $\forall X\in\gg'\subset\gg$ as desired.
\[rem.$X_M$-cohomology and singular homology\] If $M$ is a compact manifold with boundary then $H^k(M)\cong H_{k}(M)$ and $H^k(M,\partial M) \cong H_{k}(M,\partial M)$, where $H_{k}(M)$ and $H_{k}(M,\partial M)$ are the absolute and relative singular homology with real coefficients. We observe that this fact together with corollary \[coroll:Witten-fixed isomorphism\] give us the following isomorphisms $$H_{X_M}^\pm(M) \cong H_{\pm}(N(X_M))\quad \mbox{and} \quad
H_{X_M}^\pm(M,\partial M) \cong H_{\pm}(N(X_M),\partial
N(X_M)),$$ where $H_{+}(N(X_M))=\oplus_i H_{2i}(N(X_M))$ and $H_{-}(N(X_M),\partial N(X_M))=\oplus_i H_{2i+1}(N(X_M),\partial N(X_M))$, by using the map $$\label{eq.inte.}
[\omega]_{X_M}(\{c\})=\int_{c}j^*\omega,$$ where $\omega$ is $X_M$-closed $\pm$-form representing the absolute (or relative) $X_M$-cohomology class $[\omega]_{X_M}$ on $M$ and $c$ is a $\pm$-cycle representing the absolute (or relative) singular homology class $\{c\}$ on $N(X_M)$. In this light, eq. (\[eq.Stokes’ theorem\]), corollary \[coroll:Witten-fixed isomorphism\] and the bijection (\[eq.inte.\]) prove the following statement:
*An $X_M$-closed form $\omega$ is $X_M$-exact iff all the periods of $j^*\omega$ over all $\pm$-cycles of $N(X_M)$ vanish.*
Interior and boundary subspaces {#sec.style of DeTurck-Gluck}
===============================
In this section we visit some recent work of DeTurck and Gluck [@Gluck] on harmonic fields and cohomology (see also [@Clay1; @Clay2] for details), and adapt it to $X_M$-harmonic fields.
Interior and boundary subspaces after DeTurck and Gluck
-------------------------------------------------------
Given the usual manifold $M$ with boundary, there is a long exact sequence in cohomology associated to the pair $(M,\partial M)$ and one can use this to define two subspaces of $H^k(M)$ and $H^k(M,\partial M)$ as follows:
- the *interior* subspace $IH^k(M)$ of $H^k(M)$ is the kernel of $i^*:H^k(M)\to H^k(\partial M)$
- the *boundary* subspace $BH^k(M,\partial M)$ of $H^k(M,\partial M)$ is the image of $\d:H^{k-1}(\partial M)\to H^k(M,\partial M)$
Note that if $M$ has no boundary, then $IH^k=H^k$ and $BH^k=0$, as should be expected from their names.
At the level of cohomology there is no ‘natural’ definition for the boundary part of the absolute cohomology nor the interior part of the relative cohomology. However, DeTurck and Gluck [@Gluck] use the metric and harmonic representatives to provide these. Firstly the subspaces defined above are realized as $$\begin{aligned}
\mathcal{IH}_N^k &=&
\{\omega\in \mathcal{H}^k_{N}(M)\mid i^*\omega=\d\theta, \mbox{ for some }
\theta\in\Omega^{k-1}(\partial M)\}\\
\mathcal{BH}^k_D &=& \mathcal{H}^k_{D}(M)\cap\mathcal{H}^k_{\mathrm{ex}}\end{aligned}$$ respectively (these are denoted $\mathcal{E}_{\partial}\mathcal{H}^k_{N}(M)$ and $\mathcal{EH}^k_{D}(M)$ respectively in [@Gluck; @Clay1; @Clay2]). They then use the Hodge star operator to define the other spaces: $$\begin{aligned}
\mathcal{BH}_N^k &=& \mathcal{H}^{k}_{N}(M)\cap\mathcal{H}^k_{\mathrm{co}}\\
\mathcal{IH}^k_D &=& \{\omega\in \mathcal{H}^k_{D}(M):
i^*\star\omega=\d\kappa, \mbox{ for some }
\kappa\in\Omega^{n-k-1}(\partial M)\}\end{aligned}$$ (denoted $c\mathcal{EH}^k_{N}(M)$ and $c\mathcal{E}_{\partial}\mathcal{H}^k_{D}(M)$ in [@Gluck; @Clay1; @Clay2]). The first theorem of DeTurck and Gluck on this subject is
\[thm.DE-Gluck decompo.\] Both $\mathcal{H}_D^k$ and $\mathcal{H}_N^k$ have orthogonal decompositions, $$\begin{aligned}
\mathcal{H}^k_{N}(M)&=& \mathcal{IH}_N^k \oplus \mathcal{BH}_N^k\\
\mathcal{H}^k_{D}(M)&=&\mathcal{BH}_D^k\oplus \mathcal{IH}_D^k.\end{aligned}$$ Furthermore, the two boundary subspaces are mutually orthogonal inside $L^2\Omega$.
However the interior subspaces are not orthogonal, and they prove
\[De-Glu duality angles\] The principal angles between the interior subspaces $\mathcal{IH}_N^k$ and $\mathcal{IH}_D^k$ are all acute.
Part of the motivation for considering these principal angles, called *Poincaré duality angles*, is that they should measure in some sense how far the Riemannian manifold $M$ is from being closed. That these angles are non-zero follows from the fact that $\mathcal{H}_N^k\cap \mathcal{H}_D^k=0$, see [@Schwarz]. Another consequence of this, pointed out by DeTurck and Gluck is that the Hodge-Morrey-Freidrichs decomposition can be refined to a 5-term decomposition, $$\label{eq. Det&Gluck decom.}
\Omega^k(M)=\d\Omega_D^{k-1} \oplus \delta\Omega_N^{k+1}
\oplus(\mathcal{H}^k_D + \mathcal{H}^{k}_N)\oplus
\mathcal{H}^k_{\mathrm{ex,co}},$$ where $\mathcal{H}^k_{\mathrm{ex,co}}=\mathcal{H}^k_{\mathrm{ex}}\cap\mathcal{H}^k_{\mathrm{co}}$ and the symbol $+$ indicates a direct sum whereas $\oplus$ indicates an orthogonal direct sum.
In his thesis [@Clay1], Shonkwiler measures these Poincaré duality angles in interesting examples of manifolds with boundary derived from complex projective spaces and Grassmannians and shows that in these examples the angles do indeed tend to zero as the boundary shrinks to zero, see alternatively [@Clay2].
Extension to $X_M$-cohomology
-----------------------------
It seems reasonable to think that we can extend further to the style of DeTurck-Gluck, and break down the Neumann and Dirichlet $X_M$-harmonic fields into interior and boundary subspaces. If so, does the natural extension of corollary \[coroll:Witten-fixed isomorphism\] hold? The answer is affirmative and contained in the proof of theorem \[thm:refine localization\].
Answering this question will indeed give more concrete understanding of these isomorphisms and consequently will give a precise extension to Witten’s results when $\partial M\neq\emptyset$ (see Section \[sec:conclusions\]).
#### Refinement of the $X_M$-Hodge-Morrey-Friedrichs decomposition
In , we prove that $$\mathcal{H}^{\pm}_{X_{M},N}(M) \cap \mathcal{H}^{\pm}_{X_{M},D}(M)=\{0\},$$ which implies that the sum $\mathcal{H}^{\pm}_{X_{M},N}(M) + \mathcal{H}^{\pm}_{X_{M},D}(M)$ is a direct sum, and by using Green’s formula (\[eq.2.16\]), one finds that the orthogonal complement of $\mathcal{H}^{\pm}_{X_{M},N}(M) + \mathcal{H}^{\pm}_{X_{M},D}(M)$ inside $\mathcal{H}^{\pm}_{X_{M}}(M)$ is $\mathcal{H}^{\pm}_{X_{M},\mathrm{ex,co}}(M)=\mathcal{H}^{\pm}_{X_{M},\mathrm{ex}}(M) \cap \mathcal{H}^{\pm}_{X_{M},\mathrm{co}}(M)$. Therefore, we can refine the $X_M$-Friedrichs decomposition (theorem \[thm:X\_M-Friedrichs\]) into $$\mathcal{H}^{\pm}_{X_{M}}(M) = ( \mathcal{H}^{\pm}_{X_{M},N}(M)+
\mathcal{H}^{\pm}_{X_{M},D}(M) )\oplus
\mathcal{H}^{\pm}_{X_{M},\mathrm{ex,co}}(M).$$ Consequently, following DeTurck and Gluck’s decomposition (\[eq. Det&Gluck decom.\]), we can refine the $X_M$-Hodge-Morrey-Friedrichs decompositions (Corollary \[co.2.6\]) into the following five terms decomposition: $$\label{eq:5 term X_M decomposition}
\Omega^{\pm}_{G}(M) =\mathcal{E}^{\pm}_{X_M}(M)\oplus
\mathcal{C}^{\pm}_{X_M}(M)\oplus ( \mathcal{H}^{\pm}_{X_{M},N}(M)+
\mathcal{H}^{\pm}_{X_{M},D}(M) )\oplus
\mathcal{H}^{\pm}_{X_{M},\mathrm{ex,co}}(M).$$ Here as usual, $\oplus$ is an orthogonal direct sum, while $+$ is just a direct sum.
#### Interior and boundary portions of $X_M$-cohomology
Following the ordinary case described above, we can define interior and boundary portions of the $X_M$-cohomology and $X_M$-harmonic fields by $$\label{eq:interior-boundary for X_M cohomology}
\begin{array}{rcl}
IH_{X_M}^\pm(M) &=& \ker[i^*:H^\pm_{X_M}(M)\to H^\pm_{X_M}(\partial M)]\\[4pt]
BH_{X_M}^\pm(M,\partial M) &=& {\mathop{\mathrm{im}}\nolimits}[\d_{X_M}:H_{X_M}^\mp(\partial M) \to H_{X_M}^\pm(M,\partial M)].
\end{array}$$ Here $\d_{X_M}$ is the standard construction: given a closed form $\lambda$ on $\partial M$, let $\tilde\lambda$ be an extension to $M$. Then $\d_{X_M}\tilde\lambda$ defines a well-defined element of $H_{X_M}(M,\partial M)$. These spaces are realized through corollary \[coro.Hodge is often quoted\] as $$\begin{aligned}
\mathcal{IH}_{X_M,N}^\pm &=&
\{\omega\in \mathcal{H}^\pm_{X_M,N}(M)\mid i^*\omega=\d_{X_M}\theta, \mbox{ for some }
\theta\in\Omega^\mp(\partial M)\}\\
\mathcal{BH}^\pm_{X_M,D} &=& \mathcal{H}^\pm_{X_M,D}(M)\cap\mathcal{H}^\pm_{X_M,\mathrm{ex}}\end{aligned}$$ respectively. Now use the Hodge star operator to define the other spaces: $$\begin{aligned}
\mathcal{IH}^\pm_{X_M,D} &=& \{\omega\in \mathcal{H}^\pm_{X_M,D}(M):
i^*\star\omega=\d_{X_M}\kappa, \mbox{ for some }
\kappa\in\Omega^{n-\mp}(\partial M)\}\\
\mathcal{BH}_{X_M,N}^\pm &=&
\mathcal{H}^{\pm}_{X_M,N}(M)\cap\mathcal{H}^\pm_{X_M,\mathrm{co}}.\end{aligned}$$ Note that Hodge star maps boundary to boundary and interior to interior; it follows that, for example $\mathcal{BH}_{X_M,N}^\pm \cong \mathcal{BH}_{X_M,D}^{n-\pm}$.
\[thm. largest orthogonal \] The boundary subspace $\mathcal{BH}^{\pm}_{X_{M},N}(M)$ is the largest subspace of $\mathcal{H}^{\pm}_{X_{M},N}(M)$ orthogonal to all of $\mathcal{H}^{\pm}_{X_{M},D}(M)$ while the boundary subspace $\mathcal{BH}^\pm_{X_{M},D}(M)$ is the largest subspace of $\mathcal{H}^{\pm}_{X_{M},D}(M)$ orthogonal to all of $\mathcal{H}^{\pm}_{X_{M},N}(M).$
The orthogonality follows immediately from Green’s formula (\[eq.2.16\]) while the rest of the proof follow immediately from the $X_M$-Friedrichs decomposition theorem (theorem \[thm:X\_M-Friedrichs\]) (restricted to smooth invariant forms).
The main goal of this subsection is to prove the following theorem and to answer the question above.
\[thm.De- Gluck and ours decom.\] Analogous to theorem \[thm.DE-Gluck decompo.\], we have the orthogonal decompositions $$\begin{aligned}
\mathcal{H}^\pm_{X_M,N}(M)&=& \mathcal{IH}_{X_M,N}^\pm \oplus \mathcal{BH}_{X_M,N}^\pm\\
\mathcal{H}^k_{X_M,D}(M)&=&\mathcal{BH}_{X_MD}^\pm \oplus \mathcal{IH}_{X_M,D}^\pm.\end{aligned}$$
The proof by DeTurck and Gluck of the analogous result uses the duality between de Rham cohomology and singular homology. However, we do not have such a result on $M$ (though perhaps a proof using the equivariant homology described in [@MacPherson] would be possible), so we give a direct proof involving only the cohomology—the same argument can be used to prove DeTurck and Gluck’s original theorem (replacing $\pm$ by $k$ everywhere). An alternative argument can be given using the localization to the fixed point set (corollary \[coroll:Witten-fixed isomorphism\])—details of which can be found in [@my; @thesis].
The orthogonality of the right hand sides follows from Green’s formula (\[eq.2.16\]). It follows that $$\label{eq:direct sum subset}
\mathcal{IH}_{X_M,N}^\pm \oplus \mathcal{BH}_{X_M,N}^\pm \subset \mathcal{H}^\pm_{X_M,N}(M)\quad\mbox{and}\quad
\mathcal{BH}_{X_MD}^\pm \oplus \mathcal{IH}_{X_M,D}^\pm \subset \mathcal{H}^k_{X_M,D}(M).$$
Now consider the long exact sequence in $X_M$-cohomology derived from the inclusion $i:\partial M\hookrightarrow M$, $$\cdots\stackrel{i^*}{\longrightarrow} H_{X_M}^\mp(\partial M) \stackrel{\d_{X_M}}{\longrightarrow} H_{X_M}^\pm(M,\partial M) \stackrel{\rho^*}{\longrightarrow} H_{X_M}^\pm(M) \stackrel{i^*}{\longrightarrow} H_{X_M}^\pm(\partial M) \stackrel{\d_{X_M}}{\longrightarrow} H_{X_M}^\mp(M,\partial M)\longrightarrow\cdots$$ It follows from the exactness that $$IH_{X_M}^\pm(M) = {\mathop{\mathrm{im}}\nolimits}\rho^*,\quad \mbox{and} \quad BH_{X_M}^\pm(M,\partial M)=\ker\rho^*.$$ Thus, $H^\pm_{X_M}(M,\partial M) \cong BH_{X_M}^\pm(M,\partial M) + IH_{X_M}^\pm(M)$, (direct sum) or equivalently $$\label{eq:direct sum cohomology}
\mathcal{H}^\pm_{X_M,D} \cong \mathcal{BH}^\pm_{X_M,D} + \mathcal{IH}^\pm_{X_M,N}.$$ It follows from equations (\[eq:direct sum subset\]) and (\[eq:direct sum cohomology\]) that $\dim(\mathcal{IH}^\pm_{X_M,D})\leq\dim(\mathcal{IH}^\pm_{X_M,N})$. However, the Hodge star operator identifies $\mathcal{IH}_{X_M,N}^\pm$ with $\mathcal{IH}_{X_M,D}^{n-\pm}$ which implies that the inequality in dimensions is in fact an equality, and the result follows.
\[thm:refine localization\] Let $F'=N(X_M)$. We have isomorphisms, $$\begin{array}{cc}
\mathcal{IH}^\pm_{X_{M},N}(M)\cong \mathcal{IH}^\pm_{N}(F'),
&\mathcal{BH}^{\pm}_{X_{M},D}(M)\cong \mathcal{BH}^{\pm}_{D}(F'),\\ \mathcal{IH}^\pm_{X_{M},D}(M)\cong\mathcal{IH}^\pm_{D}(F'),
& \mathcal{BH}^\pm_{X_{M},N}(M)\cong \mathcal{BH}^\pm_{N}(F').
\end{array}$$
We prove the first two; the other two follow by applying the Hodge star operator (on $M$ and on $F'$). Denote by $j_X$ the inclusion of the pair, $j_X:(F',\partial F') \hookrightarrow (M,\partial M)$. Then $j_X$ induces a chain map between the long exact sequences of $X_M$ cohomology on $M$ and de Rham cohomology on $F'$, which by corollary \[coroll:Witten-fixed isomorphism\] is an isomorphism.
Since the interior part of the absolute cohomology and the boundary part of the relative cohomology are defined from these long exact sequences, it follows that $j_X$ induces isomorphisms $$IH_{X_M}(M)^\pm \cong IH^\pm(F'),\quad \mbox{and} \quad BH_{X_M}^\pm(M,\partial M) \cong BH^\pm(F',\partial F').$$ It then follows from the $X_M$-Hodge theorem \[thm:X\_M-Hodge\] that there are isomorphisms $\mathcal{IH}_{X_M,N}(M) \cong \mathcal{IH}_N^\pm(F')$ and $ \mathcal{BH}_{X_M,D}^\pm(M) \cong \mathcal{BH}_D^\pm(F').$
The analogue of Gluck and DeTurck’s theorem for the Poincaré duality angles (theorem \[De-Glu duality angles\]) also holds. The $X_M$-Poincaré duality angles are defined in the obvious way, as the principal angles between $\mathcal{IH}^\pm_{X_M,D}$ and $\mathcal{IH}^\pm_{X_M,N}$.
\[prop. our duality angles\] The $X_M$-Poincaré duality angles are all acute.
These angles can be neither 0 nor $\pi/2$, firstly because $\mathcal{H}^{\pm}_{X_{M},N}(M)\cap\mathcal{H}^{\pm}_{X_{M},D}(M)=\{0\}$ (shown in ), and secondly because of theorem \[thm. largest orthogonal \]. Hence they must all be acute.
The results above and in would allow us to extend most of the results of [@Clay1] to the context of $X_M$-cohomology and $X_M$-Poincaré duality angles but we leave this for future work.
Conclusions {#sec:conclusions}
===========
In previous sections, we began with the action of a torus $G$; here we state results for a given Killing vector field $K$ on a compact Riemannian manifold $M$ (with or without boundary), more in keeping with Witten’s original work [@Witten]. Recall that the group $\mathrm{Isom}(M)$ of isometries of $M$ is a compact Lie group, and the smallest closed subgroup $G(K)$ containing $K$ in its Lie algebra is Abelian, so a torus. Furthermore, the submanifold $N(K)$ of zeros of $K$ coincides with $\mathrm{Fix}(G(K),M).$
The equivariant cohomology constructions of Section \[sec:equivariant cohomology\] give us the proof of the following result, which extends the theorem of Witten (our Theorem \[thm:fixed point set\]) to manifolds with boundary.
\[Witten-De rham\] Let $K$ be a Killing vector field on the compact Riemannian manifold $M$ (with or without boundary), and let $N(K)$ be the submanifold of zeros of $K$. Then pullback to $N$ induces isomorphisms $$H_{K}^\pm(M) \cong H^\pm(N(K)),
\quad\mbox{and}\quad H_{K}^\pm(M,\,\partial M) \cong
H^\pm(N(K),\,\partial N(K)).$$
Apply Corollary \[coroll:Witten-fixed isomorphism\] to the equivariant cohomology for the action of the torus $G(K)$.
Furthermore, using the Hodge star operator, the Poincaré-Lefschetz duality of Theorem \[thm:X\_M-Hodge\](c) corresponds under the isomorphisms in the theorem above, to Poincaré-Lefschetz duality on the fixed point space.
Translating this theorem into the language of harmonic fields, shows $$\label{eq. witten exten.on X_M-harm.}
\mathcal{H}^\pm_{K,N}(M)\cong \mathcal{H}^\pm_{N}(N(K))
\quad\mbox{and}\quad \mathcal{H}^\pm_{K,D}(M)\cong
\mathcal{H}^\pm_{D}(N(K)).$$ where $\mathcal{H}^\pm_{N}(N(K))$ and $\mathcal{H}^\pm_{D}(N(K)) $ are the ordinary Neumann and Dirichlet harmonic fields on $N(K)$ respectively. The fact that theorem \[Witten-De rham\] and eq. (\[eq. witten exten.on X\_M-harm.\]) can be refined to the style of theorem \[thm:refine localization\] which gives a more precise meaning for these isomorphisms.
\[coro.Witt. Diri-Neum\] Given any harmonic field on $N(K)$ with either Dirichlet or Neumann boundary conditions, there is a unique $K$-harmonic field on $M$ with the corresponding boundary conditions whose restriction on $N(K)$ is cohomologous to the given field.
Note that if $\partial N(K)=\emptyset$ then the boundary condition on $N(K)$ is non-existent, and so every harmonic form (= field) on $N(K)$ has corresponding to it both a unique Dirichlet and a unique Neumann $K$-harmonic field on $M$. Moreover, since in this case there is no boundary part of the cohomology of $N(K)$, it follows from theorem \[thm:refine localization\] that $\mathcal{BH}_{X_M,N}=\mathcal{BH}_{X_M,D}=0$.
In other words, it means that all the de Rham cohomology of $N(K)$ must come only from the interior portion, i.e. $H^{\pm}(N(K))=H^{\pm}(N(K),\partial N(K))$, which shows that every interior de Rham cohomology class has corresponding to it both a unique relative and a unique absolute $K$-cohomology class on $M$.
As an application, we have the fact that theorem \[Witten-De rham\] and corollary \[coro.Witt. Diri-Neum\] can be used to extend the other results of Witten in [@Witten] and we hope that this extension will be useful in quantum field theory and other mathematical and physical applications when $\partial
M\neq\emptyset$.
#### Euler characteristics
As is well known, given a complex of $\RR[s]$ (or $\mathbb{C}[s]$) modules whose cohomology is finitely generated, the Euler characteristic of the complex is independent of $s$. This remains true for a $\ZZ_2$-graded complex, for the same reasons (briefly, the cohomology is the direct sum of a torsion module and a free module, and the torsion cancels in the Euler characteristic).
Applying this to the complexes for $X_M$-cohomology, with $X_M=sK$, it follows that $\chi(M)=\chi(N)$ and $\chi(M,\partial M) =
\chi(N,\partial N)$ (where $N=N(K)$), and furthermore applying the same arguments to the manifold $\partial M$, one has $\chi(\partial
M)=\chi(\partial N)$, i.e. $$\chi(M)=\chi(\partial M)+\chi(M,\partial M)= \chi(\partial
N)+\chi(N,\partial N)=\chi(N).$$
#### Other Applications:
We have shown that the Witten-Hodge theory can shed light to give additional equivariant geometric and topological insight. In addition, the fact that we can use the new decompositions of $L^2\Omega^{\pm}_G(M)$ given in theorem \[thm:Hodge-Morrey\] and corollary \[co.2.6\] and also the relation between the $X_M$-cohomology and $X_M$-harmonic fields (theorem \[thm:X\_M-Hodge\]) as powerful tools (under topological aspects) in the theory of differential equations on $L^2\Omega^{\pm}_G(M)$ to obtain the solubility of various <span style="font-variant:small-caps;">bvp</span>s. In particular, we can extend most of the results of chapter three of [@Schwarz] on $L^2\Omega^{\pm}_G(M)$ to the context of the operators $\d_{X_M},$ $\delta_{X_M}$ and $\Delta_{X_M}$. Moreover, the classical Hodge theory plays a fundamental role in incompressible hydrodynamics and it has applications to many other area of mathematical physics and engineering [@Marsden]. So, following these, we hope that the Witten-Hodge theory will be using as tools in these applications as well.
#### Geometric question:
Finally, we proved that $\mathcal{IH}^\pm_{X_{M},N}(M)\cong\mathcal{IH}^\pm_{N}(N(X_M))$ and $\mathcal{IH}^\pm_{X_{M},D}(M)\cong \mathcal{IH}^\pm_{D}(N(X_M))$ and that the principal angles between the corresponding interior subspaces are all acute. Hence, it would be interesting to answer the following
*How do the $X_M$-Poincaré duality angles between the interior subspaces $\mathcal{IH}^\pm_{X_{M},N}(M)$ and $\mathcal{IH}^\pm_{X_{M},D}(M)$ depend on $X$, and how do they compare to the Poincaré duality angles between the interior subspaces $\mathcal{IH}^\pm_{N}(N(X_M))$ and $\mathcal{IH}^\pm_{D}(N(X_M))$.*
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[^1]: we use real valued polynomials, though complex valued ones works just as well, and all tensor products are thus over $\RR$, unless stated otherwise
[^2]: The localized ring $R_f$ consists of elements of $R$ divided by a power of $f$ and if $K$ is an $R$-module, its localization is $K_f:=K\otimes_R R_f$; they correspond to restricting to the open set where $f$ is non-zero. See the notes by Libine [@Libine] for a good discussion of localization in this context.
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abstract: 'Blazars can be divided into two subtypes, flat spectrum radio quasars (FSRQs) and BL Lac objects, which have been distinguished phenomenologically by the strength of their optical emission lines, while their physical nature and relationship are still not fully understood. In this paper, we focus on the differences in their variability. We characterize the blazar variability using the Ornstein–Uhlenbeck (OU) process, and investigate the features that are discriminative for the two subtypes. We used optical photometric and polarimetric data obtained with the 1.5-m Kanata telescope for 2008–2014. We found that four features, namely the variation amplitude, characteristic timescale, and non-stationarity of the variability obtained from the light curves and the median of the degree of polarization (PD), are essential for distinguishing between FSRQs and BL Lac objects. FSRQs are characterized by rare and large flares, while the variability of BL Lac objects can be reproduced with a stationary OU process with relatively small amplitudes. The characteristics of the variability are governed not by the differences in the jet structure between the subtypes, but by the peak frequency of the synchrotron emission. This implies that the nature of the variation in the jets is common in FSRQs and BL Lac objects. We found that BL Lac objects tend to have high PD medians, which suggests that they have a stable polarization component. FSRQs have no such component, possibly because of a strong Compton cooling effect in sub-pc scale jets.'
author:
- 'Makoto <span style="font-variant:small-caps;">Uemura</span>'
- 'Taisei <span style="font-variant:small-caps;">Abe</span>'
- 'Yurika <span style="font-variant:small-caps;">Yamada</span>'
- 'Shiro <span style="font-variant:small-caps;">Ikeda</span>'
title: 'Feature Selection for Classification of Blazars Based on Optical Photometric and Polarimetric Time-Series Data'
---
Introduction
============
Blazars are a sub-class of active galactic nuclei (AGN) with relativistic jets that point toward us. The jet emission is amplified by the beaming effect and dominates the observed flux at almost all wavelengths ([@bla78bllac; @bla79agn; @urr95agn]). Synchrotron emission from the jet is dominant in the radio–X-ray regime. The observed X-ray–$\gamma$-ray emission is mostly due to inverse-Compton scattering by relativistic electrons in the jet. Blazars exhibit violent variability, which provides a hint for understanding the physical conditions and structure in AGN jets (e.g. [@ulr97var]).
Blazars consist of two subtypes: flat spectrum radio quasars (FSRQs) and BL Lac type objects. The former was originally defined by strong emission lines observed in the optical spectra (equivalent width $> 5$ Å; [@sti91bllac; @sto91bllac]), while the latter was defined by weaker lines or featureless spectra. In addition, FSRQs have lower peak frequencies in the synchrotron emission, $\nu_{\rm peak}\lesssim 10^{14}\;{\rm Hz}$, in their spectral energy distribution (SED), while BL Lac objects have a wide range of $\nu_{\rm peak}$ ($10^{14}\lesssim \nu_{\rm peak}\;[{\rm Hz}]\lesssim 10^{18}$) ([@abd10sed]). The luminosity of blazars has a negative correlation with $\nu_{\rm peak}$; FSRQs form the most luminous class of blazars, while BL Lac objects are less luminous. In SEDs, the relative strength of the inverse-Compton scattering component to the synchrotron component is larger in FSRQs than in BL Lac objects. These regularities are known as the “blazar sequence” ([@ghi98seq]).
In addition to the classification based on the emission line strength, a classification scheme based on $\nu_{\rm peak}$ is also used for blazars, with low synchrotron peaked (LSP) blazars for objects with $\nu_{\rm peak}\lesssim 10^{14}\;{\rm Hz}$, intermediate synchrotron peaked (ISP) blazars with $10^{14}\lesssim\nu_{\rm peak}\;[{\rm Hz}]\lesssim 10^{15}\;{\rm Hz}$, and high synchrotron peaked (HSP) blazars with $\nu_{\rm peak}\gtrsim 10^{15}\;{\rm Hz}$ ([@abd10sed]). Most FSRQs are LSP blazars. In this paper, we call LSP, ISP, and HSP BL Lac objects LBLs, IBLs, and HBLs, respectively.
The nature and links between blazar subtypes are still incompletely understood. @ghi08seq proposed that FSRQs are AGN having a radiatively efficient accretion disk (a “standard” disk; [@sha73disk]) with a high accretion rate, while BL Lac objects have a radiatively inefficient accretion flow (RIAF; [@qua01riaf; @nar95adaf]) with a low accretion rate. The accretion rate is considered to be linked to the extended radio morphology of radio galaxies, that is, the Fanaroff–Riley (FR) classification (e.g., [@bau95fr]). It is proposed that FSRQs and all or some LBLs are beamed counterparts to FR type II radio galaxies with high luminosity, and IBLs, HBLs, and possibly some LBLs are counterparts to FR type I objects with low luminosity ([@mey11type; @gio12type]). @gio12type report that known LBLs are inhomogeneous and contain both FR I and II subtypes.
The variability characteristics of the flux and polarization have also been discussed for the different subtypes of blazars, particularly for the optical waveband in which all subtypes have been frequently monitored. It is well known that the optical activity apparently depends on $\nu_{\rm peak}$; LSP blazars are more variable than HSP blazars (e.g., [@bau09var; @ike11blazar; @hov14var]). A similar $\nu_{\rm peak}$ dependence has also been reported in the polarization variations, though the number of previous studies is limited ([@ito16fermi; @ang16pol]). The mechanism of the effect of the $\nu_{\rm peak}$ on the observed flux and polarization variability is unclear. High $\nu_{\rm peak}$ objects show less activity, possibly because a large number or large area of emitting regions blur each short flare ([@mar10multi; @ito16fermi; @ang16pol]), or possibly because the jet volume fraction of a slower “sheath” component increases ([@ito16fermi; @ghi05spine]).
In this paper, we focus on blazar variability. We have performed photometric and polarimetric monitoring of blazars using the 1.5-m Kanata telescope in Hiroshima since 2008 ([@ike11blazar; @ito16fermi]). The present study has two major objectives: to establish the observational features of the flux and polarization variability for characterizing the subtypes, and thereby to investigate the nature of the subtypes, for example, whether FSRQs and LBLs have a common origin and whether the jet structure of FSRQs is different from that of BL Lac objects. Our analyses can be divided into two parts, the extraction of features from the observed time-series data and the selection of the features which are discriminative for the two subtypes. For the feature extraction, in past studies the blazar variability was occasionally characterized only by the features based on the variance of the whole data, while the variation timescale was not considered. We use the Ornstein–Uhlenbeck (OU) process to estimate both the timescale and the amplitude from the data. The OU process and more advanced models based on it have been used to characterize the variations observed in AGN and also in blazars ([@kel09agn; @kel11mixou; @rua12ou; @sob14ou]). For the feature selection, we propose a data-driven approach to select the best set of features for classifying blazars by maximizing the generalization error of a classifier.
The structure of this paper is as follows: In § 2, we describe the data (§ 2.1) and methods used in this paper, namely, the OU process for the feature extraction (§ 2.2) and sparse multinomial logistic regression for the classifier (§ 2.3). In § 3, we present the results of the feature selection. In § 4, we evaluate the classifier and discuss the implications for the selected features.
Data and methods
================
Data
----
{width="17cm"}
We used the data obtained with the Kanata telescope which was published in @ito16fermi. The data includes $V$-band time-series photometric and polarimetric data for 45 blazars from 2008–2014. Figure \[fig:samples\] shows examples of light curves and variations in the degree of polarization (PD).
Panels (a), (b), and (c) of figure \[fig:samples\] show examples of FSRQs: PKS 1510$-$089 and 3C 454.3 in 2010 and in 2008, respectively. The peak-to-peak amplitudes in the light curves are large, over 2 mag in all cases, while the light curve profiles are diverse: a solitary, short flare appears in panel (a), while a number of short flares superimposed on long outbursts appear in panels (b) and (c). The light curves change their apparent characteristics year by year even for the same object, as shown in panels (b) and (c). Panels (d), (e), and (f) of figure \[fig:samples\] show examples of BL Lac objects: BL Lac (LBL), 3C 66A (IBL), and Mrk 501 (HBL), respectively. The peak-to-peak amplitude of the light curve in panel (d) is comparable to that of FSRQs, while the light curve profile looks different. The characterization and classification of these variations are the main subjects of this paper. The variation in polarization could give rise to some interesting features. For example, PD flares are associated with FSRQ flares, though no clear correlation can be seen in the light curve and PD variations in panels (d) and (e). Panels (e) and (f) show that the variation amplitude apparently decreases from LBL to HBL, as mentioned in the previous section.
Feature extraction with the OU process
--------------------------------------
We use the OU process for our time-series analysis. The OU process is a stochastic model based on the multivariate normal distribution whose covariance between the data at time $t_i$ and $t_j$, $S_{ij}$, is given as: $$\begin{aligned}
S_{ij}=A_{\rm exp}\exp\left(-\frac{|t_i-t_j|}{\tau}\right),\end{aligned}$$ where $\tau$ and $A_{\rm exp}$ represent the characteristic variation timescale and amplitude at $\tau$, respectively ([@ou30]). For the time-series data followed by the OU process, $f(t)$, the observed data, $m(t)$, is given by $m(t)=f(t)+\mathcal{N}(0,\sigma_{\rm OU}^2)$, where the second term is the noise defined by the normal distribution having zero mean and variance $\sigma_{\rm OU}^2$. We can extract the characteristic features, $A_{\rm exp}$, $\tau$, and $\sigma_{\rm OU}^2$, from the observed time-series data using the OU process regression.
The time-series data introduced in § 2.1 have different observation periods for each object. The time-series data of each object was divided into one-year segments, each of which is regarded as a sample in this paper. For modeling the light curves with the OU process, the magnitude values were translated to fluxes on a logarithmic scale, simply dividing by $-2.5$. We assumes that the light curves are approximated with the OU process with a characteristic time-scale less than a few tens of days for our sample. The short time-scale is supported by the data in which erratic variations are detected over measurement errors in all samples. If our assumption is true, the power spectrum should be flat for frequencies ($f$) lower than the characteristic frequency, and decays as $f^{-2}$ for higher frequencies ([@kel11mixou]). A strong linear trend in the time-series data breaks this assumption because the power becomes larger in lower frequencies. We consider that the linear trend has an origin different from the short-term variations governed by the OU process. The presence of such distinct short- and long-term variations are reported in AGN ([@Arevalo2006; @McHardy2007; @kel11mixou]) and also in blazars ([@sob14ou]). Hence, we first subtracted the linear trend from the samples, and then performed the OU process regression. The slope value of the linear trend can be considered an indicator of the power at the lowest frequencies, and we use it as a feature for the classification in the next section.
The OU process is identical to the Gaussian process with an exponential kernel. We used the [python]{} package for the Gaussian process, [GPy]{}, which includes a package for the Markov chain Monte Carlo (MCMC) method for the estimation of the posterior probability distributions of the parameters. In the present work, we estimated the posterior distribution of $A_{\rm exp}$ with a flat prior probability and that of $\tau$ with a positive flat prior. We fixed $\sigma_{\rm OU}^2$ with a typical measurement error of the data. We estimated the posterior probability distributions of $A_{\rm exp}$ and $\tau$ using the MCMC method for each light-curve sample. We set $\sigma_{\rm OU}^2=10^{-5}$. Figure \[fig:good\] shows trace plots of $A_{\rm exp}$ and $\tau$, their posterior distributions, and the observed and model light curves for the sample S5 0716$+$714 between MJD 55050 and 55389. The MCMC samples converge to a stationary distribution and the posterior distributions have single-peaked profiles. In this case, we successfully obtained unique solutions of $A_{\rm exp}$ and $\tau$.
![Results of MCMC estimation of $A_{\rm exp}$ and $\tau$ for the sample S5 0716$+$714 between MJD 55050 and 55389. Panels (a) and (c) are the trace plots of the MCMC samples of $A_{\rm exp}$ and $\tau$, respectively. Panels (b) and (d) are their posterior probability distributions. Panel (e) is the observed and model light curves. The filled circles are the data and the solid line and shaded region indicate the mean of the model prediction and its 95% confidence interval, respectively. This is an example in which both $A_{\rm exp}$ and $\tau$ are uniquely determined.[]{data-label="fig:good"}](fig-Kanata-BL-12-55050-55389.pdf){width="8cm"}
On the other hand, we found that $A_{\rm exp}$ and $\tau$ were not uniquely determined for several of the samples, mainly because the data size is not large enough to make a meaningful estimate of the parameters. A significant number of the samples from @ito16fermi have only $<30$ data points. Even in the samples with larger data size, $\tau$ is not uniquely determined if it is too long. Figure \[fig:bad\] shows an example, AO 0235$+$16 between MJD 54617 and 54946. The MCMC samples do not converge to a stationary distribution and $\tau$ can be very large, reaching over 300 d. @koz17drw reports that the OU process model is degenerate when the baseline of the light-curve sample is shorter than ten times $\tau$. The result in figure \[fig:bad\] is probably an example of such a case..
In this paper, we used only samples for which $A_{\rm exp}$ and $\tau$ were uniquely determined, as in figure \[fig:good\]. This selection reduces the number of samples to 38 for 18 objects. The selected samples include 12 samples for 8 FSRQs and 26 samples for 10 BL Lac objects. The samples are listed in table \[tab:log\] in the Appendix. The designation of the objects to the subtypes FSRQ, LBL, IBL, and HBL is taken from @ito16fermi.
![As for figure \[fig:good\] with the light curve data of AO 0235$+$16 between MJD 54617 and 54946. This is an example in which $\tau$ is too long.[]{data-label="fig:bad"}](fig-Kanata-BL-06-54617-54946.pdf){width="8cm"}
Blazars occasionally exhibit large prominent flares, as shown in panel (a) of figure \[fig:samples\], which definitely arise from a non-stationary process, whereas the OU process model assumes a stationary stochastic process. In order to characterize the non-stationarity, we calculate the cross-validation error (CVE) using the OU process regression, as follows: First, the sample is divided into 25-d bins, a sub-sample of which is for validation while the others are for training. Then, the OU process regression is performed with the training subsets. Then, the log-likelihood is calculated from the validation subset and the optimized model. Using the other sub-samples as validation data, we obtained about $10$ log-likelihoods for each sample. The CVE is defined as their mean. A large CVE means that the validation data has a large deviation from the prediction of the model constructed from the training data. Hence, a large value of CVE indicates a high degree of non-stationarity.
The analysis of the time-series PD data was performed in the same manner as for the light curves, that is, dividing it into one-year-segments, converting to a logarithmic scale whose linear trends were subtracted. The OU process parameter $\sigma_{\rm OU}^2$ was fixed to $10^{-4}$. $A_{\rm exp}$ and $\tau$ were uniquely determined for the PD variations in all the samples except for seven. An example of one of the seven samples is shown in figure \[fig:pdbad\]. Although, $A_{\rm exp}$ is uniquely determined, the MCMC samples of $\tau$ do not converge, and $\tau$ can be quite small (note that the scale of $\tau$ is logarithmic in figure \[fig:pdbad\]). As a result, the model of PD is simply the mean of the data, as shown in panel (e). These results suggest that the characteristic timescale is too short to be properly determined with our data. We set $\tau=0.0$ for the seven samples. While a value of zero for $\tau$ is physically undefined, it works for training and evaluating the classifier representing very short timescales. CVEs were also calculated for the PD data. In addition, we also used the median of the PD as a feature parameter.
![As for figure \[fig:good\] with the PD data of S5 0716$+$714 between MJD 55746 and 56125. Note that the scale of $\tau$ is logarithmic. This is an example of PD analysis in which $\tau$ is too short.[]{data-label="fig:pdbad"}](fig-Kanata-PD-12-55746-56125.pdf){width="8cm"}
In total, we obtained nine features from the light curve and PD data: the four features from the light curve samples, $A_{\rm exp}$, $\tau$, the slope of the linear trend, and CVE, and five features from the PD samples, $A_{\rm exp}$, $\tau$, linear slope, CVE, and PD median. The values of the features are listed in table \[tab:log\] in the Appendix.
Sparse multinomial logistic regression
--------------------------------------
We construct a classifier for FSRQs and BL Lac objects based on the nine features described in § 2.2. We use sparse multinomial logistic regression (SMLR) to determine the classifier ([@smlr]). We consider the problem of defining an $M$-class classifier with $N$ labeled samples, each of which has a $K$-dimensional feature vector, $\bm{\theta}_i=\{\theta_{i,1},\theta_{i,2},\cdots,\theta_{i,K}\}$ $(i=1,2,\cdots,N)$. A sample that belongs to the $j$-th class can be expressed with a vector $\bm{y}=\{ y^{(1)}, y^{(2)}, \cdots , y^{(M)} \}$ such that $y^{(j)}=1$ and the other elements are $0$. Multinomial logistic regression gives the probability that a sample belongs to the $j$-th class, as follows: $$\begin{aligned}
P\left(y^{(j)}=1|\bm{\theta},\bm{w}\right)
= \frac{\exp\left( {\bm{w}^{(j)}}^T \bm{\theta} \right)}
{\sum_{j=1}^M \exp\left( {\bm{w}^{(j)}}^T \bm{\theta} \right)},\end{aligned}$$ where $\bm{w}^{(j)}$ is the weight vector for the $j$-th class. The log-likelihood function is given by the data $\bm{\theta}$ as $$\begin{aligned}
\ell(\bm{w})=\sum_{i=1}^N \log P\left(\bm{y}_i|\bm{\theta}_i,\bm{w}\right).\end{aligned}$$ Then, the solution of SMLR is expressed as $$\begin{aligned}
\hat{\bm{w}}= {\mathop{\rm arg~max}_{\bm{w}}\limits} \{ \ell(\bm{w}) - \lambda\|\bm{w}\|_1 \},\end{aligned}$$ where $\|\bm{w}\|_1$ is the $\ell_1$ norm, $\|\bm{w}\|_1=\sum_i|w_i|$, and $\lambda$ is a sparsity parameter that controls the complexity of the model.
SMLR gives a linear classifier against the observed features if it is used as $\bm{\theta}$. In this case, SMLR can select the important features because the $\ell_1$ term makes $\bm{w}$ sparse. On the other hand, a non-linear classifier can be obtained if the observed features are transformed with non-linear kernel functions. Then, we can avoid over-fitting due to the $\ell_1$ term. In the present study, the features listed in table \[tab:log\] were normalized and the feature vector of the $i$-th sample, $\bm{x}_i$, was obtained. The $j$-th element of $\bm{\theta}_i$ was obtained from $\bm{x}_i$ and $\bm{x}_j$ with the RBF kernel as follows: $$\begin{aligned}
\theta_{i,j} = \exp\left\{ -\frac{|\bm{x}_j-\bm{x}_i|^2}{2\sigma_{\rm RBF}^2} \right\}, \end{aligned}$$ where $\sigma_{\rm RBF}^2$ is the bandwidth.
As mentioned in § 2.2, the number of samples, $N$, is 38. The number of classes, $M$, is two: FSRQs and BL Lac objects. Because of the small sample size, the three subtypes LBL, IBL, and HBL are combined as one BL Lac type, while the characteristics of the subtypes are discussed in § 4.2. The classifier is evaluated from the so-called “area under the curve” (AUC), which is defined by the receiver operating characteristic (ROC) curve. The simple accuracy of the classifier is inadequate because the number of BL Lac objects is larger than that of FSRQs in our sample (12 FSRQ samples and 26 BL Lac samples). The AUC is calculated by leave-one-out cross-validation for estimating the generalization error of the classifier. Optimization of the model and the calculation of the cross-validated AUC were performed with the Java-based application [SMLR]{}[^1].
Results
=======
We investigate the features that are discriminative for FSRQs and BL Lac objects based on SMLR and cross-validated AUC using the nine features obtained from the data. SMLR has two hyper-parameters, $\sigma_{\rm RBF}^2$ and $\lambda$. We first consider appropriate values for these two parameters for our study.
A small $\sigma_{\rm RBF}^2$ leads to a complicated model with a large number of samples retained in the classifier. Such a small $\sigma_{\rm RBF}^2$ occasionally creates an island-like boundary. Figure \[fig:island\] shows examples of the probability map of BL Lac type samples calculated with $\sigma_{\rm RBF}^2=1.0$ (left) and $5.0$ (right) using two features, the light-curve CVE and PD median. We set $\lambda=0.1$ in this case. As can be seen in the left panel, the high probability region forms an “island” within the surrounding low probability area. However, it is unlikely that the two subtypes of blazars have such a complicated boundary. A linear or slightly non-linear model, like that in the right panel, is more reasonable. We confirmed that a classifier with large bandwidths ($\sigma_{\rm RBF}^2\gtrsim 5$) does not have an island-like boundary using our samples. In the following analysis, we set $\sigma_{\rm RBF}^2=5.0$.
![Examples of complicated and simple boundaries. The color map indicates the probability map of a BL Lac type sample calculated from the light-curve CVE and PD median with SMLR. The left and right panels show those obtained with bandwidth parameters of 1.0 and 5.0, respectively. The blue and red circles indicate FSRQ and BL Lac samples.[]{data-label="fig:island"}](cv-med-island.pdf){width="8cm"}
The sparsity parameter, $\lambda$, also controls the complexity of the model. We investigated the best AUC of all combinations of the nine features against various values of $\lambda$. The result is given in figure \[fig:lambda\], showing that AUC becomes maximum around $\lambda=1.0$. A model obtained with $\lambda>1$ is too simple to appropriately classify the samples. On the other hand, a small $\lambda$ ($<1$) leads to over-fitting. We set $\lambda=1.0$ in the following analysis.
![Optimal AUC against $\lambda$.[]{data-label="fig:lambda"}](lambda_cv.pdf){width="8cm"}
\[tab:auc\]
We made an exhaustive test of all the parameters to find the most important features (e.g., [@iga18ex]). The number of combinations of the nine features is $2^9-1=511$. Using SMLR, we developed 511 classifiers using models with different combinations of parameters, and calculated the AUCs for each. Table \[tab:auc\] lists the top 20 classifiers in the order of AUC values. For example, the classifier with the highest AUC ($=0.923$) uses six features, that is, the CVE, $A_{\rm exp}$, $\tau$ of the light curve, and the median, CVE, and $\tau$ of the PD. It has an accuracy of 0.842. As can be seen in the table, the correlation between the accuracy score and AUC is low in the 20 models. This is probably due to the small sample size, and indicates that a small difference in the AUC is not important. We found that CVE and $A_{\rm exp}$ of the light curve are used in all the top 20 models, and that $\tau$ of the light curve and PD median are used in 19 models. This result suggests that these four features are essential to classify FSRQs and BL Lac objects.
The probability of the BL Lac type ($P_{\rm BL}$) for each sample obtained with the classifier using the four parameters is listed in table \[tab:log\] in the Appendix. We can determine the class of each sample based on $P_{\rm BL}$. Table \[tab:err\] is the error matrix for several different decision criteria: $P_{\rm BL}=0.5$, $0.6$, and $0.7$. In the case of $P_{\rm BL}=0.5$, all the samples classified as BL Lac objects are indeed BL Lac objects (Accuracy $=1.0$). On the other hand, only six of the 12 FSRQs are correctly classified as FSRQ, while the other six samples are misidentified. The BL Lac prediction accuracy improves with increased decision criterion ($P_{\rm BL}$), while the prediction accuracy of FSRQs decreases in that case. The high rate of misidentified FSRQs suggests that a significant portion of FSRQs cannot be distinguished from BL Lac objects based on the four features.
\[tab:err\]
Discussion
==========
Significance of the classifier
------------------------------
A good classifier could incidentally be obtained in high dimensional problems even if all the features are not related to the real characteristics of the samples. We tested the significance of the obtained classifier described in the previous section using artificial data sets. The sets of artificial data consist of 12 FSRQs and 26 BL Lac objects, as for the case in table \[tab:log\], with random values for the nine features. The random numbers were uniformly distributed between $0$ and $1$. We made 100 sets of data and obtained $511\times 100=51100$ AUC values, in the same manner as described in the previous section. Figure \[fig:auchist\] shows histograms of the AUC values from the real samples (red) and from the artificial samples (black). The distribution from the real samples exhibited systematically higher AUCs than that of the random data sets. The AUC values obtained from the random data sets are concentrated in the area AUC $<0.8$. Thus, it is unlikely that the obtained best AUC values from the real data ($\sim 0.9$) are incidentally obtained.
![Histograms of the AUC obtained from real samples (red) and artificial data generated from random numbers (black). See the text for details.[]{data-label="fig:auchist"}](auc_hist.pdf){width="8cm"}
Implications from the four features
-----------------------------------
![Scatter plots of the selected features. The top, middle, and bottom panels show CVE, $\tau$, and PD median against $A_{\rm exp}$, respectively. The blue filled and open circles denote the correctly classified and misclassified FSRQ samples, respectively. The red, orange, and black triangles are LBL, IBL, and HBL samples, respectively. The gray bars represent the 95% confidence intervals.[]{data-label="fig:scat"}](scatplot.pdf){width="8cm"}
Here, we discuss the implications of the results of § 3. Figure \[fig:scat\] shows scatter plots of the four features. The correctly classified FSRQs ($P_{\rm BL}<0.5$) are indicated by the filled blue circles, while the misclassified FSRQs are indicated by the open circles. As can be seen from the top panel, the correctly classified FSRQs have high CVE and/or high $A_{\rm exp}$, while the misclassified FSRQs have values for these features comparable to those of the BL Lac objects. The high value of CVE indicates the presence of prominent non-stationary flares which deviate from the stationary OU process. The high value of $A_{\rm exp}$ indicates a large amplitude of variation at the characteristic timescale, $\tau$. We propose that FSRQs are characterized by rare and large flares which have a time-series structure distinct from ordinary variations. If the frequency of the flares is relatively high, a few times a year say, then the light curve can be reproduced by the OU process with a high $A_{\rm exp}$. If the frequency is low, such as once a year, then the light curve can be divided into two distinct periods, that is, the stationary state and the non-stationary flare, which causes a high CVE. The misclassified FSRQs may be objects in which the flare frequency was so low that no flare was detected in the year.
It is not evident that the characteristics of the light-curve CVE and $A_{\rm exp}$ originate from a different structure and/or physical condition for the jets between the blazar subtypes (e.g. [@ito16fermi]). It is possible that it is simply due to the $\nu_{\rm peak}$ effect. In order to investigate this point, we analyzed the X-ray data of the HBL Mrk 421 using the X-ray light curve presented in @yam19mrk421. The data was obtained with XRT/[*Swift*]{} from 2009 to 2014. The time-interval of the X-ray light curve is 1 d. We analyzed the data in the same manner as for the optical light curves, that is, dividing it into one-year-segments, calculating the flux density in a logarithmic scale whose linear trends were subtracted, and performing the OU process regression for each segment. Table \[tab:xrt\] shows the estimated CVE, $A_{\rm exp}$, and $\tau$ for each sample. We successfully obtained values for the four segments listed in the table. We could not obtain those values for the segment MJD 55939–56078 mainly because of the small sample size ($N=34$). The estimated values are indicated by crosses in the top and middle panels of figure \[fig:scat\]. They are definitely in the regime of FSRQs, especially regarding the large $A_{\rm exp}$. This result suggests that the large $A_{\rm exp}$ does not originate from different jet properties in the blazar subtypes, but from the $\nu_{\rm peak}$ effect.
The variation timescale of the light-curve, $\tau$, was also selected as an important feature for classification. However, as shown in the middle panel of figure \[fig:scat\], we cannot find any clear differences between the $\tau$ distributions of FSRQs and BL Lac objects. This feature was selected mainly because it is useful for the classification of only one FSRQ sample, 3C 454.3 in MJD 54542–54930. This sample has a small CVE ($=0.60$) and a not very high $A_{\rm exp}$ ($=3.15$), from which the object cannot be distinguished from BL Lac objects, but has an exceptionally large $\tau$ ($=21.30$). We consider that our analysis does not provide enough evidence to determine the importance of $\tau$.
It is proposed that the beaming factor of FSRQs is systematically larger than that of BL Lac objects. @gio12type proposed that the two subtypes have a common nature, except for the beaming factor. @ito16fermi propose that the jet volume fraction occupied by the fast “spine” should be larger in FSRQs than BL Lac objects. The difference in the beaming factor would also change the characteristics of the variability. For example, a shorter variation timescale is expected with a higher beaming factor. However, our analysis provides no strong evidence that $\tau$ of FSRQs is systematically smaller than that of BL Lac objects, while the uncertainty of $\tau$ is large.
In the bottom panel of figure \[fig:scat\], we can see a trend that BL Lac objects have high PD medians compared with FSRQs. This characteristic is stronger when HBLs (the black triangles in the figure) are neglected. The low PD medians of HBLs are probably due to a large contamination of the unpolarized emission from their host galaxies ([@sha13spec]). Figure \[fig:qusample\] shows examples of polarization variations in the Stokes $Q/I$–$U/I$ plane. The left and right panels show the data of the FSRQ PKS 1510$-$089 having a low PD median and that of the LBL OJ 287 having a high PD median, respectively. An increase in PD is occasionally associated with the flares of blazars, and in general the PD remains relatively low when the object is faint. In the left panel of figure \[fig:qusample\], most of the data points exhibit low PDs, except for a few data points with high PDs over $>20$%, which are associated with the prominent flare shown in panel (a) of figure \[fig:samples\]. The high value of the PD median in BL Lac objects indicates that PD is relatively high even in the faint state. The right panel of figure \[fig:qusample\] shows an example: the object had a relatively high PD throughout the year. It has been proposed that the LBL object BL Lac has two polarization components: short-term variations superimposed on a stable or semi-stable component ([@hag02bllac; @sak13bllac]). The fact that the PD median was selected in our analysis suggests that the presence of a stable polarization component is a characteristic feature of BL Lac objects.
![Polarization variations in the Stokes $Q/I$–$U/I$ plane of the FSRQ PKS 1510$-$089 in MJD 54759–55124 (left) and the LBL OJ 287 in MJD 55045–55451 (right).[]{data-label="fig:qusample"}](qu_sample.pdf){width="8cm"}
The origin of the PD median characteristic is unclear. The values of the PD median apparently correlate with $\nu_{\rm peak}$ in BL Lac objects, being lowest in HBLs and highest in LBLs. However, FSRQs have low PD medians, although they have the highest $\nu_{\rm peak}$. Hence, the characteristic is not due to the $\nu_{\rm peak}$ effect, but possibly due to a difference in the jet structure between the blazar subtypes. In this case, the stable polarization component should have a different emitting site or physical condition from the short-term variations since the characteristics of the short-term flux variability can be interpreted as the $\nu_{\rm peak}$ effect, as discussed above. The presence of the stable component may suggest that the accelerated electrons have a long life-time with a long cooling timescale. According to @kas05blr, the size of the broad line region (BLR) has a positive correlation with the AGN luminosity, and the AGNs with the highest luminosity have large BLRs up to sub-pc scale. FSRQs form a sub-group of blazars with the highest luminosity, not only of the jets, but also of the AGNs ([@fos98blseq; @sha13spec; @ghi17blseq]). The lack of a stable component in FSRQs may be reconciled with the presence of a strong radiation field induced by a large BLR causing strong Compton cooling of the electrons even in the sub-pc region, which is the source of the stable polarization component in BL Lac objects.
@gio12type reported that LBLs include both low luminosity FR I objects and high luminosity FR II objects. If this is the case, there may be LBLs with a low PD median. Our samples included only two LBL objects, BL Lac and OJ 287. The number of samples is so small that we cannot make conclusions about the population of LBLs. Further studies are required to understand the relationship between the presence/absence of the stable polarization component and FR types or AGN luminosity.
Features of polarization variability
------------------------------------
In this paper, we used features derived from both the light curves and PD variations, while the only PD feature selected as being useful for classifying FSRQs and BL Lac objects was the PD median. Figure \[fig:flxpd\] shows a scatter plot of $\tau$ of the light curve and of the PD variations. In this figure, we can see that the timescale of the PD variation tends to be shorter than that of the light curve. Most of the objects have a PD $\tau$ shorter than 5 d. As mentioned in § 2.2, the PD $\tau$ was too short to be uniquely determined in seven cases. These results imply that the real $\tau$ could be too short to be correctly estimated from our data. If this is the case, the PD features were not selected in our analysis possibly because they were not good indicators for the nature of the PD variability. The fact that the PD $\tau$ is significantly shorter than the light-curve $\tau$ suggests that the relaxation timescale of the ordered magnetic field is shorter than the cooling timescale of the accelerated electrons.
On the other hand, the presence of a stable polarization component in BL Lac objects can also cause a lack of PD variation features in the selected features. The observed Stokes parameters are a sum of those of multiple components. If the contamination of the stable component is strong, the PD variation of a flare is diluted by the stable component. For example, the increase in PD associated with a flare is canceled if the direction of polarization of the flare component is perpendicular to that of the stable component. This effect also causes the PD features to be poor indicators of the real PD variability. In future work, we will extract the features of the PD variation for both the short-term flares and the stable, or long-term, variation component by separating these components ([@uem10bayes]).
![Correlations of the flux and PD $\tau$. The blue circles and the red, orange, and black triangles denote the FSRQ, LBL, IBL, and HBL samples, respectively. The dashed line indicates $\tau\;{\rm (PD)}=\tau\;{\rm (Flux)}$.[]{data-label="fig:flxpd"}](flx-PD.pdf){width="8cm"}
Summary
=======
We characterized the optical variability of blazars using the OU process and investigated the features which are discriminative for the two blazar subtypes, FSRQs and BL Lac objects. Our summarized findings are as follows:
- Four features, namely, the variation amplitude, $A_{\rm exp}$, characteristic timescale, $\tau$, and non-stationarity, CVE, from the light curve and the PD median are essential to classify blazars into FSRQs and BL Lac objects.
- FSRQs are characterized by rare and large flares based on a large $A_{\rm exp}$ and/or CVE. We found that the X-ray variability of the HBL Mrk 421 also has large $A_{\rm exp}$ comparable to the optical variability of FSRQs. Hence, the characteristics of $A_{\rm exp}$ and CVE are governed not by the differences in the jet structure between the subtypes, but by the $\nu_{\rm peak}$ effect.
- The high PD median of BL Lac objects suggest that they tend to have a stable polarization component. The lack of such a component in FSRQs is possibly due to strong Compton cooling from a large BLR in sub-pc scale jets.
- The variation timescale of PD is significantly shorter than that of the light curves. This may indicate that the relaxation timescale of the ordered magnetic field is shorter than the cooling timescale of the accelerated electrons.
This work was supported by a Kakenhi Grant-in-Aid (No. 25120007) from the Japan Society for the Promotion of Science (JSPS).
Samples and features
====================
= 1mm
\[tab:log\]
\[tab:xrt\]
, A. A., [Ackermann]{}, M., [Agudo]{}, I., [Ajello]{}, M., [Aller]{}, H. D., [Aller]{}, M. F., [Angelakis]{}, E., [Arkharov]{}, A. A., [et al.]{} 2010, , 716, 30
, E., [Hovatta]{}, T., [Blinov]{}, D., [Pavlidou]{}, V., [Kiehlmann]{}, S., [Myserlis]{}, I., [B[ö]{}ttcher]{}, M., [Mao]{}, P., [et al.]{} 2016, , 463, 3365
, P., [Papadakis]{}, I. E., [Uttley]{}, P., [McHardy]{}, I. M., & [Brinkmann]{}, W. 2006, , 372, 401
, A., [Baltay]{}, C., [Coppi]{}, P., [Ellman]{}, N., [Jerke]{}, J., [Rabinowitz]{}, D., & [Scalzo]{}, R. 2009, , 699, 1732
, S. A., [Zirbel]{}, E. L., & [O’Dea]{}, C. P. 1995, , 451, 88
, R. D. & [K[ö]{}nigl]{}, A. 1979, , 232, 34
, R. D. & [Rees]{}, M. J. 1978, in BL Lac Objects, ed. A. M. [Wolfe]{} (University of Pittsburgh Press), pp 328–341
, G., [Maraschi]{}, L., [Celotti]{}, A., [Comastri]{}, A., & [Ghisellini]{}, G. 1998, , 299, 433
, G., [Celotti]{}, A., [Fossati]{}, G., [Maraschi]{}, L., & [Comastri]{}, A. 1998, , 301, 451
, G., [Righi]{}, C., [Costamante]{}, L., & [Tavecchio]{}, F. 2017, , 469, 255
, G. & [Tavecchio]{}, F. 2008, , 387, 1669
, G., [Tavecchio]{}, F., & [Chiaberge]{}, M. 2005, , 432, 401
, P., [Padovani]{}, P., [Polenta]{}, G., [Turriziani]{}, S., [D’Elia]{}, V., & [Piranomonte]{}, S. 2012, , 420, 2899
, V. A., [Larionova]{}, E. G., [Jorstad]{}, S. G., [Bj[ö]{}rnsson]{}, C. I., & [Larionov]{}, V. M. 2002, , 385, 55
, T., [Pavlidou]{}, V., [King]{}, O. G., [Mahabal]{}, A., [Sesar]{}, B., [Dancikova]{}, R., [Djorgovski]{}, S. G., [Drake]{}, A., [et al.]{} 2014, , 439, 690
,Y., [Takenaka]{}, H., [Nakanishi-Ohno]{}, Y., [Uemura]{}, M., [Ikeda]{}, S., & [Okada]{}, M. 2018, Journal of the Physical Society of Japan, 87, 44802
, Y., [Uemura]{}, M., [Sasada]{}, M., [Ito]{}, R., [Yamanaka]{}, M., [Sakimoto]{}, K., [Arai]{}, A., [Fukazawa]{}, Y., [et al.]{} 2011, , 63, 639
, R., [Nalewajko]{}, K., [Fukazawa]{}, Y., [Uemura]{}, M., [Tanaka]{}, Y. T., [Kawabata]{}, K. S., [Madejski]{}, G. M., [Schinzel]{}, F. K., [et al.]{} 2016, , 833, 77
, S., [Maoz]{}, D., [Netzer]{}, H., [Peterson]{}, B. M., [Vestergaard]{}, M., & [Jannuzi]{}, B. T. 2005, , 629, 61
, B. C., [Bechtold]{}, J., & [Siemiginowska]{}, A. 2009, , 698, 895
, B. C., [Sobolewska]{}, M., & [Siemiginowska]{}, A. 2011, , 730, 52
, S. 2017, , 597, A128
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, A. P. & [Jorstad]{}, S. G. 2010, arXiv:1005.5551
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[^1]: http://www.cs.duke.edu/\~amink/software/smlr
|
---
author:
- Joyce Jiyoung Whang
- 'David F. Gleich'
- 'Inderjit S. Dhillon'
bibliography:
- 'ref\_nise.bib'
title: |
Overlapping Community Detection Using\
Neighborhood-Inflated Seed Expansion
---
[***Index Terms—*** Community Detection, Clustering, Overlapping Communities, Seed Expansion, Seeds, Personalized PageRank.]{}
Introduction
============
Preliminaries {#sec:pre}
=============
Overlapping Community Detection Using Neighborhood-Inflated Seed Expansion {#sec:algo}
==========================================================================
Related Work
============
Experimental Results {#sec:exp}
====================
Discussion and Conclusion
=========================
Acknowledgments {#acknowledgments .unnumbered}
===============
This research was supported by NSF grants CCF-1117055 and CCF-1320746 to ID, and by NSF CAREER award CCF-1149756 to DG.
|
---
abstract: 'The new class of the non-stationary solutions to the system of N-dimensional equations for coupled gravitational and massless scalar field is found. The model represents a single (N-1)-brane in a space-time with one large (infinite) and (N-5) small (compact) space-like extra dimensions. In some particular cases the model corresponds to the gravitational and scalar field standing waves bounded by the brane. These braneworlds can be relevant in string and other higher dimensional models. 0.3cm PACS numbers: 04.50.-h; 04.20.Jb; 11.25.-w'
author:
- 'Merab Gogberashvili [^1]'
- 'Pavle Midodashvili [^2]'
- 'Giorgi Tukhashvili [^3]'
title: '**New Class of N-dimensional Braneworlds**'
---
0.5cm
Introduction
============
Solutions to the Einstein equations in a higher dimensional space-times for the 3-dimensional singular space-like surfaces with non-factorizable geometry, braneworlds [@Hi; @brane], have attracted a lot of interest recently in high energy physics (see [@reviews] for reviews). Most of these models were realized as static geometric configurations. However, there have appeared several braneworld models that assumed time-dependent metrics and fields [@S].
A key requirement for realizing the braneworld idea is that the various bulk fields be localized on the brane. For reasons of economy and avoidance of charge universality obstruction [@DuRuTi] one would like to have a universal gravitational trapping mechanism for all fields. However, there are difficulties to realize such mechanism with exponentially warped space-times used in standard brane scenario [@brane]. In the existing (1+4)-dimensional models spin $0$ and spin $2$ fields are localized on the brane with the decreasing warp factor [@brane], spin $1/2$ field can be localized with the increasing warp factor [@BaGa], and spin $1$ fields are not localized at all [@Po]. For the case of (1+5)-dimensions it was found that spin $0$, spin $1$ and spin $2$ fields are localized on the brane with the decreasing warp factor and spin $1/2$ fields again are localized with the increasing warp factor [@Od]. There exist also 6D models with non-exponential warp factors that provide gravitational localization of all kind of bulk fields on the brane [@6D-old], however, these models require introduction of unnatural sources.
It turns out that proposed recently brane model [@5D-ghost; @5D-real], with standing waves in the bulk, can provide a natural alternative mechanism for universal gravitational trapping of zero modes of all kinds of matter fields. To clarify the mechanism of localization let us remind that standing electromagnetic waves, so-called optical lattices, can provide trapping of various particles by scattering, dipole and quadruple forces[@Opt]. It is known that the motion of test particles in the field of a gravitational wave is similar to the motion of charged particles in the field of an electromagnetic wave[@Ba-Gr]. Thus standing gravitational waves could also provide confinement of matter via quadruple forces. Indeed, the equations of motion of the system of spinless particles in the quadruple approximation has the form[@Dix]: $$\label{quad}
\frac{Dp^\mu}{ds}= F^\mu = -\frac 16 J^{\alpha\beta\gamma\delta}D^\mu R_{\alpha\beta\gamma\delta}~,$$ where $p^\mu$ is the total momentum of the matter field and $J^{\alpha\beta\gamma\delta}$ is the quadruple moment of the stress-energy tensor for the matter field. The oscillating metric due to gravitational waves should induce a quadruple moment in the matter fields. If the induced quadruple moment is out of phase with the gravitational wave the system energy increases in comparison with the resonant case and the fields/particles will feel a quadruple force, $F^\mu$, which ejects them out of the high curvature region, i.e. it would localize them at the nodes.
In this paper we introduce new general class of N-dimensional braneworlds that can provide universal gravitational trapping of all matter fields on the brane. They are realized as standing wave solutions to the system of Einstein and Klein-Gordon equations in N-dimensional space-time. Some particular cases of the introduced general solutions were considered in the papers: [@domain; @GMS] (for $N=4$), [@5D-ghost] (for $N=5$ with the ghost-like scalar field), [@5D-real] (for $N=5$ with the real bulk scalar field) and [@6D] (for $N=6$).
The Model
=========
Our setup consists of a single brane and non self interacting scalar field, $\phi$, in $N$-dimensional space-time with the signature $(+, -,...,-)$. The action of the model has the form: $$\label{action}
S = \int d^Nx \sqrt {|g|} \left( \frac{M^{N-2}}{2}R + \frac \epsilon 2 g^{AB}\partial _A \phi \partial _B\phi + L_{brane} \right)~,$$ where $L_{brane}$ is the brane Lagrangian and $M$ is the fundamental scale, which is related to the N-dimensional Newton constant, $G=1/(8\pi M^{N-2})$. The sign coefficient $\epsilon$ in front of the Lagrangian of $\phi$ takes the values $+1$ and $-1$ for the real and phantom-like bulk scalar fields, respectively. Capital Latin indexes numerate $N$-dimensional coordinates, and we use the units where $c = \hbar = 1$.
Variation of the action (\[action\]) with respect to $g_{AB}$ leads to the Einstein equations: $$\label{EinsteinEquations2}
R_{AB} - \frac 12 g_{AB}R = \frac {1}{M^{N-2}}(\sigma_{AB} + \epsilon T_{AB})~,$$ where the source terms are the energy-momentum tensors of the brane, $$\label{sigma}
\sigma _B^A = M^{N-2}\delta (z)\mathrm{diag} \left[\tau_t,\tau_{x_1},...,\tau _{x_{(N - 3)}},\tau_y,\tau_z\right]~,$$ with $\tau_N$ being brane tensions, and of the bulk scalar field, $$\label{ScalarFieldEnergyMomentumTensor}
T_{AB} = \partial _A\phi \partial _B\phi - \frac 12 g_{AB} \partial ^C\phi \partial _C\phi~.$$ The Einstein equations (\[EinsteinEquations2\]) can be rewritten in the form: $$\label{EinsteinEquations4}
R_{AB}= \frac {1}{M^{N-2}}\left(\sigma_{AB}-\frac{1}{N-2}g_{AB}\sigma + \epsilon \partial_A \phi \partial_B \phi \right)~,$$ where $\sigma = g^{AB}\sigma_{AB}$.
We consider the $N$-dimensional metric [*ansatz*]{}: $$\label{MetricAnsatzGeneral}
ds^2 = (1 + k|z|)^a e^S \left(dt^2 - dz^2\right) - (1 + k|z|)^b \left[e^V \sum\limits_{i = 1}^{N - 3} dx_i^2 + e^{- (N - 3)V}dy^2 \right]~,$$ where the curvature scale $k$ and the exponents $a$ and $b$ are some constants, and the metric functions $S = S(t,|z|)$ and $V = V(t,|z|)$ depend only on time, $t$, and on the modulus of the orthogonal to the brane coordinate, $z$. The determinant of (\[MetricAnsatzGeneral\]) has the form: $$\label{DeterminantOfMetric}
\sqrt {|g|} = e^S (1 + k|z|)^{\frac{b(N - 2) + 2a}{2}}~.$$
By the metric (\[MetricAnsatzGeneral\]) we want to describe geometry of the $(N-1)$-brane placed at the origin of the large space-like extra dimension $z$. Among the $(N-2)$ spatial coordinates of the brane, three of them, $x_1$, $x_2$ and $y$, denote the ordinary infinite dimensions of our world, while the remaining ones, $x_i$ ($i = 3, ..., N-5$), can be assumed to be compact, curled up to the unobservable sizes for the present energies.
The Klein-Gordon equation for the bulk scalar field, $\phi$, in the background metric (\[MetricAnsatzGeneral\]) has the form: $$\label{BulkScalarFieldEquation}
\left\{ \partial _z^2 + \frac{b(N - 2) k ~\mathrm{sgn} (z)}{2( 1 + k|z|)} \partial _z - \partial_t^2 - (1 + k|z|)^{a - b} e^S \left[ e^{ - V}\sum \limits_{i = 1}^{N - 3} \partial _{x_i}^2 + e^{(N - 3)V}\partial _y^2 \right]\right\}\phi = 0~.$$
The non-zero components of N-dimensional Ricci tensor corresponding to (\[MetricAnsatzGeneral\]) are: $$\begin{aligned}
\label{RicciTensorComponents1}
R_{tt} &=& \left(ak + S'\right)\delta (z) + \frac 12\left[ S'' + \frac{b(N - 2)k}{2(1 + k|z|)}S' - \ddot S - \frac{(N - 2)(N - 3)}{2}\dot V^2 \right] + \nonumber \\
&+& \frac{a[b(N - 2) - 2]k^2}{4(1 + k|z|)^2}~,\nonumber\\
R_{tz} &=& \frac{b(N - 2)k~\mathrm{sgn} (z)}{4(1 + k|z|)}\dot S - \frac{(N - 2)(N - 3)~\mathrm{sgn}(z)}{4}\dot VV'~, \nonumber\\
R_{x_1x_1} &=& ... = R_{x_{(N - 3)}x_{(N - 3)}} = - \left(bk + V'\right)e^{- S + V}\delta (z) + \\
&+& \frac{e^{- S + V}}{(1 + k|z|)^{a - b}}\left\{ \frac 12\left[\ddot V - V'' - \frac{b(N - 2)k}{2(1 + k|z|)}V' \right] - \frac{b[b(N - 2) - 2]k^2}{4(1 + k|z|)^2} \right\}~ , \nonumber\\
R_{yy} &=& - \left[bk - (N - 3)V'\right]e^{- S - (N - 3)V}\delta (z) - \nonumber\\
&-& \frac{e^{- S - (N - 3)V}}{(1 + k|z|)^{a - b}}\left\{\frac{(N - 3)}{2}\left[\ddot V - V'' - \frac{b(N - 2)k}{2(1 + k|z|)}V' \right] + \frac{b[b(N - 2) - 2]k^2}{4(1 + k|z|)^2} \right\}~, \nonumber \\
R_{zz} &=& - \left\{ \left[ a + b(N - 2)\right]k + S' \right \} \delta (z) + \nonumber \\
&+& \frac 12\left[ \ddot S - S'' + \frac{b(N - 2)k}{2(1 + k|z|)}S' - \frac{(N - 2)(N - 3)}{2}V'^2 \right] + \frac{Dk^2}{4(1 + k|z|)^2}~, \nonumber\end{aligned}$$ where overdots and primes denote the derivatives with respect to $t$ and $|z|$, respectively, and to shorten the last expression we have introduced the constant: $$D = a[b(N - 2) + 2] - b(N - 2)(b - 2)~.$$
The Einstein equations (\[EinsteinEquations4\]) can be split into the system of equations for metric functions: $$\begin{aligned}
\label{SystemOfEquationsForMetricFunction}
\frac 12\left[ S'' + \frac{b(N - 2)k}{2(1 + k|z|)}S' - \ddot S - \frac{(N - 2)(N - 3)}{2}\dot V^2 \right] + \frac{a[b(N - 2) - 2]k^2}{4(1 + k|z|)^2} &=& \epsilon \frac{1}{M^{N-2}}\partial _t\phi^2~, \nonumber\\
\frac{b(N - 2)k~\mathrm {sgn}(z)}{4(1 + k|z|)}\dot S - \frac{(N - 2)(N - 3)~\mathrm {sgn}(z)}{4}\dot VV' &=& \epsilon \frac{\mathrm {sgn}(z)}{M^{N-2}}\partial _t\phi \partial _z\phi~, \nonumber \\
\frac{e^{- S + V}}{(1 + k|z|)^{a - b}}\left\{\frac 12\left[ \ddot V - V'' - \frac{b(N - 2)k}{2(1 + k|z|)V'} \right] - \frac{b[b(N - 2) - 2]k^2}{4(1 + k|z|)^2} \right\} &=& \epsilon \frac{1}{M^{N-2}}\partial_{x_1}\phi^2~, \nonumber \\
&...& \\
\frac{e^{- S + V}}{(1 + k|z|)^{a - b}}\left\{ \frac 12 \left[\ddot V - V'' - \frac{b(N - 2)k}{2(1 + k|z|)}V' \right] - \frac{b[b(N - 2) - 2]k^2}{4( 1 + k|z|)^2} \right\} &=& \epsilon \frac{1}{M^{N-2}}\partial_{x_{(N - 3)}}\phi^2~, \nonumber \\
\frac{e^{- S - (N - 3)V}}{(1 + k|z|)^{a - b}}\left\{ \frac{(N - 3)}{2}\left[V'' - \ddot V + \frac{b(N - 2)k}{2(1 + k|z|)}V' \right] - \frac{b[b(N - 2) - 2]k^2}{4(1 + k|z|)^2} \right\} &=& \epsilon \frac{1}{M^{N-2}}\partial_y\phi^2~, \nonumber \\
\frac 12 \left[\ddot S - S'' + \frac{b(N - 2)k}{2(1 + k|z|)}S' - \frac{(N - 2)(N - 3)}{2}V'^2 \right] + \frac{Dk^2}{4(1 + k|z|)^2} &=& \epsilon \frac{1}{M^{N-2}}\partial_z\phi^2~, \nonumber\end{aligned}$$ and for the brane energy-momentum tensor: $$\begin{aligned}
\label{SystemOfEquationsForBraneEnergyMomentumTensor}
\left( ak + S'\right)\delta (z) &=& \frac{1}{M^{N-2}}\left( \sigma _{tt} - \frac{1}{N - 2}~g_{tt}\sigma \right)~, \nonumber\\
- \left( bk + V'\right)e^{- S + V}\delta (z) &=& \frac{1}{M^{N-2}}\left(\sigma _{x_1x_1} - \frac{1}{N - 2}~g_{x_1x_1}\sigma \right)~, \nonumber\\
&...&\\
- \left( bk + V' \right)e^{- S + V}\delta (z) &=& \frac{1}{M^{N-2}}\left(\sigma _{x_{(N - 3)}x_{(N - 3)}} - \frac{1}{N - 2}~g_{x_{(N - 3)}x_{(N - 3)}}\sigma \right)~, \nonumber\\
- \left[bk - (N - 3)V'\right]e^{- S - (N - 3)V}\delta (z) &=& \frac{1}{M^{N-2}}\left( \sigma _{yy} - \frac{1}{N - 2}~g_{yy}\sigma \right)~, \nonumber\\
- \left\{ [a + b(N - 2)]k + S' \right\} \delta (z) &=& \frac{1}{M^{N-2}}\left( \sigma _{zz} - \frac{1}{N - 2}~g_{zz}\sigma \right)~. \nonumber\end{aligned}$$
In the following sections we present all nontrivial solutions to the system of equations (\[BulkScalarFieldEquation\]), (\[SystemOfEquationsForMetricFunction\]) and (\[SystemOfEquationsForBraneEnergyMomentumTensor\]) for our metric [*ansatz*]{} (\[MetricAnsatzGeneral\]).
Solutions with $k=0$
====================
In this section we assume $k=0$ in (\[MetricAnsatzGeneral\]) and consider the metric: $$\label{MetricAnsatzGeneral2}
ds^2 = e^S \left(dt^2 - dz^2\right) - \left[e^V \sum\limits_{i = 1}^{N - 3} dx_i^2 + e^{- (N - 3)V}dy^2 \right]~.$$ The Ricci tensor of this space-time has the $\delta$-like singularity at $z=0$, since the metric functions $S$ and $V$ are depending on of the modulus of $z$. To smooth this singularity we place the brane at the origin of the extra coordinate $z$. Note that without modulus for $z$ the metric (\[MetricAnsatzGeneral\]) will correspond to the running wave solutions, considered in [@plane] for 4D case.
When $k=0$ there exist three independent nontrivial solutions to the system (\[BulkScalarFieldEquation\]), (\[SystemOfEquationsForMetricFunction\]) and (\[SystemOfEquationsForBraneEnergyMomentumTensor\]).
The Oscillating Brane
---------------------
We start with the simplest case when the bulk scalar field and the metric function $V$ in (\[MetricAnsatzGeneral2\]) are equal to zero, $$\begin{aligned}
\phi &=& 0~, \\ \nonumber
V &=& 0~.\end{aligned}$$ In this case the solution to the system (\[BulkScalarFieldEquation\]) and (\[SystemOfEquationsForMetricFunction\]) is: $$S = \left[ C_1\sin (\Omega t) + C_2\cos (\Omega t)\right]\left[ C_3\sin (\Omega |z|) + C_4\cos (\Omega |z|) \right]~,$$ where $C_i$ ($i=1, 2, 3,4$) and $\Omega$ are some constants. The solution corresponds to the oscillating brane at $|z|=0$ in $N$-dimensional space-time.
Imposing on the metric function $S$ the boundary condition on the brane: $$S|_{|z| = 0} = 0~,$$ from the equations (\[SystemOfEquationsForBraneEnergyMomentumTensor\]) for the brane tensions we find: $$\begin{aligned}
\tau_t &=& 0~, \nonumber \\
\tau _{x_1} &=& \tau_{x_2} = ... = \tau_{x_{(N - 3)}} = - S'~, \nonumber\\
\tau_y &=& - S'~,\\
\tau_z &=& 0~.\end{aligned}$$
Gravitational and Phantom-like Scalar field Standing Waves
----------------------------------------------------------
Now we consider the case with the ghost-like scalar field, $\phi$, when the metric function $S$ is absent in (\[MetricAnsatzGeneral2\]), $$\begin{aligned}
\epsilon &=& -1~, \nonumber \\
S &=& 0~.\end{aligned}$$ Then the equations (\[BulkScalarFieldEquation\]) and (\[SystemOfEquationsForMetricFunction\]) have the following solution: $$\begin{aligned}
V &=& \left[ C_1\sin (\omega t) + C_2\cos (\omega t) \right]\left[ C_3\sin (\omega |z|) + C_4\cos (\omega |z|) \right] ~, \nonumber\\
\phi &=& \frac 12 \sqrt {M^{N-2}(N - 2)(N - 3)} \left[ C_1\sin ( \omega t) + C_2\cos (\omega t)\right]\left[ C_3\sin (\omega |z|) + C_4\cos (\omega |z|) \right],\end{aligned}$$ where $C_i$ ($i=1, 2, 3,4$) and $\omega$ are some constants. Imposing on the metric function $V$ the boundary condition: $$V |_{|z| = 0} = 0~,$$ from (\[SystemOfEquationsForBraneEnergyMomentumTensor\]) we find the brane tensions: $$\begin{aligned}
\tau_t &=& 0~, \nonumber \\
\tau_{x_1} &=& \tau_{x_2} = ... = \tau_{x_{(N - 3)}} = V'~, \nonumber \\
\tau_y &=& - (N - 3)V'~, \\
\tau _z &=& 0~. \nonumber\end{aligned}$$
Coupled Gravitational and Phantom-like Scalar Field Waves
---------------------------------------------------------
Now let us consider the case with the phantom-like scalar field, $$\begin{aligned}
\epsilon = -1~,\end{aligned}$$ when the both metric functions $S$ and $V$ are presented in the metric (\[MetricAnsatzGeneral2\]). In this case the system (\[BulkScalarFieldEquation\]) and (\[SystemOfEquationsForMetricFunction\]) has the standing wave solution of the form: $$\begin{aligned}
\label{Solution6}
V &=& \left[ C_1\sin (\omega t) + C_2\cos (\omega t) \right]\left[ C_3\sin (\omega |z|) + C_4\cos (\omega |z|) \right]~, \nonumber \\
\phi &=& \frac 12 \sqrt {M^{N-2}(N - 2)(N - 3)} \left[ C_1\sin (\omega t) + C_2\cos (\omega t) \right]\left[ C_3\sin ( \omega |z|) + C_4\cos (\omega |z|) \right], \\
S &=& \left[ C_5\sin (\Omega t) + C_6\cos (\Omega t) \right]\left[ C_7\sin ( \Omega |z|) + C_8\cos (\Omega |z|) \right]~,\nonumber\end{aligned}$$ with $C_i$ ($i=1, 2, 3,...,8$), $\Omega$ and $\omega$ being some constants.
Imposing on the metric functions $S$ and $V$ the boundary conditions on the brane: $$\begin{aligned}
S|_{|z| = 0} &=& 0~, \nonumber\\
V|_{|z| = 0} &=& 0~,\end{aligned}$$ form (\[SystemOfEquationsForBraneEnergyMomentumTensor\]) we find the brane tensions: $$\begin{aligned}
\tau_t &=& 0~, \nonumber\\
\tau_{x_1} &=& \tau_{x_2} = ... = \tau_{x_{(N - 3)}} = - S' + V'~, \nonumber \\
\tau_y &=& - S' - (N - 3)V'~,\\
\tau _z &=& 0~.\nonumber\end{aligned}$$ From (\[Solution6\]) it’s clear that there are two different frequencies associated with the metric functions $S$ and $V$ ($\Omega$ and $\omega$, respectively), and that the oscillation frequency of the phantom-like bulk scalar field standing wave coincides with the frequency of $V$.
Solutions with $k \neq 0$
=========================
In general case, when the metric curvature scale $k$ in (\[MetricAnsatzGeneral\]) is non-zero, the system of equations (\[BulkScalarFieldEquation\]) and (\[SystemOfEquationsForMetricFunction\]) is self consistent only for the following values of the exponents $a$ and $b$: $$\begin{aligned}
\label{Solution1}
a &=& - \frac{N - 3}{N - 2}~, \nonumber\\
b &=& \frac{2}{N - 2}~.\end{aligned}$$ In this case the metric (\[MetricAnsatzGeneral\]), together with the singularity at $|z|=0$, where we place the brane, and when $k > 0$, has the horizons at $|z| = -1/k$. At these points some components of Ricci tensor get infinite values, while all gravitational invariants, for example the Ricci scalar, $$\label{RicciScalar}
R = 2\left( S' + \frac{N - 1}{N - 2}k \right)e^{- S}\delta (z) + \left( 1 + k|z| \right)^{(N - 3)/(N - 2)}e^{- S}\left( S'' - \ddot S \right)~,$$ are finite. It resembles the situation with the Schwarzschild Black Hole, however, in contrast, the determinant of (\[MetricAnsatzGeneral\]) becomes zero at $|z| = -1/k$. As a result nothing can cross these horizons and for the brane observer the extra space $z$ is effectively finite.
For $k \neq 0$ there also exist three independent nontrivial braneworld solutions.
The Domain Wall in $N$ Dimensions
---------------------------------
First of all we want to mention the simplest case without the scalar field and metric functions: $$\begin{aligned}
\phi &=& 0~, \nonumber \\
V &=& 0~, \\
S &=& 0~, \nonumber\end{aligned}$$ corresponding to the $N$-dimensional generalization of the known 4D [@domain] and 5D [@5D-real] domain wall solutions: $$ds^2 = (1 + k|z|)^{- \frac{N - 3}{N - 2}}\left( dt^2 - dz^2 \right) - (1 + k|z|)^{\frac{2}{N - 2}}\left( \sum\limits_{i = 1}^{N - 3} dx_i^2 + dy^2 \right)~.$$ In this case the brane tensions in (\[sigma\]) are: $$\begin{aligned}
\tau_{t} &=& - 2k~, \nonumber \\
\tau_{x_1} &=& ... = \tau _{x_{(N - 3)}} = \tau_{y} = - \frac{N - 3}{N - 2}~k~, \\
\tau _{z} &=& 0~. \nonumber\end{aligned}$$
Gravitational and Phantom-like Scalar Field Standing Waves Bounded by the Brane
-------------------------------------------------------------------------------
Now we consider the case with the phantom-like scalar field and when the metric function $S$ is zero: $$\begin{aligned}
\epsilon &=& -1~, \nonumber \\
S &=& 0~.\end{aligned}$$ Then the metric (\[MetricAnsatzGeneral\]) reduces to: $$\begin{aligned}
\label{MetricAnsatz}
ds^2 = (1 + k|z|)^a \left(dt^2 - dz^2\right) - (1 + k|z|)^b \left[ e^V\sum\limits_{i = 1}^{N - 3}dx_i^2 + e^{- (N - 3)V}dy^2 \right]~,\end{aligned}$$ and the equations (\[BulkScalarFieldEquation\]) and (\[SystemOfEquationsForMetricFunction\]) have the following solution: $$\begin{aligned}
V &=& A \sin (\omega t)J_0(X )~, \nonumber \\
\phi &=& \frac A2 \sqrt {M^{N-2}(N - 2)(N - 3)} \sin (\omega t)J_0(X)~,\end{aligned}$$ where $A$ and $\omega$ are some constants, $J_0$ is Bessel function of the first kind, and the argument $X$ is defined as: $$\label{X}
X = \frac{|\omega|}{|k|}(1 + k|z|)~.$$ Imposing on the metric function $V$ the boundary condition: $$V |_{|z| = 0} = 0~,$$ the equations (\[SystemOfEquationsForBraneEnergyMomentumTensor\]) for the brane tensions will give the solution: $$\begin{aligned}
\tau_t &=& - 2k, \nonumber \\
\tau_{x_1} &=& \tau_{x_2} = ... = \tau_{x_{N - 3}} = -\frac{N - 3}{N - 2} k + V'~, \nonumber \\
\tau_y &=& -\frac{N - 3}{N - 2} k - (N - 3)V'~, \\
\tau_z &=& 0~. \nonumber\end{aligned}$$
Standing Wave Braneworld with the Real Scalar Field
---------------------------------------------------
Finally we introduce most realistic general solution to the equations (\[BulkScalarFieldEquation\]), (\[SystemOfEquationsForMetricFunction\]) and (\[SystemOfEquationsForBraneEnergyMomentumTensor\]) with the real scalar field, $$\epsilon = +1~,$$ and all parameters and functions in the metric (\[MetricAnsatzGeneral\]) being non-zero. Nontrivial solution to (\[BulkScalarFieldEquation\]) and (\[SystemOfEquationsForMetricFunction\]) in this case represents $N$-dimensional version of the standing wave braneworld introduced in [@5D-real] and is done by: $$\begin{aligned}
V &=& \left[ C_1 \sin (\omega t) + C_2\cos (\omega t) \right]\left[ C_3J_0(X) + C_4Y_0(X)\right]~, \nonumber \\
\phi &=& \frac 12 \sqrt {M^{N-2}(N - 2)(N - 3)} \left[ C_1\cos (\omega t) - C_2\sin (\omega t)\right]\left[ C_3J_0(X) + C_4Y_0(X) \right]~, \nonumber \\
S &=& \frac 12 (N - 2)(N - 3)X^2\left\{ C_3^2\left[ J_0(X)^2 + J_1(X)^2 - \frac 1X J_0(X)J_1(X) \right] + \right. \\
&+& C_4^2 \left[ Y_0(X)^2 + Y_1(X)^2 - \frac 1X Y_0(X)Y_1(X) \right] + \nonumber\\
&+& \left. C_3C_4 \left[ 2\left[ J_0(X)Y_0(X) + J_1(X )Y_1(X) \right] - \frac 1X \left[ J_0(X)Y_1(X) + J_1(X)Y_0(X ) \right] \right] \right\} + C_5~, \nonumber\end{aligned}$$ where $C_i$ ($i=1, 2, 3, 4, 5$) and $\omega$ are some constants, $J_0$ ($J_1$) and $Y_0$ ($Y_1$) are Bessel functions of the first and the second kind, respectively, and $X$ is defined by (\[X\]).
Imposing on the metric functions $S$ and $V$ the boundary conditions on the brane: $$\begin{aligned}
S|_{|z| = 0} &=& 0~, \nonumber \\
V|_{|z| = 0} &=& 0~,\end{aligned}$$ the system of equations (\[SystemOfEquationsForBraneEnergyMomentumTensor\]) for the brane tensions will have the following solution: $$\begin{aligned}
\tau_t &=& - 2k~, \nonumber \\
\tau_{x_1} &=& \tau _{x_2} = ... = \tau _{x_{(N - 3)}} = -\frac{N - 3}{N - 2}k - S' + V'~, \nonumber \\
\tau_y &=& -\frac{N - 3}{N - 2}k - S' - (N - 3)V'~, \\
\tau_z &=& 0~.\end{aligned}$$
Conclusions
===========
In this paper we have presented the new class of solutions to the system of the Einstein and Klein-Gordon equations in N-dimensional space-time. The model represents a single (N-1)-brane in a space-time with one large (infinite) and (N-5) small (compact) space-like extra dimensions. We also have derived and explicitly solved the corresponding junction conditions and have obtained analytical expressions for the tensions of the brane. Some cases of the general solutions, which correspond to the braneworlds in 5D and 6D with gravitational and scalar field standing waves bounded by the brane, where already studied in [@5D-ghost; @5D-real; @6D].
Considered in this paper N-dimensional models can be relevant in string and other higher dimensional theories, since they can provide universal gravitational trapping of all kinds of matter fields on the brane, and realize the natural mechanisms of dimensional reduction of multi-dimensional space-times and isotropization of initially non-symmetrically warped braneworlds [@Cosm].
Acknowledgments {#acknowledgments .unnumbered}
===============
MG was partially supported by the grant of Shota Rustaveli National Science Foundation $\#{\rm DI}/8/6-100/12$. The research of PM was supported by Ilia State University (Georgia).
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[^1]: gogber@gmail.com
[^2]: pmidodashvili@yahoo.com
[^3]: gtukhashvili10@gmail.com
|
---
abstract: 'Suppose that $X$ is a [Polish]{}space, $E$ is a countable [Borel]{}equivalence relation on $X$, and $\mu$ is an $E$-invariant [Borel]{}probability measure on $X$. We consider the circumstances under which for every countable non-abelian free group $\Gamma$, there is a [Borel]{}sequence $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ of free actions of $\Gamma$ on $X$, generating subequivalence relations $E_r$ of $E$ with respect to which $\mu$ is ergodic, with the further property that $\sequence{E_r}[r \in {\mathbb{R}}]$ is an increasing sequence of relations which are pairwise incomparable under $\mu$-reducibility. In particular, we show that if $E$ satisfies a natural separability condition, then this is the case as long as there exists a free [Borel]{}action of a countable non-abelian free group on $X$, generating a subequivalence relation of $E$ with respect to which $\mu$ is ergodic.'
address:
- |
Clinton T. Conley\
Department of Mathematical Sciences\
Carnegie Mellon University\
Pittsburgh, PA 15213\
USA
- |
Benjamin D. Miller\
Kurt Gödel Research Center for Mathematical Logic\
Universität Wien\
Währinger Stra[ß]{}e 25\
1090 Wien\
Austria
author:
- 'Clinton T. Conley'
- 'Benjamin D. Miller'
bibliography:
- 'bibliography.bib'
title: Incomparable actions of free groups
---
[^1]
Introduction {#introduction .unnumbered}
============
Suppose that $E$ and $F$ are equivalence relations on $X$ and $Y$. A [*homomorphism*]{} from $E$ to $F$ is a function $\phi {\colon}X \to Y$ sending $E$-equivalent points to $F$-equivalent points, a [*reduction*]{} of $E$ to $F$ is a homomorphism sending $E$-inequivalent points to $F$-inequivalent points, an [*embedding*]{} of $E$ into $F$ is an injective reduction, and an [*isomorphism*]{} of $E$ with $F$ is a bijective reduction. [Borel]{}reducibility plays a central role in the descriptive set-theoretic study of equivalence relations, whereas measurable isomorphism figures more prominently in the ergodic-theoretic study of equivalence relations.
We refer the reader to [@Kechris] for basic descriptive-set-theoretic definitions and notation.
The [*orbit equivalence relation*]{} generated by an action of a group $\Gamma$ on a set $X$ is given by $x \mathrel{{E_{\Gamma}^{X}}} y
\iff \exists \gamma \in \Gamma \ \gamma \cdot x = y$. Actions of groups $\Gamma$ and $\Delta$ on [Polish]{}spaces $X$ and $Y$ equipped with [Borel]{}measures $\mu$ and $\nu$ are [*orbit equivalent*]{} if there is a measure-preserving [Borel]{}isomorphism of the restrictions of their orbit equivalence relations to conull [Borel]{}sets.
Following the usual abuse of language, we say that an equivalence relation is [*countable*]{} if its classes are all countable, and [*finite*]{} if its classes are all finite. Suppose now that $E$ is a countable [Borel]{}equivalence relation on $X$. We say that a [Borel]{} probability measure $\mu$ on $X$ is [*$E$-ergodic*]{} if every $E$-invariant [Borel]{}set is $\mu$-null or $\mu$-conull. We say that $\mu$ is [*$E$-invariant*]{} if $\mu
(B) = \mu({T(B)})$, for every [Borel]{}set $B \subseteq X$ and every [Borel]{}automorphism $T {\colon}X \to X$ whose graph is contained in $E$. And we say that $\mu$ is [*$E$-quasi-invariant*]{} if the family of $\mu$-null [Borel]{}subsets of $X$ is closed under $E$-saturation. As a result of [Feldman]{}-[Moore]{}(see [@FeldmanMoore Theorem 1]) ensures that $E$ is generated by a [Borel]{}action of a countable discrete group, it follows that every $E$-invariant [Borel]{}probability measure is also $E$-quasi-invariant.
Results of [Dye]{}and [Ornstein]{}-[Weiss]{}ensure that all [Borel]{}actions of countable amenable groups equipped with suitably non-trivial ergodic invariant [Borel]{}probability measures are orbit equivalent (see, for example, [@KechrisMiller Theorem 10.7]). In contrast, the primary result of [@GaboriauPopa] ensures that for every countable non-abelian free group $\Gamma$, there are uncountably many [Borel]{}-probability-measure-preserving ergodic free [Borel]{}actions of $\Gamma$ on a [Polish]{}space which are pairwise non-orbit-equivalent. A number of simplifications of the proof and strengthenings of the result have subsequently appeared (see, for example, [@Tornquist; @Hjorth]).
We say that $E$ is [*$\mu$-reducible*]{} to $F$ if there is a $\mu$-conull [Borel]{}set on which there is a [Borel]{}reduction of $E$ to $F$. Similarly, we say that $E$ is [*$\mu$-somewhere reducible*]{} to $F$ if there is a $\mu$-positive [Borel]{}set on which there is a [Borel]{}reduction of $E$ to $F$. As the [Lusin]{}-[Novikov]{}uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections (see, for example, [@Kechris Theorem 18.10]) ensures that every countable [Borel]{}equivalence relation is [Borel]{}reducible to its restriction to any [Borel]{}set intersecting every equivalence class, it follows that when $\mu$ is $E$-ergodic, these two notions are equivalent.
In [@ConleyMiller], a variety of results were established concerning the nature of countable [Borel]{}equivalence relations at the base of the measure reducibility hierarchy, using arguments substantially simpler than those previously appearing. While the analogs of many of these results for equivalence relations generated by free [Borel]{}actions of countable non-abelian free groups trivially follow, the analog corresponding to the main results of [@GaboriauPopa; @Hjorth] also requires a generalization of a stratification result utilized in [@Hjorth; @ConleyMiller]. Here we establish the latter, and incorporate it into ideas from [@ConleyMiller] to obtain the former.
We say that a countable [Borel]{}equivalence relation is [*hyperfinite*]{} if it is the union of an increasing sequence $\sequence{F_n}[n \in {\mathbb{N}}]$ of finite [Borel]{}subequivalence relations, [*$\mu$-hyperfinite*]{} if it is hyperfinite on a $\mu$-conull [Borel]{}set, and [*$\mu$-nowhere hyperfinite*]{} if there is no $\mu$-positive [Borel]{}set on which it is hyperfinite. We say that a countable [Borel]{}equivalence relation $F$ on a [Polish]{}space $Y$ is [*projectively separable*]{} if whenever $X$ is a [Polish]{}space, $E$ is a countable [Borel]{}equivalence relation on $X$, and $\mu$ is a [Borel]{}probability measure on $X$ for which $E$ is $\mu$-nowhere hyperfinite, the pseudo-metric ${d_{\mu}}(\phi, \psi) = \mu(\set{x \in X}[\phi(x) \neq \psi(x)])$ on the space of all countable-to-one [Borel]{}homomorphisms $\phi {\colon}B \to Y$ from ${E \upharpoonright B}$ to $F$, where $B$ ranges over all [Borel]{}subsets of $X$, is separable.
In [@ConleyMiller], it is shown that the family of such equivalence relations includes the orbit equivalence relation induced by the usual action of ${\mathrm{SL}_{2}({\mathbb{Z}})}$ on ${\mathbb{T}^{2}}$, and is closed downward under both [Borel]{}reducibility and [Borel]{}subequivalence relations. Moreover, it is shown that such relations possess many of the exotic properties of countable [Borel]{}equivalence relations which were previously known to hold only of relations relatively high in the [Borel]{}reducibility hierarchy. Our primary result here strengthens one such theorem.
\[introduction:mainresult\] Suppose that $X$ is a [Polish]{}space, $E$ is a projectively separable countable [Borel]{}equivalence relation on $X$, $\mu$ is an $E$-invariant [Borel]{}probability measure on $X$, and there is a free [Borel]{}action of a countable non-abelian free group on $X$ generating a subequivalence relation of $E$ with respect to which $\mu$ is ergodic. Then for every countable non-abelian free group $\Gamma$, there is a [Borel]{}sequence $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ of free actions of $\Gamma$ on $X$, generating subequivalence relations $E_r$ of $E$ with respect to which $\mu$ is ergodic, with the further property that $\sequence{E_r}[r \in {\mathbb{R}}]$ is an increasing sequence of relations which are pairwise incomparable under $\mu$-reducibility.
The main result of [@GaboriauLyons] shows that every countable group has a free [Borel]{}action whose orbit equivalence relation satisfies the piece of our hypothesis concerning the existence of an appropriate action of an appropriate free group. It is an open question as to whether the failure of this requirement on a $\mu$-conull [Borel]{}set, or even its weakening in which ergodicity of the orbit equivalence relation is not required, is equivalent to $\mu$-hyperfiniteness (see [@KechrisMiller §28]).
Although the existence of incomparable orbit equivalence relations is primarily of interest in the presence of [Borel]{}probability measures which are both ergodic and invariant, we also establish analogous results in which these assumptions are omitted (see Theorem \[antichains:actions\]).
In §\[ergodicgenerator\], we characterize the existence of subequivalence relations induced by free [Borel]{}actions of countable non-abelian free groups in which one of the generators acts ergodically. In §\[stratification\], we use this to obtain new stratification results for free Borel actions of countable non-abelian free groups. And in §\[incomparable\], we prove our primary results.
Free actions with an ergodic generator {#ergodicgenerator}
======================================
Given a [Polish]{}space $X$, a countable [Borel]{}equivalence relation $E$ on $X$, and a [Borel]{}function $\phi {\colon}X \to {\mathbb{N}}\union \set{\infty}$, one can ask whether there is a [Borel]{}function $\psi {\colon}X \to X$, whose graph is contained in $E$, such that ${|{\psi^{-1}(x)}|} = \phi(x)$ for all $x \in X$. The uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections ensures that if $\mu$ is an $E$-invariant [Borel]{}probability measure on $X$, then the existence of such a function necessitates that $\int \phi(x)
\ d\mu(x) = 1$. We begin this section by noting that if $\mu$ is also $E$-ergodic, then this necessary condition is also sufficient (off of a $\mu$-null [Borel]{}set).
The [*support*]{} of $\phi {\colon}X \to {\mathbb{N}}\union \set{\infty}$ is given by ${\mathrm{supp}(\phi)} = X \setminus {\phi^{-1}(0)}$. We say that $\psi {\colon}X \to Y \subseteq X$ is a [*retraction*]{} if $\psi(y) = y$ for all $y \in Y$.
\[ergodicgenerator:manytoone\] Suppose that $X$ is a [Polish]{}space, $E$ is a countable [Borel]{} equivalence relation on $X$, $\phi {\colon}X \to {\mathbb{N}}\union \set{\infty}$ is [Borel]{}, and $\mu$ is an $E$-ergodic $E$-invariant [Borel]{}probability measure on $X$ for which $\int \phi(x) \ d\mu(x) = 1$. Then there is an $E$-invariant $\mu$-conull [Borel]{}set $C \subseteq X$ for which there is a [Borel]{}retraction $\psi {\colon}C \to C \intersection {\mathrm{supp}(\phi)}$ such that ${\mathrm{graph}(\psi)} \subseteq E$ and ${|{\psi^{-1}(x)}|} = \phi(x)$ for all $x \in C$.
As the fact that $\int \phi(x) \ d\mu(x) < \infty$ ensures that ${\phi^{-1}(\infty)}$ is $\mu$-null, the $E$-quasi-invariance of $\mu$ implies that so too is its $E$-saturation. By throwing out this set, we can assume that ${\phi(X)} \subseteq {\mathbb{N}}$.
By [@KechrisMiller Lemma 7.3], there is a maximal [Borel]{}set ${\mathscr{F}}$ of pairwise disjoint finite subsets $F$ of $E$-classes such that ${|F \intersection {\mathrm{supp}(\phi)}|} = 1$ and ${|F|} =
\phi(x)$, where $x \in F \intersection
{\mathrm{supp}(\phi)}$. The uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections ensures that the set $D = \union
[{\mathscr{F}}]$ is [Borel]{}, as is the function $\psi {\colon}D \to D$ given by $\psi(x)
= y \iff \exists F \in {\mathscr{F}}\ (x \in F {\text{ and }}y \in F \intersection {\mathrm{supp}(\phi)})$.
It only remains to show that there is an $E$-invariant $\mu$-conull [Borel]{}set $C \subseteq D$. As $\mu$ is $E$-quasi-invariant, it is sufficient to show that $D$ is $\mu$-conull. Suppose, towards a contradiction, that $X \setminus D$ is $\mu$-positive. As $\mu$ is $E$-invariant, the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections ensures that $\int_D \phi(x) \ d\mu(x) = \mu(D)$. The fact that $\int \phi(x) \ d\mu(x) = 1$ therefore implies that $\int_{{\mathrm{supp}(\phi)}\setminus D} \phi(x) \ d\mu(x) = \int_{X \setminus
D} \phi(x) \ d\mu(x) = \mu(X \setminus D)$. In particular, it follows that ${\mathrm{supp}(\phi)} \setminus D$ is also $\mu$-positive. As the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections ensures that the $E$-saturation of this set is [Borel]{}, the $E$-ergodicity of $\mu$ implies that this $E$-saturation is $\mu$-conull. And since the maximality of ${\mathscr{F}}$ ensures that ${|((X \setminus {\mathrm{supp}(\phi)}) \setminus D)
\intersection {[x]_{E}}|} < \phi(x) - 1$ for all $x \in {\mathrm{supp}(\phi)}
\setminus D$, the $E$-invariance of $\mu$ along with the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections implies that $\int_{X
\setminus D} \phi(x) \ d\mu(x) > \mu(X \setminus D)$, the desired contradiction.
We next examine when binary relations can be shifted along equivalence relations so as to stabilize the cardinalities of their sections.
For each $x \in X$, the [ *vertical section*]{} of a set $R \subseteq X \times Y$ is given by ${R_{x}} = \set{y \in Y}
[{(x, y)} \in R]$.
\[ergodicgenerator:matching:weak\] Suppose that $X$ and $Y$ are [Polish]{}spaces, $E$ is a countable [Borel]{}equivalence relation on $X$, $R \subseteq X \times Y$ is a [Borel]{}set with countable vertical sections, and $\mu$ is an $E$-ergodic $E$-invariant [Borel]{}probability measure on $X$ for which $\int {|{R_{x}}|} \ d\mu(x) = 1$. Then there is a [Borel]{}function $\pi {\colon}R \to X$ such that $\forall {(x, y)} \in R \ x \mathrel{E} \pi(x, y)$ and ${|{{(\pi \times {\mathrm{proj}_{Y}})(R)}_{x}}|} = 1$ $\mu$-almost everywhere.
The uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections ensures that the function $\phi {\colon}X \to {\mathbb{N}}\union \set{\infty}$ given by $\phi(x) = {|{R_{x}}|}$ is [Borel]{}. Proposition \[ergodicgenerator:manytoone\] therefore yields an $E$-invariant $\mu$-conull [Borel]{}set $C \subseteq X$ for which there is a [Borel]{}retraction $\psi {\colon}C \to
C \intersection {{\mathrm{proj}_{X}}(R)}$, whose graph is contained in $E$, such that ${|{\psi^{-1}(x)}|} = {|{R_{x}}|}$ for all $x
\in C$.
By the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections, there are [Borel]{}functions $\rho_n {\colon}X \to X$ and $\psi_n {\colon}C \to C$ such that $\sequence{\rho_n(x)}
[n < {|{R_{x}}|}]$ is an injective enumeration of ${R_{x}}$ for all $x \in X$, and $\sequence{\psi_n(x)}[n <
{|{\psi^{-1}(x)}|}]$ is an injective enumeration of ${\psi^{-1}(x)}$ for all $x \in C$. Then the function $\pi {\colon}R \to X$ given by $$\pi(x, \rho_n(x)) =
\begin{cases}
\psi_n(x) & \text{if $x \in C$ and $n < {|{R_{x}}|}$,
and} \\
x & \text{otherwise}
\end{cases}$$ is as desired.
For each $y \in Y$, the [ *horizontal section*]{} of a set $R \subseteq X \times Y$ is given by ${R^{y}} = \set{x \in X}
[{(x, y)} \in R]$.
\[ergodicgenerator:matching\] Suppose that $X$ is a [Polish]{}space, $E$ is a countable [Borel]{}equivalence relation on $X$, $R \subseteq X \times X$ is a [Borel]{}set with countable horizontal and vertical sections, and $\mu$ is an $E$-ergodic $E$-invariant [Borel]{}probability measure on $X$ with the property that $\int {|{R_{x}}|} \ d\mu(x) = \int {|{R^{x}}|} \ d\mu(x) = 1$. Then there are [Borel]{}functions $\pi_X, \pi_Y {\colon}R \to X$ such that $\forall {(x, y)} \in R \ (x \mathrel{E} \pi_X(x, y) {\text{ and }}y \mathrel{E} \pi_Y(x, y))$ and ${|{{(\pi_X \times \pi_Y)(R)}_{x}}|} = {|{{(\pi_X \times \pi_Y)(R)}^{x}}|} = 1$ $\mu$-almost everywhere.
This follows from two applications of Proposition \[ergodicgenerator:matching:weak\].
We next observe that if an invariant [Borel]{}probability measure is ergodic with respect to a subequivalence relation generated by a free [Borel]{}action of a countable non-abelian free group, then by passing to an appropriate subequivalence relation, we can assume that it is ergodic with respect to the subequivalence relation generated by one of the generators.
We say that an equivalence relation is [*aperiodic*]{} if all of its classes are infinite. If $E$ is an aperiodic countable [Borel]{}equivalence relation and $\mu$ is an $E$-invariant [Borel]{}probability measure, then the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections ensures that $\mu(A) = 0$ for every [Borel]{}set $A \subseteq X$ intersecting each $E$-class in finitely-many points.
\[ergodicgenerator:onemeasure\] Suppose that $X$ is a [Polish]{}space, $E$ is a countable [Borel]{}equivalence relation on $X$, $\mu$ is an $E$-invariant [Borel]{}probability measure on $X$, and there is a free [Borel]{}action of a countable non-abelian free group on $X$ generating a subequivalence relation of $E$ with respect to which $\mu$ is ergodic. Then for every non-abelian group $\Gamma$ freely generated by a countable set $S$ and for every $\gamma \in S$, there is a free [Borel]{}action of $\Gamma$ on $X$ generating a subequivalence relation of $E$ such that $\mu$ is ergodic with respect to the equivalence relation generated by $\gamma$.
Note that if ${(\gamma_0, \gamma_1)}$ freely generates ${\mathbb{F}_{2}}$, then $\sequence
{\gamma_1^n \gamma_0 \gamma_1^{-n}}[n \in {\mathbb{N}}]$ freely generates a copy of ${\mathbb{F}_{\aleph_0}}$ within ${\mathbb{F}_{2}}$. Thus we need only construct the desired action for ${\mathbb{F}_{2}}$, and can therefore freely discard $E$-invariant $\mu$-null [Borel]{}sets in the course of the construction.
A [*graph*]{} on $X$ is an irreflexive symmetric subset $G$ of $X \times X$. The equivalence relation [*generated*]{} by such a graph is the smallest equivalence relation on $X$ containing $G$, and a [*graphing*]{} of $E$ is a graph generating $E$.
The [*$\mu$-cost*]{} of a locally countable [Borel]{}graph $G$ on $X$ is given by ${C_{\mu}(G)} = \frac{1}{2} \int {|{G_{x}}|} \ d\mu(x)$, and the [*$\mu$-cost*]{} of $E$ is given by $${C_{\mu}(E)} = \inf \set{{C_{\mu}(G)}}[G \text{ is a {Bor\-el\xspace}graphing of } E].$$
As we can assume that $E$ is itself generated by a free [Borel]{}action of a countable non-abelian free group, [Gaboriau]{}’s formula for the cost of such a relation ensures that ${C_{\mu}(E)} \ge 2$ (see, for example, [@KechrisMiller Theorem 27.6]).
Fix an aperiodic hyperfinite [Borel]{}subequivalence relation $E_0$ of $E$ with respect to which $\mu$ is ergodic (see, for example, [@Zimmer Lemma 9.3.2]). Then there is a [Borel]{}automorphism $T_0 {\colon}X \to X$ generating $E_0$ (see, for example, \[Theorem 5.1\][DoughertyJacksonKechris]{}).
As equality on $X$ is [Borel]{}, we obtain a [Borel]{}graphing $G$ of $E$ by subtracting this relation from $E$. A [*path*]{} through a graph $H$ is a sequence of the form $\sequence{x_i}[i \le n]$, where $n \in {\mathbb{N}}$ and ${(x_i, x_{i+1})} \in H$ for all $i < n$. Such a path is a [*cycle*]{} if $n \ge 3$, $\sequence{x_i}[i < n]$ is injective, and $x_0 =
x_n$. A graph is [*acyclic*]{} if it has no cycles. The cycle-cutting lemma of [Kechris]{}-[Miller]{}and [Pichot]{}(see, for example, [@KechrisMiller Lemma 28.11]) yields an acyclic [Borel]{}subgraph $H$ of $G$ with the property that ${\mathrm{graph}(T_0)} \subseteq H$ and ${C_{\mu}(H)} \ge {C_{\mu}(E)}$, thus ${C_{\mu}(H \setminus {\mathrm{graph}(T_0^{\pm 1})})} \ge 1$.
An [*oriented graph*]{} on $X$ is an antisymmetric irreflexive subset $K$ of $X
\times X$. The graph [*induced*]{} by such an oriented graph is the smallest graph containing $K$. The fact that $X$ admits a [Borel]{}linear order easily yields a [Borel]{}oriented graph $K$ on $X$ generating $H \setminus {\mathrm{graph}(T_0^{\pm 1})}$.
The [*in-degree*]{} of $K$ at a point $y$ is given by ${|{K^{y}}|}$, whereas the [*out-degree*]{} of $K$ at a point $x$ is given by ${|{K_{x}}|}$. As the $E$-invariance of $\mu$ ensures that the average in-degree of $K$ with respect to $\mu$ equals the average out-degree of $K$ with respect to $\mu$, they must be at least one. As $\mu$ is necessarily continuous, the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections ensures the existence of a [Borel]{}oriented subgraph $L$ of $K$ with the property that the average in-degree of $L$ with respect to $\mu$ and the average out-degree of $L$ with respect to $\mu$ are exactly one.
By Proposition \[ergodicgenerator:matching\], there are [Borel]{}functions $\pi_X, \pi_Y
{\colon}L \to X$ such that $\forall {(x, y)} \in L \ (x \mathrel{E} \pi_X(x, y) {\text{ and }}y \mathrel{E} \pi_Y(x, y))$ and the in-degree and out-degree of ${(\pi_X
\times \pi_Y)(L)}$ at $\mu$-almost every point is one. As $\mu$ is $E$-quasi-invariant, the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections ensures that, after throwing out an $E$-invariant $\mu$-null [Borel]{}set, we can assume that the in-degree and out-degree of ${(\pi_X \times \pi_Y)(L)}$ at every point is one. Let $T_1$ be the unique [Borel]{}automorphism whose graph is ${(\pi_X \times \pi_Y)(L)}$. As $H$ is acyclic, the requirement that $\forall {(x, y)} \in L \ (x \mathrel{E} \pi_X(x, y) {\text{ and }}y \mathrel{E} \pi_Y(x, y))$ ensures that the action of ${\mathbb{F}_{2}}$ generated by $T_0$ and $T_1$ is free.
\[ergodicgenerator:onemeasure:remark\] After throwing out an $E$-invariant $\mu$-null [Borel]{}set, the conclusion of Proposition \[ergodicgenerator:onemeasure\] can be established from the weaker hypothesis that there is a [Borel]{}subequivalence relation $F$ of $E$, for which ${C_{\mu}(F)} > 1$, with respect to which $\mu$ is ergodic. To see this, note that we can assume that ${C_{\mu}(E)} > 1$. As $\mu$ is necessarily continuous, there is a [Borel]{}set $B \subseteq X$ for which there is a natural number $k \ge 1 / ({C_{\mu}(E)} - 1)$ such that $\mu(B) = 1 / k$, thus $\mu(B) \le {C_{\mu}(E)} - 1$. [Gaboriau]{}’s formula for the cost of the restriction of a countable [Borel]{}equivalence relation (see, for example, ) then ensures that ${C_{{\mu \upharpoonright B}}({E \upharpoonright B})} =
({C_{\mu}(E)} - 1) + \mu(B) \ge 2\mu(B)$. The proof of Proposition \[ergodicgenerator:onemeasure\] therefore yields the desired action of ${\mathbb{F}_{2}}$, albeit on $B$, not on $X$. To rectify this problem, note that by the proof of , after throwing out an $E$-invariant $\mu$-null [Borel]{}set, we can assume that there is a partition of $X$ into [Borel]{}sets $B_1, \ldots, B_k$, as well as [Borel]{}isomorphisms $\pi_i {\colon}B \to B_i$ whose graphs are contained in $E$, for all $1 \le i \le k$. Let $T_0'$ denote the [Borel]{}automorphism of $X$ which agrees with $\pi_{i+1} {\circ}{\pi_i^{-1}}$ on $B_i$, for all $1 \le i < k$, and which agrees with $\pi_1 {\circ}T_0 {\circ}{\pi_k^{-1}}$ on $B_k$. Let $T_1'$ denote the [Borel]{}automorphism of $X$ which agrees with $\pi_i {\circ}T_1^i {\circ}T_0 {\circ}T_1^{-i} {\circ}{\pi_i^{-1}}$ on $B_i$, for all $1 \le i \le k$. Then $T_0'$ and $T_1'$ yield the desired action of ${\mathbb{F}_{2}}$.
We equip the space of [Borel]{}probability measures $\mu$ on a [Polish]{}space $X$ with the (standard) [Borel]{}structure generated by the functions ${\text{eval}}_B(\mu) = \mu(B)$, where $B$ varies over all [Borel]{}subsets of $X$.
A [*uniform ergodic decomposition*]{} of a countable [Borel]{}equivalence relation $E$ on a [Polish]{}space $X$ is a sequence $\sequence{\mu_x}[x \in X]$ of $E$-ergodic $E$-invariant [Borel]{}probability measures on $X$ such that (1) $\mu_x = \mu_y$ whenever $x \mathrel{E} y$, (2) $\mu(\set{x \in X}[\mu = \mu_x]) = 1$ for every $E$-ergodic $E$-invariant [Borel]{}probability measure $\mu$ on $X$, and (3) $\mu = \int \mu_x
\ d\mu(x)$ for every $E$-invariant [Borel]{}probability measure $\mu$ on $X$. The [Farrell]{}-[Varadarajan]{}uniform ergodic decomposition theorem (see, for example, [@KechrisMiller Theorem 3.3]) ensures the existence of [Borel]{}such decompositions. We next establish an ergodicity-free analog of Proposition \[ergodicgenerator:onemeasure\], by uniformly pasting together actions obtained from the latter along such a decomposition.
\[ergodicgenerator:manymeasures\] Suppose that $X$ is a [Polish]{}space, $E$ is a countable [Borel]{}equivalence relation on $X$, $\mu$ is an $E$-invariant [Borel]{}probability measure on $X$, $\sequence
{\mu_x}[x \in X]$ is a [Borel]{}uniform ergodic decomposition of $E$, and for $\mu$-almost all $x \in X$ there is a free [Borel]{}action of a countable non-abelian free group generating a subequivalence relation of $E$ with respect to which $\mu_x$ is ergodic. Then for every non-abelian group $\Gamma$ freely generated by a countable set $S$ and for every $\gamma \in S$, there is a free [Borel]{}action of $\Gamma$ on $X$ generating a subequivalence relation of $E$ such that for $\mu$-almost all $x \in X$, the measure $\mu_x$ is ergodic with respect to the equivalence relation generated by $\gamma$.
As before, it is sufficient to construct the desired action for ${\mathbb{F}_{2}}$, freely discarding $E$-invariant $\mu$-null [Borel]{}sets as we proceed. Fix ${(\gamma_0, \gamma_1)}$ freely generating ${\mathbb{F}_{2}}$.
Fix a countable basis $\set{U_n}[n \in {\mathbb{N}}]$ for $X$ which is closed under finite unions, and define $G \subseteq \functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}} \times X$ by ${G_{\phi}} = \intersection[m \in {\mathbb{N}}][{\union[n \in {\mathbb{N}}][U_{\phi(m, n)}]}]$. By the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections, there are [Borel]{}functions $f_n {\colon}X \to X$ such that $E = \union[n \in {\mathbb{N}}]
[{\mathrm{graph}(f_n)}]$.
Define $R \subseteq \functions{{\mathbb{N}}}{(\functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}})} \times (X \times X)$ by ${R_{\phi}} = \union[n \in {\mathbb{N}}][{\mathrm{graph}({f_n \upharpoonright {G_{\phi(n)}}})}]$ for all $\phi \in \functions{{\mathbb{N}}}{(\functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}})}$, and let $B$ denote the [Borel]{}set consisting of all ${({(\phi_0, \phi_1)}, x)} \in
(\functions{{\mathbb{N}}}{(\functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}})} \times \functions{{\mathbb{N}}}{(\functions{{\mathbb{N}}\times
{\mathbb{N}}}{{\mathbb{N}}})}) \times X$ with the property that the sets ${R_{\phi_0}} \intersection
({[x]_{E}} \times {[x]_{E}})$ and ${R_{\phi_1}} \intersection ({[x]_{E}} \times {[x]_{E}})$ are graphs of functions inducing a free action of ${\mathbb{F}_{2}}$ on ${[x]_{E}}$. For each pair $\phi = {(\phi_0, \phi_1)}$ in $\functions{{\mathbb{N}}}{(\functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}})}
\times \functions{{\mathbb{N}}}{(\functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}})}$, let $\cdot_\phi$ denote the action of ${\mathbb{F}_{2}}$ on ${B_{\phi}}$ for which the graph of the function $x \mapsto
\gamma_i \cdot_\phi x$ is ${R_{\phi_i}} \intersection ({B_{\phi}} \times {B_{\phi}})$, for all $i < 2$. Let $F$ denote the equivalence relation on $B$ generated by the function ${(\phi, x)} \mapsto {(\phi, \gamma_0
\cdot_\phi x)}$, and fix a [Borel]{}uniform ergodic decomposition $\sequence{\nu_{\phi,x}}
[{(\phi, x)} \in B]$ of $F$.
A subset of a [Polish]{}space is [*analytic*]{} if it is the image of a [Borel]{}subset of a [Polish]{}space under a [Borel]{}function. We use ${\sigma(\Sigmaclass[1][1])}$ to denote the smallest $\sigma$-algebra containing all such sets, and we say that a function is [*${\sigma(\Sigmaclass[1][1])}$-measurable*]{} if pre-images of open sets are $\Sigmaclass[1][1]$. Define ${\mathrm{proj}_{X}} {\colon}X \times Y
\to X$ by ${\mathrm{proj}_{X}}(x, y) = x$. A [*uniformization*]{} of a set $R \subseteq X \times Y$ is a function $\phi {\colon}{{\mathrm{proj}_{X}}(R)} \to Y$ whose graph is contained in $R$.
Let $\Lambda$ denote the push-forward of $\mu$ through the function $x \mapsto
\mu_x$. The regularity of [Borel]{}probability measures on [Polish]{}spaces ensures that if $\nu$ is an $E$-invariant [Borel]{}probability measure on $X$ concentrating on ${B_{\phi}}$, then every free [Borel]{}action of ${\mathbb{F}_{2}}$ on an $E$-invariant $\nu$-conull [Borel]{}subset of $X$ agrees with some $\cdot_\phi$ on an $E$-invariant $\nu$-conull [Borel]{}subset of ${B_{\phi}}$. And ${\nu \upharpoonright {B_{\phi}}}$ is ergodic with respect to the equivalence relation generated by the action of $\gamma_0$ on ${B_{\phi}}$ if and only if the measures induced by $\nu$ and ${\nu_{\phi,x} \upharpoonright ({\set{\phi}}\times {B_{\phi}}})$ on ${B_{\phi}}$ agree, for $\nu$-almost all $x \in X$. As the set of pairs ${(\nu, \phi)}$ satisfying this latter property is [Borel]{}(see, for example, [@Kechris Proposition 12.4 and Theorem 17.25]), the [Jankov]{}-[von Neumann]{} uniformization theorem for analytic subsets of the plane (see, for example, [@Kechris Theorem 18.1]) yields a ${\sigma(\Sigmaclass[1][1])}$-measurable uniformization $\phi$. As ${\mathrm{dom}(\phi)}$ is analytic, and a result of [Lusin]{}’s ensures that every analytic set is $\Lambda$-measurable (see, for example, [@Kechris Theorem 21.10]), Proposition \[ergodicgenerator:onemeasure\] implies that ${\mathrm{dom}(\phi)}$ is $\Lambda$-conull. As $\phi$ is $\Lambda$-measurable, there is a $\Lambda$-conull [Borel]{}set $M \subseteq
{\mathrm{dom}(\phi)}$ on which it is [Borel]{}.
Define $C = \union[\nu \in M][{\set{x \in B_{\phi(\nu)}}[\mu_x = \nu]}]$, and observe that the action of ${\mathbb{F}_{2}}$ on $C$ given by $\gamma_i \cdot x = \gamma_i
\cdot_{\phi(\mu_x)} x$, for $i < 2$, is as desired.
\[ergodicgenerator:manymeasures:remark\] By Remark \[ergodicgenerator:onemeasure:remark\], after throwing out an $E$-invariant $\mu$-null [Borel]{}set, the conclusion of Proposition \[ergodicgenerator:manymeasures\] follows from the weaker assumption that for $\mu$-almost all $x \in X$ there is a [Borel]{}subequivalence relation $F$ of $E$, for which ${C_{\mu_x}(F)} > 1$, with respect to which $\mu_x$ is ergodic.
Stratification
==============
Here we consider various sorts of stratifications of countable [Borel]{}equivalence relations. We begin by noting that strong proper inclusion of equivalence relations often gives rise to a measure-theoretic analog.
\[stratification:properinclusion\] Suppose that $X$ is a [Polish]{}space, $E$ and $F$ are countable [Borel]{}equivalence relations on $X$ such that every $E$-class is properly contained in an $F$-class, and $\mu$ is an $E$-ergodic $F$-quasi-invariant [Borel]{}probability measure on $X$. Then for every [Borel]{}set $B \subseteq X$, the $({E \upharpoonright B})$-class of $({\mu \upharpoonright B})$-almost every point of $B$ is properly contained in an $({F \upharpoonright B})$-class.
The uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections ensures that ${[B]_{F}}$ is [Borel]{}, thus so too is the set $A = \set{x \in
{[B]_{F}}}[B \intersection {[x]_{E}} = B \intersection
{[x]_{F}}]$. As the fact that every $E$-class is properly contained in an $F$-class ensures that $A$ is contained in the $F$-saturation of its complement, the $F$-quasi-invariance of $\mu$ implies that $A$ is not $\mu$-conull. As $A$ is $E$-invariant, the $E$-ergodicity of $\mu$ therefore ensures that $A$ is $\mu$-null. It only remains to note that if $x \in B \setminus A$, then $x$ is necessarily $(F \setminus
E)$-related to some other point in $B$, thus the $({E \upharpoonright B})$-class of $x$ is properly contained in an $({F \upharpoonright B})$-class.
We say that a sequence $\sequence{E_r}[r \in {\mathbb{R}}]$ of subequivalence relations of an equivalence relation $E$ on $X$ is a [*stratification*]{} of $E$ if for all real numbers $r < s$, every $E_r$-class is properly contained in an $E_s$-class. We say that a sequence $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ of actions of a group $\Gamma$ on $X$ is a [*stratification*]{} of $E$ if the corresponding sequence $\sequence{E_r}[r \in
{\mathbb{R}}]$ of equivalence relations generated by the actions is a stratification.
We say that a stratification $\sequence{E_r}[r \in {\mathbb{R}}]$ of $E$ is a [*$\mu$-stratification*]{} if for all $\mu$-positive [Borel]{}sets $B \subseteq X$ and all real numbers $r < s$, the $({E_r \upharpoonright B})$-class of $({\mu \upharpoonright B})$-almost every point of $B$ is properly contained in an $({E_s \upharpoonright B})$-class. We say that a stratification $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ of $E$ is a [*$\mu$-stratification*]{} if the corresponding sequence $\sequence{E_r}[r \in {\mathbb{R}}]$ of equivalence relations generated by the actions is a $\mu$-stratification.
We say that a ($\mu$-)stratification $\sequence{E_r}[r \in {\mathbb{R}}]$ by equivalence relations is [*[Borel]{}*]{} if it is [Borel]{}as a sequence of subsets of $X \times X$. Analogously, a ($\mu$-)stratification $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ by actions of $\Gamma$ is [*[Borel]{}*]{} if the corresponding sequence $\sequence{{\mathrm{graph}(\cdot_r)}}[r \in {\mathbb{R}}]$ is [Borel]{}as a sequence of subsets of $(\Gamma \times X) \times X$.
\[stratification:ergodicinvariantactions\] Suppose that $X$ is a [Polish]{}space, $E$ is a countable [Borel]{}equivalence relation on $X$, $\mu$ is an $E$-invariant [Borel]{}probability measure on $X$, and there is a free [Borel]{}action of a countable non-abelian free group on $X$ generating a subequivalence relation of $E$ with respect to which $\mu$ is ergodic. Then for every countable non-abelian free group $\Gamma$, there is a [Borel]{}$\mu$-stratification of $E$ by free actions of $\Gamma$ on $X$, generating equivalence relations with respect to which $\mu$ is ergodic.
Fix a countable set $S$ freely generating $\Gamma$, as well as some $\gamma \in S$. By Proposition \[ergodicgenerator:onemeasure\], there is a free [Borel]{}action $\cdot$ of $\Gamma$ on $X$, generating a subequivalence relation of $E$, such that $\mu$ is ergodic with respect to the equivalence relation generated by $\gamma$. Fix $\delta \in S \setminus {\set{\gamma}}$, appeal to to obtain a [Borel]{}stratification $\sequence{*_r}[r \in {\mathbb{R}}]$ by actions of ${\mathbb{Z}}$ of the equivalence relation generated by $\delta$, and for each $r \in {\mathbb{R}}$, let $\cdot_r$ denote the action of $\Gamma$ on $X$ given by $\delta \cdot_r x = \delta *_r x$ and $\lambda \cdot_r x = \lambda \cdot x$, for $\lambda \in S \setminus {\set{\delta}}$. Proposition \[stratification:properinclusion\] then ensures that $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ is the desired $\mu$-stratification of $E$.
\[stratification:ergodicinvariantactions:remark\] By Remark \[ergodicgenerator:onemeasure:remark\], after throwing out an $E$-invariant $\mu$-null [Borel]{}set, the conclusion of Proposition \[stratification:ergodicinvariantactions\] follows from the weaker assumption that there is a [Borel]{}subequivalence relation $F$ of $E$, for which ${C_{\mu}(F)} > 1$, with respect to which $\mu$ is ergodic.
We next establish an analogous result in the absence of ergodicity.
\[stratification:invariantactions\] Suppose that $X$ is a [Polish]{}space, $E$ is a countable [Borel]{}equivalence relation on $X$, $\mu$ is an $E$-invariant [Borel]{}probability measure on $X$, and there is a free [Borel]{}action of a countable non-abelian free group on $X$ generating a subequivalence relation of $E$. Then for every countable non-abelian free group $\Gamma$, there is a [Borel]{}$\mu$-stratification of $E$ by free actions of $\Gamma$ on $X$.
Fix a countable set $S$ freely generating a non-abelian group $\Gamma$, as well as some $\gamma \in S$. Clearly we can assume that $E$ is itself generated by a free [Borel]{}action of a countable non-abelian free group on $X$. By the uniform ergodic decomposition theorem, there is a [Borel]{}uniform ergodic decomposition $\sequence
{\mu_x}[x \in X]$ of $E$. By Proposition \[ergodicgenerator:manymeasures\], there is a free [Borel]{}action $\cdot$ of $\Gamma$ on $X$, generating a subequivalence relation of $E$, such that for $\mu$-almost all $x \in X$, the measure $\mu_x$ is ergodic with respect to the equivalence relation generated by $\gamma$. Fix $\delta \in S \setminus
{\set{\gamma}}$, appeal to [@Miller:Incomparable Proposition 5.2] to obtain a [Borel]{}stratification $\sequence{*_r}[r \in {\mathbb{R}}]$ by actions of ${\mathbb{Z}}$ of the equivalence relation generated by $\delta$, and for each $r \in {\mathbb{R}}$, let $\cdot_r$ denote the action of $\Gamma$ on $X$ given by $\delta \cdot_r x = \delta *_r x$ and $\lambda \cdot_r x = \lambda \cdot x$, for $\lambda \in S \setminus {\set{\delta}}$. Proposition \[stratification:properinclusion\] ensures that $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ is a $\mu_x$-stratification of $E$ for $\mu$-almost all $x \in X$, and is therefore a $\mu$-stratification of $E$.
\[stratification:invariantactions:remark\] After throwing out an $E$-invariant $\mu$-null [Borel]{}set, the conclusion of Proposition \[stratification:invariantactions\] follows from the weaker assumption that for all $\mu$-positive [Borel]{}sets $B \subseteq X$, there is a [Borel]{}subequivalence relation $F$ of ${E \upharpoonright B}$ for which ${C_{{\mu \upharpoonright B}}(F)} > \mu(B)$. In light of Remark \[ergodicgenerator:manymeasures\] and the proof of Proposition \[stratification:invariantactions\], to see this, it is sufficient to show that there is a [Borel]{}subequivalence relation $F$ of $E$ with the property that if $\sequence{\mu_x}[x \in X]$ is an ergodic decomposition of $F$, then ${C_{\mu_x}(F)} > 1$ for $\mu$-almost all $x \in X$. In fact, it is enough to produce such an $F$ on an $E$-invariant $\mu$-positive [Borel]{}subset of $X$. Towards this end, fix a [Borel]{}subequivalence relation $F'$ of $E$ such that ${C_{\mu}(F')} > 1$, and appeal to the uniform ergodic decomposition theorem to produce a [Borel]{}uniform ergodic decomposition $\sequence{\mu_x}[x \in X]$ of $F'$. As [@KechrisMiller Proposition 18.1] ensures that the set $C = \set{x \in X}[{C_{\mu_x}(F')}
> 1]$ is co-analytic, and every co-analytic subset of $X$ is $\mu$-measurable, the cost integration formula (see [@KechrisMiller Corollary 18.6]) ensures that it is $\mu$-positive. Fix a $\mu$-positive [Borel]{}set $B \subseteq X$ contained in $C$. As $\mu$ is $F'$-invariant, we can assume that $B$ is $F'$-invariant. By the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections, the set ${[B]_{E}}$ is [Borel]{}and there is a [Borel]{} function $\phi {\colon}{[B]_{E}} \to B$ whose graph is contained in $E$. Then the formula for the cost of a restriction of a countable [Borel]{}equivalence relation ensures that the equivalence relation $F$ on ${[B]_{E}}$ given by $x \mathrel{F} y \iff \phi(x) \mathrel{F'}
\phi(y)$ is as desired.
A [*treeing*]{} of an equivalence relation is an acyclic graphing. We say that a [Borel]{}equivalence relation $E$ on $X$ is [*treeable*]{} if it has a [Borel]{}treeing. We say that a countable [Borel]{}equivalence relation $E$ on $X$ is [*compressible*]{} if there is a [Borel]{}injection $\phi {\colon}X
\to X$, whose graph is contained in $E$, such that $X \setminus {\phi(X)}$ intersects every $E$-class.
\[stratification:compressiblefreeaction\] Suppose that $X$ is a [Polish]{}space, $E$ is a compressible treeable countable [Borel]{} equivalence relation on $X$, and $\Gamma$ is a countable non-abelian free group. Then there is a free [Borel]{}action of $\Gamma$ on $X$ generating $E$.
A straightforward modification of the proof of [@JacksonKechrisLouveau Corollary 3.11] reveals that there is a [Borel]{}treeing $G$ of $E$ which is generated by a [Borel]{}automorphism $T_0 {\colon}X \to X$ and a [Borel]{}isomorphism $T_1 {\colon}A \to B$, where $A, B \subseteq X$ are disjoint [Borel]{}sets. Set $C = X \setminus (A \union B)$, and for each $x \in X$, let $D(x)$ denote the unique set in $\set{A, B, C}$ containing $x$.
Suppose now that $k \in \set{2, 3, \ldots, \aleph_0}$ and $\Gamma$ is freely generated by $\sequence{\gamma_i}[i < k]$. The idea is to again use compressibility to replace $E$ with $E \times {I({\mathbb{N}})}$, and to use the right-hand coordinate to accomodate the generators other than the first two, as well as the points at which $T_1$ and ${T_1^{-1}}$ are not defined.
Towards this end, fix bijections $\gamma_0^D {\colon}{\mathbb{N}}\setminus {\set{0}} \to
{\mathbb{N}}\setminus {\set{0}}$ for all $D \in \set
{A, B, C}$, $\gamma_1^A {\colon}{\mathbb{N}}\setminus {\set{0}} \to {\mathbb{N}}$, $\gamma_1^B {\colon}{\mathbb{N}}\to
{\mathbb{N}}\setminus {\set{0}}$, $\gamma_1^C {\colon}{\mathbb{N}}\to {\mathbb{N}}$, as well as $\gamma_i^D {\colon}{\mathbb{N}}\to {\mathbb{N}}$ for all $D \in \set{A, B, C}$ and $1 < i < k$, with the property that for all $D \in \set{A, B, C}$, the corresponding approximation to an action of ${\mathbb{F}_{k}}$ on ${\mathbb{N}}$, given by $$(\gamma_{s(0)}^{t(0)} \cdots \gamma_{s(n)}^{t(n)})^D \cdot x = ((\gamma_{s(0)}^D)^{t(0)}
{\circ}\cdots {\circ}(\gamma_{s(n)}^D)^{t(n)})(x),$$ is both [*free*]{} and [*transitive*]{}, in the sense that:
1. $\forall n \in {\mathbb{N}}\forall \gamma \in {\mathbb{F}_{k}} \ (\gamma^D \cdot n = n \iff \gamma = {\mathrm{id}})$.
2. $\forall m, n \in {\mathbb{N}}\exists \gamma \in {\mathbb{F}_{k}} \ (\gamma^D \cdot m = n)$.
Let $\gamma_0$ act on $X \times {\mathbb{N}}$ via $$\gamma_0 \cdot {(x, n)} =
\begin{cases}
{(T_0(x), n)} & \text{if $n = 0$, and} \\
{(x, \gamma_0^{D(x)} \cdot n)} & \text{otherwise.}
\end{cases}$$ Similarly, let $\gamma_1$ act on $X \times {\mathbb{N}}$ via $$\gamma_1 \cdot {(x, n)} =
\begin{cases}
{(T_1(x), n)} & \text{if $n = 0$ and $x \in A$, and} \\
{(x, \gamma_1^{D(x)} \cdot n)} & \text{otherwise.}
\end{cases}$$ And finally, let $\gamma_i$ act on $X \times {\mathbb{N}}$ via $\gamma_i \cdot {(x, n)} =
{(x, \gamma_i^{D(x)} \cdot n)}$, for all $1 < i < k$. This defines a free [Borel]{}action of $\Gamma$ generating $E \times {I({\mathbb{N}})}$, so the proposition follows from the fact that the compressibility of $E$ is equivalent to the existence of a [Borel]{}isomorphism between $E$ and $E \times {I({\mathbb{N}})}$ (see, for example, ).
In particular, this yields the following stratification result.
\[stratification:compressibleequivalencerelations\] Suppose that $X$ is a [Polish]{}space, $E$ is a compressible treeable countable [Borel]{} equivalence relation on $X$, and $\mu$ is a [Borel]{}probability measure on $X$ such that $E$ is $\mu$-nowhere hyperfinite. Then there is a [Borel]{}$\mu$-stratification of $E$ by $\mu$-nowhere hyperfinite equivalence relations, each of which is generated by free [Borel]{}actions of every countable non-abelian free group on $X$.
Fix a [Borel]{}isomorphism $\pi {\colon}X \to X \times {\mathbb{N}}$ of $E$ with $E \times {I({\mathbb{N}})}$, and let $\nu$ denote the push-forward of $\mu$ through $\pi$. Fix an $(E \times {I({\mathbb{N}})})$-quasi-invariant [Borel]{}probability measure $\nu' \gg \nu$ on $X \times {\mathbb{N}}$ (see, for example, the proof of [@KechrisMiller Corollary 10.2]). Define a [Borel]{}measure $\nu_0'$ on $X$ by $\nu_0'(B) = \nu'(B \times {\set{0}})$, for all [Borel]{}sets $B \subseteq X$. By [@ConleyMiller Theorem 5.7], there is a [Borel]{}$\mu$-stratification $\sequence{E_r}[r \in
{\mathbb{R}}]$ of $E$ by $\nu_0'$-nowhere hyperfinite countable [Borel]{}equivalence relations. Then the pullback of $\sequence{E_r \times {I({\mathbb{N}})}}[r \in {\mathbb{R}}]$ through $\pi$ yields a [Borel]{} stratification of $E$ by $\mu$-nowhere hyperfinite countable [Borel]{}equivalence relations. As these relations are necessarily compressible, Proposition \[stratification:compressiblefreeaction\] implies that they are induced by free [Borel]{}actions of every countable non-abelian free group on $X$.
The following ensures that [Borel]{}$\mu$-stratifications by equivalence relations generated by free [Borel]{}actions of countable groups give rise to [Borel]{}$\mu$-stratifications by actions.
\[stratification:equivalencerelationstoactions\] Suppose that $X$ is a [Polish]{}space, $E$ is a countable [Borel]{}equivalence relation on $X$, $\Gamma$ is a countable group, $\mu$ is a [Borel]{}probability measure on $X$, and $\sequence{E_r}[r \in {\mathbb{R}}]$ is a [Borel]{}$\mu$-stratification of $E$ by equivalence relations generated by free [Borel]{}actions of $\Gamma$ on $X$. Then there is a [Borel]{}$\mu$-stratification $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ of $E$ by free [Borel]{}actions of $\Gamma$ on $X$ for which there is a [Borel]{}function $\pi {\colon}{\mathbb{R}}\to {\mathbb{R}}$ such that $E_{\pi(r)}$ is the equivalence relation generated by $\cdot_r$ on an $E$-invariant $\mu$-conull [Borel]{}set, for all $r \in {\mathbb{R}}$.
Fix a countable basis $\set{U_n}[n \in {\mathbb{N}}]$ for $X$ which is closed under finite unions, and define $G \subseteq \functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}} \times X$ by ${G_{\phi}} =
\intersection[m \in {\mathbb{N}}][{\union[n \in {\mathbb{N}}][U_{\phi(m, n)}]}]$. By the uniformization theorem for [Borel]{}subsets of the plane with countable vertical sections, there are [Borel]{}functions $f_n
{\colon}X \to X$ such that $E = \union[n \in {\mathbb{N}}][{\mathrm{graph}(f_n)}]$.
Define $R \subseteq \functions{{\mathbb{N}}}{(\functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}})} \times (X \times X)$ by ${R_{\phi}} = \union[n \in {\mathbb{N}}][{\mathrm{graph}({f_n \upharpoonright {G_{\phi(n)}}})}]$, and let $B$ denote the set of all pairs ${(\phi, x)} \in \functions{\Gamma}
{(\functions{{\mathbb{N}}}{(\functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}})})} \times X$ for which the sets of the form ${R_{\phi(\gamma)}} \intersection ({[x]_{E}} \times
{[x]_{E}})$ are graphs of functions inducing a free action of $\Gamma$ on ${[x]_{E}}$. For each function $\phi {\colon}\Gamma \to \functions{{\mathbb{N}}}
{(\functions{{\mathbb{N}}\times {\mathbb{N}}}{{\mathbb{N}}})}$, let $\cdot_\phi$ denote the action of $\Gamma$ on ${B_{\phi}}$ with the property that the graph of the function $x \mapsto
\gamma \cdot_\phi x$ is ${R_{\phi(\gamma)}} \intersection
({B_{\phi}} \times {B_{\phi}})$, for all $\gamma \in \Gamma$.
The regularity of [Borel]{}probability measures on [Polish]{}spaces ensures that for all $r \in {\mathbb{R}}$, the equivalence relation generated by an action of the form $\cdot_\phi$ is $E_r$ on an $E$-invariant $\mu$-conull [Borel]{}subset of ${B_{\phi}}$. As the set of pairs ${(r, \phi)}$ satisfying this latter property is [Borel]{}, the uniformization theorem for analytic subsets of the plane yields a ${\sigma(\Sigmaclass[1][1])}$-measurable uniformization $\phi$. Fix a continuous [Borel]{}probability measure $m$ on ${\mathbb{R}}$. As $\phi$ is necessarily $m$-measurable, there is an $m$-conull [Borel]{}set $R \subseteq {\mathbb{R}}$ on which $\phi$ is [Borel]{}. As $R$ is necessarily uncountable, the proof of the perfect set theorem (see, for example, [@Kechris Theorem 13.6]) yields an order-preserving continuous embedding of ${
\functions{{\mathbb{N}}}{2}
}$ (equipped with the lexicographic order) into $R$. And by composing such an embedding with any order-preserving [Borel]{}embedding of ${\mathbb{R}}$ into ${
\functions{{\mathbb{N}}}{2}
}$, we obtain an order-preserving [Borel]{}embedding $\pi$ of ${\mathbb{R}}$ into $R$, in which case the sequence $\sequence{\cdot_{\pi(r)}}[r \in {\mathbb{R}}]$ is as desired.
Finally, we establish an analog of Proposition \[stratification:invariantactions\] without invariance.
\[stratification:actions\] Suppose that $X$ is a [Polish]{}space, $E$ is a countable [Borel]{}equivalence relation on $X$, $\mu$ is a [Borel]{}probability measure on $X$, and there is a free [Borel]{}action of a countable non-abelian free group on $X$ generating a $\mu$-nowhere hyperfinite subequivalence relation of $E$. Then for every countable non-abelian free group $\Gamma$, there is a [Borel]{}$\mu$-stratification of $E$ by free actions of $\Gamma$ on $X$ generating $\mu$-nowhere hyperfinite equivalence relations.
As a result of [Hopf]{}’s ensures that $X$ is compressible off of an $E$-invariant [Borel]{}set on which $\mu$ is equivalent to an $E$-invariant [Borel]{}probability measure (see, for example, [@Nadkarni §10]), the desired result is a consequence of Propositions \[stratification:invariantactions\], \[stratification:compressibleequivalencerelations\], and \[stratification:equivalencerelationstoactions\].
\[stratification:actions:remark\] By Remark \[stratification:invariantactions:remark\], after throwing out an $E$-invariant $\mu$-null [Borel]{}set, the conclusion of Proposition \[stratification:actions\] follows from the weaker assumption that for all $\mu$-positive [Borel]{}subsets $B \subseteq X$ such that ${\mu \upharpoonright B}$ is equivalent to an $E$-invariant [Borel]{}probability measure $\nu$, there is a [Borel]{}subequivalence relation $F$ of ${E \upharpoonright B}$ for which ${C_{{\nu \upharpoonright B}}(F)} > \nu(B)$.
Antichains {#incomparable}
==========
We are now prepared to establish our primary results. Although these can be obtained from the arguments of [@ConleyMiller] (which are themselves slight variants of arguments already appearing in [@Hjorth]) by substituting the stratification results of the previous section for those of [@ConleyMiller], we will nevertheless provide the full proofs, both for the convenience of the reader and because somewhat simpler versions are sufficient to obtain the results we consider here.
\[antichains:ergodicinvariantactions\] Suppose that $X$ is a [Polish]{}space, $E$ is a projectively separable countable [Borel]{}equivalence relation on $X$, $\mu$ is an $E$-invariant [Borel]{} probability measure on $X$, and there is a free [Borel]{}action of a countable non-abelian free group on $X$ generating a subequivalence relation of $E$ with respect to which $\mu$ is ergodic. Then for every countable non-abelian free group $\Gamma$, there is a [Borel]{} sequence $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ of free actions of $\Gamma$ on $X$, generating subequivalence relations $E_r$ of $E$ with respect to which $\mu$ is ergodic, with the further property that $\sequence{E_r}[r \in {\mathbb{R}}]$ is an increasing sequence of relations which are pairwise incomparable under $\mu$-reducibility.
By Proposition \[stratification:ergodicinvariantactions\], there is a [Borel]{}$\mu$-stratification $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ of $E$ by free actions of $\Gamma$ on $X$ generating equivalence relations $E_r$ with respect to which $\mu$ is ergodic. By replacing $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ with its pushforward through an order-preserving [Borel]{} isomorphism of the set of all real numbers with the set of positive real numbers, we can assume that $\intersection[r \in {\mathbb{R}}][E_r]$ is $\mu$-nowhere hyperfinite. Let $R$ denote the relation on ${\mathbb{R}}$ in which two real numbers $r$ and $s$ are related if $E_r$ is $\mu$-reducible to $E_s$.
\[antichains:ergodicinvariantactions:sections\] Every horizontal section of $R$ is countable.
Suppose, towards a contradiction, that there exists $t \in {\mathbb{R}}$ for which ${R^{t}}$ is uncountable. For each $s \in {R^{t}}$, fix a $\mu$-conull [Borel]{} set $B_s \subseteq X$ on which there is a [Borel]{}reduction $\phi_s {\colon}B_s \to X$ of $E_s$ to $E_t$. As each $\phi_s$ is a homomorphism from ${(\intersection
[r \in {\mathbb{R}}][E_r]) \upharpoonright B_s}$ to $E$, the $\mu$-nowhere hyperfiniteness of $\intersection[r \in {\mathbb{R}}]
[E_r]$ and the projective separability of $E$ ensure the existence of distinct $r, s \in
{R^{t}}$ for which ${d_{\mu}}(\phi_r, \phi_s) < 1$. Then $\set{x \in B_r \intersection B_s}[\phi_r(x) = \phi_s(x)]$ is a $\mu$-positive [Borel]{}set on which $E_r$ and $E_s$ coincide, a contradiction.
As $R$ is analytic (see, for example, [@ConleyMiller Proposition I.15]), a result of [Lusin]{}-[Sierpiński]{}ensures that it has the [Baire]{}property (see, for example, [@Kechris Theorem 21.6]). As the horizontal sections of $R$ are countable and therefore meager, a result of [Kuratowski]{}-[Ulam]{}implies that $R$ is meager (see, for example, [@Kechris Theorem 8.41]), in which case a result of [Mycielski]{}’s yields a continuous order-preserving embedding $\phi {\colon}{
\functions{{\mathbb{N}}}{2}
}\to {\mathbb{R}}$ such that pairs of distinct sequences in ${
\functions{{\mathbb{N}}}{2}
}$ are mapped to $R$-unrelated pairs of real numbers (see, for example, [@ConleyMiller Theorem B.5]). Fix a [Borel]{}embedding $\psi {\colon}{\mathbb{R}}\to {
\functions{{\mathbb{N}}}{2}
}$ of the usual ordering of ${\mathbb{R}}$ into the lexicographical ordering of ${
\functions{{\mathbb{N}}}{2}
}$, and observe that the sequence $\sequence{\cdot_{\pi(r)}}[r \in {\mathbb{R}}]$ is as desired, where $\pi =
\phi {\circ}\psi$.
\[antichains:ergodicinvariantactions:remark\] By Remark \[stratification:ergodicinvariantactions:remark\], after throwing out an $E$-invariant $\mu$-null [Borel]{}set, the conclusion of Theorem \[antichains:ergodicinvariantactions\] follows from the weaker assumption that there is a [Borel]{}subequivalence relation $F$ of $E$, for which ${C_{\mu}(F)} > 1$, with respect to which $\mu$ is ergodic.
In particular, this gives a simple new proof of the existence of such actions for the equivalence relation generated by the usual action of ${\mathrm{SL}_{2}({\mathbb{Z}})}$ on ${\mathbb{T}^{2}}$, a result which originally appeared in [@Hjorth].
We next establish an analogous result in the absence of ergodicity.
\[antichains:invariantactions\] Suppose that $X$ is a [Polish]{}space, $E$ is a projectively separable countable [Borel]{}equivalence relation on $X$, $\mu$ is an $E$-invariant [Borel]{}probability measure on $X$, and there is a free [Borel]{}action of a countable non-abelian free group on $X$ generating a subequivalence relation of $E$. Then for every countable non-abelian free group $\Gamma$, there is a [Borel]{}sequence $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ of free actions of $\Gamma$ on $X$, generating subequivalence relations $E_r$ of $E$, with the further property that $\sequence{E_r}[r \in {\mathbb{R}}]$ is an increasing sequence of relations which are pairwise incomparable under $\mu$-somewhere reducibility.
The proof is essentially the same as that of Theorem \[antichains:ergodicinvariantactions\]. One minor difference is that we use Proposition \[stratification:invariantactions\] in place of Proposition \[stratification:ergodicinvariantactions\] to obtain a [Borel]{}$\mu$-stratification $\sequence{\cdot_r}[r \in {\mathbb{R}}]$ of $E$ by free actions of $\Gamma$ on $X$. Another difference is that we use $R$ to denote the relation on ${\mathbb{R}}$ in which two real numbers $r$ and $s$ are related if $E_r$ is merely $\mu$-somewhere reducible to $E_s$. As $\mu$-somewhere reducibility is weaker than $\mu$-reducibility in the absence of ergodicity, we must be slightly more careful in establishing the analog of Lemma \[antichains:ergodicinvariantactions:sections\].
Every horizontal section of $R$ is countable.
Suppose, towards a contradiction, that there exists $t \in {\mathbb{R}}$ for which ${R^{t}}$ is uncountable. For each $s \in {R^{t}}$, fix a $\mu$-positive [Borel]{} set $B_s \subseteq X$ on which there is a [Borel]{}reduction $\phi_s {\colon}B_s \to X$ of $E_s$ to $E_t$. Then there exists $\epsilon > 0$ with $\mu(B_s) \ge \epsilon$ for uncountably many $s \in {R^{t}}$. As each $\phi_s$ is a homomorphism from ${(\intersection[r \in {\mathbb{R}}][E_r]) \upharpoonright B_s}$ to $E$, the $\mu$-nowhere hyperfiniteness of $\intersection[r \in {\mathbb{R}}][E_r]$ and the projective separability of $E$ ensure the existence of distinct $r, s \in {R^{t}}$ for which $\mu
(B_r), \mu(B_s) \ge \epsilon$ and ${d_{\mu}}(\phi_r, \phi_s) < \epsilon$. Then $\set{x \in B_r \intersection B_s}[\phi_r(x) = \phi_s(x)]$ is a $\mu$-positive [Borel]{}set on which $E_r$ and $E_s$ coincide, a contradiction.
One can now proceed exactly as in the proof of Theorem \[antichains:ergodicinvariantactions\] to obtain the desired sequence.
\[antichains:invariantactions:remark\] By Remark \[stratification:invariantactions:remark\], after throwing out an $E$-invariant $\mu$-null [Borel]{}set, the conclusion of Theorem \[antichains:invariantactions\] follows from the weaker assumption that for all $\mu$-positive [Borel]{}sets $B \subseteq X$, there is a [Borel]{}subequivalence relation $F$ of ${E \upharpoonright B}$ for which ${C_{{\mu \upharpoonright B}}(F)} > \mu(B)$.
Finally, we establish an analogous result in the absence of invariance.
\[antichains:actions\] Suppose that $X$ is a [Polish]{}space, $E$ is a projectively separable countable [Borel]{}equivalence relation on $X$, $\mu$ is a [Borel]{}probability measure on $X$, and there is a free [Borel]{}action of a countable non-abelian free group on $X$ generating a $\mu$-nowhere hyperfinite subequivalence relation of $E$. Then for every countable non-abelian free group $\Gamma$, there is a [Borel]{}sequence $\sequence
{\cdot_r}[r \in {\mathbb{R}}]$ of free actions of $\Gamma$ on $X$, generating subequivalence relations $E_r$ of $E$, with the further property that $\sequence{E_r}[r \in {\mathbb{R}}]$ is an increasing sequence of relations which are pairwise incomparable under $\mu$-somewhere reducibility.
This follows from the proof of Theorem \[antichains:invariantactions\], using Proposition \[stratification:actions\] in place of Proposition \[stratification:ergodicinvariantactions\].
\[antichains:actions:remark\] By Remark \[stratification:actions:remark\], after throwing out an $E$-invariant $\mu$-null [Borel]{}set, the conclusion of Proposition \[stratification:actions\] follows from the weaker assumption that for all $\mu$-positive [Borel]{}subsets $B \subseteq X$ such that ${\mu \upharpoonright B}$ is equivalent to an $E$-invariant [Borel]{}probability measure $\nu$, there is a [Borel]{}subequivalence relation $F$ of ${E \upharpoonright B}$ for which ${C_{{\nu \upharpoonright B}}(F)} > \nu(B)$.
We would like to thank the referee for his many useful suggestions.
[^1]: The authors were supported in part by FWF Grant P28153 and SFB Grant 878.
|
---
abstract: |
The dual basis of the canonical basis of the modified quantized enveloping algebra is studied, in particular for type $A$. The construction of a basis for the coordinate algebra of the $n\times
n$ quantum matrices is appropriate for the study the multiplicative property. It is shown that this basis is invariant under multiplication by certain quantum minors including the quantum determinant. Then a basis of quantum $SL(n)$ is obtained by setting the quantum determinant to one. This basis turns out to be equivalent to the dual canonical basis.
address: ' Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, P. R. China'
author:
- Hechun Zhang
title: On dual canonical bases
---
\[section\] \[Thm\][Definition]{} \[Thm\][Remark]{} \[Thm\][Lemma]{} \[Thm\][Corollary]{} \[Thm\][Proposition]{}
\[section\]
0.5cm
introduction
============
Throughout the paper, the base field is $K={\mathbb Q}(q)$, i.e., the field of quotients of polynomials in the indeterminate $q$ with rational coefficients. Let $A$ be an algebra over $K$. Two elements $b, b^\prime\in A$ are called [*equivalent*]{} (denoted by $b\sim b^\prime$) if there exists $m\in{\mathbb Z}$ such that $b^\prime=q^m b$. Two elements $b, b^\prime$ are called [*$q$-commuting*]{} if $bb^\prime\sim b^\prime b$.
Let $ g$ be the Kac-Moody algebra associated to a $n\times n$ symmetrizable Cartan matrix $A$. Let $U_q(g)$ be the quantized enveloping algebra associated to $g$, with its two usual subalgebras $U_q(n^+)$ and $U_q(n^-)$ (see section 2 for details). The dual basis of the canonical basis of $U_q(n^-)$ has been widely studied in literature. In [@k], a conjecture posed by Berenstein and Zelevinsky is stated as follows: Two elements $b_1,
b_2$ of the dual canonical basis are $q$-commuting with each other, if and only if $b_1b_2\sim b$ for some $b$ in the dual canonical basis. This property of basis is called the [*multiplicative property*]{}. By use of the Hall algebra technique, the multiplicative property of the dual canonical basis of $U_q(n^+)$ is studied in [@r]. In [@lec], counter-examples are given for the Berenstein-Zelevinsky conjecture by finding some so-called imaginary vectors. There are many connections between the irreducible representations of Hecke algebras of $A$ type and the multiplicative property of the dual canonical basis, see [@lec] and [@lnt].
Let $L(\lambda)$ be an irreducible highest weight module for $U_q(g)$ and let $L^*(\lambda)$ be its graded dual. In [@lu1], Lusztig constructed a canonical basis of the tensor product $U(\lambda, \mu):=L(\lambda)\otimes L^*(\mu)$ which can be lifted to a canonical basis $\tilde{B}$ of the so-called modified quantized enveloping algebra $\tilde{U_q}(g)$. In the present paper we will show that the module $L(\lambda)\otimes L^*(\mu)$ is absolutely indecomposable if the Kac-Moody algebra $g$ is of affine or indefinite type. Next, we focus on the case of type $A$. By constructing a basis of the coordinate algebra $O_q(M(n))$ of the $n\times n$ quantum matrices, we get a basis of $O_q(SL(n))$ which turns out to be equivalent to the dual canonical basis. A pleasant aspect of this construction is that it is appropriate to study the multiplicative property of the basis.
[**Acknowledgement**]{}The major part of the present work was done during the author visited University of Amsterdam. The author would like to thank Professor T. Koornwinder for his hospitality. The author wishes to thank J. Du, H. P. Jakobsen, E. M. Opdam and J. V. Stokman for valuable discussions.
Kashiwara’s construction
========================
Let $ g$ be the Kac-Moody algebra associated to a $n\times n$ symmetrizable Cartan matrix $A$. One can choose a bilinear form such that the integral weight lattice is an even integral lattice. Let $\Pi=\{\alpha_1,\alpha_2,\cdots,\alpha_n\}$ and $\Pi^v=\{\alpha_1^v,\alpha_2^v,\cdots,\alpha_n^v\}$ be the set of simple roots and the set of simple coroots respectively. Let $U_q( g)$ be the quantized enveloping algebra associated to $ g$ with generators $E_1, \cdots, E_n$, $F_1,\cdots, F_n$, $K_1,
K_1^{-1},\cdots, K_n, K_n^{-1}$ with the usual defining relations (see e.g. [@k1]) by replacing $q$ by $q^2$ because we do not want to use the square root of $q$ later.
Let $U_q(n^+)$ (resp. $U_q(n^-)$) be the subalgebra generated by $E_1, \cdots, E_n$ (resp. $F_1,\cdots, F_n$). For any dominant weight $\lambda$, denote by $L(\lambda)$ the irreducible highest weight module over $U_q(g)$ with the highest weight $\lambda$. Denote by $L^*(\lambda)$ the graded dual of $L(\lambda)$ which is an irreducible lowest weight module with the lowest weight $-\lambda$. Let $-$ be the automorphism of the algebra $U_q(g)$ given by $$\bar{q}=q^{-1}, \bar{E_i}=E_i, \bar{F_i}=F_i,
\bar{K_i}=K_i^{-1}$$ for all $i$. Let $v_\lambda$ (resp. $v_\mu^*$) be a highest weight vector of $L(\lambda)$ (resp. the lowest weight vector of $L^*(\mu)$). Denote also by $-$ the linear automorphism of the module $L(\lambda)$ and of the module $L^*(\mu)$ given by $$\overline{pv_\lambda}=\bar{p}v_\lambda,
\overline{pv_\mu^*}=\bar{p}v_\mu^*$$ for $p\in U_q(g)$.
Although we use $-$ to denote several different automorphism of various spaces. One can identify the meaning of the $-$ from the context.
In [@lu1], Lusztig constructed a canonical basis of the tensor product $U(\lambda, \mu):=L(\lambda)\otimes L^*(\mu)$ which can be lifted to a canonical basis $\tilde{B}$ of the modified quantized enveloping algebra $\tilde{U_q}(g)$.
We will not go in detail about the canonical basis of the module $U(\lambda,\mu)$. However, we would like to show one remarkable fact about the module $U(\lambda, \mu)$. It is known that if $g$ is of finite type, $U(\lambda, \mu)$ is of finite dimensional and is indecomposable if and only if one of $\lambda$ and $\mu$ is zero. However, if $g$ is of affine or indefinite type, the situation changes dramatically.
If $g$ is of affine or indefinite type, then $$End_{U_q(g)}U(\lambda, \mu)\cong {\mathbb Q}(q).$$ Hence, $U(\lambda, \mu)$ is absolutely indecomposable.
Clearly, if $\lambda$ or $\mu$ is trivial, then $U(\lambda,\mu)$ is a lowest weight module or a highest weight module and the theorem holds. Hence, we may assume that both $\lambda$ and $\mu$ are nontrivial.
It is known that $U(\lambda, \mu)$ is a cyclic module and is generated by $v_\lambda\otimes
v_\mu^*$. For any $\psi\in End_{U_q(g)}U(\lambda, \mu)$, then $\psi
(v_\lambda\otimes v_\mu^*) =u(v_\lambda\otimes v_\mu^*)\in
U(\lambda,\mu)_{\lambda-\mu}$ for some $u\in U_q(g)$ which is of weight zero. If $u(v_\lambda\otimes v_\mu^*)$ is not a multiple of $v_\lambda\otimes v_\mu^*$, then $$u(v_\lambda\otimes v_\mu^*)=s(v_\lambda\otimes
v_\mu^*)+\sum_iu_iv_\lambda\otimes w_iv_\mu^*$$ where $s\in{\mathbb Q}(q), u_i\in U_q(n^-), w_i\in U_q(n^+)$, for all $i$, and the set $\{w_iv_\mu^*\}_i$ is linearly independent. Choose $w_kv_\lambda^*$ such that its weight is maximal among all of the weights of $w_iv_\lambda^*$ for all $i$. Assume that $u_kv_\lambda\in L(\lambda)_\Lambda$, where $\Lambda$ must be smaller than $\lambda$.
1\. If the Cartan matrix $A$ is of indefinite type, then there exists $\alpha^v_i$ such that $<\lambda-\Lambda, \alpha_i^v><0$, i.e. $<\lambda,\alpha^v_i><<\Lambda, \alpha^v_i>$ and so $F_i^{<\lambda,\alpha^v_i>+1}u_kv_\lambda\ne 0$. However, $F_i^{<\lambda,\alpha^v_i>+1}u(v_\lambda\otimes
v_\mu^*)=\psi(F_i^{<\lambda,\alpha^v_i>+1}(v_\lambda\otimes
v_\mu^*))=0$. On the other hand, $F_i^{<\lambda,\alpha^v_i>+1}u(v_\lambda\otimes v_\mu^*)=\sum_{m,
j}c_{ij}^{(m)}F_i^{<\lambda,\alpha^v_i>+1-m}u_jv_\lambda\otimes
F_i^mw_jv_\mu^*$, where $c_{ij}^{(m)}\in{\mathbb Q}(q)$. One can see easily that $c_{ik}^{(0)}=1$. Hence, $F_i^{<\lambda,\alpha^v_i>+1}u_kv_\lambda=0$. Contradiction!
2\. Now, we may assume that the Cartan matrix $A$ is of affine type. If there exists $\alpha_i^v$ such that $<\lambda-\Lambda,\alpha^v_i><0$, then we can prove in the same way as above. If $<\lambda-\Lambda,\alpha^v_i>\ge 0$ for all $i$, then we must have $<\lambda-\Lambda,\alpha^v_i>=0$ for all $i$. As there exists $E_i$ such that $E_iu_kv_\lambda\ne 0$, we have again that $F_i^{<\lambda,\alpha^v_i>+1}u_kv_\lambda\ne 0$.
Therefore, $u(v_\lambda\otimes v_\mu^*)$ is a multiple of $v_\lambda\otimes v_\mu^*$ and so $\psi$ is a scalar endomorphism. Moreover, $U(\lambda,\mu)$ is absolutely indecomposable.
Let $U^{{\mathbb Z}}(g)$ be the integral form of the quantized enveloping algebra which is a ${\mathbb Z}[q, q^{-1}]$-subalgebra of the quantized enveloping algebra $U_q(g)$ generated by the divided powers $E_i^{(s)}:=E_i^s/[s]!,
F_i^{(s)}:=F_i^s/[s]!, K_i, K_i^{-1}$ for all $i$. The quantum integer is definied by $[s]=\frac{q^{2s}-q^{-2s}}{q^2-q^{-2}}$, (one may refer to [@jan] for terminologies and notations).
Denote by $U_q(g)^*$ the linear dual of the algebra $U_q(g)$. Since $U_q(g)$ is a $U_q(g)$ bi-module, $U_q(g)^*$ has an induced $U_q(g)$ bi-module structure. Let $$\begin{aligned}
A_q(g):&=&\{f\in
U_q(g)^*|\text{ there exists }l\ge 0\text{ such that }\\\nonumber
E_{i_1}\cdots E_{i_l}f&=& fF_{i_1}\cdots F_{i_l}=0\text{ for any
}i_1,\cdots,i_l \}.\end{aligned}$$
The quantum Peter-Weyl theorem was proved in [@k]
As $U_q(g)$ bi-modules $$A_q(g)\cong \oplus_{\lambda\in P} L(\lambda)\otimes L^*(\lambda)$$ where $u\otimes v\in L(\lambda)\otimes L^*(\lambda)$ viewed as a linear function on $U_q(g)$ as follows: $$(u\otimes v)(p)=<up, v>,\text{ for }p\in U_q(g),$$ where $L(\lambda)$ and $L^*(\lambda)$ are viewed as right $U_q(g)$ module and left $U_q(g)$ module respectively.
Let $A$ be the subring of ${\mathbb Q}(q)$ consists of the rational functions of $q$ which are regular at $q=0$. Let $ -$ be the ring endmorphism of ${\mathbb Q}(q)$ sending $q$ to $q^{-1}$.
Let $M$ be an integral $U_q(g)$ module. Then $$M=\oplus_{\lambda} F_i^{(n)}(KerE_i\cap M_\lambda).$$ We define the lower Kashiwara operators $e_i, f_i$ of $M$ by $$f_i(F_i^{(n)}u)=F_i^{(n+1)}u\text{ and
}e_i(F_i^{(n)}u)=F_i^{(n-1)}u$$ for $u\in KerE_i\cap M_{\lambda}$.
A pair $(L, B)$ is called a lower crystal base of $M$ if it satisfies the following conditions:
1. $L$ is a free sub-$A$-module of $M$ such that $M\cong
{\mathbb Q}(q)\otimes_A L$.
2. $B$ is a base of the ${\mathbb Q}$-vector space $L/qL$.
3. $e_iL\subset L$ and $f_iL\subset L$ for any $i$.
4. $e_iB\subset B\cup \{0\}$ and $f_iB\subset B\cup \{0\}$.
5. $L=\oplus_{\lambda\in P} L_{\lambda}$ and $B=\cup_{\lambda\in P} B_\lambda$, where $L_{\lambda}=L\cap
M_{\lambda}$, $B_\lambda=B\cap L_\lambda/qL_\lambda$.
6. For any $b, b^\prime\in B$, $b^\prime=f_ib$ if and only if $b=e_ib^\prime$.
The upper Kashiwara operators $e_i^\prime$ and $f_i^\prime$ are define as follows: for $u\in KerE_i\cap M_\lambda$ and $0\le n\le
<\lambda, \alpha^v>$, $$e_i^\prime(F_i^{(n)}u)=\frac{[<\lambda,
\alpha^v>-n+1]}{[n]}F_i^{(n-1)}u,$$ and $$f_i^\prime(F_i^{(n)}u)=\frac{[n+1]}{[<\lambda,
\alpha^v>-n]}F_i^{(n+1)}u.$$
We say that $(L, B)$ is an upper crystal base if $(L, B)$ satisfies the the conditions in the definition of lower crystal base with $e_i^\prime, f_i^\prime $ instead of $e_i, f_i$.
For $\lambda\in P$, we define $\psi_M\in AutM$ by $$\psi_M(u)=q^{-2(\lambda, \lambda)}u$$ for $u\in M_\lambda$. It is known that $\psi_M^{-1}e_i^\prime\psi_M$ (resp. $\psi_M^{-1}f_i^\prime\psi_M$) coincides with $e_i$ (resp. $f_i$) on $L/qL$.
In [@ka], Kashiwara proved that
$(L, B)$ is a lower crystal base if and only if $\psi_M(L, B)$ is an upper crystal base.
Let $\cal L$$(\lambda)$ be the upper crystal lattice which is the smallest $A$ submodule of $L(\lambda)$ containing $v_\lambda$ and is stable under the action of upper Kashiwara operators. Similarly, let $\cal L$$^*(\lambda)$ be the upper crystal lattice which is the smallest $A$ submodule of $L^*(\lambda)$ containing $v_\lambda^*$ and is stable under the action of upper Kashiwara operators. Set $${\cal L}(A_q(g)):=\oplus_{\lambda\in P_+}{{\cal L}}(\lambda)
\otimes {{\cal L}}^*(\lambda).$$ Define that $$<\bar{u}, p>=
\overline{<u,\bar{p}>},$$ then one can check that $\overline{u\otimes v}=\bar{u}\otimes \bar{v}$ for $u\in
L(\lambda)$ and $v\in L^*(\lambda)$. Hence $$\overline{{\cal L}(A_q(g))}=\oplus_{\lambda\in P_+}\overline{\cal L}(\lambda)
\otimes \overline{\cal L}^*(\lambda).$$
Let $$A^{\mathbb Z}_q(g)=\{f\in A_q(g)|<f,U^{\mathbb Z}(g)>\subset
{\mathbb Z}[q,q^{-1}]\}$$
Let $u\otimes v\in L(\lambda)\otimes L^*(\lambda)$, where $u$ (resp. $v$) is a weight vector of weight $\lambda_l$ (resp. $\lambda_r$). Then $u\otimes v$ is called a weight vector with left weight $\lambda_l$ and right weight $\lambda_r$. An element in $A_q(g)$ is called a refined weight vector if it is a linear combination of the elements $u\otimes v$ with the same left and right weights.
Let us recall the definition of balanced triple. Let $V$ be a vector space over ${\mathbb Q}(q)$, a $B$-lattice of $V$ is a $B$-submodule $M$ of $V$ such that $V\cong {\mathbb Q}(q)\otimes_B
M$. Let $V_{\mathbb Z}$ be a ${\mathbb Z}[q,q^{-1}]$-lattice of $V$, $L$ an $A$-lattice of $V$, and $\overline{L}$ an $\overline{A}$-lattice of $V$. In [@ka], it was proved that
Set $E=V_{\mathbb Z}\cap L\cap \overline{L}$. Then the following conditions are equivalent.
1. $E\longrightarrow V_{\mathbb Z}\cap L/V_{\mathbb Z}\cap qL$ is an isomorphism.
2. $E\longrightarrow V_{\mathbb Z}\cap \overline{L}/ V_{\mathbb
Z}\cap q^{-1}\overline{L}$ is an isomorphism.
3. $V_{\mathbb Z}\cap qL\oplus V_{\mathbb Z}\cap
\overline{L}\longrightarrow V_{\mathbb Z}$ is an isomorphism.
4. $A\otimes E\longrightarrow L$, $\overline{A}\otimes
E\longrightarrow \overline{L}$, ${\mathbb
Z}[q,q^{-1}]\otimes_{\mathbb Z} E\longrightarrow V_{\mathbb Z}$, ${\mathbb Q}(q)\otimes_{\mathbb Z} E\longrightarrow V$ are isomorphisms.
We call $(L, \overline{L},V_{\mathbb Z})$ balanced if these equivalent conditions are satisfied. Let us denote by $G$ the inverse of the isomorphism $E\longrightarrow V_{\mathbb Z}\cap
\overline{L}/ V_{\mathbb Z}\cap q^{-1}\overline{L}$. If $B$ is a base of $ V_{\mathbb Z}\cap \overline{L}/ V_{\mathbb Z}\cap
q^{-1}\overline{L}$, then $\{G(b)| b\in B\}$ is a base of $V$.
In [@k], it was proved that $(A_q^{\mathbb Z}(g), {\cal
L}(A_q(g)), \overline{{\cal L}(A_q(g))})$ is a balanced triple. Hence there is a $\mathbb Z$ basis $B^\prime$ of $$(A_q^{\mathbb
Z}(g)\cap {\cal L}(A_q(g))\cap \overline{{\cal L}(A_q(g))}).$$
In [@k1], it was shown that $B^\prime$ is the dual basis of the canonical basis of the modified enveloping algebra $\tilde{U}_q(g)$ if $g$ if of finite type.
In the following, we always assume that $g$ is of finite type. We fix a reduced expression in the longest element of the Weyl group. Let $F_{\beta_1},F_{\beta_2},\cdots,F_{\beta_N}$ be the ordered root vectors given defined according to the chosen reduced expression of the longest element in the Weyl group, where $N$ is the length of the longest element in the Weyl group. For any $I=(i_1,i_2,\cdots,i_N)\in{\mathbb Z}_+^N$, denote by $F^I$ the monomial $F_{\beta_1}^{(i_1)}F_{\beta_2}^{(i_2)}\cdots
F_{\beta_N}^{(i_N)}$ which form a PBW type basis of the subalgebra $U_q(n^-)$. The monomial $E^I$ is defined similarly which form a PBW type basis of the subalgebra $U_q(n^+)$.
Let $B^-$ and $B^+$ be the canonical basis of $U_q(n^-)$ and $U_q(n^+)$ respectively. For any dominant weight $\lambda$, denote by $$B^-_\lambda=\{b\in B^-|bv_\lambda\ne 0\}$$ and $$B^+_\lambda=\{b^\prime\in B^+|b^\prime v_\lambda^*\ne 0\}.$$ Note that each dual canonical basis element is a refined weight vector. Hence, we only need to consider the homogeneous part ${\cal L}(\lambda)_\mu\otimes {\cal L^*}(\lambda)_\gamma
\cap\overline{{\cal L}(\lambda)_\mu\otimes {\cal
L^*}(\lambda)_\gamma}
\cap A_q^{\mathbb Z}(g)$.
It is well-known that any canonical basis element $b$ in $B^-$ is of the form $$b=F^I+\sum_{I^\prime}a_{I,I^\prime}F^{I^\prime}$$ where the coefficients $a_{I, I^\prime}\in q{\mathbb Z}[q]$ and the element $b$ is $-$ invariant. $F^I$ is called the leading term of $b$. The canonical basis elements in $B^+$ have the similar form.
Let $$C_\lambda^-=\{F^I|F^I\text{ is the leading term of an
element }b\in B^-_\lambda\}$$ and let $$C_\lambda^+=\{E^I|E^I\text{ is the leading term of an element
}b^\prime\in B^+_\lambda\}.$$ Then $C_\lambda^-v_\lambda$ (resp. $C_\lambda^+v_\lambda^*$) is the PBW-basis of $L(\lambda)$ (resp. $L(\lambda)^*$) with an order given by the chosen reduced expression of the longest element in the Weyl group.
We order the PBW type basis $\bigcup_{\lambda} C_\lambda^-\otimes
C_\lambda^+$ by lexicographic ordering.
\[dual\] The basis $B^\prime$ is characterized by the following two conditions.
1. $b^\prime=F^Iv_\lambda\otimes E^{I^\prime}v_\lambda^*+\sum_{I_k,
I_K^\prime} a_{I, I^\prime}^{I_k,I_k^\prime}
F^{I_k}v_\lambda\otimes E^{I_k^\prime}v_\lambda^*$ where $a_{I,
I^\prime}^{I_k,I_k^\prime}\in q{\mathbb Z}[q]$ and $a_{I,
I^\prime}^{I_k,I_k^\prime}\ne 0$ only if $(I_k, I_k^\prime)\le (I,
I^\prime)$, for any $b^\prime\in B^\prime$.
2. $\overline{b^\prime}=b^\prime$.
Clearly, each element $bv_\lambda\otimes b^\prime v_\lambda^*$ satisfies the two conditions. The uniqueness can be proved in the same way as in [@d1] .
The following result was proved in [@k].
Let $x$ and $y$ be refined weight vectors of weights $(\lambda_l,\lambda_r)$ and $(\mu_l,\mu_r)$ respectively. Then $$\overline{xy}=q^{2(\lambda_r,\mu_r)-2(\lambda_l,\mu_l)}\bar{y}\bar{x}.$$
By using the above Proposition, one can easily verify that
The mapping $$\begin{aligned}
\phi: A_q(g) &\longrightarrow & A_q(g),\\\nonumber
q &\mapsto & q^{-1},\\\nonumber
u\otimes v &\mapsto &
q^{((\lambda_l,\lambda_l)-(\lambda_r,\lambda_r))}\bar{u}\otimes\bar{v}.\end{aligned}$$ if $u\otimes v$ is of the left weight $\lambda_l$ and right weight $\lambda_r$, extends to an algebra anti-automorphism of the algebra $A_q(g)$ over $\mathbb Q$.
Let $b^\prime\in B^\prime$ with weights $(\lambda_l,\lambda_r)$. Then the element $b=q^{\frac{1}{2}((\lambda_l,\lambda_l)-(\lambda_r,\lambda_r))}b^\prime$ is invariant under the anti automorphism $\phi$. Let $$L^*=\{b|b^\prime\in B^\prime\}.$$ Then $L^*$ is also a $\mathbb Z$$[q,q^{-1}]$ basis of $A^{\mathbb
Z}(g)$.
It is clearly that the multiplicative properties of $B^\prime$ and $L^*$ are the same.
Let $b_1,b_2\in L^*$. Assume that $b_1b_2\sim b$ for some $b\in L^*$. Then $b_1b_2\sim b_2b_1$.
Assume that $b_1b_2=q^ab$ for some $a\in\mathbb Z$. Applying the anti automorphism $\phi$, we deduce that $b_2b_1=q^{-a}b$. Hence, $b_1b_2=q^{2a}b_2b_1$.
The construction of the basis of $O_q(M(n))$
============================================
The coordinate algebra $O_{q}(M(n))$ of the quantum matrix is an associative algebra, generated by elements $Z_{ij},i,j=1,2,\cdots,n$, subject to the following defining relations: $$\begin{aligned}
\label{relations}Z_{ij}Z_{ik}&=&q^2Z_{ik}Z_{ij} \text{ if } j<k,\\
Z_{ij}Z_{kj}&=&q^2Z_{kj}Z_{ij} \text{ if }i<k,\\
Z_{ij}Z_{st}&=&Z_{st}Z_{ij}\text{ if } i>s, j<t,\\
Z_{ij}Z_{st}&=&Z_{st}Z_{ij}+(q^2-q^{-2})Z_{it}Z_{sj} \text{ if } i<s, j<t.\end{aligned}$$ For any matrix $A=(a_{ij})_{i,j=1}^n\in M_n({\mathbb Z}_+)$ ( ${\mathbb Z}_+=\{0,1,\cdots\}$) we define a monomial $Z^A$ by $$Z^A=\Pi_{i,j=1}^nZ_{ij}^{a_{ij}},$$ where the factors are arranged in the lexicographic order on $I(n)=\{(i,j)\mid i,j=1,\dots,n\}$. It is well known that the set $\{Z^A|A\in M_n({\mathbb Z}_+\}$ is a basis of the algebra $O_q(M(n))$.
From the defining relations (\[relations\]) of the algebra $O_q(M(n))$, it is easy to show the following lemma.
\(1) The mapping $$\begin{aligned}
^-:Z_{ij}&\mapsto &Z_{ij}\\\nonumber
q&\mapsto &q^{-1}\end{aligned}$$ extends to an algebra anti-automorphism of the algebra $O_{q}(M(n))$ as an algebra over ${\mathbb Q}$.
\(2) The mapping $$\sigma: Z_{ij}\mapsto Z_{ji}$$ extends to an algebra automorphism of the algebra $O_{q}(M(n))$ as an algebra over $K={\mathbb Q}(q)$.
For any $A=(a_{ij})_{n\times n}\in M_n({\mathbb Z}_+)$. Let $$ro(A)=(\sum_j a_{1j},\cdots,\sum_ja_{nj})=(r_1,r_2,\cdots,r_n)$$ which is called the row sum of $A$ and $$co(A)=(\sum_j a_{j1},\cdots,\sum_j a_{jn})=(c_1,c_2,\cdots,c_n)$$ which is called the column sum of $A$.
For any matrix $A=(a_{ij})_{i,j=1}^n\in M_n({\mathbb Z}_+)$, a monomial having the factors of $Z^A$ in arbitrary order. Then its expansion in terms of monomials $Z^B$ only involves terms where $ro(B)=ro(A)$ and $co(B)=co(A)$. Let $Pr(A, s,t)=\sum_{i\le s,
j\le t}a_{ij}$. Then $Pr(A,s,t)\ge Pr(B,s,t)$ for any $s,t\le n$ and matrix $B$ appeared in the expansion considered above.
From the defining relations (\[relations\]) of the algebra $O_q(M(n))$, we have
$$\label{bar}\overline{Z^A}=E(A)Z^A+\sum_{B}c_B(A)Z^B,$$
where $$E(A)=q^{-2(\sum_i\sum_{j>k}a_{ij}a_{ik}+\sum_i\sum_{j>k}a_{ji}a_{ki})}$$ and $B<A$, $ro(B)=ro(A)$, $co(B)=co(A)$, $c_B(A)\in{\mathbb Z}[q,q^{-1}]$, $\le$ is the lexicographic ordering.
For a pair of vectors $R,C\in{\mathbb Z}_+^n$, denote by $M(R,C)$ the subspace of $ O_q(M(n))$ spanned by $Z^A$ with $ro(A)=R,$ and $co(A)=C$. Note that $M(R,C)$ is $^-$ invariant and $ O$$_q(M(n))=\oplus_{R,C} M(R,C)$.
Let $D(A)=q^{-\sum_i \sum_{j>k}a_{ij}a_{ik}-\sum_i
\sum_{j>k}a_{ji}a_{ki}}$ and let $Z(A)=D(A)Z^A.$ Set $$L^*=\oplus_{A\in M_n({\mathbb Z}_+)}{\mathbb Z}[q]Z(A).$$
\[basis\] There is a unique basis $B^*=\{b(A)|A\in M_n({\mathbb Z}_+)\}$ of $ L^*$ determined by the following conditions:
1. $\overline{b(A)}=b(A)$ for all $A$.
2. $b(A)=Z(A)+\sum_{B<A} h_B(A)Z(B)$ where $h_B(A)\in q{\mathbb Z}$$[q]$ and $ro(B)=ro(A), co(B)=co(A)$.
We rewrite the equation (\[bar\]) in terms of $Z(A)$, then $$\overline{Z(A)}=\sum_Ba_{AB}Z(B),$$ where $a_{AA}=1$, $a_{AB}\in{\mathbb Z}[q,q^{-1}]$ and $a_{AB}=0$ unless $B\le A$, where $\le$ is the lexicographic ordering. By Theorem 1.2 of [@d1], there is an IC-basis with respect to the triple $(\{Z^A\mid A\in M_n({\mathbb Z}_+)\},
^-, \le)$ determined by the relation stated in the context of the theorem.
The quantum determinant ${\det}_q$ is defined as follows:
$${\det}_q=
\Sigma_{\sigma\in S_n}(-q^2)^{l(\sigma)}Z_{1\sigma(1)}Z_{2\sigma(2)} \cdots Z_{n\sigma(n)}.$$
It is known that $det_q$ is a central element of the algebra $ O_q(M(n))$.
For later reference we now introduce some terminology. Let $m\le n$ be a positive integer. Given any two subsets $I=\{i_1,i_2,\cdots,i_m\}$ and $J=\{j_1,j_2,\cdots,j_m\}$ of $\{1,2,\cdots,n\}$, each having cardinality $m$, it is clear that the subalgebra of $ O$$_q(M(n))$ generated by the elements $Z_{i_rj_s}$ with $r,s=1,2,\cdots,m$, is isomorphic to $ O$$_q(M(m))$, so we can talk about its determinant. Such a determinant is called a quantum minor, and will be denoted by ${\det}_q(I,J)$.
Let $I, J$ be two subsets of $\{1,2,\cdots,n\}$ with the same cardinality. Obviously, the dual canonical basis of the subalgebra generated by $Z_{ij}$ for $i\in I$ $j\in J$ is a subset of the basis $B^*$ of the algebra $O_q(M(n))$. More generally, If $(u,v)\le (s,t)$, then the subalgebra $O_q(M(n))^{(u,v)}_{(s,t)}$ generated by $Z_{i,j}$, for $(u,v)\le (i,j)\le (s,t)$, is $^-$ invariant and one can construct a basis analogous to the construction of the basis considered in Theorem \[basis\], and obviously the resulting basis of $O_q(M(n))^{(u,v)}_{(s,t)}$ is a subset of the basis $B^*$.
The quantum determinant $det_q$ is an element of the basis $B^*$. Furthermore, any quantum minor is also an element of the dual canonical basis.
We only need to show that $det_q$ is $^-$ invariant. It is well known that the center of the algebra $O_q(M(n))$ is generated by the quantum determinant [@nmy]. Note that $$\begin{aligned}
\overline{det_q}Z_{ij}&=&\overline{\overline{Z_{ij}}det_q}=\overline{det_qZ_{ij}}\\\nonumber
&=&Z_{ij}\overline{det_q},\end{aligned}$$ for any $i,j$. Hence, $\overline{det_q}$ is a polynomial of $det_q$. Therefore, $$\overline{det_q}=det_q$$ by comparing the leading terms.
\[transpose\]The basis $B^*$ is $\sigma$ invariant. More precisely,
$$\sigma(b(A))=b(A^T),$$ for all $A\in M_n({\mathbb Z}_+)$, where $A^T$ is the transposition of $A$.
Let $ b(A)$ be an element of the dual canonical basis $B^*$ of the form given in Theorem \[basis\] (2). Then it follows that all of the matrices $B$ appearing in the expansion of $b(A)$ are obtained from $A$ by a sequence of $2\times 2$ submatrix transformations of the following form:
$$\begin{pmatrix}a_{ij}&a_{it}\\a_{sj}&a_{st}\end{pmatrix}
\longrightarrow
\begin{pmatrix}a_{ij}-1&a_{it}+1\\a_{sj}+1&a_{st}-1\end{pmatrix},$$
if both $a_{ij}$ and $a_{st}$ are positive. Hence $B^T$ can be obtained from $A^T$ by a sequence of the submatrix transformations of the form:
$$\begin{pmatrix}a_{ji}&a_{ti}\\a_{js}&a_{ts}\end{pmatrix}
\longrightarrow
\begin{pmatrix}a_{ji}-1&a_{ti}+1\\a_{js}+1&a_{ts}-1\end{pmatrix}.$$
Especially, $B^T\le A^T$. Note that the monomials $Z^{B^T}$ and $\sigma(Z^B)$ have the same factors but could be in different order. However, two generators $Z_{ij}$ and $Z_{st}$ appear in the monomials but in different order must satisfy the third relation in (\[relations\]). Hence, $Z^{B^T}=\sigma(Z^B)$ $$\sigma(b(A))=Z(A^T)+\sum_{B}h_B(A)Z(B^T)$$ with $h_B(A)\in q{\mathbb Z}[q]$. Clearly,
$$\overline{\sigma(b(A)}=\sigma(b(A))$$ since $\sigma$ and $^-$ commute with each other.
Denote by $I_n$ the $n\times n$ identity matrix.
For any $A\in M_n({\mathbb Z}_+)$, $$Z(A)det_q=Z(A+I_n)\quad \bmod q L^*.$$
For $i<s, j<t$, we have
$$Z_{st}^mZ_{ij}=Z_{ij}Z_{st}^m+(q^{2-4m}-q^2)Z_{it}Z_{sj}Z_{st}^{m-1}.$$ Recall that $${\det}_q=
\Sigma_{\sigma\in S_n}(-1)^{l(\sigma)}q^{2l(\sigma)}
Z_{1\sigma(1)}Z_{2\sigma(2)} \cdots Z_{n\sigma(n)}.$$ When we compute $Z(A)det_q$, we only have to deal with those coefficients of the form $q^{-2a}$ with $a$ a positive integer. Assume that $$Z(A)det_q=\sum a_BZ(B).$$ Clearly, $a_B\in{\mathbb Z}[q,q^{-1}]$ and the leading term is $Z(A+I_n)$ and those matrix $B$ appeared in the expression has at least one nonzero entry in each row and each column. We need to compute $Z(A)(-1)^{l(\sigma)}q^{2l(\sigma)}
Z_{1\sigma(1)}Z_{2\sigma(2)} \cdots Z_{n\sigma(n)},$ for all $\sigma\in S_n$. From the expression of the quantum determinant we see that there are four possibilities to produce coefficients of the form $q^{-2a}$ with $a$ a positive integer.
Case 1. $Z_{st}^mZ_{sj}=q^{-2m}Z_{sj}Z_{st}^m$ where $t>j$ but no $Z_{it}$ behind. Then $q^{2m}$ will be absorbed by $D(B)$ where $Z(B)$ is the resulted term.
Case 2. $Z_{st}^mZ_{it}=q^{-2m}Z_{it}Z_{st}^m$ where $s>i$ but no $Z_{sj}$ appeared before. Then $q^{2m}$ will be absorbed by $D(B)$ where $Z(B)$ is the resulted term.
Case 3. Both $Z_{st}^mZ_{sj}=q^{-2m}Z_{sj}Z_{st}^m$ where $t>j$ and $Z_{st}^mZ_{it}=q^{-2m}Z_{it}Z_{st}^m$ where $s>i$ happened. Then we get $q^{-4m}$. However, we will see that it will be cancelled by a term in the next case. To this end, we need to remember that the terms we are dealing with are from $Z(A) Z_{1\sigma(1)}Z_{2\sigma(2)} \cdots Z_{n\sigma(n)}.$ Note that $l(\sigma (jt))=l(\sigma)-1$.
Case 4. $Z_{st}^mZ_{ij}= Z_{ij}Z_{st}^m+(q^{2-4m}-q^2)Z_{it}Z_{sj}Z_{st}^{m-1}$ where $s>i, t>j$. Then the coefficient $q^{2-4m}$ will be cancelled by a term in case 3.
Hence, the coefficients $a_B$ are all in $q{\mathbb Z}[q]$ except $a_A$ which is $1$.
The following proposition follows directly from the above lemma.
The basis $B^*$ is invariant under the multiplication of $det_q$. More precisely, $$b(A)det_q=b(A+I_n)$$ for all $A\in M_n({\mathbb Z}_+)$.
By using this proposition, we can determine $b(A)$, if $A$ is a diagonal matrix. Let $A=diag(a_1, a_2,\cdots,a_n)$. We may assume that $a_1\le a_2\le \cdots \le a_n$ without loss of generality. Then
$$b(A)=\Pi_{i=1}^n det_{q, i}^{a_i-a_{i-1}}$$ where $det_{q,i}$ is the quantum determinant of the subalgebra generated by $Z_{st}$ for $s,t=i,\cdots,n$, and where we put $a_0=0$.
some subalgebras
================
In this section, we study the multiplicative property of the basis $B^*$. Similar to the proof of Proposition 2.9, we get
Let $b_1, b_2\in B^*$. If $b_1b_2\sim b$ for some $b\in B^*$, then $b_1b_2\sim b_2b_1$.
Divide the matrix by a broken-line $\xi$ which consists lines determined by the equations $ax+by=m$ for $a,b\in {\mathbb Z}_+$ and $m\in\mathbb N$ (each line has non-positive slope). Recall that $I(n)=\{(i,j)\mid i,j=1,\dots,n\}$. Let $$I_1=\{(x,y)\in I(n)\mid (x,y)\text{ is in the left upper side of the broken line } \xi\},$$ and let $I_2$ be the complement of $I_1$ in $I(n)$.
Let $O_i$ be the subalgebra of $O_q(M(n))$ generated by $Z_{xy}$ for $(x,y)\in I_i$, $i=1,2$. One can easily see that $O_i$ is determined by the generators $Z_{xy}$ and relations (\[relations\]). Hence, the algebra $O_i$ is closed under the bar action and therefore there is a basis $B_i^*$ of the sub-lattice $L_i^*$ of the ${\mathbb Z}[q]$-lattice $L^*$ spanned by $\{Z(A)\mid A=(a_{xy})\in M_n({\mathbb Z}_+), a_{xy}=0 \text{
if }(x,y)\in I_{3-i}\}$. Clearly, $B^*_i$ is a subset of $B^*$ consists of those $b(A)$ for $A=(a_{xy})\in M_n({\mathbb Z}_+),
a_{xy}=0 \text{ if }(x,y)\in I_{3-i}$.
Write $$A=A^+ +A^-,$$ Where the entries of $A^+$ in the left upper side of the broken line $\xi$ are zero and the entries of $A^-$ in the right lower side (including the broken line $\xi$) are zero. Then
$b(A)\sim b(A^+)b(A^-)$ if and only if $ b(A^+)b(A^-)\sim b(A^-)b(A^+)$
If $b(A)=q^ab(A^+)b(A^-)$ for some integer $a$, then $ b(A^-)b(A^+)\sim b(A^+)b(A^-)$ by the above lemma.
For $$b(A^+)=Z(A^+)+\sum_{B^+}a_{B^+A^+}Z(B^+),$$ and $$b(A^-)=Z(A^-)+\sum_{B^-}a_{B^-A^-}Z(B^-),$$ where $ a_{B^+A^+}, a_{B^-A^-}\in q{\mathbb Z}[q]$. Assume that $b(A^+)b(A^-)=q^a b(A^-)b(A^+)$, for some integer $a$ which can be computed by only considering the leading terms. From the defining relations (\[relations\]), the integer $a$ must be even, say, $a=2m$. Then $q^{-m} b(A^+)b(A^-)$ is bar-invariant with leading term $Z(A)$. Note that the coefficients we encounter only depend on the row sums and column sums. Actually, $m=\sum_{j}(r_j^+r_j^-+c_j^+c_j^-)$ where $(r_1^+,\cdots, r_n^+)$ and $(c_1^+,\cdots,c_n^+)$ (resp. $(r_1^-,\cdots, r_n^-)$ and $(c_1^-,\cdots,c_n^-)$ are the row sum and column sum respectively of $A^+$ (resp of $A^-$). Then all term produce the same $m$. Therefore, $$b(A)=q^{-m}b(A^+)b(A^-)$$ by Theorem \[basis\].
some quantum minors
===================
Let ${\det}_q(t)={\det}_q(\{1,\cdots,t\},\{n-t+1,\cdots,n\})$, for $t=1,2,\cdots,n$.
Let $M_t^-=\{(i,j)\in \mathbb N$$^2\mid 1\le i\le t\text{ and }1\le j\le n-t\}$, $M_t^+=\{(i,j)\in \mathbb N$$^2\mid t+1\le i\le n\text{ and }n-t+1\le j\le n\}$, $M_t^l=\{(i,j)\in \mathbb N$$^2\mid t+1\le i\le n\text{ and }1\le j\le n-t \}$, and $M_t^r=\{(i,j)\in \mathbb N$$^2\mid 1\le i\le t\text{ and }n-t+1\le j\le n\}$. The following result was proved in [@jz].
For any $i, j, t$, $$\begin{aligned}
Z_{ij}{\det}_q(t)&=&{\det}_q(t)Z_{ij} \text{ if }(i,j)\in M_t^l\cup M^r_t,
\\\nonumber Z_{ij}{\det}_q(t)&=&q^2{\det}_q(t)Z_{ij}
\text{ if } (i,j)\in M_t^-,\text{ and }\\\nonumber
Z_{ij}{\det}_q(t)&=&q^{-2}{\det}_q(t)Z_{ij} \text{ if }(i,j)\in M^+_t. \end{aligned}$$
Let $E_t=\begin{pmatrix}0&I_t\\0&0\end{pmatrix}$ and let $ q^{{\mathbb Z}}$$B^*=\{q^ab(A)|\text{ for all } A \text{ and }a\in{\mathbb Z}\}$.
The set $q^{{\mathbb Z}}$$B^*$ is invariant under the multiplication of the quantum minors $det_q(t)$ and $\sigma(det_q(t))$. More precisely,
$$\label{c1}b(A)det_q(t)=q^{r_1+\cdots +r_t-c_{n-t+1}-\cdots c_n}b(A+E_t).$$
$$\label{c2}b(A)\sigma(det_q(t))=q^{c_1+c_2+\cdots +c_t-r_{n-t+1}-\cdots -r_n}b(A+E^T_t),$$
where $E^T_t$ is the transposition of the matrix $E_t$.
For any $A\in M_n({\mathbb Z}_+)$, write $$A=\begin{pmatrix}A_{11}&A_{12}\\A_{21}&A_{22}\end{pmatrix},$$ where $A_{11}$ is a $t\times (n-t)$ submatrix, $A_{12}$ is a $t\times t$ submatrix, $A_{21}$ is a $(n-t)\times (n-t)$ submatrix and $A_{22}$ is a $(n-t)\times t$ submatrix. Then two monomials $Z^A$ and $Z^{A_{11}}Z^{A_{12}}Z^{A_{21}}Z^{A_{22}}$ have the same factors but could be in different order. However, the first monomial can be obtained from the second one by applying the third defining relation of the algebra $O_q(M(n))$. Hence, $$Z^A=Z^{A_{11}}Z^{A_{12}}Z^{A_{21}}Z^{A_{22}},$$ and $$Z(A)det_q(t)=
q^{-2\sum_{i\ge t+1, j\ge n-t+1}a_{ij}}D(A)Z^{A_{11}}
Z^{A_{12}}det_q(t)Z^{A_{21}}Z^{A_{22}}.$$ Apply the above lemma to $Z^{A_{12}}det_q(t)$, then we get $$Z(A_{12})det_q(t)=Z(A_{12}+I_t)+\sum_{B_{12}}h_{B_{12}}(A_{12})Z(B_{12}),$$ where $B_{12}\le A_{12}+I_t$, $B_{12}$ and $A_{12}+I_t$ have the same row sums and column sums. Hence $$Z(A)det_q(t)=
q^{-2\sum_{i\ge t+1, j\ge n-t+1}a_{ij}}D(A)
D(A_{12})^{-1}$$ $$D(A_{12}+I_t)D(A+E_t)^{-1}
Z(A+E_t)$$ $$+\sum_{B_{12}}h_{B_{12}}(A_{12})
q^{-2\sum_{i\ge t+1, j\ge n-t+1}a_{ij}}D(A)D(A_{12})^{-1}$$ $$D(B_{12})D(\begin{pmatrix}A_{11}&B_{12}\\A_{21}&A_{22}\end{pmatrix})^{-1}
Z(\begin{pmatrix}A_{11}&B_{12}\\A_{21}&A_{22}\end{pmatrix}).$$
By direct computation, one can show that the dependence of
$$D(B_{12})D(\begin{pmatrix}A_{11}&B_{12}\\A_{21}&A_{22}\end{pmatrix})^{-1}$$ on the matrix entries of $B_{12}$ is only a dependence on the row and column sums.
Then one deduces that $$Z(A)det_q(t)=q^{r_1+\cdots+r_t-c_{n-t+1}-\cdots-c_n}(Z(A+E_t)+
\sum_{D,D<A+E_t}c(D, A)Z(D))$$ with $c(D, A)\in q{\mathbb Z}$$[q]$.
For a basis element $b(A)$ of the form given in Theorem\[basis\], we then deduce that
$$\begin{aligned}
b(A)det_q(t)& = & q^{r_1+\cdots+r_t-c_{n-t+1}-\cdots-c_n}\\\nonumber
& &[(Z(A+E_t)+\sum_{D, D<A+E_t} c_D(A)Z(D)) \\\nonumber
& &+\sum_{B, B<A}h_B(A)(Z(B+E_t)+\sum_{D,D<B+E_t}c_D(B)Z(D))]\end{aligned}$$
with $c_D(A), c_D(B)\in q{\mathbb Z}$$[q]$. By $$b(A)det_q(t)= q^{2(r_1+\cdots+r_t-c_{n-t+1}-\cdots-c_n)}det_q(t)b(A),$$ we see that $(Z(A+E_t)+\sum_{D, D<A+E_t} c_D(A)Z(D))+\sum_{B,
B<A}h_B(A)(Z(B+E_t)+ \sum_{D,D<B+E_t}c_D(B)Z(D))$ is $^-$ invariant, and it must be the basis element $b(A+E_t)$. Finally, apply the algebra automorphism $\sigma$. Then the second statement follows from Corollary \[transpose\]
Let $A=\begin{pmatrix}a_{1}&b_2&b_3&\cdots &b_{n-1}&b_{n}\\c_2&a_{2}&b_2&\cdots &b_{n-2}&b_{n-1}\\
c_3&c_2&a_{3}&\cdots &b_{n-3}&b_{n-2}\\ \quad\quad\cdots&\cdots&\cdots&\cdots\\
c_n&c_{n-1}&c_{n-2}&\cdots&c_2&a_{n}\end{pmatrix}$. Then the basis element $$b(A)\sim \Pi_{t=1}^{n-1}det_q(t)^{b_{n-t+1}}\sigma(det_q(t))^{c_{n-t+1}} b(diag(a_{1},a_{2},\cdots,a_{n})).$$
Successively peel off the off-diagonals of $A$ by (\[c1\]) and (\[c2\]).
A matrix $A=(a_{ij})\in M_n({\mathbb Z}_+)$ is called a ladder if $a_{ij}\ge a_{i+1, j+1}$ for all $i,j$.
Let $A$ be a ladder. Successively peel off the off-diagonals of $A$ by (\[c1\]) and (\[c2\]), the basis element $b(A)$ is equivalent to a product of the quantum minors $det_q(t)$ and $\sigma(det_q(t))$ and a basis element $b(\begin{pmatrix}A_{n-1}&0\\0&0\end{pmatrix})$, where $A_{n-1}$ is a ladder of size $n-1$. Repeatedly, the basis element $b(A)$ can be written as a product of some quantum minors which are $q$-commuting with each other.
the coincidence of two bases
=============================
Let $g$ be the finite dimensional simple Lie algebra of type $A_{n-1}$ and let $\Lambda_1,\cdots,\Lambda_{n-1}$ be the fundamental dominant weights. For any dominant weight $\lambda$, the irreducible highest weight module $L(\lambda)$ occurs as a sub-quotient of a suitable power of the natural representation $L(\Lambda_1)$. The simple modules $L(\Lambda_1)$ and $L(\Lambda_{n-1})$ are dual to each other and are of dimension $n$. Let $e_1,e_2,\cdots,e_n$ be the standard basis of $L(\Lambda_1)$ and let $e_1^*,e_2^*,\cdots,e_n^*$ be the dual basis of $L(\Lambda_{n-1})$. Then it is well-known that the matrix coefficients $X_{ij}=e_i^*\otimes e_j$ satisfy the following relations:
$$\begin{aligned}
X_{ij}X_{ik}&=&q^2X_{ik}X_{ij} \text{ if } j<k,\\
X_{ij}X_{kj}&=&q^2X_{kj}X_{ij} \text{ if }i<k,\\
X_{ij}X_{st}&=&X_{st}X_{ij}\text{ if } i>s, j<t,\\
X_{ij}X_{st}&=&X_{st}X_{ij}+(q^2-q^{-2})X_{it}X_{sj} \text{ if }
i<s, j<t,\\\nonumber \Sigma_{\sigma\in
S_n}&(-q^2)^{l(\sigma)}&X_{1\sigma(1)}X_{2\sigma(2)} \cdots
X_{n\sigma(n)}=1.\end{aligned}$$
Since the basis $B^*$ is invariant under the multiplication of the quantum determinant, we gets a basis $K^*$ of $O_q(SL_n)$($=A_q(g)$), by setting the quantum determinant to one . Clearly, the anti-automorphism $-$ induces the anti-automorphism $\phi$ of $O_q(SL(n))$ (see Lemma 2.9). Let $X(A)$ be the image of $Z(A)$ in $O_q(SL(n))$. Then $$\{X(A)|trA=0\}$$ is a basis of $O_q(SL(n))$.
The matrix coefficients $X_{ij}$ are both invariant under $-$ (the bar action of $A_q(g)$) and $\phi$.
It is known that $\{e_1,e_2,\cdots,e_n\}$ (resp. $\{e_1^*,e_2^*,\cdots,e_n^*\}$) is the canonical basis of $L(\Lambda_1)$ (resp. of $L(\Lambda_{n-1})$). Therefore, $e_i$ and $e_j^*$ are invariant under the bar action of $L(\Lambda_1)$ and $L(\Lambda_{n-1})$ respectively. Hence, the matrix coefficients $X_{ij}$ are $-$ invariant. Note that $\Lambda_1$ and $\Lambda_{n-1}$ are minuscule dominant weights so the left weight $\lambda_l$ (resp. the right weight $\lambda_r$) of $X_{ij}$ is conjugate to $\Lambda_1$ (resp. $\Lambda_{n-1}$) under the action of the Weyl group which implies that $(\lambda_l,\lambda_l)-(\lambda_r,\lambda_r)=(\Lambda_1,\Lambda_1)
-(\Lambda_{n-1},\Lambda_{n-1})=0$.
The basis $K^*$ can be described similarly to the theorem \[basis\] by replacing $Z_{ij}$ by $X_{ij}$ and $-$ by $\phi$.
There is a unique basis $\tilde{B^*}=\{\tilde{b(A)}|A\in
M_n({\mathbb Z}_+), trA=0\}$ of $ \tilde{L^*}=\oplus_A{\mathbb
Z}[q]X(A)$ determined by the following conditions:
1. $\phi\tilde{b(A)}=\tilde{b(A)}$ for all $A$.
2. $\tilde{b(A)}=X(A)+\sum_{B<A} h_B(A)X(B)$ where $h_B(A)\in
q{\mathbb Z}$$[q]$ and $ro(B)=ro(A), co(B)=co(A)$.
Let ${\mathbb R}^n$ be the $n$ dimensional Euclidean space with standard orthogonal basis $\epsilon_1, \epsilon_2, \cdots,
\epsilon_n$. It is well-known that the root system of type $A_{n-1}$ is a subset of ${\mathbb R}^n$ with simple roots $\alpha_i=\epsilon_i-\epsilon_{i+1}$, for $i=1,2,\cdots, n-1$.
The $U_q(g)$ bi-module structure can ba written explicitly (see also [@nym]).
For homogeneous elements $x,$ and $y$ with weights $(\lambda_l,
\lambda_r)$ and $(\mu_l, \mu_r)$ respectively.
The left action is defined by
$$E_iX_{st}=\delta_{is}X_{s-1, t}, F_iX_{st}=\delta_{i,
s+1}X_{s+1, t}, K_iX_{st}=q^{2(\epsilon_s, \alpha_i)}X_{st}$$
with Leibniz rule
$$E_i(xy)=E_i(x)y+q^{2(\lambda_l, \alpha_i)}xE_i(y),$$
$$F_i(xy)=xF_i(y)+q^{-2(\mu_l, \alpha_i)}F_i(x)y,$$
$$K_i(xy)=q^{2(\lambda_l+\mu_l, \alpha_i)}xy,$$
The right action is defined by
$$X_{st}E_i=\delta_{i,s+1}X_{s+1, t}, X_{st}F_i=\delta_{i,
s}X_{s-1, t}, X_{st}K_i=q^{2(\epsilon_s, \alpha_i)}X_{st}$$
with Leibniz rule
$$(xy)E_i=(x)E_i y+q^{2(\lambda_r, \alpha_i)}x(y)E_i,$$
$$(xy)F_i=x(y)F_i+q^{-2(\mu_r, \alpha_i)}(x)F_i y,$$
$$(xy)K_i=q^{2(\lambda_r+\mu_r, \alpha_i)}xy,$$
Denote by the same notation the image of $det_q(i)$ in $O_q(SL(n))$.
For $\lambda=m_1\Lambda_1+m_2\Lambda_2+\cdots
+m_{n-1}\Lambda_{n-1}$, where $\Lambda_1,\Lambda_2,\cdots,
\Lambda_{n-1}$ are fundamental weights. The module $L(\lambda)\otimes L^*(\lambda)$ is cyclic on $v_\lambda\otimes
v_\lambda^*$. The lattice ${\cal L}(\lambda)\otimes {\cal
L}^*(\lambda)$ is generated by $v_\lambda\otimes v_\lambda^*$ (under the actio of upper Kashiwara operators) which corresponds to $$\Pi_idet_q(i)^{m_i}$$ which is an element in the basis $K^*$. The quantum minor $det_q(t)$ is annihilated by the left action of $E_i$ for $i=1,2,\cdots, t-1$ and the right action of $F_j$ for $j=n-1, n-2, \cdots, n-t+1$.
By the action of the upper kashiwara operators, the normalized monomial $X(A)$ and the element $q^{\frac{1}{2}((\lambda_l,\lambda_l)-(\lambda_r,\lambda_r))}b^\prime$ are in the same ${\mathbb Z}[q]$-lattice. By the uniqueness of Lusztig’s construction. The bases $K^*$ and $L^*$ are coincide.
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abstract: 'This paper considers asymptotically hyperbolic manifolds with a finite boundary intersecting the usual infinite boundary – *cornered asymptotically hyperbolic manifolds* – and proves a theorem of Cartan-Hadamard type near infinity for the normal exponential map on the finite boundary. As a main application, a normal form for such manifolds at the corner is then constructed, analogous to the normal form for usual asymptotically hyperbolic manifolds and suited to studying geometry at the corner. The normal form is at the same time a submanifold normal form near the finite boundary and an asymptotically hyperbolic normal form near the infinite boundary.'
address: |
Department of Mathematics\
University of Washington\
Seattle, Washington\
USA
author:
- 'Stephen E. McKeown'
bibliography:
- 'norm.bib'
title: Exponential Map and Normal Form for Cornered Asymptotically Hyperbolic Metrics
---
Introduction
============
A foundational fact of Riemannian geometry is that the exponential map at a point of a manifold $(X,g)$ is a diffeomorphism on a neighborhood of $0$; and similarly, the normal exponential map associated to a hypersurface $\iota: Q \hookrightarrow X$ is a diffeomorphism on a neighborhood of the zero section. The Cartan-Hadamard theorem gives a spectacular global extension of the former of these in case $X$ is of nonpositive curvature and complete, to wit that the exponential is a covering map. The global situation for the normal exponential map in a negatively curved space is more subtle due to the importance of the geometry and topology of $Q$, and has developed more slowly. Thus [@her63] showed, for example, that if $X$ is complete and of nonpositive curvature, $Q$ is closed, connected, and totally geodesic, and $\iota_*\pi_1(Q) = \pi_1(X)$, then $\exp$ is a diffeomorphism. A version for level sets $Q$ of convex functions on complete nonpositively curved manifolds, among other related theorems, can be found in the expansive [@bo69], and in [@lang99], it is proved (and stated to be generally known but unpublished) that if $X$ is complete of nonpositive curvature and $Q$ totally geodesic, the normal exponential map over $Q$ is a diffeomorphism onto its image. More recently, in [@bm08] it is proved that if $X^m$ is complete and $K^m \subset X^m$ is a compact, totally convex submanifold with boundary such that $X \setminus K$ has pinched negative curvature, then the normal exponential map over $Q = \partial K$ is a diffeomorphism onto $\overline{X \setminus K}$.
Asymptotically hyperbolic (AH) manifolds $(X,M = \partial X,g)$ are complete but may have arbitrary curvature on a compact set, with curvature approaching $-1$ toward the boundary at infinity. (Henceforth, $X$ will refer to a manifold with boundary, and the metric of interest will live on the interior $\mathring{X}$). Much of the interesting geometry and analysis on these spaces occurs near the boundary, so results, such as that just mentioned in [@bm08], that allow conclusions about a collar neighborhood of the boundary can play a role analogous in this context to that of global results such as Cartan-Hadamard in the negatively curved setting. The most important of these is the existence of the geodesic normal form, first proved in [@gl91]: suppose $(M,[h])$ is the conformal infinity of $X$ and that $h \in [h]$. Then for $\varepsilon > 0$ small, there is a unique diffeomorphism $\psi$ from $[0,\varepsilon)_r \times M$ to a neighborhood of $M$ in $X$ such that $\psi^*g = \frac{dr^2 + g_r}{r^2}$ and $g_0 = h$. The normal form has been a central tool in studying the duality between the boundary and interior geometry of asymptotically hyperbolic Einstein manifolds, and has frequently been employed in studying the analysis and geometry of AH manifolds generally. In this paper we prove a theorem of hypersurface Cartan-Hadamard type near the corner for asymptotically hyperbolic manifolds that have a finite boundary in addition to the usual infinite boundary, and then use this to construct a normal form at the corner.
We define a cornered space as a manifold $X$ with two boundary components $M$ and $Q$ that meet in a codimension-two corner $S = Q \cap M \neq \emptyset$, and a cornered AH (CAH) space as a cornered space equipped on the interior with a metric $g_+$ such that $g_+$ is smooth and nondegenerate at $Q \setminus S$ but asymptotically hyperbolic at $M$. Such manifolds arose in the proof ([@bh14]) of local regularity for AH Einstein manifolds, since a small neighborhood of a boundary point on a global asymptotically hyperbolic manifold has such a structure. Such manifolds have also been studied in the physics literature in the context of a proposed AdS/CFT-type correspondence for the case when the conformal field theory lives on a space with boundary (BCFT). See [@ntu12] and the references therein. Like a usual AH metric, CAH metrics have a conformal infinity $[h]$ on $M$.
The paradigm example of such a space is a portion of hyperbolic space bounded by an umbilic hypersurface. Let $\mathbb{H}^{n + 1} = \left\{ x^0 > 0 \right\}$ be the upper half-space model of hyperbolic space, with $g_+$ the hyperbolic metric $g_+ = \frac{(dx^0)^2 + \dots + (dx^n)^2}{(x^0)^2}$. Let $\alpha \in \mathbb{R}$, and $X = \{(x^0,\dots,x^n) \in \overline{\mathbb{H}^{n + 1}}: x^n \geq \alpha x^0\}$, with $Q = \left\{ x^n = \alpha x^0\right\}$ and $M = \left\{ x^0 = 0 \text{ and } x^n \geq 0 \right\}$. The conformal infinity $[h]$ is that of the Euclidean metric on $M$. The geodesics normal to $Q$ are precisely the intersections with $X$ of the circles $(x^0)^2 + (x^n)^2 = a^2$ (where $a \in \mathbb{R}^{>0}$), $x^1 = x^2 = \dots = x^{n - 1} = const$. The corner normal form in the hyperbolic case is obtained by introducing polar coordinates $(\theta, \rho)$, in which $Q, M$, and $S$ are all given by constant coordinates: $$x^0 = \rho\sin\theta, \quad x^n = \rho\cos\theta.$$ In these coordinates, the metric takes the form $$\label{hyppolg}
g_+ = \csc^2(\theta)\left[ d\theta^2 + \frac{d\rho^2 + (dx^1)^2 + \dots + (dx^{n - 1})^2}{\rho^2}\right].$$
The appearance of polar coordinates motivates us in the general case to follow the usual expedient of blowing up $X$ along $S$, obtaining a blown up space $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$, with a blow-down map $b:{\widetilde{X}}\to X$. This has the properties that $b|_{{\widetilde{X}}\setminus {\widetilde{S}}}:{\widetilde{X}}\setminus {\widetilde{S}}\to X \setminus S$ is a diffeomorphism, as are $b|_{{\widetilde{M}}}:{\widetilde{M}}\to M $ and $b|_{{\widetilde{Q}}}:{\widetilde{Q}}\to Q$, while $b|_{{\widetilde{S}}}:{\widetilde{S}}\to S$ is a fibration with fibers diffeomorphic to the closed unit interval. We will denote such a diffeomorphism equivalence by ${\widetilde{X}}\setminus {\widetilde{S}}\approx X \setminus S$ (for example).
In applications of our normal form theorems, we will need to consider metrics smooth on the blowup but not on the base. Thus, we give results for a somewhat wider class of metrics than those of the form $b^*g_+$ for $g_+$ a smooth cornered AH metric on $X$. In Definition \[admis\] we define admissible metrics, which differ from such a pullback by a perturbation that is smooth on ${\widetilde{X}}$ and vanishes in an appropriate sense at ${\widetilde{M}}$ and ${\widetilde{S}}$. Thus, such a metric may be written $g = b^*g_+ + \mathcal{L}$ for appropriate $\mathcal{L}$. Given any admissible metric $g$, there is a well-defined angle function $\Theta$ on ${\widetilde{S}}$, which serves as a fiber coordinate.
The normal exponential map $\exp$ of $Q \setminus S \approx {\widetilde{Q}}\setminus {\widetilde{S}}$ is defined on the inward-pointing normal ray bundle $N_+({\widetilde{Q}}\setminus {\widetilde{S}})$. With $\nu$ the inward-pointing unit normal field on ${\widetilde{Q}}\setminus {\widetilde{S}}$, this bundle has a natural decomposition $N_+({\widetilde{Q}}\setminus{\widetilde{S}}) \approx [0,\infty)_t \times ({\widetilde{Q}}\setminus {\widetilde{S}})$ given by the prescription $(t,q) \mapsto t\nu_q$. We compactify $N_+({\widetilde{Q}}\setminus {\widetilde{S}})$ by adding faces corresponding to $t = \infty$ and to $[0,\infty] \times ({\widetilde{Q}}\cap {\widetilde{S}})$, and we denote the compactification by ${\overline{N_+({\widetilde{Q}}\setminus {\widetilde{S}})}}$, a manifold with corners of codimension two.
Our first main result is as follows.
\[mainthm\] Let $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$ be the blowup of a cornered space, and $g$ an admissible metric on ${\widetilde{X}}$. There is a neighborhood $V$ of ${\widetilde{Q}}\cap {\widetilde{S}}$ in ${\widetilde{Q}}$ and a neighborhood ${\widetilde{U}}$ of ${\widetilde{S}}$ in ${\widetilde{X}}$ such that $\exp$ extends to a diffeomorphism $\exp:{\overline{N_+(V\setminus{\widetilde{S}})}} \to {\widetilde{U}}$.
One of the consequences of this theorem is that near $S$ there is a distinguished representative of the conformal infinity $[h]$ on $M \setminus S$, which itself is conformally compact on $(M,S)$. To see this, simply note that $e^{-t}$ is a defining function for ${\widetilde{M}}$ via the diffeomorphism in the theorem, so that $e^{-2t}g|_{T{\widetilde{M}}}$ is a well-defined element of $(b|_{{\widetilde{M}}})^*[h]$ on ${\widetilde{M}}\approx M$ depending only on the geometry of $({\widetilde{X}},g)$. We call this the induced metric on $M$.
Theorem \[mainthm\] allows us to prove two normal form theorems. The first applies generally, while the second requires that $Q$ and $M$ make a constant angle with respect to the compactified $g_+$, but gives a normal form with better properties.
\[normform\] Let $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$ be the blowup of a cornered space, and $g$ an admissible metric on ${\widetilde{X}}$. For sufficiently small neighborhoods $V$ of ${\widetilde{Q}}\cap {\widetilde{S}}$ in ${\widetilde{Q}}$, there exist a neighborhood ${\widetilde{U}}$ of ${\widetilde{S}}$ in ${\widetilde{X}}$ and a unique diffeomorphism $\psi:[0,1]_{u} \times V
\to {\widetilde{U}}$ such that $\psi|_{ \left\{ 1 \right\} \times V} = \operatorname{id}_V$ and $$\label{normformeq}
\psi^*g = \frac{du^2 + h_u}{u^2},$$ with $h_u$ ($0 \leq u \leq 1)$ a smooth one-parameter family of smooth conformally compact metrics on $(V,{\widetilde{Q}}\cap {\widetilde{S}})$, and such that ${\widetilde{M}}= \psi(\left\{ u = 0 \right\})$ and ${\widetilde{Q}}= \psi(\left\{ u = 1 \right\})$.
It will be useful in some applications to fix ${\widetilde{M}}$ instead of ${\widetilde{Q}}$:
\[normformcor\] Let $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$ and $g$ be as in Theorem \[normform\]. For sufficiently small neighborhoods $W$ of ${\widetilde{M}}\cap {\widetilde{S}}$ in ${\widetilde{M}}$, there exist a neighborhood ${\widetilde{U}}$ of ${\widetilde{S}}$ in ${\widetilde{X}}$ and a unique diffeomorphism $\zeta:[0,1]_u \times W \to {\widetilde{U}}$ such that $\zeta|_{ \left\{ 0 \right\} \times W} = \operatorname{id}_W$ and so that $$\zeta^*g = \frac{du^2 + h_u}{u^2},$$ with $h_u$ ($0 \leq u \leq 1$) a smooth one-parameter family of smooth conformally compact metrics on $(W,{\widetilde{M}}\cap {\widetilde{S}})$, and such that ${\widetilde{M}}= \zeta(\left\{ u = 0 \right\})$ and ${\widetilde{Q}}= \zeta(\left\{ u = 1 \right\})$.
Notice that (\[normformeq\]) is in normal form in the usual asymptotically hyperbolic sense relative to ${\widetilde{M}}$, while under the subsitution $t = -\log u$, it is in the usual geodesic normal form relative to ${\widetilde{Q}}$. In particular, $t$ is the distance to ${\widetilde{Q}}$.
The metrics $h_u$ for fixed $u$ are generally not asymptotically hyperbolic. The asymptotic curvature depends on both $u$ and the angle between $Q$ and $M$ at the point of $S$ approached. However, when the two boundary components make constant angle $\theta_0$, we can make the change of variable $u = \frac{\csc\theta - \cot\theta}{\csc\theta_0 - \cot\theta_0}$ to obtain a normal form with AH slice metrics.
\[normformconst\] Let $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$ be the blowup of the cornered space $(X,M,Q)$, and $g = b^*g_+ + \mathcal{L}$ an admissible metric on ${\widetilde{X}}$. Suppose that there is some $\theta_0 \in (0,\pi)$ such that, for any defining function $\varphi$ for $M$, the boundary components $M$ and $Q$ make constant angle $\theta_0$ with respect to the compactified metric $\varphi^2g_+$.
For sufficiently small neighborhoods $V$ of ${\widetilde{Q}}\cap {\widetilde{S}}$ in ${\widetilde{Q}}$, there is a neighborhood ${\widetilde{U}}$ of ${\widetilde{S}}$ in ${\widetilde{X}}$ and a unique diffeomorphism $\psi:[0,\theta_0]_{\theta} \times V \to {\widetilde{U}}$ such that $\psi|_{ \left\{ \theta_0 \right\} \times V} = \operatorname{id}_V$ and $$\psi^*g = \frac{d\theta^2 + h_{\theta}}{\sin^2\theta},$$ where $h_{\theta}$ ($0 \leq \theta \leq \theta_0$) is a smooth one-parameter family of smooth asymptotically hyperbolic metrics on $(V,{\widetilde{Q}}\cap {\widetilde{M}})$, and such that ${\widetilde{M}}= \psi(\{\theta = 0\})$ and ${\widetilde{Q}}= \psi(\{\theta = \theta_0\})$. Moreover, $\theta|_{[0,\theta_0]\times({\widetilde{Q}}\cap {\widetilde{S}})} = \psi^*\Theta$. Also $\partial_{\theta}{\bar{h}}_{\theta}|_{\rho = 0} = 0$, where ${\bar{h}}_{\theta} = \rho^2h_{\theta}$ and $\rho$ is any defining function for ${\widetilde{Q}}\cap {\widetilde{S}}$ in $V$.
Note that the normal form here given, and $\theta_0$, depend only on $g$, and not on the decomposition $g = b^*g_+ + \mathcal{L}$.
Once again, it can be helpful to fix ${\widetilde{M}}$ instead of ${\widetilde{Q}}$.
\[normformconstcor\] Let $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$, $(X,M,Q)$, and $g$ be as in Theorem \[normformconst\], with again a constant angle $\theta_0$ between $Q$ and $M$.
For sufficiently small neighborhoods $W$ of ${\widetilde{M}}\cap {\widetilde{S}}$ in ${\widetilde{M}}$, there is a neighborhood ${\widetilde{U}}$ of ${\widetilde{S}}$ in ${\widetilde{X}}$ and a unique diffeomorphism $\zeta:[0,\theta_0]_{\theta} \times W \to {\widetilde{U}}$ such that $\zeta|_{ \left\{ 0 \right\} \times W} = \operatorname{id}_W$ and $$\zeta^*g = \frac{d\theta^2 + h_{\theta}}{\sin^2\theta},$$ where $h_{\theta}$ ($0 \leq \theta \leq \theta_0$) is a smooth one-parameter family of smooth asymptotically hyperbolic metrics on $(W, {\widetilde{M}}\cap {\widetilde{S}})$, and such that ${\widetilde{M}}= \zeta(\{\theta = 0\})$ and ${\widetilde{Q}}= \zeta(\{\theta = \theta_0\})$. Moreover, $\theta|_{[0,\theta_0] \times ({\widetilde{M}}\cap {\widetilde{S}})}
= \zeta^*\Theta$, and $\partial_{\theta}{\bar{h}}_{\theta}|_{\rho = 0} = 0$, where ${\bar{h}}_{\theta} = \rho^2h_{\theta}$ and $\rho$ is any defining function for ${\widetilde{M}}\cap {\widetilde{S}}$ in $W$.
We can put this in a yet more refined form. For the following, we let $[k] = \left\{ h|_{TS}: h \in [h] \right\}$; so $[k]$ is a conformal class of metrics on $S$.
\[polarcor\] Let $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$, $(X,M,Q)$, and $g$ be as in Theorem \[normformconst\], with again a constant angle $\theta_0$ between $Q$ and $M$. For any $k \in [k]$ and for sufficiently small $\varepsilon > 0$, there is a neighborhood ${\widetilde{U}}$ of ${\widetilde{S}}$ in ${\widetilde{X}}$ and a unique diffeomorphism $\chi:[0,\theta_0]_{\theta} \times S \times [0,\varepsilon)_{\rho} \to {\widetilde{U}}$ such that $b\circ\chi|_{ \left\{ 0 \right\}\times S \times \left\{ 0 \right\}} = \operatorname{id}_S$ and $$\chi^*g = \frac{d\theta^2 + h_{\theta}}{\sin^2\theta},$$ where $h_{\theta}$ is a smooth one-parameter family of smooth AH metrics on $S \times [0,\varepsilon)$ with $$h_0 = \frac{d\rho^2 + k_{\rho}}{\rho^2},$$ where $k_{\rho}$ is a smooth one-parameter family of smooth metrics on $S$ with $k_0 = k$, and where ${\widetilde{M}}= \chi(\left\{ \theta = 0 \right\})$, ${\widetilde{Q}}= \chi(\left\{ \theta = \theta_0 \right\})$, and ${\widetilde{S}}= \chi(\left\{ \rho = 0 \right\})$. Moreover, $\partial_{\theta}(\bar{h}_{\theta}|_{\rho = 0}) = 0$, where ${\bar{h}}= \rho^2h$.
Notice this generalizes the form of the hyperbolic metric in (\[hyppolg\]).
A normal form of a similar kind for edge spaces was constructed in [@gk12]. However, the normal form derived there corresponds to a flow transverse to the edge boundary, whereas the flow generated by $u$ in (\[normformeq\]) is tangent to the edge face ${\widetilde{S}}$. Another difference is that the normal form constructed here takes a special form at two different faces, ${\widetilde{M}}$ and ${\widetilde{Q}}$, as opposed to one.
The paper is organized as follows. In Section \[setup\] we define cornered asymptotically hyperbolic manifolds and their blowups, as well as construct a class of product decompositions that will be ubiquitous throughout the paper. We note that the blown-up face ${\widetilde{S}}$ has an edge structure in the sense of [@mel08], which meets the AH face ${\widetilde{M}}$, and we define 0-edge bundles as the natural bundles associated to this structure. We then use this to define and discuss admissible metrics. In Section \[geods\], inspired by the convexity arguments of [@bo69], we use a natural asymptotic solution to $\nabla^2_gw = wg$ to derive the central properties of the $g$-geodesics leaving ${\widetilde{Q}}$ normally. Our result shows that they approximately generalize the behavior of the analogous geodesics in hyperbolic space, namely that they do not return to ${\widetilde{Q}}$ or ${\widetilde{S}}$ and that they approach ${\widetilde{M}}$ normally. In section \[exponential\], we study the geodesic flow equations to extend the exponential map to the compactified normal bundle and show that the extended map is smooth and a local diffeomorphism. The extensive debt this paper owes to [@maz86] is especially clear here, where we regularize the flow equations using the method developed there. The final substantial step, in Section \[injectivity\], is to show that the normal exponential map is actually injective on a suitably restricted neighborhood of ${\widetilde{S}}$. Many of the previous (and elegant) Cartan-Hadamard-type proofs adapt with difficulty, if at all, to the noncomplete and local setting studied here. The homotopy-lifting approach of [@her63], however, adapts well to this setting, and it enables us to show injectivity. In section \[proofs\], the above theorems and their corollaries follow quickly.
In a sequel, we will use the normal form here constructed to study formal existence and expansion of cornered asymptotically hyperbolic Einstein metrics, after the manner of [@fg12] in the usual case.
Acknowledgments {#acknowledgments .unnumbered}
---------------
This is doctoral work under the supervision of C. Robin Graham at the University of Washington. I am most grateful to him for suggesting this and related problems, and for the really extraordinary time and attention he has given to answering questions and making suggestions large and small. I am also grateful to Andreas Karch for bringing the topic to both of our attention in the first place, to John Lee and Daniel Pollack for numerous helpful conversations, and to Hart Smith for financial support. This research was partially supported by the National Science Foundation under RTG Grant DMS-0838212 and Grant DMS-1161283.
Cornered Spaces and Blowups {#setup}
===========================
Recall that a conformally compact manifold is a smooth compact manifold $X$ with boundary $M$, equipped on the interior with a smooth metric $g$ such that, for any defining function $\varphi$ of the boundary $M$, the metric $\varphi^2g$ extends smoothly to a metric on $X$. The boundary $M$ is called the infinite boundary or boundary at infinity, and if $h = \varphi^2g|_{TM}$ for some defining function $\varphi$, then the conformal class $[h]$ on $M$ is well-defined and is called the conformal infinity. A conformally compact manifold is called *asymptotically hyperbolic* if for some (and hence any) such defining function $\varphi$, we have $|d\varphi|_{\varphi^2g} = 1$ on $M$. The name is due to the fact, shown first in [@maz86], that such manifolds have all sectional curvatures asymptotic to $-1$.
A natural generalization of a conformally compact manifold is to consider manifolds that have *finite* boundaries as well as the boundary at infinity; a simple example would be half of the Poincaré ball. In such spaces, the finite and infinite boundaries meet in a corner, which is at infinity.
We first give an intrinsic definition of this situation.
\[intrindef\] A *cornered space* is a smooth manifold with codimension-two corners, $X^{n + 1}$, such that
(i) There are submanifolds with boundary $M^n \subset \partial X$ and $Q^n \subset \partial X$ of the boundary $\partial X$, such that $\emptyset \neq S = M \cap Q$ is the mutual boundary, and is the entire codimension-two corner of $X$, and such that $\partial X = M \cup Q$; and
(ii) the corner $S \subset M$ is a smooth, compact hypersurface in $M$.
We denote a cornered space by $(X,M,Q)$, and we set $\mathring{X} = X \setminus(Q \cup M)$.
Given a cornered space $(X,M,Q)$, a smooth (resp. $C^k$) *cornered conformally compact metric* on $X$ is a smooth Riemannian metric $g_+$ on $X \setminus M$ such that, for any smooth defining function $\varphi$ for $M$, the metric $\varphi^2g_+$ extends to a smooth (resp. $C^k$) metric on $X$. We call such a metric a *cornered asymptotically hyperbolic (CAH) metric* if for some (hence any) such defining function $\varphi$, the condition $|d\varphi|_{\varphi^2g_+} = 1$ holds along $M$.
A smooth (resp. $C^k$) *cornered asymptotically hyperbolic (CAH) space* is a cornered space $(X,M,Q)$ together with a smooth (resp. $C^k$) CAH metric $g_+$. We will denote such a space by $(X,M,Q,g_+)$. The definition for cornered conformally compact space is analogous.
For a cornered conformally compact space $(X,M,Q,g_+)$, the *conformal infinity* $[h]$ is the conformal class $[\varphi^2g_+|_{TM}]$ on $M$, where $\varphi$ is a defining function for $M$. Notice that a consequence of the fact that $X$ is a manifold with corners is that the boundary components $M$ and $Q$ intersect transversely.
For each $x \in S$, we define $\theta_0(x)$ to be the angle between $M$ and $Q$ at $X$ with respect to $\varphi^2g_+$, where $\varphi$ is any smooth defining function for $M$. Plainly $\theta_0 \in C^{\infty}(S)$.
It will be important to our analysis to be able to view $X$ as a submanifold of a larger AH manifold without corner. By doubling across $Q$ ([@mel96], Chapter 1) and using partitions of unity, we may construct a global AH manifold $({\breve{X}},{\breve{g}}_+)$ with boundary ${\breve{M}}$, such that $\mathring{X}$ is an open submanifold of ${\breve{X}}$ with $\partial \mathring{X} = M \cup Q$, where $M \subset {\breve{M}}$ and $Q \subset X$ is a hypersurface in ${\breve{X}}$, and such that ${\breve{g}}_+|_{X \setminus M} = g_+$. The extension ${\breve{g}}_+$ is not canonical, of course.
As we are planning to study polar-like coordinates at the codimension-two hypersurface $S$, and since such coordinates must be singular there, we employ the usual measure of blowing up $X$ along $S$ ([@mel08]). Let $X$ be a cornered space, with $M$, $Q$, and $S$ as in the definition. For $s \in S$, define $N_sS = T_sX / T_sS$, which is a vector space of dimension two. Let $NS$ be the vector bundle $NS = \sqcup_{s \in S} N_sS$. Let $N_+S \subset NS$ be the inward-pointing normal vectors (including those tangent to $\partial X = Q \cup M$). Thus $N_+S$ is a bundle with fiber a closed cone in $\mathbb{R}^2$ and base $S$. Finally, let ${\widetilde{S}}= (N_+S\setminus \left\{ 0 \right\}) / \mathbb{R}^+$, which is the total space of a fibration over $S$ with fiber the closed interval $[0,1]$. Set $\widetilde{X} = (X \setminus S) \sqcup \widetilde{S}$, and define the *blow-down map* $b:\widetilde{X} \to X$ by $b(x) = x$ ($x \in X \setminus S$) and $b(\tilde{s}) = \pi(\tilde{s})$ ($\tilde{s} \in \widetilde{S}$), where $\pi$ is the natural projection. Then as shown in [@mel08], $\widetilde{X}$ has a unique smooth structure as a manifold with corners of codimension two such that $b$ is smooth, $b|_{{\widetilde{X}}\setminus {\widetilde{S}}}:{\widetilde{X}}\setminus {\widetilde{S}}\to X \setminus S$ is a diffeomorphism onto its image, and $db_{{\tilde{s}}}$ has rank $n$ for ${\tilde{s}}\in {\widetilde{S}}$. Moreover, polar coordinates on $X$ centered along $S$ lift to smooth coordinates. We set ${\widetilde{M}}= \overline{b^{-1}(M \setminus S)}$ and ${\widetilde{Q}}= \overline{b^{-1}(Q \setminus S)}$. Then $b|_{{\widetilde{M}}}:{\widetilde{M}}\to M$ and $b|_{{\widetilde{Q}}}:{\widetilde{Q}}\to Q$ are diffeomorphisms.
Recall that an edge structure on a manifold with boundary is a fibration of the boundary, and the associated edge vector fields are the vector fields that are tangent to the fibers at the boundary ([@maz91]). An important special case is a 0-structure ([@mm87]), for which the boundary fibers are points and the edge vector fields are those that vanish at the boundary. On our blowup space ${\widetilde{X}}$, the blown-up face ${\widetilde{S}}$ is the total space of the fibration $b|_{{\widetilde{S}}}:{\widetilde{S}}\to S$ with interval fibers, while we can view $b|_{{\widetilde{M}}}:{\widetilde{M}}\to M$ as a fibration whose fibers are points. We will refer to the structure defined by these two fibrations as a 0-edge structure, and the associated 0-edge vector fields are the smooth vector fields on ${\widetilde{X}}$ which are tangent to the fibers at ${\widetilde{S}}$, and which vanish at ${\widetilde{M}}$.
The 0-edge vector fields may be easily expressed in appropriate local coordinates. Let $\theta$ be a defining function for ${\widetilde{M}}$ whose restriction to each fiber of ${\widetilde{S}}$ is a fiber coordinate taking values in $[0,\pi)$; let $\rho$ be any defining function for ${\widetilde{S}}$; and locally let $x^s, 1 \leq s \leq n - 1$, be the lifts to ${\widetilde{X}}$ of functions on $X$ that restrict to local coordinates on $S$. Then the vector fields $$\sin\theta \frac{\partial}{\partial \theta}, \quad \rho\sin\theta \frac{\partial}{\partial x^s}, \quad \rho\sin\theta \frac{\partial}{\partial \rho}$$ span the 0-edge vector fields over $C^{\infty}({\widetilde{X}})$. As in the usual edge case, there is a well-defined vector bundle ${}^{0e}T{\widetilde{X}}$ whose smooth sections are the 0-edge vector fields. The smooth sections of the dual bundle ${}^{0e}T^*{\widetilde{X}}$ are locally spanned by $$\label{dualframe}
\frac{d\theta}{\sin\theta},\quad\frac{dx^s}{\rho\sin\theta}, \quad\frac{d\rho}{\rho\sin\theta}.$$ By a 0-edge metric we will mean a smooth positive definite section $g$ of $S^2({}^{0e}T^*{\widetilde{X}})$. This is equivalent to the condition that locally $g$ may be written as $$g = \left( \begin{array}{ccc}
\frac{d\theta}{\sin\theta},&\frac{dx^s}{\rho\sin\theta},&\frac{d\rho}{\rho\sin\theta}\end{array}\right)
G
\left(
\begin{array}{ccc}
\frac{d\theta}{\sin\theta}\\
\frac{dx^s}{\rho\sin\theta}\\
\frac{d\rho}{\rho\sin\theta}
\end{array}
\right),$$ where $G$ is a smooth, positive-definite matrix-valued function on ${\widetilde{X}}$. This allows us to define the class of metrics that we will study.
\[admis\] An admissible metric on ${\widetilde{X}}$ is a 0-edge metric $g$ on ${\widetilde{X}}$ which can be written in the form $$g = b^*g_+ + \mathcal{L},$$ where $g_+$ is a smooth cornered asymptotically hyperbolic metric on $X$ and $\mathcal{L}$ is a smooth section of $S^2({}^{0e}T^*{\widetilde{X}})$ that vanishes on ${\widetilde{S}}$ and ${\widetilde{M}}$.
The latter condition is the same as saying that $\mathcal{L} = (\rho\sin\theta)\ell$, for some smooth section $\ell$ of $S^2({}^{0e}T^*{\widetilde{X}})$. We will see below that if $g_+$ is a smooth CAH metric on $X$, then $b^*g_+$ is a 0-edge metric.
Since $b|_{{\widetilde{X}}\setminus ({\widetilde{M}}\cup {\widetilde{S}})}:{\widetilde{X}}\setminus ({\widetilde{M}}\cup {\widetilde{S}}) \to X \setminus M$ is a diffeomorphism, an admissible $g$ uniquely determines a smooth metric $g_X$ on $X \setminus M$ satisfying $b^*g_X = g$ on ${\widetilde{X}}\setminus ({\widetilde{M}}\cup {\widetilde{S}})$. Since $\mathcal{L}$ vanishes on ${\widetilde{S}}$ and ${\widetilde{M}}$, it is not hard to see that $g_X$ is a $C^0$ CAH metric on $X$. Thus we will call a metric $g_X$ on $X \setminus M$ an admissible metric on $X$ if $b^*g_X$ extends to an admissible metric on ${\widetilde{X}}$.
Observe that an admissible metric $g_X$ on $X$ determines a well-defined angle function $\Theta$ on the blown-up face ${\widetilde{S}}$, which serves as a smooth fiber coordinate. Let ${\tilde{s}}\in \widetilde{S}$, with $s = b({\tilde{s}}) \in S$. Then, under one interpretation, ${\tilde{s}}$ naturally represents a hyperplane $P_{{\tilde{s}}}$ in $T_sX$ containing $T_sS$. The angle $\Theta({\tilde{s}})$ between $P_{s}$ and $T_sM$ is well-defined. It can be computed as follows: let $\varphi$ be any defining function for $M$, and $\bar{g}_X = \varphi^2g_X$. Let $\bar{\nu}_M \in T_sM$ be normal to $T_sS$, inward pointing in $M$, and unit $\bar{g}_X$-length (this is uniquely defined and continuous, by the continuity of admissible metrics just observed). Similarly, let $\bar{\nu}_{P_{{\tilde{s}}}}$ be inward-pointing in $P_{{\tilde{s}}}$, normal to $T_sS$, and unit length. Then $\Theta({\tilde{s}}) = \cos^{-1}(\bar{g}_X(\bar{\nu}_M,\bar{\nu}_{P_{{\tilde{s}}}}))$. We could also have defined $\Theta$ using $g_+$, and in particular, it is clear that $\Theta \in C^{\infty}({\widetilde{S}})$. It is easy to show that this is defined independently of $\varphi$. Thus, $\Theta$ is well-defined.
Let $g_+$ be a smooth CAH metric on $X$. We construct a product identification on $X$ that we will use extensively throughout, and we then use it to show that $b^*g_+$ is a 0-edge metric. Choose an extension $({\breve{X}},{\breve{g}}_+)$. To each representative ${\breve{h}}$ on ${\breve{M}}$ we can associate a neighborhood ${\breve{U}}$ of ${\breve{M}}$ in ${\breve{X}}$ and a unique diffeomorphism $\chi:[0,\varepsilon)_r \times {\breve{M}}\to {\breve{U}}$ such that $\chi|_{{\widetilde{M}}} = \operatorname{id}$ and $\chi^*{\breve{g}}_+ = r^{-2}(dr^2 + {\breve{h}}_r)$, where ${\breve{h}}_0 = {\breve{h}}$. Now let $y$ be a geodesic defining function for $S$ in ${\breve{M}}$ with respect to the metric ${\breve{h}}$ – that is, a solution near $S$ on ${\breve{M}}$ to the equation $|dy|_{{\breve{h}}}^2 = 1$ with $y|_S \equiv 0$. We choose $y > 0$ on $M$. Then there is a diffeomorphism $\psi$ from $S \times (-\delta,\delta)_y$ to a neighborhood $W$ of $S$ in ${\breve{M}}$ such that $\psi^*{\breve{h}}= dy^2 + k_y$, where $k_y$ is a smooth one-parameter family of metrics on $S$. Thus, we have shown that there is a neighborhood ${\breve{U}}$ of $S$ in ${\breve{X}}$ and a unique diffeomorphism $\varphi: [0,\varepsilon)_r \times S
\times (-\delta,\delta)_y \to {\breve{U}}$, for which $\varphi|_{\{0\}\times S\times \{0\}} = \operatorname{id}_S$ and $$\label{cartform}
\varphi^*{\breve{g}}_+ = \frac{dr^2 + {\breve{h}}_r}{r^2},$$ where ${\breve{h}}_r$ is a one-parameter family of metrics on $S \times (-\delta,\delta)_y$ with $$\label{g0}
{\breve{h}}_0 = dy^2 + k_y.$$ We call this the product identification for ${\breve{g}}_+$ determined by ${\breve{h}}$, and we let $\pi_S:{\breve{U}}\to S$ be the projection onto $S$ determined by it.
In cases where $Q$ makes an obtuse angle with $M$, the values inside $X$ of the functions $r$ and $y$ just constructed will depend on ${\breve{g}}_+$ outside $X$. We will use the product identification to analyze the behavior of geodesics in $X$, which of course is independent of the extension chosen.
We obtain smooth coordinates on the blowup ${\widetilde{X}}$ near ${\widetilde{S}}$ by introducing polar coordinates on $X$. Using the coordinates defined above, these are given by $$\label{polardef}
r = \rho\sin\theta, \quad y = \rho\cos\theta,\text{ with } \rho \geq 0, \quad 0 \leq \theta < \pi.$$ Then locally, a product identification on the blowup may be given by $p \mapsto (\theta(p),\pi_S(p),\rho(p))$. Observe that $\widetilde{S}$ is given precisely by $\rho = 0$ and $\widetilde{M}$ is given by $\theta = 0$. For any admissible metric $g$ on ${\widetilde{X}}$ such that $g = b^*g_+ + \mathcal{L}$, this identification on the blowup will be called a *polar $g$-identification*, or depending on context, *polar $g$-coordinates*. Notice that by (\[cartform\]) and (\[g0\]), $\theta|_{{\widetilde{S}}} = \Theta$.
Now by (\[polardef\]), we have $$\begin{aligned}
dr &= \rho \cos \theta d\theta + \sin\theta d\rho\\
dy &= -\rho\sin\theta d\theta + \cos\theta d\rho.\end{aligned}$$ Let $\left\{ x^s \right\}_{s = 1}^{n - 1}$ be local coordinates on $S$. Extend these into ${\widetilde{X}}$ near ${\widetilde{S}}$ using the product identification. Note by (\[g0\]) that in (\[cartform\]), ${\breve{h}}_r = dy^2 + k_y + O(r)$. It is then straightforward to compute the metric in our new coordinates: $$\label{polformbetter}
b^*g_+ = \left(\begin{array}{ccc}
\frac{d\theta}{\sin\theta}, & \frac{dx^s}{\rho \sin\theta}, & \frac{d\rho}{\rho \sin\theta}\end{array}\right)
G
\left(\begin{array}{c}
\frac{d\theta}{\sin\theta} \\ \frac{dx^s}{\rho\sin\theta} \\ \frac{d\rho}{\rho\sin\theta}\end{array}\right),$$ where $$\label{polformwasbetter}
G = \left(\begin{array}{ccc}
1 + O(\rho\sin^3\theta) & O(\rho \sin^2\theta) & O(\rho\sin^2\theta)\\
O(\rho\sin^2\theta) & k_{\rho\cos\theta} + O(\rho\sin\theta) & O(\rho\sin\theta)\\
O(\rho\sin^2\theta) & O(\rho\sin\theta) & 1 + O(\rho\sin\theta)\end{array}\right)$$ Thus, $b^*g_+$ is a 0-edge metric. Notice that $k_{\rho\cos\theta} = k_{\rho} + O(\rho \sin^2\theta)$. This yields the following.
In a polar identification, an admissible metric $g$ on ${\widetilde{X}}$ takes the form $$\label{polform}
g = \frac{1}{\sin^2(\theta)}\left[ d\theta^2 + \frac{d\rho^2 + k_{\rho}}{\rho^2} \right] + (\rho\sin\theta)\ell,$$ where $k_{\rho}$ is a one-parameter family of metrics on $S$ and $\ell \in C^{\infty}(S^2({}^{0e}T^*{\widetilde{X}}))$.
We note that the statement that $g$ can be written in the form (\[polform\]) is equivalent to the statement that it can be written as $$g = \frac{1}{\sin^2(\theta)}\left[ d\theta^2 + \frac{d\rho^2 + k_{\theta,\rho}}{\rho^2} \right] + (\rho\sin\theta)\ell,$$ where $\ell$ is as before and where $k_{\theta,\rho}$ is a two-parameter family of metrics on $S$ such that $k_{\theta,0}$ is independent of $\theta$.
Notice that for the hyperbolic metric, (\[hyppolg\]) exhibits the form (\[polform\]) with $k = |dx|^2$ and $\ell = 0$.
It will be useful to have equation (\[polform\]) expressed in block form. On ${\widetilde{X}}$ in the coordinates $(\theta,x^s,\rho)$, the metric takes the form
$$\label{g}
g_{ij} = \csc^2(\theta)\left(\begin{array}{ccc}
1 + O(\rho\sin\theta) & O(\sin \theta) & O(\sin \theta)\\
O(\sin \theta) & \rho^{-2}k_{\rho} + O(\rho^{-1}\sin\theta) & O(\rho^{-1}\sin\theta)\\
O(\sin\theta) & O(\rho^{-1}\sin\theta) & \rho^{-2} + O(\rho^{-1}\sin\theta)
\end{array}\right).$$
This may also be written $$g_{ij} = \csc^2(\theta)A(\rho) \left(\begin{array}{ccc}
1 + O(\rho\sin\theta) & O(\rho\sin\theta) & O(\rho\sin\theta)\\
O(\rho\sin\theta) & k_{\rho} + O(\rho\sin\theta) & O(\rho\sin\theta)\\
O(\rho\sin\theta) & O(\rho\sin\theta) & 1 + O(\rho\sin\theta)\end{array}\right)
A(\rho),$$ where $$A(\rho) = \left(\begin{array}{ccc}
1 & &\\
& \rho^{-1} &\\
& & \rho^{-1}\end{array}\right).$$ This allows us easily to use Cramer’s rule to find that $$\label{ginv}
g^{ij} = \sin^2(\theta)\left(\begin{array}{ccc}
1 + O(\rho\sin\theta) & O(\rho^2\sin\theta) & O(\rho^2\sin\theta)\\
O(\rho^2\sin\theta) & \rho^2k_{\rho}^{-1} + O(\rho^3\sin\theta) & O(\rho^3\sin\theta)\\
O(\rho^2\sin\theta) & O(\rho^3\sin\theta) & \rho^2 + O(\rho^3 \sin\theta)\end{array}\right).$$
Notation
--------
Throughout this paper, $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$ will be the blowup of a cornered space $(X,M,Q)$, with blowdown map $b:{\widetilde{X}}\to X$. We let $g = b^*g_X = b^*g_+ + (\rho\sin\theta)\ell$ be an admissible metric on ${\widetilde{X}}$. Except where noted otherwise, $\theta$ and $\rho$ will denote the polar coordinates in a polar $g$-coordinate system. Similarly, $r = \rho\sin\theta$ will denote the geodesic defining function on $X \subset {\breve{X}}$ in terms of which they were defined. The projection onto the $S$ factor will be denoted $\pi_S$.
$X$ will be of dimension $n + 1$, where unless otherwise specified, $n \geq 2$ always.
We use index notation in polar coordinates. When doing so, $0$ will refer to the first factor $\theta$, and $n$ will refer to the last factor $\rho$. The indices $1 \leq s,t \leq n - 1$ will refer to local coordinates on the second factor, $S$, while the indices $0 \leq i, j \leq n$ will run over all $n + 1$ coordinates. Finally, $1 \leq \mu, \nu \leq n$ will run over $S$ and the last factor.
The metric $g$ will be used to raise and lower indices, except that $g^{ij}$ is the inverse metric. We write $\bar{g} = \rho^2\sin^2(\theta)g = r^2g$ for the metric compactified with respect to polar $g$-coordinates. Note that $\bar{g}$ is degenerate along ${\widetilde{S}}$, as $\left|\frac{\partial}{\partial \theta}\right|_{\bar{g}} = 0$ there.
If $a > 0$, we define $${\widetilde{Q}}_a = \left\{ q \in {\widetilde{Q}}: 0 < \rho(q) < a \right\},$$ and $${\underline{{\widetilde{Q}}_{a}}} = \left\{ q \in {\widetilde{Q}}: 0 \leq \rho(q) < a \right\}.$$ For general open $V \subseteq {\widetilde{Q}}$, we define $\underline{V} = V \cup L$, where $L$ is the interior in ${\widetilde{Q}}\cap {\widetilde{S}}$ of the set of limit points of $V$ in ${\widetilde{S}}$. We also define $${\widetilde{X}}_a = \left\{ x \in {\widetilde{X}}: 0 < \rho(x) < a \right\}.$$
We let $\nu$ be the inward-pointing unit normal vector field on ${\widetilde{Q}}\setminus {\widetilde{S}}$ with respect to $g$. If $q \in {\widetilde{Q}}\setminus {\widetilde{S}}$, we let $\gamma_q$ denote the $g$-geodesic that begins at $q$ and has initial tangent vector $\gamma_q'(0) = \nu_q$.
If $A$ is a covariant $k$-tensor, we write $A = O_g(f)$ to indicate that $|A|_g = O(f)$; or equivalently, that if $Y_1,\dots,Y_k$ are $g$-unit vector fields, then we have $A(Y_1,\dots,Y_k) = O(f)$ with constant independent of the $Y_i$. Similarly, if $Y$ is a vector field, we write $Y = O_g(f)$ to indicate $|Y|_g = O(f)$. Note that this condition is independent of the particular admissible metric $g$.
Behavior of Normal Geodesics {#geods}
============================
As a starting point to our study of the normal exponential map over the finite boundary ${\widetilde{Q}}$ of the blowup, in this section we study the basic behavior of $g$-geodesics that leave ${\widetilde{Q}}$ normally, where $g$ is an admissible metric on $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$. Throughout, we will work in a polar $g$-identification as constructed in the previous section, and we will let ${\widetilde{U}}$ be a neighborhood of ${\widetilde{S}}$ in ${\widetilde{X}}$ on which such coordinates exist.
Our first task is to study the normal field to ${\widetilde{Q}}\setminus {\widetilde{S}}$ near ${\widetilde{S}}$.
\[normlem\] Sufficiently near ${\widetilde{S}}$, the normal field $\nu$ on ${\widetilde{Q}}\setminus {\widetilde{S}}$ satisfies $$\label{qnorm}
\nu = -\sin\theta \frac{\partial}{\partial \theta} + O_g(\rho).$$ In particular, $\nu$ extends smoothly to ${\widetilde{Q}}\cap {\widetilde{S}}$.
Using the implicit function theorem and the fact that $\frac{\partial}{\partial \theta}$ is transverse to ${\widetilde{Q}}\cap {\widetilde{S}}$, we may write ${\widetilde{Q}}$ near ${\widetilde{S}}$ smoothly as $\theta = \psi(x,\rho)$. Define $f(\theta,x,\rho) = \psi(x,\rho) - \theta$. Then using (\[ginv\]), we find in local coordinates that $$\operatorname{grad}f = g^{ij}\frac{\partial f}{\partial x^i}\frac{\partial}{\partial x^j} =
(-\sin^2(\theta) + O(\rho))\frac{\partial}{\partial \theta} + O(\rho^2),$$ while $$|\operatorname{grad}f|_g = \sin \theta + O(\rho),$$ where we keep in mind that $\sin\theta$ is bounded away from $0$ on ${\widetilde{Q}}$. As $\nu = \frac{\operatorname{grad}f}{|\operatorname{grad}f|_g}$, the result follows.
It has long been the case that convex functions are important in studying spaces of negative curvature; see [@bo69]. Most of our interior analysis of the $g$-geodesics leaving $Q$ will follow from the fact that the cotangent function on a cornered AH space has a Hessian of a very special form related to convexity. This Hessian equation actually has a history of its own in the negatively curved setting and beyond; see for example [@cc96; @hpw15].
\[cotlem\] Define $w \in C^{\infty}({\widetilde{U}}\setminus {\widetilde{M}})$ by $w = \cot(\theta)$. Then $$\label{convex}
\nabla_g^2w = wg + O_g(\rho).$$
This result is motivated by the fact that, in the case of the hyperbolic upper half-space, the equation holds exactly.
We will need the Christoffel symbols $\Gamma_{ij}^0$. Computing from (\[g\]) and (\[ginv\]) using the equation $\Gamma_{ij}^k = \frac{1}{2}g^{kl}(\partial_ig_{lj} + \partial_jg_{li} - \partial_lg_{ij})$, we find that $$\Gamma^0_{ij} = \left(\begin{array}{ccc}
-\cot(\theta) + O(\rho) & O(1) & O(1)\\
O(1) & \rho^{-2}\cot(\theta)k_{st} + O(\rho^{-1}) & O(\rho^{-1})\\
O(1) & O(\rho^{-1}) & \rho^{-2}\cot(\theta) + O(\rho^{-1})\end{array}\right).$$ Now $dw = -\csc^2(\theta)d\theta$, and we can use these Christoffel computations to find that $$\begin{aligned}
\nabla^2w = \nabla(dw) &= \cot(\theta)g + \frac{\left(
d\theta, dx^s, d\rho\right)}{\sin^2\theta}\left(\begin{array}{ccc}
O(\rho) & O(1) & O(1)\\
O(1) & O(\rho^{-1}) & O(\rho^{-1})\\
O(1) & O(\rho^{-1}) & O(\rho^{-1})\end{array}\right)
\left(\begin{array}{c}d\theta \\ dx^t \\ d\rho\end{array}\right)\\
&= \cot(\theta)g\\
&\quad+\left( \frac{d\theta}{\sin\theta}, \frac{dx^s}{\rho\sin\theta}, \frac{d\rho}{\rho\sin\theta} \right)
\left(\begin{array}{ccc}O(\rho) & O(\rho) & O(\rho)\\
O(\rho) & O(\rho) & O(\rho)\\
O(\rho) & O(\rho) & O(\rho)\end{array}\right)
\left(\begin{array}{c}
\frac{d\theta}{\sin\theta}\\
\frac{dx^t}{\rho\sin\theta}\\
\frac{d\rho}{\rho\sin\theta}\end{array}\right).\end{aligned}$$ This proves the claim.
We next need two technical results.
\[complem\] Let $D = J \times \mathbb{R}_{(u,v)}^2 \subseteq \mathbb{R}^3$, where $J$ is an interval containing $[a,b)$. Let $f:D \to \mathbb{R}$ be continuous and locally Lipschitz, and weakly increasing in $u$. Define an ordinary differential operator $L$ by $Lu(x) = u''(x) - f(x,u(x),u'(x))$. Suppose $\theta, \psi:[a,b) \to \mathbb{R}$ are $C^2$ functions such that the graphs of $t \mapsto (\theta(t),\theta'(t))$ and $t \mapsto (\psi(t),\psi'(t))$ lie in $D$. If
- $\theta(a) \leq \psi(a)$; and
- $\theta'(a) \leq \psi'(a)$; and
- $L\theta \leq L\psi$ on $[a,b)$,
then $\theta(t) \leq \psi(t)$ and $\theta'(t) \leq \psi'(t)$ for all $t \in [a,b)$.
This is essentially Theorem 11.XVI in [@wal98].
\[lemcsc\] (a) Let $0 < \delta < 1$, $a > 0$, and $b \in \mathbb{R}$, and set $f(t) = a e^t + b e^{-t} + \delta$. Then there exists a continuous function $C = C(a,b,\delta) > 0$ such that, if $w(t) \geq f(t)$ for all $t \geq 0$, then $1 + w(t)^2 \geq C^{-2}e^{2t}$ for all $t \geq 0$.
\(b) Let $0 < \delta < 1$, $0 < a_1 < a_2$, and $b_1, b_2 \in \mathbb{R}$. Suppose that $a_1e^t + b_1e^{-t} + \delta \leq w(t) \leq a_2e^t + b_2e^{-t} - \delta$. Then there exists $D = D(a_1,a_2,b_1,b_2,\delta)$, continuous in its arguments, such that $1 + w(t)^2 \leq D^2e^{2t}$ for all $t \geq 0$.
\(a) There exists $T > 0$ such that, for all $t \geq T$, we have $f(t) > \frac{1}{2}a e^t$. Let $\frac{2}{a} < C$ be such that $C^{-2}e^{2T} \leq 1$. The result follows immediately.
The proof of (b) is similar.
We also recall the following result from [@maz86].
\[mazprop\] Let $(X,M,g)$ be an asymptotically hyperbolic manifold, and $\varphi$ a defining function for $M$. There exists $\varphi_0 > 0$ such that, if $p \in X$ with $0 < \varphi(p) < \varphi_0$, and if $\gamma:[0,\infty) \to
\mathring{X}$ is a geodesic ray with $\gamma(0) = p$ and $(\varphi\circ\gamma)'(0) < 0$, then $\gamma$ asymptotically approaches a well-defined point of $M$, normally with respect to $\varphi^2g$.
In the introduction, we discussed a subset $(X,M,Q)$ of hyperbolic space $\mathbb{H}^{n + 1}$ as an example of a CAH manifold, with $Q$ a Euclidean plane. Recall that geodesics leaving $Q$ normally are semi-circles that approach the infinite boundary $\mathbb{R}^n$ orthogonally. The following theorem, which is the main result of this section, shows that this behavior is approximated by the geodesics in a general cornered AH space.
First some notation. We let $d_S$ be the distance function on $S$ with respect to $k_0$, where $k_0$ is as in (\[g\]), and $\pi_S:{\widetilde{U}}\to S$ be projection onto the $S$ factor in the polar $g$-identification.
\[geodprop\] Let $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$ be the blowup of the cornered space $(X, M, Q)$, and $g$ an admissible metric. There exists $a > 0$ such that for each $q \in {\widetilde{Q}}_a$,
- $\gamma_q$ exists for all $t \geq 0$ and $\gamma_q(t) \in \mathring{{\widetilde{X}}}$ for all $t > 0$;
- the limit $\lim_{t \to \infty}\gamma_q(t)$ exists and lies in ${\widetilde{M}}\setminus {\widetilde{S}}$, and $\gamma_q$ approaches ${\widetilde{M}}$ $\bar{g}$-normally; and
- for all $t \geq 0$, $(\theta \circ \gamma_q)'(t) < 0$.
Moreover, there exist $\varepsilon > 0$ and $A_1, A_2, C > 0$ such that, for all $q \in {\widetilde{Q}}_a$ and all $t \in [0,\infty)$,
1. \[rhocond\] $\varepsilon \rho(q) < \rho(\gamma_q(t)) < C\rho(q)$;
2. $d_S(\pi_S(q),\pi_S(\gamma_q(t))) < C\rho(q)$; and\[Scond\]
3. $A_1e^{-t} < \sin(\theta(\gamma(t))) < A_2e^{-t}$.\[thetacond\]
Note that we will improve this result in Proposition \[extendprop\]; in particular, it will imply that in \[Scond\] we could write $\rho(q)^2$ instead of $\rho(q)$, and that in \[rhocond\] we can take $\frac{\varepsilon}{C} \to 1$ as $a \to 0$.
By Lemma \[normlem\], $\nu = -\sin(\theta)\frac{\partial}{\partial \theta} + O_g(\rho)$. Set $\nu_q^{\theta} = d\theta(\nu_q)$. Thus, for $q \in {\widetilde{Q}}\setminus {\widetilde{S}}$ sufficiently near ${\widetilde{S}}$, we have $-\csc^2(\theta(q))\nu^{\theta}_q = \csc(\theta(q)) + O(\rho(q))$, uniformly in $q$. Now for $\theta \in (0,\pi)$, we always have $\csc\theta > |\cot\theta|$. Let $0 < \delta < 1$ be such that there exists $\rho_0 > 0$ so that ${\widetilde{X}}_{\rho_0} \subset {\widetilde{U}}$ and such that $$\alpha := \inf_{q \in {\widetilde{Q}}_{\rho_0}}(-\csc^2(\theta(q))\nu_q^{\theta} - |\cot(\theta(q))| - \delta) > 0.$$ Such a $\delta$ exists because $\theta$ is bounded away from $0$ and $\pi$ on ${\widetilde{Q}}$. Now let $$\begin{split}
\beta :=& \sup_{q \in {\widetilde{Q}}_{\rho_0}}\left(\max\left\{ -\csc^2(\theta(q))\nu_q^{\theta} + |\cot(\theta(q))| + \delta,\right.\right.\\
&\qquad\left.\left.|\csc^2(\theta(q))\nu_q^{\theta} + \cot(\theta(q)) + \delta|\right\}\right) > 0,
\end{split}$$ which is finite because the cosecant and cotangent functions are bounded on ${\widetilde{Q}}$ near ${\widetilde{S}}$. Next, let $B > 0$ be large enough that, for all $Y \in T({\widetilde{X}}_{\rho_0})$ with $|Y|_g = 1$, we have $|d\rho(Y)| < B\rho\sin\theta$; such $B$ exists by (\[g\]).
For $q \in {\widetilde{Q}}_{\rho_0}$, define $\theta_q(t) = \theta(\gamma_q(t))$, and $\rho_q(t) = \rho(\gamma_q(t))$. Next, define $w_q(t) = \cot(\theta_q(t))$; thus $w_q$ is defined on the same domain as the geodesic $\gamma_q$. Noting that $w_q(0) = \cot(\theta(q))$ and $\dot{w}_q(0) = -\csc^2(\theta)\nu_q^{\theta}$, let $C$ be as defined in Lemma \[lemcsc\] and set $$A := \sup_{q \in {\widetilde{Q}}_{\rho_0}} C\left( \frac{1}{2}(w_q(0) + \dot{w}_q(0) - \delta),
\frac{1}{2}(w_q(0) - \dot{w}_q(0) - \delta),\delta\right) > 0,$$ which is finite. Also let $D$ be as in Lemma \[lemcsc\] and set $$\begin{aligned}
E &:= \sup_{q \in {\widetilde{Q}}_{\rho_0}}D\left( \frac{1}{2}(w_q(0) + \dot{w}_q(0) - \delta),\frac{1}{2}(w_q(0) + \dot{w}_q(0) + \delta),\right.\\
&\qquad\frac{1}{2}(w_q(0) - \dot{w}_q(0) - \delta),
\left.\frac{1}{2}(w_q(0) - \dot{w}_q(0) + \delta),\delta\right) > 0,
\end{aligned}$$ which is likewise finite. By shrinking it if necessary, we may assume that $\rho_0$ is small enough that, if $q \in {\widetilde{Q}}_{\rho_0}$, then $$\label{nubd}
\left|\nu_q + \sin(\theta)\frac{\partial}{\partial \theta}\right|_g < \frac{\alpha E^{-1}}{8}.$$ We may similarly suppose, by (\[g\]), that on ${\widetilde{Q}}_{\rho_0}$, $$\label{g00bd}
\sin(\theta)|g_{00} - \csc^2(\theta)| < \min\left\{\frac{\alpha E^{-1}}{8A\beta},1\right\},$$ and that if $Y \in T{\widetilde{X}}|_{{\widetilde{Q}}_{\rho_0}}$ with $d\theta(Y) = 0$, then $$\label{g0mubd}
\left|\left\langle\sin\theta\frac{\partial}{\partial \theta},Y\right\rangle\right| \leq \frac{\alpha E^{-1}}{8(1 + \sqrt{2}A\beta)}|Y|_g.$$
Now because $\gamma_q$ is a geodesic, $\frac{d^2}{dt^2}w(\gamma_q(t)) = (\nabla_g^2w)(\dot{\gamma}_q,\dot{\gamma}_q)$. It follows by Lemma \[cotlem\] that $$\ddot{w}_q = w_q + O(\rho_q(t)).$$ By shrinking $\rho_0$ if necessary, we assume that the $O(\rho)$ term in this equation is bounded by $\delta$ for $\rho \leq \rho_0$. Now let $0 < a < \frac{1}{2e^{AB}}\rho_0$. We henceforth assume $q \in {\widetilde{Q}}_{a}$.
Now let $f_{\pm}$ be the solutions to $\ddot{f}_{\pm} = f_{\pm} \pm \delta$, with $f_{\pm}(0) = w_q(0)$ and $\dot{f}_{\pm}(0) = \dot{w}_q(0)$. Then $$f_{\pm}(t) = \frac{1}{2}\left( w_q(0) + \dot{w}_q(0) \pm \delta \right)e^{t} + \frac{1}{2}
\left( w_q(0) - \dot{w}_q(0) \pm \delta \right)e^{-t} \mp \delta.$$ The leading coefficient is always positive, by our choice of $\delta$. Moreover, we have $$\ddot{f}_- - f_- = -\delta \leq \ddot{w}_q - w_q \leq \delta = \ddot{f}_+ - f_+,$$ so by Proposition \[complem\], we have $$\label{fineq}
f_-(t) \leq w_q(t) \leq f_+(t)$$ for all $t \geq 0$ such that $\rho_q(t) < \rho_0$ up to $t$, and so long as the geodesic continues to exist. Also by the same proposition, $$\label{dotineq}
\dot{f}_-(t) \leq \dot{w}_q(t) \leq \dot{f}_+(t),$$ subject to the same constraints. Since both bounding functions are positive, we conclude that $\dot{w}_q(t) > 0$ for all $q$, and for all $t \geq 0$ such that $\rho_q(t) < \rho_0$. This implies that $\dot{\theta} < 0$ for all such $t \geq 0$. In addition, the coefficients appearing in $f_{\pm}$ are uniformly bounded in $q$. It follows that we have shown that $\theta_q(t)$ goes to zero and $\cot(\theta_q(t))$ goes to infinity exponentially, *so long as* $\rho_q(t)$ remains bounded by $\rho_0$ and $\gamma_q$ exists.
Now by Lemma \[lemcsc\] and the definition of $f_-$, we have $$\csc^2(\theta_q(t)) = 1 + w_q(t)^2 \geq A^{-2}e^{2t}.$$ (This and the following continue, for now, to depend on the assumption that $\rho$ remains bounded by $\rho_0$.) Hence, also, $$\label{sinleq}
\sin(\theta_q(t)) \leq Ae^{-t}.$$ Also by (\[fineq\]), by definition of $E$, and by Lemma \[lemcsc\], we have $$\label{singeq}
\sin(\theta_q(t)) \geq E^{-1}e^{-t}.$$
It now follows from (\[sinleq\]) that $\left|\frac{\dot{\rho}_q}{\rho_q}\right| < ABe^{-t}$, by definition of $B$. Hence, at least as long as $\rho \leq \rho_0$, we find by integrating that $$\label{rhoineq}
e^{-AB}\rho_q(0) < \rho_q(t) < e^{AB}\rho_q(0).$$ But then, since $\rho_q(0) \leq a \leq \frac{\rho_0}{2e^{AB}}$, $\rho_q(t)$ must remain bounded by $\frac{\rho_0}{2}$; so a brief contradiction argument shows that, indeed, (\[fineq\]) – (\[rhoineq\]) hold for all time $t \geq 0$ such that $\gamma_q$ exists. We have also shown that $\gamma_q$ remains bounded away from ${\widetilde{S}}$, i.e., $\rho$ is bounded away from $0$. This shows that $\gamma_q$ never reaches ${\widetilde{S}}$, and also, with (\[rhoineq\]), yields \[rhocond\]. Also, by (\[sinleq\]) and (\[singeq\]), we have (c).
We next analyze the motion in the tangential directions along $S$. It follows from the definition and smoothness of $k_{\rho}$ in (\[g\]) that there exists $K > 0$ such that, for unit-length $Y \in T{\widetilde{X}}_{\rho_0}$, we have $|d\pi_S(Y)|_{k_{0}} < K\rho\sin(\theta)$. It then follows, using (\[sinleq\]) and (\[rhoineq\]), that $$\int_0^{\infty} |d\pi_S(\dot{\gamma}(t))|_{k_0} dt < AKe^{AB}\rho(q),$$ which yields \[Scond\].
Since $\cot(\theta)$ eventually becomes positive with $\dot{\theta}$ negative, we conclude that $\dot{r}$ is ultimately negative, and so the fact that $\gamma_q$ approaches a defined point of ${\widetilde{M}}$ normally, if it exists for all time, follows by the analysis of geodesics in the standard AH case, Proposition \[mazprop\]. Thus, we have established all desired behavior, except that the geodesic $\gamma_q$ might leave $\mathring{{\widetilde{X}}}$ and return to ${\widetilde{Q}}$, ceasing to exist. Since the geodesic is unit speed, it either exists for all time or returns to ${\widetilde{Q}}$ in finite time, and so we have only to show that the latter does not happen.
Suppose by way of contradiction that $\gamma_q$ does return to ${\widetilde{Q}}$, say at $q'$ and at time $t_1 > 0$. Then $\rho(q') < \rho_0$, and we have $\langle \dot{\gamma}_q(t_1),\nu_{q'}\rangle \leq 0$. Now by (\[dotineq\]), the definitions of $f_-$ and $\alpha$, and (\[singeq\]), we deduce that $$\label{tdotlow}
\frac{\alpha E^{-1}}{2} \leq \frac{|\dot{\theta}_q(t)|}{\sin(\theta_q(t))}$$ for all times $t \geq 0$, and in particular $t_1$. Similarly, by (\[dotineq\]), (\[sinleq\]), the definition of $f_+$, and the definition of $\beta$, we get $$\label{tdothigh}
\frac{|\dot{\theta}_q(t)|}{\sin(\theta_q(t))} \leq A\beta.$$ By (\[tdothigh\]) and (\[g00bd\]), and because $|\sin\theta| \leq 1$, we find that at $t_1$, $$\begin{aligned}
\left\langle\dot{\theta}\frac{\partial}{\partial \theta},\dot{\theta}\frac{\partial}{\partial \theta}\right\rangle &=
\frac{\dot{\theta}^2}{\sin^2\theta} + (g_{00} - \csc^2(\theta))\dot{\theta}^2\\
&\leq A^2\beta^2 + A^2\beta^2 = 2A^2\beta^2.
\intertext{Thus,}
\left|\dot{\theta}\frac{\partial}{\partial \theta}\right|_g &\leq \sqrt{2}A\beta.{\addtocounter{equation}{1}\tag{\theequation}}\label{thetaveclen}
\end{aligned}$$
By (\[nubd\])-(\[g0mubd\]), (\[tdotlow\])-(\[thetaveclen\]), Cauchy-Schwartz, and the triangle inequality, we find that $$\begin{aligned}
\langle \dot{\gamma}_q(t_1),\nu_{q'}\rangle &= \left\langle \dot{\gamma}_q(t_1),-\sin(\theta)\frac{\partial}{\partial \theta}\right\rangle
+ \left\langle \dot{\gamma}_q(t_1),\nu_{q'} + \sin(\theta)\frac{\partial}{\partial \theta}\right\rangle\\
&= \frac{|\dot{\theta}_q(t_1)|}{\sin(\theta)} - (g_{00} - \csc^2(\theta))\sin(\theta)\dot{\theta}_q(t_1)\\
&\qquad+
\left\langle \dot{\gamma}_q(t_1) -
\dot{\theta}_q(t_1)\frac{\partial}{\partial \theta},
-\sin(\theta)\frac{\partial}{\partial \theta}\right\rangle
+ \left\langle\dot{\gamma}_q(t_1),\nu_{q'} + \sin(\theta)\frac{\partial}{\partial \theta}\right\rangle\\
&\geq \frac{\alpha E^{-1}}{2} - \frac{\alpha E^{-1}}{8A\beta}(A\beta) - \frac{\alpha E^{-1}}{8(1 + \sqrt{2}A\beta)}
\left( 1 + \left|\dot{\theta}\frac{\partial}{\partial\theta}\right|_g \right)\\
&\qquad- |\dot{\gamma}_q|_g\cdot \left|\nu_q + \sin(\theta)\frac{\partial}{\partial\theta}\right|_g\\
&\geq \frac{\alpha E^{-1}}{8} > 0,
\end{aligned}$$ which is a contradiction. Hence, as desired, $\gamma_q$ does not return to ${\widetilde{Q}}$.
The Exponential Map {#exponential}
===================
We continue our analysis of the geodesics leaving ${\widetilde{Q}}$ normally now by turning our attention to the mapping properties of the exponential map on the normal bundle to ${\widetilde{Q}}$. We ultimately must prove that this map is a diffeomorphism on a suitable space. For now, we content ourselves with more local properties.
Let $N_+({\widetilde{Q}}\setminus {\widetilde{S}})$ be the inward-pointing half-closed normal *ray* bundle to ${\widetilde{Q}}\setminus {\widetilde{S}}$, so that $N_+({\widetilde{Q}}\setminus {\widetilde{S}}) \approx
[0,\infty)_t \times ({\widetilde{Q}}\setminus {\widetilde{S}})$ by the identification $t\nu_q \mapsto (t,q)$; and similarly for the normal bundle over subsets of ${\widetilde{Q}}\setminus {\widetilde{S}}$. We denote the normal exponential map by $\exp$. We have shown in Proposition \[geodprop\] that there is some $a > 0$ such that $\exp$ is defined on the entirety of $N_+{\widetilde{Q}}_a$, and takes its values in ${\widetilde{X}}\setminus ({\widetilde{M}}\cup {\widetilde{S}})$. Trivially, $\left\{ 0 \right\} \times
({\widetilde{Q}}\setminus {\widetilde{S}})$ is mapped by $\exp$ to ${\widetilde{Q}}\setminus {\widetilde{S}}$ as the identity, and Proposition \[geodprop\] also shows that $\exp|_{N_+{\widetilde{Q}}_a}^{-1}({\widetilde{Q}}_a) = \left\{ 0 \right\} \times
{\widetilde{Q}}_a$ as well. In order to show that the exponential map induces a diffeomorphism with a neighborhood of ${\widetilde{S}}$, we will have to analyze it as $q \to {\widetilde{S}}$ and as $t \to \infty$. We thus introduce a partial compactification of the normal bundle that includes faces corresponding to $t = \infty$ and to $[0,\infty] \times ({\widetilde{Q}}\cap {\widetilde{S}})$, and we will show that the exponential map is defined and a local diffeomorphism on the entire space.
Let $V$ be a neighborhood of ${\widetilde{Q}}\cap {\widetilde{S}}$ in ${\widetilde{Q}}$. Then as observed previously, $N_+(V\setminus{\widetilde{S}})$ has a natural identification, induced by $\nu$, with $[0,\infty) \times (V \setminus {\widetilde{S}})$. Letting $t$ be the coordinate on the first factor, we set $\tau = 1 - e^{-t}$, and hence obtain an identification with $[0,1) \times (V \setminus {\widetilde{S}})$. We thus define the compactification ${\overline{N_+(V\setminus {\widetilde{S}})}} = [0,1] \times V$, and we regard $N_+(V\setminus{\widetilde{S}}) \subset
{\overline{N_+(V \setminus {\widetilde{S}})}}$ as a subspace via the identification just described. Note that we have added two new faces in this compactification: one corresponding to $t = \infty$, and one corresponding to $[0,1] \times ({\widetilde{Q}}\cap{\widetilde{S}})$. We will consistently let $\tau$ be the coordinate on the first factor of ${\overline{N_+(V \setminus {\widetilde{S}})}}$. The space ${\overline{N_+(V\setminus {\widetilde{S}})}}$ has a natural smooth structure as a manifold with corners, and $T{\overline{N_+(V \setminus S)}} \cong T[0,1] \oplus TV$ canonically. We note that ${\overline{N_+(V\setminus {\widetilde{S}})}}$ is not quite a compactification, since the interior boundary of $V$ in ${\widetilde{Q}}$ is still not included.
With the compactification of the normal bundle in hand, we are ready to extend the exponential map to reach the boundary. In proving the following, we follow the approach in [@maz86]. For the statement, notice that $\theta \mapsto v(\theta) := \csc\theta - \cot\theta$ is a diffeomorphism of $(0,\pi)$ with $(0,\infty)$.
\[extendprop\] There exists $\rho_0 > 0$ such that the exponential map $\exp:N_+{\widetilde{Q}}_{\rho_0} \to \mathring{{\widetilde{X}}}$ extends smoothly to a map $\exp:{\overline{N_+{\widetilde{Q}}_{\rho_0}}} \to {\widetilde{X}}$, and the extended map is a local diffeomorphism of manifolds with corners. For $q \in {\widetilde{Q}}\cap {\widetilde{S}}$, $\exp$ maps $[0,1] \times \left\{ q \right\}$ to the $b$-fiber of ${\widetilde{S}}$ containing $q$. For such $q$, $\exp$ satisfies $$\label{ssoln}
v(\Theta(\exp(\tau,q))) = v(\Theta(q))(1 - \tau);$$ that is, in the $v$ coordinate, $\exp$ is a linear function of $\tau$.
Moreover, for $1 \leq \mu \leq n$ and $q \in {\widetilde{Q}}$ and for any $\tau$, the equation $$\label{quadest}
x^{\mu}(\exp(\tau,q)) = x^{\mu}(q) + O(\rho(q)^2)$$ holds uniformly in $\tau \in [0,1]$.
Finally, there exists $c > 0$ such that, if $Y \in T_q{\widetilde{Q}}_{\rho_0} \subset T[0,\infty) \oplus T{\widetilde{Q}}_{\rho_0}
\cong TN_+{\widetilde{Q}}_{\rho_0}$ and $t\nu_q \in N_+{\widetilde{Q}}_{\rho_0}$, then $$\label{ceq}
|d\exp_{t\nu_q}(Y)|_{\bar{g}} \geq c|Y|_{\bar{g}}.$$
We begin by recalling the equations for the geodesic flow on the cotangent bundle. Let $\left\{ x^s \right\}$ be local coordinates for $S$, so that $(\theta,x^s,\rho)$ are coordinates on some neighborhood ${\widetilde{U}}\subseteq {\widetilde{X}}$ of a fiber $F$ in ${\widetilde{S}}$. Let $\rho_1$ be small enough that Proposition \[geodprop\] holds on ${\widetilde{Q}}_{\rho_1}$, and let $V \subseteq {\underline{{\widetilde{Q}}_{\rho_1}}}$ be a sufficiently small neighborhood of the point ${\widetilde{Q}}\cap F$ that normal geodesics off points in $V \setminus {\widetilde{S}}$ remain in ${\widetilde{U}}$.
The geodesic flow off points of $V \setminus {\widetilde{S}}$ then satisfies $$\begin{aligned}
\dot{x}^i &= g^{ij}\xi_j\\
\dot{\xi}_i &= -\frac{1}{2}\frac{\partial g^{kl}}{\partial x^i}\xi_k\xi_l.
\end{aligned}$$ We also have $$\label{normedeq}
g^{ij}\xi_i\xi_j = 1.$$ We use this fact to rewrite the geodesic equations in terms of $\bar{g} = \rho^2\sin^2(\theta)g$ or, rather, $\bar{g}^{-1} = \rho^{-2}\csc^2(\theta)g^{-1}$, obtaining $$\label{floweq1}
\begin{aligned}
\dot{x}^i &= \rho^2\sin^2(\theta)\bar{g}^{ij}\xi_j\\
\dot{\xi}_i &= -\frac{1}{2}\frac{\partial}{\partial x^i}\left[ \rho^2\sin^2(\theta)\bar{g}^{kl} \right]\xi_k\xi_l\\
&= -\frac{\rho_i}{\rho} - \cot(\theta)\theta_i - \frac{1}{2}\rho^2\sin^2(\theta)\frac{\partial \bar{g}^{kl}}{\partial x^i}\xi_k\xi_l.
\end{aligned}$$ This system is obviously degenerate at both $\theta = 0$ and $\rho = 0$. We thus introduce rescaled variables, setting $$\label{rescaled}
\begin{aligned}
\bar{\xi}_{\mu} &= \rho \xi_{\mu}\quad(1 \leq \mu \leq n)\\
\bar{\xi}_0 &= \sin(\theta)\xi_0.
\end{aligned}$$ Hence, $$\label{eqmotresc}
\begin{aligned}
\dot{\bar{\xi}}_{\mu} &= \dot{\rho}\xi_{\mu} + \rho\dot{\xi}_{\mu} = \frac{\dot{\rho}}{\rho}\bar{\xi}_{\mu} + \rho\dot{\xi}_{\mu}\\
\dot{\bar{\xi}}_0 &= \cos(\theta)\dot{\theta}\xi_0 + \sin(\theta)\dot{\xi}_0 = \cot(\theta)\dot{\theta}\bar{\xi}_0 + \sin(\theta)\dot{\xi}_0.
\end{aligned}$$ Now $$\label{dottedeq}
\begin{aligned}
\dot{\rho} &= \rho^2\sin^2(\theta)\bar{g}^{nj}\xi_j = \rho\sin^2(\theta)\bar{g}^{n\mu}\bar{\xi}_{\mu} + \rho^2\sin(\theta)\bar{g}^{n0}\bar{\xi}_0\text{ and}\\
\dot{\theta} &= \rho^2\sin^2(\theta)\bar{g}^{0j}\xi_j = \rho\sin^2(\theta)\bar{g}^{0\mu}\bar{\xi}_{\mu} + \rho^2\sin(\theta)\bar{g}^{00}\bar{\xi}_0.
\end{aligned}$$ Thus, rewriting our equations of motion (\[floweq1\]) and (\[eqmotresc\]) in terms of our new variables, we get $$\label{floweq2}
\begin{aligned}
\dot{x}^i &= \rho\sin^2(\theta)\bar{g}^{i\mu}\bar{\xi}_{\mu} + \rho^2\sin(\theta)\bar{g}^{i0}\bar{\xi}_0\\
\dot{\bar{\xi}}_{\mu} &= (\sin^2(\theta)\bar{g}^{n\nu}\bar{\xi}_{\nu} + \rho\sin(\theta)\bar{g}^{n0}\bar{\xi}_0)\bar{\xi}_{\mu} - \rho_{\mu} -
\frac{1}{2}\rho\sin^2(\theta)\frac{\partial \bar{g}^{\sigma\lambda}}{\partial x^{\mu}}\bar{\xi}_{\sigma}\bar{\xi}_{\lambda}
\\&\qquad- \rho^2\sin(\theta)\frac{\partial \bar{g}^{0\sigma}}{\partial x^{\mu}}\bar{\xi}_0\bar{\xi}_{\sigma} - \frac{1}{2}\rho^3\frac{\partial\bar{g}^{00}}
{\partial x^{\mu}}\bar{\xi}_0^2\\
\dot{\bar{\xi}}_0 &= (\rho\sin(\theta)\cos(\theta)\bar{g}^{0\mu}\bar{\xi}_{\mu} + \rho^2\cos(\theta)\bar{g}^{00}\bar{\xi}_0)\bar{\xi}_0 - \cos(\theta)
- \frac{1}{2}\sin^3(\theta)\frac{\partial \bar{g}^{\sigma\lambda}}{\partial \theta}\bar{\xi}_{\sigma}\bar{\xi}_{\lambda}\\&\qquad
-\rho\sin^2(\theta)\frac{\partial \bar{g}^{0\sigma}}{\partial \theta}\bar{\xi}_0\bar{\xi}_{\sigma}-\frac{1}{2}\rho^2\sin(\theta)
\frac{\partial \bar{g}^{00}}{\partial \theta}\bar{\xi}_0^2.
\end{aligned}$$ Now $\bar{g}^{00} = O(\rho^{-2})$; otherwise $\bar{g}^{-1}$ is smooth on ${\widetilde{X}}$ by (\[ginv\]). It follows that equations (\[floweq2\]) have smooth coefficients all the way up to $\rho = 0$.
We already know that, for $\rho(q) \neq 0$, solutions exist for all time $t$. We turn to study the case when $\rho(q) = 0$, that is, when the geodesic starts from some point $q \in V \cap {\widetilde{S}}$. Using (\[ginv\]), and recalling that $\bar{g}_{st}$ is independent of $\theta$ at $\rho = 0$, note that when $\rho = 0$, the equations (\[floweq2\]) are given by $$\label{rho0eq}
\begin{aligned}
\dot{\theta} &= \sin(\theta)\bar{\xi}_0\\
\dot{x}^{\mu} &= 0\\
\dot{\bar{\xi}}_0 &= \cos(\theta)\bar{\xi}_0^2 - \cos(\theta)\\
\dot{\bar{\xi}}_{\mu} &= \sin^2(\theta)\bar{\xi}_n\bar{\xi}_{\mu} - \rho_{\mu} + \rho_{\mu}\bar{\xi}_0^2.
\end{aligned}$$ Our initial conditions for $q$, by (\[qnorm\]), (\[normedeq\]), and the above, are $$\label{flowindcond}
\begin{aligned}
\theta(0) &= \theta(q)\\
x^s(0) &= x^s(q)\\
\rho(0) &= 0\\
\bar{\xi}_0(0) &= -1\\
\bar{\xi}_{\mu}(0) &= 0.
\end{aligned}$$ Let $\psi(v)$ be the inverse of the function $\theta \mapsto \csc \theta - \cot \theta$; thus $\psi$ is defined on $(0,\infty)$ and is monotonic increasing from $0$ to $\pi$. Then observe that the solution to (\[floweq2\]) with the given initial conditions is given by $$\label{rho0sol}
\begin{aligned}
\theta(t) &= \psi((\csc\theta(q) - \cot\theta(q))e^{-t})\\
x^s(t) &\equiv x^s(0)\\
\rho(t) &\equiv 0\\
\bar{\xi}_0(t) &\equiv -1\\
\bar{\xi}_{\mu}(t) &\equiv 0.
\end{aligned}$$ This exists for all $t \geq 0$, and it satisfies the properties that $\dot\theta < 0$ for all time and that $\lim_{t \to \infty}\theta(t) = 0$. Using smooth dependence of solutions on initial conditions, we conclude that solutions to the geodesic equations may be smoothly extended to $\rho = 0$ for all time $t \geq 0$. Thus, $\exp$ is smooth on $[0,\infty) \times V$; and by compactness of ${\widetilde{S}}$, on $[0,\infty)_t \times {\underline{{\widetilde{Q}}_{\rho_0}}}$ for some $\rho_0$.
We now turn our attention to $\theta = 0$, which corresponds to $t = \infty$. We compactify the normal bundle, as above, by setting $\tau = 1 - e^{-t}$, and we wish to show that the exponential map is smooth to $\tau = 1$. It will be important throughout to understand the asymptotic behavior of $\bar{\xi}_i$. Now by (\[normedeq\]), we have $$\begin{aligned}
1 &= g^{-1}(\xi,\xi)\\
&= \left(\begin{array}{cc}
\sin(\theta)\xi_0,&\rho\sin(\theta)\xi_{\mu}
\end{array}\right)B
\left(\begin{array}{c}
\sin(\theta)\xi_0\\
\rho\sin(\theta)\xi_{\mu}\end{array}\right),\end{aligned}$$ where $B$ is a smooth, uniformly positive definite matrix on ${\widetilde{U}}$. Thus, for some $c > 0$, we get $$c(\sin^2(\theta)\xi_0^2 + \delta^{\mu\nu}\rho^2\sin^2(\theta)\xi_{\mu}\xi_{\nu}) \leq 1,$$ from which it follows that $\xi_0 = O(\csc\theta)$ and $\xi_{\mu} = O(\rho^{-1}\csc\theta)$. Hence, $\bar{\xi}_0 = O(1)$ and $\bar{\xi}_{\mu} = O(\csc\theta) = O(e^t)$, both uniformly in the starting point $q$. Putting this into (\[floweq2\]), we find that $\dot{\bar{\xi}}_{\mu} = O(1)$, from which it follows that we can improve our estimate to $\bar{\xi}_{\mu} = O(t) = O(|\log \sin \theta|)$. Finally, we put this back into (\[normedeq\]) and substitute (\[rescaled\]) to conclude that $\bar{\xi}_0^2 \to 1$ as $t \to \infty$ or $\theta \to 0$, indeed, that $\bar{\xi}_0^2 = 1 + O(e^{-t})
= 1 + O(\sin(\theta))$. Due to the sign of $\dot{\theta}$, we may likewise conclude that $$\bar{\xi}_0 = -1 + O(e^{-t}) = -1 + O(\sin(\theta)).$$ We here use that $\rho^2\bar{g}^{00} = 1 + O(\rho\sin(\theta))$.
Because $\dot{\theta} < 0$ for all $t$, we may reparametrize our geodesic equations by $\theta$. This amounts, by the chain rule, to dividing by $\dot{\theta}$, and by (\[dottedeq\]), we have $\dot{\theta} = \rho\sin^2(\theta)\bar{g}^{0\mu}\bar{\xi}_{\mu} + \rho^2\sin(\theta)\bar{g}^{00}\bar{\xi}_0$. (We recall that $\bar{g}^{00} = O(\rho^{-2})$, and regard $\rho^2\bar{g}^{00}$ as a single smooth function up to $\rho = 0$, which however does not vanish.) Changing variables on the first equation in (\[floweq2\]) we then get $$\frac{dx^{\mu}}{d\theta} = \frac{\rho\sin^2(\theta)\bar{g}^{\mu\nu}\bar{\xi}_{\nu} + \rho^2\sin(\theta)\bar{g}^{\mu 0}\bar{\xi}_0}
{\rho\sin^2(\theta)\bar{g}^{0\nu}\bar{\xi}_{\nu} + \rho^2\sin(\theta)\bar{g}^{00}\bar{\xi}_0}
= \frac{\rho\sin(\theta)\bar{g}^{\mu\nu}\bar{\xi}_{\nu} + \rho^2\bar{g}^{\mu0}\bar{\xi}_0}
{\rho\sin(\theta)\bar{g}^{0\nu}\bar{\xi}_{\nu} + \rho^2\bar{g}^{00}\bar{\xi}_0}.$$ Now, the denominator is just $\frac{\dot{\theta}}{\sin(\theta)}$, which we know by Proposition \[geodprop\] is nonzero as a function of $t$ when $t$ is finite, and thus as a function of $\theta$ when $\theta > 0$. On the other hand, when $\theta \to 0$, the denominator goes to $-1$ by our above computations of $\bar{\xi}_i$ asymptotics. Thus, the denominator is bounded away from zero, and the equation is smooth in a neighborhood of our solutions.
We next study $\frac{\partial \bar{\xi}_{\mu}}{\partial \theta}$. All but two of the terms in $\dot{\bar{\xi}}_{\mu}$ have a factor of $\sin(\theta)$ and thus yield to the same analysis we just performed. Focusing on the remaining terms, we have $$\frac{\partial \bar{\xi}_{\mu}}{\partial \theta} = (smooth) - \frac{\rho_{\mu} + \frac{1}{2}\rho^3\frac{\partial \bar{g}^{00}}{\partial x^{\mu}}\bar{\xi}_0^2}
{\rho\sin^2(\theta)\bar{g}^{0\mu}\bar{\xi}_{\mu} + \rho^2\sin(\theta)\bar{g}^{00}\bar{\xi}_0}.$$ Now if $\mu = s \neq n$, the numerator vanishes to order $\sin(\theta)$, so again the equations are smooth in a neighborhood of our solutions. If $\mu = n$, then the numerator is $1 - \bar{\xi}_0^2 + O(\sin(\theta))$. But $\bar{\xi}_0^2 = 1 + O(\sin(\theta))$, so the numerator is $O(\sin(\theta))$, and this equation is smooth in a neighborhood of our solutions. Finally we study $\frac{\partial \bar{\xi}_0}{\partial \theta}$. Once again, only two terms of $\dot{\bar{\xi}}_0$ lack a factor of $\sin(\theta)$, their sum being $\cos(\theta)(\rho^2\bar{g}^{00}\bar{\xi}_0^2 - 1)$. For the same reasons as before, this is in fact $O(\sin(\theta))$. Thus, the entire $(x^{\mu},\bar{\xi}_i)$ system is smooth up to $\theta = 0$; and so the solutions are smooth as functions of $\theta$, and depend smoothly on the initial point $q \in {\underline{{\widetilde{Q}}_{\rho_0}}}$. All of this analysis is uniform up to $\rho = 0$.
It remains to show that $\theta$ is smooth in $\tau$ up to $\tau = 1$, and depends smoothly on $q$; of course, we already know this for $\tau < 1$. We have just used $2n + 1$ of the equations in (\[floweq2\]); the remaining equation, for $\dot{\theta}$, can now be written as a scalar ODE for $\theta$ in terms of $t$, since $x^{i}$ and $\bar{\xi}_i$ depend smoothly on $\theta$. Explicitly, the equation is $$\dot{\theta} = \rho\sin^2(\theta)\bar{g}^{0\mu}\bar{\xi}_{\mu} + \rho^2\sin(\theta)\bar{g}^{00}\bar{\xi}_0.$$ The right-hand side is $O(\theta)$, so write the equation as $$\label{dotphi}
\dot{\theta} = -\theta a(\theta),$$ Here $a$ is smooth in $\theta$ all the way to $\theta = 0$. It is clear from our earlier analysis of $\bar{\xi}_0$ that $a(0) = 1$ for all $q$, and also that $a$ is nonvanishing for $\theta$ along our curves. We reparametrize $\theta$ by $\tau = 1 - e^{-t}$, and (\[dotphi\]) becomes $$(\tau - 1)\frac{d\theta}{d\tau} = \theta a(\theta).$$ This is a separable equation; if we write $a(\theta)^{-1} = 1 + \theta b(\theta)$, then the equation has solution $$\theta e^{\int_0^{\theta}b(\zeta)d\zeta} = c(1 - \tau),$$ which holds for $0 \leq \tau \leq 1$, and where $b$ is smooth in both $\theta$ and $q$. Now by the implicit function theorem, this uniquely defines $\theta$ as a function of $\tau$ and $q$ near $\tau = 1$, smoothly depending on both variables. Thus, as desired, $\theta$ – and, hence, the entire $(x^i,\bar{\xi}_i)$ system – exists and depends smoothly on both $\tau$ and $q$ for $\tau \in [0,1]$ and for $q$ up to ${\widetilde{S}}$.
We have still to show that $\exp$ is a local diffeomorphism on ${\overline{N_+{\widetilde{Q}}_{\rho_0}}}$. Now it is elementary that, given a smooth map between manifolds with corner which takes the corner to the corner, the boundary interior to the boundary interior, and the interior to the interior, it is a local diffeomorphism if and only if its differential is nowhere singular. It suffices, then, to show that $d\exp$ is everywhere a bijection, or that $\det d\exp \neq 0$. Given this last formulation, it suffices to show this on $[0,1] \times ({\widetilde{Q}}\cap {\widetilde{S}})
\subset {\overline{N_+{\widetilde{Q}}_{\rho_0}}}$, and it will then follow for $(\tau,q) \in [0,1] \times {\underline{{\widetilde{Q}}_{\rho_0}}}$ by shrinking $\rho_0$. Now it is clear from (\[rho0eq\]) that $\exp|_{[0,1] \times ({\widetilde{Q}}\cap {\widetilde{S}})}:[0,1] \times ({\widetilde{Q}}\cap {\widetilde{S}}) \to {\widetilde{S}}$ is a diffeomorphism. Moreover, by Proposition \[geodprop\](a), $(d\exp)|_{T{\overline{N_+({\widetilde{Q}}\setminus{\widetilde{S}})}}|_{[0,1]\times
({\widetilde{Q}}\cap {\widetilde{S}})}}$ takes nonzero transverse vectors to nonzero transverse vectors. Thus, $d\exp$ is an isomorphism, and $\exp$ is a local diffeomorphism on all of ${\overline{N_+{\widetilde{Q}}_{\rho_0}}}$ (for $\rho_0$ small).
Next we wish to demonstrate (\[quadest\]) using (\[floweq2\]) and the equation of variation. We consider perturbations about the solution (\[rho0sol\]) starting from $q \in {\widetilde{Q}}\cap {\widetilde{S}}$, as $q$ varies. Write (\[floweq2\]) as $(x,\bar{\xi})^{\cdot} = F(x,\bar{\xi})$; let $F^{\mu}$ be the component of $F$ corresponding to $x^{\mu}$. Then the equation of variation tells us that $$\label{eqvar}
\frac{\partial}{\partial t}\frac{\partial x^{\mu}}{\partial \rho(q)} = \frac{\partial F^{\mu}}{\partial x^i}\frac{\partial x^i}{\partial \rho}
+ \frac{\partial F^{\mu}}{\partial \bar{\xi}_i}\frac{\partial \bar{\xi}_i}{\partial \rho};$$ and because, at $t = 0$, $x^{\mu}$ are simply the coordinates of $q$, we have initial condition $\left.\frac{\partial x^{\mu}}{\partial \rho}\right|_{t = 0} = \delta^{\mu n}$. There is additionally an initial condition for $\left.\frac{\partial \bar{\xi}_i}{\partial\rho}\right|_{t = 0}$, smooth in $q$, but we do not need to write it explicitly.
We claim that the right-hand side of (\[eqvar\]) is $0$ along our solution. By (\[rho0sol\]), we have in this case $\rho = 0$ and $\bar{\xi}_{\mu} = 0$, and $\bar{\xi}_0 = -1$. First consider the first term of (\[eqvar\]), involving $\frac{\partial F^{\mu}}{\partial x^i}$. By (\[floweq2\]), $F^{\mu} = \rho\sin^2(\theta)\bar{g}^{\mu\nu}\bar{\xi}_{\nu} + \rho^2\sin(\theta)\bar{g}^{\mu 0}\bar{\xi}_0$. Because $\bar{\xi}_{\nu} = 0$ along our solution, the derivative of the first term of $F^{\mu}$ vanishes easily, and the derivative of the second term vanishes because $\rho = 0$. Very similar considerations show that the second term of (\[eqvar\]) vanish because $\rho = 0$ along the solution. Hence, the entire right-hand side of (\[eqvar\]) vanishes identically along our solution. Thus, $\frac{\partial x^{\mu}}{\partial\rho(q)} = \delta^{\mu n} + O(\rho)$, which establishes (\[quadest\]).
We now turn to the final statement. Let ${\widetilde{U}}$, $V$, and ${\widetilde{Q}}_{\rho_0}$ be as above. Notice by (\[g\]) that $\bar{g}$ extends to ${\widetilde{S}}$ as a smooth symmetric positive semidefinite tensor field, and that along ${\widetilde{S}}$, we have $\ker \bar{g} = \operatorname{span}\left\{ \frac{\partial}{\partial \theta} \right\}$. It follows that $\bar{g}|_{TV}$ is a metric. Now for any $(\tau,q) \in {\overline{N_+(V\setminus{\widetilde{S}})}}$, we have $T_{(\tau,q)}{\overline{N_+(V\setminus{\widetilde{S}})}} \cong \mathbb{R}\frac{\partial}{\partial \tau}
\oplus T_qV$ canonically. For $0 \leq \tau \leq 1$, define $\exp_{\tau}:\overline{V} \to {\widetilde{X}}$ by $\exp_{\tau}(q) = \exp(\tau,q)$. The function $f:[0,1] \times T\overline{V} \to {\widetilde{X}}$ given by $f(\tau,Y) = |d\exp_{\tau}(Y)|_{\bar{g}}$ is a smooth map. Now $\exp$ is a local diffeomorphism such that $0 \neq d\exp_{(\tau,q)}\left( \frac{\partial}{\partial \tau} \right)
\in \operatorname{span}\frac{\partial}{\partial \theta} = \ker\bar{g}$ for $q \in V \cap {\widetilde{S}}$. We conclude that $f$ is nonvanishing. Thus, it attains a positive minimum on the compact set $[0,1] \times S^1_{\bar{g}}T\overline{V}$. This yields the claim.
The above proof relied in a fundamental way on the behavior of the extension $\exp$ to the boundary ${\widetilde{S}}$ in order to show that $\exp$ is a local diffeomorphism. It is possible to give a proof on the interior that $\exp$ is a local diffeomorphism using Jacobi fields in a more general setting. The following result is unlikely to surprise practitioners in the area, but we did not find a published proof. Because of its potential applications in other settings, it may be worthwhile to record explicitly in the literature, so we state and prove the result, and then use it in Proposition \[difflem\] to give an alternate proof of the local diffeormorphism property in Proposition \[extendprop\]. The remainder of this section will not be used in subsequent sections.
\[explem\] Let $\beta > 0$ and $0 < \kappa < \sqrt{\beta}$, and let $(Z,g)$ be a Riemannian manifold with hypersurface $Q$ having unit normal field $\nu$. Suppose that $|g^{-1}K| \leq \kappa$ on $Q$, where $K$ is the second fundamental form of $Q$ and $|g^{-1}K|$ is the maximal absolute value of an eigenvalue of the shape operator. Moreover, suppose $W \subseteq N_+Q$ is an open subset of the one-sided normal bundle to $Q$ having the property that whenever $Y \in W$, $tY \in W$ for $0 \leq t \leq 1$. Finally suppose that all sectional curvatures of $g$ are bounded above by $-\beta$ on $\exp(W)$. Then $\exp$ is a local diffeomorphism on $W$, and if $\xi:(-\varepsilon,\varepsilon) \to Q$ is a smooth curve with $\nu_{\xi(s)} \in W$ and if $\Gamma:[0,a) \times (-\varepsilon,\varepsilon) \to Z$ is given by $\Gamma_t(s) := \Gamma(t,s)
= \exp(t\nu_{\xi(s)})$, then for all $t \geq 0$ and $s \in (-\varepsilon,\varepsilon)$, $|\Gamma_t'(s)|_g \geq c|\xi'(s)|_g$, where $c = \sqrt{\frac{1}{2}\left( 1 - \frac{\kappa^2}{\beta} \right)}$.
Let $\pi:N_+Q \to Q$ be the basepoint map. For convenience, we assume that $\pi(W) = Q$ (or we could just restrict $Q$). For each $p \in Q$, we let $\gamma_p$ be the geodesic in $Z$ for which $\gamma(0) = p$ and $\gamma'(0) = \nu_p$.
Let $(t_0,p) \in [0,\infty) \times Q \approx N_+Q$ be fixed. Plainly $d\exp_{(t_0,p)}\left( \frac{d}{dt} \right) = \gamma_p'(t_0) \neq 0$. For $Y \in T_pQ$ with $|Y|_g = 1$ for convenience, let $\xi:(-\varepsilon,\varepsilon) \to Q$ be a smooth curve such that $\xi(0) = p$ and $\xi'(0) = Y$. Let $a > 0$ be sufficiently small that $a\nu_{\xi(s)} \in W$ for each $s \in (-\varepsilon,\varepsilon)$ (shrinking $\varepsilon$ if necessary). For $(t,s) \in [0,a) \times (-\varepsilon,\varepsilon)$, define $\Gamma(t,s) =
\gamma_{\xi(s)}(t)$. Then $d\exp_{(t_0,p)}(Y) = \partial_s\Gamma(t_0,0)$. (We are using the identification $T_{(t_0,p)}N_{+}Q
\cong \mathbb{R} \oplus T_pQ$.) This is simply the Jacobi field along $\gamma_p$ defined by the smooth variation $\Gamma$ evaluated at $t_0$. Thus, since $Y$ is arbitrary, it suffices to show that the Jacobi field $J(t) = \partial_s\Gamma(t,0)$ is nonvanishing and is nowhere parallel to $\gamma_p'(t)$. At $t = 0$, we have $J(0) = \xi'(0) = Y \perp \nu_p$. Moreover, by the symmetry lemma we have $$D_tJ(t) = D_t\partial_s\Gamma(t,s)|_{s = 0} = D_s\partial_t\Gamma(t,s)|_{s = 0},$$ where $D_t$ denotes covariant differentiation along the curve $t \mapsto \Gamma(t,s)$, and similarly for $D_s$. At $t = 0$, this gives $$D_tJ(0) = D_s\gamma_{\xi(s)}'(0)|_{s = 0} = D_s\nu \perp \nu = \gamma_p'(0),$$ since $\nu$ is a unit vector field. Thus, at $t = 0$, both $J$ and $D_tJ$ are normal to $\gamma_p'$, which implies that $J$ is a normal Jacobi field. In particular, if nonvanishing, it is nowhere parallel to $\gamma_p'(t)$.
Set $f(t) = \langle J(t),J(t)\rangle_g$. Then $$\label{eqfprime}
f'(t) = \frac{d}{dt}\langle J,J\rangle_g = 2\langle D_tJ,J\rangle_g.$$ It follows by Cauchy-Schwartz that $$\label{eqcs}
|f'(t)| \leq 2|f(t)|^{\frac{1}{2}}|D_tJ|_g.$$ As we have seen, $D_tJ|_{t = 0} = D_s\nu$, which is simply the shape operator applied to $Y = \xi'(0) = J(0)$. It follows by our hypothesis that $\rho_0$, $\langle D_tJ,J\rangle_g|_{t = 0} \geq -\kappa$, so $$f'(0) = \frac{d}{dt}\langle J,J\rangle|_{t = 0} \geq -2\kappa.$$ Now by (\[eqfprime\]), (\[eqcs\]), and the Jacobi equation, whenever $f(t) \neq 0$ we have $$\begin{aligned}
f''(t) = \frac{d^2}{dt^2}\langle J,J\rangle_g &= 2\langle D_t^2J,J\rangle + 2\langle D_tJ,D_tJ\rangle\nonumber\\
&\geq -2R(J(t),\gamma_p'(t),\gamma_p'(t),J(t)) + \frac{1}{2}\frac{f'(t)^2}{f(t)}\nonumber\\
&> 2\beta f(t) + \frac{1}{2}\frac{f'(t)^2}{f(t)}
\end{aligned}$$ since the sectional curvature is less than $-\beta$.
We briefly pause to define weighted hyperbolic trigonometric functions. For $\eta \in \mathbb{R}$, define $\cosh_{\eta}(t) = \frac{1}{2}(e^t + \eta e^{-t})$, and $\sinh_{\eta}(t) = \frac{1}{2}(e^t - \eta e^{-t})$. It is easy to show that $\cosh_{\eta}'(t) = \sinh_{\eta}(t)$ and $\sinh_{\eta}'(t) = \cosh_{\eta}(t)$, and also that $\sinh_{\eta}^2(t) = \cosh_{\eta}^2(t) - \eta$.
Now consider the second-order differential equation given by $$h''(t) = 2\beta h(t) + \frac{1}{2}\frac{h'(t)^2}{h(t)},$$ with $h(0) = 1$ and $h'(0) = -2\kappa$. Let $A = \frac{1}{2}\left(1 - \frac{\kappa^2}{\beta}\right) > 0,
B = \frac{1}{2}\left(1 - \frac{\kappa}{\sqrt{\beta}}\right)^2 > 0$, and $\eta = \frac{(\sqrt{\beta} + \kappa)^2}{(\sqrt{\beta} - \kappa)^2} > 0$. Then a solution to our initial-value problem is $h(t) =
A + B \cosh_{\eta}(2\sqrt{\beta}t)$, as may be easily checked. We wish to apply Proposition \[complem\] to show that $f(t) \geq h(t)$, and that thus $f$ is bounded below by $A = \frac{1}{2}\left(1 - \frac{\kappa^2}{\beta}\right)$. Define $a:\mathbb{R}^2\setminus (\left\{ 0 \right\} \times \mathbb{R}) \to \mathbb{R}$ by $$a(u,v) = 2\beta u + \frac{1}{2}\frac{v^2}{u}.$$ Now define $b:\mathbb{R}^2 \to \mathbb{R}$ by $$b(u,v) = \left\{
\begin{array}{lr}
2\beta u + \frac{1}{2}\frac{v^2}{u} & |v| < 2\sqrt{\beta}|u|\\
4\beta u & |v| \geq 2\sqrt{\beta}|u|
\end{array}\right..$$ Note that $b$ is Lipschitz, and that whenever $u > 0$, we have $a \geq b$.
Now $h$ satisfies $h''(t) - a(h(t),h'(t)) = 0$; and because $|h'(t)| < 2\sqrt{\beta}|h(t)|$, it also satisfies $h''(t) - b(h(t),h'(t)) = 0$. Similarly, because $f(0) \geq 0$ and because, whenever $f \neq 0$, $f$ satisfies $f''(t) \geq a(f(t),f'(t))$, it follows that, at least up until $f$ vanishes for the first time, we have $f''(t) \geq b(f(t),f'(t))$. But $b$ is Lipschitz and is nondecreasing in $u$. Thus, by Proposition \[complem\], $f \geq h$ on any interval $I = [0,t_0]$ such that $f$ is nonvanishing on $I$. But since $f$ is continuous, and $h$ is bounded away from 0 by $A > 0$, we conclude that $f$ must be everywhere greater than $A$. This yields the claim, taking $c = \sqrt{A}
= \sqrt{\frac{1}{2}\left( 1 - \frac{\kappa^2}{\beta} \right)}$.
To apply this to our situation, we require two lemmas that will also be independently useful in applications.
Let ${\widetilde{U}}$ be a neighborhood on which a polar identification ($\theta,\pi_S,\rho)$ exists.
\[ksmoothlem\] Let $g$ be an admissible metric on $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$. If $\bar{g} = \rho^2\sin^2(\theta)g$ is the compactified metric on the interior of ${\widetilde{U}}$, then the second fundamental form $\overline{K}$ of $({\widetilde{Q}}\setminus {\widetilde{S}}) \cap {\widetilde{U}}$ with respect to $\bar{g}$ extends smoothly to ${\widetilde{Q}}\cap {\widetilde{U}}$.
Notice once again that by (\[g\]), $\bar{g}$ extends smoothly to ${\widetilde{U}}$ as a smooth tensor field (although not as a metric). Let $\bar{\nu}$ be the inward unit normal vector field on ${\widetilde{Q}}\setminus {\widetilde{S}}$ with respect to $\bar{g}$. By Lemma \[normlem\], $\bar{\nu} = -\frac{1}{\rho}\frac{\partial}{\partial \theta} + O_{\bar{g}}(\rho)$. We next wish to consider $\overline{\Gamma}_{i0j} = \frac{1}{2}(\partial_i\bar{g}_{0j} + \partial_{\theta}\bar{g}_{ij} - \partial_j\bar{g}_{i0})$. Plainly this is smooth on ${\widetilde{U}}$. But moreover, by (\[g\]), we see that it is $O(\rho)$. (Remember that $k_{\rho}$ is independent of $\theta$). All other Christoffel symbols are likewise smooth. Now using Weingarten’s equation, we have in coordinates $$\begin{aligned}
\overline{K}_{ij} &= -\bar{g}_{kj}\overline{\nabla}_i\bar{\nu}^k\\
&= -\bar{g}_{kj}\partial_i\bar{\nu}^k - \bar{\nu}^l\overline{\Gamma}_{ilj}\\
&= \bar{g}_{0j}\partial_i(\rho^{-1}) + (smooth).
\end{aligned}$$ Since $\bar{g}_{0j} = O(\rho^2)$ for any $0 \leq j \leq n$, we conclude that $\overline{K}$ extends smoothly to ${\widetilde{Q}}\cap {\widetilde{S}}$.
\[curvlem\] Let $g$ be an admissible metric on $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$ and $R$ its curvature tensor. Then $R_{ijkl} + (g_{ik}g_{jl} - g_{il}g_{jk}) = O_g(\rho\sin\theta)$.
We begin by showing that $T_{ijkl} := R_{ijkl} + (g_{ik}g_{jl} - g_{il}g_{jk}) = O_g(\rho)$, using a modification of the proof of Proposition 1.10 of [@maz86].
Let $r = \rho\sin\theta$, so that $\bar{g} = r^2g$. Now the standard formula for conformal change of the Riemann tensor shows that $$R_{ijkl} = r^{-2}\overline{R}_{ijkl} + r^{-3}\left( r_{jk}\bar{g}_{il} + r_{il}\bar{g}_{jk} -
r_{ik}\bar{g}_{jl} - r_{jl}\bar{g}_{ik}\right) - |\nabla r|_{\bar{g}}^2\left( g_{il}g_{jk} -
g_{ik}g_{jl}\right),$$ where $r_{jk}$ represents the Hessian of $r$ taken with respect to $\bar{g}$. Thus, the expression we are interested in takes the form $$\begin{gathered}
\label{Rinterest}
R_{ijkl} + (g_{ik}g_{jl} - g_{il}g_{jk}) = r^{-2}\overline{R}_{ijkl} + r^{-3}(r_{jk}\bar{g}_{il} + r_{il}\bar{g}_{jk}
- r_{ik}\bar{g}_{jl} - r_{jl}\bar{g}_{ik})\\
+ (g_{il}g_{jk} - g_{ik}g_{jl})(1 - |\nabla r|_{\bar{g}}^2).
\end{gathered}$$ Now using the fact that $r = \rho \sin \theta$ and (\[ginv\]), it follows immediately that $|\nabla r|_{\bar{g}}^2 = 1 + O(\rho)$. Thus, the last term is $O_g(\rho)$.
We next turn to computing $\overline{R}_{ijkl}$. It will be convenient to use the formula $$\overline{R}_{ijkl} = \frac{1}{2}\left( \partial_{jl}^2\bar{g}_{ik} + \partial_{ik}^2\bar{g}_{jl} -
\partial_{il}^2\bar{g}_{jk}
-\partial_{jk}^2\bar{g}_{il}\right) + \bar{g}^{pq}\left( \overline{\Gamma}_{jlp}\overline{\Gamma}_{ikq} - \overline{\Gamma}_{jkp}
\overline{\Gamma}_{ilq}\right),$$ where $\overline{\Gamma}_{ijk} = \frac{1}{2}(\partial_i\bar{g}_{jk} + \partial_j\bar{g}_{ij} - \partial_k\bar{g}_{ij})$. We thus compute these Christoffel symbols. Using our polar $g$ coordinates and (\[g\]), we find that $$\begin{array}{lll}
\overline{\Gamma}_{000} = O(\rho^3) & \overline{\Gamma}_{00u} = O(\rho^3) & \overline{\Gamma}_{00n} = -\rho + O(\rho^2)\\
\overline{\Gamma}_{0s0} = O(\rho^2) & \overline{\Gamma}_{0su} = O(\rho) & \overline{\Gamma}_{0sn} = O(\rho)\\
\overline{\Gamma}_{0n0} = \rho + O(\rho^2) & \overline{\Gamma}_{0nu} = O(\rho) & \overline{\Gamma}_{0nn} = O(\rho)\\
\overline{\Gamma}_{st0} = O(\rho) & \overline{\Gamma}_{stu} = O(1) & \overline{\Gamma}_{stn} = O(1)\\
\overline{\Gamma}_{sn0} = O(\rho) & \overline{\Gamma}_{snu} = O(1) & \overline{\Gamma}_{snn} = O(\rho)\\
\overline{\Gamma}_{nn0} = O(\rho) & \overline{\Gamma}_{nnu} = O(1) & \overline{\Gamma}_{nnn} = O(1).
\end{array}$$ Now using these computations, (\[g\]), and (\[ginv\]), it follows straightforwardly that $$\begin{aligned}
\overline{R}_{0\mu\nu0} &= O(\rho)\\
\overline{R}_{i\mu\nu\sigma} &= O(1),
\end{aligned}$$ where $1 \leq \mu \leq n$ and $0 \leq i \leq n$. It follows, since $\sin(\theta)\frac{\partial}{\partial \theta}$ and $\rho\sin(\theta)\frac{\partial}{\partial x^{\mu}}$ are a basis of approximately $g$-unit vector fields, that $r^{-2}\overline{R} = O_g(\rho)$.
Finally, we compute that $$r_{ij} = 2\rho_{(i}\theta_{j)}\cos(\theta) - \sin(\theta)\bar{g}^{nk}\overline{\Gamma}_{ijk} -
\rho\cos(\theta)\bar{g}^{0k}\overline{\Gamma}_{ijk},$$ from which it follows that $r_{\mu\nu} = O(1)$, that $r_{0\mu} = O(\rho)$, and that $r_{00} = O(\rho^2)$. Thus, the second term of (\[Rinterest\]) is also $O_g(\rho)$, which yields the claim that $T = O_g(\rho)$.
Now, near neighborhoods in ${\widetilde{M}}$ away from ${\widetilde{M}}\cap {\widetilde{S}}$, it follows from the usual curvature result for asymptotically hyperbolic spaces, given in Proposition 1.10 of [@maz86], that $T = O_g(\sin(\theta))$. Thus, the result will follow if we can show that $T$ is smooth as a section of the bundle $\otimes^4({}^{0e}T^*{\widetilde{X}})$. But this follows from what we have already done, and in particular from (\[Rinterest\]). First, $r^{-2}\overline{R}_{ijkl}$ is smooth as a section of the bundle, since no more than two of the indices can be $0$, and a smooth frame for ${}^{0e}T^*{\widetilde{X}}$ is given by (\[dualframe\]). But the second term similarly is smooth, for we have just seen that $r_{jk} = O(\rho)$ whenever either index is $0$. Thus, $T$ is a smooth section and is $O_g(\rho)$ and $O_g(\sin(\theta))$. The result follows.
We can now give the alternate proof of the local diffeomorphism property.
\[difflem\] Let $({\widetilde{X}},{\widetilde{M}},{\widetilde{Q}},{\widetilde{S}})$ be the blowup of the cornered space $(X,M,Q)$, and let $g$ be an admissible metric on ${\widetilde{X}}$. There exists $\rho_0 > 0$ such that the map $\exp:N_+{\widetilde{Q}}_{\rho_0} \to {\widetilde{X}}$ is a local diffeomorphism on the normal bundle $N_+{\widetilde{Q}}_{\rho_0}$. Moreover, there exists some $c > 0$ such that, if $\xi:(-\varepsilon,\varepsilon) \to {\widetilde{Q}}_{\rho_0}$ is a smooth curve and $\Gamma:[0,\infty)\times(-\varepsilon,\varepsilon) \to {\widetilde{X}}$ is given by $\Gamma_t(s) := \Gamma(t,s)
= \exp(t\nu_{\xi(s)})$, then for all $t \geq 0$ and $s \in (-\varepsilon,\varepsilon)$, we have $|\Gamma_t'(s)|_g \geq c|\xi'(s)|_g$.
It suffices to prove the second claim. Let $0 < \kappa < 1$ be such that $|\cos\theta| < \kappa$ on ${\widetilde{Q}}\cap {\widetilde{S}}$, which exists by compactness. Also let $\kappa < \beta < 1$.
By Lemma \[curvlem\], there is some $\rho_{\beta}$ such that the sectional curvatures of $g$ are strictly less than $-\beta$ for all $x \in \mathring{{\widetilde{X}}}_{\rho_{\beta}}$. By Proposition \[geodprop\], we can choose $\rho_0 > 0$ such that, for $q \in {\widetilde{Q}}_{\rho_0}$, $\gamma_q$ remains in ${\widetilde{X}}_{\rho_{\beta}}$.
We begin by studying the eigenvalues of the second fundamental form of ${\widetilde{Q}}$, which we denote by $K(Y,Z) =
\langle\nabla_YZ,\nu\rangle_g$ (and correspondingly for $\overline{K}$ with respect to $\bar{g}$), and to do this we first compute the compactified second fundamental form $\overline{K}$. For $r \neq 0$, the unit $\bar{g}$-normal vector field to ${\widetilde{Q}}_{\rho_0}$ is given by $\bar{\nu} = r^{-1}\nu$. (Recall that $r = \rho\sin\theta$.) A straightforward computation shows that for any vector fields $X, Y$ tangent to ${\widetilde{Q}}$, we have $$\overline{\nabla}_XY = \nabla_XY + r^{-1}\left[ dr(X)Y + dr(Y)X - \langle X,Y\rangle_{\bar{g}}\operatorname{grad}_{\bar{g}}r \right].$$ For $q \in {\widetilde{Q}}_{\rho_0}$ and $X, Y \in T_q{\widetilde{Q}}$, it follows (taking extensions where necessary) that $$\begin{aligned}
\overline{K}(X,Y) &= -\langle\overline{\nabla}_X(r^{-1}\nu),Y\rangle_{\bar{g}}\\
&= -r^{-1}\langle\nabla_X\nu + dr(X)\bar{\nu} + dr(\bar{\nu})X - \langle X,\bar{\nu}\rangle\operatorname{grad}_{\bar{g}}r - dr(X)\bar{\nu},Y\rangle_{\bar{g}}\\
&= -r^{-1}(r^2K(X,Y) - \bar{g}(X,Y)dr(\bar{\nu})).
\end{aligned}$$ Now let $Y, Z \in TQ$ be $g$-unit vectors over the same point, and let $\overline{Y} = r^{-1}Y$ and $\overline{Z} = r^{-1}Z$ be the parallel $\bar{g}$-unit vectors. It follows that $$K(Y,Z) = g(Y,Z)dr(\bar{\nu}) + r\overline{K}(\overline{Y},\overline{Z}).$$ Now $dr = \sin\theta d\rho + \rho\cos\theta d\theta$, and by Lemma \[normlem\], $\bar{\nu} = (-\frac{1}{\rho} + O(1)) \frac{\partial}{\partial \theta}
+ O(\rho)$. Thus, $|dr(\bar{\nu})| \to |\cos(\theta)| < \kappa$ as $\rho \to 0$. Since $\overline{K}$ is smooth on all of ${\underline{{\widetilde{Q}}_{\rho_0}}}$ by Lemma \[ksmoothlem\], it follows that for $\rho$ small enough, the eigenvalues of the shape operator $g^{-1}K$ are bounded in absolute value by $\kappa$: $|\lambda| < \kappa$. We restrict $\rho_0$ if necessary to ensure this condition.
The result now follows straightforwardly by applying Proposition \[explem\] with $Z = X$, with $Q = {\widetilde{Q}}_{\rho_0}$, and with $W = N_+{\widetilde{Q}}_{\rho_0}$.
Injectivity
===========
In the preceding sections, we have shown that there is a neighborhood ${\underline{{\widetilde{Q}}_{\rho_0}}}$ of ${\widetilde{S}}$ in ${\widetilde{X}}$ such that $\exp:{\overline{N_+{\widetilde{Q}}_{\rho_0}}} \to {\widetilde{X}}$ is a local diffeomorphism. The remaining step to show that $\exp$ is a diffeomorphism onto its image is to prove injectivity.
We will first work on the interior or non-compactified normal bundle $N_+{\widetilde{Q}}$ near a fixed point of ${\widetilde{Q}}\cap {\widetilde{S}}$. We will then make the result global along ${\widetilde{Q}}\cap {\widetilde{S}}$ using a compactness argument.
We prove injectivity on the interior using a homotopy lifting argument whose structure is that of Theorem 2 in [@her63]. We first prove a lifting result. We let $\pi:N_+({\widetilde{Q}}\setminus {\widetilde{S}}) \to ({\widetilde{Q}}\setminus{\widetilde{S}})$ be the basepoint map.
\[lift\] Let $c, \rho_0$ be as in Proposition \[extendprop\]. Let $W \subset {\widetilde{Q}}_{\frac{\rho_0}{2}}$ be open, $q \in W$, and let $x = \exp t_x \nu_q$ for some $t_x > 0$. Let $\alpha:[0,l] \to \mathring{{\widetilde{X}}}$ be a smooth curve such that $\alpha(0) = x$ and such that $$\label{nointercond}
\alpha([0,l]) \cap \exp(\pi^{-1}(\partial W)) = \emptyset.$$ Then there is a unique smooth curve $\sigma:[0,l] \to N_+W$ such that $\sigma(0) = t_x\nu_q$ and $\exp \sigma(s) = \alpha(s)$. Let $\xi = \pi\circ\sigma:[0,l] \to W$. Then $L_g(\xi) \leq c^{-1}L_g(\alpha)$.
Moreover, if $\alpha:[0,1] \times [0,l] \to \mathring{{\widetilde{X}}}$ is a homotopy of smooth curves such that, for each $\tau$, $\alpha(\tau,0) = x$ and the curve $s \mapsto \alpha(\tau,s)$ satisfies (\[nointercond\]), then there is a unique lift of $\alpha$ to a homotopy of curves based at $t_x\nu_q$. That is, there is a unique smooth map $\sigma:[0,1] \times [0,l] \to N_+W$ such that $\sigma(\tau,0) = t_x\nu_q$ for each $\tau$ and such that $\exp\circ\sigma = \alpha$.
We are especially interested in the special case where $W = {\widetilde{Q}}_{\frac{\rho_0}{2}}$ itself.
Let $x, \alpha$ be as in the statement. By Proposition \[extendprop\], $\exp:N_+{\widetilde{Q}}_{\frac{\rho_0}{2}} \to X$ is a local diffeomorphism. Hence, at least some opening interval of $\alpha$ may be lifted uniquely to a smooth curve $\sigma$ beginning at $t_x\nu_q$ – that is, there is some $a > 0$ and a unique smooth $\sigma:[0,a] \to N_+{\widetilde{Q}}_{\frac{\rho_0}{2}}$ such that $\sigma(0) = t_x\nu_q$ and $\exp\circ\sigma = \alpha|_{[0,a]}$. Suppose we cannot lift the entire curve, and let $b$ be the supremum of $a > 0$ such that we can uniquely lift $\alpha|_{[0,a]}$ in the preceding sense. Then there is a unique lift $\sigma$ of $\alpha|_{[0,b)}$. By continuity and (\[nointercond\]), $\sigma$ takes values in $N_+(\overline{W}\setminus{\widetilde{S}})$.
As in the statement, define $\xi:[0,b) \to W$ by $\xi = \pi\circ\sigma$. By the canonical identification $N_+W\approx [0,\infty) \times W $, we can write $\sigma(s) = (t(s),\xi(s))$. Now for each $s < b$, $\exp \sigma(s) = \alpha(s)$; it follows that, for $s < b$, $(d\exp)(\sigma'(s)) = \alpha'(s)$; and thus that $|(d\exp)(\sigma'(s))|^2_g = |\alpha'(s)|^2_g$. Under the identification $T_{(t,q)}N_+W \approx \mathbb{R}\frac{\partial}{\partial t}
\oplus T_qW$, we can write $\sigma'(s) = \dot{t}(s)\frac{\partial}{\partial t} + \xi'(s)$. Let $A = \sup_{0 \leq s \leq b}|\alpha'(s)|_g^2$. Then since $(d\exp)\left(\frac{\partial}{\partial t}\right) \perp (d\exp)(\xi'(s))$, we have $$\begin{aligned}
A &\geq |\alpha'(s)|_g^2 = \left|\dot{t}(s)(d\exp)\left( \frac{\partial}{\partial t} \right) +
(d\exp)_{t\nu_{\xi}}(\xi'(s))\right|_g^2\\
&= \dot{t}(s)^2 + |(d\exp)_{t\nu_{\xi}}(\xi'(s))|_g^2\\
&\geq \dot{t}(s)^2 + c^2|\xi'(s)|_g^2 \text{ (by Proposition \ref{difflem})}{\addtocounter{equation}{1}\tag{\theequation}}\label{liftbound}.
\end{aligned}$$ Thus, both $|\dot{t}(s)|$ and $|\xi'(s)|_g$ are bounded. It follows that $\lim_{s \to b}\xi(s)$ exists in $\overline{W}$, so $\xi$ may be extended to exist continuously on $[0,b]$ (although *a priori* $\xi(b)$ may not lie in $W$). Because $\xi:[0,b] \to \overline{W}$ has finite length by (\[liftbound\]), $\xi(b) \notin {\widetilde{S}}$.
Now also by (\[liftbound\]), $\lim_{s \to b}t(s)$ exists; so $\lim_{s \to b}\sigma(s)$ exists, and $\sigma$ may be continuously extended to $[0,b]$, possibly taking values in $N_+\overline{W} \supset N_+W$. However, by continuity we have $\exp\sigma(b) = \alpha(b)$. Because (\[nointercond\]) holds, $\xi(s) \notin \partial W$ for any $0 \leq s \leq b$. Therefore, $\xi([0,b]) \subset W$, and hence $\sigma([0,b]) \subset N_+W$. Now by Proposition \[difflem\] and because $W \subseteq {\widetilde{Q}}_{\frac{\rho_0}{2}}$, $\exp$ is a local diffeomorphism on some ball about $\sigma(b)$, so it follows that $\sigma$ can be smoothly and uniquely extended at least some distance beyond $b$. This is a contradiction, so $\sigma$ can be extended smoothly and uniquely to all of $[0,l]$.
We now turn to homotopy lifting. Suppose that $x$ is as above, and that $\alpha:[0,1] \times [0,l] \to
\mathring{{\widetilde{X}}}$ is a smooth map such that $\alpha(\tau,0) = x$ for all $\tau$ and such that, for fixed $\tau$, the curve $s \mapsto \alpha(\tau,s) =:
\alpha_{\tau}(s)$ meets condition (\[nointercond\]). We wish to show that there is a lift $\sigma:[0,1] \times [0,l] \to N_+W$ such that $\exp \sigma = \alpha$. In the following, we will also use the notation $\alpha^s(\tau) = \alpha(\tau,s)$.
Let $\sigma_0:[0,l] \to N_+W$ be a lift, as above, of $\alpha_0$ beginning at $t_x\nu_q$. For each $s \in [0,l]$, let $\tau \mapsto \sigma(\tau,s)$ be the lift, starting at $\sigma_0(s)$, of the map $\tau \mapsto \alpha(\tau,s)$. Then $\sigma:[0,1] \times [0,l] \to N_+W$ and $\alpha(\tau,s) = \exp\circ\sigma(\tau,s)$. It is plain that $\sigma$ is smooth in $\tau$. We wish to show that it is smooth in $s$.
Let $s_0 \in (0,l]$. We will construct a small strip in $[0,1] \times [0,l]$, containing $[0,1] \times \left\{ s_0 \right\}$, such that $\sigma$ is smooth on the strip. (The case $s_0 = 0$ is easy because $\exp$ is a local diffeomorphism and $\alpha_\tau(0) = x$ for all $\tau$). Let $U \subset N_+W$ be a coordinate ball, containing $\sigma(0,s_0)$, such that $\exp|_{U}$ is a diffeomorphism. Then $\alpha^{-1}(\exp(U)) \subseteq [0,1] \times [0,l]$ is an open neighborhood of $(0,s_0)$. By continuity of $\sigma_0$, there is some $\varepsilon > 0$ such that $\sigma_0([s_0 - \varepsilon,s_0 + \varepsilon]) \subset U$. Now if $s \in [s_0 - \varepsilon,s_0 + \varepsilon]$ and $a > 0$ is small enough that $[0,a] \times \left\{ s \right\} \subset \alpha^{-1}(\exp(U))$, then for $0 \leq \tau \leq a$, the map $\tau \mapsto (\exp|_U)^{-1}(\alpha(\tau,s))$ is a smooth lift of $\tau \mapsto \alpha(\tau,s)$ beginning at $\sigma_0(s)$. It follows by uniqueness of lifting that it is equal to $\tau \mapsto \sigma(\tau,s)$. Thus, on some neighborhood of $\left\{ 0 \right\} \times
[s_0 - \varepsilon,s_0 + \varepsilon]$, we have $\sigma = (\exp|_U)^{-1}\circ\alpha$, so in particular, $\sigma$ is smooth on a neighborhood of $(0,s_0)$.
Set $$b = \sup \left\{ d \geq 0: \sigma \text{ is smooth on a neighborhood of } [0,d] \times \left\{ s_0 \right\}.\right\}.$$ The preceding discussion shows that $b > 0$. Clearly $b \leq 1$, We claim $b = 1$. Suppose not, by way of contradiction. Once more, let $U$ be an open set containing $\sigma(b,s_0)$ such that $\exp|_U$ is a diffeomorphism. Then again, $\alpha^{-1}(\exp(U)) \subseteq [0,1] \times [0,l]$ is an open neighborhood of $(b,s_0)$. Moreover, because $\sigma^{s_0}$ is smooth in $\tau$, we have $\sigma(a,s_0) \in U$ for all $a$ sufficiently near $b$. Let $a_1 < b$ be sufficiently near. Then $\sigma$ is smooth on some neighborhood of $[0,a_1] \times \left\{ s_0 \right\}$ by definition of $b$. Hence by continuity, we conclude that $\sigma$ maps some neighborhood of $(a_1,s_0)$ into $U$. We can choose some $\varepsilon > 0$ such that $\sigma(\left\{ a_1 \right\} \times [s_0 - \varepsilon,
s_0 + \varepsilon]) \subset U$ and so that $\sigma$ is smooth on a neighborhood of $[0,a_1] \times [s_0 - \varepsilon,s_0 + \varepsilon]$. Thus, for $s \in [s_0 - \varepsilon,s_0 + \varepsilon]$, $\sigma(a_1,s) = (\exp|_U)^{-1}\circ\alpha(a_1,s)$. Now, by shrinking $\varepsilon$ if need be, we can choose $a_2 > b$ such that $[a_1,a_2] \times [s_0 - \varepsilon,s_0 + \varepsilon] \subseteq
\alpha^{-1}(\exp(U))$. Fix $s \in [s_0 - \varepsilon,s_0 + \varepsilon]$. The map $\tau \mapsto (\exp|_U)^{-1}(\alpha(\tau,s))$ (where $a_1 \leq \tau \leq a_2)$ is a smooth lift of the map $\tau \mapsto \alpha(\tau,s)$ beginning at $\sigma(a_1,s)$. Then by uniqueness of path lifts, we have $\sigma(\tau,s) = (\exp|_U)^{-1}(\alpha(\tau,s))$ on this rectangle. Thus, $\sigma$ is smooth on $[0,a_2] \times [s_0 - \varepsilon,s_0 + \varepsilon]$, which is a contradiction since $a_2 > b$. Thus $b = 1$. We conclude that $\sigma$ is smooth on all of $[0,1] \times [0,l]$.
It follows from the proof that, if we set $W = {\widetilde{Q}}_{\frac{\rho_0}{2}}$ and $R = \left\{ q \in {\widetilde{Q}}:
\rho(q) = \frac{\rho_0}{2}\right\}$, then the hypothesis (\[nointercond\]) could be replaced by the condition $d_{{\widetilde{Q}}}(q,R) > c^{-1}L_g(\alpha)$.
This result in hand, we may prove interior injectivity.
\[injprop\] There exists $a > 0$ such that $\exp:N_+{\widetilde{Q}}_a \to {\widetilde{X}}$ is injective.
Let $\rho_0$ be small enough that Propositions \[extendprop\] and \[lift\] hold on ${\widetilde{Q}}_{\rho_0}$, then define $R \subset {\widetilde{Q}}$ by $R = \left\{ q \in {\widetilde{Q}}: \rho(q) = \frac{\rho_0}{2}\right\}$.
We first prove a result with a topological hypothesis. By Proposition \[geodprop\], there exists $\rho_1 < \frac{\rho_0}{2}$ such that if $q \in {\widetilde{Q}}_{\rho_1}$, then $\gamma_q$ will not intersect $\exp(\pi^{-1}(R))$: for if $q^{\prime} \in R$, then by Proposition \[geodprop\](a), $\frac{\varepsilon \rho_0}{2}
< \rho(\gamma_{q'}(t))$ for all $t$, whereas $\gamma_q$ can be made to remain arbitrarily close to ${\widetilde{S}}$ by choosing $\rho_1$ sufficiently small.
We will show that if $V \subseteq {\widetilde{Q}}_{\rho_1}$ is connected, and ${\widetilde{A}}\subset {\widetilde{X}}\setminus ({\widetilde{M}}\cup {\widetilde{S}})$ is a simply connected open set such that ${\widetilde{A}}\cap \exp(\pi^{-1}(R)) = \emptyset$ and such that $\exp(N_+V) \subseteq {\widetilde{A}}$, then $\exp$ is injective on $N_+V$. Here, once again, $\pi:N_+({\widetilde{Q}}\setminus {\widetilde{S}}) \to {\widetilde{Q}}\setminus {\widetilde{S}}$ is the basepoint map.
Suppose, by way of contradiction, that $\exp$ is not injective on $N_+V$. Then there exists some $x \in {\widetilde{A}}$, $u \neq v \in N_+V$ such that $\exp(u) = x = \exp(v)$. Let $\sigma:[0,1] \to N_+V$ be a smooth path in $N_+V$ from $u$ to $v$, and let $\alpha = \exp\circ\sigma:[0,1] \to {\widetilde{A}}$. Then $\alpha$ is a smooth loop segment at $x$; and since ${\widetilde{A}}$ is simply connected, there is a smooth homotopy $\tilde{\alpha}:[0,1] \times
[0,1] \to {\widetilde{A}}$ such that $\tilde{\alpha}(0,s) = \alpha(s)$ and $\tilde{\alpha}(1,s) = x$. Now by construction, $\tilde{\alpha}(\tau,s)$ avoids $\exp(\pi^{-1}(R))$ for all $\tau, s$; so by Proposition \[lift\] with $W =
{\widetilde{Q}}_{\frac{\rho_0}{2}}$, there exists a lifted homotopy $\tilde{\sigma}:[0,1] \times [0,1] \to N_+{\widetilde{Q}}_{\frac{\rho_0}{2}}$, based at $u$, such that $\exp\circ\tilde{\sigma}
= \tilde{\alpha}$. Thus, $\exp\circ\tilde{\sigma}(\tau,1) = x$ for all $\tau$. Moreover, $\tilde{\sigma}(0,1) =
v$ and $\tilde{\sigma}(1,1) = u$ (since the lift of a constant path is constant). Thus, defining $\zeta:[0,1] \to N_+{\widetilde{Q}}_{\frac{\rho_0}{2}}$ by $\zeta(\tau) = \tilde{\sigma}(\tau,1)$, the curve $\zeta$ must be a smooth path from $v$ to $u$. On the other hand, $\exp\circ\zeta(\tau) = x$ for all $\tau$. But as $\exp$ is a local diffeomorphism, $\exp^{-1}(\left\{ x \right\})$ is discrete. Thus, $\zeta$ is a non-constant smooth map from a connected space to a discrete space, which is a contradiction. Hence, $\exp$ is injective on $N_+V$, which establishes our claim.
To allow a general topology, and in particular a full neighborhood of ${\widetilde{S}}$, first note that if $B \subseteq S$ is simply connected, then so is $b^{-1}(B)$. This is because ${\widetilde{S}}\to S$ is a trivial fibration, since $S$ is the intersection of two globally defined hypersurfaces. For such $B$, and for $\kappa > 0$, we define $${\widetilde{A}}(B,\kappa) = \left\{ (\theta,p,\rho) \in {\widetilde{X}}: p \in B \text{ and } 0 < \rho < \kappa \right\},$$ where we are using our polar identification. Notice that by taking $\kappa$ small enough, we may always assure that $\exp(\pi^{-1}(R)) \cap {\widetilde{A}}(B,\kappa) = \emptyset$. Also, ${\widetilde{A}}(B,\kappa)$ will be simply connected for $\kappa$ small. Now let $\delta > 0$ be less than the injectivity radius of $S$ with respect to $k_0 = \bar{g}|_{TS}$. For $p \in S$, let $B_{\delta}(p)$ denote the $\delta$-ball about $p$ with respect to $k_0$. Thus, for each $p \in S$, $B_{\delta}(p) \subseteq S$ is simply connected. Let $\kappa_0 > 0$ be small enough for the above conditions to hold for $B_{\delta}(p)$ at every $p \in S$. Set ${\widetilde{A}}_p = {\widetilde{A}}(B_{\delta}(p),\kappa_0)$.
By Proposition \[geodprop\], there are $\varepsilon > 0, \kappa_1 > 0$ such that for each $p \in S$, $\exp(N_+({\widetilde{A}}(B_{\varepsilon}(p),\kappa_1) \cap {\widetilde{Q}})) \subseteq {\widetilde{A}}_p$. Set $V_p = {\widetilde{A}}(B_{\frac{\varepsilon}{4}}(p),\kappa_1) \cap {\widetilde{Q}}$. Then $\left\{\underline{V_p} \right\}_{p \in S}$ covers ${\widetilde{S}}\cap {\widetilde{Q}}\approx S$, and by compactness we may take a finite subcover $\left\{ \underline{V_{p_i}} \right\}_{i = 1}^N$. We will denote $V_i = V_{p_i}$. Now since there are finitely many $V_i$, we may apply Proposition \[geodprop\](b) to conclude that by shrinking $\kappa_1$ if necessary, we may ensure that $\exp(N_+V_i) \cap \exp(N_+V_j) = \emptyset$ whenever $\overline{V_i} \cap \overline{V_j} = \emptyset$.
Set $V = \cup_{i}V_i$. We claim that $\exp$ is injective on $N_+V$. For suppose that there exist $u_1 = t_1\nu_{q_1}, u_2 = t_2\nu_{q_2} \in N_+V$ such that $\exp(u_1) = \exp(u_2)$. By what has just been said, we must have $q_1 \in V_i$ and $q_2 \in V_j$ where $\overline{V_i} \cap \overline{V_j} \neq \emptyset$. Let $p_i = \pi_S(q_i)$. Then by definition of $V_i, V_j$, we have $q_1, q_2 \in V_{\varepsilon} := {\widetilde{A}}(B_{\varepsilon}(p_1),\kappa_1) \cap {\widetilde{Q}}$; and since, by choice of $\varepsilon$, we have $\exp(V_{\varepsilon}) \subseteq {\widetilde{A}}_{p_1}$, and by the simply connected case, we may conclude that $q_1 = q_2$. Thus taking $a$ small enough that ${\widetilde{Q}}_a \subseteq V$ yields the theorem.
This result may be extended to the compactified bundle, ${\overline{N_+{\widetilde{Q}}_{a}}}$.
\[injprop2\] There exists $a > 0$ such that $\exp:{\overline{N_+{\widetilde{Q}}_{a}}} \to {\widetilde{X}}$ is injective.
Let $a$ be as in Proposition \[injprop\]. As before, we label points in ${\overline{N_+{\widetilde{Q}}_{a}}}$ by $(\tau,q) \in [0,1] \times {\underline{{\widetilde{Q}}_{a}}}
\approx {\overline{N_+{\widetilde{Q}}_{a}}}$. Throughout this proof, we regard $\exp$ as a function of $\tau \in [0,1]$ and $q \in {\widetilde{Q}}_a$. We already know that $\exp$ is injective when restricted to the set $\left\{ 0 \leq \tau < 1, \rho(q) > 0 \right\}$. We may quickly extend this to $[0,1] \times S$: the exponential map takes $[0,1] \times S$ injectively to ${\widetilde{S}}$ by the explicit solution (\[ssoln\]), and by Proposition \[geodprop\] no other points are mapped to ${\widetilde{S}}$. Moreover, since $\left\{ 1 \right\} \times {\widetilde{Q}}_a$ is mapped to ${\widetilde{M}}\setminus {\widetilde{S}}$, its image is disjoint from the image of $([0,1) \times {\widetilde{Q}}_a) \cup ([0,1] \times ({\widetilde{S}}\cap {\widetilde{Q}}))$. Therefore, we need only show that for $q_1 \neq q_2 \in {\widetilde{Q}}_a$, $\exp(1,q_1) \neq \exp(1,q_2)$.
Suppose, by way of contradiction, that $\exp(1,q_1) = \exp(1,q_2)$, with $q_1,q_2 \in {\widetilde{Q}}_a$. Let ${\widetilde{Q}}_a \supseteq
B_1 \ni q_1$ be open such that $q_2 \notin B_1$. Then $[0,1] \times B_1$ is open in ${\overline{N_+{\widetilde{Q}}_{a}}}$, and so since $\exp$ is a local diffeomorphism, $\exp([0,1] \times B_1)$ is open in ${\widetilde{X}}$. Let $\hat{\gamma}_{q_2}:[0,1] \to {\widetilde{X}}$ be the rescaled geodesic given by $\hat{\gamma}_{q_2}(\tau) = \exp(\tau,q_2)$, which in particular is continuous. Thus, $\hat{\gamma}_{q_2}^{-1}(\exp([0,1] \times B_1))$ is nonempty and is open in $[0,1]$, and so there is some $\tau_2 < 1$ such that $\hat{\gamma}_{q_2}(\tau_2) \in \exp([0,1] \times B_1)$. Since $\tau_2 < 1$, we must have $\hat{\gamma}_{q_2}(\tau) \in \mathring{{\widetilde{X}}}$, and so there is some $q_3 \in B_1$, $\tau_3 \in [0,1)$ such that $\exp(\tau_3,q_3) = \exp(\tau_2,q_2)$. This contradicts interior injectivity and thus Proposition \[injprop\].
Proofs of Theorems {#proofs}
==================
By Proposition \[extendprop\], there exists $\rho_0 > 0$ such that the exponential map $\exp:N_+{\widetilde{Q}}_{\rho_0}\to{\widetilde{X}}$ extends to a local diffeomorphism $\exp:{\overline{N_+{\widetilde{Q}}_{\rho_0}}} \to {\widetilde{X}}$. By Proposition \[injprop2\], it is injective for $\rho_0$ small enough. Taking $V = {\underline{{\widetilde{Q}}_{\rho_0}}}$ and ${\widetilde{U}}= \exp(V)$, this yields the claim.
By Theorem \[mainthm\], we may take $\rho_0 > 0$ such that $\exp:{\overline{N_+{\widetilde{Q}}_{\rho_0}}} \to {\widetilde{X}}$ is a diffeomorphism onto its image. Let $V \subset {\underline{{\widetilde{Q}}_{\rho_0}}}$ be a neighborhood of ${\widetilde{Q}}\cap {\widetilde{S}}$, and set ${\widetilde{U}}= \exp({\overline{N_+(V \setminus {\widetilde{S}})}})$. Now under the canonical decomposition $N_+(V \setminus {\widetilde{S}}) \approx \mathbb{R}_t \times (V \setminus {\widetilde{S}})$, we have $$\label{geodesicform}
(\exp|_{N_+(V \setminus {\widetilde{S}})})^*g = dt^2 + g_t,$$ where $g_t$ is a one-parameter family of metrics on $V \setminus {\widetilde{S}}$. Now let $u = 1 - \tau = e^{-t}$, so that $t = -\log u$. Then in these coordinates, $$\label{pullbackmetric}
(\exp|_{N_+(V \setminus {\widetilde{S}})})^*g = \frac{du^2 + u^2g_{-\log u}}{u^2}.$$ Set $h_u = u^2g_{-\log u}$, so that this takes the form (\[normformeq\]). Now $u$ obviously extends to a global coordinate on ${\overline{N_+(V \setminus {\widetilde{S}})}}$, and we have already observed that $\exp$ is a diffeomorphism. Taking $\psi(u,q) = \exp(1-u,q)$ and ${\widetilde{U}}= \psi({\overline{N_+(V \setminus {\widetilde{S}})}})$, we plainly have $\psi(\left\{ u = 0 \right\}) = {\widetilde{U}}\cap {\widetilde{M}}$, $\psi(\left\{ u = 1 \right\}) = {\widetilde{U}}\cap {\widetilde{Q}}$, and $\psi|_{\{1\} \times V} = \operatorname{id}$. Uniqueness of $\psi$ follows from uniqueness of the form (\[geodesicform\]), because all the intermediate steps are reversible. To prove Theorem \[normform\], it thus remains only to show that $h_u$ extends smoothly down to $u = 0$ and that each $h_u$ is a conformally compact metric on $V$.
Since $\psi$ is a diffeomorphism, we do our calculation on ${\widetilde{U}}$ and regard $u$ as a function on ${\widetilde{U}}$. For $0 \leq c \leq 1$ set $V_c =
\left\{ x \in {\widetilde{U}}: u(x) = c \right\} \approx V$. Let $\{x^s\}$ be a local coordinate system on $S$, and $(\theta,x^s,\rho)$ a polar identification. Then locally, $u = u(\theta,x^s,\rho)$, so taking the exterior derivative of both sides of the equation $u = c$, we find that along $V_c$, $$0 = du = \frac{\partial u}{\partial \theta}d\theta + \frac{\partial u}{\partial x^s}dx^s + \frac{\partial u}{\partial \rho}d\rho.$$ Now $\frac{\partial u}{\partial \theta} \neq 0$, and so for some smooth functions $a, b_s$, we have $$\label{dtheta}
d\theta = ad\rho + b_sdx^s$$ on $V_c$. Notice that $\frac{u}{\sin\theta}$ is smooth and nonvanishing on ${\widetilde{U}}$. Then by (\[polform\]), $$\begin{aligned}
u^2g &= \frac{u^2}{\sin^2(\theta)}\left( d\theta^2 + \frac{d\rho^2 + k_{\rho}}{\rho^2} \right) + (\rho u^2\sin(\theta)\ell)\\
&=\frac{u^2}{\sin^2(\theta)}\frac{\rho^2d\theta^2 + d\rho^2 + k_{\rho}}{\rho^2} + (\rho u^2\sin(\theta)\ell).
\end{aligned}$$ It is then clear by (\[dtheta\]) that the restriction of $u^2g$ to any $V_c$ gives a smooth conformally compact metric on $V$ depending smoothly on $c$ all the way up to $c = 0$. Thus Theorem \[normform\] is proved.
Now suppose that $Q$ makes a constant angle $\theta_0$ with $M$, so that ${\widetilde{Q}}\cap {\widetilde{S}}$ is given by $\left\{ \theta = \theta_0 \right\}$. Let $\alpha = (\csc\theta_0 - \cot\theta_0)^{-1}$, and define a coordinate $\phi$ on ${\overline{N_+(V\setminus{\widetilde{S}})}}$ by $u = \alpha(\csc(\phi) - \cot(\phi))$. Notice that $\frac{du}{u} = \frac{d\phi}{\sin\phi}$. Thus, the metric (\[normformeq\]) transforms to $$\label{pullbackconst}
\chi^*g = \frac{d\phi^2 + l_{\phi}}{\sin^2(\phi)},$$ where we have defined $\chi:[0,\theta_0] \times V \to {\widetilde{U}}$ by $\chi(\phi,q) = \psi(u(\phi),q)$ and $l_{\phi} = \frac{\sin^2\phi}{u^2}h_{u(\phi)}$. We view $\phi$ as a function on ${\widetilde{U}}$ via the diffeomorphism $\chi$. Now by (\[ssoln\]), we see that $\theta = \phi$ on $[0,1] \times ({\widetilde{Q}}\cap {\widetilde{S}})$; or put differently, that $\theta = \phi + O(\rho)$. Because the level sets of $\phi$ and $u$ are the same, we get (\[dtheta\]) again, and so by an identical calculation to the preceding, we find that $$\sin^2(\phi)g = \frac{\sin^2(\phi)}{\sin^2(\theta)}\frac{\rho^2d\theta^2 + d\rho^2 + k_{\rho}}{\rho^2} + (\rho\sin^2(\phi)\sin(\theta)\ell),$$ which is asymptotically hyperbolic as desired on each $V_c$ because $\frac{\sin\phi}{\sin\theta} = 1 + O(\rho)$.
We now prove the final claim. For this purpose, we define a change of coordinates $(\phi,y^{\mu})$ by setting $y^{\mu} = x^{\mu}$ on ${\widetilde{Q}}$ and extending $y^{\mu}$ to be constant along orbits of the exponential map. These are coordinates by Theorem \[mainthm\]. We wish to show that $(y^{n})^2l_{\phi}|_{\rho = 0}$ is constant in $\phi$. Now notice that it follows by (\[quadest\]) that for $1 \leq \mu \leq n$, $$\frac{\partial}{\partial y^{\mu}} = \frac{\partial}{\partial x^{\mu}} + \frac{\partial \theta}{\partial y^{\mu}}\frac{\partial}{\partial \theta} + O(\rho).$$ Since $\frac{\partial}{\partial \theta} \in \ker \bar{g}$ at $\rho = 0$, we have, for $p \in {\widetilde{U}}$, that $$\left.\bar{g}\left( \frac{\partial}{\partial y^{\mu}},\frac{\partial}{\partial y^{\nu}} \right)\right|_p = \left.\bar{g}\left( \frac{\partial}{\partial x^{\mu}},\frac{\partial}{\partial x^{\nu}} \right)\right|_p +
O(\rho(p)).$$ The right-hand side is constant in $\phi$ for $\rho = 0$ by (\[g\]). But $$(y^n)^2l_{\phi}\left(\frac{\partial}{\partial y^{\mu}},\frac{\partial}{\partial y^{\nu}} \right) =
\bar{g}\left( \frac{\partial}{\partial y^{\mu}},\frac{\partial}{\partial y^{\nu}} \right)$$ at $\rho = 0$, and as we have seen, the right-hand side is constant in $\phi$ there. This yields the claim.
Renaming $\phi$ by $\theta$, $\chi$ by $\psi$, and $l_{\phi}$ by $h_{\theta}$ yields the result.
We prove Corollary \[normformcor\]. Let $V \subset {\widetilde{Q}}$, ${\widetilde{U}}\subset {\widetilde{X}}$, and $\psi:[0,1] \times V \to {\widetilde{U}}$ be as in Theorem \[normform\]. Set $W = \psi(\left\{ 0 \right\} \times V)$, a neighborhood in ${\widetilde{M}}$ of ${\widetilde{M}}\cap {\widetilde{S}}$. Let $\phi:V \to W$ be the diffeomorphism given by $\phi(q) = \psi(0,q)$. Define $\zeta(u,m) = \psi(u,\phi^{-1}(m))$.
Uniqueness follows from uniqueness in Theorem \[normform\], the construction can be reversed to recover $\psi$ from $\zeta$.
Let $W, {\widetilde{U}}, h_{\theta},$ and $\zeta$ be as in Corollary \[normformconstcor\]. Now $h_0$ is an asymptotically hyperbolic metric. So by the existence result for the standard geodesic normal form ([@gl91]), there is a unique diffeomorphism $\varphi:S \times [0,\varepsilon)_{\rho} \to W$ such that $\varphi^*h_0 = \rho^{-2}(d\rho^2 + k_{\rho})$, where $k_{\rho}$ is a smooth one-parameter family of metrics on $S$ such that $k_0 = k$; and such that $\varphi|_{\left\{ 0 \right\} \times S} = id_S$. The desired diffeomorphism is then given by $\chi = \zeta \circ (\operatorname{id}_{[0,\theta_0]} \times \varphi)$. The claimed properties follow immediately. (Note that the $\rho$ in Corollary \[polarcor\] is not the same as that appearing in polar $g$-coordinates).
|
---
abstract: 'Recent highlights in CP violation phenomena are reviewed. B-factory results imply that CP-violation phase in the CKM matrix is the dominant contributor to the observed CP violation in K and B-physics. Deviations from the predictions of the CKM-paradigm due to beyond the Standard Model CP-odd phase are likely to be a small perturbation. Therefore, large data sample of clean B’s will be needed. Precise determination of the unitarity triangle, along with time dependent CP in penguin dominated hadronic and radiative modes are discussed. [*Null tests*]{} in B, K and top-physics and separate determination of the K-unitarity triangle are also emphasized.'
author:
- |
Amarjit Soni\
[*High Energy Theory, Department of Physics,*]{}\
[*Brookhaven National Laboratory, Upton, NY 11973-5000*]{}\
title: '**CP VIOLATION HIGHLIGHTS: CIRCA 2005**'
---
=11.6pt
B-factories help attain an important milestone: Good and bad news
=================================================================
The two asymmetric B-factories at SLAC and KEK have provided a striking confirmation of the CKM paradigm [@ckm]. Existing experimental information from the indirect CP violation parameter, $\epsilon$ for the $K_{L}\rightarrow \pi \pi$, semileptonic $b\rightarrow ue\nu$ and $\bbarb$ mixing along with lattice calculations predict that in the SM, $(\sin2\beta)\simeq.70\pm.10$ [@ans01; @ckmfit; @utfit]. This is in very good agreement with the BELLE and BABAR result [@hfag04]:\
\_[CP]{}(B\^0K\^0) = 2=.726.037 \[psik\] This leads to the conclusion that the CKM phase of the Standard Model (SM) is the dominant contributor to $\a_{CP}$. That, of course, also means that CP-odd phase(s) due to beyond the Standard Model (BSM) sources may well cause only small deviations from the SM in B-physics.\
Actually, there are several reasons to think that BSM phase(s) may cause only small deviations in B-physics. In this regard, SM itself teaches a very important lesson.
Important lesson from the CKM-paradigm
======================================
We know now that the CKM phase is 0(1) (actually, the CP violation parameter $\eta$ is 0(.3) [@ans01; @ckmfit; @utfit]). The CP effects that it causes on different observables though is quite different. In K-decays, the CP asymmetrics are $\le10^{-3}$. In charm physics, also there are good reasons to expect small observable effects. In top physics, the CKM phase causes completely negligible effects [@ehs91; @abes_pr]. Thus only in B-decays, the large asymmetries (often 0(1)) are caused by the CKM phase. So even if the BSM phase(s) are 0(1) it is unlikely that again in B-physics they will cause large effects just as the SM does.
Remember the $m_{\nu}$
======================
Situation with regard to BSM CP-odd phase(s) ($\chi_{BSM}$) is somewhat reminiscent of the neutrino mass ($m_{\nu}$) [@brwns]. There was no good reason for $m_{\nu}$ to be zero; similarly, there are none for $\chi_{BSM}$ to be zero either. In the case of $\nu^{'}s$, there were the solar $\nu$ results that were suggestive for a very long time; similarly, in the case of $\chi_{BSM}$, the fact that in the SM, baryogenesis is difficult to accomodate serves as the beacon.\
It took decades to show $m_{\nu}$ is not zero: $\Delta m^{2}$ had to be lowered from $\sim O(1-10)eV^{2}$ around 1983 down to $O(10^{-4}eV^{2})$ before $m_{\nu} \neq0$ was established via neutrino oscillations. We can hope for better luck with $\chi_{BSM}$ but there is no good reason to be too optimistic; therefore, we should not rely on luck but rather we should seriously prepare for this possibility.\
To recapitulate, just as the SM-CKM phase is 0(1), but it caused only $0(10^{-3})$ CP symmetries in K- decays, an 0(1) BSM-CP-odd phase may well cause only very small asymmetries in B-physics. To search for such small effects:\
1) We need lots and lots of clean B’s ([*i.e.*]{} $0(10^{10}$) or more)\
2) Intensive study of $B_{s}$ mesons (in addition to B’s) becomes very important as comparison between the two types of B-mesons will teach us how to improve quantitative estimates of flavor symmetry breaking effects.\
3) We also need clean predictions from theory (wherein item 2 should help).
Improved searches for BSM phase
===============================
Improved searches for BSM-CP-odd phase(s) can be subdivided into the following main categories:\
\
a) Indirect searches with theory input\
b) Indirect searches without theory input\
c) Direct searches.\
Indirect searches with theory input
-----------------------------------
Among the four parameters of the CKM matrix, $\lambda, A,\rho$ and $\eta$, $\lambda = 0.2200 \pm 0.0026$, $A \approx 0.850 \pm 0.035$ [@PDB] are known quite precisely; $\rho$ and $\eta$ still need to be determined accurately. Efforts have been underway for many years to determine these parameters. The angles $\alpha, \beta, \gamma$, of the unitarity triangle (UT) can be determined once one knows the 4-CKM parameters.
A well studied strategy for determining these from experimental data requires knowledge of hadronic matrix elements. Efforts to calculate several of the relevant matrix elements on the lattice, with increasing accuracy, have been underway for past many years. A central role is played by the following four inputs [@ans01; @ckmfit; @utfit]:
- $B_K$ from the lattice with $\epsilon$ from experiment
- $f_B\sqrt{B_B}$ from the lattice with $\Delta m_d$ from experiment
- $ \xi $ from the lattice with $\frac{\Delta m_s}{\Delta m_d}$ from experiment
- $\frac{b \to u l \nu}{b \to c l \nu}$ from experiment, along with input from phenomenology especially heavy quark symmetry as well as the lattice.
As mentioned above, for the past few years, these inputs have led to the important constraint: $ \sin 2\beta_{SM} \approx 0.70 \pm 0.10$ which is found to be in very good agreement with direct experimental determination, Eq. \[psik\].
Despite severe limitations (e.g. the so-called quenched approximation) these lattice inputs provided valuable help so that with B-Factory measurements one arrives at the very important conclusion that in $B \to \jpsi K^0$ the CKM-phase is the dominant contributor; any new physics (NP) contribution is unlikely to be greater than about 15%.
What sort of progress can we expect from the lattice in the next several years in these (indirect) determination of the UT? To answer this it is useful to look back and compare where we were to where we are now. Perhaps this gives us an indication of the pace of progress of the past several years. Lattice calculations of matrix elements around 1995 [@lat95_as] yielded (amongst other things) $\sin 2\beta \approx 0.59 \pm
0.20$, whereas the corresponding error decreased to around ${\pm 0.10}$ around 2001 [@ans01]. In addition to $\beta$, such calculations also now constrain $\gamma (\approx 60^\circ)$ with an error of around $10^\circ$ [@ans01].
There are three important developments that should help lattice calculations in the near future:\
1. Exact chiral symmetry can be maintained on the lattice. This is especially important for light quark physics.
2. Relatively inexpensive methods for simulations with dynamical quarks (esp. using improved staggered fermions [@hpqcd]) have become available. This should help overcome limitations of the quenched approximation.
3. About a factor of 20 increase in computing power is now being used compared to a few years ago.
As a specific example one can see that the error on $B_K$ with the 1st use of dynamical domain wall fermions [@rbc_nf2] now seems to be reduced by about a factor of two [@rbc_jun]. In the next few years or so errors on lattice determination of CKM parameters should decrease appreciably, perhaps by a factor of 3. So the error in $\sin 2\beta_{SM} \pm 0.10 \to
\pm 0.03$; $\gamma \pm 10^\circ \to 4^\circ$ etc. While this increase in accuracy is very welcome, and will be very useful, there are good reasons to believe, experiment will move ahead of theory in direct determinations of unitarity angles in the next 5 years. (At present, experiment is already ahead of theory for $\sin 2 \beta$).
Indirect searches without theory input: Elements of a superclean UT
-------------------------------------------------------------------
One of the most exciting developments of recent years in B-physics is that methods have been developed so that all three angles of the UT can be determined cleanly with very small theory errors. This is very important as it can open up several ways to test the SM-CKM paradigm of CP violation; in particular, the possibility of searching for small deviations. Let us very briefly recapitulate the methods in question:\
- Time dependent CP asymmetry (TDCPA) measurements in $B^0, \bar B^0
\to \psi K^0$ type of final states should give the angle $\beta$ very precisely with an estimated irreducible theory error (ITE) of $\le O(0.1\%)$ [@bmr04].
- Direct CP (DIRCP) studies in $B^{\pm} \to ``K^{\pm}" D^0, \bar D^0$ gives $\gamma$ very\
cleanly [@ans_pristine; @gsz05].
- TDCPA measurements in $B^0, \bar B^0 \to ``K^0" D^0, \bar D^0$ gives $(2 \beta + \gamma)$ [*and*]{} also $\beta$ very cleanly [@ans_b0; @bgk05].
- In addition, TDCPA measurements in $B_s \to K D_s$ type modes also gives $\gamma$ very cleanly [@pb_lhc].
- Determination of the rate for the CP violating decay $K_L \to \pi^0 \nu \bar \nu$ is a very clean way to measure the Wolfenstein parameter $\eta$, which is indeed the CP-odd phase in the CKM matrix [@ajb_nunu].
It is important to note that the ITE for each of these methods is expected to be $\le 1\%$, in fact perhaps even $\le 0.1\%$.
- Finally let us briefly mention that, TDCPA studies of $B^0, \bar B^0
\to \pi \pi$ or $\rho \pi$ or $\rho \rho$ gives $\alpha$ [@glw; @lnqs; @alnq]. However, in this case, isospin conservation needs be used and that requires, [*assuming*]{} that electro-weak penguins (EWP) make negligible contribution. This introduces some model dependence and may cause an error of order a few degrees, [*i.e.*]{} for $\alpha$ extraction the ITE may well end up being O(a few %). However, given that there are three types of final states each of which allows a determination of $\alpha$, it is quite likely that further studies of these methods will lead to a reduction of the common source of error originating from isopsin violation due to the EWP.
It is extremely important that we make use of these opportunities afforded to us by as many of these very clean redundant measurements as possible. In order to exploit these methods to their fullest potential and get the angles with errors of order ITE will, for sure, require a SUPER-B Factory(SBF) [@brwns; @sbf_kek; @mn_04; @sbf_slac].
This in itself constitutes a strong enough reason for a SBF, as it represents a great opportunity to precisely nail down the important parameters of the CKM paradigm.
### Prospects for precision determination of $\gamma$
Below we briefly discuss why the precision extraction of $\gamma$ seems so promising.
For definiteness, let us recall the basic features of the ADS method [@ads1]. In this interference is sought between two amplitudes of roughly similar size [*i.e.*]{} $B^- \to K^- D^0$ and $B^- \to K^- \bar D^0$ where the $D^0$ and $\bar D^0$ decay to common final states such as the simple two body ones like $K^+ \pi^-$, $K^+ \rho^-$, $K^+ a_1^-$, $K^{+*} \pi^-$ or they may also be multibody modes e.g. the Dalitz decay $K^+ \pi^- \pi^0$, $K^+ \pi^- \pi^+ \pi^-$ etc. It is easy to see that the interference is between a colored allowed B decay followed by doubly Cabibbo suppressed D decay and a color-suppressed B decay followed by Cabibbo allowed D decay and consequently then interference tends to be maximal and should lead to large asymmetries.
For a given (common) final state of $D^0$ and $\bar D^0$ the amplitude involves three unknowns: the color suppressed Br($B^- \to K^- \bar D^0$), which is not directly accessible to experiment [@ads1], the strong phase $\xi_{f_i}^K$ and the weak phase $\gamma$. Corresponding to each such final state (FS) there are two observables: the rate for $B^-$ decay and for the $B^+$ decay.
Thus, if you stick to just one common FS of $D^0$, $\bar D^0$, you do not have enough information to solve for $\gamma$. If you next consider two common FS of $D^0$ and $\bar D^0$ then you have one additional unknown (a strong phase) making a total of 4-unknowns with also 4-observables. So with two final states the system becomes soluble, i.e. we can then use the experimental data to solve not only the value of $\gamma$ but also the strong phases and the suppressed Br for $B^- \to K^- \bar D^0$. With N common FS of $D^0$ and $\bar D^0$, you will have 2N observables and N + 2 unknowns. We need $2N \ge (N + 2) $ i.e. $N \ge 2$. The crucial point, though, is that there are a very large number of possible common modes of $D^0$ and $\bar D^0$ which can all be used to improve the determination of $\gamma$.
Let us briefly mention some of the relevant common modes of $D^0$ and $\bar D^0$:
- The CP-eigenstate modes, originally discussed by GLW [@glw]: $K_S$ \[$ \pi^0$, $ \eta$, $ \eta'$, $ \rho^0$, $ \omega$\]; $ \pi^+ \pi^-$,....
- CP-non-eigenstates (CPNES), discussed by GLS [@gls]: $K^{*+} K^-$, $ \rho^+ \pi^-$... These are singly Cabibbo suppressed modes.
- CPNES modes originally discussed by ADS [@ads1; @ads2]: $K^{+(*)}$\[$ \pi^-$, $\rho^-$, $a_1^-$....\]
- There are also many multibody modes, such as the Dalitz $D^0$ decays: $K_S \pi^+ \pi^-$ [@ggsz] or $K^+ \pi^- \pi^0$ [@ads2] etc; and also modes such as $K^- \pi^+ \pi^- \pi^+$, $K^- \pi^+ \pi^- \pi^+ \pi^0$, or indeed $K^- \pi^+ +n \pi$ [@ans_b0; @ans_path; @ans_charm]. Furthermore, multibody modes such as $B^+ \to K_i^+ D^0 \to
(K \pi)^+ D^0$ or $(K n \pi)^+ D^0$ [@ans_path; @aegs] can also be used.
Fig. \[un\_det\] and Fig. \[over\_det\] show how combining different strategies helps a great deal. In the fig we show $\chi^2$ versus $\gamma$. As indicated above when you consider an individual final state of $D^0$ and $\bar D^0$ then of course there are 3 unknowns ( the strong phase, the weak phase ($\gamma$) and the “unmeasureable" Br) and only two observables (the rate for $B^-$ and the rate for $B^+$). So in the figure, for a fixed value of $\gamma$, we search for the minimum of the $\chi^2$ by letting the strong phase and the “unmeasureable" Br take any value they want.
Fig. \[un\_det\] and Fig. \[over\_det\] show situtation with regard to under determined and over determined cases respectively. The upper horizontal line corresponds roughly to the low luminosity i.e. comparable to the current B-factories[@sbf_kek; @sbf_slac] whereas the lower horizontal curve is relevant for a super B-factory. In Fig. \[un\_det\] in blue is shown the case when only the input from (GLW) CPES modes of $D^0$ is used; note all the CPES modes are included here. You see that the resolution on $\gamma$ then is very poor. In particular, this method is rather ineffective in giving a lower bound; its upper bound is better.
In contrast, a single ADS mode ($K^+ \pi^-$) is very effective in so far as lower bound is concerned, but it does not yield an effective upper bound (red).
Note that in these two cases one has only two observables and 3 unknowns. In purple is shown the situation when these two methods are combined. Then at least at high luminosity there is significant improvement in attaining a tight upper bound; lower bound obtained by ADS alone seems largely unaffected.
Shown in green is another under determined case consisting of the use of a single ADS mode, though it includes $K^{*-}$ as well $D^{*0}$; this again dramatically improves the lower bound. From an examination of these curves it is easy to see that combining information from different methods and modes improves the determination significantly [@ans_path].
Next we briefly discuss some over determined cases (Fig. \[over\_det\]). In purple all the CPES modes of $D^0$ are combined with just one doubly Cabibbo-suppressed (CPNES) mode. Here there are 4 observables for the 4 unknowns and one gets a reasonable solution at least especially for the high luminosity case.
The black curve is different from the purple one in only one respect; the black one also includes the $D^{0*}$ from $B^- \to K^- D^{0*}$ where subsequently the $D^{0*}$ gives rise to a $D^0$. Comparison of the black one with the purple shows considerable improvement by including the $D^{0*}$. In this case the number of observables (8) exceeds the number of unknowns (6).
Actually, the $D^{0*}$ can decay to $D^0$ via two modes: $D^{0*} \to D^0 + \pi$ or $D^0 + \gamma$. Bondar and Gershon [@bg]have made a very nice observation that the strong phase for the $\gamma$ emission is opposite to that of the $\pi$ emission. Inclusion (blue curves) of both types of emission increases the number of observables to 12 with no increase in number of unknowns. So this improves the resolving power for $\gamma$ even more.
The orange curves show the outcome when a lot more input is included; not only $K^-$, $K^{-*}$, $D^0$, $D^{0*}$ but also Dalitz and multibody decays of $D^0$ are included. But the gains now are very modest; thus once the number of observables exceeds the number of unknowns by a few (say O(3)) further increase in input only has a minimal impact.
Let us briefly recall that another important way to get these angles is by studying time-dependent CP (TDCP) (or mixing-induced CP (MIXCP)) violation via $B^0 \to D^{0(*)} ``K^0"$. Once again, all the common decay modes of $D^0$ and $\bar D^0$ can be used just as in the case of direct CP studies involving $B^{\pm}$ decays. Therefore, needless to say input from charm factory [@ans_charm; @ggr; @jr] also becomes desirable for MIXCP studies of $B^0 \to D^{0(*)} ``K^0"$ as it is for direct CP using $B^{\pm}$. It is important to stress that this method gives not only the combinations of the angles (2$\beta +\gamma \equiv \alpha - \beta + \pi$) but also in addition this is another way to get $\beta$ cleanly [@ans_b0; @bgk05]. In fact whether one uses $B^{\pm}$ with DIRCP or $B^0- \bar B^0$ with TDCP these methods are very clean with (as indicated above) the ITE of $\approx 0.1\%$. However, the TDCP studies for getting $\gamma$ (with the use of $\beta$ as determined from $\psi K_s$ ) is less efficient than with the use of DIRCP involving $B^{\pm}$. Once we go to luminosities $\ge 1 ab^{-1}$, though, the two methods for $\gamma$ should become competitive. Note that this method for getting $\beta$ is significantly less efficient than from the $\psi K_s$ studies [@ans_b0].
0.1 in
Now(0.2/ab) 2/ab 10/ab ITE
------------------ ------------------------------------------- ------------------------ ----------------- ---------------
$\sin 2 \phi_1$ 0.037 0.015 0.015(?) 0.001
$\alpha(\phi_2)$ $13^\circ$ $4^\circ$(?) $2^\circ$(?) $1^\circ$(?)
$\gamma(\phi_3)$ $ \pm 20^\circ \pm 10^\circ \pm 10^\circ$ $5^\circ$ to $2^\circ$ $ < 1^\circ(?)$ $0.05^\circ$
: *Projections for direct determination of UT.*
\[ut\_pros\]
Table \[ut\_pros\] summarizes the current status and expectations for the near future for the UT angles. With the current O(0.4/ab) luminosity between the two B-factories, $\gamma \approx (69 \pm 30 )$ degrees. Most of the progress on $\gamma$ determination so far is based on the use of the Dalitz mode, $D^0 -> K_s \pi^+ \pi^-$ [@ggsz]. However, for now, this method has a disadvantage as it entails a a modelling of the resonances involved; though model independent methods of analysis, at least in principle, exist[@ans_b0; @ggsz; @ans_path]. The simpler modes ([*e.g.*]{} $K^+ \pi^-$) require more statistics but they would not involve such modelling error as in the Dalitz method. Also the higher CP asymmetries in those modes should have greater resolving power for determination of $\gamma$. The table shows the statistical, systematic and the resonance-model dependent errors on $\gamma$ separately. Note that for now i do not think the model dependent error (around 10 degrees) ought to be added in quadrature. That is why the combined error of $\pm 30$ degrees is somewhat inflated to reflect that. The important point to note is that as more B’s are accumulated, more and more decay modes can be included in determination of $\gamma$; thus for the next several years the accuracy on $\gamma$ is expected to improve faster than $1/\sqrt(N_B)$, $N_B$ being the number of B’s.
Direct searches: Two important illustrations
--------------------------------------------
B-decays offers a wide variety of methods for searching for NP or for BSM-CP-odd phase(s). First we will elaborate a bit on the following two methods.
- Penguin dominated hadronic final states in $b \to s$ transitions.
- Radiative B-decays.
Then we will provide a brief summary of the multitude of possibilities that a SBF offers, in particular, for numerous important approximate null tests (ANTs).
Penguin dominated hadronic final states in $b \to s$ transitions
----------------------------------------------------------------
For the past couple of years, experiments at the two B-factories have been showing some indications of a tantalizing possibility [*i.e.*]{} a BSM-CP-odd phase in penguin dominated $b \to s$ transitions. Let us briefly recapitulate the basic idea.
Fig. \[hfag\_peng\] show the experimental status [@hfag04]. With about $250 \times 10^6$ B-pairs in each of the B-factories, there are two related possible indications. In particular, BABAR finds about a 3$\sigma$ deviation in $B \to \eta^{'} K_s$. Averaging over the two experiments, this is reduced to about 2.3$\sigma$. Secondly adding all such penguin dominated modes seems to indicate a 3.5$\sigma$ effect.
Since $B \to \eta^{'} K_s$ seems to be so prominently responsible for the indications of deviations in the current data sample, let us briefly discuss this particular FS. That the mixing induced CP in $\eta^{'} K_s$ can be used to test the SM was 1st proposed in [@ls97]. This was triggered in large part by the discovery of the unexpectedly large Br for $B \to \eta^{'} K_s$. Indeed ref. [@ls97] emphasized that the large Br may be very useful in determining $\sin 2 \beta$ with $B \to \eta^{'} K_s$ and comparing it with the value obtained from $B \to \psi K_s$. In fact it is precisely the large Br of $B \to \eta^{'} K_s$ that is making the error of the TDCP measurement the smallest amongst all the penguin dominated modes presently studied. Note also that there is a corresponding proposal to use the large Br of the inclusive $\eta^{'} X_s$ for searching for NP with the use of direct CP [@ans_direct; @ht_97].
Ref. [@ls97] actually suggested use of TDCP studies not just in $\eta^{'} K_s$ but in fact also $[\eta, \pi^0, \omega, \rho, \phi...]K_s$ to test the SM. These are, indeed most of the modes currently being used by BABAR & BELLE.
Simple analysis in [@ls97] suggested that in all such penguin dominated ($b \to s$) modes Tree/Penguin is small, $< 0.04$. In view of the theoretical difficulties in reliably estimating these effects, Ref [@ls97] emhasized that it would be very difficult in the SM to accomodate $\Delta S >0.10$, as a catious bound.
### Final state interaction effects
The original papers [@gw97; @rf97; @ls97] predicting,
S\_f = S\_f - S\_[K]{} 0 used naive factorization; in particular, FSI were completly ignored. A remarkable discovery of the past year is that in several charmless 2-body B-decays direct CP asymmetry is rather large. This means that FSI (CP-conserving) phase(s) in exclusive B-decays need not be small [@ccs1]. Since these are non-perturbative [@bss_pert], model dependence becomes unavoidable. Indeed characteristically these FSI phase(s) arise formally from $O(1/m_B)$ corrections:
- In pQCD [@pqcd] a phenomenological parameter $k_T$, corresponding to the transverse momentum of partons, is introduced in order to regulate the end point divergences encountered in power corrections. This in turn gives rise to sizable strong phase difference from penguin induced annihilation.
- In QCDF [@qcdf], in its nominal version, the direct CP asymmetry in many channels (e.g $B^0 \to K^+ \pi^-, \rho^- \pi^+,
\pi^+ \pi^-$.....) has the opposite sign compared to the experimental findings. Just like in the pQCD approach where the annihilation topology play an important role in giving rise to large strong phases, and for explaining the penguin-dominated VP modes, it has been suggested in [@qcdf_s4] that in a specific scenario (S4), for QCDF to agree with the Br of penguin-dominated PV modes as well as with the measured sign of the direct asymmetry in the prominent channel $B^0 \to K^+ \pi^-$, a large annihilation contribution be allowed by choosing $\rho_A =1$, $\phi_A= -55^\circ$ for PP, $\phi_A = -20^\circ$ for PV and $\phi_A = -70^\circ$ for VP modes.
- In our approach [@ccs1], QCDF is used for short-distance (SD) physics; however, to avoid double-counting, we set the above two parameters \[$\rho_A$, $\phi_A$\] as well as two additional parameters \[$\rho_H$, $\phi_H$\] that they have [@qcdf_s4] to zero. Instead we try to include long-distance ($1/m_B$) corrections by using on-shell rescattering of 2-body modes to give rise to the needed FS phases.
So, for example, color-suppressed modes such as $B^0 \to K^0 \pi^0$ gets important contributions from color allowed processes: $B^0 \to K^{-(*)} \pi^{+}(\rho^+), D_S^{-(*)} D^{+(*)}$. The coupling strengths at the three vertices of such a triangular graph are chosen to give the known rates of corresponding physical processes such as $B^0 \to D_S^{-*} D^{+(*)}$, $D^* \to D + \pi$ etc. Furthermore, since these vertices are not elementary and the exchanged particles are off-shell, form-factors have to be introduced so that loop integrals become convergent. Of course, there is no way to determine these reliably. We vary these as well as other parameters so that Br’s are in rough agreement with experiment, then we calculate the CP-asymmetries.
Recall the standard form for the asymmetries:\
=[S]{}\_f(mt)+\_f(mt)
The TDCP asymmtery ($S_f$) and direct CP asymmetry \[$A_f \equiv -C_f$ (BaBar notation)\] both depend on the strong phase. Thus measurements of direct CP asymmetry $A_f$ (in addition to $S_f$) allows tests of model calculations, though in practice its real use may be limited to those cases where the direct CP asymmetry is not small. This is the case, for example, for $\rho^0 {K_S}$ and $\omega {K_S}$ [@ccs2].
It is also important to realize that not only there is a correlation between $S_f$ and $A_f$ for FS in $B^0$ decays, but also that the model entails specific predictions for direct CP in the charged counterparts. So, for example, in our model for FSI, large direct CP asymmetry is also expected in the charged counterparts of the above two modes.
In addition to two body modes there are also very interesting 3-body modes such as $B^0 \to K^+ K^- K_S(K_L), K_S K_S K_S (K_L)$. These may also be useful to search for NP as they are also penguin dominated. We use resonance-dominance of the relevant two body channels to extend our calculation of LD rescattering phases in these decays [@ccs3].
Tables \[mix1\_tab\] and \[mix2\_tab\] summarize our results for $\Delta S$ and A for two body and 3-body modes. We find that [@ccs2; @ccs3] $B^0 \to \eta^{'} K_S$, $\phi K_S$ and 3$K_S$ are cleanest [@mb05], i.e. central values of $\Delta S$ as well as the errors are rather small, O(a few%). Indeed we find that even after including the effect of FSI, $\Delta S$ in most of these penguin-dominated modes, it is very difficult to get $\Delta S >0.10$ in the SM. Thus we can reiterate (as in [@ls97]) that $\Delta S >0.10$ would be a strong evidence for NP.
------------------------------- ---------------------------------- ------------------------- -------------------------------- -----------
\[0cm\]\[0cm\][Final State]{} SD+LD Expt SD+LD Expt
$\phi K_S$ $0.03^{+0.01+0.01}_{-0.04-0.01}$ $-0.38\pm0.20$ $-2.6^{+0.8+0.0}_{-1.0-0.4}$ $4\pm17$
$\omega K_S$ $0.01^{+0.02+0.02}_{-0.04-0.01}$ $-0.17^{+0.30}_{-0.32}$ $-13.2^{+3.9+1.4}_{-2.8-1.4}$ $48\pm25$
$\rho^0K_S$ $0.04^{+0.09+0.08}_{-0.10-0.11}$ – $46.6^{+12.9+3.9}_{-13.7-2.6}$ –
$\eta' K_S$ $0.00^{+0.00+0.00}_{-0.04-0.00}$ $-0.30\pm0.11$ $2.1^{+0.5+0.1}_{-0.2-0.1}$ $4\pm8$
$\eta K_S$ $0.07^{+0.02+0.00}_{-0.05-0.00}$ $-$ $-3.7^{+4.4+1.4}_{-1.8-2.4}$ $-$
$\pi^0K_S$ $0.04^{+0.02+0.01}_{-0.03-0.01}$ $-0.39^{+0.27}_{-0.29}$ $3.7^{+3.1+1.0}_{-1.7-0.4}$ $-8\pm14$
------------------------------- ---------------------------------- ------------------------- -------------------------------- -----------
: Direct CP asymmetry parameter $\A_f$ and the mixing-induced CP parameter $\Delta S_f^{SD+LD}$ for various modes. The first and second theoretical errors correspond to the SD and LD ones, respectively (see [@ccs2] for details). The $f_0K_S$ channel is not included as we cannot make reliable estimate of FSI effects on this decay; table adopted from [@ccs2].[]{data-label="mix1_tab"}
. \[mix2\_tab\]
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
Final State $\sin 2\beta_{\rm eff}$ Expt.
------------------------------------------------------------------------------------------------------------------------------------------------- --------------------------------------------------- ------------------------
$(K^+K^-K_S)_{\phi K_S~{\rm excluded}}$ $0.749^{+0.080+0.024+0.004}_{-0.013-0.011-0.015}$ $0.57^{+0.18}_{-0.17}$
$(K^+K^-K_S)_{CP+}$ $0.770^{+0.113+0.040+0.002}_{-0.031-0.023-0.013}$
$(K^+K^-K_L)_{\phi $0.749^{+0.080+0.024+0.004}_{-0.013-0.011-0.015}$ $0.09\pm0.34$
K_L~{\rm
excluded}}$
$K_SK_SK_S$ $0.748^{+0.000+0.000+0.007}_{-0.000-0.000-0.018}$ $0.65\pm0.25$
$K_SK_SK_L$ $0.748^{+0.001+0.000+0.007}_{-0.001-0.000-0.018}$
$\A_f(\%)$ Expt.
$(K^+K^-K_S)_{\phi $0.16^{+0.95+0.29+0.01}_{-0.11-0.32-0.02}$ $-8\pm10$
K_S~{\rm
excluded}}$
$(K^+K^-K_S)_{CP+}$ $-0.09^{+0.73+0.16+0.01}_{-0.00-0.27-0.01}$
$(K^+K^-K_L)_{\phi $0.16^{+0.95+0.29+0.01}_{-0.11-0.32-0.02}$ $-54\pm24$
K_L~{\rm
excluded}}$
$K_SK_SK_S$ $0.74^{+0.02+0.00+0.05}_{-0.06-0.01-0.06}$ $31\pm17$
$K_SK_SK_L$ $0.77^{+0.12+0.08+0.06}_{-0.28-0.11-0.07}$
------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------
: Mixing-induced and direct CP asymmetries $\sin
2\beta_{\rm eff}$ (top) and $\A_f$ (bottom), respectively, in $B^0\to K^+K^-K_S$ and $K_SK_SK_S$ decays. Results for $(K^+K^-K_L)_{CP\pm}$ are identical to those for $(K^+K^-K_S)_{CP\mp}$; table taken from [@ccs3]
Having said that, it is still important to stress that genuine NP in these penguin dominated modes must show up in many other channels as well. Indeed, on completely model independent grounds [@brwns], the underlying NP has to be either in the 4-fermi vertex (bss$\bar s$) or (bsg, $g=gluon$). In either case, it has to materialize into a host of other reactions and phenomena and it is not possible that it only effects time dependent CP in say $B \to \eta' K_s$ and/or $\phi K_s$ and/or 3$K_s$. For example, for the 4-fermi case, we should also expect non-standard effects in $B_d \to \phi(\eta^{'}) K^*$, $B^+ ->
\phi (\eta^{'}) K^{+(*)}$, $B_s \to \phi \phi (\eta^{'})$...In the second case not only there should be non-standard effects in these reactions but also in $B_{d(u)} \to X_s \gamma$, $K^* \gamma$, $B_s \to \phi
\gamma$..... and also in the corresponding $l^+ l^-$ modes. Unless corroborative evidence is seen in many such processes, the case for NP due to the non-vanishing of $\Delta S$ is unlikely to be compelling, especially if (say) $\Delta S \lsim 0.15$.
### Averaging issue
As already emphasized in [@ls97], to the extent that penguin contributions dominate in these many modes and $tree/penguin$ is only a few percent testing the SM by adding $\Sigma \Delta S_f$, where $f = K_S + \eta^{'} (\phi, \pi, \omega, \rho,
\eta, K_S K_S$...), is sensible at least from a theoretical standpoint. At the same time it is important to emphasize that a convincing case for NP requires unambiguous demonstration of significant effects (i.e. $\Delta S >0.10$) in several individual channels.
### Sign of $\Delta S$
For these penguin-dominated modes, $\Delta S_f$ is primarily proportional to the hadronic matrix element $<f | \bar u \Gamma
b \bar s \Gamma' u|B^0>$. Therefore, in the SM for several of the final states (f), $\Delta S_f$ could have the same sign. So a systematic trend of $\Delta S_f$ being positive or negative ([*and small of O(a few %)*]{}) does not necessarily mean NP.
The situation wrt to $\eta' K_S$ is especially interesting. As has been known for the past many years this mode has a very large Br, almost a factor of 7 larger than the similar two body K $\pi$ mode. This large Br is of course also the reason why the statistical error is the smallest, about a factor of two less than any other mode being used in the test. For this reason, it is gratifying that $\eta' K_S$ also happens to be theoretically very clean in several of the model calculations. This has the important repercussion that confirmation of a significant deviation from the SM, may well come 1st by using the $\eta' K_S$ mode, perhaps well ahead of the other modes [@js].
### Concluding remarks on penguin-dominated modes
Concluding this section we want to add that while at present there is no clear or compelling deviation from the SM the fact still remains that this is a very important approximate null test (ANT). It is exceedingly important to follow this test with the highest luminosity possible to firmly establish that as expected in the SM, $\Delta S_f$ is really $0.05$ and is not significantly different from this expectation. To establish this firmly, for several of the modes of interest, may well require a SBF.
Time dependent CP in exclusive radiative B-decays
=================================================
Br ($B \to \gamma X_{s(d)}$) and direct CP asymmetry $a_{cp}(B \to \gamma X_{s(d)})$ are well known tests of the SM [@hurth_rmp; @hiller_fpcp; @kn98; @ksw00]. Both of these use the inclusive reaction where the theoretical prediction for the SM are rather clean; the corresponding exclusive cases are theoretically problematic though experimentally more accessible. In 1997 another important test [@ags] of the SM was proposed which used mixing induced CP (MICP) or time-dependent CP (TDCP) in exclusive modes such as $B^0 \to K^* \gamma, \rho \gamma$..... This is based on the simple observation that in the SM, photons produced in reactions such as $B \to K^* \gamma, K_2^* \gamma, \rho \gamma$... are predominantly right-handed whereas those in $\bar B^0$ decays are predominantly left-handed. To the extent that FS of $B^0$ and $\bar B^0$ are different MICP would be suppressed in the SM. Recall, the LO $H_{eff}$ can be written as H\_[eff]{} = - G\_F F\_ + h.c. \[effective\_H\] Here $F_L^q$ ($F_R^q$) corresponds to the amplitude for the emission of left (right) handed photons in the $b_R
\to q_L \gamma_L$ ($b_L \to q_R \gamma_R$) decay, [*i.e.*]{} in the $\OL B \to \OL F
\gamma_L$ ($\OL B \to \OL F \gamma_R$) decay.
Application to ${B^0},{B_s}\to$ vector meson + photon
-----------------------------------------------------
Thus, based on the SM, [*LO $H_{eff}$*]{}, in b-quark decay (i.e. $\bar B$ decays), the amplitude for producing wrong helicity (RH) photons $\propto m_q/m_b$ where $m_q
=m_s$ or $m_d$ for $b \to s \gamma$ or $b \to d \gamma$ respectively. Consequently the TDCP asymmetry is given by,
[B\^0]{}K\^[\*0]{}&:& A(t) (2m\_s/m\_b)(2)(mt) ,\
[B\^0]{}\^0&:& A(t) 0 ,\
[B\_s]{}&:& A(t) 0 ,\
[B\_s]{}K\^[\*0]{}&:& A(t) -(2m\_d/m\_b)(2)(mt) , \[examples\] where $K^{*0}$ is observed through $K^{*0}\to K_S \pi^0$.
Interestingly not only emission of wrong-helicity photons from B decays is highly suppressed, in many extensions of the SM, [*e.g.*]{} Left-Right Symmetric models (LRSM) or SUSY [@chn03; @okada03; @chl00] or Randall-Sundrum (warped extra dimension [@aps04]) models, in fact they can be enhanced by the ratio $m_{heavy}/m_b$ where $m_{heavy}$ is the mass of the virtual fermion in the penguin-loop. In LRSM as well as some other extensions this enhancement can be around $m_t/m_b$. So while in the SM the asymmetries are expected to be very small, they can be sizeable in LRSM [@ags] (see Table \[ags\_tab\]) as well as in many other models.
---------------------------------------------------------------------------------------------------------
Process SM LRSM
------------------------ ------------------------------- ------------------------------------------------
$A(B \to K^* \gamma)$ $2\frac{m_s}{m_b} $\sin 2 \omega \cos 2 \beta \sin (\Delta m_t)$
\sin 2\beta \sin(\Delta m_t)$
$A(B \to \rho \gamma)$ $\approx 0$ $\sin 2 \omega \sin (\Delta m_t)$
---------------------------------------------------------------------------------------------------------
: Mixing-induced CP asymmetries in radiative exclusive B-decays in the SM and in the LRSM. Note $|\sin 2 \omega| \lsim 0.67$ is allowed [@ags; @brwns][]{data-label="ags_tab"}
Generalization to ${B^0}, {B_s}\to$ two pseudoscalars + photon
--------------------------------------------------------------
An important generalization was made in Ref [@aghs]. It was shown that the basic validity of this test of the SM does not require the final state to consist of a spin one meson (a resonance such as $K^*$ or $\rho$) in addition to a photon. In fact the hadronic final states can equally well be two mesons; [*e.g.*]{} ${K_S}(\pi^0, \eta', \eta, \phi...)$ or $\pi^+ \pi^-$. Inclusion of these non-resonant final states, in addition to the resonances clearly enhances the sensitivity of the test considerably. For the case when the two mesons are antiparticle of each other [*e.g.* ]{} $\pi^+ \pi^-$, then there is the additional advantage that both the magnitude and the weak phase of any new physics contribution may be determined from a study of the angular distribution [@aghs].
Theoretical subtelties
----------------------
In principle, photon emission from the initial light-quark is a non-perturbative, long-distance, contamination to the interesting signal of the short-distance dipole emission from $H_{eff}$ [@abs; @grinpir]. Fortunately, it can be shown [@aghs] that predominantly these LD photons have the same helicity as those from $H_{eff}$.
Another important source of SM contamination was recently emphasized in Ref. [@gglp] from processes such as $b \to s \gamma + $ gluon which are from non-dipole operators. Such processes do not fix the helicity of the photon and so can make a non-vanishing SM contribution to mixing induced CP.
It was emphasized in Ref [@aghs] that the presence of such non-dipole contributions can be separated from the dipole contributions, though, it may require larger amount of data, the resolution to this problem is data driven.
To briefly recapitulate, the different operator structure in $H_{eff}$ would mean, that in contrast to the pure dipole case, the time dependent CP asymmetry (S) would be a function of the Dalitz variables, the invariant mass (s) of the meson pair, and the photon angle of emission (z). A difference in the values of S for two resonances of identical $J^{PC}$ would also mean presence of non-dipole contributions. Schematically, we may write:
d S\^i/(ds dz) & = & \[A\_ + A\_0\^i\] + B\^i s + C\^i z where $A_{\sigma}$ is the “universal" contribution that one gets from the dipole operator of the $H_{eff}$ no matter if it is a resonance, or a non-resonance mode. It is distinct from the contribution of the 4-quark operators as not only it is independent of energy (s) or angle (z) Dalitz variables but also it is independent of the specific nature of the hadronic FS ([*i.e.*]{} resonant or non-resonant). The remaining contributions are all originating from 4-quark operators; not only they dependent on energy and angle but also the coefficients are expected to vary from one FS to another. In particular the 4-quark operators may give a FS dependent (energy and angle independent) constant $A_0^i$. It is easy to convince oneself that with sufficient data the important term $A_{\sigma}$, at least in principle, can be separated. Once that is done its size should be indicative of whether it is consistent with expectations from SM or requires new physics to account for it.
Approximate null tests aglore!
------------------------------
If the effects of a BSM CP-odd phase on B-physics are small, then searching for these via [***null tests***]{} becomes especially important. Since CP is not an exact symmetry of the SM, it is very difficult if not impossible to find exact null tests. Fortunately clean environment at a SBF should allow many interesting approximate null tests (ANTs); see Table \[ants\_tab\] [@brwns].
Clearly there is a plethora of powerful tests for a new CP-odd phase and /or new physics that a SBF should allow us to do. Perhaps especially noteworthy (in addition to penguin-dominated hadronic and radiative B decays) are the numerous very interesting tests pertaining to $B \to X(K,K^*..) l^+ l^-$ [@hurth_rmp; @hiller_fpcp].
Furthermore search for the transverse polarization [@aes_tau; @gl_tau] of the $\tau$ in $B \to X(D,D^*..) \tau \nu_{\tau}$ due to their unique cleanliness are extremely interesting especially in light of the discovery of neutrino mass and the potential richness of neutrinos with the possible presence of Majorana neutrinos in simple grounds-up extensions of the SM as well as in many other approaches [@abs_3g; @rnm].
Sensitivity of each of these to NP as well as theoretical cleanliness ([*i.e.*]{} how reliable SM predictions are) for each is also indicated. It should be clear that for most of these tests $> 5 \times 10^9$ B-pairs are essential, that is a SBF.
.
Final State Observable how clean how sensitive
-------------------------------------------- -------------------- ----------- ---------------
$\gamma [K_s^*,\rho,\omega]$ TDCP 5\* 5\*
$K_s[\phi,\pi^0,\omega,\eta',\eta,\rho^0]$ TDCP 4.5\* 5\*
$K^*[\phi,\rho,\omega]$ TCA 4.5\* 5\*
$[\gamma,l^+l^-] [X_s,X_d]$ DIRCP 4.5\* 5\*
same Rates 3.5\* 5\*
$\jpsi K $ TDCP, DIRCP 4\* 4\*
$\jpsi K^* $ TCA 5\* 4\*
$D (*) \tau \nu_{\tau}$ TCA ($p_t^{\tau})$ 5\* 4\*
same Rate 4\* 4\*
: Final states and observables in B - decays useful in searching for effects of New Physics. Reliability of SM predictions ([*i.e.*]{} how clean) and sensitivity to new physics are each indicated by stars ($5 = best$); table adopted from [@brwns]
\[ants\_tab\]
K-Unitarity Triangle
====================
For the past many years, effort has been directed towards constraining the UT especially the parameters $\rho$ and $\eta$ by a combination of information from K and B-physics, as mentioned briefly in Section 1. With the advent of B-factories and significant advance that has been already made (and a lot more is expected to come) it has become possible to construct the UT purely from B-physics [@ckmfit; @utfit]. In fact it may also be very interesting and important to construct a separate UT from K-decays. This could become particularly useful in search for small deviations. Reactions that are relevant for a K-UT are [@pascos03]:
- Indirect CP-violation parameter, $\epsilon_K$ with the hadronic matrix elements (parameter $B_K$) from the lattice. With the dawning of the era of dynamical simulations using discretizations that preserve chiral-flavor symmetries of the continuum [@rbc_nf2], lattice should be able to significantly reduce the errors on $B_K$ [@rbc_jun].
- Accurate measurements of the BR of $K ^+ \to \pi^+ \nu \bar \nu$ can give a clean determination of $ |V_{td}|$ [@ajb_nunu]. Important progress has been recently made in the 1st step towards an accurate determination of this Br [@e787_bnl]. Charm quark contribution in the penguin graph is difficult to reliably estimate but this is expected to be subdominant [@flp_nunu].
- Measurement of the BR of $K_L \to \pi^0 \nu \bar \nu$ can give an extremely clean value of $\eta$, [*i.e.*]{} $Im
V_{td}$. This is clearly very challenging experimentally; however, it is unique in its cleanliness, perhaps on the same footing as $\gamma$ from $BKD$ processes discussed above.
- After enormous effort, the experimentalists have determined the direct CP violation parameter $\epsilon'/\epsilon$ with considerable accuraccy [@na48; @ktev]. For theory a reliable calculation remains a very important outstanding challenge. Recently it has become clear that not only chiral symmetry on the lattice is essential for this calculation but also the quenched approximation suffers here from very serious pathology [@gp; @jlq6]. As mentioned above, since the past 2-3 years considerable effort is being expended in generation of dynamical configurations with domain wall quarks which possess excellent chiral properties. In the near future we should expect to see the application of these new generation of lattices for study of $\epsilon'/\epsilon$. It remains to be seen as to how accurately the current generation of computers can allow this important calculation to be done.
Neutron electric dipole moment: a classic ANT of the SM
=======================================================
In the SM, neutron electric dipole moment (nedm) cannot arise at least to two EW loops; thus is expected to be exceedingly small, [*i.e.*]{} $\lsim 10^{-31}ecm$. Long series of experiments over the past several decades now place a 90 %CL bound of $\lsim 6.3 \times 10^{-26}$ecm [@nedm_bound]. So the expectation from the SM is many orders of magnitude below the current experimental bound. In numerous extensions of the SM, including SUSY, warped extra dimensions etc. nedm close to or even somewhat bigger than the current experimental bound occurs [@bgk_nedm; @aps04]. Thus continual experimental improvements of this bound remains a very promising way to discover new BSM CP-odd phase(s).
Top quark electric dipole moment: another clean null test of the SM
===================================================================
The top quark is so heavy compared to the other quarks that the GIM-mechanism is extremely effective. Thus in the decays of the top-quark, in the SM, all FCNC are extremely suppressed. Once again, top quark edm cannot arise in the SM to two EW loops and is therefore expected to be extremly small. In many BSM scenarios with extra Higgs doublets [@xs; @bsp], LRSM, SUSY [@abes_pr], the top quark can acquire edm at one loop and consequently can be considerably bigger (See Table \[dipolesumtable\]). Therefore searches for the top dipole moment at the International Linear Collider will be an important goal [@abes_pr; @lc_book]. Indeed if sufficient high luminosity could be attained top quark edm of around $10^{-19}$ ecm may well be detectable (See Table \[tdm\_at\_ilc\]).
\[dipolesumtable\]
---------------------------- -------------------------- -------------- ---------------------------- ---------------------------- --
type of moment $\sqrt s$ Standard Neutral Higgs Supersymmetry
$(e-cm)~~ \Downarrow$ $({\rm GeV}) \Downarrow$ Model $m_{h}=100 -300$ $m_{\tilde g}=200-500$
500 $(4.1-2.0)\times 10^{-19}$ $(3.3-0.9)\times 10^{-19}$
$|\Im{\rm m}(d_t^\gamma)|$ $< 10^{-30}$
1000 $(0.9-0.8)\times 10^{-19}$ $(1.2-0.8)\times 10^{-19}$
500 $(0.3-0.8)\times 10^{-19}$ $(0.3-0.9)\times 10^{-19}$
$|\Re{\rm e}(d_t^\gamma)|$ $< 10^{-30}$
1000 $(0.7-0.2)\times 10^{-19}$ $(1.1-0.3)\times 10^{-19}$
500 $(1.1-0.2)\times 10^{-19}$ $(1.1-0.3)\times 10^{-19}$
$|\Im{\rm m}(d_t^Z)|$ $< 10^{-30}$
1000 $(0.2-0.2)\times 10^{-19}$ $(0.4-0.3)\times 10^{-19}$
500 $(1.6-0.2)\times 10^{-19}$ $(0.1-0.3)\times 10^{-19}$
$|\Re{\rm e}(d_t^Z)|$ $< 10^{-30}$
1000 $(0.2-1.4)\times 10^{-19}$ $(0.4-0.1)\times 10^{-19}$
---------------------------- -------------------------- -------------- ---------------------------- ---------------------------- --
: Expectations for top edm form-factor in SM and beyond; adopted from [@abes_pr]
------------------------------------ --------- ---------- ---------- --------- ---------- ----------
$P_e=0$ $P_e=+1$ $P_e=-1$ $P_e=0$ $P_e=+1$ $P_e=-1$
$\delta(\Re{\rm e}d_{t}^{\gamma})$ 4.6 0.86 0.55 1.7 0.35 0.23
$\delta(\Re{\rm e}d_{t}^{Z})$ 1.6 1.6 1.0 0.91 0.85 0.55
$\delta(\Im{\rm m}d_{t}^{\gamma})$ 1.3 1.0 0.65 0.57 0.49 0.32
$\delta(\Im{\rm m}d_{t}^{Z})$ 7.3 2.0 1.3 4.0 0.89 0.58
------------------------------------ --------- ---------- ---------- --------- ---------- ----------
: *Attainable 1-$\sigma$ sensitivities to the CP-violating dipole moment form factors in units of $10^{-18}$ e-cm, with ($P_e=\pm
1$) and without ($P_e=0$) beam polarization. $m_t=180$ GeV. Table taken from [@abes_pr; @bbo_tdm]. \[tdm\_at\_ilc\]*
Summary
=======
The new millennium marks the spectacular success of B-factories leading to a milestone in our understanding of CP-violation; in particular, for the first time CKM paradigm of CP violation is quantitatively confirmed.
Direct measurement of $\sin2\beta$ by the B-factories agrees remarkably well with the theoretical expectation from the SM to about 10%. Furthermore, first relatively crude [*direct*]{} determination of the other two angles ($\alpha$ & $\gamma$) also are consistent with theoretical expectations. While these findings are good news for the SM, at the same ime, they imply that most likely the effect of BSM CP-odd phase on B-physics is likely to be a small perturbation. Thus discovery of new BSM-CP-odd source(s) of CP violation in B-physics is likely to require very large, clean, data samples and extremely clean predictions from theory.
For the search of such small deviations approximate null tests of the SM gain new prominence.
Also important for this purpose is the drive to directly determine all three angles of the UT with highest precision possible, [*i.e.*]{} with errors roughly around the errors allowed by theory. It should be clear that to accomplish this important goal would require a Super-B Factory.
Specifically regarding penguin-dominated hadronic FS, that have been much in the recent news, the current data does not show any convincing signal for deviation from the SM; however, it is a very important and sensitive test for new physics and its of vital importance to reduce the experimental errors to O(5%); for this purpose too a SBF may well be needed.
Outside of B-Physics, K-unitarity triangle, neutron electric dipole moment and top quark dipole moment are also very important [*approximate null tests*]{} of the SM that should be pursued vigorously.
Acknowledgements
================
I want to thank the organizers (and, in particular, Mario Greco) for their kind invitation. Useful discussions with David Atwood, Tom Browder, Chun-Khiang Chua, Tim Gershon, Masashi Hazumi, Jim Smith and especially Hai-Yang Cheng are gratefully acknowledged. This research was supported in part by DOE contract No. DE-FG02-04ER41291.
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abstract: |
In this paper we show the following result: if $C$ is an n-dimensional 0-symmetric convex compact set, $f:C\rightarrow[0,\infty)$ is concave, and $\phi:[0,\infty)\rightarrow[0,\infty)$ is not identically zero, convex, with $\phi(0)=0$, then $$\frac{1}{|C|}\int_C\phi(f(x))dx\leq \frac{1}{2}\int_{-1}^1\phi(f(0)(1+t))dt,$$ where $|C|$ denotes the volume of $C$. If $\phi$ is strictly convex, equality holds if and only if $f$ is affine, $C$ is a generalized symmetric cylinder and $f$ becomes $0$ at one of the basis of $C$.
We exploit this inequality to answer a question of Francisco Santos on estimating the volume of a convex set by means of the volume of a central section of it. Second, we also derive a corresponding estimate for log-concave functions.
address: 'Departamento de Didáctica de la Matemática, Facultad de Educación, Universidad de Murcia, 30100-Murcia, Spain'
author:
- Bernardo González Merino
date:
-
-
title: 'On a generalization of the Hermite-Hadamard inequality and applications in convex geometry'
---
[^1]
The classical Jensen’s inequality [@J] states that if $(X, \Sigma, \mu)$ is a probability space, then for any concave $f:\mathbb R\rightarrow\mathbb R$ and any $\mu$-integrable function $g:X \rightarrow\mathbb R$, we have that $$\int_{X}f(g(x))d\mu(x)\leq f\left(\int_{X}g(x)d\mu(x)\right),$$ and moreover, equality holds if and only if either $f$ is affine or $g$ is independent of $x$.
Let ${\mathcal K}^n$ be the set of n-dimensional compact, convex sets. A set $K\in{\mathcal K}^n$ is *0-symmetric* if $K=-K$. Let ${\mathcal K}^n_0$ be the subset of ${\mathcal K}^n$ of 0-symmetric sets. Notice that we will consistently use $C$ (resp. $K$) in functional (resp. geometric) inequalities. For any set $K\in{\mathcal K}^n$, we denote by $|K|$ the *volume* (or Lebesgue measure) of $K$ computed in its *affine hull* $\mathrm{aff}(K)$, i.e., the smallest affine subspace containing $K$. Let $\Vert x\Vert=\sqrt{x_1^2+\cdots+x_n^2}$ be the *Euclidean norm* of $x=(x_1,\dots,x_n)\in\mathbb R^n$, let $B^n_2=\{x\in\mathbb R^n:\Vert x\Vert\leq 1\}$ be the *Euclidean unit ball* of $\mathbb R^n$, and let $\omega_n=|B^n_2|$ be its volume. For every $x,y\in\mathbb R^n$, let $\langle x,y\rangle=x_1y_1+\cdots+x_ny_n$ be the *scalar product* of $x$ and $y$. The *center of mass* of $K\in{\mathcal K}^n$ is the point $$x_K=\frac{1}{|K|}\int_{K}xdx.$$ A well-known consequence of Jensen’s inequality is the following Hermite-Hadamard inequality: for any $C\in{\mathcal K}^n$ and $f:C\rightarrow{\mathbb R}$ concave, then $$\label{eq:HH_Jensen}
\frac{1}{|C|}\int_Cf(x)dx\leq f(x_C),$$ with equality sign if and only if $f$ is affine. It was named after Hermite 1881 and Hadamard 1893, who proved independently in the 1-dimensional case. See [@DP] (and [@CalCar] or [@St]) and the references on it for other historical considerations and a comprehensive and complete view of this type of inequalities.
The mean value of $f$ measured in $C$ (the left-term in ) has repeatedly appeared during the development of different topics of Analysis and Geometry (cf. [@HLP]). For instance, Berwald [@Ber Sect. 7] studied monotonicity relations of $l_p$ means of concave functions over convex compact domains (see also [@ABG] and [@AAGJV Sect. 7] for a translation of it). Borell [@Bor] did a step further by showing some convexity relations in the same regard (Thms. 1 and 2). See also Milman-Pajor [@MP2 2.6] and [@GNT Sect. 5], or the Hardy-Littlewood Maximal Function (cf. [@Me]). Let $\mathcal L^n_i$ be the set of *i-dimensional linear subspaces* in $\mathbb R^n$. For $K\in\mathcal K^n$ and $H\in\mathcal L^n_i$, let $P_HK$ be the *orthogonal projection* of $K$ onto $H$. Moreover, let $e_1,\dots,e_n$ be the vectors of the *canonical basis* of $\mathbb R^n$. For every $A\subset\mathbb R^n$, let $\mathrm{lin}(A)$ be the *linear hull* of $A$, and let $A^\bot$ be the *orthogonal subspace* to $A$, and let $\partial A$ be the *boundary* of $A$.
In 2017 during the conference *Convex, Discrete and Integral Geometry* [^2] Francisco Santos asked the following question: What is the smallest constant $c_n > 0$ such that $$|K|\leq c_n|K\cap e_1^\bot|$$ for every $K\in\mathcal K^n$ with $P_{\mathrm{lin}\{e_1\}}K=[-e_1,e_1]$. One of our aims is to compute this constant $c_n$. A similar inequality is derived in [@IVS Lemma 5.2] where it is used to bound the volume of empty lattice 4-simplices in terms of volumes of 3-lattice polytopes. Notice that for every $K\in\mathcal K^n$ and $H\in\mathcal L^n_i$, Fubini’s theorem implies that $$|K|\leq |P_HK|\max_{x\in H^\bot}|K\cap (x+H^\bot)|.$$ There exist special subspaces for which the inequality above strengthens. In this regard Spingarn [@S] and later Milman and Pajor [@MP] proved that if $K\in{\mathcal K}^n$ and $H\in{\mathcal L}^n_i$, then $$\label{eq:volumes_and_centroid}
|K|\leq |P_HK||K\cap(x_K+H^\bot)|.$$ It is even known the worst deviation between the maximal volume section and the one passing through the centroid of $K$ (cf. [@Fr],[@MM], and further extensions in [@SY]). Surprisingly enough, a consequence of Jensen’s inequality shows that (cf. Theorem \[thm:center\_of\_Projection\] below) for every $K\in\mathcal K^n$ and $H\in\mathcal L^n_{n-1}$ then $$\label{eq:SectProjJensen}
|K|\leq |P_HK||K\cap(x_{P_HK}+H^\bot)|,$$ and this choice can *sometimes* be better than (cf. Remark \[rmk:x\_Pbetterx\_C\]). In this regard, we prove the following result, extending the inequality above when $P_HK\in\mathcal K^n_0$ and answering to the question posed at the beginning. The result below can sometimes be better than up to a linear factor in the dimension of the subspace.
\[thm:0-symm\_GeneralCase\] Let $K\in{\mathcal K}^n$ and $H\in{\mathcal L}^n_i$ be such that $P_HK=-P_HK$. Then $$|K|\leq\frac{2^{n-i}}{n-i+1}|P_HK||K\cap H^\bot|.$$ If we assume w.l.o.g. that $H={\mathbb R}^i\times\{0\}^{n-i}$, there is equality above if and only if there exist a $(n-i)\times i$ matrix $B_0$, and $u\in{\mathbb R}^i$ such that $$K\cap(x+H^\bot)=(x_1,B_0(x_1))+\lambda_x (K\cap H^\bot)\quad\text{where}\quad
\lambda_x=\frac{\langle u,x_1\rangle+|K\cap H^\bot|^\frac{1}{n-i}}{|K\cap H^\bot|^\frac{1}{n-i}},$$ for every $x=(x_1,0)\in P_HK\subset{\mathbb R}^i\times\{0\}^{n-i}$. If in addition $i\leq n-2$ then equality in the inequality implies that there exist $(x_0,0)\in{\mathbb R}^i\times{\mathbb R}^{n-i}$ and $K_0\in{\mathcal K}^{i-1}$ such that $P_HK=[(-x_0,0),(x_0,0)]+(\{0\}\times K_0\times\{0\}^{n-i})$ and $$|K\cap(x+H^\bot)|=\left(1+\frac{\langle (x_0,0),x\rangle}{\Vert x_0\Vert^2}\right)|K\cap H^\bot|,$$ for every $x\in P_HK$.
Note that Theorem \[thm:0-symm\_GeneralCase\] solves the question of Francisco Santos with optimal constant $c_n=2^n/n$, also characterizing the equality case. At the core of the proof of Theorem \[thm:0-symm\_GeneralCase\] rests a generalization of , which is the main result of the paper.
\[thm:Ineq\_Concave\_func\] Let $C\in{\mathcal K}^n_0$, let $f:C\rightarrow[0,\infty)$ be concave, and let $\phi:[0,\infty)\rightarrow[0,\infty)$ be not identically zero, convex, such that $\phi(0)=0$. Then $$\frac{1}{|C|}\int_C\phi(f(x))dx\leq \frac{1}{2}\int_{-1}^1\phi(f(0)(1+t))dt.$$ If $\phi$ is strictly convex, equality holds if and only if there after applying a suitable rotation exist $C_0\in{\mathcal K}^{n-1}_0$ and $x_0\in{\mathbb R}^n$ with $(x_0)_1>0$ such that $$C=[-x_0,x_0]+(\{0\}\times C_0)$$ and such that $f$ is an affine function with $f(-x_0+x)=0$, for every $x\in\{0\}\times{\mathbb R}^{n-1}$.
Notice that, due to the convexity of $\phi$, the term $\int_{-1}^1\phi(f(0)(1+t))dt/2$ in Theorem \[thm:Ineq\_Concave\_func\] is larger than $c\cdot\phi(f(0))$, for some constant $c\geq 1$, as in the case $\phi(t)=t^m$, $m\in\mathbb N$ (cf. Corollary \[cor:tAlpha\]). Milman and Pajor (see [@MP]) proved that if $f:{\mathbb R}^n\rightarrow[0,\infty)$ is an integrable *log-concave function* (i.e. $\log(f)$ is concave), and $\mu:{\mathbb R}^n\rightarrow[0,\infty)$ is a probability measure, then $$\label{eq:MilmanPajor}
\int_{{\mathbb R}^n}f(x)d\mu(x)\leq f\left(\int_{{\mathbb R}^n}x\frac{f(x)}{\int_{{\mathbb R}^n}f(z)d\mu(z)}d\mu(x)\right),$$ and equality holds if and only if $f(x)$ is independent of $x$. A direct consequence of this result is the following Hermite-Hadamard inequality: for any $C\in{\mathcal K}^n$, $f:C\rightarrow[0,\infty)$ concave, and $m\in{\mathbb N}$, then $$\label{eq:HHcenter_of_mass}
\frac{1}{|C|}\int_C f(x)^m dx\leq f(x_{f,m})^m,$$ where $x_{f,m}=\int_{C}x\frac{f(x)^m}{\int_{{\mathbb R}^n}f(z)^mdz}dx$ ($f^m$ is log-concave if $f$ is concave). Notice that has a better constant than in Corollary \[cor:tAlpha\]; however, only in the latter the center is *independent* of the function.
Using Theorem \[thm:Ineq\_Concave\_func\] we also derive a Hermite-Hadamard inequality as in evaluated at the center of mass of the domain. Notice that if $f(0)=f_{min}$, since $\lim_{a\rightarrow 1+}\frac{a^2-1}{\log (a^2)}=1$, the right-term below becomes $f_{min}$.
\[thm:log-concaveDirect\] Let $C\in\mathcal K^n_0$ and let $f:C\rightarrow(0,\infty)$ be log-concave and continuous. Then $$\frac{1}{|C|}\int_Cf(x)dx\leq f_{min}\frac{(f(0)/f_{min})^2-1}{\log((f(0)/f_{min})^2)},$$ where $f_{min}=\min\{f(x):x\in C\}$. Equality holds if and only if there after applying a suitable rotation exist $C_0\in{\mathcal K}^{n-1}_0$ and $x_0\in{\mathbb R}^n$ with $(x_0)_1>0$ such that $$C=[-x_0,x_0]+(\{0\}\times C_0)$$ and $f=e^u$ where $u:C\rightarrow{\mathbb R}$ is an affine function with $u(-x_0+x)=\log(f_{min})$, for every $x\in\{0\}\times{\mathbb R}^{n-1}$.
Notice that one could relax the continuity of $f$ above, by replacing it by $f(x)\geq c$ for some $c>0$. However, log-concave functions are continuous in the interior of their domain, and their integral is not affected by changes in the values attained in the boundary of its domain.
The *Brunn-Minkowski inequality* states that for any $K_1,K_2\in{\mathcal K}^n$ and $\lambda\in[0,1]$, then $$\label{eq:BandM}
|(1-\lambda)K_1+\lambda K_2|^\frac1n\geq(1-\lambda)|K_1|^\frac1n+\lambda|K_2|^\frac1n,$$ and equality holds if and only if $K_1$ and $K_2$ are dilates, or if they are lower dimensional and contained in parallel hyperplanes (see [@Ga] and the references therein for an insightful and complete study of this inequality).
We split the proofs of the results into two sections. In Section \[sec:HH\_ineqs\] we prove the functional inequality, i.e. Theorems \[thm:Ineq\_Concave\_func\] and \[thm:log-concaveDirect\]. Afterwards in Section \[sec:RS\_ineqs\] we show some volumetric inequalities solving in particular the question of Francisco Santos posed above.
Proof of the Orlicz-Jensen-Hermite-Hadamard type inequalities {#sec:HH_ineqs}
=============================================================
Let us start this section by remembering that the *Schwarz symmetrization* of $K\in{\mathcal K}^n$ with respect to $\mathrm{lin}(u)$, $u\in{\mathbb R}^n\setminus\{0\}$, is the set $$\sigma_u(K)=\bigcup_{t\in{\mathbb R}}\left(tu+r_t(B^n_2\cap u^\bot)\right),$$ where $r_t\geq 0$ is such that $|K\cap(tu+u^\bot)|=r_t^{n-1}\omega_{n-1}$. It is well-known that $\sigma_u(K)\in{\mathcal K}^n$ and that $|\sigma_u(K)|=|K|$ (cf. [@Gru Section 9.3] or [@Sch] for more details). For every $K\in\mathcal K^n$ and $x\in\mathbb R^n\setminus\{0\}$, the *support function* of $K$ at $x$ is defined by $h(K,x)=\sup\{\langle x,y\rangle:y\in K\}$.
Since $f$ is concave and non-negative in $C$, there exists an affine function $g:C\rightarrow[0,\infty)$ such that $$g(0)=f(0)\quad\text{and}\quad g(x)\geq f(x)\,\text{ for every }x\in C.$$ Since $\phi$ is convex with $\phi(0)=0$, for any $x_2>x_1>0$ we have that $$0\leq\frac{\phi(x_1)-0}{x_1-0}\leq\frac{\phi(x_2)-\phi(x_1)}{x_2-x_1},$$ i.e., $\phi$ is non-decreasing. Thus $$\label{eq:affine}
\int_C\phi(f(x))dx\leq \int_C\phi(g(x))dx.$$ Now let $H:=\mathrm{aff}(G(g))$, where $G(g)=\{(x,g(x))\in C\times\mathbb R\}$ is the graph of $g$, and observe that $H$ is an affine hyperplane in $\mathbb R^{n+1}$. Let us furthermore observe that since $H\cap(g(0)e_{n+1}+\mathrm{lin}(\{e_1,\dots,e_n\}))\neq\emptyset$, $\mathrm{dim}(H\cap(g(0)e_{n+1}+\mathrm{lin}(\{e_1,\dots,e_n\})))\geq n-1$, and there exists $L\in{\mathcal L}^n_{n-1}$ such that $$g(0)e_{n+1}+(L\times\{0\})\subset H\cap(g(0)e_{n+1}+\mathrm{lin}(\{e_1,\dots,e_n\})).$$ After a suitable rotation, we can assume that $L=\mathrm{lin}(\{e_2,\dots,e_n\})$, that $h(C,e_1)=t_0>0$, and that $$(t_0,(x_0)_2,\dots,(x_0)_n,g(0)+\delta)\in G(g),$$ for some $(t_0,(x_0)_2,\dots,(x_0)_n)\in C$ and some $\delta\geq 0$. Since $C$ is 0-symmetric and $g$ is affine, $(-t_0,-(x_0)_2,\dots,-(x_0)_{n},g(0)-\delta)\in G(g)$ too. Observe that $g(0)-\delta\geq f((-t_0,-(x_0)_2,\dots,-(x_0)_{n}))\geq 0$, i.e., $\delta\leq g(0)$.
Observe also that $g$ is constant on each affine subspace $M_t=\{(t,x_2,\dots,x_n)\in C\}$, $t\in[-t_0,t_0]$. Hence, if $(t,x_2,\dots,x_n)\in C$, let $$g(t,x_2,\dots,x_n)=g(0)+\frac{t}{t_0}\delta.$$ Using Fubini’s formula we have that $$\int_C \phi(g(x))dx=\int_{-t_0}^{t_0}\phi\left(g(0)+\frac{t\delta}{t_0}\right)|M_t|dt.$$
Let us consider now $C':=\sigma_{e_1}(C)$. If we denote by $M_t':=\{(t,x_2,\dots,x_n)\in C'\}$ for every $t\in[-t_0,t_0]$, then $$|M_t|=|M_t'|\quad\text{for every }t\in[-t_0,t_0]$$ and in particular $|C|=|C'|$. Moreover, we also have that $g(t,x_2,\dots,x_n)=g(0)+\frac{t}{t_0}\delta$ for every $(t,x_2,\dots,x_n)\in M_t'$. Therefore $$\int_C\phi(g(x))dx=\int_{-t_0}^{t_0}\phi\left(g(0)+\frac{t\delta}{t_0}\right)|M'_t|dt.$$ We now define the cylinders $$R_t:=(-te_1+M_t')+[-t_0e_1,t_0e_1]\quad\text{for every }t\in[0,t_0].$$ We now prove that $R_{t_0}\subset C'\subset R_0$. For the left inclusion, since $C'$ is 0-symmetric $-2t_0e_1+M'_{t_0}=(-t_0e_1+L)\cap C'\subset C'$ and $M_{t_0}'\subset C'$. Then, the convexity of $C'$ yields $R_{t_0}=\mathrm{conv}((-2t_0e_1+M'_{t_0})\cup M'_{t_0})\subset C'$. For the right inclusion, since $M_t'=(te_1+L)\cap C$ and $(-te_1+L)\cap C=-2te_1+M'_t$, $t\in[0,t_0]$, the convexity of $C'$ yields $$-te_1+M_t'=\frac12M_t'+\frac12(-2te_1+M'_t)\subset L\cap C'=M_0',$$ from which $C'\subset M_0'+[-t_0e_1,t_0e_1]=R_0$, as desired.
Moreover, $(R_t)_t$ is a continuously decreasing family, and thus there exists $t^*\in[0,t_0]$ such that $|R_{t^*}|=|C'|$. Let $R:=R_{t^*}$ and let $M_t'':=\{(t,x_2,\dots,x_n)\in R\}$ for $t\in[-t_0,t_0]$. Let us observe that since $R$ and $C'$ are 0-symmetric and $M_t'$ and $M_t''$ are $(n-1)$-Euclidean balls centered at $te_1$ $$M_t'\subset M_t''\text{ if }|t|\in[t^*,t_0]\text{ and }M_t''\subset M_t'\text{ if }|t|\in[0,t^*].$$ We also observe that $|C'|=|R|$ implies that $|C'\setminus R|=|R\setminus C'|$. Let us furthermore denote by $$M_t^*:=M_t'\cap M_t''\quad\text{and}\quad M_t^{**}:=(M_t'\setminus M_t'')\cup(M_t''\setminus M_t').$$ Then $$\label{eq:the_two_integrals}
\begin{split}
& \int_{-t_0}^{t_0}\phi\left(g(0)+\frac{t\delta}{t_0}\right)|M'_t|dt\\
& = \int_{-t_0}^{t_0}\phi\left(g(0)+\frac{t\delta}{t_0}\right)|M^*_t|dt+
\int_{-t^*}^{t^*}\phi\left(g(0)+\frac{t\delta}{t_0}\right)|M^{**}_t|dt.
\end{split}$$
We start bounding from above the simpler left integral in , whose domain of integration is $C'\cap R$. Let us observe that for every $a,r,\lambda\in{\mathbb R}$, $r\geq 0$, $\gamma\geq 1$, with $a-\gamma r\geq 0$, we have that $\phi(a-r)+\phi(a+r)\leq\phi(a-\gamma r)+\phi(a+\gamma r)$. In order to see this, remember that the definition of convexity of $\phi$ implies $$\frac{\phi(a+r)-\phi(a-r)}{2r},\frac{\phi(a+\gamma r)-\phi(a-\gamma r)}{2\gamma r}\leq\frac{\phi(a+\gamma r)-\phi(a-r)}{(\gamma+1)r},$$ and denote by $m_1,m_3,m_2$ these slopes, respectively. From this, if we define the lines $y_1-(1/2)(\phi(a-r)+\phi(a+r))=m_1(x-a)$, $y_2-\phi(a-r)=m_2(x-r)$ and $y_3-(1/2)(\phi(a-\gamma r)+\phi(a+\gamma r))=m_3(x-a)$, then we have that $y_1\leq y_2$ if $x\geq a-r$ and $y_2\leq y_3$ if $x\leq a+\gamma r$. In particular, $$\label{eq:convexFourFunc}
\frac12(\phi(a-r)+\phi(a+r))=y_1(a)\leq y_2(a)\leq y_3(a)=\frac12(\phi(a-\gamma r)+\phi(a+\gamma r)),$$ as desired.
Since $\phi$ is a convex function, $\delta\rightarrow\phi(g(0)+(t/t_0)\delta)$ is convex too, $\delta\in[0,g(0)]$, using with $a=g(0)$, $r=\delta t/t_0$ and $\gamma=g(0)/\delta$, we see that $$\begin{split}
& \int_{-t_0}^{t_0}\phi\left(g(0)+\frac{t\delta}{t_0}\right)|M^*_t|dt\\
& = \int_{0}^{t_0}\left(\phi\left(g(0)+\frac{t}{t_0}\delta\right)+\phi\left(g(0)-\frac{t}{t_0}\delta\right)\right)|M^*_t|dt\\
& \leq \int_{0}^{t_0}\left(\phi\left(g(0)+\frac{t}{t_0}g(0)\right)+\phi\left(g(0)-\frac{t}{t_0}g(0)\right)\right)|M^*_t|dt\\
& = \int_{-t_0}^{t_0}\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)|M^*_t|dt.
\end{split}$$
Now we focus in bounding from above the right integral in , whose domain is $(C'\setminus R)\cup(R\setminus C')$, partially using ideas from above. Using again that $\phi$ is convex, then $\delta\rightarrow\phi(g(0)+(t/t_0)\delta)$ is convex too, $\delta\in[0,g(0)]$. Hence using yields $$\label{eq:delta}
\begin{split}
& \int_{-t^*}^{t^*}\left(g(0)+\frac{t\delta}{t_0}\right)|M^{**}_t|dt\\
&= \int_0^{t^*}\left(\phi\left(g(0)+\frac{t}{t_0}\delta\right)+\phi\left(g(0)-\frac{t}{t_0}\delta\right)\right)|M_t^{**}|dt\\
& \leq \int_0^{t^*}\left(\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)+\phi\left(g(0)\left(1-\frac{t}{t_0}\right)\right)\right)|M_t^{**}|dt.
\end{split}$$ Once more since $\phi$ is convex, $t\rightarrow\phi(g(0)(1+t/t_0))$ is convex too, together with , we get that $$\label{eq:CandR}
\begin{split}
& \int_0^{t^*}\left(\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)+\phi\left(g(0)\left(1-\frac{t}{t_0}\right)\right)\right)|M_t^{**}|dt\\
& \leq \left(\phi\left(g(0)\left(1+\frac{t^*}{t_0}\right)\right)+\phi\left(g(0)\left(1-\frac{t^*}{t_0}\right)\right)\right)\int_0^{t^*}|M_t^{**}|dt \\
& = \left(\phi\left(g(0)\left(1+\frac{t^*}{t_0}\right)\right)+\phi\left(g(0)\left(1-\frac{t^*}{t_0}\right)\right)\right)\frac{|C'\setminus R|}{2} \\
& = \left(\phi\left(g(0)\left(1+\frac{t^*}{t_0}\right)\right)+\phi\left(g(0)\left(1-\frac{t^*}{t_0}\right)\right)\right)\frac{|R \setminus C'|}{2} \\
& = \left(\phi\left(g(0)\left(1+\frac{t^*}{t_0}\right)\right)+\phi\left(g(0)\left(1-\frac{t^*}{t_0}\right)\right)\right)\int_{t^*}^{t_0}|M_t^{**}|dt\\
& \leq \int_{t^*}^{t_0}\left(\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)+\phi\left(g(0)\left(1-\frac{t}{t_0}\right)\right)\right)|M_t^{**}|dt \\
& = \int_{t^*}^{t_0}\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)|M_t^{**}|dt + \int_{-t_0}^{-t^*}\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)|M_t^{**}|dt.
\end{split}$$
These two upper bounds prove from that $$\begin{split}
\int_C\phi(g(x))dx & = \int_{-t_0}^{t_0}\phi\left(g(0)+\frac{t\delta}{t_0}\right)|M_t|dt \\
& \leq \int_{-t_0}^{t_0}\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)|M^*_t|dt +
\int_{-t_0}^{-t^*}\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)|M^{**}_t|dt \\
& + \int_{t^*}^{t_0}\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)|M^{**}_t|dt = \int_R \phi(g_0(x))dx,
\end{split}$$ where $g_0(x)$ is an affine function with $g_0(0)=g(0)$ and $g_0(-t_0,x_2,\dots,x_n)=0$ for every $(x_2,\dots,x_n)\in{\mathbb R}^{n-1}$. Again by Fubini we now get that $$\begin{split}
\int_R \phi(g_0(x))dx & = \int_{-t_0}^{t_0}\phi\left(g(0)\left(1+\frac{t}{t_0}\right)\right)\frac{|R|}{2t_0}dt
= \frac12\int_{-1}^1\phi(g(0)(1+s))ds|R| \\
& = \frac12\int_{-1}^1\phi(f(0)(1+s))ds|C|,
\end{split}$$ concluding the proof of the inequality.
Let us suppose that $\phi$ is strictly convex. In the case of equality we must have equality in all inequalities above. Let us notice that the strict convexity of $\phi:[0,\infty)\rightarrow[0,\infty)$ implies strict monotonicity. Indeed, if $x_2>x_1>0$, the strict convexity of $\phi$ implies that $$0\leq\frac{\phi(x_1)-0}{x_1-0}<\frac{\phi(x_2)-\phi(x_1)}{x_2-x_1},$$ as desired. Equality in together with the strict monotonicity of $\phi$ implies that $f$ must be an affine function. Equalities in together with the strict convexity of $\phi$ force that $|M_t^{**}|=0$ for every $t\in[0,t_0]$, i.e., $C'$ has to fulfill $$|C'\cap(te_1+L)|=|R\cap(te_1+L)|=c$$ for every $t\in[-t_0,t_0]$ and some constant $c>0$. Since $C'=\sigma_{e_1}C$, we also have that $$|C\cap(te_1+L)|=|C'\cap(te_1+L)|=c.$$ Notice that $(t_0-t)/(2t_0)\in[0,1]$, and thus by the convexity of $C$ $$\left(1-\frac{t_0-t}{2t_0}\right)(C\cap(t_0e_1+L))+\frac{t_0-t}{2t_0}(C\cap(-t_0e_1+L))\subset C\cap(te_1+L).$$ Then, the Brunn-Minkowski inequality implies that $$\begin{split}
c^\frac{1}{n-1} & =|C\cap(te_1+L)|^\frac{1}{n-1}\\
& \geq
\left|\left(1-\frac{t_0-t}{2t_0}\right)C\cap(t_0e_1+L)+\frac{t_0-t}{2t_0}C\cap(-t_0e_1+L)\right|^\frac{1}{n-1} \\
& \geq \left(1-\frac{t_0-t}{2t_0}\right)|C\cap(t_0e_1+L)|^{\frac{1}{n-1}}+\frac{t_0-t}{2t_0}|C\cap(-t_0e_1+L)|^\frac{1}{n-1}\\
& =c^{\frac{1}{n-1}}
\end{split}$$ for every $t\in[-t_0,t_0]$. Thus the equality case of Brunn-Minkowski inequality implies that $C\cap(t,x_2,\dots,x_n)$ is a translation of the same $(n-1)$-dimensional set for every $t\in[-t_0,t_0]$. This is equivalent to the fact that $$C=[-x_0,x_0]+(\{0\}\times C_0),$$ where $x_0=(t_0,(x_0)_2,\dots,(x_0)_n)$ and $C_0\in{\mathcal K}^{n-1}_0$. Finally, equality in forces that $\delta=f(0)$, i.e., that $g(-t_0,x_2,\dots,x_n)=0$ for every $(x_2,\dots,x_n)\in{\mathbb R}^{n-1}$, which concludes the equality case.
Notice that if $C_0\in\mathcal K^n_0$ then $$C=[-e_1,e_1]\times C_0\quad\text{and}\quad f(x)=\langle x,e_1\rangle-1$$ attains equality in Theorem \[thm:Ineq\_Concave\_func\] for every not identically zero, convex, non-decreasing function $\phi:[0,\infty)\rightarrow[0,\infty)$ with $\phi(0)=0$.
Our first corollary follows from applying Theorem \[thm:Ineq\_Concave\_func\] to $\phi(t)=t^\alpha$. In the next section we will use it to give new estimates of the volume of a convex body in terms of the volumes of some of its sections and projections.
\[cor:tAlpha\] Let $C\in\mathcal K^n_0$, let $f:C\rightarrow[0,\infty)$ be concave, and let $\alpha\geq1$. Then $$\frac{1}{|C|}\int_Cf(x)^\alpha dx\leq \frac{2^\alpha}{\alpha+1}f(0)^\alpha.$$ If $\alpha = 1$ equality holds if and only if $f$ is affine. If $\alpha > 1$ equality holds if and only if $C$ is a generalized cylinder, $C=[-x_0,x_0]+(\{0\}\times C_0)$, for some $C_0\in\mathcal K_0^{n-1}$ with $(x_0)_1>0$, and $f$ is affine with $f(-x_0+x)=0$ for every $x\in\mathbb R^n$ with $x_1=0$.
Yet another corollary to Theorem \[thm:Ineq\_Concave\_func\] is when we apply it to $\phi(t)=e^t-1$.
Let $f(x)=e^{u(x)}$, with $u:C\rightarrow{\mathbb R}$ a concave function. Applying Theorem \[thm:Ineq\_Concave\_func\] to the function $u(\cdot)-u_0$ in $C$, where $u_0=\min_{x\in C}u(x)$, and to $\phi(t)=e^t-1$ we obtain that $$\begin{split}
\frac{1}{|C|}\int_Ce^{u(x)}dx & =\frac{e^{u_0}}{|C|}\int_C(e^{u(x)-u_0})dx \\
& =e^{u_0}\left(1+\frac{1}{|C|}\int_C\phi(u(x)-u_0)dx\right) \\
& \leq e^{u_0}\left(1+\frac12\int_{-1}^1\phi((u(0)-u_0)(1+t))dt \right)\\
& =e^{u_0}\left(1+\frac12\int_{-1}^1(e^{(u(0)-u_0)(1+t)}-1)dt\right) \\
& = e^{u_0}\left(1+\frac12\left(\frac{e^{2(u(0)-u_0)}-1}{u(0)-u_0}-2\right)\right) \\
& =\frac{e^{u_0}}{2}\frac{e^{2(u(0)-u_0)}-1}{u(0)-u_0}
\end{split}$$ which shows the result.
Since $\phi(t)=e^t-1$ is strictly convex, the equality case follows immediately from the equality case of Theorem \[thm:Ineq\_Concave\_func\].
Estimating sizes of convex sets by their marginals {#sec:RS_ineqs}
==================================================
We start this section by proving Theorem \[thm:0-symm\_GeneralCase\] as a consequence of Corollary \[cor:tAlpha\].
By Fubini’s formula, we have that $$|K|=\int_{P_HK}|K\cap(x+H^\bot)|dx.$$ By the Brunn’s Concavity Principle (see [@Giann Prop. 1.2.1], see also ) then $$f:H\rightarrow[0,\infty)\quad\text{where}\quad f(x):=|K\cap(x+H^\bot)|^{\frac{1}{n-i}}$$ is a concave function. After a suitable rigid motion, we assume that $H={\mathbb R}^i\times\{0\}^{n-i}$. Corollary \[cor:tAlpha\] then implies that $$\begin{split}
\int_{P_HK}f(x)^{n-i}dx & \leq\frac{2^{n-i}}{n-i+1}|P_HK|f(0)^{n-i}\\
& =\frac{2^{n-i}}{n-i+1}|P_HK||K\cap H^\bot|,
\end{split}$$ concluding the result.
For the equality case, we must have equality in Corollary \[cor:tAlpha\] where $f(x)=|K\cap(x+H^\bot)|^\frac{1}{n-i}$, $C=P_HK$, and $\alpha=n-i$. Hence, first of all, $f(x)$ must be an affine function. We hence can write $$f(x)=f(0)+\langle u,x\rangle,$$ for some $u\in{\mathbb R}^i\times\{0\}^{n-i}$. This means in particular that $$\begin{split}
&|K\cap((1-\lambda)x+\lambda y)|^\frac{1}{n-i}\\
&=f((1-\lambda)x+\lambda y)\\
&=f(0)+\langle u,(1-\lambda)x+\lambda y\rangle\\
&=(1-\lambda)(f(0)+\langle u,x\rangle)+\lambda(f(0)+\langle u,y\rangle)\\
&=(1-\lambda)|K\cap(x+H^\bot)|^\frac{1}{n-i}+\lambda|K\cap(y+H^\bot)|^\frac{1}{n-i}.
\end{split}$$ Hence, using Brunn-Minkowski equality case , we have that $K\cap(x+H^\bot)$ are dilates, of volume $$|K\cap(x+H^\bot)|^\frac{1}{n-i}=\langle u,x\rangle+|K\cap H^\bot|^\frac{1}{n-i}.$$ Since $K$ is convex, there exists an $n\times n$ matrix $B$ of the form $$B=\left(\begin{array}{cc}\text{I}_i & 0\\ B_0 & 0\end{array}\right),$$ where $B_0$ is an $(n-i)\times i$ matrix and $\text{I}_i$ is the $i$-dimensional identity matrix, such that $$K\cap(x+H^\bot)=B(x)+\lambda_x (K\cap H^\bot),$$ with $$\lambda_x=\frac{|K\cap(x+H^\bot)|^\frac{1}{n-i}}{|K\cap H^\bot|^\frac{1}{n-i}}
=\frac{\langle u,x\rangle+|K\cap H^\bot|^\frac{1}{n-i}}{|K\cap H^\bot|^\frac{1}{n-i}}.$$ Second, if $\alpha=n-i\geq 2$, i.e. $i\leq n-2$, then we moreover have that there exist $K_0\in{\mathcal K}^{i-1}$ and $x_0\in{\mathbb R}^i$ such that $P_HK=[-x_0,x_0]+(\{0\}\times K_0)$. Moreover, we must have also that $$f(-x_0,x_{i+1},\dots,x_n)=0\quad\text{for every }(x_{i+1},\dots,x_n)\in{\mathbb R}^{n-i},$$ i.e., that $|K\cap((-x_0,x_{i+1},\dots,x_n)+H^\bot)|=0$. Once more since $f$ is affine, this means that $$u=\frac{|K\cap H^\bot|^\frac{1}{n-i}}{\Vert x_0\Vert^2}x_0,$$ i.e., that $$|K\cap(x+H^\bot)|^\frac{1}{n-i}=\frac{|K\cap H^\bot|^\frac{1}{n-i}}{\Vert x_0\Vert^2}\langle x_0,x\rangle+|K\cap H^\bot|^\frac{1}{n-i},$$ thus concluding the equality case.
For any $C_0\in\mathcal K^{i-1}_0$, $C_1\in\mathcal K^{n-i}$, the set $$C=\left\{(t,x_2,\dots,x_n):t\in[-1,1],(x_2,\dots,x_i)\in C_0,(x_{i+1},\dots,x_n)\in (1+t)C_1\right\}$$ together with the subspace $H=\mathrm{lin}(\{e_1,\dots,e_i\})$ achieves equality in Theorem \[thm:0-symm\_GeneralCase\].
We now properly state along with the characterization of its equality cases. Notice that here we do not require $P_HK$ to be 0-symmetric.
\[thm:center\_of\_Projection\] Let $K\in{\mathcal K}^n$ and $H\in{\mathcal L}^n_{n-1}$. Then $$|K|\leq|P_HK||K\cap(x_{P_HK}+H^\bot)|.$$ If we assume w.l.o.g. that $H={\mathbb R}^{n-1}\times\{0\}$ and that $x_{P_HK}=\{0\}$, there is equality above if and only if there exist $b,u\in{\mathbb R}^{n-1}\times\{0\}$ such that $K\cap(x+H^\bot)=(x_1,\dots,x_{n-1},\langle b,x\rangle)+\lambda_x K\cap H^\bot$, where $$\lambda_x=\frac{\langle u,x\rangle+|K\cap H^\bot|}{|K\cap H^\bot|},$$ for every $x\in P_HK$.
Let us consider the function $$f:P_HK\rightarrow[0,\infty)\quad\text{with}\quad f(x)=|K\cap(x+H^\bot)|$$ which by the Brunn’s Concavity Principle, is a concave function. Hence, using Fubini’s formula and we directly obtain that $$|K|=\int_{P_HK}f(x)dx\leq|P_HK|f(x_{P_HK})=|P_HK||K\cap(x_{P_HK}+H^\bot)|,$$ as desired.
The equality case follows as the equality case of Theorem \[thm:0-symm\_GeneralCase\].
We now give two observations out of Theorem \[thm:center\_of\_Projection\].
\[rmk:x\_Pbetterx\_C\] Let us observe that Theorem \[thm:center\_of\_Projection\] sometimes gives a tighter inequality than . Indeed, if we consider the cone $K\in\mathcal K^n$ with apex at $e_n$ and basis $B^n_2\cap\mathrm{lin}(\{e_1,\dots,e_{n-1}\})$, and consider $H=\mathrm{lin}(\{e_1,\dots,e_{n-2},e_n\})$, it is straightforward to check that $P_HK=\mathrm{conv}((B^n_2\cap\mathrm{lin}(\{e_1,\dots,e_{n-2}\}))\cup\{e_n\})$, that $$x_K=\left(0,\dots,0,\frac{1}{n+1}\right)\quad\text{and}\quad x_{P_HK}=\left(0,\dots,0,\frac{1}{n}\right).$$ Therefore, since $|K\cap(x_K+H^\bot)|=\frac{2n}{n+1}>\frac{2(n-1)}{n}=|K\cap(x_{P_HK}+H^\bot)|$, $$\frac{|K|}{|P_HK||K\cap(x_K+H^\bot)|}<\frac{|K|}{|P_HK||K\cap(x_{P_HK}+H^\bot)|}<1.$$
One can combine two of those inequalities to show that any point in the line segment determined by two good choices of points (as in ), is again a good choice.
If for some $K\in{\mathcal K}^n$ and $H\in{\mathcal L}^n_i$ there exist points $x_0,x_1\in K$ such that $$\label{eq:improved_estimate}
\frac{|K|}{|P_HK|}\leq|K\cap(x_j+H^\bot)|,\quad\text{for }j=0,1,$$ then, for every $\lambda\in[0,1]$, the Brunn-Minkowski inequality gives that $$\begin{split}
|K\cap((1-\lambda)x_0 & +\lambda x_1+H^\bot)| \\
& \geq \left((1-\lambda)|K\cap(x_0+H^\bot)|^\frac{1}{n-i}+\lambda|K\cap(x_1+H^\bot)|^\frac{1}{n-i}\right)^{n-i}\\
& \geq \frac{|K|}{|P_HK|},
\end{split}$$ i.e., all points $(1-\lambda)x_0+\lambda x_1$ also fulfills the inequality , $\lambda\in[0,1]$. In particular, from Theorem \[thm:center\_of\_Projection\] and with $i=n-1$, we obtain that $c_\lambda=(1-\lambda)x_K+\lambda x_{P_HK}$ gives also an inequality of the same type.
*Acknowledgements:* I would like to thank David Alonso-Gutiérrez, Matthieu Fradelizi, Sasha Litvak, Kasia Wyczesany, Rafael Villa, and Vlad Yaskin for useful remarks and discussions, and Francisco Santos for sharing the question that motivated this article.
I would also like to thank the invaluable comments and corrections of the anonymous referee that enormously improved the presentation of this article.
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[^1]: 2010 Mathematics Subject Classification. Primary 52A20; Secondary 52A38, 52A40.\
This research is a result of the activity developed within the framework of the Programme in Support of Excellence Groups of the Región de Murcia, Spain, by Fundación Séneca, Science and Technology Agency of the Región de Murcia. Partially supported by Fundación Séneca project 19901/GERM/15, Spain, and by MICINN Project PGC2018-094215-B-I00 Spain.
[^2]: Bedlewo, Poland 2017 <http://bcc.impan.pl/17Convex/>
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abstract: 'In this letter, we examine the effect of Coulomb interactions in the normal region of a normal-superconducting (N/S) mesoscopic structure, here the change from an attractive to a repulsive coulombic interaction, at the $N/S$ interface, causes a shift in the order parameter phase. We show that this shift has a pronounced effect on Andreev bound states and demonstrate that the effect on Andreev scattering of non-zero order-parameter tails, can be used to probe the sign of the interaction in the normal region.'
address: 'Department of Physics, Lancaster University, Lancaster LA1 4YB, UK.'
author:
- 'P. Dolby, R. Seviour and C.J.Lambert'
title: 'Transport across a normal-superconducting interface: a novel probe of electron-electron interactions in the normal metal.'
---
[2]{}
Recent advances in material technology have enabled the fabrication of normal/superconducting (N/S) mesoscopic hybrid structures with well defined dimensions and interfaces [@afv:r2; @afv:r3; @afv:r4; @afv:r5]. Due to the proximity of the normal material to the superconductor the pairing field $f(x) = \langle \psi_\uparrow(x) \psi_\downarrow(x) \rangle$ , in the normal region, decays to zero on the scale of a coherence length $\xi$ [@rb]. During the past decade this proximity effect has been extensively investigated both experimentally and theoretically (see for example [@kastalsky91; @vol1; @r2; @r1]).
In contrast to the pairing field $f(x)$, the effective electron-electron interaction, $V(x)$, changes abruptly at the S/N interface, from an attractive interaction in the superconductor to either zero, a much diminished attractive interaction or to a repulsive interaction, in the normal (N) material. Consequently the order parameter, $\Delta = V(x)f(x)$, of a s-wave superconductor also changes abruptly at the $S/N$ interface, as shown in figure \[Fig.1\]. To date theoretical research into the transport properties of $N/S$ interfaces have mainly considered the order parameter in the normal region to be zero (figure \[Fig.1\]a) [@rm; @ra].
In this letter we consider the effect of an attractive or repulsive electron-electron interaction ($V(x) \ne 0$), as shown in \[Fig.1\]b and \[Fig.1\]c. The difference between figures \[Fig.1\]b and \[Fig.1\]c is the $\pi$ phase shift in $\Delta(x)$, induced by the cross-over from an attractive to a repulsive interaction at the $N/S$ interface. In what follows we examine how this phase shift affects the transport properties associated with Andreev bound states.
To investigate this regime we adopt a general scattering approach to dc transport, which was initially developed to describe phase-coherent transport in dirty mesoscopic superconductors [@c17]. For simplicity in this letter, we focus solely on the zero-voltage, zero-temperature conductance, for the structure shown in figure \[Fig.2\]. In the linear-response limit, at zero temperature, the conductance of a phase-coherent structure may be calculated from the fundamental current voltage relationship [@c18; @c19], $$I_i=\sum_{j=1}^2a_{ij}( v_j - v),
\label{a4}$$
The above expression relates the current $ I_i$ from a normal reservoir $i$ to the voltage differences $(v_j - v)$, where $v=\mu/e$ and the sum is over the 2 normal leads connected to the scattering region. The $a_{ij}$’s are linear combinations of the normal and Andreev scattering coefficients and in the absence of superconductivity satisfy $\sum_{j=1}^2a_{ij}
=\sum_{i=1}^2a_{ij} = 0 $ in which case the left hand side of equation \[a4\] becomes independent of $v$. In units of $2e^2/h$ [@c18; @c19], $a_{ii}=N_i+R_i^A-R_i^O$ and $a_{ij\ne i}=T_{ij}^A-T_{ij}^O$, where $T_{ij}^A$, $T_{ij}^O$ are Andreev and normal transmission coefficients from probe $j$ to probe $i$, $R_i^A$, $R_i^O$ are Andreev and normal reflection coefficients from probe $i$ and $N_i$ is the number of open scattering channels in lead $i$.
Setting $I_1 = -I_2 = I$ and solving equation \[a4\] for the 2 probe conductance yields [@c20],
$$G=\frac{I}{(V_{1} - V_{2})} =
T^{o}_{21}+T_{12}^{A} + {{2(R_{2}^{A} R_{1}^{A} -T_{21}^{A} T_{12}^{A})}
\over {R_{2}^{A}+R_{1}^{A}+T_{21}^{A}+T_{12}^{A}}},
\label{ce7}$$
where $G$ is the conductance in units of $\frac{2e^{2}}{h}$. As noted in [@c20] the various transmission and reflection coeffcients can be computed by solving the Bogoliubov - de Gennes equation on a tight-binding lattice of sites, where each site is labelled by an index $i$ and possess a particle (hole) degree of freedom $\psi(i)$ $(\phi(i))$. In the presence of local s-wave pairing described by a superconducting order parameter $\Delta_i$, this takes the form,
$$\begin{aligned}
\begin{array}{c c}
E\psi_i
=&\epsilon_i \psi_{i}
-\sum_{\delta} \gamma \left( \psi_{i+\delta} + \psi_{i-\delta} \right)
+ \Delta_{i} \phi_{i}\\
E\phi_i =&-
\epsilon_i \phi_{i}
+\sum_{\delta} \gamma \left( \phi_{i+\delta}+ \phi_{i-\delta}\right)
+\Delta^*_{i}\psi_{i},\\
\end{array}
\label{2}\end{aligned}$$
In what follows, the Hamiltonian of eq.(\[2\]) is used to describe the structure of figure \[Fig.2\], where for all i, the on-site energy $\epsilon_i=\epsilon_0$. In the S-region, the order parameter $\Delta_i$ is set to a constant, $\Delta_i=\Delta_0$, while in the normal region $\Delta_j$ is approximated by,
$$\Delta_j= \pm\frac{\Delta_0}{5} \left ( \tanh (j-L_n) +1 \right ).
\label{app}$$
The nearest neighbour hopping element $\gamma$ merely fixes the energy scale (ie the band-width), whereas $\epsilon_0$ determines the band-filling, and $L_n$ is the length of the normal region. In what follows we choose $\gamma=1$ and $\Delta_0 = 0.1$. By numerically solving for the scattering matrix of equation \[2\], exact results for the dc conductance can be obtained and therefore the effects of a repulsive/attractive coulomb interaction in the normal region can be examined.
For simplicity in this letter we consider the transport properties of the structure shown in figure \[Fig.2\]. Consisting of two normal, semi-infinite, crystalline leads, 20 sites wide, joined by a scattering region. The normal scattering region is 40 sites long and the superconducting region is of length $L_S$. To form Andreev bound states we create quasiparticle confinement in the area in front of the superconductor by introducing a weak point-like contact between the left lead and the scattering region, and ensuring that there is no or very little quasiparticle transmission through the superconductor. For this reason the superconductor length was set to, $L_S= 150$ sites. However care must be taken since the decay length of sub-gap states in the superconductor increases with quasiparticle energy. At energies close to the gap energy the decay length is long enough for transmission through the superconductor, see figure \[Fig.3\]. The weak contact is created by placing a potential barrier, one site wide, between the left lead and the normal scattering region, via a mean potential $U$ added to the on site energy $\epsilon_0$. Figure \[Fig.4\] shows a plot of the transmission coefficient of the barrier as a function of $U$. From this plot we see that a for low quasiparticle transmission through the barrier a high potential is needed. For these reasons the barrier potential was set at $U=20$.
These conditions produce discrete states within the normal regions. Allowing the formation of Andreev bound states, when these energy levels become populated reasonances appear in the conductance of the system, analogous to Breit-Wigner resonances[@rr]. By plotting $G$ as a function of quasiparticle energy (all energies are with respect to the Fermi energy) we are able to investigate transport resonances in the conductance due to the formation of Andreev bound states. Figure \[Fig.5\] shows the conductance as a function of quasiparticle energy for the three proximity tails corresponding to the vanishing, attractive and repulsive electron-electron interactions respectively. These represent the central result of this letter. We see that the introduction of a proximity tail has the effect of shifting the energy at which the Andreev bound states occur. A positive proximity tail causes a shift in the resonance energies to the right of the zero-tail spectrum shown in figure \[Fig.5\], whereas a negative proximity tail shifts the spectrum to the left. These results suggest a novel method for detecting the sign of the electron-electron interaction in the N-metal. Namely by suppressing the order parameter in the normal region, the resonances will either shift to higher energies indicating a repulsive interaction, or lower energies in the case of an attractive interaction. This suppression can be achieved by, for example applying a magnetic field or via a control current in the superconductor.
This work is funded by the EPSRC and the E.U. TMR programme.
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|
---
author:
- 'Erich Poppitz and M. Erfan Shalchian T.'
title: 'String tensions in deformed Yang-Mills theory'
---
Introduction
=============
Systematic ways to study the long-distance behaviour of nonabelian gauge theories, where nonperturbative phenomena set in—confinement, the generation of mass gap, and the breaking of chiral symmetries—are hard to come by. Up to date, there are only a few examples in continuum quantum field theory where theoretically-controlled analytic methods allow one to make progress. Many of those examples, such as Seiberg-Witten theory, require various amounts of supersymmetry and utilize its power.
In the past 10 years, a new direction of research into nonperturbative dynamics, applicable to a wider class of gauge theories, not necessarily supersymmetric, has emerged [@Unsal:2007jx; @01]: the study of gauge theories compactified[^1] on $\R^{1,2} \times \S^1$. The control parameter is the size of the $\S^1$-circle L. When L is taken such that NL$\Lambda \ll 1$, where N is the number of colours of an SU(N) gauge theory and $\Lambda$ its dynamical scale, it allows—as we shall review here for the theory we study—for semiclassical weak-coupling calculability. It has led to new insight into a variety of nonperturbative phenomena and has spawned new areas of research. A comprehensive list of references is, at this point, too long to include here and we recommend the recent review article [@Dunne:2016nmc] instead.
This paper studies confining strings in deformed Yang-Mills theory (dYM). dYM is a deformation of pure Yang-Mills theory, whose nonperturbative dynamics is calculable at small L. It is also believed that the dynamics is continuously connected to the large-L limit of $\R^4$, in particular that the theory exhibits confinement and has a nonzero[^2] mass gap for every size of $\S^1$. The confining mechanism in dYM is a generalization of the three dimensional Polyakov mechanism of confinement [@12], but owing to the locally four-dimensional nature of the theory many of its properties are quite distinct. As we further discuss, many features of dYM on $\R^3 \times \S^1$ can be traced back to the unbroken global center symmetry.
{width="\textwidth"}
[\[fig:0\]]{}
The properties we set out to study here are the $N$-ality dependence of the string tensions and their behaviour in the large-N limit. Renewed motivation to study the large-N limit of dYM arose from a recent intriguing observation [@Cherman:2016jtu]: in the double-scaling limit L $\rightarrow 0$, N $\rightarrow \infty$, with fixed LN$\Lambda$, the four-dimensional theory on $\R^3 \times \S^1_{L \rightarrow 0}$ dynamically generates a latticized dimension whose size grows with N. This phenomenon has superficial similarities to T-duality in string theory and is not usually expected in quantum field theory. Originally, the emergence of a discretized dimension and its properties were studied in a $\R^3 \times \S^1$ compactification and double scaling limit of ${\cal{N}}=1$ super-Yang-Mills (sYM) theory. We show here that, as already observed in sYM, in dYM string tensions also stay finite in the large-N limit while the mass gap vanishes.
Most of this rather long paper is devoted to a review of dYM and to a detailed explanation of the various methods we have developed; a guide to the paper is at the end of this Section.
The expert reader interested in the physics and not in the technical details should proceed to our “Summary of results” Section \[summary\], and to the more extended discussion in Section \[sec:5\].
Summary of results {#summary}
------------------
Here we summarize our main results, concerning both the confining string properties and the technical tools developed for their study:
1. [*k-string tension ratios:*]{} In the regime of parameters studied in this work, in particular NL$\Lambda \ll 1$, the asymptotic string tensions in dYM depend only on the $N$-ality of the representation. We argue in Section \[sec:5\] that the lowest tension stable strings between sources of $N$-ality k are sourced by quarks with charges in the highest weight of the k-index antisymmetric representation, see (\[eq:54\]).[^3] Their tensions are hence referred to as the “k-string tensions.”
Denoting by T$_{\text k}$ the k-string tension, on Fig. \[fig:0\] we show the ratio T$_{\text k}$/T$_1$ for SU(10), the largest group we studied numerically. The string tension ratio in dYM is compared to other known and much studied scaling laws, such as the Sine law and the Casimir law. It is clear from the figure that k-string tension ratios in dYM are different and do, instead, come closest to a less-known scaling, found long ago in the MIT Bag Model of the Yang-Mills vacuum: the “Square root of Casimir” scaling [@13]. In Section \[bagmodelsection\], we argue that the relation between the two is $$\label{dymscaling}
\left(T_k \over T_1\right)_{\text{dYM}} \le \;\sqrt{ k (N-k) \over N}~,$$ where the r.h.s. is the square root of the ratio of quadratic Casimirs of the k-index antisymmetric representation and the fundamental representation. The reason behind the similarity is that the model assumptions of the MIT Bag, that inside the bag the QCD chromoelectric fields can be treated classically and that the vacuum abhors chromoelectric flux, are realized almost verbatim—albeit for the Cartan components only—by the calculable confinement in dYM.
2. [*Large-N limit and $1\over N$ corrections to string tensions:*]{} As already mentioned, string tensions stay finite at large N and fixed LN$\Lambda \ll 1$, as we show using various tools in Section \[sec:largeN\]. Further, as can be inferred qualitatively from Figure \[fig:0\], and quantitatively from the analysis of Section \[sec:largeN\], k-strings in dYM are not free at large N. We show that $$\begin{aligned}
\label{dymscaling2}
{\text{T}_2 \over \text{T}_1} &=& 1.347 \pm 0.001 + (-2.7 \pm 0.2) ({1 \over N})^2 + ...,\nonumber \\
{\text{T}_3 \over \text{T}_1} &=& 1.570 \pm 0.001 + (-7.5 \pm 0.2) ({1 \over N})^2 + ... ,\end{aligned}$$ instead of approaching the free-string values T$_\text{k}$ = kT$_1$.
The large-N limit leading to the above behaviour is taken [*after*]{} the large-RT limit (RT is the Wilson loop area). As the discussion there shows, assuming large-N factorization does not always imply that k-strings are free and the way the large-RT and large-N limits are taken has to be treated with care, as we discuss in detail in Section \[sec:largeN\].[^4]
3. [*Comparing abelian confinements:*]{} We compare the properties of confining strings in dYM and in Seiberg-Witten theory [@Seiberg:1994rs], another four dimensional theory with calculable abelian confinement. We argue that the unbroken $\Z_N$ center symmetry in dYM has dramatic implications for the meson and baryon spectra. In particular there is a “baryon vertex” in dYM, leading to “Y”-type baryons, while only linear baryons exist in Seiberg-Witten theory [@15]. Thus, owing to the unbroken center symmetry, in many ways confinement in dYM is closer to the one in “real world” YM theory.[^5] For a discussion of these issues, see Section \[sec:compare\] and Figs. \[fig:dymstring\] and \[fig:SWstrings\].
4. [*“Perturbative evaluation” of string tensions:*]{} A technical tool to calculate string tensions analytically is developed in Section \[sec:4\]. We call it “perturbative,” as it utilizes a resummed all-order expansion and, at every step, requires the use of only Gaussian integrals. This method serves as a check on the computationally very intensive numerical methods that were employed in the numerical study. It also allows the large-N limit to be taken analytically, subject to the limitations discussed above, and permits us to discuss the subtleties regarding the order of limits that lead to (\[dymscaling2\]). This method can be generalized to perform a path integral expansion about a saddle point boundary value problem (e.g. a transition amplitude in quantum mechanics) using perturbation theory (Gaussian integrals) only. Further applications of these tools is the subject of work in progress [@63].
Open issues for future studies
------------------------------
As already stressed, one of our motivations is to study the peculiar large-N limit of dYM confinement, similar to the large-N limit of sYM from ref. [@Cherman:2016jtu], which shows many intriguing features that (at least superficially) resemble stringy properties. We have not yet fully addressed this limit in dYM, as there is the upper bound on N discussed above. We believe that this restriction on N is technical and more work is required to remove it.
Our study here also only briefly touched on the spatial structure of confining k-strings, noting that, upon increasing N, they become more “fuzzy” due to the decreasing mass of many of the dual photons, but retain a finite string tension due to the (also large) number of dual photons of finite mass. This spatial structure may have to do with their interacting nature and would be interesting to investigate further.
Further, in this paper, we ignored the $\theta$-angle dependence of the $k$-strings. The topological angle dependence in Yang-Mills theory has received renewed recent attention, see e.g. [@Thomas:2011ee; @Unsal:2012zj; @Anber:2013sga; @Bhoonah:2014gpa; @Gaiotto:2017yup; @Tanizaki:2017bam; @Kikuchi:2017pcp; @Gaiotto:2017tne; @Anber:2017rch]. As seen in some of the aforementioned work, the corresponding physics in dYM is also very rich and worth of future studies.
There are also the many intriguing observations of [@Aitken:2017ayq] on the nature of the dual photon, glueball, etc., bound state spectra in dYM (at arbitrary N) that await better understanding. Finally, there is the question about the (still conjectural) continuity of dYM from the calculable small $\Lambda$NL regime to the regime of large $\Lambda$NL. To this end, it would be desirable to study this theory on the lattice; for some lattice studies of related theories, see [@Cossu:2009sq; @Vairinhos:2011gv; @Bergner:2014dua; @Bergner:2015cqa].
Organization of this paper
--------------------------
Section \[sec:2\] is devoted to a review of dYM theory.[^6] In Section \[sec:2.1\] we review how dYM theory on $\R^3\times \S^1$ avoids a deconfinement transition at small L. The perturbative spectrum of dYM is discussed in Section \[sec:2.2.1\] and the nonperturbative minimal action monopole-instanton solutions—in Section \[sec:2.2.2\]. The action of a dilute gas of monopoles is discussed in at length in Section \[2.3\], with emphasis on details that often not emphasized in the literature. The derivation of the string tension action, used to calculate the semiclassical string tensions is given in Section \[sec:2.4.1\].
ection \[numericsection\] is devoted to a numerical study of the k-string tensions in dYM. The action and its discretization are studied in Sections \[sec:3.1\] and \[sec:3.3\]. The minimization procedure, the numerical methods, and the error analysis are described in Section \[sec:minimize\]. The numerical results for the k-string tensions for gauge groups up to SU(10) are summarized in Table \[table:1\], see Section \[sec:3.4\].
ection \[sec:4\] presents an analytic perturbative procedure to calculate the string tensions. We begin by explaining the main ideas with fewer technical details. In Sections \[su2analyticsection\] and \[suNanalyticsection\] we give the detailed calculations for SU(2) and general SU(N) gauge groups, respectively. The results are tabulated in Appendix \[sec:C\], demonstrating the precision of this procedure, which also serves as a check on the numerical results of Section \[numericsection\].
ection \[sec:5\] contains the discussion of our results from various points of view, including relations to other models of confinement and the behaviour of k-strings in the large-N limit:
In Section \[sec:5.1.1\] we argue that the lowest, among all weights of any given representation, semiclassical asymptotic string tensions in dYM depends only on $N$-ality $k$ of the representation and is the one obtained for quark sources with charges in the highest weight (and its $\Z_N$ orbit) of the $k$-index antisymmetric representation.
In Section \[sec:compare\] we compare confinement in dYM with confinement in Seiberg-Witten theory and point out that the unbroken $\Z_N$ center symmetry in dYM is responsible for the major differences, which make abelian confinement in dYM closer—in many aspects—to confinement in the nonabelian regime.
In Section \[bagmodelsection\] we point out the similarity, already discussed around eq. (\[dymscaling\]), of the k-string tension ratios in dYM to the ones in the MIT Bag Model and discuss the physical reasons.
In Section \[sec:5.1.4\] we compare the k-string tension scaling laws to other scaling laws considered in various theoretical models.
In Section \[sec:largeN\], we discuss the abelian large-N limit. The leading large-N terms in the k-string tension ratios, eq. (\[dymscaling2\]) above, are derived in \[sec:5.2.1\]. The fact that large-N factorization does not always imply that k-strings become free at large-N is discussed in Section \[sec:5.2.2\]. The analytic methods of Section \[sec:4\] prove indispensable in being able to track the importance of the way the large-N and large area limits are taken.
Review of dYM theory {#sec:2}
====================
In this Section we will have a brief review of dYM theory. The emphasis is on topics usually not covered in detail the literature and on topics that will be needed for the rest of the paper.
Confinement of charges in deformed Yang-Mills theory for all $\S^1$-circle sizes {#sec:2.1}
---------------------------------------------------------------------------------
Consider four-dimensional Yang-Mills theory in the Euclidean formulation with one of its dimensions compactified on the circle: $$\label{eq:2.1}
S = \int_{{\rm \R}^3 \times \S^1}d^4 x{1 \over 2g^2} \text{tr} F^2_{\mu\nu}(x)~.$$ We set the $\theta$-angle to zero in this paper, leaving the study of the $\theta$-dependence of strings’ properties for the future. Here, $T^a$ ($a=1, ..., N^2 -1$) refer to the Hermitean generators of the group $SU(N)$, $F_{\mu\nu} = F^a_{\mu\nu}T^a$, $\text{tr} (T^aT^b) = {1\over 2} \delta^{ab}$. The compactification circle $\S^1$ in pure Yang-Mills theory can either be considered as a spatial dimension of size $L$, or as a temporal one with $L = {1 / T}$ being the inverse temperature $T$. It is known, see e.g. [@04], that above a critical temperature $T_c = {1\over L_c}$ Yang-Mills theory loses confinement (i.e. the static potential between two heavy probe quarks no longer shows a linearly rising behaviour as a function of distance between the quarks). The transition from a confining to a non-confining phase, in theories with gauge groups that have a nontrivial center, is accompanied by the breaking of the center-symmetry.[^7] The critical size $L_c$ is approximately of order $\Lambda^{-1}$, with $\Lambda$ the $\overline{MS}$ strong scale of the theory. Different studies give an estimate of $200$ MeV $<T_c <300$ MeV for $SU(2)$ Yang-Mills theory in four dimensions. In what follows we shall deform Yang-Mills theory in a way that preserves confinement of charges for any circle size $L$. Due to asymptotic freedom the coupling constant is small at the compactification scale ${1 \over L}$ for small circle sizes $(L \ll \Lambda^{-1}$; as we argue below, the precise condition for $SU(N)$ gauge theories turns out to be $\Lambda L N \ll 1$). This deformation would enable us to have a model of confinement that we can study analytically in the limit of a small circle size $L$.
The expectation value of the trace of the Polyakov loop, $P(\text{x}) = \text{tr} \;{\cal{P}}\text{exp}(-i \oint_{S^1} dx_4 \text{A}_4 (\text{x},x_4))$ (where ${\cal{P}}$ denotes path ordering) serves as an order parameter for confinement [@04]:
\[eq:2.2\] &P()= 0 ,\
&P()0 . &&
On the other hand, the Polyakov loop is not invariant under a center-symmetry transformation and picks up a center element $z$, i.e. $U_zP(\text{x})U^{\dagger}_{z} = z P(\text{x})$, where we used the notation of Footnote \[center\]. Therefore for a center-symmetric vacuum $| 0 \rangle$ we have:
\[eq:2.3\] 0 | P()|0= 0|U\^\_zU\_z P()U\^\_zU\_z|0 = z0 | P()|0 P() = 0 , z 1 ,&&
indicating that a center-symmetric phase is a confined phase.
In order to show that Yang-Mills theory deconfines at high temperatures we need to show that the expectation value of the Polyakov loop at high temperatures is nonzero. The Polyakov loop is gauge invariant and the eigenvalues of the holonomy $\Omega(\text{x}) = \text{P} \text{exp}(- i \oint_{S^1} dx_4 \text{A}_4 (\text{x},x_4))$ constitute its gauge invariant content ($P = \text{tr} \Omega$). At tree level the eigenvalues of the holonomy can take any value, as there is no potential for $\Omega$ in the classical Yang-Mills Lagrangian (\[eq:2.1\]). To find an effective potential for the eigenvalues of the holonomy at one-loop, we expand around a constant diagonal $A_4$ field and evaluate the one loop contribution to the effective potential by integrating out the quadratic terms of gauge and ghost fields [@01; @06], to find:
\[eq:2.4\] V\_1\[\] = -[2\^2 L\^3]{}|\^n|\^2, = (-iLA\_4) .
From it can be seen that $V_1[\Omega]$ is minimized when $\Omega$ is an element of the centre of the gauge group, i.e. $\omega_N^k \;I$, with $I$ the unit matrix.[^8] This would imply $\langle P(\text{x})\rangle = \langle \text{tr}(\Omega)\rangle = N \omega_N^k \ne 0 $, indicating a deconfined center-symmetry broken phase of Yang-Mills theory at high temperatures, or small circle sizes $L$ (owing to asymptotic freedom, the small-$L$/high-$T$ regime is the one where the calculation leading to (\[eq:2.4\]) can be trusted).
In order to change this picture and have a model of confinement at arbitrary small circle sizes $L$ we can add a deformation potential term to Yang-Mills theory [@01; @Myers:2007vc]: $$\label{eq:2.5}
S = \int_{{\rm I\!R}^3 \times S^1}{1 \over 2g^2} \text{tr} F^2_{\mu\nu}(x) + \Delta S, \ \Delta S \equiv \int_{{\rm I\!R}^3}{1 \over L^3} P[\Omega(\text{x})], \ P[\Omega] \equiv {2 \over \pi^2} \overset{[N/2]}{\underset{n=1}{\sum}}{b_n \over n^4}|\textrm{tr}(\Omega^n)|^2~,$$ with $b_n$—sufficiently large and positive coefficients. The effect of the $\Delta S$ term is to dominate the gluonic and ghost potential in a way that the minimum of $V_1[\Omega] + {1 \over L^3}P[\Omega]$ occurs when $\text{tr}(\Omega^n) = 0$ for $n$ mod $N$ $\ne$ 0. This would imply $\langle P(\text{x})\rangle = \langle \text{tr}(\Omega)\rangle = 0$ and hence a confinement phase for deformed Yang-Mills theory at arbitrarily small circle sizes.
The $\Delta S$ deformation term in would make the theory non-renormalizable. To have a well-behaved theory at high energies, the deformation can be considered as an effective potential term generated by some renormalizable dynamics, notably $n_f$ flavors of massive adjoint Dirac fermions with periodic boundary conditions along the $\S^1$. Following [@05] for conventions on Euclidean formulation of Dirac fermions we have: $$\label{eq:2.6}
S_{dYM} = \int_{{\rm I\!R}^3 \times S^1}\{ {1 \over 2g^2} \text{tr}F^2_{\mu\nu}(x) -i \underset{i=1}{\overset{n_f}{\sum}}\bar{\psi}_i(\slashed{D}+m)\psi_i \}$$ The effective potential for the holonomy generated by the $n_f$ massive adjoint Dirac fermions is given by [@07; @08]: $$\label{eq:2.7}
V_2[\Omega] = +{2\over \pi^2 L^3}\underset{n=1}{\overset{\infty}{\sum}}n_f(nLm)^2K_2(nLm){|\textrm{tr}\Omega^n|^2 \over n^4}~,$$ where $K_2$ is the modified Bessel function of the second kind. It has to be noted that in the deformed theory the compactified dimension $S^1$ can only be a spatial dimension since the heavy fermions satisfy periodic boundary conditions along this direction.
There are two free parameters $n_f$ and $NLm$ in the effective potential .[^9] The beta function of $SU(N)$ Yang-Mills theory with $n_f$ flavours of Dirac fermions in the adjoint representation of the gauge group is, at the one loop level, $\beta(g) = - {g^3 \over (4 \pi)^2} ({11 \over 3}N - {4 \over 3}n_fN)$, hence to assure asymptotic freedom $n_f = 1 \ \text{or} \ 2$. If we allow for massive Majorana flavors, $n_f = 5/2$ is the maximum value. On the other hand, if we want the effective potential $V_2$ to dominate the gluonic potential $V_1$, $NLm$ should be of order 1 ($NLm \sim 1$; for larger values of $m$, the fermions decouple and the theory loses confinement at small $L$). To gain some intuition on how the coefficients of the potential $V_2$ behave let $c_n \equiv n_f ({n\over N}LmN)^2 K_2({n\over N}LmN)$. Choosing $n_f = 2$ ($n_f = 1$) and $NLm = 4$ ($NLm = 3$) gives $c_n \approx 4$ ($c_n \approx 2$) for $n/N \approx 0$, $c_n \approx 2$ ($c_n \approx 1.3$) for $n/N \approx 0.5$, $c_n \approx 0.56$ ($c_n \approx 0.55$) for $n/N \approx 1$ and $c_n$ approaching zero exponentially for $n/N > 1$. Minimizing[^10] the combined potential $V_1[\Omega] + V_2[\Omega]$ gives $\langle\Omega\rangle = \text{diag} (\omega_N^{N-1}, ..., \omega_N, 1)$ for odd $N$, and $\langle\Omega\rangle = e^{i {\pi \over N}} \text{diag} (\omega_N^{N-1}, ..., \omega_N, 1)$ for even $N$, which gives $\text{tr} \langle\Omega\rangle^n = 0$ for $n$ mod $N \neq 0$, hence, a confined phase for deformed Yang-Mills theory.[^11]
Perturbative and non-perturbative content of dYM {#sec:2.2}
------------------------------------------------
### Perturbative content {#sec:2.2.1}
The eigenvalues of the holonomy $\Omega (\text{x}) = {\cal{P}}\text{exp}(i\oint d\text{x}_4 A_4(\text{x},\text{x}_4))$, are the only gauge invariant content of the gauge field component in the compact direction $A_4(x)$ and are invariant under any periodic gauge transformation. Working in a gauge such that the $A_4(x)$ field assumes these eigenvalues ($A_4(x) = -i {\text{ln}(\Lambda(\text{x})) / L}$, with $\Lambda(\text{x})$—the diagonal matrix of eigenvalues of $\Omega(\text{x})$) and expanding around a center-symmetric $vev$ $$\begin{aligned}
\label{a4vev}
A^{vev}_4 &=& {1 \over N L} \text{diag}(2\pi (N-1), ..., 2\pi, 0) \; \; \text{for odd}\; N, \nonumber \\
A^{vev}_4 &=& {1 \over N L} \text{diag}(2\pi (N-1) + \pi, ..., 3\pi, \pi)\;\;\text{for even} \; N,
\end{aligned}$$ the perturbative particle content of dYM theory with action can be worked out by writing the second order Lagrangian of the modes expanded around the above center-symmetric $vev$. Clearly, the $vev$ of the “Higgs field” $A_4$ breaks the gauge symmetry $SU(N) \rightarrow U(1)^{N-1}$. The gauge fields associated with the non-compact direction can be written as: $$\label{eq:2.8}
A_i(\text{x},\text{x}_4) = {\sqrt{2} }\; A_{i,0}(\text{x}) + {}\overset{+\infty}{\underset{k = - \infty} {\sum}^\prime} A_{i,k}(\text{x})\;\text{exp}(ik{2\pi \over L}\text{x}_4) \ \ ({\sum}^\prime \text{ over}\; k \neq 0) ~,$$ with $A_{i,k}(\text{x}) = A_{i,k1}(\text{x}) + iA_{i,k2}(\text{x}) = A^{\dagger}_{i,-k}(\text{x})$ in order to ensure reality of the $A_i$ fields. It turns out that the gauge-boson field content is a non-trivial one. We work out the quadratic Lagrangian in Appendix \[app:wboson\], by substituting (\[eq:2.8\]) in the action and expanding around (\[a4vev\]).
We begin with a discussion of the abelian spectrum. The diagonal components of the gauge fields $A_i$ commute with the $vev$ $A_4^{vev}$. Hence, their zeroth Fourier modes along $\S^1$ correspond to massless 3d photons and their higher Fourier modes gain mass of ${2 \pi m \over L}$ where $m=1,2, ...\infty$ is the non-zero momentum in the compact direction. At tree level, the Lagrangian for the $N-1$ photons is simply the reduction of (\[eq:2.1\]) to the Cartan subalgebra of $SU(N)$. The leading-order coupling of the 3d $U(1)^{N-1}$ gauge theory is given by $g_3^2 = g^2/L$, where $g$ is the four-dimensional gauge coupling at the scale of the lightest $W$-boson mass, $m_W = {2\pi \over NL}$, see below.[^12]
The physical components of the “Higgs” field $A_4(x)$ are diagonal and $x_4$ independent. They are massless at the classical level but gain mass of order at least $g \over \sqrt{N} L$ via the one loop effective potential $V_1 + V_2$ generated by quantum corrections.[^13]
The massive adjoint Dirac fermions are also expanded in their Kaluza-Klein modes. Taking into account the effects of $A^{vev}_4$, it can be seen that there are massive Dirac fermions with masses $m + {2 \pi k \over L} + {2 \pi p \over LN}$ for $k = 0, 1, ...$ and $p = 0, 1, ..., N-1$. As we are interested in physics below the scale of the lightest fermion, we shall not present details of the massive fermion spectrum.
Finally, the relation from Appendix \[app:wboson\] shows that there are $W$-bosons with masses $|{2 \pi m \over L} - {2\pi |l-k| \over NL}|$ and $|{2 \pi m \over L} + {2\pi |l-k| \over NL}|$ respectively for $m =0,1,2,...$ and $1\leq l < k \leq N$. The mass of the lightest $W$-boson is $m_W = {2 \pi \over NL}$. Clearly, below that scale, there are no fields charged under the unbroken $U(1)^{N-1}$ gauge group. Thus, the gauge coupling of the $N-1$ photons is frozen at the scale $O(m_W)$. The condition that the theory be weakly coupled, therefore, is that $m_W \gg \Lambda$, or $N L \Lambda \ll 1$. This is the semiclassically calculable regime that we study in this paper.
In summary, the perturbative particle content of deformed Yang-Mills theory expanded around the center-symmetric $vev$ consists of $N-1$ photons (the diagonal Cartan components of the gauge fields, whose zero Fourier components along the $\S^1$ are massless), $N^2 - N$ massive gauge fields, charged under the $U(1)^{N-1}$ unbroken gauge symmetry, whose spectrum is given in (\[eq:2.13\]), $N-1$ massive eigenvalues of the holonomy, neutral under $U(1)^{N-1}$ and charged and uncharged massive Dirac fermions.
### Non-perturbative content: minimal action instanton solutions {#sec:2.2.2}
Finite action Euclidean configurations of pure Yang-Mills theory on $\R^3 \times \S^1$ were studied in [@06]. It was shown that they are classified by their magnetic charge $q_\alpha$, Pontryagin index $p$, and asymptotics of the $\S^1$ holonomy $\Omega$ at infinity, which are related by the following formula: $$\label{eq:2.14}
Q = p + {\text{ln} \mu_{\alpha} \over 2 \pi i}\; q_{\alpha}~.$$ Here, $Q = {1 \over 32 \pi^2}\int_{{ \R}^3 \times \S^1} d^4 x \;F_{\mu\nu}^a \widetilde{F}_{\mu\nu}^a$ is the topological charge with $\widetilde{F}_{\mu \nu} = {1\over2} \epsilon_{\mu \nu \alpha \beta} F_{\alpha \beta}$. In this Section, we use $\mu_{\alpha}$ with $\alpha = 0, ..., \kappa \leq N-1$ to label the distinct eigenvalues of the holonomy $\Omega (\text{x})$ at spatial infinity. Notice that, for finite action configurations, the eigenvalues of $\Omega$ are independent of the direction that we approach spatial infinity, and that, for the center-symmetric holonomy, $\kappa = N-1$ as all eigenvalues are distinct, given by the $N$ values of $e^{i L A_4^{vev}}$ with $A_4^{vev}$ of (\[a4vev\]): $\ln \mu_\alpha = {2 \pi i (N-1-\alpha) \over N}$ for odd $N$. The integer magnetic charges are denoted by $q_{\alpha}$, satisfy $\sum^{\kappa}_{\alpha = 0}q_{\alpha} = 0$, and will be explicitly defined further below, see paragraph after eq. (\[eq:2.21\]). The Pontryagin index $p$ is the winding number for mappings of $\S^3$ onto the full group $SU(N)$.[^14]
It is expected that for any value of the quantities $p$, $q_{\alpha}$, and $\mu_{\alpha}$ there is a separate sector of finite action configurations, with the self-dual ($F_{\mu\nu} = \widetilde{F}_{\mu\nu}$) or anti-self dual ($F_{\mu\nu} = - \widetilde{F}_{\mu\nu}$) solution corresponding to the minimum action configuration in that sector. For self-dual or anti-self-dual solutions the topological charge is proportional to the value of the action. Therefore finding configurations with the minimal non-zero topological charge is equivalent to finding the minimal non-zero action configurations. Based on the values of $\mu_{\alpha} = \text{exp}(i {2 \pi (N-1 -\alpha) \over N})$ for $\alpha = 0, ..., N-1$ in a center-symmetric vacuum, and the fact that $p$ and $q_{\alpha}$ are integers with $\sum^{N-1}_{\alpha = 0}q_{\alpha} = 0$, it can be clearly seen that the minimal non-zero topological charge is $|Q| = {1 \over N}$. Configurations of minimal $Q = {1 \over N}$ would then correspond to the following values of $q_{\alpha}$ and $p$:
\[eq:2.15\] &p = 0, q\^i\_m = (0,..,0,,-1,0,...,0), i = 1, ..., N-1,\
\[eq:2.16\] &p = 1, q\^N\_m = (-1,0,... ,0, 1),
where the $N$ components of the vector $q_m^i$ are the magnetic charges $q_\alpha$ corresponding to the $i$-th minimal action configuration. Minimal action configurations with Pontryagin indices and magnetic charges given in will be referred to as the $N-1$ $SU(N)$ BPS solutions and the minimal action configurations with Pontryagin number and magnetic charges from —as the Kaluza-Klein (KK) solution.[^15] The $\overline{\text{BPS}}$ (anti-BPS) and $\overline{\text{KK}}$ (anti-KK) configurations have the opposite sign for the magnetic charges and Pontryagin index and thus a negative topological charge $Q = -{1 \over N}$. In total this classification shows that there exist $2N$ minimum finite action non-trivial configurations. We will refer to these finite action configurations as the “non-perturbative content” of deformed Yang-Mills theory—because, as we shall see, it is these Euclidean configurations that lead to confinement of charges, to leading order in $NL\Lambda \ll 1$.
In order to construct such configurations we start from the $SU(2)$ BPS and Kaluza-Klein monopoles and embed them in $SU(N)$. For the BPS solution, this can be done in $N-1$ different ways leading to the $N-1$ different configurations in and for the Kaluza-Klein monopole this can be done in only one way. The $SU(2)$ BPS monopole solution is given by [@23; @Diakonov:2009jq]: $$\label{eq:2.17}
\begin{split}
& A^a_4 = \mp n_a \nu \mathcal{P}(\nu r)\ \ , \ \ \ \ \ \ \ \ \ \ \ \ \ \mathcal{P}(y) = \text{coth}(y) - {1 \over y} \\
& A^a_i = \epsilon_{aij} n_j {1-\mathcal{A}(\nu r) \over r}\ \ , \ \ \ \ \ \ \ \mathcal{A}(y) = {y \over \text{sinh}(y)}~,
\end{split}$$ where $n_a$ for $a = 1, 2, 3$ refer to the components of a unit vector in ${\rm I\!R}^3$ and $\nu$ is related to the eigenvalues of the holonomy at infinity. In what follows, as in the above relations, the upper sign always corresponds to the self-dual BPS solution and the lower one to the anti-self-dual $\overline{\text{BPS}}$ solution. The magnetic field strength $B_i ={1 \over 2} \epsilon_{ijk}F_{jk}$ of this solution is: $$\label{eq:2.18}
B^a_i = (\delta_{i}^a - n^an_i) \nu^2 F_1(\nu r) + n^a n_i \nu^2 F_2(\nu r)$$ The functions $F_1$ and $F_2$ are given below in . In order to embed these solutions in $SU(N)$ and in a center-symmetric vacuum $A^{vev}_4$, we first make a gauge transformation that will make the $A_4$ component diagonal in colour space along $\tau^3 /2$. For this we solve for the equation $S_- \tau^a n_a S_-^{ \dagger} = -\tau^3$ for the BPS and $S_{+} \tau^a n_a S_+^{ \dagger} = \tau^3$ for the $\overline{\text{BPS}}$. This gives: $$\label{eq:2.19}
\begin{split}
& S_+(\theta, \phi) = \text{cos} { \theta \over 2} + i \tau^2 \text{cos} \ \phi \ \text{sin} {\theta \over 2} -i \tau^1 \text{sin} \ \phi \ \text{sin} { \theta \over 2} = e^{-i\phi \tau^3 / 2} e^{i\theta \tau^2 / 2} e^{i\phi \tau^3 / 2} \\
& S_-(\theta, \phi) = - \text{sin}{ \theta \over 2} \; \text{cos} \ \phi - i\; \text{sin}{ \theta \over 2} \;\text{sin} \ \phi \ \tau^3 + i\; \text{cos}{ \theta \over 2} \tau^2 = e^{i\phi \tau^3 / 2} e^{i (\theta + \pi) \tau^2 /2 } e^{i\phi \tau^3 / 2}~.
\end{split}$$ After performing the gauge transformation $A_{\mu} \longrightarrow A^S_{\mu} = S A_{\mu} S^{\dagger} + iS \partial_{\mu} S^{\dagger}$ for $S = S_- \ \text{or} \ S_+$ we get: $$\label{eq:2.20}
\begin{aligned}
& A^S_4 = \nu \mathcal{P}(\nu r) {\tau^3 \over 2} \\
& A^S_r = 0 \\
& A^S_{\theta} = {\mathcal{A}(\nu r) \over 2r} (\pm \tau^1 \text{sin} \phi + \tau^2 \text{cos} \phi) \\
& A^S_{\phi} = {\mathcal{A}(\nu r) \over 2r} (\pm \tau^1 \text{cos} \phi - \tau^2 \text{sin} \phi) \pm \tau^3 {1 \over 2r} \text{tan} {\theta \over 2} ~,
\end{aligned}$$ where $A^S_r = \hat{r}_i A^S_i$, $A^S_{\theta} = \hat{\theta}_i A^S_i$, $A^S_{\phi} = \hat{\phi}_i A^S_i$ are the components of $A^S_i$ along the unit vectors in spherical coordinates.[^16] It has to be noted that the $A^S_{\phi}$ solution shows a singular string along $\theta = \pi$. This is a gauge artifact and does not cause any problems for to satisfy the self-duality or anti-self-duality condition. In other words, the magnetic fields evaluated from are everywhere smooth functions of the spherical coordinates, as can be seen by finding the magnetic field strength in the stringy gauge: $$\label{eq:2.21}
\begin{aligned}
& B^S_r = \mp {\nu}^2 F_2(\nu r) \tau^3 /2 , \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ F_2(y) = {1 \over \text{sinh}^2 y } - {1 \over y^2} \\
& B^S_{\theta} = \nu^2 F_1(\nu r)/2 (\mp \tau^1 \text{cos} \phi + \tau^2 \text{sin} \phi) , \ \ \ \ \ \ \ \ F_1(y) = {1 \over \text{sinh} \ y }( {1 \over y} - \text{coth} \ y) \\
& B^S_{\phi} = \nu^2 F_1(\nu r)/2 (\pm \tau^1 \text{sin} \phi + \tau^2 \text{cos} \phi) ~.
\end{aligned}$$ Using the diagonal components of the field $B^S$ at infinity, diag$(B^S)$, the magnetic charge vector for the $SU(2)$ BPS monopole solution, in the normalization of (\[eq:2.15\]) can now be defined[^17].by a surface integral at infinity, $(q_1, q_2) = \oint\limits_{S^2_\infty} d^2 \sigma \;\text{diag} (B_r^S)/(2 \pi) = (1,-1)$.[^18]
There are $N-1$ $SU(2)$ Lie subalgebras, corresponding to the elements $a_{ii}$, $a_{ii+1}$, $a_{i+1i}$, $a_{i+1i+1}$ for $i = 1, ..., N-1$, along the diagonal of an $SU(N)$ Lie algebra matrix and we can embed an $SU(2)$ BPS monopole in each of them. Only these embedded BPS monopoles will have the lowest topological charge $|Q| = {1 \over N}$. We will illustrate the embedding for the top left $SU(2)$ Lie subalgebra. We simply place the $SU(2)$ solution , with $\nu = {2\pi \over NL}$, in the top left $SU(2)$ Lie subalgebra of an $SU(N)$ Lie algebra matrix with all other elements being zero. Next, in order to make the value of $A^{S,SU(N)}_4$ ($\equiv$ $SU(2)$ $A^S_4$ solution of embedded in $SU(N)$) at infinity the same as $A^{vev}_4$ of (\[a4vev\]), we add the matrix $\bar{A} ={1 \over NL} \text{diag}(2\pi(N-1) -\pi, 2\pi(N-1) -\pi, 2\pi(N-2),...,0)$ for odd $N$ and $\bar{A} ={1 \over NL} \text{diag}(2\pi(N-1) , 2\pi(N-1) , 2\pi(N-2) + \pi,...,\pi)$ for even $N$ to $A^{S,SU(N)}_4$. Similarly, BPS monopoles can be embedded in the remaining $N-2$ diagonal $SU(2)$ subalgebras of $SU(N)$.
For the Kaluza-Klein solution[^19] we start from the BPS solution of in a vacuum where $\nu$ is replaced by $\nu \rightarrow {2 \pi \over L} - \nu$. To obtain the KK solution ($\overline{\text{KK}}$) we gauge transform the BPS solution of with $S_+$ (with $S_-$) using the upper sign (lower sign). Now the asymptotic behaviour of the $A_4$ field for both solutions is $(\nu - {2\pi \over L}) {\tau^3 \over 2}$. In order to make the asymptotics similar to the BPS $A^S_4$ field in , we perform an $x_4$-dependent gauge transformation $U(\text{x}_4) = \text{exp}(i {2 \pi \over L} \text{x}_4 {\tau^3 \over 2})$, which brings the asymptotics back to $\nu {\tau^3 \over 2}$. This gauge transformation gives a non-trivial $x_4$-dependence to the cores of the KK and $\overline{\text{KK}}$ solutions. Since the Pontryagin index $p$ of a KK monopole in relation is $p = 1$, in order to obtain the lowest topological non-zero charge (which is $|Q| = {1 \over N}$), the second term in should equal $-{N-1 \over N}$ therefore, as already discussed, there is only one way to embed an $SU(2)$ KK monopole in $SU(N)$ in a centre-symmetric $vev$ that would give the lowest action and that is to choose the $SU(2)$ subalgebra corresponding to the components $a_{11}, a_{1N}, a_{N1}, a_{NN}$ of an $SU(N)$ Lie algebra matrix (i.e. with $q_1 = - q_N = -1$, as per (\[eq:2.16\])).
This was a brief summary of the non-perturbative solutions in dYM theory that are responsible for confinement of charges to leading order in the limit $NL \Lambda \rightarrow 0$.
Action of a dilute gas of monopoles {#2.3}
-----------------------------------
The action of the (anti-)self-dual solution embedded in $SU(N)$, with $\nu = 2\pi /NL$, is given by:
\[eq:2.22\] S\_ = [2 L g\^2]{} \_[[IR]{}\^3]{} d\^3 (B\_i B\_i) = [8 L g\^2]{} \^\_0 d|[r]{} |[r]{}\^2 { [1 2]{} F\^2\_2(|[r]{}) + F\^2\_1(|[r]{})}= [4 L g\^2]{} = [8 \^2 g\^2 N]{} ,&&
where $\bar{r} = \nu r$, $r$ being the radial coordinate in spherical coordinates.
Next, we calculate the action of two far-separated BPS solutions of embedded in $SU(N)$ and living in a center-symmetric vacuum $A^{vev}_4$. We embed the first monopole (second monopole) in the $i \text{-th}$ ($j \text{-th}$) subalgebra of $SU(N)$ along the diagonal for $1 \leq i, j \leq N-1$. We work in the limit $\nu^{-1} = {NL \over 2\pi} \ll r_0 \ll d$, where $d$ denotes the distance between the centers of the monopoles and $r_0$ is the radius of a two-sphere surrounding each monopole. In constructing far separated monopole solutions, first we need to mention how the monopoles are patched together. To patch the monopoles together, we use the string gauge and first subtract $A^{vev}_4$ from the $A_4$ component of each monopole solution. The resulting configuration will have an asymptotically vanishing behaviour at infinity for all its gauge field components. Now we simply add up the fields corresponding to the various monopole configurations, with their centres being separated by a large, in the precise sense defined above, distance $d$ from each other. At the end, we add $A^{vev}_4$ to obtain the final configuration (had we simply added the two monopole configurations at a large separation $d$, asymptotically the $A_4$ component of the two monopole configuration would be $2A^{vev}_4$; this way of construction avoids this double counting of $A^{vev}_4$).
In calculating the action of far separated monopole configurations we only consider gauge invariant leading order terms in the self-energy and interaction energy of the monopoles.[^20] We write the fields as $A_{\mu} = A^{(1)}_{\mu} + A^{(2)}_{\mu}$, for $\mu =1,... , 4$. $A^{(1)}_{\mu}$ and $A^{(2)}_{\mu}$ refer to the contribution of the first and second monopole to the total $A_{\mu}$ field of the monopole configuration respectively. When $A_4$ appears in the commutator term of $F_{k4}$ the overall $A^{vev}_4$ is considered as part of $A^{(i)}_4$ in the two-sphere region of radius $r_0$ surrounding the $i$-th monopole for $i =1, 2$ and otherwise can be distributed in an arbitrary smooth way between $A^{(1)}_{4}$ and $A^{(2)}_{4}$ and for the $\partial_k A_4$ term in $F_{k4}$ the overall $A^{vev}_4$ vanishes and can be neglected. The total action of the far separated two-monopole configuration can be written as: $$\label{eq:2.23'}
\begin{split}
S^{\prime}_{2 \text{-monopole}} & = {L \over g^2} \int d^3 \text{x} \{ \text{tr} ( B_kB_k ) + \text{tr} ( F_{k4}F_{k4}) \} \\ & = \sum^{2}_{i=1}S^{(i)}_{\text{self-energy}} + S_{\text{inter.}, > r_0} + S_{\text{inter.}, < r_0} + S_{\text{non-gauge-invariant}}
\end{split}$$ Each of the above terms in will be explained and evaluated below:
1. For the self-energy, we calculate the contribution of one monopole to the action neglecting the other monopole. Similar to we find: $$\label{eq:2.23}
S^{(i)}_{\text{self-energy}} = {L \over g^2} \int d^3 \text{x} \{ \text{tr} ( B^{(i)}_kB^{(i)}_k ) + \text{tr} ( F^{(i)}_{k4}F^{(i)}_{k4}) \} = {8 \pi^2 \over g^2 N} + O(\text{exp}(-\nu r_0)) , \ \ \ \ i =1, 2$$ where $B_k^{(i)}$ and $F^{(i)}_{k4}$ refer to the magnetic and “electric” field[^21] of the $i$-th monopole respectively. The fact that the overall $A^{vev}_4$ is distributed between $A^{(1)}_{4}$ and $A^{(2)}_{4}$ outside their surrounding two-spheres of radius $r_0$ would make the monopole self-energies in to differ from by $O(\text{exp}(-\nu r_0))$.
2. The first contribution to the interaction between two monopoles at the classical level comes from the long range magnetic and electric fields of each monopole. This contribution comes from the region outside the two-spheres of radius $r_0$ surrounding each monopole (the second contribution comes from the long range electric influence that each monopole has on the other monopole inside their surrounded sphere of radius $r_0$, see next item). The magnetic and electric interactions beyond the two surrounding spheres can be evaluated as:
\[eq:2.24\] S\_[, > r\_0]{} &= [2L g\^2]{} d\^3 { ( B\^[(1)]{}\_kB\^[(2)]{}\_k ) + ( F\^[(1)]{}\_[k4]{}F\^[(2)]{}\_[k4]{}) }\
&= [2 L g\^2]{} d\^3 { [ - \_1 2| - \_1|\^3 ]{} q\^[i]{}\_[m1]{} q\^[j]{}\_[m2]{} + [ - \_1 2| - \_1|\^3 ]{} q\^[i]{}\_[e1]{} q\^[j]{}\_[e2]{} }\
& + O([L \^[-2]{} g\^2 d\^3]{}), d = |\_1 - \_2| &&
Here, $q^{i}_{m1} = q^{i}_{m}$ refers to the magnetic charge of the first monopole with $q^{i}_{m}$—an $N$-component charge vector given by relation . Similarly $q^{j}_{m2} = q^{j}_{m}$, $1 \leq i, j \leq N-1$. We substituted, from the Cartesian form of the radial terms proportional to ${1 \over r^2}$ in (\[eq:2.21\]), the magnetic field $B^{(1)}_{k,r^{-2}} \equiv \text{diag}(q^i_{m1} { \text{x}_k - \text{x}_{1k} \over 2|\text{x} - \text{x}_1|^3})$ and $B^{(2)}_{k,r^{-2}} \equiv \text{diag}(q^j_{m2} { \text{x}_k - \text{x}_{2k} \over 2|\text{x} - \text{x}_2|^3})$ (understood as a diagonal matrix with entries determined by the charge vectors $q_{m1}^i$ and $q_{m1}^j$ (\[eq:2.15\]) of the two monopoles) into the first line in (\[eq:2.24\]), and similarly for $F_{k4}$. We then replaced the trace of the product of these abelian matrices with an inner product over the vector of magnetic (or electric, i.e. scalar) charges corresponding to the diagonal elements of these abelian matrices. For a self-dual BPS solution, the electric charges are $q^i_{e1} = q^i_{m1}$ and $q^j_{e2} = q^j_{m2}$. The error term in comes from the inner product of the long range magnetic (or electric) field of one monopole with the term $\sim \hat{r}^{\text{other}} {1 \over \text{sinh}^2 \nu r}$ of magnetic (or electric) field of the other monopole in , integrated over the two-sphere of radius $r_0$ surrounding the other monopole, with $\hat{r}^{\text{other}}$ being the unit vector in the radial direction of the other monopole. After writing ${ \text{x} - \text{x}_i \over 2|\text{x} - \text{x}_i|^3} = -\nabla_{\text{x}} {1 \over 2|\text{x} - \text{x}_i|}$ for $i =1, 2$ in and integrating by parts with $\nabla^2_{\text{x}} {1 \over |\text{x} - \text{x}_i|} = -4 \pi \delta^3 (\text{x} - \text{x}_i)$ we get: $$\label{eq:2.25}
S_{\text{inter.} > r_0} = {2 \pi L \over g^2 d} \;{q^i_{m1}\cdot q^j_{m2} } + {2 \pi L \over g^2 d}\; {q^i_{e1}\cdot q^j_{e2} } + \text{O}({L \nu^{-2} \over g^2d^3})~.$$
3. The final Dirac-string independent contribution to the interaction between the two monopoles is the electric influence of the second monopole on the core of the first monopole[^22] There is also a similar electric influence from the first monopole on the core of the second one. It originates from the following cross terms in the action: $$\label{eq:2.26}
S^{2-1}_{\text{inter.}, < r_0} = -{2 iL \over g^2} \int_{< r_0 , 1} d^3 \text{x} \text{tr} ([A^{(1)}_k,A^{(2)}_4] F^{(1)}_{k4}) - {L \over g^2} \int_{< r_0 , 1} d^3 \text{x} \text{tr} ([A^{(1)}_k,A^{(2)}_4]^2 ) ~.$$ The integration region is within a sphere of radius $r_0$ centered around the first monopole. The main contribution ($\sim 1/d$) in comes from the first integral. Since we excluded the overall $A^{vev}_4$ from $A^{(2)}_4$ within the two-sphere of radius $r_0$ around the first monopole, we have[^23] $A^{(2)}_4 \approx -{1 \over d} {{\tau^3_{(j)} \over 2}}$ ; one can check that there are no other (Dirac-string independent) terms that can contribute order $1/d$ interaction terms. We can work out the integrand of the first integral, using the $A_4^{(2)}$ asymptotics just given, as: $$\label{eq:2.27}
\text{tr} ([A^{(1)}_k,A^{(2)}_4] F^{(1)}_{k4}) = \text{tr} ([F^{(1)}_{k 4}, A^{(1)}_k] A^{(2)}_4) = -{i \over 4d} ( F^{(1),1}_{k4} A^{(1),2}_k - F^{(1),2}_{k4} A^{(1),1}_k ) \; q^i_{e1} \cdot q^j_{e2} ~.$$ Here, $A^{(1),1}_k$ refers to the component of the $A^{(1)}_k$ along the first generator of the $i$-th $SU(2)$ subalgebra along the diagonal of an $SU(N)$ Lie algebra matrix (similar to $\tau^1 /2$ in $SU(2)$) and similarly for the others. The values of the fields $A^{(1)}_k$ and $F_{k4}^{(1)}$ can be read from relations and for a self-dual ($B_k = E_k$) solution. For the integral of the first term in (\[eq:2.26\]), we find $ \int_{< r_0, 1} d^3 \text{x} (F^{(1),1}_{k4} A^{(1),2}_k - F^{(1),2}_{k4} A^{(1),1}_k) = - 8 \pi \int\limits_{0}^{r_0 \nu} dy y {F_1(y) {\cal{A}}(y)} = 4 \pi ({\cal{A}}^2(0) - {\cal{A}}^2(r_0 \nu)) \simeq 4 \pi + \text{O}(e^{- 2\nu r_0})$, where we used $y F_1 = \partial_y {\cal{A}} $. Thus, going back to (\[eq:2.26\]), one obtains in total: $$\label{eq:2.28}
S_{\text{inter.}, < r_0} = S^{2-1}_{\text{inter.}, < r_0} + S^{1-2}_{\text{inter.}, < r_0} = - {2\pi L \over g^2 d}\; q^i_{e1}\cdot q^j_{e2} - {2 \pi L \over g^2 d}\; q^j_{e2}\cdot q^i_{e1} + \text{O}({L \nu^{-1} \over g^2d^2})~.$$ The $\text{O}({L \nu^{-1} \over g^2d^2})$ error term comes from the evaluation of the second integral in and the error of the first integral in coming from the variation of $A_4^{(2)}$ from its value at the center of the sphere around the first monopole over the region of integration is of order $\text{O}({L \nu^{-2} \over g^2d^3})$ which we have neglected.
Summing up the electric interactions in and we get $- {2\pi L \over g^2 d}\; q^i_{e2} \cdot q^j_{e1}$, which shows a negative potential for same-sign electric charges, hence an attractive electric force between the two monopoles with same electric charges. Since the electric interaction is mediated by the exchange of a massless (at the classical level) scalar field $A_4$, which is attractive for same sign charges, this is expected and was originally observed in [@25] using a slightly different approach. Although for simplicity we initially assumed that the solutions are BPS, eq. is general, meaning that if we had done the same calculation in with two other monopoles (e.g. a KK and a BPS) we would have reached the same relation in , but with their appropriate electric charges replaced.
4. The $S_{\text{non-gauge-invariant}}$ term in consists of any term in the action that depends on the Dirac-string singularity (in it occurs at $\theta = \pi$) or on its orientation. These non-gauge-invariant terms are unphysical and will be neglected; we were careful to only evaluate contributions that are independent of the Dirac string or its orientation. To be more specific on this matter, we notice that the $A^S_{\phi}$ component in is singular at $\theta = \pi$. Considering the commutator term $[A^{(1)}_i,A^{(2)}_j]$ in $F_{ij}$ for two far separated monopoles at the location of the string of the first monopole, we realize that the ${\text{tan} \ \theta \over 2r} {\tau^3\over 2}$ term in $A^{S, (1)}_{\phi}$ for this monopole does not commute with terms proportional to $\tau^1$ or $\tau^2$ in the components $A^{S, (2)}_{\theta}$ or $A^{S, (2)}_{\phi}$ of the second monopole therefore (even though these terms would be exponentially suppressed outside the sphere of radius $r_0$ of the second monopole) for the action they would give a term proportional to $\int^{\pi}_0 d \theta \text{sin} (\theta) \text{tan}^2 {\theta \over 2}$ which is singular when integrated near $\theta = \pi$. Or, similar to the electric interaction inside the monopole cores as in , another contribution can be evaluated for the magnetic interaction coming from the term $\approx$ ${\text{tan} {\theta \over 2} \over d} {\tau^3 \over 2}$ of $A^{S,(2)}_{\phi}$ near the center of the first monopole which would depend on the orientation of the Dirac string of the second monopole. These contributions are unphysical and a more precise treatment of the interaction of far separated monopoles, as in [@25; @29; @30], does not involve any orientation dependent contributions, at least in the $d \nu \gg 1$ limit, but will only involve interactions similar to the gauge-invariant interaction terms $S_{\text{inter.}, > r_0} + S_{\text{inter.}, < r_0}$ evaluated above.[^24]
summarize, using the relations , , and , the (Dirac-string-independent) action of two far separated monopole solutions in the limit $\nu^{-1} = {NL \over 2\pi} \ll r_0 \ll d$, with $|\text{x}_1 - \text{x}_2| = d$, can be summarized as: $$\label{eq:2.29}
S_{2 - \text{monopoles}} = 2 \times {8 \pi^2 \over g^2 N} + {2 \pi L \over g^2 d}\; {q^i_{m1} \cdot q^j_{m2}} - {2 \pi L \over g^2 d} \;{q^i_{e1} \cdot q^j_{e2}} + \text{O}({L \nu^{-1} \over g^2d^2})~.$$ Although for simplicity we assumed that both solutions are BPS, the relation is general and applies to two arbitrary monopoles or anti-monopoles. Therefore the action of a dilute gas of $n^{(i)}$ monopoles of type $i$ and $\bar{n}^{(i)}$ anti-monopoles of type $i$ for $i = 1, ..., N$, referring to the KK monopole as the monopole of type $N$, with $n = \sum^{N}_{i = 1}({n}^{(i)} +\bar{n}^{(i)})$ their total number, is given by:[^25] $$\label{eq:2.30}
S_{\text{monopole-gas}} = {8 \pi^2 \over g^2 N}n + S_{\text{int}, m} + S_{\text{int}, e} + \text{O}({n^{4/3} L \nu^{-1} \over g^2d^2})~.$$ In (\[eq:2.30\]), $S_{\text{int},m}$ ($S_{\text{int},e}$) is the sum of magnetic (“electric”) interaction terms similar to for every pair of monopoles in the gas: $$\label{eq:2.31}
S_{\text{int},m} = {2\pi L \over g^2}\big{[} {1 \over 2} \sum_{ \underset{\text{dist. pairs}}{i, j, k_i, k_j} } {q^i_m \cdot q^j_m \over |r^{(i)}_{k_i} - r^{(j)}_{k_j}|} + {1 \over 2} \sum_{ \underset{\text{dist. pairs}}{i, j, \bar{k}_i, \bar{k}_j} } {\bar{q}^i_m \cdot \bar{q}^j_m \over |\bar{r}^{(i)}_{\bar{k}_i} - \bar{r}^{(j)}_{\bar{k}_j}|} + \sum_{{i, j, k_i, \bar{k}_j }} {q^i_m \cdot \bar{q}^j_m \over |r^{(i)}_{k_i} - \bar{r}^{(j)}_{\bar{k}_j}|} \big{]}~,$$ $$\label{eq:2.32}
S_{\text{int},e} = - {2\pi L \over g^2} \big{[} {1 \over 2} \sum_{ \underset{\text{dist. pairs}}{i, j, k_i, k_j} } {q^i_e \cdot q^j_e \over |r^{(i)}_{k_i} - r^{(j)}_{k_j}|} + {1 \over 2} \sum_{ \underset{\text{dist. pairs}}{i, j, \bar{k}_i, \bar{k}_j} } {\bar{q}^i_e \cdot \bar{q}^j_e \over |\bar{r}^{(i)}_{\bar{k}_i} - \bar{r}^{(j)}_{\bar{k}_j}|} + \sum_{{i, j, k_i, \bar{k}_j }} {q^i_e \cdot \bar{q}^j_e \over |r^{(i)}_{k_i} - \bar{r}^{(j)}_{\bar{k}_j}|} \big{]}~,$$ with $1 \leq i, j \leq N$, $1 \leq k_i \leq n^{(i)}$ and $1 \leq \bar{k}_i \leq \bar{n}^{(i)}$. The summation is being performed over distinct pairs of monopole-monopole and anti-monopole-anti-monopole interactions and a factor of $1 \over 2$ has been included to cancel the double counting of pairs in these summations. Note that since the notion of anti-monopole is in regard to opposite magnetic charges, although $\bar{q}^j_m = -q^j_m$, this is not true for the electric charges, which satisfy $\bar{q}^j_e = q^j_e$.
The reader should also be reminded that the electric interaction term will not be important for us in the quantum theory since it is gapped due to the one loop effective potential for the $A_4$ field and hence is of short range ($\sim { \nu^{-1}}$). Therefore in the next Section we will be only concerned with the magnetic interaction term .
Derivation of the string tension action {#sec:2.4.1}
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The static quark-antiquark potential in a representation $r$ of the gauge group is determined by evaluating the expectation value of a rectangular Wilson loop of size $R\times T$ in representation $r$, and considering the leading exponential in the large Euclidean time $T$ limit [@04]: $$\label{eq:2.33}
\underset{T\rightarrow \infty}{\lim} \langle W_r (R,T) \rangle = \underset{T\rightarrow \infty}{\lim} \langle \text{tr}_r ({\cal{P}}\text{exp}(\int_{R\times T}A_{\mu}dx^{\mu}))\rangle \ \sim \ \text{exp}(-V_r(R)T)~.$$ In confining gauge theories in the absence of string breaking effects the potential $V_r(R)$ has a linear behaviour $V_r(R) = \sigma_r R$ at large distances,[^26] where $\sigma_r$ is referred to as the string tension for quarks in the representation $r$. At intermediate distances ($\approx \Lambda^{-1}$), the string tension can have a dependence on the particular representation $r$, it is known that the asymptotic—a few $\Lambda^{-1}$ and more—string tension, because of colour screening by gluons, depends only on the $N$-ality $k$ of the representation $r$, hence asymptotically $\sigma_r$ is referred to as the $k$-string tension $\sigma_k$.
In this Section we will be deriving an expression for the $k$-string tensions in dYM theory by evaluating . We want to calculate using the low energy degrees of freedom, to leading order in the limit of $NL \Lambda \rightarrow 0$: $$\label{eq:2.34}
\langle W_r (R,T)\rangle = {\int [D\psi][D\bar{\psi}][D A] \text{tr}_r {\cal{P}} \text{exp}(i\oint_{R \times T} dx_{\mu} A^{\mu}) \text{exp}(-S_{dYM}) \over \int [D\psi][D\bar{\psi}][A] \text{exp}(-S_{dYM}) }~.$$ To evaluate the partition function $Z = \int [D\psi][D\bar{\psi}][DA]\text{exp}(-S_{dYM})$, we expand the action around the perturbative and non-perturbative minimum action configurations (the $2N$ minimum action monopole solutions discussed in Section \[sec:2.2.2\]), including the contribution of the approximate saddle points made up of far separated monopole configurations (dilute gas of monopoles), to second order and evaluate the functional determinant in these backgrounds, using the approximate factorization of determinants around widely separated monopoles. The result is the grand canonical partition function of a multi-component Coulomb gas [@01][^27]: $$\label{eq:2.35}
Z = {Z_{\text{pert.}}} \underset{i=1} {\overset{N}{\Pi}} \{ \sum^{\infty}_{n^{(i)} = 0} {\zeta^{n^{(i)}} \over n^{(i)} !} \sum^{\infty}_{\bar{n}^{(i)} = 0} {\zeta^{\bar{n}^{(i)}} \over \bar{n}^{(i)} !} \int_{{\rm I\!R}^3}\underset{k=1} {\overset{n^{(i)}}{\Pi}} d\text{r}^{(i)}_k \int_{{\rm I\!R}^3}\underset{l=1} {\overset{\bar{n}^{(i)}}{\Pi}} d\bar{\text{r}}^{(i)}_l \} \text{exp}(- S_{\text{int},m})~,$$ where the product over $i$ implies the inclusion of the $N$ types of minimal action BPS and KK monopole-instantons (and anti-monopole-instantons) and the sum over $n^{(i)}, \bar{n}^{(i)}$ indicates that arbitrary numbers of such configurations with centers at $r_k^{(i)},\bar r_k^{(i)}$ are allowed. For any term in involving $n^{(i)}$ monopoles and $\bar{n}^{(i)}$ anti-monopoles for $i = 1, ..., N$, $S_{\text{int},m}$ is given by and the fugacity is: $$\label{eq:2.36}
\zeta = C \text{e}^{-S_0} = \bar{A} {\bar{D}_f} \; m_W^3 (g^2(m_W) N)^{-2} \text{e}^{-8 \pi^2 /N g^2(m_W)},$$ similar to the expression for the fugacity derived in [@01]. The only difference is that now ${\bar{D}_f}$ the finite part of ${D_f \equiv {\text{det}^{n_f}(\slashed{D} + m) \over \text{det}^{n_f}(\slashed{D} + M )}}$, the Pauli-Villars regulated determinant of massive adjoint fermions, is replacing $\text{e}^{- \Delta S}$ in the expression for fugacity in [@01] (instead of the $\Delta S$ term in we now have massive adjoint fermions, of mass $m \sim m_W$, in ). $\bar{A}$ is a dimensionless and $N$-independent coefficient and the finite part of $D_f$ can be absorbed in its redefinition (after taking into account its effect on coupling renormalization; we omit any details on this).
Consider now the following identity,[^28] where $\sigma$ denotes the $N$-component vector $(\sigma_1,$ $\sigma_2,...\sigma_N)$: $$\label{eq:2.37}
\begin{split}
& \int D[ \sigma] \;\text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 ) \; \text{exp}( iq^i_m \cdot \sigma (\text{x}_1)) \;\text{exp}( iq^j_m \cdot \sigma(\text{x}_2)) = \\ & {Z_{\text{pert.}}} \times \; \text{exp} (-{2 \pi L \over g^2} {q^i_m\cdot q^j_m \over |\text{x}_1 - \text{x}_2|} ) \; \text{exp} (-2 \times {2 \pi L \over g^2} \lim_{|\text{x}| \rightarrow 0} {1 \over |\text{x}|})~.
\end{split}$$ After regularizing the infinite self-energies ($\lim_{|\text{x}| \rightarrow 0} \{{1 \over |\text{x}|} -{\text{exp} (-\mu |\text{x}|) \over |\text{x}|} \} = \mu$) using the Pauli-Villars method, a typical term in , abbreviated t.t. below, using the analog of for $n$ monopoles, with $n^{(i)}$ monopoles and $\bar{n}^{(i)}$ anti-monopoles of each kind, can be written as: $$\label{eq:2.38}
\begin{split}
Z_{\text{t.t.}} & = \int D[ \sigma]\; \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 ) \; \times \\ & \underset{i=1} {\overset{N}{\Pi}} \{ {( \widetilde{N} \zeta)^{n^{(i)}} \over n^{(i)} !} { (\widetilde{N} \zeta )^{\bar{n}^{(i)}} \over \bar{n}^{(i)} !} \underset{k=1} {\overset{n^{(i)}}{\Pi}} \int_{{\rm I\!R}^3} d\text{r}^{(i)}_k \text{exp}( iq^i_m \cdot \sigma (\text{r}^{(i)}_k)) \underset{l=1} {\overset{\bar{n}^{(i)}}{\Pi}} \int_{{\rm I\!R}^3} d\bar{\text{r}}^{(i)}_l \text{exp}( i\bar{q}^i_m \cdot \sigma(\bar{\text{r}}^{(i)}_l) ) \}~,
\end{split}$$ where $\tilde{N} = \text{exp} ( + {2 \pi L \over g^2} \mu )$.
Before we continue, we pause to note that the scalar fields $(\sigma^1,...,\sigma^N)$ are the magnetic duals to the U(N) Cartan-subalgebra electric gauge fields, the so-called “dual photons.” For the purpose of the paragraph that follows, in order to elucidate the physical meaning of gradients of the $\sigma$ fields, we revert to Minkowski space. The duality relation, with $(+,-,-)$ metric, is $$\label{dualityrelation}
F_{kl}^A = - {g^2 \over 2\sqrt{2} \pi L} \epsilon_{klm} \partial^m \sigma^A~,~~A=1,...,N~.$$ The kinetic term in the Minkowski space version of (\[eq:2.38\]) is nothing but a rewriting of the first (“magnetic”) term in Minkowski space version of the action (\[eq:2.23’\]) restricted to its Cartan subalgebra and considered for a U(N) gauge group via dual variables.[^29] In order to do this in a proper way consider the Minkowski space action of the 3-dimensional low energy theory in perturbation theory with the Bianchi identity imposed as a constraint via the auxiliary field $\sigma$ to eliminate gauge degrees of freedom: $$\label{bianchi}
S = \int_{\mathbb{R}^{1,2}} \{ -{L \over 4 g^2}F^A_{kl}F^{A kl} + h \epsilon_{k l m } \partial^m F^{A kl}\sigma^A \}~, \;\;\;\;\;\; A =1, ...,N.$$ Integrating by parts the Lagrange multiplier term, completing the square of the $F^a_{kl}$ fields and integrating them out leaves an action only in terms of the dual fields $\sigma$: $S_{\text{dual}} = {2 \over L} h^2 g^2 \int_{\mathbb{R}^{1,2}} (\partial_k \sigma)^2 $. Demanding ${2 \over L} h^2 g^2 = {g^2 \over 2} {1 \over 8 \pi^2 L}$, the coefficient of the gradient of the $\sigma$ fields in , gives $h = 1/(4\sqrt{2}\pi)$. Varying the action with respect to $F_{kl}$, we obtain the duality relation . An immediate remark, relevant for the discussion in Section \[bagmodelsection\], is that the duality relation (\[dualityrelation\]) implies that spatial gradients of $\sigma$ represent perpendicular electric fields $\vec{E}^A$, i.e. $$E_i^A \equiv F_{i0}^A = {g^2 \over 2 \sqrt{2} \pi L} \epsilon_{ij} \partial_j \sigma^A~. \label{dualityelectric}$$
Returning to our main objective—obtaining the effective theory of the dYM vacuum—we sum over the contributions of all monopoles and antimonopoles in (\[eq:2.38\]), and find that the full partition function becomes: $$\label{eq:2.39}
\hspace*{-1cm}Z = \int D[ \sigma] \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 ) \sum^{\infty}_{n = 0} { (\widetilde{N} \zeta )^{n} \over n!} \{ \sum^{N}_{i = 1} \int_{{\rm I\!R}^3} d^3\text{x}( e^{iq^i_m \cdot \sigma (\text{x})} + e^{i\bar{q}^i_m \cdot \sigma (\text{x})} ) \}^n~.$$ Thus, the final form of the dual photon action reads: $$\label{eq:2.40}
Z = \int D[ \sigma] \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} \{ {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 - \widetilde{\zeta}\; \sum^{N}_{i = 1} \text{cos}(q^i_m \cdot \sigma) \} )~,$$ where $\widetilde{\zeta} = 2 \tilde{N} \zeta$.[^30]
Before working out the Wilson loop integral, we will derive the $NL \Lambda$ dependence of the fugacity and the dual photon mass, verify the dilute gas limit conjecture and discuss the hierarchy of scales in this theory. Using the one loop renormalization group invariant scale $\Lambda$ for $n_f = 1, 2$ flavours of Dirac fermions in the adjoint representation of the gauge group: $$\label{eq:2.41}
\Lambda^{b_0} = \mu^{b_0} \; \text{exp}(- {8 \pi^2 \over N g^2}), \ \ \ \ \ \ \ \ b_0 = (11 - 4n_f)/3~,$$ we can determine the leading dependence of the fugacity on $NL\Lambda$. The Pauli-Villars scale used in (\[eq:2.37\]) should be thought of as the cutoff of the long-distance theory containing no charged excitations and should be taken below the scale of any charged excitation; for the sake of definiteness, we shall take $\mu \sim {g \sqrt{N} \over N L}$, the lowest eigenvalue of the holonomy fluctuations. Then, we can neglect $\tilde{N}$ compared to $\text{exp} (- S_0)$ in the small-$NL \Lambda$ limit.[^31] The one-loop massive fermion determinant contributes some calculable constant and renormalizes the coupling of the long-distance theory, as already accounted for in (\[eq:2.41\]). From and , neglecting any $\text{log}(NL \Lambda)$ (or, equivalently, $g^2 {N}$) dependence, for the leading $NL \Lambda$ dependence of the fugacity we obtain: $$\label{eq:2.42}
\widetilde{\zeta} \sim \zeta \sim ({ 1 \over NL })^3 (NL)^{b_0} \Lambda^{b_0} = (NL\Lambda)^{b_0 - 3}\Lambda^{3}~.$$ The fugacity $\tilde{\zeta}$ or $\zeta$ is proportional to the monopole density $n_d$. For a gas with density $n_d$ the average distance between the particles in the gas is $\sim {1 \over n_d^{1/3}}$ and in order to verify the dilute gas conjecture we should have that this separation be much larger than the size of the monopoles, of order $NL$: $$\label{eq:2.43}
d \sim {1 \over \zeta^{1/3} } \gg NL \longrightarrow (NL \Lambda)^{-{b_0\over 3}} \gg 1 \longrightarrow b_0 > 0 , \ \ n_f \leq 2~ .$$ Since we are working in the limit $NL \Lambda \rightarrow 0$, the condition will be satisfied if $b_0 > 0$, which is the same condition as the asymptotic freedom condition and gives $n_f \leq 2$ (or $n_f \leq 5/2$ if Majorana masses are considered instead).
The mass of the dual photon can be read from . The coefficient of the quadratic term in the dual photon action, after expanding the cosine term and factoring out ${g^2 \over 8 \pi^2 L}$ is ${8 \pi^2 L \over 2g^2} \tilde{\zeta}$. We define the dual-photon mass scale $$\label{eq:2.44}
m^2_{\gamma} = {8 \pi^2 NL \over Ng^2} \; \widetilde{\zeta} \sim (NL\Lambda)^{b_0-2} \;\Lambda^2 ~,$$ and note that it is of order the mass of the heaviest dual photon (the dual photon mass eigenvalues, after diagonalizing the quadratic term via a discrete Fourier transform, are $ m_\gamma \sin {\pi k \over N}$ with $k=1,...,N-1$).
Thus, the hierarchy of scales in this theory can be summarized as: $$\label{eq:2.45}
{\begin{split}
m_W \sim {1 \over NL}\; &\gg \; m_H \sim {g \sqrt{N} \over NL} \sim \mu \; \gg {1 \over d} \sim ({1 \over NL \Lambda })^{1 - {b_0 \over 3}} \Lambda \; \gg \; m_{\gamma} \sim ({1 \over NL \Lambda})^{1 - {b_0 \over 2}} \Lambda~.
\end{split}}$$ For $n_f = 1$, we have ${1 \over d} \gg \Lambda \gg m_{\gamma}$ , but for $n_f = 2$: ${1 \over d} \gg m_{\gamma} \gg \Lambda$; we stress again that the scale $\Lambda$ has no physical significance in the small-$L$ theory (except that to ensure weak coupling, we must have $m_W \gg \Lambda$).
Now we will work out the Wilson loop integral. In the dilute gas limit ($NL\Lambda \rightarrow 0$) the leading contribution to the Wilson loop integral comes from the long distance abelian behaviour of the monopole gas far from the cores ($\sim NL$) of the monopoles. Therefore for a Wilson loop in the $\text{x}_1, \text{x}_2$ plane and representation $r$ with $N$-ality $k$ and for a typical monopole gas background involving $n$ monopoles we have: $$\label{eq:2.46}
\begin{split}
\hspace{-1cm}\{ \text{tr}_r \ \text{exp}( i \oint_{R\times T} A^c_m t_r^c d\text{x}^m) \}_{\text{typ. mon.}} & = \text{tr}_r \ \text{exp} (i \int_{S(R\times T)}\epsilon_{anm} \partial_n A^c_m t_r^c d\text{S}^a ) \\ & = \sum^{d(r)}_{j=1} \text{exp}( i \int_{S(R\times T)} d\text{x}_1d\text{x}_2 \overset{n}{\underset{i=1}{\sum}} \mu^j_r \cdot q^i_m\; {\text{R}^i_3 \over 2 |R^i - \text{x}|^{3}})~,
\end{split}$$ where $c=1, ..., N-1$ labels the Cartan generators of $SU(N)$ in the representation $r$ and $\mu^j_r$ is it’s $j$-th weight. On the first line above, we used Gauss’ law to rewrite the Wilson loop integral as an integral of the magnetic flux through a surface $S$ spanning the loop, and on the second line, we replaced the magnetic field by the field of $n$ monopole-instantons at positions $R^i \in \R^3 $, $i=1,...,n$.[^32] Defining the solid angle $\eta(\text{x})$ that the Wilson loop is seen at from the point $\text{x} \in \R^3$, $\eta(\text{x})$ $\equiv$ $\int_{S(R\times T)} d\text{y}_1d\text{y}_2 {\text{x}_3 \over 2 |y - \text{x}|^{3}}$, we have for the contribution to the Wilson loop expectation value of an $n$-monopole configuration: $$\label{eq:2.47}
\{ \text{tr}_r \ \text{exp}( i \oint_{R\times T} A_m d\text{x}^m) \}_{\text{typ. mon.}} = \sum^{d(r)}_{j=1} \text{exp}( i \sum^n_{i=1} \mu^j_r \cdot q^i_m \eta (\text{R}^i) )~.$$ Comparing with , we see that the effect of the Wilson loop insertion is to shift the $\sigma (\text{r}^{(i)}_k)$ field multiplying the magnetic charges in by $\mu^j_r \eta (\text{r}^{(i)}_k)$ (and similarly for $\sigma (\bar{\text{r}}^{(i)}_k)$ and the $\sigma(\text{x})$ field in ). Thus, shifting the $\sigma(\text{x})$ field by $\sigma(\text{x}) \rightarrow \sigma(\text{x}) - \mu^j_r \eta (\text{x})$ gives the final form of the expectation value of the Wilson loop in dYM theory to leading order in $NL\Lambda \rightarrow 0$, calculated using the low energy effective theory: $$\label{eq:2.48}
\langle W_r(R,T)\rangle = \int D[ \sigma] \sum^{d(r)}_{j=1} \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} \{ {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma - \mu^j_r \nabla \eta )^2 - \widetilde{\zeta} \sum^{N}_{i = 1} \text{cos}(q^i_m \cdot \sigma) \} ) / Z~.$$ The string tension action is given by making a saddle point approximation to . The Lagrange equations of motion for the contribution of the $j$-th weight of $r$ to the Wilson loop expectation value are: $$\label{eq:2.49}
\nabla^2 \sigma_i = 2 \pi (\mu^j_r)_i \theta_A(\text{x}_1, \text{x}_2) \partial_3 \delta (\text{x}_3) + m^2_{\gamma} (\text{sin}(\sigma_i - \sigma_{i+1}) + \text{sin}(\sigma_i - \sigma_{i-1})) , \ \ \ i = 1, ..., N~,$$ where $\sigma_{0} \equiv \sigma_N$, $ \sigma_{N+1} \equiv \sigma_{1}$ and $\nabla^2 \eta (\text{x}) = 2 \pi\theta_A(\text{x}_1, \text{x}_2) \partial_3 \delta (\text{x}_3)$. $\theta_{A}(\text{x}_1, \text{x}_2)$ is one for $(\text{x}_1, \text{x}_2) \in A$ and zero for $(\text{x}_1, \text{x}_2) \notin A$, with $A$ being the area of the Wilson loop $R \times T$ in the $\text{x}_1, \text{x}_2$ plane. For large Wilson loops ($R, T \rightarrow \infty$) the saddle point configuration of is zero for regions outside the Wilson loop. The solution near the boundaries would be more complicated and gives a contribution proportional to the perimeter of the Wilson loop. The solution interior to the Wilson loop far from the boundaries would depend on $\text{x}_3$ only. The corresponding one-dimensional equation is: $$\label{eq:2.50}
{\partial^2 \sigma_i \over \partial \text{x}_3 ^2} = 2 \pi (\mu^j_r)_i \theta_A(\text{x}_1, \text{x}_2) \partial_3 \delta (\text{x}_3) + m^2_{\gamma} (\text{sin}(\sigma_i - \sigma_{i+1}) + \text{sin}(\sigma_i - \sigma_{i-1})) , (\text{x}_1 , \text{x}_2) \in A~.$$ Eq. represents a boundary value problem showing a discontinuity of $2 \pi (\mu^j_r)_i$ for the $\sigma_i$ $(i=1,...,N)$ fields at $\text{x}_3 = 0$. Therefore to leading order in $NL \Lambda \rightarrow 0$ (the saddle point approximation is valid in this limit) and $R, T \rightarrow \infty$ is $$\label{eq:2.51}
\langle W_r(R,T)\rangle \sim \sum^{d(r)}_{j=1} \text{exp}(- T^j_r RT)~,$$ given by a sum over the exponential contributions of the different weights of the representation $r$. Sources in every weight have their own string tension, $T^j_r$, given by: $$\label{eq:52}
T^j_r = \underset{\sigma(\text{x}_3)}{\text{min}}\int^{+\infty}_{-\infty} d\text{x}_3 \{{1\over {2L}}{g^2 \over {8\pi^2}}({\partial \sigma \over \partial \text{x}_3})^2+ \tilde{\zeta} \overset{N}{\underset{i=1}{\sum}}[1- \text{cos}(\sigma_i-\sigma_{i+1})]\} \Big |_{\Delta \sigma(0) =2 \pi \mu^j_r}~,$$ with $\Delta \sigma(0) \equiv \sigma(0^+) - \sigma(0^-)$.
Notice that because the long-distance theory is abelian, within the abelian theory, we can insert static quark sources with charges given by any $\mu_r^j$, $j=1, ... , d(r)$. Clearly, this is not the case in the full theory, where the entire representation appears and color screening from gluons is operative. In our $N L \Lambda \rightarrow 0$ limit, the gauge group is broken, $SU(N) \rightarrow U(1)^{N-1}$, and the screening is due to the heavy off-diagonal $W$-bosons, which were integrated out to arrive at (\[eq:2.51\]). Thus, we expect that at distances $R$ such that $T_r^j R > O(m_W)$, $W$-bosons can be produced (as in the Schwinger pair-creation mechanism) causing the strings in representation $r$ with higher string tensions to decay to the string with lowest tension in $r$. Hence, we shall not study all $T^j_r$ tensions, but will focus only on the strings of lowest tension confining quarks in representation $r$.[^33]
It is shown, in Appendix \[sec:B1\], that any representation of $SU(N)$ with $N$-ality $k$ ($= 1, ..., N-1$) contains the $k$-th fundamental weight, $\mu_k$ (given by below) as one of its weights. Furthermore, in Section \[sec:5.1.1\] it is shown that this weight would give the lowest string tension action among the other weights of that representation. Therefore reduces to: $$\label{eq:2.53}
\langle W_r (R,T)\rangle \sim \text{exp}(- T_k RT)~,$$ up to pre-exponential factors and subdominant terms corresponding to the higher string tensions. The $k$-string tension $T_k$, defined by: $$\label{eq:54}
T_k=\underset{\sigma(\text{x}_3)}{\text{min}}\int^{+\infty}_{-\infty} d\text{x}_3 \{{1\over {2L}}{g^2 \over {8\pi^2}}({\partial \sigma \over \partial \text{x}_3})^2+ \tilde{\zeta} \overset{N}{\underset{i=1}{\sum}}[1- \text{cos}(\sigma_i-\sigma_{i+1})]\} \Big |_{\Delta \sigma(0) =2 \pi \mu_k}$$ will be the object of our numerical and analytical studies in the rest of this paper.
String tensions in dYM: a numerical study {#numericsection}
=========================================
String tension action {#sec:3.1}
---------------------
As derived in Section \[sec:2.4.1\] the “$k$-string tension action” is given by: $$\label{eq:3.1}
T_k=\underset{\sigma(\text{z})}{\text{min}}\int^{+\infty}_{-\infty} d\text{z}\{{1\over {2L}}{g^2 \over 8\pi^2}({\partial \sigma \over \partial \text{z}})^2+ \tilde{\zeta}\; \overset{N}{\underset{j=1}{\sum}}[1- \text{cos}(\sigma_j-\sigma_{j+1})]\} \Big |_{\Delta \sigma(0) =2 \pi \mu_k}, \ \ \sigma_{N+1}\equiv \sigma_1~,$$ [where $\mu_k$, the fundamental weights of $SU(N)$, are obtained by solving the equation ${2\alpha_i \cdot \mu_k / \left | \alpha_i \right |^2}=\delta_{ik}$ [@09], and are given by:]{} $$\label{eq:3.2}
\mu_k=({N - k \over N}, ..., \overset {\text{k-th}} {\widehat{{N-k \over N}}}, {-k \over N}, ..., {-k \over N}),\ \ \ \ \ \ 1\leq k \leq N-1,$$ Here $\alpha_i$ are the simple roots of $SU(N)$, given in their $N$-dimensional representation by: $$\label{eq:3.3}
\alpha_i=(0,..,0,\overset {\text{i-th}}{\widehat{1}},-1,0,...,0), \ \ \ \ \ \ 1\leq i \leq N-1.$$ In deriving the relation for the $\mu_k$’s, it is assumed that the $\mu_k$’s and $\alpha_i$’s span the same subspace in ${\rm I\!R}^N$, orthogonal to the vector $(1,1,1,...,1,1)$. The string tension action will be minimized when the discontinuity $\Delta \sigma(0) = 2 \pi \mu_k$ is equally split between $\sigma (0^+)$ and $\sigma (0^-)$.[^34] Therefore the value of the action would also be equally split between the positive and negative $z$-axis and we can consider half the $k$-string action. Defining $m_{\gamma}^2 \equiv {8\pi^2 \over g^2} L \tilde{\zeta}$, the parameter-free form of half of is given by making the change of variable $z ={z' {1 \over \sqrt {2} m_{\gamma}}}$:
\[eq:3.4\] & [ m\_ ]{} [T\_k 2]{} = \_[0]{}\^[+]{} d’{([f ’]{})\^2+\[1-(f\_j-f\_[j+1]{})\]} ,\
& f(+ ) = 0, f(0) = \_k f(’)=( [’ m\_]{}) .
The equations of motion for $f$ are given by: $$\label{eq:3.5}
{d^2f_j \over d\text{z}^2}={1\over 2} (\text{sin}(f_j-f_{j+1})+ \text{sin}(f_j-f_{j-1})),\ \ 1\leq j \leq N, \ f_0\equiv f_{N}, \ f_{N+1} \equiv f_{1}.$$ It is possible to solve the equations of motion directly for $SU(2)$ and $SU(3)$ and derive the exact value of .[^35] The solution for $SU(2)$ and its corresponding $\bar{T}_1$ value is: $$\label{eq:3.6}
-f_2(\text{z}) = f_1(\text{z}) = 2 \text{ arctan}(\text{exp}(-\sqrt{2}\;\text{z}))\xrightarrow{\text{after inserting in \eqref{eq:3.4}}} \bar{T}_1 = 8/\sqrt{2}~.$$ Due to charge conjugation symmetry $\bar{T}_{k} = \bar{T}_{N-k}$ [@01] hence for $SU(3)$ $\bar{T}_1 = \bar{T}_2$ and therefore it suffices to solve the equations of motion and find the action for the first fundamental weight $\mu_1$ of $SU(3)$: $$\label{eq:3.7}
-2f_2(\text{z}) = -2f_3(\text{z}) = f_1(\text{z}) = {8 \over 3} \text{ arctan}(\text{exp}(-\sqrt{3/2}\;\text{z}))\xrightarrow{\text{after inserting in \eqref{eq:3.4}}} \bar{T}_1 = 16/\sqrt{6}~>$$ In the following Sections, these exact values will be used as a check on our numerical methods.
Discretization of the string tension action {#sec:3.3}
-------------------------------------------
We will obtain the numerical value of the string tensions in deformed Yang-Mills theory by discretizing and minimizing the multivariable function obtained upon discretization. We set the boundary conditions at $f_j(0)=\pi (\mu_k)_j$ and $f_j(J)=0$ for $J > 0$.
To discretize divide the interval $[0,J]$ into $m$ partitions and consider the following array of discretized variables: $$\label{eq:3.8}
\{f_{jl}\,|\,\,\, 1\leq j \leq N\,,\,\,\,0\leq l \leq m\,,\,\,\, f_{j0}=\pi (\mu_k)_j\,,\, f_{jm}=0\}~.$$ We denote $\delta \text{z} = J/m$, and introduce the discretized functions $f_j^{(m)}$, linearly interpolated in every interval of width $\delta \text{z}$:[^36] $$\label{eq:3.9}
f^{(m)}_j(\text{z})=f_{jh}+{f_{jh+1}-f_{jh}\over \delta \text{z}}( \text{z}-h\delta \text{z})\,, \,\,\, \text{z} \in [h\delta \text{z},(h+1)\delta \text{z}]\,,\, h = 0, ..., m-1~.$$ Inserting in and performing the $\text{z}$ integration, the linearly discretized action is: $$\label{eq:3.10}
\begin{split}
\bar{T}^{m,J}_k=\bar{T}^{m,J}_{k1}+\bar{T}^{m,J}_{k2}, \
&\bar{T}^{m,J}_{k1} = \underset{j,h}{\sum}{(f_{jh+1}-f_{jh})^2\over \delta \text{z}},
\\
&\bar{T}^{m,J}_{k2} = NJ-\delta \text{z} \underset{j,h}{\sum}{\text{sin}(f_{jh+1}-f_{j+1h+1})- \text{sin}(f_{jh}-f_{j+1h})\over{(f_{jh+1}-f_{j+1h+1})-(f_{jh}-f_{j+1h})}}~,
\end{split}$$ where we split the discretized action into a kinetic ($\bar{T}^{m,J}_{k1}$) and potential ($\bar{T}^{m,J}_{k2}$)parts. Notice their different scalings with the width of partition $\delta \text{z}$ (we shall make use of this fact in Section \[bagmodelsection\] when discussing similarities with the MIT Bag Model).
Minimization of the string tension action and error analysis {#sec:minimize}
------------------------------------------------------------
In order to obtain more accurate numerical results and have control over the minimization process, a systematic method is utilized for minimizing the multivariable function . For sufficiently small $\delta \text{z}$, $\bar{T}^{m,J}_k$ has a parabolic structure along the direction of any variable $f_{lp}$ (i.e.${\partial^2\bar{T}^{m,J}_k\over \partial f_{lp}^2}>0$). The second derivative of the 1st term in with respect to $f_{lp}$ is ${4\over \delta \text{z}}$ and the second derivative with respect to the 2nd term is at least[^37] $-{4\over 3}\delta \text{z}$, hence: $$\label{eq:3.11}
{\partial^2\bar{T}^{m,J}_k\over \partial f_{jh}^2}> 0 \Longrightarrow {4\over \delta \text{z}}-{4\over 3}\delta \text{z} > 0 \Longrightarrow \delta \text{z} < \sqrt{3}~.$$ To minimize we assume $\delta z$ small enough in order to satisfy and have a parabolic structure along the direction of any variable. The string tension action and its discretized form are positive quantities with the extremum solution of being the minimum point of the action, therefore following the parabolas along the direction of any variable downward should lead us to this minimum point. In order to do this in a systematic way $R$ random points are generated in the $N\times (m-1)$ dimensional space of discretized variables and starting from each random point the multivariable function is *minimized to width* $w$ (i.e. it is minimized to a point such that moving $w$ in either direction along any variable $f_{lp}$ and keeping other variables fixed gives a higher value for the action). This process is continued by dividing $w$ in half and *minimizing to width* ${w\over 2}$ and further continued to *minimization to width* ${w\over 2^n}$ at the $n$-th step until the difference between the string tension value at step $n$ and $n-1$ is sufficiently small. Let $X_n$ denote the random variable for the value of the string tension at step $n$ obtained by this minimization process. The *minimization error* (i.e.$| \text{min}\bar{T}^{m,J}_k - \langle X_n\rangle |$) in minimizing the multivariable function is reduced to the desired accuracy if the following quantities are sufficiently small:
\[eq:3.12\] i) [\_n ]{} , ii) |X\_n-X\_[n-1]{}| ,&&
where $\sigma_n$ is the standard deviation of $X_n$ and $\langle X_n\rangle$ denotes the average of $X_n$.
The same analysis described above is done for $2m$ number of partitions and the *discretization error* (i.e. $| \text{min}\bar{T}^{m,J}_k - \text{min} \bar{T}^{\infty,J}_k|$) of the string tensions is reduced to the desired accuracy if the difference between the string tension values obtained for $m$ number of partitions and $2m$ number of partitions is small enough. We consider the difference $|\text{min}\bar{T}^{m,J}_k - \text{min} \bar{T}^{2m,J}_k|$ as an estimate for the *discretization error* of the string tension values obtained for $2m$ number of partitions.
The boundary value number $\text{z} = J$ is assumed large enough to ensure the *truncation error* (i.e. $|\text{min} \bar{T}^{\infty,J}_k - \text{min} \bar{T}^{\infty,\infty}_k(=\bar{T}_k)|$) is small enough. An upper bound estimate for the truncation error is given by \[sec:A1\]: $$\label{eq:3.13}
|\text{min} \bar{T}^{\infty,J}_k-\bar{T}_k|<2| \text{min} \bar{T}^{\infty,J}_{k1} - \text{min}\bar{T}^{\infty,J}_{k2}|.$$ The total error estimate in minimizing is given by: $$\label{eq:3.14}
\begin{split}
\text{Total Error} & = \text{Min. E.} + \text{Dis. E.} + \text{Trunc. E.}
\\
& = |\langle X_n\rangle -\langle X_{n-1}\rangle | +{\sigma_n \over \sqrt{R}}+|\text{min}\bar{T}^{m,J}_k- \text{min}\bar{T}^{2m,J}_k|
\\
&+ 2|\text{min} \bar{T}^{\infty,J}_{k1} - \text{min}\bar{T}^{\infty,J}_{k2}|~.
\end{split}$$ The analysis of the errors defined above is discussed at length in Appendix \[errorappendix\].
Numerical value of ${k}$-string tensions in dYM {#sec:3.4}
-----------------------------------------------
The numerical values of obtained by the minimization method above with their corresponding errors are listed in Table \[table:1\] below.[^38] Since the minimum value in always lies below the numerical values obtained in a numerical minimization procedure the upper bound estimate for the error has been indicated with a minus sign only.\
[|c|c|c|c|c|c|c|c|c|c|]{}
--------- ---
$SU(N)$ k
--------- ---
: The numerical values, , of half $k$-string tensions for gauge groups ranging from $SU(2)$ to $SU(10)$. The upper bound estimate for error is $-0.006$.[]{data-label="table:1"}
&1&2 &3&4&5&6&7&8 &9\
2 & 5.6576&-&-&-&-&-&-&-&-\
3 & 6.5326&6.5326&-&-&-&-&-&-&-\
4&6.8583&8.0006&6.8583&-&-&-&-&-&-\
5&7.0140&8.6602&8.6602&7.0140&-&-&-&-&-\
6&7.1001&9.0168&9.5547&9.0168&7.1001&-&-&-&-\
7&7.1526&9.2318&10.0744&10.0744&9.2318&7.1526&-&-&-\
8&7.1868&9.3713&10.4051&10.7192&10.4051&9.3713&7.1868&-&-\
9&7.2104&9.4670&10.6292&11.1455&11.1455&10.6292&9.4670&7.2104&-\
10&7.2273&9.5355&10.7882&11.4434&11.6491&11.4434&10.7882&9.5355&7.2273\
The minimization method was carried out for $J=14.0$, $m=100$ and $m=200$ number of partitions and the initial width was $w=1$. The number of random points $R$ generated initially is $R=24$. The multivariable function for $m=100$ was minimized to width $w \over 2^n$ for $n=20$. For $m=200$, was minimized to step $n=20$ for $SU(2 \leq N \leq 7)$ and to step $n=22$ for $SU(8 \leq N \leq 10)$. The numerical values listed in Table \[table:1\] refer to the numbers obtained with $m=200$ number of partitions rounded to the fourth decimal. A comparison of the known analytical results for $SU(2)$ () and $SU(3)$ () half $k$-string tensions with the numerical results is made in Table \[table:2\].
Numerical value Analytical (exact) value
--------- -------------------- --------------------------------
$SU(2)$ $5.6576_{- 0.006}$ ${8/\sqrt{2}}\approx 5.6569$
$SU(3)$ $6.5326_{-0.006}$ ${16/ \sqrt{6}}\approx 6.5320$
: $SU(2)$ & $SU(3)$ numerical and analytical half $k$-string tensions.[]{data-label="table:2"}
he same minimization process and error analysis used to derive the $SU(2)$ and $SU(3)$ half $k$-string tensions was utilized for the higher gauge groups.
A discussion of the results shown in Table \[table:1\], especially regarding the $k$-scaling of string tensions and the large-$N$ limit will be given in Section \[sec:5\].
String tensions in dYM: perturbative evaluation {#sec:4}
===============================================
Here, we will rederive the half $k$-string tensions in Table \[table:1\] by a perturbative evaluation of the saddle point. We stress that this is not an oxymoron and that, indeed, we will be using (resummed) expansions and only Gaussian integrals to compute a nonperturbative effect.
In order to explain the main ideas, we briefly summarize them now, in an attempt to divorce them from the many technical details given later. Our starting point is the partition function of the Wilson loop inserted dual photon action for a fundamental weight $\mu_k$ from Section \[sec:2.4.1\]: $$\label{eq:4.1}
Z^{\eta} =\int [D\sigma] \text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x}\{{1\over {2L}}{g^2 \over 8\pi^2}({\nabla \sigma - \nabla \eta \mu_k })^2- \tilde{\zeta} \;\overset{N}{\underset{j=1}{\sum}}\text{cos}(\sigma_j-\sigma_{j+1})\})~.$$ For a Wilson loop in the $\text{y}_1, \text{y}_2$ plane, $\eta(\text{x}) = \underset{\text{A}}{\int} \text{dy}_1\text{dy}_2 {\text{x}_3 \over 2|\text{x} -\text{y}|^3}$ with $\text{y} = (\text{y}_1,\text{y}_2, 0)$ and “A” stands for the area of the rectangular Wilson loop $R \times T$ where the integral is being evaluated. We will rewrite in a form appropriate for a perturbative evaluation. Defining ${1\over \beta} \equiv {\tilde{\zeta} \over m_{\gamma}^3}$, rescaling $\text{x}_l \rightarrow {1 \over m_{\gamma}} {\hat{\text{x}}_l}$, $\text{y}_l \rightarrow {1 \over m_{\gamma}} \hat{\text{y}}_l$ ($l = 1,2,3$) with $m^2_{\gamma} = {8 \pi^2 \over g^2} L \tilde{\zeta}$ and expanding the cosine term (neglecting the leading constant term) we have: $$\label{eq:4.222}
Z^{\eta} =\int [D\sigma] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l \sigma)^2 - (\mu_k)_j\partial_l \sigma_j \partial_l \eta + {1 \over 2} (\partial_l \eta)^2 {\mu_k}^2 + {1 \over 2} (\sigma_j-\sigma_{j+1})^2 -{1 \over 4!} (\sigma_j-\sigma_{j+1})^4 + ... \})~,$$ with $\sigma_{N+1} \equiv \sigma_1$ and an implicit sum over $j=1,..,N$ and $l = 1,2,3$. We note that based on and for the leading $NL \Lambda$ dependence of $\beta$ we have $\beta \sim (NL \Lambda)^{b_0 \over 2}$ with $b_0>0$ given by . Thus, in the regime of validity $NL \Lambda \rightarrow 0$ of the semiclassical expansion $\beta \rightarrow 0$ and the partition function (\[eq:4.222\]) can be evaluated using the saddle point approximation, which was done numerically in Section \[numericsection\] and analytically in this Section.
We shall present details for both $SU(2)$, in Section \[su2analyticsection\], and $SU(N)$, in Section \[suNanalyticsection\] below, but begin by explaining the salient points of the analytic method here. For this purpose consider for an SU(2) gauge group. Following steps from equations to we obtain: $$\label{eq:4.71}
Z^{\eta}_{g^4} =\int [Dg] \text{exp}(-\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g)^2 + {m^2 \over 2} g^2 + \beta \lambda g^4 + {1 \over 2}({b \over 2\pi})^2 (\partial_l \eta)^2 \}) \text{exp}(+ {b \over \sqrt{\beta} } \underset{\text{A}}{\int} d\hat{\text{x}}_1 d\hat{\text{x}}_2 \partial_3g)~.$$ The differences between (\[eq:4.71\]) and (\[eq:4.222\]) are that: [*i.*]{}) the integration variable, the single dual photon of $SU(2)$, is now called $g$, [*ii.*]{}) nonlinear terms higher than quartic are discarded in order to simply illustrate the procedure, [*iii.*]{}) arbitrary dimensionless constants are introduced: mass parameter $m$, quartic coupling $\lambda$ and the boundary coefficient $b$ in (\[eq:4.71\]). Naturally the values of $m, \lambda$ and $b$ are determined by the original action in (\[eq:4.222\]) (or see below equation ), but it is convenient to keep them general in order to organize the expansion. As we explain below, we perform a combined expansion in $\beta$ to all orders, and in $\lambda$ to any desired order.[^39]
To explain the procedure, from it can be seen that the $\beta$ parameter is similar to an $\hbar$ parameter and in (\[eq:4.71\]) the fields have been rescaled by this parameter for a perturbative evaluation of the saddle point. A rescaling of fields by a parameter will not change how an expansion in that parameter behaves therefore the expansion in $\beta$ is similar to an $\hbar$ expansion. In the limit of an infinite Wilson loop the $\beta$ expansion will be organized in the following way: $$\label{eq:betaexpansion}
e^{- \hat{R} \hat{T}\{ {1 \over \beta}S_0(\lambda) + f_{1\text{-loop}}(\lambda) + \beta f_{2\text{-loop}}(\lambda) + ... \} }~ ,$$ $S_0$ is the one-dimensional saddle-point action, $f_{1\text{-loop}}$, $f_{2\text{-loop}}$, etc correspond to the summation of the one loop, two loop, etc diagrams. For brevity we have only included a $\lambda$ dependence although generally they depend on $\lambda$, $m$ and $b$. In this Section we will be only concerned with the leading saddle point result of order $1 \over \beta$ in the exponent of . To carry this out we expand the Wilson loop exponent and the $g^4$ term in and look for connected terms of order $1 \over \beta$. These will be the terms that exponentiate to produce the saddle point action in . As an example consider the Wilson loop exponent expanded to second order: ${1 \over 2!}({b \over \sqrt{\beta} } \underset{\text{A}}{\int} d\hat{\text{x}}_1 d\hat{\text{x}}_2 \partial_3g)^2$. When evaluated using the free massive propagator it is a connected diagram of order $1 \over \beta$, combined with the evaluation of the non-connected terms ${1 \over 4!}({b \over \sqrt{\beta} } \underset{\text{A}}{\int} d\hat{\text{x}}_1 d\hat{\text{x}}_2 \partial_3g)^4$, etc it would exponentiate to produce the term $- {\hat{R} \hat{T} \over \beta}S_0(\lambda = 0)$ in the exponent of . The odd terms in the expansion of the Wilson loop exponent vanish due to an odd functional integral. Similarly higher order contributions in $\lambda$ to the saddle point action can be evaluated. The order $\lambda$ contribution comes from the exponentiation of the connected diagram involving one $g^4$ term and four Wilson loop terms which is of order $1 \over \beta$: $ - \beta \lambda \int_{{\rm I\!R}^3} d^3{\text{x}} g^4$ ${1 \over 4!}({b \over \sqrt{\beta}} \int d\text{x}_1 d\text{x}_2 \partial_3 g)^4$. Terms of order $\lambda^2$, etc in the expansion of the saddle point action in can also be evaluated perturbatively which would result in the perturbative expansion of the saddle point in $\lambda$ (or more precisely ${ \lambda b^2 \over m^2}$) as in . Clearly the large value of $1 \over \beta$ causes no problem for these exponentiations since the radius of convergence of an exponential function is infinite. At every order in this combined expansion, we are faced with the calculation of Gaussian integrals only—hence the “perturbative evaluation” in the title of this Section.
In this paper, we compute the leading-order contributions to the Wilson loop expectation value that behave as $$\label{wilsonperturbative}
e^{- {\hat{R} \hat{T}\over \beta}( a_1 + a_2 \lambda)}~,$$ where ${\hat{R} \hat{T}}$ is the dimensionless area of the Wilson loop, defined in Section \[su2analyticsection\], and $a_1$( $= S_0(\lambda = 0)$), $a_2$ are numerical coefficients that we compute (for $SU(2)$ we will evaluate a few higher order corrections as well).
Setting $\lambda =0$ in (\[wilsonperturbative\]) corresponds to ignoring non-linearities and is equivalent to a calculation of the saddle point action using the Gaussian approximation for the dual photon action. This was previously done in the 3d Polyakov model in [@Antonov:2003tz; @Anber:2013xfa]. However, as noted in [@Anber:2013xfa] and also follows from our results, the neglect of nonlinearities introduces an order unity error in the string tension. On the other hand, incorporating even only the leading quartic nonlinearity and setting $\lambda$ equal to the value that follows from (\[eq:4.222\]) at the end of the calculation leads to a significantly better agreement with the exact analytic or numerical data. One explanation for this is that the value of the saddle point functions approach zero quickly from its boundary value at $\text{x}_3 = 0$ which for $SU(2)$ is $\pi$ and for higher gauge groups is less than $\pi$ therefore the non-linearities will be suppressed. We show this in great detail in Appendix \[sec:C\] for a wide range of $N$ and $k$. Here, to illustrate the utility of the method, in Table \[table:71\], we only list the results for the $k=1$-string tension for gauge groups $SU(2)$—$SU(10)$, obtained via the method explained above and keeping the quartic nonlinearity only. A look at Table \[table:71\] shows that the convergence to the numerical (or exact analytic, when available) result is evident.[^40]
$SU(N)$ $a_1$ $a_2 \lambda$ $a_1 + a_2 \lambda$ Num. value
--------- -------- --------------- --------------------- -------------------
2 9.870 -2.029 7.841 8.000 [(exact)]{}
3 11.396 -2.343 9.053 9.238 [(exact)]{}
4 11.913 -2.396 9.517 9.699
5 12.150 -2.410 9.740 9.919
6 12.277 -2.415 9.862 10.041
7 12.355 -2.417 9.938 10.114
8 12.405 -2.418 9.987 10.163
9 12.439 -2.418 10.021 10.196
10 12.463 -2.418 10.045 10.221
: Comparison of $N$-ality $1$ $k$-string tensions for $SU(2 \leq N \leq 10)$, obtained using the perturbative method explained here—leading contribution $a_1$ plus first subleading $a_2 \lambda$, from eq. (\[wilsonperturbative\])—with the results of the numerical study. To avoid confusion, we note that the exact analytic values for $SU(2)$ and $SU(3)$ in the dimensionless units used here are $8$ and $9.238$, respectively which agree with the numerical values listed in Appendix \[sec:C\]; see also the end of Section \[su2analyticsection\].[]{data-label="table:71"}
Our final comment is that, in principle, this approach would also allow one to compute corrections to the leading semiclassical result. In the case at hand, this would necessitate a more precise matching of the long-distance theory to the underlying gauge theory; needless to say, any detailed study of such corrections is left for future work.
Evaluation for $\text{SU(2)}$ {#su2analyticsection}
-----------------------------
We will first demonstrate the basic ideas of the method in the simpler case of $SU(2)$ which an analytic solution to the saddle point is available, hence a direct comparison can be made with the perturbative evaluation. Defining $g_1 \equiv (\sigma_1 - \sigma_2)/\sqrt{2}$ and $g_2 \equiv (\sigma_1 + \sigma_2)/\sqrt{2}$ with $\mu_1 = (0.5,-0.5)$, for $SU(2)$ reduces to: $$\label{eq:4.3}
Z^{\eta} =\int [Dg] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g_1)^2 +{1\over 2}(\partial_l g_2)^2 - {1\over \sqrt{2}}\partial_l g_1 \partial_l \eta + {1 \over 4} (\partial_l \eta)^2 + {4 \over 2} (g_1)^2 -{8 \over 4!} (g_1)^4 + ... \})~.$$ From it is clear that $g_2$ only appears in the kinetic term hence can be neglected. In what follows we will neglect the higher order interactions and demonstrate how the method works for a $g_1^4$ interaction term only. We replace $g_1$ with $g$ and use general dimensionless parameters for the mass, the $g^4$ coupling constant and the coefficient of the Wilson loop terms: $$\label{eq:4.4}
Z^{\eta}_{g^4} =\int [Dg] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g)^2 + {m^2 \over 2} g^2 +\lambda g^4 - {b \over 2 \pi}\partial_l g \partial_l \eta + {1 \over 2}({b \over 2\pi})^2 (\partial_l \eta)^2 \})~.$$ For later use we note that the corresponding values of $m$, $\lambda$ and $b$ in are $m=2$, $\lambda = -{8 \over 4!}$ and $b = \sqrt{2} \pi$. Integrating by parts the Wilson loop term (linear term in $\partial_l g$) with: $$\label{eq:4.5}
\partial_l \partial_l \eta(\text{x}) = -2\pi \underset{\text{A}}{\int} \text{dy}_1\text{dy}_2 \partial_3 \partial^2 {1 \over 4 \pi |\text{x} -\text{y}|} = 2\pi \theta_A(\text{x}_1,\text{x}_2) \partial_3 \delta (\text{x}_3)$$ gives: $$\label{eq:4.6}
Z^{\eta}_{g^4} =\int [Dg] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g)^2 + {m^2 \over 2} g^2 +\lambda g^4 + b \partial_3 \delta (\hat{\text{x}}_3) \theta_A(\hat{\text{x}}_1, \hat{\text{x}}_2) g + {1 \over 2}({b \over 2\pi})^2 (\partial_l \eta)^2 \})~,$$ where $\theta_A(\hat{\text{x}}_1,\hat{\text{x}}_2)$ is $1$ on the Wilson loop area and zero otherwise. To evaluate perturbatively rescale $g \rightarrow \sqrt{\beta} g$: $$\label{eq:4.7}
Z^{\eta}_{g^4} =\int [Dg] \text{exp}(-\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l g)^2 + {m^2 \over 2} g^2 + \beta \lambda g^4 + {1 \over 2}({b \over 2\pi})^2 (\partial_l \eta)^2 \}) \text{exp}(+ {b \over \sqrt{\beta} } \underset{\text{A}}{\int} d\hat{\text{x}}_1 d\hat{\text{x}}_2 \partial_3g)~.$$ In what follows, we will drop the $hat$ on $\text{x}$; due to the rescaling made earlier it should be remembered that we are working with dimensionless variables.
We will first calculate the Wilson loop exponent using the quadratic terms (kinetic term + mass term) in . In expanding the exponential $\text{exp}(+ {b \over \sqrt{\beta} } \underset{\text{A}}{\int} \text{dx}_1\text{dx}_2 \partial_3g)$ the odd terms vanish due to an odd functional integral and the even terms will be organized in the form of an expansion of an exponent (hence they would exponentiate) therefore it would be sufficient to only evaluate the second order term:[^41] $$\label{eq:4.8}
\big\langle {b^2 \over 2 \beta } \underset{\text{A A}}{\iint}' \partial_3g \partial'_3g \big\rangle_{0} = {b^2 \over 2 \beta } \underset{\text{A A}}{\iint}' \partial_3 \partial'_3 P(\text{x}-\text{x}') = {b^2 \over 2 \beta } \underset{\text{A A}}{\iint}' \partial_3 \partial'_3 {\text{exp}(-m|\text{x}-\text{x}'|) \over 4 \pi |\text{x}-\text{x}'|}, \ \text{at} \ \text{x}_3= \text{x}'_3 = 0~.$$ The last expression can be evaluated as: $$\label{eq:4.9}
\underset{\text{A A}}{\iint}' \partial_3 \partial'_3 P(\text{x}-\text{x}') = \underset{\text{A A}}{\iint'} \{ (-\partial^2 + m^2)P(\text{x}-\text{x}') + (\partial^2_1+\partial^2_2)P(\text{x}-\text{x}') - m^2P(\text{x}-\text{x}') \}~.$$ For a Wilson loop in the $\text{x}_1, \text{x}_2$ plane we can bring the second term on the right hand side of on the boundaries of the Wilson loop using the identity: $$\label{eq:4.10}
\begin{split}
\underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} P(\text{x}-\text{x}') & = \{ \underset{\text{A A}}{\iint'} dS^l dS'^l \partial_n\partial'_n - \underset{\text{A A}}{\iint'} dS^l dS'^k \partial_k\partial'_l \} P(\text{x}-\text{x}') \\ & = - \underset{\text{A A}}{\iint'} d^2\text{x} d^2\text{x}' \{ \partial_1^2 + \partial_2^2 \} P(\text{x}-\text{x}')~.
\end{split}$$ Here, $b(\text{A})$ stands for the boundary of the Wilson loop area.
Using relations , , and noting that $P(\text{x}-\text{x}')$ is the Greens function of the operator $-\partial^2 + m^2$ we have: $$\label{eq:4.12}
\big\langle {b^2 \over 2 \beta } \underset{\text{A A}}{\iint}' \partial_3g \partial'_3g \big\rangle_{0} ={b^2 \over 2 \beta } \{ \delta(0) \hat{R}\hat{T} - \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} P(\text{x}-\text{x}') - {m^2} \underset{\text{A A}}{\iint'} P(\text{x} - \text{x}') \}~.$$ The subscript of zero of the expectation value refers to it being evaluated using the free theory Lagrangian (i.e. at $\lambda =0$). The first term and the infinite part of the second term on the right hand side of would cancel with the infinite parts of the ${1 \over 2} ({b \over 2 \pi})^2(\partial_l \eta)^2$ term in to give a finite perimeter law for the Wilson loop and the third term on the right hand side of would give rise to an area law in the large area limit. Evaluating the ${1 \over 2} ({b \over 2 \pi})^2(\partial_l \eta)^2$ term in using and similar methods used to evaluate gives:
\[eq:4.13\] \_[[IR]{}\^3]{} d\^3[1 2]{}([b 2]{})\^2 (\_l )\^2 & = - [1 2]{}([b 2]{})\^2 \_[[IR]{}\^3]{} d\^3 () 2\_A(\_1,\_2) \_3 (\_3) = -[b\^2 2]{} \_A d\^2\_A d\^2 \^2\_3 [1 4 | -|]{}\
& = [b\^2 2]{} { (0) - \^[l k]{}[1 4 | -|]{} } .&&
Further, , and give:
$$\label{eq:4.14'}
\hspace{-0.7cm} {\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {m^2 b^2 \over 2} \underset{\text{A A}}{\iint'} P(\text{x} - \text{x}') + {b^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~.$$
In the limit that the area of the Wilson loop goes to infinity the first term on the right hand side of can be evaluated explicitly. Consider a Wilson loop with $\hat{R} = \hat{T} \equiv a$. Rescaling $\text{x}_k \rightarrow a \text{x}_k$, $\text{x}^{\prime}_k \rightarrow a \text{x}^{\prime}_k$ for $k = 1, 2$ and considering the limit $a \rightarrow \infty$ we have:
\[eq:4.11\] ’P(-’) & = a\^3 d\^2 d\^2’ [(-am|-’|) 4 |-’|]{}\
& = a\^3 d\^2 2\^\_0 rdr [(-am) 4 ]{}\
&= [12m]{}(-m|\_3 - ’\_3|) a\^2 = [R T 2 m]{} .
To arrive at the result above, we noted that in the limit of a large Wilson loop area ($a \rightarrow \infty$) due to the exponential suppression the main contribution to the $d^2\text{x}'$ integral comes from a small circle with radius $r \sim {1 \over a}$ centered at the point $\text{x} = (\text{x}_1, \text{x}_2 , 0)$ in the Wilson loop. Therefore it can be seen that the exact value of the $d^2\text{x}'$ integral in this limit would be given when the $dr$ integral is evaluated from zero to infinity. This would imply that the $d^2\text{x}'$ integral is independent of $\text{x}$, hence the $d^2\text{x}$ integral would be a trivial one over a unit square.
Using , in the limit of a large Wilson loop becomes: $$\label{eq:4.14}
{\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {mb^2 \over 4} \hat{R}\hat{T} + {b^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~.$$ contains an area law term (first term in the exponent) and a perimeter law term (last two terms in the exponent). We note that without the use of the perturbative saddle point method the evaluation of the perimeter law term, due to the complicated behaviour of the saddle point solution near the boundaries of the Wilson loop, would have been a difficult task. The perimeter law term in is a finite quantity and proportional to $\text{Log}(a)a$ (for $a = \hat{R} = \hat{T}$) hence negligible compared to the area law term in the limit $a \rightarrow \infty$. Due to this and the fact that our main focus in this Section is the area law term we will drop this term in what follows. Another point worth mentioning, as will be seen in what follows, is that in the limit of $a \rightarrow \infty$ only the area law term in will receive $\lambda$ corrections.
The saddle point equation of motion of is given by: $$\label{eq:4.15}
\partial^2 g = m^2 g + 4 \lambda g^3 + b \partial_3 \delta (\text{x}_3) \theta_A(\text{x}_1, \text{x}_2)~.$$ For large Wilson loops, far from the boundaries of the Wilson loop the saddle point solution to obeys . For regions outside the Wilson loop the solution is zero. For regions close to the boundaries the saddle point solution will be more complicated and gives the perimeter law contribution in . For regions interior to the Wilson loop far from the boundaries the solution only depends on $\text{x}_3$ with a discontinuity of $b$ at $\text{x}_3 = 0$ and gives the area law contribution in . The corresponding one dimensional problem is given by: $$\label{eq:4.16}
{d^2 \over d\text{x}^2_3} h = m^2 h + 4 \lambda h^3 \ \ \ \ \ \ h(0^+) = {b / 2} , \ \ \ \ h(0^-) = -{b / 2}~.$$ The discontinuity should be equally split above and below the Wilson loop in order to give the lowest action. The solution to for $\lambda = 0$ is given by: $$\label{eq:4.17}
h(\text{x}_3) = \left \{
\begin{array}{c}
{b \over 2} \text{exp}(-m\text{x}_3) \ \ \ \text{for} \ \ \ \text{x}_3 > 0 \\
-{b \over 2} \text{exp}(m\text{x}_3) \ \ \ \text{for} \ \ \ \text{x}_3 < 0 \\
\end{array}
\right .$$ The action of this solution is: $$\label{eq:4.18}
S[h] = \int^{+\infty}_{-\infty} dx_3 \{ {1 \over 2} ({dh \over dx_3})^2 + {1 \over 2} m h^2 \} = {m \over 4}b^2 , \ \ \ \ \text{at} \ \ \lambda = 0~,$$ which is the same as the coefficient of the area law term in . This demonstrates the validity of the perturbative saddle point method in producing the corresponding action of the saddle point boundary value problem. In order to further verify this method we will evaluate the saddle point action for a nonzero $\lambda$ and compare it with the corresponding analytic solution. We expand the exponential of the $g^4$ term in . The order $\lambda$ term contracts with the fourth order term in the expansion of the Wilson loop exponent: $$\label{eq:4.19}
\big\langle - \beta \lambda \int d^3\text{x} g^4 ({b \over \sqrt{\beta}})^4{1\over 4!} (\int_A d^2\text{y} \partial_{\text{y}_3} g)^4 \big\rangle_{0,C} = -{4! \over 4!} {\lambda \over \beta} b^4 \int d^3\text{x} \prod^{4}_{i=1} \int_A d^2\text{y}^i-\partial_{\text{x}_3} P(\text{x}-\text{y}^i)~.$$ The subscript $C$ refers to the connected contribution. It has to be reminded that we will only be interested in connected terms of order $1 \over \beta$ since these terms would exponentiate to produce the series expansion of the saddle point action in . In the limit of $a \rightarrow \infty$ (after rescaling the variables $\text{x}_k \rightarrow a \text{x}_k, \text{y}^i_k \rightarrow a \text{y}^i_k$ for $k =1, 2$) due to the exponential suppression the $\text{x}_1$ and $\text{x}_2$ components of $\text{x}$ will be restricted to the Wilson loop and the main contribution to the integrals would come from a small circle of radius $\sim {1 \over a}$ centred at $(\text{x}_1, \text{x}_2,0)$. Following similar steps as we have:
\[eq:4.20\] \_A d\^2 (-\_[\_3]{} P(-)) & = -\_[\_3]{} \_A d\^2 P(-) -\_[\_3]{} [1 2m]{}(-m|\_3|)\
& = [(\_3) 2]{} (-m |\_3|) .&&
Then becomes: $$\label{eq:4.211}
\big\langle - {\lambda \over \beta} {b^4 \over 4!} \int d^3\text{x} g^4 (\int_A d^2\text{y} \partial_{\text{y}_3} g)^4 \big\rangle_{0,C} = - {1 \over \beta}{\lambda b^4 \over 16m} \Bbbk_1 a^2 = - {1 \over \beta}{\lambda b^4 \over 32m} \hat{R}\hat{T}~,$$ with $\Bbbk_1$ given by: $$\label{eq:4.22}
\Bbbk_1 \equiv \int^{+\infty}_{-\infty} d\text{x}_3 \text{E}(\text{x}_3)^4 = {1 \over 2}, \ \ \ \text{E}(\text{x}_3) \equiv \text{sign}(\text{x}_3) \text{exp}(- |\text{x}_3|)~.$$ We note that the $d \text{x}_1d\text{x}_2$ integral would be a trivial one over a unit square and hence only an integral over $d\text{x}_3$ would remain. In a similar way the $\lambda^2$ term can be evaluated. This term would contract with the sixth order term in the expansion of the Wilson loop exponent:
\[eq:4.23\] (d\^3 g\^4)\^2 ([b ]{})\^6[16!]{} (\_A d\^2 \_[\_3]{} g)\^6 \_[0,C]{} = &[16 6! 2 6!]{} [\^2 ]{} b\^6 d\^3d\^3’ \^[3]{}\_[i=1]{} \_A d\^2\^i\_[\_3]{} P(-\^i)\
& P(-’)\^[3]{}\_[i=1]{} \_A d\^2’\^i\_[’\_3]{} P(’-’\^i) .&
Rescaling the variables ($\text{x}_k \rightarrow a \text{x}_k , ... $ for $k = 1, 2$), considering the limit $a \rightarrow \infty$, using and we have: $$\label{eq:4.24}
\big\langle {\lambda^2 b^6 \over 2 \beta 6!} (\int d^3\text{x} g^4)^2(\int_A d^2\text{y} \partial_{\text{y}_3} g)^6 \big\rangle_{0,C} = {1 \over \beta}{8 \lambda^2 b^6 \over 128} {\Bbbk_2 \over m^3} a^2 = {1 \over \beta}{\lambda^2 b^6 \over 16} {1 \over 24 m^3} \hat{R}\hat{T}~,$$ with $\Bbbk_2$ given by:
\[eq:4.25\] \_2 d\_3d’\_3 (\_3)\^3 (-|\_3-’\_3|) (’\_3)\^3 = [1 24]{} .
Higher order terms in $\lambda$ can be calculated similarly. The $\lambda^n$ term contracts with the $2n + 2$ order term in the expansion of the Wilson loop exponent. For $n > 3$ there would be more than one way of contracting the connected diagrams hence the evaluation would be more complicated but possible in principle
Now we will directly solve for the saddle point and compare the result with the above expressions. is the one dimensional problem of interest. This is the motion of a particle moving in a potential $V(h) = -({m^2 \over 2}h^2 +\lambda h^4)$. Therefore the total energy is a constant of motion ${1\over 2}({dh \over d\text{x}_3})^2 + V(h) = C$. The minimum action corresponds to when $C=0$ therefore ${1\over 2}({dh \over d\text{x}_3})^2 =-V(h)$:
\[eq:4.29\] S\[h\] & = \^[+]{}\_[-]{} d\_3 { [1 2]{} ([dh d\_3]{})\^2 - V(h) } = 2\^[0]{}\_[b 2]{} [d\_3 dh]{} dh { - 2V(h) } = 2m \_[0]{}\^[b 2]{} dh h .&&
We expand the square root[^42] and evaluate the integral term by term. We also multiply the action by a negative sign to take into account the negative in the exponent. We find:
\[eq:4.30\] -S\[h\] & = \_[0]{}\^[b 2]{} dh { -2mh + -[2m]{}h\^3 + (-1)\^n [2 \^n m\^[2n-1]{}]{} [(2n-3)!! n!]{}h\^[2n+1]{} }\
& = -m [b\^2 4]{} - [b\^4 32 m]{} +\
& [= -[mb\^2 4]{} { 1 + [1 2]{} [b\^2 4 m\^2]{} - } ]{} . &&
The order $\lambda$ and $\lambda^2$ terms in match with the coefficients of ${\hat{R}\hat{T} \over \beta}$ in and respectively. This further demonstrates the validity of the perturbative evaluation of the saddle point. [The verification to higher orders in $\lambda$ ($n \geq 3$) can be made if the corresponding diagrams are evaluated.]{}
The higher order terms in ($g^6_1, g^8_1, ...$) and their cross terms with each other can also be evaluated similarly.
Next we will compare the $SU(2)$ $k$-string result of the perturbative evaluation of the saddle point to next to leading term, with the exact result of $SU(2)$ $k$-string in Table \[table:2\] of Section \[sec:3.4\]. The exact $SU(2)$ saddle point area law is given by , :
\[eq:4.32\] [ Z\^ Z\^[0]{} ]{} = (- T\_1 RT) = (-[2 |[T]{}\_1 ]{} ) = (- [8 ]{} ), , , 0 ,
where $R = {1 \over m_{\gamma}} \hat{R}$, $T = {1 \over m_{\gamma}} \hat{T}$, $\beta = {m^3_{\gamma} / \tilde{\zeta}} $ and, from Table \[table:2\] of Section \[sec:3.4\], $\bar{T}_1 = {8 \over \sqrt{2}}$.
Using and the perturbative saddle point method gives:
\[eq:4.33\] [ Z\^ Z\^[0]{} ]{} = (- [1 ]{}([mb\^2 4]{} + [b\^4 32m]{} + ... ) ) = (- [1 ]{}( 7.84 + ... ) ), , , 0 .&&
From the comment below equation , the values of $m=2$, $\lambda = - {8 \over 4!}$ and $b = \sqrt{2} \pi$ have been replaced. This shows the convergence of the $SU(2)$ perturbative saddle point method result to the exact value obtained by a direct calculation of the saddle point.
Evaluation for $\text{SU(N)}$ {#suNanalyticsection}
-----------------------------
Having shown how the method works for $SU(2)$, in this Section we will evaluate the $k$-string tensions perturbatively to next to leading order for $SU(N)$. We start from (recall ): $$\label{eq:4.2333}
Z^{\eta} =\int [D\sigma] \text{exp}(-{1 \over \beta}\int_{{\rm I\!R}^3} d^3\hat{\text{x}}\{{1\over 2}(\partial_l \sigma)^2 - (\mu_k)_j\partial_l \sigma_j \partial_l \eta + {1 \over 2} (\partial_l \eta)^2 {\mu_k}^2 + {1 \over 2} (\sigma_j-\sigma_{j+1})^2 -{1 \over 4!} (\sigma_j-\sigma_{j+1})^4 + ... \})~.$$ The mass term in can be diagonalized. Let $(\sigma_j-\sigma_{j+1})^2 = \sigma^{T} A \sigma$ (for $j = 1, ...,N$) where $A$ is the following $N \times N$ matrix for $N \geq 3$: $$\label{eq:4.34}
A_{ij} = 2 \ \text{for} \ i=j \ , \ A_{ij} = -1 \ \text{for} \ |i - j| = 1 \ , \ A_{1N} = A_{N1} = -1 \ , \ A_{ij} = 0 \ \text{otherwise}~,$$ while for ${SU(2)}$: $$\label{eq:4.35}
A_{11} = A_{22} = 2 \ \text{and} \ A_{12} = A_{21} = -2~.$$ The matrix $A$ is symmetric and can be diagonalized by an orthogonal transformation $D$, explicitly $D^TD = I$, $A = D \Lambda D^T$. $D = (v_1 \ v_2 \ ... \ v_N)$ with $v_q$ the eigenvectors of $A$. This gives $(\sigma_j-\sigma_{j+1})^2 = g^{T} \Lambda g$ with $g = D^T \sigma$. $A$ has an eigenvalue[^43] of zero corresponding to the eigenvector $v^{T}_N \equiv ({1 \over \sqrt{N}}, ... , {1 \over \sqrt{N}})$. The corresponding field $g_N = (\sigma_1 + ... + \sigma_N)/ \sqrt{N}$ would be a massless component which decouples from the rest of the fields and hence can be neglected. We will now express the higher order interaction terms in a form convenient for a perturbative expansion. For this define the matrix $B$ as follows: $$\label{eq:4.36}
B_{ij} = 1 \ \text{for} \ i=j \ , \ B_{ij} = -1 \ \text{for} \ j = i + 1 \ , \ B_{ij} = 0 \ \text{otherwise}~.$$ Defining $h_q \equiv \sigma_q - \sigma_{q+1}$ for $q = 1, ..., N-1$ we have $BDg = B\sigma = \big( \begin{array}{c} h \\ \sigma_N \end{array} \big )$. Also define the top left $(N-1) \times (N-1)$ block of the matrix $BD$ as $K \equiv (BD)_{N-1 \times N-1}$. Since the first $N-1$ elements of the last column of $BD$ are zero[^44] this gives: $K_{pq} g_q = h_p \equiv \sigma_p - \sigma_{p+1}$ for $p = 1, ..., N-1$. Using the previous notations and definitions, can now be rewritten as: $$\label{eq:4.37}
\begin{split}
Z^{\eta} =\int [Dg] \text{exp}( &-{1 \over \beta}\int_{{\rm I\!R}^3} d^3{\text{x}}\{ {1\over 2}(\partial_l g_N)^2+{1\over 2}(\partial_l g_q)^2 + {1 \over 2} \Lambda_q g^2_q - (\mu_k)_j D_{jq}\partial_l g_q \partial_l \eta \\ & + {1 \over 2} (\partial_l \eta)^2 {\mu_k}^2 -{1 \over 4!}\underset{p} {\sum}(K_{pq}g_q)^4 -{1 \over 4!}(\underset{p} {\sum}K_{pq}g_q)^4 + ... \})~,
\end{split}$$ where a summation over $p,q$ and $l$ is implicit. Note that $(\mu_k)_j D_{jN} = 0$, hence the massless mode $g_N$ completely decouples from the rest of the modes and interactions. Integrating by parts the Wilson loop term and rescaling $g_q \rightarrow \sqrt{\beta} g_q$, we cast it, similar to (\[eq:4.7\]), into a form appropriate for a perturbative evaluation of the saddle point:
\[eq:4.38\] Z\^ =( &-\_[[IR]{}\^3]{} d\^3{ [12]{}(\_l g\_N)\^2+[12]{}(\_l g\_q)\^2 + [1 2]{} \_q g\^2\_q + [1 2]{} ([b\_q 2]{})\^2 (\_l )\^2\
& +(K\_[pq]{}g\_q)\^4 + (K\_[pq]{}g\_q)\^4 + ... })(+ [b\_q ]{} \_A d\_1d\_2 \_3 g\_q) .
Here, we defined $b_q = 2\pi (\mu_k)_j D_{jq}$ and $\lambda = - {8 \over 4!}$. To evaluate the Wilson loop exponent using the quadratic terms, we follow steps similar to the ones leading from to . We obtain the analogue of for $SU(N)$: $$\label{eq:4.39'}
{\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b_q=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {\Lambda_q b_q^2 \over 2} \underset{\text{A A}}{\iint'} P_q(\text{x} - \text{x}') + {b_q^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P_q(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~,$$ where $P_q(\text{x}-\text{x}') = {\text{exp}(-\sqrt{\Lambda_q} |\text{x}-\text{x}'|) / (4 \pi |\text{x}-\text{x}'|} )$ and with an implicit summation over $q$. In the limit of a large Wilson loop area ($\hat{R}, \hat{T} \rightarrow \infty$), eq. , following a similar calculation as in , reduces to: $$\label{eq:4.39}
{\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b_q=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {\sqrt{\Lambda_q}b^2_q \over 4} \hat{R}\hat{T} + {b_q^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P_q(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~,$$ Using the explicit form of the eigenvectors given in Footnote \[eigenvectorfootnote\], we can analytically show that confining strings have finite tension in the large-N limit, despite the vanishing mass gap.[^45] In particular, for $k=1$ strings we find the infinite-N limit $a_1 = \lim\limits_{N \rightarrow \infty} \sum\limits_{q=1}^{N-1}{1\over 4} \sqrt{\Lambda_q}b^2_q = 4 \pi$, consistent with the results from Table \[table:71\]. For further comments on the large-N limit, see Sections \[sec:compare\] and \[sec:largeN\]. Appendix \[sec:appxproduct\] discusses the large-N limit of the string tensions in product representations and gives more details on the analytic calculations using the leading-order saddle point method of this Section.
Next, we evaluate the leading corrections to this result for $SU(N)$ and compare the values obtained with the numerical results in Table \[table:1\]. The integrals we need to do are the generalization of (\[eq:4.19\]) from the $SU(2)$ calculation:
\[eq:4.40\] &\_ - d\^3 { (K\_[pq]{}g\_q)\^4 + (K\_[pq]{}g\_q)\^4 } [14!]{}([b\_q ]{} \_A d\^2 \_[\_3]{} g\_q)\^4 \_[0,C]{}\
& = -[1 ]{}[4! 4!]{} [8]{}P\_[q\_1q\_2q\_3q\_4]{} { \^4\_[i=1]{} K\_[pq\_i]{}b\_[q\_i]{} + \^4\_[i=1]{} K\_[p\_iq\_i]{}b\_[q\_i]{} } , &&
and a summation over $p$, $q_i$, $p_i$ ($i =1,2,3,4$) from $1, ..., N-1$ is implicit. The quantities $P_{q_1q_2q_3q_4}$ are given by: $$\label{eq:4.41}
P_{q_1q_2q_3q_4} \equiv \int d^3 \text{x} \prod^4_{i=1} \int_A d^2\text{y}^i \partial_{\text{y}^i_3}P_{q_i}(\text{x} -\text{y}^i)~.$$ In the limit that $a \rightarrow \infty$ ($a = \hat{R} = \hat{T}$) using we have: $$\label{eq:4.42}
P_{q_1q_2q_3q_4} \overset{a \rightarrow \infty}{=} {a^2 \over 16} \bar{P}_{q_1q_2q_3q_4} \ \ \text{with} \ \ \bar{P}_{q_1q_2q_3q_4} = \int^{+\infty}_{-\infty} d\text{x}_3 \prod^4_{i=1} \text{E}(\sqrt{\Lambda_{q_i}} \text{x}_3)~,$$ hence $\text{I}_{\lambda}$ becomes: $$\label{eq:4.43}
\text{I}_{\lambda} \overset{a \rightarrow \infty} {=} -{1 \over \beta}{\lambda \over 128}\bar{P}_{q_1q_2q_3q_4} \{ \prod^4_{i=1} K_{pq_i}b_{q_i} + \prod^4_{i=1} K_{p_iq_i}b_{q_i} \} \hat{R}\hat{T}~,$$ where $K_{pq} = B_{pj}D_{jq}$ and $b_q = 2\pi (\mu_k)_j D_{jq}$ for $q, p = 1, ..., N-1$. The leading area law term in and its leading correction in need to be evaluated numerically with mathematical software. The results are summarized in Appendix \[sec:C\]; results for $k=1$ strings for $SU(3)-SU(10)$ were already shown in Table \[table:71\]; the inclusion of the leading order correction brings the numerical value closer to the numerical or analytic (for $SU(3)$) value.
$\mathbf{N}$-ality dependence and large-$\mathbf{N}$ behaviour of $\mathbf{k}$-strings in dYM {#sec:5}
==============================================================================================
In this Section we give a discussion on two main questions regarding the properties of $k$-string tensions: their $N$-ality dependence and large-$N$ behaviour in dYM theory. The $N$-ality of an irreducible representation of a gauge group $SU(N)$ refers to the number of boxes in the Young tableaux of the representation mod $N$ [@09] or the charge of the representation under the action of the center element $\text{exp}(-i{2 \pi \over N})\text{I}$ of the gauge group. It is believed that asymptotically the string tensions in a gauge theory depend only on the $N$-ality of that representation, see [@04]. This is due to the screening effect by gluons. A cloud of gluons would transform any charge in a representation with $N$-ality $k$ to its $k$-antisymmetric representation which carries the stable lowest energy k-string among different representations with the same $N$-ality $k$.
We will argue that this is also true in dYM theory and show that the asymptotic string tensions will only depend on the $N$-ality $k$ of the representation $k$. The screening by gluons, in the framework of dYM theory, is due to the pair production of $W$-bosons, an effect (in principle) calculable using weak coupling semiclassical methods.
We discuss qualitatively the role of the unbroken $\Z_N$ center symmetry in dYM for the confining string properties and contrast them to those in another theory with abelian confinement—Seiberg-Witten theory. We also derive an approximate analytic formula for k-string ratios in dYM theory for $N \sim 10$ and smaller and have a comparison with known scaling laws of k-string ratios.
In regards to their large-$N$ behaviour we show that dYM $k$-string ratios favour even power corrections similar to the sine law scaling and derive the leading terms in the $1/N$ expansion of $k$-string ratios in dYM theory. At the end we will argue that at large $N$ $k$-strings are not necessarily free in gauge theories; in other words, $T$$_{\text{k}}$ can remain smaller than $kT$$_{\text{1}}$ in the large-$N$ limit.
$N$-ality dependence
--------------------
### Asymptotic string tensions depend only on the $N$-ality of the representation {#sec:5.1.1}
The expectation value of the Wilson loop for charges in a representation $r$ with $N$-ality $k$ of $SU(N)$ evaluated using the low energy effective theory in dYM theory in the limit of $\hat{R}, \hat{T} \rightarrow \infty$ and $\beta (= {m^3_{\gamma} \over \tilde{\zeta}})\rightarrow 0$ using and is given by: $$\label{eq:5.1}
\langle W_r(R,T) \rangle = \sum^{d(r)}_{i = 1} \text{exp}(- T^i_rRT) = \sum^{d(r)}_{i = 1} \text{exp}(- {2 \bar{T}^i_{r} \over \sqrt{2} \beta}\hat{R}\hat{T})~,$$ where $d(r)$ refers to the dimension of the representation $r$ and $\bar{T}^i_{r}$ is given by a similar expression as but with $\mu_k$ replaced by the weight $\mu^i_r$ of representation $r$ with $N$-ality $k$:
\[eq:5.2\] [|[T]{}\^i\_r]{}= \_[0]{}\^[+]{} d{([f ]{})\^2+\[1-(f\_j-f\_[j+1]{})\]}, f(+ ) = 0, f(0) = \_r\^i .&&
Expression is the sum of exponential of area laws. The leading exponential in the limit of large $\hat{T}$ and $\hat{R}$ would give the string tension for charges in representation $r$ with $N$-ality $k$. Any representation of $SU(N)$ with $N$-ality $k$ contains the fundamental weight $\mu_k$ as one of its weights (Appendix \[sec:B1\]). Therefore in order to show that string tensions would only depend on the $N$-ality of the representation $r$ of the group $SU(N)$ we have to show that the lowest string tension action in corresponds to boundary conditions dictated by the fundamental weight $\mu_k$ among all the weights $\mu_r^i$ ($i=1, ..., d(r)$) of the representation $r$. Any weight of a representation of $SU(N)$ can be obtained from the highest weight by lowering with the simple roots [@09] and as noted above any representation with $N$-ality $k$ contains $\mu_k$ as one of its weights. Therefore any weight of a representation $r$ with $N$-ality $k$ can be obtained from the fundamental weight $\mu_k$ by adding or subtracting the simple roots.
We will now qualitatively (but convincingly) argue that adding or subtracting any simple root from $\mu_k$ would result in boundary conditions that would give a value for the minimum of the action which is equal to[^46] or more than the value obtained by boundary conditions of $\mu_k$.
The saddle point solutions $f_j$ of for $\mu_r^i = \mu_k$ start from $\pi (\mu_k)_j$ at $z = 0$ and decrease or increase monotonically to zero at $z = + \infty$. From the form of $\mu_k$ given in (\[eq:3.2\]), one sees that there are two discontinuities, as a function of $j$, in the boundary conditions for $f_j$. These occur between $j=N$ and $j=1$, since $(\mu_k)_N = -k/N$ and $(\mu_k)_1 =1 - k/N$, and between $j=k$ and $j=k+1$, as $(\mu_k)_k = 1- k/N$ and $(\mu_k)_{k+1} = -k/N$. These two discontinuities in the boundary conditions make the corresponding terms in the cosine potential $1 - \text{cos}(f_N - f_1)$ and $1 - \text{cos}(f_k - f_{k+1})$ to start from 2 at $z = 0$ and reach $0$ at $z = + \infty$ (this is in contrast to all the other terms, which start from $0$ at $z=0$ and reach $0$ again at $z = \infty$). Thus, for boundary conditions given by $\mu_k$ we would summarize the behaviour of $f_j$’s as follows. For the kinetic term in we would have $k$ functions $f_1, ..., f_k$ that start from $\pi (1 - k/N)$ at $z = 0$ and reach $0$ at $z = + \infty$ and $N - k$ functions $f_{k+1}, ..., f_N$ that start from $- \pi k/N$ at $z = 0$ and reach $0$ at $z = + \infty$. For the potential term, since only the difference between the $f_j$’s enters the cosine, the $1 - \text{cos}(f_N - f_1)$ and $1 - \text{cos}(f_k - f_{k+1})$ terms start at 2 at $z = 0$ and reach 0 at $z= + \infty$, while the rest of the terms start from 0 at $z = 0$ and reach 0 at $z= + \infty$.
Now let us ask how this picture would change if simple roots are added or subtracted from $\mu_k$. The picture of the kinetic term will either remain the same (k functions start from $\pi (1-k/N)$ and $N - k$ functions starting from $- \pi k/N$) or it would become worse (and thus increase the value of action) in a way that the boundaries values at $z=0$ become higher and result in an increase in the kinetic term (since it is the square of the derivative of the functions). The same is true for the cosine term—any addition or subtraction of the simple roots from the weight $\mu_k$ would either not change the picture of the cosine term or it would make it worse (increase the value of action) in a way that we would have more than 2 discontinuities in the boundaries that would result in more than two terms of the potential term having to start from 2 at $z = 0$, or we would still have two discontinuities but the boundary conditions would have become larger and the cosine terms corresponding to these discontinuities would oscillate between 2 and zero more than once. Both our numerical results and the simple variational ansatz of Section \[bagmodelsection\] confirm this picture.
### Comparing different abelian confinements: strings in dYM vs. softly-broken Seiberg-Witten theory {#sec:compare}
The two most-studied examples where confinement of quarks becomes analytically calculable within quantum field theory are softly-broken Seiberg-Witten theory on $\R^4$ and QCD(adj) with massive or massless adjoint fermions on $\R^3 \times \S^1$. This paper is devoted to the study of dYM theory, which belongs to the second class, QCD(adj) with massive adjoint fermions. Semiclassical calculability in dYM is achieved, as mentioned many times, by taking the $N L \Lambda \ll 1$ limit.
In both dYM and Seiberg-Witten theory confinement is “abelian:”[^47] the confining strings form in a regime where $W$-bosons are not relevant and the dynamics of confinement is described by a weakly-coupled abelian gauge theory. In Seiberg-Witten theory, this is the dual magnetic gauge theory on $\R^4$, while in dYM it is the long-distance theory on $\R^3\times \S^1$—the theory of the dual photons discussed at length in earlier Sections. In both cases, the confining dynamics involves magnetically charged—and thus nonperturbative from the point of view of the electric gauge theory—objects: the magnetic monopoles or dyons in Seiberg-Witten theory condense to break the magnetic gauge symmetry, while in dYM, the proliferation of monopole-instantons in the vacuum (which should not really be called “condensation,” the title of [@Unsal:2007jx] nothwithstanding) leads to the expulsion of electric flux.[^48] We shall see, in the next Section, that the physics of confinement in dYM has a flavor very similar to the picture of the QCD vacuum underlying the MIT Bag Model.
Here, we want to stress two aspects in which dYM confinement is distinct from Seiberg-Witten theory that have not been much discussed in the literature:
1. The presence of a global unbroken $\Z_N$ (zero-form[^49]) center symmetry in dYM vs. the fact that the Weyl group in Seiberg-Witten theory is broken [@15]. The unbroken $\Z_N$ symmetry has implications for the “meson” and “baryon” spectra of the theory, as we explain further in this Section.
2. The abelian large-$N$ behaviour: confining string tensions remain finite in dYM in the large-$N$, fixed $\Lambda N L \ll 1$ limit. This is different from their behaviour in the analogous limit of Seiberg-Witten theory, where the string tensions vanish along with the mass gap [@15]. For further discussion, see Section \[sec:largeN\].
Here we concentrate on the first point above: the unbroken $\Z_N$ center symmetry in dYM. In the long-distance theory, in the $N$-dimensional basis of dual photons we are using, this symmetry appears as a clock symmetry, taking $\sigma_i \rightarrow \sigma_{i+1}$, with $N+1 \equiv 1$. In gauge-variant terms, the action of the $\Z_N$ center symmetry resembles that of an unbroken cyclic subgroup of the Weyl symmetry of $SU(N)$, as can be seen by noting that it cyclically interchanges the $N$ monopole-instantons associated with the simple and affine root of the Lie algebra.[^50] On the other hand, in Seiberg-Witten theory, the Weyl group is spontaneously broken, as pointed out long ago [@15].
The different global symmetry realization has interesting implications for the nature of confining strings in the two theories. To illustrate the differences it suffices to consider the confinement of fundamental quarks in $SU(3)$. In dYM theory, there are degenerate “mesons” composed of quarks (introduced as static sources) of the three different colors, of weights $\nu_1 = \mu_1$, $\nu_2 = \nu_1 - \alpha_1$, and $\nu_3=\nu_2 - \alpha_2$, respectively. These mesons are confined by distinct flux tubes related by the $\Z_N$ global symmetry action (the action of $\Z_N$ on the weights of the $SU(N)$ fundamental representation is to cyclically permute them). Furthermore, the fluxes carried by these three strings add up to zero, so one can form a “baryon,” where the “baryon vertex” is a junction of three strings (a domain wall junction), as illustrated on Figure \[fig:dymstring\].
![Strings between static quarks of different colors (denoted by color circles) in $SU(3)$ dYM theory. [*Left panel:*]{} $Q_i \overline{Q}_i$ mesons in $SU(3)$ dYM are degenerate, due to the unbroken $Z_3$ center symmetry. There are three flux tubes carrying fluxes $\nu_i$ ($i=1,2,3$), one for fundamental quarks of each weight (color). [*Right panel:*]{} A “baryon vertex” in dYM is a 3-domain wall junction, which exists due to the vanishing total flux $\nu_1+\nu_2+\nu_3=0$. Similar structures persist for arbitrary number of colors in dYM theory.](dYMstrings.pdf){width="\textwidth"}
[\[fig:dymstring\]]{}
In contrast, in $SU(3)$ Seiberg-Witten theory, there are two $U(1)$ magnetic gauge groups broken by the monopole condensate, giving rise to two Abrikosov-Nielsen-Olesen (ANO) vortices. The flux of one vortex is proportional to $\mu_1$ and confines quarks in the highest weight of the fundamental representation. The other flux tube carries electric flux proportional to $\mu_2$ (the second fundamental weight of $SU(3)$) and confines quarks in the highest weight of the two-index antisymmetric representation (anti-quarks, for $SU(3)$). There is no third flux tube. The picture of “mesons” in $SU(3)$ Seiberg-Witten theory that results is shown on Figure \[fig:SWstrings\]: the lowest and highest weights of the fundamental quarks are confined by the two ANO flux tubes, while the middle-weight quark is confined by two flux tubes: one of flux $\mu_2$ and an anti-flux tube of flux $\mu_1$. The lack of a third flux tube becomes especially noticeable when baryons are considered: baryons in Seiberg-Witten theory are “linear molecules” only, as shown on Figure \[fig:SWstrings\]. This difference persists and becomes more pronounced for higher rank $SU(N)$ gauge groups.[^51]
![Strings in $SU(3)$ Seiberg-Witten theory. [*Left panel:*]{} $Q_i \overline{Q}_i$ mesons for different color quarks are non degenerate, due to existence of only two ANO flux tubes (denoted by lines with a single or double arrow) carrying electric fluxes $\mu_1$ and $\mu_2$ (notice that $\nu_2 = \mu_2-\mu_1$), respectively. [*Right panel:*]{} Only linear baryons exist in Seiberg-Witten theory. Similar pictures hold for any number of colors.](SWstrings.pdf){width="\textwidth"}
[\[fig:SWstrings\]]{}
As the $SU(3)$ example illustrates, the different symmetry realizations in dYM and Seiberg-Witten theory have implications for the spectrum of mesons and baryons. We shall not pursue this further here, but only note that in dYM one can add dynamical massive quarks and the meson, baryon (as well as glueball) spectra can be studied within weakly-coupled field theory, revealing many unusual and surprising features discussed in [@Aitken:2017ayq].
### An approximate form of $
\text{k}$-string ratios and the MIT Bag Model {#bagmodelsection}
Here, we shall derive a naive analytic upper bound for the half $k$-string tensions in Table \[table:1\] by approximating the integral in in a simple manner. We shall arrive at a simple $k$-string tension scaling law, which is in good agreement with the available data, as described further below. We shall also elaborate on the similarity between confinement in dYM and the MIT Bag Model of the Yang-Mills vacuum.
We begin by repeating the dimensionless half $k$-string tension action (recall e.g. eq. (\[eq:5.2\])):
\[eq:3.41\] [|[T\_k]{}]{}= \_[0]{}\^[+]{} d’{([ ’]{})\^2+ \_[j=1]{}\^N\[1-(f\_j-f\_[j+1]{})\]}, (+ ) = 0, (0) = \_k .
Here $\vec{f}$ represents the $N$-dimensional vector of dual photon fields (whose components are summed explicitly in the second term; we omit the arrows in what follows) and the boundary conditions at the origin and at infinity are the ones appropriate for static sources in the highest weight of the $k$-index antisymmetric tensor representation.
A simple variational ansatz for the half domain wall extremizing (\[eq:3.41\]) can be obtained by approximating the first term in the action as a linear function connecting the boundary value $\pi \mu_k$ at $z = 0$ to zero at a finite positive $z = J$. The second term is approximated by simply taking its value at $z=0$ (i.e. with $f=\pi \mu_k$) multiplied by $J$; in other words, the fields $f_i$ are taken in the vacuum (where the potential term in (\[eq:3.41\]) vanishes) outside a region of width $J$ which represents the thickness of the flux tube in our variational ansatz. As the form of $\mu_k$ and the potential term imply, for $f= \pi \mu_k$ only two terms in the sum of $N$ cosines contribute a factor of $2$ each, giving rise to second term in (\[eq:5.3\]), while the remaining $N-2$ terms do not contribute.[^52] Collecting everything, using the explicit form of the fundamental weight $\mu_k$ from (\[eq:3.2\]), we obtain the string tension as a function of the one variational parameter $J$, the flux tube thickness: $$\label{eq:5.3}
\bar{T_k}^{naive}(x) = J\{ ({\pi {N-k \over N}\over J})^2k + ({\pi {k \over N}\over J})^2(N-k) \} + 4J = {\beta_k \over J} + 4J~,$$ where the parameter $$\beta_k \equiv (\pi {N-k \over N})^2k + (\pi {k \over N})^2(N-k) = \pi^2 {(N-k)k \over N} ,
\label{betak}$$ is proportional to the quadratic Casimir of the $k$-index antisymmetric tensor. Extremizing with respect to $J$ gives $J_{k, \text{ext}} = {\sqrt{\beta_k} \over 2}$. The value of the string tension at the extremum point is: $$\label{eq:5.4}
\bar{T}_{k}^{\text{naive}} = 4\pi \sqrt{(N-k)k \over N}~.$$ Although the relation is only a naive upper bound estimate for the $k$-strings in Table \[table:1\], its ratio with the fundamental ($k=1$) $k$-string gives a good fit to the ratios of $k$-strings of Table \[table:1\]: $$\label{eq:5.5}
{\bar{T}_{k}^{\text{naive}} \over \bar{T}_{1}^{\text{naive}}} = \sqrt{(N-k)k \over N-1}~.$$
The relation is, in fact, known as the “square root of the Casimir” scaling law for $k$-string ratios. It was first seen to arise in the MIT Bag Model of the QCD vacuum a long time ago [@13].[^53] As far as we are aware, dYM theory is the only known example where this “square root of Casimir” k-string scaling has been seen to arise within a controlled approximation in quantum field theory.
We shall now discuss the physics behind (\[eq:5.3\]) and (\[eq:5.5\]) and will argue that the similarity of strings in dYM to those in the MIT Bag Model is not an accident. The first term on the r.h.s. of (\[eq:5.3\]) represents the gradient energy of the $\sigma$-field. Recall that the duality relation (\[dualityrelation\]) maps spatial gradients of the dual photon field to electric fields in the perpendicular direction (i.e. to electric flux going from the quark to the antiquark, which are here taken at infinite separation). Thus, the $\beta_k \over J$ term represents the electric field energy cost (per unit length) for a flux tube of thickness $J$. The coefficient $\beta_k$, the total electric flux, is determined by the sources—quarks in the $k$-index antisymmetric tensor representation—and is proportional to the quadratic Casimir of that representation, as in the classical MIT Bag Model of the confining string.[^54] Naturally, in order to minimize its energy, the electric flux tube wants to expand, i.e. maximize $J$—in a perturbative vacuum, the chromoelectric field would relax to the dipole field of the quarks. The second term on the r.h.s. of (\[eq:5.3\]), equal to $4 J$, represents the energy cost per unit length to “expelling the vacuum” and replacing it with electric flux in a region of width $J$. This term represents the “volume energy cost,” proportional to the bag constant parameter of the MIT Bag Model. In dYM, the vacuum is a monopole-antimonopole medium which abhors electric flux and wants to minimize $J$; the “bag constant” in dYM is not a model parameter, but is determined by the fugacity of monopole-instantons, ultimately fixed by the underlying gauge theory. The compromise between the two contributions to the energy results in $k$-strings of width $J_{k, \text{ext}} =\sqrt\beta_k/2$ and tensions given in (\[eq:5.4\]).
As we already alluded to, the agreement between the dYM and MIT Bag Model $k$-string tensions is not accidental. In the MIT Bag Model, the major assumption is that the chromoelectric fields within the confines of the (presumably small) bag can be treated classically, owing to asymptotic freedom. The “bag constant" of the YM vacuum, characterizing its abhorrence of electric flux, is introduced as a model parameter. In dYM, both the classical treatment of the Cartan electric fields and the expulsion of electric flux are dynamical features arising from the judiciously chosen deformation of YM theory and are justified in the $N \Lambda L \ll 1$ limit.
Finally, we note that while the physical picture in dYM is similar to that in the bag model, the “square root of Casimir” scaling of $k$-string tensions discussed here is not exact in dYM, as it results from a simple variational estimate. It is only an upper bound on the string tensions in dYM, see the following Section and, in particular, Figure \[fig:1\].
### Comparison with known scaling laws {#sec:5.1.4}
It is known that the asymptotic string tensions depend only on the $N$-ality $k$ of the representation of the confined charges, hence they are often referred to as the $k$-strings. Different models of confinement make different predictions for the ratios of $k$-string tensions. The main ones are the sine law and Casimir scaling. We also include the square root of Casimir scaling in the list below, due to its similarity with the $k$-string ratios in dYM theory for N $\sim$ 10 and smaller:
\[eq:5.6\] & : [T\_kT\_1]{} = [()]{},\
& : [T\_kT\_1]{} = [k (N-k) N-1]{},\
& : [T\_kT\_1]{} = .&&
In field theory calculations, usually the corresponding $k$-string tension is calculated to leading order in a small parameter expansion. It has to be noted that the above relations correspond to the leading order result in that expansion and, in each case, are subject to corrections.
![Comparison of $SU(10)$ $k$-string ratios of with dYM $k$-string ratios, labeled by “dYM”, to other $k$-string tension laws. The Sine law labeled by “sin”, the Casimir scaling by “cas”, and scaling with the Square root of the Casimir scaling by “sqrtcas”. From the known theoretical models predicting different scalings of $k$-string tensions, the ones in dYM are closest to the MIT Bag Model “square root of Casimir” $k$-string tension law. There is a clear physical reason behind this similarity, explained in Section \[bagmodelsection\].](figure_1.png){width="\textwidth"}
[\[fig:1\]]{}
The Sine law is found in Seiberg-Witten theory [@15], in MQCD [@16], in three-dimensional SU(N) gauge theories with massless Dirac or Majorana fermions [@61], and in some AdS/CFT-inspired models [@17]. Casimir scaling of string tensions refers to the relation between string tensions $T_r/T_F = C_2(r)/C_2(F)$, where $C_2(r)$ and $C_2(F)$ are the quadratic Casimir of representation $r$ and the fundamental representation, respectively ($T_r$ denotes the string tension for charges in representation $r$). This relation can be derived from the “dimensional reduction” form of the Yang-Mills vacuum wave functional [@18], from the stochastic vacuum picture [@19], and from certain supersymmetric dual models [@20].
$SU(3)$ lattice simulations have shown scaling with the Casimir of the representation $C_2(r)$ with a good accuracy [@21]: it holds at intermediate distances ($\lessapprox$ 1 fm) but at larger distances (asymptotically) gluons screen the charges down to the representation of the same non-zero $N$-ality with the lowest dimensionality which carries the most stable lowest string tension—then $C_2(r)$ is replaced by the Casimir of the $k$-antisymmetric representation which leads to the Casimir scaling relation shown in ; notice however, that for $N=3$ $T_1 = T_2$. Lattice studies of $3$-dimensional YM theory seem to also favor Casimir scaling of $k$-string tensions ratios for gauge groups up to $SU(8)$ [@Bringoltz:2008nd], while studies of $4$-dimensional YM theory (for similar number of colors) appear to favor scaling in-between the sine and Casimir laws, see [@Lucini:2012gg] for references and discussion.
The various $k$-string ratios shown in are compared with dYM $k$-string ratios for $SU(10)$ in Figure \[fig:1\]. It is clear from the figure that the square root of Casimir scaling shows most similarity with the dYM $k$-string ratios. This scaling arises in the MIT Bag Model of QCD [@13] and the reasons for the similarity was discussed in Section \[bagmodelsection\].
Large-$\text{N}$ behaviour {#sec:largeN}
--------------------------
One feature of the abelian large-N limit in dYM was already mentioned: in the $N\rightarrow \infty$, fixed-$NL \Lambda \ll 1$ double scaling limit, the mass gap vanishes, but the string tensions stay finite. This large-N behaviour is quite different from a similar abelian large-N limit of Seiberg-Witten theory, where both the string tensions and mass gap vanish [@15]. Furthermore, as observed in [@Cherman:2016jtu], in the above double-scaling limit on $\R^3\times \S^1$, where the size of the dimension $L\rightarrow 0$ and the number of colors $N\rightarrow \infty$, with $NL$-fixed, in both super Yang-Mills and dYM, the infrared theory can be viewed as a theory “living” in an emergent latticized dimension, in a manner reminiscent of T-duality in string theory. This is a behaviour not quite expected of quantum field theory and clearly deserves a better understanding.
The results of this paper show the nonvanishing of the string tension in dYM at large N. In the remainder of this Section, we study the leading large-N corrections to k-string ratios and their large-N behaviour, for a range of N that includes exponentially large values, but does not strictly extent to infinity.
The reason for this restriction, already mentioned in Section \[summary\] of the Introduction, is that our analysis has neglected the fact that at large values of N, the virtual effects of the W-bosons become important, as there is a large number of them. In particular, W-boson loops induce mixing between the Cartan algebra photons (and hence between dual photons), which were not incorporated in our effective Lagrangian. Similar to the discussion of ref. [@Cherman:2016jtu] for sYM (using the calculations of refs. [@08; @Anber:2014sda]), we estimate that these mixing terms become important when $N$ becomes comparable to $N^* = 2 \pi e^{+ {24 \pi^2 \over (11 - 4 n_f)(N g^2)}}$. This exponentially large value of $N^*$ is the one that applies to massless adjoint QCD, and uses the computations in [@vito]. The corresponding calculation for dYM (adjoint QCD with massive adjoint flavors) has not yet been performed, but we expect the appearance of a similar exponentially large $N^*$. Studying the role of these corrections in dYM is an interesting task for future work, which will allow to further study the intriguing features of the abelian large-N limit.
### Leading large-$\text{N}$ terms {#sec:5.2.1}
In this Section, we derive the leading large-N corrections to k-string ratios in dYM theory for T$_2$/T$_1$ and T$_3$/T$_1$. We will show that the k-string ratios in dYM theory favour even power corrections in $1\over N$. For this we add noise[^55] of order $0.0005$, the typical value of error of dYM k-string ratios[^56] to the exact k-string ratios of the Casimir scaling and sine law, whose scaling behaviour is known, and analyze them along with the k-string ratios in dYM theory. From Figures \[fig:2\] and \[fig:3\] it can clearly be seen that the coefficient of the linear correction term in dYM k-string ratios similar to the sine law is suppressed (whereas for the Casimir scaling law it is of same order) compared to the constant or the coefficient of the second order term therefore it can be concluded that dYM $k$-string ratios similar to the sine law disfavour a linear correction term and favour even power corrections.
To find the leading term and leading correction term, we add noise of order $0.0005$ to the exact k-string ratios of the sine law for $\text{SU}(5\leq N \leq 10)$ to generate data with errors of order of the errors of the dYM data. Next we generate n = 1000 noised data for dYM and the sine law data with noise and make even power polynomial fits: $c_0 + c_2\text{x}^2 + ... + c_p\text{x}^{p}$ for $p = 2k, k \geq 0$, with $\text{x}=1/N$. The average and standard deviation of $c_0$ and $c_2$ give estimates for the values of these coefficients and their errors. We increase $p$ and make higher order polynomial fits until consistent results are reached. Tables \[table:3\] and \[table:4\] summarize the values obtained by this analysis. It can be seen that consistent results are obtained for $p = 6$ and $p = 8$ polynomial fits. In fact the values of the $p = 6$ column for the noised sine law data are in agreement with the exact coefficients of the $1 \over N$ expansion of the sine law k-string ratios as can be seen from . This is not limited to the sine law. Any other function with even power corrections shows a similar behaviour and the results for $c_0$ and $c_2$ coefficients for an even polynomial fit with $p=6$ would agree with the true values of the coefficients in its $1/N$ expansion (e.g. doing the same analysis for a cosine(x) function with $\text{x} = {1 \over N}$). So assuming dYM k-string ratios have only even power corrections, the values of the coefficients in the $p = 6$ column would be in agreement with the true values in dYM theory.
The following relations summarize the leading large $N$ corrections in sine, Casimir, square root of Casimir and dYM k-string ratios:
\[eq:5.7\] &: [(k[N]{}) / ([N]{}) ]{} = k + (k/6 - k\^3/6)\^2 ([1 N]{})\^2 + ... ,\
&: [k(N -k) / (N-1)]{} = k + (k - k\^2) ([1 N]{}) + ... ,\
&: = k\^[1 2]{} + [1 2]{}(k\^[1 2]{} - k\^[3 2]{}) ([1 N]{}) + ... ,\
&: \_2 / \_1 = 1.347 0.001 + (-2.7 0.2) ([1 N]{})\^2 + ... ,\
& \_3 / \_1 = 1.570 0.001 + (-7.5 0.2) ([1 N]{})\^2 + ... .&&
{width="\textwidth"}
[\[fig:2\]]{}
{width="\textwidth"}
[\[fig:3\]]{}
p=2 p=4 p=6 p=8
------------------- --------------------- --------------------- ------------------- -------------------
$\text{c}_0$(sin) 1.9962 $\pm$ 0.0001 2.001 $\pm$ 0.0004 2.001 $\pm$ 0.001 1.998 $\pm$ 0.005
$\text{c}_2$(sin) -9.458 $\pm$ 0.006 -9.91 $\pm$ 0.03 -9.9 $\pm$ 0.2 -9 $\pm$ 1
$\text{c}_0$(dYM) 1.3482 $\pm$ 0.0001 1.3465 $\pm$ 0.0004 1.347 $\pm$ 0.001 1.347 $\pm$ 0.005
$\text{c}_2$(dYM) -2.822 $\pm$ 0.006 -2.65 $\pm$ 0.03 -2.7 $\pm$ 0.2 -3 $\pm$ 1
: $c_0$ and $c_2$ for even power polynomial fits of order p for $\text{T}_2 / \text{T}_1$ for noised ($\sim$ 0.0005) sine law data and dYM[]{data-label="table:3"}
p=2 p=4 p=6 p=8
------------------- --------------------- --------------------- ------------------- -------------------
$\text{c}_0$(sin) 2.9397 $\pm$ 0.0001 2.9984 $\pm$ 0.0004 3.000 $\pm$ 0.001 2.999 $\pm$ 0.005
$\text{c}_2$(sin) -33.334 $\pm$ 0.006 -39.17 $\pm$ 0.03 -39.4 $\pm$ 0.2 -39 $\pm$ 1
$\text{c}_0$(dYM) 1.5815 $\pm$ 0.0001 1.5682 $\pm$ 0.0004 1.570 $\pm$ 0.001 1.569 $\pm$ 0.005
$\text{c}_2$(dYM) -8.594 $\pm$ 0.006 -7.27 $\pm$ 0.03 -7.5 $\pm$ 0.2 -7 $\pm$ 1
: $c_0$ and $c_2$ for even power polynomial fits of order p for $\text{T}_3 / \text{T}_1$ for noised ($\sim$ 0.0005) sine law data and dYM[]{data-label="table:4"}
As a short summary of this Section, we argued that k-strings in dYM are not free at large N, i.e. $T_k/T_1 \ne k$, and leading corrections to $T_k/T_1$ are of order $1/N^2$. In the next Section, we discuss some theoretical questions behind these findings.
### Comments on free $\text{k}$-strings and large-$\text{N}$ factorization {#sec:5.2.2}
An often-discussed expected behaviour of $k$-strings at large $N$ is that they become free, meaning that the string tension with $N$-ality $k$ becomes $k$ times the tension of the fundamental $k=1$-string at large $N$ [@14; @10]. From the previous Section, in particular , it can be clearly seen that the k-string tensions in dYM theory show a different behaviour: $\underset{N \rightarrow \infty}{\text{lim}}\text{T}_2 = (1.347 \pm 0.001) \text{T}_1 < 2 \text{T}_1 $ and $\underset{N \rightarrow \infty}{\text{lim}}\text{T}_3 = (1.570 \pm 0.001) \text{T}_1 < 3 \text{T}_1$. The usual line of reasoning that leads to the conclusion that k-strings become free at large $N$ is based on large-$N$ factorization and assumes the commutativity of the large-$N$ and large Euclidean time $T$ limits. We will show that factorization and commutativity of limits should be treated more carefully.[^57]
e first briefly review the usual arguments that lead to free k-strings at large $N$:\
A correlator of two gauge invariant operators A and B can always be written as a factorized expectation value plus a connected expectation value. In the lattice strong coupling expansion and in perturbation theory in gauge theories it is known that the leading term in the large $N$ limit is the factorized one [@11]. Assuming a normalization $\langle \text{AB} \rangle\sim O(1)$ we have:
\[eq:5.8\] = + \_C , \~O(1), \_C \~O([1 N\^2]{}) . &&
In particular, we will apply this formula to the expectation value of a Wilson loop in the product representation:
\[eq:5.9\] & W\_ (U\_ ... U\_)= { (U\_ ... U\_) (U\_ ... U\_) }= W\_ W\_\
& W\_= W\_ W\_= W\_W\_+ W\_W\_\_C .&&
A subscript of a “square” (as in $W_{\square}$) refers to the fundamental representation. The product of the link matrices $U$ is being taken along a rectangular Wilson loop $R \times T$. To find the k-string tensions we take the large $T$ and $R$ limit and consider the leading exponential on the right hand side of . To consider the properties of the k-strings at large $N$ we also take the large $N$ limit. If the large $T$ and $R$ and large $N$ limits commute, then we can reverse the order of limits. Taking the large $N$ limit first makes the connected term vanish, then taking the large $T$ and $R$ limit we would find: $$\label{eq:5.10}\nonumber
\langle W_{\square \otimes \square}\rangle \sim \langle W_{\square}\rangle\langle W_{\square} \rangle, ~ \langle W_{\square \otimes \square}\rangle \sim \text{exp}(- T_2 RT), ~\langle W_{\square}\rangle \sim \text{exp}(- T_1 RT) ~~ \Longrightarrow~ ~T_2 = 2 T_1,$$ i.e. the result that the $k=2$-string tension is twice the fundamental string tension.
The line of reasoning represented above leads to the result that $k$ strings are “non-interacting” and would be correct if the large $T$ and $R$ and large $N$ limits commuted, which is not always true, as we discuss at length below (see [@62] for a discussion in a similar framework and [@Witten:1978qu] for a reminder that large-distance and large-$N$ limits’ non-commutativity has a long history). An important difference between the large area limit and large-$N$ limit relevant to their non-commutativity is the fact that the large area limit is taken in the same quantum field theory where as the large-$N$ limit is taken in different quantum field theories.
To study the general properties of field theories at large $N$, e.g. large $N$ factorization, one takes the large $N$ limit first but to study the asymptotic k-string tensions at large $N$, due to the non-commutativity of the large area and large N limits, one should not take the large $N$ limit first. For any $SU(N)$ theory, the proper way to find asymptotic $k$-string tensions at large $N$ is to first solve for the $k$-string tensions at fixed $N$, this is done by taking the large area $(RT)$ limit and considering the leading exponential in this limit. Then, once the asymptotic $k$-string tensions are determined for each $N$ from the coefficient of the area term of the leading exponential, the large-$N$ limit of $k$-string tensions can be taken.
The limits cannot be taken in reverse order as the leading exponential in the large area ($RT$) limit, which gives the $k$-string tension for the given value of $N$, can be suppressed in the large-$N$ limit compared to exponentials sub-leading in the large area limit. Let us first illustrate this important point in a toy example, similar to the way the non-commutativity of limits is realized in dYM. As the discussion of dYM is somewhat lengthy and slightly technical[^58] we prefer to first illustrate the result by the following example. Consider the function $$\label{acd}
g(A,N) = \text{exp}(-T_{N} A) + {1 \over N^p}~ \text{exp}(-T^{\prime}_{N} A)~, ~{\rm with} ~ p>0,$$ where $A$ stands in for the area of the Wilson loop. Let the large-$N$ limits of $T_N$ and $T'_N$, $\underset{N \rightarrow \infty}{\text{lim}}T^{\prime}_N = T^{\prime}$, $\underset{N \rightarrow \infty}{\text{lim}}T_N = T$, be such that $T^{\prime} < T$. For any large but fixed $N$ the leading term in the large-$A$ limit is $g(A,N) \sim \text{exp}(-T^{\prime} A)$ and for any large but fixed $A$ the leading term in the large-$N$ limit is $g(A,N) \sim \text{exp}(-TA)$. Therefore, if one is interested in the leading exponential in the large-$A$ limit, one should not take the large-$N$ limit first, as this will make the second term in (\[acd\]), which is leading in the large area limit, vanish. One would then find $g(A,N) \sim \text{exp}(-T A)$, which is an incorrect result for the leading exponential in the large-$A$ limit. A similar behaviour happens in dYM as we discuss in detail further below, see discussion after (\[eq:5.13’\]).
In what follows, we shall see that in the regime of parameters studied in this work, in particular in the framework of a bounded large-$N$ (see the comments in the beginning of Section \[sec:largeN\]), the leading exponential in the large $T$ and $R$ limits, which determines the k-string tensions, comes from the connected term, although it can be shown that for fixed $R$ and $T$ this term will be sub-leading in $N$ compared to the factorized term, similar to the toy example of eqn. (\[acd\]).
First, we argue how this can be seen more explicitly from the results of Section \[sec:5.1.1\]. There, it was argued that the lowest string tension action in the product representation of $N$-ality $2$ corresponds to boundary conditions on the dual photon fields determined by the fundamental weight $\mu_2$ ($\bar{T}^i_{\square \otimes \square}$ in relation for $\mu^i_{\square \otimes \square} = \mu_2$) and for a fundamental representation of unit $N$-ality it corresponds to $\mu_1$ ($\bar{T}^i_{\square}$ in relation for $\mu^i_{\square} = \mu_1$). Hence the leading exponential of the factorized term in the limit of large $\hat{T}$ and $\hat{R}$ is $\text{exp}(-2\times{2\bar{T}_1 \over \sqrt{2} \beta } \hat{R}\hat{T}) $ and the leading exponential for the Wilson loop in the product representation is $\text{exp}(-{2 \bar{T}_2 \over \sqrt{2} \beta } \hat{R}\hat{T})$. For large values of $N$ from equation we have $\underset{N \rightarrow \infty}{\text{lim}}2\bar{\text{T}}_2 = (1.347 \pm 0.001) 2\bar{\text{T}}_1 <2 \times 2 \bar{\text{T}}_1 $. Clearly, the factorized term can never produce this leading exponential which should, therefore, come from the connected term.
This result quoted above can also be obtained without referring to numerics, via the perturbative saddle point method developed in Section \[sec:4\], as shown in Appendix \[sec:appxproduct\].\
Next, we wish to verify the large $N$ factorization result in dYM and directly argue that, for a Wilson loop in the product representation the factorized term is leading and the connected term is sub-leading for large $N$. The discussion that we begin now becomes more transparent and explicit after reviewing the calculations of Appendix \[sec:appxproduct\].
Consider the expectation value of a Wilson loop in the product representation ${\square \otimes \square}$. Based on , for fixed but large $R$ and $T$ we have: $$\label{eq:5.13'}
\langle W_{{\square \otimes \square}}(R,T) \rangle \sim \sum^{d({\square \otimes \square})}_{h = 1} \text{exp}(- T^h_{{\square \otimes \square}}RT)~,$$ where $d({\square \otimes \square}) = N^2$ refers to the dimension of the product representation. In words, the expectation value of the Wilson loop in the product representation is given, in the abelianized dYM theory, by a sum of decaying exponentials, one for each weight $h$ of the product representation, with string tension $T^h_{{\square \otimes \square}}$ corresponding to each weight.
On the other hand, for a Wilson loop in the fundamental representation $\square$ we have: $$\label{eq:5.14'}
\langle W_{\square}(R,T)\rangle \sim \sum^{d(\square)}_{i = 1} \text{exp}(- T^i_{\square}RT) = N \text{exp}(- T_1RT)~.$$ Similar to (\[eq:5.13’\]), this is a sum of decaying exponentials, one for every weight of the fundamental representation, with the only simplification ocurring because of the unbroken $\Z_N$ center symmetry, ensuring that the string tensions for all weights of the fundamental representation have the same value $T_1$.
Now, let us study (\[eq:5.13’\]) in more detail. Our considerations from this point to eqn. are more qualitative than quantitatively rigorous (although, as already mentioned, they can be justified in the leading order perturbative evaluation of the saddle point, see Appendix \[sec:appxproduct\]). They carry similar flavor to our argument of Section \[sec:5.1.1\] that strings sourced by quarks with charges in the highest weight of the $k$-index antisymmetric representation have the smallest string tension. However, we find the considerations below quite suggestive and intuitive, supporting the large-$N$ vs. large-$RT$ limit subtlety.
The weights of the product representation are given by the sum of the weights of the fundamental representation in : $\mu^{h}_{{\square \otimes \square}} \equiv \mu^{(ij)}_{{\square \otimes \square}} = \mu^{i}_{\square} + \mu^{j}_{\square}$ for $1 \leq h \leq N^2$ and $1 \leq i, j \leq N$. These weights enter the boundary conditions of the string tension action . In what follows, we show that for large $N$ and for $|i - j| \gg 1$ and $|i - j| \ll N$ the string tensions of the product representation become approximately equal to two times the string tension of the fundamental representation at large $N$, i.e. $T^{h}_{\square \otimes \square} \equiv T^{(ij)}_{\square \otimes \square} \approx 2 T_1 $. As there are $O(N^2)$ such tensions, it will be concluded, after considering eq. , that $\langle W_{{\square \otimes \square}}(R,T) \rangle - \langle W_{\square}(R,T) \rangle \langle W_{\square}(R,T) \rangle < O(N^2)$ and therefore the connected term would be sub-leading in $N$.
Consider the fundamental string tension $T^q_\square$ given by with $r = \square$ and $q$ denoting one of the weights of the fundamental representation. The weight $\mu_\square^q$ is given in . Recall, from Section \[sec:3.3\], that the dual photon configuration extremizing the action of a given string interpolates between a value at the origin given by $\pi \mu^q_\square$ and zero at infinity. Since the $p$-th component of $\mu_\square^q$ is $(\mu_{\square }^q)_p = -{1 \over N} + \delta^{pq}$, for large values of $N$ all the components of $\sigma_a$, $1 \leq a \leq N$ at $z = 0$ approach zero except for the $q$’th component which approaches $\pi$. The fact that one component, namely $\sigma_q$, differs in its boundary conditions from the rest would result in a non-zero action for $T^q_\square$, otherwise if all components had the same boundary conditions (e.g. $-\pi/N$) at $z=0$ they would be linear functions interpolating between $-\pi/N$ at $z=0$ and zero at $z=J$ and when $J$ is taken to infinity would result in a zero action. This suggests that the main contribution to $T^q_\square$ would come from the components of $\sigma_a$ near[^59] the $q$’th (also, see Appendix \[sec:appxproduct\]). Conversely, the components farther away from the $q$’th component would approach a linear configuration, similar to the case when all boundary conditions were the same, in order to minimize the action as much as possible and will have negligible effect on the value of the string tension action $T^q_\square$, with their contribution being suppressed by a power of $1/N$.
A similar picture is true for $T^{(ij)}_{\square \otimes \square}$ with $\mu^h_{\square \otimes \square} = \mu^{(ij)}_{{\square \otimes \square}}$.[^60] Due to the $Z_N$ symmetry of the action without loss of generality we can take $i = [(N - \Delta_{ij}+1) /2]$ and $j = [(N + \Delta_{ij}+1)/2]$ with $\Delta_{ij} = |i - j| \neq 0$; the square brackets refer to the integer part. For large $N$, all components $(\mu^{(ij)}_{\square \otimes \square})_p$ approach zero, except for the $i$-th and $j$-th components, which approach $1$.
Consider now the calculation of the string tension $T^{(ij)}_{\square \otimes \square}$ for large $N$ with $|i - j| \gg 1$ and $|i - j| \ll N$. The components $\sigma_a$, $1 \leq a \leq N$ interpolate between $\pi (\mu^{(ij)}_{\square \otimes \square})_a$ at the origin and zero at infinity and therefore for large values of $N$ all the components would approach zero at $z=0$ except for $\sigma_i$ and $\sigma_j$ which approach $\pi$. Similar to the picture above for $T^q_\square$ it can be seen that the components of the dual photon fields $\sigma_a$ near the $\sigma_i$ and $\sigma_j$ components would make the main contribution to the string tension $T^{(ij)}_{\square \otimes \square}$. The components farther away from the $i$’th and $j$’th components would approach a linear configuration, similar to the case when all boundary conditions are the same, in order to minimize the action as much as possible and will have negligible effect on the values of the string tension action $T^{(ij)}_{\square \otimes \square}$, with their contribution being suppressed by a power of $1/N$. Next, we divide the string tension action of $T^{(ij)}_{\square \otimes \square}$ into two parts, one part associated with the components $\sigma_a$ for $1 \leq a \leq N /2$ and one for $N/2 < a \leq N$; at large-$N$ and $|i - j| \gg 1$, $|i - j| \ll N$, these actions become independent of each other. In each part, the components closer to the $i$’th and $j$’th components of $\sigma_a$, which are relevant to the string tension value and their boundary conditions are similar to the components near the $q$’th component of $\sigma_a$ for $T^q_\square$ for $SU([N/2])$.[^61] For large $N$, these string tensions approach—as per our numerical results of Table \[table:71\] or from the analytic study, recall paragraph after —a nonzero value $T_1$ with their differences suppressed by a power of $1/N$. From this observation, we conclude that for large $N$ and for $|i - j| \gg 1$ and $|i - j| \ll N$, $T^{(ij)}_{\square \otimes \square} \approx 2T_1$. Clearly, there are $O(N^2)$ such string tensions at large $N$.
On the other hand, the highest weight of the antisymmetric two index representation, $\mu_2$, see , which was argued and numerically found to give rise to the smallest $N$-ality two string tension, $T_2 < 2 T_1$, is obtained from $ \mu^{(ij)}_{{\square \otimes \square}}$, by taking $i = j+1$ (mod$N$). There are $O(N)$ such string tensions, including the $\Z_N$-center orbit of the highest weight of the antisymmetric two-index representation.
We now combine the results of the previous two paragraphs to conclude, recalling , that at large $N$ $$\label{eq:abd}
\langle W_{{\square \otimes \square}}(R,T) \rangle \sim \sum^{d({\square \otimes \square})}_{h = 1} \text{exp}(- T^h_{{\square \otimes \square}}RT)~ \approx O(N^2) \; e^{- 2 T_1 RT} + O(N) \;e^{- T_2 RT}.$$ The first term in the last expression above represents the contribution of the $O(N^2)$ string tensions of weights $ \mu^{(ij)}_{{\square \otimes \square}}$ with $|i - j| \gg 1$ and $|i - j| \ll N$. The second term is the contribution of the $O(N)$ $k=2$ strings in the $\Z_N$ orbit of the highest weight of the two-index antisymmetric representation. We now note that eqn. exactly mirrors the situation described and discussed earlier in eqn. , showing the subtlety of taking the large-$N$ vs. large-$RT$ limit. See also Appendix \[sec:appxproduct\], where is recovered using the leading-order perturbative saddle point, evaluated analytically for large-$N$.
The discussion in this Subsection demonstrates that in the framework of a bounded large $N$ studied in this work (recall the preamble[^62] of Section \[sec:largeN\]) large $N$ factorization would not necessarily imply free k-strings and the leading exponential in the large area limit can come from the connected term of the correlator of two Wilson loops in the fundamental representation that is sub-leading in $N$ compared to the factorized term. We remind the reader that although the connected term is sub-leading in $N$ it would still contribute to the k-string tensions at large $N$—since, as already stressed at the beginning of this Section, to find asymptotic k-string tensions, the large area limit must be taken first and the leading exponential in this limit should be considered. Thus, no matter how large $N$ is, the connected term, which contains the leading exponential in the large area limit, would contribute to the k-string tensions at large $N$.
### A comment on “holonomy-decorated” Wilson loops
Here, we want to make a point which gives additional justification of our emphasis to study $k$-strings of minimal tensions, corresponding to quark sources of a particular weight (e.g. $ \mu^{(ij)}_{{\square \otimes \square}}$, with $i = j+1$ (mod$N$) for $k=2$). So far, we only considered gauge invariant Wilson loop operators without insertions of the Higgs field (holonomy). In the small-$L$ abelianized regime of dYM theory, one can isolate the contribution of individual components of the fundamental quarks by inserting powers of the holonomy inside the trace defining the Wilson loop. This gives rise to Wilson loops in $\R^3$ “decorated” by loops winding around the $\S^1$, similar to the construction of [@Cherman:2016jtu; @Aitken:2017ayq]. The construction of these loops shows that the abelian strings of different tensions (due to quarks of a single weight) in product representations are physical, i.e. they are created by gauge invariant operators. We now define a “decorated” Wilson loop as follows. The fundamental representation holonomy around the $\S_1$ is $$\label{eq:pol1}
\Omega(x)_F = {\cal{P}} e^{i \oint\limits_{\S_1} A^a_4(x,x_4)t^a_F dx_4}~, \ \ \ \ \ \ a =1, ..., N-1.$$ The gauge invariant Wilson loop projecting on a single component of a quark field can then be written as $$\label{eq:wilson1}
W^k_F ={\rm tr}_F {\cal P} \left[{1 \over N} \sum\limits_{p=1}^N \omega_N^{-(N-k)p} (\Omega(x)_F)^p\right] \; e^{i \int\limits_{\text{x}}^{\text{x}} A_\mu d\text{x}^\mu} ~,$$ where $\omega_N = e^{i {2 \pi\over N}}$ and the integral $\int\limits_x^x$ is taken along a large $RT$ contour in $\R^3$, broken up at the point $x$ where the Higgs field is inserted. In the center symmetric vacuum at weak coupling $\Omega$ can be replaced by its vacuum expectation value, $\langle \Omega \rangle$, given (for brevity, shown below only for odd $N$ and recalling (\[a4vev\])), by $$\label{evs12}
\langle \Omega \rangle = {\rm diag}( \omega_N^{N-1}, \omega_N^{N-2},..., \omega_N, 1)~.$$ Hence, the holonomy insertion and discrete Fourier transform in project (the term in square brackets inside the trace in ) the Wilson loop to an abelian component corresponding to a source given by the $k$-th component (weight) of the fundamental quark (in the ordering of eigenvalues as in (\[evs12\])). Using , one can construct sources of various weights in product representations.
Derivation of $\mathbf{W}$-boson spectrum {#app:wboson}
=========================================
Consider two analogs of off-diagonal $SU(2)$ generators in $SU(N)$, namely $T_{(kl)}^1$ and $T_{(kl)}^2$ (analogs of $\tau^1/2$ and $\tau^2/2$ in $SU(2)$ respectively), $1 \leq k , l \leq N$, where $k \neq l$, refer to the row and column of the non-zero components of these generators. We will work out the quadratic [mass]{} terms associated with their corresponding gauge fields $A_i^1T_{(kl)}^1$ and $A_i^2T_{(kl)}^2$. The mass term comes from the $F^2_{i4}$ term in with $F_{i4} = \partial_i A_4 - \partial_4 A_i -i [A_i,A_4]$: $$\begin{aligned}
\label{eq:2.9}
\{ {1 \over 2g^2} \text{tr}F^2_{i 4}(x) \}_{\text{quad} , A_i^{1,2}} &=& {1 \over 2g^2}\{ \text{tr}(\partial_4 A^{1,2}_i)^2 - \text{tr}([A^{1,2}_i,A^{vev}_4])^2 \\ \nonumber
&& + 2i \text{tr}(\partial_4 A^{1,2}_i [A^{1,2}_i,A^{vev}_4]) \} ~,\end{aligned}$$ with $A_i^{1,2} = A_i^1T_{(kl)}^1 + A_i^2T_{(kl)}^2$. Noting that $[A^{1,2}_i,A^{vev}_4] = {2\pi |l-k| \over NL}i(A_i^2T_{kl}^1 - A_i^1T_{kl}^2)$, expanding each component in its Fourier modes using , and integrating over the compact $\text{x}_4$ direction we have:
\[eq:2.10\] \^L\_0 d\_4 { [1 2g\^2]{} F\^2\_[i 4]{}(x) }\_[ quad, A\_i\^[1,2]{}]{} & = [L 2g\^2]{} {([2 m L]{})\^2( A\_[i,m]{}\^1A\_[i,m]{}\^[1 ]{} + A\_[i,m]{}\^2A\_[i,m]{}\^[2 ]{} )\
& + ([2|l-k| NL]{})\^2 ( A\_[i,m]{}\^1A\_[i,m]{}\^[1 ]{} + A\_[i,m]{}\^2A\_[i,m]{}\^[2 ]{} )\
& +2i [2|l-k| NL]{}[2 m L]{} (A\^[1 ]{}\_[i,m]{}A\^[2]{}\_[i,m]{} - A\^[2 ]{}\_[i,m]{}A\^[1]{}\_[i,m]{}) } . &&
Expanding in real and imaginary parts of Fourier components we have:
\[eq:2.11\] \^L\_0 d\_4 { [1 2g\^2]{} F\^2\_[i 4]{}(x) }\_[ quad,A\_i\^[1,2]{}]{} & = [[L 2g\^2]{} { ([2 m L]{})\^2 \[ (A\_[i,m1]{}\^1)\^2 + (A\_[i,m2]{}\^1)\^2 + (A\_[i,m1]{}\^2)\^2 + (A\_[i,m2]{}\^2)\^2 \] ]{}\
& [+4 [2|l-k| NL]{}[2 m L]{} ( A\^[1]{}\_[i,m2]{}A\^[2]{}\_[i,m1]{} - A\^[2 ]{}\_[i,m2]{}A\^[1]{}\_[i,m1]{})]{}\
& , &&
with $A_{i,02}^1 = A_{i,02}^2 = 0$. The above mass terms can be diagonalized by defining the following fields: $$\label{eq:2.12}
\begin{split}
\bar{A}^1_{i,m} \equiv (A^{1}_{i,m1} + A^{2}_{i,m2}) / \sqrt{2}, \bar{A}^2_{i,m} \equiv (A^{1}_{i,m2} + A^{2}_{i,m1}) / \sqrt{2} \\ \bar{A}^3_{i,m} \equiv (A^{1}_{i,m2} - A^{2}_{i,m1}) / \sqrt{2} , \bar{A}^4_{i,m} \equiv (A^{1}_{i,m1} - A^{2}_{i,m2}) / \sqrt{2}~,
\end{split}$$ leading to the quadratic Lagrangian for the off-diagonal components: $$\label{eq:2.13}
\begin{split}
\int^L_0 d\text{x}_4 \{ {1 \over 2g^2} \text{tr}F^2_{i 4}(x) \}_{
quad,A_i^{1,2}} = & {L \over 2g^2} \overset{+\infty}{\underset{m=0}{\sum}} \{ ({2 \pi m \over L} - {2\pi |l-k| \over NL} )^2 {[(\bar{A}^1_{i,m})^2 + (\bar{A}^3_{i,m})^2]} \\ & + ({2 \pi m \over L} + {2\pi |l-k| \over NL} )^2 {[ (\bar{A}^2_{i,m})^2 + (\bar{A}^4_{i,m})^2 ]} \}~.
\end{split}$$ Relation shows that there are W-bosons $ {W^{\pm}_1 = (\bar{A}^1_{i,m} \pm i \bar{A}^3_{i,m})/\sqrt{2}}$ and $ {W^{\pm}_2 = (\bar{A}^4_{i,m} \pm i \bar{A}^2_{i,m}) / \sqrt{2}}$ with masses $|{2 \pi m \over L} - {2\pi |l-k| \over NL}|$ and $|{2 \pi m \over L} + {2\pi |l-k| \over NL}|$ respectively for $m =0,1,2,...$ and $1\leq l < k \leq N$.
Error analysis {#errorappendix}
==============
Truncation error {#sec:A1}
----------------
In this Section we will discuss relation . Consider at $m \rightarrow \infty$ and at its minimum solution: $$\label{eq:A1}
\bar{T}^{m,J}_{k,\text{min}}=\bar{T}^{m,J}_{k1,\text{min}}+\bar{T}^{m,J}_{k2,\text{min}}~.$$ The only explicit dependence on $J$ in is through $\Delta z = {J / m}$. Extracting this explicit dependence and suppressing the indices we have: $$\label{eq:A2}
H^J={H_1\over J} + JH_2 , \ \ \ \bar{T}^{m,J}_{k,\text{min}} \equiv H^J, \ \ \ \bar{T}^{m,J}_{k1,\text{min}} \equiv {H_1\over J}, \ \ \ \bar{T}^{m,J}_{k2,\text{min}} \equiv {JH_2}~.$$ Taking the derivative of $H^J$ with respect to $J$ gives: $$\label{eq:A3}
{dH^J\over dJ}=-{H_1\over J^2} + H_2 + {1\over J}{\partial H_1\over \partial f_{jh}}{df_{jh}\over dJ} + J{\partial H_2\over \partial f_{jh}}{df_{jh}\over dJ}~.$$ Since the partial derivatives at the minimum solution vanish we have: $$\label{eq:A4}
{dH^J\over dJ}={1\over J}(-{H_1\over J} + JH_2)~.$$ For $J_1 < J_2$ it can be shown that $H^{J_2} < H^{J_1}$. The minimum solution of $H^{J_i}$ is a path $P_i$ that connects the boundary point $\pi \mu_k$ at $\text{z} = 0$ to $0$ at $\text{z} = J_i$ for $i=1,2$ ( Section \[sec:3.3\]). If we extend path $P_1$ on the z-axis from $\text{z} = J_1$ to $\text{z} = J_2$ we would obtain a path $\tilde{P_1}$ that connects the boundary point $\pi \mu_k$ at $\text{z} = 0$ to $0$ at $\text{z} = J_2$. But the value of the action of the paths $P_1$ and $\tilde{P_1}$ is the same since the portion of the path $\tilde{P_1}$ that is on the z-axis gives zero action. On the other hand the action of $\tilde{P_1}$ should be higher than the action of $P_2$ since $P_2$ is the minimizing path of $H^{J_2}$ hence $H^{J_2} < H^{J_1}$. Due to this $ {dH^J / dJ < 0}$. From this gives ${H_1 / J} > JH_2$ and hence $\bar{T}^{m, J}_{k1,\text{min}} > \bar{T}^{m,J}_{k2,\text{min}}$ for all $0 < J < \infty$. Also we should have $\underset{{J \rightarrow \infty}}{\lim} {dH^J / dJ} = 0$ since $H^J$ is a decreasing function of $J$ and it is bounded from below ($\underset{{J \rightarrow \infty}}{\lim} {H^J} = \bar{T_k}$). This shows that $\bar{T}^{m,\infty}_{k1,\text{min}} = \bar{T}^{m,\infty}_{k2,\text{min}}$. Also in the limit of $J \rightarrow 0$ the relations $\underset{{J \rightarrow 0}}{\lim} \bar{T}^{m,J}_{k1,\text{min}} = \infty$ and $\underset{{J \rightarrow 0}}{\lim} \bar{T}^{m,J}_{k2,\text{min}} = 0$ can be easily verified from . This can also be seen from relation , $H_1$ and $H_2$ are finite quantities hence the previous limits follow. Summarizing the previous relations derived we have: $$\label{eq:A5}
\begin{aligned}
& \bar{T}^{m, J}_{k1,\text{min}} > \bar{T}^{m,J}_{k2,\text{min}},\ \ \bar{T}^{m, J}_{k,\text{min}} > \bar{T}^{m, \infty}_{k,\text{min}}, \ \ \ \ \ \ \ \ \ \ \ 0 \leq J < \infty \\
& \bar{T}^{m,0}_{k1,\text{min}} = \infty ,\ \ \ \ \ \ \ \ \bar{T}^{m,0}_{k2,\text{min}} = 0, \ \ \ \ \ \ \ \ \bar{T}^{m, \infty}_{k1,\text{min}} = \bar{T}^{m, \infty}_{k2,\text{min}} = {1\over 2} \bar{T}^{m, \infty}_{k,\text{min}}~.
\end{aligned}$$ From it can be seen that $\bar{T}^{m,J}_{k2,min}$ starts from $0$ at $J=0$ and approaches ${1\over 2} \bar{T}^{m, \infty}_{k,\text{min}}$ at $J=\infty$. We conjecture that for $0<J<\infty$, $\bar{T}^{m,J}_{k2,\text{min}}<{1\over 2} \bar{T}^{m, \infty}_{k,\text{min}}$[^63]. Also from we have ${1\over 2} \bar{T}^{m, \infty}_{k,\text{min}} < {1\over 2} \bar{T}^{m, J}_{k,\text{min}}={1\over 2}(\bar{T}^{m, J}_{k1,\text{min}} + \bar{T}^{m,J}_{k2,\text{min}}) < \bar{T}^{m, J}_{k1,\text{min}}$ so we obtain the following inequalities: $$\label{eq:A6}
\bar{T}^{m,J}_{k2,\text{min}} < {1\over 2} \bar{T}^{m, \infty}_{k,\text{min}} < {1\over 2} \bar{T}^{m, J}_{k,\text{min}} < \bar{T}^{m, J}_{k1,\text{min}}~.$$ From relation easily follows.
Sample error calculation
------------------------
We have shown data for half $k$-string tensions of $SU(10)$, k = 5 in Tables \[table:5\] and \[table:6\] to perform a sample error calculation.
m n $<T_{k1}>$ $\sigma_{k1} \over \sqrt{R}$ $<T_{k2}>$ $\sigma_{k2} \over \sqrt{R}$ $<T_{k}> $ $\sigma_{k} \over \sqrt{R}$
----- ---- ------------ ------------------------------ ------------ ------------------------------ ------------ ----------------------------- --
100 18 5.82434259 6.9E-05 5.82600776 6.2E-07 11.6503503 6.2E-07
100 19 5.82423417 1.4E-05 5.82610277 1.4E-05 11.6503369 1E-07
100 20 5.82420397 3.9E-06 5.82612949 3.9E-06 11.6503335 1.3E-08
200 20 5.82500582 1.20E-04 5.82415592 1.20E-04 11.6491617 6.90E-07
200 21 5.82486451 5.00E-05 5.82428283 5.00E-05 11.6491473 1.70E-07
200 22 5.82482467 2.30E-05 5.82431903 2.30E-05 11.6491437 4.00E-08
: $SU(10)$, $k=5$ sample data with error in the mean ($\sigma \over \sqrt{R}$)[]{data-label="table:5"}
m n $<T_{k1}>$ $\Delta_n$ $<T_{k2}>$ $\Delta_n$ $<T_{k}> $ $\Delta_n$
----- ---- ------------ ------------ ------------ ------------ ------------ ------------ --
100 18 5.82434259 - 5.82600776 - 11.6503503 -
100 19 5.82423417 -(1E-04) 5.82610277 1E-04 11.6503369 -(1E-05)
100 20 5.82420397 -(3E-05) 5.82612949 3E-05 11.6503335 -(3E-06)
200 20 5.82500582 - 5.82415592 - 11.6491617 -
200 21 5.82486451 -(1E-04) 5.82428283 1E-04 11.6491473 -1E-05
200 22 5.82482467 -(4E-05) 5.82431903 4E-05 11.6491437 -(4E-06)
: $SU(10)$, $k=5$ sample data with $\Delta_n=<X_n> - <X_{n-1}>$.[]{data-label="table:6"}
**Sample minimization error calculation:**\
Sample calculation for $SU(10)$, $k = 5$ and $m = 100$:\
Min. Error for $<T_{k2}> = |\Delta_{20}| + {\sigma_k \over \sqrt{R}}$ = 3E-05 + 3.9E-06 $\sim$ 3E-05
From Tables \[table:5\] and \[table:6\] and the sample calculation above it is clear that the minimization errors are of order $10^{-5}$ and less so they can be safely neglected in comparison to the discretization and truncation errors.\
\
**Sample discretization error calculation:**\
The difference between $<T_{k1}>$, $<T_{k2}>$ and $<T_{k}>$ for $m=100$ and $m=200$ is: $$\label{eq:diserror}
\begin{split}
&<T_k>_{200} - <T_k>_{100} = 11.6491437 - 11.6503335 = -0.0011898
\\
&<T_{k1}>_{200} - <T_{k1}>_{100} = 5.82482467 - 5.82420397 = 0.0006207
\\
&<T_{k2}>_{200} - <T_{k2}>_{100} = 5.82431903 - 5.82612949 = -0.00181046~.
\end{split}$$
Hence we predict that the continuum value of the string tensions of $SU(10)$, $k = 5$ for $J=14.0$ would be: $$\label{eq:A.8}
\begin{split}
&<T_k> = 11.6491_{-0.001}
\\
&<T_{k1}> = 5.8248^{+0.0006}
\\
&<T_{k2}> = 5.8243_{- 0.0018} ~.
\end{split}$$ **Sample truncation error calculation:**\
\
Relation gives an upper bound estimate for the truncation error. Based on : $$\label{eq:truncerror1}
\begin{split}
|<T_{k1}> - <T_{k2}>| \lessapprox 5.8248 + 0.0006 - (5.8243 - 0.0018) = 0.0029
\\
\Longrightarrow \text{Trunc.} \ E. \ \lessapprox 2\times 0.0029 = 0.0058~.
\end{split}$$ Adding the truncation and discretization error and neglecting the minimization error we have: $$\label{eq:tot}
\text{Total Error} = \text{Trunc. E.} + \text{Dis. E.} = 0.0058 + 0.001 = 0.0068 \approx 0.007~.$$
Hence we predict the value of the half string tension for $SU(10)$ and $k = 5$ is: $11.6491_{- 0.007}$. The errors obtained by this method for different half $k$-string tensions varied from 0.005 to 0.007 therefore we have considered the average as an upper bound estimate for the value of the error for all half $k$-string tensions and included it in Table \[table:1\]. It has to be noted that upper bound estimates for errors always overestimate the true value of the errors as can be seen from Table \[table:2\].
Derivation of group theory results
==================================
Any representation of ${SU(N)}$ with ${N}$-ality ${1 \leq k \leq N-1}$ contains the fundamental weight ${\mu_k}$ as one of its weights {#sec:B1}
--------------------------------------------------------------------------------------------------------------------------------------
The simple roots $\alpha_i$ and fundamental weights $\mu_k$ of $SU(N)$ are given by the following relations: $$\label{eq:B1}
\begin{split}
& \alpha_i=(0,..,0,\overset {\text{i-th}}{\widehat{1}},-1,0,...,0), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1\leq i \leq N-1 \\
& \mu_k=({N - k \over N}, ..., \overset {\text{k-th}} {\widehat{{N-k \over N}}}, {-k \over N}, ..., {-k \over N}),\ \ \ \ \ \ 1\leq k \leq N-1
\end{split}$$ An arbitrary representation of $SU(N)$ with $N$-ality $1 \leq k \leq N-1$ can be represented by its highest weight $w_k$:
\[eq:B2\] w\_k = h\_i \_i h h\_1 + 2h\_2 + ... + (N-1)h\_[N-1]{} = mN + k , m, h\_i , m, h\_i 0 , &&
where $ {h_i \geq 0}$ are the $N-1$ Dynkin indices of the representation, which determine $k$, its $N$-ality, by the mod($N$) relation given above. The proof involves two steps. First we will prove the following lemma:\
**Lemma** $w_k = \mu_k + a_i \alpha_i$ for $a_i \in \mathbb{Z} \ \text{and} \ a_i \geq 0$
It can be easily seen that: $$\label{eq:B3}
\mu_k = k \mu_1 - \beta_k \ \ \text{with} \ \ \beta_k = (k-1)\alpha_1 + (k-2)\alpha_2 + ... + \alpha_{k-1}, \ \ \beta_1 = 0$$ Hence $w_k$ can be written as: $$\label{eq:B4}
w_k = (h_1 + 2h_2 + ... + (N-1)h_{N-1}) \mu_1 - (h_2 \beta_2 + ... + h_{N-1} \beta_{N-1}) \\$$ With knowing $N \mu_1 = (N-1) \alpha_1 + (N-2) \alpha_2 + ... + \alpha_{N-1}$ and we have: $$\label{eq:B5}
\begin{split}
h \mu_1 = (mN + k) \mu_1 = & \ \mu_k + (m(N-1) + k-1) \alpha_1 + ... + (m(N-(k-1)) + 1) \alpha_{k-1} \\ & + m(N-k) \alpha_{k} + ... + m \alpha_{N-1}
\end{split}$$ Therefore using and , $w_k$ in can be written as: $$\label{eq:B6}
w_k = \mu_k + b_i \alpha_i \ \ \ \ \text{with} \ \ \ \ b_i \in \mathbb{Z}$$ We need to show that $b_i$ is greater than or equal to zero. Lets assume the contrary. First lets assume $b_k < 0$: $$\label{eq:B7}
\begin{split}
&b_k < 0 \Longrightarrow b_k \leq -1 \\
&\eqref{eq:B6} \ \& \ \eqref{eq:B2} \Longrightarrow \alpha_k \cdot w_k = 2b_k + 1 - b_{k+1} - b_{k-1} = h_k \geq 0, \\
& b_k \leq -1 \ \& \ 2b_k + 1 - b_{k+1} - b_{k-1} \geq 0 \Longrightarrow b_{k+1} \leq b_k \ \text{or} \ b_{k-1} \leq b_k~.
\end{split}$$ Lets assume $b_{k-1} \leq b_k$: $$\label{eq:B8}
\begin{split}
& \eqref{eq:B6} \ \& \ \eqref{eq:B2} \Longrightarrow \alpha_{k-1} \cdot w_k = 2b_{k-1} - b_{k} - b_{k-2} = h_{k-1} \geq 0, \\
& b_{k-1} \leq b_k \ \& \ 2b_{k-1} - b_{k} - b_{k-2} \geq 0 \Longrightarrow b_{k-2} \leq b_{k-1}~.
\end{split}$$ Similarly, it can be concluded that $0 > b_k \geq b_{k-1} \geq b_{k-2} \geq b_{k-3} \geq ... \geq b_2 \geq b_1$. Here we will clearly have a contradiction since we have: $$\label{eq:B9}
\begin{split}
& \eqref{eq:B6} \ \& \ \eqref{eq:B2} \Longrightarrow \alpha_{1} \cdot w_k = 2b_{1} - b_{2} = h_{1} \geq 0 \\
& \text{But if} \ b_1 \leq b_2 < 0 \Longrightarrow 2b_1 - b_2 < 0~.
\end{split}$$ Similarly, a contradiction occurs if it is assumed that $b_{k+1} \leq b_k$. Now, if any other $b_i < 0$ for $i \neq k$, similarly it can be argued that either $b_{i+1} \leq b_i$ or $b_{i-1} \leq b_{i}$ and concluded that $2b_1 - b_2 < 0$ or $2b_{N-1} - b_{N-2} < 0$ or $b_k \leq b_i < 0$, which would lead to contradictions similar to above.
The next step of the proof is to show that given a highest weight $w_k$ of a representation with $N$-ality $k$, it is always possible to lower with the simple roots to obtain $\mu_k$. Given a weight $\mu$ of a representation of $SU(N)$, the master formula in [@09] can be applied: $$\label{eq:B91}
{2\mu\cdot \alpha_i \over \alpha^2_i} = \mu\cdot \alpha_i = -(p_i - q_i)$$ Where $p_i \in \mathbb{Z} \ \text{and} \ p_i \geq 0$ is the number of times which we can raise $\mu$ with $\alpha_i$ and $q_i \in \mathbb{Z} \ \text{and} \ q_i \geq 0$ is the number of times which we can lower $\mu$ with $\alpha_i$. Based on the above **Lemma**, we have $w_k = \mu_k + a_i \alpha_i$. If $a_i = 0$ for all $i$ then the representation contains $\mu_k$ as one of its weights but if at least one is greater than zero then we will show that for some $\alpha_i$ which $a_i > 0$, $w_k \cdot \alpha_i > 0$ which would imply that $q_i > 0$ and hence $w_k$ can be lowered with some $\alpha_i$ which $a_i > 0$. Let $a_j = \text{Max} \{ a_i | 1 \leq i \leq N-1 \} $ for some $1 \leq j \leq N-1$. Since we assumed at least one $a_i$ is greater than zero then $a_j > 0$. If $j = 1$, $j = N-1$ or $j = k$ then $w_k \cdot \alpha_j = 2a_1 - a_2 > 0$, $w_k \cdot \alpha_j = 2a_{N-1} - a_{N-2} > 0$ or $w_k \cdot \alpha_j = 2a_k +1 - a_{k-1} - a_{k+1} > 0$ respectively, since we assumed that $a_j > 0$ and is the maximum among others. Otherwise if $a_j > a_{j+1}$ or $a_j > a_{j-1}$ then $w_k \cdot \alpha_j = 2a_j - a_{j+1} - a_{j-1} > 0$. But if $a_j = a_{j+1} = a_{j-1}$ then $w_k \cdot \alpha_j = 0$. In this case $a_{j-1}$ and $a_{j+1}$ are greater than zero and both maximum among other $a_i$. Hence we can repeat what we did for $a_j$ for $a_{j+1}$ or $a_{j-1}$ for a number of steps until $j+r$ in $a_{j+r}$ for an $r \neq 0$ becomes $j+r = 1$, $j+r = N-1$ or $j+r = k$ or we would have $a_{j+r} > a_{j+r+1}$ or $a_{j+r} > a_{j+r-1}$ which in that case $w_k$ can be lowered with $\alpha_{j+r}$. So we proved that if at least one $a_i$ is greater than zero then $w_k$ can always be lowered with some $\alpha_h$ which $a_h > 0$. Hence we continue this process until all the $\alpha_i$ are removed from the highest weight $w_k = \mu_k + a_i \alpha_i$ and we reach $\mu_k$. This shows that any representation of $SU(N)$ with $N$-ality $k$ contains $\mu_k$ as one of its weights.
Derivation of {#sec:B2}
--------------
To derive , we will work out the steps for the contribution of one monopole with magnetic charge $q^1_m$. Consider the long distance behaviour of for generators in the fundamental representation of the gauge group, located at $R \in \R^3$: $$\label{eq:B11}
\begin{split}
\{ \text{tr} \ \text{exp}( i \oint_{R\times T} A^c_m t_F^c d\text{x}^m) \}_{\text{1 mon.}} & = \text{tr} \ \text{exp} (i \int_{S(R\times T)}\epsilon_{anm} \partial_n A^c_m t_F^c d\text{S}^a ) \\ & = \text{tr} \ \text{exp}( i \int_{S(R\times T)} d\text{x}_1d\text{x}_2\; {\text{R}_3 \over 2 |R - \text{x}|^{3}} \;Q^1)~.
\end{split}$$ Here $c = 1, ..., N-1$ labels the abelian generators of $SU(N)$, $Q^1 = \text{diag}(1,-1,0, ...,0)$, and we substituted the magnetic field of a monopole, converted to Cartesian coordinates. Next, we first write $Q^1$ as a linear combination of the abelian generators of the fundamental representation $Q^1 = \bar{V}_i t^i_F = \text{diag}(\bar{V}\cdot \bar{\mu}^1_F, ...,\bar{V}\cdot \bar{\mu}^N_F)$ for $i=1, ...,N-1$; here $\bar{\mu}^j_F$ for $j = 1, ...,N$ are the $(N-1)$-dimensional weight vectors of the fundamental representation of $SU(N)$ and $\bar{V}$ is an $(N-1)$-dimensional vector. In order to transform this to an arbitrary representation $r$ we replace $t^c_F$ by its corresponding generator in the representation $r$ and write $Q^1_r {\equiv} \bar{V}_i t^i_r = \text{diag}(\bar{V}\cdot \bar{\mu}^1_r, ...,\bar{V}\cdot\bar{\mu}^{d(r)}_r)$. In order to write this in the $N$-dimensional form of weights used in this work , we note that the weight vectors of the fundamental representation of $SU(N)$ in their $N$-dimensional form are: $$\label{eq:B12}
\mu^j_{F} =(-{1 \over N}, ..., -{1 \over N}, \overset {\text{j-th}} {\widehat{1- {1 \over N}}}, - {1 \over N}, ..., - {1 \over N}),\ \ \ \ \ \ \ \ \ \ j = 1, ..., N~,$$ which can be easily verified by lowering the fundamental weight $\mu_1$ in with the simple roots. From , the $N$-dimensional form of $\bar{V}$, named $V$, can be determined by requiring: $Q^1 = \text{diag}(\bar{V}\cdot\bar{\mu}^1_F, ...,\bar{V}\cdot\bar{\mu}^N_F) = \text{diag}(V\cdot \mu^1_F, ...,V\cdot\mu^N_F)$, which gives $V = q^1_m = (1,-1,0, ...,0)$. Hence $Q^1_r$, using the $N$-dimensional form of weights, becomes $Q^1_r = \text{diag}(V\cdot\mu^1_r, ...,V\cdot \mu^{d(r)}_r) = \text{diag}(q^1_m\cdot\mu^1_r, ...,q^1_m\cdot\mu^{d(r)}_r)$. Therefore , for generators in an arbitrary representation $r$, becomes: $$\label{eq:B13}
\begin{split}
\{ \text{tr} \ \text{exp}( i \oint_{R\times T} A^c_m t_r^c d\text{x}^m) \}_{\text{1 mon.}} & = \text{tr} \ \text{exp} (i \int_{S(R\times T)}\epsilon_{anm} \partial_n A^c_m t_r^c d\text{S}^a ) \\ & = \overset{d(r)}{\underset{j = 1}{\sum}} \ \text{exp}( i \int_{S(R\times T)} d\text{x}_1d\text{x}_2 \; \mu^j_r \cdot q^1_m\; {\text{R}_3 \over 2 |R - \text{x}|^{3}})~.
\end{split}$$
Perturbative saddle point $\mathbf{k}$-strings: leading order $\mathbf{+}$ leading correction {#sec:C}
=============================================================================================
The following tables \[table:7\] - \[table:15\] compare the values of $k$-strings obtained from a perturbative saddle point calculation to their numerical values in Table \[table:1\]. The ”Leading” and ”Leading Corr.” column give values for the coefficient of $- {RT \over \beta }$ in and respectively. If $T$ represents a half $k$-string value in Table \[table:1\] and $T'$ its corresponding one in the ”Num. value” column, they are related by: $$T' = \text{ROUND}(\bar{T},3) \ \ \ \text{with} \ \ \ \bar{T} \equiv \text{ROUNDDOWN}(T,3)\times2/ \sqrt{2}$$
We multiply the half $k$-strings by $2$ to obtain the full $k$-string then we divide it by $\sqrt{2}$ to normalize them similar to the perturbative saddle point method $k$-strings as in and . There is a high chance that the numerical half $k$-strings in Table \[table:1\] will match the exact half $k$-strings rounded to the third decimal if they are rounded down to the 3rd decimal. This is due to the fact that the true value of half $k$-strings always lies below the values obtained in Table \[table:1\] and the true value of the error is of order $0.001$ or less as can be seen from Table \[table:2\].\
$SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum
--------- --------- --------------- ------------------------ ------------ ------------------
2 9.870 -2.029 7.841 8.000 0.159
3 11.396 -2.343 9.053 9.238 0.185
4 11.913 -2.396 9.517 9.699 0.182
5 12.150 -2.410 9.740 9.919 0.179
6 12.277 -2.415 9.862 10.041 0.179
7 12.355 -2.417 9.938 10.114 0.176
8 12.405 -2.418 9.987 10.163 0.176
9 12.439 -2.418 10.021 10.196 0.175
10 12.463 -2.418 10.045 10.221 0.176
: Comparison of $N$-ality $1$ $k$-strings for $SU(2 \leq N \leq 10)$[]{data-label="table:7"}
$SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum
--------- --------- --------------- ------------------------ ------------ ------------------
3 11.396 -2.343 9.053 9.238 0.185
4 13.958 -2.870 11.088 11.314 0.226
5 15.018 -2.99 12.028 12.247 0.219
6 15.568 -3.029 12.539 12.751 0.212
7 15.891 -3.045 12.846 13.055 0.209
8 16.097 -3.052 13.045 13.253 0.208
9 16.237 -3.056 13.181 13.388 0.207
10 16.337 -3.058 13.279 13.485 0.206
: Comparison of $N$-ality $2$ $k$-strings for $SU(3 \leq N \leq 10)$ []{data-label="table:8"}
$SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum
--------- --------- --------------- ------------------------ ------------ ------------------
4 11.913 -2.396 9.517 9.699 0.182
5 15.018 -2.990 12.028 12.247 0.219
6 16.449 -3.142 13.307 13.511 0.204
7 17.248 -3.197 14.051 14.247 0.196
8 17.746 -3.222 14.524 14.715 0.191
9 18.078 -3.234 14.844 15.032 0.188
10 18.311 -3.241 15.070 15.257 0.187
: Comparison of $N$-ality $3$ $k$-strings for $SU(4 \leq N \leq 10)$ []{data-label="table:9"}
$SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum
--------- --------- --------------- ------------------------ ------------ ------------------
5 12.15 -2.410 9.740 9.919 0.179
6 15.568 -3.029 12.539 12.751 0.212
7 17.248 -3.197 14.051 14.247 0.196
8 18.237 -3.262 14.975 15.159 0.184
9 18.876 -3.292 15.584 15.761 0.177
10 19.317 -3.307 16.010 16.183 0.173
: Comparison of $N$-ality $4$ $k$-strings for $SU(5 \leq N \leq 10)$ []{data-label="table:10"}
$SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum
--------- --------- --------------- ------------------------ ------------ ------------------
6 12.277 -2.415 9.862 10.041 0.179
7 15.891 -3.045 12.846 13.055 0.209
8 17.746 -3.222 14.524 14.715 0.191
9 18.876 -3.292 15.584 15.761 0.177
10 19.629 -3.326 16.303 16.474 0.171
: Comparison of $N$-ality $5$ $k$-strings for $SU(6 \leq N \leq 10)$ []{data-label="table:11"}
$SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum
--------- --------- --------------- ------------------------ ------------ ------------------
7 12.355 -2.417 9.930 10.114 0.176
8 16.097 -3.052 13.045 13.253 0.208
9 18.078 -3.234 14.844 15.032 0.188
10 19.317 -3.307 16.010 16.183 0.173
: Comparison of $N$-ality $6$ $k$-strings for $SU(7 \leq N \leq 10)$ []{data-label="table:12"}
$SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum
--------- --------- --------------- ------------------------ ------------ ------------------
8 12.405 -2.418 9.987 10.163 0.176
9 16.237 -3.056 13.181 13.388 0.207
10 18.311 -3.241 15.070 15.257 0.187
: Comparison of $N$-ality $7$ $k$-strings for $SU(8 \leq N \leq 10)$ []{data-label="table:13"}
$SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum
--------- --------- --------------- ------------------------ ------------ ------------------
9 12.439 -2.418 10.021 10.196 0.175
10 16.337 -3.058 13.279 13.485 0.206
: Comparison of $N$-ality $8$ $k$-strings for $SU(9 \leq N \leq 10)$ []{data-label="table:14"}
$SU(N)$ Leading Leading Corr. Sum(Lead.+Lead. Corr.) Num. value Num. value - Sum
--------- --------- --------------- ------------------------ ------------ ------------------
10 12.463 -2.418 10.045 10.221 0.176
: Comparison of $N$-ality $9$ $k$-strings for $SU(10)$ []{data-label="table:15"}
.\
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\
\
\
Large-$N$ limit of string tensions for product representations: a saddle point leading-order perturbative evaluation {#sec:appxproduct}
====================================================================================================================
Our starting point is . Recall that this equation gives the contribution to the expectation value of the Wilson loop of quarks of charges (weight) $\mu$, evaluated to leading order using the perturbative saddle point method. For convenience, we now reproduce the area-law part of that equation ($\hat{R}, \hat{T} \rightarrow \infty, \beta \rightarrow 0$): $$\label{eq:4.391}
{\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b_q=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {1 \over 4} {\sum\limits_{q=1}^{N-1}\sqrt{\Lambda_q}b^2_q} \hat{R} \hat{T} \} )~.$$ Here, $b_q \equiv 2\pi (\mu)_j D_{jq}$ for a representation of weight $\mu$; $(\mu)_j$ denotes the $j$-th component, $j=1,...N$, of the weight vector. Recall also that $\Lambda_q = 4 \sin^2 {\pi q \over N}$ is the dimensionless mass of the $q$-th dual photon and that the components of the matrix $D_{jq}$, $1 \le j \le N$, are $D_{jq} = \sqrt{2 \over N} \sin {2 \pi q j \over N} $, for $1 \le q < {N\over 2}$ and $D_{jq} = \sqrt{2 \over N} \cos {2 \pi q j \over N} $, for ${N\over 2} < q < {N}$; for brevity, we only give the values for odd $N$. The $q=N$ component of $D_{jq}$ does not contribute to .
The main difference compared to the discussion in the main text is that we now consider also weights corresponding to product representations, for concreteness the ${\square \otimes \square}$ representation. Recall, from , that the expectation value of the Wilson loop in the product representation is given, in the abelianized regime of this paper, by a sum of exponentials, one for each weight of the product representation: $$\label{eq:abd1}
\langle W_{{\square \otimes \square}}(R,T) \rangle \sim \sum^{d({\square \otimes \square})}_{h = 1} \text{exp}(- T^h_{{\square \otimes \square}}RT) = \sum^{d({\square \otimes \square})}_{h = 1} \text{exp}(- {1 \over \beta} \hat{T}^h_{{\square \otimes \square}}\hat{R}\hat{T}) ~.$$ Where we have also written it in its dimensionless form (recall the relations $R = \hat{R} / m_{\gamma}$, $T = \hat{T} / m_{\gamma}$, $\beta = m_{\gamma}^3/ \tilde{\zeta}$ from the comment below ). Comparing with $\hat{T}^h_{{\square \otimes \square}}$ to leading order (l.o.) is given by: $$\label{eq:apxe1}
\hat{T}^h_{{\square \otimes \square},\text{l.o.}}= {1 \over 4} {\sum\limits_{q=1}^{N-1}\sqrt{\Lambda_q}b^2_q}~, ~~ b_q \equiv 2\pi \sum\limits_{j=1}^N (\mu^h)_j D_{jq} ~,$$ with $\mu^h$—the $h$-th weight of the ${\square \otimes \square}$ representation.
The goal of this Appendix is to evaluate for all weights of the ${\square \otimes \square}$ product representation, to leading order in the analytic perturbative saddle point method and in the large-$N$ limit. We shall see that the leading-order analytic considerations support the findings discussed qualitatively after around Eqns. (\[acd\]) and of the main text of the behaviour of the product-representation Wilson loop at large $N$.
We begin by noting that the weights of the ${\square \otimes \square}$ representation are labeled by two integers $a, b=1,...N$ (there are $N^2$ weights) and are given by $$\label{eq:weight}
(\mu^h)_j \rightarrow (\mu^{ab})_j = \delta^a_j + \delta^b_j -{2 \over N} \approx \delta^a_j + \delta^b_j .$$ The last equality is valid for sufficiently large $N$. From , recalling that we consider odd-$N$, we find an explicit expression for the tension of strings sourced by quarks with weight $\mu^{ab}$, , of the product representation at leading order:[^64]
\[eq:appxe2\] \^[ab]{}\_[,]{} = 4 &&
The sum in can be evaluated exactly for arbitrary $N$, but to illustrate our point it suffices to consider [*i.*]{}) the results of a numerical evaluation and [*ii.*]{}) the evaluation of at infinite $N$ by replacing the sum by an integral.
We begin with a discussion of the numerical results for the $N^2$ product representation string tensions shown on Figure \[fig:productstrings\]. The $N^2$ string tensions for $N=21$ are evaluated numerically using . As the plot shows, most of the $N^2$ string tensions are of order $2 T_{1,\text{l.o.}}$, while $2N$ of them are approximately equal to the minimal value $T_{2,\text{l.o.}}$, and $N$ are equal to approximately $4 T_{1,\text{l.o.}}$. Clearly, this is conforming to the discussion in the main text, Section \[sec:5.2.2\]..
{width="\textwidth"}
[\[fig:productstrings\]]{}
We can also evaluate in the infinite-$N$ limit by replacing the sum by an integral, for $a,b$ fixed, i.e.,
\[eq:appxe3\] [\^[ab]{}\_[,]{} 4 ]{} = \_[0]{}\^[[2]{}]{} d x x ( + )\^2 + \_[[2]{}]{}\^d x x ( + )\^2 .
Thus, the product representation string tensions, normalized to the fundamental string tension (equal to $4\pi$ in the leading saddle point approximation), becomes in the large-N limit $$\label{eq:E.7}
{\hat{T}^{ab}_{{\square \otimes \square},\text{l.o.}}\over 4 \pi} = 2 - {2\over 4 (a - b)^2-1}~.$$ Due to the $Z_N$ symmetry of the string tension action we expect to obtain the same tensions for $|a-b| = n$ and $|a-b| =N - n$ for $1 \leq n \leq [N/2]$. This symmetry is lost in due to the infinite $N$ limit, therefore it is best to use this relation for $|a-b| \leq [N/2]$ only at large N and for $|a-b| > [N/2]$ make the replacement $|a-b| \rightarrow N - |a-b|$ in . In the limit $|a-b|\gg 1$, this relation approaches the value of $2$, while for $|a-b|=1$, we obtain the value ${4\over 3} \approx 1.33$; the value of $4$ for $a=b$ is also obtained. This distribution of the $N^2$ string tensions in the infinite-$N$ limit is consistent with the numerical result shown for $N=21$ and with the general discussion of Section \[sec:5.2.2\].
At the end, we also acknowledge an additional subtlety one might be worried about. The calculation that led to eqn. —see as well as eqns. (\[eq:4.7\]–\[eq:4.14\]) which directly lead to it—assumes that $RT$ is larger than the inverse mass squared of all dual photons, including the lightest one. Thus, strictly speaking one expects to pertain to the order of limits we advocated for here: infinite area at fixed N, followed by $N \rightarrow \infty$ which is the proper order of limits necessary for calculating k-strings at large $N$.
However for the discussion of large $N$ factorization in gauge theories the large $N$ limit is taken first. Now, if $RT$ is smaller than the mass of some dual photons, the area law due to these photons should be replaced with a perimeter law contribution. This remark is relevant because if the large-N limit is taken first, the masses of some dual photons vanish—recall that their masses are scale as $\sqrt{\Lambda_q} =2 \sin {\pi q \over N}$—and these dual photons do not lead to an area law. To take this into account, consider the integral and omit contributions of dual photons of (dimensionless) mass $ 2 \sin {\pi q\over N} = 2 \sin x < {1\over \sqrt{\hat R \hat T}}$, as they do not give rise to area law. Thus, the region of integration in , instead of $(0, {\pi \over 2})$ and $({\pi\over 2}, \pi)$, should be replaced by, respectively, $(\epsilon, {\pi \over 2})$ and $({\pi\over 2}, \pi-\epsilon)$, with $\epsilon \sim {1\over \sqrt{\hat{R} \hat{T}}}$. In the further large $\hat{R}\hat{T}$ limit, we have that $\epsilon \rightarrow 0$, showing that the contributions to the string tension of dual photons of mass vanishing at large-$N$ is negligible. Thus, we expect that if the order of limits is taken as described now ($N$ to infinity first, large area next), the factorization result analyzed above in terms of string tensions is recovered.
The discussion of large-$N$ factorization above and in the 2nd half of Section \[sec:5.2.2\] was carried out in terms of string tensions, since its more explicit and intuitive and allows for a qualitative analysis of large-$N$ factorization in terms of the full saddle point as was done in Section \[sec:5.2.2\]. For this analysis, the large area limit had to be taken to isolate the area law contribution and find the string tensions, as done in the previous paragraph. However one can show the large-$N$ factorization result in a more general and abstract setting without the need to refer to any large area limits or expressions for string tensions. Consider eq. , which is a general expression for the saddle point at leading order, without reference to any large area limit: $$\label{eq:E9}
{\{ Z^{\eta}_{g^4} \}_{\lambda =0} \over \{ Z^{\eta}_{g^4} \}_{b_q=\lambda =0} } = \text{exp}(- {1 \over \beta} \{ {\Lambda_q b^2_q \over 2} \underset{\text{A} \ \text{A}}{\iint} d^2 \text{x} d^2 \text{x}' P_q(\text{x} - \text{x}') + {b_q^2 \over 2} \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P_q(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|}) \} )~,$$ Using and noting that the integrals in are finite quantities and a function of $\hat{R}, \hat{T}$ and $\sqrt{\Lambda_q} =2 \sin {\pi q \over N}$ with $x \equiv {\pi q \over N}$, the large $N$ limit of the leading saddle point (s.p.) result in reduces to: $$\label{eq:E10}
\hspace{-0.5cm} \text{s.p.}_{\square \otimes \square,\text{l.o.}} = \int^{\pi /2}_0 dx ( \sin{2 a x} + \sin{2 b x} )^2 F_{\hat{R}, \hat{T}}(\sin x) + \int^{\pi}_{\pi /2} dx ( \cos{2 a x} + \cos{2 b x} )^2 F_{\hat{R}, \hat{T}}(\sin x)~,$$ where $F_{\hat{R},\hat{T}}(\sin {\pi q \over N})$ is given by: $$\label{eq:E11}
\hspace{-1mm} F_{\hat{R}, \hat{T}}(\sin {\pi q \over N}) \equiv 4 \pi {\Lambda_q} \underset{\text{A} \ \text{A}}{\iint} d^2 \text{x} d^2 \text{x}' P_q(\text{x} - \text{x}') + 4 \pi \underset{b(\text{A}) \ b(\text{A})}{\iint'} d\text{x}^l d\text{x}'^k \delta^{l k} ( P_q(\text{x}-\text{x}') - {1 \over 4 \pi |\text{x} -\text{x}'|})$$
The expression corresponding to in the fundamental representation of $SU(N)$ is: $$\label{eq:E12}
\hspace{-0.5cm} \text{s.p.}_{\square,\text{l.o.}} = \int^{\pi /2}_0 dx ( \sin{2 a x})^2 F_{\hat{R}, \hat{T}}(\sin x) + \int^{\pi}_{\pi /2} dx ( \cos{2 a x})^2 F_{\hat{R}, \hat{T}}(\sin x)$$ Making the change of variable $x \rightarrow \pi - x$ in the second integral of and , they can be simplified to:
\[eq:E13\] &\_[,]{} = 2\^[/2]{}\_0 dx F\_[, ]{}(x) + 2\^[/2]{}\_[0]{} dx (2a x - 2b x) F\_[, ]{}(x) ,\
\[eq:E14\] & \_[,]{} = \^[/2]{}\_0 dx F\_[, ]{}(x) .
When $|a-b| \gg 1$ (and $|a-b| \ll N$ if the discrete form of is considered for a finite but large $N$), the second integral in , due to the rapid oscillations of $\cos (2a x - 2b x)$, is near zero therefore $O(N^2)$ weights of the product representation give approximately twice the value of the fundamental representation string tension in the leading saddle point approximation from , which has the same value for all weights of the fundamental representation. Therefore relations and clearly show the large $N$ factorization result in dYM theory at leading order of the saddle point without any reference to a large area limit.
Although the calculations in this Appendix were done at the leading order saddle point level the same ideas and methods can be applied to show large $N$ factorization regarding the corrections (as in ) to these leading order saddle point results.
We thank Mohamed Anber and Aleksey Cherman for comments on the manuscript and Adi Armoni for many enlightening discussions of k-strings. We acknowledge support by a NSERC Discovery Grant. The computations for this paper were performed on the gpc supercomputer at the SciNet HPC Consortium. SciNet is funded by: the Canada Foundation for Innovation under the auspices of Compute Canada; the Government of Ontario; Ontario Research Fund - Research Excellence; and the University of Toronto.
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[^1]: Hereafter, as most of our studies are Euclidean, we shall denote the spacetime manifold simply by $\R^3 \times \S^1$, but we use $\R^{1,2}\times \S^1$ here in order to stress that $\S^1$ is a spatial circle and the object of our study is not finite-temperature theory.
[^2]: Apart for the large-N limit, see below.
[^3]: There is a plethora of metastable strings that can also be studied using the tools developed here. An evaluation of their tensions and decay rates is left for future work. See Appendix E for a calculation of some metastable string tensions at leading order.
[^4]: An important additional subtlety is that the values of N for which the relations (\[dymscaling2\]) have been derived, while numerically large, are bounded above by an exponentially large number $N \ll 2 \pi e^{c\over \lambda}$, where $\lambda \sim |\log \Lambda N L|^{-1}$ is the arbitrarily small ’t Hooft coupling and $c$ is an ${\cal{O}}(1)$ coefficient. Preliminary estimates suggest that the effect of the W-boson induced mixing on the string tensions (whose neglect is the source of the upper bound on N, see Section \[sec:largeN\]) will not qualitatively change the large-$N$ limit. However, we prefer to defer further discussion until the relevant calculations for dYM have been performed.
[^5]: Some of these points were, without elaboration, made earlier in [@Anber:2015kea]. We also note that the glueball spectra in dYM, as well as the mesonic and baryonic spectra with quarks added as in [@Cherman:2016hcd], exhibit many intriguing properties and are the subject of the more quantitative recent study [@Aitken:2017ayq].
[^6]: The reader already familiar with dYM and interested in our numerical and analytic metnods can proceed to Sections \[numericsection\] and \[sec:4\] and the discussion in Section \[sec:5\].
[^7]: \[center\]Center symmetry transformations are global symmetries that can be loosely thought as “gauge” transformations periodic up to the centre of the gauge group. For example, for an $SU(2)$ gauge group, the center-symmetry transformation periodic up to the nontrivial $\Z_2$ center element $z=-1$ can be represented by $U_{-1}(\text{x},x_4) = \text{exp}(i {\pi \over L} x_4 \sigma_3)$, with $U_{-1}(\text{x},0) = - U_{-1}(\text{x},L)$ with $\sigma_3$ the third Pauli matrix and $x_4$—the $\S^1$ coordinate. See [@04] for a proper definition of center symmetry as a global symmetry on the lattice and [@Gaiotto:2014kfa] for a continuum point of view.
[^8]: We defined $\omega_N = e^{ 2 \pi i \over N}$.
[^9]: The $n_f = 1/2$ massless case leads to vanishing potential, as is clear by comparing the massless limit of (\[eq:2.7\]) with (\[eq:2.4\]). This case corresponds to the minimally supersymmetric Yang-Mills theory in four dimensions.
[^10]: This has been explicitly performed for the above choices of parameter up to $SU(10)$ and with considering the effective potentials up to $n=20$.
[^11]: General $SU(N)$ theories with semiclassically calculable dynamics at small-$L$ have been classified in [@Anber:2017pak].
[^12]: At subleading order, threshold corrections from the $W$-bosons cause the $N-1$ photons (and consequently, the dual photons) to mix. These mixing effects are expected to be similar to the ones in super-Yang-Mills [@08] and QCD(adj) [@Anber:2014sda; @vito]. They become important in the abelian large-$N$ limit [@Cherman:2016jtu], where dYM has a curious “emergent dimension” representation. The mixing between the $N-1$ photons is also expected to affect the $k$-string tensions in the abelian large-$N$ limit. In this paper, we have not taken these effects into account.
[^13]: The $N$-dependence of the lightest $A_4$ shows that the mass scale of the holonomy fluctuations remains fixed in the abelian large-$N$ limit, where $g^2 N$ and $m_W \gg \Lambda$ remain fixed.
[^14]: After any twist of $\Omega$ at infinity associated with the magnetic charges $q_{\alpha}$ is removed by a (singular) gauge transformation, the resulting field may be regarded as a mapping of compactified three space (or $\S^3$) onto the group $SU(N)$, leading to the familiar Pontryagin index. More details regarding the definitions of these quantities can be found in [@06].
[^15]: This terminology is adopted for historical reasons. In the limit when the mass of the physical holonomy fluctuations is neglected, both our BPS and KK solutions satisfy a BPS bound and can be found by solving first-order equations.
[^16]: Our convention for spherical coordinates is $r(\text{sin} \ \theta \ \text{cos} \ \phi,\text{sin} \ \theta \ \text{sin} \ \phi,\text{cos} \ \theta) = (\text{x}_1,\text{x}_2,\text{x}_3 )$).
[^17]: This definition applies to when the eigenvalues of the holonomy ${\Omega}$ at infinity are distinct. For the general definition of magnetic charges that would also apply to holonomies with degenerate eigenvalues at infinity refer to relation (B.6) in [@06]
[^18]: A direct calculation of $Q$ for the $SU(2)$ BPS solution yields $Q=1/2$, thus verifying explicitly (\[eq:2.14\]) with $p=0$ and the appropriate expression for $\mu_\alpha$.
[^19]: More details regarding this solution and its explicit form can be found in, e.g. [@23]. These “twisted" solutions were first found in [@Lee:1997vp; @Kraan:1998sn] using different techniques.
[^20]: While we use energetics terminology, motivated by the electro-/magneto-static analogy, we clearly mean Euclidean action. Also by gauge-invariant terms we refer to any terms in the action of two far separated monopoles that are independent of the Dirac string (singularity of the solutions at $\theta = \pi$ in ) or its orientation.
[^21]: At the classical level, the $A_4$-field, mediating the so-called “electric” interactions, is massless hence it is of long range. We stress that the term “scalar interaction” is the precise one, within the framework of spatial-$\S^1$ compactifications; for brevity, we continue calling these interactions “electric” and omit the quotation marks in what follows. Furthermore, as already explained, at the quantum level the $A_4$ field gains mass hence the electric interaction is short range and not important in the derivation of the string tension action. We only discuss the electric interaction here for the sake of mentioning some points not usually explicitly discussed with regard to the classical interaction of monopole-instantons.
[^22]: For another discussion on the core interaction between dyons refer to [@60].
[^23]: $\tau^3_{(j)}$ refers to the $\tau^3$ Pauli matrix placed in the j-th Lie subalgebra of $SU(N)$ along the diagonal.
[^24]: This also implies that a more precise treatment (as opposed to simply summing far separated monopoles) for the construction of far separated monopole solutions is required, in particular, one that will not involve any non-gauge-invariant contributions. The construction of the monopole gas by summing far separated monopole solutions is appealing due to its simplicity and the fact that its leading gauge-invariant interaction terms reproduce results consistent with the more accurate far separated solutions, as studied in [@25; @29; @30].
[^25]: The $n^{4/3}$ power in the last term is an attempt at a better than naive estimate of the error. Naively, one could imagine the correction scaling as $n^2$, with $d$ being the typical separation between monopoles, but it is clear that not all monopoles are separated by the same distance. Assuming a uniform distribution of monopoles, with $d$ the closest distance between a given monopole and its neighbors, one can arrive at the estimate given (one expects some power $n^p$ with $1 < p < 2$). Note also the fact that not all $N$ types of monopoles have classical interactions, is not taken into account in writing the last term in (\[eq:2.30\]).
[^26]: At distances $R$ $\gtrapprox \Lambda^{-1}$ with $\Lambda$ being the strong scale of the theory.
[^27]: [More details regarding the derivation of this partition function can be found in [@01]. In this Section we will use this partition function to derive the Wilson loop inserted dual photon action for the evaluation of the $k$-string tensions. $Z_{\text{pert.}}$ refers to the perturbative contribution of the effective dual photon action: $Z_{\text{pert.}} = { \int D[ \sigma] \;\text{exp}(-\int_{{\rm I\!R}^3} d^3\text{x} {1 \over 2} {g^2 \over 8 \pi^2 L}( \nabla \sigma )^2 )}$.]{}
[^28]: For simplicity it has been written for only two insertions of $e^{i q \cdot \sigma}$.
[^29]: See Footnote \[modes1\] for the relation between the $U(N)$ and $SU(N)$ Cartan fields and further comments on the duality.
[^30]: \[modes1\]The remarks that follow are tangential to our exposition, but serve to convince the reader of the consistency of our dual action coefficients with charge quantization (our Eq. (\[eq:2.40\]) was derived solely by demanding that the long-distance interactions between monopole-instantons from Section \[2.3\] are correctly reproduced) and to correct minor typos in expressions that have appeared previously in the literature. Integrating the duality relation (\[dualityelectric\]), we obtain $${g^2 \over \sqrt{2} L}\; \; \oint_C { d \sigma^A \over 2 \pi} = \oint_C d \vec{n} \cdot \vec{E}^A,
\label{dual12}$$ representing the fact that a static electric charge inside $C$ generates flux through $C$ ($\vec{n}$ is an outward unit normal to $C$) which duality relates to the $\sigma$-field monodromy around $C$. Eq. (\[dual12\]) implies that the $\sigma^A$ fields have periodicities determined by the fundamental electric charges. To find them, we begin with the relation between the $N$ Cartan field strengths $F^A_{kl}$ that first appeared in (\[dualityrelation\], \[bianchi\]) and the original $N-1$ $SU(N)$ fields $F^a_{kl}$ in (\[eq:2.1\]). The reader can convince themselves that it is given by $F^A_{kl} = {1 \over \sqrt{N}} F_{kl}^0 + \sum\limits_{a=1}^{N-1} F_{kl}^a \lambda^{a A}$, with $\lambda^{aA} \equiv (\theta^{aA} - a \delta^{a+1, A})/\sqrt{a(a+1)}$, where $\theta^{aA} = 1$ for $a \ge A$ and $\theta^{aA}=0$ otherwise. The spectator $U(1)$ field $F^0_{kl}$ is not coupled to dynamical sources. The relations $\sum\limits_{A=1}^N \lambda^{a A} \lambda^{b A} = \delta^{ab}$ and $ \sum\limits_{A=1}^N \lambda^{a A} =0$ help establish that $\sum\limits_{A=1}^N (F^A)^2 = \sum\limits_{a=1}^{N-1} (F^a)^2 + (F^0)^2$. A fundamental static charge is represented by the insertion of a static Wilson loop in the fundamental representation. Early on, see (\[eq:2.1\]), we stated that our fundamental representation generators are normalized as tr($T^a T^b) = \delta^{ab}/2$. Thus, using the definitions just made, it follows that the fundamental representation Cartan generators are $T^a = {\rm diag}(\lambda^{a 1},..., \lambda^{a N})/\sqrt{2}$. A fundamental static Wilson loop is then represented by insertions of $\int dt A_0^a(\vec{r},t) \lambda^{aA}/\sqrt{2}$ ($A$ labels the Wilson loop eigenvalues) in the path-integral action. Considering one of the eigenvalues of the Wilson loop (one component of the fundamental static quark), the corresponding electric flux is found by solving the static equation of motion, ${L \over g^2} \nabla^2 A_0^a = {\lambda^{a A}\over \sqrt{2}}\delta(\vec{r})$, thus $ \oint_{C_A} d \vec{n}\cdot \vec{E}^a = - \lambda^{a A}{g^2 \over \sqrt{2} L}$. From the earlier relations, we also have that $\vec{E}^a = \sum\limits_{A=1}^N \lambda^{a A} \vec{E}^A$, thus $\sum\limits_{B=1}^N \lambda^{a B} \oint_{C_A} d \vec{n} \cdot \vec{E}^B = {g^2 \over \sqrt{2} L} \lambda^{a A}$. Finally, from (\[dual12\]), this leads to $\sum\limits_{B=1}^N \lambda^{a B} \oint_{C_A} { d \sigma^B \over 2 \pi} = \lambda^{a A}$. It can be already seen, from the explicit form of $\lambda^{a A}$, that this relation implies that the periodicity (monodromies) of differences of $\sigma^A$’s have to be proportional to $2 \pi$. Even more explicitly, from the relation $\sum\limits_{a=1}^{N-1} \lambda^{a A} \lambda^{a B} = \delta^{AB} - {1 \over N}$, one finds that the monodromies of the dual photons are given by $2 \pi$ times the weights of the fundamental representation. This is consistent with the periodicities of the potential terms in (\[eq:2.39\]) and with the dual photon actions given in e.g. [@Simic:2010sv; @23; @08].
[^31]: We note that with this choice the effect of $\mu$ on $\tilde\zeta$ is comparable to the effect of finite $A_4$ mass on the classical monopole action, an effect that we have neglected throughout. Matching between the UV theory, valid at scales $\ge m_W$, and the IR theory, valid at scales $\ll m_W$, to better precision that has been attempted so far is needed to properly account for these effects. We also note that in the supersymmetric case, the only case where the determinants in the monopole-instanton backgrounds have actually been computed, where $\sigma$ is replaced by a chiral superfield and the monopole-instantons are “localized in superspace,” this ambiguity is absent [@08]—in super-Yang-Mills, divergent self energies of monopoles due to electric and magnetic charge cancel out in the analogue of (\[eq:2.37\]).
[^32]: A more detailed derivation of is done in Appendix \[sec:B2\].
[^33]: The order of magnitude of the string tension is $g^2 N m_\gamma m_W$. Thus $W$-boson production takes place once $R \sim {\rm O}(1/(g^2 N m_\gamma))$ and Higgs production (recall $m_H \sim g \sqrt{N} m_W$) when $R \sim {\rm O}(1/(g \sqrt{N} m_\gamma))$. Notice that the values on the r.h.s., owing to small coupling, are much larger than the Debye screening length $1/m_\gamma$.
[^34]: This can be seen as follows. Consider the boundary conditions $\sigma (0^+) = \pi \mu_k + \bar{\text{z}}$ and $\sigma (0^-) = - \pi \mu_k + \bar{\text{z}}$. We will show that the minimum of is when $\bar{\text{z}} = 0$. If $\sigma(\text{z})$ is an extremum solution of so is $\sigma(-\text{z})$ and $- \sigma(\text{z})$, therefore it can be seen that $\bar{\text{z}} = 0$ is an extremum point. It is a minimum since otherwise the kinetic term will increase if we make the magnitude of the boundaries larger than $\pi (\mu_k)_j$ on either side of $\text{z} = 0^+$ or $\text{z} = 0^-$.
[^35]: That an ansatz with a single exponential works for $SU(3)$ is a consequence of the existence of only a single mass scale in the dual-photon theory, a fact that only holds for $N=2,3$.
[^36]: The superscript $(m)$ indicates that this is the discretization with $m$ partitions of the interval.
[^37]: Minimizing the second derivative of $\bar{T}^{m,J}_{k,2}$ with respect to $f_{lp}$, gives $-{\delta \text{z} \over 3}$ for each time the variable $f_{lp}$ appears in the sum over $j$ and $h$. Since it appears 4 times when replacing each variable $f_{jh+1}$, $f_{j+1h+1}$, $f_{jh}$ and $f_{j+1h}$ in the expression and the minimum value is the same for all 4 cases, this gives $-{4\over 3} \delta \text{z}$ for a lower bound on the 2nd derivative of the second term.
[^38]: Numerical computations of the string tensions were performed on the gpc supercomputer at the SciNet HPC Consortium [@22]. Due to a high number of $k$-string calculations ($>1000$) with most of them involving minimization of multivariable functions with more than 500 variables, using a cluster that could perform many $k$-string computations at the same time in parallel was necessary.
[^39]: As is evident from equation , the expansion parameter is $ {\lambda b^2\over 4 m^2}$; as discussed there, convergence of the perturbative expansion of the saddle point for a $g^4$ interaction term only requires that this parameter be less than $ {1/2}$. This condition is met in dYM theory, but not in QCD(adj) [@Anber:2015kea], for the choices of parameters following from the underlying action (In dYM from below equation ), although not strictly required since the full potential in both theories includes higher non-linearities and in taking these into account the perturbative series evaluation of the saddle point would be a convergent one.
[^40]: This agreement can be further improved, as we have verified for $SU(2)$. Summing only contributions to (\[wilsonperturbative\]) to order $\lambda$ we obtain the value $7.84$ shown in Table \[table:71\]. Including the higher-order correction terms show an oscillatory convergence: Including the first order correction due to the $g_1^6$ term in gives $8.285$ and including order $\lambda^2$ of the quartic term expansion, we obtain $8.007$, to be compared with the exact value $8$.
[^41]: We have to note that since we are summing to all orders such a perturbative expansion is justified although ${1 \over \sqrt{\beta}}$ becomes large as $\beta \rightarrow 0$.
[^42]: \[footnotetaylor\]The Taylor series expansion of $\sqrt{1+ \text{x}}$ converges for $|\text{x}| < r = 1$. Evaluating $\text{x} = {2\lambda h^2 / m^2}$ for $h = {b / 2}$ with values of parameters from below equation gives $|\text{x}| = {\pi^2 / 12} < 1$ which lies within the radius of convergence. [As a reminder we mention that the condition $|\text{x}| = |{2\lambda h^2 / m^2}| < 1$ is not strictly required in dYM since the full potential is cosine which would allow for a wider range of these parameters.]{}
[^43]: The diagonalization matrix $D$ has the effect of an $\Z_N$ Fourier transform and the eigenvalues of $A$ are $\Lambda_q = 4 \sin^2{ \pi q\over N}$, $q = 1, ..., N-1$ and $\Lambda_N = 0$. As discussed in [@Cherman:2016jtu], this is the spectrum of a latticized emergent dimension of $N$ sites.
[^44]: \[eigenvectorfootnote\]Because $D= (v_1,...,v_N)$, where $v_q$, $q=1,...N-1$ are the eigenvectors of $A$ with eigenvalues $\Lambda_q = 4 \sin^2{ \pi q \over N}$ and $v_N ={1\over \sqrt{N}} (1,1,1...,1)^T$ is the zero eigenvector. For use below, the other $N-1$ eigenvectors, for brevity shown for odd $N$ only, with components $v_q^l$ are: $v_{q < {N\over 2}}^l = \sqrt{2\over N} \sin{2 \pi q l\over N}$ and $v_{ {N\over 2}<q<N}^l = \sqrt{2\over N} \cos{2 \pi q l\over N}$.
[^45]: Note that the string tension remains fixed at large-N, despite the vanishing mass gap, as there is a number of dual photons of nonzero mass ($\sim m_\gamma$) whose flux is confined, as well as a number of dual photons approaching zero mass ($\sim m_\gamma/N$) whose flux spreads out. Thus the finite tension confining string in the gapless abelian large-N limit is a rather fuzzy object. We defer a further study until the large-N corrections, discussed in [@Cherman:2016jtu] for sYM, are better understood in the dYM case.
[^46]: Degenerate string tensions will occur when the corresponding weights are related by the unbroken $\Z_N$ center symmetry. For example, for the fundamental representation all weights have the same string tension, see Section \[sec:compare\]. For higher $N$-ality representations, the dim($r$) weights fall into distinct $\Z_N$ orbits, each of which has degenerate string tensions.
[^47]: This should not be taken to mean that the nonabelian nature of the theory is not relevant: on the contrary, it is crucial in both examples.
[^48]: There are hints that the two confinement mechanisms are related, see [@Poppitz:2011wy].
[^49]: In the terminology of [@Gaiotto:2014kfa].
[^50]: In terms independent of the choice of basis vectors of the root lattice, the $\Z_N$ center acts on the dual photons ${\sigma}$ as the ordered product of Weyl reflections with respect to all simple roots, see [@Anber:2015wha].
[^51]: See [@16] for a description of confining strings in softly-broken Seiberg-Witten theory within its $M$-theory embedding.
[^52]: As discussed in Section \[sec:5.1.1\], one of the qualitative reasons why charges $\mu_k$ are confined by strings of the lowest tension (for every representation) is that adding or subtracting any root from $\mu_k$ leads to higher “vacuum energy” cost.
[^53]: Ref. [@13] studied a rotating string solution, but a simpler static one exists, see discussion below and ref. [@Hasenfratz:1977dt], which also contains a review of the physical picture underlying the MIT Bag Model of the Yang-Mills vacuum.
[^54]: The classical chromoelectric flux of static sources in a given representation is proportional to the quadratic Casimir, see Section 3.3 in [@Hasenfratz:1977dt]. Also note that the “square root of Casimir” scaling is obtained in the Bag Model without surface tension and that introducing additional Bag Model parameters, e.g. bag surface tension, modifies the scaling with the Casimir of the representation.
[^55]: Noise of order $\epsilon$ refers to a random fluctuation of order $\epsilon$ imposed on the data. The fluctuation can be a Gaussian, uniform, etc., distribution of width $\epsilon$ centred on the data point. We have used a uniform distribution.
[^56]: We consider half of the upper bound estimate of the error in Table \[table:1\] ($-0.006/2 = -0.003$) as the value of error for k-strings. Hence for k-string ratios as a typical example we get: $T_2/T_1 = 8.0006_{-0.003}/6.8583_{-0.003} \approx 1.1666^{+ 0.0005}_{- 0.0004} $. The reader has to be reminded that an error of $-0.003$ is still a high confidence interval for the true value of k-strings.
[^57]: For another discussion on the non-commutativity of the large-$N$ and large-$T$ limits refer to [@62].
[^58]: See further below the discussion of this Section (between eqs. and ) as well as the explicit calculations in Appendix \[sec:appxproduct\].
[^59]: Note that due to the $Z_N$ symmetry of the $N$ components of $\sigma_a$, $1 \leq a \leq N$ can be considered similar to $N$ points on a circle corresponding to angles $\theta = {2 \pi a / N }$. The components near the $q$’th component are defined as the points (components) close to the $q$’th point on this circle.
[^60]: In components, the weights of the product representation are $(\mu^{(ij)}_{\square \otimes \square})_p = \delta^i_p + \delta^j_p -2/N$. The $\Z_N$ symmetry acts as $\mu^{(ij)}_{\square \otimes \square} \rightarrow \mu^{(i+1({\rm mod} N),j+1({\rm mod} N))}_{\square \otimes \square}$, i.e. the $N^2$ weights of the product representation fall into $N$ $\Z_N$ orbits.
[^61]: In this regard, notice that the components of $\mu^{(ij)}_{\square \otimes \square}$ with $i \ne j$ can also be written as $-1/(N/2)$ or $1- 1/(N/2)$, similar to the components of $\mu^i_\square$ for $SU([N/2])$ (when $N$ is odd the difference would be clearly negligible).
[^62]: At larger values of $N$, as mentioned in the preamble of Section \[sec:largeN\], the virtual effects of the W-bosons become important which has not been taken into account in this work. We speculate, based on preliminary results, that with taking these effects into account the same picture, i.e. large $N$ factorization and interacting k-strings, persists at large $N$.
[^63]: This behaviour has been verified in the numerical simulations up to $J=14$. It has also been verified for cases when an analytic solution is possible. For example expanding the cosine term and keeping only the quadratic term. For this case it would be possible to solve the saddle point analytically for a finite boundary condition at $\text{z} = J$ and see that $\bar{T}^{\infty,J}_{k2,min}$ starts from $0$ at $\text{z}=0$ and increases monotonically to ${1\over 2} \bar{T}^{\infty, \infty}_{k,\text{min}}$ at $\text{z} = \infty$.
[^64]: To obtain , we noted that for large and odd $N$, for $1 \le q < {N\over 2}$: $$\label{eq:E.6}
\begin{split}
{b_q \over 2\pi} = \sum\limits_{j=1}^N (\mu^h)_j D_{jq} = \sqrt{{2 \over N}} \{ \sin{2 \pi a q \over N} + \sin{2 \pi b q \over N} - \sum\limits_{j=1}^N {2 \over N} \sin{2 \pi j q \over N} \} \approx \sqrt{{2 \over N}} \{ \sin{2 \pi a q \over N} + \sin{2 \pi b q \over N} + 0 \},
\end{split}$$ and used the fact that the last term in the first line of for large $N$ can be approximated by $\approx -{1 \over \pi}\int_0^{2 \pi}d y \sin q y = 0$. Also, a similar expression can be written for ${N\over 2} < q < N$.
|
---
abstract: 'Colloids immersed in a critical or near-critical binary liquid mixture and close to a chemically patterned substrate are subject to normal and lateral critical Casimir forces of dominating strength. For a single colloid we calculate these attractive or repulsive forces and the corresponding critical Casimir potentials within mean-field theory. Within this approach we also discuss the quality of the Derjaguin approximation and apply it to Monte Carlo simulation data available for the system under study. We find that the range of validity of the Derjaguin approximation is rather large and that it fails only for surface structures which are very small compared to the geometric mean of the size of the colloid and its distance from the substrate. For certain chemical structures of the substrate the critical Casimir force acting on the colloid can change sign as a function of the distance between the particle and the substrate; this provides a mechanism for stable levitation at a certain distance which can be strongly tuned by temperature, i.e., with a sensitivity of more than $200 \textrm{nm}/\textrm{K}$.'
author:
- 'M. Tr[ö]{}ndle'
- 'S. Kondrat'
- 'A. Gambassi'
- 'L. Harnau'
- 'S. Dietrich'
date: 'May, 6 2010'
nocite:
- '[@krech:book]'
- '[@brankov:book]'
- '[@krech:9192all]'
- '[@evans:1994]'
- '[@diehl:2006]'
- '[@zandi:2007]'
- '[@schmidt:2008]'
- '[@mohry:2009]'
- '[@garcia:9902all]'
- '[@ganshin:2006]'
- '[@fukuto:2005]'
- '[@rafai:2007]'
- '[@hucht:2007]'
- '[@vasilyev:2007]'
- '[@vasilyev:2009]'
- '[@Burkhardt:1995]'
- '[@Eisenriegler:1995]'
- '[@hanke:1998]'
- '[@Schlesener:2003]'
- '[@Eisenriegler:2004]'
- '[@troendle:2009]'
- '[@soyka:2008]'
- '[@sprenger:2006]'
- '[@troendle:2008]'
title: Critical Casimir effect for colloids close to chemically patterned substrates
---
Introduction
============
Since the discovery of the Casimir effect in quantum electrodynamics [@casimir:1948; @kardar:1999] it is well-known that the inherent fluctuations of a medium lead to an effective force acting on its confining boundaries. In soft matter physics, the analogue of the vacuum fluctuations in quantum electrodynamics are the thermal fluctuations of the order parameter $\phi$ of a fluid. These occur on the length scale of the bulk correlation length $\xi$ which is generically of molecular size. However, upon approaching a critical point at the temperature ${T=T_c}$, the correlation length $\xi$ increases with an algebraic singularity and attains *macroscopic* values. The confinement of these long-ranged fluctuations results in the so-called critical Casimir force acting on a length scale set by $\xi$ [@fisher:1978]. Since the correlation length diverges as ${\xi(T\to T_c)\propto|T-T_c|^{-\nu}}$, where $\nu$ is a standard bulk critical exponent, the range of the critical Casimir force (and therefore its strength at a certain distance) can be controlled and tuned by minute temperature changes (see, e.g., Refs. ). The characteristic energy scale of the critical Casimir effect is given by $k_B T_c$, which allows for a direct measurement of the critical Casimir forces, in particular if the critical point is located at ambient thermodynamic conditions [@hertlein:2008; @gambassi:2009].
The attractive or repulsive character of the critical Casimir force can be controlled by suitable treatments of the confining surfaces. Generically, the surfaces which confine a binary liquid mixture preferentially adsorb one of its two components (or the gas or liquid phase in the case of a one-component fluid). This can be described by effective, symmetry breaking surface fields, which lead to a preference for either positive $[(+)]$ or negative $[(-)]$ values of the scalar order parameter $\phi$, corresponding to the difference between the local concentrations of the two species (or the deviation of the density of the one-component fluid from its critical value). The critical Casimir force strongly depends on the effective boundary conditions (BC) at the walls (see, e.g., Refs. and references therein). It is attractive for equal symmetry breaking $(\pm,\pm)$ BC and repulsive for opposing $(\pm,\mp)$ BC. Inter alia, this latter feature qualifies critical Casimir forces to be a tool to overcome the problem of “stiction” which occurs in micro- and nano-mechanical devices. (The quantum electrodynamic Casimir force is typically attractive and thus responsible for stiction; turning it to be repulsive requires a careful choice of the fluid and of the bulk materials of the confinement [@Munday:2009].) The theoretical description of the critical Casimir forces is particularly challenging due to the non-Gaussian character of the order parameter fluctuations, which contrasts with the intrinsically Gaussian nature of the low energy fluctuations of the electromagnetic field; in addition, the critical Casimir effect is also particularly rich as it allows, inter alia, symmetry breaking boundary conditions, which do not occur for electromagnetic fields.
The critical Casimir effect exhibits universality, i.e., the critical Casimir force expressed in terms of suitable scaling variables depends only on the universality class of the bulk critical point and on the type of boundary conditions, whereas it is independent of the microscopic structure and of the material properties of the specific fluid medium involved. In our present theoretical analysis we focus on the Ising universality class which encompasses the experimentally relevant classical binary liquid mixtures and simple fluids.
The existence of the critical Casimir effect has been experimentally confirmed and its strength has been first measured [indirectly]{} for wetting films [@garcia:9902all; @fukuto:2005; @ganshin:2006; @rafai:2007]. The first [direct]{} measurement of this effect has been performed at the sub-micrometer scale for a spherical *colloid* immersed in a (near) critical binary liquid mixture close to a laterally homogeneous and planar substrate [@hertlein:2008; @gambassi:2009]. The corresponding Monte Carlo simulation data for the *film* geometry are in very good quantitative agreement with all available experimental data [@hertlein:2008; @gambassi:2009; @hucht:2007; @vasilyev:2007; @vasilyev:2009; @Hasenbusch:2009]. Theoretical studies of the critical Casimir effect acting on [colloidal]{} particles involve spherically [@Burkhardt:1995; @Eisenriegler:1995; @hanke:1998; @Schlesener:2003; @Eisenriegler:2004] or ellipsoidally [@kondrat:2009] shaped colloids adjacent to *homogeneous* substrates.
![ Sketch of a spherical colloid immersed in a near-critical binary liquid mixture (not shown) and close to a (patterned) planar substrate. The sphere with $(b)$ boundary condition (BC) and radius $R$ is located at a surface-to-surface distance $D$ from the substrate and its center has a lateral coordinate $x=X$ with the substrate pattern being translationally invariant in all other directions. The following four different types of substrate surfaces are considered: homogeneous substrate \[Sec. \[sec:homog\]\], a chemical step \[${\ensuremath{\textrm{s}}}$; Sec. \[sec:step\]\], a single chemical lane \[${\ensuremath{\ell}}$; Sec. \[sec:stripe\]\], and a periodically patterned substrate \[${\ensuremath{\textrm{p}}}$; Sec. \[sec:period\]\]. (Note that for a four-dimensional system, which we also consider, this is a three-dimensional cut of the system, which is invariant along the fourth direction; the sphere thus corresponds to a hypercylinder in four dimensions.) For later reference, the box on the left side summarizes the definitions of the various scaling variables which the scaling functions of the critical Casimir force depend on for the listed geometrical configurations. On the right, $(a)$, $(a_\gtrless)$, $(a_{{\ensuremath{\ell}}})$, $(a_1)$, and $(a_2)$ indicate the boundary conditions corresponding to the various chemical patterns. []{data-label="fig:sketch"}](all_sub)
Besides their wide use as model systems in soft matter physics, colloids have applications at the micro- and nanometer scale. In this context, they are widely used in micro- and nano-mechanical devices. Therefore, one may utilize the critical Casimir forces acting on colloids because their strength and their direction can be tuned in a controlled way. Suitably designed chemically or geometrically structured substrates generate *lateral* critical Casimir forces acting on colloidal particles [@soyka:2008; @troendle:2008; @troendle:2009; @sprenger:2006]. Current techniques allow one to endow solid surfaces with precise structures on the nano- and micrometer-scale. Hence, the critical Casimir effect can be used to create laterally confining potentials for a single colloid, which can be tuned by temperature [@soyka:2008].
Recently, the critical Casimir potential of a colloid close to a substrate with a pattern of parallel chemical stripes with laterally alternating adsorption preference has been measured [@soyka:2008]. In our corresponding theoretical study [@troendle:2009], we have calculated the normal and lateral critical Casimir forces acting on a colloid close to such a patterned substrate as well as the corresponding potentials. We have used our theoretical predictions for the universal scaling functions of the critical Casimir potential in order to interpret the available experimental data in Ref. . It has turned out that an agreement between theory and experiment can be achieved only if one takes into account the geometrical details of the chemical substrate pattern. This demonstrates that the critical Casimir effect is very sensitive to the details of the imprinted structures and that it can resolve them.
Here we generalize our previous analysis [@troendle:2009] to various substrate patterns. In particular we study the critical Casimir effect for a *three*-dimensional sphere close to a homogeneous substrate \[Sec. \[sec:homog\]\], a chemical step \[Sec. \[sec:step\]\], a single chemical lane \[Sec. \[sec:stripe\]\], and periodic patterns of chemical stripes of alternating adsorption preference \[Sec. \[sec:period\]\] \[see [Fig. \[fig:sketch\]]{}\]. For completeness, we also consider a *cylinder* which is aligned with the chemical pattern \[Sec. \[sec:cylinder\]\]. We provide quantitative predictions for the scaling functions of the critical Casimir forces, pursuing a two-pronged approach: (i) We calculate the force using the full three-dimensional numerical analysis of the appropriate mean-field theory (MFT). (ii) We use the so-called Derjaguin approximation (DA) based on the scaling functions for the critical Casimir force in the film geometry either obtained analytically within MFT [@krech:1997] or obtained from Monte Carlo simulations [@vasilyev:2007; @vasilyev:2009], which allows us to predict the critical Casimir force in the physically relevant three-dimensional case. Inter alia, we determine the range of validity of the DA within MFT, which provides guidance concerning its applicability in three spatial dimensions $d=3$. This is an important information because presently available Monte Carlo simulations are far from being able to capture complex geometries [@vasilyev:2007; @vasilyev:2009].
Currently, the possibility of realizing stable levitation of particles by means of the electrodynamic Casimir forces has been the subject of intense theoretical investigation [@leonhardt:2007; @rodriguez:2008; @rodriguez:2009; @rahi:2009; @rahi:2009a; @zhao:2009]. Our results presented in Secs. \[sec:period\] and \[sec:cylinder\] show that for suitable choices of the geometry of the chemical pattern of the substrate, the critical Casimir forces can be used to levitate a colloid above the substrate at a height which can be tuned by temperature. This levitation is stable against perturbations because it corresponds to a minimum of the potential of the critical Casimir force acting on the colloid.
In Sec. \[sec:background\] we briefly introduce the necessary terminology related to finite-size scaling and we discuss briefly the corresponding MFT. Section \[sec:homog\] is devoted to the well-studied case of a colloid close to a homogeneous substrate. (In $d=4$, as appropriate for MFT, the three-dimensional colloid is extended to the fourth dimension as a hypercylinder, for which we also present the results of our analysis.) As mentioned above, the various patterns and setups are considered in Secs. \[sec:step\]–\[sec:cylinder\]. We conclude and summarize our findings in Sec. \[sec:summary\]. Certain important technical details concerning the calculation of the Derjaguin approximation are presented in the Appendices \[app:step\]–\[app:cylinder\].
Theoretical background \[sec:background\]
=========================================
Finite-size scaling
-------------------
According to the theory of finite-size scaling, the normal and lateral critical Casimir forces and the corresponding potentials can be described by *universal* scaling functions, which are independent of the molecular details of the system but depend only on the gross features of the system, i.e., on the bulk universality class (see, e.g., Refs. and references therein) of the associated critical point. Here, we focus on the Ising universality class (which is characterized by a scalar order parameter $\phi$) in spatial dimensions $d=3$ and $d=4$. In addition, the critical Casimir force depends on the type of effective boundary conditions at the walls, which we denote by $(a)$ and $(b)$, and by the geometry of the confining surfaces [@binder:1983; @diehl:1986; @diehl:1997]. Note that $(a)$ and $(b)$ can represent the various symmetry preserving fixed-point BC (the so-called ordinary, special, periodic, or antiperiodic boundary conditions [@krech:book; @brankov:book]) in addition to the symmetry breaking cases $(\pm)$ we are mainly interested in, and which describe the adsorption of fluids at the confining walls.
Inspired by the experiments described in Ref. we consider binary liquid mixtures with their consolute critical point approached by varying the temperature $T$ towards $T_c$ at fixed pressure and critical composition. We first study the *film* geometry in which the fluid undergoing the continuous phase transition is confined between two parallel, infinitely extended walls at distance $L$. According to renormalization group theory the normal critical Casimir force $f_{(a,b)}$ per unit area which is acting on the walls scales as [@krech:9192all] $$\label{eq:planar-force}
f_{(a,b)}(L,T)=k_BT \frac{1}{L^d}k_{(a,b)}( \operatorname{sign}(t)\, L/\xi_\pm),$$ where $(a,b)$ denotes the pair of boundary conditions $(a)$ and $(b)$ characterizing the two walls. The scaling function $k_{(a,b)}$ depends only on a single scaling variable given by the sign of the reduced temperature distance $t$ from the critical point ($\pm$ for $t\gtrless0$) and the film thickness $L$ in units of the bulk correlation length $\xi_\pm(t\to0^\pm)=\xi_0^\pm|t|^{-\nu}$, where $\nu\simeq0.63$ in $d=3$ and $\nu=1/2$ in $d=4$ [@pelissetto:2002]. (Clearly, one has $f_{(a,b)}(L,T)=f_{(b,a)}(L,T)$.) Positive values of $t$, $t>0$, correspond to the disordered (homogeneous) phase of the fluid, whereas negative values of $t$, $t<0$, correspond to the ordered (inhomogeneous) phase, where phase separation occurs. Typically, the homogeneous phase is found at high temperatures, and one has $t=(T-T_c)/T_c$. However, many experimentally relevant binary liquid mixtures exhibit a *lower* critical point, for which the homogeneous phase corresponds to the low-temperature phase and one has $t=-(T-T_c)/T_c$ [@hertlein:2008; @gambassi:2009]. The two non-universal amplitudes $\xi_0^\pm$ of the correlation length are of molecular size and characterized by the universal ratio $\xi_0^+/\xi_0^-\simeq1.9$ in $d=3$ [@pelissetto:2002; @privman:1991] and $\xi_0^+/\xi_0^-=\sqrt{2}$ in $d=4$ [@tarko:all]; $\xi_\pm$ is determined by the exponential spatial decay of the two-point correlation function of the order parameter $\phi$ in the bulk.
At the critical point $T=T_c$, the correlation length diverges, $\xi_\pm\to\infty$, and the scaling function of the critical Casimir force acting on the two planar walls attains a universal constant value referred to as the critical Casimir amplitude [@krech:book; @brankov:book]: $$\label{eq:delta-ab}
k_{(a,b)}(L/\xi_\pm=0)={\ensuremath{\Delta_{{\ensuremath{(a,b)}}}}}.$$
Away from criticality, the critical Casimir force decays exponentially as a function of $L/\xi_\pm$. For the specific case of symmetry breaking BC $a,b\in\{+,-\}$ and for $t>0$ one expects for ${L/\xi_+\gg1}$ a *pure* exponential decay of ${\ensuremath{f_{{\ensuremath{(+,\pm)}}}}}$ (see, e.g., Refs. and footnote $3$ in Ref. , i.e., a decay without an algebraic prefactor to the exponential and without a numerical prefactor to $L/\xi_+$ in the argument of the exponential) corresponding to $$\label{eq:exponential-decay}
k_{(+,\pm)}(L/\xi_+\gg1)=A_\pm \left(\frac{L}{\xi_+}\right)^d \exp(-L/\xi_+),$$ where $A_\pm$ are universal constants [@gambassi:2009]. Note that, in the absence of symmetry-breaking fields inside the film, the scaling functions for $(+,+)$ BC are the same as for $(-,-)$ BC.
Mean-field theory \[sec:MFT\]
-----------------------------
The standard Landau-Ginzburg-Wilson fixed-point effective Hamiltonian describing critical phenomena of the Ising universality class is given by [@binder:1983; @diehl:1986] $$\label{eq:hamiltonian}
\mathcal{H}[\phi]=\int_V\,{{\ensuremath{\textrm{d}}}}^d{\mathbf{r}}\,\left\{
\frac{1}{2}(\nabla\phi)^2
+\frac{\tau}{2}\phi^2
+\frac{u}{4!}\phi^4
\right\},$$ where $\phi({\mathbf{r}})$ is the order parameter describing the fluid, which completely fills the volume $V$ in $d$-dimensional space. The first term in the integral in [Eq. ]{} penalizes local fluctuations of the order parameter. The parameter $\tau$ in [Eq. ]{} is proportional to $t$, and the coupling constant $u$ is positive and provides stability of the Hamiltonian for $t<0$. The mean-field order parameter profile $m{\ensuremath \mathrel{\mathop:}=}u^{1/2}\langle\phi\rangle$ minimizes the Hamiltonian, i.e., $\updelta \mathcal{H}[\phi]/\updelta\phi|_{\phi=u^{-1/2}m}=0$. In the bulk the mean-field order parameter is spatially constant and attains the values $\langle\phi\rangle=\pm a|t|^\beta$ for $t<0$ and $\langle\phi\rangle=0$ for $t>0$, where, besides $\xi_0^+$, $a$ is the only additional independent non-universal amplitude appearing in the description of bulk critical phenomena [@binder:1983; @diehl:1986], and $\beta(d=4)=1/2$ is a standard critical exponent. Within MFT $\tau=t (\xi_0^+)^{-2}$ and $u=6a^2(\xi_0^+)^{-2}$. In a finite-size system the bulk Hamiltonian $\mathcal{H}[\phi]$ is supplemented by appropriate surface and curvature (edge) contributions [@binder:1983; @diehl:1986]. In the strong adsorption limit [@burkhardt:1994; @diehl:1993], these contributions generate boundary conditions for the order parameter such that $\phi\big|_{\text{surface}}=\pm\infty$. For binary liquid mixtures these fixed-point $(\pm)$ BC are the experimentally relevant ones. (Note that a *weak* adsorption preference might lead to a crossover between various kinds of effective boundary conditions for the order parameter $\phi$ [@mohry:2009; @schmidt:2008; @gambassi:2009].)
We have minimized numerically $\mathcal{H}[\phi]$ using a $3d$ finite element method in order to obtain the (spatially inhomogeneous) profile $m({\mathbf{r}})$ for the geometries under consideration \[see [Fig. \[fig:sketch\]]{}\]. The normal and the lateral critical Casimir forces are calculated directly from these mean-field order parameter profiles using the stress tensor [@krech:1997; @kondrat:2009]. This allows one to infer the universal scaling functions of the critical Casimir forces at the upper critical dimension $d=4$ up to an overall prefactor $\propto u^{-1}$ and up to logarithmic corrections. The corresponding critical Casimir potential is obtained by the appropriate integration of the normal or of the lateral critical Casimir forces.
In the case of planar walls the MFT scaling functions for the critical Casimir force can be determined analytically [@krech:1997] and one finds \[see [Eq. ]{}\] for the case of symmetry breaking boundary conditions the following critical Casimir amplitudes: ${\ensuremath{\Delta_{{\ensuremath{(+,+)}}}}}={\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}}=24[K(1/\sqrt{2})]^4/u\simeq-283.61\times u^{-1}$, where $K$ is the complete elliptic integral of the first kind, and ${\ensuremath{\Delta_{{\ensuremath{(+,-)}}}}}=-4{\ensuremath{\Delta_{{\ensuremath{(+,+)}}}}}$ \[see Ref. and Eq. (27) and Ref. \[49\] in Ref. \].
In ${d=4}$ (corresponding to MFT) the three-dimensional sphere is a hypercylinder and the physical properties are invariant along the fourth dimension. Accordingly, the MFT results for the force and the potential given below are those per length along this additional direction.
Homogeneous substrate \[sec:homog\]
===================================
We first consider a three-dimensional sphere of radius $R$ with $(b)$ BC facing a chemically *homogeneous* substrate with $(a)$ BC at a surface-to-surface distance $D$ as shown in [Fig. \[fig:sketch\]]{}, denoting this combination by $(a,b)$. The critical Casimir force ${\ensuremath{F_{{\ensuremath{(a,b)}}}}}(D,R,T)$ *normal* to the substrate surface and the corresponding critical Casimir potential ${\ensuremath{\Phi_{{\ensuremath{(a,b)}}}}}(D,R,T)=\int_D^\infty {{\ensuremath{\textrm{d}}}}z\; {\ensuremath{F_{{\ensuremath{(a,b)}}}}}(z,R,T)$ take the scaling forms [@hanke:1998; @hertlein:2008; @troendle:2009; @gambassi:2009] $$\begin{aligned}
\label{eq:force-homog}
{\ensuremath{F_{{\ensuremath{(a,b)}}}}}(D,R,T)&=k_BT\frac{R}{D^{d-1}}{\ensuremath{K_{{\ensuremath{(a,b)}}}}}(\Theta,\Delta)\\
\intertext{and}
\label{eq:potential-homog}
{\ensuremath{\Phi_{{\ensuremath{(a,b)}}}}}(D,R,T)& =k_BT\frac{R}{D^{d-2}}{\ensuremath{\vartheta_{{\ensuremath{(a,b)}}}}}(\Theta,\Delta),\end{aligned}$$ where $\Delta={D}/{R}$ and $\Theta=\operatorname{sign}(t)\,{D}/{\xi_\pm}$ (for $t\gtrless 0$) are the scaling variables corresponding to the distance $D$ in units of the radius $R$ of the colloid and of the correlation length $\xi_\pm$, respectively. The case $d=4$ corresponds to the MFT solution up to logarithmic corrections, which we shall neglect here. Equations and describe a force and an energy, respectively, per $D^{d-3}$, which for $d=4$ corresponds to considering ${\ensuremath{F_{{\ensuremath{(a,b)}}}}}$ and ${\ensuremath{\Phi_{{\ensuremath{(a,b)}}}}}$ per length $L_4$ of the extra translationally invariant direction of the hypercylinder.
Derjaguin approximation \[sec:homog-da\]
----------------------------------------
The Derjaguin approximation (DA) is based on the idea of decomposing the surface of the spherical colloid into infinitely thin circular rings of radius $\rho$ and area ${{\ensuremath{\textrm{d}}}}S (\rho)= 2\pi\rho{{\ensuremath{\textrm{d}}}}\rho$ which are parallel to the opposing substrate surface [@derjaguin:1934; @hanke:1998; @hertlein:2008; @gambassi:2009; @troendle:2009]. (Here we do not multiply $2\pi\rho{{\ensuremath{\textrm{d}}}}\rho$ by the linear extension $L_4$ of the hypercylinder along its axis in the fourth dimension, because the critical Casimir force is eventually expressed in units of $L_4$, which therefore drops out from the final expressions.) The distance $L$ of a ring with radius $\rho$ from the substrate is given by $$\label{eq:da-L}
L(\rho)= D+R\left(1-\sqrt{1-{\rho^2}/{R^2}}\right).$$ Assuming *additivity* of the forces and neglecting edge effects, the normal critical Casimir forces ${{\ensuremath{\textrm{d}}}}F(\rho)$ acting on these rings can be expressed in terms of the force acting on parallel plates \[[Eq. ]{}\]: $$\label{eq:da-dF}
\frac{{{{\ensuremath{\textrm{d}}}}F(\rho)}}{k_BT} = \frac{{{\ensuremath{\textrm{d}}}}S}{\left[L(\rho)\right]^{d}} {\ensuremath{k_{{\ensuremath{(a,b)}}}}}(\operatorname{sign}(t)\, L(\rho)/\xi_\pm).$$ Finally, in order to calculate the total force ${\ensuremath{F_{{\ensuremath{(a,b)}}}}}$ acting on the colloid, one sums up the contributions of the rings, which yields $$\label{eq:da-def}
\frac{{\ensuremath{F_{{\ensuremath{(a,b)}}}}}(D,R,T)}{k_BT}\simeq2\pi \int_0^R {{\ensuremath{\textrm{d}}}}\rho \rho \left[L(\rho)\right]^{-d} {\ensuremath{k_{{\ensuremath{(a,b)}}}}}(\operatorname{sign}(t)\, L(\rho)/\xi_\pm).$$ (For $d=3$, ${\ensuremath{F_{{\ensuremath{(a,b)}}}}}$ is the force on a sphere whereas in $d=4$ it is the force on a hypercylinder per length of its axis.)
![ (a) Scaling functions ${\ensuremath{K_{{\ensuremath{(\mp,-)}}}}}$ for the normal critical Casimir force \[[Eq. ]{}\] acting on a three-dimensional sphere with $(b)={\ensuremath{(-)}}$ BC close to a homogeneous substrate with $(a)=(\mp)$ BC \[[Fig. \[fig:sketch\]]{}\]. The suitably normalized scaling functions ${\ensuremath{K_{{\ensuremath{(\mp,-)}}}}}$ are shown as a function of the scaling variable $\Theta=\operatorname{sign}(t) D/\xi_\pm$ for $t\gtrless0$, where $t$ is the reduced deviation from the critical temperature and ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}(0,0)$ is the value of the critical Casimir force scaling function within the DA at $T=T_c$ for $(-,-)$ BC. The solid lines correspond to the Derjaguin approximation (DA, $\Delta=D/R\to0$) within mean-field theory (MFT, $d=4$) whereas the dotted lines correspond to the DA obtained by using Monte Carlo (MC) results for films in $d=3$ the systematic uncertainties of which are not indicated [@mcdata]. The normalization implies that at $\Theta=0$ both the solid and dotted lines pass through $-1$ for ${\ensuremath{(-,-)}}$ BC whereas the solid line passes through $4$ for ${\ensuremath{(+,-)}}$ BC. The symbols correspond to the full numerical MFT results obtained for $\Delta=1/3$ and $\Delta=1$, the size of which indicates the estimated numerical error. (For ${\ensuremath{(+,-)}}$ BC and $t<0$ we have not been able to calculate the corresponding scaling functions with adequate precision due to severe numerical difficulties in obtaining the full three-dimensional order parameter profile in the presence of two “competing” bulk values.) Since within the DA the dependence of ${\ensuremath{K_{{\ensuremath{(\mp,-)}}}}}$ on $\Delta$ drops out, the difference between the symbols $\boxdot$ and $\odot$ and the solid lines measures the accuracy of the DA in $d=4$. (b) Difference $\Delta\vartheta={\ensuremath{\vartheta_{{\ensuremath{(+,-)}}}}}-{\ensuremath{\vartheta_{{\ensuremath{(-,-)}}}}}$ of the scaling functions for the Casimir potentials \[[Eq. ]{}\] for ${\ensuremath{(+,-)}}$ and ${\ensuremath{(-,-)}}$ BC, suitably normalized by ${\ensuremath{\vartheta_{{\ensuremath{(-,-)}}}}}(0,0)$. The solid line corresponds to the DA within MFT and the symbols correspond to the full MFT results for $\Delta=1/3$ and $\Delta=1$; the dotted line is the DA for $d=3$. Due to the normalization the solid line reaches $5$ for $\Theta=0$. []{data-label="fig:homog"}](homog_force "fig:")\
![ (a) Scaling functions ${\ensuremath{K_{{\ensuremath{(\mp,-)}}}}}$ for the normal critical Casimir force \[[Eq. ]{}\] acting on a three-dimensional sphere with $(b)={\ensuremath{(-)}}$ BC close to a homogeneous substrate with $(a)=(\mp)$ BC \[[Fig. \[fig:sketch\]]{}\]. The suitably normalized scaling functions ${\ensuremath{K_{{\ensuremath{(\mp,-)}}}}}$ are shown as a function of the scaling variable $\Theta=\operatorname{sign}(t) D/\xi_\pm$ for $t\gtrless0$, where $t$ is the reduced deviation from the critical temperature and ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}(0,0)$ is the value of the critical Casimir force scaling function within the DA at $T=T_c$ for $(-,-)$ BC. The solid lines correspond to the Derjaguin approximation (DA, $\Delta=D/R\to0$) within mean-field theory (MFT, $d=4$) whereas the dotted lines correspond to the DA obtained by using Monte Carlo (MC) results for films in $d=3$ the systematic uncertainties of which are not indicated [@mcdata]. The normalization implies that at $\Theta=0$ both the solid and dotted lines pass through $-1$ for ${\ensuremath{(-,-)}}$ BC whereas the solid line passes through $4$ for ${\ensuremath{(+,-)}}$ BC. The symbols correspond to the full numerical MFT results obtained for $\Delta=1/3$ and $\Delta=1$, the size of which indicates the estimated numerical error. (For ${\ensuremath{(+,-)}}$ BC and $t<0$ we have not been able to calculate the corresponding scaling functions with adequate precision due to severe numerical difficulties in obtaining the full three-dimensional order parameter profile in the presence of two “competing” bulk values.) Since within the DA the dependence of ${\ensuremath{K_{{\ensuremath{(\mp,-)}}}}}$ on $\Delta$ drops out, the difference between the symbols $\boxdot$ and $\odot$ and the solid lines measures the accuracy of the DA in $d=4$. (b) Difference $\Delta\vartheta={\ensuremath{\vartheta_{{\ensuremath{(+,-)}}}}}-{\ensuremath{\vartheta_{{\ensuremath{(-,-)}}}}}$ of the scaling functions for the Casimir potentials \[[Eq. ]{}\] for ${\ensuremath{(+,-)}}$ and ${\ensuremath{(-,-)}}$ BC, suitably normalized by ${\ensuremath{\vartheta_{{\ensuremath{(-,-)}}}}}(0,0)$. The solid line corresponds to the DA within MFT and the symbols correspond to the full MFT results for $\Delta=1/3$ and $\Delta=1$; the dotted line is the DA for $d=3$. Due to the normalization the solid line reaches $5$ for $\Theta=0$. []{data-label="fig:homog"}](homog_pot "fig:")
One expects the DA to describe the actual behavior accurately if the colloid is very close to the substrate, i.e., for $\Delta=D/R\to0$. In this limit, [Eq. ]{} can be approximated by $L(\rho)=D\alpha$ where $\alpha=1+\rho^2/(2RD)$, so that one finds for the scaling function of the force [@hanke:1998; @gambassi:2009] $$\label{eq:da-force}
{\ensuremath{K_{{\ensuremath{(a,b)}}}}}(\Theta,\Delta\to0)=2\pi\int_{1}^{\infty}{{\ensuremath{\textrm{d}}}}\alpha\alpha^{-d}{\ensuremath{k_{{\ensuremath{(a,b)}}}}}(\alpha\Theta),$$ and, accordingly, for the scaling function of the potential [@hertlein:2008; @troendle:2009; @gambassi:2009] $${\ensuremath{\vartheta_{{\ensuremath{(a,b)}}}}}(\Theta,\Delta\to0)=\\
2\pi\int_{1}^{\infty}{{\ensuremath{\textrm{d}}}}\beta\left( \frac{1}{\beta^{d-1}}-\frac{1}{\beta^d} \right){\ensuremath{k_{{\ensuremath{(a,b)}}}}}(\beta\Theta).
\label{eq:derjaguinpot}$$ At the bulk critical point, using [Eq. ]{}, one finds the well known values ${{\ensuremath{K_{{\ensuremath{(a,b)}}}}}(0,0)} = {2\pi{\ensuremath{\Delta_{{\ensuremath{(a,b)}}}}}/(d-1)}$ and ${{\ensuremath{\vartheta_{{\ensuremath{(a,b)}}}}}(0,0)} = {2\pi{\ensuremath{\Delta_{{\ensuremath{(a,b)}}}}}/[(d-2)(d-1)]}$. We note that the DA implies that the dependence of ${\ensuremath{F_{{\ensuremath{(a,b)}}}}}$ and ${\ensuremath{\Phi_{{\ensuremath{(a,b)}}}}}$ on the size $R$ of the sphere reduces to the proportionality $\propto R$ indicated explicitly in Eqs. and .
Before proceeding further one first has to assess the accuracy of the DA, which will carried out below within MFT ($d=4$). We expect the range of validity of the DA to be similar for $d=3$, so that within that range one can use the DA based on scaling functions for the film geometry obtained from Monte Carlo simulations [@mcdata] in order to calculate the critical Casimir force acting on a colloid in $d=3$.
Scaling functions for the normal critical Casimir force and the potential
-------------------------------------------------------------------------
The expressions obtained above within the DA hold for general boundary conditions $(a)$ and $(b)$ and are valid beyond the cases we consider in the following, i.e., $a\in\{+,-\}$ and $b=-$. Figure \[fig:homog\](a) shows the full numerical MFT ($d=4$) results for the scaling functions ${\ensuremath{K_{{\ensuremath{(\pm,-)}}}}}$ with $\Delta=1$ and $\frac{1}{3}$ compared with the corresponding DA results based on the suitable numerical integration \[[Eq. ]{}\] of the analytic (MFT) expression for ${\ensuremath{k_{{\ensuremath{(\pm,-)}}}}}$ [@krech:1997]. Moreover, in [Fig. \[fig:homog\]]{}, the corresponding DA results for $d=3$ are shown; they are obtained from the film scaling functions determined by MC simulations [@mcdata] and by using the corresponding ratio of the correlation lengths above and below $T_c$ [@pelissetto:2002]. In [Fig. \[fig:homog\]]{}(b) we report the difference $\Delta\vartheta(\Theta,\Delta){\ensuremath \mathrel{\mathop:}=}{\ensuremath{\vartheta_{{\ensuremath{(+,-)}}}}}(\Theta,\Delta)-{\ensuremath{\vartheta_{{\ensuremath{(-,-)}}}}}(\Theta,\Delta)$ computed for the various cases reported in [Fig. \[fig:homog\]]{}(a), which will be useful for describing the case of a chemically patterned substrate. The scaling functions in $d=4$ are reasonably well reproduced by the DA for $\Delta\lesssim0.4$ and we expect this to hold for $d=3$ as well. The fact that for increasing values of $\Delta$ the magnitude of the actual scaling functions becomes larger compared with those within the DA (corresponding to $\Delta\to0$) is in agreement with earlier results obtained for a $d$-dimensional hypersphere (see, e.g., Ref. ).
It has been shown that the scaling functions obtained within the DA for $d=3$ agree very well – within the experimental accuracy – with the ones obtained from direct measurements of the critical Casimir potential [@hertlein:2008; @gambassi:2009] corresponding to $\Delta\lesssim0.35$ (see also Ref. \[48\] in Ref. ).
Chemical step (${\ensuremath{\textrm{s}}}$) \[sec:step\]
========================================================
The basic building block of a chemically patterned substrate of the type we consider here, i.e., with translational invariance in all directions but one ($x$), is a chemical step (${\ensuremath{\textrm{s}}}$) realized by a substrate with $(a_\gtrless)$ BC for $x\gtrless0$ at its surface. In this section we analyze the critical Casimir force if such a substrate is approached by a colloid with $(b)$ BC with its center located at the lateral position $x=X$ (see [Fig. \[fig:sketch\]]{} and Ref. for experimental realizations). We denote this configuration by $(a_<|a_>,b)$. The normal critical Casimir force ${\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}$ is described by the scaling form [@troendle:2009] $$\label{eq:step-force}
{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(X,D,R,T)= k_BT \frac{R}{D^{d-1}} \; {\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}(\Xi,\Theta,\Delta),$$ where $\Xi=X/\sqrt{RD}$ is the scaling variable corresponding to the lateral position of the colloid. It is useful to write the scaling function ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}$ as $$\begin{gathered}
\label{eq:step-K}
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}(\Xi,\Theta,\Delta) =
\frac{K_{\ensuremath{ {(a_<,b)}}}+K_{\ensuremath{ {(a_>,b)}}}}{2}
\\
+\frac{K_{\ensuremath{ {(a_<,b)}}}-K_{\ensuremath{ {(a_>,b)}}}}{2}
\psi_{(a_<|a_>,b)}(\Xi,\Theta,\Delta)
,$$ $$\label{eq:step-K}
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}(\Xi,\Theta,\Delta) =
\frac{K_{\ensuremath{ {(a_<,b)}}}+K_{\ensuremath{ {(a_>,b)}}}}{2}
+\frac{K_{\ensuremath{ {(a_<,b)}}}-K_{\ensuremath{ {(a_>,b)}}}}{2}
\psi_{(a_<|a_>,b)}(\Xi,\Theta,\Delta)
,$$ where the scaling functions of the laterally homogeneous substrates $K_{(a_\gtrless,b)}$ depend on $\Theta$ and $\Delta$ only \[[Eq. ]{}\], and the scaling function $\psi_{(a_<|a_>,b)}$ varies from $+1$ at $\Xi\to-\infty$ to $-1$ at $\Xi\to+\infty$, such that the laterally homogeneous cases are recovered far from the step. Accordingly, the corresponding critical Casimir potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{s}}}}}}(X,D,R,T)=\int_D^\infty {{\ensuremath{\textrm{d}}}}z\;{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(X,z,R,T)$ can be cast in the form [@troendle:2009] $$\label{eq:step-pot}
{\ensuremath{\Phi_{{\ensuremath{\textrm{s}}}}}}(X,D,R,T)=k_BT\frac{R}{D^{d-2}}{\ensuremath{\vartheta_{{\ensuremath{\textrm{s}}}}}}(\Xi,\Theta,\Delta),$$ and $$\begin{gathered}
\label{eq:step-vartheta}
{\ensuremath{\vartheta_{{\ensuremath{\textrm{s}}}}}}(\Xi,\Theta,\Delta)=\frac{\vartheta_{\ensuremath{ {(a_<,b)}}}+\vartheta_{\ensuremath{ {(a_>,b)}}}}{2}
\\
+\frac{\vartheta_{\ensuremath{ {(a_<,b)}}}-\vartheta_{\ensuremath{ {(a_>,b)}}}}{2}
\omega_{(a_<|a_>,b)}(\Xi,\Theta,\Delta)
,\end{gathered}$$ $$\label{eq:step-vartheta}
{\ensuremath{\vartheta_{{\ensuremath{\textrm{s}}}}}}(\Xi,\Theta,\Delta)=\frac{\vartheta_{\ensuremath{ {(a_<,b)}}}+\vartheta_{\ensuremath{ {(a_>,b)}}}}{2}
+\frac{\vartheta_{\ensuremath{ {(a_<,b)}}}-\vartheta_{\ensuremath{ {(a_>,b)}}}}{2}
\omega_{(a_<|a_>,b)}(\Xi,\Theta,\Delta)
,$$ where $\vartheta_{(a_\gtrless,b)}$ depend on $\Theta$ and $\Delta$ only \[[Eq. ]{}\], and $\omega_{(a_<|a_>,b)}(\Xi=\pm\infty,\Theta,\Delta)=\mp1$. Note that the scaling functions $\psi_{(a_<|a_>,b)}$ and $\omega_{(a_<|a_>,b)}$ are independent of the common prefactor $\propto u^{-1}$ \[see Sec. \[sec:MFT\]\], which is left undetermined by the analytical and numerical mean-field calculation of ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}$ and ${\ensuremath{\vartheta_{{\ensuremath{\textrm{s}}}}}}$.
Derjaguin approximation
-----------------------
If the sphere is close to the substrate, i.e., $\Delta\to0$, the DA can be applied, and one finds for the scaling function of the critical Casimir force \[see Appendix \[app:step\]\] $$\begin{gathered}
\label{eq:step-psi-da}
\psi_{(a_<|a_>,b)}(\Xi\gtrless0,\Theta,\Delta\to0)=
\mp 1 \\
\pm\frac{
4
\int_{1+\Xi^2/2}^{\infty}{{\ensuremath{\textrm{d}}}}\alpha\;
\alpha^{-d}\arccos\left({|\Xi|}(2\alpha-2)^{-1/2}\right)
\Delta k( \alpha\Theta)}
{
K_{(a_<,b)}(\Theta,\Delta\to0)-K_{(a_>,b)}(\Theta,\Delta\to0)
},\end{gathered}$$ $$\label{eq:step-psi-da}
\psi_{(a_<|a_>,b)}(\Xi\gtrless0,\Theta,\Delta\to0)=
\mp 1
\pm\frac{
4
\int_{1+\Xi^2/2}^{\infty}{{\ensuremath{\textrm{d}}}}\alpha\;
\alpha^{-d}\arccos\left({|\Xi|}(2\alpha-2)^{-1/2}\right)
\Delta k( \alpha\Theta)}
{
K_{(a_<,b)}(\Theta,\Delta\to0)-K_{(a_>,b)}(\Theta,\Delta\to0)
},$$ where $\Delta k(\Theta) = k_{\ensuremath{ {(a_<,b)}}}(\Theta) - k_{\ensuremath{ {(a_>,b)}}}(\Theta)$ is the difference between the scaling functions for the critical Casimir forces acting on two planar walls with ${\ensuremath{ {(a_<,b)}}}$ and with ${\ensuremath{ {(a_>,b)}}}$ boundary conditions, respectively. We note that according to Eqs. and within the DA $\psi_{(a_<|a_>,b)}$ can be determined from the knowledge of the film scaling functions ${\ensuremath{k_{{\ensuremath{(a,b)}}}}}(\Theta)$ \[[Eq. ]{}\] only. Due to the assumption of additivity which underlies the DA, (i) $\psi_{(a_<|a_>,b)}$ vanishes at $\Xi=0$ for all $\Theta$ and it is an antisymmetric function of $\Xi$ and (ii) $\psi_{(a_<|a_>,b)}=\psi_{(a_>|a_<,b)}$; within the DA both of these properties are valid irrespective of the type of boundary conditions on both sides of the chemical step. (However, the actual scaling function $\psi_{(a_<|a_>,b)}$ as, e.g., obtained from full numerical MFT calculations may violate this symmetry because the actual critical Casimir forces are non-additive.) At the bulk critical point one has $\Theta=0$ so that \[see Appendix \[app:step-crit\]\], $$\begin{gathered}
\label{eq:step-psi-da-crit}
\psi_{(a_<|a_>,b)}(\Xi,\Theta=0,\Delta\to0)=\\
\Xi^{2d-7}\left(\tfrac{15}{2}(3-d)+(3-2d)\Xi^2-\Xi^4\right)\left(2+\Xi^2\right)^{-(d-\frac{3}{2})}\end{gathered}$$ $$\label{eq:step-psi-da-crit}
\psi_{(a_<|a_>,b)}(\Xi,\Theta=0,\Delta\to0)=\\
\Xi^{2d-7}\left(\tfrac{15}{2}(3-d)+(3-2d)\Xi^2-\Xi^4\right)\left(2+\Xi^2\right)^{-(d-\frac{3}{2})}$$ independent of $k_{(a_\gtrless,b)}$. Similarly, within the DA one finds for the scaling function $\omega$ of the critical Casimir potential \[see Appendix \[app:step\] and Ref. \] $$\begin{gathered}
\label{eq:step-omega-da}
\omega_{(a_<|a_>,b)}(\Xi\gtrless0,\Theta,\Delta\to0)=\mp1
\\
\pm
\frac{\Xi^4
\int_{1}^{\infty}{{\ensuremath{\textrm{d}}}}s
\frac{
s\arccos\left( s^{-1/2}\right)-\sqrt{s-1}}
{(1+\Xi^2s/2)^{d}}
\Delta k
\left( \Theta[1+\Xi^2 s/2] \right)}
{
\vartheta_{(a_<,b)}(\Theta,\Delta\to0)-\vartheta_{(a_>,b)}(\Theta,\Delta\to0)
}.\end{gathered}$$ $$\label{eq:step-omega-da}
\omega_{(a_<|a_>,b)}(\Xi\gtrless0,\Theta,\Delta\to0)=
\mp1\pm
\frac{\Xi^4
\int_{1}^{\infty}{{\ensuremath{\textrm{d}}}}s
\frac{
s\arccos\left( s^{-1/2}\right)-\sqrt{s-1}}
{(1+\Xi^2s/2)^{d}}
\Delta k
\left( \Theta[1+\Xi^2 s/2] \right)}
{
\vartheta_{(a_<,b)}(\Theta,\Delta\to0)-\vartheta_{(a_>,b)}(\Theta,\Delta\to0)
}.$$ This yields $\omega_{(a_<|a_>,b)}(\Xi=0,\Theta,\Delta\to0)=0$, as expected from the underlying assumption of additivity; within full MFT this only holds in the limit $\Delta\to0$. At the critical point we find \[see Appendix \[app:step-crit\]\] $$ \label{eq:step-omega-da-crit}
\omega_{(a_<|a_>,b)}(\Xi,\Theta=0,\Delta\to0) =
{\Xi\left(1-d-\Xi^2\right)}{\left(\Xi^2+2\right)^{-3/2}}.$$
![ (a) Scaling function $\omega_{(+|-,-)}$ \[[Eq. ]{}\] for the critical Casimir potential of a spherical colloid with ${\ensuremath{(-)}}$ BC across a chemical step $(+|-)$ as a function of $\Xi \equiv X/\sqrt{R D}$ for various (positive) values of $\Theta=D/\xi_+$ [@troendle:2009]. Within the DA $\omega_{(+|-,-)}$ is an antisymmetric function of $\Xi$ \[[Eq. ]{}\] whereas within full MFT this antisymmetry is slightly violated, in particular for small $\Theta$. (b) Corresponding scaling function ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ \[[Eq. ]{}\] of the *lateral* critical Casimir force, normalized by the amplitude ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}(0,0)=2\pi{\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}}/(d-1)$ of the normal critical Casimir force at $T=T_c$ acting on a colloid with ${\ensuremath{(-)}}$ BC close to a homogeneous substrate with ${\ensuremath{(-)}}$ BC within the DA \[Sec. \[sec:homog-da\]\]. For both (a) and (b) the full numerical MFT results obtained for $\Delta=1/3$ are shown as symbols (the symbol size represents the estimated numerical error) whereas the lines show the corresponding results obtained within the DA (i.e., $\Delta\to0$); the dotted lines refer to $d=3$ and are obtained by using Monte Carlo simulation data [@mcdata] and the solid lines refer to $d=4$. The lines for $\Theta=0$ are obtained by using [Eq. ]{} and [Eq. ]{}, respectively; for $\Theta=3.2,4.7,8.1$ the DA lines de facto coincide with the asymptotic results obtained for symmetry breaking BC and $\Theta\gg1$ \[[Eq. ]{} and [Eq. ]{}, respectively\] and thus are indeed independent of $d$. The DA ($d=4$) provides a good approximation for the full numerical MFT data, in particular for $\Theta\gtrsim3$. ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel>0$ implies that the colloid moves to the right where it enjoys an attractive potential versus a repulsive one for $\Xi<0$. Within the DA ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ is a symmetric function of $\Xi$ \[Eqs. and \] whereas within full MFT this symmetry is slightly violated, in particular for small $\Theta$. []{data-label="fig:lateral"}](step_temp "fig:"){width="8cm"}\
![ (a) Scaling function $\omega_{(+|-,-)}$ \[[Eq. ]{}\] for the critical Casimir potential of a spherical colloid with ${\ensuremath{(-)}}$ BC across a chemical step $(+|-)$ as a function of $\Xi \equiv X/\sqrt{R D}$ for various (positive) values of $\Theta=D/\xi_+$ [@troendle:2009]. Within the DA $\omega_{(+|-,-)}$ is an antisymmetric function of $\Xi$ \[[Eq. ]{}\] whereas within full MFT this antisymmetry is slightly violated, in particular for small $\Theta$. (b) Corresponding scaling function ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ \[[Eq. ]{}\] of the *lateral* critical Casimir force, normalized by the amplitude ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}(0,0)=2\pi{\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}}/(d-1)$ of the normal critical Casimir force at $T=T_c$ acting on a colloid with ${\ensuremath{(-)}}$ BC close to a homogeneous substrate with ${\ensuremath{(-)}}$ BC within the DA \[Sec. \[sec:homog-da\]\]. For both (a) and (b) the full numerical MFT results obtained for $\Delta=1/3$ are shown as symbols (the symbol size represents the estimated numerical error) whereas the lines show the corresponding results obtained within the DA (i.e., $\Delta\to0$); the dotted lines refer to $d=3$ and are obtained by using Monte Carlo simulation data [@mcdata] and the solid lines refer to $d=4$. The lines for $\Theta=0$ are obtained by using [Eq. ]{} and [Eq. ]{}, respectively; for $\Theta=3.2,4.7,8.1$ the DA lines de facto coincide with the asymptotic results obtained for symmetry breaking BC and $\Theta\gg1$ \[[Eq. ]{} and [Eq. ]{}, respectively\] and thus are indeed independent of $d$. The DA ($d=4$) provides a good approximation for the full numerical MFT data, in particular for $\Theta\gtrsim3$. ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel>0$ implies that the colloid moves to the right where it enjoys an attractive potential versus a repulsive one for $\Xi<0$. Within the DA ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ is a symmetric function of $\Xi$ \[Eqs. and \] whereas within full MFT this symmetry is slightly violated, in particular for small $\Theta$. []{data-label="fig:lateral"}](lateral_step "fig:"){width="8cm"} ![ (a) Scaling function $\omega_{(+|-,-)}$ \[[Eq. ]{}\] for the critical Casimir potential of a spherical colloid with ${\ensuremath{(-)}}$ BC across a chemical step $(+|-)$ as a function of $\Xi \equiv X/\sqrt{R D}$ for various (positive) values of $\Theta=D/\xi_+$ [@troendle:2009]. Within the DA $\omega_{(+|-,-)}$ is an antisymmetric function of $\Xi$ \[[Eq. ]{}\] whereas within full MFT this antisymmetry is slightly violated, in particular for small $\Theta$. (b) Corresponding scaling function ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ \[[Eq. ]{}\] of the *lateral* critical Casimir force, normalized by the amplitude ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}(0,0)=2\pi{\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}}/(d-1)$ of the normal critical Casimir force at $T=T_c$ acting on a colloid with ${\ensuremath{(-)}}$ BC close to a homogeneous substrate with ${\ensuremath{(-)}}$ BC within the DA \[Sec. \[sec:homog-da\]\]. For both (a) and (b) the full numerical MFT results obtained for $\Delta=1/3$ are shown as symbols (the symbol size represents the estimated numerical error) whereas the lines show the corresponding results obtained within the DA (i.e., $\Delta\to0$); the dotted lines refer to $d=3$ and are obtained by using Monte Carlo simulation data [@mcdata] and the solid lines refer to $d=4$. The lines for $\Theta=0$ are obtained by using [Eq. ]{} and [Eq. ]{}, respectively; for $\Theta=3.2,4.7,8.1$ the DA lines de facto coincide with the asymptotic results obtained for symmetry breaking BC and $\Theta\gg1$ \[[Eq. ]{} and [Eq. ]{}, respectively\] and thus are indeed independent of $d$. The DA ($d=4$) provides a good approximation for the full numerical MFT data, in particular for $\Theta\gtrsim3$. ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel>0$ implies that the colloid moves to the right where it enjoys an attractive potential versus a repulsive one for $\Xi<0$. Within the DA ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ is a symmetric function of $\Xi$ \[Eqs. and \] whereas within full MFT this symmetry is slightly violated, in particular for small $\Theta$. []{data-label="fig:lateral"}](step_temp "fig:")\
![ (a) Scaling function $\omega_{(+|-,-)}$ \[[Eq. ]{}\] for the critical Casimir potential of a spherical colloid with ${\ensuremath{(-)}}$ BC across a chemical step $(+|-)$ as a function of $\Xi \equiv X/\sqrt{R D}$ for various (positive) values of $\Theta=D/\xi_+$ [@troendle:2009]. Within the DA $\omega_{(+|-,-)}$ is an antisymmetric function of $\Xi$ \[[Eq. ]{}\] whereas within full MFT this antisymmetry is slightly violated, in particular for small $\Theta$. (b) Corresponding scaling function ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ \[[Eq. ]{}\] of the *lateral* critical Casimir force, normalized by the amplitude ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}(0,0)=2\pi{\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}}/(d-1)$ of the normal critical Casimir force at $T=T_c$ acting on a colloid with ${\ensuremath{(-)}}$ BC close to a homogeneous substrate with ${\ensuremath{(-)}}$ BC within the DA \[Sec. \[sec:homog-da\]\]. For both (a) and (b) the full numerical MFT results obtained for $\Delta=1/3$ are shown as symbols (the symbol size represents the estimated numerical error) whereas the lines show the corresponding results obtained within the DA (i.e., $\Delta\to0$); the dotted lines refer to $d=3$ and are obtained by using Monte Carlo simulation data [@mcdata] and the solid lines refer to $d=4$. The lines for $\Theta=0$ are obtained by using [Eq. ]{} and [Eq. ]{}, respectively; for $\Theta=3.2,4.7,8.1$ the DA lines de facto coincide with the asymptotic results obtained for symmetry breaking BC and $\Theta\gg1$ \[[Eq. ]{} and [Eq. ]{}, respectively\] and thus are indeed independent of $d$. The DA ($d=4$) provides a good approximation for the full numerical MFT data, in particular for $\Theta\gtrsim3$. ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel>0$ implies that the colloid moves to the right where it enjoys an attractive potential versus a repulsive one for $\Xi<0$. Within the DA ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ is a symmetric function of $\Xi$ \[Eqs. and \] whereas within full MFT this symmetry is slightly violated, in particular for small $\Theta$. []{data-label="fig:lateral"}](lateral_step "fig:")
For symmetry breaking $(\mp,-)$ BC and $\Theta\gg1$ the critical Casimir force ${\ensuremath{f_{{\ensuremath{(\mp,-)}}}}}(D,T)$ acting on two planar walls at a distance $D$ decays $\propto\exp(-\Theta)$ \[Eqs. and \], which within the DA leads to the same *$d$-independent* result for the scaling functions $\psi_{(+|-,-)}$ and $\omega_{(+|-,-)}$ \[see Appendix \[app:step-gg\]\]: $$\begin{gathered}
\label{eq:erf}
\psi_{(+|-,-)}(\Xi,\Theta\gg1,\Delta\to0) =\\
\omega_{(+|-,-)}(\Xi,\Theta\gg1,\Delta\to0) =\\
- \operatorname{erf}\left( \sqrt{\Theta/2}\;\Xi\right),\end{gathered}$$ $$\label{eq:erf}
\psi_{(+|-,-)}(\Xi,\Theta\gg1,\Delta\to0) =
\omega_{(+|-,-)}(\Xi,\Theta\gg1,\Delta\to0) =
- \operatorname{erf}\left( \sqrt{\Theta/2}\;\Xi\right),$$ where $\operatorname{erf}$ is the error function.
Figure \[fig:lateral\](a) compares the scaling function $\omega_{(a_<|a_>,b)}$ for the critical Casimir potential of a sphere with ${\ensuremath{(-)}}$ BC in front of a $(+|-)$ step, as obtained within the DA for $d=4$ \[[Eq. ]{}\], with the one obtained numerically within full MFT for $\Delta=1/3$. For $\Delta\lesssim1/3$ the DA captures the scaling function very well, in particular for $\Theta\gtrsim3$ [@troendle:2009]. The scaling function $\omega_{(a_<|a_>,b)}$ obtained within the DA ($d=3$) on the basis of the Monte Carlo data of Ref. , which is also shown in [Fig. \[fig:lateral\]]{}(a), has been used successfully in order to interpret the experimental data of Ref. , for which the analysis in terms of separate, independent, and consecutive chemical steps turned out to be accurate. Moreover, the critical Casimir forces turned out to be a sensitive probe of the chemical pattern and its geometric design [@troendle:2009].
Lateral critical Casimir force \[sec:lateral\]
----------------------------------------------
The *lateral* critical Casimir force is given by ${\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}^\parallel=-\partial_X{\ensuremath{\Phi_{{\ensuremath{\textrm{s}}}}}}$ and can be cast in the scaling form $$\label{eq:step-lateral-force}
{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}^\parallel(X,D,R,T)=k_BT\,\frac{R}{D^{d-1}}\left(\frac{D}{R}\right)^{1/2}\,{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel(\Xi,\Theta,\Delta),$$ where ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ is a universal scaling function. ${\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}^\parallel$ and ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ vanish far from the chemical step, i.e., for $|\Xi|\to\infty$. In [Eq. ]{} the prefactors in terms of $R$ and $D$ and their exponents are chosen such that ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ is regular and non-vanishing for $\Delta\to0$. We note that the same holds for the normal critical Casimir forces and the corresponding potentials \[see Eqs. , , , , and the considerations following below\].
Within the DA ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ can be calculated from Eqs. and : $$\begin{gathered}
\label{eq:step-22new}
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel(\Xi,\Theta,\Delta\to0)=\\-\frac{1}{2}
\left[\vartheta_{\ensuremath{ {(a_<,b)}}}(\Theta,\Delta\to0)-\vartheta_{\ensuremath{ {(a_>,b)}}}(\Theta,\Delta\to0)\right]\times\\
\partial_\Xi\omega_{(a_<|a_>,b)}(\Xi,\Theta,\Delta\to0).\end{gathered}$$ $$\label{eq:step-22new}
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel(\Xi,\Theta,\Delta\to0)=-\frac{1}{2}
\left[\vartheta_{\ensuremath{ {(a_<,b)}}}(\Theta,\Delta\to0)-\vartheta_{\ensuremath{ {(a_>,b)}}}(\Theta,\Delta\to0)\right]
\partial_\Xi\omega_{(a_<|a_>,b)}(\Xi,\Theta,\Delta\to0).$$ At bulk criticality $\Theta=0$ one finds with [Eq. ]{} \[see [Eq. ]{}\] $$\label{eq:step-lateral-force-crit}
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel(\Xi,\Theta=0,\Delta\to0)=\pi\Delta k(0)\left(2+\Xi^2\right)^{-(d-\frac{3}{2})}.$$ For $(\mp,-)$ BC and $\Theta\gg1$ Eqs. , , and lead to $$\begin{gathered}
\label{eq:step-lateral-force-erf}
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel(\Xi,\Theta\gg1,\Delta\to0)=\\
\left[{\ensuremath{\vartheta_{{\ensuremath{(+,-)}}}}}(\Theta,\Delta)-{\ensuremath{\vartheta_{{\ensuremath{(-,-)}}}}}(\Theta,\Delta)\right]\sqrt{\frac{\Theta}{2\pi}}\;\exp\left\{-\frac{\Theta\Xi^2}{2}\right\},\end{gathered}$$ $$\label{eq:step-lateral-force-erf}
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel(\Xi,\Theta\gg1,\Delta\to0)=\\
\left[{\ensuremath{\vartheta_{{\ensuremath{(+,-)}}}}}(\Theta,\Delta)-{\ensuremath{\vartheta_{{\ensuremath{(-,-)}}}}}(\Theta,\Delta)\right]\sqrt{\frac{\Theta}{2\pi}}\;\exp\left\{-\frac{\Theta\Xi^2}{2}\right\},$$ for *both* $d=3$ and $d=4$. \[The prefactor $\Delta\vartheta(\Theta,\Delta)={\ensuremath{\vartheta_{{\ensuremath{(+,-)}}}}}(\Theta,\Delta)-{\ensuremath{\vartheta_{{\ensuremath{(-,-)}}}}}(\Theta,\Delta)$ in [Eq. ]{} is shown in [Fig. \[fig:homog\]]{}(b).\]
Figure \[fig:lateral\](b) shows the comparison between the normalized lateral critical Casimir force obtained within the DA (solid lines) and the full MFT data obtained for $\Delta=1/3$ (symbols). We infer that not only the shape of ${\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^\parallel$ as a function of $\Xi$ but also its *amplitude* is described well by the DA \[Eqs. and \] for $\Delta\lesssim1/3$, and in particular for $\Theta\gtrsim3$. We expect this feature to hold in $d=3$, too, as well as for the normal critical Casimir force and the critical Casimir potential. The lateral critical Casimir forces for $d=3$ obtained within the DA on the basis of Monte Carlo simulation data for the film geometry [@mcdata] are shown in [Fig. \[fig:lateral\]]{}(b) as dashed lines. Compared with the previous curves, these ones have similar shapes but their overall amplitudes in units of the normal critical Casimir force at $\Theta = 0$ are significantly different for $\Theta = 0$ and $\Theta = 3.2$. This difference reflects the analogous one observed in the normalized difference between the corresponding critical Casimir potentials for $(+,-)$ and $(-,-)$ BC, reported in [Fig. \[fig:homog\]]{}(b).
Single chemical lane (${\ensuremath{\ell}}$)\[sec:stripe\]
==========================================================
In this section we consider the case of a colloid with $(b)$ BC close to a substrate with a single chemical lane (${\ensuremath{\ell}}$) with $(a_{{\ensuremath{\ell}}})$ BC and width $2L$ in the lateral $x$ direction and which is invariant along the other lateral direction(s). The remaining parts of the substrate are two semi-infinite planes at $|x|>L$ with $(a)$ BC \[see [Fig. \[fig:sketch\]]{}\]. The lateral coordinate $X$ of the center of mass of the sphere along the $x$ direction is chosen to vanish in the center of the chemical lane. One expects that for “broad” lanes a description in terms of two subsequent chemical steps is appropriate \[Sec. \[sec:step\] and Ref. \], whereas for “narrow” lanes the effects of the two subsequent chemical steps interfere. We find that in addition to the variables characterizing the chemical step \[[Eq. ]{}\], a further scaling variable $\Lambda=L/\sqrt{RD}$ emerges naturally, which corresponds to the width of the lane. Accordingly, the normal critical Casimir force ${\ensuremath{F_{{\ensuremath{\ell}}}}}$ acting on the colloid can be cast in the form $$\label{eq:stripe-force}
{\ensuremath{F_{{\ensuremath{\ell}}}}}(L,X,D,R,T)=k_BT \frac{R}{D^{d-1}} {\ensuremath{K_{{\ensuremath{\ell}}}}}\left(\Lambda,\Xi,\Theta,\Delta\right),$$ where ${\ensuremath{K_{{\ensuremath{\ell}}}}}$ is the corresponding universal scaling function. The critical Casimir potential scales as $$\label{eq:stripe-pot}
{\ensuremath{\Phi_{{\ensuremath{\ell}}}}}(L,X,D,R,T)=k_BT \frac{R}{D^{d-2}} {\ensuremath{\vartheta_{{\ensuremath{\ell}}}}}\left(\Lambda,\Xi,\Theta,\Delta\right),$$ with ${\ensuremath{\vartheta_{{\ensuremath{\ell}}}}}$ as the universal scaling function for the potential of a sphere close to a single chemical lane. Analogously to Eqs. and we define ${\ensuremath{\psi_{{\ensuremath{\ell}}}}}$ and ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ according to $$\begin{gathered}
\label{eq:stripe-psi}
{\ensuremath{K_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta)=
\frac{K_{(a,b)}+K_{(a_{{\ensuremath{\ell}}},b)}}{2}\\
+\frac{K_{(a,b)}-K_{(a_{{\ensuremath{\ell}}},b)}}{2}
{\ensuremath{\psi_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta),\end{gathered}$$ $$\label{eq:stripe-psi}
{\ensuremath{K_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta)=
\frac{K_{(a,b)}+K_{(a_{{\ensuremath{\ell}}},b)}}{2}+
\frac{K_{(a,b)}-K_{(a_{{\ensuremath{\ell}}},b)}}{2}
{\ensuremath{\psi_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta),$$ and $$\begin{gathered}
\label{eq:stripe-omega}
{\ensuremath{\vartheta_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta)=
\frac{\vartheta_{(a,b)}+\vartheta_{(a_{{\ensuremath{\ell}}},b)}}{2}\\
+\frac{\vartheta_{(a,b)}-\vartheta_{(a_{{\ensuremath{\ell}}},b)}}{2} {\ensuremath{\omega_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta),\end{gathered}$$ $$\label{eq:stripe-omega}
{\ensuremath{\vartheta_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta)=
\frac{\vartheta_{(a,b)}+\vartheta_{(a_{{\ensuremath{\ell}}},b)}}{2}+\\
\frac{\vartheta_{(a,b)}-\vartheta_{(a_{{\ensuremath{\ell}}},b)}}{2} {\ensuremath{\omega_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta),$$ so that far from the lane ${\ensuremath{\psi_{{\ensuremath{\ell}}}}}\left(\Lambda,|\Xi|\gg\Lambda,\Theta,\Delta \right)=
{\ensuremath{\omega_{{\ensuremath{\ell}}}}}\left(\Lambda,|\Xi|\gg\Lambda,\Theta,\Delta \right)=1$. On the other hand, only for a “broad” lane the scaling functions at the center of the chemical lane approach their limiting value ${\ensuremath{\psi_{{\ensuremath{\ell}}}}}\left(\Lambda\to\infty,\Xi=0,\Theta,\Delta \right)=-1=
{\ensuremath{\omega_{{\ensuremath{\ell}}}}}\left(\Lambda\to\infty,\Xi=0,\Theta,\Delta \right)$, corresponding to the homogeneous case with $(a_{{\ensuremath{\ell}}},b)$ BC.
Derjaguin approximation
-----------------------
Using the underlying assumption of additivity of the forces, within the DA ($\Delta\to0$) we find for the scaling functions of the critical Casimir force and of the critical Casimir potential \[see Appendix \[app:stripe\]\] $$\begin{gathered}
\label{eq:stripe-psi-da}
{\ensuremath{\psi_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta\to0)=\\
1+\psi_{(a_{{\ensuremath{\ell}}}|a,b)}(\Xi+\Lambda,\Theta,\Delta\to0)-\psi_{(a_{{\ensuremath{\ell}}}|a,b)}(\Xi-\Lambda,\Theta,\Delta\to0)\end{gathered}$$ $$\label{eq:stripe-psi-da}
{\ensuremath{\psi_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta\to0)=
1+\psi_{(a_{{\ensuremath{\ell}}}|a,b)}(\Xi+\Lambda,\Theta,\Delta\to0)-\psi_{(a_{{\ensuremath{\ell}}}|a,b)}(\Xi-\Lambda,\Theta,\Delta\to0)$$ and $$\begin{gathered}
\label{eq:stripe-omega-da}
{\ensuremath{\omega_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta\to0)=\\
1+\omega_{(a_{{\ensuremath{\ell}}}|a,b)}(\Xi+\Lambda,\Theta,\Delta\to0)-\omega_{(a_{{\ensuremath{\ell}}}|a,b)}(\Xi-\Lambda,\Theta,\Delta\to0),\end{gathered}$$ $$\label{eq:stripe-omega-da}
{\ensuremath{\omega_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta\to0)=
1+\omega_{(a_{{\ensuremath{\ell}}}|a,b)}(\Xi+\Lambda,\Theta,\Delta\to0)-\omega_{(a_{{\ensuremath{\ell}}}|a,b)}(\Xi-\Lambda,\Theta,\Delta\to0),$$ respectively. Thus, within the DA, from the knowledge of the scaling functions $\psi_{(a_{{\ensuremath{\ell}}}|a,b)}$ \[[Eq. ]{}\] and $\omega_{(a_{{\ensuremath{\ell}}}|a,b)}$ \[[Eq. ]{}\] for the chemical step with the appropriate BC, one can directly calculate the corresponding scaling functions for the chemical lane configuration. Accordingly, in the limit $\Delta \rightarrow 0$ and for symmetry breaking BC, ${\ensuremath{\psi_{{\ensuremath{\ell}}}}}$ and ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ can be analytically calculated on the basis of Eqs. and by taking advantage of Eqs. , , and .
Scaling function for the critical Casimir potential
---------------------------------------------------
![ Scaling function ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ \[[Eq. ]{}\] describing the lateral variation of the critical Casimir potential of a colloid across a single chemical lane of width $2L$ as a function of the lateral position $X$ of the colloid in units of the half width of the lane \[see [Fig. \[fig:sketch\]]{}; $\Xi=X/\sqrt{RD}$, $\Lambda=L/\sqrt{RD}$, $\Theta=D/\xi_+$\]. Here, ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ has been obtained within the DA ($\Delta \rightarrow 0$) in $d=3$ and $4$ \[[Eq. ]{}\]. In (a) the curves correspond to $\Theta=0$ \[[Eq. ]{}\], whereas in (b) they correspond to $\Theta=7.7$ and $a,a_{{\ensuremath{\ell}}},b\in\{+,-\}$ BC \[[Fig. \[fig:sketch\]]{}\]. For $\Theta\gg1$ \[(b)\] the corresponding scaling functions obtained from Monte Carlo simulation data [@mcdata] in $d=3$ and from analytic MFT results [@krech:1997] in $d=4$ de facto coincide and their asymptotic expressions are given by Eqs. and . ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}=1$ corresponds to the laterally homogeneous critical Casimir potential for $(a,b)$ BC outside the chemical lane, whereas ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}=-1$ corresponds to the value of the critical Casimir potential for the homogeneous case with $(a_{{\ensuremath{\ell}}},b)$ BC as within the chemical lane. For large values of $\Lambda$ the critical Casimir potential is the same as for two independent chemical steps, and ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ reaches its limiting value $-1$ in the center of the lane at $\Xi=0$ \[see the main text\]. In (b), for $\Theta\gg1$, ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ attains $-1$ in the center of the chemical lane already for smaller values of $\Lambda$ due to the exponential decay of the critical Casimir force. We note that the DA results for $\Theta=0$ (i.e., at the critical point) are independent of the actual boundary conditions which, accordingly, were not specified in (a). []{data-label="fig:stripe"}](omega_34 "fig:"){width="8.4cm"}\
![ Scaling function ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ \[[Eq. ]{}\] describing the lateral variation of the critical Casimir potential of a colloid across a single chemical lane of width $2L$ as a function of the lateral position $X$ of the colloid in units of the half width of the lane \[see [Fig. \[fig:sketch\]]{}; $\Xi=X/\sqrt{RD}$, $\Lambda=L/\sqrt{RD}$, $\Theta=D/\xi_+$\]. Here, ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ has been obtained within the DA ($\Delta \rightarrow 0$) in $d=3$ and $4$ \[[Eq. ]{}\]. In (a) the curves correspond to $\Theta=0$ \[[Eq. ]{}\], whereas in (b) they correspond to $\Theta=7.7$ and $a,a_{{\ensuremath{\ell}}},b\in\{+,-\}$ BC \[[Fig. \[fig:sketch\]]{}\]. For $\Theta\gg1$ \[(b)\] the corresponding scaling functions obtained from Monte Carlo simulation data [@mcdata] in $d=3$ and from analytic MFT results [@krech:1997] in $d=4$ de facto coincide and their asymptotic expressions are given by Eqs. and . ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}=1$ corresponds to the laterally homogeneous critical Casimir potential for $(a,b)$ BC outside the chemical lane, whereas ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}=-1$ corresponds to the value of the critical Casimir potential for the homogeneous case with $(a_{{\ensuremath{\ell}}},b)$ BC as within the chemical lane. For large values of $\Lambda$ the critical Casimir potential is the same as for two independent chemical steps, and ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ reaches its limiting value $-1$ in the center of the lane at $\Xi=0$ \[see the main text\]. In (b), for $\Theta\gg1$, ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ attains $-1$ in the center of the chemical lane already for smaller values of $\Lambda$ due to the exponential decay of the critical Casimir force. We note that the DA results for $\Theta=0$ (i.e., at the critical point) are independent of the actual boundary conditions which, accordingly, were not specified in (a). []{data-label="fig:stripe"}](omega_77 "fig:"){width="8.4cm"}
In [Fig. \[fig:stripe\]]{}(a) we show the scaling function ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ for the critical Casimir potential obtained within the DA for $d=3$ and $d=4$ (MFT) at the bulk critical point $T=T_c$ \[Eqs. and \] for various values of $\Lambda=L/\sqrt{RD}$ as a function of the lateral coordinate of the colloid. One can infer from [Fig. \[fig:stripe\]]{} that, at bulk criticality, the critical Casimir potential varies less pronounced in $d=3$ than in $d=4$. As expected, for small values of $\Lambda$ (i.e., “narrow” chemical lanes), the potential does not reach the limiting homogeneous value $-1$ in the center of the chemical lane. On the other hand for large values of the scaling variable $\Lambda$ (i.e., “broad” chemical lanes), ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ does attain the value $-1$ in the center of the chemical lane and the critical Casimir potential flattens. In this case the potential is adequately described by two independent chemical steps. However, the criterion for being a sufficiently “broad” lane depends sensitively on $\Theta$ and $d$. Indeed, from Eqs. and we find that at criticality ($\Theta=0$) the critical Casimir potential at the center of the chemical lane ($\Xi=0$) reaches the limiting value corresponding to the colloid facing a homogeneous substrate by up to $1\%$ for $\Lambda\gtrsim3.3$ in $d=4$ and for $\Lambda\gtrsim10$ in $d=3$. We note that the curves in [Fig. \[fig:stripe\]]{}(a) as well as these bounds are *independent* of the actual boundary conditions because for all kinds of BC the scaling function of the normal critical Casimir force is constant at the critical point \[see [Eq. ]{}\].
Below we shall discuss some properties which are specific for BC with $a,a_{{\ensuremath{\ell}}},b\in\{+,-\}$, which exhibit the feature that the normal critical Casimir force ${\ensuremath{f_{{\ensuremath{(\mp,-)}}}}}$ acting on two planar walls decays *purely* exponentially \[see the text preceding [Eq. ]{}\] as a function of their distance expressed in units of the bulk correlation length \[see Eqs. and \]. In [Fig. \[fig:stripe\]]{}(b) the scaling functions ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ in $d=3$ and $d=4$ obtained from Monte Carlo simulation data [@mcdata] and analytic MFT results [@krech:1997], respectively, within the DA \[see Eqs. and \] are shown for the same values of $\Lambda$ as in [Fig. \[fig:stripe\]]{}(a) but off criticality. For $\Theta=7.7$ the curves for $d=3$ and $d=4$ are indistinguishable from each other and from their common asymptotic expression given in [Eq. ]{} \[see also Ref. \]. For $\Theta\gg1$, the critical Casimir potential attains its limiting homogeneous value in the center of the lane for values of $\Lambda$ which are smaller than the ones for $\Theta=0$ due to the shorter range of the forces. That is, for both $d=3$ and $d=4$ the single chemical lane is almost equally well approximated by two independent chemical steps for $\Lambda\gtrsim1.5$ at $\Theta=3.3$ (data not shown) and for $\Lambda\gtrsim1.0$ at $\Theta=7.7$ \[[Fig. \[fig:stripe\]]{}(b)\].
![ Test of the performance of the DA for the scaling function ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ \[[Eq. ]{}\] of the critical Casimir potential for a sphere with ${\ensuremath{(-)}}$ BC close to a single chemical lane with ${\ensuremath{(-)}}$ BC embedded in a substrate with ${\ensuremath{(+)}}$ BC. The MFT ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ is evaluated at bulk criticality $\Theta=0$ in $d=4$ both on the basis of the DA (lines, $\Delta\to0$) and of the full numerical MFT (symbols, $\Delta=1/3$). There is good agreement between the DA and the full MFT results, even for small values of $\Lambda=L/\sqrt{RD}$. Nonlinear effects, which are inherently present in the theory, do not strongly affect the potential. For $\Delta\to0$ the assumption of additivity of the critical Casimir forces underlying the DA is reliable even for small $\Lambda$. []{data-label="fig:omega_cl"}](omega_cl)
In [Fig. \[fig:omega\_cl\]]{} we compare the MFT ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}$ obtained within the DA ($\Delta\to0$) at $\Theta=0$ \[Eqs. and \] with the scaling function obtained from the full numerical MFT calculations for $\Delta=1/3$. We find a rather good agreement even for small values of $\Lambda$ (i.e., “narrow” chemical lanes). This shows that for the geometry of a colloid close to a single chemical lane, nonlinearities, which are actually present in the critical Casimir effect and potentially invalidate the assumption of additivity underlying the DA, do not affect the resulting potential for small values of $\Delta$. We expect this property to hold beyond MFT in $d=3$ as well, in particular off criticality, i.e., for $\Theta\ne0$.
Periodic chemical patterns (${\ensuremath{\textrm{p}}}$) \[sec:period\]
=======================================================================
In this section we consider a pattern of chemical stripes which are alternating *periodically* along the $x$ direction. The pattern consists of stripes of width $L_1$ with $(a_1)$ BC joined with stripes of width $L_2$ with $(a_2)$ BC, such that the periodicity is given by $P=L_1+L_2$. Thus, the geometry of the substrate pattern is characterized by the two variables $L_1$ and $P$ \[see [Fig. \[fig:sketch\]]{}\]. The coordinate system is chosen such that the lateral coordinate $X$ of the center of the sphere is zero at the center of a $(a_1)$ stripe. The normal critical Casimir force $F_{{\ensuremath{\textrm{p}}}}$ acting on the colloidal particle and its corresponding potential $\Phi_{{\ensuremath{\textrm{p}}}}$ take on the following scaling forms: $$\begin{aligned}
\label{eq:period-force}
{\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}(L_1,P,X,D,R,T)=&k_BT \frac{R}{D^{d-1}} {\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta)\\
\intertext{and}
\label{eq:period-pot}
{\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}(L_1,P,X,D,R,T)=&k_BT\frac{R}{D^{d-2}} {\ensuremath{\vartheta_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta),\end{aligned}$$ where $\Pi=P/\sqrt{RD}$ is the scaling variable characterizing the periodicity of the pattern and $\lambda=L_1/P$ is the scaling variable chosen to correspond to the relative width of the stripe with $(a_1)$ BC. ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ and ${\ensuremath{\vartheta_{{\ensuremath{\textrm{p}}}}}}$ are universal scaling functions for the normal critical Casimir force and the critical Casimir potential, respectively. For $\lambda=1$ or $0$ the force and the potential correspond to the homogeneous cases with $(a_1,b)$ BC or $(a_2,b)$ BC, respectively \[see Sec. \[sec:homog\]\]. As before it is useful to define scaling functions ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ and ${\ensuremath{\omega_{{\ensuremath{\textrm{p}}}}}}$ which vary for $\lambda\in[0,1]$ within the range $[-1,1]$ and describe the lateral behavior of the critical Casimir effect: $$\begin{gathered}
\label{eq:period-psi}
{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta)=
\frac{K_{(a_2,b)}+K_{(a_1,b)}}{2}\\
+\frac{K_{(a_2,b)}-K_{(a_1,b)}}{2}
{\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta)\end{gathered}$$ $$\label{eq:period-psi}
{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta)=
\frac{K_{(a_2,b)}+K_{(a_1,b)}}{2}+
\frac{K_{(a_2,b)}-K_{(a_1,b)}}{2}
{\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta)$$ and $$\begin{gathered}
\label{eq:period-omega}
{\ensuremath{\vartheta_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta)=
\frac{\vartheta_{(a_2,b)}+\vartheta_{(a_1,b)}}{2}\\
+\frac{\vartheta_{(a_2,b)}-\vartheta_{(a_1,b)}}{2}
{\ensuremath{\omega_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta).\end{gathered}$$ $$\label{eq:period-omega}
{\ensuremath{\vartheta_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta)=
\frac{\vartheta_{(a_2,b)}+\vartheta_{(a_1,b)}}{2}+
\frac{\vartheta_{(a_2,b)}-\vartheta_{(a_1,b)}}{2}
{\ensuremath{\omega_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta).$$
Derjaguin approximation\[sec:period-da\]
----------------------------------------
Taking advantage of the assumption of additivity of the forces underlying the DA, one finds for the scaling function of the normal critical Casimir force in the limit $\Delta\to0$ \[see Appendix \[app:period\]\] $$\begin{gathered}
\label{eq:period-psi-da}
{\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta\to0)=\\
1+\sum_{n=-\infty}^{\infty}
\left\{\psi_{(a_1|a_2,b)}(\Xi+\Pi(n+\tfrac{\lambda}{2}),\Theta,\Delta\to0)\right.\\
\left.-\psi_{(a_1|a_2,b)}(\Xi+\Pi(n-\tfrac{\lambda}{2}),\Theta,\Delta\to0)\right\}.\end{gathered}$$ $$\begin{gathered}
\label{eq:period-psi-da}
{\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta\to0)=\\
1+\sum_{n=-\infty}^{\infty}
\left\{\psi_{(a_1|a_2,b)}(\Xi+\Pi(n+\tfrac{\lambda}{2}),\Theta,\Delta\to0)
-\psi_{(a_1|a_2,b)}(\Xi+\Pi(n-\tfrac{\lambda}{2}),\Theta,\Delta\to0)\right\}.\end{gathered}$$ Thus, the knowledge of the scaling function $\psi_{(a_1|a_2,b)}$ for a single chemical step with the appropriate BC \[Sec. \[sec:step\]\] is sufficient to calculate directly the corresponding scaling function of the critical Casimir force acting on a colloid close to a periodic pattern of chemical stripes. As expected, from [Eq. ]{} one recovers the values ${{\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda=0,\Pi,\Xi,\Theta,\Delta)=1}$ and ${{\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda=1,\Pi,\Xi,\Theta,\Delta)=-1}$, i.e., the cases of a colloid with $(b)$ BC facing a homogeneous substrate with $(a_2)$ BC and $(a_1)$ BC, respectively \[see Appendix \[app:period\]\].
In the limit $\Pi\to0$, i.e., for a pattern with a very fine structure compared to the size of the colloid, the sum in [Eq. ]{} turns into an integral \[see Appendix \[app:period\]\] and, as expected, ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ becomes independent of $\Xi$, i.e., of the lateral position of the colloid: $$\label{eq:small-period}
{\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi\to0,\Xi,\Theta,\Delta\to0)=1-2\lambda.$$ Accordingly, in the limit $\Pi\to0$ the force acting on the colloid – within the DA – is the average of the ones corresponding to the two boundary conditions weighted by the corresponding relative stripe width \[see Eqs. and \]: $$\begin{gathered}
\label{eq:small-period-K}
{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi\to0,\Xi,\Theta,\Delta\to0)=\\
\frac{L_1}{L_1+L_2} K_{(a_1,b)}(\Theta,\Delta\to0)+\frac{L_2}{L_1+L_2}K_{(a_2,b)}(\Theta,\Delta\to0).\end{gathered}$$ $$\label{eq:small-period-K}
{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi\to0,\Xi,\Theta,\Delta\to0)=
\frac{L_1}{L_1+L_2} K_{(a_1,b)}(\Theta,\Delta\to0)+\frac{L_2}{L_1+L_2}K_{(a_2,b)}(\Theta,\Delta\to0).$$
For the scaling function of the critical Casimir potential the results are completely analogous to Eqs. – \[see Appendix \[app:period\]\].
Scaling function for the normal critical Casimir force
------------------------------------------------------
![ MFT ($d=4$) scaling function ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] of the normal critical Casimir force acting on a colloidal sphere with $(b)={\ensuremath{(-)}}$ BC which is close to a periodically patterned substrate \[[Fig. \[fig:sketch\]]{}\] with $(a_1)={\ensuremath{(-)}}$ BC on one kind of stripes \[shaded areas\] and $(a_2)={\ensuremath{(+)}}$ BC on the other kind of stripes. Due to this choice of the BC the colloid is attracted by the shaded stripes and repelled by the others. ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ is shown as a function of the lateral position of the colloid $X/P$ with $P=L_1+L_2$ and at the bulk critical point $\Theta=0$. The geometry of the pattern is characterized by $\Pi=P/\sqrt{RD}$ and $\lambda=L_1/P$, for which we have chosen the values (a) $\lambda=0.5$ and (b) $\lambda=0.2$. The lines are the results for ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ as obtained within the DA for $d=4$ \[Eqs. and \], whereas the symbols represent the full numerical data obtained within MFT for $\Delta=1/3$ for various values of $\Pi$. For patterns which are finely structured on the scale of the colloid size, i.e., $\Pi\lesssim2$, the actual results deviate from the approximate ones obtained within the DA due to the strong influence (in this context) of the inherent nonlinear effects. []{data-label="fig:period_normal"}](period_normal_05 "fig:")\
![ MFT ($d=4$) scaling function ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] of the normal critical Casimir force acting on a colloidal sphere with $(b)={\ensuremath{(-)}}$ BC which is close to a periodically patterned substrate \[[Fig. \[fig:sketch\]]{}\] with $(a_1)={\ensuremath{(-)}}$ BC on one kind of stripes \[shaded areas\] and $(a_2)={\ensuremath{(+)}}$ BC on the other kind of stripes. Due to this choice of the BC the colloid is attracted by the shaded stripes and repelled by the others. ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ is shown as a function of the lateral position of the colloid $X/P$ with $P=L_1+L_2$ and at the bulk critical point $\Theta=0$. The geometry of the pattern is characterized by $\Pi=P/\sqrt{RD}$ and $\lambda=L_1/P$, for which we have chosen the values (a) $\lambda=0.5$ and (b) $\lambda=0.2$. The lines are the results for ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ as obtained within the DA for $d=4$ \[Eqs. and \], whereas the symbols represent the full numerical data obtained within MFT for $\Delta=1/3$ for various values of $\Pi$. For patterns which are finely structured on the scale of the colloid size, i.e., $\Pi\lesssim2$, the actual results deviate from the approximate ones obtained within the DA due to the strong influence (in this context) of the inherent nonlinear effects. []{data-label="fig:period_normal"}](period_normal_02 "fig:")
![ (a) The same as in [Fig. \[fig:period\_normal\]]{}, but for $\lambda=0.8$. Also in this case, the DA turns out to be accurate for $\Pi\gtrsim2$ while it fails to describe quantitatively the full numerical data for smaller values of $\Pi$. (b) Comparison between the scaling functions ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ in $d=3$ (dotted lines) and $d=4$ (solid lines), at $T=T_c$, for $\lambda=0.8$, and within the DA. At the critical point the expression for this scaling function ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ is known analytically \[see Eqs. and \], and the corresponding plot presented here shows that the lateral variation of the normal critical Casimir force is less pronounced in $d=3$ than in $d=4$. (We note that for $\Pi\to0$ we expect that also in $d=3$ the DA fails to describe quantitatively the actual behavior; however, we nonetheless present the curve for $\Pi=0.57$ in order to show that the critical Casimir force obtained within the DA practically does not change laterally for such small values of $\Pi$.) []{data-label="fig:normal_3d"}](period_normal_08 "fig:")\
![ (a) The same as in [Fig. \[fig:period\_normal\]]{}, but for $\lambda=0.8$. Also in this case, the DA turns out to be accurate for $\Pi\gtrsim2$ while it fails to describe quantitatively the full numerical data for smaller values of $\Pi$. (b) Comparison between the scaling functions ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ in $d=3$ (dotted lines) and $d=4$ (solid lines), at $T=T_c$, for $\lambda=0.8$, and within the DA. At the critical point the expression for this scaling function ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ is known analytically \[see Eqs. and \], and the corresponding plot presented here shows that the lateral variation of the normal critical Casimir force is less pronounced in $d=3$ than in $d=4$. (We note that for $\Pi\to0$ we expect that also in $d=3$ the DA fails to describe quantitatively the actual behavior; however, we nonetheless present the curve for $\Pi=0.57$ in order to show that the critical Casimir force obtained within the DA practically does not change laterally for such small values of $\Pi$.) []{data-label="fig:normal_3d"}](period_normal_3d "fig:")
Figure \[fig:period\_normal\] shows the scaling function ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] as a function of $\Xi/\Pi=X/P$, describing the lateral variation of the normal critical Casimir force at $\Theta=0$ as obtained within the DA for $d=4$ \[[Eq. ]{} with [Eq. ]{}; solid lines\] compared with the one obtained from the full numerical MFT calculation \[$\Delta=1/3$; symbols\] for symmetry breaking boundary conditions $(a_1)={\ensuremath{(-)}}$, $(a_2)={\ensuremath{(+)}}$, and $(b)={\ensuremath{(-)}}$ \[[Fig. \[fig:sketch\]]{}\]. From this comparison for $\lambda=0.5$ \[[Fig. \[fig:period\_normal\]]{}(a)\] and $\lambda=0.2$ \[[Fig. \[fig:period\_normal\]]{}(b)\] and for various values of $\Pi$ one can infer that for $\Delta\to0$ and $\Pi\gg1$, i.e., $L_1+L_2\gg\sqrt{RD}$ the DA describes well the actual behavior of the scaling function, even if the force scaling function does not attain its limiting homogeneous values ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}=\pm1$ in the center of the stripes. However, for $\Pi\lesssim2$ (in $d=4$ at $T=T_c$) the DA does not quantitatively describe the actual behavior and the scaling function ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ obtained from the full numerical MFT calculations deviates from the one obtained within the DA. Within both the DA and the full numerical MFT calculation, for $\Pi\to0$ the normal critical Casimir force loses its lateral dependence on $\Xi$. But from the full numerical calculation we find that the corresponding constant value which is attained by ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ differs from the one obtained within DA \[[Eq. ]{}\]. This shows that for small periodicities $P\lesssim\sqrt{RD}$ nonlinearities inherent in the critical Casimir effect strongly affect the resulting scaling functions of the force and the potential, so that in this respect the assumption of additivity of the force and thus the use of the DA are not justified.
Figure \[fig:normal\_3d\](a) shows the same comparison as [Fig. \[fig:period\_normal\]]{} but for $\lambda=0.8$, which corresponds to an areal occupation of 80% of the substrate surface with ${\ensuremath{(-)}}$ BC and 20% with ${\ensuremath{(+)}}$ BC. Due to the fact that at the critical point $\psi_{(a_1|a_2,b)}(\Xi,\Theta=0,\Delta\to 0)$ is actually independent of the BC, ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda=0.8,\Pi,\Xi,\Theta=0,\Delta\to 0)$ in [Fig. \[fig:normal\_3d\]]{}(a) is, within the DA, complementary to the one for $\lambda = 0.2$ in [Fig. \[fig:period\_normal\]]{}(b), i.e., it is obtained from the latter by a reflection with respect to ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}=0$ followed by a shift in $\Xi/\Pi$ of $0.5$. Instead, the full numerical data in [Fig. \[fig:normal\_3d\]]{}(a) and [Fig. \[fig:period\_normal\]]{}(b) show a different behavior as they clearly tend to assume the value $-1$ corresponding to the homogeneous case with $(-,-)$ BC. By contrast, for the case $\lambda=0.2$ shown in [Fig. \[fig:period\_normal\]]{}(b), the full numerical data do not reach as closely the value $+1$ corresponding to $(+,-)$ BC, although the substrate area is covered by 80% with ${\ensuremath{(+)}}$ BC. This feature is addressed in more detail in Sec. \[sec:cylinder\]. Figure \[fig:normal\_3d\](b) compares the scaling function ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ of the normal critical Casimir force at $T=T_c$ and for $\lambda=0.2$ as obtained within the DA for $d=4$ (solid lines) with the corresponding one for $d=3$ (dotted lines). At $T=T_c$, ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ is determined by Eqs. and from which one can infer that the lateral variation of the normal Casimir force is less pronounced for $d=3$ than for $d=4$. This qualitative feature holds for all values of $\lambda$ (not shown). However, off criticality, $\Theta\gg1$, \[according to Eqs. and \] the DA scaling functions both for $d=3$ as obtained from MC simulation data and for $d=4$ as obtained from MFT de facto coincide (not shown), similarly to the case of a single chemical lane in [Fig. \[fig:stripe\]]{}(b).
![ Normalized scaling function $\Delta^{ph}$ of the critical Casimir force at criticality acting on a homogeneous *planar* wall with $(+)$ BC opposite to a periodically patterned planar substrate with stripes of alternating $(+)$ and $(-)$ BC as a function of $v=S_-/S_+$, where $S_+$ and $S_-$ are the respective widths. The symbols correspond to the MFT ($d=4$) data presented in Fig. 12 of Ref. for various values of $S_+/L$ (note that $\Delta_0^{++}={\ensuremath{\Delta_{{\ensuremath{(+,+)}}}}}/(d-1)$ in Fig. 12 of Ref. ). The dashed and dotted lines which join the data points are a guide to the eye. The solid line corresponds to the DA result given in [Eq. ]{} which assumes additivity of the forces and turns out to be independent of the ratio $S_+/L$. One can immediately infer from the graph that here the assumption of additivity is not justified, which is the limiting configuration of the sphere-wall geometry for $\Pi\to0$. []{data-label="fig:kraft"}](kraft)
Although one would expect the DA to be valid for large radii $R$, the lateral variation of the boundary conditions at the surface of the patterned substrate on a scale $P \lesssim \sqrt{R D}$ – corresponding to the limit $\Pi\to0$ – renders the DA less accurate, as it clearly emerges from the numerical data presented in Figs. \[fig:period\_normal\] and \[fig:normal\_3d\]. The fact that a large colloid radius $R$ does not guarantee the validity of the DA can be understood by noting that such a discrepancy between the full numerical calculation and the result of the DA approximation already emerges in the *film* geometry (formally corresponding to the limit $R\to\infty$), i.e., for a chemically [*p*]{}atterned wall opposite to a laterally [*h*]{}omogeneous flat wall. This “$ph$” configuration has been studied in Ref. within MFT for laterally alternating chemical stripes of width $L_1=S_+$ and $L_2=S_-$ with $(+)$ and $(-)$ BC, respectively, opposite to a homogeneous substrate with $(+)$ BC a distance $L$ apart \[see [Fig. \[fig:sketch\]]{} and the inset of [Fig. \[fig:kraft\]]{}\]. Indeed, by using the assumption of additivity of the critical Casimir forces underlying the DA and neglecting edge effects, the normal critical Casimir force $f^{ph}_{\rm (DA)}(S_+,S_-,L,T)$ per unit area acting on the walls is predicted to be given by $$\begin{gathered}
\label{eq:kraft-da}
f^{ph}_{\rm (DA)}(S_+,S_-,L,T)=\\
\frac{S_+}{S_+ + S_-}{\ensuremath{f_{{\ensuremath{(+,+)}}}}}(L,T)+\frac{S_-}{S_+ + S_-}{\ensuremath{f_{{\ensuremath{(+,-)}}}}}(L,T),\end{gathered}$$ $$\label{eq:kraft-da}
f^{ph}_{\rm (DA)}(S_+,S_-,L,T)=\frac{S_+}{S_+ + S_-}{\ensuremath{f_{{\ensuremath{(+,+)}}}}}(L,T)+\frac{S_-}{S_+ + S_-}{\ensuremath{f_{{\ensuremath{(+,-)}}}}}(L,T),$$ where $f_{(+,\pm)}$ refer to homogeneous parallel walls, as in [Eq. ]{}. At the bulk critical point the critical Casimir force is given in general by [@sprenger:2006] $$\label{eq:kraft-def}
f^{ph}(S_+,S_-,L,T=T_c)=k_B T_c \frac{d-1}{L^d}\Delta^{ph}\left(v=\frac{S_-}{S_+},\frac{S_+}{L}\right).$$ Using [Eq. ]{} together with Eqs. and one finds within the DA that $$\label{eq:kraft-da-delta}
(d-1)\Delta^{ph}_{\rm (DA)}\left(v,\frac{S_+}{L}\right)=\frac{v{\ensuremath{\Delta_{{\ensuremath{(+,-)}}}}}+{\ensuremath{\Delta_{{\ensuremath{(+,+)}}}}}}{1+v},$$ which renders the rhs of [Eq. ]{} to be independent of the scaling variable $S_+/L$. Within MFT as studied in Ref. ($d=4$), one has ${\ensuremath{\Delta_{{\ensuremath{(+,-)}}}}}=-4{\ensuremath{\Delta_{{\ensuremath{(+,+)}}}}}>0$ \[see the end of Sec. \[sec:MFT\]\] so that $$\label{eq:kraft-da-delta-fin}
\Delta^{ph}_{\rm (DA)}\left(v,\frac{S_+}{L}\right)=\frac{|{\ensuremath{\Delta_{{\ensuremath{(+,+)}}}}}|}{3}\;\frac{4v-1}{1+v}.$$ In [Fig. \[fig:kraft\]]{} we show the comparison between the actual scaling function $\Delta^{ph}$ (data points, obtained numerically as reported in Fig. 12 of Ref. ) and $\Delta^{ph}_{\rm (DA)}$ ([Eq. ]{}, solid line) derived by assuming additivity of the forces and neglecting edge effects. Figure \[fig:kraft\] clearly shows that the actual behavior of the critical Casimir force in the film geometry is not properly predicted within these assumptions. This is expected to be due to the presence of nonlinear effects and of edge effects in this context. This explains why in the limit $\Pi\to0$ the DA ($R\gg D$) used here does not capture the behavior of the critical Casimir force acting on a colloid close to periodically patterned substrate.
![ Scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] of the normal critical Casimir force acting on a spherical colloid with ${\ensuremath{(-)}}$ BC located at $X=0$ ($\Xi=X/\sqrt{RD}$) close to a periodically chemically patterned substrate \[see [Fig. \[fig:sketch\]]{}\]. ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ is suitably normalized by the absolute value of the force scaling function ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}(0,0)=2\pi{\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}}/(d-1)$ for the homogeneous $(-,-)$ case at criticality and within the DA \[Sec. \[sec:homog-da\]\]. The lateral position of the center of the colloid is fixed at the center of a stripe with $(a_1)=(-)$ BC and width $L_1=\lambda P$, which it is attracted to, in contrast to the second type of stripes with $(a_2)=(+)$ BC and width $L_2=(1-\lambda)P$, which it is repelled from. The scaling variable corresponding to the periodicity of the substrate pattern is (a) $\Pi=P/\sqrt{RD}=2.7$ and (b) $\Pi=0.57$, whereas the relative area fraction of the ${\ensuremath{(-)}}$ stripes changes from $\lambda=L_1/(L_1+L_2)=0$ to $\lambda=1$ (top to bottom: fully repulsive to fully attractive). In (a) and (b) the lines represent the result for the MFT critical Casimir force within the DA \[$\Delta\to0$, $d=4$, see [Eq. ]{}\], whereas the symbols represent the full numerical MFT data obtained for $\Delta=1/3$. The DA agrees reasonably well with the full data for $\Pi=2.3$ \[(a)\] and $\Theta\gtrsim1$, but for $\Pi=0.57$ \[(b)\] it fails to describe the actual behavior within the ranges $\Theta\lesssim4$ and $0.3\lesssim\lambda\lesssim0.9$ where the nonlinear effects strongly affect the resulting scaling function. In (c) ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ is shown for $\Pi=0.57$ and $2.3$, as obtained for $d=3$ within the DA on the basis of the Monte Carlo simulation data for the film geometry [@mcdata]. (We note, however, that we do not expect that the curves shown for $\Pi=0.57$ are quantitatively reliable.) []{data-label="fig:comparesystem"}](comparesystem_b "fig:"){width="7.8cm"}\
![ Scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] of the normal critical Casimir force acting on a spherical colloid with ${\ensuremath{(-)}}$ BC located at $X=0$ ($\Xi=X/\sqrt{RD}$) close to a periodically chemically patterned substrate \[see [Fig. \[fig:sketch\]]{}\]. ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ is suitably normalized by the absolute value of the force scaling function ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}(0,0)=2\pi{\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}}/(d-1)$ for the homogeneous $(-,-)$ case at criticality and within the DA \[Sec. \[sec:homog-da\]\]. The lateral position of the center of the colloid is fixed at the center of a stripe with $(a_1)=(-)$ BC and width $L_1=\lambda P$, which it is attracted to, in contrast to the second type of stripes with $(a_2)=(+)$ BC and width $L_2=(1-\lambda)P$, which it is repelled from. The scaling variable corresponding to the periodicity of the substrate pattern is (a) $\Pi=P/\sqrt{RD}=2.7$ and (b) $\Pi=0.57$, whereas the relative area fraction of the ${\ensuremath{(-)}}$ stripes changes from $\lambda=L_1/(L_1+L_2)=0$ to $\lambda=1$ (top to bottom: fully repulsive to fully attractive). In (a) and (b) the lines represent the result for the MFT critical Casimir force within the DA \[$\Delta\to0$, $d=4$, see [Eq. ]{}\], whereas the symbols represent the full numerical MFT data obtained for $\Delta=1/3$. The DA agrees reasonably well with the full data for $\Pi=2.3$ \[(a)\] and $\Theta\gtrsim1$, but for $\Pi=0.57$ \[(b)\] it fails to describe the actual behavior within the ranges $\Theta\lesssim4$ and $0.3\lesssim\lambda\lesssim0.9$ where the nonlinear effects strongly affect the resulting scaling function. In (c) ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ is shown for $\Pi=0.57$ and $2.3$, as obtained for $d=3$ within the DA on the basis of the Monte Carlo simulation data for the film geometry [@mcdata]. (We note, however, that we do not expect that the curves shown for $\Pi=0.57$ are quantitatively reliable.) []{data-label="fig:comparesystem"}](comparesystem_d "fig:"){width="7.8cm"}\
![ Scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] of the normal critical Casimir force acting on a spherical colloid with ${\ensuremath{(-)}}$ BC located at $X=0$ ($\Xi=X/\sqrt{RD}$) close to a periodically chemically patterned substrate \[see [Fig. \[fig:sketch\]]{}\]. ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ is suitably normalized by the absolute value of the force scaling function ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}(0,0)=2\pi{\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}}/(d-1)$ for the homogeneous $(-,-)$ case at criticality and within the DA \[Sec. \[sec:homog-da\]\]. The lateral position of the center of the colloid is fixed at the center of a stripe with $(a_1)=(-)$ BC and width $L_1=\lambda P$, which it is attracted to, in contrast to the second type of stripes with $(a_2)=(+)$ BC and width $L_2=(1-\lambda)P$, which it is repelled from. The scaling variable corresponding to the periodicity of the substrate pattern is (a) $\Pi=P/\sqrt{RD}=2.7$ and (b) $\Pi=0.57$, whereas the relative area fraction of the ${\ensuremath{(-)}}$ stripes changes from $\lambda=L_1/(L_1+L_2)=0$ to $\lambda=1$ (top to bottom: fully repulsive to fully attractive). In (a) and (b) the lines represent the result for the MFT critical Casimir force within the DA \[$\Delta\to0$, $d=4$, see [Eq. ]{}\], whereas the symbols represent the full numerical MFT data obtained for $\Delta=1/3$. The DA agrees reasonably well with the full data for $\Pi=2.3$ \[(a)\] and $\Theta\gtrsim1$, but for $\Pi=0.57$ \[(b)\] it fails to describe the actual behavior within the ranges $\Theta\lesssim4$ and $0.3\lesssim\lambda\lesssim0.9$ where the nonlinear effects strongly affect the resulting scaling function. In (c) ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ is shown for $\Pi=0.57$ and $2.3$, as obtained for $d=3$ within the DA on the basis of the Monte Carlo simulation data for the film geometry [@mcdata]. (We note, however, that we do not expect that the curves shown for $\Pi=0.57$ are quantitatively reliable.) []{data-label="fig:comparesystem"}](period_3d "fig:"){width="7.8cm"}
In [Fig. \[fig:comparesystem\]]{} we show the behavior of scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] of the normal critical Casimir force acting on the colloid in $d=4$ with $(b)={\ensuremath{(-)}}$ BC as a function of $\Theta=D/\xi_+$ (i.e., as a function of the normal distance of the colloid from the substrate in units of the bulk correlation length) and for various values of $\lambda$ and $\Pi$. In [Fig. \[fig:comparesystem\]]{} the scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ is evaluated at $X=0$ \[see [Fig. \[fig:sketch\]]{}\] which corresponds to the most preferred lateral position of the colloid in which the normal force is least repulsive or most attractive \[see [Fig. \[fig:period\_normal\]]{}\]. From [Fig. \[fig:comparesystem\]]{} one can infer that the DA does not provide an accurate estimate of ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ in the whole range of $\Theta$ for $\Pi=0.57$ \[panel (b)\], whereas it does so for $\Pi=2.3$ \[panel (a)\]. Indeed, for $\Pi=0.57$ the discrepancy between the DA and the numerical data is already significant for $\Theta\lesssim4$ and $0.3\lesssim\lambda\lesssim0.9$, whereas for $\Pi=2.3$ agreement is found for all values of $\lambda$ except for $\Theta\lesssim1$ \[[Fig. \[fig:comparesystem\]]{}(a)\]. This fact suggests that for relatively small periodicities $\Pi\lesssim2$ non-additive and edge effects become important. On the other hand, for large values of $\Theta\gg1$ the DA describes the behavior of ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ rather well for all values of $\Pi$ due to the exponential decay of the critical Casimir force for $\Theta\gg1$ \[[Eq. ]{}\]. Figure \[fig:comparesystem\](c) shows the scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ for $d=3$ within the DA as obtained from Monte Carlo simulation data for the film geometry [@mcdata]. The qualitative features of the behavior of ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ in $d=3$ and $d=4$ are similar.
From our analysis in $d=4$ we conclude that the DA describes quantitatively well the behavior of the actual critical Casimir force for $\Pi\gtrsim2$ for all values of $\Theta$. For smaller values of $\Pi$, the DA is only quantitatively reliable for large values of $\Theta$ (at which the force decays exponentially). For example, for $\Pi\gtrsim0.5$ the DA result is quantitatively correct for $\Theta\gtrsim4$. We expect these properties to be carried over to $d=3$.
Critical Casimir levitation \[sec:levitation\]
----------------------------------------------
Rather remarkably, within a certain range of values of $\lambda$, ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ *changes sign* as a function of $\Theta = D/\xi_+$ \[[Fig. \[fig:comparesystem\]]{}\]. In this context it is convenient to introduce for later purposes another scaling variable $\Psi=\Pi|\Theta|^{1/2}=P/\sqrt{R\xi_\pm}$ which is independent of $D$ and therefore does not vanish in the DA limit $D\ll R$ (i.e., $\Delta\to0$). Due to this change of sign of ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$, there exists a certain value $\Theta = \Theta_0(\Psi,\lambda,\Xi,\Delta)$ at which the normal critical Casimir force ${\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}$ acting on the colloid vanishes. This implies that in the absence of additional forces the colloid levitates at a height $D_0$ determined by $\Theta_0$ and $\xi_+$, which can be tuned by changing the temperature. Since for fixed geometrical parameters $R$, $X$, and $P$ the scaling variables $\Theta$, $\Pi$, $\Xi$, and $\Delta$ depend on $D$, one has to consider the behavior of ${\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}$ as a function of $D$ near $D_0$ in order to assess whether the levitation is stable against perturbations of $D$ or not. Stability requires $\partial_D {\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}|_{D=D_0}<0$ (so that for $D<D_0$ the colloid is repelled from the patterned substrate, whereas for for $D>D_0$ it is attracted). According to [Eq. ]{} one has $$\begin{gathered}
\label{eq:levitation-1}
\partial_D {\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}=
k_BT \frac{R}{D^d}\times\left\{-(d-1)\right.\\\left.-\tfrac{1}{2}\Pi\partial_\Pi-\tfrac{1}{2}\Xi\partial_\Xi+\Theta\partial_\Theta
+\Delta\partial_\Delta\right\}{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta).\end{gathered}$$ $$\label{eq:levitation-1}
\partial_D {\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}=k_BT \frac{R}{D^d}\left\{-(d-1)-\tfrac{1}{2}\Pi\partial_\Pi-\tfrac{1}{2}\Xi\partial_\Xi+\Theta\partial_\Theta
+\Delta\partial_\Delta\right\}{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta).$$ The laterally preferred position is always at $X=X_0=0$, corresponding to $\Xi=\Xi_0=0$, so that within the DA ($\Delta\to0$) one has $$\begin{gathered}
\label{eq:levitation-2}
\operatorname{sign}\left(\partial_D {\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}\big|_{D=D_0,X=X_0,{\rm DA}}\right)=\\
\operatorname{sign}\left(\left\{-\tfrac{1}{2}\Pi\partial_{\Pi}+\Theta\partial_\Theta\right\}
{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi=0,\Theta,\Delta\to 0)\big|_{\Theta=\Theta_0}\right),\end{gathered}$$ $$\label{eq:levitation-2}
\operatorname{sign}\left(\partial_D {\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}\big|_{D=D_0,X=X_0,{\rm DA}}\right)=
\operatorname{sign}\left(\left\{-\tfrac{1}{2}\Pi\partial_{\Pi}+\Theta\partial_\Theta\right\}
{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi=0,\Theta,\Delta\to 0)\big|_{\Theta=\Theta_0}\right),$$ where we have used the implicit equation ${\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}|_{D=D_0}=0$ so that ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}|_{D=D_0}=0$. (Equation assumes that $\partial_\Delta{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ does not diverge $\propto\Delta^{-1}$ for $\Delta\to0$.) In the following we only consider $\Theta\ge0$ and BC $(a_1)=(-)$, $(a_2)=(+)$, and $(b)=(-)$.
Within the DA we find that both $\partial_{\Pi}{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}|_{\Theta=\Theta_0,\Xi=\Xi_0}$ and $\partial_{\Theta}{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}|_{\Theta=\Theta_0, \Xi=\Xi_0}$ are negative, so that according to [Eq. ]{} the sign of $\partial_D{\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}|_{D=D_0,X=X_0,{\rm DA}}$ can vary and depends on their values as well as on $\Theta_0$ and $\Pi$. However, at criticality ($\Theta=0$) the second term of the rhs of [Eq. ]{} vanishes. Thus, at the bulk critical point $T=T_c$ the derivative $\partial_D{\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}$ evaluated at $D=D_0$ and $X=X_0=0$ is always positive so that one cannot achieve stable levitation. On the other hand, for ${\Theta>0}$ it is always possible to find geometrical configurations for which the colloid exhibits stable levitation, as described in the following.
![ Values of the scaling variable $\Theta_0$ at which within the DA ($\Delta\to0$) the normal critical Casimir force ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ shown in [Fig. \[fig:comparesystem\]]{} vanishes as a function of $\Psi$ for (a) $d=4$ and (b) $d=3$ on the basis of Monte Carlo simulation data [@mcdata] and for various values of $\lambda=L_1/P$. The solid lines correspond to values of $\Theta_0$ for which the levitation of the colloid at a height $D_0$ above the substrate is *stable* against perturbations of $D$ \[$\partial_D{\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}|_{D=D_0}<0$, see [Eq. ]{}\]. The shaded region and the dashed lines indicate those values of $\Theta_0$ for which $\partial_D{\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}|_{D=D_0}>0$ and thus do *not* correspond to stable levitation. For $\lambda>\lambda_0$ with $\lambda_0(d=4)=4/5$ and $\lambda_0(d=3)\simeq0.88$, $\Theta_0$ ceases to exist, i.e., ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ does not exhibit a zero. For $\lambda<\lambda_1$ with $\lambda_1(d=4)=1/2$ and $\lambda_1(d=3)\simeq0.545$, $\Theta_0(\Psi\searrow\Psi_0(\lambda))$ diverges. (The values for $\Psi_0(\lambda)$ are indicated by upward arrows.) For any $\lambda<\lambda_0$, $\Theta_0$ exists for $\Psi<\Psi^*(\lambda)$. (From the analysis in [Fig. \[fig:comparesystem\]]{} we expect the DA to be quantitatively reliable only for $\Psi\gtrsim2\sqrt{\Theta_0}$ for $\Theta_0\lesssim4$ and for $\Psi\gtrsim0.5\sqrt{\Theta_0}$ for $\Theta_0\gtrsim4$, which implies $\lambda\lesssim0.7$ in $d=3$ and $\lambda\lesssim0.6$ in $d=4$.) []{data-label="fig:levitation"}](lev_new_4d_c "fig:")\
![ Values of the scaling variable $\Theta_0$ at which within the DA ($\Delta\to0$) the normal critical Casimir force ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ shown in [Fig. \[fig:comparesystem\]]{} vanishes as a function of $\Psi$ for (a) $d=4$ and (b) $d=3$ on the basis of Monte Carlo simulation data [@mcdata] and for various values of $\lambda=L_1/P$. The solid lines correspond to values of $\Theta_0$ for which the levitation of the colloid at a height $D_0$ above the substrate is *stable* against perturbations of $D$ \[$\partial_D{\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}|_{D=D_0}<0$, see [Eq. ]{}\]. The shaded region and the dashed lines indicate those values of $\Theta_0$ for which $\partial_D{\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}|_{D=D_0}>0$ and thus do *not* correspond to stable levitation. For $\lambda>\lambda_0$ with $\lambda_0(d=4)=4/5$ and $\lambda_0(d=3)\simeq0.88$, $\Theta_0$ ceases to exist, i.e., ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ does not exhibit a zero. For $\lambda<\lambda_1$ with $\lambda_1(d=4)=1/2$ and $\lambda_1(d=3)\simeq0.545$, $\Theta_0(\Psi\searrow\Psi_0(\lambda))$ diverges. (The values for $\Psi_0(\lambda)$ are indicated by upward arrows.) For any $\lambda<\lambda_0$, $\Theta_0$ exists for $\Psi<\Psi^*(\lambda)$. (From the analysis in [Fig. \[fig:comparesystem\]]{} we expect the DA to be quantitatively reliable only for $\Psi\gtrsim2\sqrt{\Theta_0}$ for $\Theta_0\lesssim4$ and for $\Psi\gtrsim0.5\sqrt{\Theta_0}$ for $\Theta_0\gtrsim4$, which implies $\lambda\lesssim0.7$ in $d=3$ and $\lambda\lesssim0.6$ in $d=4$.) []{data-label="fig:levitation"}](lev_new_3d_c "fig:")
Figure \[fig:levitation\] shows the values of $\Theta_0$ at which the normal critical Casimir force acting on a colloid vanishes as a function of the new scaling variable $\Psi$ introduced at the beginning of this subsection, for various $\lambda$, for $\Xi=0$, and within the DA ($\Delta\to0$) for (a) $d=4$ and (b) $d=3$. The corresponding sign of $\partial_D {\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}\big|_{D=D_0}$ \[according to [Eq. ]{}\] is also indicated: $\Theta_0$ drawn as a solid line indicates $\partial_D {\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}\big|_{D=D_0} < 0$, i.e., stable levitation of the colloid; a dashed line, instead, indicates $\partial_D {\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}\big|_{D=D_0}>0$ and therefore a local maximum of the critical Casimir potential with respect to $D$, which occurs within the shaded regions in [Fig. \[fig:levitation\]]{}. For a given value of $\lambda$ (with $\lambda_1<\lambda<\lambda_0$ as we shall discuss in detail further below), e.g., $\lambda =0.60$ in [Fig. \[fig:levitation\]]{}(a), the corresponding curve for $\Theta_0$ shows a bifurcation at $\Psi = \Psi^*(\lambda)$ such that a vertical line drawn in [Fig. \[fig:levitation\]]{} at a certain $\Psi$ intersects this curve in two points $\Theta_{0,u}$ and $\Theta_{0,s}>\Theta_{0,u}$ if $\Psi < \Psi^*(\lambda)$, whereas it has no intersection for $\Psi >\Psi^*(\lambda)$. In the former case $\Theta_{0,u}$ and $\Theta_{0,s}$ correspond to a local maximum and to a local minimum of the critical Casimir potential at distances $D_{0,u} = \xi_+ \Theta_{0,u}$ and $D_{0,s} = \xi_+ \Theta_{0,s}$, respectively, i.e., to an [*u*]{}nstable and a [*s*]{}table levitation point for the colloid, respectively. Instead, for $\Psi>\Psi^*(\lambda)$, the critical Casimir force has no zero at any finite value of $D$. We note that $D=0$ (stiction) and thus $\Theta=0$ always corresponds to the global minimum of the potential because for $D\to 0$ the critical Casimir potential is strongly attractive. The corresponding geometrical configuration into which the colloid is finally attracted by the substrate \[due to $(a_1) =(-)$, $(b)=(-)$, and $X=0$, see Fig.1\] is stabilized by the steric repulsion of the wall. We note that within the DA the critical Casimir potential for $X=0$ is attractive at sufficiently small distances, even if the major part of the substrate is characterized by $(+)$ BC, i.e., even if $0\ne\lambda \ll 1$. Indeed, in this case the potential of the colloid at $X=0$ and close to a periodically patterned substrate can be approximated by the one due to a single chemical lane centered at $X=0$, which has been discussed in Sec. \[sec:stripe\]. For given colloid radius $R$ and width $L_1=\lambda P>0$ of the attractive stripe, the scaling variable $\Lambda = L_1/(2\sqrt{RD})$ diverges as $D\to 0$, so that the scaling function ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}(\Lambda,\Xi,\Theta,\Delta)$ which characterizes the potential of the lane \[see [Eq. ]{}\] attains the value $-1$ corresponding to the case of homogeneous, attractive $(-,-)$ BC \[see [Fig. \[fig:stripe\]]{}\]. Within this approximation and for $D\ll\xi_\pm$ the critical Casimir force becomes attractive if ${\ensuremath{\vartheta_{{\ensuremath{\textrm{p}}}}}}\simeq {\ensuremath{\vartheta_{{\ensuremath{\ell}}}}}< 0$ which, due to Eqs. , , and , yields the condition ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}(\Lambda,\Xi=0,\Theta\to0,\Delta\to0)<1-2{\ensuremath{\Delta_{{\ensuremath{(+,-)}}}}}/({\ensuremath{\Delta_{{\ensuremath{(+,-)}}}}}-{\ensuremath{\Delta_{{\ensuremath{(+,+)}}}}})$, i.e., ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}<-0.6$ in $d=4$ [@krech:1997] and ${\ensuremath{\omega_{{\ensuremath{\ell}}}}}\lesssim-0.76$ in $d=3$ [@mcdata]; this occurs for $\Lambda > \Lambda_0 = 1.1$ in $d = 4$, and $\Lambda>\Lambda_0=2.7$ in $d = 3$, respectively \[see also [Fig. \[fig:stripe\]]{}(a)\]. Accordingly, at distances $D<\lambda^2P^2/(4R\Lambda_0^2)$ (together with $D\ll\xi_\pm$) the critical Casimir potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ is negative and diverges to $-\infty$ for $D \to 0$. (However, for very small values of $\lambda$ this would occur at distances of microscopic scale such that the scaling limit and thus the form of ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ do no longer hold). Thus the bifurcation of $\Theta_0$ at $\Psi^*(\lambda)$ corresponds to a transition from (metastable) levitation at $D=D_{0,s}$ for $\Psi<\Psi^*(\lambda)$ to stiction at $D=0$ for $\Psi>\Psi^*(\lambda)$. For $\Psi<\Psi^*(\lambda)$ the metastable levitation minimum at $D_{0,s}$ is shielded from the global minimum at $D=0$ by a potential barrier the height of which vanishes for $\Psi\nearrow\Psi^*(\lambda)$ \[see [Fig. \[fig:levitation\_example\]]{}\]. Experimentally, one typically varies the value of $\xi_+$ by changing the temperature [@hertlein:2008; @gambassi:2009; @nellen:2009; @soyka:2008] and leaves the geometry ($\lambda$, $P$, and $R$) unchanged, which results in a change of $\Psi$ via varying $T$. Thus, experimentally, the transition at $\Psi^*(\lambda)$ corresponds to a de facto irreversible transition from separation to stiction of the colloid as a function of temperature.
Moreover, from [Fig. \[fig:levitation\]]{} one can infer that for both $d=3$ and $d=4$ there is a $\lambda_0$ such that, for $1\ge\lambda>\lambda_0$, ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ has no zero for any choice of $\Psi$ (i.e., there is no solution $\Theta_0$) and the critical Casimir force is attractive at all distances. Within the DA, $\lambda_0={\ensuremath{\Delta_{{\ensuremath{(+,-)}}}}}/({\ensuremath{\Delta_{{\ensuremath{(+,-)}}}}}-{\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}})$ \[see also [Eq. ]{}\], which renders the values $\lambda_0=0.80$ in $d=4$ [@krech:1997] and $\lambda_0\simeq0.88$ in $d=3$ [@mcdata]. In addition, from [Fig. \[fig:levitation\]]{} one can infer that for $\lambda_0>\lambda>\lambda_1 \simeq 0.5$ and $\Psi \lesssim 1$, $\Theta_{0,s}$ effectively does no longer depend on $\Psi$ but solely on $\lambda$. Accordingly, the distance $D_{0,s} \propto \xi_+$ at which the colloid stably levitates can be tuned by temperature upon approaching criticality. However, for $\lambda < \lambda_1 \simeq 0.5$, $\Theta_{0,s}$ diverges at $\Psi = \Psi_0(\lambda) < \Psi^*(\lambda)$ such that for $\Psi_0(\lambda)<\Psi<\Psi^*(\lambda)$ the colloid exhibits critical Casimir levitation at a local minimum of the potential, whereas within this range of $\lambda$ values for $\Psi<\Psi_0(\lambda)$ the critical Casimir potential has only a local (positive) maximum at $D_{0,u}$; it is repulsive for $D>D_{0,u}$ and therefore for large values of $D$ (i.e., $\Theta \gg 1$ and $\Pi \ll 1$) it approaches zero from positive values. This qualitative change in the behavior of the critical Casimir potential occurs at $\lambda = \lambda_1$. The value of $\lambda_1$ is close to 0.5 because the repulsive and attractive forces for $(+,-)$ and $(-,-)$ BC, respectively, have similar strengths but opposite signs for $\Theta \gg 1$, i.e., ${\ensuremath{k_{{\ensuremath{(+,-)}}}}}(\Theta \gg 1)\simeq-{\ensuremath{k_{{\ensuremath{(-,-)}}}}}(\Theta\gg 1)$ for both $d=3$ and $d=4$ \[see [Eq. ]{}, where $|A_-/A_+|\simeq 1.2$ in $d=3$ [@gambassi:2009] and $|A_-/A_+|=1$ in $d=4$ [@krech:1997]\]. Accordingly, depending on $\lambda$ being larger or smaller than $\lambda_1\simeq 0.5$, the area covered by one of the two BC prevails and the resulting force is asymptotically (i.e., $\Theta\gg1$) attractive or repulsive, respectively \[see the remark at the end of Sec. \[sec:period-da\] and Eqs. and \]. Taking into account the slight difference in the strength of the asymptotic forces for $(+,-)$ and $(-,-)$ BC one finds $\lambda_1 = (1-A_+/A_-)^{-1} $ which renders $\lambda_1=1/2$ in $d=4$ and $\lambda_1\simeq 0.545$ in $d=3$. The asymptotic behavior of the force at large distances can be inferred from the asymptotic behavior of ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi =\Psi\Theta^{-1/2},\Xi=0,\Theta\gg1,\Delta\to 0) \simeq {\mathcal A}(\Psi,\lambda)\; \Theta^{d-1} e^{-\Theta}$, which can be obtained from Eqs. , , , , and . Accordingly, the value $\Psi_0(\lambda)$ at which $\Theta_{0,s}$ diverges is characterized by the fact that ${\mathcal A}(\Psi \lessgtr \Psi_0(\lambda),\lambda)\gtrless 0$ so that the force approaches zero from above or from below depending on having $\Psi<\Psi_0(\lambda)$ or $\Psi>\Psi_0(\lambda)$, respectively. The condition ${\mathcal A}(\Psi_0(\lambda),\lambda) = 0$ yields the following implicit equation for $\Psi_0(\lambda)$: $$\label{eq:psi0}
2\lambda_1=\sum_{n=-\infty}^{\infty}
\operatorname{erf}\left\{\tfrac{\Psi_0(\lambda)}{\sqrt{2}}(n+\tfrac{\lambda}{2})\right\}
-\operatorname{erf}\left\{\tfrac{\Psi_0(\lambda)}{\sqrt{2}}(n-\tfrac{\lambda}{2})\right\}.$$ For $\lambda\ll1$ the sum on the rhs of [Eq. ]{} can be approximated by the term $n=0$ alone and one finds $\Psi_0(\lambda\ll1)\simeq2^{3/2}\lambda^{-1}\operatorname{erf}^{-1}(\lambda_1)$, where $\operatorname{erf}^{-1}$ is the inverse error function, which yields the relations $\Psi_0(\lambda\ll1)\simeq1.49/\lambda$ for $d=3$ and $\Psi_0(\lambda\ll1)\simeq1.35/\lambda$ for $d=4$. On the other hand, in the marginal case one expects $\Psi_0(\lambda=\lambda_1)=0$. However, as argued above, *at* the critical point ($\Theta=0$) the colloid does not exhibit stable levitation for any geometrical configuration; this is in accordance with [Fig. \[fig:levitation\]]{} because for $T\to T_c$, the levitation minimum of the potential moves to large $D$ ($D_{0,s}=\Theta_{0,s}\xi_+\to\infty$) and disappears at $T=T_c$.
In summary, as function of $\lambda$ there are three distinct levitation regimes:
- $\lambda>\lambda_0$ with $\lambda_0(d=3)\simeq0.88$ and $\lambda_0(d=4)=4/5$: There is no levitation and the critical Casimir force is attractive at all distances for any temperature.
- $\lambda_0>\lambda>\lambda_1$ with $\lambda_1(d=3)\simeq0.545$ and $\lambda_1(d=4)=1/2$: Sufficiently close to $T_c$, i.e., for $\Psi=P/\sqrt{R\xi_+}<\Psi^*(\lambda)$ there is a local critical Casimir levitation minimum. Upon approaching $T_c$ its position $D_{0,s}=\Theta_{0,s}\xi_+$, with $\Theta_{0,s}(\xi_+\to\infty)$ finite, moves to macroscopic values proportional to the bulk correlation length.
- $\lambda_1>\lambda$: As in (ii) there is a local critical Casimir levitation minimum sufficiently close to $T_c$, i.e., for $\Psi<\Psi^*(\lambda)$. In general the onset of its appearance occurs further away from $T_c$ upon lowering $\lambda$. Upon approaching $T_c$ the position $D_{0,s}$ of this minimum diverges at a distinct nonzero reduced temperature given by $\Psi_0(\lambda)$, i.e., at $\xi_+=P^2/[R\Psi_0^2(\lambda)]$: $D_{0,s}=\Theta_{0,s}\xi_+$ with $\Theta_{0,s}(\Psi\searrow\Psi_0(\lambda))\to\infty$.
We note that, according to Figs. \[fig:period\_normal\], \[fig:normal\_3d\], \[fig:kraft\] and \[fig:comparesystem\], we expect that for $\Pi\lesssim2$ and $\Theta\lesssim4$ and for $\Pi\lesssim0.5$ and $\Theta\gtrsim4$, the DA does not provide a quantitatively reliable description of the actual behavior of ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ and therefore of ${\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}$; thus, for values of $\Psi\lesssim2\sqrt{\Theta_0}$ for $\Theta_0\lesssim4$, and $\Psi\lesssim0.5\sqrt{\Theta_0}$ for $\Theta_0\gtrsim4$, we expect quantitative discrepancies between the actual behavior and the one predicted by the DA shown in [Fig. \[fig:levitation\]]{}. Nonetheless our results demonstrate that the geometric arrangement of the chemical patterns allows one to design the normal critical Casimir force over a wide range.
![ Critical Casimir potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] in $d=3$ of a colloid of radius $R=1.35\mu$m close to a periodically patterned substrate as a function of $D$ and for various values of $\xi_+$ for $P=1\mu$m with $\lambda=0.4$ in (a) and $P=0.4\mu$m with $\lambda=0.65$ in (b) and (c). The values of $P$, $\lambda$, and $\xi_+$ are chosen as to be experimentally accessible in a colloidal suspension exhibiting critical Casimir forces [@nellen:2009; @hertlein:2008; @gambassi:2009; @soyka:2008]. The critical Casimir potential for the colloid close to a patterned substrate may exhibit – depending on the value of $\xi_+$, and, thus, on the temperature – a local minimum corresponding to stable levitation. In (c) an electrostatic potential $\Phi_{\textrm{el}}$ \[[Eq. ]{}\] is added to ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$, which refers to actual experimental data [@nellen:2009]. The shaded area indicates the ranges of the positions and the depths of the local minima of the total potential occurring if the substrate is laterally homogeneous and purely attractive, i.e., for $\lambda=1$ ($(-,-)$ BC) for the range $14$nm $<\xi_+<75$nm leading to potential depths between $0.5k_BT$ and $70k_BT$ (indicated by the shaded arrow); for $\lambda=1$ the preferred colloid position is dictated by the electrostatic repulsion and restricted to the range of $50$nm to $75$nm, whereas the colloid position $D_{0,s}=\Theta_{0,s}\xi_+$ due to critical Casimir levitation can be much larger and tuned by temperature. Moreover, whereas for $\lambda=1$ and upon approaching $T_c$ the minima monotonically become deeper, the levitation minima first deepen and move to smaller values of $D$ followed by a decrease of the depth, by becoming more shallow, and moving to larger values of $D$. Reducing the range and strength of the electrostatic repulsion by adding salt to the solvent is expected to provide access to even more details of the critical Casimir levitation potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ shown in (b). []{data-label="fig:levitation_example"}](lev_ex_1 "fig:"){width="6.5cm"}\
![ Critical Casimir potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] in $d=3$ of a colloid of radius $R=1.35\mu$m close to a periodically patterned substrate as a function of $D$ and for various values of $\xi_+$ for $P=1\mu$m with $\lambda=0.4$ in (a) and $P=0.4\mu$m with $\lambda=0.65$ in (b) and (c). The values of $P$, $\lambda$, and $\xi_+$ are chosen as to be experimentally accessible in a colloidal suspension exhibiting critical Casimir forces [@nellen:2009; @hertlein:2008; @gambassi:2009; @soyka:2008]. The critical Casimir potential for the colloid close to a patterned substrate may exhibit – depending on the value of $\xi_+$, and, thus, on the temperature – a local minimum corresponding to stable levitation. In (c) an electrostatic potential $\Phi_{\textrm{el}}$ \[[Eq. ]{}\] is added to ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$, which refers to actual experimental data [@nellen:2009]. The shaded area indicates the ranges of the positions and the depths of the local minima of the total potential occurring if the substrate is laterally homogeneous and purely attractive, i.e., for $\lambda=1$ ($(-,-)$ BC) for the range $14$nm $<\xi_+<75$nm leading to potential depths between $0.5k_BT$ and $70k_BT$ (indicated by the shaded arrow); for $\lambda=1$ the preferred colloid position is dictated by the electrostatic repulsion and restricted to the range of $50$nm to $75$nm, whereas the colloid position $D_{0,s}=\Theta_{0,s}\xi_+$ due to critical Casimir levitation can be much larger and tuned by temperature. Moreover, whereas for $\lambda=1$ and upon approaching $T_c$ the minima monotonically become deeper, the levitation minima first deepen and move to smaller values of $D$ followed by a decrease of the depth, by becoming more shallow, and moving to larger values of $D$. Reducing the range and strength of the electrostatic repulsion by adding salt to the solvent is expected to provide access to even more details of the critical Casimir levitation potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ shown in (b). []{data-label="fig:levitation_example"}](lev_ex_2 "fig:"){width="6.5cm"}\
![ Critical Casimir potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ \[[Eq. ]{}\] in $d=3$ of a colloid of radius $R=1.35\mu$m close to a periodically patterned substrate as a function of $D$ and for various values of $\xi_+$ for $P=1\mu$m with $\lambda=0.4$ in (a) and $P=0.4\mu$m with $\lambda=0.65$ in (b) and (c). The values of $P$, $\lambda$, and $\xi_+$ are chosen as to be experimentally accessible in a colloidal suspension exhibiting critical Casimir forces [@nellen:2009; @hertlein:2008; @gambassi:2009; @soyka:2008]. The critical Casimir potential for the colloid close to a patterned substrate may exhibit – depending on the value of $\xi_+$, and, thus, on the temperature – a local minimum corresponding to stable levitation. In (c) an electrostatic potential $\Phi_{\textrm{el}}$ \[[Eq. ]{}\] is added to ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$, which refers to actual experimental data [@nellen:2009]. The shaded area indicates the ranges of the positions and the depths of the local minima of the total potential occurring if the substrate is laterally homogeneous and purely attractive, i.e., for $\lambda=1$ ($(-,-)$ BC) for the range $14$nm $<\xi_+<75$nm leading to potential depths between $0.5k_BT$ and $70k_BT$ (indicated by the shaded arrow); for $\lambda=1$ the preferred colloid position is dictated by the electrostatic repulsion and restricted to the range of $50$nm to $75$nm, whereas the colloid position $D_{0,s}=\Theta_{0,s}\xi_+$ due to critical Casimir levitation can be much larger and tuned by temperature. Moreover, whereas for $\lambda=1$ and upon approaching $T_c$ the minima monotonically become deeper, the levitation minima first deepen and move to smaller values of $D$ followed by a decrease of the depth, by becoming more shallow, and moving to larger values of $D$. Reducing the range and strength of the electrostatic repulsion by adding salt to the solvent is expected to provide access to even more details of the critical Casimir levitation potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ shown in (b). []{data-label="fig:levitation_example"}](lev_ex_2_el "fig:"){width="6.5cm"}\
Figures \[fig:levitation\_example\](a) and (b) show the critical Casimir potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ as a function of $D$ in $d=3$ within the DA based on Monte Carlo simulation data for the film geometry [@mcdata] for a variety of specifically chosen values of the parameters $P$, $L_1$, $R$, and $\xi$. The choice of these values is motivated by the typical experimental parameters which characterize recent investigations of the critical Casimir force acting on colloids immersed in binary liquid mixtures [@nellen:2009; @soyka:2008; @hertlein:2008; @gambassi:2009]. In particular, concerning the colloid radius we focus on the data of Ref. , corresponding to $R=1.35\mu$m, while for the pattern we have chosen a periodicity $P=1\mu$m with $\lambda=0.4$ (i.e., $L_1=400$nm and $L_2=600$nm) \[[Fig. \[fig:levitation\_example\]]{}(a)\], or $P=0.4\mu$m with $\lambda=0.65$ (i.e., $L_1=260$nm and $L_2=140$nm) \[[Fig. \[fig:levitation\_example\]]{}(b)\]. A chemically patterned substrate with these characteristics appears to be realizable with presently available preparation techniques [@soyka:2008; @vogt:2009a; @vogt:2009]. \[We note that ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ as shown in [Fig. \[fig:levitation\_example\]]{}(a) and (b) is expected to describe the actual interaction potential in the scaling regime characterized by values of $D$ and $\xi_+$ much larger than microscopic length scales (such as $\xi_0^+\simeq0.2$nm [@hertlein:2008; @gambassi:2009]) so that this prediction for ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ is valid only for $D,\xi_+\gtrsim5$nm.\] With this choice of parameters we have calculated ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}$ for various values of $\xi_+$ within an experimentally accessible range [@nellen:2009; @soyka:2008; @hertlein:2008; @gambassi:2009]. From Figs. \[fig:levitation\_example\](a) and \[fig:levitation\_example\](b) one can infer that for small values of $\xi_+$ (corresponding to large values of $\Psi>\Psi^*(\lambda)$) the critical Casimir potential is always attractive with a monotonic dependence on $D$ \[see also [Fig. \[fig:levitation\]]{}\]. Upon approaching criticality, i.e., for increasing values of $\xi_+$ and decreasing values of $\Psi<\Psi^*(\lambda)$, a local maximum and a local minimum of the potential develop, so that for very small as well as for large $D$ the colloid is attracted to the patterned substrate, whereas within an intermediate range of values for $D$ it is repelled from it \[see also [Fig. \[fig:levitation\]]{}\]. Thus, the colloid stably levitates at a distance $D_{0,s}$ corresponding to a local minimum of the potential. The depth of this minimum ranges between a few $k_BT$ \[[Fig. \[fig:levitation\_example\]]{}(a)\] up to several $k_BT$ \[[Fig. \[fig:levitation\_example\]]{}(b)\]. Upon increasing $\xi_+$, $D_{0,s}$ increases as well, i.e., the colloid position is shifted away from the patterned substrate with the potential minimum becoming more shallow. In [Fig. \[fig:levitation\_example\]]{}(a) $\lambda=0.4$ and we find $\Psi^*(\lambda=0.4)\simeq4.65$ and $\Psi_0(\lambda=0.4)\simeq3.71$ \[see [Fig. \[fig:levitation\]]{}(b)\] so that for $\Psi<\Psi_0(\lambda=0.4)$, i.e., for $\xi_+\gtrsim53.5$nm \[[Fig. \[fig:levitation\_example\]]{}(a)\] the colloid does not exhibit stable levitation and the critical Casimir potential has a local maximum only. The levitation minimum moves to macroscopic values of $D$ upon approaching the temperature corresponding to $\xi_+\simeq53.5$nm. In [Fig. \[fig:levitation\_example\]]{}(b) $\lambda=0.65$ and one has $\Psi^*(\lambda=0.65)\simeq2.63$; here $\Theta_{0,s}$ remains finite for $\Psi\to0$ in contrast to the case $\lambda<0.545$ \[[Fig. \[fig:levitation\]]{}(b)\]. Thus, within the DA, for the case shown in [Fig. \[fig:levitation\_example\]]{}(b) stable levitation of the colloid is preserved for all finite values of $\xi_+>P^2/[R\,\,(\Psi^*(\lambda=0.65))^2]\simeq17$nm. In this case upon approaching $T_c$ the levitation minimum moves to macroscopic values of $D$ proportional to the bulk correlation length $\xi_+$.
The discussion above focuses on the position of mechanical equilibrium of the colloid, corresponding to the point at which the forces acting on the particle vanish and the associated potential $\Phi$ has a local minimum $\Phi_{\rm min}$. However, due to the thermal fluctuations of the surrounding near-critical fluid at temperature $T$, the colloid undergoes a Brownian diffusion which allows it to explore randomly such regions in space where the potential $\Phi$ is typically larger than $\Phi_{\rm min}$ for at most few $k_BT$. As a result, a position of mechanical equilibrium is stable against the effect of thermal fluctuations only if the potential depth of the minimum is larger than few $k_BT$. In particular, if the potential barrier $\Phi(L_1,P,0,D=D_{0,u},R,T) - \Phi(L_1,P,0,D=D_{0,s},R,T)$, which separates the position of the local minimum at distance $D=D_{0,s}$ (levitation) from the global one at $D=0$ (stiction), is not sufficiently large \[see, e.g., the curves corresponding to $\xi_+ = 36$nm in [Fig. \[fig:levitation\_example\]]{}(a) or corresponding to $\xi_+\lesssim 18$nm in [Fig. \[fig:levitation\_example\]]{}(b)\], a de facto irreversible transition from levitation to stiction may occur as a consequence of thermal fluctuations.
In [Fig. \[fig:levitation\_example\]]{}(c) we show the resulting total potential of the forces acting on the colloid in the presence of an additional electrostatic repulsion which is experimentally practically unavoidable, in order to study its effect on critical Casimir levitation. We assume that the electrostatic repulsion is laterally homogeneous and that it can be simply added to the critical Casimir potential [@troendle:2009; @gambassi:2009; @nellen:2009] \[see also Sec. \[sec:summary\] below\]. Concerning the spatial dependence of the electrostatic repulsion we consider the one of Ref. , which corresponds to a colloid of radius $R=1.35\mu$m immersed in a near-critical water-lutidine mixture and close to a substrate exhibiting critical adsorption of water or lutidine [@nellen:2009]: $$\label{eq:phitot}
\Phi_{\textrm{el}}(D)/k_BT = \exp\{-\kappa(D-D_0)\},$$ where $D_0=88$nm and $\kappa^{-1}=11$nm [@nellen:2009]. (Formally, $\Phi_{\textrm{el}}$ in [Eq. ]{} is finite for $D\to0$, and thus ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}+\Phi_{\textrm{el}}$ is negative for $D\lesssim2$nm and has a global minimum at $D=0$ because ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}\to-\infty$ for $D\to0$. However, [Eq. ]{} is actually the asymptotic form of the electrostatic interaction which is valid for distances larger than the electrostatic screening length, i.e., $D\gg\kappa^{-1}$. The corresponding total potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{p}}}}}}+\Phi_{\textrm{el}}$ is therefore not accurate for small values of $D$ and is reported in [Fig. \[fig:levitation\_example\]]{}(c) for $D>50$nm only.) As in [Fig. \[fig:levitation\_example\]]{}(b) we choose $P=0.4\mu$m, $\lambda=0.65$, and experimentally accessible values of $\xi_+$. Figure \[fig:levitation\_example\](c) provides a realistic comparison of the critical Casimir potential with other forces as they typically occur in actual experimental systems. One can infer from the graph reported in [Fig. \[fig:levitation\_example\]]{}(c) that for this choice of parameters the critical Casimir levitation exhibited by the colloid is rather pronounced even in the presence of electrostatic interaction. Far from the critical point ($\xi_+=10$nm) the interaction of the colloid with the substrate is completely dominated by electrostatic repulsion. Upon approaching criticality ($10$nm $\lesssim\xi_+\lesssim35$nm) a minimum in the total potential develops and becomes deeper due to the increasing critical Casimir attraction working against the electrostatic repulsion. For this latter range of values of $\xi_+$ the local minimum of the critical Casimir potential corresponding to levitation is located at distances $D_{0,s}\lesssim60$nm at which the electrostatic repulsion still strongly contributes to the resulting total potential \[see [Fig. \[fig:levitation\_example\]]{}(c)\]. Closer to the critical point ($\xi_+\gtrsim45$nm) the levitation minimum of the critical Casimir potential occurs at distances $D_{0,s}\gtrsim100$nm \[see [Fig. \[fig:levitation\_example\]]{}(b)\] at which the electrostatic force acting on the colloid is weak. Thus, here the critical Casimir effect dominates and the position of the minimum of the total potential increases with increasing values of $\xi_+$, which allows for measurements of the critical Casimir potential for distances at which the precise form of $\Phi_{\textrm{el}}$ is not important. Moreover, the depth of the minimum *decreases* upon approaching criticality and the minimum becomes more shallow. This behavior of the levitation minimum is distinct from the critical Casimir effect acting on a colloid close to a *homogeneous* substrate: a local minimum also occurs in the latter case if the critical Casimir force is purely attractive ($\lambda=1$, $(-,-)$ BC) and works against the electrostatic repulsion [@hertlein:2008; @gambassi:2009], due to the competition of different forces with opposite sign. (We note that the critical Casimir levitation described above emerges from the critical Casimir force alone, i.e., it is a feature of a *single* force contribution.) However, in this homogeneous case the preferred colloid position $D_{0,(-,-)}$ depends crucially on the form of the electrostatic interaction and is almost constant ($50$nm $<D_{0,(-,-)}< 75$nm). Moreover, the depths of these latter minima monotonically increase as a function of of $\xi_+$ and become much larger than those shown in [Fig. \[fig:levitation\_example\]]{}(c) (see, e.g., Fig. 2(a) and Fig. 2(c) in Ref. and Fig. 3 in Ref. ). In [Fig. \[fig:levitation\_example\]]{}(c) this is indicated by the shaded area and the shaded arrow, which corresponds to the area of the graph within which minima of the total potential in the homogeneous case $\lambda=1$ occur for $14$nm $<\xi_+<75$nm corresponding to potential depths of $0.5k_BT$ up to $70k_BT$. On the other hand, the colloid position $D_{0,s}$ due to critical Casimir levitation can be much larger, can reach values of several $\xi_+$, and can be tuned by temperature according to $D_{0,s}=\Theta_{0,s}\xi_+$. In conclusion, the examples presented in [Fig. \[fig:levitation\_example\]]{} strongly suggest that the critical Casimir levitation of a colloid close to a patterned substrate is experimentally accessible.
By patterning the substrate, one introduces an additional (*lateral*) length scale into the system, which, according to our results presented above, can finally lead to stable levitation. Introducing an additional length scale along the *normal* direction by stacking different materials on top of each other may lead to levitation due to *quantum-electrodynamic* Casimir forces [@rodriguez:2010]. The behavior of the stable levitation distance shows a bifurcation and irreversible transitions from separation to stiction [@rodriguez:2010] similarly to the ones described above \[see [Fig. \[fig:levitation\]]{}\]. In that context great importance has been given to the temperature dependence of the position $D_{0,s}$ of stable quantum Casimir levitation [@rodriguez:2010], which is quantified by the value of $\frac{d}{dT}D_{0,s}$. In the critical Casimir case presented here, for an estimate of $\frac{d}{dT}D_{0,s}$ we pick as an example the stable levitation positions for $\xi_+=18$nm and $\xi_+=60$nm as reported in [Fig. \[fig:levitation\_example\]]{} (a different choice would lead to similar results). The results reported in [Fig. \[fig:levitation\_example\]]{} correspond to the experimentally relevant water-lutidine mixture with $\xi_0^+=0.2$nm and $T_c\simeq307$K [@nellen:2009; @hertlein:2008; @gambassi:2009]. Therefore, according to $\xi_+/\xi_0^+=|(T-T_c)/T_c|^{-\nu}$, the difference in temperature required to move from $\xi_+=18$nm to $\xi_+=60$nm is $\Delta T\simeq0.2$K. Thus we find $\frac{d}{dT}D_{0,s}\simeq560\text{nm K}^{-1}$ for the average temperature dependence of critical Casimir levitation \[[Fig. \[fig:levitation\_example\]]{}(b)\], and $\frac{d}{dT}D_{0,s}\simeq230\text{nm K}^{-1}$ by additionally taking electrostatics into account \[[Fig. \[fig:levitation\_example\]]{}(c)\]. We note that in the present critical case $\frac{d}{dT}D_{0,s}$ can become arbitrarily large at temperatures corresponding to the transition from separation to stiction and the emergence of the local minimum and the local maximum of the critical Casimir potential \[see [Fig. \[fig:levitation\]]{} and the curves for $\xi_+=34$nm and $\xi_+=36$nm in [Fig. \[fig:levitation\_example\]]{}(a)\]. This shows that the critical Casimir levitation is strongly temperature dependent, even near room temperature, with the variation of stable separation $\frac{d}{dT}D_{0,s}$ being two orders of magnitude larger than the one predicted for the *quantum-electrodynamic* Casimir effect in Ref. . In general the colloid will not only be exposed to the critical Casimir force and to an electrostatic force but also to gravity and to laser tweezers, which generate a linearly increasing potential contribution. This attractive contribution tends to reduce the potential barriers shown in [Fig. \[fig:levitation\_example\]]{} and can eliminate small barriers altogether. Thus these external forces can be used to switch levitation on and off (compare a similar discussion related to the quantum-electrodynamic Casimir levitation in Ref. ).
Cylinder \[sec:cylinder\]
=========================
Currently, there is an increasing experimental interest in *elongated* colloidal particles which have a typical diameter of up to several $100$ nm and a much larger length (see, e.g., Refs. and references therein). These types of colloids resemble *cylinders* rather than spheres. The description of their behavior in confined critical solvents calls for a natural extension of the studies presented in Secs. \[sec:homog\]–\[sec:period\]. Hence, in the present section we consider the case of a $3d$ cylinder with ${\ensuremath{(-)}}$ BC which is adjacent and parallel aligned to a periodically chemically patterned substrate consisting of alternating ${\ensuremath{(-)}}$ and ${\ensuremath{(+)}}$ stripes as the ones discussed in Sec. \[sec:period\]. Accordingly, the axis of rotational invariance of the cylinder is perpendicular to both the $x$ direction \[[Fig. \[fig:sketch\]]{}\] and the direction normal to the substrate, and it is parallel to the direction of spatial translational invariance of the chemical stripes forming the pattern. As compared with the case of the sphere the analysis for the cylinder is technically simpler because the system as a whole is invariant along all directions but two, the lateral one, $x$, and the one normal to the substrate. (For the sphere its finite extension in the second lateral direction, which is normal to the $x$-axis, matters and thus leads to a basically three-dimensional problem. Accordingly, here we do not consider short cylinders, for which this finite length matters, too.) This reduction of the number of relevant dimensions allows us to perform numerical calculations of adequate precision for a range of various pattern geometries which is wider than in the case of the sphere. (Here, we do not consider a cylindrical colloid which is not perfectly aligned with the pattern and which would, therefore, experience a critical Casimir torque [@kondrat:2009].) Even though the expressions derived in Appendix \[app:cylinder\] can be used to study the case of a cylinder having its axis laterally displaced by an arbitrary amount $X$ from the chemical step, our numerical calculations for the case of a chemical stripe address only the case $X=0$. This corresponds to a lateral position of the symmetry axis of the cylinder which coincides with the center of an attractive ${\ensuremath{(-)}}$ stripe.
In Appendix \[app:cylinder\] we briefly derive the scaling behavior of the normal critical Casimir force acting on the cylinder and compare it with the case of a sphere. Then, we adapt the Derjaguin approximation appropriate for the geometry of the cylinder. On this basis, we have calculated the scaling function of the normal critical Casimir force acting on the cylinder in $d=3$ and $d=4$ on the basis of the Monte Carlo simulation data for the film geometry [@mcdata] and of the analytic MFT expression for the critical Casimir force for the film geometry [@krech:1997], respectively. In addition, within the same approach as the one of Sec. \[sec:MFT\] we have calculated numerically the MFT scaling functions corresponding to $\Delta\ne 0$, in order to assess the performance of the DA.
![ Normalized scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ \[see Appendix \[app:cylinder\], including expressions for ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(0,0)$\] of the normal critical Casimir force acting on a *cylindrical* colloid close to and parallel to a periodically patterned substrate. The cylinder axis is aligned with the striped pattern and positioned above the center of a $(-)$ stripe which has the same adsorption preference as the cylinder (analogous to [Fig. \[fig:comparesystem\]]{} for a spherical colloid). In (a) for $\Pi=1.92$ the appropriate DA describes the actual MFT data rather well, and for $0.3\lesssim\lambda\lesssim0.7$ there is a change of sign of the force. In (b), instead, apart from the limiting homogeneous cases $\lambda=0$ and $\lambda=1$, for $\Pi=0.29$ the DA fails to describe quantitatively the actual behavior \[see the main text\]. In (c) ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ is shown for $d=3$ within the DA based on the Monte Carlo simulation data for the film geometry [@mcdata] for the two cases $\Pi=0.29$ and $\Pi=1.92$. We expect that also in $d=3$ the DA for $\Pi=0.29$ is not quantitatively reliable. []{data-label="fig:cylinder"}](compare_cylinder_1_92 "fig:"){width="7.8cm"}\
![ Normalized scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ \[see Appendix \[app:cylinder\], including expressions for ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(0,0)$\] of the normal critical Casimir force acting on a *cylindrical* colloid close to and parallel to a periodically patterned substrate. The cylinder axis is aligned with the striped pattern and positioned above the center of a $(-)$ stripe which has the same adsorption preference as the cylinder (analogous to [Fig. \[fig:comparesystem\]]{} for a spherical colloid). In (a) for $\Pi=1.92$ the appropriate DA describes the actual MFT data rather well, and for $0.3\lesssim\lambda\lesssim0.7$ there is a change of sign of the force. In (b), instead, apart from the limiting homogeneous cases $\lambda=0$ and $\lambda=1$, for $\Pi=0.29$ the DA fails to describe quantitatively the actual behavior \[see the main text\]. In (c) ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ is shown for $d=3$ within the DA based on the Monte Carlo simulation data for the film geometry [@mcdata] for the two cases $\Pi=0.29$ and $\Pi=1.92$. We expect that also in $d=3$ the DA for $\Pi=0.29$ is not quantitatively reliable. []{data-label="fig:cylinder"}](compare_cylinder_0_29 "fig:"){width="7.8cm"}\
![ Normalized scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ \[see Appendix \[app:cylinder\], including expressions for ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(0,0)$\] of the normal critical Casimir force acting on a *cylindrical* colloid close to and parallel to a periodically patterned substrate. The cylinder axis is aligned with the striped pattern and positioned above the center of a $(-)$ stripe which has the same adsorption preference as the cylinder (analogous to [Fig. \[fig:comparesystem\]]{} for a spherical colloid). In (a) for $\Pi=1.92$ the appropriate DA describes the actual MFT data rather well, and for $0.3\lesssim\lambda\lesssim0.7$ there is a change of sign of the force. In (b), instead, apart from the limiting homogeneous cases $\lambda=0$ and $\lambda=1$, for $\Pi=0.29$ the DA fails to describe quantitatively the actual behavior \[see the main text\]. In (c) ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ is shown for $d=3$ within the DA based on the Monte Carlo simulation data for the film geometry [@mcdata] for the two cases $\Pi=0.29$ and $\Pi=1.92$. We expect that also in $d=3$ the DA for $\Pi=0.29$ is not quantitatively reliable. []{data-label="fig:cylinder"}](cylinder_3d "fig:"){width="7.8cm"} ![ Normalized scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ \[see Appendix \[app:cylinder\], including expressions for ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(0,0)$\] of the normal critical Casimir force acting on a *cylindrical* colloid close to and parallel to a periodically patterned substrate. The cylinder axis is aligned with the striped pattern and positioned above the center of a $(-)$ stripe which has the same adsorption preference as the cylinder (analogous to [Fig. \[fig:comparesystem\]]{} for a spherical colloid). In (a) for $\Pi=1.92$ the appropriate DA describes the actual MFT data rather well, and for $0.3\lesssim\lambda\lesssim0.7$ there is a change of sign of the force. In (b), instead, apart from the limiting homogeneous cases $\lambda=0$ and $\lambda=1$, for $\Pi=0.29$ the DA fails to describe quantitatively the actual behavior \[see the main text\]. In (c) ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ is shown for $d=3$ within the DA based on the Monte Carlo simulation data for the film geometry [@mcdata] for the two cases $\Pi=0.29$ and $\Pi=1.92$. We expect that also in $d=3$ the DA for $\Pi=0.29$ is not quantitatively reliable. []{data-label="fig:cylinder"}](compare_cylinder_1_92 "fig:")\
![ Normalized scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ \[see Appendix \[app:cylinder\], including expressions for ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(0,0)$\] of the normal critical Casimir force acting on a *cylindrical* colloid close to and parallel to a periodically patterned substrate. The cylinder axis is aligned with the striped pattern and positioned above the center of a $(-)$ stripe which has the same adsorption preference as the cylinder (analogous to [Fig. \[fig:comparesystem\]]{} for a spherical colloid). In (a) for $\Pi=1.92$ the appropriate DA describes the actual MFT data rather well, and for $0.3\lesssim\lambda\lesssim0.7$ there is a change of sign of the force. In (b), instead, apart from the limiting homogeneous cases $\lambda=0$ and $\lambda=1$, for $\Pi=0.29$ the DA fails to describe quantitatively the actual behavior \[see the main text\]. In (c) ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ is shown for $d=3$ within the DA based on the Monte Carlo simulation data for the film geometry [@mcdata] for the two cases $\Pi=0.29$ and $\Pi=1.92$. We expect that also in $d=3$ the DA for $\Pi=0.29$ is not quantitatively reliable. []{data-label="fig:cylinder"}](compare_cylinder_0_29 "fig:")\
![ Normalized scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ \[see Appendix \[app:cylinder\], including expressions for ${\ensuremath{K_{{\ensuremath{(-,-)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(0,0)$\] of the normal critical Casimir force acting on a *cylindrical* colloid close to and parallel to a periodically patterned substrate. The cylinder axis is aligned with the striped pattern and positioned above the center of a $(-)$ stripe which has the same adsorption preference as the cylinder (analogous to [Fig. \[fig:comparesystem\]]{} for a spherical colloid). In (a) for $\Pi=1.92$ the appropriate DA describes the actual MFT data rather well, and for $0.3\lesssim\lambda\lesssim0.7$ there is a change of sign of the force. In (b), instead, apart from the limiting homogeneous cases $\lambda=0$ and $\lambda=1$, for $\Pi=0.29$ the DA fails to describe quantitatively the actual behavior \[see the main text\]. In (c) ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ is shown for $d=3$ within the DA based on the Monte Carlo simulation data for the film geometry [@mcdata] for the two cases $\Pi=0.29$ and $\Pi=1.92$. We expect that also in $d=3$ the DA for $\Pi=0.29$ is not quantitatively reliable. []{data-label="fig:cylinder"}](cylinder_3d "fig:")
Here we focus on the comparison between the DA appropriate for the cylinder and the full numerical MFT data for the scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}(\lambda,\Pi,\Xi=0,\Theta,\Delta)$ which characterizes the normal critical Casimir force in the presence of a periodically patterned substrate; $\lambda$, $\Pi$, $\Xi$, $\Theta$, and $\Delta$ are defined as in the case of the sphere \[see Sec. \[sec:period\] and Appendix \[app:cylinder\]\]. Figure \[fig:cylinder\] shows the scaling function of the normal critical Casimir force acting on a cylinder as a function of $\Theta$ as obtained from the DA ($\Delta\to0$) in $d=4$ and from the full numerical MFT calculations for $\Delta=1/3$. Besides the quantitative differences in the scaling function as a function of $\Theta$, the *qualitative* features of the behavior of the force acting on a cylinder, which is reported in [Fig. \[fig:cylinder\]]{} for various values of $\lambda$, are similar to the ones for the sphere \[compare [Fig. \[fig:comparesystem\]]{}\]. For $\Pi=1.92$ \[[Fig. \[fig:cylinder\]]{}(a)\] the DA describes the actual behavior of the critical Casimir force rather well, in particular for $\Theta\gtrsim2$, even for most values of $\lambda$. As in [Fig. \[fig:comparesystem\]]{}, for a certain range of values of $\lambda$ the normal critical Casimir force changes sign at $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}(\Pi,\lambda,\Xi=0,\Delta)$. On the other hand for small periodicities ($\Pi=0.29$ in [Fig. \[fig:cylinder\]]{}(b)) the DA in $d=4$ fails to describe quantitatively the actual behavior of the force as obtained from the full numerical MFT calculations. These strong deviations from the DA \[[Fig. \[fig:cylinder\]]{}(b)\] indicate the relevance of effects caused by the actual non-additivity of critical Casimir forces.
For $\lambda\gtrsim0.6$ the scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ of the normal critical Casimir force obtained numerically and represented by symbols in [Fig. \[fig:cylinder\]]{}(b) is very close (much closer than within the DA) to the one corresponding to the homogeneous case with $(-,-)$ BC (corresponding to $\lambda=1$) and does not show a change of sign. This means that, even if the substrate is not homogeneous but chemically patterned – but such that the larger part of the surface still corresponds to ${\ensuremath{(-)}}$ BC, i.e., $\lambda\gtrsim0.5$ – the resulting critical Casimir force acting on the colloid with ${\ensuremath{(-)}}$ BC resembles the behavior for laterally homogeneous $(-,-)$ BC. This can be understood in terms of the fixed point Hamiltonian in [Eq. ]{} which penalizes spatial variations of the order parameter at short scales. Thus the system tries to smooth out spatial inhomogeneities of the order parameter profile, biased by the preference of the colloidal particle. If the pattern is very finely structured, i.e., $\Pi=(L_1+L_2)/\sqrt{RD}\ll1$, regions with a positive order parameter close to the narrow ${\ensuremath{(+)}}$ stripes ($\lambda\simeq 1^-$, i.e., $L_2\ll L_1$) extent only very little into the direction normal to the substrate and the resulting order parameter profile at a distance from the substrate remains negative only [@sprenger:2005], so that the force resembles the one corresponding to the homogeneous case. (Note that within the DA, the corresponding order parameter profile would simply consist of a patchwork of the order parameter profiles corresponding to the film geometry, with no smoothing taking place at the edges of the various spatial regions.) Similarly, but in a weaker manner due to the opposite order parameter preference at the colloid, the curves in [Fig. \[fig:cylinder\]]{}(b) for $\lambda\lesssim0.5$ approach the corresponding homogeneous one for the case $(+,-)$ (i.e., $\lambda=0$). Thus, the fact that both in [Fig. \[fig:cylinder\]]{}(a) and [Fig. \[fig:cylinder\]]{}(b) the curves for $\lambda=1/5$ are less close to their limiting ones for $\lambda=0$ than the curves for $\lambda=4/5$ are close to the ones for $\lambda=1$ – although the portions of the minority part of the surface are the same – is due to the fact that an order parameter profile with $(+,-)$ boundary conditions is energetically less preferred than the one with $(-,-)$ boundary conditions because in the $(+,-)$ case an interface emerges between the two phases. For broad stripes, i.e., in contrast to the case $\Pi\to0$, the energy costs for a similar behavior are seemingly larger: the full numerical MFT data for $\lambda=1/5$ and $\lambda=4/5$ are less close to the corresponding limiting homogeneous cases $\lambda=0$ and $\lambda=1$, respectively, for $\Pi=1.92$ than for $\Pi=0.29$.
![ Values of the scaling variable $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ at which the normal critical Casimir force ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ acting on a cylinder close to a periodically patterned substrate vanishes as a function of $\Psi=P/\sqrt{R\xi_+}$ \[compare [Fig. \[fig:levitation\]]{} for the case of a sphere\] within the DA. The region indicated by solid lines corresponds to the one in which the levitation of the cylinder at a height $D=D_0=\Theta_0\xi_+$ is stable against small perturbations of $D$, wheres in the shaded region indicated by dashed lines there is no such stable levitation although the normal critical Casimir force acting on the colloid vanishes. For $\lambda>\lambda_0$ with $\lambda_0(d=4)=4/5$ and $\lambda_0(d=3)\simeq0.88$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ ceases to exist, i.e., ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ does not exhibit a zero. For $\lambda<\lambda_1$ with $\lambda_1(d=4)=1/2$ and $\lambda_1(d=3)\simeq0.545$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}(\Psi\searrow\Psi_0(\lambda))$ diverges. (The values for $\Psi_0(\lambda)$ are indicated by upward arrows.) For any $\lambda<\lambda_0$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ exists for $\Psi<\Psi^*(\lambda)$. We expect the DA to be quantitatively reliable only for $\Psi/\sqrt{\Theta_0}\gtrsim2$ for $\Theta_0\lesssim4$ and for $\Psi/\sqrt{\Theta_0}\gtrsim0.5$ for $\Theta_0\gtrsim4$. []{data-label="fig:levitation_cylinder"}](levitation_4d_cylinder_c "fig:"){width="8.4cm"}\
![ Values of the scaling variable $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ at which the normal critical Casimir force ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ acting on a cylinder close to a periodically patterned substrate vanishes as a function of $\Psi=P/\sqrt{R\xi_+}$ \[compare [Fig. \[fig:levitation\]]{} for the case of a sphere\] within the DA. The region indicated by solid lines corresponds to the one in which the levitation of the cylinder at a height $D=D_0=\Theta_0\xi_+$ is stable against small perturbations of $D$, wheres in the shaded region indicated by dashed lines there is no such stable levitation although the normal critical Casimir force acting on the colloid vanishes. For $\lambda>\lambda_0$ with $\lambda_0(d=4)=4/5$ and $\lambda_0(d=3)\simeq0.88$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ ceases to exist, i.e., ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ does not exhibit a zero. For $\lambda<\lambda_1$ with $\lambda_1(d=4)=1/2$ and $\lambda_1(d=3)\simeq0.545$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}(\Psi\searrow\Psi_0(\lambda))$ diverges. (The values for $\Psi_0(\lambda)$ are indicated by upward arrows.) For any $\lambda<\lambda_0$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ exists for $\Psi<\Psi^*(\lambda)$. We expect the DA to be quantitatively reliable only for $\Psi/\sqrt{\Theta_0}\gtrsim2$ for $\Theta_0\lesssim4$ and for $\Psi/\sqrt{\Theta_0}\gtrsim0.5$ for $\Theta_0\gtrsim4$. []{data-label="fig:levitation_cylinder"}](levitation_3d_cylinder_c "fig:"){width="8.4cm"} ![ Values of the scaling variable $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ at which the normal critical Casimir force ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ acting on a cylinder close to a periodically patterned substrate vanishes as a function of $\Psi=P/\sqrt{R\xi_+}$ \[compare [Fig. \[fig:levitation\]]{} for the case of a sphere\] within the DA. The region indicated by solid lines corresponds to the one in which the levitation of the cylinder at a height $D=D_0=\Theta_0\xi_+$ is stable against small perturbations of $D$, wheres in the shaded region indicated by dashed lines there is no such stable levitation although the normal critical Casimir force acting on the colloid vanishes. For $\lambda>\lambda_0$ with $\lambda_0(d=4)=4/5$ and $\lambda_0(d=3)\simeq0.88$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ ceases to exist, i.e., ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ does not exhibit a zero. For $\lambda<\lambda_1$ with $\lambda_1(d=4)=1/2$ and $\lambda_1(d=3)\simeq0.545$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}(\Psi\searrow\Psi_0(\lambda))$ diverges. (The values for $\Psi_0(\lambda)$ are indicated by upward arrows.) For any $\lambda<\lambda_0$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ exists for $\Psi<\Psi^*(\lambda)$. We expect the DA to be quantitatively reliable only for $\Psi/\sqrt{\Theta_0}\gtrsim2$ for $\Theta_0\lesssim4$ and for $\Psi/\sqrt{\Theta_0}\gtrsim0.5$ for $\Theta_0\gtrsim4$. []{data-label="fig:levitation_cylinder"}](levitation_4d_cylinder_c "fig:")\
![ Values of the scaling variable $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ at which the normal critical Casimir force ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ acting on a cylinder close to a periodically patterned substrate vanishes as a function of $\Psi=P/\sqrt{R\xi_+}$ \[compare [Fig. \[fig:levitation\]]{} for the case of a sphere\] within the DA. The region indicated by solid lines corresponds to the one in which the levitation of the cylinder at a height $D=D_0=\Theta_0\xi_+$ is stable against small perturbations of $D$, wheres in the shaded region indicated by dashed lines there is no such stable levitation although the normal critical Casimir force acting on the colloid vanishes. For $\lambda>\lambda_0$ with $\lambda_0(d=4)=4/5$ and $\lambda_0(d=3)\simeq0.88$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ ceases to exist, i.e., ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ does not exhibit a zero. For $\lambda<\lambda_1$ with $\lambda_1(d=4)=1/2$ and $\lambda_1(d=3)\simeq0.545$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}(\Psi\searrow\Psi_0(\lambda))$ diverges. (The values for $\Psi_0(\lambda)$ are indicated by upward arrows.) For any $\lambda<\lambda_0$, $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ exists for $\Psi<\Psi^*(\lambda)$. We expect the DA to be quantitatively reliable only for $\Psi/\sqrt{\Theta_0}\gtrsim2$ for $\Theta_0\lesssim4$ and for $\Psi/\sqrt{\Theta_0}\gtrsim0.5$ for $\Theta_0\gtrsim4$. []{data-label="fig:levitation_cylinder"}](levitation_3d_cylinder_c "fig:")
Figure \[fig:cylinder\](c) shows the scaling function ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{{\ensuremath{{ \textrm{cyl}}}}}$ of the normal critical Casimir force for $d=3$ within the DA as obtained by using Monte Carlo simulation data for the film geometry [@mcdata]. One can infer from [Fig. \[fig:cylinder\]]{}(c) that the qualitative features of the MFT scaling function as described above, such as the change of sign, are carried over to $d=3$.
As discussed in the previous section, the vanishing of the normal critical Casimir force corresponds to a stable levitation of the colloid at a distance $D_0$ from the substrate only if $\partial_D{\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}|_{D=D_0}<0$. Within the DA and at the laterally stable position $\Xi=0$ the sign of $\partial_D{\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ is given by [Eq. ]{} with ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}$ replaced by ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$. The behavior of $\Theta_0^{\ensuremath{{ \textrm{cyl}}}}$ as a function of $\Psi$ and the demarcation of the regions where levitation is stable against perturbations of $D$ is shown in [Fig. \[fig:levitation\_cylinder\]]{}, where the solid and the dashed lines correspond to stable and unstable levitation, respectively. The behavior for the normal critical Casimir force acting on the cylinder is qualitatively similar to the one for the sphere shown in [Fig. \[fig:levitation\]]{}. Analogously to the case of a sphere discussed in Sec. \[sec:levitation\], no stable levitation is found at $T=T_c$ or for $\lambda>\lambda_0={\ensuremath{\Delta_{{\ensuremath{(+,-)}}}}}/({\ensuremath{\Delta_{{\ensuremath{(+,-)}}}}}-{\ensuremath{\Delta_{{\ensuremath{(-,-)}}}}})$, where $\lambda_0=0.80$ in $d=4$ and $\lambda_0\simeq0.88$ in $d=3$. On the other hand, for $\Theta>0$, and $\lambda<\lambda_0$, it is always possible to find values of $P$ and $R$ such that stable levitation of the cylinder occurs at a certain distance from the substrate. The values of $\lambda_1$ below which one has a finite value $\Psi_0(\lambda)$ at which $\Theta_0$ diverges remain the same as for the case of a sphere, i.e., $\lambda_1(d=4)=1/2$ and $\lambda_1(d=3)\simeq0.545$; also the corresponding values of $\Psi_0(\lambda)$ remain the same \[see [Eq. ]{}\].
Summary and conclusions \[sec:summary\]
=======================================
We have investigated the universal properties of the normal and lateral critical Casimir forces acting on a spherical or cylindrical colloidal particle close to a chemically structured substrate with laterally varying adsorption preferences for the species of a (near) critical classical binary liquid mixture (at its critical composition) in which the colloid is immersed. Within the Derjaguin approximation (DA) \[see [Fig. \[fig:lat\_color\]]{}\] in spatial dimensions $d=3$ and $d=4$ we have derived analytic expressions for the corresponding universal scaling functions of the forces and the potentials for general fixed-point boundary conditions (BC) in terms of the scaling function of the critical Casimir force acting on two parallel, homogeneous plates. These expressions are given explicitly analytically at the bulk critical point $T=T_c$ and – for symmetry breaking boundary conditions – far away from the critical point. These relations enable one to obtain predictions for actual three-dimensional systems with a sphere-inhomogeneous plate geometry (for which currently computations are not possible) based on the scaling function for the parallel homogeneous plate geometry, for which, e.g., Monte Carlo simulation data in $d=3$ are available. Moreover, results within mean-field theory (MFT, corresponding to $d=4$) and symmetry-breaking boundary conditions \[Sec. \[sec:MFT\]\] have been obtained fully numerically and have been compared with the approximate results of the DA, which allows us to explore the limits of validity of the latter. We have studied several relevant situations \[see [Fig. \[fig:sketch\]]{}\] and our main findings are the following:
1. First, we have studied a spherical colloid immersed in a binary liquid mixture close to a chemically *homogeneous* substrate which has, compared to the colloid, the same ${\ensuremath{(-)}}$ or a different ${\ensuremath{(+)}}$ adsorption preference for one of the species of the mixture \[Sec. \[sec:homog\]\]. Close to the bulk critical point at $T=T_c$ the critical Casimir force induced by the confinement of the order parameter (e.g., the concentration difference in a binary liquid mixture) can be described in terms of universal scaling functions depending on the surface-to-surface distance $D$ of the colloid from the substrate scaled by the bulk correlation length, $\Theta=\operatorname{sign}( (T-T_c)/T_c ) D/\xi_\pm$, and its ratio with the radius of the colloid, $\Delta=D/R$ \[Eqs. and \]. The scaling functions obtained within the DA \[Eqs. and \] are valid for $\Delta\to0$. From the comparison with the full numerical MFT results \[[Fig. \[fig:homog\]]{}\] we find that in $d=4$ the DA describes the actual behavior quite well for $\Delta\lesssim0.4$. Based on Monte Carlo simulation data for the scaling function of the critical Casimir force between parallel, homogeneous plates and within the DA we have obtained also the scaling function for the critical Casimir force on a spherical colloid close to a homogeneous substrate in $d=3$ \[[Fig. \[fig:homog\]]{}\].
2. The basic building block of a chemically patterned substrate is a *chemical step*, which we have studied in Sec. \[sec:step\]. Due to the broken translational invariance in one lateral direction ($x$) the critical Casimir forces and potentials acquire a dependence on the additional scaling variable $\Xi=X/\sqrt{RD}$, which corresponds to the lateral distance $X$ of the center of the spherical colloid from the position of the chemical step along the plane \[Eqs. , , and \]. Due to the different boundary conditions on both sides of the chemical step a *lateral* critical Casimir force emerges, which leads to a laterally varying potential for the colloid. In the limit $\Delta\to0$ both the scaling function for the potential and for the lateral critical Casimir force as obtained within the DA are in agreement with the full numerical data \[[Fig. \[fig:lateral\]]{}\]. We have derived the corresponding scaling functions within the DA also in $d=3$ by using Monte Carlo data for the parallel plate geometry \[[Fig. \[fig:lateral\]]{}\]. The preceding results have been partly presented in Ref. as well as their suitable comparison with corresponding experimental results [@soyka:2008], which revealed that the critical Casimir effect is rather sensitive to the geometrical details of the substrate patterns.
3. Section \[sec:stripe\] deals with the critical Casimir forces and the corresponding potential acting on a spherical colloid in front of a *single chemical lane* of width $2L$, which additionally depends on a fourth scaling variable $\Lambda=L/\sqrt{RD}$ \[Eqs. and \]. It turns out that within the DA the scaling functions for the critical Casimir force and the critical Casimir potential across a chemical lane can be expressed in terms of the ones for the chemical step \[Eqs. and \]. For large values of $\Lambda$ the resulting potential can be described as a suitable superposition of chemical steps, whereas for $\Lambda\lesssim 3$ one has explicitly to account for the finite width of the chemical stripe \[[Fig. \[fig:stripe\]]{}\]. Comparing the results of the DA with the ones obtained by a full numerical analysis, one finds that the DA describes the actual behavior quite well for $\Delta\lesssim0.4$, even for small $\Lambda$. Seemingly, in this respect, the nonlinearities inherent in the critical Casimir effect and edge effects do not considerably affect the resulting scaling functions \[[Fig. \[fig:omega\_cl\]]{}\].
4. On the basis of the results of Sec. \[sec:stripe\], in Sec. \[sec:period\] we have studied the universal scaling functions of the critical Casimir force and the corresponding potential for a sphere opposite to a *periodically patterned substrate* with laterally alternating chemical stripes of different adsorption preferences \[Sec. \[sec:period\]\]. These scaling functions \[Eqs. and \] depend, besides the scaling variables $\Theta$, $\Delta$, and $\Xi$, on two additional scaling variables $\Pi=P/\sqrt{RD}$ and $\lambda=L_1/P$, which correspond to the period $P=L_1+L_2$ of the pattern and to the width $L_1\le P$ of the stripes with the same adsorption preference as the colloid. The scaling function for the normal critical Casimir force obtained within the DA can be expressed in terms of the one for the chemical step and describes the actual behavior well for $\Pi\gtrsim2$ \[[Eq. ]{} and Figs. \[fig:period\_normal\], \[fig:normal\_3d\](a) and \[fig:comparesystem\](a)\]. However, for $\Pi\to0$ \[[Eq. ]{}\] the DA fails to capture quantitatively the numerically obtained behavior within MFT, reflecting the importance of nonlinearities and edge effects in this context, which are not accounted for by the DA \[Figs. \[fig:period\_normal\], \[fig:normal\_3d\](a) and \[fig:comparesystem\](b)\]. The failure of the DA in the limit $\Pi\to0$ can be traced back to the fact that for the *film* geometry of a patterned wall next to a laterally homogeneous flat wall, additivity of the critical Casimir forces does not hold \[[Fig. \[fig:kraft\]]{}\].
5. The MFT scaling function of the normal critical Casimir force acting on a colloid close to a periodically patterned substrate shows a remarkable behavior as a function of $\Theta=D/\xi_+$. Within a certain range of values of $\Pi$ and $\lambda$ the critical Casimir force vanishes at $\Theta_0$ corresponding to a distance $D=D_0$ between the colloid and the substrate. We have analyzed the sign of the derivative of the critical Casimir force with respect to $D$ at $D_0$, which is negative if for $D<D_0=D_{0,s}$ the colloid is repelled from the substrate whereas for $D>D_0=D_{0,s}$ it is attracted to the substrate \[[Fig. \[fig:levitation\]]{}\]. This means that in the absence of other forces the colloid can levitate above the substrate at a stable distance which can be tuned by temperature. Stable levitation points are found also in $d=3$, within the DA and on the basis of the Monte Carlo data for the parallel plate geometry \[Figs. \[fig:normal\_3d\](b), \[fig:comparesystem\](c), and \[fig:levitation\](b)\]. Our analysis shows that *at* the critical point $T=T_c$ levitation is not possible, whereas off criticality a geometrical configuration leading to stable levitation can always be found. For fixed geometrical parameters, the critical Casimir potential as a function of $D$ changes from a monotonic behavior to a non-monotonic one upon approaching criticality; a local maximum and a local minimum, the latter corresponding to stable levitation, occur \[[Fig. \[fig:levitation\_example\]]{}(a) and (b)\]. Experimentally, this corresponds to a de facto irreversible transition from separation to stiction of a colloid and a patterned substrate. The depths of these potential minima can be up to several $k_BT$ so that the levitation is stable against Brownian motion of the colloid. The critical Casimir levitation can be rather pronounced and robust even in the presence of electrostatic interactions \[[Fig. \[fig:levitation\_example\]]{}(c)\]. The levitation height is proportional to the bulk correlation length and thus can be tuned by varying temperature. Depending on the geometric parameter $\lambda$ we have identified two distinct types of temperature dependences of the levitation height $D_{0,s}$. In both cases it exhibits a high temperature sensitivity $\frac{d}{dT}D_{0,s}$ which, for realistic examples at room temperature, is of the order of several $100\text{nm K}^{-1}$. These results show that the periodic patterning of the substrate enables one to design critical Casimir forces over a wide range of properties.
6. This behavior is also observed for a *cylindrical* colloid which lies parallel to the substrate such that its axis is aligned with the translationally invariant direction of the stripes \[Sec. \[sec:cylinder\] and Appendix \[app:cylinder\]\]. The main features of the scaling function for the corresponding normal critical Casimir force are similar to the ones for the spherical colloid: the DA describes well the actual behavior as obtained from full numerical MFT calculations for large values of $\Pi$, but fails quantitatively for $\Pi\lesssim2$ \[[Fig. \[fig:cylinder\]]{}\]. The numerical studies for $\Pi\to0$ indicate that a substrate with a very fine pattern, dominated by one of the two BC as far as the corresponding covered area is concerned, leads to a normal critical Casimir force which resembles the one for a homogeneous substrate characterized by the dominating BC \[[Fig. \[fig:cylinder\]]{}(b)\]. Based on Monte Carlo data for the parallel plate geometry we calculated within the DA the critical Casimir force acting on a cylinder in $d=3$ \[[Fig. \[fig:cylinder\]]{}(c)\]. Above a chemically patterned substrate, also for a cylinder stable levitation is possible for a wide range of parameters \[[Fig. \[fig:levitation\_cylinder\]]{}\].
Typically, in experiments with a colloidal suspension one has to consider also other forces, such as electrostatics, gravitation, and van der Waals forces which act on the colloidal particles in addition to the critical Casimir forces. The total force is approximately the sum of these contributions [@dantchev:2006; @dantchev:2007] \[see [Fig. \[fig:levitation\_example\]]{}(c)\]. Upon approaching the critical point in the phase diagram, experiments [@soyka:2008; @hertlein:2008; @gambassi:2009] and theory (see, e.g., Refs. ) highlight the importance and the relevance of the critical Casimir effect in comparison with these other forces.
The lateral critical Casimir forces occurring for patterned substrates as discussed here are highly sensitive to the details of the geometry of the pattern. A detailed comparison with available experimental data [@soyka:2008] has to take this into account [@troendle:2009]. This sensitivity even allows for an independent determination of the geometry of a chemically structured substrate by means of the critical Casimir effect. This is useful in cases in which it is difficult to infer the geometry of the chemical pattern directly [@troendle:2009; @soyka:2008]. Concerning the comparison with experiments for chemically structured substrates, the theoretical predictions for the critical Casimir force are in agreement with the presently available data [@troendle:2009; @soyka:2008], for which the description in terms of independent chemical steps \[Sec. \[sec:step\]\] turns out to be sufficient [@troendle:2009]. In order to test our specific predictions obtained for narrow single chemical lanes and for periodic chemical stripes, structures on the nanometer scale are needed. Preliminary experimental data in this direction are encouraging [@vogt:2009; @vogt:2009a].
In view of present basic research efforts and potential applications, it is important to study the effect of *weak* critical adsorption of the fluid at the confining surfaces, corresponding to *finite* surface fields. Such weak surface fields can be realized by applying suitable surface chemistry and they influence the resulting behavior of the critical Casimir effect strongly [@mohry:2009; @nellen:2009]. Another approach to create an effective reduction of the surface adsorption is to create fine periodic chemical patterns with different (strong) adsorption preferences as discussed here. However, our results for $\Pi\to0$ \[Figs. \[fig:comparesystem\](b) and \[fig:cylinder\](b)\] show that a fine patterning of the substrate with alternating boundary conditions does not necessarily lead to an effective reduction of the surface adsorption at *short* distances because in this range the critical Casimir force for a inhomogeneous adsorption preference resembles the one for a *homogeneous* substrate corresponding to strong adsorption. On the other hand, at *large* distances a periodically patterned substrate does lead to an effective BC corresponding to a weak adsorption preference, and for $\lambda=1/2$ the surface fields even cancel out, leading to an effective BC resembling the so-called ordinary BC [@sprenger:2006]. This offers the interesting perspective to study, at least asymptotically, critical Casimir forces with Dirichlet BC by using classical fluids instead of superfluid quantum fluids [@krech:9192all; @garcia:9902all; @ganshin:2006; @maciolek:2007].
A patterning on the *molecular* scale is *not* captured by the continuous approach pursued here, which gives the universal features of the critical Casimir effect. Nonetheless, a molecular patterning of the substrates may provide another means for an effective reduction of the adsorption of the corresponding fluid at the surface. However, on a molecular scale the patterning is more likely to lead to randomly distributed surface fields which opens a new challenge in the context of critical Casimir forces.
S. K. and L. H. gratefully acknowledge support by grant HA 2935/4-1 of the Deutsche Forschungsgemeinschaft. A. G. is supported by MIUR within the program “Incentivazione alla mobilit[à]{} di studiosi stranieri e italiani residenti all’estero”.
Derjaguin approximation for a chemical step\[app:step\]
=======================================================
In this appendix we first calculate within the DA the normal critical Casimir force ${\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(X,D,R,T)$ \[[Eq. ]{}\] acting on a spherical colloid of radius $R$ facing a chemical step by using the DA. (We cannot directly calculate the lateral critical Casimir force ${\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}^\parallel(X,D,R,T)$ within the DA because for two parallel homogeneous plates such a force vanishes.) In a second step we derive the critical Casimir potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{s}}}}}}(X,D,R,T)=\int_D^\infty{{\ensuremath{\textrm{d}}}}z\; {\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(X,z,R,T)$ by integrating this result for the normal critical Casimir force. In a third step the lateral critical Casimir force is obtained as ${\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}^\parallel(X,D,R,T)=-\partial_X{\ensuremath{\Phi_{{\ensuremath{\textrm{s}}}}}}(X,D,R,T)=
-\int_D^\infty{{\ensuremath{\textrm{d}}}}z\; \partial_X{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(X,z,R,T)$ \[see Sec. \[sec:lateral\]\].
In the spirit of the DA, the surface of the spherical colloid with $(b)$ BC is thought of as being made of a pile of (infinitely thin) rings parallel to the opposing substrate and with an area ${{\ensuremath{\textrm{d}}}}S(\rho)=2\pi\rho{{\ensuremath{\textrm{d}}}}\rho$, where $\rho$ is the radius of the ring. Each of these rings is partly facing (in normal direction) the surface with $(a_<)$ BC, with an extension ${{\ensuremath{\textrm{d}}}}S_<(\rho)$, and partly facing the surface with $(a_>)$ BC on the other side of the chemical step \[[Fig. \[fig:lat\_color\]]{}\], with an extension ${{\ensuremath{\textrm{d}}}}S_>(\rho)$, such that ${{\ensuremath{\textrm{d}}}}S(\rho)={{\ensuremath{\textrm{d}}}}S_<(\rho)+{{\ensuremath{\textrm{d}}}}S_>(\rho)$. For an assigned $\rho$, ${{\ensuremath{\textrm{d}}}}S_\gtrless(\rho)$ depend, inter alia, on the lateral position $X$ of the colloid. Using the *assumption of additivity* of the forces underlying the DA we suppose that the contribution ${{\ensuremath{\textrm{d}}}}{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(\rho)$ of the ring to the total critical Casimir force ${\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}$ is given by the *sum* of the critical Casimir forces which would act, in a film, on portions of areas ${{\ensuremath{\textrm{d}}}}S_<$ and ${{\ensuremath{\textrm{d}}}}S_>$ in the presence of $(a_<,b)$ and $(a_>,b)$ BC, respectively. According to [Eq. ]{} this leads to the following expression for the force acting on a single ring: $$\begin{gathered}
\label{eq:app-step-force-ring}
\frac{{{\ensuremath{\textrm{d}}}}{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(\rho)}{k_BT}=
\frac{{{\ensuremath{\textrm{d}}}}S_<(\rho)}{L^{d}(\rho)} k_{\ensuremath{ {(a_<,b)}}}(\operatorname{sign}(t)\, L(\rho)/\xi_\pm)\\
+
\frac{{{\ensuremath{\textrm{d}}}}S_>(\rho)}{L^{d}(\rho)} k_{\ensuremath{ {(a_>,b)}}}(\operatorname{sign}(t)\, L(\rho)/\xi_\pm),\end{gathered}$$ $$\label{eq:app-step-force-ring}
\frac{{{\ensuremath{\textrm{d}}}}{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(\rho)}{k_BT}=
\frac{{{\ensuremath{\textrm{d}}}}S_<(\rho)}{L^{d}(\rho)} k_{\ensuremath{ {(a_<,b)}}}(\operatorname{sign}(t)\, L(\rho)/\xi_\pm)
+
\frac{{{\ensuremath{\textrm{d}}}}S_>(\rho)}{L^{d}(\rho)} k_{\ensuremath{ {(a_>,b)}}}(\operatorname{sign}(t)\, L(\rho)/\xi_\pm),$$ where $L(\rho)$ is the substrate-ring distance \[[Fig. \[fig:lat\_color\]]{}\] as given in [Eq. ]{}, and $k_{(a_\gtrless,b)}$ are the scaling functions of the critical Casimir force in the film geometry with $(a_>,b)$ and $(a_<,b)$ BC, respectively \[see [Eq. ]{}\]. This assumption neglects all edge effects along the boundary between the areas ${{\ensuremath{\textrm{d}}}}S_>(\rho)$ and ${{\ensuremath{\textrm{d}}}}S_<(\rho)$, which might actually be relevant in view of the spatial variation of the order parameter profile. It is therefore important to test the validity of this assumption at least in some relevant cases. This is carried out in Sec. \[sec:step\] for $d=4$, i.e., within MFT.
![ Sketch concerning the Derjaguin approximation for the critical Casimir force acting on a colloid opposite to a chemical step. The critical Casimir force is subdivided into contributions from rings parallel to the substrate. The projection of the area ${{\ensuremath{\textrm{d}}}}S(\rho)$ of a ring onto the substrate is separated into the areal contributions ${{\ensuremath{\textrm{d}}}}S_<$ and ${{\ensuremath{\textrm{d}}}}S_>$ which emerge as the intersection of the projected ring with the half-planes carrying $(a_<)$ and $(a_>)$ BC, respectively \[see the main text\]. The sphere has a surface-to-surface distance $D$ from the substrate and its center has a lateral distance $X$ from the chemical step. []{data-label="fig:lat_color"}](lateraltest "fig:"){width="7cm"} ![ Sketch concerning the Derjaguin approximation for the critical Casimir force acting on a colloid opposite to a chemical step. The critical Casimir force is subdivided into contributions from rings parallel to the substrate. The projection of the area ${{\ensuremath{\textrm{d}}}}S(\rho)$ of a ring onto the substrate is separated into the areal contributions ${{\ensuremath{\textrm{d}}}}S_<$ and ${{\ensuremath{\textrm{d}}}}S_>$ which emerge as the intersection of the projected ring with the half-planes carrying $(a_<)$ and $(a_>)$ BC, respectively \[see the main text\]. The sphere has a surface-to-surface distance $D$ from the substrate and its center has a lateral distance $X$ from the chemical step. []{data-label="fig:lat_color"}](lateraltest "fig:")
Without loss of generality in the following we assume $X>0$, i.e., that the normal projection of the center of the sphere falls on the part of the substrate with $(a_>)$ BC \[Figs. \[fig:sketch\] and \[fig:lat\_color\]\]. The results for $X<0$ are obtained by exchanging in the formulas below $a_< \leftrightarrow a_>$ and $X \leftrightarrow -X$. Taking into account that ${{\ensuremath{\textrm{d}}}}S(\rho)={{\ensuremath{\textrm{d}}}}S_<(\rho)+{{\ensuremath{\textrm{d}}}}S_>(\rho)$ one can rewrite [Eq. ]{} as $$\begin{gathered}
\label{eq:app-step-force-ring-separation}
\frac{{{\ensuremath{\textrm{d}}}}{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(\rho)}{k_BT}=
\frac{{{\ensuremath{\textrm{d}}}}S(\rho)}{L^{d}(\rho)} k_{\ensuremath{ {(a_>,b)}}}(\operatorname{sign}(t)\,{L(\rho)}/{\xi_\pm})
\\+
\frac{{{\ensuremath{\textrm{d}}}}S_<(\rho)}{L^{d}(\rho)} \Delta k(\operatorname{sign}(t)\,{L(\rho)}/{\xi_\pm}),\end{gathered}$$ $$\label{eq:app-step-force-ring-separation}
\frac{{{\ensuremath{\textrm{d}}}}{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(\rho)}{k_BT}=
\frac{{{\ensuremath{\textrm{d}}}}S(\rho)}{L^{d}(\rho)} k_{\ensuremath{ {(a_>,b)}}}(\operatorname{sign}(t)\,{L(\rho)}/{\xi_\pm})
+
\frac{{{\ensuremath{\textrm{d}}}}S_<(\rho)}{L^{d}(\rho)} \Delta k(\operatorname{sign}(t)\,{L(\rho)}/{\xi_\pm}),$$ where $\Delta k(\Theta) = k_{\ensuremath{ {(a_<,b)}}}(\Theta) - k_{\ensuremath{ {(a_>,b)}}}(\Theta)$. Summing up all force contributions from the rings of different radii $\rho$, one finds for the *total* normal force acting on the sphere $$\label{eq:app-step-force-separation}
{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(X,D,R,T)=F_{\ensuremath{ {(a_>,b)}}}(D,R,T)+\Delta F(X,D,R,T),$$ where $F_{\ensuremath{ {(a_>,b)}}}$ is the force acting on a sphere close to a *homogeneous* substrate with $(a_>)$ BC and is given by [Eq. ]{} or by Eqs. and . This term does not contribute to the *lateral* critical Casimir force experienced by the colloid near the chemical step, because it does not depend on the lateral coordinate of the colloid. The second term $\Delta F$ in [Eq. ]{} corresponds to the integration of the force differences $\Delta k$ in the region of overlap between the projection of the sphere onto the substrate plane and that part of the substrate with $(a_<)$ BC. For each ring this area is given by \[see [Fig. \[fig:lat\_color\]]{}\] $$\label{eq:}
{{\ensuremath{\textrm{d}}}}S_<(\rho)=
\begin{cases}
0, &\rho<X,\\
2 \arccos(X/\rho)\rho{{\ensuremath{\textrm{d}}}}\rho,&X\le\rho\le R.
\end{cases}$$ This leads to $$\label{eq:app-step-delta-f-int}
\frac{\Delta F (X,D,R,T)}{k_BT} =
2\int\limits_X^R{{\ensuremath{\textrm{d}}}}\rho\,\rho\,\arccos\left(\frac{X}{\rho}\right)
\,\frac{\Delta k(\operatorname{sign}(t)\, L(\rho)/\xi_\pm)}{ L^{d}(\rho)}.$$ In the spirit of the DA, the radius of the sphere is taken to be large compared to its distance to the substrate, i.e., $\Delta=D/R\ll1$, and the contributions from the rings closest to the substrate dominate. Therefore, it is well justified and in accordance with the DA to assume $X/R\ll1$ because the contributions of rings with large radii do not change the behavior of the force in the Derjaguin limit. Within these two limits we can use the *parabolic* approximation for the distance of the rings to the substrate \[[Eq. ]{}\], $L(\rho)\simeq D\alpha$, with $\alpha=1+{\rho^2}/{2RD}$. Changing the integration variable in [Eq. ]{} we directly find $$\label{eq:app-step-delta-f-int-rewritten-2}
\Delta F (X,D,R,T) = k_B T\frac{R}{D^{d-1}}\Delta K(\Xi,\Theta,\Delta),$$ where $\Delta K$ is a universal scaling function given by $$\label{eq:app-step-delta-K}
\Delta K(\Xi,\Theta,\Delta\to0) = 2\!\!\!\!
\int\limits_{1+\Xi^2/2}^{\infty}\!\!\!\!{{\ensuremath{\textrm{d}}}}\alpha\;
\alpha^{-d}\arccos\left(\tfrac{\Xi}{\sqrt{2(\alpha-1)}}\right)
\Delta k( \alpha\Theta).$$ Note that the relevant scaling variable $\Xi=X/\sqrt{RD}$ can take on arbitrary values, irrespective of the two assumptions $D/R\ll1$ and $X/R\ll1$. From [Eq. ]{} one finds with Eqs. and directly the expression for the scaling function $\psi_{(a_<|a_>,b)}$ given in [Eq. ]{}.
The critical Casimir potential ${\ensuremath{\Phi_{{\ensuremath{\textrm{s}}}}}}(X,D,R,T)=\int_D^\infty{{\ensuremath{\textrm{d}}}}l {\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}(X,l,R,T)$ can be separated analogously to [Eq. ]{}, i.e., $$\label{eq:app-step-pot-separation}
{\ensuremath{\Phi_{{\ensuremath{\textrm{s}}}}}}(X,D,R,T)=\Phi_{\ensuremath{ {(a_>,b)}}}(D,R,T)+\Delta\Phi(X,R,D,T)$$ with $$\label{eq:app-step-pot-delta}
\Delta\Phi(X,R,D,T)=
\int\limits_D^\infty{{\ensuremath{\textrm{d}}}}l \Delta F(X,l,R,T){\ensuremath =\mathrel{\mathop:}}k_BT\frac{R}{D^{d-2}}\Delta\vartheta(\Xi,\Theta,\Delta).$$
Using [Eq. ]{}, the scaling function $\Delta\vartheta$ is given by $$\label{eq:app-step-delta-vartheta-1}
\Delta\vartheta(\Xi,\Theta,\Delta)=
2\int\limits_1^\infty{{\ensuremath{\textrm{d}}}}y \frac{1}{y^{d-1}}
\int\limits_{1+\Xi^2/(2y)}^{\infty}\!\!\!\!{{\ensuremath{\textrm{d}}}}\alpha\;
\frac{1}{\alpha^{d}}\arccos\left(\frac{\Xi}{\sqrt{2y(\alpha-1)}}\right)
\Delta k( y\alpha \Theta).$$ By changing the integration variable $\alpha\mapsto z{\ensuremath \mathrel{\mathop:}=}2y(\alpha-1)/\Xi^2$ followed by $y\mapsto v{\ensuremath \mathrel{\mathop:}=}y+\Xi^2z/2$ one obtains $$\label{eq:app-step-delta-vartheta-2}
\Delta\vartheta(\Xi,\Theta,\Delta)=
\Xi^2\int_1^\infty{{\ensuremath{\textrm{d}}}}z \int_{1+z\Xi^2/2}^\infty{{\ensuremath{\textrm{d}}}}v\frac{1}{v^d}\arccos(1/\sqrt{z})\Delta k(v\Theta).$$ After changing the order of integration $$\label{eq:app-step-delta-vartheta-3}
\int_1^\infty{{\ensuremath{\textrm{d}}}}z
\int_{1+z\Xi^2/2}^\infty{{\ensuremath{\textrm{d}}}}v
=
\int_{1+\Xi^2/2}^\infty{{\ensuremath{\textrm{d}}}}v
\int_1^{2(v-1)/\Xi^2}{{\ensuremath{\textrm{d}}}}z,$$ and using the primitive [@note:1] $$\label{eq:app-step-delta-vartheta-3b}
\int {{\ensuremath{\textrm{d}}}}z \arccos(1/\sqrt{z})=z\arccos(1/\sqrt{z})-\sqrt{z-1}+c,$$ one obtains after a final change of variables $v\mapsto w{\ensuremath \mathrel{\mathop:}=}2(v-1)/\Xi^2$ $$\label{eq:app-step-delta-vartheta-4}
\Delta\vartheta(\Xi,\Theta,\Delta)=
\frac{\Xi^4}{2}\int_1^\infty{{\ensuremath{\textrm{d}}}}s \frac{1}{(1+\Xi^2s/2)^d}\left[s\arccos(s^{-1/2})-\sqrt{s-1}\right]
\Delta k (\Theta[1+\Xi^2s/2]).$$ From [Eq. ]{} together with [Eq. ]{} one obtains the final expression for the scaling function of the critical Casimir potential as given in [Eq. ]{}.
Bulk critical point: $\Theta=0$ \[app:step-crit\]
-------------------------------------------------
In order to calculate the critical Casimir force acting on the colloid at the *bulk critical point* one inserts [Eq. ]{} into [Eq. ]{} and obtains $$\begin{aligned}
\label{eq:app-step-delta-K-crit}
\Delta K(\Xi,\Theta=0,\Delta) &=& 2\left(\Delta_{\ensuremath{ {(a_<,b)}}}-\Delta_{\ensuremath{ {(a_>,b)}}}\right)
\int_{1+\Xi^2/2}^{\infty}\!\!\!\!{{\ensuremath{\textrm{d}}}}\alpha\;
\alpha^{-d}\arccos\left(\frac{\Xi}{\sqrt{2\alpha-2}}\right)\\
\label{eq:app-step-I-def}
&{\ensuremath =\mathrel{\mathop:}}& \Xi^2\left(\Delta_{\ensuremath{ {(a_<,b)}}}-\Delta_{\ensuremath{ {(a_>,b)}}}\right)\; I_d(\Xi^2/2),\nonumber\end{aligned}$$
where $\Delta_{(a,b)} = k_{(a,b)}(0)$ \[see [Eq. ]{}\], and with the substitution $\alpha\mapsto z=\Xi/\sqrt{2(\alpha-1)}$ for $d>1$, $$\label{eq:app-step-I}
I_d(a)=2\int_0^1{{\ensuremath{\textrm{d}}}}z \frac{z^{2d-3}}{(z^2+a)^d}\arccos(z).$$ For $I_d(a)$ the recursion relation $$\label{eq:app-step-I-recursion}
I_{d+1}(a)=\frac{1}{d}a^{1-d}\frac{{{\ensuremath{\textrm{d}}}}}{{{\ensuremath{\textrm{d}}}}a}[a^d I_d(a)]$$ holds, so that $I_4$ and $I_3$ can be expressed in terms of $I_2$. Performing the integration we find [@note:2] $$\label{eq:app-step-I-2}
I_2(a)=\frac{\pi}{2a}\left[1-\frac{a^{1/2}}{(1+a)^{1/2}}\right],$$ and therefore with [Eq. ]{} $$\label{eq:app-step-I-3}
I_3(a) =\frac{\pi}{4a}\left[1-\frac{\frac{3}{2}a^{1/2}+a^{3/2}}{(1+a)^{3/2}}\right],$$ and $$\label{eq:app-step-I-4}
I_4(a)=\frac{\pi}{6a}\left[1-\frac{\frac{15}{8}a^{1/2}+\frac{5}{2}a^{3/2}+a^{5/2}}{(1+a)^{5/2}}\right].$$ Thus, from Eqs. , , and together with the expression for $K_{(a_\gtrless,b)}(0,0)=2\pi\Delta_{(a_\gtrless,b)}/(d-1)$ \[Sec. \[sec:homog-da\]\] and [Eq. ]{}, one finds the expression for the scaling function $\psi_{(a_<|a_>,b)}$ given in [Eq. ]{}. The critical Casimir potential at $\Theta=0$ for $d=3$ and $4$ can be found from [Eq. ]{} together with [Eq. ]{}: $$\label{eq:app-step-pot-1a}
\Delta\vartheta(\Xi,\Theta=0,\Delta)=\Xi^2\left(\Delta_{\ensuremath{ {(a_<,b)}}}-\Delta_{\ensuremath{ {(a_>,b)}}}\right)
\int_1^{\infty}{{\ensuremath{\textrm{d}}}}y\;y^{-d} I_d\left(\tfrac{\Xi^2}{2y}\right),$$ and from a change of variable $y\mapsto a=\Xi^2/(2y)$ one finds $$\label{eq:app-step-pot-1}
\Delta\vartheta(\Xi,0,\Delta)=\frac{2^{d-2}}{\Xi^{2d-4}}\left(\Delta_{\ensuremath{ {(a_<,b)}}}-\Delta_{\ensuremath{ {(a_>,b)}}}\right)
\int_0^{\Xi^2/2}{{\ensuremath{\textrm{d}}}}a\;a^{d-2} I_d(a).$$ Using [Eq. ]{} and the limiting behavior $I_d(a\to0)=\pi/( 2(d-1) a)$, we find $$\label{eq:app-step-pot-2}
\Delta\vartheta(\Xi,0,\Delta)=\frac{\Xi^{2}}{2(d-1)}\left(\Delta_{\ensuremath{ {(a_<,b)}}}-\Delta_{\ensuremath{ {(a_>,b)}}}\right)
I_{d-1}(\Xi^2/2).$$ From Eqs. , , and together with $\vartheta(0,0)$ as given in Sec. \[sec:homog-da\] one obtains [Eq. ]{} for the scaling function of the critical Casimir potential at $T_c$.
Far from criticality: $\Theta\gg1$ \[app:step-gg\]
--------------------------------------------------
Far from the critical point, i.e., for $\Theta\gg1$, and for *symmetry breaking* boundary conditions $(a_<)=(+)$, $(a_>)=(-)$, and $(b)=(-)$ [Eq. ]{} holds and the integrals in Eqs. and can be calculated analytically. For $\Theta\gg1$ [Eq. ]{} turns into $$\label{eq:app-step-K-gg-1}
\Delta K(\Xi,\Theta\gg1,\Delta)=
2(A_- - A_+)\Theta^d\int_{1+\Xi^2/2}^\infty
{{\ensuremath{\textrm{d}}}}\alpha \arccos\left(\tfrac{\Xi}{\sqrt{2(\alpha-1)}}\right)
e^{-\alpha\Theta}.$$ Substituting $\alpha\mapsto \beta=2(\alpha-2)/\Xi^2$ one has $$\label{eq:app-step-K-gg-2}
\Delta K(\Xi,\Theta\gg1,\Delta)=
\Xi^2(A_- -A_+)\Theta^de^{-\Theta}\int_1^\infty
{{\ensuremath{\textrm{d}}}}\beta \arccos(\beta^{-1})
e^{-\Xi^2\Theta\beta/2}.$$ Integrating by parts leads to $$\label{eq:app-step-K-gg-3}
\Delta K(\Xi,\Theta\gg1,\Delta)=
(A_- -A_+)\Theta^{d-1} e^{-\Theta}
\int_1^\infty
{{\ensuremath{\textrm{d}}}}\beta \frac{1}{\beta\sqrt{\beta-1}}
e^{-\Xi^2\Theta\beta/2}.$$ By using the relation [@note:3] $$\label{eq:app-erfc}
\int_1^\infty
{{\ensuremath{\textrm{d}}}}\beta \frac{1}{\beta\sqrt{\beta-1}}
e^{-a^2\beta}=\pi\operatorname{erfc}(a),$$ where $a>0$ and $\operatorname{erfc}(a)=1-\operatorname{erf}(a)=2\pi^{-1/2}\int_a^\infty{{\ensuremath{\textrm{d}}}}t \exp(-t^2)$ is the complementary error function, we finally arrive at $$\label{eq:app-step-K-gg-4}
\Delta K(\Xi,\Theta\gg1,\Delta)=
\pi(A_- -A_+)\Theta^{d-1} e^{-\Theta}
\operatorname{erfc}(\Xi\sqrt{\Theta/2}).$$ The scaling function ${\ensuremath{K_{{\ensuremath{(\mp,-)}}}}}$ for $\Theta\gg1$ in the homogeneous case \[Sec. \[sec:homog\]\] is given by [@gambassi:2009] $$\label{eq:app-step-K-homog}
{\ensuremath{K_{{\ensuremath{(\mp,-)}}}}}(\Theta\gg1,\Delta\to0)=2\pi A_\pm \Theta^{d-1}e^{-\Theta}$$ and from Eqs. , , and one obtains the expression for $\psi_{(-|+,-)}$ as given in [Eq. ]{}. Similarly, after rewriting [Eq. ]{} for $\Theta\gg1$ as $$\label{eq:app-step-vartheta-gg-1}
\Delta \vartheta(\Xi,\Theta\gg1,\Delta)=
(A_- -A_+)\Theta^{d} e^{-\Theta}\frac{\Xi^4}{2}
\int_1^\infty
{{\ensuremath{\textrm{d}}}}s \left(s\arccos(s^{-1/2})-\sqrt{s-1}\right)
e^{-\Xi^2\Theta s/2},$$ one can integrate by parts, which yields $$\label{eq:app-step-vartheta-gg-2}
\Delta \vartheta(\Xi,\Theta\gg1,\Delta)=
(A_- -A_+)\Theta^{d-2} e^{-\Theta}
\int_1^\infty
{{\ensuremath{\textrm{d}}}}s \frac{1}{\sqrt{s-1}}\left[\frac{1}{s}+\frac{\Theta\Xi^2}{2}\left(1+\Theta\Xi^2\right)-\frac{\Theta^2\Xi^4}{2}s\right]
e^{-\Xi^2\Theta s/2}.$$ Using [Eq. ]{} and the relations \[which follow from taking successive derivatives $-d/d(a^2)$ of [Eq. ]{}\] $$\label{eq:app-step-vartheta-gg-3}
\int_1^\infty{{\ensuremath{\textrm{d}}}}s\frac{1}{\sqrt{s-1}}e^{-a^2s}=\frac{\sqrt{\pi}}{a}e^{-a^2},\qquad
\int_1^\infty{{\ensuremath{\textrm{d}}}}s\frac{s}{\sqrt{s-1}}e^{-a^2s}=\frac{\sqrt{\pi}}{2a^3}\left(1+2a^2\right)e^{-a^2},$$ one ends up with $$\label{eq:app-step-vartheta-gg-4}
\Delta \vartheta(\Xi,\Theta\gg1,\Delta)=
\pi (A_- -A_+)\Theta^{d-2} e^{-\Theta}
\operatorname{erfc}(\Xi\sqrt{\Theta/2}).$$ Together with the expression for the homogeneous case \[see Sec. \[sec:homog\] and Ref. \], $$\label{eq:app-step-vartheta-homog}
{\ensuremath{\vartheta_{{\ensuremath{(\mp,-)}}}}}(\Theta\gg1,\Delta\to0)=2\pi A_\pm \Theta^{d-2}e^{-\Theta},$$ one obtains the expression for $\omega_{(-|+,-)}$ given in [Eq. ]{}.
Derjaguin approximation for a single chemical lane\[app:stripe\]
================================================================
Based on the assumption of additivity which underlies the Derjaguin approximation one can use the results presented in Sec. \[sec:step\] for a chemical step in order to study a chemical lane. The chemical lane configuration can be regarded as the superposition of two chemical steps, one $(A)$ being a chemical step located at $x=-L$ with $(a|a_{{\ensuremath{\ell}}})$ BC, and the other one $(B)$ being a chemical step located at $x=L$ with $(a_{{\ensuremath{\ell}}}|a)$ BC. This superposition overcounts a contribution corresponding to a homogeneous substrate with $(a_{{\ensuremath{\ell}}})$ BC which must be subtracted \[see [Eq. ]{}\]: $$\begin{aligned}
{8}
\label{eq:testdia}
\left.
\begin{aligned}
(A):\;\;\stackrel{(a)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}\!\!\!\!\!\!
\mathop{|}_{-L}\!\!\!\!\!\!
\stackrel{(a_{{\ensuremath{\ell}}})}{\photon\photon}
\\
\fbox{$+$}\qquad\qquad
\\
(B):\;\; \stackrel{(a_{{\ensuremath{\ell}}})}{\photon\photon}\!\!\!\!
\mathop{|}_{L}\!\!\!\!
\stackrel{(a)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}
\end{aligned}
\right\}&\fbox{$-$}&\;\;
\photon\!\!
\stackrel{(a_{{\ensuremath{\ell}}})}{\photon}\!\!
\photon
&=&\;\;
\stackrel{(a)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}\!\!\!\!\!\!
\mathop{|}_{-L}\!\!\!\!\!\!
\stackrel{(a_{{\ensuremath{\ell}}})}{\photon}\!\!\!\!
\mathop{|}_{L}\!\!\!\!
\stackrel{(a)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}
\nonumber\\
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^{(A)}\,+\;{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^{(B)}\qquad
&-&
K_{(a_{{\ensuremath{\ell}}},b)}\qquad
&=&
{\ensuremath{K_{{\ensuremath{\ell}}}}},\qquad\end{aligned}$$ where $$\label{eq:configA}
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^{(A)} (\Lambda,\Xi,\Theta,\Delta)
= \frac{K_{(a,b)}+K_{(a_{{\ensuremath{\ell}}},b)}}{2}\\
+\frac{K_{(a,b)}-K_{(a_{{\ensuremath{\ell}}},b)}}{2}
\psi_{(a|a_{{\ensuremath{\ell}}},b)}(\Xi+\Lambda,\Theta,\Delta)$$ and $$\label{eq:configB}
{\ensuremath{K_{{\ensuremath{\textrm{s}}}}}}^{(B)} (\Lambda,\Xi,\Theta,\Delta) = \frac{K_{(a,b)}+K_{(a_{{\ensuremath{\ell}}},b)}}{2}\\
+\frac{K_{(a_{{\ensuremath{\ell}}},b)}-K_{(a,b)}}{2}
\psi_{(a_{{\ensuremath{\ell}}}|a,b)}(\Xi-\Lambda,\Theta,\Delta).$$ Since within the DA $\psi_{(a_{{\ensuremath{\ell}}}|a,b)}=\psi_{(a|a_{{\ensuremath{\ell}}},b)}$, Eqs. – and [Eq. ]{} lead directly to [Eq. ]{}. The procedure for calculating the critical Casimir potential is analogous to the one discussed here for the force and leads to [Eq. ]{}.
Derjaguin approximation for periodic chemical patterns\[app:period\]
====================================================================
In order to obtain the scaling function for the critical Casimir force and the potential of a sphere close to a periodic chemical pattern one can follow a procedure analogous to the one presented in Appendix \[app:stripe\]. Indeed, in order to form a lane ${\ensuremath{\ell}}'$ with $(a_1)$ BC on an otherwise homogeneous *portion* of a substrate with $(a_2)$ BC and lateral extension $P$, one can proceed as follows:
- superimpose onto the substrate the single chemical lane ${\ensuremath{\ell}}$ studied in Sec. \[sec:stripe\], with $a_\ell=a_1$, $a=a_2$, suitably positioned in space such that it coincides with the lane $\ell'$ to be formed.
- subtract the contribution of a homogeneous substrate with $(a_2)$ BC, which is overcounted in the previous superposition. After this subtraction, the contribution to the force resulting from that part – marked by $(?)$ in [Eq. ]{} – of the original substrate which is not affected by the formation of the extra lane is unchanged.
$$\begin{aligned}
{8}
\label{eq:diaper12}
\begin{aligned}
\phantom{(A):\;\fbox{$+$}\,}
\stackrel{(?)}{\gluon}\!\!{\overbrace{\stackrel{(a_2)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}}^{P}}\!\!\stackrel{(?)}{\gluon}
\\[3mm]
(A):\;\fbox{$+$}\, \stackrel{(a_2)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}\!\!\!\!\!\!\!\!\!\!\!\!
\mathop{|}_{X'-\frac{L_1}{2}} \!\!\!\!\!\!\!\!\!\!\!
\stackrel{(a_1)}{\photon} \!\!\!\!\!\!\!\!\!\!\!\!
\mathop{|}_{X'+\frac{L_1}{2}} \!\!\!\!\!\!\!\!\!\!\!\!
\stackrel{(a_2)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}
\\[1mm]
(B):\;\fbox{$-$}\,
\stackrel{(a_2)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}
\\[2mm]
\phantom{(A):\;}\;
=
\;\,\stackrel{(?)}{\gluon}
\stackrel{(a_2)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}\!\!\!\!\!\!\!\!\!\!\!\!
\mathop{|}_{X'-\frac{L_1}{2}}\!\!\!\!\!\!\!\!\!\!\!
\stackrel{(a_1)}{\photon}\!\!\!\!\!\!\!\!\!\!\!\!
\mathop{|}_{X'+\frac{L_1}{2}}\!\!\!\!\!\!\!\!\!\!\!\!
\stackrel{(a_2)}{{\ensuremath{\mbox{\rule[1mm]{32pt}{0.5pt}}}}}
\stackrel{(?)}{\gluon}
\ifTwocolumn
\\[-1mm]
\else
\\[-3mm]
\fi
\qquad\qquad\qquad\text{\small lane $\ell'$}\qquad\qquad\;\;\;\;\;\;\,
\end{aligned}\end{aligned}$$
The contribution $\Delta F$ to the critical Casimir force experienced by a colloid close to such a substrate and due to the addition of the lane is characterized by the scaling function \[see [Eq. ]{}\] $$\begin{gathered}
\label{eq:pippo}
\Delta K(\lambda,\Pi,\Xi-\Xi',\Theta,\Delta\rightarrow 0) = K_{\ensuremath{\ell}}(\Pi\tfrac{\lambda}{2},\Xi-\Xi',\Theta) - K_{(a_2,b)}
=\frac{K_{(a_2,b)}-K_{(a_1,b)}}{2}\\
\times \left[\psi_{(a_1|a,b)}(\Xi-\Xi' + \Pi \tfrac{\lambda}{2},\Theta,\Delta\rightarrow 0)-\psi_{(a_1|a,b)}(\Xi-\Xi' - \Pi \tfrac{\lambda}{2},
\Theta,\Delta\rightarrow 0)\right]\end{gathered}$$ where we have used the relation $(L_1/2)/\sqrt{RD} = \Pi \lambda/2$ and have introduced $\Xi' \equiv X'/\sqrt{RD}$, with $X'$ as the position of the center of the added lane ${\ensuremath{\ell}}'$. The force resulting from a periodic pattern can now be obtained by starting out with a homogeneous substrate with $(a_2)$ BC and by iterating the procedure discussed above which adds progressively displaced lanes at positions $X' = n P$, i.e., $\Xi' = n\Pi$, with $n\in \mathbb{Z}$. The resulting force is characterized by the scaling function $${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta\to0) = K_{(a_2,b)} + \sum_{n=-\infty}^{+\infty} \Delta K(\lambda,\Pi,\Xi-n\Pi,\Theta,\Delta\to0)$$ which, together with [Eq. ]{}, yields immediately [Eq. ]{} for ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$.
For $\lambda=0$ or $\lambda=1$ one recovers from [Eq. ]{} the homogeneous cases with $(a_2,b)$ BC or $(a_1,b)$ BC, respectively. Obviously, for $\lambda=0$, the sum in [Eq. ]{} vanishes, and one is left with ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda=0,\Pi,\Xi,\Theta,\Delta\to0)=1$, corresponding to $(a_2,b)$ BC. On the other hand for $\lambda=1$, the sum in [Eq. ]{} can be easily evaluated \[see [Eq. ]{} for $|\Xi|\to\infty$\]: $$\begin{gathered}
\label{eq:app-period-sum-1}
\lim_{M,N\to\infty} \sum_{n=-M}^{N}
\left\{\psi_{(a_1|a_2,b)}(\Xi+\Pi(n+\tfrac{1}{2}),\Theta,\Delta)
-\psi_{(a_1|a_2,b)}(\Xi+\Pi(n-\tfrac{1}{2}),\Theta,\Delta)\right\}\\
=
\lim_{M,N\to\infty}
\left\{\psi_{(a_1|a_2,b)}(\Xi+\Pi(N+\tfrac{1}{2}),\Theta,\Delta)
-\psi_{(a_1|a_2,b)}(\Xi+\Pi(-M-\tfrac{1}{2}),\Theta,\Delta)\right\}=-2,\end{gathered}$$ where we have used the fact that $\psi_{(a_1|a_2,b)}(\Xi=\pm\infty,\Theta,\Delta)=\mp1$. Accordingly, ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}(\lambda=1,\Pi,\Xi,\Theta,\Delta\to0)=-1$, which corresponds to the homogeneous case with $(a_1,b)$ BC.
In the limit $\Pi\to0$ (i.e., for very fine patterns compared with $\sqrt{RD}$), the sum in [Eq. ]{} turns into an integral: $$\begin{gathered}
\label{eq:app-period-sum-2}
\sum_{n=-\infty}^{\infty}
\left\{\psi_{(a_1|a_2,b)}(\Xi+\Pi(n+\tfrac{\lambda}{2}),\Theta,\Delta)
-\psi_{(a_1|a_2,b)}(\Xi+\Pi(n-\tfrac{\lambda}{2}),\Theta,\Delta)\right\}\\
\xrightarrow[\Pi\to0]{}
\frac{1}{\Pi}
\int_{-\infty}^{\infty}{{\ensuremath{\textrm{d}}}}\eta
\left\{\psi_{(a_1|a_2,b)}(\Xi+\eta+\tfrac{\Pi\lambda}{2},\Theta,\Delta)
-\psi_{(a_1|a_2,b)}(\Xi+\eta-\tfrac{\Pi\lambda}{2},\Theta,\Delta)\right\}\\
=
\int_{-\infty}^\infty {{\ensuremath{\textrm{d}}}}\eta\;\lambda\;\frac{{{\ensuremath{\textrm{d}}}}}{{{\ensuremath{\textrm{d}}}}\eta}
\psi_{(a_1|a_2,b)}(\Xi+\eta,\Theta,\Delta)\\
=
\lambda \left\{\psi_{(a_1|a_2,b)}(+\infty,\Theta,\Delta)-
\psi_{(a_1|a_2,b)}(-\infty,\Theta,\Delta)\right\}=-2\lambda,\end{gathered}$$ and finally one finds [Eq. ]{}.
For completeness, we provide the corresponding expression for the scaling function of the critical Casimir potential ${\ensuremath{\omega_{{\ensuremath{\textrm{p}}}}}}$ within the DA: $$\label{eq:period-omega-da}
{\ensuremath{\omega_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta\to0)=
1+\sum_{n=-\infty}^{\infty} \left\{\omega_{(a_1|a_2,b)}(\Xi+\Pi(n+\tfrac{\lambda}{2}),\Theta,\Delta\to0)
-\omega_{(a_1|a_2,b)}(\Xi+\Pi(n-\tfrac{\lambda}{2}),\Theta,\Delta\to0)\right\}.$$ $$\begin{gathered}
\label{eq:period-omega-da}
{\ensuremath{\omega_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi,\Xi,\Theta,\Delta\to0)=\\
1+\sum_{n=-\infty}^{\infty} \left\{\omega_{(a_1|a_2,b)}(\Xi+\Pi(n+\tfrac{\lambda}{2}),\Theta,\Delta\to0)
-\omega_{(a_1|a_2,b)}(\Xi+\Pi(n-\tfrac{\lambda}{2}),\Theta,\Delta\to0)\right\}.\end{gathered}$$ In the limit $\Pi\to0$, ${\ensuremath{\omega_{{\ensuremath{\textrm{p}}}}}}$ reduces to $$\label{eq:small-period-omega}
{\ensuremath{\omega_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi\to0,\Xi,\Theta,\Delta\to0)=1-2\lambda.$$ Accordingly, within the DA and in the limit $\Pi\to0$ the critical Casimir potential is the average of the ones corresponding to the two boundary conditions, weighted with the corresponding relative stripe width: $$\label{eq:small-period-vartheta}
{\ensuremath{\vartheta_{{\ensuremath{\textrm{p}}}}}}(\lambda,\Pi\to0,\Xi,\Theta,\Delta\to0)=
\lambda \vartheta_{(a_1,b)}(\Theta,\Delta\to0)+(1-\lambda)\vartheta_{(a_2,b)}(\Theta,\Delta\to0).$$
Cylinder close to a patterned substrate\[app:cylinder\]
=======================================================
Derjaguin approximation for a homogeneous substrate
---------------------------------------------------
Similarly to the case of a sphere discussed before, the critical Casimir force ${\ensuremath{F_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ per unit length acting on a (three-dimensional) cylinder of radius $R$ with $(b)$ BC close to and parallel to a substrate with $(a)$ BC at a surface-to-surface distance $D$ can be expressed in terms of a universal scaling function $K^{\ensuremath{{ \textrm{cyl}}}}$: $$\label{eq:app-cyl-force}
{\ensuremath{F_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(D,R,T)=k_BT\frac{R^{1/2}}{D^{d-1/2}}{\ensuremath{K_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(\Theta,\Delta),$$ with $\Theta=\operatorname{sign}(t)\,D/\xi_\pm$ and $\Delta=D/R$ as before. Equation describes a force divided by a length and per $D^{d-3}$ which for $d=4$ corresponds to considering ${\ensuremath{F_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ per length of its axis and per length of the *extra* translationally invariant direction of a hypercylinder \[compare [Eq. ]{}\]. The geometric prefactor in [Eq. ]{}, however, differs from the one for the sphere \[[Eq. ]{}\] because it is chosen such that within the DA ($\Delta\to0$) the scaling function ${\ensuremath{K_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ attains a nonzero and finite limit, as discussed before. The DA can be implemented along the lines of Sec. \[sec:homog-da\] for the sphere. Here the surface of the cylindrical colloid is decomposed into pairs of infinitely narrow stripes of combined area ${{\ensuremath{\textrm{d}}}}S=2 M {{\ensuremath{\textrm{d}}}}\rho$, positioned parallel to the substrate at a distance $L(\rho)$ from it \[[Eq. ]{}\] and each at a distance $\rho$ from the symmetry plane of the configuration. $M$ is the length of the cylinder and drops out from ${\ensuremath{F_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ which follows analogously from Eqs. and : $$\label{eq:app-cyl-force-da-def}
{\ensuremath{F_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(D,R,T)/k_BT\simeq 2 \int_0^R {{\ensuremath{\textrm{d}}}}\rho \left[L(\rho)\right]^{-d} {\ensuremath{k_{{\ensuremath{(a,b)}}}}}(\operatorname{sign}(t)\, L(\rho)/\xi_\pm),$$ where $L(\rho)$ is given in [Eq. ]{}. Finally, in the limit $\Delta\to0$ we obtain $$\label{eq:app-cyl-force-da}
{\ensuremath{K_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(\Theta,\Delta\to0)=\sqrt{2}\int\limits_1^\infty{{\ensuremath{\textrm{d}}}}\alpha\,(\alpha-1)^{-\frac{1}{2}}\,\alpha^{-d}\,{\ensuremath{k_{{\ensuremath{(a,b)}}}}}(\Theta\alpha).$$ At the bulk critical point $\Theta=0$ one finds ${\ensuremath{K_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(0,0)=\sqrt{2\pi}[\Gamma(d-\frac{1}{2})/\Gamma(d)]{\ensuremath{\Delta_{{\ensuremath{(a,b)}}}}}$ so that ${\ensuremath{K_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(0,0)=[3\pi/(4\sqrt{2})]{\ensuremath{\Delta_{{\ensuremath{(a,b)}}}}}\simeq 1.66 \times{\ensuremath{\Delta_{{\ensuremath{(a,b)}}}}}$ for $d=3$ and ${\ensuremath{K_{{\ensuremath{(a,b)}}}}}^{\ensuremath{{ \textrm{cyl}}}}(0,0)=[5\pi/(8\sqrt{2})]{\ensuremath{\Delta_{{\ensuremath{(a,b)}}}}}\simeq 1.38 \times{\ensuremath{\Delta_{{\ensuremath{(a,b)}}}}}$ for $d=4$.
Derjaguin approximation for a chemical step
-------------------------------------------
Here, we assume that the axis of the cylinder is parallel to the chemical step, i.e., perpendicular to the $x$ direction \[[Fig. \[fig:sketch\]]{}\], as well as parallel to the substrate. The projection of the position of the axis of the cylinder with respect to the $x$ direction is denoted by $X$, so that at $X=0$ the cylinder is positioned directly above the chemical step \[[Fig. \[fig:sketch\]]{}\]. Accordingly, the problem is effectively two-dimensional and the corresponding DA can be performed much easier than in Appendix \[app:step\]. Following an approach analogous to the one adopted for the sphere in Sec. \[sec:step\] and in Appendix \[app:step\], we rewrite the normal critical Casimir force per unit length acting on the cylinder as in [Eq. ]{}: $$\label{eq:app-cyl-F-step}
{\ensuremath{F_{{\ensuremath{\textrm{s}}}}}}^{\ensuremath{{ \textrm{cyl}}}}(X,D,R,T)=F_{\ensuremath{ {(a_>,b)}}}^{\ensuremath{{ \textrm{cyl}}}}(D,R,T)+\Delta F^{\ensuremath{{ \textrm{cyl}}}}(X,D,R,T).$$ Within the DA we find for $\Delta\to0$ \[compare [Eq. ]{}\] $$\label{eq:app-cyl-delta-F}
\Delta F^{\ensuremath{{ \textrm{cyl}}}}(X,D,R,T)=k_BT\frac{R^{1/2}}{D^{d-1/2}}\Delta K^{\ensuremath{{ \textrm{cyl}}}}(\Xi,\Theta,\Delta\to0),$$ where \[compare [Eq. ]{}\] $$\label{eq:app-cyl-delta-K}
\Delta K^{\ensuremath{{ \textrm{cyl}}}}(\Xi,\Theta,\Delta\to0)=
\sqrt{2}\int_{1+\Xi^2/2}^\infty{{\ensuremath{\textrm{d}}}}\alpha\,(\alpha-1)^{-\frac{1}{2}}\,\alpha^{-d}\Delta k(\Theta\alpha).$$ Using [Eq. ]{} and [Eq. ]{} we find for the whole range of values of $\Xi$ the scaling function $\psi_{(a_<|a_>,b)}^{\ensuremath{{ \textrm{cyl}}}}$ which is defined completely analogous to [Eq. ]{} \[compare [Eq. ]{}\]: $$\begin{gathered}
\label{eq:app-cyl-psi-step}
\psi_{(a_<|a_>,b)}^{\ensuremath{{ \textrm{cyl}}}}(\Xi\gtrless0,\Theta,\Delta\to0)=
\mp 1\\
\pm
\frac{\sqrt{2}
\int_{1+\Xi^2/2}^\infty{{\ensuremath{\textrm{d}}}}\alpha\,(\alpha-1)^{-\frac{1}{2}}\,\alpha^{-d}\Delta k(\Theta\alpha)}
{{\ensuremath{K_{{\ensuremath{ {(a_<,b)}}}}}}^{\ensuremath{{ \textrm{cyl}}}}(\Theta,\Delta\to0)-{\ensuremath{K_{{\ensuremath{ {(a_>,b)}}}}}}^{\ensuremath{{ \textrm{cyl}}}}(\Theta,\Delta\to0)}.\end{gathered}$$ $$\label{eq:app-cyl-psi-step}
\psi_{(a_<|a_>,b)}^{\ensuremath{{ \textrm{cyl}}}}(\Xi\gtrless0,\Theta,\Delta\to0)=
\mp 1
\pm
\frac{\sqrt{2}
\int_{1+\Xi^2/2}^\infty{{\ensuremath{\textrm{d}}}}\alpha\,(\alpha-1)^{-\frac{1}{2}}\,\alpha^{-d}\Delta k(\Theta\alpha)}
{{\ensuremath{K_{{\ensuremath{ {(a_<,b)}}}}}}^{\ensuremath{{ \textrm{cyl}}}}(\Theta,\Delta\to0)-{\ensuremath{K_{{\ensuremath{ {(a_>,b)}}}}}}^{\ensuremath{{ \textrm{cyl}}}}(\Theta,\Delta\to0)}.$$
Derjaguin approximation for a periodic chemical pattern
-------------------------------------------------------
The derivation of the scaling function for the critical Casimir force acting on the cylinder close to and aligned with a periodic chemical pattern as studied in Sec. \[sec:cylinder\] is analogous to the one for the sphere described in Appendix \[app:period\]. The final formula for ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ is the same as in [Eq. ]{} with $\psi_{(a_1|a_2,b)}$ replaced by $\psi_{(a_1|a_2,b)}^{\ensuremath{{ \textrm{cyl}}}}$ given by [Eq. ]{}. This renders the critical Casimir force per unit length $${\ensuremath{F_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}(L_1,P,X,D,R,T)=k_BT\frac{R^{1/2}}{D^{d-1/2}}{\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}(\lambda,\Pi,\Xi,\Theta,\Delta)$$ where ${\ensuremath{K_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$ is defined as in [Eq. ]{} with $K_{(a_1,b)}$ and $K_{(a_2,b)}$ replaced by $K_{(a_1,b)}^{\ensuremath{{ \textrm{cyl}}}}$ and $K_{(a_2,b)}^{\ensuremath{{ \textrm{cyl}}}}$, respectively, which are given by [Eq. ]{}, and with ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}$ replaced by ${\ensuremath{\psi_{{\ensuremath{\textrm{p}}}}}}^{\ensuremath{{ \textrm{cyl}}}}$. The corresponding results are shown in [Fig. \[fig:cylinder\]]{}.
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5[[[AdS]{}\_5]{}]{} §4[[[S]{}\_4]{}]{}
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October 2001\
[ **De Sitter space versus Nariai Black Hole: stability in d5 higher derivative gravity.** ]{}
[Shin’ichi NOJIRI]{}[^1] and [Sergei D. ODINTSOV]{}$^{\spadesuit}$[^2]\
[*Department of Applied Physics\
National Defence Academy, Hashirimizu Yokosuka 239-8686, JAPAN*]{}
[*$\spadesuit$ Instituto de Fisica de la Universidad de Guanajuato,\
Lomas del Bosque 103, Apdo. Postal E-143, 37150 Leon,Gto., MEXICO*]{}
[**ABSTRACT**]{}
d5 higher derivative gravity on the Schwarzschild-de Sitter (SdS) black hole background is considered. Two horizons SdS BHs are not in thermal equilibrium and Hawking-Page phase transitions are not expected there, unlike to the case of AdS BHs. It is demonstrated that there exists the regime of d5 theory where Nariai BH which is extremal limit of SdS BH is stable. It is in the contrast with Einstein gravity on such background where only pure de Sitter space is always stable. Speculating on the applications in proposed dS/CFT correspondence, these two (de Sitter and Nariai) stable spaces may correspond to confining-deconfining phases in dual CFT.
PACS numbers: 04.50.+h, 04.70.Dy, 11.25.Db
The increasing evidence indicating a positive cosmological constant for our universe calls to better study of de Sitter gravity. One of the fundamental questions in de Sitter gravity is related with holographic principle which is presumably realized there in the form of dS/CFT correspondence suggested in ref.[@strominger] (for earlier attempts on dS/CFT duality see [@earlier]). Despite the number of efforts [@ds/cft; @r2] (for a recent review, see [@lecture]) still dS/CFT correspondence is not understood on the same level as AdS/CFT correspondence. Moreover, the corresponding consistent dual CFT is not yet formulated. Nevertheless, the hunt for dS/CFT correspondence continues. It is expected that analogy with AdS/CFT may be often helpful in such investigation.
It has been observed quite long ago by Hawking and Page [@HP] that AdS black hole (BH) thermodynamics admits phase transitions in the following way: low temperature BHs are not stable and they decay into global AdS space. High temperature BHs are more stable than global AdS space. Hawking-Page phase transitions for 5d AdS BHs are very important in frames of AdS/CFT. Indeed, they were interpreted by Witten [@witten] as corresponding to a deconfinement-confinement transition in the large-$N$ limit of an ${\cal N}=4$ $SU(N)$ super Yang-Mills theory living on the boundary of 5d AdS BH.
The interesting question appears: Can the similar effect be expected for proposed dS/CFT correspondence? From first look it seems that the answer is completely negative. The reason is that there are two Hawking temperatures in Schwarzschild-de Sitter (SdS) BH (which is the natural analog of AdS BH) since there are two horizons, black hole one and cosmological one. Then SdS spacetime is not in the thermal equilibrium unlike to AdS BH. Moreover, it is known that pure de Sitter space is the only stable space among SdS backgrounds for Einstein gravity. There is, however, second exception. It is the Nariai space, which is the extremal limit where the black hole horizon coincides with the cosmological one.
In the present Letter we discuss the question of stability of dS and Nariai BH in higher derivative gravity (for general introduction to such theory, see [@BOS]). We will show that despite the absence of Hawking-Page phase transitions, there is some theory regime (defined by coefficients of higher derivative terms) where Nariai BH is stable and does not decay into pure de Sitter space. Brief speculation of the relevance of this observation to proposed dS/CFT correspondence is made.
The general action of $d+1$ dimensional $R^2$-gravity is given by \[Svi\] S=d\^[d+1]{} x {a R\^2 + b R\_ R\^ + c R\_ R\^ + [1 \^2]{} R - } . For simplicity, we only consider the $c=0$ case, for a while. Then Schwarzschild-anti de Sitter or Schwarzschild-de Sitter spacetime is an exact solution: \[SAdS\] && ds\^2=G\_dx\^dx\^ =-\^[2\_0]{}dt\^2 + \^[-2\_0]{}dr\^2 + r\^2\_[i,j]{}\^[d-1]{} g\_[ij]{}dx\^i dx\^j ,&& \^[2\_0]{}=[1 r\^[d-2]{}]{}(-+ [kr\^[d-2]{} d-2]{} + [r\^d L]{}) . Here $g_{ij}$ is the metric of the $(d-1)$-dimensional Einstein manifold, which is defined by $R_{ij}=kg_{ij}$, where $R_{ij}$ is the Ricci tensor defined by $g_{ij}$ and $k$ is a constant. For example, we have $k>0$ for the sphere, $k<0$ for the hyperboloid, and as a special case, flat space for $k=0$. The parameter $L$ in (\[SAdS\]) is related with the length parameter of the asymptotic AdS or dS spacetime and is found solving the equation \[ll\] 0=[d\^2(d+1)(d-3) a L\^2]{} + [d\^2(d-3) b L\^2]{} - [d(d-1) \^2 L]{}- . If $L>0$, the spacetime is SAdS and if $L<0$, SdS. Here we consider the case of $L<0$ and $d=4$. In dS, we cannot embedd the hyperbolic or the flat space as the surface with constant $r$. Then one only discusses the case that $g_{ij}$ (\[SAdS\]) is the metric of the unit sphere \[us\] \_[i,j]{}\^3 g\_[ij]{}dx\^i dx\^j = d\_3\^2 . In this case $k=2$. Defining the length parameter $l$ by \[Ni\] l\^2 = -L , $\e^{2\rho}$ in the metric (\[SAdS\]) has the following form \[Nii\] \^[2\_0]{}=[1 r\^[d-2]{}]{}(-+ r\^2 - [r\^4 l\^2]{}) . Then there are two horizons, where $\e^{2\rho_0}=0$, at \[Niii\] r=r\_[c,bh]{}\^2 , where the plus sign corresponds to the cosmological horizon $r=r_c$ and the minus to the black hole one $r=r_{bh}$. The corresponding Hawking temperature $T_H$ is given by \[Niv\] T\_H=[1 4]{}|.[d(\^[2\_0]{}) dr]{}|\_[r=r\_[c,bh]{}]{}| = [1 2]{}|[r\_[c,bh]{}\^3]{} - [r\_[c,bh]{} l\^2]{}| . For the pure dS case, where $\mu=0$, we find \[Nv\] r\_[bh]{}=0 ,r\_c=l ,T\_H=[1 2l]{} and for the Nariai space [@N], where $\mu={l^2 \over 4}$, \[Nvi\] r\_[bh]{}=r\_c=[l ]{} , T\_H=[15 2l]{} . We should note that when $c\neq 0$ in (\[Svi\]), the general Schwarzschild-(anti) de Sitter spacetime is not the exact solution. Nevertheless, even if $c\neq 0$, the pure dS and the Nariai space are exact solutions since the Riemann curvature is covariantly constant: \[Riemann\] \_ = [1l\^2]{}( \_ \_ -\_\_ ). Here, instead of (\[ll\]), the length parameter $l^2$ is given by \[ll2\] 0&=&[a l\^4]{}(d+1)d\^2(d-3) + [b l\^4]{}d\^2(d-3) && + [2c l\^4]{}d(d-3) + [d(d-1) \^2 l\^2]{} - . Then in what follows we also consider the case of $c\neq 0$.
Before going forward, we give some remarks about the Nariai space, which is given in the Nariai limit $\mu\rightarrow {l^2 \over 4}$. Before taking the limit, one changes the coordinate $(t,r)$ to $(\tau,\theta)$ by \[Nvii\] t=[l ]{} , r\^2 = [l\^2 2]{}- . Then the black hole horizon corresponds to $\theta=0$ and the cosmological one to $\theta=\pi$. In the coordinates $(\tau,\theta)$, the metric of SdS is rewritten by \[SdS\] ds\^2 = - [\^2 2(l\^2 - )]{}d\^2 + [l\^2 2]{}d\^2 + ([l\^2 2]{} - )d\_3\^2 . Then by taking the Nariai limit $\mu\rightarrow {l^2 \over 4}$, one finds \[Nr\] ds\_[Nariai]{}\^2 = - [\^2 2]{}d\^2 + [l\^2 2]{}d\^2 + [l\^2 2]{} d\_3\^2 . If we Wick-rotate the time coordinate $\tau$ by \[Nviii\] il , the metric (\[Nr\]) is the direct product of S$_2$ and $S_3$ with the radius ${l \over \sqrt{2}}$ and we find $\tilde \tau$ should have the periodicity of $2\pi$.
In 4 dimensional ($d=3$) Nariai space, there occurs a very interesting phenomenon called “anti-evaporation”, which was first found in [@BH] by using 2d trace anomaly induced effective action including dilaton [@BHano; @NOano; @A]. It corresponds to quantum expansion of BH. (It is not yet completely clear if this is fundamental or just transitionary effect). This phenomenon has been confirmed by using 4d trace anomaly induced effective action [@NOnari].
Let us discuss the free energies of the pure dS and the Nariai space. Since for these cases, the scalar, Ricci and Riemann curvatures are given by \[rr\] = [20 l\^2]{} ,\_= [4l\^[2]{}]{}G\_ ,\_ = [1l\^2]{}( \_ \_ -\_\_ ) . the action (\[Svi\]) with $d=4$ is given by \[Nix\] S&=&([400a l\^4]{} + [80 b l\^4]{} + [40 c l\^4]{} + [20 \^2]{} - ) d\^5 x &=&([320a l\^4]{} + [64 b l\^4]{} + [32 c l\^4]{} + [8 l\^2\^2]{} ) d\^5 x . In the second line of (\[Nix\]), Eq. (\[ll\]) is used. For pure de Sitter space, one gets \[Nx\] V\_[dS]{}=d\^5 x =[V\_3 T\_H]{}\_0\^[r\_c=l]{} dr r\^3 = [V\_3l\^4 4 T\_H]{} =[V\_3 4T\_H(2T\_H)\^4]{} , and for the Nariai space \[Nxi\] V\_[Nariai]{}=d\^5 x =V\_2V\_3 ([l\^2 2]{} )\^[5 2]{}=[815\^5 (2)\^[5 2]{}]{}[V\_3 4T\_H(2T\_H)\^4]{} , Here $V_2=4\pi$ and $V_3$ are the volumes of 2 sphere and 3 sphere, respectively. In (\[Nxi\]) it is assumed $\tilde\tau$ in (\[Nviii\]) has the period $2\pi$. As it follows from (\[Nvii\]) and (\[Nviii\]) \[Nxib\] i= [t l\^2]{} . Then if $t$ has the periodicity of ${1 \over T_H}$ (near the Nariai limit) when Wick-rotated, the periodicity $\tilde P$ of $\tilde\tau$ should be \[Nxic\] P = [ T\_H l\^2]{} . Then the volume $V_{\rm Nariai}$ of the Nariai space in (\[Nxi\]) should be modified as \[Nxid\] V\_[Nariai]{}V\_[Nariai]{} , which, we should note, vanishes in the Nariai limit $\mu\rightarrow {l^2 \over 4}$ since $\tilde P$ vanishes.
Using (\[Nix\]) and (\[Nx\]), we find that the free energy $F=-T_H S$ for the pure dS is given by \[Nxii\] F\_[dS]{}=-([320a l\^4]{} + [64 b l\^4]{} + [32 c l\^4]{} + [8 l\^2\^2]{} ) [V\_3 4(2T\_H)\^4]{} , Using (\[Nix\]), (\[Nxi\]) and (\[Nxid\]), one gets the free energy of the Nariai space: \[Nxiii\] F\_[Nariai]{}=-([320a l\^4]{} + [64 b l\^4]{} + [32 c l\^4]{} + [8 l\^2\^2]{} ) [815\^5 (2)\^[5 2]{}]{} [V\_3 4(2T\_H)\^4]{}[P 2]{} 0 . Therefore the pure dS is more stable than the Nariai space if \[Nxv\] [320a l\^4]{} + [64 b l\^4]{} + [32 c l\^4]{} + [8 l\^2\^2]{}>0 , but the Nariai space becomes stable if \[Nxvi\] [320a l\^4]{} + [64 b l\^4]{} + [32 c l\^4]{} + [8 l\^2\^2]{}<0 . Therefore there is a critical point (or surface) at \[Nxvii\] [320a l\^4]{} + [64 b l\^4]{} + [32 c l\^4]{} + [8 l\^2\^2]{}=0 . It is easily seen that for pure Einstein gravity, the above equation has no solution. Hence, dS space is always stable there!
The expressions of the free energies (\[Nxii\]) and (\[Nxiii\]) seem to be strange since the fourth inverse power of the temperature appears in the expressions. As we are considering the limits, Nariai limit and the vanishing mass limit, the temperature does not depend on the black hole mass but only depends on the length parameter $l$ as in (\[Nv\]) and (\[Nvi\]). The parameter $l$ is not the dynamical variable. Of course, when we include the scalar fields $\phi_i$ with potential $V(\phi_i)$, the length parameter can be regarded as a dynamical variable by replacing the cosmological constant $\Lambda$ in the action (\[Svi\]) by the potential $\Lambda\rightarrow V(\phi_i)$. One can also consider the condensation of the anti-symmetric tensor fields as in the usual AdS$_5$/CFT$_4$ scenario. If $l$ is not the dynamical variable, we should consider the radius of the black hole as a dynamical variable but when the radius changes from one of the (Nariai and the vanishing mass) limits, the system is not in the thermal equilibrium. Then one cannot define the heat capacity and (or) to use the other thermodynamical stability conditions as those developed in ref.[@CG]. This makes difficult to argue about the local thermal stability. One can conjecture that two limits are connected with each other by some thermal inequilibrium process.
From the viewpoint of the WKB approximation of the path integral, the partition function $Z$ can be given by the classical action $S_{\rm cl}$, where the classical solution is substitued into the action (\[Svi\]), \[Nxviii\] Z\~\^[S\_[cl]{}]{} . The expression (\[Nxiii\]) is valid even if the system is not in the thermal equilibrium. For the general SdS spacetime, one gets \[Nxix\] V\_[SdS]{}=d\^5 x =[V\_3 T\_H]{}\_[r\_[bh]{}]{}\^[r\_c]{} dr r\^3 = [V\_3l\^4 4 T\_H]{} . Here we assume the time variable $t$ has a period ${1 \over T_H}$ although the system is in the thermal inequilibrium. When $c=0$, the action $S_{\rm SdS}$ for SdS is also given by (\[Nix\]) \[NixB\] S\_[SdS]{}=([320a l\^4]{} + [64 b l\^4]{} + [8 l\^2\^2]{} ) V\_[SdS]{} . The actions for the pure dS and Nariai space are just given by replacing $V_{\rm SdS}$ with $V_{\rm dS}$ in (\[Nx\]) and ${\tilde P \over 2\pi}V_{\rm Nariai}$ in (\[Nxi\]) and (\[Nxid\]), respectively. Since $\sqrt{1 - {4\mu \over l^2}}<1$, we find \[Nxx\] V\_[dS]{}>V\_[SdS]{}>[P 2]{}V\_[Nariai]{}=0 . Then the classical action for dS is larger than that for SdS if the condition (\[Nxv\]) is satisfied. Therefore dS is stable even locally. On the other hand, if the condition (\[Nxvi\]) is satisfied, the classical action for the Nariai space is larger than that for SdS. Then the Nariai space becomes stable even locally. Hence, we demonstrated that there is regime of d5 higher derivative gravity where Nariai BH does not decay into pure dS space.
In case of AdS$_5$/CFT$_4$ correspondence higher derivative terms like $R^2$-terms appear as next-to-leading, ${1 \over N}$ correction. The ${\cal N}=2$ theory with the gauge group $Sp(N)$ arises as the low-energy theory on the world volume on $N$ D3-branes sitting inside 8 D7-branes at an O7-plane [@Sen]. The string theory dual to this theory has been conjectured to be type IIB string theory on ${\rm AdS}_5\times {\rm X}^5$ where ${\rm X}_5={\rm S}^5/Z_2$ [@FS], whose low energy effective action is given by \[bng3\] S=\_[[AdS]{}\_5]{} d\^5x {[N\^2 4\^2]{} R- + [6N 2416\^2]{}R\_ R\^} . Then $R^2$-term appears as $1/N$ correction. We should note that the coupling constants are chosen to be dimensionless by the proper redefinitions. Then one can identify \[bng4\] [1 \^2]{}=[N\^2 4\^2]{} , c=[6N 2416\^2]{} , In the model (\[bng\]), $\Lambda$ is negative and the spacetime is always asymptotically AdS.
Here as a toy model, we consider the case that $\Lambda$ is positive and is given by \[bng5\] =[12N\^2 4\^2]{} - [6N 2416\^2]{} , Suppose that $c$ is negative: \[bng6\] [1 \^2]{}=[N\^2 4\^2]{} , c=-[6N 2416\^2]{} , It has been demonstrated in ref.[@r2] that in frames of dS/CFT correspondence such de Sitter higher derivative gravity reproduces the holographic conformal anomaly for above $Sp(N)$ super Yang-Mills theory. Then Eq.(\[ll2\]) tells $l^2=1$ and Eq.(\[Nxvii\]) gives the critical point at \[bng\] N=[1 4]{} Then from Eq.(\[Nxv\]), the pure de Sitter space is stable when $N\geq 1$. Of course, since this model is not realistic dual CFT model one can expect that there will be and dS and Nariai phases when realistic dual CFT will be proposed. (Moreover, even in above model the next powers of the curvatures may qualitatively change the situation). Then presumably one of these two phases will correspond to confinement, while another one to deconfinement in dual CFT. In any case, the fact that Nariai BH may be preferrable vacuum state for some dS gravitational theory looks quite attractive.
[**Acknoweledgements.**]{} We thank M. Cvetič and V. Tkach for helpful discussion. The research by SDO has been supported in part by CONACyT (CP, Ref.990356).
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[^1]: nojiri@cc.nda.ac.jp
[^2]: On leave from Tomsk State Pedagogical University, 634041 Tomsk, RUSSIA.\
odintsov@ifug5.ugto.mx, odintsov@mail.tomsknet.ru
|
---
author:
- 'A. Dutrey, S. Guilloteau , V. Piétu , E. Chapillon, V. Wakelam, E. Di Folco , T. Stoecklin, O. Denis-Alpizar, U. Gorti, R. Teague, T. Henning, D. Semenov, and N. Grosso'
bibliography:
- 'biblio-fls.bib'
date: '2017 / 2017'
title: |
The Flying Saucer:\
Tomography of the thermal and density gas structure of an edge-on protoplanetary disk.
---
[Determining the gas density and temperature structures of protoplanetary disks is a fundamental task to constrain planet formation theories. This is a challenging procedure and most determinations are based on model-dependent assumptions. ]{} [We attempt a direct determination of the radial and vertical temperature structure of the Flying Saucer disk, thanks to its favorable inclination of 90 degrees.]{} [We present a method based on the tomographic study of an edge-on disk. Using ALMA, we observe at 0.5$''$ resolution the Flying Saucer in CO J=2-1 and CS J=5-4. This edge-on disk appears in silhouette against the CO J=2-1 emission from background molecular clouds in $\rho$ Oph. The combination of velocity gradients due to the Keplerian rotation of the disk and intensity variations in the CO background as a function of velocity provide a direct measure of the gas temperature as a function of radius and height above the disk mid-plane.]{} [The overall thermal structure is consistent with model predictions, with a cold ($< 15-12 $ K), CO-depleted mid-plane, and a warmer disk atmosphere. However, we find evidence for CO gas along the mid-plane beyond a radius of about 200au, coincident with a change of grain properties. Such a behavior is expected in case of efficient rise of UV penetration re-heating the disk and thus allowing CO thermal desorption or favoring direct CO photo-desorption. CO is also detected up to 3-4 scale heights while CS is confined around 1 scale height above the mid-plane. The limits of the method due to finite spatial and spectral resolutions are also discussed.]{} [This method appears to be very promising to determine the gas structure of planet-forming disks, provided that the molecular data have an angular resolution which is high enough, of the order of $0.3 - 0.1''$ at the distance of the nearest star forming regions. ]{}
Introduction
============
Protoplanetary disks orbiting young pre-main sequence stars are the sites of planetary system formation. In these disks, gas represents about 99$\%$ of the mass and is mostly in the form of H$_2$. Since a minimum mass of 0.01 ${\,\mathrm{M}_\odot}$ has been determined by @Weidenschilling+1977 for the Proto-solar Nebula based on the current Solar System, models of planetary systems formation have drastically evolved. Observational determinations of vertical and radial mass distribution of protoplanetary disks provide key constraints for planet formation models. Furthermore, studying the gas and dust distributions in protoplanetary disks found around low-mass TTauri stars, recognized as young analogs to the Solar System, has become a major challenge to understand how planetary systems form and evolve. With the advent of ALMA, many new results such as the observation of narrow dust rings in the HL Tau dust disk [@Alma+etal_2015] are changing our views on these objects. A series of studies of the disk associated with TW Hydrae - the closest T Tauri star - has significantly improved our understanding of disk physics and chemistry. This disk is seen almost face-on, maximizing its surface, and the dust and gas distributions have been intensively observed and modeled [@Qi+etal_2004; @Andrews+etal_2012; @Rosenfeld+etal_2012; @Qi+etal_2013; @Andrews+etal_2016; @van_Boekel+etal_2017; @Teague+etal_2017]. The J=1-0 transition of HD, the hydrogen deuteride, has been also detected by @Bergin+etal_2013 who determined the gas mass of the disk to be $> 0.056 {\,\mathrm{M}_\odot}$, a value ranging at the upper end of previous estimates based on indirect mass tracers ( 5$\cdot 10^{-4}$ - 0.06 ${\,\mathrm{M}_\odot}$, [@Thi+etal_2010; @Gorti+etal_2011]. More recently @Teague+etal_2016 used simple molecules such as CO, CN or CS to determine the turbulence inside the disk and found that the turbulent line broadening is less than $0.05\,{\,{\rm km\,s}^{-1}}$. @Schwarz+etal_2016 re-analyzed the HD observations and confirmed the high mass of the disk. However, both studies remain limited by the knowledge of the disk vertical structure, in particular the thermal profile of the gas, and have to make assumptions on the vertical location of molecules, since this cannot be directly recovered in a face-on disk.
Contrary to a face-on object, the disk around HD163296, a Herbig Ae star of 2${\,\mathrm{M}_\odot}$, is inclined by about 45$^\circ$ along the line of sight. Such an inclination is enough to partially reveal the vertical location of the molecular layer, confirming that a significant fraction of the mid-plane is devoid of CO emission [@Gregorio-Monsalvo+etal_2013; @Rosenfeld+etal_2013]. In the case of IM Lupi, a 1${\,\mathrm{M}_\odot}$ star surrounded by a disk inclined by about 45$^\circ$, a multi-line CO analysis coupled to a study of the dust disk (images and SED) allowed @Cleeves+etal_2016 to provide a coherent picture of the gas and dust disk. However, due to the combination of Keplerian shear and inclination, a complete determination of the vertical structure is challenging because at a given velocity can correspond to several locations (radii) inside the disk [see @Beckwith+Sargent_1993]. In other words, there is no perfect correspondence between a radius and a velocity and this generates degeneracies, which are particularly important when the spatial resolution is limited. A purely edge-on disk can allow the retrieval of the full vertical structure of the molecules from which the density and temperature vertical gradients can be derived, provided the angular resolution is high enough and the molecular transitions are adequately selected.
To test the ability of deriving the disk vertical structure from an edge-on disk, we submitted the ALMA project 2013.1.00387.S dedicated to the study of the Flying Saucer. The () is an isolated, edge-on disk in the outskirts of the $\rho$ Oph dark cloud L1688 [@Grosso+etal_2003] with evidence for 5-10 $\mu$m-sized dust grains in the upper layers [@Pontoppidan+etal_2007]. @Grosso+etal_2003 resolved the light scattered by micron-sized dust grains in near-infrared with the NTT and the VLT and estimated from the nebula extension dust a disk radius of $2.15''$, which is about 260 au for the adopted distance of 120 pc [@Loinard+etal_2008]. The detection of the CN N=2-1 line [@Reboussin+etal_2015] indicated the existence of a large gas disk. The $\rho$ Oph region is crowded with molecular clouds that are are strongly emitting in CO lines. The low extinction derived by @Grosso+etal_2003 toward the Flying Saucer suggests it lies in front of these clouds, and this is confirmed by the CO study of @Guilloteau+etal_2016a.
We observed CO J=2-1, CS J=5-4, CN N=2-1. This is a set of standard lines which has been extensively used to retrieve disk structures [@Dartois+etal_2003; @Pietu+etal_2007; @Chapillon+etal_2012; @Rosenfeld+etal_2013].
The dust and CO emissions detected in this ALMA project were partly discussed in @Guilloteau+etal_2016a where we analyzed the absorption of the CO background cloud by the dust disk, deriving a dust temperature of about 7 K in the dust disk mid-plane at 100 au. This second paper deals with the retrieval of the gas temperature and density structures based on the analysis of the low angular resolution CO and CS lines. After showing the results and the analysis of the data, we then discuss the ability of using edge-on disks to determine the vertical structure of gas disks.
Observations
============
{height="11.65cm"}
{height="6.5cm"}
Imaging observations were performed with the Atacama Large mm/submm Array (ALMA) in a moderately compact configuration. The project 2013.1.00387.S was observed on 17 and 18 May 2015 under excellent weather conditions. The correlator was configured to deliver very high spectral resolution with a channel spacing of 15 kHz (and an effective velocity resolution of 40 ms${-1}$). We observed simultaneously CO J=2-1, all the most intense hyperfine components of the CN N=2-1 transition, and the CS J=5-4 line.
Data was calibrated via the standard ALMA calibration script in the CASA software package (Version 4.2.2). Titan was used as a flux calibrator. The calibrated data was regridded in velocity to the LSR frame using the “cvel” task, and exported through UVFITS format to the GILDAS package for imaging and data analysis. Atmospheric phase errors were small, providing high dynamic range continuum images and thermal noise limited spectral line data. The total continuum flux is 35 mJy at 242 GHz (with about 7% calibration uncertainty). With robust weighting, the $uv$ coverage provided by the $\sim 34$ antennas yields a nearly circular beam size **close to 0.5$''$. The CS images were produced at an effective spectral resolution of 0.1 kms$^{-1}$; the rms noise is 3 mJy/beam, i.e. about 0.27 K given the beam size of $0.48'' \times 0.46''$ at PA $53^\circ$. For CO, a spectral resolution of 0.08 kms$^{-1}$ was used, and the rms noise is 4 mJy/beam, i.e. about 0.37 K given the beam size of $0.51'' \times 0.48''$ at PA $54^\circ$.**
Figures \[fig:co-map\] and \[fig:cs-map\] present the CO and CS channel maps, respectively, **spectrally smoothed to a $0.4\,{\,{\rm km\,s}^{-1}}$ for clarity.** Figure \[fig:wholeII\] is a summary of the emission from CO, CS and the dust continuum.
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In addition, a CO J=2-1 spectrum of the clouds along the line of sight was obtained with the IRAM 30-m telescope, as described in @Guilloteau+etal_2016a.
Results
=======
The CO J=2-1 and continuum results were partially reported by @Guilloteau+etal_2016a, who used them to measure the dust temperature.
Images
------
Figure \[fig:wholeII\] clearly shows that the disk is viewed close to edge-on. It also reveals a vertical stratification of the dust and molecules.
The modeling of the continuum image by @Guilloteau+etal_2016a leads for the (millimeter) dust grains to a scale height of $12.7\pm0.3$ au at 100 au, increasing with a $0.34\pm0.04$ exponent. For comparison, the modeling of the near-infrared images by @Grosso+etal_2003 leads for the (micron) dust grains to a larger scale height of $22.5\pm1.5$au at 100 au when adopting the same definition and distance, and increasing more rapidly with a 1.25 exponent. Therefore, there is a clear indication of dust settling in this disk, with large grains preferentially close to the disk mid-plane.
The integrated intensity maps (Fig.\[fig:wholeII\]) show that CS is significantly more confined towards the disk mid-plane than CO. As mentioned in @Guilloteau+etal_2016a, CO is contaminated by background emission from extended molecular clouds at four different velocities, which affect the derivation of the integrated emission and result in apparent asymmetries. The CS molecular emission extends at least up to radius of about 300 au, and slightly more ($\sim 330$ au) for CO. On the contrary, the dust emission is confined within 200 au. The apparent distributions may be more a result of temperature gradients, excitation conditions and line opacities than reflecting different abundance gradients for these molecules. The CO J=2-1 line is much more optically thick than the CS line, and thus more sensitive to the (warmer) less dense gas high above the disk mid-plane. Along the mid-plane, self-absorption by colder, more distant gas, can result in lower apparent brightness. However, at the disk edges, this effect should be small, so the higher brightness above the disk plane likely indicates a vertical temperature gradient, with warmer gas above the plane.
The aspect of the iso-velocity contours (Fig.\[fig:wholeII\]) is exactly what is expected from a Keplerian flared disk seen edge-on. In such a configuration, at an altitude $z$ above the disk plane, the disk only extends inwards to an inner radius depending on $z/H$, where $H$ is the scale height, so that the maximum velocity reached at altitude $z$ is limited by this inner radius. Thus the mean velocity decreases from the mid-plane to higher altitude, resulting in the “butterfly” shape of the iso-velocity contours. The effect is however less pronounced for CO, as its high optical depth allows us to trace the emission well above the disk plane (iso-velocity contours would be parallel for a cylindric distribution).
Simple determination of the disk parameters {#sub:diskfit}
-------------------------------------------
To constrain the basic parameters of the disk, we make a simple model of the CS J=5-4 emission with DiskFit [@Pietu+etal_2007] assuming power laws for the CS surface density ($\Sigma_{CS}(r) = \Sigma_0 (r/\mathrm{100 au})^{-p}$) and temperature $T_{ex}(r) = T_0 (r/\mathrm{100 au})^{-q}$. **The disk is assumed to have a sharp outer edge at $R_\mathrm{out}$.** The vertical density profile is assumed Gaussian [see Eq.1 @Pietu+etal_2007]), with the scale height a (free) power law of the **radius: $h(r) = H_0 (r/\mathrm{100 au})^{-h}$. The line emission is computed assuming a (total) local line width $dV$ independent of the radius and LTE (i.e. $T_0$ represent the rotation temperature of the level population distribution).**
**Besides the above intrinsic parameters, the model also includes geometric parameters: the source position $x_0,y_0$, the inclination $i$ and the position angle of the rotation axis $PA$, and the source systemic velocity $V_\mathrm{LSR}$ relative to the LSR frame.**
Results are given in Table \[tab:cs\]. This simple model allows us to determine the overall disk orientation, the systemic velocity and the stellar mass, and gives an idea of the temperature required to provide sufficient emission. **The apparent scale height $H_0$ derived at 100 au would correspond to a temperature of 53 K, much larger than $T_0$. The difference may indicate that CS is substantially sub-thermally excited or, more likely, that CS emission only originates from above one hydrostatic scale height.**
Parameter Value (at 100 au) Unit
------------------ --------------------------------- ------------------------ --------------------------
$PA$ $3.6 \pm 0.4$ $^\circ$ PA of disk rotation axis
$i$ $85.4 \pm 0.5$ $^\circ$ Inclination
$V_\mathrm{LSR}$ $3.755 \pm 0.003$ kms$^{-1}$ Systemic velocity
$M_*$ $0.58 \pm 0.01$ ${\,\mathrm{M}_\odot}$ Star mass (a)
$R_\mathrm{out}$ $290 \pm 7$ au Outer radius
$dV$ $0.17 \pm 0.01$ kms$^{-1}$ Local line width (b)
$\Sigma_0$ $4.3\,10^{13} \pm 0.3\,10^{13}$ cm$^{-2}$ CS Surface density
$p$ $2.71 \pm 0.03$ Surface density exponent
$T_0$ $18.0 \pm 0.5$ K CS temperature
$q$ $-0.18 \pm 0.03$ temperature exponent
$H_0$ $ 25.8 \pm 0.3 $ au Scale height of CS (c)
$h$ $-1.40 \pm 0.03$ exponent of scale height
: CS disk modeling results.
\
\[tab:cs\] (a) Assuming Keplerian rotation. (b) assumed constant with $r$. Errors are formal errorbars from the fit. **(c) apparent scale height (see section \[sub:diskfit\]).**
PV-diagrams {#sub:diagram}
-----------
A more detailed understanding of the disk properties can be derived from the position-velocity diagram shown in Figures \[fig:co21\] and \[fig:cs54\] where several altitudes $z$ are shown. In such diagrams, radial straight lines (i.e. lines with $v(x) - V_\mathrm{disk} \propto
x$, where $x$ is the impact parameter, $v$ the velocity and $V_\mathrm{disk}$ the disk systemic velocity) trace a constant radius $r$ (see Appendix \[app:rv\] for details). The blue straight lines indicate the outer radius ($R_\mathrm{out}
\simeq 330$ au). The blue curve is the Keplerian rotation curve $\sqrt{G M_*/r}$, with a stellar mass of $0.57 {\,\mathrm{M}_\odot}$. The black line is the apparent inner radius $R_\mathrm{in}$, and white-over-black line delineates the radius $r_\mathrm{dip} \simeq {185\,\mathrm{au}}$, where a dip in emission is observed at low altitudes both in CO and in CS.
The CO J=2-1 line provides a direct view into the thermal structure of the disk. Like for the continuum emission, the background provided by the four extended molecular clouds identified in the 30-m spectrum modulates the apparent brightness of the disk, since the ALMA array only measures the difference in emission between the disk and the background clouds. Because the CO J=2-1 line is essentially optically thick in disks, we can simply recover a corrected CO PV diagram by adding the background spectrum obtained with the IRAM 30m, [see @Guilloteau+etal_2016a their Fig.1] to the observed CO emission, at least within the disk boundaries (in position and velocity). The result is given in Fig.\[fig:co21\] (bottom).
{height="12.0cm"} {height="12.0cm"}
{height="12.0cm"}
In the disk plane, while CO appears to extend down to very small radii ($< 15$au), CS may have an inner radius around 25 au.
At a given location, the impact of the finite beamsize depends on the brightness temperature gradients which are different for CO and CS (because of different opacities and excitation conditions) leading therefore to different apparent structures. Nevertheless, all apparent inner radii increase with height above the disk plane, as described before for the analysis of iso-velocity curves. This happens because the disk is flaring due to the hydrostatic equilibrium, hence its vertical thickness is increasing with radius.
The disk mid-plane is also clearly colder than the brightest background molecular cloud at 2.8 kms$^{-1}$, which has $J_\nu(T) = 11$K, beyond about 100 au radius, being almost as warm as the second brightest cloud at 4.2 kms$^{-1}$, with $J_\nu(T) = 8$K, at radii around ${185\,\mathrm{au}}$. From this simple consideration, we safely constrain the mean disk mid-plane temperature, averaged over one beam, to be 13 K near 180 au, after proper conversion of the brightness temperature outside of the Rayleigh-Jeans domain. It would rise to about 18 K at 100 au.
The PV diagrams above the disk plane indicate a warmer temperature in the upper layers, since no absorption is visible for $z>40$au or so. Because of our limited linear resolution (about 56 au), with such a vertical temperature gradient the CO PV diagram only gives an upper limit to the disk mid-plane temperature because the scale height (about 10 au at 100 au) is substantially smaller than the linear resolution except at the disk edge.
We also note that both CS and CO shows a drop in the emission intensity at radius $r_\mathrm{dip} \simeq {185\,\mathrm{au}}$. This drop is somewhat more difficult to identify in the CO PV diagrams because of the background clouds. In CO, the emission drop seems to disappear at an height of 40 au, suggesting it occurs only below about 30-40 au given the limited angular resolution. Beyond a radius of 220 au, CO emission is observed again. In CS, the deficit of emission near ${185\,\mathrm{au}}$ extends somewhat higher, up to a height of 60 au.
Figure \[fig:co-cs-over\] also reveals a north-south asymmetry, the north side being brighter in CS and in CO.
A new method of Analysis
========================
Deriving the brightness distribution from the PV-diagram {#sub:pv}
--------------------------------------------------------
**For an homogeneous medium, the measured brightness temperature $T_b$ is given by $$T_b = (1-\exp(-\tau)) (J_\nu(T) - J_\nu(T_{bg}))$$ where $J_\nu$ is the Planck function multiplied by $c^2/2k\nu^2$, $$J_\nu(T) = \frac{h \nu}{k} \frac{1}{\exp(h\nu/(kT))-1} . $$ Since** in a PV diagram a radial line represents a constant radius, we can recover the disk temperature from the thermalized and optically thick CO J=2-1 transition by averaging **the observed brightness** along such radial lines in regions where the signal is sufficiently resolved spectrally and spatially. **This averaging process yields the mean radiation temperature, $J_\nu(T)-J_\nu(T_\mathrm{bg})$, from which the temperature $T$ is derived.** The disk being seen edge-on, cuts at various altitudes $z$ provide a direct visualisation of the gas temperature versus radius $T(r,z)$, although only at the angular resolution of the observations. **The 2-D image resulting from the application of this averaging process is called hereafter the *tomographically reconstructed distribution* or TRD**.
**While the CO J=2-1 TRD is just the temperature distribution**, for optically thinner **or non thermalized** lines, the interpretation is more complex because the **TRD** is a function of both the temperature and local density. Nevertheless, it also provides a direct measurement of the altitude of the molecular layer.
We use the PV-diagrams of CO and CS transitions to derive **their respective TRDs** and show them in Fig.\[fig:co-cs-over\]. The derivations were performed onto data cubes without continuum subtraction because the subtraction, which was needed to compute the iso-velocity contours, may lead to substantial problems near the peak of the continuum. For CO, which is optically thick, the TRD$(r,z)$ map is obtained using the PV-diagram with the CO emission from the background clouds added, since that emission is fully resolved out by the ALMA observations. This explains why the derived CO temperature brightness is of the order of 15 K around the disk, in agreement with the values derived from the CO spectrum taken with the IRAM 30-m radiotelescope [@Guilloteau+etal_2016a].
In each case, we calculated a map of the mean, median and maximum brightness along each radius. For CO, the mean and maximum brightness are contaminated by the background clouds, and the median is expected to be a better estimator. Furthermore, for CS, we found that the mean and the median always give results which differ by less than 1 K inside the disk at $r < 350$ au (see Fig.\[fig:co-cs-over\], panel (d) which shows the difference between the mean and the median for CS 5-4). This indicates that the derivations are robust and do not suffer from significant biases. The observed **TRDs** clearly confirm the location of the molecular layer above the mid-plane but the CS emission peaks at a lower temperature and is located below the CO emission. The ratio of the CS **TRD** over the CO **TRD** confirms these behaviors (see panel (a) of Fig.\[fig:co-cs-over\]).
The Keplerian shear implies that the spatial averaging of the derived brightness is not the same everywhere inside the disk. Indeed, at a radius $r$ corresponding to a velocity $v(r)$, the smearing due to the Keplerian shear is given by $dr = 2r dv/v(r)$ where $dv$ is the local linewidth, which is due to a combination of thermal and turbulent broadening. This limits the radial resolution which can be obtained in the disk outer parts. For instance, for CO, at 20 au, assuming a temperature of 30 K and a turbulent boadening ($0.05 {\,{\rm km\,s}^{-1}}$) similar to that observed in TW Hydrae disk by @Teague+etal_2016, the $dr$ would be of the order of $\sim 2$ au, a value which would not affect studies at spatial resolution down to $0.1''$ (12 au) or so. On the contrary, at radius 200 au, the $dr$ would be of the order of 35 au for the same line width, dropping to 20 au assuming the same turbulence but a (mid-plane) temperature of 7 K. This limits the gain obtained with high angular resolution only at large radii. It also explains why the apparent extent of the **tomographically reconstructed distribution** in Fig.\[fig:co-cs-over\] exceeds the disk outer radius more than expected from the angular resolution only.
Nevertheless, this smearing is purely radial and the vertical structure is not affected: the smearing is the same above and below the mid-plane at a given radius (assuming there is no vertical temperature gradient).
This occurs because the disk is edge-on but it is no longer true in less inclined disks. In such disks, the smearing resulting from the local line width will limit the effective resolution radially and vertically.
Finally, we note that this direct method of analysis, specific to edge-on disks, is complementary to the classical channel maps studies by providing a more synthetic but direct view of the vertical disk structure.
{height="10.0cm"}
CO Modeling using DiskFit {#sect:mod}
-------------------------
To go beyond the resolution-limited information provided in Fig.\[fig:co-cs-over\], we study here the impact of several key parameters of the disk by performing grids of models to better constrain the disk geometry and structure. We use the ray-tracing model DiskFit [see Section \[sub:diskfit\], @Pietu+etal_2007]. For simplicity, we assume LTE conditions, which is appropriate for CO. **The temperature structure is here more complex than the simple vertically isothermal model of Section \[sub:diskfit\].** The atmosphere temperature is given by $$T_{atm}(r) = T_{atm}^0 \left(\frac{r}{r_0}\right)^{-q_{atm}}$$ and the mid-plane temperature is given by $$T_{mid}(r) = \min \left(T_{atm}(r),T_{mid}^0 \left(\frac{r}{r_0}\right)^{-q_{mid}}\right)$$ In between for an altitude of $z < z_q$ , the temperature is defined by $$T(r) = \left(T_{atm}(r)-T_{mid}(r)\right) \left(\cos\left(\frac{\pi z}{2 z_q H(r)}\right)\right)^{2\delta} + T_{mid}(r)$$ where $H(r)$ is the hydrostatic scale height (defined by $T_{mid}(r)$). Provided $q_{atm} > q_{mid}$, there is a radius $R_q$ beyond which the temperature becomes vertically isothermal. Note that for $q_{mid} = 0$, this definition is identical to that used by @Dartois+etal_2003.
The models of the molecular emission were performed together with the continuum emission not subtracted. The continuum model uses, in particular for the density and temperature, the parameters defined in Table 1 of @Guilloteau+etal_2016a. We take into account the spatial resolution by convolving all models by a $0.5''$ circular beam. In all models, the outer radius $r_{out}$ is taken at 330 au, in agreement the value derived from the PV-diagram.
We find that a small departure from edge-on inclination by 2-3 degrees is sufficient to explain the small North-South brightness asymmetry visible in Fig.\[fig:co-cs-over\]. The most probable value is $i = 87^\circ$ (see Sect.\[sec:disk-structure\]), a value used for all further models.
In a first series of models, we assume a CO vertical distribution which follows the H$_2$ density distribution, i.e. assuming a constant abundance. For the H$_2$ density distribution, we assume either power laws or exponentially tapered distribution following the prescription given in @Guilloteau+etal_2011. We explored CO surface densities ranging from 10$^{16}$ to 10$^{19}$ cm$^{-2}$ at 100 au. To account for the observed brightness at high altitudes above the disk plane, we find that the CO surface density at 100 au must be at least of the order of 5 $10^{17}$ cm$^{-2}$, with a power law index of the order of $p=1.2$ for the radial distribution. We also explored the impact of the temperature distribution $z_q$, $\delta$, $T_{atm}^0$, $q_{atm}$, $T_{mid}^0$, $q_{mid}$ and $R_q$. Best runs are obtained for mid-plane temperatures $T_{mid} \approx 10$ K at 100 au, $q_{mid} =
0.4$ leading to about 6 K at the outer disk radius and 17 K at 26 au (CO snowline location), $T_{atm}^0 = $50 K, $q_{atm}=0-0.2$, $\delta = 2$ and $z_q = 1.3 - 2$.
However none of these models, which assume CO molecules are present everywhere in the disk, properly reproduce the CO depression observed around the mid-plane (see panel (b) of Fig.\[fig:co-cs-over\]). This remains true even if the mid-plane temperature is set to values 5-7K, matching the temperature of large grains measured by @Guilloteau+etal_2016a.
We attempted to perform more realistic models by assuming complete molecular depletion (abundance ${X_p(\mathrm{CO})}= 0$) in the disk mid-plane. In this model, the zone in which CO is present is delimited upwards by a depletion column density $\Sigma_{dep}$ and downwards by the CO condensation temperature. CO molecules are present (with a constant abundance ${X_u(\mathrm{CO})}=10^{-4}$) when the H$_2$ column density from the current $(r,z)$ point upwards (i.e. towards ($r,\infty$)) exceeds a given threshold $\Sigma_{dep}$, to reflect the possible impact of photo-desorption of molecules, or when the temperature $T(r,z)$ is above 17K. Such a model takes into account the possible presence of CO in the inner disk mid-plane inside the CO snowline radius, and also at large radii because as soon as the surface density becomes low enough, CO emission can again be located onto the mid-plane. In this model, the CO surface density radial profile $\Sigma(r) \times {X_u(\mathrm{CO})}$ (where $\Sigma(r)$ is the H$_2$ surface density profile) is constrained, because of the need to provide sufficient opacity for the CO J=2-1 at high altitudes above the disk mid-plane to reproduce the observed brightness. The derived H$_2$ densities are then strictly inversely proportional to the assumed ${X_u(\mathrm{CO})}$.
Table \[tab:model\] gives the parameters of the best model we found with this approach. **Unfortunately the **current** angular resolution of the data limits the analysis. Because of this limited angular resolution, parameters $T_{atm}^0,\delta,z_q$ are strongly coupled. $z_q$ also depends implicitely on $T_{mid}$, because it is the number of hydrostatic scale heights at which the atmospheric temperature is reached. In practice, Table \[tab:model\] only confirms a low mid-plane temperature ($10$ K at 100 au), and temperatures at least a factor 2 larger than this in the CO rich region, consistent with the (spatially averaged) values derived in Section \[sub:diagram\]. Only higher spatial resolution data would allow us to break the degeneracy and accurately determine the vertical temperature gradient (see Section \[sub:spatial\]).**
Furthermore, even this model only qualitatively reproduces the brightness distribution of the CO emission around the mid-plane. In particular, the shape of the depletion zone is difficult to evaluate from the current data. We also fail to reproduce the rise of the CO brightness after 250 au, most likely because our model does not include an increase of the temperature in the outer part.
Most chemical models [e.g. @Reboussin+etal_2015] predict that there is some CO at low abundance (${X_p(\mathrm{CO})}\sim 10^{-8} - 10^{-6}$) in the mid-plane, depending on the grain sizes and the local dust to gas ratio, contrary to our simple assumption of ${X_p(\mathrm{CO})}=0$. Such low abundances would not impact our determination of the CO surface density, which only relies on the need to have sufficient optical thickness in the upper layers. However, CO could start being sufficiently optically thick around the mid-plane, diminishing the contrast between the mid-plane and the molecular layer. We estimated through modelling that this happens for ${X_p(\mathrm{CO})}\geq 3\,10^{-8}$. In such cases, the brightness distribution becomes similar to that of the undepleted case. Observations of a less abundant isotopologue would be a better probe of the mid-plane depletion.
We thus conclude that the current data are insufficient to disentangle between molecular depletion and a very cold mid-plane, but indicate a rise in mid-plane temperature beyond 200 au.
![Superimposition of the CO and CS median **TRD** to a standard disk model. (a): CO **TRD (colour)** and the H$_2$ volume density superimposed in **labelled** contours from $10^{5}$ up to $10^{8}$ cm$^{-3}$. (b): CO **TRD** with the scale height superimposed in black contours from 1 to 4 scale heights. (c) and (d) as (a) and (b) but for CS. The model corresponds to the parameters given in the Table \[tab:model\].[]{data-label="fig:structure"}](structure-crop.pdf){height="8.0cm"}
In Fig.\[fig:structure\], we overlay the brightness temperature derived from the observations to the structure of the model given in Table\[tab:model\]. The brightness temperature from the model is compared to the observations and also shown in Fig.\[fig:resol\] at three different angular resolutions of $0.5'', 0.3''$ and $0.1''$.
Parameter Value Unit Parameters at 100 au
------------------- ---------------- ------------------------ -------------------------------
$T_{atm}^0 $ 50 K Atmosphere temperature
$q_{atm} $ 0.4
$z_q $ 3
$\delta $ 2
$T_{mid}^0 $ 10 K Mid-plane temperature
$q_{mid} $ 0.4
$\Sigma_0$ $10^{23}$ cm$^{-2}$ H$_2$ Surface density
$p$ $0.5 $ Surface density parameter
$R_{C}$ $50 $ au Radius for exponential decay
$H_0$ 11.3 au Scale height
$h$ -1.3 exponent of scale height
$R_\mathrm{out}$ $330 $ au Outer radius
$i$ $87$ $^\circ$ Inclination
$\Sigma_{dep}$ $10^{22}$ cm$^{-2}$ Surface density for depletion
$T_{dep}$ 17 K Depletion temperature
$X_u($CO$)$ 10$^{-4}$ CO abundance in upper layers
$X_p($CO$)$ 0 as above, in mid-plane
$M_\mathrm{disk}$ $0.7\,10^{-3}$ ${\,\mathrm{M}_\odot}$ Disk mass
: Model derived from the CO observations
\[tab:model\]
{width="18.0cm"}
Discussion
===========
Overall disk structure {#sec:disk-structure}
----------------------
The analysis of the CO and CS brightness temperature patterns shows that there is no significant departure from a simple disk geometry at the linear resolution of 60 au, with the exception of the North-South brightness difference. This may be due to an intrinsic assymetry. However, the same effect can also be produced if the disk is slightly inclined by a few degrees from edge-on. Due to the flaring and radial temperature gradient, the optically thick emission of the far side always originates from slightly warmer gas @Guilloteau+Dutrey_1998. Indeed, we find that an inclination angle of 87$^\circ$ is enough to account for the North-South dissymmetry, with the Southern part of the disk being closer to us. This inclination (and orientation) is in very good agreement with that of $86 \pm 1^\circ$ derived by @Grosso+etal_2003 from near InfraRed (NIR) observations. The brightness distributions of the NIR images also show that the southern part of the disk is closer to us. Finally, there is no apparent sign of warp beyond a radius of about 50 au.
Figure \[fig:structure\] shows the CO emission is above one scale height, while the CS emission appears slightly below and less extended vertically. This is not surprising. CS J=5-4 is excited only at the high densities near the mid-plane while CO J=2-1 is easy to thermalize at density as low as a few 10$^{3}$ cm$^{-3}$ (see panels a-c of Fig.\[fig:structure\] which trace the density distribution). As a consequence, the convolution by the $\sim 56$au beam leads to different positions of the peak of the emission layer. The CO layer is vertically resolved and extended, while the CS emission is vertically unresolved and peaks just below one scale height. These vertical locations for the CO and CS layers are consistent with predictions by chemical models [see @Dutrey+etal_2011 Fig.8 where CS peaks between 1 and 2.5 scale heights]. A CO layer above the mid-plane was already observed in the disk of HD163296 [@Gregorio-Monsalvo+etal_2013; @Rosenfeld+etal_2013].
We also estimate for the first time the amplitude of the vertical temperature gradient between the cold mid-plane, the molecular layer and CO atmosphere at radii between 50 and 300 au. The gas vertical temperature gradient which is derived at 100 au is in agreement with that predicted by thermo-chemical models [e.g. @Cleeves+etal_2016] with a mid-plane at about 8-10 K and a temperature of 25 K reached at two scale-heights. The main limitation here is the linear resolution of 60 au. Nevertheless, the observed pattern of the brightness distribution suggests the existence of warmer gas inside a radius of about 50-80 au. Any inner hole of radius $<15$au cannot be seen in these data due to our sensitivity limit.
Radial profile of CO near the mid-plane
---------------------------------------
#### Rise of the CO brightness a large radius:
While in the upper layers, the CO brightness decreases smoothly with radius, in the mid-plane, we observe a rise in CO brightness beyond a radius of about 200 au. The transition radius coincides with the mm-emitting dust disk outer boundary, $\sim 187$ au in the simple model from @Guilloteau+etal_2016a, while in scattered light, the disk is nearly as extended as the CO emission [@Grosso+etal_2003]. This suggests that a change of grain size distribution may also play a role here. With the dust composed of mostly micron-sized grains better coupled with the gas [see @Pontoppidan+etal_2007], more stellar light may be intercepted, resulting in a more efficient heating of the gas, as suggested by recent chemical models [@Cleeves+2016]. Moreover, at the expected densities in the disk mid-plane, the dust and gas temperatures should be strongly coupled, and a dust temperature rise is also naturally expected when the disk becomes optically thin to the incident radiation and for the re-emission, as shown e.g. by @Dalessio+etal_1999.
Also, as the $\rho$ Oph region is bathed in a higher-than-average UV field due the presence of several B stars, this effect may be reinforced by additional heating due to a stronger ambient UV field. However, the Flying Saucer is located on the Eastern side of the dark cloud, whose dense clouds absorb the UV from the B stars, located mainly on the Western side. Fig. 1a and 4a of @Lim+2015 show that there is a clear dip in the FUV emission due to the dark cloud silhouette. Therefore at the location of the Flying Saucer (l=353.3, b=16.5) the ambient FUV field cannot be very large.
#### Apparent gap at radius ${185\,\mathrm{au}}$:
The presence of an apparent gap at $\sim$ 185 au both in CO and CS is puzzling. Its existence in CO however indicates that a change in temperature and/or beam dilution is the primary cause for this deficit, as CO is easily thermalized and optically thick. This suggests that the disk mid-plane warms up beyond this radius, before the emission fades again near the disk edge (of the order of $\sim$ 330 au), where CO becomes optically thin and CS unexcited.
An alternative explanation is that it is the result of real gap in the molecular distribution, smoothed out by our limited angular resolution. The optically thick CO line traces material both at low and high altitude (up to 3-4 scale heights above the mid-plane) while CS J=5-4, a high density tracer, is more optically thin and only observed in high density regions (at typically one scale height). Contrary to a face-on or inclined disk, the gap can be seen in $^{12}$CO because the disk is edge-on.
The brightness minimum at ${185\,\mathrm{au}}$ is only seen at low altitudes, typically between the mid-plane and an altitude of 40 au in CO, and slightly higher in CS, so the putative gap cannot extend up in the disk. This morphology remains consistent with expectations for gaps created by planets, provided the Hills radius is smaller than the disk scale height.
Deriving the gas density from CS excitation conditions
------------------------------------------------------
In CS J=5-4, we observe a similar North-South asymmetry than in CO J=2-1. This indicates that the emission has a substantial optical thickness along the line of sight.
The brightness ratio of CS over CO is within the range 0.5-0.7 until a radius of about 250 au and drops quickly beyond. As indicated by the iso-density contours shown in Figure \[fig:structure\], this is consistent with the decreasing H$_2$ density with radius, because the CS J=5-4 transition is thermalized at a few $10^6$ cm$^{-3}$ [@Denis-Alpizar+etal_2017].
On the contrary, the observed ratio of 0.5-0.7 in the inner disk mid-plane cannot be explained by excitation conditions. A simple LVG calculation using the CO freeze out temperature $T_k = 17$K (which is in agreement with the CS temperature derived from the simple analysis), a surface density of $4\,10^{13}$cm$^{-2}$ and a local line width of 0.3 kms$^{-1}$ (obtained from a simple analysis with DiskFit using a power law surface density distribution) yields $T_{ex}
= 12$K for a density of $10^6$ cm$^{-3}$. This **density** is much lower than expected in the disk given the dust emission observed at mm wavelengths, and than the values derived from our CO modelling.
The ratio is better explained as resulting from different beam dilutions in CO and CS. At 200 au, while the CO emission is spatially resolved, the region emitting in CS J=5-4 must fill only 50% of the synthesized beam, i.e. must have a thickness of only $\sim 30$au. The emission peak being located $\sim 45$ au above the plane at this radius, the CS emitting layer must be confined between about 30 and 60 au there. Furthermore, the density at $r=200$au and $z=60$au is about $10^6$ cm$^{-3}$ in our fiducial disk model, so that the upper layers are no longer dense enough to excite the J=5-4 line. Hence the observed CS/CO brightness ratio is roughly consistent with a molecular layer extending above one scale height, with sub-thermal excitation of the CS J=5-4 transition truncating the CS brightness distribution upwards.
In this interpretation, the CS layer is expected to be thinner for higher J transitions. An angular resolution around $0.2''$ would be needed to resolve the CS layer.
At smaller radii, the CO and CS layer become both unresolved vertically, but the ratio of their respective thickness is not expected to change significantly, leading to the nearly constant brightness ratio.
Limits: angular resolution, local line width and inclinations. {#sub:spatial}
--------------------------------------------------------------
Figure \[fig:resol\] shows the TRD of the model obtained with DiskFit assuming CO depletion around the mid-plane at 3 different angular resolutions, $0.5''$, $0.3''$ and $0.1''$. The impact of the resolution on this apparent brightness distribution is striking. **For comparison, the intrinsic temperature $T(r,z)$ distribution is shown in panel (g) for all points where the H$_2$ density exceeds $10^4$cm$^{-3}$. Comparing panels (f) and (g) shows the impact of the small Keplerian shear compared to the local line width at the disk edge (which spreads the TRD beyond the outer disk radius, see Section \[sub:pv\]), and of the small deviation from a pure edge-on disk (which result in a top/bottom asymmetry).** For this specific disk structure, the peak brightness is lowered by a factor 2 when degrading the angular resolution from $0.1''$ to $0.5''$. The inner CO disk, the radius of the CO snowline and the whole gas distribution can be resolved at $0.1''$, but not at $0.5''$. The stratification of the molecular layer can only be studied at $0.1''$ resolution, **down to about 30 au, where the scale height becomes too small compared to the linear resolution for a direct measurement**. We also observe a displacement of the peak brightness towards larger radii at $0.5''$ resolution compared to its (true) location at $0.1''$ resolution. This is due to the fact that the vertical extent of the CO emission at large radii is larger due to the flaring. Finally, at $0.5''$, we find that it is not possible to determine the shape of the (partly) depleted zone around the mid-plane beyond the CO snowline radius. Even a resolution of $0.3''$ would allow the measurement of the depletion factor (and estimate of the CO/dust ratio) around the mid-plane while an angular resolution of $0.1''$ would in addition provide the determination of the shape of this area.
**Besides angular resolution and local line width, inclination is another limitation of the method. To first order, the disk should be edge-on to within about $h(r)/r$ for the TRD to be directly useable, i.e. in the range $80-90^\circ$ given the typical $h(r)/r$ of disks at 100-300 au. However, even for somewhat lower inclination, the TRD can give insight onto the location of the molecular layer. We illustrate this in Appendix \[app:incli\].**
Such simulations clearly demonstrate how powerful ALMA can be to characterize the structure of an edge-on protoplanetary disk, provided its distance is reasonable.
Summary
=======
We report an analysis of the CO J=2-1 and CS J=5-4 ALMA maps of the Flying Saucer, a nearly edge-on protoplanetary disk orbiting a T Tauri star located in the $\rho$ Oph molecular cloud. At the angular resolution of $0.5''$ (60 au at 120 pc) and in spite of some confusion in CO due to the background molecular clouds, we find that:
- the disk is in Keplerian orbit around a 0.57 ${\,\mathrm{M}_\odot}$ star and nearly edge-on (inclined by $87^\circ$). It does not exhibit significant departure from symmetry, neither in CO nor in CS,
- direct evidence for a vertical temperature gradient is demonstrated by the CO emission pattern. Quantitative estimates are however limited by the spatial resolution.
- disentangling between CO depletion and very low temperatures is not possible because of the limited angular resolution. Models with CO depletion in the mid-plane only agrees marginally better, and the mid-plane temperature cannot be significantly larger than 10 K at 100 au.
- the CO emission is observed between 1 and 3 scale heights while the CS emission is located around one scale height. Sub-thermal excitation of CS may explain this apparent difference.
- CO is also observed beyond a radius of 230-260 au, in agreement with models predicting a secondary increase of temperature due to higher UV flux penetration in the outer disk. However, the limited angular resolution does not rule out an alternate explanation with a molecular gap near ${185\,\mathrm{au}}$.
Finally, our results demonstrate that observing an edge-on disk is a powerful method to directly sample the vertical structure of protoplanetary disks provided the angular resolution is high enough. At least one data set with angular resolution around $0.1-0.2''$ is needed for a source located at 120-150 pc.
We thank the referee for constructive comments. This work was supported by “Programme National de Physique Stellaire” (PNPS from INSU/CNRS.) This research made use of the SIMBAD database, operated at the CDS, Strasbourg, France. This paper makes use of the following ALMA data: ADS/JAO.ALMA\#2013.1.00387.S. ALMA is a partnership of ESO (representing its member states), NSF (USA), and NINS (Japan), together with NRC (Canada), NSC and ASIAA (Taiwan), and KASI (Republic of Korea) in cooperation with the Republic of Chile. The Joint ALMA Observatory is operated by ESO, AUI/NRAO, and NAOJ. This paper is based on observations carried out with the IRAM 30-m telescope. IRAM is supported by INSU/CNRS (France), MPG (Germany), and IGN (Spain). VW’s research is founded by the European Research Council (Starting Grant 3DICE, grant agreement 336474).
\[app:rv\]
PV diagram for an edge-on Keplerian disk
========================================
![Definition of notations.[]{data-label="fig:notations"}](notations-crop.pdf){width="6.0cm"}
![Radius as a function of Position and Velocity for an edge-on Keplerian disk[]{data-label="fig:pv-b2"}](pv-radius-crop.pdf){width="6.0cm"}
Let $x$ be the impact parameter in the disk, and $y$ the coordinate along the line of sight, $r$ the radial distance $$r(y) = \sqrt{x^2+y^2}$$ where $y < y_m$, with $y_m$ given by $$y_m = \sqrt{R_d^2-x^2}$$ The angle $\phi$ is defined such that $x = r \sin{\phi}$ (see Fig.\[fig:notations\]) The projected velocity along the line of sight is $$V_y = \sqrt{GM/r} \sin{\phi} = \sqrt{GM/r} \frac{x}{r} = \sqrt{GM} \frac{x}{r^{3/2}}
\label{eq:vy}$$ Thus we simply recover $r$ $$r = \left[ GM \left( \frac{x}{V_y} \right)^2 \right] ^{1/3} = R_d \left( \frac{X}{V} \right)^{2/3}$$ where $X= x/R_d$,$V=V_y/V_d$ and $V_d = \sqrt{GM/R_d}$ is the rotation velocity at the outer disk radius. Since $x \leq r \leq R_d$, the above equation has a solution provided $X < V < 1/\sqrt{X}$ (i.e. $V_y > V_d x/R_d$, and $V_y < \sqrt{GM/x}$). So, for any given velocity $V_y$ and impact parameter $x$, we can solve for $r$, and then recover along the line of sight. As a consequence, in the PV-diagram (showing functions of $(x,V)$) of a Keplerian disk, any line starting from ($x=0, V=V_\mathrm{sys}$) represent locii of constant radius. This is illustrated in Fig.\[fig:pv-b2\]. We use this property to directly recover the temperature as a function of radius (by taking the mean or the median for any given $r$) and altitude (by making cuts in the PV diagrams for different altitudes).
For optically thick lines, this procedure yields the (beam averaged) excitation temperature, and thus the kinetic temperature if the line is thermalized.
For optically thin lines, the opacity is a function of $(r,x)$ because of the Keplerian shear which breaks the rotational symmetry. The above procedure thus yields a more complex function of the temperature and density, whose value however cannot exceed the excitation temperature at any radius $r$.
Inclination effects {#app:incli}
===================
{width="18.0cm"}
**Fig.\[fig:incli\] shows the expected TRDs of our fiducial disk model (Table 2) for different disk inclinations. Since in this model, CO emits mostly 1 or 2 scale heights above the disk mid-plane, when the inclination differs from edge-on by more $h(r)/r$, the two opposite layers can project on the same side compared to the mid-plane projection. The farthest part of the disk projects to positive altitudes in Fig.\[fig:incli\]. It appears warmer than the projection of the nearest part, because for the same impact parameter in altitude, the line-of-sight intercepts first warm gas due to the disk flaring, while for the nearest part, the warm gas is hidden behind the forefront colder regions [see @Dartois+etal_2003]. The depletion in the mid-plane make a clear distinction between the two emitting cones in the disk, resulting in a bright double layer at positive altitudes for $i=70-75^\circ$. At lower inclinations, the projected velocity gradient becomes insufficient to clearly separate these two layers on the TRD. For $i=80-85^\circ$, the two layers no longer project on the same side: a small lukewarm “finger” of emission appears at an altitude around $-20$ au.**
|
\#1[(see Fig. \#1.)]{}
1.4cm
[**Local $E_{11}$**]{}
[Fabio Riccioni and Peter West]{}
[*Department of Mathematics\
King’s College London\
Strand London WC2R 2LS\
UK*]{}
1.5cm
addtoreset[equation]{}[section]{}
Introduction
============
It has been conjectured in [@1] that eleven dimensional supergravity could be extended so as to have a non-linearly realised infinite-dimensional Kac-Moody symmetry called $E_{11}$, whose Dynkin diagram is shown in Fig. \[fig1\]. In a non-linear realisation the algebra used to construct it is realised as a rigid symmetry. However, in the eleven dimensional supergravity theory all the symmetries are local. In this paper we will propose a non-linear realisation in which $E_{11}$ symmetries become local. To put this work in context it will be useful to list some of the main developments of the $E_{11}$ programme which are relevant for this paper.
Eleven dimensional supergravity itself can be formulated as a non-linear realisation based on an algebra that includes generators with non-trivial Lorentz character [@2]. To find the precise dynamics one takes the simultaneous non-linear realisation of this algebra with the conformal group. This naturally gives rise to both a 3-form and a 6-form fields and the resulting field equations are first order duality relations, whose divergence reproduces the 3-form second-order field equations of 11-dimensional supergravity provided on chooses one constant. The eleven-dimensional gravity field describes non-linearly $GL(11,\mathbb{R})$, which is a subalgebra of this algebra. Indeed, gravity in $D$ dimensions can be described as a non-linear realisation of the closure of the group $GL(D,\mathbb{R})$ with the conformal group [@2], as was originally shown in the four dimensional case in [@3].
$E_{11}$ first arose as the smallest Kac-Moody algebra which contains the algebra found in the non-linear realisation above. This $E_{11}$ algebra is infinite-dimensional, and the $E_{11}$ non-linear realisation contains an infinite number of fields with increasing number of indices. The first few fields are the graviton, a three form, a six form and a field which has the right spacetime indices to be interpreted as a dual graviton. This is the field content of eleven dimensional supergravity, and keeping only the first three of these fields one finds that the non-linear realisation of $E_{11}$ reduces to the construction discussed in the first point and so results in the dynamics of this theory [@1]. Theories in $D$ dimensions arise from the $E_{11}$ non-linear realisation by choosing a suitable $GL(D,\mathbb{R})$ subalgebra, which is associated with $D$-dimensional gravity. The $A_{D-1}$ Dynkin diagram of this subalgebra, called the gravity line, must include the node labelled 1 in the Dynkin diagram of Fig. \[fig1\]. In ten dimensions there are two possible ways of constructing this subalgebra, and the corresponding non-linear realisations give rise to two theories that contain the fields of the IIA and IIB supergravity theories and their electromagnetic duals [@1; @4]. Below ten dimensions, there is a unique choice for this subalgebra, and this corresponds to the fact that massless maximal supergravity theories in dimensions below ten are unique. Again, the non-linear realisation in each case describes, among an infinite set of other fields, the fields of the corresponding supergravity and their electromagnetic duals. In each dimension, the part of the $E_{11}$ Dynkin diagram which is not connected to the gravity line corresponds to the internal hidden symmetry of the $D$ dimensional theory. This not only reproduces all the hidden symmetries found long ago in the dimensionally reduced theories, but it also gives an eleven-dimensional origin to these symmetries.
All the maximal supergravity theories mentioned so far are massless in the sense that no other dimensional parameter other than the Planck scale is present. In fact, even this parameter can be absorbed into the fields such that it is absent from the equations of motion. There are however other theories that are also maximal, [*i.e.*]{} invariant under 32 supersymmetries, but are massive in the sense that they possess additional dimensionful parameters. These can be viewed as deformations of the massless maximal theories. However, unlike the massless maximal supergravity theories they can not in general be obtained by a process of dimensional reduction and in each dimension they have been determined by analysing the deformations that the corresponding massless maximal supergravity admits. With the exception of the one deformation allowed for type IIA supergravity in ten dimensions, called Roman’s theory, all the massive maximal supergravities possess a local gauge symmetry carried by vector fields that is a subgroup of the symmetry group $G$ of the corresponding maximal supergravity theory, and are therefore called gauged supergravities. In general these theories also have potentials for the scalars fields which contain the dimensionful parameters as well as a cosmological constant. In recent years there have been a number of systematic searches for gauged maximal supergravity theories and in particular in nine dimensions and in dimension from seven to three all such theories have been classified [@5; @6; @7].
It will be useful to recall how $E_{11}$ has from a very different perspective lead to the classification of gauged supergravities that agrees with these results and how the $E_{11}$ formulation of the gauged supergravity theories has lead to new work in these theories. The cosmological constant of ten-dimensional Romans IIA theory [@8] can be described as the dual of a 10-form field-strength [@9], and the supersymmetry algebra closes on the corresponding 9-form potential [@10]. The Romans theory was found to be a non-linear realisation [@11] which includes all form fields up to and including a 9-form with a corresponding set of generators. This 9-form is automatically encoded in the non-linear realisation of $E_{11}$ [@12]. From the eleven dimensional $E_{11}$ theory it arises as the dimensional reduction of the eleven-dimensional field $A_{a_1 \dots a_{10} , (bc)}$ in the irreducible representation of $GL(11,\mathbb{R})$ with ten antisymmetric indices $a_1 \dots a_{10}$ and two symmetric indices $b$ and $c$. Therefore $E_{11}$ not only contains Romans IIA, but it also provides it for the first time with an eleven-dimensional origin [@13].
By studying the eleven-dimensional fields of the $E_{11}$ non-linear realisation, one can determine all the forms, [*i.e.*]{} fields with completely antisymmetric indices, that arise from dimensional reduction to any dimension [@14]. In particular, in addition to all the lower rank forms, this analysis gives all the $D-1$-forms and the $D$-forms in $D$ dimensions. The list of all form fields obtained in this way for all supergravity theories is given in table \[Table1\]. The $D-1$ and $D$-forms predicted by $E_{11}$ can also be derived in each dimension separately [@15]. The $D-1$-forms have $D$-form field strengths, that are related by duality to the mass deformations of gauged maximal supergravities, and the $E_{11}$ analysis shows perfect agreement with the complete classification of gauged supergravities performed in [@6; @7]. Therefore $E_{11}$ not only contains all the possible massive deformations of maximal supergravities in a unified framework, but it also provides an eleven-dimensional origin to all of them. Indeed, while some gauged supergravities were known to be obtainable using dimensional reduction of ten or eleven dimensional supergravities, this was not generically the case. As a result the gauged supergravities were outside the framework of M-theory as it is usually understood.
One striking feature of the $E_{11}$ formulation of massless or massive supergravity theories is that it includes fields together with all their dual fields. The presence of the dual forms is essential to formulate the field equations as duality relations. Some dual forms have been introduced in the past in an ad-hoc way beginning with [@16], but it is only with $E_{11}$ that they have arisen from an underlying principle. Indeed, the forms of table \[Table1\] were proposed in [@14; @15] to play a crucial role in gauged supergravities, the $D-1$ forms classifying the gauged supergravities and the lower forms providing a chain of form fields that occur in the duality relations. This is compatible with the structure of the gauge algebra arising in gauged supergravities, in which one is forced to introduce a $p+1$ form to close the gauge algebra of a $p$ form, thus determining a hierarchy of forms [@17]. For the cases in which this latter method has been subsequently used to compute the hierarchy of forms, the results are precisely in agreement with $E_{11}$ [@18], and indeed the presence of the forms given in table \[Table1\] has now been systematically adopted by those studying gauged supergravities.
[|c|c||c|c|c|c|c|c|c|c|c|c|]{}
------------------------------------------------------------------------
D & G & 1-forms & 2-forms & 3-forms & 4-forms & 5-forms & 6-forms & 7-forms & 8-forms & 9-forms & 10-forms\
------------------------------------------------------------------------
& & & & & & & & & & & ${\bf 1}$\
& & & & & & & & & & & ${\bf 1}$\
------------------------------------------------------------------------
& & & & & & & & & & & ${\bf 4}$\
& & & & & & & & & & & ${\bf 2}$\
------------------------------------------------------------------------
& & ${\bf 2}$ & & & & & ${\bf 2}$ & ${\bf 3}$ & ${\bf 3}$ & ${\bf 4}$\
& & & & & & & & & & ${\bf 2}$\
& & ${\bf 1}$ & & & & & ${\bf 1}$ & ${\bf 1}$ & ${\bf 2}$ & ${\bf 2 }$\
------------------------------------------------------------------------
& & & & & & & & & ${\bf (15,1)}$\
& & & & & & & ${\bf (8,1)}$ & ${\bf (6,2)}$ & ${\bf (3,3)}$\
& & & & & & & ${\bf (1,3)}$ & ${\bf (\overline{3},2)}$ & ${\bf (3,1)}$\
& & & & & & & & & ${\bf (3,1)}$\
------------------------------------------------------------------------
& & & & & & & ${\bf \overline{40}}$ & ${\bf 70}$\
& & & & & & & & ${\bf 45}$\
& & & & & & & ${\bf \overline{15}}$ & ${\bf 5}$\
------------------------------------------------------------------------
& & & & & & & ${\bf 320}$\
& & & & & & & ${\bf \overline{126}}$\
& & & & & & & ${\bf 10}$\
------------------------------------------------------------------------
& & & & & & ${\bf \overline{1728}}$\
& & & & & & ${\bf \overline{27}}$\
------------------------------------------------------------------------
& & & & & ${\bf 8645}$\
& & & & & ${\bf 133}$\
------------------------------------------------------------------------
& & & ${\bf 3875}$ & [**147250**]{}\
& & & & [**3875**]{}\
& & & ${\bf 1}$ & [**248**]{}\
All in all there is considerable evidence for an $E_{11}$ symmetry in the low energy limit of what is often called M theory. The above evidence concerns the adjoint representation of $E_{11}$, or the part of the non-linear realisation that involves the fields associated with the $E_{11}$ generators. However, there is also the question of how space-time is encoded in the theory. In the non-linear realisations mentioned above the generator of space-time translations $P_a$ was introduced by hand in order to encode the coordinates of space-time. From the beginning it was understood that this was an ad-hoc step that did not respect the $E_{11}$ symmetry. It was subsequently proposed [@19] that one could include an $E_{11}$ multiplet of generators which had as its lowest component the generator of space-time translations. This is just the fundamental representation of $E_{11}$ associated with the node labelled 1 in the Dynkin diagram of Fig. \[fig1\] and it is denoted by $l$. A method of constructing the gauged supergravities was given in reference [@20] using $E_{11}$ and the $l$ multiplet of generators. Indeed as an example all the gauged supergravity generators in five dimensions were derived from this viewpoint. This reference also contains a review of the evidence for the $l$ multiplet as the multiplet of brane charges and a table of its low level content in dimensions three and above. In this context there has been a recent interesting paper [@21] which keeps the scalar charges in the $l$ multiplet for the seven-dimensional maximal supergravity and still finds diffeomorphism invariance in seven dimensions.
What was not clear from this method was how the global $E_{11}$ symmetries would become local and this is the subject of this paper. In the context of purely gravity this was achieved long ago in reference [@3] by taking the simultaneous non-linear realisation of $IGL(4,\mathbb{R} )$ with the conformal group in four dimensions. As mentioned above, if one took the non-linear realisation of $E_{11}$ at low levels, that is to include the six form generator, took only the Lorentz group as the local subgroup and the simultaneous non-linear realisation with the conformal group, then the dynamics predicted by the non-linear realisation is just the maximal supergravity theory in eleven dimensions. This can be seen by realising that the low level $E_{11}$ algebra [@1] is just that used in reference [@2] to construct the eleven dimensional supergravity theory as a non-linear realisation once one includes the conformal group. The effect of latter is that it makes not only the space-time translations a local symmetry but also turns the shifts associated with form fields into gauge transformations [@2]. However, it is not clear how to combine the conformal group with the algebra formed from $E_{11}$ and the $l$ multiplet. In particular how to extend the action of the conformal group on the usual coordinates of space-time to include the other coordinates encoded in the $l$ multiplet.
In this paper we will not use the conformal group, but rather add the generators that the closure of this algebra with $E_{11}$ would generate. We will also not use the generators from the $l$ multiplet, but only the space-time translations $P_a$. The prototype example of this mechanism was given long ago for the case of Yang-Mills theory [@22]. Essentially one takes an algebra that contains the generators $P_a$ and the Yang Mills generators $Q^\alpha$, as well as the generators $R^{a,\alpha}$ for which the gauge fields are Goldstone bosons and an infinite number of generators $K^{a_1\ldots a_n,\alpha}$, symmetric in their spacetime indices, which do not commute with $P_a$ and whose role is to make the rigid symmetry generated by $R^{a,\alpha}$ local. We will review this construction later on in this introduction.
We will first show the analogous mechanism for pure gravity. In particular, we will show how to construct Einstein’s theory of gravity using a non-linear realisation which takes as its underlying algebra one that consists of $IGL(D,\mathbb{R} )$ and an infinite set of additional generators whose effect will to promote the rigid $IGL(D,\mathbb{R})$ to be local. The generators $P_a$ lead in the non-linear realisation to the coordinates of space-time while the Goldstone boson for $GL(D, \mathbb{R})$ is the vierbein which is subject to local Lorentz transformations. The infinite number of additional generators lead to local translations, that is general coordinate transformations, but to no new fields in the final theory as their Goldstone fields are solved in terms of the graviton field using a set of invariant constraints placed on the Cartan forms. This is an example of what has been called the inverse Higgs effect [@23]. The unique theory resulting from this non-linear realisation with only two space-time derivatives is Einstein’s theory up to a possible cosmological term. In this case one can see that the additional generators we have added are just those found by taking the closure of $IGL(D,\mathbb{R})$ with the conformal group.
We will then generalise this procedure to $E_{11}$ at low levels. We take the algebra, called $E_{11}^{local}$ consisting of non-negative level $E_{11}$ generators, the generators $P_a$ and an infinite number of additional generators. While the latter lead in the final result to no new Goldstone fields they do result in all the low level $E_{11}$ symmetries becoming local, thus we find general coordinate transformations and gauge transformations for all the form fields. For the eleven dimensional theory, space-time arises in the group element due to the $P_a$ generators, however, for lower dimensional theories we will take space-time to be not only the translation operator $P_a$ for that dimension but also certain other Lorentz scalar charges that include the translation operators for the dimensionally reduced generators, in effect we take only the Lorentz scalar part of the $l$ multiplet. As we add just the spacetime translations rather than the whole $l$ multiplet we will take $P_a$ to commute with the non-negative level generators of $E_{11}$. The price for proceeding in this way is that we are working with only the non-negative level generators of $E_{11}$ and we have essentially thrown out the negative level generators. We show that the non-linear realisation of the algebra $E_{11}^{local}$ describes at low levels in eleven dimensions the 3-form and the 6-form of the eleven dimensional supergravity theory with all their gauge symmetries. This can be thought of as equivalent to taking the non-linear realisation of $E_{11}$ at low levels and taking the simultaneous non-linear realisation with the conformal group as was discussed earlier [@2; @1], but here the procedure is more transparent.
We then consider the formulation of lower dimensional maximal gauged supergravity theories from the viewpoint of the enlarged algebra $E_{11,D}^{local}$. The $D$ refers to the fact that although we take the same non-negative level $E_{11}$ generators and generators $P_a$, the infinite number of additional generators we take vary from dimension to dimension. We first consider as a toy model the Scherk-Schwarz dimensional reduction of the IIB supergravity theory from this viewpoint. We begin with an algebra consisting of $E_{11,10B}^{local}$ and take the ten dimensional space-time to arise from an operator $\tilde Q$ which is constructed from $Q=P_{9}$ and part of the $SL(2,\mathbb{R})$ symmetry of the theory. This means that the 10th direction of space-time is twisted to contain a part in the $SL(2,\mathbb{R})$ coset symmetry of the theory. This non-linear realisation gives a nine dimensional gauged supergravity. We observe that not all of the algebra $E_{11,10B}^{local}$ is essential for the construction of the gauged supergravity in nine dimensions, but only an algebra which we call $\tilde E_{11, 9}^{local}$ which is the subalgebra of $E_{11,10B}^{local}$ that commutes with $\tilde Q$. Its generators are non-trivial combinations of $E_{11}$ generators and the additional generators and in general the generators of $\tilde
E_{11, 9}^{local}$ have non-trivial commutation relations with nine dimensional space-time translations. Although the subalgebra $\tilde E_{11, 9}^{local}$ appears to be a deformation of the original $E_{11}$ algebra and the space-time translations we have not changed the original commutators, but rather the new algebra arises due to the presence of the additional generators which are added to the $E_{11}$ generators.
However, we then show that one can find the algebra $\tilde E_{11,
9}^{local}$ without carrying out all the above steps. Given the non-trivial relation between the lowest non-trivial positive level generator of $\tilde E_{11,9}^{local}$ and the nine dimensional space-time translations one can derive the rest of the algebra $\tilde E_{11, 9}^{local}$ simply using Jacobi identities. This algebra determines uniquely all the field strengths of the theory, and thus one finds a very quick way of deriving the gauged supergravity theory.
This picture applies to all gauged supergravity theories, as one can easily find the algebra $\tilde E_{11, D}^{local}$ without using its derivation from $E_{11}^{local}$ and this provides a very efficient method of constructing all gauged supergravities. We illustrate how this works by constructing the massive IIA theory as well as all the gauged maximal supergravities in five dimensions.
Finally, we consider how this construction generalises to the fields with mixed symmetry, [*i.e.*]{} not completely antisymmetric, of $E_{11}$ and in general of any non-linear realisation of a very-extended Kac-Moody algebra. We will consider as a prototype of such fields the dual graviton in four dimensions, which is a field $A_{ab}$ symmetric in its two spacetime indices. We will show that if one tries to promote the global shift symmetry of the dual graviton field to a gauge symmetry, one finds that this is not compatible with the $E_{11}$ algebra. The solution of this problem is that actually $E_{11}$ forces to include additional generators, whose role is to enlarge the gauge symmetry of the dual graviton so that one can gauge away the field completely. We show this first for the simpler case of the non-linear realisation of the Kac-Moody algebra $A_1^{+++}$ in four dimensions. We then consider the case of $E_{11}$ in four dimensions. For simplicity in this case we neglect the gravity generators, and we still find that even considering consistency conditions involving only the generators associated to the form fields and those associated to the dual graviton, one is forced to include additional generators for the dual graviton that generate a local symmetry that gauges away the dual graviton completely. We claim that this picture generalises to all mixed symmetry fields in any dimension. It is important to stress that the dynamics is compatible with this result. Indeed, while the field strengths of the antisymmetric fields are first order in derivatives, and therefore one needs fields and dual fields to construct duality relations which are first order equations for these fields, the gravity Riemann tensor is at second order in derivatives and thus there is no need of a dual field to construct its equation of motion.
It will be helpful to recall some facts about non-linear realisations. A non-linear realisation of a group $G$ with respect to a subgroup $H$ is by definition a theory invariant under the two separate transformations $$g (x) \to g_0g(x) ,\ \ \ g (x) \to g(x) h(x) \label{1.1}$$ where $g\in G$, $g_0\in G$ while $h\in H$. The dependence on the generic symbol $x$ signifies which group elements dependence on the coordinates of the space-time. For the case of an internal symmetry the space-time dependence is incorporated by hand. However, in this paper space-time will arise naturally in that its associated generators are part of the Lie algebra of the group $G$. Indeed, a part of the group element is just space-time viewed as a coset. We note that the $h$ transformations depend on the space-time coordinates so can be said to be local, unlike the rigid $g_0$ transformations. Working with the most general group element $g$ we must then find a theory that is invariant under both $g_0$ and local $h$ transformations.
It is often more transparent to use the $h$ transformations to choose $g(x)$ to be of a particular form, that is choose coset representatives. If one does this then when making a rigid $g_0$ transformation one finds a group element $ g_0 g$ which is in general not one of the chosen coset representatives. To rectify this one must make a compensating $h_c$ that depends on $g_0$ and the original coset representative $g(x)$. That is $g\to g'=g_0gh_c^{-1}$ where both $g$ and $g'$ are chosen coset representatives.
The problem of finding the invariant dynamics is most often solved by using the Cartan forms ${\cal V}=g^{-1}dg$. This is obviously invariant under rigid $g_0$ transformations and transforms as $${\cal V}\to h^{-1}{\cal V} h+h^{-1}dh
\label{1.2}$$ under local $h$ transformations. We note that $g^{-1}dg= dx\cdot
g^{-1}\partial g$ is invariant but $g^{-1}\partial g$ is not as the coordinates of space-time $x$ transform under $g_0$ transformations. To be more explicit we consider a group that contains the generators $L_N$ and we denote the remaining generators by the generic symbol $T^*$. We will assume that the generators $L_N$ from a representation of the $T^*$’s. The general group element is of the form $$g=e^{x\cdot L}e^{\phi(x)\cdot T} \quad . \label{1.3}$$ We recognise $x$ as the coordinates and $\phi$ as the fields. The local subgroup can be used to set some of the fields $\phi$ to zero. The discussion below holds if one makes this choice or work with the general group element. The Cartan forms can be written as $${\cal V}= g^{-1}d g= dx^\Pi E_\Pi{}^N L_N+ dx^\Pi G_{\Pi,*} T^*
\quad . \label{1.4}$$ Since $ {\cal V}$ is invariant under $g\to g_0g$ it follows that each of the coefficients of the above generators is invariant, that is $dx^\Pi E_\Pi{}^N$ and $dx^\Pi G_{\Pi, *}$ are invariant. However, $dx^\Pi$ does transform under $g_0$ and so $E_\Pi{}^N$ and $G_{\Pi, *}$ are not invariant. To find quantities that only transform under the local subalgebra we can rewrite $ {\cal V}$ as $${\cal V}= g^{-1}d g= dx^\Pi E_\Pi{}^N( L_N+ G_{N,*} T^*) \quad ,
\label{1.5}$$ where we recognise that $G_{N,*}=(E^{-1})_N{}^\Pi G_{\Pi,*}$. It follows that $G_{N,*}$ are inert under $g_0$ transformations and just transform under local transformations. As such they are useful quantities with which to construct the dynamics as one must now only solve the problem of finding objects which are invariant under the local symmetry. We may think of $G_{N,*}$ as covariant derivatives of the fields $\phi$.
There is one subtle point that is sometimes worth remembering if one chooses coset representatives. Although $G_{N,*}$ is naively invariant under $g_0$ transformations it is not invariant under the required compensating $h_c$ transformation under which $G_{N,*}$ transforms as in eq. (\[1.2\]) with $h$ replaced by $h_c^{-1}$. However, having found a set of dynamics that is invariant under $h$ transformations it is of course also invariant under the compensating transformations.
Realising Yang-Mills theory as a non-linear realisation was first given by Ivanov and Ogievetsky [@22] and we now summarise this approach as it will serve as a prototype model for the later sections of this paper. We begin with the algebra $$P_a \ , \quad J_{ab} \ , \quad Q^\alpha \ , \quad
R^{a, \alpha} \ , \quad K^{a_1 a_2 , \alpha} \ ,\quad K^{a_1 a_2 a_3 ,\alpha} \
, \ ... \ K^{a_1 ... a_n ,\alpha} \
\ldots \label{1.6}$$ which will generate the group $G$ of the non-linear realisation. The generators $P_a$ and $J_{ab}$ are those of the Poincare group while the $Q^{\alpha}$’s will become identified with those of the gauge group. The generator $R^{a, \alpha}$ is the generator associated to the gauge vector in the non-linear realisation, while the generators $K^{a_1 ... a_n ,\alpha}$ are symmetric in the spacetime indices and will be responsible for the symmetry of the vectors to be promoted to a gauge symmetry. The $Q^\alpha$ generators obey the commutators $$[Q^{\alpha}, Q^{\beta}]=g f^{\alpha\beta}{}_\gamma Q^\gamma \quad
, \label{1.7}$$ where $g$ is the coupling constant. The remaining commutation relations are given by $$[K^{a_1\ldots a_n, \alpha}, P_b]=n \delta_b^{(a_1} K^{a_2\ldots a_n),
\alpha} \ ,\ \ \ [K^{a_1\ldots a_n, \alpha}, K^{b_1\ldots b_m,
\beta}] = g f^{\alpha\beta}{}_\gamma K^{a_1\ldots a_n b_1\ldots b_m ,
\gamma} \quad . \label{1.8}$$ Although the $K^{a_1 \dots a_n, \alpha}$ generators have at least two indices, the commutation relations of $Q^\alpha$ and $R^{a ,
\alpha}$ with all the generators are encoded in the equation above making the identification $K^{a ,\alpha} = R^{a,\alpha}$ and $K^{\alpha} = Q^\alpha$. The Lorentz generators $J_{ab}$ have the usual commutators with the above generators. The local sub-group $H$ is generated by the $Q^\alpha$ and the $J_{ab}$. As a result we may choose the group element to be of the form $$g=e^{x^a P_a} \dots e^{\Phi_{a_1 a_2 a_3,\alpha}(x) K^{a_1 a_2 a_3
,\alpha}} e^{\Phi_{a_1 a_2,\alpha}(x) K^{a_1
a_2,\alpha}} e^{A_{a,\alpha}(x) R^{a,\alpha}} \quad .
\label{1.9}$$
Computing the Cartan forms we find that $$\begin{aligned}
g^{-1} d g & =& d x^a [ P_a + G_{a,b ,\alpha} R^{b ,\alpha} +
G_{a,bc , \alpha} K^{bc , \alpha} - A_{a ,\alpha} Q^\alpha + ... ]
\nonumber \\
& =& d x^a [ P_a + ( \partial_a A_{b , \alpha} - {1
\over 2} g A_{a , \beta} A_{b , \gamma} f^{\beta \gamma}{}_\alpha -
2 \Phi_{ab , \alpha} ) R^{b , \alpha} \nonumber \\
& + & ( \partial_a \Phi_{bc , \alpha} - {1 \over 6} g^2 A_{a ,
\epsilon} A_{b , \beta} A_{c, \gamma} f^{\epsilon \beta}{}_\delta
f^{\delta \gamma}{}_\alpha - 2 g\Phi_{ab ,\beta} A_{c , \gamma}
f^{\beta \gamma}{}_\alpha \nonumber \\
& +& {1 \over 2} g \partial_a A_{b , \beta} A_{c , \gamma} f^{\beta
\gamma}{}_\alpha -3 \Phi_{abc , \alpha} ) K^{bc ,\alpha }- A_{a
,\alpha} Q^\alpha + ... ] \quad , \label{1.10}
\end{aligned}$$ where the dots denote $K$ generators with more than two spacetime indices. Only the last term in eq. (\[1.10\]) is in the local sub-algebra and as such we can identify $A_{a , \alpha}$ as the connection, [ *i.e.*]{} the gauge field, for the gauge group generated by $Q^{\alpha}$. Each of the other terms separately transform covariantly under the local subgroup and so we can place constraints on them and still preserve all the symmetries. In particular we can set $$G_{(a,b), \alpha}=0 \quad , \label{1.11}$$ which implies $$2\Phi_{ab , \alpha}=\partial_{(a} A_{b) , \alpha} \quad ,
\label{1.12}$$ and also $$G_{(a,bc)\alpha}=0\quad , \label{1.13}$$ which implies $$3\Phi_{abc, \alpha}=\partial_{(a} \Phi_{bc), \alpha} -2 g \Phi_{(ab, \beta}
A _{c),\gamma} f^{\beta\gamma}{}_\alpha + {1 \over 2} g
\partial_{(a} A_{b , \beta} A_{c),\gamma} f^{\beta \gamma}{}_\alpha
\quad . \label{1.14}$$ Indeed one can solve in this way for all the $\Phi$ fields leaving only with the field $A_{a, \alpha}$. The elimination of some fields using constraints on the Cartan forms that preserve the symmetries is sometimes called the inverse Higgs mechanism [@23].
Substituting the above solutions for the $\Phi$ fields into the Cartan forms one finds expressions that contain $A_{a, \alpha}$ alone which are given by $$g^{-1}d g= dx^a [ P_a + F_{ab , \alpha} R^{b , \alpha} +{2\over 3}
D_b F_{ac, \alpha} K^{bc, \alpha} + ... - A_{a , \alpha}Q^\alpha ]
\quad , \label{1.15}$$ where $F_{ab,\alpha}= \partial _{[a}A_{b]\alpha} - {1\over 2} g
A_{a,\beta} A_{b,\gamma} f^{\beta\gamma}{}_\alpha$ and $D_a$ is the expected covariant derivative. We recognise this as the Yang-Mills field strength and the higher Cartan form as its covariant derivatives. The object invariant under the symmetries of the non-linear realisation, which is lowest order in derivatives, is just the usual Yang-Mills action.
In fact only the lowest order Cartan form $G_{a,b,\alpha}$ was evaluated in reference [@22], but it is interesting to realise that the Cartan forms do contain all the gauge covariant derivatives of the field strength.
One way to arrive at the above set of generators of eq. (\[1.6\]) is to write the Yang-Mills gauge parameter as a Taylor expansion $$\lambda_\alpha(x)= a_\alpha +
a_{a, \alpha}x^a+a_{ab , \alpha}x^ax^b+\ldots
\label{1.16}$$ where the parameters $a$ do not depend on space-time. The usual Yang-Mills transformation can then be interpreted as a an infinite set of rigid transformations whose generators are just those of eq. (\[1.6\]) with the commutation relations of eqs. (\[1.7\]) and (\[1.8\]). Indeed carrying rigid transformations $e^{a\cdot R}$ and $e^{a \cdot K}$ on the group element of eq. (\[1.9\]) one finds the same result that a Yang-Mills transformation would produce if the gauge parameter were expanded as in eq. (\[1.16\]).
In a tribute to Ogievetsky’s important contributions to the theory of non-linear realisations we will call the additional generators $K^{a_1 \ldots a_n, \alpha}$ Ogievetsky generators (Og for short) and similarly for their associated fields. They will be used throughout this paper and they are the generators that make the original symmetry, in this case that of the $R^{a,\alpha}$, local. We can systematically assign a grade to the generators, in particular $Q^\alpha$ and $P_a$ have grade -1, $R^{a,\alpha}$ has grade 0 and $K^{a_1\ldots a_{n+1}, \alpha}$ have grade $n$. The coupling constant $g$ has grade -1. We denote the Og generator of grade $n$ as Og $n$. The algebra of eq. (\[1.8\]) can then schematically be written as $$[G, {\rm Og} \ n ]= g \ {\rm Og} \ n \quad [{ \rm Og}\ n ,P_a ]={\rm
Og} \ (n-1) \quad [{\rm Og}\ n ,{\rm Og}\ m ]= g \ {\rm Og}\ (m+n+1) \
. \label{1.17}$$
It will be instructive to consider the dimensional reduction of the above non-linear realisation in $D$ dimensions on a circle with coordinate $y$. For simplicity we will just consider the abelian case here, and we will therefore drop the index $\alpha$. After dimensional reduction, the vector field becomes $A_a,
A_\star=\varphi$, while the Og 1 field becomes $\Phi_{ab},
\Phi_{a\star}, \Phi_{\star\star}$ and similarly for the higher grade Og fields. Here $\star$ denotes the $y$th, [*i.e.*]{} circle, components and $a,b=0,\ldots , D-2$. Neglecting for simplicity the contribution along the Og 1 generator, the Cartan form of eq. (\[1.10\]) becomes $$\begin{aligned}
g^{-1}d g & = & dx^a P_a + dy P_\star + dx^a (\partial_a
A_{b}-2\phi_{ab} )R^{b} + dx^a(\partial_a \varphi -2\Phi_{a \star
})R^{\star} \nonumber \\
& + & dy(\partial_\star A_{a}-2\Phi_ {\star a})R^{a} + dy
(\partial_\star \varphi -2\Phi_{\star \star})R^{\star}
-dx^a A_{a} Q-dy \varphi Q \ . \label{1.18}
\end{aligned}$$ We now take all the fields to be independent of $y$. Imposing that the Cartan form in the $dy$ direction vanishes, apart from the term in the local subalgebra, we find that $\Phi_{\star
a}=\Phi_{\star\star}=0$. This generalises to all the Og fields of any grade having at least one index in the internal direction. Solving for the remaining Cartan forms as above one finds that one is then left with the fields $A_{a}$ and $\varphi$ with the expected dynamics. The net effect of these steps is that from the original set of generators in the higher dimension we take only those that commute with $Q$, the generator of $y$ transformations, and construct the non-linear realisation from the sub-algebra formed by these generators. Since the $\Phi$ fields are related to the derivatives of the usual fields it is to be expected that some of the Ogievetsky fields will vanish in dimensional reduction on a circle.
The non-linear realisation of the Yang-Mills theory will be the prototype example of all the analysis that we will perform throughout this paper. The paper is organised as follows. Section 2 discusses the non-linear realisation of gravity, while section 3 is devoted to the analysis of the 3-form and the 6-form of eleven-dimensional supergravity from $E_{11}$. In section 4 we show how to derive from $E_{11}$ the Scherk-Schwarz reduction of the IIB theory to nine dimensions. Sections 5 and 6 are devoted to the $E_{11}$ derivation of the massive IIA theory of Romans and of gauged five-dimensional maximal supergravities respectively. In section 7 we discuss the dual graviton in four dimensions, considering first the algebra of the dual graviton alone, and then the cases of gravity and dual gravity in $A_1^{+++}$ in four dimensions and of dual graviton coupled to vectors in $E_{11}$ in four dimensions. Finally, section 8 contains the conclusions.
Gravity as a non-linear realisation
===================================
It was shown long ago by Borisov and Ogievetsky that four-dimensional gravity could be formulated as a non-linear realisation [@3]. These authors showed that gravity in four dimensions could be formulated as the non-linear realisation of $IGL(4,\mathbb{R})$ with local subgroup $SO(4)$ if taken together with the simultaneous realisation of the four dimensional conformal group $SO(2,4)$ with local subgroup $SO(4)$. The first non-linear realisation possesses coset representatives $g=e^{x\cdot P}
e^{h\cdot K}$ that contain the coordinates of spacetime $x^\mu$ as coefficients of the spacetime translation generator $P_a$ and the field $h_a{}^b$, which was taken to depend on $x^\mu$, and are associated with the generators $K^a{}_b$ of $GL(4,\mathbb{R})$. The non-linear realisation of the conformal group has coset representatives $g=e^{x\cdot P} e^{\phi D} e^{\phi_a K^a}$ that are labelled by the coordinates of space-time $x^\mu$ and the fields $\phi$ and $\phi_a$ associated with the dilation generator $D$ and special conformal generator $K^a$. The field $\phi_a$ can be eliminated using the inverse Higgs mechanism, that is by setting constraints on the Cartan forms that preserve all the symmetries. The simultaneous non-linear realisation of the two groups is achieved by constructing the dynamics from only the Cartan forms of $IGL(4,\mathbb{R})$ which also transform covariantly under the conformal group. The transformations of the two groups are linked in that the dilation generator $D$ and the trace of the $GL(4,\mathbb{R})$ generators $K^a{}_a$ generate the same scaling of the coordinates $x^\mu$ and so their corresponding Goldstone fields $\phi$ and $h^a{}_a$ must be identified with an appropriate proportionality constant. Although a little complicated the result of this procedure is Einstein’s theory if one restricts one’s attention to terms that are second order in spacetime derivatives. Taking only the non-linear realisation of $IGL(4,\mathbb{R})$ one can also find Einstein’s theory from the Cartan forms provided one fixes a number of coefficients in a way not determined by the symmetries of $IGL(4,\mathbb{R})$ alone. The results can be generalised to $D$ dimensions [@2]. However, this latter reference did not use the Lorentz group to make a particular choice of coset representative and introduces a vierbein rather than a metric.
The derivation of gravity as a non-linear realisation was anticipated by an earlier paper of Ogievetsky’s [@24] that showed that the closure of $IGL(4,\mathbb{R})$ and the conformal group as realised on the coordinates of space-time $x^\mu$ in the well known way is equivalent to just considering all infinitesimal general coordinates transformations $x^\mu\to x^\mu +f^\mu (x)$ where $f^\mu (x)$ is an arbitrary function of $x^\mu$. Thus the closure of the two groups is an infinite dimensional group that is just the group of general coordinate transformations. We note that the starting point [*i.e.*]{} the well known transformations on $x^
\mu$ are just those found by taking space-time to be a coset or equivalently a non-linear realisation in which the fields are absent.
As such an equivalent more straightforward approach would be take the non-linear realisation of the infinite group which is the closure of the two groups, that is the algebra of general coordinate transformations. This calculation is the subject of this section. Such an approach was adopted by Pashnev [@25], however, although we will begin from the same starting point our method will depart in some important ways, some of which are discussed in [@kirsch; @boulanger], that are explained below.
Let us begin with the infinite dimensional algebra that contains the generators $$P_a, K^a{}_b, K^{ab}{}_c,\ldots , K^{a_1\ldots a_n}{}_c,\ldots
\label{2.1}$$ where $K^{a_1\ldots a_n}{}_c=K^{(a_1\ldots a_n )}{}_c$. These generators obey the relations $$[ K^{a_1\ldots a_n}{}_c, P_b]= (n-1) \delta_b^{(a_1}K^{a_2\ldots
a_n)}{}_c \label{2.2}$$ and $$[K^{a_1\ldots a_n}{}_c, K^{b_1\ldots b_m}{}_d]= (n+m-1 )[{1 \over n}
\delta_c^{(b_1|}
K^{a_1\ldots a_n|b_2\ldots b_m)}{}_d - {1 \over m}
\delta_d^{(a_1}K^{a_2\ldots
a_n)b_1\ldots b_m}{}{}_c ] \ . \label{2.3}$$ The generators $P_a, K^a{}_b$ are those of $IGL(D,\mathbb{R})$ while the special conformal transformations are contained in $K^{ab}{}_c$. Indeed the entire algebra can be generated by $P_a, K^a{}_b$ and $
K^{ab}{}_c$. We note that one can assign grade to the generators; $K^{a_1\ldots a_{n+1}}{}_c$ has grade $n$, $P_a$ has grade $-1$ and $K^a{}_b$ has grade zero. This notion of grade is preserved by the above commutation relations. In terms of our previous notation we call the additional generators Ogievetsky, or Og, generators. In particular, $K^{a_1\ldots a_{n+1}} {}_c$ is an Og $n$ generator. The commutators of eqs. (\[2.2\]) and (\[2.3\]) can thus schematically be written as $$[ {\rm Og} \ n, {\rm Og } \ m ] = {\rm Og } \ (n+m) \quad ,
\label{2.4}$$ which includes all possible commutators provided that we denote with Og (-1) the momentum operator and with Og 0 the $GL(D,\mathbb{R})$ generators.
We now carry out the non-linear realisation of the group based on the algebra of eqs. (\[2.2\]) and (\[2.3\]) taking as our local subgroup the Lorentz group which has the generators $J_{ab}= \eta_{[
a \vert c \vert }K^{c}{}_{b]}$. As such we may choose our group element, or coset representative to be given by $$g=e^{x^aP_a}\ldots
e^{\Phi_{a_1\ldots a_n}^b (x) K^{a_1\ldots a_n}{}_b}\ldots e^{\Phi _{a_1 a_2}^b (x)
K^{a_1 a_2}{}_b} e^{
h_a{}^b (x) K^a{}_b}\equiv e^{x^aP_a}g_\phi g_h \quad .\label{2.5}$$ In fact this is the most general group element as we have not used the Lorentz group to make any choice. The Cartan forms are given by $$\begin{aligned}
g^{-1}d g& =& g_h^{-1} g_\phi^{-1}dx^a P_a g_\phi g_h +
g_h^{-1}(g_\phi^{-1}d g_\phi) g_h +g_h^{-1}d g_h
\nonumber \\
& =& dx^\mu (e_\mu{}^a P_a+ G_{\mu , b}{}^c K^b{}_c +G_{\mu ,a b}{}^c
K^{ab}{}_c+\ldots ) \quad .\label{2.6}
\end{aligned}$$ A straightforward calculation gives $$e_\mu{}^a= (e^h)_\mu{}^a,\ \ G_{\mu , b}{}^c = (e^{-1}\partial _\mu
e)_b{}^c -\Phi _{\mu \rho}{}^\kappa (e^{-1})_b{}^\rho
e_\kappa{}^c,\ \ \label{2.7}$$ $$G_{\mu , a b}{}^c= (\partial_\mu \Phi _{\rho\kappa}^\lambda -2
\Phi_{\mu\rho\kappa}^\lambda - \Phi_{\mu
(\rho}^\tau \Phi_{\kappa ) \tau}^\lambda + {1 \over 2} \Phi_{\rho \kappa}^\tau
\Phi_{\mu \tau}^\lambda ) (e^{-1})_a{}^\rho (e^{-1})_b{}^\kappa
e_\lambda{}^c, \ldots \label{2.8}$$ In deriving these expressions no conversion of indices on the objects has taken place but the indices have been relabelled with curved or flat indices suitable for their latter interpretation. The factors of $e$ come from the final factor of $g_h$ in the group element. Indeed, carrying out a local Lorentz transformation the Cartan forms transform as in eq. (\[1.2\]) and one sees that the $e_\mu{}^a$ are rotated on their $a$ index by a local Lorentz rotation allowing us to interpreted $e_\mu{}^a$ as the vierbein.
The part of the Cartan form involving the local subalgebra is contained in the second term of eq. (\[2.7\]) which we may write as $$G_{\mu ,a }{}^b K^{a}{}_b= G_{\mu , (a }{}^{b)} K^{(a}{}_{b)}
+\omega_{\mu a}{}^b J^{a}{}_b \label{2.9}$$ where $$\omega_{\mu a}{}^b= (e^{-1}\partial _\mu e)_{[a}{}^{b]} -\Phi _{\mu
\rho}{}^\kappa (e^{-1})_{[a}{}^\rho e_\kappa{}^{b]} \quad .\label{2.10}$$ We note that although the algebra of eqs. (\[2.2\]) and (\[2.3\]) is formulated in terms of the generators of $GL(D,\mathbb{R} )$ and other generators that are representations of $GL(D,\mathbb{R})$ the choice of the local sub-algebra to be $SO(1,D-1)$ allows us to introduce the tangent space metric $\eta_{ab}$ with which we may raise and lower indices to achieve the above (anti-)symmetrisations.
Thus far we agree with the paper of Pashnev [@25]. However, in this reference it was proposed that the Maurer Cartan equations $d
{\cal V}+{\cal V}\wedge {\cal V}=0$, which are identities, would place constraints on the fields. Imposing inverse Higgs conditions to find the Christoffel symbol in terms of the metric was correctly carried out in [@kirsch; @boulanger].
From now on we follow a different path. The dynamics are constructed in the way explained in the introduction with the Cartan forms transforming as in eq. (\[1.2\]). Recalling our discussion in the introduction we conclude that $G_{a,\star}\equiv (e^{-1})_a{}^\mu
G_{\mu\star}$, where $\star$ stands for any form except those lying in the Poincare algebra, transform under the Lorentz group as its indices suggest. As such we can place constraints on these Cartan forms and preserve all the symmetries, that is use the inverse Higgs mechanism. Indeed, we can set $$G_{c,(a }{}^{b)}=(e^{-1})_c{}^\mu G_{\mu (a }{}^{b)}
=(e^{-1})_c{}^\mu (e^{-1}\partial _\mu e)_{(a}{}^{b)} - \Phi _{\mu
\rho}{}^\kappa (e^{-1})_{(a}{}^\rho e_\kappa{}^{b)}=0 \quad .\label{2.11}$$ The effect of this is to solve for $\Phi _{\mu \rho}{}^\kappa=\Phi _
{(\mu \rho)}{}^\kappa$ in terms of the $e_\mu{}^a$. The result is [@kirsch] $$\Phi _{\mu \nu}{}^\kappa= \Gamma _{\mu \nu}{}^\kappa \equiv {1\over
2}g^{\kappa \tau}(\partial_\nu g_{\tau\mu}+
\partial_\mu g_{\tau\nu}-\partial_\tau g_{\mu\nu}) \quad .
\label{2.12}$$ We define $g_{\mu\nu}=e_\mu{}^ae_\nu{}^b\eta_{ab}$ and recognise $
\Gamma _{\mu \nu}{}^\kappa$ as the usual Christoffel connection of general relativity. A quick check of this result is to verify that eq. (\[2.11\]) implies that $2 g_{\lambda\kappa} \Phi _{\mu
\nu}{}^\kappa = \partial_\mu g_{\lambda \nu}$. Substituting into eq. (\[2.10\]) we find that $$\begin{aligned}
\omega_{\mu a}{}^b & = & (e^{-1}\partial _\mu e)_{[a}{}^{b]} -
\Gamma_{\mu \rho}{}^\kappa (e^{-1})_{[a}{}^\rho e_\kappa{}^{b]} ={1\over
2} e_a{}^\tau(\partial_\mu e_\tau{}^b -\partial_\tau e_\mu{} ^b )
\nonumber \\
& -& {1\over 2} \eta^{bc}e_c{}^\tau(\partial_\mu e_\tau{}^a
-\partial_\tau e_\mu{}^a ) -{1\over 2} e_a{}^\tau
\eta^{bc}e_c{}^\sigma (\partial_\tau e_\sigma{}^d -\partial_\sigma
e_\tau{}^d )e_\mu{}^d \quad , \label{2.13}
\end{aligned}$$ which is the well known formula for the spin connection.
At the next level we can covariantly set $$G_{(d,ab)}{}^c=0 \quad , \label{2.14}$$ where $G_{d,ab}{}^c\equiv e_d{}^\mu G_{\mu,ab}^c\equiv e_d{}^\mu
e_a{}^\rho e_b{}^\kappa G_{\mu,\rho\kappa}{}^\lambda e_\lambda{}^c$. This solves for the field $\Phi_{\mu\nu\rho}{}^\lambda=
\Phi_{(\mu\nu\rho )}{}^\lambda$ in terms of $\Phi_{\mu\nu}{}^\lambda$ by imposing symmetrisation in the obvious way. Substituting the solution into the part of this Cartan form that remains we find that $$2G_{\mu,\rho\kappa}{}^\lambda= R_{\mu\rho}{}^\lambda{}_\kappa\equiv
\partial_\mu \Gamma _{\rho\kappa}{}^\lambda-\partial_\rho
\Gamma_{\mu\kappa}{}^\lambda +\Gamma_{\mu\tau}{}^\lambda \Gamma_{\rho
\kappa}{}^\tau - \Gamma_{\rho\tau}{}^\lambda \Gamma_{\mu
\kappa}{}^\tau \label{2.15}
\quad ,$$ which we recognise as the well known expression of the Riemann tensor.
At higher orders one imposes covariant constraints on the Cartan forms so as to solve for all the Og fields $\Phi$ to leave only the field $h_a{}^b$ or equivalently $ e_\mu{}^a= (e^h)_\mu{}^a$. Substituting the solutions back into the Cartan forms we find that $$g^{-1}dg= dx^\mu (e_\mu{}^a P_a+ \omega_{\mu a}{}^b
J^a{}_b+{1\over 2}R_{\mu\rho}{}^\lambda{}_\kappa e_a{}^\rho
e_b{}^\kappa e_c {}^\lambda K_c^{ab}+\dots )
\quad , \label{2.16}$$ where $+\dots$ denotes terms which contain covariant derivatives of the Riemann tensor.
The Og generators play the role of turning $GL(D,\mathbb{R})$ into a local symmetry and one can verify that carrying out a general rigid group transformation $g\to g_0g$ on the group element of eq. (\[2.5\]) we recover the usual general coordinate transformations of general relativity on the vierbein $e_\mu{}^a$.
We will now summarise the above discussion. We started with the group $GL(D,\mathbb{R})$, generators $K^a{}_b$ and the translations $P_a$ to which we assigned grades $0$ and $-1$ respectively. To these we added an infinite number of Ogievetsky generators $K^{a_1\ldots a_{n+1}}{}_c$ each with grade $n$. These obey the Lie algebra of eqs. (\[2.2\]) and (\[2.3\]). We then placed covariant constraints on the Cartan forms solving for all the Ogievetsky fields whereupon the remaining parts of the Cartan form contain the spin connection at lowest grade and then the Riemann tensor and its covariant derivatives. The introduction of the Ogievetsky generators leads in the non-linear realisation to general coordinate invariance. As such we find Einstein’s theory in a completely systematic way from the viewpoint of non-linear realisations.
We now consider the dimensional reduction of this non-linear realisation that is equivalent to the usual dimensional reduction on a circle. Let us denote by $y$ the coordinate of the circle, $*$ the components in this direction and let $Q=P_*$. Dimensionally reducing the Cartan forms of eq. (\[2.6\]) we find $$\begin{aligned}
g^{-1}d g& =& dx^\mu (e_\mu{}^a P_a+e_\mu{}^*Q+ G_{\mu ,b}{}^c K^b{}_c
+G_{\mu , *}{}^c K^*{}_c+G_{\mu , b}{}^* K^b{}_*+G_{\mu ,*}{}^*
K^*{}_* \nonumber \\
& +&G_{\mu , a b}{}^c K^{ab}{}_c+2G_{\mu , * b}{}^c K^{*b}{}_c +\ldots
) \nonumber \\
&+&dy (e_*{}^a P_a+e_*{}^*Q+ G_{*, b}{}^c K^b{}_c+G_{*, *}{}^c K^*{}_c
+G_{*, b}{}^* K^b{}_*+G_{*, *}{}^* K^*{}_* \nonumber \\
&+&G_{* ,a b}{}^c K^{ab}{}_c+2G_{* ,* b}{}^c K^{*b}{}_c +G_{* ,a
b}{}^* K^{ab}{}_*+\ldots ) \quad .\label{2.17}
\end{aligned}$$ The coefficients $G$ can be read off from eqs. (\[2.7\]) and (\[2.8\]). We now take all the fields not to depend on $y$ and imposing the inverse Higgs constraint on all $G_{*,\bullet}$ where ${\bullet }$ is any index, that is set the part of the Cartan form in the $dy$ direction to zero. We find that all the Ogievetsky fields that contain a lower $*$ index vanish. Thus all the Ogievetsky generators that do not commute with $Q$ disappear from the group element and so the Cartan form. The only Ogievetsky fields left are $\Phi_{ab}{}^c$ and $\Phi_{ab}^*$. The later field occurs in the Cartan form in the term $$dx^\mu G_{\mu , b}{}^* K^b{}_*=((e^{-1}\partial _\mu e)_b{}^*
-\Phi_{\mu \rho}{}^* (e^{-1})_b{}^\rho e_*{}^* ) K^b{}_*
\label{2.18} \quad .$$ In the dimensionally reduced theory we set the coefficient of $dx^\mu$ lying in $K^{(a}{}_{b)}$ of the Cartan form to zero and solve for $\Phi_{ab}{}^c$ which just plays the role of the Ogievetsky field of gravity in the lower dimension. Setting the part of the Cartan form of eq. (\[2.18\]) $(e^{-1})_{(a}{}^\mu G_{\mu
, b )}{}^*=0$ we solve for $\Phi_{ab}^*$ in terms of $e_\mu{}^*$. The latter field is just the vector field that arises in this dimensional reduction and so this step is as we found for the case of the vector studied earlier. In the dimensionally reduced theory we have as our local symmetry only the Local Lorentz group in the lower dimension. Substituting for $\Phi_{ab}{}^c$ in the part of the Cartan form in this part of the algebra we find the spin connection for the lower dimensional theory. There remains, however, the term containing $(e^{-1})_{[a}{}^\mu G_{\mu , b ]}{}^*$, but this we recognise as just the field strength for the vector.
$E_{11}$ and eleven-dimensional supergravity
============================================
In this section we want to repeat the analysis of the previous section for the non-linear realisation based on the very-extended Kac-Moody algebra $E_{11}$, whose Dynkin diagram is shown in fig. \[fig1\].
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The decomposition of the adjoint representation of $E_{11}$ with respect to the subalgebra $GL(11, \mathbb{R})$ corresponding to nodes from 1 to 10 in the diagram leads to the generators $K^a{}_b$ of $GL(11, \mathbb{R})$ and $R^{abc}$ and $R_{abc}$ in the completely antisymmetric representations of $GL(11, \mathbb{R})$, together with an infinite set of generators which can be obtained by multiple commutators of the generators $R^{abc}$ and $R_{abc}$ subject to the Serre relations. Defining the level $l$ as the number of times the generator $R^{abc}$ occurs in such multiple commutators, one obtains for instance at level 2 the generator $R^{a_1 \dots a_6}$ with completely antisymmetric indices and at level 3 the generator $R^{a,b_1 \dots b_8}$ antisymmetric in the indices $b_1 \dots b_8$ and with $R^{[a,b_1 \dots b_8]} =0$. The generator $R^{abc}$ itself has level 1, while the generator $R_{abc}$ has level -1 and correspondingly multiple commutators of this generator have negative level [@1].
In the last section we have shown how spacetime arises in the nonlinear realisation based on the algebra $GL(D, \mathbb{R})$ in $D$ dimensions. This corresponds to introducing the momentum operator $P_a$, together with an infinite set of Og $n$ operators $K^{a_1 ... a_{n+1}}{}_b$. In $E_{11}$ the momentum operator arises as the lowest component of the $E_{11}$ representation corresponding to $\lambda_1 =1$, where $\lambda_1$ is the Dynkin index associated to node 1 in fig. \[fig1\], and called the $l$ multiplet [@19]. In this paper we will consider a different approach, that is we will consider the momentum operator as commuting with all the positive level generators. This approach has the advantage that one can naturally introduce the Og operators for each positive level generator of $E_{11}$, although it has the disadvantage of breaking $E_{11}$ to Borel $E_{11}$, or more precisely to the subgroup of $E_{11}$ generated by $GL(11, \mathbb{R})$ and all the positive level generators. The corresponding local subalgebra is $SO(11)$, or $SO(10,1)$ in Minkowski signature. We denote with $E_{11}^{local}$ the algebra generated by the momentum operator, the non-negative level $E_{11}$ operators and the Og operators.
In the non-linear realisation, the fields associated to $R^{abc}$ and $R^{a_1 \dots a_6}$ correspond to the 3-form and its dual 6-form of eleven dimensional supergravity. The field associated to the generator $R^{a,b_1 \dots b_8}$ has the right indices to be associated to the dual graviton, and we will call it the dual graviton for short. In this section we will concentrate on the 3-form and 6-form, while section 7 will be devoted to the dual graviton, although not in eleven dimensions but in the simpler four dimensional case.
Following the analysis of the previous section, we take Og operators for the 3-form and the 6-form in the representations obtained adding symmetrised indices to the set of 3 or 6 antisymmetric indices respectively. The Young Tableaux corresponding to the first three Og operators is shown in fig. \[fig2\]. In particular, the Og 1 operators $K_1^{a, b_1 b_2 b_3}$ and $K_1^{a, b_1 \dots b_6}$ belong to the $GL(11,\mathbb{R})$ representations defined by $$\begin{aligned}
& & K_1^{a, b_1 b_2 b_3} = K_1^{a,[ b_1 b_2 b_3 ]} \qquad K_1^{[a, b_1 b_2 b_3
]}=0 \nonumber \\
& & K_1^{a, b_1 \dots b_6} = K_1^{a,[ b_1 \dots b_6 ]} \qquad K_1^{[a, b_1 \dots
b_6 ]}=0 \label{3.1}
\end{aligned}$$ and we take their commutation relation with $P_a$ to be $$\begin{aligned}
& & [ K_1^{a, b_1 b_2 b_3} , P_c] = \delta^a_c R^{b_1 b_2 b_3} - \delta^{ [ a}_c
R^{b_1 b_2 b_3 ]}
\nonumber \\
& & [ K_1^{a, b_1 \dots b_6} , P_c] = \delta^a_c R^{b_1 \dots b_6} - \delta^{ [ a}_c
R^{b_1 \dots b_6 ]} \quad . \label{3.2}
\end{aligned}$$ The Og 2 operator for the 3-form $K_2^{a,b,c_1 c_2 c_3}$ belongs to the representation defined by $$K_2^{a,b,c_1 c_2 c_3} = K_2^{( a,b ),c_1 c_2 c_3} = K_2^{a,b,[ c_1 c_2 c_3
]} \qquad K_2^{a,[b,c_1 c_2 c_3]} = 0 \label{3.3}$$ and we take its commutation relation with $P_a$ to be $$[ K_2^{a,b,c_1 c_2 c_3}, P_d ] = \delta^a_d K_1^{b, c_1 c_2 c_3}
+ \delta^b_d K_1^{a, c_1 c_2 c_3} + {3 \over 4} \delta^{ [c_1 }_d K_1^{ \vert a , b \vert c_2 c_3] }
+ { 3 \over 4 } \delta^{ [c_1 }_d K_1^{ \vert b , a \vert c_2 c_3] } \quad
, \label{3.4}$$ and similarly for the Og 2 operator for the 6-form $K_2^{a,b,c_1
\dots c_6}$. Proceeding this way one can write down the representation and the commutation relation with $P_a$ of the next Og operators. Denoting with $n$ the grade of the Og operators, [*i.e.*]{} the Og 1 operators have grade 1, then the commutator of an Og $n$ operator with $P_a$ gives an Og $(n-1)$ operator.
(380,140)
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(60,-10)[(0,1)[140]{}]{}
(70,80)[$K_1^{a,b_1 \dots b_3}$:]{}
(120,80)[(1,0)[20]{}]{} (120,90)[(1,0)[20]{}]{} (120,70)[(1,0)[10]{}]{} (120,60)[(1,0)[10]{}]{} (120,60)[(0,1)[30]{}]{} (130,60)[(0,1)[30]{}]{} (140,80)[(0,1)[10]{}]{}
(70,40)[$K_1^{a,b_1 \dots b_6}$:]{}
(120,50)[(1,0)[20]{}]{} (120,40)[(1,0)[20]{}]{} (120,30)[(1,0)[10]{}]{} (120,20)[(1,0)[10]{}]{} (120,10)[(1,0)[10]{}]{} (120,0)[(1,0)[10]{}]{} (120,-10)[(1,0)[10]{}]{} (120,-10)[(0,1)[60]{}]{} (130,-10)[(0,1)[60]{}]{} (140,40)[(0,1)[10]{}]{}
(150,-10)[(0,1)[140]{}]{}
(160,80)[$K_2^{a,b,c_1 \dots c_3}$:]{}
(220,80)[(1,0)[30]{}]{} (220,90)[(1,0)[30]{}]{} (220,70)[(1,0)[10]{}]{} (220,60)[(1,0)[10]{}]{} (220,60)[(0,1)[30]{}]{} (230,60)[(0,1)[30]{}]{} (240,80)[(0,1)[10]{}]{} (250,80)[(0,1)[10]{}]{}
(160,40)[$K_2^{a,b,c_1\dots c_6}$:]{}
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(260,-10)[(0,1)[140]{}]{}
(270,80)[$K_3^{a,b,c,d_1 \dots d_3}$:]{}
(335,80)[(1,0)[40]{}]{} (335,90)[(1,0)[40]{}]{} (335,70)[(1,0)[10]{}]{} (335,60)[(1,0)[10]{}]{} (335,60)[(0,1)[30]{}]{} (345,60)[(0,1)[30]{}]{} (355,80)[(0,1)[10]{}]{} (365,80)[(0,1)[10]{}]{} (375,80)[(0,1)[10]{}]{}
(270,40)[$K_3^{a,b,c,d_1 \dots d_6}$:]{}
(335,50)[(1,0)[40]{}]{} (335,40)[(1,0)[40]{}]{} (335,30)[(1,0)[10]{}]{} (335,20)[(1,0)[10]{}]{} (335,10)[(1,0)[10]{}]{} (335,0)[(1,0)[10]{}]{} (335,-10)[(1,0)[10]{}]{} (335,-10)[(0,1)[60]{}]{} (345,-10)[(0,1)[60]{}]{} (355,40)[(0,1)[10]{}]{} (365,40)[(0,1)[10]{}]{} (375,40)[(0,1)[10]{}]{}
The 6-form generator occurs in the commutator $$[ R^{a_1 a_2 a_3} , R^{a_4 a_5 a_6} ] = 2 R^{a_1 \dots a_6} \quad
. \label{3.5}$$ Using this relation and eq. (\[3.2\]) one can then determine the commutation relations between the Og 1 generators and $R^{abc}$ requiring that the Jacobi identities are satisfied. This gives $$[ K_1^{a, b_1 b_2 b_3 } , R^{c_1 c_2 c_3} ] = 2 K_1^{a , b_1 b_2 b_3
c_1 c_2 c_3 } - 2 K_1^{[ a , b_1 b_2 b_3 ] c_1 c_2 c_3 } \quad .
\label{3.6}$$ Neglecting higher level generators and the gravity contribution, as well as higher Og generators, we can thus write down the group element as $$g = e^{x \cdot P} e^{\Phi_{\rm Og} K_1^{\rm Og}} e^{A_{a_1 \dots
a_6 } R^{a_1 \dots a_6}} e^{A_{a_1 \dots a_3} R^{a_1 \dots a_3}}
\quad , \label{3.7}$$ where we have denoted with $\Phi_{\rm Og}$ the Og 1 fields for both the 3-form and the 6-form, and similarly $K_1^{\rm Og}$ denotes collectively the Og 1 operators for the 3-form and the 6-form. One can then compute the Maurer Cartan form, which is $$\begin{aligned}
g^{-1} \partial_\mu g & = & P_\mu + ( \partial_\mu A_{a_1 a_2 a_3} -
\Phi_{\mu , a_1 a_2 a_3 } ) R^{a_1 a_2 a_3 }
+ ( \partial_\mu
A_{a_1 \dots a_6} + \partial_\mu A_{a_1 a_2 a_3 } A_{a_4 a_5
a_6} \nonumber \\
& - & \Phi_{\mu , a_1 \dots a_6} - 2 \Phi_{\mu , a_1 a_2 a_3} A_{a_4 a_5 a_6}
) R^{a_1 \dots a_6} + ... \label{3.8}
\end{aligned}$$ The inverse Higgs mechanism allows one to express the Og 1 fields in terms of the 3-form and the 6-form in such a way that only the completely antisymmetric expressions are left in (\[3.8\]). This corresponds to $$\begin{aligned}
\Phi_{\mu , a_1 a_2 a_3} &=& \partial_\mu A_{a_1 a_2 a_3} -
\partial_{[\mu} A_{a_1 a_2 a_3 ]}
\nonumber \\
\Phi_{\mu ,a_1 \dots a_6} &=& \partial_\mu A_{a_1 \dots a_6} -
\partial_{[\mu} A_{a_1 \dots a_6 ]} - \partial_\mu A_{a_1 a_2 a_3 }
A_{a_4 a_5 a_6} \nonumber \\
&-& \partial_{[ \mu} A_{a_1 a_2 a_3 } A_{a_4 a_5
a_6]} + 2 \partial_{[\mu} A_{a_1 a_2 a_3 ]} A_{a_4 a_5
a_6} \quad , \label{3.9}
\end{aligned}$$ where antisymmetry in the $a$ indices is understood. These relations are all invariant with respect to the local subalgebra. Plugging this into the Maurer-Cartan form one gets $$g^{-1} \partial_\mu g = P_\mu + F_{\mu a_1 a_2 a_3 } R^{a_1 a_2
a_3} + F_{\mu a_1 \dots a_6} R^{a_1 \dots a_6 } + ... \quad ,
\label{3.10}$$ where $$\begin{aligned}
& & F_{a_1 a_2 a_3 a_4 } = \partial_{[ a_1} A_{a_2 a_3 a_4 ]}
\nonumber \\
& & F_{a_1 \dots a_7} = \partial_{[a_1} A_{a_2 \dots a_7 ]} +
F_{[a_1 \dots a_4 } A_{a_5 a_6 a_7]}
\label{3.11}
\end{aligned}$$ are the field strengths of the 3-form and its dual 6-form of 11-dimensional supergravity.
The Maurer-Cartan form is invariant under transformations in the Borel subalgebra, and we use this to derive transformations for the fields. In the particular case discussed in this section, where we have restricted the group element to be as in eq. (\[3.7\]), we consider the action of $$g_0 = e^{a_{a_1 a_2 a_3} R^{a_1 a_2 a_3 } } e^{a_{a_1 \dots a_6}
R^{a_1 \dots a_6 } } e^{b_{a, b_1 b_2 b_3} K_1^{a, b_1 b_2 b_3 } }
e^{b_{a, b_1 \dots b_6} K_1^{a, b_1 \dots b_6 } } \label{3.12}$$ from the left. Taking the parameters $a$ and $b$ to be infinitesimal, we derive the transformations of the fields to be $$\begin{aligned}
& & \delta A_{a_1 a_2 a_3} = a_{a_1 a_2 a_3} + x^b b_{b ,a_1 a_2 a_3
} \nonumber \\
& & \delta A_{a_1 \dots a_6} = a_{a_1 \dots a_6} + a_{a_1 a_2
a_3}A_{a_4 a_5 a_6} + x^b b_{b , a_1 \dots a_6} + x^b b_{b , a_1
a_2 a_3} A_{a_4 a_5 a_6} \nonumber \\
& & \delta \Phi_{b , a_1 a_2 a_3} = b_{b , a_1 a_2 a_3} \nonumber \\
& & \delta \Phi_{b , a_1 \dots a_6} = b_{b , a_1 \dots a_6} -2 \Phi_{b ,
a_1 a_2 a_3} a_{a_4 a_5 a_6} -2 \Phi_{ b ,
a_1 a_2 a_3} x^c b_{c , a_4 a_5 a_6} \quad . \label{3.13}
\end{aligned}$$ The eqs. (\[3.9\]) and the field strengths of eqs. (\[3.11\]) are separately invariant under these transformations, and in particular the transformations of the 3-form and the 6-form can be written as $$\begin{aligned}
& & \delta A_{a_1 a_2 a_3} = \partial_{[ a_1 } \Lambda_{a_2 a_3 ]}
\nonumber \\
& & \delta A_{a_1 \dots a_6} = \partial_{[ a_1 } \Lambda_{a_2 \dots a_6 ]}
+ \partial_{[ a_1 } \Lambda_{a_2 a_3 } A_{a_4 a_5 a_6 ]}
\label{3.14}
\end{aligned}$$ with gauge parameters $$\begin{aligned}
& & \Lambda_{a_1 a_2} = x^b a_{b a_1 a_2} + {3 \over 4} x^b x^c
b_{ b ,c a_1 a_2} \nonumber \\
& & \Lambda_{a_1 \dots a_5} = x^b a_{b a_1 \dots a_5} + {6 \over
7} x^b x^c b_{b,c a_1 \dots a_5} \quad . \label{3.15}
\end{aligned}$$ Including higher order Og generators corresponds to higher powers of $x$ in the equations above. The full gauge invariance is obtained including all the Og generators.
It is worth mentioning that the normalisation used here is different from the one used in the original $E_{11}$ paper [@1]. This is for consistency with the normalisation used in the rest of this paper. Going from this normalisation to the original one in [@1] corresponds to making the field redefinitions $$\begin{aligned}
& & A_{a_1 a_2 a_3 } \rightarrow {1 \over 3!} A_{a_1 a_2 a_3}
\nonumber \\
& & A_{a_1 \dots a_6} \rightarrow {1 \over 6!} A_{a_1 \dots a_6}
\quad , \label{3.16}
\end{aligned}$$ as can be deduced from eq. (2.6) in [@1].
It is also instructive to consider the Maurer-Cartan form at the next order in the Og generators. For simplicity we will now perform this analysis only for the 3-form, so that we can neglect the contributions coming from commutators of Og generators among themselves. The generalisation to include the 6-form generators is straightforward, although technically more complicated. We thus consider the group element as only containing the 3-form generators in the $E_{11}$ sector and including the Og 1 operator of eq. (\[3.1\]) and the Og 2 operator of eq. (\[3.3\]), that is $$g = e^{x \cdot P} e^{\Phi_{a,b,c_1 c_2 c_3} K_2^{a,b,c_1 c_2 c_3}}
e^{\Phi_{a,b_1 b_2 b_3} K_1^{a, b_1 b_2 b_3}} e^{A_{a_1 a_2 a_3 }
R^{a_1 a_2 a_3 }} \quad . \label{3.17}$$ Using eq. (\[3.4\]), as well as eq. (\[3.2\]), one gets $$\begin{aligned}
g^{-1} \partial_\mu g & = & P_\mu + ( \partial_\mu A_{a_1 a_2 a_3} -
\Phi_{\mu , a_1 a_2 a_3 } ) R^{a_1 a_2 a_3 } \nonumber \\
& + & (\partial_\mu \Phi_{a , b_1 b_2 b_3 } - {5 \over 2}
\Phi_{\mu , a , b_1 b_2 b_3}) K_1^{a , b_1 b_2 b_3}+ \dots \quad
. \label{3.18}
\end{aligned}$$ Using the inverse Higgs mechanism one solves for the Og 1 field in terms of the derivative of the 3-form, and the Og 2 field in terms of the derivative of the Og 1 field. Plugging this into the group element leads to $$g^{-1} \partial_\mu g = P_\mu + F_{\mu a_1 a_2 a_3 } R^{a_1 a_2
a_3 } + \partial_a F_{\mu b_1 b_2 b_3 } K_1^{a , b_1 b_2 b_3}+ ...
\quad . \label{3.19}$$
This is an example of the general picture, in which after applying the inverse Higgs mechanism one is left with the field strengths of the 3-form and the 6-form together with infinitely many derivatives of those, without breaking any of the original symmetries. These fields are the only forms that arise in the decomposition of $E_{11}$ with respect to $GL(11, \mathbb{R} )$. Indeed, in [@26] it was shown that all the positive level 11-dimensional generators of $E_{11}$ can be cast in generators of the form $R^{9,9,...,9,3}$, $R^{9,9,...,9,6}$ and $R^{9,9,...,9,8,1}$, together with generators with at least one set of 10 or 11 completely antisymmetric indices (here we are using a shortcut notation, in which each number corresponds to the number of antisymmetric indices; for example the Og 2 generator for the 6-form is written as $K_2^{6,1,1}$ in this notation). The fields associated to the former generators (the ones with sets of 9 antisymmetric indices) were interpreted in [@26] as being all the possible dual formulations of the 3-form and the graviton, while the latter were interpreted as giving rise to non-propagating fields. In section 7 we will consider the case of $E_{11}$ fields with mixed symmetries, focusing in particular on the case of the dual graviton in four dimensions, while in the next section we will show that the introduction of the Og generators is crucial to understand and derive the algebra that describes gauged supergravity theories.
Scherk-Schwarz reduction of IIB supergravity from $E_{11}$
==========================================================
In this section we will show how to dimensionally reduce maximal supergravities in the context of their $E_{11}$ formulation including the Og extension. We will in particular focus on the case of ten-dimensional IIB reduced to nine dimensions and study both the dimensional reduction on a circle and the Scherk-Schwarz reduction.
We will first introduce the Og generators required to encode the gauge symmetries of the ten-dimensional theory. This gives rise to the algebra $E_{11,10B}^{local}$. We will then express the $E_{11}$ and Og generators of the IIB theory in a nine-dimensional set-up. The consistency of the truncation from ten-dimensional IIB supergravity to maximal supergravity in nine dimensions corresponds to the fact that within the algebra $E_{11,10B}^{local}$ of the ten-dimensional $E_{11}$ and Og generators one can find a sub-algebra $E_{11,9}^{local}$ appropriate to the nine-dimensional theory. This indeed corresponds to a maximal supergravity theory in nine dimensions, which is a compactification of the ten-dimensional IIB theory on a coordinate $y$. If one takes the ten-dimensional group element not to depend on $y$ apart from the momentum contribution $e^{y Q}$, where $Q$ is the internal momentum, then this corresponds to standard, [*i.e.*]{} massless, dimensional reduction on a circle parametrised by $y$, and the form of the group element is preserved by the sub-algebra $E_{11,9}^{local}$ of the ten-dimensional algebra $E_{11,10B}^{local}$ of $E_{11}$ plus Og generators appropriate to massless dimensional reduction. One can also consider a ten-dimensional group element with a suitable $y$ dependence, which we show to give rise in nine dimensions to the massive theory corresponding to the Scherk-Schwarz reduction of the IIB theory [@27]. This different form of the group element is preserved by a different sub-algebra of the ten-dimensional algebra $E_{11,10B}^{local}$ of $E_{11}$ and Og generators that we call $\tilde{E}_{11,9}^{local}$. We show how to construct this subalgebra corresponding to Scherk-Schwarz reduction. The mass parameter mixes $E_{11}$ and Og generators, and from the nine-dimensional perspective this corresponds to a deformation of the massless $E_{11}$ algebra. The occurrence of a deformed $E_{11}$ algebra associated to massive theories was shown for the first time in [@11] for the case of the ten-dimensional massive IIA theory. In that case the occurrence of a mass parameter for the 2-form was shown to arise from requiring that the commutator of the 2-form generator with momentum does not vanish, but is instead equal to the vector generator times the Romans mass parameter.
We now consider the decomposition of the $E_{11}$ generators appropriate to the IIB theory, that arises from deleting node 9 in the Dynkin diagram of fig. \[fig1\]. The $GL(10, \mathbb{R})$ subalgebra associated to the non-linear realisation of gravity corresponds to nodes from 1 to 8 and node 11, while node 10 corresponds to the internal $SL(2 , \mathbb{R} )$ symmetry of the IIB theory. We denote tangent spacetime indices in ten dimensions with $\hat{a}, \hat{b}, ...$ and curved spacetime indices with $\hat{\mu}, \hat{\nu},...$, where the indices go from 1 to 10. One constructs the positive level generators as multiple commutators of the 2-form generator $R^{\hat{a} \hat{b} ,\alpha}$, $\alpha = 1,2$, which is a doublet of $SL(2 , \mathbb{R} )$. Together with the $GL(10, \mathbb{R} )$ generators $K^{\hat{a}}{}_{\hat{b}}$ and the $SL(2, \mathbb{R} )$ generators $R^i$, $i=1,2,3$ at level zero, one has the doublet of 2-form generators at level 1, a 4-form generator $R^{\hat{a}\hat{b}\hat{c}\hat{d}}$ at level 2, and then a doublet of 6-forms at level 3, a triplet of 8-forms at level 4 and a doublet and a quadruplet of 10-forms at level 5, together with an infinite set of generators with mixed, [*i.e.*]{} not completely antisymmetric, indices. We consider the positive level generators as commuting with the momentum operator $P_{\hat{a}}$.
We now want to write down the relevant algebra in ten dimensions. For simplicity, we consider a level truncation and we therefore only consider in ten dimensions the 2-form generators $R^{\hat{a}\hat{b} , \alpha}$, together with the $GL(10,\mathbb{R})$ generators $K^{\hat{a}}{}_{\hat{b}}$ and the $SL(2, \mathbb{R})$ generators $R^i$. We have the commutation relations $$\begin{aligned}
& & [ R^i , R^j] = f^{ij}{}_k R^k \nonumber \\
& & [ R^i , R^{\hat{a}\hat{b},\alpha } ]= D^i_\beta{}^\alpha R^{\hat{a}\hat{b} ,\beta}
\label{4.1}
\end{aligned}$$ where $D^i_\beta{}^\alpha$ are the generators of $SL(2, \mathbb{R})$ satisfying $$[ D^i , D^j ]_\beta{}^\alpha = f^{ij}{}_k D^k_\beta{}^\alpha
\label{4.2}$$ and $f^{ij}{}_k$ are the structure constants of $SL(2,\mathbb{R})$. In terms of Pauli matrices, a choice of $D^i_\beta{}^\alpha$ is $$D_1 = {\sigma_1 \over 2} \qquad D_2 = {i \sigma_2 \over 2} \qquad D_3 = {\sigma_3 \over
2} \quad . \label{4.3}$$
We now add the Og generators to the $E_{11}$ formulation of ten-dimensional IIB. In this way we encode all the local gauge symmetries of the ten-dimensional IIB theory. The procedure is much like the one discussed in the previous sections for other cases. The Og 1 operator for the 2-form is a doublet of generators $K^{\hat{a},\hat{b}\hat{c} , \alpha}$, satisfying $$K^{\hat{a},\hat{b}\hat{c} , \alpha} =K^{\hat{a},[\hat{b}\hat{c}] ,
\alpha} \qquad K^{[\hat{a},\hat{b}\hat{c}] , \alpha} =
0 \quad , \label{4.4}$$ and whose commutation relation with the momentum operator $P_{\hat{a}}$ is $$[ K^{\hat{a},\hat{b}\hat{c} , \alpha} , P_{\hat{d}} ] = \delta^{\hat{a}}_{\hat{d}} R^{\hat{b}\hat{c} , \alpha} -
\delta^{[\hat{a}}_{\hat{d}} R^{\hat{b}\hat{c} ] , \alpha} \quad .
\label{4.5}$$ Ignoring for simplicity the gravity contribution, the non-linear realisation can be constructed from the group element $$g = e^{x \cdot P} e^{\Phi_{\hat{a},\hat{b}\hat{c} , \alpha} K^{\hat{a},\hat{b}\hat{c} , \alpha}}
e^{A_{\hat{a}\hat{b} ,\alpha} R^{\hat{a}\hat{b} ,\alpha}} e^{\phi_i R^i} \quad
, \label{4.6}$$ and the corresponding Maurer-Cartan form gives $$g^{-1} d g = dx^{\hat{\mu}} [ P_{\hat{\mu}} + ( \partial_{\hat{\mu}} A_{\hat{a}\hat{b} , \alpha} -
\Phi_{{\hat{\mu}} , \hat{a}\hat{b} , \alpha})e^{- \phi_i R^i} R^{\hat{a}\hat{b} , \alpha} e^{\phi_i R^i} +
e^{-\phi_i R^i} \partial_{\hat{\mu}} e^{\phi_i R^i} + ...] \quad
. \label{4.7}$$ The inverse Higgs mechanism then fixes $\Phi_{{\hat{\mu}} ,
\hat{a}\hat{b} , \alpha}$ in terms of $\partial_{\hat{\mu}}
A_{\hat{a}\hat{b} , \alpha}$ so that the $R^{\hat{a}\hat{b},\alpha}$ term becomes proportional to $$F_{\hat{a}\hat{b}\hat{c} , \alpha} = \partial_{[ \hat{a}} A_{\hat{b}\hat{c} ] ,\alpha} \quad
, \label{4.8}$$ which is the field strength for the 2-form. This procedure is completely consistent because the inverse Higgs mechanism preserves entirely the local subalgebra, which is $SO(9,1) \times SO(2)$.
We now consider a generic compactification of the IIB theory in the above $E_{11}$ formulation to nine dimensions. This will include the derivation of both the massless theory and the Scherk-Schwarz reduction, which both have maximal supersymmetry. We thus split the ten-dimensional coordinates in $x^\mu$, $\mu = 1, ..., 9$, and the 10th coordinate $y$. Correspondingly, the momentum operator splits in $P_a$ and $Q$, where $Q=P_y$. As we did in ten-dimensions, we consider a level truncation and thus we are only interested in 1-forms and 2-forms in nine dimensions. The doublet of 2-form generators in ten dimensions gives a doublet of 2-forms $R^{ab ,
\alpha}$ and a doublet of 1-forms $R^{a , \alpha} = R^{ay ,
\alpha}$. One also obtains a 1-form from the $GL(10 , \mathbb{R})$ generators, namely $R^a = K^a{}_y$, whose commutator with $P_a$ is $$[ R^a , P_b ] = - \delta^a_b Q \quad .
\label{4.9}$$ One also has the $SL(2,\mathbb{R})$ triplet of scalar generators $R^i$, as well as the singlet scalar generator $R = K^y{}_y$, satisfying $$[ R, Q ] = -Q \quad .
\label{4.10}$$ The commutator between $R^a$ and $R^{a ,\alpha}$ is $$[ R^a , R^{b , \alpha}] = - R^{ab ,\alpha} \quad ,
\label{4.11}$$ while the non-vanishing commutators with the scalars are $$\begin{aligned}
& & [ R, R^a ] = -R^a \qquad \qquad \ \ [ R , R^{a , \alpha} ] = R^{a ,
\alpha} \nonumber \\
& & [ R^i , R^{a , \alpha} ] = D^i_\beta{}^\alpha R^{a ,\beta}
\qquad [ R^i , R^{ab , \alpha} ] = D^i_\beta{}^\alpha R^{ab
,\beta} \quad . \label{4.12}
\end{aligned}$$ The commutator of $R^a$ with itself and the commutator of $R^{a
,\alpha}$ with itself vanish, $$[ R^a , R^b ] =0 \qquad [ R^{a , \alpha} , R^{b , \beta} ] = 0
\quad . \label{4.13}$$ These are all the $E_{11}$ commutators we need consider at the level we are analysing. At the end of this section we will also consider the 3-form generator $R^{abc}$ and 4-form generator $R^{abcd}$. The first arises from the 4-form generator of IIB with one index in the internal direction, $R^{abcy}$, while the latter is just the 4-form of IIB with all indices along the nine-dimensional spacetime.
Just as for the $E_{11}$ generators, we also rewrite the ten-dimensional Og generators as decomposed in $GL(9 , \mathbb{R})$ representations. The generator $K^{\hat{a},\hat{b} \hat{c} ,
\alpha}$ thus gives rise to $K^{a,bc , \alpha}$, $K^{[ab],\alpha}$, $K^{(ab), \alpha}$ and $K^{a , \alpha}$, where $$K^{[ab], \alpha} = K^{y, ab , \alpha} - K^{[a,b]y , \alpha} \qquad
K^{(ab), \alpha} = K^{(a,b)y , \alpha} \qquad K^{a , \alpha}
= K^{y, a y , \alpha} \quad . \label{4.14}$$ The commutation relations of these operators with $P_a$ and $Q$ are $$\begin{aligned}
& [ K^{a,bc , \alpha} , P_d ] = \delta^a_d R^{bc , \alpha} -
\delta^{[a}_d R^{bc ] , \alpha} \qquad & [ K^{a,bc , \alpha} , Q ] =
0 \nonumber \\
& \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\! \! \! \! \! \! \! [ K^{[ab], \alpha} , P_c ] = - \delta^{[a}_c R^{b],\alpha}
& [ K^{[ab], \alpha} , Q ] = R^{ab , \alpha} \nonumber \\
& \! \! \! \! \! \! \! \!\! \! \! \! \! \! \! \! \! \! \! \! \! \!\! \! \! \! \! \! \!
\! \! \! \! \! \! \![ K^{(ab), \alpha} , P_c ] = \delta^{(a}_c R^{b),\alpha}
& [ K^{(ab), \alpha} , Q ] = 0 \nonumber \\
& \! \! \! \! \! \! \! \!\! \! \! \! \! \! \!\! \! \! \! \! \! \!\! \! \! \! \! \! \!
\! \! \! \! \! \! \!\! \! \! \! \! \! \!\! \! \! \! \! \! \!
\! \! \! \! \! \! \![ K^{a ,\alpha} , P_b ] = 0 & [ K^{a ,\alpha} , Q ] =
R^{a , \alpha} \quad . \label{4.15}
\end{aligned}$$ Similarly, the dimensional reduction of the gravity Og 1 operator gives the Og operators $K^{(ab)}$, $K^a$ and $K$, that satisfy $$\begin{aligned}
& & [K^{(ab)} , P_c ] = \delta^{(a}_c R^{b )} \qquad [ K^{(ab)} , Q ]
=0 \nonumber \\
& & [ K^a , P_b ] = \delta^a_b R \qquad \quad \ \ [ K^a , Q ] = R^a
\nonumber \\
& & [ K , P_a ] = 0 \qquad \qquad \quad \ [ K , Q ] = R \quad .
\label{4.16}
\end{aligned}$$
We now write down the group element. For simplicity we will neglect Og 2 contributions, and therefore we will consider the Og 1 generators as commuting among themselves. We will denote with $\Phi_{\rm Og}$ and $K^{\rm Og}$ the whole set of Og 1 fields and generators. Thus the group element is $$g = e^{x \cdot P} e^{y Q} e^{\Phi_{\rm Og} K^{\rm Og}}
e^{A_{ab ,\alpha} R^{ab ,\alpha}} e^{A_{a , \alpha} R^{a , \alpha}}
e^{A_a R^a} e^{\phi R} e^{\phi_i R^i} \quad , \label{4.17}$$ where all the fields are taken to depend on $x$ and $y$. We now compute the Maurer-Cartan form. The result is $$\begin{aligned}
g^{-1} d g & = & dx^\mu [ P_\mu + A_\mu e^\phi Q +
( \partial_\mu A_{ab , \alpha} - \partial_\mu A_{a , \alpha} A_b -
\Phi_{\mu , ab , \alpha} + \Phi_{(\mu a ) , \alpha} A_b -
\Phi_{[\mu a ] , \alpha} A_b \nonumber \\
& +& \Phi_{(\mu a)} A_{b , \alpha} + \Phi_\mu A_{a , \alpha} A_b
)e^{- \phi_i R^i} R^{ab , \alpha} e^{\phi_i R^i}
+ (\partial_\mu A_{a , \alpha} - \Phi_{(\mu a ) , \alpha} +
\Phi_{[\mu a ] , \alpha} \nonumber \\
&-& \Phi_\mu A_{a , \alpha}) e^{-\phi} e^{- \phi_i R^i} R^{a , \alpha} e^{\phi_i
R^i}
+ (\partial_\mu A_a - \Phi_{(\mu a )} + \Phi_\mu A_a ) e^\phi R^a
+ ( \partial_\mu \phi - \Phi_\mu ) R \nonumber \\
&+&
e^{-\phi_i R^i} \partial_\mu e^{\phi_i R^i} + ...]
+ d y [ e^\phi Q +
(\partial_y A_{ab , \alpha} - \partial_y A_{a , \alpha} A_b
- \Phi_{[ab ] , \alpha} +
\Phi_{ a , \alpha} A_b \nonumber \\
&+& \Phi A_{a ,\alpha} A_b )
e^{- \phi_i R^i} R^{ab , \alpha} e^{\phi_i R^i} + ( \partial_y
A_{a , \alpha}- \Phi_{a , \alpha} - \Phi A_{a ,\alpha} ) e^{-\phi} e^{- \phi_i R^i} R^{a , \alpha}
e^{\phi_i R^i}\nonumber \\
& +& ( \partial_y A_a - \Phi_a + \Phi A_a ) e^\phi R^a
+ ( \partial_y \phi - \Phi ) R
+ e^{-\phi_i R^i} \partial_y e^{\phi_i
R^i} + \dots ]
\quad , \label{4.18}
\end{aligned}$$ where the dots denote contributions from higher level $E_{11}$ generators and Og generators.
Before discussing the Scherk-Schwarz reduction of the IIB theory, we first consider the derivation of the massless nine-dimensional supergravity. We take all the fields in the group element of eq. (\[4.17\]) not to depend on $y$, and using the inverse Higgs mechanism we set the part of the Maurer-Cartan form of eq. (\[4.18\]) in the $dy$ direction and proportional to the $E_{11}$ and Og generators to zero. This imposes $$\Phi_{[ab],\alpha} = \Phi_{a ,\alpha} = \Phi_a = \Phi = 0 \quad . \label{4.19}$$ Considering the $d x^\mu$ part and imposing the inverse Higgs mechanism on the remaining Og fields one finds that the Maurer-Cartan form gives the field strengths $$\begin{aligned}
& & F_{abc ,\alpha} = \partial_{[a} A_{bc ] , \alpha} -
\partial_{[a} A_{b , \alpha} A_{c ]} \nonumber \\
& & F_{ab , \alpha} = \partial_{[a} A_{b ] , \alpha} \nonumber \\
& & F_{ab} = \partial_{[a} A_{b ]} \quad , \label{4.20}
\end{aligned}$$ which are invariant under the gauge transformations $$\delta A_{ab , \alpha} = \partial_{[a} \Lambda_{b ] , \alpha} -
\partial_{[a} \Lambda A_{b ] , \alpha} \qquad \delta A_{a ,
\alpha} = \partial_{a} \Lambda_{\alpha} \qquad \delta A_a =
\partial_{a} \Lambda \quad . \label{.4.21}$$ This construction is consistent because the relations that the inverse Higgs mechanism imposes are invariant under the local subalgebra, which is $SO(1,8) \otimes SO(2)$. The field-strengths and gauge transformations we have derived are those of the 1-forms and 2-forms of massless maximal nine-dimensional supergravity.
In the above, we have set to zero the Og fields corresponding to the generators $K^{[ab],\alpha}$, $K^{a , \alpha}$, $K^a$ and $K$. Implementing this in the group element, and so the Cartan form, we find that these generators in fact play no role. These generators are indeed the only ones in eqs. (\[4.15\]) and (\[4.16\]) that do not commute with $Q$. As such, one is left with the original $E_{11}$ generators and a subset of the Og generators, all of which commute with $Q$ apart from the scalar generator $R$, and all fields which do not depend on $y$. We note that the operator $Q$ appears in the commutation relations of eq. (\[4.9\]) and (\[4.10\]). However, $Q$ commutes with every operator in the theory other that $R$, and the commutator of $R$ with $Q$ is proportional to $Q$, and so one can consistently set the commutator of $R^a$ with $P_a$ to zero and ignore $Q$ in the algebra. Correspondingly, one can ignore the presence of $Q$ in the group element of eq. (\[4.17\]), which corresponds to no $y$ dependence at all. Thus one finds a non-linear realisation that is the one that arises if one constructs the massless nine-dimensional theory using the formulation of $E_{11}$ appropriate to nine dimensions, which corresponds to deleting nodes 9 and 11 in the Dynkin diagram in fig. \[fig1\] and decomposing $E_{11}$ in terms of the $GL(9, \mathbb{R})$ subalgebra. The Og generators that are left are the Og generators that encode the gauge symmetries of the nine-dimensional theory, and they form with the non-negative level $E_{11}$ generators the algebra $E_{11,9}^{local}$. To summarise, the massless nine-dimensional theory arises from taking the subset of Og generators that commute with $Q$. This implies that one can consistently remove $Q$ from the algebra, which can be used to construct the non-linear realisation. This in the nine-dimensional $E_{11}$ formulation of massless maximal supergravity.
We now describe the Scherk-Schwarz dimensional reduction of the ten-dimensional IIB supergravity theory to nine dimensions in an analogous way. We take the same starting point, namely the $E_{11}$ formulation of the IIB theory in ten dimensions together with the Og generators and corresponding fields. We consider the group element $$g = e^{x \cdot P} e^{y ( Q + m_i R^i ) } e^{\Phi_{\rm Og} (x) K^{\rm Og}}
e^{A_{ab ,\alpha}(x) R^{ab ,\alpha}} e^{A_{a , \alpha} (x) R^{a , \alpha}}
e^{A_a (x) R^a} e^{\phi(x) R } e^{\phi_i (x) R^i} \quad . \label{4.22}$$ We thus take the dependence on the coordinate $y$ in the group element to be in the form $e^{y (Q + m_i R^i )}$. This is equivalent to taking the theory to be defined on the conventional nine-dimensional spacetime tensored with a manifold that is a circle constructed form the usual ten-dimensional circle of spacetime and a circle, or one parameter subgroup of $SL(2,\mathbb{R})$, which is specified by the mass parameter $m_i$. The factor $e^{y (Q + m_i R^i
)}$ occurs at the beginning of the group element in the usual place for the introduction of spacetime, and the fields are taken to not depend on $y$, however we can rearrange the group element by taking the $e^{ y m_i R^i}$ factor to the right whereupon the fields acquire a $y$ dependence, that is $$g = e^{x \cdot P} e^{y Q} e^{\Phi_{\rm Og} (x,y) K^{\rm Og}}
e^{A_{ab ,\alpha} (x,y) R^{ab ,\alpha}} e^{A_{a , \alpha} (x,y) R^{a , \alpha}}
e^{A_a (x) R^a} e^{\phi(x) R } e^{y m_i R^i} e^{\phi_i (x) R^i} \quad
. \label{4.23}$$ The $y$ dependence of the fields in this last expression can be derived using the relation $$e^A e^B e^{-A} = e^{e^A B e^{-A}} \quad . \label{4.24}$$ This implies in particular that any $SL(2, \mathbb{R})$ doublet acquires the same $y$ dependence. For instance for the 2-form this is $$A_{ab , \alpha} (x,y) = ( e^{y m_i D^i} )_\alpha{}^\beta A_{ab
,\beta} (x) \quad , \label{4.25}$$ and similarly for any doublet, including the Og fields, while the $SL(2,\mathbb{R})$ singlets acquire no $y$ dependence. This is the $y$ dependence that results in the Scherk-Schwarz dimensional reduction, which consists in compactifying the ten-dimensional theory to nine dimensions on a circle of coordinate $y$, while performing a $y$-dependent $SL(2, \mathbb{R})$ transformation [@27].
From the group element in (\[4.23\]) one obtains the Maurer-Cartan form of eq. (\[4.18\]). It is instructive to write this down to show explicitly the $y$ dependence. The result is $$\begin{aligned}
g^{-1} d g & = & dx^\mu [ P_\mu + A_\mu e^\phi Q +
( \partial_\mu A_{ab , \alpha} - \partial_\mu A_{a , \alpha} A_b -
\Phi_{\mu , ab , \alpha} + \Phi_{(\mu a ) , \alpha} A_b -
\Phi_{[\mu a ] , \alpha} A_b \nonumber \\
& +& \Phi_{(\mu a)} A_{b , \alpha} + \Phi_\mu A_{a , \alpha} A_b
)e^{- \phi_i R^i} e^{-y m_i R^i} R^{ab , \alpha} e^{-y m_i R^i} e^{\phi_i
R^i} \nonumber \\
& + & (\partial_\mu A_{a , \alpha} - \Phi_{(\mu a ) , \alpha} +
\Phi_{[\mu a ] , \alpha}
- \Phi_\mu A_{a , \alpha}) e^{-\phi} e^{- \phi_i R^i} e^{-y m_i R^i} R^{a , \alpha} e^{y m_i R^i} e^{\phi_i
R^i} \nonumber \\
& + & (\partial_\mu A_a - \Phi_{(\mu a )} + \Phi_\mu A_a ) e^\phi R^a
+ ( \partial_\mu \phi - \Phi_\mu ) R
+
e^{-\phi_i R^i} \partial_\mu e^{\phi_i R^i} + ...] \nonumber \\
& + & d y [ e^\phi Q +
(\partial_y A_{ab , \alpha} - \partial_y A_{a , \alpha} A_b
- \Phi_{[ab ] , \alpha} +
\Phi_{ a , \alpha} A_b \nonumber \\
& + & \Phi A_{a ,\alpha} A_b )
e^{- \phi_i R^i} e^{-y m_i R^i} R^{ab , \alpha} e^{y m_i R^i} e^{\phi_i R^i} + ( \partial_y
A_{a , \alpha}- \Phi_{a , \alpha} \nonumber \\
&-& \Phi A_{a ,\alpha} ) e^{-\phi} e^{- \phi_i R^i} e^{-y m_i R^i} R^{a ,
\alpha} e^{y m_i R^i}
e^{\phi_i R^i}
+ ( \partial_y A_a - \Phi_a + \Phi A_a ) e^\phi R^a \nonumber \\
&+& ( \partial_y \phi - \Phi ) R
+ e^{-\phi_i R^i } m_i R^i e^{\phi_i R^i} + \dots ]
\quad . \label{4.26}
\end{aligned}$$ Alternatively, one can compute the Maurer-Cartan form with the group element written as in eq. (\[4.22\]). In this way of writing down the group element, the fields have no $y$ dependence and the $dy$ part of eq. (\[4.26\]) results from passing $m_i R^i$ through the group element. Indeed it can be shown that the two ways of computing the Maurer-Cartan form are identical using the $y$ dependence given as in eq. (\[4.25\]).
As for the massless case, we now use the inverse Higgs mechanism to impose that all the terms in $dy$ proportional to positive level generators vanish, and we get $$\begin{aligned}
& & \Phi_{a , \alpha} (x) = ( m_i D^i )_\alpha{}^\beta A_{a , \beta}
(x) \qquad \Phi_{[ab] , \alpha} (x) = ( m_i D^i )_\alpha{}^\beta A_{ab ,
\beta} (x) \nonumber \\
& & \Phi = \Phi_a =0 \quad . \label{4.27}
\end{aligned}$$ This does not apply to the scalars $\phi_i$, and indeed the $m_i
R^i$ term in the $dy$ part of eq. (\[4.26\]) is not affected by the inverse Higgs mechanism. This will be discussed later. We put the relations of eq. (\[4.27\]) back in the group element and so the Cartan form, and we use the inverse Higgs mechanism on the remaining Og fields such that the $d x^\mu$ part of the Cartan form gives the field-strengths $$\begin{aligned}
& & F_{abc ,\alpha} = \partial_{[a} A_{bc ] , \alpha} -
\partial_{[a} A_{b , \alpha} A_{c ]} -
( m_i D^i )_\alpha{}^\beta A_{[ab , \beta} A_{c ]}
\nonumber \\
& & F_{ab , \alpha} = \partial_{[a} A_{b ] , \alpha}
+( m_i D^i )_\alpha{}^\beta A_{ab , \beta} \quad , \nonumber \\
& & F_{ab} = \partial_{[a} A_{b ]} \quad ,
\label{4.28}
\end{aligned}$$ which transform covariantly under the gauge transformations $$\begin{aligned}
& & \delta A_{ab ,\alpha} = \partial_{[a} \Lambda_{b ] , \alpha} -
\partial_{[a} \Lambda A_{b ] , \alpha} + \Lambda
( m_i D^i )_\alpha{}^\beta A_{ab ,\beta} \nonumber \\
& & \delta A_{a , \alpha} = \partial_{a} \Lambda_{\alpha}
+ \Lambda ( m_i D^i )_\alpha{}^\beta A_{a , \beta} -
( m_i D^i )_\alpha{}^\beta \Lambda_{a , \beta} \quad ,\nonumber \\
& & \delta A_a = \partial_a \Lambda \quad .
\label{4.29}
\end{aligned}$$ These are the field strengths and gauge transformations of the 1-forms and 2-forms of the nine-dimensional gauged maximal supergravity that arises from Scherk-Schwarz reduction of the IIB theory [@27].
We now discuss the scalar sector. One obtains the correct covariant derivative for the scalars observing that the metric that results from the Maurer-Cartan form in eq. (\[4.26\]) is $$\left( \begin{array}{cc}
e_\mu{}^a & A_\mu e^\phi \\
0 & e^\phi \end{array} \right) \label{4.30}$$ as the coefficients of the generators $P_a$ and $Q$ (for simplicity we are actually not considering the gravity contribution in nine dimensions and therefore the coefficient of $P_a$ in eq. (\[4.26\]) is the diagonal metric). The corresponding inverse metric is $$\left( \begin{array}{cc}
e^\mu{}_a & - A_a \\
0 & e^{-\phi} \end{array} \right) \quad . \label{4.31}$$ Taking account of having applied the inverse Higgs mechanism, the only other part of the Maurer-Cartan form along $dy$ is $$G_{y i} R^i = m_i e^{-\phi_i R^i } R^i e^{\phi_i R^i} \quad ,
\label{4.32}$$ while the $R^i$ term along $d x^\mu$ is $$G_{\mu i} R^i = e^{-\phi_i R^i } \partial_\mu e^{\phi_i R^i} \quad
. \label{4.33}$$ Therefore the covariant derivative for the scalar is given by $$e^{\mu}{}_a G_{\mu ,i } - A_a G_{y , i} \quad , \label{4.34}$$ which reads $$e^{-\phi_i R^i} \partial_a e^{\phi_i R^i} - A_a m_i e^{-\phi_i R^i} R^i e^{\phi_i
R^i} \quad . \label{4.35}$$ This analysis therefore gives all the covariant quantities of the nine-dimensional theory corresponding to the Scherk-Schwarz reduction of IIB.
As we observed, eq. (\[4.27\]) expresses some Og fields in terms of $E_{11}$ fields. Similarly, requiring that the Og 1 operators $K^{[ab], \alpha}$ and $K^{a , \alpha}$ have vanishing coefficients in the $dy$ direction relates $\Phi_{[ab ], \alpha}$ and $\Phi_{a ,
\alpha}$ to Og 2 fields carrying the same spacetime and $SL(2,
\mathbb{R})$ representations. Iterating this one obtains for any $n$ an Og $n$ generator identified with $A_{ab ,\alpha}$ times the $n$th power of the mass parameter, and similarly for $A_{a ,\alpha}$. This generalises to all the fields in the theory. Putting these solutions into the original group element of eq. (\[4.22\]) we find that it takes the form $$g = e^{x \cdot P} e^{y ( Q + m_i R^i )} e^{\Phi_{\rm Og} (x) \tilde{K}^{\rm Og}}
e^{A_{ab ,\alpha}(x) \tilde{R}^{ab ,\alpha}} e^{A_{a , \alpha} (x) \tilde{R}^{a , \alpha}}
e^{A_a (x) R^a} e^{\phi(x) R } e^{\phi_i (x) R^i} \quad ,
\label{4.36}$$ where $$\begin{aligned}
& & \tilde{R}^{a , \alpha} = {R}^{a , \alpha} + m_i D^i_\beta{}^\alpha
K^{a, \beta} + ... \nonumber \\
& & \tilde{R}^{ab , \alpha} = {R}^{ab , \alpha} + m_i D^i_\beta{}^\alpha
K^{[ab], \beta} + ... \quad , \label{4.37}
\end{aligned}$$ where the dots correspond to higher powers in $m_i$ multiplying higher grade Og generators, and $\tilde{K}$ denotes deformed Og generators associated with nine-dimensional gauge transformations. The group element of eq. (\[4.36\]) resembles the group element corresponding to the massless nine-dimensional theory, in the sense that each generator in eq. (\[4.36\]) corresponds to a generator with identical index structure of the massless nine-dimensional theory. As such we can interpret the $\tilde{R}$ generators as deformed $E_{11}$ generators. In particular, we claim that although the expansions in eq. (\[4.37\]) are non-polynomial in $m_i$, all the commutation relations involving these operators, or the commutation relations between these operators and momentum, only contain terms at most linear in $m_i$. In particular, the commutator of $\tilde{R}^{ab ,\alpha}$ with $P_a$ is $$[ \tilde{R}^{ab ,\alpha}, P_c ] = - (m_i D^i
)_\beta{}^\alpha \delta^{[a}_c \tilde{R}^{b ] , \beta} \quad ,
\label{4.38}$$ while the commutator of $\tilde{R}^{a ,\alpha}$ with $P_a$ vanishes, as can be seen from eq. (\[4.15\]).
The deformed $E_{11}$ and indeed the deformed Og generators have a simple algebraic classification. They are the operators that commute with the operator $\tilde{Q}$ defined as $$\tilde{Q} = Q + m_i R^i \quad . \label{4.39}$$ Indeed, the commutation relation of $R^a$, $R^{a, \alpha}$ and $R^{ab, \alpha}$ with $\tilde{Q}$ is $$\begin{aligned}
& & [ \tilde{Q} , R^a ] = 0 \nonumber \\
& & [ \tilde{Q} , R^{a , \alpha} ] = m_i D^i_\beta{}^\alpha R^{a ,
\beta} \nonumber \\
& & [ \tilde{Q} , R^{ab , \alpha} ] = m_i D^i_\beta{}^\alpha R^{ab ,
\beta} \quad . \label{4.40}
\end{aligned}$$ We thus have to deform the operators $R^{a ,\alpha}$ and $R^{ab
,\alpha}$, and from eq. (\[4.15\]) one gets that the deformed operators given in eq. (\[4.37\]) satisfy $$[ \tilde{Q} , \tilde{R}^{a , \alpha} ] = [ \tilde{Q} , \tilde{R}^{ab , \alpha} ] = 0
\quad . \label{4.41}$$ The Og generators of the nine-dimensional theory are also redefined in order to commute with $\tilde{Q}$. One thus constructs the generators $\tilde{K}^{a, bc , \alpha}$ and $\tilde{K}^{(ab) ,
\alpha}$, which are Og 1 generators followed by an expansion in $m_i$ of higher grade Og generators. The Og 1 generator $K^{(ab)}$ of the singlet vector $R^a$ is not modified as it commutes with $\tilde{Q}$.
Having introduced the operator $\tilde{Q}$, we can write down the group element of eq. (\[4.36\]) as $$g = e^{x \cdot P} e^{y \tilde{Q}} e^{\Phi_{\rm Og} (x) \tilde{K}^{\rm Og}}
e^{A_{ab ,\alpha}(x) \tilde{R}^{ab ,\alpha}} e^{A_{a , \alpha} (x) \tilde{R}^{a , \alpha}}
e^{A_a (x) R^a} e^{\phi(x) R } e^{\phi_i (x) R^i} \quad .
\label{4.42}$$ Indeed, we now show that using the operator $\tilde Q$ rather than $Q$ one obtains the field strengths, including the covariant derivative of the scalars, in a straightforward way. Calculating the Cartan forms from the group element of eq. (\[4.42\]) and using eq. (\[4.38\]) and the fact that all the positive level operators commute with $\tilde{Q}$ we find $$\begin{aligned}
g^{-1} d g & = & dx^a [ P_a + (\partial_a A_{bc , \alpha} -
\partial_a A_{b ,\alpha} A_c - (m_i D^i )_\alpha{}^\beta A_{ab ,
\beta} A_c + ...) e^{-\phi_i R^i} \tilde{R}^{bc , \alpha} e^{\phi_i
R^i} \nonumber \\
&+& (\partial_a A_{b ,\alpha} + (m_i D^i )_\alpha{}^\beta A_{ab ,
\beta} +...) e^{-\phi} e^{-\phi_i R^i} \tilde{R}^{b , \alpha} e^{\phi_i
R^i} + (\partial_a A_b + ...) e^\phi R^b \nonumber \\
&+& \partial_a \phi R + A_a e^{-\phi_i R^i} e^{-\phi R} \tilde{Q}
e^{\phi R} e^{\phi_i R^i} + e^{-\phi_i R^i} (\partial_a - A_a m_i
R^i ) e^{\phi_i R^i} ] \nonumber \\
& + & dy e^{-\phi_i R^i} e^{-\phi R} \tilde{Q}
e^{\phi R} e^{\phi_i R^i} \quad , \label{4.43}
\end{aligned}$$ where the dots in each term denote the Og field contributions, whose role is to cancel the non-antisymmetric terms in the Cartan form using the inverse Higgs mechanism. As explained above $g^{-1}dg$ is invariant under $g\to g_0 g$ and so all the coefficients of the generators in the above equation are invariant. Hence, in particular the two terms $$dx^a P_a \quad , \qquad
dx^a e^{-\phi_i R^i}(\partial_a -A_a m_i R^i)e^{\phi_i R^i}
\label{4.44}$$ are separately invariant under $g_0$ transformations. Hence we can identify the covariant derivative of the scalars as $$e^{-\phi_i R^i}(\partial_a -A_a m_i R^i )e^{\phi_i R^i}
\label{4.45}$$ which now only transforms under the local transformations. The infinite number of rigid $g_0$ transformations constitute the gauge transformations and so this covariant derivative is also covariant in the conventional sense.
We note that the operator $\tilde Q$ and the variable $y$ although important for the logic of the result did not appear explicitly in the calculation of the terms that lead to this covariant derivative. Indeed one could have written down the group element without any $\tilde Q$ or $y$ dependence and the Cartan forms would give the correct covariant derivatives and so gauge invariant quantities. Dropping the operator $\tilde{Q}$, one obtains in particular the commutation relation $$[ R^a , P_b ] = \delta^a_b m_i R^i \quad
. \label{4.46}$$ We now consider eq. (\[4.46\]) as our starting point to define the nine-dimensional algebra $\tilde{E}_{11,9}^{local}$. This is the algebra that describes the deformed nine-dimensional theory considered in this section, and contains the generators $m_i R^i$, $P_a$ and all the positive level deformed generators, including the deformed Og generators. In the remaining of this section we will show that all the results obtained so far can be derived simply requiring the closure of the Jacobi identities in $\tilde{E}_{11,9}^{local}$ starting from eq. (\[4.46\]). This approach is entirely nine-dimensional, and one never makes use of the fact that the theory has a ten-dimensional origin. As we will see, this provides an extremely fast method of deriving the field strengths of all gauged maximal supergravities.
We start considering the Jacobi identity involving $R^a$, $\tilde{R}^{a , \alpha}$ and $P_a$. The commutator between $R^a$ and $\tilde{R}^{b , \alpha}$ is a deformation of the commutator in eq. (\[4.11\]), and the most general expression we can write with the generators at our disposal is $$[ R^a , \tilde{R}^{b , \alpha}] = - \tilde{R}^{ab ,\alpha} + a m_i D^i_\beta{}^\alpha
\tilde{K}^{(ab), \alpha} \quad
, \label{4.47}$$ with $a$ to be determined, and where $\tilde{K}^{(ab), \alpha}$ is the modified Og 1 generator satisfying $$[ \tilde{K}^{(ab), \alpha} , P_c ] = \delta^{(a}_c \tilde{R}^{b),
\alpha} \quad . \label{4.48}$$ We also demand that the commutator between $\tilde{R}^{ab, \alpha}$ and $P_c$ be of the form $$[ \tilde{R}^{ab ,\alpha}, P_c ] = b (m_i D^i
)_\beta{}^\alpha \delta^{[a}_c \tilde{R}^{b ] , \beta} \quad ,
\label{4.49}$$ with the parameter $b$ to be determined. The Jacobi identity involving $R^a$, $\tilde{R}^{a , \alpha}$ and $P_a$ is satisfied provided that the values of $a$ and $b$ are $$a = 1 \qquad \qquad b = -1 \quad . \label{4.50}$$ To summarise, we have obtained the relations $$\begin{aligned}
& & [ R^a , \tilde{R}^{b , \alpha}] = - \tilde{R}^{ab ,\alpha} + m_i D^i_\beta{}^\alpha
\tilde{K}^{(ab), \alpha} \nonumber \\
& & [ \tilde{R}^{ab ,\alpha}, P_c ] = - (m_i D^i
)_\beta{}^\alpha \delta^{[a}_c \tilde{R}^{b ] , \beta} \quad ,
\label{4.51}
\end{aligned}$$ and in particular the second relation coincides with eq. (\[4.38\]).
Proceeding this way, one can determine all the commutation relations of the modified $E_{11}$ generators among themselves and with the momentum operator $P_a$. For instance, the Jacobi identity involving the operators $R^a$, $\tilde{R}^{ab , \alpha}$ and $P_a$ requires the cancellation of terms linear in $m_i$ as well as terms quadratic in $m_i$. The latter are cancelled by requiring that also the commutator of $\tilde{K}^{a,bc ,\alpha}$ with $P_d$ receives a correction at order $m_i$. The result is $$[ R^a , \tilde{R}^{bc ,
\alpha} ] = {3 \over 2} (m_i D^i
)_\beta{}^\alpha \tilde{K}^{a , bc , \beta}
\label{4.52}$$ and $$[ \tilde{K}^{a,bc ,\alpha} , P_d ] = \delta^a_d \tilde{R}^{bc ,
\alpha} - \delta^{[a}_d \tilde{R}^{bc ] , \alpha} - { 1 \over 3} (m_i D^i
)_\beta{}^\alpha ( \delta^b_d \tilde{K}^{ac , \beta} - \delta^c_d \tilde{K}^{ab ,
\beta} )
\quad . \label{4.53}$$ Using the definition of the operator $\tilde{R}^{ab ,\alpha}$ in eq. (\[4.37\]), and eq. (\[4.14\]), one can for instance recover eq. (\[4.52\]), that we have obtained requiring the closure of the Jacobi identities, directly using the ten-dimensional commutation relations. Indeed, at lowest order in the mass parameter, one gets $$[ R^a , \tilde{R}^{bc ,
\alpha} ] = (m_i D^i
)_\beta{}^\alpha [ K^a{}_y , K^{y,bc , \beta} - K^{[b,c]y ,
\beta} ] = {3 \over 2} (m_i D^i
)_\beta{}^\alpha K^{a ,bc ,\beta} \quad . \label{4.54}$$
In order to show the power of this method, we now determine the field-strengths for the 3-form and the 4-form of the nine-dimensional massive theory without using its ten-dimensional origin. We first write down the relevant commutators of the massless theory. We thus add to the commutators of eqs. (\[4.11\]) and (\[4.12\]) all the commutators that involve generators up to the 4-form included. We only write down the non-vanishing commutators, that are $$\begin{aligned}
& & [ R , R^{abc} ] = R^{abc} \qquad \qquad \qquad [ R^a , R^{bcd} ] = - R^{abcd}
\nonumber \\
& & [ R^{a , \alpha} , R^{bc , \beta} ] = \epsilon^{\alpha \beta}
R^{abc} \quad \qquad \ [ R^{ab , \alpha} , R^{cd , \beta} ] = \epsilon^{\alpha \beta}
R^{abcd} \quad , \label{4.55}
\end{aligned}$$ where $\epsilon^{\alpha \beta}$ is the invariant antisymmetric tensor of $SL(2, \mathbb{R})$. Starting from these relations and using eq. (\[4.46\]) we can determine all the commutation relations involving such deformed generators by imposing the closure of the Jacobi identities. Denoting with $\tilde{R}^{abc}$ and $\tilde{R}^{abcd}$ the deformed generators, one can show that the only commutation relation that needs to be modified with respect to eq. (\[4.55\]) is the commutator between two deformed 2-form generators, which becomes $$[ \tilde{R}^{ab ,\alpha} , \tilde{R}^{cd , \beta} ] =
\epsilon^{\alpha \beta} \tilde{R}^{abcd} + 2 (m_i D^i )^{\alpha
\beta} \tilde{K}^{[a , b] cd} \quad , \label{4.56}$$ where $\tilde{K}^{a , b cd}$ is the deformed Og 1 generator associated to the deformed 3-form generator $\tilde{R}^{abc}$, satisfying $$[ \tilde{K}^{a , b_1 b_2 b_3} , P_c ] = \delta^a_c \tilde{R}^{b_1
b_2 b_3} - \delta^{ [a}_c \tilde{R}^{b_1 b_2 b_3 ]} \label{4.57}$$ and we have used $\epsilon^{\alpha \beta}$ to raise the $SL(2,
\mathbb{R})$ index, that is $$D^{i \ \alpha \beta} = \epsilon^{\alpha \gamma} D^i_\gamma{}^\beta
\quad , \label{4.58}$$ and $D^{i \ \alpha \beta}$ is symmetric. The deformed Og 1 generator for the 4-form is $\tilde{K}^{a , b_1 b_2 b_3}$, satisfying $$[ \tilde{K}^{a , b_1 ...b_4} , P_c ] = \delta^a_c \tilde{R}^{b_1
... b_4} - \delta^{ [a}_c \tilde{R}^{b_1 ... b_4 ]} \quad .
\label{4.59}$$ Both the deformed 3-form and the deformed 4-form commute with the momentum operator (actually neither the 3-form nor the 4-form generator are really deformed, but this is not relevant for this analysis).
We now consider the group element $$g = e^{x \cdot P} e^{\Phi_{\rm Og} \tilde{K}^{\rm Og}}
e^{A_{abcd} \tilde{R}^{abcd}} e^{A_{abc} \tilde{R}^{abc}}
e^{A_{ab ,\alpha} \tilde{R}^{ab ,\alpha}} e^{A_{a , \alpha} \tilde{R}^{a , \alpha}}
e^{A_a R^a} e^{\phi R } e^{\phi_i R^i} \quad ,
\label{4.60}$$ which only depends on the nine-dimensional coordinates $x^a$. Computing the Maurer-Cartan form and applying the inverse Higgs mechanism, one can show that all the terms which are not antisymmetric are set to zero by fixing the Og 1 fields in terms of the $E_{11}$ fields, and one is left with completely antisymmetric terms. These are the field-strengths of the 1-forms and 2-forms given in eq. (\[4.28\]), as well as the field-strengths $$\begin{aligned}
& & F_{a_1 \dots a_4} = \partial_{[a_1} A_{a_2 a_3 a_4 ]} +
\epsilon^{\alpha \beta} \partial_{[a_1} A_{a_2 a_3 , \alpha}
A_{a_4 ] , \beta} - {1 \over 2} (m_i D^i )^{ \alpha \beta} A_{[
a_1 a_2 , \alpha} A_{a_3 a_4 ] , \beta} \nonumber \\
& & F_{a_1 \dots a_5} = \partial_{[a_1} A_{a_2 ... a_5 ]} -
\partial_{[a_1} A_{a_2 a_3 a_4} A_{a_5 ]}+ {1 \over 2} \epsilon^{\alpha \beta}
\partial_{[a_1} A_{a_2 a_3 , \alpha } A_{a_4 a_5 ] , \beta} \nonumber \\
& & \qquad \quad -
\epsilon^{\alpha \beta} \partial_{[a_1} A_{a_2 a_3 , \alpha}
A_{a_4 , \beta} A_{a_5 ]} + {1 \over 2} (m_i D^i )^{ \alpha \beta} A_{[
a_1 a_2 , \alpha} A_{a_3 a_4 , \beta} A_{a_5 ]} \label{4.61}
\end{aligned}$$ for the 3-form and the 4-form. These are indeed the field-strengths of the 3-form and its dual 4-form of the massive nine-dimensional supergravity. The gauge transformations of the fields arise in the non-linear realisation as rigid transformations of the group element with the Og generators included. One obtains the transformations of the 2-forms and 1-forms given in eq. (\[4.29\]) as well as the transformations $$\begin{aligned}
& & \delta A_{abc} = \partial_{[a} \Lambda_{bd ]} + \epsilon^{\alpha
\beta} \partial_{[a} \Lambda_{\alpha} A_{bc ] , \beta} + {1 \over
2} \epsilon^{\alpha \beta} \partial_{[a} \Lambda A_{b , \alpha}
A_{c] , \beta} + (m_i D^i )^{ \alpha \beta} \Lambda_{[a , \alpha}
A_{bc ], \beta} \nonumber \\
& & \delta A_{a_1 ... a_4} = \partial_{[a_1} \Lambda_{a_2 a_3 a_4 ]}
+ {1 \over 2}\epsilon^{\alpha
\beta} \partial_{[a_1} \Lambda_{a_2 , \alpha} A_{a_3 a_4 ] , \beta}
- \partial_{[a_1} \Lambda A_{a_2 a_3 a_4 ]} \nonumber \\
& & \qquad \qquad
+ {1 \over
2} \epsilon^{\alpha \beta} \partial_{[a_1} \Lambda A_{a_2 , \alpha}
A_{a_3 a_4 ] , \beta} \label{4.62}
\end{aligned}$$ of the 3-form and the 4-form.
Observe that although the operators $R^i$ other than $m_i R^i$ do not belong to the algebra $\tilde{E}_{11,9}^{local}$, one can nonetheless use the group element of eq. (\[4.60\]). Indeed, the covariant derivative for the scalars is also obtained from the nine-dimensional group element of eq. (\[4.60\]). Indeed, the Maurer-Cartan form contains the terms $$e^{-\phi_i R^i }
\partial_\mu e^{\phi_i R^i }
- A_\mu e^{-\phi_i R^i } m_i R^i e^{\phi_i R^i } \quad , \label{4.63}$$ which we recognise as the covariant derivative of the scalars.
To summarise, we have found a general pattern for carrying out dimensional reduction to obtain gauged supergravities. The higher dimensional coordinates have a generator ($Q$ in the massless case and $\tilde{Q}$ in the Scherk-Schwarz case) which is associated with the space being reduced on. From the set on $E_{11}$ and Og generators, we can find a set of deformed $E_{11}$ generators which are just those that commute with the preferred generator associated with the reduction ($Q$ or $\tilde{Q}$). The field strengths can then just be deduced from this deformed $E_{11}$ algebra. We will see that this method transcends dimensional reduction and in fact applies to all gauged maximal supergravities. This will be the focus on the next two sections.
$E_{11}$ and massive IIA
========================
In the last section we have analysed the massless and Scherk-Schwarz reductions to nine dimensions of ten-dimensional IIB supergravity from an $E_{11}$ perspective. Starting from the algebra $E_{11,10B}^{local}$ of $E_{11}$ plus the Og generators that encodes all the gauge symmetries of the ten-dimensional theory, the massless dimensional reduction corresponds to taking the $E_{11}$ generators together with the subset of Og generators that commute with the momentum operator in the internal direction. On the other hand, the Scherk-Schwarz dimensional reduction corresponds to choosing operators that commute with a twisted internal momentum operator, and the twist is such that these operators are combinations of the ten-dimensional $E_{11}$ and Og generators. It is important to stress that the content of the sets of generators in the massless and the Scherk-Schwarz theory are exactly the same, and the two theories differ because the commutation relations are different. In particular, the set of non-negative level $E_{11}$ generators in the massless theory is the same as the set of operators in the Scherk-Schwarz reduction case that are obtained by adding to the $E_{11}$ generators suitable Og generators of the ten-dimensional theory multiplied by powers of the mass deformation parameter $m_i$, where $i$ is an $SL(2, \mathbb{R})$ triplet index. From the nine-dimensional perspective, these operators look like $E_{11}$ generators, but their commutation relation receives a correction at order $m_i$. Therefore, the algebra appears from the nine-dimensional perspective as a deformation of the original $E_{11}$ algebra. This deformation is such that the commutator of two positive level generators gives the standard $E_{11}$ result at zero order in $m_i$ together with an order $m_i$ deformation proportional to the Og generators of the nine-dimensional theory. Correspondingly, the commutator of the deformed positive level $E_{11}$ generators with the nine-dimensional momentum is proportional to the deformed $E_{11}$ generators times the mass parameter. Starting from the commutation relation (\[4.46\]), the entire algebra of the nine-dimensional deformed theory can be determined by requiring that the Jacobi identities close.
In this section we consider the case of the massive deformation of the IIA theory, discovered by Romans in [@8]. In this case the theory does not arise as a dimensional reduction of eleven-dimensional supergravity, and therefore one cannot deform the $E_{11}$ generators adding eleven-dimensional Og generators. Nonetheless, we will show that from the ten-dimensional perspective one can still consider deformed $E_{11}$ and Og generators, and the corresponding algebra $\tilde{E}_{11,10A}^{local}$, which appears as a deformation of the massless ten-dimensional algebra, determines all the field-strengths of the theory. In [@11] it was shown that the massive IIA theory can be recovered from an $E_{11}$ perspective by adopting a non-trivial commutation relation between the momentum operator and the positive level generators. In particular the commutator of the 2-form generator with momentum gives the 1-form generator multiplied by the mass deformation parameter. The resulting algebra [@11] though has a problem of consistency because the corresponding Jacobi identities do not close. In [@28] it was shown that if one insists on requiring the consistency of the algebra for the lower-rank forms, the commutator of two 2-forms cannot vanish in the massive theory, but instead is proportional to an operator in the (3,1) representation of $GL(10, \mathbb{R})$. This operator is indeed the Og 1 operator for the 3-form. We show that the whole algebra corresponding to the massive IIA theory is determined starting from the deformed commutation relation of the 2-form with momentum and requiring the closure of all the Jacobi identities. A different approach, based on the Kac-Moody algebra $E_{10}$ [@29], has recently been given in [@30].
We start by writing down the algebra associated to the massless IIA theory. The massless IIA theory arises from the dimensional reduction of eleven-dimensional supergravity. The corresponding algebra arises from a decomposition of the $E_{11}$ algebra in terms of $GL(10,\mathbb{R})$ as relevant for the IIA theory, which corresponds to deleting nodes 10 and 11 in the Dynkin diagram in fig. \[fig1\]. In this section we denote with $a,b,...$ the tangent spacetime indices in ten dimensions. In deriving the $E_{11}$ generators in terms of their $GL(10,\mathbb{R})$ IIA representations it is useful to consider the eleven-dimensional generators and denote with $y$ the internal 11th coordinate. One then obtains that the theory contains a scalar $R$, which is the $GL(11,\mathbb{R})$ generator $K^y{}_y$, a vector $R^a$ corresponding to the eleven-dimensional $K^a{}_y$, a 2-form $R^{ab}$ which arises from the eleven-dimensional 3-form with one index in the internal direction $R^{aby}$, and then a 3-form $R^{abc}$, a 5-form $R^{a_1 \dots a_5}$, a 6-form $R^{a_1 \dots a_6}$, a 7-form $R^{a_1 \dots a_7}$, an 8-form $R^{a_1 \dots a_8}$, a 9-form $R^{a_1
\dots a_9}$ and two 10-forms $R^{a_1 \dots a_{10}}$ and $R^{\prime
a_1 \dots a_{10}}$, together with an infinite set of generators with mixed, [*i.e.*]{} not completely antisymmetric, indices [@12].
We now write down the part of the $E_{11}$ algebra that involves these completely antisymmetric generators. This was first derived in [@11], but we use different normalisations for the generators, that make the eleven-dimensional origin of the algebra more transparent. For simplicity, we will neglect the contribution from the 10-form generators, that is we will consider a level truncation only involving generators up to the 9-form included. The algebra is $$\begin{aligned}
& & [ R, R^a ] = - R^a \qquad \qquad \quad \ \ \ [ R , R^{ab} ] = R^{ab}\nonumber \\
& & [ R ,
R^{a_1 \dots a_5} ] = R^{a_1 \dots a_5} \qquad \ \quad [ R , R^{a_1 \dots a_7} ]
= 2 R^{a_1 \dots a_7} \nonumber \\
& & [ R , R^{a_1 \dots
a_8} ] = R^{a_1 \dots a_8} \qquad \quad \ [ R, R^{a_1 \dots a_9} ] = 3 R^{a_1 \dots
a_9} \nonumber \\
& & [ R^a , R^{bc} ] = R^{abc} \qquad \qquad \quad \ \ [ R^{a_1} , R^{a_2 \dots
a_6} ] = - R^{a_1 \dots a_6} \nonumber \\
& & [ R^{a_1} , R^{a_2 \dots
a_8} ] = 3 R^{a_1 \dots a_8} \qquad \ [ R^{a_1 a_2} , R^{a_3 \dots
a_5} ] = - 2R^{a_1 \dots a_5} \nonumber \\
& & [ R^{a_1 a_2} , R^{a_3 \dots
a_7} ] = R^{a_1 \dots a_7} \qquad [ R^{a_1 a_2} , R^{a_3 \dots a_8} ] = - 2 R^{a_1 \dots a_8} \nonumber
\\
& & [ R^{a_1 a_2} , R^{a_3 \dots a_9} ] = R^{a_1 \dots a_9} \qquad
[ R^{a_1 \dots a_3} , R^{a_4 \dots
a_6} ] = 2 R^{a_1 \dots a_6}\nonumber \\
& & [ R^{a_1 \dots a_3} , R^{a_4 \dots
a_8} ] = R^{a_1 \dots a_8} \quad , \label{5.1}
\end{aligned}$$ with all the other commutators vanishing or giving generators with mixed symmetries. One can show that all the Jacobi identities involving these operators are satisfied. Following the results of section 4, one can then obtain the Og generators of the ten-dimensional theory by decomposing the Og generators of the eleven-dimensional theory in terms of representations of $GL(10,\mathbb{R})$. The subset of such generators that commute with the momentum operator along the 11th direction are the Og generators of the massless IIA theory. These are exactly the operators that are needed to encode all the gauge symmetries of the fields of the massless IIA theory. This corresponds to the fact that the massless IIA theory arises as a circle dimensional reduction of eleven-dimensional supergravity. In particular, for each $n$-form $E_{11}$ generator the corresponding Og 1 operator is $K^{a, b_1
\dots b_n}$ satisfying $K^{[ a, b_1 \dots b_n ]}=0$.
We now consider the deformation of the massless IIA algebra giving rise to massive IIA. This theory was constructed by Romans in [@8], and it corresponds to a Higgs mechanism in which the 2-form acquires a mass by absorbing the vector. In [@11] this mechanism was recovered from an $E_{11}$ perspective by adopting a non-trivial commutation relation between the 2-form generator and momentum. Following the results of section 4, we interpret this as a redefinition of the $E_{11}$ generators. We thus denote all the generators of the massive theory with a tilde. These generators, although forming a set identical to the one corresponding to the $E_{11}$ generators of the massless theory, have different commutation relations. These commutation relations make the corresponding algebra look like a deformation of the massless algebra involving the mass parameter. We thus write down the commutation relation between the 2-form and momentum as $$[ \tilde{R}^{ab} , P_c ] = - m \delta^{[a}_c \tilde{R}^{b]} \quad ,
\label{5.2}$$ where $m$ is the Romans mass parameter. Our strategy is to use eq. (\[5.2\]) as our starting point, and to derive all the commutation relations of the deformed theory from it imposing the closure of the Jacobi identities. We will show that this will fix all the field-strengths and gauge transformations of the forms in the theory. In [@31] it was shown that the supersymmetry algebra of IIA closes on all the forms predicted by $E_{11}$, and the field-strengths and gauge transformations of the form fields were derived imposing the closure of the supersymmetry algebra. We will show that the field-strengths and gauge transformations as obtained using supersymmetry exactly coincide, up to field redefinitions, with the ones obtained here from $E_{11}$.
The Jacobi identity involving the operators $\tilde{R}$, $\tilde{R}^{ab}$ and $P_a$ imposes that $$[ \tilde{R} , P_a ] =-2 P_a \quad . \label{5.3}$$ Introducing the Og 1 operator for the deformed 3-form, defined as $$[ \tilde{K}^{a , b_1 b_2 b_3} , P_c ] = \delta^a_c \tilde{R}^{b_1 b_2 b_3}
- \delta^{[a}_c \tilde{R}^{b_1 b_2 b_3 ]} \label{5.4}$$ one can then show that the Jacobi identity between two 2-forms and momentum imposes [@28] $$[ \tilde{R}^{ab} , \tilde{R}^{cd} ] = - 2m \tilde{K}^{[a,b]cd}
\label{5.5}
\quad .$$ One can then show that the Jacobi identities involving the scalar operator $\tilde{R}$ require a non-trivial commutation relation between the deformed 7-form generator and the momentum operator, while the commutator between the 2-form and the 5-form generator has to be modified by a term proportional to the Og 1 $\tilde{K}^{a ,
b_1 \dots b_6}$ the 6-form, which satisfies $$[ \tilde{K}^{a , b_1 \dots b_6} , P_c ] = \delta^a_c \tilde{R}^{b_1 \dots b_6}
- \delta^{[a}_c \tilde{R}^{b_1 \dots b_6 ]} \quad . \label{5.6}$$ The result is $$[ \tilde{R}^{a_1 \dots a_7} , P_b ] = m \delta^{[a_1}_b \tilde{R}^{a_2 \dots a_7
]} \label{5.7}$$ and $$[ \tilde{R}^{a_1 a_2} , \tilde{R}^{b_1\dots b_5} ] = \tilde{R}^{a_1 a_2 b_1 \dots b_5}
+ m \tilde{K}^{[ a_1 , a_2 ] b_1 \dots b_5}
\label{5.8}
\quad .$$ Finally, the Jacobi identities also impose that the commutator between the 9-form and momentum, as well as the commutator between the 2-form and the 7-form, must be modified. The result is $$[ \tilde{R}^{a_1 \dots a_9} , P_b ] = - 5 m \delta^{[a_1}_b \tilde{R}^{a_2 \dots a_9
]} \label{5.9}$$ and $$[ \tilde{R}^{a_1 a_2} , \tilde{R}^{b_1\dots b_7} ] = \tilde{R}^{a_1 a_2 b_1 \dots b_7}
- {17 \over 7}
m \tilde{K}^{[ a_1 , a_2 ] b_1 \dots b_7}
\label{5.10}
\quad ,$$ where $\tilde{K}^{a, b_1 \dots b_8}$ is the Og 1 operator for the 8-form, satisfying $$[ \tilde{K}^{a , b_1 \dots b_8} , P_c ] = \delta^a_c \tilde{R}^{b_1 \dots b_8}
- \delta^{[a}_c \tilde{R}^{b_1 \dots b_8 ]} \quad . \label{5.11}$$ All the other commutators are not modified, and they are as in eq. (\[5.1\]) with all operators replaced by deformed operators.
To summarise, we have shown that starting from the $E_{11}$ algebra of eq. (\[5.1\]) and introducing the deformed 2-form generator which satisfies the commutation relation of eq. (\[5.2\]), the Jacobi identities determine completely the rest of the algebra. In particular, once the algebra is expressed in terms of the tilde generators, the only commutators that are modified with respect to eq. (\[5.1\]) are those of eqs. (\[5.5\]), (\[5.8\]) and (\[5.10\]), while the additional non-trivial commutation relations with $P_a$ are given is eqs. (\[5.3\]), (\[5.7\]) and (\[5.9\]).
We now consider the group element $$g = e^{x \cdot P} e^{\Phi_{\rm Og} \tilde{K}^{\rm Og}}
e^{A_{a_1 \dots a_9} \tilde{R}^{a_1 \dots a_9}} \dots
e^{A_{a_1 \dots a_5} \tilde{R}^{a_1 \dots a_5}}
e^{A_{a_1 \dots a_3} \tilde{R}^{a_1 \dots a_3}} e^{A_{a_1 a_2} \tilde{R}^{a_1
a_2}}
e^{A_{a} \tilde{R}^{a}} e^{\phi \tilde{R}}
\label{5.12}$$ where we denote with $\Phi_{\rm Og}$ the whole set of Og 1 field of the ten-dimensional massive IIA theory. Similarly we denote with $\tilde{K}^{\rm Og}$ the whole set of deformed ten-dimensional Og 1 operators, which we treat as commuting because we are ignoring the contribution of Og 2 generators for simplicity. One can compute the Maurer-Cartan form that results from the group element in eq. (\[5.12\]). The result is $$\begin{aligned}
g^{-1} \partial_\mu g & = & e^{2\phi} P_\mu + \partial_\mu \phi \tilde{R} + (\partial_\mu A_a +
m A_{\mu a} + ...) e^{\phi} \tilde{R}^a + (\partial_\mu A_{a_1 a_2} + ...)
e^{- \phi}
\tilde{R}^{a_1 a_2}\nonumber \\
& + & (\partial_\mu A_{a_1 a_2 a_3} - \partial_\mu A_{a_1
a_2} A_{a_3} + {m \over 2} A_{\mu a_1} A_{a_2 a_3} +...) \tilde{R}^{a_1
... a_3} \nonumber \\
& +& (\partial_\mu A_{a_1 ...a_5} +2 \partial_\mu A_{a_1
a_2 a_3} A_{a_4 a_5} + {m \over 3} A_{\mu a_1} A_{a_2 a_3} A_{a_4 a_5}+...)
e^{- \phi} \tilde{R}^{a_1
... a_5} \nonumber \\
& + & (\partial_\mu A_{a_1 ...a_6} - \partial_\mu A_{a_1 ...a_5}
A_{a_6} -2 \partial_\mu A_{a_1 a_2 a_3} A_{a_4 a_5} A_{a_6} +
\partial_\mu A_{a_1 a_2 a_3 } A_{a_4 a_5 a_6}\nonumber \\
& -& {m \over 3} A_{\mu a_1} A_{a_2 a_3} A_{a_4 a_5} A_{a_6} - m A_{\mu a_1 ...a_6} +...)
\tilde{R}^{a_1
... a_6} \nonumber \\
& +& (\partial_\mu A_{a_1 ...a_7} - \partial_\mu A_{a_1 ...a_5} A_{a_6
a_7} - \partial_\mu A_{a_1 a_2 a_3} A_{a_4 a_5 } A_{a_6 a_7}
- {m \over 12} A_{\mu a_1} A_{a_2 a_3} A_{a_4 a_5} A_{a_6 a_7} \nonumber \\
&+& ...)
e^{- 2 \phi} \tilde{R}^{a_1
... a_7} + ( \partial_\mu A_{a_1 ...a_8} + 3 \partial_\mu A_{a_1
...a_7} A_{a_8} +2 \partial_\mu A_{a_1 ...a_6} A_{a_7 a_8} \nonumber \\
&+&
\partial_\mu A_{a_1 ...a_5} A_{a_6 ...a_8} -3 \partial_\mu A_{a_1
...a_5} A_{a_6 a_7} A_{a_8} -3 \partial_\mu A_{a_1 a_2 a_3} A_{a_4
a_5 }A_{a_6 a_7} A_{ a_8} \nonumber \\
&+& 2 \partial_\mu A_{a_1 ..a_3} A_{a_4 ...a_6} A_{a_7 a_8} -{m
\over 4} A_{\mu a_1} A_{a_2 a_3} A_{a_4 a_5} A_{a_6 a_7} A_{a_8} -
2m A_{\mu a_1 ...a_6} A_{a_7 a_8} \nonumber \\
&+& 5m A_{\mu a_1 ...a_8}+ ...) e^{- \phi} \tilde{R}^{a_1 ... a_8} + (
\partial_\mu A_{a_1 ...a_9} - \partial_\mu A_{a_1 ...a_7} A_{a_8
a_9}\nonumber \\
& + &{1 \over 2} \partial_\mu A_{a_1 ...a_5} A_{a_6 a_7} A_{a_8
a_9} + {1 \over 3} \partial_\mu A_{a_1 a_2 a_3} A_{a_4 a_5} A_{a_6
a_7} A_{a_8 a_9} \nonumber \\
&+& {m \over 60} A_{\mu a_1 } A_{a_2 a_3 } A_{a_4 a_5} A_{a_6 a_7}
A_{a_8 a_9} +...) e^{- 3 \phi} \tilde{R}^{a_1 ...a_9} + ... \quad .
\label{5.13}
\end{aligned}$$ The dots at the end denote terms proportional to the higher level deformed $E_{11}$ generators as well as all the Og generators, while the dots in each bracket denote the contributions from the Og 1 fields, which we did not write down explicitly because their contribution vanishes after antisymmetrisation of the $\mu$ index with the other indices. Indeed, the Og 1 fields in the group element of eq. (\[5.12\]) are $\Phi_{a, b_1 \dots b_n}$, for $n=1,2,3,5,...$, satisfying $\Phi_{[ a, b_1 \dots b_n ]}=0$. The inverse Higgs mechanism relates these Og fields to the deformed $E_{11}$ fields in such a way that only the completely antisymmetric terms in (\[5.13\]) survive. These terms are $$\begin{aligned}
& & F_{a_1 a_2} = \partial_{[a_1} A_{a_2]} +
m A_{a_1 a_2} \nonumber \\
& & F_{a_1 a_2 a_3} = \partial_{[a_1} A_{a_2 a_3 ] }\nonumber \\
& & F_{a_1 ... a_4} = \partial_{[a_1} A_{a_2 a_3 a_4 ]} - \partial_{[a_1}
A_{a_2
a_3} A_{a_4 ]} + {m \over 2} A_{[ a_1 a_2} A_{a_3 a_4 ]} \nonumber
\\
& & F_{a_1 ... a_6} = \partial_{[a_1} A_{a_2 ...a_6 ]} +2 \partial_{[a_1}
A_{a_2
a_3 a_4} A_{a_5 a_6 ]} + {m \over 3} A_{[ a_1 a_2} A_{a_3 a_4} A_{a_5 a_6
]} \nonumber \\
& & F_{a_1 ... a_7} =
\partial_{[ a_1} A_{a_2 ...a_7 ]} - \partial_{[ a_1} A_{a_2 ...a_6} A_{a_7 ]}
-2 \partial_{[ a_1} A_{a_2 a_3 a_4} A_{a_5 a_6} A_{a_7 ]} +
\partial_{[a_1} A_{a_2 a_3 a_4 } A_{a_5 a_6 a_7 ]}\nonumber \\
& & \qquad \quad -{m \over 3} A_{[ a_1 a_2} A_{a_3 a_4} A_{a_5 a_6} A_{a_7 ]} - m A_{a_1
...a_7} \nonumber \\
& & F_{a_1 ...a_8} = \partial_{[a_1} A_{a_2 ...a_8 ]} -
\partial_{[ a_1 } A_{a_2 ...a_6} A_{a_7 a_8 ]}
- \partial_{[a_1} A_{a_2 a_3 a_4} A_{a_5 a_6 } A_{a_7 a_8
]}\nonumber \\
& & \qquad \quad
- {m \over 12} A_{[ a_1 a_2} A_{a_3 a_4} A_{a_5 a_6} A_{a_7 a_8 ]}
\nonumber \\
& & F_{a_1 ...a_9} = \partial_{[a_1} A_{a_2 ...a_9 ]} +
3 \partial_{[a_1} A_{a_2
...a_8} A_{a_9 ]} +2 \partial_{[a_1} A_{a_2 ...a_7} A_{a_8 a_9 ]}
+
\partial_{[a_1} A_{a_2 ...a_6} A_{a_7 ...a_9 ]} \nonumber \\
& & \qquad \quad -3 \partial_{[a_1}
A_{a_2
...a_6} A_{a_7 a_8} A_{a_9 ]} -3 \partial_{[a_1} A_{a_2 a_3 a_4}
A_{a_5
a_6 }A_{a_7 a_8} A_{ a_9 ]}
+ 2 \partial_{[a_1} A_{a_2 ..a_4} A_{a_5 ...a_7} A_{a_8 a_9 ]}\nonumber \\
& & \qquad \quad -{m
\over 4} A_{[ a_1 a_2} A_{a_3 a_4} A_{a_5 a_6} A_{a_7 a_8} A_{a_9 ]} -
2m A_{[ a_1 ...a_7} A_{a_8 a_9 ]}
+ 5m A_{a_1 ...a_9} \nonumber \\
& & F_{a_1 ... a_{10}} =
\partial_{[a_1} A_{a_2 ...a_{10}]} - \partial_{[a_1} A_{a_2 ...a_8}
A_{a_9
a_{10}]}
+ {1 \over 2} \partial_{[a_1} A_{a_2 ...a_6} A_{a_7 a_8} A_{a_9
a_{10}]} \nonumber \\
& & \qquad \quad + {1 \over 3} \partial_{[a_1} A_{a_2 a_3 a_4} A_{a_5 a_6}
A_{a_7
a_8} A_{a_9 a_{10}]}
+ {m \over 60} A_{[ a_1 a_2} A_{a_3 a_4 } A_{a_5 a_6} A_{a_7 a_8}
A_{a_9 a_{10}]} \quad . \label{5.14}
\end{aligned}$$ These are the field-strengths of the fields of the massive IIA theory. Out of these field-strengths one can construct the field equations, which are duality relations between the various field-strengths. In particular, the 2-form $F_{a_1 a_2}$ is dual to the 8-form $F_{a_1 ... a_8}$, the 3-form $F_{a_1 a_2 a_3}$ is dual to the 7-form $F_{a_1 ... a_7}$ and the 4-form $F_{a_1 ... a_4}$ is dual to the 6-form $F_{a_1 ... a_6}$, while the 9-form $F_{a_1 ...
a_9}$ is dual to the derivative of the scalar and the 10-form $F_{a_1 ... a_{10}}$ is dual to the mass parameter $m$. All these relations are covariant under the local subalgebra of the non-linear realisation, which is $SO(9,1)$.
The gauge transformations of the fields arise in the non-linear realisation as rigid transformations of the group element, $g
\rightarrow g_0 g$, as long as one includes the Og generators. One obtains $$\begin{aligned}
& & \delta A_{a} = \partial_a \Lambda - m \Lambda_{a} \nonumber
\\
& & \delta A_{a_1 a_2} = \partial_{[a_1} \Lambda_{a_2 ]}
\nonumber \\
& & \delta A_{a_1 a_2 a_3} = \partial_{[a_1} \Lambda_{a_2 a_3 ]}
+ \partial_{[ a_1} \Lambda A_{a_2 a_3 ]} - m \Lambda_{[a_1} A_{a_2 a_3
]} \nonumber \\
& & \delta A_{a_1 ... a_5} = \partial_{[a_1} \Lambda_{a_2 ... a_5
]} - \partial_{[a_1} \Lambda A_{a_2 a_3 } A_{a_4 a_5 ]} -2
\partial_{[a_1} \Lambda_{a_2} A_{a_3 a_4 a_5 ]} + m \Lambda_{[a_1}
A_{a_2 a_3 } A_{a_4 a_5 ]} \nonumber \\
& & \delta A_{a_1 ... a_6} = \partial_{[a_1} \Lambda_{a_2 ... a_6
]} + \partial_{[a_1} \Lambda_{a_2 a_3} A_{a_4 a_5 a_6 ]} -
\partial_{[a_1} \Lambda A_{a_2 ...a_6] } - \partial_{[a_1}
\Lambda A_{a_2 a_3} A_{a_4 a_5 a_6 ]} \nonumber \\
& & \qquad \quad + m \Lambda_{a_1 ... a_6} + m \Lambda_{[a_1} A_{a_2
... a_6 ]} + m \Lambda_{[a_1} A_{a_2 a_3} A_{a_4 a_5 a_6 ]}
\nonumber \\
& & \delta A_{a_1 ... a_7} = \partial_{[a_1} \Lambda_{a_2 ... a_7
]} - {1 \over 3} \partial_{[a_1} \Lambda A_{a_2 a_3} A_{a_4 a_5}
A_{a_6 a_7 ]} + \partial_{[a_1} \Lambda_{a_2} A_{a_3 ... a_7 ]} \nonumber \\
& & \qquad \quad +
{1 \over 3} m \Lambda_{[a_1} A_{a_2 a_3} A_{a_4 a_5}
A_{a_6 a_7 ]}\nonumber \\
& & \delta A_{a_1 ... a_8} = \partial_{[a_1} \Lambda_{a_2 ... a_8
]} + \partial_{[a_1} \Lambda_{a_2 a_3} A_{a_4 ...a_8]} + 3 \partial_{[a_1}
\Lambda A_{a_2 ...a_8 ]} + \partial_{[a_1} \Lambda A_{a_2 a_3 a_4}
A_{a_5 a_6} A_{a_7 a_8 ]} \nonumber \\
& & \qquad \quad - 2 \partial_{[a_1} \Lambda_{a_2} A_{a_3 ...
a_8 ]} - 5m \Lambda_{a_1 ... a_8} - 3 m \Lambda_{[a_1}
A_{a_2 ...a_8 ]} - m \Lambda_{[a_1} A_{a_2 a_3 a_4}
A_{a_5 a_6} A_{a_7 a_8 ]}\nonumber \\
& & \delta A_{a_1 ... a_9} = \partial_{[a_1} \Lambda_{a_2 ... a_9
]} - {1 \over 12} \partial_{[a_1} \Lambda A_{a_2 a_3} A_{a_4 a_5}
A_{a_6 a_7 } A_{a_8 a_9]}+ \partial_{[a_1} \Lambda_{a_2} A_{a_3 ... a_9 ]} \nonumber \\
& & \qquad \quad +
{1 \over 12} m \Lambda_{[a_1} A_{a_2 a_3} A_{a_4 a_5}
A_{a_6 a_7 }A_{a_8 a_9 ]}\quad . \label{5.15}
\end{aligned}$$
In [@31] the supersymmetry transformations of all the forms and dual forms of the massive IIA theory where determined. The supersymmetry algebra closes on all the local symmetries of the theory, and this was used to determine all the gauge transformations and the field-strengths of the various forms, as well as their duality relations. These forms are exactly those predicted by $E_{11}$. One can show that the field strengths and gauge transformations of [@31] coincide with those given in eqs. (\[5.14\]) and (\[5.15\]) up to field redefinitions. The fact that using simple algebraic techniques one can easily determine these quantities proves the power of the $E_{11}$ formulation of maximal supergravities and of the methods explained in this paper. In the next section we will apply these methods to the case of maximal gauged supergravity in five dimensions, deriving again the results of [@20].
$E_{11}$ and gauged five-dimensional supergravity\[D=5section\]
===============================================================
In section 4 we derived the algebra $\tilde{E}_{11,9}^{local}$ associated to the Scherk-Schwarz dimensional reduction of the IIB theory to nine dimensions. After deriving the field-strengths of the theory from a ten-dimensional group element with a given dependence on the 10th coordinate $y$, we have shown that the same results can be obtained directly in nine dimensions. Indeed from the nine-dimensional perspective the fact that the ten-dimensional group element has a non-trivial $y$ dependence translates in having generators of the nine-dimensional theory that are deformed with respect to the massless case. We have shown that the algebra of these deformed generators is uniquely fixed by the Jacobi identities, and in deriving the deformed algebra in this way one never makes use of the fact that the theory has a ten-dimensional origin. This approach was indeed taken in the previous section, where we derived the algebra $\tilde{E}_{11,10A}^{local}$ corresponding to the massive IIA theory by requiring the closure of the Jacobi identities. From this algebra we have then derived the field-strengths of all the forms in the theory.
In this section we will perform precisely the same analysis for the case of maximal gauged supergravity in five dimensions. We will derive the algebra $\tilde{E}_{11,5}^{local}$ and from it we will determine the field-strengths of the forms in the theory. The analysis follows exactly the same steps as we have shown in the previous section for the case of the massive IIA theory in ten dimensions. We will first review the $E_{11}$ algebra as decomposed with respect to its $GL(5, \mathbb{R}) \otimes E_6$ subalgebra [@14] which is relevant for the five-dimensional analysis. This corresponds to deleting node 5 in the Dynkin diagram of fig. \[fig1\]. We will then consider the algebra of the deformed generators which occur in the description of the gauged theory from the $E_{11}$ perspective. The commutation relations of these generators are completely fixed by imposing Jacobi identities. The resulting algebra is such that the non-linear realisation determines completely all the field-strengths of gauged maximal supergravity in five dimensions. A different approach to gauge supergravities, based on $E_{10}$, was presented in [@3dime10] for the three-dimensional case.
We now review the $E_{11}$ commutation relations of the form generators up to the 4-form included that occur in the decomposition of $E_{11}$ with respect to $GL(5, \mathbb{R}) \otimes E_6$ [@14]. These generators are $$R^\alpha \qquad R^{a , M} \qquad R^{ab}{}_M \qquad R^{abc , \alpha} \qquad R^{abcd}{}_{[MN]}
\quad , \label{6.1}$$ where $R^\alpha$, $\alpha = 1 ,\dots ,78$ are the $E_6$ generators, and an upstairs $M$ index, $M= 1 ,\dots , 27$, corresponds to the ${\bf \overline{27}}$ representation of $E_6$, a downstairs $M$ index to the ${\bf 27}$ of $E_6$ and a pair of antisymmetric downstairs indices $[MN]$ correspond to the ${\bf \overline{351}}$. The commutation relations for the $E_6$ generators is $$[R^\alpha , R^\beta ]= f^{\alpha\beta}{}_{\gamma} R^\gamma \quad ,
\label{6.2}$$ where $f^{\alpha\beta}{}_{\gamma}$ are the structure constants of $E_6$. The commutation relations of $R^\alpha$ with all the other generators is determined by the $E_6$ representations that they carry. This gives $$\begin{aligned}
& & [R^\alpha , R^{a,M} ]= (D^\alpha )_N{}^M R^{a, N} \nonumber \\
& & [R^\alpha , R^{ab}{}_M ]= -(D^\alpha )_M{}^N R^{ab}{}_N
\nonumber \\
& & [R^\alpha , R^{abc ,\beta} ]= f^{\alpha\beta}{}_{\gamma}
R^{abc,\gamma} \nonumber \\
& & [R^\alpha , R^{abcd}{}_{[MN]} ]= -(D^\alpha )_M{}^P
R^{abcd}{}_{[PN]}-(D^\alpha )_N{}^P R^{abcd}{}_{[MP]}\quad ,
\label{6.3}
\end{aligned}$$ where $(D^\alpha )_N{}^M $ obey $$[D^\alpha , D^\beta ]_M{}^N = f^{\alpha\beta}{}_\gamma (D^\gamma )_M{}^N \quad
. \label{6.4}$$ The commutation relations of all the other generators are $$\begin{aligned}
& & [R^{a,M} , R^{b,N} ]= d^{MNP} R^{ab}{}_P \nonumber \\
& & [R^{a,N} , R^{bc}{}_M ]= g_{\alpha\beta} (D^\alpha )_M{}^N R^{abc, \beta}
\nonumber \\
& & [R^{ab}{}_M , R^{cd}{}_N ]= R^{abcd}{}_{[MN]} \nonumber \\
& & [R^{a, P}, R^{bcd , \alpha} ]= S^{\alpha P[MN]} R^{abcd}{}_{[MN]} \quad
, \label{6.5}
\end{aligned}$$ where $d^{MNP}$ is the symmetric invariant tensor of $E_6$ and $g_{\alpha\beta}$ is the Cartan-Killing metric of $E_6$. $S^{\alpha
P[MN]}$ is also an invariant tensor, which the Jacobi identities fix to be $$S^{\alpha P[MN]} = -{1 \over 2} D^\alpha_Q{}^{[M} d^{N]QP } \quad , \label{6.6}$$ and which satisfies the further identity $$g_{\alpha\beta} D^\alpha_Q{}^{(P} S^{\beta R)[MN]} = -{1 \over 2} \delta_{Q}^{[M} d^{N]PR}
\quad . \label{6.7}$$ One can show that all the Jacobi identities involving the generators in eq. (\[6.1\]) are satisfied using the commutators listed above.
To obtain the field-strengths of the massless theory, one introduces the Og generators, that encode the gauge transformations of all the fields. We focus in particular on the Og 1 generators for the $E_{11}$ generators listed in eq. (\[6.1\]), that are $$K^{a,b , M} \qquad K^{a , b_1 b_2}{}_M \qquad K^{a, b_1 b_2 b_3 ,
\alpha} \qquad K^{a , b_1 ... b_4}{}_{[MN]} \quad ,
\label{6.8}$$ and whose commutators with the momentum operator are $$\begin{aligned}
& & [ K^{a,b , M} , P_c ] = \delta^{(a}_c R^{b),M} \nonumber \\
& & [ K^{a , b_1 b_2}{}_M , P_c ] = \delta^a_c R^{b_1 b_2 }{}_M -
\delta^{[a}_c R^{b_1 b_2 ] }{}_M \nonumber \\
& & [ K^{a , b_1 b_2 b_3 , \alpha} , P_c ] = \delta^a_c R^{b_1 b_2 b_3 , \alpha}-
\delta^{[a}_c R^{b_1 b_2 b_3] , \alpha}\nonumber \\
& & [ K^{a , b_1 ... b_4}{}_{[MN]} , P_c ] = \delta^a_c R^{b_1 ... b_4}{}_{[MN]} -
\delta^{[a}_c R^{b_1 ... b_4 ] }{}_{[MN]} \quad . \label{6.9}
\end{aligned}$$ We then write down the group element $$g = e^{x \cdot P} e^{ \Phi_{\rm Og} K^{\rm Og}} e^{A_{a_1
...a_4}^{[MN]} R^{a_1 ... a_4}_{[MN]} } e^{A_{a_1 a_2 a_3, \alpha}
R^{a_1 a_2 a_3, \alpha}} e^{A_{a_1 a_2}^M R^{a_1 a_2}_M } e^{A_{a
,M} R^{a , M}} e^{\phi_\alpha R^\alpha} \quad , \label{6.10}$$ where we denote with $K^{\rm Og}$ all the Og 1 generators listed in eq. (\[6.8\]) and with $\Phi_{\rm Og}$ their corresponding fields. One can then compute the Maurer-Cartan form, and use the inverse Higgs mechanism to fix all the Og 1 fields in terms of derivatives of the $E_{11}$ fields, in such a way that only the completely antisymmetric terms in the Maurer-Cartan form survive. These quantities are the gauge-invariant field-strengths of the massless theory obtained in [@20], which we list here $$\begin{aligned}
& & F_{a_1 a_2 ,M} = \partial_{[a_1} A_{a_2 ] ,M} \nonumber \\
& & F^{M}_{a_1 a_2 a_3} = \partial_{[a_1} A^{M}_{a_2 a_3 ]} + {1 \over 2} \partial_{[a_1} A_{a_2 ,N}
A_{a_3 ] ,P} d^{MNP} \nonumber \\
& & F^{\alpha}_{a_1 a_2 a_3 a_4} = \partial_{[a_1} A^{\alpha}_{a_2 a_3 a_4 ]} - {1 \over 6}
\partial_{[a_1}
A_{a_2 ,M} A_{a_3 ,N} A_{a_4 ] ,P} d^{MNQ} D^\alpha_Q{}^P - \partial_{[a_1}
A^{M}_{a_2
a_3} A_{a_4 ],N} D^\alpha_M{}^N \nonumber \\
& & F^{MN}_{a_1 \dots a_5} = \partial_{[a_1} A^{MN}_{a_2 \dots a_5 ]} - {1 \over 24}
\partial_{[a_1}
A_{a_2 , P} A_{a_3 , Q} A_{a_4 , R} A_{a_5 ] , S} d^{PQT} D^\alpha_T{}^R
S^{\alpha S [MN]} \nonumber \\
& & \qquad \qquad - {1 \over 2} \partial_{[a_1} A^{P}_{a_2 a_3 } A_{a_4 , Q} A_{a_5 ], R}
D^\alpha_P{}^Q S^{\alpha R[MN]} + {1 \over 2} \partial_{[a_1} A^{[M}_{a_2 a_3 } A^{N]}_{a_4 a_5 ]
}\nonumber \\
& & \qquad \qquad +
\partial_{[a_1} A^{\alpha}_{a_2 a_3 a_4} A_{a_5 ] , P} S^{\alpha P[MN]}
\label{6.11}
\end{aligned}$$ for completeness.
We now consider the deformed case. We take as our set of generators that of eqs. (\[6.1\]) and (\[6.8\]), but to indicate that these generators themselves have been deformed, we denote them with a tilde. The commutation relations receive order $g$ corrections with respect to the massless ones, where $g$ is the deformation parameter. We start from the commutation relation between the deformed vector generator and the momentum operator. We impose this to be $$[ \tilde{R}^{a , M} , P_b ] = - g \delta^a_b \Theta^M_\alpha
{R}^\alpha \quad . \label{6.12}$$ The quantity $\Theta^M_\alpha$ turns out to be the embedding tensor [@7], and the generators $\Theta^M_\alpha R^\alpha$ are the generators of the subgroup $G$ of $E_6$ that is gauged. These generators belong to the algebra $\tilde{E}_{11,5}^{local}$. We now show that all the commutation relations of the deformed operators among themselves and with the momentum operators are uniquely fixed by Jacobi identities. From the resulting algebra we construct the non-linear realisation whose Maurer-Cartan form gives the field-strengths of the gauged theory.
We first consider the Jacobi identity between $\Theta^M_\alpha
{R}^\alpha$, $\tilde{R}^{a , N}$ and $P_b$. Defining $$X^{MN}_P = \Theta^M_\alpha D^\alpha_P{}^N \label{6.13}$$ one gets $$\Theta^M_\alpha \Theta^N_\beta f^{\alpha \beta}{}_\gamma -
\Theta^P_\gamma X^{MN}_P =0 \quad , \label{6.14}$$ which turns out to be the condition that the embedding tensor is invariant under the gauge group. We then write the commutator of the 2-form $\tilde{R}^{ab}{}_M$ with $P_c$ as $$[\tilde{R}^{ab}{}_M , P_c ] = -2 g W_{MN} \delta^{[a}_c
\tilde{R}^{ b ] , N} \quad , \label{6.15}$$ which defines the antisymmetric tensor $W_{MN}$. The Jacobi identity involving $\tilde{R}^{ab}{}_M$, $P_c$ and $P_d$ gives $$W_{MN} \Theta^N_\alpha =0 \quad . \label{6.16}$$ The Jacobi identity involving the operators $\tilde{R}^{a , M}$, $\tilde{R}^{b , N}$ and $P_c$ gives $$[ \tilde{R}^{a ,M} , \tilde{R}^{ b ,N}] = d^{MNP}
\tilde{R}^{ab}{}_P - 2 g X^{[MN]}_P \tilde{K}^{a,b,P} \quad ,
\label{6.17}$$ using the fact that the Og 1 for the vector $\tilde{K}^{a,b,M}$ is symmetric in $ab$ and satisfies $$[ \tilde{K}^{a,b,M}, P_c ] = \delta^{(a}_c \tilde{R}^{b ) , M}
\quad . \label{6.18}$$ To get eq. (\[6.17\]) one has also to impose $$X^{(MN)}_P = - W_{PQ} d^{QMN} \quad . \label{6.19}$$
The Jacobi identity between $\tilde{R}^{ab}{}_M$, $\Theta^N_\alpha
R^\alpha$ and $P_c$ gives $$X^{MN}_{[P} W_{Q]N} =0 \quad , \label{6.20}$$ which is the condition that the tensor $W_{MN}$ is invariant under the gauge subgroup $G$. The Jacobi identity involving $\tilde{R}^{a,
M}$, $\tilde{R}^{bc}{}_N$ and $P_d$ gives $$[ \tilde{R}^{a ,M} , \tilde{R}^{bc}{}_N ] = ( D^\alpha )_N{}^M
\tilde{R}^{abc}{}_\alpha + {3 \over 2} g (X^{MP}_N + {1\over 3}
X^{PM}_N ) \tilde{K}^{a ,bc}{}_P \label{6.21}$$ and $$[ \tilde{R}^{abc}{}_\alpha , P_d ] = - g \Theta^M_\alpha
\delta^{[a}_d \tilde{R}^{bc ]}{}_M \quad , \label{6.22}$$ where the Og 1 operator $\tilde{K}^{a, bc}{}_M$ satisfies $$[ \tilde{K}^{a, bc}{}_M , P_d ] = \delta^a_d \tilde{R}^{bc}{}_M -
\delta^{[a}_d \tilde{R}^{bc ]}{}_{M} - {2 \over 3} g W_{MN} (
\delta^b_d \tilde{K}^{a ,c , N} - \delta^c_d \tilde{K}^{a,b,N} )
\quad . \label{6.23}$$ Proceeding this way, one can determine all the commutators requiring the closure of the Jacobi identities. This gives $$[ \tilde{R}^{ab}{}_M , \tilde{R}^{cd}{}_N ] =
\tilde{R}^{abcd}{}_{MN} - 4 g W_{(M \vert P \vert }
D^\alpha_{N)}{}^P \tilde{K}^{[a,b]cd}{}_\alpha \quad ,
\label{6.24}$$ $$[ \tilde{R}^{a , M} , \tilde{R}^{bcd}{}_\alpha ] =
S^{M[NP]}_\alpha \tilde{R}^{abcd}{}_{MN} - g ( f^{\beta
\gamma}{}_\alpha \Theta^M_\gamma + {1 \over 3} D^\beta_P{}^M
\Theta^P_\alpha ) \tilde{K}^{a , bcd}{}_\beta \label{6.25}$$ and $$[ \tilde{R}^{a_1 ... a_4}{}_{MN} , P_b ] = - 4 g W_{[M\vert P
\vert } D^\alpha_{N]}{}^P \delta^{[ a_1}_b \tilde{R}^{a_2 a_3 a_4
]}{}_\alpha \quad , \label{6.26}$$ where we introduce the Og 1 generator for the 3-form $\tilde{K}^{a,
b_1 b_2 b_3}{}_\alpha$, satisfying $$[ \tilde{K}^{a, b_1 b_2 b_3}{}_\alpha , P_c ] = \delta^a_c
\tilde{R}^{b_1 b_2 b_3}{}_\alpha - \delta^{[a}_c
\tilde{R}^{b_1 b_2 b_3 ]}{}_\alpha - {3 \over 4} g \Theta^M_\alpha
\delta^{[ b_1}_c \tilde{K}^{\vert a \vert , b_2 b_3 ]}
\quad . \label{6.27}$$ In order to get these results one has to impose an additional constraint $$f^{\alpha\beta}{}_\gamma \Theta^Q_\beta - D^\alpha_P{}^Q \Theta^P_\gamma = 4 D^\alpha_M{}^P W_{PN} g_{\beta
\gamma} S^{\beta Q[MN]} \quad , \label{6.28}$$ which shows that the embedding tensor and $W_{MN}$ are related by the invariant tensor $S^{\alpha P[MN]}$, and thus belong to the same representation of $E_6$, which is the ${\bf \overline{351}}$.
To summarise, we have determined the commutation relations satisfied by the deformed $E_{11}$ $p$-form generators, the corresponding deformed Og 1 generators and the momentum operator of the five-dimensional massive theory starting from eq. (\[6.12\]) and imposing the closure of the Jacobi identities. We will now determine the field-strengths of the 1-forms, 2-forms and 3-forms of the theory using these results. We consider the group element $$g = e^{x \cdot P} e^{ \Phi_{\rm Og} \tilde{K}^{\rm Og}} e^{A_{a_1
...a_4}^{[MN]} \tilde{R}^{a_1 ... a_4}_{[MN]} } e^{A_{a_1 a_2 a_3, \alpha}
\tilde{R}^{a_1 a_2 a_3, \alpha}} e^{A_{a_1 a_2}^M \tilde{R}^{a_1 a_2}_M } e^{A_{a
,M} \tilde{R}^{a , M}} e^{\phi_\alpha R^\alpha} \quad ,
\label{6.29}$$ where as in the massless case we denote with $\tilde{K}^{\rm Og}$ all the deformed Og 1 generators and with $\Phi_{\rm Og}$ the corresponding fields. One can then compute the Maurer-Cartan form, and use the inverse Higgs mechanism to fix all the Og 1 fields in such a way that only the completely antisymmetric terms in the Maurer-Cartan form survive. To compute the field-strengths, it is thus sufficient to consider only the $\tilde{R}$ operators and $P_a$ in the group element above. The final result is $$\begin{aligned}
& & \! \! \! \! \! \! \! F_{a_1 a_2 ,M} = \partial_{[a_1} A_{a_2 ] ,M} + {1 \over 2} g
X^{[NP]}_M A_{[a_1 ,N} A_{a_2], P} - 2 g W_{MN}A_{a_1
a_2}^N \nonumber \\
& & \! \! \! \! \! \! \! F^{M}_{a_1 a_2 a_3} = \partial_{[a_1} A^{M}_{a_2 a_3 ]} + {1 \over 2} \partial_{[a_1} A_{a_2 ,N}
A_{a_3 ] ,P} d^{MNP} - 2 g X^{(MN)}_P A_{[a_1 a_2}^P A_{a_3 ], N}
\nonumber \\
& & +{1 \over 6} g X^{[NP]}_R d^{RQM} A_{[a_1 , N} A_{a_2 , P} A_{[a_3 ] , Q} + g \Theta^M_\alpha A_{a_1 a_2
a_3 }^\alpha \nonumber \\
& & \! \! \! \! \! \! \!
F^{\alpha}_{a_1 \dots a_4} = \partial_{[a_1} A^{\alpha}_{a_2 \dots a_4 ]} - {1 \over 6} \partial_{[a_1}
A_{a_2 ,M} A_{a_3 ,N} A_{a_4 ] ,P} d^{MNQ} D^\alpha_Q{}^P - \partial_{[a_1} A^{M}_{a_2
a_3} A_{a_4 ],N} D^\alpha_M{}^N \nonumber \\
& & + g D^\alpha_M{}^P \Theta^M_\beta A_{[a_1 , P} A^\beta_{a_2 \dots a_4 ]} + 4 g D^\alpha_M{}^P W_{PN}
A^{MN}_{a_1 \dots a_4} - g D^\alpha_M{}^P W_{PN} A^M_{[a_1 a_2} A^N_{a_3 a_4
]} \nonumber \\
& & - g D^\alpha_M{}^P X^{(MR)}_Q A_{[a_1 , P} A_{a_2 , R} A_{a_3 a_4 ]}^Q - {1 \over 24} g X^{[MN]}_R d^{RPS}
D^\alpha_S{}^Q A_{[a_1 , M} A_{a_2 , N} A_{a_3 , P} A_{a_4 ] , Q}
\quad \label{6.30}
\end{aligned}$$ These are the field-strengths of the five-dimensional gauged maximal supergravity [@20]. One can also derive the gauge transformations of the fields from the non-linear realisation as they arise as rigid transformations of the group element, $g
\rightarrow g_0 g$, as long as one includes the Og generators. The result is $$\begin{aligned}
& & \delta A_{a , N}= \partial_a \Lambda_N -g \Lambda_S X^{SM}_N A_{aM}+ 2 gW_{MP}
\Lambda^P_a \nonumber \\
& & \delta A^N_{a_1a_2}= \partial_{[a_1}\Lambda^N_{a_2]} +{1\over 2} \partial_{[a_1} \Lambda_S A_{a_2 ]T}
d^{STN} +g \Lambda_S X^{SN}_M A_{a_1a_2}^M +2W_{SP} \Lambda^P_{[a_1} A_{a_2
]T}d^{STN}\nonumber \\
& & \quad - g \Lambda_{a_1a_2}^\alpha \Theta _\alpha^N \nonumber
\\
& & \delta A_{a_1a_2a_3}^\alpha= \partial_{[a_1} \Lambda_{a_2 a_3 ]}^\alpha + \partial_{[a_1} \Lambda_M
A^{N}_{a_2 a_3 ]} D^\alpha_N{}^M + {1 \over 6}
\partial_{[a_1} \Lambda_M A_{a_2 , N} A_{a_3 ], P} d^{MNQ}
D^\alpha_Q{}^P \nonumber \\
& & \quad
- g \Lambda_P \Theta^P_\beta f^{\alpha\beta}{}_\gamma A^\gamma_{a_1 a_2
a_3} + 2 g W_{MP} \Lambda_{[a_1}^P D^\alpha_N{}^M A_{a_2 a_3 ]}^N
+ {1 \over 3} g W_{MR} \Lambda_{[a_1}^R d^{MNQ}
D^\alpha_Q{}^P A_{a_2 , N} A_{a_3 ] } \nonumber \\
& & \quad - 4 g D^\alpha_M{}^P W_{PN} \Lambda_{a_1 a_2 a_3}^{MN}
\quad . \label{6.31}
\end{aligned}$$ In [@20] these transformations were derived both from $E_{11}$ and from requiring the closure of the supersymmetry algebra. Indeed, the commutator of two supersymmetry transformations on these fields gives the gauge transformations above provided that the fields are related by dualities. In particular the 1-forms are dual to 2-forms while the 3-forms are dual to scalars in five dimensions. The field-strength of the 4-form is dual to the mass parameter. The field strengths and gauge transformations obtained here precisely agree with those obtained by supersymmetry.
In the above we have taken Jacobi identities with all the deformed generators of eqs. (\[6.1\]) and (\[6.8\]) with the exception of $R^\alpha$, but we have instead restricted our use to $\Theta^M_\alpha R^\alpha$. Using the Jacobi identities for $R^\alpha$ would lead to results that are too strong. A solution to this problem, at least at low levels, requires adding to spacetime the scalar charges in the $l$ multiplet, as was done in [@20]. In this case the Jacobi identities are automatically satisfied. Adding the higher charges in the $l$ multiplet may also resolve this problem at higher level.
To summarise, we have thus shown that the methods explained in this paper give an extremely fast way of computing the field strengths of all the forms and dual forms of five-dimensional gauged maximal supergravity. These methods can be easily generalised to any dimension, providing a remarkably efficient way of determining the gauge algebra of any massless or massive theory with maximal supersymmetry.
The dual graviton
=================
Any very-extended Kac-Moody algebra, when decomposed in terms of a $GL(D, \mathbb{R})$ subalgebra which one associates to the non-linear realisation of gravity, contains a generator with $D-2$ indices in the hook Young Tableaux irreducible representation with $D-3$ completely antisymmetric indices, that is $R^{a, b_1 \dots
b_{D-3}}$ with $R^{[a, b_1 \dots b_{D-3}]}=0$ (in the case of $E_{11}$ decomposed in terms of $GL(11, \mathbb{R})$, this generator is $R^{a, b_1 \dots b_8}$). The field associated to this generator in the non-linear realisation has the degrees of freedom of the dual graviton. The Kac-Moody algebra therefore describes together the graviton and the dual graviton. In this section we will consider the Og operators for the dual graviton. We will focus on the four-dimensional case, in which the dual graviton generator $R^{ab}$ is symmetric in its two indices.
In subsection 7.1 we will first consider the case of the dual graviton in flat space. This corresponds to considering the dual graviton generator by itself, together with its corresponding Og generators. This does not arise from any very-extended Kac-Moody algebra. A field theory description of a linearised dual graviton is known to exist, and its field equations in four dimensions were first obtained by Curtright [@32]. For subsequent developments see [@1; @33].
We then consider the dual graviton coupled to gravity. The simplest very-extended Kac-Moody algebra whose non-linear realisation gives rise to a four-dimensional theory is the algebra $A_1^{+++}$, whose Dynkin diagram is shown in fig. \[fig5\]. The corresponding spectrum does not contain any form. We will show that there is no consistent solution of the inverse Higgs mechanism that leaves a propagating dual graviton. We will also consider the case of $E_{11}$ in four dimensions, which corresponds to deleting node 4 in the diagram of fig. \[fig1\], leading to the internal symmetry algebra $E_7$. In this case we will show that even considering linearised gravity, that is neglecting the $GL(4,\mathbb{R})$ generators and the corresponding Og generators, and only considering interactions of the dual graviton with matter, one is left with no consistent field strength for the dual graviton. This result is consistent with [@34], where it was shown that it is impossible to write down a dual Riemann tensor in the presence of matter even when gravity is treated at the linearised level.
In the first subsection we will only consider the algebra of $R^{ab}$ and all the corresponding Og generators. The Maurer-Cartan form, that can be thought as the Maurer-Cartan form of $A_1^{+++}$ or $E_{11}$ truncated to this sector, leads to invariant quantities that can be constrained by means of the inverse Higgs mechanism to generate the Riemann tensor for the linearised dual graviton. In the second subsection we will then consider the case of dual graviton coupled to gravity, which corresponds to the algebra $A_1^{+++}$, and in the third subsection we will consider the $E_{11}$ case of the dual graviton coupled to vectors with linearised gravity.
The dual graviton in four dimensions
------------------------------------
In this subsection we want to consider the dual graviton alone, that is without introducing the generator associated to the graviton or any other matter generator. We want to show that one can introduce suitable Og generators for the dual graviton in such a way that gives rise to a consistent field strength and consistent gauge transformations. The dual graviton generator in four dimensions is a generator with 2 symmetric indices $R^{ab}$. Following the notation of the previous section, we define two ${\rm Og} \ 1$ generators $K_1^{a,bc}$ and $\tilde{K}_1^{abc}$ in the irreducible $GL(4,\mathbb{R})$ representations defined as $$\begin{aligned}
& & K_1^{a,bc}= K_1^{a,(bc)} \qquad K_1^{(a,bc)}=0 \nonumber \\
& & \tilde{K}_1^{abc}= \tilde{K}_1^{(abc)} \quad , \label{7.1}
\end{aligned}$$ and whose corresponding Young Tableaux are shown in fig. \[fig3\]. Note that the sum of these two representations corresponds to an object with three indices, symmetric under the exchange of two of them and with no further constraint. These operators satisfy the commutation relations $$\begin{aligned}
& & [ K_1^{a,bc} , P_d ]= \delta^a_d R^{bc} - \delta^{(a}_d R^{bc)}
\nonumber \\
& & [ \tilde{K}_1^{abc} , P_d ]= \delta^{(a}_d R^{bc)} \quad ,
\label{7.2}
\end{aligned}$$ while we take $R^{ab}$ as commuting with $P_a$. We also take $R^{ab}$ as commuting with itself because we are considering the dual graviton alone (for instance in $A_1^{+++}$ the dual graviton is a generator at level 1, and therefore the commutator of two dual graviton generators leads to an operator at level 2). Also all the dual graviton Og generators are taken to commute with each other, and to commute with $R^{ab}$ as well.
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We now consider the ${\rm Og} \ 2$ operators. These are $K_2^{a,bcd}$ and $\tilde{K}_2^{abcd}$ in the $GL(4,\mathbb{R})$ representations $$\begin{aligned}
& & K_2^{a,bcd}= K_2^{a,(bcd)} \qquad K_2^{(a,bcd)}=0 \nonumber \\
& & \tilde{K}_2^{abcd}= \tilde{K}_2^{(abcd)} \quad ,\label{7.3}
\end{aligned}$$ whose Young Tableaux are shown in fig. \[fig3\], and their commutation relation with $P_a$ is $$\begin{aligned}
& & [ K_2^{a,bcd} , P_e ]= \delta^{(b}_e K_1^{\vert a \vert ,cd)} +
{2 \over 3} (\delta^a_e \tilde{K}_1^{bcd} - \delta^{(a}_e
\tilde{K}_1^{bcd)} )
\nonumber \\
& & [ \tilde{K}_2^{abcd} , P_e ]= \delta^{(a}_e \tilde{K}_1^{bcd)} \quad
. \label{7.4}
\end{aligned}$$ The coefficient ${2 \over 3}$ in the first commutator can be obtained from the Jacobi identity involving $K_2^{a,bcd}$ and two $P$’s.
We will now compute the Maurer-Cartan form, and we will first consider only the contribution from dual graviton and the Og 1 fields, while the Og 2 fields will be included later. We thus consider the group element $$g = e^{x \cdot P} e^{\Phi^1_{a,bc} K_1^{a,bc}}
e^{\tilde{\Phi}^1_{abc} \tilde{K}_1^{abc}} e^{A_{ab}R^{ab}} \quad
, \label{7.5}$$
from which one computes the Maurer-Cartan form $$g^{-1} \partial_\mu g = P_\mu + (\partial_\mu A_{ab} - \Phi^1_{\mu,ab}
- \tilde{\Phi}^1_{\mu ab} ) R^{ab} + ... \quad , \label{7.6}$$ which is invariant under $$\begin{aligned}
& & \delta A_{ab} = a_{ab} + x^c b_{c,ab} + x^c
\tilde{b}_{abc} \nonumber\\
& & \delta \Phi^1_{a,bc} = b_{a,bc} \nonumber\\
& & \delta \tilde{\Phi}^1_{a,bc} = \tilde{b}_{a,bc} \quad .
\label{7.7}
\end{aligned}$$ The first of eqs. (\[7.7\]) is reproducing the gauge transformation for the dual graviton in flat space, $$\delta A_{ab} = \partial_{(a} \Lambda_{b)} \label{7.8}$$ at linear order in $x$, that is quadratic order in $x$ for the gauge parameter $\Lambda_a$, $$\Lambda_a = a_{ab} x^b - b_{a, bc} x^b x^c + {1 \over 2}
\tilde{b}_{abc} x^b x^c \quad . \label{7.9}$$
One can solve for inverse Higgs in such a way that the whole Maurer-Cartan form proportional to $R^{ab}$ vanishes compatibly with the symmetries. This corresponds to fixing $$\Phi^1_{\mu,ab} = \partial_\mu A_{ab} - \partial_{ (\mu} A_{ab)}
\label{7.10}$$ and $$\tilde{\Phi}^1_{\mu,ab} = \partial_{ (\mu} A_{ab)} \quad .
\label{7.11}$$ The fact that reproducing the gauge transformations for $A_{ab}$ at linear order in $x$ allows one to eliminate completely the Maurer-Cartan form proportional to $R^{ab}$ by means of the inverse Higgs mechanism corresponds to the fact that one cannot write a gauge invariant quantity at linear order in the derivatives. Note that there is a crucial difference here with respect to the non-linear realisation of gravity discussed in section 2. In that case the part of the Maurer-Cartan form proportional to the generators of the local subalgebra $SO(D)$ gives the $SO(D)$ connection, which becomes the spin connection once the inverse Higgs mechanism is applied. In this case the dual graviton field is already symmetric, and thus there is no corresponding local Lorentz symmetry. It is for this reason that at this level the Maurer-Cartan form vanishes once the inverse Higgs mechanism is applied.
We now consider the contribution from the Og 2 fields. We write the group element as $$g = e^{x \cdot P} e^{\Phi^2_{a,bcd} K_2^{a,bcd}}
e^{\tilde{\Phi}^2_{abcd} \tilde{K}_2^{abcd}} e^{\Phi^1_{a,bc} K_1^{a,bc}}
e^{\tilde{\Phi}^1_{abc} \tilde{K}_1^{abc}} e^{A_{ab}R^{ab}} \quad ,
\label{7.12}$$ and obtain the corresponding Maurer-Cartan form $$\begin{aligned}
g^{-1} \partial_\mu g &= & P_\mu + (\partial_\mu A_{ab} - \Phi^1_{\mu,ab}
- \tilde{\Phi}^1_{\mu ab} ) R^{ab} \nonumber \\
& +& (\partial_\mu \Phi^1_{a,bc} - \Phi^2_{a, \mu bc} ) K_1^{a,bc}
\nonumber \\
& +& (\partial_\mu \tilde{\Phi}^1_{abc} - {2 \over 3} \Phi^2_{\mu , abc}
- \tilde{\Phi}^2_{\mu abc } )
\tilde{K}_1^{abc} + ... \quad . \label{7.13}
\end{aligned}$$ Having introduced the ${\rm Og} \ 2$ operators, the transformations of the field that leave the Maurer-Cartan form invariant acquire additional contributions, and in particular there is a term in the variation of $A_{ab}$ which is quadratic in $x$. The result is $$\begin{aligned}
& & \delta A_{ab} = a_{ab} + x^c b_{c,ab} + x^c
\tilde{b}_{abc} +{5 \over 6} x^c x^d c_{c, abd} - {1 \over 2}x^c x^d c_{(c, ab)d}
+ {1 \over 2} x^c x^d \tilde{c}_{abcd} \nonumber\\
& & \delta \Phi^1_{a,bc} = b_{a,bc} + x^d c_{a, bcd} - x^d c_{(a ,bc)d}\nonumber\\
& & \delta \tilde{\Phi}^1_{a,bc} = \tilde{b}_{a,bc} +{2 \over 3} x^d c_{d,abc} + x^d
\tilde{c}_{abcd}\nonumber \\
& & \delta \Phi^2_{a,bcd} = c_{a,bcd}\nonumber \\
& & \delta \tilde{\Phi}^2_{a,bcd} = \tilde{c}_{a,bcd} \quad .
\label{7.14}
\end{aligned}$$ In particular the first of these variations is the most general gauge transformation for the field $A_{ab}$ of the form (\[7.8\]) up to terms cubic in $x$.
We now apply the inverse Higgs mechanism, solving for the fields $\Phi^2_{a,bcd}$ and $\tilde{\Phi}^2_{abcd}$ in terms of $A_{ab}$. The result is $$\begin{aligned}
\Phi^2_{a,bcd} & =& {1 \over 4} [ \partial_a \partial_b A_{cd} + \partial_a \partial_c
A_{db} + \partial_a \partial_d A_{bc} - \partial_b \partial_c
A_{ab} - \partial_b \partial_d A_{ac} - \partial_c \partial_d
A_{ab} ] \nonumber \\
\tilde{\Phi}^2_{abcd} & =& \partial_{( a} \partial_b A_{cd )} \quad
. \label{7.15}
\end{aligned}$$ Plugging this into the Maurer-Cartan form, one notices that there is a non-vanishing term proportional to $K_1^{a,bc}$, that is $$g^{-1} \partial_\mu g = P_\mu + ( {1 \over 3} \partial_\mu
\partial_a A_{bc} - {1 \over 3} \partial_\mu \partial_b A_{ac} -
{1\over 3} \partial_a \partial_b A_{\mu c} + {1 \over 3} \partial_b
\partial_c A_{\mu a} ) K_1^{a,bc} + ... \quad . \label{7.16}$$ This is indeed the Riemann tensor of linearised gravity $D_{ab,cd}$, which is a tensor in the window-like Young Tableaux representation.
One can introduce in the same way the higher ${\rm Og}$ generators, constructing in this way gauge invariant quantities which are derivatives of the Riemann tensor. The end result is thus $$g^{-1} \partial_\mu g = P_\mu + D_{\mu a ,bc} K_1^{a,bc} + ...
\quad , \label{7.17}$$ where the dots correspond to derivatives of the dual graviton Riemann tensor $$D_{ab ,cd} = {1 \over 3} \partial_a
\partial_b A_{cd} - {1 \over 6} \partial_a \partial_c A_{bd} - {1
\over 6} \partial_a \partial_d A_{bc} - {1 \over 6}
\partial_b \partial_c A_{ad} - {1 \over 6} \partial_b \partial_d A_{ac}+ {1 \over 3}
\partial_c \partial_d A_{ab} \label{7.18}$$ contracted with higher order Og generators. This shows that the linearised dual graviton admits a description in terms of $R^{ab}$ and Og generators, and the corresponding Maurer-Cartan form contains the correct Riemann tensor, which can be used to construct the dynamics.
We now want to perform a dimensional reduction on a circle of coordinate $y$. We thus take the dual graviton and all the Og fields to be $y$ independent. The representations of $GL(3,\mathbb{R})$ that arise in the three-dimensional compactified theory are shown in fig. \[fig4\] for the dual graviton field and the first two Og fields. The circle dimensional reduction corresponds to the assumption that neither the dual graviton field nor the Og fields depend on $y$.
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The dual graviton $A_{ab}$ in four dimensions has two symmetric indices, and after dimensional reduction it leads to an object with two symmetric indices $A_{ab}$, a vector $A_a = A_{a y}$ and a scalar $A = A_{yy}$. As the figure shows, the Og 1 fields can be divided in three sets. The first one contains the Og fields for the field with two symmetric indices $A_{ab}$ and the vector $A_{a}$. This is precisely what we want in order to obtain the correct gauge transformations for the three-dimensional fields. The second set contains the same representations as the dimensionally reduced fields. The $dy$ part of the Maurer-Cartan form contains these fields summed to the $y$ derivative of the dimensionally reduced fields. Thus, from the requirement that the fields do not depend on $y$ it follows that these Og fields can be put to zero using the inverse Higgs mechanism. Finally, the third set contains a field with two antisymmetric indices and a vector, and we call these fields the $\overline{\rm Og}$ 1 fields for the vector and the scalars. These fields are the ones of interest to us in the following. The dimensional reduction of the Og 2 fields gives the Og 2 fields for the field with two symmetric indices and the vector, together with a set of fields in the same representations as the dimensionally reduced Og 1 fields, and again the $dy$ part of the Maurer-Cartan form contains these Og 2 fields summed to the $y$ derivative of all the Og 1 fields. Using $y$ independence and the inverse Higgs mechanism one thus sets to zero these Og 2 fields.
We now explain the occurrence of the $\overline{\rm Og}$ 1 fields and generators in the dimensional reduction. In the four-dimensional theory, once the inverse Higgs mechanism is applied the Maurer-Cartan form is given in eq. (\[7.17\]), and the first non-vanishing term is the dual graviton Riemann tensor, which is at second order in derivatives. Using the $y$-independence of the fields, the dimensional reduction of the Riemann tensor leads to the Riemann tensor for $A_{ab}$ in three dimensions together with $$\begin{aligned}
& & D_{ab,c y} = {1 \over 2} \partial_a F_{bc} + {1 \over 3}
\partial_{(b} F_{\vert a \vert c )} \nonumber \\
& & D_{ab, yy} = {1 \over 3} \partial_a \partial_b A \quad ,
\label{7.19}
\end{aligned}$$ while $D_{ay,yy}$ vanishes. Here we have denoted with $F_{ab} =
\partial_{[a} A_{b ]}$ the field-strength of the vector. As eq. (\[7.19\]) shows, the Maurer-Cartan form in three dimensions thus contains the Riemann tensor of $A_{ab}$ together with the derivative of the field-strength of the vector and the double derivative of the scalar. This implies that among the rest, the Maurer-Cartan form is invariant under the transformations $$\delta A_a = x^b b_{[ba]} \qquad \delta A = b_a x^a \quad ,
\label{7.20}$$ which indeed lead to $$\delta F_{ab} = b_{[ab]} \qquad \delta ( \partial_a A ) = b_a \quad
. \label{7.21}$$ Such transformations cannot be written as standard gauge transformations for the corresponding fields, and indeed they do not leave the field strength invariant, although they are symmetries of the dimensionally reduced Riemann tensor. They are generated by the operators associated to the $\overline{\rm Og}$ 1 fields in fig. \[fig4\], and in general we define the $\overline{\rm Og}$ generators as those producing transformations that cannot be written as gauge transformations. The $\overline{\rm Og}$ 1 fields in fig. \[fig4\], together with the standard Og 1 fields for $A_{ab}$ and $A_a$, are such that all the terms with one derivative of the fields in the Maurer-Cartan form vanish once the inverse Higgs mechanism is applied. The standard gauge transformations of the fields are obtained by performing a truncation that projects out the $\overline{\rm Og}$ 1 generators, and once this truncation is performed one can no longer use the inverse Higgs mechanism to cancel the one derivative terms completely, which indeed give $F_{ab}$ and the derivative of the scalar.
The dual graviton in $A_1^{+++}$ in four dimensions
---------------------------------------------------
The non-linear realisation based on the algebra $A_1^{+++}$, whose Dynkin diagram is shown in fig. \[fig5\], has the particular feature of only containing in four dimensions the graviton and its duals, which are fields with two symmetric indices together with an arbitrary number of blocks of two antisymmetric indices, as well as generators with sets of 3 or 4 antisymmetric indices. This in particular means that the spectrum does not contain any forms, that is fields with completely antisymmetric indices.
(140,30) (10,10)(40,0)[4]{} (15,10)(40,0)[2]{}[(1,0)[30]{}]{} (95,12)[(1,0)[30]{}]{} (95,8)[(1,0)[30]{}]{} (8,-8)[$1$]{} (48,-8)[$2$]{} (88,-8)[$3$]{} (128,-8)[$4$]{}
Decomposing the adjoint representation of $A_1^{+++}$ in representations of $GL(4,\mathbb{R})$ one gets $K^a{}_b$ at level zero, which are the generators of $GL(4,\mathbb{R})$, and $R^{ab}$ at level one. The generators at higher level can be obtained as multiple commutators of $R^{ab}$ subject to the Serre relations, and the number of indices of a generator at level $l$ is $2l$. We will ignore all the generators of level higher than 1 in this subsection. The commutation relation between $K^a{}_b$ and $R^{ab}$ is $$[ K^a{}_b , R^{cd} ] = \delta^c_b R^{ad} + \delta^d_b R^{ca}
\quad . \label{7.22}$$
We now want to introduce the Og generators for both the graviton and the dual graviton, in order to reproduce the correct general coordinate transformation for the fields, as well as the expected gauge transformation for the dual graviton. As in the previous subsection, we have $$[R^{ab} , P_c ] =0 \quad . \label{7.23}$$ We thus obtain the commutator between the gravity Og 1 operator $K^{ab}{}_c$ and $R^{ab}$ by the Jacobi identity with $P_a$. The result is $$[ K^{ab}{}_c , R^{de}] = \delta^d_c K_1^{e,ab} + \delta^e_c
K_1^{d,ab} -2 (\delta^d_c \tilde{K}_1^{abe} +\delta^e_c
\tilde{K}_1^{abd} ) \quad . \label{7.24}$$ Similarly, imposing the Jacobi identity between $P_a$, $K^a{}_b$ and the Og 1 dual graviton operators gives $$\begin{aligned}
& & [ K^a{}_b , K_1^{c,de} ] = \delta^c_b K_1^{a,de} + \delta^d_b
K_1^{c,ae} + \delta^e_b K_1^{c,da} \nonumber \\
& & [ K^a{}_b , \tilde{K}_1^{cde} ] = 3 \delta^{(c}_b
\tilde{K}_1^{de)a} \label{7.25}
\end{aligned}$$ as expected from the index structure of the operators.
We would expect that the commutator between the Og 1 gravity operator $K^{ab}{}_c$ and the dual graviton Og 1 operators gave the Og 2 dual graviton operators in fig. \[fig3\] if a description of both gravity and dual gravity were possible. This turns out to be impossible, [*i.e.*]{} one can show that the Jacobi identity between $K^{ab}{}_c$, $P_a$ and either $K_1^{a,bc}$ or $\tilde{K}_1^{abc}$ is not satisfied if the commutator between $K^{ab}{}_c$ and the dual graviton Og 1 operators gives dual graviton Og 2 operators. This is indeed the problem that one typically encounters when trying to construct a dual Riemann tensor.
One can define an operator $\bar{K}_2^{ab,cd}$ satisfying $$\bar{K}_2^{ab,cd} = \bar{K}_2^{(ab),cd} = \bar{K}_2^{ab,(cd)} =
\bar{K}_2^{cd,ab} \qquad \bar{K}_2^{a(b,cd)} = 0 \quad ,
\label{7.26}$$ whose corresponding Young Tableaux is shown in the last column in fig. \[fig3\]. We define the commutation relation of this operator with $P_a$ to be $$[ \bar{K}_2^{ab,cd} , P_e ] = {1 \over 2} \delta^{(a}_e K_1^{b
),cd} + {1 \over 2} \delta^{(c}_e K_1^{d ),ab} \quad .
\label{7.27}$$ This is indeed the most general result compatible with the symmetries, and one can show that the Jacobi identity with a further $P_a$ operator is satisfied.
If one adds the term ${\rm exp}(\bar{\Phi}^2_{ab,cd}
\bar{K}_2^{ab,cd})$ to the group element of eq. (\[7.12\]), one obtains that a transformation $$\delta \bar{\Phi}^2_{ab,cd} = \bar{c}_{ab,cd} \label{7.28}$$ implies an $x^2$ transformation for $A_{ab}$ of the form $$\delta A_{ab} = {1 \over 2} \bar{c}_{ab,cd} x^c x^d \quad .
\label{7.29}$$ This transformation cannot be written as a gauge transformation of eq. (\[7.8\]) for the linearised graviton. Following the arguments of the previous subsection, we refer to $\bar{K}_2^{ab,cd}$ as an ${\overline{\rm Og}}$ operator. The inverse Higgs mechanism allows to gauge away completely all the terms at most quadratic in $x$ in the field, with this still being compatible with all the symmetries.
Having introduced the operator $\bar{K}_2^{ab,cd}$, one obtains that the commutator between $K^{ab}{}_c$ and the dual graviton Og 1 operators can now be made compatible with the Jacobi identity with $P_a$. The result is $$\begin{aligned}
& & [ K^{ab}{}_c , K_1^{d,ef}] = -6 \delta^{(e}_c K_2^{\vert d
\vert , f) a b} + 6 \delta^{(d}_c K_2^{e , f) a b} +2 \delta^d_c
\tilde{K}_2^{efab} - 2 \delta^{(d}_c \tilde{K}_2^{ef) ab} \nonumber \\
& &\qquad \qquad \qquad \ \ - 4 \delta^d_c
\bar{K}_2^{ef,ab} + 4 \delta^{(d}_c \bar{K}_2^{ef), ab}
\nonumber\\
& & [ K^{ab}{}_c , \tilde{K}_1^{def} ] = 3
\delta^{(d}_c K_2^{e,f)ab} - 10 \delta^{(d}_c \tilde{K}_2^{ef)ab} +
2 \delta^{(d}_c \bar{K}_2^{ef),ab} \quad . \label{7.30}
\end{aligned}$$ The fact that $\bar{K}_2^{ab,cd}$ must appear on the right hand side of this commutation relation is the main result of this section. This shows that the only inverse Higgs mechanism compatible with the symmetries is the one that gauges away the dual graviton completely. We expect that once all the ${\overline{\rm Og}}$ operators for the dual graviton are introduced together with the Og operators for both the graviton and the dual graviton, the resulting algebra is well defined. We conjecture that the same applies to all the generators of $A_1^{+++}$ with positive level. As a consequence of this, after having applied the inverse Higgs mechanism, the Maurer-Cartan form will contain only the graviton Riemann tensor and its derivatives.
It is interesting to discuss the dimensional reduction to three dimensions in this case as we have done in the previous subsection. Following arguments similar to that case, one can show that the dimensional reduction of all the generators in fig. \[fig3\] contains the ${\overline{\rm Og}}$ 1 and ${\overline{\rm Og}}$ 2 generators for the scalar and the vector, and more generally the dimensional reduction of all the Og and ${\overline{\rm Og}}$ dual graviton generators leads to all the Og and ${\overline{\rm Og}}$ generators for the dimensionally reduced fields. The algebra of the dimensionally reduced theory can be truncated in such a way that the ${\overline{\rm Og}}$ generators for the scalar and the vector that arise from the reduction of the dual graviton can be consistently projected out, so that the corresponding Maurer-Cartan form results in the field-strengths for this fields, as well as their derivatives, once the inverse Higgs mechanism is applied.
The dual graviton in $E_8^{+++}$ in four dimensions
---------------------------------------------------
In this subsection we want to discuss the case in which the dual graviton couples to matter. We will discuss the case of the non-linear realisation of $E_8^{+++}$, [*i.e.*]{} $E_{11}$, in four dimensions, which corresponds to the bosonic sector of four-dimensional maximal supergravity. The Dynkin diagram of $E_{11}$ is shown in fig. \[fig1\], and the four dimensional theory is obtained deleting node 4 in the diagram. The internal symmetry is $E_{7}$, and the spectrum contains among the rest vectors in the ${\bf 56}$ of $E_7$. We will show that even neglecting couplings to gravity, it is impossible to make the gauge transformation of the dual graviton compatible with that of the vector. The situation is exactly as in the previous subsection: the commutator of two Og 1 operators for the vector generate the operator $\bar{K}_2^{ab,cd}$, which is an ${\overline{\rm Og}}\ 2$ operator for the dual graviton.
Decomposing the adjoint representation of $E_{11}$ in representations of $GL(4, \mathbb{R})$ one gets at level zero the gravity generators $K^a{}_b$ and the $E_7$ generators $R^\alpha$, while at level 1 one gets $R^{a, M}$, where $M$ denotes the ${\bf
56}$ of $E_7$. The higher level generators can be obtained as multiple commutators of $R^{a, M}$. In particular at level 2 one gets $$[ R^{a, M} , R^{b, N} ] = D_{\alpha}^{MN} R^{[ab] , \alpha} +
\Omega^{MN} R^{ab} \quad , \label{7.31}$$ where $R^{ab , \alpha}$ is the 2-form generator in the adjoint of $E_7$ and $R^{ab}$ is the dual graviton generator. We have also introduced $$D^{\alpha MN} = \Omega^{MP} D^\alpha_P{}^N \label{7.32}$$ which is symmetric in $MN$, and $D^\alpha_M{}^N$ are the generators in the ${\bf 56}$. Finally $\Omega^{MN}$ is the antisymmetric invariant tensor of $E_7$. The field associated to the generator $R^{[ab] , \alpha}$ is related to the scalars by duality. In the rest of this section we will ignore the 2-form contribution to the commutator of eq. (\[7.31\]), and we will only consider the dual graviton contribution, $$[ R^{a, M} , R^{b, N} ] =
\Omega^{MN} R^{ab} \quad . \label{7.33}$$ This truncation of the algebra is consistent because the Jacobi identities close independently on the 2-form generators and on the dual graviton generators.
The Og 1 generator for the vector $R^{a, M}$ is a generator $K^{ab ,
M}$ symmetric in $ab$, whose commutation relation with $P_a$ is $$[ K^{ab , M} , P_c ] = \delta^{(a}_c R^{b), M} \quad .
\label{7.34}$$ The commutation relation of $K^{ab , M}$ with $R^{a, M}$ can be obtained by imposing the Jacobi identity of these operators with $P_a$ and using eqs. (\[7.2\]), eq. (\[7.33\]) and eq. (\[7.34\]), as well as the fact that $R^{a, M}$ commutes with $P_a$. The result is $$[ R^{a, M} , K^{bc , N} ] = - {1 \over 2 } \Omega^{MN} K_1^{a, bc}
+ \Omega^{MN} \tilde{K}_1^{abc} \quad . \label{7.35}$$
We can now write the group element up to Og 2 generators, $$g = e^{x \cdot P} e^{\Phi^1_{a,bc} K_1^{a,bc}}
e^{\tilde{\Phi}^1_{abc} \tilde{K}_1^{abc}} e^{\Phi_{ab ,M} K^{ab ,M}}
e^{A_{ab}R^{ab}} e^{A_{a ,M} R^{a, M}}
\quad , \label{7.36}$$ which leads to the Maurer-Cartan form $$\begin{aligned}
g^{-1} \partial_\mu g & = & P_\mu + (\partial_\mu A_{a , M} - \Phi_{\mu
a , M} ) R^{a , M}
+ (\partial_\mu A_{ab}
+ {1 \over 2} \Omega^{MN} \partial_\mu
A_{a, M} A_{b ,N} \nonumber \\
&-& \Phi^1_{\mu,ab}
- \tilde{\Phi}^1_{\mu ab} - \Phi_{\mu a , M} A_{b ,N} \Omega^{MN} ) R^{ab} + ... \quad
. \label{7.37}
\end{aligned}$$ The inverse Higgs mechanism then leaves the field strength for the vector, while the term contracting $R^{ab}$ is put to zero by solving for $\Phi^1_{\mu,ab}$ and $\tilde{\Phi}^1_{\mu ab}$ in terms of $A_{ab}$ and $A_{a ,M}$.
We now compute the commutator of two Og 1 operators $K^{ab ,M}$ for the vector, and we determine which Og 2 generators are needed to satisfy the Jacobi identities. It turns out that because of the symmetry of the commutator, it is not possible to generate the Og 2 dual graviton operator $K_2^{a,bcd}$, and the Jacobi identity with $P_a$ imposes that this actually closes on $\tilde{K}_2^{abcd}$ and $\bar{K}_2^{ab,cd}$. The result is $$[ K^{ab ,M} , K^{cd ,N} ] = 2 \Omega^{MN} \tilde{K}_2^{abcd} -
\Omega^{MN} \bar{K}_2^{ab,cd} \quad . \label{7.38}$$ Thus exactly as in the case of the dual graviton coupled to gravity of the previous subsection we have found here that the commutator of two Og operators generates an ${\overline{\rm Og}}$ operator for the dual graviton, which means that a gauge invariant field strength for the dual graviton is not compatible with vector gauge invariance.
We claim that this is a generic feature of $E_{11}$ positive level generators with spacetime indices with mixed symmetry. The algebra of their Og generators does not close, and one is forced to introduce ${\overline{\rm Og}}$ generators for all these mixed symmetry generators. Only for the gravity generator, which has level zero, and for the generators with completely antisymmetric indices the Og algebra closes. As a consequence only for these fields can one use the inverse Higgs mechanism and be left with a non-vanishing field-strength. The fact that the positive level mixed symmetry generators require the introduction of the Og and ${\overline{\rm
Og}}$ generators implies instead that the corresponding fields do not allow a gauge invariant field strength and the inverse Higgs mechanism gauges away these fields completely. To show this one computes Jacobi identities involving positive level $E_{11}$ generators, Og generators and the momentum operator $P_a$. Thus this result deeply relies on the structure of the $E_{11}$ algebra. The dimensional reduction allows a further truncation of the algebra in the case in which a mixed symmetry generator gives rise to a generator with completely antisymmetric indices. Indeed in this case, as was shown in the previous subsections, the ${\overline{\rm
Og}}$ generators can be consistently projected out.
It is important to stress that the dynamics is compatible with this construction. The field strengths of the antisymmetric fields are first order in derivatives, and therefore one needs fields and dual fields to construct duality relations which are first order equations for these fields. The gravity Riemann tensor instead is at second order in derivatives and thus there is no need of a dual field to construct its equation of motion.
Conclusions
===========
In this paper we have given a method of obtaining field strengths and gauge transformations of all the massless and massive maximal supergravity theories starting from $E_{11}$. The global $E_{11}$ transformations of the fields are promoted to gauge transformations by the inclusion in the algebra of additional generators.
We have first shown how this mechanism works for pure gravity. We have constructed Einstein’s theory of gravity using a non-linear realisation which takes as its underlying algebra one that consists of $IGL(D,\mathbb{R} )$ and an infinite set of additional generators whose effect is to promote the rigid $IGL(D,\mathbb{R})$ to be local. This infinite number of additional generators lead to local translations, that is general coordinate transformations, but to no new fields in the final theory as their Goldstone fields are solved in terms of the graviton field using a set of invariant constraints placed on the Cartan forms. This is an example of what has been called the inverse Higgs effect [@23].
We have then generalised this procedure to $E_{11}$ at low levels. We have taken the algebra, called $E_{11}^{local}$ consisting of non-negative level $E_{11}$ generators, the generators $P_a$ and an infinite number of additional generators, whose role is to promote all the low level $E_{11}$ symmetries to gauge symmetries. Again, as in the gravity case these generators do not lead to new Goldstone fields. We have shown that the non-linear realisation of the algebra $E_{11}^{local}$ describes at low levels in eleven dimensions the 3-form and the 6-form of the eleven dimensional supergravity theory with all their gauge symmetries.
We have then considered in general the formulation of $D$-dimensional maximal gauged supergravity theories from the viewpoint of the enlarged algebra $\tilde{E}_{11,D}^{local}$. We have first considered as a toy model the Scherk-Schwarz dimensional reduction of the IIB supergravity theory from this viewpoint. One starts from the algebra $E_{11,10B}^{local}$ corresponding to the IIB theory and take the ten dimensional space-time to arise from an operator $\tilde Q$ which is constructed from $Q=P_{9}$ and part of the $SL(2,\mathbb{R})$ symmetry of the theory. This means that the 10th direction of space-time is twisted to contain a part in the $SL(2,\mathbb{R})$ coset symmetry of the theory. This non-linear realisation gives a nine dimensional gauged supergravity. We have observed that not all of the algebra $E_{11,10B}^{local}$ is essential for the construction of the gauged supergravity in nine dimensions, but only an algebra which we call $\tilde E_{11,
9}^{local}$ which is the subalgebra of $E_{11,10B}^{local}$ that commutes with $\tilde Q$. Its generators are non-trivial combinations of $E_{11}$ generators and the additional generators and in general the generators of $\tilde E_{11, 9}^{local}$ have non-trivial commutation relations with nine dimensional space-time translations. Although the subalgebra $\tilde E_{11, 9}^{local}$ appears to be a deformation of the original $E_{11}$ algebra and the space-time translations we have not changed the original commutators, but rather the new algebra arises due to the presence of the additional generators which are added to the $E_{11}$ generators.
However, we have then shown that one can find the algebra $\tilde
E_{11, 9}^{local}$ without carrying out all the above steps. Given the non-trivial relation between the lowest non-trivial positive level generator of $\tilde E_{11,9}^{local}$ and the nine dimensional space-time translations one can derive the rest of the algebra $\tilde E_{11, 9}^{local}$ simply using Jacobi identities. This algebra determines uniquely all the field strengths of the theory, and thus one finds a very quick way of deriving the gauged supergravity theory.
This picture applies to all gauged supergravity theories, as one can easily find the algebra $\tilde E_{11, D}^{local}$ without using its derivation from $E_{11}^{local}$ and this provides a very efficient method of constructing all gauged supergravities. We have illustrated how this works by constructing the massive IIA theory as well as all the gauged maximal supergravities in five dimensions.
Finally, we have considered how this construction generalises to the fields with mixed symmetry, [*i.e.*]{} not completely antisymmetric, of $E_{11}$ and in general of any non-linear realisation of a very-extended Kac-Moody algebra. We have considered as a prototype of such fields the dual graviton in four dimensions. If one tries to promote the global shift symmetry of the dual graviton field to a gauge symmetry, one finds that this is not compatible with the $E_{11}$ algebra. The solution of this problem is that actually $E_{11}$ forces to include additional generators, whose role is to enlarge the gauge symmetry of the dual graviton so that one can gauge away the field completely. This also applies if one only restricts his attention to the compatibility of the dual graviton with matter fields, that is if one neglects the gravity generators. This result agrees with the field theory analysis of [@34]. More generally, this agrees with the no-go theorems of [@35] on the consistency of self-interactions for the dual graviton. Recently, an alternative approach to the construction of an action for the dual graviton has been taken [@36], in which the metric only appears via topological couplings, and an additional shift gauge field is included.
As we have mentioned in the introduction it is not obvious how to to implement the conformal group, or equivalently, add the Og fields in the presence of the generators of the $l$ multiplet. The rational for introducing the $l$ multiplet was that it would allow an $E_{11}$ way of encoding space-time. However, in this paper we have chosen to take only the first component of the $l$ multiplet, namely the space-time translations and we have taken this to commute with the positive level $E_{11}$ generators. As a result we have had to discard the negative level $E_{11}$ generators. This is unsatisfactory as $E_{11}$ is defined from its Chevalley generators and relations and there is no definition that uses only the positive levels. For this reason the content of the adjoint representation and the $l$ multiplet also rely on the negative root generators. However, we know that many of the generators, and so fields in the non-linear realisation, in the former and brane charges in the latter are in very convincing agreement with what one might expect in M theory. One example being the classification of all gauged supergravities using the $D-1$ forms found in the adjoint representation of $E_{11}$. How to reconcile local symmetries, space-time and the full $E_{11}$ algebra is for future work.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work is supported by the PPARC rolling grant PP/C5071745/1, the EU Marie Curie research training network grant MRTN-CT-2004-512194 and the STFC rolling grant ST/G000/395/1.
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abstract: 'We study a likelihood ratio test for the location of the mode of a log-concave density. Our test is based on comparison of the log-likelihoods corresponding to the unconstrained maximum likelihood estimator of a log-concave density and the constrained maximum likelihood estimator where the constraint is that the mode of the density is fixed, say at $m$. The constrained estimation problem is studied in detail in the companion paper, Doss and Wellner (2016b). Here the results of that paper are used to show that, under the null hypothesis (and strict curvature of $-\log f$ at the mode), the likelihood ratio statistic is asymptotically pivotal: that is, it converges in distribution to a limiting distribution which is free of nuisance parameters, thus playing the role of the $\chi_1^2$ distribution in classical parametric statistical problems. By inverting this family of tests we obtain new (likelihood ratio based) confidence intervals for the mode of a log-concave density $f$. These new intervals do not depend on any smoothing parameters. We study the new confidence intervals via Monte Carlo methods and illustrate them with two real data sets. The new intervals seem to have several advantages over existing procedures. Software implementing the test and confidence intervals is available in the R package `logcondens.mode`.'
address:
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School of Statistics\
University of Minnesota\
Minneapolis, MN 55455\
\
- |
Department of Statistics, Box 354322\
University of Washington\
Seattle, WA 98195-4322\
\
author:
-
-
bibliography:
- 'ModeConstrained.bib'
title: 'Inference for the mode of a log-concave density'
---
Introduction and overview: inference for the mode {#sec:Intro}
=================================================
Let ${\cal P}$ denote the class of all log-concave densities $f$ on ${{\mathbb R}}$. It is well-known since Ibragimov (1956) that all log-concave densities $f$ are strongly unimodal, and conversely; see [@DJ1988UCA] for an exposition of the basic theory. Of course, “the mode” of a log-concave density $f$ may not be a single point. It is, in general, the modal interval $MI (f) \equiv \{ x \in {{\mathbb R}}: \ f(x) = \sup_{y \in {{\mathbb R}}} f(y) \}$, and to describe “the mode” completely we need to choose a specific element of $MI(f)$, for example $M(f) \equiv \inf \{ x \in MI (f) \}$. For a large sub-class of log-concave densities the set reduces to a single point. Our focus here is on the latter case and, indeed, on inference concerning $M(f)$ based on i.i.d. observations $X_1, \ldots , X_n$ with density $f_0 \in {\cal P}$. We have restricted to log-concave densities for several reasons:
(a)
: It is well known that the MLE over the class of all unimodal densities does not exist; see e.g. Birgé (1997).
(b)
: On the other hand, MLE’s do exist for the class ${\cal P}$ of log-concave densities if $n\ge 2$: see, for example, [@MR2459192], [@RufibachThesis], [@DR2009LC].
(c)
: Moreover the MLE’s for the class of log-concave densities have remarkable stability and continuity properties under model miss-specification: see e.g. [@DSS2011LCreg].
In terms of estimation, [@BRW2007LCasymp] showed that if $f_0 = e^{\varphi_0}$ where the concave function $\varphi_0$ satisfies $\varphi_0^{\prime\prime} (m_0) < 0$ at the mode $m_0 = M(f_0)$ of $f_0$, then the MLE $M(\widehat{f}_n)$ satisfies $$\begin{aligned}
n^{1/5} \left ( M(\widehat{f}_n ) - M(f_0) \right ) \rightarrow_d \left ( \frac{(4!)^2 f_0 (m_0)}{f_0^{\prime \prime} (m_0)^2} \right )^{1/5} M (H_2 ^{(2)} )
\label{LimitDistribMLEofMode}\end{aligned}$$ where $M(H_2^{(2)})$ has a universal distribution (not depending on $f_0$) defined in terms of two-sided Brownian motion processes: briefly it is the mode (or argmax) of the estimator $H_2^{(2)}$ of the (“unknown”) concave function $g_0 (t) \equiv -12t^2$ in the “white noise” model where one observes $$\begin{aligned}
d X(t) = g_0 (t) dt + d W(t)
\label{WhiteNoiseCanonicalConcave}\end{aligned}$$ where $W$ is two-sided Brownian motion on ${{\mathbb R}}$. The limit distribution (\[LimitDistribMLEofMode\]) gives useful information about the behavior of $M(\widehat{f}_n)$, but it is somewhat difficult to use for inference because of the constant $( (4!)^2 f_0 (m_0) / f_0^{\prime \prime} (m_0)^2 )^{1/5}$ which involves the unknown density through $f_0^{\prime\prime} (m_0)$. This can be estimated via smoothing methods, but because we wish to avoid the consequent problem of choosing bandwidths or other tuning parameters, we take a different approach to inference here.
Instead, we first consider the following testing problem: test $$\begin{aligned}
H : \ M(f) = m \ \ \ \mbox{versus} \ \ \ K: \ M(f) \not= m\end{aligned}$$ where $m \in {{\mathbb R}}$ is fixed. To construct a likelihood ratio test of $H$ versus $K$ we first need to construct both the unconstrained MLE’s $\widehat{f}_n$ and the mode-constrained MLE’s $\widehat{f}_n^0$. The unconstrained MLE’s $\widehat{f}_n$ are available from the results of [@MR2459192], [@RufibachThesis], and [@DR2009LC] cited above. Corresponding results concerning the existence and properties of the mode-constrained MLE’s $\widehat{f}_n^0$ are given in the companion paper [@Doss-Wellner:2016ModeConstrained]. Global convergence rates for both estimators are given in [@DossWellner:2016a]. Once both the unconstrained estimators $\widehat{f}_n$ and the constrained estimators $\widehat{f}_n^0$ are available, then we can consider the natural likelihood ratio test of $H$ versus $K$: reject the null hypothesis $H$ if $$\begin{aligned}
2 \log \lambda_n \equiv 2 \log \lambda_n (m) \equiv 2 n {{\mathbb P}}_n ( \log \widehat{f}_n - \log \widehat{f}_n^0 )
= 2 n {{\mathbb P}}_n (\widehat{\varphi}_n - \widehat{\varphi}_n^0)
\end{aligned}$$ is “too large” where $\widehat{f}_n = \exp ( \widehat{\varphi}_n ) $, $\widehat{f}_n^0 = \exp ( \widehat{\varphi}_n^0 ) $, ${{\mathbb P}}_n = \sum_{i=1}^n \delta_{X_i} / n$ is the empirical measure, and ${{\mathbb P}}_n(g) = \int g d{{\mathbb P}}_n$. To carry out this test we need to know how large is “too large”; i.e. we need to know the (asymptotic) distribution of $2 \log \lambda_n$ when $H $ is true. Thus the primary goal of this paper is to prove the following theorem:
\[LRasympNullDistribution\] If $X_1, \ldots , X_n$ are i.i.d. $f_0 = e^{\varphi_0}$ with mode $m$ where $\varphi_0$ is concave and $\varphi_0^{\prime\prime}(m) < 0$, then $$\begin{aligned}
2 \log \lambda_n \rightarrow_d {{\mathbb D}}\end{aligned}$$ where ${{\mathbb D}}$ is a universal limiting distribution (not depending on $f_0$); thus $2 \log \lambda_n$ is asymptotically pivotal.
With Theorem \[LRasympNullDistribution\] in hand, our likelihood ratio test with (asymptotic) size $\alpha \in (0,1)$ becomes: “reject $H$ if $2 \log \lambda_n > d_{\alpha}$” where $d_{\alpha}$ is chosen so that\
$P({{\mathbb D}}> d_{\alpha} ) = \alpha$. Furthermore, we can then form confidence intervals for $m$ by inverting the family of likelihood ratio tests: let $$\begin{aligned}
J_{n,\alpha} \equiv \{ m \in {{\mathbb R}}: \ 2\log \lambda_n (m) \le d_{\alpha} \} .
\label{LRConfInterval}\end{aligned}$$ Then it follows that for $f_0 \in {\cal P}_m = \{ f \in {\cal P} : \ M(f ) = m \}$ with $(\log f_0)''<0$, we have $$P_{f_0} ( m \in J_{n,\alpha} ) \rightarrow P( {{\mathbb D}}\le d_{\alpha} ) = 1- \alpha.$$ This program is very much analogous to the methods for pointwise inference for nonparametric estimation of monotone increasing or decreasing functions developed by [@MR1891743] and [@MR2341693]. Those methods have recently been extended to include pointwise inference for nonparametric estimation of a monotone density by [@MR3375875].
\[rem:rem:DD-formula\] Theorem \[LRasympNullDistribution\] says that $\mathbb{D}$ is pivotal (this is sometimes known as the Wilks phenomenon) over the class of all log-densities $\vvo$ satisfying $\vvo^{(2)}(m) < 0$. We can specify more about the limit random variable ${{\mathbb D}}$. Let $\widehat{\varphi}$ and $\widehat{\varphi}^0$ denote the least squares (and hence also maximum likelihood) unconstrained estimator and mode-constrained estimator of the canonical concave function in the Gaussian white noise problems described in Subsections \[ssec:UnconstrainedWhiteNoiseProblem\] and \[ssec:ConstrainedWhiteNoiseProblem\] below. Then under the assumptions of Theorem \[LRasympNullDistribution\], the proof of the theorem shows that $${{\mathbb D}}= \int_{-\infty}^\infty (\widehat{\varphi}^2(u) -(\widehat{\varphi}^0)^2(u))du.$$ The form of this random variable is the same as that found in [@MR1891743] and [@MR2341693], if we replace our $\widehat{\varphi}$ and $\widehat{\varphi}^0$ with the corresponding random functions studied in the monotone case.
A secondary goal of this paper is to begin a study of the likelihood ratio statistics $2 \log \lambda_n$ under fixed alternatives. Our second theorem concerns the situation when $f \in {\cal P}$ has mode $M(f) \not= m$.
\[LR-limit-FixedAlternative\] Suppose that $f_0 \in {\cal P}$ with $m \notin MI(f_0)$. Then $$\begin{aligned}
\frac{2}{n} \log \lambda_n (m)
& \rightarrow_p & 2 K(f_0, f_m^0 ) \nonumber \\
& = & 2 \inf \{ K(f_0, g) : \ g \in {\cal P}_m \} > 0 \label{LR-limit-FixedAlt}\end{aligned}$$ where $ f_m^0 \in {\cal P}_m$ achieves the infimum in and $$\begin{aligned}
K(f,g) \equiv \left \{ \begin{array}{c l} \int f (x) \log \frac{f(x)}{g(x)} d x , & \ \ \mbox{if} \ \ f \prec \prec g \\ \infty, & \ \ \mbox{otherwise} .
\end{array} \right .\end{aligned}$$
Here $f \prec \prec g$ means $f = 0$ whenever $g = 0$ except perhaps on a set of Lebesgue measure $0$. The proof of Theorem \[LR-limit-FixedAlternative\] is sketched in Subsection \[ssec:PfSketchThm2\] with all the details given in Subsection \[subsec:thm-alternative\], and relies on the methods used by [@MR2645484] and [@DSS2011LCreg], in combination with the results of [@DossWellner:2016a]. Theorem \[LR-limit-FixedAlternative\] implies consistency of the likelihood ratio test based on the critical values from Theorem \[LRasympNullDistribution\]. That is: let $d_{\alpha}$ satisfy $P(
{{\mathbb D}}> d_{\alpha}) = \alpha$ for $0 < \alpha < 1$, and suppose we reject $H: \
M(f) = m$ if $2 \log \lambda_n (m) > d_{\alpha}$.
If the hypotheses of Theorem \[LR-limit-FixedAlternative\] hold, then the likelihood ratio test “reject $H$ if $2\log \lambda_n(m) > d_{\alpha}$” is consistent: if $f \notin {\cal P}_m$, then $$\begin{aligned}
P_f ( 2 \log \lambda_n (m) > d_{\alpha} ) \rightarrow 1 .\end{aligned}$$
Here is an explicit example:
Suppose that $f $ is the Laplace density given by $$f(x) = (1/2) \exp ( - | x |) .$$ First we note that $M(f) = 0$ so that $f \notin {\cal P}_1$. Thus for testing $H : \ M(f) = 1$ versus $K : M(f) \not= 1$, the Laplace density $f$ satisfies $f \in {\cal P} \setminus {\cal P}_1$. So we have (incorrectly) hypothesized that $M(f) = 1 \equiv m$. In this case the constrained MLE $\widehat{f}_n^0$ satisfies $\int | \widehat{f}_n^0 - f^0 | dx \rightarrow_{a.s.} 0$ where $f^0 \equiv g^* \in {\cal P}_1$ is determined by Theorem \[DSS-thm2.7m\] below, which is the population analogue of Theorem 2.10 of [@Doss-Wellner:2016ModeConstrained]. It also satisfies (\[LR-limit-FixedAlt\]) in Theorem 1.2. In the present case, $g^* = g_{a^*} $ where $\{ g_a : \ a \in (0,1]\}$ is the family of densities given by $$\begin{aligned}
g_a (x) = \left \{ \begin{array}{l c} (1/2) e^x, & - \infty < x \le -a \\ (1/2) e^{-a} , & -a \le x \le 1, \\ (1/2) e^{-a} e^{-c(x-1)}, & \ \ 1 \le x < \infty ,
\end{array} \right .\end{aligned}$$ where $c \equiv c(a) = 1/(2e^a - (2+a))$ is chosen so that $\int g_a (x)dx =1$. Here it is not hard to show that $a^* \approx .490151 \ldots $ satisfies $c(a^*)^2 = \exp ( - (a^* -1))$, while $K(f,f^0) = K(f,g_{a^*} ) \approx 0.03377\ldots $.
Although the basic approach here has points in common with the developments in [@MR1891743] and [@MR2341693], the details of the proofs require several new tools and techniques due to the relative lack of development of theory for the mode-constrained log-concave MLEs. To prove Theorem \[LRasympNullDistribution\] we first prepare the way by reviewing the local asymptotic distribution theory for the unconstrained estimators $\widehat{f}_n$ and $\widehat{\varphi}_n$ developed by [@BRW2007LCasymp] and some of the relevant background from [@MR1891741]. We then develop corresponding local asymptotic distribution theory for the mode-constrained estimators $\widehat{f}_n^0$ and $\widehat{\varphi}_n^0$. These results are stated in Subsections 3.1 and 3.2. Our proofs of Theorems 1.1 and 1.2 are each outlined in Section \[sec:PfSketches\]. The complete details of the proof of Theorem 1.1 are deferred to Subsections \[ssec:ProofsLocalRTs\] and \[ssec:GlobalRTs\]. In Subsection \[ssec:ProofsLocalRTs\] we treat remainder terms in a local neighborhood of the mode $m$, while remainder terms away from the mode are treated in Subsection \[ssec:GlobalRTs\]. Our proofs in Subsections \[ssec:ProofsLocalRTs\] and \[ssec:GlobalRTs\] rely heavily on the theory developed for the constrained estimators in [@Doss-Wellner:2016ModeConstrained]. The proof of Theorem 1.2 is outlined in Subsection \[ssec:PfSketchThm2\] with further details given in Subsection \[subsec:thm-alternative\].
In Section 5 we present Monte-Carlo estimates of quantiles of the distribution of ${{\mathbb D}}$ and provide empirical evidence supporting the universality of the limit distribution (under the assumption that $\varphi_0^{\prime \prime}(m) < 0$). We illustrate the likelihood ratio confidence sets with two data sets and also present some Monte-Carlo evidence concerning the length and coverage probabilities of our proposed intervals. Section 6 gives a brief description of further problems and potential developments. We also discuss connections with the results of [@MR947566], [@MR964293], [@MR1105839], and [@MR1671670; @MR1747496].
The white noise problem and limit processes {#sec:WhiteNoise-LimitProcesses}
===========================================
Unconstrained white noise problem {#ssec:UnconstrainedWhiteNoiseProblem}
---------------------------------
Our goal here is to introduce the limit processes that arise in the asymptotic (or white noise) version of the unconstrained estimation problem. First, let $W$ denote two-sided Brownian motion, starting at $0$. We then define $$\begin{aligned}
Y(t) \equiv \int_0^t W(s) ds - t^4, & \ \ \mbox{for} \ \ t \in {{\mathbb R}};
\label{TwiceIntegratedWhiteNoise}\end{aligned}$$ here, for $t<0$, $\int_0^t W(s) ds = - \int_t^0 W(s)ds = - \int_0^{-t} W(-r) dr $. This is just the white noise relation (\[WhiteNoiseCanonicalConcave\]) integrated twice. The [*lower invelope*]{} $H$ of the process $Y$ is the unique process satisfying:\
(i) $H(t) \le Y(t) $ for all $t \in {{\mathbb R}}$;\
(ii) $H^{(2)} $ is concave;\
(iii) $H(t) = Y(t)$ if the slope of $H^{(2)}$ decreases strictly at $t$; equivalently $$\begin{aligned}
\int_{-\infty}^{\infty} (Y(t) - H(t)) dH^{(3)} (t) = 0 .\end{aligned}$$ We think of $H^{(2)}\equiv \widehat{\varphi}$ as the estimator of the (canonical) concave function $ - 12 t^2$. $\widehat{\varphi}$ is piecewise linear, although it is not known if its knots are separated; they do collectively have Lebesgue measure $0$. This will not affect our proofs. For the proof of the existence and uniqueness of $H$ see [@MR1891741]. \[page:unconstrainedWhiteNoiseProblem\]
Constrained white noise problem {#ssec:ConstrainedWhiteNoiseProblem}
-------------------------------
Now we introduce the limit processes that arise in the asymptotic (or white noise) version of the mode constrained estimation problem. As before $W$ denotes a two-sided Brownian motion starting at $0$. Let $\widehat{\varphi}^0$ be a concave function with argmax at $0$. Define its knot points $\Sa$ by $ (\Sa)^c := {\left\{} \newcommand{\rb}{\right\}}t \in {{\mathbb R}}: (\vva)^{(2)}(t \pm ) = 0 \rb.$ Next, define $ \TauL = \sup \Sa \cap (-\infty, 0)$ and $\TauR = \inf \Sa \cap (0, \infty).$ We then set $$\begin{aligned}
\begin{array}{l l l}
F_L (t) = \int_t^{\tau_L} \widehat{\varphi}^0(v) dv, \qquad & H_L (t) = \int_t^{\tau_L} F_{L} (v) dv , \\ F_R (t) = \int_{\tau_R}^t \widehat{\varphi}^0(v) dv, \qquad & H_R (t) = \int_{\tau_R}^t F_{R} (v) dv , \end{array}\end{aligned}$$ and, with $X(t) \equiv Y^{(1)} (t) = W(t) - 4 t^3 $, we define $$\begin{aligned}
\begin{array}{l l l}
X_{L} (t) \equiv \int_t^{\tau_L} d X (v) , \qquad & Y_L (t) \equiv \int_t^{\tau_L} X_L (v) dv, \\ X_{R} (t) \equiv \int_{\tau_R}^t d X(v) , \qquad & Y_R (t) \equiv \int_{\tau_R}^t X_R(v) dv,
\end{array}\end{aligned}$$ with all definitions holding for all $t \in {{\mathbb R}}$. Finally let $$\begin{aligned}
\tau_-^0 = \sup {\left\{} \newcommand{\rb}{\right\}}t \in \Sa : (\vva)'(t-{\varepsilon}-) > 0 \mbox{ for all }
{\varepsilon}> 0\rb \le 0, \\
\tau_+^0 = \inf {\left\{} \newcommand{\rb}{\right\}}t \in \Sa : (\vva)'(t+ {\varepsilon}+) < 0 \mbox{ for all }
{\varepsilon}> 0 \rb \ge 0.\end{aligned}$$ These variables satisfy $\tau_+^0 \le \tau_R$ and $\tau_L \le \tau_-^0 $, but the inequalities will be strict in some cases where $\tau_-^0$ or $\tau_+^0$ equals $0$. We have the following theorem.
\[ConstrainedWhiteNoiseProblem\] ([@Doss:2013]; [@Doss-Wellner:2016ModeConstrained] Theorem 5.6). There exists a unique pair of processes $H_L$ and $H_R$ such that $H_L^{(2)}(t) = H_R^{(2)}(t) \equiv \vva(t)$ for $t \in {{\mathbb R}}$, such that $\vva$ is concave with mode at $0$, and such that:\
(i) $H_L (t) - Y_L (t) \le 0$ for all $t \le 0$, and $H_R (t) - Y_R (t) \le 0$ for all $t \ge 0$.\
(ii) $\int_{(-\infty, \tau_{-}^0]} (H_L (v) - Y_L (v) ) d (\widehat{\varphi}^0)^{\prime} (v) = 0$\
$\phantom{blab}$ and $ \int_{[\tau_{+}^0, \infty)} (H_R (v) - Y_R (v) ) d ( \widehat{\varphi}^0 )^{\prime} (v) = 0$.\
(iii) $\int_{\tau_L}^{\tau_R} ( \widehat{\varphi}^0 (v)dv - d X (v)) = 0 $.
The proof of Theorem \[ConstrainedWhiteNoiseProblem\] can be found in [@Doss-Wellner:2016ModeConstrained].
We note that condition (iii) in Theorem 2.1 is an analogue in the white noise problem of the condition $\int_{-\infty}^\infty d ( { {\mathbb F}}_n (v) - \widehat{F}_n^0 (v)) = 0$ in the finite-sample case. See [@Doss:2013], Remark 5.1.11, pages 126 - 127 for a discussion.
Local asymptotic distribution theory near the mode {#sec:AsymptoticDistTheorNearMode}
==================================================
Preparation: limit processes and scaling relations {#ssec:PrepLimitProcessesScaling}
--------------------------------------------------
From Sections 2.1 and 2.2 we know that the processes $H$ and $H^{(2)} = \widehat{\varphi}$ and $H^0$ and $(H^0)^{(2)} = \widehat{\varphi}^0$ exist and are unique in the limiting Gaussian white noise problem described by (\[WhiteNoiseCanonicalConcave\]). Before proceeding to statements concerning convergence of the local processes we introduce further notation and basic scaling results. As in [@MR1891741] Appendix A, Proposition A.1, and Theorem 4.6 of [@BRW2007LCasymp] (noting the corrections indicated at the end of this section), let $\sigma \equiv 1/\sqrt{f_0 (m)}$, $a = | \varphi_0^{(2)} (m)/4!$, and let $$\begin{aligned}
&& Y_{a,\sigma}(t) \equiv \sigma \int_0^t W(s) ds - a t^4 \stackrel{d}{=} \sigma (\sigma/a)^{3/5} Y( (a/\sigma)^{2/5} t) , \\
&& Y_{a,\sigma}^{(1)} (t) = \sigma W(t) - 4a t^3 \stackrel{d}{=} \sigma (\sigma/a)^{1/5} Y^{(1)} ( (a/\sigma)^{2/5} t),\end{aligned}$$ where $Y \equiv Y_{1,1}$. These processes arise as the limits of appropriate (integrated) localized empirical processes. Similar relations are satisfied by the unconstrained and constrained invelope processes $H_{a,\sigma}$, $H_{a,\sigma}^0$, and their derivatives: with $H\equiv H_{1,1}$ and $H^0 \equiv H_{1,1}^0$, where $H^0$ can be either $H_L$ or $H_R$, $$\begin{aligned}
&& H_{a,\sigma}(t) \stackrel{d}{=} \sigma (\sigma/a)^{3/5} H( (a/\sigma)^{2/5} t) , \\
&& H_{a,\sigma}^0 (t)\stackrel{d}{=} \sigma (\sigma/a)^{3/5} H^0 ( (a/\sigma)^{2/5} t), \\
&& H_{a,\sigma}^{(1)} (t) \stackrel{d}{=} \sigma (\sigma/a)^{1/5} H^{(1)} ( (a/\sigma)^{2/5} t),\\
&& (H_{a,\sigma}^0 )^{(1)} (t) \stackrel{d}{=} \sigma (\sigma/a)^{1/5} (H^0)^{(1)} ( (a/\sigma)^{2/5} t),\\\end{aligned}$$ and $$\begin{aligned}
\widehat{\varphi}_{a,\sigma}
& = & H_{a,\sigma}^{(2)} \stackrel{d}{=} \sigma^{4/5} a^{1/5} H^{(2)} ((a/\sigma)^{2/5} \cdot ) \nonumber \\
& = & \frac{1}{\gamma_1 \gamma_2^2} H^{(2)} ( \cdot / \gamma_2)
\equiv \frac{1}{\gamma_1 \gamma_2^2} \widehat{\varphi} (\cdot / \gamma_2), \label{ScalingRelationUnConstrained}\end{aligned}$$ and, similarly, $$\begin{aligned}
\widehat{\varphi}_{a,\sigma}^0
& = & (H_{a , \sigma}^0)^{(2)} \stackrel{d}{=} \sigma^{4/5} a^{1/5} (H^0)^{(2)} ((a/\sigma)^{2/5} \cdot ) \nonumber \\
& = & \frac{1}{\gamma_1 \gamma_2^2} (H^0)^{(2)} ( \cdot / \gamma_2)
\equiv \frac{1}{\gamma_1 \gamma_2^2} \widehat{\varphi}^0 (\cdot / \gamma_2),
\label{ScalingRelationConstrained}\end{aligned}$$ Here $$\begin{aligned}
&& \gamma_1 = \left ( \frac{f_0(m)^4 | \varphi_0^{(2)} (m) |^3}{(4!)^3} \right )^{1/5} = \frac{1}{\sigma} \left ( \frac{a}{\sigma} \right )^{3/5} , \\
&& \gamma_2 = \left ( \frac{(4!)^2}{f_0 (m) | \varphi_0^{(2)} (m) |^2 } \right )^{1/5} = \left ( \frac{\sigma}{a} \right )^{2/5},
\label{GammaDefnsLogConcaveAtMode}\end{aligned}$$ and we note that $$\begin{aligned}
&& \gamma_1 \gamma_2^{3/2} = \sigma^{-1} = \sqrt{f_0 (m)}, \ \ \ \gamma_1 \gamma_2^4 = a^{-1} = \frac{4!}{| \varphi_0^{(2)} (m)|} ,
\label{GammaRelationsPart1} \\
&& \gamma_1 \gamma_2^2 = \frac{1}{C(m, \varphi_0)} \equiv \left ( \frac{4! f_0 (m)^2}{| \varphi_0^{(2)} (m)|} \right )^{1/5} .
\label{GammaRelationsPart2}\end{aligned}$$
Unconstrained and Constrained local limit processes
---------------------------------------------------
Now we are ready for the statement of the convergence theorem for the local unconstrained and constrained estimator processes. As an extension of Theorem 2.1 of [@BRW2007LCasymp] we have the following theorem describing the joint limiting behavior of the estimators $(\widehat{\varphi}_n (x_0) , \widehat{\varphi}_n^0(x_0))$ at a fixed point $x_0 \not= m$ with $\varphi_0^{\prime\prime} (x_0) < 0$, and the joint limiting behavior of the local processes defined in neighborhoods of $m$ under the assumption of curvature at $m$. Here are the assumptions we will make for this theorem.
\[CurvatureAtTheMode\] (Curvature at $m$) Suppose that $X_1, \ldots , X_n$ are i.i.d. $f_0 = e^{\varphi_0} \in {\cal P}_m$ and that $f_0$ and $\varphi_0$ are twice continuously differentiable at $m$ with $\varphi_0^{\prime \prime} (m) < 0$.
\[CurvatureAwayFromMode\] (Curvature at $x_0 \not= m$) Suppose that $X_1, \ldots , X_n$ are i.i.d. $f_0 = e^{\varphi_0} \in {\cal P}_m$ and that $f_0$ and $\varphi_0$ are twice continuously differentiable at $x_0 \not= m$ with $\varphi_0^{\prime \prime} (x_0) < 0$ and $f_0 (x_0) > 0$.
\[JointLimitingDistributions\] A. (At a point $x_0 \not= m$). Suppose that $\varphi_0$ and $f_0$ satisfy Assumption \[CurvatureAwayFromMode\]. Then $$\begin{aligned}
\left ( \begin{array}{l} n^{2/5} ( \widehat{\varphi}_n (x_0) - \varphi_0 (x_0)) \\
n^{2/5} ( \widehat{\varphi}_n^0 (x_0) - \varphi_0 (x_0))
\end{array} \right ) \rightarrow_d
\left ( \begin{array}{l} {{\mathbb V}}\\ {{\mathbb V}}\end{array} \right )
\end{aligned}$$ where ${{\mathbb V}}\equiv C(x_0 , \varphi_0 ) H^{(2)} (0)$ and where $C(x_0, \varphi_0 )$ is as given in (\[GammaRelationsPart2\]) (but with $m$ replaced by $x_0$): $$\begin{aligned}
C(x_0 , \varphi_0 ) = \left ( \frac{ | \varphi_0^{(2)} (x_0)|}{4!f_0 (x_0)^2} \right )^{1/5} .
\label{ConstantReducToCanonical}
\end{aligned}$$ Consequently $$\begin{aligned}
n^{2/5} ( \widehat{\varphi}_n (x_0) - \widehat{\varphi}_n^0 (x_0) ) \rightarrow_p 0 .
\end{aligned}$$ B. (In $n^{-1/5}-$neighborhoods of $m$) Suppose $\varphi_0$ and $f_0$ satisfy Assumption \[CurvatureAtTheMode\]. Define processes $X_n$ and $X_n^0 $ by $$\begin{aligned}
X_n (t) \equiv n^{2/5} (\widehat{\varphi}_n (m + n^{-1/5} t) - \varphi_0 (m) ) \\
X_n^0 (t) \equiv n^{2/5} ( \widehat{\varphi}_n^0 (m + n^{-1/5} t) - \varphi_0 (m)).
\end{aligned}$$ Then the finite-dimensional distributions of $(X_n (t), X_n^0 (t))$ converge in distribution to the finite-dimensional distributions of the processes $$\begin{aligned}
(\widehat{\varphi}_{a,\sigma} (t), \widehat{\varphi}_{a,\sigma}^0 (t) )\stackrel{d}{=}
\frac{1}{\gamma_1 \gamma_2^2} ( \widehat{\varphi} (t/\gamma_2) , \widehat{\varphi}^0 (t/\gamma_2) ) \equiv ({{\mathbb X}}(t), {{\mathbb X}}^0 (t))\end{aligned}$$ where $H, H^0, H^{(2)} = \widehat{\varphi}$, and $(H^0)^{(2)} = \widehat{\varphi}^0$ are as described in Subsections \[ssec:UnconstrainedWhiteNoiseProblem\] \[ssec:ConstrainedWhiteNoiseProblem\], and \[ssec:PrepLimitProcessesScaling\]. Furthermore, for $p\ge 1$ $$(X_n (t) , X_n^0 (t) ( \rightarrow_d ({{\mathbb X}}(t) , {{\mathbb X}}^0(t) )$$ in ${\cal L}^p [-K,K] \times {\cal L}^p [-K,K]$ for each $K>0$.
Theorem \[JointLimitingDistributions\] is a consequence of a general theorem describing a vector of local processes in $( {\cal D}_K \times {\cal C}_K^3 )^3 \times ({\cal D}_K\times {\cal C}_K )^3$ where the various coordinates describe:\
(a) The localized processes $\tilde{H}_{n}^{(3)}, \tilde{H}_{n}^{(2)}, \tilde{H}_{n}^{(1)}, \tilde{H}_{n}$ in the unconstrained problem; here $\tilde{H}_n^{(2)}$ corresponds to $X_{n}(t)$, and the other three processes are the derivative, the integral and the double integral of $\tilde{H}_n^{(2)}$, respectively.\
(b) The localized L processes $\tilde{H}_{n,L}^{(3)}, \tilde{H}_{n,L}^{(2)}, \tilde{H}_{n,L}^{(1)}, \tilde{H}_{n,L}$ and R processes $\tilde{H}_{n,R}^{(3)},$ $\tilde{H}_{n,R}^{(2)},$ $\tilde{H}_{n,R}^{(1)},$ $\tilde{H}_{n,R}$ in the constrained problem. Here, it turns out we need different processes for $t \le 0$ and $ t \ge 0$, so the L processes correspond to $t \le 0$ and the R processes to $t \ge 0$. $\tilde{H}_{n,L}^{(2)}$ and $\tilde{H}_{n,R}^{(2)}$ correspond to $X_{n}^0(t)$.\
(c) And then the corresponding “driving” empirical processes $$\begin{aligned}
( \tilde{{{\mathbb Y}}}_{n,L}^{(1)}, \tilde{{{\mathbb Y}}}_{n,L}, \tilde{{{\mathbb Y}}}_{n,R}^{(1)}, \tilde{{{\mathbb Y}}}_{n,R} , \tilde{{{\mathbb Y}}}_{n}^{(1)} , \tilde{{{\mathbb Y}}}_n ) ,\end{aligned}$$ where $\tilde{{{\mathbb Y}}}_{n,L}, \tilde{{{\mathbb Y}}}_{n,R},$ and $\tilde{{{\mathbb Y}}}_{n}$ are localized integrated empirical processes, and again the L and R processes for the constrained problem correspond to $t \ge 0$ and $t \le 0$. This theorem is given in [@Doss-Wellner:2016ModeConstrained]. The most important coordinates for the proof of Theorem \[LRasympNullDistribution\] are $\tilde{H}_{n,L}^{(2)}, \ \tilde{H}_{n,R}^{(2)}$ (which together give $(\tilde{H}_n^0 )^{(2)}$), and $\tilde{H}^{(2)}_n$.
Corrections for [@BRW2007LCasymp] {#ssec:BRW2009Corrections}
---------------------------------
In (4.25) of [@BRW2007LCasymp], replace $$Y_{k,a,\sigma} (t) := a \int_0^t W(s) ds - \sigma t^{k+2}$$ by $$Y_{k,a,\sigma} (t) := \sigma \int_0^t W(s) ds - a t^{k+2}$$ to accord with [@MR1891741], page 1649, line -4, when $k=2$. In (4.22) of [@BRW2007LCasymp], page 1231, replace the definition of $\gamma_1$ by $$\begin{aligned}
\gamma_1 = \left ( \frac{f_0 (x_0)^{k+2} | \varphi_0 (x_0)|^3}{(4!)^3} \right )^{1/(2k+1)} .\end{aligned}$$ In (4.23) of [@BRW2007LCasymp], page 1321, replace the definition of $\gamma_2$ by $$\begin{aligned}
\gamma_2 = \left ( \frac{((k+2)!)^2}{f_0 (x_0) | \varphi_0^{(k)} (x_0) |^2 } \right )^{1/(2k+1)} .\end{aligned}$$ When $k=2$ and $x_0 = m$, these definitions of $\gamma_1, \gamma_2$ reduce to $\gamma_1 $ and $\gamma_2$ as given in (\[GammaDefnsLogConcaveAtMode\]). One line above (4.25) of [@BRW2007LCasymp], page 1321, change $Y_{a,k,\sigma} $ to $Y_{k,a,\sigma}$.
Proof sketches for Theorems \[LRasympNullDistribution\] and \[LR-limit-FixedAlternative\] {#sec:PfSketches}
=========================================================================================
Proof sketch for Theorem \[LRasympNullDistribution\] {#ssec:PfSketchThm1}
----------------------------------------------------
To begin our sketch of the proof of Theorem \[LRasympNullDistribution\] we first give the basic decomposition we will use. We begin by using $\int_{{{\mathbb R}}} \widehat{f}_n (u) du = 1 = \int_{{{\mathbb R}}} \widehat{f}_n^0 (u) du$ to write $$\begin{aligned}
2 \log \lambda_n
& = & 2 n {{\mathbb P}}_n (\widehat{\varphi}_n - \widehat{\varphi}_n^0 ) \nonumber \\
& = & 2 n \int_{{{\mathbb R}}} ( \widehat{\varphi}_n - \widehat{\varphi}_n^0 ) d { {\mathbb F}}_n - \int_{{{\mathbb R}}} (\widehat{f}_n (u) - \widehat{f}_n^0 (u) ) du \nonumber \\
& = & 2 n \int_{[X_{(1)}, X_{(n)} ]} \left ( \widehat{\varphi}_n d \widehat{F}_n - \widehat{\varphi}_n^0 d \widehat{F}_n^0 \right )
\label{LR-DecompStepone} \\
&& \qquad \ - \ 2 n \int_{[X_{(1)}, X_{(n)} ]} \left ( e^{\widehat{\varphi}_n (u)} - e^{\widehat{\varphi}_n^0 (u)} \right ) du \nonumber\end{aligned}$$ where we have used the characterization Theorems 2.2 and 2.8 of [@Doss-Wellner:2016ModeConstrained] with $\Delta = \pm \ \widehat{\varphi}_n$ and $\Delta = \pm \ \widehat{\varphi}_n^0$ respectively. As we will see, inclusion of the second term in (\[LR-DecompStepone\]) will be of considerable help in the analysis.
Now we split the integrals in (\[LR-DecompStepone\]) into two regions: let $D_n \equiv [t_{1}, t_{2}]$ for some $t_{1} < m < t_{2}$ and then let $D_n^c \equiv [X_{(1)}, X_{(n)}] \setminus D_n$. The set $D_n$ is the region containing the mode $m$; here the unconstrained estimator $\widehat{\varphi}_n$ and the constrained estimator $\widehat{\varphi}_n^0$ tend to differ. On the other hand, $D_n^c$ is the union of two sets away from the mode, and on both of these sets the unconstrained estimator $\widehat{\varphi}_n$ and the constrained estimator $\widehat{\varphi}_n^0$ are asymptotically equivalent (or at least nearly so). Sometimes we will take the $t_i$, $i=1,2$, to be constant in $n$, sometimes to be fixed or random sequences approaching $m$ as $n \to \infty$. We will sometimes suppress the dependence of $D_n \equiv D_{n, t_{1}, t_{2}}$ on $t_{i}$, and will emphasize it when it is important. Now, from , we can write $$\begin{aligned}
2 \log \lambda_n
& = & 2 n \left \{
\int_{D_n} \left ( \widehat{\varphi}_n d \widehat{F}_n - \widehat{\varphi}_n^0 d \widehat{F}_n^0 \right ) \right .
\left . - \ \int_{D_n} \left ( e^{\widehat{\varphi}_n (u)} - e^{\widehat{\varphi}_n^0 (u)} \right ) du \right . \\
&& \left . + \ \int_{D_n^c} \left ( \widehat{\varphi}_n d \widehat{F}_n - \widehat{\varphi}_n^0 d \widehat{F}_n^0 \right ) \right .
\left . - \ \int_{D_n^c} \left ( e^{\widehat{\varphi}_n (u)} - e^{\widehat{\varphi}_n^0 (u)} \right ) du \right \} \\
& = & 2 n \left \{
\int_{D_n} \left ( ( \widehat{\varphi}_n - \varphi_0 (m)) d \widehat{F}_n
- (\widehat{\varphi}_n^0 - \varphi_0 (m)) d \widehat{F}_n^0 \right ) \right . \\
& & \left . - \ \int_{D_n} \left ( ( e^{\widehat{\varphi}_n (u)} - e^{\varphi_0 (m)}) - ( e^{\widehat{\varphi}_n^0 (u)} - e^{\varphi_0(m)} ) \right ) du \right \} \\
&& \hspace{2em} + \ 2 n (R_{n,1} + R_{n,1}^c )\end{aligned}$$ where $$\begin{aligned}
R_{n,t_{1},t_2} \equiv & R_{n,1} \equiv \int_{D_n}\varphi_0 (m) ( \widehat{f}_n (x) - \widehat{f}_n^0 (x) ) dx , \label{RemainTermOne}\\
R_{n,t_{1},t_2}^c \equiv & R_{n,1}^c \equiv \int_{D_n^c} \left ( \widehat{\varphi}_n d \widehat{F}_n - \widehat{\varphi}_n^0 d \widehat{F}_n^0 \right )
- \ \int_{D_n^c} \left ( e^{\widehat{\varphi}_n (u)} - e^{\widehat{\varphi}_n^0 (u)} \right ) du . \label{RemainTermOneCompl}\end{aligned}$$ Now we use an expansion of the exponential function to rewrite the second part of the main term: since $$e^b - e^a = e^{a} \{ e^{b-a} -1 \} = e^a \{ (b-a) + \frac{1}{2} (b-a)^2 + \frac{1}{6} e^{a^*} (b-a)^3 \}$$ where $|a^* - a | \le | b -a |$, we have $$\begin{aligned}
\lefteqn{\int_{D_n} \left ( ( e^{\widehat{\varphi}_n (u)} - e^{\varphi_0 (m)} ) - ( e^{\widehat{\varphi}_n^0 (u)} - e^{\varphi_0(m)} ) \right ) du } \\
& = & \int_{D_n} e^{\varphi_0 (m)} \left ( (\widehat{\varphi}_n (u) - \varphi_0 (m))
+ \frac{1}{2} ( \widehat{\varphi}_n (u) - \varphi_0(m))^2 ) \right ) du + R_{n,2} \\
&& \ - \ \int_{D_n} e^{\varphi_0 (m)} \left ( (\widehat{\varphi}_n^0 (u) - \varphi_0 (m))
+ \frac{1}{2} ( \widehat{\varphi}_n^0 (u) - \varphi_0(m))^2 ) \right ) du - R_{n,2}^0\end{aligned}$$ where $$\begin{aligned}
&& R_{n,2} \equiv \int_{D_n} \frac{1}{6} f_0 (m) e^{\tilde{x}_{n,2} (u)} (\widehat{\varphi}_n (u) - \varphi_0 (m))^3 du,
\label{RemainTermTwoUnconstrained}\\
&& R_{n,2}^0 \equiv \int_{D_n} \frac{1}{6} f_0 (m) e^{\tilde{x}_{n,2}^0 (u)} (\widehat{\varphi}_n^0 (u) - \varphi_0 (m))^3 du .
\label{RemainTermTwoConstrained}\end{aligned}$$ Thus $$\begin{aligned}
\lefteqn{2 \log \lambda_n } \label{LR-SecondDecomposition}\\
& = & n \left \{
2 \int_{D_n} \left ( ( \widehat{\varphi}_n (u)- \varphi_0 (m)) (d \widehat{F}_n(u) - f_0 (m) du) \right . \right . \nonumber \\
&& \qquad \left . \left .
- \ 2\int_{D_n} (\widehat{\varphi}_n^0 (u) - \varphi_0 (m)) ( d \widehat{F}_n^0 (u) - f_0 (m) du) \right ) \right .\nonumber \\
&& \qquad \left .
- \ \int_{D_n} \left ( (\widehat{\varphi}_n (u) - \varphi_0 (m))^2 - (\widehat{\varphi}_n^0 (u) - \varphi_0(m) )^2 \right ) f_0(m) du \right \}
\nonumber \\
&& + \ 2 n (R_{n,1} + R_{n,1}^c + R_{n,2} - R_{n,2}^0 ). \nonumber\end{aligned}$$ Now we expand the first two terms in the last display, again using a two term expansion, $e^{b} - e^{a} = (e^{b-a} -1)e^a = e^a \{ (b-a) + \frac{1}{2} (b-a)^2 e^{a^*} \}$, to find that $$\begin{aligned}
\lefteqn{\int_{D_n} ( \widehat{\varphi}_n (u)- \varphi_0 (m)) (d \widehat{F}_n(u) - f_0 (m) du ) }\\
& = & \int_{D_n} ( \widehat{\varphi}_n (u)- \varphi_0 (m)) ( e^{\widehat{\varphi}_n (u) - \varphi_0 (u)} -1 ) f_0 (m) du \\
& = & \int_{D_n} ( \widehat{\varphi}_n (u)- \varphi_0 (m)) \left ( ( \widehat{\varphi}_n (u)- \varphi_0 (m))
+ e^{\tilde{x}_{n,3} (u)} \frac{1}{2} ( \widehat{\varphi}_n (u)- \varphi_0 (m))^2 \right ) du \\
& = & \int_{D_n} ( \widehat{\varphi}_n (u)- \varphi_0 (m))^2 f_0 (m) du + R_{n,3}\end{aligned}$$ where $$\begin{aligned}
R_{n,3,t_1,t_2} \equiv
R_{n,3} = \int_{D_n} \frac{1}{2} f_0 (m) e^{\tilde{x}_{n,3} (u)} ( \widehat{\varphi}_n (u)- \varphi_0 (m))^3 du .
\label{RemainTermThreeUnconstrained}\end{aligned}$$ Similarly, $$\begin{aligned}
\lefteqn{
\int_{D_n} ( \widehat{\varphi}_n^0 (u)- \varphi_0 (m)) (d \widehat{F}_n^0 (u) - f_0 (m) du ) }\\
& = & \int_{D_n} ( \widehat{\varphi}_n^0 (u)- \varphi_0 (m))^2 f_0 (m) du + R_{n,3}^0\end{aligned}$$ where $$\begin{aligned}
R_{n,3,t_1,t_2}^0 \equiv
R_{n,3}^0 = \int_{D_n} \frac{1}{2} f_0 (m) e^{\tilde{x}_{n,3}^0 (u)} ( \widehat{\varphi}_n^0 (u)- \varphi_0 (m))^3 du .
\label{RemainTermThreeConstrained}\end{aligned}$$ If we let $t_1 = t_{n,1} = m - bn^{-1/5}$ and $t_2 = t_{n,2} = m+ bn^{-1/5}$ for $ b > 0$, then from (\[LR-SecondDecomposition\]) we now have $$\begin{aligned}
2 \log \lambda_n
& = & n \int_{D_n} f_0 (m) \left \{
(\widehat{\varphi}_n (u) - \varphi_0 (m))^2 - (\widehat{\varphi}_n^0 (u) - \varphi_0(m) )^2 \right \} du \nonumber \\
&& \qquad + \ 2 n \left (R_{n,1} + R_{n,1}^c + R_{n,2} - R_{n,2}^0 + R_{n,3} - R_{n,3}^0 \right ) \nonumber \\
& \equiv & {{\mathbb D}}_{n,b} + R_n .
\label{BasicDecompositionFirstForm}\end{aligned}$$ Now we sketch the behavior of ${{\mathbb D}}_{n,b}$. Let $u = m + v n^{-1/5}$; with this change of variables and the definition of $t_{n,i}$, $i=1,2$, we can rewrite ${{\mathbb D}}_{n,b}$ as $$\begin{aligned}
\MoveEqLeft {{\mathbb D}}_{n,b}
= \frac{n^{4/5}}{a^{2}} \int_{-b}^{b}
\bigg \{
\left (\widehat{\varphi}_n (m + n^{-1/5} v) - \varphi_0 (m) \right )^2 \\
& \hspace{2.3cm} - \left (\widehat{\varphi}_n^0 (m + n^{-1/5} v) - \varphi_0 (m) \right )^2
\bigg \} dv.\end{aligned}$$ By Theorem 3.1.B this converges in distribution to $$\begin{aligned}
f_0 (m) \int_{-b}^{b} \left \{ \left (\widehat{\varphi}_{a,\sigma} (v) \right )^2 - \left (\widehat{\varphi}_{a,\sigma}^0 (v) \right )^2 \right \} dv ,
\label{LimitInDistributionMainTerm}\end{aligned}$$ where the processes $(\widehat{\varphi}_{a,\sigma} , \widehat{\varphi}_{a,\sigma}^0 )$ are related to $ ( \widehat{\varphi} , \widehat{\varphi}^0)$ by the scaling relations (\[ScalingRelationUnConstrained\]) and (\[ScalingRelationConstrained\]). We conclude that the limiting random variable in (\[LimitInDistributionMainTerm\]) is equal in distribution to $$\begin{aligned}
\lefteqn{f_0 (m) \int_{-b}^{b} \left \{ \left ( \frac{1}{\gamma_1 \gamma_2^2}
\widehat{\varphi} ( v/\gamma_2) \right )^2
- \left ( \frac{1}{\gamma_1
\gamma_2^2}
\widehat{\varphi}^0 (
v/\gamma_2) \right )^2
\right \} dv } \nonumber \\
& = & \frac{f_0 (m)}{\gamma_1^2 \gamma_2^3} \frac{1}{\gamma_2 }
\int_{-b}^{b} \left \{ \widehat{\varphi} ( v/\gamma_2)^2
- \widehat{\varphi}^0
( v/\gamma_2)^2 \right
\} dv \nonumber \\
& = & \int_{-b/\gamma_2}^{b/\gamma_2} \left \{ \widehat{\varphi} (s)^2
- \widehat{\varphi}^0
(s)^2 \right \}
ds \label{eq:DDb-formula-gammas}\end{aligned}$$ in view of (\[GammaRelationsPart1\]). This is not yet free of the parameter $\gamma_2$, but it will become so if we let $b \to \infty$. If we show that this is permissible and we show that the remainder term $R_n$ in (\[BasicDecompositionFirstForm\]) is negligible, then the proof of Theorem \[LRasympNullDistribution\] will be complete. For details, see sections \[sec:ProofsPart2\].
Proof sketch for Theorem \[LR-limit-FixedAlternative\] {#ssec:PfSketchThm2}
------------------------------------------------------
Recall ${\cal P}_m = {\left\{} \newcommand{\rb}{\right\}}f \in {\cal P} : M(f) = m \rb$. Let $$\begin{aligned}
f_m^0
& = \mbox{argmin}_{g \in {\cal P}_m} \left \{ - \int \log g f_0 d\lambda \right \}
\label{eq:defn:fm0} \\
& = \mbox{argmin}_{g\in {\cal P}_m} \int f_0 \left ( \log f_0 - \log g \right ) d \lambda
= \mbox{argmin}_{g \in {\cal P}_m} K(f_0,g) , \nonumber\end{aligned}$$ where we will make rigorous later, in Theorem \[DSS-thm2.2m\] in Subsection \[subsec:thm-alternative\]. Let ${{\mathbb P}}_n = \sum_{i=1}^n \delta_{X_i} /n$ be the empirical measure and for a function $g$ let ${{\mathbb P}}_n(g) = \int g d{{\mathbb P}}_n$. We now have $$\begin{aligned}
n^{-1} \log \lambda_n (m)
& = {{\mathbb P}}_n ( \log \widehat{f}_n / \widehat{f}_n^0 )
= {{\mathbb P}}_n \left \{ \log \frac{\widehat{f}_n}{f_0} \cdot \frac{f_0}{f^0_m}\cdot \frac{f^0_m}{\widehat{f}_n^0} \right \} \nonumber \\
&= {{\mathbb P}}_n \left \{ \log \frac{\widehat{f}_n}{f_0} \right \} + {{\mathbb P}}_n \left \{ \log \frac{f_0}{f_m^0} \right \}
- {{\mathbb P}}_n \left \{ \log \frac{\widehat{f}_n^0 }{f^0_m} \right\}. \label{eq:TLLR-alt-3terms}
\end{aligned}$$ From this we will conclude that as $n \to \infty$, $$\begin{aligned}
n^{-1} 2 \log \lambda_n(m) = O_p (n^{-4/5})
+ 2 {{\mathbb P}}_n \left \{ \log \frac{f_0}{f^0_m} \right \} - o_p (1)
\rightarrow_p 2 K(f_0, f_m^0).\end{aligned}$$ That ${{\mathbb P}}_n \left \{ \log \frac{\widehat{f}_n}{f_0} \right \} = O_p(n^{-4/5})$ follows from [@DossWellner:2016a], Corollary 3.2, page 962. The convergence of $2 {{\mathbb P}}_n \left \{ \log \frac{f_0}{f^0_m} \right \}$ to $K(f_0, f_m^0)$ follows from the weak law of large numbers. The indicated negligibility of the third term in follows from Theorem \[DSS-thm2.15m\] below (which is a constrained analogue of Theorem 2.15 of [@DSS2011LCreg]); we elaborate on this in Subsection \[subsec:thm-alternative\].
Simulations: some comparisons and examples {#sec:Illustration-Simul}
==========================================
Monte Carlo estimates of the distribution of $\ {{\mathbb D}}$ {#subsec:DD}
--------------------------------------------------------------
To implement our likelihood ratio test and the corresponding new confidence intervals, we first conducted a Monte-Carlo study of the limiting distribution of the limit random variable ${{\mathbb D}}$ (under the null hypothesis). We did this by simulating $M = 5\times 10^3$ samples of $n = 10^4$ from the following distributions satisfying the key hypothesis ($\varphi_0^{\prime\prime} (m)< 0$) of Theorem \[LRasympNullDistribution\]: Gamma$(3,1)$, Beta$(2,3)$, Weibull$(3/2,1)$. The results are shown in Figure \[fig:figure1\]. Figure \[fig:figure1\] also includes: (i) a plot of the empirical distribution of $2 \log \lambda_n$ for $M = 5\times 10^3$ samples of size $n=10^4$ drawn from the Laplace density $2^{-1} \exp ( - | x |)$ for which the assumption of Theorem 1 fails; (ii) a plot of the known d.f. of a chi-square random variable with $1$ degree of freedom, (which is the limiting distribution of the likelihood ratio test statistic for testing a one-dimensional parameter in a regular parametric model). In keeping with Theorem \[LRasympNullDistribution\], the empirical results for all three distributions satisfying Theorem \[LRasympNullDistribution\] are tightly clustered, in spite of the fact that the various constants associated with these distributions are quite different, as shown in Table 1; in the next to last column $C(f_0) \equiv \left ((4!)^2 f_0 (m)/ (f_0^{\prime\prime} (m))^2) \right )^{1/5}$ from , and in the last column SLC stands for “strongly log-concave” (see e.g. [@Wellner:2013gq]).
distribution $m$ $f_0 (m)$ $f_0^{\prime \prime} (m)$ $\varphi_0^{\prime\prime} (m)$ $C(f_0)$ SLC
------------------ ------------ ------------------------------- --------------------------- -------------------------------- ----------- -----
N(0,1) $0$ $(2\pi)^{-1/2} = .3989\ldots$ $-(2\pi)^{-1/2}$ $-1$ $4.28$ Y
Gamma$(3,1)$ $2$ $2e^{-2}$ $-e^{-2}$ $-1/2$ $6.109$ Y
Weibull$(3/2,1)$ $3^{-2/3}$ $\frac{3^{2/3}}{2 e^{1/3}}$ $- \frac{27}{8 e^{1/3}}$ $ - \frac{9\cdot 3^{1/4}}{4}$ $2.36$ N
Beta$(2,3)$ $3^{-1}$ $\frac{16}{9} $ $-24$ $- \frac{27}{2} $ $1.12$ Y
Logistic $0$ $1/4$ $-1/8$ $-1/2$ $6.207$ N
Gumbel $0$ $e^{-1}$ $-e^{-1}$ $-1$ $4.3545 $ N
$\chi_4^2 $ $2$ $\frac{1}{2 e}$ $- \frac{1}{8 e}$ $-\frac{1}{4} $ $8.7091$ N
: Numerical characteristics of the distributions in the null hypothesis Monte-Carlo study[]{data-label="tab:1"}
![Empirical distributions of $2 \log \lambda_n$ for three distributions, $n=10^4$, $M = 5\times 10^3$ replications[]{data-label="fig:figure1"}](Figures/LRasymptotics_all.pdf){width=".9\textwidth"}
Now for $\alpha \in (0,1)$ let $d_{\alpha}$ satisfy $P ( {{\mathbb D}}> d_{\alpha} ) = \alpha $. The following table gives
$\alpha $ $.25 $ $.20$ $.15$ $.10$ $.05$ $.01$
-------------- -------- ------- ------- ------- -------- --------
$d_{\alpha}$ $.40$ $.49$ $.61$ $.79$ $1.11$ $1.92$
: Estimated critical values $d_{\alpha}$[]{data-label="tab:2"}
a few estimated values for $d_{\alpha}$: These are based on $350,000$ Monte Carlo simulations each based on simulating $1 \times 10^6$ observations from a standard normal. These values, and the simulated critical values for all $\alpha \in (0,1)$, are available in the `logcondens.mode` package [@doss:logcondens.mode] in R [@R-core].
[@MR1891743] study a likelihood ratio test in the context of constraints based on monotonicity, and find a universal limiting distribution, denote it ${{\mathbb D}}_{\text{mono}}$, for their likelihood ratio test. Comparison of the values in Table \[tab:2\] with Table 2 of [@MR1891743] (particularly Method 2 in column 3 of that table) suggest, perhaps surprisingly, that $P( 2 {{\mathbb D}}\le t) \approx P( {{\mathbb D}}_{\text{mono}} \le t)$ for $ t \in {{\mathbb R}}$. It would be quite remarkable if this held exactly, and we suspect that exact equality fails.
Comparisons via simulations
---------------------------
We now test and compare our procedure via a small simulation study. Code to compute the mode-constrained log-concave MLE, implement a corresponding test, and invert the family of tests to form confidence intervals is available in the `logcondens.mode` package [@doss:logcondens.mode].
[@MR964293] proposed and investigated several methods of forming confidence intervals for the mode of a unimodal density. His estimators of the mode and confidence intervals were based on the classical kernel density estimators of the density $f$ going back to [@MR0143282]. One method, which Romano called the “normal approximation method”, was based on the limiting normality of the kernel density estimator of the mode, followed in a straightforward way by estimation of the asymptotic variance. Romano’s second method involved bootstrapping the mode estimator, and involved the choice of two bandwidths, one for the initial estimator to determine the mode, and a second (larger) bandwidth for the bootstrap sampling.
To facilitate comparison of our results to those of [@MR964293], we first briefly describe his results. Let $X_1, \ldots , X_n$ be i.i.d. with distribution function $F_0$ and density $f_0$, let ${ {\mathbb F}}_n $ be the empirical d.f. of the $X_i$’s. Romano (1988b) then defines $$\begin{aligned}
\widehat{f}_{n,h_n} (x) & \equiv & \hat{f}_{n,h_n} (x; X_1, \ldots , X_n ) \equiv \frac{1}{h_n} \int_{{{\mathbb R}}} k \left ( \frac{x - y}{h_n} \right ) d { {\mathbb F}}_n (y), \\
\widetilde{f}_{n,h_n} (x) & \equiv & \frac{1}{h_n} \int_{{{\mathbb R}}} k \left ( \frac{x - y}{h_n} \right ) d F_0 (y) .\end{aligned}$$ where the bandwidth $h_n$ may depend on $X_1, \ldots , X_n$ and $k$ is a kernel (density) function. Let $\widehat{\theta}_{n,h_n} \equiv M( \widehat{f}_{n,h_n} )$ and $\widetilde{\theta}_{n,h_n} \equiv M( \widetilde{f}_{n,h_n} ) $.
--------------------------- --------- --------- --------- ---------
$80\%$ $90\%$ $95\%$ $99\%$
$h=0.3$ $0.695$ $0.795$ $0.849$ $0.933$
$h=0.4$ $0.831$ $0.899$ $0.941$ $0.976$
$h=0.5$ $0.881$ $0.951$ $0.980$ $0.996$
$h=0.6$ $0.939$ $0.976$ $0.990$ $0.998$
$h=0.7$ $0.956$ $0.989$ $0.997$ $0.999$
$h=0.8$ $0.962$ $0.991$ $0.996$ $1.000$
JR Parzen, CP $h=0.4$ $0.831$ $0.899$ $0.941$ $0.976$
[DW Parzen, CP]{} $h=0.4$ $0.835$ $0.902$ $0.941$ $0.978$
[DW Parzen, length]{} $0.966$ $1.240$ $1.478$ $1.942$
[DW LR, CP ]{} 0.785 0.888 0.941 0.983
[DW LR, length]{} 0.966 1.178 1.346 1.663
--------------------------- --------- --------- --------- ---------
: Estimated coverage probabilities based on normal approximation 1000 simulations: bandwidth $ = h \cdot$ sample standard deviation, data $N(0,1)$, $n=100$; from Table 1, Romano (1988b), page 576[]{data-label="tab:3"}
![Coverage probabilities, Romano’s Table 1 compared with LR coverage probabilities, Normal approximation confidence intervals, Gaussian data (blue); LR confidence intervals, Gaussian data (magenta)[]{data-label="fig:figure2"}](Figures/Plots/PlotComparisonRomanoT1){height="2.8in"}
Suppose that $F_0$ has density $f_0$ satisfying:\
(A) $m=M(f_0)$ is unique with $\sup_{\{t : |t - m| > \delta \}} f_0 (t) < f_0 (m)$ for all $\delta>0$; $f_0^{(3)} $ is continuous in some neighborhood of $m$; and $f_0^{(2)} (m)< 0$.\
Suppose $k$ satisfies:\
(B) $k$ is symmetric with a continuous derivative of bounded variation and, for some $r>0$, $k^{2+r}, \ |k^{(1)}|^{2+r}, \ k^3 , \ z^2 | k^{(2)} (z)|^2$ are integrable.\
Then, as Romano shows in his Theorem 2.1, if $h_n \rightarrow 0$ and $n h_n^5/\log n \rightarrow \infty$, $$\begin{aligned}
(nh_n^3)^{1/2} ( \widehat{\theta}_{n,h_n} - \widetilde{\theta}_{n,h_n} ) \rightarrow_d N(0, V(f_0,k))\end{aligned}$$ where, with $J(k) \equiv \int [k'(z)]^2 dz$, $$V(f,k) = \frac{f_0(m)}{[ f_0^{\prime \prime} (m)]^2} \cdot J(k) .$$ Note that if $f_c(x) \equiv c^{-1} f_0 (x/c)$ for $c>0$, then $V ( f_c, k ) = c^5 V (f_0, k)$ while $C(f_c) = c C(f_0)$. From we see that the variance of the limiting distribution of $\widehat{M} \equiv M(\widehat{f}_n)$ automatically scales as $c^2$ while the asymptotic variance of $\sqrt{n h_n^3} (\widehat{\theta}_{n,h_n} -
\widetilde{\theta}_{n,h_n})$ scales as $c^5$. If $h$ is chosen proportional to $c$ then Var$(\widehat{\theta}_{n,h})$ will scale with $c^2$, and this is why Romano \[1988a\] chose bandwidths proportional to the sample standard deviation.
Tables \[tab:3\]-\[tab:6\] below reproduce Tables 1-4 of [@MR964293]. The estimated coverage probabilities of Romano’s “Normal approximation” based confidence intervals for the mode are given in Tables \[tab:3\]-\[tab:4\], with rows corresponding to the choice of bandwidth $h$. Note that the bandwidths chosen for Table 1 (our Table \[tab:3\]) with a $N(0,1)$ population are different than the bandwidths chosen for Table 2 (our Table \[tab:4\]) for a $\chi_4^2$ population. The estimated coverage probabilities of Romano’s bootstrap confidence intervals for the mode are given in Tables 5-6, with rows corresponding to choices of the two bandwidths $h$ and $b$.
In our Figures \[fig:figure2\]-\[fig:figure3\] we plot the estimated coverage probabilities of Romano’s normal approximation confidence intervals (blue) together with the target (ideal) coverage probabilities (green line), and the estimated coverage probabilities of our Likelihood Ratio (LR) based confidence intervals. As can be seen the estimated coverage probabilities of our LR based procedure are reasonably close to the nominal target values in both Figures \[fig:figure2\] and \[fig:figure3\] without the necessity of any bandwidth choice.
In Figures \[fig:figure4\] and \[fig:figure5\] we plot the estimated coverage probabilities of Romano’s bootstrap confidence intervals (blue) together with the target (ideal) coverage (green line) and the estimated coverage probabilities of our Likelihood Ratio based confidence intervals: these are the same as in as in Figures \[fig:figure2\] and \[fig:figure3\], and hence reasonably close to the nominal target values.
Methods of bandwidth selection for various problems have received considerable attention in the period since [@MR964293]; see especially [@MR1089472], [@MR1450020], [@MR1049312], [@MR1160479], [@MR1839003], [@MR1394655], [@MR1618191], and [@MR2662359]. Although bandwidth selection in connection with mode estimation is mentioned briefly by [@MR1089472] (see their last paragraph, page 734), we are not aware of any specific proposal or detailed study of bandwidth selection methods in the problem of confidence intervals for the mode of a unimodal density. For this reason, we have not undertaken a full comparative study of possible methods here.
---------------------------- --------- --------- --------- ---------
$80\%$ $90\%$ $95\%$ $99\%$
$h=0.25$ $0.759$ $0.834$ $0.890$ $0.944$
$h=0.28$ $0.796$ $0.870$ $0.914$ $0.958$
$h=0.32$ $0.817$ $0.899$ $0.940$ $0.975$
$h=0.35$ $0.804$ $0.889$ $0.938$ $0.981$
$h=0.39$ $0.782$ $0.889$ $0.841$ $0.986$
$h=0.42$ $0.705$ $0.889$ $0.914$ $0.976$
$h=0.46$ $0.634$ $0.826$ $0.918$ $0.983$
$h=0.50$ $0.531$ $0.766$ $0.880$ $0.979$
JR Parzen, CP $h=0.35$ $0.804$ $0.899$ $0.938$ $0.981$
[DW Parzen, CP]{} $h=0.35$ $0.794$ $0.898$ $0.936$ $0.980$
[DW Parzen, length]{} $1.771$ $2.273$ $2.709$ $3.560$
[DW LR, CP]{} 0.789 0.909 0.952 0.986
[DW LR, length]{} 1.957 2.376 2.720 3.341
---------------------------- --------- --------- --------- ---------
: Estimated coverage probabilities based on normal approximation 1000 simulations: bandwidth $ = h \cdot$ sample standard deviation, data $\chi_4^2$, $n=100$; from Table 2, Romano (1988b), page 577[]{data-label="tab:4"}
![Coverage probabilities, Romano’s Table 2 compared with LR coverage probabilities, Normal approximation confidence intervals, $\chi_4^2$ data (blue); LR confidence intervals, Gaussian data (magenta)[]{data-label="fig:figure3"}](Figures/Plots/PlotComparisonRomanoT2){height="2.8in"}
------------------ --------- --------- --------- ---------
$80\%$ $90\%$ $95\%$ $99\%$
$h=0.2$, $b=0.0$ $0.710$ $0.887$ $0.962$ $1.000$
$h=0.2$, $b=0.2$ $0.682$ $0.786$ $0.898$ $0.956$
$h=0.2$, $b=0.3$ $0.672$ $0.868$ $0.942$ $0.976$
$h=0.2$, $b=0.4$ $0.724$ $0.882$ $0.952$ $0.984$
$h=0.4$, $b=0.0$ $0.718$ $0.880$ $0.942$ $0.980$
$h=0.4$, $b=0.4$ $0.646$ $0.852$ $0.924$ $0.978$
$h=0.4$, $b=0.6$ $0.728$ $0.876$ $0.938$ $0.982$
$h=0.4$, $b=0.8$ $0.764$ $0.898$ $0.960$ $0.988$
$h=0.6$, $b=0.9$ $0.660$ $0.848$ $0.962$ $0.982$
$h=0.6$, $b=0.6$ $0.628$ $0.846$ $0.918$ $0.982$
$h=0.6$, $b=0.9$ $0.718$ $0.852$ $0.916$ $0.980$
$h=0.6$, $b=1.2$ $0.758$ $0.902$ $0.944$ $0.984$
$h=0.8$, $b=0.0$ $0.754$ $0.830$ $0.906$ $0.980$
$h=0.8$, $b=0.8$ $0.704$ $0.842$ $0.914$ $0.980$
$h=0.8$, $b=1.2$ $0.786$ $0.894$ $0.946$ $0.992$
$h=0.8$, $b=1.6$ $0.776$ $0.888$ $0.960$ $0.998$
------------------ --------- --------- --------- ---------
: Estimated coverage probabilities based on bootstrap, 500 simulations, 200 bootstrap replications: bandwidth $ = h \cdot$ sample standard deviation, resampling bandwidth $ = b\cdot $ sample standard deviation; data $N(0,1)$, $n=100$; from Table 3, Romano (1988b), page 577[]{data-label="tab:5"}
![Coverage probabilities, Romano’s Table 3 compared with LR coverage probabilities, Bootstrap confidence intervals, $N(0,1)$ data (blue); LR confidence intervals, Gaussian data (magenta)[]{data-label="fig:figure4"}](Figures/Plots/PlotComparisonRomanoT3){height="2.8in"}
-------------------- --------- --------- --------- ---------
$80\%$ $90\%$ $95\%$ $99\%$
$h=0.25$, $b=0.00$ $0.758$ $0.872$ $0.928$ $0.986$
$h=0.25$, $b=0.25$ $0.790$ $0.910$ $0.966$ $1.000$
$h=0.25$, $b=0.37$ $0.822$ $0.924$ $0.972$ $0.996$
$h=0.25$, $b=0.50$ $0.818$ $0.926$ $0.968$ $0.992$
$h=0.32$, $b=0.00$ $0.644$ $0.786$ $0.854$ $0.964$
$h=0.32$, $b=0.32$ $0.824$ $0.940$ $0.980$ $1.000$
$h=0.32$, $b=0.48$ $0.816$ $0.922$ $0.956$ $0.992$
$h=0.32$, $b=0.64$ $0.820$ $0.920$ $0.962$ $0.994$
$h=0.39$, $b=0.00$ $0.582$ $0.764$ $0.862$ $0.970$
$h=0.39$, $b=0.39$ $0.770$ $0.876$ $0.950$ $0.996$
$h=0.39$, $b=0.59$ $0.792$ $0.916$ $0.954$ $0.990$
$h=0.39$, $b=0.78$ $0.804$ $0.908$ $0.956$ $0.992$
$h=0.46$, $b=0.00$ $0.538$ $0.688$ $0.786$ $0.918$
$h=0.46$, $b=0.46$ $0.744$ $0.856$ $0.894$ $0.980$
$h=0.46$, $b=0.69$ $0.784$ $0.924$ $0.964$ $0.992$
$h=0.46$, $b=0.92$ $0.792$ $0.918$ $0.954$ $0.990$
-------------------- --------- --------- --------- ---------
: Estimated coverage probabilities based on bootstrap, 500 simulations, 200 bootstrap replications: bandwidth $ = h \cdot$ sample standard deviation, resampling bandwidth $ = b\cdot $ sample standard deviation; data $\chi_4^2$, $n=100$; from Table 4, Romano (1988b), page 578[]{data-label="tab:6"}
![Coverage probabilities, Romano’s Table 4 compared with LR coverage probabilities, Bootstrap confidence intervals, $\chi_4^2$ data (blue); LR confidence intervals, Gaussian data (magenta)[]{data-label="fig:figure5"}](Figures/Plots/PlotComparisonRomanoT4){height="2.8in"}
But we have made an initial effort to compare the lengths of Romano’s “normal approximation” confidence intervals with our LR confidence intervals, which we now describe. In Table \[tab:3\], the first row of the second group of rows (starting with “JR Parzen”) repeats the row with bandwidth $h=0.4$ from the first group of rows (taken from Romano’s Table 1). This is the “best/oracle bandwidth” (of the bandwidths in the simulation study), achieving minimal total error in coverage probabilities summed across the four different coverage levels (for this given distribution and sample size). It is an “oracle” because it is unknown in practice. The second row in this group (starting with “DW Parzen CP”) give our simulated coverage probabilities for Romano’s method with $h=0.4$ across the different coverage levels. These seem to be reasonably close to Romano’s estimated coverage probabilities, so it seems reasonable to compare average confidence interval lengths for the two methods. The third row (“DW Parzen, length”) gives our estimated average lengths of Romano’s “normal approximation” type intervals, while the last row of Table \[tab:3\] (“DW, LR, length”) gives our estimated lengths of the Likelihood Ratio based confidence intervals. We see that the estimated average lengths for the LR confidence intervals are shorter or equal to those of the best/oracle bandwidth confidence intervals at all nominal coverage levels. Similarly, in Table \[tab:4\], the first row of the second group of rows (starting with “JR Parzen”) repeats the row with bandwidth $h=0.35$ from the first group of rows (taken from Romano’s Table 2). This is the “best/oracle bandwidth” for total error in coverage probabilities across the four different coverage levels. The second row in this group (starting with “DW Parzen CP”) give our coverage probabilities for Romano’s method with $h=0.35$ across the different coverage levels. These again seem to be reasonably close to Romano’s estimated coverage probabilities, so it might be reasonable to compare average confidence interval lengths for the two methods. The third row (“DW Parzen, length”) gives our estimated average lengths of Romano’s “normal approximation” type intervals, while the last row of Table \[tab:4\] (“DW, LR, length”) gives our estimated lengths of the Likelihood Ratio based confidence intervals. For this population ($\chi_4^2$), the estimated average lengths for the LR confidence intervals are longer than those of the best/oracle bandwidth confidence intervals at nominal coverage levels of $80\%$, $90\%$, and $95\%$, but perhaps somewhat shorter at nominal coverage level $99\%$.
Further comparisons of our LR based confidence intervals with methods based on kernel density estimates of the type studied by [@MR964293] but incorporating current state of the art bandwidth selection procedures will be of interest.
We close this section with two quotes from [@MR964293]. From the abstract of [@MR964293]:
> “In summary, the results are negative in the sense that a straightforward application of a naive bootstrap yields invalid inferences. In particular the bootstrap fails if resampling is done from the kernel density estimate.”
[@MR964293] notes in summarizing his simulation results:
> “... but the problem of constructing a confidence interval for the mode for smaller sample sizes remains a challenging one. In summary, the simulations reinforce the idea that generally automatic methods like the bootstrap need mathematical and numerical justification before their use can be recommended.”
Comparisons via examples {#subsec:examples}
------------------------
### Bright Star Catalogue {#subsubsec:brightstar}
We will now examine a few real examples. We first consider the rotational velocities of $3933$ stars from the Bright Star Catalogue which lists all stars of stellar magnitude $6.5$ or brighter (which is approximately all stars visible from earth) [@HW1991brightstar]. See [@Owen:2001wh], page 8 (also [@Doss:2013] page 217) for a brief description of the data. A star’s rotational velocity is affected by other physical quantities in a galaxy, and the distribution of rotational velocities can provide evidence for or against various models of galaxy behavior. In Figure \[fig:brightstar-rotational\_KDE-LC\], we present the data, as hashes along the bottom of the plot, and some density estimates. We plot a kernel density estimate, the log-concave MLE, the mode-constrained log-concave MLE with mode fixed at $16$, and the confidence interval for the mode given by our likelihood ratio statistic. There is an extensive literature on using boundary kernels for density estimation when support is bounded, but for simplicity we just use the bandwidth chosen by Silverman’s rule-of-thumb [@Silverman:1986uh page 48], which gives $14.3$. Note that the log-concave estimators automatically adapt to the boundary of the support without any extra work needed by the end user. The $95$% confidence interval given by our likelihood ratio test and plotted in red in Figure \[fig:brightstar-rotational\_KDE-LC\], is $(0, 19.6)$. The interval is somewhat large for a sample size this large. This is caused by the fact that the unconstrained MLE is very flat on $[0,16]$. This relatively flat interval is the reason we also plotted the mode-constrained MLE with mode fixed at $16$. The unconstrained and mode-constrained MLEs are visually quite similar, so it is plausible that the true density indeed has an approximately flat modal region.
![Rotational velocity of $3933$ stars with stellar magnitude brighter than $6.5$.[]{data-label="fig:brightstar-rotational_KDE-LC"}](Figures/brightstar_rotational_KDE-LC-MC-CI)
### S&P $500$
Next, we consider the $1006$ daily log returns for the S&P $500$ stock market index from January $1$, 2003 to December $29$, 2006. In Figure \[fig:SP500\_2002\_KDE-LC\] we plot the data, a kernel density estimate with bandwidth chosen by the method of [@Sheather:1991tp] to be $.13$, the log-concave MLE, and the $95$% confidence interval for the mode given by our likelihood ratio statistic. The log-concave mode estimate is $0.17$, and the $95$% confidence interval is $(0.10, 0.21)$. We also plot the maximum likelihood Gaussian density estimate, for comparison. The mean of the data (i.e. the Gaussian density MLE for the mean, median, and mode) is $0.04$ with a Wald normal-approximation confidence interval of $(-0.004,0.09)$. This latter confidence interval is plotted as two green diamonds along the bottom of the plot. Note that our confidence interval for the mode excludes $0$ and does not intersect with the confidence interval for the mean of the data. Thus, our procedure highlights some interesting features of the data and provide evidence for its non normality. We also note that despite the potentially very different theoretical rates of convergence for mode estimators and mean estimators, the two confidence intervals are similar in length.
![$1006$ S&P $500$ daily log returns for the years $2003$–$2006$[]{data-label="fig:SP500_2002_KDE-LC"}](Figures/SP500_03-06_KDE-LC-N-CI)
Further Problems and potential developments: outlook
====================================================
Uniformity and rates of convergence
-----------------------------------
There is a long line of research giving negative results concerning nonparametric estimation, starting with [@MR0084241], [@MR0205414], [@MR0155394; @MR0232487], and continuing with [@Donoho1988oneSidedInference] and [@MR1671670; @MR1747496]. In particular, [@MR1747496] considers a general setting involving estimators or confidence limits with optimal convergence rate $n^{-\rho}$ with $0 < \rho < 1/2$. He shows, under weak additional conditions, that: (i) there do not exist estimators which converge locally uniformly to a limit distribution; and (ii) there are no confidence limits with locally uniform asymptotic coverage probability. As an example he considers the mode of probability distributions $P$ on ${{\mathbb R}}$ with corresponding densities $p$ having a unique mode $M(p)$ and continuous second derivative in a neighborhood of $M(p)$. [@MR1747496] also reproves the result of [@Hasminskii1979mode] to the effect that the optimal rate of convergence of a mode estimator for such a class is $n^{-1/5}$. In this respect, we note that [@BRW2007LCasymp] established a comparable lower bound for estimation of the mode in the class of log-concave densities with continuous second derivative at the mode; they obtained a constant which matches (up to absolute constants) the pointwise (fixed $P$) behavior of the plug-in log-concave MLE of the mode. [@MR1747496] also notes the results of [@MR1105839], Theorem 5.5, page 653, concerning uniformity of convergence of kernel density estimates of the mode. Apparently a considerable portion of their difficult proof is given in [@DonohoLiu:87] (which we have not seen). In spite of the negative results of [@MR1747496], we have proceeded here with efforts to construct reasonable confidence intervals with pointwise (in $P$ or density $p$) correct coverage without proof of any local uniformity properties. In view of the recent uniform rate results of [@Kim-Samworth:16] we suspect that our new confidence intervals [*will*]{} (eventually) be shown to have some uniformity of convergence in their coverage probabilities over appropriate subclasses of the class of log-concave densities, but we leave the uniformity issues to future work.
Some further directions and open questions
------------------------------------------
But we now turn to discussion of some difficulties and potential for further work.
### Relaxing the second derivative assumption:
As noted in the previous subsection, most of the available research concerning inference for $M(f)$ has assumed $f \in C^2 (m, \mbox{loc})$ and $f^{(2)} (M(f)) <0$. Second derivative type assumptions of this type are made in [@MR0143282], [@Hasminskii1979mode], [@Eddy1980mode], [@MR1105839], [@MR964293; @MR947566], and [@MR1747496]. Exceptions include [@MR1015137], [@Ehm1996cuspMode], [@HerrmannZiegler2004mode], [@BRW2007LCasymp].
What happens if the second derivative curvature assumption does not hold, but instead is replaced by something either stronger or weaker, such as $$\begin{aligned}
f(m) - f(x) \le C |x - m|^r\end{aligned}$$ for some $C$ where $1 \le r < 2$ (in the “stronger” case) or $2 < r < \infty$ (in the “weaker” case)? It is natural to expect that it is easier to form confidence intervals for $m$ when $1\le r< 2$ holds, but that it is harder to form confidence intervals for $m$ when $2<r< \infty$. In fact, [@BRW2007LCasymp] page 1313 gives the following result: if $f = \exp (\varphi)$ with $\varphi$ concave and where $\varphi^{(j)} (m) = 0$ for $j=2, \ldots , k-1$ but $\varphi^{(k)} $ exists and is continuous in a neighborhood of $m$ with $\varphi^{(k)} (m) \not= 0$, then $$n^{1/(2k+1)} ( \widehat{M}_n - m ) \rightarrow_d C_k (f(m), \varphi^{(k)} (m) ) M( H_k^{(2)} ) .$$ Thus the convergence rate of the log-concave MLE of the mode is slower as $k$ increases. \[On the other hand, by Theorem 2.1 of [@BRW2007LCasymp], page 1305, the convergence rate of of the MLE $\widehat{f}_n $ of $f$ at $m$ (and in a local neighborhood of $m$) is [*faster*]{}: $$n^{k/(2k+1)} ( \widehat{f}_n (m) - f(m)) \rightarrow_d c_k (m, f) H_k^{(2)} (0) .]$$ Furthermore the sketch of the proof of the limiting distribution of the likelihood ratio statistic in Section \[ssec:PfSketchThm1\] (ignoring any remainder terms) together with the results of [@BRW2007LCasymp], suggest that $2\log \lambda_n \rightarrow_d {{\mathbb D}}_k $ under $f \in {\cal P}_m \cap {\cal S}_k$ where $${\cal S}_k = \{ f \in {\cal P} : \ \varphi^{(j)} (m) = 0, \ j=2, \ldots , k-1, \ \varphi^{(k)} (m) \not= 0,\ \varphi \in C^k (m , \mbox{loc} ) \}$$ and where with $ \varphi_k$ and $\varphi_k^0$ denoting the local limit processes in the white noise model (\[WhiteNoiseCanonicalConcave\]) with drift term $g_0 (t) = - 12t^2$ replaced by $- (k+2)(k+1) | t |^k$ (or $t^4$ in \[TwiceIntegratedWhiteNoise\] replaced by $t^{k+2}$), $${{\mathbb D}}_k \equiv \int \{ (\widehat{\phi}_k (v))^2 - (\widehat{\phi}_k^0(v) )^2 \} dv .$$ Figure \[fig:LRasymptotics-all\] gives some evidence in favor of this conjecture. Note that the (Monte-Carlo estimator of) the distributions of ${{\mathbb D}}_3$ and ${{\mathbb D}}_r$ seem to be stochastically larger than the (Monte-Carlo estimator) of the distribution of ${{\mathbb D}}_2 \equiv {{\mathbb D}}$, and that the distribution of ${{\mathbb D}}_4$ is apparently stochastically larger than that of ${{\mathbb D}}_3$.
![Empirical distributions of $2 \log \lambda_n$ for $f \in {\cal P}_m \cap {\cal S}_k$, $k \in \{ 2,3,4 \}$. $n=10^4$, $M = 5\times 10^3$ replications[]{data-label="fig:LRasymptotics-all"}](Figures/LRasymptotics_all-1-2-3-4.pdf){width=".9\textwidth"}
This raises several possibilities:
Option 1:
: It seems likely that by choosing a critical value from the distribution of ${{\mathbb D}}_6$ (say), that the resulting confidence intervals will have correct coverage for $f \in {\cal P} \cap {\cal S}_6$ with conservative coverage if we happen to have $f \in {\cal P}\cap {\cal S}_2$ (in which case critical points from ${{\mathbb D}}= {{\mathbb D}}_2$ would have sufficed), and anti-conservative coverage if the true $f$ belongs to $ {\cal P}\cap ({\cal S}_k \setminus {\cal S}_6)$ for some $k \ge 8$.
Option 2:
: Try to construct an adaptive procedure which first estimates $k$ (the degree of “flatness”) of the true $f \in {\cal P}$ (by $\hat{k}$ say), and then choose a critical point from the distribution of ${{\mathbb D}}_{\hat{k}}$.
We leave the investigation of both of these possibilities to future work.
### Beyond dimension $d=1$:
It seems natural to consider generalizations of the present methods to the case of multivariate log-concave and $s-$concave densities. While there is a considerable amount of work on estimation of multivariate modes, mostly via kernel density estimation, much less seems to be available in terms of confidence sets or other inference tools. For some of this, see e.g. [@MR1051586], [@MR1985502], [@MR1327618], [@MR2112688], [@MR0336874], [@MR491553], [@MR0331618]. On the other hand apparently very little is known about the multivariate mode estimator $M(\widehat{f}_n)$ where $\widehat{f}_n$ is the log-concave density estimator for $f $ on ${{\mathbb R}}^d$ studied by [@MR2758237] and [@MR2645484]. Further study of this estimator will very likely require considerable development of new methods for study of the pointwise and local properties of the log-concave density estimator $\widehat{f}_n$.
Proofs {#sec:ProofsPart2}
======
It remains to deal with the remainder terms defined in (\[BasicDecompositionFirstForm\]) in the course of our “proof sketch” for Theorem \[LRasympNullDistribution\] in Section \[sec:Intro\]. We first deal with the “local” remainder terms $R_{n,j}, \ R_{n,j}^0 $ with $j \in \{ 2, 3 \}$ in Subsection \[ssec:ProofsLocalRTs\]. Subsection \[ssec:GlobalRTs\] is dedicated to the proofs for the “non-local” remainder terms.
The local remainder terms $R_{n,j}, R_{n,j}^0$, $j\in \{ 2, 3\}$ {#ssec:ProofsLocalRTs}
----------------------------------------------------------------
We first deal with the (easy) local remainder terms.
\[prop:localerrorterms\] Let $t_{n,1} = m - M n^{-1/5}$ and $t_{n,2} = m + M n^{-1/5}$ for $M >
0$. Then the remainder terms $R_{n,2}$, $R_{n,2}^0$, $R_{n,3}$, and $R_{n,3}^0$ satisfy $n R_{n,j} = o_p (1)$ and $nR_{n,j}^0 = o_p (1)$ for $j
\in \{2,3\}$.
Recall that the remainder terms $R_{n,2}$, $R_{n,2}^0$, $R_{n,3}$, and $R_{n,3}^0$ given by (\[RemainTermTwoUnconstrained\]), (\[RemainTermTwoConstrained\]), (\[RemainTermThreeUnconstrained\]), and (\[RemainTermThreeConstrained\]) are all of the form a constant times $$\begin{aligned}
&& \tilde{R}_n \equiv \int_{D_n} e^{\tilde{x}_{n} (u) } (\widehat{\varphi}_n (u) - \varphi_0 (m) )^3 du, \ \ \ \mbox{or}\\
&& \tilde{R}_n^0 \equiv \int_{D_n} e^{\tilde{x}_{n}^0 (u) } (\widehat{\varphi}_n^0 (u) - \varphi_0 (m) )^3 du,\end{aligned}$$ where $D_n$ is a (possibly random) interval of length $O_p (n^{-1/5})$ and $\tilde{x}_{n,j}$ converges in probability, uniformly in $u \in D_n$, to zero. But by Assumption 1 and by [@DR2009LC] (see also Theorem 4.6.B of [@Doss-Wellner:2016ModeConstrained]), it follows that for any $M>0$ we have $$\begin{aligned}
\lefteqn{\sup_{|t| \le M} | \widehat{\varphi}_n ( m + n^{-1/5} t) - \varphi_0 (m) | } \\
& \le & \sup_{|t| \le M} \left \{ | \widehat{\varphi}_n ( m + n^{-1/5} t) - \varphi_0 (m + n^{-1/5} t) |
+ | \varphi_0 (m + n^{-1/5} t) - \varphi_0 (m) | \right \}\\
& = & O_p ( (n^{-1} \log n)^{2/5} ) + O ( n^{-2/5} ) = O_p ( ( n^{-1} \log n)^{2/5} ),\end{aligned}$$ and hence $$\begin{aligned}
n | \tilde{R}_n | = n O_p ( (n^{-1} \log n)^{6/5} \cdot n^{-1/5} ) = O_p ( n^{-2/5} (\log n)^{6/5} ) = o_p (1) .\end{aligned}$$ by Assumption 1 and by Theorem 4.7.B of [@Doss-Wellner:2016ModeConstrained]), it follows that for any $M>0$ we have $$\begin{aligned}
n | \tilde{R}_n^0 | = n O_p ( (n^{-1} \log n)^{6/5} \cdot n^{-1/5} ) = O_p ( n^{-2/5} (\log n)^{6/5} ) = o_p (1) .\end{aligned}$$ This completes the proof of negligibility of the local error terms $R_{n,j}, R_{n,j}^0$, $j\in \{ 2, 3\}$.
The global remainder terms $R_{n,1}$ and $R_{n,1}^c$ {#ssec:GlobalRTs}
----------------------------------------------------
Recall that the remainder terms $R_{n,1}$ and $R_{n,1}^c$ are given by (\[RemainTermOne\]) and (\[RemainTermOneCompl\]). Note that the integral in the definition of (\[RemainTermOneCompl\]) is over $[X_{(1)}, X_{(n)}]
\setminus D_n$, and hence this term in particular has a global character. We will see later that $R_{n,1}$ also can be seen as having a global nature.
[**Outline:**]{} From now on, we will focus our analysis on the portion of $R_{n,1,\twoArgs}^c$ given by integrating over the left side, $[X_{(1)},
t_1]$. Arguments for the integral over $[t_2, X_{(n)}]$ are analogous. Thus, by a slight abuse of notation, define the one-sided counterpart to $\RnTwo^c$ from for any $t < m $ by $$\label{eq:defn:LRS-remainder-4}
\begin{split}
R_{n,1,\oneArg}^c & \equiv \int_{[X_{(1)}, t]} \vvn d\FFn - \vvna d\FFna -
\int_{[X_{(1)}, t]} (e^{\vvn} - e^{\vvna}) \, d\lambda.
\end{split}$$ Here $\lambda$ is Lebesgue measure (and is unrelated to the likelihood ratio $\lambda_n$). The analysis of $R_{n,1,t}^c$ is the greatest difficulty in understanding $2 \log
\lambda_n$. The proof that $R_{n,1,t_n}^c$ is $o_p(n^{-1})$ when $b \to \infty$ where $t_n = m - b n^{-1/5}$ is somewhat lengthy so we provide an outline here.
1. \[item:rem:outline-1\] [**Step 1, Decomposition of $R_{n,1,t}^c$:**]{} Decompose $R_{n,1,t}^c$, to see that $$R_{n,1,t}^c = A_{n,t}^1 + E_{n,t}^1 - T_{n,t}^1
= A_{n,t}^2 + E_{n,t}^2 + T_{n,t}^2,$$ where the summands $A_{n,t}^i, E_{n,t}^i, T_{n,t}^i$ are defined below (see and the preceding text).
2. \[item:rem:outline-2\] [**Step 2, Global $O_p(n^{-1})$ conclusion:**]{} In this section we use the fact that away from the mode, the characterizations of $\vvn $ and $\vvna$ are identical to study $T_n^i$, $i=1,2$, which are related to $\int_{[X_{(1)}, t]} (\vvn - \vvna)^2 \ffn d\lambda$. and $\int_{[X_{(1)}, t]} (\vvn - \vvna)^2 \ffna d\lambda$. We will show ${T}_n^i = O_p(n^{-1})$, $i=1,2$. Note $O_p(n^{-1})$ would be the size of the integral if it were over a local interval of length $O_p(n^{-1/5})$ (under our curvature assumptions), but here the integral is over an interval of constant length or larger, so this result is global in nature.
3. \[item:rem:outline-3\] [**Step 3, Convert global $O_p$ to local $o_p$ to global $o_p$:**]{} Convert the global $O_p(n^{-1})$ conclusion over $T_{n}^i$ into an $o_p(n^{-1})$ conclusion over a interval of length $O_p(n^{-1/5})$ local to $m$. Feed this result back into the argument in Step \[item:rem:outline-2\], yielding $T_{n,t}^i = o_p(n^{-1}),$ $i=1,2$. Apply Lemma \[lem:rem:local-to-global-square-integral\] to show additionally that there exist knots of $\vvn$ and $\vvna$ that are $o_p(n^{-1/5})$ apart in an $O_p(n^{-1/5})$ length interval on which $\Vert
\vvna - \vvn \Vert = o_p(n^{-2/5})$, $\Vert
\ffna - \ffn \Vert = o_p(n^{-2/5})$, and $\Vert \FFna - \FFn \Vert =
o_p(n^{-3/5})$.
4. \[item:rem:outline-4\] [**Step 4, Concluding arguments:**]{} Return to the decomposition of $R_{n,1,t}^c$ given in Step \[item:rem:outline-1\]; the terms given there depend on $\vvna-\vvn$, $\ffna -\ffn$, and $\FFna-\FFn$. Thus, using the results of Step \[item:rem:outline-3\] we can show $n R_{n,1,t}^c = o_p(1)$ as desired.
To finalize the argument, in Section \[subsubsec:final-arguments\], we take $t_n = m - bn^{-1/5}$, but we also need to let $b \to \infty$. Thus, the $O_p$ and $o_p$ statements above need to hold uniformly in $b$.
### Decomposition of $R_{n,1,t}^c$
We begin by decomposing $R_{n,1,t}^c$ for fixed $t < m$. By with $\vp_{1n} = \vvn$ and $\vp_{2n} = \vvna$, we see that $$\begin{aligned}
R_{n,1,t}^c
& =
\int_{[X_{(1)}, t]} {\left(}\vvn \ffn - \vvna \ffna
- {\left(}\vvn - \vvna + \frac{(\vvn - \vvna)^2}{2} e^{{\varepsilon}_{n}^1} {\right)}\ffna
{\right)}\, d \lambda \nonumber \\
& = \int_{[X_{(1)}, t]} {\left(}\vvn \ffn - \vvn \ffna
- \frac{(\vvn - \vvna)^2}{2} e^{{\varepsilon}_{n}^1} \ffna {\right)}d\lambda \nonumber
\\
& = \int_{[X_{(1)}, t]} {\left(}\vvn (\ffn - \ffna )
- \frac{(\vvn - \vvna)^2}{2} e^{{\varepsilon}_{n}^1} \ffna {\right)}\, d\lambda,
\label{eq:rem:Rn1tc-ffna-decomp1}\end{aligned}$$ where $\lambda $ is Lebesgue measure and ${\varepsilon}_{n}^1(x)$ lies between $0
$ and $\vvn(x)- \vvna(x)$. Again applying now with $\vp_{1n} = \vvna$ and $\vp_{2n} =
\vvn$, we see that $$\begin{aligned}
R_{n,1,t}^c
& =
\int_{[X_{(1)}, t]} {\left(}\vvn \ffn - \vvna \ffna
+ {\left(}e^{\vvna - \vvn} - 1 {\right)}\ffn
{\right)}\, d \lambda \nonumber \\
& = \int_{[X_{(1)}, t]} {\left(}\vvn \ffn - \vvna \ffna
+ {\left(}\vvna - \vvn + \frac{(\vvna - \vvn)^2}{2} e^{{\varepsilon}_{n}^2} {\right)}\ffn
{\right)}\, d \lambda \nonumber \\
& = \int_{[X_{(1)}, t]} {\left(}\vvna (\ffn - \ffna)
+ \frac{(\vvna - \vvn)^2}{2} e^{{\varepsilon}_{n}^2} \ffn {\right)}\,
d\lambda, \label{eq:rem:Rn1tc-ffn-decomp1}\end{aligned}$$ where ${\varepsilon}_n^2$ lies between $0$ and $\vvna(x) - \vvn(x)$. For a function $f(x)$, recall the notation $f_{s}(x) = f(x) -
f(s)$ for $x \le s$ and $f_s(x)=0$ for $x \ge s$. Now define $A^i_{n,t}
\equiv A^i_n$, $i=1,2$ by $$\begin{aligned}
A_n^1
\equiv \int_{[X_{(1)}, t]} \widehat{\varphi}_{n,t} \, d{\left(}{ {\mathbb F}}_n - \FFna {\right)}\quad \mbox{ and } \quad
A_n^2 \equiv
\int_{[X_{(1)}, t]} \widehat{\varphi}_{n,t}^0 d{\left(}\FFn - { {\mathbb F}}_n {\right)}\end{aligned}$$ and define $ E_{n,t}^1 \equiv E_n^1 $ to be $$\begin{split}
\MoveEqLeft \int_{(\tau,t]} \widehat{\varphi}_{n,t} \, d{\left(}\FFn - \Fn {\right)}+ \vvn(t) ( \FFn(t) - \FFna(t) ) \\
& + (\vvn(\tau) - \vvn(t)) ( \FFn(\tau) - \Fn(\tau) ) \label{eq:rem:defn:Ent1}
\end{split}$$ and $ E_{n,t}^2 \equiv E_n^2 $ to be $$\begin{split}
\MoveEqLeft \int_{(\tau^0, t]} \widehat{\vp}_{n,t}^0 d{\left(}\Fn - \FFna {\right)}+ \vvna(t) ( \FFn(t) - \FFna(t)) \\
& + (\vvna(\tau^0)
- \vvna(t)) (\Fn(\tau^0) - \FFna(\tau^0)), \label{eq:rem:defn:Ent2}
\end{split}$$ where $\tau \equiv \tau_-(t) = \sup S(\vvn) \cap (-\infty,t]$ and $\tau^0 \equiv \tau_-^0(t) = \sup S(\vvna) \cap (-\infty, t]$. We will assume that $$\tau \le \tau^0$$ without loss of generality, because the arguments are symmetric in $\vvn$ and $\vvna$, since we will be arguing entirely on one side of the mode.
Our next lemma will decompose the first terms in and , into $A_{n}^i + E_n^i$, $i=1,2$. The crucial observation is that $A_n^1 \le 0 $ and $A_n^2 \ge 0$, by taking $\Delta = \vvnt{t}$ and $\Delta =
\vvnat{t}$ in the characterization theorems for the constrained and unconstrained MLEs, Theorems 2.2 and 2.8 of [@Doss-Wellner:2016ModeConstrained]. \[page:Ani-inequalities\] Note that since $t \le m$, $\vvnt{t}$ has modal interval containing $m$.
Let all terms be as defined above. We then have $$\label{eq:rem:rn1c:1}
\begin{split}
\int_{[X_{(1)}, t]} \vvn ( \ffn - \ffna ) d\lambda
& = A_{n,t}^1
+ E_{n,t}^1
\end{split}$$ and $$\label{eq:rem:rn1c:2}
\begin{split}
\int_{[X_{(1)}, t]} \vvna (\ffn - \ffna) d\lambda
& = A_{n,t}^2
+ E_{n,t}^2.
\end{split}$$
We first show . We can see $ \int_{[X_{(1)}, t]} \vvn ( \ffn - \ffna)$ equals $$\begin{aligned}
\int_{[X_{(1)}, t]} ( \widehat{\varphi}_{n,\tau_-} + \vvn -
\widehat{\varphi}_{n,\tau_-} ) \ffn d\lambda
- \int_{[X_{(1)}, t]} ( \widehat{\vp}_{n,t} + \vvn - \widehat{\vp}_{n,t}
)
\ffna d\lambda, \nonumber
\end{aligned}$$ and since $\int \widehat{\varphi}_{n,\tau_-} \, d{\left(}\Fn - \FFn {\right)}= 0$, this equals $$\begin{aligned}
\MoveEqLeft \int_{[X_{(1)}, \tau_-]} \widehat{\vp}_{n,\tau_-} d\Fn
+ \int_{[X_{(1)}, \tau_-]} (\vvn - \widehat{\vp}_{n,\tau_-} ) \ffn
d\lambda
+ \int_{(\tau_-,t]} \vvn \ffn d\lambda \nonumber \\
& \quad - {\left(}\int_{[X_{(1)}, t]} \widehat{\vp}_{n,t} \ffna d\lambda
+ \vvn(t) \FFna(t) {\right)}\nonumber \\
& = \int_{[X_{(1)}, t]} \widehat{\vp}_{n,t} d\Fn
+ \int_{[X_{(1)}, \tau_-]} (\widehat{\vp}_{n,\tau_-} -
\widehat{\vp}_{n,t} ) d\Fn
- \int_{(\tau_-,t]} \widehat{\vp}_{n,t} d\Fn \nonumber \\
& \quad + \int_{[X_{(1)}, \tau_-]} \vvn(\tau_-) \ffn d\lambda
+ \int_{(\tau_-,t]} \vvn \ffn d\lambda
- {\left(}\int \widehat{\vp}_{n,t} \ffna d\lambda
+ \vvn(t) \FFna(t) {\right)}\nonumber \\
& = \int_{[X_{(1)}, t]} \widehat{\vp}_{n,t} \, d{\left(}\Fn - \FFna {\right)}+ {\left(}\vvn(t) - \vvn(\tau_-) {\right)}\Fn(\tau_-)
- \int_{(\tau_-, t]} \vvn d\Fn \nonumber \\
& \quad + \vvn(t) ( \Fn(t) - \Fn(\tau_-))
+ \vvn(\tau_-) \FFn(\tau_-) + \int_{(\tau_-, t]} \vvn \ffn d\lambda
- \vvn(t) \FFna(t), \nonumber
\end{aligned}$$ which equals $$\begin{split}
\MoveEqLeft \int \widehat{\vp}_{n,t} d {\left(}\Fn - \FFna {\right)}+ \int_{(\tau_-, t]} \vvn d {\left(}\FFn - \Fn {\right)}\\
& + \vvn(t) ( \Fn(t) - \FFna(t))
+ \vvn(\tau_-) ( \FFn(\tau_-) - \Fn(\tau_-))
\end{split}$$ which equals $$\begin{split}
\MoveEqLeft \int \widehat{\vp}_{n,t} d{\left(}\Fn - \FFna {\right)}+ \int_{(\tau_-,t]} \widehat{\vp}_{n,t} d {\left(}\FFn-\Fn {\right)}\\
& + \vvn(t) (\FFn(t) - \FFna(t))
+ (\vvn(\tau_-) - \vvn(t)) ( \FFn(\tau_-) - \Fn(\tau_-)),
\end{split}$$ as desired.
Now we show . We see $\int_{[X_{(1)}, t]} \vvna (\ffn - \ffna) d\lambda$ equals $$\begin{aligned}
\int_{[X_{(1)}, t]} {\left(}\vvnat{t} + \vvna - \vvnt{t} {\right)}\ffn d\lambda
- \int_{[X_{(1)}, t]} {\left(}\vvnat{\tau^0_-} + \vvna - \vvnat{\tau^0_-} {\right)}\ffna d\lambda
\end{aligned}$$ and since $\int \vvnat{\tau^0_-} d(\Fn - \FFna) = 0$, this equals $$\begin{aligned}
\MoveEqLeft
\int_{[X_{(1)}, t]} \vvnat{t}\ffn d\lambda
+ \vvna(t) \FFn(t)
- \bigg[ \int \vvnat{\tau^0_-} d \Fn
+ \int_{[X_{(1)}, \tau^0_-]}
\vvna(\tau^0_-) \ffna d\lambda
\\
& \hspace{13em} + \int_{(\tau^0_-, t]} \vvna \ffna
d\lambda \bigg],
\end{aligned}$$ which equals $$\begin{aligned}
\MoveEqLeft \int_{[X_{(1)}, t]} \vvnat{t} \ffn d\lambda
+ \vvna(t) \FFn(t)
- \bigg[ \int_{[X_{(1)}, t]} \vvnat{t} d \Fn \\
& + \int_{[X_{(1)}, t]} (\vvnat{\tau^0_-} - \vvnat{t} ) d\Fn
+ \vvna(\tau^0_-) \FFna(\tau^0_-)
+ \int_{(\tau^0_-, t]} \vvna \ffna d\lambda
\bigg]
\end{aligned}$$ which equals $$\begin{aligned}
\MoveEqLeft \int_{[X_{(1)}, t]} \vvnat{t} d(\FFn - \Fn )
+ \vvna(t) \FFn(t)
- \bigg[
\int_{[X_{(1)}, \tau^0_-]} (\vvna(t) - \vvna(\tau^0_-) ) d\Fn \\
& \quad - \int_{(\tau^0, t]} \vvna d\Fn
+ \int_{(\tau^0_-, t]} \vvna(t) d\Fn
+ \vvna(\tau^0_-) \FFna(\tau^0_-)
+ \int_{(\tau^0_-, t]} \vvna \ffna d\lambda
\bigg]
\end{aligned}$$ which equals $$\begin{aligned}
\MoveEqLeft \int_{[X_{(1)}, t]} \vvnat{t} d(\FFn - \Fn)
+ \int_{(\tau^0_-, t]} \vvna d (\Fn - \FFna)
+ \vvna(t) \FFn(t)
+ \vvna(\tau^0_-) \Fn(\tau^0_-) \\
& \quad - \vvna(t) \Fn(\tau^0_-)
- \vvna(t) (\Fn(t) - \Fn(\tau^0_-))
- \vvna(\tau^0_-) \FFna(\tau^0_-)
\end{aligned}$$ which equals $$\begin{aligned}
\MoveEqLeft
\int_{[X_{(1)}, t]} \vvnat{t} d(\FFn - \Fn) + \int_{(\tau^0_-,t]} \vvna
d(\Fn - \FFna) \\
& + \vvna(t) (\FFn(t) - \Fn(t)) + \vvna(\tau^0_-) (
\Fn(\tau^0_-) - \FFna(\tau^0_-))
\end{aligned}$$ which equals $$\begin{aligned}
\MoveEqLeft \int_{[X_{(1)}, t]} \widehat{\vp}_{n,t}^0 d {\left(}\FFn - \Fn {\right)}+
\int_{(\tau^0_-, t]} \widehat{\vp}_{n,t}^0 d{\left(}\Fn - \FFna {\right)}+
\vvna(t) ( \FFn(t) - \FFna(t)) \\
& \quad + (\vvna(\tau^0_-) - \vvna(t) ) (\Fn(\tau^0_-) -
\FFna(\tau^0_-)),
\end{aligned}$$ as desired.
Define $T_{n,t}^i \equiv T_n^i$, $i=1,2$, by \[rem:defn:Tni\] $$\begin{aligned}
\label{eq:rem:defn:Tni}
T_n^1 =
\int_{[X_{(1)}, t]} \frac{ (\vvn - \vvna)^2}{2}
e^{{\varepsilon}_{n}^1} \ffna
d\lambda
\quad \mbox{ and } \quad
T_n^2 =
\int_{[X_{(1)}, t]} \frac{(\vvn - \vvna)^2}{2}
e^{{\varepsilon}_{n}^2} \ffn d\lambda,\end{aligned}$$ so that $$\label{eq:rem:Rn1tc-main-decomposition}
R_{n,1,t}^c = A_{n,t}^1 + E_{n,t}^1 - T_{n,t}^1
= A_{n,t}^2 + E_{n,t}^2 + T_{n,t}^2.$$ by and . Recall (from page ) that $A_n^1 \le 0 \le A_n^2$. Thus $$\label{eq:rem:main-An2-ineq}
E_n^1 - E_n^2 \ge E_n^1 - E_n^2 - T_n^2 - T_n^1
= A_n^2 - A_n^1 \ge
\begin{cases}
& A_n^2 \ge 0,\\
& -A_n^1 \ge 0.
\end{cases}$$ To see that $R_{n,1}^c = O_p(n^{-1})$ we need to see that $E_n^i, A_n^i,$ and $T_n^i$ are each $O_p(n^{-1})$ (for, say, $i=1$). We can see already that $E_n^1 - E_n^2 = O_p(n^{-1})$ (by direct analysis of the terms in $E_n^1-E_n^2$ from and ), which yields that $A_n^1$ and $A_n^2$ are both $O_p(n^{-1})$. However, it is clear that we also need to analyze $T_n^1 +
T_n^2$ to understand $R_{n,1}^c$. We need to show that $T_{n,t}^1 +
T_{n,t}^2$ is $O_p(n^{-1})$ to see that $R_{n,1,t}^c = O_p(n^{-1})$; but we will also be able to use that $T_{n,t}^1+T_{n,t}^2=O_p(n^{-1})$ to then find $t^*$ values such that $T_{n,t^*}^1+T_{n,t^*}^2=o_p(n^{-1})$, which will allow us to argue in fact that $E_{n,t^*}^1+E_{n,t^*}^2 = o_p(n^{-1})$ (rather than just $O_p(n^{-1})$), and thus that $R_{n,1,t^*}^c =
o_p(n^{-1})$, as is eventually needed. Thus, we will now turn our attention to studying $T_n^1 + T_n^2$. Afterwards, we will study $$\label{eq:rem:Rn1tc-average-decomp}
R_{n,1,t}^c = (A_{n,t}^1 + E_{n,t}^1 - T_{n,t}^1
+ A_{n,t}^2 + E_{n,t}^2 + T_{n,t}^2)/2,$$ from . From seeing $T_{n,t^*}^1+T_{n,t^*}^2 = o_p(n^{-1})$, we will be able to conclude that $A_{n,t^*}^1+A_{n,t^*}^2$ and $E_{n,t^*}^1+E_{n,t^*}^2$ are also $o_p(n^{-1})$, as desired. Then we can conclude $R_{n,1, t^*}^c = o_p(n^{-1})$.
### Show ${T}_n^i = O_p(n^{-1})$, $i=1,2$
The next lemma shows that terms that are nearly identical to $T_{n}^i$ are $O_p(n^{-1})$. The difference between the integrand in the terms in the lemma and the integrand defining $T_n^i$ is that ${\varepsilon}_n^i$ is replaced by a slightly different $\tilde {\varepsilon}_n^i$. Previously, we considered $t$ to be fixed, whereas now we will have it vary with $n$.
\[lem:rem:step2:Tni-Opn-1\] Let $t_n < m $ be a (potentially random) sequence such that $$\label{eq:rem:2}
t_n \le \max
{\left(}S(\vvn) \cup S(\vvna) {\right)}\cap (-\infty, m).$$ For $t < m$, let $$\label{eq:rem:defn:tilde-Tni}
\tilde T_{n,t}^1 = \int_{[X_{(1)}, t_n]}
{\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^1} \ffna d
\lambda
\; \mbox{ and } \;
\tilde T_{n,t}^2
= \int_{[X_{(1)}, t_n]} {\left(}\vvna - \vvn {\right)}^2 e^{ \tilde {\varepsilon}_{n}^2} \ffn d
\lambda,$$ where $\tilde {\varepsilon}_{n}^1(x)$ lies between $\vvn(x)-\vvna(x)$ and $0$, and $\tilde {\varepsilon}_{n}^2(x)$ lies between $\vvna(x)-\vvn(x)$ and $0$, and are defined in in the proof. Then we have $$\label{eq:rem:3}
\tilde T_{n,t_n}^1 =
O_p(n^{-1})
= \tilde T_{n,t_n}^2.
$$
For a function $f(x)$, recall the notation $f_{s}(x) = f(x) - f(s)$ for $x \le s$ and $f_s(x)=0$ for $x \ge s$. Let $\tau \in S(\vvn)$ and $\tau^0 \in S(\vvna)$, and assume that $$\label{eq:rem:defn:tau-tau0}
\tau \le \tau^0 < m.$$ (The argument is symmetric in $\vvn$ and $\vvna$, so we may assume this without loss of generality.) We will show the lemma holds for the case $t_n =
\tau^0$, and then the general $t_n \le \tau^0$ case follows since the integral is increasing in $t_n$. Now, because $\vvnat{\tau}$ is concave, for ${\varepsilon}\le 1$, the function $\vvn(x) + {\varepsilon}(\vvnat{\tau}(x) -
\vvnt{\tau}(x))$ is concave. So, by Theorem 2.2, page 43, of [@DR2009LC], we have $$\label{eq:rem:old-I}
\int (\vvnat{\tau} - \vvnt{\tau}) d(\Fn - \FFn) \le 0.$$ Similarly, if $\tau^0 $ is a knot of $\vvna$ and is less than the mode, then since $\vvna(x) + {\varepsilon}{\left(}\vvnt{\tau^0}(x) -\vvnat{\tau^0}(x)
{\right)}$ is concave with mode at $m$ for ${\varepsilon}$ small (since $\vvnt{\tau^0}(x) -\vvnat{\tau^0}(x)$ is only nonzero on the left side of the mode), by the characterization Theorem 2.8 of [@Doss-Wellner:2016ModeConstrained], we have $$ \int (\vvnt{\tau^0} - \vvnat{\tau^0}) d(\Fn-\FFna) \le 0.$$ Then setting \[page:def-II-L-tau\] $II_{n,\tau^0}^L := \int_{[X_{(1)}, \tau^0]} {\left(}\vvn - \vvna {\right)}d{\left(}\Fn - \FFna {\right)}$, we have $$\begin{aligned}
\label{eq:rem:4}
0 & \ge \int_{[X_{(1)}, \tau^0]} {\left(}\vvnt{\tau^0} - \vvnat{\tau^0}{\right)}d{\left(}\Fn -
\FFna {\right)}\\
& = II_{n,\tau^0}^L
- {\left(}\vvn(\tau^0) - \vvna(\tau^0) {\right)}{\left(}\Fn(\tau^0) - \FFna(\tau^0)
{\right)}. \label{eq:rem:5}
\end{aligned}$$ And setting $I_{n,\tau^0}^L := \int_{[X_{(1)}, \tau^0]} {\left(}\vvna - \vvn {\right)}d{\left(}\Fn -
\FFn {\right)}$, we have $$\label{eq:rem:decompose-IL-tau}
\begin{split}
I_{n,\tau^0}^L
& = \int_{[X_{(1)}, \tau]} {\left(}\vvna(u) - \vvna(\tau) {\right)}d {\left(}\Fn - \FFn
{\right)}(u) \\
& \quad - \int_{[X_{(1)}, \tau]} {\left(}\vvn(u)-\vvn(\tau) {\right)}d{\left(}\Fn-\FFn{\right)}(u) \\
& \quad + {\left(}\vvna(\tau) - \vvn(\tau) {\right)}\int_{[X_{(1)}, \tau]} d {\left(}\Fn - \FFn {\right)}\\
& \quad + \int_{(\tau, \tau^0]} {\left(}\vvna - \vvn {\right)}d{\left(}\Fn - \FFn {\right)},
\end{split}$$ and, since the first two summands together yield the left hand side of , we have $$\label{eq:rem:InL-upperbound}
I_{n,\tau^0}^L \le
{\left(}\vvna(\tau) - \vvn(\tau) {\right)}\int_{[X_{(1)}, \tau]} d {\left(}\Fn - \FFn {\right)}+ \int_{(\tau, \tau^0]} {\left(}\vvna - \vvn {\right)}d{\left(}\Fn - \FFn {\right)}.$$ \[page:new-analysis-squared-integral\] Now, we apply of Lemma \[lem:f-to-varphi-taylor\] to see that $$\begin{aligned}
I_{n,\tau^0}^L +II_{n,\tau^0}^L
& =
\int_{[X_{(1)}, \tau^0]} {\left(}\vvna - \vvn {\right)}d {\left(}\FFna - \FFn
{\right)}\nonumber \\
& =
\begin{cases}
& \int_{X_{(1)}}^{\tau^0} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^1} \ffna d
\lambda \ge 0, \\
& \int_{X_{(1)}}^{\tau^0} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^2} \ffn d
\lambda \ge 0,
\end{cases} \label{eq:rem:second-I-plus-II}
\end{aligned}$$ where $\tilde {\varepsilon}_{n}^2(x)$ lies between $\vvna(x)-\vvn(x)$ and $0$ and $\tilde {\varepsilon}_{n}^1(x)$ lies between $\vvn(x)-\vvna(x)$ and $0$. By and , is bounded above by $$\label{eq:rem:I-plus-II-error}
\begin{split}
& {\left(}\vvna(\tau) - \vvn(\tau) {\right)}\int_{[X_{(1)}, \tau]} d {\left(}\Fn - \FFn {\right)}+ \int_{(\tau, \tau^0]} {\left(}\vvna - \vvn {\right)}d{\left(}\Fn - \FFn {\right)}\\
& \quad + {\left(}\vvn(\tau^0) - \vvna(\tau^0) {\right)}{\left(}\Fn(\tau^0) - \FFna(\tau^0) {\right)}.
\end{split}$$ By Proposition 5.2 of [@Doss-Wellner:2016ModeConstrained], $\sup_{t \in [\tau, \tau^0]}
\left| \vvna(t)-\vvn(t) \right| = O_p(n^{-2/5})$. By Corollary 2.5 of [@DR2009LC], $$\left| \int_{[X_{(1)}, \tau]} d {\left(}\Fn - \FFn {\right)}\right| \le 1/n,$$ so the first term in the above display is $O_p(n^{-7/5})$. Similarly, by Corollary 2.12 of [@Doss-Wellner:2016ModeConstrained], $\left| \Fn(\tau^0) - \FFna(\tau^0) \right| \le 1/n$, so the last term in the previous display is $O_p(n^{-7/5})$. \[page:Op-seven-fifths\] We can also see that the middle term in the previous display equals $$\label{eq:rem:square-integral-errorterms}
\begin{split}
\MoveEqLeft ( \widehat{\varphi}_n^0 - \widehat{\varphi}_n )(\tau^0) ( { {\mathbb F}}_n - \widehat{F}_n )(\tau^0)
- ( \widehat{\varphi}_n^0 - \widehat{\varphi}_n )(\tau) ( { {\mathbb F}}_n - \widehat{F}_n )(\tau)\\
& \ \ - \ \int_{(\tau, \tau^0]} ( { {\mathbb F}}_n - \widehat{F}_n ) ( \widehat{\varphi}_n^0 - \widehat{\varphi}_n )^{\prime} d \lambda.
\end{split}$$ Now the middle term in the previous display is $O_p(n^{-7/5})$. For the last two terms, we apply Lemma \[lem:rem:FFn-Fn-empirical-proc-arg\] taking $I = [\tau, \tau^0]$ to see that $$\sup_{t \in (\tau, \tau^0]} n^{3/5} \left| \int_{(\tau,t]} d {\left(}\Fn
- \FFn {\right)}\right| = O_p(1).$$ Thus, again using Proposition 5.2 of [@Doss-Wellner:2016ModeConstrained], $$\label{eq:rem:lemma-error-integ-by-parts}
\int_{(\tau, \tau^0]} ( { {\mathbb F}}_n - \widehat{F}_n )
( \widehat{\varphi}_n^0 - \widehat{\varphi}_n )^{\prime} d \lambda
= O_p(n^{-4/5}) \int_{(\tau,\tau^0]} d\lambda = O_p(n^{-1}),$$ so we have now shown that is $O_p(n^{-1})$, so the middle term in is $O_p(n^{-1})$. Thus, is $O_p(n^{-1})$, and since bounds we can conclude that $$\label{eq:rem:6}
\int_{X_{(1)}}^{\tau^0} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^1} \ffna d
\lambda =
\int_{X_{(1)}}^{\tau^0} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^{2}} \ffn d
\lambda = O_p(n^{-1}),$$ and so we are done.
Note that if we computed the integrals in over an interval of length $O_p(n^{-1/5})$, by using that the corresponding integrand is $O_p(n^{-4/5})$ (under smoothness/curvature assumptions), the integrals would be $O_p(n^{-1})$. However, shows that the integrals are $O_p(n^{-1})$ over a larger interval whose length is constant or larger, with high probability. Thus we can use to show that $\vvna - \vvn$ must be of order smaller than $
O_p(n^{-2/5})$ somewhere, and this line of reasoning will in fact show that $T_{n,t}^1$ and $T_{n,t}^2$ are $o_p(n^{-1})$ for certain $t$ values.
Having shown , it may seem that we can easily find a subinterval over which the corresponding integrals are $o_p(n^{-1})$ (or smaller), and that this should allow us to quickly finish up our proof. There is an additional difficulty, though, preventing us from naively letting $|t| \to \infty$: we need to control the corresponding integrals actually within small neighborhoods of $m$ (of order $O_p(n^{-1/5})$), not just arbitrarily far away from $m$. This is because our asymptotic results for the limit distribution take place in $n^{-1/5}$ neighborhoods of $m$.
To connect the result about $\tilde{T}_n^i$ to the title of this section (which states $T_{n}^i = O_p(n^{-1})$), note that by Lemma \[lem:rem:epsilon-inequality\], $0 \le T_n^i \le 2 \tilde{T}_n^i =
O_p(n^{-1})$.
### Local and Global $o_p(n^{-1})$ Conclusion
We will now find a subinterval $I$ such that $$\int_{I} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^2} \ffn d
\lambda = o_p(n^{-1}).$$ We will argue by partitioning a larger interval over which the above integral is $O_p(n^{-1})$ into smaller subintervals. Let ${\varepsilon}> 0$. We will later identify a subinterval $J^*$ of length $L n^{-1/5} >0$, which we specify to be the length such that intervlas of length $L n^{-1/5}$ whose endpoints converge to $m$ contain a knot from each of $\vvn$ and $\vvna$ with probability $ 1- {\varepsilon}$. Also let $\delta > 0$ and $\zeta = \delta / L$ which we take without loss of generality to be the reciprocal of an integer. By Proposition 5.1 of [@Doss-Wellner:2016ModeConstrained], fix $M \ge L $ large enough such that with probability $1 - {\varepsilon}\zeta$ for any random variable $\xi_n \to m$, $[\xi_n - M n^{-1/5}, \xi_n + M n^{-1/5}]$ contains knots of both $\vvn$ and of $\vvna$. Now, each of the intervals $$I_{jn} := ( \tau^0 - Mjn^{-1/5}, \tau^0 - M(j-1)n^{-1/5}] \, \mbox{ for
} \, j=1, \ldots, 1/\zeta$$ contains a knot of $\vvn$ and of $\vvna$ by taking $\xi_n$ to be $j n^{-1/5}$. There are $1 / \zeta $ such intervals so the probability that all intervals contain a knot of both $\vvn$ and $\vvna $ is $1 - {\varepsilon}$. Now, let $K= O_p(1)$ be such that $\int_{X_{(1)}}^{\tau^0}
{\left(}\vvna - \vvn
{\right)}^2 e^{\tilde {\varepsilon}_{n}^2} \ffn d \lambda \le K n^{-1}$ for $\tau^0 < m$, by Lemma \[lem:rem:step2:Tni-Opn-1\]. In particular, $$\label{eq:rem:14}
\int_{I_{j^*}} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^2} \ffn d
\lambda :=
\min_{j=1,\ldots,1/\zeta} \int_{I_j} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^2} \ffn d
\lambda \le \zeta K n^{-1}.
$$ We next conclude by Lemma \[lem:rem:local-to-global-square-integral\], since $\zeta = \delta / L$, that there exists a subinterval $J^* \subset I_{j^*}$ containing knots $\eta \in
S(\vvn)$ and $\eta^0 \in S(\vvna)$, such that $$\label{eq:rem:local-op-bound}
\sup_{x \in J^*} |\vvn(x)-\vvna(x)| \le c \delta_1 n^{-2/5}
\quad \mbox{ and } \quad
| \eta^0 - \eta | \le c \delta_1 n^{-1/5}$$ for a universal constant $c>0$ and where $\delta_1 \to 0$ as $\delta \to 0$.
We can now re-apply the proof of Lemma \[lem:rem:step2:Tni-Opn-1\], this time taking as our knots $\eta$ and $\eta^0$, and again assuming without loss of generality $\eta \le \eta^0$. We again see that is bounded above by , and the middle term of is bounded by . Using , we can conclude by that is bounded by $\delta_2
O_p(n^{-1})$, so is also, and so is also, where $\delta_2 \to 0$ as $\delta
\to 0$. We can conclude $$ \int_{X_{(1)}}^{\eta^0} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^2} \ffn d
\lambda
\le \tilde \delta n^{-1}.$$ Now $\eta^0 \ge \tau^0 - M n^{-1/5} / \zeta$, the endpoint of $I_{1/\zeta,n}$. Thus, take $b n^{-1/5} \ge \tau^0 - M n^{-1/5} / \zeta$, let $t_n =
m - bn^{-1/5}$ and now let $J^* = [t_n - \tilde L n^{-1/5}, t_n]$ where $\tilde L = \max( L , 8 D /
\vvo^{(2)}(m))$, chosen so that we can apply Lemma \[lem:rem:local-to-global-square-integral\]. Then $$\label{eq:rem:Tni-op-epsilon-tildes}
\int_{X_{(1)}}^{t_n} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^2} \ffn d
\lambda
\le \tilde \delta n^{-1}$$ with high probability. Analogously, $$\label{eq:rem:Tn1-op-epsilon-tildes}
\int_{X_{(1)}}^{t_n} {\left(}\vvna - \vvn {\right)}^2 e^{\tilde {\varepsilon}_{n}^1} \ffna d
\lambda
\le \tilde \delta n^{-1}$$ as $n \to \infty$ with high probability. And we can apply Lemma \[lem:rem:local-to-global-square-integral\] to the interval $J^*$ to see $$\begin{aligned}
\Vert \vvn - \vvna \Vert_{J^*} = \delta O_p(n^{-2/5}) , &
\quad
\quad
& \Vert \ffn - \ffna
\Vert_{J^*} \le \delta K n^{-2/5},
\label{eq:rem:vv-ff-op-statements} \\
\Vert \FFn - \FFna \Vert_{J^*} \le
\delta K n^{-3/5}, &
\quad
\mbox{ and }
\quad
& |\tau - \tau^0 | \le \delta
K n^{-1/5} , \label{eq:rem:FF-knot-op-statements}\end{aligned}$$ where $\tau \in S(\vvn) \cap J^*$ and $\tau^0 \in S(\vvna) \cap
J^*$, \[page:rem:defn:tau-tau0\] and $$\label{eq:rem:vv-prime-op-statements}
\Vert (\vvn - \vvna)' \Vert_{[\max(\tau,\tau^0) + \delta O_p(n^{-1/5}), t_n
- \delta O_p(n^{-1/5})]} = \delta K n^{-1/5};$$ here, $K = O_p(1)$ and depends on ${\varepsilon}$ and $\tilde L \equiv \tilde
L_{\varepsilon}$, but not on $\delta$ or $t_n$. Thus when we eventually let $\tilde \delta \to 0$, so $b
\equiv b_{\tilde \delta} \to \infty$, we can still conclude $\tilde \delta K \to 0$.
Note we cannot assume that $\vvn$ or $\vvna$ are linear on $[\max(\tau,\tau^0), \sup J^*]$.
We also continue to assume, without loss of generality, that $$\tau \le \tau^0.$$ Thus, here is the sense in which we mean $o_p$, for the remainder of the proof: if we say, e.g., $E_{n,t_n}^1-E_{n,t_n}^2=o_p(n^{-1})$ we mean for any $\tilde
\delta > 0$, we may set $t_n = m-bn^{-1/5}$ and choose $b$ large enough that $| E_{n,t_n}^1-E_{n,t_n}^2| \le \tilde \delta K n^{-1}$ where $K$ does not depend on $t_n$.
We can now conclude that $$\begin{aligned}
\label{eq:rem:tilde-Tni-op}
\tilde T_{n,t_n}^i =
o_p(n^{-1})
\quad \mbox{ for } \quad
i=1,2,\end{aligned}$$ The difference in the definitions of $T_n^i$ (defined in ) and $\tilde T_n^i$ (defined in ), for $i=1,2$, is only in the $e^{{\varepsilon}^i_n}$’s and $e^{\tilde {\varepsilon}_n^i}$’s. These arise from Taylor expansions of the exponential function. The definition of $T_n^i$ arises from the expansions of $R_{n,1,t}^c$ (see and ). Thus, if we let $ e^x = 1 + x + 2^{-1} x^2 e^{{\varepsilon}(x)}$ we can see that ${\varepsilon}_n^1(x) = {\varepsilon}( \vvn(x) - \vvna(x))$ and ${\varepsilon}^2_n(x) =
{\varepsilon}(\vvna(x)-\vvn(x))$. Let $ e^x = 1 + x e^{\tilde {\varepsilon}(x)}.$ Then we can see that $\tilde {\varepsilon}_n^1(x) =\tilde {\varepsilon}( \vvn(x) -
\vvna(x))$ and $\tilde {\varepsilon}^2_n(x) = \tilde {\varepsilon}(\vvna(x)-\vvn(x))$. Now, by Lemma \[lem:rem:epsilon-inequality\], for all $x \in {{\mathbb R}}$, $
e^{{\varepsilon}(x)} \le 2 e^{\tilde {\varepsilon}(x)},$ so that $$\label{eq:rem:Tni-op-switch-epsilons}
0 \le T_{n,t_n}^i \le 2 \tilde T_{n,t_n}^i = o_p(n^{-1}), \mbox{ for } i =1,2,$$ by .
### Return to $R_{n,1,t}^c$
We take $t_n$ and $J^*$ as defined at the end of the previous section. Now, if we could show that $E_{n,t_n}^1 - E_{n,t_n}^2 = o_p(n^{-1})$ then from we could conclude that $A_{n,t_n}^i$, $i=1,2$, are both $o_p(n^{-1})$. If, in addition, we can show $E_{n,t_n}^1 + E_{n,t_n}^2 =
o_p(n^{-1})$, then since $$\label{eq:rem:2}
R_{n,1,t_n}^c = (E_{n,t_n}^1 + E_{n,t_n}^2 + A_{n,t_n}^1 + A_{n,t_n}^2 + T^2_{n,t_n} -
T_{n,t_n}^1 ) / 2$$ by , we could conclude $R_{n,1,t_n}^c =
o_p(n^{-1})$. Unfortunately it is difficult to get any results about $E_{n,t_n}^1-E_{n,t_n}^2$. We can analyze $E_{n,t_n}^1 + E_{n,t_n}^2$, though. The next lemma shows that the difficult terms in $E_{n,t_n}^1 + E_{n,t_n}^2$ are $o_p(n^{-1})$.
\[lem:rem:Fni-op\] Let all terms be as defined above. For any $t < m $ let $F_{n,t}^1 =
\int_{(\tau, t]} \vvnt{t} d(\FFn - \Fn)$ and $F_{n,t}^2 = \int_{(\tau^0,t]}
\vvnat{t} d(\Fn - \FFna)$. Then $$F_{n,t_n}^1 + F_{n,t_n}^2 = o_p(n^{-1}).$$
For the proof, denote $t \equiv t_n$ and recall that we assume $\tau \le
\tau^0$. We see $$\begin{aligned}
\MoveEqLeft \int_{(\tau, t]} \vvnt{t} d(\FFn - \Fn) + \int_{(\tau^0, t]}
\vvnat{t} d(\Fn - \FFna) \\
& = \int_{(\tau^0,t]} \vvnt{t} d(\FFn - \Fn) + \int_{(\tau, \tau^0]}
\vvnt{t} d(\FFn -\Fn) \\
& \quad - {\left(}\int_{(\tau^0,t]} \vvnt{t} d(\FFna - \Fn)
+ \int_{(\tau^0,t]} (\vvnat{t}-\vvnt{t} ) d(\FFna-\Fn) {\right)}\end{aligned}$$ which equals $$\begin{aligned}
\label{eq:int_tau0-t-vvntt}
\int_{(\tau^0, t]} \vvnt{t} (\ffn-\ffna) d\lambda
- \int_{(\tau^0, t]} (\vvnat{t} - \vvnt{t} ) d(\FFn - \Fn)
+ \int_{(\tau, \tau^0]} \vvnt{t} d(\FFn - \Fn).
\end{aligned}$$ Note $\Vert \vvnat{t} \Vert_{J^*} = O_p(n^{-2/5})$. This follows because $n^{1/5}(\tau^0 - t) = O_p(1)$ by Proposition 5.1 of [@Doss-Wellner:2016ModeConstrained], and because $\Vert (\vvna)'
\Vert_{J^*} = n^{-1/5} O_p(1)$ by Corollary 5.4 of [@Doss-Wellner:2016ModeConstrained], since $\vvo'(m)=0$. In both cases the $O_p(1)$ does not depend on $t_n$. Thus, the first term in is $o_p(n^{-1})$, since $ \Vert \ffn - \ffna
\Vert_{J^*} = o_p(n^{-2/5})$. We will rewrite the other two terms of with integration by parts. The negative of the middle term, $ \int_{(\tau^0,t]} (\vvnat{t} - \vvnt{t}) d(\FFn - \Fn)$, equals $$\label{eq:ffn-fnvvn-vvnttt}
((\FFn-\Fn)(\vvnat{t}-\vvnt{t}))(\tau^0,t]
- \int_{(\tau^0,t]} (\FFn - \Fn ) ((\vvna)' - \vvn')
d\lambda.$$ Note $\Vert \FFn - \Fn \Vert_{J^*} = O_p(n^{-3/5})$ by Lemma \[lem:rem:FFn-Fn-empirical-proc-arg\]. Thus the first term in is $o_p(n^{-1})$ because $\Vert \vvna - \vvn
\Vert_{J^*} = o_p(n^{-2/5})$. The second term in is $o_p(n^{-1})$ because implies that $\int_{[\tau^0,t]} |
(\vvn - \vvn)'| d\lambda = o_p(n^{-2/5})$, and, as already noted, $\Vert
\FFn - \Fn \Vert_{J^*} = O_p(n^{-3/5})$. Thus is $o_p(n^{-1})$.
We have left the final term of . This can be bounded by $$\begin{aligned}
\label{eq:lv-ffn-fn}
{\left\vert}(\FFn - \Fn)( \vvnt{t}) (\tau, \tau^0]
- \int_{(\tau, \tau^0]} (\FFn - \Fn) \vvn' d\lambda {\right\vert},
\end{aligned}$$ and the second term above bounded by $\Vert \FFn - \Fn \Vert_{[\tau,
\tau^0]} \vvn'(\tau+) (\tau^0-\tau) = o_p(n^{-1})$ because $\Vert \FFn -
\Fn \Vert_{J^*} = O_p(n^{-3/5})$, $\vvn'(\tau+) = O_p(n^{-1/5})$ (since $\vvo'(m)=0$, and recall $\vvn$ is linear on $[\tau, \tau^0]$), and $(\tau^0-\tau) = o_p(n^{-1/5})$. The first term of is $o_p(n^{-1})$ because in fact $\Vert \FFn - \Fn \Vert_{[\tau, \tau^0]} =
o_p(n^{-3/5})$, by Lemma \[lem:rem:FFn-Fn-empirical-proc-arg\] (since $|\tau-\tau^0| = o_p(n^{-1/5})$), and $\Vert \vvnt{t} \Vert_{[\tau,
\tau^0]} = O_p(n^{-2/5})$. Thus is $o_p(n^{-1})$ so we are done.
For $t < m$, define $$\label{eq:rem:defn:Gnt1}
G_{n,t}^1 =
\vvn(t) ( \FFn(t) - \FFna(t) )
+ (\vvn(\tau) - \vvn(t)) ( \FFn(\tau) - \Fn(\tau) )$$ and $$\label{eq:rem:defn:Gnt2}
G_{n,t}^2 =
\vvna(t) ( \FFn(t) - \FFna(t))
+ (\vvna(\tau^0)
- \vvna(t)) (\Fn(\tau^0) - \FFna(\tau^0)),$$ so that $G_{n,t_n}^i = E_{n,t_n}^i -F_{n,t_n}^i$ for $i=1,2$ (recalling the definitions of $E_{n,t_n}^i$ in and ). The key idea now is that the first term in $G_{n,t}^i$ matches up with $R_{n,1,\twoArgs}$. To make this explicit, we need to define a one-sided version of $R_{n,1,\twoArgs}$. Since both $\ffn$ and $\ffna$ integrate to $1$, note for any $t_1 \le m \le t_2$, that $$\label{eq:rem:Rn1-integral-identity}
R_{n,1,\twoArgs} = - \vvo(m) \int_{D_{n,\twoArgs}^c} ( \ffn - \ffna) d\lambda;$$ thus, define $$\label{eq:rem:defn:Rn1t-one-sided}
R_{n,1,t_1} = - \vvo(m) (\FFn(t_1) - \FFna(t_1)).$$ The corresponding definition for the right side is $-\vvo(m)
\int_{t_2}^{X_{(n)}} (\ffn - \ffna ) d\lambda$, which when summed with yields $R_{n,1,\twoArgs}$.
\[lem:rem:Gni-Rn1-op\] Let all terms be as defined above. We then have for $i=1,2$, $$G_{n,t_n}^i + R_{n,1,t_n} = o_p(n^{-1}).$$
The second terms in the definitions of $G_{n,t_n}^i$, $i=1,2$, are both $O_p(n^{-7/5})$ since $ \vvn(\tau) - \vvn(t)$ and $\vvna(\tau^0) -
\vvna(t)$ are both $O_p(n^{-2/5})$ by Lemma 4.5 of [@BRW2007LCasymp] and Corollary 5.4 of [@Doss-Wellner:2016ModeConstrained], and the terms $\FFn(\tau) -\Fn(\tau)$ and $\FFna(\tau^0) - \Fn(\tau^0)$ are both $O_p(n^{-1})$ by Corollary 2.4 and Corollary 2.12 of [@Doss-Wellner:2016ModeConstrained].
Thus we consider the first terms of $G_{n,t_n}^i$, in sum with $R_{n,1,t_n}$. Consider the case $i=1$; we see that $$\label{eq:rem:vvnt_n-ffnt_n-ffnat}
\vvn(t_n) (\FFn(t_n) - \FFna(t_n)) + R_{n,1,t_n}
= (\vvn(t_n) - \vvo(m)) (\FFn(t_n) - \FFna(t_n)),$$ and $(\vvn(t_n) - \vvo(m)) = O_p(n^{-2/5})$ by Lemma 4.5 of [@BRW2007LCasymp] since $\vvo'(m)=0$. Crucially, we are not making a claim that $\vvn$ is close to $\vvo(m)$ uniformly over an interval, just a claim at the point $t_n$ satisfying $t_n \to m$, so the $O_p$ statement does not depend on $b$. Since $\FFn(t_n) - \FFna(t_n) =
o_p(n^{-3/5})$ by , we conclude that is $o_p(n^{-1})$. Identical reasoning applies to the case $i=2$, using Corollary 5.4 of [@Doss-Wellner:2016ModeConstrained]. Thus we are done.
Thus by Lemmas \[lem:rem:Fni-op\] and \[lem:rem:Gni-Rn1-op\], $$\label{eq:rem:E1-+-E2-op}
R_{n,1,t_n} + (E^1_{n,t_n} + E^2_{n,t_n})/2 = o_p(n^{-1}).$$ Now we decompose the $A^i_{n,t_n}$ terms. Let $$\begin{aligned}
& B_{n,t_n}^1 = \int \vvnt{\tau} d(\Fn - \FFna), \\
& B_{n,t_n}^2 = \int \vvnat{\tau^0} d(\FFn - \Fn).\end{aligned}$$ where $\tau$ and $\tau^0$ are as previously defined (on page ) and $$\begin{aligned}
& C_{n,t_n}^1 = \int \vvn'(t_n-) (x-t_n)_- \, d(\Fn - \FFna) \\
& C_{n,t_n}^2 = \int (\vvna)'(t_n-) (x-t_n)_- \, d(\FFn - \Fn).\end{aligned}$$ Then, for $i=1,2$, by the definitions of $\tau$ and $\tau^0$, $$\begin{aligned}
A_n^i = B_n^i + C_n^i,\end{aligned}$$ and note that $$\label{eq:rem:1}
B_{n,t_n}^1 = \int (\vvnt{\tau} - \vvnat{\tau^0}) d(\Fn - \FFna)
\quad \mbox{ and } \quad
B_{n,t_n}^2 = \int (\vvnat{\tau^0} - \vvnt{\tau} ) d(\FFn - \Fn),$$ by the characterization theorems, Theorem 2.2 of [@Doss-Wellner:2016ModeConstrained] with $\Delta = \pm \vvnat{\tau^0}$ and Theorem 2.8 of [@Doss-Wellner:2016ModeConstrained] with $\Delta = \pm
\vvnt{\tau}$. Perhaps strangely, it seems it is easier to analyze $B_n^1-
B_n^2$ than $B_n^1+B_n^2$, and $C_n^1 + C_n^2$ rather than $C_n^1 - C_n^2$. Perhaps more strangely, this will suffice. Again by Theorem 2.2 of [@Doss-Wellner:2016ModeConstrained] with $\Delta = \vvnt{\tau}$ and Theorem 2.8 of [@Doss-Wellner:2016ModeConstrained] with $\Delta =
\vvnat{\tau^0}$, $B_{n,t_n}^1 \le 0$ and $ B_{n,t_n}^2 \ge 0 ,$ so $$\label{eq:b_n-t_n1-b_n}
B_{n,t_n}^1 - B_{n,t_n}^2 \le
\begin{cases}
-B_{n,t_n}^2 \le 0 \\
B_{n,t_n}^1 \le 0.
\end{cases}$$ Thus, if we can show $B_{n,t_n}^1 - B_{n,t_n}^2 = o_p(n^{-1})$ then $B_{n,t_n}^i=o_p(n^{-1})$, $i=1,2,$ so $B_{n,t_n}^1+B_{n,t_n}^2 = o_p(n^{-1})$. We do this in the following lemma.
\[lem:rem:Bni-op\] With all terms as defined above, $$B_{n,t_n}^1 - B_{n,t_n}^2 = o_p(n^{-1}),$$ and thus $$B_{n,t_n}^1+B_{n,t_n}^2 = o_p(n^{-1}).$$
Now by $$\begin{split}
B_{n,t_n}^1-B_{n,t_n}^2 & = \int (\vvnt{\tau} - \vvnat{\tau} ) d(\FFn - \FFna)
+ \int (\vvnat{\tau} - \vvnat{\tau^0}) d( \FFn - \FFna)
\end{split}$$ which equals $$\label{eq:b_n1-b_n2-T-type-term}
\int (\vvnt{\tau} - \vvnat{\tau}) d(\FFn - \FFna)
+ (\vvna(\tau^0)-\vvna(\tau)) (\FFn - \FFna)(\tau)
+ \int_\tau^{\tau^0} \vvnat{\tau^0} d(\FFn - \FFna).$$ The first term in equals, applying , $$\label{eq:int_x_1tau-vvn-vvna2}
\int_{X_{(1)}}^{\tau} (\vvn - \vvna)^2
e^{\tilde{{\varepsilon}}_n^1} \ffna d\lambda -
((\vvn-\vvna)(\FFn-\FFna))(\tau),$$ where $\tilde{{\varepsilon}}_n^1$ is identical to $\tilde{{\varepsilon}}_n^1$ in Lemma \[lem:rem:step2:Tni-Opn-1\], and thus by the first term in is $o_p(n^{-1})$. The second term is also $o_p(n^{-1})$ since $\Vert \vvn - \vvna \Vert_{J^*} = o_p(n^{-2/5})$ and $\Vert \FFn - \FFna \Vert_{J^*} = o_p(n^{-3/5})$. This also shows that the middle terms in is $o_p(n^{-1})$. To see the last term is $o_p(n^{-1})$, recall $\Vert \vvnat{\tau^0} \Vert_{J^*} =
O_p(n^{-2/5})$ by Corollary 5.4 of [@Doss-Wellner:2016ModeConstrained], using that $\vvo'(m)=0$. Since $|\tau^0 - \tau| = o_p(n^{-1/5})$ and $\Vert
\ffn - \ffna \Vert_{J^*} = o_p(n^{-2/5})$ we see the last term of is $o_p(n^{-1})$, so is $o_p(n^{-1})$, so $B^1_{n,t_n} -
B^2_{n,t_n} = o_p(n^{-1})$. By , $B^1_{n,t_n} +
B^2_{n,t_n} = o_p(n^{-1})$, so we are done.
We now turn our attention to $ C_{n,t_n}^1 + C_{n,t_n}^2$.
With all terms as defined above, $$C^1_{n,t_n} + C^2_{n,t_n} = o_p(n^{-1}).$$
Note that $ C_{n,t_n}^1 + C_{n,t_n}^2$ equals $$\begin{aligned}
& \int (\vvn'(t_n-) - (\vvna)'(t_n-)) (x-t_n)_- \, d(\Fn - \FFna) \\
&\quad + \int (\vvna)'(t_n-) (x-t_n)_- \, d(\Fn - \FFna)
+ \int (\vvna)'(t_n-) ( x-t_n)_- \, d(\FFn - \Fn)
\end{aligned}$$ which equals $$\label{eq:rem:Cn1-plus-Cn2-first-identity}
\int (\vvn'(t_n-) - (\vvna)'(t_n-)) (x-t_n)_- \, d(\Fn - \FFna)
+ \int (\vvna)'(t_n-) (x-t_n)_- \, d(\FFn - \FFna).$$ Since $(\FFn-\FFna)(X_{(1)}) = 0$, the second term in equals $$\label{eq:rem:Cn1-plus-Cn2-second-term-1}
-(\vvna)'(t_n-) \int_{-\infty}^{t_n} (\FFn - \FFna) d\lambda
= -(\vvna)'(t_n-) \int_{\upsilon}^{t_n} (\FFn - \FFna) d\lambda$$ for a point $\upsilon \in [\tau, \tau^0]$ which exists by the proof of Proposition 2.13 of [@Doss-Wellner:2016ModeConstrained]. By , since $t_n - \upsilon = O_p(n^{-1/5})$, $$\label{eq:rem:int_-ffn-ffna}
\int_{\upsilon}^{t_n} (\FFn - \FFna) d\lambda = o_p(n^{-4/5}).$$ Since $t_n \to m$, by Corollary 5.4 of [@Doss-Wellner:2016ModeConstrained], $(\vvna)'(t_n-) = O_p(1) n^{-1/5}.$ As in previous cases, by taking $t_n = \xi_n$ and $C = 0$ in that corollary, the $O_p(1)$ does not depend on $t_n$. Thus we have shown is $o_p(n^{-1})$.
Since $\Fn(-\infty)-\FFna(-\infty)=0$, the first term in equals $$- (\vvn - \vvna)'(t_n-) \int_{X_{(1)}}^{t_n} (\Fn - \FFna) d\lambda,$$ which equals $$\label{eq:rem:-vvn-vvnat}
-(\vvn - \vvna)'(t_n-) \int_{\tau^0}^{t_n} (\Fn - \FFna) d\lambda$$ because $\widehat{H}_{n,L}^0(\tau^0) = \mathbb{Y}_{n,L}(\tau^0)$ by Theorem 2.10 of [@Doss-Wellner:2016ModeConstrained]. The absolute value of is bounded above by $$\label{eq:rem:Cn1-plus-Cn2-first-term-bound}
|\vvn'(t_n-) - (\vvna)'(t_n-)| (t_n - \tau^0) \sup_{u \in [\tau^0, t_n]} |\Fn(u) - \FFna(u)|.$$ We know $ |\vvn'(t_n-) - (\vvna)'(t_n-)| = O_p(n^{-1/5})$ but unfortunately it is not necessarily $o_p(n^{-1/5})$. However, $$|\vvn'(t_n-) - (\vvna)'(t_n-)| (t_n - \tau^0)
= | (\vvn-\vvna)(t_n) - (\vvn-\vvna)(\tau^0)|
= o_p(n^{-2/5}),$$ so is $o_p(n^{-1})$, and thus so also is ; that is, $C_{n,t_n}^1 + C_{n,t_n}^2 = o_p(n^{-1})$.
Thus we have shown that $C^1_{n,t_n} + C^2_{n,t_n} = o_p(n^{-1})$ and $B^1_{n,t_n} + B^2_{n,t_n} = o_p(n^{-1})$, so we can conclude $$A^1_{n,t_n} +
A^2_{n,t_n} = o_p(n^{-1}).$$ Together with , , and , we can conclude that $$\label{eq:rem:Rn1c-op}
R_{n,t_n} + R_{n,1,t_n}^c = o_p(n^{-1}).$$
Proof completion / details: the main result {#subsubsec:final-arguments}
-------------------------------------------
The preceding one-sided arguments apply symmetrically to the error terms on the right side of $m$. Thus, we now return to handling simultaneously the two-sided error terms. We have thus shown for any $\delta > 0$ we can find a $b \equiv b_\delta$, such that, letting $t_{n,1} = m - b n^{-1/5}$, $t_{n,2} = m + b n^{-1/5}$, we have $$\label{eq:rem:16}
|R_{n,1,t_{n,1}, t_{n,2}} + R_{n,1,t_{n,1}, t_{n,2}}^c | \le \delta K n^{-1}$$ where $K = O_p(1)$ does not depend on $b$ (i.e., on $\delta$). Now by Proposition \[prop:localerrorterms\] $$ |R_{n,2,t_{n,1},t_{n,2}}| + |R_{n,2,t_{n,1},t_{n,2}}^0|
+ |R_{n,3,t_{n,1},t_{n,2}}| + |R_{n,3,t_{n,1},t_{n,2}}^0| = o_p(n^{-1}).$$ Let $$\begin{aligned}
R_{n,t_{n,1},t_{n,2}} & \equiv
2 n(R_{n,1,t_{n,1},t_{n,2}} + R_{n,1,t_{n,1},t_{n,2}}^c +
R_{n,2,t_{n,1},t_{n,2}} \\
& \qquad - R_{n,2,t_{n,1},t_{n,2}}^0 + R_{n,3,t_{n,1},t_{n,2}} -R_{n,3,t_{n,1},t_{n,2}}^0).\end{aligned}$$ Then by , write $2 \log \lambda_n =
\mathbb{D}_{n,\twoArgs[t_{n,1},t_{n,2}]} + R_{n,t_{n,1},t_{n,2}}$ (slightly modifying the form of the subscripts). Now fixing any subsequence of ${\left\{} \newcommand{\rb}{\right\}}n
\rb_{n=1}^\infty$, we can find a subsubsequence such that $R_{n,t_{n,1},t_{n,2}} \to_d \delta R$ for a tight random variable $R$ by . For $b > 0$ let $ \mathbb{D}_b \equiv \int_{-b/ \gamma_2}^{b/\gamma_2}
(\widehat{\varphi}^2(u) -(\widehat{\varphi}^0)^2(u))du$, as in , which lets us conclude that $$\label{eq:rem:17}
2 \log \lambda_n = \mathbb{D}_{n, t_{n,1},t_{n,2}} + R_{n,t_{n,1},t_{n,2}}
\to_d \mathbb{D}_{b} + \delta R$$ along the subsubsequence. Taking say $\delta = 1$ shows that there exists a (tight) limit random variable, which we denote by ${{\mathbb D}}$. Then, since $R$ does not depend on $\delta$, we can let $\delta \searrow 0$ so $b_\delta \equiv b
\nearrow \infty$, and see that $\lim_{b \to \infty} {{\mathbb D}}_b = {{\mathbb D}}$, which can now be seen to be pivotal. Thus along this subsubsequence, $ 2 \log
\lambda_n \to_d {{\mathbb D}}.$ This was true for an arbitrary subsubsequence, and so the convergence in distribution holds along the original sequence. Thus, $$2 \log \lambda_n \to_d \mathbb{D}
\quad \mbox{ as } \quad
n \to \infty.$$
Proof of Theorem \[LR-limit-FixedAlternative\] {#subsec:thm-alternative}
----------------------------------------------
In this subsection we complete the proof of Theorem \[LR-limit-FixedAlternative\], which was sketched in Subsection \[ssec:PfSketchThm2\]. It remains to justify the definition given in , and to show that the third term of is $o_p(1)$, under the assumptions of Theorem \[LR-limit-FixedAlternative\]. We first state three theorems. These are mode-constrained analogues of Theorems 2.2, 2.7, and 2.15 of [@DSS2011LCreg], and are proved with methods similar to the methods used in [@DSS2011LCreg]. The full proofs will be given in a separate work on estimation and inference for modal regression functions. Now, much as in [@DSS2011LCreg], we set $$\begin{aligned}
L(\varphi , Q) \equiv \int \varphi dQ - \int e^{\varphi} d\lambda +1\end{aligned}$$ and define $$\begin{aligned}
L_m (Q) \equiv \sup_{\varphi \in {\cal C}_m} L( \varphi , Q)\end{aligned}$$ where $$\begin{aligned}
{\cal C}_m
\equiv \{ \varphi : \ m \in MI (\varphi), \varphi \mbox{ concave} \}\end{aligned}$$ and recall $MI (\varphi ) \equiv \{ x \in {{\mathbb R}}: \ \varphi (x) = \sup_{y \in {{\mathbb R}}} \varphi (y) \} .$ If for fixed $Q$ there exists $\psi_m \in {\cal C}_m$ such that $$L(\psi_m , Q ) = L_m (Q) \in {{\mathbb R}},$$ then $\psi_m$ will automatically satisfy $\int \exp ( \psi_m (x) ) dx = 1$: note that $\phi+c \in {\cal C}_m$ for any fixed $\phi \in {\cal C}_m$ and $c \in {{\mathbb R}}$. On the other hand, $$\begin{aligned}
\frac{\partial}{\partial c} L({\cal C} +c , Q)
& = & \frac{\partial}{\partial c} \left \{ \int ( \phi +c) dQ - e^c \int e^{\phi} d\lambda +1 \right \} \\
& = & 1 - e^c \int e^{\phi} d \lambda \end{aligned}$$ if $L ( \phi , Q) \in {{\mathbb R}}$. Thus $L(\phi+c, Q)$ is maximal for $c = - \log \int e^{\phi} d\lambda $.\
For the next theorem we need to define $$\mbox{csupp}_m (Q) = \bigcap {\left\{} \newcommand{\rb}{\right\}}C \subseteq {{\mathbb R}}^d : C \mbox{ closed, convex }, Q(C) = 1 , m \in C\rb.$$
\[DSS-thm2.2m\] Let $Q$ be a measure on ${{\mathbb R}}^d$. The value of $L_m (Q)$ is real if and only if $\int |x | dQ (x) < \infty$ and $\mbox{int}( \mbox{csupp}_m (Q) ) \neq \emptyset$. In that case there exists a unique $\psi_m \equiv \psi_m (\cdot | Q) \in \mbox{argmax}_{\varphi \in {\cal C}_m} L( \varphi , Q)$. This function $\psi_m$ satisfies $\int e^{\psi_m} d \lambda =1$ and $$\mbox{int}( \mbox{csupp}_m (Q) )
\subseteq \mbox{dom} (\psi_m) \subseteq \mbox{csupp}_m (Q)$$ where $\mbox{dom} (\psi_m) \equiv \{ x \in {{\mathbb R}}^d : \ \psi_m (x) > -\infty \} $.
Theorem \[DSS-thm2.2m\] justifies rigorously the definition given in , since $f_0 \in {\cal P}$ has a finite mean and is non-degenerate. Now, for $\psi_m \in {\cal C}_m $, let $$\begin{aligned}
{\cal S}(\psi_m) =
\{ x \in \mbox{dom} ( \psi_m ) : \ \psi_m (x) > 2^{-1} \left ( \psi_m (x- \delta) + \psi_m (x+ \delta) \right ) \ \ \mbox{for all} \ \delta > 0 \} ,\end{aligned}$$ and $$\begin{aligned}
{\cal S}_L(\psi_m) &= {\left\{} \newcommand{\rb}{\right\}}x \in {\cal S}(\psi_m) : \psi_m'(x-) > 0 \rb \subseteq (-\infty, m], \\
{\cal S}_R(\psi_m) &= {\left\{} \newcommand{\rb}{\right\}}x \in {\cal S}(\psi_m) : \psi_m'(x+) < 0 \rb \subseteq [m, \infty).\end{aligned}$$ It is possible for $m$ to be an element of either of the sets ${\cal S}_L(\psi_m)$ and ${\cal S}_R(\psi_m)$ without being a member of the other. The following theorem is the population analogue of Theorem 2.10 of [@Doss-Wellner:2016ModeConstrained].
\[DSS-thm2.7m\] Let $Q$ be a non-degenerate distribution on ${{\mathbb R}}$ with finite first moment and distribution function $G$. Let $F_m$ be a distribution function with log-density $\varphi_m \in {\cal C}_m$. Then $\varphi_m = \psi( \cdot | Q)$ if and only if $$\begin{aligned}
&& \int_{- \infty}^x F_m (y) dy \le \int_{-\infty}^x G (y) dy \ \ \ \mbox{for all} \ \ x \le m, \ \ \mbox{and} \label{eq:pop-charzn-L} \\
&& \int_{x}^\infty (1-F_m (y)) dy \le \int_x^{\infty} (1-G (y)) dy \ \ \ \mbox{for all} \ \ x \ge m , \label{eq:pop-charzn-R} \end{aligned}$$ with equality in if $x \in {\cal S}_L(\varphi_m)$ and equality in if $x \in {\cal S}_R(\varphi_m)$.
Thus, again much as in [@DSS2011LCreg], for $x \in {\cal S} ( \psi_m (\cdot | Q)) $, $x \le m$, and (small) $\delta>0$, $$\begin{aligned}
&& 0 \le \frac{1}{\delta} \int_{x-\delta}^x (F_m (y) - G(y)) dy \rightarrow F_m (x) - G(x-) \ \ \mbox{as} \ \delta\searrow 0, \\
&& 0 \ge \frac{1}{\delta} \int_x^{x+\delta} F_m (y) - G(y)) dy \rightarrow F_m (x) - G(x) \ \ \mbox{as} \ \ \delta\searrow 0,\end{aligned}$$ and hence $G(x-) \le F_m (x) \le G(x)$ for all $x \in {{\mathbb R}}$.
Now we need to understand the properties of the maps $Q \mapsto L_m (Q)$ and $Q \mapsto \psi_m( \cdot | Q)$ on ${\cal Q}_1 \cap {\cal Q}_{0,m}$, where we let ${\cal Q}_1 = {\left\{} \newcommand{\rb}{\right\}}Q : \int | x | dQ < \infty \rb$ and ${\cal Q}_{0,m} = {\left\{} \newcommand{\rb}{\right\}}Q : \mbox{int} (\mbox{csupp}_m (Q)) \ne \emptyset \rb$. As in [@DSS2011LCreg] we show that these are both continuous with respect to Mallows distance $D_1$: $$D_1 (Q, Q') \equiv \inf_{(X,X')} E | X - X' |$$ where the infimum is taken over all pairs $(X,X')$ of random variables $X \sim Q$ and $X' \sim Q'$ on a common probability space. Convergence of $Q_n$ to $Q$ in Mallows distance is equivalent to having $\int |x| dQ_n \to \int |x| dQ$ and $Q_n \Rightarrow Q$ [@Mallows:1972vy].
\[DSS-thm2.15m\] Let $\{ Q_n \}$ be a sequence of distributions on ${{\mathbb R}}^d$ such that $D_1 (Q_n , Q) \rightarrow 0$ for some $Q \in {\cal Q}_1$. Then $$L_m (Q_n )\rightarrow L_m(Q).$$ If $Q \in {\cal Q}_{0,m} \cap {\cal Q}_1$ the probability densities $f^0 \equiv \exp ( \psi_m (\cdot | Q))$ and $f_n^0 \equiv \exp ( \psi_m (\cdot | Q_n))$ are well-defined for large $n$ and satisfy $$\begin{aligned}
\lim_{n \rightarrow \infty, x \rightarrow y} f_n^0 (x) & = & f^0 (y) \ \ \mbox{for all} \ \ y \in {{\mathbb R}}^d \setminus \partial \{ f^0 > 0 \}, \\
\limsup_{n\rightarrow \infty , x \rightarrow y} f_n^0 (x) & \le &f^0(y) \ \ \mbox{for all} \ \ y \in \partial \{ f^0 > 0 \} , \\
\int | f_n^0 (x) - f^0 (x) | dx & \rightarrow & 0 .\end{aligned}$$
We can now show that the third term of is $o_p(1)$ under the assumptions of Theorem \[LR-limit-FixedAlternative\]. First note that ${{\mathbb P}}_n $ converges weakly to $P$, the measure corresponding to $f_0 \in {\cal P}$, with probability $1$ and ${{\mathbb P}}_n (|x|) = n^{-1} \sum_{i=1}^n |X_i | \rightarrow_{a.s.} \int |x| dP(x)$ by the strong law of large numbers. Thus $D_1 ({{\mathbb P}}_n ,P) \rightarrow_{a.s.} 0$. It follows from Theorem \[DSS-thm2.15m\] that $$\begin{aligned}
\log \widehat{f}_n^0
& = & \mbox{argmax} _{\varphi \in {\cal C}_m} \left \{ {{\mathbb P}}_n \varphi - \int e^{\varphi} d \lambda +1 \right \} \\
& = & \psi_m (\cdot | {{\mathbb P}}_n ) = \widehat{\varphi}_n^0 \end{aligned}$$ where, by the last part of Theorem \[DSS-thm2.15m\], $\int | \widehat{f}_n^0 - f_m^0 | d \lambda \rightarrow_{a.s.} 0$ and by the continuity $$\begin{aligned}
L_m ({{\mathbb P}}_n)
& = & L( \psi_m ( \cdot | {{\mathbb P}}_n ) , {{\mathbb P}}_n ) = {{\mathbb P}}_n \log \widehat{f}_n^0 \nonumber \\
& \rightarrow_{a.s.} & L ( \psi_m ( \cdot | P), P) = P( \log f_m^0 ). \label{CruxContinuity}\end{aligned}$$ But then $$\begin{aligned}
{{\mathbb P}}_n \log \frac{\widehat{f}_n^0 }{f_m^0 }
& = & {{\mathbb P}}_n \log \widehat{f}_n^0 - P \log f_m^0 - ({{\mathbb P}}_n \log f_m^0 - P\log f_m^0 ) \\
& \rightarrow_{a.s.} & 0 - 0 = 0 \end{aligned}$$ by (\[CruxContinuity\]) for the first term and the strong law of large numbers for the second term (using that $-\infty < L_m(P) < \infty$ by Theorem \[DSS-thm2.2m\]).
Lemmas
======
Below are some necessary lemmas.
\[lem:f-to-varphi-taylor\] Let $f_{in} = e^{\varphi_{in}}$ for $i=1,2$, and let $x$ be such that $|f_{1n}(x) - f_{2n}(x)| \to 0 $ as $n \to \infty$. Then $$\begin{aligned}
f_{1n}(x)-f_{2n}(x)
& = {\left(}\varphi_{1n}(x) - \varphi_{2n}(x) {\right)}e^{\tilde {\varepsilon}_{n}(x)} e^{\varphi_{2n}(x)} \label{eq:f-to-varphi-taylor-1} \\
& = {\left(}\varphi_{1n}(x) - \varphi_{2n}(x) + \frac{{\left(}\varphi_{1n}(x) -
\varphi_{2n}(x) {\right)}^2 }{2} e^{{\varepsilon}_n(x)} {\right)}e^{\varphi_{2n}(x)}
\label{eq:f-to-varphi-taylor-2}
\end{aligned}$$ where $\tilde {\varepsilon}_n(x)$ and ${\varepsilon}_n(x)$ both lie between $0$ and $\varphi_{1n}(x) - \varphi_{2n}(x)$, and thus converge to $0$ as $n \to
\infty$.
Taylor expansion shows $$\begin{aligned}
f_{1n}(x)-f_{2n}(x) = e^{\varphi_{1n}(x)} - e^{\varphi_{2n}(x)}
& = {\left(}e^{\varphi_{1n}(x) -\varphi_{2n}(x)} - 1 {\right)}e^{\varphi_{2n}(x)}\\
& = {\left(}\varphi_{1n}(x) - \varphi_{2n}(x) {\right)}e^{{\varepsilon}_n(x)} e^{\varphi_{2n}(x)},
\end{aligned}$$ yielding . The second expression, , follows from a similar (two-term) expansion.
\[lem:rem:local-to-global-square-integral\] Assume $\vvo$ is twice continuously differentiable in a neighborhood of $m$ and $\vvo''(m) < 0$. Let $I$ be a random interval whose endpoints are in an $O_p(n^{-1/5})$ neighborhood of $m$. Let $D$ be such that for any $\xi_n \to m$, $|(\vvna)'(\xi_n)- \vvo'(\xi_n)| \le D n^{-1/5}$ with probability $1- {\varepsilon}$ for large $n$ by Corollary 5.4 of [@Doss-Wellner:2016ModeConstrained]. for ${\varepsilon}> 0$. Assume $n^{1/5} \lambda(I) \ge 8 D / \vvo^{(2)}(m)$. Let $L>0$, ${\varepsilon}> 0$, and $\check \delta > 0$. Suppose there exists $\check K>0$ such that $$\label{eq:knot-lemma:integral-ineq}
\int_{I} (\vvna - \vvn)^2 d\lambda \le \frac{\check \delta}{L} \check K n^{-1}$$ with probability $1- {\varepsilon}$ for $n$ large. Then for any interval $J
\subset I$ where $\lambda(J) = Ln^{-1/5}$, we have with probability $1 -
{\varepsilon}$ for $n$ large
(A) \[item:lem:integral-to-sup:vv\] $\Vert \vvn - \vvna \Vert_{J} \le \delta O_p(n^{-2/5})$,
(B) \[item:lem:integral-to-sup:ff\] $\Vert \ffn - \ffna \Vert_J \le \delta O_p(n^{-2/5})$,
(C) \[item:lem:integral-to-sup:FF\] $\Vert \FFn - \FFna \Vert_J \le \delta O_p(n^{-3/5})$, and
(D) \[item:lem:integral-to-sup:knots\] there exist knots $\eta \in
S(\vvn) \cap J$ and $\eta^0 \in S(\vvna) \cap J$ such that $| \eta - \eta^0 | \le \delta O_p(n^{-1/5})$,
(E) \[item:lem:integral-to-sup:vv-prime\] and, letting $K = [\max(\eta, \eta^0) + \delta O_p(n^{-1/5}), \sup J -
\delta O_p(n^{-1/5})]$, we have $\Vert \vvn' - (\vvna)'
\Vert_K \le \delta O_p(n^{-1/5})$,
where $ \delta \to 0$ as $\check \delta \to 0$.
First we prove the first three statements. By Taylor expansion, $\vvo'(x) = \vvo^{(2)}(\xi) (x - m)$ where $\xi$ is between $x$ and $m$. Let $J = [j_1, j_2] \subseteq I$. Then $\vvo'(j_2) -
\vvo'(j_1) \ge |\vvo^{(2)}(m)| L n^{-1/5} 2$ for $n$ large enough since $\vvo^{(2)}$ is continuous near $m$. Note with probability $1 - {\varepsilon}$, $|(\vvna)'(j_1) - \vvo'(j_1)|$ and $|(\vvna)'(j_2) - \vvo'(j_2) | $ are less than $n^{-1/5} D$ by applying Corollary 5.4 of [@Doss-Wellner:2016ModeConstrained] twice taking $\xi_n = j_1$ and $\xi_n = j_2$, and $C=0$ (not $C = \lambda(J)$). Here $(\vvna)'$ may be the right or left derivative. Now apply Lemma \[lem:rem:L2-to-sup-1\] (taking $I$ in that lemma to be our $J$) with $\varphi_U' - \varphi_L' = n^{-1/5}{\left(}2D + |\vvo^{(2)}(m)| L /2 {\right)}$ and ${\varepsilon}= \check \delta \check K n^{-1} / L$. Then for small enough $\check \delta$, $${\left(}\frac{ \check \delta \check K}{L (2D + |\vvo^{(2)}(m)|L / 2)^2}
{\right)}^{1/3} n^{-1/5}
\le L n^{-1/5} = \lambda(J),$$ as needed. Thus, Lemma \[lem:rem:L2-to-sup-1\] allows us to conclude $$\Vert \vvna - \vvn \Vert
\le {\left(}8 \frac{ n^{-6/5} \check \delta \check K }{L}
{\left(}2D + \vvo^{(2)}(m) L / 2 {\right)}{\right)}^{1/3}.$$ Thus taking $\check \delta $ so that $\check \delta \check K D \to 0$, we see that Lemma \[lem:rem:local-to-global-square-integral\] (\[item:lem:integral-to-sup:vv\]) holds. Then (\[item:lem:integral-to-sup:ff\]) follows by the delta method (or Taylor expansion of $\exp$). Note that $\check K$ and $D$ depend only on ${\varepsilon}$, not on $I$.
We show (\[item:lem:integral-to-sup:FF\]) and (\[item:lem:integral-to-sup:knots\]) next. Note that (\[item:lem:integral-to-sup:FF\]) follows from (\[item:lem:integral-to-sup:knots\]) and (\[item:lem:integral-to-sup:ff\]). This is because $$\FFn(x) - \FFna(x) = \FFn(\eta) - \FFna(\eta) + \int_{\eta}^x
(\ffn(x)-\ffna(x)) dx.$$ By (\[item:lem:integral-to-sup:ff\]), the second term above is $\delta
O_p(n^{-3/5})$ since $x \in J$ satisfies $|x - \eta| \le L n^{-1/5}$. We can next see that the first term in the previous display is $\delta^{1/2}
O_p(n^{-3/5})$. Notice that $\sup_{t \in [\eta,\eta^0]} |\int_\eta^t
d(\FFn(u) - \Fn(u))| = \delta^{1/2} O_p(n^{-3/5})$ by Lemma \[lem:rem:FFn-Fn-empirical-proc-arg\], where the random variable implicit in the $O_p$ statement depends on $J$ only through $L$. Since $|\FFn(\eta) - \Fn(\eta)| \le 1/n$ by Corollary 2.4 of [@Doss-Wellner:2016ModeConstrained], we see that $\sup_{t \in
[\eta,\eta^0]} | \FFn(t) - \Fn(t)| = \delta^{1/2} O_p(n^{-3/5})$. Similarly, since $|\FFna(\eta^0) - \Fn(\eta^0)| \le 1/ n$ by Corollary 2.12 of [@Doss-Wellner:2016ModeConstrained], $\sup_{t \in [\eta,\eta^0]} |
\FFna(t) - \Fn(t)| = \delta^{1/2} O_p(n^{-3/5})$ by analogous computations. Together, these let us conclude that $\sup_{t \in [\eta,\eta^0]} | \FFna(t)
- \FFn(t)| = \delta^{1/2} O_p(n^{-3/5})$.
an alternative proof approach would be just to use the point $s$ where $\FFn(s) = \FFna(s)$
Now we prove (\[item:lem:integral-to-sup:knots\]). Let $ \delta > 0$ and define $i_{1}$ and $i_{2}$ by $I = [i_1 - \delta n^{-1/5}, i_2 + \delta
n^{-1/5}]$, taking $\delta$ small enough that $i_1 < i_2$. Let $ i_2-i_1 = \tilde M n^{-1/5}$. Then, by the Taylor expansion of $\vvo'$ (see the beginning of this proof), $\vvo'( i_2 ) - \vvo'(i_1) \ge
|\vvo^{(2)}(m)| \tilde M n^{-1/5} / 2$ for $\delta$ small and $n$ large enough, since $\vvo^{(2)}$ is continuous. Additionally, by applying Corollary 5.4 of [@Doss-Wellner:2016ModeConstrained] twice, taking $\xi_n =
i_1$ and $\xi_n = i_2$ and $C=0$ (not $C = \tilde M$), we find $D$ (independent of $\lambda(I)$) such that $|(\vvna)'(i_1) - \vvo'(i_{1})|$ and $|(\vvna)'(i_{2}) - \vvo'(i_{2})|$ are, with probability $1 -
{\varepsilon}$, less than $n^{-1/5} D$. Here $(\vvna)'$ may be the right or left derivative. For $\delta $ small enough, by assumption $D <
|\vvo^{(2)}(m)| \tilde M / 8$, and we then have $(\vvna)'(i_{1}+) -
(\vvna)'(i_{2}-) \ge |\vvo^{(2)}(m)| \tilde M n^{-1/5} / 4$. We do not know a priori how much $(\vvna)'$ decreases at any specific knot in $S(\vvna)$, but by partitioning $[i_1,i_2]$ into intervals of a fixed length, we can find one such interval on which $(\vvna)'$ decreases by the corresponding average amount. That is, there exists a subinterval of $[i_1,i_2]$, denoted $J^* = [l^*, r^*],$ of length $2 \delta n^{-1/5}$, such that $$ (\vvna)'(l^*+) - (\vvna)'(r^*-)
\ge \frac{|\vvo^{(2)}(m)| \tilde M n^{-1/5} / 4}{ (i_2-i_1) / 2\delta n^{-1/5} }
= \frac{|\vvo^{(2)}(m)| \delta n^{-1/5}}{2}
\equiv K n^{-1/5}$$ since $i_2-i_1 = \tilde M n^{-1/5}.$ Let $x^*_l = \sup {\left\{} \newcommand{\rb}{\right\}}x \in
J^* : (\vvna - \vvn)'(x) \ge 0 \rb$, and let $x^*_l = l^*$ if the set is empty, and let $x^*_r = \inf {\left\{} \newcommand{\rb}{\right\}}x \in J^* : (\vvna - \vvn)'(x) \le 0 \rb$, and $x^*_r = x^*_l = r^*$ if the set is empty. Now $(\vvna - \vvn)'$ decreases by at least $K n^{-1/5} / 2$ on either $[l^*, x_l^*]$ or on $[x_r^* , r^*]$ (since $(\vvna- \vvn)'$ is constant on $[x^*_l, x_r^*]$). Let $\eta^0_l = \inf S(\vvna) \cap J^*$ and $\eta^0_r = \sup S(\vvna) \cap
J^*$. Now by assumption $\vvn$ is linear on $[\eta^0_l - \delta n^{-1/5},
\eta^0_r + \delta n^{-1/5}]$ (since $\vvn$ is linear on $J^*$ and within $\delta n^{-1/5}$ of any knot of $\vvna$), and so $$\begin{aligned}
(\vvna - \vvn)'(u) & \ge \frac{K n^{-1/5}}{2}
& \mbox{ for } u \in [\eta^0_l
- \delta n^{-1/5}, \eta^0_l], &
\quad \mbox { or }
\label{eq:knot-lemma:slopeineq-1} \\
(\vvna - \vvn)'(u) & \le - \frac{ Kn^{-1/5}}{2}
& \mbox{ for } u \in
[\eta^0_r, \eta^0_r + \delta n^{-1/5}], & \label{eq:knot-lemma:slopeineq-2}
\end{aligned}$$ depending on whether $ (\vvna - \vvn)'$ decreases by $K n^{-1/5}/2$ on $[l^*,
x^*_l]$ (in which case $\eta^0_l \le x^*_l$) or on $[x^*_r, r^*]$ (in which case $x^*_r \le \eta^0_r$). (In the former case, holds because $(\vvna - \vvn)'$ is nonincreasing and its decrease to $0$ on $[l^*, x_l^*]$ actually happens on $[\eta^0_l, x_l^*]$ since $\eta^0_l$ is the last knot of $\vvna$ in $J^*$. Similar reasoning in the latter case yields .) If holds then $$\sup_{u \in [\eta^0_l -
\delta n^{-1/5}, \eta^0_l]} | \vvna(u) - \vvn(u)| \ge ( Kn^{-1/5}/2)
(\delta n^{-1/5}/2) = |\vvo^{(2)}(m) | n^{-2/5} \delta^2 / 8,$$ and if holds then $\sup_{u \in [\eta^0_r,
\eta^0_r + \delta n^{-1/5}]} | \vvna(u) - \vvn(u)| \ge |\vvo^{(2)}(m)|
n^{-2/5} \delta^2 / 8$. This allows us to lower bound $\int_{I} ( \vvna(u)
- \vvn(u))^2$, to attain a contradiction with .
Assume that holds. The case where holds is shown analogously. Let $ z =
\operatorname*{argmin}_{u \in [i_1,i_2]} | \vvna(u) - \vvn(u)|$. Let $L$ be the affine function such that $L(z) = \vvna(z) - \vvn(z)$ and $L$ has slope $K
n^{-1/5} / 2$. Then for $x \in [\eta^0_l - \delta n^{-1/5}, \eta^0_l]$, $|L(x)| \le | \vvna(x) - \vvn(x)|$. For $z \le x \le \eta^0_l$, this is because $$\begin{aligned}
| \vvna(x) - \vvn(x)|
= \vvna(x) - \vvn(x)
& = \vvna(z) - \vvn(z)
+ \int_z^x (\vvna - \vvn)' d\lambda \\
& \ge L(z) + \int_z^x L' d\lambda = |L(x)|,
\end{aligned}$$ where the first equality holds since $\vvna - \vvn$ is increasing on $[\eta^0_l - \delta n^{-1/5}, \eta^0_l]$ (by ), so by the definition of $z$, $\vvna(x)
- \vvn(x) \ge 0$ for $x \ge z$, and the last is similar. For $\eta^0_l -
\delta n^{-1/5} \le x \le z$, $$\begin{aligned}
- |\vvna(x)-\vvn(x)|
= (\vvna(x) -\vvn)(x)
& = (\vvna-\vvn)(z) - \int_x^z (\vvna - \vvn)'d\lambda \\
& \le L(z) - \int_x^z L' d\lambda
= - |L(x)|,
\end{aligned}$$ where the first and last equalities follow because for $\eta_l^0 \le x \le
z$, since $\vvna - \vvn$ is increasing it must be negative by the definition of $z$. Thus, $|L(u)| \le | \vvna(u) - \vvn(u)|$ on $[\eta_l^0 - \delta n^{-1/5}, \eta_l^0]$ so $$\frac{\delta^3 K^2 n^{-1}}{3 \cdot 2^5}
\le \int_{[\eta_l^0 - \delta n^{-1/5}, \eta_l^0]} L^2 d\lambda
\le \int_{[\eta_l^0 - \delta n^{-1/5}, \eta_l^0]} (\vvna - \vvn)^2 d\lambda
\le \int_{I} (\vvna - \vvn)^2 d\lambda,$$ where the quantity on the far left is $\int_0^{\delta n^{-1/5}/2} (x K
n^{-1/5}/2)^2 dx$. This is a contradiction if $\delta$ is fixed and we let $\check \delta \to 0$, since $[\eta_l^0 - \delta n^{-1/5}, \eta_r^0 +
\delta n^{-1/5}] \subset I$ by the definition of $[i_1,i_2]$, and then $ \int_{I} (\vvna - \vvn)^2 d\lambda \le \check \delta \check K n^{-1}$. A similar inequality can be derived if holds. Thus $\delta \to 0$ as $\check \delta \to 0$.
Finally, we show holds with similar logic. Let $\xi_1 < \xi_2$ be points such that $\vvn$ is linear on $[\xi_1, \xi_2]$, and let $\delta > 0$. Then if all knots $\xi^0$ of $\vvna$ satisfy $| \xi_i - \xi^0 | > \sqrt{\delta} n^{-1/5}$, $i =1 ,2$, then we can see that $\Vert \vvn' - (\vvna)' \Vert_{[\xi_1,\xi_2]} \le
\sqrt{\delta} n^{-1/5}$. This is because at any $x \in [\xi_1 + \sqrt{
\delta} n^{-1/5}, \xi_2 - \sqrt{ \delta} n^{-1/5}]$, if $(\vvna - \vvn)'(x) > \sqrt{ \delta } n^{-1/5}$, then $(\vvna - \vvn)' >
\sqrt{\delta} n^{-1/5}$ on $[x- \sqrt{\delta}n^{-1/5}, x]$ (since $\vvn$ is linear on a $\sqrt{\delta} n^{-1/5}$ neighborhood of $x$), so $(\vvna - \vvn)(x
- \sqrt{\delta} n^{-1/5}, x] > \delta n^{-2/5}$, a contradiction. Here we use the notation $g(a,b] = g(b) - g(a)$. Similarly if $(\vvna - \vvn)'(x)
< \sqrt{\delta}n^{-1/5}$ then $(\vvna - \vvn)(x , x+\sqrt{\delta} n^{-1/5}]
< - \delta n^{-2/5}$, a contradiction. We have thus shown that if we take $\eta$ and $\eta^0$ to be the largest knot pair within $\sqrt{\delta} n^{-1/5}$ (meaning $\max(\eta, \eta^0)$ is largest among such pairs) then $\Vert (\vvn - \vvna)' \Vert_{[\max(\eta, \eta^0), \sup J
- \sqrt{\delta}n^{-1/5}]} \le \sqrt{ \delta} n^{-1/5}$, by Part and by partitioning $[\max(\eta,
\eta^0), \sup J]$ into intervals on which $\vvn$ is linear.
The proofs of Parts and in the previous lemma could also be completed with the roles of $\vvna$ and $\vvn$ reversed. The next lemma provides the calculation used in Lemma \[lem:rem:local-to-global-square-integral\] to translate an upper bound on $\int_{I} {\left(}\vvna - \vvn {\right)}^2 d \lambda$ into an upper bound on $\sup_{x \in I} {\left(}\vvna(x) - \vvn(x)
{\right)}$ for an appropriate interval $I$.
\[lem:rem:L2-to-sup-1\] Let ${\varepsilon}> 0$. Assume $\varphi_i$, $i=1,2$, are functions on an interval $I$, where $$\label{eq:6}
\varphi_L' \le \varphi_i'(x) \le \varphi_U' \, \mbox{ for } x \in I, i=1,2,$$ where $\varphi_i'$ refers to either the left or right derivative and $\varphi_L',\varphi_U'$ are real numbers. Assume $I$ is of length no smaller than ${\left(}{\varepsilon}/ {\left(}\varphi_U'-\varphi_L'{\right)}^2 {\right)}^{1/3}$. Assume that $\int_I
{\left(}\varphi_1(x) - \varphi_2(x) {\right)}^2 dx \le {\varepsilon}$. Then $$\sup_{x \in I} | \varphi_1(x)-\varphi_2(x) | \le {\left(}8
{\varepsilon}{\left(}\varphi_U' - \varphi_L' {\right)}{\right)}^{1/3}.$$
Assume that $x$ is such that $\varphi_1(x) - \varphi_2(x) = \delta$, and without loss of generality, $\delta > 0$. Then by , if $y$ is such that $|x - y | \le (\delta/2) / (\varphi_U'-\varphi_L')$ then $\varphi_1(y) - \varphi_2(y) \ge \delta / 2$. Thus, if $I$ is an interval whose length is no smaller than $(\delta /2) / (\varphi_U'-\varphi_L')$ then if $\varphi_1(x) - \varphi_2(x) \ge \delta$ for any $x \in J \subseteq I$ where the length of $J$ is equal to $(\delta /2) / (\varphi_U'-\varphi_L')$, then $$\int_I {\left(}\varphi_1(x) - \varphi_2(x) {\right)}^2 dx
\ge \int_J {\left(}\varphi_1(x) - \varphi_2(x) {\right)}^2 dx
\ge \frac{\delta^2}{2^2} \lambda(J)
= \frac{1}{8} \frac{\delta^3}{\varphi_U'-\varphi_L'} .$$ Thus, substituting ${\varepsilon}= \frac{1}{8}
\frac{\delta^3}{\varphi_U'-\varphi_L'}$, we see that for $x \in I$, $|\varphi_1(x)-\varphi_2(x) | \le {\left(}8
{\varepsilon}{\left(}\varphi_U'-\varphi_L'{\right)}{\right)}^{1/3}$, as desired.
If we set ${\varepsilon}= \frac{\tilde{f}}{8}
\frac{\delta^3}{\varphi_U'-\varphi_L'}$, then $\delta = {\left(}8 {\varepsilon}{\left(}\varphi_U'-\varphi_L'{\right)}/ \tilde f {\right)}^{1/3}$, and then $$\frac{\delta/2}{ \varphi_U'-\varphi_L'} = \frac{1}{2}
\frac{8^{1/3}}{\tilde
f^{1/3}} {\varepsilon}^{1/3} {\left(}\varphi_U'-\varphi_L' {\right)}^{-2/3}
= \frac{{\varepsilon}^{1/3} {\left(}\varphi_U'-\varphi_L'{\right)}^{-2/3}}{ \tilde f^{1/3}}$$ is the length of $I$. (Note that $\tilde f$ is now replaced by $1$ in the lemma. –2016 sep 09)
\[lem:rem:FFn-Fn-empirical-proc-arg\] Let either Assumption \[CurvatureAtTheMode\] hold at $x_0 = m$ or Assumption \[CurvatureAwayFromMode\] hold at $x_0 \ne m$, and let $\FFna$ and $\FFn$ be the log-concave mode-constrained and unconstrained MLEs of $\FFo$. Let $I = [v_1, v_2]$ be a random interval whose dependence on $n$ is suppressed and such that $n^{1/5}(v_j - x_0) = O_p(1), j = 1,2$. Then $$\label{eq:rem:dFFn-dFn}
\left \{
\begin{array}{l} \sup_{t \in I} {\left\vert}\int_{[v_1,t]} d(\FFn(u) - \Fn(u)) {\right\vert}\\
\ \\
\sup_{t \in I} {\left\vert}\int_{[v_1,t]} d(\FFna(u) - \Fn(u)) {\right\vert}\end{array} \right \}
= \sqrt{ \lambda(I)} O_p(n^{-2/5}).$$ The random variables implicit in the $O_p$ statements in depend on $I$ through its length (in which they are increasing) and not the location of its endpoints.
We analyze $ \sup_{t \in I} {\left\vert}\int_{v_1}^t
d(\FFn(u) - \Fn(u)) {\right\vert}$ first. Note $$\label{eq:rem:7}
\begin{split}
\MoveEqLeft
\sup_{t \in I} \left| \int_{[v_1,t]} d {\left(}\Fn
- \FFn {\right)}\right| \\
& \le \sup_{t \in I}
\bigg| \int_{v_1}^{t} {\left(}\ffn(u) - \ffo(v_1) - (u- v_1)
\ffo'(v_1) {\right)}du \\
& \qquad \qquad \qquad - \int_{v_1}^{t} {\left(}\ffo(u) - \ffo(v_1) - (u- v_1) \ffo'(v_1) {\right)}du \\
& \qquad \qquad \qquad - \int_{[v_1,t]} d{\left(}\Fn-\FFo{\right)}\bigg|.
\end{split}$$ By (the proof of) Lemma 7.20 of [@Doss-Wellner:2016ModeConstrained] $$\sup_{t \in I} {\left\vert}\int_{[v_1, t]} d{\left(}\Fn - \FFo {\right)}{\right\vert}=
\sqrt{\lambda(I)} O_p(n^{-2/5}).$$ The supremum over the middle term in is $\lambda(I)
O_p(n^{-2/5})$ by a Taylor expansion of $\ffo$, and applying in addition Lemma 4.5 of [@BRW2007LCasymp], we see that the supremum over the first term in is also $\lambda(I) O_p(n^{-2/5})$.
The same analysis, using Proposition 5.3 or Corollary 5.4 of [@Doss-Wellner:2016ModeConstrained], applies to $ \sup_{t \in I} {\left\vert}\int_{[v_1,t]} d(\FFna(u) - \Fn(u)) {\right\vert}$. Note in all cases that the random variables implicit in the $O_p$ statements depend on $I$ only through its length (and they are increasing in the length) and not the location of its endpoints, since $\Vert \sqrt{\ffo} \Vert \le \ffo(m)$ and since $\ffo^{(2)}$ is continuous and so uniformly bounded in a neighborhood of $x_0$.
When we apply the previous lemma, the length of $I$ will depend on $ {\varepsilon}$ which gives the probability bound implied by our $o_p$ statements whereas its endpoints will depend on $\delta$, which gives the size bound implied by our $o_p$ statements.
\[lem:rem:epsilon-inequality\] Let ${\varepsilon}(x)$ and $\tilde {\varepsilon}(x)$ be defined by $e^x = 1 + x +
2^{-1} x^2 e^{{\varepsilon}(x)}$ and $ e^x = 1 + x e^{\tilde {\varepsilon}(x)}.$ Then $$ e^{{\varepsilon}(x)} \le 2 e^{\tilde {\varepsilon}(x)}.$$
We can see $$\begin{aligned}
e^{{\varepsilon}(x)} = \frac{e^{x}-1 -x}{(x^2/2)} = \frac{2}{x^2} \sum_{k=2}^\infty \frac{x^k}{k!} = \sum_{k=0}^\infty \frac{2 x^k}{(k+2)!} .
\end{aligned}$$ Similarly, $$\begin{aligned}
e^{\tilde{{\varepsilon}}(x)} = \frac{e^{x}-1 }{x} = \frac{1}{x} \sum_{k=1}^\infty \frac{x^k}{k!} = \sum_{k=0}^\infty \frac{x^k}{(k+1)!} .
\end{aligned}$$ Comparing coeffients in the two series, we see that $$\begin{aligned}
\frac{2}{(k+2)!} \le \frac{1}{(k+1)!} \ \ \mbox{for all} \ k\ge 0
\end{aligned}$$ since $k+2 \ge 2$ for $k\ge 0$. It follows that $e^{{\varepsilon}(x)} \le e^{\tilde{{\varepsilon}} (x)} $ for all $x \ge 0$. This implies that ${\varepsilon}(x) \le \tilde{{\varepsilon}}(x)$ for all $x \ge 0$.
Now for $x \le 0$ we have $$\begin{aligned}
2e^{\tilde{{\varepsilon}} (x)} - e^{{\varepsilon}(x)}
& = \frac{2}{x} \left \{ e^x -1 - \frac{1}{x} (e^x -1 -x ) \right \} \\
& = \frac{2}{x} \left \{ \sum_{k=1}^ \infty \frac{x^k}{k!} - \sum_{k=1}^\infty
\frac{ x^k}{(k+1)!} \right \}
= \frac{2}{x} \sum_{k=1}^\infty \frac{k}{(k+1)!} x^k
\end{aligned}$$ where the infinite sum is negative for all $x<0$ since the first term is negative. It follows that $$\begin{aligned}
2e^{\tilde{{\varepsilon}} (x)} - e^{{\varepsilon}(x)} \ge 0
\end{aligned}$$ for all $x\le 0$. Combined with the result for $x\ge 0$ the claimed result holds: $e^{{\varepsilon}(x)} \le 2 e^{\tilde{{\varepsilon}}(x)}$ for all $x \in {{\mathbb R}}$.
|
---
abstract: 'In this work we establish some polynomials and entire functions have only real zeros. These polynomials generalize q-Laguerre polynomials $L_{n}^{(\alpha)}(x;q)$, while the entire functions are generalizations of Ramanujan’s entire function $A_{q}(z)$, q-Bessel functions $J_{\nu}^{(2)}(z;q)$, $J_{\nu}^{(3)}(z;q)$ and confluent basic hypergeometric series.'
author:
- Ruiming Zhang
title: Zeros of Ramanujan Type Entire Functions
---
[^1]
Introduction
============
In [@Ismail2] Ismail and Zhang defined an entire function $$A_{q}^{\left(\alpha\right)}\left(a;z\right)=\sum_{n=0}^{\infty}\frac{\left(a;q\right)_{n}q^{\alpha n^{2}}z^{n}}{\left(q;q\right)_{n}},\label{eq:1.1}$$ where $\alpha>0$, $0<q<1$, and for any positive integer $m\in\mathbb{N}$ and all complex numbers $a,a_{1},\dots,a_{m}\,n\in\mathbb{C}$, [@Andrews1; @Ismail1] $$(a;q)_{\infty}=\prod_{k=0}^{\infty}\left(1-aq^{k}\right),\ (a;q)_{n}=\frac{\left(a;q\right)_{\infty}}{\left(aq^{n};q\right)_{\infty}},\ (a_{1},\dots,a_{m};q)_{n}=\prod_{j=1}^{m}(a_{j};q)_{n}.\label{eq:1.2}$$ Clearly, $A_{q}^{\left(\alpha\right)}\left(a;z\right)$ is a kind of entire function treated in Lemma 14.1.4 of [@Ismail1], and by the lemma it must have infinitely many zeros if $aq^{n}\neq1$ for any nonnegative integers $n\in\mathbb{N}_{0}$. Furthermore, in [@Hayman] Hayman proved an asymptotic expansion for the $n$-th zero of this class of entire function under the general condition that the parameter $q$ is strictly inside the unit disk $|q|<1$.
We observe that
$$A_{q}^{\left(1/2\right)}\left(q^{-n};z\right)=\sum_{k=0}^{\infty}\frac{\left(q^{-n};q\right)_{k}q^{k^{2}/2}z^{k}}{\left(q;q\right)_{k}}=\left(q;q\right)_{n}S_{n}\left(zq^{1/2-n};q\right),\label{eq:1.3}$$
and
$$A_{q}^{\left(1\right)}\left(0;z\right)=\sum_{n=0}^{\infty}\frac{q^{n^{2}}z^{n}}{(q;q)_{n}}=A_{q}\left(-z\right),\ A_{q}^{\left(1\right)}\left(q;z\right)=\sum_{n=0}^{\infty}q^{n^{2}}z^{n},\label{eq:1.4}$$
where $A_{q}(z)$ and $S_{n}(z;q)$ are the Ramanujan entire function and Stieltjes-Wigert polynomial respectively, see [@Ismail1], for example. Since $A_{q}^{\left(\alpha\right)}\left(a;z\right)$ generalizes both $A_{q}(z)$ and $S_{n}(z;q)$, and it is well-known that both of them have all real positive zeros. Thus it is natural to ask under what conditions the zeros of the entire function $A_{q}^{\left(\alpha\right)}\left(a;z\right)$ are all real. In this work we shall present a partially answer to this question and more.
Preliminaries
=============
In the proofs we need the Vitali’s theorem [@Titchmarsh1] and Hurwitz’s theorem [@Ahlfors1]. For our convenience we list them here. The first is the Vitali’s theorem:
\[thm:Vitali\] Let $\left\{ f_{n}(z)\right\} $ be a sequence of functions analytic in a domain $D$ and assume that $f_{n}(z)\to f(z)$ point-wise in $D$. Then $f_{n}(z)\to f(z)$ uniformly in any subdomain bounded by a contour $C$, provided that $C$ is contained in $D$.
Here is the Hurwitz’s theorm:
\[thm:Hurwitz\] If the functions $f_{n}(z)$ are analytic and $\neq0$ in a region $\Omega$, and if $f_{n}(z)$ converges to $f(z)$, uniformly on every compact subset of $\Omega$, then $f(z)$ is either identically zero or never equal to zero in $\Omega$.
It is known that if an entire function has a finite order that it is not a positive integer, then $f(z)$ has infinitely many zeros, [@Boas; @Ismail1]. Given an entire function of order $0$ with $f(0)\neq0$, if $-z_{k},\ k\in\mathbb{N}$ are all the roots, then by Hadamard’s canonical representation of entire functions we have $$f(z)=f(0)\prod_{k=1}^{\infty}\left(1+\frac{z}{z_{k}}\right),\label{eq:2.2}$$ where ${\displaystyle \sum_{k=1}^{\infty}}|z_{k}|^{-1}<\infty.$ A real entire function $f(z)$ is of Laguerre-Pólya class if [@Dimitrov1; @Karlin] $$f(z)=cz^{m}e^{-\alpha z^{2}+\beta z}\prod_{k=1}^{\infty}\left(1+\frac{z}{z_{k}}\right)e^{-z/z_{k}},\label{eq:2.3}$$ where $c,\beta,\,z_{k}\in\mathbb{R}$, $\alpha\ge0$, $m\in\mathbb{Z}^{+}$, and ${\displaystyle \sum_{k=1}^{\infty}}z_{k}^{-2}<\infty$. Clearly, given a real entire function $f(z)$ of order $0$ that has nonnegative Taylor coefficients and satisfies $f(0)\neq0$, if it is also in the Laguerre-Pólya class then it must have the factorization (\[eq:2.2\]) with $z_{k}>0$.
A real sequence $\left\{ a_{n}\right\} _{n=0}^{\infty}$ is called a Pólya frequence (PF) sequence if the infinite matrix $\left(a_{j-i}\right)_{i,j=0}^{\infty}$ is totally positive, i.e. all its minors are nonnegative, where we follow the usual convention that $a_{n}=0$ if $n<0$. [@Aissen1; @Driver1; @Karlin]
\[thm:aesw\] The sequence $\left\{ b_{k}\right\} _{k=0}^{\infty}$ is a PF sequence if and only if $$cz^{m}e^{\gamma z}\prod_{k=1}^{\infty}\frac{1+\alpha_{k}z}{1-\beta_{k}z}=\sum_{k=0}^{\infty}b_{k}z^{k},\label{eq:2.5}$$ where $c\ge0,\ \gamma\ge0,\ \alpha_{k}\ge0,\ \beta_{k}\ge0$, $m\in\mathbb{Z}^{+}$, and ${\displaystyle \sum_{k=1}^{\infty}}\left(\alpha_{k}+\beta_{k}\right)<\infty$.
In the case $b_{k}=0$ for $k=n+1,\dots$, then the right hand series becomes a polynomial of degree at most $n$, then we have the following, [@Aissen1; @Driver1; @Karlin; @Pitman]
\[thm:pf\]The PF sequences have the following properties:
1. Let $b_{k}\ge0,\ k\ge0$. Then, the sequence $\left\{ b_{k}\right\} _{k=0}^{n}$ is a PF sequence if and only if the polynomial ${\displaystyle \sum_{k=0}^{n}}b_{k}x^{k}$ has all nonpositive zeros.
2. Let $a_{k},\,b_{k}\ge0,\ k\ge0$. If $\left\{ a_{k}\right\} _{k=0}^{m}$ and $\left\{ b_{k}\right\} _{k=0}^{n}$ are PF sequences, then so is $\left\{ a_{k}b_{k}\right\} _{k\ge0}$.
3. Let $b_{0},\,b_{1},\,\dots,\,b_{n}\ge0$ and $\left\{ b_{k}\right\} _{k=0}^{n}$ be a PF sequence, then so is $\left\{ b_{k}/k!\right\} _{k=0}^{n}$ .
4. Let $\left\{ a_{k}\right\} _{k=0}^{m}$ and $\left\{ b_{k}\right\} _{k=0}^{n}$ be PF sequences such that $a_{k},\,b_{k}\ge0,\ k\ge0$, then $\left\{ k!a_{k}b_{k}\right\} _{k\ge0}$ is also a PF sequence.
It was proved in that [@Carnicer1; @Dimitrov1] $$\sum_{n=0}^{\infty}\frac{q^{n^{2}}}{n!}x^{n}\label{eq:2.6}$$ is in Laguerre-Pólya class for $q\in(-1,1)$. For $0<q<1$. This function is clearly a real entire function of order $0$ with positive Taylor coefficients, hence the sequence $\left\{ q^{n^{2}}/n!\right\} _{n=0}^{\infty}$ is a PF sequence by Theorem \[thm:aesw\].
For $a\ge0$ and $0<q<1$, from the $q$-Binomial theorem we have [@Andrews1; @Ismail1] $$\frac{(-az;q)_{\infty}}{(bz;q)_{\infty}}=\sum_{n=0}^{\infty}\frac{(-a;q)_{n}}{(q;q)_{n}}z^{n},\label{eq:2.7}$$ and $$\left(-q^{n}z;q\right)_{n}=\sum_{k=0}^{n}\frac{\left(q^{-n};q\right)_{k}\left(-z\right)^{k}}{(q;q)_{k}}.\label{eq:2.8}$$ Then by Theorem \[thm:aesw\] both$\left\{ (-a;q)_{k}/(q;q)_{k}\right\} _{k=0}^{\infty}$ and $\left\{ \left(-1\right)^{k}\left(q^{-n};q\right)_{k}/(q;q)_{k}\right\} _{k=0}^{\infty}$ are PF sequences.
The $q$-Bessel functions $J_{\nu}^{(1)}(z;q)$, $J^{(2)}(z;q)$ and $J^{(3)}(z;q)$ are defined by [@Ismail1]
$$\frac{2^{\nu}(q;q)_{\infty}J_{\nu}^{(1)}(z;q)}{\left(q^{\nu+1};q\right)_{\infty}z^{\nu}}=\sum_{n=0}^{\infty}\frac{\left(-z^{2}/4\right)^{n}}{\left(q,q^{\nu+1};q\right)_{n}},\label{eq:2.9}$$
$$\frac{2^{\nu}(q;q)_{\infty}J_{\nu}^{(2)}(z;q)}{\left(q^{\nu+1};q\right)_{\infty}z^{\nu}}=\sum_{n=0}^{\infty}\frac{q^{n^{2}}\left(-z^{2}q^{\nu}/4\right)^{n}}{\left(q,q^{\nu+1};q\right)_{n}},\label{eq:2.10}$$
and $$\begin{aligned}
\frac{2^{\nu}(q;q)_{\infty}J_{\nu}^{(3)}(z;q)}{\left(q^{\nu+1};q\right)_{\infty}z^{\nu}} & = & \sum_{n=0}^{\infty}\frac{q^{\binom{n+1}{2}}\left(-z^{2}/4\right)^{n}}{\left(q,q^{\nu+1};q\right)_{n}}\label{eq:2.11}\end{aligned}$$ respectively.
For $\nu>-1$ and $0<q<1$, it is known that $J^{(2)}(z;q)$ is an entire function of order $0$ such that $$\frac{2^{\nu}(q;q)_{\infty}J_{\nu}^{(2)}(z;q)}{\left(q^{\nu+1};q\right)_{\infty}z^{\nu}}=\prod_{n=0}^{\infty}\left(1-\frac{z^{2}}{j_{\nu,n}(q)^{2}}\right),\label{eq:2.13}$$ where $$0<j_{\nu,1}(q)<j_{\nu,2}(q)<\dots\label{eq:2.14}$$ are the positive zeros of $J^{(2)}(z;q)$ satisfying $$\sum_{n=1}^{\infty}\frac{1}{j_{\nu,n}(q)^{2}}<\infty.\label{eq:2.15}$$ For $\nu>-1$ and $0<q<1$, from (\[eq:2.13\]) and the relation [@Ismail1] $$J_{\nu}^{(1)}(z;q)=\frac{J_{\nu}^{(2)}(z;q)}{\left(-z^{2}/4;q\right)_{\infty}}\label{eq:2.16}$$ we get $$\sum_{n=0}^{\infty}\frac{z^{n}}{\left(q,q^{\nu+1};q\right)_{n}}=\frac{{\displaystyle \prod_{n=0}^{\infty}}\left(1+z/j_{\nu,n}(q)^{2}\right)}{\left(z/4;q\right)_{\infty}}.\label{eq:2.17}$$ From (\[eq:2.10\]), (\[eq:2.13\]) and (\[eq:2.17\]) we see that both $\left\{ 1/\left(q,q^{\nu+1};q\right)_{k}\right\} _{k=0}^{\infty}$ and $\left\{ q^{k^{2}}/\left(q,q^{\nu+1};q\right)_{k}\right\} _{k=0}^{\infty}$ are PF sequences by Theorem \[thm:aesw\].
It is worth noticing that these $q$-Bessel functions are much more than merely being $q$-analogues to their classical counterpart $$J_{\nu}(z)=\frac{\left(z/2\right)^{\nu}}{\Gamma(\nu+1)}\sum_{n=0}^{\infty}\frac{\left(-z^{2}/4\right)^{n}}{n!(\nu+1)_{n}},\label{eq:2.18}$$ where $$(a)_{n}=\frac{\Gamma(a+n)}{\Gamma(a)},\quad a,n\in\mathbb{Z}\label{eq:2.19}$$ with $-a-n\notin\mathbb{N}_{0}$ and $\Gamma(z)$ is the Euler’s Gamma function. They have many interesting properties that the classical $J_{\nu}(z)$ does not possess. Here we only mention two of them. First of them is that $J_{\nu}^{(2)}\left(2iq^{-n/2};q\right),\ n\in\mathbb{N}_{0}$ can be evaluated explicitly, see [@Ismail3]. The second is that there are many symmetries among themselves.
Let $$j_{\nu}^{(k)}(z;q)=\frac{2^{\nu}(q;q)_{\infty}J_{\nu}^{(k)}(z;q)}{\left(q^{\nu+1};q\right)_{\infty}z^{\nu}},\quad k=1,2,3.\label{eq:2.20}$$ Then for $q>1$ and $|z|<1$, we have $$j_{\nu}^{(1)}\left(z;\frac{1}{q}\right)=j_{\nu}^{(2)}\left(\sqrt{q}z;q\right),\ j_{\nu}^{(2)}\left(z;\frac{1}{q}\right)=j_{\nu}^{(1)}\left(\sqrt{q}z;q\right),\ j_{2}^{(3)}\left(z;\frac{1}{q}\right)=j_{\nu}^{(3)}\left(q^{\frac{\nu}{2}}z;q\right).\label{eq:2.21}$$ The above relations are very typical among basic hypergeometric series, but it is seldom explored in the literature.
Main Results
============
We follow the usual convention that an empty sum is zero but an empty product is 1.
\[thm:poly\] Let $n\in\mathbb{N}$, $0<q<1$, and $\alpha\ge0$.
1. The polynomial $$A_{q}^{(\alpha)}\left(q^{-n};x\right)=\sum_{k=0}^{n}\frac{\left(q^{-n};q\right)_{k}q^{\alpha k^{2}}x^{k}}{\left(q;q\right)_{k}}\label{eq:3.2}$$ has all positive zeros.
2. For $m,\ell\in\mathbb{N}_{0}$, $n_{j}\in\mathbb{N},\ 0<q_{j}<1,\ 1\le j\le m$, and $\beta_{r}>0,\ 0<q_{r}<1,\ 1\le r\le\ell$, the polynomial $$\sum_{k=0}^{\min\left\{ n_{j}\vert1\le j\le m\right\} }\prod_{j=1}^{m}\frac{\left(q_{j}^{-n_{j}};q_{j}\right)_{k}}{(q_{j};q_{j})_{k}}\frac{q^{\alpha k^{2}}\left((-1)^{m}x\right)^{k}}{{\displaystyle \prod_{r=1}^{\ell}}\left(q_{r},q_{r}^{\beta_{r}};q_{r}\right)_{k}}\label{eq:3.3}$$ has all negative zeros.
3. For $m,\ell\in\mathbb{N}_{0}$, $n_{j}\in\mathbb{N},\ 1\le j\le m$, and $\beta_{r}>0,\ 1\le r\le\ell$, the polynomial $$\sum_{k=0}^{\min\left\{ n_{j}\vert1\le j\le m\right\} }\frac{\prod_{j=1}^{m}(-n_{j})_{k}}{\prod_{r=1}^{\ell}\left(\beta_{r}\right)_{k}}\frac{\left((-1)^{m}x\right)^{k}}{(k!)^{m+\ell}}\label{eq:3.4}$$ has all negative zeros.
The following results are for the entire functions in the first parameter range:
\[thm:func1\] Let $0<q<1$ and $\alpha>0$. Then,
1. For $a\ge0$, the entire function $$A_{q}^{\left(\alpha\right)}\left(-a;z\right)=\sum_{n=0}^{\infty}\frac{\left(-a;q\right)_{n}q^{\alpha n^{2}}z^{n}}{\left(q;q\right)_{n}}\label{eq:3.5}$$ has infinitely many zeros and all of them are negative.
2. For $m,\ell\in\mathbb{N}_{0}$, $a_{j}\ge0,\ 0<q_{j}<1,\ 1\le j\le m$, and $\beta_{r}>0,\ 0<q_{r}<1,\ 1\le r\le\ell$, the entire function $$\sum_{k=0}^{\infty}\prod_{j=1}^{m}\frac{\left(-a_{j};q_{j}\right)_{k}}{(q_{j};q_{j})_{k}}\frac{q^{\alpha k^{2}}z^{k}}{{\displaystyle \prod_{r=1}^{\ell}}\left(q_{r},q_{r}^{\beta_{r}};q_{r}\right)_{k}}\label{eq:3.6}$$ has infinitely many zeros and all of them are negative.
3. For $m\in\mathbb{N}_{0},\ \ell\in\mathbb{N}$ and $\beta_{r}\ge1,\ 1\le r\le\ell$, the entire function $$\sum_{k=0}^{\infty}\frac{z^{k}}{\left(k!\right)^{m+\ell}{\displaystyle \prod_{r=1}^{\ell}\left(\beta_{r}\right)_{k}}}\label{eq:3.7}$$ has infinitely many zeros and all of them are negative.
For $r,\,s\in\mathbb{N}_{0}$ and $0<q<1$, let $a_{1},\dots,\,a_{r},\ b_{1},\dots,\,b_{s}\in\mathbb{C}$ $$_{r}A_{s}^{(\alpha)}(a_{1},\dots a_{r};\ b_{1},\dots,\ b_{s}\ ;\,q;\,z)=\sum_{n=0}^{\infty}\frac{(a_{1},\dots,\,a_{r};\,q)_{n}}{\left(b_{1},\dots,b_{s};\,q\right)_{n}}q^{\alpha n^{2}}z^{n},\label{eq:3.8}$$ then it is clear that for $\alpha>0$ and $0<q<1$ we have $$A_{q}^{(\alpha)}(a;z)=_{1}A_{1}^{(\alpha)}(a,\,q;z),\label{eq:3.9}$$ and for $z\in\mathbb{C}$ and $s+1>r$ we have $$\begin{aligned} & _{r}A_{s}^{\left((s+1-r)/2\right)}\left(a_{1},\dots a_{r};\ q,\,b_{1},\dots,\ b_{s}\ ;\,q;\,(-1/\sqrt{q})^{s+1-r}z\right)\\
& ={}_{r}\phi_{s}\left(\begin{array}{cc}
\begin{array}{c}
a_{1},\dots,\,a_{r}\\
b_{1},\dots,\,b_{s}
\end{array} & \bigg|q,z\end{array}\right),
\end{aligned}$$ where the basic hypergeometric series $_{r}\phi_{s}$ is defined by [@Andrews1; @Ismail1] $$\begin{aligned}_{r}\phi_{s}\left(\begin{array}{cc}
\begin{array}{c}
a_{1},\dots,\,a_{r}\\
b_{1},\dots,\,b_{s}
\end{array} & \bigg|q,z\end{array}\right) & =\sum_{n=0}^{\infty}\frac{(a_{1},\dots,\,a_{r};q)_{n}}{(q,b_{1},\dots,\,b_{s};q)_{n}}\left(-q^{(n-1)/2}\right)^{n(s+1-r)}z^{n}.\end{aligned}
\label{eq:61}$$ Finally, here is the result for entire functions in another parameter range:
\[thm:func2\] For $r,\,s\in\mathbb{N}_{0}$, let $\alpha>0,\ 1>a_{j},\,b_{k}>0,\ 1\le j\le r,\ 1\le k\le s$ such that $$\prod_{j=1}^{r}\left(1-a_{j}\right)\prod_{k=1}^{s}\left(1-b_{k}q\right)\ge4q^{2\alpha},\label{eq:3.10}$$ then the entire function $_{r}A_{s}^{(\alpha)}(a_{1},\dots a_{r};\ b_{1},\dots,\ b_{s}\ ;\,q;\,z)$ has all negative zeros.
Proofs
======
Proof of Theorem \[thm:poly\]
-----------------------------
For $n\in\mathbb{N}$, $0<q<1$ and $\alpha>0$, since the sequences $\left\{ \left(q^{-n};q\right)_{k}\left(-1\right)^{k}/(q;q)_{k}\right\} _{k=0}^{n}$ and $\left\{ q^{\alpha k^{2}}/k!\right\} _{k=0}^{n}$are PF sequences, then apply property 4 of Theorem \[thm:pf\] to get the PF sequence $\left\{ \left(q^{-n};q\right)_{k}\left(-1\right)^{k}q^{\alpha k^{2}}/(q;q)_{k}\right\} _{k=0}^{n}$. Then by property 1 of Theorem \[thm:pf\] we know the polynomial $$\sum_{k=0}^{n}\frac{\left(q^{-n};q\right)_{k}\left(-1\right)^{k}q^{\alpha k^{2}}z^{k}}{(q;q)_{k}}=A_{q}^{(\alpha)}\left(q^{-n};-z\right)$$ has all negative zeros, which is equivalent to that $A_{q}^{(\alpha)}\left(q^{-n};z\right)$ has all positive zeros.
For $m,\ell\in\mathbb{N}$, $n_{j}\in\mathbb{N},\ 0<q_{j}<1,\ 1\le j\le m$, and $\nu_{r}>-1,\ 0<q_{r}<1,\ 1\le r\le\ell$, first by property 2, then by property 4 of Theorem \[thm:pf\] we know that $$\left\{ \prod_{j=1}^{m}\frac{\left(q_{j}^{-n_{j}};q_{j}\right)_{k}}{(q_{j};q_{j})_{k}}\frac{\left(-1\right)^{km}}{{\displaystyle \prod_{r=1}^{\ell}}\left(q_{r},q_{r}^{\nu_{r}+1};q_{r}\right)_{k}}\right\} _{k\ge0}$$ and $$\left\{ \prod_{j=1}^{m}\frac{\left(q_{j}^{-n_{j}};q_{j}\right)_{k}}{(q_{j};q_{j})_{k}}\frac{\left(-1\right)^{km}q^{\alpha k^{2}}}{{\displaystyle \prod_{r=1}^{\ell}}\left(q_{r},q_{r}^{\nu_{r}+1};q_{r}\right)_{k}}\right\} _{k\ge0}$$ are all PF sequences. Then for $\alpha\ge0$ the polynomial (\[eq:3.3\]) has all negative zeros.
For $m,\ell\in\mathbb{N}$, take $$q_{j}=q,\ 1\le j\le m,\quad q_{r}=q,\ 1\le r\le\ell,\quad z=\left(1-q\right)^{2\ell}x,$$ and let $q\uparrow1$ in $$\sum_{k=0}^{\min\left\{ n_{j}\vert1\le j\le m\right\} }\prod_{j=1}^{m}\prod_{s=1}^{k}\frac{\left(1-q^{s-1-n_{j}}\right)}{\left(1-q^{s}\right)}\frac{\left(-1\right)^{km}q^{\alpha k^{2}}\left(1-q\right)^{2k\ell}x^{k}}{{\displaystyle \prod_{r=1}^{\ell}}{\displaystyle \prod_{t=1}^{k}}\left(1-q^{t}\right)\left(1-q^{\nu_{r}+t}\right)}$$ to get $$\begin{aligned} & \sum_{k=0}^{\min\left\{ n_{j}\vert1\le j\le m\right\} }\prod_{j=1}^{m}\prod_{s=1}^{k}\frac{s-1-n_{j}}{s}\frac{\left(-1\right)^{km}x^{k}}{{\displaystyle \prod_{r=1}^{\ell}}{\displaystyle \prod_{t=1}^{k}}t(\nu_{r}+t)}\\
& =\sum_{k=0}^{\min\left\{ n_{j}\vert1\le j\le m\right\} }\frac{\prod_{j=1}^{m}(-n_{j})_{k}}{{\displaystyle \prod_{r=1}^{\ell}}\left(\nu_{r}+1\right)_{k}}\frac{\left(-1\right)^{km}x^{k}}{(k!)^{m+\ell}}
\end{aligned}$$ for each $z\in\mathbb{C}$. Then this limit is also uniform on any compact subset of $\mathbb{C}$ by Vitali’s theorem \[thm:Vitali\]. Thus the polynomial (\[eq:3.4\]) has all negative zeros by applying Hurwitz’s theorem \[thm:Hurwitz\].
Proof of Theorem \[thm:func1\]
------------------------------
For $n\in\mathbb{N}$, $0<q<1$, $a\ge0$ and $\alpha>0$, since the sequences $\left\{ \left(-a;q\right)_{k}/(q;q)_{k}\right\} _{k=0}^{n}$ and $\left\{ q^{\alpha k^{2}}/k!\right\} _{k=0}^{n}$are PF sequences, then apply property 4 of Theorem \[thm:pf\] to get the PF sequence $\left\{ \left(-a;q\right)_{k}q^{\alpha k^{2}}/(q;q)_{k}\right\} _{k=0}^{n}$. Then by property 1 of Theorem \[thm:pf\] we know the polynomial $$\sum_{k=0}^{n}\frac{\left(-a;q\right)_{k}q^{\alpha k^{2}}}{(q;q)_{k}}z^{k}$$ has all negative zeros. For $0<q<1$, $a\ge0$ and $\alpha>0$ and for each $n\in\mathbb{N}$ and $z\in\mathbb{C}$ we have, $$\left|\sum_{k=0}^{n}\frac{\left(-a;q\right)_{k}q^{\alpha k^{2}}}{(q;q)_{k}}z^{k}\right|\le\sum_{k=0}^{\infty}\frac{\left(-a;q\right)_{k}q^{\alpha k^{2}}}{(q;q)_{k}}\left|z\right|^{k}<\infty.$$ Now we first apply Vitali’s theorem, then apply the Hurwitz’s theorem to show the entire function $A_{q}^{(\alpha)}(-a;z)$ has no zeros outside the set $(-\infty,0)$. Since by Lemma 14.1.4 of [@Ismail1] we know that $A_{q}^{(\alpha)}(-a;z)$ has infinitely many zeros, then we have proved that $A_{q}^{(\alpha)}\left(-a;z\right)$ has infinitely many zeros and all of them are negative.
For $m,\ell\in\mathbb{N}_{0},\ \alpha>0,\ 0<q<1$, $a_{j}\ge0,\ 0<q_{j}<1,\ 1\le j\le m$, and $\nu_{r}>-1,\ 0<q_{r}<1,\ 1\le r\le\ell$, similar to the polynomial case, we first apply property 2, then apply property 4 of Theorem \[thm:pf\] we know that for each $n\in\mathbb{N}_{0}$, then sequence $$\left\{ \prod_{j=1}^{m}\frac{\left(-a_{j};q_{j}\right)_{k}}{(q_{j};q_{j})_{k}}\frac{q^{\alpha k^{2}}}{{\displaystyle \prod_{r=1}^{\ell}}\left(q_{r},q_{r}^{\nu_{r}+1};q_{r}\right)_{k}}\right\} _{k=0}^{n}$$ is PF, thus the polynomial $$\sum_{k=0}^{n}\prod_{j=1}^{m}\frac{\left(-a_{j};q_{j}\right)_{k}}{(q_{j};q_{j})_{k}}\frac{q^{\alpha k^{2}}z^{k}}{{\displaystyle \prod_{r=1}^{\ell}}\left(q_{r},q_{r}^{\nu_{r}+1};q_{r}\right)_{k}}$$ has all negative zeros. For each $n\in\mathbb{N}_{0}$ and $z\in\mathbb{C}$ we have $$\begin{aligned} & \left|\sum_{k=0}^{n}\prod_{j=1}^{m}\frac{\left(-a_{j};q_{j}\right)_{k}}{(q_{j};q_{j})_{k}}\frac{q^{\alpha k^{2}}z^{k}}{{\displaystyle \prod_{r=1}^{\ell}}\left(q_{r},q_{r}^{\nu_{r}+1};q_{r}\right)_{k}}\right|\\
& \le\sum_{k=0}^{\infty}\prod_{j=1}^{m}\frac{\left(-a_{j};q_{j}\right)_{k}}{(q_{j};q_{j})_{k}}\frac{q^{\alpha k^{2}}\left|z\right|^{k}}{{\displaystyle \prod_{r=1}^{\ell}}\left(q_{r},q_{r}^{\nu_{r}+1};q_{r}\right)_{k}}<\infty.
\end{aligned}$$ Now we first apply Vitali’s theorem, then apply Hurwitz’s theorem to prove that the entire function (\[eq:3.6\]) has no zeros outside $(-\infty,0)$. Since by Lemma 14.1.4 of [@Ismail1] we know that the entire function (\[eq:3.6\]) has infinitely many zeros, then the assertion for on (\[eq:3.6\]) is proved.
Let $$m\ge0,\ \ell\ge1,\ \alpha=\left(\ell+\frac{m}{2}\right),\ \nu_{r}\ge0,\ 1\le r\le\ell,$$ and $$a_{j}=0,\ q_{j}=q,\ 1\le j\le m,\quad q_{r}=q,\ 1\le r\le\ell$$ in (\[eq:3.6\]), then for each $z\in\mathbb{N}$ and each $k\in\mathbb{N}_{0}$, by $$\lim_{q\uparrow1}\frac{(1-q)^{k}}{(q;q)_{k}}=\frac{1}{k!},\ \lim_{q\uparrow1}\frac{(1-q)^{k}}{\left(q^{\nu_{r}+1};q\right)_{k}}=\frac{1}{\left(\nu_{r}+1\right)_{k}}$$ we have $$\lim_{q\uparrow1}\frac{q^{\alpha k^{2}}z^{k}\left(1-q\right)^{(m+2\ell)k}}{(q;q)_{k}^{m+\ell}{\displaystyle \prod_{r=1}^{\ell}}\left(q^{\nu_{r}+1};q\right)_{k}}=\frac{z^{k}}{\left(k!\right)^{m+\ell}{\displaystyle \prod_{r=1}^{\ell}\left(\nu_{r}+1\right)_{k}}}.$$ Since for $n\in\mathbb{N}$ we have $$nq^{n-1}\le\frac{1-q^{n}}{1-q}=1+q+\dots+q^{n-1}\le n,$$ and $$n!q^{\binom{n}{2}}\frac{(q;q)_{n}}{\left(1-q\right)^{n}}\le n!.$$ Then, $$\frac{1}{n!}\le\frac{(1-q)^{n}}{(q;q)_{n}}\le\frac{q^{-\binom{n}{2}}}{n!}<\frac{q^{-\frac{n^{2}}{2}}}{n!}.$$ We also observe that for $\beta\ge1$ we have $$\frac{(1-q)^{n}}{\left(q^{\beta};q\right)_{n}}\le\frac{(1-q)^{n}}{(q;q)_{n}}.$$ Then, $$\left|\frac{q^{\alpha k^{2}}z^{k}\left(1-q\right)^{(m+2\ell)k}}{(q;q)_{k}^{m+\ell}{\displaystyle \prod_{r=1}^{\ell}}\left(q^{\nu_{r}+1};q\right)_{k}}\right|\le\frac{\left|z\right|^{k}}{\left(k!\right)^{m+2\ell}}.$$ Hence $$\begin{aligned}\lim_{q\uparrow1}\sum_{k=0}^{\infty}\frac{q^{\alpha k^{2}}z^{k}\left(1-q\right)^{(m+2\ell)k}}{(q;q)_{k}^{m+\ell}{\displaystyle \prod_{r=1}^{\ell}}\left(q^{\nu_{r}+1};q\right)_{k}} & =\sum_{k=0}^{\infty}\frac{z^{k}}{\left(k!\right)^{m+\ell}{\displaystyle \prod_{r=1}^{\ell}\left(\nu_{r}+1\right)_{k}}}\end{aligned}$$ converges uniformly for $z$ in any compact subset of $\mathbb{C}$. Then by Hurwitz’s theorem we know that the entire function $$g(z)=\sum_{k=0}^{\infty}\frac{z^{k}}{\left(k!\right)^{m+\ell}{\displaystyle \prod_{r=1}^{\ell}\left(\nu_{r}+1\right)_{k}}}$$ has no zeros outside $(-\infty,0)$.
Let $$a_{k}=\frac{1}{\left(k!\right)^{m+\ell}{\displaystyle \prod_{r=1}^{\ell}\left(\nu_{r}+1\right)_{k}}},$$ then by the Stirling’s formula [@Andrews1], $$\log\Gamma(x+1)=x\log x+\mathcal{O}(x),\quad x\to\infty,$$ we have $$-\log a_{k}=(m+2\ell)k\log k+\mathcal{O}(k)$$ as $k\to\infty$. Thus the order of $g(z)$ is given by [@Boas] $$\rho=\limsup_{k\to\infty}\frac{k\log k}{-\log|a_{k}|}=\frac{1}{m+2\ell},$$ which is a positive number less than $\frac{1}{2}$. Hence $g(z)$ has infinitely many zeros, and the claim on (\[eq:3.7\]) is proved.
Proof of Theorem \[thm:func2\]
------------------------------
By Lemma 14.1.4 of [@Ismail1] we know that $_{r}A_{s}^{(\alpha)}(a_{1},\dots a_{r};\ b_{1},\dots,\ b_{s}\ ;\,q;\,z)$ has infinitely many zeros.
Let $$c_{n}=\frac{(a_{1},\dots,\,a_{r};\,q)_{n}}{\left(b_{1},\dots,b_{s};\,q\right)_{n}}q^{\alpha n^{2}},$$ then for $n\ge1$ we have $$\begin{aligned}
\frac{c_{n}^{2}}{c_{n+1}c_{n-1}} & = & q^{-2\alpha}\prod_{j=1}^{r}\frac{1-a_{j}q^{n-1}}{1-a_{j}q^{n}}\cdot\prod_{k=1}^{s}\frac{1-b_{k}q^{n}}{1-b_{k}q^{n-1}}\\
& \ge & q^{-2\alpha}\prod_{j=1}^{r}\left(1-a_{j}\right)\prod_{k=1}^{s}\left(1-b_{k}q\right)\ge4.\end{aligned}$$ Our assertion follows from Theorem B of [@Dimitrov1] or Theorem A of [@Hutchinson].
[10]{} G. E. Andrews, R. Askey and R. Roy, *Special Functions,* Cambridge University Press, Cambridge, 1999.
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K. Driver, K. Jordaan and A. Martinínez-Finkelshtein, Pólya frequency sequences and real zeros of some $_{3}F_{2}$ polynomials, J. Math. Anal. 332(2007) 1045-1055.
W. K. Hayman On the zeros of a q-Bessel function Complex Analysis and Dynamical Systems, Contemp. Math., Amer. Math. Soc., Providence, RI (2006), 205216.
J. I. Hutchinson, On a remarkable class of entire functions, Trans. Amer. Math. Soc. 25 (1923) 325332.
S. Karlin, Total Positioity, Vol. 1, Stanford University Press, 1968.
M. E. H. Ismail, *Classical and Quantum Orthogonal Polynomials in One Variable,* Cambridge University Press, Cambridge, 2005.
M. E. H. Ismail and R. Zhang, \$q\$-Bessel Functions and Rogers-Ramanujan Type Identities, Proceedings of American Mathematical Society, in production.
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[^1]: This research is partially supported by National Natural Science Foundation of China, grant No. 11371294 and Northwest A&F University.
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---
abstract: 'The formation rate of luminous galaxies seems to be roughly constant from $z\sim 2$ to $\sim 4$ from the recent observations of Lyman break galaxies (LBGs) (Steidel 1999). The abundance of luminous quasars, on the other hand, appears to drop off by a factor of more than twenty from $z\sim 2$ to $z\sim 5$ (Warren, Hewett, & Osmer 1994; Schmidt, Schneider, & Gunn 1995). The difference in evolution between these two classes of objects in the overlapping, observed redshift range, $z=2-4$, can be explained naturally, if we assume that quasar activity is triggered by mergers of luminous LBGs and one quasar lifetime is $\sim 10^{7-8}~$yrs. If this merger scenario holds at higher redshift, for the evolutions of these two classes of objects to be consistent at $z>4$, the formation rate of luminous LBGs is expected to drop off at least as rapidly as $\exp\left(-(z-4)^{6/5}\right)$ at $z>4$.'
author:
- Renyue Cen
title: Evolution of Lyman Break Galaxies Beyond Redshift Four
---
\#1 [[[\#1]{}[,]{} ]{}]{}
Introduction
============
Observations of galaxies in the rest frame UV band ([@Lilly96]1996; [@Madau96]1996; [@Connolly97]1997; [@Sawicki97] 1997; [@Treyer98]1998; [@Pascarelle98] 1998) indicate that the galaxy formation rate rises steeply from $z=0$ to $z\sim 1$, with a nearly constant rate thereafter up to $z\sim 4$ ([@Steidel99]1999). While at low redshift ($z<2$) the evolution of luminous quasar abundance resembles that of luminous galaxies (e.g., Sanders & Mirabel 1996; Boyle & Terlevich 1998), at high redshift ($z>2$) the two classes of objects do not seem to parallel one another, with the luminous quasar formation rate (e.g., Warren 1994; Schmidt 1995) dropping off more steeply than that of luminous galaxies.
In this [*Letter*]{} a phenomenological approach is taken to relate the [*observed*]{} formation rate of luminous LBGs to the [*observed*]{} abundance evolution of luminous quasars at $z>2$. It is shown that, [*if 1) quasar activity is triggered by LBG mergers*]{} and [*2) each quasar period lasts $\sim 10^{7-8}$yrs*]{}, then the apparent difference, [*in both shape and amplitude*]{}, between the evolutions of bright LBGs and bright quasars from $z=2$ to $z=4$ can be explained quantitatively. The first assumption finds its support from both the observational evidence that a significant fraction of quasar hosts have disturbed morphologies or ongoing galaxy-galaxy interactions (e.g., Boyce 1996; Bahcall 1997; Boyce, Disney, & Bleaken 1999) and the theoretical consideration that merger of two (spiral) galaxies seems to provide a natural mechanism to fuel the central black hole (e.g., Barnes & Hernquist 1991). The second assumption is also theoretically well motivated (Rees 1984, 1990) and now strongly implied or required by the mounting observational evidence that most nearby massive galaxies seem to harbor inactive black holes at their centers (e.g., Richstone 1998).
The primary purpose of this work is to use this merger model to infer the LBG formation rate at higher redshift $z>4$. Given the precipitous drop-off of luminous quasar ($M_B<-26.0$) abundance from $z=2$ to $z=5$ the formation rate of luminous ($M_{AB}\geq -23$ to $-22$) LBGs at higher redshift ($z>4$) is predicted to drop off as least as fast as $\propto \exp\left(-(z-4)^{6/5}\right)$, if the merger scenario holds. A cosmological model with $q_0=0.5$ and Hubble constant $H_0=50$km/sec/Mpc is assumed for the analysis presented here.
It is noted that this simple merger model would probably fail at $z<2$ without having taken into account the evolution of gaseous fuel supply to the central black holes in galaxies (Kauffmann & Haehnelt 1999; Haiman & Menou 1999).
Galaxy Merger Rate and Quasar Abundance
=======================================
Denoting $f(z)$ as the galaxy formation rate (galaxy formation per unit time per unit comoving volume) as a function of redshift, then the (cumulative) number density of formed galaxies (number of galaxies per unit comoving volume) is (ignoring the small fraction of galaxies that merge) $$g(z)=\int_\infty^z f(z^\prime) {dt\over dz^\prime} dz^\prime.$$ For simplicity $\Omega_0=1$ will be assumed, which should be a good approximation at high redshift ($z>2$) for the range of cosmological models of current interest ($\Omega_0>0.2$). The merger rate for a galaxy with an internal one-dimensional velocity dispersion $\sigma_i(z)$ in a cluster/group with galaxy number density $d(z)$ (assuming all galaxies under consideration are identical) and one-dimensional velocity dispersion $\sigma_e(z)$ has been computed by Makino & Hut (1997, equation 33) to be: $$P(z)= {18\over \sqrt{\pi}} {1\over x(z)^3} d(z) r_v(z)^2 \sigma_i(z) R(x),$$ where $R(x)$ is a dimensionless function of $x(z)\equiv \sigma_e/\sigma_i$ which depends on the galaxy model and $r_v(z)$ is the virial radius of a galaxy. Makino & Hut (1997) demonstrate that $R(x)$ is a constant ($\sim 11-14$) to good accuracy for $x>2$ for several different galaxy models.
Clearly, not all galaxies participate in merging at any given time; most galaxies have merger time scales much longer than the Hubble time. Rather, only galaxies in dense environments such as groups or clusters of galaxies have significant probability to merge with others. To make the problem more tractable it is assumed that a fraction, $\beta(z)$, of all galaxies \[$g(z)$\] under consideration at any given time is in dense environments (i.e., typical groups/clusters at $z$) where most mergers occur, and the remainder of galaxies (i.e., field galaxies) have zero probability of merger. Then, the total merger rate is $$M(z) = \beta(z) g(z) P(z)$$ and the quasar abundance at any given redshift $z$ is
$$Q(z) = M(z) t_Q(z)$$
where $t_Q(z)$ is the assumed quasar lifetime (assuming that $t_Q$ is much less than the Hubble time, which turns out to be necessary for the model to be viable).
There are two significantly uncertain remaining parameters, $d(z)$ and $\sigma_e(z)$, which need to be specified. It is noted that quasar activities at high redshift seem to occur mostly in regions with galaxy number density typical of present-day clusters/groups of galaxies. This information is provided by observations of quasar companions which have a typical separation from a quasar of a few hundred comoving kiloparsecs (e.g., Djorgovski 1999). At redshift $z=1-2$ there is evidence from larger observational data sets that quasars reside in cluster-like environment (Hall & Green 1998). It thus appears that $d(z)$ may be a weak function of redshift and is assumed to be constant here (more discussion on this later). The velocity dispersion of characteristic systems (groups/clusters in this case), $\sigma_e(z)$, should be a decreasing function of redshift in any hierarchical cosmological model. Here we take advantage of the insight of Kaiser (1986) and use the solution for simple power-law models: $$\sigma_e(z) = \sigma_e(0) (1+z)^{{1\over 2} {n-1\over n+3}},$$ where $n$ is the power index of the primordial density fluctuation spectrum at the relevant scales for clusters/groups. For cold dark matter like models or from observations of local large scale structure $n$ is expected to be $\sim -1$.
The purpose of estimating LBG formation rate at $z>4$ is met by finding $f(z)$ at $z>4$ that matches the observed quasar abundance in the range $z>2$. For the present analysis a simple functional form of LBG formation rate is adopted: $$\begin{aligned}
%&f(z)&=A\left(1-0.9(1-z)\right) \quad \quad \quad \quad\hbox{for} \quad \quad \quad z<1\nonumber\\
&f(z)&=A\quad \quad\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad\hbox{for} \quad \quad \quad 2<z<4\nonumber\\
&f(z)&=A\exp \left(-(z-4)^{6/5}\right) \quad\quad \quad \quad\hbox{for} \quad \quad \quad z>4,\end{aligned}$$ consistent with the latest LBG observations at high redshift up to $z=4$ (Steidel 1999), where $A$ is a normalization constant. At $z>4$, where observations are unavailable, a simple form is proposed so as to provide an adequate fit to the observed quasar abundance at $z>4$ (see Figure 1 below). Using equations (1-3,5-6), we find $Q(z)$ (equation 4), shown as the heavy solid curve in Figure 1. Here, for the shown $Q(z)$ we use $n=-1.0$, $\sigma_e(0)=10^3$km/s, $\sigma_i=100$km/s, $\beta=0.025$ (being constant which is consistent with the adoption of $n=-1$ powerlaw model), $d=40.0~h^{3}$Mpc$^{-3}$, $r_v=200h^{-1}$kpc, $R(x)=12$ and $t_{Q}=3\times 10^{7}$yrs. A cosmological model with $q_0=0.5$ and Hubble constant $H_0=50$km/sec/Mpc is assumed. Also shown as symbols are observational data of bright quasars ($M_B<-26.0$): open circles are from Warren (1994) and solid dots are from Schmidt (1995). The open square from Kennefick, Djordovski, & de Carvalho (1995) for $M_B<-26.7$ quasars is shown to indicate the steepness of quasar luminosity function near the absolute magnitude $M_B\sim -26.0$. It is seen that the merger model provides an adequate fit to the observed luminous quasar abundance in the entire redshift range considered ($z>2$). The dashed curve in Figure 1 shows $f(z)$ with arbitrary vertical units. The dotted curve in Figure 1 shows $g(z)$, normalized to be $1.0\times 10^{-4}h^3$Mpc$^{-3}$ at $z\sim 3$. Note that Figure 5 of Steidel (1999) shows the differential luminosity function of UV bright LBG galaxies (i.e., star-forming galaxies), calling it $g_d(z)$, while here, $g(z)$ is the cumulative density of formed galaxies. Roughly speaking, if $f(z)$ is constant, then $g(z)/g_d(z) = t_H(z)/t_{SF}$, where $t_H(z)$ is the Hubble time at redshift $z$ and $t_{SF}$ is the star (burst) formation duration (i.e., LBG phase) of a galaxy. Since $t_H(z)/t_{SF}\approx 10^9 yrs/10^8 yrs \approx 10$, the above normalization roughly corresponds to LBGs with $g_d(z)\sim 10^{-5}h^3$Mpc$^{-3}$, which in turn corresponds to LBGs with $M_{AB}=-23$ to $-22$ (Figure 5 of Steidel 1999).
Discussion
==========
On one hand, as $Q(z)$ at $z<4$ does not depend sensitively on the form of $f(z)$ at $z>4$, the good agreement between $Q(z)$ and the observed quasar abundance in the redshift range $z=2-4$ (where both types of objects are observed) suggests that the merger scenario of luminous LBGs provides a quantitatively viable model for bright quasar formation. On the other hand, $Q(z)$ at $z>4$ does depend sensitively on the adopted form of $f(z)$ at $z>4$. The fact that the proposed model yields an overall shape at $z=2-5$ that fits observations implies that the luminous LBG formation rate should drop off at $z>4$ as indicated by $f(z)$ in eq. 1, [*if merger scenario holds at $z>4$*]{}. But to have a secure estimate of $f(z)$ at $z>4$, it is vital to understand the dependences of $Q(z)$ on various other parameters, namely, $Q(z)\propto \beta(z) d(z) \sigma_i^4(z) r_v^2(z) t_{Q}(z)(1+z)^{-{3\over 2}{n-1\over n+3}}$. We have set each of the parameters constant (independent of redshift), which is considered to [*conservative*]{} in the following discussions [*if*]{} a more likely redshift dependence of the quoted parameter (holding all other parameters constant) would require an even steeper decreasing function for $f(z)$ at $z>4$ than indicated by equation (6). Let us now examine each parameter to assess how each parameter may vary with redshift.
First, it seems that $\sigma_i(z)$, $r_v(z)$ and $t_{Q}$ are likely to decrease with redshift, making the assumption of their being constant [*conservative*]{}.
Second, $\beta=0.025$ is equivalent to the assumption of mergers taking place in galaxy systems corresponding roughly to $2\sigma$ peaks and has implications for the correlation function of quasars. The bias factor of halos over mass is $b=1+(\nu^2-1)/\delta_c$ (Mo & White 1996), equal to $2.91$ for $\nu=2$ and $\delta_c=1.57$. If the cluster-cluster correlation function has a shape $\propto r^{-2}$ (close to the usual slope of $-1.8$), then the correlation length of clusters is $b r_{m}$, where $r_{m}$ is the correlation length of the underlying mass and evolves as $\propto (1+z)^{-1}$ (Kaiser 1986) for $n=-1$ and $\Omega_0=1$. Our choice of $\beta=0.025$ consequently implies a correlation length for quasars of approximately $2.91 r_{m}(0) /(1+z)$, which is equal to $\sim 5h^{-1}$ comoving Mpc at $z\sim 2$ (using $r_{m}(0)\sim 5.0h^{-1}$Mpc), in agreement with what is observed for quasars (e.g., Kundic 1997; Boyle 1998). In any case, it is unlikely that $\beta$ decreases with redshift. Therefore, setting $\beta(z)$ constant is [*conservative*]{}. An important implication of this model is that the comoving correlation length of luminous quasars should [*decrease*]{} with redshift no faster than $(1+z)^{-1}$ at $z>2$, a potentially testable prediction. Stephens (1997) give a correlation length of $z>2.7$ quasars of $17.5\pm 7.5h^{-1}$Mpc. It will be very valuable to determine the correlation length of high redshift quasars with significantly smaller errorbars.
Third, observations may have indicated that $d(z)$ may be an increasing function of redshift at $z>4$ (Djorgovski 1997; Djorgovski 1999). Therefore, assuming $d(z)$ to be constant is [*conservative*]{}.
Finally, for a plausible power spectrum (such as CDM like) $n$ is likely to be smaller at smaller scales thus smaller at higher redshift. Thus, assuming $n$ to be a constant is [*conservative*]{}. Overall, our assumption of constancy for various parameters seems [*conservative*]{}; i.e., $f(z)$ should decrease at least as rapidly as indicated by equation (6) at $z>4$.
All the analyses so far have been based on the available (optical) observations of quasars, which appears to indicate a sharp drop-off of quasar abundance at $z>4$. Dust obscuration effects are often invoked to explain the apparent drop-off of quasar abundances at high redshift (e.g., Ostriker & Heiler 1984; Pei 1995). However, recent radio surveys of high redshift quasars seem to indicate that the drop-off of the number density of bright radio quasars is very similar to that from optical surveys (e.g., Hook, Shaver, & McMahon 1998) with the implication that the effect of dust on the observed drop-off of bright quasars at $z>2$ may be small.
One potential problem with the merger model is that observations show that a large fraction of quasar hosts at low redshift ($z<0.5$) appear to be quite normal looking, i.e., without disturbed appearances. But one would expect that, if galaxy-galaxy merger time scale is longer than the proposed quasar lifetime, all quasar hosts should display appearances of some interaction. One possible solution to this problem is that quasar formation is delayed, i.e., a quasar does not start to shine until the galaxy-galaxy merger is nearly complete. In other words, the time it takes to set up the central (BH) region for quasar activity during galaxy merger may be comparable to the time that it takes for the two galaxies to merger.
Conclusions
===========
In an early classic paper Efstathiou & Rees (1988) show that quasar abundance at high redshift can be accounted for in the standard cold dark matter model [*if massive halos are related to the formation of black holes*]{}, with an intriguing prediction that the abundance of luminous quasars should decrease rapidly beyond $z=5$ (for a more recent treatment see Haehnelt & Rees 1993). \[The evolution of low-luminosity quasars, of course, does not necessarily have to follow that of their luminous counterparts (e.g., Haiman & Loeb 1998)\].
In this [*Letter*]{} a different approach is taken by directly relating the [*observed*]{} evolution of luminous LBGs to the [*observed*]{} evolution of luminous quasars at high redshift ($z>2$). With a set of seemingly reasonable parameter values, it is shown that consistency between the two classes of objects at $z=2-4$, where both classes are observed, can be achieved, if one assumes that 1) [*Lyman break galaxies merger to trigger quasar activity*]{} and 2) [*quasar lifetime is $\sim 10^{7-8}$yrs*]{}. At $z>4$, consistency can be achieved, [*only if additionally*]{} the formation rate of luminous LBGs drops off as $\exp(-(z-4)^{6/5})$ or faster, a prediction that may be tested by future observations. One implication from this model is that LBGs with $M_{AB}\geq -23$ to $-22$ merge to form quasars with $M_B<-26.0$ at $z>2$. Correlation analysis of relevant LBGs and quasars should shed light on this.
At lower redshift additional, more model dependent assumptions regarding the supply of available gas to fuel black holes would be required to make qualitatively viable predictions. Kauffmann & Haehnelt (1999; see also Haiman & Menou 1999) have presented a detailed model, based also on merger scenario, to unify the evolution of galaxies and quasars in the cold dark matter model under several plausible assumptions concerning the evolution of fuel gas to the central black holes. The success of the model of Kauffmann & Haehnelt (1999) at low redshift ($z<2$) and the model presented here at high redshift ($z>2$), both based on galaxy merger scenario, suggests that [*galaxy merger may play an indispensable role in quasar formation.*]{}
The work is supported in part by grants AST9318185 and ASC9740300. I thank Xiaohui Fan, Zoltan Haiman, Jerry Ostriker, Michael Strauss and David Weinberg for many useful discussions. An anonymous referee is acknowledged for helpful comments.
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abstract: 'The Chiral Constituent Quark Model (CCQM) interactions that bind the $H$ dibaryon and $_{\Lambda\Lambda}^{~~3}{\rm H}$ overbind $_{\Lambda\Lambda}^{~~6}{\rm He}$ by more than 4 MeV, thus outdating the CCQM in the $\cal S$=$-$2 sector.'
author:
- 'A. Gal'
title: 'Comment on recent strangeness $\cal S$=$-$2 predictions'
---
In a recent Letter, Garcilazo and Valcarce [@GV12] reported on a $\Lambda\Lambda N$–$\Xi NN$ coupled-channel three-body Faddeev calculation that binds $_{\Lambda\Lambda}^{~~3}{\rm n}$ and $_{\Lambda\Lambda}^{~~3}{\rm H}$ by about 0.5 MeV below the corresponding $\Lambda\Lambda N$ thresholds. This contrasts with [*ab initio*]{} $A\leq 6$ few-body coupled-channel calculations associating a loosely bound $_{\Lambda\Lambda}^{~~4}{\rm H}$ with the onset of $\Lambda\Lambda$ hypernuclear binding [@Nemura05]. Here I argue that the $\cal S$=$-$2 Chiral Constituent Quark Model (CCQM) interactions [@VGC10] that bind $_{\Lambda\Lambda}^{~~3}{\rm H}$ [@GV12], as well as the unobserved $H$ dibaryon [@CV12], overbind the uniquely identified NAGARA emulsion event of $_{\Lambda\Lambda}^{~~6}
{\rm He}$ [@NAGARA] by more than 4 MeV, casting doubts on the predictive power of the CCQM for $\cal S$=$-$2.
\[h\]
[@FG02] [@FG02] [@FG02] [@FG02] [@FG02] [@CAG97] [@CAG97]
------------------------------ --------- --------- --------- --------- --------- --- ---------- ----------
$-a_{\Lambda\Lambda}$ 0.31 0.77 2.81 5.37 10.6 1.90 21.0
$r_{\Lambda\Lambda}$ 3.12 2.92 2.95 2.40 2.23 3.33 2.54
$\Delta B_{\Lambda\Lambda}$ 0.79 1.51 2.91 3.91 4.51 4.12 8.29
: $\Delta B_{\Lambda\Lambda}({_{\Lambda\Lambda}^{~~6}{\rm He}})$=$
B_{\Lambda\Lambda}({_{\Lambda\Lambda}^{~~6}{\rm He}})$$-$$2B_{\Lambda}
({_{\Lambda}^{5}{\rm He}})$ (in MeV) from $\alpha\Lambda\Lambda$ calculations [@FG02; @CAG97] with no $\Lambda\Lambda$–$\Xi N$ coupling, and scattering lengths $a_{\Lambda\Lambda}$ and effective ranges $r_{\Lambda\Lambda}$ (in fm) of the input $\Lambda\Lambda$ interaction $V_{\Lambda\Lambda}$. $\Delta
B^{\rm exp}_{\Lambda\Lambda}({_{\Lambda\Lambda}^{~~6}{\rm He}})$=0.67$\pm$0.17 MeV [@Nakazawa10].[]{data-label="tab:1"}
Listed in Table \[tab:1\] are $\Delta B_{\Lambda\Lambda}({_{\Lambda\Lambda}^
{~~6}{\rm He}})$ values obtained in two sets of $\alpha\Lambda\Lambda$ three-body calculations [@FG02; @CAG97] which use identical $V_{\Lambda
\alpha}$; $V_{\Lambda\Lambda}$ from Ref. [@CAG97] are softer than $V_{\Lambda\Lambda}$ from Ref. [@FG02]. Within each set $\Delta B_{\Lambda
\Lambda}$ increases with increasing the strength of $V_{\Lambda\Lambda}$, as represented by the listed values of $-a_{\Lambda\Lambda}$. For $a^{\rm CCQM}_{\Lambda\Lambda}$=$-$3.3 fm, corresponding to the decoupled $V^{\rm CCQM}_{\Lambda\Lambda}$ [@Gar12], interpolation within the first set [@FG02] suggests that $\Delta B^{\rm CCQM}_{\Lambda\Lambda}
({_{\Lambda\Lambda}^{~~6}{\rm He}})$=3.2$\pm$0.1 MeV, at variance with $\Delta B^{\rm exp}_{\Lambda\Lambda}({_{\Lambda\Lambda}^{~~6}{\rm He}})
$=0.67$\pm$0.17 MeV [@Nakazawa10]. Interpolation within the second set [@CAG97] results in a value larger by at least 1 MeV. Since $V^{\rm CCQM}_
{\Lambda\Lambda}$ [@CV12] is softer than $V_{\Lambda\Lambda}$[@CAG97], which is softer than $V_{\Lambda\Lambda}$[@FG02], $\Delta B^{\rm CCQM}_
{\Lambda\Lambda}({_{\Lambda\Lambda}^{~~6}{\rm He}})$ should be even larger. Furthermore, the inclusion of the Pauli-suppressed $\Lambda\Lambda$–$\Xi N$ coupling increases $\Delta B_{\Lambda\Lambda}({_{\Lambda\Lambda}^{~~6}
{\rm He}})$ by another 0.2–0.5 MeV [@CAG97], and by much more in the CCQM owing to its stronger coupling effects. Altogether I estimate conservatively $\Delta B^{\rm CCQM}_{\Lambda\Lambda}({_{\Lambda\Lambda}^{~~6}{\rm He}})>4.7
$$\pm$0.5 MeV, overbinding $_{\Lambda\Lambda}^{~~6}{\rm He}$ by more than 4.0$\pm$0.5 MeV and thereby destroying the consistency among the bulk of $\Lambda\Lambda$ hypernuclear data [@GM11].
The CCQM $\Lambda\Lambda$–$\Xi N$ coupled-channel interactions used in Ref. [@GV12] are not unambiguously constrained by the scarce, imprecise free-space scattering data [@Ahn06]. Figure 5 in Ref. [@HM12] shows a variety of $\cal S$=$-$2 interactions satisfying such constraints. In particular, there are no $\Lambda\Lambda$ scattering data to constrain $a_{\Lambda\Lambda}$. Recent analysis of the $\Lambda\Lambda$ invariant mass from the in-medium reaction $^{12}$C($K^-,K^+\Lambda\Lambda X$) [@Yoon07] results in $a_{\Lambda\Lambda}$=$-$1.2$\pm$0.6 fm [@GHH12], consistently with $a_{\Lambda\Lambda}$$\sim$$-$0.5 fm from $_{\Lambda\Lambda}^{~~6}{\rm He}$ [@FG02; @VRP04], in disagreement with $a^{\rm CCQM}_{\Lambda\Lambda}$=$-$3.3 fm. Furthermore, the very strong CCQM $\Lambda\Lambda$–$\Xi N$ coupling interaction which leads to a bound $H$ below the $\Lambda\Lambda$ threshold [@CV12] and is also responsible for binding $_{\Lambda\Lambda}^{~~3}{\rm H}$, is at odds with the latest HAL QCD lattice-simulation analysis which locates the $H$ dibaryon near the $\Xi N$ threshold [@HALQCD12]. For all these reasons, foremost for heftily overbinding $_{\Lambda\Lambda}^{~~6}{\rm He}$, the predictive power of the CCQM for $\cal S$=$-$2, including the prediction of a $_{\Lambda\Lambda}^{~~3}
{\rm H}$ bound state [@GV12], is questionable.
[00]{}
H. Garcilazo and A. Valcarce, PRL [**110**]{}, 012503 (2013).
H. Nemura, S. Shinmura, Y. Akaishi, and K.S. Myint, PRL [**94**]{}, 202502 (2005).
A. Valcarce, H. Garcilazo, and T.F. Caramés, PLB [**693**]{}, 305 (2010).
T.F. Caramés and A. Valcarce, PRC [**85**]{}, 045202 (2012).
H. Takahashi [*et al.*]{}, PRL [**87**]{}, 212502 (2001).
I.N. Filikhin and A. Gal, NPA [**707**]{}, 491 (2002); see also I.N. Filikhin, A. Gal, and V.M. Suslov, PRC [**68**]{}, 024002 (2003), who discuss $\Lambda\Lambda$–$\Xi N$ coupling.
S.B. Carr, I.R. Afnan, and B.F. Gibson, NPA [**625**]{}, 143 (1997); I.R. Afnan and B.F. Gibson, PRC [**67**]{}, 017001 (2003). Both include $\Lambda\Lambda$–$\Xi N$ coupling.
K. Nakazawa, NPA [**835**]{}, 207 (2010).
H. Garcilazo (private communication, Dec. 2012).
A. Gal and D.J. Millener, PLB [**701**]{}, 342 (2011).
J.K. Ahn [*et al.*]{}, PLB [**633**]{}, 214 (2006).
J. Haidenbauer and U.-G. Mei[ß]{}ner, NPA [**881**]{}, 44 (2012).
C.J. Yoon [*et al.*]{} (KEK E522), PRC [**75**]{}, 022201 (2007).
A.M. Gasparyan, J. Haidenbauer, and C. Hanhart, PRC [**85**]{}, 015204 (2012); see also A. Ohnishi [*et al.*]{} (ExHIC Collab.), NPA (2013) arXiv:1301.7261, who deduce from recent RHIC-STAR data that $a_{\Lambda\Lambda}>-$1.25 fm.
I. Vidaña, A. Ramos, and A. Polls, PRC [**70**]{}, 024306 (2004), who include $\Lambda\Lambda$–$\Xi N$–$\Sigma\Sigma$ coupling.
T. Inoue [*et al.*]{} (HAL QCD Collab.), NPA [**881**]{}, 28 (2012), who consider SU(3) flavor breaking, realistic ${\cal S}$=$-$2 thresholds and coupled-channels effects.
|
---
abstract: 'The anisotropies in the pressure obtained from the energy-momentum tensor are studied for magnetized quark matter within the su(3) Nambu-Jona-Lasinio model for both $\beta$-equilibrium matter and quark matter with equal quark chemical potentials. The effect of the magnetic field on the particle polarization, magnetization and quark matter constituents is discussed. It is shown that the onset of the $s$-quark after chiral symmetry restoration of the $u$ and $d$-quarks gives rise to a special effect on the magnetization in the corresponding density range: a quite small magnetization just before the $s$ onset is followed by a strong increase of this quantity as soon as the $s$ quark sets in. It is also demonstrated that for $B<10^{18}$ G within the two scenarios discussed, always considering a constant magnetic field, the two components of pressure are practically coincident.'
author:
- 'Débora P. Menezes'
- 'Marcus B. Pinto'
- Constança Providência
title: Anisotropy in the EoS of Magnetized Quark Matter
---
The structure of the QCD phase diagram is of utmost importance in understanding many physical aspects of nature, ranging from the early universe to possible nuclear liquid-gas and hadronic quark matter phase transitions to the physics of compact objects [@reviews]. Early analyzes performed within the Nambu–Jona-Lasinio model (NJL) framework indicate that when strongly interacting matter is subject to intense magnetic fields the QCD phase diagram boundaries are modified [@qcd+b]. Some of the most important changes concern the size and location of the first order chiral transition region since the results show that a strong magnetic field favors this type of transition. At the same time, at low temperatures, the value of the coexistence chemical potential decreases as $B$ increases in accordance with the inverse magnetic catalysis phenomenon (ICM) [@preis].
The low-$T$/high-$\mu$ region where a first order type transition is expected to occur is currently unavailable to lattice QCD evaluations (LQCD). However, the region high-$T$/low-$\mu$, has already been exploited using LQCD simulations which indicate, in accordance with most model predictions, that the crossover observed at $B=0$ persists when $B \ne 0$ [@earlylattice; @preussker; @lattice1; @lattice2]. On the other hand, a major disagreement between recent LQCD results [@lattice1; @lattice2] and model calculations regards the dependence of crossover pseudocritical temperature, $T_{\rm pc}$, on the strength $B$ of the magnetic field. Specifically, the lattice results of Refs. [@lattice1; @lattice2], performed with $2+1$ quark flavors and physical pion mass values, predict an inverse catalysis, with $T_{\rm pc}$ decreasing with $B$, while effective models predict an increase of $T_{\rm pc}$ with $B$. This problem has been recently addressed by different groups [@teoriaIMC; @catalysis-ours] which basically agree that the different results stem from the fact that most effective models miss back reaction effects (the indirect interaction of gluons and $B$) as well as the QCD asymptotic freedom phenomenon. At the same time, other important aspects of the effects of strong magnetic fields on the QCD phase diagram have already been studied including the behavior of the coexistence chemical potential and the location of the critical end point (CEP) [@CEP1], the dependence of the CEP on strangeness, isospin and charge asymmetry [@CEP2] and also the internal structure of the phase diagram [@structure].
Regarding physical observables, the understanding of magnetized quark matter is particularly important at low densities and high temperatures, which is the relevant regime for the present heavy ion collisions experiments [@kharzeev], as well as at low temperatures and high densities, which is the regime concerning magnetars [@magnetars].
As far as heavy ion-collisions are concerned the presence of a strong magnetic field most certainly plays a role despite the fact that, in principle, the field intensity should decrease very rapidly lasting for about 1-2 fm/c only [@kharzeev]. The possibility that this short time interval may [@tuchin] or may not [@mclerran] be affected by conductivity remains under dispute. The effects of strong magnetic fields and their relation with the impact parameters have also been discussed [@event] while the particle yields dependence on a constant external magnetic field has been investigated in a naive approach [@paoli]. Another aspect related to the presence of strong magnetic fields at the early stages of the collisions is the anisotropy of photon production in heavy ion collisions at the RHIC energies [@photon_anisotropy]. The new PHENIX data brings some doubts on the conventional picture of thermalization and subsequent hydrodynamics or infers the possibility that a new photon production mechanism is possible. These works tell us that there is much to be done if a complete understanding of the effects of hadronic matter subject to strong magnetic fields is expected.
When we look at the recent literature on magnetars, the controversy is already present at the level of calculating the energy-momentum tensor. While some of the first works advocated that the pressure, having in mind the thermodynamical pressure obtained from the thermodynamical potential that relates pressure to density, should be isotropic [@prc1; @prc2; @luizlaercio; @rudiney], based on an interpretation given in Ref. [@blandford], other works were based on the fact that the energy-momentum tensor gives different contributions for the parallel and perpendicular pressure [@sedrakian; @jorge; @aurora] (see Ref. [@gregor] for a discussion based on LQCD). If two different pressures are indeed present in the system, the usual way of using the equation of state as input to the Tolman-Oppenheimer-Volkoff equations (TOV) [@tov], which determine the structure of a spherically symmetric body of isotropic material in static gravitational equilibrium, to obtain compact objects macroscopic properties, as radii and masses, has to be done with care. What is normally done is to observe at what value of the magnetic fields, the two pressures start to deviate and then use the TOV equations up to this strength, so that in principle, the EoS used as input is [*practically*]{} isotropic [@jorge; @aurora; @veronica]. Another important aspect is related to the contribution of the electromagnetic interaction to the pressure(s) and energy density, a term proportional do $B^2$, where $B$ is the magnetic field strength. Since no field larger then $10^{16}$G has been observed at the surface of a magnetar, but according to the virial theorem one could expect fields as strong as $\sim 10^{18}$G in the interior, an [*ad hoc*]{} exponential density-dependent magnetic field was proposed in ref. [@Pal] and widely adopted in subsequent works [@Mao; @Rabhi; @prc2; @rudiney; @Ryu; @Rabhi2; @Mallick; @luizlaercio; @Dex; @njlv; @Mallick2; @Ro]. This [*ansatz*]{}, however, violates Maxwell equations. Another similar prescription for an [*energy density dependent*]{} magnetic field was proposed in Ref. [@chaotic].
To avoid the use of the TOV equations, the authors of Ref. [@Mallick2] treated the anisotropic pressure as a perturbation in a way similar to the Hartle-Thorn method, generally used for slowly rotating neutron stars. In a more complete treatment, the authors consider the anisotropy in solving Einstein’s field equations in axisymmetric regime with a fully general relativistic formalism [@Bocquet; @Cardall; @Micaela]. In both cases, once the macroscopic properties are obtained, a small increase of the maximum mass is found in contrast with the other previous works. Let us comment that the results computed from the LORENE modified code in Ref. [@Micaela], generate a magnetic field in the central regions of the star of the order of $10^{18}$ G, which seems reasonable but at the surface, its value is of the order of $10^{17}$ G, which is too large in contrast with observational results that point to a maximum value of $10^{15}$ G.
According to classical books on gravitation [@Misner; @Zel], when anisotropies are present, the concept of pressure is not so well defined. Based on the concepts discussed in these two books, in Ref. [@chaotic] a small-scale chaotic field is used and the stress tensor is modified, so that the resulting EoS is also isotropic. A curious outcome is that the increase in the maximum stellar mass is also very small, as found in Ref. [@Mallick2; @Micaela]. Hence, it is clear that there is no unique way of computing magnetic field effects on compact stars.
An estimation of the maximum magnetic field intensity supported by a star before magnetic field stresses give rise to the formation of a black hole may be obtained equating the magnetic field energy of an uniform field in a sphere with the star radius $R$ to the gravitational binding energy. A maximum field of the order of $10^{18}$ G is obtained in agreement with the maximum fields obtained in the framework of a relativistic magneto-hydrostatic formalism, of the order of $\sim 5\times 10^{18}$G Ë™ with a nucleonic EOS [@Cardall], or $\sim 3 \times
10^{18}$ G in Ref. [@broderick2002] with an hyperonic EOS. It was suggested that a disordered field with $\langle B^2\rangle> \langle \vec
B\rangle^2$ could possibly give rise to larger fields in still stable stars. The previous estimations referred to stars that are bound by gravitation. For self-bound stars larger fields could in principle exist [@noronha2007]. Taking these numbers as indicative we next consider fields $B\le 1.5 \times 10^{19}$ G.
In the present work we investigate, within the su(3) version of the Nambu- Jona-Lasinio [@njl] model the quark matter polarization and magnetization, the thermodynamical pressure, and the parallel and perpendicular pressure contributions obtained from the energy-momentum tensor. We consider both $\beta$-equilibrium matter and quark matter with equal chemical potentials for the three flavors, [ which we call symmetric quark matter or matter with isochemical potentials throughout the text.]{} The first scenario applies to neutron stars while the second is relevant to heavy ion collisions investigations. Some of these quantities are inputs for numerical codes that calculate the structure of neutron stars subject to strong magnetic fields.
The paper is organized as follows: In Sections I and II, the general formalism and the resulting equation of state, developed in Refs. [@prc1; @prc2] are revisited. In Section III, the expressions for the magnetization and the anisotropic pressures are shown, with some of the details given in Appendix A. In Section IV the results are shown and discussed and in Section IV the final conclusions are drawn.
General formalism
=================
In order to consider [ both symmetric quark matter and stellar quark matter]{} in $\beta$ equilibrium with strong magnetic fields we introduce the following Lagrangian density $${\cal L} = {\cal L}_{f}+{\cal L}_{l} - \frac {1}{4}F_{\mu
\nu}F^{\mu \nu} \,,$$ [ which contains a quark sector, ${\cal L}_f$, a leptonic sector, ${\cal L}_l$, and the electromagnetic contribution.]{} The quark sector is described by the su(3) version of the Nambu–Jona-Lasinio model $${\cal L}_f = {\bar{\psi}}_f \left[\gamma_\mu\left(i\partial^{\mu}
- {\hat q}_f A^{\mu} \right)-
{\hat m}_c \right ] \psi_f ~+~ {\cal L}_{sym}~+~{\cal L}_{det}~.
\label{njl}$$ The ${\cal L}_{sym}$ and ${\cal L}_{det}$ terms are given by: $${\cal L}_{sym}~=~ G \sum_{a=0}^8 \left [({\bar \psi}_f \lambda_ a \psi_f)^2 + ({\bar \psi}_f i\gamma_5 \lambda_a
\psi_f)^2 \right ] ~,
\label{lsym}$$ $${\cal L}_{det}~=~-K \left \{ {\rm det}_f \left [ {\bar \psi}_f(1+\gamma_5) \psi_f \right] +
{\rm det}_f \left [ {\bar \psi}_f(1-\gamma_5) \psi_f \right] \right \} ~,
\label{ldet}$$ where $\psi_f = (u,d,s)^T$ represents a quark field with three flavors, ${\hat m}_c= {\rm diag}_f (m_u,m_d,m_s)$ with $m_u=m_d \ne m_s$ is the corresponding (current) mass matrix while ${\hat q}_f={\rm diag}(q_u,q_d,q_s)$ is the matrix that represents the quark electric charges. In the same equation, $\lambda_0=\sqrt{2/3}I$ where $I$ is the unit matrix in the three flavor space, and $0<\lambda_a\le 8$ denote the Gell-Mann matrices. The t’Hooft interaction term (${\cal L}_{det}$) represents a determinant in flavor space which, for three flavor, gives a six-point interaction [@buballa] and ${\cal L}_{sym}$, which is symmetric under global $U(N_f)_L\times U(N_f)_R$ transformations and corresponds to a four-point interaction in flavor space. The model is non renormalizable, and as a regularization scheme for the divergent ultraviolet integrals we use a sharp cut-off $\Lambda$ in three-momentum space. The parameters of the model, $\Lambda$, the coupling constants $G$ and $K$ and the current quark masses, $m_u$ and $m_s$, are determined by fitting $f_\pi$, $m_\pi$ , $m_K$ and $m_{\xi'}$ to their empirical values. We adopt the parametrization of the model proposed in [@hatsuda]: $\Lambda = 631.4 \, {\rm MeV}$ , $m_u= m_d=\, 5.5\, {\rm MeV}$, $m_s=\, 135.7\, {\rm MeV}$, $G \Lambda^2=1.835$ and $K \Lambda^5=9.29$.
The leptonic sector is described by $$\mathcal{L}_l=\bar \psi_l\left[\gamma_\mu\left(i\partial^{\mu} - q_l A^{\mu}
\right) -m_l\right]\psi_l \,\,,
\label{lage}$$ where $l=e,\mu$. One recognizes this sector as being represented by the usual QED type of Lagrangian density. As usual, $A_\mu$ and $F_{\mu \nu }=\partial
_{\mu }A_{\nu }-\partial _{\nu }A_{\mu }$ are used to account for the external magnetic field. We are interested in a static and constant magnetic field in the $z$ direction and hence, we choose the gauge $A_\mu=\delta_{\mu 2} x_1 B$.
The EoS
=======
We now need to evaluate the thermodynamical potential for the three flavor quark sector, $\Omega_f$, which as usual can be written as $$\Omega_f = -P_f = {\cal E}_f - T {\cal S}_f - \sum_f \mu_f \rho_f,$$ where $P_f$ represents the pressure, ${\cal E}_f$ the energy density, $T$ the temperature, ${\cal S}_f$ the entropy density, $\mu_f$ the chemical potential, and $\rho_f$ the quark number density. A similar expression can be written for the leptonic sector.
The total pressure for three flavor in $\beta$ equilibrium is then given by $$P(\mu_f,\mu_l,B)= P_f |_{M_f}+ P_l |_{m_l} \pm \frac{B^2}{2}
\,\,,$$ where our notation means that $P_f$ is evaluated in terms of the quark effective mass, $M_f$, which is determined in a (nonperturbative) self consistent way while $P_l$ is evaluated at the leptonic bare mass, $m_l$. The term $B^2/2$, arises due to the kinetic term of the electromagnetic field, $F_{\mu \nu}F^{\mu \nu}/4$, in the original Lagrangian density. [ Within the formalism used in the present work, the sign of this term comes from the stress tensor and is shown in the section where we discuss anisotropy.]{} Remark also that in the sequel our results will be presented in terms of (vacuum) subtracted pressures ($\Delta P$) such that $P_f=0$ at $\mu_f=0$ ($f=u,s,d$) and $P_l=0$ at $\mu_l=0$ ($l=e,\mu$). With this normalization choice only the magnetic pressure, ($ \pm B^2/2)$, survives at vanishing chemical potentials. Again, a similar expression can be written for the leptonic sector, apart from the color index. The lepton masses are $m_e=0.511 \, {\rm MeV}$ and $m_\mu=105.66\, {\rm MeV}$.
In the mean field approximation the pressure can be written as $$P_f = \theta_u+\theta_d+\theta_s
-2G(\phi_u^2+\phi_d^2+\phi_s^2) + 4K \phi_u \phi_d \phi_s \,\,,$$ where the free gas type of term is $$\theta_f=-\frac{i}{2} {\rm tr} \int \frac {d^4 p}{(2\pi)^4} \ln \left(-p^2 + M_f^2 \right ),$$ while the scalar condensates, $\phi_f$ are given by $$\phi_f= \langle {\bar \psi}_f \psi_f \rangle= -i \int \frac {d^4 p}{(2\pi)^4} {\rm tr}\frac{1}{(\not \!
p - M_f+i\epsilon)}
\label{cond}$$ According to standard Feynman rules for this model, all the traces are to be taken over color ($N_c=3$) and Dirac space, but not flavor.
The effective quark masses can be obtained self consistently from $$M_i=m_i - 4 G \phi_i + 2K \phi_j \phi_k,
\label{mas}$$ with $(i,j,k)$ being any permutation of $(u,d,s)$. So, to determine the EOS for the su(3) NJL at finite density and in the presence of a magnetic field we need to know the condensates, $\phi_f$, as well as the contribution from the gas of quasiparticles, $\theta_f$. Both quantities, which are related by $\phi_f \sim d \theta_f /
dM_f$, have been evaluated with great detail in [@prc1; @prc2] so that here we just quote the results:
$$\theta_f= \left (\theta^{vac}_f+\theta^{mag}_f + \theta^{med}_f \right )_{M_f}\,\,,
\label{pressBmu2}$$
where the vacuum contribution reads $$\theta^{vac}_{f}=- \frac{N_c }{8\pi^2} \left \{ M_f^4 \ln \left [
\frac{(\Lambda+ \epsilon_\Lambda)}{M_f} \right ]
- \epsilon_\Lambda \, \Lambda\left(\Lambda^2 + \epsilon_\Lambda^2 \right ) \right \},$$ and where we have also defined $\epsilon_\Lambda=\sqrt{\Lambda^2 + M_f^2}$ with $\Lambda$ representing a non covariant ultraviolet cut off, the finite magnetic contribution is given by $$\theta^{mag}_f= \frac {N_c (|q_f| B)^2}{2 \pi^2} \left [ \zeta^{\prime}(-1,x_f) - \frac {1}{2}( x_f^2 - x_f) \ln x_f +\frac {x_f^2}{4} \right ]\,\,,$$ with $x_f = M_f^2/(2 |q_f| B)$ while $\zeta^{\prime}(-1,x_f)= d
\zeta(z,x_f)/dz|_{z=-1}$ where $\zeta(z,x_f)$ is the Riemann-Hurwitz zeta function. To take further derivatives, as well as for numerical purposes, it is useful to use the following representation for this quantity $$\zeta^{\prime}(-1,x_f)=\zeta^{\prime}(-1,0)+\frac{x_f}{2}[x_f-1-\ln(2\pi) + \psi^{(-2)}(x_f)] \;,$$ where $\psi^{(m)}(x_f)$ is the $m$-th polygamma function and the $x_f$ independent constant is $\zeta^{\prime}(-1,0)=-1/12$. The medium contribution can be written as $$\theta^{med}_f=T \sum_{k=0}^{k_{f,max}} \alpha_k \frac {|q_f| B N_c }{4 \pi^2}
\int_{-\infty}^{+\infty} dp \left[ \ln\left(1 + \exp[-(E^*_f - \mu_f)/T]\right)
+ \ln\left(1 + \exp[-(E^*_f + \mu_f)/T]\right) \right],
\label{PmuB}$$ with $\alpha_0=1,\,\alpha_{k>0}=2$. In the above equation we have defined the energy dispersion $$E^*_f= \sqrt{p^2 + s_f(k,B)^2} ~ , \qquad s_f(k,B)= \sqrt {M_f^2 + 2 |q_f| B k}\;.$$ When considering just the zero temperature case, Eq.(\[PmuB\]) becomes: $$\theta^{med}_f=\sum_{k=0}^{k_{f,max}} \alpha_k\frac {|q_f| B N_c }{4 \pi^2} \left [ \mu_f \sqrt{\mu_f^2 - s_f(k,B)^2} -
s_f(k,B)^2 \ln \left ( \frac { \mu_f +\sqrt{\mu_f^2 -
s_f(k,B)^2}} {s_f(k,B)} \right ) \right ] ,
\label{PmuBt0}$$ where $s_f(k,B) = \sqrt {M_f^2 + 2 |q_f| B k}$. At $T=0$, the upper Landau level (or the nearest integer) is defined by $$k_{f, max} = \frac {\mu_f^2 -M_f^2}{2 |q_f|B}= \frac{p_{f,F}^2}{2|q_f|B}.
\label{landaulevels}$$
The condensates $\phi_f$ entering the quark pressure at finite density and in the presence of an external magnetic field can be written as
$$\phi_f=(\phi_f^{vac}+\phi_f^{mag}+\phi_f^{med})_{M_f}$$
where
$$\begin{aligned}
\phi_f^{vac} &=& -\frac{ M_f N_c }{2\pi^2} \left [
\Lambda \epsilon_\Lambda -
{M_f^2}
\ln \left ( \frac{\Lambda+ \epsilon_\Lambda}{{M_f }} \right ) \right ]\,\,,\end{aligned}$$
$$\phi_f^{mag}
= -\frac{ M_f |q_f| B N_c }{2\pi^2}\left [ \ln \Gamma(x_f) -\frac {1}{2} \ln (2\pi) +x_f -\frac{1}{2} \left ( 2 x_f-1 \right )\ln (x_f) \right ] \,\,,$$
and $$\begin{aligned}
\phi_f^{med}&=&
\sum_{k=0}^{k_{f,max}} \alpha_k \frac{ M_f |q_f| B N_c }{4 \pi^2}
\int_{-\infty}^{+\infty} dp \frac{(f_+ + f_-)}{E^*_f}\,,\label{MmuB}\end{aligned}$$ where the Fermi distribution functions are $$f_{\pm}= {1}/\{1+\exp[ (E^*_f \mp \mu_f)/T]\}\;.
\label{dfq}$$ The quark density reads $$\rho_f= \sum_{k=0}^{k_{f,max}} \alpha_k \frac{|q_f| B N_c }{2 \pi^2} \int_{-\infty}^{+\infty} dp (f_+ - f_-)
\label{densq}.$$
At $T=0$, Eqs.(\[MmuB\]) and (\[densq\]) become: $$\phi_f^{med}=
\sum_{k=0}^{k_{f,max}} \alpha_k \frac{ M_f |q_f| B N_c }{2 \pi^2}\left [\ln \left ( \frac { \mu_ f +\sqrt{\mu_f^2 -
s_f(k,B)^2}} {s_f(k,B)} \right ) \right ]\,\,,
\label{MmuBt0}$$ and $$\rho_f=
\sum_{k=0}^{k_{f,max}} \alpha_k \frac{|q_f| B N_c }{2 \pi^2} k_{F,f} \,\,,$$ where $k_{F,f}=\sqrt{\mu_f^2 - s_f(k,B)^2}$. The entropy density ${\cal S}_f=-(\partial \Omega/\partial T)$ is $${\cal S}_f =-\sum_f \sum_{k=0}^{k_{f,max}} \alpha_k \frac {|q_f| B N_c }{4 \pi^2} \int_{-\infty}^{+\infty} dp
\left[f_+ \ln\left(f_+\right)
+\left(1-f_+\right) \ln\left(1-f_+\right)+(f_+\leftrightarrow f_-)
\right]\;.$$ The corresponding leptonic contributions can be trivially obtained from the above quantities by replacing $M_f \to m_l$, $|q_f| \to |q_l|$ and $\mu_f \to \mu_l$. Since the leptonic masses are unaffected by strong interactions one considers their bare values which do not depend on $T$, $\mu$ and $B$ as opposed to the effective $M_f$ masses. Therefore, the only piece which effectively contributes to the subtracted pressure, defined as $\Delta P= P(T,\mu,B) - P(0,0,B)$, is $$\begin{aligned}
P_l^{med}= T\sum_{k=0}^{k_{f,max}} \alpha_k \frac {|q_l| B}{4 \pi^2}
\int_{-\infty}^{+\infty} dp \left[ \ln\left(1 + \exp[-(E_l - \mu_l)/T]\right)
+ \ln\left(1 + \exp[-(E_l + \mu_l)/T]\right) \right]\;,
\label{Pl}\end{aligned}$$ from which the leptonic density can be written as $$\rho_l= \sum_{k=0}^{k_{f,max}} \alpha_k \frac{|q_l| B}{2 \pi^2} \int_{-\infty}^{+\infty} dp (l_+ - l_-)
\,\,,$$ where $\mu_l$ represents the leptonic chemical potential. The quantities $E_l$ and $l_\pm$ can be obtained from their quark counterparts by the replacements already mentioned.
At vanishing temperatures the above expressions [become]{}:
$$P_l^{med}=\sum_{l=e}^\mu \sum_{k=0}^{k_{l,max}} \alpha_k\frac {|q_l| B }{4 \pi^2} \left [ \mu_l \sqrt{\mu_l^2 - s_l(k,B)^2} -s_l(k,B)^2 \ln \left ( \frac { \mu_l +\sqrt{\mu_l^2 - s_l(k,B)^2}}
{s_l(k,B)} \right ) \right ]~.\nonumber \\$$
and $$\rho_l = \sum_{k=0}^{k_{l, max}}\alpha_k \frac{ |q_l| B }{2 \pi^2} k_{F,l}(k,s_l) \,\,,
\label{rholmuB}$$ where $k_{F,l}(k,s_l) =\sqrt{\mu_l^2 - s_l(k,B)^2}$.
The anisotropy in the pressure
================================
The parallel and the perpendicular components of the pressure can be written in terms of the magnetization, ${\cal M} = {\partial \Delta
P}/{\partial B}$, as [@blandford; @veronica; @Mallick]:
$$P_{\parallel}=\Delta P - \frac{B^2}{2} \quad {\rm and} \quad P_{\perp}=\Delta P -{\cal
M}B + \frac{B^2}{2} \,
\label{eq:press}$$
where $\Delta P$ stands for the already defined subtracted pressure. [For a magnetic field in the $z$ direction, the stress tensor has the form: diag$(B^2/2,B^2/2,-B^2/2)$ and this explains the difference in sign appearing in the parallel and perpendicular pressures.]{} For the leptonic sector one easily gets $$\frac{ dP_l^{ med}}{dB}=\frac{P_l^{med}}{B}- \frac{ |q_l|B}{4\pi^2} \sum_{k=0}^{k_{f,max}} \alpha_k(k|q_l|) \int_{-\infty}^{+\infty} dp \frac{1}{E_l}[l_+ + l_-] \;,$$ which, at $T=0$ becomes: $$\frac{d P^{ med}_l}{dB}= \frac{P_l^{med}}{B}- \frac{B |q_l|}{2\pi^2} \sum_{k=0}^{k_{max}} \alpha_k \ln \left ( \frac{ \mu_l + \sqrt{\mu_l^2-s_l^2}}{s_f} \right ) (k|q_l|) \;,$$ whereas for the quark sector one obtains
$$\frac{d P_f}{d B} = \theta_u^\prime+\theta_d^\prime+\theta_s^\prime
-4G(\phi_u \phi_u^\prime+\phi_d \phi_d^\prime+\phi_s \phi_s^\prime) + 4K (\phi_u^\prime \phi_d \phi_s +\phi_u \phi_d^\prime \phi_s+ \phi_u \phi_d \phi_s^\prime) \,\,,$$
where $$\theta^\prime_f=(\theta^{\prime\, vac}_f+\theta^{\prime \, mag}_f+\theta^{\prime \, med}_f)_{M_f} \,\,\,,$$ and $$\phi^\prime_f=(\phi^{\prime \, vac}_f+\phi^{\prime \, mag}_f+ \phi^{\prime \, med}_f)_{M_f} \,\,\,.$$ The primes denote derivatives with respect to $B$ and the explicit form of each term can be found in the appendix A.
Results and discussion
======================
In this section we present and discuss results concerning several properties of $\beta$-equilibrium quark matter and symmetric quark matter (with equal chemical potentials). We first discuss some general properties of quark matter, in particular, its particle content and particle polarization. We then investigate how the magnetization of quark matter changes with the magnetic field intensity and finally we discuss the parallel and perpendicular pressure contributions obtained from the energy-momentum tensor.
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-- --
In Fig. \[fig1\] it is shown how the magnetic field affects $\beta$-equilibrium matter (left panels) and symmetric quark matter (right panels), in particular, the quark and electron polarizations, their densities and onset of total polarization. The $\beta$-equilibrium matter onset of the $s$ quark occurs just below 0.7 fm$^{-3}$, see Fig. \[fig1:c\], and, therefore, close to these densities the $s$ quark density is small and feels strongly the magnetic field, total polarization being attained with $B<10^{18}$ G. In denser matter, the larger densities of $s$ quarks require larger magnetic field intensities for a total polarization. In symmetric quark matter the $s$ quarks set in above 0.9 fm$^{-3}$, a density that presently is not attained in the laboratory. In $\beta$-equilibrium matter $u$ and $d$ quarks are not totally polarized for fields below 10$^{19}$ G, however in symmetric quark matter $u$ and $d$ quark total polarization occurs at small densities that do not exist in quark stellar matter with a surface baryonic density which is of the order of 0.3 fm$^{-3}$. In Fig. \[fig1:a\] we also show information on the electron polarization. The magnetic field increases the electron content and for $10^{18}$G their density is practically constant and equal to $\sim
0.01$ fm$^{-3}$, see Fig. \[fig1:c\]. At the onset of the $s$ quark the density of electron has always a maximum, above which the electron fraction decreases. This means that electrons are totally polarized for fields $B\gtrsim 9\times 10^{17}$G.
For totally polarized matter, all particles lie on the lowest Landau level (LLL). In this case, the dependence of the pressure on the magnetic field intensity of a gas of free particles occurs only through a multiplicative factor that defines the LLL degeneracy, and the magnetization is independent of $B$. For the NJL model the interaction terms give a nonlinear dependence to the pressure even above for totally polarized matter. This contribution becomes small when the chiral symmetry is restored.
-- --
-- --
In order to determine the two pressure contributions, the magnetization is obtained using Eq. (\[eq:mag\]). In Fig. \[fig2\] this quantity is plotted for $\beta$-equilibrium matter, Figs. \[fig2:a\] and \[fig2:c\], and quark matter with equal chemical potentials, Figs. \[fig2:b\] and \[fig2:d\]. For each case we have considered a set of densities of interest: a) the surface baryonic density of a quark star is $\sim 2\rho_0$ and in the interior we may have densities larger than $5\rho_0$; b) in heavy ion collisions we may have densities below $\rho_0$ and do not expect densities as large as 5$\rho_0$. The magnetization has a term that explodes whenever $\mu_f=s_f$, i.e., whenever $B$ approaches a $n\ne 0$ Landau level from below, see Eq. (\[phimed\]) and the corresponding discussion in Ref. [@sedrakian]. The spikes in this figure occur precisely at this values of $B$. A small number of spikes occurs if only a few LL are occupied, e.g. if the magnetic field is very strong with respect to the Fermi momentum of the particle. In Figs. \[fig2:a\] and \[fig2:b\] for the densities represented this occurs for the larger fields. For the symmetric matter at smaller densities, fewer spikes are obtained for larger magnetic fields. It is interesting to notice that for $\beta$-equilibrium matter there is a range of densities between 0.5 and 0.7 fm$^{-3}$ where for all the fields shown the number of oscillations is small: this occurs before the onset of the $s$ quark and after the $u$ and $q$ quark restoration of chiral symmetry. [ In Fig. \[fig2:c\], for densities lower than 0.3 fm$^{-3}$ and in between 0.8 and 1.1 fm$^{-3}$, one can clearly see that oscillations with high frequencies are modulated by smaller frequency oscillations. The same is seen in Fig. \[fig2:d\], for densities larger than 0.3 fm$^{-3}$. This superposition of fluctuations with different frequencies]{} is due to a mixing of particles with different charges and masses. In $\beta$-equilibrium matter the oscillations defined by the $u$ quark are wider apart because the $u$ quark density is smaller and its charge larger, both effects adding to a reduction of the number of occupied LL, see Fig. \[fig2:c\]. This same effect if also present in Fig. \[fig2:d\] where, for similar $u$ and $d$ quark densities, the charge of the $u$ explains the difference.
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-- --
In Fig. \[fig3\] the two pressure contributions defined Eq. (\[eq:press\]), except for the terms proportional to $B^2$, are plotted for both $\beta$-equilibrium matter (left panels) and symmetric matter (right panels). As in the previous discussion for each scenario we take the same representative values of the magnetic field intensity or baryonic density.
For fields $B< 10^{18}$ G both contributions, the parallel and the perpendicular pressures, are practically coincident. However, due to the magnetization contribution entering the perpendicular pressure, discontinuities occur whenever a new LL level is reached, giving rise to a series of discontinuities that would correspond to unstable regions, since $d{\cal M}/dB<0$. [ Let us point out that in the region $0.5<\rho<0.7$ fm$^{-3}$, and for $B<10^{18}$G, these discontinuous contributions to the perpendicular pressure contribution are practically zero, because, as seen before, the magnetization is very small in this range of densities. ]{} As compared with calculations performed with the MIT bag model [@veronica], the behavior is the same, except that the deviation of the two pressures takes place at even larger magnetic fields with the MIT bag model. Nevertheless, had we considered not only the matter contribution but also the contributions from the the terms proportional to $B^2$, the deviation would start at lower magnetic field intensities.
One should bear in mind that the discontinuities seen in Fig. \[fig3\] are washed out at finite temperatures as the ones that we expect in heavy ion collisions [@ours_finite]. As suggested in Refs. [@blandford; @noronha2007; @sedrakian] the unstable regions may give rise to domains with disordered fields that would allow to continuous phase transitions between regions with a different number of LL.
Final remarks
=============
In this work we have examined the effects of strong magnetic fields in quark matter, as described by the NJL model, paying special to effects due to pressure anisotropy. Two scenarios were investigated: matter in $\beta$-equilibrium, as the one possibly present in quark stars or in the core of hybrid stars and symmetric matter with isochemical potentials, as the one present in heavy ion collisions. For the first case, large densities and magnetic fields up to the order of $10^{18}$ G are of interest whereas for symmetric matter, the relevant densities are somewhat smaller while the magnetic fields can be larger.
Part of our work has been devoted to the analysis of quark matter constituents, the onset of $s$ quarks and how much polarized matter can be present when different field intensities are considered since these are important ingredients for stellar modeling.
Magnetization, which is an important quantity for the description of magnetized matter, has also been considered in the present work and by investigating this quantity we conclude that the amount of spikes, related to the filling of the LL, depends quite substantially on the density and on the scenario examined. Of particular interest is the fact that very few spikes are seen for densities between 0.5 and 0.7 fm$^{-3}$ in $\beta$-equilibrium matter, no matter the intensity of the magnetic field, because the particles that constitute matter occupy only a few LL. For symmetric matter, the same pattern is found for densities larger than 0.5 fm$^{-3}$, when the number of occupied LL are small for the three different quarks.
This effect in $\beta$-equilibrium matter is due to the late onset of que $s$ quark, precisely for $\rho\gtrsim 0.7$fm$^{-3}$. Due to its large constituent mass magnetic fields satisfying $B\lesssim 10^{18}$G, as the ones possibly existing inside magnetars, give rise to the filling of many LL and large contributions to the magnetization and perpendicular pressure, in contrast to the behavior at densities just below the $s$ quark onset. This effect is clearly seen comparing the pressure for $\rho=3\rho_0$ corresponding to $u$ and $d$ quark matter with restored chiral symmetry with the pressure obtained for either $\rho=2\rho_0$ or $\rho=5\rho_0$ as a function of $B$: in the first case chiral symmetry is still not completely restored and in the second the onset of $s$ quark has occured. If a quark model with complete chiral symmetry restored such as the MIT bag model, or a model without the strangeness degree of freedom such as the su(2) NJL is used to describe $\beta$-equilibrium quark matter, a different behavior will probabily occur for $B\le 10^{18}$G and $\rho>0.5$ fm$^{-3}$ right until the star center, e. g. $\rho\sim 1.2$ fm$^{-3}$ [@prc2; @melrose] , in particular, both pressure contributions will be coincident.
Therefore within the su(3) NJL model, a non negligible effect of the magnetic field on the quark star structure close to the surface and in the interior is expected even for $B\sim 10^{18}$ G. Taking as reference the calculation done in [@rezzolla2012] using pure toroidal magnetic field equilibrium models of relativistic stars for both non-rotating and rotating hadronic stars, fields as large as $10^{18}$ G were obtained on the equatorial plane deep inside the star. In this calculation, however, the magnetic field effects on the EOS have been neglected.
We have then looked at the pressure anisotropy obtained from the components of the energy-momentum tensor that define the parallel and the perpendicular pressure contributions and examined the relation between parallel and perpendicular pressures for both scenarios described above when the magnetic field is fixed and also when the baryonic density is kept constant. We have observed that the larger the magnetic field intensity, the larger the discontinuities in the perpendicular pressure. It is important to note that the magnetization is not responsible for the complete picture, because it comes multiplied by the magnetic field in the calculation of the perpendicular pressure. In agreement with Refs. [@jorge; @veronica] we have observed that when the densities are fixed, the parallel and perpendicular pressures are practically coincident up to very large magnetic fields, what could justify the use of isotropic matter [hydrostatic equations]{} in stellar calculations. However, the integration of the full relativistic hydrostatic equations still requires the inclusion of the magnetic field contributions which involve a $B^2$ term that gives rise to a quite large effect. D.P.M. and M.B.P. are partially supported by Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq-Brazil) and by Fundação de Amparo à Pesquisa e Inovação do Estado de Santa Catarina (FAPESC-Brazil), under project 2716/2012. C.P. acknowledges financial support by Project No. PEst-OE/FIS/UI0405/2014 developed under the initiative QREN financed by the UE/FEDER through the program COMPETE $-$ “Programa Operacional Factores de Competitividade.
Appendix A - Derivatives with respect to $B$
============================================
As already shown in the text, the magnetization is given by:
$${\cal M}=-\left(\frac{d\Omega}{dB}
\right)_\mu=-\left(\frac{\partial\Omega}{\partial B}
\right)_\mu- \sum_f\left(\frac{\partial\Omega}{\partial
M_f}\right)_\mu
\frac{d M_f}{dB},$$ but in equilibrium $$\left(\frac{\partial\Omega}{\partial M_f} \right)_\mu=0\;,$$ then $$M=-\left(\frac{\partial\Omega}{\partial B}
\right)_\mu\;.
\label{eq:mag}$$ Thus, in the following expressions we do not need to include terms containing $\frac{d M_f}{dB}$ which also means that the vacuum contributions $\theta^{\prime \,vac}_f$ and $\phi^{\prime \,vac}_f$ trivially vanish. Then,
$$\theta^{\prime \, mag}_f= 2 \frac{\theta^{mag}_f}{B} -
\frac{N_c |q_f| B}{2\pi ^2} \frac{M_f^2}{2B} \left[ \ln
\Gamma(x_f)-\frac{1}{2} ln (2 \pi) + x_f - (x_f - \frac{1}{2}) \ln(x_f) \right]\; ,$$
and $$\theta^{\prime \, med}_f=\frac{\theta_f^{med}}{B}- \frac{N_c
|q_f|B}{4\pi^2} \sum_{k=0}^{k_{f,max}} \alpha_k k|q_f| \int_{-\infty}^{+\infty} dp \frac{1}{E_f^*}[f_+ + f_-]\,\,.$$ For $T=0$ the above relation becomes: $$\theta^{\prime \, med}_f= \frac{\theta_f^{med}}{B}- \frac{N_c B |q_f|}{2\pi^2} \sum_{k=0}^{k_{max}} \alpha_k \ln \left ( \frac{ \mu_f + \sqrt{\mu_f^2-s_f^2}}{s_f} \right ) k|q_f|\,\,.$$ A straightforward evaluation yields
$$\begin{aligned}
\phi^{\prime \, mag}_f=\frac{\phi_f^{mag}}{B} - \frac{N_c}{4\pi^2} \left ( M_f^\prime M_f - \frac{M_f^2}{2B} \right ) \left \{ |q_f| B +M_f^2 \left[ \psi^{(0)}(x_f) - \ln(x_f)\right] \right \}\,\,,\end{aligned}$$
where $\psi^0(x_f)=\frac{\Gamma^{\prime}(x_f)}{\Gamma(x_f)}$ is the digamma function. The in medium contribution reads
$$\begin{aligned}
\phi^{\prime \, med}_f&=& \frac{\phi_f^{med}}{B} - \frac{N_c |q_f|B}{4\pi^2} \sum_{k=0}^{k_{f,max} }\alpha_k(k|q_f|+M_f M_f^\prime) \int_{-\infty}^{+\infty} dp \left \{\frac{f_+}{(E^*_f)^2}\left [
\frac{1}{E^*_f} +\frac{f_+}{T}\exp[(E_f^*-\mu_f)/T]\right ] \right . \nonumber \\
&+& \left . (\mu \leftrightarrow -\mu \;\;'\;\; f_+ \leftrightarrow f_- ) \right \} \;,\end{aligned}$$
which, at $T=0$, can be written as $$\phi^{\prime \, med}_f= \frac{\phi_f^{med}}{B} + \phi_f^{med}\frac{M_f^\prime}{M} -\frac{ N_c |q_f|B}{2 \pi ^2}
\sum_{k=0}^{k_{f,max}} \alpha_k \frac{
\mu_f M_f \left(k |q_f|+M_f
M_f^\prime\right)}{s_f(k,B)^2 \sqrt{\mu_f^2-s_f(k,B)^2}} .
\label{phimed}$$
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|
revised 15 September, 2000\
accepted for publication in [*Physical Review D*]{}\
[Graviton Production in Relativistic\
Heavy-Ion Collisions]{}\
\
\
[*P.O. Box 413, Milwaukee, Wisconsin 53201, USA.*]{}\
(email: sahern@uwm.edu; norbury@uwm.edu; poyser@uwm.edu)\
PACS numbers: 25.75.-q, 25.20.Lj, 04.50.+h\
ABSTRACT
We study the feasibility of producing the graviton of the novel Kaluza-Klein theory in which there are $d$ large compact dimensions in addition to the $4$ dimensions of Minkowski spacetime. We calculate the cross section for producing such a graviton in nucleus-nucleus collisions via t-channel $\gamma \gamma$ fusion using the semiclassical Weizsäcker-Williams method and show that it can exceed the cross section for graviton production in electron-positron scattering by several orders of magnitude.
. INTRODUCTION
Until recently, it has been believed that the role of gravity in particle interactions only becomes important at the experimentally inaccessible energy scale of the Planck mass, $M_P=1.2\times10^{19} \ GeV$. Recent advances in M-Theory, a Kaluza-Klein (KK) theory in which there are 11 spacetime dimensions, suggest that there might be an effective Planck mass $M_D$ much lower than $M_P$ (perhaps as low as $O(1 \ TeV)$) at which gravitational effects become important (cf. [@Ark; @Han; @Atw] and references therein). Traditional KK theories contain the usual four dimensions of space and time, plus additional compact dimensions which form an unobservably small (perhaps Planck length sized) manifold. These recent models propose that there are $d$ extra compact dimensions, all of roughly the same size $R$, but $R$ might possibly be much larger than the Planck length — perhaps as large as a millimeter. The size $R$ of these $d$ dimensions is related to $M_P$ and $M_D$ via $$R^d \sim \frac{M_P^2}{M_D^{d+2}}.
\label{one}$$ $R$ also represents the scale at which the Newtonian inverse-square force law is expected to fail. If $d=1$ and $M_D$ is taken to be $1 \ TeV$ then $R\sim10^{10} \ km$, which implies there should be deviations from Newtonian gravity over solar system distances. This case is clearly ruled out, but if $d\ge2$ then $R<1 \ mm$ — a possibility that is not in conflict with any current experimental data [@Ark; @Atw]. While gravity has not been tested at distances smaller than a millimeter, Standard Model (SM) interactions have been accurately tested down to distances of about $10^{-16} \ cm$ [@Pol]. Hence the $4+d$ dimensional graviton is conjectured to propagate in the $4+d$ dimensional spacetime bulk, while the SM particles are confined to a $4$ dimensional submanifold (Minkowski spacetime).
Recently it has been shown (cf. [@Atw]) that such a graviton might be detected as missing energy in electron-positron scattering. We propose here a new way that the graviton might be discovered, namely in a peripheral (near-miss) collision of two heavy nuclei. In Section II the basics of this novel KK scenario are summarized. In Section III, the calculation of the cross section for graviton production in electron-positron scattering is summarized using the Feynman rules in the Weizsäcker-Williams leading log approximation, as outlined in [@Atw]. Then, in Section IV, the process is generalized in the semiclassical Weizsäcker-Williams method to graviton production via nucleus-nucleus collisions.
. SUMMARY OF THE NEW KALUZA-KLEIN SCENARIO
The starting point of the novel KK scheme is a $4+d$ dimensional spacetime action that describes the $4+d$ dimensional graviton fields. The action is extremized and the theory is expanded around a vacuum metric that is the product of Minkowski spacetime with a $d$ dimensional torus. The dependence on the compact dimensions $x^d$ of the $4+d$ dimensional graviton is Fourier expanded in a complete set of plane waves. Due to the topology of the torus, all $d$ compact dimensions are periodic and consequently the wave numbers $k_n$ of the modes on the torus are all quantized: $k_n=2\pi n/R$, where $n$ is an integer that labels the modes and the sizes of the $d$ compact dimensions are assumed to all be $\sim R$. The $n=0$ modes, which are identically the coefficients of the normal modes, are confined to Minkowski spacetime and naturally divide into a massless spin-2 graviton (that gives rise to Newtonian gravity), $d$ massless $U(1)$ gauge bosons and $d(d+1)/2$ massless scalar bosons. The $n\ne0$ modes reorganize themselves at each KK level $n$ into a massive spin-2 boson, $(d-1)$ massive vector bosons and $d(d-1)/2$ massive scalar bosons, all of which have the same mass-squared, $m_n^2=4\pi^2n^2/R^2$. The reorganization of these modes is associated with spontaneous symmetry breaking where, like in the Higgs mechanism, the massless spin-2 graviton fields absorb the spin-1 and spin-0 fields and become massive. This KK formalism is to be regarded as an effective theory, and an ultraviolet cutoff $\Lambda \sim M_D$ is imposed on the tower of KK modes, so that $m_n^2 < \Lambda^2$ for all $n$ [@Ark; @Han]. In practical calculations, such as the one in this study, a density $\rho$ of modes (i.e., the differential number $dN$ per unit mass-squared $dm_n^2$) is used. This function is derived in [@Han] and is given by $$\rho (m_n^2) \equiv \frac{dN}{dm_n^2 } = \frac{R^d m_n^{d-2}}{(4\pi)^{d/2}\Gamma (d/2)},
\label{two}$$ where $\Gamma$ is the Gamma function. As pointed out in [@Han], it is this function that is to be convoluted with a physical amplitude or cross section for a mode with mass $m_n$. While the coupling of any individual mode to SM matter is Planck-mass suppressed (viz., $\propto M_P^{-2}$), the “end-of-the-day” coupling is only $M_D$ suppressed (viz., $\propto M_D^{-(d+2)}$) due to the factor of $M_P^2$ in $\rho$ (plug (\[one\]) into (\[two\])) that multiplies each individual mode coupling term in the overall sum. This enhancement is simply the result of the high multiplicity of graviton states, as described by $\rho$, within the same mass interval.
. GRAVITON PRODUCTION IN ELECTRON-POSITRON SCATTERING
The process considered in this section is graviton G production in electron-positron scattering via t-channel $\gamma \gamma$ fusion (i.e., $e^+e^- \to e^+e^-
\gamma \gamma \to e^+e^-G$), which, according to [@Atw], is a promising mechanism by which the particle might be produced and detected. The t-channel process is more significant than the s-channel one, wherein the initial electron and positron mutually annihilate, because the photons are produced almost collinearly with the electrons, and are more likely to directly interact with one another than are the electron and positron with each other. One way to determine the cross section for this process is to use the Feynman rules with the Weizsäcker-Williams leading log approximation. The calculation is performed in [@Atw] and reduced to the following one dimensional integral: $$\sigma_{\gamma\gamma} (e^+e^- \to e^+e^-G) =\frac{\alpha^2}{8s}\frac{\pi^{d/2-1}}{\Gamma (d/2)}\left[{\frac{\sqrt s}{M_D}} \right]^{d + 2}F_{d/2} \log ^2 \left[{\frac{s}{4m_e^2 }}\right].
\label{three}$$ Here $\alpha \approx 1/137$ is the fine structure constant, $\sqrt s$ is the center-of-mass energy of the collision and $F_k\equiv \int_0^1 d\omega {f(\omega )\omega ^k}$, where $f(\omega)=-\left[{(2+\omega)^2\log (\omega)+2(1-\omega)(3+\omega)}\right]/\omega$. $\sigma_{\gamma\gamma}(e^+e^- \to e^+e^-G)$ is a function of input parameters $\sqrt s$ and $d$. A plot of this function vs. $\sqrt s$ at three different values of $d$ ($2$, $4$ and $6$) is shown in Fig. \[eeG\], along with an indication of the operational energy of LEP2 (200 GeV).
. GRAVITON PRODUCTION IN NUCLEUS-NUCLEUS COLLISIONS
We now consider the possibility of producing a graviton in the peripheral collision of two relativistic heavy ions, via t-channel $\gamma\gamma$ fusion (i.e., $A_1A_2 \to A_1A_2 \gamma\gamma \to A_1A_2G$). The colliding nuclei are taken to be identical and are described by the atomic number $Z$ (the number of protons) and the mass number $A$ (the number of protons and neutrons). Our interest is motivated by the fact that cross sections for such reactions scale as $Z^4$, where (for heavy ions) $Z$ can be on the order of 10–100. The Feynman rules are of course an extremely accurate way of determining cross sections for simple processes, but are impractical for processes such as nucleus-nucleus collisions, where many particles are interacting with each other simultaneously. For these types of interactions, certain approximations must be made in order to simplify the calculations. For this study, we use one such approximation scheme — the semiclassical Weizsäcker-Williams method, wherein it is assumed that the colliding particles are ultrarelativistic and follow straight-line classical trajectories, and the photons mediating the interactions are real (“on-shell”). In using this approach, the $\gamma\gamma \to G_n$ subprocess cross section for one KK mode is folded with the two separate Weizsäcker-Williams photon spectra $N(\omega_1)$ and $N(\omega_2)$, and the resulting quantity is summed over all contributing modes. The $\gamma\gamma \to G_n$ subprocess cross section, denoted $\sigma_{\gamma_1\gamma_2 \to G_n}(\omega_1,\omega_2)$, is the cross section for the production of one mode of mass $m_n$ via the fusion of two photons of angular frequencies $\omega_1$ and $\omega_2$. This function, which is easily derived from the Feynman rules, is found to be [@Bar]: $$\sigma_{\gamma_1 \gamma_2 \to G_n}(\omega_1,\omega_2) = \frac{10\pi^2 }{m_n^2 }\Gamma_{G_n \to \gamma \gamma} \delta (\sqrt{\hat s}-m_n).
\label{four}$$ $\Gamma_{G_n \to \gamma\gamma}$ is the partial decay width for one graviton mode to decay into two photons, $\delta$ is the Dirac delta function and $\sqrt{\hat s}$ is the center of mass energy of the two-photon system. $\Gamma_{G_n \to \gamma\gamma}$ is given in [@Han], as: $$\Gamma _{G_n \to \gamma \gamma} = \frac{1}{20}\frac{m_n^3}{M_P^2}.
\label{five}$$ The function $N(\omega)$ gives the number of virtual photons per unit photon frequency outside the nucleus and can be derived from the Feynman rules or via a classical analysis [@Ter; @Kra; @Vid; @Jac; @Eic]. Here we use a classical formula, derived for example in [@Jac] and [@Eic]: $$N(\omega) = \frac{2}{\pi}\frac{Z^2\alpha}{\omega \beta^2}\left\{{\xi K_0(\xi)K_1(\xi) - \frac{1}{2}\beta^2 \xi^2 \left[{K_1^2(\xi)-K_0^2(\xi)} \right]} \right\}.
\label{six}$$ The functions $K_0$ and $K_1$ are modified Bessel functions of the second kind, of order zero and one, respectively. The argument $\xi$ of these functions is defined as $\xi \equiv \frac{\omega b_{\min}}{\gamma \beta}$, where $\beta$ is the speed of either ion, $\gamma \equiv \frac{1}{\sqrt{1-\beta^2}}$ and $b_{\min}$ is the minimum impact parameter of the collision, which is the distance of closest approach between the center of either nucleus and the point of G production. We take $b_{\min}$ to be the nuclear radius so as to trigger against the strong interaction effects which completely dominate electromagnetic effects when the nuclei overlap [@Nor]. For a nucleus of mass number $A$, the nuclear radius is $r \approx 1.2A^{1/3}\ fm$ [@Bau]. For our applications, we consider lead ($Z=82$ and $A=208$) and calcium ($Z=20$ and $A=40$) nuclei; thus $b_{\min} \approx 7.11\ fm$ for the former and $b_{\min} \approx 4.10\ fm$ for the latter. Lead is interesting because it is one of the more common stable nuclei with a high $Z$ value, and calcium is interesting because it has a relatively high luminosity when used as a heavy ion beam [@Bra]. The total cross section for the production of one mode of mass $m_n$ is: $$\sigma_{\gamma \gamma} (A_1A_2 \to A_1A_2G_n) = \int {d\omega_1 \int {d\omega_2 \ N(\omega_1) N(\omega_2)}} \sigma_{\gamma_1 \gamma_2 \to G_n}(\omega_1,\omega_2).
\label{seven}$$ The limits of integration can be derived from conservation of 4-momentum; an elegant version of such a calculation is presented in [@Cah]. As used here, $m_n^2/2 \sqrt s \le \omega_1 \le \sqrt s/2$ and $m_n^2/4 \omega_1 \le \omega_2 \le \sqrt s/2+m_n^2/2 \sqrt s-\omega_1$, but because of the delta function in (\[four\]), only the limits on $\omega_1$ are needed. Finally, the total cross section for graviton production in this process is found by summing (\[seven\]) over all contributing modes: $$\sigma_{\gamma \gamma}(A_1A_2 \to A_1A_2G) = \int {dm_n^2 \ \rho(m_n^2)} \sigma_{\gamma \gamma}(A_1A_2 \to A_1A_2G_n).
\label{eight}$$ $\rho(m_n^2)$ was given in (\[two\]) and $m_n^2$ ranges from 0 to the smaller of $s$, as demanded by conservation of energy and momentum, and $\Lambda^2$, which is the absolute upper limit on $m_n^2$. In its simplest form, the cross section is given by: $$\sigma_{\gamma \gamma}(A_1A_2 \to A_1A_2G) = const. \int {dm_n \int
{d\omega_1 \ \frac{m_n^{d-1}}{\omega_1}f(\omega_1)f \left({\frac{m_n^2}{4\omega_1}} \right)}},
\label{nine}$$ where $$const. \equiv \frac{8}{(4 \pi)^{d/2} \Gamma(d/2)}\left({\frac{Z^2 \alpha}
{\beta^2}} \right)^2 \frac{1}{M_D^{d+2}}
\label{ten}$$ and $$f(\xi) \equiv \xi K_0(\xi)K_1(\xi) - \frac{1}{2}\beta^2 \xi^2
\left[{K_1^2(\xi)-K_0^2(\xi)} \right]
\label{eleven},$$ and the limits of integration and values of parameters are as specified above.
The cross section curves are shown in Fig. \[pbpbG\] (for $PbPb \to PbPbG$) and Fig. \[cacaG\] (for $CaCa \to CaCaG$) for values of $d$ = $2$, $4$ and $6$, along with indications of the machine energies of the planned RHIC (1 TeV/A/beam) and LHC (2.76 TeV/A/beam). There is clearly an enhancement of several orders of magnitude compared to the cross section for the same process via electron-positron scattering, at least for certain ranges of parameters. However, it must be pointed out that the work presented in this paper neglected to take into account various complicating factors that might potentially be of great significance. The most obvious is mentioned in [@Atw] — that we do not have a proper quantum theory of gravity to work with, so there are necessarily uncertainties in this regard, particularly in utilizing the density of modes function (Eq. (\[two\])). Another one is that a definitive identification of the graviton signature is precluded by the oversimplified nature of the formulation we used. We assumed the usual Weizsäcker-Williams scenario, wherein the scattering angles of the interacting particles are always negligibly small, which means that the nuclei contributing to graviton production cannot be distinguished from the other nuclei in the accelerator beams. Since the graviton couples only very weakly to ordinary matter, the signature for the overall reaction would be missing mass-energy, and therefore a definitive experimental signature cannot be predicted. This problem can be possibly remedied by relaxing the assumption that the photons are on-shell. Presumably, though, the resulting calculations (which would involve such concepts as nuclear form factors, partons within quarks, and the hadronic structure of photons) would yield much smaller signal cross sections. Furthermore, for nucleus-nucleus collisions, one must also take into account limitations due to other effects such as luminosity and background. The $\gamma\gamma$ luminosity ${\mathcal L}$ in a heavy ion collider is generally suppressed by several orders of magnitude compared to that in an electron-positron collider; compare ${\mathcal L} \sim 10^{32} \ cm^{-2}s^{-1}$ for $e^+e^-$ scattering at LEP2 to ${\mathcal L} \sim5 \times 10^{26} \ cm^{-2}s^{-1}$ for Pb-Pb collisions at LHC and ${\mathcal L} \sim5 \times 10^{30}\ cm^{-2}s^{-1}$ for Ca-Ca collisions at LHC [@Hen; @Vid]. In addition, electron-positron collisions are expected to be much cleaner experimentally compared to nucleus-nucleus collisions because of the hadronic debris accompanying processes of the latter type [@Vid]. Although, one could always trigger against multiplicity to reject this background.
. CONCLUSIONS
In summary, we investigated graviton production via two different processes. The first production mechanism was through $\gamma\gamma$ fusion in electron-positron scattering, and we summarized a calculation that used the Feynman rules in the Weizsäcker-Williams leading log approximation. The second process was graviton production via $\gamma\gamma$ fusion in peripheral nucleus-nucleus collisions, where we considered both $^{208}$Pb and $^{40}$Ca ions. We calculated the cross sections for these reactions using the semiclassical Weizsäcker-Williams method and found them to be comparable to, and in some cases substantially greater than, that for the previous process. But, we note that because of the oversimplified nature of our analysis, there are potentially great uncertainties in our results.
ACKNOWLEDGEMENTS
This work was supported in part by a graduate school dissertation fellowship from UW-Milwaukee and by funding from the National Space Grant College and Fellowhsip Program through the Wisconsin Space Grant Consortium.
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---
title: Energy and composition sensitivity of geosynchrotron radio emission from EAS
---
Introduction
============
It is our current understanding that radio emission from extensive air showers is dominated by “geosynchrotron radiation” [@HuegeFalcke2003a] emitted by secondary shower electrons and positrons being deflected in the earth’s magnetic field. The geosynchrotron model has by now been implemented in a sophisticated Monte Carlo code called REAS2 [@HuegeUlrichEngel2007a; @HuegeIcrc2007b], which itself relies on CORSIKA [@HeckKnappCapdevielle1998] for the simulation of the relevant extensive air shower (EAS) properties. In this article, we analyse the sensitivity of REAS2-simulated geosynchrotron radio emission on the primary cosmic ray energy and mass. The determination of these parameters on a shower-to-shower basis is one of the main goals of measuring radio emission from EAS, and we demonstrate that radio measurements of inclined showers indeed provide relatively direct access to these parameters.
Methodology
===========
Results gathered so far point to inclined air showers with zenith angles above $45^{\circ}$ as particularly promising targets for radio measurements of EAS [@PetrovicApelAsch2006]. Inclined air showers have a large radio “footprint” [@HuegeFalcke2005b] and thus allow radio antennas to be spaced relatively far apart, an important prerequisite for the instrumentation of large effective areas at moderate cost. Concentrating on the energy range relevant to the Pierre Auger Observatory, we have thus performed an analysis of air showers with $60^{\circ}$ and $45^{\circ}$ zenith angles at primary particle energies of $10^{18}$ eV, $10^{19}$ eV and $10^{20}$ eV. For each of these zenith angles and energies, we have simulated 25 iron-induced and 25 proton-induced air showers. 25 gamma-induced air showers per zenith angle were simulated for $10^{18}$ eV and $10^{19}$ eV, but not for $10^{20}$ eV, where pre-showering in the geomagnetic field would have to be taken into account. The simulation chain consisted of a CORSIKA 6.502 run using the QGSJETII-03 and UrQMD1.3.1 interaction models, Argentinian magnetic field, fixed azimuth angle (showers coming from south), $10^{-6}$ optimised thinning, and an observer height 1400 m a.s.l. followed by a REAS2 simulation with antenna locations between 25 m and 925 m from the shower core (in ground-based coordinates); cf. also [@HuegeUlrichEngel2007a]. For each of the simulated radio events we then determined the peak amplitude of the electric field pulses filtered using idealised rectangle filters from 16 to 32 MHz, 32 to 64 MHz and 64 to 128 MHz, respectively. In the following, we discuss the $60^{\circ}$ zenith angle case. The qualitative behaviour at $45^{\circ}$ is completely analogue. Other shower azimuth angles do not change the qualitative behaviour either.
Signal information content
==========================
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{width="4.80cm"}
The lateral slope of the radio signal is known to exhibit a dependence on the depth of the shower maximum ($X_{\mathrm{max}}$) and consequently, contains information on the primary particle energy and mass [@HuegeFalcke2005b; @HuegeIcrc2005a]. To date, however, there had been no detailed investigation how this information content could be exploited in practice.
![The ratio of field strengths in the [*steep region*]{} vs. the [*flat region*]{} of $10^{19}$ eV showers with $60^{\circ}$ distinguishes between primary particle types.[]{data-label="fig:ratio"}](icrc0889_fig03.eps){width="6.0cm"}
The detailed analysis presented here reveals that for a given shower zenith angle, a suitable lateral distance exists where the radio signal is relatively independent of $X_{\mathrm{max}}$ (hereafter called [*flat region*]{}), in analogy to the surface detector quantity $S(1000)$ of the Pierre Auger Observatory. This behaviour is illustrated for the $60^{\circ}$ zenith angle case in Fig. \[fig:flat1e19\]: the electric field strength at 275 m north from the shower core is relatively constant regardless of primary particle type and shower $X_{\mathrm{max}}$. In contrast, at a distance of 725 m north as shown in Fig. \[fig:steep1e19\], there is a clear dependence of signal strength on $X_{\mathrm{max}}$ (hereafter called [*steep region*]{}).
A combination of measurements in these two regions can therefore differentiate between different types of primaries (Fig. \[fig:ratio\]).
Signal scaling with $N_{\mathrm{max}}$
======================================
{width="4.80cm"}
{width="4.80cm"}
When the measured radio field strength is divided by the number of electrons plus positrons in $X_{\mathrm{max}}$, hereafter called $N_{\mathrm{max}}$, it becomes clear that the radio signal scales linearly with $N_{\mathrm{max}}$. The reason for this is that most of the radio emission stems from the particles close to the shower maximum [@HuegeUlrichEngel2007a]. (The intensity of optical fluorescence light, in contrast, scales with the calorimetric energy deposited in the atmosphere by the shower.) The clean linear scaling is illustrated in Fig. \[fig:flatnorm\], where the results from all energies and particle species yield an approximately constant electric field strength per $N_{\mathrm{max}}$ in the [*flat region*]{}. The electric field strength per $N_{\mathrm{max}}$ in the [*steep region*]{} is also constant over the different particle types and energies for a given $X_{\mathrm{max}}$.
{width="4.80cm"}
{width="4.80cm"}
Optimum parameter determination
===============================
{width="4.80cm"}
{width="4.80cm"}
Histogramming the distribution of electric field strengths per $N_{\mathrm{max}}$ over all particle species and energies yields an RMS of only 5% in the [*flat region*]{} (Fig. \[fig:rms\]). In spite of shower to shower fluctuations, radio measurements could thus determine the $N_{\mathrm{max}}$ of individual EAS to very high precision. The position of the [*flat region*]{} for a given zenith angle is determined by the observing frequency band. As illustrated in Fig. \[fig:rmsdep\], for $60^{\circ}$ zenith angle and 32 to 64 MHz observing frequency, $N_{\mathrm{max}}$ can be determined to highest precision around 275 m distance to the north (or south). At 16 to 32 MHz, 5% precision in the $N_{\mathrm{max}}$ determination can be reached anywhere between around 200 m and 500 m from the core. Low frequencies are thus particularly well suited for an energy determination of EAS primaries, if the technical difficulties involved with measurements at these frequencies can be overcome. In contrast, at 64 to 128 MHz the range for $N_{\mathrm{max}}$ determination becomes much smaller and resides at smaller distances. (The peculiar steps in the 64 to 128 MHz RMS curve point to coherence effects starting to play a significant role at these high frequencies.) For the determination of $X_{\mathrm{max}}$ using measurements in the [*steep region*]{}, a lateral distance should be chosen that shows a large RMS spread in electric field strength per $N_{\mathrm{max}}$. For 32 to 64 MHz and $60^{\circ}$ zenith angle, 725 m constitutes a suitable compromise between a good handle on the $X_{\mathrm{max}}$ value and detectable absolute signal levels. (The absolute signal strength drops quickly with lateral distance, cf. [@HuegeFalcke2005b]).
Conclusions
===========
Simulations of geosynchrotron radio emission with the REAS2 Monte Carlo code reveal that information contained in the lateral profile of the radio signal can be exploited for a direct determination of $N_{\mathrm{max}}$ and $X_{\mathrm{max}}$ on a shower-to-shower basis. These quantities can in turn be related to the energy and mass of the primary cosmic ray particle.
For a given zenith angle and observing frequency band, distinct distance regimes denoted here as the [*flat region*]{} and the [*steep region*]{} can be identified. Measurements in the [*flat region*]{} directly yield the $N_{\mathrm{max}}$ of the shower, with an RMS deviation of only 5%. A comparison of peak field strengths in the [*steep region*]{} and [*flat region*]{} for a given observing bandwidth provides a direct estimate of the shower $X_{\mathrm{max}}$. Alternatively, measurements at a fixed distance but in two different observing frequency bands provide comparable information.
The position of the [*flat region*]{} in case of $60^{\circ}$ zenith angles for the observing frequency band from 32 to 64 MHz lies at approximately 275 m and would thus require an antenna spacing of that order. (The given number is valid for antenna positions along the shower axis; in the perpendicular direction the scales are smaller due to azimuthal asymmetries of the radio footprint.) If technical problems arising in measurements at lower frequencies such as 16 to 32 MHz can be overcome, antenna spacings of up to 500 m will allow measurements in the [*flat region*]{}. The qualitative behaviour at $45^{\circ}$ zenith angle is analogue, yet at lower lateral distances. At larger zenith angles, the scales will be larger, but consistent simulations require the implementation of curved atmospheres for zenith angles much larger than $60^{\circ}$, which is currently being prepared.
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D. [Heck]{}, J. [Knapp]{}, J. N. [Capdevielle]{}, G. [Schatz]{}, and T. [Thouw]{}. FZKA Report 6019, Forschungszentrum Karlsruhe, 1998.
T. [Huege]{}, W. D. [Apel]{}, A. F. [Badea]{} [et al.]{} In [*Proc. of the 29th ICRC, Pune, India*]{}, volume 7, page 107, 2005. astro-ph/0507026.
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---
abstract: 'We consider the tensor formulation of the non-linear O(2) sigma model and its gauged version (the compact Abelian Higgs model), on a $D$-dimensional cubic lattice, and show that tensorial truncations are compatible with the general identities derived from the symmetries of these models. This means that the universal properties of these models can be reproduced with highly simplified formulations desirable for implementations with quantum computers or for quantum simulations experiments. We discuss the extensions to global non-Abelian symmetries, discrete symmetries and pure gauge Abelian models.'
author:
- Yannick Meurice$^1$
title: 'Examples of symmetry-preserving truncations in tensor field theory'
---
Introduction
============
There has been a lot interest for tensorial formulations of lattice models in the context of the renormalization group method [@nishino96; @Levin2006; @Gu:2009dr; @PhysRevLett.103.160601; @PhysRevB.86.045139; @prb87; @prd88; @pre89; @prd89; @PhysRevLett.115.180405; @Shimizu:2014fsa; @Shimizu:2014fsa; @Takeda:2014vwa; @Shimizu:2017onf; @PhysRevB.98.235148; @Bal:2017mht; @Nakamura:2018enp; @Kuramashi:2018mmi; @Yoshimura:2017jpk; @Kadoh:2018hqq; @juyu1; @juysu2]. Tensor formulations provide a new approach of lattice models that we call tensor field theory (TFT). TFT should not be confused with theories involving fields that in the continuum have more than one Lorentz index, for instance the Kalb-Ramond field [@kalbramond], and are often called “tensor fields". For theories with compact fields like the nonlinear sigma models and Wilson lattice gauge theories, the tensor reformulation relies on character expansions and is always discretized [@prd88]. This is suitable for quantum computations or quantum simulations [@pra90; @ahm; @prl121]. In practical situations such as Tensor Renormalization Group (TRG) calculations, truncations of infinite sums appearing in the TFT formulation of models with continuous symmetries are necessary. This can be achieved by discarding contributions to the partition function or observable averages that involve tensor indices larger than some cut-off value $n_{max}$. Concrete examples will be given in Secs. \[sec:o2\] and \[sec:cahm\].
A truncation procedure can be understood as a regularization and we need to ask if the regularization is compatible with the symmetries of the theory or if it generates what we call anomalies. As far as the universal behavior is concerned, we expect that if truncations preserve the symmetries, one should be able to obtain the properties associated with the universality classes by taking the contiunuum limit using a considerably simplified microscopic formulation. In other words, we could use drastic truncations of the sums such that at each site, link or plaquettes only a few values of the indices are kept. This is very important when the computational units available to represent the local degrees of freedom, such as qubits or trapped atoms, are in limited supply.
In the following, we discuss identities associated with global and local symmetries in the Lagrangian approach of lattice models and examine their compatibility with truncations. We focus on two related examples with a continuous Abelian symmetry: the O(2) nonlinear sigma model and the compact Abelian Higgs model. We also connect with the Hamiltonian formulation by taking the time continuum limit. In the Hamiltonian approach, it is sufficient to check that the generators of symmetry groups commute with the Hamiltonian. We want to emphasize that the Lagrangian approach used in the TRG and followed here is more general and that we will not rely on infinitesimal transformations as in the traditional Noether’s approach. The compatibility of the symmetries with truncations in TFT is a frequently asked question and we think that it is important to collect basic results about this question in situations where compact field integrations are replaced by discrete sums.
The article is organized as follows. In Sec. \[sec:sym\], we introduce simple identities that are valid for global or local symmetries appearing in generic lattice models. In Sec. \[sec:o2\], we discuss the nonlinear O(2) sigma model in arbitrary dimension. This is an example of a model with a continuous global Abelian symmetry. In Sec. \[sec:cahm\], we consider the gauged version of the O(2) model, the compact Abelian Higgs model. In both cases, we find conclusive evidence that truncations fully preserve the symmetries of the model. Extensions to discrete symmetries, global non-Abelian symmetries and pure gauge Abelian theories are discussed in Sec. \[sec:ext\]. In the conclusions, we summarize the results, provide an intuitive picture and emphasize the practical implications of the results.
Implications of symmetries for lattice models {#sec:sym}
=============================================
In this section, we consider a generic lattice model with action $S[\Phi]$, where $\Phi$ denotes a field configuration of fields $\phi_\ell$ attached to locations $\ell$ which can be sites, links, plaquettes or higher dimensional objects. Additional indices possibly attached to the fields are kept implicit. The partition function reads Z=\^[-S\[\]]{}, with ${\mathcal D}\Phi$ the measure of integration over the fields. The average value of a function of the fields $f(\Phi)$ is defined as f() =f()[e]{}\^[-S\[\]]{}/Z. \[eq:sym\]
We define symmetries as field transformations \_\_’= \_+\_, that preserve the action and the integration measure: ’ =[D]{} [and ]{} S\[’\]=S\[\]. These symmetries can be global or local. In all the examples we know, these symmetries form a group and the invariance is valid for any group element and not only for infinitesimal transformations. Changing variable from $\Phi$ to $\Phi'$ and using the symmetry properties of Eq. (\[eq:sym\]), we find the intuitively clear result: f() =f(+) . \[eq:inv\]
Can this simple expression of the symmetries be used to derive the existence of conserved quantities for global continuous symmetries as in Noether’s theorem? In classical mechanics, if a transformation $\delta q_i$ of generalized coordinates $q_i$ leaves the action invariant, then after using the equation of motion, we obtain conservation law: (q\_i)=0. The use of the equations of motion guarantees that the variation $\delta q_i$ has no effect [*except*]{} at the initial and final times where unlike what is done in the variational procedure $\delta q_i$ are not required to vanish. Consequently, the two individual surface terms do [*not*]{} vanish and are equal to the conserved quantity.
In field theory, a similar procedure leads to a relativistically invariant current conservation \_J\^(x)= .+/t=0, \[eq:cont\]which has the form of a continuity equation. By considering its integration between two time slices with spatial boundary conditions such that the spatial current does not flow outside the region of integration, one obtains that the integral of the charge density over a time slice is a constant of motion.
In the following, we will show that Eq. (\[eq:inv\]) can actually be obtained as a global consequence of a continuity equation encoded in the local tensors used in the reformulation. We will not need to use the equations of motion explicitly. In the generic formulation used above, the equations of motions are obtained by varying a single local variable $\phi_\ell$: \_\_’= \_+. Assuming that the ${\mathcal D}\Phi$ is invariant under this shift and that the action changes by an amount $\Delta_{\ell,\alpha}S$, we obtain that \^[-\_[,]{}S]{}=1. Taking the derivative with respect to $\alpha$ and setting $\alpha=0$, we obtain the lattice equation of motion S/\_=0.
Example 1: the O(2) model {#sec:o2}
=========================
The model and its symmetry
--------------------------
As a first example we consider a lattice model with a global continuous Abelian symmetry: the non linear O(2) sigma model. This is a generalization of the Ising model where the spins are two-dimensional vectors of length one. We parametrize them with an angle $\varphi$ where 0 and $2\pi$ are identified. We use a $D$-dimensional (hyper) cubic Euclidean space-time lattice. For instance, for $D=2$, we use a square lattice. The sites are denoted $x=(x_1, x_2,\dots x_D)$, with $x_D=\tau$, the Euclidean time direction. The total number of sites is denoted $V$ and we assume periodic or open boundary conditions. If we take the time continuum limit, we obtain a quantum Hamiltonian formulation in $D-1$ spatial dimensions.
In terms of the generic notations introduced in Sec. \[sec:sym\], the field configurations are $\Phi=\{\varphi_x\}_x$. The integration measure is normalized to one and reads =\_x\_[-]{}\^, and the action S\[\]=-\_[x,i]{} (\_[x+]{}-\_x),where $\hat{i}$ denotes a unit vector in the positive $i$-th direction. The invariance requirements for the action and measure of Eq. (\[eq:sym\]) are satisfied for the global shift \_x’=\_x+. \[eq:phivar\] This implies that for a function $f$ of $N$ variables f(\_[[x]{}\_1]{},…,\_[[x]{}\_N]{})= f(\_[[x]{}\_1]{}+, …,\_[[x]{}\_N]{}+). Since $f$ is $2\pi$-periodic in its variables and can be expressed in terms Fourier modes, this can be reduced to $$\begin{aligned}
\label{eq:insert}
&\langle& \exp(i(n_1\varphi_{{x}_1}+\dots n_N\varphi_{{x}_N}))\rangle =\\ \nonumber
&\ &\exp((n_1+\dots n_N)\alpha)\langle \exp(i(n_1\varphi_{{x}_1}+\dots n_N\varphi_{{x}_N}))\rangle. \end{aligned}$$ This implies that if \[eq:selection\]\_[n=1]{}\^N n\_i 0, then (i(n\_1\_[[x]{}\_1]{}+…+n\_N\_[[x]{}\_N]{}))=0.We will show that this selection rule can be explained by a microscopic continuity equation that is manifest in the tensor formulation that we proceed to discuss.
The tensor formulation
----------------------
The basic aspects of the tensor reformulation of the O(2) model have been discussed in Refs. [@prd88; @pre89; @prd89]. We briefly review the main results. It borrows tools from duality constructions [@RevModPhys.52.453]. At each link, we use the Fourier expansion\^[(\_[x+]{}-\_x)]{} = \_[n\_[x,i]{}=-]{}\^[+]{} [e]{}\^[i n\_[x,i]{}(\_[x+]{}-\_x)]{} I\_[n\_[x,i]{}]{}() , \[eq:fou\] where the $I_n$ are the modified Bessel functions of the first kind. This attaches an index $n_{x,i}$ at each link coming out of $x$ in the positive $i$-th direction. It is then possible to integrate over the $\varphi_x$ and rewrite the partition function as the trace of a tensor product: Z=I\_0\^V() [Tr]{} \_x T\^x\_[(n\_[x-,1]{}, n\_[x,1]{},…,n\_[x,D]{})]{}. \[eq:trace\] The local tensor $T^x$ has $2D$ indices. The explicit form is $$\begin{aligned}
\label{eq:tensor}
T^x_{(n_{x-\hat{1},1}, n_{x,1},\dots,n_{x-\hat{D},D},n_{x,D})}&=&\\ \nonumber
\sqrt{t_{n_{x-\hat{1},1}} t_{n_{x,1}},\dots,t_{n_{x-\hat{D},D}}t_{n_{x,D}} }&\times&\delta_{n_{x,out},n_{x,in}},\end{aligned}$$ with the definitions$$\begin{aligned}
\nonumber
t_n&\equiv& I_n(\beta)/I_0(\beta)\\ \label{eq:defs}
n_{x,in}&\equiv&\sum_in_{x-\hat{i},i} \\ \nonumber n_{x,out}&\equiv&\sum_in_{x,i}, \end{aligned}$$ where the sums over $i$ run from 1 to $D$. The Kronecker delta in Eq. (\[eq:tensor\]) \_i(n\_[x,i]{}-n\_[x-,i]{})=0, \[eq:noether\] is a discrete version of Noether current conservation Eq. (\[eq:cont\]) if we interpret the $n_{x,i}$ with $i<D$ as spatial current densities and $n_{x,D}$ as a charge density.
The insertion of various ${\rm e}^{in_Q\varphi_x}$ is required in order to calculate the averages function of Eq. (\[eq:insert\]). This can be done by inserting an “impure" tensor instead of the usual one at the location $x$. This tensor only differs from the “pure" tensor of Eq. (\[eq:tensor\]) by the Kronecker symbol replacement \_[n\_[x,out]{},n\_[x,in]{}]{} \_[n\_[x,out]{},n\_[x,in]{}+n\_Q]{}. \[eq:impure\]
In Eq. (\[eq:trace\]), the trace is a sum over all the link indices. We need to specify the boundary conditions. Periodic boundary conditions (PBC) allow us to keep a discrete translational invariance. As a consequence the tensors themselves are translation invariant and assembled in the same way at every site. Open boundary conditions (OBC) can also be implemented by introducing new tensors that can be placed at the boundary. Their construction is similar to the tensors in the bulk. The only difference is that there are some links which could be attached at sites on the boundary and are missing. With the normalization introduced in Eq. (\[eq:tensor\]) the indices carrying a zero index carry a unit weight and we can take into account the missing links at the boundary by setting their corresponding indices to zero. At finite $\beta$, the ratios of Bessel functions $t_n$ defined in Eq. (\[eq:defs\]) decay rapidly with $n$ and it is justified to introduce a truncation. If any of the indices in a tensor element is larger in magnitude than a certain value $n_{max}$, we approximate the tensor by zero. The main question addressed here is to decide if this type of truncation is compatible with the symmetries.
Microscopic explanation of the selection rule
---------------------------------------------
In this subsection, we provide a microscopic derivation of the selection rule Eq. (\[eq:selection\]). In absence of insertions of ${\rm e}^{in_Q\varphi_x}$, the Kronecker delta at the sites can be interpreted as a divergence-free condition. If we enclose a site $x$ in a small $D$-dimensional cube, the sum of indices corresponding to positive directions ($n_{x,out}$) is the same as the sum of indices corresponding to negative directions ($n_{x,in}$). For instance in two dimensions, the sum of the left and bottom indices equals the sum of the right and top indices. We can “assemble" such elementary objects by tracing over indices corresponding to their interface and construct an arbitrary domain. Each tracing automatically cancels an in index with an out index and consequently, at the boundary of the domain, the sum of the in indices remains the same as the sum of the out indices.
We can now repeat this procedure with insertions of ${\rm e}^{in_Q\varphi_x}$. Each insertion adds $n_Q$, which can be positive or negative, to the sum of the out indices. We can apply this bookkeeping on an existing tensor configuration until we have gathered all the insertions and we reach the boundary of the system. For PBC, this means that all the in and out indices get traced in pairs at the boundary. This is only possible if the sum of the inserted charges is zero. Eq. (\[eq:selection\]) tells us that when it is not the case, the average is zero. For OBC, all the boundary indices are zero and the same conclusions apply.
In summary we have shown that the selection rule in Eq. (\[eq:selection\]) is a consequence of the Kronecker delta appearing in the tensor and is independent of the particular values taken by the tensors. So if we set some of the tensor elements to zero as we do in a truncation, this does not affect the selection rule.
Hamiltonian formulation
-----------------------
The transition from the Lagrangian formulation considered above, to the quantum Hamiltonian formulation can be achieved by using the transfer matrix. As shown in Ref. [@pra90], the transfer matrix can be constructed by taking all the tensors on a time slice and tracing over the spatial indices. With either PBC or OBC, there is no flow of indices in the spatial directions. Consequently the sum of the time indices going in the time slice equals the sum of the indices going out. This conserved quantity can be identified as the charge of the initial or final state and the transfer matrix commutes with the charge operator which counts the sum of the in or out indices. Consequently, setting some matrix elements to zero if some of the local indices exceeds some value $n_{max}$ in absolute value will not affect this property. The transfer matrix can be used to define an Hamiltonian by taking an anisotropic limit where $\beta$ becomes large on time links and the Hamiltonian will inherit the properties of the transfer matrix.
In the rest of this subsection, we restrict the discussion to $D=1$ where the operator formalism is transparent. In addition we impose periodic boundary conditions in the Euclidean time direction. The tensor reads T\_[n\_x,n\_[x-1]{}]{}=t\_[n\_x]{}()\_[n\_x,n\_[x-1]{}]{}, and represents the diagonal transfer matrix. In the limit of large $\beta$, $t_n(\beta)\simeq 1-n^2/2\beta$ and if we identify the time lattice spacing with $1/\beta$, we find the rotor spectrum with energies $E_n=n^2/2$. The value of the conserved charge $n$ is often called the angular momentum of the rotor. For periodic boundary conditions, the partition function is the trace of the $N_\tau$ power of the transfer matrix. If we insert ${\rm e}^{i\varphi_x}$ in the functional integral, the charge $n$ increases by 1 and the trace is zero unless we insert ${\rm e}^{-i\varphi_{x+y}}$ or a product of having the same effect. So for $D=1$, the selection rule Eq. (\[eq:selection\]) is immediate. For visualization purpose, the transfer matrix evolves an initial state which is placed on the right of the operator as a ket vector and the left indices refer to the future.
In the Hamiltonian formalism, we introduce the angular momentum eigenstates which are also energy eigenstates $$\begin{aligned}
\hat{L}\ket{n}&=&n\ket{n},\\ \nonumber
\hat{H}\ket{n}&=&\frac{n^2}{2}\ket{n}.\end{aligned}$$ We assume that $n$ can take any integer value from $-\infty$ to $+\infty$. As $\hat{H}=(1/2)\hat{L}^2$, it is obvious that =0. \[eq:commute\] The insertion of ${\rm e}^{i\varphi_x}$ in the path integral, translates into an operator $\widehat{{\rm e}^{i\varphi}}$ which raises the charge as in Eq. (\[eq:impure\]) =, while its Hermitean conjugate lowers it ()\^=.
This implies the commutation relations =, \[L,\^\]=-\^, \[eq:eu1\] and \[,\^\]=0. \[eq:zero\]
We now discuss the effect of a truncation on these algebraic results. By truncation we mean that there exists some $\nmax$ for which =0, [and]{} ()\^=0.
If we now study the commutation relation with this restriction, we see that the only changes are $$\begin{aligned}
&\ &\bra{\nmax}[\cre,\cre^\dagger ]\ket{\nmax}=1,\\ \nonumber
&\ &\bra{-\nmax}[\cre,\cre^\dagger ]\ket{-\nmax}=-1,\\ \nonumber\end{aligned}$$ instead of 0. The important point is that the truncation does not affect the basic expression of the symmetry in Eq. (\[eq:commute\]). It only affects matrix elements involving the $\cre$ operators but not in a way that contradicts charge conservation. For a related discussion of the algebra for the O(3) model see Ref. [@falko3p]. Related deformations of the original Hamiltonian algebra appear in the quantum link formulation of lattice gauge theories [@qlink2]. It should also be noticed that Eqs. (\[eq:eu1\]) and (\[eq:zero\]) correspond to the $M(2)$ algebra, the rotations and translations in a plane. Its representations are infinite dimensional with matrix elements given in terms of Bessel functions [@vilenkinspecial].
Example 2: the compact Abelian Higgs model {#sec:cahm}
==========================================
The model and its symmetries
----------------------------
Having shown that the truncation preserve the symmetries of the O(2) model, we now proceed to discuss the question in its gauged version, the “compact Abelian Higgs model". By “compact" we mean that both the gauge field and the matter field are compact fields. On the matter side, the Brout-Englert-Higgs mode has been decoupled and the Nambu-Goldstone mode is $\varphi_x$ as in the O(2) model. For more details about the decoupling of the Brout-Englert-Higgs field see Ref. [@ahm]. The gauge fields are located on the links and are denoted $A_{x,\hat{i}}$. The integration measure becomes =\_x\_[-]{}\^\_[x,i]{}\_[-]{}\^. \[eq:gaugemeasure\] The action splits into a matter part \[eq:smatter\] S\_[matter]{}\[\]=-\_[x,i]{} (\_[x+]{}-\_x+A\_[x,i]{}),and a gauge part S\_[gauge]{}=-\_p\_[x,i<j]{} (A\_[x,i]{}+A\_[x+,j]{}-A\_[x++,i]{}-A\_[x,j]{}). \[eq:gauge\] The symmetry of the $O(2)$ model becomes local \_x’=\_x+\_x \[eq:varphix\] and these local changes in $S_{matter}$ are compensated by the gauge field changes A\_[x,i]{}’=A\_[x,i]{}-(\_[x+]{}-\_x), which also leave $S_{gauge}$ invariant. The measure in Eq. (\[eq:gaugemeasure\]) is invariant under these local shifts.
The general consequence of symmetries expressed by Eq. (\[eq:inv\]) can again be applied to Fourier modes. We find that for [*every*]{} site $x$, if we have indices such that n+\_i m\_i -\_i \_i 0,then (i(n\_x+\_i m\_i A\_[x,i]{}+\_i \_iA\_[x-,i]{})=0. \[eq:gaugeselection\] This is nothing but the statement that non gauge-invariant observables have a zero expectation value. By applying this restriction to every site, we end up with observables such as Wilson loops or Wilson lines attached to suitable powers of ${\rm e}^{i\varphi_x}$. Even though we might not want to calculate the average of non gauge-invariant observable, it is legitimate to ask if truncations could generate non-zero average values for gauge-variant observables.
Tensor formulation {#subsec:tensor}
------------------
The tensor formulation of this model has been discussed extensively in Ref. [@ahm] and used to propose cold atom simulations for the model [@prl121]. In the following we focus on aspects relevant to a possible symmetry breaking. In order to calculate the partition function, we expand all the Boltzmann weights using Eq. (\[eq:fou\]) and keeping the fields [*with exactly the same signs*]{} as in the cosine functions in the action. This introduces discrete quantum numbers $n_{x,i}$ for the links, just the same as for O(2), and additional quantum numbers $m_{x,i,j}$ associated with the plaquette with corners $(x,x+\hat{i},x+\hat{i}+\hat{j},x+\hat{j})$ and $i<j$. Comparing with Eq. (\[eq:gauge\]), we see that the gauge fields on the lowest numbered positive direction coming out of $x$ come with a positive sign and those with the largest numbered positive direction with a minus sign. We now integrate over the gauge fields. If we use the convention m\_[x,i,j]{}=-m\_[x,j,i]{}, \[eq:sign\] when $i>j$ and in addition $m_{x,i,i}$=0, then it is clear that \_[i,j]{}=m\_[x,i,j]{}=0. \[eq:sumzero\] We can write the selection rules in a very compact way: n\_[x,i]{}=\_[j]{}(m\_[x,j,i]{}-m\_[x-,j,i]{}). \[eq:dmufmunu\] If we plug this relation in $\sum_i(n_{x,i}-n_{x-\hat{i},i})$, it is automatically zero because of Eq. (\[eq:sumzero\]) and we recover the discrete version of Noether current conservation for the O(2) model. This is a discrete version of $\partial_\mu\partial_\nu F^{\mu\nu}=0$.
Eq. (\[eq:dmufmunu\]) shows that the quantum numbers associated with the links ($n_{x,i}$) are completely determined by the quantum numbers of the plaquettes ($m_{x,i,j}$) which play the role of dual variables [@RevModPhys.52.453] but with additional interactions given by $S_{gauge}$. The states of the Hilbert space for the transfer matrix and the associated Hamiltonian when we take the time continuum limit depend only on the $m_{x,i,j}$.
So far we have only performed the integration over the gauge fields. However, the matter field $\phi_x$ appears in exponentials multiplied by $\sum_i(n_{x,i}-n_{x-\hat{i},i})$ which we just argued is zero because of Eq. (\[eq:dmufmunu\]). Consequently, the integration over the matter fields is trivial and produces a factor 1. Note that we did not fix the gauge and that the procedure is manifestly gauge invariant. The fact that the matter fields play no role here can be interpreted as a consequence of the fact that they can eliminated from the action by a gauge transformation, but we did not fix the gauge.
Interpretation of the selection rule
------------------------------------
In the case of the global symmetry previously discussed, we found that if the sum of the inserted charges in the full $D$-dimensional space-time volume is non zero, then there is a flow at the boundary clashing with PBC or OBC and the average can only be zero. In the case of the local symmetry, the selection rule is microscopic and applies to a unit $D$-dimensional cube enclosing any site.
The reason gauge-variant expressions are zero is simple. For instance, it is easy to show that \^[i\_x]{}=0, in agreement with Elitzur’s theorem [@elitzurt]. We proceed as before and integrate over the gauge fields, and all the $\varphi$’s except for $\varphi_x$. If we now insert ${\rm e}^{i\varphi_x}$ in the functional integral, this is the only part that contains $\varphi_x$ since we just explained that other dependence on $\varphi_x$ disappears and the integration over $\varphi_x$ produces 0 in agreement with Eq. (\[eq:gaugeselection\]). In order to cancel ${\rm e}^{i\varphi_x}$, we need to insert another contribution, for instance ${\rm e}^{-i(A_{x,1}+\varphi_{x+\hat{1}})}$, which allows us to escape the consequences of Eq. (\[eq:gaugeselection\]) at $x$ and $x+\hat{1}$. This modifies the gauge integration and introduces non-zero values for $\sum_i(n_{x,i}-n_{x-\hat{i},i})$ which cancel the insertions of $\varphi_{x}$ and $\varphi_{x+\hat{1}}$. This mechanism persists after truncation of the Hilbert space parametrized in terms of the $m_{x,i,j}$: Eq. (\[eq:dmufmunu\]) and its consequence that we just discussed remain valid for a restricted set of $m_{x,i,j}$. Numerical studies of truncations in Lagrangian and Hamiltonian forms can be found in Refs. [@prl121; @prd98].
Extensions of the results {#sec:ext}
=========================
Discrete symmetries
-------------------
The results presented in Secs. \[sec:o2\] and \[sec:cahm\] extend easily to the case of discrete Abelian symmetries like $Z_n$ where the shifts $\alpha$ in Eqs. (\[eq:phivar\]) and (\[eq:varphix\]) are restricted to integer multiples of $2\pi/n$. With that restriction, some product of Fourier modes that must have a zero expectation value for the full $U(1)$ symmetry may become non-zero if the sum of the Fourier mode vanishes modulo $n$. In a similar way, the Kronecker deltas apply modulo $n$.
More generally, we never used infinitesimal transformations and as explained in Sec. \[sec:sym\], the measure and the action are invariant under the entire group of symmetry. The main difference in the treatment of discrete subgroups is that the sums are already finite in the original theory.
Non-Abelian global symmetries
-----------------------------
For the O(3) model, the Fourier modes are replaced by spherical harmonics. For a specific global rotation $R$, Eq. (\[eq:insert\]) becomes $$\begin{aligned}
\label{eq:inserto3}\nonumber
&\langle& Y_{\ell_1 m_1}(\theta_{{x}_1},\varphi_{{x}_1})\dots Y_{\ell_N m_N}(\theta_{{x}_N},\varphi_{{x}_N}) \rangle =D^{\ell_1}_{m_1m_1'} (R)
\\ \nonumber
& &
\dots D^{\ell_N}_{m_N m_N'} (R)\langle Y_{\ell_1 m_1'}(\theta_{{x}_1},\varphi_{{x}_1})\dots Y_{\ell_N m_N'}(\theta_{{x}_N},\varphi_{{x}_N})\rangle, \nonumber \end{aligned}$$ where the $D^{\ell}_{m m'} (R)$ are the matrices corresponding to the $\ell$ representation and the $m_i'$ indices are summed from $-\ell_i$ to $\ell_i$. By using iteratively the Clebsch-Gordan series, the expectation value can be decomposed into a sum of irreducible representations, and only the singlets are allowed to get a non-zero expectation value.
Arbitrary truncations are likely to generate non-zero expectation values for the non-singlets. However, if we keep irreducible representations at each link, in other words, if we keep all the $m$’s corresponding to a given $\ell\leq \ell _{max}$ , Eq. (3.12) of Ref. [@prd88] shows that the truncation in $\ell$ respects the global symmetries. This is because \_[m=-]{}\^Y\^\_[m]{}(,) Y\_[m]{}(’,’), is invariant under global rotations. It seems possible to extend the argument beyond this special example.
Pure gauge Abelian models
-------------------------
The pure gauge $U(1)$ model can be obtained by taking the limit $\beta \rightarrow 0$ in Eq. (\[eq:smatter\]). The $\varphi_x$ fields disappear from the action and their integration results in a factor 1. In the compact Abelian Higgs model, the link indices $n_{x,i}$ associated with the $\phi$ interactions are completely determined by the plaquette indices $m_{x,i,j}$ as shown in Eq. (\[eq:dmufmunu\]). When we insert ${\rm e}^{iA_{x,i}}$ in the functional integral, an additional term is introduced in Eq. (\[eq:dmufmunu\]) and it conflicts with $\sum_i(n_{x,i}-n_{x-\hat{i},i})=0$ which is independently enforced by the $\varphi_x$ integration. Consequently, for the compact Abelian Higgs model we have \[eq:elit2\] \^[iA\_[x,i]{}]{} =0, in agreement with Elitzur’s theorem [@elitzurt].
Extra work is needed in order to show that a similar equation is true in the pure gauge limit and that it is respected by truncations. This can be achieved by assembling tensors surrounding a given site $x$ in a way that is compatible with a selection rule. Following Ref. [@prd88], we use $2D$ $A$-tensors with $2(D-1)$ legs. Each $A$-tensor is associated with a link coming out of the site $x$ and its legs are orthogonal to this link. We assemble these $A$-tensors by connecting them with $B$-tensors in the middle of the plaquettes attached to $x$. Geometrically, the $A$-tensors form the boundaries of a $D$-dimensional cube. Graphical representations can be found in Ref. [@prd88]. The $A$-tensors provide a Kronecker delta that is a discrete version of $\partial _\mu F^{\mu \nu}=0$. It is expressed with a specific sign convention in Eq. (\[eq:dmufmunu\]) with $n_{x,i}=0$. The weight $I_m(\beta_{pl})$ appearing in the Fourier expansion of the Boltzmann weights of the plaquette interactions can be moved to the $B$-tensor and plays no role in the discussion.
We can now imitate the procedure of Sec. \[sec:o2\] and assign “in” and “out" qualities to the legs of the $A$-tensors. For a given pair of directions $i$ and $j$, there are 8 types of legs for the $A$-tensors that we label $[(x,i),\pm \hat{j}]$, $[(x-\hat{i},i),\pm \hat{j}]$, $[(x,j),\pm \hat{i}]$, and $[(x-\hat{j},j),\pm \hat{i}]$. The pair of indices appearing first refers to the links where the $A$-tensor is attached and the second index to the direction of the leg which can be positive or negative. The $[(x,i), \hat{j}]$ with $i<j$ are given an out assignment. There are three operations that swap in and out: changing $(x,i)$ into $(x-\hat{i},i)$, changing $\hat{j}$ into $-\hat{j}$ and interchanging $i$ and $j$. A detailed inspection shows that this assignment gives consistent in-out assignments at the $B$ tensors and that the assignment is compatible with the sign partition used in Eq. (\[eq:dmufmunu\]). Consequently, the Kronecker delta appearing at any link is independently enforced by the Kronecker deltas on the $2D-1$ other links attached to $x$ and if we insert ${\rm e}^{iA_{x,i}}$ the conditions become incompatible which implies Eq. (\[eq:elit2\]). Again the argument is based on the selection rules and is independent of the specific values of the tensors for any set of allowed indices.
Conclusions
===========
In summary, we have discussed the way symmetries are implemented in TFT for two models with a continuous Abelian symmetry. In both cases, we found that the truncation of the tensorial sums are compatible with the general identities reflecting the symmetries. By approximating some of the tensors with high indices by zero, we do not break these symmetries. The only way to do that would be to introduce new tensors which explicitly break the conservations laws at the sites or links. For numerical calculations, this implies for instance that it is possible to get a zero magnetization in the symmetric phase when a symmetry breaking term is set to zero. This is illustrated in Fig. 4 of Ref. [@pre89].
For the models considered here, the symmetry is encoded in Kronecker deltas build in the tensors and located at the vertices of graphs that cover either the entire space-time lattice for global symmetries, or are enclosed in a $D$-dimensional cube for local symmetries. An intuitive picture of the way the Abelian symmetries are realized can be obtained by considering the sampling of the tensor configurations that can be performed using the worm algorithm [@PhysRevLett.87.160601; @Banerjee:2010kc; @pra90; @pre93]. In this sampling algorithm, the worm carries a discrete charge which is conserved at each vertex following the Kronecker delta prescription. Restricting options at the vertices does not conflict with the charge conservation.
Unlike Nother’s standard field theoretical construction, our construction does not rely on taking infinitesimal symmetry transformations. The character expansions require the full group. Consequently everything we did applies to discrete subgroups. For global non-Abelian symmetries, the truncation must keep a certain number of irreducible representations and combine the weights in a way that is manifestly invariant before the field integrations are performed, as we showed explicitly for the O(3) sigma model. It seems possible to extend this construction in more general circumstances.
Fermions are more complicated, because if we try to derive equations similar to Eq. (\[eq:dmufmunu\]), the indices associated to the fermions only take a finite number of values. As fermionic theories are under construction in the tensor language [@Shimizu:2014fsa; @Shimizu:2014fsa; @Takeda:2014vwa; @Shimizu:2017onf; @Yoshimura:2017jpk; @Kadoh:2018hqq], this is work for the future. The TFT formulation of the non-Abelian Higgs model has been recently discussed and used for numerical purposes [@juysu2]. It would be interesting to try to generalize the construction of Sec. \[sec:cahm\] for $SU(2)$. Another question of interest would be to understand the relationship of truncated tensor methods with quantum link models [@qlink97; @qlink2] or matrix product states [@Banuls:2018jag].
The fact symmetries are preserved by truncations means that it is advantageous to keep these symmetries exactly in numerical formulations for instance in TRG calculations. A simple example where it is possible is given in Ref. [@prb87] for the Ising model where sectors of different charges can be separated explicitly. In quantum computations and quantum simulations experiments, it is desirable to have formulations with a minimal numbers of local degrees of freedom compatible with the symmetries. One can than expect to recover the result characterizing the universality class in the continuum limit. In noisy quantum computations, symmetry breaking is expected to occur generically and mix the energy sectors. If this symmetry breaking represents a relevant direction of the renormalization group flows and can be varied, results for different levels of noise and different size systems could be analyzed using finite size scaling. Alternatively, one might try to design qubits assignments such that the mixing of the energy sectors is impossible.
We thank R. Brower, E. Gustafson, S. Lloyd, W. Polyzou, J. Unmuth-Yockey, and F. Verstraete for stimulating questions. This work was supported in part by the U.S. Department of Energy (DOE) under Award Numbers DE-SC0010113, and DE-SC0019139.
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---
abstract: 'In this letter, taking the well known (2+1)-dimensional soliton systems, Davey-Stewartson (DS) model and the asymmetric Nizhnik-Novikov-Veselov (ANNV) model, as two special examples, we show that some types of lower dimensional chaotic behaviors may be found in higher dimensional soliton systems. Especially, we derive the famous Lorenz system and its general form from the DS equation and the ANNV equation. Some types of chaotic soliton solutions can be obtained from analytic expression of higher dimensional soliton systems and the numeric results of lower dimensional chaos systems. On the other hand, by means of the Lax pairs of some soliton systems, a *lower dimensional chaos system may have some types of *higher dimensional Lax pairs. An explicit (2+1)-dimensional Lax pair for a (1+1)-dimensional chaotic equation is given.**'
author:
- |
Sen-yue Lou$^{1,2,3,4}$[^1], Xiao-yan Tang$^{2,3}$ and Ying Zhang$^{3}$\
**CCAST (World Laboratory), PO Box 8730, Beijing 100080, P. R. China\
**Physics Department of Shanghai Jiao Tong University, Shanghai 200030, P. R. China[^2]\
*$^{3}$Abdus Salam International Centre for Theoretical Physics, Trieste, Italy*****
title: |
**Chaos of soliton systems\
and special Lax pairs for chaos systems**
---
= 16truecm = 23truecm
= -1truecm = -2truecm
.1in
[**PACS numbers: 02.30.Ik, 05.45.-a, 05.45.Ac, 05.45.Jn**]{}
.1in
In the past three decades, both the solitons ${\cite{soliton}}$ and the chaos${\cite{chaos}}$ have been widely studied and applied in many natural sciences and especially in almost all the physics branches such as the condense matter physics, field theory, fluid dynamics, plasma physics and optics etc. Usually, one considers that the solitons are the basic excitations of the integrable models, and the chaos is the basic behavior of the nonintegrable models. Actually, the above consideration may not be complete especially in higher dimensions. When one says a model is integrable, one should emphasize two important facts. The first one is that we should point out the model is integrable under what special meaning(s). For instance, we say a model is Painlevé integrable if the model possesses the Painlevé property, and a model is Lax or IST (inverse scattering transformation) integrable if the model has a Lax pair and then can be solved by the IST approach. An integrable model under some special meanings may not be integrable under other meanings. For instance, some Lax integrable models may not be Painlevé integrable$\cite{Eilbeck,
lsytxy}$. The second fact is that for the general solution of a higher dimensional integrable model, say, a Painlevé integrable model, there exist some lower dimensional *arbitrary functions, which means any lower dimensional chaotic solutions can be used to construct exact solutions of higher dimensional integrable models.*
In this letter, we will show that an IST integrable model and/or a Painlevé integrable model may have some lower dimensional reductions with chaotic behaviors and then a lower dimensional chaos system may have some higher dimensional Lax pairs.
To show our conclusions, we use the (2+1)-dimensional Davey-Stewartson (DS) equation${\cite{DS}}$ $$\begin{aligned}
&&i u_t+2^{-1}(u_{xx}+u_{yy})+\alpha |u|^2u-uv=0,\\ &&v_{xx}-
v_{yy}-2\alpha (|u|^2)_{xx}=0,\end{aligned}$$ as a concrete example at first. The DS equation is an isotropic Lax integrable extension of the well known (1+1)-dimensional nonlinear Schrödinger (NLS) equation. The DS system is the shallow water limit of the Benney-Roskes equation${\cite{DS}}$, where $u$ is the amplitude of a surface wave-packet and $v$ characterizes the mean motion generated by this surface wave. The DS system (1) and (2) can also be derived from the plasma physics${\cite{NAS}}$ and the self-dual Yang-Mills field. The DS system has also been proposed as a 2+1 dimensional model for quantum field theory${\cite{SAB}}$. It is known that the DS equation is integrable under some *special meanings, namely, it is IST integrable and Painlevé integrable$\cite{Boiti}$. Many other interesting properties of the model like a special bilinear form, the Darboux transformation, finite dimensional integrable reductions, infinitely many symmetries and the rich soliton structures${\cite{Boiti, Lou}}$ have also been revealed.*
To select out some chaotic behaviors of the DS equation, we make the following transformation $$\begin{aligned}
&&v =v_0-f^{-1}(f_{x'x'}+f_{y'y'}+2f_{x'y'})
+f^{-2}(f_{x'}^2+2f_{y'}f_{x'}+f_{y'}^2), \\ &&u = gf^{-1}+u_0\end{aligned}$$ with real $f$ and complex $g$, where $x'=(x+y)/\sqrt{2},\
y'=(x-y)/\sqrt{2}$, and $\{u_0,\ v_0\}$ is an arbitrary seed solution of the DS equation. Under the transformation (3) and (4), the DS system (1) and (2) is transformed to a general bilinear form: $$\begin{aligned}
&&(D_{x'x'}+D_{y'y'}+2iD_t)g\cdot f
+u_0(D_{x'x'}+2D_{x'y'}+D_{y'y'})f\cdot f\nonumber\\ &&
\qquad+2\alpha u_0gg^* +2\alpha u_0^2g^*f-2v_0gf+G_1fg=0\end{aligned}$$ $$\begin{aligned}
2(D_{x'y'}+\alpha |u_0|^2)f\cdot f +2\alpha gh+2\alpha g f
u_0^*+2\alpha u_0g^*f-G_1ff=0,\end{aligned}$$ where $D$ is the usual bilinear operator${\cite{Hirota}}$ defined as $D_{x}^mA\cdot B\equiv (\partial_x-\partial_{x_1})^m
A(x)B(x_1)|_{x_1=x}$, and $G_1$ is an arbitrary solution of $-16\alpha(u_{0x'}+u_{0y'})(u_{0x'}^*+u_{0y'}^*)
+G_{1x'x'}+G_{1y'y'}+2G_{1x'y'}
-4\alpha(D_{x'x'}+D_{y'y'}+2D_{x'y'})u_0\cdot u_0^*=0.$ For the notation simplicity, we will drop the “primes" of the space variables later.
To discuss further, we fix the seed solution $\{u_0,\ v_0\}$ and $G_1$ as $$\begin{aligned}
u_0=G_1=0, \qquad v_0=p_0(x,t)+q_0(y,t),\end{aligned}$$ where $p_0\equiv p_0(x,t)$ and $q_0\equiv q_0(y,t)$ are some functions of the indicated variables.
To solve the bilinear equations (5) and (6) with (7) we make the ansatz $$\begin{aligned}
f=C+p+q,\ g=p_1q_1\exp(ir+is),\end{aligned}$$ where $p \equiv p(x,\ t) ,\ q\equiv q(y,\ t) ,\ p_1\equiv p_1(x,\
t) ,\ q_1\equiv q_1(y,\ t) ,\ r\equiv r(x,\ t),\ s\equiv s(y,\ t)
$ are all real functions of the indicated variables and $C$ is an arbitrary constant. Substituting (8) into (5) and (6), and separating the real and imaginary parts of the resulting equation, we have $$\begin{aligned}
2p_xq_y-\alpha p_1^2q_1^2=0.\end{aligned}$$ $$\begin{aligned}
&&(q_1 p_{1xx} + p_1 q_{1yy} -p_1 q_1 (2r_t+2s_t +2 (p_0+q_0)
+s_{y}^2 +r_{x}^2 )) (C+p+q) \nonumber\\ && + q_1 (p_1 p_{xx}-2
p_{1x} p_{x}) + p_1 (q_1 q_{yy}-2 q_{1y} q_{y})=0\end{aligned}$$ $$\begin{aligned}
&&( -q_1 (2r_{x} p_{1x} +2 p_{1t} + p_1r_{xx}) - p_1 (2s_{y}
q_{1y} +2 q_{1t}+q_1 s_{yy}))(C+p+q) \nonumber\\ && +2 q_1 p_1
(q_t+s_{y} q_{y}) +2 q_1 p_1 (r_{x} p_{x}+p_t)=0\end{aligned}$$
Because the functions $p_0,\ p,\ p_1$ and $r$ are only functions of $\{x,\ t\}$ and the functions $q_0,\ q,\ q_1$ and $s$ are only functions of $\{y,\ t\}$, the equation system (9), (10) and (11) can be solved by the following variable separated equations: $$\begin{aligned}
p_1=\delta_1\sqrt{2\alpha^{-1}c_1^{-1}p_x},\
q_1=\delta_2\sqrt{c_1q_y},\ (\delta_1^2=\delta_2^2=1),\end{aligned}$$ $$\begin{aligned}
p_t=-r_{x}p_{x}+c_2,\ q_t=-s_{y}q_{y}-c_2,\end{aligned}$$ $$\begin{aligned}
&& 4(2r_t+r_{x}^2+2p_0)p_{x}^2+p_{xx}^2
-2p_{xxx}p_{x}+c_0p_{x}^2=0,\\
&&4(2s_t+s_{y}^2+2q_0)q_{y}^2+q_{yy}^2
-2q_{y}q_{yyy}-c_0q_{y}^2=0.\end{aligned}$$ In Eqs. (12)–(15), $c_1,\ c_2$ and $c_0$ are all arbitrary functions of $t$.
Generally, for a given $p_0$ and $q_0$ the equation systems {(12), (14)} and {(13), (15)} may not be integrable. However, because of the arbitrariness of $p_0$ and $q_0$, we may treat the functions $p$ and $q$ are arbitrary while $p_0$ and $q_0$ are determined by (14) and (15). Because $p$ and $q$ are arbitrary functions, in addition to the stable soliton selections, there may be various chaotic selections. For instance, if we select $p$ and $q$ are solutions of ($\tau_1\equiv x+\omega_1 t,\ \tau_2\equiv
x+\omega_2 t$) $$\begin{aligned}
p_{\tau_1\tau_1\tau_1}=\left(p_{\tau_1\tau_1}p_{\tau_1}+(c+1)p_{\tau_1}^2\right)p^{-1}
-(p^2+bc+b)p_{\tau_1}-(b+c+1)p_{\tau_1\tau_1}+pc(ba-b-p^2),\
\\
q_{\tau_2\tau_2\tau_2}=\left(q_{\tau_2\tau_2}q_{\tau_2}+(\gamma+1)q_{\tau_2}^2\right)q^{-1}
-(q^2+\beta\gamma+\beta)q_{\tau_2}-(\beta+\gamma+1)q_{\tau_2\tau_2}+qc(\beta\alpha-\beta-q^2),\end{aligned}$$ where $\omega_1,\ \omega_2,\ a,\ b,\ c,\ \alpha,\ \beta$ and $\gamma$ are all arbitrary constants, then $$\begin{aligned}
&& c_0=c_2=0,\ r=-\omega_1 (x+\omega_1 t/2),\ s=-\omega_2
(y+\omega_2 t/2), \\
&&p_0=-4^{-1}\left({cp^3}{p_{\tau_1}^{-1}}+p^2-{bc}{p_{\tau_1}^{-1}}(a-1)p
+b(c+1)+(b+c+1){p_{\tau_1\tau_1}}{p_{\tau_1}^{-1}}\right.\nonumber\\
&&\qquad \left.+{p_{\tau_1\tau_1}^2}{2^{-1}p_{\tau_1}^{-2}}
-p^{-1}(p_{\tau_1}(c+1)+p_{\tau_1\tau_1})\right),\\
&&q_0=-4^{-1}\left({\gamma
q^3}{q_{\tau_2}^{-1}}+q^2-{\beta\gamma}{q_{\tau_2}}^{-1}(\alpha-1)q
+\beta(\gamma+1)+(\beta+\gamma+1){q_{\tau_2\tau_2}}{q_{\tau_2}^{-1}}\right.\nonumber\\
&&\qquad \left.+{q_{\tau_2\tau_2}^2}{2^{-1}q_{\tau_2}^{-2}}
-q^{-1}(q_{\tau_2}(\gamma+1)+q_{\tau_2\tau_2})\right).\end{aligned}$$ Substituting (8) with (12)–(20) into (3) and (4), we get a general solution of the DS equation $$\begin{aligned}
&&u={\delta_1\delta_2\sqrt{2\alpha^{-1}p_{\tau_1}q_{\tau_2}}\exp(-i(\omega_1x
+\omega_2y+\frac12(\omega_1^2+\omega_2^2)t)} {(C+p+q)^{-1}},\\
&&v=p_0+q_0-{(q_{\tau_2\tau_2}
+p_{\tau_1\tau_1})}{(C+p+q)^{-1}}+{(q_{\tau_2}^2
+2q_{\tau_2}p_{\tau_1}+p_{\tau_1}^2)}{(C+p+q)^{-2}}\end{aligned}$$ where $p_0$ and $q_0$ are determined by (19) and (20), while $p$ and $q$ are given by (16) and (17).
It is straightforward to prove that (16) (and (17)) is equivalent to the well known chaos system, the Lorenz system${\cite{Lorenz}}$: $$\begin{aligned}
p_{\tau_1}=-c(p-g),\ g_{\tau_1}=p(a-h)-g,\ h_{\tau_1}=pg-bh.\end{aligned}$$ Actually, after canceling the functions $g$ and $h$ in (23), one can find (16) immediately.
From the above discussions, some interesting things are worth emphasizing:
Firstly, because of the arbitrariness of the functions $p$ and $q$, any types of other lower dimensional systems may be used to construct the exact solutions of the DS system.
Secondly, the lower dimensional chaotic behaviors may be found in many other higher dimensional soliton systems. For instance, by means of the direct substitution or the similar discussions as for the DS equation, one can find that ($\tau_1\equiv x+\omega_1t$) $$\begin{aligned}
u={2p_{x}w_y}{(p+w)^{-2}},\end{aligned}$$ $$\begin{aligned}
v=\frac{2p_{x}^2}{(p+w)^2}+\frac{(c+1)p_{x}}{3p}
-\frac{(5p-w)p_{xx}}{3p(p+w)}-\frac13 p^2
-\frac1{3p_{x}}[(1+b+c)p_{xx}+cp^3-cb(a-1)p],\end{aligned}$$ with $w$ being an arbitrary function of $y$, and $p$ being determined by the (1+1)-dimensional extension of the Lorenz system $$\begin{aligned}
p_t=-p_{xxx}+p^{-1}[p_{xx}p_x+(c+1)p_x^2]-p^2p_x-(b+c+1)p_{xx}-pc(b-ba+p^2),\end{aligned}$$ solves the following IST and Painlevé integrable KdV equation which is known as the ANNV model${\cite{BLMP}}$ $$\begin{aligned}
u_t+u_{xxx}-3(uv)_x=0,\ u_x=v_y.\end{aligned}$$ It is clear that the Lorenz system (16) is just a special reduction of (27) with $ p=p(x+b(c+1)t)\equiv p(\tau_1). $ Actually, $p$ of equations (24) may also be arbitrary function of $\{x,\ t\}$ after changing (25) appropriately $\cite{BLMP}$. In other words, any lower dimensional chaotic behavior can also be used to construct exact solutions of the ANNV system.
The third thing is more interesting. The Lax pair plays a very important and useful role in integrable models. Nevertheless, there is little progress in the study of the possible Lax pairs for chaos systems like the Lorenz system. In Ref. $\cite{Eilbeck}$, Chandre and Eilbeck had given out the Lax pairs for two discrete non-Painlevé integrable models. In Ref. $\cite{lsytxy}$, we have found some Lax pairs for some non-integrable Schwarzian equations. Now from the above discussions, we know that both the Lax pairs of the DS equation and those of the ANNV system may be used as the special higher dimensional Lax pairs of *arbitrary chaos systems like the Lorenz system (16) and/or the generalized Lorenz system (26) by selecting the fields appropriately like (21)-(22) and/or (24)-(25). For instance, the (1+1)-dimensional generalized Lorenz system (26) has the following Lax pair $$\begin{aligned}
\psi_{xy}=u\psi,\qquad \psi_t=-\psi_{xxx}+3v\psi_{x}\end{aligned}$$ with $\{u,\ v\}$ being given by (24) and (25). From (24)-(27), we know that a *lower dimensional chaos system can be considered as a consistent condition of a *higher dimensional linear system. For example, $\psi_{xyt} =\psi_{txy} $ of (28) just gives out the generalized Lorenz system (26).***
Now a very important question is what the effects of the lower dimensional chaos to the higher dimensional soliton systems are. To answer this question, we use the numerical solutions of the Lorenz system to see the behaviors of the corresponding solution (24) of the ANNV equation by taking $ w=200+\tanh(y-y_0)\equiv
w_s$ and $p=p(\tau_1)\equiv p(X)$ as the numerical solution of the Lorenz system (16). Under the selection $w=w_s$, (24) is a line soliton solution located at $y=y_0$. Due to the entrance of the function $p$, the structures of the line soliton become very complicated. For some types of the parameters, the solutions of the Lorenz system have some kinds of periodic behavior, then the line soliton solution (24) with $w=w_s$ becomes an $x$ periodic line soliton solution that means the solution is localized in $y$ direction (not identically equal to zero only near $y=y_0$) and periodic in $x$ direction. Fig. 1 shows the behavior of the periodic two line soliton solution and $p$ being a periodic two solution of the Lorenz system (16) with $a=350,\ b=8/3$ and $c=10$. In some other types of the parameter ranges, the solution of the Lorenz system becomes chaotic and then the line soliton of the ANNV system becomes chaotic line soliton which is localized in $y$ direction and chaotic in $x$ direction. Fig. 2 displays the chaotic behavior of the amplitude of the line soliton located at $y=y_0$ with $a=60, b=8/3$ and $c=10$. The parameters we used here are the same as those used in literature$\cite{Lorenz}$.
epsf
epsf
In summary, though some (2+1)-dimensional soliton systems, like the DS equation and the ANNV equation, are Lax and IST integrable, and some special types of soliton solutions can be found by IST and other interesting approaches${\cite{Boiti}}$, any types of chaotic behaviors may still be allowed in some special ways. Especially, the famous chaotic Lorenz system and its (1+1)-dimensional generalization are derived from the DS equation and the ANNV equation. Using the numerical results of the lower dimensional chaotic systems, we may obtain some types of nonlinear excitations like the periodic and chaotic line solitons for higher dimensional soliton systems. On the other hand, the lower dimensional chaos systems like the generalized Lorenz system may have some particular Lax pairs in higher dimensions.
It is also known that both the ANNV system and the DS systems are related to the Kadomtsev-Petviashvili (KP) equation while the DS and the KP equation are the reductions of the self-dual Yang-Mills (SDYM) equation. So both the KP and the SDYM equations may possess arbitrary lower dimensional chaotic behaviors induced by arbitrary functions. From the results of this paper, various important and interesting problems should be studied further. For instance, what on earth is the *complete integrability and what kinds of information about chaos can be obtained from some types of special higher dimensional Lax pairs?*
.2in The author is in debt to thanks the helpful discussions with the professors Q. P. Liu and G-x Huang. The work was supported by the National Outstanding Youth Foundation of China (No.19925522), the Research Fund for the Doctoral Program of Higher Education of China (Grant. No. 2000024832) and the National Natural Science Foundation of Zhejiang Province of China.
.2in
[99]{} Y. S. Kivshar and B. A. Malomend, Rev. Mod. Phys. 61, 765 (1989). J. P. Gollub and M. C. Cross, Nature, 404, 710 (2000); R. A. Jalabert and H. M. Pastawski, Phys. Rev. Lett. 86, 2490 (2001). C. Chandre and J. C. Eilbeck, Does the existence of a Lax pair imply integrability, Preprint (1997). S-y Lou, X-y Tang, Q-P Liu and T. Fukuyama, nlin.SI/0108045 (2001). A. Davey and K. Stewartson, Proc. R. Soc. A, 338, 101 (1974). K. Nishinari and K. Abe and J. Satsuma, J. Phys. Soc. Japan, 62, 2021 (1993). C. L. Schultz, M. J. Ablowitz and D. BarYaacov, Phys. Rev. Lett. 59, 2825 (1987). M. Boiti, J. J. P. Leon, L. Martina and F. Penpinelli, Phys. Lett. 132A, 432 (1988); A. S. Fokas and P. M. Santini, Phys. Rev. Lett. 63, 1329 (1989); R. A. Leo, G. Mancarella, G. Soliani and L. Solombrino, J. Math. Phys. 29 (1988) 2666. J. Hietarinta, Phys. Lett. 149A, 133 (1990); V. B. Matveev and M. A. Salle, *Darboux Transformation and Solitons, (Springer-Verlag 1991); Z-x Zhou, Inverse Problems, 14, 1371 (1998); S-y Lou and X-b Hu, J. Phys. A: Math. Gen. 27, L207 (1994); S-y Lou, and Lu J-z, J. Phys. A: Math. Gen. 29, 4209 (1996). R. Hirota, Phys. Rev. Lett. 27, 1192 (1971). C. Sparrow, *The Lorenz Equations: Bifurcations, Chaos, and Strange Attractors, Appl. Math. Sci. 41, (1982 Springer-Verlag New York). M. Boiti, J. J. P. Leon, M. Manna and F. Penpinelli, Inverse Problems, 2, 271(1986); S-y Lou and H-y Ruan, ibid. 34, 305 (2001).**
[^1]: Email: sylou@mail.sjtu.edu.cn
[^2]: Mailing address
|
---
abstract: 'A gaseous pixel readout module with four GridPix chips, called the quad, has been developed as a building block for a large time projection chamber readout plane. The quad module has dimensions and an active surface coverage of 68.9%. The GridPix chip consists of a Timepix3 chip with integrated amplification grid and have a high efficiency to detect single ionisation electrons, which enable to make a precise track position measurement. A quad module was installed in a small time projection chamber and measurements of electrons were performed at the ELSA accelerator in Bonn, where a silicon telescope was used to provide a reference track. The error on the track position measurement, both in the pixel plane and drift direction, is dominated by diffusion. The quad was designed to have minimum electrical field inhomogeneities and distortions, achieving systematics of better than in the pixel plane. The resolution of the setup is , where the total systematic error of the quad detector is .'
address:
- 'Nikhef, Science Park 105, 1098 XG Amsterdam, The Netherlands'
- 'Physikalisches Institut, University of Bonn, Nussallee 12, 53115 Bonn, Germany'
author:
- 'C. Ligtenberg'
- 'Y. Bilevych'
- 'K. Desch'
- 'H. van der Graaf'
- 'M. Gruber'
- 'F. Hartjes'
- 'K. Heijhoff'
- 'J. Kaminski'
- 'P.M. Kluit'
- 'N. van der Kolk'
- 'G. Raven'
- 'T. Schiffer'
- 'J. Timmermans'
bibliography:
- 'mybibfile.bib'
title: Performance of the GridPix detector quad
---
Micromegas,gaseous pixel detector,micro-pattern gaseous detector,Timepix,GridPix,time projection chamber
Introduction
============
In drift chambers charged particles are identified through ionisation in the gas. For the readout of a time projection chamber (TPC) the finest granularity is offered by pixel readouts. In particular, a GridPix is a CMOS pixel readout chip with an integrated amplification grid added by MEMS postprocessing techniques [@Colas:2004ks; @Campbell:2004ib]. As a result, single ionisation electrons can be detected with great precision, allowing an excellent track position measurement and an estimate of the number of clusters for an energy loss (dE/dx) measurement for particle identification.
The original GridPix using the Timepix chip [@Kaminski:2017bgj] has recently been succeeded by a GridPix based on the Timepix3 chip [@Poikela:2014joi]. This newer chip offers superior timing, faster readout speed and the possibility to apply time walk corrections. The first results of a single chip detector have been analysed and published in [@Ligtenberg:2018a]. Electron diffusion in gas was found to be the dominant error on the track position measurement and systematics in the pixel plane remained below . Using a truncated sum, an energy loss (dE/dx) resolution of was found for an effective track length of . The single chip detector was operated reliably in a test beam experiment. However, equipping a large detector surface poses an entirely new challenge.
In order to cover a large detector surface, it is practical to subdivide it into a number of standardized modules. Here we present the design of a quad module with four Timepix3 chips. Because the quad module has all services under the active area, it can be tiled to cover arbitrarily large areas. The performance of a TPC, read out by a single quad module was tested at the ELSA test beam facility in Bonn. Possible applications are in TPCs at future electron-positron colliders, other particle physics experiments and medical imaging such as proton therapy [@Kaminski:2017bgj; @Krieger:2017yax].
Quad detector design and construction
=====================================
The Timepix3 based GridPix
--------------------------
Here the GridPix consists of a Timepix3 chip with an integrated grid. Directly on the surface of the Timepix3 chip, a thick silicon-rich silicon nitride protection layer is deposited in order to prevent damage of the readout chip from discharges. On top of this high SU8 pillars are attached that support the thick aluminium grid that has diameter circular holes aligned to the pixel input pads. The grid and dykes design was reoptimized: at four sides the grid ends on a solid SU8 dyke for which on two sides three readout columns were given up. The Timepix3 chip has a low equivalent noise charge ($\approx$70 e$^-$) and allows for a simultaneous measurement of the Time of Arrival (ToA) and the Time over Threshold (ToT) using a TDC (clock frequency ) per pixel. For the readout, one out of the eight available links per chip is connected to a speedy pixel detector readout (SPIDR) board [@Visser:2015bsa] at a speed of 80 Mbps. The hardware allows reading out at twice this speed.
The quad module design and assembly
-----------------------------------
In order to cover large areas, the quad module shown in figure \[fig:quadPicture\] was developed. Because of the complexity of the GridPix technology and the fragility of the grids, a small number of four chips per module was chosen. The chips are mounted on a common cooled base plate (COCA). They are electrically connected by wire bonds to a wide PCB between the two pairs of chips. This allows the control and output lines to be directed to the backside of the quad to maximize the sensitive detection area. A short Kapton cable at the other side of the wire bond PCB provides a low impedance connection to the low voltage (LV) regulator. The grids are connected by insulated copper wires to a high voltage (HV) filtering board. The connection to the common HV input uses a 100 M$\Omega$ resistor for each grid to rapidly quench a micro-discharge. To support and cool the LV regulator board and the HV filtering board, a U-shaped support is attached by thermally conductive glue under the carrier plate. Finally, the wire bonds of the quad are covered by a 10 mm wide central guard electrode located 1.1 mm above the grids to maintain a homogeneous drift field.
The external quad dimensions are of which 68.9% is active. In the present design the support components are made of aluminum contributing substantially to the material budget. In the future the material budget can be further minimized by replacing the aluminium by carbon based materials. During low rate operation, the module consumes of power of which in the LV regulator.
1 ![Picture of the quad detector with four Timepix3 GridPixes (TPX3) mounted on a cold carrier plate (COCA). The central guard was omitted to show the wire bond PCB, and its operating position is indicated with a transparent rectangle. On the left the Low Voltage (LV) regulator and flexible Kapton cable are visible.[]{data-label="fig:quadPicture"}](img_gray/figure1.pdf "fig:"){width="60.00000%"} ![Picture of the quad detector with four Timepix3 GridPixes (TPX3) mounted on a cold carrier plate (COCA). The central guard was omitted to show the wire bond PCB, and its operating position is indicated with a transparent rectangle. On the left the Low Voltage (LV) regulator and flexible Kapton cable are visible.[]{data-label="fig:quadPicture"}](img/figure1.pdf "fig:"){width="60.00000%"}
The quad detector
-----------------
The quad module was embedded in a TPC consisting of a steel box and a 40 mm high field cage to provide a homogeneous drift field. The sides of the field cage are formed by CuBe wires with a pitch to facilitate UV laser beam measurements. The field cage is terminated on one end by the quad module fitted in a closely surrounding coppered frame at the grid potential, and on the other end by a solid cathode plate. The whole structure was put in a gas-tight container with Kapton windows on two sides to minimize the material traversed by the beam.
Test beam measurement
=====================
The device was tested in October 2018 at the ELSA test beam facility in Bonn. The ELSA accelerator provided electrons at a rate of approximately during spills of in beam cycles of . The whole quad detector was mounted on a remotely controlled slider stage. To provide a precise reference track, the quad TPC was sandwiched between 2 $\times$ 3 planes of a [@Jansen2016], see figure \[fig:testBeamSetup\]. Each plane consists of a MAPS detector with $1152\times576$ pixels of size .
A scintillator provided a trigger signal to the Trigger Logic Unit (TLU) [@TLU] which numbers the triggers and subsequently directs them to both the SPIDR and telescope readout. The telescope hits were collected in time frames of . Due to the high beam intensity, the telescope frames often contain hits from more than one track. The Timepix3 was operated in data driven mode with the trigger data merged in.
Because of the chosen limited link speed between the Timepix3 chips and the SPIDR a maximum of could be read out. This caused some hits to arrive late at the SPIDR readout, acquiring a wrong course timestamp. As a work-around hits, up to 200 timestamps of after the trigger were collected and analysed. The first track hit had to arrive no more than 5 timestamps late, and the average course timestamp should not deviate more than 150 timestamps. During data taking the gas volume was flushed at a rate of with premixed T2K TPC gas. This gas is a mixture consisting of , , and suitable for large TPCs because of the relatively high drift velocity and the low diffusion in a magnetic field. The temperature and pressure were relatively stable at and . The gas mixture contained a 814 ppm contamination and a contamination, primarily due to the high gas permeability of a silicon rubber cable feed-through.
The cathode and guard voltages were set such that the electric field was , which is close to the maximum drift velocity for the contaminated gas. The grid voltage was set to , at which there is limited secondary emission from the grid by UV photons produced in avalanches. The threshold level being a trade off between noise, sensitivity and time walk, was set to about . The gain depends on the beam rate, because of charging up of the protection layer. During the high beam rate, the effective gain was approximately 1000. Some of the relevant run parameters are summarized in table \[tab:runParameters\].
![Setup of the quad with telescope at the ELSA test beam facility.[]{data-label="fig:testBeamSetup"}](img/figure2.pdf){width="80.00000%"}
--------------------------- ----------------------
Runs duration 10 minutes
Triggers per run
$V_\text{grid}$
$E_\text{drift}$
Threshold
Temperature $(300.5\pm0.13)$ K
Pressure $(1011\pm0.16)$ mbar
Oxygen concentration
Water vapor concentration
--------------------------- ----------------------
: Parameters of the three analyzed runs. The error on the temperature and pressure indicates the spread in the time the three runs were taken.
\[tab:runParameters\]
Track reconstruction and event selection
========================================
Track reconstruction procedure
------------------------------
Tracks are reconstructed as straight lines. The $y$-axis is defined roughly in the direction of the beam, and the $x$-axis and $z$-axis are in the horizontal and vertical direction respectively. For the telescope, the $y$-coordinate is taken from the plane position, and the $x$-coordinate and $z$-coordinate correspond to the columns and rows. Apart from a small rotation, the $x$ and $y$-coordinates correspond to the columns and rows of the GridPixes. The $z$-coordinate is the drift length calculated from the ToA and the drift velocity. Tracks are fitted using a linear regression fit with hit errors in the two directions perpendicular to the beam $\sigma_x$ and $\sigma_z$. The expressions for the error values are given in sections \[sec:hitResolutionx\] and \[sec:hitResolutionz\].
The detectors are aligned using the data. First, the telescope is independently aligned. The positions in the $y$-direction along the beam are measured and kept fixed. Taking one plane as a reference, the other five planes can be rotated. These rotations and additionally two shifts in $x$-direction and $z$-direction for four of the planes are iteratively determined from the fitted tracks. Next, the quad detector is aligned. Using iterative alignment each chip has three rotations and two shifts in the $x$ and $z$-directions. Additionally, each chip has one parameter describing the angle in the $xz$-plane between the drift direction and the pixel plane.
An example event with a telescope track is shown in figure \[fig:eventDisplay\].
1 ![An event with 148 quad detector hits and the corresponding telescope track. The positions of the chips are outlined in blue.[]{data-label="fig:eventDisplay"}](img_gray/figure3.pdf "fig:"){width="60.00000%"} ![An event with 148 quad detector hits and the corresponding telescope track. The positions of the chips are outlined in blue.[]{data-label="fig:eventDisplay"}](img/figure3.pdf "fig:"){width="60.00000%"}
Selections
----------
In the telescope a stringent selection is made to acquire a sample of clean tracks. At least 5 planes should have a hit and the hits should be within from the track. By requiring the slope difference of the track in the first three planes and in the last three planes to be smaller than , scattered tracks are rejected.
GridPix hits are considered if their ToA is within of a trigger and their ToT is at least . The hits are collected using a track detected by the telescope as a seed. Outliers are rejected by requiring the residuals $r$ (pulls $r/\sigma$) with respect to the telescope track to be smaller than () in the $x$-direction and () in the $z$-direction.
A track is rejected if it has less than 20 hits. Moreover the average position of all Gridpix hits must be within in the $x$-direction and $z$-direction of the telescope track. Given the high beam rate, the TPC often contains multiple tracks overlapping in time. To suppress overlapping tracks and to reject tracks with delta electrons, of the hits within of a track are required to lie within a distance of .
The selections are summarized in table \[tab:selections\] and the total efficiency for events is about 12%. Most events are rejected, because there are less than 20 GridPix hits corresponding to the telescope track.
Telescope
--------------------------------------------------------------------
Number of planes hits $\ge$ 5
Reject outliers ($r_{x,z} <$ )
Slope difference between sets of planes $<$
GridPix hit selection
$\SI{-500}{ns} < t_\text{hit}-t_\text{trigger} < \SI{500}{ns} $
Hit ToT $>$
Reject outliers ( $r_x<\SI{1.5}{mm}, r_z<\SI{2}{mm}$ )
Reject outliers ( $r_x<2\sigma_x, r_z<3\sigma_z$ )
Event Selection
$ N_\text{hits}\ge20 $
$ ( N_{r_x < 1.5 \text{mm}}$ $/$ $N_{r_x < 5 \text{mm} } )$ $>0.8$
$|x_\text{Timepix}-x_\text{telescope}|<\SI{0.3}{mm}$
$|z_\text{Timepix}-z_\text{telescope}|<\SI{0.3}{mm}$
: Table with selection cuts
\[tab:selections\]
Results
=======
Number of hits
--------------
The distribution of the number of track hits per chip and the total number of track hits are shown in figure \[fig:nHits\]. The most probable number of hits per chip varies between 52 and 65 hits, and the mean varies between 65 and 80 hits. The most probable number of hits per quad is 131 and the mean number of track hits is 146 for an effective track length of approximately . This is significantly below the calculated most probable value of 225 electron-ion pairs for a 2.5 GeV electron with this track length [@Garfield]. This is due to the too low effective grid voltage and possibly due to readout problems. Because of the low single electron efficiency, no energy loss (dE/dx) results were extracted.
1 ![Distribution of the number of hits after track selection in total and per chip.[]{data-label="fig:nHits"}](img_gray/figure4.pdf "fig:"){width="60.00000%"} ![Distribution of the number of hits after track selection in total and per chip.[]{data-label="fig:nHits"}](img/figure4.pdf "fig:"){width="60.00000%"}
Hit time corrections
--------------------
To increase precision in the drift direction, the hit times were corrected. To correct for the double column structure and power distribution deformations of the Timepix3 chip, a ToT factor per column was extracted by injecting test pulses for each pixel. Furthermore, a ToA correction offset was determined from the test beam data based on the underlying substructure of pixels due to the clock distribution. In addition one ToA correction offset per column and one offset per row was applied. The ToT corrections are of $\mathcal{O}(10\%)$ and the ToA corrections are of $\mathcal{O}(\SI{1}{ns})$.
Time walk correction
--------------------
A hit is registered when the collected charge reaches the threshold. Since it takes longer for a small signal to reach the threshold than it does for a large signal, the measured ToA depends on the magnitude of the signal. This effect is called time walk and can be corrected by using the ToT as a measure of signal magnitude. In figure \[fig:fittedTW\] the mean of $z$-residuals is shown as a function of the ToT for all four chips. The relation can be parametrized using the time walk $\delta z_\text{tw}$ as a function of the ToT $t_\text{ToT}$: $$\delta z_\text{tw} = \frac{c_1}{t_\text{ToT} + t_0},
\label{eq:timewalk}$$ where $c_1$ and $t_0$ are constants determined from a fit per chip.
1 ![Mean z-residual without time walk correction as function of ToT, fitted with equation . The right axis is given in units of ns using a drift velocity of .[]{data-label="fig:fittedTW"}](img_gray/figure5.pdf "fig:"){width="60.00000%"} ![Mean z-residual without time walk correction as function of ToT, fitted with equation . The right axis is given in units of ns using a drift velocity of .[]{data-label="fig:fittedTW"}](img/figure5.pdf "fig:"){width="60.00000%"}
Hit resolution in the pixel plane {#sec:hitResolutionx}
---------------------------------
The resolution of the single electrons in the transverse plane ($xy$) was measured as a function of the predicted drift position ($z$). Figure \[fig:resolutionx\] displays this relation for tracks crossing a fiducial region in the center of the chip. The resolution for the detection of ionisation electrons $\sigma_x$ is given by: $$\sigma_x^2=\frac{d_\text{pixel}^2}{12} +D_T^2(z-z_0),
\label{eq:sigmax}$$ where $d_\text{pixel}$ is the pixel pitch size, $z_0$ is the position of the grid, and $D_T$ is the transverse diffusion coefficient. The resolution at zero drift distance $d_\text{pixel}/\sqrt{12}$ was fixed to . Tracks with a $z$-position around are given a larger error because they scatter on the central guard. Fitting expression to the data gives a transverse diffusion coefficient $D_T$ of with negligible statistical uncertainty. The measured value is larger than the value of $\pm$ 3% predicted by the gas simulation software Magboltz [@Biagi:1999nwa]. Probably this is due to an inaccuracy in the gas mixing, which caused the content to be lower than intended.
To compare the precision of the GridPix readout with the precision of conventional pad based TPC readouts, the resolution can be calculated over the length of one pad row. For example, at a drift distance of the resolution of a single ionisation electron is approximately , so the resolution of a track segment which has on average 32 electrons is therefore about .
1 ![Measured hit resolution in the pixel plane (blue points) fitted with the resolution function according to equation (red line).[]{data-label="fig:resolutionx"}](img_gray/figure6.pdf "fig:"){width="60.00000%"} ![Measured hit resolution in the pixel plane (blue points) fitted with the resolution function according to equation (red line).[]{data-label="fig:resolutionx"}](img/figure6.pdf "fig:"){width="60.00000%"}
Hit resolution in the drift direction {#sec:hitResolutionz}
-------------------------------------
The measured $z$-position is directly related to the drift velocity. Using the predicted positions from the telescope, the drift velocity is measured to be , which is slightly lower than the value of expected by Magboltz. Both values have negligible statistical uncertainties.
The resolution for the detection of ionisation electrons $\sigma_z$ is given by: $$\sigma_z^2=\sigma_{z0}^2+D_L^2(z-z_0),
\label{eq:sigmaz}$$ where $\sigma_{z0}$ is the resolution at zero drift distance. The resolution as function of the drift distance is shown in figure \[fig:resolutionz\] for tracks crossing the fiducial region. Since tracks with a $z$-position around scatter on the central guard, these data points are given a larger error. The longitudinal diffusion coefficient $D_L$ was determined to be with negligible statistical uncertainty, which is equal to the expected value $\pm$ 3% from a Magboltz calculation.
1 ![Measured hit resolution in drift direction split by ToT. The hits with a ToT above (blue points) are fitted with the resolution according to equation (red line). In the legend the fractions of hits in both selections are given.[]{data-label="fig:resolutionz"}](img_gray/figure7.pdf "fig:"){width="60.00000%"} ![Measured hit resolution in drift direction split by ToT. The hits with a ToT above (blue points) are fitted with the resolution according to equation (red line). In the legend the fractions of hits in both selections are given.[]{data-label="fig:resolutionz"}](img/figure7.pdf "fig:"){width="60.00000%"}
Deformations in the pixel plane {#sec:deformationsPixelPlane}
-------------------------------
It is important to measure possible deformations in the pixel plane ($xy$), because for applications in a TPC this affects the momentum resolution. Because of limited statistics, the mean transverse ($x$) residuals are calculated in bins of pixels over the quad plane using the tracks defined by the telescope, see figure \[fig:deformationsx\]. Only bins with more than 800 entries are shown.
A distortion is present near the edges of the chips. The cause is twofold; firstly there is a geometrical bias at the edge of the detector because only part of the ionisation cloud can be detected. Secondly, the grounded region at the edge of the Timepix3 die causes a non-uniformity of the electric field.
An empirically selected function of four Cauchy (Breit-Wigner) functions can be fitted to the geometrical bias and the non-uniformity of the field. Near the top and bottom edges the size of the deformations is different, as such - while keeping the other parameters fixed - a $4^\text{th}$ order polynomial function in $y$ was fitted in a second step to set the scale. All in all, the fitted function is given by: $$\delta x_\text{deformations} = \sum_{j=0}^4 \left(\frac{1}{\pi}\frac{\gamma_j}{(x-d_{j})^2+\gamma_j^2} \sum_{i=0}^{4} \left( c_{ij} y^i \right) \right),
\label{eq:correction}$$ where $d_j$ and $\gamma_j$ are the location and scale parameters of the Cauchy distributions. $c_{ij}$ are the parameters of the fourth order polynomial.
The outlines of the fitted function are shown in figure \[fig:deformationsx\]. The fitted function can be used as a correction by subtracting it from the mean residuals. The result of this procedure is shown in figure \[fig:deformationsxcorrected\].
The r.m.s. of the distribution of the measured mean residual over the surface - or the systematic error for a measurement before the correction in the quad plane - is . After subtraction of the fitted correction function , the r.m.s. of the mean values is over the whole plane and in the selected region from the edges indicated by a black outline. The distribution of the mean $x$-residuals after correction are shown in figure \[fig:deformationsFreq\]. The distortions could be further reduced by improving the homogeneity of the electric field near the dyke e.g. by adding a field wire above the quad detector at the boundaries between the neighbouring chips.
1 ![Mean residuals in the pixel plane ($x$-residuals) at the expected hit position, fitted with equation (red contours).[]{data-label="fig:deformationsx"}](img_gray/figure8.pdf "fig:"){width="60.00000%"} ![Mean residuals in the pixel plane ($x$-residuals) at the expected hit position, fitted with equation (red contours).[]{data-label="fig:deformationsx"}](img/figure8.pdf "fig:"){width="60.00000%"}
1 ![mean residuals in the pixel plane ($x$-residuals) at the expected hit position after subtracting the fitted edge deformations using equation . []{data-label="fig:deformationsxcorrected"}](img_gray/figure9.pdf "fig:"){width="60.00000%"} ![mean residuals in the pixel plane ($x$-residuals) at the expected hit position after subtracting the fitted edge deformations using equation . []{data-label="fig:deformationsxcorrected"}](img/figure9.pdf "fig:"){width="60.00000%"}
Deformations in the drift direction {#sec:deformationsDrift}
-----------------------------------
A similar measurement is done for distortions in the drift direction. In figure \[fig:deformationsz\] the mean longitudinal ($z$) residuals are shown in bins of pixels over the quad plane using the tracks defined by the telescope. Only bins with more than 800 entries are shown. As shown in figure \[fig:deformationsFreq\], the r.m.s. of the distortion is () and () in the black outlined central area from the edges.
1 ![Distribution of the mean residuals from $4\times4$ bins in figure \[fig:deformationsxcorrected\] ($x$-residuals) and figure \[fig:deformationsz\] ($z$-residuals)](img_gray/figure10.pdf "fig:"){width="60.00000%"} ![Distribution of the mean residuals from $4\times4$ bins in figure \[fig:deformationsxcorrected\] ($x$-residuals) and figure \[fig:deformationsz\] ($z$-residuals)](img/figure10.pdf "fig:"){width="60.00000%"}
. \[fig:deformationsFreq\]
1 ![Mean residuals in the drift direction ($z$-residuals) at the expected hit position.[]{data-label="fig:deformationsz"}](img_gray/figure11.pdf "fig:"){width="60.00000%"} ![Mean residuals in the drift direction ($z$-residuals) at the expected hit position.[]{data-label="fig:deformationsz"}](img/figure11.pdf "fig:"){width="60.00000%"}
Quad detector resolution
------------------------
The overall accuracy of a track position measurement using the quad detector can be tested by comparing the quad track to the telescope track. The difference will be a combination of statistical errors, systematic errors and multiple scattering contributions. Here it is important to estimate the systematical error, because multiple scattering occurs primarily outside the fiducial gas volume, and in applications with multiple quad modules the statistical errors will be further reduced.
Figure \[fig:outOfSyncBG\] displays the difference between the mean position of all quad track hits and the telescope track in the fiducial region. The distribution has long tails, which are in part from unrelated background tracks that are erroneously matched. The number of background tracks is estimated by shifting the telescope timing by 1000 frames and is shown for comparison. In the fit, these tracks are accounted for by introducing a constant offset.
A fit with a Gaussian function with constant offset yields a standard deviation $\sigma_x^\text{quad}$ of . This value is the result of various contributions. Firstly, the statistical precision of a position measurement is acquired from a track fit of the quad hits with hit errors. This statistical precision of the position at the center of the quad is . Furthermore, there is a systematic deviation of in the pixel plane in the fiducial region after the correction as discussed in section \[sec:deformationsPixelPlane\]. Additionally, there is a systematic deviation in the $x$-direction of along the drift direction most likely due to electric field inhomogeneities. This is the $x$-deviation as a function of $z$-position. which should not be confused with the $z$-deviation as a function of $x$ and $y$-position that was mentioned before in section \[sec:deformationsDrift\].
In addition, the precision is limited due to multiple scattering in the setup. The precision was calculated with a simple simulation of the setup using the approach to multiple scattering suggested by reference [@Patrignani:2016xqp]. The setup has multiple scattering contributions from the telescope planes (0.075% $X_{0}$ per plane) [@Jansen2016], the air (0.084% $X_{0}$), the TPC gas (0.09% $X_{0}$) and the two Kapton foils (0.035% $X_{0}$) [@Patrignani:2016xqp]. By comparing the track angle in the first three telescope planes and the second three telescope planes, the total radiation length of the setup is estimated to be 0.82% $X_{0}$ (0.66% $X_{0}$ expected). From the simulation the multiple scattering contribution at the position of the quad center is estimated to be .
An overview of the contributing errors is given in table \[tab:errors\]. In the end, there is still a small unidentified systematic error of .
1 ![difference between mean hit position and track prediction in the pixel plane, fitted with a Gaussian plus a flat background (solid red line). In comparison the background tracks are shown (dashed red), acquired by offsetting the telescope by 1000 frames. []{data-label="fig:outOfSyncBG"}](img_gray/figure12.pdf "fig:"){width="60.00000%"} ![difference between mean hit position and track prediction in the pixel plane, fitted with a Gaussian plus a flat background (solid red line). In comparison the background tracks are shown (dashed red), acquired by offsetting the telescope by 1000 frames. []{data-label="fig:outOfSyncBG"}](img/figure12.pdf "fig:"){width="60.00000%"}
Observed standard deviation $\sigma_x^\text{quad}$
---------------------------------------------------- --
Statistical quad detector error
Statistical telescope error
Systematics over the pixel plane (corrected)
Systematics along the drift direction
Multiple scattering contribution
Remaining systematic error
: Overview of the errors on the difference between mean hit position and track prediction in the pixel plane
\[tab:errors\]
Conclusion and outlook
======================
A quad module with four Timepix3 based GridPixes has been designed and realised. The module has dimensions of and an active surface of 68.9%. The quad module was embedded in a TPC detector and operated at the ELSA test beam facility. The single electron resolutions in the transverse and longitudinal planes are similar to the results obtained for the single-chip detector [@Ligtenberg:2018a] and primarily limited by diffusion. It is shown that a systematic error from the quad detector for the distortions over the pixel plane of ( in the central region) has been achieved. The demonstrated resolution of the setup is , of which the statistical error is , the error caused by multiple scattering in the setup is and the total systematic error is .
The next step is to demonstrate a large detection area with the quad module as a building block and confirm the potential of the GridPix technology for large detectors. A new detector with 8 quad modules carrying a total of 32 Timepix3 Gridpix chips is under construction.
Acknowledgements {#acknowledgements .unnumbered}
================
This research was funded by the Netherlands Organisation for Scientific Research NWO. The authors want to thank the support of the mechanical and electronics departments at Nikhef and the accelerator group at the ELSA facility in Bonn. Their gratitude is extended to the Bonn SiLab group for providing the beam telescope.
|
---
abstract: 'This paper is motivated by the need to enhance today’s Automatic Generation Control (AGC) for ensuring high quality frequency response in the changing electric power systems. Renewable energy sources, if not controlled carefully, create persistent fast and often large oscillations in their electric power outputs. A sufficiently detailed dynamical model of the interconnected system which captures effects of fast nonlinear disturbances created by the renewable energy resources is derived for the first time. Consequently, the real power flow interarea oscillations, and the resulting frequency deviations are modeled. The modeling is multi-layered, and the dynamics of each layer (component level (generator); control area (control balancing authority), and the interconnected system) is expressed in terms of internal states and the interaction variables (IntV) between the layers and within the layers. E-AGC is then derived using this model to show how these interarea oscillations can be canceled. Simulation studies are carried out on a 5-bus system.'
author:
-
-
title: 'Enhanced Automatic Generation Control (E-AGC) for Electric Power Systems with Large Intermittent Renewable Energy Sources [^1] [^2] '
---
Introduction and motivation {#Sec:Intro}
===========================
A high quality of electricity service requires near-ideal nominal frequency, which is achieved by maintaining instantaneous supply-demand power balance. System operation under off-nominal frequency can deteriorate electric equipment, degrade the performance of electric load and even lead to wide-spread system failures and blackouts [@blackout]. Recently, the industrial concerns regarding frequency quality have grown as the increasing Renewable Energy Sources (RES) presence. The RES which are inherently intermittent can lead to continuous supply-demand mismatch and drive the system frequency varying around the desired nominal value with unacceptable quality of response (QoR).
To secure power system operations, the unacceptable frequency excursion must be regulated close to zero in real time by means of automated feedback control. The AGC is widely implemented for this purpose [@AGC1; @AGC2]. However, AGC is mainly designed based on steady state concepts. When AGC is applied to a system with RES, the fast persistent disturbances caused by the RES can drive the system dynamically varying around the equilibrium such that the assumptions of the AGC could become invalid and the AGC might not be as effective as expected. Therefore, the frequency regulation needs to be enhanced and the new approach should extend the modeling and control of AGC from steady state to dynamics.
In the past decades, to improve the performance of AGC, a concept of Area Control Error (ACE) Diversity Interchange (ADI) was proposed in the industry practice [@ADI]. However, it is still based on steady state concepts and it is not economic efficient as no coordination exists between different area. A LQR-based full state feedback control was proposed in [@LQR] for load frequency control. Thereafter, many follow-up works have been done. One limitation of LQR-based approaches is that they are centralized and requiring overly complicated sensing and communication. To improve QoR and system level coordination without complicated sensing and communication infrastructure, the authors introduce Enhanced AGC (E-AGC) concept in [@Qixing2012; @QixingThesis]. However, it should be pointed out that almost all of these design utilize the linearized model which is not valid for large disturbances. Thus, a new approach is needed to consider the tradeoff between the control performance and the complexity.
There are two main contributions of this paper. First, a sufficiently detailed nonlinear dynamical model which captures effects of fast and large nonlinear disturbances is derived for the first time in Section \[Sec:model\]. Notably, most of existing frequency regulation methods ignore the network dynamics. In Section \[Sec:simulation\], we show that the fast network dynamics should not be neglected in today’s electric power systems due to high RESs penetration. Second, we adopt the merits of [@Qixing2012; @QixingThesis] and then propose a general multi-layered control using the proposed nonlinear interconnected system model in Section \[Sec:ControlDesign\]. We have also provably shown that the proposed approach is capable of canceling fast network inter-area dynamical oscillations and achieving system-level coordination. Unlike [@Qixing2012], the proposed method no longer utilizes the small signal model and the routinely made assumptions such as the network dynamics are non-oscillatory. Thus, the contributions of this paper differ substantially from [@Qixing2012; @QixingThesis].
The paper is organized as follows. Section \[Sec: problemformulation\] and \[Sec:model\] provide problem formulation and dynamic model. The proposed multi-layered control is explained in Section \[Sec:ControlDesign\]. Case studies are given in Section \[Sec:simulation\]. Section \[Sec:Conclusion\] concludes the paper.
Dynamic Modeling and Problem Formulation {#Sec: problemformulation}
=========================================
Power electronic devices have been widely installed in the field for voltage regulation. Thus, it is reasonable to make:
The voltage magnitude of each bus is bounded.
It should be noted that Assumption 1 is a mild assumption as we do not fix the voltage. It can vary within a range. In fact, most of existing frequency control approaches neglect the voltage dynamics, i.e., voltage is constant. Notice that the network dynamics is considered. As the interconnected system is treated as a dynamical system, each bus is equivalent. In other words, traditional bus classification (PV, PQ bus) is no longer suitable.
Dynamical model of system components
------------------------------------
### Generation component
Generators have similar role contributing to frequency dynamics, regardless of their types. Thus, we choose a nonlinear non-reheat generator with a primary governor controller embedded as [@QixingThesis]: $$\begin{split}
\label{Eqn:SM}
\dot \delta_G &= \omega_0(\omega_G-\omega^{ref})\\\
M\dot \omega_G &= P_m + P_m^{ref} - D(\omega_G - \omega_0) - P_e \\
T_u \dot{P}_m &= -P_m + K_t a \\
T_g \dot{a} &= -ra - (\omega_G -\omega^{ref}) + u_{AGC}
\end{split}$$ State variables $x_G = [\delta_G, \omega_G, P_m, a]^T$ represent the rotor angle, rotational speed, mechanical power injection, and steam valve position, respectively. $\omega_0$ is the rated angular velocity. $M$, $D$, $K_t$, $T_u$, $T_g$ and $r$ are machine parameters.
It should be pointed out that $P_e$ is the source of the nonlinearity for . $P_e = f_1(\delta_G, x_{TL}, x_{L})$ represents the sum of the real power transfered on its connecting transmission lines, which is a nonlinear function of $\delta_G$, line states $x_{TL}$ and load states $x_{L}$.
### Load component
The load is modeled in the network reference frame as: $$\begin{split}
\label{Eqn:load}
L_{L} \dot i_{Ld} = -R_{L} i_{Ld} + \omega L_{L}i_{Lq} + V_{Ld} \\
L_{L} \dot i_{Lq} = -R_{L} i_{Lq} - \omega L_{L}i_{Ld} + V_{Lq}
\end{split}$$ State variables $x_L = [i_{Ld}, i_{Lq}]^T$ represent the d-axis and q-axis load current, respectively. $L_{L}$ and $R_L$ stand for the load inductance and resistance. $\omega$ denotes the grid frequency. $V_{Ld}$ and $V_{Lq}$ are the d-axis and q-axis of the terminal voltage. $V_{Ld}:=V\cos\theta_V$ and $V_{Lq} := V\sin\theta_V$ are nonlinear function of the terminal voltage angle $\theta_V$. Note that $\theta_V =\delta_G$ when a generator is connected at the same bus. The load satisfies:
\[prop:1\] Given Assumption 1, state variables $x_{L}$ of load component are bounded.
It can be seen that is asymptotically stable if $V_{Ld} = 0$ and $V_{Lq} = 0$. In addition, system matrix $A_L$ is Hurwitz. Thus, using Corollary 5.2 in [@Khaili], we know that is $\mathcal{L}_p$ stable. Notice that voltage magnitude $V = \sqrt{V_{Ld}^2+V_{Lq}^2}$. Provided Assumption 1, nonlinear inputs $V_{Ld}$ and $V_{Lq}$ are bounded, which yields that $x_{L}$ are bounded.
### Network component (transmission line)
Transmission line component is modeled in the network reference frame as: $$\label{Eqn:TL}
\begin{split}
L_{TL} \dot i_{TLd} = -R_{TL} i_{TLd} + \omega L_{TL}i_{TLq} + V_{d,L} - V_{d,R} \\
L_{TL} \dot i_{TLq} = -R_{TL} i_{TLq} - \omega L_{TL}i_{TLd} + V_{q,L} - V_{q,R}
\end{split}$$ State variables $x_{TL} = [i_{TL,d}, i_{TL,q}]^T$ represent the d and q-axis line current. $R_{TL}$ and $L_{TL}$ are resistance and inductance of the line. $(V_{d,L}, V_{q,L})$ and $ (V_{d,R}, V_{q,R})$ denote the left and right port voltage, respectively. The nonlinearity of is introduced by its port voltages.
\[prop:2\] Given Assumption 1, state variables $x_{TL}$ of network dynamics are bounded.
The proof is similar as that of Proposition \[prop:1\]. Detail derivation is omitted for brevity.
Modeling of disturbances
------------------------
Disturbances are characterized as exogenous hard-to-predict inputs to the system. Since disturbances can enter the system through different components, we group them into a vector of external disturbances $d_{ext}$ as seen by components.
Dynamical model of interconnected systems
-----------------------------------------
The overall interconnected system dynamics can be obtained by combining components together as: $$\label{Eqn:system}
\begin{split}
\dot{x}_G & = A_Gx_G + B_Gu_{AGC} + F_Gf_1(x_G,x_{TL},x_{L}, d_{ext})\\
\dot{x}_{TL} &= A_{TL}x_{TL} + F_{TL}f_2(x_{TL},x_G,x_{L}, d_{ext}) \\
\dot{x}_{L} &= A_Lx_L + F_Lf_3(x_{L},x_{TL},x_G, d_{ext})
\end{split}$$
Notably, $A_G$ is rank 1 deficiency due to the conservation of power, while $A_{TL}$ and $A_{L}$ are Hurwitz matrices. $F_G$, $F_{TL}$ and $F_L$ are the input matrices corresponding to nonlinear coupling $f_1$, $f_2$ and $f_3$, respectively. Network coupling between different components are implicitly shown in $f_1$, $f_2$ and $f_3$.
Problem formulation
-------------------
The problem considered in this paper can be posed as:
- Given: interconnected system dynamical model
- Design: AGC control input $u_{AGC}$
- Objectives: both state variables $[x_G, x_{TL},x_L]^T$ and nonlinear interaction $[f_1, f_2, f_3]$ are stabilized and regulated.
Multi-layered dynamical model of interconnected systems {#Sec:model}
=======================================================
In this section the multi-layered model of the interconnected systems is derived. Notice that variations of $f_2$ and $f_3$ are indeed driven by $f_1$, due to the fact that generators are the only active components that produce power. In addition, we have shown that $x_{TL}$ and $x_{L}$ are bounded by $f_2$ and $f_3$ (see Proposition 1 and 2). Therefore, if $f_1$ can be controlled, $f_2$ and $f_3$ can be indirectly controlled, which further ensures the system performance.
To achieve this goal, the definition of the interaction variable (IntV), which was proposed in [@IntV2; @QixingThesis], is revisited and a new interpretation is proposed for the nonlinear systems .
Given a dynamic component (subsystem), its IntV $z$ is an output variable in terms of the local states of the component (subsystem) and it satisfies: $$z \equiv const$$ when the component (subsystem) is free of any conserved net power imbalance.
An IntV is generally defined to capture the non-zero conserved net power imbalance of a component (subsystem). In what follows, the multi-layered dynamic model based on IntV is provided.
We first decompose $u_{AGC}$ of into component-level, area-level, and system-level control signal as: $$\label{Eqn:uAGC}
u_{AGC} = u_{AGC, c} + u_{AGC,r} + u_{AGC,s}$$ These control components will later appear at different layers.
Component-level dynamical model
-------------------------------
Component-level IntV dynamical model has the form: $$\label{Eqn:componentIntV}
\dot{z}_c = P_m^{ref} -P_e + \frac{K_t}{r}u_{AGC,c} \quad z_c(t_0) =z_{c0}$$ It can been seen that $\dot{z}_c$ captures the conserved net power imbalance of the component. It is worthwhile mentioning that $z_c$ simply depends on its own states, i.e., no assumption about the strength of the external interconnection is needed. This fact has been proved for linearized models [@IntV2; @QixingThesis].
Area-level dynamical model
--------------------------
Similarly, a new IntV $z_r$ is introduced for the control areas. Recall the steady-state concept ACE. The dynamics of $z_r$ can be therefore considered as a dynamic version of ACE.
The dynamics of IntV $z_r^C$ is: $$\label{Eqn:areaInt}
\dot{z}_r^C = \sum_{i=1}^{N_c^r}\dot z_{c,i} = B_r \mathbf{u}_{AGC,r} ~~~ z_r(t_0) =z_{r0},~B_r = \mathbf{1}^{N_c^r\times 1}$$ $N_c^r$ stands for the number of generators inside the control area.
System-level dynamical model
----------------------------
We can then apply the same procedure at the interconnected system level. Thus, the dynamic of system-level IntV $z_s$ is: $$\label{Eqn:sysInt}
\dot{z}_s^R= \sum_{i=1}^{N_s^R}\dot z_{r,i}^C = B_s \mathbf{u}_{AGC,s}~~~ z_s(t_0) =z_{s0},~B_s = \mathbf{1}^{N_s^r\times 1}$$ where $N_s^R$ is the number of control areas.
Design of Enhanced AGC(E-AGC) for complex electric power system dynamics {#Sec:ControlDesign}
========================================================================
Primarily, the objective of the E-AGC is to ensure an acceptable QoR of frequency dynamics. This is achieved through controlling the IntVs at different levels. First, $z_c$ is controlled at constant in order to eliminate the real-time net power imbalance. Second, $z_r^C$ and $z_s^R$ need to be regulated to zero in order to maintain the variation of total inadvertent power exchange around zero. Through the coordination of widely dispersed control resources, the inexpensive ones can be fully utilized so that the system-level control cost can be reduced in comparison to today’s AGC approach.
Component-level design
----------------------
The component level IntV dynamics is utilized. In order to have constant $z_c$, we design $u_{AGC,c}$ as: $$\label{Eqn:u_c}
u_{AGC,c} = \frac{r}{K_t}(P_e - P_m^{ref})$$
Substituting Eqn. into Eqn., we obtain the closed-loop generator model, which is provably stable under certain conditions. The result is given below.
\[lemma1\] With control design Eqn., the generator module is stable in the sense of Lyapunov if the following condition is satisfied: $$||P_e - P_m^{ref}||_2 \leq \frac{K_t}{r}u_{max}$$ where $u_{max}$ denotes the saturation limit of the control input.
Proof of Lemma \[lemma1\] is organized in the appendix.
Area-level coordination
-----------------------
The objective of this layer is to eliminate the conserved net power imbalance of each area by optimally controlling $z_r^C$. Within each control area, in order to obtain the optimal coordinated law among participating generators, we design the following LQR problem: $$\begin{aligned}
\label{Eqn:u_r}
\min_{u_{AGC,r}} &J = \int_{t_0}^{\infty}[(z_r^C)^TQ_rz_r^C + (u_{AGC,r})^TR_ru_{AGC,r}]~d\tau \notag\\
s.t. \quad & \dot{z}_r^C = u_{AGC,r} \quad z_r^C(t_0) = z_{r0} \end{aligned}$$ $R_r$ specifies the weight of control cost of each generator. In practice, these two matrices are tunable under the constraint that $Q_r$ and $R_r$ are positive definite matrices.
System-level coordination
-------------------------
The objective of this layer is to eliminate the conserved net power imbalance of the overall system by optimally controlling $z_s^R$. The control areas are coordinated through exchanging their IntVs and controlling the IntVs that are collected. Similarly, we apply the LQR technique to optimize the following objective function: $$\begin{aligned}
\label{Eqn:u_s}
\min_{u_{AGC,s}} &J = \int_{t_0}^{\infty}[(z_s^R)^TQ_sz_s^R + (u_{AGC,s})^TR_su_{AGC,s}] ~d\tau \notag\\
s.t. \quad & \dot{z}_s^R = u_{AGC,s} \quad z_s^R(t_0) = z_{s0}\end{aligned}$$ $R_s$ defines the relative control cost between different control areas. In operation, $Q_s$ and $R_s$ can be tuned accordingly.
Main theoretical result of the E-AGC
------------------------------------
In this section, we give the main theoretical result of the proposed E-AGC approach.
Given Assumption 1 and the composite control design - , the interconnected dynamical system will be stabilized and the frequency of each generator will be regulated if the following condition is satisfied: $$||P_e - P_m^{ref}||_2 \leq \frac{K_t}{r}u_{max}$$
Given the page limit, we provide a sketch for the proof:
Notice that that $z_s$ and $z_r$ can be provably regulated via LQR problems. Thus, as $t \to \infty$, $z_r \to 0$ and $z_s \to 0$. Recall the IntV definition and Lemma 1. It can be concluded that $z_c\to 0$ which indicates $\omega \to \omega^{ref}$.
Communication and implementation discussion
-------------------------------------------
The communication infrastructures shown in Fig.\[fig:5bus\] enable the measurement and also the exchange of IntVs for implementing the E-AGC.
![Information exchange of the E-AGC on a 5 bus system []{data-label="fig:5bus"}](5bus){width="35.00000%"}
First, each generator measure its local state variables and then use to obtain the IntV. Once an IntV is locally computed, a synchronized time-stamp should be added, and then sent to its control area (doted blue line). The control area has to compute its IntV and then compute the coordinated control signals using and . Similarly, after receiving the IntV from control areas (doted black line), the central coordinator computes the system-level coordinated control signals using and then distributed back to the control areas (solid black line). Each control area further provides each generator with a control signal comprised of the control signals from different levels (solid blue line). It should be emphasized that only the IntVs are exchanged between different layers. Thus, we minimize the required information exchange, which is also safe from the cyber security perspective.
Note that the proposed control does not change existing governor control. Different layer signals will be added using and then be applied to as a composite control. Governor set points $\omega^{ref}$ are fixed. In addition, the proposed approach does not require predefined area power set points, unlike some hierarchical control approaches where the secondary layer needs tertiary level power set points. Therefore, the proposed control indeed achieves both stabilization and regulation.
Illustration of the E-AGC on a 5-Bus System {#Sec:simulation}
============================================
In this section, simulation studies on a 5-bus (two-area) test system (Fig.\[fig:5bus\]) are carried out. The nonlinear system including network dynamics is simulated using SEPSS at MIT[@SEPSS]. The purpose are twofold: to show the importance of network dynamics in frequency regulation and to illustrate the effectiveness of the proposed E-AGC.
System description and the test scenario
----------------------------------------
The total capacity of the system is 25 MW with 20% of the electric energy provided by the RES installed at bus 3. As shown in Fig.\[fig:5bus\], two areas are interconnected via two transmission lines. We assume that two control areas are strongly connected, while components are weakly connected within the area.
In order to show the effect of network dynamics, we only consider the step changes at all loads. For this scenario, the conventional AGC is supposed to restore the frequency.
In what follows, the simulation results of the cases with no AGC, the conventional AGC, and the proposed E-AGC are shown. As the low frequency oscillations are observed in operation and our simulations, we then provide an explanation of why they have not been captured by classic methods.
Simulation results and discussion
---------------------------------
### Performance with the conventional AGC
[0.22]{} {width="\textwidth"}
[0.2]{} {width="\textwidth"}
We first disable the AGC and simulate the system with primary controllers only. Frequency responses are organized in Fig.\[fig:NoAGC\]. It can be seen that the frequency of the generators in Area 1 settles around 1.1 $p.u.$ but has $2-5~Hz$ oscillations. Similarly, the frequency of Area 2 is oscillating around 1.05 $p.u$.
Next, we activate the conventional AGC [@Kundur]. Corresponding frequency responses are shown in Fig.\[fig:AGC\]. It can be seen that steady-state errors are greatly reduced. However, the frequency of Area 1 is higher than the nominal value, while Area 2 is slightly lower. This indicates that the inter-area oscillation exists between two areas. It is because the RES provides more power than what Area 1 needs.
It should be also noted that the low frequency oscillations observed in Fig.\[fig:NoAGC\] still exist in Fig.\[fig:AGC\]. It is worthwhile mentioning that such low frequnecy oscillations never show up in classic analysis but system operators do observe similar phenomena in operation. This is because most of conventional approaches are designed based on the quasi-static ACE and the network dynamics is ignored. However, we only assume that voltage magnitude is bounded in our model. In other words, d-q axis voltage can vary over time. As shown in the line dynamics , the varying voltage angle may act as disturbances to the component. Recall Proposition 1 and 2. They both explain why we observe oscillatory but bounded behavior in simulations. Therefore, it is important to consider network dynamics into frequency analysis. otherwise such unobserved oscillations are likely large enough to trigger protection devices.
Zooming into control design, we notice that neither the primary control nor the conventional AGC has feedback with respect to rotor angle, i.e., rotor angle (voltage angle) is not directly controlled. Hence, voltage angle may interact with $x_{TL}$ and then start to oscillate. In other words, real and reactive power produced at one bus are interacting with the energy stored in the line when the oscillation occurs. Consequently, disturbances at one bus may spread out to other buses through line dynamics, which further cause oscillatory behavior in the entire system. Fig.\[fig:AGC\] also supports the fact that there is no guarantee that output feedback can stabilize the rotor angle.
One argument for ignoring network dynamics is that the network has much smaller time constant compared to generators. However, transmission line dynamics cannot change instantaneously in reality and the argument is no longer true in microgrids. The rate of real and reactive power entered from two ends of the line are not necessary to be the same. Thus, fast disturbances introduced by RESs may excite the fast network dynamics, resulting in accumulated effects on the slow dynamics (frequency dynamics). This will become a critical issue if more and more RESs are integrated.
It should be mentioned that the performance can potentially be improved if rotor angle deviation is considered in the feedback design. It is equivalent to design a PI controller. However, there are several challenges in implementing this solution. First of all, it is hard to get accurate rotor angle reference. It may not be realistic to run centralized optimization (such as AC OPF) after every change in the system. Second, the feedback gain with respect to rotor angle needs to be carefully design. The gain should be tunned so that it can tolerate large and fast-varying disturbances. Improperly tunned integrator can destabilize the system. Last but not the least, it is challenging to measure rotor angle accurately without large delay.
### Performance with the proposed E-AGC
Simulation results of the proposed E-AGC are given in Fig.\[fig:EAGC\].
![Frequency responses with E-AGC[]{data-label="fig:EAGC"}](EAGC){width="22.00000%"}
In comparison to the conventional AGC, frequency of each generators are regulated to the nominal value. More importantly, low frequency oscillations no longer exist. This indicates that the imbalances within and between control areas are limited around zero, i.e., no inter-area oscillations. It is because the proposed E-AGC is designed based on nonlinear dynamic systems. Network dynamics is preserved in the dynamics of the IntV at different levels. At regulation stage, instead of requiring hard-to-get angle reference, area-level and system-level control are using the IntV as feedback signals. These two layers not only coordinate the resources, but also act as an integrator, which eventually eliminates the low frequency oscillations observed in Fig.\[fig:AGC\].
From economic point of view, the proposed E-AGC significantly reduce the systematic regulation cost via area-level and system-level coordination, due to the proposed formulation. Considering that the E-AGC requires much less information exchange, we argue that this proposed control scheme could be very cost-effective. In addition, we believe that the IntV information is one potential communication protocol for the future grid operation, grid control, etc.
Conclusions {#Sec:Conclusion}
===========
In this paper we revisit the frequency regulation problem for future electric energy systems. We summarize the emerging practical problems of applying the conventional AGC, especially when network dynamics and highly variable RESs are presented in the system. The E-AGC approach is thus introduced as an alternative solution. The regulation cost can be systematically reduced by using little information exchange. Simulations show that the E-AGC outperforms the conventional approaches. Simulations for large-scale systems will be given in our future publications.
Proof Sketch for Lemma 1
------------------------
$T_1 =
[\frac{D+Kt/r}{M}~M~T_u~ \frac{T_gK_t}{r}
]$, $T_2 = [
0~M~T_u~\frac{T_gK_t}{r}]^T$ and $P = (T_{2}T_{1})^T(T_2T_1)$. It is easy to check that $P\in \mathbb{R}_{+}^{4\times4}$. Thus, if we choose a Lyapunov function $V = x_G^TPx_G$, the rest is straightforward to show using the procedures in [@Khaili].
[99]{} D. Davidson, et al.,“Long term dynamic response of power systems:An analysis of major disturbances,” IEEE Trans.Power App. and Syst.,1975. N. Cohn, Control of Generation and Power Flow on Interconnected Systems. New York: Wiley, 1961. Y. G. Rebours, D. S. Kirschen, et al., “A survey of frequency and voltage control ancillary services” IEEE Trans. on Power Systems, 2007. A. Oneal, “A simple method for improving control area performance: area control error (ace) diversity interchange adi,” Power Systems, IEEE Transactions on, vol. 10, no. 2, pp. 1071–1076, 1995. C. E. Fosha and O. I. Elgerd, “The megawatt-frequency control problem: A new approach via optimal control theory,” IEEE Trans. Power App. and Syst., 1970. Q. Liu and M. Ilic. “Enhanced automatic generation control (E-AGC) for future electric energy systems.” PES General Meeting, 2012 IEEE.
Q. Liu, “A Large-Scale Systems Framework for Coordinated Frequency Control of Electric Power Systems”, Ph.D. dissertation, CMU, 2013.
M. Ilić, R. Jaddivada, and X. Miao. “Modeling and analysis methods for assessing stability of microgrids.” IFAC-PapersOnLine 2017 Khalil, Hassan K. “Noninear systems.” Prentice-Hall, New Jersey (1996)
X. Liu, “Structural Modeling and Hierarchical Control of Large-Scale Electric Power Systems,” Ph.D. dissertation, MIT, April 1994. M. Ilic, R. Jaddivada, X. Miao, “Scalable Electric Power System Simulator”, ISGT-Europe, October 2018 Kundur, Prabha, Neal J. Balu, and Mark G. Lauby. Power system stability and control. Vol. 7. New York: McGraw-hill, 1994.
[^1]: The authors greatly appreciate partial funding by NIST. Also, discussions with Ms. Rupamathi Jaddivada and the use of Scalable Electric Power System Simulator (SEPSS) at MIT [@SEPSS] are greatly acknowledged.
[^2]: Accepted to the IEEE PES GM 2019. 2019 IEEE.
|
---
abstract: 'We establish a new theoretical framework, based on a time-dependent mean field approach, to address the dynamics of the driven Dicke model. The joint evolution of both mean fields and quantum fluctuations gives rise to a rich and generally non-linear dynamics, featuring a normal (stable) regime and an unstable, super-radiant one. Various dynamical phenomena emerge, such as the spontaneous amplification of vacuum fluctuations, or the appearance of special points around which the mean-field amplitudes rotate during driven time evolution, signalling a dynamical symmetry breaking. We also provide a characterization of the driving-induced photon production in terms of the work done by the driving agent, of the non-adiabaticity of the process and of the entanglement generated between the atomic system and the cavity mode.'
author:
- 'G. Francica'
- 'S. Montangero'
- 'M. Paternostro'
- 'F. Plastina'
title: 'The driven Dicke Model: time-dependent mean field and quantum fluctuations in a non-equilibrium quantum many-body system'
---
The dynamical behavior of quantum critical systems displays interesting features concerning defects or excitations production [@polkormp; @eisertnatphys], which occurs when the system is driven across its critical point [@varikz]. The Dicke model is a paradigmatic example in this context, embodying a quantum many-body system with highly non-trivial critical features [@Brennecke; @pnas], where an electromagnetic mode is coupled to a collection of $N$ identical two-levels atoms [@dicke]. Indeed, strong correlations are set both among atoms and with the field, which in turn result in significant non-classical behavior of the radiation and in cooperative effects giving rise to both atomic and photon squeezing.
These tantalising features persist in the case of external driving of the dynamics, as remarkably shown experimentally in Ref. [@Brennecke], where the corresponding spontaneous symmetry-breaking effect induced by an adiabatic crossing of the quantum critical point of the model has been demonstrated. Non-adiabatic driving has also been the focus of substantive theoretical and experimental investigation [@varie]. In particular, for a periodic driving of the atom–field coupling strength, Ref. [@Bastidas12PRL108] has shown the emergence of new metastable phases in the driven Dicke model, whose phase diagram appears to be substantially different from the static one, both qualitatively and quantitatively.
In this paper we discuss symmetry breaking and photon generation in the driven Dicke model, where the frequency of the field, the energy of the atoms, [*and*]{} their mutual coupling are all allowed to vary in time. While the dynamical behavior of the single-atom case has been recently studied [@plenion=1], we are interested in the thermodynamic limit. By non-adiabatically driving the system, and thoroughly analysing its stability conditions, we highlight implications that non-adiabaticity has on the evolution of both the mean fields and the residual quantum fluctuations. Indeed, the [*macroscopic*]{} trajectories followed by the mean fields are shown to dynamically select one of the symmetry broken configurations each time the critical point is crossed; while the [*microscopic*]{} features of the fluctuations are shown to be crucial for the characterization of the temporal behavior of key observables of the system, such as the photon number. Moreover, we study the thermodynamic work produced by driving the system, and the associated degree of irreversibility, thus addressing the non-adiabatic production of photons from the electromagnetic vacuum, a phenomenon akin to the dynamical Casimir effect [@Vacanti12PRL108], and the associated generation of atom-field entanglement, from a genuinely non-equilibrium perspective.
For a negligible mutual atomic interaction and for an atomic system occupying a linear dimension much smaller than the electromagnetic wavelength, the atom field system is described by the collective-spin Hamiltonian $\hat H= \hat H_0 + \hat H_{int}$ with $$\label{hami}
\hat H_0 = \omega_a \hat a^\dag \hat a + \omega_b \hat J_z \, , \quad \hat H_{int}= {2
g} \left( \hat a^\dag + \hat a \right) \hat J_x/\sqrt N.$$ Here $\hat a$ ($\hat a^\dag$) is a bosonic annihilation (creation) operator, and $\hat {\bm J} = (
\hat J_x, \hat J_y, \hat J_z )$ is the collective spin operator, with $\hat{\bm J}
= \sum_{i=1}^N {\hat{\bm \sigma_i}}/{2}$ and $\hat{\bm \sigma_i}$ is the vector of Pauli spin operators.
As mentioned, we will be interested in the case where $\omega_{a,b}$, and $g$ are all functions of time (in order to avoid notational clutter, and unless otherwise specified, we will avoid writing explicitly any time dependence). In fact, as we discuss below, the key ingredient in the dynamics of the system is the time dependence of the parameter $\mu={\omega_a \omega_b}/(4
g^2)$. We will treat the atoms as indistinguishable, and consider a fully symmetric initial atomic state. This feature is preserved during time evolution as the symmetric subspace is dynamically invariant. We now consider the Holstein-Primakoff transformation of $\hat{\bm J}$ restricted to such subspace and thus introduce the bosonic operators $\hat b$ and $\hat b^{\dag}$ such that $$\label{HP}
\hat J_z =\hat b^\dag \hat b - N/2 \, , \quad
\hat J_+ \equiv \hat J_x + i \hat J_y =\hat b^\dag \sqrt{N - \hat b^\dag \hat b}.$$ As a result, within the symmetric subspace $\hat H_{int}$ takes the form $$\label{hamiltonian HN}
\hat H_{int}=
g \left(\hat a^\dag +\hat a \right)\left(\hat b^\dag\sqrt{1-\frac{\hat b^\dag \hat b}{N}} + \sqrt{1-\frac{\hat b^\dag \hat b}{N}}\hat b \right).$$ In the time-independent case and at the thermodynamic limit, $\hat
H$ can be diagonalized by isolating from $\hat b$ a macroscopic ($\sim\sqrt N$) mean contribution [@Emary03PRE67] and retaining only the leading terms of a $1/N$ expansion of the square roots in Eq. , thus obtaining a quadratic Hamiltonian. This procedure implies subtracting a [*static*]{} mean field chosen in order to approximate the Hamiltonian as accurately as possible at low energies, or, loosely speaking, chosen in such a way as to minimize residual quantum fluctuations near the ground state. Our approach to the investigation of the driven model is based on an analogous idea: we will isolate [*time-dependent*]{} mean fields, chosen so as to minimize residual quantum fluctuations around the [*instantaneous*]{} state vector ${\left\vert\psi,t\right\rangle}$, whose evolution is generated by an approximately quadratic time-dependent Hamiltonian.
In the thermodynamic limit and for $\mu > \mu_c=1$ \[$\mu<1$\], $\hat H$ admits a normal (N) \[super-radiant (SR)\] quantum phase. At $\mu=1$, a second-order phase transition is found. As we discuss below, the driven system correspondingly displays two dynamical regimes [@Bastidas12PRL108]. In order to perform a quantitative analysis, we start by shifting the field and atomic operators $\hat a$ and $\hat b$ by their time-dependent mean values $\langle \hat a \rangle = \sqrt{N}
\alpha$ and $\langle \hat b \rangle = \sqrt{N} \beta$, thus introducing new operators describing deviations from the averages, $\hat c = \hat a - \sqrt N \alpha$, and $\hat d = \hat b - \sqrt N
\beta$ (with $\alpha,\beta\in\mathbb{C}$). We require any macroscopic contribution to be ascribed to the mean fields, and thus assume that fluctuations remain very small. Therefore, after the rescaling by $\sqrt{N}$ performed above, we expect $\alpha$ and $\beta$ to remain ${\cal O}(1)$ as $N\rightarrow \infty$ (as it is the case for a static $\hat H$ [@Emary03PRE67]), see [@nota] for details. We thus have $$\label{expansion}
\sqrt{1-\frac{\hat b^\dag \hat b}{N}}
\simeq \sqrt{\Gamma}\bigg[ 1 - \frac{\beta \hat d^\dag + \beta^* \hat d}{2 \Gamma \sqrt{N} } -\frac{d^\dag \hat d}{2N \Gamma} -\frac{(\beta \hat d^\dag + \beta^* \hat d)^2}{8 N
\Gamma^2}\bigg]$$ with $\Gamma = 1-{\left\vert\beta\right\vert}^2$. Using the leading terms only, one can derive equations for the bosonic operators and their averages as [@nota] $$\label{eqs}
i \dot{\alpha} = \omega_a \alpha + 2 g \sqrt{\Gamma} \beta_{r},\,\, i \dot{\beta} = \omega_b \beta + 2 g \sqrt{\Gamma}
\alpha_{r}\left(1-{\beta \beta_{r}}/{\Gamma}\right).
$$ Here $s_r=\text{Re}(s)$ and $s_i=\text{Im}(s)$ with $s=\alpha,\beta$. Eq. are explicitly nonlinear. However, by starting from low-energy conditions, the dynamics can be well approximated by linear equations of motion up until the time $t_{\text{lin}}$ within which $|\alpha|,|\beta|\ll
1$ [@nota]. When the mean fields acquire macroscopic values, their dynamics become fully non linear and the whole of Eq. (\[eqs\]) should be retained. For a constant Hamiltonian and any value of $\mu$, these equations admit the stationary solution $\alpha^n=\beta^n=0$, corresponding to the ground state of the N phase [@Emary03PRE67]. In the SR-phase, with $\mu <1$, two other stationary points appear, corresponding to states with broken (parity) symmetry, $\alpha^{sr}_{\pm} = \pm \frac{g}{\omega_a}\sqrt{1-\mu^2}$, and $\beta^{sr}_{\pm} = \mp \sqrt{\frac{1-\mu}{2}}$. For a driven system, $\mu$ depends on time and so do $\alpha^{sr}$ and $\beta^{sr}$, while the normal value remains stationary. As a result, if the initial state has null mean fields, the condition $\alpha=\beta=0$ will hold at all times and the dynamics never exit the linear transient. However, if the system is instantaneously brought into the SR region, such normal stationary state can become unstable.
![(Color online) [**(a)**]{} Stability diagram in the driving frequency vs. coupling plane, signalling in white the regions with positive instability rate, $\gamma^*>0$. The plot is drawn at the static resonance, $\lambda_0=1$, with $\lambda=\frac{1}{2}$. The red line marks the static critical coupling $\mu_0=1$, while the green one corresponds to $\mu_{min}=1$; so that, for a point under the green line $\mu(t)>1
\forall t$, while $\mu(t)$ periodically goes below unity above it. Finally, the black line corresponds to $\mu_{max}=1$. For $g\rightarrow 0$ the white zones open for $\eta_k= 2 \omega_a /k$, $k\in Z$. In the limit $\eta \rightarrow \infty$ the first blue triangle on the left tends to fill the whole region under the red line, while for $\eta \rightarrow 0$ it is the region under the green line that becomes blue. The inset shows the stability diagram for larger values of the driving frequency. [**(b)**]{}-[**(d)**]{} Trajectories of the mean field $\alpha$ corresponding to the three points in panel [**(a)**]{}. We have assumed a slow periodic driving ($\omega_a = 11 \eta$) and evaluated $\alpha(t)$ up to $t
= 60 \eta^{-1}$ by numerically solving Eqs. (\[eqs\]) for the parameters indicated by the black dots in the central panel, with $2 g /\eta = 9, 12.5, 14$. Initially, the radiation field has been taken in the coherent state $|{\sqrt{N}\epsilon}\rangle$ with $\epsilon = 10^{-2}$, with all of the atoms in their ground states. []{data-label="Fig1"}](NewStabilita){width="\linewidth"}
In the following, we focus on a periodically driven system. We specifically assume the atomic frequency to be sinusoidally perturbed, $\omega_b/\omega_a = \lambda_0 + \lambda \sin(\eta t)$, as in [@Vacanti12PRL108], implying an harmonic time dependence for $\mu(t)$, which oscillates with frequency $\eta$, between $\mu_{min}=\mu_0 (1- \lambda/\lambda_0)$ and $\mu_{max}=
\mu_0 (1+ \lambda/\lambda_0)$, where $\mu_0=\lambda_0 \omega_a^2/4
g^2$. For such a periodic driving, we can make use of Floquet theory [@Floq] to study the dynamics of the system. In particular, the stability of the solutions of Eqs. (\[eqs\]) can be characterized by an instability rate $\gamma^*$ defined as the largest positive Floquet exponent of the linearized equations [@Floq], which embodies the growing rate of the mean fields in the linear regime. While the details of this analysis are given in the supplementary material [@nota], the result is reported in Fig. \[Fig1\], where we see that $\gamma^*>0$ if $\mu(t)< 1$ at all times (above the black line in Fig. \[Fig1\]), while driving-induced instabilities appear even in the (static) N region (below the red line in Fig. \[Fig1\]). For small couplings, this occurs near the parametric resonance points $\eta_k = \omega_a(1+\lambda_0)/k$ ($k\in\mathbb{Z}$). These are the so called Arnold instability tongues, discussed in [@Bastidas12PRL108]. Although our treatment is valid in general, the discussion below will mainly focus on the case of a slow (but non-adiabatic) driving, therefore Fig. \[Fig1\] explicitly displays the case $\eta \ll \omega_a, \omega_b$.
Although the dynamics can exit the linear regime when the mean fields acquire large values (which can occur quite quickly, e.g., for initial coherent states with very small amplitudes), the diagram in Fig. \[Fig1\] still helps classifying the dynamical behavior of the mean fields, identifying those values of frequency and coupling for which $\alpha$ and $\beta$ grow exponentially in time, from those for which they stay bounded. As shown below, the very same diagram will help in the study of quantum fluctuations (cfr. the discussion after Eq. \[wveceq\]).
Solutions of Eqs. (\[eqs\]) in the different regimes are reported in Fig. \[Fig1\] [**(b)**]{}-[**(d)**]{}, where various examples of trajectories of the photon mean field $\alpha$ are shown ($\beta(t)$ follows similar paths). At $t=0$ the electromagnetic mode is taken in the coherent state $|{\sqrt{N}\epsilon}\rangle$ (with $\epsilon \ll 1$), while the atoms are in their ground states. If the system’s parameters are chosen in the unstable region, the mean fields grow exponentially and, once out of the linear regime, get macroscopic values. For a set of parameters in the stable zone, instead, the trajectory is bounded, with $|\alpha(t)|$ remaining $ \sim \epsilon$. This behavior is quite general and does not depend on the value of the driving frequency.
For a slow driving, the trajectories show some regularity as, after an initial amplification stage, the mean fields remarkably tend to be [*attracted*]{} towards the nearest of the equilibrium points; namely, either $(\alpha^{n}, \beta^{n})$, or one of the two broken symmetry points $(\alpha^{sr}_{\pm},
\beta^{sr}_{\pm})$, which is then ‘followed’ as its value changes due to the driving. More specifically, when $\mu (t)>1$, the two mean fields circulate around their N stationary point, $\alpha^{n}$ and $\beta^{n}$, respectively. A switching occurs once $\mu(t)$ crosses its critical value, with the mean field trajectory that dynamically breaks the parity symmetry, “selects a sign”, and moves towards either the positive or the negative real axis, to begins rotating around one of SR values. When, later, we get $\mu(t) >1$ again, the mean fields are attracted back by the N fixed point, to move again towards one of the SR points further on. Such a sequence of switching events occurs if the driving parameters and the coupling strength are such that $\mu_0 \in [(1+\lambda/\lambda_0)^{-1},1]$ whose boundaries correspond to the black and red lines in Fig. \[Fig1\] [**(a)**]{}, respectively. This is the case, e.g., of Fig. \[Fig1\] [**(c)**]{}. A different behavior is found for $\mu_0 \in (1,(1-\lambda/\lambda_0)^{-1}]$, where the right boundary corresponds to the green line in Fig. \[Fig1\] [**(a)**]{}. In this case, although $\mu<1$ for some $t$, the trajectory keeps rotating around the N equilibrium point, as in Fig. \[Fig1\] [**(b)**]{}. For parameters taking us below the green line, the driving is not able to bring the system to criticality and the solution of Eqs. (\[eqs\]) never exit the linear regime. Correspondingly, the values of the mean fields remain of order $\epsilon$. Above the black line, on the other hand, only the two SR stationary points come into play. One further particular trajectory for the mean photon field, in the regime in which both the N and the SR points are relevant, is analyzed in detail in Fig. (\[Fig mean field\]), which shows how the field follows the instantaneous equilibrium point, dynamically breaking the equivalence between the two SR values when entering the regime with $\mu < 1$.
![ (Color online): We show the driving cycle \[panel [**(a)**]{}\], the mean field trajectory [**(b)**]{}, the photon number fluctuations $\rho_\infty$ [**(c)**]{}, and the two-mode squeezing parameter $r^*$ [**(d)**]{}. In panels [**(b)**]{}-[**(d)**]{}, segments of a given color refer to the corresponding part of the driving cycle \[panel [**(a)**]{}\]. We set $\eta = 0.1 \omega_a$ and $\lambda=0.5$, with $\lambda_0=1$, and $g= 0.55 \omega_a$. Initial conditions are $\alpha(0)=\alpha^{sr}(0)$ and $\beta(0)=\beta^{sr}(0)$. []{data-label="Fig mean field"}](dy.pdf){width="\linewidth"}
The dynamic mean fields are not sufficient to obtain the average value of a generic observable. This is the case, for instance, of the photon number. A complete quantum description requires the knowledge of the (operator) fluctuations around the mean fields. The associated equations of motion for the fluctuations are explicitly derived in [@nota]. They are best displayed in terms of the quadrature vector $\hat {\bm Q}(t)= (\hat q_c,\hat
q_d,\hat p_c,\hat p_d)$ with $\hat q_k= (\hat k+\hat
k^{\dag})/\sqrt 2$, $\hat p_k= i (\hat k^{\dag}-\hat k)/\sqrt 2$ ($k=c,d$), and take the form $$\dot{\hat{\bm Q}}(t)= {\bm M}_{\alpha, \, \beta} \, \hat{\bm
Q}(t), \label{wveceq}$$ where matrix ${\bm M}_{\alpha, \, \beta}$ is a non-linear function of the instantaneous mean field values $\alpha$ and $\beta$ [@nota]. Although the dynamics of the mean fields discussed above is independent of the fluctuations, the reverse is not true. Moreover, for $\alpha, \beta \ll 1$ (i.e. when the mean fields are in the linear regime), ${\bm M}_{\alpha, \, \beta}$ reduces to the very same dynamical kernel ruling Eqs. (\[eqs\]) for $t < t_{\text{lin}}$. This implies that the stability analysis of Fig. \[Fig1\] applies to the quantum fluctuations as well. In particular, for a parameter set in the white region, the dynamics of the fluctuations becomes chaotic: two very close initial states exponentially diverge in time, at a rate $\gamma^*$. Eq. (\[wveceq\]) are formally solved as $\hat{\bm Q}(t) =
\Phi(t) \hat{\bm Q}(0)$, where matrix $\Phi$ is such that $\dot{\Phi} = {\bm M}_{\alpha, \, \beta} \Phi$ and satisfies the boundary condition $\Phi(0) = \mathds{1}$. The first moments of the quadratures are zero at all times (as the averages are given by the mean fields), while the second moments form the covariance matrix $W$ with elements $ W_{i\,j} = \langle\{\hat Q_i,\hat Q_j\}
\rangle/2 - \langle \hat Q_i \rangle \langle \hat Q_j\rangle$, [@Weedbrook2012; @Simon1994; @Ferraro05arxiv]. The time-evolved covariance matrix is then given by $W(t) = \Phi(t) W(0) \Phi(t)^T
$. In general, both the mean fields and the quantum fluctuations contribute to the evolution of a physical observable, an interesting example being given by the average photon number $$\label{photons number}
n_a = \langle \hat a^{\dag} \hat a\rangle \equiv N
{\left\vert\alpha\right\vert}^2+ \left( W_{11} + W_{33} -1\right)/2.$$ Covariance matrix contributions dominate for initial mean fields values $\lesssim N^{-1/2}$, and, in particular, if one takes the vacuum of both modes as initial state. In fact, in virtue of the previous discussion, this implies $\alpha=\beta=0 \, \forall t$, so that photons are generated, in this case, by the exponential amplification of the initial fluctuations, due to the instability of the system under sinusoidal perturbation signalled by a positive instability rate $\gamma^*$. For a very fast perturbation, $\eta \gg \omega_a, \omega_b$, this requires $\mu_0<1$. On the other hand, if $\mu_0 >1$, fluctuations are bounded in time. Differently, for a very slow perturbation, photon production occurs if we can make $\mu(t) <1$ in some time interval, while fluctuations are bounded only if $\mu (t)
>1\,\forall t$. Photon generation from the vacuum is related to the thermodynamic work done on the system by the driving agent [@work]. The average work performed at time $t$ is $$\begin{aligned}
\langle w \rangle &= \langle\hat H\rangle_t-\langle\hat H\rangle_0= \omega_a n_a(t) + {N}\omega_b(0)/2+2gW_{12}(t)\\
&+ {\omega_b(t)}\left[ W_{22}(t) + W_{44}(t) -(N+1) \right]/2,
\end{aligned}
\label{dipew12}$$ which depends not only on the local energies of the two modes involved, but also on their correlations through the $W_{12}$ term. Eq. (\[dipew12\]) shows that not all of the energy pumped into the system is used for photon production in light of the non-adiabatic nature of the driving. Part of such energy goes to the atoms and part is stored as interaction energy. The non-adiabaticity of the driving process can be quantitatively studied using specifically designed thermodynamic figures of merit for irreversibility [@work]. Among them is the [*inner friction*]{} [@Plastina14PRL113], which is defined as the non-adiabatic part of the work. In our case, assuming the coupling to be switched on at $t=0$, it reads $ \langle w_{fric} \rangle =
\langle w \rangle - E^t_{GS} + E^0_{GS}$, where $E^t_{GS}$ is instantaneous energy of the ground state ${\left\vertGS\right\rangle}$ of the system, which becomes non-analytic for $\mu=1$. The behavior of $\langle
w_{fric} \rangle$ crucially depends on the driving amplitude $\lambda$.If $\lambda$ is such that $\mu_{min}>1$,the system never exit the N region and $\langle w_{fric} \rangle$ never gets macroscopic values. If $\mu_{min}=1$(i.e., $\mu=1$ at times $\eta t=(3/2 + 2 k )\pi$, $k\in
\mathbb{Z}^*$), the inner friction becomes non analytic (as the energy gap closes), yet remaining ${\cal O}(1)$ as $N\rightarrow \infty$. If $\lambda$ is such that $\mu_{min} <1$,instead, the system enters the SR region for $t \in [(2 k-1) \pi +
\tilde t, 2 k \pi -\tilde t]$, where $\eta \tilde t = \arcsin
(\lambda_c / \lambda) $ ($k\in\mathbb{Z}^*$). Then, as the evolved state becomes macroscopically different from the instantaneous ground state, inner friction gets macroscopic values. Altogether, the inner friction per atom in the thermodynamic limit is $$\lim_{N\rightarrow \infty}{\langle w_{fric} \rangle}/{N} =
\begin{cases}
0 & \text{for}~\mu(t) \geq 1, \\
{\omega_b(t)}[1-1/\mu(t)]^2 /4& \text{for}~\mu(t)<1.
\end{cases}$$ Besides the mean photon number and the average (non-adiabatic) work, we can characterize the photon statistics by the variance, $\sigma_a^2 = \langle (a^\dag a)^2 \rangle - n_a^2$, and the Mandel parameter $\rho = \frac{\sigma_a^2 }{n_a}$. The latter signals a sub- or super-Poissonian statistics, with $\rho=1$ for a coherent state. For very large $N$, in the regime in which the mean fields dominate with respect to quantum fluctuations (which is the case for initial mean fields larger than $N^{-1/2}$), and neglecting $ {\cal O}(\sqrt{N})$ terms, the variance is $$\label{photons variance}
\sigma_a^2 = 2 N \left( \alpha_{r}^2 W_{11} + \alpha_{i}^2 W_{33} + 2 \alpha_{r} \alpha_{i} W_{13} \right)$$ so that, as $N \rightarrow \infty$, the Mandel parameter becomes $$\label{ratio poisson}
\rho_{\infty} = {2}(\alpha_{r}^2 W_{11} + \alpha_{i}^2 W_{33}
+ 2 \alpha_{r} \alpha_{i} W_{13})/|\alpha|^2$$ In general terms, the time behavior of the quantum fluctuations is very different depending on wether $\mu$ is larger or smaller than $1$. This is reflected in the covariance matrix and witnessed by the parameter $\rho_{\infty}$ (see Fig. \[Fig mean field\]). Roughly, this is an oscillating function of time, which oscillates faster as the mean fields get larger, with an amplitude that suddenly increases whenever $\mu$ crosses $\mu_c$ to enter in the SR region. While $n_a$ and $\rho$ describe the reduced photon-state only, we can also characterize global state correlations by evaluating the degree of two-mode squeezing. With this aim, we consider the parameter $r_{opt}(t)$ optimizing the fidelity [@Marian12PRA86] between ${\left\vert\psi,t\right\rangle}$ and the two-mode squeezed coherent state ${\left\vert\psi_{\alpha, \, \beta}(r,t)\right\rangle}$ having coherent amplitudes given by the mean fields and (real) squeezing degree $r$, i.e. $|{\psi_{\alpha, \beta}(r,t)}\rangle= e^{r ( c^\dag d^\dag - c d)
} {\left\vert\sqrt N \, \alpha(t)\right\rangle}_a{\left\vert\sqrt N \, \beta(t)\right\rangle}_b$. Regardless of $t$, we find a value $r_{opt}(t)$ for which the fidelity is $\ge 0.9999$ [@nota]. Therefore, $r_{opt}$ itself can be thought as a good (although approximate) descriptor of the photon-atom entanglement, [@Horodecki96PLA96; @Simon00PRL84]. During time evolution, it turns out that $r_{opt}$ can either grow exponentially at short times (if $\gamma^*>0$), or remains very close to its initial value (for $\gamma^*=0$). An example of the behavior of $r_{opt}$ is reported in Fig. (\[Fig mean field\]), for the various stages of the driving induced dynamics. After the initial fast increase, $r_{opt}$ displays small ‘jumps’ whenever the driving brings the system in the SR region.
[*Concluding remarks.-*]{}We have provided a dynamical mean field-based description of the driven Dicke model, discussing the non-linear evolution of the mean fields, as well as that of quantum fluctuations. Both are needed to determine the time behavior of physical observables such as the photon number. Our approach is exact in the thermodynamic limit. However, for finite $N$, our description is accurate provided that fluctuations do not become macroscopic (otherwise the expansion in Eq. (\[expansion\]) breaks down). This gives a time limit $t_{max}$ (generically increasing with $N$) within which the analysis is meaningful. Remarkably $t_{max}$, which is estimated in [@nota], may be different in the various dynamical regimes. Within such limit, we have discussed the phenomenon of dynamical breaking of the parity symmetry in the mean field evolution under driving, analysed photon generation from the vacuum, using out-of-equilibrium thermodynamical tools to characterize it, and described the generation of two-mode squeezing and entanglement between field and atoms.
[*Acknowledgements*]{} We acknowledge support from the Collaborative Projects QuProCS (Grant Agreement 641277), and TherMiQ (Grant Agreement 618074), the John Templeton Foundation (grant number 43467), the Julian Schwinger Foundation (grant number JSF-14-7-0000), and the UK EPSRC (grant number EP/M003019/1). We acknowledge partial support from COST Action MP1209.
Supplementary material
======================
Holstein-Primakoff trasformation
--------------------------------
Since the total spin $|\vec J|^2$ is conserved, the full Hilbert space can be decomposed into invariant subspaces labelled by the index $j$ describing its eigenvalues, $j=1,2,\cdots,\frac{N}{2}$ for an even $N$, or $j=\frac{1}{2},\cdots,\frac{N}{2}$ if $N$ is odd. Defining the projector $P_j$ onto $j$-th subspace, we can rewrite the Hamiltonian as $H = \sum_j P_j H P_j \equiv \sum_j H^{(j)} $
In the main text we focus on the case of a fully symmetric initial atomic state, which implies selecting the $j=\frac{N}{2}$ invariant subspace. We therefore performed the Holstein-Primakoff transformation on the spin operators projected in this sector alone. The procedure, however, could have been repeated for each $j$. To start with, one has to express the projected spin $\vec{J}^{(j)}=P_{j} \vec{J} P_{j}$ through boson operators $b_{j}, b_j^{\dag}$: $$J_z^{(j)} = b_j^\dag b_j - j \, , \quad
J_+^{(j)} = b_j^\dag \sqrt{2 j - b_j^\dag b_j}$$ so that the projected Hamiltonian is $$\begin{aligned}
\label{hamiltonian HN}
H^{(j)} &=& \omega_a a^\dag a + \omega_b \left( b_j^\dag b_j - j \right) + \\ \nonumber
&& g \left( a^\dag + a \right)\left(b_j^\dag\sqrt{1-2 \frac{b_j^\dag b_j}{j}} + \sqrt{1-2\frac{b_j^\dag b_j}{j}}b_j \right)\end{aligned}$$ In the main text, $j=N/2$ is taken and all of the subscripts are erased.
Equations of motion
-------------------
The core of our approximate analysis is the expansion of the non linear term $$\label{rootapp}
\sqrt{1-\frac{b^\dag b}{N}} = \sqrt{\Gamma}
\sqrt{1-\frac{\sqrt{N}(\beta d^\dag + \beta^* d) + d^\dag d}{N}}
\, .$$ In the limit $N\rightarrow \infty$, this can be expanded into a power series as $$\begin{aligned}
\label{expansionapp}
\sqrt{1-\frac{b^\dag b}{N}} &=& \sqrt{\Gamma} \bigg( 1 - \frac{\beta d^\dag + \beta^* d}{2 \sqrt{N} \Gamma} -\frac{d^\dag d}{2N \Gamma} \\
\nonumber && -\frac{\left(\beta d^\dag + \beta^* d\right)^2}{8 N
\Gamma^2} \bigg)+ O\left(N^{-\frac{3}{2}}\right)\end{aligned}$$ with $\alpha(t)$ and $\beta(t)$ assumed to stay order $O(1)$ so that the lowest order term gives already a good approximation once inserted in the Hamiltonian.
Using the leading contribution only, the Hamiltonian becomes $$\begin{aligned}
\nonumber H^{(N/2)} &=& \omega_a c^\dag c + \left[ \omega_b - 2 g
\frac{\alpha_{r} \beta_{r}}{\sqrt{\Gamma}}
\left(2+\frac{|\beta|^2}{2\Gamma}\right)
\right] d^{\dag} d + \\
&& g \sqrt{\Gamma}(c^\dag+c)\left[ \left(1-\frac{\beta^* \beta_{r}}{\Gamma}\right)d+h.c. \right]- \nonumber \\
&& g \frac{\alpha_{r}}{\sqrt{\Gamma}}\left[\beta^*\left(1+\frac{\beta^*\beta_{r}}{2 \Gamma}\right) d^2 + h.c.\right] + \nonumber \\
&& \sqrt{N}\left( \Delta_{c}^* c + \Delta_{d}^* d + h.c. \right) +
\Lambda_N + O\left(\frac{1}{\sqrt{N}}\right) \label{hamiltonian
evo}\end{aligned}$$ where we defined $\Delta_{c}$ and $\Delta_{d}$ as $$\label{delta vec}
\vec{\Delta} = \left(
\begin{array}{c}
\Delta_{c} \\
\Delta_{d} \\
\end{array}
\right) = \left(
\begin{array}{c}
\omega_a \alpha + 2 g \sqrt{\Gamma} \beta_{r} \\
\omega_b \beta + 2 g \sqrt{\Gamma} \alpha_{r}\left(1-\frac{\beta \beta_{r}}{\Gamma}\right) \\
\end{array}
\right)$$ while $\Lambda_N$ is the c-number $$\begin{aligned}
\Lambda_N &=& N \left\{ \omega_a |\alpha|^2 + \omega_b \left(
|\beta|^2-
\frac{1}{2} \right )+4g \sqrt{\Gamma} \alpha_{r} \beta_{r} \right \}- \nonumber \\
&& g\alpha_{r}\beta_{r} \frac{|\beta|^2}{2 \sqrt{\Gamma}}
\label{Lambda N}\end{aligned}$$ In the strict thermodynamic limit, thus, the Hamiltonian becomes quadratic, so that the time evolution can be described by a Gaussian propagator.
We describe the dynamics in the Heisenberg picture, and obtain the following equation for the annihilation operators:
$$\begin{aligned}
i \frac{d}{dt} c^{(H)}(t) &=& -
U^\dag(t,0)\left[H^{(N/2)}(t),c\right]U(t,0)
- i \sqrt N \frac{d \alpha(t)}{ dt} \label{heis eq} \\
\nonumber &=& \omega_a c^{(H)} + g\sqrt{\Gamma}\left[\left(1-\frac{\beta^* \beta_{r}}{\Gamma}\right) d^{(H)} + h.c.\right] \\
\nonumber && + \sqrt{N} \Delta_{c}- i \sqrt N \frac{d \alpha}{ dt}
+ O\left(\frac{1}{\sqrt{N}}\right)
\end{aligned}$$
In order for the operator $c$ to stay of order $O(1)$, the two terms $\sim \sqrt N$ above should compensate each other. To this end, we require the mean field $\alpha$ to satisfy
$$\label{eq diff alfa}
i \frac{d\alpha}{dt} = \Delta_{c}$$
In this way, the Heisenberg equation becomes
$$\label{eq Heis c}
i \frac{d}{dt}c^{(H)} = \omega_a c^{(H)} + g\sqrt{\Gamma}\left[\left(1-\frac{\beta^* \beta_{r}}{\Gamma}\right) d^{(H)} + h.c.\right] + O\left(\frac{1}{\sqrt{N}}\right)$$
In the limit $N \rightarrow \infty$ the terms $O\left(\frac{1}{\sqrt{N}}\right)$ do not bring any contribution to the dynamics of the operator $c^{(H)}$. For finite $N$, on the other hand, in order to give a more accurate description of the dynamics of the fluctuations, one should consider further terms in the expansion . Even if this is done, however, the dynamics of the mean field will remain unchanged, as it is determined by the $\sim \sqrt N$ terms only.
After a similar analysis is carried out for the operator $d$, we find the differential equation for $\beta$
$$\label{eq diff beta}
i \frac{d\beta}{dt} = \Delta_{d}$$
and the Heisenberg equations
$$\begin{aligned}
\nonumber i \frac{d}{dt}d^{(H)} &=& \left( \omega_b - 2 g \frac{\alpha_{r}\beta_{r}}{\sqrt{\Gamma}}\left(2+\frac{|\beta|^2}{2 \Gamma}\right)\right) d^{(H)}\\
\nonumber && + g \sqrt{\Gamma}\left( 1 - \frac{\beta \beta_{r}}{\Gamma} \right)(c^{(H)} + h.c.) \\
\label{eq Heis d} && - 2g \frac{\alpha_{r}\beta}{\sqrt{\Gamma}}\left( 1 + \frac{\beta \beta_{r}}{2 \Gamma} \right) {d^{(H)}}^\dag + O\left(\frac{1}{\sqrt{N}}\right)
\end{aligned}$$
Time evolution of the mean fields
---------------------------------
Explicitly, the equations for the real and imaginary parts of the re-scaled mean fields are
$$\begin{cases}
\dot{\alpha}_{r} = \omega_a \alpha_{i} \\
\dot{\alpha}_{i} = -\omega_a \alpha_{r} -2 g \sqrt{\Gamma} \beta_{r} \\
\dot{\beta}_{r} = \omega_b \beta_{i} - 2g \alpha_{r} \frac{\beta_{r} \beta_{i}}{\sqrt{\Gamma}} \\
\dot{\beta}_{i} = -\omega_b \beta_{r} - 2g\sqrt{\Gamma} \alpha_{r}(1-\frac{\beta_{r}^2}{\Gamma})
\end{cases}
\label{system}$$
If we regard $\alpha_{r}$ and $\beta_{r}$ as the generalized coordinates, and $\alpha_{i}$ and $\beta_{i}$ as their conjugate momenta, these equations can be derived from the Hamiltonian function
$$H(\alpha,\beta) = \frac{\omega_a }{2}|\alpha|^2 + \frac{\omega_b }{2}|\beta|^2 + 2 g \sqrt{\Gamma}\beta_{r}\alpha_{r}$$
They can be considered as purely classical equations, but should be solved with initial conditions that comes from the choice of an initial quantum state:
$$\label{initial conditions}
\left(
\begin{array}{c}
\alpha(0) \\
\beta(0) \\
\end{array}
\right) = \frac{1}{\sqrt{N}}\left(
\begin{array}{c}
\langle a(0) \rangle \\
\langle b(0) \rangle \\
\end{array}
\right)$$
As our treatment is based on the requirement that fluctuations stay of lower order than averages, we need to restrict our-selves to initial quantum states that fulfill this very same condition.
The system can be linearized if the initial conditions are such that $|\alpha(0)|,|\beta(0)| \ll 1$. Then, until the time $t_{lin}$ such that $\alpha$ and $\beta$ are of order one, the dynamics of the mean fields can be described by the linearized system
$$\label{system diff linear}
\frac{d}{dt}\left(
\begin{array}{c}
\alpha_{r} \\
\beta_{r} \\
\alpha_{i} \\
\beta_{i} \\
\end{array}
\right) \approx M_{0}\left(
\begin{array}{c}
\alpha_{r} \\
\beta_{r} \\
\alpha_{i} \\
\beta_{i} \\
\end{array}
\right)$$
where
$$\nonumber
M_{0}=\left(
\begin{array}{cccc}
0 & 0 & \omega_a & 0 \\
0 & 0 & 0 & \omega_b \\
-\omega_a & -2g & 0 & 0 \\
-2g & -\omega_b & 0 & 0 \\
\end{array}
\right)$$
Time evolution of the fluctuations
----------------------------------
As for the quantum fluctuations, we can recast the Heisenberg equations and in a more compact form using the quadrature operators $\vec Q = (q_c,q_d,p_c,p_d)$ as introduced in the main text. In the limit $N\rightarrow \infty$, we find
$$\nonumber
\dot{\vec{Q}}^{(H)} = M_{\alpha,\beta} \vec{Q}^{(H)}$$
where $M_{\alpha,\beta}$ is the matrix
$$\label{matrix M}
M_{\alpha,\beta} = \left(
\begin{array}{cccc}
0 & 0 & \omega_a & 0 \\
-\frac{2 g \beta_{r}\beta_{i} }{\sqrt{\Gamma}} & - \frac{2g
\alpha_{r} \beta_{i}}{\sqrt{\Gamma}}\left( 1 +
\frac{\beta_{r}^2}{\Gamma}\right) &
0 & \omega_b - \frac{2 g \alpha_{r} \beta_{r}}{\sqrt{\Gamma}}\left( 1+ \frac{\beta_{i}^2}{\Gamma} \right) \\
-\omega_a & -2g\sqrt{\Gamma}\left(1-\frac{\beta_{r}^2}{\Gamma}\right) & 0 & 2g\frac{\beta_{r}\beta_{i}}{\sqrt{\Gamma}} \\
-2g\sqrt{\Gamma}\left(1-\frac{\beta_{r}^2}{\Gamma}\right) &
-\omega_b + \frac{2g\alpha_{r}\beta_{r}}{\sqrt{\Gamma}}\left( 3 +
\frac{\beta_{r}^2}{\Gamma} \right) &
0 & \frac{2 g \alpha_{r} \beta_{i}}{\sqrt{\Gamma}}\left( 1+\frac{\beta_{r}^2}{\Gamma} \right) \\
\end{array}
\right)$$
We can define the fundamental matrix $\Phi$ as the solution of the differential equation $$\label{phi super}
\dot{\Phi} = M_{\alpha,\beta} \Phi$$ with the initial condition $\Phi(0) = \mathds{1}$. Then the solution of the Heisenberg equation is $$\nonumber
\vec{Q}^{(H)}(t) = \Phi(t) \vec{Q}$$
In our description the first moments are zero for every time, i.e. $\langle \vec{Q} \rangle = \vec{0}$. The second moments are given by the covariance matrix $W$, as mentioned in the main text. If the initial state is Gaussian, then the covariance matrix completely characterizes the fluctuations around the mean fields.
In the linear transient regime, $t < t_{lin}$, we find that $M_{\alpha,\beta} = M_0 $, so that the dynamics of the fluctuations are the same as the mean field ones. Furthermore, if the initial state is such that $\alpha(0)=\beta(0)=0$, then the mean fields will remain zero $\forall t$ and, thus the dynamics of the fluctuations is described by $M_0$ at all times.
In general, $M_{\alpha, \beta}$ has a parametric time dependence, as it contains $\omega_b(t)$. Therefore, it inherits from $\omega_b$ a periodicity of $T=2 \pi/\eta$. Then, by the Floquet theorem, we can write $$\Phi(t) = e^{-\frac{B t }{T}} \, P(t) \, ,$$ with a constant $B$ and a periodic $P(t)$. For us, $\Phi
(0)=\mathds{1}$, therefore also $P(0) = P(T) = \mathds{1}$. As a result, we have that the so called monodromy matrix is $\Phi(T) =
e^{- B}$.
Limits of validity for finite $N$
---------------------------------
Our description of the dynamics becomes exact for every finite time $t$ in the limit $N\rightarrow \infty$. Anyway, it’s crucial to understand what the limits of applicability of our approach are for finite $N$. We observe that, during the time evolution, fluctuations (assumed to be $O(1)$ at $t=0$) can become very large and even unbounded. When this happens, the expansion of the square root in the Hamiltonian may become incorrect. In other words, for finite but large $N$ our description stays accurate until the n-th moments become of order $N^{n/2}$. To ensure that this is not the case, it is enough to require that all of the elements of the matrix $\Phi$ are small compared to $\sqrt{N}$. Therefore, our description of the dynamics is accurate until a time $t_{max}$, defined as the first instant for which
$$\label{condition time}
max_{i\,j} \left\{ \Phi_{ij}(t_{max}) \right\} \approx \sqrt{N}$$
In the limit $N\rightarrow \infty$, we expect $t_{max} \rightarrow
\infty$.
In order to characterize and estimate $t_{max}$, we need to consider the initial conditions for the mean fields, $\alpha(0) =
\alpha_0$ and $\beta(0) = \beta_0$. In particular, if we take $\delta= max\left\{|\alpha_0| , |\beta_0|\right\} \ll 1$, the first part of the dynamics is included in the linear transient. This implies that, for $0\leq t <t_{lin}$, the time evolution of both the fluctuations and of the mean-fields is determined by the linearized matrix $M_0$.
In the absence of driving and in the linear transient, the dynamics would be characterized by the eigenvalues of the matrix $M_0$. Two of them are always purely imaginary, i.e. $\pm i
\lambda_1$. The other two are $\pm i \lambda_2$, where $\lambda_2$ is real if $\mu>1$, it is equal to $\lambda_1$ if $\mu=1$ and it is purely imaginary if $\mu<1$. This means that if $\mu>1$ the fluctuations and the mean fields stay always of the same order, i.e $t_{lin}\rightarrow \infty$ and $t_{max}\rightarrow \infty$. At the transition point, $\mu=1$, fluctuations and mean-fields grow linearly in time; while in the super-radiant phase $\mu<1$, the fluctuations and the mean fields can experience an exponential growth, with an [*instability rate*]{} given by $\gamma^*=|\lambda_2|$. This is true until $t\lesssim t_{lin}$. After $t_{lin}$, the nonlinear terms cannot be neglected anymore and one has to consider the full non-linear equations. Thus, within the linear regime, we can define the characteristic time $\tau^*$ as the inverse of the instability rate, i.e.
$$\nonumber
{\tau^*}^{-1} = \gamma^*=
\sqrt{\sqrt{\left(\frac{\omega_a^2-\omega_b^2}{2}\right)^2+4\omega_a\omega_b
g^2}-\frac{\omega_a^2+\omega_b^2}{2}}$$
We can estimate $t_{lin}$ as the time for which one of the mean fields (either $\alpha$ or $\beta$) becomes of order one, $$\label{time Tstar}
t_{lin} \approx \tau^* \ln\left(\delta^{-1}\right)$$ Since $\delta \ll 1$, we expect $t_{lin}$ to be much larger than $\tau^*$. The crucial point, however, is wether or not all of the linear transient is contained within our limit of validity. Indeed, for this description to be valid for times of the order of $t_{lin}$, we have to require that
$$\nonumber
e^{\gamma^* t_{lin}} \ll \sqrt{N}$$
from which it follows that
$$\nonumber
t_{lin} \ll \frac{\tau^*}{2} \ln(N)$$
and that
$$\nonumber
\delta \gg \frac{1}{\sqrt{N}}$$
This means that if the initial state is not so close to the vacuum (the difference from zero of $\alpha$ and $\beta$ being larger than $\frac{1}{\sqrt{N}}$), then $t_{lin} \ll t_{max}$. In this case, our description makes full sense even outside the linear transient, and can be used even when the mean-fields take macroscopic values. In this case, $t_{lin}$ can indeed be estimated by and is finite. On the other hand, for finite $N$, it is not possible to give a simple expression for $t_{max}$, as it depends on the non-linear terms appearing in the time evolution, and the only way to check the validity of our approach is to check the condition .
Instead, if both of the mean fields start too close to zero, specifically if $\delta \lesssim \frac{1}{\sqrt{N}}$, then our description ceases to be good before the mean fields $\alpha$ and $\beta$ reach macroscopic values. In this case, in fact, $t_{max}$ is inside the linear transient, and, from , we can make the simple estimate
$$\nonumber
t_{max} = \frac{\tau^*}{2}\ln(N)$$
These estimates and reasoning can be adapted also to the case of a driven system, since they are simply a consequence of the fact that the dynamics of the mean fields and that of the fluctuations are the same in the linear transient.
For instance, for a periodically driven Hamiltonian, we can use the largest positive Floquet exponent of the linearized matrix $M_0$ in order to estimate $\tau^*$; and then use again the equations above to estimate $t_{lin}$ and $t_{max}$. Specifically, if $M_0$ is periodic in time, with period $T$, then the monodromy matrix (in the Floquet description) is given by $\mathcal{M}=\Phi(T)$. The eigenvalues of $\mathcal{M}$ are the Floquet multipliers $\{\rho_i\}_{i=1}^4$. From these, we can calculate the Floquet exponents, that are the complex numbers $\nu_i = \frac{\ln(\rho_i)}{T}$. So we can define the instability rate $\gamma^*$ as the maximum among zero and the real parts of the Floquet exponents, i.e.
$$\label{time tau star}
\gamma^* = max \left\{ 0,\, \left\{\mathds{R}e\{\nu_i \} \right\}_{i=1}^4\right\}$$
As a result, the arguments above can be applied in this case too, even if, in a strict sense, the stability of the dynamics cannot be fully characterized by the instantaneous eigenvalues of the matrix $M_0$.
Optimal degree of two-mode squeezing
------------------------------------
As discussed in the main text, we compare the instantaneous state of the global (atom+field) system with the two-mode squeezed coherent state ${\left\vert\psi_{\alpha,\beta}(r,t)\right\rangle}$ obtained by applying the unitary squeeze operator of (real) degree $r$, $$S(r)= e^{r ( c^\dag d^\dag - c d) } \, ,$$ to the two-mode coherent state obtained by taking the instantaneous mean fields $\alpha(t)$ and $\beta(t)$ as amplitudes, $${\left\vert\psi_{\alpha,\beta}(r,t)\right\rangle}=S(r){\left\vert\sqrt N \,
\alpha(t)\right\rangle}_a{\left\vert\sqrt N \, \beta(t)\right\rangle}_b$$ The fidelity that we obtain by optimizing the parameter $r$, is very close to unity, as reported in Fig. (\[Fig fidelity\]).
![ The fidelity $F(t) = |\left \langle
\psi_{\alpha(t),\beta(t)}(r_{opt},t) | \Psi, t \right \rangle
|^2$, evaluated for the same parameter values used in Fig. 2 of the main text. $F(t)$ stays very close to one, tending to slowly decrease for long times.[]{data-label="Fig fidelity"}](fidelity.png){width="\linewidth"}
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---
abstract: 'The present work considers the optimal control of a convective Cahn-Hilliard system, where the control enters through the velocity in the transport term. We prove the existence of a solution to the considered optimal control problem. For an efficient numerical solution, the expensive high-dimensional PDE systems are replaced by reduced-order models utilizing proper orthogonal decomposition (POD-ROM). The POD modes are computed from snapshots which are solutions of the governing equations which are discretized utilizing adaptive finite elements. The numerical tests show that the use of POD-ROM combined with spatially adapted snapshots leads to large speedup factors compared with a high-fidelity finite element optimization.'
author:
- 'Carmen Gräßle, Michael Hinze'
- Nicolas Scharmacher
title: 'POD for optimal control of the Cahn-Hilliard system using spatially adapted snapshots'
---
Introduction
============
The optimal control of two-phase systems has been studied in various papers, see e.g. [@HRV10], [@HW12] and [@RS15]. In this paper, we concentrate our investigations on the diffuse interface approach, where we assume the existence of interfacial regions of small width between the phases. This has the advantage that topology changes like droplet collision or coalescence can be handled in a natural way. Many degrees of freedom are needed in the interfacial regions in order to resemble the steep gradients well, whereas in the pure phases a small number of degrees of freedom is sufficient. Thus, in order to make numerical computations feasible, we utilize adaptive finite element methods. However, the optimization of a phase field model is still a costly issue, since a sequence of large-scale systems has to be solved repeatedly. For this reason, we replace the high-dimensional systems by low-dimensional POD approximations. This has been done in e.g. [@Vol01] for uniformly discretized snapshots.\
We perform POD based optimal control using spatially adapted snapshots. The combination of POD with adaptive finite elements has been investigated for time-dependent problems in [@URL16] and [@GH17].
Convective Cahn-Hilliard system
===============================
We consider the Cahn-Hilliard system which was introduced in [@CH58] as a model for phase transitions in binary systems. In a bounded and open domain , ${\mathtt{d} \in \{2,3\}}$, with Lipschitz boundary $\partial \Omega$, we assume two substances $A$ and $B$ to be given. In order to describe the spatial distribution over time $I=(0,T]$ with fixed end time $T>0$, a phase field variable $\varphi$ is introduced which fulfills $\varphi(t,{{\boldsymbol x}}) = +1$ in the pure $A$-phase and $\varphi(t,{{\boldsymbol x}}) = -1$ in the pure $B$-phase. Values of $\varphi$ between $-1$ and $+1$ represent the interfacial area between the two substances. Introducing the chemical potential $\mu$, the Cahn-Hilliard equations can be written as a coupled system of second-order in space $$\label{CHcoupled_nocontrol}
\left\{
\begin{array}{rcll}
\varphi_t + \mathtt{v} \cdot \nabla \varphi - b\Delta \mu & = & 0 & \quad \text{in } I \times \Omega,\\
-\sigma \varepsilon \Delta \varphi + \frac{\sigma}{\varepsilon} \mathcal{F}'(\varphi) & = & \mu & \quad \text{in } I \times \Omega,\\
\partial_n \varphi = \partial_n \mu & = & 0 & \quad \text{on } I \times \partial \Omega,\\
\varphi(0,\cdot) & = & \varphi_0 & \quad \text{in } \Omega.
\end{array}
\right.$$
By $b>0$ we denote a constant mobility, $\sigma > 0$ describes the surface tension and $\varepsilon >0$ is a parameter which is related to the interface width. For the free energy $\mathcal{F}$, we consider the smooth polynomial free energy (see e.g. [@EZ86]) $$ \mathcal{F}(\varphi) = \frac{1}{4}(1-\varphi^2)^2.$$
A possible flow of the mixture at a given velocity field $\mathtt{v}$ is modeled in by the transport term which, in the context of multiphase flow, represents the coupling to the Navier-Stokes system, see e.g. [@HH77] and [@AGG12]. We use the following notations and assumptions:\
*Notations 2.1*\
We denote by $H_{(0)}^1(\Omega)$ the space of functions in $H^1(\Omega)$ with zero mean value and by $L_\sigma^2(\Omega)^\mathtt{d} = \{f \in L^2(\Omega)^\mathtt{d}: \text{div}f = 0, f \cdot {{\boldsymbol n}}_\Omega |_{\partial \Omega} = 0\}$ the space of solenoidal vector fields, for which we refer to [@Soh01] for details about well-definedness. We use as the solution space for the phase field variable $W(0,T) = \{f \in L^2(0,T;H_{(0)}^1(\Omega)) : f_t \in L^2(0,T;H_{(0)}^{-1}(\Omega))\}$.\
[*Assumptions 2.2*]{}
1. The initial phase field $\varphi_0 \in H_{(0)}^1(\Omega)$ fulfills $E_0 = E(\varphi_0)<\infty$ with Ginzburg-Landau free energy $$ E(\varphi) = \int_\Omega \frac{\sigma \varepsilon}{2} |\nabla \varphi|^2 + \frac{\sigma}{\varepsilon} \mathcal{F}(\varphi)d{{\boldsymbol x}}.$$
2. The velocity $\mathtt{v}$ fulfills $ \mathtt{v} \in L^\infty(0,T;L_\sigma^2(\Omega)^\mathtt{d})
\cap L^2(0,T;H^1(\Omega)^\mathtt{d})$.\
[*Remark 2.3*]{}\
It is shown in [@Abe07 Theorem 4.1.1] that there exists a unique solution $(\varphi, \mu)$ to with $\varphi \in
W(0,T)\cap L^2(0,T;H^2(\Omega))$, $\mu \in L^2(0,T;H^1(\Omega))$. This solution satisfies $$\label{est-1}
\|\varphi \|_{L^2(0,T;H^2(\Omega))}^2 +
\|\varphi_t\|_{L^2(0,T;H_{(0)}^{-1}(\Omega))}^2 \leq C \left( E_0 + \| \mathtt{v}\|_{L^2(0,T;L^2(\Omega)^\mathtt{d})}^2\right)$$ where $C$ is independent of $\mathtt{v}$ and $ \varphi_0$.
Optimal control of Cahn-Hilliard
================================
We investigate the minimization of the quadratic objective functional $$ J(\varphi,u) = \frac{\beta_1}{2} \|\varphi - \varphi_d\|_{L^2(0,T;L^2(\Omega))}^2 + \frac{\beta_2}{2} \|\varphi-\varphi_T\|_{L^2(\Omega)}^2 + \frac{\gamma}{2} \|u\|_U^2$$ where $\beta_1,\beta_2 \geq 0$ are given constants, $\varphi_d \in L^2(0,T;L^2(\Omega))$ is the desired phase field, $\varphi_T \in L^2(\Omega)$ is the target phase pattern at final time, $\gamma > 0$ is the penalty parameter and $u \in U = L^2(0,T;\mathbb{R}^m)$, with $m \in \mathbb{N}$, denotes the control variable which is a time-dependent variable and in particular independent of the current spatial discretization. The goal of the optimal control problem is to steer a given initial phase distribution $\varphi_0$ to a given desired phase pattern. This problem can also be interpreted as an optimal control of a free boundary which is encoded through the phase field variable. We consider distributed control which enters through the transport term: $$\label{CHcoupled_withcontrol}
\left\{
\begin{array}{rcll}
\varphi_t + ({{\mathcal B}}u) \cdot \nabla \varphi- b\Delta \mu & = & 0 & \quad \text{ in } I \times \Omega,\\
-\sigma \varepsilon \Delta \varphi + \frac{\sigma}{\varepsilon} \mathcal{F}'(\varphi) & = & \mu & \quad \text{ in } I \times \Omega.\\
\partial_n \varphi = \partial_n \mu & = & 0 & \quad \text{on } I \times \partial \Omega,\\
\varphi(0,\cdot) & = & \varphi_0 & \quad \text{in } \Omega.
\end{array}
\right.$$ The control operator ${{\mathcal B}}$ is defined by ${{\mathcal B}}: U \to L^2(0,T;H^1(\Omega)^\mathtt{d}), ({{\mathcal B}}u)(t) =
\sum_{i=1}^m u_i(t) \chi_i$ where $\chi_i \in L_\sigma^2(\Omega)^\mathtt{d} \cap H^1(\Omega)^\mathtt{d}, 1 \leq i
\leq m$, represent given control shape functions. The admissible set of controls is $$ {{U_{\text{ad}}}}= \{u \in U \; | \; u_a(t) \leq u(t) \leq u_b(t) \text{ in } \mathbb{R}^m \text{ a.e. in } [0,T]\}$$ with $u_a, u_b \in L^\infty(0,T;\mathbb{R}^m), u_a(t) \leq u_b(t)$ almost everywhere in $[0,T]$. The inequalities between vectors are understood componentwise. Then, the optimal control problem can be expressed as $$\label{ocp}
\min_u \hat{J}(u) = J(\varphi(u),u) \quad \text{ s.t. } \quad (\varphi(u),u) \text{ satisfies } \eqref{CHcoupled_withcontrol} \quad \text{ and } \quad u\in {{U_{\text{ad}}}}.$$
Problem admits a solution $\bar{u} \in {{U_{\text{ad}}}}$.
The infimum $\inf_{u \in {{U_{\text{ad}}}}} \hat{J}(u)$ exists due to $\hat{J} \geq 0$ and ${{U_{\text{ad}}}}\neq \emptyset$. Let $\{u_n\}_{n \in \mathbb{N}} \subset {{U_{\text{ad}}}}$ be a minimizing sequence and $\{\varphi_n\}_{n \in \mathbb{N}}$ the corresponding sequence of states $\varphi_n = \varphi(u_{n})$. Since ${{U_{\text{ad}}}}$ is closed, convex and bounded in $L^2(0,T;\mathbb{R}^m)\supset L^{\infty}(0,T;\mathbb{R}^m)$, we can extract a subsequence (denoted by the same name), which converges weakly to some $\bar{u}\in {{U_{\text{ad}}}}$. Weak convergence ${{\mathcal B}}u_n \rightharpoonup {{\mathcal B}}\bar{u}$ in $L^2(0,T;H^1(\Omega)^{\mathtt{d}})$ follows from the linearity and boundedness of ${{\mathcal B}}$. Due to the energy estimate there exists a constant $M>0$ such that for all $n\in\mathbb{N}$ we have $$\|\varphi_n \|_{L^2(0,T;H^2(\Omega))}^2 + \|\varphi_{n,t}\|_{L^2(0,T;H_{(0)}^{-1}(\Omega))}^2 \leq M.$$ Since $W(0,T)\cap L^2(0,T;H^2(\Omega))$ is reflexive, there exists another subsequence (denoted by the same name) that converges weakly to some $\bar{\varphi}
\in W(0,T)\cap L^2(0,T;H^2(\Omega))$. It remains to show, that the pair $(\bar{\varphi},\bar{u})$ is admissible, i.e. $\bar{\varphi}=\varphi(\bar{u})$. While passing to the limit in the weak formulation is clear for the linear terms, the nonlinear ones require further investigation. Since $W(0,T)\cap L^2(0,T;H^2(\Omega))$ is compactly embedded in $L^2(0,T;H^1(\Omega))$ (see [@Sim86 Sect. 8, Corr. 4]), the sequence $\{\varphi_n\}_{n \in \mathbb{N}}$ converges strongly to $\bar{\varphi}$ in $L^2(0,T;H^1(\Omega))$. For the control term we have for $v\in H^1(\Omega)$ the splitting $$\begin{gathered}
\int_0^T ( {{\mathcal B}}u_n \cdot\nabla \varphi_n - {{\mathcal B}}\bar{u}\cdot\nabla
\bar{\varphi},v )_{L^2(\Omega)} \,dt =
\\ \int_0^T ( {{\mathcal B}}u_n \cdot \nabla
(\varphi_n - \bar{\varphi}),v )_{L^2(\Omega)} \,dt +
\int_0^T ( {{\mathcal B}}(u_n-\bar{u})\cdot\nabla \bar{\varphi},v )_{L^2(\Omega)} \,dt.
\end{gathered}$$ Due to $\nabla \bar{\varphi} \in L^2(0,T;H^1(\Omega)^{\mathtt{d}})$, the product $v\nabla \bar{\varphi} \in L^2(0,T;L^2(\Omega)^{\mathtt{d}})$ gives rise to a continuous linear functional on $L^2(0,T;H^1(\Omega)^{\mathtt{d}})$. Hence, the right term vanishes for $n\rightarrow\infty$ by definition of weak convergence. For the left term we estimate $$\begin{gathered}
\left| \int_0^T ( {{\mathcal B}}u_n \cdot \nabla (\varphi_n -
\bar{\varphi}),v )_{L^2(\Omega)} \,dt \right| \leq \\ \| {{\mathcal B}}u_n
\|_{L^2(0,T;H^1(\Omega)^{\mathtt{d}})} \| \varphi_n - \bar{\varphi} \|_{L^2(0,T;H^1(\Omega))} \| v \|_{H^1(\Omega)},
\end{gathered}$$ which also vanishes for $n\rightarrow \infty$. For the nonlinearity $\mathcal{F}'$ we infer from $$\begin{aligned}
| \mathcal{F}' (\varphi) - \mathcal{F}'(\psi) | \leq C (\varphi ^2 + \psi ^2) \left| \varphi - \psi \right|
\end{aligned}$$ for all $\varphi,\psi \in \mathbb{R}$ and some $C>0$ the estimate $$\begin{gathered}
\left| \int_0^T ( \mathcal{F}'(\varphi_n)-\mathcal{F}'(\bar{\varphi}),v )_{L^2(\Omega)}
\, dt \right| \leq \\
C (\| \varphi_n^2 \|_{L^2(0.T;L^2(\Omega))} + \| \bar{\varphi}^2 \|_{L^2(0,T;L^2(\Omega))}) \| \varphi_n - \bar{\varphi} \|_{L^2(0,T;H^1(\Omega))} \| v \|_{H^1(\Omega)},
\end{gathered}$$ which gives the desired convergence due to $L^{\infty}(0,T;H^1(\Omega))\subset L^4(0,T;L^4(\Omega))$. Finally, the lower semi-continuity of $J$ yields $$\hspace*{4cm} J(\bar{\varphi},\bar{u})=\inf_{u \in {{U_{\text{ad}}}}} \hat{J}(u). \hspace{4cm} \qed$$
Problem is a non-convex programming problem, so that different minima might exist. Numerical solution methods will converge in a local minimum which is close to the initial point. In order to compute a locally optimal solution to , we consider the first-order necessary optimality condition given by the variational inequality $$\label{firstordercond}
\langle \hat{J}'(\bar{u}),u-\bar{u} \rangle_{U',U} \geq 0 \quad \forall u \in {{U_{\text{ad}}}}.$$ Following the standard adjoint techniques, we derive that is equivalent to $$\label{vi}
\int_0^T \sum_{i=1}^m \left( \gamma \bar{u}_i(t) + \int_\Omega (\chi_i({{\boldsymbol x}}) \cdot
\nabla \varphi(t,{{\boldsymbol x}}))\bar{p}(t,{{\boldsymbol x}}) d{{\boldsymbol x}}\right)(u_i(t) - \bar{u}_i(t)) dt \geq 0$$ for all $u \in {{U_{\text{ad}}}}$ where the function $\bar{p}$ is a solution to the adjoint equations $$\label{adjoint}
\left\{
\begin{array}{r c l l}
-p_t - \sigma \varepsilon \Delta q + \frac{\sigma}{\varepsilon} \mathcal{F}''(\bar{\varphi})q
- \mathcal{B}u \cdot \nabla p & = & -\beta_1(\bar{\varphi}-\varphi_d) & \; \text{in } I \times \Omega,\\
-q - b\Delta p & = & 0, & \; \text{in } I \times \Omega,\\
\partial_n p = \partial_n q & = & 0, & \; \text{on } I \times \partial \Omega, \\
p(T, \cdot) & = & -\beta_2(\bar{\varphi}(T,\cdot)-\varphi_T), & \; \text{in } \Omega.
\end{array}
\right.$$ The variable $\bar{\varphi}$ in denotes the solution to associated with an optimal control $\bar{u}$.
POD-ROM using spatially adapted snapshots
=========================================
The optimal control problem is discretized by adaptive finite elements and solved by a standard projected gradient method with an Armijo line search rule. In order to replace the resulting high-dimensional PDEs by low-dimensional approximations, we make use of POD-ROM, see e.g. [@HLBR12] or [@Vol13]. The nonlinearity is treated using DEIM, cf [@CS10]. In order to combine POD-ROM with spatially adapted snapshots, we follow the ideas in [@URL16] and [@GH17].
Numerical results
=================
We consider the unit square $\Omega = (0,1)\times(0,1)$, the end time $T=0.0125$ and utilize a uniform time grid with time step size $\Delta t = 2.5 \cdot 10^{-5}$. The mobility is $b = 2.5 \cdot 10^{-5}$, the surface tension is $\sigma = 25.98$ and the interface parameter is set to $\varepsilon = 0.02$. In the cost functional we use $\gamma = 0.0001$, $\beta_1 = 20$ and $\beta_2 = 20$. We use $m=1$ control shape function given by $\chi(x) = (\text{sin}(\pi x_1)\text{cos}(\pi x_2), -\text{sin}(\pi x_2)\text{cos}(\pi x_1))^T$. The desired state is shown in Figure \[fig:c\_desired\]. The initial state $\varphi_0$ coincides with $\varphi_d(0)$. In order to fulfill the Courant-Friedrichs-Lewy (CFL) condition, we impose the control constraints $u_a = 0, u_b = 50$ and demand $h_{\min} > 0.00177$.\
The optimization is initialized with an input control $u=0 \in {{U_{\text{ad}}}}$. We compute the POD basis with respect to the $L^2(\Omega)$-inner product for the snapshot ensemble formed by the desired phase field $\varphi_d$, which is discretized using adaptive finite elements. Figure \[fig:c\] shows the finite element solution and the POD solution for the phase field using $\ell=10$ and $\ell=20$ POD modes, respectively. It turns out that a large number of POD modes is needed in order to smoothen out oscillations due to the convection.
![*Desired phase field at $t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c_desired"}](cdesiredt0.png "fig:") ![*Desired phase field at $t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c_desired"}](cdesiredthalf.png "fig:") ![*Desired phase field at $t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c_desired"}](cdesiredtend.png "fig:")
![*Finite element (top) and POD-DEIM optimal solution with $\ell=10$ (middle) and $\ell=20$ (bottom) of the phase field $\varphi$ at $t=t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c"}](cFEt0.png "fig:") ![*Finite element (top) and POD-DEIM optimal solution with $\ell=10$ (middle) and $\ell=20$ (bottom) of the phase field $\varphi$ at $t=t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c"}](cFEthalf.png "fig:") ![*Finite element (top) and POD-DEIM optimal solution with $\ell=10$ (middle) and $\ell=20$ (bottom) of the phase field $\varphi$ at $t=t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c"}](cFEtend.png "fig:")\
![*Finite element (top) and POD-DEIM optimal solution with $\ell=10$ (middle) and $\ell=20$ (bottom) of the phase field $\varphi$ at $t=t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c"}](cromell10t0.png "fig:") ![*Finite element (top) and POD-DEIM optimal solution with $\ell=10$ (middle) and $\ell=20$ (bottom) of the phase field $\varphi$ at $t=t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c"}](cromell10thalf.png "fig:") ![*Finite element (top) and POD-DEIM optimal solution with $\ell=10$ (middle) and $\ell=20$ (bottom) of the phase field $\varphi$ at $t=t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c"}](cromell10tend.png "fig:")\
![*Finite element (top) and POD-DEIM optimal solution with $\ell=10$ (middle) and $\ell=20$ (bottom) of the phase field $\varphi$ at $t=t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c"}](cromell20t0.png "fig:") ![*Finite element (top) and POD-DEIM optimal solution with $\ell=10$ (middle) and $\ell=20$ (bottom) of the phase field $\varphi$ at $t=t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c"}](cromell20thalf.png "fig:") ![*Finite element (top) and POD-DEIM optimal solution with $\ell=10$ (middle) and $\ell=20$ (bottom) of the phase field $\varphi$ at $t=t_0, t_{250}, t_{500}$ with adaptive meshes*[]{data-label="fig:c"}](cromell20tend.png "fig:")
Table 1 (left) summarizes the iteration history for the finite element projected gradient method where we used the stopping criterion $\|\hat{J}'(u^k)\|_{U_h} < 0.01\cdot\|\hat{J}'(u^0)\|_{U_h} + 0.01. $ Table 1 (right) tabulates the POD-ROM optimization. Note that the value of the POD cost functional $\hat{J}_\ell(u^k)$ stagnates due to the POD error. The value of the full-order cost functional at the POD solution is $\hat{J}(\bar{u}_{\text{POD}}) = 7.31 \cdot 10^{-4}$. If $\ell=20$ POD modes are used, the relative $L^2(0,T;L^2(\Omega))$-error between the finite element and the POD solution for the phase field is err$_\varphi = 7.19 \cdot 10^{-3} $; for POD-DEIM it is err$_\varphi = 7.38 \cdot 10^{-3} $.
$k$ $\hat{J}(u^k)$ $\| \hat{J}'(u^k) \|_{U_h}$ $s_k$
----- ---------------------- ----------------------------- --------
0 $8.61 \cdot 10^{0}$ $2.85 \cdot 10^0$ $1.0$
1 $6.48 \cdot 10^{-1}$ $2.32 \cdot 10^0$ $0.25$
2 $1.90 \cdot 10^{-2}$ $4.56 \cdot 10^{-1}$ $0.25$
3 $3.82 \cdot 10^{-3}$ $1.93 \cdot 10^{-1}$ $0.25$
4 $1.18 \cdot 10^{-3}$ $8.45 \cdot 10^{-2}$ $0.25$
5 $6.80 \cdot 10^{-4}$ $3.67 \cdot 10^{-2}$
: *Iteration history finite element optimization (left) and POD optimization (right) with $\ell=20$. The Armijo step size is denoted by $s_k$.*
$k$ $\hat{J}_\ell(u^k)$ $\| \hat{J}_{\ell}'(u^k) \|_{U_h}$ $s_k$
----- ---------------------- ------------------------------------ --------
0 $8.77 \cdot 10^{0}$ $2.81 \cdot 10^0$ $1.0$
1 $7.98 \cdot 10^{-1}$ $2.41 \cdot 10^0$ $0.25$
2 $5.79 \cdot 10^{-2}$ $3.67 \cdot 10^{-1}$ $0.25$
3 $5.02 \cdot 10^{-2}$ $1.63 \cdot 10^{-1}$ $0.25$
4 $4.76 \cdot 10^{-2}$ $7.45 \cdot 10^{-2}$ $0.25$
5 $4.76 \cdot 10^{-2}$ $3.48 \cdot 10^{-2}$
: *Iteration history finite element optimization (left) and POD optimization (right) with $\ell=20$. The Armijo step size is denoted by $s_k$.*
\[tab:iter\_hist\]
In Table 2 the computational times for the uniform FE, adaptive FE, POD and POD-DEIM optimization are listed. The offline costs for POD when using spatially adapted snapshots are as follows: the interpolation of the snapshots takes 212 seconds, the POD basis computation costs 40 seconds and the computations for DEIM take 30 seconds. In comparison, the use of uniformly discretized snapshots leads to the computational time of 243 seconds for POD basis computation and 193 seconds for the DEIM computations.
uniform FE adaptive FE POD POD-DEIM
------------------------------ ------------ ------------- --------- ----------
optimization 36868 sec 5805 sec 675 sec 0.3 sec
$\to$ solve each state eq. 1660 sec 348 sec 42 sec 0.02 sec
$\to$ solve each adjoint eq. 761 sec 121 sec 16 sec 0.01 sec
: *Computational times for the FE, POD and POD-DEIM optimization.*
\[tab:iter\_hist\]
Outlook
=======
In future work, we intend to embed the optimization of Cahn-Hilliard in a trust-region framework in order to adapt the POD model accuracy within the optimization. We further want to consider a relaxed double-obstacle free energy which is a smooth approximation of the non-smooth double-obstacle free energy. We expect that more POD modes are needed in this case to get similar accuracy results as in the case of a polynomial free energy. Moreover, we intend to couple the smoothness of the model to the trust-region fidelity.
Acknowledgments {#acknowledgments .unnumbered}
===============
We like to thank Christian Kahle for providing many libraries which we could use for the coding. The first author gratefully acknowledges the financial support by the DFG through the priority program SPP 1962. The third author gratefully acknowledges the financial support by the DFG through the Collaborative Research Center SFB/TRR 181.
[99]{} 1.0ex
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|
---
author:
- |
Zhuo Hui and Aswin C. Sankaranarayanan[^1]\
ECE Department, Carnegie Mellon University, Pittsburgh, PA
bibliography:
- 'refs.bib'
title: |
A Dictionary-based Approach for Estimating Shape and\
Spatially-Varying Reflectance
---
[**Keywords.**]{} Photometric stereo, BRDF estimation, Dictionaries, Spatially varying BRDF.
Introduction {#sec:intro}
============
Prior work {#sec:prior}
==========
Problem setup {#sec:overview}
=============
Surface normal estimation {#sec:Normal}
=========================
Reflectance estimation {#sec:BRDF}
======================
Results {#sec:results}
=======
Discussions {#sec:discuss}
===========
[^1]: The authors were supported in part by the NSF grant CCF-1117939. Email: [{zhui, saswin}@andrew.cmu.edu]{}
|
---
author:
- 'Richard Evan Schwartz [^1]'
title: 'Outer Billiards, the Arithmetic Graph, and the Octagon'
---
[^1]: 5 pt Supported by N.S.F. Research Grant DMS-0072607
|
---
abstract: |
The recent finding by Chevalier & Ilovaisky (1998) from [*Hipparcos*]{} observations that OB-supergiant X-ray binaries have relatively large runaway velocities (mean peculiar tangential velocity[^1] ${\mbox{$\langle v_{\rm tr} \rangle$}}= 42 \pm 14$ ), whereas Be/X-ray binaries have low runaway velocities (${\mbox{$\langle v_{\rm tr} \rangle$}}= 15 \pm
6$), provides confirmation of the current models for the formation of these two types of systems. These predict a difference in runaway velocity of this order of magnitude. This difference basically results from the variation of the fractional helium core mass as a function of stellar mass, in combination with the conservation of orbital angular momentum during the mass transfer phase that preceded the formation of the compact object in the system. This combination results into: (i) Systematically narrower pre-supernova orbits in the OB-supergiant systems than in the Be-systems, and (ii) A larger fractional amount of mass ejected in the supernovae in high-mass systems relative to systems of lower mass. Regardless of possible kick velocities imparted to neutron stars at birth, this combination leads to a considerable difference in average runaway velocity between these two groups. If one includes the possibility for non-conservative mass transfer the predicted difference between the runaway velocity of the two groups becomes even more pronounced. The observed low runaway velocities of the Be/X-ray binaries confirm that in most cases not more than 1 to 2 was ejected in the supernovae that produced their neutron stars. This, in combination with the –on average– large orbital eccentricities of these systems, indicates that their neutron stars must have received a velocity kick in the range 60 - 250 at birth. The considerable runaway velocity of Cygnus X-1 (${\mbox{$v_{\rm tr}$}}= 50\pm15$ ) shows that also with the formation of a black hole considerable mass ejection takes place.
author:
- 'E. P. J. van den Heuvel'
- 'Simon F. Portegies Zwart[^2]'
- Dipankar Bhattacharya
- Lex Kaper
date: 'Received; accepted: 04.05.2000'
title: 'On the origin of the difference between the runaway velocities of the OB-supergiant X-ray Binaries and the Be/X-ray Binaries'
---
\#1[[**\[\#1 – Simon\]**]{}]{} \#1[[**\[\#1 – Simon\]**]{}]{} \#1[[**\[\#1 – Ed\]**]{}]{} \#1[[**\[\#1 – Ed\]**]{}]{} \#1[[**\[\#1 – Dipankar\]**]{}]{} \#1[[**\[\#1 – Dipankar\]**]{}]{}
Introduction
============
A high-mass X-ray binary (HMXB) consists of a massive OB-type star and a compact X-ray source, a neutron star or a black hole. The X-ray source is powered by accretion of wind material, though in some systems mass transfer takes place through Roche-lobe overflow; the compact stars in the latter systems are surrounded by an accretion disk. Since wind accretion plays an important role, in practice only an OB supergiant or a Be-star companion have a strong enough stellar wind to result in observable X-ray emission. In a Be/X-ray binary the X-ray source is only observed when the neutron star moves through the dense Be-star disk at periatron passage. About 75% of the known HMXBs are Be/X-ray binaries, although this is a lower limit given their transient character.
Chevalier & Ilovaisky (1998) derived the proper motions for a sample of HMXBs from [*Hipparcos*]{} measurements. The four OB-supergiant HMXBs for which proper motions are available (0114+65, 0900-40 \[Vela X-1\], 1700-37 and Cyg X-1) have relatively large peculiar tangential velocities. Some corrections to the values given by these authors are needed (cf. Steele et al. 1998, Kaper et al. 1999). Taking these into account (Table 1) the mean peculiar velocity of these systems is $42
\pm 14$. It was already known that the OB-supergiant system of 1538-52 (QV Nor) has a peculiar radial velocity of about 90 with respect to its local standard of rest (Crampton et al. 1978; Gies & Bolton 1986; van Oijen 1989). For the 13 Be/X-ray binaries with measured proper motions Chevalier & Ilovaisky found peculiar tangential velocities ranging from ${\mbox{$v_{\rm tr}$}}= 3.3 \pm 0.7$ to $21 \pm
7.4$ , with an average of ${\mbox{$\langle v_{\rm tr} \rangle$}}= 11.4 \pm 6.6$. Again, after corrections (see Sect.2) and excluding the Oe systems X Per (0352+309) and V725 Tau (0535+262), one finds for the genuine Be/X-ray binary a slightly higher value of ${\mbox{$v_{\rm tr}$}}= 15 \pm 6$. We would like to point out here that these mean values are in good agreement with the runaway velocities of these two types of systems predicted on the basis of simple “conservative” evolutionary models (van den Heuvel 1983, 1985, 1994; Habets 1985; van den Heuvel & Rappaport 1987) and even better agreement is obtained when mass is not conserved in the transfer process (Portegies Zwart 2000). The effect of sudden mass loss during the supernova explosion is taken into account and in a massive binary this is the dominant contribution to the runaway velocity; a random kick velocity of a few hundred imparted to the neutron star at birth (see e.g. Hartman 1997) has only a small effect, as the kick’s impulse has to be distributed over the entire massive ($\apgt 15$) system. (See Portegies Zwart & van den Heuvel 1999, for arguments in favor of kicks). Therefore, in first-order approximation, these kicks can be neglected in calculating the runaway velocities of HMXBs, but [*not*]{} in calculating their orbital eccentricities (see Sects. 3.4 and 3.5).
The aim of the present paper is to give a quantitative assessment of the above-mentioned conjectures. It should be noted here that five Be-star systems in the Be/X-ray binary sample studied by Chevalier & Ilovaisky (1998) are of spectral type B4 Ve or later (masses $\leq 6
M{_\odot}$). The companions of these stars might be white dwarfs instead of neutron stars. Therefore, a supernova explosion is not necessarily the reason for their (excess) space velocity, which, in any case, is relatively small. It may be due to the typical random velocities observed in young stellar systems. Leaving these late-type Be/X-ray binaries out does not result in a significant change in the observed mean peculiar velocity of the Be-systems. Furthermore, there is some doubt concerning the use of the distances based on [*Hipparcos*]{} parallaxes of several of the other Be-systems, as these distances differ very much from the distances determined in other ways, e.g. by using reddening etc. (Steele et al. 1998). In Sect. 2 we therefore critically examine the distances and proper motions of all the systems with Be companions. In Sects. 3.1 and 3.2 we present an analytical calculation of the expected runaway velocities and orbital eccentricities of typical OB-supergiant and Be HMXBs, on the basis of the standard evolutionary models for these systems, adopting conservative mass transfer during phases of mass exchange, and including the effects of stellar-wind mass loss for the OB-supergiant systems. In Sect. 4 we discuss the effect of non-conservative mass transfer on the runaway velocity and in Sect. 5.1 for the Be/X-ray binaries with known orbital eccentricities. We calculate which kick velocities should be imparted to the neutron stars of Be/X-ray binaries in order to produce their, on average, large orbital eccentricities (since the mass-loss effects alone cannot produce these). In Sect. 5.2, as an alternative, we compare the observed runaway velocities and orbital eccentricities of the Be/X-ray binaries with those expected on the basis of symmetric mass ejection and show that without kicks their combination of high orbital eccentricities and low space velocities cannot be explained. Our conclusions are summarized in Sect. 6.
The observed peculiar tangential velocities of HMXBs
====================================================
The 4 OB-supergiant systems in the [*Hipparcos*]{} sample of Chevalier & Ilovaisky (1998) have distances larger than 1 kpc, which is too remote for a reliable parallax determination. For these systems they estimated the distances based on the spectral type, visual magnitude and reddening, and eventually the strength and velocity of interstellar absorption features, etc. After correcting for the peculiar solar motion and differential galactic rotation (see also Moffat et al. 1998) the [*Hipparcos*]{} proper motions result in the peculiar tangential velocities listed in Table 1. Chevalier & Ilovaisky give a mean peculiar tangential velocity of ${\mbox{$v_{\rm tr}$}}= 41.5 \pm
15$. We derive a similar value of $42 \pm 14$ .
For the Be-systems, Chevalier & Ilovaisky use the [*Hipparcos*]{} parallaxes to determine the distances. For some systems this leads to very surprising results. In particular, Steele et al. (1998) point out that for the system of 0236+610 (LSA + $61^{\circ} 303$) the [*Hipparcos*]{} parallax leads to a ten times smaller distance than the distance estimated from the spectral type and reddening. These authors convincingly show that for this system the distance estimate based on the [*Hipparcos*]{} parallax cannot be correct; the distance of the system must be of order 1.8 kpc instead of the 177 pc determined from the [*Hipparcos*]{} parallax. Similarly, from a variety of criteria they find that for A0535+262 the distance must be $>1.3$ kpc, instead of the 300 pc determined from the [*Hipparcos*]{} parallax. Steele et al. point out that for both systems the [*Hipparcos*]{} parallaxes are smaller than 3 times their probable (measurement) error, and are therefore not reliable. In such a case one cannot reliably use the [*Hipparcos*]{} parallax to determine the distance. With the [*Hipparcos*]{} distances the OB-star companions of 0236+610 and 0535+262 would become highly underluminous for their spectral types, and would be very peculiar stars, as was already noticed by Chevalier & Ilovaisky (1998). On the other hand, using alternative distance criteria, their absolute luminosities become perfectly normal for their spectral types. This gives confidence that the latter distances are more reliable.
The systems including a Be star with spectral type later than B4 V (mass $\leq 6 M_{\odot}$) may well have white dwarfs instead of neutron stars as companions (Portegies Zwart 1995). Therefore, their space velocity is not necessarily caused by a supernova explosion, which is the scenario we exploit in this paper. Excluding these systems, the observed mean peculiar velocity hardly changes ($\langle
{\mbox{$v_{\rm tr}$}}\rangle = 14.4 \pm 6.6$ in stead of $15 \pm 6$, excluding X Per and V725 Tau), and since the nature of their compact companions is not known anyway (e.g. no X-ray pulsations observed which would identify the compact star as a neutron star), we decided to leave them in the calculation of the mean peculiar velocity. The peculiar tangential velocities of X Per (0352+309, O9 III-IVe, 27 ) and V725 Tau (0535+262, O9.7 IIe, 97 ) are relatively high; their early spectral types suggest that they have masses comparable to those of the OB supergiants, so that, like the OB-supergiant systems, they would also originate from relatively massive binary systems. In Table 1 we list the peculiar tangential velocity for each individual system and calculate the average for different subsamples. We left out the Be star $\gamma$ Cas, because its X-ray binary nature is not clear; furthermore, its X-ray spectrum is consistent with that of a white dwarf (Haberl 1995).
For the systems 0236+610, 0535+262, 1036-565 and 1145-619 the [*Hipparcos*]{} parallaxes yield absolute visual magnitudes very different from those expected on the basis of the OB-spectral types of the stars. In these cases, the [*Hipparcos*]{} parallax measurements are less than three times their probable errors and thus not reliable. For these stars we therefore used the distances determined from spectral type and reddening, which yield absolute visual magnitudes consistent with their spectral types.
We rederived the peculiar tangential velocities relative to the local restframe from the [*Hipparcos*]{} proper motions (cf. Kaper et al. 1999). Table 1 lists the peculiar tangential velocity corrected for the peculiar solar motion and differential galactic rotation for three different distances ($d/1.4$, $d$, $1.4d$, following Gies & Bolton 1986). The uncertainty in distance (and thus in peculiar motion) is difficult to estimate; therefore, we calculated the space velocity for different values of the distance. The peculiar tangential velocities for the HMXBs discussed in Clark & Dolan (1999) are identical to ours for the OB-supergiant systems, though they find different values for the Be/X-ray binaries X Per ($15 \pm 3$ , $d=700$ pc), V725 Tau ($57 \pm 14$ , $d=2$ kpc), and 1145-619 ($17
\pm 7$ , $d=510$ pc). Obviously, the precise values for the peculiar motion depend on the adopted model for the galactic rotation; we used the formalism employed in Comerón et al. (1998). For the OB-supergiant systems in the sample of Chevalier & Ilovaisky (1998) also the radial velocities are available from literature. This is not the case for the Be/X-ray binary systems. Therefore, we only consider the two components of the tangential velocity for the comparison of the kinematic properties of the two groups. The table shows that, leaving the two O-emission systems out, the Be/X-ray binaries have low space velocities: $15 \pm 6$ .
-------------- --------------------- --------------- -------------- --------- ------ --------
X-ray source Name Spectral type Distance $d$
\[kpc\] $d/1.4$ $d$ $1.4d$
0114+650 V662Cas B0.5Ib 3.8 14.2 26.4 43.6
0900-403 GP Vel B0.5Ib 1.8 22.7 34.0 50.0
1700-377 V884Sco O6.5If+ 1.7 46.0 58.0 74.8
1956+350 V1357Cyg O9.7Iab 2.5 35.3 47.4 64.5
0236+610 LSI+$61^{\circ}$303 B0IIIe 1.8 2.6 9.0 19.4
0352+309 X Per O9III-IVe 0.8 23.6 27.3 33.4
0521+373 HD34921 B0IVpe 1.05 18.4 23.4 32.1
0535+262 V725 Tau O9.7IIe 2.0 73.2 96.5 129.1
0739-529 HD63666 B7IV-Ve 0.52 9.7 12.2 16.9
0749-600 HD65663 B8IIIe 0.4 6.9 9.9 16.0
1036-565 HD91188 B4IIIe 0.5 18.3 20.4 23.4
1145-619 V801 Cen B1Ve 1.1 7.9 5.8 6.1
1249-637 BZ Cru B0IIIe 0.3 12.9 13.6 16.3
1253-761 HD109857 B7Ve 0.24 17.6 23.7 33.8
1255-567 $\mu^{2}$ Cru B5Ve 0.11 12.5 13.5 16.9
-------------- --------------------- --------------- -------------- --------- ------ --------
Runaway velocities expected on the basis of models with conservative mass transfer and symmetric mass ejection.
===============================================================================================================
Change of orbital period due to mass transfer
---------------------------------------------
We only consider here so-called case B mass transfer since for the evolution of massive close binaries this is the dominant mode of mass transfer (cf. Paczynski 1971; van den Heuvel 1994, but see Wellstein & Langer 1999). In case B the mass transfer starts after the primary has terminated core-hydrogen burning, and before core-helium ignition. After the mass transfer in this case the remnant of the primary star is its helium core, while its entire hydrogen-rich envelope has been transferred to the secondary, which due to this became the more massive component of the system. There is a simple relation between the mass of the helium core and that of its progenitor (see for example van der Linden 1982; Iben & Tutukov 1985). We adopt here the relation given by Iben & Tutukov (1985):
$${\mbox{$M_{\rm He}$}}= 0.058 {\mbox{$M_\circ$}}^{1.57},
\label{Eq:1}$$
which results in a fractional helium core mass $p$ given by: $$p = {\mbox{$M_{\rm He}$}}/{\mbox{$M_\circ$}}= 0.058{\mbox{$M_\circ$}}^{0.57}.
\label{Eq:2}$$
The change in orbital period of the system in case of conservative mass transfer (i.e.: conservation of total system mass and orbital angular momentum $J$) and initially circular orbits is (Paczynski 1971; van den Heuvel 1994): $${{\mbox{$P_f$}}\over {\mbox{$P_\circ$}}} = \left( {{\mbox{$M_\circ$}}{\mbox{$m_\circ$}}\over {\mbox{$M_f$}}{\mbox{$m_f$}}}
\right)^3,
\label{Eq:3}$$ where , and ${\mbox{$m_\circ$}}= {\mbox{$M_{\rm tot}$}}- {\mbox{$M_\circ$}}$ denote the orbital period and component masses before the mass transfer, and , and ${\mbox{$m_f$}}= {\mbox{$M_{\rm tot}$}}-{\mbox{$M_f$}}$ are the orbital period and component masses after the transfer. The transformation between orbital separation and the orbital period is given by Kepler’s third law.
Introducing the initial mass ratio ${\mbox{$q_\circ$}}= {\mbox{$m_\circ$}}/{\mbox{$M_\circ$}}$ and using equation \[Eq:2\], equation \[Eq:3\] can be written as $${{\mbox{$P_f$}}\over {\mbox{$P_\circ$}}} = \left( \frac {{\mbox{$q_\circ$}}} {p(q_{o}+1-p)} \right)^3.
\label{Eq:4}$$
Since according to equation \[Eq:2\], $p$ increases for increasing stellar mass, one observes that, due to the third power in equation \[Eq:4\], for the same the orbital period of a very massive system increases much less as a result of the mass transfer, than for systems of lower mass. (The term $(q_{o} +1-p)$ changes much less than $p$ itself for increasing stellar mass, so for a given $q_{o}$ this term has only a modest effect.) This is the main reason for the systematically longer orbital periods of the Be/X-ray binaries (always $> 16$days) relative to those of the OB-supergiant HMXBs (in all but one case: between 1.4 days and 11 days, cf. van den Heuvel, 1983, 1985, 1994). This is illustrated in Table\[Tab:model\] where we list the relative post-mass-transfer periods ${\mbox{$P_f$}}/{\mbox{$P_\circ$}}$ for typical Be/X-ray binary progenitor systems, with ${\mbox{$M_\circ$}}= 10$ and 12, respectively, and for two typical OB-supergiant HMXB progenitors with ${\mbox{$M_\circ$}}= 25$ and 35, respectively, for values ranging from 0.4 through 0.8.
Possible effects of further mass transfer and stellar winds on the orbits
-------------------------------------------------------------------------
### Case BB mass transfer
The helium cores left by the 10 and the 12 stars have masses of 2.15 and 2.87, respectively. During helium-shell burning, when these stars have CO-cores, their outer layers may expand to dimensions of a few to several tens of solar radii, and a second, so-called Case BB, mass transfer may ensue before their cores collapse to neutron stars (Habets 1985,1986ab). However, since the radius of the 2.87 helium star will not exceed 5 $R_{\odot}$, a second mass transfer phase is unlikely to occur here. In the case of the 2.15 helium star, which does attain a large radius, the amount of mass that is in the extended envelope is not more than 0.1. For these reasons, we will neglect here the effects of Case BB mass transfer, and will assume that these helium stars do not lose any mass before their final supernova explosion. This means, that we will slightly overestimate the imparted runaway velocities (as the orbits at the time of the explosion will be slightly wider than we assume, and the ejected amounts of mass will be somewhat smaller than we assume).
### Stellar-wind mass loss in massive stars
Since wind mass-loss rates from Wolf-Rayet (WR) stars –massive helium stars– are much larger (viz.: $\apgt 10^{-5}$) than the mass-loss rates of lower-mass main-sequence stars, for the sake of argument (in order to include only the largest effects) we take into account the effects of the wind mass loss during the WR phase. The effects of these winds are: (1) to widen the orbits, and (2) to considerably decrease the mass of the helium ($\equiv$ WR) star before its core collapses. In the cases of the 25 and 35 primary stars, the masses of the helium cores are 9.1and 15.4, respectively. Such stars live $9 \times 10^5$ years and $7.5 \times 10^5$ years, respectively, and are expected to lose about 4.0 and 7.4 through their wind during this phase of their evolution, respectively[^3] Thus, at the moment of the supernova explosion the collapsing cores of these stars will have masses of 5 and 8.0, respectively. To keep the same notation we will express the relative mass loss in the stellar wind with $\delta = \Delta M_{\rm wind} / {\mbox{$M_\circ$}}$. The value for $\delta$ is 0.16 for a primary with a mass of 25 and 0.21 for a 35 primary star. In the cases of no wind mass loss (in lower mass primaries): $\delta = 0$.
The wind mass loss will change the post-mass-transfer orbits as follows (van den Heuvel 1994): $$d \log a = - d \log {\mbox{$M_{\rm tot}$}},
\label{Eq:5}$$ and $$d \log P = - 2d \log {\mbox{$M_{\rm tot}$}},
\label{Eq:6}$$
where $a$ is the orbital separation and ${\mbox{$M_{\rm tot}$}}$ the total system mass.
Eq. (\[Eq:6\]) results in: $$P/P' = \left( {{\mbox{$M'_{\rm tot}$}}\over {\mbox{$M_{\rm tot}$}}} \right)^2
\, ,
\label{Eq:7}$$ where $P$ and $P'$ correspond to the system masses ${\mbox{$M_{\rm tot}$}}$ and , respectively. ${\mbox{$M_{\rm tot}$}}$ is the total mass at the beginning of the WR phase and the total system mass at the end of this phase, just prior to the supernova explosion of the WR Star. The orbital separation after mass transfer and additional WR mass loss phase is expressed as: $${a' \over {\mbox{$a_\circ$}}} = {{\mbox{$a_f$}}\over {\mbox{$a_\circ$}}} {a' \over {\mbox{$a_f$}}}
\equiv \left( {{\mbox{$q_\circ$}}\over p(1+{\mbox{$q_\circ$}}-p)} \right)^2
\left(1 - {\delta \over 1+{\mbox{$q_\circ$}}} \right)^{-1}
\label{Eq:8}$$
Runaway velocities induced by symmetric supernova mass ejection
---------------------------------------------------------------
The runaway velocity imparted to the system by the supernova mass loss is calculated from the loss of momentum of the system during the explosion: $-{\mbox{$V_{{\rm orb}, 1}$}}{\mbox{$\Delta M_{\rm sn}$}}$, where is the orbital velocity of the helium star prior to the explosion and is the amount of mass ejected in the supernova.
We assume all compact remnants to be a neutron star. Then $\Delta M_{\rm sn}$ is given by $\Delta M_{\rm sn} = (p-\delta) M_{o} -1.4
M_{\odot}$. The remaining mass of the system is: $${\mbox{$M''_{\rm tot}$}}= {\mbox{$m_f$}}+ 1.4{\mbox{${\rm M}_\odot$}}= (q_{o} +1-p) M_{o} + 1.4 M_{\odot}.
\label{Eq:10}$$
This yields a recoil velocity (or runaway velocity) of the system of: $$\begin{aligned}
{\mbox{$V_{\rm rec}$}}& = & V_{\rm orb,1}
{{\mbox{$\Delta M_{\rm sn}$}}\over (q_{o}+1-p) M_{o} + 1.4 M_{\odot}}
\label{Eq:11}\end{aligned}$$ Here the second term in the right argument is simply the post supernova eccentricity and we may write simply $${\mbox{$V_{\rm rec}$}}= e V_{\rm orb,1}
\label{Eq:11b}$$
The relative orbital velocity before the explosion is $\sqrt{G
{\mbox{$M'_{\rm tot}$}}/a'}$. One therefore has: $$V_{\rm orb,1} = \left( {GM_{o} \over a_{o}} \right)^{1/2}
{ p(1+q_{o}-p)^{2} \over q_{o}(1+q_{o})^{1/2}}
\label{Eq:12}$$
Substitution of Eq. (11) into Eq. (10) results now in $$\begin{aligned}
{\mbox{\em V}}_{\rm rec} &=& \left({GM_{o} \over a_{o}}\right)^{1/2} \nonumber \\
&\times& {p(1+q_{o}-p)^{2} \over q_{o}(1+q_{o})^{1/2}} \;
{(p-\delta - 1.4M_{\odot}/M_{o}) \over (q_{o} +1 -p+1.4
M_{\odot}/M_{o})}
\label{Eq:14}\end{aligned}$$ which in numerical form becomes:
$$\begin{aligned}
V_{\rm rec} &=& 212.9 [{\mbox{${\rm km~s}^{-1}$}}] \left( {M_{o} \over [M_{\odot}]}
{[{\rm days}] \over P_{o}}
\right)^{1/3} \nonumber \\
&\times& {p(q_{o}+1-p)^{2}
(p- \delta - 1.4M_{\odot}/M_{o})
(1+q_{o}-\delta) \over q_{o}(1+q_{o}
)^{2/3} (q_{o} +1-p+1.4M_{\odot}/M_{o}) }
\label{Eq:15}\end{aligned}$$
Fig. 1 shows for ${\mbox{$P_\circ$}}= 5$days the values of as a function of for the four primary masses of Table 2, using the and as given above. The figure shows that for the “Be-systems” (initial primary masses 10 and 12 yielding Be-star masses ranging from 11.85 to 18.7) the expected recoil velocities range from 5 to 21 , whereas for the “OB-supergiant systems” (with OB-companions between 25 and 40) they range between 21 and $> 80$, respectively. These velocities correspond to transverse velocities that are $\pi/4$ times these values, i.e.: 3.9 to 17 for the Be-systems, and 16.5 to $>71$ for the OB-supergiant systems with neutron stars. Thus one expects average transverse velocities of order 10.5 and 45 for the Be/X-ray binaries and OB-supergiant systems, respectively. For both the Be/X-ray binaries and the OB-supergiant systems Be/X-ray the predicted and observed mean transverse runaway velocities agree well: $15 \pm 6$ and $42 \pm 14$, respectively.
As Eq. (\[Eq:15\]) shows, the dependence of the recoil velocity on is rather weak, so for initial orbital periods between a few days and 10days these results don’t change by more than a factor 1.5. Therefore, certainly qualitatively, Fig. 1 is representative for the two types of systems. Eq. (\[Eq:15\]) further shows that the large difference in runaway velocity between the two types of systems is due to a combination of two factors, as follows: (1) the larger fractional helium core masses ($p$) in the more massive systems, which cause their pre-supernova orbital periods to be shorter and thus their pre-supernova orbital velocities to be larger than those of the lower-mass systems; and (2) the much lower amounts of mass ejected ($\Delta M_{\rm sn}$) in the lower mass systems compared to the systems of higher mass, which leads to a lower recoil effect.
Relaxing the assumption that mass is conserved during the phase of mass transfer changes little, which we will discuss now.
---- ----- ------- ------ ------- ------- ------- ------ -------
10 0.8 12.93 15.8 5.33 0.022 0.045 14.2 0.045
0.6 8.18 13.8 6.64 0.027 0.051 9.1 0.050
0.4 3.86 11.8 9.18 0.036 0.060 4.4 0.060
0.3 2.13 10.8 11.64 0.046 0.065 2.4 0.065
12 0.8 9.84 18.7 10.20 0.040 0.073 11.4 0.073
0.6 6.26 16.3 12.70 0.048 0.083 7.4 0.083
0.4 2.99 13.9 17.55 0.065 0.096 3.7 0.096
0.3 1.65 10.7 22.25 0.082 0.104 2.0 0.104
25 0.8 3.61 35.9 22.70 0.069 0.098 5.3 0.098
0.6 2.38 30.9 28.01 0.083 0.114 3.8 0.115
0.4 1.20 25.9 38.23 0.111 0.135 2.0 0.136
0.3 0.69 23.4 48.07 0.140 0.148 1.2 0.149
35 0.8 2.39 47.6 38.00 0.112 0.136 4.1 0.137
0.6 1.62 40.6 46.52 0.139 0.158 2.9 0.160
0.4 0.85 33.6 62.76 0.201 0.190 1.8 0.193
0.3 0.50 30.1 78.29 0.250 0.211 1.2 0.213
---- ----- ------- ------ ------- ------- ------- ------ -------
\[Tab:model\]
The effects of non-conservative mass transfer {#non_conservative}
=============================================
In the above it was assumed that the case B mass transfer was conservative in all systems. For the Be-systems this “conservative” assumption seems confirmed quite straightforwardly as the Be nature is interpreted by the accretion of angular momentum and thus of mass. On the other hand, for the OB-supergiant X-ray binaries several authors (starting with Flannery and Ulrich 1977 for the Cen X-3 system) have pointed out that certainly in part of the systems the mass transfer has been non-conservative, and there is a considerable evolution for massive close binaries altogether (De Loore & De Greeve 1992). Indeed, close Wolf-Rayet binaries with high mass ratios $q =
M_{\rm WR}/M_{OB}$ such as CQ Cep ($P=1.64$ days, $q=1.19$) and CX Cep ($P=2.22$ days, $q=.44$) cannot have been produced by conservative evolution, and just these systems are the progenitors of the OB-supergiant X-ray binaries (cf. van den Heuvel 1994).
The amount of mass lost from the system during the transfer will depend on the initial mass ratio of the system. For small the companion will accrete little and most of the envelope mass of the primary will be lost from the system. On the other hand, for large little mass will be lost from the system. Therefore, in order to study the effect of mass and angular momentum loss on the runaway velocity, we assume as a first approximation that the fraction $f$ of the primary’s envelope which is accreted by the companion star is proportional to the initial mass ratio (Portegies Zwart 1995) $$f = {\mbox{$q_\circ$}}.$$ After mass transfer the secondary mass then becomes $${\mbox{$m_f$}}= {\mbox{$M_\circ$}}+ f {\mbox{$M_\circ$}}(1-p)
\equiv {\mbox{$M_\circ$}}{\mbox{$q_\circ$}}(2-p).$$
The gas lost by the donor leaves with low velocity but gains angular momentum via the interaction with the companion star. It finally leaves the binary system via the second Lagrangian point $L_2$, carrying specific angular momentum with it (de Loore & De Greve 1992). The specific angular momentum of this lost matter is considerably larger than what is lost in the specific amount of angular momentum in the stellar wind (given by Eq.\[Eq:5\]), see for example Soberman et al. (1997).
We assume that the mass that leaves the system carries a fraction $\beta$ of the specific angular momentum of the binary. We can then write the change in orbital separation due to mass transfer as $${a' \over {\mbox{$a_\circ$}}} =
\left( {{\mbox{$M_f$}}{\mbox{$m_f$}}\over {\mbox{$M_\circ$}}{\mbox{$m_\circ$}}} \right)^{-2}
\left( {{\mbox{$M_f$}}+{\mbox{$m_f$}}\over {\mbox{$M_\circ$}}+{\mbox{$m_\circ$}}} \right)^{2\beta +1}.
\label{Eq:12b}$$ and use it as an alternative for Eq. (\[Eq:7\]). Following Portegies Zwart (1995) we use $\beta = 3$.
Eq.(\[Eq:8\]) then becomes: $${a' \over {\mbox{$a_\circ$}}} = \left( {1 \over {\mbox{$q_\circ$}}p(2-p)} \right)^2
\left( {1+{\mbox{$q_\circ$}}\over p + {\mbox{$q_\circ$}}(2-p)}
\right)^{-2\beta -1}
\left(1 - {\delta \over 1+{\mbox{$q_\circ$}}} \right)^{-1}
\label{Eq:12c}$$
The result of this calculation is presented as the dotted lines in Fig. 1. The small number near each $\circ$ indicates the mass of the visible component, which is smaller than if mass transfer would proceed conservatively. One observes that for the same mass of the visible component of the binary, the runaway velocity of the OB-system is between 50 and 100 per-cent larger than in the conservative case. The higher velocity of the binary is mainly caused by the smaller orbital separation at the moment of the supernova. We thus see, from this simple numerical experiment, that non-conservative mass transfer makes the difference in runaway velocity between the two types of high mass X-ray binaries considerably larger.
Predicted and observed orbital eccentricities of the Be/X-ray binaries: evidence for kicks
==========================================================================================
Orbital eccentricities of Be/X-ray binaries in case of symmetric ejection
-------------------------------------------------------------------------
In the case of spherically symmetric mass ejection the orbital eccentricity induced by the mass loss is (cf. Hills 1983): $$e = \frac{{\mbox{$\Delta M_{\rm sn}$}}}{{\mbox{$M'_{\rm tot}$}}- {\mbox{$\Delta M_{\rm sn}$}}}
\equiv \frac{{\mbox{$\Delta M_{\rm sn}$}}}{{\mbox{$M''_{\rm tot}$}}}.
\label{Eq:13}$$ One expects that because of the extensive mass transfer and the fact that before the mass transfer the primary was a (sub)giant, the orbits just prior to the explosions are circular. Hence, in case of spherically symmetric mass ejection, one expects the eccentricities of the Be/X-ray binaries simply to be given by Eq. (19).
Table 3 shows that for the Be/X-ray binaries resulting from systems with an initial primary mass of 10, the orbital eccentricities expected on the basis of Eq. (\[Eq:13\]) range from 0.045 to 0.060. For systems resulting from binaries with primaries of 12, the eccentricities range from 0.073 to 0.096. It should be noted that these are, in fact, overestimates, since we ignored case BB mass transfer, which would still have somewhat reduced these values.
Orbital eccentricities are known for only five Be/X-ray binaries, as is listed in Table 3. They range from 0.3 to $> 0.7$, with an average of about 0.5. For the two long-period systems, 1145-619 and 1258-613, the orbital eccentricities have not yet been measured, but a rough estimate of their values can be made as follows. Both systems are recurrent transients, with outbursts occurring once per orbit, when the Be star is active, presumably when the stars are near periastron. The same is true for the systems 0115+63, V0331+53 and EX02030+375 when their Be-stars are in an active phase. For the latter systems one calculates from their orbital periods and eccentricities that within 20 percent their periastron distances are the same. Apparently, this is the periastron distance required for triggering an outburst when the Be-star is in an active phase. It thus seems reasonable to assume that the same is true for 1145-619 and 1258-613. Using this, one finds the latter systems to have eccentricities of between 0.75 and 0.83, and between 0.70 and 0.80, respectively. To be conservative, we have indicated this in Table 3 as: $e \geq 0.70$.
Two more systems consisting of a B-star and a neutron star are known: the binary radio pulsars PSRJ 1259-63 and PSRJ 0045-7319. These have very eccentric orbits as indicated in Table 3. So, in total we have nine B-star plus neutron star systems with measured or estimated orbital eccentricities. References to the orbital parameters of these systems are indicated in the table.
Observations show that all binaries –including detached ones– with orbital periods shorter than 10 days have circular orbits, whereas detached systems with longer orbital periods do not. This suggests that in systems with orbital periods shorter than 10 days tidal forces are effective in circularizing the orbits on a timescale considerably shorter than the lifetimes of the components of the binary, whereas in wider systems they apparently are not. Since the Be/X-ray binaries are detached systems (cf. van den Heuvel & Rappaport 1987; van den Heuvel 1994) and have orbital periods longer than 16 days, it is not surprising that their orbits have not yet been circularized.
The lifetime of a Be/X-ray binary is expected to be of the order of a few million years up to about 10Myr, the lifetime of the Be companion of the neutron star. The timescale for tidal circularization for main-sequence binaries with orbital periods $> 16$ days is at least a few tens of Myr (see Zahn 1977, Kochanek 1992). Therefore it is unlikely to catch the binary in the circularization process. Therefore, we expect that the eccentricities for the Be/X-ray binaries in Table 3 are still close to those just after the supernova explosion. The orbits of the high-mass X-ray binaries with orbital periods $< 10$ days are all practically circularized by tidal effects.
It should be noted that if the eccentricities of the Be-systems had resulted from spherically symmetric supernova mass ejection, the amounts of mass ejected in their supernovae should have been very large, of order 4 to over 7 solar masses (see for example Iben & Tutukov 1998). Since in the case of symmetric mass ejection the orbital eccentricity and runaway velocity are directly proportional to each other (see Eqs. \[10\] and \[14\]), also the induced runaway velocities should have been much larger than observed. For example, induction of an eccentricity 0.5 with a symmetric explosion requires 1/3 of the system mass to be ejected in the explosion. With a Be star of 12, as is representative for a typical B0.5 Ve star, and a neutron star mass of 1.4, the initial system mass in this case must have been 20.1, implying an ejected amount of mass of 6.7. In order to obtain a post-supernova orbital period of about 30days, as is typical for many Be/X-ray binaries, the initial orbital period in this example must have been around 11days. With this initial period, and 6.7 explosively ejected, the induced runaway velocity would be $\simeq 87$(see the equations in Sect. 3.3), which is advariant with the observed velocities.
Similarly, if the induced eccentricity would be 0.3, one finds that for the same final system mass and orbital period, the runaway velocity induced by the explosion would have been about 45.
As these velocities are some 5, respectively 2.5 times larger than the mean excess space velocity of 19 \[$(4/\pi) \times 15\,{\mbox{${\rm km~s}^{-1}$}}$\] of the Be/X-ray binaries, it is clear that the orbital eccentricities of the Be/X-ray binaries cannot be due purely to symmetric mass ejection in the supernova explosion.
The only way to obtain both a low runaway velocity of the system and the high orbital eccentricities listed in Table 3 is by having a small amount of mass ejected in the supernova, in combination with a velocity kick of order 60 to 250 imparted to the neutron star at birth. We describe below how these required kick velocities were calculated. The randomly directed kick hardly changes the runaway velocity of the system, as the impulse of the kick imparted to the neutron star is shared by the entire system (with a mass of order 15 solar masses in the case of the Be/X-ray binaries), and thus the kick velocity is “diluted” to an extra velocity of the system of only 4 to 16, in a random direction. Adding this velocity quadratically (because of its random direction) to the velocity of between 5 and 21 imparted to the systems purely by the mass loss (Fig. 1), one obtains mean runaway velocities of between 6 and 21 for a 60 kick and between 17 and 26 for a 250 kick.
These values are in good agreement with the observed mean excess space velocities of Be/X-ray binaries of $19 \pm 8$($\pi/4$ times their average peculiar tangential velocities).
We calculated the minimum kick velocities that have to be imparted to the neutron star during the supernova explosion in order to obtain the presently observed orbital eccentricities of the Be-systems in Table 3. We used the equations derived by Wijers et al. (1992). The minimum required kick velocitiy is the one that is imparted in the orbital plane in the direction of motion of the pre-supernova star (assuming the initial orbit was circular). We assumed in these calculations that the B stars have a mass of $15\,{\mbox{${\rm M}_\odot$}}$, as corresponds to a B0-1 main-sequence star, and that the neutron star has a mass of $1.4\,{\mbox{${\rm M}_\odot$}}$. (For B-star masses in the range $10-20\,{\mbox{${\rm M}_\odot$}}$ the required minimum runaway velocities do not differ by more than $\pm$ 10 per cent from the values for $15{\mbox{${\rm M}_\odot$}}$). The table shows that the required minimum kick velocities range from about 50 to about 200 . Assuming the real kick velocities to be randomly distributed, the required kicks become $\sqrt{3/2}$ times larger, and range from about 60 to about 250 .
We conclude from the above that the combination of low mean space velocity of the Be/X-ray binaries and large mean orbital eccentricity provides unequivocal evidence for the existence of velocity kicks imparted to neutron stars at their birth.
An alternative way to approach the problem of the orbital eccentricities is to calculate, from the measured mean runaway velocities of Be/X-ray systems, what orbital eccentricity these systems should have had, were this runaway velocity imparted by purely symmetric mass ejection. This is the topic of the next section.
[lrcrrccrr]{} & & & & & & &\
& & & & & & & &\
& & & & & & & &\
X0115+634 & 24.3 & 0.34 &60.1 & 182.8 & 0.06 & 0.11 & 66 & (1)\
X0331$+$53 & 34.3 & 0.31 &53.5 & 162.7 & 0.06 & 0.12 & 53 & (1,3)\
X0535$+$26 & 111.0 & $\geq 0.4$ &42.8 & 108.4 & 0.09 & 0.18 & 47 & (1)\
X0535$-$67 & 16.7 & $\geq 0.7$ &184 & 207 & 0.05 & 0.10 & 203 & (1)\
X1145$-$619 & 187.5 & $\geq 0.7\footnote{Estimated in the text}$ &82.3 & 92.4 & 0.11 & 0.21 & 90.5 & (1)\
X1258$-$613 & 133 & $\geq 0.7^{c}$ &92.7 & 103.9 & 0.10 & 0.19 & 102.0 & (2,3)\
EXO2030$+$375 & 46 & 0.38 &55.1 & 147.6 & 0.07 & 0.13 & 60.6 & (4,5)\
PSRJ1259$-$63 & 1236.8 & 0.87 &79.1 & 49.3 & 0.20 & 0.39 & 87 & (6)\
PSRJ0045$-$7319 & 51.17 & 0.81 &179 & 142.5 & 0.07 & 0.14 & 195.4 & (7,8,9)\
\[Tab:observed\_v\]
Predicted relation between orbital eccentricity and runaway velocity expected in case of symmetric explosions - comparison with observations
--------------------------------------------------------------------------------------------------------------------------------------------
Eq. (14) yields: $${e \over 1 + e} = {{\mbox{$\Delta M_{\rm sn}$}}\over {\mbox{$M'_{\rm tot}$}}},
\label{Eq:17}$$ Combination of Eqs (\[Eq:11\]) and (\[Eq:17\]) yields: $${\mbox{$V_{\rm rec}$}}= \sqrt{ {G{\mbox{$M''_{\rm tot}$}}\over a'} } \frac{{\mbox{$m_f$}}}{{\mbox{$M''_{\rm tot}$}}}
\frac{e}{1+e},
\label{Eq:18}$$ where $a'$ is the pre-supernova orbital radius. The semi-major axis after the supernova $a''$ follows from: $${a' \over a''} = 1 - {{\mbox{$\Delta M_{\rm sn}$}}\over {\mbox{$M''_{\rm tot}$}}},$$ and by writing $${\mbox{$M'_{\rm tot}$}}= {\mbox{$M''_{\rm tot}$}}+{\mbox{$\Delta M_{\rm sn}$}}= {\mbox{$M''_{\rm tot}$}}(1 + {{\mbox{$\Delta M_{\rm sn}$}}\over {\mbox{$M''_{\rm tot}$}}}).
\label{Eq:20}$$ one obtains after insertion of Eq. (18) in Eq. (16): $${\mbox{$V_{\rm rec}$}}^2 = \frac{G{\mbox{$M''_{\rm tot}$}}}{a''}
\frac{1+{\mbox{$\Delta M_{\rm sn}$}}/{\mbox{$M''_{\rm tot}$}}}{1-{\mbox{$\Delta M_{\rm sn}$}}/{\mbox{$M''_{\rm tot}$}}}
\left(\frac{{\mbox{$m_f$}}}{{\mbox{$M''_{\rm tot}$}}} \right)^2
\left(\frac{e}{1+e} \right)^2.
\label{Eq:21}$$
Defining now the presently observed mean orbital velocity by $$\langle {\mbox{$V_{\rm orb}$}}\rangle^2 = {G{\mbox{$M''_{\rm tot}$}}\over a''},
\label{Eq:22}$$ and substituting ${\mbox{$\Delta M_{\rm sn}$}}/{\mbox{$M''_{\rm tot}$}}$ from Eq. (\[Eq:13\]) one obtains: $${{\mbox{$V_{\rm rec}$}}\over \langle {\mbox{$V_{\rm orb}$}}\rangle}
{{\mbox{$M''_{\rm tot}$}}\over {\mbox{$m_f$}}} = \frac{e}{(1-e^2)^{1/2}}.
\label{Eq:23}$$ This defines, in the case of symmetric supernova-mass ejection the relation that is expected to be found between the observed system runaway velocity and the observed orbital eccentricity $e$, for a system with a Be/X-ray star of mass , and observed mean orbital velocity $\langle {\mbox{$V_{\rm orb}$}}\rangle$. Since ${\mbox{$M''_{\rm tot}$}}= {\mbox{$m_f$}}+1.4$, and since in general ${\mbox{$m_f$}}>10$, the quantity ${\mbox{$M''_{\rm tot}$}}/{\mbox{$m_f$}}$ is close to unity. Defining: $$f_v \equiv \frac{{\mbox{$V_{\rm rec}$}}}{{\mbox{$V_{\rm orb}$}}} \, {{\mbox{$M''_{\rm tot}$}}\over {\mbox{$m_f$}}} = \frac {e}{(1-e^{2})^{1/2}},
\label{Eq:24}$$ one obtains a simple relation between $f_v$ and $e$, the plotted curve in Fig. 2. In the case of symmetric supernova mass ejection, the observed value of $f_v$ of a Be/X-ray binary should be related to the observed orbital eccentricity according to this curve, which shows that large eccentricities correspond to large runaway velocities.
In Fig. 2 we also plotted the values of $f_{v}$ and $e$ for the nine systems with observed orbital periods and eccentricities (see Table 3), taking runaway velocities $V_{\rm rec}$ in the observed range $19 \pm 8$ for the Be X-ray binaries. We assumed a Be-star mass of $15{\mbox{${\rm M}_\odot$}}$. The figure shows that all systems fall far below the curve expected for symmetric supernova mass ejection. This again shows that the combination of low runaway velocities and large orbital eccentricities as observed in the Be/X-ray binaries cannot be obtained by symmetric mass ejection in the supernovae, and that a velocity kick imparted to the neutron stars at birth is absolutely required.
Conclusions
===========
The measured tangential velocities of the Be/X-ray binaries and OB-supergiant X-ray binaries by the [*Hipparcos*]{} satellite confirm the expectations from the evolution of massive close binaries in which little mass is lost from the binary systems during the first mass transfer phase. The much higher tangential velocities of supergiant X-ray binaries than those of the Be-systems follow from a combination of (1) the much larger fractional helium core masses in the progenitors of the OB-supergiant systems which cause their pre-supernova orbital periods to be shorter, and thus their pre-supernova orbital velocities to be much larger than those of the less massive Be-systems, and (2) the much lower amounts of mass ejected during the supernova explosion in the lower-mass Be-systems compared to the OB-supergiant systems.
The combination of a high orbital eccentricity with a low space velocity observed for the Be type X-ray binaries can only be understood if a kick with appreciable velocity –in the range 60 to 250 – is imparted to the newly born neutron star. Such a kick tends to only slightly affect the space velocity of the binary system since the neutron star has to drag along its massive companion. The orbital eccentricity, however, is strongly affected by such a asymmetric velocity kick. If the supernova explosions in these systems had been symmetric, the high orbital eccentricities observed in the class of Be X-ray binaries are impossible to reconcile with their on average low runaway velocities.
Chevalier C. Ilovaisky, S., A. 1998, A&A 330, 201 Clark, L.L., Dolan, J.F. 1999, A&A 350, 1085 Comerón, F., Torra, J., Gómez, A.E. 1998 A&A 330, 975 Corbet, R.H.D., Smale, A.P., Menzies, J.W., Branduardi Raymont, G., Charles, P.A., Mason, K.O., Booth, L. 1986, MNRAS 221, 961 Crampton, D., Hutchings, J.B., Cowley, A.P. 1978 ApJ 225, L63 A&A 142, 367 de Loore, C., De Greve, J. P. 1992, A&ASS 94, 453 Ergma, E., van den Heuvel, E. P. J. 1998 A&A 331, L29 Flannery, B. P., Ulrich, R. K., 1977, ApJ 212, 533 Gies, D.R., Bolton, C.T. 1986, ApJS 61, 419 Haberl, F. 1995, A&A 296, 685 Habets, G. M. H. J. 1985, PhD Thesis, U. Amsterdam Habets, G. M. H. J. 1986a, A&A 165, 95 Habets, G. M. H. J. 1986b, A&A 167, 61 Hartman, J. W., 1997, A&A 322, 127 Hills, J. 1983, ApJ 267, 322 Iben I. J., Tutukov, A. V. 1985, ApJS 58, 661 Iben I. J., Tutukov, A. V. 1998, ApJ 501, 263 Johnston, J., Manchester, R.R., Lyme, A.G., Bailes, M., Kaspi, V.M., Guojun, Quand, d‘Amico, N. 1992, ApJ 387, L37 Kaper, L., Comerón, F., Barziv, O. 1999, in Proc. IAU Symp. 193, p. 316 Kaspi, V. Johnston, S., Bell, J.F, Manchester, R.N., Bailes, M., Bessel, M., Lyne, A.G., d‘Amico, N. 1994, ApJ 243, L43 Kaspi, V.M., Tauris, T.M., Manchester, R.N. 1996a, ApJ 459, 717 Kaspi, V.M., Bailes, M., Manchester, R.N, Stappers, B.V., Bell, J.F. 1996b, Nat 381, 584 Kochanek, C., S. 1992, ApJ 385, 604 Leitherer, C., Chapman, J.M., Koribalski, B. 1995 ApJ 450, 289 Moffat, A.F.J., Marchenko, S.V., Van der Hucht, K.A., et al. 1998, A&A 331, 949 Paczyński, B. 1971, Acta. Astron. 21, 417 Paczyński, B. 1990 ApJ 348, 485 Parmar, A. N., White, N. E., Stella, L., Izzo, C., Ferri, P. 1989, ApJ 338, 359 Parmar, A. N., White, N. E., Stella, L. 1989, ApJ 338, 373 Portegies Zwart, S.F., 1995 A&A 296, 691 Portegies Zwart, S.F., & van den Heuvel, E.P.J. 1999, New Astron 4, 355 Portegies Zwart, S.F., 2000 ApJ [*in press*]{} (astro-ph/0005021) Priedhorsky, W.C., Terrel, J. 1983, ApJ 273, 709 Steele, I. A., Negueruela, I., Coe, M. J., Roche, P. 1998, MNRAS 297, L5 Soberman, G. E., Phinney, E., S., van den Heuvel, E. P. J., 1997, A&A 327, 620 van den Heuvel, E. P. J. 1983, in: Accretion-driven stellar X-ray sources P. 308 van den Heuvel, E. P. J. 1985, in: Birth and evolution of massive stars and stellar groups; Proceedings of the Symposium, Dwingeloo, Netherlands, (Dordrecht, D. Reidel) p. 107 van den Heuvel, E. P. J. 1994, A&A 291, L39 van den Heuvel, E. P. J., Rappaport, S. 1987, in: Physics of Be stars; Proceedings of the Ninety-second IAU Colloquium, (Cambridge University Press, 1987) p. 291 van den Linden, Th. 1982, Ph. Thesis, University of Amsterdam van Oijen, J.G.J. 1989, A&A 217, 115 van Paradijs, J. 1995, in: “X-ray Binaries”, (eds. W.H.G. Lewin, J.A. van Paradijs and E.P.J. van den Heuvel), Cambridge University Press, 536-577 Wellstein, S., Langer, N. 1999, A&A 350, 148 Woosley, S. E., Langer, N., Weaver, T. A. 1995, ApJ 448, 315 Wijers, R. A. M. J., van Paradijs, J., van den Heuvel, E. P. J. 1992, A&A 261, 145 Zahn, J., P. 1977, A&A 57, 383
[^1]: The values given here are not identical (though similar) to those listed in Chevalier & Ilovaisky (1998). The corrections we applied are outlined below.
[^2]: Hubble Fellow
[^3]: We assumed here wind mass-loss rates of $0.5 \times 10^{-5}$ for the 9.1 star and $10^{-5}$ for the 15.4 star, respectively. These rates are in good agreement with observed WR-wind mass-loss rates (cf. Leitherer et al. 1995), but are lower than the rates adopted by Woosley et al. (1995), which [**may**]{} overestimate the real mass-loss rates, since they give for all initial helium star masses, final masses before core collapse of only about 4.
|
---
abstract: 'Recently, Andrews and Berkovich introduced a trinomial version of Bailey’s lemma. In this note we show that each ordinary Bailey pair gives rise to a trinomial Bailey pair. This largely widens the applicability of the trinomial Bailey lemma and proves some of the identities proposed by Andrews and Berkovich.'
author:
- |
S. Ole Warnaar[^1]\
\
*Department of Mathematics, University of Melbourne\
*Parkville, Victoria 3052, Australia**
date: 'February, 1997'
title: 'A note on the trinomial analogue of Bailey’s lemma'
---
The trinomial Bailey lemma {#the-trinomial-bailey-lemma .unnumbered}
--------------------------
In a recent paper, Andrews and Berkovich (AB) proposed a trinomial analogue of Bailey’s lemma [@AB]. As starting point AB take the following definitions of the $q$-trinomial coefficients $$\binom{L;B;q}{A}_2 = \sum_{j=0}^{\infty} \frac{q^{j(j+B)}(q)_L}
{(q)_j(q)_{j+A}(q)_{L-2j-A}}$$ and $$T_n(L,A,q) = q^{\frac{L(L-n)-A(A-n)}{2}} \binom{L;A-n;q^{-1}}{A}_2.$$ Here $(a)_{\infty}=\prod_{n=0}^{\infty}(1-aq^n)$ and $(a)_n=(a)_{\infty}/(aq^n)_{\infty}$, $n\in \mathbb{Z}$. To simplify equations it will also be convenient to introduce the notation $$Q_n(L,A,q) = T_n(L,A,q)/(q)_L.$$ We note that the $q$-trinomial coefficients are non-zero for $-L\leq A \leq
L$ only.
A pair of sequences $\tilde\alpha=\{\tilde\alpha_L\}_{L\geq 0}$ and $\tilde\beta=\{\tilde\beta_L\}_{L\geq 0}$ is said to form a trinomial Bailey pair relative to $n$ if $$\tilde\beta_L=\sum_{r=0}^L
Q_n(L,r,q)\, \tilde\alpha_r .$$ The trinomial analogue of the Bailey lemma is stated as follows [@AB].
If $(\tilde\alpha,\tilde\beta)$ is a trinomial Bailey pair relative to $0$, then $$\sum_{L=0}^{\infty} (-1)_L \, q^{L/2} \tilde\beta_L =
(-1)_{M+1} \sum_{L=0}^{\infty}
\frac{\tilde\alpha_L}{q^{L/2}+q^{-L/2}} \: Q_1(M,L,q).$$ Similarly, if $(\tilde\alpha,\tilde\beta)$ is a trinomial Bailey pair relative to $1$, then $$\begin{gathered}
\sum_{L=0}^{\infty} \big(-q^{-1}\big)_L\, q^L \tilde\beta_L \\
= (-1)_M \sum_{L=0}^{\infty} \tilde\alpha_L \biggl\{ Q_1(M,L,q)
-\frac{Q_1(M-1,L+1,q)}{1+q^{-L-1}}
-\frac{Q_1(M-1,L-1,q)}{1+q^{L-1}} \biggr\}.\end{gathered}$$
As a corollary of their lemma, AB obtain the identities $$\label{Cor1}
\frac{1}{2} \sum_{L=0}^{\infty} (-1)_L \, q^{L/2} \tilde\beta_L =
\frac{(-q)^2_{\infty}}{(q)^2_{\infty}} \sum_{L=0}^{\infty}
\frac{\tilde\alpha_L}{q^{L/2}+q^{-L/2}},$$ for a trinomial Bailey pair relative to $0$, and $$\label{Cor2}
\frac{1}{2}\sum_{L=0}^{\infty} \big(-q^{-1}\big)_L\, q^L \tilde\beta_L
= \frac{(-q)^2_{\infty}}{(q)^2_{\infty}} \sum_{L=0}^{\infty}
\tilde\alpha_L
\biggl\{\frac{1}{1+q^{L+1}} -\frac{1}{1+q^{L-1}}\biggr\},$$ for a trinomial Bailey pair relative to $1$.
From binomial to trinomial Bailey pairs {#from-binomial-to-trinomial-bailey-pairs .unnumbered}
---------------------------------------
In ref. [@AB], the equations and are used to derive several new $q$-series identities. As input AB take trinomial Bailey pairs obtained from polynomial identities which on one side involve $q$-trinomial coefficients. Among these identities is an identity by the author which was stated in ref. [@W] without proof, and therefore AB conclude “We have checked that his conjecture implies” followed by their equation (3.21), which is an identity for the characters of the $N=2$ superconformal models $SM(2p,(p-1)/2)$.
We now point out that equation (3.21) is a simple consequence of lemma \[lemma\] stated below. First we recall the definition of the ordinary (i.e., binomial) Bailey pair. A pair of sequences $(\alpha,\beta)$ such that $$\label{BP}
\beta_L=\sum_{r=0}^L
\frac{\alpha_r}{(q)_{L-r}(aq)_{L+r}}$$ is said to form a Bailey pair relative to $a$.
\[lemma\] Let $(\alpha,\beta)$ form a Bailey pair relative to $a=q^{\ell}$, where $\ell$ is a non-negative integer. For $n=0,1$, the following identity holds: $$\label{eqlemma}
\sum_{\substack{s=0 \\ s \equiv L+\ell {\; (\bmod\, 2)}}}^{L-\ell}
\frac{q^{s(s-n)/2}}{(q)_{\ell} (q)_s} \beta_{(L-s-\ell)/2} =
\sum_{r=0}^{\infty} Q_n(L,2r+\ell,q) \, \alpha_r .$$
For $\ell>L$ the above of course trivializes to $0=0$.
Before proving lemma \[lemma\] we note an immediate consequence.
\[cor\] Let $(\alpha,\beta)$ form a Bailey pair relative to $a=q^{\ell}$ with non-negative integer $\ell$. Then $(\tilde\alpha,\tilde\beta)$ defined as $$\begin{aligned}
\tilde\alpha_0,\dots,\tilde\alpha_{\ell-1}=0, &\qquad
\tilde\alpha_{2L+\ell} = \alpha_L, \qquad
\tilde\alpha_{2L+\ell+1} = 0, \qquad L\geq 0 \notag \\
\tilde\beta_0,\dots,\tilde\beta_{\ell-1}=0, & \qquad
\tilde\beta_{L+\ell} =
\sum_{\substack{s=0 \\ s \equiv L {\; (\bmod\, 2)}}}^L
\frac{q^{s(s-n)/2}}{(q)_{\ell}(q)_s} \beta_{(L-s)/2}, \qquad L\geq 0\end{aligned}$$ forms a trinomial Bailey pair relative to $n=0,1$.
The proof is trivial once one adopts the representation of the $q$-trinomial coefficients as given by equations (2.58) and (2.59) of ref. [@ABa], $$Q_n(L,A,q) = \frac{T_n(L,A,q)}{(q)_L} =
\sum_{\substack{s=0 \\ s\equiv L+A {\; (\bmod\, 2)}}}^{\infty}
\frac{q^{s(s-n)/2}}
{(q)_{\frac{L-A-s}{2}}(q)_{\frac{L+A-s}{2}}(q)_s}\: , \qquad n=0,1.$$
Now take the defining relation of a Bailey pair with $a=q^{\ell}$ and make the replacement $L\to (L-s-\ell)/2$ where $s$ is an integer $0\leq s \leq L-\ell$ such that $s\equiv L+\ell {\; (\bmod\, 2)}$. After multiplication by $q^{s(s-n)/2}/(q)_s$ this becomes $$\frac{q^{s(s-n)/2}}{(q)_s} \beta_{(L-s-\ell)/2} =
(q)_{\ell} \sum_{r=0}^{\infty}
\frac{\alpha_r q^{s(s-n)/2}}
{(q)_{\frac{L-s-\ell}{2}-r} (q)_{\frac{L-s+\ell}{2}+r} (q)_s}.$$ Summing over $s$ yields equation .
Returning to AB’s paper, we note that their equation (3.21) simply follows from corollary \[cor\] and the “$M(p-1,p)$ Bailey pairs” which arises from the $M(p-1,p)$ polynomial identities proven in refs. [@B; @W]. Of course, an equivalent statement is that the “conjecture of ref. [@W]”, is proven using lemma \[lemma\] and the $M(p-1,p)$ Bailey pairs. To make this somewhat more explicit we consider the special case $p=3$. Then the $M(2,3)$ Bailey pairs are nothing but the entries A(1) and A(2) of Slater’s list [@Sl]. Specifically, A(1) contains the following Bailey pair relative to $1$: $$\alpha_L = \begin{cases}
q^{6j^2-j}, & L=3j \geq 0 \\
q^{6j^2+j}, & L=3j >0 \\
-q^{6j^2-5j+1}, & L=3j-1 >0 \\
-q^{6j^2+5j+1}, & L=3j+1 >0
\end{cases} \qquad \text{and} \qquad
\beta_L = \frac{1}{(q)_{2L}}.$$ By application of corollary \[cor\] this gives the trinomial Bailey pair $$\tilde\alpha_L = \begin{cases}
q^{6j^2-j}, & L=6j \geq 0\\
q^{6j^2+j}, & L=6j >0 \\
-q^{6j^2-5j+1}, & L=6j-1 >0 \\
-q^{6j^2+5j+1}, & L=6j+1 >0
\end{cases} \qquad \text{and} \qquad
\tilde\beta_L = \!
\sum_{\substack{s=0 \\ s \equiv L {\; (\bmod\, 2)}}}^L \!
\frac{q^{s(s-n)/2}}{(q)_s (q)_{L-s}}.$$ Likewise, using entry A(2), we get $$\tilde\alpha_L = \begin{cases}
q^{6j^2-j},& L=6j-1 >0 \\
q^{6j^2+j},& L=6j+1 >0 \\
-q^{6j^2-5j+1}, & L=6j-3 >0 \\
-q^{6j^2+5j+1}, & L=6j+3 >0
\end{cases} \qquad \text{and} \qquad
\tilde\beta_L = \!
\sum_{\substack{s=0 \\ s \not\equiv L {\; (\bmod\, 2)}}}^L \!
\frac{q^{s(s-n)/2}}{(q)_s(q)_{L-s}}.$$ Setting $n=0$ and summing up both trinomial Bailey pairs, we arrive at the trinomial Bailey pair of equations (3.18) and (3.19) of [@AB]. (Unlike the case $p\geq 4$, this trinomial Bailey pair was actually proven by AB, using theorem 5.1 of ref. [@A].) Very similar results can be obtained through application of Slater’s A(3) and A(4), A(5) and A(6), and A(7) and A(8).
Conclusion {#conclusion .unnumbered}
----------
We conclude this note with several remarks. First, it is of course not true that each trinomial Bailey pair is a consequence of an ordinary Bailey pair. The pairs given by equations (3.13) and (3.14) of ref. [@AB] being examples of irreducible trinomial Bailey pairs. Second, if one replaces $Q_n(L,r,q)$ by its $q$-multinomial analogue [@S; @Wb] and takes that as the definition of a $q$-multinomial Bailey pair, it becomes straightforward to again construct multinomial Bailey pairs out of ordinary ones. Finally, it is worthwhile to note that the Bailey flow from the minimal model $M(p,p+1)$ to the $N=2$ superconformal model $SM(2p,(p-1)/2)$ as concluded by AB could now be replaced by $M(p-1,p)\to N=2~SM(2p,(p-1)/2)$. Perhaps better though would be to write $M(p-1,p)\to M(p,p+1)\to
N=2~SM(2p,(p-1)/2)$, where the first arrow indicates the flow induced by corollary \[cor\] and the second arrow the flow induced by and .
Acknowledgements {#acknowledgements .unnumbered}
----------------
I thank Alexander Berkovich for helpful comments. This work is supported by the Australian Research Council.
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[^1]: e-mail: [warnaar@maths.mu.oz.au]{}
|
---
abstract: 'We present an intuitive formalism for implementing cellular automata on arbitrary topologies. By that means, we identify a symmetry operation in the class of elementary cellular automata. Moreover, we determine the subset of topologically sensitive elementary cellular automata and find that the overall number of complex patterns decreases under increasing neighborhood size in regular graphs. As exemplary applications, we apply the formalism to complex networks and compare the potential of scale-free graphs and metabolic networks to generate complex dynamics.'
author:
- Carsten
- 'Marc-Thorsten'
title: 'Outer-totalistic cellular automata on graphs'
---
*Introduction*—Cellular automata (CA) on graphs in principle provide the possibility to monitor systematic changes of dynamics under variation of network topology. In practice, however, unambiguously studying the relation between topology and dynamics with CA is conceptually difficult, since changes in topology inevitably induce changes in the rule space. Proposed by @neumann63 as a model system for biological self-reproduction, a surge of research activity from the 80’s onwards [@wolfram83] established them as the standard tool of complex systems theory and spatio-temporal pattern formation [@deutsch05] on regular grids. Another discrete and binary modeling approach for complex biological systems are random Boolean networks (RBNs), introduced by @kauffman69. While the CA framework introduces one rule for all regularly ordered cells with bi-directional links, the original RBNs consist of randomly and directionally linked nodes with individual rules. Here, we present a formalism that generalizes CA to arbitrary architectures. It allows (i) the establishment of a general correspondence between CA and isotropic RBNs and (ii) the comparison of the potential of different topologies to generate complex dynamics. As applications we examine the topological sensitivity of elementary CA, monitor the number of complex rules of CA under increasing neighborhood size, and compare the dynamic potential of scale-free graphs and representations of metabolism as substrate graphs.
*The formalism*—Within the CA framework, the discrete (binary) state $x_i \in \Sigma = \{0,1\}$ of a node $i$ at time $t+1$ solely depends on its own state and the states of its $d$ neighboring nodes at time $t$. All cells are updated synchronously by the same, time-independent rule $f: \Sigma^{d+1} \rightarrow \Sigma$. To implement CA on a directed or undirected graph $G$, we have to account for different neighborhood sizes $d_i$ due to the heterogeneous connectivity and thus, in general, to allow for individual rules $f_i$. Our strategy instead is to impose constraints on the rule space, motivated by simple physical requirements, in order to obtain a set of discrete rules, implementable on arbitrary topologies:
- Homogeneity $f_i=f \; \forall \; i$, i.e. the same rule applies to all nodes in the graph.
- Isotropy $f=f(x_i, \rho_i)$, i.e. rules may not depend on the order of neighboring states and are thus functions of the density of neighboring states, $\rho_i(t) = \frac{1}{d_i} \sum_j A_{ij}
x_j(t)$. Here, $G$ is represented by the adjacency matrix ${\mathbf}A$: If a link connects node $j$ to node $i$, $A_{ij} = 1$, and we call $j$ an input node of $i$. The number of all input nodes is called the in-degree of node $i$, $d_i= \sum_j A_{ij}$.
- Functional simplicity, i.e. the rule $f$ is a piecewise constant function of the density $\rho_i$.
*Elementary Cellular Automata*—The simplest CA, termed elementary CA (ECA) [@wolfram83], are defined on a one-dimensional grid with minimal neighborhood size, $d=2$, and a binary state space, $\Sigma = \{0,1\}$. The $2^3 = 8$ different neighborhood configurations $x_{i-1},x_i,x_{i+1}$ result in $2^{2^3}=256$ possible rules. In this set, $2^6=64$ rules fulfill the conditions mentioned above and depend only on the state $x_i$ ($0$ or $1$) and on the density $\rho_i$ of neighboring states (0, 1/2, or 1). These 64 rules are called outer-totalistic [@wolfram83] and are now parametrized with the rule parameter set ($\alpha$, $\beta$, $\gamma$): $$x_i(t+1) =
\begin{cases}
\alpha \,, & \rho_i = 0 \\
\beta \,, & \rho_i = 1/2\\
\gamma \,, & \rho_i = 1
\end{cases}
\label{eq:2}$$ We distinguish the following cases for the rule parameters $\alpha,\beta,\gamma$: The state $x_i(t+1)$ may be $0$ or $1$ independently of the state $x_i(t)$ itself, or it may remain unchanged ($+$) or be flipped ($-$), $\alpha,\beta,\gamma \in
\{0,1,+,-\}$. The frequently used majority rule [@crutchfield95_proceedings-of-the-n; @moreira04; @amaral04; @nochomovitz06], for example, where a node $i$ is mapped onto $0/1$ if the density $\rho_i$ is below$/$above $0.5$, and stays in its state otherwise, is described in our formalism by $(\alpha,\beta,\gamma) = (0,+,1)$. For $\alpha,\beta,\gamma \in \{0,1\}$, the corresponding CA rules are called totalistic [@wolfram83], since $x_i (t+1)$ depends exclusively on the density $\rho_i$ of the input states. Only these rules have strict RBN rule equivalents (see Table \[tab:gca\]).
Aside from the initial system state ${\mathbf}{x}(0) := (x_1, x_2, \dots,
x_N)$ at $t=0$, the patterns of rule $(0,0,0)$ and rule $(1,1,1)$ are perfectly symmetric under the action of the operator $\mathcal{T}: \xi
\mapsto 1-\xi, \; \xi \in \{0,1\}$. The operator $\mathcal{T}$ exchanges all 0s and 1s in an array of elements, which can be both a pattern consisting of 0’s and 1’s or a set of rule parameters. Note that the elements $\{+,-\}$ remain unaffected under the action of $\mathcal{T}$. Generally, the symmetric rule to $(\alpha,\beta,\gamma)$ is rule $\mathcal{T} (\gamma,\beta,\alpha)$. The patterns emerging from the action of a rule onto an initial state, written as $(\alpha, \beta, \gamma) \cdot {\mathbf}{x}(0)$, are identical to the inverted patterns emerging from the inverted initial state $\mathcal{T} {\mathbf}{x}(0)$ due to $\mathcal{T} (\gamma,\beta,\alpha)$: $(\alpha,\beta,\gamma) \cdot {\mathbf}{x}(0) = \mathcal{T}
(\gamma,\beta,\alpha) \cdot \mathcal{T} {\mathbf}{x}(0)$. Explicitly, the symmetric rule to $(0,1,+)$, corresponding to the ECA with rule number 218 [@wolfram83], is $(+,0,1)$ with ECA rule number 164 (see Table \[tab:gca\] for more examples). Some rules, like the majority rule $(0,+,1)$, are self-symmetric. After elimination of all symmetric counterparts, 34 different ECA rules remain.
![Rules $(1,0, -)$, $(+, +, -)$, $(1, +, -)$, and $(1, 0,
+)$ (in ECA terms 37, 108, 109, and 133), on graphs with $N=100$ nodes and $d=2$. (a) The upper row shows patterns from a regular ECA architecture, the lower row from randomized counterparts. Time runs downwards. (b) The Shannon entropy $S$ changes differently for every one of the four rules (colored lines) under randomization and is not trivially correlated with the variation of the diameter of the graph (dotted black line), as a prominent topological observable. The results are qualitatively independent of the network size.[]{data-label="patterns"}](sensitiveRules.pdf){width="0.8\columnwidth"}
Which of these 34 rules are topologically sensitive? That is, which lead to patterns of considerably different complexity when implemented on the regular ECA grid and the RBN architecture? Wolfram classified CA heuristically according to the complexity of the emerging patterns into the four Wolfram classes [@wolfram84]. On graphs, this classification inevitably fails because of a lacking natural node order. Instead, we apply two entropy-like measures, the Shannon entropy $S$ and the word entropy $W$, which we have previously shown to provide a feasible framework for the quantification of pattern complexity [@marr06_physics-letters-a; @marr05_physica-a; @marr07_pre]. The Shannon entropy $S$ serves as a measure for the homogeneity of the spatio-temporal pattern, by averaging over all nodes: $S = \frac{1}{N}
\sum_{i=1}^N - (p^0_i \log_2 p^0_i + p^1_i \log_2 p^1_i )$. The probabilities $p^0_i$ and $p^1_i$ denote the ratios of 0’s and 1’s in the time series of node $i$. The word entropy $W$ serves as a complexity measure beyond single time steps. It quantifies the irregularity of a time series by counting the number of words, i.e. blocks of constant states confined by the respective different state: $W = \frac{1}{N} \sum_{i=1}^N \left( -\sum_{l=1}^{t} p_i^l
\log_2 p_i^l \right)$. The probability $p_i^l$ is the number of words of length $l$ divided by the number of all words found in the time series of node $i$. The maximal possible word length is given by the length $t$ of the time series analyzed.
We compare $10^3$ random initial conditions on the regular architecture with $10^3$ samples of a randomized graph, where the number of incoming and outgoing links of every node is preserved and kept to $d=2$, but the link architecture has been randomized [@trusina04]. Notably, for a considerably large number of randomization steps, we generate random regular graphs [@wormald99_randreggraphs] rather than Poisson-distributed random graphs [@erdos59].
We classify rules as topologically sensitive, if the difference of the mean entropies for regular and randomized architectures is beyond the standard deviation of the difference. Out of the 34 rules, 14 rules fulfill that condition for at least one observable, $S$ or $W$. These can be divided into three groups: (1) For some rules, the ratio of constant and oscillating nodes changes under randomization, but complex or chaotic patterns never occurs. (2) Others exhibit chaotic patterns on both topologies, but the amount of complexity varies. (3) The most interesting rules are those, for which the change of topology leads to a fundamental change in the complexity of the resulting patterns, i.e., a change in the Wolfram class. These rules are $(1, 0,
-)$, $(+, +, -)$, $(1, +, -)$, and $(1, 0, +)$ (in ECA terms rules 37, 108, 109, and 133). Typical time evolutions of these four rules on regular and random architectures are shown in Figure \[patterns\](a). Moreover, these four rules react specifically to topological changes. Figure \[patterns\](b) shows the Shannon entropy against the number of randomization steps performed. While the patterns of rules $(1,0,-)$ and $(+,+,-)$ change already when a small number of shortcuts are introduced into the system, $(1,+,-)$ stays constant in this regime but shows higher $S$ and large variations for strongly disordered topologies. Finally, the Shannon entropy of the patterns emerging from rule $(1,0,+)$ grows monotonously with the randomization depth.
![Number of complex patterns (solid colored lines) and number of rules (dashed line) for regular graphs of different networks size against degree $d$. The absolute number of possible rules increases due to the enlarged possible set of $\kappa$ values. Strikingly, the number of single threshold rules with complex patterns is maximal at $d=4$.[]{data-label="neighbors"}](hoodEnlarge){width="0.8\columnwidth"}
*Regular graphs*—How much complexity is possible on regular graphs? With growing neighborhood size $d$, the number of possible densities $\rho$ and therefore the number of possible rules increases. For the sake of simplicity, we restrict our investigation to a binary state space and to rules with a single threshold: $\kappa$: $$x_i(t+1) =
\begin{cases}
\alpha \,, & \rho_i < \kappa \\
\beta \,, & \rho_i = \kappa \\
\gamma \,, & \rho_i > \kappa
\end{cases}
\label{eq:3}$$ For networks with $d=12$, 11 different threshold parameters $\kappa
\in \{1/12, 2/12, \ldots, 11/12\}$ lead to 336 different rules, where symmetric rules are considered only once. To estimate the number of rules with complex (that is in our context: non-trivial) patterns, we calculate $W$ for time evolutions. The word entropy is a feasible complexity measure for individual time evolutions. It however fails to disentangle periodic patterns (Wolfram class II) from complex (Wolfram class IV) ones. We therefore supplement our classification with a detrended fluctuation analysis (DFA) [@peng94]. This method characterizes the time correlations of a signal with a single scaling exponent, by calculating the variance of the signal from its trend in a time window for different window sizes. The DFA exponent is the slope of the mean variance against the window size and lies between 0.5 and 1.5 for white and Brownian noise, respectively. Applied to the time evolution of the system’s state density $\bar{\rho}(t) = \sum_i
x_i(t)$ as, e.g., in [@amaral04], it can be used to discriminate stationary and periodic patterns from complex ones. We count patterns as complex, if the DFA exponent is positive and the word entropy $W>1$. However, the exact values of these thresholds do not alter our results qualitatively. As shown in Figure \[neighbors\], the number of possible rules (dashed line) increases linearly with $d$, while a maximum of complex patterns (full lines) occurs for $d=4$. The striking overall reduction of complexity for neighborhood enlargement, seen as the most dominant effect in Figure \[neighbors\] can be understood qualitatively from a homogeneity rationale: In the limit of a fully connected graph, all nodes see nearly the same neighborhood and thus follow the same dynamics (see [@marr05_physica-a] for a more detailed explanation of a corresponding phenomenon).
![Entropy signature difference plot for 25 scale-free graphs with 200 nodes and 400 links. Both positive and negative entropy differences ($\Delta E = E_\textrm{H} - E_\textrm{R}$) occur for the comparison of hierarchized vs. randomized graphs.[]{data-label="sfGraphs"}](gcaSfDifference){width="0.8\columnwidth"}
*Complex networks*—The formalism of Eq. (\[eq:3\]) can be transferred to networks of arbitrary topology. Compared to regular graphs with global neighborhood size $d$, the case where $\rho_i =
\kappa$ will rarely occur in graphs with heterogeneous connectivity. We thus simplify the set of possible rules with single threshold: By setting $\alpha = \beta$ in Eq. (\[eq:3\]), the rule space is condensed and a rule is now defined by the rule parameters $(\alpha,\gamma)$ and the threshold parameter $\kappa$.
As a first exemplary application, we consider scale-free graphs, generated by the incremental Barabási-Albert model [@barabasi99], with a power law degree distribution, a property often found in real-life networks [@albert02; @newman03]. Due to their pivotal topological property, the existence of hubs, scale-free graphs have been used frequently as model graphs. They have also been used as a starting point to investigate the relation of degree-degree correlations in complex networks [@maslov02_science; @trusina04; @weber08_europhysics-letters]. Here we want to study how degree correlations in a scale-free graph affect its ability to generate complex patterns. We implement all resulting rules on randomized, hierarchized and anti-hierarchized variants of scale-free graphs with 200 nodes and 400 links. Hierarchization and anti-hierarchization means the gradual randomization towards positive and negative degree-degree correlations, respectively [@trusina04]. For each graph type, we calculate the entropy signatures, given by the Shannon entropy and word entropy of the emerging patterns, for all rules. Figure \[sfGraphs\] shows the entropy signature difference plot, where $(S,W)$ of the randomized graphs (R) has been subtracted from the entropy signature of the hierarchized graphs (H), $\Delta E = E_\textrm{H} -
E_\textrm{R}$. Here, $E$ stands for $S$ and $W$ respectively. Most rules are insensitive to degree-degree correlations. Positive and negative entropy signature differences occur preferentially for $(\alpha,\beta) = (+,-)$ or $(\alpha,\beta) = (-,+)$ with $\kappa \in
[0.3,0.6]$. Notably, rule $(+,-)$ is a condensed form of the topology-sensitive ECA rule 108, appearing in Figure \[patterns\]. For the anti-hierarchized graph with negative degree-degree correlations, a similar picture emerges (data not shown).
![Entropy signature difference plot for the metabolic networks of *H. sapiens* $(N=625, M=779)$, *S. cerevisiae* (448, 564), *E. coli* (563, 709), and *B. subtilis* (501, 612). $N$ denotes the number of nodes, $M$ the number of links in the largest connected component of the substrate graphs of the respective species. Notably, predominantly negative differences $\Delta S$ and $\Delta W$ occur.[]{data-label="metNets"}](fourMetnets){width="0.8\columnwidth"}
As a second example, we consider the topology of metabolic networks, which abstracts the wiring architecture of the set of enzyme-catalyzed reactions in a specific species. Substrate graphs, where nodes represent metabolites and links represent a reaction between connected substrates can be generated from genomic data [@ma03]. In [@marr07_pre], we recently studied the impact of the topology of metabolic networks on a specific dynamics. There we implemented and studied only a single rule, namely $(+,-)$ as a dynamic probe and interpreted the enhanced regularizing capacity of real networks compared to randomized null models as a possible topological contribution to the reliable establishment of metabolic steady-states and to the effective dampening of fluctuations. To show that the results presented in [@marr07_pre] are valid over the whole range of dynamics discussed in the present paper, we now implement all possible rules of the form $(\alpha,\gamma)$ on substrate graphs and analyze the entropy signature differences. Figure \[metNets\] shows the results for *Homo sapiens*, *Saccharomyces cerevisiae*, *Escherichia coli*, and *Bacillus subtilis*. We find that while the entropy signatures of most rules do not discriminate between real and randomized topologies, a few rules are topologically sensitive. These rules comply with $(\alpha,\beta) =(+,-)$ or $(\alpha,\beta) = (-,+)$ while $\kappa \in
[0.35,0.55]$. For all these rules, the entropy signature of real graphs is significantly smaller compared to the null model topologies. This is also true for hierarchized and anti-hierarchized null models, as well as for all other species investigated in [@marr07_pre]. We believe that the application of dynamic probes is a particularly helpful tool for studying dynamical constraints imposed by topology.
------------------------------------------------ --------------------------- ------------------------ ------------------------------------------------------------------------------
Outer-totalistic CA ECA RBN Ref.
\[.1cm\] $(0, 0, 0) \leftrightarrow (1, 1, 1)$ $0 \leftrightarrow 255$ $1 \leftrightarrow 16$
$(1, 0, 0) \leftrightarrow (1, 1, 0)$ $5 \leftrightarrow 95$ $2 \leftrightarrow 8$ [@paul06_physical-review-e]
$(1, -, +) \leftrightarrow (+, -, 0)$ $22 \leftrightarrow 151$ - [@matache04; @nagler05_pre]
$(0, 1, 0) \leftrightarrow (1, 0, 1)$ $90 \leftrightarrow 165$ $7 \leftrightarrow 10$ [@nagler05_pre]
$(+, +, -) \leftrightarrow (-, +, +)$ $108 \leftrightarrow 201$ - [@marr06_physics-letters-a; @marr05_physica-a; @marr07_pre]
$(+, 1, -) \leftrightarrow (-, 0, +)$ $126 \leftrightarrow - [@matache04; @nagler05_pre]
129$
$(0, 0, 1) \leftrightarrow (0, 1, 1)$ $160 \leftrightarrow $9 \leftrightarrow 15$ [@nochomovitz06]
250$
$(0, 1, +) \leftrightarrow (+, 0, 1)$ $164 \leftrightarrow - [@nagler05_pre]
218$
$(0, +, 1)$ $232$ - [@crutchfield95_proceedings-of-the-n; @moreira04; @amaral04; @nochomovitz06]
\[.1cm\]
------------------------------------------------ --------------------------- ------------------------ ------------------------------------------------------------------------------
: Examples of symmetric outer-totalistic CA as defined in Eq. (\[eq:2\]), corresponding ECA and RBN rule numbers, as defined in [@wolfram83] and [@kauffman69], respectively, and references where these rules have been discussed previously. \[tab:gca\]
*Discussion*—Our formalism can be used to describe outer-totalistic CA and isotropic RBN rules in a common framework. It allows the comprehensive discussion of previously introduced rule sets on diverse topologies, like the selection of Boolean rules presented in [@amaral04] or variations of the majority rule as used in [@moreira04; @nochomovitz06]. It moreover formalizes previous attempts to generalize CA to graphs [@osullivan01; @darabos07_adv-complex-sys], and is easily extensible, e.g. by introducing more than just one threshold parameter or by using a larger state space. With the presented framework, the often huge rule space of a discrete dynamical system can be intuitively parametrized and systematically analyzed. The finding of symmetric rules, for example, helps complementing specific CA classes. The set of rules exhibiting power law spectra, as introduced in [@nagler05_pre], can thus be completed with the corresponding symmetric counterparts. Also, many coarse-graining transitions between CA rules, as presented in [@israeli06_phys.rev.e], can be immediately understood with a symmetry rational. Specific rule generalizations, as discussed and analytically analyzed in [@matache04] may be reconsidered from the more general perspective provided in this letter. As a specific example, the application of ECA rule 22 to arbitrary graphs, stated as an open question in [@matache04], is straightforward with the presented formalism. Limitations arise as soon as individual node characteristics are to be taken into account. Still, the isotropic subset of canalyzing Boolean rules, as discussed in [@paul06_physical-review-e], can be represented with our approach. Table \[tab:gca\] shows some examples of symmetric rules in our formalism, the corresponding ECA and RBN rule number and references where these rules have been previously applied.
An analysis of topologically sensitive rules with analytical tools as developed in [@drossel05_phys.rev.lett.] or the recently introduced basin entropy [@krawitz07_physical-review-lett] may reveal state space changes associated with topological modifications. Such analyses can elucidate dynamic properties also relevant for regulatory dynamics of biological networks, which have been successfully modeled with CA approaches [@bornholdt00_proc-biol-sci; @li04_pnas; @davidich08_plos-one]. From this perspective, our framework provides a means to comprehensively study the sensitivity of a system to topological perturbations and associated rule space modifications.
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---
abstract: 'This paper considers coordinated multicast beamforming in a multi-cell wireless network. Each multiantenna base station (BS) serves multiple groups of single antenna users by generating a single beam with common data per group. The aim is to minimize the sum power of BSs while satisfying user-specific SINR targets. We propose centralized and distributed multicast beamforming algorithms for multi-cell multigroup systems. The NP-hard multicast problem is tackled by approximating it as a convex problem using the standard semidefinite relaxation method. The resulting semidefinite program (SDP) can be solved via centralized processing if global channel knowledge is available. To allow a distributed implementation, the primal decomposition method is used to turn the SDP into two optimization levels. The higher level is in charge of optimizing inter-cell interference while the lower level optimizes beamformers for given inter-cell interference constraints. The distributed algorithm requires local channel knowledge at each BS and scalar information exchange between BSs. If the solution has unit rank, it is optimal for the original problem. Otherwise, the Gaussian randomization method is used to find a feasible solution. The superiority of the proposed algorithms over conventional schemes is demonstrated via numerical evaluation.'
author:
- |
Harri Pennanen, Dimitrios Christopoulos, Symeon Chatzinotas and Björn Ottersten\
SnT - securityandtrust.lu, University of Luxembourg\
e-mail: {harri.pennanen, dimitrios.christopoulos, symeon.chatzinotas, bjorn.ottersten}@uni.lu
bibliography:
- 'strings\_all.bib'
---
Distributed optimization, physical layer multigroup multicasting, multi-cell coordination, primal decomposition, sum power minimization.
Introduction {#sec:intro}
============
Transmit beamforming (or equivalently precoding) is a signal processing technique that aims at improving the performance of a communication system by efficiently exploiting the spatial domain of a wireless multi-antenna channel. Advanced multi-antenna beamforming techniques can increase spectral efficiency significantly, if properly designed. However, without proper interference coordination between neighboring cells, inter-cell interference may limit the system performance. In this respect, coordinated beamforming, where inter-cell interference coordination is involved in the design of multi-antenna techniques, has been recognized as a powerful approach to improve the performance of wireless systems, especially at cell-edge areas [@Gesbert-10]. In coordinated beamforming, each data stream is linearly precoded in the spatial domain and transmitted from a single base station (BS). To control interference, precoded data transmissions are jointly designed among BSs such that a practical network design target is achieved while predetermined constraints imposed on users and BSs are satisfied. The performance of coordinated beamforming schemes rests on the availability of channel state information (CSI) at the BSs. Coordinated beamforming techniques can be implemented either in a centralized or a decentralized manner. Centralized algorithms require the knowledge of the channels between all BSs and all users in the system, i.e., global CSI. Distributed approaches rely on the availability of local CSI, i.e., the knowledge of the channels between a BS and all users in the system. Throughout this paper, the acquired global or local CSI is assumed to be perfect. In general, decentralized schemes are often more practically realizable than the centralized ones due to possibly reduced signaling overhead and lower computational requirements per processing unit. Coordinated beamforming has been extensively studied for various system design objectives, such as sum power minimization [@Rashid-Farrokhi-98a], minimum SINR maximization [@Huang-12] and sum rate maximization [@Shi-11]. In the classical sum power minimization problem [@Rashid-Farrokhi-98a], the goal is to minimize the sum power of the BSs while satisfying user-specific SINR targets. This system design objective is of practical interest for wireless applications which have stringent data rate and delay constraints. In the literature, centralized and distributed beamforming algorithms have been proposed in [@Rashid-Farrokhi-98a; @Bengtsson-99; @Bengtsson-01; @Wiesel-06] and [@Rashid-Farrokhi-98a; @Dahrouj-10; @Tolli-11; @Pennanen-11; @Pennanen-14a], respectively. These algorithms are either based on standard convex optimization techniques or exploitation of uplink-downlink duality.
In the hitherto presented literature, independent data is addressed to each user. This transmission strategy is known as unicast beamforming. When a symbol is addressed to more than one user, however, a more elaborate multicasting problem arises. Physical layer multicasting has the potential to efficiently address the nature of future traffic demands, e.g., to support demanding video broadcasting applications. A physical layer multicasting problem was originally proposed in [@Sidiropoulos-06], proven NP-hard and accurately approximated by the semidefinite relaxation (SDR) and Gaussian randomization techniques. In [@Karipidis-08], a unified framework was derived for physical layer multigroup multicasting, where independent sets of common data are transmitted to different interfering groups of users. Therein, the sum power minimization and the minimum SINR maximization problems, also known as the Quality of Service (QoS) and the max-min fair problems, were formulated, proven NP-hard and accurately approximated for a multicast multigroup system with a sum power constraint. In [@Christopoulos-14; @Christopoulos-14b], a consolidated solution was derived for the weighted max-min fair multigroup multicast beamforming under per-antenna power constraints. This work was extended for the sum rate maximization problem in [@Christopoulos-14c]. In [@Xiang-13], a distributed algorithm was proposed for a multi-cell multicast system with a single group per cell. Both QoS and max-min fair problems were studied. The energy efficiency maximization problem was recently considered in [@He-15] for a multi-cell multigroup case, and a centralized algorithm was derived. In the literature, however, there is a lack of generic centralized and distributed algorithms for the sum power minimization problem in a multi-cell multicast system with multiple groups per cell.
In the present contribution, in contrast to existing works, centralized and distributed beamforming designs are proposed for a multi-cell multigroup multicast system. The target is to minimize the total transmission power of the system while providing the guaranteed minimum SINRs for active users. This non-convex problem is first approximated as a convex one via the SDR. The resulting semidefinite program (SDP) can be efficiently solved via centralized processing, requiring global channel knowledge. In order to obtain a distributed implementation, the primal decomposition method is used to reformulate the one-level SDP into two optimization levels. In the higher level, upper bounding constraints for inter-cell interference powers are optimized while the lower level is in charge of optimizing the beamformers for a given set of interference power constraints. Distributed processing requires only local CSI at each BS and the exchange of scalar information with other BSs via low-rate backhaul links. With rank-one solution, the SDR is optimal for the original problem. Otherwise, the Gaussian randomization method is utilized to provide a sub-optimal, but feasible, beamforming solution. The effectiveness of the proposed centralized and distributed beamforming schemes is demonstrated via numerical examples.
The paper is organized as follows. In Section \[sec:System\_Model\], the multi-cell multicast system is introduced, and the corresponding sum power minimization problem is formulated. In sections \[sec:CentralizedAlg\] and \[sec:DecentralizedAlg\], the centralized and distributed beamforming algorithms are derived, respectively. The performance of the proposed algorithms are examined in Section \[sec:SimulationResults\] via numerical examples. Finally, conclusions are drawn in Section \[sec:Conclusion\]. The following notation is used. Bold face lower case and upper case characters denote column vectors and matrices, respectively. The operators $\left(\cdot\right)\herm$ and $\mathrm{Tr}(\cdot)$ correspond to the conjugate transpose and the trace of a matrix. ${\mathbb R}_{++}^{N}$ denotes the set of $N$-dimensional positive real vectors, while ${\mathbb C}^{M}$ represents the set of $M$-dimensional complex vectors.
System Model and problem formulation {#sec:System_Model}
====================================
Consider a multi-cell multigroup multicasting system with $B$ BSs, $G$ groups and $U$ users. The corresponding sets of BSs, groups and users are denoted by $\mathcal{B}=\{1,\ldots,B\}$, $\mathcal{G}=\{1,\ldots,G\}$ and $\mathcal{U}=\{1,\ldots,U\}$, respectively. Each BS is equipped with $A$ transmit antennas, whereas each user has only one receive antenna. An independent data stream is transmitted to each group of users from a single serving BS. Thus, there exists inter-group interference between the groups of a serving BS (i.e., intra-cell interference) and inter-group interference between the groups that belong to the different BSs (i.e., inter-cell interference). The set of groups served by BS $b$ is given by $\mathcal{G}_b$. The number of groups in set $\mathcal{G}_b$ is denoted by $G_b$. The set of users in group $g$ is denoted by $\mathcal{U}_g$, and the corresponding number of users is given by $U_g$. Since each user belongs to only one group, the sets of users belonging to different groups are disjoint, i.e., $\mathcal{U}_i \cap \mathcal{U}_j = \oslash$, $\forall i,j \in \mathcal{G}, i \neq j$. The received signal at user $u$ is given by $$\begin{aligned}
\label{eq:RxSignal}
{y}_{u} &=& \overbrace{{\vec h}_{b,u}\herm {\vec w}_{g} {s}_{g}}^\text{desired signal} + \overbrace{\sum\limits_{i \in \mathcal{G}_{b} \setminus \{g\}} {\vec h}_{b,u}\herm {\vec w}_{i} {s}_{i}}^\text{intra-cell interference} \nonumber \\
&& + \underbrace{\sum\limits_{j \in \mathcal{B} \setminus \{b\}} \sum\limits_{k \in \mathcal{G}_j} {\vec h}_{j,u}\herm {\vec w}_{k} {s}_{k}}_\text{inter-cell interference} + {n}_{u}, \nonumber \\
&& \forall b \in \mathcal{B}, \forall g \in \mathcal{G}_b, \forall u \in \mathcal{U}_g\end{aligned}$$ where ${\vec h}_{b,u} \in \C^{A}$ is the channel vector from BS $b$ to user $u$, ${\vec w}_{g} \in \C^{A}$ is the transmit beamforming vector of group $g$, ${s}_{g} \in \C$ is the corresponding normalized data symbol and ${n}_{u} \ \sim \mathcal{C} \mathcal{N} (0, \sigma_{u}^{2})$ is the complex white Gaussian noise sample with zero mean and variance $\sigma_{u}^{2}$.
The system optimization objective is to minimize the total transmission power of all BSs while guaranteeing minimum SINR target for each active user. The mathematical expression of the problem is given by $$\label{eq:SPMinMulticast}
\begin{array}{ll}
\displaystyle \underset{\{{\vec w}_{g}\}_{g \in \mathcal{G}}}{\mathrm{min.}} & \displaystyle \sum\limits_{g \in \mathcal{G}} {\rm Tr} \left({\vec w}_{g} {\vec w}_{g}\herm \right)\\
{\mathrm{s.\ t.}}
& \displaystyle \frac{|{\vec h}_{b,u}\herm {\vec w}_{g}|^{2}}{{\sigma_{u}^{2} + \sum\limits_{j \in \mathcal{B}} \sum\limits_{k \in \mathcal{G}_j \setminus \{g\}} |{\vec h}_{j,u}\herm {\vec w}_{k}|^{2}}} \geq \gamma_{u}, \\
& \forall b \in \mathcal{B}, \forall g \in \mathcal{G}_b, \forall u \in \mathcal{U}_g
\end{array}$$ where $\gamma_{u}$ is the minimum SINR target for user $u$. Problem can be infeasible in some channel conditions and system settings, e.g., the predetermined SINR targets and/or the number of active users are too high. In general, it is the duty of admission control to handle infeasible cases by relaxing the system requirements, i.e., by decreasing the SINR targets and/or reducing the number of users [@Stridh-06]. Feasibility was discussed for unicast and multicast beamforming systems in [@Wiesel-06] and [@Xiang-13], respectively. In the rest of this paper, is assumed to be feasible. Problem is non-convex and NP-hard since it is a more generic version of an NP-hard single-cell multicast problem [@Sidiropoulos-06]. Thus, cannot be solved in its current form.
Centralized beamforming design {#sec:CentralizedAlg}
==============================
Problem can be approximated as a convex problem, which can be efficiently solved. In this respect, the SDR method is applied by replacing ${\vec w}_{g}{\vec w}_{g}\herm$ with a semidefinite matrix ${\vec W}_{g}$, $\forall g \in \mathcal{G}$. The relaxation lets the rank of ${\vec W}_{g}$ be arbitrary. The resulting convex SDP is expressed as $$\label{eq:SPMinMulticastApproximated}
\begin{array}{ll}
\displaystyle \underset{\{{\vec W}_{g}\}_{g \in \mathcal{G}}}{\mathrm{min.}} & \displaystyle \sum\limits_{g \in \mathcal{G}} {\rm Tr} \left({\vec W}_{g}\right)\\
{\mathrm{s.\ t.}}
& \displaystyle \frac{{\rm Tr} \left({\vec H}_{b,u}{\vec W}_{g}\right)}{{\sigma_{u}^{2} + \sum\limits_{j \in \mathcal{B}} \sum\limits_{k \in \mathcal{G}_j \setminus \{g\}} {\rm Tr} \left({\vec H}_{j,u} {\vec W}_{k} \right)}} \geq \gamma_{u}, \\
& \forall b \in \mathcal{B}, \forall g \in \mathcal{G}_b, \forall u \in \mathcal{U}_g \\
& {\vec W}_{g} \succeq 0, \forall g \in \mathcal{G}
\end{array}$$ where ${\vec H}_{b,u}={\vec h}_{b,u}{\vec h}_{b,u}\herm$. Problem can be solved in a centralized way if global CSI is available at a central controlling unit or at each BS. An optimal solution of is not necessarily optimal for the original non-convex problem . If the solution is rank-one, i.e., all the optimal transmit covariance matrices $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}}$ have unit ranks, then the solution is also optimal for the original problem. In this case, the optimal beamformers $\{{\vec w}_{g}^{*}\}_{g \in \mathcal{G}}$ can be extracted from $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}}$ by using the eigenvalue decomposition. The resulting beamformers are given by ${\vec w}_{g}^{*}=\sqrt{\lambda_{g}}{\vec u}_{g}$, $\forall g \in \mathcal{G}$, where $\lambda_{g}$ and ${\vec u}_{g}$ are the principal eigenvalue and eigenvector of ${\vec W}_{g}^{*}$.
For specific optimization problems, the SDR provides optimum solutions. The most prominent example of this case is the optimal unicast beamforming solution in [@Bengtsson-01]. Nevertheless, due to the NP-hardness of the multicast problem, the relaxed problems do not necessarily yield unit rank matrices. Consequently, one can apply a rank-one approximation over the higher rank solution. The Gaussian randomization method is reported to give the highest accuracy in the multicast beamforming case [@Luo-10]. Let the symmetric positive semidefinite matrices $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}}$ constitute a solution of the relaxed problem. Then, a candidate rank-one beamforming solution to the original problem can be generated as a complex Gaussian vector with zero mean and covariance equal to ${\vec W}_{g}^{*}$, i.e. $\hat{{\vec w}}_g \ \sim \mathcal{C} \mathcal{N}(0, {\vec W}_{g}^{*} )$, $\forall g \in \mathcal{G}$. Next, an intermediate step is required between generating a Gaussian instance with the statistics obtained from the relaxed solution and creating a feasible candidate instance of the original problem since the feasibility of the original problem is not guaranteed. In this respect, an additional power minimization problem needs to be solved. For a given set of candidate beamformers $\{\hat{{\vec w}}_{g}\}_{g \in \mathcal{G}}$, the transmission powers $\{p_g\}_{g \in \mathcal{G}}$ are minimized while satisfying the user-specific SINR targets $\{\gamma_u\}_{u \in \mathcal{U}}$. The resulting linear program (LP) is given by $$\label{eq:GR_PowOpt_Centr}
\begin{array}{ll}
\displaystyle \underset{\{p_{g}\}_{g \in \mathcal{G}}}{\mathrm{min.}} & \displaystyle \sum\limits_{g \in \mathcal{G}} p_{g}\\
{\mathrm{s.\ t.}}
& \displaystyle \frac{p_g \left|{\vec h}_{b,u}\hat{{\vec w}}_{g}\right|^{2}}{{\sigma_{u}^{2} + \sum\limits_{j \in \mathcal{B}} \sum\limits_{k \in \mathcal{G}_j \setminus \{g\}} p_k \left|{\vec h}_{j,u}\hat{{\vec w}}_{k}\right|^{2}}} \geq \gamma_{u}, \\
& \forall b \in \mathcal{B}, \forall g \in \mathcal{G}_b, \forall u \in \mathcal{U}_g.
\end{array}$$ By solving , a set of beamformers is defined by ${\vec w}_{g}=\sqrt{p_{g}^{*}} \hat{{\vec w}}_{g}$, $\forall g \in \mathcal{G}$, where $p_{g}^{*}$ is the optimal power associated with fixed candidate beamformer $\hat{{\vec w}}_{g}$. The beamformers $\{{\vec w}_{g}\}_{g \in \mathcal{G}}$ are sub-optimal, but feasible, for the original problem. Finally, after generating a predetermined number of candidate solutions, the one that yields the lowest objective value of the original problem is chosen. The accuracy of this approximate solution is measured by the distance of the approximate objective value and the optimal value of the relaxed problem. This accuracy increases with the increasing number of Gaussian randomizations. The proposed centralized multicast approach is summarized in [*Algorithm \[alg:SPMinMulticastAlgCentr\]*]{}. With global CSI, [*Algorithm \[alg:SPMinMulticastAlgCentr\]*]{} is performed at a central controlling unit or at BS $b$, for all $b$ in parallel.
\[tbp!\]
Compute optimal transmit covariance matrices $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}}$ by solving the relaxed problem as an SDP . Check whether the ranks of $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}}$ are all one or not. If the ranks are one, apply eigenvalue decomposition for $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}}$ to find optimal beamformers $\{{\vec w}_{g}^{*}\}_{g \in \mathcal{G}}$ for the original problem. Otherwise, apply Gaussian randomization with power optimization to find feasible, but sub-optimal, beamformers $\{{\vec w}_{g}\}_{g \in \mathcal{G}}$.
Distributed beamforming design {#sec:DecentralizedAlg}
==============================
In this section, a primal decomposition-based distributed beamforming approach is proposed. Primal decomposition method can be used to facilitate distributed implementation since it decouples the problem at each iteration. This method can be applied to an optimization problem which has such coupling constraints that by fixing them, the problem decouples. In the following, we first reformulate the centralized SDR problem, and then apply primal decomposition. By using primal decomposition, the one-level optimization problem is divided into two levels, i.e., the lower level subproblems and the higher level master problem. The solution method for this two-level optimization is derived. The conditions for the optimality of the obtained solution with respect to the original problem are described. Gaussian randomization method is presented in case the solution is not optimal for the original problem. Finally, the distributed approach is summarized by a step-by-step algorithm, and its practical properties are discussed.
Reformulation of the centralized relaxed problem {#sec:ReformulatedProblem}
------------------------------------------------
In order to apply primal decomposition, needs to be reformulated by adding auxiliary variables. In this respect, we separate interference power to intra-cell and inter-cell terms, and add auxiliary variables to denote the inter-cell interference terms. Now, the coupling is transferred from beamformers to inter-cell interference variables. The reformulated problem is expressed as $$\label{eq:SPMinMulticastApproximatedReform}
\begin{array}{ll}
\displaystyle \underset{\{{\vec W}_{g}\}_{g \in \mathcal{G}}, {\boldsymbol \theta} }{\mathrm{min.}} & \displaystyle \sum\limits_{g \in \mathcal{G}} {\rm Tr} \left({\vec W}_{g}\right)\\
{\mathrm{s.\ t.}}
& \displaystyle \hspace{-0.5cm} \frac{{\rm Tr} \left({\vec H}_{b,u}{\vec W}_{g}\right)}{{\sigma_{u}^{2} + \sum\limits_{j \in \mathcal{B} \setminus \{b\}} \theta_{j,u} + \sum\limits_{k \in \mathcal{G}_{b} \setminus \{g\}} {\rm Tr} \left({\vec H}_{b,u} {\vec W}_{k} \right)}} \geq \gamma_{u}, \\
& \hspace{-0.5cm} \forall b \in \mathcal{B}, \forall g \in \mathcal{G}_b, \forall u \in \mathcal{U}_g \\
& \hspace{-0.5cm} \sum\limits_{i \in \mathcal{G}_{b}} {\rm Tr} \left({\vec H}_{b,u} {\vec W}_{i} \right) \leq \theta_{b,u}, \forall b \in \mathcal{B}, \forall u \in \mathcal{U} \setminus \mathcal{U}_b \\
& \hspace{-0.5cm} {\vec W}_{g} \succeq 0, \forall g \in \mathcal{G}_b, \forall b \in \mathcal{B}
\end{array}$$ where $\theta_{b,u}$ is the inter-cell interference from BS $b$ to user $u$ and the vector ${\boldsymbol \theta}$ consists of all inter-cell interference variables. Since the inequality constraints are met with equality at the optimal solution, yields the same solution than .
Two-level optimization via primal decomposition {#sec:TwoLevelProblem}
-----------------------------------------------
By applying primal decomposition, is divided into BS-specific subproblems for beamforming design with fixed inter-cell interference levels, and a network wide master problem in charge of optimizing the interference levels. The resulting subproblem for BS $b$ is given by $$\label{eq:SPMinMulticastSubproblem}
\begin{array}{ll}
\displaystyle \underset{\{{\vec W}_{g}\}_{g \in \mathcal{G}_b}}{\mathrm{min.}} & \displaystyle \sum\limits_{g \in \mathcal{G}_b} {\rm Tr} \left({\vec W}_{g}\right)\\
{\mathrm{s.\ t.}}
& \displaystyle \hspace{-0.4cm} \frac{{\rm Tr} \left({\vec H}_{b,u}{\vec W}_{g}\right)}{{\sigma_{u}^{2} + \sum\limits_{j \in \mathcal{B} \setminus \{b\}} \theta_{j,u} + \sum\limits_{k \in \mathcal{G}_{b} \setminus \{g\}} {\rm Tr} \left({\vec H}_{b,u} {\vec W}_{k} \right)}} \geq \gamma_{u}, \\
& \hspace{-0.4cm} \forall g \in \mathcal{G}_b, \forall u \in \mathcal{U}_g \\
& \hspace{-0.4cm} \sum\limits_{i \in \mathcal{G}_{b}} {\rm Tr} \left({\vec H}_{b,u} {\vec W}_{i} \right) \leq \theta_{b,u}, \in \mathcal{U} \setminus \mathcal{U}_b \\
& \hspace{-0.4cm} {\vec W}_{g} \succeq 0, \forall g \in \mathcal{G}_b,
\end{array}$$ Problem can be optimally solved since it is an SDP. The master problem is given by $$\label{eq:SPMinMulticastMasterProblem}
\begin{array}{cl} \underset{\{{\boldsymbol \theta}_{b}\}_{b \in \mathcal{B}}}{\mathrm{min.}} & \sum\limits_{b \in \mathcal{B}} f^{\star}_{b} ({\boldsymbol \theta}_{b}) \\
{\mathrm{s.\ t.}} & {\boldsymbol \theta}_{b} \in \R^{L}_{++}, \forall b \in \mathcal{B} \\
\end{array}$$ where $f^{\star}_{b} ({\boldsymbol \theta}_{b})$ denotes the optimal objective value of for given ${\boldsymbol \theta}_{b}$. The vector ${\boldsymbol \theta}_{b}$ with length $L$ is composed of BS $b$ specific inter-cell interference terms. The master problem can be solved for the inter-cell interference variables $\{\theta_{b,u}\}_{b \in \mathcal{B}, u \in \mathcal{U} \setminus \mathcal{U}_b}$ by using the projected subgradient method $$\begin{aligned}
\label{eq:SubgradientMethod}
\theta_{b,u}^{(r+1)} & = & \mathcal{P} \left \{ \theta_{b,u}^{(r)} - \sigma^{(r)} s_{b,u}^{(r)} \right \}, b \in \mathcal{B}, u \in \mathcal{U} \setminus \mathcal{U}_b\end{aligned}$$ where $\mathcal{P}$ is the projection onto a positive orthant, $r$ is the iteration index, $\sigma^{(r)}$ is the step-size and $s_{b,u}^{(r)}$ is the subgradient of at point $\theta_{b,u}^{(r)}$. Due to the convexity of problem , the subgradient $s_{b,u}^{(r)}$ can be defined via the dual problem of by using similar derivation as in [@Pennanen-14a]. The resulting subgradient at point $\theta_{b,u}^{(r)}$ is given by $s_{b,u}^{(r)} = \lambda_{b,u}^{(r)} - \mu_{j,u}^{(r)}$, where $\lambda_{b,u}^{(r)}$ is the dual variable associated with $\theta_{b,u}^{(r)}$ in the SINR constraint of user $u$ at its serving BS $b$ (i.e., in subproblem $b$) and $\mu_{j,u}^{(r)}$ is the dual variable associated with $\theta_{b,u}^{(r)}$ in the inter-cell interference constraint of user $u$ at the interfering BS $j$ (i.e., in subproblem $j$). Since is convex, the optimal dual variables can be obtained as side information (i.e., a certificate for optimality) by solving using standard SDP solvers. An alternative and explicit way to find the dual variables is to formulate and solve the dual problem of .
The master problem can be optimally solved if the step-size of the projected subgradient method is properly chosen [@Palomar-06]. If local CSI is available and a small amount of information exchange is allowed between the BSs, a distributed implementation is possible, es explained in Section \[sec:SummaryAlg\]. If all the optimal covariance matrices $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}_b, b \in \mathcal{B}}$ have unit ranks, then this solution is also optimal for the original problem . In this case, the optimal beamformers $\{{\vec w}_{g}^{*}\}_{g \in \mathcal{G}_b, b \in \mathcal{B}}$ are obtained from $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}_b, b \in \mathcal{B}}$ by applying the eigenvalue decomposition, i.e., ${\vec w}_{g}^{*}=\sqrt{\lambda_{g}}{\vec u}_{g}$, $\forall g \in \mathcal{G}_b$, $\forall b \in \mathcal{B}$.
Gaussian randomization {#sec:GR}
----------------------
If at least one of $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}_b, b \in \mathcal{B}}$ has a rank higher than one, the solution of the SDR problem is not optimal for the original problem . In this case, feasible, but sub-optimal, rank-one beamformers can be found via Gaussian randomization method. A candidate beamforming solution $\hat{{\vec w}}_{g}$, $\forall g \in \mathcal{G}_b$, $\forall b \in \mathcal{B}$, is generated as a Gaussian random variable with zero mean and covariance ${\vec W}_{g}^{*}$. Since the candidate beamformers may not be feasible to the original problem as such, an additional power optimization problem needs to be solved at each BS. At BS $b$, powers $\{p_g\}_{g \in \mathcal{G}_b}$ are optimized for a given set of fixed candidate beamformers $\{\hat{{\vec w}}_g\}_{g \in \mathcal{G}_b}$ while the SINR targets $\{\gamma_u\}_{u \in \mathcal{U}_b}$ need to be satisfied. This problem can be expressed as the following LP $$\label{eq:GR_PowOpt}
\begin{array}{ll}
\displaystyle \underset{\{p_{g}\}_{g \in \mathcal{G}_b}}{\mathrm{min.}} & \displaystyle \sum\limits_{g \in \mathcal{G}_b} p_{g}\\
{\mathrm{s.\ t.}}
& \displaystyle \frac{p_g \left|{\vec h}_{b,u}\hat{{\vec w}}_{g}\right|^{2}}{{\sigma_{u}^{2} + \sum\limits_{j \in \mathcal{B} \setminus \{b\}} \theta_{j,u} + \sum\limits_{k \in \mathcal{G}_{b} \setminus \{g\}} p_k \left|{\vec h}_{b,u}\hat{{\vec w}}_{k}\right|^{2}}} \geq \gamma_{u}, \\
& \forall g \in \mathcal{G}_b, \forall u \in \mathcal{U}_g \\
& \sum\limits_{i \in \mathcal{G}_{b}} p_i \left|{\vec h}_{b,u}\hat{{\vec w}}_{i}\right|^{2} \leq \theta_{b,u}, \in \mathcal{U} \setminus \mathcal{U}_b. \\
\end{array}$$ After solving , BS $b$ can define its beamformers by ${\vec w}_{g}=\sqrt{p_{g}^{*}} \hat{{\vec w}}_{g}$, $\forall g \in \mathcal{G}_b$, where $p_{g}^{*}$ is the optimal power associated with the candidate beamformer $\hat{{\vec w}}_{g}$. The resulting beamformers are sub-optimal for the original problem. After generating a predefined number of candidate solutions, the one that gives the lowest objective value of the original problem is selected. Solving does not require any information exchange between the BSs since the inter-cell interference variables are fixed while only the powers are optimized. The fixed values are taken from the optimal solution of . An alternative problem formulation is possible where both powers and inter-cell interference variables are optimized simultaneously with the aid of iterative primal decomposition method. Solving this problem requires scalar information exchange among the BSs via backhaul.
Distributed implementation {#sec:SummaryAlg}
--------------------------
The distributed implementation of the beamforming design is enabled if each BS acquires local CSI and scalar information exchange between the BSs is allowed via low-rate backhaul links. More precisely, the subproblem $b$ in and the corresponding part of the master problem in , i.e., the update of ${\boldsymbol \theta}_{b}$, are solved independently at BS $b$, for all $b \in \mathcal{B}$ in parallel. At subgradient iteration $r$, the backhaul information exchange is performed by BS $b$ as follows. BS $b$ signals the dual variables associated with the SINR constraints, i.e., $\{\lambda_{b,u}\}_{u \in \mathcal{U}_{b}}$, to all the interfering BSs. Whereas the dual variables associated with the inter-cell interference constraints, i.e., $\{\mu_{b,u}\}_{u \in \mathcal{U} \setminus \mathcal{U}_{b}}$, are signaled to the BS of which user is being interfered by BS $b$. Assuming a fully connected network and an equal number of users at each cell (i.e., $U_b=U/B$, $\forall b \in \mathcal{B}$), the total amount of the required backhaul signaling at each subgradient iteration $r$ is the sum of the real-valued terms exchanged between the coupled BS pairs. Thus, the total number of exchanged scalar values per iteration is given by $2B(B-1)(U/B)$. After solving the SDR problem via iterative distributed optimization, each BS needs to know if the covariance matrices of other BSs are all rank-one. This is easily handled in a distributed manner by each BS sending one-bit feedback to other BSs. If the Gaussian randomization procedure needs to be used, extra backhaul signaling is required. More precisely, the BS-specific powers for each Gaussian randomization instance need to be shared among other BSs in order to select the best one in a distributed manner. The overall distributed approach is summarized in [*Algorithm \[alg:SPMinMulticastAlg\]*]{}. [*Algorithm \[alg:SPMinMulticastAlg\]*]{} is performed at BS $b$, for all $b$ in parallel.
\[tbp!\]
Set $r=0$. Initialize inter-cell interference powers $\boldsymbol{\theta}_{b}^{(0)}$. Compute optimal transmit covariance matrices $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}_b}$ and dual variables $\{{\lambda}_{b,u}\}_{u \in \mathcal{U}_b}$, $\{{\mu}_{b,u}\}_{u \in \mathcal{U} \setminus \mathcal{U}_b}$ by solving the relaxed subproblem $b$ as an SDP . Communicate dual variables $\{{\lambda}_{b,u}\}_{u \in \mathcal{U}_b}$, $\{{\mu}_{b,u}\}_{u \in \mathcal{U} \setminus \mathcal{U}_b}$ to the coupled BSs via backhaul. Update inter-cell interference variables $\boldsymbol {\theta}_{b}^{(r+1)}$ via projected subgradient method . Set $r=r+1$. Check whether the ranks of $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}_b}$ are all one or not. Share this one-bit information among other BSs via backhaul. If the ranks are one for all $g \in \mathcal{G}_b, b \in \mathcal{B}$, apply eigenvalue decomposition for $\{{\vec W}_{g}^{*}\}_{g \in \mathcal{G}_b}$ to find optimal beamformers $\{{\vec w}_{g}^{*}\}_{g \in \mathcal{G}_b}$ for the original problem. Otherwise, apply Gaussian randomization with power optimization to find feasible, but sub-optimal, beamformers $\{{\vec w}_{g}\}_{g \in \mathcal{G}_b}$.
Practical considerations {#sec:PracticalConsiderations}
------------------------
To acquire optimal performance, [*Algorithm \[alg:SPMinMulticastAlg\]*]{} needs to be run until convergence, and provided that the obtained covariance matrices are all rank-one. However, this is somewhat impractical since the more iterations are run, the higher the signaling/computational load and the longer the caused delay. In this respect, [*Algorithm \[alg:SPMinMulticastAlg\]*]{} naturally lends itself to a practical design where it can be stopped after a limited number of iterations to reduce delay and signaling load. Since the inter-cell interference levels are fixed at each iteration, feasible beamformers can be computed via the eigenvalue decomposition or the Gaussian randomization procedure, depending on the rank properties of the covariance matrices. Limiting the number of iterations comes at the cost of increased sum power.
In Table \[tab:SignalingDistributed\], the backhaul signaling overhead of the centralized and distributed algorithms are compared under different system settings. In the centralized algorithm, it is assumed that each BS exchanges its local CSI with all other BSs via backhaul links. Thus, global CSI is made available for each BS. Assuming equal number of users at each cell, the total backhaul signaling load in terms of scalar-valued channel coefficients in the centralized system is given by $2AU(B-1)B$. Here, one complex channel coefficient is considered as two real-valued coefficients. For the distributed algorithm, the total backhaul signaling load is presented per subgradient iteration. In Table \[tab:SignalingDistributed\], the values inside the brackets denote the percentage of the signaling load required per distributed iteration, compared with the overall signaling load required by the centralized algorithm. One can see that the distributed algorithm requires notably less amount of backhaul signaling per iteration compared to the centralized approach. The difference gets greater with the increasing network size. In conclusion, backhaul signaling overhead can be significantly reduced by limiting the number of iterations.
The distributed approach allows some special case designs where the number of optimization variables is reduced, leading to a lower computational load and even a further decreased signaling overhead. These special case designs come at the cost of somewhat decreased performance. Some of the possible special cases are presented below:
- Common interference constraint: $\theta_{b,u}=\theta, \forall b \in \mathcal{B}, \forall u \in \mathcal{U} \setminus {\mathcal{U}}_b$.
- Fixed interference constraints: $\theta_{b,u}=c_{b,u}$, $\forall b \in \mathcal{B}, \forall u \in \mathcal{U} \setminus {\mathcal{U}}_b$, where $c_{b,u}$ is a predefined constant. Does not require any backhaul signaling.
- Inter-cell interference nulling, i.e., $\theta_{b,u}=0$, $\forall b \in \mathcal{B}, \forall u \in \mathcal{U} \setminus {\mathcal{U}}_b$. Does not require any backhaul signaling.
Centralized Distributed
------------------------- ------------- --------------
$\{B,U,A\}=\{2,8,8\}$ 256 16 (6.3$\%$)
$\{B,U,A\}=\{3,12,12\}$ 1728 48 (2.8$\%$)
$\{B,U,A\}=\{4,16,16\}$ 6144 96 (1.6$\%$)
: Total backhaul signaling load (per iteration).[]{data-label="tab:SignalingDistributed"}
Simulation results {#sec:SimulationResults}
==================
In this section, the performance of the proposed centralized and distributed algorithms is evaluated via numerical examples. First, the convergence behavior of the distributed algorithm is examined, and its performance after limited number of iterations is compared to the centralized approach. Then, the use of coordinated multicast beamforming (i.e., the proposed centralized algorithm) is justified by showing its superiority over conventional transmission schemes. The performance of the centralized algorithm is also studied against the lower bound solution under different system settings. Finally, the tightness of the SDR method and the properties of the higher rank solutions are also examined. The used simulation model consists of $B$ BSs, each of which is equipped with $A$ transmit antennas and serves $G$ groups of $U$ single antenna users. The number of user per each group is given by $U/G$. In the figures hereafter, the main system parameters are given by $\{B,G,U,A\}$. We assume frequency-flat Rayleigh fading channel conditions with uncorrelated channel coefficients between antennas. The SINR constraints are set equal for all users, i.e., $\gamma_u=\gamma$, $\forall u \in \mathcal{U}$. The simulation results are achieved by averaging over $100$ channel realizations. In the case of higher rank covariance matrices, $100$ Gaussian randomizations are generated.
![Convergence behavior of distributed algorithm.[]{data-label="fig:Convergence"}](Fig1){width="6cm"}
![Comparison of centralized and distributed algorithms.[]{data-label="fig:Centr_vs_Distr"}](Fig2){width="6cm"}
In Fig. \[fig:Convergence\], the convergence behavior of the distributed algorithm is examined under different system settings. In this example, the speed of convergence is relatively fast. Especially, the first few iterations improve the performance significantly, and after $10$ iterations the algorithms has almost converged. In Fig. \[fig:Centr\_vs\_Distr\], sum power is plotted against independent channel realizations for the centralized and distributed algorithms. The number of iterations is limited for the distributed algorithm. The main system parameters are given by $\{B,G,U,A\}=\{2,4,8,8\}$. The results demonstrate that the performance of the distributed algorithm with $10$ subgradient iterations is very close to that of the centralized scheme. It can be seen that performance is relatively good even with $1$ iteration. For Figs. \[fig:Convergence\] and \[fig:Centr\_vs\_Distr\], the SINR target was set to $0$ dB. All the covariance matrices in these results were rank-one.
In Fig. \[fig:PvsSINR\], average sum power is illustrated against SINR target for various transmission schemes under different system settings. The following schemes are compared:
- Single-cell beamforming with orthogonal access (extension of unicast case in [@Tolli-09c] to multicast)
- Coordinated beamforming with inter-cell interference nulling (proposed special case design in Section \[sec:PracticalConsiderations\])
- Coordinated beamforming with inter-cell interference optimization (proposed centralized design in Section \[sec:CentralizedAlg\])
In the orthogonal access scheme, each BS uses independent time or frequency slot to optimize the beamformers for its own users leading to an inter-cell interference free communication scenario. However, the rate target of each user needs to be $B$ times higher as in the non-orthogonal multi-cell case in order to guarantee the same SINR targets. The inter-cell interference nulling scheme forces interference towards other cells’ users to be zero via spatial processing. For simplicity, the results for coordinated beamforming in Fig. \[fig:PvsSINR\] were obtained via centralized processing. However, the same results can be achieved via distributed algorithm if it is let to converge. The numerical results show that the proposed coordinated beamforming method outperforms the conventional transmission schemes. Significant performance gains over the interference nulling scheme are witnessed mainly for low and medium SINR targets. The gain diminishes with the increasing SINR target. On the other hand, the superiority against the orthogonal access scheme is greatly emphasized as the SINR target or the number of BSs increases.
![Sum power versus SINR target for different transmission schemes.[]{data-label="fig:PvsSINR"}](Fig3){width="6cm"}
![Sum power versus SINR target.[]{data-label="fig:PvsSINR2"}](Fig4){width="6cm"}
![Sum power versus the number of users per group.[]{data-label="fig:PvsUsersPerGroup"}](Fig5){width="6cm"}
In Fig. \[fig:PvsSINR2\], the centralized algorithm is compared to the lower bound solution of the relaxed problem. When the solution is rank-one, the SDR is optimal and gives the lower bound. Otherwise, the Gaussian randomization process needs to be applied to get a feasible rank-one solution, which is then compared to the lower bound higher rank solution. The results imply that the SDR is (usually) optimal when the number of users per group is low (i.e., $U/G=2$) irrespective of the SINR target. If the number of users per group is high (i.e., $U/G=6$), some solutions of the SDR problem have higher rank than one. Hence, the Gaussian randomization method needs to be used leading to a small gap between the feasible rank-one result and the lower bound solution. In Fig. \[fig:PvsUsersPerGroup\], the effect of increasing the number of users per group is further studied. Specifically, sum power is presented against the number of users per group. For low number of users, it seems that the SDR is optimal since it gives the same solution as the lower bound. However, the performance degrades as the number of users increases, and the gap between the approximation method and the lower bound gets larger. The SINR target was set to $10$ dB.
Table \[tab:Rank1\] presents the probability that the solution of is rank-one for the increasing number of users per group and for different SINR target values. The results were obtained by averaging over $5000$ channel realizations. The system parameters are given by $\{B,G,U,A\}=\{2,4,4-24,24\}$. One can see that the probability depends heavily on the number of users per group, while the SINR target has less impact. More precisely, the probability of rank-one solution decreases as the number of users per group increases. For example, the probability is $100 \%$ for $U/G=1$ and $U/G=2$, while it is less than $25 \%$ for $U/G=6$. In Table \[tab:AveRank\], the average ranks of the higher rank solutions of are illustrated. The parameter setting is identical with Table \[tab:Rank1\]. The average rank is calculated by summing the ranks of all transmit covariance matrices and dividing it by the number of groups $G$, and then averaging it over $5000$ channel realizations. It can be seen that the average rank slightly increases as the number of users per group increases. Since the dimension of each transmit covariance matrix is 24, the maximum rank could be 24. However, the results demonstrate that the average ranks are relatively low, i.e., always below $1.5$.
$U/G$ 1 2 3 4 5 6
---------------- ----- ----- ------- ------ ------ ------
$\gamma=0$ dB 100 100 99.78 79.0 48.4 22.8
$\gamma=10$ dB 100 100 99.86 78.7 45.6 24.2
$\gamma=20$ dB 100 100 99.98 75.2 41.3 19.7
: Probability of rank-one solutions ($\%$).[]{data-label="tab:Rank1"}
$U/G$ 1 2 3 4 5 6
---------------- --- --- ------ ------ ------ ------
$\gamma=0$ dB - - 1.25 1.27 1.32 1.40
$\gamma=10$ dB - - 1.25 1.27 1.32 1.39
$\gamma=20$ dB - - 1.25 1.28 1.34 1.42
: Average rank of higher rank solutions.[]{data-label="tab:AveRank"}
Conclusions {#sec:Conclusion}
===========
In this paper, coordinated multicast beamforming algorithms were proposed for a multi-cell multigroup network, where each BS sends independent sets of common data to distinct groups of users. The optimization objective is to minimize the total transmission power while guaranteeing the user-specific quality of service constraints. This non-convex problem is approximated as a convex one via the SDR method. In the case of higher rank solution, the Gaussian randomization method is used to provide feasible rank-one beamformers. In addition to a centralized approach, we proposed a novel primal decomposition-based distributed algorithm which relies only on local CSI and limited backhaul information exchange. The numerical results showed that the proposed beamforming coordination is beneficial compared to the conventional transmission schemes. The results also demonstrated that the distributed algorithm obtains performance close to that of the centralized approach even after few iterations.
Acknowledgements {#acknowledgements .unnumbered}
================
This work was partially supported by the National Research Fund, Luxembourg, under the projects $\mathrm{SATSENT}$ and $\mathrm{SEMIGOD}$, and by the European Commission, H2020, under the project $\mathrm{SANSA}$.
|
---
date: March 2005
title: 'Shuffling a Stacked Deck: The Case for Partially Randomized Ranking of Search Engine Results '
---
Introduction\[sec:intro\]
=========================
Search engines are becoming the predominant means of discovering and accessing content on the Web. Users access Web content via a combination of following hyperlinks (browsing) and typing keyword queries into search engines (searching). Yet as the Web overwhelms us with its size, users naturally turn to increased searching and reduced depth of browsing, in relative terms. In absolute terms, an estimated $625$ million search queries are received by major search engines each day [@searchEngineWatch].
Ideally, search engines should present query result pages in order of some intrinsic measure of *quality*. Quality cannot be measured directly. However, various notions of *popularity*, such as number of in-links, PageRank [@pagerank], number of visits, etc., can be measured. Most Web search engines assume that popularity is closely correlated with quality, and rank results according to popularity.
The Entrenchment Problem
------------------------
Unfortunately, the correlation between popularity and quality is very weak for newly-created pages that have few visits and/or in-links. Worse, the process by which new, high-quality pages accumulate popularity is actually inhibited by search engines. Since search engines dole out a limited number of clicks per unit time among a large number of pages, always listing highly popular pages at the top, and because users usually focus their attention on the top few results [@joachims; @lempel], newly-created but high-quality pages are “shut out.” This increasing “entrenchment effect” has witnessed broad commentary across political scientists, the popular press, and Web researchers [@sherman; @power; @bias; @contro; @googlearchy; @define] and even led to the term *Googlearchy*. In a recent study, Cho and Roy [@impact] show that heavy reliance on a search engine that ranks results according to popularity can delay widespread awareness of a high-quality page by a factor of over $60$, compared with a simulated world without a search engine in which pages are accessed through browsing alone.
Even if we ignore the (contentious) issue of fairness, there are well-motivated economic objectives that are penalized by the entrenchment effect. Assuming a notion of intrinsic page quality as perceived by users, a hypothetical ideal search engine would bias users toward visiting those pages of the highest quality at a given time, regardless of popularity. Relying on popularity as a surrogate for quality sets up a vicious cycle of neglect for new pages, even as entrenched pages collect an increasing fraction of user clicks. Given that some of these new pages will generally have higher quality than some entrenched pages, pure popularity-based ranking clearly fails to maximize an objective based on average quality of search results seen by users.
Entrenchment Problem in Other Contexts
--------------------------------------
The entrenchment problem may not be unique to the Web search engine context. For example, consider recommendation systems [@kum-recom], which are widely used in e-commerce [@berk-cob]. Many users decide which items to view based on recommendations, but these systems make recommendations based on user evaluations of items they view. This circularity leads to the well-known *cold-start* problem, and is also likely to lead to entrenchment.
Indeed, Web search engines can be thought of as recommendation systems that recommend Web pages. The entrenchment problem is particularly acute in the case of Web search, because the sheer size of the Web forces large numbers of users to locate new content using search engines alone. Therefore, in this paper, we specifically focus on diminishing the entrenchment bias in the Web search context.
![Improvement in overall quality due to rank promotion in live study.[]{data-label="fig:intbarQPC"}](barQPC){width="180pt"}
Our Key Idea: Rank Promotion \[sec:intro:jokes\]
-------------------------------------------------
We propose a very simple modification to the method of ranking search results according to popularity: promote a small fraction of unexplored pages up in the result list. A new page now has some chance of attracting clicks and attention even if the initial popularity of the page is very small. If a page has high quality, the rank boost gives the page a chance to prove itself. (Detailed definitions and algorithms are given later in the paper.)
As an initial test for effectiveness, we conducted a real-world study, which we now describe briefly (a complete description is provided in Appendix \[sec:jokes\]). We created our own small Web community consisting of several thousand Web pages, each containing a joke/quotation gathered from online databases. We decided to use “funniness” as a surrogate for quality, since users are generally willing to provide their opinion about how funny something is. Users had the option to rate the funniness of the jokes/quotations they visit. The main page of the Web site we set up consisted of an ordered list of links to individual joke/quotation pages, in groups of ten at a time, as is typical in search engine responses. Text at the top stated that the jokes and quotations were presented in descending order of funniness, as rated by users of the site.
A total of $962$ volunteers participated in our study over a period of $45$ days. Users were split at random into two user groups: one group for which a simple form of rank promotion was used, and one for which rank promotion was not used. The method of rank promotion we used in this experiment is to place new pages immediately below rank position 20. For each user group we measured the ratio of funny votes to total votes during this period. Figure \[fig:intbarQPC\] shows the result. The ratio achieved using rank promotion was approximately $60\%$ larger than that obtained using strict ranking by popularity.
Design of Effective Rank Promotion Schemes {#sec:introDesign}
------------------------------------------
In the search engine context it is probably not appropriate to insert promoted pages at a consistent rank position (lest users learn over time to avoid them). Hence, we propose a simple *randomized rank promotion* scheme in which promoted pages are assigned randomly-chosen rank positions.
Still, the question remains as to how aggressively one should promote new pages. Many new pages on the Web are not of high quality. Therefore, the extent of rank promotion has to be limited very carefully, lest we negate the benefits of popularity-based ranking by displacing pages known to be of high quality too often. With rank promotion there is an inherent tradeoff between [*exploration*]{} of new pages and [*exploitation*]{} of pages already known to be of high quality. We study how to balance these two aspects, in the context of an overarching objective of maximizing the average quality of search results viewed by users, amortized over time. In particular we seek to answer the following questions:
- Which pages should be treated as candidates for exploration, i.e., included in the rank promotion process so as to receive transient rank boosts?
- Which pages, if any, should be exploited unconditionally, i.e., protected from any rank demotion caused by promotion of other pages?
- What should be the overall ratio of exploration to exploitation?
Before we can begin to address these questions, we must model the relationship between user queries and search engine results. We categorize the pages on the Web into disjoint groups by [*topic*]{}, such that each page pertains to exactly one topic. Let $\mathcal{P}$ be the set of pages devoted to a particular topic $T$ (e.g., “swimming” or “Linux”), and let $\mathcal{U}$ denote the set of users interested in topic $T$. We say that the users $\mathcal{U}$ and pages $\mathcal{P}$ corresponding to topic $T$, taken together make up a [*Web community*]{}. (Users may participate in multiple communities.) For now we assume all users access the Web uniquely through a (single) search engine. (We relax this assumption later in Section \[sec:mixedbrowsing\].) We further assume a one-to-one correspondence between queries and topics, so that each query returns exactly the set of pages for the corresponding community. Although far from perfect, we believe this model preserves the essence of the dynamic process we seek to understand.
Communities are likely to differ a great deal in terms of factors like the number of users, the number of pages, the rate at which users visit pages, page lifetimes, etc. These factors play a significant role in determining how a given rank promotion scheme influences page popularity evolution. For example, communities with very active users are likely to be less susceptible to the entrenchment effect than those whose users do not visit very many pages. Consequently, a given rank promotion scheme is bound to create quite different outcomes in the two types of communities. In this paper we provide an analytical method for predicting the effect of deploying a particular randomized rank promotion scheme in a given community, as a function of the most important high-level community characteristics.
Experimental Study
------------------
We seek to model a very complex dynamical system involving search engines, evolving pages, and user actions, and trace its trajectory in time. It is worth emphasizing that even if we owned the most popular search engine in the world, “clean-room” experiments would be impossible. We could not even study the effect of different choices of a parameter, because an earlier choice would leave large-scale and indelible artifacts on the Web graph, visit rates, and popularity of certain pages. Therefore, analysis and simulations are inescapable, and practical experiments (as in Section \[sec:intro:jokes\]) must be conducted in a sandbox.
Through a combination of analysis and simulation, we arrive at a particular recipe for randomized rank promotion that balances exploration and exploitation effectively, and yields good results across a broad range of community types. Robustness is desirable because, in practice, communities are not disjoint and therefore their characteristics cannot be measured reliably.
Outline
-------
In Section \[sec:definitions\] we present our model of Web page popularity, describe the exploration/exploitation tradeoff as it exists in our context, and introduce two metrics for evaluating rank promotion schemes. We then propose a randomized method of rank promotion in Section \[sec:randomized\], and supply an analytical model of page popularity evolution under randomized rank promotion in Section \[sec:TBP\]. In Sections \[sec:experiments\]–\[sec:mixedbrowsing\] we present extensive analytical and simulation results, and recommend and evaluate a robust recipe for randomized rank promotion.
Related Work \[sec:relwork\]
============================
The entrenchment effect has been attracting attention for several years [@sherman; @power; @bias; @contro; @googlearchy; @define], but formal models for and analysis of the impact of search engines on the evolution of the Web graph [@prefer] or on the time taken by new pages to become popular [@impact] are recent.
A few solutions to the entrenchment problem have been proposed [@estqual; @agebased; @timepage]. They rely on variations of PageRank: the solutions of [@agebased; @timepage] assign an additional weighting factor based on page age; that of [@estqual] uses the derivative of PageRank to forecast future PageRank values for young pages.
Our approach, randomized rank promotion, is quite different in spirit. The main strength of our approach is its simplicity—it does not rely on measurements of the age or PageRank evolution of individual Web pages, which are difficult to obtain and error-prone at low sample rates. (Ultimately, it may make sense to use our approach in conjunction with other techniques, in a complementary fashion.)
The exploration/exploitation tradeoff that arises in our context is akin to problems studied in the field of reinforcement learning [@rein-survey]. However, direct application of reinforcement learning algorithms appears prohibitively expensive at Web scales.
Model and Metrics {#sec:definitions}
=================
In this section we introduce the model of Web page popularity, adopted from [@impact], that we use in the rest of this paper. (For convenience, a summary of the notation we use is provided in Table \[tab:symbols\].) Recall from Section \[sec:introDesign\] that in our model the Web is categorized into disjoint groups by topic, such that each page pertains to exactly one topic. Let $\mathcal{P}$ be the set of pages devoted to a particular topic $T$, and let $\mathcal{U}$ denote the set of users interested in topic $T$. Let $n = |\mathcal{P}|$ and $u = |\mathcal{U}|$ denote the number of pages and users, respectively, in the community.
Page Popularity
---------------
In our model, time is divided into discrete intervals, and at the end of each interval the search engine measures the popularity of each Web page according to in-link count, PageRank, user traffic, or some other indicator of popularity among users. Usually it is only possible to measure popularity among a minority of users. Indeed, for in-link count or PageRank, only those users who have the ability to create links are counted. For metrics based on user traffic, typically only users who agree to install a special toolbar that monitors Web usage, as in [@Alexa], are counted. Let $\mathcal{U}_m \subseteq \mathcal{U}$ denote the set of [*monitored users*]{}, over which page popularity is measured, and let $m = |\mathcal{U}_m|$. We assume $\mathcal{U}_m$ constitutes a representative sample of the overall user population $\mathcal{U}$.
[**Symbol**]{} [**Meaning**]{}
----------------- ----------------------------------------------------------
$\mathcal{P}$ Set of Web pages in community
$n$ $= |\mathcal{P}|$
$\mathcal{U}$ Set of users in community
$u$ $= |\mathcal{U}|$
$\mathcal{U}_m$ Set of monitored users in community
$m$ $= |\mathcal{U}_m|$
$P(p,t)$ Popularity among monitored users of page $p$ at time $t$
$V_u(p,t)$ Number of user visits to page $p$
during unit time interval at $t$
$V(p,t)$ Number of visits to $p$ by monitored users at $t$
$v_u$ Total number of user visits per unit time
$v$ Number of visits by monitored users per unit time
$A(p,t)$ Awareness among monitored users of page $p$ at time $t$
$Q(p)$ Intrinsic quality of page $p$
$l$ Expected page lifetime
: Notation used in this paper.
\[tab:symbols\]
Let the total number of user visits to pages per unit time be fixed at $v_u$. Further, let $v$ denote the number of visits per unit time by monitored users, with $v = v_u \cdot \frac{m}{u}$. The way these visits are distributed among pages in $\mathcal{P}$ is determined largely by the search engine ranking method in use; we will come back to this aspect later. For now we simply provide a definition of the visit rate of a page $p \in \mathcal{P}$.
(Visit Rate) The visit rate of page $p$ at time $t$, $V(p, t)$, is defined as the number of times $p$ is visited by any monitored user within a unit time interval at time $t$.
Similarly, let $V_u(p, t)$ denote the number of visits by any user in $\mathcal{U}$ (monitored and unmonitored users alike) within a unit time interval at time $t$. We require that $\forall t, \sum_{p \in \mathcal{P}} V_u(p, t) = v_u$ and $\forall t, \sum_{p \in \mathcal{P}} V(p, t) = v$. Once a user visits a page for the first time, she becomes “aware” of that page.
(Awareness) The awareness level of page $p$ at time $t$, $A(p,t)$, is defined as the fraction of monitored users who have visited $p$ at least once by time $t$.
We define the popularity of page $p$ at time $t$, $P(p,t) \in [0,1]$, as follows:
$$\begin{aligned}
\label{eqn:aware-to-popularity}
P(p,t) = A(p,t) \cdot Q(p)\end{aligned}$$
where $Q(p)\in [0,1]$ ([*page quality*]{}) denotes the extent to which an average user would “like” page $p$ if she was aware of $p$.
In our model page popularity is a monotonically nondecreasing function of time. Therefore if we assume nonzero page viewing probabilities, for a page of infinite lifetime $\lim_{t \to \infty} P(p,t) = Q(p)$.
Rank Promotion
--------------
If pages are ranked strictly according to current popularity, it can take a long time for the popularity of a new page to approach its quality. Artificially promoting the rank of new pages can potentially accelerate this process. One important objective for rank promotion is to minimize the time it takes for a new high-quality page to attain its eventual popularity, denoted [*TBP*]{} for “time to become popular.” In this paper we measure TBP as the time it takes for a high-quality page to attain popularity that exceeds $99\%$ of its quality level.
![Exploration/exploitation tradeoff.[]{data-label="fig:keyFig"}](keyFig){width="230pt"}
Figure \[fig:keyFig\] shows popularity evolution curves for a particular page having very high quality created at time $0$ with lifetime $l$, both with and without rank promotion. (It has been shown [@impact] that popularity evolution curves are close to step-functions.) Time is plotted on the x-axis. The y-axis plots the number of user visits per time unit. Note that while the page becomes popular earlier when rank promotion is applied, the number of visits it receives once popular is somewhat lower than in the case without rank promotion. That is because systematic application of rank promotion inevitably comes at the cost of fewer visits to already-popular pages.
Exploration/Exploitation Tradeoff and Quality-Per-Click Metric
--------------------------------------------------------------
The two shaded regions of Figure \[fig:keyFig\] indicate the positive and negative aspects of rank promotion. The [*exploration benefit*]{} area corresponds to the increase in the number of additional visits to this particular high-quality page during its lifetime made possible by promoting it early on. The [*exploitation loss*]{} area corresponds to the decrease in visits due to promotion of other pages, which may mostly be of low quality compared to this one. Clearly there is a need to balance these two factors. The TBP metric is one-sided in this respect, so we introduce a second metric that takes into account both exploitation and exploitation: [*quality-per-click*]{}, or QPC for short. QPC measures the average quality of pages viewed by users, amortized over a long period of time. We believe that maximizing QPC is a suitable objective for designing a rank promotion strategy.
We now derive a mathematical expression for QPC in our model. First, recall that the number of visits by any user to page $p$ during time interval $t$ is denoted $V_u(p,t)$. We can express the cumulative quality of all pages in $\mathcal{P}$ viewed at time $t$ as $\sum_{p \in \mathcal{P}} V_u(p,t) \cdot Q(p)$. Taking the average across time in the limit as the time duration tends to infinity, we obtain:
$$\lim_{t \to \infty} \sum_{t_l = 0}^{t}{\sum_{p \in \mathcal{P}} \big( V_u(p,t_l) \cdot Q(p) \big)}$$ By normalizing, we arrive at our expression for QPC:
$$\begin{aligned}
\mathit{QPC} & = & \lim_{t \to \infty} \frac{ \sum_{t_l=0}^{t}{\sum_{p \in \mathcal{P}} {\big( V_u(p,t_l) \cdot Q(p) \big)}} }
{ \sum_{t_l=0}^{t}{ \big( \sum_{p \in \mathcal{P}} V_u(p,t_l) \big) } }\end{aligned}$$
Randomized Rank Promotion {#sec:randomized}
=========================
We now describe our simple randomized rank promotion scheme (this description is purely conceptual; more efficient implementation techniques exist).
Let $\mathcal{P}$ denote the set of $n$ responses to a user query. A subset of those pages, $\mathcal{P}_p \subseteq \mathcal{P}$ is set aside as the [*promotion pool*]{}, which contains the set of pages selected for rank promotion according to a predetermined rule. (The particular rule for selecting $\mathcal{P}_p$, as well as two additional parameters, $k \ge 1$ and $r \in [0,1]$, are configuration options that we discuss shortly.) Pages in $\mathcal{P}_p$ are sorted randomly and the result is stored in the ordered list $\mathcal{L}_p$. The remaining pages ($\mathcal{P} - \mathcal{P}_p$) are ranked in the usual deterministic way, in descending order of popularity; the result is an ordered list $\mathcal{L}_d$. The two lists are merged to create the final result list $\mathcal{L}$ according to the following procedure:
1. The top $k-1$ elements of $\mathcal{L}_d$ are removed from $\mathcal{L}_d$ and inserted into the beginning of $\mathcal{L}$ while preserving their order.
2. The element to insert into $\mathcal{L}$ at each remaining position $i = k, k+1, \ldots, n$ is determined one at a time, in that order, by flipping a biased coin: with probability $r$ the next element is taken from the top of list $\mathcal{L}_p$; otherwise it is taken from the top of $\mathcal{L}_d$. If one of $\mathcal{L}_p$ or $\mathcal{L}_d$ becomes empty, all remaining entries are taken from the nonempty list. At the end both of $\mathcal{L}_d$ and $\mathcal{L}_p$ will be empty, and $\mathcal{L}$ will contain one entry for each of the $n$ pages in $\mathcal{P}$.
The configuration parameters are:
- [**Promotion pool ($\mathcal{P}_p$):**]{} In this paper we consider two rules for determining which pages are promoted: (a) the [*uniform*]{} promotion rule, in which every page is included in $\mathcal{P}_p$ with equal probability $r$, and (b) the [*selective*]{} promotion rule, in which all pages whose current awareness level among monitored users is zero (i.e., $\mathcal{A}(p,t) = 0$) are included in $\mathcal{P}_p$, and no others. (Other rules are of course possible; we chose to focus on these two in particular because they roughly correspond to the extrema of the spectrum of interesting rules.)
- [**Starting point ($k$):**]{} All pages whose natural rank is better than $k$ are protected from the effects of promoting other pages. A particularly interesting value is $k=2$, which safeguards the top result of any search query, thereby preserving the “feeling lucky” property that is of significant value in some situations.
- [**Degree of randomization ($r$):**]{} When $k$ is small, this parameter governs the tradeoff between emphasizing exploration (large $r$) and emphasizing exploitation (small $r$).
Our goal is to determine settings of the above parameters that lead to good TBP and QPC values. The remainder of this paper is dedicated to this task. Next we present our analytical model of Web page popularity evolution, which we use to estimate TBP and QPC under various ranking methods.
Analytical Model {#sec:TBP}
================
Our analytical model has these features:
- Pages have finite lifetime following an exponential distribution (Section \[sec-lifetime\]). The number of pages and the number of users are fixed in steady state. The quality distribution of pages is stationary.
- The expected awareness, popularity, rank, and visit rate of a page are coupled to each other through a combination of the search engine ranking function and the bias in user attention to search results (Sections \[sec:aware\] and \[sec:g\]).
Given that (a) modern search engines appear to be strongly influenced by popularity-based measures while ranking results, and (b) users tend to focus their attention primarily on the top-ranked results [@joachims; @lempel], it is reasonable to assume that the expected visit rate of a page is a function of its current popularity (as done in [@impact]): $$\begin{aligned}
\label{eqn:ran-vtop-mapping}
V(p,t) &=& F(P(p,t))\end{aligned}$$ where the form of function $F(x)$ depends on the ranking method in use and the bias in user attention. For example, if ranking is completely random, then $V(p,t)$ is independent of $P(p,t)$ and the same for all pages, so $F(x) = v \cdot \frac{1}{n}$. (Recall that $v$ is the total number of monitored user visits per unit time.) If ranking is done in such a way that user traffic to a page is proportional to the popularity of that page, $F(x) = v \cdot \frac{x}{\phi}$, where $\phi$ is a normalization factor; at steady-state, $\phi = \sum_{p \in \mathcal{P}} P(p,t)$. If ranking is performed the aforementioned way $50\%$ of the time, and performed randomly $50\%$ of the time, then $F(x) = v \cdot \big( 0.5 \cdot \frac{x}{\phi} + 0.5 \cdot \frac{1}{n} \big)$. For the randomized rank promotion we introduced in Section \[sec:randomized\] the situation is more complex. We defer discussion of how to obtain $F(x)$ to Section \[sec:g\].
Page Birth and Death \[sec-lifetime\]
-------------------------------------
The set of pages on the Web is not fixed. Likewise, we assume that for a given community based around topic $T$, the set $\mathcal{P}$ of pages in the community evolves over time due to pages being created and retired. To keep our analysis manageable we assume that the rate of retirement matches the rate of creation, so that the total number of pages remains fixed at $n = |\mathcal{P}|$. We model retirement of pages as a Poisson process with rate parameter $\lambda$, so the expected lifetime of a page is $l = \frac{1}{\lambda}$ (all pages have the same expected lifetime[^1]). When a page is retired, a new page of equal quality is created immediately, so the distribution of page quality values is stationary. When a new page is created it has initial awareness and popularity values of zero.
Awareness Distribution \[sec:aware\]
------------------------------------
We derive an expression for the distribution of page awareness values, which we then use to obtain an expression for quality-per-click (QPC). We analyze the steady-state scenario, in which the awareness and popularity distributions have stabilized and remain steady over time. Our model may not seem to indicate steady-state behavior, because the set of pages is constantly in flux and the awareness and popularity of an individual page changes over time. To understand the basis for assuming steady-state behavior, consider the set $\mathcal{C}_t$ of pages created at time $t$, and the set $\mathcal{C}_{t+1}$ of pages created at time $t+1$. Since page creation is governed by a Poisson process the expected sizes of the two sets are equal. Recall that we assume the distribution of page quality values remains the same at all times. Therefore, the popularity of all pages in both $\mathcal{C}_t$ and $\mathcal{C}_{t+1}$ will increase from the starting value of $0$ according to the same popularity evolution law. At time $t+1$, when the pages in $\mathcal{C}_t$ have evolved in popularity according to the law for the first time unit, the new pages in $\mathcal{C}_{t+1}$ introduced at time $t+1$ will replace the old popularity values of the $\mathcal{C}_t$ pages. A symmetric effect occurs with pages that are retired, resulting in steady-state behavior overall. In the steady-state, both popularity and awareness distributions are stationary.
The steady-state awareness distribution is given as follows.
\[th:awareness\] Among all pages in $\mathcal{P}$ whose quality is $q$, the fraction that have awareness $a_i = \frac{i}{m}$ (for $i = 0, 1, \dots, m$) is: $$\label{eqn:recursion5}
f(a_{i}|q) = \frac{\lambda}{(\lambda + F(0)) \cdot (1-a_{i})} \prod_{j=1}^{i}
\frac{F(a_{j-1} \cdot q)}{\lambda + F(a_{j} \cdot q)}$$ where $F(x)$ is the function in Equation \[eqn:ran-vtop-mapping\].
See Appendix \[apx:thm1Proof\].
{width="200pt"} {width="200pt"}
Figure \[fig:awareness\] plots the steady-state awareness distribution for pages of highest quality, under both nonrandomized ranking and selective randomized rank promotion with $k=1$ and $r=0.2$, for our default Web community characteristics (see Section \[sec:defaultSettings\]). For this graph we used the procedure described in Section \[sec:g\] to obtain the function $F(x)$.
Observe that if randomized rank promotion is used, in steady-state most high-quality pages have large awareness, whereas if standard nonrandomized ranking is used most pages have very small awareness. Hence, under randomized rank promotion most pages having high quality spend most of their lifetimes with near-$100\%$ awareness, yet with nonrandomized ranking they spend most of their lifetimes with near-zero awareness. Under either ranking scheme pages spend very little time in the middle of the awareness scale, since the rise to high awareness is nearly a step function.
Given an awareness distribution $f(a|q)$, it is straightforward to determine expected time-to-become-popular (TBP) corresponding to a given quality value (formula omitted for brevity). Expected quality-per-click (QPC) is expressed as follows:
$$QPC = \frac{
\sum_{p \in \mathcal{P}} \sum_{i=0}^{m} f(a_i|Q(p)) \cdot F(a_i \cdot Q(p)) \cdot Q(p)
}{
\sum_{p \in \mathcal{P}} \sum_{i=0}^{m} f(a_i|Q(p)) \cdot F(a_i \cdot Q(p))
}$$
where $a_i = \frac{i}{m}$. (Recall our assumption that monitored users are a representative sample of all users.)
Popularity to Visit Rate Relationship \[sec:g\]
-----------------------------------------------
In this section we derive the function $F(x)$ used in Equation \[eqn:ran-vtop-mapping\], which governs the relationship between $P(p,t)$ and the expectation of $V(p,t)$. As done in [@impact] we split the relationship between the popularity of a page and the expected number of visits into two components: (1) the relationship between popularity and rank position, and (2) the relationship between rank position and the number of visits. We denote these two relationships as the functions $F_1$ and $F_2$ respectively, and write: $$\begin{aligned}
F(x) &=& F_2(F_1(x))\end{aligned}$$ where the output of $F_1$ is the rank position of a page of popularity $x$, and $F_2$ is a function from that rank to a visit rate. Our rationale for splitting $F$ in this way is that, according to empirical findings reported in [@joachims], the likelihood of a user visiting a page presented in a search result list depends primarily on the rank position at which the page appears.
We begin with $F_2$, the dependence of the expected number of user visits on the rank of a page in a result list. Analysis of AltaVista usage logs [@impact; @lempel] reveal that the following relationship holds quite closely[^2]:
$$\begin{aligned}
\label{eqn:rtop-mapping}
F_2(x) &=& \theta \cdot x^{-3/2}\end{aligned}$$
where $\theta$ is a normalization constant, which we set as: $$\begin{aligned}
\theta &=& \frac{v}{\sum_{i=1}^n i^{-3/2}}\end{aligned}$$ where $v$ is the total number of monitored user visits per unit time.
Next we turn to $F_1$, the dependence of rank on the popularity of a page. Note that since the awareness level of a particular page cannot be pinpointed precisely (it is expressed as a probability distribution), we express $F_1(x)$ as the [*expected*]{} rank position of a page of popularity $x$. In doing so we compromise accuracy to some extent, since we will determine the expected number of visits by applying $F_2$ to the expected rank, as opposed to summing over the full distribution of rank values. (We examine the accuracy of our analysis in Sections \[sec:analysisValidation\] and \[sec:analysisValidation2\].)
Under nonrandomized ranking, the expected rank of a page of popularity $x$ is one plus the expected number of pages whose popularities surpass $x$. By Equation \[eqn:aware-to-popularity\], page $p$ has $P(p,t)>x$ if it has $A(p,t) > x/Q(p)$. From Theorem \[th:awareness\] the probability that a randomly-chosen page $p$ satisfies this condition is:
$$\sum_{i = 1 + \lfloor m \cdot x/Q(p) \rfloor }^{m} f \left(\left. \frac{i}{m} \right| Q(p) \right)$$
By linearity of expectation, summing over all $p \in \mathcal{P}$ we arrive at:
$$\label{eq:popularity}
F_1(x) \approx 1 + \sum_{p \in \mathcal{P}}
\left(
\sum_{i = 1 + \lfloor m \cdot x/Q(p) \rfloor }^{m} f\left(\left.\frac{i}{m}\right|Q(p) \right)
\right)$$
(This is an approximate expression because we ignore the effect of ties in popularity values, and because we neglect to discount one page of popularity $x$ from the outer summation.)
The formula for $F_1$ under uniform randomized ranking is rather complex, so we omit it. We focus instead on selective randomized ranking, which is a more effective strategy, as we will demonstrate shortly. Under selective randomized ranking the expected rank of a page of popularity $x$, when $x > 0$, is given by:
$$\label{eq:sel-popularity}
F'_1(x) \approx \left\{ \begin{array}{ll}
F_1(x) & \textrm{if $F_1(x) < k$} \\
F_1(x) + \min \{\frac{r \cdot (F_1(x)-k+1)}{(1-r)}, z\} & \textrm{otherwise}
\end{array}
\right.$$
where $F_1$ is as in Equation \[eq:popularity\], and $z$ denotes the expected number of pages with zero awareness, an estimate for which can be computed without difficulty under our steady-state assumption. (The case of $x = 0$ must be handled separately; we omit the details due to lack of space.)
The above expressions for $F_1(x)$ or $F'_1(x)$ each contain a circularity, because our formula for $f(a|q)$ (Equation \[eqn:recursion5\]) contains $F(x)$. It appears that a closed-form solution for $F(x)$ is difficult to obtain. In the absence of a closed-form expression one option is to determine $F(x)$ via simulation. The method we use is to solve for $F(x)$ using an iterative procedure, as follows.
We start with a simple function for $F(x)$, say $F(x) = x$, as an initial guess at the solution. We then substitute this function into the right-hand side of the appropriate equation above to produce a new $F(x)$ function in numerical form. We then convert the numerical $F(x)$ function into symbolic form by fitting a curve, and repeat until convergence occurs. (Upon each iteration we adjust the curve slightly so as to fit the extreme points corresponding to $x = 0$ and $x = 1$ especially carefully; details omitted for brevity.) Interestingly, we found that using a quadratic curve in log-log space led to good convergence for all parameter settings we tested, so that:
$$ \log F =
\alpha \cdot (\log x)^2 + \beta \cdot \log x + \gamma$$
where $\alpha$, $\beta$, and $\gamma$ are determined using a curve fitting procedure. We later verified via simulation that across a variety of scenarios $F(x)$ can be fit quite accurately to a quadratic curve in log-log space.
Effect of Randomized Rank Promotion and Recommended Parameter Settings {#sec:experiments}
======================================================================
In this section we report our measurements of the impact of randomized rank promotion on search engine quality. We begin by describing the default Web community scenario we use in Section \[sec:defaultSettings\]. Then we report the effect of randomized rank promotion on TBP and QPC in Sections \[sec:analysisValidation\] and \[sec:analysisValidation2\], respectively. Lastly, in Section \[sec:efConfigs\] we investigate how to balance exploration and exploitation, and give our recommended recipe for randomized rank promotion.
Default Scenario {#sec:defaultSettings}
----------------
For the results we report in this paper, the default[^3] Web community we use is one having $n = 10,000$ pages. The remaining characteristics of our default Web community are set so as to be in proportion to observed characteristics of the entire Web, as follows. First, we set the expected page lifetime to $l = 1.5$ years (based on data from [@new]). Our default Web community has $u = 1000$ users making a total of $v_u = 1000$ visits per day (based on data reported in [@SIMS], the number of Web users is roughly one-tenth the number of pages, and an average user queries a search engine about once per day). We assume that a search engine is able to monitor $10\%$ of its users, so $m = 100$ and $v = 100$.
As for page quality values, we had little basis for measuring the intrinsic quality distribution of pages on the Web. As the best available approximation, we used the power-law distribution reported for PageRank in [@impact], with the quality value of the highest-quality page set to $0.4$. (We chose $0.4$ based on the fraction of Internet users who frequent the most popular Web portal site, according to [@searchEngineWatch].)
Effect of Randomized Rank Promotion on TBP {#sec:analysisValidation}
------------------------------------------
Figure \[fig:onepopevolana\] shows popularity evolution curves derived from the awareness distribution determined analytically for a page of quality $0.4$ under three different ranking methods: (1) nonrandomized ranking, (2) randomized ranking using uniform promotion with the starting point $k=1$ and the degree of randomization $r=0.2$, and (3) randomized ranking using selective promotion with $k=1$ and $r=0.2$. This graph shows that, not surprisingly, randomized rank promotion can improve TBP by a large margin. More interestingly it also indicates that selective rank promotion achieves substantially better TBP than uniform promotion. Because, for small $r$, there is limited opportunity to promote pages, focusing on pages with zero awareness turns out to be the most effective method.
Figure \[fig:tbpanasim\] shows TBP measurements for a page of quality $0.4$ in our default Web community, for different values of $r$ (fixing $k=1$). As expected, increased randomization leads to lower TBP, especially if selective promotion is employed.
To validate our analytical model, we created a simulator that maintains an evolving ranked list of pages (the ranking method used is configurable), and distributes user visits to pages according to Equation \[eqn:rtop-mapping\]. Our simulator keeps track of awareness and popularity values of individual pages as they evolve over time, and creates and retires pages as dictated by our model. After a sufficient period of time has passed to reach steady-state behavior, we take measurements.
These results are plotted in Figure \[fig:tbpanasim\], side-by-side with our analytical results. We observe a close correspondence between our analytical model and our simulation.[^4]
Effect of Randomized Rank Promotion on QPC {#sec:analysisValidation2}
------------------------------------------
We now turn to quality-per-click (QPC). Throughout this paper (except in Section \[sec:mixedbrowsing\]) we normalize all QPC measurements such that $QPC = 1.0$ corresponds to the theoretical upper bound achieved by ranking pages in descending order of quality. The graph in Figure \[fig:qpcanasim\] plots normalized QPC as we vary the promotion rule and the degree of randomization $r$ (holding $k$ fixed at $k=1$), under our default Web community characteristics of Section \[sec:defaultSettings\]. For a community with these characteristics, a moderate dose of randomized rank promotion increases QPC substantially, especially under selective promotion.
Balancing Exploration, Exploitation, and Reality {#sec:efConfigs}
------------------------------------------------
We have established a strong case that selective rank promotion is superior to uniform promotion. In this section we investigate how to set the other two randomized rank promotion parameters, $k$ and $r$, so as to balance exploration and exploitation and achieve high QPC. For this purpose we prefer to rely on simulation, as opposed to analysis, for maximum accuracy.
![Quality-per-click (QPC) for default Web community as degree of randomization ($r$) is varied.[]{data-label="fig:qpcanasim"}](defaultQPCAnaSim){width="200pt"}
The graph in Figure \[fig:kandr\] plots normalized QPC as we vary both $k$ and $r$, under our default scenario (Section \[sec:defaultSettings\]). As $k$ grows larger, a higher $r$ value is needed to achieve high QPC. Intuitively, as the starting point for rank promotion becomes lower in the ranked list (larger $k$), a denser concentration of promoted pages (larger $r$) is required to ensure that new high-quality pages are discovered by users.
For search engines, we take the view that it is undesirable to include a noticeable amount of randomization in ranking, regardless of the starting point $k$. Based on Figure \[fig:kandr\], using only $10\%$ randomization ($r=0.1$) appears sufficient to achieve most of the benefit of rank promotion, as long as $k$ is kept small (e.g., $k=1$ or $2$). Under $10\%$ randomization, roughly one page in every group of ten query results is a new, untested page, as opposed to an established page. We do not believe most users are likely to notice this effect, given the amount of noise normally present in search engine results.
A possible exception is for the topmost query result, which users often expect to be consistent if they issue the same query multiple times. Plus, for certain queries users expect to see a single, “correct,” answer in the top rank position (e.g., most users would expect the query “Carnegie Mellon” to return a link to the Carnegie Mellon University home page at position $1$), and quite a bit of effort goes into ensuring that search engines return that result at the topmost rank position. That is why we include the $k=2$ parameter setting, which ensures that the top-ranked search result is never perturbed.
[**Recommendation:**]{} [*Introduce $10\%$ randomization starting at rank position $1$ or $2$, and exclusively target zero-awareness pages for random rank promotion.*]{}
![Qualitiy-per-click (QPC) for default Web community under selective randomized rank promotion, as degree of randomization ($r$) and starting point ($k$) are varied.[]{data-label="fig:kandr"}](defaultQPCSel){width="200pt"}
Robustness Across Different Community Types {#sec:robustness}
===========================================
In this section we investigate the robustness of our recommended ranking method (selective promotion rule, $r=0.1$, $k \in \{ 1, 2 \}$) as we vary the characteristics of our testbed Web community. Our objectives are to demonstrate: (1) that if we consider a wide range of community types, amortized search result quality is never harmed by our randomized rank promotion scheme, and (2) that our method improves result quality substantially in most cases, compared with traditional deterministic ranking. In this section we rely on simulation rather than analysis to ensure maximum accuracy.
Influence of Community Size
---------------------------
Here we vary the number of pages in the community, $n$, while holding the ratio of users to pages fixed at $u/n = 10\%$, fixing the fraction of monitored users as $m/u = 10\%$, and fixing the number of daily page visits per user at $v_u/u = v/m = 1$. Figure \[fig:page\] shows the result, with community size $n$ plotted on the x-axis on a logarithmic scale. The y-axis plots normalized QPC for three different ranking methods: nonrandomized, selective randomized with $r=0.1$ and $k=1$, and selective randomized with $r=0.1$ and $k=2$. With nonrandomized ranking, QPC declines as community size increases, because it becomes more difficult for new high-quality pages to overcome the entrenchment effect. Under randomized rank promotion, on the other hand, due to rank promotion QPC remains high and fairly steady across a range of community sizes.
Influence of Page Lifetime {#sec:ans-life}
--------------------------
Figure \[fig:life\] shows QPC as we vary the expected page lifetime $l$ while keeping all other community characteristics fixed. (Recall that in our model the number of pages in the community remains constant across time, and when a page is retired a new one of equal quality but zero awareness takes its place.) The QPC curve for nonrandomized ranking confirms our intuition: when there is less churn in the set of pages in the community (large $l$), QPC is penalized less by the entrenchment effect. More interestingly, the margin of improvement in QPC over nonrandomized ranking due to introducing randomness is greater when pages tend to live longer. The reason is that with a low page creation rate the promotion pool can be kept small. Consequently new pages benefit from larger and more frequent rank boosts, on the whole, helping the high-quality ones get discovered quickly.
Influence of Visit Rate
-----------------------
The influence of the aggregate user visit rate on QPC is plotted in Figure \[fig:view\]. Visit rate is plotted on the x-axis on a logarithmic scale, and QPC is plotted on the y-axis. Here, we hold the number of pages fixed at our default value of $n=10,000$ and use our default expected lifetime value of $l=1.5$ years. We vary the total number of user visits per day $v_u$ while holding the ratio of daily page visits to users fixed at $v_u/u = 1$ and, as always, fixing the fraction of monitored users as $m/u = 10\%$. From Figure \[fig:view\] we see first of all that, not surprisingly, popularity-based ranking fundamentally fails if very few pages are visited by users. Second, if the number of visits is very large ($1000$ visits per day to an average page), then there is no need for randomization in ranking (although it does not hurt much). For visit rates within an order of magnitude on either side of $0.1 \cdot n = 1000$, which matches the average visit rate of search engines in general when $n$ is scaled to the size of the entire Web, [^5] there is significant benefit to using randomized rank promotion.
Influence of Size of User Population
------------------------------------
Lastly we study the affect of varying the number of users in the community $u$, while holding all other parameters fixed: $n=10,000$, $l=1.5$ years, $v_u = 1000$ visits per day, and $m/u = 10\%$. Note that we keep the total number of visits per day fixed, but vary the number of users making those visits. The idea is to compare communities in which most page visits come from a core group of fairly active users to ones receiving a large number of occasional visitors. Figure \[fig:user\] shows the result, with the number of users $u$ plotted on the x-axis on a logarithmic scale, and QPC plotted on the y-axis. All three ranking methods perform somewhat worse when the pool of users is large, although the performance ratios remain about the same. The reason for this trend is that with a larger user pool, a stray visit to a new high-quality page provides less traction in terms of overall awareness.
Mixed Surfing and Searching \[sec:mixedbrowsing\]
=================================================
The model we have explored thus far assumes that users make visit to pages only by querying a search engine. While a very large number of surf trails start from search engines and are very short, nonnegligible surfing may still be occurring without support from search engines. We use the following model for mixed surfing and searching:
- While performing [*random surfing*]{} [@pagerank], users traverse a link to some neighbor with probability $(1-c)$, and jump to a random page with probability $c$. The constant $c$ is known as the [*teleportation probability*]{}, typically set to 0.15 [@glen].
- While browsing the Web, users perform random surfing with probability $x$. With probability $(1-x)$ users query a search engine and browse among results presented in the form of a ranked list.
We still assume that there is only one search engine that every user uses for querying. However, this assumption does not significantly restrict the applicability of our model. For our purposes the effect of multiple search engines that present the same ranked list for a query is equivalent to a single search engine that presents the same ranked list and gets a user traffic equal to the sum of the user traffic of the multiple search engines.
Assuming that page popularity is measured using PageRank, under our mixed browsing model the expected visit rate of a page $p$ at time $t$ is given by: $$\begin{aligned}
V(p,t) &=& (1-x) \cdot F(P(p,t)) \\
&+& x \cdot \bigg(\big((1-c) \cdot \frac{P(p,t)}{\sum_{p' \in
\mathcal{P}}P(p',t)} + c \cdot \frac{1}{n} \big) \bigg)\end{aligned}$$
![Influence of the extent of random surfing.[]{data-label="fig:mixed"}](mixedComp){width="200pt"}
Figure \[fig:mixed\] shows absolute QPC values for different values of $x$ (based on simulation). Unlike with other graphs in this paper, in this graph we plot the absolute value of QPC, because the ideal QPC value varies with the extent of random surfing ($x$). Recall that $x=0$ denotes pure search engine based surfing, while $x=1$ denotes pure random surfing. Observe that for all values of $x$, randomized rank promotion performs better than (or as well as) nonrandomized ranking. It is interesting to observe that when $x$ is small, random surfing helps nonrandomized ranking, since random surfing increases the chances of exploring unpopular pages (due to the teleportation probability). However, beyond a certain extent, it does not help as much as it hurts (due to the exploration/exploitation tradeoff as was the case for randomized rank promotion).
Summary {#sec:concl}
=======
The standard method of ranking search results deterministically according to popularity has a significant flaw: high-quality Web pages that happen to be new are drastically undervalued. In this paper we first presented results of a real-world study which demonstrated that diminishing the bias against new pages by selectively and transiently promoting them in rank can improve overall result quality substantially. We then showed through extensive simulation of a wide variety of Web community types that promoting new pages by partially randomizing rank positions (using just $10\%$ randomization) consistently leads to much higher-quality search results compared with strict deterministic ranking. From our empirical results we conclude that randomized rank promotion is a promising approach that merits further study and evaluation. To pave the way for further work, we have developed new analytical models of Web page popularity evolution under deterministic and randomized search result ranking, and introduced formal metrics by which to evaluate ranking methods.
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Real-World Effectiveness of Rank Promotion {#sec:jokes}
==========================================
In this section we describe a live experiment we conducted to study the effect of rank promotion on the evolution of popularity of Web pages.
Experimental Procedure
----------------------
For this experiment we created our own small Web community consisting of several thousand Web pages containing entertainment-oriented content, and nearly one thousand volunteer users who had no prior knowledge of this project.
[**Pages:**]{} We focused on entertainment because we felt it would be relatively easy to attract a large number of users. The material we started with consisted of a large number of jokes gathered from online databases. We decided to use “funniness” as a surrogate for quality, since users are generally willing to provide their opinion about how funny something is. We wanted the funniness distribution of our jokes to mimic the quality distribution of pages on the Web. As far as we know PageRank is the best available estimate of the quality distribution of Web pages, so we downsampled our initial collection of jokes and quotations to match the PageRank distribution reported in [@impact]. To determine the funniness of our jokes for this purpose we used numerical user ratings provided by the source databases. Since most Web pages have very low PageRank, we needed a large number of nonfunny items to match the distribution, so we chose to supplement jokes with quotations. We obtained our quotations from sites offering insightful quotations not intended to be humorous. Each joke and quotation was converted into a single Web page on our site.
[**Overall site:**]{} The main page of the Web site we set up consisted of an ordered list of links to individual joke/quotation pages, in groups of ten at a time, as is typical in search engine responses. Text at the top stated that the jokes and quotations were presented in descending order of funniness, as rated by users of the site. Users had the option to rate the items: we equipped each joke/quotation page with three buttons, labeled “funny,” “neutral,” and “not funny.” To minimize the possibility of voter fraud, once a user had rated an item the buttons were removed from that item, and remained absent upon all subsequent visits by the same user to the same page.
[**Users:**]{} We advertised our site daily over a period of $45$ days, and encouraged visitors to rate whichever jokes and quotations they decided to view. Overall we had $962$ participants. Each person who visited the site for the first time was assigned at random into one of two user groups (we used cookies to ensure consistent group membership across multiple visits, assuming few people would visit our site from multiple computers): one group for which rank promotion was used, and one for which rank promotion was not used. For the latter group, items were presented in descending order of current popularity, measured as the number of funny votes submitted by members of the group.[^6] For the other group of users, items were also presented in descending order of popularity among members of the group, except that all items that had not yet been viewed by any user were inserted in a random order starting at rank position $21$ (This variant corresponds to selective promotion with $k=21$ and $r=1$.). A new random order for these zero-awareness items was chosen for each unique user. Users were not informed that rank promotion was being employed.
[**Content rotation:**]{} For each user group we kept the number of accessible joke/quotation items fixed at $1000$ throughout the duration of our $45$-day experiment. However, each item had a finite lifetime of less than $45$ days. Lifetimes for the initial $1000$ items were assigned uniformly at random from $[1,30]$, to simulation a steady-state situation in which each item had a real lifetime of $30$ days. When a particular item expired we replaced it with another item of the same quality, and set its lifetime to $30$ days and its initial popularity to zero. At all times we used the same joke/quotation items for both user groups.
Results
-------
First, to verify that the subjects of our experiment behaved similarly to users of a search engine, we measured the relationship between the rank of an item and the number of user visits it received. We discovered a power-law with an exponent remarkably close to $-3/2$, which is precisely the relationship between rank and number of visits that has been measured from usage logs of the AltaVista search engine (see Section \[sec:g\] for details).
We then proceeded to assess the impact of rank promotion. For this purpose we wanted to analyze a steady-state scenario, so we only measured the outcome of the final $15$ days of our experiment (by then all the original items had expired and been replaced). For each user group we measured the ratio of funny votes to total votes during this period. Figure \[fig:intbarQPC\] shows the result. The ratio achieved using rank promotion was approximately $60\%$ larger than that obtained using strict ranking by popularity.
Proof of Theorem 1 {#apx:thm1Proof}
==================
Because we consider only the pages of quality $q$ and we focus on steady-state behavior, we will drop $q$ and $t$ from our notation unless it causes confusion. For example, we use $f(a)$ and $V(p)$ instead of $f(a|q)$ and $V(p,t)$ in our proof.
We consider a very short time interval $dt$ during which every page is visited by at most one monitored user. That is, $V(p) dt < 1$ for every page $p$. Under this assumption we can interpret $V(p) dt$ as the probability that the page $p$ is visited by one monitored user during the time interval $dt$.
Now consider the pages of awareness $a_i = \frac{i}{m}$. Since these pages are visited by at most one monitored user during $dt$, their awareness will either stay at $a_i$ or increase to $a_{i+1}$. We use $\mathcal{P}_S(a_i)$ and $\mathcal{P}_I(a_i)$ to denote the probability that that their awareness remains at $a_i$ or increases from $a_i$ to $a_{i+1}$, respectively. The awareness of a page increases if a monitored user who was previously unaware of the page visits it. The probability that a monitored user visits $p$ is $V(p) dt$. The probability that a random monitored user is aware of $p$ is $(1-a_i)$. Therefore,
$$\begin{aligned}
\label{eq:PI}
\mathcal{P}_I(a_i) &= V(p) dt (1-a_i) = F(P(p)) dt (1-a_i)\notag\\
&= F(q a_i) dt (1-a_i)
\end{aligned}$$
Similarly, $$\label{eq:PS}
\mathcal{P}_S(a_i) = 1 - \mathcal{P}_I(a_i) = 1 - F(q a_i) dt (1-a_i)$$
We now compute the fraction of pages whose awareness is $a_i$ after $dt$. We assume that before $dt$, $f(a_i)$ and $f(a_{i-1})$ fraction of pages have awareness $a_i$ and $a_{i-1}$, respectively. A page will have awareness $a_i$ after $dt$ if (1) its awareness is $a_i$ before $dt$ and the awareness stays the same or (2) its awareness is $a_{i-1}$ before $dt$, but it increases to $a_{i}$. Therefore, the fraction of pages at awareness $a_i$ after $dt$ is potentially $$f(a_i)\mathcal{P}_S(a_{i}) + f(a_{i-1})\mathcal{P}_I(a_{i-1}).$$ However, under our Poisson model, a page disappears with probability $\lambda dt$ during the time interval $dt$. Therefore, only $(1 - \lambda dt)$ fraction will survive and have awareness $a_i$ after $dt$: $$[f(a_i)\mathcal{P}_S(a_{i}) + f(a_{i-1})\mathcal{P}_I(a_{i-1})](1-\lambda dt)$$ Given our steady-state assumption, the fraction of pages at $a_i$ after $dt$ is the same as the fraction of pages at $a_i$ before $dt$. Therefore, $$\label{eq:fa}
f(a_i) = [f(a_i)\mathcal{P}_S(a_{i}) + f(a_{i-1})\mathcal{P}_I(a_{i-1})](1-\lambda dt).$$ From Equations \[eq:PI\], \[eq:PS\] and \[eq:fa\], we get $$\frac{f(a_i)}{f(a_{i-1})}
= \frac{(1-\lambda dt) F(q a_{i-1}) dt (1-a_{i-1})}{
(\lambda + F(q a_i)) dt (1-a_i)}\notag$$ Since we assume $dt$ is very small, we can ignore the second order terms of $dt$ in the above equation and simplify it to $$\frac{f(a_i)}{f(a_{i-1})}
= \frac{F(q a_{i-1}) (1-a_{i-1})}{
(\lambda + F(q a_i)) (1-a_i)}$$ From the multiplication of $\frac{f(a_i)}{f(a_{i-1})} \times
\frac{f(a_{i-1})}{f(a_{i-2})} \times \dots \times
\frac{f(a_{1})}{f(a_{0})}$, we get $$\label{eq:recurrence}
\frac{f(a_i)}{f(a_{0})}
= \frac{1-a_0}{1-a_i}
\prod_{j=1}^{i} \frac{F(q a_{j-1})}{\lambda + F(q a_j)}$$
We now compute $f(a_0)$. Among the pages with awareness $a_0$, $\mathcal{P}_S(a_0)$ fraction will stay at $a_0$ after $dt$. Also, $\lambda dt$ fraction new pages will appear, and their awareness is $a_0$ (recall our assumption that new pages start with zero awareness). Therefore, $$f(a_0) = f(a_0)\mathcal{P}_S(a_0)(1-\lambda dt) + \lambda dt$$ After rearrangement and ignoring the second order terms of $dt$, we get $$\label{eq:a0}
f(a_0) = \frac{\lambda}{F(q a_0) + \lambda}
= \frac{\lambda}{F(0) + \lambda}$$ By combining Equations \[eq:recurrence\] and \[eq:a0\], we get $$\begin{aligned}
f(a_i)
&= f(a_0)\frac{1-a_0}{1-a_i}
\prod_{j=1}^{i} \frac{F(q a_{j-1})}{\lambda + F(q a_j)}\\
& = \frac{\lambda}{(\lambda + F(0)) (1-a_{i})} \prod_{j=1}^{i}
\frac{F(q a_{j-1})}{\lambda + F(q a_{j})}
\end{aligned}$$
[^1]: In reality, lifetime might be a positively correlated with popularity. If so, popular pages would remain entrenched for a longer time than under our model, leading to even worse TBP than our model predicts.
[^2]: User views were measured at the granularity of groups of ten results in [@lempel], and later extrapolated to individual pages in [@impact].
[^3]: We supply results for other community types in Section \[sec:robustness\].
[^4]: Our analysis is only intended to be accurate for small values of $r$, which is why we only plot results for $r < 0.2$. From a practical standpoint only small values of $r$ are of interest.
[^5]: According to our rough estimate based on data from [@SIMS].
[^6]: Due to the relatively small scale of our experiment there were frequent ties in popularity values. We chose to break ties based on age, with older pages receiving better rank positions, to simulate a less discretized situation.
|
---
abstract: 'A closed Teichmüller geodesic in the moduli space $\M_g$ of Riemann surfaces of genus $g$ is called [*$L$-short*]{}, if it has length at most $L/g$. We show that for any $L > 0$ there exist $\epsilon_2 > \epsilon_1 > 0$, independent of $g$, so that the $L$-short geodesics in $\M_g$ all lie in the intersection of the $\epsilon_1$-thick part and the $\epsilon_2$-thin part. We also estimate the number of $L$-short geodesics in $\M_g$, bounding this from above and below by polynomials in $g$ whose degrees depend on $L$ and tend to infinity as $L$ does.'
address:
- |
Dan Margalit\
School of Mathematics\
Georgia Institute of Technology\
686 Cherry St.\
Atlanta, GA 30332\
margalit@math.gatech.edu
- |
Christopher J. Leininger\
Dept. of Mathematics, University of Illinois at Urbana-Champaign\
273 Altgeld Hall, 1409 W. Green St.\
Urbana, IL 61802\
clein@math.uiuc.edu
author:
- 'Christopher J. Leininger and Dan Margalit'
bibliography:
- 'geography.bib'
title: On the number and location of short geodesics in moduli space
---
[^1]
Introduction
============
Given $g \geq 1$, let $\M_g$ denote the moduli space of Riemann surfaces of genus $g$ equipped with the Teichmüller metric. For any $L > 0$, we define $$\G_g(L) = \{ \text{closed geodesics in } \M_g \text{ of length at most } L/g \}.$$ We refer to the elements of $\G_g(L)$ as *$L$-short geodesics*, or *short geodesics* for short.
Ivanov [@Iv] and Arnoux–Yoccoz [@AY] showed that the set $\G_g(L)$ is finite for every $g \geq 1$ and $L > 0$. Penner [@Pe] proved that there exists an $L_0$ so that $\G_g(L)$ is nonempty for all $g \geq 1$ and $L > L_0$. In fact, Hironaka [@Hir] showed that we can take $L_0 = \log((3+\sqrt{5})/2) \approx 0.962$ for sufficiently large $g$; see also [@AD; @KT].
Given an interval $I \subset (0,\infty)$, let $\M_{g,I}$ be the subset of $\M_g$ consisting of those hyperbolic surfaces (Euclidean surfaces in the case $g=1$) in which the length of the shortest essential closed curve lies in $I$. For example, the sets $\M_{g,(0,\epsilon]}$ and $\M_{g,[\epsilon,\infty)}$ are often called the *$\epsilon$-thin part* and *$\epsilon$-thick part* of $\M_g$, respectively.
Our first theorem provides a coarse description of the location of the set of short geodesics in $\M_g$.
\[T:location\] Given $L > 0$ there exists $\epsilon_2 > \epsilon_1 > 0$ so that each element of $\G_g(L)$ lies in $\M_{g,[\epsilon_1,\epsilon_2]}$ for all $g \geq 1$.
Our second and third theorems concern the number of short geodesics in $\M_g$, counted as a function of $g$: the number of $L$-short geodesics in $\M_g$ is bounded from above and below by polynomials in $g$ whose degrees depend on $L$ and tend to infinity as $L$ does.
\[T:count\] Given $L > 0$ there exists a polynomial $P_L(g)$ so that $$\left| \G_g(L) \right| \leq P_L(g)$$ for all $g \geq 1$.
\[T:count2\] Given $d > 0$, there exists a polynomial $Q_d(g)$ of degree $d$, with positive leading coefficient, and $L>0$ so that $$\left| \G_g(L) \right| \geq Q_d(g)$$ for all $g \geq 1$.
[**Short geodesics as small-dilatation pseudo-Anosov mapping classes.**]{} Let $S_g$ denote a closed, connected, orientable surface of genus $g$. We will now give an interpretation of $\G_g(L)$ that is intrinsic to $S_g$. For more details on pseudo-Anosov homeomorphisms, the mapping class group, and Teichmüller space, see [@primer].
A homeomorphism $\phi : S_g \to S_g$ is *pseudo-Anosov* if there are measured singular foliations $(\F_+,\mu_+)$ and $(\F_-,\mu_-)$, called the stable and unstable measured foliations, and a real number $\lambda(\phi) > 1$, called the *dilatation*, so that $$\phi(\F_+,\mu_+) = \lambda(\phi)(\F_+,\mu_+) \quad \text{and} \quad \phi(\F_-) = \lambda(\phi)^{-1}(F_-,\mu_-).$$ When $g=1$, the foliations $\F_+$ and $\F_-$ are nonsingular, and $\phi$ is usually called *Anosov*. For ease of exposition, we will consider Anosov homeomorphisms to also be pseudo-Ansoov.
The mapping class group $\Mod(S_g)$ is the group of homotopy classes of homeomorphisms of $S_g$. An element of $\Mod(S_g)$ is pseudo-Anosov if it has a pseudo-Anosov representative.
There is a natural action of $\Mod(S_g)$ on Teichmüller space $\T(S_g)$, the space of isotopy classes of hyperbolic metrics on $S_g$, and the quotient is nothing other than moduli space: $$\M_g = \T(S_g)/\Mod(S_g).$$ The Teichmüller distance between two points of $\T(S_g)$ is $\log(K)/2$, where $K$ is the quasiconformal distortion between the two corresponding metrics on $S_g$, minimized over all representatives of the respective isotopy classes. The group $\Mod(S_g)$ acts on $\T(S_g)$ properly discontinuously by isometries, and so there is an induced metric on $\M_g$, as above.
Each pseudo-Anosov $[\phi]$ in $\Mod(S_g)$ acts on $\T(S_g)$ by translation along a geodesic axis. The translation length of $[\phi]$ is precisely $\log(\lambda(\phi))$ and the quotient of the axis descends to a closed geodesic in $\M_g$ of length $\log(\lambda(\phi))$. Furthermore, conjugate pseudo-Anosov mapping classes define the same closed geodesic, and, moreover, every closed geodesic in $\M_g$ arises in this way.
We define the set of small dilatation pseudo-Anosov mapping classes as $$\Psi_g(L) = \{ [\phi] \in \Mod(S_g) \mid \phi \mbox{ is pseudo-Anosov and } \log(\lambda(\phi)) \leq L/g \}.$$ By the previous paragraph, there is a bijection between $\G_g(L)$ and $\Psi_g(L)/\Mod(S_g)$, the set of $\Mod(S_g)$-conjugacy classes of elements of $\Psi_g(L)$: $$\G_g(L) \leftrightarrow \Psi_g(L)/\Mod(S_g).$$ As such, both of our main theorems can be rephrased as statements about the set of real numbers $\lambda$ that arise as dilatations of pseudo-Anosov homeomorphisms of $S_g$.
[**Small dilatations and 3-manifolds.**]{} Theorems \[T:location\] and \[T:count\] will be deduced from a finiteness result proven by the authors with Benson Farb, which we now recall. Consider the set of all small dilatation pseudo-Anosov mapping classes of all closed surfaces: $$\Psi(L) = \bigcup_{g \geq 1} \Psi_g(L).$$ For each element of $\Psi(L)$, we define a new pseudo-Anosov homeomorphism by removing the singularities of the stable and unstable foliations and taking the restriction. Let $\TT(L)$ denote the set of mapping tori that arise from these modified pseudo-Anosov maps, considered up to homeomorphism.
We have the following theorem; see [@FLM Theorem 1.1] and [@Ag Theorem 6.2].
\[T:universal\] For all $L > 0$, the set $\TT(L)$ is finite.
Because of Theorem \[T:universal\], it is enough to prove Theorems \[T:location\] and \[T:count\] for the pseudo-Anosov homeomorphisms corresponding to a single element of $\TT(L)$. We can then take maxima and minima of all of the resulting bounds in order to obtain the theorems.
[**Acknowledgments.**]{} We would like to thank Jayadev Athreya, Greg Blekherman, Martin Bridgeman, Jeff Brock, Dick Canary, Benson Farb, Richard Kent, Curt McMullen, and Maryam Mirzakhani for helpful conversations.
The Thurston norm {#S:thurston norm}
=================
Theorem \[T:universal\] allows us to realize $\G_g(L)$ as the union of a finite number of sets, namely, the short geodesics arising from the different fibers of the finite set of 3-manifolds $\TT(L)$. In order to leverage this theorem effectively, we will need a way of organizing the elements of $\G_g(L)$ coming from a particular 3-manifold $M$ of $\TT(L)$. The Thurston norm on $H^1(M;\R)$ is well-suited to this purpose.
Let $M \in \TT(L)$. By definition, $M$ is equal to the mapping torus $M_\phi$, where $\phi : S \to S$ is a pseudo-Anosov homeomorphism of a punctured surface $S$: $$M_\phi = S \times [0,1]/((\phi(x),0) \sim (x,1)).$$ We refer to the surface $S \subset M$ as a fiber, since it is a fiber in a fibration of $M$ over the circle. A deep theorem of Thurston states that the mapping torus of any pseudo-Anosov homeomorphism, hence $M$, admits a finite-volume hyperbolic structure [@Ot]. In this section, we will only use the observation that $M$ is atoroidal.
Fibers of $M$ represent elements of $H^1(M;\R)$. More precisely, each fiber in a fibration of $M$ over $S^1$ determines and is determined up to isotopy by a homology class which is Poincaré dual to an integral element of $H^1(M;\R)$. Furthermore, primitive integral elements of $H^1(M;\R)$ correspond to connected fibers. In this way, we identify the set of isotopy classes of fibers in $M$ with a subset of the integral elements of $H^1(M;\R)$.
Let $M$ be any finite-volume hyperbolic 3-manifold. Thurston defined a norm $$\| \cdot \| : H^1(M;\R) \to \R,$$ now called the *Thurston norm*, and proved that the set of all fibers of the mapping torus $M_\phi$ has a convenient description in terms of $|| \cdot ||$. We summarize the properties of the Thurston norm in the following theorem [@ThNorm].
\[T:thurston\] Suppose $M$ is a finite volume hyperbolic 3-manifold.
- The unit ball in $H^1(M;\R)$ with respect to the Thurston norm is a compact polyhedron $B$.
- There is a set of open top-dimensional faces $F_1,\dots,F_n$ of $B$ so that the fibers of $M$ exactly correspond to the integral elements of the union of the open cones $\R_+ \cdot F_i$.
- The restriction of $\| \cdot \|$ to any cone $\R_+ \cdot F_i$ is equal to the restriction of a homomorphism $\psi_i:H^1(M;\R) \to \R$ with the property that $\psi_i(H^1(M;\Z)) \subseteq 2\Z$.
- If $S$ is a fiber, then $\|S\| = -\chi(S)$.
The open faces $F_1,\ldots,F_n$ in Theorem \[T:thurston\] are called the *fibered faces* of $M$. We will often abuse notation by writing $S \in \R_+ \cdot F_i$ to mean that the cohomology class dual to the fiber $S$ lies in the cone over the fibered face $F_i$.
The homomorphisms $\psi_i$ can be described as follows [@ThNorm Theorem 3]. Given a fiber $S \in \R_+ \cdot F_i$, the union of all fibers of the fibration defines a codimension $1$ foliation of $M$. The tangent spaces to the leaves form a $2$-plane bundle $\tau_i$ on $M$, whose homotopy class only depends on $F_i$. The relative Euler class $e(\tau_i)$, relative to the inward-pointing vector in a neighborhood of the cusp, is dual to an element of $H_1(M;\R)$ which, by pairing with $H^1(M;\R)$, defines a homomorphism to $\R$. This is precisely $-\psi_i$: $$\psi_i(\eta) = - e(\tau_i) \cdot \eta$$ for all $\eta \in H^1(M;\R)$.
Counting short geodesics I
==========================
We now apply Theorems \[T:universal\] and \[T:thurston\] in order to prove Theorem \[T:count\], which states that, given $L > 0$ there exists a polynomial $P_L(g)$ so that $$\left| \G_g(L) \right| \leq P_L(g)$$ for all $g \geq 1$.
Recall from the introduction that $$|\Psi_g(L)/\Mod(S_g)| = |\G_g(L)|.$$ Thus, given $L > 0$, it suffices find a polynomial $P_L(g)$ so that $$|\Psi_g(L)/\Mod(S_g)| \leq P_L(g)$$ for all $g$.
According to Theorem \[T:universal\], the set of 3-manifolds $\TT(L)$ is finite. For any $M \in \TT(L)$, let $b_1(M) = \dim(H^1(M ;\R))$ be the first Betti number of $M$.
Let $B(r)$ denote the closed ball of radius $r$ around 0 in $H^1(M ;\R)$ with respect to the Thurston norm. There is a polynomial $p_M(r)$ of degree $b_1(M)$ so that $$| H^1(M ;\Z) \cap B(r) | \leq p_M(r).$$
Let $(\phi:S \to S) \in \Psi_g(L)$. By the Poincaré–Hopf index theorem, the number of singular points of the stable foliation for $\phi$ is at most $4g-4$. Thus, if $S'$ denotes the surface obtained from $S$ by deleting these singular points, we have $$|\chi(S')| \leq 6g-6.$$ Let $\phi' : S' \to S'$ denote the restriction of $\phi$ to $S'$. The map $\phi$ is completely determined up to conjugacy by the conjugacy class of $\phi'$, so it suffices to count the number of conjugacy classes of maps $\phi'$ arising from elements of $\Psi_g(L)$. By the last statement of Theorem \[T:thurston\], we have $$\|S'\| \leq 6g-6.$$ In other words, each $\phi \in \Psi_g(L)$ is, after deleting singular points, the monodromy of some fiber in the ball of radius $6g-6$ with respect to the Thurston norm of some $M \in \TT(L)$. Thus, setting $$P_L(g) = \sum_{M \in \TT(L)} p_M(6g-6),$$ it follows that $$|\Psi_g(L)/\Mod(S_g)| \leq P_L(g),$$ as desired.
Two theorems of Fried about fibered faces {#S:more thurston norm}
=========================================
Let $\phi:S \to S$ be a pseudo-Anosov homeomorphism. The [*suspension flow*]{} $\phi_t$ determined by $S$ and $\phi$ is a flow on $M_\phi$ defined using the coordinates $$M_\phi = S \times [0,1]/(x,1) \sim (\phi(x),0),$$ and extending the local flow $(x,s) \mapsto (x,s+t)$ on $S \times [0,1]$ to $M_\phi$. If $b_1(M_\phi) \geq 2$, then $M_\phi$ fibers in infinitely many ways and we obtain infinitely many different suspension flows on $M_\phi$.
We note that $\phi_t$ is transverse to $S$ and the first return map to $S$ is precisely the monodromy $\phi$. The (unmeasured) stable and unstable foliations $\F_\pm$ for $\phi$ can therefore be suspended. The result is a pair of $\phi_t$-invariant singular foliations on $M$ which we denote $\F_\pm^M$.
Fried [@Fr Theorem 7 and Lemma] proved that monodromy of any other fiber in $\R_+ \cdot F$, the cone containing $S$, has the following description (see also [@LO]).
\[T:friedtransverse\] Let $\phi:S \to S$ be a pseudo-Anosov homeomorphism with stable and unstable foliations $\F_\pm$. Let $\phi_t$ denote the suspension flow on $M_\phi$ determined by $S$ and $\phi$, and let $\F_\pm^M$ denote the $\phi_t$-suspensions of $\F_\pm$. Let $F$ be the fibered face of $M_\phi$ with $S \in \R_+ \cdot F$. Then for any fiber $\Sigma \in \R_+ \cdot F$, we can modify $\Sigma$ by isotopy so that
1. the fiber $\Sigma$ is transverse to $\phi_t$ and the first return map $\Sigma \to \Sigma$ is precisely the pseudo-Anosov monodromy associated to $\Sigma$, and
2. the intersections $\F_+^M \cap \Sigma$ and $\F_-^M \cap \Sigma$ are the stable and unstable foliations for $\varphi$, respectively.
In Theorem \[T:friedtransverse\], the foliations $\F_+^M$ and $\F_-^M$ are only topological foliations of $M_\phi$, and not transversely measured foliations. In particular, the intersections $\F_+^M \cap \Sigma$ and $\F_-^M \cap \Sigma$ are only the topological stable and unstable foliations for a fiber $\Sigma$, and not the transversely measured foliations. We will return to this issue in Section \[S:continuity\].
Let $M$ be a finite-volume hyperbolic 3-manifold, and let $F$ be a fibered face of $M$. Fried proved that there is a continuous function $$\Lambda_F : \R_+ \cdot F \to (1,\infty),$$ with the property that for each fiber $S \in \R_+ \cdot F$, $\Lambda_F(S) = \lambda(\phi)$, where $\phi : S \to S$ is the monodromy. We summarize the properties of $\Lambda_F$ in the following; see [@Fr0 Theorem F] and [@Mc Section 5].
\[T:frieddilatation\] Let $M$ be a finite-volume hyperbolic 3-manifold. For each fibered face $F$ of $M$, there exists a continuous function $$\Lambda_F:\R_+ \cdot F \to \R$$ with the following properties:
- For every $\eta \in \R_+ \cdot F$ and every $t > 0$, we have $$\Lambda_F(t \eta) = \Lambda_F(\eta)^{1/t}.$$
- For any fiber $S \in \R_+ \cdot F$ with monodromy $\phi$, we have: $$\Lambda_F(S) = \lambda(\phi).$$
- For any sequence $\{ \eta_i \} \subset \R_+ \cdot F$ with a nonzero limit outside the open cone $\R_+ \cdot F$, we have $$\Lambda_F(\eta_i) \to \infty.$$
As observed by McMullen [@Mc], Theorem \[T:frieddilatation\] can be used to prove Penner’s theorem that there is an $L_0$ so that $\G_g(L_0)$ (equivalently, $\Psi_g(L_0)$) is nonempty for all $g \geq 1$. To see this, let $\phi : S_2 \to S_2$ be a pseudo-Anosov homeomorphism whose action on $H_1(S_2;\R)$ fixes a nontrivial element and consider the mapping torus $M_\phi$. Because $\phi$ has a nonzero fixed vector, we have $b_1(M_\phi) \geq 2$. Let $F$ be the fibered face of $M_\phi$ with $S_2 \in \R_+ \cdot F$.
Recall from Theorem \[T:thurston\] that the restriction of the Thurston norm to the cone over $F$ is given by the restriction of a homomorphism $\psi : H^1(M_\phi;\R) \to \R$ with $\psi(H^1(M_\phi;\Z)) \subseteq 2\Z$. Since every integral class has even norm, and $\psi(S_2) = ||S_2|| = 2$, we have $\psi(H^1(M_\phi;\Z)) = 2\Z$. Let $\Sigma \in H^1(M_\phi;\Z)$ be an element of the kernel of $\psi$ which, together with $S_2$, is part of a basis for $H^1(M_\phi;\Z)$. For large $g$, the primitive cohomology class $$\Sigma_g = (g-1) \cdot S_2 + \Sigma$$ lies in the cone $\R_+ \cdot F$. Also, applying the last statement of Theorem \[T:thurston\], we have $$\|\Sigma_g\| = \psi(\Sigma_g) = \psi((g-1) \cdot S_2 + \Sigma) = (g-1)\psi(S_2) = (g-1)\|S_2\| = 2g-2.$$ As $M_\phi$ is closed, each fiber $\Sigma_g$ is a closed surface. Moreover, since each $\Sigma_g$ represents a primitive cohomology class, it is a connected surface. Since $\|\Sigma_g\|=2g-2$, it follows from Theorem \[T:thurston\] that $\Sigma_g$ has genus $g$.
According to Theorems \[T:thurston\] and \[T:frieddilatation\], the function $$\eta \mapsto \|\eta\| \log(\Lambda_F(\eta))$$ is continuous on $\R_+ \cdot F$ and is constant on rays from the origin. Furthermore, for every $\Sigma_g$, we have $$\|\Sigma_g\| \log(\Lambda_F(S_g)) = (2g-2)\log(\lambda(\phi_g)),$$ where $\phi_g:\Sigma_g \to \Sigma_g$ is the monodromy.
The rays through the $\Sigma_g$ limit to the ray through $S_2$. Thus, by the previous paragraph, $$(2g-2)\log(\lambda(\phi_g)) \to 2 \log(\lambda(\phi)) < \infty.$$ It follows that $(2g-2)\log(\lambda(\phi_g))$ is bounded from above by some constant $L_0$, independent of $g$, and thus $\log(\lambda(\phi_g)) < L_0/g$ for all sufficiently large $g$. By increasing $L_0$ if necessary, we can accommodate the finitely many genera not covered by this construction, and Penner’s theorem follows.
[**Remark.**]{} By investigating the monodromies of specific finite-volume fibered hyperbolic $3$-manifolds, Hironaka [@Hir], Aaber–Dunfield [@AD], and Kin–Takasawa [@KT] showed that $L_0 = \log((3+\sqrt{5})/2)$ suffices for all sufficiently large $g$, as mentioned in the introduction.
Counting short geodesics II
===========================
In this section we prove Theorem \[T:count2\], which states that, given $d > 0$, there exists a polynomial $Q_d(g)$ of degree $d$, with positive leading coefficient, and $L>0$ so that $$\left| \G_g(L) \right| \geq Q_d(g)$$ for all $g \geq 1$.
First, we require a lemma.
\[L:big bundle small norm\] For any $g \geq 2$, there exists a pseudo-Anosov homeomorphism $\phi:S_g \to S_g$ that acts trivially on $H_1(S_g;\R)$ and has the following property: if $\R_+ \cdot F \subset H^1(M;\R)$ is the cone on the fibered face containing $S_g$, and $\psi:H^1(M_\phi;\R) \to \R$ is the homomorphism that restricts to the Thurston norm on $\R_+ \cdot F$, then $\psi(H^1(M;\Z)) = 2 \Z$.
Assume first that $g \geq 6$, and let $$\alpha_0,\ldots,\alpha_{m},\beta_0,\ldots,\beta_{n},\gamma$$ be the simple closed curves in $S_g$ shown here for the case $g=8$:
\[ \] at 120 58 \[ \] at 95 100 \[ \] at 95 -7 \[ \] at 22 70 \[ \] at 165 15 \[ \] at 200 17 \[ \] at 255 16 \[ \] at 334 22 \[ \] at 349 75 \[ \] at 445 22 \[ \] at 291 80 
(when $g$ is odd, $m=(g+1)/2$ and $n=(g-1)/2$, and when $g$ is even, $m=n=g/2$).
Consider the product of Dehn twists: $$\phi = (T_{\beta_0}^{-1}T_{\alpha_0})(T_{\alpha_1} T_{\alpha_2}\cdots T_{\alpha_m})(T_{\beta_1}^{-1}T_{\beta_2}^{-1} \cdots T_{\beta_n}^{-1}).$$ By Thurston’s theorem [@ThMCG Theorem 7], the conjugate $T_{\beta_0}\phi T_{\beta_0}^{-1}$ is isotopic to a pseudo-Anosov homeomorphism (see also [@Pe2]), hence $\phi$ is.
The action of $\phi$ on $H_1(S_g;\R)$ is trivial. Indeed, since each $\alpha_i$ and $\beta_i$ is separating in $S_g$ for $i > 0$, each of these $T_{\alpha_i}$ and $T_{\beta_i}$ acts trivially on $H_1(S_g;\R)$. Also, since $\alpha_0$ and $\beta_0$ can be oriented so that they represent the same element of $H_1(S_g;\Z)$, the twists $T_{\alpha_0}$ and $T_{\beta_0}$ have the same action on $H_1(S_g;\R)$.
We will require one further property of $\phi$, which is that $\gamma$ and $\phi(\gamma)$ cobound an embedded genus $1$ surface $\Sigma_0$ in $S_g$. To check this, note that $\phi(\gamma) = T_{\beta_0}^{-1}T_{\alpha_0}(\gamma)$.
We can construct similar configurations of curves, and hence a similar $\phi$, when $g$ is 3, 4, or 5. Indeed, for $g=3$, we can simply use the curves $\alpha_0$, $\beta_0$, $\alpha_1$, and $\beta_1$. The cases of $g=4$ and $g=5$ require nontrivial modifications. However, since these cases are not logically needed for the proof of Theorem \[T:count2\], we leave the constructions to the reader. For the last case, $g=2$, any pseudo-Anosov $\phi$ acting trivially on $H_1(S_2;\Z)$ suffices.
The surface $\Sigma_0 \subset S_g \subset M_\phi = M$ is transverse to the suspension flow $\phi_t$ on $M$ since $S_g$ is. We can push $\Sigma_0$ along flow lines to construct a closed embedded surface of genus $2$ transverse to $\phi_t$ as follows (compare \[CLR\], for example). Let $N(\gamma) \subset \Sigma_0$ be a collar neighborhood of the boundary component $\gamma$. Let $\eta:\Sigma_0 \to [0,1]$ be a smooth function supported on $N(\gamma)$, where $\eta^{-1}(1) = \gamma$, and where the derivative of $\eta$ vanishes on $\gamma$. Define $f:\Sigma_0 \to M_\phi$ by $f(x)= \phi_{\eta(x)}(x)$. The map $f$ is an embedding on the interior of $\Sigma_0$ and has $f(\gamma) = \phi(\gamma)$. The image is a closed genus 2 surface $\Sigma$ which is transverse to $\phi_t$.
Let $\tau$ denote the $2$-plane bundle on $M$ defined by the tangent space to the fibers of the fibration $M \to S^1$, and let $e(\tau)$ denote its Euler class. The restriction of $\tau$ to $\Sigma$ is homotopic to the tangent plane bundle, and hence $\psi([\Sigma]) = - e(\tau)\cdot [\Sigma] = -\chi(\Sigma) = 2$. Since $\psi$ is a homomorphism, its image contains $2\Z$ as desired.
Without loss of generality, suppose $d \geq 4$ is even. Let $S$ be a closed surface of genus $d/2$, let $\phi : S \to S$ be a pseudo-Anosov homeomorphism as in Lemma \[L:big bundle small norm\], and set $M = M_\phi$. Since $\phi$ acts trivially on $H_1(S;\R)$, we have $b_1(M) = d + 1$. Let $F$ be the fibered face of $M$ with $S \in \R_+ \cdot F$ and let $\psi:H^1(M;\R) \to \R$ be the homomorphism agreeing with the Thurston norm on $\R_+ \cdot F$. By Lemma \[L:big bundle small norm\], we have $\psi(H^1(M;\Z)) = 2\Z$.
Choosing a basis for $H^1(M;\Z)$ induces an isomorphism $H^1(M;\Z) \cong \Z^{d+1}$ which extends to an isomorphism $H^1(M;\R) \cong \R^{d+1}$. We choose a basis for $H^1(M;\Z)$ so that, with respect to this isomorphism, $\psi$ is given by $$\psi(x_0,\ldots,x_d) = 2 x_0$$ It follows that the face $F$ is contained in the hyperplane $x_0 = 1/2$.
Let $K$ be a closed $d$-cube in $F$. If $K$ is centered at $(1/2,t_1,\ldots,t_d)$ and has side length $2r$, then $$K = \{ (1/2,x_1,\ldots,x_d) \in \R^{d+1} \mid \max_{j=1,\ldots,d} {|x_j - t_j| \leq r} \}.$$ Since $K$ is compact, the function $\Lambda_F$ from Theorem \[T:frieddilatation\] attains a maximum $C$ on $K$. Since $\| \cdot \|\log(\Lambda_F(\cdot))$ is constant on rays (Theorems \[T:thurston\] and \[T:frieddilatation\]), the function $\| \cdot \| \log(\Lambda_F(\cdot))$ restricted to $\R_+ \cdot K$ also has some maximum $L$. Thus, the monodromy of every primitive integral point in $\R_+ \cdot K$ is an element of $\Psi(L)$, and so corresponds to an element of $\cup_g \G_g(L)$.
Now, given $g \geq 2$ the set $$\Omega_g = \{ v \in \Z^{d+1} \cap (2g-2) \cdot K \mid v \mbox{ primitive} \}$$ determines a set of fibers of $M$ with monodromies defining geodesics in $\G_g(L)$. Two different fibers may define the same geodesic in $\G_g(L)$, but only if the monodromies are conjugate. In this case there is a self-homeomorphism of $M$ that sends one fiber to the other. Such a homeomorphism induces an nontrivial isometry of the Thurston norm on $H^1(M;\R)$. Since the unit ball is a polyhedron, this symmetry group is finite of some order $N$ (compare [@ThNorm Corollary of Theorem 1]), and so the map from $\Omega_g$ to $\G_g(L)$ is at most $N$ to $1$.
It remains to show that $|\Omega_g|$ is bounded from below by a degree $d$ polynomial with positive leading coefficient. This is a standard counting argument (cf. [@Ap Theorem 3.9]), and so we content ourselves to explain the idea. Before we begin, we notice that it is enough to check this for large $g$; the statement for arbitrary $g$ is then obtained by subtracting a constant from the polynomial.
We can explicitly describe the $d$-cube $(2g-2) \cdot K$ as: $$(2g-2) \cdot K = \left \{ (g-1,n_1,\ldots,n_d) \mid \max_{j=1,\ldots,d} |n_j - (2g-2)t_j| \leq (2g-2)r \right \}.$$ For $g$ large enough, the number of integral points in this cube is approximately $((4g-4)r)^d$.
If an integral vector $(g-1,n_1,\ldots,n_d) \in (2g-2) \cdot K$ is imprimitive, it must be divisible by one of the prime factors $p$ of $g-1$, and hence must be the $p$th multiple of an integral point in the $d$-cube $(2g-2)/p \cdot K$, which, for large $g$, contains approximately $$\left (\frac{(4g-4)r}{p}\right)^d.$$ integral points. Now, if $p_1,\ldots,p_m$ are the prime divisors of $g-1$, it follows that $$\begin{aligned}
|\Omega_g| &\sim ((4g-4)r)^d - \sum_{i=i}^m \left( \frac{(4g-4)r}{p_i} \right)^d \\
&= ((4g-4)r)^d \left( 1 - \sum_{i=i}^m \frac{1}{p_i^d} \right) \\
&\geq ((4g-4)r)^d \left( 1 - \sum_{n=i}^\infty \frac{1}{n^d} \right) \\
&= C ((4g-4)r)^d.\end{aligned}$$ Since $d \geq 4$, we have $C > 0$, and we are done.
Comparing quadratic differentials {#S:continuity}
=================================
Let $M$ be a finite-volume hyperbolic 3-manifold, and let $F$ be a fibered face. To prove Theorem \[T:location\] we will need to see how the $3$-manifold $M$ influences the geometry of the surfaces lying over the axis for the monodromy $\phi$ of a fiber $S \in \R_+ \cdot F$. We will need uniform control on the geometry of these surfaces as we vary the fibers.
Each monodromy of $M$ acts on Teichmüller space by translation along an axis. Each such axis is defined by a quadratic differential on some Riemann surface. The goal of this section is to describe a construction of McMullen [@Mc] that provides a bridge between the $3$-manifold $M$ and the quadratic differentials corresponding to its various fibers.
Let $\Gamma = \pi_1(M)$ and $\Gamma_0 \triangleleft \Gamma$ be the kernel of the abelianization, modulo torsion: $$1 \to \Gamma_0 \to \pi_1(M) \to H_1(M;\Z)/\text{torsion} \to 1 .$$ Let $\widetilde M \to M$ denote the cover of $M$ associated to $\Gamma_0$. Let $S \in \R_+ \cdot F$ be a connected fiber with monodromy $\phi:S \to S$. The fibration $S \to M \to S^1$ lifts to a fibration over the universal covering $\mathbb R \to S^1$: $$\xymatrix{ \widetilde S \ar[r] \ar[d] & \widetilde M \ar[r]\ar[d] & \mathbb R \ar[d] \\
S \ar[r] & M \ar[r] & S^1}$$ The fiber $\widetilde S$ is a connected cover of $S$—in fact, it is precisely the cover corresponding to the $\phi$-invariant subspace of $H^1(S;\Z)$.
Let $\widetilde \phi_t$ denote the lift to $\widetilde M$ of the suspension flow on $M$ associated to $\phi$. There is a product structure $$\widetilde M \cong \widetilde S \times \mathbb R;$$ indeed, the map $(x,t) \mapsto \widetilde \phi_t(x)$ gives a homeomorphism $\widetilde S \times \mathbb R \to \widetilde M$.
Pulling back the foliations $\F_\pm$ produces foliations $\widetilde \F_\pm$ on $\widetilde S$, and we can suspend these by $\widetilde \phi_t$ to produce foliations $\widetilde \F_\pm^M$ on $\widetilde M$. Alternatively, $\widetilde \F_\pm^M$ is obtained by pulling back $\F_\pm^M$ to $\widetilde M$.
Let $\pi:\widetilde M \to \widetilde S$ denote the map obtained by collapsing each flow line of $\widetilde \phi_t$ to a point: $$\pi(\widetilde \phi_t(x)) = x.$$
Let $\Sigma$ be a fiber in $\R_+ \cdot F$. By Theorem \[T:friedtransverse\], we can assume that $\Sigma$ is transverse to $\phi_t$. Next, let $\widetilde \Sigma$ be one component of the preimage of $\Sigma$ in $\widetilde M$. The first return map of $\phi_t$ is the monodromy $\varphi:\Sigma \to \Sigma$, and from this one can show that $\pi|_{\widetilde \Sigma}:\widetilde \Sigma \to \widetilde \Sigma$ is a homeomorphism; see [@CLR Corollary 3.4]. Since the stable and unstable foliations for $\Sigma'$ are obtained by intersecting $\Sigma$ with $\F_\pm^M$, it follows that this homeomorphism sends the leaves of the lifts of the stable and unstable foliations on $\widetilde \Sigma$ to those on $\widetilde S$. That is, for every connected fiber $\Sigma \in \R_+ \cdot F$, we have identified the cover $\widetilde \Sigma$ homeomorphically with a fixed connected covering $\widetilde S$ of $S$ so that the preimages of the stable and unstable foliations for $\varphi$ under this identification pull back to $\widetilde \F_\pm$.
The monodromy $\varphi$ for $\Sigma$ does determine a pair of transverse measures $\mu_\pm(\Sigma)$ (unique up to scaling) on the stable and unstable foliations, respectively. This defines a complex structure on $\Sigma$ and a holomorphic quadratic differential $q(\Sigma)$ for which the vertical and horizontal measured foliations are $\mu_\pm(\Sigma)$, respectively. Furthermore, $q(\Sigma)$ defines the axis for $\varphi$ on Teichmüller space $\T(\Sigma)$. Pulling $q(\Sigma)$ back to $\widetilde S$, we have a complex structure and holomorphic quadratic differential we denote $\widetilde q(\Sigma)$ on $\widetilde S$ whose vertical and horizontal foliations are precisely $(\widetilde \F_\pm,\widetilde \mu_\pm(\Sigma))$, where $\widetilde \mu_\pm(\Sigma)$ are the measures $\mu_\pm(\Sigma)$ pulled back to $\widetilde S$.
McMullen extends this construction of a complex structure and quadratic differential in a continuous way to every point of $\R_+ \cdot F$, not just the fibers [@Mc]. More precisely, let $Q(\widetilde S,\widetilde{\F}_\pm)$ denote the set of pairs consisting of a complex structure on $\widetilde S$ together with a holomorphic quadratic differential for which the horizontal and vertical foliations are $\widetilde \F _\pm$. We denote a point of $Q(\widetilde S,\widetilde \F_\pm)$ by $\widetilde q$, suppressing the complex structure in the notation. An element $\widetilde q \in Q(\widetilde S,\widetilde \F_\pm)$ determines a Euclidean cone metric for which the leaves of $\widetilde{\F}_\pm$ are geodesics (with the leaves of $\widetilde {\F}_+$ orthogonal to those of $\widetilde {\F}_-$), and by an abuse of notation we denote this metric $\widetilde q$. We topologize $Q(\widetilde S,\widetilde{\F}_\pm)$ with the topology of locally uniform convergence of these metrics. Specifically, a sequence $\{q_n\} \subset Q(\widetilde S)$ converges to $q \in Q(\widetilde S)$ if for any compact set $K \subset \widetilde S$, $q_n:K \times K \to \mathbb R$ converges uniformly to $q:K \times K \to \mathbb R$.
The main consequence of McMullen’s work that we will need is the following.
\[T:mcmullen\] There is a continuous map $\widetilde \q:\R_+ \cdot F \to Q(\widetilde S,\widetilde{\F}_\pm)$ which is constant on rays, and has the property that for every fiber $S \in \R_+ \cdot F$, $\widetilde \q(S) = \widetilde q(S)$, up to scaling and Teichmüller deformation.
The map $\widetilde \q$ is given in [@Mc Theorem 9.3], though it is only defined up to scaling and Teichmüller deformation, and so one must make some choices to obtain a well-defined map. This can be done, for example, by choosing a rectangle with sides in $\widetilde {\F}_\pm$, and then for any $\eta \in \R_+ \cdot F$, we normalize the quadratic differential $\widetilde \q(\eta)$ by requiring that the side lengths are both $1$. Continuity follows from the description in terms of train tracks: the horizontal and vertical foliations are carried by train tracks $\widetilde \tau_\pm$ on $\widetilde S$, and the weights on the branches determined by the vertical and horizontal foliations for $\widetilde \q$ are given by eigenvectors of a continuously varying family of Perron-Frobenius matrices and appropriate equivariance conditions; see the proof of [@Mc Theorem 8.1]. (Our normalization convention can be chosen to correspond to a normalization in the Perron-Frobenius eigenvectors). Since the charts for the Euclidean metric are obtained by integrating these two measures, and since the measures vary continuously, so do the metrics.
Dehn filling {#S:df}
============
Let $M$ be a finite-volume cusped hyperbolic 3-manifold with $r$ cusps. Let $\widehat M$ denote the manifold obtained by removing the interiors of the cusps, so that $\widehat M$ is a compact manifold with $r$ boundary components $\partial_1 \widehat M,\ldots,\partial_r \widehat M$, each homeomorphic to a torus.
A *slope* on $\partial_i \widehat M$ is either the isotopy class of an unoriented essential simple closed curve in $\partial_i \widehat M$ or $\infty$. If we choose a basis for $\pi_1(\partial_i \widehat M) \cong \Z^2$, then a slope $\beta_i \neq \infty$ corresponds to a coprime pair of integers $\beta_i = (p_i,q_i)$, unique up to sign.
Suppose $\beta = (\beta_1,\ldots,\beta_r)$ is a choice of slopes in $\partial_1 \widehat M,\ldots,\partial_r \widehat M$, respectively. The $\beta$-*Dehn filling* of $M$ is the 3-manifold $M(\beta)$ obtained from $\widehat M$ by the following procedure:
- For each $i$ with $\beta_i \neq \infty$, we glue a solid torus $S^1 \times D^2$ to $\partial_i \widehat M$ so that the curve $\{*\} \times \partial D^2$ represents $\beta_i$.
- For each $i$ with $\beta_i = \infty$, we reglue the original cusp (or, what is the same thing, we can leave that cusp alone from the start).
The homeomorphism type of $M(\beta)$ depends only on $\beta$.
We can view the set of slopes on $\partial_i \widehat M$ as points in $\Z^2 \cup \{\infty\} \subset \R^2 \cup \{\infty\} \cong S^2$. We say that a sequence of slopes $\{\beta_i^n\}_{n=1}^\infty$ on $\partial_i \widehat M$ tends to $\infty$ if it does in $\R^2 \cup \{\infty\}$.
The inclusion $\widehat M \to M$ induces an isomorphism $\pi_1(M) \cong \pi_1(\widehat M)$. If we compose this isomorphism with the homomorphism $\pi_1(\widehat M) \to \pi_1(M(\beta))$ induced by inclusion, we obtain a canonical homomorphism $\pi_1(M) \to \pi_1(M(\beta))$. By Van Kampen’s theorem, this is surjective.
The next result, Thurston’s Dehn surgery theorem [@Th Theorem 5.8.2], states that Dehn filling on a hyperbolic $3$-manifold usually produces a hyperbolic $3$-manifold.
\[T:Dehn surgery theorem\] Let $M$ be a finite-volume hyperbolic 3-manifold with $r$ cusps. Suppose $\{ \beta^n\}_{n=1}^\infty$ is a sequence of $k$-tuples of slopes with $\beta^n = (\beta^n_1,\ldots,\beta^n_r)$ and $\beta_i^n$ a slope on $\partial_i \widehat M$ for all $i$ and $n$. Assume that for all $i$, $\beta_i^n$ tends to $\infty$. Then $M(\beta^n)$ is hyperbolic for all but finitely many $n$.
Moreover, for appropriate choices of the holonomy homomorphisms $\pi_1(M),\pi_1(M(\beta^n)) \to \PSL_2(\C)$ within the respective conjugacy classes, the composition $$\pi_1(M) \to \pi_1(M(\beta^n)) \to \PSL_2(\C)$$ converges pointwise to $$\pi_1(M) \to \PSL_2(\C).$$
We will require the following simple application of Theorem \[T:Dehn surgery theorem\].
\[C:Dehn Surgery Corollary\] Let $M$ be a finite-volume hyperbolic 3-manifold with $r$ cusps and let $\alpha \in \pi_1(M)$ be any nontrivial element. For each $i = 1,\ldots,r$, there are finitely many slopes $\beta_i^1,\ldots,\beta_i^{s_i}$ on $\partial_i \widehat M$, $\beta_i^{j} \neq \infty$ for all $j$, so that if $\beta_1,\ldots,\beta_r$ are slopes with $\beta_i \neq \beta_i^j$ for each $i = 1,\ldots,r$ and $j = 1,\ldots,s_i$, then $\alpha$ represents a nontrivial element of $\pi_1(M(\beta_1,\ldots,\beta_r))$.
Lengths of curves
=================
In this section we recall three notions of length for a simple closed curve in a surface $S$ equipped with a complex structure $X$, and we recall various well-known relationships between them. Here, and throughout, we say that a simple closed curve is *essential* if it is homotopic neither to a point nor a puncture. For further details, see [@Ah1; @Ah2; @Ma; @Wo].
A Borel metric on $S$ with respect to $X$ is a metric that is locally given by $\rho(z)|dz|$, where $\rho \geq 0$ is a Borel measurable function and $z$ is a local coordinate for the complex structure $X$.
Let $\alpha$ be a simple closed curve in $S$. The [*extremal length*]{} of $\alpha$ with respect to $X$ is $$\ext_X(\alpha) = \sup_{\rho} \frac{L_\rho(\alpha)^2}{\Area(\rho)},$$ where the supremum is over all Borel metrics $\rho$ in the conformal class of $X$, $L_\rho(\alpha)$ is the infimum of $\rho$-lengths of closed curves in the homotopy class of $\alpha$, and $\Area(\rho)$ is the area of $S$ with respect to $\rho$.
Another number associated to $\alpha$ with respect to $X$ is the [*modulus*]{}. Recall that if an annulus $A$ is conformally equivalent to $\{ z \in \C : 1 < |z| < R \}$, then the modulus of $A$ is $m_X(A) = \log(R)/2\pi$. The modulus of a simple closed curve $\alpha$ is defined as $$m_X(\alpha) = \sup_{A \supset \alpha} m_X(A),$$ where the supremum is taken over all embedded annuli in $S$ containing a curve homotopic to $\alpha$. When $\alpha$ is inessential, then $m_X(\alpha)=\infty$.
We can alternatively define the modulus of $A$ via extremal lengths: $$m_X(A) = \sup_{\rho} \frac{L_\rho(A)^2}{\Area(\rho)},$$ where the supremum is over all Borel metrics $\rho$ in the conformal class of $X$, and where $L_\rho(A)$ is the infimum of the $\rho$-lengths of all paths in $A$ connecting distinct boundary components.
The relationship between modulus and extremal length is provided by the following; see, for example, [@Ah2 Section 1.D].
\[P:extremal and modulus\] Let $\alpha$ be a simple closed curve in $S$ and $X$ a complex structure on $S$. We have $$\ext_X(\alpha) = 1/m_X(\alpha).$$
There is a third measurement associated to a closed curve $\alpha$ with respect to $X$. Suppose $(S,X)$ can by uniformized as a quotient of the hyperbolic plane (for example, if $\chi(S) < 0$), that is, there is a conformal homeomorphism between $(S,X)$ and a quotient of the hyperbolic plane by a discrete, torsion-free subgroup of the orientation-preserving isometry group. In a hyperbolic surface, every essential closed curve has a unique geodesic representative. The length of the geodesic representative of $\alpha$ is thus an invariant of $\alpha$ that we denote $\ell_X(\alpha)$. This is called the *hyperbolic length* of $\alpha$.
Keen’s collar lemma [@Ke] provides a quantitative lower bound on the width of an annular neighborhood of a simple closed geodesic in a hyperbolic surface. From this one obtains lower bounds on the length of a curve intersecting the given curve. This is stated conveniently in terms of the geometric intersection number $i(\alpha,\beta)$ for a pair of simple closed curves $\alpha$ and $\beta$.
\[L:collar\] There is a function $F:\R_+ \to \R_+$ that satisfies $$\displaystyle{\lim_{x \to 0} F(x) = \infty}$$ and also satisfies the following property: if $\alpha$ and $\beta$ are simple closed curves in $S$, then for any $X \in \T(S)$ we have $$\ell_X(\beta) \geq i(\alpha,\beta)F(\ell_X(\alpha)).$$
In fact, we can take the function $F$ from Lemma \[L:collar\] to be $$F(x) = 2\sinh^{-1}\left( \frac{1}{\sinh(x/2)} \right);$$ see, for example, [@primer Lemma 13.6].
Since a hyperbolic metric on $S$ is Borel, the hyperbolic length of a curve can be related to its extremal length directly from the definition. The following result of Maskit gives stronger bounds, independent of the topology of $S$ [@Ma].
\[P:hyperbolic and extremal\] For $\alpha$ an essential closed curve in $S$ we have $$\frac{\ell_X(\alpha)}{\pi} \leq \ext_X(\alpha) \leq \frac{\ell_X(\alpha)}{2} e^{\ell_X(\alpha)/2}.$$
The following result of Wolpert [@Wo Lemma 3.1] relates the distance in Teichmüller space to distortion of hyperbolic lengths.
\[P:wolpert\] Given $X,Y \in \T(S)$ and $\alpha$ an essential closed curve in $S$, we have $$\ell_X(\alpha) \leq e^{d_\T(X,Y)} \ell_Y(\alpha).$$
The next fact, sometimes called the Schwarz–Pick–Ahlfors lemma, states that a holomorphic mapping is a contraction with respect to the hyperbolic metrics on domain and range [@Ah1 Theorem A].
\[T:Schwarz-Pick\] If $f:S \to S'$ is a holomorphic mapping with respect to complex structures $X$ and $Y$ on surfaces $S$ and $S'$, respectively, then $f$ is a contraction with respect to the hyperbolic metrics on the domain and range. In particular, $$\ell_Y(f(\alpha)) \leq \ell_X(\alpha)$$ for any closed curve $\alpha$ in $S$.
Location of short geodesics {#S:locationProof}
===========================
We are now ready to prove Theorem \[T:location\], which states that, given $L > 0$, there exists $\epsilon_2 > \epsilon_1 > 0$ so that, for each $g \geq 1$, we have $$\G_g(L) \subset \M_{g,[\epsilon_1,\epsilon_2]}.$$
Propositions \[P:thick\] and \[P:thin\] below give the containments $\G_g(L) \subset \M_{g,[\epsilon_1,\infty)}$ and $\G_g(L) \subset \M_{g,(0,\epsilon_2]}$, respectively.
\[P:thick\] Let $L > 0$. There exists $\epsilon > 0$ so that $$\G_g(L) \subset \M_{g,[\epsilon,\infty)}.$$
First of all, since $\G_1(L)$ is finite, it suffices to prove the proposition for $g \geq 2$. Indeed, we can take $\epsilon$ to be the minimum of the $\epsilon$’s obtained for $g=1$ and for $g \geq 2$, respectively.
Let $[\phi] \in \Psi_L(S)$ and let $X \in \T(S)$ be a point on the axis for $\phi$. Let $\gamma$ denote the essential closed curve with shortest length $\ell_X(\gamma)$. We must find a uniform lower bound $\epsilon$ for $\ell_X(\gamma)$.
Let $F(x)$ be the function from Lemma \[L:collar\], and let $\epsilon > 0$ be such that $$F(x) > e^{3L}x$$ for every $x < \epsilon$. We will show that $\ell_X(\gamma) \geq \epsilon$.
Say that the genus of $S$ is $g \geq 2$. Any collection of pairwise disjoint, homotopically distinct, essential simple closed curves in $S$ has cardinality at most $3g-3$. Thus, for some $k \leq 3g-2$ we have $i(\phi^k(\gamma),\gamma) \neq 0$. By Lemma \[L:collar\], $$\ell_X(\phi^k(\gamma)) \geq F(\ell_X(\gamma)).$$ On the other hand, by Proposition \[P:wolpert\], we have $$\ell_X(\phi^k(\gamma)) \leq \lambda(\phi^k) \ell_X(\gamma) = \lambda(\phi)^k \ell_X(\gamma) \leq \lambda(\phi)^{3g-2} \ell_X(\gamma)$$ Combining the last two displayed inequalities with the fact that $3g-2 < 3g$ and the assumption that $[\phi] \in \Psi_L(g)$, we have $$\begin{aligned}
F(\ell_X(\gamma)) \leq \ell_X(\phi^k(\gamma)) \leq \lambda(\phi)^{3g-2} \ell_X(\gamma) < \\ \lambda(\phi)^{3g}\ell_X(\gamma) \leq (e^{L/g})^{3g} \ell_X(\gamma) \leq e^{3L} \ell_X(\gamma).\end{aligned}$$ By the definition of $\epsilon$, this implies that $\ell_X(\gamma) \geq \epsilon$, as desired.
The second half of Theorem \[T:location\] is more involved. As the proof is in terms of pseudo-Anosov homeomorphisms rather than geodesics in $\M_g$, we explain a complete translation to that language.
Given a finite volume $3$-manifold $M$ and any subset $K \subset F$ of an open fibered face $F$ of $M$, let $\Psi(L,K)$ denote the set of closed-surface pseudo-Anosov homeomorphisms $(\phi:S \to S) \in \Psi(L)$ such that, after removing some $\phi$-invariant subset of the singular points of the stable foliation, the resulting surface $S'$ is a fiber in $\R_+ \cdot K$ with monodromy $\phi' = \phi|_{S'}$. We emphasize that $S'$ is obtained by removing none, some, or all of the singular points of the closed surface $S$.
Given $X \in \T(S)$, let $\inj(X)$ denote the $X$-hyperbolic length of the shortest essential closed curve. Given a pseudo-Anosov homeomorphism $\phi:S \to S$, write $\inj(\phi)$ to denote the maximum of $\inj(X)$ as $X$ varies over all complex structures in $\T(S)$ lying on the axis for $\phi$. For any pseudo-Anosov $\phi : S_g \to S_g$ with associated geodesic $\gamma_\phi \subset \M_g$ we have: $$\inj(\phi) \leq \epsilon \Longleftrightarrow \gamma_\phi \subset \M_{g,(0,\epsilon]}.$$ Finally, we define $$\Psi_{(0,\epsilon]} = \{ \phi: S \to S : S \text{ any surface}, \inj(\phi) \leq \epsilon \}.$$ This notation should remind the reader of $\M_{g,(0,\epsilon]}$, as each $\phi : S_g \to S_g$ in $\Psi_{(0,\epsilon]}$ corresponds to a geodesic contained in $\M_{g,(0,\epsilon]}$.
\[P:thin\] For every $L > 0$, there exists $\epsilon > 0$ so that, for every $g \geq 1$, we have $$\G_g(L) \subset \M_{g,(0,\epsilon]}.$$ Equivalently, $$\Psi(L) \subset \Psi_{(0,\epsilon]}.$$
Again, since $\G_1(L)$ is finite, it is enough to prove the proposition for $g \geq 2$. Fix $L > 0$. We will prove the following statement by induction on $r$.
> *Let $M$ be a hyperbolic 3-manifold with $r \geq 0$ cusps. There is an $\epsilon(M)$ so that, for each open fibered face $F$ of $M$, we have $\Psi(L,F) \subseteq \Psi_{(0,\epsilon(M)]}$.*
The proposition then follows by taking $\epsilon$ to be the maximum of $\epsilon(M)$, where $M$ ranges over the finite set of manifolds $\mathcal{T}(L)$ given by Theorem \[T:universal\].
We first treat the case $r = 0$, that is, the case where $M$ is closed. Besides serving as the base case for the induction, this case will also explain the main ideas for the more complicated inductive step.
Fix a closed, fibered, hyperbolic $M$. Since $M$ has finitely many fibered faces (Theorem \[T:thurston\]), it suffices to show that, given some such face $F$, there is an $\epsilon(F)$ so that $$\Psi(L,F) \subseteq \Psi_{(0,\epsilon(F)]}.$$
It follows from Theorem \[T:frieddilatation\] that there is a compact subset $K$ of the open face $F$ with the property that: $$\Psi(L,F) = \Psi(L,K).$$ Thus, it suffices to show that, for any such $K$, there is an $\epsilon(K)$ so that $$\Psi(L,K) \subseteq \Psi_{(0,\epsilon(K)]}.$$
Fix a fibered face $F$ of $M$, and let $\Gamma_0$, $\Gamma$, $\widetilde M$, $\widetilde S$, $\widetilde{\F}_\pm$, and $\widetilde \q:\R_+ \cdot F \to Q(\widetilde S,\widetilde{\F}_\pm)$ be the objects associated to $F$ as in Section \[S:continuity\]. As above, fix $K \subseteq F$ so that $\Psi(L,F)=\Psi(L,K)$.
Let $\alpha$ be any essential simple closed curve in $\widetilde S$. We take an annular neighborhood $A$ of $\alpha$ in $S$. By Theorem \[T:mcmullen\], $\widetilde \q(K)$ is a compact subset of $Q(\widetilde S,\widetilde{\F}_\pm)$. Thus, we obtain a uniform lower bound on the $\widetilde \q(\eta)$-distance between the boundary components of $A$ and a uniform upper bound on the $\widetilde \q(\eta)$-area, for every $\eta \in \R_+ \cdot K$. Consequently, the modulus $m_{\widetilde\q(\eta)}(A)$ is uniformly bounded below, and hence so is $m_{\widetilde \q(\eta)}(\alpha)$. By Proposition \[P:extremal and modulus\], the $\widetilde \q(\eta)$-extremal length of $\alpha$ is then uniformly bounded from above. Then, by Proposition \[P:hyperbolic and extremal\], the $\widetilde \q(\eta)$-hyperbolic length is bounded from above. Denote this uniform bound by $C$.
Suppose $(\phi :S \to S) \in \Psi(L,K)$, where $S=S_g$. (Since $M$ is closed, we do not remove any points of $S$ in order to obtain a fiber in $\R_+ \cdot K$.) Let $p: \widetilde S \to S$ be the covering map. The quadratic differential $\widetilde \q(S)$ descends to the quadratic differential $q(S)$ on $S$ defining the axis for $\phi$ in $\T(S)$. Let $X_0$ be the point on the axis corresponding to the underlying complex structure of $q(S)$. It follows that $p(\alpha)$ is a closed essential curve in $S$ with $X_0$-hyperbolic length at most $C$.
Since $\phi \in \Psi_g(L)$, we have $$\log(\lambda(\phi)) \leq L/g.$$ If $X \in \T(S)$ is any point along the axis of $\phi$ between $X_0$ and $\phi(X_0)$, then $$d(X_0,X) \leq d(X_0,\phi(X_0)) \leq L/g,$$ and hence Propostion \[P:wolpert\] implies $$\ell_X(\alpha) \leq e^{L/g} \ell_{X_0}(\alpha) \leq e^{L/g} C \leq e^L C.$$ For any other point $X \in \T(S)$ on the axis for $\phi$, there is an $n$ so that $\phi^n(X)$ lies between $X_0$ and $\phi(X_0)$, and hence $$\ell_X(\phi^{-n}(\alpha)) = \ell_{\phi^n(X)}(\alpha) \leq e^L C.$$ Thus, $\inj(\phi) \leq e^L C$. As this bound is independent of the choice of $\phi : S \to S$ in $\Psi(L,K)$, we can set $\epsilon(F) = \epsilon(K) = e^L C$, and this completes the proof in the base case.\
We are now ready for the inductive step. Let $M$ be a 3-manifold with $r > 0$ cusps. As in the base case, it suffices to focus on a single fibered face $F$ and a compact subset $K \subset F$ with $\Psi(L,F) = \Psi(L,K)$.
Let $\widetilde S$ be the common cover for all fibers in $\R_+ \cdot K$, as in Section \[S:continuity\]. We can carry out the same argument as in the base case in order to find an essential curve $\alpha$ in $\widetilde S$ with $\widetilde \q(\eta)$-hyperbolic length at most $C$ for all $\eta \in \R_+ \cdot K$. Then, for any $(\phi:S \to S) \in \Psi(L,K)$ with $S' \in \R_+ \cdot K$ the punctured fiber, if we let $X' \in \T(S')$ denote the underlying complex structure for $q(S')$, then $\ell_{X'}(p(\alpha)) \leq C$. Moreover, if $X \in \T(S)$ is the complex structure on $S$ obtained by filling in the punctures and extending $X'$, then by Theorem \[T:Schwarz-Pick\] $$\ell_X(p(\alpha)) \leq C.$$ Thus, as long as $p(\alpha)$ remains an essential curve after filling in the punctures, we can argue just as in the base case and prove $\inj(\phi) \leq e^L C$. In other words, we have shown that if $(\phi:S \to S) \in \Psi(L,F)$ and $p(\alpha)$ is essential in $S$, then $\phi \in \Psi_{(0,N]}$, for $N = e^L C$. It remains to deal with the cases where $p(\alpha)$ is inessential in $S$. That is, we must find $N'$ so that any $(\phi:S \to S) \in \Psi(L,F)$ with $p(\alpha)$ inessential in $S$ is contained in $\Psi_{(0,N']}$. Then we may set $\epsilon(F) = \max\{N,N'\}$.
We will first define $N'$, and then prove that it satisfies the above statement. For each $i = 1,\ldots,r$, let $\beta_i^1,\ldots,\beta_i^{s_i}$ be the slopes from Corollary \[C:Dehn Surgery Corollary\], and define Dehn fillings $$M_i^j = M(\infty,\ldots,\infty,\beta_i^j,\infty,\ldots,\infty).$$ The manifold $M_i^j$ has $r-1$ cusps. Therefore, by induction, there are real numbers $\epsilon(M_i^j)$, so that if $F$ is any fibered face of $M_i^j$, then $\Psi(L,F) \subset \Psi_{(0,\epsilon(M_i^j)]}$. Let $$N' = \max \{ \epsilon(M_1^1),\ldots,\epsilon(M_r^{s_r}) \}.$$
Let $\phi:S \to S$ be an element of $\Psi(L,F)$ with $p(\alpha)$ inessential in $S$. We must show that $\phi \in \Psi_{(0,N']}$. The idea is to show that, up to removing singularities, $\phi$ is the monodromy for some $M_i^j$.
We can view the mapping torus $M_\phi$ as being obtained from $M$ by Dehn filling: $$M_\phi = M(\beta_1,\ldots,\beta_r).$$ Let $S'$ denote the fiber in the cone over $F$ corresponding to $S$; recall that $S'$ is obtained by removing from $S$ a set of singular points of the foliations for $\phi$.
Each slope $\beta_i \neq \infty$ is the intersection of $S'$ with the corresponding boundary component of the truncated manifold $\widehat M$; see Section \[S:df\]. In particular it makes sense to write $\beta_i = \beta_i(S')$.
If $p(\alpha)$ is not essential in $S$, then $p(\alpha)$ must be trivial in $$M_\phi = M(\beta_1(S'),\ldots,\beta_r(S'))$$ and hence $\beta_i(S') = \beta_i^j$ for at least one $i \in \{1,\ldots,r\}$ and some $j \in \{1,\ldots,s_i\}$. It follows that the manifold $M_i^j$ defined above fibers with fiber $S''$ where $S' \subset S'' \subset S$, and $S''$ is obtained from $S'$ by adding in the $\phi$-orbit of the singular point corresponding to the $i$th cusp of $M$.
Suppose $F_i^j$ is the open face of $M_i^j$ with $S'' \in \R_+ \cdot F_i^j$. Since $\phi \in \Psi(L,F)$, it follows that $$\phi \in \Psi(L,F_i^j) \subset \Psi_{(0,\epsilon(M_i^j)]} \subset \Psi_{(0,N']},$$ as desired.
*(1)* It is conceivable that one might be able to find a single curve $\alpha \in \widetilde S$ which when projected to any fiber remains essential after filling in the missing singular points, simplifying the proof, though it is not clear how to find such a curve.
*(2)* It can happen that a punctured surface has a hyperbolically short essential closed curve, while the filled in surface has no short curves. For example, start with a closed surface; it has some shortest essential curve. Next, puncture the surface at two points. By taking these points to be close together, a curve surrounding these two punctures can have an annular neighborhood of arbitrarily large modulus, and so this curve is arbitrarily short on the punctured surface. This short curve must become inessential when the punctures are filled back in.\
[^1]: The authors gratefully acknowledge support from the National Science Foundation and the Sloan Foundation
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---
abstract: 'We present a variational treatment of the ground state of the 2-leg $t$-$J$ ladder, which combines the dimer and the hard-core boson models into one effective model. This model allows us to study the local structure of the hole pairs as a function of doping. A second order recursion relation is used to generate the variational wave function, which substantially simplifies the computations. We obtain good agreement with numerical density matrix renormalization group results for the ground state energy in the strong coupling regime. We find that the local structure of the pairs depends upon whether the ladder is slightly or strongly dopped.'
address:
- 'Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106 '
- ' Departamento de Física Teórica I, Universidad Complutense, 28040 Madrid, Spain '
- 'Instituto de Estructura de la Materia, C.S.I.C., 28006 Madrid, Spain '
- 'Department of Physics and Astronomy, University of California, Irvine, CA 92697 '
- ' Department of Physics, University of California, Santa Barbara, CA 93106 '
author:
- 'Germán Sierra[^1]'
- 'Miguel Angel Martín-Delgado[^2]'
- 'Jorge Dukelsky[^3]'
- 'Steven R. White[^4]'
- 'D.J. Scalapino[^5]'
title: ' The dimer-hole-RVB state of the 2-leg t-J ladder: A Recurrent Variational Ansatz'
---
Introduction {#introduction .unnumbered}
============
The 2-leg, $t-J$ ladder represents one of the simplest systems which exhibits some of the phenomena associated with high $T_c$ cuprate superconductivity [@DRS; @BDRS; @GRS; @WNS; @TTR94; @TTR96]. The ground state of the undoped system, a 2-leg Heisenberg ladder, is a spin liquid with a finite spin gap and exponentially decaying antiferromagnetic spin-spin correlations. Upon doping, the spin gap remains and there appear power law CDW and singlet superconducting pairing correlations. In addition, the pairing correlations have an internal $d_{x^2-y^2}$-like symmetry with a relative sign difference between the leg and rung singlets which make up a pair. Despite all of the numerical and analytical work which has been done on this system, we still lack a picture of the ground state which accommodates all of these physical properties. There are, however, many hints of what that picture may look like. It is the purpose of this paper to take one step further in that direction.
Short-range resonating valence bonds (RVB) provide a useful basis for representing the ground state of spin liquids[@KRS; @LDA]. For the $t-J$ ladder, a $0^{\rm th}$-order picture has been provided by the study of the strong coupling limit where the exchange coupling constant along the rungs, $J'$, is much larger than any other scale in the problem. The other coupling constants of the model are, $J$: the exchange coupling constant along the legs, and $t$ and $t'$: the hopping parameters along the legs and the rungs respectively. In the limit $J' >> J, t, t'$, the ground state of the undoped ladder is simply given by the coherent superposition of singlets across the rungs. Addition of one hole requires the breaking of one of these singlets, in which case the hole gets effectively bound to the unpaired spin, becoming a quasiparticle with spin $1/2$ and charge $|{\rm e}|$. Addition of another hole leads to the binding of two holes in the same rung in order to minimize the cost in energy. In this picture there is no spin-charge separation, a fact that remains valid down to intermediate and weak couplings, as confirmed by various numerical and analytical studies. Based on this picture it is possible to construct an effective theory describing the motion and interactions of the hole pairs [@TTR96]. It is given by a hard-core boson model (HCB) characterized by an effective hopping parameter $t^*$ and interaction $V^*$ of the hole pairs. The HCB model describes the doped ladder as a Luther-Emery liquid, with gapped spin excitations and gapless charge collective modes, which are responsible for the CDW and SC power law correlations. We summarize the $0^{\rm th}$-order picture in Figure 1, which shows a typical state of HCB’s, as well as the two building blocks that are used its construction.
In order to go beyond this picture, we need to consider the fluctuations of the states of the HCB model. To lowest order in perturbation theory they are shown in Fig. 2. The admixture of the state shown in Fig. 2(a) is of order $J/J'$ and represents a resonance of two nearest neighbor rung singlets. According to the standard RVB scenario, this resonance effect leads to a substantial lowering of the ground state energy. The state in Fig. 2(b) is of order $t/J'$, and it can be though of as a bound state of two quasiparticles, whose characteristic feature is the diagonal frustrating bond across the holes. From the RVB point of view, Fig. 2(b) is a resonance of a singlet and a hole pair. The importance of this state, even for intermediate couplings such as $J=J'=0.5 t$, was emphasized in the DMRG study of reference [@WS], where it was shown to be the most probable configuration of two dynamical holes in a 2-leg ladder. In the HCB model of [@TTR96], the states of the form of Fig. 2(b) are taken into account as intermediate or virtual states, which lead to the effective hoping, $t^*$ and interaction, $V^*$ between the hole pairs. It is clear however that “integrating out" the diagonal states through perturbation theory, erases the internal structure of the hole pairs. Here we want to extend the HCB description to include the internal structure of the hole pairs.
In order to define an effective model which would retain the degrees of freedom associated with the internal structure of the hole pairs, we need to consider the states that appear in second order in the strong coupling expansion. They are given in Fig. 3. Let us comment on them. The state of Fig. 3(a) is of order $(J/J')^2$ and it is a higher order RVB state, whose contribution to the ground state of the undoped ladder was studied in [@SMD]. In this reference it was shown that its inclusion in a variational ansatz improves the numerical results, but does not change the qualitative picture obtained using the dimer ansatz [@WNS; @FM]. The state of Fig. 3(b), which is in fact first order in $t'$, can be seen as a bound state of two quasiparticles, while 3(c) and 3(d) are higher order corrections to the diagonal state shown in Fig. 2(b). For these reasons it seems consistent to keep the state 3(b) on an equal footing with the states 2(a) and 2(b). To give further support to this choice, we notice that the exact solution for two holes on the $2\times 2$ cluster requires a superposition of the states shown in 2(b) and 3(b) along with 1(a) and 1(b) (see Fig. 4) [@WS].
In summary, we conjecture that in order to discuss the nature of the superconducting order parameter of the doped 2-leg, $t-J$ ladder, in the strong coupling regime, it is sufficient to consider states built up from 5 possible local configurations, given by rung-singlet-bonds (Fig. 1(a)), rung-hole-pairs (Fig. 1(b)), two-leg-bonds (Fig. 2(a)), hole-pairs with a singlet diagonal bond (Fig. 2(b)) and hole-pairs with a singlet leg bond (Fig. 3(b)). A typical state constructed using these building blocks is shown in Fig. 5. We shall call these types of states [*dimer-hole-RVB*]{} states. The effective model that governs their dynamics will be called the [*dimer hard-core boson model*]{} (DHCB) and its Hamiltonian can be determined by considering the fluctuations of the dimer-hole states, in a manner similar to the one considered above for the HCB states. The DHCB model contains spin and charge degrees of freedom, together with their couplings, and in that sense is an interesting model to study the interplay between the two types of degrees of freedom, although here we will focus on the variational ground state of the model.
The mathematical formulation of the DHCB model involves an interesting but complicated combination of vertex and Interaction Round a Face (IRF) models. The latter terminology is borrowed from Statistical Mechanics [@B]. The vertex variables describe the number of electrons per rung, i.e. $n_i =0,1,2$, while the IRF variables describe the number and type of bonds connecting two rungs, i.e. $\ell_{i,i+1} = 0, 1_d, 1_h, 2$, where the subindices $d$, $h$ indicate the diagonal or horizontal nature of the bond. The only allowed configurations for two consecutive IRF variables $(\ell_{i,i+1}, \ell_{i+1,i+2})$ are: $(0,0), (1_d,0), (1_h,0), (2,0)$ together with their permutations. Moreover the vertex variables are subject to certain constraints imposed by the IRF ones. Namely, A) if $\ell_{i,i+1}= 1_d $ or $1_h$ then $n_i = n_{i+1} = 1,$ and B) if $\ell_{i,i+1}= 2$ then $n_i = n_{i+1} = 2$. Only if $\ell_{i,i+1}=0$ can $n_i$ and $n_{i+1}$ take any value, i.e. 0, 1 or 2.
It is beyond the scope of this work to present a full account of the DHCB model. Instead, we shall try to uncover some of its physics, by means of a combination of two approaches, namely the Density Matrix Renormalization Group[@W] and the Recurrence Relation Method (RRM)[@SMD]. While the DMRG is a powerful numerical technique, which in many cases yields the exact answer, the RRM is essentially analytic, lacking the numerical precision of the DMRG, but sharing with it some features, as for example the Wilsonian way of growing the system by the addition of sites at the boundary. In the RRM one begins with an assumption about the local configurations through which the system grows. Then one may test whether the state that is generated gives results in agreement with the essentially exact DMRG results.
The variational wave function {#the-variational-wave-function .unnumbered}
==============================
The Hamiltonian of the 2-leg, $t-J$ ladder is given by,
$$\begin{aligned}
& {\cal H} = {\cal H}_S + {\cal H}_K =
\sum_{\langle i,j \rangle} J_{ij} \;
( {\bf S}_i \cdot {\bf S}_j - \frac{1}{4} n_i n_j) & \nonumber \\
&- \sum_{\langle i,j \rangle , s} t_{ij} \; P_G \;
( c^\dagger_{i , s} c_{j,s}
+ c^\dagger_{j , s} c_{i,s} ) \; P_G &
\label{1}\end{aligned}$$
where $J_{ij}, t_{ij} = J, t$ or $J', t'$, depending on whether the link $\langle ij \rangle$ is along the legs or the rungs respectively. $P_G$ is the Gutzwiller projection operator which forbids double occupancy. The rest of the operators appearing in (\[1\]) are standard (we use the conventions of reference [@WS]). Each site $i$ is labelled by the coordinates $(x,y)$ with $x= 1, \dots , N$ and $y= 1,2$. We choose open boundary conditions along the legs of the ladder.
The pair field operator which creates a pair of electrons, at the sites $i$ and $j$, out of the vacuum is given by,
$$\Delta^\dagger_{i,j} = \frac{1}{\sqrt{2}}
( c^\dagger_{i , \uparrow} c^\dagger_{j,\downarrow}
+ c^\dagger_{j , \uparrow} c^\dagger_{i,\downarrow} )
\label{2}$$
As explained in the introduction, we want to built up an ansatz for the ground state based on the 5 local configurations of the DHCB model. The explicit realization of these configurations in terms of pair field operators are given by ( see Fig.6),
$$\begin{array}{rl}
| \phi_{1,1} \rangle_x = & |0 \rangle_x \\
| \phi_{1,0} \rangle_x = & \Delta^\dagger_{(x,1) (x,2)}\,\, |0\rangle_x \\
| \phi_{2,0} \rangle_{x,x+1} =& -u\,\, \Delta^\dagger_{(x,1) (x+1,1)} \,\,
\Delta^\dagger_{(x,2) ( x+1, 2)} \,\, |0\rangle_{x,x+1} \\
| \phi_{2,1} \rangle_{x,x+1} = &
[ b\,\, ( \Delta^\dagger_{(x,1) (x+1, 2)}
+ \Delta^\dagger_{(x,2) ( x+1, 1)} ) \\
& + c\,\,
( \Delta^\dagger_{(x,1) ( x+1, 1)} + \Delta^\dagger_{(x,2) ( x+1, 2)} ) ]
\,\, |0 \rangle_{x,x+1}
\end{array}
\label{3}$$
where $|0\rangle_x$ is the Fock vacuum associated with the rung labelled by the coordinate $x$ ( $|0\rangle_{x,x+1} = |0\rangle_x \otimes |0\rangle_{x+1} $). The states $|\phi_{n,p} \rangle$, involve $n=1,2$ rungs and $p=0,1$ pairs of holes. The variational parameter $u$ gives the amplitude of the resonance of a pair of bonds between vertical and horizontal positions [@SMD], while $b$ and $c$ are the variational parameters associated with the diagonal and horizontal configurations of two holes respectively. In the strong coupling limit, $J' >> J, t, t'$, we expect to find $ u \sim J/J', \,\,b \sim t/J' $ and $ c \sim t t'/J'^2$.
Let us call $|N,P \rangle$ the ground state of a ladder with $N$ rungs and $P$ pairs of holes. Of course we should be in a regime of the coupling constants where there is binding of two holes. The state $|N,P \rangle$ will be in general a linear superposition of the dimer-hole states of Fig.5, which suggests that working with this sort of states could be a formidable task. Fortunately, we can apply the method developed in [@SMD] to generate $|N,P \rangle$ in a recursive manner, in terms of the states of the ladders with $N-1$ and $N-2$ rungs, and $P$ and $P-1$ pairs of holes. In [@SMD] it was shown that $|N ,P =0 \rangle$, which is in fact a dimer-RVB state [@WNS; @FM], can be generated by a second order recursion relation. Then by a simple procedure one can compute overlaps and expectation values of different operators using recursion formulas, whose thermodynamic limit can be studied analytically.
Following the strategy of considering first the HCB states and then the DHCB ones, we shall give the rule that generates the former type of states. It is given by the first order recursion relation,
$$|N+1,P+1 \rangle = |N,P+1 \rangle \, |\phi_{1,0} \rangle_{N+1}
+ \, |N,P \rangle \, |\phi_{1,1} \rangle_{N+1}
\label{4}$$
supplemented with the initial conditions,
$$\begin{aligned}
& |1,0 \rangle = |\phi_{1,0}\rangle & \nonumber \\
& |1,1 \rangle = |\phi_{1,1}\rangle & \label{5} \\
& |N, P \rangle = 0, \,\,\,{\rm for} \,\, N < P & \nonumber\end{aligned}$$
Calling $F^{HCB}_{N,P}$ the number of linearly independent states contained in $|N,P\rangle$, we deduce from Eq.(\[4\]) the recursion relation,
$$F^{HCB}_{N+1, P+1} = F^{HCB}_{N, P+1} + F^{HCB}_{N, P}
\label{6}$$
whose solution is given by the combinatorial number,
$$F^{HCB}_{N,P} = \left( \begin{array}{c} N \\ P \end{array} \right)
\label{7}$$
Eq. (\[7\]) is the dimension of the Hilbert space of the HCB model with $N$ sites and $P$ pair of holes. We have not introduced variational parameters in Eqs. (\[5\]), but if we did, then all states of the Hilbert space of the HCB model would be generated by the first order recursion relation. It may be worthwhile to recall that the HCB model is essentially equivalent to the spinless fermion model or the XXZ model [@TTR96].
Turning now to the DHCB model, the key point is to realize that the dimer-hole states can be generated by the following second order recursion relation, involving the local configurations given by eq.(\[3\]),
$$\begin{aligned}
& |N+2, P+1 \rangle = |N+1, P+1 \rangle \,\, | \phi_{1,0} \rangle_{N+2}
& \nonumber \\
& + |N+1, P \rangle \,\, | \phi_{1,1} \rangle_{N+2} +
|N, P+1 \rangle \,\, | \phi_{2,0} \rangle_{N+1,N+2} +
|N, P \rangle \,\, | \phi_{2,1} \rangle_{N+1,N+2} & \label{8} \end{aligned}$$
with the initial conditions (\[5\]). See Fig.7 for a graphical representation of (\[8\]).
[**Counting dimer-hole states**]{}
Let $F_{N,P}$ denote the number of dimer-hole states of a 2-leg ladder with $N$ rungs containing $P$ pairs of holes. According to (\[8\]) they satisfy the recursion relation
$$F_{N+2,P+1} =F_{N+1,P+1} + F_{N,P+1}+ F_{N+1,P} +4 F_{N,P}
\label{9}$$
with the initial conditions
$$F_{N,N} = 1, \,\, F_{N,P} =0 \,\,\, {\rm for } \,\,\, \, N < P
\label{10}$$
From (\[9\]) and (\[10\]) we deduce that $F_{N,0}$ satisfies the well known Fibonacci recursion formula [@SMD], and that in the limit of very large $N$ it grows exponentially,
$$F_{N,0} \sim \Phi_0^N, \,\,\,( N>>1)
\label{11}$$
where $\Phi_0 = \frac{1}{2} ( 1 + \sqrt{5})$ is the golden ratio. Using generating function methods[@SMD] one can easily solved the recursion relation (\[9\]), together with the initial condition (\[10\]). The result is given by the contour integral,
$$F_{N,P} = \oint \frac{d z}{2 \pi i}
\frac{ z^{N+1} \, ( z +4)^P }{ (z^2 - z -1)^{P+1} }
\label{12}$$
where the contour encircles the singularities of the integrand. For $P=0$ the integrand has two simple poles at the zeros of the polynomial $z^2-z-1$, the largest of which is precisely the golden ratio $\Phi_0$. In this way one gets Eq.(\[11\]). For a finite number of holes the residue formula applied to (\[12\]) yields, to leading order in $N$
$$F_{N,P} \sim N^P\,\, \Phi_0^N, \,\,\,\, N >> 1, \,\,
P : {\rm finite}
\label{13}$$
where the proportionality constant depends only on $P$. Let us finally consider the limit where both $N$ and $P$ go to infinity, while keeping their ratio fixed,
$$x = \frac{ {\rm Number }\,{\rm of } \,{\rm holes} }{
{\rm Number }\,{\rm of } \,{\rm sites}}=
\frac{P}{N}\ , \quad 0 \leq x \leq 1\
\label{14}$$
Here $x$ can be identified with the hole doping factor of the state $|N,P\rangle$. The saddle point method applied to (\[12\]) gives the asymptotic behaviour of the number of dimer-hole states for a finite density of holes,
$$F_{N,P} \sim f(x)^N ,\,\,\,\,\,\,\, f(x) =
\frac{ \Phi ( \Phi + 4)^x }{ (\Phi^2 - \Phi -1)^x}
\label{15}$$
where $\Phi = \Phi(x)$ is the highest root of the following equation
$$x = \frac{ (\Phi^2 - \Phi - 1)( \Phi + 4)}{ \Phi ( \Phi^2 + 8 \Phi -3)}
\label{16}$$
The function $f(x)$ is depicted in Fig. 8. Observe that $\Phi(0) = \Phi_0$. The effect of a finite density of holes is that of moving a singularity. This phenomena also occurs in the computation of the energy, and other observables.
[**Ground State Energy**]{}
The parameters $u,b,c$ are found by the standard minimization of the mean value of the energy $\langle N,P| H_N | N,P\rangle/\langle N,P|N,P \rangle $, where $H_N$ denotes the Hamiltonian of the ladder with $N$ rungs. The usefulness of Eq.(\[8\]) is that it implies that the wave function and energy overlaps also satisfy recursion relations. Let us define the following quantities,
$$\begin{array}{cl}
Z_{N,P} = & \langle N,P | N,P \rangle \\
Y_{N,P} = & _N \langle \phi_{1,0}| \langle N-1, P| N,P \rangle \\
E_{N,P} = & \langle N,P| H_N | N,P \rangle \\
D_{N,P} = & _N \langle \phi_{1,0}| \langle N-1, P| H_N |N,P \rangle \\
W_{N,P} = & \langle N,P| n_N | N,P \rangle
\end{array}
\label{17}$$
where $n_N$ is the number operator acting on the rung $N$. The off-diagonal overlaps arise from the cross terms when applying (\[8\]) to the ket and the bras in $\langle N+2,P+1|N+2,P+1 \rangle
$ and $\langle N+2,P+1|H_{N+2}|N+2,P+1 \rangle $. The recursion relations satisfied by (\[17\]) are given by,
$$\begin{array}{rl}
Z_{N+2, P+1} = & Z_{N+1, P+1} + u^2 \,\, Z_{N, P+1} + u \,\,
Y_{N+1, P+1} + \,\, Z_{N+1, P} + 2(b^2 +c^2) \,\, Z_{N, P} \\
Y_{N+2, P+1} = & Z_{N+1, P+1} + u/2 \,\, Y_{N+1, P+1} \\
E_{N+2,P+1} = & E_{N+1,P+1} - J' \, Z_{N+1,P+1} + u^2 \,
E_{N,P+1} - (2 J + J'/2) u^2 \, Z_{N,P+1} + \, E_{N+1,P} \\
& + 2 ( b^2 + c^2) E_{N,P}
- (2 J c^2 +4 b t + 8 b c t') \, Z_{N,P}
+ u D_{N+1,P+1} \\
&-2 u (J +J'/2) Y_{N+1, P+1} - 4 t b Y_{N+1,P}\\
& - \frac{1}{4} J \, W_{N+1, P+1} - \frac{1}{4} J u^2 W_{N, P+1}
- \frac{1}{4} J ( b^2 + c^2) W_{N,P} \\
D_{N+2,P+1} = & E_{N+1,P+1} - J' Z_{N+1, P+1}
+u/2 D_{N+1, P+1} - u (J +J'/2) Y_{N+1, P+1} \\
& - 2 t b Z_{N,P} - \frac{1}{4} J\, W_{N+1, P+1} \\
W_{N+2, P+1} = & 2 Z_{N+1, P+1} + 2 u^2 \, Z_{N,P+1}
+ 2 ( b^2 + c^2) Z_{N,P} + 2 u Y_{N+1, P+1}
\end{array}
\label{18}$$
The initial conditions read,
$$\begin{array}{ll}
Z_{0,0} =1,& Y_{0,0} = E_{0,0}= D_{0,0}= W_{0,0} =0 \\
X_{N,P} = 0, & {\rm for }\,\,\,\, N < P \,\,\, {\rm and}\,\,
X = Z, Y, E, D, W \end{array}
\label{19}$$
For finite values of $N$ and $P$, and given choices of $u,b,c$, one can iterate numerically the recursion relation (\[18\]) using the initial conditions (\[19\]) and look for the minimum of the ground state energy $E_{N,P}/Z_{N,P}$. We give below the results obtained using this variational method for a $2\times 32$ ladder and compare them with the corresponding results obtained with the DMRG.
The RRM wave function versus the DMRG: numerical results {#the-rrm-wave-function-versus-the-dmrg-numerical-results .unnumbered}
========================================================
As explained in the introduction the DHCB model is the appropiate framework to study the strong coupling limit of the 2-leg ladder, if one wishes to take into account the local structure of the hole pairs. To check the validity of this asumption we have studied the cases where the coupling constants takes the following values, $t=t'=1, J =0.5$ and $J'= 0.5, 1,2,3,4$ and 5. In this manner we go from the intermediate coupling regime, i.e. $J' \sim 1$ to the strong coupling regime $J' \gtrsim 3.5$. We are always working in a non-phase-separated region.
In Figure 9 we show the ground state energy of the 2 x 32 ladder, for the previous choices of parameters, computed with the RRM for all dopings and the DMRG for $x= 1/8, 1/2$ and 7/8. One sees that the results obtained with the RRM wave function agree reasonably well with those of the DMRG and their accuracy improves as $J'$ increases.
The kinetic energy of the ladder is shown in Fig.10. It has the pattern expected for a collective charge mode, as described by the HCB and the DHCB models. The similarity between this figure and Fig.8 have a common origin. They both correspond to holes moving collectively through the spins in a complicated many body state.
Fig.10 shows the existence of an optimal doping for which the kinetic energy is a minimum. The existence and position of this optimal doping depends on the values of the coupling constants.
The nature of this many body state is clarified by figures 11, 12 and 13 where we show the values of the variational parameters $u,b$ and $c$ as functions of the doping $x$ for different coupling constants. The parameter $u$ starts from a positive value corresponding to the undoped ladder [@SMD], and it decreases upon doping until a critical value $x_c(J/J')$, where it vanishes. For higher dopings $u$ becomes negative. For the undoped ladder the parameter $u$ can be interpreted as the square of the RVB amplitude $h_{{\rm RVB}}$ for having a bond along the legs[@SMD]. The analogue amplitude for a bond along the rungs has been implicitly normalized to 1. For low doping, i.e. $x < x_c$, since $u(x) > 0$, we can similarly define a doping dependent amplitude for a leg-bond as
$$u(x) = h_{{\rm RVB}}^2(x) > 0, \;\;\;\;( x < x_c)
\label{20}$$
In order to fulfill the Marshall theorem for the undoped ladder one requires the RVB amplitude $h_{{\rm RVB}}(0)$ to be positive [@LDA], which explains why $u(0)$ is also positive. Actually for the positivity of $u(0)$ one just need $h_{{\rm RVB}}(0)$ to be a real number. At $x=0$ $h_{{\rm RVB}}(0)$ increases with $J/J'$ due to the resonance between rung and leg singlets, according to the RVB scenario. Upon doping, however, the holes give rise to destructive interference which degrades progressively the aforementioned resonance mechanism. This explains why $u(x)$ and $h_{{\rm RVB}}(x)$ decrease with $x$. For $x < x_c$ the ground state is dominated by the resonating valence bonds and the RVB picture remains qualitatively correct.
For $x > x_c$ the interference due to the holes has driven $u$ negative and it is no longer appropiate to interpret $u(x)$ as the square of $ h_{{\rm RVB}}$. Rather, the physical interpretation of the overdoped region comes from the solution of the Cooper problem in the $t-J$, 2-leg ladder, and its BCS extension. It can be shown analytically that two electrons in the latter system form a bound state only under certain conditions (details will be given elsewhere). For $J=0.5, t=t'=1$ one must have $J' > 3.3048$, ( note that the binding of two electrons in the $t-J$ chain requires $J/2t >1$ [@O]). The exact solution for 4 or more electrons is difficult to construct, but we expect it to be given essentially by a Gutzwiller projected BCS like wave function. A short range version of the latter type of wave function can be generated from the recursion relation (\[8\]), with $u$ a negative parameter, which can be written as
$$u(x) = - h_{{\rm BCS}}^2(x) < 0, \;\;\;\;( x > x_c)
\label{21}$$
where $h_{{\rm BCS}}$ is the BCS amplitude for finding two electrons at distance 1 along the legs. Of course this interpretation of $u$ as minus the square of a BCS amplitude requires it to be negative. As we put more electrons into the ladder the value of $h_{{\rm BCS}}$ decreases and for electron densities larger than $1 - x_c$, we switch into the RVB regime.
The difference between the underdoped and overdoped regimens can be attributed to two different internal structures of the pairs. In the low doping regime $x < x_c $, holes doped into the spin-liquid RVB state form pairs with an internal $d_{x^2-y^2}$-like structure relative to the undoped system. However for $x > x_c$ one moves into the low density limit characterized by electrons doped into an internal $s$-wave like symmetry. This issue will be discussed in detail in a separate publication.
Let us now comment on Figs. 12 and 13. Both are very similar and show that for $x \sim 1/2$, $b$ and $c$ reach their maximum. At $x=1/2$ there are as many electrons as holes, and in a certain sense the ground state of the ladder is a large scale reproduction of the microscopic ground state of the 2 x 2 cluster given in Fig. 4. Indeed for $J=J'=0.5, t=t'=1$ the ratio $b/a$ of the parameters appearing in Fig. 4 is given by 1.30, which is very close to the value of $b$ at its maximum. For $x < 0.7$ and $J'=0.5$ the parameter $b$ is larger than 1 and it is always larger than $c$ for all dopings and couplings. This is in agreement with the DMRG results of [@WS], which show the importance of the diagonal frustrating bonds above the horizontal or vertical ones for $J/t=J'/t=0.5$.
Finally Fig.14 is a $J/t-n$ diagram which shows the boundary of phase separation obtained by means of the DMRG and the RRM in the case where $J=J', t=t'=1$. Observe that this is not the strong coupling case we have been discussing so far, and hence the validity of the RRM is more questionable. In any case, we see an overall agreement between both results (see references [@TTR96; @HP; @S] for comparisons with other numerical results). In the two-leg $t$-$J$ model, phase separation is controlled by $J$, rather than $J'$, so the strongest coupling we have considered above, $J'/t = 5$, $J/t=0.5, t'/t=1$, does not phase separate.
Conclusions {#conclusions .unnumbered}
===========
In this paper we have proposed an extension of the effective hard-core boson model (HCB) of the 2-leg ladder of reference [@TTR96], in order to include the local structure of the hole pairs. The extended effective model, called the DHCB model, contains both dimer bonds, hard core bosons and various combinations between bonds and holes, whose relevance have been studied previously with DMRG [@WS]. Generalizing the methods of reference [@SMD] to the case with holes, we study a variational ansatz for the ground state of the DHCB model, which depends only on three variational parameters. The resulting dimer-hole state is generated by a second order recursion formula, which also leads to recursion formulas for the overlaps necessary to compute the energy of the ansatz. We give the results of the energy minimization for the 2 x 32 ladder and compare them with those obtained with the DMRG method in the strong coupling region. The recursion relations we have derived for the ground state energy can be solved analytically in the thermodynamic limit and the minimization can be then done numerically. Finally we give a physical interpretation of the behaviour of the variational parameters with doping.
Acknowledgments {#acknowledgments .unnumbered}
===============
GS would like to thanks the organizers of the ITP program “Quantum Field Theory in Low Dimensions: From Condensed Matter to Particle Physics" for the warm hospitality. MAMD thanks the organizers of the Benasque Center of Physics 1997 for their support and hospitality. GS acknowledges support from the NSF under Grant No. PHY94-07194 and the Dirección General de Enseñanza Superior, MAMD acknowledges support from the CICYT under contract AEN93-0776, JD acknowledges support from the DIGICYT under contract No. PB95/0123, SRW acknowledges support from the NSF under Grant No. DMR-9509945, and DJS acknowledges support from the NSF under Grant numbers PHY-9407194 and DMR-9527304.
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Figure captions {#figure-captions .unnumbered}
===============
[**Figure 1:**]{}The $0^{\rm th}$ order picture of the Hard Core Boson model: a) The vertical bond, b) the vertical hole-pair singlet, c) a typical state of the HCB model.
[**Figure 2 :**]{} The two lowest order states in the strong coupling limit $J^{\prime} \gg J, t, t^{\prime}$ of the HCB model. They represent the first order contribution to the DHCB model. a) the resonance of two vertical bonds, b) bound state of two quasiparticles.
[**Figure 3 :**]{} Higher order strong coupling states contributing to the DHCB model. a) a higher order RVB state, b) a bound state of two quasiparticles, c) and d) higher order corrections to the diagonal state 2(b)).
[**Figures 4 :**]{} The exact ground state for a single plaquette with two holes[@WS](case $N=2$ and $P=1$).
[**Figure 5 :**]{} A typical dimer-hole-RVB state.
[**Figure 6 :**]{}Elementary building block states of the RRM used in the construction of the dimer-hole states.
[**Figure 7 :**]{}A pictorical representation of Eq. (\[8\]).
[**Figure 8 :**]{}The function $f(x)$ appearing in (\[15\]). The maximum appears at $x=0.44$.
[**Figure 9 :**]{} Ground state energy per site of the 2 x 32 ladder with $J=0.5, t=t'=1$ and $J'=0.5,1,2,3,4,5$. The remaining data given below in figures 10-13 also corresponds to these choices of couplings. The continuum curves are obtained with the RRM, while the special symbols are the DMRG data corresponding to $x=1/8, 1/2 $ and 7/8 respectively.
[**Figure 10 :**]{} Kinetic energy per site.
[**Figure 11 :**]{} The variational parameter $u$ as function of the doping.
[**Figure 12 :**]{} The variational parameter $b$ as function of the doping.
[**Figure 13 :**]{} The variational parameter $c$ as function of the doping.
[**Figure 14 :**]{} Boundary of the phase separation region in the case where $J=J', t=t'$, computed with DMRG and the RRM.
[^1]: On leave from Instituto de Matemáticas y Física Fundamental, C.S.I.C., 28006 Madrid, Spain. em: sierra@sisifo.imaff.csic.es
[^2]: em: mardel@eucmax.sim.ucm.es
[^3]: em: emduke@iem.csic.es
[^4]: em: srwhite@uci.edu
[^5]: em: djs@spock.physics.ucsb.edu
|
---
abstract: |
A spherical dust cloud which is initially at rest and which has a monotonously decaying density profile collapses and forms a shell-focussing singularity. Provided the density profile is not too flat, meaning that its second radial derivative is negative at the center, this singularity is visible to local, and sometimes even to global observers. According to the strong cosmic censorship conjecture, such naked singularities should be unstable under generic, nonspherical perturbations of the initial data or when more realistic matter models are considered.
In an attempt to gain some understanding about this stability issue, in this work we initiate the analysis of a simpler but related problem. We discuss the stability of test fields propagating in the vicinity of the Cauchy horizon associated to the naked central singularity. We first study the high-frequency limit and show that the fields undergo a blueshift as they approach the Cauchy horizon. However, in contrast to what occurs at inner horizons of black holes, we show that the blueshift is uniformly bounded along incoming and outgoing null rays. Motivated by this boundedness result, we take a step beyond the geometric optic approximation and consider the Cauchy evolution of spherically symmetric test scalar fields. We prove that under reasonable conditions on the initial data a suitable rescaled field can be continuously extended to the Cauchy horizon. In particular, this result implies that the physical field is everywhere finite on the Cauchy horizon away from the central singularity.
author:
- Néstor Ortiz and Olivier Sarbach
bibliography:
- 'refs\_collapse.bib'
title: 'Cauchy horizon stability in a collapsing spherical dust cloud I: geometric optic approximation and spherically symmetric test fields'
---
Introduction {#Sec:Intro}
============
The simplest model describing the gravitational collapse of a massive body in general relativity is the one of Tolman-Bondi (TB), which describes the total collapse of a spherical dust cloud (see, for instance, Ref. [@MTW-Book]). Although the underlying assumptions of spherical symmetry and zero pressure are hardly satisfied in a realistic collapse, the advantage of this model relies in the fact that it can be described by closed-form expressions for the spacetime metric and energy density when co-moving, synchronous coordinates are employed. This facilitates the analysis of the physical properties of the model, which has already played an important role in the understanding of black hole formation in the seminal work by Oppenheimer and Snyder [@jOhS39]. Under rather reasonable assumptions on the initial data, namely the initial density and velocity profiles, one can show that a shell-focusing singularity forms, where the density and hence also the curvature blow up. Despite the fact that the metric is known in explicit form, the analysis of the causal structure in the vicinity of the shell-focusing singularity is a nontrivial task since the understanding of the behaviour of the light rays (even those with zero angular momentum) requires the solution of singular, nonlinear ordinary equations. The first systematic study of the radial light rays has been undertaken by Christodoulou [@dC84]. Interestingly, his work (and also earlier numerical work by Eardley and Smarr [@dElS79]) revealed a picture that is rather different than the one obtained from the simple Oppenheimer-Snyder scenario of homogeneous density. Indeed, the analysis in [@dC84] showed that for regular, time-symmetric generic initial data there exist infinitely many radial light rays emanating from the central singularity, implying that the later is visible to local observers. Furthermore, it was shown in [@dC84] that depending on the initial density profile, some of these light rays arrive at the surface of the cloud before the horizon forms, so that they extend all the way to null infinity. In this case the shell-focusing singularity is “globally naked” in the sense that it is visible to observers that are located arbitrarily far from the dust cloud. For generalizations of these results to time-asymmetric initial data, see Refs. [@rN86; @iDpJ92; @Joshi-Book]. For a self-contained exposition of these results we refer to our recent work in [@nOoS11], where we also provide a sufficient condition on the initial data for the occurrence of a globally naked singularity and present an algorithm for numerically generating conformal diagrams.
In this article we initiate a detailed analysis of test fields propagating on the fixed (but dynamical) spacetime geometry given by the TB collapse model. We are particularly interested in the behaviour of such test fields in a vicinity of the central naked singularity and the Cauchy horizon, corresponding to the first light ray emanating from the central singularity. Assuming regular initial data for the test field on a Cauchy surface, a relevant question is whether or not the field or its energy density grow arbitrarily large when they approach the Cauchy horizon. Such a divergent behaviour would suggest that the Cauchy horizon is unstable when the self-gravity of the field is taken into account. Indeed, such an instability was found at the inner horizon of a Reissner-Nordström black hole, where early numerical work [@mSrP73] evolving test fields indicated an instability of the Cauchy (inner) horizon, a fact that was later confirmed by self-consistent calculations [@ePwI90; @mD05] resulting in the famous mass inflation scenario.
There has been some previous work regarding the stability of the Cauchy horizon associated to naked singularities in the TB dust collapse model. In the geometric optic approximation of test fields, Christodoulou [@dC84] showed that, for time-symmetric initial data, a radial light ray emitted from the center of the cloud -even close to the singularity- does not undergo an infinite redshift. Based on a similar approximation, Waugh and Lake [@bWkL89] studied the stability of the Cauchy horizon for the particular case of self-similar collapse and did not find evidence for an instability. Duffy and Nolan took a step beyond the geometric limit approximation and studied the stability of the Cauchy horizon in the case of self-similar, marginally bound collapse under odd- [@eDbN11b] and even-parity [@eDbN11] linear metric perturbations. In the odd-parity sector, they found that the perturbations remain finite at the Cauchy horizon. In contrast, in the even-parity sector, they showed that the perturbations diverge on the Cauchy horizon. Other progress concerning the marginally bound case include the work by Iguchi, Harada and Nakao [@hItHkN98] who analyzed numerically the stability of the Cauchy horizon under odd-parity linear gravitational perturbations. When the matter perturbations vanish, they show that the metric perturbations do not diverge at the Cauchy horizon. On the other hand, they also found that the metric perturbations diverge when the matter perturbations are not zero [@hItHkN99]. Nevertheless, this divergence occurs only at the central singularity and does not propagate along the Cauchy horizon. The same authors extended their numerical studies to even-parity perturbations and concluded that they diverge near the Cauchy horizon while the energy flux keeps bounded [@hItHkN00], suggesting that the singularity is not a strong source of gravitational waves.
The main purpose of our work is to derive rigorous results in order to obtain a deeper insight into the physical effects that occur near the naked central singularity and the Cauchy horizon. For this reason, we analyze the generic TB collapse, without restricting ourselves to particular cases as marginally boundedness or self-similarity. The remaining of this article is organized as follows. In the next section we review the basic properties of the TB collapse model, state our assumptions on the initial data, recall important results on the propagation of radial light rays emanating from the central singularity and state some preliminary results that are relevant for the analysis that follows. Next, in section \[Sec:BlueShift\] we compute the gravitational redshift factor for a light ray that is emitted and received by two observers which are co-moving with the dust shells. We first show that for observers which are sufficiently close to the central singularity, this factor is always negative, implying a gravitational *blueshift*, even along outgoing null rays. As we show, the fact that there is a blueshift and not a redshift is an effect which is due to the collapse of the cloud, implying that outgoing photons move towards a region of *stronger* gravity. As the path of the light ray moves closer and closer to the central singularity, the blueshift becomes more pronounced, as expected. However, we show in section \[Sec:BlueShift\] that the total blueshift along such a light ray is uniformly bounded. Motivated by these considerations, in section \[Sec:SphSym\] we analyze the dynamics of a spherically symmetric scalar field $\Phi$ on the TB background. The propagation of such a field is described by an effective, two-dimensional wave equation with potential $V$ for the rescaled field $\psi = r\Phi$, where $r$ denotes the areal radius. We prove that under a suitable integrability condition on the effective potential $V$, the rescaled field $\psi$ can be continuously extended to the Cauchy horizon. Then, we verify that this condition on $V$ is satisfied and conclude that the physical field $\Phi$ must be bounded by a constant divided by $r$ in the vicinity of the Cauchy horizon. This implies, in particular, that $\Phi$ is finite everywhere on the Cauchy horizon away from the central singularity. Therefore, if $\Phi$ should diverge at the central singularity, this divergence cannot propagate along the Cauchy horizon. A discussion of our results and an outlook are presented in section \[Sec:Conclusions\]. More general results concerning the propagation of linear and nonlinear test fields in the vicinity of the Cauchy horizon of a TB dust collapse spacetime will be presented elsewhere.
Tolman-Bondi collapse: review, preliminaries and notation {#Sec:Model}
=========================================================
In this section we first review the TB collapse model, which describes the complete gravitational collapse of a spherical dust cloud. Then, we summarize our assumptions on the initial data. These assumptions are taken from our earlier work [@nOoS11] and essentially impose physically “reasonable” conditions on the initial density and velocity profiles. Furthermore, our conditions guarantee that no singularities occur at a finite radius, therefore avoiding shell-crossing singularities. After stating our assumptions we review the results in [@nOoS11] regarding the propagation of radial light rays which are relevant for this work. Finally, we introduce some notation and derive some preliminary results used later in this article.
The spherically symmetric solutions of the field equations describing a self-gravitating dust configuration can be explicitly parametrized in terms of co-moving, synchronous coordinates $(\tau,R,\vartheta,\varphi)$. Here, $R=const.$ describes the world surfaces of the collapsing dust shells, where the label $R$ is chosen such that it coincides with the shells’ areal radius at initial time $\tau=0$. $\tau$ is the proper time measured by a radial observer moving along a collapsing dust shell, and $(\vartheta,\varphi)$ are standard polar coordinates on the invariant two-spheres. The metric is determined by the function $r(\tau,R)$ which describes the areal radius at the event $(\tau,R,\vartheta,\varphi)$. Therefore, for fixed $R$, the function $\tau\mapsto r(\tau,R)$ describes the evolution of the dust shell labeled by $R$, and according to the definition of $R$, $r(0,R) = R$. As a consequence of Einstein’s field equations (see, for example, Ref. [@MTW-Book]), this function satisfies the following one-dimensional mechanical system, $$\frac{1}{2} \dot{r}(\tau,R)^2 + V(r(\tau,R), R) = E(R),\qquad
V(r,R) := -\frac{m(R)}{r},
\label{Eq:1DMechanical}$$ where the dot denotes partial derivative with respect to $\tau$, and where $m(R)$ denotes the Misner-Sharp mass function [@cMdS64].[^1] The mass and energy profiles $m(R)$ and $E(R)$ are determined by the initial density and velocity profiles $(\rho_0(R),v_0(R))$ according to $$m(R) = 4\pi G\int\limits_0^R \rho_0(\bar{R})\bar{R}^2 d\bar{R},\qquad
E(R) = \frac{1}{2} v_0(R)^2 - \frac{m(R)}{R},$$ with Newton’s constant $G$. Once the function $r(\tau,R)$ is known, the spacetime metric ${\bf g}$, the four-velocity ${\bf u}$ and the energy density $\rho$ are obtained by means of the following explicit formulae: $$\begin{aligned}
{\bf g} &=& -d\tau^2 + \frac{dR^2}{\gamma(\tau,R)^2}
+ r(\tau,R)^2(d\vartheta^2 + \sin^2\vartheta\, d\varphi^2),\qquad
\gamma(\tau,R) := \frac{\sqrt{1 + 2E(R)}}{r'(\tau,R)},
\label{Eq:MetricSol}\\
{\bf u} &=& \frac{\partial}{\partial\tau}\; , \qquad
\rho(\tau,R) = \rho_0(R)\left( \frac{R}{r(\tau,R)} \right)^2\frac{1}{r'(\tau,R)},
\label{Eq:FluidSol}\end{aligned}$$ where the prime denotes partial differentiation with respect to $R$. It is convenient to introduce the two functions $$c(R) := \frac{2m(R)}{R^3},\qquad
q(R) := \sqrt{E(R)/V(R,R)} = \sqrt{1 - \frac{R v_0(R)^2}{2m(R)}},$$ describing (up to a numerical factor) the mean density within the dust shell $R$ and the square root of the ratio between the total and initial potential energy. In terms of these quantites, the assumptions in [@nOoS11] can be summarized as follows:
1. $\rho_0$ and $v_0$ have even and odd $C^\infty$-extensions, respectively, on the real axis (regular, smooth initial data),
2. $\rho_0(R) > 0$ for all $0\leq R < R_1$ and $\rho_0(R) = 0$ for $R\geq R_1$ (finite, positive density cloud)
3. $c'(R)\leq 0$ for all $R > 0$ (monotonically decreasing mean density),
4. $2m(R)/R < 1$ for all $R > 0$ (absence of trapped surfaces on the initial slice),
5. $v_0(R)/R < 0$ for all $R\geq 0$ (collapsing cloud),
6. $(v_0(R)/R)^2 < 2m(R)/R^3$ for all $R\geq 0$ (bounded collapse),
7. $q'(R)\geq 0$ for all $R > 0$ (exclusion of shell-crossing singularities),
8. For all $R\geq 0$, we have $q'(R)/R > 0$ whenever $c'(R)/R = 0$ (non-degeneracy condition).
Notice that condition (i) ensures that the functions $c$ and $q$ have even $C^\infty$-extensions on the real axis, and that condition (vi) implies that $q(R) > 0$ for all $R\geq 0$. Condition (viii) implies the existence of a null portion of the singularity which is visible at least to local observers. Explicit four-parameter families of initial data $(\rho_0(R),v_0(R)$ satisfying all of these conditions except (i) have been constructed in Ref. [@nOoS11] (see Eq. (28)) and in Ref. [@nO12] (see Eq. (4)). In fact, these families also satisfy condition (i) except at the surface of the cloud $R = R_1$, where $\rho$ and $v_0$ are only continuous. The lack of smoothness at the surface of the cloud does not affect our results below, which mostly refer to regions close to the central singularity.
Under the conditions (i)–(viii) the solution of the mechanical system (\[Eq:1DMechanical\]) is given by the explicit formula $$r(\tau,R)
= \frac{R}{q(R)^2} \left[f^{-1}\left( f(q(R)) + \sqrt{c(R)}q(R)^3\tau \right) \right]^2,
\label{Eq:Sol}$$ where $f$ is the strictly decreasing function $$f: [0,1] \to [0,\pi/2],\quad x\mapsto x\sqrt{1 - x^2} + \arccos(x),
\label{Eq:fDef}$$ which is $C^\infty$-differentiable on the interval $[0,1)$ and whose first derivative is $f'(x) = -2x^2/\sqrt{1 - x^2}$, $0\leq x < 1$. The function $r(\tau,R)$ is well-defined for all $R\geq 0$ and $0\leq\tau < \tau_s(R)$, where the boundary $\tau = \tau_s(R)$ parametrizes the location of the shell-focusing singularity, for which $r/R$ vanishes. From Eq. (\[Eq:Sol\]) one obtains $$\tau_s(R) = \frac{\frac{\pi}{2} - f(q(R))}{\sqrt{c(R)}q(R)^3},\qquad R\geq 0.
\label{Eq:taus}$$ Since the energy density $\rho$ diverges, Einstein’s field equations imply that the Ricci scalar diverges at the shell-focusing singularity. Therefore, the boundary points $\tau = \tau_s(R)$ represent a curvature singularity. The tidal forces are much stronger near such points than in the case of shell-crossing singularities [@pSaL99]. Outside the cloud, $R > R_1$, where $\rho_0=0$, the spacetime is isometric to a subset of the Schwarzschild-Kruskal manifold according to Birkhoff’s theorem, see for example Ref. [@Straumann-Book].
For the following, it is convenient to use new local coordinates $(y,R)$ instead of $(\tau,R)$, with $y$ defined by $y:=\sqrt{r(\tau,R)/R}$. As shown in [@nOoS11] this facilitates the analysis of the light rays in several ways. First, in these coordinates the spacetime region inside the collapsing dust cloud is the rectangular region $(y,R)\in (0,1)\times (0,R_1)$, with the initial surface and the singularity corresponding to the lines $y=1$ and $y=0$, respectively. Second, using $y$ instead of $\tau$, allows one to get around computing the inverse of the function $f$ defined in Eq. (\[Eq:fDef\]). The equation for the outgoing radial null geodesics in these coordinates is $$\frac{dy}{dR} = \frac{1}{2}\sqrt{1-q(R)^2y^2}\left[ \frac{R\Lambda(y,R)}{y^2}
\left( 1 - \frac{R Q(R)}{ y}\sqrt{1 - q(R)^2y^2} \right) - Q(R) \right],
\label{Eq:dy/dR}$$ where the functions $\Lambda: [0,1)\times [0,R_1] \to {\mathbb{R}}$, $Q: [0,R_1]\to {\mathbb{R}}$, and $g,h: (0,1)\times [0,1)\to{\mathbb{R}}$ are defined as $$\begin{aligned}
\Lambda(y,R) &:=& 2\frac{q'(R)}{Rq(R)} h(q(R),y)
- \frac{c'(R)}{2Rc(R)} g(q(R),y),\\
Q(R) &:=& \sqrt{\frac{c(R)}{1 - R^2 q(R)^2 c(R)}},\\
g(q,y) &:=& \frac{ f(qy) - f(q) }{q^3},\\
h(q,y) &:=& \frac{1}{\sqrt{1-q^2}} - \frac{y^3}{\sqrt{1- q^2y^2}} - \frac{3}{2}g(q,y).\end{aligned}$$ It follows from Lemma 1 of Ref. [@nOoS11] that these functions are strictly positive and $C^\infty$-differentiable on their domain. Before we proceed, let us mention two limiting cases which can be included in our analysis below with the following adjustments:
1. $q=1$ (time-symmetric initial data)\
This case corresponds to zero initial velocity, and here $g(q,y) = f(y)$, while the function $\Lambda$ is $$\Lambda(y,R) = -\frac{c'(R)}{2Rc(R)} f(y),$$ while the function $h$ is void.
2. $q=0$ (marginally bound case)\
This corresponds to the case where each shell has zero energy, $E(R)=0$. Here, the functions $g(q,y)$ and $\Lambda(y,R)$ have the following expressions: $$g(q,y) = \frac{2}{3}(1-y^3),\qquad
\Lambda(y,R) = -\frac{c'(R)}{3R c(R)} (1-y^3).$$
In terms of the new coordinates $(y,R)$ the metric coefficient $\gamma$ in Eq. (\[Eq:MetricSol\]) and the energy density $\rho$ given in Eq. (\[Eq:FluidSol\]) can be computed according to the following formulae: $$\begin{aligned}
\gamma(y,R) &=& \frac{\sqrt{1 - R^2 q(R)^2 c(R)}}{r'(y,R)},
\label{Eq:gamma}\\
\rho(y,R) &=& \frac{\rho_0(R)}{y^4 r'(y,R)},
\label{Eq:rho}\end{aligned}$$ where $\rho_0(R) = 2m'(R)/(8\pi G R^2) = [R^3 c(R)]'/(8\pi G R^2)$ and the partial derivative of $r(\tau,R)$ with respect to $R$ is $$r'(y,R) = y^2\left( 1 + \frac{R^2}{y^3}\sqrt{1 - q(R)^2y^2} \Lambda(y,R) \right).
\label{Eq:rprime}$$ In addition, if follows from Eq. (\[Eq:Sol\]) that the proper time at $(y,R)$ is $$\tau(y,R) = \frac{g(q(R),y)}{\sqrt{c(R)}}.
\label{Eq:tau}$$ The causal structure of the spacetime described by Eqs. (\[Eq:MetricSol\],\[Eq:FluidSol\]) has been analyzed in Refs. [@dC84; @rN86], and more recently in Ref. [@nOoS11], where conformal diagrams are constructed based on local analysis and numerical integration of Eq. (\[Eq:dy/dR\]). The common feature is the presence of a locally naked central singularity and an associated Cauchy horizon that emanates from it. Depending on whether or not the Cauchy horizon intersects the surface of the cloud before or after the apparent horizon, the central singularity is globally naked or hidden inside a black hole, see figure \[Fig:Hidden-Naked\] for two examples taken from [@nOoS11].
![\[Fig:Hidden-Naked\] Spacetime diagrams in conformal coordinates $(T,X)$ describing a TB dust collapse with the initial data given in Eq. (28) of Ref. [@nOoS11]. Left panel: The singularity is hidden inside the black hole region. Right panel: A portion of the null singularity is globally naked. The lines denoted by “AH”, “EH” and “CH” refer to the apparent, event and Cauchy horizons, respectively. Details on the numerical construction of these diagrams are given in Ref. [@nOoS11]. The parameters used correspond to the ones described in figures 3 and 4 of that reference.](hidden.pdf "fig:"){width="9.4cm"} ![\[Fig:Hidden-Naked\] Spacetime diagrams in conformal coordinates $(T,X)$ describing a TB dust collapse with the initial data given in Eq. (28) of Ref. [@nOoS11]. Left panel: The singularity is hidden inside the black hole region. Right panel: A portion of the null singularity is globally naked. The lines denoted by “AH”, “EH” and “CH” refer to the apparent, event and Cauchy horizons, respectively. Details on the numerical construction of these diagrams are given in Ref. [@nOoS11]. The parameters used correspond to the ones described in figures 3 and 4 of that reference.](naked.pdf "fig:"){width="9.4cm"}
In the following, for definiteness we will only consider the case of a globally naked singularity, although most of our results also hold in the black hole case. We will focus our attention to the region $D$ of spacetime describing the maximal development of the initial surface $\tau=0$, see figure \[Fig:D\_epsilon\]. In terms of the coordinates $(y,R)$ this region is described by $$D := \{ (y,R) : R\geq 0, y_{CH}(R) < y \leq 1 \},$$ where the curve $y_{CH}(R)$ describes the Cauchy horizon, that is, the first light ray emanating from the singularity. For $0 < R < \delta$ small enough it was shown in Proposition 2 of Ref. [@nOoS11] that $y_{CH}(R)$ has the following form: $$y_{CH}(R)^3 = \frac{3\Lambda_0}{4} R^2\left[ 1 + z(R^{1/3}) \right],\qquad
R\in [0,\delta),$$ where $\Lambda_0 := \Lambda(0,0) > 0$ and $z: [0,\delta)\to{\mathbb{R}}$ is a $C^\infty$-function satisfying $z(0)=0$. Besides the spacetime region $D$, we will also consider for each $\varepsilon \in (0,1)$ the small spacetime regions $D(\varepsilon)\subset D$ near the central singularity $(0,0)$ defined as $$D(\varepsilon) := \{ (y,R) : 0 \leq R \leq R(\varepsilon), y_{CH}(R) < y \leq y(\varepsilon) \},
\label{Eq:DepsDef}$$ where the functions $R(\varepsilon) \in (0,\delta)$ and $y(\varepsilon) := y_{CH}(R(\varepsilon))$ are chosen such that they converge to zero when $\varepsilon\to 0$ and such that
1. $z(R^{1/3}) \geq -\varepsilon$ for all $0\leq R \leq R(\varepsilon)$
2. $\Lambda(y,R)\leq (1 + \varepsilon)\Lambda_0$ for all $0\leq R\leq R(\varepsilon)$ and $0\leq y\leq y(\varepsilon)$,
see figure \[Fig:D\_epsilon\].
![\[Fig:D\_epsilon\] Conformal diagram illustrating the maximal development $D$ of the initial slice and the small subsets $D(\varepsilon)$ close to the central singularity. Here “CH” denotes the Cauchy horizon.](D_epsilon.pdf){width="9cm"}
It follows for each $(y,R)\in D(\varepsilon)$ that $$0\leq \frac{R^2}{y^3}\sqrt{1 - q(R)^2y^2} \Lambda(y,R)
\leq \frac{R^2}{y_{CH}(R)^3}\Lambda(y,R)
= \frac{4}{3}\frac{1}{1 + z(R^{1/3})}\frac{\Lambda(y,R)}{\Lambda_0}
\leq \frac{4}{3}\frac{1 + \varepsilon}{1 - \varepsilon}.
\label{Eq:R2y3Estimate}$$ This and Eq. (\[Eq:rprime\]) imply that inside the region $D(\varepsilon)$, $$y^2\leq r'(y,R) \leq \left( 1 + \frac{4}{3}\frac{1 + \varepsilon}{1 - \varepsilon} \right)y^2.
\label{Eq:y2Estimat}$$ In view of Eqs. (\[Eq:gamma\],\[Eq:rho\]) this yields the following behaviour for the metric coefficient $\gamma$ and the energy density $\rho$ in the vicinity of the central singularity:
Let $\varepsilon\in (0,1)$. Then there are constants $m < M$ such that for all $(y,R)\in D(\varepsilon)$, $$\frac{m}{y^2} \leq \gamma(y,R) \leq \frac{M}{y^2},\qquad
\frac{m}{y^6} \leq \rho(y,R) \leq \frac{M}{y^6}.$$
Therefore, $\gamma$ diverges as $1/y^2$ and $\rho$ as $1/y^6$ as the singularity is approached from within the maximal development $D$ of the initial data surface. In the next two sections we analyze the propagation of test fields on the background spacetime $(D,{\bf g})$ and show that despite these divergences, the fields do not behave “too badly” as the central singularity $(0,0)$ and the Cauchy horizon are approached.
In the following, the radial vector fields ${\bf u}$, ${\bf w}$, ${\bf k}$ and ${\bf l}$ play an important role. ${\bf u} = \partial_\tau$ is the four-velocity of the radial observers co-moving with the collapsing dust shells, see Eq. (\[Eq:FluidSol\]), ${\bf w} := \gamma(\tau,R)\partial_R$ is the unit outward radial vector orthogonal to ${\bf u}$, and ${\bf k} := {\bf u} + {\bf w}$, ${\bf l} := {\bf u} - {\bf w}$ are radial out- and ingoing null vector fields.
Boundedness of the blueshift of light rays and null dust {#Sec:BlueShift}
========================================================
In this section we start by analyzing the behaviour of test fields on the background spacetime $(D,{\bf g})$ in the geometric optics approximation. In this limit, the propagation of test fields is described by null geodesics of $(D,{\bf g})$. This leads to the identification of the vector fields ${\bm \xi}\in {\cal X}(D)$ satisfying ${\bf g}({\bm \xi},{\bm \xi}) = 0$ and $\nabla_{\bm \xi} {\bm \xi} = 0$. Here, for simplicity, we restrict ourselves to radial null geodesics. Then, we consider an emitter $(e)$ which sends a high-frequency signal to an observer $(obs)$ and compute the frequency shift due to gravitational redshift effects. We assume that both the emitter and the observer are co-moving with the dust particles and thus have four-velocity ${\bf u}$, see figure \[Fig:Blueshift\] for possible scenarios.
![\[Fig:Blueshift\] Left panel: A free-falling emitter ($e$) sends an outgoing radial null ray, which is received by a free-falling observer ($obs$). Middle panel: Same situation, but now ($e$) emits an ingoing radial null ray which is received by the observer ($obs$). Right panel: Similar situations, but now the observer is located at a point which is diametrically opposed to the emitter.](Redshift.pdf "fig:"){height="4.6cm"} ![\[Fig:Blueshift\] Left panel: A free-falling emitter ($e$) sends an outgoing radial null ray, which is received by a free-falling observer ($obs$). Middle panel: Same situation, but now ($e$) emits an ingoing radial null ray which is received by the observer ($obs$). Right panel: Similar situations, but now the observer is located at a point which is diametrically opposed to the emitter.](Redshift_II.pdf "fig:"){height="4.6cm"} ![\[Fig:Blueshift\] Left panel: A free-falling emitter ($e$) sends an outgoing radial null ray, which is received by a free-falling observer ($obs$). Middle panel: Same situation, but now ($e$) emits an ingoing radial null ray which is received by the observer ($obs$). Right panel: Similar situations, but now the observer is located at a point which is diametrically opposed to the emitter.](Redshift_extra.pdf "fig:"){height="4.6cm"}
Under these assumptions the frequency shift is given by (see, for instance, Ref. [@Straumann-Book]) $$\frac{\nu_{obs}}{\nu_e} = \frac{\left. {\bf g}({\bf u},{\bm \xi}) \right|_{obs}}
{\left. {\bf g}({\bf u},{\bm \xi})\right|_e}
\label{Eq:RedShift}$$ with ${\bm \xi}$ the generator of the null rays. We are particularly interested in the case where the signal is emitted or received in a region that lies arbitrarily close to the central singularity, since an unboundedly large blueshift would indicate a large concentration of energy and thus an instability of the Cauchy horizon. In fact, there is also another interesting effect an unboundedly large blueshift could have.[^2] Namely, when the self-gravity of the test field is taken into account, a large blueshift along ingoing light rays could lead to the formation of a trapped surface in the self-consistent calculation. The presence of such a trapped surface would effectively censor the singularity [@mD05b], as occurs in the spherically symmetric collapse of a self-gravitating scalar field [@dC87; @dC94; @dC99].
In order to discuss the frequency shift, we first identify the radial vector fields ${\bm\xi}$ generating null geodesics. They must be parallel to either one of the two null vectors ${\bf k}$ and ${\bf l}$ defined at the end of the last section, so all that needs to be determined is the factor between ${\bm \xi}$ and ${\bf k}$ or ${\bm \xi}$ and ${\bf l}$.
\[Lem:Geodesics\] The affinely parametrized future-directed radial null geodesics of $(M,{\bf g})$ are generated by the vector fields ${\bm \xi}_+ = f_+ {\bf k}$ and ${\bm \xi}_- = f_- {\bf l}$, where the functions $f_\pm$ are determined by the advection equations $${\bf k}[f_+] = \frac{\dot{\gamma}}{\gamma} f_+,\qquad
{\bf l}[f_-] = \frac{\dot{\gamma}}{\gamma} f_-.$$
[[**Proof.** ]{}]{}With respect to the local coordinates $(\tau,R,\vartheta,\varphi)$ we have for ${\bm \xi} = {\bm \xi}_+$ or ${\bm \xi} = {\bm \xi}_-$: $$0 = \nabla_{\bm\xi} \xi_b = \xi^a\nabla_a\xi_b = \xi^a(\nabla_a\xi_b - \nabla_b\xi_a)
= \xi^a(\partial_a\xi_b - \partial_b\xi_a),
\label{Eq:xiGeod}$$ where we have used the fact that ${\bm \xi}$ is null. Using the expressions $$\xi_\pm^\tau = f_\pm,\quad \xi_\pm^R = \pm\gamma f_\pm,\quad
\xi_\pm^\vartheta = \xi_\pm^\varphi = 0,$$ and $$\xi_{\pm\tau} = -f_\pm,\quad \xi_{\pm R} = \pm\gamma^{-1} f_\pm,\quad
\xi_{\pm\vartheta} = \xi_{\pm\varphi} = 0,$$ for the contravariant and covariant components of ${\bm \xi}_\pm$ we obtain form Eq. (\[Eq:xiGeod\]) the equation $$\left( \frac{1}{\gamma}\frac{\partial}{\partial \tau}
\pm \frac{\partial}{\partial R} \right) f_\pm = \frac{\dot{\gamma}}{\gamma^2} f_\pm,
\label{Eq:fpm}$$ which is equivalent to the claim. [$\fbox{\hspace{0.3mm}}$ ]{}
An explicit solution representation for the functions $f_\pm$ can be obtained by the method of characteristics. For this, let $\tau_\pm(R)$ be a solution of $$\frac{d}{dR}\tau_\pm(R) = \pm \frac{1}{\gamma(\tau_\pm(R),R)},$$ describing an out- or ingoing radial null ray. Define $F_\pm(R) := f_\pm(\tau_\pm(R),R)$. Then, by virtue of Eq. (\[Eq:fpm\]), $F_\pm$ satisfy the ordinary differential equations $$\frac{d}{dR} F_\pm(R)
= \pm \frac{\dot{\gamma}}{\gamma^2}(\tau_\pm(R),R) F_\pm(R).$$ For the following we introduce the propagators $${\cal P}_\pm(R_2,R_1) := \exp\left[ \pm \int\limits_{R_1}^{R_2} \frac{\dot{\gamma}}{\gamma^2}(\tau_\pm(R),R) dR \right],
\label{Eq:Propagators}$$ which propagate the values of $f_\pm$ along the radial light rays $(\tau_\pm(R),R)$ from $R = R_1$ to $R = R_2$. Using these propagators, Lemma \[Lem:Geodesics\] and the observation that ${\bf g}({\bf u},{\bf k}) = {\bf g}({\bf u},{\bf l}) = -1$, Eq. (\[Eq:RedShift\]) yields the following expression for the frequency shift due to gravitational redshift effects: $$\frac{\nu_{obs}}{\nu_e} = {\cal P}_\pm(R_{obs},R_e),
\label{Eq:Blueshift}$$ where the plus sign refers to the case where an outgoing signal is sent and the minus sign to the case of the ingoing signal.
Therefore, the question about the boundedness of the frequency shift is reduced to the analysis of the propagators (\[Eq:Propagators\]). We are particularly interested in their properties close to the central singularity. For this, we first note the following property of the integrand $\dot{\gamma}/\gamma^2$:
\[Lem:gamma\] For $\varepsilon > 0$ small enough there exists a constant $C_1 > 0$ such that on $D(\varepsilon)$, $$0 < y\frac{\dot{\gamma}}{\gamma^2} \leq C_1.$$
[[**Proof.** ]{}]{}We start with the explicit computation of $\dot{\gamma}/\gamma^2$, based on Eq. (\[Eq:gamma\]) which can be rewritten as $$\frac{1}{\gamma} = \frac{r'}{\sqrt{1 - R^2 q(R)^2 c(R)}}.$$ Taking a partial derivative with respect $\tau$ on both sides of this equation gives $$-\frac{\dot{\gamma}}{\gamma^2} = \frac{Q(R)}{\sqrt{c(R)}} \dot{r}',$$ where the function $Q(R)$ was defined in the previous section. In order to evaluate $\dot{r}'$ we use the equation $$\dot{r} = -\sqrt{2E(R) + \frac{2m(R)}{r}} = -\frac{R\sqrt{c(R)}}{y}
\sqrt{1 - q(R)^2 y^2},$$ which follows from the energy conservation law, see Eq. (\[Eq:1DMechanical\]). Together with $$y' = \frac{R}{2y^2}\sqrt{1 - q(R)^2 y^2}\Lambda(y,R),$$ we finally obtain $$\frac{\dot{\gamma}}{\gamma^2} = \frac{Q(R)}{y} H(y,R),
\label{Eq:gamma_dot/gamma_sqrd}$$ with the function $$H(y,R) := \frac{1}{\sqrt{1-q(R)^2y^2}}
\left[ 1 + R\frac{c'(R)}{2c(R)} - q(R)^2 y^2\left( 1 + R\frac{c'(R)}{2c(R)} + R\frac{q'(R)}{q(R)} \right) \right]
-\frac{1}{2}\frac{R^2}{y^3}\Lambda(y,R).$$ The first term on the right-hand side is a continuous function of $(y,R)$ which converges to $1$ as $(y,R) \to (0,0)$. As for the second term, we use a similar estimate than Eq. (\[Eq:R2y3Estimate\]) which shows that inside region $D(\varepsilon)$, $$0\leq \frac{R^2}{y^3}\Lambda(y,R) \leq \frac{4}{3}\frac{1+\varepsilon}{1-\varepsilon}.$$ It follows that for small enough $\varepsilon > 0$, $0 < H(y,R) < 4/3$ for all $(y,R)\in D(\varepsilon)$. Since $Q$ is strictly positive and continuous the lemma follows. [$\fbox{\hspace{0.3mm}}$ ]{}
A first important consequence of Lemma \[Lem:gamma\] is the positivity of ${\cal P}_+(R_{obs},R_e)$ for small enough $R_{obs} > R_e > 0$ and the positivity of ${\cal P}_-(R_{obs},R_e)$ for small enough $R_e > R_{obs} > 0$. This implies that the photons with zero angular momentum, as observed by co-moving observers close to the singularity, are *blueshifted*. For the ingoing case, this might be expected since the photons move towards the central singularity. However, for the outgoing case the fact that the photons are blueshifted and not redshifted might come as a surprise since in this case the photons move away from the singularity. In fact, it was recently shown [@lKdMcB13] for the marginally bound case that the blueshift of photons which are emitted by the dust particles in the interior of the cloud can be larger than their redshift in the exterior region so that in this case, a distant observer measures a blueshift.
In order to clarify why outgoing photons are blueshifted close to the singularity, we consider the compactness ratio $$C(\tau,R) := \frac{2m(R)}{r(\tau,R)}$$ as a measure for the strength of the gravitational field at $(\tau,R)$. Its variation along outgoing null geodesics is given by $$k[C] = -\frac{2m}{r^2} k[r] + \frac{\gamma}{r} 2m'.$$ As long as the null ray lies outside the apparent horizon, $k[r] > 0$, and the first term is negative. Since we assume positive mass density, the second term is positive inside the cloud. Outside the cloud, it vanishes and the photons are redshifted in the outside region, provided they arrive at the surface of the cloud before the horizon forms. In order to see which term dominates inside the cloud we use $2m = R^3 c$ and $r = Ry^2$ and rewrite $$k[C] = \frac{2m}{r^2}\left[ -k[r] + \left( 3 + \frac{Rc'}{c} \right)\gamma y^2 \right].
\label{Eq:kC}$$ Since $$\begin{aligned}
k[r] &=& \dot{r} + \gamma r' = -\frac{R\sqrt{c}}{y}\sqrt{1 - q^2 y^2} + \sqrt{1 - R^2 q^2 c},
\\
\gamma y^2 &=& \frac{\sqrt{1 - R^2 q^2 c}}{1 + \frac{R^2}{y^3}\sqrt{1 - q^2 y^2}\Lambda},\end{aligned}$$ we see that $k[r]$ and $\gamma y^2$ both converge to one when $R\to 0$ and $y$ is bounded away from zero. Consequently, the expression inside the square parenthesis on the right-hand side of Eq. (\[Eq:kC\]) converges to $2$ and since $2m/r^2 > 0$ it follows that $k[C] > 0$ is positive for outgoing radial light rays close to the center. Using the estimate (\[Eq:R2y3Estimate\]) it is not difficult either to show that $k[C]$ is positive in the interior of the region $D(\varepsilon)$ for sufficiently small $\varepsilon > 0$. In this sense the outgoing photons move towards a region of stronger gravity when they are close to the center of the cloud or close to the central singularity. This provides an explication for their blueshift in these regions.
A further important consequence of Lemma \[Lem:gamma\] concerns the *boundedness* of the blueshift. We formulate this result in terms of the redshift factor $$z := \frac{\nu_e}{\nu_{obs}} - 1 = {\cal P}_\pm(R_e,R_{obs}) - 1
= \exp\left[ \mp \int\limits_{R_e}^{R_{obs}} \frac{\dot{\gamma}}{\gamma^2}(\tau_\pm(R),R) dR \right] - 1.
\label{Eq:zDef}$$
\[Prop:Blueshift\] Consider the redshift factor $z$ along in- and outgoing radial geodesics in the region $D$ of spacetime. Then, $|z|$ is uniformly bounded inside the dust cloud, and $z$ is negative in a region of the form $D(\varepsilon)$ close to the central singularity.
[[**Proof.** ]{}]{}Let $D(\varepsilon)$ be such that the inequality in Lemma \[Lem:gamma\] holds. Since $\dot{\gamma}/\gamma^2$ is a smooth function on $D$ it is sufficient to prove that $|z|$ is uniformly bounded on $D(\varepsilon)$. For this, we use a similar estimate than Eq. (\[Eq:R2y3Estimate\]) in order to find $$\frac{R^2}{y^3} \leq M^3 := \frac{4}{3\Lambda_0}\frac{1}{1-\varepsilon}$$ on $D(\varepsilon)$. Therefore, it follows from Lemma \[Lem:gamma\] that inside $D(\varepsilon)$, $$0 < \frac{\dot{\gamma}}{\gamma^2} \leq \frac{C_1}{y} \leq \frac{C_1 M}{R^{2/3}},$$ implying that $$0 < \int\limits_{R_e}^{R_{obs}} \frac{\dot{\gamma}}{\gamma^2} dR
Ê\leq C_1 M\int\limits_{R_e}^{R_{obs}} \frac{dR}{R^{2/3}} \leq 3C_1 M R_{obs}^{1/3} < \infty$$ for $R_{obs} \geq R_e > 0$ sufficiently small. This proves that $z$ is uniformly bounded and negative on $D(\varepsilon)$. [$\fbox{\hspace{0.3mm}}$ ]{}
To summarize the results of this section, we have found that photons emitted and received by radial observers which are co-moving with the dust particles are blueshifted in a vicinity of the central singularity. The blueshift can be computed from the explicit expression in Eq. (\[Eq:zDef\]). Since $\dot{\gamma}/\gamma^2$ diverges as $1/y$ near the singularity, see Lemma \[Lem:gamma\], the blueshift is more pronounced near the singularity, as expected. However, Proposition \[Prop:Blueshift\] shows that this blueshift is uniformly bounded and thus cannot be arbitrarily large even when the light ray passes arbitrarily close to the singularity. In particular, this implies that photons that are sent from a static observer outside the cloud, traverse the collapsing cloud and are received by another static observer outside the cloud cannot gain an arbitrarily large amount of energy, even if they pass close to the central singularity. In this respect, an observer outside the cloud will notice nothing particular as he reaches the Cauchy horizon.
Boundedness of spherically symmetric effective test fields {#Sec:SphSym}
===========================================================
In the last section we discussed the propagation of fields on the collapsing dust background in the geometric optics approximation. In this section we relax the high-frequency assumption leading to this approximation and consider instead the exact Cauchy evolution of massless and massive test scalar fields $\Phi$ from regular initial data on the Cauchy surface $\tau=0$. The dynamics of $\Phi$ is governed by an effective wave equation with potential $V$ for the rescaled field $\psi = r\Phi$ on the two-dimensional spacetime $(\tilde{M},\tilde{\bf g})$, where $\tilde{M} = \{ (\tau,R) : R \geq 0, 0\leq \tau < \tau_s(R) \}$, $\tilde{\bf g} = -d\tau^2 + dR^2/\gamma(\tau,R)^2$ and $r$ is the areal radius. Considering a solution $\psi$ on the domain of dependence $D$ of the initial surface $\tau=0$ satisfying suitable regularity conditions at $R=0$ we prove in Theorem \[Thm:psiBoundedness\] that $\psi$ can be continuously extended to the closure $\overline{D}$ of $D$, provided the potential $V$ is $L^1$-integrable on compact subsets of $(\tilde{M},\tilde{\bf g})$ and in a vicinity of the form $D(\varepsilon)$ of the naked singularity, see figure \[Fig:D\_epsilon\]. This means that the field $\psi$ can be continuously extended to the Cauchy horizon, including the first singular point from which it emanates. In particular, this result implies that $\psi$ is uniformly bounded in the part $D_c$ of $D$ lying inside the cloud. For a spherically symmetric scalar field $\Phi$ of arbitrary mass we prove in Lemma \[Lem:Int\_V\_bounded\] that the integrability conditions on $V$ are satisfied, and thus we conclude that $r\Phi$ can be extended to the Cauchy horizon and is uniformly bounded on $D_c$.
When decomposed into spherically harmonics, $$\Phi(\tau,R,\vartheta,\varphi)
= \frac{1}{r}\sum\limits_{\ell=0}^\infty\sum\limits_{m=-\ell}^\ell
\psi_{\ell m}(\tau,R) Y^{\ell m}(\vartheta,\varphi)$$ the Klein-Gordon equation on an arbitrary spherically symmetric background spacetime reduces to a family of effective wave equations on $(\tilde{M},\tilde{\bf g})$ (see, for instance, Ref. [@eCnOoS13]) $$\tilde{g}^{ab}\tilde{\nabla}_a\tilde{\nabla}_b\psi_{\ell m} = V_\ell\psi_{\ell m},$$ with potential $$V_\ell = \frac{\ell(\ell+1)}{r^2} + \frac{\tilde{g}^{ab}\tilde{\nabla}_a\tilde{\nabla}_b r}{r}
+ \mu^2,
\label{Eq:EffectivePotential}$$ where $\tilde{\nabla}$ denotes the Levi-Civita connection associated to $(\tilde{M},\tilde{\bf g})$ and $\mu$ is the inverse Compton length of the field $\Phi$. We stress that the effective potential $V_\ell$ diverges at the central singularity, even for $\ell=0$, see Eq. (\[Eq:VEffective\]) below. Therefore, one would naively expect that the scalar field diverges at the central singularity and that this divergence could propagate along the Cauchy horizon. Surprisingly, this is not the case as follows from the arguments below.
Our result can be stated as follows:
\[Thm:psiBoundedness\] Let $(\tilde{M},\tilde{\bf g})$ be the two-dimensional spacetime manifold $\tilde{M} = \{ (\tau,R) : R \geq 0, 0\leq \tau < \tau_s(R) \}$ with metric $\tilde{\bf g}$ obtained from the one in Eq. (\[Eq:MetricSol\]) by fixing the angular coordinates. Assume the conditions (i)–(viii) on the initial density and velocity profiles hold. Furthermore, suppose $V$ is a function on $(\tilde{M},\tilde{\bf g})$ which is locally integrable and such that for some $\varepsilon\in (0,1)$, $$\int\limits_{D(\varepsilon)} |V(x)| \sqrt{|\det\tilde{g}(x)|} d^2 x < \infty.
\label{Eq:VIntegrability}$$
Then, any solution $\psi$ of the effective wave equation $$\tilde{g}^{ab}\tilde{\nabla}_a\tilde{\nabla}_b\psi = V\psi,
\label{Eq:EffectiveWave}$$ on the domain of dependence $D$ belonging to smooth initial data on the initial surface $\tau=0$ and which satisfies the regularity condition $\psi(\tau,R=0) = 0$, $0\leq \tau < \tau_s(0)$, at the center can be continuously extended to the closure $\overline{D}$ of $D$. In particular, $\psi$ is uniformly bounded on the region $D_c$, that is, there exists a constant $C > 0$ such that for all $x\in D_c$ $$|\psi(x)| \leq C.$$
[**Remark**]{}: It is interesting to note that the extension principle of $\psi$ only requires the integrability condition (\[Eq:VIntegrability\]), and not any further conditions on the metric $\tilde{\bf g}$. The reason for this is that in terms of double null coordinates $(u,v)$ the two-metric has the form $\tilde{\bf g} = -\Omega^2 du dv$, where all singularities of $\tilde{\bf g}$ are contained in the positive conformal factor $\Omega$. In terms of these coordinates, Eq. (\[Eq:EffectiveWave\]) reads $$-2\partial_u\partial_v\psi = \sqrt{|\det\tilde{g}(x)|} V(x)\psi,
\label{Eq:EffectiveWaveConfCoord}$$ and we see that the singularity only enters the combination $\sqrt{|\det\tilde{g}(x)|} V(x)$.\
[[**Proof.** ]{}]{}We first note that according to standard uniqueness and existence theorems, the solution $\psi$ must be at least continuous on $D$. The proof of Theorem \[Thm:psiBoundedness\] is based on the integration of Eq. (\[Eq:EffectiveWaveConfCoord\]) over a characteristic rectangle $E := [u_1,u_2]\times [v_1,v_2] \subset D$, which yields $$-2 \left[ \psi(u_2,v_2) - \psi(u_1,v_2) - \psi(u_2,v_1) + \psi(u_1,v_1) \right]
= \int\limits_E V(x)\psi(x)\sqrt{|\det\tilde{g}(x)|} d^2x.
\label{Eq:ParallelIdentity}$$
![\[Fig:Domain\_D\] Illustration of the regions used in the proof of the extension principle.](Domain_D.pdf){width="9cm"}
We first show that $\psi$ is uniformly bounded inside the region $D(\varepsilon)$. For this, we choose $E$ such that the point $(u_2,v_1)$ lies on the center of the cloud (see figure \[Fig:Domain\_D\]). Then, since $\psi = 0$ at the center, we obtain the implicit relation $$\psi(u_2,v_2) = \psi(u_1,v_2) - \psi(u_1,v_1(u_2))
- \frac{1}{2} \int\limits_E V(x)\psi(x)\sqrt{|\det\tilde{g}(x)|} d^2 x
\label{Eq:ParallelIdentityBis}$$ for $\psi$, where $v_1(u_2)$ indicates that $v_1$ depends on $u_2$. This yields the estimate $$|\psi(u_2,v_2)| \leq |\psi(u_1,v_2)| + |\psi(u_1,v_1(u_2))|
+ \frac{1}{2}\int\limits_E |V(x)| |\psi(x)| \sqrt{|\det\tilde{g}(x)|} d^2x.
\label{Eq:EstimateScal1}$$
Next, we fix $\varepsilon > 0$ and $u_1$, and choose a closed subset $D'\subset D(\varepsilon)$ of the form depicted in figure \[Fig:Domain\_D\]. Moreover, we introduce the quantities $$\begin{aligned}
A &:=& 2\sup\{ |\psi(u_1,v) | : \hbox{$v$ such that $(u_1,v)\in D'$} \},\\
B &:=& \frac{1}{2} \int\limits_{D(\varepsilon)} |V(x)| \sqrt{|\det\tilde{g}(x)|} d^2x,\end{aligned}$$ which are finite according to the hypothesis of the theorem. With this notation the estimate (\[Eq:EstimateScal1\]) implies that $$|\psi(u_2,v_2)| \leq A + B\sup\limits_{(u,v)\in D'} |\psi(u,v)|
\label{Eq:EstimateScal2}$$ for all $(u_2,v_2)\in D'$. Taking the supremum over $(u_2,v_2)\in D'$ on both sides yields the simple inequality $$x \leq A + Bx,\qquad x := \sup\limits_{(u,v)\in D'} |\psi(u,v)|.$$ Finally, since $B$ is finite, we can make it arbitrarily small by choosing $\varepsilon > 0$ sufficiently small. (This follows from the dominated convergence theorem, see for instance [@Royden-Book].) In particular, if $B < 1$ we conclude that $$x\leq \frac{A}{1 - B} < \infty,$$ which proves that $\psi$ is uniformly bounded on the union of all closed subsets $D'\subset D(\varepsilon)$ of the form depicted in figure \[Fig:Domain\_D\] with fixed $u_1$. Since according to the assumptions of the theorem $|V|$ is integrable on any compact subset $K\subset \tilde{M}$ we can use similar arguments and conclude that $\psi$ is bounded on any subset of the form $K\cap D$.
Next, we prove the extension result. For this, let $(u_c,v_c)$ be a point on the Cauchy horizon, and let $(u^{(n)},v^{(n)})$ be a sequence in $D$ which converges to $(u_c,v_c)$. Evaluating the implicit relation (\[Eq:ParallelIdentityBis\]) at $(u_2,v_2) = (u^{(k)},v^{(k)})$ and $(u_2,v_2) = (u^{(n)},v^{(n)})$ and taking the difference we obtain the estimate $$\begin{aligned}
| \psi(u^{(k)},v^{(k)}) - \psi(u^{(n)},v^{(n)}) | &\leq&
| \psi(u_1,v^{(k)}) - \psi(u_1,v^{(n)}) |
+ | \psi(u_1,v_1(u^{(k)})) - \psi(u_1,v_1(u^{(n)})) |
\nonumber\\
&+& M\int\limits_{E^{(k)}\bigtriangleup E^{(n)}} |V(x)| \sqrt{|\det\tilde{g}(x)|} d^2x,
\label{Eq:EstimateScal3}\end{aligned}$$ where $E^{(k)}\bigtriangleup E^{(n)}$ denotes the symmetric difference of $E^{(k)}$ and $E^{(n)}$, that is, the set of points which are in either of the two sets but not in their intersection. In deriving this estimate we have used the previous boundedness result on $\psi$. Since $\psi$ is continuous on $D$ and because of the integrability condition on $V$, it follows that the right-hand side of Eq. (\[Eq:EstimateScal3\]) vanishes in the limit $k,n\to\infty$. Consequently, it follows from Eq. (\[Eq:EstimateScal3\]) that $\psi(u^{(n)},v^{(n)})$ is a Cauchy sequence in ${\mathbb{R}}$, and the limit $$\lim\limits_{n\to\infty} \psi(u^{(n)},v^{(n)})$$ exists in ${\mathbb{R}}$. The limit is independent of the sequence that converges to $(u_c,v_c)$, because if $(\tilde{u}^{(n)},\tilde{v}^{(n)})$ is another sequence that converges to $(u_c,v_c)$, it follows by replacing $(u^{(k)},v^{(k)})$ with $(\tilde{u}^{(n)},\tilde{v}^{(n)})$ in Eq. (\[Eq:EstimateScal3\]) and by taking the limit $n\to\infty$ that $$\lim\limits_{n\to\infty} \psi(\tilde{u}^{(n)},\tilde{v}^{(n)})
= \lim\limits_{n\to\infty} \psi(u^{(n)},v^{(n)}) =: \psi(u_c,v_c).$$ This defines a continuous extension of $\psi$ on $\overline{D}$. In particular, $\psi$ vanishes at the central naked singularity since $\psi$ is zero at the center. [$\fbox{\hspace{0.3mm}}$ ]{}
The next lemma shows that the integrability conditions on $V_\ell$ are satisfied in the spherically symmetric case $\ell=0$.
\[Lem:Int\_V\_bounded\] For $\ell = 0$ the effective potential $V_0$ defined in Eq. (\[Eq:EffectivePotential\]) is locally integrable, and for all $\varepsilon\in (0,1)$ it satisfies $$\int\limits_{D(\varepsilon)} |V_0(x) | \sqrt{|\det\tilde{g}(x)|} d^2x < \infty.
\label{Eq:B_lt_infty}$$
[[**Proof.** ]{}]{}In terms of the coordinates $(y,R)$ the effective potential (\[Eq:EffectivePotential\]) is given by $$V_\ell(y,R) = \frac{\ell(\ell+1)}{R^2 y^4} + \frac{c(R)}{y^6}
- \frac{Rc'(R) + 3c(R)}{2y^4 r'} + \mu^2,
\label{Eq:VEffective}$$ and the weight in the volume element belonging to the two-metric is $$\sqrt{|\det\tilde{g}(y,R)|} = \frac{1}{\sqrt{c(R)}\sqrt{1 - q(R)^2 y^2}}
\frac{2y^2 r'}{\sqrt{1-R^2 q(R)^2 c(R)}}.$$ Therefore, the lemma reduces to the verification of the integrability of the function $$V_\ell\sqrt{|\det\tilde{g}|} = \frac{1}{\sqrt{c}\sqrt{1 - q^2 y^2} \sqrt{1-R^2 q^2 c}}\left[
\frac{2\ell(\ell+1)A}{R^2} + \frac{2c A}{y^2} - \frac{Rc' + 3c}{y^2} + \mu^2 y^4 A \right],
\label{Eq:Vldet}$$ where we have introduced the function $$A(y,R) := \frac{r'}{y^2} = 1 + \frac{R^2}{y^3}\sqrt{1 - q(R)^2 y^2}\Lambda(y,R).$$
First, we note that this function is continuous on $D$, implying the local integrability of $V_0$. Next, we note that the function $A(y,R)$ is bounded in $D(\varepsilon)$ according to the estimate (\[Eq:y2Estimat\]). Since the factor in front of the square parenthesis in Eq. (\[Eq:Vldet\]) is bounded, it follows from this that exists a constant $k > 0$ such that $$| V_0 |\sqrt{|\det\tilde{g}|} \leq \frac{k}{y^2}$$ on $D(\varepsilon)$. Now the lemma is a consequence of the integrability of the function $1/y^2$ over the region $D(\varepsilon)$. In order to show this, recall the definition (\[Eq:DepsDef\]) of $D(\varepsilon)$, which implies in particular that $y_{CH}(R)^3 \geq \frac{3\Lambda_0}{4}(1-\varepsilon) R^2$ for all $(y,R)\in D(\varepsilon)$. Therefore, $$\begin{aligned}
\int\limits_{D(\varepsilon)} \frac{1}{y^2} dy dR
&=& \int\limits_0^{R(\varepsilon)} \left(
\int\limits_{y_{CH}(R)}^{y(\varepsilon)} \frac{dy}{y^2} \right) dR\\
&=& \int\limits_0^{R(\varepsilon)}
\left( \frac{1}{y_{CH}(R)} - \frac{1}{y(\varepsilon)} \right) dR\\
&\leq& \left( \frac{4}{3\Lambda_0(1-\varepsilon)} \right)^{1/3}
\int\limits_0^{R(\varepsilon)}\frac{dR}{R^{2/3}} - \frac{R(\varepsilon)}{y(\varepsilon)}\\
&=& 3\left( \frac{4}{3\Lambda_0(1-\varepsilon)} \right)^{1/3} R(\varepsilon)^{1/3}
- \frac{R(\varepsilon)}{y(\varepsilon)} < \infty,\end{aligned}$$ which is finite.
In contrast, the function $1/R^2$ is not integrable over $D(\varepsilon)$ and consequently, the statement of the lemma cannot be generalized to the case $\ell > 0$. In this case, bounds have to be obtained via energy estimates. [$\fbox{\hspace{0.3mm}}$ ]{}
As a consequence of Theorem \[Thm:psiBoundedness\] and Lemma \[Lem:Int\_V\_bounded\] we have:
Consider a smooth, spherically symmetric solution $\Phi$ of the Klein-Gordon equation on the domain $D$ with smooth and bounded initial data for $\Phi$ and $\dot{\Phi}$ on the initial surface $\tau = 0$. Then, there exists a continuous extension of $\Phi$ on $\overline{D}\setminus \{(0,0)\}$ and there is a constant $C$ such that $$|\Phi(x)| \leq \frac{C}{r}
\label{Eq:PhiBound1}$$ for all $x\in D_c$ in the interior of the cloud.
It should be stressed that this bound does not guarantee the finiteness of $\Phi$ at the naked central singularity, since it allows $\Phi$ to diverge as $r\to 0$. However, the importance of this result is that it rules out an infinite field propagating along the Cauchy horizon. In other words, even if $\Phi$ were to diverge at the central singularity, this divergence could not propagate along the Cauchy horizon.
Discussion {#Sec:Conclusions}
==========
In this work we have analyzed physical effects that occur in the vicinity of a central singularity arising in the TB dust collapse. The initial data for the density and velocity profiles satisfy a set of reasonable assumptions, but are generic otherwise. In particular, we do not restrict ourselves to the marginally bound case nor to the self-similar collapse. The first effect we analyzed is the gravitational redshift along radial light rays, as measured by two observers which are co-moving with the dust particles. We showed that in the vicinity of the singularity there is a blueshift along in- and outgoing rays. As we explained, this effect is due to the collapse of the cloud, which implies that outgoing photons move towards a region of stronger gravity, and thus they gain energy. However, we also showed that the blueshift is uniformly bounded close to the singularity and the Cauchy horizon. An infinite blueshift would have indicated an instability of the Cauchy horizon, as is the case at inner horizons of black holes [@mSrP73; @ePwI90; @mD05]. The fact that we could exclude an infinite blueshift does not, of course, prove that the Cauchy horizon is stable, but it indicates that its stability properties might be much more subtle than in the case of inner horizons.
Motivated by the results in the geometric optic approximation, we considered a spherically symmetric test scalar field $\Phi$ propagating on the background geometry of the TB dust collapse. Our main result is the extension principle proven in Theorem \[Thm:psiBoundedness\] which states that the rescaled field $r\Phi$ can be continuously extended to the Cauchy horizon. This result has several interesting consequences. First, it implies that it is possible to extend the scalar field continuously beyond the Cauchy horizon, provided appropriate data is given on an ingoing characteristic surface excising the null portion of the singularity. Second, the extension principle immediately implies that $\Phi$ is bounded by a constant divided by the areal radius $r$, close to the singularity. An important consequence of this bound is that it shows that even if $\Phi$ diverged at the central singularity, this divergence could not propagate along the Cauchy horizon. Although the techniques used to prove this bound cannot be extended to non-spherical fields, such as electromagnetic or linearized gravitational waves, this result is interesting in view of the results reported by numerical studies for gravitational perturbations, where it was found that a naked singularity could not be an important source of radiation [@hItHkN98; @hItHkN99; @hItHkN00].
In a follow up work we derive uniform energy bounds for more general fields propagating on the dust collapse background. In particular, we analyze the stability of linearized gravitational and fluid perturbations of the TB model.
We thank Mihalis Dafermos and Thomas Zannias for fruitful and stimulating discussions. This work was supported in part by CONACyT Grants No. 46521 and 101353 and by a CIC Grant to Universidad Michoacana.
[^1]: $m(R)$ is also the Hawking mass [@sH68] associated to the invariant two-spheres.
[^2]: We are thankful to M. Dafermos for explaining this point to us.
|
---
abstract: 'In this paper, we propose bi-directional cooperative non-orthogonal multiple access (NOMA). Compared to conventional NOMA, the main contributions of bi-directional cooperative NOMA can be explained in two directions: 1) The proposed NOMA system is still efficient when the channel gains of scheduled users are almost the same. 2) The proposed NOMA system operates well without accurate channel state information (CSI) at the base station (BS). In a two-user scenario, the closed-form ergodic capacity of bi-directional cooperative NOMA is derived and it is proven to be better than those of other techniques. Based on the ergodic capacity, the algorithms to find optimal power allocations maximizing user fairness and sum-rate are presented. Outage probability is also derived, and we show that bi-directional cooperative NOMA achieves a power gain over uni-directional cooperative NOMA and a diversity gain over non-cooperative NOMA and orthogonal multiple access (OMA). We finally extend the bi-directional cooperative NOMA to a multi-user model. The analysis of ergodic capacity and outage probability in two-user scenario is numerically verified. Also, simulation results show that bi-directional cooperative NOMA provdes better data rates than the existing NOMA schemes as well as OMA in multi-user scenario.'
author:
- 'Minseok Choi, Dong-Jun Han, Jaekyun Moon,'
title: |
Bi-Directional Cooperative NOMA\
without Full CSIT
---
Non-orthogonal multiple access, Inaccurate CSI, Cooperative NOMA, Ergodic capacity, Outage probability, Power allocation, User fairness problem, Max-sum-rate problem
Introduction {#sec:Introduction}
============
Non-orthogonal multiple access (NOMA) based on power multiplexing has been introduced to utilize radio resources efficiently for a massive number of user terminals [@NOMA_basic:Docomo-Saito]. The 4G networks mainly operate based on orthogonal multiple access (OMA), allocating orthogonal resources to multiple users. However, as a massive number of various devices is deployed in the network, OMA is no longer able to maximize resource efficiency and to serve all devices simultaneously. For 5G communication systems, much higher data rates are expected compared to 4G, and efficient and flexible uses of energy and spectrum have become critical issues [@5G:Andrews; @5G:Wunder]. To this end, NOMA has been actively researched as a promising technology in 5G networks [@NOMA_5G:Dai; @NOMA_5G:Islam]. NOMA superposes the multi-user signals within the same frequency, time or spatial domain. The advanced receivers with successive interference cancellation (SIC) are typically considered to detect non-orthogonally multiplexed signals. In theory, NOMA provides a significant benefit in improving the cell throughput by using perfect SIC [@Book:Tse]. Performance analysis has also been conducted to examine the effectiveness of NOMA in practical environments [@NOMA:PerformanceAnalysis:Ding; @NOMA:PerformanceAnalysis:Saito]. User scheduling for non-orthogonally multiplexed signaling has been studied in [@NOMA:UserPairing:TVT-Ding]. Optimal power allocation at the BS for NOMA users is an important issue [@NOMA:PowerAllocation:TWC-Choi], for several system goals, e.g., sum-rate maximization [@NOMA:PowerAllocation:TWC-Yang] and user fairness [@NOMA:UserFairness:SPL-Timotheou]. Recently, joint optimization of power allocation and user scheduling has been also studied for NOMA systems [@NOMA:URLLC:ArXiv-Choi]. NOMA has been extensively researched in conjunction with various technologies. There have been some studies on the system applying NOMA to MIMO [@MIMO-NOMA:TWC-Ding; @MIMO-NOMA:TWC-Ding2] and on analyzing ergodic capacity of MIMO-NOMA system [@MIMO-NOMA:WCommLetter-Sun]. NOMA has been also considered to increase the data rate of the cell-edge user in coordinated multipoint (CoMP) [@NOMA-CoMP:CommLetter-Choi] and to maximize user fairness and sum-rate in distributed antenna systems [@NOMA-CoMP:ICC-Han]. Recently, the application of NOMA to simultaneous wireless information and power transfer (SWIPT) [@NOMA-SWIPT:TWC-PD; @NOMA:uni-cN-SWIPT:Liu] and physical security [@NOMA:security:CommLetter-Zhang] have been studied.
This paper proposes bi-directional cooperative NOMA which targets two practical channel environments: 1) when the BS knows statistical CSI but channel gain differences among users are not large, and 2) when the BS does not know CSI at all. The proposed NOMA scheme is a kind of cooperative NOMA [@NOMA:uni-cN:Ding], which allows cooperation among users via short-range communications, based on instantaneous CSI at transmitter (CSIT). The cooperative NOMA system of [@NOMA:uni-cN:Ding] improves the outage probability performance compared to conventional NOMA and it is applicable to relay communication [@NOMA:relay-uni-cN:Ding] and SWIPT [@NOMA:uni-cN-SWIPT:Liu]. However, it is difficult to figure out which user has the better instantaneous channel gain and which user transmits the cooperation signal to others, when only statistical CSIT or no CSIT is available. In bi-directional cooperative NOMA system, the direction of cooperation among users can be figured out by allowing users to exchange channel information via short-range communications.
The main contributions of this paper are shown below:
- The closed-form ergodic capacity of bi-directional cooperative NOMA is derived, especially for a two-user scenario. Also, the ergodic capacity of bi-directional cooperative NOMA is shown to be better than those of other existing NOMA schemes and OMA, even when the channel variances between the users are small.
- Based on the ergodic capacity analysis, this paper presents the optimal power allocation algorithms to maximize user fairness and sum-rate for bi-directional cooperative NOMA.
- Outage probabilities of bi-directional cooperative NOMA are derived in a two-user scenario. For the non-SIC user, it is shown that the proposed system has a power gain over the existing NOMA schemes. For the SIC user, our scheme is shown to have a diversity gain over conventional NOMA and OMA.
- The extension of bi-directional cooperative NOMA to the multi-user model is presented by performing cooperation on signal-by-signal basis, not on user-by-user [@NOMA:uni-cN:Ding]. The cooperation on signal-by-signal basis does not require additional power allocations for the cooperation phases.
- Simulation results verify the analysis of ergodic capacity and outage probability. Moreover, bi-directional cooperative NOMA is shown to provide better data rates than other NOMA schemes and OMA even without enough CSI, necessarily required for conventional NOMA.
We first propose the two-user scenario of bi-directional cooperative NOMA in Section \[sec:system\_model\]. Ergodic capacity analysis is performed in Section \[sec:ergodic\_capacity\_analysis\] and the optimal power allocation algorithms to maximize user fairness and sum-rate maximization are presented in Section \[sec:power\_allocation\]. In Section \[sec:outage\_prob\], outage probability of the proposed system is analyzed. Bi-directional cooperative NOMA is extended to the multi-user model in Section \[sec:multi-user\], and simulation results are shown in Section \[sec:simulation\]. Lastly, we conclude the paper in Section \[sec:conclusion\].
System Model {#sec:system_model}
============
Channel Model {#subsec:channel_model}
-------------
Consider cellular downlink communications in which a BS transmits signals to two users simultaneously. Extension to the multi-user model will be shown in Section \[sec:multi-user\]. The Rayleigh fading channel from the BS to user $i$ is defined as $h_i = \sqrt{L_i}g_i$ for $i=1,2$. $L_{i}=1/d_i^2$ denotes the slow fading, where $d_i$ is the distance from the BS to user $i$ and $g_i$ is a fast fading component with a complex Gaussian distribution, $g_i \sim CN(0,1)$.
In this paper, two cases are considered in terms of the CSI knowledge: only statistical CSIT and no CSIT. Here, the statistical CSIT means that the BS knows only users’ channel variances. The BS usually allocates more power to the user having the smaller channel variance. The user having a larger channel variance performs SIC first to subtract inter-user interference, and then decodes its data. However, if the distances from the BS to two users are similar, then randomly generated channels cannot guarantee that the user of the larger variance experiences the stronger channel gain. In this case, the performance gain of NOMA over OMA mostly vanishes.
In practice, there exists harsh environments where CSI is hardly known at the BS. For example, in an IoT environment, a clumsy device acts as a transmitter but cannot handle substantial processing tasks, i.e., channel tracking and elaborate user scheduling, so CSI is not available at the transmitter side. In this no-CSIT case, the BS cannot judge which user has a larger or smaller channel variance, so the system should arbitrarily decide the given user to perform SIC or not. Also, optimal power allocation cannot be found without any CSI, so fixed power ratios for the users will be assumed. The problem for the no-CSIT case occurs when the user with the weaker channel gain is selected to perform SIC. In this case, there is no merit of employing conventional NOMA. This paper proposes a new cooperative NOMA system for reliable downlink transmission for both statistical-CSIT only and no-CSIT cases.
\[h!\] ![Bi-directional NOMA model[]{data-label="fig:bi-cN_model"}](2-user_model.pdf "fig:"){width="48.00000%"}
Direct Transmission Phase
-------------------------
Denote $\gamma_i$ as the power ratio allocated to user $i\in \{1,2\}$, satisfying $\gamma_1+\gamma_2 = 1$. The received signal of user $i$ is given by $$r_i = h_i(\sqrt{\gamma_1}s_1 + \sqrt{\gamma_2}s_2) + n_i, \label{eq:received:user}$$ where $s_i$, and $n_i$ are transmitted symbol and noise at user $i$, respectively, and $n_i \sim CN(0,\sigma_n^2)$. $\sigma_n^2$ is the normalized noise variance. Assume a normalized unit power at the BS, $\mathbb{E}[|s_i|^2]=1$. Throughout the paper, users 1 and 2 are the non-SIC user and the SIC user, respectively. Let $V_{i,k}$ be the SINR of user $i$ to decode $s_k$. Then, the received SINRs at both users become $$\begin{aligned}
V_{1,1} &=& \frac{|h_1|^2 \gamma_1}{|h_1|^2 \gamma_2 +\sigma_n^2} \\
V_{2,2} &=& \frac{|h_2|^2 \gamma_2}{\sigma_n^2}, \end{aligned}$$ and SINR for SIC at user 2 is given by $$V_{2,1} = \frac{|h_2|^2 \gamma_1}{|h_2|^2 \gamma_2 + \sigma_n^2}.$$ Here, user 2 performs SIC for $s_1$ first with $V_{2,1}$, and decodes $s_2$ with $V_{2,2}$.
Channel Information Exchange Phase {#subsec:channel_exchange_phase}
----------------------------------
Since both users decode $s_1$ in the direct transmission phase, cooperation between users for improved decoding of $s_1$ is possible. Although both users’ decoding processes at the direct transmission phase can be reliable, the risk is that all users receive and exploit the cooperation signal, when decoding of the user with the weaker channel gain fails. Therefore, the system allows only the user $i_0$, satisfying $i_0 = \underset{i\in \{1,2\}}{\arg \max} |h_i|^2$, to transmit the cooperation signal. This indicates that transmission of the cooperation signal can be bi-directional, but actual cooperation at each time is performed at only the user of the weaker channel gain. To find user $i_0$, users exchange their CSI or just the received channel power in this phase. We assume that all users are located nearby and the exchange of CSI is performed via short-range communications, so this phase would not take too much time. A highly crowded stadium is one example, where the distances between the BS and users do not differ greatly so the BS with statistical CSIT only and no CSIT would hardly determine the direction of cooperation appropriately. *Remark*: Obviously, CSI exchange among users allow the BS not to collect all users’ CSI at the expense of the additional delay and signaling overhead. Therefore, the proposed scheme can be more advantageous than existing cooperative NOMA with full CSIT, only when the CSI exchange step requires less time and overhead than transmission of CSI feedbacks. In the case of statistical CSIT only or no CSIT, CSI feedbacks are not required, but much better data rates can be obtained by allowing the exchanges of CSI among users at the expense of the additional delay and overhead, as shwon in Section \[sec:simulation\]. One more thing to remark is that even if the CSI exchange step incurs longer delays than transmission of CSI feedbacks, the proposed technique can be still beneficial over conventional schemes, especially when the channel coherence time is very short. In the proposed scheme, since all users already received NOMA signals from the BS and obtained the desired CSI at the direct transmission phase, even though channel conditions change during the phase of CSI exchange, users can find the appropriate direction of cooperation. On the other hand, in conventional NOMA where the BS should collect the users’ CSI feedbacks, channel conditions at the time when users send CSI feedbacks to the BS, could be changed when the BS transmits the NOMA signal to all users. Also, as responsibility for determining the direction of cooperation signals is shifted to user sides, the BS does not need to handle substantial processing tasks for estimating the exact CSI, e.g., channel tracking.
Bi-Directional Cooperative Phase {#subsec:bi-directional_cooperative_phase}
--------------------------------
In this phase, the cooperation signal is transmitted from the user with stronger channel gain to the user with weaker gain. The cooperation signal can help user 1 to decode its data, or user 2 to perform SIC better. The received cooperation signal at user $i$ is given by $$c_{i} = g_{k,i} s_c + n_{c,i}$$ where $i \neq k$, $g_{k,i}$ is a Rayleigh fading channel coefficient from user $k$ to user $i$, and $s_c=s_1$ here. As mentioned in the channel information exchange phase, when $|h_1|^2 > |h_2|^2$, only $c_2$ exists, and when $|h_1|^2 < |h_2|^2$, only $c_1$ is transmitted from user 2. The received SINR at user $i$ is $$W_{i} = \frac{|g_{k,i}|^2}{\sigma_n^2},$$ and SINR for decoding $s_1$ is given by $$Z_{\text{cN},1}^{\text{bi}} =
\begin{cases}
\min \{ \max\{V_{1,1}, W_{1}\}, V_{2,1} \} & \text{if }|h_1|^2 < |h_2|^2 \\
\min \{ V_{1,1}, \max\{V_{2,1}, W_{2}\} \} & \text{otherwise}
\end{cases}.
\label{eq:bi-cN:R1}$$ Here, if a certain user receives the cooperation signal, she chooses the better of the signals received in the direct transmission and bi-directional cooperative phases. When maximal-ratio combining is exploited [@NOMA:uni-cN:Ding; @NOMA:relay-uni-cN:Ding], we achieve $$Z_{\text{cN},1}^{\text{bi}} =
\begin{cases}
\min \{ V_{1,1} + W_{1}, V_{2,1} \} & \text{if }|h_1|^2 < |h_2|^2 \\
\min \{ V_{1,1}, V_{2,1}+ W_{2} \} & \text{otherwise}
\end{cases}.
\label{eq:bi-cN:R1-MRC}$$ Since only $s_1$ is shared for cooperation, SINR for decoding $s_2$ is $Z_{\text{cN},2}^{\text{bi}} = V_{2,2}$.
Assume that both users are located close to each other, i.e., $W_{1}, W_{2} \gg V_{1,1}, V_{2,1}, V_{2,2}$. Then, $Z_{\text{cN},1}^{\text{bi}} \approx V_{2,1}$ when $|h_1|^2 < |h_2|^2$ or $Z_{\text{cN},1}^{\text{bi}} \approx V_{1,1}$ otherwise, and both (\[eq:bi-cN:R1\]) and (\[eq:bi-cN:R1-MRC\]) are simplified to $$\tilde{Z}_{\text{cN},1}^{\text{bi}} \simeq \max \{V_{1,1}, V_{2,1}\}.
\label{eq:bi-cN:R1-near-user}$$ This assumption is used throughout the paper. The data rate of $s_i$ in bi-directional cooperative NOMA becomes $R_{\text{cN},i}^{\text{bi}} = \log_2(1+Z_{\text{cN},i}^{\text{bi}})$.
The difference of our work from [@NOMA:uni-cN:Ding] is that the direction of cooperation is determined at user sides by exchanging CSI among users, especially when the BS knows only statistical CSIT or no CSIT at all. Cooperative NOMA in [@NOMA:uni-cN:Ding] is based on instantaneous CSIT, and the BS can determine the user with stronger channel gain as the SIC user and the other one with weaker gain as the non-SIC user. This makes the direction of cooperation to be always from the SIC user to the non-SIC user in the cooperative NOMA scheme of [@NOMA:uni-cN:Ding]. In the statistical-CSIT only or no CSIT cases, however, there is no guarantee that the SIC user’s instantaneous channel is better than that of the non-SIC user. Therefore, exchanging the channel information among users is necessary for bi-directional cooperative NOMA to force the user with stronger channel gain transmit the cooperation signal. We consider for comparison purposes uni-directional cooperative NOMA where direction of cooperation is always from the SIC user to the non-SIC user.
Since uni-directional cooperative NOMA only allows the SIC user to transmit the cooperation signal to the non-SIC user, SINR for decoding $s_1$ is $Z_{\text{cN},1}^{\text{uni}} = \min\{ \max\{V_{1,1}, W_{1} \}, V_{2,1} \}$, and $\tilde{Z}_{\text{cN},1}^{\text{uni}} \simeq V_{2,1}$ is obtained with the assumption that both users are located close to each other. Also, SINR for decoding $s_1$ in conventional NOMA is $Z_{1}^{\text{N}} = \min \{V_{1,1}, V_{2,1}\}$ [@NOMA:PerformanceAnalysis:Ding]. Comparing $\tilde{Z}_{\text{cN},1}^{\text{bi}}$ with $\tilde{Z}_{\text{cN},1}^{\text{uni}}$ and $Z_1^{\text{N}}$, we can find that bi-directional cooperative NOMA exploits channel diversity. It is clear that $\tilde{R}_{\text{cN},1}^{\text{bi}}$ is better than or equal to $\tilde{R}_{\text{cN},1}^{\text{uni}}$ and $R_{\text{N},1}$. On the other hand, since the cooperation is only helpful for $s_1$, the data rate of $s_2$ is the same for all considered schemes, i.e., $R_{\text{cN},2}^{\text{bi}} = R_{\text{cN},2}^{\text{uni}} = R_{\text{N},2}$. The sum rate is obtained by $R_{\text{cN}}^{\text{bi}} = R_{\text{cN},1}^{\text{bi}} + R_{\text{cN},2}^{\text{bi}}$.
When targeted data rates are already determined, the outage event of a certain user is the criterion for determining whether the other user should receive the cooperation signal or not. Let $\epsilon_{1}$ and $\epsilon_{2}$ be SINR thresholds for decoding $s_1$ and $s_2$, respectively. Then, in this two-user scenario, even though $|h_1|^2 < |h_2|^2$, user 1 cannot receive the cooperation signal from user 2 when $V_{2,1} < \epsilon_{1}$. Since $|h_1|^2 < |h_2|^2$, it is also clear $V_{1,1} < \epsilon_{1}$, so decoding of $s_1$ fails at both user sides. On the other hand, when $V_{2,1} > \epsilon_{1}$, user 1 can decode $s_1$ by using cooperation from user 2, even if $V_{1,1} < \epsilon_{1}$. The analysis of outage probability is given in Section \[sec:outage\_prob\].
Ergodic Capacity Analysis {#sec:ergodic_capacity_analysis}
=========================
When users’ data rates are opportunistically determined by their Quality of Service (QoS) requirements, ergodic capacity analysis is important. Some key lemmas are established first in deriving the closed-form ergodic capacity of bi-directional cooperative NOMA.
For real constants $a, b>0$ and a chi-square random variable $X$, an expected value of the function $\log_2(1+\frac{aX}{b})$ becomes $$\mathbb{E}\bigg[\log_2 \Big(1+ \frac{aX}{b} \Big)\bigg] = C_1 \Big( \frac{2a}{b} \Big),$$ where $C_1(x) = \frac{1}{\ln 2} e^{1/x} \int_{1}^{\infty} \frac{1}{t} e^{-t/x} \mathrm{d}t$, for $x>0$ \[lemma:expected\_log\_chi\]
Let the nonnegative random variable $Z = \log_2(1+aX/b)$; then $\mathbb{E}[Z] = \int_{0}^{\infty} P[Z \geq z] \mathrm{d}z$ is satisfied. Therefore, $$\mathbb{E}[Z] = \int_{0}^{\infty} \bigg( 1-P[Z \leq z] \mathrm{d}z \bigg) = \int_{1}^{\infty} \frac{1}{t \ln 2} e^{-\frac{b}{2a}(t-1)} \mathrm{d}t = C_1 \Big(\frac{2a}{b}\Big),$$ where $t=2^z$.
For real constants $a>0$, $b$, and a chi-square random variable $X$, $$\int_{0}^{\infty} e^{-bx} \log_2(1+ax) \mathrm{d}x
= \frac{1}{b} C_1 \Big(\frac{a}{b}\Big)$$ \[lemma:integral\_chi\]
$$\begin{aligned}
\int_{0}^{\infty} e^{-bx} \log_2(1+ax) \mathrm{d}x &=& \Big[-\frac{1}{b}e^{-bx}\log_2(1+ax)\Big]_0^{\infty} + \int_{0}^{\infty} \frac{1}{\ln 2}\cdot \frac{a}{b(1+ax)} e^{-bx} \mathrm{d}x \\
&=& \int_{1}^{\infty} \frac{1}{b \ln 2} \cdot \frac{1}{t} e^{-\frac{b}{a}(t-1)} \mathrm{d}t = \frac{1}{b} C_1 \Big(\frac{a}{b}\Big),
\end{aligned}$$
where $t=1+ax$.
$C_1 (x)$ is an increasing function of $x>0$.
$$\frac{\mathrm{d}}{\mathrm{d}x} C_1 (x) = -\frac{1}{x^2 \ln 2} e^{1/x} \int_1^{\infty} \frac{e^{-t/x}}{t} \mathrm{d}t + \frac{1}{x \ln 2} > -\frac{1}{x^2 \ln 2} \ln (1+x) + \frac{1}{x \ln 2},$$
where the last inequality is satisfied according to $e^{-1/x}\ln (1+x) > \int_1^{\infty} \frac{e^{-t/x}}{t} \mathrm{d}t$ [@EiFunction]. Since $x>0$ and $x>\ln (1+x)$, $\frac{\mathrm{d}}{\mathrm{d}x} C_1 (x) > 0$, $C_1 (x)$ is an increasing function of $x>0$.
With the assumption that both users are located nearby, Theorem \[thm:closed\_ergodic\_bi-cN\] gives the closed-form ergodic capacity of the bi-directional cooperative NOMA system. Also, Theorem \[thm:ergodic\_comparison\_NOMA\] shows that the ergodic capacity of the bi-directional cooperative NOMA is larger than those of uni-directional cooperative NOMA and conventional NOMA, no matter which user’s channel gain is larger.
Assuming that both users are located close to each other, the closed-form ergodic capacity of two-user bi-directional cooperative NOMA is $$\mathbb{E}[\tilde{R}_{\text{cN}}^{\text{bi}}] = C_1 \Big(\frac{L_1}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L_1}{\sigma_n^2}\Big) + C_1 \Big(\frac{L_2}{\sigma_n^2}\Big) - C_1 \Big(\frac{L_1 L_2}{(L_1+L_2)\sigma_n^2}\Big) + C_1 \Big(\frac{\gamma_2 L_1 L_2}{(L_1 + L_2)\sigma_n^2}\Big),
\label{eq:closed_ergodic_cN_bi}$$ \[thm:closed\_ergodic\_bi-cN\]
See Appendix \[appendix:thm1\].
Assuming that both users are located close to each other, $$\mathbb{E}[\tilde{R}^{\text{bi}}_{\text{cN}}] \geq \mathbb{E}[\tilde{R}^{\text{uni}}_{\text{cN}}] \geq \mathbb{E}[R_{\text{N}}]$$ \[thm:ergodic\_comparison\_NOMA\]
See Appendix \[appendix:thm2\].
To verify that bi-directional cooperative NOMA is applicable, comparison with OMA is also necessary. Theorem \[theorem:ergodic\_inequality\_oma\] shows that the ergodic capacity of bi-directional cooperative NOMA is better than that of OMA when the channel variances of two users are identical. Lemma \[lemma:Cx-Cax\] is introduced first before stating Theorem \[theorem:ergodic\_inequality\_oma\].
$C_1(x) - C_1(\beta x)$ is an increasing function of $x>0$, for any $0<\beta<1$. \[lemma:Cx-Cax\]
According to (\[eq:R1\_uni-cN\_closed\]) and (\[eq:R2\_closed\]), the closed-form ergodic capacity of the uni-directional cooperative NOMA system becomes $$\mathbb{E}[\tilde{R}_{\text{cN}}^{\text{uni}}] = C_1 \Big(\frac{L_1}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L_1}{\sigma_n^2}\Big) + C_1 \Big(\frac{\gamma_2 L_2}{\sigma_n^2}\Big).
\label{eq:closed_ergodic_cN_omni}$$
According to (\[eq:closed\_ergodic\_cN\_bi\]) and (\[eq:closed\_ergodic\_cN\_omni\]), the following inequality holds by Theorem \[thm:ergodic\_comparison\_NOMA\], $$\begin{aligned}
&& C_1 \Big(\frac{L_1}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L_1}{\sigma_n^2}\Big) + C_1 \Big(\frac{L_2}{\sigma_n^2}\Big) - C_1 \Big(\frac{L_1 L_2}{(L_1+L_2)\sigma_n^2}\Big) + C_1 \Big(\frac{\gamma_2 L_1 L_2}{(L_1 + L_2)\sigma_n^2}\Big) \nonumber \\
&&~~~~~\geq C_1 \Big(\frac{L_1}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L_1}{\sigma_n^2}\Big) + C_1 \Big(\frac{\gamma_2 L_2}{\sigma_n^2}\Big) \\
&\Leftrightarrow& C_1 \Big(\frac{L_2}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L_2}{\sigma_n^2}\Big) - \bigg\{ C_1 \Big(\frac{L_1 L_2}{(L_1+L_2)\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L_1 L_2}{(L_1 + L_2)\sigma_n^2}\Big) \bigg\} \geq 0.
\label{eq:temp3}
\end{aligned}$$ Equation (\[eq:temp3\]) holds for any $L_1, L_2>0$ and $0<\gamma_2<1$. Thus, $C_1(x) - C_1(\beta x)$ is an increasing function of $x>0$.
When $L_1 = L_2 = L$, $\mathbb{E}[\tilde{R}_{\text{cN}}^{\text{bi}}] > \mathbb{E}[R_{\text{O}}]$, provided $\frac{L}{\sigma_n^2}, \frac{\gamma_1 L}{\alpha_1 \sigma_n^2},\frac{\gamma_2 L}{\alpha_2 \sigma_n^2}>1$. \[theorem:ergodic\_inequality\_oma\]
See Appendix \[appendix:thm3\].
It is reasonable that $\mathbb{E}[\tilde{R}_{\text{cN}}^{\text{bi}}]$ becomes much larger than $\mathbb{E}[R_{\text{O}}]$ as $L_2$ increases above $L_1$. Also, $\mathbb{E}[\tilde{R}_{\text{cN}}^{\text{bi}}] - \mathbb{E}[R_{\text{O}}] > 0$ when $L_1=L_2$ by Theorem \[theorem:ergodic\_inequality\_oma\], so it can also be expected that $\mathbb{E}[\tilde{R}_{\text{cN}}^{\text{bi}}]$ could be still larger than $\mathbb{E}[R_{\text{O}}]$ when $L_1 = L_2+\delta$ for small $\delta>0$. In Section \[sec:simulation\], numerical results show that bi-directional cooperative NOMA still has a rate gain compared to OMA even when the SIC user (user 2) experiences the weaker channel than the non-SIC user (user 1).
Optimal Power Allocation Rule {#sec:power_allocation}
=============================
Based on ergodic capacity analysis, we present the optimal power allocation rule for bi-directional cooperative NOMA. Two optimization goals are considered: user fairness and sum-rate. Note that the BS should know statistical CSI at least for the optimal power allocation, and we do not consider the no-CSIT case here. Assume $L_1 < L_2$ in this section.
User Fairness Problem
---------------------
As in [@NOMA:UserFairness:SPL-Timotheou; @NOMA-CoMP:ICC-Han], the max-min optimization problem is formulated for user fairness as $$\gamma_{2}^{*}=\underset{0<\gamma_2 <1}{\arg \max}~\min (\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}], \mathbb{E}[R_{\text{cN},2}^{\text{bi}}]),
\label{eq:min-max-prob}$$ where $\gamma_2^{*}$ is the optimal power ratio for user 2 and recall that $\gamma_1 + \gamma_2 =1$. The following lemma helps to solve the above optimization problem.
$\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}]$ is a decreasing function of $\gamma_2$ and $\mathbb{E}[R_{\text{cN},2}^{\text{bi}}]$ is an increasing function of $\gamma_2$. \[lemma:R\_1\_R\_2\]
$\tilde{R}_{\text{cN},1}^{\text{bi}} = \max\{Z_1, Z_2\}$, where $Z_1 = \log_2 (1+ \frac{ |h_1|^2 \gamma_1 }{ |h_1|^2 \gamma_2 + \sigma_n^2}) = \log_2(1+\frac{|h_1|^2}{\sigma_n^2}) - \log_2(1+ \frac{|h_1|^2 \gamma_2}{\sigma_n^2})$, and $Z_2 = \log_2 ( 1+ \frac{ |h_2|^2 \gamma_1 }{ |h_2|^2 \gamma_2 + \sigma_n^2}) = \log_2(1+\frac{|h_2|^2}{\sigma_n^2}) - \log_2(1+ \frac{|h_2|^2 \gamma_2}{\sigma_n^2})$. Since $\log_2(1+\frac{|h_i|^2\gamma_2}{\sigma_n^2})$ is an increasing function of $\gamma_2$, $Z_1$ and $Z_2$ are decreasing functions so $\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}]$ is a decreasing function of $\gamma_2$. Also, $R_{\text{cN},2}^{\text{bi}} = \log_2(1+\frac{|h_2|^2 \gamma_2}{\sigma_n^2})$, so $\mathbb{E}[R_{\text{cN},2}^{\text{bi}}]$ is an increasing function of $\gamma_2$.
By Lemma \[lemma:R\_1\_R\_2\], the optimal solution of (\[eq:min-max-prob\]) is directly obtained when $\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}] = \mathbb{E}[R_{\text{cN},2}^{\text{bi}}]$. Since $\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}], \mathbb{E}[{R}_{\text{cN},2}^{\text{bi}}] \geq 0$ for any $\gamma_1, \gamma_2 \in [0,1]$, $\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}]=0$ at $\gamma_1=0$, and $\mathbb{E}[{R}_{\text{cN},2}^{\text{bi}}]=0$ at $\gamma_2=0$, the solution of $\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}] = \mathbb{E}[{R}_{\text{cN},2}^{\text{bi}}]$ would satisfy $0\leq \gamma_2^{*} \leq 1$. However, the closed-form solution of $\gamma_2^{*}$ is difficult to derive because of the expectation operations. Therefore, the bisection method is used to solve (\[eq:min-max-prob\]). Algorithm \[algo:user\_fairness\] shows the detail.
Max-Sum-Rate Problem
--------------------
To maximize the sum-rate of NOMA system, allocating all power to the strong user is a simple solution. However, it destroys user-fairness completely; the sum-rate performance is usually studied under a minimum rate constraint as in [@NOMA-CoMP:ICC-Han]. The problem can be formulated as $$\begin{aligned}
&\gamma_{2}^{*}=\underset{0<\gamma_2 <1}{\arg \max}~\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}] + \mathbb{E}[R_{\text{cN},2}^{\text{bi}}] \label{eq:max-sum-rate_prob}\\
&\text{s.t.} \ \text{min}(\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}], \mathbb{E}[R_{\text{cN},2}^{\text{bi}}])\geq R_{t}
\label{eq:max-sum-rate_const}\end{aligned}$$ where $R_t$ is the minimum data rate constraint for user fairness. Lemma \[lemma:R\_1+R\_2\_NearUserAssumption\] is introduced for solving the above optimization problem.
$\mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}] + \mathbb{E}[{R}_{\text{cN},2}^{\text{bi}}]$ is a decreasing function of $\gamma_2$, for any $L_1, L_2>0$. \[lemma:R\_1+R\_2\_NearUserAssumption\]
According to (\[eq:closed\_ergodic\_cN\_bi\]), $$\begin{aligned}
\frac{\mathrm{d}}{\mathrm{d}\gamma_2}\big( \mathbb{E}[\tilde{R}_{\text{cN},1}^{\text{bi}}] + \mathbb{E}[{R}_{\text{cN},2}^{\text{bi}}] \big) = \frac{\mathrm{d}}{\mathrm{d}\gamma_2} \bigg\{ - C_1 \Big(\frac{\gamma_2 L_1}{\sigma_n^2}\Big) + C_1 \Big(\frac{\gamma_2 L_1 L_2}{(L_1 + L_2)\sigma_n^2}\Big) \bigg\} < 0,
\end{aligned}$$ by Lemma \[lemma:Cx-Cax\] and $0<\frac{L_1}{L_1+L_2}<1$.
By Lemmas \[lemma:R\_1\_R\_2\] and \[lemma:R\_1+R\_2\_NearUserAssumption\], the optimal solution of (\[eq:max-sum-rate\_prob\]) is obtained when $\mathbb{E}[{R}_{\text{cN},2}^{\text{bi}}] = R_t$. Similar to the user fairness problem, the bisection method can be used to find $\gamma_2^{*}$ of (\[eq:max-sum-rate\_prob\]). However, if the assumption that both users are located nearby is not satisfied, Lemma \[lemma:R\_1+R\_2\_NearUserAssumption\] does not hold anymore. In this case, we consider some special cases depending on the relative amounts of $W_{1}$ and $W_{2}$ compared to $V_{1,1}$ and $V_{2,1}$. Suppose $V_{1,1} < V_{2,1}$, then Lemma \[lemma:R\_1+R\_2\_NearUserAssumption\] holds when $V_{2,1} < W_1$. However, when $V_{1,1} < W_1 < V_{2,1}$, $R_{\text{cN},1}^{\text{bi}}$ becomes $\log_2(1+\frac{|g_c|^2}{\sigma_n^2})$, so $\mathbb{E}[R_{\text{cN},1}^{\text{bi}}]$ does not depend on the power allocation ratio. Then, $\mathbb{E}[R_{\text{cN},1}^{\text{bi}}]+\mathbb{E}[R_{\text{cN},2}^{\text{bi}}]$ becomes an increasing function of $\gamma_2$ because $\mathbb{E}[R_{\text{cN},2}^{\text{bi}}]$ does. On the other hand, when $W_1 < V_{1,1}$, $R_{\text{cN},1}^{\text{bi}}$ becomes $\log_2(1+\frac{\gamma_1|h_1|^2}{\gamma_2|h_2|^2 + \sigma_n^2})$, and it can be proven that $\mathbb{E}[R_{\text{cN},1}^{\text{bi}}]+\mathbb{E}[R_{\text{cN},2}^{\text{bi}}]$ is a decreasing function of $\gamma_2$ in a way similar to the proof of Lemma \[lemma:R\_1+R\_2\_NearUserAssumption\].
The situation where $V_{1,1} > V_{2,1}$ can be also considered similar to $V_{1,1} < V_{2,1}$. However, this case is not applied to solve the max-sum-rate problem (\[eq:max-sum-rate\_prob\]). The reason is that $\mathbb{E}[V_{1,1}], \mathbb{E}[V_{2,1}], \mathbb{E}[W_1]$ and $\mathbb{E}[W_2]$ are used instead of $V_{1,1}, V_{2,1}, W_1$ and $W_2$ in the statistical CSIT case. This approximation does not consider the case of $V_{1,1} > V_{2,1}$, because we assume $L_2 > L_1$ first so $\mathbb{E}[V_{1,1}] < \mathbb{E}[V_{2,1}]$ always. Thus, this approximation makes $\gamma_2^{*}$ of (\[eq:max-sum-rate\_prob\]) a suboptimal solution.
In summary, if $V_{1,1} < W_1 < V_{2,1}$, $\mathbb{E}[R_{\text{cN},1}^{\text{bi}}] + \mathbb{E}[R_{\text{cN},2}^{\text{bi}}]$ is an increasing function of $\gamma_2$, so the solution is obtained when $\mathbb{E}[R_{\text{cN},1}^{\text{bi}}] = R_t$. If not, $\mathbb{E}[R_{\text{cN},1}^{\text{bi}}] + \mathbb{E}[R_{\text{cN},2}^{\text{bi}}]$ is a decreasing function of $\gamma_2$, and $\gamma_2^{*}$ is found when $\mathbb{E}[R_{\text{cN},2}^{\text{bi}}] = R_t$. Based on these behaviors, the suboptimal bisection method for maximizing the sum-rate of bi-directional cooperative NOMA in the statistical CSIT case is presented in Algorithm \[algo:max\_sum\_rate\]. Note that the outage event occurs when the minimum rate constraint (\[eq:max-sum-rate\_const\]) is not satisfied. In addition, according to Lemma \[lemma:R\_1\_R\_2\], we can recognize that $R_0 = \mathbb{E}[R^{\text{bi}}_{\text{cN},1}] = \mathbb{E}[R^{\text{bi}}_{\text{cN},2}]$ should be larger than $R_t$; otherwise, the system cannot avoid outage.
Outage Probability {#sec:outage_prob}
==================
When the targeted data rates, $R_{t,1}$ and $R_{t,2}$, are determined by the users’ QoS requirements, the outage probability is an important performance criterion. If the outage event occurs at the non-SIC user, the SIC user does not use the cooperation signal, and outage of SIC user does not allow the cooperation from the SIC user to the non-SIC user. The outage probability at the non-SIC user (user 1) in bi-directional cooperative NOMA is given by $$P_{\text{cN},1}^{\text{bi}} = P\{V_{1,1} < \epsilon_{1},~V_{2,1} < \epsilon_{1}\} + P\{ \max\{ V_{1,1},~W_1 \} < \epsilon_{1},~V_{2,1} > \epsilon_{1} \},
\label{eq:outage1_bi-cN}$$ where $\epsilon_i = 2^{R_{t,i}}-1$.
Again, $|h_i|^2=L_i X_i/2$ for $i \in \{1,2\}$. The first term of (\[eq:outage1\_bi-cN\]) becomes $$P\bigg\{ \frac{\gamma_1 L_1 X_1}{\gamma_2 L_1 X_1 + 2\sigma_n^2} < \epsilon_1 \bigg\} \cdot P\bigg\{ \frac{\gamma_1 L_2 X_2}{\gamma_2 L_2 X_2 + 2\sigma_n^2} < \epsilon_1 \bigg\}
=
\begin{cases}
1 & \text{if } \frac{\gamma_1}{\gamma_2}<\epsilon_1 \\
(1- e^{-\xi / L_1}) (1- e^{ - \xi / L_2 } ) & \text{otherwise} \\
\end{cases}
\label{eq:outage1_bi-cN_temp1}$$ where $\epsilon_i = 2^{R_{ti}}-1$ and $\xi = \frac{\sigma_n^2 \epsilon_1}{\gamma_1 - \epsilon_1 \gamma_2}$, and the second term of (\[eq:outage1\_bi-cN\]) becomes $$\begin{aligned}
&&P\bigg\{ \frac{\gamma_1 L_1 X_1}{\gamma_2 L_1 X_1 + 2\sigma_n^2} < \epsilon_1 \bigg\} \cdot P\bigg\{ \frac{L_c X_c}{2\sigma_n^2} < \epsilon_1 \bigg\} \cdot P\bigg\{ \frac{\gamma_1 L_2 X_2}{\gamma_2 L_2 X_2 + 2\sigma_n^2} > \epsilon_1 \bigg\} \\
&&~~~=
\begin{cases}
0 & \text{if } \frac{\gamma_1}{\gamma_2}<\epsilon_1 \\
( 1- e^{ - \xi/L_1} ) ( 1- e^{ -\sigma_n^2 \epsilon_1/L_c } ) e^{ -\xi / L_2} & \text{otherwise}
\end{cases}
\label{eq:outage1_bi-cN_temp2}\end{aligned}$$
By (\[eq:outage1\_bi-cN\]), (\[eq:outage1\_bi-cN\_temp1\]) and (\[eq:outage1\_bi-cN\_temp2\]), $P_{\text{cN},1}^{\text{bi}}$ is given by $$P_{\text{cN},1}^{\text{bi}} =
\begin{cases}
1 & \text{if } \frac{\gamma_1}{\gamma_2}<\epsilon_1 \\
(1- e^{ - \xi / L_1} ) (1- e^{ -\sigma_n^2 \epsilon_1/L_c - \xi / L_2} ) & \text{otherwise}
\end{cases}.$$ $P_{\text{cN},1}^{\text{bi}}$ conditioned on $\gamma_1 / \gamma_2>\epsilon_1$ is approximated in the high SNR region by $$P_{\text{cN},1}^{\text{bi}} \approx
\frac{\sigma_n^2 \epsilon_1}{L_1 (\gamma_1 - \epsilon_1 \gamma_2)} \cdot \Big( \frac{\sigma_n^2 \epsilon_1}{L_c} + \frac{\sigma_n^2 \epsilon_1}{L_2(\gamma_1-\epsilon_1\gamma_2)} \Big),
\label{eq:outage1_bi-cN_highSNR}$$ and it indicates user 1 achieves a diversity order of 2.
Here, the outage probability of user 1 in bi-directional cooperative NOMA is the same as that of uni-directional cooperative NOMA, i.e., $P_{\text{cN},1}^{\text{bi}} = P_{\text{cN},1}^{\text{uni}}$, because user 1 receives the cooperation signal from user 2 also in uni-directional cooperative NOMA. On the other hand, conventional NOMA and OMA are different. For conventional NOMA, the outage probability of user 1 of conventional NOMA, $P_{\text{N},1}$, is given by $$\begin{aligned}
P_{\text{N},1} = P\Big\{ \frac{\gamma_1 L_1 X_1}{\gamma_2 L_1 X_1 + 2\sigma_n^2} < \epsilon_1 \Big\} &=& 1 - e^{ -\xi / L_1} \label{eq:outage1_conv-N} \\
&\approx& \frac{\sigma_n^2 \epsilon_1}{L_1(\gamma_1 - \epsilon_1\gamma_2)},
\label{eq:outage1_conv-N_highSNR}\end{aligned}$$ and it just has a diversity order of 1. Equation (\[eq:outage1\_conv-N\_highSNR\]) is achieved by a high-SNR approximation.
Likewise, user 1 of OMA also achieves a diversity order of 1, as shown in (\[eq:outage1\_OMA\]) and (\[eq:outage1\_OMA\_highSNR\]). $$\begin{aligned}
P_{\text{O},1} = P\Big\{ \frac{\gamma_1L_1X_1}{2\alpha_1 \sigma_n^2} < \epsilon_{O,1} \Big\} &=& 1-\exp\Big\{ -\frac{\alpha_1 \sigma_n^2 \epsilon_{O,1}}{\gamma_1L_1} \Big\} \label{eq:outage1_OMA}\\
&\approx& \frac{\alpha_1 \sigma_n^2 \epsilon_{O,1}}{\gamma_1L_1},
\label{eq:outage1_OMA_highSNR}\end{aligned}$$ where $\epsilon_{O,i} = 2^{R_{t,i}/\alpha_i}-1$ for $i \in \{1,2\}$. We can easily note that bi- and uni-directional cooperative NOMA systems can achieve multiuser diversity.
Next, consider the outage probability of user 2 in bi-directional cooperative NOMA system. If user 1 avoids the outage event, user 2 can use the cooperation signal transmitted from user 1 for its SIC process. However, if the user 1’s data rate is less than $R_{t,1}$, the cooperation from user 1 to user 2 cannot be performed. The outage probability of user 2 in bi-directional cooperative NOMA is given by $$\begin{aligned}
P_{\text{cN},2}^{\text{bi}} &=& P\{ (V_{2,2} < \epsilon_{2} \cup V_{2,1} < \epsilon_{1}),~V_{1,1} < \epsilon_{1} \} \nonumber \\
&&~~+ P\{ (V_{2,2} < \epsilon_{2} \cup \max \{V_{2,1},~W_2\} < \epsilon_{1}),~V_{1,1} > \epsilon_{1} \}
\label{eq:outage2_bi-cN}\end{aligned}$$
The first term of (\[eq:outage2\_bi-cN\]) becomes $$\begin{aligned}
&&(1- P\{Z_2 > \epsilon_{2},~ V_{2,1} > \epsilon_{1} \})P\{V_{1,1} < \epsilon_{1}\} \\
&&~~~= \bigg(1- P\Big\{ \frac{\gamma_2L_2X_2}{2\sigma_n^2} > \epsilon_2,~ \frac{\gamma_1 L_2 X_2}{\gamma_2 L_2 X_2 + 2\sigma_n^2} > \epsilon_1 \Big\} \bigg) P\Big\{ \frac{\gamma_1 L_1 X_1}{\gamma_2 L_1 X_1 + 2\sigma_n^2} < \epsilon_1 \Big\}
\label{eq:outage2_bi-cN_temp1}\end{aligned}$$
Similarly, the second term of (\[eq:outage2\_bi-cN\]) becomes
$$\begin{aligned}
&&\big[ P\{ V_{2,2} < \epsilon_{2}\} + P\{\max\{V_{2,1},~W_2\} < \epsilon_{1} \} \nonumber \\
&&~~~- P\{ V_{2,2} < \epsilon_{2} \cap \max\{V_{2,1},~W_2\} < \epsilon_{1} \} \big] \cdot P\{V_{1,1} > \epsilon_{1}\},
\label{eq:outage1_bi-cN_2ndterm}\end{aligned}$$
where $$\begin{aligned}
&&P\{\max\{V_{2,1},~W_2\} < \epsilon_{1} \} = P\{V_{2,1} < \epsilon_{1}\} P\{W_2 < \epsilon_{1}\} \\
&&~~~~~~~~=
\begin{cases}
1 & \text{if } \frac{\gamma_1}{\gamma_2}<\epsilon_1 \\
( 1- e^{ -\xi / L_2} ) ( 1- e^{ -\sigma_n^2\epsilon_1/L_c } ) & \text{otherwise} \\
\end{cases}
\label{eq:outage2_bi-cN_temp2}\end{aligned}$$ and $$P\{ V_{2,2} < \epsilon_{2} \cap \max\{V_{2,1},~W_2\} < \epsilon_{1} \}
=
\begin{cases}
1 & \text{if } \frac{\gamma_1}{\gamma_2}<\epsilon_1 \\
( 1- e^{ -\xi / L_2} ) ( 1- e^{ -\sigma_n^2\epsilon_1/L_c } ) & \text{else if } \frac{\epsilon_2}{\gamma_2} > \frac{\epsilon_1}{\gamma_1-\epsilon_1\gamma_2} \\
( 1- e^{ -\sigma_n^2 \epsilon_2/\gamma_2 L_2 } ) ( 1- e^{ -\sigma_n^2\epsilon_1/L_c } ) & \text{else}
\end{cases}
\label{eq:outage2_bi-cN_temp3}$$
Thus, according to (\[eq:outage2\_bi-cN\]), (\[eq:outage2\_bi-cN\_temp1\]), (\[eq:outage2\_bi-cN\_temp2\]), and (\[eq:outage2\_bi-cN\_temp3\]), $P_{\text{cN},2}^{\text{bi}}$ is given by $$P_{\text{cN},2}^{\text{bi}} =
\begin{cases}
1 & \text{if } \frac{\gamma_1}{\gamma_2}<\epsilon_1 \\
1 - e^{ -\sigma_n^2 \epsilon_2 / \gamma_2 L_2 } \approx \frac{\sigma_n^2 \epsilon_2}{\gamma_2 L_2}
& \text{else if } \frac{\epsilon_2}{\gamma_2} > \frac{\epsilon_1}{\gamma_1-\epsilon_1\gamma_2} \\
( 1- e^{ - \xi / L_2 } ) - e^{ - \sigma_n^2 \epsilon_1 / L_c - \xi / L_1 } ( e^{ - \sigma_n^2 \epsilon_2 / \gamma_2 L_2 } - e^{ - \xi / L_2 } ) \\
\approx \frac{\sigma_n^2 \epsilon_1}{L_2(\gamma_1 - \epsilon_1 \gamma_2)} - \Big( \frac{\sigma_n^2 \epsilon_1}{L_2(\gamma_1 - \epsilon_1 \gamma_2)} - \frac{\sigma_n^2 \epsilon_2}{\gamma_2 L_2} \Big) \Big( 1 - \frac{\sigma_n^2 \epsilon_1}{L_c} - \frac{\sigma_n^2 \epsilon_1}{L_1(\gamma_1 - \epsilon_1 \gamma_2)} \Big)
& \text{else}
\end{cases}. \label{eq:outage2_bi-cN_closed}$$ The approximations in (\[eq:outage2\_bi-cN\_closed\]) are obtained in the high SNR region. Unlike $P_{\text{cN},1}^{\text{bi}}$, $P_{\text{cN},2}^{\text{bi}}$ has a diversity order of 1. The reason is that user 2 should perform SIC before its data decoding, even though the cooperation signal from user 1 could help its SIC process. Uni-directional cooperative NOMA does not allow user 2 to receive the cooperation signal, so its outage probability is obtained by
$$\begin{aligned}
P_{\text{cN},2}^{\text{uni}} &=& P\{ V_{2,2} < \epsilon_{2} \cup V_{2,1} < \epsilon_{1} \} = 1 - P\{ V_{2,2} > \epsilon_{2},~ V_{2,1} > \epsilon_{1} \} \\
&=& \begin{cases}
1 & \text{if } \frac{\gamma_1}{\gamma_2}<\epsilon_1 \\
1 - \exp\{-\frac{\sigma_n^2 \epsilon_2}{\gamma_2 L_2}\} \approx \frac{\sigma_n^2 \epsilon_2}{\gamma_2 L_2} & \text{else if }\frac{\epsilon_2}{\gamma_2} > \frac{\epsilon_1}{\gamma_1-\epsilon_1 \gamma_2}\\
1 - \exp\{ -\frac{\sigma_n^2 \epsilon_1}{L_2 (\gamma_1 - \epsilon_1 \gamma_2)} \} \approx \frac{\sigma_n^2 \epsilon_1}{L_2 (\gamma_1 - \epsilon_1 \gamma_2)} & \text{else}
\end{cases}.
\label{eq:outage2_uni-cN_closed}\end{aligned}$$
The high SNR approximation in (\[eq:outage2\_uni-cN\_closed\]) shows user 2 in uni-directional cooperative NOMA realizes a diversity order of 1, the same as bi-directional cooperative NOMA. Conditioned on $\frac{\gamma_1}{\gamma_2}>\epsilon_1$, $P_{\text{cN},2}^{\text{bi}} = P_{\text{cN},2}^{\text{uni}}$ when $\frac{\epsilon_2}{\gamma_2} > \frac{\epsilon_1}{\gamma_1-\epsilon_1 \gamma_2}$, but $P_{\text{cN},2}^{\text{bi}}$ has a power gain compared to $P_{\text{cN},2}^{\text{uni}}$ when $\frac{\epsilon_2}{\gamma_2} < \frac{\epsilon_1}{\gamma_1-\epsilon_1 \gamma_2}$. Since there is no cooperation signal from user 1 to user 2 in uni-directional cooperative NOMA, $P_{\text{cN},2}^{\text{uni}}$ is the same as the outage probability of user 2 of conventional NOMA, $P_{\text{cN},2}^{\text{uni}} = P_{\text{N},2}$. Meanwhile, the outage probability of the user 2 of OMA is given by $$P_{\text{O},2} = 1 - \exp \Big\{ -\frac{\alpha_2 \sigma_n^2 \epsilon_{O,2}}{\gamma_2 L_2} \Big\} \approx \frac{\alpha_2 \sigma_n^2 \epsilon_{O,2}}{\gamma_2 L_2},$$ and it also realizes a diversity order of 1.
In summary, bi-directional cooperative NOMA achieves a power gain for the SIC user (user 2) conditioned on $\frac{\epsilon_2}{\gamma_2} < \frac{\epsilon_1}{\gamma_1-\epsilon_1 \gamma_2}$ and $\frac{\gamma_1}{\gamma_2} > \epsilon_1$, compared to uni-directional cooperative NOMA and conventional NOMA. On the other hand, the non-SIC user of bi- or uni-directional cooperative NOMA scheme achieves better diversity order than that of conventional NOMA and OMA, as shown by (\[eq:outage1\_bi-cN\_highSNR\]).
Extension to Multi-User Scenario {#sec:multi-user}
================================
Thus far, we considered the two-user model for bi-directional cooperative NOMA. However, the proposed system can be extended to the multi-user scenario. Assume that the BS serves $K$ users by NOMA. Bi-directional cooperative NOMA consists of $K+2$ phases. The first and the second phases correspond to direct transmission and channel information exchange, respectively, and others are for cooperation. In the statistical-CSIT case, $L_1 < L_2 < \cdots < L_K$ is assumed, and suppose that $\gamma_1 > \gamma_2 > \cdots > \gamma_K$ for both cases of statistical CSIT and no CSIT. All phases of bi-directional cooperative NOMA in the multi-user scenario are explained below:
### Direct Transmission Phase
The BS transmits the superpositioned signal to all users. The received signal at the user $i$ is $$r_i = h_i \sum_{k=1}^{K} \sqrt{\gamma_k}s_k + n_i.$$
### Channel Information Exchange Phase {#channel-information-exchange-phase}
Users exchange their channel information to determine the direction of cooperation. As we will see in the $j$-th cooperative phase, the cooperation signal is transmitted at the user $i_0$, whose channel gain is the strongest among users $j,\cdots,K$. Therefore, users should know the order of channel gains of all users and this phase requires $K$ time slots. For each slot, a user delivers its channel information to the others via short-range communications. Note that in the proposed scheme in $K$-user scenario, each time slot is actually reduced for exchange of channel information when the users are crowded.
*Remark*: As mentioned earlier in Section \[subsec:channel\_exchange\_phase\], the CSI exchange step causes additional delay and overhead, and those penalties grow as $K$ increases. However, even without the CSI exchange step, this problem also arises in the existing systems where the BS receives CSI feedbacks. Large $K$ also causes a huge computational burden for SIC and requires large power budget to enable multiple steps of SIC. Therefore, only two or four-user NOMA signaling has been considered in practical system, and the proposed scheme can be applied well with appropriate $K$ for practical scenarios.
### The $j$-th Cooperative Phase
The cooperation phase consists of $K$ phases. Cooperation is performed on signal-by-signal basis, i.e., decoding of $s_j$ is performed in the $j$-th cooperative phase. Users $j,\cdots,K$ decode $s_j$, and this phase corresponds to one of the SIC steps, especially for users $j+1,\cdots,K$. Let $V_{i,j}$ be the SINR for decoding $s_j$ at user $i$, where $i\geq j$, and $$V_{i,j} = \frac{|h_i|^2 \gamma_j}{|h_i|^2\sum_{k=j+1}^{K}\gamma_k + \sigma_n^2}.$$ Therefore, cooperation among users $j,\cdots,K$ for improved decoding of $s_j$ is possible in the $j$-th cooperative phase. Among users $j,\cdots,K$, the system allows one whose channel condition is the best to transmit the cooperation signal. Let user $i_0$ be the strongest one, i.e., $i_0 = \underset{j \leq i \leq K}{\arg \max} |h_{i}|^2 = \underset{j \leq i \leq K}{\arg \max} |V_{i,j}|^2$. Then, user $i$, where $j \leq i \leq K$ and $i \neq i_0$, receives the cooperation signals $c^j_{i}$ to help decoding of $s_j$ from user $i_0$, and user $i_0$ does not receive any cooperation signal. $$c^j_{i} = g_{i,i_0} s_{j} + n_i,~~\forall i\in \{j, \cdots, K\},~i \neq i_0$$ where $g_{i,i_0}$ is channel fading from user $i_0$ to user $i$. Let $W_{i,i_0}$ be the SINR of the cooperation signal from user $i_0$ to user $i$, as written by $$W_{i,i_0} = \frac{|g_{i,i_0}|^2}{\sigma_n^2}.$$
The $j$-th cooperation step can increase the data rate of user $j$, and/or help other users $j+1,\cdots,K$ to perform SIC better. The SINR for decoding $s_j$ at user $i$ is denoted by $Z_{\text{cN},i,j}^{\text{bi}}$ and obtained by $$Z_{\text{cN},i,j}^{\text{bi}} =
\begin{cases}
V_{i,j} & i=i_0 \\
\max \{ V_{i,j}, W_{i,i_0} \} & i \neq i_0
\end{cases},~~\forall i \in \{j, \cdots, K\}.$$ Therefore, the total SINR for decoding $s_j$ becomes as follows: $$Z_{\text{cN},j}^{\text{bi}} = \min \{ Z_{\text{cN},j,j}^{\text{bi}}, \cdots, Z_{\text{cN},K,j}^{\text{bi}} \},$$ With the assumption that all users are located nearby, the cooperation signal is much stronger than the signal from the BS, i.e., $W_{i,i_0} \gg V_{i,j}$, $Z_{\text{cN},j}^{\text{bi}}$ can be approximated by $$Z_{\text{cN},j}^{\text{bi}} \approx \tilde{Z}_{\text{cN},j}^{\text{bi}} = \max \{ V_{j,j}, \cdots, V_{K,j} \}.$$ On the other hand, in uni-directional cooperative NOMA, the direction of cooperation is already determined, so all users except for user $K$ receive the cooperation signals from user $K$, whose channel variance is the largest in the statistical-CSIT case, or who are arbitrarily determined in the no-CSIT case. Therefore, the SINRs are given by $${Z}_{\text{cN},i,j}^{\text{uni}} =
\begin{cases}
V_{i,j} & i=K \\
\max \{ V_{i,j}, W_{j,K} \} & i \neq i_0
\end{cases},~~\text{for}~ j\leq i\leq K,$$ $$\begin{aligned}
{Z}_{\text{cN},j}^{\text{uni}} &= \min \{ {Z}_{\text{cN},j,j}^{\text{uni}}, \cdots, {Z}_{\text{cN},K,j}^{\text{uni}} \} \\
&\approx \tilde{Z}_{\text{cN},j}^{\text{uni}} = V_{K,j}.\end{aligned}$$ Conventional NOMA does not allow any cooperation, so its SINR for decoding $s_j$ is given by $$Z_{\text{N},j} = \min \{ V_{j,j}, V_{j+1,i}, \cdots, V_{K,i} \}.$$
Mathematical analysis of ergodic capacity and outage probability for the multi-user model is omitted here, but we verify the advantages of bi-directional cooperative NOMA in the multi-user model by simulation. Section \[subsec:capacity\_randomlyusers\] shows that bi-directional cooperative NOMA gives still better data rates than uni-directional cooperative NOMA, conventional NOMA and OMA for randomly generated multiple users.
*Remark:* Complexity for performing SIC is an important issue in the NOMA system. For conventional NOMA, $k-1$ times of SIC processes are required for the $k$-th strongest user and SIC processes of all users are performed independently. The number of required SIC processes for bi-directional cooperative NOMA is the same as conventional one. Also, the cooperation phases are performed on signal-by-signal basis, so the SIC step for decoding $s_j$ is performed at every user $i\in\{j,\cdots,K\}$ in parallel. Therefore, bi-directional cooperative NOMA does not require the additional time slots for SIC processes of different users, as long as the $j$-th cooperation phase is successfully completed after SIC of $s_j$. On the other hand, the $k$-th strongest user of cooperative NOMA in [@NOMA:uni-cN:Ding] should wait for the others with the better channel conditions to finish the SIC processes and to transmit the cooperation signals.
Simulation Results {#sec:simulation}
==================
The simulation model is based on Fig. \[fig:bi-cN\_model\], and we assume $R=50$. Without loss of generality, users 1 and 2 are assume to be the non-SIC user and the SIC user, respectively. Short-hand notations ‘Bi-cN’, ‘Uni-cN’, and ‘NOMA’ denote bi-directional cooperative NOMA, uni-directional cooperative NOMA, and conventional NOMA, respectively, in the figures.
Ergodic Capacity under Statistical-CSIT only {#subsec:ergodicCap_statistical}
--------------------------------------------
First, consider the statistical-CSIT case. For bi-directional cooperative NOMA, Algorithms \[algo:user\_fairness\] and \[algo:max\_sum\_rate\] are used to find the optimal power allocations for user fairness and sum-rate maximization, respectively. For other schemes, uni-directional cooperative NOMA, conventional NOMA and OMA, the optimal power ratios are numerically found. The ergodic capacity plots versus $d_2$ are obtained with the fixed position of user 1, $d_1=40$. Since user 2 is the SIC user, $L_1 \leq L_2$, i.e., $d_2 \leq d_1$, in the statistical-CSIT case.
![Sum-rate in statistical CSIT-case, SNR=10dB[]{data-label="fig:SumRate_statisticalCSI_10dB_d1=40"}](Fairness_statisticalCSI_10dB_d1=40.pdf){width="\linewidth"}
![Sum-rate in statistical CSIT-case, SNR=10dB[]{data-label="fig:SumRate_statisticalCSI_10dB_d1=40"}](SumRate_statisticalCSI_10dB_d1=40.pdf){width="\linewidth"}
Figs. \[fig:Fairness\_statisticalCSI\_10dB\_d1=40\] and \[fig:SumRate\_statisticalCSI\_10dB\_d1=40\] show user-fairness and sum-rate performances in the two-user scenario with the transmit SNR of 10 dB, respectively. $R_{t,1} = R_{t,2} = 0.8$ is assumed for sum-rate results. Each figure includes both ergodic capacity and numerically obtained data rates, and we can easily see that both are almost the same as $d_2$ approaches to $d_1=40$. This indicates the assumption that the users are located nearby is reliable, when $d_1 \approx d_2$. When $d_2$ is small, the ergodic capacity and numerically obtained sum-rate of bi-directional cooperative NOMA are somewhat different, because Algorithm \[algo:max\_sum\_rate\] is suboptimal, as mentioned earlier. In Figs. \[fig:Fairness\_statisticalCSI\_10dB\_d1=40\] and \[fig:SumRate\_statisticalCSI\_10dB\_d1=40\], bi-directional cooperative NOMA gives better performances of both user fairness and sum-rate than other schemes. As $d_2$ approaches to $d_1$, the channel gain difference between the users decreases, so the capacity gain of NOMA compared to OMA also decreases. Therefore, uni-directional cooperative NOMA has the same performances as OMA and conventional NOMA becomes even worse than OMA, when $d_1 \approx d_2$. Whereas, bi-directional cooperative NOMA is still better than OMA, so it can be said that bi-directional cooperative NOMA is useful even when the channel gain difference of users is not large.
Ergodic Capacity under No-CSIT {#subsec:ergodicCap_noCSIT}
------------------------------
In this subsection, we consider the situation where the BS does not know users’ CSI at all. Again, the two-user scenario is considered with $d_1=40$. Since no CSI is available at the BS, the BS arbitrarily determines users 1 and 2 as the non-SIC user and the SIC user, respectively, so $d_2>d_1$ is possible. It is impossible to find the optimal power allocation, so the fixed power allocation is used. $\gamma_1 = 0.8$ is assumed for NOMA schemes, and $\gamma_1 = 0.5$ and $\alpha_1 = 0.5$ are used for OMA, because fair power allocation is preferable for OMA without any CSI. Also, only sum-rate performances are investigated in the no-CSIT case, because the trends of user-fairness performances depend largely on the power allocation ratios.
\[h!\] ![Sum-rate in no-CSIT case, SNR=10dB[]{data-label="fig:SumRate_noCSI_10dB_d1=40"}](SumRate_noCSI_10dB_d1=40.pdf "fig:"){width="42.00000%"}
Fig. \[fig:SumRate\_noCSI\_10dB\_d1=40\] shows the sum-rate graphs with no CSIT and a transmit SNR of 10 dB. In Fig. \[fig:SumRate\_noCSI\_10dB\_d1=40\], it is easily noticeable that bi-directional cooperative NOMA gives better capacity than other NOMA schemes and OMA. When $d_2<d_1$, capacity gains of bi-directional cooperative NOMA over other NOMA schemes are not large, but its gain over OMA is large due to SIC. When $d_2 \geq d_1$, since the channel gain of the SIC user usually becomes smaller than that of the non-SIC user, the advantage of NOMA and SIC vanishes, so uni-directional cooperative NOMA and conventional NOMA become worse than OMA. However, bi-directional cooperative NOMA still has a capacity gain compared to OMA even when $d_2 \geq d_1$. These results are consistent with Theorems \[thm:ergodic\_comparison\_NOMA\] and \[theorem:ergodic\_inequality\_oma\]. Especially when $d_1=d_2=40$, bi-directional cooperative NOMA shows a capacity increase of 15% compared to OMA. As $d_2$ increases much, bi-directional cooperative NOMA would give a smaller sum-rate than OMA, but its gap is relatively small, compared to the region of $d_2 \leq d_1$.
Outage Probability {#outage-probability}
------------------
![Outage probabilities when $d_1=40, d_2=20, R_{t1}=1.5, R_{t2}=0.7$[]{data-label="fig:Outage_d1=40_d2=20_Rt1=15_Rt2=07_p1=075"}](Outage_d1=40_d2=20_Rt1=07_Rt2=15_p1=075.pdf){width="\linewidth"}
![Outage probabilities when $d_1=40, d_2=20, R_{t1}=1.5, R_{t2}=0.7$[]{data-label="fig:Outage_d1=40_d2=20_Rt1=15_Rt2=07_p1=075"}](Outage_d1=40_d2=20_Rt1=15_Rt2=07_p1=075.pdf){width="\linewidth"}
![Outage probabilities when $d_1=40, d_2=60, R_{t1}=1.5, R_{t2}=0.7$[]{data-label="fig:Outage_d1=40_d2=60_Rt1=15_Rt2=07_p1=075"}](Outage_d1=40_d2=60_Rt1=07_Rt2=15_p1=075.pdf){width="\linewidth"}
![Outage probabilities when $d_1=40, d_2=60, R_{t1}=1.5, R_{t2}=0.7$[]{data-label="fig:Outage_d1=40_d2=60_Rt1=15_Rt2=07_p1=075"}](Outage_d1=40_d2=60_Rt1=15_Rt2=07_p1=075.pdf){width="\linewidth"}
As shown in Section \[sec:outage\_prob\], the outage probability is computed by channel variances and power allocation ratios. Power allocation and $R_{t,1}$ are appropriately chosen to satisfy $\frac{\gamma_1}{\gamma_2} > \epsilon_1$, and $\gamma_1=0.75$ is assumed here. Figs. \[fig:Outage\_d1=40\_d2=20\_Rt1=07\_Rt2=15\_p1=075\] and \[fig:Outage\_d1=40\_d2=20\_Rt1=15\_Rt2=07\_p1=075\] give outage probability performances when $d_1=40$ and $d_2=20$. Fig. \[fig:Outage\_d1=40\_d2=20\_Rt1=07\_Rt2=15\_p1=075\] assumes $R_{t,1}=0.7$ and $R_{t,2}=1.5$ and Fig. \[fig:Outage\_d1=40\_d2=20\_Rt1=15\_Rt2=07\_p1=075\] is obtained with $R_{t,1}=1.5$ and $R_{t,2}=0.7$. In other words, Fig. \[fig:Outage\_d1=40\_d2=20\_Rt1=07\_Rt2=15\_p1=075\] satisfies the condition of $\frac{\epsilon_2}{\gamma_2} > \frac{\epsilon_1}{\gamma_1 - \epsilon_1\gamma_2}$, so bi- and uni-directional cooperative NOMA schemes show exactly the same outage probabilities for both users. Also, it is easily noted by the slopes of graphs that bi- and uni-directional cooperative NOMA schemes provide a better diversity order than conventional NOMA and OMA for user 1, but not for user 2. Difference in diversity order is also observed in Fig. \[fig:Outage\_d1=40\_d2=20\_Rt1=15\_Rt2=07\_p1=075\], and two cooperative NOMA schemes still have the same outage probability of user 1. However, since $\frac{\epsilon_2}{\gamma_2} < \frac{\epsilon_1}{\gamma_1 - \epsilon_1\gamma_2}$ is satisfied in Fig. \[fig:Outage\_d1=40\_d2=20\_Rt1=15\_Rt2=07\_p1=075\], the outage probability of user 2 of bi-directional cooperative NOMA has a power gain compared to uni-directional cooperative NOMA.
Figs. \[fig:Outage\_d1=40\_d2=60\_Rt1=07\_Rt2=15\_p1=075\] and \[fig:Outage\_d1=40\_d2=60\_Rt1=15\_Rt2=07\_p1=075\] give outage probability performances obtained with $d_1=40$ and $d_2=60$. Since $L_1>L_2$ in Figs. \[fig:Outage\_d1=40\_d2=60\_Rt1=07\_Rt2=15\_p1=075\] and \[fig:Outage\_d1=40\_d2=60\_Rt1=15\_Rt2=07\_p1=075\], those results only correspond to the no-CSIT case. The plots in Figs. \[fig:Outage\_d1=40\_d2=60\_Rt1=07\_Rt2=15\_p1=075\] and \[fig:Outage\_d1=40\_d2=60\_Rt1=15\_Rt2=07\_p1=075\] show almost same trends with those in Figs. \[fig:Outage\_d1=40\_d2=20\_Rt1=07\_Rt2=15\_p1=075\] and \[fig:Outage\_d1=40\_d2=20\_Rt1=15\_Rt2=07\_p1=075\], except for a little bit of power gain differences. The power gain of bi-directional cooperative NOMA over uni-directional cooperative NOMA and the diversity order gains over conventional NOMA and OMA are still guaranteed even when $d_1=40$ and $d_2=60$.
\[h!\] ![Cellular model of randomly generated multiple users[]{data-label="fig:RandomUserModel"}](SimulationModel.pdf "fig:"){width="48.00000%"}
Capacity of Randomly Generated Users {#subsec:capacity_randomlyusers}
------------------------------------
This section considers the cellular model of randomly positioned multiple users, as shown in Fig. \[fig:RandomUserModel\]. In Fig. \[fig:RandomUserModel\], $K$ users are uniformly placed in the outer ring of the cell of radius $R=50$, i.e., the region between inner and outer circles whose radii are $R-\Delta$ and $R$, respectively, so $d_k \in [R-\Delta,R],~\forall k=\{1,\cdots,K\}$. When $\Delta$ is very small, the only cell-edge users are chosen, and as $\Delta$ increases, almost all of the cell region is covered. Also, $K$ users are separated by an angle smaller than $\theta$ to guarantee that all users are close to one another. It makes the exchange of CSI among users easier and the cooperation more helpful. We only presents an optimal power allocation rule for the two-user model, so optimal power allocations for $K$ users are numerically obtained in the statistical-CSIT case. On the other hand, in the case of no CSIT, fixed power allocation is applied. Assume that $\gamma_{k} = 2\gamma_{k+1}$ for NOMA schemes, and $\gamma_1 = \cdots = \gamma_K = 0.5$ and $\alpha_1 = \cdots = \alpha_K = 0.5$ are used for OMA. Similar to Section \[subsec:ergodicCap\_noCSIT\], the sum-rate performance is only considered in the no-CSIT case.
![Capacity for sum-rate in cellular model of randomly generated users[]{data-label="fig:CapacityRandomEdgeUser_SumRate"}](CapacityRandomEdgeUsers_UserFairness2.pdf){width="\linewidth"}
![Capacity for sum-rate in cellular model of randomly generated users[]{data-label="fig:CapacityRandomEdgeUser_SumRate"}](CapacityRandomEdgeUsers_SumRate2.pdf){width="\linewidth"}
Figs. \[fig:CapacityRandomEdgeUser\_UserFairness\] and \[fig:CapacityRandomEdgeUser\_SumRate\] show plots of data rates versus $\Delta$ obtained from user fairness and sum-rate problems, respectively, with $K=4$ randomly located users. Solid and dashed graphs correspond to the cases of statistical CSIT and no CSIT, respectively. All graphs show increasing trends over $\Delta$, because users are likely to have stronger channel gains with a larger $\Delta$. We can easily see that bi-directional cooperative NOMA gives the best data rates among comparison schemes both in the statistical-CSIT and the no-CSIT cases. Especially in the statistical-CSIT case, when $\Delta$ is small, channel gain differences among users are not large enough to show the advantage of NOMA schemes compared to OMA, so the performance of uni-directional cooperative NOMA is similar to that of OMA, and conventional NOMA is worse than OMA. On the other hand, bi-directional cooperative NOMA is still better than OMA even with small $\Delta$, in terms of both user fairness and sum-rate.
The sum-rate performances in the no-CSIT case are much worse than those in the statistical-CSIT case as $\Delta$ increases. The reason is that when $\Delta$ is large, the situation in which users with weaker channel conditions perform SIC for decoding the signal of users with stronger channel gains so the advantage of NOMA vanishes happens frequently. In this situation, smaller power is allocated to weaker user than stronger one, so uni-directional cooperative NOMA and conventional NOMA give worse sum-rates than OMA in the no-CSIT case. However, the sum-rates of bi-directional cooperative NOMA are still better than other NOMA schemes as well as OMA in the most values of $\Delta$, even when the BS does not have accurate knowledge of users’ CSI and thus arbitrarily schedules the users for signal transmission. This means that bi-directional cooperative NOMA is useful when there is little need for channel tracking or elaborate user scheduling.
Conclusion {#sec:conclusion}
==========
This paper proposes bi-directional cooperative NOMA, in which NOMA users cooperate with each other by channel information exchange, with statistical CSIT only or no CSIT. Performance analysis has been conducted in terms of ergodic capacity and outage probability. The closed-form ergodic capacity of bi-directional cooperative NOMA in the two-user model is derived, and it is shown to be better than those of uni-directional cooperative NOMA, conventional NOMA, and OMA even when the scheduled users have similar channel gains, under only statistical CSI or no CSI at the BS. Based on the ergodic capacity, algorithms to find the optimal power allocations are presented for user fairness and max-sum-rate problems. In addition, we show that the outage probability of the SIC user of bi-directional cooperative NOMA has a power gain over that of uni-directional cooperative NOMA, and bi- and uni-directional cooperative NOMA schemes have a diversity gain over conventional NOMA and OMA. Also, the multi-user model of bi-directional cooperative NOMA is presented by using the cooperations among users on signal-by-signal basis. Simulation results verify the above performance analyses, and that bi-directional cooperative NOMA works well with multiple users in statistical- and no-CSIT cases. The proposed system is beneficial in some important practical scenarios: a highly crowded stadium in which many users experience similar channel gains, and an IoT environment in which an inexpensive transmitter should serve a massive number of machine-type devices but cannot handle substantial processing tasks, so not enough CSI is available at the BS.
Proof of Theorem \[thm:closed\_ergodic\_bi-cN\] {#appendix:thm1}
===============================================
$|h_1|^2$ and $|h_2|^2$ are chi-square distributed with variances of $L_1/2$ and $L_2/2$, respectively, so $|h_i|^2 = L_i X_i / 2$, for $i=\{1,2\}$, where $X_i$ is a chi-square random variable of unit variance. According to (\[eq:bi-cN:R1-near-user\]), $$\begin{aligned}
\mathbb{E}[\tilde{R}_{cN,1}^{bi}] &=& \iint p(x_1,x_2) \tilde{R}_{cN,1}^{bi} \mathrm{d}x_1 \mathrm{d}x_2 \\
&=& \iint_{|h_1|^2 > |h_2|^2} p(x_1,x_2) Z_1 \mathrm{d}x_1 \mathrm{d}x_2 + \iint_{|h_1|^2 < |h_2|^2 } p(x_1,x_2) Z_{2,SIC} \mathrm{d}x_1 \mathrm{d}x_2.
\label{eq:exp_cN1}\end{aligned}$$
Since $X_1$ and $X_2$ are independent, $p(x_1, x_2) = p(x_1)p(x_2)$, the first term of (\[eq:exp\_cN1\]) becomes $$\begin{aligned}
&&\int_{0}^{\infty} \int_{0}^{\frac{L_1}{L_2}x_1} \frac{1}{4} e^{-\frac{x_1}{2}} e^{-\frac{x_2}{2}} \bigg\{ \log_2\Big(1+ \frac{\gamma_1 L_1 x_1}{\gamma_2 L_1 x_1 + 2\sigma_n^2}\Big) \bigg\} \mathrm{d}x_1 \mathrm{d}x_2 \\
&=& \int_{0}^{\infty} \frac{1}{2} (e^{-\frac{x_1}{2}} - e^{-\frac{L_1+L_2}{2L_2}x_1}) \bigg\{ \log_2\Big(1+ \frac{L_1 x_1}{2\sigma_n^2}\Big) - \log_2\Big(1+ \frac{\gamma_2 L_1 x_1}{2\sigma_n^2}\Big) \bigg\} \mathrm{d}x_1 \\
&=& C_1\Big(\frac{L_1}{\sigma_n^2}\Big) - C_1\Big(\frac{\gamma_2 L_1}{\sigma_n^2}\Big) - \frac{L_2}{L_1 + L_2} \bigg\{ C_1\Big(\frac{L_1 L_2}{(L_1+L_2)\sigma_n^2}\Big) - C_1\Big(\frac{\gamma_2 L_1 L_2}{(L_1+L_2)\sigma_n^2}\Big) \bigg\}.
\label{eq:temp1}\end{aligned}$$ The last equation (\[eq:temp1\]) holds by Lemma \[lemma:integral\_chi\].
Likewise, the second term of (\[eq:exp\_cN1\]) is $$\begin{aligned}
&&\int_{0}^{\infty} \int_{0}^{\frac{L_2}{L_1}x_2} \frac{1}{4} e^{-\frac{x_1}{2}} e^{-\frac{x_2}{2}} \log_2 \Big( 1 + \frac{\gamma_1 L_2 x_2}{\gamma_2 L_2 x_2 + 2\sigma_n^2} \Big) \mathrm{d}x_1 \mathrm{d}x_2 \\
&=& C_1 \Big(\frac{L_2}{\sigma_n^2}\Big) - C_1\Big(\frac{\gamma_2 L_2 }{\sigma_n^2}\Big) - \frac{L_1}{L_1 + L_2} \bigg\{ C_1 \Big(\frac{L_1 L_2}{(L_1+L_2)\sigma_n^2}\Big) - C_1\Big(\frac{\gamma_2 L_1 L_2}{(L_1+L_2)\sigma_n^2}\Big) \bigg\}\end{aligned}$$
Therefore, $$\mathbb{E}[\tilde{R}_{cN,1}^{bi}] = C_1\Big(\frac{L_1}{\sigma_n^2}\Big) - C_1\Big(\frac{\gamma_2 L_1}{\sigma_n^2}\Big) + C_1\Big(\frac{L_2}{\sigma_n^2}\Big) - C_1\Big(\frac{\gamma_2 L_2}{\sigma_n^2}\Big) - C_1\Big(\frac{L_1 L_2}{(L_1+L_2)\sigma_n^2}\Big) + C_1\Big(\frac{\gamma_2 L_1 L_2}{(L_1+L_2)\sigma_n^2}\Big)$$
Next, $$\begin{aligned}
\mathbb{E}[R_{cN,2}^{bi}] &=& \int_{0}^{\infty} \int_{0}^{\infty} \frac{1}{4} e^{-\frac{x_1}{2}} e^{-\frac{x_2}{2}} \log_2 \Big(1+\frac{\gamma_2 L_2 x_2}{2\sigma_n^2}\Big) \mathrm{d}x_1 \mathrm{d}x_2 \\
&=& \int_{0}^{\infty} \frac{1}{2} e^{-\frac{x_2}{2}} \log_2 \Big(1+\frac{\gamma_2 L_2 x_2}{2\sigma_n^2}\Big) \mathrm{d}x_2 = C_1 \Big(\frac{\gamma_2 L_2}{\sigma_n^2}\Big).
\label{eq:temp2}\end{aligned}$$ Equation (\[eq:temp2\]) holds by Lemma \[lemma:integral\_chi\]. Thus, closed-form ergodic capacity (\[eq:closed\_ergodic\_cN\_bi\]) can be obtained.
Proof of Theorem \[thm:ergodic\_comparison\_NOMA\] {#appendix:thm2}
==================================================
$$\begin{aligned}
\mathbb{E}[\tilde{R}^{bi}_{cN,1}] &=& \mathbb{E}\bigg[\max \Big\{\log_2 \Big(1 + \frac{|h_1|^2 \gamma_1}{|h_1|^2 \gamma_2 +\sigma_n^2}\Big),~\log_2\Big(1 + \frac{|h_2|^2 \gamma_1}{|h_2|^2 \gamma_2 +\sigma_n^2}\Big)\Big\}\bigg]\\
&\geq& \max \bigg\{ \mathbb{E}\Big[\log_2 \Big(1 + \frac{|h_1|^2 \gamma_1}{|h_1|^2 \gamma_2 +\sigma_n^2}\Big)\Big],~\mathbb{E}\Big[\log_2\Big(1 + \frac{|h_2|^2 \gamma_1}{|h_2|^2 \gamma_2 +\sigma_n^2}\Big)\Big] \bigg\} \\
&=& \max \bigg\{ C_1 \Big(\frac{L_1}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2L_1}{\sigma_n^2}\Big),~C_1 \Big(\frac{L_2}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2L_2}{\sigma_n^2}\Big) \bigg\}
\label{eq:R1_bi-cN_closed_lowerbound}\end{aligned}$$
The last equation (\[eq:R1\_bi-cN\_closed\_lowerbound\]) holds by Lemma \[lemma:expected\_log\_chi\].
For uni-directional cooperative NOMA system, according to \[lemma:expected\_log\_chi\], $$\begin{aligned}
\mathbb{E}[\tilde{R}_{cN,1}^{uni}] &=& \mathbb{E} \bigg[\log_2\Big(1 + \frac{|h_2|^2 \gamma_1}{|h_2|^2 \gamma_2 +\sigma_n^2}\Big)\bigg]= C_1 \Big(\frac{L_2}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2L_2}{\sigma_n^2}\Big)
\label{eq:R1_uni-cN_closed}\end{aligned}$$
Likewise, the upper bound on ergodic capacity of conventional NOMA system becomes $$\begin{aligned}
\mathbb{E}[R_{N,1}] &=& \mathbb{E}\bigg[\min \Big\{\log_2 \Big(1 + \frac{|h_1|^2 \gamma_1}{|h_1|^2 \gamma_2 +\sigma_n^2}\Big),~\log_2\Big(1 + \frac{|h_2|^2 \gamma_1}{|h_2|^2 \gamma_2 +\sigma_n^2}\Big)\Big\}\bigg]\\
&\leq& \min \bigg\{ \mathbb{E}\Big[\log_2 \Big(1 + \frac{|h_1|^2 \gamma_1}{|h_1|^2 \gamma_2 +\sigma_n^2}\Big)\Big],~\mathbb{E}\Big[\log_2\Big(1 + \frac{|h_2|^2 \gamma_1}{|h_2|^2 \gamma_2 +\sigma_n^2}\Big)\Big] \bigg\} \\
&=& \min \bigg\{ C_1 \Big(\frac{L_1}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2L_1}{\sigma_n^2}\Big),~C_1 \Big(\frac{L_2}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2L_2}{\sigma_n^2}\Big) \bigg\}
\label{eq:R1_noma_closed_upperbound}\end{aligned}$$
Note that the data rates of $s_2$ of three schemes are all the same, $$\mathbb{E}[R_{cN,2}^{bi}] = \mathbb{E}[R_{cN,2}^{uni}] = \mathbb{E}[R_{N,2}] = \mathbb{E}\bigg[\log_2\Big(1+\frac{|h_2|^2 \gamma_2 }{\sigma_n^2}\Big)\bigg] = C_1 \Big(\frac{\gamma_2 L_2}{\sigma_n^2}\Big)
\label{eq:R2_closed}$$ Since $\max\{x,y\} \geq x \geq \min \{x,y\},~\forall x,y \in \mathbb{R}$, $\mathbb{E}[\tilde{R}^{bi}_{cN}] \geq \mathbb{E}[\tilde{R}^{uni}_{cN}] \geq \mathbb{E}[R_{N}]$ is satisfied according to (\[eq:R1\_bi-cN\_closed\_lowerbound\]), (\[eq:R1\_uni-cN\_closed\]), (\[eq:R1\_noma\_closed\_upperbound\]), and (\[eq:R2\_closed\]).
Proof of Theorem \[theorem:ergodic\_inequality\_oma\] {#appendix:thm3}
=====================================================
First, concavity of $C_1(x)$ for $x>1$ is proved. Differentiating $C_1(x)$ twice, $$\frac{\mathrm{d}^2}{\mathrm{d}x^2} C_1(x) = \Big( \frac{2}{x^3}+\frac{1}{x^4} \Big) C_1(x) - \frac{1}{\ln 2} \Big( \frac{1}{x^3} + \frac{1}{x^2} \Big)$$
Let $f(x) = x^4 \frac{\mathrm{d}^2}{\mathrm{d}x^2} C_1(x) = (2x+1)C_1(x) - \frac{1}{\ln 2}(x^2+x)$. Then, $$\begin{aligned}
x^2 \frac{\mathrm{d}}{\mathrm{d}x}f(x) &=& (2x^2-x-1)\Big( C_1(x) - \frac{x}{\ln2} \Big) - x C_1(x) \\
&<& \frac{1}{\ln 2} (2x^2-x-1)(\ln(1+x)-x) - xC_1(x)
\label{eq:temp4}\end{aligned}$$ Since $2x^2-x-1 > 0$ when $x>1$ and $x > \ln (1+x)$, equation (\[eq:temp4\]) is smaller than 0 when $x>1$. $f(1)<0$ and $\frac{\mathrm{d}}{\mathrm{d}x}f(x) < 0$, so $f(x)$ is a strictly decreasing function of $x$ for $x>1$. Therefore, $C_1(x)$ is a strictly concave function of $x$ for $x>1$.
Then, $$\begin{aligned}
\mathbb{E}[\tilde{R}_{cN}^{bi}] - \mathbb{E}[R_O] &=& C_1 \Big(\frac{L}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L}{\sigma_n^2}\Big) - \bigg\{ C_1 \Big(\frac{L}{2\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L}{2\sigma_n^2}\Big) \bigg\} \nonumber \\
&&~~+C_1 \Big(\frac{L}{\sigma_n^2}\Big) - \alpha_1 C_1 \Big(\frac{\gamma_1 L}{\alpha_1 \sigma_n^2}\Big) - \alpha_2 C_1 \Big(\frac{\gamma_2 L}{\alpha_2 \sigma_n^2}\Big)\end{aligned}$$ By Lemma \[lemma:Cx-Cax\], $$C_1 \Big(\frac{L}{\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L}{\sigma_n^2}\Big) \geq C_1 \Big(\frac{L}{2\sigma_n^2}\Big) - C_1 \Big(\frac{\gamma_2 L}{2\sigma_n^2}\Big),$$ and since $C_1(x)$ is strictly concave for $x>1$, $$C_1 \Big(\frac{L}{\sigma_n^2}\Big) > \alpha_1 C_1 \Big(\frac{\gamma_1 L}{\alpha_1 \sigma_n^2}\Big) + \alpha_2 C_1 \Big(\frac{\gamma_2 L}{\alpha_2 \sigma_n^2}\Big),$$ according to Jensen’s inequality, if $\frac{L}{\sigma_n^2}, \frac{\gamma_1 L}{\alpha_1 \sigma_n^2},\frac{\gamma_2 L}{\alpha_2 \sigma_n^2}>1$. Theorem \[theorem:ergodic\_inequality\_oma\] is proved.
[1]{}
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[Minseok Choi]{} received the B.S. and M.S. degree in electrical engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea, in 2011 and 2013, respectively. He is currently pursuing the Ph.D degree in KAIST. His research interests include wireless caching network, NOMA, 5G Communications, and stochastic network optimization.
[Dong-Jun Han]{} received the B.S. degrees in Mathematics and Electrical Engineering from the Korea Advanced Institute of Science and Technology (KAIST), Daejeon, South Korea, in 2016, where he is currently pursuing the M.S. degree. His research interests include 5G wireless communications and machine learning.
[Jaekyun Moon]{} received the Ph.D degree in electrical and computer engineering at Carnegie Mellon University, Pittsburgh, Pa, USA. He is currently a Professor of electrical enegineering at KAIST. From 1990 through early 2009, he was with the faculty of the Department of Electrical and Computer Engineering at the University of Minnesota, Twin Cities. He consulted as Chief Scientist for DSPG, Inc. from 2004 to 2007. He also worked as Chief Technology Officier at Link-A-Media Devices Corporation. His research interests are in the area of channel characterization, signal processing and coding for data storage and digital communication. Prof. Moon received the McKnight Land-Grant Professorship from the University of Minnesota. He received the IBM Faculty Development Awards as well as the IBM Partnership Awards. He was awarded the National Storage Industry Consortium (NSIC) Technical Achievement Award for the invention of the maximum transition run (MTR) code, a widely used error-control/modulation code in commercial storage systems. He served as Program Chair for the 1997 IEEE Magnetic Recording Conference. He is also Past Chair of the Signal Processing for Storage Technical Committee of the IEEE Communications Society. He served as a guest editor for the 2001 IEEE JSAC issue on Signal Processing for High Density Recording. He also served as an Editor for IEEE TRANSACTIONS ON MAGNETICS in the area of signal processing and coding for 2001-2006. He is an IEEE Fellow.
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abstract: |
We have conducted a deep radio survey with the Very Large Array at 1.4 GHz of a region containing the Hubble Deep Field. This survey overlaps previous observations at 8.5 GHz allowing us to investigate the radio spectral properties of microjansky sources to flux densities greater than 40 $\mu$Jy at 1.4 GHz and greater than 8 $\mu$Jy at 8.5 GHz. A total of 371 sources have been catalogued at 1.4 GHz as part of a complete sample within 20 of the HDF. The differential source count for this region is only marginally sub-Euclidean and is given by $n(S) = (8.3 \pm 0.4) S^{-2.4 \pm 0.1}$ sr$^{-1}$Jy$^{-1}$. Above about 100 $\mu$Jy the radio source count is systematically lower in the HDF as compared to other fields. We conclude that there is clustering in our radio sample on size scales of 1- 40.
The 1.4 GHz selected sample shows that the radio spectral indices are preferentially steep ($\overline{\alpha } _{1.4} = 0.85 $) and the sources are moderately extended with average angular size $\theta$ = 1.8. Optical identification with disk-type systems at $z \sim $ 0.5-0.8 suggests that synchrotron emission, produced by supernovae remnants, is powering the radio emission in the majority of sources. The 8.5 GHz sample contains primarily moderately flat spectrum sources ($\overline{\alpha } _{8.5} = 0.35$), with less than 15% inverted. We argue that we may be observing an increased fraction of optically thin bremsstrahlung over synchrotron radiation in these distant star-forming galaxies.
author:
- 'E. A. Richards'
title: 'The Nature of Radio Emission from Distant Galaxies: The 1.4 GHz Observations'
---
=cmb10 scaled =cmr10 scaled =cmss10 scaled =cmti10 scaled =cmtt10 scaled i\
[$\backslash$]{} ß[[$^{s}$]{}]{} i[[$i$]{} ]{}
Introduction
============
Deep radio surveys show a surface density of radio objects approaching 60 arcmin$^{-2}$ down to 1 $\mu$Jy (as inferred from fluctuation analyses on the most sensitive VLA images at 1.4 and 8.4 GHz), or equal to the surface density of B = 27 magnitude galaxies (Fomalont [*et al.*]{} 1991, Windhorst [*et al.*]{} 1993, Richards 1996, Richards [*et al.*]{} 1998, hereinafter Paper I). Despite their modest average luminosity of L$\simeq$ 10$^{22.5}$ W/Hz, the sheer number of microjansky sources implies that they dominate the radio luminosity budget of the universe at centimeter wavelengths. However, surprisingly little is known about the physical origin of these objects or their nature.
Some generalizations from studies involving deep 5 and 8.5 GHz VLA fields imaged with HST and ground-based telescopes are possible (Hammer [*et al.*]{} 1995, Windhorst [*et al.*]{} 1995, Fomalont [*et al.*]{} 1997, Richards [*et al.*]{} 1998). At least half of microjansky sources are associated with morphologically peculiar, merging and/or interacting galaxies with evidence for active star-formation (blue colors, infra-red excess, HII-like optical spectra). The remaining identifications are composed of low-luminosity FR Is, ellipticals, Seyferts, LINERs, and luminous star-forming field spirals. Thus a variety of physical mechanisms may be driving the observed evolution among microjansky radio sources, including non-thermal radiation from AGN activity, synchrotron emission associated with diffuse supernova remnants, and thermal emission from HII regions.
Another clue is available from the observed distribution of radio spectral indices. While at millijansky levels, the average spectral index for radio sources is about $\alpha \sim$ 0.8 (Donnelly [*et al.*]{} 1987), microjansky sources selected at high frequencies ($\nu \geq$ 5 GHz) have a surprisingly flat spectra of $\alpha$ = 0.3 $\pm$ 0.2 (Fomalont [*et al.*]{} 1991, Windhorst [*et al.*]{} 1993). Several explanations for this observed flattening compared with sources selected at higher flux density levels are possible, including, free-free absorption, an increasing number of synchrotron self-absorbed AGN among the microjansky population, and/or a rising component of thermal radiation from active star-formation.
The purpose of this study is to enlarge the faint radio sample and investigate the radio spectral and morphological properties for a statistically significant sample of microjansky sources. In addition, examination of the optical properties of the identifications may shed insight into the nature of the radio emission in these sources. To further this goal we have observed a region of the sky at 1.4 GHz and 8.5 GHz centered on the Hubble Deep Field, where excellent wide field optical data are available. In Paper I (Richards et al. 1998) we presented the optical identifications for a complete sample of twenty-nine 8.5 GHz selected radio sources in our radio survey of the Hubble Deep Field. The principal conclusion is that the majority ($\sim$70% ) of the identifications are with relatively bright (R$\sim$ 22 mag.) disk systems, many at moderate redshifts ($z$ = 0.4-1). In this paper (Paper II) we present the 1.4 GHz observations. In section two we describe our observations and data reduction techniques. Section three presents our source list, while in section four we calculate the spectral index distribution. In §5 we discuss the spatial clustering of sources in our catalog. Finally, in §6 we summarize our findings and give conclusions.
The 1.4 GHz Radio Observations and Data Reduction
=================================================
In November 1996, we observed a field centered on the Hubble Deep Field ($\alpha$ = 123649.4ß and $\delta$ = 621258.00 (J2000)) for a total of 50 hours at 20 cm in the A-configuration of the VLA. In order to minimize chromatic aberration, we observed in a pseudo-continuum, spectral line mode with 7 $\times$ 3.125 MHz channels centered on intermediate frequencies 1365 MHz and 1435 MHz, frequency windows known to be relatively free of radio frequency interference. Each frequency channel was composed of two independent circular polarizations. All knowledge of linear-polarized intensities was lost due to the limited number of VLA correlator channels. Visibility data were recorded every 3.3 sec from the correlator.
We monitored the point source 1313+675 (S$_{1.4}$ = 2.40 Jy), located 6.5 degrees from the HDF, every 40 minutes to provide amplitude, phase, and bandpass calibration. Daily observations of 3C286 with assumed flux densities of 14.91 Jy at 1385 MHz and 14.62 Jy at 1435 MHz provided the absolute flux density scale.
The calibrator observations allowed us to identify baselines with systematic phase or amplitude errors. A few baselines were found to have recurrent amplitude and/or phase closure errors greater than five percent and/or five degrees. All data associated with these suspicious baselines were discarded. To remove bursts of radio frequency interference, we excised all visibilities with flux densities greater than about 10 $\sigma$ above the expected rms value of 0.08 Jy. This amounted to about 2% of the data.
After time averaging the $(u,v)$ data from 3.3 to 13 sec, we made preliminary 10 resolution maps which cover the field out to the first sidelobe of the primary beam about 0.8 from the phase center. We searched these images for bright, confusing sources whose sidelobes might contaminate the noise properties of the inner portion of field. All sources above 0.5 mJy were catalogued.
We then imaged and heavily CLEANed these sources using the full unweighted $(u,v)$ data set. Because the primary beam response changes significantly over our 44 MHz bandpass (about 3%), it was necessary to deconvolve each of the confusing sources using each 3.125 MHz channel. In addition, the confusing sources were independently deconvolved in each of the circular polarizations (right and left) to account for the ’beam squint’ of the VLA antenna. Their CLEAN components were Fourier transformed and then subtracted from the visibility data. Using these ”strong source subtracted data sets”, we then imaged the inner few arcmin of the field. With this procedure the rms noise was found to be about 50% higher than expected from receiver noise alone. In particular, a few sidelobes from particularly strong sources (S $>$ 10 mJy) located near the half power point and first null of the primary beam were still apparent.
By examining the images made from 30 minute segments of the data, we isolated a few time intervals where the visibility data appeared to be corrupted, possibily due to low level interference. For any of these 30 minute snap-shot images with a a rms noise greater than 50% of the mean value, the corresponding visibility data were deleted from the final analysis. These data amounted to about seven hours of time; thus, in all, about 42 hours of of high quality data were used to construct the final images. The final combined data set has a rms noise of 7.5 $\mu$Jy, whereas we expected a noise closer to 5 $\mu$Jy.
Construction of the 1.4 GHz Images
----------------------------------
Our goal is to map the 40 field of the 1.4 GHz observations out to the 20% response of the VLA antennas. There are two complications which make this a difficult task.
First is the sky curvature. While in practice the VLA is often treated as a two dimensional array, in reality the instrumental response to radio emission from the sky is a three dimensional complex function. For observations of short duration, small fields of view, or low resolution, most sources can be adequately deconvolved without reference to the so called 3-D effect. However, the isoplanic assumption fails for sources located further than $\theta \simeq 1/n_{syn}$ radians from the phase tracking center, where $n_{syn}$ is the phase center distance in synthesized beam widths. For the A-array at 1.4 GHz this corresponds to a patch size of about 18 arcmin across.
Thus we chose to approximate the one degree primary beam of our observations by using a number of independent and equidistant facet images. Each facet is constructed from the Fourier transform of the data which has been phase shifted to its tangent point on the celestial sphere. In all, 16 facets were used. This technique is known as polyhedral imaging and is discussed at length by Cornwell & Perley (1992).
One further complication comes from the practical limitation of the extensive computations needed to reduce these these wide field observations. Our entire calibrated and edited data set consists of over $10^7$ discrete complex visibilities (after 13 sec time averaging) which must be directly Fourier transformed to compute the sky brightness distribution over approximately the same number of pixels. Because we are interested in radio emission over the entire primary beam, we must properly deconvolve sources within the entire field of view. Although only sources fainter than 0.5 mJy remain in the visibility data after the subtraction of the confusing sources, their collective sidelobes can limit the dynamic range of the final images.
The most accurate method for deconvolution of the faint sources is to CLEAN each of the 16 images in parallel, subtracting each source’s sidelobe contribution from all the image facets simultaneously. In this manner, one would recover the sky brightness function free from sidelobe contamination over the entire primary beam. However, in practice, this is a prohibitive computational task. Therefore we opted to CLEAN each of the 16 facets in series using the much more efficient Clark-Hogbom algorithm (Clark 1980). The price paid is that sidelobes from the multitude of sources less than 0.5 mJy are only properly removed locally, within the individual image facet which contains each source.
In order to examine what effect this might have on the rms noise near the center of the image, we performed a simple test. First we made an image of the inner seven arcmin of the field center, heavily CLEANed using the Clark-Hogbom method. Then we made a similar image twice the size of the former, again heavily CLEANed. Comparison of the two images, one relatively free of sidelobe contamination from the far field sources with S $\leq$ 0.5 mJy, the other not, yields a first order approximation of the unCLEANed sidelobes left in the center of the field due to our deconvolution method outlined above. About 0.8 $\mu$Jy of flux density remains per independent beam, and thus increases the noise about 10% over the thermal noise.
We produced the final image by CLEANing each of the sixteen 2048 $\times$ 2048 pixel fields (0.4$\arcsec$ pixels) with 10,000 iterations and a 10% loop gain. This produced an image with a natural resolution of $2.0\arcsec \times
1.8\arcsec $ and P. A. = $-86\deg.$ The rms noise near the center of the field was 7.5 $\mu$Jy, approximately 30% higher than the value expected from thermal noise alone based on our system temperature (37 K on average), integration time, and bandwidth. The noise in each of the 16 images was found to be fairly uniform, although it increased to as much as 10 $\mu$Jy in regions near the brightest millijansky sources.
Residual ringlike structure from unCLEANable sidelobes around these strong sources was evident, suggesting that our images are dynamic range limited at about 5000:1. This is typical of blank field deep imaging where the data cannot be effectively self-calibrated. We clipped out regions immediately around these sources, as well as a larger region around one particularly strong source (S = 35 mJy) located at the half-power point of the primary beam. We attribute the dynamic range limitation in this image to pointing fluctuations, which are typically 15 rms for the VLA. This effect causes amplitude fluctuations in the apparent brightness of the radio sources, which are of order 1% at the half power point, and hence induces unCLEANable sidelobes into the image.
Examination of the stronger deconvolved sources in the field with $S _p$ 5 mJy showed evidence of a radial sidelobe oriented towards the phase tracking center, or center of our images. These sidelobes appeared to be symmetric about these stronger millijansky sources in the field and with first order amplitude of about 10%. Their width was approximately that of the delay beam.
These sidelobes were also apparent during our test observations of 1400+621, and only appear around the off axis observations. Other observations taken at the VLA have not shown similar artifacts and our suspicion is that the problem was an online system recording error with the 3.3 sec visibility data that we chose to use. Examination of VLA data of the same test field taken in an identical observing mode, except with 10 sec visibility integration, showed no sign of the radial sidelobes.
The cause of these artifacts is unknown. They do not appear to be present around the near field, weaker sources (less than about 1 mJy). As a test of whether these sidelobes might adversely effect our measurements of the flux densities of the stronger millijansky sources, we examined the off-axis flux density measurements of 1400+621 in images with and without the presence of the radial sidelobes. The measurements agree to within the standard flux density errors. It is more difficult to assess if these radial sidelobes adversely affect the general rms noise in the final 1.4 GHz images.
The Complete Source List
========================
In order to minimize the introduction of spurious radio sources into our sample, we examined the distribution of the $negative$ pixels in the images to estimate its completeness limit. In general most regions appeared to reflect well behaved Gaussian noise with the most negative peak being about five times the local rms noise. However, one region with a local rms of 7.5 $\mu$Jy contained an unresolved negative feature with $S_p$ = -63 $\mu$Jy. This feature appeared to be quite isolated and located several arcmin from any of the brighter millijansky confusing sources. No other strange artifacts (e.g., rings, streaks, residual sidelobes) were apparent in the vicinity of this negative ’source’; hence its presence remains enigmatic. The next most negative pixel value of -40 $\mu$Jy is located near a strong confusing source. We therefore adopt 40 $\mu$Jy to be the formal completeness limit over our entire one degree field. Even if there are positive counterparts to the -63 $\mu$Jy ’source’, there should be no more than a few if the negative and positive pixel amplitude distribution are fairly symmetric about zero. However, we note that the probability of finding a –9$\sigma$ source within our field is much less than one percent, and demonstrates that the noise properties of this image are not entirely Gaussian.
Next we searched our images for all pixels with S $\geq$ 40 $\mu$Jy and fit these sources with elliptical Gaussians to determine their peak flux densities and positions using the automated AIPS task SAD. In all 314 sources were found within 20 arcmin of the center of the image (20% power contour).
Determination of Discrete Source Angular Sizes and Flux Densities
-----------------------------------------------------------------
Gaussian fitting routines such as SAD are subject to noise dependent biases which cause significant overestimation of source sizes and flux densities (Windhorst et al. 1984, Condon 1997). In order to estimate the effects of population and noise bias in our images, we performed a series of Monte Carlo simulations. Our basic technique was to inject a number of point sources (100) of known flux density into the CLEANed sky images of this field. Then these sources were recovered from the images using SAD and their measured peak and integrated flux densities compared to the input model. We used the ratio $S_p /S_i$ to determine if the simulated source was significantly broadened by population and/or noise bias.
Because our goal is to set a resolution criteria in the presence of these fitting biases, we performed this simulation as a function of flux density, or alternatively, signal to noise. By examining the distribution of $S_p /S_i$ at different flux densities, we obtained some idea about at what level we could confidently believe that a given source was resolved in our real sky images. For a source of intrinsic dimension $\theta _{maj} \times \theta _{min}$ resolved by a beam of extent $a _{maj} \times a _{min}$, the integrated and peak flux densities are related by $S_i = S_p (1 + \frac{\theta _{maj} \times \theta _{min}}{
a _{maj} \times a _{min}})$, thus allowing us to relate the ratio $S_p /S_i$ directly to a source angular size. By examining the distribution of modeled $S_p /S_i$ to input $S_p /S_i$, we were able to adopt a 95% confidence in the resolution criteria as a function of signal to noise. The resulting best fit data yields $S_p /S_i$ (95%) $< 1 - \frac{2.3}{\sigma _{snr}}$ where $\sigma _{snr}$ is the signal to noise ratio of the radio source. Thus at the detection limit of our images (40 $\mu$Jy), a source must have a $S_p /S_i$ value less than 0.57 to satisfy our resolution criterion, or an angular size greater than about 2.7. It follows that we can resolve only those sources stronger than 70 $\mu$Jy integrated flux in our complete sample.
Next, we used these same simulations with the sources of angular size that we could have just detected at each signal to noise ratio (e.g., 2.7 at $S_p$ = 40 $\mu$Jy). Comparison of the input to recovered peak and integrated flux densities, and angular sizes yields an estimate of the bias induced by Gaussian fitting techniques in the presence of population and noise bias. We corrected our real sky source parameters to account for these biases. In general the peak flux densities of the fitting routines were in good agreement with the models, while the integrated flux densities and angular sizes from the Gaussian fits were overestimated by up to a factor of 1.5.
As a check on the integrated flux densities for apparently resolved sources in our complete sample obtained from the Gaussian fitting algorithms, we examined the distribution of peak flux densities as measured in various resolution images. Images were constructed at 1, 3.5, and 6 resolution in addition to our nominal 2 naturally weighted image. Table 1 gives the parameters of each image. For those sources which satisfied our initial resolution criterion, we checked that the peak flux density of the source increased with decreasing resolution consistent with the fitted angular size. Those sources which did not were considered unresolved and their adopted peak and integrated flux densities were measured on the the 2 image. Figure 1 shows a greyscale of the inner 10 of the 1.4 GHz image. In Figure 2, we show the greyscale of the 8.5 GHz mosaic image (see §6).
We also searched the lower resolution 3.5 and 6 images for resolved sources not detected in the untapered 2 image above our completeness limit of $S_p$ = 40 $\mu$Jy. This search yielded 57 additional sources in the tapered images above completeness limits of 50 $\mu$Jy at 3.5 resolution and 75 $\mu$Jy at 6 resolution. Because the angular sizes are uncertain in these low signal to noise ratio detections (and in many cases may be instrumentally broadened), the adopted peak and integrated flux densities were set equal to the peak pixel values in the tapered images. These additional sources were added to the complete source list which in total contains 371 sources. The complete sample of all radio sources detected at 1.4 GHz within 20 of the phase center is presented in Table 2.
A description of Table 2 is as follows. All uncertainties are given at the one sigma level.
[*Column*]{} (1) — The Right Ascension in J2000 coordinates with one sigma uncertainty.
[*Column*]{} (2) — The Declination in J2000 coordinates with one sigma uncertainty.
[*Column*]{} (3) — The [*deconvolved*]{} (FWHM) major axis of the best Gaussian fit to the source, $\Theta$, is given in arcsec.
[*Column*]{} (4) — The signal to noise ratio of the detection calculated from $S_p /\sigma $ where $S_p$ is the peak flux density as measured in either the 2, 3.5, or 6 image, and $\sigma$ equals the rms noise in that image.
[*Column*]{} (5) — The integrated sky flux density (S$_{1.4}$) after correction for the instrumental gain, (see §3.2),
with corresponding one sigma errors.
Instrumental Corrections
------------------------
We must also correct the derived source parameters for various instrumental effects. In order to measure the off axis response of the VLA to a point source, we observed the 4.2 Jy point source 1400+621 in each of the cardinal directions at positions of 5, 10, 15, 20 and 25 arcmin from the nominal phase reference center.
There are four principal effects which reduce point source response as a function of radial distance from the phase center. In their degree of increasing importance they are 1) 3-D smearing, 2) finite time visibility sampling (time delay smearing), 3) chromatic aberration (bandwdith smearing), and 4) primary beam attenuation.
1\. We have approximated the curvature of the celestial sphere with a number of two-dimensional facets. However, smearing may still be present at a reduced level near the edges of each of our individual facet images. The effect of a small amount of 3-D smearing are such that the integral flux density of a given source is preserved while its peak signal is reduced by an amount dependent on distance from the tangent point on the celestial sphere. In order to determine the amplitude of 3-D smearing in our data, we performed a series of simulations inserting point sources at a variety of distances from the phase tracking center into visibility data with our exact $(u,v)$ coverage. These data were then imaged in the same manner as the true sky images. In this manner we determined the amount of point source degradation as a function of distance from the celestial sphere tangent point or image facet center. At the maximum possible distance of a source from a tangent point ($\sim$ 600) the peak degradation is less than 20%. All values of the peak flux density ($S_p$) were corrected for this effect according to our empirically fit polynomial.
2\. The calibrated $(u,v)$ data set used to construct our images consisted of 13 sec averaged visibility data. Because the actual radio sources rotate in the sky during this sampling time, their flux density is smeared in the image plane. The analytic calculation of this smearing is complicated by the aspects of the observing geometry. However, at the North Celestial Pole the effect reduces to a tangential smearing in the image plane and its amplitude is given by $S_p/S_t$ = 1 –2.06 $\times 10^{-9}$ $\theta$/$\theta _{syn}$ where $S_t$ is the integrated flux density of a source located an angular distance $\theta$ from the phase tracking center and $\theta _{syn}$ is the size of the synthesized beam. We used this approximation to correct the peak flux densities as measured in our images.
3\. Although we planned our observations of the HDF field to minimize the effects of chromatic aberration, for far-field sources this effect can still be important. It is especially crucial to understand the effects of smearing on the completeness level of the images. We measured the off-axis response of 1400+621 due to chromatic aberration by examining the ratio of the peak to integrated flux density, $S_p /S_i $, as a function of distance from the image phase center. Bandwidth smearing is governed by the observing frequency, width and shape of the bandpass and the $(u,v)$ coverage, all of which define the synthesized beam. In theory, these are known functions and the final beam response can be calculated analytically as a function of position in the image. However, in practice such uncertainties as non-uniform $(u,v)$ coverage, central intermediate frequency offsets, and imperfect bandpass filters make this impossible. We chose instead to fit a function to the empirical data of the form $S_p /S_i $ = (1+(r/k)$^2$)$^{-0.5}$ where r is the distance from the phase center and k is a constant which absorbs the uncertainties discussed above. Our least squares fit to the empirical data is shown in Figure 3 where k = 16.19 arcmin. This is within 3% of the theoretical value assuming $\delta \nu /\nu$ = 3.125/1400.5 and a $\theta _{beam}
$ = 2.1. For purposes of correcting $S_p$ in our images, we use the above equation scaled by the appropriate beam size. The integrated flux density, $S_i$ is preserved and needs no correction for smearing.
4\. The primary beam attenuation at 1.40 GHz has been measured to about 1% (Condon 1997) to the first sidelobe (-10 dB). Each of the peak and integrated flux densities in the source list were corrected for the primary beam attenuation. The uncertainty in the correction is the standard rms pointing error of the VLA elements (about 15) multiplied by the differential log of the primary beam response.
In Figure 3 we plot the point source response of a source as a function of distance from the phase center. We define the instrumental gain factor as the product of bandwidth smearing, time averaging smearing, and the primary beam correction.
Positional Accuracy
-------------------
The assumed position of our phase calibrator 1400+621 is $\alpha$ = 140028.6526ß and $\delta$ = 62$^\circ$1538.526 (J2000). We observed systematic, monotonically increasing shifts in both RA and DEC of comparable amplitude as a function of $r$. It is a small effect and the difference between the actual radial distance from the field center and that measured in the image plane is $\delta$r = -0.042 arcsec at 25 arcmin distance from the field center. This small systematic term can be explained by the so called annual aberration effect (Fomalont et al. 1992). The predicted scale contraction from annular aberration in the direction of the HDF during the Julian epoch 1996.6 is 0.9999783. The good agreement between the observed and predicted scaling factor for 1400+621 yields confidence that our images are free from significant distortions due to asymmetric bandpasses or IF offsets.
We also corrected all the radio source positions in our catalog for position offsets induced by the 3-D effect as discussed in §3.2. By phase shifting the image tangent points to the location of individual sources, their true angular positions on the sky were measured. The difference between the apparent position as measured on the nominal sky maps and the corrected positions is typically only 0.2 in the north-south direction.
After correction for annular aberration and the 3-D term, the relative positional accuracy for sources across the field in the limit of infinite signal to noise should approach about 0.02. The absolute positional accuracy depends on the translation of our phase calibrator position to that of the HDF and is about 0.02. Thus we estimate our radio catalog to be within 0.03 of the J2000/FK5 coordinate grid. The single coordinate rms position errors as given in Table 2 are defined as $\sigma = \sqrt{(1.8\arcsec /2\sigma
_{snr})^2 + (0.03\arcsec)^2}$.
Survey Completeness and Source Counts
=====================================
Because the completeness of the radio source sample is defined in terms of peak image flux, $S_p$, corrections must be made for the the instrumental response and biases inherent in our detection algorithm. Although we have corrected the individual sources in Table 2 for these effects, we now calculate the fraction of sky sources which remain undetected in our survey due to the finite angular size of the sources.
Angular Size Distribution
-------------------------
Previous high resolution studies of the microjansky radio population suggested that the median angular size ($\theta _{med}$) for submillijansky radio sources is approximately 2 and almost independent of flux density between 80 - 1000 $\mu$Jy (Windhorst et al. 1993, Fomalont et al. 1991, Oort 1988). The resolution of our present microjansky survey is thus well suited to study the angular size distribution for a statistically large and well defined sample of microjansky radio sources.
Of the 151 radio sources in our complete sample with 70 $\mu$Jy $ < S_i < 1000 ~\mu$Jy for which we have angular size information, only 77 (50%) are resolved with our typical 2 resolution limit. We divided these sources into two flux density bins, containing approximately equal numbers of resolved sources. Considering only the number of radio sources with angular size greater than 2.7 (the angular size detection limit of the weakest radio sources in our sample as defined in §3.1), we find twice as many resolved sources with $S_i >$ 250 $\mu$Jy (52% ) as opposed to those with $S_i \leq $ 250 $\mu$Jy (25% ). This suggests that $\theta _{med}$ may be a [*decreasing*]{} function of flux density. Figure 4 shows our measurements of angular size as a function of source intensity. In order to estimate the mean angular size of this sample, we applied the survival analysis techniques of Feigelson & Nelson (1985) using the statistical package ASURV (Rev. 1.2; Isobe & Feigelson 1992). This technique incorporates upper limits in the calculation of the mean of a distribution, which is particularly important in our sample which is dominated by non-detections (i.e. we are measuring the tail of the angular size distribution). The technique assumes a symmetric Gaussian model, hence the mean and median are equal. At $S_{1.4}$ = 370 $\mu$Jy, $\theta _{med}$ = 2.6 $\pm$ 0.4, and at $S_i$ = 100 $\mu$Jy , $\theta _{med}$ = 1.6$\pm$ 0.3. The errors are based on the number of angular size measurements (not limits).
In Fig. 5 , we compare our determinations of $\theta _{med}$ with previous measurements made at 1.4 GHz. Because of the uncertain selection effects inherent in the higher frequency deep radio surveys, particularly their bias towards flat-spectrum, compact AGN, we chose not to include these points. With the notable exception of the discrepant Condon & Coleman (1985) point, there is general agreement amongst the different data. Because the median angular size is known to change rather sharply below a few millijansky at 1.4 GHz (presumably due to the emergence of an increasing population of starburst galaxies among radio sources) from $\sim$10 to a few arcsec, we suggest that the high flux density point of Oort (1988) is too low, possibly due to resolution biases in his A-array snapshots.
We believe the decrease in angular size at lower flux densities to be real. Thus for the purposes of modeling the median angular size-flux density relationship, we fit a function of the form:
$\theta = frac{3}{4} \times 0.175 S_{1.4} ^{0.5}+ frac{1}{4} \times 0.1$ arcsec
This fit is also shown in Fig. 5. For completeness, we also plot a straight line, with $\theta $ = 2.0 and independent of flux density.
Completeness
------------
In order to investigate the combined effects of noise, population, and resolution bias on the completeness level of our survey, we used a series of Monte Carlo simulations similar to the ones described in §3.1. In particular we want to determine how many sources with $S_i \geq$ 40 $\mu$Jy, but with $S_p <$ 40 $\mu$Jy we missed based on our peak flux density detection limit. We randomly populated our sky images with 100 sources of finite angular size assuming an angular size distribution as found in §4.1. This simulation was repeated at a variety of flux densities from 40 to 1000 $\mu$Jy. In each flux density interval the ratio of the number of sources recovered from the images with $S_p \geq$ 40 $\mu$Jy to the number originally injected in the model was tabulated. This was taken to be the effective correction factor needed to account for the combined effects of resolution, population and noise bias in our images (although resolution bias is always the dominant term). At 80 $\mu$Jy which is the average flux density source detected in our survey (weighted by $S^{-2.4}$) and where the count will most accurately be determined, this correction factor is 1.05. As resolution bias is the dominant source of incompleteness in our survey, we estimate that we have detected approximately 95% of the microjansky sources in the HDF region to this flux density limit. The principal uncertainty in the correction factor is the uncertainity in the angular size distribution of the microjansky radio sources. If we had assumed a constant $\theta$ = 2.0 model, our corrections would have increased by over a factor of two.
From the complete source list of Table 2, we then binned sources in flux density intervals such that each bin had at least 50 radio sources (except for the highest flux density bin). The differential count was then calculated based on the number of sources in each bin interval. Bandwidth smearing, time averaging smearing, and the primary beam response decreases the effective area over which a source of given $S_p$ can be detected. Thus when counting the number of sources in each bin, care must be taken to weight the contribution of each source to the count by the effective area over which it could have been detected. This factor can be calculated by solving for the image radius where a source of amplitude $S_p$ would have just been missed by our peak detection limit (Katgert 1973). Table 3 presents the differential source counts for our complete flux limited sample. The counts from this survey are compared to other microjansky surveys in Figure 6a and 6b (Mitchell & Condon 1985, Oort & Windhorst 1985, Hopkins [*et al.*]{} 1998) normalized to a Euclidean geometry n/$n_o = n(s) /S^{-2.5}$. In general the agreement is reasonable and in agreement with the errors and possible field to field variations. The best fit to the source counts in this field in the range 40 - 1000 $\mu$Jy is $n(S) = (8.25\pm 0.42) S^{-2.38 \pm 0.13}$ ster$^{-1}$ Jy$^{-1}$.
The counts in the HDF appear systematically lower than those of other fields above 100 $\mu$Jy. This effect could be due either to 1) real field to field variations on the degree scale as a result of large-scale clustering of radio sources, or 2) survey incompleteness due to the finite angular size of the radio sources. Without complementary, low resolution observations, it is difficult to discriminate between these two possibilities. We note that if the mean angular size does not decrease significantly below 100 $\mu$Jy, the radio sky will become forever naturally confused at the level of a few hundred nanojansky, perhaps providing a natural limitation to the sensitivity of the next generation of centimeter radio telescopes (Windhorst et al. 1993).
Spatial Clustering
==================
In order to test for the presence of two dimensional spatial clustering among the radio sources in the Hubble Deep Field, we calculated the two-point correlation function for the sources in Table 2. First, we compiled a table of angular separations by considering the separation of each individual source with all other sources in the catalog. These provide our $DD$ estimate (Peebles, 1980). Next, we generated random catalogs of sources according to the source count of §4.2 , and distributed randomly across a 40 VLA primary beam. The peak flux densities of these sources have been attenuated by the instrumental corrections discussed in §3.2. Angular pairs were calculated for these catalogs and form the basis of our $RR$ measurement. We define the correlation function of our catalog to be $w(\theta )$ = DD/RR -1 . The correlation function of our catalog on scales of 0- 40 is presented in Table 4.
We calculated the errors in our clustering measurement by following the bootstrap method of Ling et al. (1986). These agree well with the Poissonian error estimate of $\delta w(\theta )$ = 1 + $w(\theta)$)/N$_{DD}$ where N$_{DD}$ is the number of independent data pairs in a given bin. We find evidence for an excess of radio sources on scales of approximately 1-10, while on scales much larger than this there are fewer radio sources in our catalog than expected from a random distribution. Figure 7 shows the correlation function for radio sources in the HDF, compared to the correlation function of a somewhat shallower 1.4 GHz survey of Oort (1987) complete to 100 $\mu$Jy. The amplitudes are comparable in the two separate surveys.
Spatial clustering at higher flux density levels and lower amplitudes has been reported by Cress et al. (1996) and Magliocchetti et al. (1998). More recently, Hopkins et al. (1999) claim fluctuations in field to field source counts at similar completeness levels to ours, possibly indicating the presence of large-scale radio source spatial variations. Thus it is plausible that there are both fewer radio sources in the HDF region than the average field, and that these sources are clustered on arcmin scales amongst themselves.
Radio Spectral Indices
======================
The HDF has been observed previously with the VLA at 8.5 GHz to a one sigma sensitivity of 1.8 $\mu$Jy (Paper I). In June 1997 we observed the HDF region for an additional 40 hours at 8.5 GHz. We mosaiced an area defined by four separate pointings offset 2.7 from the center of the HDF (the half-power scale of the primary beam response) in each of the cardinal directions for about 10 hours duration each. The observing technique and data reduction are discussed in Paper I. The final combined 8.5 GHz images have an effective resolution of 3.5 and a completeness limit of 8 $\mu$Jy. The sensitivity of this mosaic to sky emission is a sharp function of distance from the nominal pointing center because the observations were heavily weighted towards imaging the central HDF region.
Because the size of the VLA primary beam scales inversely with frequency, our sensitivity at 8.5 GHz is limited to the inner 6.6 arcmin (HWHM) of the 1.4 GHz field. This is the point where the maximum beam attenuation at 8.5 GHz is equal to 0.2 (while at 1.4 GHz the attenuation is only 0.9). Within this region there are 109 sources contained in the 1.4 GHz complete sample. We measured the 8.5 GHz flux density at the location of each of these sources. When a source was not clearly detected ($S_p <
3 \sigma$ at 8.5 GHz), we calculated a conservative upper limit to its 8.5 GHz flux density equal to three times the rms noise corrected by the antenna gain. If a 1.4 GHz radio source had a peak flux value 3$\sigma < S_p
< 5 \sigma$, its flux limit was taken as $S_p$ also corrected by the primary beam. This ensures that our 1.4 GHz selected spectral index sample is complete and free from uncertain weak source biases. Based on this criteria, 30 sources from the 1.4 GHz sample had clear counterparts in the 8.5 GHz image. Using the 1.4 GHz and 8.5 GHz flux density values as measured in their respective 3.5 convolved images , we calculated individual spectral indices using the convention $S_{\nu} \propto \nu ^{-\alpha}$. In the following discussion, steep spectrum sources are defined as those with $\alpha \geq$ 0.50, while flat spectrum as those with $\alpha < $ 0.50.
The 1.4 GHz to 8.5 GHz spectral index distribution of the 1.4 GHz selected sample is shown in Figure 8 (only those with meaningful lower limits, $\alpha \geq$ 0.50 are shown for clarity). Because of the large number of spectral index limits for the 1.4 GHz sample (79/109), we chose to only consider those sources with $S_{1.4} > $ 100 $\mu$Jy when calculating the mean of the sample. The mean as calculated from both the detections and lower limits (using ASURV) for sources with $S_{1.4} > $ 100 $\mu$Jy is $\overline{\alpha } _{1.4}$ = 0.85 $\pm$ 0.16. For those sources with $S_{1.4} > $ 100 $\mu$Jy the fraction of steep spectrum sources is $\alpha$ = 0.62. We also calculated the median of the spectral index distribution for the entire 1.4 GHz sample. We did not consider spectral index lower limits which were weaker than ${\alpha } _{1.4} >$ 0.3, to avoid a bias in the median calculation. The median of the 1.4 GHz selected sample is 0.63.
We now consider those radio sources within the central 6.6 (HWHM) detected on the basis of the 8.5 GHz data alone. There are 29 sources in the complete sample 8.5 GHz sample of Paper I. Eleven additional sources were detected in the mosaiced regions with $S_p \geq 5 \sigma$. These 40 sources comprise a complete sample of radio sources detected at 8.5 GHz within 6.6 of the HDF center. We measured the 1.4 GHz flux densities at their positions in the 3.5 image. All but 10 of these sources are contained in the complete 1.4 GHz sample in Table 2. Upper limits to the 1.4 GHz flux density were calculated as three times the rms normalized by the antenna gain.
The spectral index distribution is shown in Figure 9. The mean spectral index for the 8.5 GHz selected sample is $\overline{\alpha } _{8.5}$ = 0.35 $\pm$ 0.07 while the median spectral index of the detections is 0.41. The fraction of flat spectrum sources in the 8.5 GHz selected sample is 0.60. Table 5 gives individual spectral indices or limits (where meaningful limits are available) for both the 1.4 and 8.5 GHz samples.
The Nature of Flat Spectrum Sources: AGN vs. Starbursts
-------------------------------------------------------
It has been noted by previous authors that below a few milljansky, the median spectral index for high frequency selected samples ($\nu \geq$ 5 GHz) flattens from a value of 0.7 to about 0.3-0.4 and then remains constant for at least two decades in flux density (Windhorst et al. 1993 and references therein). Our spectral index study confirms this trend.
This raises the question of what physical mechanism is responsible for the flattening of the high frequency selected microjansky population. One clue comes from the optical identification of the sources. Of the 26 flat spectrum sources presented here, only 4 can be reliably associated with elliptical galaxies, the majority (70%) residing in mergers, interacting disk systems, or isolated field spirals (Paper I, Richards et al. 1998). Thus the flattening of the spectral index distribution for the microjansky population is unlikely to be due to radio evolution of the elliptical population.
If the 8.5 GHz sample preferentially selects out self-absorbed AGN cores from the microjansky population, then we might expect these sources to have a smaller angular size on average as compared to a 1.4 GHz selected sample. The mean angular size for the 26 flat ($\alpha < 0.5$) spectrum sources of Table 5, is $\theta $ = 1.7 $\pm$ 0.6, as compared to $\theta $ = 1.8 $\pm$ 0.5, for the 47 steep ($\alpha \geq 0.5$) spectrum sources. Thus there is no evidence for a significant change in source size between the flat and steep spectrum population. Interestingly, there are only four inverted spectrum sources among the flat spectrum sample, indicating that strongly self-absorbed systems are rare among the microjansky population.
In Paper I we considered two possibilities for the origin of microjansky radio emission in distant disk galaxies, 1) increased radio activity associated with a central engine (e.g., Seyfert and LINER AGN), and 2) radio emission excited by star-formation. Both are capable of producing flat radio spectra through synchrotron self-absorption in the case of the former, and through increasing amounts of thermal radio emission in the later. These two different physical mechanisms take place on very different physical scales. For the case of star formation these scales correspond from approximately 0.1-10 kpc as observed in the local starburst population (Condon 1989). Thus the cosmological microjansky population at a mean redshift of 0.8 (Paper I) should have an angular extent of 0.01-1 if star formation is the ultimate source of energy powering the radio emission. On the other hand if the radio emission has its origin in an AGN then the flattening of the spectral index distribution can be attributed to partial self-absorption. In this case the observed flux density of the source makes calculation of a minimum angular size for synchrotron self-absorption possible (Pacholczyk 1970). For a 100 $\mu$Jy source with a critical absorption frequency of 1.4 GHz and an assumed magnetic field strength no larger than $10^{-4}$ Gauss, the characteristic angular size scale is of order $10^{-5}$ arcsec.
Because the radio spectral index is such a sensitive function of absorption varying from -2.5 in the case of pure synchrotron self-absorption, to 0.8 for a standard transparent, non thermal spectrum, the observation of low dispersion in the flat spectral indices of the microjansky population (i.e., very few inverted spectrum sources) would suggest that we are seeing very nearly the same fraction of absorbed radiation in all sources. Perhaps a more natural explanation for the observed spectral index distribution of flat spectrum microjansky radio sources is that we are observing varying ratios of thermal to synchrotron emission causing the spectral indices to vary from -0.1 to 0.8 (cf., Condon 1992). This could be due to the combined effect of a radio K-correction which serves to bring a greater fraction of bremsstrahlung radiation into the observed radio window for sources at appreciable redshifts, as well as a steepening of the synchrotron radiation itself as a result of synchrotron and Compton losses off the microwave background. A flatenning of the relativistic electron energy spectrum could flatten the observed synchrotron spectral index as well. Sub-arcsecond radio observations of the microjansky population are necessary to discriminate between these possibilities (Muxlow et al. 1998). High frequency observations would also be useful to determine the slope of the radio spectrum in these distant radio sources, but will not be feasible until the commisioning of the Millimeter Array or until the expansion of the high frequency capabilities of the VLA.
Conclusions
===========
We have presented a complete catalog of 371 radio sources brighter than 40 $\mu$Jy at 1.4 GHz in a 0.3 deg$^2$ field centered on the Hubble Deep Field. This is the most sensitive survey available at this resolution (2) and frequency. For a subsample of these sources we have calculated two point spectral indices based on 8.5 GHz mosaic observations.
The principal results of this study are:
1\. We have extended the direct source count at 1.4 GHz to 40 $\mu$Jy, confirming that the differential slope for radio sources remains steep at $\gamma $ = -2.4 to this level.
2\. The average angular size for the microjansky population is observed to decrease as a function of flux density. The mean size for radio sources between 40-1000 $\mu$Jy is 2.0, consistent with their association with large disk galaxies at $z \sim $ 1.
3\. Microjansky radio sources appear to be clustered on scales of 1- 40, corresponding to projected distances of 0.5 - 20 Mpc .
4\. The average spectral index for a 1.4 GHz selected subsample is $\overline{\alpha } _{1.4} = 0.85
\pm 0.2$, indicating optically thin synchrotron emission as the dominant radio emission mechanism. For a 8.5 GHz selected sample, the mean is $\overline{\alpha } _{8.5} = 0.4 \pm 0.1.$ This flattening of the spectral index distribution over that of samples selected above a few millijansky is consistent with either the cosmological evolution of the disk galaxy AGN population (LINERs and Seyferts) or of their star-formation activity. We suggest that we are observing increasing amounts of bremsstrahlung radiation in these sources, causing the observed decrease in the spectral index distribution at 8.5 GHz.
I thank my collaborators Ed Fomalont, Ken Kellermann, Bruce Partridge, Rogier Windhorst, and Tom Muxlow for their help with this project. This study would not have been possible without the expert assistance of the NRAO staff in the planning, execution, and analysis of these observations, especially F. Owen and M. Rupen. This work benefited from suggestions by J. Condon and J. Wall. I also thank L. Cowie and A. Barger for pointing out a positional inconsistency in an earlier version of this work.
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Figure Captions
===============
1\. A greyscale of the inner 9.5 $\times$ 9.5 of the 1.4 GHz image. The pixel range is from -10 - 20 $\mu$Jy. The effective resolution is 1.8 with an rms noise of 7.5 $\mu$Jy. The image has been corrected by the primary beam attenuation.
2\. Here we show the same 9.5$\times$ 9.5 area at 8.5 GHz. The greyscale extends from -2 - 6 $\mu$Jy. The effective resolution is 3.5 with an rms noise of 1.6 $\mu$Jy. This image has been corrected by the primary beam attenuation and hence the noise is not uniform.
3\. The total off-axis response of a point source due to the combined effects of bandwidth smearing, time averaging smearing, and the primary beam correction (gain). The relative amplitude decrease due to time average smearing (tsmear), bandwidth smearing (bwsmear), and the primary beam attenuation (pbcor) are also shown.
4\. The angular size distribution for the 1.4 GHz complete sample. Notice the increasing number of upper limits at lower flux density levels.
5\. The 1.4 GHz median angular size vs. flux density relation for microjansky radio sources. Points from Oort (1988; O88), Coleman & Condon (1985; CC), and this study are plotted. The curve shown has been calculated according to the model given in §4 . The broken line represents a median angular size independent of flux density.
6a. The 1.4 GHz source counts from this field and other deep radio surveys are shown. The count is presented in differential form normalized to the counts expected in a Euclidean geometry. For comparison, counts from Mitchell & Condon (1985; MC), the Phoenix Deep Field (Hopkins et al. 1998; PDF), Oort & Windhorst (1985, OW85), and the ELAIS survey (Cillegi et al. 1998; ELAIS) are plotted. The best fit to the earlier deep survey compilation by Katgert et al. (1988) is shown as a solid line.
6b. A blow-up of the sub-millijansky counts. The solid line is the best fit to the current survey data. There is good evidence for field to field fluctuations above that of the statistical noise.
7\. The correlation function of the HDF radio sruvey is shown as heavy dots. The correlation function in the Lynx3 survey of Oort (1987) is shown for comparison.
8\. The spectral index flux density distribution for the 1.4 GHz selected sample. The mean of the sample as calculated by survival analysis is shown as the broken line.
9\. The spectral index flux density distribution for the 8.5 GHz selected sample. The mean of the sample as calculated by survival analysis is shown as the broken line.
|
---
abstract: 'Bone adapts in response to its mechanical environment. This evolution of bone density is one of the most important mechanisms for developing fracture resistance. A finite element framework for simulating bone adaptation, commonly called bone remodelling, is presented. This is followed by a novel method to both quantify fracture resistance and to simulate fracture propagation. The authors’ previous work on the application of configurational mechanics for modelling fracture is extended to include the influence of heterogeneous bone density distribution. The main advantage of this approach is that configurational forces, and fracture energy release rate, are expressed exclusively in terms of nodal quantities. This approach avoids the need for post-processing and enables a fully implicit formulation for modelling the evolving crack front. In this paper density fields are generated from both (a) bone adaptation analysis and (b) subject-specific geometry and material properties obtained from CT scans. It is shown that, in order to correctly evaluate the configurational forces at the crack front, it is necessary to have a spatially smooth density field with higher regularity than if the field is directly approximated on the finite element mesh. Therefore, discrete density data is approximated as a smooth density field using a Moving Weighted Least Squares method. Performance of the framework is demonstrated using numerical simulations for bone adaptation and subsequent crack propagation, including consideration of an equine 3^rd^ metacarpal bone. The degree of bone adaption is shown to influence both fracture resistance and the resulting crack path.'
address:
- 'Glasgow Computational Engineering Centre, The James Watt School of Engineering, University of Glasgow, Glasgow, G12 8QQ, UK.'
- 'Weipers Centre Equine Hospital, School of Veterinary Medicine, University of Glasgow, Glasgow, G61 1QH, UK,'
author:
- Karol Lewandowski
- '[Ł]{}ukasz Kaczmarczyk'
- Ignatios Athanasiadis
- 'John F. Marshall'
- 'Chris J. Pearce'
bibliography:
- 'bibfile.bib'
title: Numerical investigation into fracture resistance of bone following adaptation
---
Finite element analysis,bone remodelling ,fracture ,3rd metacarpal ,moving weighted least squares ,configurational mechanics ,heterogeneity
Introduction
============
This paper presents a framework for the computational modelling of bone adaptation (commonly referred to as bone remodelling) and bone fracture, and their inter-relationship. Bone adaptation is the on-going biological process of replacing old bone tissue with new bone, thus repairing fatigue damage [@hughes2017role]. This ability to repair bone micro-damage caused by cyclic loading is essential for maintaining mechanical integrity. Consequently, there is a strong correlation between stress fractures and the adaptation process [@hughes2017role]. Furthermore, bone repair can be overwhelmed by load-induced bone densification that also increases brittleness and reduces fracture resistance [@loughridge2017qualitative].
One of the first mathematical theories for bone adaptation [@cowin1976bone], based on open system thermodynamics, has its foundation in the theory of poroelasticity. In this approach (unlike classical closed systems), energy, mass, momentum and entropy can be exchanged with the environment. It has been adopted and enhanced over the years [@harrigan1996bone; @jacobs1995numerical; @weinans1992behavior]. This process of density evolution requires a mechanical stimuli as a trigger for bone adaptation. This stimulus may take the form of stress [@beaupre1990approach; @carter1996mechanical; @doblare2002anisotropic], strains [@cowin1976bone] or strain energy density [@weinans1992behavior; @kuhl2003theory; @kaczmarczyk2011efficient; @Connor2017bone].
The use of computational tools to describe bone behaviour has gained a tremendous importance over the last decade. In particular, the Finite Element Method (FEM) has been used to improve understanding of the fracture behaviour of bones and the relationships between load conditions and bone architecture [@podshivalov2014road; @poelert2013patient]. However, there are only a few examples of the numerical analysis of both bone adaptation and fracture, e.g. [@hambli2013integrated]. This paper presents a new computational framework based on FEM in order to predict bone density profiles (bone adaptation) due to exercise, quantify fracture resistance and simulate fracture propagation. This will improve understanding of the interrelationship between these phenomena and enable subject-specific simulations to be undertaken.
A schematic of the modelling framework is presented in Figure \[fig:framework\]. This paper extends the authors’ previous work on modelling fracture propagation [@kaczmarczyk2014three; @kaczmarczyk2017energy] to incorporate the influence of spatially varying bone density. Furthermore, it combines this with bone adaptation [@kaczmarczyk2011efficient; @lewandowski2017].
Although this work is generic in nature and applicable to both human and animal bone, this paper focuses the numerical examples on equine bone.
To the best of the authors’ knowledge, to date there is only one report of equine bone adaptation in a FEM framework [@Wang2016]. In that work, a mechanostat micro-scale model of three-dimensional cortical bone remodelling, informed by *in vivo* equine data, was presented. The model used the von Mises stress as a stimulus to control microstructural cortical bone remodelling. In contrast, the current paper presents a full macro-scale model of equine bone response to mechanical loading, testing a hypothesis that micro-damage and fracture can be modelled at the macroscale by using clinically available CT-scanning data. The motivation for this work is to generate subject-specific simulations to acquire meaningful insight into bone resistance for veterinary practitioners.
This article is structured as follows. After establishing the kinematic preliminaries in Section \[preliminaries\], the mathematical framework for bone adaptation is briefly presented in Section \[sec:bone\_remodel\]. Section \[sec:fracture\] extends the authors’ previous work for evolving crack propagation in the context of configurational mechanics. The method is utilised to calculate fracture resistance and crack propagation under quasi-static loading during different stages of adaptation. Section \[sec:fem\_modelling\] describes the finite element method implementation and Section \[sec:fem\_modelling\] describes a special element for capturing the singular stress field at the crack front. All the above components are brought together into a single framework and its performance is demonstrated using a series of numerical examples in Section \[sec:numerical\_examples\].
at (0,4) [![Framework for estimating bone fracture resistance in MoFEM [@mofem2017]. a) Density derived from Quantitative Computed Tomography (qCT) is mapped onto finite element mesh; (b) bone adaptation analysis; (c) assessment of fracture resistance and crack propagation analysis.[]{data-label="fig:framework"}](Figures/framework.png "fig:"){width="14cm"}]{}; at(0,0.6)[ a) Density mapping. b) Bone adaptation. c) Crack propagation analysis. ]{};
Preliminaries
=============
Figure \[fig:domains4\] shows a section of bone with an initial crack in the reference domain $\mathscr{B}_{0}$. As a result of loading, the crack extends and the body deforms elastically. Working within the framework of configurational mechanics [@kaczmarczyk2014three; @kienzler2014configurational], it is convenient to decompose this behaviour into an extension of the crack in the material domain $\mathscr{B}_t$ followed by elastic deformation in the spatial domain $\Omega_t$. The former is described by the mapping from the reference domain to the material domain ${\boldsymbol\Xi}$, whilst the latter is described by the mapping from the material to the spatial domain ${\boldsymbol\varphi}$ - Figure \[fig:domains4\].
![Kinematics of crack propagation in elastically deforming bone.[]{data-label="fig:domains4"}](Figures/domains5.pdf){width="10cm"}
The material coordinates $\mathbf{X}$ are mapped onto the spatial coordinates $\mathbf{x}$ via the familiar deformation map $\boldsymbol\varphi(\mathbf{X},t)$. The physical displacement is: $$\mathbf{u}=\mathbf{x}-\mathbf{X}$$ The reference material domain describes the body before crack extension. ${\boldsymbol\Xi}(\boldsymbol\chi,t)$ maps the reference material coordinates $\boldsymbol\chi$ on to the current material coordinates $\mathbf{X}$, representing a configurational change, i.e. extension of the crack due to advancement of the crack front. ${\boldsymbol\Phi}$ maps the reference material coordinates $\boldsymbol\chi$ on to the spatial coordinates $\mathbf{x}$. The current material and spatial displacement fields are given as $$\mathbf{W} = \mathbf{X} - {\boldsymbol\chi}\quad\textrm{and}\quad
\mathbf{w} = \mathbf{x} - {\boldsymbol\chi}$$ $\mathbf{H}$ and $\mathbf{h}$ are the gradients of the material and spatial maps and $\mathbf{F}$ the deformation gradient [@kaczmarczyk2014three], defined as: $$\mathbf{H}=\frac{\partial {\boldsymbol\Xi}}{\partial {\boldsymbol\chi}},\quad\mathbf{h}=\frac{\partial {\boldsymbol\Phi}}{\partial {\boldsymbol\chi}},\quad\mathbf{F} = \frac{\partial \boldsymbol\varphi}{\partial \mathbf{X}} = \mathbf{h}\mathbf{H}^{-1}$$
The time derivative of the physical displacement $\mathbf{u}$ and the deformation gradient $\mathbf{F}$ (material time derivative) are given as [@kaczmarczyk2014three]: $$\label{eq:phy_vel}
\dot{\mathbf{u}}= \dot{\mathbf{w}}-\mathbf{F}\dot{\mathbf{W}} \qquad
\dot{\mathbf{F}} = \nabla_\mathbf{X} \dot{\mathbf{x}} = \nabla_\mathbf{X} \dot{\mathbf{u}} =
\nabla_\mathbf{X} \dot{\mathbf{w}} - \mathbf{F} \nabla_\mathbf{X} \dot{\mathbf{W}}$$
Bone adaptation {#sec:bone_remodel}
===============
In this paper, the modelling of bone adaptation is based on the work of Kuhl and Steinmann [@kuhl2003theory] in which bone is considered an elastic porous material. The model is stable [@kuhl2003computational], efficient [@kaczmarczyk2011efficient] and capable of producing bone mineral density profiles that are quantitatively comparable with DEXA scans following gait analysis [@pang2012computational]. Using this approach, bone adaptation in human scapula [@liedtke2017computational], tibia [@pang2012computational], humerus [@taylor2009phenomenon] and femur with various surgical implants [@ambrosi2011perspectives; @Connor2017bone] have been simulated and its potential in topology optimization [@waffenschmidt2012application] has been explored. One of the advantages of such a phenomenological approach is that only a small number of parameters are required, which can be experimentally determined from, for example, CT imaging [@zadpoor2013open].
Conservation of mass
--------------------
Following Kuhl and Steinmann [@kuhl2003computational], it is assumed that the rate of change of the time-dependent material density is in equilibrium with mass flux, expressed as: $$\label{eq:mass_balance}
\frac{\partial\rho}{\partial t} + \dot{\mathbf{W}} \cdot \nabla_\mathbf{X} \rho =
\nabla_{\boldsymbol {\rm X}} \cdot \mathbf{R} + \mathcal{R}_0$$ where $\rho$ is mass density and $\mathcal{R}_0$ is the locally created mass. Furthermore, $\mathbf{R}$ is the mass flux defined as:
$$\mathbf{R} = \mathcal{R} \nabla_\mathbf{X} \rho
\label{eq:mass_flux}$$
where $\mathcal{R}$ is mass conductivity. The term on the right hand side of Eq. \[eq:mass\_balance\] is the material time derivative associated with the evolving current material configuration. However, in the present work it is assumed that the mass flux $\mathbf{R}$ is zero and hence only the local mass source $\mathcal{R}_0$ contributes to the changes in density.
### Constitutive relationship for bone adaptation {#sec:constitutive_eq}
Following Harrigan and Hamilton [@Harrigan1993], the constitutive relation for the mass source is: $$\mathcal{R}_{0}=c\left[\Biggl[\frac{\rho}{\rho_{0}^{\ast}}\Biggr]^{-m}\Psi
-\Psi^{\ast}\right]
\label{eq:mass_source}$$ where $\rho_0^\ast$ and $\Psi^\ast$ represent reference values of the density, $\rho$ and free energy, $\Psi$, respectively. The driving term $\left[ \rho / \rho_0^\ast \right]^{-m}\Psi$ tends to converge to $\Psi^\ast$ (see Eq. (\[eq:mass\_source\])) when density saturation is achieved and local generation of bone ceases. The exponent $m$ is a dimensionless scalar introduced to guarantee uniqueness and stability [@Harrigan1993] . The coefficient $c$ controls the rate of the adaptation process with units $[\rm{s/cm^2}]$. As proposed in [@Waffenschmidt2012], it can be beneficial to prescribe an upper and lower bound for bone density, thereby avoiding spurious or non-physical values. In this paper, the parameter $c$, which is conventionally considered to be constant, is replaced by a bell function defined as: $$\begin{aligned}
c(\rho) = & \frac{1}{1 + \left[ (\rho - \rho^{\mathrm{mid}}) /
(\rho{^\mathrm{max}} - \rho{^\mathrm{mid})} \right]^{2 b}}\\
& \mathrm{with} \quad \rho^{\mathrm{mid}} =
\frac{\rho{^\mathrm{max}} + \rho{^\mathrm{min}}}{2}
\end{aligned}
\label{eq:bell_function}$$ $\rho^\mathrm{max}$ and $ \rho{^\mathrm{min}}$ where $\rho^\mathrm{max}$ and $\rho^\mathrm{min}$ are the maximum and minimum values of $\rho$, and $\rho^{\rm {mid}}$ is their average. The bell function (\[eq:bell\_function\]) is illustrated in Figure \[fig:bell\_func\] for different values of the integer exponent, $b$. Its application and influence on the overall results are elaborated in Section \[sec:numerical\_examples\].
Elastic constitutive relationship
---------------------------------
As an elastic porous material, the free energy $\Psi (\mathbf F, \rho)$, for bone is taken as $$\Psi=\left[\frac{\rho}{\rho_{0}^{\ast}}\right]^{n}\Psi^{\mathrm{neo}},
\label{eq:free_energ}$$ where $\Psi^{\rm {neo}}$ is the Helmholtz free energy for a Neo-hookean material, which is expressed in terms of the right Cauchy-Green deformation tensor $\boldsymbol{\rm{C}}$: $$\Psi^{\mathrm{neo}}=\frac{\mu}{2}\left[\textrm{tr}(\mathbf{C})-3\right]-\mu\ln(\sqrt{\det\mathbf{C}})+\frac{\lambda}{2}\ln^{2}(\sqrt{\det\mathbf{C}})$$ where $\mu$ and $\nu$ are the Lamé constants. Moreover, the exponent $n$ is a non-physical parameter that typically varies as $1 \leq n \leq 3.5$, depending on the porosity of the material [@Gibson2005]. Bone adaptation is a mechanically driven process, whereby the density field evolves in response to the mechanical environment. Likewise, the material stiffness is directly dependent on the density and this, in turn, influences the mechanical response. Therefore, the equation for conservation of mass \[eq:mass\_balance\] is coupled with the equation for linear momentum balance: $$\label{eq:linear_momentum}
\nabla_{\mathbf X} \cdot \mathbf P = 0$$ where $\mathbf P$ is the the first Piola-Kirchhoff stress: $$\mathbf P = \frac{\partial \Psi (\mathbf F, \rho)}{\partial \mathbf F}$$
This coupled system of equations is solved using the finite element method - see Section \[sec:numerical\_examples:bone\_adap\].
Fracture resistance and fracture propagation {#sec:fracture}
============================================
Various theories exist in the literature regarding failure criteria for bone tissue and it is now common practice for researchers to estimate fracture resistance within the framework of FEM. In particular, subject-specific FEM models can potentially quantify the risk of failure under a given loading scenario. However, this still remains an open challenge.
In recent years, the main focus in bone mechanics was in the use of different strength criteria for the onset of failure. The most commonly adopted ones were based on stress [@keyak2005predicting] or strain measures [@schileo2008subject] assuming bone failure is determined by a yield criterion [@yosibash2010predicting]. Experimental validation of such simplified models show that there is a significant spread in the predicted failure, with errors between 10% and 20% [@van2014accurately]. This variation is explained perhaps by the focus on the local initiation of failure, rather than the complete failure mechanism. The fracture process of bone is very important, particularly in the case of fatigue fractures [@gupta2008fracture]. Limitations in previous studies (e.g. use of 2D geometry [@bettamer2017using], assuming homogeneous bone properties [@gasser2007numerical]), can also explain why an appropriate model for bone fracture has not been developed previously.
This paper builds on the authors’ computational framework [@kaczmarczyk2017energy] for brittle fracture within the context of configurational mechanics, extending it here to include the influence of heterogeneous materials such as bone. The concept of configurational forces was originally introduced by Eshelby [@eshelby1951force]. Unlike physical forces, configurational forces act on the material manifold and represent the tendency of imperfections like cracks, voids or material inhomogeneities to move relative to the surrounding material. The past two decades have seen a growing interest in this approach for analysis of material imperfections [@maugin2016configurational] and in particular for evaluating the forces driving crack advancement [@kaczmarczyk2017energy; @steinmann2001application; @ozencc2016configurational]. However, until recently this approach has never been used to effectively assess configurational forces in heterogeneous bodies with cracks.
In the current study, additional configurational forces arising from inhomogeneities [@kienzler2014configurational] associated with spatially varying bone density are introduced into the formulation. This allows for the accurate assessment of the likelihood of a crack to propagate and to simulate the subsequent propagation of fractures in bone. An additional goal is to investigate bone fracture at different stages of bone adaptation, utilising either the results from bone adaptation analysis (Section \[sec:bone\_remodel\]) or data directly taken from CT scans. Similar concepts of combined adaptation and fracture analyses has been presented before [@hambli2013integrated]. However, it utilised a different adaptation model and continuum damage mechanics approach for fracture, both of which require many more parameters to calibrate.
First and second laws of thermodynamics
---------------------------------------
The first law of thermodynamics can be expressed as $$\label{eq::first_law}
\int_{\partial \mathcal B_t} {\dot {\boldsymbol {\rm u} } }
\cdot \mathbf t \mathrm d S = \int_{\partial \Gamma } \gamma \dot A_\Gamma +
{\frac{\mathrm d}{\mathrm d t }}
\int_{\mathcal B_t} \Psi(\mathbf F, \rho) \mathrm d V$$ where the left hand side is the power of external work, the first term on the right hand side is the rate of crack surface energy and the last term is the rate of internal energy. $\mathbf t$ is the external traction vector, $\gamma $ is the surface energy $[ {\rm{N m}}^{-1} ]$, $\dot{A}_\Gamma$ is the change in the crack surface area and $\Psi$ is the volume specific free energy. The crack surface $\Gamma$ comprises two crack faces and a crack front $\partial\Gamma$ - see Figure \[fig:crac\_surf\_construct\].
In [@kaczmarczyk2017energy], a kinematic relationship between the change in the crack surface area $\dot{A}_\Gamma$ and the crack front velocity $\dot{\mathbf{W}}$ was derived that is given as: $$\label{eq::Agamma2}
\dot{A}_\Gamma
=
\int_{\partial\Gamma}
\mathbf{A}_{\partial\Gamma} \cdot \dot{\mathbf{W}} \textrm{d}L$$ where $\mathbf{A}_{\partial\Gamma}$ is a dimensionless kinematic state variable that defines the orientation of the current crack front that can be considered a unit vector normal to the crack front and tangential to the crack surface. In deriving this expression, it was recognised that any change in the crack surface area $\dot{A}_\Gamma$ in the current material space can only occur due to motion of the crack front.
Making use of Equations (\[eq:phy\_vel\]) and (\[eq::Agamma2\]), and given that $\mathrm d \dot V = \nabla _{\mathbf X} \cdot \mathbf{\dot W}
\mathrm d V$, Eq (\[eq::first\_law\]) can be reformulated as:
$$\label{eq:crack_first_law}
\int_{\partial \mathcal B_t} \big( {\dot {\boldsymbol {\rm w} } } \cdot \mathbf t -
{\dot {\boldsymbol {\rm W} } } \cdot \mathbf F^{\rm T} \mathbf t \big) \mathrm d S =
\int_{\partial \Gamma } \gamma \mathbf A_{\partial \Gamma}
\cdot {\dot {\boldsymbol {\rm W} } } \mathrm d L + \int_{\mathcal B_t}
\big( \mathbf P\, \colon \nabla_{\mathbf X} {\dot {\boldsymbol {\rm w} } } +
{\bm \Sigma}\, \colon \nabla_{\mathbf X} {\dot {\boldsymbol {\rm W} } }
+ \mathbf f^{\mathrm {inh}} \cdot \dot{\mathbf{W}}
\big) \mathrm d V$$
where $${\bm {\Sigma}} = \Psi(\mathbf F, \rho) \mathbf 1 - \mathbf F^{\rm T}
\mathbf P(\mathbf F, \rho),
\, \mathrm{and} \quad
\mathbf f^{\mathrm {inh}} =
\left.
\frac{\partial \Psi}{\partial \rho}
\right|_{(\mathbf{F} = {\rm{const}})}
\frac{\partial \rho}{\partial \mathbf X}$$ ${\bm {\Sigma}}$ is the Eshelby stress tensor and $\mathbf f^{\mathrm {inh}}$ is an additional fictitious force that arises from variations in the density field and drives the crack front from dense to less dense material.
The spatial conservation law of linear momentum balance is repeated here: $$\label{eq:linear_momentum2}
\nabla_{\mathbf X} \cdot \mathbf P = 0
\;
\forall \mathbf{X}\in\mathcal B_t,
\quad
\mathbf{P}\mathbf{N} = \mathbf{t}\;
\forall \mathbf{X}\in\partial\mathcal B_t^\sigma$$ where $\partial\mathcal B_t^\sigma$ is the region of the boundary where tractions are applied.
The equivalent material momentum balance is expressed as: $$\nabla_{\mathbf X } \cdot {\bm {\Sigma}}= \mathbf f^{\mathrm {inh}}
\;
\forall \mathbf{X}\in\mathcal B_t,
\quad
{\bm {\Sigma}}\mathbf{N} = \mathbf{F}^\textrm{T}\mathbf{t}\;
\forall \mathbf{X}\in\partial\mathcal B_t^\sigma$$ It is important to note that $\mathbf f^{\mathrm {inh}}=\mathbf{0}$ in the case of homogeneous materials, with uniform density distribution.
After applying the divergence theorem to Eq. (\[eq:crack\_first\_law\]) and recognising the momentum balance laws, we follow [@kaczmarczyk2017energy] to establish a local form of Eq. (\[eq:crack\_first\_law\]), which represents an expression for equilibrium of the crack front as $$\label{eq:crack_local_first_law}
{\dot {\boldsymbol {\rm W} } } \cdot
\left( \gamma \mathbf A_{\partial \Gamma} - \mathbf G \right) = 0$$ where the configurational force $\mathbf{G}$ is the driving force for crack propagation: $$\mathbf G = \lim_{|\mathcal{ L }|\to 0}
\int_{ {{\mathcal L}_{\rm n}} } {\bm {\Sigma}}\mathbf{N}\, \mathrm d L
\label{eq:crack_configuration_force}$$ From this equation, it is clear that the crack front is in equilibrium when the crack is not propagating, i.e. material velocity $\dot{\mathbf{W}}$ at the crack front is zero, or when the crack front is propagating and the configurational force is in equilibrium with the material resistance $\gamma \mathbf A_{\partial \Gamma}$.
It should be noted that crack front equilibrium is unaffected by material heterogeneities and does not depend on $\mathbf f^{\mathrm {inh}}$. All terms in Eq. \[eq:crack\_local\_first\_law\] are only evaluated at the crack front. However, it will be shown in Section \[sec:fem\_fracture\_prop\] that, in a discrete setting, calculation of the nodal configurational forces involves a volume integral of the density gradient.
Since Eq. (\[eq:crack\_local\_first\_law\]) has more than one solution at equilibrium, depending on whether the crack does or does not propagate, the formulation is supplemented by a straightforward criterion for crack growth, equivalent to Griffith’s criterion [@kaczmarczyk2017energy]: $$\label{eq:grif1}
\phi(\mathbf{G}) =
\mathbf{G} \cdot \mathbf{A}_{\partial\Gamma} - g_c/2 \leq 0$$ where $g_c=2\gamma$ is a material parameter specifying the critical threshold of energy release per unit area of the crack surface $\Gamma$, also known as the Griffith energy. For a point on the crack front to satisfy the crack growth criterion, either $\phi<0$ and $\dot{\mathbf{W}}=0$, or $\phi=0$, $\dot{\mathbf{W}}\ne 0$ and $\gamma\mathbf{A}_{\partial\Gamma}=\mathbf{G}$.
The direction of fracture propagation is constrained by the second law of thermodynamics. Here we assume that fracture takes place relatively fast compared to the process of adaptation (with no healing), such that non-negative dissipation at the crack front can be expressed as $$\mathcal{D} = \gamma \dot{\mathbf{W}} \cdot \mathbf{A}_{\partial\Gamma}= \dot{\mathbf{W}} \cdot \mathbf{G}\ge0$$
It should be noted that the well-established stress intensity factors are not applicable in the case of heterogeneous materials, since it requires the existence of an analytical solution for the stress field in the vicinity of the crack front that is independent of arbitrary distribution of density. Similarly, the use of J-integrals requires integration over the closed surface without inhomogeneities (including heterogeneous density distribution), except for the crack front itself and therefore not applicable in this case. Finally, it is worth noting that the current framework is formulated within the realm of large displacements and large strains, hence it is generally valid under any assumption for strains and displacements.
Density field {#sec:dens_mapping}
-------------
The previous subsections have shown that fracture modelling of bone is influenced by the density distribution in the material configuration. This density field can be generated from either (a) a bone adaptation analysis, solving both Eqs (\[eq:mass\_balance\]) and (\[eq:linear\_momentum2\]), or (b) subject-specific data (geometry and material properties) available from, for example, computed tomography (CT) scans. Previous examples in the literature of subject-specific modelling to assess the stresses and fracture resistance of bones can be found in [@poelert2013patient; @Helgason2008b; @Yosibash2010]. Most algorithms that use voxel data have simply averaged [@zannoni1999material] or integrated data onto finite elements, thereby supplying a constant density within their volume [@taddei2007material; @schileo2008subject]. In this paper, radiopacity associated with each 3D voxel from CT scan data is spatially approximated.
In the numerical examples described later, both sources of density data are used. It will be shown in the next section that, in order to evaluate the configurational forces at the crack front, it is necessary to have a spatially smooth density field. Therefore, discrete density data will need to be approximated as a smooth density field, and this will be achieved by adopting the Moving Weighted Least Squares (MWLS) method. This mapping approach was chosen since it offers higher regularity (i.e. higher derivatives exist) than when the field is directly approximated on the finite element mesh. Full details are given in [@karol_lewandowski_moving_2019].
Finite element modelling {#sec:fem_modelling}
========================
This section considers the sequential analysis of bone adaptation and fracture propagation, although it is recognised that the density field could be obtained directly from subject-specific data, in which case it may not be necessary to undertake the bone adaptation analysis. A sequential approach is justified since the process of bone adaptation takes place at a much longer time scales than fracture.
Three-dimensional domains are discretised with tetrahedral finite elements. Fields are approximated in the current material and current spatial spaces with hierarchical basis functions of arbitrary polynomial order, following the work of Ainsworth and Coyle [@Ainsworth2003]. $$\begin{aligned}
\rho^h({\boldsymbol {\rm \upchi}},t) = \pmb\Phi({\boldsymbol {\rm {\upchi}}}) {\tilde{\pmb\uprho}}(t) \\
\mathbf{ X}^h({\boldsymbol {\rm \upchi}},t) = \pmb\Phi({\boldsymbol {\rm {\upchi}}}) \tilde {\mathbf{ X}}(t),
\quad \mathbf{x}^h({\boldsymbol {\rm \upchi}},t) = \pmb\Phi({\boldsymbol {\upchi}}) {\tilde{\mathbf{x}}}(t) \\
\mathbf{ W}^h({\boldsymbol {\rm \upchi}},t) = \pmb\Phi({\boldsymbol {\rm \upchi}}) {\dot { \tilde {\boldsymbol {\rm W} } }}(t),
\quad \mathbf{w}^h({\boldsymbol {\rm \upchi}},t) = \pmb\Phi({\boldsymbol {\upchi}}){\dot { \tilde {\boldsymbol {\rm w} } }}(t)
\label{eq:discretisation}\end{aligned}$$ where $\mathbf{\Phi}$ are shape functions, superscript $h$ indicates approximation and $(\tilde \cdot)$ nodal values. Moreover, the smoothed density field is approximated by MWLS shape functions $$\rho^{h,\textrm{MWLS}}(\mathbf{X},t) = \Phi^\textrm{MWLS}(\mathbf{X})
\tilde{\pmb\uprho}^h(\Xi({\boldsymbol {\rm \upchi}}),t)$$ It should be noted that shape functions $\Phi^\textrm{MWLS}(\mathbf{X})$ are evaluated at current material points, $\mathbf{X}$, rather than reference points, $\pmb\upchi$, as presented in Eq. (\[eq:discretisation\]) with the property of partition of unity. Since the density field is evaluated at $\mathbf{X}$, the approximation is independent of changes of the material configuration (i.e. changing mesh).
Bone adaptation {#bone-adaptation}
---------------
The bone adaptation problem is solved with a staggered approach, the material configuration is fixed such that: $$\tilde {\mathbf{ X}}(t) = \tilde {{\boldsymbol {\rm \upchi}}} = \textrm{const}
\;\;\textrm{and}\;\;
\dot{\tilde {\mathbf{X}}}(t) = \dot{\tilde {\mathbf{W}}}(t) = 0$$ where $\tilde {{\boldsymbol {\rm \upchi}}}$ is vector of nodal positions. The semi-discrete form of equations (\[eq:mass\_balance\]) and (\[eq:linear\_momentum\]) take the form of residuals $$\left\{
\begin{split}
\mathbf{r}^\rho(\tilde{\pmb\uprho}(t), \tilde{\mathbf{x}}(t)) =
\int_{\mathcal{B}^h_t} \pmb{\Phi}^\textrm{T}\dot{\tilde{\pmb\uprho}}^h({\boldsymbol {\rm \upchi}},t)
\textrm{d}V
+
\int_{\mathcal{B}^h_t} \nabla_\mathbf{X} \pmb{\Phi}^\textrm{T}
\mathcal{R} \nabla_\mathbf{X}\pmb{\Phi} \tilde{\pmb\uprho}
\textrm{d}V
-
\int_{\mathcal{B}^h_t} \pmb{\Phi}^\textrm{T} \mathcal{R}^h_0
\textrm{d}V
-
\int_{\partial\mathcal{B}^h_t} \pmb{\Phi}^\textrm{T}
\mathbf{q}^\textrm{external}
\textrm{d}S
= \mathbf{0}\\
\mathbf{r}^x(\tilde{\pmb\uprho}(t), \tilde{\mathbf{x}}(t)) =
\int_{\mathcal{B}^h_t} \nabla_\mathbf{X}\pmb{\Phi}^\textrm{T}\mathbf{P}^h\textrm{d}V
-
\int_{\partial\mathcal{B}^h_t} \pmb{\Phi}^\textrm{T}\mathbf{f}^\textrm{ext, adapt}\textrm{d}S
= \mathbf{0}
\end{split}
\right.$$ where $\mathbf{r}^\rho$ is the vector of residuals related to mass density flux equilibrium, $\mathbf{q}^\textrm{external}$ is influx of mass across the boundary, $\mathbf{r}^x$ is the vector of residuals associated with balance of linear momentum and $\mathbf{f}^\textrm{ext, adapt}$ are averaged long term forces mimicking mechanical load on the bone over long time period and $\mathcal{R}$ is mass conductivity. A truncated Taylor series expansion leads to the semi-discrete form, expressed as: $$\left[
\begin{array}{c}
\mathbf{r}^\rho(\tilde{\pmb\uprho}_i(t), \tilde{\mathbf{x}}_i(t))\\
\mathbf{r}^x(\tilde{\pmb\uprho}_i(t), \tilde{\mathbf{x}}_i(t))
\end{array}
\right]
+
\left[
\begin{array}{cc}
\mathbf{M}_{\rho\rho} & \mathbf{0}\\
\mathbf{0} & \mathbf{0}
\end{array}
\right]
\left\{
\begin{array}{c}
\delta\dot{\tilde{\pmb\uprho}}_{i+1}(t)\\
\delta\dot{\tilde{\mathbf{x}}}_{i+1}(t)
\end{array}
\right\}
+
\left[
\begin{array}{cc}
\mathbf{K}_{\rho\rho} & \mathbf{K}_{\rho x}\\
\mathbf{K}_{x\rho} & \mathbf{K}_{xx}
\end{array}
\right]
\left\{
\begin{array}{c}
\delta\tilde{\pmb\uprho}_{i+1}(t)\\
\delta\tilde{\mathbf{x}}_{i+1}(t)
\end{array}
\right\}
=
\left[
\begin{array}{c}
\mathbf{0}\\
\mathbf{0}
\end{array}
\right]
\label{eq:tangent_remodelling}$$ with $$\begin{split}
\mathbf{M}_{\rho\rho} =
\left.
\frac{\partial \mathbf{r}^\rho}{\partial \dot{\tilde{\pmb\uprho}}}
\right|_{(\tilde{\pmb\uprho}_i(t),\tilde{\mathbf{x}}_i(t))},\,
\mathbf{K}_{\rho\rho} =
\left.
\frac{\partial \mathbf{r}^\rho}{\partial {\tilde{\pmb\uprho}}}
\right|_{(\tilde{\pmb\uprho}_i(t),\tilde{\mathbf{x}}_i(t))},\,
\mathbf{K}_{\rho x} =
\left.
\frac{\partial \mathbf{r}^\rho}{\partial {\tilde{\mathbf{x}}}}
\right|_{(\tilde{\pmb\uprho}_i(t),\tilde{\mathbf{x}}_i(t))},\,\\
\mathbf{K}_{x \rho} =
\left.
\frac{\partial \mathbf{r}^x}{\partial {\tilde{\pmb\uprho}}}
\right|_{(\tilde{\pmb\uprho}_i(t),\tilde{\mathbf{x}}_i(t))},\,
\mathbf{K}_{x x} =
\left.
\frac{\partial \mathbf{r}^x}{\partial {\tilde{\mathbf{x}}}}
\right|_{(\tilde{\pmb\uprho}_i(t),\tilde{\mathbf{x}}_i(t))},
\end{split}$$ and $$\tilde{\pmb\uprho}_{i+i}(t) =
\tilde{\pmb\uprho}_{i}(t) + \delta\tilde{\pmb\uprho}_{i+i}(t),\;
\tilde{\mathbf{x}}_{i+i}(t) =
\tilde{\mathbf{x}}_{i}(t) + \delta\tilde{\mathbf{x}}_{i+i}(t)$$ where $(\cdot)_i$ is quantity at Newton iteration $i$. Finally, the above semi-discrete problem is discretised in time using implicit Euler scheme: $$\dot{\tilde{\pmb\uprho}}_{i+i}^{n+1}(t) =
\frac{
\tilde{\pmb\uprho}_{i+i}^{n+1}-\tilde{\pmb\uprho}^{n}
}{\Delta t}$$ where $\Delta t$ is length of time step, and $n$ is time step number.
Note that the density field variables are approximated using polynomial bases functions that are one order less than those used for the spatial position variables, thereby ensuring a stable solution without oscillations.
The discretised balance equations are solved iteratively using the Newton-Raphson method for the displacements and density.
Fracture propagation {#sec:fem_fracture_prop}
--------------------
Given that the bone adaptation and fracture propagation problems have different boundary conditions and geometry they are solved as a staggered coupled problem. Initially bone adaptation is simulated under long-term effective loads applied without initial crack. Subsequently, an initial crack is inserted to compute the effect of short-term loads or extreme cyclic loading. The two different meshes for bone adaptation and fracture propagation are tailored for the specific analysis at hand. As a consequence, the approximated Piola stress tensor for bone adaptation is expressed as follows: $$\mathbf{P}^\textrm{h} =
\mathbf{P}(\mathbf{F}^\textrm{h}, \rho^h)$$ whereas, for the fracture propagation problem, it is approximated as $$\mathbf{P}^{\textrm{h},\textrm{MWLS}} =
\mathbf{P}(
\mathbf{F}^\textrm{h}, \rho^{h,\textrm{MWLS}})$$
The residual force vector in the discretised spatial domain is expressed in the classical way as: $$\label{spatial_residual}
\mathbf{r}_\textrm{s}^\textrm{h}(\tilde{\pmb\uprho}(t), \tilde{\mathbf{x}}(t)) = \tau\mathbf{f}^\textrm{h}_\textrm{ext,s}-\mathbf{f}^\textrm{h}_\textrm{int,s}=\tau \int_{\partial\mathcal{B}^h_t} \pmb{\Phi}^\textrm{T}
\mathbf{f}^\textrm{ext}
\textrm{d}S-
\int_{\mathcal{B}^h_t} \nabla_\mathbf{X} \pmb{\Phi}^\textrm{T}
\mathbf{P}^{h,\textrm{MWLS}}\textrm{d}V=\mathbf{0}$$ where $\tau$ is the unknown scalar load factor, $\mathbf{f}^\textrm{h}_\textrm{ext,s}$ is the vector of externally applied forces and $\mathbf{f}^\textrm{h}_\textrm{int,s}$ is the vector of internal forces.
Discretisation of Eq. \[eq:crack\_local\_first\_law\] establishes the material counterpart to Eq. \[spatial\_residual\], expressed as $$\label{material_residual}
\mathbf{r}_\textrm{m}^\textrm{h}(\tilde{\pmb\uprho}(t), \tilde{\mathbf{x}}(t)) = \mathbf{f}^\textrm{h}_\textrm{res}-\tilde{\mathbf{G}}^\textrm{h}=\mathbf{0}$$ $\tilde{\mathbf{G}}^\textrm{h}$ is the vector of nodal configurational forces only on nodes on the crack front, with the integration restricted to elements adjacent to the crack front: $$\tilde{\mathbf{G}}^\textrm{h} =
\int_{\mathcal{B}^h_t}
\nabla_\mathbf{X}\pmb{\Phi}^\textrm{T} {\pmb\Sigma}^{h,\textrm{MWLS}}
\textrm{d}V
+
\int_{\mathcal{B}^h_t}
\pmb{\Phi}^\textrm{T} \dfrac{\partial {\Psi}^{h,\textrm{MWLS}} }{\partial \rho^{h,\rm{MWLS}}}
\left(
\frac{\partial
\rho^{h,\textrm{MWLS}}}{\partial \mathbf{X}}
\right)
\textrm{d}V
\label{eq:mat_int_front}$$ These configurational forces are the driving force for crack propagation. It should be noted that the second term of $\tilde{\mathbf{G}}^\textrm{h}$ reflects the influence of the spatially varying density. In the case of a homogeneous material, this second term would be zero. It should also be noted that this is only the case for the discretised configurational forces and that the continuum equivalent (Eq. \[eq:crack\_configuration\_force\]) is unaffected by variation in the density field.
$\mathbf{f}^\textrm{h}_\textrm{res}$ is the vector of nodal material resistance forces, given as: $$\mathbf{f}^\textrm{h}_\textrm{res}=\frac{1}{2}\left(\tilde{\mathbf{A}}_\Gamma^\textrm{h}\right)^\textrm{T}\mathbf{g}_\textrm{c}$$ where $\mathbf{g}_\textrm{c}=\mathbf{1}g_\textrm{c}$ is a vector of size equal to the number of nodes on the crack front. $\tilde{\mathbf{A}}_\Gamma^\textrm{h}$ defines the current orientation of the crack front and is a matrix comprising direction vectors along the crack front that are normal to the crack front and tangent to the crack surface: $$\tilde{\mathbf{A}}_\Gamma^\textrm{h} =
\int_{S^h_\Gamma}
\pmb{\Phi}^\textrm{T}
\frac{\partial {A}^h_{\Gamma}}{
\partial \tilde{\mathbf{X}}}
\textrm{d}L$$ $\tilde{\mathbf{A}}_\Gamma^\textrm{h}$ is evaluated by only integrating over $S_\Gamma^\textrm{h}$ that defines the area of those triangular finite elements that discretise the crack surface $\Gamma^\textrm{h}$ adjacent to the crack front $\partial\Gamma^\textrm{h}$. $A^\textrm{h}_{\Gamma}$ is calculated as: $$A^\textrm{h}_{\Gamma} =
\| \mathbf{N}(\tilde{\mathbf{X}}) \|
=
\left\|
\epsilon_{ijk}
\frac{\partial \Phi^\alpha_p}{\partial \xi_i}
\frac{\partial \Phi^\beta_r}{\partial \xi_j}
\tilde{X}^\alpha_p
\tilde{X}^\beta_r
\right\|$$ where $\alpha, \beta \in \{0, \dots, N_{\rm{base}}\}$ are numbers of base functions, $i,j,k,l,p,r \in \{0,1,2\}$ are material indices and $\boldsymbol{\upepsilon}$ is the Levi-Civita tensor. Moreover, the total number of degrees of freedom on element is $3(N_{\rm{base}}+1)$ and the units of $\tilde{\mathbf{A}}_\Gamma^\textrm{h}$ are $[{\rm m}^{-1}]$. $\mathbf{N}$ are the normals to the crack surface $\Gamma$.
Singularity element {#sec:singularity}
===================
For the purposes of determining parameters such as stress intensity factors, it can be useful to reproduce the singular stress field at the crack front. However, conventional finite elements that adopt polynomial approximation functions are unable to do this. In this paper a new type of finite element with hierarchical approximation functions that overcome this problem is briefly presented. This is inspired by the so-called quarter-point elements, originally developed in the 1970s, whereby the mid-node of all edges connected to the crack tip node were shifted to the quarter-point [@barsoum1976use; @henshell1975crack]. In this work, all bodies are discretised using 3D tetrahedral elements. However, for simplicity, we present the main attributes in this paper in 1D.
For elements adjacent to the crack tip in the material configuration, the approximated material displacement field, using hierarchical shape functions (up to 2nd order), is expressed as: $$\label{eq:hierarchical_u}
W(\xi) = \sum_{a=0}^2 N_a (\xi) W^{(a)} = (1 -\xi)W^{(0)} + \xi W^{(1)} + \kappa (1 - \xi) \xi W^{(2)}$$ where the natural coordinate $0\le \xi \le 1$ and $N_2 = \kappa N_0 N_1 = \kappa\xi(1 - \xi)$. The parameter $\kappa$ is introduced, resulting in a nonlinear mapping between the natural and physical coordinates and leading to the desired singular stress and strain field at the crack tip.
Adopting an isoparametric formulation, the element geometry can also be interpolated using the same approximation functions as for $W$. Thus, the physical distance from the crack tip is expressed as: $$\label{eq:hierarchical_r}
r_{\rm q}(\xi) = \sum_{a=0}^2 N_a (\xi) r_{\rm q}^{(a)} = \xi l + \kappa \xi(1-\xi) l$$ where $r_{\rm q}(\xi=0)=0$ at the crack tip and $r_{\rm q}(\xi=1)=l$. Setting $\kappa=-1$ results in the following relationship: $$r_{\rm q}= \xi l - \xi(1-\xi)l= \xi l \quad \Rightarrow \quad \xi = \sqrt{\frac{r_{\rm q}}{l}},$$ This yields the following radial dependence for displacements and strains: $$\label{eq:singstrain}
\begin{aligned}
W(r_{\rm q}) &= W^{(0)} + \left( -W^{(0)} + W^{(1)} - W^{(2)} \right) \sqrt \frac{r_{\rm q}}{l} - W^{(2)} \frac{r_{\rm q}}{l}\\
\varepsilon(r_{\rm q}) &= \frac{\partial W}{\partial r_{\rm q}} = \left( W^{(0)} + W^{(1)} - W^{(2)} \right) \frac{1}{2} \sqrt \frac{l}{r_{\rm q}} + W^{(2)} \frac{1}{l}
\end{aligned}$$ Eq. (\[eq:singstrain\]) has the necessary terms to reproduce rigid body motion and pass the patch tests, as well as the desired singularity at the crack tip due the existence of the term $1 / \sqrt r_{\rm q}$. This will enable the elements adjacent to the crack front to reproduce the strain singularity resulting in an accurate finite element solution [@nejati2015use]. The influence of this approach for tetrahedral elements will be investigated in Section \[sec:release\_energy\_rate\].
Numerical examples {#sec:numerical_examples}
==================
Several numerical examples are presented to illustrate each aspect of the proposed framework. The first set of analyses, presented in Subsection \[sec:numerical\_examples:bone\_adap\], considers bone adaptation, using an equine 3rd metacarpal bone as a case study. The performance of the singularity element formulation is demonstrated in Subsection \[sec:release\_energy\_rate\] using a finite plate with through thickness crack subjected to uniaxial stress. In the penultimate subsection, the framework is used to investigate the likelihood of fracture at different stages of adaptation. The final example considers fracture propagation at different stages of adaptation.
Bone adaptation examples {#sec:numerical_examples:bone_adap}
------------------------
This subsection considers the bone adaptation of an equine 3rd metacarpal bone. In the UK, approximately 60% of horse fatalities at racecourses are directly or indirectly associated with a fracture, with the distal limb the most commonly affected site [@parkin2004risk]. Most of these fractures occur due to the accumulation of tissue fatigue, as a result of repetitive loading [@Parkin2005], rather than a specific traumatic event. Intense exercise and excessive loading of the metacarpal bones results in maladaptation. The location of 3rd metacarpal fractures is remarkably consistent across a large number of racehorses, with crack initiation presenting from the lateral para-sagittal groove of the distal condyle of the leading forelimb [@jacklin2012frequency; @parkin2006analysis]. Despite considerable research in the field, including applying diagnostic methods such as radiography [@bogers2016quantitative; @crijns2014intramodality; @loughridge2017qualitative], magnetic resonance imaging [@tranquille2017MRI] and biomarkers [@mcilwraith2005use], it still remains a challenge to accurately predict the fracture risk and prevent this type of significant injury.
Three cases are studied, using the material parameters presented in Table \[tab:parameters\_mc3\]. Stiffness and porosity values are derived from mechanical tests [@Les1994], whereas other values are from previous studies of human tibia [@Pang2012; @Waffenschmidt2012]. Each case considers a different function for the parameter $c$ that defines the rate of bone adaptation and is used to compute the mass source, $\mathcal{R}_0$, according to Eq. (\[eq:mass\_source\]). In Case 1, $c$ is constant. For Case 2 and Case 3 different bell functions (Eq. (\[eq:bell\_function\])) are used. The parameters for each case are presented in Table \[tab:three\_cases\]. The finite element mesh used in all cases comprises 17041 tetrahedral elements and was generated by discretising the segmented geometry of a full-scale model of an equine 3rd metacarpal bone derived from CT scan data - see Figure \[fig:mc3\_BC\].
For each analysis, the initial density is chosen to be homogeneous since, in the thermodynamic-based model, the starting density does not have a significant effect on the final bone density distribution (similar to other models at biological equilibrium [@kuhl2003theory]). Boundary conditions are simplified to two representative forces (5 $\text{[kN]}$ each) spanning over a small area based on pressure film studies [@Brama2001], as demonstrated in Figure \[fig:mc3\_BC\]. The two forces are often considered in the literature as an equivalent of joint peak force at the mid-stance of a horse gait. An adaptive time stepping scheme (using PETSc [@petsc-web]) is used in all the simulations with an initial time step $\Delta t = 0.5$ $[\text{days (d)}]$, maximum time step of $\Delta t_{\text{max}} = 50$ $[\text d]$ and minimum of $\Delta t_{\text {min}} = 0.05$ $[\text d]$.
Parameter Description Value
------------------ ------------------------- -------------------------------------------------
$E $ Young’s modulus $4700 \,\mathrm{ [MPa]}$ [@Les1994]
$\nu $ Poisson ratio $0.3 \,\mathrm{ [-]}$
$\rho_0 ^\ast $ Initial density $1.0 \,\mathrm{[ g/cm^{3}]}$
$\psi_{0}^\ast $ Target energy density $0.0275\,\mathrm{ [MPa]}$ [@Waffenschmidt2012]
$c$ Density growth velocity $1.0 \,\mathrm{ [d/cm^{2}]}$
$m$ Algorithmic exponent $ 3.25 \,\mathrm{ [-]}$
$n$ Porosity exponent $2.25 \,\mathrm{ [-]}$ [@Les1994]
: Material parameters used for the simulations of 3rd metacarpal bone adaptation.[]{data-label="tab:parameters_mc3"}
Case $c$ $b$ $\rho^\mathrm{max}$ $\rho^\mathrm{min}$
------ ----------------------------- ------ ------------------------- -------------------------
1 1 - - -
2 Eq. (\[eq:bell\_function\]) 1000 2.5 $[{\text{g/cm}}^3]$ 0.3 $[{\text{g/cm}}^3]$
3 Eq. (\[eq:bell\_function\]) 30 1.8 $[{\text{g/cm}}^3]$ 1.0 $[{\text{g/cm}}^3]$
: Presentation of three cases input parameters for the evaluation of coefficient $c$ to compute mass source, $\mathcal{R}_0$, as presented in Eq. (\[eq:mass\_source\]). All cases have common material input parameters presented in Table \[tab:parameters\_mc3\].[]{data-label="tab:three_cases"}
at (0,0) [![Finite element mesh of the equine 3rd metacarpal bone. The mesh consists of 14,041 quadratic tetrahedral elements and 70,901 degrees of freedom. To simulate the peak load of a gallop, 5 kN forces are applied on the lateral and medial side of the distal condyle.[]{data-label="fig:mc3_BC"}](Figures/mc3_BC.png "fig:"){width="8cm"}]{}; at(-2.5,-1.8)[$F = 5 \mathrm{kN}$ ]{}; at(-3.1,-0.3)[ $F = 5 \mathrm{kN}$ ]{}; at(2.8,-0.3)[ Fixed end ]{};
Results of Case 1 are presented in Figure \[fig:mc3\_density\] where density maps at five different points in time $(\text{0, 10, 40, 100, 700})$ $[\text d]$ are visualised. Significant densification occurred immediately after reaching the maximum level of the loading, particularly in the proximity of the applied forces, associated with high levels of strain energy. Conversely, areas with low levels of strain energy experience a reduction in density. After 100 $[{\text d}]$, biological equilibrium was achieved and no further changes in density took place. The resulting maximum density is 2.8 $[{\text {g/cm}}^3]$ and the minimum is close to zero.
[0.475]{}
at (0,0)
; at(0.5,1.0)[ ![Change in bone mass over time for 3 cases (see Table \[tab:three\_cases\]). Density distribution contours in 3rd metacarpal bone at five snapshots in time in (a) and at the last converged step for (b) and (c).[]{data-label="fig:density_changes_"}](Figures/tikz/mc3_density_fig.png "fig:"){width="4.5cm"}]{};
[0.475]{}
at (0,0)
; at(1.5,1.0)[ ![Change in bone mass over time for 3 cases (see Table \[tab:three\_cases\]). Density distribution contours in 3rd metacarpal bone at five snapshots in time in (a) and at the last converged step for (b) and (c).[]{data-label="fig:density_changes_"}](Figures/tikz/density_bell1_imag.png "fig:"){width="2cm"}]{};
[0.475]{}
at (0,0)
; at(1.5,1.0)[ ![Change in bone mass over time for 3 cases (see Table \[tab:three\_cases\]). Density distribution contours in 3rd metacarpal bone at five snapshots in time in (a) and at the last converged step for (b) and (c).[]{data-label="fig:density_changes_"}](Figures/tikz/density_bell2_imag.png "fig:"){width="2cm"}]{};
[0.475]{}
at(0,0)[ ]{} ; at(3,4.5)
;
The maximum density for Case 1 is noticeably higher than in the actual equine bones [@yamada2015experimental] and the minimum density is unrealistically close to zero. This justifies the proposed bell shape function (Eq. (\[eq:bell\_function\])) used for the next two analyses (Case 2 and Case 3).
The results for Case 2 are plotted in Figure \[fig:density\_bell1\]. The last converged step takes place at $t=93$ $[{\text{d}}]$. By setting a high value of $b$, the transition between densities is very sharp and the algorithm encounters convergence difficulties, even with adaptive time-stepping, and biological equilibrium cannot be achieved in this case. However, by observing the range of densities obtained, it is evident that they slowly converge to the same solution as Case 1 (Figure \[fig:mc3\_density\]).
For Case 3, a more moderate value for the exponent in the bell function was chosen along with a narrower density range than those chosen for Case 2 (see Table \[tab:three\_cases\]). The plot presented in Figure \[fig:density\_bell2\] demonstrates how these values influence the results of the analysis. It is evident that with a much lower value for exponent, $b$, the algorithm no longer has problems converging. Furthermore, reducing the range between the upper and lower bounds of density has a significant impact on the results. The dense cortical shaft on the dorsal side of the bone is less dense and covers a much larger region. Furthermore, unrealistically low values of densities have been eliminated. However, as with the previous case, the overall solution converges to the same mass as in Case 1, albeit requiring significantly more time steps.\
Stress intensity calculations {#sec:release_energy_rate}
-----------------------------
To examine the calculation of configurational forces at the crack front in bodies with both homogeneous and heterogeneous density distributions, five numerical examples are presented. First, a simple quasi-two-dimensional plate with homogeneous material distribution is considered. The convergence study utilises an analytical solution as a reference. Second, the proposed singularity elements are included for the same plate problem and their influence on the rate of convergence is presented. Third, the same problem is considered again but with a heterogeneous material distribution. The final two examples demonstrate the calculation of configurational forces for a more representative bone.
### Finite plate with a horizontal crack {#sec:plate_section}
A finite plate with height, $h_{\rm {pl}} = 10$, thickness $t_{\rm {pl}}=1$ and half width $b_{\rm{pl}} = 2.5$ and a horizontal through-thickness crack with half width $a_{\rm {pl}} = 1$, as presented in Figure \[fig:plate\_load\_mesh\](a), is considered. All input parameters presented are dimensionless. The plate is spatially discretised using 1384 tetrahedral elements and subjected to uniaxial stress in the longitudinal direction, as indicated in Figure \[fig:plate\_load\_mesh\](b). Displacements are constrained on three vertices of the plate to prevent rigid body motion.
[c]{}
\
The purpose of this analysis is to calculate the Mode I stress intensity factor $ K_{\rm I} $ directly from the configurational and compare with the analytical solution [@rooke1976compendium] for an infinite plate: $$\label{eq:frac_analytical}
K_{\rm I}=\sigma \sqrt{\pi a_{\rm {pl}}} \left[ \frac{1 - \frac{a_{\rm {pl}}}{2b_{\rm {pl}}} + 0.326 (\frac{a_{\rm {pl}}}{b_{\rm {pl}}})^2 }{\sqrt{1-\frac{a_{\rm {pl}}}{b_{\rm {pl}}}}} \right]$$ where $\sigma $ is the applied stress. Young’s modulus $E$ and Poisson’s ratio $\nu$ are $1000$ and $0.3$, respectively.
Hierarchical approximation functions allow for global and local p-refinment without changing the mesh. In general, all tetrahedrons of the mesh have a global order of approximation, $p_{\rm g}$, with some elements subjected to local refinement of order $p_{\rm{l}}$. All analyses presented were run using the same mesh with p-refinement varying from 1^st^-order to 6^th^-order so that $p_{\rm l} + p_{\rm g} \leq 7$. The Mode I stress intensity factor, $K_{\rm I}$, was calculated directly from the output configurational forces as: $$K_{\rm I} = \sqrt{GE}$$ where $G$ is the change of elastic strain energy per unit area of crack growth. From Figure \[fig:plate\_conv\](a), it is evident that, for the same coarse mesh and number of nodes, the solution can improve drastically when the order of approximation is increased. The well known shear locking associated with first-order approximation is observed. The minimum error achieved is $0.50\%$ for all the cases with total order of approximation $p_{\rm l} + p_{\rm g} = 7$. Therefore, it can be observed that using a low order of global approximation plus local p-refinement can achieve the same level of accuracy as using high order approximation globally, but with fewer degrees of freedom and lower computational cost.
at (0,0)
; at (-1.2,-1.1)[$0.50\%$]{}; at (0,0) ;
at (0,0)
; at (-1.2,-1.0)[$0.028\%$]{};
(a)(b)
Based on the results in Figure \[fig:plate\_conv\](b), it is evident that using singularity elements improves the convergence rate significantly and lowers the error by an order of magnitude, from 0.50% down to 0.028%. However, it can also be seen that for each combination of p-refinement, the error increases with further refinement after it reaches the minimum value. This suggests that the solution cannot be further improved by enhancing the order of approximation alone. Reducing the the elements size and increasing the plate width (to better replicate the infinite plate used to determine the analytical solution), the error would probably decrease further. Nevertheless, the results are considered sufficiently accurate for the purpose at hand.
Overall, these results indicate that it is of great benefit to use the singularity elements, since they improve the accuracy of the solution with no extra cost. Furthermore, the difference in execution time for the analysis with and without their inclusion was negligible.
### Heterogeneous material
So far the numerical examples have assumed homogenous material properties. Here we consider the effect of a heterogeneous density distribution. Considering the same problem of the finite plate with horizontal crack, a density field $\mathbf{\rho}(x,y,z) = 0.125y + 1$ is directly assigned to the integration (Gauss) points of each tetrahedral element. As expected, configurational forces are induced at the crack tip under load and, as explained in Section \[sec:fem\_fracture\_prop\], these forces are influenced by the non-uniform density distribution. However, the stress intensity factor loses its meaning in the case of heterogeneous materials and there is no agreed approach to validate either configurational forces or stress intensity factors for such cases (except for the special case of functionally graded materials [@kim2002finite]). A straightforward verification can be performed by using a central difference numerical integration. The energy release rate for crack growth can be calculated as the change in elastic strain energy per unit area of crack growth [@Griffith163]: $$G = \frac{\partial \psi}{\partial a_{\rm {pl}}}$$ where $\psi$ is the elastic energy of the system, and $a_{\rm{pl}}$ is the crack length. This derivative can be approximated as: $$\frac{\partial \psi}{\partial a_{\rm {pl}}} = \lim_{\Delta a_{\rm {pl}} \to 0} \frac{\psi(a_{\rm {pl}} + \Delta a_{\rm {pl}}) - \psi(a_{\rm {pl}} -\Delta a_{\rm {pl}})}{2\Delta a_{\rm {pl}}}$$ where the elastic strain energies $\psi(a_{\rm {pl}} \pm \Delta a_{\rm {pl}})$ is obtained from two additional analyses with horizontal cracks of lengths: ($a + \Delta a_{\rm {pl}}$) and ($a - \Delta a_{\rm {pl}}$), where $\Delta a_{\rm {pl}}$ is a very small value. Next, knowing the resulting release energy with the crack length of $a_{\rm {pl}}$, a relative error can be calculated. Twenty-four analyses, for different levels of $p$ - refinement and values of $\Delta a_{\rm {pl}}$, have been undertaken in order to determine the error in the release energy. The results are presented in Figure \[fig:covergencefdm\], where it is apparent that the error in fracture energy release rate is converging to 0.3% with increasing levels of refinement. It is worth noting that a similar level of accuracy was attained for the homogeneous case. Achieving higher precision with this means of validation is difficult due to the accumulation of truncation, approximation and discretisation errors. Therefore, it can be concluded that the proposed estimation of fracture energy release rate for heterogeneous materials is obtained with a satisfactory level of accuracy.
at (0,0)
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### Fracture energy release rate for metacarpal bone {#sec:mc3_release_eng}
This numerical example considers the same bone as presented in Section \[sec:numerical\_examples:bone\_adap\]. An initial crack was generated in the mesh using a cutting plane, as shown in Figure \[fig:bone\_ct\_mesh\_cut\]. A notch is situated at the origin of the most common location of a lateral condyle fracture [@jacklin2012frequency]. The numerical analyses were undertaken using three meshes consisted of 6069, 10032 and 21189 tetrahedrons and repeated for 1^st^, 2^nd^ and 3^rd^-order of global $p$ - refinement and local $p$ - refinement at the crack tip. Boundary conditions and material parameters remain the same as in Table \[tab:parameters\_mc3\]. Using a $\mathrm {K_2 HPO_4}$ calibration phantom, grey scale values from CT scans are converted to bone mineral density using five tubes with reference densities. The mechanical material properties were mapped onto the integration points of the mesh of the metacarpal bone using the MWLS method described earlier. The application of load induces configurational forces at the crack front, as shown in Figure \[fig:crackfrontforce\]. The direction of the vectors also indicates the direction of crack propagation. The values of numerically predicted maximal nodal fracture energy release rates in Mode I (crack opening) for subsequent meshes are plotted in Figure \[fig:max\_g1\_convergece\]. It can be seen that, for the same mesh, as the order of approximation increases, the energy release rate converges.
A crack will propagate when the energy release rate $G$ equals the material’s resistance to crack extension, $g_{\rm c}$. Assuming $g_{\rm c} = 2.0\,[\mathrm{ kJ/m^2}]$ [@gasser2007numerical] it can be estimated that this particular metacarpal bone with this initial crack can sustain loading of approximately 2.2 times greater before a fracture starts to propagate.
### Fracture energy release rate for adapted bone
The previous example is extended to investigate the likelihood of fracture in an equine metacarpal bone at different phases of adaptation during training. However, this time, densities from a bone adaptation analysis (Section \[sec:numerical\_examples:bone\_adap\]) are mapped onto the coarse mesh, as shown in Figure \[fig:frackmeshcutting\].
The resulting energy release rate at different points in time of bone adaptation are illustrated in Figure \[fig:crackmc3release\] for three different local $p$ - refinements. It can be seen that the variation in energy release rate for increased orders of approximation at the crack front is very small.
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;
It can be seen that there is a trend of increasing release energy rate over time and that by introducing a notch in the resorption zone, where no loading is applied, the configurational force attains larger values. This indicates that over time the bone becomes more prone to fracture in this specific region.
Crack propagation in bone
-------------------------
In this section we move the analysis further by simulating the process of crack propagation for different levels of bone adaptation. The magnitude of applied forces (Figure \[fig:mc3\_BC\]) is controlled by the increment in crack area during each load step using an arc-length technique. The initial finite element mesh is the same as previously, although it is locally refined as the crack front advances. The fracture energy is $2.0\,[\mathrm{ kJ/m^2}]$ for the entire domain. All five cases (time snapshots) are solved using 2nd-order approximation functions. The numerically predicted crack paths are shown in Figure \[fig:crack\_snapshots\]. It can be seen that the crack has an initially planar shape and then curves towards the lateral side of the bone. This simulated crack path compares well with fractures observed in radiographs [@whitton2010third], especially considering the simplified loading conditions. The load factor versus crack area plots are shown in Figure \[fig:crack\_remodel\_frac\_compar\]. Consistent with the previous analysis in Section \[sec:mc3\_release\_eng\], the metacarpal bone shows a decreased resistance to fracture - i.e. for the same crack area, the remodelled bone requires much lower force (load factor) to induce crack propagation. Low density levels at biological equilibrium ($t=90$ and $t=200$) also influences the crack path, with the crack curving earlier than in the initial stages of remodelling.
at(0.,0)
; at (0.25,1.8) [ ![Load factor versus crack area for different moments in time during bone adaptation analysis. Bone density distribution influenced both load factor and the resulting crack surface.[]{data-label="fig:crack_remodel_frac_compar"}](Figures/pngs/crack_surfaces_adapt.png "fig:"){width="7cm"}]{}; at (1.5,0) [ Crack surfaces]{};
at (0,0) [ ![Crack surface evolution in equine 3rd metacarpal.[]{data-label="fig:crack_snapshots"}](Figures/pngs/crack_spatial_conf.png "fig:"){width="12cm"} ]{}; at(1,-2.5)[a) b) c) d) ]{};
As previously demonstrated (Figure \[fig:plate\_conv\](b)) modelling singularity can significantly improve the accuracy of the configurational forces at the crack front. In the next example we considered bone with heterogeneous density distribution mapped from CT scan data. In the Figure \[fig:load\_factor1\](a) results from crack analysis with and without Quarter Point elements are depicted. It is evident that accurate stress state at the tip has a negligible impact on the full crack propagation analysis and the resulting load factor.
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\(a) (b)
From the load-crack area curves in Figure \[fig:load\_factor1\](b) it can be observed that including density data from CT scans have a significant impact on the predicted load factor and crack path as well.
Finally, we investigated the h and p convergence. The results presented on the Figures \[fig:load\_factor2\](a) and \[fig:load\_factor2\](b) show good numerical convergence for consecutive refinements. It can be concluded that our formulation predicts crack path accurately with minimal effect from original mesh or order of approximation.
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\(a) (b)
Discussion {#sec:discussion}
==========
This paper has presented a FEM computational modelling framework to investigate the influence of bone adaptation, and associated bone density distribution, on fracture resistance and fracture propagation. The influence of the heterogeneous density distribution was captured using an extension of the authors’ previous work on configurational mechanics for fracture. Configurational forces are the driver for crack propagation and it was shown that in order to evaluate correctly these forces at the crack front it is necessary to have a spatially smooth density field, with higher regularity than if the field is directly approximated on the finite element mesh. Therefore, density data is approximated as a smooth field using a Moving Weighted Least Squares method. In this paper, the bone density field was generated from both bone adaptation analyses and from subject-specific geometry and material properties obtained from CT scans. It is important to note that the adoption of configurational mechanics avoids the need for post-processing, since configurational forces, and the fracture energy release rate, are expressed exclusively in terms of nodal quantities.
The constitutive model for bone adaptation included a bell function to define the rate of adaptation. This did not enforce rigid bounds on density levels, but merely slowed down the rate of convergence to biological equilibrium. This approach will be useful when trying fit model parameters to the actual density data form CT scans in defined periods of time. It is also possible to enforce bounds on density levels by introducing and calibrating mass influx in the mass balance equation [@sharma2013adaptive].
Numerical examples demonstrated the performance and accuracy of the proposed framework. Numerical convergence was demonstrated for all examples and the use of singularity elements was shown to further improve the rate of convergence. However, it was also confirmed that improved accuracy of the stress at the tip had no impact on the crack propagation analysis and the resulting crack path. The final example, demonstrated how mechanical loading and subsequent adaptation influence the resistance to bone fracture. Therefore, this framework will be a useful tool in understanding fractures in bone and ultimately preventing catastrophic fractures.
All analyses were undertaken using MoFEM [@mofem2017] that has been developed to support scalability and ensure robustness. The entire framework can be executed on parallel computer systems. Supplementary data (CT scans, mesh files, command lines) necessary to reproduce the results of all numerical examples can be found in [@karol_lewandowski_2019_dataset]. The bone adaptation and fracture mechanics are both submodules in the MoFEM library [@mofem2017], which can be installed using the flexible package manager, Spack [@spack2015].
|
---
author:
- |
[**Shuangjian Guo$^{1}$, Xiaohui Zhang$^{2}$, Shengxiang Wang$^{3}$** ]{}\
[1. School of Mathematics and Statistics, Guizhou University of Finance and Economics]{}\
[Guiyang 550025, P. R. of China]{}\
[2. School of Mathematical Sciences, Qufu Normal University]{}\
[Qufu 273165, P. R. of China]{}\
[3. School of Mathematics and Finance, Chuzhou University]{}\
[Chuzhou 239000, P. R. of China]{}
title: '**On split regular Hom-Leibniz-Rinehart algebras**'
---
****
ABSTRACT
In this paper, we introduce the notion of the Hom-Leibniz-Rinehart algebra as an algebraic analogue of Hom-Leibniz algebroid, and prove that such an arbitrary split regular Hom-Leibniz-Rinehart algebra $L$ is of the form $L=U+\sum_{\g}I_\g$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\g}$, a well described ideal of $L$, satisfying $[I_\g, I_\d]= 0$ if $[\g]\neq [\d]$. In the sequel, we develop techniques of connections of roots and weights for split Hom-Leibniz-Rinehart algebras respectively. Finally, we study the structures of tight split regular Hom-Leibniz-Rinehart algebras.
[**Key words**]{}: Hom-Leibniz-Rinehart algebra; root space; weight space; decomposition; simple ideal.
[**2010 Mathematics Subject Classification:**]{} 17A32; 17A60; 17B22; 17B60
INTRODUCTION {#introduction .unnumbered}
============
The notion of the Lie-Rinehart algebra plays an important role in many branches of mathematics. The idea of this notion goes back to the work of Jacobson to study certain field extensions. It was also appeared in some different names in several areas which includes differential geometry and differential Galois theory. In [@Mackenzie05], Mackenzie provided a list of 14 different terms mentioned for this notion. Huebschmann viewed Lie-Rinehart algebras as an algebraic counterpart of Lie algebroids defined over smooth manifolds. His work on several aspects of this algebra has been developed systematically through a series of articles namely [@Huebschmann90; @Huebschmann98; @Huebschmann99; @Huebschmann04] .
The notion of Hom-Lie algebras was first introduced by Hartwig, Larsson and Silvestrov in [@Hartwig], who developed an approach to deformations of the Witt and Virasoro algebras basing on $\sigma$-deformations. In fact, Hom-Lie algebras include Lie algebras as a subclass, but the deformation of Lie algebras twisted by a homomorphism.
Mandal and Mishra defined modules over a Hom-Lie-Rinehart algebra and studied a cohomology with coefficients in a left module. They presented the notion of extensions of Hom-Lie-Rinehart algebras and deduced a characterisation of low dimensional cohomology spaces in terms of the group of automorphisms of certain abelian extensions and the equivalence classes of those abelian extensions in the category of Hom-Lie-Rinehart algebras in [@Mandal2017]. The concept of a Hom-Lie-Rinehart algebra has a geometric analogue which is nowadays called a Hom-Lie algebroid in [@Cai17] and [@Laurent-Gengoux]. See also [@Castiglioni18; @Mandal2018; @Mandal17; @Mandal18; @zhang18] for other works on Hom-Lie-Rinehart algebras.
The class of the split algebras is specially related to addition quantum numbers, graded contractions and deformations. For instance, for a physical system which displays a symmetry, it is interesting to know the detailed structure of the split decomposition, since its roots can be seen as certain eigenvalues which are the additive quantum numbers characterizing the state of such system. Determining the structure of split algebras will become more and more meaningful in the area of research in mathematical physics. Recently, in [@Aragon2015]-[@Cao18], the structure of different classes of split algebras have been determined by the techniques of connections of roots. Recently, we studied the structures of split regular Hom-Lie Rinehart algebras in [@Wang19]. The purpose of this paper is to consider the structure of split regular Hom-Leibniz-Rinehart algebras by the techniques of connections of roots based on some work in [@Cao18] and [@Wang19] .
This paper is organized as follows. In Section 2, we introduce the notion of the Hom-Leibniz-Rinehart algebra and prove that such an arbitrary split regular Hom-Leibniz-Rinehart algebras $L$ is of the form $L=U+\sum_{\g}I_\g$ with $U$ a subspace of a maximal abelian subalgebra $H$ and any $I_{\g}$, a well described ideal of $L$, satisfying $[I_\g, I_\d]= 0$ if $[\g]\neq [\d]$. In Section 3 and Section 4, we develop techniques of connections of roots and weights for split Hom-Leibniz-Rinehart algebras respectively. In the last section, we study the structures of tight split regular Hom-Leibniz-Rinehart algebras.
Preliminaries
=============
Let $R$ denote a commutative ring with unity, $\mathbb{Z}$ the set of all integers and $\mathbb{N}$ the set of all nonnegative integers, all algebraic systems are considered of arbitrary dimension and over an arbitrary base field $\mathbb{K}$. And we recall some basic definitions and results related to our paper from [@Makhlouf10] and [@Mandal2017].
Given an associative commutative algebra $A$, an $A$-module $M$ and an algebra endomorphism $\phi: A \rightarrow A$, we call an $R$-linear map $\delta: A \rightarrow M$ a $\phi$-derivation of $A$ into $M$ if it satisfies the required identity: $$\begin{aligned}
\delta(ab)=\phi(a)\delta(b)+\phi(b)\delta(a), ~~~\mbox{for any $a, b\in A$}.\end{aligned}$$
Let us denote by $Der_{\phi}(A)$ the set of $\phi$-derivations of $A$ into itself.
A Hom-Leibniz algebra $L$ is an algebra $L$, endowed with a bilinear product $$\begin{aligned}
[\c,\c]: L\times L\rightarrow L,\end{aligned}$$ and a homomorphism $\psi: L\rightarrow L$ $$\begin{aligned}
&&[\psi(x), [y, z]]=[[x,y],\psi(z)]+[\psi(y), [x,z]]~~\mbox{(Hom-Leibniz~identity)}\end{aligned}$$ holds for any $x, y, z\in L$.
If $\psi$ is furthermore an algebra automorphism, that is, a linear bijective on such that $\psi([x,y])=[\psi(x),\psi(y)]$ for any $x,y\in L$, then $L$ is called a regular Hom-Leibniz algebra.
A Hom-Lie-Rinehart algebra over $(A, \phi)$ is a tuple $(A, L, [\c, \c], \phi, \psi, \rho)$, where $A$ is an associative commutative algebra, $L$ is an $A$-module, $[\c, \c] : L \times L \rightarrow L$ is a skew symmetric bilinear map, $\phi: A\rightarrow A$ is an algebra homomorphism, $\psi: L\rightarrow L$ is a linear map satisfying $\psi([x, y])=[\psi(x), \psi(y)]$, and the $R$-map $\rho: L\rightarrow Der_{\phi}(A)$ such that following conditions hold.\
(1) The triple $(L, [\c, \c], \psi)$ is a Hom-Lie algebra.\
(2) $\psi(a\c x)=\phi(a)\c \psi(x)$ for all $a\in A, x \in L$.\
(3) $(\rho, \phi)$ is a representation of $(L, [\c, \c], \psi)$ on $A$.\
(4) $\rho(a\c x)=\phi(a)\c \rho(x)$ for all $a\in A, x \in L$.\
(5) $[x, a\c y] = \phi(a)\c [x, y] + \rho(x)(a)\psi(y)$ for all $a\in A, x, y\in L$.
A Hom-Lie-Rinehart algebra $(A, L, [\c, \c], \phi, \psi, \rho)$ is said to be regular if the map $\phi: A\rightarrow A$ is an algebra automorphism and $\psi: L\rightarrow L$ is a bijective map.
Decomposition
=============
In this section, we introduce the notion of the Hom-Leibniz-Rinehart algebra as an algebraic analogue of Hom-Leibniz algebroid. In a sequel, we introduce the class of split algebras in the framework of Hom-Leibniz-Rinehart algebras.
A Hom-Leibniz-Rinehart algebra over $(A, \phi)$ is a tuple $(A, L, [\c, \c], \phi, \psi, \rho)$, where $A$ is an associative commutative algebra, $L$ is an $A$-module, $[\c, \c] : L\times L \rightarrow L$ is a skew symmetric linear map, $\phi: A\rightarrow A$ is an algebra homomorphism, $\psi: L\rightarrow L$ is a linear map satisfying $\psi([x, y])=[\psi(x), \psi(y)]$, and the $R$-map $\rho: L\rightarrow Der_{\phi}(A)$ satisfying the following conditions.\
(1) The triple $(L, [\c, \c], \psi)$ is a Hom-Leibniz algebra.\
(2) $\psi(a\c x)=\phi(a)\c \psi(x)$ for all $a\in A, x \in L$.\
(3) $(\rho, \phi)$ is a representation of $(L, [\c, \c], \psi)$ on $A$.\
(4) $\rho(a\c x)=\phi(a)\c \rho(x)$ for all $a\in A, x\in L$.\
(5) $[x, a\c y] = \phi(a)\c [x, y] + \rho(x)(a)\psi(y)$ for all $a\in A, x, y\in L$.
We denote it by $(L,A)$ or just by $L$ if there is not any possible confusion. A Hom-Leibniz-Rinehart algebra $(A, L, [\c, \c], \phi, \psi, \rho)$ is said to be regular if the map $\phi: A\rightarrow A$ is an algebra automorphism and $\psi: L\rightarrow L$ is a bijective map.
A Leibniz-Rinehart algebra $L$ over $A$ with the linear map $[\c, \c] : L\times L \rightarrow L$ and the $R$-map $\rho: L\rightarrow Der(A)$ is a Hom-Leibniz-Rinehart algebra $(A, L, [\c, \c], \phi, \psi, \rho)$, where $\psi=Id_L, \phi=Id_A$ and $\rho: L\times L\rightarrow Der_{\phi}(A)=Der(A)$.
A Hom-Leibniz algebra $(L, [\c, \c], \psi)$ structure over an $R$-module $L$ gives the Hom-Leibniz-Rinehart algebra $(A, L, [\c, \c], \phi, \psi, \rho)$ with $A=R$, the algebra morphism $\phi=id_{R}$ and the trivial action of $L$ on $R$.
If we consider a Leibniz-Rinehart algebra $L$ over $A$ along with an endomorphism $$\begin{aligned}
(\phi, \psi):(A, L)\rightarrow (A, L)\end{aligned}$$ in the category of Leibniz-Rinehart algebras, then we get a Hom-Leibniz-Rinehart algebra $(A, L, [\c, \c]_\psi, \phi, \psi, \rho_{\phi})$ as follows:\
(1) $[x, y]_{\psi}=\psi([x, y])$ for any $x, y\in L$.\
(2) $\rho_{\phi}(x)(a)=\phi(\rho(x)(a))$ for all $a\in A, x\in L$.
A Hom-Leibniz algebroid is a tuple $(\xi, [\c, \c], \phi, \psi, \rho)$, where $\xi: A\rightarrow M$ is a vector bundle over a smooth manifold $M$, $\phi: M\rightarrow M$ is a smooth map, $[-, -] : \Gamma(A)\times \Gamma(A) \rightarrow \Gamma(A)$ is a bilinear map, the map $\rho: \phi^{!}A\rightarrow \phi^{!}TM$ is called the anchor and $\psi:\Gamma(A) \rightarrow \Gamma(A)$ is a linear map such that following conditions are satisfied .\
(1) The triplet $(\Gamma(A), [-, -], \psi)$ is a Hom-Lie algebra.\
(2) $\psi(fX)=\phi^{\ast}(F)\psi(X)$ for all $X\in \Gamma(A), f \in C^{\infty}(M)$.\
(3) $(\rho, \phi^{\ast})$ is a representation of $(\Gamma(A), [-, -], \psi)$ on $C^{\infty}(M)$.\
(5) $[X, fY] = \phi^{\ast}(f) [X, Y] + \rho(X)[f]\psi(Y)$ for all $X, Y\in \Gamma(A), f\in C^{\infty}(M)$.
A Hom-Leibniz algebroid provides a Hom-Leibniz-Rinehart algebra $(C^{\infty}(M), \Gamma(A),$\
$[\c, \c], \phi^{\ast}, \psi, \rho)$, where $\Gamma(A)$ is the space a sections of the underline vector bundle $A\rightarrow M$ and $\phi^{\ast}: C^{\infty}(M)\rightarrow C^{\infty}(M)$ is canonically defined by the smooth map $\phi: M\rightarrow M$.
Let $(A, L, [\c, \c]_L, \phi, \psi_L, \rho_L)$ and $(A, M, [\c, \c]_M, \phi, \psi_M, \rho_M)$ be two Hom-Leibniz-Rinehart algebras over $(A, \phi)$. We consider $$\begin{aligned}
L\times_{Der_{\phi}A} M=\{(l, m)\in L\times M: \rho_L(l)=\rho_M(m)\}.\end{aligned}$$ Then $(A, L\times_{Der_{\phi}A} M, [\c, \c], \phi, \psi, \widetilde{\rho})$ is a Hom-Leibniz-Rinehart algebra, where\
(1) The linear bracket $[\c, \c]$ is given by $$\begin{aligned}
[(l_1,m_1),(l_2,m_2)]:=([l_1, l_2], [m_1, m_2]),\end{aligned}$$ for any $l_1,l_2 \in L$ and $m_1,m_2\in M$.\
(2) The map $\psi: L\times_{Der_{\phi}A} M\rightarrow L\times_{Der_{\phi}A} M$ is given by $$\begin{aligned}
\psi(l, m):=(\psi_L(l), \psi_M(m)),\end{aligned}$$ for any $l\in L$ and $m\in M$.\
(3) The action of $L\times_{Der_{\phi}A} M$ on $A$ is given by $$\begin{aligned}
\widetilde{\rho}(l, m)(a):=\rho_L(l)(a)=\rho_M(m)(a),\end{aligned}$$ for any $l\in L, m\in M$ and $a\in A$.
Next we define homomorphisms of Hom-Leibniz-Rinehart algebras.
Let $(A, L, [\c, \c]_L, \phi, \psi_L, \rho_L)$ and $(B, L', [\c, \c]_{L'}, \psi, \psi_{L'}, \rho_{L'})$ be two Hom-Leibniz-Rinehart algebras, then a Hom-Leibniz-Rinehart algebra homomorphism is defined as a pair of maps $(g, f)$, where the map $g: A \rightarrow B$ is an $R$-algebra homomorphism and $f: L\rightarrow L'$ is an $R$-linear map such that following identities hold:\
(1) $f(a\c x) =g(a)\c f(x)$, for all $x\in L$ and $a\in A$.\
(2) $f([x, y]_L)=[f(x), f(y)]_{L'}$, for all $x, y\in L$.\
(3) $f(\psi_L(x))=\psi_{L'}(f(x))$, for all $x\in L$.\
(4) $g(\phi(a))=\psi(g(a))$, for all $a\in A$.\
(5) $g(\rho_L(x)(a))=\rho_{L'}(f(x))(g(a))$, for all $x\in L$ and $a\in A$.
A subalgebra $(S,A)$ of $(L,A)$ is called a *Hom-Leibniz subalgebra*, if $(S,A)$ satisfies $AS\subset S$ such that $S$ acts on $A$ via the composition $S\hookrightarrow L\rightarrow Der_{\phi} (A)$. A Hom-Leibniz subalgebra $(I,A)$ of $(L, A)$ is called an *ideal*, if $I$ is a Hom-Leibniz ideal of $L$ such that $\rho(I)(A)L\subseteq I.$
The ideal $J$ generated by $$\begin{aligned}
\{[x, y]+[y, x]: x, y\in L\}\end{aligned}$$ plays an important role in mathematics since it determines the non-super Lie character of $L$. From Hom-Leibniz identity, it is straightforward to check that this ideal satisfies $$\begin{aligned}
[L, J]=0.\end{aligned}$$
Let us introduce the class of split algebras in the framework of Hom-Leibniz algebras from [@Cao18]. Denote by $H$ a maximal abelian subalgebra of a Hom-Leibniz algebra $L$. For a linear functional $$\begin{aligned}
\g:H\rightarrow \mathbb{K},\end{aligned}$$ we define the root space of $L$ associated to $\g$ as the subspace $$\begin{aligned}
L_{\g}:=\{v_{\a}\in L:[h,\psi(v_{\g})]=\a(h)\psi(v_{\g}), \mbox{for any $h\in H$}\}.\end{aligned}$$ The elements $\g:H\rightarrow \mathbb{K}$ satisfying $L_{\g}\neq 0$ are called roots of $L$ with respect to $H$ and we denote $\Gamma:=\{\g\in H^{\ast}\setminus\{0\}:L_{\g}\neq 0\}$. We call that $L$ is a split regular Hom-Leibniz algebra with respect to $H$ if $$\begin{aligned}
L=H\oplus \bigoplus_{\g\in \Gamma}L_{\g}.\end{aligned}$$ We also say that $\Gamma$ is the root system of $L$.
A *split regular Hom-Leibniz-Rinehart algebra* (with respect to a MASA $H$ of the regular Hom-Leibniz algebra $L$, here MASA means maximal abelian subalgebra) is a regular Hom-Leibniz-Rinehart algebra $(L,A)$ in which the Hom-Leibniz algebra $L$ contains a splitting Cartan subalgebra $H$ and the algebra $A $ is a weight module (with respect to $H$) in the sense that $A$ can be written as the direct sum $A=A_0\oplus (\bigoplus_{\alpha\in \Lambda}A_{\alpha})$ with $\phi(A_\alpha)\subset A_\alpha$, where $$\begin{aligned}
A_{\alpha}:=\{a_{\alpha}\in A|\rho(h)(a_{\alpha})=\alpha(h)\phi(a_{\alpha}),~\forall h\in H\},\end{aligned}$$ for a linear functional $\alpha\in H^{\ast}$ and $\Lambda:=\{\alpha\in H^{\ast}\backslash\{0\}: A_\a\neq 0\}$ denotes the weights system of $A$. The linear subspace $ A_{\alpha}$ , for $\alpha\in \Lambda$, is called the *weight space of $A$ associate to $\alpha$*, the element $\alpha\in \Lambda\cup \{0\}$ are called *weights* of $A$.
Let $(L,A)$ be a Leibniz-Rinehart algebra, where $L=H\oplus (\bigoplus_{\a\in \Gamma}L_{\g}), ~A=A_0\oplus (\bigoplus_{\alpha\in \Lambda}A_{\alpha})$ , $\psi: L\rightarrow L, \phi: A\rightarrow A$ are two automorphisms such that $\psi(H)=H, \phi(A_0)=A_0$ and $\phi(A_\alpha)\subset A_\alpha$. By Example 2.3, we know that $(L, A)$ is a regular Hom-Leibniz-Rinehart algebra. Then we have $$\begin{aligned}
L=H\oplus (\bigoplus_{\a\in \Gamma}\mathfrak{L}_{\a\psi^{-1}}),~~~ A=A_0\oplus (\bigoplus_{\alpha\in \Lambda}A_{\alpha\phi^{-1}}),\end{aligned}$$ which make the regular Hom-Leibniz-Rinehart algebra $(L, A)$ being the roots system $\Gamma'=\{\a\psi^{-1}:\a\in \Gamma\}$ and weights system $\Lambda':=\{\alpha\phi^{-1}:\a\in \Lambda\}$.
The following lemma is analogous to the results of [@Wang19].
For any $ \gamma,\xi\in\Gamma\cup \{0\}$ and $ \alpha,\beta\in\Lambda\cup \{0\}$, the following assertions hold.
\(1) $L_0=H.$
\(2) $\psi(L_{\g})=L_{\g\psi^{-1}}$ and $\psi^{-1}(L_{\g})=L_{\g\psi}$.
\(3) If $[L_{\gamma},L_{\xi}]\neq 0$, then $ \gamma\psi^{-1}+\xi\psi^{-1}\in\Gamma\cup \{0\}$ and $[L_{\gamma},L_{\xi}]\subset L_{\gamma\psi^{-1}+\xi\psi^{-1}}$.
\(4) If $A_{\alpha}A_{\beta}\neq 0$, then $\alpha+\beta\in\Lambda\cup \{0\}$ and $A_{\alpha}A_{\beta}\subset A_{\alpha+\beta}$.
\(5) If $A_{\alpha}L_{\gamma}\neq 0$, then $\alpha+\gamma\in\Gamma\cup \{0\}$ and $A_{\alpha}L_{\gamma}\subset L_{\alpha+\gamma}$.
\(6) If $\rho(L_{\gamma})A_{\alpha}\neq 0$, then $\alpha+\gamma\in\Lambda\cup \{0\}$ and $\rho(L_{\gamma})A_{\alpha}\subset A_{\alpha+\gamma}$.
Connections of roots
====================
In what follows, $L$ denotes a split regular Hom-Leibniz-Rinehart algebra and $$\begin{aligned}
L=H\oplus (\bigoplus_{\g\in \Gamma}L_{\g}), \ \ \ A=A_0\oplus (\bigoplus_{\alpha\in \Lambda}A_{\alpha}).\end{aligned}$$ Given a linear functional $\gamma: H\rightarrow \mathbb{K}$, we denote by $-\gamma: H\rightarrow \mathbb{K}$ the element in $H^{\ast}$ defined by $(-\gamma)(h):=-\gamma(h)$ for all $h\in H.$ We also denote $-\Gamma:=\{-\gamma: \gamma\in \Gamma\}$. In a similar way we can define $-\Lambda:=\{-\alpha: \alpha\in\Lambda\}$. Finally, we denote $\pm\Gamma:=\Gamma\cup -\Gamma $ and $\pm\Lambda:=\Lambda\cup -\Lambda.$
Let $\gamma,\xi\in \Gamma$, we say that $\gamma$ is connected to $\xi$ if
$\bullet$ Either $\xi=\epsilon\gamma\psi^{z}$ for some $z\in \mathbb{Z}$ and $\epsilon\in \{1,-1\}$.
$\bullet$ Either there exists a family $\{\zeta_1,\zeta_2,\ldots,\zeta_n\}\subset\pm\Lambda\cup\pm\Gamma$, with $n\geq 2$, such that
\(1) $\zeta_1\in\{\gamma\psi^{k}|k\in \mathbb{Z}\}$.
\(2) $\zeta_1\psi^{-1}+\zeta_2\psi^{-1}\in\pm\Gamma$,
$~~~~~\zeta_1\psi^{-2}+\zeta_2\psi^{-2}+\zeta_3\psi^{-1}\in\pm\Gamma$,
$~~~~~\zeta_1\psi^{-3}+\zeta_2\psi^{-3}+\zeta_3\psi^{-2}+\zeta_4\psi^{-1}\in\pm\Gamma$,
$~~~~~~~~~~\cdots\cdots\cdots$
$~~~~~\zeta_1\psi^{-i}+\zeta_2\psi^{-i}+\zeta_3\psi^{-i+1}+\cdots+\zeta_{i+1}\psi^{-1}\in\pm\Gamma$,
$~~~~~~~~~~\cdots\cdots\cdots$
$~~~~~\zeta_1\psi^{-n+2}+\zeta_2\psi^{-n+2}+\zeta_3\psi^{-n+3}+\cdots+\zeta_{n-1}\psi^{-1}\in\pm\Gamma$.
\(3) $\zeta_1\psi^{-n+1}+\zeta_2\psi^{-n+1}+\zeta_3\psi^{-n+2}+\cdots+\zeta_{n}\psi^{-1}\in\{\pm\xi\psi^{-m}|m\in \mathbb{Z}\}$.
We will also say that $\{\zeta_1,\zeta_2,\ldots,\zeta_n\}$ is a *connection from $\gamma$ to $\xi$*.
The proof of the next result is analogous to the one of [@Cao18].
The relation $\sim$ in $\Gamma$ is an equivalence relation, where $\gamma\sim\xi$ if and only if $\gamma$ is connected to $\xi$.
By Proposition 3.2 we can consider the quotient set $$\begin{aligned}
\Gamma/\sim=\{[\g]:\g\in \Gamma\},
\end{aligned}$$ with $[\g]$ being the set of nonzero roots which are connected to $\g$. Our next goal is to associate an ideal $I_{[\g]}$ to $[\g]$. Fix $[\g]\in \Gamma/\sim$, we start by defining $$\begin{aligned}
L_{0,[\g]}:=(\bigoplus_{\xi\in [\g], -\xi\in \Lambda}A_{-\xi}L_{\xi})+(\bigoplus_{\xi\in [\g]}[L_{-\xi}, L_{\xi}]).\end{aligned}$$ Now we define $$\begin{aligned}
L_{[\g]}:=\bigoplus_{\xi\in [\gamma]}L_{\xi}.\end{aligned}$$ Finally, we denote by $I_{[\gamma]}$ the direct sum of the two subspaces above: $$\begin{aligned}
I_{[\g]}:=L_{0,[\g]}\oplus L_{[\g]}.\end{aligned}$$
For any $[\g]\in \Lambda/\sim$, the following assertions hold.
\(1) $[I_{[\g]},I_{[\g]}]\subset I_{[\g]}$.
\(2) $\psi(I_{[\g]})=I_{[\g]}$.
\(3) $AI_{[\g]}\subset I_{[\g]}$.
\(4) $\rho(I_{[\g]})(A)L\subset I_{[\g]}$.
\(5) For any $[\g]\neq [\delta]$, we have $[I_{[\g]}, I_{[\delta]}]=0$.
[**Proof.**]{} (1) First we check that $[I_{[\g]},I_{[\g]}]\subset I_{[\g]}$, we can write $$\begin{aligned}
[I_{[\g]},I_{[\g]}]&=&[L_{0,[\g]}\oplus L_{[\g]},L_{0,[\g]}\oplus L_{[\g]}]\nonumber\\
&\subset&[L_{0,[\g]}, L_{[\g]}]+[L_{[\g]}, L_{0,[\g]}]+[L_{[\g]}, L_{[\g]}].\end{aligned}$$ Given $\delta\in [\g]$, we have $[L_{0,[\g]}, L_{\delta}]\subset L_{\delta}\subset L_{[\g]}$. By a similar argument, we get $[L_{\delta}, L_{0,[\g]}]\subset L_{[\g]}$.
Next we consider $[L_{[\g]},L_{[\g]}]$. If we take $\delta,\eta\in [\g]$ such that $[L_{\delta}, L_{\eta}]\neq 0$, then $[L_{\delta}, L_{\eta}]\subset L_{\delta+\eta}$. If $\delta\psi^{-1}+\eta\psi^{-1}=0$, we get $[L_{\delta}, L_{-\delta}]\subset L_{0,[\g]}$ . Suppose that $\delta\psi^{-1}+\eta\psi^{-1}\in \Gamma$. We infer that $\{\delta,\eta\}$ is a connection from $\delta$ to $\delta\psi^{-1}+\eta\psi^{-1}$. The transitivity of $\sim$ now gives that $\delta\psi^{-1}+\eta\psi^{-1}\in [\g]$ and so $[L_{\delta}, L_{\eta}]\subset L_{[\g]}$. Hence $$\begin{aligned}
[L_{[\g]},L_{[\g]}]\in I_{[\g]}.\end{aligned}$$ From (3.1) and (3.2), we get $[I_{[\g]},I_{[\g]}]\subset I_{[\g]}$.
\(2) It is easy to check that $\psi(I_{[\g]})=I_{[\g]}$.
\(3) and (4) similar to [@Wang19].
\(5) We will study the expression $[I_{[\g]},I_{[\delta]}]$. Notice that $$\begin{aligned}
[I_{[\g]},I_{[\delta]}]&=&[L_{0,[\g]}\oplus L_{[\g]},L_{0,[\delta]}\oplus L_{[\delta]}]\nonumber\\
&\subset&[L_{0,[\g]}, L_{[\delta]}]+[L_{[\g]}, L_{0,[\delta]}]+[L_{[\g]}, L_{[\delta]}].\end{aligned}$$
First we consider $[L_{[\gamma]}, L_{[\delta]}]$ and suppose that there exist $\g_1\in [\g], \d_1\in [\d]$ such that $[L_{\g_1}, L_{\d_1}]\neq 0$. As necessarily $\g_1\psi^{-1}\neq-\d_1\psi^{-1}$, then $\g_1\psi^{-1}+\d_1\psi^{-1}\in \Gamma$. So $\{\g_1,\d_1, -\g_1\psi^{-1}\}$ is a connection between $\g_1$ and $\d_1$. By the transitivity of the connection relation we see $\g\in [\d]$, a contradicition. Hence $[L_{\g_1}, L_{\d_1}]=0$, and so $$\begin{aligned}
[L_{[\gamma]}, L_{[\delta]}]=0.\end{aligned}$$ By the definition of $L_{0,[\gamma]}$, we have $$[L_{0,[\gamma]},L_{[\delta]}]=[(\sum_{\gamma_1\in [\gamma],-\gamma_1\in\Lambda}A_{-\gamma_1}L_{\gamma_1})
+(\sum_{\gamma_{1}\in [\gamma]}[L_{-\gamma_1},L_{\gamma_1}]),L_{[\delta]}].$$
Suppose that there exist $\g_1\in [\g]$ and $\d_1\in [\d]$ such that $$\begin{aligned}
[L_{\d_1}, [L_{\g_1}, L_{-\g_1}]]= 0.\end{aligned}$$ Suppose that $
[L_{\d_1}, [L_{\g_1}, L_{-\g_1}]]\neq 0,
$ then Hom-Leibniz identity gives $$\begin{aligned}
0&\neq& [ \psi\psi^{-1}(L_{\d_1}), [L_{\g_1}, L_{-\g_1}]]\\
&\subset&[[\psi^{-1}(L_{\d_1}), L_{\g_1}], \psi(L_{-\g_1})]+[[\psi^{-1}(L_{\d_1}), L_{-\g_1}], \psi(L_{\g_1})].\end{aligned}$$ Hence $$\begin{aligned}
[\psi^{-1}(L_{\d_1}), L_{\g_1}]+[\psi^{-1}(L_{\d_1}), L_{-\g_1}]\neq 0,\end{aligned}$$ which contradicts (3.4). Therefore, $[L_{\d_1}, [L_{\g_1}, L_{-\g_1}]]=0$.
For the expression $[A_{-\gamma_1}L_{\gamma_1},L_{[\delta]}]$, suppose there exists $\delta_1\in [\delta]$ such that $[A_{-\gamma_1}L_{\gamma_1},L_{\delta_1}]\neq 0$. By Definition 2.1, we have $$[A_{-\gamma_1}L_{\gamma_1},L_{\delta_1}]
=[L_{\delta_1},A_{-\gamma_1}L_{\gamma_1}]
\subset \phi(A_{-\gamma_1})[L_{\delta_1},L_{\gamma_1}]+\rho(L_{\delta_1})(A_{-\gamma_1})L_{\delta_{1}\psi^{-1}}.$$ By the discussion above, we get $[L_{\delta_1},L_{\gamma_1}]=0$. Since $[A_{-\gamma_1}L_{\gamma_1},L_{[\delta]}]\neq 0$, it follows that $0 \neq\rho(L_{\delta_1})(A_{-\gamma_1})L_{\delta_{1}\psi^{-1}}\subset A_{\delta_{1}-\gamma_{1}}L_{\delta_{1}\psi^{-1}}$. Thus $A_{\delta_{1}-\gamma_{1}}\neq 0$ and $\delta_{1}-\gamma_{1}\in \Lambda\cup\{0\}$. $\delta_{1}\sim\gamma_{1}$, a contradiction. So $[A_{-\gamma_1}L_{\gamma_1},L_{[\delta]}]=0$. Therefore, we have $[L_{0,[\gamma]},L_{[\delta]}]=0$. In a similar way we can prove $[L_{[\g]}, L_{0,[\delta]}]=0$, we conclude $[I_{[\g]},I_{[\delta]}]=0$. $\square$
A Hom-Leibniz-Rinehart algebra $(L,A)$ is *simple* if $[L,L]\neq 0,AA\neq 0,AL\neq 0$ and its only ideals are $\{0\}, J, L $ and the kernel of $\rho$.
The following assertions hold.
\(1) For any $[\g]\in \Gamma/\sim$, the linear space $I_{[\g]}=L_{0,[\g]}+L_{[\g]}$ of $L$ associated to $[\g]$ is an ideal of $L$.
\(2) If $L$ is simple, then there exists a connection from $\g$ to $\delta$ for any $\g,\delta\in \Gamma$ and $$\begin{aligned}
H=(\sum_{\gamma\in \Gamma,-\gamma\in\Lambda}A_{-\gamma}L_{\gamma})+(\sum_{\gamma\in \Gamma}[L_{-\gamma},L_{\gamma}]).\end{aligned}$$ .
[**Proof.**]{} (1) Since $[I_{[\g]},H]+[H, I_{[\g]}]\subset I_{[\g]}$, by Proposition 3.3, we have $$\begin{aligned}
[I_{[\g]}, L]=[I_{[\g]}, H\oplus(\bigoplus_{\xi\in [\g]} L_{\xi})\oplus (\bigoplus_{\delta\notin [\g]} L_{\delta})]\subset I_{[\g]}\end{aligned}$$ and $$\begin{aligned}
[L, I_{[\g]},]=[ H\oplus(\bigoplus_{\xi\in [\g]} L_{\xi})\oplus (\bigoplus_{\delta\notin [\g]} L_{\delta}), I_{[\g]}]\subset I_{[\g]}.\end{aligned}$$ Furthermore, $$\begin{aligned}
[I_{[\g]}, L]+[L, I_{[\g]}]
=[I_{[\g]}, H\oplus(\bigoplus_{\xi\in [\g]} L_{\xi})\oplus (\bigoplus_{\delta\notin [\g]} L_{\delta})]
+[H\oplus(\bigoplus_{\xi\in [\g]} L_{\xi})\oplus (\bigoplus_{\delta\notin [\g]} L_{\delta}), I_{[\g]}]\subset I_{[\g]}.\end{aligned}$$ As we also have $\psi(I_{[\g]})=I_{[\g]}$. So we show that $I_{[\gamma]}$ is a Hom-Leibniz ideal of $L$, we also have that $I_{[\gamma]}$ is an $A$-module, the we conclude $I_{[\gamma]}$ is an ideal of $L$.
\(2) The simplicity of $L$ implies $I_{[\g]}\in \{J, L$, ker$\rho\}$. If some $\g \in \Gamma$ is such that $I_{[\g]}=L$, then $[\g]=\Gamma$. Otherwise, if $I_{[\g]}=J$ for any $\a \in \Gamma$, then $[\g]=[\delta]$ for any $\g,\delta\in \Gamma$, and so $[\g]=\Gamma$. Otherwise, if $I_{[\g]}= ker \rho$ for any $\g \in \Gamma$, then $[\g]=[\delta]$ for any $\g,\delta\in \Gamma$, and so $[\g]=\Gamma$. Thus $H=(\sum_{\gamma\in \Gamma,-\gamma\in\Lambda}A_{-\gamma}L_{\gamma})+(\sum_{\gamma\in \Gamma}[L_{-\gamma},L_{\gamma}])$. $\square$
We have $$\begin{aligned}
L=U+\sum_{[\g]\in \Lambda/\sim}I_{[\g]},\end{aligned}$$ where $U$ is a linear complement in $H$ of $(\sum_{\gamma\in \Gamma,-\gamma\in\Lambda}A_{-\gamma}L_{\gamma})+(\sum_{\gamma\in \Gamma}[L_{-\gamma},L_{\gamma}])$ and any $I_{[\g]}$ is one of the ideals of $L$ described in Theorem 3.5, satisfying $[I_{[\g]},I_{[\delta]}]=0$ if $[\g]\neq[\delta]$.
[**Proof.**]{} $I_{[\g]}$ is well defined and is an ideal of $L$ and it is clear that $$\begin{aligned}
L=H\oplus\sum_{[\g]\in \Gamma} L_{[\g]}=U+\sum_{[\g]\in \Gamma/\sim}I_{[\g]}.
\end{aligned}$$ Finally, Proposition 3.3 gives us $[I_{[\g]},I_{[\delta]}]=0$ if $[\g]\neq[\delta]$.$\square$
The annihilator of a Hom-Leibniz-Rinehart algebra $L$ is the set $$\begin{aligned}
Z(L):=\{v\in L:[v,L]+[L, v]=0\mbox{~and~} \rho(v)=0\}.\end{aligned}$$
If $Z(L)=0$ and $H=(\sum_{\gamma\in \Gamma,-\gamma\in\Lambda}A_{-\gamma}L_{\gamma})+(\sum_{\gamma\in \Gamma}[L_{-\gamma},L_{\gamma}])$. Then $L$ is the direct sum of the ideals given in Theorem 3.6, $$\begin{aligned}
L=\bigoplus_{[\g]\in \Gamma/\sim}I_{[\g]},\end{aligned}$$ Furthermore, $[I_{[\g]},I_{[\delta]}]=0$ if $[\g]\neq[\delta]$.$\square$.
[**Proof.**]{} Since $H=(\sum_{\gamma\in \Gamma,-\gamma\in\Lambda}A_{-\gamma}L_{\gamma})+(\sum_{\gamma\in \Gamma}[L_{-\gamma},L_{\gamma}])$, it follows that $L=\sum_{[\g]\in \Gamma/\sim}I_{[\g]}$. To verify the direct character of the sum, take some $v\in I_{[\g]}\cap(\sum_{[\delta]\in\Gamma/\sim,[\delta]\neq[\g]}I_{[\delta]})$. Since $v\in I_{[\g]}$, the fact $[I_{[\g]},I_{[\delta]}]=0$ when $[\g]\neq[\delta]$ gives us $$\begin{aligned}
[v, \sum_{[\delta]\in\Gamma/\sim,[\delta]\neq[\g]}I_{[\delta]}]+[\sum_{[\delta]\in\Gamma/\sim,[\delta]\neq[\g]}I_{[\delta]},v]=0.\end{aligned}$$ In a similar way, $v\in \sum_{[\delta]\in\Gamma/\sim,[\delta]\neq[\g]}I_{[\delta]}$ implies $[v,I_{[\g]}]+[I_{[\g]}, v]=0$. It is easy to obtain that $ \rho(v)=0$. That is $v\in Z(L)$ and so $v=0$. $\square$
Decompositions of $A$
=====================
We will discuss the weight spaces and decompositions of $A$ similar to [@Wang19] and omit the proof.
Let $\alpha,\beta\in \Lambda$ we say that $\alpha$ and $\beta$ are *connected* if
$\bullet$ either $\beta=\varepsilon\alpha$ for some $\varepsilon\in \{1,-1\}$;
$\bullet$ either there exists a family $\{\sigma_1,\sigma_2,\ldots,\sigma_n\}\subset\pm\Lambda\cup\pm\Gamma$, with $n\geq 2$, such that
\(1) $\sigma_1=\alpha$.
\(2) $\sigma_1+\sigma_2\in\pm\Lambda\cup\pm\Gamma$,
$~~~~~\sigma_1+\sigma_2+\sigma_3\in\pm\Lambda\cup\pm\Gamma$,
$~~~~~~~~~~\cdots\cdots\cdots$
$~~~~~\sigma_1+\sigma_2\cdots+\sigma_{n-1}\in\pm\Lambda\cup\pm\Gamma$,
\(3) $\sigma_1+\sigma_2\cdots+\sigma_{n}\in\{\beta,-\beta\}$.
We will also say that $\{\sigma_1,\sigma_2,\ldots,\sigma_n\}$ is a *connection* from $\alpha$ to $\beta$.
The relation $\approx$ in $\Lambda$ is an equivalence relation, where $\alpha\approx\beta$ if and only if $\alpha$ is connected to $\beta$.
By Proposition 4.2, we can consider the quotient set $$\Lambda/\approx:=\{[\alpha]|\alpha\in \Lambda\},$$ where $[\alpha]$ denotes the set of nonzero weights of $A$ which are connected to $\alpha$. In the following we will associate an adequate ideal $\mathcal{A}_{[\alpha]}$ to any $[\alpha]$. For a fixed $\alpha\in \Lambda$, we define $$\begin{aligned}
A_{0,[\alpha]}:=\bigl(\sum_{\beta\in [\alpha],-\beta\in\Lambda}\rho(L_{-\beta})(A_{\beta})
+(\sum_{\beta\in [\alpha]}A_{-\beta},A_{\beta})\bigr)\subset A_0,~~
A_{[\alpha]}:=\bigoplus_{\beta\in [\alpha]}A_{\beta}.
\end{aligned}$$ Then we denote by $\mathcal{A}_{[\alpha]}$ the direct sum of the two subspaces above, $$\begin{aligned}
\mathcal{A}_{[\alpha]}:=A_{0,[\alpha]}\oplus A_{[\alpha]}.
\end{aligned}$$
For any $\alpha,\beta\in \Lambda$, the following assertions hold.
\(1) $\mathcal{A}_{[\alpha]}\mathcal{A}_{[\alpha]}\subset \mathcal{A}_{[\alpha]}$.
\(2) If $[\alpha]\neq[\beta]$, then $\mathcal{A}_{[\alpha]}\mathcal{A}_{[\beta]}=0$.
Let $A$ be a commutative and associative algebra associated to a Hom-Leibniz-Rinehart algebra $L$. Then the following assertions hold.
\(1) For any $[\alpha]\in\Lambda/\approx$, the linear space $\mathcal{A}_{[\alpha]}=A_{0,[\alpha]}\oplus A_{[\alpha]}$ of $A$ associated to $[\alpha]$ is an ideal of $A$.
\(2) If $A$ is simple, then all weights of $\Lambda$ are connected. Furthermore, $$A_{0}=\sum_{-\alpha\in \Gamma,\alpha\in\Lambda}\rho(L_{-\alpha})(A_{\alpha})
+(\sum_{\alpha\in \Lambda}A_{-\alpha}A_{\alpha}).$$
Let $A$ be a commutative and associative algebra associated to a Hom-Leibniz-Rinehart algebra $L$. Then $$\begin{aligned}
A=V+\sum_{[\alpha]\in\Lambda/\approx}\mathcal{A}_{[\alpha]},\end{aligned}$$ where $V$ is a linear complement in $A_0$ of $\sum_{-\alpha\in \Gamma,\alpha\in\Lambda}\rho(L_{-\alpha})(A_{\alpha})
+(\sum_{\alpha\in \Lambda}A_{-\alpha}A_{\alpha})$ and any $A_{[\alpha]}$ is one of the ideals of $A$ described in Theorem 4.4-(1), satisfying $\mathcal{A}_{[\alpha]}\mathcal{A}_{[\beta]}=0$, whenever $[\alpha]\neq[\beta]$.
Let us denote by $Z(A)$ the center of $A$, that is, $Z(A):=\{a\in L| aA=0\}.$
Let $(L,A)$ be a Hom-Leibniz-Rinehart algebra. If $Z(A)=0$ and $$A_{0}=\sum_{-\alpha\in \Gamma,\alpha\in\Lambda}\rho(L_{-\alpha})(A_{\alpha})
+(\sum_{\alpha\in \Lambda}A_{-\alpha}A_{\alpha}),$$ then $A$ is the direct sum of the ideals given in Theorem 4.5, that is, $$\begin{aligned}
A=\sum_{[\alpha]\in\Lambda/\approx}\mathcal{A}_{[\alpha]},\end{aligned}$$ satisfying $\mathcal{A}_{[\alpha]}\mathcal{A}_{[\beta]}=0$, whenever $[\alpha]\neq[\beta]$.
The simple components
=====================
In this section we focus on the simplicity of split regular Hom-Leibniz-Rinehart algebra $(L,A)$ by centering our attention in those of maximal length. From now on we always assume that $\Lambda$ is symmetric in the sense that $\Lambda=-\Lambda$.
Let $(L,A)$ be a split regular Hom-Leibniz-Rinehart and $I$ an ideal of $L$. Then $I=(I\cap H)\oplus(I\cap \bigoplus_{\gamma\in\Gamma}L_\gamma)$.
[**Proof.**]{} Since $(L,A)$ is split, we get $L=H\oplus(\bigoplus_{\gamma\in\Gamma}L_\gamma)$. By the assumption that $I$ is an ideal of $L$, it is clear that $I$ is a submodule of $L$. Since a submodule of a weight module is again a weight module. Thus $I$ is a weight module and therefore $I=(I\cap H)\oplus(I\cap \bigoplus_{\gamma\in\Gamma}L_\gamma)$.$\hfill \Box$
Let $(L,A)$ be a split regular Hom-Leibniz-Rinehart algebra with $Z(L)=0$ and $I$ an ideal of $L$. If $I \subset H$, then $I=\{0\}$.
[**Proof.**]{} Since $I \subset H$, $[I,H]+[H,I]\subset [H,H]=0$. It follows that $[I,L]+[L, I]=[I,\bigoplus_{\gamma\in\Gamma}L_\gamma]+[\bigoplus_{\gamma\in\Gamma}L_\gamma, I]\subset H\cap (\bigoplus_{\gamma\in\Gamma}L_\gamma)=0$. So $I \subset Z(L)=0$. $\hfill \Box$
Observe that if $L$ is of maximal length, then we have $$\begin{aligned}
I=(I\cap H)\oplus (\bigoplus_{\g\in\Gamma^{I}} L_{\g})),\end{aligned}$$ where $\Gamma^{I}=\{\g\in \Gamma: I\cap L_{\g}\neq 0\}$.
In particular, in case $I=J$, we get $$\begin{aligned}
J=(J\cap H)\oplus (\bigoplus_{\g\in \Gamma^{J}} L_{\g})\end{aligned}$$ with $\Gamma^{J}=\{\g\in \Gamma: J\cap L_{\g}\neq 0\}=\{\g\in \Gamma: 0\neq L_{\g}\subset J\}$.
From here, we can write $$\begin{aligned}
\Gamma=\Gamma^{J}\cup \Gamma^{\neg J},\end{aligned}$$ where $
\Gamma^{\neg J}=\{\g\in \Gamma: L_{\g}\neq 0 ~~\mbox{and}~~ J\cap L_{\g}= 0\}.
$ Therefore, we can write $$\begin{aligned}
L=H\oplus (\bigoplus_{\g\in \Gamma^{J}} L_{\g})\oplus (\bigoplus_{\delta\in \Gamma^{\neg J}} L_{\delta}).
\end{aligned}$$
Let us introduce the notion of root-multiplicativity in the framework of split regular Hom-Leibniz-Rinehart algebras of maximal length, in a similar way to the ones for split regular Hom-Lie Rinehart algebras in [@Wang19].
A split regular Hom-Leibniz-Rinehart algebra $(L,A)$ is called *root-multiplicative* if the following conditions hold:
\(1) Given $\gamma,\delta\in\Gamma^{\neg J}$ such that $\gamma\psi^{-1}+\delta\psi^{-1}\in\Gamma$, then $[L_{\gamma},L_{\delta}]\neq 0.$
\(2) Given $\gamma \in\Gamma^{J}, \delta\in\Gamma^{\neg J}$ such that $\gamma\psi^{-1}+\delta\psi^{-1}\in \Gamma^{J}$, then $[L_{\gamma},L_{\delta}]\neq 0.$
\(3) Given $\a\in \Lambda, \gamma \in\Gamma$ such that $\alpha+\gamma\in\Gamma$, then $A_\alpha L_{\gamma}\neq 0.$
\(4) If $\alpha+\beta\in\Lambda$, then $A_\alpha A_\beta\neq 0.$
A split regular Hom-Leibniz-Rinehart $(L,A)$ is called *of maximal length* if dim $L_{\gamma}$=dim $A_\alpha=1$ for any $\gamma\in\Gamma$ and $\alpha\in\Lambda$.
Suppose $ H=(\sum_{\gamma\in\Gamma^{\neg J},-\gamma\in\Lambda}A_{-\gamma}L_{\gamma})+(\sum_{\gamma\in \Gamma^{\neg J}}[L_{-\gamma},L_{\gamma}])$, $Z_{Lie}(L)=0$ and root-multiplicative. If $\Gamma^{\neg J}$ has all of its roots $\neg J$-connected, then any ideal $I$ of $L$ such that $I\nsubseteq H\oplus J$, then $I=L$.
[**Proof.**]{} By (5.1) and (5.3), we can write $$\begin{aligned}
I=(I\cap H)\oplus (\bigoplus_{\g\in \Gamma^{\neg J,I}} L_{\g}),\end{aligned}$$ where $\Gamma^{\neg J,I}=\Gamma^{\neg J}\cap \Gamma^{I}$ and $\Gamma^{J,I}=\Gamma^{J}\cap \Gamma^{I}$. Since $I\nsubseteq H\oplus J$, there exists $\g_0\in \Gamma^{\neg J}$ such that $$\begin{aligned}
0\neq L_{\g_0} \subset I.\end{aligned}$$ By Lemma 2.11, $\psi(L_{\g_0})=L_{\g_0\psi^{-1}}$. Equation(5.5) gives us $\psi(L_{\g_0}) \subset \psi(I)=I$. So $L_{\g_0\psi^{-1}}\subset I$. Similarly we get $$\begin{aligned}
L_{\g_0\psi^{-n}}\subset I, ~~\mbox{for}~ n\in \mathbb{N}.\end{aligned}$$ For any $\delta\in \Gamma^{\neg J}$, $\b\notin \pm \g_0\psi^{-n}$, for $n\in \mathbb{N}$, the fact that $\g_0$ and $\g$ are $\neg J$-connected gives us a $\neg J$-connection $\{\g_1,\g_2,...,\g_n\}\subset \Gamma^{\neg J}$ from $\g_0$ to $\delta$ such that\
$\g_1=\g_0\in\Gamma^{\neg J}, \g_k\in \Gamma^{\neg J}$, for $k=2,...,n$,\
$\g_1\psi^{-1}+\g_2\psi^{-1}\in \Gamma^{\Upsilon}$,\
$\cdots\cdots\cdots\cdots$\
$\g_1\psi^{-n+1}+\g_2\psi^{-n+1}+\g_3\psi^{-n+2}+\cdot\cdot\cdot+\g_{n-1}\psi^{-2}+\g_{n}\psi^{-1}\in \Gamma^{\Upsilon}$,\
$\g_1\psi^{-n+1}+\g_2\psi^{-n+1}+\g_3\psi^{-n+2}+\cdot\cdot\cdot+\g_{i}\psi^{-n+i-1}+\cdot\cdot\cdot+\g_{n-1}\psi^{-2}+\g_{n}\psi^{-1} \in \{\pm \b\phi^{-m}:m\in \mathbb{N}\}.$
Taking into account $\g_1=\g_0\in\Gamma^{\neg J}$, if $\g_2\in \Lambda$ (respectively $\g_2\in \Gamma^{\neg J}$), the root-multiplicativity and maximal length of $(L, A)$ allow us to assert $$\begin{aligned}
0\neq A_{\g_1}L_{\g_2}=L_{\g_1+\g_2} ~(\mbox{respectively}, 0\neq [L_{\g_1}, L_{\g_2}]=L_{\g_1\psi^{-1}+\g_2\psi^{-1}}).
\end{aligned}$$ By (5.6), we have $$\begin{aligned}
0\neq L_{\g_1\psi^{-1}+\g_2\psi^{-1}}\subset I.
\end{aligned}$$ We can discuss in a similar way from $\g_1\psi^{-1}+\g_2\psi^{-1}\in \Gamma^{\neg J}, \g_3\in \Lambda \cup \Gamma^{\neg J}$ and $\g_1\psi^{-2}+\g_2\psi^{-2}+\g_3\psi^{-1}\in \Gamma^{\neg J}$ to get $$\begin{aligned}
0\neq [L_{\g_1\psi^{-1}+\g_2\psi^{-1}}, L_{\g_3}]=L_{\g_1\psi^{-2}+\g_2\psi^{-2}+\g_3\psi^{-1}}.
\end{aligned}$$ Thus we have $$\begin{aligned}
0\neq L_{\g_1\psi^{-2}+\g_2\psi^{-2}+\g_3\psi^{-1}}\subset I.
\end{aligned}$$ Following this process with the $\neg J$-connection $\{\g_1,\g_2,...,\g_n\}$ we obtain that $$\begin{aligned}
0\neq L_{\g_1\psi^{-n+1}+\g_2\psi^{-n+1}+\g_3\psi^{-n+2}+...+\g_n\psi^{-1}}\subset I.
\end{aligned}$$ It follows that either $$\begin{aligned}
L_{\delta\psi^{-m}}\subset I ~~\mbox{or}~~L_{-\delta\psi^{-m}}\subset I
\end{aligned}$$ for any $\delta\in \Gamma^{\neg J}, m\in \mathbb{N}$. Moreover, we have $$\begin{aligned}
\delta\psi^{-m}\in \Gamma^{\neg J}.
\end{aligned}$$ Since $ H=(\sum_{\gamma\in\Gamma^{\neg J},-\gamma\in\Lambda}A_{-\gamma}L_{\gamma})+(\sum_{\gamma\in \Gamma^{\neg J}}[L_{-\gamma},L_{\gamma}])$, by (5.7) and (5.8), we get $$\begin{aligned}
H\subset I.\end{aligned}$$ Now, for any $\Upsilon\in \{J, \neg J\}$, given any $\delta\in \Gamma^{J}$, the facts $\delta\neq0, H\subset I$ and the maximal length of $(L, A)$ show that $$\begin{aligned}
L_{\delta}=[L_{\delta\psi}, H]\subset I.\end{aligned}$$ The decomposition of $L$ in (5.4) finally gives us $H=I$. $\square$
Another interesting notion related to a split Hom-Leibniz-Rinehart algebra of maximal length $(L, A)$ is Lie annihilator. Write $L=H\oplus (\bigoplus_{\g\in \Gamma^{\neg J}} L_{\g})\oplus (\bigoplus_{\delta\in \Gamma^{ J}} L_{\delta})$.
The Lie-annihilator of a split Hom-Leibniz-Rinehart algebra of maximal length $(L, A)$ is the set $$\begin{aligned}
Z_{Lie}(L):=\{v\in L:[v,H\oplus (\bigoplus_{\g\in \Gamma^{\neg J}} L_{\g})]+[H\oplus (\bigoplus_{\g\in \Gamma^{\neg J}} L_{\g}), v]=0\mbox{~and~} \rho(v)=0\}.\end{aligned}$$
Observe that $Z(L)\subset Z_{Lie}(L)$.
In the following, we will discuss the relation between the decompositions of $L$ and $A$ of a Hom-Leibniz-Rinehart algebra $(L,A)$.
A split regular Hom-Leiniz-Rinehart algebra $(L,A)$ is tight if $Z_{Lie}(L)=0,Z(A)=0,AA=A,AL=L$ and $$\begin{aligned}
H=(\sum_{\gamma\in \Gamma^{\neg J}, -\g\in \Lambda}A_{-\gamma}L_{\gamma})+(\sum_{\gamma\in \Gamma^{\neg J}}[L_{-\gamma},L_{\gamma}]),
~A_{0}=(\sum_{-\alpha\in \Gamma^{\neg J},\alpha\in\Lambda}\rho(L_{-\alpha})(A_{\alpha}))
+(\sum_{\alpha\in \Lambda}A_{-\alpha}A_{\alpha}).\end{aligned}$$
Let $(L,A)$ be a tight split regular Hom-Leibniz-Rinehart algebra, then $$\begin{aligned}
L=\sum_{[\gamma]\in\Gamma^{\neg J}/\sim}I_{[\gamma]},~A=\sum_{[\alpha]\in\Lambda/\approx}\mathcal{A}_{[\alpha]},\end{aligned}$$ with any $I_{[\gamma]}$ an ideal of $L$ verifying $[I_{[\gamma]},I_{[\delta]}]=0$ if $ [\gamma]\neq[\delta] $ and any $\mathcal{A}_{[\alpha]}$ an ideal of $A$ satisfying $\mathcal{A}_{[\alpha]}\mathcal{A}_{[\beta]}=0$ if $ [\alpha]\neq[\beta].$
Let $(L,A)$ be a tight split regular Hom-Leibniz-Rinehart algebra, then for any $[\gamma]\in\Gamma^{\neg J}/\sim$ there exists a unique $[\alpha]\in\Lambda/\approx$ such that $\mathcal{A}_{[\alpha]}I_{[\gamma]}=0$.
[**Proof.**]{} Similar to Proposition 4.2 in [@Cao18]. $\hfill \Box$
Let $(L,A)$ be a tight split regular Hom-Leibniz-Rinehart algebra, then $$\begin{aligned}
L=\sum_{i\in\Gamma^{\neg J}/ I}L_{i},~A=\sum_{j\in K}A_{j},\end{aligned}$$ with any $L_i$ a nonzero ideal of $L$ and any $A_j$ a nonzero ideal of $A$. Furthermore, for any $i\in I$ there exists a unique $\tilde{j}\in K$ such that $A_{\tilde{j}}L_i=0$.
Let $(L, A)$ be a tight split regular Hom-Leibniz-Rinehart algebra of maximal length and root multiplicative. If $\Gamma^{J}$, $\Gamma^{\neg J}$ are symmetric and $\Gamma^{\neg J}$ has all of its roots $\neg J$-connected, then any ideal $I$ of $L$ such that $I\subseteq J$ satisfies either $I=J$ or $ J = I \oplus I'$ with $I'$ an ideal of $L$.
[**Proof.**]{} By (5.1), we can write $$\begin{aligned}
I=(I\cap H)\oplus (\bigoplus_{\g\in\Gamma^{J I}} L_{\g})),\end{aligned}$$ and with $\Gamma^{J I}\subset \Gamma^{J}$. For any $\g\notin \Gamma^{J}$, we have $$\begin{aligned}
[J\cap H, L_{\g}]+[L_{\g}, J\cap H]\subset L_{\g}\subset J.\end{aligned}$$ Hence, in case $[J\cap H, L_{\g}]+[L_{\g}, J\cap H]\neq 0$ we have $\g\in \Gamma^{J}$, a contradiction. Hence $[J\cap H, L_{\g}]+[L_{\g}, J\cap H]= 0$, and so $$\begin{aligned}
J\cap H \subset Z_{Lie}(L).\end{aligned}$$ Taking into account $I\cap H\subset J\cap H=0$, we also write $$\begin{aligned}
I=\bigoplus_{\delta\in \Gamma^{J I}} L_{\delta},\end{aligned}$$ with $\Gamma^{J I}\subset \Gamma^{J}$. Hence, we can take some $\delta_0\in \Gamma^{I}$ such that $$\begin{aligned}
0\neq L_{\delta_0}\subset I.\end{aligned}$$ Now, we can argue with the root-multiplicativity and the maximal length of $L$ as in Proposition 5.5 to conclude that given any $\delta \in \Gamma^{J}$, there exists a $\neg J$-connection $\{\delta_1,\delta_2, ..., \delta_r\}$ from $\delta_0$ to $\delta$ such that $$\begin{aligned}
0\neq [[...[L_{\delta_0}, L_{\delta_2}],...], L_{\delta_r}]\in L_{\delta\psi^{-m}}, ~\mbox{for}~m\in \mathbb{N}\end{aligned}$$ and so $$\begin{aligned}
L_{\epsilon\delta\psi^{-m}}\subset I, ~~\mbox{for~some}~\epsilon\in \pm1, m\in \mathbb{N}.\end{aligned}$$ Note that $\delta\in \Gamma^{J}$ indicates $L_{\delta}\in J$. By Lemma 2.11, $\psi(L_{\delta})=L_{\delta\psi^{-1}}$. Since $L$ is of maximal length, we have $\psi(L_{\delta}) \subset \psi(J)=J$. So $L_{\delta\psi^{-1}}\subset I$. Similarly we get $$\begin{aligned}
L_{\delta\psi^{-m}}\in J, ~\mbox{for}~m\in \mathbb{N}.\end{aligned}$$ Hence we can argue as above with the root-multiplicativity and maximal length of $L$ from $\delta$ instead of $\delta_0$, to get that in case $\epsilon \delta_0\phi^{-m}\in \Gamma^{J}$ for some $\epsilon\in \pm1$, then $0\neq L_{\epsilon\delta_0\psi^{-m}}\in I$.
The decomposition of $J$ in (5.12) finally gives us $I=J$.
Now suppose there is not any $\delta_0\in \Gamma^{J I}$ such that $-\delta_0\in \Gamma^{J I}$. Then we have $$\begin{aligned}
\Gamma^{J}=\Gamma^{J I}\dot{\cup} -\Gamma^{J I},\end{aligned}$$ where $-\Gamma^{J I}:=\{-\gamma|\gamma\in\Gamma^{J I}\}$. Define $$\begin{aligned}
I':=(\sum_{-\gamma\in -\Gamma^{ J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma})\oplus(\bigoplus_{-\gamma\in -\Gamma^{J I}}L_{-\gamma}).\end{aligned}$$
First, we claim that $I'$ is a Hom-Leibniz-ideal of $L$. In fact, By Lemma 2.11, $\psi(L_{-\gamma})\subset L_{-\gamma\psi^{-1}}$, $-\gamma\psi^{-1}\in -\Gamma^{J I}$ and $\psi(A_{\gamma}L_{-\gamma})\subset \psi(L_0)\subset L_0$ if $A_{\gamma}L_{-\gamma}\neq 0$ (otherwise is trivial). So $\psi(I')\subset I'$.
Since $A_{\gamma}L_{-\gamma}\subset L_0$, by (5.15), we have $$\begin{aligned}
&&[L,I']\nonumber\\
&=&[H\oplus(\bigoplus_{\delta\in\Gamma^{\neg J}}L_\delta),
(\sum_{-\gamma\in -\Gamma^{ J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma})\oplus(\bigoplus_{-\gamma\in -\Gamma^{J I}}L_{-\gamma})]\subset\nonumber\\
&&[\bigoplus_{\delta\in\Gamma^{\neg J}}L_\delta,(\sum_{-\gamma\in -\Gamma^{ J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma})]
+[\bigoplus_{\delta\in\Gamma}L_\delta,\bigoplus_{-\gamma\in -\Gamma^{J I}}L_{-\gamma}]
+\sum_{-\gamma\in -\Gamma^{J I}}L_{-\gamma}.~~\end{aligned}$$
For the expression $[\bigoplus_{\delta\in\Gamma^{\neg J}}L_\delta,(\sum_{-\gamma\in -\Gamma^{J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma})]$ in (5.16), if some $[L_\delta,A_{\gamma}L_{-\gamma}]\neq 0,$ we have that in case $\delta=-\gamma$, $[L_{-\gamma},A_{\gamma}L_{-\gamma}]\subset L_{-\gamma\psi^{-1}}\subset I'$, and in case $\delta=\gamma$, since $I$ is a Hom-Leibniz-ideal of $L$, $-\gamma\notin \Gamma^{J I}$ implies $[L_{-\gamma},A_{-\gamma}L_{\gamma}]=0$. By the maximal length of $L$ and the symmetry of $\Gamma$, we have $[L_{\gamma},A_{\gamma}L_{-\gamma}]=0$. Suppose $\delta\notin \{\gamma,-\gamma\}.$ By Definition 2.3, $$\begin{aligned}
[L_\delta,A_{\gamma}L_{-\gamma}]
\subset \phi(A_{\gamma})[L_\delta, L_{-\gamma}]+\rho(L_\delta)(A_{\gamma})L_{-\gamma}.\end{aligned}$$ Since $(L,A)$ is regular, $\phi(A_{\gamma})\subset A_{\gamma}.$ As $[L_\delta,A_{\gamma}L_{-\gamma}]\neq 0,$ we get $A_{\gamma}[L_\delta, L_{-\gamma}]\neq 0$ or $\rho(L_\delta)(A_{\gamma})L_{-\gamma}\neq 0$. By the maximal length of $L$, either $A_{\gamma}[L_\delta, L_{-\gamma}]=L_{\gamma+(\delta-\gamma)\psi^{-1}}$ or $\rho(L_\delta)(A_{\gamma})L_{-\gamma}=L_{\delta}$. In both cases, since $\gamma\in\Gamma^{J I}$, by the root-multiplicativity of $L$, we have $L_{-\delta}\subset I$ and therefore $-\delta\in\Gamma^{J I}$. That is, $L_{\delta}\subset I'$. So $[\bigoplus_{\delta\in\Gamma^{\neg J}}L_\delta,(\sum_{-\gamma\in -\Gamma^{J I},\gamma\in\Lambda}A_{\gamma}$\
$L_{-\gamma})]\subset I'$.
For the expression $[\bigoplus_{\delta\in\Gamma^{\neg J}}L_\delta,\bigoplus_{-\gamma\in -\Gamma^{J I}}L_{-\gamma}]$ in (5.16), if some $[L_\delta,L_{-\gamma}]\neq 0,$ then $[L_\delta,L_{-\gamma}]=L_{(\delta-\gamma)\psi^{-1}}$. On the one hand, let $\delta-\gamma\neq 0$. Since $\gamma\in\Gamma^{J I}$, by the root-multiplicativity of $L$, we have $[L_\gamma,L_{-\delta}]=L_{(\gamma-\delta)\psi^{-1}}\subset I$. So $(\delta-\gamma)\psi^{-1}\in\Gamma_I$ and therefore $L_{(\delta-\gamma)\psi^{-1}}\subset I'$. On the other hand, let $\delta-\gamma=0$. Suppose $[L_\gamma,L_{-\gamma}]\neq 0$, since $\gamma\in\Gamma^{J I}$, we get $[L_\gamma,L_{-\gamma}]\subset I$. Thus $L_{-\gamma}=[[L_\gamma,L_{-\gamma}],L_{-\gamma\psi}]\subset I$. According to the discussion above, $\gamma,-\gamma\in\Gamma^{J I}$, a contradiction with (5.14). So $[\bigoplus_{\delta\in\Gamma^{\neg J}}L_\delta,\bigoplus_{-\gamma\in -\Gamma^{J I}}L_{-\gamma}]\subset I'$.
Second, we claim that $\rho(I')(A)L\subset I'$. In fact, by Definition 2.1, we have $$\begin{aligned}
\rho(I')(A)L\subset [I',AL]+A[I',L]
\end{aligned}$$ Since $I'$ is a Hom-Leibniz-ideal of $L$, we get $[I',AL]\subset I', [I',L]\subset I'$. So it is sufficient to verify that $AI'\subset I'$. For this, we calculate $$\begin{aligned}
AI'&=&(A_0\oplus(\bigoplus_{\alpha\in\Lambda}A_\alpha))
(\sum_{-\gamma\in -\Gamma^{J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma})\oplus(\bigoplus_{-\gamma\in -\Gamma^{J I}}L_{-\gamma})\subset\nonumber\\
&&I'+(\bigoplus_{\alpha\in\Lambda}A_\alpha)(\sum_{-\gamma\in -\Gamma^{J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma})
+(\bigoplus_{\alpha\in\Lambda}A_\alpha)(\bigoplus_{-\gamma\in -\Gamma^{J I}}L_{-\gamma}).
\end{aligned}$$
For the expression $(\bigoplus_{\alpha\in\Lambda}A_\alpha)(\sum_{-\gamma\in -\Gamma^{J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma})$ in (5.17), if some $A_\alpha(A_{\gamma}L_{-\gamma})\neq 0$, we have that in case $\alpha=-\gamma$, clearly $A_\alpha(A_{\gamma}L_{-\gamma})=A_{-\gamma}(A_{\gamma}L_{-\gamma})\subset L_{-\gamma}\subset I'$. In case of $\alpha=\gamma$, since $-\gamma\notin \Gamma_I$, we get $A_{-\gamma}(A_{-\gamma}L_{\gamma})=0$. By the the maximal length of $L$, we have $A_\alpha(A_{\gamma}L_{-\gamma})=A_{\gamma}(A_{\gamma}L_{-\gamma})=0$. Suppose that $\alpha\notin\{\gamma,-\gamma\}$, by the the maximal length of $L$, we have $A_\alpha(A_{\gamma}L_{-\gamma})=L_\alpha$. Since $\gamma\in\Gamma^{J I}$, by the root-multiplicativity of $L$, we have $L_{-\gamma}\subset I$, that is, $-\alpha\in\Gamma^{J I}$. So $\alpha\in-\Gamma^{J I}$ and $L_\alpha\subset I'$. Thus $(\bigoplus_{\alpha\in\Lambda}A_\alpha)(\sum_{-\gamma\in -\Gamma^{J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma})\subset I'$.
For the expression $(\bigoplus_{\alpha\in\Lambda}A_\alpha)(\bigoplus_{-\gamma\in -\Gamma^{J I}}L_{-\gamma})$ in (5.17), if some $A_\alpha L_{-\gamma}\neq 0$, in case $\alpha-\gamma\in \Gamma^{J I}$, by the root-multiplicativity of $L$, we have $A_{-\alpha} L_{\gamma}\neq 0$. Again by the maximal length of $L$, we have $A_{-\alpha} L_{\gamma}=L_{-\alpha+\gamma}$. So $ -\alpha+\gamma\in \Gamma^{J I}$, a contradiction. Thus $ \alpha-\gamma\in -\Gamma^{J I}$ and therefore $(\bigoplus_{\alpha\in\Lambda}A_\alpha)(\bigoplus_{-\gamma\in -\Gamma^{J I}}L_{-\gamma})\subset I'$.
By the discussion above, we have shown that $\rho(I')(A)L\subset I'$ and therefore $I'$ is an ideal of $(L,A)$.
Finally, we will verify that $L=I\oplus I'$ with ideals $I,I'$. Since $[I',I]=0$, by the commutativity of $H$, we get $\sum_{\gamma\in\Gamma}[L_{\gamma},L_{-\gamma}]=0$, so $H$ must has the form $$\begin{aligned}
H=(\sum_{\gamma\in \Gamma^{J I},-\gamma\in\Lambda}A_{\gamma}L_{-\gamma})\oplus(\sum_{-\gamma\in -\Gamma^{J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma}).\end{aligned}$$ In order to show that the sum in (5.19) is direct, we take any $h\in(\sum_{\gamma\in \Gamma^{J I},-\gamma\in\Lambda}A_{\gamma}L_{-\gamma})\oplus(\sum_{-\gamma\in -\Gamma^{J I},\gamma\in\Lambda}A_{\gamma}L_{-\gamma}).$ Suppose $h\neq 0$, then $h\notin Z(L)$. Since $L$ is split, there is $v_\delta\in L_\delta, \delta\in \Gamma$ satisfying $[h,v_\delta]=\delta(h)\psi(v_\delta)\neq 0$. By Proposition 3.3, $0\neq\delta(h)\psi(v_\delta)\in L_{\delta\psi^{-1}}$. While $L_{\delta\psi^{-1}}\subset I\cap I'=0$, a contradiction. So $h=0$, as required. And this finishes the proof. $\hfill \Box$
Let $(L,A)$ be a tight split regular Hom-Leibniz-Rinehart algebra of maximal length and root-multiplicative. If $\Gamma^{J}$, $\Gamma^{\neg J}$ are symmetric and $\Gamma^{\neg J}$ has all of its roots $\neg J$-connected, and $\Lambda$ have all its nonzero weights connected. Then $$\begin{aligned}
L=\bigoplus_{i\in I}L_i,~~A=\bigoplus_{j\in K}A_j,\end{aligned}$$ where any $L_i$ is a simple ideal of $L$ having all of its nonzero roots connected satisfying $[L_i,L_{i'}]=0$ for any $i'\in I$ with $i\neq i'$, and any $A_j$ is a simple ideal of $A$ satisfying $A_j A_{j'}=0$ for any $j'\in K$ such that $j'\neq j$.
Furthermore, for any $i\in I$ there exists a unique $\overline{j}\in K$ such that $A_{\overline{j}}L_i\neq 0$. We also have that any $L_i$ is a split regular Hom-Leibniz-Rinehart algebra over $A_{\overline{j}}$.
[**Proof.**]{} It is analogous to Theorem 5.7 in [@Wang19]. $\hfill \Box$
[**ACKNOWLEDGEMENT**]{}
The paper is supported by the NSF of China (Nos. 11761017 and 11801304), the Youth Project for Natural Science Foundation of Guizhou provincial department of education (No. KY\[2018\]155) and the Anhui Provincial Natural Science Foundation (Nos. 1908085MA03 and 1808085MA14).
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---
abstract: 'We discuss a cosmology in which cold dark matter begins to decay into relativistic particles at a recent epoch ($z < 1$). We show that the large entropy production and associated bulk viscosity from such decays leads to an accelerating cosmology as required by observations. We investigate the effects of decaying cold dark matter in a $\Lambda = 0$, flat, initially matter dominated cosmology. We show that this model satisfies the cosmological constraint from the redshift-distance relation for type Ia supernovae. The age in such models is also consistent with the constraints from the oldest stars and globular clusters. Possible candidates for this late decaying dark matter are suggested along with additional observational tests of this cosmological paradigm.'
author:
- 'G. J. Mathews$^1$, N. Q. Lan$^{1,2}$ and C. Kolda$^1$'
title: 'Late Decaying Dark Matter, Bulk Viscosity and the Cosmic Acceleration'
---
INTRODUCTION
============
Understanding the nature and origin of both the dark energy [@garnavich] and cold dark matter [@Feng06] constitutes a significant challenge to modern cosmology. The simplest particle physics explanation for the cold dark matter is, perhaps, that of the lightest supersymmetric particle, an axion, or a heavy (e.g. “sterile”) neutrino. The dark energy, on the other hand is generally attributed to a cosmological constant, or possibly a vacuum energy in the form of a “quintessence” scalar field [@wetterich; @ratra] or $k$-essence [@zlatev; @steinhardt99] which must be slowly evolving along an effective potential. See [@Barger06] for a review. In addition to these explanations, however, the simple coincidence that both of these unknown entities currently contribute comparable mass energy toward the closure of the universe begs the question as to whether they could be different manifestations of the same physical phenomenon. Indeed, suggestions along this line have been made by many [@Fabris; @Chaplygin; @Abramo04; @Umezu]. See [@Bean05] for a recent review.
In Ref. [@Wilson07] yet another mechanism was considered by which a dark-matter particle could produce the cosmic acceleration. In that work it was shown that entropy production and an associated bulk viscosity could result from a decaying dark-matter particle. Moreover, that bulk viscosity would act as a negative pressure similar to a cosmological constant or quintessence. However, in that paper it was shown the decay alone was not sufficient to produce the observed cosmic acceleration. It was proposed, however, [@Wilson07] that some, but not all of the desired cosmic acceleration could be accounted for if the particle decay was delayed by proceeding thorough a cascade of long lived intermediate states before the final entropy-producing decay.
In this paper we expand on the hypothesis that the dark energy could be produced from a delayed decaying dark-matter particle. Here, we show that a dark-matter particle which, though initially stable, begins to decay to relativistic particles near the time of the present epoch will produce a cosmology consistent with the observed cosmic acceleration deduced from the type Ia supernova distance-redshift relation without the need for a cosmological constant. Hence, this paradigm has the possibility to account for the apparent dark energy without the well known fine tuning and smallness problems [@Bean05] associated with a cosmological constant. Also, for the model proposed herein, the apparent acceleration is a temporary phenomenon. This avoids the difficulties [@hellerman] in accommodating a cosmological constant in string theory.
The idea of delayed dark matter decay is not new. It was previously introduced [@Turner] as a means to provide an $\Omega_M = 0.1-0.3$ without curvature ( $\Omega_{tot} = 1$) by hiding matter in weakly interacting relativistic particles. Here, we point out that such a cosmology not only allows for a flat cosmology with low apparent cold dark matter matter content, but can produce an accelerating cosmology consistent with observations. We show that the bulk viscosity produced during the decay will briefly accelerate the cosmic expansion as matter is being converted from nonrelativistic to relativistic particles. This model thus shifts the dilemma in modern cosmology from that of explaining dark energy to one of explaining how an otherwise stable heavy particle might begin to decay at a late epoch.
In the next section we summarize the cosmology of late decaying dark matter and its associated bulk viscosity. In the following section we discuss candidate particles for such decays, and in Section IV we present fits to the supernova magnitude-redshift relation which show that these data can be reasonably well fit in a flat $k=0$, $\Lambda = 0$ cosmology with recent dark-matter decay. We summarize the constraints that the supernova data places on the properties of the decay along with independent constraint from the ages of oldest stars and globular clusters.
Cosmology with Bulk Viscosity
=============================
Numerous papers have appeared in recent years which deal with the subject of Bulk viscosity as a dark energy [@Sawyer06; @Fabris]. What is needed, however, is a physical model for the generation of the bulk viscosity. Below, we consider one possible means to produce a bulk viscosity in the cosmic fluid by decaying dark matter particles. To begin with, however, we first examine the effects on the cosmic acceleration of the bulk viscosity. For this purpose we utilize a flat ($k = 0$, $\Lambda = 0$) cosmology in a comoving Friedman-Robertson-Walker metric. $$g_{\mu \nu} dx^\mu dx^\nu = -dt^2 + a(t)^2 \biggl[
{dr^2} + r^2 d \theta^2 + r^2 \sin^2{\theta} d\phi^2\biggr]~~,
\label{RW}$$ for which $U^0 = 1$, $U^i = 0$, and $U^{\lambda}_{~; \lambda} = 3 \dot a/a$.
We consider a fluid with total mass-energy density $\rho$ given by, $$\rho = \rho_{DM} + \rho_b + \rho_{h} + \rho_\gamma + \rho_{l}~~,
\label{rhotot}$$ where $\rho_b$ is the baryon density, while $\rho_{DM}$ is the contribution from stable dark matter, $\rho_\gamma$ is the energy density in the usual stable relativistic particles (i.e. photons, neutrinos, etc.), and we denote the relativistic particles specifically produced by decaying dark matter as $\rho_{l}$ (although these products may well be normal neutrinos).
Prior to decay, by fiat $\rho_l = \rho_l(0) = 0$ and the other terms in Eq. (\[rhotot\]) obey the usual relations as given by the the conservation condition $T^{\mu \nu}_{~~;\nu} = 0$ and their respective equations of state, i.e., $$\frac{d \rho_i}{d t} = - 3 \frac{\dot a}{a}(\rho_i + p_i ) ~~,
\label{drhom}$$ where $p_i$ the partial pressure from each species, so that $$\rho_{DM} = \frac{ \rho_{DM} (0)}{a^3}~~,~~
\rho_{h} = \frac{ \rho_{h}(0) }{a^3}~~,~~
\rho_{\gamma} = \frac{ \rho_{\gamma}(0) }{a^4}~~,$$ where $\rho_i(0)$ denotes the initial energy densities at some arbitrary start time.
We begin our models well into the time of the matter dominated epoch. Hence, $\rho_\gamma(0) = aT_\gamma^4 \approx 0$ and the cosmology is nearly pressureless. However, once decay begins, the total energy density in relativistic particles $\rho_\gamma + \rho_{l}$ is not negligible even at the present epoch, neither is the pressure. Thus, we have: $$p = \frac{1}{3}[\rho_\gamma + \rho_{l}]~~.$$
With the introduction of bulk viscosity produced (as described below) from decaying dark-matter particles, the energy momentum tensor then becomes [@Wilson07] $$\begin{aligned}
T_{0 0} &=& \rho \\
T_{0 i} &=& 0 \\
T_{i j} &=& \biggl(p - 3 \zeta \frac{\dot a}{a}\biggr) g_{i j}~~.
\label{TFRW}
\end{aligned}$$ where this last equation shows that the effect of bulk viscosity is to replace the fluid pressure with an effective pressure given by, $$p_{\rm eff} = p - \zeta 3 \frac{\dot a}{a}~~.
\label{peff}$$ Thus, for large $\zeta$ it is possible for the negative pressure term to dominate and an accelerating cosmology to ensue. It is necessary, therefore, to clearly define the bulk viscosity for the system of interest.
The Friedmann equation derives from the $\mu \nu = 0 0 $ component of the Einstein equation. Therefore, it does not depend upon the effective pressure and is exactly the same as for a non dissipative cosmology, i.e. $$H^2 = \frac{\dot a^2}{a^2}=\frac{8}{3}\pi G \rho ~~,
\label{Friedmann}$$ where $\rho$ represents to total mass-energy density from matter and relativistic particles (Eq. \[rhotot\]). Even so, the bulk viscosity from particle decays can briefly affect the cosmic acceleration by producing a temporary condition of nearly constant $\rho$.
Once decay begins, $\rho$ remains as given above in Eq. (\[rhotot\]). However, the conservation equations give new equations for energy densities in decaying matter and produced relativistic particles. For $\rho_l$ and $\rho_h$ we then have,
$$\frac{d \rho_h}{d t} = - 3 \frac{\dot a}{a}(\rho_h +p_{eff}) - \lambda \rho_h ~~,
\label{drhoh}$$
and $$\frac{d \rho_l}{d t} = - 4\frac{\dot a}{a}(\rho_l + p_{eff}) + \lambda \rho_h ~~.
\label{drhor}$$ Denoting $t_d$ as the time at which decays begin, for $t > t_{d}$ we have the following analytic solutions: $$\rho_h = \frac{1}{a^3} \rho_{h} (t_d) e^{- (t-t_d)/\tau}~~.$$ and $$\rho_l = \frac{1}{a^4} \biggl[ \rho_{l} (t_d) + \frac{ \rho_{h} (t_d) }{\tau} \int_{t_d}^t e^{- (t' - t_d)/\tau}a(t')dt'
+ \rho_{BV} \biggr]~~,$$ where $\rho_{BV}$ is an effective dissipated energy in light relativistic species due to the cosmic bulk viscosity, $$\rho_{BV}= 9 \int_{t_d}^t \zeta(t') \biggl(\frac{\dot a}{a}\biggr)^2 a(t')^4 dt'~~.$$
The total density for the Friedmann equation will then include not only terms from heavy and light dark matter, but a dissipated energy density in bulk viscosity. The introduction of this $ \rho_{BV}$ term can lead to a cosmic acceleration as we shall see, but first, for completeness, we summarize the derivation of the bulk viscosity coefficient $\zeta$ given in [@Wilson07] and show how it results from the delayed decay of interest here.
Bulk Viscosity Coefficient
--------------------------
Bulk viscosity can be thought of [@Landau; @Weinberg71; @Okumura03; @Hoover80] as a relaxation phenomenon. It derives from the fact that the fluid requires time to restore its equilibrium pressure from a departure which occurs during expansion. The viscosity coefficient $\zeta$ depends upon the difference between the pressure $\tilde p$ of a fluid being compressed or expanded and the pressure $p$ of a constant volume system in equilibrium. Of the several formulations [@Okumura03] the basic non-equilibrium method [@Hoover80] is identical [@Weinberg71] with Eq. (\[peff\]). $$\zeta 3 \frac{\dot a}{a} = \Delta p~~,
\label{zeta1}$$ where $\Delta p = \tilde p - p$ is the difference between the constant volume equilibrium pressure and the actual fluid pressure.
In Ref. [@Weinberg71] the bulk viscosity coefficient was derived for a gas in thermodynamic equilibrium at a temperature $T_M$ into which radiation is injected with a temperature $T$ and a mean pressure equilibration time $\tau_{\rm e}$. The solution for the relativistic transport equation [@Thomas30] can then be used to infer [@Weinberg71] the bulk viscosity coefficient. More specifically, the form of the pressure deficit and associated bulk viscosity can be deduced from Eq. (2.31) of Ref. [@Weinberg71] which we have generalized slightly and write as, $$\Delta p \sim \biggl(\frac{\partial p}{\partial T}\biggr)_n (T_M - T) = \frac{4 \rho_\gamma \tau_{\rm e}}{3}
\biggl[ 1 - \biggl(\frac{3\partial p}{\partial \rho}\biggr)\biggr] \frac{\partial U^\alpha}{\partial x^\alpha} ~~,
\label{Weinbv}$$ where the subscript $n$ denotes a partial derivative at fixed comoving number density. The factor of 4 on the [*r.h.s.*]{} comes from the derivative of the radiation pressure $p \sim T^4$ of the injected relativistic particles, and the term in square brackets derives from a detailed solution to the linearized relativistic transport equation [@Thomas30]. This term guarantees that no bulk viscosity can exist for a completely relativistic gas (for which $\partial p/\partial \rho = 1/3$). In the cosmic fluid, however, we must consider a total mass-energy density $\rho$ given by both nonrelativistic and relativistic components.
Pressure Equilibration Time
---------------------------
The timescale $\tau_e$ to obtain pressure equilibrium in an expanding cosmology from an initial pressure deficit of $\Delta p(0)$ can be determined [@Okumura03] from, $$\tau_{\rm e} = \int_0^\infty \frac{\Delta p(t)}{\Delta p(0)} dt ~~.
\label{taue}$$
As in Ref. [@Weinberg71], in the present context we also have a thermalized gas into which relativistic particles at some effective temperature are injected. There are, however, some differences. For one, Eq. (\[Weinbv\]) was derived under the assumption of a short relaxation time $\tau_e$ so that only the terms of linear order in $\tau_e$ were retained in the solution to the transport equation. In what follows we will keep this form of the solution even for long relaxation times $\tau_e$ with the caveat that this is only deduced from a leading order approximation to the full transport solution. We will, however, also consider a phenomenological analysis of the effects of terms of higher order in $\tau_e$.
Another difference in the present approach involves the nature of the pressure equilibration time $\tau_e$. Indeed, this term contains the essential physics of the bulk viscosity. There are in principle two ways in which pressure equilibrium can be restored. One is from particle collisions and the other is for simply all of the unstable particles to decay. That is, we can write an instantaneous pressure restoration lifetime $\tau$ as, $$\frac{1}{\tau} = \frac{1}{\tau_{decay}} + \frac{1}{\tau_{coll}}~~.$$
For the cosmological application of interest here the mean collision time $\tau_{coll} = 1/(n \sigma c)$ for weakly interacting (or electromagnetic) particles is very long (many Hubble times) and can be ignored. Hence, one only need consider the timescale to restore pressure equilibrium from the decay of unstable nonrelativistic dark matter. That is, at any time in the cosmic expansion the pressure deficit will be 1/3 of the remaining mass-energy density of unstable heavy particles. Hence, we replace $\rho_\gamma/3$ with the pressure deficit from remaining (undecayed) mass energy $\rho_{\rm h} /3$ in Eq. (\[Weinbv\]) and write, $$\Delta p = \frac{4{\rho_{\rm h} \tau_{\rm e}}}{3}
\biggl[ 1 - \biggl(3 \frac{\partial p}{\partial \rho}\biggr)\biggr] \frac{\partial U^\alpha}{\partial x^\alpha}~~.
\label{Weinbv2}$$ A form for the equilibration time $\tau_{\rm e}$ in the expanding cosmology can then be obtained from Eq. (\[taue\]) by setting $\tau = \tau_{decay}$, and approximating $H = \dot a/a \approx $ constant. This gives, $$\tau_{\rm e} =\frac{\tau}{[1 + 3 (\dot a/a)\tau]} ~~.$$ Note, that the factor in the denominator acts as a limiter to prevent unrealistically large bulk viscosity in the limit of a large $\tau$.
Following the derivation in [@Weinberg71], and inserting Eq. (\[Weinbv2\]) in place of Eq. (\[Weinbv\]), the form for the bulk viscosity of the cosmic fluid due to particle decay was deduced in [@Wilson07] to be, $$\zeta = \frac{4 \rho_{\rm h} \tau_{\rm e}}{3} \biggl[ 1 - \frac{\rho_{\rm l} + \rho_{\gamma}}{\rho} \biggr]^2
~~,
\label{zeta}$$ where the square of the term in brackets comes from inserting Eq. (\[Weinbv2\]) into the linearized relativistic transport equation of Ref. [@Thomas30]. Note that equation (\[zeta\]) implies a non-vanishing bulk viscosity even in the limit of long (infinite) relaxation times $\tau$ as long as the total mass energy density $\rho$ in the denominator is comprised of a mixture of relativistic and nonrelativistic particles so that the term in square brackets does not vanish. This apparent contradiction arises from keeping only the linear terms in the transport equation. Hence, one should be cautious about using this linearized approximation in the long pressure relaxation-time limit. Even so, a more general derivation has been made [@Xinzhong01] which shows that, even in the limit of interest here of a long radiation equilibration time there is a non-vanishing bulk viscosity consistent with experimental determinations. As an indicator of possible higher order effects in the relaxation time, we consider replacing $\tau_e$ in Eq. \[zeta\] with $$\tau_e \rightarrow (\tau_e + a\tau_e^2) = C(\tau) \tau_e~~,$$ where $a$ or $C$ is a parameter to be adjusted to fit the cosmological data.
Evolution of Energy Densities
-----------------------------
To illustrate how an accelerating cosmology arises we plot the evolution of various energy densities in Figure \[rhoplot\] for a model with $\tau = 20$ Gyr and the onset of dark-matter decay at $z_d = 0.3$ ($t_d \approx 8$ Gyr). In this figure the dashed line shows the evolution of the energy density in relativistic particles denoted $\rho_r = \rho_\gamma + \rho_l $. The dot-dashed line shows the evolution of the total matter density $\rho_M = \rho_{DM} + \rho_b + \rho_h$. From this figure it is evident that once the late decay begins, the sudden increase in radiation plus dissipative bulk viscosity leads to a finite period of nearly constant total mass-energy density from the onset until well past the present epoch (at $t = 11$ Gyr). This mimics a $\Lambda$-dominated ($\rho \approx~{\rm constant}$) cosmology until nearly all of the unstable nonrelativistic dark matter has been converted to radiation. Afterward, the total energy density continues to diminish as a radiation-dominated gas and eventually becomes a simple flat cosmology dominated by the remaining stable dark matter and baryons.
![Evolution of various densities as labeled as a function of time for a cosmology in which the dark matter begins to decay at a redshift $z_d = 0.3$ ($t_d \approx 8$ Gyr) with a decay lifetime of $\tau = 20$ Gyr. The dashed line shows the evolution of relativistic particles $\rho_r = \rho_\gamma + \rho_l $. The dot-dashed line shows the evolution of total matter $\rho_M = \rho_{DM} + \rho_b + \rho_h$. The dot-dot-dashed line shows the combined evolution of radiation and $\rho_{BV}$. The solid line shows the evolution of the total mass-energy density . The central dotted line is to aid the eye to identify and also the flattening of the total density during the decay epoch. This flattening leads to cosmic acceleration and apparent dark energy. The present age for this model ($t = 11$ Gyr) occurs at ($\rho/\rho_c({\rm now}) = 1.0$. []{data-label="rhoplot"}](rhobvzp3t20.ps){width="3.5in"}
Candidates for Late Decaying Dark Matter
========================================
To avoid observational constraints, the decay products of any cold dark-matter particle must not be in the form of observable photons or charged particles. Otherwise the implied background in energetic photons would have been easily detectable [@noem]. Neutrinos or some other light weakly interacting particle would thus be the most natural products from such decay. We now summarize some suggestions for this type of decaying particle.
A good candidate considered in [@Wilson07] is that of a heavy right-handed (sterile) neutrino. Such particles could decay into into light $\nu_e,~\nu_\mu,~\nu_\tau$ “active” neutrinos [@Abazajian01]. In this case, the limitation of decays to non-detectable neutrinos places some constraints on the sterile neutrino as summarized in [@Wilson07].
Even so, various models have been proposed in which singlet “sterile” neutrinos $\nu_s$ mix in vacuum with active neutrinos ($\nu_e$, $\nu_\mu$, $\nu_\tau$). Such models provide both warm and cold dark matter candidates. By virtue of the mixing with active neutrino species, the sterile neutrinos are not truly “sterile” and, as a result, can decay. In most of these models, however, the sterile neutrinos are produced in the very early universe through active neutrino scattering-induced de-coherence and have a relatively low abundance. As pointed out in [@Wilson07], however, this production process could be augmented by medium enhancement stemming from a significant lepton number. Here we speculate that a similar medium effect at late times might also induce a late time enhancement of the decay rate.
There are several other ways by which such a heavy neutrino might be delayed from decaying until the present epoch. The possibility of delayed decay by a cascade of intermediate decays prior to the final bulk-viscosity generating decay was explored in [@Wilson07]. However, this possibility was found to be incapable of accounting for all of the cosmic acceleration. Here we propose that the decay of heavy neutrinos at late times requires one of two other possibilities: 1) A late low-temperature cosmic phase transition whereby a new ground state causes the previously stable dark matter to become stable; or 2) a time varying effective mass for either the decaying particle or its decay products whereby a new ground state appears due to a level crossing at a late epoch.
The first possibility was considered in Ref. [@Turner]. A suitable late decaying heavy neutrino could be obtained if the decay is caused by some horizontal interaction (e.g. as in the Majoron [@majoron] or familion [@familion] models). The decay within the familion model can be written; $$\nu \rightarrow \nu' + f~~,$$ where $f$ is a massless Nambu-Goldstone boson associated with a spontaneously broken “family symmetry.” The lifetime for the heavy neutrino decay then becomes: $$\tau_\nu \approx (10^9{\rm~yr}) [100 {\rm~eV}/m_\nu)^3 [F/(10^9 {\rm ~GeV})]^2 ~~,$$ where $F$ is the scale of the spontaneous symmetry breaking and $m_\nu$ is the neutrino mass. An appropriately chosen symmetry breaking scale is required [@Turner] to induce the decay at the desired epoch.
Regarding the second possibility, a number of papers have been written [@MaVaNs] which consider dark matter neutrinos with a time varying mass of (the so-called MaVaN model). Those models were proposed [@MaVaNs] as a means to account for dark energy from self interaction among dark matter neutrinos. In our context we would require the self interaction of the neutrino to produce a time-dependent heavy neutrino mass such that the lifetime for decay of an initially unstable long-lived neutrino becomes significantly shorter at late times.
Another possibility might be a more generic long-lived dark-matter particle $\psi$ whose rest mass increases with time [@VAMP]. This could be achieved (for example in scalar-tensor theories of gravity [@Casas]) by having the rest mass derive from the expectation value of a scalar field $\phi$. If the potential for $\phi$ depends upon the number density of $\psi $ particles then the mass of the particles could increase naturally with the cosmic expansion. This could lead to a late-time instability to decay.
In another proposal [@Bellido] it has been pointed out that in string effective theories there can exist a dilaton scalar field which couples to gravity matter and radiation. In general, different particle masses will have different dilaton couplings. This can lead to dark matter with variable mass [@Bellido]. Moreover, the dilaton couples to radiation in the form of a variable gauge coupling. This could also lead to a variable decay rate of a long-lived dark-matter particle such that an initially neglegible decay rate quickly accelerates at some late epoch.
Another possible candidate may be from supersymmetric dark matter. It is now popular to presume that the initially produced dark matter relic must be a superWIMP in order to produce the correct relic density [@Feng06]. This superWIMP then decays to a lighter stable dark-matter particle. One interpretation of a candidate for late decaying dark matter here, then might be a decaying superWIMP with time-dependent couplings and hence a variable lifetime. Alternatively, the light supersymmetric particle, might also be unstable with a variable decay lifetime. It has been proposed [@Hamaguchi] for example that there are discrete gauge symmetries (e.g. ${\bf Z}_{10}$) which naturally protect heavy $X$ gauge particles from decaying into ordinary light particles so that the $X$ particles become a candidate for long-lived dark matter. The lifetime of the $X$, however is strongly dependent upon the ratio of the cutoff scale ($M_* \approx 10^{18}$ GeV) to the mass of the $X$. $$\tau_X \sim \biggl( \frac{M_*}{M_X}\biggr)^{14} \frac{1}{M_X} = 10^2 - 10^{17}~~{\rm Gyr}~~.$$ From this it is apparent that even a small variation in either $M_X$ or $M_*$ could lead to a drastic speed up in the decay lifetime.
Supernova Distance-Redshift analysis
====================================
Having defined the cosmology of interest we now examine the magnitude-redshift relation for type Ia supernovae (SNIa). The apparent brightness of the type Ia supernova standard candle with redshift is given [@Carroll] by a simple relation for a flat $\Lambda = 0$ cosmology. The luminosity distance in the present cosmological model can be written, $$\begin{aligned}
D_L &=& \frac{c (1+z)}{ H_0 } \biggl\{ \int_0^z dz'
\biggl[\Omega_\gamma (z') + \Omega_{\rm l} (z')
\nonumber \\
&&
+ \Omega_{\rm DM}(z')
+ \Omega_{\rm b}(z') + \Omega_{\rm h}(z')
\biggr]^{-1/2} \biggr\}~~,\end{aligned}$$ where $H_0$ is the current Hubble parameter. The $\Omega_i$ are the energy densities normalized by the present critical density, i.e. $\Omega_i(z) = {8 \pi G \rho_i(z) / 3 H_0^2}$. $\Omega_{\rm h}$ is the closure contribution from the decaying heavy cold dark-matter particles. Their decay is taken here to produce light neutrinos or other noninteracting relativistic particles $ \Omega_{\rm l}$, while $\Omega_\gamma$ is the contribution from normal relativistic matter. Note that $\Omega_{\rm h}$, $\Omega_\gamma$ and $ \Omega_{\rm l}$ each have a nontrivial redshift dependence due to particle decays, while $\Omega_\gamma (z)$ varies as $(1+z)^{-4}$ and stable dark matter and baryons $\Omega_{\rm DM}(z)
+ \Omega_{\rm b}(z) $ obey the usual $(1+z)^{-3}$ scaling.
Figures \[fig:mag\] and \[fig:magc\] compare various cosmological models with some of the combined data from the High-Z Supernova Search Team and the Supernova Cosmology Project [@garnavich; @Riess]. These figures shows the evolution of the relative distance modulus as given in the usual way $\Delta (m - M) = 5 \log{[D_L/D_L(\Omega_k=1)]}$, where the K-corrected magnitudes $m = M + 5 \log{ D_L} + 25$ are plotted relative to to a fiducial $\Omega_k = 1/(a_0 H_0)^2 = 1$ open cosmology, for which $$\begin{aligned}
D_L(\Omega_k=1)&=& \frac{c (1+z)}{ 2 H_0 } \biggl[z + 1 - \frac{1}{(z+1)}\biggr]~~.\end{aligned}$$
![Fit to the SNIa magnitude-redshift relation for an un-enhanced bulk viscosity from late decaying dark matter (solid line) compared with the observations [@Riess] (points). The fit corresponds to $z_d$ = 0.3 ($t_d \approx 8$ Gyr), and $\tau \ge 20$ Gyr with an age of 11 Gyr. Data and lines are plotted relative to a fiducial $\Omega_k = 1$ open cosmology. For comparison, the dashed line is for a flat $\Omega_M = 1$ matter-dominated cosmology. The dot-dashed line shows a flat standard $\Lambda$CDM cosmology with $\Omega_\Lambda = 0.7$ and $\Omega_M = 0.3$. The Dash-double-dot line shows a model as in [@Wilson07] with undelayed particle decay. []{data-label="fig:mag"}](demzdecay.ps){width="8.cm"}
![Same as Figure \[fig:mag\], but for additional parameter $C = 1 + a \tau_e$ due to possible higher order effects in the bulk viscosity $\zeta$. The solid line corresponds to $z_d$ = 0.3 ($t_d \approx 9$ Gyr), and $C = 1.6$ with $\tau = 20$ Gyr and an age of 12 Gyr. The dash-dash-dotted line corresponds to $z_d$ = 0.26 ($t_d \approx 9$ Gyr), and $C = 4.0$ with $\tau =5$ Gyr and an age of 12 Gyr. []{data-label="fig:magc"}](demzbv8zp3t20c1p6.ps){width="8.cm"}
As a measure of the quality of the different models, Table 1 summarizes the relevant model parameters and reduced $\chi^2$ goodness of fit. The quantity $\Omega_M$ in column 4 is defined as the present sum of nonrelativistic matter, i.e. $$\Omega_{\rm M} = \Omega_{\rm h}(z=0) + \Omega_{\rm DM}(z=0) + \Omega_{\rm b}(z=0 ~~.$$
[lcccccc]{} $\tau$ (Gyr) & $z_d$ & C & $\Omega_{M}$ & $\Omega_{\Lambda}$ & $\chi_r^2$ & Age (Gyr)\
$> 20$ & 0.30$\pm0.04$ & 1. & $0.50 \pm 0.03$ & 0. & 1.32 & $11.0\pm 0.1$\
($\Lambda$CDM) & - & 1. & 0.31 & 0.69 & 1.14 & 13.0\
(CDM) & - & 1. & 0. &1. & 3.23 & 9.2\
20 & (no delay) & 1. & 0.16 & 0. & 3.93 & 8.7\
20 & 0.3 & 1.6 & 0.30 & 0. & 1.13 & 12.0\
5 & 0.26 & 4.0& 0.27 & 0. & 1.13 & 11.9\
\
The solid line on Figure \[fig:mag\] shows a fit to the SNIa data for a model with a fixed bulk viscosity coefficient as given in Eq. \[zeta\]. The solid line and dash-dash-dotted lines on Figure \[fig:magc\] show fits to the data when an enhanced $\zeta$ is allowed due to possible higher order terms in the transport equation. The dotted line on both figures shows the fiducial $\Omega_k = 1$ open cosmology. For comparison, the dashed line on both figures is for a flat $\Omega_M = 1$ matter-dominated cosmology, while the dot-dashed line shows a flat standard $\Lambda + $cold dark matter ($\Lambda$CDM) cosmology with $\Omega_\Lambda = 0.7$ and $\Omega_M = 0.3$.
The fit with the bare bulk viscosity coefficient (Eq. \[zeta\]) on Figure \[fig:mag\] corresponds to $z_d$ = 0.3 ($t_d \approx 8$ Gyr), and $\tau > 20$ Gyr with an age of 11 Gyr. The Dash-double-dot line shows a model as in [@Wilson07] with undelayed particle decay. As discussed in [@Wilson07], a model without delayed decay actually produces a worse fit than even a matter-dominated model. The reason is simply that the formation of a radiation-dominated universe by particle decay causes the energy density to diminish even faster with scale factor than a matter dominated cosmology, even with the help of the bulk-viscosity term. On the other hand, by delaying the onset of particle decay and the associated bulk viscosity until late, it is possible to produce some of the features of a $\Lambda$CDM model without invoking a $\Lambda$.
As a practical matter, it turns out that with the un-enhanced bulk viscosity in Eq. \[zeta\], the best fit requires the largest bulk viscosity. From Eq. (\[zeta\]), however, it is clear that large $\zeta$ occurs for large $\tau_e$ in the limit of $\tau >> (3H)^{-1}$. On the other hand, an infinite lifetime should imply no viscosity as there is no decay. We resolve this dilemma by imposing an arbitrary cutoff in the decay lifetime of a couple of Hubble times ($\sim 20$ Gyr). There is not much change in the goodness of fit for larger times. The decay lifetime then is not a parameter, it is simply large ($> 20$ Gyr). The only parameter in this model is therefore the time (or redshift) at which the decay begins.
On the other hand, when the enhancement factor $C$ is introduced as shown of Figure \[fig:magc\], equivalent fits can be obtained with almost any value for $\tau$. So again, $\tau$ is not a parameter, or more precisely, it has a degeneracy with $C$, such that any decrease in $\tau$ can be offset by an increase in $C$. Moreover, because the total bulk viscosity can be increased, the goodness of the fit is substantially improved, even after allowing for the introduction of an additional degree of freedom.
One interesting feature of the bulk-viscosity models apparent in both Figures \[fig:mag\] and \[fig:magc\] is that the magnitude-redshift relation decreases more rapidly (brighter apparent magnitude) than in a standard $\Lambda$CDM cosmology. This can be traced to the higher matter content in the past which leads to a more rapid deceleration during the matter-dominated epoch. In principle, this feature could allow one to distinguish between the two cosmologies as more SNIa data is accumulated at highr redshift. For now, the data slightly favor the bulk-viscosity models.
Of particular interest regarding Figures \[fig:mag\], \[fig:magc\], and Table 1 is the fact that a $\Lambda = 0$ model can produce a reduced $\chi^2$ which is as good as the best fit standard $\Lambda$CDM cosmology. This is based, however, upon a parameterization of the effects from higher-order terms in the transport equation, and needs to be verified. Even so, it is at least possible that this late-decaying model for bulk viscosity realizes the present cosmic acceleration without the need for a cosmological constant, and hence, is a viable alternative to the $\Lambda$CDM model based upon fits to the SNIa data.
The cosmic age and the growth of large-scale structure place additional constraints on a late-decaying cosmology [@Turner]. However, for the optimum fits derived here the onset of the decay occurs close enough to the time when a normal $\Lambda$CDM cosmology begins to become $\Lambda$ dominated that there is not much difference in the implied cosmic age. The ages deduced here are lower ( 11-12 Gyr) than the age ($13.8^{+0.1}_{-0.2}$ Gyr) deduced from an analysis [@WMAP] of WMAP data based upon a $\Lambda$CDM model. However, a direct comparison with the WMAP age is not meaningful unless the CMB analysis is redone using the present cosmological model. This we plan to do in a future work. Nevertheless, the present model is consistent with the independent constraints from the age from the oldest stars ($13 \pm 2$ [@Cowan]) and globular clusters ($11 \pm 3$ [@Carretta]). Moreover, even if the age is low, this problem could be alleviated in open cosmological models with bulk viscosity [@Fabris].
The formation of large scale structure in the present model, however, might be a more serious problem. Even though the expansion rate is not much different than in the best fit $\Lambda$CDM model, in this model the dark matter content is higher in the past. This may lead to excess of large scale structure at early times. Also, the current matter content in some of the fits, $\Omega_M = 0.27-0.50$, is high compared to the value derived from the WMAP [@WMAP] analysis $\Omega_M = 0.26^{+0.01}_{- 0.03}$. Both of these issues could be resolved, however, by considering an open cosmology. Clearly, this is something which needs to be investigated and in a future work we will examine these issues. We do note, however, another positive feature of cosmic structure in these models [@Wilson07]. The formation of large scale structure in a cosmology with decaying dark matter can lead to a flattening of the dark matter density profiles consistent with observations [@DMflat].
Conclusion
==========
We have considered models in which the present cosmic acceleration derives from the temporary insertion of dissipative mass energy due to the bulk viscosity created by the recent decay of a cold dark-matter particle into light (undetectable) relativistic species. As illustrative examples we have considered initially matter dominated flat ($\Lambda=0$) cosmologies plus late-time particle decay. We find that models with bulk viscosity from late-time dark-matter decay are consistent with observations of the supernova magnitude-redshift relation, and ages from the oldest stars and globular clusters. We argue that it will likely satisfy other constraints as well.
Moreover, there is a difference in the SNIa magnitude-redshift relation for this cosmology compared to the standard $\Lambda$CDM model. This is because the deceleration is faster at high redshift due to a higher matter content during the matter-dominated epoch. Thus, as more data are accumulated at the highest redshifts, it may be possible to distinguish between this cosmology and a standard $\Lambda$CDM model. For now, however, there is sufficient success in the present model to motivate further work. In a subsequent paper we plan to consider higher order terms in the transport equation in detail as well as the effects of such late decays on the observed power spectrum of the cosmic microwave background [@WMAP], and the growth of large scale structure.
Ultimately, of course, one must decide whether the dilemma of a cosmological constant with all of its difficulties is less palatable than the dilemma of a bulk viscosity produced by the delayed onset of dark-matter decay. For now, however, our purpose has simply been to establish that such a possibility exists and that it warrants further investigation.
Work at the University of Notre Dame supported by the U.S. Department of Energy under Nuclear Theory Grant DE-FG02-95-ER40934. One of the authors (NQL) wishes to also acknowledge partial support from the Joint Institute for Nuclear Astrophysics (JINA) at the University of Notre Dame, and also to acknowledge support from the CERN Theory Division as a Visiting Fellow along with useful discussions with John Ellis.
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KEK-TH-508
KEK Preprint 96-159
January 1997
---------------------
PHENOMENOLOGY OF THE HIGGS SECTOR IN
SUPERSYMMETRIC STANDARD MODEL [^1]
YASUHIRO OKADA
National Laboratory for High Energy Physics (KEK)
Oho 1-1, Tsukuba 305, Japan
Introduction
============
After the discovery of top quark at Fermilab and precise measurements of electroweak interaction at LEP and SLC experiments it has been more and more evident that the elementary particle physics is described by the Standard Model (SM). This model is based on two physical principles, [*i.e.*]{} the gauge principle and the Higgs mechanism. Although we can understand most of the experimental results by the $SU(3)\times SU(2)\times U(1)$ gauge symmetry, little is known about the dynamics behind the electroweak symmetry breaking. Therefore exploring the Higgs sector is the most important issue of the current high energy physics and the primary objective of future collider experiments.
Study of the Higgs sector is not only important to establish the SM but also crucial to search for physics beyond the SM. In this respect the mass of the Higgs boson itself gives us important information. For example, a heavy Higgs boson suggests that the dynamics of the electroweak symmetry breaking is governed by strong interaction. On the other hand if we assume that fundamental interactions are described by perturbation theory up to the Planck scale or a scale close to it, the Higgs boson is expected to exist below 200 GeV. Grand unified theory (GUT) and supersymmetric (SUSY) extension of the SM are examples of the latter case.
In this talk I would like to discuss phenomenological aspects of the supersymmetric standard model, especially some issues related to the Higgs sector of the SUSY model. Among various extensions of the SM, the SUSY SM is now supposed to be the most promising candidate of physics beyond the SM. Since early 80’s SUSY extensions of the SM and GUT have been extensively studied because SUSY is the unique symmetry to ensure the smallness of electroweak scale compared to the Planck scale by cancelling the quadratic divergence of scalar mass renormalization. More recently, SUSY theories attracted much attention since three gauge coupling constants measured precisely at LEP and SLC experiments are consistent with SUSY GUT although the simplest non-SUSY GUT is excluded experimentally.
In the following sections, I first discuss the Higgs sector of the minimal supersymmetric standard model (MSSM). It is shown that the upper bound of the lightest CP-even Higgs boson in this model is given by about 130 GeV, which is a prime target of future experiments at LHC and $e^+e^-$ linear colliders. Then the Higgs sector of extended version of the SUSY SM is reviewed. Finally, I show how the measurements of various Higgs decay branching ratios are useful to determine MSSM parameters in future $e^+e^-$ linear collider experiments.
The Higgs Sector in the MSSM
============================
In order to construct a SUSY version of the SM we need to introduce a SUSY partner for each particle of the SM. For quarks and leptons their scalar partners, squarks and sleptons, are introduced. Corresponding to the $SU(3)$, $SU(2)$ and $U(1)$ gauge fields we need spin $1/2$ gauge fermions called gluino, wino and bino respectively. Unlike the minimal SM the $SU(2)$-doublet Higgs field giving masses to up-type quarks and that to down-type quarks and leptons have to be introduced separately in SUSY models, therefore the Higgs sector contains at least two doublet Higgs fields. In the minimal SUSY extension, [*i.e.*]{} the MSSM, we introduce two Higgs doublets and their SUSY partners, higgsinos. The winos, bino and higgsinos can have mixings due to the electroweak symmetry breaking and form four neutral Majorana fermions (neutralinos) and two charged Dirac fermions (charginos). Therefore the MSSM is characterized as a two doublet-Higgs SM with scalar superpartners (squarks/sleptons) and fermionic superpartners (neutralinos/charginos).
Let us first discuss the MSSM Higgs sector. The most important feature is that the Higgs-self-coupling constant at the tree level is completely determined by the $SU(2)$ and $U(1)$ gauge coupling constants. After electroweak symmetry breaking, the physical Higgs states include two CP-even Higgs bosons ($h, H$), one CP-odd Higgs boson ($A$) and one pair of charged Higgs bosons ($H^\pm$) where we denote by $h$ and $H$ the lighter and heavier Higgs bosons respectively. Although at the tree level the upper bound on the lightest CP-even Higgs boson mass is given by the $Z^0$ mass, the radiative corrections weaken this bound.[@OYY] The Higgs potential is given by
$$\begin{aligned}
V_{Higgs} & = & m^2_1|H_1|^2 + m_2^2|H_2|^2 - m_3^2
(H_{1}\cdot H_{2}+\bar{H_{1}}\cdot\bar{ H_{2}})
\nonumber\\
& & +\frac{g_2^2}{8}(\bar{H_1}\tau^aH_1
+ \bar{H_2}\tau^aH_2)^2
+ \frac{g_1^2}{8}(|H_1|^2 - |H_2|^2)^2
\nonumber\\
& & + \Delta V,\end{aligned}$$
where $\Delta V$ represents the contribution from one-loop diagrams. Since the loop correction due to the top quark and its superpartner, the stop squark, is proportional to the fourth power of the top Yukawa coupling constant and hence is large, the Higgs self-coupling constant is no longer determined only by the gauge coupling constants. The upper bound on the lightest CP-even Higgs mass ($m_h$) can significantly increase for a reasonable choice of the top-quark and stop-squark masses. Figure 1 shows the upper bound on $m_h$ as a function of top-quark mass for several choices of the stop mass and the ratio of two Higgs-boson vacuum expectation values ($\tan\beta = \frac{<H_2^0>}{<H_1^0>})$.
We can see that, in the MSSM, at least one neutral Higgs-boson should exist below 130 - 150 GeV depending on the top and stop masses.
The Higgs boson in this mass range is a target of the future collider experiments both at LHC and $e^+ e^-$ linear colliders. In the coming experiment at LEP II the SM Higgs boson is expected to be discovered if its mass is below 95 GeV. Since the upper bound exceeds the discovery limit of LEP II many efforts are made to clarify the discovery potential of the SUSY Higgs in LHC experiments. In this mass range the main decay mode of the SM Higgs boson is $h\rightarrow b \bar{b}$. Unfortunately because of QCD backgrounds we cannot use this mode in the LHC experiments and we have to rely on the two photon mode whose branching ratio is $O(10^{-3})$. In the SUSY case its branching ratio can be even smaller, and the search may be more difficult. Recent study shows that it is probably possible to get at least one signal of the SUSY Higgs sector in almost all parameter space but we may have to wait for several years before we find the signal.[@LHC] On the other hand an $e^+ e^-$ linear collider with $\sqrt{s}\sim300 - 500$ GeV is a suitable place to study the Higgs boson in this mass region. Here we can not only discover the Higgs boson easily but also measure various quantities, [*i.e.*]{} production cross sections and branching ratios related to the Higgs boson.[@Janot; @Kawagoe; @JLC; @Hildreth] These measurements are very important to clarify nature of the discovered Higgs boson and distinguish the SM Higgs boson from Higgs particles associated with some extensions of the SM like the MSSM.
Other Higgs states, namely the $H, A, H^\pm$, are also important to clarify the structure of the model. Their existence alone is proof of new physics beyond the SM, but we may be able to distinguish the MSSM from a general two-Higgs model through the investigation of their masses and couplings. In the MSSM the Higgs sector is described by four independent parameters for which we take the mass of the CP-odd Higgs boson ($m_A$), $\tan\beta,$ the top-quark mass $(m_t$) and the stop mass ($m_{stop}$). The top and stop masses enter through radiative corrections to the Higgs potential. Speaking precisely, there are left- and right-handed stop states which can mix to form two mass eigenstates; therefore more than just one parameter is required to specify the stop sector. In Figure 2, the masses for the $H, A$, and $ H^\pm$ are shown as a function of $m_A$ for several choices of $\tan\beta$ and $m_{stop}$=1 TeV.
We can see that, in the limit of $m_A \rightarrow \infty$, $m_h$ approaches a constant value which corresponds to the upper bound in Figure 1. Also in this limit the $H, A$ and $H^\pm$ become degenerate in mass.
The neutral Higgs-boson couplings to gauge bosons and fermions are determined by the ratio of vacuum expectation values $\tan{\beta}$ and the mixing angle $\alpha$ of the two CP-even Higgs particles defined as $$\begin{aligned}
ReH_1^0 &=& \frac{1}{\sqrt{2}}
(\upsilon\cos\beta - h\sin\alpha + H\cos\alpha)
\nonumber \\
ReH_2^0 &=& \frac{1}{\sqrt{2}}
(\upsilon\sin\beta + h\cos\alpha + H\sin\alpha).\end{aligned}$$ For Higgs-boson production, the Higgs-bremsstrahlung process $e^+e^- \rightarrow Zh$ or $ZH$ and the associated production $e^+e^- \rightarrow Ah$ or $AH$ play complimentary roles. Namely $e^+e^- \rightarrow Zh~(ZH)$ is proportional to $\cos(\beta-\alpha)(\sin(\beta-\alpha))$, and $e^+e^- \rightarrow Ah~(AH)$ is proportional to $\sin(\beta-\alpha)(\cos(\beta-\alpha))$, so at least one of the two processes has a sizable coupling. It is useful to distinguish the following two cases when we discuss the properties of the Higgs particles in the MSSM, namely (i) $m_A {\mbox{ \raisebox{-1.0ex}{$\stackrel{\textstyle <}
{\textstyle \sim}$ }}}150$ GeV, (ii) $m_A \gg 150$ GeV. In case (i), the two CP-even Higgs bosons can have large mixing, and therefore the properties of the neutral Higgs boson can be substantially different from those of the minimal SM Higgs. On the other hand, in case (ii), the lightest CP-even Higgs becomes a SM-like Higgs, and the other four states, $H, A, H^\pm$ behave as a Higgs doublet orthogonal to the SM-like Higgs doublet. In this region, $\cos(\beta-\alpha)$ approaches unity and $\sin(\beta-\alpha)$ goes to zero so that $e^+e^- \rightarrow Zh$ and $e^+e^- \rightarrow AH$ are the dominant production processes.
Scenarios for the Higgs physics at a future $e^+e^-$ linear collider are different for two cases. In case (i) it is possible to discover all Higgs states with $\sqrt{s} = 500$ GeV, and the production cross-section of the lightest Higgs boson may be quite different from that of the SM so that it may be clear that the discovered Higgs is not the SM Higgs. On the other hand, in case (ii), only the lightest Higgs may be discovered at the earlier stage of the $e^+e^-$ experiment, and we have to go to a higher energy machine to find the heavier Higgs bosons. Also, since the properties of the lightest Higgs boson may be quite similar to those of the SM Higgs boson we need precision experiments on the production and decay of the particle in order to investigate possible deviations from the SM.
The Higgs sector in extended versions of the SUSY SM
====================================================
Although the MSSM is the most widely studied model, there are several extensions of the SUSY version of the SM. If we focus on the structure of the Higgs sector, the MSSM is special because the Higgs self-couplings at the tree level are completely determined by the gauge coupling constants. It is therefore important to know how the Higgs phenomenology is different for models other than the MSSM.
A model with a gauge-singlet Higgs boson is the simplest extension.[@singlet] This model does not destroy the unification of the three gauge coupling constants since the new light particles do not carry the SM quantum numbers. Moreover, we can include a term $W_\lambda = \lambda NH_1H_2$ in the superpotential where $N$ is a gauge singlet superfield. Since this term induces $\lambda^2|H_1H_2|^2$ in the Higgs potential, the tree-level Higgs-boson self-coupling depends on $\lambda$ as well as the gauge coupling constants. There is no definite upper-bound on the lightest CP-even Higgs-boson mass in this model unless a further assumption on the strength of the coupling $\lambda$ is made. If we require all dimensionless coupling constants to remain perturbative up to the GUT scale we can calculate the upper-bound of the lightest CP-even Higgs-boson mass.[@singletmass] In Figure 3, the upper bound of the Higgs-boson mass is shown as a function of the top-quark mass.
In this figure we have taken the stop mass as 1 TeV and demanded that no dimensionless coupling constant may blow up below the GUT scale ($\sim 10^{16}$ GeV). We can see that the upper bound is given by 130 $\sim$ 140 GeV for this choice of the stop mass. The top-quark-mass dependence is not significant compared to the MSSM case because the maximally allowed value of $\lambda$ is larger (smaller) for a smaller (larger) top mass.
From this figure we can see that the lightest Higgs boson is at least kinematically accessible at an $e^+e^-$ linear collider with $\sqrt{s}\sim 300 - 500$ GeV. This does not, however, mean that the lightest Higgs boson is detectable. In this model the lightest Higgs boson is composed of one gauge singlet and two doublets, and if it is singlet-dominated its couplings to the gauge bosons are significantly reduced, hence its production cross-section is too small. In such a case the heavier neutral Higgs bosons may be detectable since these bosons have a large enough coupling to gauge bosons. In fact we can put an upper-bound on the mass of the heavier Higgs boson when the lightest one becomes singlet-dominated. By quantitative study of the masses and the production cross-section of the Higgs bosons in this model, we can show that at least one of the three CP-even Higgs bosons has a large enough production cross-section in the $e^+e^- \rightarrow Zh^o_i$ $(i =1, 2, 3)$ process to be detected at an $e^+e^-$ linear collider with $\sqrt{s}\sim300 - 500$ GeV.[@KOT] For this purpose we define the minimal production cross-section, $\sigma_{min}$, as a function of $\sqrt{s}$ such that at least one of these three $h^0_i$ has a larger production cross section than $\sigma_{min}$ irrespective of the parameters in the Higgs mass matrix. The $\sigma_{min}$ turns to be given by one third of the SM production cross-section with the Higgs boson mass equal to the upper-bound value. In Figure 4 we show $\sigma_{min}$ as a function of $\sqrt{s}$ for $m_{stop} = 1$ TeV .
From this figure we can conclude that the discovery of at least one neutral Higgs boson is guaranteed at an $e^+e^-$ linear collider with an integrated luminosity of 10 fb$^{-1}$.
Determination of the heavy Higgs mass scale from branching measurements in the MSSM
===================================================================================
In the previous section we have discussed the detectability of the Higgs boson in the SUSY SM with a gauge singlet Higgs field. Since the situation is better for the case of the MSSM we can show that at least one CP-even Higgs boson of the MSSM can be discovered at the first stage of an $e^+e^-$ linear collider experiment where the CM energy is $\sim 300 - 500$ GeV.
If a Higgs boson is discovered, the next question is to determine whether this boson is the SM Higgs boson or a Higgs boson associated with some extension of the SM. For this purpose it is important to know to what extent the non-minimality of the Higgs boson can be detected through the investigation of the production cross-section and decay branching ratios.[@Janot; @Kawagoe; @JLC; @Hildreth] Here we would like to consider this problem in the context of the MSSM, that is, we would like to know whether the parameters in the Higgs sector are determined by various observable quantities related to the Higgs boson. Although it is possible to discover all five Higgs states at the first stage of the linear collider experiment, we may at first be able to find only one CP-even Higgs boson. In such a case it is important to determine the heavy Higgs mass scale because the heavy Higgs bosons become targets of the second stage of the $e^+e^-$ linear-collider experiments after the beam energy is increased.
In the following analysis let us assume that only one CP-even Higgs boson is discovered at the $e^+e^-$ linear-collider experiment. The free parameters required to specify the Higgs sector in the MSSM can be taken to be the CP-odd Higgs-boson mass ($m_A$), the ratio of two vacuum expectation values ($\tan\beta$) and masses of the top quark and the stop squark. The latter two parameters ($m_t, m_{stop}$) are necessary to evaluate the Higgs potential at the one-loop level. Suppose that the lightest CP-even Higgs boson is discovered such that its mass ($m_h$) is precisely known. Then we can solve for one of the free parameters, for example, $\tan\beta$, in terms of the other parameters. Assuming the top-quark mass is well determined by the time when the $e^+e^-$ linear collider is under operation, the unknown parameters for the Higgs sector are then $m_A$ and $ m_{stop}$. The question is, to what extent these parameters are constrained from observable quantities such as the production cross-section and the various branching ratios.
We show that one particular ratio of two branching ratios, $$R_{br}\equiv \frac{Br(h \rightarrow c\bar{c}) + Br(h \rightarrow gg)}
{Br(h \rightarrow b\bar{b})},$$ is especially useful to constrain the heavy Higgs mass scale.[@Kamoshita] In the MSSM, each of the two Higgs doublets couples to either up-type or down-type quarks. Therefore, the ratio of the Higgs couplings to up-type quarks and to down-type quarks is sensitive to the parameters of the Higgs sector, [*i.e.*]{} the angles $\alpha$ and $\beta$ in Section 2. Since the gluonic width of the Higgs boson is generated by a one-loop diagram with an internal top-quark, the Higgs-gluon-gluon coupling is essentially proportional to the Higgs-top coupling. Then $R_{br}$ is proportional to square of the ratio of the up-type and down-type Yukawa coupling constants. Since the up-type (down-type) Yukawa coupling constant contains a factor $\frac{\cos\alpha}{\sin\beta}$, $(-\frac{\sin\alpha}{\cos\beta})$ compared to the SM coupling constant, $R_{br}$ is proportional to $(\tan{\alpha} \tan{\beta})^{-2}$. In Figure 5 $R_{br}$ is shown as a function of $m_A$ for $m_{susy}(\equiv m_{stop})=1,5$ TeV.
From this figure we can see that $R_{br}$ is almost independent of $m_{stop}$. In fact, it can be shown that $R_{br}$ in the MSSM, normalized by $R_{br}$ in the SM, is approximately given by, $$\frac{R_{br}(MSSM)}{R_{br}(SM)}\approx \left(\frac{m_h^2 - m_A^2}
{m_Z^2 + m_A^2}\right)^2$$ for $m_A \gg m_h \sim m_Z$. Measuring this quantity to a good accuracy is therefore important for constraining the scale of the heavy Higgs mass. Note that $R_{br}$ approaches the SM value in the large $m_A$ limit. We can see that $R_{br}$ is reduced by 20$\%$ even for $m_A = 400$ GeV. By simulation study for $e^+e^-$ linear collider experiments it is shown that the sum of the charm and gluonic branching ratios can be determined reasonably well. The statistical error in the determination of $R_{br}$ after two years at an $e^+e^-$ linear collider with $\sqrt{s} = 300$ GeV is 17$\%$.[@Nakamura] We also need to know the theoretical ambiguity of the calculation of the branching ratios in $h \rightarrow b\bar{b}, c\bar{c}, gg$. At the moment the theoretical error in the calculation of $R_{br}$ is estimated to be rather large ($\sim$ 20$\%$) due to uncertainties in $\alpha_s$ and $m_c$.[@Kamoshita; @DSZ] But these uncertainties can be reduced in future from both theoretical and experimental improvements.
Conclusions
===========
I have reviewed some aspects of the Higgs physics in the SUSY SM. I have shown that an future $e^+ e^-$ linear collider is an ideal place to study the SUSY Higgs sector. At earlier stage of the experiment with $\sqrt{s}\sim$ 300 - 500 GeV, it is easy to find a light Higgs boson predicted in SUSY standard models. In particular, both in the MSSM and the SUSY SM with a gauge singlet Higgs, at least one of neutral Higgs bosons is detectable. More importantly, detailed study on properties of the Higgs boson is possible at an $e^+ e^-$ linear collider through measurements of various production cross-sections and branching ratios. As an example we show that the measurement of Higgs couplings to $c\bar{c}$/$gg$/$b\bar{b}$ gives us important information on the Higgs sector of the MSSM. It is therefore very important to build an $e^+ e^-$ linear collider along with LHC, then combining both results we will be able to clarify the Higgs sector of the SM and explore physics beyond the SM such as the SUSY SM.
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[^1]: Talk presented at International Workshop on Frontiers in Quantum Field Theory, August 11-18, 1996, Urumqi, China.
|
---
abstract: |
An extended $U(3)_L\bigotimes U(3)_R$ chiral theory which includes pseudoscalar and vector meson nonets as dynamic degrees of freedom is presented. We combine a hidden symmetry approach with a general procedure of including the $\eta'$ meson into chiral theory. The $U(3)_L\bigotimes U(3)_R$ and the $SU(3)$ symmetries are broken by the mechanism based on quark mass matrix. Meson radiative decay widths are parameterized in terms of a single and real $\eta-\eta'$ mixing angle $\theta_P$, a $U(3)_V$ symmetry breaking scale parameter $c_W$, and the radiative decay constants $F_\pi,~~ F_K, ~~F_\eta, ~~ F_{\eta'}$, for the $\pi , \eta$, K, $\eta '$ mesons, respectively. Taking $F_\pi = 93 MeV$, a global fit to decay width data yields $$\begin{aligned}
F_K/F_\pi = 1.16 \pm 0.11~,&~ F_\eta/F_\pi = 1.14\pm 0.04~,&
~F_{\eta'}/F_\pi = 1.09 \pm 0.04\\
c_W = -0.19 \pm0.03~,& ~ \theta_P =-(15.4 \pm 1.8)^o~. &\end{aligned}$$\
Key Words : Meson radiative decays, Meson decays, Chiral Perturbation Theory,
address: ' Department of Physics, Ben Gurion University, 84105, Beer Sheva, Israel'
author:
- 'E.Gedalin[^1], A.Moalem[^2] and L.Razdolskaya[^3]'
title: Effective Chiral Theory for Radiative Decays of Mesons
---
\
Introduction
============
Meson decays of pseudoscalar and vector mesons have been discussed by several groups[@gilman87; @bramon90; @bramon97; @ball96; @bramon99], using phenomenological approaches based on effective field theory. In particular, the value of the $\eta$-$\eta '$ mixing angle, $\theta_P$, was deduced from the analysis of electromagnetic decays of pseudoscalar and vector mesons, $J/\psi$ decays into a vector and a pseudoscalar meson, and some other transitions. Gilman and Kauffman[@gilman87] assumed SU(3) symmetry and often the stronger condition of nonet symmetry in order to relate the SU(3)-octet wave function to that of the SU(3) singlet, and obtained a value of $\theta_P \cong -20^o$. Less negative a value was extracted by Bramon and Scadron[@bramon90; @bramon97] from a rather similar analysis which takes into account small departure from the $\omega$-$\phi$ ideal mixing. Somewhat different approach was taken by Ball, Frère and Tytgat[@ball96] by relating vector meson decays to the Quantum Electrodynamics (QED) triangle anomaly. More recently, Bramon, Escribano and Scadron [@bramon99] have extracted a value $\theta_P = 15.5^o
\pm 1.3^o$ from a rather exhaustive analysis of data including strong decays of tensor and higher-spin mesons.
Spontaneously broken chiral symmetry plays a major role in low energy hadron physics. The Quantum Chromodynamics (QCD) Lagrangian exhibits an $SU(3)_L\bigotimes SU(3)_R$ chiral symmetry which breaks down spontaneously to $SU(3)_V$, giving rise to a light Goldstone boson octet of pseudoscalar mesons. The corresponding effective field theory (EFT) exhibits the symmetry properties of QCD and involves both of the pseudoscalar meson octet and vector meson nonet as dynamical field variables (see, for example, Ref.[@bramon95]). The axial $U(1)$ symmetry of the QCD Lagrangian is broken by the anomaly. Though considerably heavier than the octet states, it is rather well accepted that the $\eta '$ meson is the most natural candidate for the corresponding pseudoscalar singlet. In this context, we shall introduce the $\eta'$ meson also as a dynamical field variable. We combine the “hidden symmetry approach” of Bando et al.[@bando] with a general procedure of including the $\eta'$ meson into chiral theory[@gasser85; @leut96; @leut97; @herera97] to construct a most general chiral effective Lagrangian with broken $U(3)_L\bigotimes U(3)_R$ local symmetry. To this aim we introduce in section II the whole nonents of pseudoscalar and vector mesons interacting with external electroweak fields. In section III we apply this approach to study radiative decays for anomalous processes, like $V^0 \to P^0 \gamma$, $P^0 \to V^0 \gamma$ and $P^0 \to \gamma \gamma$, with $P^0 = \pi ,\eta ,\eta '$ and $V^0 =
\rho ,\omega ,\phi$. The numerical values of the radiative decay constants, $F_i~ ( i=\pi,\eta,K,\eta ')$ and the $\eta$-$\eta '$ mixing angle, $\theta_P$ are fixed by fitting to experimental rates of these processes. We shall summarize and conclude in section IV.
The effective Lagrangian
========================
In order to include the $\eta'$ meson into chiral effective Lagrangian one has to extend the $SU(3)_L\bigotimes SU(3)_R$ local symmetry of the QCD Lagrangian into $U(3)_L\bigotimes U(3)_R$ local symmetry. This can be achieved by adding to the QCD Lagrangian (herein denoted $L_{QCD}$) a term proportional to the topological charge operator, i.e., $$L = L_{QCD} -
\Theta(x)\frac{g^2}{16\pi^2}Tr_c\left(G_{\mu\nu}\tilde{G}^{\mu\nu}\right)~,
\label{lqcd}$$ where $\Theta(x)$ represents an auxiliary external field, the so called QCD vacuum angle. Here, $\tilde{G}^{\mu\nu} = \epsilon^{\mu\nu\alpha\beta}G_{\alpha\beta};
{G}_{\mu\nu} = \partial_\mu G_\nu - \partial_\nu G_\mu +i[G_\mu, G_\nu]
$, $G_\mu$ being the gluon field, and $Tr_c$ stands for the trace over color indices. Obviously, the Lagrangian $L$ of Eqn. \[lqcd\] has $SU(3)_L\bigotimes SU(3)_R$ local symmetry. It can be shown [@gasser85; @leut96; @leut97; @herera97; @herera98], that $L$ would also have $U(3)_L\bigotimes U(3)_R$ local symmetry provided $\Theta(x)$ transforms under axial $U(1)$ rotations as, $$\Theta(x) \rightarrow \Theta'(x) = \Theta(x) - 2N_f\alpha~,$$ where $N_f$ represents the number of flavors and $\alpha$ the axial $U(1)$ transformation parameter. Indeed, the term generated by the anomaly in the fermion determinant is compensated by the shift in $\Theta(x)$, so that the overall change in the Lagrangian amounts to a total derivative, giving rise to the well known anomaly Wess-Zumino term. An effective field theory Lagrangian which involves an integrated form of this anomaly term would also have this same feature. For more details see Ref. [@bramon95].
We now turn to construct a general chiral effective Lagrangian with $U(3)_L\bigotimes U(3)_R$ local symmetry for pseudoscalar and vector meson nonets interacting with external electroweak fields. As a non-linear representation of a Goldstone nonet we take [@gasser85; @leut96], $$U(P,\eta_0+F_0\Theta) = \xi^2(P,\eta_0+F_0\Theta) =
\exp \left\{i\frac{\sqrt{2}}{F_8}P +
i\sqrt{\frac{2}{3}}\frac{1}{F_0}(\eta_0 + F_0 \Theta){\bf
1}\right\}~,
\label{pmfield}$$ where $P$ is the Goldstone pseudoscalar octet, $$P = \left(\begin{array}{ccc}
\frac{\pi^0}{\sqrt{2}}+\frac{\eta_8}{\sqrt{6}} & \pi^+ &K^+ \\
\pi^- & - \frac{\pi^0}{\sqrt{2}}+\frac{\eta_8}{\sqrt{6}}
&K^0 \\
K^- & \bar{K}^0 &-\frac{2\eta_8}{\sqrt{6}}
\end{array}\right)~.
\label{poctet}$$ and $\eta_0$ the pseudoscalar singlet. The unimodular part of the field $U$ contains the octet degrees of freedom while the phase $detU = \exp {i X} = \exp\left\{i\sqrt{6}(\eta_0/F_0 +
\Theta)\right\}$ describes that of the singlet. The auxiliary field $\Theta(x)$ ascertains that $detU$ is invariant under $U(3)_L\bigotimes U(3)_R$ transformations [@gasser85; @leut96]. The $U(3)_L\bigotimes U(3)_R$ group does not have a dimension-nine irreducible representation; the quantity in the curly bracket of Eqn. \[pmfield\] does not exhibit a nonet symmetry so that the octet ($F_8$) and singlet ($F_0$) decay constants are not necessarily identical. As in Refs. [@gasser88; @krause90; @bernard95] we define vector type $\Gamma _\mu$ and axial-vector type $\Delta_\mu$ covariants $$\begin{aligned}
&& \Gamma_\mu = \frac{i}{2}
\left[\xi^\dagger {\cal D}_\mu \xi - \xi {\cal D}_\mu \xi^\dagger \right] =
\frac{i}{2}\left[\xi^\dagger ,\partial_\mu\xi\right]
+\frac{1}{2}\left(\xi^\dagger r_\mu \xi + \xi l_\mu\xi^\dagger\right) ~,
\label{gammac}
\\
&&\Delta_{\mu} =
\frac{i}{2}\left(\xi^\dagger{\cal D}_{\mu}\xi +
\xi{\cal D}_{\mu}\xi^\dagger\right)=
\frac{i}{2}\left\{\xi^\dagger ,\partial_{\mu}\xi\right\}
+\frac{1}{2}\left(\xi^\dagger r_\mu\xi - \xi l_\mu\xi^\dagger\right) ~,
\label{deltaf}
\end{aligned}$$ with, $$D_\mu \xi = \partial_\mu \xi +i r_\mu \xi - i \xi l_\mu~.$$ Here $r_\mu$ and $l_\mu$ are the relevant external gauge fields of the standard model; $l_\mu= v_\mu + a_\mu$ and $ \quad l_\mu = v_\mu -
a_\mu$, with $v_\mu$ and $a_\mu$ being the vector and axial vector external electroweak fields, respectively. Electroweak interactions are contained in the covariant derivative $D_\mu\xi$. For pure electromagnetic interactions these fields are related to the quark charge operator $Q=diag (2/3,-1/3,-1/3)$ and the photon filed $A_\mu$; $l_\mu = r_\mu = -eQA_\mu$.
Under $U(3)_L\bigotimes U(3)_R$ the field, Eqn. \[pmfield\], transforms as, $$U' = RUL^\dagger,$$ with $R\in U(3)_R, \quad L\in U(3)_L$. The vector ($\Gamma _\mu$) and axial-vector ($\Delta_\mu$) like quantities transform, respectively, as a gauge and matter fields, i.e., $$\begin{aligned}
& & \Gamma '_\mu = K \Gamma_\mu K^\dagger + iK\partial_\mu
K^\dagger ~,
\\
& & \Delta' _\mu = K \Delta_\mu K^\dagger~,
\end{aligned}$$ where $K(U,R,L)$ is a compensatory field representing an element of conserved vector subgroup $U(3)_V$.
The dynamical gauge bosons are defined as a $3 \times 3$ matrix vector field $V_\mu$ which transforms as, $$V'_\mu = KV_\mu K^\dagger + \frac{i}{g} K\partial_\mu K^\dagger \ .$$ Clearly, the vectors $\Gamma_\mu -g V_\mu$ and $\Delta_\mu$ transform homogeneously and at lowest order the Lagrangian can be constructed from the traces $Tr\Delta^2_\mu$ , $Tr(\Gamma_\mu - gV_\mu)^2$, $Tr\Delta_\mu$, $Tr(\Gamma_\mu - gV_\mu)$ and arbitrary functions of the variable $X = \sqrt{2N_f}\eta_0/F_0 +\Theta(x)$, all being invariant under $U(3)_L\bigotimes U(3)_R$ transformations. We may thus conclude that a most general lowest order (i.e. with the smallest number of derivatives) effective chiral Lagrangian can be written in the form [@bando], $$L = L_A + aL_V - \frac{1}{2}Tr(V_{\mu\nu}V^{\mu\nu})~ ,$$ where, $$\begin{aligned}
& & L_A = W_1(X)Tr(\Delta_\mu\Delta^\mu)
+W_4(X)Tr(\Delta_\mu)Tr(\Delta^\mu) +
\nonumber\\
&& W_5(X)Tr(\Delta_\mu)D_\mu\Theta +
W_6(X)D_\mu\Theta D^\mu \Theta~,
\\
& & L_V =
\tilde{W}_1(X)Tr([\Gamma_\mu - gV_\mu][\Gamma^\mu - gV^\mu])+
\nonumber\\
& & \tilde{W}_4(X)Tr(\Gamma_\mu - gV_\mu)Tr(\Gamma^\mu -
gV^\mu)~,
\label{laginv}\end{aligned}$$ and, $$\begin{aligned}
& & D_\mu \Theta = \partial_\mu \Theta + Tr(r_\mu - l_\mu)~,
\\
& & V_{\mu\nu} = \partial_\mu V_\nu - \partial_\nu V_\mu -ig
[V_\mu,V_\nu] ~.
\label{vmunu}\end{aligned}$$ All three terms of the lagrangian $L$ in Eqn. 12 are invariant under $U(3)_L\bigotimes U(3)_R$ transformations. Though in form this Lagrangian is similar to that of Bando et al. [@bando], the expressions for $L_A$ and $L_V$ are different. Namely, the inclusion of the $\eta '$ meson as a dynamical variable involves additional terms with $Tr (\Delta^{\mu})$, $D^{\mu} \Theta$ and $Tr(\Gamma_{\mu} - g V_{\mu})$ and coefficient functions $W_i(X)$ which are absent in the $SU(3)$ limit. We note though that as in Bando et al. [@bando], the Lagrangian $L_A + aL_V$ contains, amongst other contributions, a vector meson mass term $\sim V_\mu V^\mu$, a vector-photon conversion factor $\sim V_\mu A^\mu$ and coupling of both vectors and photons to pseudoscalar pairs. The latter can be eliminated by choosing $a=2$, a choice which allows incorporating the conventional vector-dominance in electromagnetic form-factors of pseudoscalar mesons[@bando].
The mass degeneracy is removed via the additional of pseudoscalar mass term. The most general expression of a (local) $U(3)_L\bigotimes U(3)_R$ symmetry violating term reads [@gasser85; @leut96; @herera97], $$L_m = -W_0(X) + W_2(X)Tr \chi_+ +iW_3(X) Tr\chi_-~,
\label{maslag}$$ with, $$\begin{aligned}
\chi_\pm = 2B_0(\xi{\cal M^\dagger}\xi
\pm \xi^\dagger {\cal M} \xi^\dagger)~,
&& \qquad B_0 = m_\pi^2 / (m_u +m_d)~, \end{aligned}$$ and ${\cal M} = diag (m_u, m_d, m_s)$ is the quark mass matrix. Parity conservation implies that all $W_i$ and $\tilde{W}_i$ are even functions of the variable $X$ except $W_3$ which is odd. The correct normalization of the $U(3)_L\bigotimes U(3)_R$ invariant kinetic term requires that $W_1(0) = F^2_8, \quad W_4(0 )= (F^2_0 - F^2_8)/3$ and $W_2(0) = F^2_8$ to ensure the standard $\chi$PT pion mass term.
One possible way to incorporate $SU(3)$ symmetry breaking is to introduce a universal matrix $B$ proportional to $\chi_+$, i.e., $$B = \frac{1}{4B_0(2m + m_s)} \chi_+~.
\label{breakm}$$ For simplicity we take the exact isospin symmetry limit $m_u = m_d = m$. Symmetry breaking terms to be added to $L_A$ and $L_V$ can be constructed either as conserving, or alternatively, as non-conserving the quadratic form of the Golstone meson kinetic terms. In what follows we develop the former alternative by including terms which break the octet symmetry only. The latter procedure is worked out in the Appendix. Let, $$U_8 = \xi^2_8 = \exp(i \frac{\sqrt{2}}{F_8}P)
\label{u8}$$ be the pure octet matrix and let, $$\bar{\Delta}_\mu =
\frac{i}{2}\left\{\xi^\dagger_8 ,\partial_{\mu}\xi_8\right\}
+\frac{1}{2}\left(\xi^\dagger_8 r_\mu\xi_8 -
\xi_8 l_\mu\xi^\dagger_8\right) ~,
\label{bdelta}$$ be the octet covariant. Then general $SU(3)$ symmetry breaking Lagrangians $\bar{L}_A$ and $\bar{L}_V$ would be, $$\begin{aligned}
& &\bar{L}_A =
\nonumber\\
&& W_1(X)\left( c_A Tr (\{B, \bar{\Delta}_\mu\}
\bar{\Delta}^\mu) +
d_A Tr (B \bar{\Delta}_\mu B\bar{\Delta}^\mu)\right)+
\nonumber \\
&& W_4(X)
d_A Tr (B \bar{\Delta}_\mu) Tr (B\bar{\Delta}^\mu)~,
\label{labreak} \\
& & \bar{L}_V =
\bar{W}_1(X)\left(c_V Tr (B[\Gamma_\mu -gV_\mu]
[\Gamma^\mu -gV^\mu])\right. +
\nonumber\\
&& \left. d_V Tr (B[\Gamma_\mu-gV_\mu] B [\Gamma^\mu -gV^\mu])\right) +
\nonumber\\
&&\tilde{W}_4(X)\left(c_VTr(\Gamma_\mu - gV_\mu)Tr(B[\Gamma^\mu -
gV^\mu])\right.
\nonumber\\
&& +\left. d_VTr(B[\Gamma_\mu - gV_\mu])Tr(B[\Gamma^\mu -
gV^\mu])\right)~.
\label{lvbreak}\end{aligned}$$ where $c_A, c_V, d_A, d_V$ are arbitrary constants. We stress that $\bar{L}_A$ and $\bar{L}_V$ differ from those of Bramon et al.[@bramon95] and Bando et al. [@bando]. First, like our symmetric $L_A$ and $L_V$ the asymmetric $\bar{L}_A$ and $\bar{L}_V$ parts involve additional terms which are absent in the $SU(3)$ limit. Secondly, the terms proportional to $d_A$ and $d_V$ were included by Bando et al. [@bando] but with $d_i =
c_i^2$. Thirdly, our symmetry breaking matrix B is not constant as in ref. [@bando] though similar (but not identical) to that of Bramon et al.[@bramon95].
The full Lagrangian may now be written in the form, $$L = L_A + \bar{L}_A + a(L_V + \bar{L}_V) + L_m + L_{WZW} -
\frac{1}{2}Tr(V_{\mu\nu}V^{\mu\nu}) ~,
\label{elag}$$ where we included the well known Wess-Zumino-Witten term $L_{WZW}$. This corresponds to the action defined as [@witten83; @callan84] $$\begin{aligned}
& & S_{WZW} = -\frac{i}{80\pi^2}\int_{M^2} d^5x
\epsilon^{ijklm}Tr(\Sigma^L_i\Sigma^L_j\Sigma^L_k\Sigma^L_l\Sigma^L_m)
\nonumber\\
& & -\frac{i}{16\pi^2}\int d^4x \epsilon^{\mu\nu\alpha\beta}
\left[W(U,l,r)_{\mu\nu\alpha\beta} -
W(1,l,r)_{\mu\nu\alpha\beta}\right]~,\end{aligned}$$ with, $$\begin{aligned}
& & W(U,l,r)_{\mu\nu\alpha\beta} =
Tr[U l_\mu l_\nu l_\alpha l_\beta +
\frac{1}{4}U l_\mu U^\dagger r_\nu U l_\alpha U^\dagger r_\beta
\nonumber\\
& & + iU\partial_\mu l_\nu l_\alpha U^\dagger r_\beta
+ iU\partial_\mu r_\nu l_\alpha U^\dagger r_\beta
- i\Sigma^L_\mu l_\nu U^\dagger r_\alpha U l_\beta
\nonumber\\
& & + \Sigma^L_\mu U^\dagger \partial_\nu r_\alpha U l_\beta
- \Sigma^L_\mu \Sigma^L_\nu U^\dagger r_\alpha U l_\beta
+ \Sigma^L_\mu l_\nu\partial_\alpha l_\beta
+ \Sigma^L_\mu \partial_\nu l_\alpha l_\beta
\nonumber\\
& & - i\Sigma^L_\mu l_\nu l_\alpha l_\beta
+ \frac{1}{2} \Sigma^L_\mu l_\nu \Sigma^L_\alpha l_\beta
- i\Sigma^L_\mu \Sigma^L_\nu \Sigma^L_\alpha l_\beta -
(L \leftrightarrow R)]~,
\label{wzw}\end{aligned}$$ where $\Sigma^L_\mu = U^\dagger \partial_\mu U$, $\Sigma^R_\mu = U\partial_\mu U^\dagger$, and $(L \leftrightarrow R)$ stands for a similar expression with $U$, $l$ and $\Sigma$ interchanged according to, $$(U \leftrightarrow U^\dagger),\qquad (l \leftrightarrow r),
\qquad (\Sigma^L_\mu \leftrightarrow \Sigma^R_\mu )~.$$ Note that this expression involves Lagrangian terms up to fifth chiral order, only. Other terms which accounts for the regularization of the one loop contributions are listed in Refs.[@gasser85; @bijnens90; @bijnens901].
For the purpose of treating radiative decays, one can safely neglect the auxiliary field $\Theta(x)$ and the quantities $W_i$ and $\bar{W}_i$ become functions of $\eta_0$ only. To lowest order their expansions read, $$\begin{aligned}
& & W_0 = const + F^4_8 w_0 \frac{\eta^2_0}{F^2_0} + \ldots~,
\label{wco} \\
& & W_1 = F^2_8(1 + w_1 \frac{\eta^2_0}{F^2_0} + \ldots)~,
\label{wc1} \\
& & W_2 = \frac{F^2_8}{4}(1 + w_2 \frac{\eta^2_0}{F^2_0}+ \ldots)~,
\label{wc2} \\
& & W_3 = \frac{ F^2_8}{2}(w_3 \frac{\eta_0}{F_0} + \ldots)~,
\label{wc3} \\
& & W_4 = \frac{F^2_0 - F^2_8}{3}(1 + w_4 \frac{\eta^2_0}{F^2_0}
+ \ldots)~,
\label{wc4} \\
& & \tilde{W}_1 = F^2_8(1 + \tilde{w}_1\frac{\eta^2_0} {F^2_0}+
\ldots)~,
\\label{wwc1}¥ \\
&&\tilde{W}_4 = F^2_8(\tilde{w}_4 + \tilde{w}'_4\frac{\eta^2_0}{F^2_0}+
\ldots~).
\label{wwc4}
\end{aligned}$$ By substituting these expressions into Eqn. \[elag\], the kinetic terms of the pseudoscalar mesons is, $$\begin{aligned}
& & L_{kin} = \frac{1}{2}\left(1 + c_A \frac{2m}{2m+m_s}
+ d_A \frac{m^2}{(2m+m_s)^2}\right)(\partial_\mu \vec{\pi})^2+
\nonumber\\
& & \frac{1}{2}\left(1 + c_A \frac{m+m_s}{2m+m_s}
+ d_A \frac{mm_s}{(2m+m_s)^2}\right)\sum_{i}(\partial_\mu K_i)^2+
\nonumber\\
& & \frac{1}{2}\left(1 + c_A\frac{1}{3} \frac{m+2m_s}{2m+m_s}
+ d_A \frac{1}{3}\frac{m^2+2m^2_s}{(2m+m_s)^2}\right)
(\partial_\mu \eta_8)^2+
\frac{1}{2}(\partial_\mu \eta_0)^2~.
\label{lkin}
\end{aligned}$$ To restore the standard normalization of the kinetic term we rescale the pseudoscalar fields according to, $$\pi\Rightarrow z_\pi\pi,\qquad K\Rightarrow z_s K,\qquad \eta_8
\Rightarrow z_8\eta_8~,
\label{rescal}$$ with, $$\begin{aligned}
& & z_\pi \equiv \frac {F_8}{F_\pi}
= \frac{1}{\sqrt{1 + c_A \frac{2m}{2m+m_s}
+ d_A \frac{m^2}{(2m+m_s)^2}}}~,
\label{zpi} \\
& & z_s \equiv \frac {F_8}{F_K}
= \frac{1}{\sqrt{1 + c_A \frac{m+m_s}{2m+m_s}
+ d_A \frac{mm_s}{(2m+m_s)^2}}}~,
\label{zs} \\
& & z_8 \equiv \frac {F_8}{F_\eta}
= \frac{1}{\sqrt{1 + c_A\frac{1}{3} \frac{m+2m_s}{2m+m_s}
+ d_A \frac{1}{3}\frac{m^2+2m^2_s}{(2m+m_s)^2}}}~.
\label{zet}\end{aligned}$$ After some algebraic manipulations the octet field matrix can be written in the form, $$P = \left(\begin{array}{ccc}
\frac{\pi^0}{\sqrt{2}}+\bar{z}_8\frac{\eta_8}{\sqrt{6}}
& \pi^+ &\bar{z}_sK^+ \\
\pi^- & - \frac{\pi^0}{\sqrt{2}}+\bar{z}_8\frac{\eta_8}{\sqrt{6}}
&\bar{z}_sK^0 \\
\bar{z}_sK^- & \bar{z}_s\bar{K}^0 &-\bar{z}_8\frac{2\eta_8}{\sqrt{6}}
\end{array}\right)~,
\label{poctet}$$ where, $$\bar{z}_8 = z_8/z_\pi, \qquad \bar{z}_s = z_s/z_\pi ~.$$ In addition to the usual quadratic terms $\eta^2_8$ and $\eta^2_0$, the quantity $W_2(X)Tr\chi_+ +iW_3(X)Tr\chi_-$ in the mass term, Eqn. \[maslag\] gives rise to a mixing term $\sim \eta_8\eta_0$ which violates the orthogonality of the $\eta_8$ and $\eta_0$ states. The mass matrix is diagonalized via the usual unitary transformation [@pdg98] $$\begin{aligned}
& & \eta_8 = \eta \cos \theta_P + \eta'\sin \theta_P~,
\\
& & \eta_0 = - \eta \sin \theta_P + \eta' \cos \theta_P~, \end{aligned}$$ where $ \theta_P$ is the $\eta$-$\eta '$ mixing angle.
In terms of the physical fields $\eta$ and $\eta'$, the nonlinear representation of the pseudoscalar particles can now be written as, $$U = \exp{i\frac{\sqrt{2}}{F_\pi}{\cal P}}~,
\label{ufield}$$ where $\cal P$ stands for the pseudoscalar nonet matrix, $${\cal P} = \left(\begin{array}{ccc}
\frac{\pi^0}{\sqrt{2}}+\frac{1}{\sqrt{6}}(X_\eta \eta +
X_{\eta'}\eta') & \pi^+ &\bar{z}_sK^+ \\
\pi^- & - \frac{\pi^0}{\sqrt{2}}+ \frac{1}
{\sqrt{6}}(X_\eta \eta +X_{\eta'}\eta') &\bar{z}_sK^0 \\
\bar{z}_sK^- & \bar{z}_s\bar{K}^0 &\frac{1}{\sqrt{6}}
(Y_\eta \eta + Y_{\eta'}\eta')
\end{array}\right)~,
\label{pnonet}$$ with, $$\begin{aligned}
X_\eta = \cos\theta_P(\bar{z}_8 - \sqrt{2}\bar{r}\tan\theta_P)~, &
\quad &X_{\eta'} = \cos\theta_P(\bar{z}_8\tan\theta_P +\sqrt{2}\bar{r})~,
\nonumber \\
Y_\eta = \cos\theta_P(-2\bar{z}_8 -
\sqrt{2}\bar{r}\tan\theta_P)~,
&\quad &Y_{\eta'} = \cos\theta_P(-2\bar{z}_8\tan\theta_P +
\sqrt{2}\bar{r})~,
\label{xy}
\end{aligned}$$ and $\bar{r} = F_8 / F_0~$.
Similarly, the vector nonet with ideal mixing has the form[@pdg98], $$V = \left(\begin{array}{ccc}
\frac{\rho^0}{\sqrt{2}}+\frac{\omega}{\sqrt{2}} & \rho^+ & K^{*+} \\
\rho^- & - \frac{\rho^0}{\sqrt{2}}+\frac{\omega}{\sqrt{2}} & K^{*0} \\
K^{*-} & \bar{K}^{*0} &\phi
\end{array}\right)~.
\label{vnonet}$$ In order to account for a strange (non strange) admixture in $\omega (\phi)$ we substitute $$\begin{aligned}
& &\omega \rightarrow \omega - \epsilon' \phi ~, \\
& &\phi \rightarrow \phi +\epsilon' \omega ~.\end{aligned}$$
Radiative Decay Widths
=======================
We now turn to calculate radiative decay widths for $P^0 \rightarrow \gamma \gamma$, $V^0 \rightarrow P^0 \gamma$ and $P^0 \rightarrow V^0 \gamma$, with $P^0 = \pi ,\eta ,\eta ' , K$ and $V^0 = \rho ,\omega ,\phi , K^*$ using the formalism outlined above. We generalize the treatment of Ref.[@bramon95] by incorporating “indirect” symmetry breaking effects via pseudoscalar and vector nonet matrices Eqns. \[pnonet\] and \[vnonet\] and “direct” symmetry breaking terms ( such as $\bar{L}_A$ and $\bar{L_V}$). The Lagrangian is factorized in the form, $$\begin{aligned}
& & L_{P\gamma\gamma} = L^{(s)}_{P\gamma\gamma} +
c_W L^{(b)}_{P\gamma\gamma},
\\
& & L_{VP\gamma} = L^{(s)}_{VP\gamma} +
c_W L^{(b)}_{VP\gamma}~,
\label{dlag}\end{aligned}$$ where $L^{(s)}$ and $L^{(b)}$ are generic for indirect and direct symmetry breaking contributions, and $c_W$ is a symmetry breaking parameter. In order to write these terms explicitly, we consider first the indirect anomalous Lagrangian[@bijnens90], $$L^{(s)}_{anomalous} = L^{(0)}_{VVP} +
L_{WZW}(P\gamma\gamma)~.
\label{dlag0}$$ From the four covariants $\Delta_\mu$, $\Gamma_\mu-gV_\mu$, $V_{\mu \nu}$ and $ \Gamma_{\mu\nu} = \partial_\mu \Gamma_\nu -
\partial_\nu \Gamma_\mu - i[\Gamma_\mu, \Gamma_\nu]$, the Lagrangian $L^{(0)}_{VVP}$ can have at most six terms, $$\begin{aligned}
&& L^{(0)}_{VVP} = g_1\epsilon^{\mu\nu\alpha\beta}
Tr(V_{\mu\nu}
[V_\alpha -\frac{1}{g}\Gamma_\alpha]\Delta_\beta) +
g_2\epsilon^{\mu\nu\alpha\beta}Tr(\Gamma_{\mu\nu}
[V_\alpha -\frac{1}{g}\Gamma_\alpha]\Delta_\beta) +
\nonumber \\
&& g_3\epsilon^{\mu\nu\alpha\beta}Tr(V_{\mu\nu})
Tr([V_\alpha -\frac{1}{g}\Gamma_\alpha]\Delta_\beta) +
g_4 \epsilon^{\mu\nu\alpha\beta}Tr(\Gamma_{\mu\nu})
Tr([V_\alpha -\frac{1}{g}\Gamma_\alpha]\Delta_\beta) +
\nonumber \\
&& g_5\epsilon^{\mu\nu\alpha\beta}Tr(V_{\mu\nu})
Tr(V_\alpha -\frac{1}{g}\Gamma_\alpha )Tr(\Delta_\beta) +
\nonumber \\
&& g_6\epsilon^{\mu\nu\alpha\beta}Tr(\Gamma_{\mu\nu})
Tr(V_\alpha -\frac{1}{g}\Gamma_\alpha )Tr(\Delta_\beta)~,
\label{vvplag}\end{aligned}$$ where $g_i,~~~i=1,...6$ are arbitrary functions of the variable X. We recall that $\Gamma_\mu$ involves a term proportional to the photon field $A_\mu$. By rearranging contributions to $VP\gamma$ and $P\gamma\gamma$ interaction terms we obtain, $$\begin{aligned}
& & L^{(s)}_{VP\gamma}=
g_V \frac {e}{F_\pi}\epsilon^{\mu\nu\alpha\beta}\partial_\mu
A_\nu Tr( Q\{\partial_\alpha V_\beta, {\cal P}\})~,
\label{vpgamma}\\
& & L^{(s)}_{P\gamma\gamma} =
g_P\frac{e^2}{2F_\pi}\epsilon^{\mu\nu\alpha\beta}\partial_\mu
A_\nu\partial_\alpha A_\beta Tr(\left\{ Q^2,{\cal P}\right\})~.
\label{pgamma} \end{aligned}$$ For convenience we have introduced coupling constants $g_V$ and $g_P$ which incorporate all relevant contributions to $ L^{(s)}_{VP\gamma}$ and $ L^{(s)}_{P\gamma\gamma}$. It is now rather easy to obtain the direct symmetry breaking terms by introducing the quantity B, Eqn. \[breakm\], as described in the previous section, i.e., $$\begin{aligned}
& & L^{(b)}_{VP\gamma} =
g_V \frac {e}{F_\pi}\epsilon^{\mu\nu\alpha\beta}\partial_\mu
A_\nu Tr( Q \{B,\{\partial_\alpha {V}_\beta ,{\cal P}\}\})~,
\label{vpgammab} \\
& & L^{(b)}_{P\gamma\gamma}=
g_P\frac{e^2}{2F_\pi}\epsilon^{\mu\nu\alpha\beta}\partial_\mu
A_\nu\partial_\alpha A_\beta Tr(\left\{ Q^2,\{B,{\cal P}\}\right\})~.
\label{ptwogammab}\end{aligned}$$
The $V\to P \gamma$ and $P \to V \gamma$ Processes
--------------------------------------------------
The relevant vertices are, $$V(VP\gamma) =
- i g_V\frac{e}{F_\pi}w(VP)\epsilon^{\mu\nu\alpha\beta}k_\mu
e^{(\gamma)}_\nu p_\alpha e^{(V)}_\beta ~,
\label{vpgamma}$$ where $ e^{(V)}_\nu$ ($p$) and $ e^{(\gamma)}_\nu$ ($k$) are the polarization (four-momentum) of the vector meson and final photon, respectively. With the quark mass ratios advocated by Weinberg[@wein77] $m_u:m_d:m_s= 0.55:1.0:20.3$, the ratio $m:m_s=(m_u+m_d):2m_s= 0.038$ is rather small and terms proportional to $c_W m/(2m+m_s)$ can be neglected.[^4]. Then for the ${\cal P}$- matrix of Eqns. \[pnonet\], \[xy\] one obtains, $$\begin{aligned}
w(\rho \pi) = \frac{1}{3}~,& \quad w(\rho \eta) =
\frac{1}{\sqrt{3}} X_\eta ~, &\quad w(\rho \eta') = \frac{1}{\sqrt{3}}
X_{\eta'}~,
\label{rhop}\\
w(\omega \pi) = 1~,& \quad w(\omega \eta) = - \frac{1}{3\sqrt{3}}
X_\eta~,& \quad w(\omega \eta') = \frac{1}{3\sqrt{3}} X_{\eta'}~,
\label{omegap}\\
w(\phi \pi) = \epsilon'~,&\qquad w(\phi \eta) = -
\frac{\sqrt{2}}{3\sqrt{3}}
Y_\eta (1+ c_W \frac{m_s}{2m+m_s})~,&
\nonumber\\
& \qquad w(\phi \eta') = - \frac{\sqrt{2}}{3\sqrt{3}}
Y_{\eta'}(1 + c_W\frac{m_s}{2m+m_s})~,&
\label{phip}\end{aligned}$$ $$\begin{aligned}
&& w(K^{*0}K^0) = w(\bar{K}^{*0}\bar{K}^0)
= -\frac{2}{3}\bar{z}_s (1+\frac{1}{2}c_W \frac{m_s}{2m+m_s})~,
\label{kstk0}\\
&& w(K^{*+}K^+) = w(K^{*-}K^-)
= \frac{1}{3}\bar{z}_s (1-c_W \frac{m_s}{2m+m_s})~.
\label{kstpkp} \end{aligned}$$
In terms of these vertices the decay widths of $V \rightarrow P \gamma$ and $P \rightarrow V \gamma$ are, $$\begin{aligned}
& & \Gamma (VP\gamma) = G\frac{(m^2_V - m^2_P)^3}{m^3_V F^2_8}
|w(VP)|^2~,
\label{gvp} \\
& & \Gamma (P V\gamma) =3 G\frac{(m^2_{P} - m^2_V)^3}
{m^3_{P} F^2_8}|w(V P)|^2~,
\label{pvg}
\end{aligned}$$ with, $$G = \frac{e^2}{4\pi}\frac{g^2_V}{24}~.$$
The $P\rightarrow \gamma\gamma$ decays
--------------------------------------
The relevant vertices are, $$V(P\gamma\gamma) =
- 2i g_P\frac{e^2}{F_8}\bar{w}(P)\epsilon^{\mu\nu\alpha\beta}k_{1\mu}
e^{(\gamma)}_\nu k_{2\alpha} e^{(\gamma)}_\beta ~,$$ where $ e^{(\gamma)}_\nu$ and $ e^{(\gamma)}_\alpha$ are the polarizations of the final photons , $k_1$ and $k_2$ are their corresponding four-momenta. For the $\pi , \eta $ and $\eta '$ one has, $$\begin{aligned}
& & \bar{w}(\pi) = \frac{3}{\sqrt{2}}~,
\label{wpi}¥ \\
& & \bar{w}(\eta) =
\nonumber\\
&&\frac{3\cos \theta_P}{\sqrt{6}}\left[\bar{z}_8(1-\frac{4}{3}c_W
\frac{m_s}{2m+m_s})
-2\sqrt{2}(1+\frac{1}{3}c_W \frac{m_s}{2m+m_s})\bar{r}\tan \theta_P
\right]~,
\label{weta} \\
& & \bar{w}(\eta') =
\nonumber \\
&&\frac{3\cos\theta_P}{\sqrt{6}}\left[\bar{z}_8(1-\frac{4}{3}c_W
\frac{m_s}{2m+m_s})\tan\theta_P
+2\sqrt{2}(1+\frac{1}{3}c_W \frac{m_s}{2m+m_s}) \bar{r}\right]~.
\label{weta'}¥
\end{aligned}$$ With these vertices the decay rate is given by, $$\Gamma (P\gamma\gamma) = \bar{G}\frac{ m^3_P}{ F^2_8}|\bar{w}(P)|^2~,
\label{p2gamma}¥$$ with, $$\bar{G} = \frac{\pi}{2}\left(\frac{e^2}{4\pi}\right)^2\left(\frac{g_P }
{9}\right)^2~.$$
Numerical Analysis and Results
------------------------------
The decay width of the anomalous processes mentioned above are described in terms of coupling constants ($g_V ,g_P$), pseudoscalar singlet-octet mixing angle ($\theta_P$), relative radiative decay constants (${\bar r}, {\bar z}_s, {\bar z}$) and direct $U(3)_V$ symmetry breaking scale ($c_W$). The numerical values of these parameters can be fixed from experimental decay rates. The value of $g_V$ is determined from the $\omega \rightarrow \pi \gamma$ decay, $\Gamma(\omega\pi\gamma) = G (m^2_\omega - m^2_\pi)^3/
(m^3_\omega F^2_8) = (716 \pm 43) KeV$, to have, $$\begin{aligned}
G=(1.44\pm0.04)\cdot10^{-5}~, &\qquad &
g_V = 0.22 \pm 0.006~.
\label{gvec}
\end{aligned}$$ From the $\rho \rightarrow \pi \gamma$ decay, $\Gamma(\rho\pi\gamma) = 76\pm 10
KeV$ one obtains practically identical value for $g_V$. Similarly, the decay $\pi \rightarrow \gamma \gamma$ can serve to fix $g_P$. From the experimental decay width, $ \Gamma(\pi^0\gamma\gamma) = 9 \bar{G} m^3_\pi/2F^2_8
= (7.8 \pm 0.55) eV$, one obtains, $$\begin{aligned}
\bar{G}=(4.9\pm0.07)\cdot10^{-8}, &\qquad &
g_P = 0.073 \pm 0.001~.
\label{gps}
\end{aligned}$$ The value of the symmetry breaking scale $c_W$ can be fixed from the ratio, $$\begin{aligned}
&&\frac{\Gamma (K^{*0}K^0\gamma)}{\Gamma (K^{*+}K^+\gamma)} =
4\left[\frac{1+\frac{1}{2}c_W }{1-c_W}\right]^2 =
\frac{(117\pm10)KeV}{(50\pm 5)KeV} = 2.34\pm 0.43~,
\label{cwpar}
\end{aligned}$$ which yields $c_W = -0.19 \pm 0.04$. From equating $\Gamma(K^{*0}K^0\gamma)$ to its experimental value one finds $\bar{z}_s = 0.86 \pm 0.08$ a value corresponding to $F_K = (1.16 \pm 0.11)F_\pi$. In fact, from the ratio, $$\begin{aligned}
& & \frac{\Gamma(\phi\pi^0\gamma)}{\Gamma(\omega\pi^0\gamma)} =
\epsilon'^2 \left[\frac{(m^2_\phi - m^2_\pi)m_\omega}
{(m^2_\omega - m^2_\pi)m_\phi}\right]^3 =
\frac{(5.8 \pm 0.6)KeV}{(716\pm 43) KeV} = 0.008 \pm 0.001\end{aligned}$$ one obtains, $ |w(\phi \pi^0)| = 0.059 \pm 0.005$, a value identical to that quoted previously by Bramon et al. [@bramon95] from using the vector meson dominance.
The remaining parameters $\bar{z}$, $\bar{r}$ and $\phi_P$ can now be calculated using Eqns. \[gvp\], \[pvg\], \[p2gamma\] and the experimental decay widths $\Gamma(\eta\gamma\gamma)$, $\Gamma(\phi\eta\gamma)$ and $\Gamma(\eta'\gamma\gamma)$. From the ratios $\Gamma(\eta\gamma\gamma)/ \Gamma(\phi\eta\gamma)$, $\Gamma(\eta\gamma\gamma)/\Gamma(\pi^0\gamma\gamma)$, and $\Gamma(\eta'\gamma\gamma)/\Gamma(\eta\gamma\gamma)$, one obtains, $$\begin{aligned}
\bar{z} = 0.92\pm0.06~, &\quad \bar{r} = 0.97 \pm0.06~,
& \quad\theta_P = -(15 \pm 2.4)^o~.
\label{¥}
\end{aligned}$$ These values (hereafter we refer to as solution I) are listed in Table \[fit\]. Rather similar values (solution II of Table \[fit\]) one deduced from a global fit of the data listed in Table \[vpgpred\]. Predictions of decay widths as obtained with the solutions I and II of Table \[fit\] are summarized in Table \[vpgpred\].
From Eqns. \[rhop\]-\[phip\] and Eqns. \[wpi\] - \[weta’\] one can see that for fixed $c_W$ the width ratios $\Gamma(\eta'\gamma\gamma)/ \Gamma(\eta\gamma\gamma)$, $\Gamma(\phi\eta'\gamma)/ \Gamma(\phi\eta\gamma)$, $\Gamma(\eta'\omega\gamma)/ \Gamma(\omega\eta\gamma)$ and $\Gamma(\eta'\rho\gamma)/ \Gamma(\rho\eta\gamma)$ depend on $\tan\theta_P$ and $r/z$. Precision measurements of these ratios would be very useful to obtain more accurate values for $\theta_P$ and $r/z$. Perhaps even more attractive quantities are the ratios $\Gamma(\eta'\gamma\gamma)/ \Gamma(\eta\gamma\gamma)$, $\Gamma(\phi\eta'\gamma)/ \Gamma(\phi\eta\gamma)$ which can now be determined at DA$\Phi$NE with high precision.
Our set of parameters agree with the results of Bramon et al.[@bramon95] and Venugopal et al. [@veno98] except for the mixing angle which differs significantly from Bramon et al.[@bramon95] and Venugopal et al. [@veno98] but agrees with more recent analysis of Bramon et al.[@bramon97] and Escribano et al.[@escrib99].
How significant are the departure of these parameters from their values in the limit exact $U(3)_L\bigotimes U(3)_R$ symmetry? Clearly, the exact $SU(3)$ limit, i.e., $c_W =0, F_\pi = F_K = F_8$ is inconsistent with the ratio $\Gamma (K^{*0}K^{0}\gamma)/
\Gamma (K^{*+}K^{+}\gamma)$; a value of $c_W = 0$ predicts a ratio equals 4 as opposed to the experimental value of $2.34 \pm 0.43$. Furthermore, with $c_W = - 0.19$ and with $F_K = F_\pi$ one obtains $\Gamma
(K^{*0}K^{0}\gamma)$ about 35$\%$ higher than experimental value and far beyond the measurement accuracy. We may thus conclude that data requires $SU(3)$ symmetry to be broken directly ($c_W \neq 0$) and indirectly ($F_K \neq F_\pi$). If either direct or indirect symmetry breaking is not included, the quality of the fit deteriorates significantly. The value of the mixing angle $\theta_P$ is rather sensitive to direct symmetry breaking. At the limit $F_0 = F_8$ the mixing angle varies from $\theta_P = -23^o $ at $c_W = 0$ to $\theta_P = -16^o $ at $c_W =
-0.19$. The mixing angle is less sensitive to indirect symmetry breaking. Indeed, a global fit which neglects indirect symmetry breaking,i.e., with $F_K = F_\pi = F_8 = F_0$ gives $c_W = -0.22$ and $\theta_P = -16.2^o$. Also, a global fit which assumes broken SU(3) symmetry but with nonet symmetry , i.e., with $F_0 = F_8 =F_K = F$ but $F \neq F_\pi$ yields $F = 1.1 F_\pi$, $c_w = -0.20$ and $\theta_P = -14.6^o$, rather close to the values of solution II. Upon concluding we stress that confidence criterion favorables our solution II, i.e., with direct and indirect symmetry breaking.
Summary and Discussion
======================
¥
In this paper, using the hidden symmetry approach of Bando et al.[@bando] combined with general procedure of including the $\eta'$ meson into $\chi$PT[@gasser85; @leut96; @leut97; @herera97] we have constructed an effective Lagrangian which incorporates pseudoscalar and vector meson nonets as dynamical degrees of freedom interacting with external electroweak fields. At lowest order the Lagrangian $L$ is a linear combination of three parts $L_A$, $L_V$ and a vector nonet “kinetic” term $\frac{1}{2}Tr(V_{\mu\nu}V^{\mu\nu})$, all of which possessing a $U(3)_L\bigotimes U(3)_R$ and a local (hidden) $U(3)_V$ symmetry. The $L_A$ and $L_V$ parts involve the pseudoscalar and vector meson fields and their interactions with external electroweak fields, respectively. Though in form this division of the Lagrangian is identical to that of Bando et al.[@bando], the expressions for $L_A$ and $L_V$ are different, as they include the $\eta '$ meson as a dynamical variable also. The symmetry breaking effects are included via the pseudoscalar meson mass term as well as direct symmetry breaking terms ${\bar
L}_A$ and ${\bar L}_V$ in a fashion similar to that proposed by Bramon et al. [@bramon95]. These terms are constructed by introducing a universal matrix $B$ which is proportional to pseudoscalar meson mass matrix into our general expressions for $L_A$ and $L_V$. The symmetry breaking leads to the mass splitting for the pseudoscalar and vector meson nonets, $\eta-\eta'$ and $\omega-\phi$ mixing effects, $F_\pi\neq F_K\neq F_\eta \neq F_{\eta'}$ etc. We may thus conclude that the Lagrangian of Eqn.\[elag\] provides a basis for an effective perturbative chiral theory capable to describe interacting pseudoscalar and vector mesons. To demonstrate that, we have considered anomalous radiative decay processes within our approach. Namely, the decay widths of anomalous processes are calculated by taking into account indirect as well as direct symmetry breaking effects. The widths were parameterized in terms of five parameters, including a symmetry breaking scale $c_W$, pseudoscalar meson weak decay constants $F_K,~~~F_\eta,~~~F_{\eta'}$ and the $\eta-\eta'$ meson mixing angle $\theta_P$. Our analysis show that the value of the mixing angle $\theta_P$ is rather sensitive to the presence of a direct symmetry breaking. The best solution was obtained with $c_W = -0.19$ suggesting a value $\theta_P \approx -(15.4 \pm 1.8)^o$. This agrees with the value extracted by Bramon et al. [@bramon99] from rather exhaustive analysis of data. Our analysis provides evidence for a broken $U(3)$ symmetry with $F_0
\neq F_8$ and $F_K \neq F_\eta \neq F_\pi$.
This work was supported in part by the Israel Ministry of Absorption.
Solution I Solution II From Ref.[@bramon95] From Ref.[@veno98]
----------------- ------------------- -------------------- ---------------------- --------------------
$c_W$ $-0.19\pm0.04$ $-0.19\pm 0.03 $ $-0.2 \pm 0.06$
$F_K/F_\pi$ $1.16\pm 0.11$ $1.16\pm 0.05$ $1.22\pm 0.02$ $1.38\pm 0.22$
$F_\eta/ F_\pi$ $1.09\pm 0.07$ $1.14\pm 0.04 $ $1.06\pm 0.08$
$F_0/F_\pi$ $1.03\pm 0.06$ $1.11\pm 0.04 $ $1.06 \pm 0.03$
$\theta_P$ $-(15.\pm 2.4)^o$ $-(15.4\pm 1.8)^o$ $ - 19.5^o$ $-(22\pm 3.3)^o$
: Values of parameters.
\
\[fit\]
----------------------------------- -------------------------- ---------------------- ----------------------
Decay $\Gamma_{exp}$(KeV) $\Gamma_{calc}$(KeV) $\Gamma_{calc}$(KeV)
solution I solution II
$\rho\rightarrow \pi\gamma$ $76\pm10$ $76\pm 12$ $76\pm 12$
$\omega\rightarrow \pi\gamma$ $716\pm43$ \* $716\pm43$ \* $716\pm43$
$\rho\rightarrow \eta\gamma$ $58\pm11$ $47\pm 11$ $42\pm 10$
$\omega\rightarrow \eta\gamma$ $7.0\pm1.8$ $6.\pm 1.6$ $5.5\pm 1.5$
$\phi\rightarrow \eta \gamma$ $56.7\pm2.8$ $61.4\pm 3$ $56.2 \pm 2.8$
$\phi\rightarrow \eta'\gamma$ $\dagger\quad 0.54 \pm $0.5\pm 0.14$ $0.44\pm 0.14$
\begin{array}{c}
+0.29 \\
-0.23
\end{array}$
$\eta'\rightarrow \rho\gamma$ $60.7 \pm 7.4$ $74\pm 12$ $62\pm 9.1 $
$\eta'\rightarrow \omega\gamma$ $6.07 \pm 0.74$ $6.8\pm 0.8$ $6.0\pm 0.8$
$\pi^0\rightarrow \gamma\gamma$ $7.8 \pm 0.55$ \* $7.8 \pm 0.55$ \* $7.8 \pm 0.55$
$\eta\rightarrow \gamma\gamma$ $460 \pm 40$ $550\pm 70$ $490\pm 65$
$\eta'\rightarrow \gamma\gamma$ $4290 \pm 190$ $4430 \pm 280$ $4050 \pm 260$
$K^{*0}\rightarrow K^0\gamma$ $117\pm 10$ $117\pm 10$ $117\pm 10$
$K^{*\pm}\rightarrow K^\pm\gamma$ $ 50 \pm 5$ $ 50 \pm 5$ $ 50 \pm 5$
----------------------------------- -------------------------- ---------------------- ----------------------
: Calculated decay widths with the parameter set solution I and solution II of Table \[fit\]. Widths marked with asterisk were used to fix $g_P$ and $g_V$. Data are taken from [@pdg98]. A value from the KLOE collaboration[@pc], $\Gamma_{exp} (\phi\rightarrow \eta ' \gamma) =
(0.36 \pm 0.12)$ KeV yield practically the same results.
¥
\[vpgpred\]
Appendix
========
The symmetry breaking terms ${\bar L}_A$ and ${\bar L}_V$ can be constructed by using the nonet (rather than the octet) covariant ${\Delta}_\mu$. We demonstrate that for $L_A$. We write, $$\begin{aligned}
& &\bar{L}_A =
\nonumber\\
&& W_1(X)\left( c_A Tr (\{B, \Delta_\mu\} \Delta^\mu) +
d_A Tr (B \Delta_\mu B\Delta^\mu)\right) +
\nonumber\\
&&W_4(X) \left(c_A Tr (B \Delta_\mu) Tr (\Delta^\mu) +
d_A Tr (B \Delta_\mu) Tr (B\Delta^\mu)\right)~,
\label{labreak}\end{aligned}$$ From this expression the contributions of the $\eta_0$ and $\eta_8$ to the kinetic term is, $$\begin{aligned}
L_{kin}^{08} =
&&\kappa_{88}(\partial_\mu\eta_8)^2 +
\kappa_{00}(\partial_\mu\eta_0)^2 +
\kappa_{80}\partial_\mu\eta_8\partial^\mu\eta_0 +
\nonumber \\
&& m^2_{88}\eta^2_8 + m^2_{00}\eta^2_0 + 2m^2_{80}\eta_8\eta_0,
\label{}
\end{aligned}$$ where the matrix $\kappa$ depends on the parameters $c_A$ and $d_A$. This expression gives rise to twofold $\eta -\eta '$ mixing, one from the kinetic term and one from the nondiagonal mass matrix. We first diagonalize the matrix $\kappa$ using the unitary transformation $$\left(\begin{array}{c}
\eta_8 \\
\eta_0
\end{array}\right) =
\left(\begin{array}{cc}
\cos\lambda & \sin\lambda \\
-\sin\lambda & \cos\lambda
\end{array}\right) \left(
\begin{array}{c}
\bar{\eta}_8 \\
\bar{\eta}_0
\end{array}\right) = \Upsilon\left(\begin{array}{c}
\bar{\eta}_8 \\
\bar{\eta}_0
\end{array}\right)$$ This leads to, $$L_{kin}^{08} =
\kappa_8(\partial_\mu\bar{\eta}_8)^2 + \kappa_0
(\partial_\mu\bar{\eta}_0)^2 + (\bar{\eta}_8,
\bar{\eta}_0)
\Upsilon^{-1}{\cal M}^2\Upsilon
\left(\begin{array}{c}
\bar{\eta}_8 \\
\bar{\eta}_0
\end{array}\right)
\label{}$$ where $\kappa_i$ are the eigenvalues of the matrix $\kappa$. Now to restore the standard normalization of the kinetic term we rescale the pseudoscalar fields $$\pi\Rightarrow z_\pi\pi,\qquad K\Rightarrow z_s K,\qquad
\bar{\eta}_8
\Rightarrow z\hat{\eta}_8~,\qquad \bar{\eta}_0 \Rightarrow
f\hat{\eta}_0
\label{rescal}$$ where $z=1/\sqrt{\kappa_8}, f=1/\sqrt{\kappa_0}$. In other words the fields $\bar{\eta}_i$ and $\hat{\eta}_i$ are related by nonunitary transformation (matrix) $R = diag(z,f)$. Therefore, the $\hat{\eta}$ mass matrix has the (nondiagonal) form $$\hat{{\cal M}}^2 = R\Upsilon^{-1}{\cal M}^2\Upsilon R
\label{}$$ and the Goldstone field kinetic term now reads $(1/2)[(\partial_\mu
\pi)^2 + (\partial_\mu K)^2 +(\partial_\mu \hat{\eta}_8)^2 +
(\partial_\mu\hat{\eta}_0)^2]$. As a last step we relate the $\hat{\eta}_i$ to the physical fields $\eta$ and $\eta'$ which are eigenvectors of the mass matrix $\hat{{\cal M}}^2$, $$\left(
\begin{array}{c}
\hat{\eta}_8 \\
\hat{\eta}_0
\end{array}
\right) =\Omega \left(
\begin{array}{c}
\eta \\
\eta '
\end{array}
\right) ~,
\label{}$$ where, $$\Omega = \left(
\begin{array}{cc}
\cos\chi & \sin\chi \\
-\sin\chi & \cos\chi
\end{array}
\right)~.
\label{}$$ The relations between the $\eta_8$ and $\eta_0$ and physical fields $\eta,~ \eta'$ is then given by, $$\left(
\begin{array}{c}
\eta_8 \\
\eta_0
\end{array}
\right) = \Theta\left(
\begin{array}{c}
\eta \\
\eta'
\end{array}
\right)~.
\label{}$$ where, $$\Theta = \left(
\begin{array}{cc}
z\cos\lambda\cos\chi-f\sin\lambda\sin\chi &
z\cos\lambda\sin\chi+f\sin\lambda\cos\chi \\
-z\sin\lambda\cos\chi-f\cos\lambda\sin\chi &
-z\sin\lambda\sin\chi+f\cos\lambda\cos\chi
\end{array}
\right)~.
\label{tetatr}$$ Clearly, the transformation $\Theta$ is nonunitary and can not be written in the so called two angle form of Refs.[@bramon97; @escrib99] since $\Theta^2_{i1} + \Theta^2_{i2} \neq 1$.
The pseudoscalar meson matrix has the form of Eqn.\[pnonet\] but with $X$ and $Y$ defined as, $$\begin{aligned}
&& X_\eta = z\cos\lambda\cos\chi-f\sin\lambda\sin\chi +
\sqrt{2}r(-z\sin\lambda\cos\chi-f\cos\lambda\sin\chi)~,
\nonumber \\
&&X_{\eta'} = z\cos\lambda\sin\chi+f\sin\lambda\cos\chi +
\sqrt{2}r(-z\sin\lambda\sin\chi+f\cos\lambda\cos\chi)~,
\nonumber \\
&&Y_\eta = -2(z\cos\lambda\cos\chi-f\sin\lambda\sin\chi) +
\sqrt{2}r(-z\sin\lambda\cos\chi-f\cos\lambda\sin\chi)~,
\nonumber \\
&&Y_{\eta'} =
-2(z\cos\lambda\sin\chi+f\sin\lambda\cos\chi) +
\sqrt{2}r(-z\sin\lambda\sin\chi+f\cos\lambda\cos\chi)~.
\label{nxy}\end{aligned}$$ Clearly, for $\lambda = 0$ and $f = 1$ the expressions \[nxy\] reduce to the ones in Eqns. \[xy\].
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[^1]: gedal@bgumail.bgu.ac.il
[^2]: moalem@bgumail.bgu.ac.il
[^3]: ljuba@bgumail.bgu.ac.il
[^4]: The Weinberg’s ratios give apparently the lowest limit for $m_s/m$. The current algebra prediction is $m_s/m = (2m^2_K -m^2_\pi)/m^2_\pi=25.6$ while recent estimations[@leut97] give the value $m_s/m \approx 26.6$.
|
=5000
For two-dimensional electrons in a perpendicular magnetic field $B_{\perp}$, independent electron eigenstates occur in manifolds known as Landau levels with macroscopic degeneracy $AB_{\perp}/\Phi_0$, where $A$ is the sample area and $\Phi_0$ is the magnetic flux quantum. The zero-width energy bands are responsible for a tremendous variety of many-body physics that has been observed in the quantum Hall regime [@prangegirvin; @sarmapinczuk]. Quantum Hall ferromagnetism, of interest here, occurs when two different Landau levels distinguished by the cyclotron energy, spin, or quantum well subband labels of their orbitals are brought into energetic alignment and the Landau level filling factor $\nu$ is close to an integer. Neglecting charge fluctuations, low-energy states of quantum Hall ferromagnets (QHFs) are specified by assigning to each orbital in the Landau level a two-component spinor $(\cos\theta/2,
e^{i\varphi}\sin\theta/2)$ corresponding to a pseudospin oriented along a general unit vector $\hat{m}=(\sin\theta\cos\varphi, \sin\theta\sin\varphi,
\cos\theta)$. (The influence of remote Landau levels can be captured perturbatively as necessary.) While ordered states can occur when any two Landau levels simultaneously approach the chemical potential, the nature of the ground state is sensitive to the microscopic character of the crossing Landau levels [@jungwirthprl98; @jungwirthprb01]. Isotropic [@tycko; @girvinmacd], XY [@girvinmacd; @eisenstein; @spielman] and Ising QHFs [@jungwirthprl98; @piazza; @eom; @giuliani; @daneshvar] are now well established. Our work is motivated by the recent observation [@depoortere] of hysteretic transport and unexplained resistance spikes when Landau levels with different quantized kinetic (cyclotron) energies cross. We argue that the resistance spikes are due to charge transport in the 1D quasiparticle systems of long domain wall loops and establish a correspondence between their occurrence and vanishing domain-wall free-energy density at the Ising transition temperature, $T_c$.
The dependence of the uniform QHF state energy per electron on pseudospin orientation has the form [@jungwirthprb01]: $E[\hat{m}]=-bm_z-Jm_z^2$, where $b$ is an effective magnetic field that includes both single-particle Landau level splitting and interaction contributions [@jungwirthprb01] and $J >0$ is an effective Ising interaction parameter. At $b =0$, the $m_z = 1$ (pseudospin $\uparrow$) and $m_z = -1$ (pseudospin $\downarrow$) states are degenerate. In the following we establish an association between $b=0$ and the experimental resistance spikes, and propose an explanation for the spike origin. For tilted magnetic fields and variable 2D electron densities, the $b=0$ condition at a given filling factor is achieved along a continuous line in the two-dimensional $(B_{tot}-B_{\perp})$ space, which can be explored experimentally by tilting the field away from the sample normal. ($B_{tot}$ is the total magnetic field.)
=3.3in
In Fig. \[prlfig1\] we compare our theoretical[@jungwirthprl98; @jungwirthprb99] $b=0$ line for $\nu=3$, based on numerical self-consistent-field calculations for the geometry of De Poortere [*et al.*]{}’s sample and on many-body RPA/Hartree-Fock theory, with the line along which resistance spikes were observed. The coincidence of these two curves strongly suggests that the spikes occur when $b=0$. The same calculations[@jungwirthprb01] yield the estimate $J= 0.018 e^2/\epsilon \ell/k_B \sim
2 {\rm K}$, where $\ell$ is the magnetic length defined by $2 \pi \ell^2 B_{\perp} = \Phi_0$.
The ground state of an Ising QHF has $m_z = 1$ for $b>0$ and $m_z = -1$ for $b <0$. At finite temperatures, non-trivial pseudospin magnetization configurations become important. For Ising QHFs, an elementary calculation shows that spin-wave collective excitations have a gap $\Delta/k_B = 4J/k_B
\sim 8 {\rm K}$. Since the hysteretic resistance spikes occur only for $T < 0.5 {\rm K}$, spin-wave excitations cannot play a role. Instead, as we now explain, the important thermal fluctuations in Ising QHFs involve domain walls between $m_z=1$ and $m_z=-1$ regions of the sample.
In classical 2D Ising models the critical temperature can be understood as a competition between unfavorable near-neighbor-spin interaction energy along a domain wall, $L\gamma$, and the wall configurational entropy, $Ls_c=L/\xi\, k_B\ln (3)$, where $\xi$ is the domain wall persistence length. Both give free-energy contributions proportional to wall length, $L$, with the former effect favoring short walls and the latter contribution, which is proportional to temperature, favoring long walls. For $T > T_c$ the system free energy is lowered when domain walls expand to the sample perimeters, destroying magnetic order. The structure of domain walls is more complicated in Ising QHFs. As the domain wall is transversed, the local pseudospin orientation goes from the north pole ($m_z=1$) to the south pole ($m_z=-1$), at a fixed orientation $\varphi$ of its $\hat x-\hat y$ plane projection. We have evaluated the energy per unit length $\gamma$ of an infinite domain wall by solving self-consistent Hartree-Fock equations: $$\tan\theta(X)=\frac{-2\big(b-H^F_{\uparrow,\downarrow}(X)\big)}
{H^H_{\uparrow,\uparrow}(X)+H^F_{\uparrow,\uparrow}(X)
-H^H_{\downarrow,\downarrow}(X)-H^F_{\downarrow,\downarrow}(X)}\; ,
\label{hf}$$ where the Hartree energy is given by $$\begin{aligned}
& &H^H_{\sigma,\sigma^{\prime}}(X)=\sum_{X^{\prime}}\int d^3\vec{r}_1
\int d^3\vec{r}_2 V(\vec{r}_1-\vec{r}_2)\,\times\nonumber \\
& &\psi^{\ast}_{\sigma,X}(\vec{r}_1)
\psi_{\sigma^{\prime},X}(\vec{r}_1)
\psi^{\ast}_{\hat{m}(X^{\prime}),X^{\prime}}(\vec{r}_2)
\psi_{\hat{m}(X^{\prime}),X^{\prime}}(\vec{r}_2)
\label{hartree}\end{aligned}$$ and the exchange energy by $$\begin{aligned}
& &H^F_{\sigma,\sigma^{\prime}}(X)=-\sum_{X^{\prime}}\int d^3\vec{r}_1
\int d^3\vec{r}_2 V(\vec{r}_1-\vec{r}_2)\,\times\nonumber \\
& &\psi^{\ast}_{\sigma,X}(\vec{r}_1)
\psi_{\sigma^{\prime},X}(\vec{r}_2)
\psi^{\ast}_{\hat{m}(X^{\prime}),X^{\prime}}(\vec{r}_1)
\psi_{\hat{m}(X^{\prime}),X^{\prime}}(\vec{r}_2)\; .
\label{exchange}\end{aligned}$$ In Eqs. (\[hartree\]) and (\[exchange\]), $V(\vec{r}_1-\vec{r}_2)$ is the RPA-screened Coulomb potential and the self-consistent-field one-particle orbitals, $\psi_{\sigma,X}(\vec{r})$, are extended along the domain wall and localized near wavevector $k$ dependent guiding centers $X=k \ell^2$. The energy density $\gamma$ is proportional to the increase in Hartree-Fock quasiparticle energies, integrated across the domain wall. We find that the domain wall width is typically several magnetic lengths and for the $\nu=3$ coincidence we find that $\gamma \ell =
0.009 e^2/\epsilon \ell$.
A unique property of QHFs is the proportionality between electron charge density and pseudospin topological index density. It is this property that is responsible for the fascinating skyrmion physics extensively studied in the isotropic case [@tycko; @girvinmacd]. In the case of Ising QHFs, the proportionality implies a local excess charge per unit length along a domain wall $\rho_{\parallel} = e \nabla \varphi \cdot \hat n/(2 \pi) $ where $\hat n$ specifies the local direction along the domain wall. Single-valuedness of the magnetization requires that the winding number of the angle $\varphi$ around a domain wall loop be quantized in units of $2 \pi$ and hence that the excess charge of a domain wall loop be quantized in units of the the electron charge $e$. The free-energy associated with the classical $\varphi$ field fluctuations within a domain wall, $$f_{\varphi}=\frac{1}{L}k_BT\ln Z ;\;
Z=\int{\cal D}\varphi\exp\big(E_c[\varphi]/k_BT\big)\; ,
\label{phifree}$$ is controlled by the Coulomb interaction energy $E_c$ due to the consequent charge fluctuations.
Assuming a domain wall persistence length $\xi\approx\ell$, the free energy density of Ising QHF domain wall loops is given by $f=k_BT\ln(3)/\ell+\gamma+f_{\varphi}$ and equals zero at $T=T_c$. For the $\nu=3$ QHF, these considerations imply that infinitely long domain walls proliferate and order is lost for $T$ larger than the transition temperature $T_c\approx 500$ mK. The close correspondence between this $T_c$ estimate, and the maximum temperature ($430$ mK) at which hysteretic resistance spikes are observed[@depoortere] strongly supports our contention that the unusual transport phenomena are a consequence of the existence of long domain wall loops in these materials. In the following we first discuss the $b$-field, Landau level filling factor, and temperature dependence of the system’s domain-wall soup and then demonstrate that this picture can account for many details of the transport observations.
Domain wall loops are characterized by their length and by their charge, with infinitely long loops appearing only for $b=0$ and $T > T_c$. For finite size loops with a typical radius larger than the domain wall width we can use our Hartree-Fock self-consistent results for $\theta(X)$ to estimate the Coulomb self-interaction energy. A two-dimensional charge density of a circular loop with excess charge $e$ distributed uniformly along the domain wall is given by $$\rho_{2D}(r)=\frac{e}{4\pi r}\frac{d}{dr}\cos\theta(r)\;.
\label{rho2d}$$ Since the corresponding Coulomb self-interaction energy is proportional to the square of the charge and approximately inversely proportional to the length, charged domain wall loops have a higher energy and, at integer filling factors, will always be less common than neutral domain wall loops. Resistance spikes are generically observed slightly away from integer filling factors, however, and here the situation changes because the domain wall loops can exchange charge with the rest of the 2D electron system.
=3.3in
The lowest energy elementary charged excitations of the $\nu=3$ ground state are ordinary Hartree-Fock electron and hole quasiparticles [@lilliehook], not charged domain wall loops. In systems with no disorder, the chemical potential lies in the middle of the Hartree-Fock gap when $\nu$ is an integer but moves quickly (by $\delta \mu$) toward the electron quasiparticle energy for $\nu >3$ and toward the hole quasiparticle energy for $\nu < 3$. These chemical potential shifts will be reduced by disorder which broadens the quasiparticle bands. The change in chemical potential favors charge $Q$ over neutral skyrmions by a large factor $\exp (Q|\delta \mu||/k_B T)$. This factor can be estimated quantitatively using the experimental value of the quasiparticle excitation gap ($\sim 2$ K) [@depoortere].
Also important in controlling the domain wall soup is the effective field $b$, which measures the distance from Landau level coincidence. For $b \ne 0$ the energy of a domain wall loop has a contribution proportional to $b$ and the number of condensate electrons contained within the loop: $$E_b=-\frac{b}{2\ell^2}\int dr\, r\,[\cos\theta(r)-1]\; .
\label{eb}$$ This contribution will decrease the number of large domain wall loops enclosing the minority phase and is independent of the charge carried by the loop. Summing up $E_b$, the Coulomb self-interaction and chemical potential contributions and the Hartree-Fock domain wall energy of a circular loop of radius $R$, $2\pi R\gamma$, we can estimate statistical weights of neutral and charged domain wall loops in the sample of De Poortere [*et al*]{}. In Fig. \[prlfig2\] we plot our results for temperature near $T_c$ and $\nu > 3$. For non-zero $b$-fields, small neutral domain wall loops dominate, while typical loops at the coincidence are large and carry an excess charge.
We now address characteristic features of the measured resistive hysteresis loop. Dissipation can occur in Ising QHFs as a result of Hartree-Fock quasiparticle diffusion, charged domain-wall-loop diffusion, or as a result of charge diffusion within domain-wall loops. It is clear that the resistance spikes, which appear only for small $b$ and $T < T_c$, are associated with the appearance in the sample of large domain-wall loops. Even though these loops tend to be charged at the spike maximum, we expect that they will be immobile because of their large size and that dissipation due to their motion is small. Instead we propose that mobile quasiparticles inside domain walls are responsible for the increase of dissipation. In Fig. \[prlfig3\] we plot the Hartree-Fock quasiparticle energies in a cross-section of a domain wall, obtained from Eqs. (\[hf\])-(\[exchange\]). In the center of the domain wall the quasiparticle gap is reduced by nearly 50%. Away from integer filling factors, the bottom of these 1D quasiparticle bands will lie below the chemical potential which is pinned to the bulk quasiparticle energies. We note that, unlike quantum Hall edge states, counter propagating states exist within each loop. At $\nu\approx 3$, for example, the 1D states have a nearly parabolic dispersion characterized by an effective mass $m^*\approx
2 m_e$. The particles can cross the sample by scattering between overlapping loops. For $\nu=3$ and $T\approx
T_c$, it follows from inset of Fig. \[prlfig1\] and from Fig. \[prlfig2\] that the characteristic loop radius at the spike edge is $3\ell$. At low temperatures or small magnetic lengths (high 2D electron gas densities), domain wall loops become small and dilute, loops do not overlap, and charge diffusion within domain walls cannot contribute to dissipation. This explains the absence of resistance spikes at Landau level coincidence [@depoortere] under these circumstances.
=3.3in
The above mechanism also explains the different resistance spike heights observed in up and down field sweeps [@depoortere] near $\nu=3$. As shown in the inset of Fig. \[prlfig1\], the majority pseudospin Landau level for the up-sweep is the $n=2$ spin-up level while for the down-sweep it is the $n=0$ spin-down Landau level. This difference in Landau level configurations in the two sweep directions alters remote Landau level screening in the sample which has a marked effect on the quasiparticle energy spectrum. The reduction of the quasiparticle gap in the domain wall, relative to its bulk value, is stronger in the up-sweep case, leading to more domain-wall quasiparticles and more dissipation, as seen in experiment. Similar agreement between the measured hysteresis loop properties and domain wall quasiparticle spectra applies for $\nu=4$ [@depoortereunpubl]. In Fig. \[prlfig3\] we also plot energy spectra at $\nu=5$ which are nearly identical for the up or down majority pseudospin orientations. This explains the absence of peak-height asymmetry in the hysteresis measurement at this filling factor $\nu=5$.[@depoortereunpubl]
We thank Etienne De Poortere, Herbert Fertig, and Mansour Shayegan for many important discussions. Our work was supported by R.A. Welch Foundation, by Minsistry of Education of the Czech Republic under Grant OC P5.10, and by the Grant Agency of the Czech Republic under Grant 202/01/0754.
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Self-consistent local-spin-density approximation calculations were carried out to establish that for the 15 nm wide AlAs quantum wells studied in these experiments, orbital effects of the in-plane field are negligibly small. In the calculation we assumed Landé g-factor $g=1.9$, electron effective mass $m^*=0.41 m_e$, and average heterostructure dielectric constant $\epsilon = 11.5$.
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---
abstract: 'We present a class of supersymmetric models in which flavor symmetries are broken dynamically, by a set of composite flavon fields. The strong dynamics that is responsible for confinement in the flavor sector also drives flavor symmetry breaking vacuum expectation values, as a consequence of a quantum-deformed moduli space. Yukawa couplings result as a power series in the ratio of the confinement to Planck scale, and the fermion mass hierarchy depends on the differing number of preons in different flavor symmetry-breaking operators. We present viable non-Abelian and Abelian flavor models that incorporate this mechanism.'
---
LBNL-40331\
UCB-PTH-97/23\
.25in
[**New Mechanism of Flavor Symmetry Breaking\
from Supersymmetric Strong Dynamics**]{}[^1]
0.3in
Christopher D. Carone$^1$, Lawrence J. Hall$^{1,2}$, and Takeo Moroi$^1$
0.1in
[$^1$ *Theoretical Physics Group\
Ernest Orlando Lawrence Berkeley National Laboratory\
University of California, Berkeley, California 94720*]{}
0.1in
[$^2$ *Department of Physics\
University of California, Berkeley, California 94720*]{}
.1in
[**Disclaimer**]{}
.2in
> This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor The Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial products process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or The Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof, or The Regents of the University of California.
[*Lawrence Berkeley Laboratory is an equal opportunity employer.*]{}
Introduction {#sec:intro}
============
Symmetry is a powerful tool for understanding the physical world, even when the symmetry in question is known to be broken. However, many candidate fundamental theories are incomplete, or flawed, because we do not know how their symmetries are broken — the origin of symmetry breaking is perhaps the greatest gap in our understanding of nature.
The spontaneous breaking of approximate light-quark flavor symmetries in QCD, leading to light pions and kaons, is the only case in nature where we know the underlying theory of symmetry breaking [@Georgi]. The origin of SU(2)$_L\times$U(1)$_Y$ electroweak symmetry breaking, leading to the $W$ and $Z$ masses, and of the U(3)$^5$ flavor symmetry breaking, leading to the quark and lepton masses, is unknown. There are only a few candidate field theory mechanisms for such symmetry breakings. Symmetries are apparently easily broken by the vacuum expectation values of elementary scalar fields [@Higgs], but this alone is unsatisfactory, as it does not provide an understanding for the mass scale of the associated symmetry breaking. Without such information, we do not have an understanding of the basic mass scales of nature.
The only known way to generate symmetry breaking mass scales in quantum field theory is by dimensional transmutation, frequently, but not always, involving strongly interacting dynamics. Examples of such dynamical symmetry breaking are provided by QCD, and by theories of dynamical supersymmetry breaking. In supersymmetric theories, once soft scalar masses are induced from supersymmetry breaking, gauge and global symmetries may be broken by having further interactions which evolve these squared masses negative, thus dynamically generating new symmetry breaking [@RadiativeBreaking]. For example, the large top Yukawa coupling has been used to drive the Higgs mass-squared negative, breaking electroweak symmetry. Much model building has centered around this two stage breaking of symmetries: first supersymmetry is broken to generate the soft squared masses, then further, non-gauge interactions give radiative corrections so that the squared masses become negative.
In view of the importance of symmetry breaking, it is striking that certain strong supersymmetric gauge interactions necessarily force a direct breaking of symmetries [@seiberg]. This does not require supersymmetry breaking, nor any other interactions beyond the supersymmetric gauge interactions.[^2] For example, in supersymmetric QCD with an equal number of flavors and colors, the strong gauge interaction forms bound state mesons and baryons, $T$ and $B$, and induces vevs for some of their scalar components. This direct forcing of symmetry breaking offers a new avenue for exploring the origins of gauge and flavor symmetry breaking. In this paper, we use this strong dynamics to construct realistic theories of flavor.
In supersymmetric extensions of the standard model, flavor symmetries are in general broken by squark and slepton mass matrices, $m^2$, as well as by Yukawa matrices, $h$, which generate the quark and lepton masses. In this paper we study theories where the form of both $m^2$ and $h$ are governed by some fundamental global flavor symmetry group, $G_F$, and its breaking pattern. We take the preons, $p$, and the bound states, $T$ and $B$, of some new strong gauge force to transform non-trivially under $G_F$. The theory contains the most general set of interactions which are gauge and $G_F$ invariant, both F and D terms, including non-renormalizable operators, scaled by inverse powers of the cutoff $M_*$, which we take to be the reduced Planck scale $M_{Pl}/\sqrt{8\pi}\simeq 2.4\times 10^{18}$ GeV. In the fundamental theory, the scalar mass and Yukawa matrices can be written as field dependent polynomials, $m^2(p/M_*)$ and $h(p/M_*)$, where $p$ is a preon field. At the scale $\Lambda$ of the new strong force, these matrices become polynomials in the meson and baryon fields, $$m^2 = m^2 \left({\Lambda T \over M_*^2}, {\Lambda^{N-1} B \over M_*^N}\right)
\label{m^2}$$ $$h = h \left({\Lambda T \over M_*^2}, {\Lambda^{N-1} B \over M_*^N}\right),
\label{h}$$ where $N$ is the number of preons in a baryon. The new strong force constrains $T$ and $B$ to acquire vevs so that these fields become flavon fields, spontaneously breaking the flavor group $G_F$. However, there is a large vacuum degeneracy, so that $m^2$ and $h$ become functions on the moduli space. The main phenomenological problem is to lift this vacuum degeneracy, so that for a certain choice of $T, B$ and $G_F$, (\[m\^2\]) and (\[h\]) give realistic masses.
In the next section we elaborate on the framework for symmetry breaking and solving the vacuum alignment problem. In Sections 3 and 4 we give explicit realistic theories of flavor, based on non-Abelian and Abelian $G_F$, respectively. Our vacuum alignment mechanism results in all non-zero vevs of $T$ and $B$ being of order $\Lambda$. The flavor group is broken at a single scale – there is no hierarchy of symmetry breaking scales – so that all the small parameters of $m^2$ and $h$ are derived from $\Lambda/M_*$. For example, a term in (\[m\^2\]) or (\[h\]) involving $n_T$ meson fields and $n_B$ baryon fields leads to a dimensionless coefficient of size $(\Lambda/M_*)^{2n_T + Nn_B}$. The hierarchy of quark and lepton masses arises because of the small value of $\Lambda/M_*$, because mesons and baryons contain different numbers of preons, and because the $G_F$ quantum number assignments lead to interactions with differing numbers of mesons and baryons.
Framework {#sec:frame}
=========
In this section we outline our general approach for breaking flavor symmetries dynamically in models with composite flavon fields. We give explicit examples of viable models that incorporate this mechanism in the following section.
The sector of the theory that is responsible for confinement is a supersymmetric SU(N) gauge theory with $N$ flavors. The nonanomalous global symmetries of the theory are $G=$SU(N)$_p\times$SU(N)$_{\overline{p}}\times$U(1)$_B\times$U(1)$_R$, where the first U(1) factor is the analog of baryon number in ordinary QCD, and the second U(1) is an R-symmetry. The transformation properties of the preons and their bound states under the global symmetries $G$ are shown in Table \[table1\]. Notice that there are $N^2$ meson fields with zero baryon number, transforming as an $(N,N)$ under the two global SU(N) groups, and a baryon-antibaryon pair that are singlets under the two SU(N)s.
SU(N) SU(N)$_p$ SU(N)$_{\overline{p}}$ U(1)$_B$ U(1)$_R$
------------------ ---------------------- ----------- ------------------------ ---------- ----------
$p$ $\Box$ $\Box$ $1$ $1$ $0$
$\overline{p}$ $\stackrel{-}{\Box}$ $1$ $\Box$ $-1$ $0$
$p\overline{p}$ $1$ $\Box$ $\Box$ $0$ $0$
$p^N$ $1$ $1$ $1$ $N$ $0$
$\overline{p}^N$ $1$ $1$ $1$ $-N$ $0$
: SU(N) with $N$ flavors. \[table1\]
This confining theory has two features that are particularly relevant to model building. First, an SU(N) gauge theory with $N$ flavors has no dynamically generated superpotential. This follows from the fact that all the preons in Table \[table1\] have R-charge $0$, so that it is not possible to write down an invariant combination of the fields that have R-charge $2$. Secondly, the vacuum manifold of the theory is distorted by quantum mechanical effects so that the origin of field space is excluded [@seiberg]. Classically, we have the identity $$\det (p \overline{p}) - p^N \overline{p}^N = 0$$ which we can rewrite in terms of canonically normalized meson and baryon fields as $$\det M - \Lambda^{N-2} B \overline{B} = 0 .$$ Quantum mechanically, this relation is modified, and becomes $$\det M - \Lambda^{N-2} B \overline{B} = \Lambda^N \,\,\, .
\label{eq:qcon}$$ Notice that there is no symmetry which prevents the right-hand side of Eq. (\[eq:qcon\]) from becoming nonzero. Furthermore this modified constraint is necessary if we are to properly recover the Affleck-Dine-Seiberg superpotential [@ads] when we decouple one flavor, beginning with the SU(N) theory with equal numbers of flavors and colors.
We learn from Eq. (\[eq:qcon\]) that some of the meson and baryon fields acquire vevs, breaking the original global symmetry $G$. If the preons transform nontrivially under a flavor symmetry group $G_F$, then meson and baryon vevs may break the flavor symmetry as well. If we interpret $G$ as an accidental symmetry of the sector responsible for confinement, while $G_F$ is respected by all the interactions of the theory, then some of the mesons and baryons may couple to ordinary matter and serve as flavon fields. Yukawa couplings may arise via Planck-suppressed operators, as described in Section 1, so that the small parameter that characterizes flavor symmetry breaking is the ratio of the confinement scale $\Lambda$ to the reduced Planck mass $M_*$.
The ambiguity that must now be resolved is the precise set of composites that actually acquire vevs. For example, Eq. (\[eq:qcon\]) is satisfied by a point in field space where the baryons $B$ and $\overline{B}$ acquire confinement-scale vevs, while the mesons remain at the origin. This vacuum would not be particularly useful if we were to construct a model in which only the mesons coupled to ordinary matter. In a viable model, we must sufficiently reduce this vacuum degeneracy so that the flavor symmetry breaking fields which couple to ordinary matter are forced to get vevs. The models that we present in the next section achieve this in two steps:
First, we introduce additional fields $X_{j}$ $(j=0,1,2,\ldots)$, that couple to the preons via nonrenormalizable superpotential interactions. Since the preon fields have U(1)$_R$ charge $0$, we will take the fields $X$ to have $R$-charge $2$. We impose the U(1)$_R$ symmetry so that all of the preonic operators involve one of the $X$ fields[^3]. We will assume that $X_0$ is a singlet under the non-$R$ symmetries shown in Table \[table1\], while the remaining $X_j$ transform nontrivially under $G_F$. The $X$ fields will be responsible for restricting the moduli space such that the desired set of mesons and baryons develop vacuum expectation values when the scalar potential is minimized.
The F-flatness conditions for the $X$ fields significantly reduce the original supersymmetric vacuum degeneracy. Consider the superpotential interactions for the field $X_0$. In the models of interest, these will be of the form $$W_0 = \left(\frac{1}{M_*}\right)^{2N-2}
\left(\frac{\Lambda}{M_*}\right)^N
\left[\sum_j G_j + \Lambda^{N-2} B \overline{B} \right] X_0 \,\,\,
\label{eq:w0}$$ where the $G_i$ represent all possible flavor-group invariant combinations of the meson fields involving $2N$ preons. In the models we will consider, these interactions will be the ones of lowest order in $1/M_*$ that are allowed by the flavor symmetry. Other interactions, such as direct $G_F$-invariant couplings between the baryons and mesons, will arise at higher order, and will be suppressed. Note that we have omitted a Planck-scale linear term for $X_0$, which can be forbidden by imposing an anomalous discrete symmetry, as we will see explicitly in the next section. Notice that the F-flatness condition for $X_0$ together with the quantum-modified constraint (\[eq:qcon\]) yield two restrictions on the set of invariants $G_i$, $B\overline{B}$. Thus, we have succeeded in reducing the vacuum degeneracy by one degree of freedom. The $X_0$ field orients the vacuum so that at least some of the mesons have non-vanishing vevs.
Now we introduce additional fields $X_j$, that transform nontrivially under the flavor group $G_F$. These lead to additional superpotential couplings of the form $$W_j = X_j \sum_i G'_i$$ where the $G'_i$ represent all possible baryon and meson interactions with the appropriate quantum numbers to couple to $X_j$. We have absorbed powers of $\Lambda$ and $M_*$ into the definition of the $G'_i$ for notational convenience. In a successful model, we introduce enough nontrivial constraints in this way such that the flavor invariant combinations of the mesons and baryons shown in brackets in Eq. (\[eq:w0\]) acquire vevs [*individually*]{}, $$\begin{aligned}
& B\overline{B} \sim \Lambda^2 & \nonumber \\
& G_1 \sim \Lambda^N & \nonumber \\
& G_2 \sim \Lambda^N & \nonumber \\
& \mbox{etc.} & ,
\label{eq:morevevs}\end{aligned}$$ while all the $X$ field vevs vanish. This result should remain valid provided that the Kähler metric is positive definite in the region of field space where we have located the minimum. Since the Kähler potential is not calculable for field amplitudes of the same order as the confinement scale, we take this positivity requirement as a mild assumption. Note also that if too many F-flatness constraints are added, it is possible that the resulting superpotential may break supersymmetry. This would lead to direct, flavor-dependent couplings of fields with large F components to ordinary matter, which would not be desirable. In all the models we consider, supersymmetry will remain unbroken after the effects of the $X$ fields are taken into account.
Once we have arranged for each gauge invariant combination of the mesons and baryons to acquire vevs, we must lift the remaining vacuum degeneracy. Notice that given any point in the moduli space defined by Eq. (\[eq:morevevs\]), we can reach another point by transforming the fields under the complexification of the flavor group. If we include positive soft supersymmetry-breaking squared masses for the the composite flavon fields, the complexified symmetry will be broken, and this last flat direction will be lifted. (We justify this procedure below.) To make this point concrete, imagine we have a theory with three flavons, $\phi_0$, $\phi_+$ and $\phi_-$, where the subscript indicates the charge under some U(1) symmetry. Now assume that the moduli space is constrained such that $\phi_0^3 \sim \Lambda^3$ and $\phi_0\,\phi_+\phi_- \sim \Lambda^3$. The remaining flat direction corresponds to a rescaling of $\phi_+$ and $\phi_-$, which is generated by the complexification of the U(1) symmetry. The complexified symmetry is explicitly broken by the soft mass terms $V_{soft} = m^2_{soft} (|\phi_0|^2 + |\phi_+|^2+|\phi_-|^2)$, and minimization of the full potential then yields $\phi_+ \sim \phi_- \sim \phi_0 \sim \Lambda$, as desired.
This last step may be questioned since the form of the soft supersymmetry breaking interactions in the confining theory are not determined by any symmetry argument. However, we may justify our qualitative result by considering the behavior of the theory in the limit of large field amplitudes. Our constraints on the gauge-invariant products of the fields $G_i \sim \Lambda^N$ imply that varying any moduli field away from $\Lambda$ forces some field to acquire a vev greater than $\Lambda$. In the limit of large field amplitudes, this corresponds to at least some of the preons $p$ acquiring large expectation values as well, $p > \Lambda$. In the same limit we expect there will be soft supersymmetry breaking squared masses for the preon fields, so $V_{soft} \sim m^2_{soft} |p|^2$. Again assuming positive $m^2_{soft}$, the potential grows as we take any $p$ larger than $\Lambda$, and we conclude that our previous result is energetically favored. This conclusion is consistent with the assumption made in Ref. [@peskin] that minimization of a potential that includes soft supersymmetry breaking masses for the composite fields should lead to correct qualitative results, even when field amplitudes are of the same order as the confinement scale. Therefore, for the purpose of calculation, we will assume soft masses for the composite fields, but the reader should keep in mind that the results are supported by this more general argument.
Finally, we will make the simplifying assumption that trilinear scalar interactions (A-terms) can be neglected in the potential. Any minimum of the potential that we find in the absence of A-terms will remain at least a local minimum for small but nonvanishing A parameters. This will be sufficient for our purposes. We will not attempt to find the explicit conditions implied by vacuum stability on the possible trilinear scalar interactions in the flavor sector when the $A$ parameters are large.
After taking into account both the F-flatness conditions for the $X$ fields, and the effect of soft supersymmetry-breaking scalar masses, it is often the case that the desired composite fields will each be forced to acquire a vev of order $\Lambda$. We will now present two complete models that successfully incorporate the flavor symmetry breaking mechanism described in this section.
Non-Abelian Model {#sec:model_su(3)}
=================
The models we present in this and the next section are based on SU(3) supersymmetric QCD with three flavors. The global symmetries of the strong interaction are $$G = {\rm SU(3)}_p\times {\rm SU(3)}_{\overline{p}}\times {\rm U(1)}_B
\times {\rm U(1)}_R$$ and $$G_A = {\rm U(1)}_A \,\,\, ,$$ where $G_A$ is the anomalous U(1) symmetry corresponding to axial phase rotations on $p$ and $\overline{p}$. In each of the models we present, the action of the flavor group $G_F$ on the preon fields will be isomorphic to a subgroup of $G \times G_A$. However, one should keep in mind that the ordinary fermions will transform under $G_F$ even though they do not transform under the global symmetries of the confining flavor sector.
[l| l l]{}\
& [${\rm G_{SM}}$]{} & [${\rm G_F}$]{}\
$Q^i$ & $(\Box, \Box, \frac{1}{6})$ & $(\stackrel{-}{\Box},-1,2,-)$\
$Q^3$ & $(\Box, \Box , \frac{1}{6})$ & $({\bf 1},0,0,+)$\
$U^i$ & $(\stackrel{-}{\Box}, {\bf 1}, -\frac{2}{3})$ & $(\stackrel{-}{\Box},-1,2,-)$\
$U^3$ & $(\stackrel{-}{\Box}, {\bf 1}, -\frac{2}{3})$ & $({\bf 1},0,0,+)$\
$D^i$ & $(\stackrel{-}{\Box}, {\bf 1}, \frac{1}{3})$ & $(\stackrel{-}{\Box},-1,2,-)$\
$D^3$ & $(\stackrel{-}{\Box}, {\bf 1}, \frac{1}{3})$ & $({\bf 1},2,2,-)$\
\
\
& [${\rm G_{SM}}$]{} & [${\rm G_F}$]{}\
$\phi_i$ & $({\bf 1}, {\bf 1}, 0)$ & $(\Box,1,-2,-)$\
$\tilde{\phi}_i$ & $({\bf 1}, {\bf 1}, 0)$ & $(\Box,-2,1,-)$\
$A$ & $({\bf 1}, {\bf 1}, 0)$ & $({\bf 1},1,1,-)$\
$S_{ij}$ & $({\bf 1}, {\bf 1}, 0)$ & $(\Box\hspace{-0.04in}\Box,1,1,-)$\
$\sigma$ & $({\bf 1}, {\bf 1}, 0)$ & $({\bf 1},-2,-2,-)$\
$B$ & $({\bf 1}, {\bf 1}, 0)$ & $({\bf 1},1,-2,-)$\
$\bar{B}$ & $({\bf 1}, {\bf 1}, 0)$ & $({\bf 1},-1,2,+)$\
The flavor group of the first model is $$G_F= {\rm SU(2)}_F\times {\rm U(1)}_F\times {\rm U(1)}_{F'}\times Z_2
\,\,\, .$$ The transformation properties of the MSSM superfields (as well as those of the composite states discussed later) are shown in Table \[table:tp\]. The lighter two generations of the matter fields ($Q^i$, $U^i$, and $D^i$, with $i=1,2$) transform as doublets under SU(2)$_F$, while the third generation fields ($Q^3$, $U^3$, and $D^3$) are singlets. The ${\bf 2}+{\bf 1}$ representation structure provides a natural degeneracy between squark masses of the first and second generations in the flavor symmetric limit [@dlk]. This leads to a suppression of flavor changing neutral current effects when the flavor symmetries are broken. The remaining group factors, U(1)$_F\times$U(1)$_{F'}$, are used to obtain realistic Yukawa textures. The fields $Q^i$, $U^i$, $D^i$, and $D^3$ transform nontrivially under the two flavor U(1) factors, while $Q^3$, $U^3$ and the ordinary Higgs fields are $G_F$ invariant. The top quark Yukawa coupling is invariant under the flavor symmetry, and hence can be of order one, while the other Yukawa elements will be suppressed by the ratios of flavon vevs to $M_*$.
If we consider the preonic sector alone, the flavor symmetry can be identified with a subgroup of $G\times G_A$. We first decompose each SU(3) factor into its SU(2)$\times$U(1) subgroup: $${\rm SU(3)}_p\times{\rm SU(3)}_{\overline{p}} \times {\rm U(1)}_B
\rightarrow \left[ {\rm SU(2)}\times{\rm U(1)}\right]_p \times
\left[ {\rm SU(2)}\times{\rm U(1)}\right]_{\overline{p}} \times
{\rm U(1)}_B$$ The flavor SU(2) is simply the diagonal subgroup of SU(2)$_p\times$SU(2)$_{\overline{p}}$. The two flavor U(1) factors are different linear combinations of U(1)$_B$, U(1)$_p$ and U(1)$_{\overline{p}}$. The charges under the flavor U(1)s are defined by $$Q_{\rm F} = 2\sqrt{3}Q_p + \frac{1}{3}Q_{\rm B}
\,\,\,\,\, \mbox{ and } \,\,\,\,\,
Q_{\rm F'} = 2\sqrt{3}Q_{\bar{p}} - \frac{2}{3}Q_{\rm B} \,\,\, ,$$ where $Q_p$ and $Q_{\overline{p}}$ are the eigenvalues of the $T^8$ generators of SU(3)$_p$ and SU(3)$_{\overline{p}}$, respectively.
The quantum numbers of the preons $p$ and $\overline{p}$ under SU(3)$\times G_F$ are given by $$\begin{aligned}
p^i &\sim & (\Box,\Box,\frac{4}{3},-\frac{2}{3},-) \\
p &\sim & (\Box,{\bf 1},-\frac{5}{3},-\frac{2}{3},-) \\
\bar{p}^i & \sim & (\stackrel{-}{\Box},\Box,-\frac{1}{3},\frac{5}{3},+) \\
\bar{p} & \sim & (\stackrel{-}{\Box},{\bf 1},-\frac{1}{3},\frac{4}{3},+).\end{aligned}$$ Notice that the $Z_2$ factor is a symmetry under which all the preons are odd and all anti-preons are even; this is a discrete subgroup of U(1)$_A\times$U(1)$_B$. Once the SU(3) gauge group becomes strong at the scale $\Lambda$, the preons form composite states: $$\begin{aligned}
&& S_{ij}\sim \Lambda^{-1} (p_{i}\bar{p}_{j}+p_{j}\bar{p}_{i}),
\label{S_ij} \\ &&
A\sim \Lambda^{-1} \epsilon^{ij}(p_{i}\bar{p}_{j}),
\\ &&
\phi_i\sim\Lambda^{-1} (p_i\bar{p}),
\\ &&
\tilde{\phi}_i\sim \Lambda^{-1} (p\bar{p}_i),
\\ &&
\sigma \sim \Lambda^{-1} (p\bar{p}),
\\ &&
B \sim \Lambda^{-2} \epsilon^{ij} (p_i p_j p),
\\ &&
\bar{B} \sim \Lambda^{-2} \epsilon^{ij} (\bar{p}_i \bar{p}_j \bar{p}),
\label{B-bar}
\end{aligned}$$ where $i$ and $j$ are SU(2)$_F$ flavor indices. The composite fields have been given canonical mass dimension by including appropriate powers of $\Lambda^{-1}$. The transformation properties of the composite states under the flavor symmetry are also summarized in Table \[table:tp\].
Given these quantum number assignments, and our assumption that Planck-scale physics induces all operators that are consistent with the symmetries, some of the composite states above can serve as flavon fields. The $G_F$-allowed couplings that can contribute to the Yukawa matrices are summarized as follows: $$h_u \sim \left(
\begin{array}{cc|c}
0 & B^2 & 0 \\
- B^2 & \phi_2\phi_2 & \phi_2 \\
\hline
0 & \phi_2 & 1 \end{array}\right)
\label{eq:hu}$$ $$h_d \sim \left(
\begin{array}{cc|c}
0 & B^2 & 0 \\
-B^2 & \phi_2\phi_2 & \phi_2 \sigma \\
\hline
0 & \phi_2 & \sigma \end{array}\right) \,\,\, .
\label{eq:hd}$$ We have not shown couplings to $\phi_1$, since we will always work in a basis where the $\phi_1$ vev vanishes. We have also temporarily suppressed the factors of $\Lambda$ and $M_*$ in each entry, which depend on the dimensionality of the original preonic interaction. Note that if a composite field above acquires a vev of order $\Lambda$, then the size of the corresponding Yukawa entry will be $(\Lambda/M_*)^n$, where $n$ is the total number of preons involved in the preonic higher-dimension operator. To obtain a realistic theory we need only lift the vacuum degeneracy such that $B, \phi$ and $\sigma$ are all forced to acquire vevs of order $\Lambda$. There may be many ways to accomplish this; below we provide an explicit example.
Since the confining sector of our model is of the type described in Section \[sec:frame\], the moduli space of the composite states is restricted by a quantum-modified constraint [@seiberg]. The important point is that the origin of field space is excluded, so that the flavor symmetries are guaranteed to break. The constraint is realized by a dynamically generated Lagrange multiplier term in the superpotential $$\begin{aligned}
W_{\rm dyn}&=& \eta \left[ C_{M^3}
( \epsilon^{ij}A\phi_i\tilde{\phi}_j
+ A^2 \sigma
+ \epsilon^{ij}\epsilon^{kl}
S_{ik}\phi_j\tilde{\phi}_l \right.
\nonumber \\ &&
\left. + \epsilon^{ij}\epsilon^{kl}S_{ik}S_{jl}\sigma)
+ C_{B\bar{B}} \Lambda B\bar{B}
- \Lambda^3 \right],
\label{W_dyn}
\end{aligned}$$ where $\eta$ is the Lagrange multiplier field, and $C$’s are ${\cal
O}(1)$ coefficients that arise from the dynamics of confinement.
As we described in Section \[sec:frame\], the constraint equation alone leaves us with a rather large vacuum degeneracy, and the possibility that we will not obtain a viable pattern of flavor symmetry breaking. We will remove most of these flat directions by introducing several $X$ fields, to place additional constraints on the composite states. Perhaps the simplest set of $X$ fields is given by $$\begin{aligned}
X_0 & \sim & ({\bf 1},0,0,-) \nonumber \\
X_1 & \sim & ({\bf 1},1,1,+) \nonumber \\
X_2 & \sim & (\Box\hspace{-0.04in}\Box,-1,-1,-) \,\,\, ,\end{aligned}$$ where we have shown the transformation properties under ${\rm G_F}$ in parentheses. We can now write down the following interaction terms for the preons: $$\begin{aligned}
W_0 &=& \frac{1}{M_*^4} X_0
[ (\epsilon^{ij}p_{i}\bar{p}_{j})
\epsilon^{kl} (p_k\bar{p})(p\bar{p}_l)
+ (\epsilon^{ij}p_{i}\bar{p}_{j})^2 (p\bar{p})
+ \epsilon^{ij}\epsilon^{kl} (p_{i}\bar{p}_{k}+p_{k}\bar{p}_{i})
(p_j\bar{p}) (p\bar{p}_l) \label{eq:w0eq}
\nonumber \\ &&
+ \epsilon^{ik}\epsilon^{jl}
(p_{i}\bar{p}_{j}+p_{j}\bar{p}_{i})
(p_{k}\bar{p}_{l}+p_{l}\bar{p}_{k}) (p\bar{p})
+ (\epsilon^{ij} p_i p_j p)
(\epsilon^{kl} \bar{p}_k \bar{p}_l \bar{p}) ]
\\
W_1 & = &
\frac{1}{M_*^2} X_1
[(\epsilon^{ij}p_{i}\bar{p}_{j}) (p\bar{p})
+ \epsilon^{ij} (p_i\bar{p})(p\bar{p}_j)] \\
W_2 & = &
\epsilon^{ik}\epsilon^{jl}
X_{2,ij} (p_{k}\bar{p}_{l}+p_{l}\bar{p}_{k}),\end{aligned}$$ where we have omitted unknown ${\cal O}(1)$ coefficients. Notice that there is no linear term in $X_0$ due to the discrete $Z_2$ symmetry. Without this symmetry, the interaction $M_*^2X_0$ would also be allowed, and the argument presented below would break down. Note that the $Z_2$ symmetry has no significant effect on the mass matrix textures that we obtain in either of our models.
After confinement, F-flatness conditions for the Lagrange multiplier field $\eta$ and the fields $X_j$ give us the following four equations of motion for composite states: $$\begin{aligned}
&&
C_{M^3} \epsilon^{ij}A\phi_i\tilde{\phi}_j
+ C_{M^3} A^2 \sigma + C_{B\bar{B}} \Lambda B\bar{B}
- \Lambda^3 = 0,
\label{F_X0=0} \\ &&
C'_{A\phi\tilde{\phi}} \epsilon^{ij}A\phi_i\tilde{\phi}_j
+ C'_{A^2\sigma} A^2 \sigma + C'_{B\bar{B}} \Lambda B\bar{B}
- \Lambda^3 = 0,
\label{F_X1=0} \\ &&
C''_{\phi\tilde{\phi}} \epsilon^{ij}\phi_i\tilde{\phi}_j
+ C''_{A\sigma} A \sigma = 0,
\label{F_X2=0} \\ &&
S_{ij} = 0.
\label{F_X3=0}
\end{aligned}$$ Here the $C'$ and $C''$ are also ${\cal O}(1)$ coefficients[^4]. Note that we have dropped the terms which depend on $S_{ij}$ in Eqs.(\[F\_X0=0\]) – (\[F\_X2=0\]) by using eq.(\[F\_X3=0\]). We can easily solve eqs.(\[F\_X0=0\]) – (\[F\_X2=0\]), and we obtain $$\begin{aligned}
&& A^2 \sigma \sim \Lambda^3,
\label{A^2*sigma} \\
&& \epsilon^{ij}A\phi_i\tilde{\phi}_j \sim \Lambda^3,
\label{a*phi*phi} \\
&& B\bar{B} \sim \Lambda^2,
\label{BB}
\end{aligned}$$ neglecting the possibility of accidental cancellations.
At this stage, the remaining flat directions correspond to rescaling of the composite fields, since only their products are constrained by Eqs. (\[A\^2\*sigma\]) – (\[BB\]). One might suspect that these flat directions can be lifted by including yet higher order corrections to the superpotential. However, this can never be the case because the remaining flat directions are protected by a symmetry that is respected by all F-term contributions to the potential in the supersymmetric limit. Since our model has an SU(2)$_F\times$U(1)$_F\times$U(1)$_{F'}$ global symmetry, and the superpotential is holomorphic in the fields, the actual symmetry of the superpotential before supersymmetry breaking is the complexification of the flavor group. Since SU(2)$\times$U(1)$\times$U(1) has $5$ generators, this symmetry corresponds to 5 complex degrees of freedom in the moduli space. We began with $9$ meson and $2$ baryon fields, and imposed $6$ F-flatness conditions (for the fields $\eta$, $X_0$, $X_1$, and the three components of $X_2$) leaving $5$ complex degrees of freedom. Thus, we have lifted all the flat directions that are not protected by the complexified symmetry.
To lift the flat directions defined by Eqs. (\[A\^2\*sigma\]) – (\[BB\]), we include the soft supersymmetry-breaking scalar masses for the composite states, $$\begin{aligned}
V_{\rm soft} = m_{\rm soft}^2
(|A|^2 + |S|^2 + |\phi|^2 + |\tilde{\phi}|^2 + |\sigma|^2
+ |B|^2 + |\bar{B}|^2)
\label{eq:smasses}\end{aligned}$$ with $m^2_{{\rm soft}} > 0$[^5]. We now minimize the potential above subject to the constraints (\[A\^2\*sigma\]) – (\[BB\]). The qualitative result is easy to understand. We first use the ${\rm SU(2)_F}$ symmetry to work in the basis where $\phi_1=0$. In this basis, $\tilde{\phi}_2$ appears in Eq. (\[eq:smasses\]), but not in any of the constraints, and is therefore driven to zero. The non-vanishing elements ($\phi_2$, $\tilde{\phi}_1$, $A$, $\sigma$, $B$, and $\bar{B}$) must all be of ${\cal O}(\Lambda)$, so that the constraints (\[A\^2\*sigma\]) – (\[BB\]) are satisfied and $V_{\rm soft}\sim m_{\rm
soft}^2\Lambda^2$. If any of the fields were to have a vev smaller than $\Lambda$, the constraint equations assure that another composite have a vev larger than $\Lambda$, and we would obtain $V_{\rm soft}> m_{\rm
soft}^2\Lambda^2$. Therefore, up to an ${\rm SU(2)_F}$ rotation, the minimum is at $$\begin{aligned}
\phi\sim
\left( \begin{array}{c} 0 \\ \Lambda \end{array} \right),~~~
\tilde{\phi}\sim
\left( \begin{array}{c} \Lambda \\ 0 \end{array} \right),~~~
A \sim \sigma \sim B \sim \bar{B} \sim \Lambda,~~~
S_{ij}=0.
\label{eq:themin}
\end{aligned}$$ This result can be verified by explicit minimization of the potential, taking into account all the order one parameters. However, the estimate in Eq. (\[eq:themin\]) will be sufficient for our purposes.
The vevs in Eq. (\[eq:themin\]) are exactly what we require to obtain viable textures from Eqs. (\[eq:hu\]) and (\[eq:hd\]). If we fix the ratio $$\lambda\equiv\frac{\Lambda}{M_*}<1,$$ we obtain $$\begin{aligned}
h_u \sim
\left( \begin{array}{ccc}
0 & \lambda^6 & 0 \\
\lambda^6 & \lambda^4 & \lambda^2 \\
0 & \lambda^2 & 1 \\
\end{array} \right),~~~
h_d \sim
\left( \begin{array}{ccc}
0 & \lambda^6 & 0 \\
\lambda^6 & \lambda^4 & \lambda^4 \\
0 & \lambda^2 & \lambda^2 \\
\end{array} \right).
\label{Yud}
\end{aligned}$$ All the elements in the Yukawa matrices are predicted in terms of one small parameter $\lambda$, up to unknown coefficients of order one. As we will see below, Eq. (\[Yud\]) results in a realistic pattern of the quark masses and mixing angles, if $\lambda\sim 0.2-0.3$, and the unknown ${\cal O}(1)$ coefficients are chosen appropriately.
We will now consider the pattern of quark masses and mixing angles more carefully, beginning with the up sector. The largest element in the up quark Yukawa matrix is the (3,3) entry, which is of order $1$, while the other elements are suppressed by powers of $\lambda$. Thus, $h_u$ has an eigenvalue close to one which we can identify with the top quark Yukawa coupling. Next, we consider the 2-3 block, since the remaining elements are much more suppressed. The determinant of this block is of ${\cal O}(\lambda^4)$ indicating there is an eigenvalue of the same order, which we identify as the charm quark Yukawa coupling. The rotation angle involved in the diagonalization of this block is ${\cal O}(\lambda^2)$, and hence $V_{cb}\sim \lambda^2$, if the up sector gives the dominant contribution. Finally, we see that $\det h_u \sim
{\cal O}(\lambda^{12})$, which implies that the smallest eigenvalue is ${\cal O}(\lambda^8)$. We identify this with the up quark Yukawa coupling. Thus, in our model we find $$m_u : m_c : m_t \sim \lambda^8 : \lambda^4 : 1
\,\,\,\,\, \mbox{ and } \,\,\,\,\,
V_{cb} \sim \lambda^2.$$ As far as the mass eigenvalues and $V_{cb}$ are concerned, the result for $h_u$ in our model works fairly well. The only problem is that the mixing between first and second generations, i.e. the Cabibbo angle, is ${\cal O}(\lambda^2)$, which is too small if $\lambda\sim
0.2-0.3$. ($V_{ub}$ on the other hand would be ${\cal O}(\lambda^4)$, which is acceptable.) Thus, the Cabibbo angle should have its origin in the down sector. We will come back to this point later.
We may now analyze $h_d$ in the same way. From the 2-3 block, we obtain the two larger eigenvalues of $h_d$, which are ${\cal
O}(\lambda^2)$ and ${\cal O}(\lambda^4)$. We identify these as the Yukawa coupling of the bottom and strange quarks, respectively. Notice that, with the choice of $\tan\beta\sim 2$, we obtain the correct value of the ratio of $m_b/m_t$. The 2-3 mixing angle is again ${\cal
O}(\lambda^2)$, and is consistent with the value of $V_{cb}$ for $\lambda\sim 0.2-0.3$. Finally, we must evaluate the down quark Yukawa coupling as well as the 1-2 mixing. Our results in Eq. (\[Yud\]) imply naively that the down quark Yukawa coupling is of ${\cal
O}(\lambda^8)$ and the the 1-2 mixing angle of ${\cal O}(\lambda^2)$, both of which are too small to be consistent with observation. To fix this problem, we must take into account the possible fluctuations of the unknown order one coefficients. If we allow the couplings giving the (1,2) and (2,1) elements of Eq. (\[eq:hd\]) to be enhanced by a factor of $1/\sqrt{\lambda}\sim 2$, and the (2,2) and (3,2) elements to be suppressed by the same amount, we will obtain a Cabibbo angle of ${\cal O}(\lambda)$, and a down quark coupling ${\cal}(\lambda^6
\sqrt{\lambda})$. Note that the predicted ratio $m_d/m_b \sim
\lambda^4 \sqrt{\lambda}$ is consistent with recent lattice estimates of the down quark mass [@lattice].
With this choice for the ${\cal O}(1)$ coefficients, the Yukawa matrix elements are given more accurately by $$\begin{aligned}
h_u \sim
\left( \begin{array}{ccc}
0 & \lambda^6 & 0 \\
\lambda^6 & \lambda^4 & \lambda^2 \\
0 & \lambda^2 & 1 \\
\end{array} \right),~~~
h_d \sim
\left( \begin{array}{ccc}
0 & \lambda^{11/2} & 0 \\
\lambda^{11/2} & \lambda^{9/2} & \lambda^4 \\
0 & \lambda^{5/2} & \lambda^2 \\
\end{array} \right).
\label{Yukawa_with_O(1)}\end{aligned}$$ We will diagonalize the results shown in Eq. (\[Yukawa\_with\_O(1)\]) when we need to evaluate a squark mass matrix in the quark mass eigenstate basis.
Finally, we present the textures for the squark mass matrices. The soft supersymmetry-breaking masses originate from D-terms interactions, which are not required to be holomorphic functions of the flavon fields. For example, the leading contributions to the left-handed squark masses are given by the operators $$\begin{aligned}
V_{\tilde{Q}\tilde{Q}^*} &\sim&
\tilde{m}^2 \Bigg[ c_0 |\tilde{Q}^1|^2 + c_0 |\tilde{Q}^2|^2
+ c_3 |\tilde{Q}^3|^2
\nonumber \\ &&
+ \frac{\Lambda^2}{M_*^4}
(\phi_i\tilde{Q}^i) (\phi_j\tilde{Q}^j)^*
+ \frac{\Lambda^2}{M_*^4}
(\tilde{\phi}_i\tilde{Q}^i) (\tilde{\phi}_j\tilde{Q}^j)^*
\nonumber \\ &&
+ \frac{\Lambda^2}{M_*^4}
(\epsilon_{ij}\phi^{*i}\tilde{Q}^j)
(\epsilon_{kl}\phi^{*k}\tilde{Q}^l)^*
+ \frac{\Lambda^2}{M_*^4}
(\epsilon_{ij}\tilde{\phi}^{*i}\tilde{Q}^j)
(\epsilon_{kl}\tilde{\phi}^{*k}\tilde{Q}^l)^*
\nonumber \\ &&
+ \frac{\Lambda^6}{M_*^{10}}
\left\{
(\phi_i\tilde{Q}^i) (\epsilon_{jk}\phi^{*j}\tilde{Q}^k)^* B^{*2}
+ h.c. \right\}
\nonumber \\ &&
+ \frac{\Lambda^5}{M_*^8}
\left\{
(\epsilon_{ij}\phi^{*j}\tilde{Q}^k) \tilde{Q}^{3*}
B \overline{B}^*
+ h.c. \right\}
\nonumber \\ &&
+ \frac{\Lambda^2}{M_*^4}
\left\{ (\tilde{\phi}_i\tilde{Q}^i) \tilde{Q}^{3*} + h.c. \right\}
\Bigg] \,\,\, ,
\end{aligned}$$ where $\tilde{m}$ is the typical scale of the squark masses. We have only shown order one coefficients explicitly in the flavor-invariant terms ($c_0$ and $c_3$) to remind the reader that the first two generation scalars are degenerate in the flavor symmetric limit, while the third generation scalar is unconstrained. After flavor symmetry breaking, the operators above lead to the texture $$\begin{aligned}
(\tilde{M}_q^2)^0_{LL} \sim
\tilde{m}^2 \left( \begin{array}{ccc}
c_0 + \lambda^4 & \lambda^{10} & \lambda^{8} \\
\lambda^{10} & c_0 + \lambda^4 & \lambda^2 \\
\lambda^{8} & \lambda^2 & c_3
\end{array}\right),
\end{aligned}$$ where the powers of $\lambda$ indicate the correction to the flavor invariant result with ${\cal O}(1)$ coefficients suppressed. It is straightforward to repeat this analysis for the right-handed squarks, and we obtain $$\begin{aligned}
(\tilde{M}_u^2)^0_{RR} \sim
\tilde{m}^2 \left( \begin{array}{ccc}
c'_0 + \lambda^4 & \lambda^{10} & \lambda^{8} \\
\lambda^{10} & c'_0 + \lambda^4 & \lambda^2 \\
\lambda^{8} & \lambda^2 & c'_3
\end{array}\right)\end{aligned}$$ and $$\begin{aligned}
(\tilde{M}_d^2)^0_{RR} \sim
\tilde{m}^2 \left( \begin{array}{ccc}
c''_0 + \lambda^4 & \lambda^{10} & \lambda^{10} \\
\lambda^{10} & c''_0 + \lambda^4 & \lambda^4 \\
\lambda^{10} & \lambda^4 & c''_3
\end{array}\right).\end{aligned}$$
We may now consider the bounds from flavor changing neutral current processes. We define the parameters $$\begin{aligned}
&&
(\delta_{ij}^q)_{XX} \equiv
|(\tilde{M}_q^2)_{XX,ij}| / \tilde{m}^2~~~(X=L,R),
\\ &&
\overline{\delta}_{ij}^q \equiv
\{(\delta_{ij}^q)_{LL} (\delta_{ij}^q)_{RR}\}^{1/2},
\end{aligned}$$ where $q=u,d$. Note that the absence of the superscript $0$ above $\tilde{M}$ indicates that the scalar mass matrices are to be evaluated in the [*quark mass eigenstate basis*]{}. The $\delta$ parameters corresponding to 1-2 and 1-3 scalar mass matrix elements are constrained by neutral pseudoscalar meson mixing to be less than $10^{-1}$ – $10^{-3}$, depending on the superparticle mass spectrum. Typical upper bounds are given in Table \[table:delta\].
We see that the off-diagonal elements are small enough to satisfy the experimental constraints, with the parameter $\zeta \equiv c''_3/c''_0
=1.3$. This ratio is not constrained by the flavor symmetry, and must be mildly adjusted (at the 30% level) because of the large right-handed 2-3 mixing angle in the down quark Yukawa matrix. This tuning is so mild, we will not let it concern us further. However, one should keep in mind that $\zeta$ may be naturally close to one if the model is embedded into a larger non-Abelian flavor group at a high scale. Finally, we note that the constraint on 2-3 mixing from $b\rightarrow s\gamma$ is very weak; $(\delta_{23}^d)_{LL,RR}\sim {\cal O}(1)$ is allowed [@NPB477-321]. We conclude that the non-Abelian model presented in this section is consistent with the flavor changing neutral current constraints. Note that the model can be extended trivially to the lepton sector by choosing the lepton transformation properties to be identical to those of the down quarks. Then the differences between the down quark and lepton masses can be explained by fluctuations in the order one coefficients.
[$(\delta_{12}^d)_{LL}$]{} [$(\delta_{13}^d)_{LL}$]{} [$(\delta_{12}^u)_{LL}$]{}
----------------------------- ----------------------------------- ----------------------------------- -------------------------------- -- -- --
[Exp. upper bound]{} [$4.0\times 10^{-2}$]{} [$9.8\times 10^{-2}$]{} [$1.0\times 10^{-1}$]{}
[Prediction of the model]{} [$\lambda^{5}$]{} [$\lambda^{3}$]{} [$\lambda^{6}$]{}
[$\sim 5.2\times 10^{-4}$]{} [$\sim 1.1\times 10^{-2}$]{} [$\sim 1.1\times 10^{-4}$]{}
[$(\delta_{12}^d)_{RR}$]{} [$(\delta_{13}^d)_{RR}$]{} [$(\delta_{12}^u)_{RR}$]{}
[Exp. upper bound]{} [$4.0\times 10^{-2}$]{} [$9.8\times 10^{-2}$]{} [$1.0\times 10^{-1}$]{}
[Prediction of the model]{} [$(\zeta-1)\lambda^{2}$]{} [$(\zeta-1)\lambda^{3/2}$]{} [$\lambda^{6}$]{}
[$\sim 1.5\times 10^{-2}$]{} [$\sim 3.1\times 10^{-2}$]{} [$\sim 1.1\times 10^{-4}$]{}
[$\overline{\delta}_{12}^d$]{} [$\overline{\delta}_{13}^d$]{} [$\overline{\delta}_{12}^u$]{}
[Exp. upper bound]{} [$2.8\times 10^{-3}$]{} [$1.8\times 10^{-2}$]{} [$1.7\times 10^{-2}$]{}
[Prediction of the model]{} [$\sqrt{\zeta-1}\lambda^{7/2}$]{} [$\sqrt{\zeta-1}\lambda^{9/4}$]{} [$\lambda^{6}$]{}
[$\sim 2.7 \times 10^{-3}$]{} [$\sim 1.8 \times 10^{-2}$]{} [$\sim 1.1 \times 10^{-4}$]{}
: Upper bounds on $(\delta_{ij}^q)_{LL,RR}$ and $\overline{\delta}_{ij}^q$ . Here, we take all the squark and gluino masses to be 500 GeV. For comparison, we also show the prediction of our model with $\lambda =0.22$ and $\zeta=c''_3/c''_0=1.3$.[]{data-label="table:delta"}
Abelian Model {#sec:abel}
=============
We have seen that it is possible to construct models with non-Abelian flavor group factors in which flavor symmetries are broken via the dynamics of confinement. Non-Abelian theories greatly alleviate the supersymmetric flavor-changing problem by imposing a natural degeneracy between the first two generation scalar masses in the flavor symmetric limit. In this section, we show that models based on Abelian flavor symmetries can also incorporate our mechanism. Such models solve the supersymmetric flavor problem by arranging an alignment of the quark and squark mass matrices, so that the squark masses are nearly diagonal in the quark mass eigenstate basis. The alignment is strongest in the down quark sector, where the phenomenological constraints are most powerful, and the Cabibbo angle originates in the up quark sector. In this section, we will not present an exhaustive phenomenological analysis, but simply show that our symmetry-breaking mechanism can be combined with the prototypical alignment models of Nir and Seiberg [@NS].
The important feature of the models of Ref. [@NS], as well as similar models in Ref. [@align], is the presence of two flavon fields, that transform under two independent U(1) flavor symmetries: $S_1(-1,0)$ and $S_2(0,-1)$. These fields are assumed to acquire vevs $$\langle S_1 \rangle = \epsilon_1 \sim \lambda^2
\,\,\,\,\,\,\,\, \mbox{ and } \,\,\,\,\,\,\,\,
\langle S_2 \rangle= \epsilon_2 \sim \lambda^3 \,\,\, .
\label{eq:svevs}$$ Two models using these flavons are presented in Ref. [@NS] (models A and B) which differ only in the flavor quantum number assignments of the matter fields. We will explicitly consider model A below.
The most elegant way of embedding this flavor sector into the SU(3) theory described in the previous section is to choose $S_1$ to be one of the meson fields, and $S_2$ to be one of the baryons. Since the baryon $S_2$ has one additional preon, its symmetry breaking effect will be suppressed relative to the meson $S_1$ by one factor of $\Lambda/M_*$. If we again choose this ratio to be the Cabibbo angle $\lambda$, we account for Eq. (\[eq:svevs\]) in a natural way. The two U(1) factors can be taken such that $$\begin{aligned}
Q_I = \sqrt{3} \, Q_{\overline{p}} \nonumber \\
Q_{II} = - Q_B \end{aligned}$$ where the charges $Q_{\overline{p}}$ and $Q_B$ are defined as in the previous section. We can then make the identification $$\begin{aligned}
S_1(-1,0) \equiv \sigma \nonumber \\
S_2(0,-1) \equiv B \end{aligned}$$ The remaining composites $S$, $A$, $\phi$, $\tilde{\phi}$ and $\bar{B}$, also have flavor quantum numbers, and may alter the Yukawa matrices slightly from the form presented in Ref. [@NS]. However, we will now show that the quark-squark alignment remains unaffected. We will assume that the flavor SU(2) of the previous section is a good flavor symmetry (even though the matter fields are SU(2) singlets). Since the matter fields will have integral charges under the two U(1) factors, the lowest order combinations of the remaining composites that can contribute to the Yukawa textures are: $$\bar{B}(0,+1) \sim \epsilon_2 \,\,\,\,\, \mbox{ and } \,\,\,\,\,
A^2 (+1,0) \sim \epsilon_1^2$$ Note that we have neglected terms involving $S$ which does not acquire a vev at lowest order. The combination $(\phi \tilde{\phi})^2\sim (-1,0)$ couples in the same way as $S_1$, but is of higher order in $1/M_*$ and can also be neglected. In model A, the matter fields are assigned charges $$\begin{aligned}
Q_1(3,-1,+) & Q_2(1,0,-) & Q_3(0,0,+) \nonumber \\
U_1(-3,3,+) & U_2(-1,1,+) & U_3(0,0,+) \nonumber \\
D_1(-3,3,+) & D_2(1,0,-) & D_3(1,0,-) \end{aligned}$$ where the third entry is the charge under our anomalous $Z_2$ factor, defined in the previous section. The original textures for Model A in Ref. [@NS] $$h_u \sim \left(\begin{array}{ccc}
\epsilon_2^2 & \epsilon_1^2 & 0 \\
0 & \epsilon_2 & \epsilon_1 \\
0 & 0 & 1 \end{array}\right) \,\,\,\,\,\,\,\,\,\,\,
h_d \sim \left(\begin{array}{ccc}
\epsilon_2^2 & 0 & 0 \\
0 & \epsilon_1^2 & \epsilon_1^2 \\
0 & \epsilon_1 & \epsilon_1 \end{array}\right)$$ become $$h_u \sim \left(\begin{array}{ccc}
\epsilon_2^2 & \epsilon_1^2 & \epsilon_1^6 \epsilon_2 \\
\epsilon_1^4\epsilon_2^3 & \epsilon_2 & \epsilon_1 \\
\epsilon_1^9 \epsilon_2^3 & \epsilon_1^5\epsilon_2 & 1
\end{array}\right) \,\,\,\,\,\,\,\,\,\,\,
h_d \sim \left(\begin{array}{ccc}
\epsilon_2^2 & \epsilon_1^7\epsilon_2 & \epsilon_1^7\epsilon_2 \\
\epsilon_1^4\epsilon_2^3 & \epsilon_1^2 & \epsilon_1^2 \\
\epsilon_1^9 \epsilon_2^3 & \epsilon_1 & \epsilon_1 \end{array}\right)
\,\,\, .
\label{eq:newent}$$ The scalar mass matrices are not holomorphic functions of the flavon fields, so their textures remain unchanged: $$\frac{(\tilde{M^2_q})^0_{LL}}{\tilde{m}^2} \sim \left(\begin{array}{ccc}
1 & \epsilon_1^2 \epsilon_2 & \epsilon_1^3 \epsilon_2 \\
\epsilon_1^2 \epsilon_2 & 1 & \epsilon_1 \\
\epsilon_1^3 \epsilon_2& \epsilon_1 & 1 \end{array}\right)
\,\,\,\,\,\,\,\,\,\,\,
\frac{(\tilde{M^2_u})^0_{RR}}{\tilde{m}^2} \sim \left(\begin{array}{ccc}
1 & \epsilon_1^2\epsilon_2^2 & \epsilon_1^3\epsilon_2^3 \\
\epsilon_1^2\epsilon_2^2 & 1 & \epsilon_1 \epsilon_2 \\
\epsilon_1^3\epsilon_2^3 & \epsilon_1\epsilon_2 & 1 \end{array}\right)$$$$\frac{(\tilde{M^2_d})^0_{RR}}{\tilde{m}^2} \sim \left(\begin{array}{ccc}
1 & \epsilon_1^4\epsilon_2^3 & \epsilon_1^4\epsilon_2^3 \\
\epsilon_1^4\epsilon_2^3 & 1 & 1 \\
\epsilon_1^4\epsilon_2^3 & 1 & 1 \end{array}\right) \,\,\, .$$ If we now go to the quark mass eigenstate basis, all the rotations on the left-handed quark fields that are induced by the additional entries in Eq. (\[eq:newent\]) do not alter the order of magnitude of any off-diagonal squark mass matrix elements. Only the 1-2 rotation in the right-handed down sector is large enough to change the (1,2) entry of $(\tilde{M^2_d})_{RR}$ from $\epsilon_1^4\epsilon_2^3$ to $\epsilon_1^2 \epsilon_2^3 \sim 10^{-9}$. The bound on this entry from flavor changing neutral currents is of order $10^{-2}$, and is still easily satisfied in the modified model.
Thus, the presence of additional flavons implied by our symmetry-breaking mechanism does not disturb the quark-squark alignment. One can easily verify that the same is true for Model B of Ref. [@NS] as well.
Conclusions
===========
Supersymmetric theories have two sets of small dimensionless flavor parameters: one describes the quark and lepton mass ratios and mixing angles, while the other describes squark and slepton non-degeneracies and mixings, which are constrained from flavor-changing processes. We have described a general framework of theories with a flavor symmetry, and given two explicit realistic models, where
- Flavor symmetry breaking is forced by strong supersymmetric gauge interactions.
- All non-zero vevs have a magnitude of order the $\Lambda$ parameter of the new strong gauge force. All flavor symmetry breaking occurs at a single scale, and there is a single small dimensionless parameter, $\Lambda/M_*$, where $M_*$ is the cutoff for the theory.
- The flavor symmetry allows certain higher dimension F and D operators coupling quarks ($q$), Higgs ($H$) and preons ($p$), $$[\bar{q} q H (p/M_*)^n]_F \,\,\,\,\, \mbox{ and }
\,\,\,\,\,
[q^\dagger q (p/M_*)^m]_D \,\,\, ,$$ generating small entries in the quark and squark mass matrices of order $(\Lambda / M_*)^n$ and $(\Lambda / M_*)^m$ respectively.
- While flavor symmetry breaking is forced, there is a large vacuum degeneracy — the quark and squark mass matrices are functions on moduli space. This degeneracy can be lifted in a favorable direction by the combined use of the $X$ fields and soft, positive supersymmetry breaking squared masses.
[**Acknowledgments**]{}
We would like to thank N. Arkani-Hamed, H. Murayama and J. Terning for useful discussions. This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098. LJH was also supported in part by the National Science Foundation under grant PHY-95-14797.
[99]{}
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[^1]: This work was supported in part by the Director, Office of Energy Research, Office of High Energy and Nuclear Physics, Division of High Energy Physics of the U.S. Department of Energy under Contract DE-AC03-76SF00098. LJH was also supported in part by the National Science Foundation under grant PHY-95-14797.
[^2]: In certain other theories, symmetry breaking can occur by the combination of supersymmetric gauge interactions and superpotential interactions. These have recently been used to study the breaking of grand unified symmetries [@cheng].
[^3]: We assume that this symmetry is spontaneously broken in the hidden sector, so that we generate gaugino masses, trilinear scalar interactions, etc.
[^4]: In writing down the low energy description of the operator $X_0[ppp\bar{p}\bar{p}\bar{p}]/M^4_*$, we have included a linear term $X_0 \Lambda^6/M_*^4$. This term can be justified by treating $\Lambda$ as a spurion under the anomalous axial U(1) symmetry of the dynamical sector, and including the most general set of invariant interactions. However, nothing in our analysis changes if such a term is absent.
[^5]: Here, we do not assume universal soft supersymmetry-breaking masses. The $m^2_{{\rm soft}}$ in Eq. (\[eq:smasses\]) is understood to be different for each of the terms shown.
|
---
author:
- 'H.M. Günther'
- 'J.H.M.M. Schmitt'
- 'J. Robrade'
- 'C. Liefke'
bibliography:
- '../articles.bib'
date: 'Received 23 May 2006 / accepted 21 February 2007'
title: 'X-ray emission from classical T Tauri stars: Accretion shocks and coronae? [^1]'
---
Introduction
============
T Tauri stars are young ($<10\;\mathrm{Myr}$), low mass ($M_*<3M_{\sun}$), pre-main sequence stars exhibiting strong H$\alpha$ emission. The class of “Classical T Tauri stars” (CTTS) are surrounded by an accretion disk and are actively accreting material from the disk. The disks do not reach all the way to the stellar surface, rather they are truncated in the vicinity of the corotation radius. Infrared (IR) observations typically yield inner radii of 0.07-0.54 AU [@2003ApJ...597L.149M], consistent with the corotation radius. Disk material is ionised by energetic stellar radiation and – somehow – loaded onto the stellar magnetic field lines, traditionally assumed to be dipole-like [but see @Valenti04; @2006MNRAS.371..999G]. Along the magnetic field accretion funnels or curtains develop and matter impacts onto the star at nearly free-fall velocity [@1984PASJ...36..105U; @1991ApJ...370L..39K]. This process can remove angular momentum from the star [@1994ApJ...429..781S]. Observationally the accretion can be traced in the optical in the H$\alpha$ line profile [@2000ApJ...535L..47M], in the IR [@2001ApJ...551.1037B] and in the ultraviolet UV [@2005AJ....129.2777H]. Further support for this scenario comes from the measurement of magnetic fields in some CTTS using the technique of spectropolarimetry and Zeeman-broadening [@1999ApJ...516..900J; @2003RMxAC..18...38J; @2005MNRAS.358..977S]. The kinetic energy of the accretion stream is released in one or several hot spots close to the stellar surface, producing the observed veiling continuum, and also line emission in the UV and X-rays. The emitted UV continuum radiation was previously calculated by @calvetgullbring and detailed models of the accretion geometry prove that stable states with two or more accretion spots on the surface can exist [@2004ApJ...610..920R]. The UV emission has been also used to estimate mass accretion rates in CTTS with typical values of $10^{-8}M_{\sun}$ year$^{-1}$ [@1999MNRAS.304L..41G; @2000ApJ...539..815J].
T Tauri stars are copious emitters of X-ray emission. Specifically, X-ray emission from quite a few CTTS was detected with the *Einstein* [@1981ApJ...243L..89F; @1989ApJ...338..262F] and *ROSAT* satellites . The origin of the detected X-ray emission is usually interpreted as a scaled-up version of coronal activity as observed for our Sun, and the data and their interpretation prior to the satellites XMM-Newton and *Chandra* are summarised in a review by . However, recent observations of the CTTS [@2002ApJ...567..434K; @twhya], [@bptau] and [@v4046] with the grating spectrometers onboard XMM-Newton and *Chandra* indicate very high plasma densities in the X-ray emitting regions much higher than those observed in typical coronal sources. This finding suggests a different origin of at least the soft part of the X-ray spectrum in CTTS. Simple estimates show that X-rays can indeed be produced in the accretion spot of a typical CTTS [@lamzin]. We present here a more detailed accretion shock model, which predicts individual emission lines and can thus be directly confronted with observations to determine model parameters such as the maximum plasma temperature and mass accretion rate.
Unfortunately, only a few CTTS have so far been studied in detail using high-resolution X-ray spectroscopy with sufficient signal-to-noise ratio. It is therefore unknown at present whether the observed low forbidden to intercombination line ratios in He-like ions as measured for the CTTS TW Hya, BP Tau and V4046 Sgr are typical for CTTS as a class. Lower resolution studies of CTTS show significant differences between individual stars, possibly caused by considerable individual coronal activity, and in general it is difficult to disentangle coronal and accretion contributions [@Rob0507] in the X-ray spectra of CTTS. Other suggested sources of X-rays include dense clumps in the stellar or disk wind of CTTS, heated up by shock waves. Simulations by @2002ApJ...574..232M show sufficiently dense regions in the stellar magnetosphere.
The goal of this paper is to consider the maximally possible accretion shock contribution, to determine the physical shock parameters, compute the emitted X-ray spectrum and assess to what extent other emission components are necessary to account for the observed X-ray spectra. The detailed plan of our paper is as follows: In Sect. \[obs\] the observations used in this study are described briefly, in Sect. \[model\] we present our model and give the main assumption and limitations, in Sect. \[res\] the results of the simulation are shown and applied to observational data, followed by a short discussion of the main points in Sect. \[dis\].
Observations {#obs}
============
Stellar parameters
-------------------
With a distance of only 57 pc [@1998MNRAS.301L..39W] TW Hya is the closest known CTTS; it is not submerged in a dark, molecular cloud like many other CTTS. Photometric observations show variability between magnitude 10.9 and 11.3 in the V-band . Broad H$\alpha$ profiles (FWHM $\sim 200$ km s$^{-1}$) were observed by @2000ApJ...535L..47M, and TW Hya apparently belongs to a group of objects with similar age, the so-called TW Hydrae association [TWA; @1999ApJ...512L..63W]. TW Hydrae’s mass and radius are usually quoted as $M_*$=0.7 $M_{\sun}$, $R_*$=1.0 $R_{\sun}$, and its age as 10 Myr from @1999ApJ...512L..63W. Alternative values are given by @2002ApJ...580..343B, who place TW Hya on the HR diagram by and determine stellar parameters from the optical spectrum, which fits an older (30 Myr) and smaller star ($R_*$=0.8 $R_{\sun}$). The spectral type of TW Hya is K7$\;$V-M1$\;$V [@1999ApJ...512L..63W; @2002ApJ...580..343B] and the system is seen nearly pole-on [@1997Sci...277...67K; @2000ApJ...534L.101W; @2002ApJ...571..378A]. Moreover, TW Hya displays variations in line profiles and veiling, which have been interpreted as signatures of accretion spot rotation [@2002ApJ...571..378A; @2002ApJ...580..343B]. TW Hya has been observed in the UV with *IUE*, *FUSE* and *HST/STIS* [@2002ApJ...572..310H], revealing a wealth of H$_2$ emission lines, consistent with the origin in the surface of a irradiated disk, and in X-rays with *ROSAT* by , *Chandra*/HETGS [@2002ApJ...567..434K] and XMM-Newton/RGS [@twhya], where the grating data indictes a significant accretion shock contribution.
X-ray observations {#obs_data}
------------------
We use high resolution spectra obtained with the *Chandra* and XMM-Newton. TW Hya was observed for 48 ks with the *Chandra* HETGS on July 18, 2000 (Chandra ObsId 5). @2002ApJ...567..434K report atypically high densities measured from the and triplets, and a very high neon abundance as observed for many active coronal sources in combination with a low iron abundance; the anomalously high neon abundance of TW Hya was investigated in more detail by @2005ApJ...627L.149D. A different approach to assess the plasma density of the emitting material by means of iron line ratios was performed by @Ness0510. First-order grating spectra were extracted applying standard CIAO 3.2 tools, positive and negative orders were added up. Individual emission lines in the HEG and MEG spectra were analysed with the CORA line fitting tool [@2002AN....323..129N], assuming modified Lorentzian line profiles with $\beta = 2.5$. A flare occurring in the second half of the observation was already mentioned by @2002ApJ...567..434K; for our global fitting approach we excluded the flaring period to avoid contamination due to the probable coronal origin of the flare.
Another X-ray spectrum was taken with XMM-Newton on July 9, 2001 with an exposure time of 30 ks (Obs-ID 0112880201) using the RGS as prime instrument. An analysis of this observation was presented in @twhya. We newly reduced also this dataset with the XMM-Newton Science Analysis System (SAS) software, version 6.0 and applied the standard selection criteria. The X-ray spectral analysis was carried out using XSPEC V11.3 [@1996ASPC..101...17A], and CORA for line-fitting. Because the line widths are dominated by instrumental broadening we keep them fixed at $\Delta\lambda=0.06$ Å. The RGS spectra cover a larger wavelength range than HETGS spectra and include, in addition to the He-like triplets of Ne and O, also the N triplet. Both observations show the observed helium-like triplets to be incompatible with the low-density limit and an emission measure analysis indicates the presence of plasma with a few million degrees emitting in the soft X-ray region [@2002ApJ...567..434K; @twhya].
The model {#model}
=========
In the currently accepted accretion paradigm the material follows the magnetic field lines from the disk down to the surface of the star. Here we only model the base of the accretion column, where the infalling material hits the stellar surface, is heated up in a shock and cools down radiatively. A sketch of the envisaged accretion scenario is shown in Fig. \[funnel\_names\].
Calculations of the excess continuum produced in this region were carried out by @calvetgullbring. @lamzin already computed the emerging soft X-ray emission from such shocks, but due to bin sizes of 50 Å his results do not resolve individual lines, thus concealing much of the valuable diagnostic information. Nevertheless his models show that the hot spot produced by accretion can possibly produce not only the veiling, but also the soft X-ray emission. In our modelling we resolve all individual emission lines, allowing us to use line ratios as sensitive tracers of the density and temperature in the emitting region, and determine the elemental abundances of the emitting regions. Additionally we explicitly consider non-equilibrium ionisation states and distinguish ion and electron temperatures.\
We assume a one-dimensional, plane parallel geometry for the accretion column. This is reasonable because the filling factor $f$, expressing the ratio between total spot size $A_{\mathrm{spot}}$ and stellar surface $A_*=4\pi R_*^2$, is quite small as will be demonstrated a posteriori. The magnetic field is assumed to be perpendicular to the stellar surface and the material flows along the magnetic field lines. All turbulent fluxes are neglected and our model further assumes a stationary state. We use a two-fluid approximation attributing different temperatures to the atom/ion $T_{\rm{ion}}$ and to the electron components $T_{\rm{e}}$ of the plasma. However, both components move with the same bulk velocity $v$ because they are strongly coupled by the microscopic electric fields. We ignore energy transport by heat conduction which is justified again a posteriori (see Sect. \[heat\_cond\]) and any radiative transport (see Sect. \[opticaldepth\]).
Calculation of shock front
--------------------------
Once the infalling material impacts onto the stellar surface, a shock forms, defining the origin of the depth coordinate $z$ (see Fig. \[funnel\_names\]). The shock front itself is very thin, only of the order of a few mean free paths [@raizerzeldovich], thus contributing only very marginally to the total emission. Therefore it is not necessary to numerically resolve its internal structure, rather it can be treated as a mathematical discontinuity and the total change in the hydrodynamic variables over this discontinuity is given by the Rankine-Hugoniot conditions [@raizerzeldovich chap. 7, § 15], which transform supersonic motion into subsonic motion in one single step. To simplify the numerical treatment we assume the direction of flow to be exactly parallel the magnetic field lines, so the Lorentz force does not influence the dynamics; we also expect the magnetic field to suppress heat conduction. Following the treatment by @calvetgullbring and @lamzin we assume the gas to expand into a vacuum behind the shock front. Because of larger viscous forces the strong shock formation occurs only in the ionic component, while the electron component is first only adiabatically compressed and subsequently heated through electron-ion collisions. In order to calculate the state of the ionic plasma behind the shock front in terms of the pre-shock conditions only the fluxes of the conserved quantities mass, momentum and energy are required. Marking the state in front of the shock front by the index 0, that behind the shock by index 1, the Rankine-Hugoniot conditions become $$\begin{aligned}
\rho_0 v_0 &=& \rho_1 v_1 \label{RH1}\\
P_0+\rho_0 v_0^2 &=& P_1+\rho_1 v_1^2 \label{RH2}\\
\frac{5 P_0}{2\rho_0}+\frac{v_0^2}{2}&=&\frac{5 P_1}{2\rho_1}+\frac{v_1^2}{2} \ ,\label{RH3}\end{aligned}$$ where $\rho$ denotes the total mass density of the gas and $P$ its pressure.
Requiring that the electric coupling between ions and electrons leads an adiabatic compression of the electron component, implies a temperature rise for the electronic plasma component of $$T_{e_1}=T_{e_0} \left(\frac{\rho_1}{\rho_0}\right)^{(\gamma-1)}$$ with $\gamma=5/3$ denoting the adiabatic index. Because the time scale for heat transfer from ions to electrons is much larger than that of the ions passing through the shock front, ions and electrons leave the shock with vastly different temperatures. Numerical evaluation of the above equations behind the accretion shock front results in electron temperatures orders of magnitude lower than the ion temperature. Furthermore, the ions pass the shock front so fast that other degrees of freedom than kinetic are not excited and therefore kinetic and ionisation temperatures of the ions substantially differ.
Structure of the post-shock region
----------------------------------
In the following section we compute how the originally different kinetic temperatures of ions and electrons as well as the ionisation temperature equilibrate and calculate the emitted X-ray spectrum.
### Momentum balance {#hydrodyn}
In the post-shock region heat is transferred from the ionic to the electronic component, at the same time the gas radiates and cools down, so the energy of the gas is no longer conserved. However, the particle number flux $j$ of ions (and atoms) $$j=nv\label{j_n}$$ is conserved, where $n$ is the ion/atom number density; the electron number density is denoted by $n_{\mathrm{e}}$. The total momentum flux $j_p$ is conserved, since we ignore the momentum loss by radiation; it consists of the ion and the electron momentum as follows: $$\begin{aligned}
j_p&=&\mu m_{\mathrm{H}} n v^2+P_{\mathrm{ion}}+m_{\mathrm{e}} n_{\mathrm{e}} v^2+P_{\mathrm{e}} \nonumber \\
&=&\mu m_{\mathrm{H}} n v^2+nkT_{\mathrm{ion}}+m_{\mathrm{e}} n_{\mathrm{e}} v^2+n_{\mathrm{elec}}kT_{\mathrm{e}} \nonumber\\
&\approx&\mu m_{\mathrm{H}} n v^2+nkT_{\mathrm{ion}}+n_{\mathrm{elec}}kT_{\mathrm{e}} \ (m_{e}\ll m_{\mathrm{H}})\label{j_p}\end{aligned}$$ with $P_{\mathrm{ion}}$ and $P_{\mathrm{e}}$ denoting the thermodynamic pressure of the ion and the electron gas respectively, $T_{\mathrm{ion}}$ and $T_{\mathrm{e}}$ their temperatures; $m_{\mathrm{H}}$ denotes the mass of a hydrogen atom, $m_{\mathrm{e}}$ the electron mass and $\mu$ is the dimensionless atomic weight.\
### Energy balance {#energybalance}
Let us next consider the energy balance in the post-shock region. Both for the ions and the electrons the evolution of the energy per particle is described by an ordinary differential equation (ODE). Starting from the thermodynamic relation $$\label{tsminuspdvisdu} T d\Sigma -P dV=dU$$ where $\Sigma$ denotes the entropy and $U$ the internal energy of the plasma per heavy particle we will derive this ODE for the ionic component. In eqn. \[tsminuspdvisdu\] the quantity $T d\Sigma=dQ$ denotes the heat flux through the boundaries of the system. The system “ions” looses heat by collisions with the colder electrons. Heat transfer is most efficient for the lighter ions and especially protons have a much larger collision cross sections than neutral hydrogen and they are by far the most abundant species in the plasma. To describe the heat transfer between ions and electrons we follow @raizerzeldovich [Chapter VII, §10], who give the heat flow $\omega_{ei}$ per unit volume per unit time as (in cgs-units) $$\omega_{ei}=\frac{3}{2}k n_{\mathrm{\ion{H}{ii}}} n_{\mathrm{e}} \frac{T_{\mathrm{ion}}-T_{\mathrm{e}}}{T_{\mathrm{e}}^{3/2}}\frac{\Lambda}{252} \textnormal{ (T in K)}$$ where $k$ is Boltzmann’s constant and $\Lambda$ is the Coulomb-logarithm $$\Lambda\approx 9.4+1.5 \ln T_{\mathrm{e}} -0.5 \ln n_{\mathrm{e}}\ .
\label{hyd_coulomblogarithm}$$
The number density of hydrogen ions is the number density of all heavy particles $n$ multiplied by the abundance $\xi_{\mathrm{H}}$ of hydrogen and the hydrogen ionisation fraction $x_{\mathrm{H}}^1$ : $$\label{omegaei}
\omega_{ei} =\frac{3}{2}k\xi_H x_H^1 n_{\mathrm{e}} \frac{T_{\mathrm{ion}}-T_{\mathrm{e}}}{T_{\mathrm{e}}^{3/2}}\frac{\Lambda}{252}\textnormal{ (T in K).}$$ Transforming Eq. (\[tsminuspdvisdu\]) written per heavy particle and taking the time derivative results in $$\frac{dU}{dt}+P\frac{dV}{dt}=T\frac{d\Sigma}{dt}=\frac{dQ}{dt}=\omega_{ei} \ .$$ Using the stationarity condition $\frac{d}{dt}=\frac{\partial}{\partial t}+\frac{\partial z}{\partial t}\frac{\partial}{\partial z}=v\frac{\partial}{\partial z}$ transforms this into an ODE with the dependent variable $z$, measured from the shock front inwards (see Fig. \[funnel\_names\]); differentiation with respect to $z$ will be indicated by $'$. Thus $$v U'+PvV'=\omega_{ei} \ .$$ The internal energy $U$ is in this case the thermal energy $U=\frac{3}{2}kT$, the pressure $P$ can be rewritten using the equation of state for a perfect gas. The specific volume $V$ is the inverse of the number density $V=\frac{1}{n}$. We thus obtain $$\label{energyion}
v\left(\frac{3}{2}k T_{\mathrm{ion}}\right)'+v n k T_{\mathrm{ion}} \left(\frac{1}{n}\right)'=-\omega_{ei} \ .$$ The derivation of a corresponding equation for the electron component is similar. Since the heat loss of the ion gas is a gain for the electrons this term enters with opposite sign, and an additional loss term $Q_{col}$ appears representing energy losses by collisions which excite or ionise a heavy particle that in turn radiates.\
It is convenient to write the electron number density as with $x_e$ denoting the number of electrons per heavy particle. $$\label{energyelec}
v\left(\frac{3}{2}x_e k T_{\mathrm{e}}\right)'+v x_e n k T_{\mathrm{e}} \left(\frac{1}{n}\right)'=\omega_{ei}-Q_{col} x_e n,$$ because $\omega_{ei}$ already includes the factor $x_e n$ by definition in eqn \[omegaei\]. Thus we are left with four independent variables ($n$, $v$, $T_{\mathrm{ion}}$ and $T_{\mathrm{e}}$); therefore, given $x_e$, the system of the four hydrodynamic equations (\[j\_n\], \[j\_p\], \[energyion\] and \[energyelec\]) is closed and can be solved by numerical integration.
According to @spitzer [Chapter 5.3] particle velocities reach a Maxwellian distribution after a few mean free path lengths. Evaluating the conditions behind the shock front we find that after a few hundred meters such distributions are established separately for both the ions and the electrons. We therefore assume that both ions and electrons each have their own individual Maxwellian velocity distributions throughout our simulation. This allows us to define an effective kinetic temperature for electron-atom/ion collisions. Usually collisions are treated using the ion rest frame as reference frame, and we fold the kinetic velocities of ion and electron and write the resulting Maxwellian distribution with the effective temperature $T_{\mathrm{eff}}=T_{\mathrm{e}}+T_{\mathrm{ion}}\frac{m_{\mathrm{e}}}{\mu m_{\mathrm{H}}}$. This effective collision temperature is then used to calculate the radiative loss term $Q_{col}$ with the CHIANTI 4.2 code [@CHIANTI; @CHIANTIVI]. Because the gas may be in a non-equilibrium ionisation state the tables produced by the built-in CHIANTI rad\_loss procedure, which assumes kinetic and ionisation temperatures equilibrated, are not valid in this case, instead a spectrum with the current state of ionisation and effective collision temperature is calculated and integrated over all contributing wavelengths to determine the instantaneous radiative losses. For fitting purposes it is further necessary to perform the simulations with different abundances because lower metallicity significantly lowers the cooling rate and therefore the extent of the post-shock cooling zone. In this approach it has to be assumed that $Q_{col}$ is constant over each time step.
### Microscopic physics
The calculation of the ionisation state is completely decoupled from the hydrodynamic equations given above. In each time step the density and the temperatures of both plasma components are fed from the hydrodynamic into the microscopic equations below. We assume changes in ionisation to occur only through collisions with electrons. Ions are ionised by electron collisions (bound-free) and recombine by electron capture (free-bound). So the number density of ionisations $r_{i\rightarrow i+1}$ per unit time from state $i$ to $i+1$ is proportional to the number density of ions $n_i$ in ionisation state $i$ and the number density of electrons $n_{\mathrm{e}}$; for convenience we leave out a superscript identifying the element in question here: $$\label{mp_ionisation}
r_{i\rightarrow i+1}=R_{i\rightarrow i+1} n_{\mathrm{e}} n_i \ .$$ Recombination is the reverse process: $$\label{mp_recombination}
r_{i\rightarrow i-1}=R_{i\rightarrow i-1}n_{\mathrm{e}} n_i \ .$$ The quantity $R_{i\rightarrow{} j}$ is the rate coefficient describing ionisation for $j=i+1$ and recombination for $j=i-1$. For element $z$ $R_{1\rightarrow{} 0}=R_{z+1\rightarrow{} z+2}\equiv0$ because $1$ represents the neutral atom, which cannot recombine any further, and $z+1$ the completely ionised ion which cannot lose any more electrons. The cross section $\sigma$ for each process depends not only on the ion, but also on the relative velocity of ion and electron. On the one hand, the number of ions in state $i$ decreases by ionisation to state $i+1$ or recombination to state $i-1$, on the other hand, it increases by ionisations from $i-1$ to $i$ and by recombination from $i+1$. For an element with atomic number $Z$ there is thus a set of $Z+1$ equations $$\begin{aligned}
\label{mp_rateeqn}
\frac{d{n_i}}{dt}=n_{\mathrm{e}}&(R_{i-1\rightarrow{} i} \,n_{i-1}-(R_{i\rightarrow{} i+1}\,+R_{i\rightarrow{}i-1})\,n_i\nonumber\\
&+R_{i+1\rightarrow{} i}\,n_{i+1}) \ .\end{aligned}$$
Through the electron number density $n_{\mathrm{e}}$ the equations for all elements are coupled and together with the condition of number conservation they provide a complete system of differential equations. This system can be simplified considerably under the assumption that $n_{\mathrm{e}}$ is constant during each time step $\Delta t$. Because hydrogen and helium, the main donors of electrons, are completely ionised $n_{\mathrm{e}}$ is mainly given by the hydrodynamics. This assumption leads to one independent set of equations per element. Dividing by the number density of the element in question and using that the number density $n_Z^A$ of ions in ionisation stage $Z$ for element $A$ is $n_Z^A=n \xi^A x_Z^A$ with abundance $\xi^A$ of element A and the ionisation fraction $x_Z^A$, finally leads to $$\begin{aligned}
\label{mp_rateeqn1}
\frac{d{x_i}}{dt}=n_{\mathrm{e}} & (R_{i-1\rightarrow{} i}\, x_{i-1}-(R_{i\rightarrow{} i+1}\,+R_{i\rightarrow{}i-1})\,x_i \nonumber \\
&+R_{i+1\rightarrow i}\,x_{i+1})\ .\end{aligned}$$
The rate coefficients $R_{i\rightarrow j}$ are taken from @mazzottaetal for dielectronic recombination, for the radiative recombination and the ionisation (collisional and auto-ionisation) rate we use a code from D.A. Verner, which is available in electronic form on the web[^2]. We calculate only the elements with Z=1-28, considering all ionisation states for each of them. The model is implemented in IDL (Interactive data language) and the ODEs are independently integrated using ’lsode’, an adaptive stepsize algorithm, which is provided in the IDL distribution.
Verification
------------
In order to check our computational procedures we considered several special cases with known analytical solutions. This includes a pure hydrogen gas with different electron and ion temperatures to test the heat transfer treatment and the ionisation of hydrogen at constant temperature. We compare our calculations for the collisional ionisation equilibrium to @mazzottaetal, who use the same data sources as we do without the corrections from . Cl is the element where the largest differences occur, for all important elements all differences are marginal at best. In addition to these physical tests we examined which spatial resolution is possible. We use an adaptive step size sampling regions with steep gradients sufficiently densely so that none of our physical variables changes by more than 5%. In order to keep the computation time at a reasonable level, we also enforced a minimum step size of 1 m.
Heat conduction {#heat_cond}
---------------
Thermal conduction tends to smooth out temperature gradients. It is not included in our simulation and we use the models’ temperature gradients to estimate its importance. According to @spitzer [Chapter 5.5] the thermal heat flux $F_{\mathrm{cond}}$ is $$\label{res_thermal_cond}
F_{\mathrm{cond}}=\kappa_0 T^{5/2} \frac{dT}{dz} \ .$$ Here $\kappa_0 =2\times 10^{-5}{\Lambda}^{-1}\;\textnormal{erg K}^{7/2} \mathrm{s}^{-1} \mathrm{cm}^{-1}$ is the coefficient of thermal conductivity. Comparing the thermal heat flux according to this equation to the energy flux carried by the bulk motion, it never exceeds more than a few percent of the bulk motion energy transport in the main part of the cooling zone except for the lowest density cases. Small scale chaotic magnetic fields in the plasma are possible; they would be frozen in and expected to further suppress thermal conduction.
Optical depth effects {#opticaldepth}
---------------------
Our simulation assumes all lines to be optical thin. The continuum opacity in the soft X-ray region is small, however, we need to check line opacities. The optical depth $\tau(\lambda)$ for a given line can be expressed as $$\tau(\lambda)=\int \kappa(\lambda) dl \ ,$$ where $l$ measures distance along the photon path and $\kappa(\lambda)$ is the local absorption coefficient, which can be computed from the oscillator strength $f$ of the line in question, the number density $n_{\mathrm{low}}$ of ions in the lower state and the line profile function $\Phi(\lambda)$: $$\kappa(\lambda,z)= \frac{\pi e^2}{m_{\mathrm{e}} c}f n_{\mathrm{low}}(z) \Phi(\lambda,z) \ ,$$ with $e$ being the electron charge and $c$ the speed of light. We approximate the line profile to follow a Gaussian distribution law with the normalisation $\int_0^{\infty} \Phi(\lambda,z) d\lambda=1$ at all $z$, centred at $\lambda_0$ with the width $\Delta \lambda_b(z)$: $$\Phi(\lambda,z)=\frac{1}{\sqrt{\pi}}\frac{\lambda_0(z)^2}{c\Delta \lambda_b(z)}\exp\left(-\left(\frac{\lambda-\lambda_0(z)}{\Delta \lambda_b(z)}\right)^2\right) \ ;$$ $\lambda_0(z)$ is the wavelength at line centre. Because the shocked gas is moving into the star, but decelerating, it is Doppler-shifted with the bulk velocity $v(z)$ at depth $z$ according to $\lambda_0(z)=\lambda_{\mathrm{rest}}\left(1+v(z)/c\right)$, where $\lambda_{\mathrm{rest}}$ is the rest wavelength. The broadening $\Delta \lambda_b(z)$ is in the case of purely thermal broadening $$\Delta \lambda_b(z)=\frac{\lambda_0(z)}{c}\sqrt{\frac{2kT(z)}{m_{\mathrm{ion}}}} \ ,$$ but turbulent broadening $\Delta \lambda_t(z)$ may additionally contribute. This cannot be calculated from the 1D-hydrodynamics in our approach, it can only be included in an [*ad-hoc*]{} fashion. On the one hand, at the boundaries of the accretion shock region typical turbulent flows might reach velocities comparable to the bulk motion and thus significantly broaden the observed lines, on the other hand, the magnetic fields presumed to be present should tend to suppress flows perpendicular to the field lines. For the calculation of the optical depth we chose $\Delta \lambda_t(z)=10$ km s$^{-1}$. If the turbulent broadening is larger, the line profiles get wider and the optical depth decreases. All lines considered in this study are excited from the ground state. Since in collisionally-dominated plasmas almost all excited ion states decay relatively fast, we assume that all ions are in their ground state. $n_{low}$ is then the product of the ionisation fraction for the line producing ion, the abundance of that element and the total ion number density. It is important to consider that the line centre depends on depth because of the Doppler shift due to the bulk velocity. Photons emitted at line centre in deeper regions end up in the wings of the profile in higher layers. Since our simulation does not include radiative transfer all lines have to be checked for optical depth effects. The exact geometry and position of the accretion shock is still unkown, but of substantial importance for estimates the line optical depth. The radiation could either escape through the boundaries of the post-shock funnel perpendicular to the direction of flow or through the shock and the pre-shock region. If the spatial extent of any single accretion funnel is small and it is located high up in the stellar atmosphere, the optical depth in the first scenario is small and the accretion contribution to the total stellar emission is large. We select this scenario in the further discussion. If, however, the shock is buried deep in the stellar atmosphere, the radiation can only escape through the shock and the thin pre-shock gas. In this case, denpending in the infall conditions, the optical depth of resonance lines can be large compared to unity.
Limitations by 1D
-----------------
Since our model is 1D, all emitted photons can travel only up or down, they cannot leave the accretion region sideways. To a first approximation half of the photons is emitted in either direction. The downward emitted photons will eventually be absorbed by the surrounding stellar atmosphere which will be heated by this radiation. The influence of the surrounding atmosphere on the shock region is expected to be small, since the temperature and hence the energy flux from the surrounding atmosphere into the shock region is much lower. We expect the shock structure to be well represented in 1D, but the size of the hot spots could be underpredicted because, depending on the geometrical extend of each spot, less than half of the emission escapes.
Boundary conditions and limitations of the model
------------------------------------------------
All our simulations start with a pre-shock temperature of 20000 K and and the corresponding (stationary) equilibrium ionisation state [@ar85]. This choice of temperature is motivated by studies of the photoionisation in the accretion stream by the post-shock emission [@calvetgullbring] and analytical considerations about the electron heat flux [@raizerzeldovich]. While we ignore heat conduction in the post-shock region, across the shock front the temperature gradient is of course large. The electrons have then mean free path lengths much larger than the shock front extent will therefore heat up the inflowing gas. We tested the influence of different initial conditions and found that the post-shock zone depends only marginally on the chosen initial ionisation state. We terminate our simulations when the temperature drops below 12000 K. Here the opacity begins to play an important role and the energy flux from the central core of the star is no longer negligible compared to the accretion flux. Here and just behind the shock front the accuracy is affected because the step size reaches its lower limit and rapid ionisation or recombination processes are not resolved. So the model is expected to be more accurate for ions which exist at high temperatures e.g. in the formation region of or , which are precisely those observed at X-ray wavelengths.
Model grids
-----------
The free fall velocity onto a star with mass $M_*$ and radius $R_*$ is $$v_{\textnormal{free}}=\sqrt{\frac{2GM_*}{R_*}} = 617 \sqrt{\frac{M_*}{M_\odot}}\sqrt{\frac{R_\odot}{R_*}} \frac{\textnormal{km}}{\textnormal{s}}\ . \label{infallvel}$$ Typically CTTS have masses comparable to the Sun and radii between $R_*=1.5\;R_{\sun}$ and $R_*=4\;R_{\sun}$ [@2003ApJ...597L.149M], because they have not yet finished their main sequence contraction. Most CTTS have inner disk radii of 10-90 solar radii [@2003ApJ...597L.149M], for the specific case of TW Hya @2006ApJ...637L.133E find an inner disk radius of $\approx12R_{\sun}$, thus the actual infall speed can almost reach the free-fall speed. Previous analyses indicate particle number densities of the infalling gas of about $10^{12}$ cm$^{-3}$. We therefore calculated a grid of models with infall velocities $v_0$ varying between $200$ km s$^{-1}$ and $600$ km s$^{-1}$ in steps of $25$ km s$^{-1}$, and infall densities $n_0$ varying between $10^{10}$ cm$^{-3}$ and $10^{14}$ cm$^{-3}$ with 13 points equally spaced on a logarithmic density scale. For each model in the grid we then calculate the emissivity for selected lines in the X-ray region using the version 5.1 of CHIANTI (@CHIANTI, @CHIANTIVII). We start out with abundances from @1998SSRv...85..161G and iterate the model fits until convergence (see Sect. \[res\_tw\_abund\]).
Results {#res}
=======
Structure of the post-shock region
----------------------------------
Fig. \[res\_gen\_tn\] shows typical model temperature and density profiles. In the (infinitesimally thin) shock, defined at depth 0 km, the ion temperature suddenly rises and cools down directly behind the shock front because the ion gas transfers energy to the electrons. After a few kilometers both ions and electrons have almost identical temperatures and henceforth there is essentially only radiative heat loss. This region we refer to as the post-shock cooling zone, where most of the X-ray emission originates (see Fig. \[funnel\_names\]).
During radiative cooling the density rises and because of momentum conservation the gas slows down at the same time (Eq. \[j\_p\]). As more and more energy is lost from the system, the density and the energy loss rate increase and the plasma cools down very rapidly in the end. In the example shown in Fig. \[res\_gen\_tn\], the shock reaches a depth of about 950 km, which is much smaller than a stellar radius, thus justifying our simplifying assumption of a planar geometry [*a posteriori\]*]{}.\
The region where the electron and ion temperature substantially differ from each other is much smaller than this maximum depth, thus a two-fluid treatment is not strictly necessary for most parts of the shock. In Fig. \[res\_gen\_t\_ausgl\] both temperatures are plotted in comparison. At the shock front the electrons stay relatively cool because they are only compressed adiabatically. Behind the shock front the energy flows from the ions to the electrons, and already at a depth of 5 km ions and electrons have almost identical temperatures.
In Fig. \[neemission\] we show the depth dependence of the ionisation state of neon, which produces strong lines observed by [*Chandra*]{} and XMM-Newton. On passing through the shock front neon is nearly instantaneously ionised up to , in the following $\approx 50$ km the mean ionisation rises until the equilibrium is reached. The plasma then contains a few percent nuclei, about 40% of the is in the form of , the rest comes as . Further away from the shock the ions recombine because of the general cooling of the plasma, following the local equilibrium closely, so the fraction of rises. At the maximum depth the ions quickly recombine to lower ionisation states.\
The maximal temperatures are proportional to $v_0^2$ as can be seen analytically from the equation of state for a perfect gas: $$T_1\sim \frac{P_1}{n_1} \sim \frac{n_0 v_0^2}{n_1} \sim v_1 u_0 \sim v_0^2 \label{res_t_v}$$ These estimates are obtained using Eq. \[RH1\], \[RH2\], \[RH3\] and neglecting the initial pressure.\
A higher infall velocity leads to a deeper post-shock cooling zone since the material reaches higher temperatures and consequently needs longer to cool down, so it flows for longer times and penetrates into deeper regions. Secondly, lower infall densities result in shocks with a larger spatial extent. Since the energy losses roughly scale with the square of the density, a lower density will increase the cooling time of the gas to cool down to photospheric temperatures. Fig. \[res\_maxdepth\] shows that cooling zone lengths between 1 km and 10000 km can be reached depending on the chosen model parameters.
Our model assumes a free plasma flow during cooling without any direct influence from the surrounding stellar atmosphere. As long as the infalling gas has a sufficiently large ram pressure, the model assumption should be applicable, since the surrounding atmosphere is pushed away and mixes with the accreted material only after cooling. For thin gases the ram pressure is lower and the gas needs more time for cooling down. Therefore in the deeper and denser layers the approximation of a freely flowing gas is no longer valid and the whole set of model assumptions breaks down. The depth, where this happens, depends on the stellar parameters. The shock front forms where its ram pressure $$\label{p_ram} p_\mathrm{ram}=\rho v_0^2 \ .$$ roughly equals the gas pressure of the stellar atmosphere. The pressure of the stellar atmosphere rises exponentially with depth, so, independent of the starting point, only shocks with small cooling length can be described by the hydrodynamic modelling used here. We place a cut-off at $z=1000$ km, where the pressure of the surrounding atmosphere will be larger by about an order of magnitude already.
Optical depth
-------------
We find that in all reasonable cases the total column density over the whole simulated region is small, in the best-fit solution if turns out to $N_H=10^{21}$ cm$^{-2}$, so the continuum opacity is small as assumed. Because emission originates at all depths the mean absorption column is even less. The line opacity along the direction of flow reaches values considerably above unity, so if the radiation passes through the whole post-shock region and does not escape through the boundaries the emssion in the resonance lines is considerably reduced (e.g. in Fig. \[netriplet\]). However, as we show in the following, the observed line ratios can be consistently explained in the accretion shock model, so the assumption of a geometry allowing most photons to escape, seems to be realistic.
Application to TW Hya {#res_twhya}
---------------------
In order to model the actual X-ray data available for TW Hya we applied a two-step process: First we used only the line fluxes of selected strong lines detected in the X-ray spectra and determined the best-fit shock model in an attempt to assess the maximally possible shock contribution. Then, in a second step, we performed a global, simultaneous fit to all available - lower resolution - data allowing for possible additional coronal contributions.
### Fit to line fluxes {#res_tw_nv}
Ratios between emission lines of the same element allow a determination of the best model parameters independent of the elemental abundances. Specifically, the XMM-Newton data contain the helium-like triplets of N,O and Ne, which strongly emit at plasma temperatures of a few MK (see Table \[res\_tab\_chi\_lines\]). For these three elements the corresponding Ly$\alpha$ lines are also measured, while we do not use any of the Ly$\beta$ lines because they provide relatively little additional temperature sensitivity and are substantially weaker than the Ly$\alpha$ lines. For each model we therefore compute three line ratios for each of the elements N, O and Ne, i.e., the so-called R- and G-ratios defined from the He-like triplets through $R = f/i$ and $G = (f+i)/r$ respectively as well as the ratio of the Ly$\alpha$ to the He-like r-lines. These nine line ratios are compared to the data via the $\chi^2-$ statistics; the resulting contour plot of $\chi^2$ as a function of the model parameters $n_0$ and $v_0$ is shown in Fig. \[res\_fit\_twhya\]. Correcting for the absorption does not alter the results since $N_H$ is small [$N_H=3.5\cdot10^{20}\textnormal{ cm}^{-2}$; @Rob0507]. The best model is found for the parameters $v_0=525$ km s$^{-1}$ and $n_0=10^{12}$ cm$^{-3}$ with an unreduced $\chi^2=31.9$ (7 degrees of freedom). One has to be careful here in interpreting the absolute value of the $\chi^2$ because it is derived from very few highly significant numbers with non-Gaussian errors. The strong neon lines confine the fit most effectively because their values have the smallest statistical uncertainties. The density is mainly restricted by R-ratios, the velocity by the Ly$\alpha$/r-ratios, which are temperature-sensitive. The Ne G-ratio deviates from the best fit parameters significantly (see Fig. \[negratio\]).
Its observed value is $1.1\pm0.13$, which points to infall velocities between $300$ km s$^{-1}$ and $400$ km s$^{-1}$; fitting only the remaining eight ratios we obtain the same best fit model as before. The fit does not depend on the chosen background radiation temperature in the range 6000 K to 10000 K.
Line ratio XMM/RGS Chandra/HETGS $v_0=525$ km s$^{-1}$ $v_0=575$ km s$^{-1}$
------------------------ ------------------- ------------------- ----------------------- -----------------------
N R-ratio $ 0.33 \pm 0.24 $ n\. a. 0.00 0.00
N G-ratio $ 0.88 \pm 0.31 $ n\. a. 0.77 0.75
N Ly$\alpha$/ r $ 4.04 \pm 1.00 $ n\. a. 2.40 2.82
O R-ratio $ 0.06 \pm 0.05 $ $ 0.04 \pm 0.06$ 0.02 0.02
O G-ratio $ 0.51 \pm 0.07 $ $ 0.82 \pm 0.22$ 0.73 0.71
O Ly$\alpha$/ r $ 2.01 \pm 0.18 $ $2.19 \pm 0.43 $ 1.49 1.97
Ne R-ratio $ 0.50 \pm 0.07 $ $0.54 \pm 0.08 $ 0.32 0.31
Ne G-ratio $ 1.10 \pm 0.10 $ $0.94 \pm 0.09 $ 0.80 0.75
Ne Ly$\alpha$/ r $ 0.27 \pm 0.04 $ $0.62 \pm 0.06 $ 0.26 0.49
(17.09Å+17.05Å)/16.78Å n\. a. $3.32 \pm 0.88 $ 2.25 2.22
17.09Å/17.05Å n\. a. $0.58 \pm 0.16 $ 0.81 0.79
red. $\chi^2$ $4.6$ $1.4$
The available *Chandra* data include a somewhat different wavelength interval. Only the triplets from O and Ne are covered, but the *Chandra* MEG has better spectral resolution, allowing to reliably measure several iron lines. We use the three lines of at 16.78 Å, 17.05 Å and 17.09 Å, and calculate the ratio of the two doublet members and the total doublet to the third line (see Table \[res\_tab\_chi\_lines\]). The best fit model for this data set has $v_0=575$ km s$^{-1}$, slightly higher than found for the XMM-Newton data, and $n_0=10^{12}$ cm$^{-3}$, with an unreduced value $\chi^2$ of 8.3 for 6 degrees of freedom. Again there is no difference between 6000 K and 10000 K taken as temperature for a black-body radiation background. The neon G-ratio points again to lower infall velocities. Both fits have a tight lower bound on the density $n_0=10^{12}$ cm$^{-3}$ of the infalling gas. We estimate the error as one grid point, i.e., $\pm 25$ km s$^{-1}$ for $v_0$ and $\pm0.33$ for $\log(n_0)$. The free-fall velocity for TW Hya is between 500 km s$^{-1}$ to 550 km s$^{-1}$, setting a tight upper bound and we adopt $v_0=525$ km s$^{-1}$ as the best value. We thus conclude that the same shock model is capable of explaining the line ratios observed in both the XMM-Newton RGS and [*Chandra*]{} HETGS grating spectra of TW Hya.
### Global fit {#globalfit}
In our second step we implement the shock emission as XSPEC table model and proceed with a normal XSPEC analysis of the medium and high resolution XMM-Newton spectra. The analysis of both the [*Chandra*]{} and XMM-Newton data [@2002ApJ...567..434K; @twhya] suggested the presence of higher temperature plasma possibly from an active corona; the flare observed in the *Chandra* observation may be a signature of activity.
To account for these additional contributions we add up to three thermal VAPEC models and include interstellar extinction. These additional thermal components are calculated in the low density limit and are meant to represent a coronal contribution. The elemental abundances for all components are coupled. It would be interesting to check if the accreting plasma is grain depleted compared to the corona, but the data quality is not sufficient to leave more abundances as free fit parameters. Because of the parameter degeneracy between the emission measure of the cool component and the interstellar absorption column $N_H$, we kept the latter fixed at $N_H=3.5\times10^{20}$ cm$^{-2}$, a value suggested by @Rob0507 [ where a detailed discussion is given]. The fit uses the data from one of the *XMM* EPIC MOS detectors, RGS1 and RGS2 and *Chandra’s* HEG and MEG simultaneously in the energy range from 0.2 keV to 10 keV. The normalisation between the instruments is left independent to allow for calibration uncertainties and possible brightness variations between the observations. Because of the much larger count rates the lower resolution MOS detectors tend to dominate the $\chi^2$ statistic. To balance this we include all available grating information, but only one (MOS1) low resolution spectrum in the global fit.
The fit results for our different models are presented in Table \[chi\]. Model A represents the best fitting pure accretion shock, while models B-C include additional temperature components; the main improvement is obviously brought about by the introduction of a high temperature ($kT \approx 1.30$ keV) component, representing the emission from a hot corona. “Normal” coronae usually have emission measure distributions that can be described by a two-temperature model [@2003MNRAS.345..714B] and this motivates us to add cool low-density plasma component in model C. Although the reduced $\chi^2$ is only marginally smaller, we regard this as a better model, because it can be naturally interpreted in terms of a stellar corona. A third low-density component does clearly not improve the fit any further (model D). In Fig. \[epic\] we show the recorded EPIC MOS1 low resolution spectrum, our best-fit model and separately the accretion and coronal contributions. An inspection of Fig. \[epic\] shows that a energies below $\approx$ 1.2 keV the overall emission is dominated by the shock emission, while at higher energies the coronal contribution dominates because of the thermal cut-off of the shock emission. A distinction between high and low density plasma is possible only by examining the line ratios in the He-like triples or in iron lines. In broad band spectra cool coronal and shock plasma exhibit the same signatures. The ratio of their emission measures was therefore taken from the RGS modelling alone (model C and D). Our global fit reproduces the triplet ratios quite satisfactorily as shown in Fig. \[netriplet\] for neon and in Fig. \[otriplet\] for the oxygen triplet.
The resulting infall velocity for model C is $530^{+8}_{-4}$ km s$^{-1}$ and the density $n_0=(1\times10^{12}\pm 3\times10^{11})$ cm$^{-3}$ (errors are statistical only). In this scenario the total flux is dominated by the accretion shock ($3.7\times10^{-12}$ ergs cm$^{-2}$ s$^{-1}$) which is about four times stronger than the cool corona ($1.0\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$) and five times stronger than the hot corona ($0.8\times10^{-12}$ erg cm$^{-2}$ s$^{-1}$) in the 0.3-2.5 keV band.
Model $kT_1$ $kT_2$ $kT_3$ $N_H$ red. $\chi^2$ (dof)
------- -------- -------- -------- ------- ---------------------
A - - - =3.5 2.8 (584)
B - - 1.33 =3.5 1.63 (580)
C 0.27 - 1.35 =3.5 1.57 (577)
D 0.27 0.72 1.26 =3.5 1.56 (573)
: \[chi\]Reduced $\chi^2$ values for models with zero to three VAPEC components with temperature $kT_1$ to $kT_3$ in \[keV\] and one shock component. The absorption column is given in units of $[10^{20}\frac{1}{\mathrm{cm}^2}]$. The shock model converges to $v_0 \approx 530$ km s$^{-1}$ and $n_0 \approx 10^{12} \mathrm{ cm}^{-3}$ in all cases except A ($v_0$ pegs at the free-fall velocity of $\approx 575$ km s$^{-1}$).
### Elemental abundances {#res_tw_abund}
The abundance fitting has to be performed recursively until the abundances converges, because different abundances lead to different cooling functions and thus change the whole shock structure. Specifically, we start from the set of elemental abundances determined by @Rob0507 and iterate. Our global fit procedure yields abundance values (Table \[abund\]) relative to solar abundances from [@1998SSRv...85..161G], the errors given are purely statistical ($1\sigma$ range), while we believe the systematic error to be about 15%. As a cross-check we compare the intensities of lines from different elements in our pure shock model from Sect. \[res\_tw\_nv\] and find that the abundance ratios estimated in this way roughly agree. The final abundance estimates (Table \[abund\]) show a metal depleted plasma with the exception of neon, which is enhanced by about a factor of ten compared to the other elements and nitrogen, which is enhanced by a factor of two.
Element abundance FIP \[eV\]
--------- ------------------------ ------------
C $0.20^{+0.03}_{-0.03}$ 11.3
N $0.51^{+0.05}_{-0.04}$ 14.6
O $0.25^{+0.01}_{-0.01}$ 13.6
Ne $2.46^{+0.06}_{-0.04}$ 21.6
Mg $0.37^{+0.10}_{-0.06}$ 7.6
Si $0.17^{+0.07}_{-0.07}$ 8.1
S $0.02^a$ 10.4
Fe $0.19^{+0.01}_{-0.01}$ 7.9
: \[abund\]Abundance of elements relative to and first ionisation potentials (FIP). The errors are statistical only, we estimate the systematic error to 15%.
$^a$ formal $2\sigma$ limit
Metal depletion has also been observed in the wind of TW Hya by @2004AstL...30..413L and was noted by @twa5 using X-ray observations of the non-accreting quadruple system TWA 5 in the vicinity of TW Hya. @twhya interpret the abundances as a sign of grain depletion, where the grain forming elements condensate and mainly those elements are accreted, which stay in the gas phase like the noble gas neon. This is discussed in more detail by @2005ApJ...627L.149D, who collect evidence that metal depletion can be also seen in the the infrared and UV, where the spectral distribution indicates well advanced coagulation into larger orbiting bodies, which may resist the inward motion of the accreted gas. On the other hand, stars with active corona often show an enhancement of elements with a high first ionisation potential (IFIP), which also leads to an enhanced neon abundance [@Brinkman_HR_1099].
### Filling factor and mass accretion rate {#res_tw_fm}
A comparison of the observed energy flux $f_{\mathrm{obs}}$ (at the distance $d$ to the star) and the simulated flux $f_{\mathrm{sim}}$ per unit area allows to calculate the accretion spot size $A_{\mathrm{spot}}$ through $$\label{res_aspot}
A_{\mathrm{spot}}=\frac{f_{\mathrm{obs}}(d)}{f_{\mathrm{sim}}(R_*)}4\pi d^2 \ .$$ The filling factor $f$ is the fraction of the stellar surface covered by the spot: $$\label{res_f}
f=\frac{A_{\mathrm{spot}}}{4\pi R_*^2}=\frac{f_{\mathrm{obs}}(d)}{f_{\mathrm{sim}}(R_*)}\frac{d^2}{R_*^2}$$ and the mass accretion rate is the product of the spot size and the mass flux per unit area, which in turn is the product of the gas density $\rho_0=\mu m_{\mathrm{H}} n_0$ and the infall velocity $v_0$: $$\label{res_mdot}
\frac{dM}{dt}=A_{\mathrm{spot}}\rho v_0=A_{\mathrm{spot}} \mu m_{\mathrm{H}} n_0 v_0 \ .$$ We assume that half of the emission is directed outward and can be observed, the other half is directed inwards, where it is absorbed. For model C the spot size is $1\times 10^{20}$ cm$^2$, yielding filling factors of 0.15% and 0.3% for $R_*=1\ R_{\sun}$ and $R_*=0.8\ R_{\sun}$ respectively and accretion rates of about $2\times10^{-10}\ M_{\sun}\; \mathrm{ yr}^{-1}$.
Shock position in the stellar atmosphere
----------------------------------------
We now compare the ram pressure of the infalling gas to the stellar atmospheric pressures as calculated from `PHOENIX`; we specifically use a density profile from AMES-cond-v2.6 with effective temperature $T_\mathrm{eff}=4000$ K, surface gravity $\log g=4.0$ and solar metalicity (@2005tdug.conf..565B based on @2001ApJ...556..357A). The chosen stellar parameters resemble those of typical CTTS. The shock front is expected to form, where the ram pressure approximately equals the stellar atmospheric pressure, which increases exponentially inwards. In a strict 1D-geometry photons emitted upwards out of the cooling zone will be absorbed by an infinite accretion column, but in a more realistic geometry they can pass either through the stellar atmosphere or the pre-shock gas as can be seen in Fig. \[funnel\_names\]. We estimate a lower limit for the hydrogen column density of $N_H=10^{20}$ cm$^{-2}$, the actually measured column density is $3.5\times 10^{20}$ cm$^{-2}$ [@Rob0507] by adding the column density between shock front and emitting ion to the the column density of the pre-shock gas, which the photons penetrate before escaping from the accretion funnel outside the stellar atmosphere. The optical depth of the stellar atmosphere is far higher than our lower limit for the pre-shock gas. This estimate proves that shocks, as described by our model, are actually visible; for a contrary view on this subject matter see @2005CSSS.519.D.
Discussion {#dis}
==========
The best fit parameters of our shock model to match the X-ray observations of TW Hya are obtained by using a shock with the parameters $v_0\approx525$ km s$^{-1}$ and $\log n_0\approx12$. Previously infall velocities closer to $\sim300$ km s$^{-1}$ were reported by a number of authors [@lamzin; @calvetgullbring]. Other observational evidence also suggests lower values; in the UV emission is found in highly ionised emission lines (, and ) extending up to $\approx 400$ km s$^{-1}$ and in cool ions in absorption () against a hot continuum likely emerging from an accretion spot [@2004AstL...30..413L]. Since the gas strongly accelerates close to the stellar surface, the density will be lower in the high velocity region because of particle number conservation, so, depending on the geometry of the accretion funnel, the emission measure in this region may well be very small. In this case the observed lines will have weak wings extending to larger velocities, which are difficult to identify observationally. We therefore regard these observations only as a lower bound; an upper bound is provided by the free-fall velocity of $\sim500-550$ km s$^{-1}$.\
Measurements of TW Hya in different wavelength regions lead to conspicuously distinct mass accretion rates. Generally, the published estimates far exceed the results of our simulation, which gives an accretion rate of $\approx (2\pm0.5)\times10^{-10}\ M_{\sun}\; \mathrm{ yr}^{-1}$ and filling factors of 0.2%-0.4%. @2002ApJ...571..378A and @2002ApJ...580..343B use optical spectroscopy and photometry and state a mass accretion rate between $10^{-9}$ and $10^{-8}\ M_{\sun} \;\mathrm{ yr}^{-1}$ and a filling factor of a few percent. In the UV the picture is inconsistent. On the one hand, two empirical relations for line intensities as accretion tracers [@2000ApJ...539..815J] indicate mass accretion rates above $3\times 10^{-8}\ M_{\sun}\; \mathrm{ yr}^{-1}$ (data from @2000ApJS..129..399V, evaluated by @2002ApJ...567..434K), on the other hand fitting blackbodies on the UV-veiling by @2000ApJ...535L..47M suggests a significantly lower value: $4\times 10^{-10}\ M_{\sun}\; \mathrm{ yr}^{-1}$. A similar procedure has been earlier applied by with a much larger filling factor. The previous X-ray analyses by @2002ApJ...567..434K [ $\dot{M}=10^{-8}\ M_{\sun}\; \mathrm{ yr}^{-1}$] and @twhya [ $\dot{M}=10^{-11}\ M_{\sun}\; \mathrm{ yr}^{-1}$] suffer from the problem that they use filling factors extracted from UV-measurements (from and @2000ApJ...535L..47M respectively) and a post-shock density calculated from X-ray observations which does not necessarily represent the same region. Our simulation now is the first attempt to rely solely on the X-ray measurements. The different values for mass accretion rates are summarised in Fig. \[dis\_mdot\_fig\].\
A physical explanation for these differences goes as follows: At longer wavelengths one observes plasma at cooler temperatures and in general the filling factor and mass accretion rates are higher, because the spot is inhomogeneous with different infall velocities. Fast particles would be responsible for a shock region with high temperatures which is observed in X-rays, whereas in other spectral bands cooler areas can be detected, and therefore the total area and the observed total mass flux is larger. Accretion spots with these properties are predicted by the magneto-hydrodynamic simulations recently performed by @2004ApJ...610..920R and @2006MNRAS.371..999G. Very probably some of the difference can be attributed to intrinsically changing accretion rates. Simulations of the inner flow region often show highly unstable configurations [@2006AN....327...53V].
We showed that basic properties of the X-ray spectra from CTTS can be naturally explained by accretion on a hot spot, but this is not the only X-ray emission mechanism. To understand especially the high energy tail we had to introduce two thermal components which fit a corona as it is expected in late-type stars and suggested by observations of activity; in the case of TW Hya the shock can dominate the overall X-ray emission. Forthcoming high-resolution observations will hopefully allow to extend the sample of CTTS, where a similar analysis is feasible.
CHIANTI is a collaborative project involving the NRL (USA), RAL (UK), MSSL (UK), the Universities of Florence (Italy) and Cambridge (UK), and George Mason University (USA).\
H.M.G., J.R and C.L. acknowledge support from DLR under 50OR0105.
[^1]: Based on observations obtained with XMM-Newton, an ESA science mission with instruments and contributions directly funded by ESA Member States and NASA.
[^2]: http://www.pa.uky.edu/$\sim$verner/fortran.html
|
---
abstract: |
By recognising that tensors are fundamentally related to gravitation in spacetime it is argued that the classical electromagnetic properties of a simple polarisable medium may be parameterised in terms of a constitutive tensor whose properties can in principle be determined by experiments in non-inertial (accelerating) frames and in the presence of weak but variable gravitational fields. After establishing some geometric notation, discussion is given to basic concepts of stress, energy and momentum [*in the vacuum*]{} where the useful notion of a [*drive*]{} form is introduced in order to associate the conservation of currents involving the flux of energy, momentum and angular momentum with spacetime isometries. The definition of the tensor is discussed with particular reference to its symmetry based on its role as a source of relativistic gravitation. General constitutive properties of [*material continua*]{} are formulated in terms of spacetime tensors including those that describe [*magneto-electric phenomena*]{} in moving media. This leads to a formulation of a self-adjoint constitutive tensor describing, in general, inhomogeneous, anisotropic, magneto-electric bulk matter in arbitrary motion. The question of an invariant characterisation of intrinsically magneto-electric media is explored. An action principle is established to generate the phenomenological Maxwell system and the use of variational derivatives to calculate tensors is discussed in some detail. The relation of this result to tensors proposed by Abraham and others is discussed in the concluding section where the relevance of the whole approach to experiments on matter in non-inertial environments with variable gravitational and electromagnetic fields is stressed.
MSC codes: 83D05, 83C40, 83C35
Keywords: Stress-energy-momentum tensor, constitutive relations, variational, electromagnetic, polarisable, magneto-electric, gravitation,\
Maxwell’s equations.
author:
- |
T. Dereli[^1]\
[Department of Physics, Koç University]{}\
[34450 Sar[i]{}yer, İstanbul, Turkey]{}\
\
J. Gratus[^2]\
[Department of Physics, Lancaster University and the Cockcroft Institute]{}\
[Lancaster LA1 4YB, UK]{},\
\
R. W. Tucker[^3]\
[Department of Physics, Lancaster University and the Cockcroft Institute]{}\
[Lancaster LA1 4YB, UK]{}
title: New Perspectives On the Relevance of Gravitation for the Covariant Description of Electromagnetically Polarizable Media
---
Introduction {#ch_intro}
============
The laws of quantum-electrodynamics have been devised to describe the electromagnetic interactions with matter according to the tenets of relativistic quantum field theory. However Maxwell’s classical equations remain mandatory for the description of a vast amount of natural phenomena. This versatility is in part due to the supplementary constitutive relations that are necessary to accommodate the wide range of materials that respond to electromagnetic fields. Although in principle such relations can be derived from the underlying quantum description of matter, in many practical situations one must rely on experimental guidance to ascertain the classical response of materials to such fields.
Once the unifying power of a spacetime formulation of physical phenomena became apparent with Einstein’s relativistic world view, the natural mathematical tool for describing constitutive responses became the total tensor for all matter and fields. Early suggestions by Minkowski [@mink] and Abraham [@abr] for the structure of its component in simple media initiated a long debate involving both theoretical and experimental contributions that continues to the current time (see e.g. [@peierls1]), [@peierls], [@gordon], [@brevik], [@loudon], [@ob_hehl], [@feigel], [@bowyer], [@cambell]). Although it is widely recognised that this controversy is an argument about definitions [@feld] and that the relative merits of alternative definitions are undecidable without a complete (experimentally verifiable) covariant description of relativistic continuum mechanics for matter and fields, it remains important to clarify the many conflicting arguments that have appeared over the years and to offer new insights that may help in modelling the electromagnetic properties of moving media in the absence of a viable or complete description of field-particle interactions at a more fundamental level.
Some way towards this goal is offered by (covariant) averaging methods [@degrootbook], [@degroot]. These however yield non-symmetric tensors for fields in simple media. If the total is to remain symmetric this implies that other asymmetric contributions must compensate and no guidance is offered to account for such material induced asymmetries. The need for a symmetric total tensor is often attributed to conservation of total angular momentum despite the fact that such global conservation laws may not exist in arbitrary gravitational fields. Although the magnitude of gravitational interactions may be totally insignificant compared with the scale of those due to electromagnetism, gravity does have relevance in establishing the general framework (via the geometry of spacetime) for classical field theory and in particular this framework [@israel], [@mikura] offers the most cogent means to define the total tensor as the source of relativistic gravitation. This in turn may be related to a variational formulation [@israelstewart] of the fully coupled field system of equations that underpin the classical description of interacting matter in terms of tensor (and spinor) fields on spacetime.
In this article tensors are defined as variational derivatives and it is argued that the classical properties of a simple polarisable medium may be parameterised in terms of a constitutive tensor whose properties can in principle be determined by experiments in non-inertial (accelerating) frames and in the presence of weak but variable gravitational fields.
There has been a rapid development in recent years in the construction of traps" for confining collective states of matter on scales intermediate between macro- and micro-dimensions. Cold atoms and nano-structures offer many new avenues for technological development when coupled to probes by fields. The constitutive properties of such novel material will play an important role in this development. Space science is also progressing rapidly and can provide new laboratory environments with variable gravitation and controlled acceleration in which the properties of such states of matter may be explored. It will be shown below that the response of electromagnetically polarisable media to such novel experimental environments offers a means to describe their constitutive properties and hence gain insight into the stresses induced by fields in such media. Supplemented with additional data based on their mechanical and elasto-dynamic responses one thereby gains a more confident picture of the total phenomenological for media than that based on previous ad-hoc choices.
Throughout this article the formulation will be expressed in terms of tensor fields on spacetime with an arbitrary metric. Attention will be drawn to conservation laws when this metric admits particular symmetries. Thus the results have applicability to simple media in arbitrary gravitational fields and accommodate both media and observers with arbitrary velocities.
After establishing some geometric notation, section 2 relates the electromagnetic 1-forms $\Me,\Mb,\Md,\Mh$ to the 2-forms $F$ and $G$ that enter into Maxwell’s phenomenological covariant field equations in the presence of matter. Section 3 discusses stress, energy and momentum [*in the vacuum*]{} and introduces the useful notion of a [*drive*]{} form that can be used to calculate electromagnetically induced currents involving the flux of electromagnetic energy, momentum and angular momentum in Minkowski spacetime. In section 4 the definition of the tensor is discussed with particular reference to its symmetry based on its role as a source of relativistic gravitation. The constitutive properties of the media considered in this paper are delineated in section 5 in terms of a constitutive tensor on spacetime. This includes an account of general [*magneto-electric continua*]{} and leads in section 6 to a formulation of a self-adjoint constitutive tensor describing, in general, inhomogeneous, anisotropic, magneto-electric matter in arbitrary motion. The question of an invariant characterisation of magnto-electric media is mentioned in section 7. In section 8 an action principle is established to generate the phenomenological Maxwell system and the use of variational derivatives to calculate tensors is discussed in section 9. The computation of the electromagnetic tensor, based on the action of section 8, is non-trivial for general media exhibiting anisotropy and magneto-electric properties in arbitrary motion and is presented in some detail. The relation of this result to tensors proposed by Abraham and others is discussed in the concluding section where the relevance of the whole approach to experiments on matter in non-inertial environments with variable gravitational and electromagnetic fields is stressed.
Notations follow standard conventions with spacetime modelled as a 4-dimensional, orientable, manifold $\man$ with a metric tensor field $g$ of Lorentzian signature $(-,+,+,+)$. $\GamTM$ denotes the set of vector fields and $\GamLamM{p}$ the set of $p-$form fields on $\man$. The set $\Set{e^0,e^1,e^2,e^3}$ denotes a local $g$-orthonormal coframe (a linearly independent collection of $1-$ forms) with dual frame $\Set{X_0,X_1,X_2,X_3}$. If $g_{ab}=g(X_a,X_b)$, the interior contraction operator $i_{X_a}$ with respect to $X_a$ is written $i_a$ with $i^a=g^{ab}i_{X_b}$, $e_b= g_{ac} e^c$ and summation over $0,1,2,3$. Metric duals with respect to $g$ are written with a tilde so that $\dual{X}=g(X,-)\in\GamLamM{1}$ for $X\in\GamTM$ and $\dual{\alpha}=g^{-1}(\alpha,-)\in\GamTM$ for $\alpha\in\GamLamM{1}$. The Hodge dual map associated with $g$ is denoted $\star$. The following standard identities will be used repeatedly in subsequent sections to simplify expressions. $$\begin{aligned}
\Phi\wedge\Psi = (-1)^{pq} \Psi\wedge\Phi \qquad&\text{for}\quad
\Phi\in\GamLamM{p},\ \Psi\in\GamLamM{q} \label{id_wedge}\\
\Phi\wedge\star\Psi=\Psi\wedge\star\Phi \qquad&\text{for}\quad
\Phi,\Psi\in\GamLamM{p} \label{id_star_pivot}
\\
i_X\star \Phi=\star(\Phi\wedge\tilde X)
\qquad&\text{for}\quad
X\in\GamTM,\
\Phi\in\GamLamM{p}
\label{id_iX_star}
\\
\star\, i_X\Phi=-\star\Phi\wedge\tilde X \qquad&\text{for}\quad
X\in\GamTM,\ \Phi\in\GamLamM{p} \label{id_star_iX}
\\
\star\star\Phi=(-1)^{p+1}\Phi
\qquad&\text{for}\quad
\Phi\in\GamLamM{p}
\label{id_star_star}
\\
i_X \Phi\wedge\Psi =
(-1)^{p+1} \Phi\wedge i_X\Psi
\quad&\text{for}\quad
\Phi\in\GamLamM{p},\
\Psi\in\GamLamM{q},\
p+q\ge5
\label{id_iX_move}
\\
d \Phi\wedge\Psi = (-1)^{p+1} \Phi\wedge d\Psi
\qquad&\text{for}\quad \Phi\in\GamLamM{p},\ \Psi\in\GamLamM{q},\
p+q\ge4 \label{id_d_move}\end{aligned}$$
Electromagnetic Fields {#ch_fields}
======================
Maxwell’s equations for an electromagnetic field in an arbitrary medium can be written $$\begin{aligned}
d\,F=0 \qquadand d\,\star\, G =j
\label{intro_Maxwell}\end{aligned}$$ where $F\in\GamLamM{2}$ is the Maxwell 2-form, $G\in\GamLamM{2}$ is the excitation 2-form and $j\in\GamLamM{3}$ is the 3-form electric current source [^4]. In general, the effects of gravitation and electromagnetism on matter are encoded in this system in $\star G$ and $j$. This dependence may be non-linear and non-local. To close this system, “constitutive relations” relating $G$ and $j$ to $F$ are necessary. In the following the medium will be considered as containing polarisable (both electrically and magnetically) matter with $G$ restricted to a real point-wise linear function of $F$, thereby ignoring losses and spatial and temporal material dispersion in all frames. Continua endowed with such properties will be termed “simple” here. The electric 4-current $j$ will be assumed to describe (free) electric charge and plays no role in subsequent discussions.
The electric field $\Me\in\GamLamM{1}$ and magnetic induction field $\Mb\in\GamLamM{1}$ associated with $F$ are defined with respect to an arbitrary [*unit*]{} future-pointing timelike $4-$velocity vector field $U\in\GamTM$ by $$\begin{aligned}
\Me = \iuF \qquadand \cc\Mb = \iustarF \label{intro_e_b}\end{aligned}$$ Since $g(U,U)=-1$ $$F=\Me\wedge \dualu - \star\,(\cc\Mb\wedge \dualu) \label{intro_F}$$ The field $U$ may be used to describe an [*observer frame*]{} on spacetime and its integral curves model idealised observers.
Likewise the displacement field $\Md\in\GamLamM{1}$ and the magnetic field $\Mh\in\GamLamM{1}$ associated with $G$ are defined with respect to $U$ by $$\begin{aligned}
\Md = \iu G\,, \qquadand \Mh/\cc = \iu\star G\,. \label{Media_d_h}\end{aligned}$$ Thus $$\begin{aligned}
G=\Md\wedge \dualu - \star\,((\Mh/\cc)\wedge \dualu)
\label{Media_G}\end{aligned}$$ It will be assumed that a material medium has associated with it a future-pointing timelike unit vector field $V$ which may be identified with the bulk $4-$velocity field of the medium in spacetime. Integral curves of $V$ define the averaged world-lines of identifiable constituents of the medium. A [*comoving observer frame with $4-$velocity $U$*]{} will have $U=V$.
Electromagnetic Stress, Energy and Momentum in the Vacuum {#em_stress}
=========================================================
The historical development of Newtonian continuum mechanics led to the notion of a stress tensor in Euclidean $3-$space that entered into the balance laws for momentum and angular momentum. With the advent of relativistic concepts this was generalised to a stress-energy-momentum tensor in spacetime giving rise to conserved quantities in situations where the metric admits symmetries.
The basic properties of the electromagnetic stress-energy-momentum tensor in the vacuum[^5] can be succinctly discussed in terms of a set of drive$3-$forms. In vacuo the Maxwell field system with a $3-$form electric current source $j$ satisfies $$\begin{aligned}
d\,F=0 \qquadand \ee d\,\star\, F =j. \label{intro_Maxwell_vacuum}\end{aligned}$$ For any vector field $\VV$ on spacetime and any Maxwell solution $F$ to this system one can introduce a “drive” 3-form associated with $Y$ and $F$ $$\begin{aligned}
{\tau^{EM}}_\VV=\frac{\epsilon_0}{2c}(i_\VV F \wedge \star\, F -
i_\VV\star\, F \wedge F)\label{ddrive}\end{aligned}$$ This $3-$form can be used to generate different types of conserved quantities when the vector field $\VV$ generates [*(conformal) isometries*]{} on spacetime.[^6] If $\KK$ is any (conformal) Killing vector on a domain of spacetime it then follows simply from the vacuum Maxwell-system above that $$\begin{aligned}
d\,{\tau^{EM}}_\KK=-\frac{1}{\cc}\,i_\KK\,F\wedge j\end{aligned}$$ Thus for each (conformal) Killing vector field these equations describe a “local conservation equation” ( $d\tau_\KK=0$) in a source-free region $(j=0)$.
For $K$ any [*unit timelike*]{} Killing vector one has from (\[ddrive\]) $$\begin{aligned}
{\tau^{EM}}_K=\frac{1}{c^2} \Me \wedge \Mh \wedge \tilde K -
\frac{1}{2c} \{\ee g(\tilde\Me,\tilde\Me) +\mu_0\,
g(\tilde\Mh,\tilde\Mh)\} \star\,\dual{K}\end{aligned}$$ where $\Mh=\mu_0^{-1}\Mb$, $\Mb,\Me$ are defined with respect to $U=K$ and $\mu_0\equiv \frac{1}{c^2\ee}$. The spatial $2-$form $
\Me\wedge \Mh$ was identified by Poynting in a source-free region as proportional to the local field energy transmitted normally across unit area per second (field energy current or power) and $\frac{1}{2} \{\ee g(\tilde\Me,\tilde\Me) +
\mu_0\,g(\tilde\Mh,\tilde\Mh)\}$ proportional to the local field energy density. More precisely $\int_\Sigma \tau_K$ is the field energy associated with the spacelike 3-chain $\Sigma$ and $\int_{S^2}\,i_K\tau_K$ is the power flux across an oriented spacelike 2-chain $S^2$.
If $X$ is a [*unit spacelike*]{} Killing vector generating spacelike translations along open integral curves then with the split: $$\begin{aligned}
{\tau^{EM}}_X=\mu_X \wedge \tilde V + {{\cal G}_X}\end{aligned}$$ where $\iv{{\cal G}_X}=0$, the Maxwell stress 2-form $\mu_X$ may be used to identify mechanical forces produced by a flow of field momentum current or pressure with momentum density 3-form ${{\cal G}_X}$ [@liu].
It is important to stress that different timelike Killing vectors give rise to physically distinct notions of conserved energy. For completeness the interpretation of energy" requires further information related to its mode of detection. The existence of timelike [*parallel*]{} Killing vector fields (including those whose integral curves are geodesics) are further conditions that single out particular classes that may have priority in establishing appropriate notions of conserved energy.
In general, in the absence of Killing vectors one loses strictly conserved currents (closed $3-$forms) but a set of four local $3-$form currents ${\tau^{EM}}_c\equiv {\tau^{EM}}_{X_c}$ can be defined in any local coframe. In any frame $\{X_a\}$ with dual coframe $\{e^b\}$ the 16 functions $T^{EM}_{ab}=i_{X_b} \star\,
{\tau^{EM}}_{a}$ may be used to construct the tensor $$\begin{aligned}
T^{EM}=T^{EM}_{ab} \,e^a\otimes e^b\end{aligned}$$ usually referred as the [*stress-energy-momentum tensor*]{} associated with the above drive forms [^7] .
The relationships between any stress-energy tensor $T$ and the associated drive forms $\tau_a$ are given by $$\begin{aligned}
\tau_a = \star (T(X_a,-)) \qquadand T = \star(\tau_a\wedge e_b)
e^a\otimes e^b \label{intro_def_tau_a}\end{aligned}$$ In terms of the $3-$forms $\tau_c$ the symmetry condition $
T_{bc}=T_{cb}$ is $$e_c{\wedge}\tau_b=e_b {\wedge}\tau_c$$
The total tensor {#total_SEM}
================
When spacetime contains domains with matter (where $j$ may or may not be zero) such regions will in general have physical properties distinct from vacuum domains.
If a coupled system of electromagnetic, gravitational and matter fields has a [*total*]{} stress-energy-momentum tensor $$\begin{aligned}
T^{Total}=T^{EM}+ T^{m+I}\end{aligned}$$ where $ T^{m+I} $ describes matter and its interactions not included in $T^{EM}$, then on general grounds, if $T^{Total}$ is symmetric, one has: $${\nabla\cdot T^{Total}=0 }$$ in terms of a (Koszcul) connection $\nabla$ on spacetime. Different authors partition the total stress-energy tensor into a sum of partial stress-energy tensors in different ways. The divergences of certain partial stress-energy tensors are sometimes called [*pondermotive*]{} forces.
If the connection $\nabla$, induced from a connection on the bundle of linear frames over spacetime, is both metric compatible and torsion free, with gravitational fields satisfying Einstein’s equations, then $T^{Total}$ must give rise to a symmetric stress-energy tensor $T^{Total}_{ab}=T^{Total}_{ba}$. However any such symmetric tensor can be partitioned into non-symmetric partial tensors in infinitely many ways. Such a partition is then an expediency without fundamental significance.
In theories of gravitation based on non-pseudo-Riemannian geometries the [*natural connection*]{} may have torsion. For example in an Einstein-Cartan theory with matter that gives rise to a connection with torsion, the generalised Einstein tensor $Ein^{EC}$, determined by varying the generalised Einstein-Hilbert action with respect to orthonormal coframes, is non-symmetric and hence the source tensor $T^{EC}$ defined by $${ Ein^{EC}=T^{EC}\label{EC}}$$ is similarly non-symmetric. However for some forms of gravitational-matter couplings the variation of the total action with respect to the connection gives rise to algebraic equations for the connection.[^8] In principle these can be solved for the connection which can always be decomposed into a sum containing the torsion-free metric-compatible (Levi-Civita) connection used in Einstein’s pseudo-Riemannian description of gravitation. The generalised Einstein tensor $Ein^{EC}$ can then be written $Ein+S$ in terms of the Einstein tensor $Ein$ and (\[EC\]) becomes $${Ein=T^{E} \label{Ein}}$$ where $T^{E}\equiv T^{EC}-S$ is symmetric and divergenceless with respect to the Levi-Civita divergence. In such cases one may define the total stress-energy tensor as the source tensor for Einstein’s equations (\[Ein\]). It is then by definition symmetric. If the natural connection $\nabla$ (determined by a connection variation of the total action) gives rise to dynamic torsion, determined by a partial differential system involving all fields, the reduction to a geometrical formulation in terms of a metric and Levi-Civita connection becomes an impracticality. In such a situation the definition of the stress-energy tensor is best left as $T^{EC}$. This has two distinct divergences with respect to $\nabla$ since it is not symmetric.
Such general considerations offer guidance in the construction of phenomenological partial stress-energy tensors based on either coarse-graining detailed interactions between fields or the introduction of effective degrees of freedom [@nelson]. Indeed such phenomenological stress-energy tensors are often of greater value than actions based on “fundamental fields” since they can often be related more directly to experiment. Thus although in this article gravitation will be regarded as a background interaction the electromagnetic properties of a simple medium will be accommodated into certain constitutive tensors that respond to gravitation. We then demand that an action describing such a medium in the absence of free charges give rise by variation to Maxwell’s phenomenological equations for a simple medium [*and*]{} a symmetric stress-energy tensor.
The constitutive tensor for simple media {#ch_Media}
========================================
In general $G$ may be a functional of $F$ and properties of the medium[^9]. $$\begin{aligned}
G = {\cal Z}[F,\ldots] \label{Media_Nonline_Constitutive}\end{aligned}$$ Such a functional induces, in general, non-linear and non-local relations between $\Md, \Mh$ and $\Me, \Mb$. These relations may be explored either empirically or by coarse graining a suitable macroscopic model. For general [*linear continua*]{} one may have for some positive integer $N$ and collection of [*constitutive tensor fields*]{} $Z^{\,(r)}$ on spacetime the relation $$\begin{aligned}
G = \Sigma_{r=0}^{N}{ Z^{\,{(r)}}}(\nabla^{\,r} F,\ldots)
\label{Media_General_Constitutive}\end{aligned}$$ in terms of some spacetime connection $\nabla$. Additional arguments refer to variables independent of $F$ and its derivatives. In this article, for the simple linear media under consideration, we restrict to $$G=Z(F)$$ for some [ constitutive tensor field]{} $Z$ . In the vacuum $G=\epsilon_0 F$.
A particularly simple linear isotropic medium may be described by a bulk $4-$velocity field $V$, a relative permittivity scalar field $\epsilon$ and a non-vanishing relative permeability scalar field $\mu$. In this case $Z$ follows from $$\begin{aligned}
\frac{G}{\ee} = &\epsilon \,\ivF\wedge\dualv -
\mu^{-1}\star(\ivstarF\wedge\dualv)\\= & (\epsilon -
\frac{1}{\mu})\, i_V F \wedge\dualv - \frac{1}{\mu} \, F
\label{Media_scalar_medium_G}\end{aligned}$$ In a comoving frame with $U=V$ (\[Media\_scalar\_medium\_G\]) becomes $$\begin{aligned}
\Md = \ee\epsilon\, \Me \qquadand \Mh = (\mu_0\mu)^{-1} \Mb \,
\label{Media_scalar_medium_dh}\end{aligned}$$ For a non-magneto-electric but anisotropic medium, the relative permittivity $\epsilon$ and inverse relative permeability $\mu^{-1}$ become [*spatial tensor fields*]{} on spacetime. Thus $\epsilon:\GamLamM{1}\to\GamLamM{1}$ and $\mu^{-1}:\GamLamM{1}\to\GamLamM{1}$ for all $\alpha\in
\GamLamM{1}$ where $$\begin{aligned}
\epsilon(\dualv)=0\,,\quad i_V\epsilon(\alpha)=0\,,\quad
\mu^{-1}(\dualv)=0\quadand i_V\mu^{-1}(\alpha)=0\,.
\label{Media_eps_mu_V}\end{aligned}$$ The more general constitutive relation is then given by $$\begin{aligned}
\frac{G}{\ee} = \epsilon(\ivF)\wedge\dualv -
\star(\mu^{-1}(\ivstarF)\wedge\dualv)
\label{Media_tensor_medium_G}\end{aligned}$$ which in the comoving frame with $U=V$ becomes $$\begin{aligned}
\Md = \ee\epsilon(\Me) \qquadand \mu_0\Mh = \mu^{-1}(\Mb) \,.
\label{Media_tensor_medium_dh}\end{aligned}$$ Based on standard thermodynamic arguments the inverse relative permeability and relative permittivity tensors are symmetric with respect to the metric $g$: $$\begin{aligned}
i_{\dual{\alpha}}\epsilon(\beta)=i_{\dual{\beta}}\epsilon(\alpha)
\quadand
i_{\dual{\alpha}}\mu^{-1}(\beta)=i_{\dual{\beta}}\mu^{-1}(\alpha)
\quad\text{for}\quad \alpha,\beta \in \GamLamM{1}\end{aligned}$$ In general, the electromagnetic fields may be related by $$\begin{aligned}
\Md =& \PermDE(\Me) + \PermDB(\Mb)
\\
\Mh =& \PermHE(\Me) + \PermHB(\Mb)
\end{aligned}
\label{Media_Constitutive_dh}$$ where $\PermDE,\PermDB,\PermHE,\PermHB:\GamLamM{1}\to\GamLamM{1}$ are spatial tensors satisfying: $$\begin{aligned}
&\PermGen(\dualv)=
\iv(\PermGen(\alpha))=0
\label{Media_iv_Perm_0}\end{aligned}$$ and therefore $$\begin{aligned}
\PermGen(\piV(\alpha))=
\piV(\PermGen(\alpha))=\PermGen(\alpha)
\label{Media_piV_Perm_0}\end{aligned}$$ for $\PermGen=\PermDE,\PermDB,\PermHE,\PermHB$ and for all $\alpha\in \GamLamM{1}$, where $\piV$ projects spacetime 1-forms to spatial 1-forms with respect to $V$, on spacetime $$\begin{aligned}
\piV:\GamLamM{1}\to\GamLamM{1}\,,\qquad
\piV=\Id + \dualv\otimes V
\label{Media_def_pi_V}\end{aligned}$$ From (\[Media\_tensor\_medium\_dh\]), (\[Media\_Constitutive\_dh\]) it follows that if $\PermHE=\PermDB=0$ in some frame then $\PermDE=\ee\epsilon$ and $\PermHB=(\mu_0\mu)^{-1}$ in that frame. For such materials,however, one cannot assert that $\PermHE,
\PermDB$ remain zero in all frames. Media with constitutive relation (\[Media\_Constitutive\_dh\]) are often referred to as [*magneto-electric*]{} [@odell]. We prefer to use this term to describe intrinsic magneto-electric media and will return to this point in section \[ch\_Post\].
The tensor fields $\PermDE$, $\PermDB$, $\PermHE$ and $\PermHB$ are encoded into the tensor $\PermTen:\GamLamM{2}\to\GamLamM{2}$ such that $G=Z(F)$. Since $$\begin{aligned}
\PermTen(\alpha+\beta)=\PermTen(\alpha)+\PermTen(\beta) \qquadand
\PermTen(\lambda \alpha)=\lambda\PermTen(\alpha)
\label{Media_Z_linear}\end{aligned}$$ for all $\lambda\in\GamLamM{0}$ and $\alpha,\beta\in\GamLamM{2}$ the constitutive relation may be expanded in a local co-frame field $\Set{e^0,e^1,e^2,e^3}$ as $$\begin{aligned}
\tfrac12 G_{a b}e^a\wedge e^b = \tfrac14 \PermTen^{c d}{}_{a b}
F_{c d} e^a\wedge e^b\end{aligned}$$ where $$\begin{aligned}
\PermTen^{c d}{}_{a b} = - \PermTen^{c d}{}_{b a} = - \PermTen^{d
c}{}_{a b} = \PermTen^{d c}{}_{b a}
\label{Media_Z_abcd_Z_bacd}\end{aligned}$$ These conditions alone imply that the tensor $\PermTen$ has 36 independent components, although additional symmetry conditions given below will reduce these to 21. From the definition of $G$ in terms of comoving fields and (\[Media\_Constitutive\_dh\]), the relationship between $\PermTen$ and $\Set{\PermDE,\PermDB,\PermHE,\PermHB}$ follows as $$\begin{aligned}
\PermTen(F) =& \PermDE(\ivF)\wedge\dualv +
\PermDB(\ivstarF)\wedge\dualv
\\&
- \star(\PermHE(\ivF)\wedge\dualv) - \star(\PermHB(\iv \star
F)\wedge\dualv)
\end{aligned}
\label{Media_Z_decomp}$$ and hence by contraction with $V$ $$\begin{aligned}
& \PermDE(\xi) = \iv \PermTen(\xi\wedge\dualv) \,,\quad
\PermDB(\xi) = - \iv \PermTen(\star(\xi\wedge\dualv)) \,,\quad
\\&
\PermHE(\xi) = \iv\star \PermTen(\xi\wedge\dualv) \,,\quad
\PermHB(\xi) = - \iv\star \PermTen(\star(\xi\wedge\dualv))
\end{aligned}
\label{Media_def_Z_emme}$$
Symmetry of the constitutive tensor. {#Z_SYM}
====================================
The adjoint of any tensor $T:\GamLamM{p}\to\GamLamM{p}$, is the tensor $T^\dagger:\GamLamM{p}\to\GamLamM{p}$ defined by $$\begin{aligned}
\alpha\wedge\star T(\beta) = \beta\wedge\star T^\dagger(\alpha)
\qquad\text{for }\alpha,\beta\in\GamLamM{p}
\label{Media_def_adjoint}\end{aligned}$$ Clearly $T^{\dagger\dagger}=T$. If $p=1$, (\[Media\_def\_adjoint\]) gives $$\begin{aligned}
i_{\dual{\alpha}} T(\beta) = i_{\dual{\beta}} T^\dagger(\alpha)
\qquad\text{for }\alpha,\beta\in\GamLamM{1}
\label{Media_def_adjoint_1forms}\end{aligned}$$ The symmetry conditions for the relative permittivity and inverse permeability tensors imply that $\PermDE$ and $\PermHB$ are self adjoint. This symmetry is generalised to magneto-electric media: $$\begin{aligned}
\PermDE^\dagger=\PermDE \,,\quad \PermHB^\dagger=\PermHB \quadand
\PermDB^\dagger=-\PermHE \label{Media_Zeta_adjoint}\end{aligned}$$ i.e. $\PermTen$ is assumed self-adjoint $$\begin{aligned}
\PermTen=\PermTen^\dagger\, \label{Media_Z_adjoint}\end{aligned}$$ or, raising indices with the metric $$\begin{aligned}
Z^{abcd}=Z^{cdab}
\label{Media_Z_adjoint_abcd}\end{aligned}$$ Using sequentially (\[Media\_def\_Z\_emme\]), (\[id\_star\_iX\]), (\[Media\_Z\_adjoint\]), (\[id\_iX\_star\]), (\[id\_iX\_move\]), (\[Media\_def\_Z\_emme\]), (\[id\_wedge\]), (\[id\_star\_pivot\]), (\[Media\_def\_adjoint\]) this condition yields $$\begin{aligned}
\alpha\wedge\star\PermDB(\beta) =& -\alpha\wedge\star\iv
\PermTen(\star(\beta\wedge\dualv)) = \alpha\wedge\dualv\wedge\star
\PermTen(\star(\beta\wedge\dualv))
\\
=& {\star}(\beta\wedge\dualv)\wedge\star
\PermTen(\alpha\wedge\dualv) =
\iv{\star}\beta\wedge{\star}\PermTen(\alpha\wedge\dualv) =
\star\beta\wedge\iv{\star}\PermTen(\alpha\wedge\dualv)
\\
=& \star\beta\wedge\PermHE(\alpha) =
-\PermHE(\alpha)\wedge\star\beta =
-\beta\wedge\star\PermHE(\alpha)= -\alpha\wedge\star
\PermHE^{\dagger}(\beta)\end{aligned}$$ i.e. $\PermDB=-\PermHE^\dagger$. The remaining equations in (\[Media\_Zeta\_adjoint\]) follow similarly.
It follows from (\[Media\_Z\_abcd\_Z\_bacd\]) and (\[Media\_Z\_adjoint\_abcd\]) that the number of independent components of $Z$ reduce from 36 to 21.
Intrinsic magneto-electric media and the Post constraint. {#ch_Post}
=========================================================
A constitutive tensor $\PermTen$ describes a [*non intrinsic-magneto-electric medium*]{} if there exists a velocity field $V$ for the medium such that $\PermDB=0$ and $\PermHE=0$. Thus a constitutive tensor $\PermTen$ is intrinsically magneto-electric if there does not exists a velocity field $V$ such that $\PermDB=0$ and $\PermHE=0$. If $Z(F)$ is decomposed with respect to an arbitrary frame $U\ne V$ one may find all tensors $\zeta$ non-zero, even for media that are not intrinsically magneto-electric. For a general constitutive tensor, it is a matter of linear algebra to decide whether it describes an intrinsically magneto-electric medium or not.
A useful characterisation of magneto-electric media may be given in terms of invariants constructed from $Z$ and the metric. One such invariant introduced by Post [@post], [@ob_hehl_post], is $$\begin{aligned}
\chi = \ia\ib \star (\PermTen(e^a\wedge e^b))
\label{Media_Post_def}\end{aligned}$$ In terms of spatial tensors with respect the the medium velocity $V$ $$\begin{aligned}
\chi
=&
\ia\ib \star (\PermTen(e^a\wedge e^b))
\\
=&
\ia\ib{\star}(\PermDE(\iv (e^a\wedge e^b))\wedge\dualv)
+ \ia\ib \star(\PermDB(\iv{\star}(e^a\wedge e^b))\wedge\dualv)
\\&
+ \ia\ib (\PermHE(\iv(e^a\wedge e^b))\wedge\dualv)
+ \ia\ib (\PermHB(\iv{\star}(e^a\wedge e^b))\wedge\dualv)
\\
=& \ia\ib \star(\PermDB(\iv{\star}(e^a\wedge e^b))\wedge\dualv)
+ \ia\ib (\PermHE(\iv(e^a\wedge e^b))\wedge\dualv)
\end{aligned}
\label{Media_expand_chi}$$ since $$\begin{aligned}
\ia\ib{\star}(\PermDE(\iv (e^a\wedge e^b))\wedge\dualv) =
\ia\ib (\PermHB(\iv{\star}(e^a\wedge e^b))\wedge\dualv) = 0\end{aligned}$$ Using $$\begin{aligned}
\star(e^a\wedge e^b\wedge \dualv)
= \star(e^a\wedge e^b\wedge e^c)V_c
= V_c \star(e^a\wedge e^b\wedge e^c\wedge e^d) e_d
= - \varepsilon^{abcd} V_c e_d\end{aligned}$$ and $$\begin{aligned}
\star(\xi\wedge\dualv\wedge e_b\wedge e_a)
= \star(e_a \wedge e_b \wedge \dualv \wedge\xi)
= \star(e_a \wedge e_b \wedge e_e \wedge e_f) V^e \xi^f
= -\varepsilon_{abef} V^e\xi^f\end{aligned}$$ with $\varepsilon_{abef} \varepsilon^{abcd} =
\delta^d_e\delta^c_f-\delta^c_e\delta^d_f$, the first term on the last line of (\[Media\_expand\_chi\]) yields $$\begin{aligned}
\ia\is \star(\PermDB(\iv{\star}(e^a\wedge e^s))\wedge\dualv) =&
\star(\PermDB(\star(e^a\wedge e^s\wedge \dualv))
\wedge\dualv\wedge e_s\wedge e_a)
\\
=& -\star(\PermDB(e_r)\wedge\dualv\wedge e_s\wedge
e_a)\varepsilon^{ascr} V_c
\\
=& i^f\PermDB(e_r) \varepsilon_{asef} \varepsilon^{ascr}V^e V_c =
2 \ia\PermDB(e^a)\end{aligned}$$ while the second term is $$\begin{aligned}
\ia\ib (\PermHE(\iv(e^a\wedge e^b))\wedge\dualv)
&=
\ia\ib(\PermHE(v^a e^b)\wedge\dualv) -\ia\ib(\PermHE(v^be^a)\wedge\dualv)
\\
&=
2\iv\ib (\PermHE(e^b)\wedge\dualv)
=
-2\ib\PermHE(e^b)\end{aligned}$$ Hence using (\[Media\_Zeta\_adjoint\]) $$\begin{aligned}
\chi = 4\ia \PermDB(e^a) = -4\ia\PermHE(e^a)
\label{Media_Post_Zdb}\end{aligned}$$ Thus since $V$ is an arbitrary medium velocity, a sufficient condition for a medium to be intrinsically magneto-electric is that $\chi\ne 0$.
However some intrinsically magneto-electric media may have $\chi=0$. For example, consider the self-adjoint constitutive tensor given, in some local orthonormal coframe $\{e^0, e^1,e^2,e^3\}$, by $$\begin{aligned}
\PermTen(F) =
F_{23} e^0\wedge e^1
+ F_{13} e^0\wedge e^2 - F_{02} e^1\wedge e^3 - F_{01}
e^2\wedge e^3\end{aligned}$$ Then with $V=X_0$ $$\begin{aligned}
\PermDB(\xi) = (i_1 \xi) e^1 - (i_2 \xi) e^2
\qquadand
\PermHE(\xi) = -(i_1 \xi) e^1 + (i_2 \xi) e^2\end{aligned}$$ and $\chi=0$. However one easily verifies that $\PermDB\ne0$ with respect to any arbitrary unit timelike $V$. Hence $Z$ describes an intrinsically magneto-electric medium.
A minimal set of invariants whose non-vanishing is a necessary and sufficient condition for a medium to be intrinsically magneto-electric is not known to the authors.
Action for Source Free Electromagnetic Fields in a Simple Medium {#ch_Action_Max}
================================================================
The classical equations describing the total system of matter and fields will be considered as arising from the extremum of some [*total action functional*]{} under suitable variations with compact support. This action should be constructed from an action density 4-form on spacetime in terms of (pull-backs) of sections (and their derivatives) of field bundles carrying representations of local symmetry (gauge) groups and maps between them. Observed local symmetries in nature arise in such a formalism by ensuring that the action $4-$form is a scalar under local changes of section. To maintain these covariances appropriate connections are required to define tensorial (and spinorial) covariant derivatives of sections. In addition the action may depend on tensor-valued functions of these sections. All variational principles require a specification of what objects in the action are to be varied and these then constitute the dynamical variables of the theory. In the following we concentrate on a contribution $\Lambda$ to the total action arising from the effects of the electromagnetic field and gravitation in different types of media". We exclude from this $\Lambda$ the interaction with charged matter and the dynamics of the gravitational field itself. Included is the effect of the electromagnetic field on a polarisable and magnetisable medium assumed to be described in terms of a particular constitutive tensor $Z$. In particular we explore how the response of the medium to gravitation as well as the electromagnetic field can be used to establish the stress-energy-momentum tensor associated with different choices of constitutive tensor. Thus the action 4-form $\Lambda$ will be taken to depend only on the spacetime metric and the class of Maxwell 1-form potentials $A$ with $F=dA$. The dependence of the tensor field $Z$ on these variables will be explored in some detail below.
We have insisted that in the absence of free charge the electromagnetic fields $F$ and $G$ for a simple medium in any spacetime metric must satisfy $$d\,F=0 \qquadand d\star\, G=0
\label{Action_Max_Maxwells_equations}$$ Before generating an electromagnetic tensor from a particular contribution to the total action it is necessary to verify that these field equations arise by suitable variation. Consider then the contribution $S[A,g]=\int_M \Lambda$ where $F=d\,A$ ,$G=\PermTen(F)$ with $\PermTen=\PermTen^\dagger$ and $$\cc\Lambda = \tfrac{1}{2}\,F \wedge \star\, G = \tfrac12\,F \wedge
\star\, \PermTen(F) \, \label{Action_Max_action}$$ If a prime denotes the variation with respect to $A$, then working modulo $d$: $$\begin{aligned}
\cc\Lambda' =& \tfrac12 \Big( d A'\wedge\star\PermTen( d A) + d
A\wedge\star\PermTen( d A') \Big)
\\
=& d A'\wedge\star\PermTen( d A) =
A'\wedge d \star \PermTen( d A)
= A'\wedge d\star G\end{aligned}$$ Hence the source-free Maxwell equations (\[Action\_Max\_Maxwells\_equations\]) follow by variation with respect to $A$ from the action (\[Action\_Max\_action\]). Note that the symmetry condition (\[Media\_Z\_adjoint\]) of the tensor $\PermTen$ is essential in this variation.
Variational derivatives and tensors {#ch_Metric}
===================================
To effect the metric variations of the above action functional let $t\to\gT$ be a curve in the space of Lorentzian signatured metrics, with $\gO\equiv \gT|_{t=0}$. The “tangent” to the the curve $t\to\gT$ at the point $t=0$ is written $\gD$: $$\begin{aligned}
\gT = \gO + t \gD + O(t^2) \label{Metric_g_t_0_dot}\end{aligned}$$ For a general object $K$ which may be a tensor or a map which depends on the metric $g$, write similarly $t\to K_t$ as the one parameter set of objects encoding the dependence of $K$ on $\gT$, $K_0=K_t|_{t=0}$ and $\dot{K}=\tfrac{d}{dt}K_t|_{t=0}$, so $$\begin{aligned}
K_t=K_0 + t \dot{K} + O(t^2) \label{Metric_general_lift}\end{aligned}$$ $K_t$ will be referred to as the metric induced [*[lift]{}*]{} of $K$.
One may represent the local variation $\gT$ in different ways. One way is to vary the components of $\gT$ with respect to a fixed local co-frame $\{e^a_0\}$, i.e. $$\begin{aligned}
\gT=(\gT)_{ab} e_0^a\otimes e_0^b \qquad\text{where}\quad
(\gT)_{ab} = \gT\big((X_0)_a,(X_0)_b\big) \label{Metric_gt_ab}\end{aligned}$$ One can set the fixed frame to be orthonormal with respect to the unvaried metric so that $(\gO)_{ab}=\eta_{ab}=\text{diag}(-1,+1,+1,+1)$. The derivative $\gD$ is therefore given by $$\begin{aligned}
\gD=\gD_{ab} e_0^a\otimes e_0^b \label{Metric_gD_ab}\end{aligned}$$ Alternatively one may vary the co-frame: i.e. choose a one parameter set of coframes $t\to e_t^a$ for $a=0,..,3$ such that $e_t^a|_{t=0}=e_0^a$ and $$\begin{aligned}
\gT=\eta_{ab} e_t^a \otimes e_t^b \label{Metric_et_a}\end{aligned}$$ The derivative of $t\to e_t^a$ at $t=0$ follows from $$\begin{aligned}
e^a_t = e^a_0 + t\,\deltae^a + O(t^2) \label{Metric_eD_a}\end{aligned}$$ The derivative $\gD$ may also therefore be written $$\gD= \eta_{ab}\left( \dot e^a \otimes e^b_0 + e^a_0 \otimes \dot
e^b\right) \label{Metric_gD_eD}$$ The drive 3-forms $\tau_a$ associated with any action 4-form $\Lambda$ are defined by the variation of $\Lambda$ with respect to the orthonormal coframe as $$\begin{aligned}
\dot{\Lambda} = \deltae^a \wedge \tau_a \label{Lift_def_tau_a}\end{aligned}$$ If the variation of $\Lambda$ with respect to the ortho-normal coframe is induced entirely from the metric $g$ (and the metric compatible torsion-free Levi-Civita connection) then $$\begin{aligned}
\tau_a = 2 i_{X_b} \left(\frac{\delta\Lambda}{\delta
g_{bc}}\right) \eta_{ac} \label{Metric_g_var_tau_a}\end{aligned}$$
This follows immediately by equating (\[Metric\_gD\_ab\]) and (\[Metric\_gD\_eD\]): $$\begin{aligned}
\gD_{ab} e_0^a\otimes e_0^b= \eta_{ab}\left( \dot e^a \otimes
e^b_0 + e^a_0 \otimes \dot e^b\right)\end{aligned}$$ so $$\begin{aligned}
\gD_{ab} = \eta_{cd}\left( \dot e^c(X_a)\delta^d_b + \delta^c_a
\dot e^d(X_b)\right) = \dot e_a(X_b) + \dot e_b(X_a) =i_b\dot e_a
+ i_a\dot e_b\end{aligned}$$ since one may drop the $0$ subscript here without ambiguity: $X_a=(X_0)_a$. Then $$\begin{aligned}
\dot\Lambda =& \deltae^a \wedge \tau_a =
\frac{\delta\Lambda}{\delta g_{ab}} \gD_{ab} =
\frac{\delta\Lambda}{\delta g_{ab}} (\dot e_a(X_b) + \dot
e_b(X_a)) = 2 \frac{\delta\Lambda}{\delta g_{ab}} \dot e_a(X_b)
\\
=& 2 \dot e_a \wedge i_b \left(\frac{\delta\Lambda}{\delta
g_{ab}}\right)\end{aligned}$$ By (\[intro\_def\_tau\_a\]) the tensor associated to the $\tau_a$ is given by $$\begin{aligned}
T =-2\star \left(\frac{\delta\Lambda}{\delta g_{ab}}\right)
e_a\otimes e_b \label{Metric_g_var_tau_a_eb}\end{aligned}$$ and is manifestly symmetric.
In the following it is necessary to make explicit the metric dependence of various elements that enter in the action $4-$form $\Lambda$ and in particular to pass between vector fields and forms using the varied metric $g_t$. Thus the notations $\gT:\GamTM\to\GamLamM{1}$, $X\mapsto\gT(X)$ and $\gT^{-1}:\GamLamM{1}\to\GamTM$, $\alpha\mapsto\gT^{-1}(\alpha)$ for the metric dual of vectors and 1-forms with respect to the metric $\gT$ are used. For vectors or 1-forms which already have a subscript $0$ or $t$ we continue to use the tilde notation without ambiguity so that, for example, the $\gT$ metric dual of the vector $X_t$ can be written $\dual{X}_t\equiv\gT(X_t)$.
Following (\[Metric\_general\_lift\]) one has the maps $\starO$, $\starT$ and $\starD$ and from the Leibnitz rule (evaluated at $t=0$) $$\begin{aligned}
(\star\alpha)\dot{}=\starD\alpha+\star\dot\alpha
\label{Metric_star_alpha_dot}\end{aligned}$$ for all $\alpha_t\in\GamLamM{p}$. It follows simply (see appendix) that $$\begin{aligned}
\starD\alpha = \deltae^a\wedge\ia\star\alpha -
\star(\deltae^a\wedge\ia\alpha) \qquad\text{for}\quad
\alpha_t\in\GamLamM{p} \label{Lift_starD}\end{aligned}$$ Taking the derivative of $\Phi\wedge\starT\Psi=\Psi\wedge\starT\Phi$ with respect to $t$ gives $$\begin{aligned}
\Phi\wedge\starD\Psi=\Psi\wedge\starD\Phi
\qquad&\text{for}\quad
\Phi,\Psi\in\GamLamM{p}
\label{id_starT_pivot}\end{aligned}$$
Thus with the metric induced lift of the constitutive tensor $\PermTen$: $$\begin{aligned}
\PermTenT = \PermTenO + t\,\PermTenD + O(t^2)
\label{Lift_Z_t_0_dot}\end{aligned}$$ one writes: $$\begin{aligned}
(\star Z_t(F))\dot{} = \dot\star Z(F) + \star\dot Z(F)
\label{RWT1}\end{aligned}$$ Since there is a one parameter set of Hodge duals, we need to distinguish $\daggerT$ and $\daggerO$. Furthermore (\[Media\_def\_adjoint\]) becomes $$\begin{aligned}
\alpha\wedge\starT T(\beta) = \beta\wedge\starT T^\daggerT(\alpha)
\label{Media_def_adjoint_t0}\end{aligned}$$ for all $\alpha,\beta\in\GamLamM{p}$ and (\[Media\_def\_adjoint\_1forms\]) becomes $$\begin{aligned}
i_{\gT^{-1}\alpha} T(\beta) = i_{\gT^{-1}\beta} T^\daggerT(\alpha)
\label{Media_def_adjoint_t0_1forms}\end{aligned}$$ for all $\alpha,\beta\in\GamLamM{1}$.
Computation of the tensor {#ch_Mink}
==========================
In this section the variation of the above action (\[Action\_Max\_action\] )is explored for a particular choice of the metric dependence for $Z_t$, corresponding to a perturbative response of the medium to gravitation.
For a general lift the action $4-$form (\[Action\_Max\_action\]) is written $$\begin{aligned}
\cc\Lambda_t = \tfrac12 F \wedge \starT Z_t (F)
\label{Mink_Lagrange}\end{aligned}$$ hence $$\begin{aligned}
\cc\dot\Lambda = \tfrac12\big( F\wedge\starD \PermTen(F) + F
\wedge \star \dot{Z} (F) \big)\end{aligned}$$ From (\[Lift\_starD\]) $$\begin{aligned}
F\wedge\starD \PermTen(F) =& F\wedge\starD G = F \wedge \deltae^a
\wedge \ia \star G - F \wedge \star (\deltae^a \wedge \ia G)
\\
=& \deltae^a \wedge F \wedge \ia \star G - \deltae^a \wedge \ia G
\wedge \star F
\\
=& \deltae^a \wedge \big( F \wedge \ia \star G - \ia G \wedge
\star F \big)
\end{aligned}
\label{Mink_tau_a_Mink}$$ and $$\begin{aligned}
F \wedge \star \dot{Z} (F) = 2F\wedge \star \frac{\delta
Z}{\delta g_{ab}}(F) i_b\deltae^a = 2\deltae^a\wedge
i_b\left(F\wedge \star \frac{\delta Z}{\delta
g_{ab}}(F)\right)\end{aligned}$$
Therefore the drive forms are given by: $$\begin{aligned}
\cc\tau_a = \tfrac12F \wedge \ia \star G - \tfrac12\ia G \wedge
\star F + i_b \left( F\wedge\star\frac{\delta Z}{\delta
g_{ab}}(F)\right) \label{Mink_general_tau_a}\end{aligned}$$
For a physical medium with bulk motion that can sustain elastic stresses associated with its atomic constituents one expects that the history of such bulk motion should have some influence on the constitutive properties via some associated 4-velocity field[^10]. To include the possible dependence of the stress-energy-momentum tensor on such bulk motion of the medium one requires $Z$ to depend on this motion in some manner. In (\[Media\_Constitutive\_dh\]) $Z$ is specified in terms of electromagnetic fields measured in the comoving frame $V$ of the medium. It is therefore natural to prescribe a lift of this expression involving the lifts of $V_0$ and $\Set{\PermDEO,\PermDBO,\PermHEO,\PermHBO}$. The natural lift of the medium velocity $\vO$ is $$\begin{aligned}
\vT = \frac{\vO}{\sqrt{-\gT(\vO,\vO)}} \label{Lift_V_t}\end{aligned}$$ The metric dual of $\vT$ is given by $\dualvT = \gT(\vT)$ and the projection $\piV$ (\[Media\_def\_pi\_V\]) is lifted to $$\begin{aligned}
\piT=\Id + \dualvT\otimes\vT
\label{Lift_pi_t}\end{aligned}$$ The decomposition (\[Media\_def\_Z\_emme\]) of $\PermTenO$ and $\PermTenT$ with respect to the medium velocities $\vO$ and $\vT$ is given by $\Set{\PermDEO,\PermDBO,\PermHEO,\PermHBO}$ and $\Set{\PermDET,\PermDBT,\PermHET,\PermHBT}$ respectively, following the notation (\[Metric\_general\_lift\]).
The lifted tensors $\PermDET$, $\PermDBT$, $\PermHET$ and $\PermHBT$ will be now chosen to satisfy three properties:
For all $t$ in the neighbourhood of $t=0$ $$\begin{aligned}
\PermGenT|_{t=0} = \PermGenO \qquad\text{for}\quad
\PermGenT=\PermDET,\PermDBT,\PermHET,\PermHBT
\label{Lift_Perm_t_0}\end{aligned}$$
For all $t$ in the neighbourhood of $t=0$ they map the vector space that is $g_t-$orthogonal to $\dualvT$ to itself. This is achieved by lifting (\[Media\_iv\_Perm\_0\]): $$\begin{aligned}
&\PermGenT(\dualvT)=0 \quadand i_{\vT}\PermGenT(\alpha)=0
\label{Lift_iv_Perm_0}\end{aligned}$$ for $\PermGenT=\PermDET,\PermDBT,\PermHET,\PermHBT$ and all $\alpha\in\GamLamM{1}$. The corresponding lift of (\[Media\_piV\_Perm\_0\]) is $$\begin{aligned}
\PermGenT(\piT(\alpha))=
\piT(\PermGenT(\alpha))=\PermGenT(\alpha)
\label{Lift_piV_Perm_0}\end{aligned}$$
For all $t$ in the neighbourhood of $t=0$ they retain the adjoint conditions (\[Media\_Zeta\_adjoint\]). $$\begin{aligned}
(\PermDET)^{\daggerT}=\PermDET \,,\quad
(\PermHBT)^{\daggerT}=\PermHBT \quadand
(\PermDBT)^{\daggerT}=-\PermHET \label{Lift_Perm_adjoint}\end{aligned}$$
These requirements are all satisfied by setting $$\begin{aligned}
i_X\PermGenT(\alpha) =\tfrac12 \left( i_X\PermGenO(\piT\alpha) +
i_{\gT^{-1}\alpha}(\PermGenO)^{\daggerO}(\piT{{g_t}}X) \right)
\label{Lift_PermGen_t_res2}\end{aligned}$$ for all $\alpha\in\GamLamM{1}$ and $X\in\GamTM$, i.e.[^11]
$$\begin{aligned}
i_X\PermDET(\alpha) =& \tfrac12\left( i_X\PermDEO(\piT\alpha) +
i_{\gT^{-1}\alpha}\PermDEO(\piT\gT X) \right)\,,
\\
i_X\PermHBT(\alpha) =& \tfrac12\left( i_X\PermHBO(\piT\alpha) +
i_{\gT^{-1}\alpha}\PermHBO(\piT\gT X) \right)\,,
\\
i_X\PermDBT(\alpha) =& \tfrac12\left( i_X\PermDBO(\piT\alpha) -
i_{\gT^{-1}\alpha}\PermHEO(\piT\gT X) \right)\,,
\\
i_X\PermHET(\alpha) =& \tfrac12\left( i_X\PermHEO(\piT\alpha) -
i_{\gT^{-1}\alpha}\PermDBO(\piT\gT X) \right)
\end{aligned}
\label{Lift_PermGen_t_res3}$$
To verify that (\[Lift\_PermGen\_t\_res2\]) obeys (\[Lift\_Perm\_t\_0\]) note that from (\[Media\_def\_pi\_V\]) $$\begin{aligned}
\gO^{-1}\piO\gO X = X + \gO(X,\vO)\vO\end{aligned}$$ and hence from (\[Media\_def\_adjoint\_t0\_1forms\]) and (\[Media\_iv\_Perm\_0\]) $$\begin{aligned}
i_{\gO^{-1}\alpha}(\PermGenO)^{\daggerO}(\piO\gO X)
= i_{\gO^{-1}\piO\gO X}\PermGenO(\alpha) = i_X\PermGenO(\alpha)\end{aligned}$$ Thus at $t=0$ (\[Lift\_PermGen\_t\_res2\]) becomes $$\begin{aligned}
i_X\PermGenT(\alpha)|_{t=0} =& \tfrac12 \left(
i_X\PermGenO(\piO\alpha) +
i_{\gO^{-1}\alpha}(\PermGenO)^{\daggerO}(\piO\gO X) \right)
=i_X\PermGenO(\alpha)\end{aligned}$$ using (\[Media\_piV\_Perm\_0\]).
To verify that (\[Lift\_PermGen\_t\_res2\]) obeys (\[Lift\_iv\_Perm\_0\]) observe that $$\begin{aligned}
i_X\PermGenT(\dualvT) =\tfrac12 \left( i_X\PermGenO(\piT\dualvT) +
i_{\gT^{-1}\dualvT}(\PermGenO)^{\daggerO}(\piT{{g_t}}X) \right)\end{aligned}$$ Now $\piT\dualvT=0$ so the first term vanishes. Also using (\[Media\_def\_adjoint\_t0\_1forms\]) $$\begin{aligned}
i_{\gT^{-1}\dualvT}(\PermGenO)^{\daggerO}(\piT{{g_t}}X)
=&
\ivT (\PermGenO)^{\daggerO}(\piT{{g_t}}X)
=
\sqrt{-\gT(\vO,\vO)}\
i_{\vO}(\PermGenO)^{\daggerO}(\piT{{g_t}}X)
\\
=&
\sqrt{-\gT(\vO,\vO)}\ i_{\gT^{-1}\piT\gT X}\PermGenO(\dualvO)
=0\end{aligned}$$ Hence $\PermGenT(\dualvT)=0$. Likewise $$\begin{aligned}
\ivT \PermGenT(\alpha) = \tfrac12 \left( \ivT
\PermGenO(\piT\alpha) +
i_{\gT^{-1}\alpha}(\PermGenO)^{\daggerO}(\piT\dualvT) \right) = 0\end{aligned}$$ Finally to verify that (\[Lift\_PermGen\_t\_res3\]) obeys (\[Lift\_Perm\_adjoint\]) use (\[Media\_def\_adjoint\_t0\_1forms\]) and (\[Lift\_PermGen\_t\_res3\]) twice $$\begin{aligned}
i_{\gT^{-1}\alpha}(\PermDBT)^{\daggerT}(\beta) =&
i_{\gT^{-1}\beta}\PermDBT(\alpha) = \tfrac12\left(
i_{\gT^{-1}\beta}\PermDBO(\piT\alpha) -
i_{\gT^{-1}\alpha}\PermHEO(\piT\beta) \right)
\\
=& - i_{\gT^{-1}\alpha}\PermHET(\beta)\end{aligned}$$ In a similar way it follows that $(\PermDET)^{\daggerT}=\PermDET$ and $(\PermHBT)^{\daggerT}=\PermHBT$. Thus (\[Lift\_PermGen\_t\_res3\]) provide natural conditions for the lifts (\[Lift\_Perm\_t\_0\]) to (\[Lift\_Perm\_adjoint\])[^12].
Inserting the relations (\[Lift\_PermGen\_t\_res3\]) into $Z(F)$, (\[Media\_Z\_decomp\]), the action $4-$form (\[Mink\_Lagrange\]) becomes $$\begin{aligned}
2\cc\LagT =& F\wedge\starT(\PermDET(\ivT F)\wedge\dualvT) +
F\wedge\starT(\PermDBT(\ivT\starT F)\wedge\dualvT)
\\
&+ F\wedge\PermHET(\ivT F)\wedge\dualvT + F\wedge\PermHBT(\ivT
\starT F)\wedge\dualvT \,
\end{aligned}
\label{Lift_Lag_T}$$ To ease the density of notation in the following, the symbol $\MbT$ now stands for $\cc\MbT$ and $\MhT$ stands for $\frac{\MhT}{c}$. The lifts $$\begin{aligned}
\MeT=\ivT F \,,\quad \MbT=\ivT\starT F \,,\quad
\dualMeT=\gT^{-1}(\ivT F) \quadand \dualMbT=\gT^{-1}(\ivT\starT F)
\label{lift_eT_bT}\end{aligned}$$ satisfy $$\begin{aligned}
\piT\MeT=\MeT
\qquadand
\piT\MbT=\MbT
\label{lift_piT_eT_bT}\end{aligned}$$ Sequentially using (\[id\_iX\_star\]), (\[id\_iX\_move\]), (\[id\_star\_iX\]), (\[Lift\_PermGen\_t\_res3\]), (\[lift\_piT\_eT\_bT\]), (\[id\_star\_iX\]), (\[id\_iX\_move\]), (\[id\_iX\_star\]), the first term on the right hand side of (\[Lift\_Lag\_T\]) becomes $$\begin{aligned}
F\wedge\starT(\PermDET(\MeT)\wedge\dualvT)
=&
F\wedge \ivT\starT\PermDET(\MeT)
=
-\MeT\wedge \starT\PermDET(\MeT)
=
-(\starT1) i_{\dualMeT}\PermDET(\MeT)
\\
=&
-\tfrac12 (\starT1) \left( i_{\dualMeT}\PermDEO(\piT\MeT) +
i_{\dualMeT}\PermDEO(\piT\MeT) \right)
=
-(\starT1)
i_{\dualMeT}\PermDEO(\MeT)
\\
=&
- \MeT \wedge\starT \PermDEO(\MeT)
=
F \wedge\ivT\starT
\PermDEO(\MeT) = F \wedge\starT(\PermDEO(\MeT)\wedge\dualvT)\end{aligned}$$
Similarly sequentially using (\[id\_iX\_star\]), (\[id\_iX\_move\]), (\[id\_star\_iX\]), (\[Lift\_PermGen\_t\_res3\]), (\[lift\_piT\_eT\_bT\]), (\[id\_star\_iX\]), (\[id\_iX\_move\]), (\[id\_iX\_star\]), (\[id\_star\_pivot\]), (\[id\_star\_star\]) the second term on the right hand side of (\[Lift\_Lag\_T\]) gives $$\begin{aligned}
\lefteqn{F\wedge\starT(\PermDBT(\MbT)\wedge\dualvT)} \qquad&
\\
=&
F\wedge \ivT\starT\PermDBT(\MbT)
=
- \MeT\wedge\starT\PermDBT(\MbT)
=
-(\starT1) i_{\dualMeT}\PermDBT(\MbT)
\\
=&
-\tfrac12 (\starT1) \left( i_{\dualMeT}\PermDBO(\piT\MbT)
- i_{\dualMbT}\PermHEO(\piT\MeT) \right)
=
-\tfrac12 (\starT1) \left(i_{\dualMeT}\PermDBO(\MbT) -
i_{\dualMbT}\PermHEO(\MeT) \right)
\\
=&
-\tfrac12 \MeT\wedge\starT\PermDBO(\MbT) + \tfrac12 \MbT \wedge
\starT\PermHEO(\MeT)
=
\tfrac12 F\wedge\ivT \starT\PermDBO(\MbT) -
\tfrac12 \starT F \wedge \ivT\starT\PermHEO(\MeT)
\\
=&
\tfrac12 F\wedge\starT(\PermDBO(\MbT)\wedge\dualvT) -
\tfrac12 \starT(\PermHEO(\MeT)\wedge\dualvT) \wedge \starT F
\\
=&
\tfrac12 F\wedge\starT(\PermDBO(\MbT)\wedge\dualvT) -
\tfrac12 F \wedge \starT \starT(\PermHEO(\MeT)\wedge\dualvT)
\\
=&
\tfrac12 F\wedge\starT(\PermDBO(\MbT)\wedge\dualvT) +
\tfrac12 F \wedge \PermHEO(\MeT)\wedge\dualvT\end{aligned}$$ It is useful to record from this calculation that $$\begin{aligned}
-\tfrac12 (\starT1) \left(i_{\dualMeT}\PermDBO(\MbT) -
i_{\dualMbT}\PermHEO(\MeT) \right)
=
\tfrac12 F\wedge\starT(\PermDBO(\MbT)\wedge\dualvT) +
\tfrac12 F \wedge \PermHEO(\MeT)\wedge\dualvT
\label{lift_intermediate_step}\end{aligned}$$ Sequentially using (\[id\_star\_star\]), (\[id\_star\_iX\]), (\[id\_star\_pivot\]), (\[id\_star\_iX\]), (\[Lift\_PermGen\_t\_res3\]), (\[lift\_piT\_eT\_bT\]), (\[lift\_intermediate\_step\]) the third term on the right hand side of (\[Lift\_Lag\_T\]) yields $$\begin{aligned}
\lefteqn{F\wedge\PermHET(\MeT)\wedge\dualvT} \qquad&
\\
=&
-\PermHET(\MeT)\wedge\dualvT\wedge\starT\starT F
=
\PermHET(\MeT)\wedge\starT\MbT
=
\MbT\wedge\starT\PermHET(\MeT)
\\
=&
(\starT1)i_{\dualMbT}\PermHET(\MeT)
=
\tfrac12(\starT1)\left(i_{\dualMbT}\PermHEO(\piT\MeT) -
i_{\dualMeT}\PermDBO(\piT\MbT) \right)
\\
=&
\tfrac12(\starT1)\left(i_{\dualMbT}\PermHEO(\MeT) -
i_{\dualMeT}\PermDBO(\MbT) \right)
\\
=&
\tfrac12 F\wedge\starT(\PermDBO(\MbT)\wedge\dualvT) +
\tfrac12F \wedge \PermHEO(\MeT)\wedge\dualvT\end{aligned}$$ Finally on sequential use of (\[id\_star\_star\]), (\[id\_star\_iX\]), (\[id\_star\_pivot\]), (\[id\_star\_iX\]), (\[Lift\_PermGen\_t\_res3\]), (\[lift\_piT\_eT\_bT\]) $$\begin{aligned}
F\wedge\PermHBT(\MbT)\wedge\dualvT
=&
-\PermHBT(\MbT)\wedge\dualvT\wedge\starT\starT F
=
\PermHBT(\MbT)\wedge\starT\MbT
=
\MbT \wedge\starT \PermHBT(\MbT)
\\
=&
(\starT1)i_{\dualMbT} \PermHBT(\MbT)
=
\tfrac12 (\starT1) \left(
i_{\dualMbT} \PermHBO(\piT\MbT) + i_{\dualMbT}
\PermHBO(\piT\MbT)\right)
\\
=&
(\starT1)i_{\dualMbT} \PermHBO(\MbT)\end{aligned}$$ and so by reversing this sequence of steps $$\begin{aligned}
F\wedge\PermHBT(\MbT)\wedge\dualvT
=
F\wedge\PermHBO(\MbT)\wedge\dualvT\end{aligned}$$ Hence (\[Lift\_Lag\_T\]) simplifies to $$\begin{aligned}
2\cc\LagT =& F\wedge\starT(\PermDEO(\ivT F)\wedge\dualvT) +
F\wedge\starT(\PermDBO(\ivT\starT F)\wedge\dualvT)
\\
&+ F\wedge\PermHEO(\ivT F)\wedge\dualvT +
F\wedge\PermHBO(\ivT\starT F)\wedge\dualvT
\end{aligned}
\label{Lift_Lag_T_res}$$ i.e. the constitutive tensors $\zeta_t$ in the action may be replaced by $\zeta_0$ and hence the metric dependence of $\Lambda_t$ is seen to reside solely in $\star_t$, $V_t$ and $\tilde V_t$.
The derivative of (\[Lift\_Lag\_T\_res\]) at $t=0$ is given by $$\begin{aligned}
\cc\LagD =& F\wedge\starD(\PermDE(\ivF)\wedge\dualv)
+F\wedge\star(\PermDE(\ivD F)\wedge\dualv)
+F\wedge\star(\PermDE(\ivF)\wedge\dualvD)
\\ &
+F\wedge\starD(\PermDB(\ivstarF)\wedge\dualv)
+F\wedge\star(\PermDB(\ivD\star F)\wedge\dualv)
+F\wedge\star(\PermDB(\iv\starD F)\wedge\dualv)
\\ &
+F\wedge\star(\PermDB(\ivstarF)\wedge\dualvD) +F\wedge\PermHE(\ivD
F)\wedge\dualv +F\wedge\PermHE(\ivF)\wedge\dualvD
\\ &
+F\wedge\PermHB(\ivD \star F)\wedge\dualv +F\wedge\PermHB(\iv
\starD F)\wedge\dualv +F\wedge\PermHB(\iv \star F)\wedge\dualvD
\end{aligned}
\label{Stress_LagD_1}$$ where the subscript $0$ is omitted on the right hand side.
To determine the drive forms, observe that there are three different types of term in (\[Stress\_LagD\_1\]) which contain $\starD$, $\vD$ or $\dualvD$. Since $\dualvD=(g(V))\dot{}=g(\dot{V})+\gD(V)$, terms in $\starD$, $\vD$ and $\gD(V)$ can be collected to give: $$\begin{aligned}
2\cc\LagD = \LagD_{\starD} + \LagD_{\vD} + \LagD_{\gD(V)}
\label{Stress_LagD_split}\end{aligned}$$ where $$\begin{aligned}
2\cc\LagD_{\starD} =& F\wedge\starD(\PermDE(\ivF)\wedge\dualv)
+F\wedge\starD(\PermDB(\ivstarF)\wedge\dualv)
\\&
+F\wedge\star(\PermDB(\iv\starD F)\wedge\dualv)
+F\wedge\PermHB(\iv \starD F)\wedge\dualv \,,
\end{aligned}
\label{Stress_LagD_starD}$$ $$\begin{aligned}
2\cc\LagD_{\vD} =& F\wedge\star(\PermDE(\ivD F)\wedge\dualv)
+F\wedge\star(\PermDE(\ivF)\wedge g(\vD))
\\&
+F\wedge\star(\PermDB(\ivD\star F)\wedge\dualv)
+F\wedge\star(\PermDB(\ivstarF)\wedge g(\vD))
\\&
+F\wedge\PermHE(\ivD F)\wedge\dualv +F\wedge\PermHE(\ivF)\wedge
g(\vD)
\\&
+F\wedge\PermHB(\ivD \star F)\wedge\dualv +F\wedge\PermHB(\iv
\star F)\wedge g(\vD)
\end{aligned}
\label{Stress_LagD_vD}$$ and $$\begin{aligned}
2\cc\LagD_{\gD(V)} =& F\wedge\star(\PermDE(\ivF)\wedge\gD(V))
+F\wedge\star(\PermDB(\ivstarF)\wedge\gD(V))
\\&
+F\wedge\PermHE(\ivF)\wedge\gD(V) +F\wedge\PermHB(\iv \star
F)\wedge\gD(V)
\end{aligned}
\label{Stress_LagD_gDv}$$ The third term on the right hand side of (\[Stress\_LagD\_starD\]) may be expressed as $$\begin{aligned}
F\wedge\star(\PermDB(\iv\starD F)\wedge\dualv) =&
F\wedge\iv\star(\PermDB(\iv\starD F)) =
-\ivF\wedge\star\PermDB(\iv\starD F)
\\
=& \iv\starD F\wedge\star \PermHE(\ivF) = -\starD F\wedge\iv\star
\PermHE(\ivF)
\\
=& -\iv\star \PermHE(\ivF)\wedge\starD F =
-F\wedge\starD\star(\PermHE(\ivF)\wedge\dualv)\end{aligned}$$ using sequentially (\[id\_iX\_star\]), (\[id\_iX\_move\]), (\[Media\_Zeta\_adjoint\]), (\[id\_iX\_move\]), (\[id\_wedge\]), (\[id\_starT\_pivot\]). The fourth term on the right hand side of (\[Stress\_LagD\_starD\]) may be expressed as $$\begin{aligned}
F\wedge\PermHB(\iv \starD F)\wedge\dualv =&
-F\wedge\dualv\wedge\PermHB(\iv \starD F) =
-F\wedge\dualv\wedge\star\star\PermHB(\iv \starD F)
\\
=& -\star\PermHB(\iv \starD F)\wedge\star(F\wedge\dualv) =
\ivstarF\wedge\star\PermHB(\iv \starD F)
\\
=& \iv \starD F\wedge\star\PermHB(\ivstarF) = -\starD
F\wedge\iv\star\PermHB(\ivstarF)
\\
=& -\iv\star\PermHB(\ivstarF)\wedge\starD F = -F\wedge\starD
\star(\PermHB(\ivstarF)\wedge\dualv)\end{aligned}$$ using sequentially (\[id\_wedge\]), (\[id\_star\_star\]), (\[id\_starT\_pivot\]), (\[id\_iX\_star\]), (\[Media\_Zeta\_adjoint\]), (\[id\_iX\_move\]), (\[id\_wedge\]), (\[id\_starT\_pivot\]).
Hence from (\[Mink\_tau\_a\_Mink\]) $$\begin{aligned}
2\cc\LagD_{\starD} =& F\wedge\starD \Big(
\PermDE(\ivF)\wedge\dualv +\PermDB(\ivstarF)\wedge\dualv
-\star(\PermHE(\ivF)\wedge\dualv)
-\star(\PermHB(\ivstarF)\wedge\dualv) \Big)
\\
=& F\wedge\starD G
\\
=& \deltae^a\wedge\big(F \wedge \ia \star G -\ia G \wedge \star
F\big)
\end{aligned}
\label{Stress_tau_starD}$$
To collect terms in $\LagD_{\vD}$ observe that by differentiating (\[Lift\_V\_t\]), $\vD=\lambda V$ where $\lambda=\deltae^a(V)\,V_a$ one has $$\begin{aligned}
2\cc\LagD_{\vD} =& 2\lambda\big(
F\wedge\star(\PermDE(\ivF)\wedge\dualv)
+F\wedge\star(\PermDB(\ivstarF)\wedge\dualv)
\\&
+F\wedge\PermHE(\ivF)\wedge\dualv +F\wedge\PermHB(\iv \star
F)\wedge\dualv \big)
\\
=& 2 \lambda F\wedge\star G
= 2 \iv\deltae^a\ V_a F\wedge\star G
= 2 \deltae^a\wedge V_a\iv(F\wedge\star G)\end{aligned}$$
The first two terms on the right hand side of (\[Stress\_LagD\_gDv\]) become $$\begin{aligned}
\lefteqn{ F\wedge\star(\PermDE(\ivF)\wedge\gD(V))
+F\wedge\star(\PermDB(\ivstarF)\wedge\gD(V)) }\qquad&
\\
=&F \wedge \star (\iv G\wedge \dot{g}(V)) = \iv G\wedge \dot{g}(V)
\wedge \star F = -\dot{g}(V)\wedge \iv G\wedge \star F\end{aligned}$$ and the last two terms on the right hand side of (\[Stress\_LagD\_gDv\]) become $$\begin{aligned}
F\wedge\PermHE(\ivF)\wedge\gD(V) +F\wedge\PermHB(\iv \star
F)\wedge\gD(V) = -\dot{g}(V)\wedge F \wedge \iv\star G\end{aligned}$$ so using $\gD(V)
= \deltae^a(V)\ e_a + \deltae^a \ e_a(V) = 2\deltae^a \ V_a
+\iv(\deltae_a\wedge e^a)$ one has $$\begin{aligned}
2\cc\LagD_{\gD(V)} =& -\dot{g}(V)\wedge (\iv G\wedge \star F + F
\wedge \iv\star G)
\\
=& -(2\deltae^a \ V_a +\iv(\deltae_a\wedge e^a))\wedge (\iv
G\wedge \star F + F \wedge \iv\star G)
\\
=& -2\deltae^a\wedge (\iv G\wedge \star F + F \wedge \iv\star G) +
\deltae_a\wedge e^a\wedge \iv(\iv G\wedge \star F + F \wedge
\iv\star G)
\\
=& \deltae\wedge\big( -2(\iv G\wedge \star F + F \wedge \iv\star
G) + e^a\wedge(\ivF \wedge \iv\star G-\iv G\wedge \star F) \big)\end{aligned}$$ Adding this to $2\LagD_{\vD}$ gives $$\begin{aligned}
2\cc\LagD_{\vD}+2\cc\LagD_{\gD(V)} =& \deltae^a\wedge\big(
2V_a(\ivF\wedge\star G - \iv G\wedge \star F) + \deltae^a\wedge
e^a\wedge(\ivF \wedge \iv\star G-\iv G\wedge \star F) \big)
\\
=& 2 V_a \deltae^a\wedge(\ivF\wedge\star G - \iv G\wedge \star F)
- \deltae^a\wedge e^a\wedge\iv(\ivF\wedge\star G - \iv G\wedge
\star F)\end{aligned}$$ Using the relation $\star G=\iv\star G\wedge\dualv+\iv\star\iv G$ and the similar relation for $\star F$, the combination above may be writen $$\begin{aligned}
\lefteqn{\ivF\wedge\star G - \iv G\wedge \star F} \qquad\qquad&
\\
=& \ivF\wedge\ivstarG\wedge\dualv + \ivF\wedge\iv{\star}\ivG -
\ivG\wedge\ivstarF\wedge\dualv - \ivG\wedge\iv{\star}\ivF
\\
=& \star\GenPoyn - \iv(\ivF\wedge\star\ivG - \ivG\wedge\star\ivF)
= \star\GenPoyn\end{aligned}$$ where the 1-form $$\begin{aligned}
\GenPoyn=\star\left(\ivF\wedge\ivstarG\wedge\dualv +
\ivstarF\wedge\ivG\wedge\dualv\right)
\label{intro_general_poyntin}\end{aligned}$$ Hence $$\begin{aligned}
2\cc\LagD_{\vD}+2\cc\LagD_{\gD(V)} =& \deltae^a\wedge \big(2 V_a
{\star}\GenPoyn - e^a\wedge\iv{\star}\GenPoyn\big)
\label{Stress_tau_dualv}\end{aligned}$$ Adding together (\[Stress\_tau\_starD\]) and (\[Stress\_tau\_dualv\]) gives finally: $$\begin{aligned}
2\cc\LagD=\deltae^a\wedge \big(F \wedge \ia \star G -\ia G \wedge
\star F + 2 V_a {\star}\GenPoyn - e^a\wedge\iv{\star}\GenPoyn
\big)\end{aligned}$$ Hence the drive forms are given by $$\begin{aligned}
\cc\tau_a =& \tfrac12 \big(F \wedge \ia \star G -\ia G \wedge
\star F \big) + V_a\star\GenPoyn - \tfrac12
e_a\wedge\iv\star\GenPoyn \label{intro_Abraham_Stress_forms}\end{aligned}$$ with associated stress-energy-momentum tensor: $$\begin{aligned}
T= \tfrac12 \Big(i_a F\otimes i^a G + i_a G\otimes i^a F -
\star(F\wedge\star G) g + \dualv \otimes \GenPoyn + \GenPoyn
\otimes \dualv \Big) \label{intro_Abraham_Stress_T}\end{aligned}$$ The tensor $T$ above coincides in Minkowski spacetime with that attributed historically to Abraham. It is derived here in a considerably wider context.
In terms of comoving fields the drive forms can be written: $$\begin{aligned}
\cc\tau_{a} =& V_a \iv(\Me \wedge \star\Md + \Mh \wedge \star\Mb)
- \tfrac{1}{2}(\Me \wedge \ia\star\Md + \ia \Md \star\Me )
\\ &
- \tfrac{1}{2}(\Mh \wedge \ia\star\Mb + \ia \Mb \star\Mh )
+ 2v_a \Me \wedge \Mh \wedge \dualv + e_a \wedge \Me \wedge \Mh
\end{aligned}
\label{Abraham_tau_a}$$ and hence $$\begin{aligned}
T =& -\tfrac{1}{2} (\Me \otimes \Md + \Md \otimes \Me)
-\tfrac{1}{2} (\Mh \otimes \Mb + \Mb \otimes \Mh)
\\ &+
\tfrac{1}{2} (
g(\tilde{\Me},\tilde{\Md})
+ g(\tilde{\Mh},\tilde{\Mb})) ( g + 2 \dualv \otimes \dualv ) +
(\dualv \otimes \tilde{S} + \tilde{S} \otimes{V})
\end{aligned}
\label{Abraham_T_ebdh}$$ where the Poynting 1-form $$\begin{aligned}
\tilde{S} = \star( \dualv \wedge \Me \wedge \Mh)\end{aligned}$$
One may express the expressions above in terms of comoving polarisation $1-$forms ${{\mathbf p}}$ and magnetisation $1-$forms ${{\mathbf m}}$, defined in terms of comoving electromagnetic fields by [[$${{\mathbf d}}={{\mathbf e}}+{{\mathbf p}}$$]{}]{} [[$${{\mathbf h}}={{\mathbf b}}-{{\mathbf m}}$$]{}]{} Thus [[$$G=F + {{\cal P}}$$]{}]{} where [[$${{\cal P}}={{\mathbf p}}{\wedge}\tilde{V} + \star\,({{\mathbf m}}{\wedge}\tilde{V})$$]{}]{} Then one finds $$\begin{aligned}
{ \tau_c=\tau^{1}_c + \tau^{2}_c + \tau^{3}_c + \tau^{4}_c }\end{aligned}$$ where $$\begin{aligned}
2\cc\tau^{1}_c =& i_c {\star\,}G {\wedge}F - i_c G {\wedge}{\star\,}F \,,
\\
2\cc\tau^{2}_c =& V_c\,\left( {{\mathbf p}}{\wedge}{\star\,}F + {{\mathbf m}}{\wedge}F
-({{\mathbf p}}{\wedge}\Mb + {{\mathbf m}}{\wedge}\Me) {\wedge}\tilde V \right) \,
\\
2\cc\tau^{3}_c =& -(i_c F) {\wedge}({{\mathbf m}}{\wedge}\tilde V) - (i_c{\star\,}F)
{\wedge}{\star\,}({{\mathbf m}}{\wedge}\tilde V)
\\
2\cc\tau^{4}_c =& -V_c\,({{\mathbf p}}{\wedge}{\star\,}F + {{\mathbf m}}{\wedge}F) -
V_c\,\tilde V {\wedge}({{\mathbf p}}{\wedge}\Mb + {{\mathbf m}}{\wedge}\Me) - e_c{\wedge}({{\mathbf p}}{\wedge}\Mb + {{\mathbf m}}{\wedge}\Me)\end{aligned}$$ The above are valid for all simple media in arbitrary gravitational fields. For a simple medium, which may be inhomogeneous, anisotropic and intrinsically magneto-electric, at rest in an inertial frame in [*Minkowski spacetime*]{} with Minkowski coordinates $\{t, \vec{x}\}$ one has (in Euclidean notation) $$\begin{aligned}
\dualv= -dt \quad , \quad \Me = \vec{E} \cdot d\vec{x} \quad ,
\quad \Mb = \vec{B} \cdot d\vec{x} \quad , \quad \Mh = \vec{H}
\cdot d\vec{x} \quad , \quad \Md = \vec{D} \cdot d\vec{x}\end{aligned}$$ and $$\begin{aligned}
g(\tilde{\Me},\tilde{\Md}) = \vec{E}\cdot \vec{D} \quad , \quad
g(\tilde{\Mh},\tilde{\Mb}) = \vec{H} \cdot \vec{B}\end{aligned}$$ $$\begin{aligned}
\tilde{S} = - (\vec{E} \times \vec{H}) \cdot d\vec{x}\end{aligned}$$ The coordinate components of the stress-energy-momentum tensor follow as $$\begin{aligned}
T_{00} &= \tfrac{1}{2} (\vec{E}\cdot \vec{D} + \vec{H} \cdot
\vec{B})
\\
T_{ij} &= -\tfrac{1}{2} ( E_i D_j + E_j D_i )
-\tfrac{1}{2} ( H_i B_j + B_j H_i ) + \tfrac{1}{2} \delta_{ij} (
\vec{E}\cdot \vec{D} + \vec{H} \cdot \vec{B})
\\
T_{0k} &= T_{k0} = -(\vec{E} \times \vec{H})_k
\end{aligned}
\label{Abraham_T_components}$$
Conclusions {#concl}
============
Natural assumptions made above for the dependence of the constitutive tensor $Z$ on the normalised 4-velocity of a simple medium have led via a non-trivial variational argument to a contribution to the stress-energy-momentum tensor (involving phenomenological electromagnetic interactions with bulk matter) that coincides with that suggested by Abraham under more restricted circumstances. Although natural, the assumptions based on physical considerations are not, however, necessarily the simplest to make.
If $\PermTen$ is chosen to be independent of the metric and hence $\tilde V$, with $\PermTenT=\PermTenO$ and $\PermTenD=0$ so that $G=Z_0(F)$ in all gravitational fields, one obtains immediately from the above variational calculations (\[Mink\_general\_tau\_a\]) the drive forms $$\begin{aligned}
\cc\tau_a = \tfrac12 \big( F \wedge \ia \star G - \ia G \wedge
\star \,F \big) \label{intro_sym_Minkowski_Stress_forms}\end{aligned}$$ and the associated stress-energy-momentum tensor $$\begin{aligned}
T= \tfrac12 i_a G\otimes i^a F + \tfrac12 i_a F\otimes i^a G -
\tfrac12 \star(F\wedge\star\, G) g\end{aligned}$$ showing clearly its independence of the 4-velocity of the medium. It is of interest to note that such a tensor coincides with that obtained by symmetrising the one proposed by Minkowski.
In the absence of a generally accepted relativistic covariant description of deformable matter interacting with electromagnetic fields, the adoption of a particular tensor for the electromagnetic field alone in polarisable (and possible magneto-electric) media must remain a matter of expediency. However, useful models for the total tensor for such systems can benefit from the use of sufficiently general phenomenological descriptions of the electromagnetic properties of moving media compatible with relativistic covariance. For example a thermodynamically inert (pressureless, cold) fluid can be modelled by adding the electromagnetic stress-energy-momentum tensor (\[intro\_Abraham\_Stress\_T\]) to the matter stress-energy-momentum tensor $\frac{m_0}{c\epsilon_0} \,{\nn}
\widetilde {V} \otimes \widetilde {V} $ where $\nn$ is a scalar number density field, $m_0$ some constant with the dimensions of mass and $V$ the unit time-like 4-velocity field of the fluid. Supplemented with continuity conditions, the vanishing divergence of such a combination yields the dynamics of the system and with prescribed boundary conditions at an interface separating such media with different properties one may compute bulk forces and torques.
A review has also been given of the symmetry constraints expected of the total tensor particularly when this is considered to be a source of relativistic gravitation. This led to a definition in terms of a variational derivative and a consideration of the response of the electromagnetic constitutive properties to gravitational perturbations. It is suggested that tensors parameterised by a self-adjoint constitutive tensor $Z$ offer a viable means to explore the electromagnetic properties of a range of inhomogeneous, anisotropic and possibly magneto-electric continua, at least in regions where dispersion and losses can be ignored to a first approximation. This formulation suggests a method to determine the properties of $Z$ by exploring its phenomenological response to electromagnetic fields in arbitrarily moving reference frames and variable gravitational fields. It opens up the possibility of performing such experiments in new environments such as those carried out under terrestrial free-fall or space station situations or in astrophysical contexts.
[**Acknowledgements**]{} The authors are grateful D. Burton and A. Noble for helpful discussions and to the EPSRC and Framework 6 (FP6-2003-NEST-A) for financial support for this research.
Appendix
=========
Using the notation established in the text, this appendix derives the useful formula (\[Lift\_starD\]) relating $(\star \Psi)\DOT{}$ to $\star \dot\Psi$ where $\Psi\in
\GamLamM{p}$. Let $I$ denote a multi-index constructed from the single indices $a,b,c\ldots $ in the range $0,1,2,3$ where the components of the metric tensor $g$ and $\Psi$ in an $g$-orthonormal basis $\{e^c\}$ are respectively $\eta_{ab}$ and $\Psi_I$. Thus
$$\begin{aligned}
\Psi=&\Psi_I \, e^I\end{aligned}$$
and $$\begin{aligned}
\dot\Psi=&\dot\Psi_I\,e^I +
\Psi_I\,{(e^I)}\DOT\label{A1}\end{aligned}$$ Since $e^I$ is the exterior product of $p$ 1-forms $$\begin{aligned}
(e^I)\DOT=\dot e^c\wedge i_c(e^I)\end{aligned}$$ Similarly, since the basis is orthonormal $$\begin{aligned}
(\star e^I)\DOT=\dot e^c\wedge i_c(\star e^I)\end{aligned}$$ Thus using (\[A1\]) $$\begin{aligned}
\dot\Psi_I e^I=\dot\Psi - \dot e^c\wedge i_c\Psi\end{aligned}$$ Applying $\star$ to this gives $$\begin{aligned}
\dot\Psi_I \star e^I=\star\dot\Psi - \star(\dot e^c\wedge
i_c\Psi)\label{A2}\end{aligned}$$ But $$\begin{aligned}
(\star\Psi)\DOT=&(\Psi_I\star e^I)\DOT\\=&\dot\Psi_I\star e^I +
\Psi_I(\star e^I)\DOT\\=& \dot\Psi_I\,\star e^I + \Psi_I\,\dot
e^c\wedge i_c(\star e^I)\\=&\dot\Psi_I\,\star e^I + \dot e^c
\wedge i_c(\star \Psi)\end{aligned}$$ Substituting from (\[A2\]) yields the relation $$\begin{aligned}
(\star \Psi)\DOT= \dot e^c \wedge i_c(\star \Psi) - \star(\dot e^c
\wedge i_c\Psi) + \star \dot\Psi\end{aligned}$$
[99]{}
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[^1]: E.mail: tdereli@ku.edu.tr
[^2]: E.mail: j.gratus@lancaster.ac.uk
[^3]: E.mail: r.tucker@lancaster.ac.uk
[^4]: All tensors in this article have dimensions constructed from the SI dimensions $[M], [L], [T], [Q]$ where $[Q]$ has the unit of the Coulomb in the MKS system. We adopt $[g]=[L^2], [G]=[j]=[Q],\,[F]=[Q]/\ee$ where the permittivity of free space $\epsilon_0$ has the dimensions $ [
Q^2\,T^2 M^{-1}\,L^{-3}] $ and $c$ denotes the speed of light in vacuo
[^5]: The notion of a classical vacuum here corresponds to spacetime devoid of all material ($j=0$) although if $j$ has compact support one can refer to vacuum domains" where $j=0$. All regions can admit non-zero and gravitational fields.
[^6]: i.e. In terms of the Lie derivative ${\cal L}_Y$, ${\cal L}_Y g=\lambda\,g$ for some scalar $\lambda$. $Y$ is a Killing field when $\lambda=0$. Angular momentum currents follow in terms Killing vector fields that generate rotational diffeomorphisms.
[^7]: In view of the above comments on the role of particular timelike and spacelike Killing vectors in constructing conserved energy-power and momentum-force currents a more coherent label for $T$ might be the drive tensor
[^8]: For example, locally $SL(2,C)$ covariant couplings of spinor fields to gravitation fall into this category.
[^9]: e.g. electrostriction and magnetostriction arise from the dependence of $ {\cal Z} $ on the elastic deformation tensor of the medium [@antoci].
[^10]: Relativistic strings and membranes with dynamics that arise from re-parameterisation independent actions are an exception since, without constituents", no preferred parametrisation of their histories should be identified.
[^11]: For an isotropic, non-magneto-electric medium (\[Media\_scalar\_medium\_G\]) and (\[Media\_scalar\_medium\_dh\]), the lifts (\[Lift\_PermGen\_t\_res3\]) reduces to the lifts $$\begin{aligned}
\PermDET=\epsilon\piT \,,\quad \PermHBT=\mu^{-1}\piT \,,\quad
\PermDBT=0 \quadand \PermHET=0\end{aligned}$$ which in a comoving frame yield the relations $$\begin{aligned}
\MdT = \ee\epsilon\, \MeT \qquadand \MhT = (\mu_0\,\mu)^{-1} \MbT
\,\end{aligned}$$ where the scalars $\epsilon$ and $\mu^{-1}$ are independent of the ambient metric.
[^12]: The requirements (\[Lift\_Perm\_t\_0\]-\[Lift\_Perm\_adjoint\]) are not meant to be exhaustive. Other lifts $\PermGenT$ could involve gradients of the spacetime metric corresponding to gravitational tidal effects on the constitutive tensor. For example if $\cal R$ is the curvature scalar associated with the Levi-Civita connection then the lifts $$\begin{aligned}
i_X\PermGenT(\alpha) =\tfrac12({\cal R}_t - {\cal R}_0 + 1) \left(
i_X\PermGenO(\piT\alpha) +
i_{\gT^{-1}\alpha}(\PermGenO)^{\daggerO}(\piT{{g_t}}X) \right)\end{aligned}$$ also satisfy (\[Lift\_Perm\_t\_0\]-\[Lift\_Perm\_adjoint\]).
|
---
abstract: 'We have shown earlier that hyperfine spectroscopy in a vapor cell using co-propagating pump-probe beams has many advantages over the usual technique of saturated-absorption spectroscopy using counter-propagating beams. The main advantages are the absence of crossover resonances, the appearance of the signal on a flat (Doppler-free) background, and the higher signal-to-noise ratio of the primary peaks. Interaction with non-zero-velocity atoms causes additional peaks, but only one of them appears within the primary spectrum. We first illustrate the advantages of this technique for high-resolution spectroscopy by studying the $D_2$ line of Rb. We then use an acousto-optic modulator (AOM) for frequency calibration to make precise hyperfine-interval measurements in the first excited $P_{3/2}$ state of $^{85,87}$Rb and $^{133}$Cs.'
author:
- 'Alok K. Singh, Sapam Ranjita Chanu, Dipankar Kaundilya and Vasant Natarajan'
title: 'Hyperfine spectroscopy using co-propagating pump-probe beams'
---
Introduction
============
High-resolution laser spectroscopy has been revolutionized in the last two decades with the advent of low-cost tunable diode lasers [@WIH91]. These diodes, when placed in an external cavity with optical feedback, have frequency uncertainty of about 1 MHz, which is small enough for hyperfine transitions in atoms to be resolved [@MSW92; @BRW01]. Hyperfine spectroscopy, particularly in the low-lying electronic states of alkali-metal atoms, plays an important role in fine-tuning atomic wavefunctions used in theoretical calculations. This is because comparison between theoretical and experimental determinations of hyperfine structure provides a stringent test of atomic calculations in the vicinity of the nucleus [@SJD99]. In addition, hyperfine structure in these multielectron atoms is sensitive to core polarization and core correlation effects [@AIV77].
Many of the alkali-metal atoms have transitions to the first-excited state (so-called $D$ lines) which are in the near infrared, and therefore accessible with diode lasers. They also have a high-enough vapor pressure near room temperature that the spectroscopy can be done in a vapor cell. The thermal motion of the atoms inside the cell causes [*Doppler broadening*]{}, which is typically 100 times larger than the natural width of the hyperfine transitions. The standard technique to overcome the first-order Doppler effect is to use a [*counter-propagating*]{} pump beam to saturate the transition for zero-velocity atoms, in what is called saturated-absorption spectroscopy (SAS) [@DEM82].
Most atoms also have several closely-spaced hyperfine levels within the Doppler profile. In these cases, it is well known that SAS also produces spurious crossover resonances in between each pair of hyperfine transitions. They occur because, for some non-zero-velocity group, the pump drives one transition while the probe drives the other. In earlier work [@BAN03], we have shown that the use of co-propagating pump and probe beams overcomes the problem of crossover resonances. Closely-spaced levels that are not resolved in SAS can be resolved by this technique. Probe transmission in such multilevel atoms is caused by the phenomenon of electromagnetically induced transparency (EIT) [@HAR97; @DAN05], and population depletion due to optical pumping. Furthermore, by scanning only the pump beam, the signal appears on a flat background without the underlying Doppler profile seen in SAS. This is advantageous for applications such as laser locking or laser frequency measurement.
In this work, we present a complete study of the spectra taken with the co-propagating technique in the $D_2$ line of the two isotopes of Rb. In particular, we show that the effect of non-zero velocity groups is to cause additional peaks. However, in contrast to SAS, only one of these spurious peaks appears within the spectrum, and the real peaks are unaffected. Interestingly, the spurious peak within the spectrum is almost negligible for transitions starting from the upper ground hyperfine level, but highly prominent for transitions starting from the lower level. Such differences between the two levels have also been seen in EIT, arising from the fact that the closed transition is $F \rightarrow (F+1)$ for the upper hyperfine set and $F
\rightarrow (F-1)$ for the lower hyperfine set.
For a second set of experiments, we have used a single laser along with an acousto-optic modulator (AOM) to produce the pump-probe beams with a precisely-controlled frequency offset. We then obtain the entire spectrum by scanning the frequency of the AOM. The scan axis is guaranteed to be linear because it is determined by the frequency of the rf oscillator driving the AOM. A curve fit to the observed spectrum yields the hyperfine interval. The measurements have an accuracy of 20 kHz, which is comparable to the accuracy of other techniques.
In recent work from our laboratory, we have reported high-accuracy values for the hyperfine constants in the $D$ lines of all alkali atoms [@DAN08]. The hyperfine intervals in that work were obtained (with an accuracy of 6 kHz) by [*locking*]{} the AOM to the neighboring transition. One of the uncertainties when locking the AOM is whether the lock point is exactly at the center of the peak, since any shift would cause a systematic error in the measurement. Though we had done experiments to verify that this error was less than 2 kHz, we wanted to repeat the measurements with another technique that was not at all susceptible to errors arising from lock-point uncertainty. As we will see below, the co-propagating technique achieves precisely this. In addition, by measuring the entire spectrum and looking at the symmetry of the line shape, we can be sure that other sources of error are not significant. The current set of measurements in $^{85}$Rb and $^{133}$Cs, although having slightly smaller precision, are consistent with our earlier work.
Spectroscopy on the $D_2$ line of rubidium
==========================================
The schematic for the first set of experiments is shown in Fig. \[schema\]. The pump and probe beams are derived from two home-built frequency-stabilized diode laser systems [@BRW01] operating on the 780 nm $D_2$ line of Rb. The linewidth of the lasers after stabilization is of the order of 1 MHz. The output beams are elliptical with $1/e^2$ size of $2 \times 4$ mm and powers of around 10 $\mu$W each. Part of the probe laser is sent into a Rb SAS cell and the laser is locked to a hyperfine transition using fm modulation spectroscopy. The pump laser is scanned around the same set of transitions. The beams are mixed in a polarizing beamsplitter cube (PBS) and [*copropagate*]{} through a room-temperature vapor cell (5 cm long) with orthogonal linear polarizations. Halfwave retardation plates in the path of each beam allow precise control of their powers. The probe beam is separated using a second PBS, and its transmitted signal is detected with a photodiode. The PBS’s have extinction ratios of $1000:1$, ensuring good purity of the detected signal.
Spectrum in $^{87}$Rb, $F=2 \rightarrow F'=1,2,3$
-------------------------------------------------
As mentioned before, the main advantage of the co-propagating configuration is the absence of spurious crossover resonances. This difference is seen clearly in Fig. \[satabs\]. In (a), we show the usual saturated-absorption spectrum for the $F=2 \rightarrow
F'=1,2,3$ transitions in $^{87}$Rb. The spectrum is Doppler corrected, which is necessary because probe absorption through a vapor cell will show a broad Doppler profile when the probe addresses a velocity group different from that resonant with the pump. When the pump and probe are resonant with the same velocity class, we get transmission peaks. As expected, there are three hyperfine peaks and three crossovers. The crossovers are more prominent than the actual peaks because two velocity classes contribute to each crossover resonance, compared to one (zero-velocity) class for each hyperfine peak. Probe transparency is primarily caused by two effects: (i) saturation of absorption caused by the strong pump beam, and (ii) optical pumping into the $F=1$ ground hyperfine level for open transitions (i.e. those involving the $F'=1$ and 2 excited levels). In addition, there will be population redistribution among the magnetic sublevels, which can cause increased absorption or transparency depending on the $F$ values of the levels. The linewidth of the peaks in the figure is about 12 MHz, compared to the natural linewidth of 6 MHz. This increase is typical in SAS and arises due to a misalignment angle between the beams and power broadening.
(a)\
(b)\
(c)
Now let us consider the spectrum shown in (b) taken with the co-propagating configuration. The probe is locked to the $F=2 \rightarrow 3$ transition and the pump is scanned across the set of $F=2 \rightarrow F'=1,2,3$ transitions. Since the probe is locked, its transmitted signal primarily corresponds to absorption by zero-velocity atoms (i.e.atoms moving perpendicular to the laser beam) making transitions to the $F'=3$ level. The signal remains flat (or Doppler free) until the pump also comes into resonance with a transition for the same zero-velocity atoms. Thus there are three transmission peaks at the locations of the hyperfine transitions, with no crossover resonances in between. The hyperfine peaks are located at $-423.600$ MHz, $-266.657$ MHz, and 0 [@DAN08], all measured with respect to the frequency of the locked probe laser. The linewidth of the peaks is about 19 MHz, which is only 50% larger than the linewidth obtained in the saturated-absorption spectrum. The primary cause for the transparency peaks is the phenomenon of EIT in this V-type system. The pump laser causes an [*AC Stark shift of the ground level*]{} (creation of dressed states [@COR77]) and hence reduces probe absorption at line center. In addition, there are effects of saturation and optical pumping, but these are less important than the EIT effects.
Since the experiments are done in a vapor cell with the full Maxwell-Boltzmann distribution of velocities, we have to consider that there will be two additional velocity classes that absorb from the locked probe: both moving in the same direction as the probe but with velocities such that one drives transitions to the $F'=2$ level (266.657 MHz lower) and the second to the $F'=1$ level (423.600 MHz lower). Each of these will cause three additional transparency peaks from the mechanisms discussed above. The first velocity class moves at 208 m/s and will cause peaks at $-156.943$ MHz, 0, and $+266.657$ MHz, i.e., a set of peaks shifted up by 266.657 MHz. The second velocity class moves at 330 m/s and will cause peaks at 0, $156.943$ MHz, and $+423.600$ MHz, i.e., a set of peaks shifted up by 423.600 MHz. Thus there will be 7 peaks in all, with 3 real peaks and 4 spurious ones. However, only the peak at $-156.943$ MHz will appear within the spectrum, caused by the probe driving the $F=2 \rightarrow F'=2$ transition and the pump driving the $F=2 \rightarrow F'=1$ transition. The other three spurious peaks will lie outside the spectrum to the right. This is indeed what is observed in Fig.\[satabs\](b): there is a small peak at $-157$ MHz within the spectrum.
The above explanation is borne out by the calculated spectrum shown in Fig. \[satabs\](c). Using a density-matrix formulation, we can calculate the absorption of a probe laser in a V-type system [@DAN05]. The calculation is done for multiple hyperfine levels with full thermal averaging. The only adjustable parameters are the relative amplitudes of the three EIT resonances. As seen from the figure, the calculation reproduces the locations of the peaks in the measured spectrum. The observed linewidth is slightly larger than the calculated one, but this could be because of a small misalignment angle between the beams, which is known to broaden the EIT resonance [@CAT04]. The calculation shows that there is only one spurious peak within the spectrum. However, if we extend the calculation up to $+500$ MHz, we see all the seven peaks mentioned in the previous paragraph. The extended calculation is shown in Fig. \[theory\].
The advantages of this scheme are quite clear from Fig.\[satabs\](b). The spectrum appears on a flat background, obviating the need for Doppler subtraction as in the case of SAS. There are no crossover resonances, which often swamp the true peaks. And there is only one additional peak within the spectrum due to absorption by non-zero velocity atoms. In Fig. ref[coprop]{}, we show the effect of pump power on the peaks. As the power is varied from $0.33$ to $1.66$ times the probe power, the three main peaks remain quite prominent with good signal-to-noise ratio. The additional peak at $-157$ MHz increases in height, but not significantly. By comparison, a good saturated-absorption spectrum requires the pump-probe power ratio to be accurately controlled to a value of $3:1$, with loss in signal at lower pump powers and power broadening at higher powers. In Fig. \[coprop\](b), we show a multipeak Lorentzian fit to the spectrum measured with a pump power of 15 $\mu$W. The residuals show that the line shape of all the peaks is Lorentzian.
(a)\
(b)
Spectrum in $^{87}$Rb, $F=1 \rightarrow F'=0,1,2$
-------------------------------------------------
The same advantages are seen in the spectrum of transitions starting from the lower ground hyperfine level ($F=1
\rightarrow F'=0,1,2$) shown in Fig. \[rb10\]. The Doppler-subtracted saturated-absorption spectrum on top has 6 peaks including the 3 crossover resonances. The spectrum with the co-propagating beams shown below is taken with the probe locked to the $F=1 \rightarrow 0$ transition. The beam powers are 9 $\mu$W (probe) and 15 $\mu$W (pump). It appears on a flat background and shows the 3 hyperfine peaks without any crossovers in between. The 3 hyperfine peaks are located at 0, $+72.223$ MHz, and $+229.166$ MHz. Two of the additional peaks are seen, one at $-72.223$ MHz (outside the spectrum) and the other at $+156.943$ MHz (within the spectrum). The additional peak within the spectrum is due to atoms moving with a velocity of 52 m/s such that the probe drives the $F=1 \rightarrow F'=1$ transition and the pump drives the $F=1 \rightarrow F'=2$ transition. This spurious peak is more prominent compared to transitions starting from the upper ground level \[see Fig. \[satabs\](b)\] (though the real peaks still have high signal-to-noise ratio). The difference arises due to the fact that the closed transition for this set is the $F=1 \rightarrow F'=0$ transition, which has fewer magnetic sublevels in the excited state compared to the ground state. This leads to population trapping in the $m_F = \pm
1$ sublevels, and the relative importance of EIT effects in causing probe transparency (which is the same for all peaks) increases.
Spectrum in $^{85}$Rb, $F=3 \rightarrow F'=2,3,4$
-------------------------------------------------
The improvement with this technique is much more dramatic in the spectra of the other isotope, $^{85}$Rb. For transitions starting from the upper ground level ($F=3
\rightarrow F'=2,3,4$) shown in Fig. \[rb34\], the hyperfine peaks corresponding to $F'=2$ and 4 in the saturated-absorption spectrum are barely visible. In the co-propagating spectrum shown below, the peaks become prominent. The hyperfine intervals [@DAN08] are such that the real peaks are at $-184.390$ MHz, $-120.966$ MHz, and 0, while the additional peaks are at $-63.424$ MHz, $+63.424$ MHz, $+120.966$ MHz, and $+184.390$ MHz. The additional peak within the spectrum (at $-63.424$ MHz) is almost negligible, as was observed for upper-level transitions in $^{87}$Rb. The beam powers are 9 $\mu$W (probe) and 15 $\mu$W (pump).
Spectrum in $^{85}$Rb, $F=2 \rightarrow F'=1,2,3$
-------------------------------------------------
For transitions starting from the lower ground level ($F=2
\rightarrow F'=1,2,3$) shown in Fig. \[rb21\], the hyperfine interval between $F'=1$ and 2 is so small that the crossover resonance in the saturated-absorption spectrum completely swamps the $F'=1$ peak. However, the spectrum with the co-propagating technique shows the peak well resolved. The real peaks are located at 0, $+29.268$ MHz, and $+92.692$ MHz, while the additional peaks are at $-92.692$ MHz, $-63.424$ MHz, $-29.268$ MHz, and $+63.424$ MHz [@DAN08]. The additional peak within the spectrum (at $+63.424$ MHz) is quite prominent as in the case of transitions starting from the lower hyperfine level in $^{87}$Rb, again because the closed $F=2 \rightarrow F'=1$ transition has population trapping in the $m_F = \pm 2$ sublevels. There are two additional peaks appearing outside the spectrum to the left, which are closer because of the smaller hyperfine intervals. The beam powers are 9 $\mu$W (probe) and 15 $\mu$W (pump).
Hyperfine measurements using an acousto-optic modulator
=======================================================
The above experiments were done using separate pump and probe lasers. However, it is possible to do the experiment with just one laser by using an AOM to produce the scanning pump beam. The scan range of an AOM is limited to about 20 MHz, but this is large enough to scan across a hyperfine peak. The main advantage of using an AOM is that the frequency-scan axis (with respect to the probe beam) is both linear and calibrated by the rf frequency of the driver powering the AOM, thus allowing the hyperfine interval to be measured accurately. By measuring the entire peak, potential systematic errors due to locking of the pump laser to a peak are avoided. In addition, if there is a systematic shift in the lock point of the probe laser, this will not cause an error in the interval because the frequency which brings the pump into resonance will also be shifted by the same amount, and hence the AOM offset (with respect to the probe) for the spectrum will remain the same.
We have therefore used a single laser and a scanning AOM to measure hyperfine intervals in the $D_2$ lines of Rb and Cs. The measurements are motivated by the fact that there are several experimental values reported in the literature that are somewhat discrepant from each other. In many cases, we feel that a potential source of error is the uncertainty in locking to a peak. In our current technique, measuring the entire spectrum avoids such errors, as discussed before.
The experimental schematic for this second set of experiments is shown in Fig. \[schema2\]. As before, the primary laser is a frequency-stabilized diode laser. The probe beam is derived after locking the laser to a hyperfine transition using SAS. The scanning pump beam is frequency offset from the probe using a double-passed AOM. The frequency is adjusted so that the pump is resonant with a nearby hyperfine transition whose interval has to be measured. The pump intensity is stabilized to better than 1% in a servo-loop by controlling the rf power driving the AOM. The two beams co-propagate through a vapor cell kept inside a magnetic shield. The residual field (measured with a three-axis fluxgate magnetometer) is below 5 mG. The beams have orthogonal linear polarizations and are mixed and separated using PBS’s. The beam powers are about 15 $\mu$W each and adjusted to get good signal-to-noise ratio in the spectrum.
The experiment proceeds as follows. An rf frequency generator whose timebase is referenced to an ovenized quartz clock (uncertainty less than $10^{-8}$) is used to drive the AOM. A computer program is used to set the rf frequency, and the probe signal is measured and recorded. The frequency is changed in steps of 0.1 MHz over a range of 15 MHz to obtain the complete spectrum. A curve fit to the spectrum yields the AOM frequency at the peak center, which is the hyperfine interval.
Error analysis
--------------
Systematic errors can arise due to one of the following reasons.
1. [*Radiation-pressure effects.*]{} Radiation pressure causes velocity redistribution of the atoms in the vapor cell. In the SAS technique, the opposite Doppler shifts for the counter-propagating beams can result in asymmetry of the observed lineshape. However, with co-propagating beams, the effects are less important because the Doppler shift will be the same for both beams and will not affect the hyperfine interval, similar to how the interval is insensitive to any detuning of the probe from resonance.
2. [*Effect of stray magnetic fields.*]{} The primary effect of a magnetic field is to split the Zeeman sublevels and broaden the line without affecting the line center. However, line shifts can occur if there is asymmetric optical pumping into Zeeman sublevels. For a transition $\ket{F,m_F}
\rightarrow \ket{F',m_{F'}}$, the systematic shift of the line center is $\mu_B(g_{F'}m_{F'} -
g_Fm_F)B$, where $\mu_B=1.4$ MHz/G is the Bohr magneton, $g$’s denote the Landé $g$ factors of the two levels, and $B$ is the magnetic field. The selection rule for dipole transitions is $\Delta m
= 0,\pm 1$, depending on the direction of the magnetic field and the polarization of the light. Thus, if the beams are linearly polarized, there will be no asymmetric driving and the line center will not be shifted. We therefore minimize this error in two ways. First, we use polarizing cubes to ensure that the beams have near-perfect linear polarization. Second, we use a magnetic shield around the cell to minimize the field.
The experiment is repeated by reversing the scan direction to check for errors that might depend on which direction the rf generator is scanned. Another source of error is whether the intensity stabilization servo-loop stays locked. But if this loses lock, it shows up in the spectrum as an asymmetry of the line shape. Indeed, both the sources of error discussed above also show up as asymmetry of the line shape. Thus a symmetric line shape is a good indication that the measurement proceeded correctly. From the residual asymmetry, we estimate the systematic errors to be 20 kHz.
Measurements in the $5P_{3/2}$ state of $^{87}$Rb
-------------------------------------------------
The first set of measurements were done in $^{87}$Rb. The different values in the literature are consistent with each other, and have an accuracy of 10 kHz. Therefore, our main motivation was to see if the scanning-AOM technique worked well and our error budget was proper.
A typical spectrum with the probe locked to the $F=2
\rightarrow F'=3$ transition and pump scanning across the $F=2 \rightarrow F'=2$ transition is shown in Fig.\[rb22\]. We saw earlier that the line shape was well described by a Lorentzian. We therefore fit a Lorentzian curve to the spectrum and extract the peak center. Note the symmetry of the spectrum and the high signal-to-noise ratio. The $\{ 3-2 \}$ interval is twice the center frequency (because the AOM is double passed). For technical reasons, there is an additional AOM with a fixed frequency in the path of the probe, and this offset has to be added to obtain the interval.
The average value from 14 individual measurements is listed in Table \[t1\]. The standard deviation of the set is 32 kHz, which means the expected error in the mean is $32/\sqrt{14} = 8.6$ kHz, less than our estimated error of 20 kHz. This value is compared to other values reported in the literature. The two most accurate measurements [@DAN08; @YSJ96] have uncertainties below 10 kHz and have overlapping error bars. The more recent measurement [@DAN08] is also from our laboratory and used an AOM to measure the interval, but the AOM was locked to the peak. The current measurement obtained by measuring the entire spectrum is consistent with this value, thus giving confidence in the current technique. The only slightly discrepant measurement is from the work in Ref.[@BGR91], which is $1.7 \sigma$ away.
$\{ 3-2 \} $ Interval Reference
----------------------- -----------
266.653(20) This work
266.657(8) [@DAN08]
266.650(9) [@YSJ96]
266.503(84) [@BGR91]
: Comparison of measurements of hyperfine intervals in the $5P_{3/2}$ state of $^{87}$Rb to previous results. The last row is calculated from the $A$ and $B$ coefficients reported therein. All values in MHz.
\[t1\]
Measurements in the $5P_{3/2}$ state of $^{85}$Rb
-------------------------------------------------
With the reliability of the technique established with measurements in $^{87}$Rb, we turned to the other isotope, namely $^{85}$Rb. The probe was locked to the $F=3
\rightarrow F'=4$ transition, and the pump was scanned either across the $F=3 \rightarrow F'=3$ transition or the $F=3 \rightarrow F'=2$ transition. Typical spectra for the two cases are shown in Fig. \[rb33\]. As before, Lorentzian fits to the measured spectra were used to determine the peak center, and thus the $\{ 4-3 \}$ and $\{
3-2 \}$ intervals.
\(a) (b)
The average values for the two intervals are listed in Table \[t2\]. For the $\{ 4-3 \}$ interval, the standard deviation from a set of 14 measurements is 35 kHz. For the $\{ 3-2 \}$ interval, the standard deviation from 11 measurements is 34 kHz. These values are also compared to other values in the literature. There are two non-overlapping sets for the $\{ 4-3 \}$ interval. The value from Ref. [@BGR91] is $3.7 \sigma$ away from the other two values, which are both from our laboratory and both of which relied on AOM locking. Ref. [@BGR91] is also the work in which the value in $^{87}$Rb was discrepant by $1.7 \sigma$. The current measurement is consistent with our previous ones. All the values for the $\{ 3-2 \}$ interval are consistent with each other.
$\{ 4-3 \} $ Interval $\{ 3-2 \} $ Interval Reference
----------------------- ----------------------- -----------
120.958(20) 63.436(20) This work
120.966(8) 63.424(6) [@DAN08]
120.960(20) 63.420(31) [@RKN03]
120.506(124) 63.402(93) [@BGR91]
: Comparison of measurements of hyperfine intervals in the $5P_{3/2}$ state of $^{85}$Rb to previous results. The last row is calculated from the $A$ and $B$ coefficients reported therein. All values in MHz.
\[t2\]
Measurements in the $6P_{3/2}$ state of $^{133}$Cs
--------------------------------------------------
The next set of measurements was done on the $D_2$ line in $^{133}$Cs at 852 nm. For this, the probe was locked to the $F=4 \rightarrow F'=5$ transition and the pump was scanned either across the $F=4 \rightarrow F'=4$ transition or the $F=4 \rightarrow F'=3$ transition. Representative spectra for the two cases are shown in Fig. \[cs44\]. Lorentzian fits to the spectra yielded the line center and hence the $\{ 5-4 \}$ and $\{ 4-3 \}$ intervals.
\(a) (b)
The average value for the $\{ 5-4 \}$ interval is 251.031(20) MHz, as listed in Table \[t3\]. This was obtained from a set of 27 independent measurements with a standard deviation of 21 kHz. We concentrated on this interval because two of the values reported in the literature, 251.092(2) MHz from Ref. [@GDT03] and 251.000(20) from Ref. [@TAW88], differ by $4.5
\sigma$. The more recent measurement of the two [@GDT03] was done using a frequency comb. An earlier measurement from our laboratory using the AOM locking technique [@DAN05] yielded a result of 251.037(6) MHz, which was consistent with the earlier value at the $1.5
\sigma$ level, but totally inconsistent with the frequency-comb result (difference of $9\sigma$). Our current value vindicates our earlier result since it is consistent with the work in Ref. [@TAW88] but not with the frequency-comb result.
$\{ 5-4 \} $ Interval $\{ 4-3 \} $ Interval Reference
----------------------- ----------------------- -----------
251.031(20) 201.260(20) This work
251.037(6) 201.266(6) [@DAN05]
251.092(2) 201.287(1) [@GDT03]
251.000(20) 201.240(20) [@TAW88]
: Comparison of measurements of hyperfine intervals in the $6P_{3/2}$ state of $^{133}$Cs to previous results. All values in MHz.
\[t3\]
For the $\{ 4-3 \}$ interval, we obtain an average value of 201.260(20) MHz from a set of 10 measurements with a standard deviation of 33 kHz. The value from the work in Ref. [@TAW88] was 201.240(20) MHz, while the more recent frequency-comb work in Ref. [@GDT03] reported a value of 201.287(1) MHz. The inconsistency of $2.4 \sigma$ is smaller but still quite significant. Again, our previous result of 201.266(6) MHz obtained with AOM locking [@DAN05] overlapped with the earlier value but was inconsistent with the frequency-comb result. Our new value, though with a larger error bar, gives confidence in the previous measurement.
Conclusions
===========
In summary, we have shown that hyperfine spectroscopy with co-propagating beams in a vapor cell has several advantages over conventional saturated-absorption spectroscopy. In addition to the usual mechanisms responsible for probe transparency, there are EIT effects that enhance the peaks, which is supported by density-matrix calculations. As a result, the primary peaks are more prominent and appear with good signal-to-noise ratio. The transmitted signal appears on a flat background (Doppler-free) and does not have the problem of crossover resonances in between hyperfine transitions (which are stronger and often swamp the true peaks). Absorption by non-zero velocity groups causes additional peaks, but only one of them appears within the spectrum. These observations are again supported by density-matrix calculations taking the thermal velocity distribution into account. An important difference between transitions starting from the upper ground level and transitions starting from the lower ground level, is that the additional peak is almost negligible in the first case and quite prominent in the second case. This difference arises because of the difference in number of magnetic sublevels for the closed transition in each set.
We have adapted this technique to make measurements of hyperfine intervals by using one laser along with an AOM to produce the scanning pump beam. We measure intervals in the $D_2$ lines of Rb and Cs with 20 kHz precision. By measuring the entire spectrum and looking at the symmetry of the line shape, we avoid several potential sources of systematic error. The measurements are consistent with earlier results from our laboratory obtained by locking the AOM to the frequency difference, and show that our earlier error budget was reasonable.
This work was supported by the Department of Science and Technology, India. V.N. acknowledges support from the Homi Bhabha Fellowship Council and A.K.S. from the Council of Scientific and Industrial Research, India.
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---
abstract: 'We formulate a generic concept model for the deformation of a locally disordered, macroscopically homogeneous material which undergoes irreversible strain softening during plastic deformation. We investigate the influence of the degree of microstructural heterogeneity and disorder on the concomitant strain localization process (formation of a macroscopic shear band). It is shown that increased microstructural heterogeneity delays strain localization and leads to an increase of the plastic regime in the macroscopic stress-strain curves. The evolving strain localization patterns are characterized and compared to models of shear band formation published in the literature.'
author:
- Dániel Tüzes
- Michael Zaiser
- Péter Dusán Ispánovity
bibliography:
- 'references.bib'
title: 'Disorder is good for you: The influence of local disorder on strain localization and ductility of strain softening materials'
---
Introduction
============
Strain softening, loosely defined as a decrease of load carrying capability with increasing plastic deformation of a material, leads to strain localization (formation of shear bands) which in turn may lead to catastrophic failure of a material. If the width of the shear band is small as compared to the specimen dimensions, the macroscopic strain associated with the localized deformation may be small and failure occurs immediately after the material enters the softening regime. In materials where irreversible softening occurs shortly after yield, this may lead to a brittle appearance of the stress strain curves even though the failure mode is actually ductile. The most prominent example of this type of behavior are metallic glasses – a class of materials with potentially outstanding mechanical properties [@Ashby2006] but whose application is hindered by a propensity to fail shortly after yield by catastrophic shear band formation. The softening mechanism is in this case most likely associated with a shear-induced increase in free volume [@Steif1982] though thermal softening associated with localized, adiabatic heating has been discussed as an alternative explanation (see e.g. [@Wright2001]).
Metallic glasses are an obvious example of materials which exhibit local structural disorder – in this case down to the atomic scale. However, if one looks at defect microstructures, even crystalline solids exhibit (micro)structural disorder on scales well below the scale of a typical macroscopic specimen. On even larger scales, microstructural disorder is present in solid foams. In all these materials, one may legitimately ask how the macroscopic deformation behavior is influenced by the microstructural disorder and the associated length scales – which in the examples given may range from nanometers (for metallic glasses) up to millimetres for solid foams. For the case of transient softening, as observed in compression of metallic foams, it has been shown that increasing the microstructural heterogeneity may actually lead to a more homogeneous distribution of deformation on the macroscopic scale [@Zaiser2013]. In the present paper we consider a generic model which accounts for heterogeneity and randomness in the material microstructure and microstructure evolution, in conjunction with strain softening. The model builds upon the scalar plasticity model of Zaiser and Moretti [@Zaiser2005] which was originally introduced for single-slip deformation of crystals with disordered dislocation microstructure, but has recently been used by many authors to model the inception of shear bands in amorphous materials and the associated avalanche phenomena (see e.g. [@Talamali2012; @Budrikis2013; @Sandfeld2015; @Lin2015]). We generalize this model to explicitly introduce a strain softening mechanism. We first describe the model and then use it to study how the simulated deformation behavior depends on the degree of microstructural disorder (scatter of the distribution of local flow stresses). In particular we study the strain localization process and the concomitant stress strain curves, which demonstrate that increasing the disorder can delay strain localization and thus lead to a significant increase in macroscopic ductility.
The stochastic continuum plasticity model
=========================================
The model was originally formulated for single slip crystal plasticity. Accordingly, the plastic strain is characterized by a scalar shear strain variable $\gamma$. Plastic deformation is assumed to proceed in discrete, localized events which occur once the local stress in a volume element exceeds a threshold value. An elementary slip event at ${{{\mbox{\boldmath$r$}}}}$ creates a localized plastic Eigenstrain $\Delta{{{\mbox{\boldmath$\epsilon$}}}}^{\rm pl}({{{\mbox{\boldmath$r$}}}}) = \Delta\gamma^{\rm pl} {{{\mbox{\boldmath$M$}}}}\delta({{{\mbox{\boldmath$r$}}}})$ which we model as a point-like Eshelby inclusion [@Eshelby1957]. As a consequence of such an event, the internal stress field in the specimen volume changes. Increases in local stress in parts of the volume may trigger further localized events, leading to an avalanche which only terminates once the local stresses in all volume elements fall below the respective thresholds.
In the present work we consider a 2D system where an infinitely extended specimen mimicked by periodic boundary conditions is, by remote boundary displacements, subject to a pure shear stress in the $xy$ plane. Thus, the stress tensor $\mathbf{\sigma}$ has only one independent component $\tau \left( {{{{\mbox{\boldmath$r$}}}}} \right) := {\sigma _{xy}}\left( {{{{\mbox{\boldmath$r$}}}}} \right)$. As stated above we assume that also the plastic strain tensor has only one independent component, hence ${{{{\mbox{\boldmath$\epsilon$}}}}^{\rm pl}}({{{\mbox{\boldmath$r$}}}}) = \gamma^{\rm pl}\left( {{{{\mbox{\boldmath$r$}}}}} \right){{{\mbox{\boldmath$M$}}}}$ where $\gamma^{\rm pl}\left( {{{{\mbox{\boldmath$r$}}}}} \right)$ is the local plastic strain field and ${{{\mbox{\boldmath$M$}}}}= \left({{{{\mbox{\boldmath$e$}}}}_{y}} \otimes {{{{\mbox{\boldmath$e$}}}}_{x}} + {{{{\mbox{\boldmath$e$}}}}_{x}} \otimes {{{{\mbox{\boldmath$e$}}}}_{y}}\right)/2$. The stress acting on a volume element at ${{{\mbox{\boldmath$r$}}}}$ can then be evaluated as the sum of the external stress and the internal stress associated with the inhomogeneous plastic strain field, ${\tau ^{{\rm loc}}}\left( {{{{\mbox{\boldmath$r$}}}}} \right) = {\tau ^{\operatorname{int} }}\left( {{{{\mbox{\boldmath$r$}}}}} \right) + {\tau ^{{\rm ext}}}$. For an infinite body, the internal stress can be evaluated as the convolution of the plastic strain with an elastic Green’s function $G^E$, ${\tau ^{\operatorname{int} }}\left( {{{\mbox{\boldmath$r$}}}}\right) = \left( {{G^E} * {\gamma ^{{\text{pl}}}}} \right)\left( {{{\mbox{\boldmath$r$}}}}\right)$. Discrete plastic strain increments occur when the local stress reaches a local yield threshold $\tau^{\rm c}({{{\mbox{\boldmath$r$}}}})$, hence the elastic domain is defined by the inequality $$\label{eq:prop}
{\tau ^{{\text{th}}}}({{{\mbox{\boldmath$r$}}}},t) = {\tau ^{\text{c}}}({{{\mbox{\boldmath$r$}}}},t) - \left| {{\tau ^{{\text{ext}}}}(t) + \left( {{G^E} * {\gamma ^{{\text{pl}}}}(t)} \right)\left( {{{\mbox{\boldmath$r$}}}}\right)} \right| \geqslant 0.$$ The quantity $\tau^{\rm th}$ quantifies the distance of a given site from its yield stress. As long as this quantity has a positive value, the site behaves elastically. Before specifying the evolution of plastic strain which occurs once the inequality \[eq:prop\] is violated, and the concomitant rules for assigning and evolving the local yield threshold $\tau^{\rm c}({{{\mbox{\boldmath$r$}}}},t)$, we first need to specify the implementation of the model on a discrete lattice and explain the manner how stresses are evaluated.
Discretisation and stress evaluation
------------------------------------
The Eq. (\[eq:prop\]) is space-discretised on a square lattice of size $L \times L$ with periodic boundary conditions, where the edges of the square unit cell of size $d \times d$ are oriented along the $x$ and $y$ directions. To each cell we assign a single value of the local strain, the local yield stress and the local stress. Where it is not noted otherwise, distances are henceforth measured in the unit of $d$. $L$ is always an integer, and in this paper, a power of two: $L = {2^n}$.
The stress and strain fields generated by an elementary slip event are calculated as follows (Fig. \[fig:stress\_field\_calc\] illustrates the calculation.) The cell under deformation is cut along the $x$ and $y$ direction. The upper part is moved by a distance $b$ along the $x$ direction and the right side is moved by $b$ along the $y$ direction according to the sign of the shear stress acting on the cell. Then, the 4 parts are glued back together. Next, an elastic deformation is applied which transforms the cell back to its original shape so it fits its original place in the sample. The cell is placed back to its original position and the sample is elastically relaxed. The average plastic strain generated by this process in the cell is $\Delta \gamma^{\rm pl} = 2b/d$.
![\[fig:stress\_field\_calc\] In the elementary slip event, a cell is cut into 4 pieces which are displaced according to the acting shear stress and then glued back together. This cell is inserted back into the original lattice and forced elastically to fit, generating an internal stress field.](Figures/arrangement.pdf)
The process is equivalent to adding four edge dislocations[@Hirth1982] with the respective Burgers vectors $b{{{{\mbox{\boldmath$e$}}}}_{x}}$, $b{{{{\mbox{\boldmath$e$}}}}_{y}}$, $-b{{{{\mbox{\boldmath$e$}}}}_{x}}$, $-b{{{{\mbox{\boldmath$e$}}}}_{y}}$ at the centerpoints of the right, top, left and bottom sides of the cell. Accordingly, the stress field can be evaluated as the superposition of the stress fields of these four dislocations. Periodic boundary conditions are implemented by adding to the stress fields of the four dislocations those of their periodic images which form an infinite lattice of period $L$ (for details of the method used for evaluating the lattice sum, see [@Bako2006]). We evaluate stresses at the cell centerpoints, hence, the stress field induced by a elementary slip event $\Delta {\gamma ^{{\text{pl}}}}$ at the centerpoint of the active cell is $G_{0,0}^E\Delta {\gamma ^{{\text{pl}}}} = - 2\mu \Delta {\gamma ^{{\text{pl}}}}/\left[ {\pi \left( {1 - \nu } \right)} \right]$ where $\mu$ is the shear modulus and $\nu$ is Poisson’s ratio. The overall internal stress acting in an arbitrary cell $\left( {i,j} \right)$ is evaluated as $$\tau _{i,j}^{\operatorname{int}}(t) = \sum\limits_{k,l = 1}^{L} {G_{k - i,l - j}^E \gamma^{\rm pl}_{k,l}(t)},\quad
\gamma_{k,l}^{\rm pl}(t) = \sum_{t_i < t} \Delta \gamma^{\rm pl}_{k,l}(t_i).$$ Here $\gamma _{k,l}^{\rm pl}\left( t \right)$ is the plastic shear strain in the cell $\left( {k,l} \right)$ which is the sum of all local strain increments that have occurred in this cell up to time $t$. The kernel $G_{k,l}^E$ is the sum of the stress fields of four dislocations as detailed above, evaluated at the cell centrepoints. This method of evaluation of the internal stress field ensures that the average of the internal stress is zero, as required by stress equilibrium in an infinite body. Numerical values of the kernel $G_{k,l}^E$ in units of $\left| {G_{0,0}^E} \right|$ are shown in Fig. \[Fig:kernel\] for $L=32$.
![\[fig:stress\_field\] The stress field of a unit slip event located in the origin in the units of $\left| {G_{0,0}^E} \right|$, assuming periodic boundary conditions with $L = 32$. Note the symmetry in the $x$ and $y$ directions and the logarithmic color scale. In the upper right corner we show a magnification for $k \in \left[ {0,2} \right], l\in \left[ {0,2} \right]$.[]{data-label="Fig:kernel"}](Figures/periodic_dipole_stress_field32x32symm_bin_dat_tilized_multiplot2.pdf)
The external stress is controlled by remote displacements acting on the system which impose a total (elastic and plastic) shear strain $\gamma^{\rm tot}$. Since the average of all internal stresses is by construction zero, stress equilibrium requires that $$\label{eq:tauext}
\tau^{\operatorname{ext}} = \mu \left(\gamma^{\rm tot} - \gamma^{\rm pl} \right),\quad
\gamma^{\rm pl} = \frac{1}{L^2} \sum\limits_{k,l = 1}^{L} \gamma^{\rm pl}_{k,l} .$$
Stochastic flow rule
--------------------
We consider a quantized, discrete plasticity model where plastic strain increases locally in discrete, finite amounts whenever the inequality, Eq. (\[eq:prop\]) is locally violated. If this is the case for any site $\left( {k,l} \right)$, we increase the plastic strain $\gamma_{k,l}^{\rm pl}$ at this site instantaneously by $$\label{eq:plastic_unit}
\Delta \gamma_{k,l}^{\rm pl} = \min \left( {\Delta\gamma_0,\Delta \gamma_{k,l}^* } \right), \quad
\Delta \gamma_{k,l}^* = \Delta\gamma_0 \frac{\tau _{k,l}^{\operatorname{int}} + \tau ^{{\rm ext}}}{\left| {G_{0,0}^E} \right|}.$$ This means that the strain in increased by a value that sets the local stress to zero if this value is less than $\Delta\gamma_0$, or otherwise by $\Delta\gamma_0$.
Local structural disorder is taken into account in terms of random variations of the local flow threshold ${\tau ^c}({{{\mbox{\boldmath$r$}}}})$. We assume that the system is statistically homogeneous and that the size of a cell is larger than the spatial correlation range of the microstructural disorder that gives rise to local flow stress variations. Hence, the local flow thresholds are considered as independent, identically distributed random variables ${\tau ^c}_{k,l}$ which we take to be Weibull distributed with exponent $\beta$ and mean value $\tau^{c}_0$. Independent values of ${\tau ^c}_{k,l}$ are initially assigned to all sites. Plasticity-induced changes in the local flow threshold are taken into account by assigning, after each local strain increment occurring at a cell, to this cell a new local flow threshold. Specifically, we draw a new value from the same distribution with average $\tau^{c}_0$ and multiply this with a strain dependent factor $F(\gamma_{k,l}^{\rm pl}) = 1 - f \gamma_{k,l}^{\rm pl}$ where $f < 0$ (called softening parameter), thus implementing a linear strain softening.
Simulation protocol
-------------------
We non-dimensionalize the model by measuring all stresses in units of the mean flow threshold $\tau^c_0$, all strains in units of $\tau^c_0/\mu$ (elastic strain needed to reach the mean flow threshold, divided by the shear modulus), and spatial coordinates in units of the cell size $d$. The model behaviour is then, in addition to the Weibull parameter $\beta$, controlled by a single numerical parameter $I = 2 \mu \Delta\gamma_0/[\pi(1-\nu)\tau^c_0]$ (henceforth: ‘coupling constant’) which controls the magnitude of the internal-stress re-distribution after a deformation event relative to the average flow stress. In the following we make the simplifying assumption that $\nu = 0.353$ in which case $I=\mu \Delta \gamma_0/\tau^c_0$ equals the scaled local strain increment. The local stress reduction at the site of a unit deformation event is then $I$ and the external stress reduction associated with the same event is $I/L^2$.
Simulations are performed as follows: We assign initial flow thresholds to all sites according to the prescribed Weibull distribution with exponent $\beta$ and mean 1. We then determine the site with the lowest threshold and increase the total strain $\gamma^{\rm tot}$ such that the concomitant stress increase as given by Eq. (\[eq:tauext\]) exactly matches the threshold, triggering the first deformation event. After the event, which is supposed to occur instantaneously, we re-compute all stresses while keeping $\gamma^{\rm tot}$ fixed, evaluate the local threshold stresses $\tau^{\rm th}_{k,l}$ for all sites, and check whether there are additional sites which become unstable ($\tau^{\rm th}_{k,l} < 0$). If yes we increase, still at fixed $\gamma^{\rm tot}$, the strain at the unstable site with the lowest value of $\tau^{\rm th}_{k,l}$, thus implementing an extremal dynamics. We repeat this until there are no more unstable sites (the avalanche has terminated). The plastic strain and the stress at this point are evaluated from Eq. (\[eq:tauext\]). We then determine again the site with the smallest threshold, increase $\gamma^{\rm tot}$ such that the concomitant stress increase as given by Eq. (\[eq:tauext\]) makes this site unstable and triggers the next avalanche. We repeat this cycle of avalanche triggerings until the local strain of at least one site reaches the value $\gamma_{k,l}^{\rm pl} = 1/f$ such that the strength of this site becomes zero. This is tantamount to the nucleation of a microcrack which we take as a signature of impending system failure. The concomitant average plastic strain defines the system failure strain $\gamma^{\rm pl}_{\rm f}$.
Results
=======
Simulations were performed for Weibull shape parameters $\beta = 1, 2, 4, \text{ and } 8$, coupling constants $I = 0.125,0.25,0.5 \text{, and } 1$, and for system sizes $L = 32, 64, 128, 256 \text{ and } 512$. In each case ensembles of 512 simulations with statistically independent initial conditions were performed. The softening parameter $f$ was kept fixed at $f = 1/16$.
Stress-strain curves
--------------------
Average stress-strain curves were obtained by averaging the external stress at a given deformation over the simulations as shown in Fig. \[fig:ssc\].
![\[fig:ssc\] Stress-total strain curves for two different yield-stress distributions (Weibull exponents $\beta= 1$ and $\beta=4$) and different system sizes; other parameters $I=1; f=1/16$.](Figures/linS_1_4_1000-1512_s_tsc_DsG.pdf)
The curves exhibit three different regimes: An initial quasi-elastic loading regime is followed by a transition to a plastic deformation regime where the stress increases with strain (hardening regime). The elastic and hardening regimes are system size independent. The hardening regime is followed by a transition to a softening part where the stress decreases with macroscopic strain. The simulations are terminated once microcrack nucleation occurs as indicated by a complete loss of strength at one or more sites. The corresponding failure strains are much below the expectation $\gamma_f = 16$ for a homogeneous system, indicating a significant degree of deformation localization. We also observe that the softening regime is system size dependent: The stress decrease occurs more rapidly and failure occurs at lower strains in larger systems. Such system size dependence again indicates some kind of deformation localization. We therefore proceed to investigate the strain patterns that emerge in the different deformation stages.
Patterns in the strain maps
---------------------------
Figure \[fig:pattern\] illustrates the changes in the strain patterns that occur during the softening regime. At the peak stress before the onset of softening, deformation is macroscopically homogeneous but exhibits mesoscale patterns in the form of numerous diffuse shear bands which follow the planes of maximum shear stress, here aligned with the $x$ and $y$ directions. These patterns are more pronounced with increasing degree of disorder. Note that the peak stress is reached later in the more disordered sample (top left graph in Fig. \[fig:pattern\]), hence the overall strain is bigger. During the softening regime we observe a qualitative change in the patterns as most of the additional strain accruing during the softening regime is localized in a single shear band which also contains the location where microcrack nucleation takes place. This shear band is sharper and more pronounced in the sample with less disorder (bottom right graph in Fig. \[fig:pattern\]).
![\[fig:pattern\] Strain patters at the highest external stress just before the onset of softening (left) and at the end of the simulation (right); $\beta=1$ (top) and $\beta=4$ (bottom); other parameters: $I = 1, f=1/16, L=256$.](Figures/loc-strain-maps-1-4-highest-stress-and-strain_mod.png)
The formation of a localized shear band is in line with the ideas of classical continuum mechanics which predicts localization to occur, in a system without boundary constraints and under pure shear loading, at the transition from strain hardening to strain softening regimes. To better characterize this behavior we now seek to define a quantitative measure for the strain localization process.
Deformation localisation
------------------------
In order to quantify strain localisation we investigate the spatial distribution of the incremental strain. We divide the average stress-strain curve into $n=50$ intervals, the $k\text{th}$ interval is defined by ${\gamma ^{{\text{pl}}}} \in \left[ {{\gamma ^{{\text{pl}},k}},{\gamma ^{{\text{pl}},k + 1}}} \right),\quad {\gamma ^{{\text{pl}},k}} = k\left\langle {\gamma _{\text{f}}^{{\text{pl}}}} \right\rangle /n$. The plastic strain increment occurring at the site $(i,j)$ during strain interval $k$ is denoted as $\Delta \gamma _{i,j}^{{\rm pl},k}$.
We now use that a shear band has a planar shape. For any given plane $\cal P$ we can define a scalar measure of distance which characterizes the distribution of the incremental strain with respect to the plane. To this end we denote the distance between site $(i,j)$ and the plane $\cal P$ as $d_{i,j}^{\cal P,\perp}$. (Because of the periodic boundary conditions used, we evaluate $d_{i,j}^{\cal P,\perp}$ as the minimum distance between the site $(i,j)$ and any of the periodic images of $\cal P$). We now define $$d_{\gamma,k}^{\cal P} = \frac{\sum_{ij} \Delta \gamma _{i,j}^{{\rm pl},k} \cdot d_{i,j}^{\cal P \perp}}{\sum_{ij} \Delta \gamma _{i,j}^{{\rm pl},k}}.$$ For a completely homogeneous distribution of the plastic strain increment, we have $d_{\gamma,k}^{\cal P} = L/4$ for all planes $\cal P$. For a heterogeneous distribution we identify the plane for which $d_{\gamma,k}^{\cal P}$ has the smallest value and define a localization parameter $\eta$ as $$\eta_k = 1- \frac{4 d_{\rm min,k}}{L} \quad,\quad d_{\rm min,k} = \min_{\cal P} d_{\gamma,k}^{\cal P}$$ This parameter takes the value $\eta_k = 0$ for a statistically homogeneous distribution of the plastic strain increment, and the value $\eta_k = 1$ if the incremental strain is completely localized on a single plane.
![\[fig:radius\] Stress strain curves and strain evolution of the localization parameter $\eta$ for different degrees of disorder (Weibull parameter $\beta \in [8,4,2,1]$. The upper figure corresponds to $I=0.125$, the lower to $I=1$.](Figures/radius_and_ssc_256_DG_125.pdf "fig:") ![\[fig:radius\] Stress strain curves and strain evolution of the localization parameter $\eta$ for different degrees of disorder (Weibull parameter $\beta \in [8,4,2,1]$. The upper figure corresponds to $I=0.125$, the lower to $I=1$.](Figures/radius_and_ssc_256_DG1.pdf "fig:")
It can be seen in Fig \[fig:radius\] that in all simulations the localization parameter $\eta$ starts at $\eta = 0$ and then gradually increases during the hardening regime. Immediately after the peak stress is reached and the system enters the macroscopic softening regime, $\eta$ increases rapidly towards $\eta = 1$, indicating the localization of deformation in a single shear band. The comparison of the strain evolution of $\eta$ and $\tau^{\rm ext}$ clearly demonstrates the correlation. It is equally evident that an increasing degree of disorder (decrease of the Weibull exponent from $\beta = 8$ to $\beta = 1$), even though it leads to an earlier onset of plastic flow, extends the hardening regime to larger strains and delays the onset of deformation localization. The role of the coupling constant $I$, which reflects the magnitude of the local strain increment, is more ambiguous: for small disorder, large values of $I$ seen to promote localization whereas for large disorder, the opposite is the case.
![\[fig:shearband\] Evolution of the distribution of strain around the final failure plane for Weibull parameters $\beta = 1$ (top) and $\beta=8$ (bottom).](Figures/shearbandsW1.pdf "fig:") ![\[fig:shearband\] Evolution of the distribution of strain around the final failure plane for Weibull parameters $\beta = 1$ (top) and $\beta=8$ (bottom).](Figures/shearbandsW8.pdf "fig:")
We now look at the distribution of incremental strain around the [*final*]{} failure plane. This is shown in Fig. \[fig:shearband\] which depicts shear band profiles for large disorder ($\beta = 1$) and for small disorder ($\beta = 8$), recorded for different values of the localization parameter $\eta$. The width of the shear band is almost the same in both cases, however, localization of deformation around the final failure plane happens later in case of large disorder. This looks strange at first glance, given that the curves compare situations with equal value of $\eta$, however, the reason is simple: In case of large disorder, deformation first localizes in a transient manner (i.e., on slip planes that are in general [*not*]{} close to the final failure plane) and localization on the final failure plane happens after extensive deformation activity has occurred elsewhere. In case of small disorder, by constrast, deformation localizes on the final failure plane almost from the onset.
Mean strain to failure
----------------------
The beneficial effect of disorder on ductility is also borne out when we consider the mean strain at failure.
. \[fig:strain\_to\_failure\_size\]
This strain is system size dependent and decreases with increasing system size. This dependency can be rationalized by making the simplifying assumption that the system deforms homogeneously during the hardening regime, accumulating a homogeneous strain $\gamma_h$, whereas all strain accruing during the subsequent softening regime is localized in a slip band of finite width $d_{\beta}$. Failure occurs once the plastic strain in the band reaches the value $\gamma_{\rm f}^{\rm loc}$. Then the mean strain at failure is $\gamma_f = \gamma_h + (\gamma_{\rm f}^{\rm loc} - \gamma_h)(d/L)$. We can thus fit the system size dependence as $\gamma_f = c_1 + c_2/L$ where the parameter $c_1$ defines the homogeneous strain $\gamma_h$ which is also the failure strain in the infinite system limit. This strain is plotted in Fig. \[fig:strain\_to\_failure\_shape\] as a function of the Weibull exponent $\beta$. Again we see that larger microstructural disorder leads to an increase in ductility of the strain softening material. For large systems $(L \to \infty)$ the increase is quite dramatic - between Weibull exponent $\beta =8$, corresponding to a coefficient of variation of 0.148, and Weibull exponent $\beta =1$ (coefficient of variation 1), the strain to failure increases in this limit by a factor of about 60.
![Mean strain at failure as a function of the Weibull shape parameter $\beta$, for different system sizes. The value for infinitely large system size is obtained from the parameter $c_1$ of the fit curves in Figure \[fig:strain\_to\_failure\_size\].[]{data-label="fig:strain_to_failure_shape"}](Figures/strain_to_failure_vs_shape.pdf)
A local criterion for shear band growth
---------------------------------------
Failure of a macroscopic system occurs once the first macroscopic (system spanning) shear band forms and deformation localizes there. However, embryonic shear bands are present already before the onset of softening (Fig. \[fig:pattern\]). At this point we ask whether we can establish a criterion which allows us to better understand the conditions for the emergence of a macroscopic shear band. To this end we assume a pre-existing shear band of some extension and investigate its growth. The stress concentration at the tip of a straight band with a width of one cell can be estimated as $$\tau_{\rm tip} = \sum_{k=1}^{\infty} G^E_{k,0} \approx 0.385 G^E_{0,0}$$ The band expands if this stress concentration triggers a deformation event at one of the sites ahead of either tip. We denote the stress needed to activate a site with $\Delta\tau$ (we might also call this the residual strength of the site), and the probability that a randomly chosen site is activated by a stress increment $\Delta \tau < \tau^*$ as $P(\Delta \tau < \tau^*)$. We now investigate the evolution of the probability $P(\Delta \tau < \tau_{\rm tip})$ (i.e, the probability that an advance of a band triggers another advance straight ahead) as a function of strain. Figure \[fig:trigger\] indicates that this probability increases continually with increasing strain until it reaches a level of about $P(\Delta \tau < \tau_{\rm tip}) \approx 0.3$ which does not depend on the model parameters (disorder parameter $\beta$, coupling constant $I$). At this critical value, $P(\Delta \tau < \tau_{\rm tip})$ suddenly drops. This drop coincides with the formation of a system spanning shear band where deformation localizes: the associated stress drop reduces the value of $P(\Delta \tau < \tau_{\rm tip})$ everywhere except in the b and itself where the local strain softening maintains it at the critical level for sustained shear band operation.
![\[fig:trigger\] Evolution of the triggering probability $P(\Delta \tau < \tau_{\rm tip})$ at the tip of an incipient shear band. (a) $I=0.125$, (b) $I=0.25$, (c) $I=0.5$ (d) $I=1$.](Figures/fig_0411_DG0125.pdf)
(0,0) (-205,144)
![\[fig:trigger\] Evolution of the triggering probability $P(\Delta \tau < \tau_{\rm tip})$ at the tip of an incipient shear band. (a) $I=0.125$, (b) $I=0.25$, (c) $I=0.5$ (d) $I=1$.](Figures/fig_0411_DG025.pdf)
(0,0) (-205,144)
![\[fig:trigger\] Evolution of the triggering probability $P(\Delta \tau < \tau_{\rm tip})$ at the tip of an incipient shear band. (a) $I=0.125$, (b) $I=0.25$, (c) $I=0.5$ (d) $I=1$.](Figures/fig_0411_DG05.pdf)
(0,0) (-205,144)
![\[fig:trigger\] Evolution of the triggering probability $P(\Delta \tau < \tau_{\rm tip})$ at the tip of an incipient shear band. (a) $I=0.125$, (b) $I=0.25$, (c) $I=0.5$ (d) $I=1$.](Figures/fig_0411_DG1.pdf)
(0,0) (-205,144)
Despite significant variations in the distributions $P(\Delta \tau < \tau^*)$ which depend strongly on the disorder parameter $\beta$, we observe that critical value of $P(\Delta \tau < \tau_{\rm tip})$ is quite universal. This leads us to the conclusion that shear band growth is driven by processes occurring at the tip of an incipient shear band, and that it depends on the interaction of the shear band tip with the local distribution of the residual strength whether or not shear band growth can occur. Indeed, if we account for the fact that shear bands may grow at both tips, and that growth may not necessarily be constrained to expansion straight ahead but may occur via sideways deflection (thus, at each tip there may be three sites available for continuing growth), we can estimate that a critical probability of the order of 1/3 for triggering a site at the crack tip may be sufficient for sustained growth of a shear band.
Thus our analysis leads us to envisaging shear band formation as essentially a two-stage process: In a first stage, local yielding and the concomitant stress re-distribution lead to a system-wide re-shuffling of the residual strength distribution in such a manner that the triggering probability $P(\Delta \tau < \tau_{\rm tip})$ at the tip of an incipient shear band gradually increases. During this stage, deformation is macroscopically homogeneous, even though shear band nuclei are continuously forming and getting again inactivated. The duration of this latency stage depends on the degree of disorder and increases with increasing disorder parameter $\beta$. As soon as the triggering probability reaches a critical value $P(\Delta \tau < \tau_{\rm tip}) \approx 0.3$, a transition to a second stage occurs where the flow process is governed by the rapid formation of a system-spanning shear band where deformation localizes. In large systems, this leads to near-instantaneous failure.
It may be noted in passing that the residual strength probability distribution $P(\Delta \tau < \tau^*)$, notably its behavior near the edge of stability, $\Delta \tau \to 0$, has recently been conjectured by Wyart and co-workers [@Lin2014; @Lin2015] to play a crucial role in the dynamics of similar models as the present one but [*without*]{} softening. In these works it is argued that the spatial organization of slip in shear bands is largely irrelevant during the approach to failure, and that the non-local elastic kernel can be approximated by a random stress re-distribution in a mean-field model which by construction destroys any spatial correlations. Our observations, by contrast, point to the importance of correlated shear band growth in controlling system stability as soon as some degree of softening is introduced into the model.
Discussion and Conclusions
==========================
We studied the deformation and failure behavior of microstructurally disordered model materials which exhibit irreversible strain softening and therefore fail by shear band formation. Contrary to the intuitive idea that increased microstructural heterogeneity may facilitate shear band nucleation and therefore have a negative impact on deformability, we find a strong positive effect of increased heterogenity and randomness on the deformation properties. Increased microstructural heterogeneity indeed leads to an earlier onset of deformation in the form of diffuse shear bands – an effect that is easily understood within the classical paradigm of weakest-link statistics (for a discussion in the plasticity context see e.g. [@Ispanovity2013]. However, the same heterogeneity prevents the spreading of shear bands and their coalescence into a system spanning macro-shear-band. The earlier onset of deformation is matched by an extended hardening regime, associated with the elimination of weak regions from the microstructure. This hardening can be envisaged as a survival-bias-hardening (easily deformable configurations are eliminated, stronger configurations survive) and becomes more pronounced with increasing scatter in local strength. Only after the survival-bias-hardening is exhausted, structural softening takes over and promotes macroscopic deformation localization. In line with classical concepts of continuum mechanics of homogeneous materials, the onset of macroscopic localization neatly coincides with the peak stress where the system enters the softening regime of the stress strain curve.
Our findings indicate that, in microstructurally disordered materials where ductility is limited by shear band formation, it may be a good idea to [*increase*]{} the degree of microstructural heterogeneity - an increase which results both in an increase in strength and in a very significant increase in ductility. In the context of metallic glasses, our findings match well with ideas to increase the ductility of metallic glasses by introducing a second interface phase [@Adibi2013] or by embedding nano-crystallites into a glassy matrix [@Das2005] – ideas which are tantamount to increasing the scatter of local deformation properties within a generally disordered microstructure.
Of course, it is a well established idea that strong-yet-ductile materials can be engineered by combining weak-but-ductile and strong-but-brittle components into heterogeneous composite microstructures. However, this is not what we are studying in the present work: As manifest from the constitutive relation of our model material, volume elements of different strength are assumed to fail at the same local strain. If we investigate the evolution of the final, macroscopic shear band, then we can see little difference between weakly disordered and strongly disordered microstructures (Fig. \[fig:shearband\]). Nevertheless, the overall deformation behavior is radically different in both cases, because the disorder extends the hardening regime and prevents local shear band nuclei from coalescing into a macroscopic shear band. To understand this behavior it is necessary to go beyond the investigation of averaged, effective materials properties and move towards an understanding of the manner how fluctuations emerge and extend across scales. This viewpoint is corroborated by investigations of models similar to the present one which demonstrate the emergence of scale-free, system spanning correlations in the internal stress and local strain patterns [@Zaiser2006; @Kapetanou2015]. As a consequence of such correlations, the emergent macroscopic materials behavior can neither be inferred from local statistics (e.g. using weakest-link arguments) nor can it be related to the properties of small, circumscribed representative volume element. Thus, novel conceptual tools may be needed to exploit the possibilities created for improving materials performance by exploiting the manner in which local fluctuations in materials properties may not only influence, but qualitatively change the macroscopic behavior of materials.
Acknowledgements {#acknowledgements .unnumbered}
================
Financial support of the Hungarian Scientific Research Fund (OTKA) under contract number PD-105256 and of the European Commission under grant agreement No. CIG-321842 are also acknowledged. PDI is also supported by the János Bolyai Scholarship of the Hungarian Academy of Sciences.
|
---
author:
- 'German Ros\*, Jose M. Álvarez, and Julio Guerrero[^1][^2]'
bibliography:
- 'sections/paperbib.bib'
title: Motion Estimation via Robust Decomposition with Constrained Rank
---
[Ros et.al : Motion Estimation via Robust Decomposition with Constrained Rank]{}
[German Ros]{} received his B.Sc (Hons.) in computer science from Universidad de Murcia (Spain) in 2010 and his M.Sc degree from Kingston University of London (UK) in 2011. In 2012 he obtained a second M.Sc degree from Universitat Autònoma de Barcelona, where he is currently pursuing a Ph.D. His research interests span computer vision, robotics and visual perception, with special interest in manifold optimization, robust estimation and visual geometry. He is a student member of the IEEE.
[Jose M. Álvarez]{} is currently a senior researcher at NICTA and a research fellow at the Australian National University. Previously, he was a postdoctoral researcher at the Computational and Biological Learning Group at New York University. During his Ph.D. he was a visiting researcher at the University of Amsterdam and VolksWagen research. His main research interests include road detection, color, photometric invariance, machine learning, and fusion of classifiers. He is associate editor for the IEEE Trans. on Intelligent Transportation Systems and member of the IEEE.
[Julio Guerrero]{} is an Associate Professor in the Department of Applied Maths at Universidad de Murcia (Spain) since 2000. Previously he was a postdoctoral researcher at Naples University, and visiting researcher at Durhan University (UK) and Syracure Univerity (USA). His research interests include, among others, applied maths and computer vision, with special interest in manifold optimization and robust estimation.
[^1]: G. Ros is with the Computer Vision Center at Universitat Autònoma de Barcelona, Spain, e-mail: gros@cvc.uab.es .
[^2]: J. Guerrero is with the Department of Applied Mathematics at Universidad de Murcia, Spain, e-mail: juguerre@um.es .
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---
abstract: 'The Li abundances of the two components of the very metal-poor (\[Fe/H\]$=-2.5$) double-lined spectroscopic binary G 166–45 (BD+26$^{\circ}$2606) are determined separately based on high resolution spectra obtained with the Subaru Telescope High Dispersion Spectrograph and its image slicer. From the photometric colors and the mass ratio the effective temperatures of the primary and secondary components are estimated to be 6350$\pm 100$ K and 5830$\pm 170$ K, respectively. The Li abundance of the primary ($A$(Li)$=2.23$) agrees well with the Spite plateau value, while that of the secondary is slightly lower ($A$(Li)$=2.11$). Such a discrepancy of the Li abundances between the two components is previously found in the extremely metal-poor, double-lined spectroscopic binary CS 22876–032, however, the discrepancy in G 166–45 is much smaller. The results agree with the trends found for Li abundance as a function of effective temperature (and of stellar mass) of main-sequence stars with $-3.0<$\[Fe/H\]$<-2.0$, suggesting that the depletion of Li at [[$T_{\rm eff}$]{}]{}$\sim 5800$ K is not particularly large in this metallicity range. The significant Li depletion found in CS 22876–032B is a phenomenon only found in the lowest metallicity range (\[Fe/H\]$<-3$).'
author:
- Wako Aoki
- Hiroko Ito
- Akito Tajitsu
title: 'Examination of the mass-dependent Li depletion hypothesis by the Li abundances of the very metal-poor double-lined spectroscopic binary G 166–45'
---
Introduction
============
Lithium is an important element as a diagnostic of the structure and evolution of low-mass stars. Since Li is easily destroyed by nuclear fusion in the hot interiors of stars where the temperature is above 2.5 $\times 10^{6}$ K, stellar internal processes can be examined from observed Li abundances at stellar surface. Li depletion processes also play an important role in reconciling the serious discrepancy between the observed constant Li abundance, the so-called Spite plateau, of metal-poor turnoff stars ($A$(Li) = $\log$(Li/H)+12 $\sim$ 2.2: e.g., @spite82 [@asplund06]) and the primordial Li abundance predicted from standard Big Bang nucleosynthesis (SBBN) models adopting the baryon density determined from observations of the cosmic microwave background (CMB) by the WMAP satellite ($A$(Li) $= 2.72$: e.g. Cyburt et al. 2008). The discrepancy seems to be larger at lower metallicity [@bonifacio07; @aoki09; @sbordone10]. If the stellar intrinsic Li depletion is in operation in the atmospheres of metal-poor turnoff stars, the observed Li abundance is lower than the initial value, which may have been closer to the CMB and SBBN prediction.
Although no Li depletion is inferred for metal-poor stars on the main-sequence-turn-off by standard stellar evolution theories because of their shallow convection zones, evolutionary models that include effects of atomic diffusion [e.g., @richard02a; @richard02b] predict significant decline of surface Li abundances for metal-poor dwarfs. @richard05 additionally introduced parametrized effects of turbulent mixing to their evolution model for \[Fe/H\] $= -2.31$ [^1], though the understanding of the physics of the turbulent mixing is still insufficient. @korn06 and @korn07 investigated effects of these processes for the globular cluster NGC 6397 by examining small variations of Mg, Ca, Ti and Fe abundances in main-sequence turnoff, subgiant and giant stars. They suggested diffusion process during main-sequence phase and attributed the slightly lower Li abundances in turn-off stars than in subgiants to these effects.
The most convincing evidence of stellar Li depletion in metal-poor dwarfs found so far is the large difference between Li abundances of the primary and secondary dwarf stars in the extremely metal-poor (\[Fe/H\]$ = -3.6$) double-lined spectroscopic binary CS 22876–032, which is about 0.4 dex [@gonzalezhernandez08]. Since the two stars should have shared the same chemical compositions at their birth, it is reasonable to presume that depletion processes have made the Li abundance difference during the stellar evolution. Such depletion is not expected for a star with the effective temperature of the secondary ([[$T_{\rm eff}$]{}]{}$=5900$ K) from previous studies of less metal-poor stars. Given that the mass of the secondary star is about 0.9 times that of the primary, Li depletion may operate mass-dependently, which is consistent with the predictions of @richard05, as discussed by @melendez10.
The best approach to examine the hypothesis of the mass-dependent lithium depletion is accurate determinations of Li abundances for several metal-poor binary stars. Although @melendez10 suggested a correlation between Li abundance and stellar mass for metal-poor single stars, investigation of single stars which have different ages and distances would suffer from uncertainties in mass estimates.
In this Letter, we report our chemical abundance analysis, in particular the Li abundances, of the two components of the very metal-poor, double-lined spectroscopic binary G 166–45. This object is selected from the sample of @goldberg02 who investigated the orbital parameters of 34 binary stars found by the Carney-Latham proper motion sample [@carney94]. The chemical abundances of individual components of this object have not yet been investigated. We discuss in particular the implication of the small difference of the Li abundance between the two components found from the present analysis of high resolution spectra.
Observations and measurements
=============================
A high-resolution spectrum of G 166–45 was obtained with the Subaru Telescope High Dispersion Spectrograph [HDS; @noguchi02] on July 22, 2011. We applied the image slicer recently installed in the spectrograph, which provides the resolving power of $R=110,000$ with high efficiency [@tajitsu12]. The wavelength range from 5100 to 7800 [Å]{} is covered by the standard setup of StdRa. The exposure time is 60 minutes, resulting in the S/N ratio (per 0.9 [[ km s$^{-1}$]{}]{} pixel) of 350 at 6700 [Å]{}.
According to the orbital parameters of the binary determined by @goldberg02, the velocity difference between the two components of this object is expected to be 15–20 [[ km s$^{-1}$]{}]{} at this epoch. Indeed, the separation of the absorption features in our spectrum is 18.5 [[ km s$^{-1}$]{}]{} (see below), agreeing well with the above estimate.
Standard data reduction was carried out with the IRAF echelle package[^2]. Spectra are extracted from individual sliced images and combined for the slice numbers 2–5 in the definition of @tajitsu12 to obtain the best quality spectrum.
Equivalent widths ($W$’s) for isolated absorption lines were measured by fitting Gaussian profiles for each component of the binary. The equivalent widths are used in the following chemical abundance analysis by dividing the fraction of the contribution of each component to the continuum flux estimated from colors and mass ratio of the two components (§ \[sec:ana\]).
Analysis and results {#sec:ana}
====================
Atmospheric Parameters
----------------------
In order to determine the atmospheric parameters of the two components of the binary system and their contributions to the continuum flux, we adopt the mass ratio of the two components ($q=m_{2}/m_{1}=0.89\pm
0.04$), where $m_{1}$ and $m_{2}$ are the primary and secondary masses, respectively, determined by @goldberg02, and calculate the colors ($B-V$, $V-R$, and $V-I$) as a function of the primary mass using the $Y^{2}$ isochrone [@demarque04] for the age of 12 Gyr. The metallicity is assumed to be \[Fe/H\]$=-2.5$, which is consistent with the results obtained from the following analysis.
The upper panel of Figure \[fig:param\] shows the three colors calculated using the isochrone. Solid and dotted lines indicate the cases of $q=0.89$ and those with 0.04 higher or lower, respectively. Very accurate colors of G 166–45 are available, because this object is a Landolt’s photometric standard star. The colors measured by their recent work [@landolt07] with a reddening correction are shown by the horizontal lines in the figure. The reddening correction ($E(B-V)=0.02$) is estimated from the interstellar D$_{2}$ line detected in our spectrum (the equivalent width of 80 m[Å]{}).
Comparisons of these colors with the calculations provide constraints on the primary mass of this binary system (as well as the secondary mass). The primary mass estimated from the three colors is $0.757\pm
0.007$M$_{\odot}$, in which the error corresponds to the differences of the results from the three colors. Even if the photometry error (0.015 mag) for G 166–45 estimated by @landolt07 (see the erratum of the paper) and the uncertainty of the mass ratio are included, the error size of the primary mass does not change significantly. The other solution that satisfies the colors and the mass ratio is the case that the primary is a subgiant ($m_{1}\sim
0.82~$M$_{\odot}$) and the secondary is a main-sequence star with similar temperatures (see Fig. \[fig:param\]). This solution is, however, excluded because the luminosity difference of the two components can not be so large, given the strengths of the absorption features of the two components.
The lower panel of Figure \[fig:param\] shows the effective temperatures of the primary and secondary, as a function of the primary mass (assuming the mass ratio), adopted from the $Y^{2}$ isochrone. The effective temperature of the primary is well determined to be 6350$\pm 100$ K. The error reflects the uncertainty of the mass determination from different color indices and photometric errors, but does not include the uncertainty of the temperature scale (depending on model atmospheres).
By contrast, the uncertainty of the effective temperature of the secondary is relatively large (170 K) due to the uncertainty of the mass ratio. We adopt 5830 K for the secondary as the best estimate in the following analysis, but also conduct the analyses for [[$T_{\rm eff}$]{}]{} = 5650 and 6000 K to estimate the errors due to the uncertainty of the mass ratio. The surface gravities of the primary and the secondary are estimated from the isochrone to be $\log g=4.4$ and 4.6, respectively.
The fraction of the contributions of each component to the continuum flux in the $R$-band ($f$: $f_{\rm A}$ and $f_{\rm B}$ for the primary and secondary, respectively), in which the resonance line exists, is also estimated from the isochrone for the three choices of the secondary’s effective temperature.
The adopted parameter sets ([[$T_{\rm eff}$]{}]{}, [[$\log g$]{}]{}, and $f_{\rm B}$) are (5830 K, 4.6, 0.27), (5650 K, 4.6, 0.21) and (6000 K, 4.6, 0.34) for the secondary, while [[$T_{\rm eff}$]{}]{} = 6350 K, [[$\log g$]{}]{} = 4.4, and $f_{\rm A}=1-f_{\rm B}$ are adopted for the primary (Table 1).
For comparison purposes, we applied the above procedure to the double-lined spectroscopic binary CS 22876–032 adopting the mass ratio of 0.91 [@gonzalezhernandez08] and colors given by @norris00. The results agree well with those determined by @gonzalezhernandez08, i.e., 6500 K and 5900 K for the primary and secondary, respectively, with uncertainties of about 100 K.
Chemical abundances
-------------------
Chemical compositions of the two components of this binary system are determined separately based on the standard LTE analysis of the equivalent widths obtained by dividing the measured equivalent widths from the spectrum by the $f$. The ATLAS NEWODF model atmosphere grid [@castelli03] assuming $\alpha$ elements excesses is used in the analysis.
The micro-turbulent velocity ([[$v_{\rm micro}$]{}]{}) is determined by demanding no dependence of the abundance results for individual lines on their strengths (equivalent widths). The derived [[$v_{\rm micro}$]{}]{} is 1.5 [[ km s$^{-1}$]{}]{} for the primary and 1.0 [[ km s$^{-1}$]{}]{} for the secondary.
The results of the abundance analyses are given in Table \[tab:abundance\] for eight elements (along with the Li abundance determined separately) for the standard case of the stellar parameters. We also give the results of Fe abundances for the other parameter sets in the table. The errors given in Table \[tab:abundance\] include random errors and those due to uncertainties of atmospheric parameters. We estimate the random errors in the measurements to be $\sigma/N$, where $\sigma$ is the standard deviation of derived abundances from individual lines and $N$ is the number of lines used. When only a few lines are available, the $\sigma$ of is adopted in the estimates. The errors due to the uncertainty of the atmospheric parameters ($\delta$[[$T_{\rm eff}$]{}]{}$= 100$ K, $\delta$[[$\log g$]{}]{}$=0.3$, and $\delta$[[$v_{\rm micro}$]{}]{}$= 0.2$ [[ km s$^{-1}$]{}]{}) are also estimated and added in quadrature to the random errors. The errors for the secondary could be slightly larger if the the uncertainty of effective temperature due to the uncertainty of the mass ratio ($\sim 170$ K) is included.
The Fe abundances derived from individual lines show no statistically significant dependence on their lower excitation potentials for the three choices of the stellar parameters. The Fe abundance derived from lines is slightly higher than that from lines, suggesting the [[$\log g$]{}]{} values adopted in the analysis are systematically too high (by 0.1–0.3 dex), or the lines suffer from non-LTE effects [e.g., @asplund05]. We note that the Li line analysis reported in the following subsection is not sensitive to the changes of the adopted [[$\log g$]{}]{} and [[$v_{\rm micro}$]{}]{}.
The chemical compositions of the two components agree very well with each other. The changes of the Fe abundance from lines by changing the choice of stellar parameter set are only 0.05 dex for the primary. The change for the secondary is even smaller, because the effect of the change of [[$T_{\rm eff}$]{}]{} on the derived Fe abundance is partially compensated by the change of the contribution of the secondary to the continuum flux (adopting a higher [[$T_{\rm eff}$]{}]{} results in a higher Fe abundance in general, while a larger $f_{\rm B}$ results in smaller equivalent widths and a lower abundance).
Li abundances
-------------
The Li abundances of the two components of G 166–45 are determined by fitting synthetic spectra to the observed one. The synthetic spectra are calculated using the same model atmospheres used in the above subsection for both components, and are combined taking their $f$’s into account and the velocity shift between them (18.5 [[ km s$^{-1}$]{}]{}). Comparisons of the synthetic spectra with the observed one are depicted in Figure \[fig:li\], where the doppler correction is made for the primary. In the fitting process, the $\chi^{2}$ minimum is searched for each component: the wavelength ranges 6707.6–6708.0 [Å]{} and 6708.0–6708.4 [Å]{} are used to determine the Li abundances of the primary and secondary, respectively. The $2 \sigma$ error in the fitting is 0.02 and 0.03 dex for the primary and secondary, respectively.
The results for the three choices of mass ratios are given in Table \[tab:abundance\]. Interestingly, the Li abundance of the secondary is insensitive to the choice of the mass ratio for the same reason for the behavior of the Fe abundance mentioned above.
The derived Li abundance of the secondary is 0.05–0.15 dex lower than that of the primary. In the following discussion on the Li abundance difference between the two components, we take the fitting errors (0.02–0.03 dex) and the errors due to the uncertainty of the mass ratio (0.02–0.05 dex) into consideration. We note that this does not include the error due to the uncertainty of [[$T_{\rm eff}$]{}]{} scales, which is as large as 0.1 dex in $A$(Li), but is a systematic error affecting the results for both components.
Discussion and concluding remarks
=================================
The Li abundance of the primary derived by the above analysis ($A$(Li)$=2.23\pm0.05$) well agrees with the Spite plateau value, while that of the secondary is slightly lower. Figure \[fig:disc\] (top panel) depicts the Li abundances of main-sequence ([[$\log g$]{}]{}$>4.0$: filled symbols) and subgiant ([[$\log g$]{}]{}$<4.0$: open symbols) stars as a function of \[Fe/H\]. The data other than G 166–45 are adopted from @gonzalezhernandez08 for CS 22876–032 and @melendez10 for other stars determined by LTE analyses. The possible Li depletion in the main-sequence phase can be inspected from filled symbols shown here. While large scatter of Li abundances is found in \[Fe/H\]$>-2$, which is due to the lower abundances in stars with lower [[$T_{\rm eff}$]{}]{} (see below), there is no distinctive scatter in $-3<$\[Fe/H\]$<-2$ in this sample. This is not changed even if subgiants ([[$\log g$]{}]{}$<4$) are included. @melendez10 suggested the existence of two plateaus with slightly higher and lower Li abundances in \[Fe/H\]$>-2.5$ and $<-2.5$, respectively. The Li abundance of G 166–45A is apparently in agreement with the lower plateau. This result is, however, not definitive, given a possible systematic offset in Li abundance determinations (as large as 0.1 dex) mostly due to [[$T_{\rm eff}$]{}]{} scales used. Among the stars in this metallicity range, the Li abundance of G 166–45B is relatively low.
Some scatter of Li abundances is also found in \[Fe/H\]$<-3$, though that is smaller than that in \[Fe/H\]$>-2$. The discrepancy of the two components of CS 22876–032 is consistent with the size of the scatter.
The middle panel of Figure \[fig:disc\] shows the Li abundances as a function of [[$T_{\rm eff}$]{}]{}. A clear dependence of Li abundances on [[$T_{\rm eff}$]{}]{} is found in [[$T_{\rm eff}$]{}]{}$<6000$ K. It should be noted that all stars in this temperature range, except for CS 22876–032B and two subgiants, are objects with \[Fe/H\]$>-2$. The difference of Li abundances between objects with [[$T_{\rm eff}$]{}]{}$\sim 6350$ K and those with $\sim$5830 K is as large as 0.4 dex. The discrepancy of the Li abundances found between the two components of G 166–45 is much smaller than this value. This suggests that the depletion of Li in stars with [[$T_{\rm eff}$]{}]{}$\sim 5800$ K is much smaller at this metallicity (\[Fe/H\]$\sim -2.5$) than that in \[Fe/H\]$>-2$. This might be related to the shallower convection zone in stars with lower metallicity, however, other causes for Li depletion would be required as mentioned by @richard05, who demonstrated that the temperature of the bottom of the convective zone is almost independent of metallicity. It should also be noted that subgiants ([[$\log g$]{}]{}$<4.0$) with [[$T_{\rm eff}$]{}]{}$<6000K$ have Li abundances as high as stars with [[$T_{\rm eff}$]{}]{}$>6000$ K, hence, the correlation between the Li abundance and [[$T_{\rm eff}$]{}]{} discussed above is only found in main-sequence stars with high \[Fe/H\] ($>-2$). The exception is CS 22876–032B, which shows a Li abundance along with the correlation found for stars with \[Fe/H\]$>-2$. The reason for the Li depletion in this object is, however, likely different from the [[$T_{\rm eff}$]{}]{}-dependent depletion found for \[Fe/H\]$>-2$ (see below).
The bottom panel of Figure \[fig:disc\] shows Li abundances as a function of stellar mass. Li depletion in lower mass stars is discussed by @melendez10. Excluding the stars with \[Fe/H\]$>-2$ (indicated by triangles), which show a tight correlation between Li abundances and [[$T_{\rm eff}$]{}]{}, the dependence of the Li abundance on stellar mass is not very clear. Indeed, @melendez10 reported that the slopes of Li abundances as a function of stellar masses is significant at only 1$\sigma$ level in \[Fe/H\]$<-2.5$, which increases to the 3$\sigma$ level by including the binary CS 22876–032. However, if stars with lowest metallicity (\[Fe/H\]$<-3$) are selected, a correlation seems to appear again. In order to demonstrate this, stars in this metallicity range are indicated by over-plotting open circles (except for CS 22876–032) in the bottom panel of the figure.
On the other hand, no clear correlation is found between the Li abundances and stellar masses for objects with $-3<$\[Fe/H\]$<-2$ (squares without open circles). Our study of the Li abundance of G 166–45 B extends this trend to a mass below 0.7 M$_{\odot}$.
The dependence of Li abundances on stellar mass and metallicity is nonlinear. In \[Fe/H\]$>-2$, Li abundances are lower in main-sequence stars with lower [[$T_{\rm eff}$]{}]{} (lower mass) in [[$T_{\rm eff}$]{}]{}$\lesssim 6000$ K ($M\lesssim 0.8$M$_{\odot}$). A similar dependence is also found in \[Fe/H\]$<-3$, possibly in stars with even higher [[$T_{\rm eff}$]{}]{} ($\lesssim
6300$ K, which corresponds to $M\lesssim 0.8$M$_{\odot}$). On the other hand, such a dependence is not clearly seen in $-3<$\[Fe/H\]$<-2$. The double-lined spectroscopic binaries CS 22876–032 and G 166–45 most clearly represent these trends in \[Fe/H\]$<-3$ and $-3<$\[Fe/H\]$<-2$, respectively. Since the initial abundances of the two components of a binary system can be assumed to be the same, the discrepancy of the Li abundances found between the two components indicates a depletion of Li in the secondary. This suggests that the scatter of Li found in the lowest metallicity range is due to depletion of Li in some object, which is not clearly seen in $-3<$\[Fe/H\]$<-2$.
[*Facilities:*]{} .
We are grateful to M. Parthasarathy for his useful comments for the manuscript. W.A. was supported by the Grants-in-Aid for Science Research of JSPS (20244035).
natexlab\#1[\#1]{}
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Tajitsu, A., Aoki, W., & Yamamuro, T. 2012, PASJ, in press, arXiv:1203.5568
[lcccccccc]{} $q=0.89$ & &\
Species & $\log\epsilon$ & \[X/Fe\](\[Fe/H\]) & $n$ & error & $\log\epsilon$ & \[X/Fe\](\[Fe/H\]) & $n$ & error\
Fe I & 5.05 & $-2.45$ & 35 & 0.10 & 5.10 & $-2.40$ & 39 & 0.11\
Fe II & 5.15 & $-2.35$ & 3 & 0.14 & 5.30 & $-2.20$ & 2 & 0.15\
Na I & 3.74 & $-0.05$ & 2 & 0.09 & 3.76 & $-0.08$ & 2 & 0.09\
Mg I & 5.44 & $0.29 $ & 3 & 0.09 & 5.31 & $0.11 $ & 3 & 0.12\
Ca I & 4.22 & $0.34 $ & 8 & 0.08 & 4.20 & $0.26 $ & 8 & 0.08\
Ti II & 3.00 & $0.50 $ & 3 & 0.13 & 3.01 & $0.46 $ & 3 & 0.13\
Cr I & 3.13 & $-0.06$ & 3 & 0.10 & 3.14 & $-0.10$ & 3 & 0.10\
Ni I & 3.90 & $0.13 $ & 1 & 0.13 & 4.17 & $0.35 $ & 2 & 0.13\
Ba II &$-$0.47&$-0.20$ & 1 & 0.14 & & & &\
Li I & 2.23 & & 1 & 0.02 & 2.11 & & 1 & 0.03\
$q=0.93$ & &\
Fe I & 5.11 & $-2.39$ & 35 & 0.10 & 5.09 & $-2.42$ & 39 & 0.11\
Fe II & 5.13 & $-2.37$ & 3 & 0.14 & 5.19 & $-2.31$ & 2 & 0.15\
Li I & 2.27 & & 1 & 0.02 & 2.12 & & 1 & 0.03\
$q=0.85$ & &\
Fe I & 5.00 & $-2.50$ & 35 & 0.10 & 5.14 & $-2.36$ & 39 & 0.11\
Fe II & 5.04 & $-2.46$ & 3 & 0.14 & 5.43 & $-2.07$ & 2 & 0.15\
Li I & 2.18 & & 1 & 0.02 & 2.13 & & 1 & 0.03\
[^1]: \[A/B\] = $\log(N_{\rm
A}/N_{\rm B}) -\log(N_{\rm A}/N_{\rm B})_{\odot}$, and $\log\epsilon_{\rm A}
=\log(N_{\rm A}/N_{\rm H})+12$ for elements A and B.
[^2]: IRAF is distributed by the National Optical Astronomy Observatories, which is operated by the Association of Universities for Research in Astronomy, Inc. under cooperative agreement with the National Science Foundation.
|
---
abstract: 'This work presents swarm parameters of electrons (the bulk drift velocity, the bulk longitudinal component of the diffusion tensor, and the effective ionization frequency) in , with $n =$ , measured in a scanning drift tube apparatus under time-of-flight conditions over a wide range of the reduced electric field, $\SI{1}{Td}\leq E/N\leq \SI{1790}{Td}$ ( = ). The effective steady-state Townsend ionization coefficient is also derived from the experimental data. A kinetic simulation of the experimental drift cell allows estimating the uncertainties introduced in the data acquisition procedure and provides a correction factor to each of the measured swarm parameters. These parameters are compared to results of previous experimental studies, as well as to results of various kinetic swarm calculations: solutions of the electron Boltzmann equation under different approximations (multiterm and density gradient expansions) and Monte Carlo simulations. The experimental data are consistent with most of the swarm parameters obtained in earlier studies. In the case of , the swarm calculations show that the thermally excited vibrational population should not be neglected, in particular, in the fitting of cross sections to swarm results.'
address:
- '$^1$ Instituto de Plasmas e Fusão Nuclear, Instituto Superior Técnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal'
- '$^2$ Leibniz Institute for Plasma Science and Technology, Felix-Hausdorff-Str. 2, 17489 Greifswald, Germany'
- '$^3$ Institute for Solid State Physics and Optics, Wigner Research Centre for Physics, 1121 Budapest, Konkoly Thege Miklós str. 29-33, Hungary'
- '$^4$ Department of Electrical Engineering and Information Science, Ruhr-University Bochum, D-44780, Bochum, Germany'
- '$^5$ Institute of Physics, University of Belgrade, Pregrevica 118, 11080 Belgrade, Serbia'
author:
- 'N R Pinhão$^1$, D Loffhagen$^2$, M Vass$^3$, P Hartmann$^3$, I Korolov$^4$, S Dujko$^5$, D Bošnjaković$^5$ and Z Donkó$^3$'
bibliography:
- 'C2HmBib.bib'
title: 'Electron swarm parameters in , and : measurements and kinetic calculations'
---
[*Keywords*]{}: electron swarm parameters, drift tube measurements, kinetic theory and computations.
Introduction {#sec:intro}
============
Acetylene (), ethylene () and ethane () are relatively simple hydrocarbons useful in specialized applications such as plasma-assisted combustion [@Adamovich:2014; @Starikovskiy:2013; @kosarev2009; @kosarev2013; @kosarev2015; @kosarev2016], the fabrication of diamond-like films [@Robertson:2002], graphene and carbon nanostructures [@Kumar:2010], and particle detectors [@Fonte:2010]. They are also involved in various chemical reactions in fusion plasmas [@VonKeudell:2001], the Earth’s atmosphere [@Varanasi:1983] and in planetary atmospheric chemistry [@Courtin:1984].
Knowledge on both electron collision cross sections and electron swarm parameters is needed for the quantitative modelling of plasmas. However, with the exception of the drift velocity, which was measured e.g. in [@Hasegawa_Date:2015; @Nakamura:2010; @Cottrell:1968; @Bowman:1967; @Cottrell:1965] for , in [@Hasegawa_Date:2015; @Takatou:2011; @Schmidt:1992; @Bowman:1967; @Wagner:1967; @Christophorou:1966; @Cottrell:1965; @Hurst:1963; @Bortner:1957] for , and in [@Hasegawa_Date:2015; @Shishikura:1997; @Kersten:1994; @Schmidt:1992; @Cottrell:1968; @Bowman:1967; @Cottrell:1965] for , further experimental transport and ionization coefficients have less frequently been reported for these hydrocarbon gases. Measurements of the longitudinal component of the diffusion tensor under time-of-flight (TOF) conditions were additionally reported in [@Nakamura:2010] for , [@Takatou:2011; @Schmidt:1992; @Wagner:1967] for , and [@Shishikura:1997; @Schmidt:1992] for . Hasegawa and Date [@Hasegawa_Date:2015] also determined the effective ionization coefficient by the steady-state Townsend (SST) method for seven organic gases including acetylene, ethylene, and ethane. In addition to the drift velocity for , Kersten [@Kersten:1994] measured the effective ionization coefficient under TOF conditions for a narrow range of the reduced electric field, $E/N$. Furthermore, measured data for the effective SST ionization coefficient have been reported e.g. in [@Heylen:1963] for , in [@Heylen:1978; @Heylen:1963] for , and in [@Watts:1979; @Heylen:1975; @LeBlanc:1960] for .
The aim of this work is (i) to determine the electron transport and ionization coefficients in , and gases in a wide range of $E/N$, (ii) to compare these results with those obtained in earlier investigations of these gases, and (iii) to compare the experimental data with those obtained from kinetic calculations and simulations using up-to-date electron collision cross section sets.
![Graphical representation of the work reported in this article. The red arrows indicate the path from the measurements to the ’Experimental’ transport coefficients and ionization frequencies via fitting of the measured ’swarm maps’. Another ’Corrected’ set of experimental data is also derived based on a correction procedure which is aided by simulations of the experimental setup and related data acquisition (indicated by blue arrows) and by kinetic computations of the swarm parameters. The results of these calculations (’Computed’ transport coefficients) are also compared to the experimental data (green arrows).[]{data-label="fig:flow"}](figure1.pdf){width="0.5\linewidth"}
The workflow of our studies can be followed with the aid of figure \[fig:flow\]. The red arrows show the path to the ’Experimental transport coefficients’ including the effective ionization frequencies. The first step along this path consists of the measurements carried out with our scanning drift tube apparatus. This is a pulsed system, which is described in section \[Sec Exp\]. It records current traces generated by electrons collected from clouds that arrive after having flown over the drift region. The results of the experiments are the so-called ’swarm maps’ which are collections of these current traces for a number of drift gap length values. The swarm parameters are derived from the measured swarm maps via a fitting procedure that assumes that the current measured in the experiment is proportional to the electron density. For the fitting we use the theoretical form of the electron density in the presence of an electric field pointing in the $-z$ direction and under TOF conditions: $$n_{\rm e}(z,t) = \frac{n_0}{(4 \pi D_{\rm L} t)^{1/2}} \exp
\biggl[ \nu_{\rm eff} t - \frac{(z - Wt)^2}{4 D_{\rm L} t} \biggr].
\label{eq:n}$$ This is the solution of the spatially one-dimensional (1D) continuity equation and represents a Gaussian pulse drifting along the $z$ direction with the bulk drift velocity, $W$, and diffuses along the centre-of-mass according to the bulk longitudinal component of the diffusion tensor $D_{\rm L}$. Here $n_0$ is the electron density at $z=0$ at time $t=0$, and $\nu_{\rm eff}$ is the effective ionization frequency. From the fitting procedure we obtain $W$, $D_\mathrm{L}$, and $\nu_\mathrm{eff}$. The application of the relation [@Blevin:1984] $$\frac{1}{\alpha_\mathrm{eff}} = \frac{W}{2 \nu_{\rm eff}} +
\sqrt{\left( \frac{W}{2 \nu_{\rm eff}} \right)^2 - \frac{D_{\rm L}}{\nu_{\rm eff}}}
\label{eq:Blv1}$$ allows us to derive the effective SST ionization coefficient, $\alpha_\mathrm{eff}$, as well.
The assumption that the measured current is proportional to the electron density is, in fact, an approximation, due to two reasons. First, the measured current is generated by moving charges in the detector of the system (see later). In our previous work [@Donko:2019] we have found that the detection sensitivity depends on the gas pressure and the collision cross sections, which both influence the free path of the electrons. This means that any variation of the energy distribution along the $z$ direction in the electron cloud may results in a distortion of the detected pulse and a deviation from the analytical fitting function (\[eq:n\]) assumed. Second, the measured current is proportional to the electron flux consisting of the advective and diffusive component. The advective component is proportional to the electron density, where the coefficient of proportionality is the flux drift velocity, and the diffusive component is proportional to the gradient of the electron density. Using Ramo’s theorem [@R], it can be shown that for the experimental conditions considered in the present work, the contribution of the diffusive component to the current is negligible compared to the contribution of the advective component, except in the early stage of the swarm development when the spatial gradients of the electron density are more significant.
The errors introduced by the first effect mentioned above can be quantified by a procedure, which is marked by blue arrows in figure \[fig:flow\]. We carry out a (Monte Carlo) simulation of the electrons’ motion in the experimental system. This simulation generates the *same* type of swarm maps, which are obtained in the experiments, and a set of swarm parameters is derived via the *same* fitting procedure as in the case of experimental swarm maps. The transport coefficients and ionization frequencies obtained in this way are compared with the ’Computed’ ones, originating from kinetic swarm calculations. We note that (i) this comparison does not include any experimental data, (ii) the system’s simulations use the *same* cross section set as in the kinetic swarm calculations, and (iii) uncertainties of the used collision cross sections have little influence on the outcome of the comparison of the parameter sets obtained by swarm calculations and simulations of the experimental system. The result of this comparison is gas- and $E/N$-dependent correction factors that are applied to the experimental data, providing sets of ’Corrected’ experimental transport and ionization coefficients. Details are given below in section \[sec:correction\].
The two (raw and corrected) sets of experimental results are compared with swarm parameters derived from kinetic calculations based on solutions of the electron Boltzmann equation and on Monte Carlo simulations as described in detail in section \[Sec 3\]. The application of these different approaches allows us to mutually verify the accuracy of the different methods and test the assumptions used by each method. The ’flow’ of this process is indicated by the green arrows in figure \[fig:flow\].
The manuscript is organized as follows: in section \[Sec Exp\] we give a concise description of our experimental setup. A discussion of the various computational methods and the resulting swarm parameters is presented in section \[Sec 3\], and section \[sec:correction\] describes the correction procedure applied to the experimental data. It is followed by the discussion of the results in section \[Sec:results\]. This section comprises the presentation of the present experimental results for each gas and their comparison with previously available measured data as well as the comparison between transport parameters and ionization coefficients computed using the various numerical methods and the present experimental data. Section \[Sec 6\] gives our concluding remarks and in the appendix we provide tabulated values of our experimental results.
Experimental system {#Sec Exp}
===================
The experiments are based on a ’scanning’ drift tube apparatus, of which the details have been presented in [@Korolov:2016]. This apparatus has already been applied for the measurements of transport and ionization coefficients of electrons in various gases: argon, synthetic air, methane, deuterium [@Korolov_2016] and carbon dioxide [@Vass:2017]. In contrast to previously developed drift tubes, our system allows for recording of ’swarm maps’ that show the spatio-temporal development of electron clouds under TOF conditions. The simplified scheme of our experimental apparatus is shown in figure \[fig:exp\].
![Simplified schematic of the scanning drift tube apparatus[]{data-label="fig:exp"}](figure2){width="0.8\linewidth"}
The drift cell is situated within a vacuum chamber made of stainless steel. The chamber can be evacuated by a turbomolecular pump backed with a rotary pump to a level of about . The pressure of the working gases inside the chamber is measured by a Pfeiffer CMR 362 capacitive gauge.
Ultraviolet light pulses (, ) of a frequency-quadrupled diode-pumped YAG laser enter the chamber via a feedthrough with a quartz window and fall on the surface of a Mg disk used as photoemitter. This disk is placed at the centre of a stainless steel electrode with 105mm diameter that serves as the cathode of the drift cell. The detector that faces the cathode at a distance $L_1$ consists of a grounded nickel mesh (with $T = \SI{88}{\percent}$ ’geometric’ transmission and 45 lines/inch density) and a stainless steel collector electrode that is situated at a distance of $L_2 = \SI{1}{\milli\metre}$ behind the mesh.
Electrons emitted from the Mg disk fly towards the collector under the influence of an accelerating voltage that is applied to the cathode. This voltage is established by a BK Precision 9185B power supply. Its value is adjusted according to the required $E/N$ for the given experiment and the actual value of the gap ($L_1$) during the scanning process, where $E/N$ is ensured to be fixed. The current of the detector system is generated by the moving charges within the mesh-collector gap: according to the Shockley-Ramo theorem [@R; @S; @SH] the current induced by an electron moving in a gap between two plane-parallel electrodes with a velocity $v_z$ perpendicular to the electrodes is $I = - e_0 v_z / L$, where $-e_0$ is the charge of the electron and $L$ is the distance between the electrodes ($L=L_2$ in our case). Accordingly, in our setting the measured current at a given time $t$ is $$I(t) = c \sum_{k} v_{z,k}(t) \, ,
\label{eq:ramo}$$ where $c$ is a constant. The summation goes over all electrons being present in the volume bounded by the mesh and the collector at time $t$, and $v_{z,k}$ is the velocity component of the $k$-th electron in $z$ direction.
Electrons entering the detector region (the gap between the nickel mesh and the collector) contribute to the measured current until their first collisions with the gas molecules, as these collisions randomise the angular distribution of their velocities. Therefore, the free path of the electrons plays a central role in the magnitude of the current. For conditions when this free path is longer than the detector gap, the electron sticking property of the collector surface plays a crucial role too, as reflected electrons generate a current contribution with an opposite sign with respect to that generated by the ’incoming’ electrons. According to the above effects, which have been explored to some details in [@Donko:2019], the sensitivity of the detector changes with the nature of the gas (magnitudes and energy dependence of the electron collision cross sections), the pressure, as well as the energy distribution of the electrons. This dependence is the primary reason which calls for a correction of the measured transport and ionization coefficients as discussed in more details in section \[sec:correction\].
The collector current is amplified by a high speed current amplifier (type Femto HCA-400M) connected to the collector, with a virtually grounded input, and is recorded by a digital oscilloscope (type Picoscope 6403B) with sub-ns time resolution. Data collection is triggered by a photodiode that senses the laser light pulses. The low light pulse energy necessitates averaging over typically pulses. The experiment is fully controlled by a computer using LabView software.
![(a-c) Swarm maps recorded in for different values of $E/N$, as indicated. (d) Vertical cuts of the swarm map of (b), which are the measured current traces at the drift length values given in the legend. The pulses have nearly Gaussian shapes. The ’shift’ of the pulses with increasing drift length ($L_1$) is the manifestation of the drift, while their widening is due to (longitudinal) diffusion. As ionization in is weak at $E/N$ = , the amplitude of the pulses decreases with increasing $L_1$ due to the widening of the pulse.[]{data-label="fig:maps"}](figure3a.png "fig:"){width="0.4\linewidth"} ![(a-c) Swarm maps recorded in for different values of $E/N$, as indicated. (d) Vertical cuts of the swarm map of (b), which are the measured current traces at the drift length values given in the legend. The pulses have nearly Gaussian shapes. The ’shift’ of the pulses with increasing drift length ($L_1$) is the manifestation of the drift, while their widening is due to (longitudinal) diffusion. As ionization in is weak at $E/N$ = , the amplitude of the pulses decreases with increasing $L_1$ due to the widening of the pulse.[]{data-label="fig:maps"}](figure3b.png "fig:"){width="0.4\linewidth"}\
![(a-c) Swarm maps recorded in for different values of $E/N$, as indicated. (d) Vertical cuts of the swarm map of (b), which are the measured current traces at the drift length values given in the legend. The pulses have nearly Gaussian shapes. The ’shift’ of the pulses with increasing drift length ($L_1$) is the manifestation of the drift, while their widening is due to (longitudinal) diffusion. As ionization in is weak at $E/N$ = , the amplitude of the pulses decreases with increasing $L_1$ due to the widening of the pulse.[]{data-label="fig:maps"}](figure3c.png "fig:"){width="0.4\linewidth"} ![(a-c) Swarm maps recorded in for different values of $E/N$, as indicated. (d) Vertical cuts of the swarm map of (b), which are the measured current traces at the drift length values given in the legend. The pulses have nearly Gaussian shapes. The ’shift’ of the pulses with increasing drift length ($L_1$) is the manifestation of the drift, while their widening is due to (longitudinal) diffusion. As ionization in is weak at $E/N$ = , the amplitude of the pulses decreases with increasing $L_1$ due to the widening of the pulse.[]{data-label="fig:maps"}](figure3d.png "fig:"){width="0.4\linewidth"}\
During the course of the measurements current traces are recorded for several values of the gap length. The grid and the collector are moved together by a step motor connected to a micrometer screw mounted via a vacuum feedthrough to the vacuum chamber. The distance between the cathode and the mesh, i.e. the ’drift length’, can be set within a range of $L_1$ = . Here, we use 53 positions within this domain.
Sequences of the measured current traces are subsequently merged to form ’swarm maps’, which provide information about the spatio-temporal development of the electron cloud. Figure \[fig:maps\](a)-(c) illustrates such swarm maps, obtained in experiments on , for different values of the reduced electric field. The qualitative behaviour of the swarm is directly seen in these pictures: the slope of the region with appreciable current indicates the drift of the cloud, the widening of this region is related to (longitudinal) diffusion, while an increasing amplitude (as seen in panel (c)) is an indication of ionization. Figure \[fig:maps\](d) displays vertical ’cuts’ of the map shown in panel (b), for $E/N$ = . These cuts are, actually, the current traces recorded in the measurements at different gap length values.
Simulation of the electron swarm {#Sec 3}
================================
The experimental studies of the electron transport are supplemented by numerical modelling and simulation. In addition to Monte Carlo (MC) simulations, three different methods are applied to solve the Boltzmann equation (BE) for electron swarms in a background gas with density $N$ and acted upon by a constant electric field, $\vec{E}$: a multiterm method for the solution of the time-independent Boltzmann equation under spatially homogeneous and SST conditions, respectively, and the $S_n$ method applied to a density gradient expansion of the electron velocity distribution function (EVDF). They differ in their initial physical assumptions and in the numerical algorithms used and provide different properties of the electron swarms.
Details of the different Boltzmann equation methods, as well as main aspects of the MC simulation have been discussed in [@Vass:2017], and we just provide a brief discussion below.
In the following, the electric field is parallel to the $z$ axis and points in the negative direction, $\vec{E} = -E\vec{e}_z$, and $\theta$ is the angle between $\vec{v}$ and $\vec{E}$. Moreover, we assume that the spatial and time scales, respectively, exceed the energy relaxation length and time, such that the transport properties of the electrons do not change with time $t$ and distance $z$ any longer. That is, the electrons have reached a hydrodynamic regime characterizing a state of equilibrium of the system where the effects of collisions and forces are dominant and the EVDF, $f(\vec{r},\vec{v},t)$, has lost any memory of the initial state.
We base our studies on the electron collision cross section sets from Song *et al.* [@Song:2017] for acetylene, Fresnet *et al.* [@Fresnet:2002] for ethylene and Shishikura *et al.* [@Shishikura:1997] for ethane. The cross sections for acetylene and ethane were extended to electron kinetic energies, $\epsilon$, of by fitting a function with a $\log(\epsilon)/\epsilon$ dependence, according to the Born-Bethe high-energy approximation, to the tail of the original cross sections.
The data set includes the momentum transfer cross section for elastic collisions, three vibrational cross sections for single quanta excitation of modes $v_1/v_3$, $v_4/v_5$ and $v_2$ (the first two unresolved) and one vibrational cross section for two quanta excitation of $v_4+v_5$, three electronic excitation cross sections, the total electron-impact ionization cross section and the total dissociative electron attachment cross section for leading to the formation of , and , respectively.
The data set includes the momentum transfer cross section, two lumped vibrational cross sections with thresholds at , three electronic excitation cross sections, the total electron ionization cross section and a collision cross section for electron attachment.
Finally, the set of collision cross sections includes the momentum transfer cross section, three lumped vibrational cross sections with thresholds at , two electronic excitation cross sections, the total electron ionization cross section and an electron attachment cross section.
All of the above cross section sets were developed neglecting the population of thermally excited vibrational states and superelastic processes. The implications of this approximation are discussed in section \[Sec: VibEx\].
Boltzmann equation methods
--------------------------
### Multiterm method for spatially homogeneous conditions {#Sec: BE0D}
In this approach, to describe $f(\vec{r}, \vec{v}, t)$ (abbreviated by BE 0D in the figures shown in section \[Sec:results\]), we consider that the EVDF is spatially homogeneous (0D) and the electron density changes exponentially in time according to $n_\mathrm{e}(t) \propto \exp(\nu_\mathrm{eff}t)$ at the scale of the swarm. Here, the effective ionization frequency $\nu_\mathrm{eff}=\nu_{\rm i}-\nu_{\rm a}$ is the difference of the ionization ($\nu_{\rm i}$) and attachment ($\nu_{\rm a}$) frequencies. In this case we can neglect the dependence of $f$ on the space coordinates and write the EVDF under hydrodynamic conditions as $$f(\vec{v},t) = \hat{F}(\vec{v})n_\mathrm{e}(t).$$ The corresponding microscopic and macroscopic properties of the electrons are determined by the time-independent, spatially homogeneous Boltzmann equation for $\hat{F}(\vec{v})$. As this distribution is symmetric around the field direction, it can be expanded with respect to the angle $\theta$ in Legendre polynomials $P_n(\cos\theta)$ with $n\geq0$. Substituting this expansion in the Boltzmann equation leads to a hierarchy of partial differential equations for the coefficients $\hat{f}_n(v)$ of this expansion. The resulting set of equations with typically eight expansion coefficients is solved employing a generalized version of the multiterm solution technique for weakly ionized steady-state plasmas [@Leyh:1998] adapted to take into account the ionizing and attaching electron collision processes.
Using the computed expansion coefficients $\hat{f}_n(v)$, we obtain the effective ionization frequency, $\nu_\mathrm{eff}$, and the *flux* drift velocity $$w = -\mu E\,,\label{Eq:vd}$$ where $\mu$ is the *flux* mobility. Explicit formulas of these transport parameters obtained by the BE 0D method can be found in [@Vass:2017].
### Multiterm method for SST conditions {#Sec: BESST}
This approach to describe the EVDF (abbreviated by BE SST in the figures shown in section \[Sec:results\]) takes into account that $f(\vec{r},\vec{v},t)$ has reached SST conditions so that the mean transport properties of the electrons are time-independent, do not vary with position any longer, and the electron density assumes an exponential dependence on the distance according to $n_\mathrm{e}(z) \propto \exp(\alpha_\mathrm{eff}z)$. Thus, we can neglect the dependence of $f$ on time and write the EVDF under SST conditions as $$f(z,\vec{v}) = \tilde{F}^{(\rm S)}(\vec{v})n_\mathrm{e}(z),$$ where the upper index (S) denotes SST conditions. In accordance with the procedure described in section \[Sec: BE0D\], the corresponding microscopic and macroscopic properties of the electrons are determined by the steady-state, spatially homogeneous Boltzmann equation for $\tilde{F}^{(\rm S)}(\vec{v})$. Since this distribution is again symmetric around the direction of the field, it can be expanded in Legendre polynomials $P_n(\cos\theta)$ with $n \ge0$. The substitution of this expansion into the Boltzmann equation leads to a set of partial differential equations for the expansion coefficients $\tilde{f}^{(\rm S)}_n(v)$, which is solved efficiently by a modified version of the multiterm method [@Leyh:1998] adapted to treat SST conditions [@Vass:2017].
In this approach, the effective SST ionization coefficient is directly given by $$\alpha_\mathrm{eff}=\frac{\nu_\mathrm{eff}^{(\rm S)}}{v_\mathrm{m}^{(\rm S)}} .
\label{eq:alphaeffBESST}$$ Here, $\nu_\mathrm{eff}^{(\rm S)}$ and $v_\mathrm{m}^{(\rm S)}$ are the effective ionization frequency and mean velocity at SST conditions, respectively, which are calculated by means of the computed expansion coefficients $\tilde{f}_n^{(\rm S)}(v)$ [@Vass:2017].
### Density gradient representation {#Sec: BE DG}
When ionization or attachment processes become important in TOF experiments, the electron swarm can no longer be considered homogeneous and the electron density gradients become significant.
This approach to describe the electron swarm at hydrodynamic conditions (labelled as BE DG below) is based on an expansion of the EVDF with respect to space gradients of the electron density $n_\mathrm{e}$, of consecutive order. In this case, $f$ depends on $(\vec{r}, t)$ only via the density $n_\mathrm{e}(\vec{r}, t)$ and can be written as an expansion on the gradient operator $\nabla$ according to $$f(\vec{r},\vec{v},t) = \sum_{j=0} F^{(j)}(\vec{v})\stackrel{j}\odot
(-\nabla)^{j}n_\mathrm{e}(\vec{r},t)\,,$$ where the expansion coefficients $F^{(j)}(\vec{v})$ are tensors of order $j$ depending only on $\vec{v}$, and $\stackrel{j}\odot$ indicates a $j$-fold scalar product [@Kumar:1980]. Note that the first coefficient $F^{(0)}(\vec{v})$ corresponds to the function $\hat{F}(\vec{v})$ above, for spatially homogeneous conditions (cf. section \[Sec: BE0D\]).
The expansion coefficients $F^{(j)}$ of order $j$ are obtained from a hierarchy of equations for each component, which all have the same structure and depend on the previous orders. In particular, to obtain the transport coefficients measured in TOF experiments, a total of five equations are required, namely for the expansion coefficients $F^{(0)}$, $F_z^{(1)}$, $F_T^{(1)}$, $F_{zz}^{(2)}$ and $F_{TT}^{(2)}$. In the present study, these equations are solved using a variant of the finite element method given in [@Segur:1983] in a $(v, \cos\theta)$ grid.
From the above expansion coefficients we obtain two sets of transport coefficients: the *flux* coefficients, neglecting the contribution of non-conservative processes and equivalent to those obtained by the BE 0D approach described in section \[Sec: BE0D\], and the *bulk* coefficients including a contribution from ionization and attachment. The latter are, the *bulk* drift velocity, $$W = w + \int \tilde{\nu}_\mathrm{eff}(v)F_z^{(1)}(\vec{v})\mathrm{d}\vec{v}
\label{Eq: Wd}$$ with $\tilde{\nu}_\mathrm{eff}(v) = vN[\sigma^{\rm i}(v)-\sigma^{\rm a}(v)]$ where $\sigma^{i}$ and $\sigma^{a}$ are, respectively, the ionization and attachment cross sections; and the longitudinal and transverse components of the diffusion tensor, $$\begin{aligned}
D_{\rm L} &= \int v_z F_z^{(1)}(\vec{v})\mathrm{d}\vec{v} + \int \tilde{\nu}_\mathrm{eff}(v)F_{zz}^{(2)}(\vec{v})\mathrm{d}\vec{v} \\
D_{\rm T} &= \frac{1}{2}\left\{ \int v_T F_T^{(1)}(\vec{v})\mathrm{d}\vec{v} + \int \tilde{\nu}_\mathrm{eff}(v)F_{TT}^{(2)}(\vec{v})\mathrm{d}\vec{v} \right\} \label{Eq: DT}
% D_{\rm T} &= \frac{1}{2} \int v_T F_T^{(1)}(v)dv + \frac{1}{2}\int \tilde{\nu}_\mathrm{eff}(v)F_{TT}^{(2)}(v)dv\end{aligned}$$
Note that the first terms of the right-hand side of equations (\[Eq: Wd\]-\[Eq: DT\]) are the *flux* component. Further details can be found in [@Vass:2017].
The effective or apparent Townsend ionization coefficient $\alpha_\mathrm{eff}$, as determined in SST experiments, can be computed from the TOF parameters using the relation [@Blevin:1984] $$\alpha_\mathrm{eff} = \frac{W}{2D_{\rm L}}-\sqrt{\left(\frac{W}{2D_{\rm L}}\right)^2
- \frac{\nu_\mathrm{eff}}{D_{\rm L}}}, \label{eq:Blv2}$$ which is an equivalent way of writing equation (\[eq:Blv1\]).
Monte Carlo technique
---------------------
In the MC simulation technique, we trace the trajectories of the electrons in the external electric field and under the influence of collisions. As the degree of ionization under the swarm conditions considered here is low, only electron-background gas molecule collisions are taken into account. The motion of the electrons with mass $m_\mathrm{e}$ between collisions is described by their equation of motion $$m_\mathrm{e}\frac{\mathrm{d}^2\vec{r}}{\mathrm{d}t^2}=-e_0\vec{E}.
\label{Eq: Newton}$$ The electron trajectories between collisions are determined by integrating this equation numerically over time steps of duration $\Delta t$ ranging between for the various conditions. While this procedure is totally deterministic, the collisions are handled in a stochastic manner. The probability of the occurrence of a collision is computed after each time step, for each of the electrons, as $$P(\Delta t) = 1-\exp{[-N\nu\sigma^{\rm T}(v)\Delta t]}.$$ The occurrence of a collision is determined by comparing $P(\Delta t)$ with a random number with a uniform distribution over the $(0,1)$ interval. The type of collision is also selected in a random manner taking into account the values of the cross sections of all possible processes at the energy of the colliding electron. For a more detailed description see [@Vass:2017].
The transport parameters (labeled as MC below) are determined as $$W = \frac{\mathrm{d}}{\mathrm{d}t}\left[\frac{\sum_{j=1}^{N_\mathrm{e}(t)}z_j(t)}
{N_\mathrm{e}(t)}\right]\label{Eq: W_MC}$$ and $$w = \frac{1}{N_\mathrm{e}(t)}
\sum_{j=1}^{N_\mathrm{e}(t)}\frac{\mathrm{d}z_j(t)}{\mathrm{d}t},$$ respectively for the *bulk* and *flux* drift velocities, where $N_\mathrm{e}(t)$ is the number of electrons in the swarm at time $t$. The bulk longitudinal and transverse components of the diffusion tensor are $$\begin{aligned}
D_{\rm L} &= \frac{1}{2}\frac{\mathrm{d}}{\mathrm{d}t}\left[\langle z^2(t)\rangle-\langle z(t)\rangle^2\right]\label{Eq: DL_MC} \\
D_{\rm T} &= \frac{1}{4}\frac{\mathrm{d}}{\mathrm{d}t}\left[\langle x^2(t)+y^2(t)\rangle\right]\,,\end{aligned}$$ and the effective ionization frequency is given by $$\nu_\mathrm{eff} = \frac{\mathrm{d}\ln{(N_\mathrm{e}(t))}}{\mathrm{d}t}.\label{Eq: nu_MC}$$
Furthermore, the effective SST ionization coefficient $\alpha_\mathrm{eff}$ is also calculated according to relation (\[eq:Blv2\]) using (\[Eq: W\_MC\]), (\[Eq: DL\_MC\]) and (\[Eq: nu\_MC\]).
All results of calculated electron swarm parameters presented in this work were additionally verified by independent Monte Carlo simulations and calculations based on multi-term solutions of the electron Boltzmann equation developed by the Belgrade group [@Dujko:2010; @Dujko:2011]. For clarity, these results are not included in the figures shown in the next sections, but are available from the authors on request.
As it was already mentioned in the Introduction and is discussed in somewhat more detail in the next section, Monte Carlo simulations are also applied in the simulation of the electrons’ motion in the experimental system, assisting a correction procedure of the experimental data.
Correction of the experimental results {#sec:correction}
======================================
To quantify the effect caused by the variations of the electron energy distribution along the swarm, that in turn makes the detection sensitivity time-dependent, Monte Carlo simulations of the experimental system have been carried out for most of the sets of conditions $(p, E/N)$ in the experiments. These simulations generate swarm maps, similarly to those measured, and a set of swarm parameters is derived from these maps via exactly the same fitting procedure as in the case of the experimental data. The transport parameters and ionization frequencies obtained from the simulations of the setup are compared with those obtained from kinetic swarm calculations based on the solution of the electron Boltzmann equation, where the same electron collision cross section sets are used. Good agreement between the two sets of swarm parameters implies that the assumption made in the fitting of the experimental data, i.e. the use of the theoretical form (\[eq:n\]) of $n_{\rm e}(z,t)$ as a fit to the measured data, is acceptable. In contrast, strong deviations indicate that this assumption is not applicable for the given condition. We note that no experimental data are involved in this procedure.
![Deviations of the results between the swarm parameters obtained from the simulations of the experimental systems vs. the theoretical values. (a) Bulk drift velocity, (b) longitudinal component of the diffusion tensor, (c) effective ionization frequency, and (d) effective SST ionization coefficient. Applying these correction factors to the experimental results leads to the set of ’Corrected’ transport coefficients.[]{data-label="fig:validation"}](figure4a_4d){width="0.9\linewidth"}
Results of this procedure for each of the gases and for the whole domain of $E/N$ are presented in figure \[fig:validation\]. The panels correspond to the swarm parameters $W$, $D_{\rm L}$, $\nu_{\rm eff}$, and $\alpha_{\rm eff}$, respectively, and show the differences of each parameter derived by the simulation of the experimental system with respect to its theoretical value obtained from the BE solution.
In the case of the bulk drift velocity (figure \[fig:validation\](a), the error is in the few % range for most of the conditions, and it approaches $\approx$ at the highest $E/N$ values. This indicates that the determination of the bulk drift velocity values from the experimental data is quite relyable.
The situation turns out to be much worse for the longitudinal component of the diffusion tensor (figure \[fig:validation\](b)). Here, the error ranges from $\approx-$ to $\approx+$, depending on $E/N$. The $D_{\rm L}$ data can be considered to be acceptably accurate at intermediate $E/N$ values only. The much larger error of $D_\mathrm{L}$ with respect to that of $W$ can be explained by the fact that the distribution of the average electron energy along the swarm is inhomogeneous. In the close vicinity of the maximum of the spatial distribution of the electron density, the variation of the average energy along the swarm is comparatively small. However, by moving away from this maximum, the spatial variations of the average energy along the swarm increase. As the drift velocity is primarily determined by the position of the maximum of the spatial profile of the electron density while the diffusion is predominantly determined by the width of this distribution, it is clear that the width of the distribution is more affected by non-uniform sensitivity of the detector with respect to the average electron energy than the position of the maximum.
Regarding the effective ionization frequency (figure \[fig:validation\](c)) and the strongly related SST ionization coefficient (figure \[fig:validation\](d)), we observe small errors at high $E/N$ values, where ionization is appreciable. The error, on the other hand, grows high when $E/N$ approaches $\approx$ , where both $\nu_{\rm eff}$ and $\alpha_{\rm eff}$ drop rapidly.
Results and discussion {#Sec:results}
======================
The electron swarm parameters have been measured in a wide range of the reduced electric field, between at a gas temperature $T$ of . In the following, results of our measurements are presented for the three hydrocarbon gases , , and . Besides the transport parameters and ionization coefficients resulting from the experiments via the fitting procedure described in section \[sec:intro\], we also present the corrected values of these data resulting from the procedure introduced in section \[sec:correction\]. For each swarm parameter, we compare the present measured data with previous experimental results and with the results of the kinetic computations based on the solution of the electron Boltzmann equation or on MC simulations, obtained with the selected electron collision cross sections. The results for the *flux* parameters obtained by methods , and MC overlap, and so do the *bulk* parameters obtained from the and MC methods.
Electron mobility
-----------------
![Mobility in , and obtained from drift velocity results: Bortner *et al* [@Bortner:1957], Hurst *et al* [@Hurst:1963], Cottrell and Walker [@Cottrell:1965], Christophorou *et al* [@Christophorou:1966], Bowman and Gordon [@Bowman:1967], Wagner *et al* [@Wagner:1967], Cottrell *et al* [@Cottrell:1968], Schmidt and Roncossek [@Schmidt:1992], Kersten [@Kersten:1994], Shishikura *et al* [@Shishikura:1997], Nakamura [@Nakamura:2010], Takatou *et al* [@Takatou:2011], Hasegawa and Date [@Hasegawa_Date:2015] and present measurements. The figures share the same *E/N* scale. ’Present experiment’ corresponds to the uncorrected experimental data. The corrected data are not shown here because of the small correction factors for the bulk drift velocity and the mobility.[]{data-label="fig: C2Hm-Mob_Exp"}](figure5){width="0.60\linewidth"}
We start by comparing the values of the reduced mobility, $N\,\mu$, derived from the bulk drift velocity, with previous experimental data for the three hydrocarbon gases in figure \[fig: C2Hm-Mob\_Exp\]. Our experimental results for the (uncorrected and corrected) bulk drift velocity are compiled in table \[tab:WC2Hnwithneq246\] in \[App A\]. We estimate the maximum experimental error of these values to be around .
Except for the high values of $E/N$, our measured *bulk* drift velocity and mobility results are in excellent agreement with all previous results. In , however, at low $E/N$ we find two distinct sets of results: the present results are consistent with the measurements of Bowman and Gordon [@Bowman:1967], while the results of Cottrell and Walker [@Cottrell:1965] are in accordance with those of Nakamura [@Nakamura:2010]. Note that the latter results were used to obtain the recommended electron collision cross sections for [@Song:2017] used in the present modelling and simulation. At high $E/N$ the present results deviate from those of Hasegawa and Date [@Hasegawa_Date:2015] in and . However the latter results are obtained from the mean arrival-time velocity defined in [@Kondo:1990] and are not easily comparable with the present TOF results in the presence of reaction processes.
In figure \[fig: C2Hm-Mob\_Mod\] we compare the results of the present measurements with the kinetic computation results. In this figure the *E/N* scale is common to the three gases but the $N\mu$ scale and data for and have been shifted upwards to avoid overlapping of the curves. Above the contribution of non-conservative processes becomes visible and the mobility results are split into a *bulk* branch (for MC and *bulk* mobilities and the present measurements) and *flux* values (respectively for , MC and *flux* mobilities). Here our measured data show some differences to the MC and bulk results for all three gases. In case of , as the electron collision cross sections used are based on the swarm results of Nakamura [@Nakamura:2010], the modelling results deviate from the present experimental results below . Note that below the modelling results also deviate from the measurements of Bowman and Gordon [@Bowman:1967] as well as of Cottrell and Walker [@Cottrell:1965] in figure \[fig: C2Hm-Mob\_Exp\].
![Mobility in , and : present experiment and modelling results. The results and $N\mu$ scale for and have been shifted. ’Present experiment’ corresponds to the uncorrected experimental data. The corrected data are not shown here because of the small correction factors for the bulk drift velocity / mobility.[]{data-label="fig: C2Hm-Mob_Mod"}](figure6){width="0.9\linewidth"}
Diffusion tensor
----------------
The present experimental results for the gas number density times the longitudinal component of the diffusion tensor, $N D_{\rm L}$, for , and are given in \[App A\], table \[tab:NDLC2Hnwithneq246\]. They are shown in figure \[fig: C2Hm-ND\] together with previously measured data as well as with the kinetic computation values for the *bulk* longitudinal and transverse components of the diffusion tensor for each gas. The present measured values of $N D_{\rm L}$ exhibit larger scattering, which is explained by the higher uncertainty of the determination of $D_{\rm L}$ in the experiments ($\approx\SI{10}{\percent}$) compared to that of the drift velocity.
Above there is reasonable agreement of the present measurements with previous experimental data and the modelling results, for the three gases. Below however, the present measurements evidence the same qualitative behaviour but are systematically above previous measurements. Note that the application of the correction procedure, detailed in section \[sec:correction\], to our experimental results leads to much better agreement with previously measured data, in particular for and .
The modelling results for $D_{\rm L}$ in and below and , respectively, also deviate from all experimental results indicating that the corresponding cross section sets require improvement. In each of the three gases, the values of the transverse component of the diffusion tensor, $D_{\rm T}$, obtained by the kinetic computations, are very different from the longitudinal component, $D_{\rm L}$. The measurement of data of this component can provide additional tests for the fitting of the electron collision cross sections.
![Longitudinal and transverse *bulk* components of the diffusion tensor in , and . Experimental results: present experiment, Wagner *et al* [@Wagner:1967], Schmidt and Roncossek [@Schmidt:1992], Shishikura *et al* [@Shishikura:1997], Nakamura [@Nakamura:2010], Takatou *et al* [@Takatou:2011]. Modelling results: MC and BE DG ($ND_{\rm L}$ and $ND_{\rm T}$). The figures share the same $E/N$ scale. The panels show both the uncorrected and corrected experimental results of this study.[]{data-label="fig: C2Hm-ND"}](figure7){width="0.62\linewidth"}
Effective ionization frequency and SST ionization coefficient
-------------------------------------------------------------
The experimental and modelling results for the reduced effective ionization frequency, $\nu_\mathrm{eff}/N$, for the three gases studied are displayed in figure \[fig: C2Hm-Nui\_eff\]. To our best knowledge this is the first report of $\nu_\mathrm{eff}$ in these three gases for an extended range of $\SI{100}{Td} \le E/N \le \SI{1790}{Td}$, for which the estimated experimental error of the data is $\le \SI{8}{\percent}$. Our measured data are also listed in \[App A\], table \[tab:nueffpNC2Hnwithneq246\]. In order to accommodate the results on the same figure, all gases share a same $\nu_\mathrm{eff}/N$ axis but the $E/N$ scales for and have been shifted to the right.
Good agreement between our measured and calculated results is generally found for $E/N$ values larger than about , indicating that the electron collision cross section sets for the three gases are reasonably well adapted to allow for an appropriate determination of the rate coefficients for ground state ionization. Certain differences are obvious for lower $E/N$ values. These differences seem to result from the measurement and/or, more likely, from the fitting procudure (see figure \[fig:validation\]).
![Reduced effective ionization frequency in , and : present experiment and modelling results. The results and $E/N$ scale for and are shifted horizontally. ’Present experiment’ corresponds to the uncorrected data.[]{data-label="fig: C2Hm-Nui_eff"}](figure8){width="0.9\linewidth"}
Our experimental data for the reduced effective SST ionization coefficient, $\alpha_\mathrm{eff}/N$, obtained using equation (\[eq:Blv1\]), are compared with previous measurements and the kinetic computation results in figure \[fig: C2Hm-alpha\]. As $\alpha_\mathrm{eff}$ is derived from the set of parameters $\{W, D_{\rm L}, \nu_\mathrm{eff}\}$, these results have a higher uncertainty than $\nu_\mathrm{eff}$ with an estimated experimental error of $\le$ . Notice that the kinetic computation results using method do not include the approximations involved in equation (\[eq:Blv1\]), but are directly obtained by solving the electron Boltzmann equation at SST conditions according to (\[eq:alphaeffBESST\]). In this respect, their comparison with the and MC results can indicate the range of validity of equation (\[eq:Blv1\]) or (\[eq:Blv2\]). Our experimental results of $\alpha_\mathrm{eff}/N$ are compiled in \[App A\], table \[tab:alphaeffpNC2Hnwithneq246\] as well.
![Reduced effective Townsend ionization coefficient in , and . Experimental results: present experiment, Heylen [@Heylen:1963; @Heylen:1975], Watts and Heylen [@Watts:1979], Kersten [@Kersten:1994] and Hasegawa and Date [@Hasegawa_Date:2015]. Modelling results: BE SST, MC and BE DG. ’Present experiment’ corresponds to the uncorrected data.[]{data-label="fig: C2Hm-alpha"}](figure9){width="0.65\linewidth"}
Except for the low values of $E/N$, our results for the effective SST ionization coefficient are in excellent agreement with all previous results and the kinetic computations. At values close to the threshold, however, the present results are higher than previous measurements. Notice that Kersten’s effective Townsend ionization coefficient was measured under TOF conditions and corresponds to $\nu_\mathrm{eff}/W$ [@Kersten:1994]. Thus, it represents the effective SST ionization coefficient $\alpha_\mathrm{eff}$ according to (\[eq:Blv1\]) only in the absence of diffusion, i.e., $D_{\rm L}=0$.
Effect of the vibrationally excited population {#Sec: VibEx}
----------------------------------------------
The cross sections sets used above were obtained considering only electron collisions with the ground state of the molecules. However, as polyatomic molecules have multiple vibrational modes and these modes can be degenerated, in these gases we can find a significant fraction of molecules in thermally excited vibrational states at room temperature. In addition to their contribution to energy losses due to elastic, exciting, ionizing and attaching collision processes, these excited states contribute to electron energy gains due to superelastic collisions and influence the EVDF and transport parameters, mainly at low to medium $E/N$ field values. The importance of their effect increases with the energy associated with the collision and the fractional population of thermally excited states with that energy. This population, however, decreases exponentially with energy. From the combination of these two factors, the effect on the EVDF should be maximum for a given energy value.
Taking into account the equations for the fractional populations and statistical weights of polyatomic molecules in \[App B\], we can estimate the populations of the different states of these gases.
Acetylene
: has five vibrational modes, with the two bending modes ($v_4$ and $v_5$) double degenerated and with energies of, respectively, and [@Sh1972NBS]. At a gas temperature of , the vibrational states with fractional population above are indicated in table \[tab:FracPop\]. At this temperature only around of the acetylene molecules are in the ground state and the vibrational population in excited states of modes $v_4$ and $v_5$ is significant.
[cccSS]{} Vibr. state & short notation & g & &\
(00000) & $v_0$ & 1 & 0.0 & 85.37\
(10000) & $v_1$ & 1 & 0.421 & 5.5e-6\
(01000) & $v_2$ & 1 & 0.245 & 5.3e-3\
(00100) & $v_3$ & 1 & 0.411 & 8.3e-6\
(00010) & $v_4$ & 2 & 0.075 & 8.47\
(00020) & & 3 & 0.150 & 0.63\
(00001) & $v_5$ & 2 & 0.0905 & 4.75\
(00002) & & 3 & 0.180 & 0.20\
(00011) & $v_4+v_5$ & 4 & 0.165 & 0.47\
Ethylene:
: In contrast to , none of the twelve ethylene vibrational modes [@Sh1972NBS] is degenerated, where the lowest threshold energy for vibrational excitation to $v_{10}$ is and, at the same temperature, more than of the molecules are in the ground state.
Ethane:
: All the degenerated vibrational modes of ethane [@Sh1972NBS] have energies above and at room temperature their fractional population is small. Overall, however, only of ethane molecules are in the ground state as mode $v_4$ has an excitation energy of only . Molecules in the two first excited vibrational states of this mode represent of the total. On the other hand, as the excitation energy of the $v_4$ mode transitions is very small, the effect on the EVDF and transport parameters is also small.
Of the three gases analysed, the impact of the thermally excited vibrational population on the EVDF should be largest in . The vibrational excitation cross section set for [@Song:2017] is also more complete than the vibrational cross section sets for and used in this study. For these reasons we study the effect of the thermally excited vibrational states only for acetylene.
Our goal is to single out the contribution of the vibrationally excited molecules due to superelastic collisions and we will change the electron collision cross sections in such a way that, if we neglect these collisions, we obtain the same results as before. Starting from the recommended cross section set for ethylene [@Song:2017], we introduce the following modifications:\
a) We split the lumped cross sections for the vibrational excitation of modes $v_1/v_3$ and $v_4/v_5$ into individual cross sections for each modes, with a value of half of the original cross section. That is $\sigma_{v_1} = \sigma_{v_3} =\frac{1}{2}\sigma_{v_1/v_3}$ and $\sigma_{v_4} = \sigma_{v_5} =\frac{1}{2}\sigma_{v_4/v_5}$.\
b) The threshold for the excitation of modes $v_1$ and $v_3$ and of modes $v_4$ and $v_5$ is set at the same value as before of, respectively, and .\
c) We assume that all molecules are in one of the three states $(00000)$, $(00010)$ and $(00001)$, with the fractional population, $\delta$, of the last two states in thermal equilibrium with the gas and the ground state fraction given by $\delta_{00000}=(1-\delta_{00010}-\delta_{00001})$.\
d) We consider the following vibrational excitation processes for electron collisions with the ground state $(00000)$: $$\begin{array}{lcl}
\ce{e} + \ce{C2H2}(00000) & \to & \ce{e} + \ce{C2H2}(10000)\\
\ce{e} + \ce{C2H2}(00000) & \to & \ce{e} + \ce{C2H2}(01000)\\
\ce{e} + \ce{C2H2}(00000) & \to & \ce{e} + \ce{C2H2}(00100)\\
\ce{e} + \ce{C2H2}(00000) & \leftrightarrow & \ce{e} + \ce{C2H2}(00010)\\
\ce{e} + \ce{C2H2}(00000) & \leftrightarrow & \ce{e} + \ce{C2H2}(00001)\\
\ce{e} + \ce{C2H2}(00000) & \leftrightarrow & \ce{e} + \ce{C2H2}(00011)
\end{array}$$ where reactions with double-arrows include superelastic collisions.\
f) We additionaly include the following vibrational excitation processes on collisions with states $(00010)$ and $(00001)$: $$\begin{array}{lcl}
\ce{e} + \ce{C2H2}(00010) & \to & \ce{e} + \ce{C2H2}(10010)\\
\ce{e} + \ce{C2H2}(00010) & \to & \ce{e} + \ce{C2H2}(01010)\\
\ce{e} + \ce{C2H2}(00010) & \to & \ce{e} + \ce{C2H2}(00110)\\
\ce{e} + \ce{C2H2}(00010) & \leftrightarrow & \ce{e} + \ce{C2H2}(00020)\\
\ce{e} + \ce{C2H2}(00010) & \leftrightarrow & \ce{e} + \ce{C2H2}(00011)\\
\ce{e} + \ce{C2H2}(00010) & \to & \ce{e} + \ce{C2H2}(00021)
\end{array}$$ and $$\begin{array}{lcl}
\ce{e} + \ce{C2H2}(00001) & \to & \ce{e} + \ce{C2H2}(10001)\\
\ce{e} + \ce{C2H2}(00001) & \to & \ce{e} + \ce{C2H2}(01001)\\
\ce{e} + \ce{C2H2}(00001) & \to & \ce{e} + \ce{C2H2}(00101)\\
\ce{e} + \ce{C2H2}(00001) & \leftrightarrow & \ce{e} + \ce{C2H2}(00011)\\
\ce{e} + \ce{C2H2}(00001) & \leftrightarrow & \ce{e} + \ce{C2H2}(00002)\\
\ce{e} + \ce{C2H2}(00001) & \to & \ce{e} + \ce{C2H2}(00012)
\end{array}$$ adopting for these processes the same cross sections as the corresponding excitations from the ground state.\
e) We further assume that the electron collision cross sections for momentum transfer, electronic excitation, ionization and attachment with the vibrational states $(00010)$ and $(00001)$ are the same as for state $(00000)$.
Note that if we neglect superelastic collisions, the EVDF and swarm parameters obtained with these modified cross sections and electron collision reactions are exactly the same as with the original set [@Song:2017] and are independent of the fractional population of levels $(00010)$ and $(00001)$.
The influence of superelastic collisions is ilustrated in figure \[fig: C2H2-eedf\] which shows the isotropic component $\hat{f}_0(\epsilon)$ of the EVDF as a function of the electron kinetic energy, $\epsilon=m_\mathrm{e}v^2/2$, calculated at $E/N$ values of , respectively, with and without the inclusion of superelastic processes. Pronounced differences between the corresponding isotropic distributions $\hat{f}_0(\epsilon)$ are found at $E/N = \SI{1}{Td}$, while the impact of superelastic electron collision processes is comparatively small at . This finding is not only reflected by the isotropic distribution but also by different macroscopic properties.
![Isotropic component of the EVDF in at for , with and without superelasic collision processes included.[]{data-label="fig: C2H2-eedf"}](figure10){width="0.65\linewidth"}
The influence of superelastic collisions is mostly visible in the drift velocity and mobility as shown in figure \[fig: C2H2-mobSE\]. This figure compares the values of mobility and the longitudinal and transverse *bulk* components of the diffusion tensor obtained with the original cross sections set with the results obtained using the modified set with and without the inclusion of superelastic processes. As predicted, the results of the modified set neglecting superelastic collisions are the same as those obtained with the original set. Superelastic collisions are responsible for a reduction of the electron mobility in the range of low reduced field, visible up to approximately . The influence on the components of the diffusion tensor is overall smaller than that on the mobility with the largest differences in the longitudinal component around .
![(a) Mobility and longitudinal and (b) transverse *bulk* components of the diffusion tensor in at : modelling results obtained with the electron collision cross sections from [@Song:2017] without considering superelastic processes and with a modified set with and without superelastic processes.[]{data-label="fig: C2H2-mobSE"}](figure11){width="0.65\linewidth"}
As the impact of superelastic collisions decreases remarkably above about , their influence on the effective ionization frequency and Townsend ionization coefficient is negligible.
Concluding remarks {#Sec 6}
==================
We have investigated electron swarm parameters in , and experimentally using a scanning drift tube, as well as computationally by solutions of the electron Boltzmann equation and via Monte Carlo simulation, corresponding to both time-of-flight and steady-state Townsend conditions. The measured data made it possible to derive the bulk drift velocity, the bulk longitudinal component of the diffusion tensor and the effective ionization frequency of the electrons, for the wide range of the reduced electric field from 1 to . The measured TOF transport parameters as well as the effective SST ionization coefficient, deduced from the TOF swarm parameters, have been compared to experimental data obtained in previous studies. Here, generally good agreement with most of the transport parameters and the effective SST ionization coefficients obtained in these earlier studies was found. In the case of the drift velocity or the mobility, respectively, and the longitudinal component of the diffusion tensor we found disagreements at low or high values of $E/N$.
The experimental data have undergone a correction procedure, which was supposed to quantify the errors caused by the dependence of the sensitivity of the detector of the drift cell on the energy distribution of the electrons in the swarm that may have a spatial dependence.
In particular, in case of our measured drift velocities at low $E/N$ agree well with previous data of Bowman and Gordon [@Bowman:1967] but not with the results of Cottrell and Walker [@Cottrell:1965] as well as of Nakamura [@Nakamura:2010]. Further measurements in this range are required to clarify this contradiction.
The comparison of the experimental data was also carried out with swarm parameters resulting from various kinetic computations, which used the most recently recommended cross section sets [@Song:2017; @Fresnet:2002; @Shishikura:1997]. Here, excellent agreement between electron Boltzmann equation and MC simulation results verifies the computational approaches and data for the three gases. The agreement of the computed data with the present and previously measured values of the reduced effective ionization frequency and SST ionization coefficient was generally good. However, certain differences between kinetic computational and measured results found for the drift velocities and, especially, for the longitudinal component of the diffusion tensor illustrate the need for an improvement of the existing collision cross section sets for the three hydrocarbon gases considered.
We have also studied the influence of the thermally excited vibrational populations on the transport parameters. In the case of we have found that this population has a significant value and superelastic collisions influence the drift velocity and the components of the diffusion tensor up to . The fitting of electron collision cross sections for this gas using swarm experiments should include these processes.
This work was partially supported by the Portuguese FCT–Fundação para a Ciência e a Tecnologia, under project UID/FIS/50010/2013, by the Hungarian Office for Research, Development and Innovation (NKFIH) grants K119357, K115805, by the ÚNKP-19-3 New National Excellence Program of the Ministry for Innovation and Technology, and funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 327886311. SD and DB are supported by Grants No. OI171037 and III41011 from the Ministry of Education, Science and Technological Development of the Republic of Serbia. We thank Prof. Y. Nakamura for providing numerical values of measured electron transport parameters in and Mr T. Szűcs for his contributions to the construction of the experimental apparatus.
Tables of experimental data {#App A}
===========================
[S\[table-number-alignment=right\]SS S\[table-number-alignment=right\]SS S\[table-number-alignment=right\]SS]{} & & [$E/N$]{} & & [$E/N$]{} &\
(r)[2-3]{}(r)[5-6]{}(r)[8-9]{} & *exptl.* & *cor.* & & *exptl.* & *cor.* & & *exptl.* & *cor.*\
\
3 & 1.43 & 1.43 & 125 & 8.36 & 8.33 & 450 & 29.6 & 30.3\
5 & 2.36 & 2.35 & 145 & 9.38 & 9.35 & 480 & 31.9 & 32.7\
9 & 3.48 & 3.47 & 160 & 10.1 & 10.2 & 540 & 36.0 & 37.1\
21 & 4.88 & 4.86 & 170 & 10.8 & 10.9 & 600 & 40.4 & 41.7\
27 & 5.14 & 5.12 & 190 & 11.9 & 11.9 & 720 & 50.2 & 52.2\
34 & 5.38 & 5.36 & 220 & 13.6 & 13.7 & 830 & 58.4 & 61.0\
47 & 5.75 & 5.72 & 230 & 14.0 & 14.2 & 1043 & 73.5 & 77.8\
62 & 6.15 & 6.12 & 250 & 15.3 & 15.5 & 1120 & 77.9 & 82.8\
74 & 6.37 & 6.35 & 270 & 16.8 & 17.0 & 1330 & 93.7 & 108\
85 & 6.57 & 6.55 & 310 & 19.4 & 19.7 & 1530 & 108 & 117\
96 & 7.17 & 7.15 & 360 & 23.0 & 23.4 & 1560 & 111 & 121\
110 & 7.80 & 7.77 & 410 & 26.9 & 27.4 & 1790 & 122 & 135\
\
1 & 2.33 & 2.27 & 50 & 5.69 & 5.53 & 310 & 19.8 & 19.4\
1.2 & 2.80 & 2.72 & 62 & 6.24 & 6.06 & 350 & 22.1 & 21.7\
1.5 & 2.89 & 2.81 & 69 & 6.60 & 6.41 & 410 & 25.6 & 25.3\
2 & 3.68 & 3.57 & 85 & 7.44 & 7.23 & 455 & 28.0 & 27.7\
2.5 & 3.99 & 3.88 & 100 & 8.51 & 8.27 & 480 & 29.7 & 29.4\
3.2 & 4.43 & 4.30 & 110 & 9.09 & 8.86 & 550 & 34.0 & 33.7\
4 & 4.74 & 4.60 & 125 & 10.2 & 9.96 & 600 & 37.0 & 36.7\
5 & 4.90 & 4.76 & 145 & 11.0 & 10.7 & 710 & 43.8 & 43.6\
7 & 5.02 & 4.88 & 160 & 11.8 & 11.5 & 840 & 50.9 & 50.9\
10 & 4.88 & 4.74 & 180 & 12.7 & 12.4 & 1040 & 64.7 & 65.2\
15 & 4.70 & 4.57 & 200 & 13.9 & 13.5 & 1140 & 70.5 & 71.2\
17 & 4.62 & 4.48 & 220 & 14.7 & 14.4 & 1330 & 82.6 & 83.9\
20 & 4.61 & 4.48 & 230 & 15.3 & 15.0 & 1510 & 82.1 & 84.0\
29 & 4.77 & 4.63 & 250 & 16.3 & 16.0 & 1570 & 86.1 & 88.3\
34 & 4.96 & 4.82 & 270 & 17.6 & 17.3 & 1760 & 93.6 & 96.6\
\
1 & 3.23 & 3.14 & 48 & 5.36 & 5.22 & 350 & 20.3 & 20.1\
1.5 & 3.97 & 3.86 & 60 & 5.55 & 5.42 & 410 & 24.0 & 23.9\
2 & 4.58 & 4.45 & 70 & 5.68 & 5.55 & 470 & 27.8 & 27.7\
2.5 & 4.84 & 4.71 & 82 & 5.93 & 5.80 & 550 & 33.3 & 33.4\
3 & 5.02 & 4.88 & 100 & 6.46 & 6.32 & 610 & 37.3 & 37.6\
4 & 5.06 & 4.93 & 125 & 7.52 & 7.36 & 710 & 43.2 & 43.7\
5 & 5.31 & 5.17 & 145 & 8.44 & 8.28 & 715 & 42.4 & 43.0\
6 & 5.39 & 5.25 & 160 & 9.18 & 9.01 & 836 & 49.3 & 50.3\
7 & 5.49 & 5.34 & 180 & 10.2 & 10.0 & 900 & 53.8 & 55.0\
8.5 & 5.53 & 5.38 & 200 & 11.3 & 11.1 & 1002 & 60.3 & 62.0\
10 & 5.54 & 5.39 & 225 & 12.7 & 12.5 & 1170 & 71.8 & 74.5\
15 & 5.45 & 5.30 & 250 & 14.3 & 14.1 & 1398 & 77.0 & 80.9\
26 & 5.33 & 5.18 & 270 & 15.6 & 15.4 & 1606 & 82.9 & 88.1\
36 & 5.31 & 5.17 & 310 & 17.7 & 17.5 & & &\
[S\[table-number-alignment=right\]SS S\[table-number-alignment=right\]SS S\[table-number-alignment=right\]SS]{} & & [$E/N$]{} & & [$E/N$]{} &\
(r)[2-3]{}(r)[5-6]{}(r)[8-9]{} & *exptl.* & *cor.* & & *exptl.* & *cor.* & & *exptl.* & *cor.*\
\
21 & 0.84 & 0.53 & 160 & 1.10 & 1.02 & 480 & 2.68 & 3.00\
27 & 0.63 & 0.42 & 170 & 1.17 & 1.10 & 540 & 2.82 & 3.24\
34 & 0.48 & 0.34 & 190 & 1.31 & 1.28 & 600 & 3.03 & 3.75\
47 & 0.51 & 0.38 & 220 & 1.44 & 1.43 & 720 & 3.46 & 4.28\
62 & 0.47 & 0.37 & 230 & 1.53 & 1.53 & 830 & 3.70 & 4.78\
74 & 0.46 & 0.38 & 250 & 1.67 & 1.68 & 1043 & 4.26 & 5.95\
85 & 0.55 & 0.50 & 270 & 1.81 & 1.84 & 1120 & 4.36 & 6.25\
96 & 0.67 & 0.57 & 310 & 1.97 & 2.04 & 1330 & 4.73 & 7.27\
110 & 0.80 & 0.70 & 360 & 2.39 & 2.54 & 1530 & 4.98 & 8.14\
125 & 0.83 & 0.74 & 410 & 2.39 & 2.59 & 1560 & 4.97 & 8.20\
145 & 1.12 & 1.03 & 450 & 2.50 & 2.76 & 1790 & 5.35 & 9.43\
\
1 & 1.35 & 0.94 & 50 & 0.54 & 0.48 & 310 & 1.97 & 1.91\
1.2 & 1.36 & 0.96 & 62 & 0.68 & 0.60 & 350 & 2.14 & 2.09\
1.5 & 1.23 & 0.88 & 69 & 0.77 & 0.68 & 410 & 2.28 & 2.27\
2 & 1.33 & 0.97 & 85 & 0.88 & 0.79 & 455 & 2.40 & 2.41\
2.5 & 1.04 & 0.77 & 100 & 1.09 & 0.99 & 480 & 2.45 & 2.49\
3.2 & 0.94 & 0.70 & 110 & 1.16 & 1.05 & 550 & 2.69 & 2.78\
4 & 0.80 & 0.61 & 125 & 1.22 & 1.12 & 600 & 2.96 & 3.09\
5 & 0.66 & 0.51 & 145 & 1.30 & 1.19 & 710 & 2.77 & 2.99\
7 & 0.48 & 0.38 & 160 & 1.37 & 1.27 & 840 & 3.06 & 3.41\
10 & 0.41 & 0.33 & 180 & 1.45 & 1.35 & 1040 & 3.64 & 4.26\
15 & 0.40 & 0.33 & 200 & 1.51 & 1.41 & 1140 & 3.99 & 4.77\
17 & 0.42 & 0.35 & 220 & 1.54 & 1.45 & 1330 & 4.49 & 5.61\
20 & 0.42 & 0.35 & 230 & 1.68 & 1.59 & 1510 & 3.33 & 4.32\
29 & 0.42 & 0.36 & 250 & 1.69 & 1.60 & 1570 & 3.43 & 4.51\
34 & 0.43 & 0.37 & 270 & 1.85 & 1.77 & 1760 & 3.70 & 5.06\
\
1 & 2.41 & 1.89 & 48 & 0.71 & 0.65 & 350 & 2.29 & 2.33\
1.5 & 2.10 & 1.68 & 60 & 0.72 & 0.66 & 410 & 1.80 & 1.87\
2 & 1.85 & 1.49 & 70 & 0.73 & 0.67 & 470 & 2.58 & 2.74\
2.5 & 1.57 & 1.28 & 82 & 0.78 & 0.72 & 550 & 2.89 & 3.15\
3 & 1.42 & 1.17 & 100 & 0.94 & 0.87 & 610 & 3.31 & 3.67\
4 & 1.16 & 0.96 & 125 & 1.24 & 1.17 & 710 & 3.08 & 3.53\
5 & 1.03 & 0.86 & 145 & 1.44 & 1.36 & 715 & 3.28 & 3.76\
6 & 0.94 & 0.79 & 160 & 1.55 & 1.48 & 836 & 3.45 & 4.10\
7 & 0.88 & 0.74 & 180 & 1.69 & 1.62 & 900 & 3.66 & 4.43\
8.5 & 0.83 & 0.71 & 200 & 1.79 & 1.73 & 1002 & 3.73 & 4.65\
10 & 0.79 & 0.68 & 225 & 1.90 & 1.85 & 1170 & 4.30 & 5.61\
15 & 0.74 & 0.64 & 250 & 2.01 & 1.98 & 1398 & 4.75 & 6.58\
26 & 0.73 & 0.65 & 270 & 2.09 & 2.07 & 1606 & 4.10 & 5.98\
36 & 0.73 & 0.65 & 310 & 1.61 & 1.62 & & &\
[S\[table-number-alignment=right\] S\[table-column-width=20pt,table-number-alignment=left\] S\[table-column-width=20pt,table-number-alignment=left\] S\[table-number-alignment=right\] S\[table-column-width=15pt,table-number-alignment=left\] S\[table-column-width=15pt,table-number-alignment=left\] S\[table-number-alignment=right\] S\[table-column-width=10pt,table-number-alignment=left\] S\[table-column-width=10pt,table-number-alignment=left\]]{} & & [$E/N$]{} & & [$E/N$]{} &\
(r)[2-3]{}(r)[5-6]{}(r)[8-9]{} & *exptl.* & *cor.* & & *exptl.* & *cor.* & & *exptl.* & *cor.*\
\
145 & 0.00137 & 0.00136 & 310 & 0.0419 & 0.0422 & 830 & 0.853 & 0.879\
160 & 0.00263 & 0.00264 & 360 & 0.0792 & 0.0800 & 1043 & 1.34 & 1.39\
170 & 0.00325 & 0.00325 & 410 & 0.122 & 0.123 & 1120 & 1.50 & 1.56\
190 & 0.00487 & 0.00488 & 450 & 0.158 & 0.161 & 1330 & 2.25 & 2.37\
220 & 0.0109 & 0.0109 & 480 & 0.201 & 0.204 & 1530 & 3.06 & 3.25\
230 & 0.0111 & 0.0111 & 540 & 0.263 & 0.268 & 1560 & 3.36 & 3.57\
250 & 0.0171 & 0.0171 & 600 & 0.365 & 0.372 & 1790 & 3.80 & 4.07\
270 & 0.0235 & 0.0237 & 720 & 0.603 & 0.619 & & &\
\
125 & 0.00106 & 0.00105 & 270 & 0.0241 & 0.0238 & 710 & 0.501 & 0.501\
145 & 0.00194 & 0.00191 & 310 & 0.0388 & 0.0385 & 840 & 0.696 & 0.698\
160 & 0.00310 & 0.00305 & 350 & 0.0592 & 0.0587 & 1040 & 1.20 & 1.21\
180 & 0.00463 & 0.00458 & 410 & 0.0881 & 0.0876 & 1140 & 1.36 & 1.37\
200 & 0.00682 & 0.00675 & 455 & 0.122 & 0.121 & 1330 & 1.83 & 1.85\
220 & 0.0101 & 0.00999 & 480 & 0.153 & 0.152 & 1510 & 2.30 & 2.33\
230 & 0.0138 & 0.0136 & 550 & 0.234 & 0.233 & 1570 & 2.54 & 2.59\
250 & 0.0186 & 0.0184 & 600 & 0.265 & 0.265 & 1760 & 2.90 & 2.97\
\
100 & 0.000673 & 0.000675 & 270 & 0.0429 & 0.0430 & 715 & 0.571 & 0.581\
125 & 0.00196 & 0.00197 & 310 & 0.0861 & 0.0863 & 836 & 0.776 & 0.792\
145 & 0.00385 & 0.00386 & 350 & 0.0988 & 0.0991 & 900 & 0.929 & 0.949\
160 & 0.00521 & 0.00522 & 410 & 0.221 & 0.222 & 1002 & 1.20 & 1.23\
180 & 0.00667 & 0.00669 & 470 & 0.230 & 0.233 & 1170 & 1.61 & 1.65\
200 & 0.0137 & 0.0137 & 550 & 0.356 & 0.361 & 1398 & 1.46 & 1.51\
225 & 0.0214 & 0.0215 & 610 & 0.417 & 0.423 & 1606 & 1.82 & 1.89\
250 & 0.0309 & 0.0310 & 710 & 0.657 & 0.669 & & &\
[S\[table-number-alignment=right\]SS S\[table-number-alignment=right\]SS S\[table-number-alignment=right\]SS]{} [$E/N$]{} & & [$E/N$]{} & & [$E/N$]{} &\
(r)[2-3]{}(r)[5-6]{}(r)[8-9]{} & *exptl.* & *cor.* & & *exptl.* & *cor.* & & *exptl.* & *cor.*\
\
145 & 0.0146 & 0.0145 & 310 & 0.221 & 0.219 & 830 & 1.63 & 1.65\
160 & 0.0260 & 0.0258 & 360 & 0.358 & 0.356 & 1043 & 2.08 & 2.15\
170 & 0.0301 & 0.0299 & 410 & 0.473 & 0.470 & 1120 & 2.19 & 2.28\
190 & 0.0413 & 0.0411 & 450 & 0.561 & 0.558 & 1330 & 2.80 & 3.00\
220 & 0.0809 & 0.0805 & 480 & 0.667 & 0.663 & 1530 & 3.37 & 3.74\
230 & 0.0796 & 0.0791 & 540 & 0.778 & 0.775 & 1560 & 3.61 & 4.07\
250 & 0.113 & 0.112 & 600 & 0.974 & 0.973 & 1790 & 3.71 & 4.33\
270 & 0.143 & 0.142 & 720 & 1.32 & 1.33 & & &\
\
125 & 0.0104 & 0.0105 & 270 & 0.138 & 0.140 & 710 & 1.24 & 1.26\
145 & 0.0177 & 0.0178 & 310 & 0.200 & 0.202 & 840 & 1.50 & 1.53\
160 & 0.0264 & 0.0267 & 350 & 0.275 & 0.278 & 1040 & 2.10 & 2.16\
180 & 0.0366 & 0.0370 & 410 & 0.355 & 0.358 & 1140 & 2.21 & 2.28\
200 & 0.0495 & 0.0500 & 455 & 0.453 & 0.457 & 1330 & 2.57 & 2.69\
220 & 0.0690 & 0.0698 & 480 & 0.539 & 0.544 & 1510 & 3.21 & 3.36\
230 & 0.0905 & 0.0915 & 550 & 0.730 & 0.738 & 1570 & 3.41 & 3.59\
250 & 0.115 & 0.117 & 600 & 0.765 & 0.772 & 1760 & 3.62 & 3.85\
\
100 & 0.0104 & 0.0107 & 270 & 0.286 & 0.291 & 715 & 1.52 & 1.57\
125 & 0.0262 & 0.0268 & 310 & 0.511 & 0.519 & 836 & 1.80 & 1.86\
145 & 0.0460 & 0.0470 & 350 & 0.518 & 0.524 & 900 & 2.00 & 2.07\
160 & 0.0573 & 0.0585 & 410 & 0.994 & 1.01 & 1002 & 2.32 & 2.42\
180 & 0.0662 & 0.0675 & 470 & 0.906 & 0.925 & 1170 & 2.68 & 2.82\
200 & 0.123 & 0.126 & 550 & 1.19 & 1.22 & 1398 & 2.20 & 2.29\
225 & 0.173 & 0.177 & 610 & 1.26 & 1.29 & 1606 & 2.50 & 2.60\
250 & 0.223 & 0.227 & 710 & 1.74 & 1.79 & & &\
Statistical weights and statistical sums {#App B}
========================================
The fractional populations for the levels of a polyatomic molecule with $n_v$ modes and vibrational quantum numbers $(v_1 v_2 v_3\ldots)$ are given by $$\delta_{(v_1 v_2 v_3\ldots)} = \frac{g_{(v_1 v_2 v_3\ldots)}}{Q_v}
\exp\left(-\frac{\epsilon_{(v_1 v_2 v_3\ldots)}}{k_B T}\right)$$ where $\epsilon_{(v_1 v_2 v_3\ldots)}$ is the level energy and $g$ the total statistical weight, $$g_{(v_1 v_2 v_3\ldots)} = \prod_{n=1}^{n=n_v}\frac{(v_n+d_n-1)!}{v_n!(d_n-1)!}$$ where $d_n$ is the degeneracy multiplicity for mode $n$, and $Q_v$ the vibrational statistical sum which, in the harmonic oscilator approximation for the vibrational states, is $$Q_v = \prod_{n=1}^{n=n_v}(1-Z_n)^{-d_n}\,,\quad Z_n = \exp\{-hc\nu_n/k_B T\}$$ where $\nu_n$ are the vibrational frequencies.
References {#references .unnumbered}
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abstract: |
Recall that combinatorial $2s$-designs admit a classical lower bound $b \ge \binom{v}{s}$ on their number of blocks, and that a design meeting this bound is called tight. A long-standing result of Bannai is that there exist only finitely many nontrivial tight $2s$-designs for each fixed $s \ge 5$, although no concrete understanding of ‘finitely many’ is given. Here, we use the Smith Bound on approximate polynomial zeros to quantify this asymptotic nonexistence. Then, we outline and employ a computer search over the remaining parameter sets to establish (as expected) that there are in fact no such designs for $5 \le s \le 9$, although the same analysis could in principle be extended to larger $s$. Additionally, we obtain strong necessary conditions for existence in the difficult case $s=4$.\
\
MSC Primary 05B05, 33D45; Secondary 05E30\
*Key words and phrases*: tight design, symmetric design, orthogonal polynomials, Delsarte theory
author:
- |
Peter Dukes and Jesse Short-Gershman[^1]\
Mathematics and Statistics\
University of Victoria\
Victoria, BC V8W 3R4\
Canada\
[dukes@uvic.ca, jesseasg@uvic.ca]{}
title: |
**Nonexistence Results for Tight\
Block Designs**
---
Introduction
============
Let $v \ge k \ge t$ be positive integers and ${\lambda}$ be a nonnegative integer. A $t$-$(v,k,{\lambda})$ *design*, or simply a $t$-*design*, is a pair $(V,\mathcal B)$ where $V$ is a $v$-set and $\mathcal B$ is a collection of $k$-subsets of $V$ such that any $t$-subset of $V$ is contained in exactly ${\lambda}$ elements of $\mathcal B$. The elements of $V$ are *points* and the elements of $\mathcal B$ are *blocks*. Since $t$-designs are also $i$-designs for $i \le t$, the parameter $t$ is typically called the *strength*. The number of blocks is usually denoted $b$ and an easy double-counting argument shows $b={\lambda}\binom{v}{t}/\binom{k}{t}$.
Suppose $(V,\mathcal B)$ is a $t$-$(v,k,{\lambda})$ design. Generalizing Fisher’s Inequality, Ray-Chaudhuri and Wilson [@RW] showed that if $t$ is even, say $t=2s$, and $v \ge k+s$, then $b \ge \binom{v}{s}$. If equality holds in this bound, we say $(V,\mathcal B)$ is *tight*. The *trivial* tight $2s$-designs are those with $v = k+s$, where each of the $\binom{v}{k} = \binom{v}{s}$ $k$-subsets of $V$ is a block. The case of odd strength is investigated in [@IS]; however, it is impossible for $(2s-1)$-designs to be tight in the sense of having $\binom{v}{s-1}$ blocks.
Returning to even strength, the full set of parameters $v$ and $k$ for which a tight $2s$-design exists has only been determined for $s=2,3$. Note that, when $s=1$, tight 2-designs have $b=v$ and are the ‘symmetric’ designs; see [@Ionin; @Lander] for surveys of this rich (yet very challenging) topic. In 1975, Ito [@Ito] published a proof that the only nontrivial tight 4-designs are the Witt 4-(23,7,1) design and its complementary 4-(23,16,52) design, but his proof was found to be incorrect. A few years later, Enomoto, Ito, and Noda [@EIN] proved the weaker result that there are finitely many nontrivial tight 4-designs, though still believing Ito’s initial claim to be true. Finally, in 1978, Bremner [@Bremner] successfully settled $s=2$ by reaffirming Ito’s result. Peterson [@Peterson] proved in 1976 that there exist no nontrivial tight 6-designs.
Bannai [@B] proved that there exist only finitely many nontrivial tight $2s$-designs for each $s \ge 5$. The case $s=4$ is quite open, and the ‘finitely many’ for $s \ge 5$ is not explicit and potentially grows with $s$. However, it is probably the case that there are no unknown tight $2s$-designs for $s \ge 2$.
Central to these negative results is the following strong condition, discovered first by Ray Chaudhuri and Wilson [@RW], and also implicitly by Delsarte [@Delsarte].
[([@Delsarte; @RW])]{} \[psi\] If there exists a tight $2s$-$(v,k,{\lambda})$ design, then the zeros of the following degree $s$ polynomial $\Psi_s(x)$ are the intersection numbers of the design, and hence they must all be nonnegative integers: $$\label{psi-def}
\Psi_s(x)=\sum_{i=0}^s(-1)^{s-i}\frac{\binom{v-s}{i}\binom{k-i}{s-i}\binom{k-1-i}{s-i}}{\binom{s}{i}}\binom{x}{i}.$$
The polynomials $\Psi_s$ are variants of the Hahn polynomials, [@Hahn].
Since a $2s$-design with $v \ge k+s$ induces at least $s$ intersection numbers [@RW], it follows that the zeros of $\Psi_s$ must additionally be distinct integers for tight designs. Note also that $\Psi_s$ has no dependence on $\lambda$; indeed, for tight designs $\lambda = \binom{v}{s}\binom{k}{2s} \binom{v}{2s}^{-1}$ and is therefore uniquely determined by $v$ and $k$.
Analogously, the Lloyd polynomials $L_e(x)$ are important for the characterization of perfect $e$-error-correcting codes; see [@vL]. It is interesting that this characterization of perfect codes was completed long ago, while the open problems mentioned before Proposition \[psi\] remain for tight designs. Our goal here is to revive the interest in tight designs and take a modest step toward the full characterization of their parameters.
The outline is as follows. In Section 2, we review the work of Bannai in [@B] on the asymptotic structure of the zeros of $\Psi_s$. Extending this, we obtain some exact bounds relevant to this analysis. Section 3 summarizes the techniques for exhausting small cases $s \ge 5$, and Section 4 is devoted to a partial analysis of the case $s=4$. An appendix of tables following the main text will prove useful to the interested reader.
Bannai’s analysis and the Smith bound
=====================================
Notation
--------
Assuming a tight design, let $x_i$, for $i = {-{\left\lfloor\tfrac{s}{2}\right\rfloor},\dotsc,(0),\dotsc,{\left\lfloor\tfrac{s}{2}\right\rfloor}}$, denote the zeros of $\Psi_s$ listed in increasing order. For example, the zeros of $\Psi_4$ and $\Psi_5$ are denoted $x_{-2}<x_{-1}<x_1<x_2$ and $x_{-2}<x_{-1}<x_0<x_1<x_2$, respectively.
An important parameter is the arithmetic mean of the zeros of $\Psi_s(x)$, which we denote by $\overline{\alpha}$. From the coefficient of $x^{s-1}$, we have $$\label{alpha-bar}
\overline{\alpha} = \frac{(k-s+1)(k-s)}{v-2s+1}+\frac{s-1}{2}.$$ Now define, as in [@B], $$\alpha = \frac{(k-s+1)(k-s)}{v-2s+1},$$ so that $\overline{\alpha} = \alpha+(s-1)/2$. Also, following Bannai’s notation, let us redefine the parameter $t$ as $$t= \frac{v-2s+1}{k-s+1}.$$ Note $t=2$ implies $v=2k+1$. Moreover, if $v<2k$, we may complement blocks, replacing $k$ with $v-k$ and obtain $v>2k$. This is discussed further in Section \[t-sym\].
Finally, put $\beta = \left(1-\tfrac{1}{t}\right)\sqrt{\alpha}$. In terms of $v$ and $k$, $$\beta = \frac{(v-k-s)\sqrt{(k-s+1)(k-s)}}{(v-2s+1)^{3/2}}.$$ So, in particular, $\beta = 0$ if and only if $v=k+s$. In some sense $\beta$ can be seen as measuring the ‘nontriviality’ of a (tight) $2s$-design. Note also that $$\begin{aligned}
\label{k}
k &=& t^3(t-1)^{-2} \beta^2 + s,~{\rm and}\\
\label{v}
v &=& t^4(t-1)^{-2} \beta^2 +t+2s-1.\end{aligned}$$
Bannai’s proof of the existence of only finitely many nontrivial tight $2s$-designs, $s \ge 5$, is divided into cases according to this parameter $\beta$. In particular, he proves
- for any $\beta_0$, there are only finitely many tight $2s$-designs with $\beta \le \beta_0$; and
- there exists $\beta_0$ (depending only on $s$), such that there are no nontrivial tight $2s$-designs with $\beta > \beta_0$.
Here, our main goal is to compute such a $\beta_0$ explicitly for $5 \le s \le 9$ and, by searching across all pairs $(v,k)$ for which $\beta \le \beta_0$, show that there are in fact zero nontrivial tight $2s$-designs for these $s$.
Symmetry with respect to the parameter $t$ {#t-sym}
------------------------------------------
In the analytic work which follows, it is helpful to obtain a lower bound on $t$. As discussed above, we may complement blocks to assume $v \ge 2k$. The following was mentioned but not fully proven in [@B].
\[vnot2k\] Let $s \ge 1$. There are no tight $2s$-designs with $v = 2k$.
[[*Proof:* ]{}]{}Suppose $v = 2k$. Then from (\[alpha-bar\]), $s\overline{\alpha}-\binom{s}{2} = s\alpha = \frac{s(k-s+1)(k-s)}{2(k-s)+1}$. Without too much effort, it can be seen that the least residue of $s(k-s+1)(k-s) \pmod{2(k-s)+1}$, denoted here by $r_{k,s}$, satisfies $$r_{k,s}=
\begin{cases}
2(k-s)-\frac{s-4}{4} & \text{if $s \equiv 0 \imod{4}$};\\
k-s-\frac{s-2}{4} & \text{if $s \equiv 2 \imod{4}$};\\
\frac{k-s}{2}-\frac{s-1}{4} & \text{if $s \equiv 1 \imod{4}$ and $k$ is odd} \\
& \text{~~~or $s \equiv 3 \imod{4}$ and $k$ is even};\\
\frac{3}{2}(k-s)-\frac{s-3}{4} & \text{if $s \equiv 3 \imod{4}$ and $k$ is odd}\\
& \text{~~~or $s \equiv 1 \imod{4}$ and $k$ is even}.\\
\end{cases}$$ Since $k-s \ge s$, it follows that in all cases $r_{k,s}$ is an integer lying strictly between 0 and $2(k-s)+1$, so $s\overline{\alpha}-\binom{s}{2}$ is not an integer. But the integrality of $s\overline{\alpha}$ is necessary for the existence of a tight design since it is the sum of the zeros of $\Psi_s(x)$; therefore there are no tight $2s$-designs with $v = 2k$.
Now, we are able to justify assuming that $t \ge 2$ for nonexistence of tight designs.
\[t\] Let $s \ge 1$. If there exists a nontrivial tight $2s$-design with $t < 2$, then there also exists a nontrivial tight $2s$-design with $t \ge 2$.
[[*Proof:* ]{}]{}Suppose $\mathcal D$ is a nontrivial tight $2s$-$(v,k,{\lambda})$ design with $t < 2$. This means $k \le v \le 2k-1$ because $v \neq 2k$ by Lemma \[vnot2k\], and so the complementary $2s$-$(v,v-k,{\lambda}')$ design of $\mathcal D$ is a nontrivial tight $2s$-design with $t \ge 2$.
Incidentally, Bannai and Peterson ruled out the case $t=2$, observing that it yields symmetric zeros of $\Psi_s$ about their mean $\overline{\alpha}$. This is a key observation.
\[v2k+1\] [([@B; @Peterson])]{} There does not exist any tight $2s$-design with $v=2k+1$.
Hermite polynomials
-------------------
Let $H_s(x)$ denote the normalized Hermite polynomial of degree $s$ defined recursively by $H_0(x) = 1$, $H_1(x) = x$, and for $s \ge 2$, $$H_s(x) = xH_{s-1}(x)-(s-1)H_{s-2}(x).$$ Furthermore, let $\xi_i$, $i = {-{\left\lfloor\tfrac{s}{2}\right\rfloor},\dotsc,(0),\dotsc,{\left\lfloor\tfrac{s}{2}\right\rfloor}}$, denote the zeros of $H_s(x)$ listed in increasing order. It is easily seen that $\xi_{-i} = -\xi_i$ for each $i$. See Appendix \[hzeros\] for a table of zeros of $H_s(x)$, $1 \le s \le 10$. In particular, for the analytical work in Section 3, we will make use of the following known estimates.
\[xi\]
1. If $s$ is odd and $\ge 5$, then $\xi_1^2 < \sqrt{3}$.
2. If $s$ is even and $\ge 8$, then $\xi_2^2-\xi_1^2 < \sqrt{3}$.
3. If $s=6$, then $1.0 < \frac{\xi_2^2-\xi_1^2}{3} < 1.1$, $3.5 < \frac{\xi_3^2-\xi_1^2}{3} < 3.6$, and $3.34634 < \frac{\xi_3^2-\xi_1^2}{\xi_2^2-\xi_1^2} < 3.34635$.
[[*Proof:* ]{}]{}Items (i) and (ii) are referenced in Bannai’s Proposition 13 and proven on page 126 of [@Szego]. Item (iii) can be verified numerically. See Appendix \[hzeros\]. (Note that Bannai’s Proposition 13 (iii) actually contains an error).
A useful identity is $$\label{Hermite-derivative}
H_s'(x) = sH_{s-1}(x).$$
For later reference we define, again as in [@B], $$\label{lam-i-def}
{\lambda}_i={\lambda}_i(t) = \left(1-\tfrac{2}{t} \right)^2 \left(\tfrac{\xi_i^2}{6} - \tfrac{s-1}{6}\right).$$ Informally, Proposition 16 in [@B] states that as $\beta\to\infty$, the zeros $x_i$ of $\Psi_s(x)$ approach $\overline{\alpha}+\beta\xi_i+{\lambda}_i$. That is, when suitably normalized, $\Psi_s$ behaves like $H_s$ for large $\beta$ and fixed $t$.
The Smith bound
---------------
We now state a useful result for explicitly finding $\beta_0$. Sometimes known as the Smith bound, it is a consequence of the Gershgorin circle theorem.
[([@Smith])]{} \[smithbound\] Let $P(z)$ be a monic polynomial of degree $n$ and let $\xi_1,\dots,\xi_n$ be distinct points approximating the zeros of $P(z)$. Define the circles $$\Gamma_i = \left\{z : |z - \xi_i| \le \frac{n|P(\xi_i)|}{|Q'(\xi_i)|}\right\},$$ where $Q(z)$ is the monic polynomial of degree $n$ with zeros $\xi_1,\dots,\xi_n$. Then the union of the circular regions $\Gamma_i$ contains all the zeros of $P(z)$, and any connected component consisting of just $k$ circles $\Gamma_i$ contains exactly $k$ zeros of $P(z)$.
Let $s \ge 1$. For each $i \in \{{-{\left\lfloor\tfrac{s}{2}\right\rfloor},\dotsc,(0),\dotsc,{\left\lfloor\tfrac{s}{2}\right\rfloor}}\}$, define the monic degree $s$ polynomial (in $z$) $$\label{G-def}
G_s^{(i)}(z) = \frac{s!}{\beta^s \binom{v-s}{s}} \Psi_s(\overline{\alpha} + \beta z + {\lambda}_i),$$ and put $z_i = (x_i-\overline{\alpha}-{\lambda}_i)/\beta$, the zero of $G_s^{(i)}(z)$ corresponding to $x_i$.
We will see from Propositions \[smithtight\] and \[beta1\] that the $z_i$ are well-approximated by the $\xi_i$ as $\beta\to\infty$, independently of $t$.
\[smithtight\] Let $s \ge 1.$ Then $$|z_i - \xi_i| \le \frac{|G_s^{(i)}(\xi_i)|}{|H_{s-1}(\xi_i)|}.$$
[[*Proof:* ]{}]{}Simply apply Theorem \[smithbound\] to the polynomial $G_s^{(i)}(z)$, letting $Q(z) = H_s(z)$, to get $$|z_i - \xi_i| \le \frac{ s |G_s^{(i)}(\xi_i)| }{ |H_s'(\xi_i)| }.$$ The result now follows from (\[Hermite-derivative\]).
Bounding $G_s$ in terms of $\beta$
----------------------------------
In the next proposition, it is helpful to think of the $G_s^{(i)}(\xi_i)$ as functions of $\beta$ and $t$.
\[beta1\] Let $s \ge 2$. For each $i \in \{{-{\left\lfloor\tfrac{s}{2}\right\rfloor},\dotsc,(0),\dotsc,{\left\lfloor\tfrac{s}{2}\right\rfloor}}\}$, there exist constants $B_i,C_i$ such that whenever $\beta > B_i$, $$|G_s^{(i)}(\xi_i)| < \frac{C_i}{\beta^2}$$ for all $t \ge 2$.
The necessary ingredients for this result were proved in [@B], although the bound was not directly stated in this form. Therefore, we omit the proof and instead focus on how to (carefully) obtain $B_i$ and $C_i$ for small $s$ using some basic computer algebra.
\[al\] For fixed $s$ and $i$, we may obtain constants $B_i$ and $C_i$ in Proposition \[beta1\] by the following procedure.
1. Using (\[psi-def\]), substitute (\[alpha-bar\]), (\[k\]), (\[v\]) and (\[lam-i-def\]) into (\[G-def\]). To defer floating-point precision issues, we first replace $\xi_i$ in (\[lam-i-def\]) by a symbolic parameter $r$.
2. This results in an expression for $G_s^{(i)}(r)$ as a rational function of $\beta$, say $$G_s^{(i)}(r)(\beta,t) = \frac{p(r,\beta,t)}{q(\beta,t)}.$$ Here, the denominator is $$\label{den-def}
q(\beta,t) = \beta^s \tbinom{v-s}{s} = \beta^s \binom{t^4(t-1)^{-2} \beta^2 +t+s-1}{s}.$$
3. Observe that $q$ is positive for $\beta>0$ and $t \ge 2$, and that a lower bound on $q$ is $$\tilde{q}(\beta,t) = \tfrac{1}{s!} \beta^{3s} t^{4s} (t-1)^{-2s}.$$ This is obtained by replacing each factor in the falling factorial of (\[den-def\]) by $t^4(t-1)^{-2} \beta^2$.
4. The numerator $p(r,\beta,t)$ is, for general $r$, a polynomial of degree $3s$ in $\beta$. However, for $r=\xi_i$, Proposition \[beta1\] shows the two top coefficients, namely of $\beta^{3s}$ and $\beta^{3s-1}$, vanish. Again, to maintain symbolic algebra, we artifically replace these coefficients by zero and call this polynomial $\tilde{p}(r,\beta,t)$.
5. We have $$\beta^2 G_s^{(i)}(r)(\beta,t) \le \frac{\beta^2 \tilde{p}(r,\beta,t)}{\tilde{q}(\beta,t)}.$$ Note that for $r=\xi_i$, the right hand side is a polynomial in $\beta^{-1}$.
6. Consider the coefficient $\kappa_j(r,t)$ of $\beta^{3s-j}$ in $\beta^2 \tilde{p}(r,\beta,t)$. With $r=\xi_i$, compute (or upper-bound) the maxima $$M_{j}= \sup_{t \ge 2} \frac{|\kappa_j(\xi_i,t)|}{\tfrac{1}{s!}(t-1)^{-2s} t^{4s}}.$$ Then, estimating term-by-term, $$|\beta^2 G_s^{(i)}(\xi_i)(\beta,t)| \le M_0+M_1 \beta^{-1} + M_2 \beta^{-2} + \dots$$ for all $t \ge 2$.
7. Construct $B_i,C_i$ so that $\beta > B_i$ implies $M_0+ M_1\beta^{-1} + M_2 \beta^{-2} + \dots \le C_i$. Note that with sufficiently large $B_i$ and a safe choice of $C_i$, it suffices to estimate the first few coefficients $M_j$.
We should remark that for small $s$, Algorithm \[al\] – even the calculation of all $3s-1$ coefficient maxima $M_j$ – is essentially instantaneous on today’s personal computers. Moreover, deferring the use of floating-point arithmetic to step 5 – when $t$ is eliminated – makes our subsequent use of floating-point numbers $M_j$ quite mild. Indeed, there is virtually no loss in taking $M_j$ as (integer) ceilings of the suprema, so that estimating for $C_i$ can be performed in $\mathbb{Q}$.
See Appendix \[C\_is\] for the results of this calculation for each $5 \le s \le 9$ and all relevant indices $i$.
Bounding the zeros
------------------
We are now ready for our main result of this section. This is in Bannai’s paper [@B], but with no attempt to control $\beta$.
\[beta2\] Fix a positive integer $s$ and $i \in \{{-{\left\lfloor\tfrac{s}{2}\right\rfloor},\dotsc,(0),\dotsc,{\left\lfloor\tfrac{s}{2}\right\rfloor}}\}$. Put $y_i = x_i - \overline{\alpha} - \beta \xi_i,$ where recall $x_i$ and $\xi_i$ are corresponding roots of $\Psi_s$ and $H_s$, respectively. Let $\epsilon > 0$ and define $$\widehat{\beta}(i,\epsilon) = \max\left\{B_i,\frac{C_i}{\epsilon D_i}\right\},$$ where $D_i=|H_{s-1}(\xi_i)|$. Then for all $\beta > \widehat{\beta}$ and all $t \ge 2$, $$|y_i - {\lambda}_i| < \epsilon.$$
[[*Proof:* ]{}]{}Observe that $|y_i-{\lambda}_i| = \beta|z_i-\xi_i|$, since $$x_i=\overline{\alpha}+\beta\xi_i+y_i=\overline{\alpha}+\beta z_i+{\lambda}_i.$$ The estimate now follows easily from Propositions \[smithtight\] and \[beta1\].
The case $s \ge 5$
==================
Estimates for large $\beta$
---------------------------
The goal here is to provide formulas for the smallest $\beta_0$ possible (see the end of Section 2.1) using the $B_i$ and $C_i$ constructed in Algorithm \[al\]. This task is simplified under the conditions that $B_i$ is independent of $i$ and $C_i=C_{-i}$. There is no loss of generality in assuming this because we can simply take $\overline{B}$ to be the maximum of the $B_i$ and $\overline{C_i}=\max\{C_i,C_{-i}\}$, and then redefine each $B_i = \overline{B}$ and $C_i = C_{-i} = \overline{C_i}$. In fact, this is not necessary for our explicit constructions because the constants in Appendix \[C\_is\] satisfy the above conditions.
Again, for convenience, we denote $|H_{s-1}(\xi_i)|$ by $D_i$ in the following proofs.
\[sOdd\] Let $s \ge 5$ be odd.
1. There exists $\beta_1$ such that, whenever $\beta > \beta_1$, $$|y_1+y_{-1}-2y_0| < 1.$$
2. There exists $\beta_2$ such that, whenever $\beta > \beta_2$, $$|y_i+y_{-i}-y_{i-1}-y_{-(i-1)}| < 1+\frac{\xi_i^2-\xi_{i-1}^2}{\xi_{i-1}^2-\xi_{i-2}^2}|y_{i-1}+y_{-(i-1)}-y_{i-2}-y_{-(i-2)}|$$ for $2\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$.
3. There exists $\beta_0(s)$ such that, whenever $\beta > \beta_0$ and $y_i+y_{-i}-y_{i-1}-y_{-(i-1)}$ is an integer for $1\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$, it is necessarily the case that $y_i+y_{-i}-y_{i-1}-y_{-(i-1)} = 0$ for each $i$.
[[*Proof:* ]{}]{}
1. Observe that since $t \ge 2$, $$\label{sOdd-est}
0 \le 2({\lambda}_1-{\lambda}_0) = (1-\tfrac{2}{t})^2\tfrac{\xi_1^2}{3} <\tfrac{\xi_1^2}{3}.$$ Define $$\epsilon_0 = \frac{1}{2}\left(1-\frac{\xi_1^2}{3}\right)\left(1+\frac{C_1D_0}{C_0D_1}\right)^{-1}\hspace{1mm}\text{and}\hspace{3mm}\beta_1 = \widehat{\beta}(0,\epsilon_0).$$ If $\epsilon_1=\epsilon_0\tfrac{C_1D_0}{C_0D_1}$, then $$\widehat{\beta}(1,\epsilon_1)=\beta_1\hspace{3mm}\text{and}\hspace{3mm}2\epsilon_0+2\epsilon_1=1-\frac{\xi_1^2}{3}.$$ Hence for $\beta > \beta_1$, $$\begin{aligned}
|y_1+y_{-1}-2y_0-2({\lambda}_1-{\lambda}_0)| &\le& |y_1-{\lambda}_1|+|y_{-1}-{\lambda}_{-1}|+2|y_0-{\lambda}_0|, \\
& < & 2\epsilon_0+2\epsilon_1 = 1-\frac{\xi_1^2}{3},\end{aligned}$$ By (\[sOdd-est\]), $$-\left(1-\tfrac{\xi_1^2}{3}\right) < y_1+y_{-1}-2y_0 < 1$$ and the claim follows.
2. For $2\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$, let $$a_i=\frac{\xi_i^2-\xi_{i-1}^2}{\xi_{i-1}^2-\xi_{i-2}^2}\hspace{3mm}\text{and}\hspace{3mm}\epsilon_i = \frac{1}{2}\left(1+(1+a_i)\frac{C_{i-1}D_i}{C_iD_{i-1}}+a_i\frac{C_{i-2}D_i}{C_iD_{i-2}}\right)^{-1}.$$ Note if $2\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$, then $$\label{lam-rel}
({\lambda}_i-{\lambda}_{i-1}) = ({\lambda}_{i-1}-{\lambda}_{i-2})a_i.$$ Define $\beta_2 = \max\left\{\widehat{\beta}(i,\epsilon_i):2\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}\right\}$. For $\beta > \beta_2$ and working as in (i), $$\label{xx}
|y_i+y_{-i}-y_{i-1}-y_{-(i-1)}-2({\lambda}_i-{\lambda}_{i-1})| < 2\epsilon_i \left( 1+\tfrac{C_{i-1}D_i}{C_iD_{i-1}} \right).$$ Using (\[lam-rel\]) and (\[xx\]) again with $i-1$ replacing $i$, $$\nonumber
\begin{split}
|y_i+y_{-i}-&y_{i-1}-y_{-(i-1)}| \\
&< 2\epsilon_i \left(1+\tfrac{C_{i-1}D_i}{C_iD_{i-1}}\right) + 2 \epsilon_i a_i \left(\tfrac{C_{i-1}D_i}{C_iD_{i-1}} + \tfrac{C_{i-2}D_i}{C_iD_{i-2}} \right) \\
&~~~~~~~~~~ + a_i|y_{i-1}+y_{-(i-1)}-y_{i-2}-y_{-(i-2)}| \\
&=1+a_i|y_{i-1}+y_{-(i-1)}-y_{i-2}-y_{-(i-2)}|,
\end{split}$$ as required.
3. Set $\beta_0(s) = \max\{\beta_1,\beta_2\}$ and assume that $\beta > \beta_0(s)$ and $y_i+y_{-i}-y_{i-1}-y_{-(i-1)}$ is an integer for $1\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$. By (i), $y_1+y_{-1}-2y_0 = 0$ since it is an integer whose absolute value is less than 1. Assume that $y_{i-1}+y_{-(i-1)}-y_{i-2}-y_{-(i-2)} = 0$ for some $2\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$. Then (ii) gives that $|y_i+y_{-i}-y_{i-1}-y_{-(i-1)}|$ is also less than one and hence equal to 0 since it is an integer, so by induction $y_i+y_{-i}-y_{i-1}-y_{-(i-1)} = 0$ for $1\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$, and so the proof is complete.
\[sEven\] Let $s \ge 8$ be even.
1. There exists $\beta_1$ such that, whenever $\beta > \beta_1$, $$|y_2+y_{-2}-y_1-y_{-1}| < 1.$$
2. There exists $\beta_2$ such that, whenever $\beta > \beta_2$, $$|y_i+y_{-i}-y_{i-1}-y_{-(i-1)}| < 1+\frac{\xi_i^2-\xi_{i-1}^2}{\xi_{i-1}^2-\xi_{i-2}^2}|y_{i-1}+y_{-(i-1)}-y_{i-2}-y_{-(i-2)}|$$ for $3\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$.
3. There exists $\beta_0(s)$ such that, whenever $\beta > \beta_0(s)$ and $y_i+y_{-i}-y_{i-1}-y_{-(i-1)}$ is an integer for $2\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$, it is necessarily the case that $y_i+y_{-i}-y_{i-1}-y_{-(i-1)} = 0$ for each $i$.
[[*Proof:* ]{}]{}
1. Since $t \ge 2$, $$\label{sEven-est}
0 \le 2({\lambda}_2-{\lambda}_1) = (1-\tfrac{2}{t})^2\tfrac{\xi_2^2-\xi_1^2}{3} < \tfrac{\xi_2^2-\xi_1^2}{3}.$$ Define $$\epsilon_1 = \frac{1}{2}\left(1-\frac{\xi_2^2-\xi_1^2}{3}\right)\left(1+\frac{C_2D_1}{C_1D_2}\right)^{-1}\hspace{1mm}\text{and}\hspace{3mm}\beta_1 = \widehat{\beta}(1,\epsilon_1).$$ If $\epsilon_2=\epsilon_1\tfrac{C_2D_1}{C_1D_2}$, then $$\widehat{\beta}(2,\epsilon_2)=\beta_1\hspace{3mm}\text{and}\hspace{3mm}2\epsilon_1+2\epsilon_2=1-\frac{\xi_2^2-\xi_1^2}{3}.$$ Hence for $\beta > \beta_1$, $$|y_2+y_{-2}-y_1-y_{-1}-2({\lambda}_2-{\lambda}_1)| < 2\epsilon_1+2\epsilon_2=1-\frac{\xi_2^2-\xi_1^2}{3}.$$ By (\[sEven-est\]), $$-\left(1-\tfrac{\xi_2^2-\xi_1^2}{3}\right) < y_2+y_{-2}-y_1-y_{-1} < 1$$ and the claim follows.
2. Define $\epsilon_i$ and $\beta_2$ in as in the proof of Proposition \[sOdd\] (ii), but omit $i=2$.
3. Imitate the proof of Proposition \[sOdd\] (iii).
In the case $s=6$, $\tfrac{\xi_2^2-\xi_1^2}{3} > 1$. Hence it is impossible to choose a $\beta_1$ to guarantee that $y_2+y_{-2}-y_1-y_{-1} = 0$ whenever it is an integer and $\beta > \beta_1$.
\[sEquals6\] Let $s=6$. There exists $\beta_0(6)$ such that, whenever $\beta > \beta_0(6)$ and $(y_2+y_{-2}-y_1-y_{-1})$, $(y_3+y_{-3}-y_1-y_{-1})$ are both integers, it is necessarily the case that $y_2+y_{-2}-y_1-y_{-1} = y_3+y_{-3}-y_1-y_{-1} = 0$.
[[*Proof:* ]{}]{}Observe $$0 \le 2({\lambda}_2-{\lambda}_1) < \tfrac{\xi_2^2-\xi_1^2}{3} < 1.1~~\text{ and}~~
0 \le 2({\lambda}_3-{\lambda}_1) < \tfrac{\xi_3^2-\xi_1^2}{3} < 3.6$$ by Proposition \[xi\] (iii). Let $a = \tfrac{\xi_3^2-\xi_1^2}{\xi_2^2-\xi_1^2}$ and define $$\epsilon_1 = \frac{1}{2}\left(a-3\right)\left(1+a+a\frac{C_2D_1}{C_1D_2}+\frac{C_3D_1}{C_1D_3}\right)^{-1}\hspace{1mm}\text{and}\hspace{3mm}\beta_0(6) = \widehat{\beta}(1,\epsilon_1).$$ Then, with $$\epsilon_2 = 2\epsilon_1\left(1+\frac{C_2D_1}{C_1D_2}\right)\hspace{3mm}\text{and}\hspace{3mm}\epsilon_3 = 2\epsilon_1\left(1+\frac{C_3D_1}{C_1D_3}\right),$$ we have $$\epsilon_2a+\epsilon_3 = a-3,$$ $$0 < \epsilon_2 < (a-3)/a\approx 0.10350,\hspace{3mm}\text{and}$$ $$0 < \epsilon_3 < a-3\approx 0.34635.$$ Assume $\beta > \beta_0(6)$. Then $|y_2+y_{-2}-y_1-y_{-1}-2({\lambda}_2-{\lambda}_1)| < \epsilon_2$ implies $$y_2+y_{-2}-y_1-y_{-1} \in \{0,1\}.$$ Likewise, $|y_3+y_{-3}-y_1-y_{-1}-2({\lambda}_3-{\lambda}_1)| < \epsilon_3$ implies $$\label{0123}
y_3+y_{-3}-y_1-y_{-1} \in \{0,1,2,3\}.$$ If $y_2+y_{-2}-y_1-y_{-1} = 0$, then $2({\lambda}_2-{\lambda}_1) < \epsilon_2$ and so $0 \le 2({\lambda}_3-{\lambda}_1) < \epsilon_2a$. Hence $y_3+y_{-3}-y_1-y_{-1} < \epsilon_2a+\epsilon_3 = a-3\approx 0.34635$, and so $y_3+y_{-3}-y_1-y_{-1} = 0$. On the other hand, suppose $y_2+y_{-2}-y_1-y_{-1} = 1$. Then $2({\lambda}_2-{\lambda}_1) > 1-\epsilon_2$ and so $2({\lambda}_3-{\lambda}_1) > (1-\epsilon_2)a = a-\epsilon_2a$. Hence $y_3+y_{-3}-y_1-y_{-1} > a-\epsilon_2a-\epsilon_3 = 3$, a contradiction to (\[0123\]).
It follows that $y_2+y_{-2}-y_1-y_{-1} = y_3+y_{-3}-y_1-y_{-1} = 0$.
To summarize, we have the following reworking of Proposition 17 in [@B], but with explicit $\beta_0$.
\[bannai\] For each $s \ge 5$, there are no tight $2s$-designs with $\beta > \beta_0(s)$.
[[*Proof:* ]{}]{}Suppose $x_{-{\left\lfloor\tfrac{s}{2}\right\rfloor}} < \cdots < x_{{\left\lfloor\tfrac{s}{2}\right\rfloor}}$ are the intersection numbers of a tight $2s$-design with $\beta > \beta_0(s)$. By Proposition \[t\], we may assume $t \ge 2$. Then, since $\xi_{-i}=-\xi_i$ and ${\lambda}_{-i}={\lambda}_i$, we have $x_i+x_{-i}-x_j-x_{-j} = y_i+y_{-i}-y_j-y_{-j}$, and this implies that $y_i+y_{-i}-y_j-y_{-j}$ is an integer for each $i,j \in \{(0),1,2,\dots,{\left\lfloor\tfrac{s}{2}\right\rfloor}\}$. By Propositions \[sOdd\] (iii), \[sEven\] (iii) and \[sEquals6\], these integers must vanish. Specifically,
$s$ is odd and $\ge 5$ implies $y_i+y_{-i}-y_{i-1}-y_{-(i-1)} = 0$ for $1\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$.\
$s$ is even and $\ge 8$ implies $y_i+y_{-i}-y_{i-1}-y_{-(i-1)} = 0$ for $2\le i\le{\left\lfloor\tfrac{s}{2}\right\rfloor}$.\
$s=6$ implies $y_2+y_{-2}-y_1-y_{-1} = y_3+y_{-3}-y_1-y_{-1} = 0$.
In each case, the $x_i$ are symmetric about their arithmetic mean $\overline{\alpha}$. By Proposition 2 in [@B], this implies $v=2k+1$. Proposition \[v2k+1\] says this is impossible, and the proof is therefore complete.
Searching over small $\beta$
----------------------------
We now turn to small values of $\beta$, for which the problem becomes finite.
\[search\] To exclude tight $2s$-designs with $\beta \le \beta_0$, we may implement the following steps.
1. Compute $\beta_0$ from the $B_i,C_i$ as in the previous section.
2. By Propositions \[t\] and \[v2k+1\], we may restrict attention to $t>2$. Since $\alpha = \beta^2/(1-\frac{1}{t})^2$, it follows that $\alpha < 4\beta_0^2$. Now, since $\alpha = \left(s\overline{\alpha}+\binom{s}{2}\right)/s$ and $s\overline{\alpha}$ is an integer, we have $\alpha \in \tfrac{1}{s} \mathbb{Z}$. This gives an explicit finite number of admissible $\alpha$, as Bannai observed in [@B].
3. Note that, under the assumption of a tight design, the expression $$\label{coef-integer}
\binom{s}{2}\alpha\left(\alpha+\frac{2\alpha t-\alpha+2}{\alpha t^2+t+1}\right)$$ is an integer. This is because Proposition 5 in [@B] asserts that the coefficient of $x^{s-2}$ in the monic polynomial $s!\Psi_s(x)/\binom{v-s}{s}$ is $$\binom{s}{2}\alpha\left(\alpha+\frac{2\alpha t-\alpha+2}{\alpha t^2+t+1}\right)+\binom{s}{3}\left(3\alpha+\frac{3s-1}{4}\right),$$ and the latter term is always an integer.
4. Fix $\alpha$ as in Step 2. Put $n = k-s = \alpha t$ and define $$g_{\alpha}(n) := \binom{s}{2}\alpha^2\left(1+\frac{2n-\alpha+2}{n^2+n+\alpha}\right)$$ as in (\[coef-integer\]). As $t>2$, we may take a lower bound $n_{\min}(\alpha) = \max\{s,\lfloor 2\alpha \rfloor+1\}$.
5. Since $g'_{\alpha}(n) < 0$ for all $n \ge n_{\min}(\alpha)$, it suffices to loop on integers $n$ from $n_{\min}(\alpha)$ until $n_{\max}(\alpha)$, where $g_\alpha(n_{\max}(\alpha)) \le \lfloor \binom{s}{2}\alpha^2 \rfloor+1.$ Any pairs $(k,v)$ which give integral $g_\alpha(n)$ are obtained by $k = n+s$ and $v = \frac{n^2+n}{\alpha}+2s-1$.
6. In principle, at this point the zeros of $\Psi_s$ for these pairs $(k,v)$ can be analyzed. However, in practice we found it sufficient in all cases to merely see that ${\lambda}= \binom{v}{s}\binom{k}{2s}/\binom{v}{2s}$ was never even an integer.
We wrote a C program that implements Algorithm \[search\] for a given $s$ and $\beta_0$, but with an important optimization. For $n$ near $n_{\max}(\alpha)$, $|g'_{\alpha}(n)|$ is very small so it would be inefficient to loop over $n$ in this region. Therefore, the program loops over integer values of $g_\alpha(n)$ from $\lceil g_\alpha(n_{\max}(\alpha))\rceil$ and checks the integrality of the corresponding $n$ until the derivative becomes larger than a certain threshold (in absolute value), at which point it begins looping over $n$ to a much smaller $n_{\max}$. The program is available by contacting the authors.
Our calculations of $\beta_0$ in step 1 are displayed in Appendix \[C\_is\]. We can report that the method succeeds for $5 \le s \le 9$, and probably much higher $s$. We have chosen to avoid continued searches for $s>9$ until new ideas are obtained. In particular, it would be interesting if $s \ge s_0$ could be excluded for nontrivial tight $2s$-designs.
\[none\] For each $5 \le s \le 9$, there are no nontrivial tight $2s$-designs.
The case $s = 4$
================
The same analytic approach that is successful for $s \ge 5$ fails when $s=4$. We can only guarantee that $$0 \le 2({\lambda}_2-{\lambda}_1) < \frac{\xi_2^2-\xi_1^2}{3} = \sqrt{8/3} \approx 1.63299$$ when $t \ge 2$, and so there does not exist $\beta_0$ such that $|y_2+y_{-2}-y_1-y_{-1}| < 1$ for all $\beta > \beta_0$.
However, it is possible to bound $|y_2+y_{-2}-y_1-y_{-1}|$ away from 2. Let $$\epsilon_1=\frac{1}{2}\left(2-\sqrt{8/3}\right)\left(1+\frac{C_2D_1}{C_1D_2}\right)^{-1}\hspace{1mm}\text{and}\hspace{3mm}\beta_\star(4)=\widehat{\beta}(1,\epsilon_1).$$ Then the existence of a tight 8-design with $\beta > \beta_\star(4)$ and $t \ge 2$ implies $y_2+y_{-2}-y_1-y_{-1} = 1$ and $2({\lambda}_2-{\lambda}_1) \approx 1$, for which $$t = \tfrac{v-7}{k-3} \approx \frac{2}{1-\sqrt[4]{3/8}} \approx 9.1971905725.$$
We are able to obtain more precise conditions in the following result.
\[sEquals4\] If there exists a nontrivial tight $8$-design with parameters $v$ and $k$, then $k >$ [25,000]{} and $f(k,v) = 0$, where $f(k,v)$ is as in Appendix \[fgkv\].
[[*Proof:* ]{}]{}We first used Algorithm \[search\] to find that there are no nontrivial tight 8-designs with $\beta \le \beta_\star(4)$. Thus, any tight 8-design with $t \ge 2$ must have $x_2+x_{-2}-x_1-x_{-1} = y_2+y_{-2}-y_1-y_{-1} = 1$. Consider the monic and root-centered polynomial $$F(x) = 24\Psi_4(x+\overline{\alpha})/\binom{v-4}{4} = x^4+p_1 x^3+p_2 x^2 + p_3 x + p_4.$$ By Equation (15) in [@Peterson], we have $$\begin{aligned}
p_2 &= -\frac{5}{2}-\frac{6(k-3)(k-4)(v-k-3)(v-k-4)}{(v-6)(v-7)^2},\nonumber\\
\label{p-intermsof-kv}
p_3 &= \frac{-4(k-3)(k-4)(v-k-3)(v-k-4)(v-2k+1)(v-2k-1)}{(v-5)(v-6)(v-7)^3},\\
p_4 &= \frac{9}{16}+\frac{3}{2}\cdot\frac{(k-3)(k-4)(v-k-3)(v-k-4)g(k,v)}{(v-4)(v-5)(v-6)(v-7)^4},\nonumber\end{aligned}$$ where $g(k,v)$ is as in Appendix \[fgkv\]. Assuming $x_2+x_{-2}-x_1-x_{-1} = 1$, the roots of $F(x)$ must be $r_1-1/4, r_2+1/4, -r_1-1/4, -r_2+1/4$ where $r_1-1/4 = x_1-\overline{\alpha}$ and $r_2+1/4 = x_2-\overline{\alpha}$ (note that $r_1, r_2 \in \tfrac{1}{4} \mathbb{Z}$). Expanding, $$x^4+p_2x^2+p_3x+p_4 = x^4+\left(-\tfrac{1}{8}-r_1^2-r_2^2\right)x^2+\left(\tfrac{r_1^2}{2}-\tfrac{r_2^2}{2}\right)x+\left(r_1^2-\tfrac{1}{16}\right)\left(r_2^2-\tfrac{1}{16}\right)\text{,}$$ which yields $$\label{s4identity}
p_4 = \left(\frac{p_2}{2}+\frac{1}{8}\right)^2-p_3^2.$$ Substituting (\[p-intermsof-kv\]) into (\[s4identity\]) results in the equation $f(k,v) = 0$. An easy computer search shows that there are no integer solutions $v$ to $f(k,v)=0$ for $9 \le k \le$ 25,000.
By reducing $f(k,v)$ modulo some primes, one may obtain infinite classes of both $k$ and $v$ which admit no soultions. Some more (very easy) computing is required here.
There is no nontrivial tight $8$-$(v,k,{\lambda})$ design with parameters in any of the following congruence classes: $$\begin{array}{l}
k \equiv 12 \pmod{13} \\
k \equiv 5,9,11,12 \pmod{17}
\end{array}
~
\begin{array}{l}
v \equiv 3 \pmod{7} \\
v \equiv 3 \pmod{11} \\
v \equiv 1,3 \pmod{13} \\
v \equiv 0,1,8,12,13,14 \pmod{17}
\end{array}$$
Despite these strict conditions on hypothetical tight $8$-designs, it remains open whether there are a finite number of nontrivial such designs.
To loosely summarize our work, we have shown that any unknown tight $2s$-design with $s>1$, if it exists, must have
- large $s$ and small $\beta$, or
- $s=4$ with very large $k$ and $v$ satisfying strict conditions.
Appendices {#appendices .unnumbered}
==========
{#hzeros}
$s$ $H_s(x)$
----- ---------------------------------
1 $x$
2 $x^2-1$
3 $x^3-3x$
4 $x^4-6x^2+3$
5 $x^5-10x^3+15x$
6 $x^6-15x^4+45x^2-15$
7 $x^7-21x^5+105x^3-105x$
8 $x^8-21x^6+210x^4-420x^2+105$
9 $x^9-36x^7+378x^5-1260x^3+945x$
Note: In the following table, the values of $D_i=|H_{s-1}(\xi_i)|$ are rounded down. $$\begin{array}{|l|l|l|l|}
\hline
s & i & \xi_i~\text{for}~H_s(x) & D_i \ge \\
\hline
1 & 0 & 0 & 1 \\
\hline
2 & 1 & 1 & 1 \\
\hline
\multirow{2}{*}{3} & 0 & 0 & 1 \\
& 1 & \sqrt{3} & 2 \\
\hline
\multirow{2}{*}{4} & 1 & \sqrt{3-\sqrt{6}} = 0.7420 & 1.817 \\
& 2 & \sqrt{3+\sqrt{6}} = 2.3344 & 5.718 \\
\hline
\multirow{3}{*}{5} & 0 & 0 & 3 \\
& 1 & \sqrt{5-\sqrt{10}} = 1.3556 & 4.649 \\
& 2 & \sqrt{5+\sqrt{10}} = 2.8570& 20.64 \\
\hline
\multirow{3}{*}{6} & 1 & 0.61670659019 & 6.994 \\
& 2 & 1.88917587775 & 15.02 \\
& 3 & 3.32425743355 & 88.46 \\
\hline
\end{array}
~
\begin{array}{|l|l|l|l|}
\hline
s & i & \xi_i & D_i \ge \\
\hline
\multirow{4}{*}{7} & 0 & 0 & 15 \\
& 1 & 1.1544 & 20.69 \\
& 2 & 2.3668 & 57.82 \\
& 3 & 3.7504 & 433.1 \\
\hline
\multirow{4}{*}{8} & 1 & 0.5391 & 41.09 \\
& 2 & 1.6365 & 73.30 \\
& 3 & 2.8025 & 255.7 \\
& 4 & 4.1445 & 2365 \\
\hline
\multirow{5}{*}{9} & 0 & 0 & 105 \\
& 1 & 1.0233 & 135.4 \\
& 2 & 2.0768 & 299.5 \\
& 3 & 3.2054 & 1267 \\
& 4 & 4.5127 & 14159 \\
\hline
\end{array}$$
{#C_is}
Notes: For convenience, $B_i$ was chosen independently of $i$ and $C_i$ was taken with $C_i = C_{-i}$.
$s$ $B_i$ $C_i$, $i=(0),1,\dots,{\left\lfloor\tfrac{s}{2}\right\rfloor}$ $\beta_0(s)$ $\beta_\star(s)$
----- ------- ---------------------------------------------------------------- -------------- ------------------
4 10 $2,14$ 19.35
5 10 $1,12,88$ 33.76
6 100 $11,63,558$ 156.96
7 10 $6,93,458,4649$ 86.55
8 100 $100,501,2561,30779$ 106.77
9 100 $9,773,3186,17732,247789$ 146.37
{#fgkv}
$f(k,v) = -3408102864+1506333312k^{2}+974873344k^{4}-488998144k^{6}+62323584k^{8}-3309568k^{10}+65536k^{12}+9310949028v-1506333312kv-4733985888k^{2}v-1949746688k^{3}v-1015706784k^{4}v+1466994432k^{5}v+511604992k^{6}v-249294336k^{7}v-49810560k^{8}v+16547840k^{9}v+1744896k^{10}v-393216k^{11}v-16384k^{12}v-11097146016v^{2}+4733985888kv^{2}+6922441360k^{2}v^{2}+2031413568k^{3}v^{2}-1428764528k^{4}v^{2}-1534814976k^{5}v^{2}+209662720k^{6}v^{2}+199242240k^{7}v^{2}-21567744k^{8}v^{2}-8724480k^{9}v^{2}+786432k^{10}v^{2}+98304k^{11}v^{2}+7281931941v^{3}-5947568016kv^{3}-4944873072k^{2}v^{3}+412538336k^{3}v^{3}+1856597696k^{4}v^{3}+243542016k^{5}v^{3}-293538048k^{6}v^{3}-13016064k^{7}v^{3}+17194752k^{8}v^{3}-327680k^{9}v^{3}-253952k^{10}v^{3}-2755473732v^{4}+3929166288kv^{4}+1497511456k^{2}v^{4}-1155170432k^{3}v^{4}-582955856k^{4}v^{4}+183266304k^{5}v^{4}+58253568k^{6}v^{4}-16432128k^{7}v^{4}-1102464k^{8}v^{4}+368640k^{9}v^{4}+544096980v^{5}-1459281552kv^{5}+28759472k^{2}v^{5}+469164960k^{3}v^{5}-7038496k^{4}v^{5}-59703552k^{5}v^{5}+6536960k^{6}v^{5}+2050560k^{7}v^{5}-328320k^{8}v^{5}-18769932v^{6}+293023248kv^{6}-127930016k^{2}v^{6}-58917568k^{3}v^{6}+27050224k^{4}v^{6}+1258752k^{5}v^{6}-1642240k^{6}v^{6}+182784k^{7}v^{6}-14780538v^{7}-24513072kv^{7}+27560816k^{2}v^{7}-2875616k^{3}v^{7}-2296192k^{4}v^{7}+698880k^{5}v^{7}-61184k^{6}v^{7}+2961396v^{8}-764688kv^{8}-1582560k^{2}v^{8}+772608k^{3}v^{8}-143664k^{4}v^{8}+10752k^{5}v^{8}-191952v^{9}+203472kv^{9}-52816k^{2}v^{9}+7520k^{3}v^{9}-640k^{4}v^{9}+972v^{10}-2352kv^{10}+336k^{2}v^{10}+45v^{11}$.\
\
$g(k,v) = 2k^{4}v-26k^{4}-4k^{3}v^{2}+52k^{3}v+2k^{2}v^{3}-20k^{2}v^{2}-120k^{2}v+258k^{2}-6kv^{3}+120kv^{2}-258kv+v^{4}-23v^{3}+123v^{2}-433v+764$.
Acknowledgement {#acknowledgement .unnumbered}
===============
The authors would like to thank Jane Wodlinger for helpful discussions.
[99]{}
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[^1]: Research of the authors is supported by NSERC.
|
---
abstract: 'The classical theory of *monstrous moonshine* describes the unexpected connection between the representation theory of the monster group $M$, the largest of the sporadic simple groups, and certain modular functions, called Hauptmoduln. In particular, the $n$-th Fourier coefficient of Klein’s $j$-function is the dimension of the grade $n$ part of a special infinite dimensional representation $V^{\natural}$ of the monster group. More generally the coefficients of Hauptmoduln are graded traces $T_g$ of $g \in M$ acting on $V^{\natural}$. Similar phenomena have been shown to hold for the Mathieu group $M_{24}$, but instead of modular functions, *mock modular forms* must be used. This has been conjecturally generalized even further, to *umbral moonshine*, which associates to each of the 23 Niemeier lattices a finite group, infinite dimensional representation, and mock modular form. We use *generalized Borcherds products* to relate monstrous moonshine and umbral moonshine. Namely, we use mock modular forms from umbral moonshine to construct via generalized Borcherds products rational functions of the Hauptmoduln $T_g$ from monstrous moonshine. This allows us to associate to each pure $A$-type Niemeier lattice a conjugacy class $g$ of the monster group, and gives rise to identities relating dimensions of representations from umbral moonshine to values of $T_g$. We also show that the logarithmic derivatives of the Borcherds products are $p$-adic modular forms for certain primes $p$ and describe some of the resulting properties of their coefficients modulo $p$.'
address:
- 'Mathematics Institute, University of Cologne, Gyrhofstr. 8b, 50931 Cologne, Germany'
- 'Department of Mathematics, Emory University, Emory, Atlanta, GA 30322'
- 'Department of Mathematics, Emory University, Emory, Atlanta, GA 30322'
author:
- 'Ken Ono, Larry Rolen, Sarah Trebat-Leder'
bibliography:
- 'moonshine\_paper.bib'
title: 'Classical and Umbral Moonshine: Connections and $p$-adic Properties'
---
[^1]
Introduction
============
*Monstrous moonshine* begins with the surprising connection between the coefficients of the modular function $$J(\tau):=j(\tau)-744 = \frac{(1 + 240 \sum_{n = 1}^\infty \sum_{d \mid n}d^3 q^n)^3}{q \prod_{n = 1}^\infty (1 - q^n)^{24}}-744 = \frac{1}{q} + 196884 q + 21493760 q^2 + \dots \; \;$$ and the representation theory of the monster group $M$, which is the largest of the simple sporadic groups. Here $q:=e^{2\pi i\tau}$ and $\tau\in\H:=\{z\in\C\colon\Im z>0\}$. McKay noticed that $196884$, the $q^1$ coefficient of $J(\tau)$, can be expressed as a linear combination of dimensions of irreducible representations of the monster group $M$. Namely, $$196884 = 196883 + 1.$$ McKay saw that the same was true for other Fourier coefficients of $J(\tau)$. For example, $$21493760 = 21296876 + 196883 + 1.$$ In [@Thompson:1979tm], McKay and Thompson conjectured that the $n$-th Fourier coefficient of $J(\tau)$is the dimension of the grade $n$ part of a special infinite-dimensional graded representation $V^{\natural}$ of $M$.
This was later expanded into the full monstrous moonshine conjecture by Thompson, Conway, and Norton [@Conway:1979ul; @Thompson:1979ev]. Since the graded dimension is just the graded trace of the identity element, they looked at the graded traces $T_g(\tau)$ of nontrivial elements $g$ of $M$ acting on $V^{\natural}$ and conjectured that they were all expansions of principal moduli, or Hauptmoduln, for certain genus zero congruence groups $\Gamma_g$ commensurable with $\SL_2(\Z)$. Note that these $T_g$ are constant on each of the 194 conjugacy classes of $M$, and therefore are class functions, which automatically have coefficients which are $\C$-linear combinations of irreducible characters of $M$. Part of the task of proving monstrous moonshine was showing that they were in fact $\Z_{\geq 0}$-linear combinations.
By way of computer calculation, Atkin, Fong, and Smith [@Smith:1985hf] verified the existence of a virtual representation of M. Then using vertex-operator theory, Frenkel, Lepowsky, and Meurman [@Frenkel:1984uj] finally constructed a representation $V^{\natural}$ of $M$ thereby providing a beautiful algebraic explanation for the original numerical observations of McKay and Thompson. Borcherds [@Borcherds:1986ui] further developed the theory of vertex-operator algebras, which he then used in [@Borcherds:1992ua] to prove the full conjectures as given by Conway and Norton.
Monstrous moonshine provides an example of coefficients of modular functions enjoying distinguished properties. Moreover, their values at Heegner points have also been considered important. A *Heegner point* $\tau$ of discrimant $d < 0$ is a complex number of the form $\tau = \frac{-b \pm \sqrt{b^2 - 4 ac}}{2a}$ with $a, b, c \in \Z$, $\gcd(a, b, c) = 1$, and $d = b^2 - 4 a c$. The values of principal moduli at such points are called *singular moduli*. As an example of their importance, it is a classical fact that the singular moduli of $j(\tau)$ generate Hilbert class fields of imaginary quadratic fields. Moreover, the other McKay-Thompson series arising in monstrous moonshine satisfy analogous properties [@ChenYui]. It is natural to ask what other interesting properties the values of the Hauptmoduln $T_g(\tau)$ could possess. We show that some of these values are related to another kind of moonshine, called *umbral moonshine*.
Recently, it was shown that phenomena similar to monstrous moonshine occur for other $q$-series and groups. In particular, the Mathieu group $M_{24}$ exhibits moonshine [@Eguchi:2011ux; @Gannon:2012uv], with the role of the $j$-invariant played by a *mock modular form* of weight $1/2$, denoted $H^{(2)}(\tau)$. A mock modular form is the holomorphic part of a *harmonic weak Maass form*. Cheng, Duncan, and Harvey conjecture in [@Cheng:2013vr] that this is a special case of a more general phenomenon, which they call *umbral moonshine*. For each of the 23 Niemeier lattices $X$ they associate a vector-valued mock modular form $H^X(\tau)$, a group $G^X$, and an infinite-dimensional graded representation $K^X$ of $G^X$ such that the Fourier coefficients of $H^X$ encode the dimensions of the graded components of $K^X$.
In particular, if $c^X(n, h)$ is the $n$-th Fourier coefficient of the $h$-th component of $H^X$, then $$\label{coeff-dim}
c^X(n, h)
= \left\{
\begin{array}{l l}
a^X \dim_{K^X_{h, -D/4m}} & \text{if $n = -D/4m$ where $D \in \Z, D = h^2 \pmod{4m}$,}\\
0 & \text{otherwise,} \\
\end{array}
\right.$$ where $a^X \in \{ 1, 1/3\}$ and $$K^X = \bigoplus_{h \pmod{2m}} \bigoplus_{\substack{D \in \Z \\ D = h^2 \pmod{4m}}} K^X_{h, -D/4m}.$$ For more information on umbral moonshine see Section \[sec:umbral\] and for a definition of $H^X$ see Section \[sec:definitions\].
Using *generalized Borcherds products* (see [@Bruinier:2010ff]), we describe a connection between the mock modular forms $H^X(\tau)$ of umbral moonshine and the McKay-Thompson series $T_g(\tau)$ of monstrous moonshine. *Generalized Borcherds products* are a method to produce modular functions as infinite products of rational functions whose exponents come from the coefficients of mock modular forms, and they can be viewed as generalizations of the automorphic products in Theorem 13.3 of [@Borcherds].
We focus on the Niemeier lattices $X$ whose root systems are of pure $A$-type according to the ADE classification. They are listed in Table \[intro-umbral\], along with their Coxeter numbers $m(X)$ and the notation we will use for the mock modular form $H^X$.
$$\arraycolsep=10pt\def\arraystretch{1.5}
\begin{array}{| c | c | c |}
\hline
\text{Root System $X$} & \text{Coxeter Number $m(X)$} &\text{mock modular Form $H^X$} \\
\hline \hline
A_1^{24} & 2 & H^{(2)}(\tau) \\
\hline
A_2^{12} & 3 & H^{(3)}(\tau) \\
\hline
A_3^{8} & 4 & H^{(4)}(\tau)\\
\hline
A_4^{6} & 5 & H^{(5)}(\tau)\\
\hline
A_6^{4}& 7 & H^{(7)}(\tau)\\
\hline
A_8^{3} & 9 & H^{(9)}(\tau)\\
\hline
A_{12}^{2} & 13 & H^{(13)}(\tau)\\
\hline
A_{24}^{1} & 25 & H^{(25)}(\tau)\\
\hline
\end{array}$$ \[intro-umbral\]
Table \[monster\] gives the monstrous moonshine dictionary for the conjugacy classes $g$ which correspond to pure $A$-type cases of umbral moonshine[^2]. Note that $\eta(\tau)$ is the *Dedekind eta function*, defined by $$\eta(\tau) := q^{1/24}\prod_{n = 1}^\infty (1 - q^n).$$ All of our Hauptmoduln are normalized so that they have the form $q^{-1} + O(q)$, which is why all of the $\eta$-quotients in the table have a constant added to them.
$$\arraycolsep=10pt\def\arraystretch{1.5}
\begin{array}{| c | c | c |}
\hline
\text{Monster Conjugacy Class $g$} & \text{Congruence Subgroup $\Gamma_g$} & \text{McKay-Thomspon Series $T_g(\tau)$} \\
\hline
\hline
2B & \Gamma_0(2) & \eta(\tau)^{24}/\eta(2\tau)^{24} + 24 \\
\hline
3B &\Gamma_0(3) & \eta(\tau)^{12}/\eta(3\tau)^{12} + 12\\
\hline
4C & \Gamma_0(4) & \eta(\tau)^{8}/\eta(4\tau)^{8} + 8 \\
\hline
5B & \Gamma_0(5) & \eta(\tau)^{6}/\eta(5\tau)^{6} + 6 \\
\hline
7B & \Gamma_0(7) & \eta(\tau)^{4}/\eta(7\tau)^{4} + 4 \\
\hline
9B & \Gamma_0(9) & \eta(\tau)^{3}/\eta(9\tau)^{3} + 3 \\
\hline
13B & \Gamma_0(13) & \eta(\tau)^{2}/\eta(13\tau)^{2} + 2\\
\hline
(25 Z) & \Gamma_0(25) & \eta(\tau)/\eta(25\tau) + 1\\
\hline
\end{array}$$ \[monster\]
There is an evident correspondence between the pure $A$-type lattices $X$ in Table \[intro-umbral\] and the conjugacy classes $g$ in Table \[monster\]. We give this correspondence in Table \[correspondence\].
We show that for a pure $A$-type Niemeier lattice $X$ and its corresponding conjugacy class $g := g(X)$, the “Galois (twisted) traces” of the CM values of the McKay-Thompson series $T_g(\tau)$ are the coefficients of the mock modular form $H^X$. To more precisely state this, we set up the following notation.
Let $X$ be a pure $A$-type Niemeier lattice with Coxeter number $m := m(X)$ and corresponding conjugacy class $g := g(X)$. We call a pair $(\Delta, r)$ *admissible* if $\Delta$ is a negative fundamental discriminant and $r^2 \equiv \Delta \pmod{4m}$. We also let $e(a):=e^{2\pi i a}$.
\[mainthm\] Let $c^+(n, h)$ be the $n$-th Fourier coefficient of the $h$-th component of $H^X$. Let $(\Delta, r)$ be an admissible pair for $X$. Then the twisted generalized Borcherds product $$\Psi_{\Delta, r}(\tau, H^X):= \prod_{n = 1}^\infty P_\Delta(q^n)^{c^+\pa{\frac{|\Delta| n^2}{4 m}, \frac{rn}{2m}}},$$ where $$P_\Delta(x) := \prod_{b \in \Z/|\Delta|\Z} [1 - e(b/\Delta)x]^{\pa{\frac{\Delta}{b}}}$$ is a rational function in $T_g(\tau)$ with a discriminant $\Delta$ Heegner divisor.
We consider only the pure $A$-type cases, because these are the ones for which the harmonic Maass form transforms under the Weil representation. See Section \[sec:definitions\] for more information.
The next result gives a precise description of the rational functions in Theorem \[mainthm\]. In particular, it gives a “twisted” trace function for the values of $T_g$ at points in the divisor and the coefficients $c^+$ of the mock modular forms $H^X$. It is often the case that coefficients of automorphic forms can be expressed in terms of singular moduli (see e.g., [@Bringmann:2007fk; @Bruinier:2006iy; @Duke:2011vb; @Zagier:2002tp]).
$$\arraycolsep=10pt\def\arraystretch{1.5}
\begin{array}{| c | c |}
\hline
\text{Root System $X$} & \text{Conjugacy Class $g(X)$} \\
\hline \hline
A_1^{24} & 2B \\
\hline
A_2^{12} & 3B \\
\hline
A_3^{8} & 4C \\
\hline
A_4^{6} & 5B\\
\hline
A_6^{4}& 7B\\
\hline
A_8^{3} & 9B\\
\hline
A_{12}^{2} & 13B\\
\hline
A_{24}^{1} & (25Z) \\
\hline
\end{array}$$ \[correspondence\]
\[twisted\_trace\] By Theorem \[mainthm\], we can write $$\Psi_{\Delta, r}(\tau, H^X) = \prod_i \pa{T_g(\tau) - T_g(\alpha_i)}^{\gamma_i}$$ for some discriminant $\Delta$ Heegner points $\alpha_i$. Then we have that $$c^+\pa{\frac{|\Delta|}{4 m}, \frac{r}{2m}} = \frac{1}{\epsilon_\Delta} \sum_i \gamma_i \cdot T_g(\alpha_i),$$ where $$\epsilon_\Delta = \sum_{b \in \Z/|\Delta|\Z} e(b/\Delta) \cdot \pa{\frac{\Delta}{b}}.$$
Assuming the umbral moonshine conjecture, the previous corollary implies the following “degree” formula in traces of singular moduli for classical moonshine functions: $$\frac{1}{\epsilon_\Delta} \sum_i \gamma_i \cdot T_g(\alpha_i) = c^+\pa{\frac{|\Delta|}{4 m}, \frac{r}{2m}} = a^X \dim_{K^X_{r, |\Delta|/4m}}.$$
In the case where $m = 2$, the relationship between the coefficients of the mock-modular form and the dimensions of the graded components of the representation has been proven by Gannon [@Gannon:2012uv], and so our work implies the following: $$\frac{1}{\epsilon_\Delta} \sum_i \gamma_i \cdot T_{2B}(\alpha_i) = c^+\pa{\frac{|\Delta|}{8}, \frac{r}{4}} = \dim_{K^{(2)}_{r, |\Delta|/8}}.$$
\[borcherds-example\] Let $X = A_1^{24}$, so $m(X) = 2$ and $g(X) = 2B$. Then the corresponding McKay-Thompson series is $$T_g(\tau) = \frac{\eta(\tau)^{24}}{\eta(2 \tau)^{24}} + 24 = \frac{1}{q} + 276 q + \dots.$$ We pick the admissible pair $(\Delta, r) = (-7, 1)$. In Section \[relating\], we will show that $$\begin{aligned}
\Psi_{\Delta, r}(\tau, H^X)&=& \frac{\pa{T_{g}(\tau) - T_g(\alpha_1)}^2}{\pa{T_{g}(\tau) - T_g(\alpha_2)}^2} = \frac{\pa{T_{g}(\tau) - \frac{1 - 45 \sqrt{-7}}{2}}^2}{\pa{T_{g}(\tau) - \frac{1 + 45 \sqrt{-7}}{2}}^2}\\
&=& 1 + 90 \sqrt{-7} q + (28350 + 45 \sqrt{-7}) q^2 + \dots,\end{aligned}$$ where $\alpha_1 := \frac{-1 + \sqrt{-7}}{4}$ and $\alpha_2 := \frac{1 + \sqrt{-7}}{4}$. Note that $T_g(\alpha_1)$ and $T_g(\alpha_2)$ are algebraic integers of degree 2 which form a full set of conjugates. Their twisted trace is $$2[T_g(\alpha_1) - T_g(\alpha_2)] = -90\sqrt{-7},$$ which matches the $q^1$ Fourier coefficient above. To check Corollary \[twisted\_trace\], we note that $$\epsilon_\Delta = \sum_{b \in \Z/7\Z}e(-b/7) \cdot \pa{\frac{-7}{b}} = -\sqrt{-7}$$ and $$\frac{1}{\epsilon_\Delta} \sum_i \gamma_i T_{g}(\alpha_i) = 90 = c^+\pa{7/8, 1/4} = \dim_{K^{(2)}_{1, 7/8}}.$$
As a second example, again consider $X = A_1^{24}$, so $m(X) = 2$ and $g(X) = 2B$. We pick the admissible pair $(\Delta, r) = (-15, 1)$. Let $\rho_1, \rho_2, \rho_3,\rho_4$ be the roots of $$x^4-47 x^3+192489 x^2-9012848 x+122529840,$$ with $\rho_1, \rho_2$ having positive imaginary parts. Then $$\Psi_{-15, 1} = \frac{(T_g(\tau) - \rho_1)^2 (T_g(\tau) - \rho_2)^2}{(T_g(\tau) - \rho_3)^2 (T_g(\tau) - \rho_4)^2}.$$ We get that $$\epsilon_{-15} = \sqrt{-15},$$ and $$\frac{1}{\epsilon_\Delta} \sum_i \gamma_i T_{g}(\alpha_i) = 462 = c^+\pa{15/8, 1/4} = \dim_{K^{(2)}_{1, 15/8}}.$$
In view of this correspondence, it is clear that the mock modular forms of umbral moonshine have important properties. The congruence properties of their coefficients have just begun to be studied. For example, [@Creutzig:2012vx] examines the parity of the coefficients of the McKay-Thompson series for Mathieu moonshine in relation to a certain conjecture in [@Cheng:2012ue], which in our case corresponds to $X = A_{1}^{24}$. Congruences modulo higher primes were also considered in [@Miezaki:ux].
Let $\Theta:= q \frac{d}{dq} = \frac{1}{2 \pi i} \frac{d}{d\tau}$. Given the product expansion of a generalized Borcherds product, it is natural to consider its logarithmic derivative. It turns out that this logarithmic derivative has nice arithmetic properties. This idea was also used in [@Bruinier:2010ff] and [@Ono:2010ws].
\[mero-modform\] Fix a pure $A$-type Niemeier lattice $X$ with Coxeter number $m$. Let $(\Delta, r)$ be an admissible pair. Consider the logarithmic derivative $$f_{\Delta, r}(\tau)= \sqrt{\Delta}\sum a_{\Delta, r}(n)q^n := \sqrt{\Delta} \sum_{n} \sum_{ij = n} i c^+\pa{\frac{|\Delta|i^2}{4m}, \frac{ri}{2m}}\pa{\frac{\Delta}{j}}q^n$$ of $\Psi_{\Delta, r}(\tau) = \Psi_{\Delta, r}(\tau, H^X)$. Then $f_{\Delta, r}(\tau)$ is a meromorphic weight $2$ modular form.
When $p$ is inert or ramified in $\Q(\sqrt{\Delta})$, it turns out that $f_{\Delta, r}(\tau)$ is more than just a meromorphic modular form; it is a $p$-adic modular form. Essentially, a $p$-adic modular form is a $q$-series which is congruent modulo any power of $p$ to a holomorphic modular form; we refer the reader to Section \[sec:padic-def\] for the definition.
\[padic\] Let $X$ be a pure $A$-type Niemeier lattice with Coxeter number $m$. Let $(\Delta, r)$ be admissible and suppose $p$ is inert or ramified in $\Q(\sqrt{\Delta})$. Then $f_{\Delta, r}$ is a $p$-adic modular form of weight $2$.
We will use this result to study the $p$-divisibility of the coefficients $a_{\Delta, r}(n)$.
\[general-modp-cor\] Let $X, \Delta, r, p$ be as above. Then for all $k \geq 1$ there exists $\alpha_k > 0$ such that $$\#\{n \leq x: a_{\Delta, r}(n) \not \equiv 0 \pmod{p^k}\} = O \pa{\frac{x}{(\log x)^{\alpha_k}}}.$$ In particular, if we let $$\pi_{\Delta, r}(x; p^k) := \#\{n \leq x: a_{\Delta, r}(n) \equiv 0 \pmod{p^k}\},$$ then $$\lim_{x \to \infty}\frac{\pi_{\Delta, r}(x; p^k)}{x} = 1.$$
Corollary \[general-modp-cor\] also applies to any constant multiple of $f_{\Delta, r}$ with integral coefficients. In the example below, we consider the coefficients of $$\frac{f_{-7,1}(\tau)}{90\sqrt{-7}} = q + O(q^2).$$ However, it is not always the case that the analogous normalization has integral coefficients.
We illustrate Corollary \[general-modp-cor\] for $X = A_{1}^{24}$, $\Delta = -7$, $r = 1$. Note that this is the same case considered in Example \[borcherds-example\]. The first few coefficients of the normalized logarithmic derivative are given by $$\frac{f_{-7,1}(\tau)}{90\sqrt{-7}}=:\sum_{n\geq1}a_{-7, 1}(n)q^n=q+q^2-4371q^3+q^4+17773755q^5+\ldots$$
The prime $p = 2$ is split in $\Q(\sqrt{-7})$, and so Theorem \[padic\] and Corollary \[general-modp-cor\] do not apply. Therefore, we expect the coefficients $a_{-7, 1}(n)$ to be equally distributed modulo 2, but cannot prove anything about them. The prime $p = 3$ is inert, so Corollary \[general-modp-cor\] tell us that, asymptotically, 100% of the coefficients $a_{-7, -1}(n)$ are divisible by 3. We illustrate this behavior in Table \[PadicNumerics\].
\[PadicNumerics\]
$x$ $\pi_2(x)/x$ $\pi_3(x)/x$
---------- -------------- --------------
50 0.38 0.64
100 0.45 0.68
150 0.47 0.69
200 0.49 0.71
250 0.48 0.71
300 0.49 0.72
$\infty$ .5? 1
: Divisibility of $a_{-7, 1}(n)$ by $p=2,3$
Umbral Moonshine {#sec:umbral}
================
In this section, we summarize the main objects and conjectures of umbral moonshine. However, we first briefly describe Mathieu moonshine, which umbral moonshine generalized.
Mathieu Moonshine
-----------------
In 2010, the study of a new form of moonshine commenced, called Mathieu moonshine. Let $\mu(z, \tau) := \mu(z, z, \tau)$ be Zwegers’ famous function from his thesis [@Zwegers:2008wk], which is defined in the appendix. Let $H^{(2)}(\tau)$ be the $q$-series $$H^{(2)}(\tau) := -8 \sum_{\omega \in \{\frac{1}{2}, \frac{1 + \tau}{2}, \frac{\tau}{2}\}} \mu(\omega, \tau) = 2 q^{-1/8}(-1 + 45 q + 231 q^2 + \dots),$$ which occurs in the decomposition of the elliptic genus of a K3 surface into irreducible characters of the $N = 4$ superconformal algebra. This is a mock-modular form, and plays the role of $J(\tau)$ in Mathieu moonshine. Eguchi, Ooguri, and Tachikawa conjectured that the Fourier coefficients encode dimensions of irreducible representations of the Mathieu group $M_{24}$ [@Eguchi:2011ux]. This was extended to the full Mathieu moonshine conjecture by [@Cheng:2010va; @Eguchi:2011uo; @Gaberdiel:2010wu; @Gaberdiel:2010wx], which included providing mock modular forms $H_g^{(2)}$ for every $g \in M_{24}$. The existence of an infinite dimensional $M_{24}$ module underlying the mock modular forms was shown by Gannon in 2012 [@Gannon:2012uv].
The Objects of Umbral Moonshine
-------------------------------
Cheng, Duncan, and Harvey generalized even further - conjecturing that Mathieu moonshine is but one example of a more general phenomenon which they call umbral moonshine [@Cheng:2013vr].
For each of the 23 Niemeier root systems $X$, which are unions of irreducible simply-laced root systems with the same Coxeter number, they associate many objects, including a group $G^X$ (playing the role of $M$), a mock modular form $H^X(\tau)$ (playing the role of $j(\tau)$), and an infinite dimensional graded $G^X$ module $K^X$ (playing the role of the $M$-module $V^{\natural}$) Table \[umbral\_objects\] gives a more complete list of the associated objects.
$L^X$ The Niemeier lattice corresponding to $X$
------------------------ ----------------------------------------------------------------------------
$m$ The Coxeter number of all irreducible components of $X$
$W^X$ The Weyl group of $X$
$G^X := \Aut(L^X)/W^X$ The umbral group corresponding to $X$
$\pi^X$ The (formal) product of Frame shapes of Coxeter elements of
irreducible components of $X$
$\Gamma^X$ The genus zero subgroup attached to $X$
$T^X$ The normalized Hauptmodul of $\Gamma^X$, whose eta-product expansion
corresponds to $\pi^X$
$\ell$ The lambency. A symbol that encodes the genus zero group $\Gamma^X$.
Sometimes used instead of $X$ to denote which case of umbral moonshine
is being considered.
$\psi^X$ The unique meromorphic Jacobi form of weight 1 and index $m$
satisfying certain conditions.
$H^X$ The vector-valued mock modular form of weight 1/2 whose $2m$ components
furnish the theta expansion of the finite part of $\psi^X$.
Called the umbral mock modular form.
$S^X$ The vector-valued cusp form of weight 3/2 which is the shadow of $H^X$.
Called the umbral shadow.
$H_g^X$ The umbral McKay-Thompson series attached to $g \in G^X$.
It is a vector-valued mock modular form of weight 1/2,
and equals $H^X$ when $g$ is the identity.
$S_g^X$ The vector-valued cusp form [*conjectured*]{} to be the shadow of $H_g^X$.
$K^X$ The [*conjectural*]{} infinite dimensional graded $G^X$-module whose
graded super-dimension is encoded by $H^X$.
: This table gives the objects associated to a Niemeier root system $X$
\[umbral\_objects\]
The ADE classification of simply laced Dynkin diagrams allows us to classify the irreducible components of the Niemeier root systems $X$. We will focus on the simplest cases - the root systems of pure $A$-type, i.e. $X = A_{m - 1}^{24/(m - 1)}$, where $(m - 1) \mid 24$. In these cases, the lambency $\ell$ is an integer and equals $m$, and $\Gamma^X = \Gamma_0(m)$. The case $X = A_1^{24}$ corresponds to Mathieu moonshine, with $G^X = M_{24}$ and $H^X = H^{(2)}$, as defined above. We will generally refer to $H^X$ , $S^X$, $\psi^X$, and $T^X$ as $H^{(m)}$, $S^{(m)}$, $\psi^{(m)}$, and $j_m$ respectively. These are the main quantities from Table \[umbral\_objects\] that we will work with, and we will only define them for pure $A$-type. This is done in Section \[sec:definitions\].
The Conjectures of Umbral Moonshine
-----------------------------------
The main conjectures of umbral moonshine are as follows:
1. The mock modular form $H^X$ encodes the graded super-dimension of a certain infinite-dimensional, $\Z/2m \Z \times \Q$-graded $G^X$-module $K^X$.
2. The graded super-characters $H_g^X$ arising from the action of $G^X$ on $K^X$ are vector-valued mock modular forms with concretely specified shadows $S_g^X$.
3. The umbral McKay-Thompson series $H_g^X$ are uniquely determined by an optimal growth property which is directly analogous to the genus zero property of monstrous moonshine.
vector-valued Modular Forms
===========================
In this section, we follow [@Bruinier:2010ff] in giving the needed background on vector-valued modular forms, though we state results in less generality.
A Lattice related to $\Gamma_0(m)$
----------------------------------
We will define a lattice $L$ and a dual lattice $L'$ related to $\Gamma_0(m)$ such that the components of our vector-valued modular forms are labeled by the elements of $L'/L$.
We consider the quadratic space $$V:= \{X \in \Mat_2(\Q): \tr(X) = 0\}$$ with the quadratic form $P(X) := m \det(X)$.[^3] The corresponding bilinear form is then $(X, Y) := -m \tr(XY)$. Let $L$ be the lattice $$L:=\left\{ \begin{pmatrix} b& -a/m \\ c&-b \end{pmatrix}; \quad a,b,c\in\Z \right\}.$$ The dual lattice is then given by $$L':=\left\{ \begin{pmatrix} b/2m& -a/m \\ c&-b/2m \end{pmatrix}; \quad a,b,c\in\Z \right\}.$$ We will switch between viewing elements of $L'$ as matrices and as quadratic forms, with the matrix $$X = \begin{pmatrix} b/2m& -a/m \\ c&-b/2m \end{pmatrix}$$ corresponding to the integral binary quadratic form $$Q = [mc, b, a] = mcx^2 + bxy + c y^2.$$ Note that then $P(X) = -\Disc(Q)/4m$.
We identify $L'/L$ with $(\frac{1}{2m}\Z)/\Z$, and the quadratic form $P$ with the quadratic form $\frac{h}{2m} \mapsto \frac{-h^2}{4m}$ on $\Q/\Z$. We will also occasionally identify $\frac{h}{2m} \in \Q/\Z$ with $h \in \Z/2m\Z$.
For a fundamental discriminant $D$ and $r/2m \in L'/L$ with $r^2 \equiv D \pmod{4N}$, let $$\label{QDR}
Q_{D, r} := \{Q = [mc, b, a]: a, b, c \in \Z, \Disc(Q) = D, b \equiv r \pmod{2m}\}.$$ The action of $\Gamma_0(m)$ on this set is given by the usual action of congruence subgroups on binary quadratic forms. We will later be working with $Q_{D, r}/\Gamma_0(m)$.
The Weil representation
-----------------------
By $\Mp_2(\Z)$ we denote the integral metaplectic group. It consists of pairs $(\gamma, \phi)$, where $\gamma = {\smallabcd \in \Sl_2(\Z)}$ and $\phi:\h\rightarrow \C$ is a holomorphic function with $\phi^2(\tau)=c\tau+d$. The group $\widetilde{\G}:=\Mp_2(\Z)$ is generated by $S:=(\smallSmatrix,\sqrt{\tau})$ and $T:=(\smallTmatrix, 1)$.
We consider the *Weil representation* $\rho_L$ of $\Mp_2(\Z)$ corresponding to the discriminant form $L'/L$. We denote the standard basis elements of $\C[L'/L]$ by $\e_h$, $h/2m \in L'/L$. Then the Weil representation $\rho_L$ associated with the discriminant form $L'/L$ is the unitary representation of $\widetilde{\Gamma}$ on $\C[L'/L]$ defined by
$$\rho_L(T) \e_h = e(h^2/4m) \e_h,$$
and $$\rho_L(S) \e_h = \frac{e(-1/8)}{\sqrt{2m}}
\sum_{h' \in \Z/2m\Z} e(hh'/2m)\e_{h'}.$$
Harmonic weak Maass forms
-------------------------
If $f\colon \H \to \C[L'/L]$ is a function, we write $$f = \sum_{h \in \Z/2m\Z} f_h \e_h$$ for its decomposition into components. For $k \in \frac{1}{2} \Z$, let $M_{k, \rho_L}^!$ denote the space of $\C[L'/L]$ valued weakly holomorphic modular forms of weight $k$ and type $\rho_L$ for the group $\widetilde{\Gamma}$. The subspaces of holomorphic modular forms (resp. cusp forms) are denoted by $M_{k, \rho_L}$ (resp. $S_{k, \rho_L}$). Now, assume that $k \leq 1$. A twice continuously differentiable function $f: \H \to \C[L'/L]$ is called a *harmonic weak Maass form* (of weight $k$ with respect to $\widetilde{\Gamma}$ and $\rho_L$) if it satisfies:
1. $f(M\tau) = \phi(\tau)^{2k} \rho_L(M, \phi) f(\tau)$ for all $(M, \phi) \in \widetilde{\Gamma}$;
2. $\Delta_k f = 0$;
3. There is a polynomial $$P_f(\tau) = \sum_{h \in \Z/2m\Z} \sum_{\substack{n \in \Z - \frac{h^2}{4m}, \\ -\infty << n \leq 0}} c^+(n, h)e(n\tau) \e_h$$ such that $$f(\tau)-P_f=O(e^{-\epsilon v})$$ for some $\epsilon >0$ as $v\to +\infty$.
Note here that $$\Delta_k := -v^2\pa{\frac{\partial^2}{\partial u^2} + \frac{\partial^2}{\partial v^2}} + i k v \pa{\frac{\partial}{\partial u} + i \frac{\partial}{\partial v}}$$ is the usual weight $k$ hyperbolic Laplace operator, and that $\tau = u + i v$. We denote the vector space of these harmonic weak Maass forms by $\calH_{k, \rho_L}$. The Fourier expansion of any $f \in \calH_{k, \rho_L}$ gives a unique decomposition $f = f^+ + f^-$, where $$\begin{aligned}
f^+(\tau) &=& \sum_{h \in \Z/2m\Z} \sum_{\substack{n \in \Z - \frac{h^2}{4m}, \\ -\infty << n}} c^+(n, h)e(n\tau) \e_h, \\
f^-(\tau) &=& \sum_{h \in L'/L} \sum_{\substack{n \in \Q, \\ n < 0}} c^-(n, h)W(2 \pi n v) e(n \tau) \e_h, \end{aligned}$$ and $W(x) := \int_{-2x}^\infty e^{-t}t^{-k} dt = \Gamma(1 - k, 2|x|)$ for $x < 0$. Then $f^+$ is called the *holomorphic part* and $f^-$ the *nonholomorphic part* of $f$. The polynomial $P_f$ is also uniquely determined by $f$ and is called its *principal part*. We define a *mock modular form* of weight $k$ to be the holomorphic part $f^+$ of a harmonic weak Maass form $f$ of weight $k$ which has $f^- \neq 0$. Its weight is just the weight of the harmonic weak Maass form.
Recall that there is an antilinear differential operator defined by $$\xi_k: \calH_{k, \overline{\rho}_L} \to S_{2 - k, \rho_L}, \; \; f(\tau) \mapsto \xi_k(f)(\tau) := 2iy^k\overline{\frac{\partial}{\partial\overline{\tau}}},$$ where $\overline{\rho}_L$ is the complex conjugate representation. The Fourier expansion of $\xi_k(f)$ is given by $$\xi_k(f) = -\sum_{h \in \Z/2m\Z} \sum_{n \in \Q, n > 0}(4 \pi n)^{1 - k} \overline{c^-(-n, h)} q^n \e_h.$$ The kernel of $\xi_k$ is equal to $M^!_{k, \overline{\rho}_L}$, and we have the following exact sequence: $$0 \to M^!_{k, \overline{\rho}_L} \to \calH_{k, \overline{\rho}_L} \to S_{2 - k, \rho_L} \to 0.$$ We call $\xi_k(f)$ the *shadow* of $f$. Note that $\xi_k(f)$ uniquely determines $f^-$, but the $f^+$ is only determined up to the addition of a weakly holomorphic modular form.
Defining the Umbral mock modular Forms {#sec:definitions}
======================================
In this section we define the mock modular forms $H^{(m)}$ from umbral moonshine, as well as their shadows $S^{(m)}$ and non-holomorphic parts. Note that we only give definitions for the pure $A$-type cases - see [@Cheng:2013vr] for a more detailed and general definition. We also refer the reader to the appendix for definitions of $\varphi_1^{(m)}(\tau, z), \mu_{m, 0}(\tau, z), \theta_{m, r}(\tau, z), \text{ and } R(u; \tau)$.
For each lambency $m \in \{2, 3, 4, 5, 7, 9, 13, 25\}$, which correspond to the pure $A$-type cases, define the Jacobi form $\psi^{(m)}$ by $$\psi^{(m)}(\tau, z) := c_m \varphi_1^{(m)}(\tau, z) \mu_{1, 0}(\tau, z),$$ where $c_m = 2$ for $m = 2, 3, 4, 5, 7, 13$ and $c_m = 1$ for $m = 9, 25$. We can break up $\psi^{(m)}$ into a finite part $\psi_F^{(m)}$ and a polar part $\psi_P^{(m)}$. The polar part is given by $$\psi_P^{(m)}(\tau, z) = \frac{24}{m - 1} \mu_{m, 0}(\tau, z).$$ Then the mock modular form $H^{(m)}$ is defined by $$\psi_F^{(m)}(\tau, z) = \psi^{(m)}(\tau, z) - \psi_P^{(m)}(\tau, z) = \sum_{h \in \Z/2m\Z} H_h^{(m)}(\tau) \theta_{m, h}(\tau, z),$$ where $$\theta_{m, h}(\tau, z) := \sum_{n \equiv h \pmod{2m}} q^{n^2/4m} y^k.$$ Note that $\psi^{(m)}$ satisfies an optimal growth condition, which is that $$\label{optimal}
q^{1/4m} H_h^X(\tau) = O(1)$$ as $\tau \to i \infty$ for all $h \in \Z/2m\Z$.
We also define the shadow $S^{(m)}(\tau)$, the non-holomorphic part $F_r^{(m)}(\tau)$, and the harmonic weak Maass form $\widehat{H}^{(m)}(\tau)$ corresponding to the mock modular form $H^{(m)}$ via their components: $$\begin{aligned}
S_h^{(m)}(\tau) &:= \sum_{n \equiv h \pmod{2m}} n q^{n^2/4m}, \\
F_h^{(m)}(\tau) &:= \int_{-\overline{\tau}}^{i \infty} \frac{S_h^{(m)}(z)}{\sqrt{-i(z + \tau)}} dz \\
& = -2 m q^{-(h-m)^2/4m} R\left(\frac{h - m}{2m} (2m\tau) + \frac{1}{2}; 2m\tau \right), \text{ and} \nonumber\\
\widehat{H}_h^{(m)}(\tau) &:= H_h^{(m)}(\tau) + F_h^{(m)}(\tau)\end{aligned}$$
Note that by definition, $S_h^{(m)}(\tau) = -S_{-h}^{(m)}(\tau)$. Therefore, $S_0^{(m)} = S_m^{(m)} = 0$. The same is true of $H_h^{(m)}$. We can write this in terms of Shimura’s theta functions as $S_{h}^{(m)}(\tau) = \theta(\tau; h, 2m, 2m, x)$ [@Shimura:1973uk]. Then using the transformation laws for his $\theta$-functions, we get that $S^{(m)}$ transforms as follows: $$\begin{aligned}
S^{(m)}_h(\tau + 1) &= e(h^2/4m)S^{(m)}_h(\tau), \text{ and}\\
S^{(m)}_h(-1/\tau) &= \tau^{3/2} \frac{e(-1/8)}{\sqrt{2m}} \sum_{k \pmod{2m}} e(kh/2m) S^{(m)}_k(\tau).\end{aligned}$$ Thus, we have $$\begin{aligned}
S^{(m)}(\tau + 1) &= \rho_L(T) S^{(m)}(\tau), \text{ and} \\
S^{(m)}(-1/\tau) &= \tau^{3/2} \rho_L(S) S^{(m)}(\tau).\end{aligned}$$
From these transformations, we see that $S^{(m)}(\tau): \H \to \C[L'/L]$ is a weight 3/2 vector-valued modular form transforming under the Weil representation $\rho_L$, i.e. an element of the space $M_{3/2, \rho_L}$. From [@Cheng:2013vr], we know that $H^{(m)}$ is a mock modular form with shadow $S^{(m)}$.[^4] This gives us the following theorem.
We have that $\widehat{H}^{(m)}(\tau): \H \to \C[L'/L]$ is a weight 1/2 vector-valued harmonic weak Maass form transforming under the Weil representation $\overline{\rho}_L$, i.e., it is an element of $\calH_{1/2, \overline{\rho}_L}$. Moreover, it has shadow $S^{(m)}(\tau)$, non-holomorphic part $F^{(m)}$, and principal part $P(\tau) = - 2 q^{-1/4m} (\e_1 - \e_{2m - 1})$.
The reason we focus on the lattices of pure $A$-type is because this theorem is not true for the other cases - the vector-valued harmonic weak Maass forms no longer transform under the Weil representation.
Relating umbral and monstrous moonshine {#relating}
=======================================
In this section, we explain the relationship between the mock modular forms $H^{(m)}$ from umbral moonshine and the Hauptmoduln $T_g$ from monstrous moonshine.
Twisted Generalized Borcherds Products {#subsec:borcherds}
--------------------------------------
We begin by giving the theorem of Bruinier and Ono we will use.
Let $c^+(n, h)$ be the $n$-th Fourier coefficient of $H_h^{(m)}$. Let $(\Delta, r)$ be an admissible pair, so that $\Delta$ is a negative fundamental discriminant and $r^2 \equiv \Delta \pmod{4m}$. Let $\Psi_{\Delta, r}(\tau, \widehat{H}^{m})$ be the twisted generalized Borcherds product defined in Theorem \[mainthm\].
(Theorem 6.1 in [@Bruinier:2010ff]) \[Bruinier-thm\] We have that $\Psi_{\Delta, r}(\tau, \widehat{H}^{(m)})$ is a weight 0 meromorphic modular function on $\Gamma_0(m)$ with divisor $Z_{\Delta, r}(\widehat{H}^{(m)}).$
For this theorem to make sense, we need to define the twisted Heegner divisor $Z_{\Delta, r}(\widehat{H}^{(m)})$ associated to $\widehat{H}^{(m)}$. It is defined by $$Z_{\Delta, r}(\widehat{H}^{(m)}) := \sum_{h \in \Z/2m\Z} \sum_{n < 0} c^+(n, h) Z_{\Delta, r}(n, h).$$ Since the principal part of $\widehat{H}^{(m)}$ is $- 2 q^{-h^2/4m} (\e_1 - \e_{2m - 1})$, this means that $$Z_{\Delta, r}(\widehat{H}^{(m)}) = 2 Z_{\Delta, r}\pa{\frac{-1}{4m}, \frac{-1}{2m}} - 2 Z_{\Delta, r}\pa{\frac{-1}{4m}, \frac{1}{2m}}.$$ Now, we just have to compute the divisors $Z_{\Delta, r}\pa{\frac{-1}{4m}, \frac{h}{2m}}$. They are defined as follows. $$Z_{\Delta, r}\left(\frac{-1}{4m}, \frac{h}{2m}\right) := \sum_{Q \in Q_{\Delta, hr}/\Gamma_0(m)} \frac{\chi_{\Delta}(Q)}{w(Q)} \alpha_Q,$$ where $w(Q) = 2$ for $\Delta < -4$, $\chi_\Delta$ is the generalized genus character defined in Gross-Kohnen-Zagier, and $\alpha_Q$ is the unique root of $Q(x, 1)$ in $\H$.
Proofs of Theorem \[mainthm\] and Corollary \[twisted\_trace\]
--------------------------------------------------------------
Theorem \[Bruinier-thm\] gives us that $\Psi_{\Delta, r}(\tau, \widehat{H}^{(m)})$ is a weight 0 meromorphic modular function on $\Gamma_0(m)$ with specified divisor, which is a discriminant $\Delta$ Heegner divisor. For all of our $m$, $\Gamma_0(m)$ has genus zero. Therefore, $\Psi_{\Delta, r}(\tau, \widehat{H}^{(m)})$ is a rational function in the Hauptmodul for $\Gamma_0(m)$. The normalized Hauptmodul, which we call $j_m(\tau)$, is defined by $$j_m(\tau) := \frac{\eta(\tau)^{24/(m - 1)}}{\eta(m\tau)^{24/(m - 1)}} + \frac{24}{m - 1}.$$ But using Table \[intro-umbral\], we see that $j_m(\tau)$ is equal to $T_{g(X)}(\tau)$, the graded trace of $g(X) \in M$ on $V$.
From Theorem \[mainthm\], we have that $$\prod_{n = 1}^\infty P_\Delta(q^n)^{c^+\pa{\frac{|\Delta| n^2}{4 m}, \frac{rn}{2m}}} = \prod_i (T_g(\tau) - T_g(\alpha_i))^{\gamma_i}.$$ We equate the $q^1$ Fourier coefficients of each side, using Table \[monster\] to get the Fourier expansion $$T_g(\tau) = \frac{1}{q} + O(q).$$
Examples
--------
For each pure A-type case $X$ with coxeter number $m$, we illustrate how to write $\Psi_{\Delta, r}(\tau, \widehat{H}^{(m)})$ as a rational function in $j_m$. Note that here $\Delta < 0$ is a fundamental discriminant and $r \in \Z$ is such that $\Delta \equiv r^2 \pmod{4m}$.
First we work out an example for $m = 2$ in some detail, then list one example for each $m$. In Section \[subsec:quadratic\_forms\], we explain how to find representatives of $Q_{\Delta, r}/\Gamma_0(m)$ using a method of Gross, Kohen, and Zagier.
Consider the case $m = 2, \Delta = -7, r = 1$. Using the method of Section \[subsec:quadratic\_forms\], we compute that $Q_{-7, 1}/\Gamma_0(2) = \{Q_1, Q_2\}$ and that $Q_{-7, -1}/\Gamma_0(2) = \{-Q_1, -Q_2\}$, where the quadratic forms $Q$, their Heenger points $\alpha_Q$, and their generalized genus characters $\chi_\Delta(Q)$ are given in Table \[quadratic\_form\_example\]. We also include the value of $j_2$ at each Heegner point.
$$\arraycolsep=10pt\def\arraystretch{2.2}
\begin{array}{| c | c | c | c |}
\hline
\text{quadratic form} = Q & \alpha_Q & \chi_{\Delta}(Q) & j_2(\alpha_Q) \\
\hline \hline
Q_1 = [ 2, 1, 1] & \alpha_1 = \frac{-1 + \sqrt{-7}}{4} & 1 &\gamma_1 := \frac{1 + 45 \sqrt{-7}}{2} \\
\hline
Q_2 = [-2, 1, -1]& \alpha_2 = \frac{1 + \sqrt{-7}}{4} & - 1 & \gamma_2 := \frac{1 - 45 \sqrt{-7}}{2} \\
\hline
-Q_2 & \alpha_2 & 1 & \gamma_2\\
\hline
-Q_1 & \alpha_1 & - 1& \gamma_1 \\
\hline
\end{array}$$ \[quadratic\_form\_example\]
Using the table, the divisor of $\Psi_{-7, 1}(\tau)$ is given by: $$(-\alpha_1 + \alpha_2) -(\alpha_1 - \alpha_2) = 2 \alpha_2 - 2 \alpha_1.$$ Therefore, $$\Psi_{-7, 1}(\tau, \widehat{H}^{(2)}) = \frac{(j_2(\tau) - \gamma_2)^2}{(j_2(\tau) - \gamma_1)^2}.$$
Similarly, for each value of $m$ corresponding to a pure A-type case, we demonstrate in Table \[borcherd\_examples\] how to write $\Psi_{\Delta, r}(\tau, \widehat{H}^{(m)})$ as a rational function in $j_m$ for some nice choice of $\Delta, r$. In all the examples we consider, $$\Psi_{\Delta, r}(\tau, \widehat{H}^{(m)}) =\frac{(j_m(\tau) - \gamma_2)^2}{(j_m(\tau) - \gamma_1)^2}$$ for some $\gamma_1, \gamma_2 \in \calO_{\Q(\sqrt{\Delta})}$. Note that $\Psi_{\Delta, r}$ will not always be a rational function of this particular form - we always picked $\Delta$ with class number 1.
$$\arraycolsep=10pt\def\arraystretch{2.2}
\begin{array}{| c | c | c | c | c |}
\hline
m & \Delta & r & \gamma_1 & \gamma_2 \\
\hline \hline
2 & -7 & 1 & \frac{1 + 45 \sqrt{-7}}{2} & \frac{1 - 45 \sqrt{-7}}{2} \\
\hline
3 & -11 & 1 & 17 + 8 \sqrt{-11} & 17- 8 \sqrt{-11} \\
\hline
4 & -7 & 3 & \frac{-15 +3\sqrt{-7}}{2} & \frac{-15 - 3 \sqrt{-7}}{2} \\
\hline
5 & -11 & 3 & -3 + 2 \sqrt{-11}& -3 - 2 \sqrt{-11} \\
\hline
7 & -19 & 3 & \frac{3 + 3 \sqrt{-19}}{2}& \frac{3 - 3 \sqrt{-19}}{2}\\
\hline
9 & -11 & 5 & -1 + \sqrt{-11} & -1 - \sqrt{-11} \\
\hline
13 & -43 & 3 & \frac{7 + \sqrt{-43}}{2} & \frac{7 - \sqrt{-43}}{2} \\
\hline
25 & -19 & 9 & \frac{\sqrt{-19}}{2} & \frac{- \sqrt{-19}}{2} \\
\hline
\end{array}$$
Computing the elements in $Q_{\Delta, r}/\Gamma_0(m)$ {#subsec:quadratic_forms}
-----------------------------------------------------
In this section, we explain how to compute $Q_{\Delta, r}/\Gamma_0(m)$, following [@Gross:1987ul].
Let $Q_{\Delta, r}^0$ be the subset of primitive forms. Then we have a $\Gamma_0(m)$-invariant bijection of sets $$Q_{\Delta, r} = \bigcup_{\ell^2 \mid \Delta} \left( \bigcup_{h \in S(\ell)} \ell Q_{\Delta/\ell^2, h}^0\right),$$ where $S(\ell) := \{h \in \Z/2m \Z: h^2 \equiv \Delta/\ell^2 \pmod{4m}, \ell h \equiv r \pmod{2m}\}.$ Since we pick $\Delta$ to be a fundamental discriminant, the only possible prime we need to worry about is $\ell = 2$. In our examples, we always choose $\Delta, r$ such that $S(2) = \emptyset$. In this case, we just need to work with $Q_{\Delta, r}^0$.
Now, let $n := \left(m, r, \frac{r^2 - \Delta}{4m}\right)$. Then for $Q = [mc, b, a] \in Q_{\Delta, r}^0$, define $n_1 := (m, b, a), n_2 :=(m, b, c)$, which are coprime and have product $n$. We have the following result:
(Section 1.1 of [@Gross:1987ul]) Define $n$ as above and fix a decomposition $n = n_1 n_2$ with $n_1, n_2$ positive and relatively prime. Then there is a 1:1 correspondence between the $\Gamma_0(m)$-equivalence classes of forms $[cm, b, a] \in Q_{\Delta, r}^0$ satisfying $(m, b, a) = n_1, (m, b, c) = n_2$ and the $\SL_2(\Z)$ equivalence classes of forms in $Q_{\Delta}^0$ given by $Q = [mc, b, a] \mapsto \tilde{Q} = [c m_1, b, a m_2]$, where $m_1 \cdot m_2$ is any decomposition of $m$ into coprime positive factors satisfying $(n_1, m_2) = (n_2, m_1) = 1$. In particular, $|Q_{\Delta, r}^0/\Gamma_0(m)| = 2^v |Q_\Delta^0/\SL_2(\Z)|$, where $v$ is the number of prime factors of $n$.
Note that $|Q_\Delta^0/\SL_2(\Z)|$ equals $2h(\Delta)$ for $\Delta < 0$, where the factor of 2 arises because $Q_\Delta^0$ also contains negative semi-definite forms.
In our examples, we always choose $\Delta, r$ such that $n = 1$, so that $|Q_{\Delta, r}^0/\Gamma_0(m)| = |Q_\Delta^0/\SL_2(\Z)| = 2 h(\Delta)$, where $h(\Delta)$ is the class number of $\Q(\sqrt{\Delta})$. The theory of reduced forms allows us to easily compute $Q_\Delta^0/\SL_2(\Z)$.
$p$-adic properties of the logarithmic derivative
=================================================
$p$-adic modular forms {#sec:padic-def}
----------------------
For each $i \in \N$, let $f_i = \sum a_i(n) q^n$ be a modular form of weight $k_i$ with $a_i(n) \in \Q$. If for each $n$, the $a_i(n)$ converge $p$-adically to $a(n) \in \Q_p$, then $f := \sum a(n)q^n$ is called a $p$-adic modular form. For $p \neq 2$, we define the weight space $$W := \varprojlim_t \Z/\phi(p^t)\Z = \Z_p \times \Z/(p -1)\Z.$$ For $p = 2$, we define $$W := \varprojlim_t \Z/2^{t - 2}\Z = \Z_2.$$ Then the $k_i$ converge to an element $k \in W$, which we call the weight of $f$. We identify integers by their image in $\Z_p \times \{0\}$.
Proof of Theorem \[mero-modform\]
---------------------------------
By Theorem \[mainthm\], $\Psi_{\Delta,r}(\tau)$ is a meromorphic modular function, so that $\Theta(\Psi_{\Delta, r}(\tau))$ is a weight 2 meromorphic modular form on $\Gamma_0(m)$. Thus, the logarithmic derivative $\frac{\Theta(\Psi_{\Delta, r}(\tau))}{\Psi_{\Delta, r}(\tau)}$ is a weight 2 meromorphic modular form on $\Gamma_0(m)$ whose poles are simple and are supported on Heegner points of discriminant $\Delta.$
Proof of Theorem \[padic\] and its corollary
--------------------------------------------
We show that if $(\Delta, r)$ is an admissible pair and $p$ is inert or ramified in $\Q(\sqrt{\Delta})$, that $$f_{\Delta, r} := \frac{\Theta(\Psi_{\Delta, r}(\tau))}{\Psi_{\Delta, r}(\tau)}$$ is a $p$-adic modular form of weight 2. Say $f$ has poles at $\alpha_1, \dots, \alpha_n$, all of which are CM points of discriminant $\Delta$. For each $\alpha_i$, there is some zero $\beta_i$ of $E_{p -1}$ such that $j(\tau) - j(\alpha_i) \equiv j(\tau) - j(\beta_i)$ (see Theorem 1 of [@Kaneko:1998wt]). Then let $$\mathcal{E}:= E_{p - 1} \prod_{i} \frac{(j(\tau) - j(\alpha_i))}{(j(\tau) - j(\beta_i))}.$$ This has weight $p - 1$, is congruent to 1 modulo $p$, has zeros at $\alpha_1, \dots, \alpha_n$, and has no poles. Let $f_t := f \mathcal{E}^{(p^t)}$. Then $f_t \equiv f \pmod{p^t}$ and is a modular form of weight $k_t = 2 + (p - 1)p^t \equiv 2 \pmod{\phi(p^{t + 1})}$, so $f$ is a $p$-adic modular form of weight $2$.
This corollary follows directly for the coefficients of any $p$-adic modular form using the following beautiful result, proven by Serre [@Serre:1974tp] using the theory of Galois representations.
Let $K$ be a number field and $\mathcal O_K$ the ring of integers of $K$. Suppose $f(\tau)=\sum_{n\geq0}a_nq^n\in\mathcal O_K[[q]]$ is a modular form of integer weight $k\geq1$ on a congruence subgroup. For any prime $p$, let $\mathfrak p$ be a prime above $p$ in $\mathcal O_K$. Let $m \geq 1$. Then there exists a positive constant $\alpha_m$ such that $$\#\left\{n\leq X\colon a_n\not\equiv0\pmod{\mathfrak p}^m\right\}=O\left(\frac{X}{(\log X)^{\alpha_m}}\right).$$
Appendix: Definitions of jacobi forms, theta functions, etc. {#sec:appendix}
============================================================
We define the Jacobi theta functions $\theta_i(\tau, z)$ as follows for $q := e(\tau)$ and $y := e(z)$.
$$\begin{aligned}
\theta_2(\tau, z) &:=& q^{1/8}y^{1/2} \prod_{n = 1}^\infty (1 - q^n)(1 + y q^n)(1 + y^{-1}q^{n - 1}) \\
\theta_3(\tau, z) &:=& \prod_{n = 1}^\infty (1 - q^n)(1 + y q^{n - 1/2})(1 + y^{-1}q^{n - 1/2}) \\
\theta_4(\tau, z) &:=& \prod_{n = 1}^\infty (1 - q^n)(1 - y q^{n - 1/2})(1 - y^{-1}q^{n - 1/2}) \\\end{aligned}$$
We use them to define weight zero index $m - 1$ weak Jacobi forms $ \varphi_1^{(m)}$ as follows. Let $$\begin{aligned}
\varphi_1^{(2)} &:= 4 (f_2^2 + f_3^2 + f_4^2),\\
\varphi_1^{(3)} &:= 2(f_2^2 f_3^2 + f_3^2 f_4^2 + f_4^2 f_2^2), \\
\varphi_1^{(4)} &:= 4 f_2^2 f_3^2 f_4^2, \\
\varphi_1^{(5)} &:= \frac{1}{4} \pa{\varphi_1^{(4)} \varphi_1^{(2)} - (\varphi_1^{(3)})^2} \\
\varphi_1^{(7)} &:= \varphi_1^{(3)} \varphi_1^{(5)} - (\varphi_1^{(4)})^2 \\
\varphi_1^{(9)} &:= \varphi_1^{(3)} \varphi_1^{(7)} - ( \varphi_1^{(5)})^2 \\
\varphi_1^{(13)} &:= \varphi_1^{(5)} \varphi_1^{(9)} - 2 (\varphi_1^{(7)})^2 \\\end{aligned}$$ where $f_i(\tau, z) := \theta_i(\tau, z)/\theta_i(\tau, 0)$ for $i = 2, 3, 4$.
For the remaining positive integers $m$ with $m \leq 25$, we define $\varphi_1^{(m)}$ recursively.\
For $(12, m - 1) = 1$ and $m > 5$ we set $$\varphi_1^{(m)} = (12, m - 5) \varphi_1^{(m - 4)} \varphi_1^{(5)} + (12, m - 3) \varphi_1^{(m - 2)} \varphi_1^{(3)} - 2(12, m - 4) \varphi_1^{(m - 3)} \varphi_1^{(4)}.$$ For $(12, m - 1) = 2$ and $m > 10$ we set $$\varphi_1^{(m)} = \frac{1}{2}\pa{(12, m - 5) \varphi_1^{(m - 4)} \varphi_1^{(5)} + (12, m - 3) \varphi_1^{(m - 2)} \varphi_1^{(3)} - 2 (12, m - 4) \varphi_1^{(m - 3)} \varphi_1^{(4)}}.$$ For $(12, m - 1) = 3$ and $m > 9$, we set $$\varphi_1^{(m)} = \frac{2}{3}(12, m - 4) \varphi_1^{(m - 3)} \varphi_1^{(4)} + \frac{1}{3} (12, m - 7) \varphi_1^{(m - 6)} \varphi_1^{(7)} - (12, m - 5) \varphi_1^{(m - 4)} \varphi_1^{(5)}.$$ For $(12, m - 1) = 4$ and $m > 16$ we set $$\varphi_1^{(m)} = \frac{1}{4} \pa{(12, m -13) \varphi_1^{(m - 12)} \varphi_1^{(13)} + (12, m - 5) \varphi_1^{(m - 4)} \varphi_1^{(5)} - (12, m - 9) \varphi_1^{(m - 8)} \varphi_1^{(9)}}.$$ For $(12, m - 1) = 6$ and $m > 18$ we set $$\varphi_1^{(m)} = \frac{1}{3}(12, m - 4) \varphi_1^{(m - 3)} \varphi_1^{(4)} + \frac{1}{6} (12, m - 7) \varphi_1^{(m - 6)} \varphi_1^{(7)} - \frac{1}{2} (12, m - 5) \varphi_1^{(m - 4)} \varphi_1^{(5)}.$$ For $m = 25$, we set $$\varphi_1^{(25)} = \frac{1}{2} \varphi_1^{(21)} \varphi_1^{(5)} - \varphi_1^{(19)} \varphi_1^{(7)} + \frac{1}{2} ( \varphi_1^{(13)})^2.$$
See the appendix of [@Cheng:2013vr] for more information on the space of weight zero Jacobi forms.
We use two versions of an Appell-Lerch sum. The first is the generalized Appell-Lerch sum $\mu_{m, 0}$, defined as in [@Cheng:2013vr]. It is given by $$\mu_{m, 0}(\tau, z) := -\sum_{k \in \Z}q^{m k^2} y^{2 m k} \frac{1 + y q^k}{1 - y q^k},$$ and is the holomorphic part of a weight 1 index $m$ “real-analytic Jacobi form”.
Zwegers [@Zwegers:2008wk] uses a slightly different version of the Appell-Lerch sum. He first defines the theta function $$\vartheta(z, \tau) := \sum_{\nu \in 1/2 + \Z} q^{\nu^2/2} y^{\nu} e(\nu/2).$$ Then he defines $$\mu(u, v; \tau) := \frac{e(u/2)}{\vartheta(v; \tau)} \sum_{n \in \Z} \frac{(-1)^n q^{(n^2 + n)/2} e(n v)}{1 - q^n e(u)}.$$ This is completed to a “real-analytic Jacobi form” $\tilde{\mu}(u, v; \tau)$ of weight 1/2 by letting $$\tilde{\mu}(u, v; \tau) := \mu(u, v; \tau) + \frac{i}{2}R(u - v; \tau),$$ where $$R(z, \tau) := \sum_{\nu \in 1/2 + \Z} \left\{ \text{sgn}(\nu) - E(\nu + a) \sqrt{2t}\right\} (-1)^{\nu - 1/2} q^{-\nu^2/2} y^{-\nu},$$ $t := \Im(\tau)$, $a := \frac{\Im(u)}{\Im(\tau)}$, and $E(z) := 2 \int_0^z e^{-\pi u^2} du$.
[^1]: The first author thanks the NSF and the Asa Griggs Candler Fund for their generous support. The second author thanks the University of Cologne and the DFG for their generous support via the University of Cologne postdoc grant DFG Grant D-72133-G-403-151001011, funded under the Institutional Strategy of the University of Cologne within the German Excellence Initiative. The third author thanks the NSF for its support. The authors began jointly discussing this work at the mock modular forms, moonshine, and string theory conference in Stony Brook, August 2013 and are grateful for the good hospitality and excellent conference. The authors are also grateful to John Duncan and Jeffery Harvey for useful comments which improved the quality of exposition.
[^2]: The case $X=A_{24}$ corresponds to $g(X) = (25Z)$, which is what Conway and Norton call a “ghost element”. This means that $\Gamma_0(25)$ is the only genus zero $\Gamma_0(N)$ that does not correspond to a conjugacy class of the monster group. The parentheses are used to indicate a ghost element.
[^3]: Note that this corrects a typo in [@Bruinier:2010ffz].
[^4]: In fact, it is the only vector-valued mock modular form with shadow $S^{(m)}$ satisfying the optimal growth condition in \[optimal\].
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abstract: 'The model is built in which the main global properties of classical and quasi-classical black holes become local. These are the event horizon, “no-hair”, temperature and entropy. Our construction is based on the features of a quantum collapse, discovered when studying some quantum black hole models. But our model is purely classical, and this allows to use selfconsistently the Einstein equations and classical (local) thermodynamics and explain in this way the “$\log 3$”-puzzle.'
author:
- Victor Berezin
date: |
Institute for Nuclear Research, Russian Academy of Sciences,\
60th October Anniversary pr., 7-a, 117312 Moscow, Russia\
[ ]{}\
and\
[ ]{}\
Institut des Hautes Études Scientifiques, 35 route de Chartres,\
F-91440 Bures-sur-Yvette, France [ ]{}\
[ ]{}\
e-mail: berezin@ms2.inr.ac.ru
title: 'Classical analog of quantum Schwarzschild black hole: local vs global, and the mystery of $\log 3$'
---
Introduction and preliminaries {#sec1}
==============================
Classical definition of the black hole is based on the existence of the event horizon [@HE] – the boundary of a space-time region from which the light cannot escape to infinity. The very notion of the event horizon is global and requires the knowledge of the whole history, both past and future.
Classical “black hole has no hair” [@RW] and is described by only few parameters, mass, Coulomb-like charge and angular momentum. The Schwarzschild black hole has only the mass, the Reissner-Nordstrom one – mass and charge, the Kerr black one – mass and angular momentum. The most general type – Kerr-Newman black hole – has all three parameters. This resembles the body in thermal equilibrium. The process of becoming bold is also global, its duration, formally, is infinite, like the process of coming to thermal equilibrium. It goes through radiating of all possible (scalar, vector, spinor, tensor) perturbations and governed by Schroedinger-like wave equation, first derived in [@ReW]. The results of many numerical studies for a long period (two decades) were summarized in the book [@Ch]. It appeared that such perturbation modes have discrete spectra with complex frequencies $w$. They received the name “quasi-normal modes” and “quasi-normal frequencies”, respectively [@Pr]. The imaginary parts are equidistant indicating that the decaying modes are radiating away in a manner reminiscent of the last pure dying tones (infinitely many overtones) of a ringing bell, and the higher the overtone, the shorter its lifetime. The real part of quasi-normal frequencies tends to some constant value which depends on the black hole type. For Schwarzschild black holes, we are interested in here, $$\label{QNMN}
Gm \, w_n = 0.0437123 - \frac{i}{4} \left( n+\frac{1}{2} \right) + O [(n+1)^{-1/2}] \, , \quad n \to \infty$$ where $m$ is the mass, and $G$ is the Newton’s constant. All that shows that black holes have some inherent frequency. Therefore, they are not “dead” but have some “private life”, encoded in some features of their horizons. Evidently, this is also the global property because it does not depend on what is going on inside.
Investigation of the processes near an event horizon showed that they can be reversible and irreversible like in thermodynamics [@C; @CR]. The assimilation of a point (classical) particle by a (non extremal[^1]) black hole reversible if it is injected at the event horizon from a radial turning point of its motion. In this case the black hole (horizon) area remains unchanged, and the change in other parameters (mass, charge and angular momentum) can be undone by another suitable (reversible) process. In all other cases the horizon area $A$ increases. Thus, for classical black holes $$\label{HA}
dA \geq 0 \, .$$
The new area in black hole physics started with the seminal paper by J.D. Bekenstein [@B], where he presented serious physical arguments that the Schwarzschild black hole should be described by a certain amount of entropy which is proportional to the area of event horizon. Such a strict proportionality could appear to be playing games with symbols with only one parameter, black hole mass, but it was then confirmed by J.M. Bardeen, B. Carter and S.W. Hawking [@BCH] who proved the four laws of thermodynamics for the general class of Kerr-Newman black holes. Moreover, it was shown that the role of the temperature is played by the surface gravity $\varkappa$ at the event horizon (up to some numerical factor), which is constant there. And only after discovering by S.W. Hawking the black hole evaporation [@H] this thermodynamical analogy became the real physical phenomenon. He considered the quantum theory of massless scalar field on the Schwarzschild static space-time background and found that the specific boundary conditions – only infalling waves in the vicinity of the horizon – result in a thermal behavior of the wave functions and nonvanishing energy flow to the infinity. It appeared that the spectrum of such a radiation is Planckian with the temperature $$\label{TH}
T_H = \frac{\varkappa_H}{2 \, \pi} \, ,$$ where $\varkappa$ is the surface gravity at the event horizon. It follows then, that the black hole entropy is exactly one fourth of dimensionless horizon area $$\label{E}
S = \frac{1}{4} \, \frac{A}{\ell^2_{p_{\ell}}} \, ,$$ where $\ell_{p_{\ell}} = \sqrt{\frac{\hbar G}{c^3}} \sim 10^{-33} cm$ is the Planckian length ($\hbar$ is the Planck constant, $c$ is the speed of light, and $G$ is the Newton’s gravitational constant). We will use the units $\hbar = c = k = 1$ ($k$ is the Boltzmann constant), so $\ell_{p_{\ell}} = \sqrt G$ and the Planckian mass is $m_{p_{\ell}} = \sqrt{\frac{\hbar c}{G}} = 1/\sqrt G \sim 10^{-5} gr$.
The nature of Hawking radiation and its black body spectrum lies in the nontrivial causal structure of the space-times containing black holes. The crucial point is the existence of the event horizons. The same takes place in the Rindler space-time. This space-time is obtained by transforming the two-dimensional Minkowski flat space-time from the “ordinary” coordinates $(t,x)$ and metric $ds^2 = dt^2 - dx^2$ related to the set of inertial observers, to the so-called Rindler coordinates $(\eta , \xi)$ and metric $$t = \frac{1}{a} \, e^{a\xi} \, \sinh a \eta \, , \quad x = \pm \frac{1}{a} \, e^{a\xi} \cosh a\eta \quad (x \gtrless 0) \, ,$$ $$-\infty < t < \infty \, , \quad -\infty < \xi < \infty$$
$$\label{R}
ds^2 = e^{2a\xi} (d\eta^2 - d\eta^2) \, .$$
Thus, the Rindler space-time is static and locally flat but differs from the two-dimensional Minkowski space-time globally, because it covers only one half of the latter and, in addition, possesses the event horizon at $t = \pm \, x$ ($\eta = \pm \, \infty$, $\xi = {\rm const}$). The Rindler observers at $\xi = {\rm const}$ are uniformly accelerated. The norm of the acceleration vector $a^{\mu}$ equals $$\label{acc}
\alpha = \sqrt{-a^{\mu} \, a_{\mu}} = ae^{-a\xi} \, .$$ Considering a quantum scalar field in the Rindler space-time, W.G. Unruh found [@U], in fact, the finite temperature quantum field theory with the temperature $$\label{UT}
T_U = \frac{a}{2 \, \pi} \, .$$ We see that this temperature is proportional to the acceleration of the Rindler observer sitting a $\xi = 0$ with $g_{00} = 1$. But, all of them are equivalent (we can always shift the spatial coordinate $\xi \to \xi - \xi_0$). The temperature is not an invariant, but it is a temporal component of a heat vector. This means that each observer measures the Unruh temperature when using its proper time $\tau$ ($ds = d\tau$). If the same observer uses the local clocks that show the local time $t$ ($ds = \sqrt{g_{00}} \, dt$), the local temperature measured by him equals $$\label{LUT}
T_{\rm loc} = \frac{T_U}{\sqrt{g_{00}}} = \frac{a}{2 \, \pi} \, e^{-a\xi} = \frac{\alpha}{2 \, \pi} \, ,$$ which is proportional to the local acceleration $\alpha$. The very fact that the uniformly accelerated observer ($=$ detector) will detect the real particles in the vacuum, was known to people doing quantum electrodynamics long ago. It was understood as a change of a vacuum state due to the external forces that cause such an acceleration. The same happens in the space-time with event horizons. But that the spectrum is thermal appeared to be new and purely relativistic feature. We know from the university course of thermodynamics (see, e.g. [@LL]) that the condition for thermal equilibrium in static space-times is $T_{\rm loc} \, \sqrt{g_{00}} = {\rm const}$. Thus, all the Rindler observers are in thermal equilibrium with each other. Is the Rindler space-time unique in this sense? To answer, let us consider some general two-dimensional static space-time with a metric $$\label{tdm}
ds^2 = e^{\nu} \, dt^2 - d \rho^2 = e^{\nu} \, dt^2 - e^{\lambda} \, dq^2 \, .$$ In the Rindler case $\rho = \frac{1}{a} \, e^{a \xi}$, $e^{\nu} = a^2 \rho^2 = g_{00}$. The static observer undergoes a constant acceleration with the invariant $\alpha = \frac{1}{2} \left\vert \frac{d\nu}{d\rho} \right\vert = \frac{1}{2} \left\vert \frac{d\nu}{dq} \right\vert e^{-\frac{\lambda}{2}}$, and the (now local) Rindler parameter $a(\rho)$, which is called “the surface gravity $\varkappa$”, equals $$\label{sgr}
\varkappa = \frac{1}{2} \left\vert \frac{d\nu}{dq} \right\vert e^{\frac{\nu - \lambda}{2}} = \frac{1}{2} \left\vert \frac{d\nu}{d\rho} \right\vert e^{\frac{\nu}{2}} \, .$$ The thermal equilibrium requires $\varkappa = {\rm const}$, therefore, $g_{00} = C\rho^2$, and this proves that the Rindler space-time is the only one which static observers are in the mutual thermal equilibrium.
By the Einstein equivalence principle we can extend all we learned studying Rindler space-times, to the static gravitational fields, especially to the spherically symmetric ones, because after fixing spherical angles $\theta$ and $\varphi$ the latter become, in fact, the two-dimensional pseudo-surfaces. Of course, in general these surfaces are curved, the equivalence principle holds only locally, and the static observers will not be in thermal equilibrium with each other. Such a temperature is observer-dependent and cannot be considered as an intrinsic property of a given space-time. But for the black hole space-times the position of the event horizon is absolute and does not depend on the observer. So, its temperature does serve an important characteristic of space-time itself. To know the temperature we just need to compute the surface gravity value at the event horizon, $\varkappa_H$. For the Schwarzschild black hole with the famous metric $$\begin{aligned}
\label{Sm}
ds^2 &= &F \, dt^2 - \frac{1}{F} \, dr^2 - r^2 (d\theta^2 + \sin^2 \theta \, d\varphi^2) \, , \nonumber \\
F &= &1-\frac{2 \, Gm}{r} \, ,\end{aligned}$$ where $m$ is the black hole mass, and $r$ is the radius of a sphere (in that sense that its area equals $4 \, \pi \, r^2$), the horizon is located at the Schwarzschild radius $r_g = 2 \, Gm$, and the surface gravity equals $$\label{sgrs}
\varkappa_H = \frac{1}{2} \left\vert \frac{d\nu}{dr} \right\vert \, e^{\frac{\nu-\lambda}{2}} = \frac{1}{2} \, F' (r_H) = \frac{Gm}{r^2} \biggl\vert_{r_g} = \frac{1}{4 \, Gm} \, .$$ Therefore, the Hawking temperature is just the Unruh temperature at the event horizon measured by distant observers (at infinity). The same is true also for Kerr-Newman black holes. Note that outside the event horizon $r > r_g$ the Schwarzschild observers are not in thermal equilibrium with each other, and this is a thermodynamical explanation of the Hawking radiation and, thus, evaporation of black holes. It should be stressed that both the black hole temperature and entropy are global features because their very appearance is due to the existence of the event horizon.
Evaporating, black holes become smaller and smaller and will reach eventually a Planckian size where the still unknown quantum gravity should play an important role. Since the radiation is quantized, the black hole mass have to be quantized as well. Of course the relation is not direct because a black hole is not necessarily transformed into black hole again, but the new black hole will eventually be formed only by radiation. To the black hole mass there may contribute not only the rest masses and kinetic energy of particles, including the total angular momentum, but also Coulomb and magnetic energies of their electric and gauge charges and all kinds of other physical fields confined under the event horizon. But the common feature for all types of black holes is their entropy with its universal relation (\[E\]) to the horizon area. Thus, the black hole quantization means the quantization of its entropy. Moreover, the thermodynamical description is possible only if the jump in the temperature due to quantization of mass, charge and angular momentum during black hole evaporation is negligible compared to its absolute value, while the notion of the entropy as a measure of the information, hidden or ignored, is still valid. This latter feature gives rise to common believe that the black hole quasiclassical quantization can shade light on the structure of the future full quantum gravity, or, at least, will provide us with some selection rules in the attempts to construct such a theory. The quantization of a black hole as whole was proposed long ago by J. Bekenstein [@B1]. The idea was based on the remarkable observation that the horizon area of non-extremal black holes behaves as a classical adiabatic invariant. The Bohr-Sommerfeld quantization rule then predicts the equidistant spectrum for the horizon area and thus, for the black hole entropy. The gedanken experiments show that, due to the quantum effects, the minimal increase in the horizon area in the processes of capturing a neutral [@B2] or electrically charged [@DR] particle approximately equals $$\label{Amin}
\Delta \, A_{\rm min} \approx 4 \, \ell^2_{p_{\ell}} \, .$$ This suggests for the black hole entropy $$\label{edsp}
S_{BH} = \gamma_0 \, N \, , \quad N = 1,2,\ldots$$ where $\gamma_0$ is of order of unity. In their famous work on the black hole spectroscopy J.D. Bekenstein and V.F. Mukhanov [@BM] related the black hole entropy to the number $g_n$ of microstates that corresponds to the particular external macrostate through the well-known formula in statistical physics $g_n = \exp [S_{BH} (n)]$, i.e., $g_n$ is the degeneracy of the $n$-th area eigenvalue. Since $g_n$ should be integer, they deduce that $$\label{gz}
\gamma_0 = \log k \, , \quad k=2,3,\ldots$$ In the spirit of the information theory and the famous claim by J.A. Wheeler “It from Bit” the value of $\log 2$ seems most suitable one.
The logarithmic behavior of the spacing coefficient $\gamma_0$ comes also from the Loop Quantum Gravity. It was shown in [@ABCK], [@ABK], that the entropy of the Schwarzschild black hole is proportional to the horizon area as well as a numerical constant called the Barbero-Immirzi parameter. To fit the Bekenstein-Hawking relation (\[E\]) and the possible value for $\gamma_0$ (\[gz\]) this parameter should equal $\log 2 / (\pi \sqrt 3)$ if the fundamental group in LQG is $SU(2)$, and $\log 3 / (2\pi \sqrt 2)$ if it is $SU(3)$. The choice of the value for $\gamma_0$ leads to minimal possible change in the black hole mass. S. Hod [@Hod], using famous Bohr’s correspondence principle (1923): “Transition frequencies at large quantum numbers should equal classical oscillation frequencies”, deduced that the real part of the complex quasi-normal frequencies for the Schwarzschild black hole should be proportional to $\log 3$. And, indeed, $$\label{Rep}
Gm \, Re \, w = 0.0437123 = \frac{\log 3}{8 \, \pi} \, .$$ The value of $\gamma_0$ as well as that of Barbero-Immirzi parameter and, thus, the choice of the fundamental group in LQG, must be universal. Therefore, it is not surprising that people tried to find some analytical methods for evaluating the quasi-normal frequencies for different types of black holes. By using rather sophisticated tools from the general theory of ordinary differential equations, L. Molt and A. Neitzke showed [@M], [@MN] that for the scalar and tensor perturbations around Schwarzschild black holes the value $\log 3$ is exact. For more general types of black holes the corresponding calculations were fulfilled in [@RS]. It appeared that the simple value $\log 3$ for the spacing coefficient $\gamma_0$ is by no means universal, but exceptional. That is why we use the expression “the mystery of $\log 3$”.
Below we construct a model which is not really a black hole, but possesses its main features. It has an event horizon – but local, the temperature – but local. Then, we develop the local thermodynamics for such a model and show how the mystery of $\log 3$ can be solved.
Quantum thin shells
===================
The geodesically complete Schwarzschild space-time has a geometry of non-transversable wormhole (it is also called an eternal black hole). There are two asymptotically flat regions with spatial infinities connected by the Einstein-Rosen bridge (the throat). Two sides of the bridge are causally disconnected and separated by (past and future) event horizons. The gravitating source is concentrated at two space-like (momentarily existing) singular surfaces at zero radius. In a sense, there is nothing to quantize. To get physically meaningful results we need to introduce some dynamical gravitating source. The simplest generalization of the point mass is the spherically symmetric self-gravitating thin dust shell.
The theory of thin shells in General Relativity was developed by W. Israel [@I] and applied then to various problems by many authors. In this approach the whole space-time is divided into three parts: interior (in), exterior (out) and the hypersurface in-between, called “shell” or “brane” (short for “membrane”). The matter source on the shell is proportional to $\delta$-function and described by a surface energy momentum tensor. The equations of motion are obtained by integrating the Einstein equations along the normal direction to the shell from “in” to “out”. In the case of spherical symmetry the only dynamical variable is the shell radius $\rho (\tau)$ as a function of the proper time of the observer sitting on the shell. Our shell consists of dust. This means that it is characterized by the bare mass $M$ (which is just the sum of masses of the constituent particles), and the only non-zero component of the surface energy momentum tensor is $S_0^0 = \frac{M}{4 \, \pi \, \rho^2}$. Thus, we need only one equation, and this is the energy constraint $\left( {0 \atop 0} \right)$-Einstein equation. We assume that both inside and outside the shell there are vacuum Schwarzschild metrics with the masses, correspondingly $m_{\rm in}$ and $m_{\rm out}$. Then the shell equation reads as follows (a dot denotes the time derivative) $$\label{sheq}
\sigma_{\rm in} \sqrt{\dot\rho^2 + F_{\rm in}} - \sigma_{\rm out} \sqrt{\dot\rho^2 + F_{\rm out}} = \frac{GM}{\rho}$$ $$F = 1 - \frac{2 \, Gm}{\rho} \, .$$ Here $\sigma = \pm \, 1$ is the sign function that indicates whether the radii increase in the normal outward direction $(\sigma = +1)$, or they decrease $(\sigma = -1)$. When $\sigma_{\rm in} = -1$, this means that in the interior we have a semiclosed world, then for $\sigma_{\rm out} = -1$ we obtain $M < 0$, i.e., the shell consists of particles with negative mass what is too exotic. If $\sigma_{\rm out} = -1$ and $M > 0$, then in this case outside the shell we would have no spatial infinity at all but the curvature singularity at zero radius instead. We exclude the case $\sigma_{\rm in} = -1$ for physical reasons. Therefore, the global geometry of the exterior region is determined by the sign of $\sigma_{\rm out} = \sigma$. The sign of $\sigma$ is dynamically changed at the point where $\dot\sigma^2 + F_{\rm out} = 0$, so $F_{\rm out} < 0$, that is, beyond the event horizon $r_g = 2 \, Gm_{\rm out}$, but outside it is a constant of motion. Thus, the value of $\sigma$ distinguishes between two different types of shells. If $\sigma = +1$ outside the horizon, the shell moves on “our” side of the Einstein-Rosen bridge. In the opposite case it forms the semiclosed world and does not appear at all in “our” part of the whole space-time manifold. This depends on the relation between the mass inside $m_{\rm in}$, the bare mass $M$ of the shell itself and on the total mass of the system $m$ that includes the gravitational mass defect and is determined by the shell initial data (position and velocity). It is not difficult to show that $$\begin{aligned}
\label{SS}
\sigma = +1 &{\rm if} &\frac{\Delta m}{M} > \frac{1}{2} \left( \sqrt{\frac{m_{\rm in}^2}{M^2} + 1} - \frac{m_{\rm in}}{M} \right) \nonumber \\
\sigma = -1 &{\rm if} &\frac{\Delta m}{M} < \frac{1}{2} \left( \sqrt{\frac{m_{\rm in}^2}{M^2} + 1} - \frac{m_{\rm in}}{M} \right) \, .\end{aligned}$$ Here $\Delta m$ is the total mass of the shell, including the mass defect, $\Delta m = m_{\rm out} - m_{\rm in}$. In what follows we will be interested in bound motion only, so $\frac{\Delta m}{M} < 1$. The above inequalities were derived using the explicit expression for the turning point $\rho_0$. In the case of quantum shells there is no trajectories and no turning point, the latter being replaced by a principal quantum number $n$. So, the type of the shell should be determined in another way. At the turning point $\dot\rho = 0$ and our shell equation becomes $$\label{etp}
\sqrt{1-\frac{2 \, Gm_{\rm in}}{\rho_0}} - \sigma \sqrt{1 - \frac{2 \, Gm_{\rm out}}{\rho_0}} = \frac{GM}{\rho_0} \, .$$ We see that the sign of a derivative $\partial \Delta m / \partial M$ for fixed $m_{\rm in}$ and $\rho_0$ coincides with the sign of $\sigma$. For quantum shell we, thus, have $$\label{qs}
\sigma = {\rm sign} \, \frac{\partial \Delta m}{\partial M} \biggl\vert_{n,m_{\rm in}} \, .$$
The first attempt to quantize a spherically symmetric thin dust shell, using the squared version of Eqn. (\[sheq\]) and for flat $(m_{\rm in} = 0)$ interior, were made in [@BKKT]. The total mass $m_{\rm out} = m_{\rm tot} = \Delta m$ was considered as the energy $E$ of the system, and the squared equation – as the pre-Hamiltonian $E(\dot\rho , \rho)$, from which the conjugate momentum $p$ and the Hamiltonian $H(p,\rho)$ was derived. In this approach there remained no trace of $\sigma$ and, thus, no trace of the non-trivial causal structure of the geodesically complete Schwarzschild space-time. The same straightforward quantization of the original equation, again for $m_{\rm in} = 0$, was fulfilled in [@Ber]. The result was the following Schroedinger-like equation in finite differences $$\label{me}
\Psi (S + i\zeta) + \Psi (S-i\zeta) = \frac{2-\frac{1}{\sqrt S} - \frac{M^2}{4 \, m^2 S}}{\sigma \sqrt{1-\frac{1}{\sqrt S}}} \, \Psi (S)$$ where $m=m_{\rm out} = m_{\rm tot}$, $M$ is the bare mass of the shell, $S = \frac{\rho}{4 \, G^2 m^2}$, $\zeta = \frac{1}{2 \, G m^2} = \frac{m^2_{p_{\ell}}}{2 \, m^2}$. This equation in the above-written form is valid only outside the horizon. But, if we consider $\sqrt{ \ }$ as a complex function $( \ )^{1/2}$ of one complex variable, then it can be continued beyond the horizon into two other regions (of inevitable contraction – black hole region, and of inevitable expansion – white hole region) of the whole Schwarzschild space-time, the horizon being at the branching point $S=1$. The physically acceptable solutions for the wave function $\Psi$ should be exponentially damped in both asymptotically flat regions (at both spatial infinities). Then, for large enough black holes $(\zeta \ll 1)$ it appears that for the appropriate choice of the pass around the branching point, the leading term in the region of inevitable contraction is an ingoing wave, while in the region of inevitable expansion it is an outgoing wave, as it should be expected for the quasi-classical motion of the shell.
The ADM-formalism for spherically symmetric space-times with a thin shell was developed in [@BBN] whith the Wheeler-DeWitt (Schroedinger-like) equation of the same type as Eqn. (\[me\]), but for general case $m_{\rm in} \ne 0$: $$\begin{aligned}
\label{oe}
\Psi (S + i\zeta) + \Psi (S - i\zeta) &= &\frac{F_{\rm in} + F_{\rm out} - \frac{M^2}{4 \, m^2 S}}{\sigma \sqrt{F_{\rm in}} \, \sqrt{F_{\rm out}}} \, \Psi (S) \, , \nonumber \\
F \left( {{\rm in} \atop {\rm out}} \right) &= &1 - \frac{2 \, G m \left( {{\rm in} \atop {\rm out}} \right)}{\rho} \, .\end{aligned}$$ Note that the shift in the argument is purely imaginary. Therefore, the “good” solutions should be analytical functions. Besides, there are branching points. So, the wave functions should be analytical on some Riemann surface. The physical reason for this is the following. In quantum theory there are no trajectories. Thus, if a shell has parameters $\Delta m$ and $M$ corresponding, say, to $\sigma = +1$ and to motion on “our” side of the Einstein-Rosen bridge, its wave function is nonzero on the other side as well and “feels” both infinities where we have to impose the appropriate boundary conditions. Comparing the behavior of the solutions to the Eqn. (\[oe\]) in vicinities of singular points (infinities and singularities) and around branching points, the following quantum conditions were found for a discrete mass spectrum in the case of bound motion: $$\begin{aligned}
\label{dsp}
\frac{2 \, (\Delta m)^2 - M^2}{\sqrt{M^2 - (\Delta m)^2}} &= &\frac{2 \, m^2_{p_{\ell}}}{\Delta m + m_{\rm in}} \, n \nonumber \\
M^2 - (\Delta m)^2 &= &2 \, (1+2p) \, m^2_{p_{\ell}} \, ,\end{aligned}$$ where $n$ and $p$ are integers. The appearance of two quantum numbers instead of one as in conventional quantum mechanics is due to the nontrivial causal structure of the complete Schwarzschild manifold. The principal quantum number $n$ comes from the boundary condition at “our” infinity, while the new, second, quantum number $p$ – from the infinity on the other side of the Einstein-Rosen bridge.
Wave functions corresponding to this discrete spectrum form the two-parameter family $\Psi_{n,p}$. But for the quantum Schwarzschild black holes we expect a one-parameter family of solutions because quantum black holes should have “no hairs”, otherwise there will be no smooth classical limit. This means that our spectrum is not a quantum black hole spectrum, and corresponding quantum shells do not collapse, like an electron in hydrogen atom. Physically, it is quite understandable, because the radiation was not included into consideration. Since two quantum numbers determine completely the parameters of the shell, $\Delta m$ and $M$, for fixed $m_{\rm in}$, the quantum collapse goes through decrease in $m_{\rm out}$, due to radiation, and increase in $m_{\rm in}$, due to creation new shells, thus diminishing $\Delta m$. Of course, our picture is extremely simplified, but in the real collapse, during which there can be deviations from the strict spherical symmetry, the tendency is the same. This process can go in many different ways, so, the quantum collapse is accompanied with the loss of information, thus converting an initially pure quantum state into some thermal mixed one. But how could quantum collapse be stopped? The natural limit is the crossing of the Einstein-Rosen bridge, since such a transition requires (at least in a quasi-classical regime) insertion of infinitely large volume, with, of course, zero probability. Computer simulations show that the process of quantum collapse for our shells stops when the principal quantum number becomes zero, $n=0$.
The point $n=0$ in our spectrum is very special. In this case the shell does not “feel” not only the outer region (what is natural for the spherical configuration) but it does not know anything about what is going on inside. It “feels” only itself. Such a situation reminds the “no hair” property of a classical black hole. Finally, when all the shells (both the primary one and newly born) are in the corresponding states $n_i = 0$, the whole system does not “remember” its own history. Then it is this “no-memory” state that can be called “the quantum black hole”. Note that the total masses of all the shells obey the relation $$\label{sqrt2}
\Delta \, m_i = \frac{1}{\sqrt 2} \, M_i \, .$$ The subsequent quantum Hawking radiation can proceed via some collective excitations.
Classical analog of quantum Schwarzschild\
black hole$^2$
==========================================
The final state of quantum gravitational collapse can be viewed as some stationary matter distribution. Therefore, we may hope that for massive enough quantum black hole such a distribution is described approximately by a classical static spherically symmetric perfect fluid with energy density $\varepsilon$ and (effective) pressure $p$ obeying classical Einstein equations. This is what we call a classical analog of a quantum black hole. Of course, such a distribution has to be very specific. To study its main features, let us consider the situation in more details.
Any static spherically symmetric metric can be written in the form $$\label{ssm}
ds^2 = e^{\nu} \, dt^2 - e^{\lambda} \, dr^2 - r^2 (d\theta^2 + \sin^2 \theta \, d\varphi^2) \, .$$ Here $r$ is the radius of a sphere with the area $A = 4 \, \pi \, r^2$, $\nu = \nu (r)$, $\lambda = \lambda (r)$. The Einstein equations are (prime denotes differentiation in $r$): $$\begin{aligned}
\label{ee1}
-e^{-\lambda} \left( \frac{1}{r^2} - \frac{\lambda'}{r} \right) + \frac{1}{r^2} &= &8 \, \pi \, G \varepsilon \, , \nonumber \\
- e^{-\lambda} \left( \frac{1}{r^2} + \frac{\nu'}{r} \right) + \frac{1}{r^2} &= &-8 \, \pi \, G p \, , \nonumber \\
-\frac{1}{2} \left( \nu'' + \frac{\nu'^2}{2} + \frac{\nu' - \lambda'}{r} - \frac{\nu' \lambda'}{2} \right) &= &-8 \, \pi \, G p \, .\end{aligned}$$ We see that there are three equations for four unknown functions. But, even we would know an equation of state for our perfect fluid, $p=p(\varepsilon)$, the closed (formally) set of equations would have too many solutions. We need, therefore, some selection rules in order to single out the classical analog of quantum black hole. Surely, the “no hair” feature should be the main criterium. Thus, we have to adjust our previous definition of the “no-memory” state to the case of a continuum matter distribution.
For this, let us integrate the first of Eqns. (\[ee1\]): $$\label{int}
e^{-\lambda} = 1 - \frac{2 \, G m(r)}{r} \, ,$$ where $$\label{mf}
m(r) = 4 \, \pi \int_0^r \varepsilon \, \tilde r^2 \, d\tilde r$$ is the mass function that should be identified with $m_{\rm in}$. Now, the “no-memory” principle is readily formulated as the requirement, that $m(r) = ar$, i.e.$^3$, $$\begin{aligned}
\label{nm}
e^{-\lambda} &= &1-2 \, Ga = {\rm const}, \nonumber \\
\varepsilon &= &\frac{a}{4 \, \pi \, r^2} \, .\end{aligned}$$ We can also introduce a bare mass function $M(r)$ (the mass of the system inside a sphere of radius $r$ without gravitational mass defect): $$\label{bmf}
M(r) = \int \varepsilon \, d \, V = 4 \, \pi \int_0^r \varepsilon \, e^{\frac{\lambda}{2}} \, \tilde r^2 \, d\tilde r = \frac{ar}{\sqrt{1-2 \, Ga}} \, .$$ The remaining two equations can now be solved for $p(r)$ and $e^{\nu} (r)$. The general solution is rather complex, but the correct non-relativistic limit for the pressure $p(r)$ (we are to reproduce the famous equation for hydrostatic equilibrium) is given by only the following one-parameter family: $$\label{p}
p(r) = \frac{b}{4 \, \pi \, r^2} \, ,$$ where $$\label{b}
b = \frac{1}{G} \left( 1-3 \, Ga - \sqrt{1-2 \, Ga} \, \sqrt{1-4 \, Ga} \right) \, .$$ We see that the solution exists only for $a \leq \frac{1}{4 \, G}$, then $b \leq a$. The physical meaning of these inequalities is that the speed of sound cannot exceed the speed of light, $v_{\rm sound}^2 = \frac{b}{a} \leq 1 = c^2$, the equality being reached just for $a=b=\frac{1}{4 \, G}$. Finally, for the temporal metric coefficient $g_{00} = e^{\nu}$ we get $$e^{\nu} = C_0^2 \, r^{\frac{4b}{a+b}} = C_0^2 \, r^{2G \frac{a+b}{1-2Ga}} \, .$$ Thus, demanding the “no-memory” feature and the existence of the correct non-relativistic limit, we obtained the two-parameter family of static solutions. But, we need a one parameter family, so we have to continue our search.
Evidently, the point $r=0$ is singular both for matter distribution and $g_{00}$ metric coefficient. To examine what kind of singularity we are dealing with, one should calculate the Riemann curvature tensor. It appears that for $b < a$ this tensor is, indeed, divergent at $r=0$. But, if $a=b=\frac{1}{4 \, G}$, we are witnessing a miracle, the (before) divergent components become zero. Thus, demanding, in addition to the previous two very natural requirements, the third one (also natural), namely, the absence of the real (curvature) singularity at $r=0$, we arrive at the following one-parameter family of solutions to the Einstein equations (\[ee1\]) $$\begin{aligned}
\label{fs}
g_{00} &= &e^{\nu} = C_0^2 \, r^2 \, , \nonumber \\
g_{11} &= &-e^{\lambda} = -2 \, , \nonumber \\
\varepsilon &= &p = \frac{1}{16 \, \pi \, G r^2} \, .\end{aligned}$$ So, the equation of state of our perfect fluid is the stiffest possible one. The constant of integration $C_0$ can be determined by matching the interior and exterior metrics at some boundary value of radius, $r=r_0$. Let us suppose that for $r > r_0$ the space-time is empty, so, the interior should be matched to the Schwarzschild metric with the mass parameter $m$. Of course, to compensate the jump in the pressure $\Delta p$ $( = p(r_0) = p_0)$ we must include in our model a surface tension $\Sigma$, so, actually, we are dealing with some sort of liquid. It is easy to check that $$C_0^2 = \frac{1}{2 \, r_0^2} \ ; \qquad \Delta p = \frac{2 \, \Sigma}{\sqrt 2 \, r_0} \ ;$$ $$e^{\nu} = \frac{1}{2} \left( \frac{r}{r_0} \right)^2 \ ; \qquad p_0 = \varepsilon_0 = \frac{1}{16 \, \pi \, G r_0^2} \ ;$$ $$\label{mc}
m = m_0 = \frac{r_0}{4 \, G} \, .$$ Note that the bare mass $M = \sqrt 2 \, m$, the relation is exactly the same as for the shell “no-memory” state, Eqn. (\[sqrt2\]), and $r_0 = 4 \, G m_0$, so, the size of our analog model is twice as that for a classical black hole of the same mass.
Now, how about the special point in our solution, $r=0$? It is not a trivial coordinate singularity, like in a three-dimensional spherically symmetric space, because $$\label{hor}
ds^2 \, (r=0) = 0 \, .$$ This looks like an event horizon. Indeed, the two-dimensional $(t-r)$-part of our metric describes a locally flat manifold. Since the static observers at $r = {\rm const}$ are, in fact, uniformly accelerated, this is a Rindler space-time with the event horizon at $r=0$. By definition, the surface of zero radius cannot be crossed, and this is just in this sense the generally global event horizon becomes local. The corresponding Rindler parameter which in more general case is called the “surface gravity”, equals $$\label{sgr1}
\varkappa = \frac{1}{2} \left\vert \frac{d\nu}{dr} \right\vert e^{\frac{\nu - \lambda}{2}} = \frac{C_0}{\sqrt 2} = \frac{1}{2 \, r_0} \, .$$ Therefore, the Unruh temperature in our model is $$\label{UT1}
T_U = \frac{1}{4 \, \pi \, r_0} = \frac{1}{16 \, \pi \, Gm} \, ,$$ what is twice less than the Hawking temperature for the Schwarzschild black hole, $$\label{HUT}
T_H = \frac{1}{8 \, \pi \, Gm} = 2 \, T_U \, .$$ Note that the local observer measures the temperature $$\label{loct}
T_{\rm loc} = \frac{1}{2 \, \sqrt 2 \, \pi \, r}$$ which does not depend on the boundary value $r_0$, and is, in fact, universal.
To clarify the situation with the temperature, let us consider the general form of spherically symmetric line element with the Rindler two-dimensional part: $$\label{R4}
ds^2 = a^2 \, \rho^2 \, dt^2 - d\rho^2 - R^2 (t,\rho) (d\theta^2 + \sin^2 \theta \, d\varphi^2) \, ,$$ where $R(t,\rho)$ is the radius, and $a={\rm const}$ (the possible dependence $a(t)$ can always be absorbed by redefinition of the time coordinate). Then the Einstein equations read as follows $$\begin{aligned}
\label{ee2}
G_0^0 &= &- \frac{2R''}{R} + \frac{\dot R^2}{a^2 \, \rho^2 \, R^2} + \frac{1-R'^2}{R^2} = 8 \, \pi \, G \, \varepsilon \nonumber \\
G_{01} &= &-2 \, \frac{\dot R'}{R} + 2 \, \frac{\dot R}{\rho \, R} = 8 \, \pi \, G \, T_{01} \nonumber \\
G_1^1 &= &\frac{2 \, \ddot R}{a^2 \, \rho^2 \, R} + \frac{\dot R^2}{a^2 \, \rho^2 \, R^2} - 2 \, \frac{R'}{\rho \, R} + \frac{1-R'^2}{R^2} = - 8 \, \pi \, G \, p_r \nonumber \\
G_2^2 &= &\frac{\ddot R}{a^2 \, \rho^2 \, R} - \frac{R''}{R} - \frac{R'}{\rho \, R} = - 8 \, \pi \, G \, p_t \, .\end{aligned}$$ Then, demanding $T_{01} = 0$ because in thermal equilibrium there is no energy (heat) flow, we obtain $$\label{rho}
R = \alpha (t) \, \rho \, .$$ For static space-times $\alpha = {\rm const}$, and we recover the “no-memory” condition. We see that, in general, the radial and tangential pressures are not equal, $p_r \ne p_t$, and our liquid is anisotropic. But in the absence of external fields or some other influence (Coulomb force, angular momentum and so on) the local observer should see the isotropic surroundings. So, our assumption about perfect fluid seems correct.
Thermodynamics
==============
We saw that all the parts of our matter distribution are in thermal equilibrium. This is also reflected in the following remarkable feature. If one removes some outer layer, nothing would be changed inside. In this way the analog model resembles the (quasi-) classical black hole, where only the surface of the horizon is responsible for everything. Another feature is the existence of the intrinsic black hole frequency (resonance frequency) that follows from studying of quasi-normal modes. This forces us to consider the whole system as the set of quasi-particles, black hole phonons. The number of these phonons will be one of the thermodynamical extensive parameters, together with the entropy and the volume.
We are going to derive the local thermodynamical relations for our system. They should be distinguished from the global ones observed at infinity. The local observer deals with the bare mass $M$ ($=$ energy $E$) defined as the following integral over some volume $V$: $$\label{defM}
E = M = \int T^{0\lambda} \, \xi_{\lambda} \, dV = \int T_0^0 \, \xi^0 \, dV = \int \varepsilon \, dV \, ,$$ where $T_{\nu}^{\lambda}$ is the energy-momentum tensor, $\xi^{\mu}$ is the Killing vector normalized as $\xi^0 = 1$. In relativistic theory both the temperature $T$ and the entropy $S$ are temporal components of corresponding four-vectors. We will be using the local temperature $T_{\rm loc}$ and the invariant quantity $S = \xi_{\mu} \, S^{\mu}$, where $S^{\mu}$ is the entropy flow. With this in mind we can write the first law of thermodynamics as $$\label{th1}
dM = \varepsilon \, dV = T_{\rm loc} \, dS - p \, dV + \mu \, dN \, ,$$ $\mu$ is the chemical potential related to the number of black hole phonons (this is how the integer number enters our model), it ought to be included because in the model all the distributions are universal and the only parameter that changes is the boundary value of radius $r_0$, and this means automatical changing of all the extensive variables, $M$, $S$, $V$ and $N$. Dividing the above expression by the volume element $dV$ we get the first law in its local form $$\label{th2}
\varepsilon (r) = T_{\rm loc} (r) \, s(r) - p(r) + \mu (r) \, n(r) \, ,$$ where $s$ and $n$ are the entropy and particle densities, respectively. In our model $\varepsilon = p$, but how about $s$? The local observer cannot calculate it without knowing the corresponding microscopic structure, but he can ask his global counterpart who is educated enough (read proper books) and knows that the total entropy of the black hole is $S = \frac{1}{4 \, G} \, A_{\rm hor}$, what for the Schwarzschild black hole gives $(A_{\rm hor} = 4 \, \pi \, r_g^2)$ $S = \frac{\pi}{G} \, r_g^2 = \frac{\pi \, r_0^2}{4 \, G}$. Having this information, our local observer can deduce that $$\label{ed}
s(r) = \frac{1}{8 \, \sqrt 2 \, Gr}$$ and $$\label{Ts}
T_{\rm loc} (r) \, s(r) = \frac{1}{32 \, \pi \, Gr^2} \, .$$ Remembering now that $\varepsilon = \frac{1}{16 \, \pi \, Gr^2}$, we obtain $$T_{\rm loc} (r) \, s(r) = \frac{1}{2} \, \varepsilon \, , \quad \mu (r) \, n(r) = \frac{3}{2} \, \varepsilon \, .$$
To construct a thermodynamical potential of energy $E = E(S,V,N)$, we make use of its additivity property, $$\begin{aligned}
\label{add}
&E = N\varphi (x,y) \, , &x = \frac{S}{N} \, , \ y = \frac{N}{V} \, , \nonumber \\
&\varepsilon = y \varphi (x,y) \, , &p = - \frac{\partial \, E}{\partial \, V} = y^2 \, \frac{\partial \, \varphi}{\partial \, y} \, .\end{aligned}$$ Since in our model $\varepsilon = p$, we obtain $$\label{epsp}
\varphi = n \, \alpha (x) = y \, \alpha (x) \qquad p = n^2 \alpha (x) = y^2 \alpha (x) \, .$$ Now, $T = \frac{\partial \, E}{\partial \, S} = \frac{\partial \, \varphi}{\partial \, x} = y \, \frac{\partial \, \alpha}{\partial \, x}$, $s = xy$, so $Ts = y^2 \, x \, \frac{\partial \, \alpha}{\partial \, x}$. Then, from the relation $Ts = \frac{1}{2} \, \varepsilon$ we get $$\label{alpha}
x \, \frac{\partial \, \alpha}{\partial \, x} = \frac{1}{2} \, \alpha \, , \qquad \alpha = \alpha_0 \, \sqrt x \, ,$$ where $\alpha_0$ is some universal constant. Eventually, we have $$\begin{aligned}
&\varepsilon = \alpha_0 \, s^{1/2} \, n^{3/2} \, , &p = \alpha_0 \, s^{1/2} \, n^{3/2} = \varepsilon \, , \nonumber \\
&T = \frac{\alpha_0}{2} \, \frac{n^{3/2}}{s^{1/2}} \, , &\mu = \frac{3}{2} \, \alpha_0 \, s^{1/2} \, n^{1/2} \, . \nonumber\end{aligned}$$ $$\label{thp}
E = \alpha_0 \, \frac{\sqrt{SN^3}}{V} \, .$$
In what follows we will need the expression for the free energy $F$: $$\begin{aligned}
\label{fe}
F &= &\int f \, dV \nonumber \\
f &= &\varepsilon - T_{\rm loc} \, s = \frac{1}{2} \, \varepsilon \nonumber \\
F &= &\frac{1}{2} \, M \, .\end{aligned}$$ It is known that the thermal equilibrium conditions for the systems in static gravitational field are (see, i.e., [@LL]) $$\begin{aligned}
\label{tmu}
T \, \sqrt{g_{00}} &= &{\rm const} \, , \nonumber \\
\mu \, \sqrt{g_{00}} &= &{\rm const} \, .\end{aligned}$$ The constants on the right-hand sides are universal for our model – they do not depend on the boundary value $r_0$. Therefore, their ratio is also a universal constant. Thus, we have $$\label{gamma}
\frac{\mu}{T} = 3 \, \frac{s}{n} = 3 \, \frac{S}{N} = 3 \, \gamma_0 \, .$$ Hence, the entropy is naturally quantized: $$\label{entrsp}
S = \gamma_0 \, N \, , \qquad N = 1,2 \ldots$$ The constant $\gamma_0$ is, therefore, universal. It can be easily related to $\alpha_0$ that appeared in thermodynamical relations, Eqn. (\[thp\]). Indeed, $$\label{calog}
\varepsilon = \alpha_0 \left( \frac{n}{s} \right)^{3/2} \, s^2 = \frac{\alpha_0}{\gamma_0^{3/2}} \, s^2 \, .$$ Since $\varepsilon = \frac{1}{16 \, \pi \, Gr^2}$, and $s = \frac{1}{8 \, \sqrt 2 \, Gr}$, then $$\label{calca}
\alpha_0 = \frac{8 \, G}{\pi} \, \gamma_0^{3/2} \, .$$
Solving the mystery of $\log 3$
===============================
In order to calculate the spacing coefficient $\gamma_0$ we have to make some assumption about the microscopic structure of our model. We assume that the interior matter distribution consists of $N$ black hole phonons with the equidistant spectrum of excitations $$\label{phon}
\varepsilon_n = \omega \, n \, , \qquad n = 1,2 \ldots$$ In this case the partition function for the whole system is the product of that ones for each phonons, and $$\begin{aligned}
\label{pf1}
Z_{\rm tot} &= &(Z_1)^N \nonumber \\
Z_1 &= &\sum_n e^{-\frac{\varepsilon_n}{T}} = \sum_n \left( e^{-\frac{\omega}{T}} \right)^n = \frac{e^{-\frac{\omega}{T}}}{1-e^{-\frac{\omega}{T}}} \, .\end{aligned}$$ It is natural to suppose that $\omega$ is just the black hole resonance frequency, its existence follows from the properties of quasi-normal modes (as was already explained earlier). Of course, $\omega$ is a temporal component of a four-vector, but the temperature $T$ also does, so their ratio does not depend on the choice of the clocks by local static observers. We accept that the observers are using their proper time, so $T$ is just the Unruh temperature $T_U$ which is constant in the whole interior. The partition function is an invariant, and we can calculate it in another way, using thermodynamical relations. Indeed, we can consider some small volume element $dV$ and the corresponding partition function $Z_{\rm small}$. Then, using the well-known formula for the free energy $F = -T \log Z$, and writing it for the volume element $$\label{pf2}
dF = f \, dV = - T_{\rm loc} \log Z_{\rm small} \, ,$$ where, as before, we use the local intrinsic quantities in thermodynamical relations. From this we have $$\label{pf3}
\int \frac{f}{T_{\rm loc}} \, dV = - \sum \log Z_{\rm small} = -\log Z_{\rm tot} \, .$$ The left-hand side equals $$\begin{aligned}
\label{lhs}
\int \frac{f}{T_{\rm loc}} \, dV &= &\int \frac{\varepsilon - T_{\rm loc} \, s}{T_{\rm loc}} \, dV = \frac{1}{2} \int \frac{\varepsilon}{T_{\rm loc}} \, dV \nonumber \\
&= &2 \, \sqrt 2 \, \pi \int_0^{r_0} \frac{\varepsilon}{T_{\rm loc}} \, r^2 \, dr = \frac{\pi \, r_0^2}{4 \, G} = \frac{\pi \, r_g^2}{G} = S \, .\end{aligned}$$ Here $r_g$ is the Schwarzschild radius, and $S$ is the total black hole entropy. Eventually, we obtain the important relation $$\label{SZ}
e^{-S} = Z_{\rm tot} = (Z_1)^N \, ,$$ from which it follows that $$\begin{aligned}
\label{Sgamma}
\frac{e^{-\frac{\omega}{T}}}{1-e^{-\frac{\omega}{T}}} &= &e^{- \frac{S}{N}} = e^{-\gamma_0} \, , \nonumber \\
e^{\gamma_0} &= &e^{\frac{\omega}{T}} - 1 \, . \end{aligned}$$
To go further, let us consider the irreversible process of converting the mass (energy) of the system into radiation from a thermodynamical point of view. In our model such a process takes place just at the boundary $r=r_0$, and the thin shell with zero surface energy density and surface tension $\Sigma$ serves as a converter supplying the radiation with extra energy and extra entropy, this resembles the “brick wall” model [@tH]$^4$. One can imagine that the near-boundary layer of thickness $\Delta \, r_0$ is converting into radiation, thus decreasing the boundary of the inner region to $(r_0 - \Delta \, r_0)$. Its energy equals $\Delta \, M = \varepsilon \, \Delta \, V$. To this we should add the energy released from the work done by the surface tension due to its shift, which is equal exactly to $\sum d (4 \, \pi \, r_0^2) = p \, d \, \Delta \, V = \varepsilon \, \Delta \, V = \Delta \, M$. Therefore, both the energy and the temperature in the converter becomes two times higher than that for any inner layer of the same thickness. And this double energy is gained by radiating quanta. Clearly, they have double frequency and exhibit double temperature, so $$\label{log3}
\frac{R e \, w}{T_H} = \frac{\omega}{T_U} = \log 3 \, ,$$ as follows from the spectrum of quasi-normal modes for the Schwarzschild black holes. Substituting this into Eqn. (\[Sgamma\]) and remembering that $$\label{3-2}
3-1=2$$ we obtain $$\label{log2}
\gamma_0 = \log 2 \, .$$ Since the radiated energy is thermalized, the interpretation of $dm$ as equal to $R e \, w$ is an improper procedure. This resolves the “$\log 3$-paradox”.
Acknowledgments
===============
The author is greatly indebted to the Institut des Hautes Études Scientifiques for kind hospitality extended to him. He would like to thank Thibault Damour, Pierre Vanhove, Maxim Kontsevich, Alexey Smirnov, Valentin Zagrebnov and Galina Eroshenko, Carlo Rovelli and members of his group for helpful discussions.
I am much thankful to Cécile Gourgues for careful typing the manuscript. Special thanks to my wife Anastasia Kupriyanova and to D.O., D.O. and N.O. Ivanovs for the permanent and strong moral support.
This work was supported by the grant No. 10-02-00635-a from the Russian Foundation of Fundamental Investigations (RFFI).
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[^1]: If a black hole has more than one parameter, then for fixed value of other, than mass, parameters there exists the minimal value of mass (critical, or extreme), below which the event horizon (and, thus, the black hole itself) does not exist.
|
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abstract: 'In the study of $CP$ violation signals in $\O \rightarrow\pi\Xi$ nonleptonic decays, the strong $J$=3/2 $P$ and $D$ phase shifts for the $\pi\Xi$ final-state interactions are needed. These phases are calculated using an effective Lagrangian model, that considers $\Xi$, $\Xi^*$(1530), $\rho$ and the $\sigma$-term, in the intermediate states. The $\sigma$-term is calculated in terms of the scalar form factor of the baryon.'
author:
- 'C.C. Barros Jr.'
title: '$\pi\Xi$ phase shifts and $CP$ Violation in ${ \Omega\rightarrow\pi\Xi}$ Decay'
---
introduction
============
In the search for physics beyond the standard model, the observation of $CP$ violation could be a very usefull tool. Many models shows $CP$ nonconservation effects, as, for example, the superweak model [@sw], the Kobayashi-Maskawa model [@KM], the penguin model [@pen] or the Weinberg-Higgs model [@WH].
Today, there are three systems where $CP$ violation has been observed. The first one was the $K_L^0\rightarrow \pi^+\pi^-$ decay, where it was shown [@dec] that $K_2^0$ is not a pure eigenstate of $CP$, and then, the parameter $\epsilon$ is nonzero. Later, direct $CP$ violation in $K \rightarrow \pi \pi$ decays has been observed. Very recently, $CP$ violation has also been observed in the $B \rightarrow J/ \Psi K_S$ and other related modes (for a review, see [@nir]).
In 1957, Okubo [@OK] noted that $CP$ violation could cause differences in the branching ratios of the $\S$ and ${\overline\S}^+$ decays. Pais [@PA] extended this proposal also to $\L$ and $\overline\L$ decays. In more recent works the $CP$ violation signs were also investigated using nonleptonic hyperon decays [@don]-[@meis] where $\Xi$ decays were stiudied too. In [@ta2] the study has been made in the framework of the standard model, and in [@xh1], [@xh2], new physics was considered. At the experimental level, there are experiments searching for $CP$ violation in hyperon decays [@luk].
The nonleptonic $\O$ decays has only been studied in [@ta3] where the the strong phases were estimated at leading order in heavy-baryon chiral perturbation theory. The aim of this paper is to calculate the $\Xi\pi$ strong phases using an effective Lagrangian model, without the heavy-baryon approximation, and their effects to the asymmetry parameters in the $\Omega^-$ decays.
The size of the $CP$ violation depends on the final-state strong interaction between the produced particles. So, in order to perform the calculations, the strong phase shifts are needed. At the moment, the situation of the $\Lambda\rightarrow\pi N$ and $\Sigma\rightarrow\pi N$ decays is very confortable, the $\pi N$ strong phase shift analysis is very well known experimentally [@PS]. At the theoretical level, this system is very well described, and, at least at low energies, the chiral perturbation theory is very precise [@LG], [@PiN].
However, in the decays that produce hyperons, the situation is not so good, no experimental data is available to the $\pi Y$ interactions. In fact, some information can be obtained in the study of hyperonic atoms (see, for example, [@LOI], where the $\pi\L\S$ coupling constant is estimated), but it is not enough to fully understand the $\pi \L$ interaction. Thus, to investigate the $\pi Y$ interactions, the only way is to use a model.
As it was said, the chiral perturbation theory is very accurate when applied to the $\pi N$ interactions, so, we hope that it works in the $\pi Y$ system too. In [@Kam]-[@meis], [@keis] that was done to the $\Xi\rightarrow\pi\L$ decay, and the calculated phases were very small.
In this work, the $\O\rightarrow\pi\Xi$ is studied, and the strong phase shifts for $\pi \Xi$ interactions are calculated using the model presented in [@BH]. In [@BH], chiral lagrangians are used, describing processes with $\Xi$, $\Xi^*$ and $\rho$ in the intermediate states. The $\sigma$-term is also included, but not only as a parametrization (as it was done in [@BH]), but relating it with the scalar form factor $\sigma(t)$, based in the results of [@MAN], [@CM].
This paper will show the following contents: In section II it will be shown the $\Omega^-$ decay and how to calculate the observables. In section III we will calculate the phase shifts in the $\pi\Xi$ interactions. The results and conclusions are in section IV.
Nonleptonic ${\bf \Omega^-}$ Decay
==================================
In the $\Omega^-$ ($J^p={3\over 2}^+$) decays, the transitions are of the form 3/2 0 + [spin]{} 1/2 ,
and the contributing phases are the $J={3\over 2}$ $P$(parity conserving) and $D$(parity violating) waves in the $\Omega$ rest frame. The $\Delta S=1$ $\pi\Xi $ nonleptonic decays are $\Omega^-\rightarrow\Xi^0\pi^-$ and $\Omega^-\rightarrow\Xi^-\pi^0$.
The experimental observables are the total rate $\Gamma$, and the asymmetry parameters, that can be written as =2 [Re]{}(P\^\*D)/(|P|\^2+|D|\^2) =2 [Im]{}(P\^\*D)/(|P|\^2+|D|\^2) =(|P|\^2-|D|\^2)/(|P|\^2+|D|\^2)
and obeys the relation \^2+\^2+\^2=1 .
For antihyperons decays the expressions are =2 [Re]{}([P]{}\^\* [D]{})/ (|[P]{}|\^2+|[D]{}|\^2) =2 [Im]{}([P]{}\^\* [D]{})/ (|[P]{}|\^2+|[D]{}|\^2) .
The $P$ and $D$ amplitudes can be parametrized as P=\_Ia\_[2I]{}|P\_[2I]{}|e\^[i(\_P\^[2I]{}+\_P\^[2I]{})]{} \[2.12\] D=\_Ia\_[2I]{}|D\_[2I]{}|e\^[i(\_D\^[2I]{}+\_D\^[2I]{})]{} , \[2.13\]
where $I$ is the isospin state, $\delta_l^{2I}$ are the strong phase shifts and $\phi_l^{2I}$ are the weak $CP$ violating phases. The respective $CP$ conjugated amplitudes are =\_Ia\_[2I]{}|P\_[2I]{}|e\^[i(\_P\^[2I]{}-\_P\^[2I]{})]{} =-\_Ia\_[2I]{}|D\_[2I]{}|e\^[i(\_D\^[2I]{}-\_D\^[2I]{})]{} ,
Using eq. (\[2.12\]), (\[2.13\]) for the $\Omega^-\rightarrow\Xi^0\pi^-$ decay we have P(\_-\^-)=-P\_[1]{}e\^[i(\_P\^1+\_P\^1)]{}+ P\_[3]{}e\^[i(\_P\^3+\_P\^3)]{} D(\_-\^-)=-D\_[1]{}e\^[i(\_D\^1+\_D\^1)]{}+ D\_[3]{}e\^[i(\_D\^3+\_D\^3)]{}
and for $\Omega^-\rightarrow\Xi^-\pi^0$ P(\_0\^-)=P\_[1]{}e\^[i(\_P\^1+\_P\^1)]{}+ P\_[3]{}e\^[i(\_P\^3+\_P\^3)]{} D(\_0\^-)=D\_[1]{}e\^[i(\_D\^1+\_D\^1)]{}+ D\_[3]{}e\^[i(\_D\^3+\_D\^3)]{} .
In the limit of $CP$ conservation, the $CP$ asymmetry parameters A=[+-]{}
and B=[+-]{}
vanish, since $\alpha$=-${\overline \alpha}$ and $\beta$=-${\overline \beta}$. In hyperon decays, the $\Delta I$=3/2 amplitudes are much smaller then the $\Delta I$=1/2, then, in the first order in $\Delta I$=3/2 amplitudes, A(\_-\^-)&=&-[tan]{}(\_P\^1-\_D\^1)[tan]{}(\_P\^1-\_D\^1) { 1 +\
&& +[1]{}[P\_3P\_1]{} .\
&& . - [(\_P\^3-\_D\^1)(\_P\^3-\_D\^1)(\_P\^1-\_D\^1)(\_P\^1-\_D\^1)]{}\
&& +[1]{}[D\_3D\_1]{} .\
&& . -[(\_P\^3-\_D\^1)(\_P\^3-\_D\^1)(\_P\^1-\_D\^1)(\_P\^1-\_D\^1)]{} } \[2.10\]\
B(\_-\^-)&=&[cot]{}(\_P\^1-\_D\^1)[tan]{}(\_P\^1-\_D\^1) { 1 +\
&& +[1]{}[P\_3P\_1]{} .\
&& . -[(\_P\^3-\_D\^1)(\_P\^3-\_D\^1)(\_P\^1-\_D\^1)(\_P\^1-\_D\^1)]{}\
&& +[1]{}[D\_3D\_1]{} .\
&& . -[(\_P\^3-\_D\^1)(\_P\^3-\_D\^1)(\_P\^1-\_D\^1)(\_P\^1-\_D\^1)]{} } \[2.11\]
and similar expressions for the $\Omega\rightarrow \pi^0\Xi^-$ decays (replacing the factors $1/\sqrt{2}$ for $-\sqrt{2}$ in the expressions (\[2.10\]), (\[2.11\])). At leading order, A(\_-\^-)=A(\_-\^0)= -[tan]{}(\_P\^1-\_D\^1)[tan]{}(\_P\^1-\_D\^1) , \[1.30\]
and B(\_-\^-)=B(\_-\^0)= [cot]{}(\_P\^1-\_D\^1)[tan]{}(\_P\^1-\_D\^1) . \[1.31\]
In the next section we will calculate the phase shifts $\delta_l^{2I}$, that are needed to estimate $A$ and $B$.
Low energy ${\pi\Xi}$ interaction
=================================
In order to describe the low energy $\pi\Xi$ interaction, a reliable way is to use effective Lagrangians, as it was done in a previous work [@BH]. A very important feature of this model is to allow the inclusion of spin 3/2 ressonances in the intermediate states. In the low energy $\pi^+ P$ interactons, for example, the $\Delta$(1232) dominates almost completely the total cross section (when $\sqrt{s}$ is near the $\Delta$ mass). Consequently, it is expected that in some reactions of $\pi Y$ scattering, the same behaviour will occur [@BH]. The lagrangians to be considered are $$\begin{aligned}
{\cal{L}}_{\Xi\pi\Xi} &=& {g_{\Xi\pi\Xi}\over 2m_\Xi}\lbrack {\overline\Xi}
\gamma_\mu \gamma _{5} \vec \tau\Xi \rbrack .\partial^\mu \vec \phi \\
{\cal{L}}_{\Xi\pi\Xi^*} &=& g_{\Xi\pi\Xi^*} \lbrace {{\overline{\Xi}^*}^\mu}
\lbrack g_{\mu\nu} - (Z+{1\over 2})\gamma_\mu \gamma _\nu
\rbrack \vec\tau\Xi \rbrace .\partial ^\nu \vec \phi + H.c. \nonumber \\
&& \label{3.1} \\
{\cal{L}}_{\Xi\rho\Xi} &=& {\gamma_0\over 2}\lbrack {\overline \Xi}\gamma_\mu
\vec\tau\Xi \rbrack \vec\rho^\mu \nonumber \\
&&
+ {\gamma_0\over 2}\lbrack {\overline \Xi} ({\mu_{\Xi^0}-\mu_{\Xi^-}
\over 4m_\Xi})i
\sigma_{\mu\nu}\vec\tau\Xi \rbrack .(\partial ^\mu \vec{\rho^\nu} -
\partial ^\nu \vec{\rho^\mu}) \\
{\cal{L}}_{\rho\pi\pi} &=& \gamma_0\vec\rho_\mu.(\vec\phi\times
\partial^\mu \vec \phi) \nonumber \\
&& - {\gamma_0\over 4m_\rho^2}(\partial _\mu
\vec\rho_\nu - \partial_\nu\vec\rho_\mu).(\partial^\mu \vec\phi\times
\partial^\nu \vec \phi)
\ \ , \end{aligned}$$
where $\Xi$, $\Xi^*$, $\vec\phi$ and $\vec\rho$ are the cascade, the resonance $\Xi^*$(1530), the pion and the rho fields. $Z$ is the off-shell parameter [@PiN], $\mu_{\Xi^0}$ and $\mu_{\Xi^-}$ are the magnetic moments.
The lagrangians are almost the same as the $\pi N$ ones [@PiN], because the $\pi N$ system is very similar to the $\pi \Xi$. $N$ and $\Xi$ are particles with isospin 1/2, the only difference is that $\Delta$(1232) has isospin 3/2 and $\Xi^*$(1530), 1/2, so, a $\vec\tau$ matrix is included in (\[3.1\]).
The spin 3/2 propagator for a mass $M$ particle, is then G\^(p) &=& - g\^ - .\
& &. - + - . \[2.3\]
=80.mm
The contributing diagrams are shown in the Fig. 1 (we show only the direct diagrams, but in the calculations, the crossed diagrams are also included). The scattering matrix will have the general form T\_\^[ba]{}&=& ()A\^+ + [(k + k’)2]{}B\^+\_[ba]{}\
&& + A\^- + [(k + k’)2]{}A\^- i\_[bac]{}\^cu(p) ,
where $k$ and $k'$ are the initial and final $\pi$ momenta. Calculating the amplitudes from the diagrams, the contributions from Fig. 1(a) (intermediate $\Xi$) are $$\begin{aligned}
& &A_\Xi^+ = {g_{\Xi\pi\Xi}^2\over m_\Xi} \nn \\
& &A_\Xi^-=0 \nn \\
& &B_\Xi^+ = g_{\Xi\pi\Xi}^2\lb {1\over u-m_\Xi^2}-
{1\over s-m_\Xi^2}\rb
\nn \\
& &B_\Xi^- =-{g_{\Xi\pi\Xi}^2\over 2 m_\Xi}
-g_{\Xi\pi\Xi}^2\lb {1\over u-m_\Xi^2}+
{1\over s-m_\Xi^2}\rb \ \ . \end{aligned}$$
Fig 1(d), the $\rho$ exchange, gives $$\begin{aligned}
& &A_\rho^+ = B_\rho^+ = 0 \nn \\
& &A_\rho^- = -{\gamma_0^2\over m_\rho^2}(\mu_{\Xi^0} - \mu_{\Xi^-})\nu
{1 - {t/ 4m_\rho^2}\over 1 - {t/ m_\rho^2}} \nn \\
& &B_\rho^- = {\gamma_0^2\over m_\rho^2}(1 + \mu_{\Xi^0} - \mu_{\Xi^-})
{1 - {t/ 4m_\rho^2}\over 1 - {t/ m_\rho^2}} \ \ . \end{aligned}$$
The contribution from Fig. 1(b), the interaction with the intermediate $\Xi^*$, is $$\begin{aligned}
A_{\Xi^*}^+ &=& {g_{\Xi\pi\Xi^*}^2\over 3m_\Xi}\lbrace {\nu_r \over
\nu_r^2 - \nu^2}\hat A \nonumber \\
&&
- {m_\Xi^2+m_\Xi m_{\Xi^*}\over m_{\Xi^*}^2}(2m_{\Xi^*}^2
m_\Xi m_{\Xi^*}-m_\Xi^2+2\mu^2) \nonumber \\
& & +{4m_\Xi\over m_{\Xi^*}^2}\lbrack (m_\Xi+m_{\Xi^*})Z +
(2m_{\Xi^*}+m_\Xi)Z^2 \rbrack k.k' \rbrace \nonumber \\
&& \\
A_{\Xi^*}^- &=& {g_{\Xi\pi\Xi^*}^2\over 3m_\Xi}\lbrace {\nu \over
\nu_r^2 - \nu^2}\hat A +
{8m_\Xi^2\nu\over m_{\Xi^*}^2}\lbrack (m_\Xi+m_{\Xi^*})Z \nonumber \\
&&
+ (2m_{\Xi^*}+m_\Xi)Z^2 \rbrack \rbrace \\
B_{\Xi^*}^+ &=& {g_{\Xi\pi\Xi^*}^2\over 3m_\Xi}\lbrace {\nu \over
\nu_r^2 - \nu^2}\hat B - {8m_\Xi^2\nu Z^2\over m_{\Xi^*}^2} \rbrace \\
B_{\Xi^*}^- &=& {g_{\Xi\pi\Xi^*}^2\over 3m_\Xi}\lbrace {\nu_r \over
\nu_r^2 - \nu^2}\hat B
-{4m_\Xi\over m_{\Xi^*}^2}\lbrack (2m_\Xi^2 \nonumber \\
&&
+2m_\Xi m_{\Xi^*}-2\mu^2)Z
+ (2m_\Xi^2+4m_\Xi m_{\Xi^*})Z^2 \rbrack \nonumber \\
&&
+ {(m_\Xi+m_{\Xi^*})^2 \over m_{\Xi^*}^2}
- {4m_\Xi Z^2\over m_{\Xi^*}^2}k.k' \rbrace \ \ ,\end{aligned}$$
with $$\begin{aligned}
\hat A &=& {(m_{\Xi^*}+m_\Xi)^2-\mu^2\over 2m_{\Xi^*}^2}\lbrack
2m_{\Xi^*}^3-2m_\Xi^3-
2m_\Xi m_{\Xi^*}^2 \nonumber \\
&&
-2m_\Xi^2m_{\Xi^*}+\mu^2(2m_\Xi-m_{\Xi^*}) \rbrack +
\nonumber \\
& &+ {3\over 2}(m_\Xi+m_{\Xi^*})t \\
\hat B &=& {1\over 2m_{\Xi^*}^2}\lbrack (m_{\Xi^*}^2-m_\Xi^2)^2
-2m_\Xi m_{\Xi^*}(m_\Xi^*+m_\Xi)^2 \nonumber \\
&&
+6\m^2m_\Xi (m_\Xi^*+m_\Xi)
-2\mu^2(m_\Xi^*+m_\Xi)^2+\mu^4\rbrack + {3\over 2}t \ , \nonumber \\
&& \label{3.2} \end{aligned}$$
where $\mu$ is the pion mass, and $\n$ and $\n_r$ are defined in the appendix A. One must remark that eq. (\[3.2\]) is different from the expression of [@BH], where there was a mistake. The correct expression is presented here.
In [@BH], [@PiN] the $\sigma$ term (diagram 1.c) was simply considered as a parametrization $$\begin{aligned}
A_\sigma &=& a+bt \nonumber \\
B_\sigma &=& 0 \ \ .\end{aligned}$$
In fact, the $\sigma$ term represents the exchange of a scalar isoscalar system in the $t$-channel. This contribution is related to the scalar form factor of the baryon, and at large distances is dominated by triangle diagrams (Figure 2) involving the exchange of 2 pions [@LW]. In the $\pi \Xi$ interaction, this contribution is associated with two triangle diagrams, with $\Xi$ and $\Xi^*$ intermediate states, as it was calculated in [@CM].
=90.mm
The scalar form factor for a spin $1/2$ baryon B is defined as $ <B(p') | - {\cal L}_{sb} | B(p) > \equiv \s(t) \; \ub(\bp') \;u(\bp) $, where ${\cal L}_{sb}$ is the chiral symmetry breaking lagrangian.
The contribution of an intermediate particle of spin $s$ and mass $M$ to the scalar form factor is given by && \_s(t;M) u = - \^2 \^2 T\_a\^T\_a\
&& , \[2.4\]
where $Q$ is the average of the internal pion momenta, $T$ is a vertex isospin matrix and && \[ \_[1/2]{}u\] = -(m+M) + .\
&& . + 1 + u , \[2.6\]\
&& \[\_[3/2]{} u\] = - (m+M) (\^2-t/2) ...\
&& ... -\^4 ..\
&&.. + -1 (m+M) (2 M - m ) + 2\^2 ..\
&&.. -+ (\^2-t/2) +(m+M)...\
&&... - - + -1 ..\
&& .. M\^2 + 2m M - m\^2 + 2\^2 - u \[2.7\].
Calculating $\sigma(t)$ in the $\pi\Xi$ interaction, and using the loop integrals $\P$ defined in appendix B, we obtain && \_[1/2]{}(t) = \^2 (m\_+m\_[\^\*]{}) ¶\_[cc]{}\^[(000)]{}- .\
&&. ¶\_[c]{}\^[(000)]{} - ¶\_[c]{}\^[(001)]{} , \[2.8\]\
&& \_[3/2]{}(t) = \^2 - (m\_+m\_[\^\*]{})\^2 ..\
&&.. (2m\_[\^\*]{}-m\_) + 2 \^2 (m\_+m\_[\^\*]{}) + (\^2-t/2) m\_¶\_[cc]{}\^[(000)]{}.\
&& . - 2 \^2 m\_ \_[cc]{}\^[(000)]{} + (m\_\^2-m\_[\^\*]{}\^2) (m\_+m\_[\^\*]{})\^2 (2m\_[\^\*]{}-m\_)..\
&& .. + 2\^2 (m\_+m\_[\^\*]{})(m\_\^2-m\_[\^\*]{}\^2) ..\
&& .. + 6 (\^2- t/2) m\_[\^\*]{}\^2 (m\_+m\_[\^\*]{}) -\^4 (2m\_[\^\*]{}+m\_) .\
&& . + (m\_+m\_[\^\*]{})\^2 (4m\_m\_[\^\*]{}-m\_\^2-m\_[\^\*]{}\^2) + 6 m\_[\^\*]{}\^2 (\^2- t/2) ..\
&& .. - 2 \^2 (m\_+m\_[\^\*]{}) (2m\_[\^\*]{}-m\_) - \^4 . \[2.9\]
More details, as, for example regularization and the determination of $\sigma(t=0)$ can be found in [@CM].
The partial wave amplitudes are obtained summing the contributions from the diagrams of fig. 1 and making a straightforward application of the expressions found in appendix A. This can be done, for example, in the $\pi\Xi$ center-of-mass frame, where $\kappa$ is the momentum and $x=cos\ \theta$, where $\theta$ is the scattering angle. One notes that the $a_{l\pm}$ amplitudes are real, and, so, the corresponding $S$ matrix is not unitary. To unitarize the amplitudes, we reinterpret them as elements of the $K$ matrix [@EB], and then a\_[l]{}\^U=[a\_[l]{}1-i a\_[l]{}]{} ,
where $U$ means unitarized. Now the phase shifts are \_[l]{} = [tg]{}\^[-1]{}( f\_[l]{}) .
The parameters used are the same that were used in [@BH] and are $m_\Xi$=1.318 GeV, $m_{\Xi^*}$=1.533 GeV, $\mu_{\Xi^0}=-1.25$, $\mu_{\Xi^-}=0.349$, $g_{\Xi\pi\Xi}$=4 and $\gamma_0^2/m_\rho^2$=$1/(2f_\p^2)$, with $f_\p$=93 MeV. The coupling constant $g_{\Xi\pi\Xi^*}$ can be calculated comparing the ressonant $\delta_P^{1}$ phase with the Breit-Wigner expression \_[l]{}= where $\kappa_0$ is the center-of-mass momentum at the peak of the resonance. The obtained value is 4.54 ${\rm GeV}^{-1}$.
The numerical results of the phase shifts at $\sqrt{s}=m_{\Omega}$ are \_[P]{}\^[1]{}= -10.173\^o [and]{} \_[D]{}\^[1]{}= 0.208\^o \[3.20\] \_[P]{}\^[3]{}= 0.106\^o [and]{} \_[D]{}\^[3]{}= -0.078\^o . \[3.21\]
Summary and conclusions
=======================
We have calculated the strong $P$ and $D$ phase shifts for the $\Omega^-$ decay at its mass including the contributions from the diagrams of Fig. 1. The numerical values of the phases are shown in the expressions (\[3.20\]), (\[3.21\]). The respective values calculated in [@ta3] are $\delta_{P}^{1}= -12.8^o$ and $\delta_{P}^{1}= 1.1^o$, that are greater in magnitude than the ones obtained in this work. One must remark that the calculations presented here have no heavy-baryon approximation and the $\rho$-exchange and the $\sigma$-term are included, that are the sources of the differences. As we can see in eq. (\[3.20\]), (\[3.21\]), the $D$ phases are much smaller than the $P$ ones, as it is expected at low energies, and can even be neglected in a first approximation. We expect that the same pattern occurs for the weak phases.
In the other hyperon decays, the weak phases are of the order $\phi_P\sim 10^{-3}$ in the Weinberg-Higgs model and $\phi_P\sim 10^{-4}$ in the Kobayashi-Maskawa model [@don]. In [@ta3], using the KM model, the estimated value was $\phi_P\sim 10^{-3}$, which is significantly larger when compared with the other hyperons.
The asymmetry parameter $A$, eq. (\[1.30\]), in the decays $\Omega\rightarrow \Xi\p$ depends on the factor $\tan(\delta_{P}^{1}-\delta_{D}^{1})\sim$-0.18. In the $\Lambda\rightarrow p\p^-$ decay, the factor is $\tan(\delta_{S}^1-\delta_{P}^1)\sim$-0.12, and using the results of [@Kam], for the $\p\Lambda$ phases, in the $\Xi\rightarrow\p\Lambda$ decay, $\tan(\delta_{S}-\delta_{P})\sim$0.05. One notes that the greatest value happens in the $\Omega$ decay (and the weak phases are also larger), so, we conclude that the $A$ parameter must be the largest in this decay. On the other hand, considering the $B$ parameter eq. (\[1.31\]), the $\Omega$ decay shows a smaller term, from $\cot(\delta_{P}^{1}-\delta_{D}^{1})$, but the term that depends on the weak phases is larger. So, the $B$ parameter is probably of the same order of the one that appears in the other hyperon decays. The $B$ parameter seems to be the one where the $CP$ violation would be most evident.
I wish to tank professor Y.Hama and professor M. R. Robilotta, for many helpful discussions. This work was supported by FAPESP and CNPq.
Basic formalism
===============
In this paper $p$ and $p^{\prime}$ are the initial and final hyperon 4-momenta, $k$ and $k^{\prime}$ are the initial and final pion 4-momenta, so the Mandelstam variables are $$\begin{aligned}
s &=& (p+k)^2=(p^{\prime}+k^{\prime})^2 \\
t &=& (p-p^{\prime})^2=(k-k^{\prime})^2 \\
u &=& (p^{\prime}-k)^2=(p-k^{\prime})^2 \ \ .\end{aligned}$$ With these variables, we can define $$\begin{aligned}
\nu &=& {\frac{s-u}{4m}} \\
\nu_0 &=& {\frac{2\m^2-t}{4m}} \\
\nu_r &=& {\frac{m_r^2-m^2-k.k^{\prime}}{2m}} \ \ ,\end{aligned}$$ where $m$, $m_r$ and $\m$ are, respectively, the hyperon mass, the resonance mass and the pion mass. The scattering amplitude for an isospin $I$ state is T\_I=(p)A\^I + B\^Iu(p) ,
where $A_I$ and $B_I$ are calculated using the Feynman diagrams. So the scattering matrix is M\_I\^[ba]{} = = f\_I() + .n g\_I() = f\_1\^I + f\_2\^I , with $$\begin{aligned}
& &f_1^I(\theta) = {\frac{(E+m)}{8\pi\sqrt{s}}} \lbrack A_I + (\sqrt{s}%
-m)B_I\rbrack\ \ , \\
& & f_2^I(\theta) = {\frac{(E-m)}{8\pi\sqrt{s}}} \lbrack -A_I + (\sqrt{s}%
+m)B_I\rbrack \ \ ,\end{aligned}$$ where $E$ is the hyperon energy, and A\^[12]{}=A\^++2A\^- , A\^[32]{}=A\^+-A\^- ,
and similar expressions holds to $B^I$. The partial-wave decomposition is done with a\_[l]{} = \_[-1]{}\^[1]{}P\_l(x)f\_1(x) + P\_[l1]{}(x)f\_2(x) dx .
In our calculation (tree level) $a_{l\pm}$ is real. With the unitarization, as explained in Section III, we obtain a\_[l]{}\^U = e\^[2i\_[l]{}]{} -1= e\^[i\_[l]{}]{}[sen]{}(\_[l]{})a\_[l]{} .
loop integrals
==============
The basic loop integrals needed in order to perform the calculations of Fig. 2 are
&& I\_[cc]{}\^ = ,\
&& \[a‘\]\
&& I\_[c]{}\^= .\
&& \[a2\]
The integrals are dimensionless and have the following tensor structure && I\_[cc]{} = ¶\_[cc]{}\^[(000)]{}, \[a3\]\
&& I\_[cc]{}\^ = ¶\_[cc]{}\^[(200)]{} + g\^\_[cc]{}\^[(000)]{}, \[a4\]\
&& I\_[c]{} = ¶\_[c]{}\^[(000)]{} , \[a5\]\
&& I\_[c]{}\^ = ¶\_[c]{}\^[(001)]{}. \[a6\]
Thus, the $\Pi$ integrals that appear in the text are && ¶\_[cc]{}\^[(n00)]{} = - \_0\^1d a (1/2-a)\^n , \[a7\]\
&& \_[cc]{}\^[(000)]{}= - \_0\^1 d a , \[a8\]\
&& ¶\_[c]{}\^[(00n)]{}= - 2m /\^[n+1]{} \_0\^1 d aa \_0\^1 d b ,\
&& \[a9\]
with D\_[cc]{} &=& -a(1-a)q\^2 + \^2 ,\
D\_[c]{} &=& -a(1-a)(1-b)q\^2\
&& + \[\^2 -ab(\^2+m\^2-\^2) + a\^2b\^2 m\^2\] .
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|
---
abstract: 'We theoretically explore the annihilation of vortex dipoles, generated when an obstacle moves through an oblate Bose-Einstein condensate, and the possible reasons for the annihilation. We show that the grey soliton, which results from the vortex dipole annihilation, is lower in energy than the vortex dipole. We also investigate the annihilation events numerically and observe that the annihilation occurs only when the vortex dipole overtakes the obstacle. Furthermore, we find that the noise reduces the probability of annihilation events. This may explain the lack of annihilation events in experimental realizations.'
author:
- Shashi Prabhakar
- 'R. P. Singh'
- 'S. Gautam'
- 'D. Angom'
title: 'Annihilation of vortex dipoles in an Oblate Bose-Einstein Condensate'
---
\[sec:introduction\]Introduction
================================
One of the important developments in recent experiments on atomic Bose-Einstein condensates (BECs) is the creation of vortices and the study of their dynamics [@Anglin2002; @PhysRevLett.83.2498]. Equally important is the very recent experimental observation of a vortex dipole, which consists of a vortex-antivortex pair, when an obstacle moves through a BEC [@PhysRevLett.104.160401] and [*in situ*]{} observation of vortex dipoles produced through phase imprinting [@Freilich03092010]. In superfluids, the vortices carry quantized angular momenta and are the topological defects, which often serve as the conclusive evidence of superfluidity. In a vortex dipole, vortices of opposite charges cancel each other’s angular momentum and thus carry only linear momentum. This is the cause of several exotic phenomena like leap frogging, snake instability [@PhysRevA.65.043612], orbital motion [@PhysRevA.84], trapping [@PhysRevX.1], and others. The effects of vortices are widespread in classical fluid flow [@CambridgeJournals:392656] and optical manipulation [@Grier2003]. A good description of vortices in superfluids is given in Ref. [@pethick] and review articles [@rmp.81.647; @rmp.59.87]. More detailed discussion of vortices is given in Ref. [@okulov].
Among the important phenomena associated with the Bose-Einstein condensate (BEC), the creation, dynamics, and annihilation of vortex dipoles carry lots of information associated with the system. Several methods have been suggested to nucleate vortices. Recently, nucleation of the vortices have been observed experimentally by passing a Gaussian obstacle through the BEC with a speed greater than some critical speed [@PhysRevLett.104.160401]. The trajectories of these vortex dipoles are ring-structured as described in Refs. [@PhysRevA.61.013604; @PhysRevA.77.053610]. Annihilation of vortices in the BEC has been speculated in a number of studies. However, no extensive study has been done so far. Moreover, it has not been observed in the experiments done with an oblate BEC. The study of vortex dipole annihilation will shed light on the process that leads to minimum separation between vortex-antivortex and conditions for annihilation along with other phenomena arising due to dynamics of vortex dipoles.
In this article, we present analytical and numerical results related to vortex dipole annihilation for an oblate BEC at zero temperature. The results have been obtained using Gross-Pitaevskii (GP) equation. In the Section. II of the paper, we provide a brief description of the two-dimensional (2D) GP equation and vortex dipole solutions. Approximate analytic solutions of 2D GP equation with a vortex dipole or a grey soliton, expressed as linear combinations of the eigenstates of trapping potential, are given in Section \[sec:vdannhilation\]. Stability of the solutions are analysed from the energies of the solutions. The numerical results, confirming the analytic results, are discussed in Section \[sec:result\], and we then conclude.
\[sec:theory\]Superfluid vortex dipole
======================================
In the mean-field approximation, the dynamics of the dilute BEC is very well described by the GP equation $$\label{hamiltonian}
i\hbar \partial_{t}\Psi(\textbf{r},t)=[{\cal H}+U|\Psi(\textbf{r},t)|^{2}]
\Psi(\textbf{r},t),$$ where $\Psi$, ${\cal H}$, and $U$ are normalized wave function of the condensate, single-particle Hamiltonian and interaction strength respectively. The complete single-particle Hamiltonian ${\cal H}$ consists of a kinetic-energy operator, an axis-symmetric harmonic trapping potential, and a Gaussian obstacle potential, i.e., $$\label{gpeqn}
{\cal H}=-\frac{\hbar^{2}}{2m}\nabla^2+\frac{m\omega^2}{2}(x^2+
\alpha^2y^2+\beta^2z^2)+V_{\rm obs}(x,y,t),$$ where $\alpha$ and $\beta$ are the anisotropies along $y$ and $z$ axis respectively, and $V_{\rm obs}(x,y,t)$ is the repulsive Gaussian obstacle potential. Experimentally, obstacle potential is produced by a laser beam which is blue-detuned with respect to the frequency of the atomic transition and can be written as $$\label{vobs}
V_{\rm obs}(x,y,t)=V_{0}(t)\exp\left[-2\frac{(x-vt)^{2}+y^{2}}
{w_{0}^{2}} \right],$$ where $V_{0}(t)$ is the potential at the center of the Gaussian obstacle at time $t$, $v$ is the velocity of the obstacle along $x$-axis, and $w_0$ is the width of the obstacle potential. In the present work, we consider the motion of obstacle along $x$-axis only. Defining the oscillator length of the trapping potential $a_{\rm osc}=\sqrt{\hbar/(m \omega)}$, and considering $\hbar\omega$ as the unit of energy, we can then rewrite the equations in dimensionless form with transformations $\tilde{\mathbf{r}}= \mathbf{r}/a_{\rm osc}$, $\tilde{t}=t
\omega$, and the transformed order parameter $$\phi(\tilde{\mathbf{r}},\tilde{t})=\sqrt{\frac{a_{\rm osc}^3}{N}}
\Psi(\mathbf{r},t),$$ where $N$ is the number of atoms in the condensate. For the sake of notational simplicity, hereafter we denote the scaled quantities without tilde in the rest of the manuscript. In the pancake-shaped traps ($\alpha=1$ and $\beta\gg 1$), the order parameter $$\phi(\mathbf{r},t)=\psi(x,y,t)\zeta(t)\exp(-i\beta t/2),$$ where $\zeta=(\beta/(2\pi))^{1/4}\exp(-\beta z^2/4)$ is the order parameter in the axial direction. The Eq. (\[hamiltonian\]) can then be reduced to the two dimensional form $$\begin{aligned}
\label{gp_2d}
\left[-\frac{1}{2}\left(\frac{\partial^2}{\partial x^2}
+\frac{\partial^2}{\partial y^2}\right)+\frac{x^2+y^2}{2}
+V_{\rm obs}(x,y,t) \right. \nonumber \\
\left.+u|\psi({\mathbf r},t)|^2-i\frac{\partial}{\partial t}\right]
\psi ({\mathbf r},t) = 0,\end{aligned}$$ here $u=2aN\sqrt{2\pi\beta}/a_{\rm osc}$, where $a$ is the $s$-wave scattering length, is the modified interaction strength. In the present work, we consider condensate consisting of $^{87}$Rb atoms in $F=1$, $m_F=1$ state with $s$-wave scattering length equal to $99a_0$ [@prl88]. We have neglected a constant term corresponding to energy along axial direction as it only shifts the energies and chemical potentials by a constant without affecting the dynamics. This equation can be solved numerically by using Crank-Nicholson method [@Muruganandam20091888].
Generation of vortex-dipole pairs
---------------------------------
There are several theoretical proposals to generate vortices in non-rotating traps. These include stirring of the condensate using blue-detuned laser or several laser beams [@PhysRevLett.104.160401; @PhysRevA.61.013604], adiabatic passage [@PhysRevLett.80.2972], Raman transitions in bicondensate systems [@PhysRevLett.79.4728], laser beam vortex guiding [@springerlink.s003400000337], laser beam diffraction on a helical light grating [@springerlink.s003400050601], and phase imprinting [@PhysRevA.64.043601]. Among these methods, the easiest one to nucleate vortex dipoles is by stirring a BEC with a blue-detuned laser beam. When the velocity of the laser beam exceeds a critical velocity, vortex-antivortex pairs are released from the localized dip in the number density created due to the laser beam. These vortex dipoles then move through the BEC and exhibit various interesting dynamics [@Freilich03092010; @PhysRevA.61.013604; @PhysRevA.83.011603]. The critical velocity depends on the number density, width and intensity of the laser beam, and the frequency of the trapping potential. This nucleation process exhibits a high degree of coherence and stability, allowing us to map out the annihilation of the dipoles. In an axis symmetric trap, a vortex dipole is a metastable state of superfluid flow with long lifetime.
Trajectory of a vortex dipole
-----------------------------
The motion of a vortex dipole in a trapped BEC may be understood in terms of two contributions to the velocity of each vortex in the dipole. First, the precession due to the inhomogeneity of the condensate and secondly, the velocity induced by the other vortex. In our case, we have considered highly oblate BEC with a few well separated vortex dipoles, hence the contribution to velocity field due to the other vortex dipoles can be neglected. Moreover, as the obstacle is moved along $x$-axis, it leads to creation of the vortex dipoles located symmetrically about this axis. The velocity of the each vortex in the dipole is then given by $$\mathbf{v}=\omega_p \hat{\mathbf{k}} \times \mathbf{r}
+ \frac{1}{2y}\hat{\mathbf i},$$ where $\omega_p$ is the precession frequency, $\hat{\mathbf{k}}$ is the direction of circulation at the location of vortex, and $2y$ is the separation between vortex and antivortex. This equation can be reduced to two equations in $xy$-plane as $$\begin{aligned}
\frac{dx}{dt} & = &-\omega_{p}y+\frac{1}{2y} \label{eomx}, \\
\frac{dy}{dt} & = &\omega_{p}x \label{eomy}.\end{aligned}$$
These two equations govern the motion of a vortex dipole. The solutions of these equations with different initial conditions are shown in Fig. \[fig1\]. From the Fig. \[fig1\], one can observe that the vortex dipole come closer only along the $x$-axis of the BEC. They can even be closer than the coherence length. For this, the vortex needs to be created close to the obstacle. Hence, the annihilation will occur only along the diameter of BEC.
![The trajectories of the vortex dipole in oblate BEC calculated numerically from the equation of motion Eq. (\[eomx\],\[eomy\]). The trajectories depend on the initial conditions.[]{data-label="fig1"}](EOM.pdf){width="4.8cm"}
\[sec:vdannhilation\]vortex dipole annihilation
===============================================
To analyse the vortex dipole annihilation, we consider a model system where the vortex and antivortex are static. However, we vary the distance of separation and examine the energy of the total system.
The vortex and anti-vortex annihilation occurs when the separation is less than $\xi$. A simplistic model of a vortex dipoles in the BEC of trapped dilute atomic gases is the superposition of harmonic oscillator eigenstates. The minimalist wave function which supports a vortex and antivortex at cordinates $(-a/c,-\sqrt{b/d})$ and $(-a/c,\sqrt{b/d})$ is $$\psi(x, y) = \left (ia - b + i xc +dy^2 \right) e ^{-(x^2 + y^2)/f},
\label{vdip_sol}$$ where $a$, $b$, $c$, $d$, and $f$ are positive variational parameters. The wave function is a superposition of the scaled ground state and the first and the second excited states of harmonic oscillator along the $x$ and $y$-axes, respectively. The wave function is ideal for weakly interacting condensates, however, the qualitative descriptions remain unaltered for strongly interacting condensates. The normalization condition, considering correction for the excited states being negligible, is $a^2 + b^2 = 2/(f\pi)$. This is a constraint equation.
With a slight modification, the trajectory of the vortex dipole can be represented in terms of the time-dependent parameters. For example, the wave function when the vortex dipole is moving along $x$-axis $$\psi(x,y,t) = e^{-i\mu t}\left[-b(t)+ic(x-vt)+dy^2\right]e^{-(x^2 + y^2)/f},$$ where $v$ is the velocity with which the vortex dipole moves along the $x$-axis, and $\mu$ is the chemical potential of the system.
Diametric vortex dipole
-----------------------
Consider that the vortex and antivortex are located on the diameter of the condensate. Without loss of generality, we consider the diameter as coinciding with the $y$-axis, which is equivalent to $a=0$ in Eq. \[vdip\_sol\]. Such an assumption does not modify the qualitative descriptions, but expressions are far less complicated. The wave function is then $$\psi(x,y,t) = e^{-i\mu t}\left[-b(t)+icx+dy^2\right]e^{-(x^2 + y^2)/f}.$$ The nontrivial phase of the wave function $\theta$ is discontinuous along $x=0$ line for $-\sqrt{b/d}\leqslant y\leqslant\sqrt{b/d}$. Across the discontinuity, there is a phase change from $-\pi$ to $\pi$ as we traverse along $x$-axis from $0^-$ to $0^+$. To assign phase and elucidate the nature of the phase discontinuity, consider $y=\delta$, a line parallel to $x$-axis, where $\delta
<\sqrt{b/d}$ is a very small positive constant. Along the line $y=\delta$, $(dy^2-b(t))<0$ and at $x\gg0$, $cx/(dy^2-b(t))\rightarrow-\infty$ and $\theta$ is $\approx\pi/2$. When we decrease $x$, $\theta$ increases and it is $\approx
\pi$ when $x\rightarrow 0^+$. On the other side of the $y$-axis, at $x\ll 0$, $cx/(dy^2-b(t))\rightarrow \infty$ and $\theta$ is $\approx -\pi/2$. However, on increasing $x$ the phase $\theta $ tends to $-\pi$. So, there is a discontinuity across the $y$-axis. This is the typical phase pattern associated with vortex dipoles. For the present case, the ground state wave function is $$\psi_{\rm g}(x, y, t) = -e^{-i\mu t}b(t) e ^{-(x^2 + y^2)/f},$$ and from the normalization condition $\int_{-\infty}^\infty\int_{-\infty}^{
\infty}|\psi_{\rm g}|^2 dxdy = 1$, we get the constraint equation $$b^2 = \frac{2}{f\pi}.
\label{norm_vort}$$ For general considerations, rewrite the additional term as $$\delta \psi(x, y, t) = e^{-i\mu t}\left (i cx e ^{-i\omega t}
+ dy^2 e^{i\omega t} \right ) e ^{-(x^2 + y^2)/f}.$$ So that the total wave function $\psi = \psi_{\rm g} + \delta\psi $, where $\delta\psi$ represents an elementary excitation of the condensate. From the Bogoliubov theory, it satisfies the normalization condition $$\int_{-\infty}^{\infty}\int_{-\infty}^{\infty} \left (
c^2x^2 - d^2 y^4\right ) e^{-2(x^2 + y^2)/f} dxdy = 1,$$ which after evaluation is $$\frac{\pi f^2}{4}(2 c^2 - 3 d^2 f) = 1.
\label{norm_exci}$$ The limits, in reality, must be the physical size of the condensate clouds. However, due to the exponential decay and relatively large extent of the condensate clouds there is little difference in the results. The Eq. (\[norm\_exci\]), like Eq. (\[norm\_vort\]), is an additional constraint equation to the wave function. We can calculate the total energy of the system, without the obstacle potential, as $$\begin{aligned}
E_{\rm vd} &= & \int_{-\infty}^{\infty}\int_{\infty}^{\infty}
\bigg [ \frac{1}{2}|\nabla_{\perp}\psi (x, y)|^2
+ \frac{x^2+y^2}{2}|\psi(x, y)|^2 \nonumber \\
&& + u|\psi(x, y)|^4 \bigg ] dx dy .\end{aligned}$$ This is the energy of condensate with a vortex dipole with the assumption that it is a weakly interacting system. Energy without the vortex may be calculated trivially [@pethick]. In general, the energy added to the system due to the vortex dipole is not large compared to the total and for obvious reason, the angular momentum of the condensate is still zero.
Grey soliton {#grey_sol}
------------
A slight modification to the wave function can describe a solitonic solution along $y$-axis. The form of the modifed wave function is $$\psi(x, y) = \left [b(t) + i cx +dy^2 \right ] e ^{-(x^2 + y^2)/f},
\label{soli_sol}$$ where except for the change in the sign of $b$, all the terms remain unaltered as in Eq. (\[vdip\_sol\]). It is a [*grey*]{} soliton as the density $n\propto (b+dy^2)^2+(a+cx)^2$ has a dip but is different from zero.
Like in vortex dipole, we examine the nature of the phase around the points $(0,-\sqrt{b/d})$ and $(0,\sqrt{b/d})$. The nontrivial phase of the wave function $\theta $ has no discontinuinities. Consider again the line $y=\delta\ll\sqrt{b/d}$, which is parallel to $x$-axis. Along the line $(dy^2+
b(t))> 0$ at all values of $x$ and $-\pi/2 \leqslant\theta\leqslant\pi/2$. In the $x\gg 0$ domain, $cx/(dy^2 + b(t))\rightarrow \infty$ and $\theta\approx\pi/2$. When we decrease $x$, $\theta$ decreases and it is $\approx 0$ when $x\rightarrow 0^+$. On the other side of the $y$-axis, at $x\ll0$ the phase $\theta\approx-\pi/2$. However, on decreasing $x$ the phase $\theta$ tends to $0$. So, the phase varies smoothly from $-\pi/2$ to $\pi/2$ along the normal to the line which connects $(0,-\sqrt{b/d})$ and $(0,\sqrt{b/d})$.
Using the wave function in Eq. (\[soli\_sol\]), we can then evaluate the total energy of the system $E_{\rm gs}$. We define the energy difference between the two states of the condensates as $$\Delta E = E_{\rm vd} - E_{\rm gs},$$ which after evaluation is $$\Delta E = \frac{b d f^2\pi}{256} \left [ 64 b^2 u + 15 d^2 f^2 u
+ 8 f (8 + c^2 u)\right].$$ The most general solution is when all the constants are positive, then $\Delta
E>0$ and the grey soliton is lower in energy. This shows that when the vortex-antivortex collides, it is energetically favourable for them to decay into grey soliton. As discussed in the results section, this is confirmed in the numerical calculations.
The analysis so far is for an ideal system at zero temperature, where we have neglected the thermal fluctuations and perturbations from imperfections. In addition, there is dissipation from three body collision losses in the condensates of dilute atomic gases.
\[sec:result\]Numerical Results
===============================
The specific paremeters we consider for the numerical studies are: the species is chosen as $^{87}$Rb with $N=2\times10^6$ atoms. The trapping potential and obstacle laser potential parameters are same as those considered in Ref. [@PhysRevLett.104.160401], i.e., $\omega/(2\pi)= 8$ Hz, $\beta=11.25$, $V_0(0)=93.0~\hbar\omega$ and $w_0=10~\mu$m. The obstacle potential $V_{\rm obs}$ is initially located at $-12.5~a_{\rm osc}$. To nucleate vortices, we move the obstacle along the $x$ direction at a constant velocity with decreasing intensity until $V_{\rm obs}$ vanishes at $5.18~a_{\rm osc}$.
\[subsec:nucleation\]Vortex dipole nucleation
---------------------------------------------
We study the nucleation of vortices by $V_{\rm obs}$ with the translation speed $v$ ranging from $80~\mu$m/s to $200~\mu$m/s. Vortices are not nucleated when the speed is $80~\mu$m/s. However, a vortex-antivortex or a vortex dipole is nucleated when the speed is in the range $90~\mu$m/s $<v<140~\mu$m/s. Increasing the speed of obstacle generates two pairs in $140~\mu$m/s $\leqslant
v<160~\mu$m/s and more than two when $v\geqslant 160~\mu$m/s. In other words, the number of vortex dipoles created can be controlled with the speed of the obstacle. Creation of vortex dipoles above a critical speed $v_c$ is natural as the vortex nucleation must satisfy the Landau criterion [@Lifshitz]. The density and phase of the condensate after the nucleation of vortex dipole for $v=160~\mu$m/s is shown in Fig. \[160\_pdf\]. The figure clearly shows nucleation dynamics of the vortex dipoles.
Through a series of calculations, we determine $v_c\approx 90~\mu$m/s. This is, however, less than the acoustic velocity of the medium $s$, which depends on the local condensate density $s=\sqrt{nU/m} $. This also explains the reason for the predominant vortex dipole nucleation around the edge of the condensate where the density is lower and the acoustic speed is accordingly lower.
\[subsec:nucleation\]Vortex dipole annihilation
-----------------------------------------------
![ A vortex dipole is nucleated when the obstacle potential traverses the condensate at a speed of $120~\mu$m/s. The vortex dipole, however, passes through and overtakes the obstacle. Later, as seen in (e), the vortex dipole annihilates and generates a grey soliton. The figures in the left panel show the density distribution and those on the right show the phase pattern of the condensate. []{data-label="120_pdf"}](120.pdf){width="8.3cm"}
It is observed that the vortex dipole annihilation is critically dependent on the initial conditions of the nucleation. For this reason, the annihilation events are observed only for specific range of $v$. As an example, the annihilation event when $v$ is $120~\mu$m/s is shown in Fig. \[120\_pdf\]. In Fig. \[120\_pdf\], we can notice the density minima arising from the annihilation and propagating away from the $V_{\rm obs}$.
One observation, which is common to all the vortex dipoles getting annihilated is the nature of their trajectory. All of them traverse through $V_{\rm obs}$, whereas the ones which do not get annihilated avoid $V_{\rm obs}$. This is again related to the initial conditions. The vortex dipoles are generally nucleated at the aft region of the $V_{\rm obs}$ where there is a trailling superflow. When nucleated very close to each other and with high velocity, the mutual force further increases the velocity of the vortex dipoles. At the same time, it decreases the distance separating vortex and antivortex. So, the kinetic energy is high enough to surpass $V_{\rm obs}$. Later, at some point vortex and antivortex separation is less than $\xi$, and they annihilate.
![A vortex dipole is nucleated as the obstacle potential traverses the BEC with a speed of $160~\mu$m/s. The figures in the left panel shows the density with time, where time progresses from top to bottom. Figures on the right panel show the phase pattern of the condensate.[]{data-label="160_pdf"}](160.pdf){width="8.3cm"}
![Density variation at the core of the vortex with time (in scaled unit). After the vortex dipole annihilation, density increases till it reaches the bulk value. The values correspond to $120~\mu$m/s obstacle speed.[]{data-label="density_core"}](Density_Core.pdf){width="7.5cm"}
A reliable and qualitative way to describe occurrence of annihilation could be achieved by observing the density at the cores of vortex and antivortex which form the dipole. For the vortex, the matter density at the core when $v$ is $120~\mu$m/s is shown in Fig \[density\_core\]. In the plot, at Time $\approx
900$ (arbitrary units), the core density starts increasing. This is because the core starts to fill in with the atoms from around the vortex after the annihilation.
As discussed in Section. \[grey\_sol\], annihilation can occur only when it is energetically favorable. In other words, the state with the grey soliton must have lower energy than the vortex dipole. This is clearly seen in the change of total energy of the system, which is shown in Fig. \[energy\]. After the annihilation, the energy of the system decreases and continues to do so at a steady rate. Although not shown in the plot, before the annihilation the energy is on an average constant. The energy change, can thus be considered as a very good signature of the annihilation of vortex dipoles.
![Variation in the energy of the system with time (in scaled units). There is a decrease in energy after the vortex dipole annihilation, and the plot is based on the results of calculations in which a vortex dipole is imprinted at ($-2.0~a_{\rm osc}$, $\pm0.1~a_{\rm osc}$). It is then allowed to annihilate and evolve in time.[]{data-label="energy"}](energy.pdf){width="8cm"}
It is to be mentioned that for the parameters considered in the present work, the speed of sound is $2190~\mu$m/s, which is similar to the speed observed by Neely et. al. [@PhysRevLett.104.160401]. The coherence length of the system is $\sim0.25~\mu$m and the minimum distance between the vortex dipole pair is $\sim0.3~\mu$m. The position of annihilation determined from the equation of motion (Eq. \[eomx\]-\[eomy\]) matches with those obtained from the simulation.
\[subsec:noise\]Effect of noise and dissipation
-----------------------------------------------
In the numerical studies, the annihilation events are not rare. But, this is in contradiction with the experimental results of Neely and collaborators [@PhysRevLett.104.160401], they observed no signatures of annihilation events. One possible reason is that our numerical calculations are too ideal, and one immediate remedy is to include fluctuations. For this we introduce white noise during the real time evolution. One immediate outcome is, the symmetry in the trajectory of the vortex and antivortex is lost. The superflow around the vortex is no longer a mirror reflection of the antivortex, which was nearly the case without the white noise. The deviations are shown for an example case in the Fig. \[noise\]. The change in path leads to the suppression of annihilation of vortex dipoles.
![The trajectory of a vortex dipole in the presence of white noise. There is a lack of symmetry in the trajectory of the vortex and antivortex. This reduces the possibility of an annihilation event significantly. The figures in the left (right) panel show the density (phase) of the condensate and time increases from top to bottom figures of each panel. The speed of the obstacle is $180~\mu$m/s, and the white noise is at the level of 0.01%.[]{data-label="noise"}](noise.pdf){width="8.3cm"}
The other important effect is the loss of atoms from the trap. We have examined the effect of loss terms, which arise from inelastic collisions in the condensate. There are two types of inelastic collisions that lead to the loss of atoms from the trap: two body inelastic collision loss and the three body loss. To model the effect of loss of atoms from the trap, we add the loss terms $$\frac{-i\hbar}{2} \left( K_{2} |\Psi(\mathbf r,t)|^2 +K_3 |\Psi(\mathbf
r,t)|^4 \right),$$ to the Hamiltonian ${\cal H}$. Based on the previous work [@PhysRevLett.80.2097] for $^{87}$Rb, the inelastic two-body loss rate coefficient $K_{2}=4.5\times10^{-17}~{\rm cm}^3~{\rm s}^{-1}$, and inelastic three-body loss rate coefficient $K_3=3.8\times10^{-29}~{\rm cm}^6~
{\rm s}^{-1}$. With trap loss, the annihilation events continue to occur. However, during the destructive time of flight observations in the experiments, the decreased atom numbers may lower the contrast and reduce the possibility of observing an annihilation event. \
\
\[sec:conclusions\]Conclusions
==============================
When an obstacle moves through a condensates above a critical speed, it nucleates vortex dipoles and the number of dipoles seeded depends on the obstacle velocity. Depending on the initial condition of nucleation, vortex and antivortex annihilation events occur under ideal conditions: at zero temperature, no loss, and without noise. However, the noise destroys the superflow reflection symmetry around the vortex and antivortex. This reduces the possibility of annihilation events and may explain the lack of annihilation events in experimental observation in Ref. [@PhysRevLett.104.160401].
\
The numerical calculations reported in this paper have been performed on 3 TeraFlop high-performance cluster (HPC) at Physical Research Laboratory (PRL), Ahmedabad.
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|
One of the most interesting questions in glassy physics is whether [*localized spatial heterogenities*]{} are generated in supercooled liquids and glasses. [@Sillescu]
In most supercooled liquids, the linear response to small external perturbations is nonexponential in the time-difference $\tau$. Within the “heteregeneous scenario”, the stretching is due to the existence of dynamically distinguishable entities in the sample, each of them relaxing exponentially with its own characteristic time. A different interpretation is that the macroscopic response is intrinsically nonexponential. In the glass phase, the relaxation is nonstationary and the dependence in $\tau$ is also much slower than exponential.
The heterogeneous regions, if they exist, are expected to be nanoscopic. The development of experimental techniques capable of giving evidence for the existence of such distinguishable spatial regions has been a challenge for experimentalists.
With non-resonant spectral hole-burning (NSHB) techniques one expects to probe, selectively, the microscopic responses.[@Bohmer1] The method is based on a wait, pump, recovery and probe scheme depicted in Fig. \[schema\]. The amplitude of the ac perturbation is sufficiently large to pump energy in the sample, modifying the response as a linear function of the absorbed energy. The step-like perturbation $\delta$ is very weak and serves as a probe to measure the integrated linear response of the full system. The large ac and small dc fields can be magnetic, electric, or any other perturbation relevant for the sample studied. The idea behind the method is that the comparison of the modified (perturbed by the oscillation) and unmodified (unperturbed) integrated responses yield information about the microscopic structure of the sample. On the one hand, a spatially homogeneous sample will absorb energy uniformly and its modified integrated response is expected to be a simple translation towards shorter time-differences $\tau$ of the unmodified one. On the other hand, in a heterogeneous sample, the degrees of freedom that respond near the pump frequency $\Omega$ are expected to absorb an important amount of energy and a maximum difference in the relaxation (equivalently, a spectral hole) is expected to generate around $t\sim 1/\Omega$.
The NSHB technique has been first applied to the study of supercooled liquids. The polarization response of dielectric samples, glycerol and propylene carbonate, was measured after being modified by an ac electric field.[@Bohmer1] More recently, ion-conducting glasses like CKN [@Bohmer2], relaxor ferroelectrics (90PMN-10PT ceramics) [@Bohmer3] and spin-glasses (5% Au:Fe) [@Chamberlin] were studied with similar methods. The results have been interpreted as evidence for the existence of spatial heterogeneities. We show here that their main features can be reproduced by a system with [*no spatial structure*]{}. We use one model, out of a family, that captures many of the experimentally observed features of super-cooled liquids and glasses as, for instance, a two-step equilibrium relaxation close and above $T_c$ [@Gotze], aging effects below $T_c$ [@Cuku], etc. The model is the $p$ spherical spin-glass [@Crso], that is intimately related to the $F_{p-1}$ mode-coupling model [@Kith]. It can be interpreted as a system of $N$ fully-connected continuous spins or as a model of a particles in an infinite dimensional random environment. [@review] In both cases, no reference to a geometry in real space nor any identification of spatially distinguishable regions can be made.
In the presence of a uniform field, the model is $$H_J[{\bf s}] = \sum_{i_1 \leq \dots \leq i_p} J_{i_1\dots i_p} \; s_{i_1} \dots s_{i_p}
+
h \; \sum_{i=1}^N s_i
\; .
\label{ham_pspin}$$ The interactions $J_{i_1\dots i_p}$ are quenched independent random variables taken from a Gaussian distribution with zero mean and variance $[J^2_{i_1\dots i_p}]_J={\tilde J}^2 p!/(2 N^{p-1})$. $p$ is a parameter and we take $p=3$. Hereafter $[\;\;]_J$ represents an average over $P[J]$ and $\tilde J=1$. The continuous variables $s_i$ are constrained spherically $\sum_{i=1}^N s_i^2=N$. A stochastic evolution is given to ${\bbox s}$, $
\dot s_i(t) = -\delta_{s_i(t)} H_J[{\bbox s}] + \xi_i(t)
$ with $\xi_i$ a white noise with $\langle \xi_i\rangle =0$ and $
\langle \xi_i(t) \xi_i(t') \rangle = 2 T \delta(t-t')
$. When $N\to \infty$, standard techniques lead to a set of coupled integro-differential equations for the autocorrelation $
NC(t,t') \equiv \sum_{i=1}^N [\langle s_i(t) s_j(t') \rangle]_J
$ and the linear response $
R(t,t') \equiv \left. \sum_{i=1}^N
\delta [\langle s_i(t) \rangle]_J /\delta
\delta_i(t') \right|_{\delta=0}
$, with $\delta_i(t')$ an infinitesimal perturbation modifying the energy at time $t'$ according to $H \to H - \sum_i \delta_i s_i$. The dynamic equations read [@Cuku3] $$\begin{aligned}
& & \partial_t C(t,t')
=
-z(t) \, C(t,t') +
\frac{p}{2}
\int_0^{t'} dt'' C^{p-1}(t,t'') R(t',t'')
\nonumber\\
& &
\;\;\;\;\;\;\;
+
\frac{p (p-1)}{2}\int_0^t dt'' C^{p-2}(t,t'') R(t,t'') C(t'',t')
\nonumber\\
& &
\;\;\;\;\;\;\;
+ 2T R(t',t) + h(t) \int_0^{t'} dt'' h(t'') R(t',t'')
\; ,
\label{eqC}
\\
& &
\partial_t R(t,t')
=
-z(t) \, R(t,t')
\nonumber \\
& &
\;\;\;\;\;\;\;+ \frac{p (p-1)}{2}
\int_{t'}^t dt'' C^{p-2}(t,t'') R(t,t'') R(t'',t')
\; ,
\label{eqR}
%\\
%& &
%d_t m(t)
%=
%-z(t) m(t) + h(t)
%\nonumber\\
%& &
%\;\;\;\;\;\;\;+ \frac{p (p-1)}{2}
% \int_{0}^t dt'' C^{p-2}(t,t'') R(t,t'') m(t'')
%\; ,
%\nonumber\\
%& &
%z(t)=
%T +
%\frac{p^2}{2}
%\int_0^t dt'' C^{p-1}(t,t'') R(t,t'')
%+
%h(t) m(t)
%\; .
%\nonumber\end{aligned}$$ The Lagrange multiplier $z(t)$ enforces the spherical constraint and an integral equation for it follows from Eq. (\[eqC\]) and the condition $C(t,t)=1$. In deriving these equations, a random initial condition at $t_0=0$ has been used. It corresponds to an infinitely fast quench from equilibrium at $T=\infty$ to the working temperature $T$. The evolution continues in isothermal conditions.
In the absence of energy pumping, these models have a dynamic phase transition at a ($p$-dependent) critical temperature $T_c$, $T_c \sim 0.61$ for $p=3$. When an external ac-field is applied, it drives the system out-of-equilibrium and stationarity and FDT do not necessarily hold at [*any*]{} temperature. The question as to whether the clearcut dynamic transition survives under an oscillatory field is open and we do not address it here. We simply study the dynamics close to the critical temperature in the absence of the field by constructing a numerical solution to Eqs. (\[eqC\]) and (\[eqR\]) with a constant grid algorithm of spacing $\epsilon$. We present data for small spacings, typically $\epsilon=0.02$, to minimize the numerical errors. Due to the fact that Eqs. (\[eqC\]) and (\[eqR\]) include integrals ranging from $t_0=0$ to present time $t$, the algorithm is limited to a maximum number of iterations of the order of $8000$ that imposes a lower limit $\Omega \sim 2\pi/(8000 \epsilon) \sim 0.1$ to the frequencies we use.
A word of caution concerning the scheme in Fig. \[schema\] and the times involved is in order. For the purpose of collecting the data for each reference unmodified integrated response, the sample is prepared at the working temperature $T$ at $t_0=0$ and let freely evolve during a total waiting time $t_w+t_1+t_r$. Depending on $T$, this interval may or may not be enough to equilibrate the sample. ($t_1$ is chosen as $t_1=2\pi n_c/\Omega$ with $\Omega$ the angular velocity of the field that will be used to record the modified curve.) A constant infinitesimal probe $\delta$ is applied after $t_w+t_1+t_r$ to measure $$\begin{aligned}
\Phi(\tau) \equiv \int_0^\tau d\tau' \, R(t_w+t_1+t_r+\tau, t_w+t_1+t_r+\tau')
\; .\end{aligned}$$ As an abuse of notation we explicitate only the $\tau$ dependence and eliminate the possible $t_w+t_1+t_r$ dependence. The modified integrated response $\Phi^*$ is measured after waiting $t_w$, applying $n_c$ oscillations of duration $t_1=2\pi n_c/\Omega$, further waiting $t_r$, and only then applying the probe $\delta$. The effect of the ac perturbation is then quantified by studying the difference: $$\Delta \Phi \equiv \Phi^* - \Phi
\; .$$
We have examined $\Delta \Phi$ at $T=0.8 > T_c$ and $T=0.59< T_c$. We pump one oscillation with $h_F=0.1$ and later check that this field is small enough to provoke a spectral modification that is linear in the absorbed energy (see Fig. \[checklinearity\] below). For simplicity, we start by choosing $t_w=t_r=0$. In Fig. \[Delta\_Phi\_1\] we show $\Delta \Phi$ against $\log \tau$ for different $\Omega$ at $T=0.8$. All the curves are bell-shaped and vanish both at short and long times. In panel a, the $\Omega$s are larger than a threshold value $\Omega_c \sim 1$. The height of the peak $\Delta\Phi_m \equiv \max ( \Delta\Phi)$ decreases with increasing frequency reaching the limit $\Delta\Phi_m= 0$ for $\Omega\to\infty$. In addition, the location of the peak $t_m$ moves towards longer times when $\Omega$ decreases. In panel b, $\Omega< \Omega_c$ and the behaviour of the height of the peak is the opposite, it decreases when $\Omega$ decreases and, within numerically errors, its position is either independent of $\Omega$ or it very smoothly moves towards shorter times for increasing $\Omega$. The nonmonotonic behaviour of $\Phi_m$ with $\Omega$ is a consequence of the interplay between $t_\alpha$, the $\alpha$ relaxation time, and $2\pi/\Omega$ the period of the oscillation. The term $\int_0^{\min(t',t_1)} dt'' h(t'') R(t',t'')$ in Eq. (\[eqC\]) controls the effect of the field and, clearly, vanishes in the limits $\Omega\to\infty$ and $\Omega\to 0$. The inversion then occurs at a frequency $\Omega_c$ that is of the order of $2\pi/t_\alpha$. These results qualitatively coincide with the measurements of the electric relaxation in CKN at $T<T_g$ in Fig 1 a and b of Ref. [@Bohmer2]. In Fig. \[Delta\_Phi\_2\] we show $\Delta \Phi$ against $\log \tau$ for different $\Omega$ at $T=0.59$. For all $\Omega$ we reproduced the situation of panel a in Fig. \[Delta\_Phi\_1\], as if $\Omega > \Omega_c$. We have not found a threshold $\Omega_c$, that has gone below the minimum $\Omega$ reachable with the algorithm.
The maximum modification of the relaxation $\Delta\Phi_m$ increases quadratically with the square of the amplitude of the pumping field $h_F$, and hence linearly in the absorbed energy, as long as $h_F \leq 1$. In Fig. \[checklinearity\] we display the relation $\Delta\Phi_m\propto h_F^2$ in a log-log scale for the two temperatures explored. The amplitude $h_F=0.1$ used in Figs. \[Delta\_Phi\_1\] and \[Delta\_Phi\_2\] is in the linear regime.
The effect of the pump diminishes with increasing recovery time $t_r$. A convenient way of displaying this result is to plot the normalized maximum deviation $\Delta\Phi_m(t_r)/\Delta\Phi_m(0)$ [*vs*]{} $\Omega t_r$. Using several frequencies and recovery times, we verified that this scaling holds for $T=0.59$ but does not hold for $T=0.8$, as shown in Fig. \[recovery\]. This simple saling holds very nicely in the relaxor ferroelectric [@Bohmer3] and in the spin-glass [@Chamberlin] but it is very different from the $\Omega$-independence of the propylene carbonate [@Bohmer1].
Up to now, the effect of a single cycle of different frequencies has been studied. Another procedure can be envisaged. Since $t_1=2\pi n_c/\Omega$, we can change $t_1$ by applying different numbers of cycles $n_c$ while keeping $\Omega$ fixed. In Fig. \[ncyc\] we show the distortion due to $n_c=10,2,1$ cycles with $\Omega=10$ at $T=0.8$. The qualitative dependence on $n_c$ is indeed the same as the dependence on $1/\Omega$: the peaks are displaced towards longer times with increasing $n_c$ (longer $t_1$). This behaviour is similar to the results obtained for propylene carbonate in Fig. 11 of Ref. [@Bohmer1]b. though we do not reach the expected saturation within our accessible time window.
Below $T_c$ the nonperturbed model never equilibrates and the relaxation depends on $t_w$. Indeed, $t_\alpha$ is an approximately linear function of $t_w$[@Cuku; @review] and the distortion might depend on $t_w$. We compare $\Delta\Phi$ vs $\log\tau$ for two $t_w$’s in Fig. \[tw-dep\].
Finally, we checked that the effect of one or many pump oscillations on the difference $\Delta C
\equiv C^*(t_w+t_1+t_r+\tau,t_w+t_1+t_r)-C(t_w+t_1+t_r+\tau,t_w+t_1+t_r)$ is very similar to the one observed in $\Delta\Phi$. This observation is interesting since it is easier to compute numerically correlations than responses. Figure \[corr\] shows the modification observed at $T=0.8$ and $\Omega > \Omega_c$ (to be compared to Fig. \[Delta\_Phi\_1\]).
We conclude by stressing that we do not claim that spatial heterogeneities do not exist in real glassy systems. We just wish to stress that the ambiguities in the interpretation of experimental results have to be eliminated in order to have unequivocal evidence for them. The detailed comparison of the experimental measurements to the behaviour of glassy models [*with and without space*]{} will certainly help us refine the experimental techniques. Numerical simulations can play an important role in this respect.
LFC and JLI thank the Dept. of Phys. (UNMDP) and LPTHE (Jussieu) for hospitality, and ECOS-Sud, CONICET and UNMDP for financial support. We thank R. Böhmer, H. Cummins, G. Diezemann, M. Ediger, J. Kurchan and G. Mc Kenna for very useful discussions and T. Grigera, N. Israeloff and E. Vidal-Russel for introducing us to the hole-burning experiments.
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|
---
abstract: |
We consider a firm that sells a large number of products to its customers in an online fashion. Each product is described by a high dimensional feature vector, and the market value of a product is assumed to be linear in the values of its features. Parameters of the valuation model are unknown and can change over time. The firm sequentially observes a product’s features and can use the historical sales data (binary sale/no sale feedbacks) to set the price of current product, with the objective of maximizing the collected revenue. We measure the performance of a dynamic pricing policy via regret, which is the expected revenue loss compared to a clairvoyant that knows the sequence of model parameters in advance.
We propose a pricing policy based on projected stochastic gradient descent (PSGD) and characterize its regret in terms of time $T$, features dimension $d$, and the temporal variability in the model parameters, $\delta_t$. We consider two settings. In the first one, feature vectors are chosen antagonistically by nature and we prove that the regret of PSGD pricing policy is of order $O(\sqrt{T} + \sum_{t=1}^T \sqrt{t}\delta_t)$. In the second setting (referred to as stochastic features model), the feature vectors are drawn independently from an unknown distribution. We show that in this case, the regret of PSGD pricing policy is of order $O(d^2 \log T + \sum_{t=1}^T t\delta_t/d)$.
author:
- |
Adel Javanmard\
Department of Data Sciences and Operations\
Marshall School of Business\
University of Southern California, Los Angeles, CA 90089\
bibliography:
- 'dynamicpricing.bib'
title: |
Perishability of Data: Dynamic Pricing\
under Varying-Coefficient Models
---
Introduction {#sec:intro}
============
Motivated by the prevalence of online marketplaces, we consider the problem of a firm selling a large number of products, that are significantly differentiated from each other, to customers that arrive over time. The firm needs to price the products in a dynamic manner, with the objective of maximizing the expected revenue.
The majority of work in dynamic pricing assume that a retailer sells *identical* items to its customers [@BesbesZ09; @farias2010dynamic; @broder2012dynamic; @den2013simultaneously; @WangDY14]. Recently, feature-based models have been used to model the products differentiation by assuming that each product is described by vectors of high-dimensional features. These models are suitable for business settings where there are an enormous number of distinct products. One important example is online ad markets. In this context, products are the impressions (user view) that are sold by the web publisher to advertisers. Due to the ever-growing amount of data that is available on the Internet, for each impression there is large number of associated features, including demographic information, browsing history of the user, and context of the webpage. Many other online markets, such as Airbnb, eBay and Etsy also have a similar setting in which products to be sold are highly differentiated. For example, in the case of Aribnb, the products are “stays" and each is characterized by a large number of features including space properties, location, amenities, house rules, as well as arrival dates, events in the area, availability of near-by hotels, etc [@Airbnb].
Here, we consider a feature-based model that postulates a linear relation between the market value of each product and its feature values. Further, from the firm’s perspective, we treat distinct buyers independently, and hereafter focus on a single buyer. Put it formally, we start with the following model for the buyer’s valuation: $$\begin{aligned}
\label{eq:model1}
v(x_t) = \<x_t, \th\> + z_t\,,\end{aligned}$$ where $x_t\in \reals^d$ denotes the product feature vector, $\th$ represents the buyer’s preferences and $z_t$, $t\ge 1$ are idiosyncratic shocks, referred to as noise, which are drawn independently and identically from a zero mean distribution. For two vectors $a,b$, we write $\<a,b\>$ to refer to their inner product. Feature vectors $x_t$ are observable, while model parameter $\th$ is a-priori unknown to the firm (seller). Therefore, the buyer’s valuation $v(x_t)$ is also hidden from the firm.
Parameters of the above model represents how different features are weighted by the buyer in assessing the product. Considering such model, a firm can use historical sales data to estimate parameters of the valuation model, while concurrently collecting revenue from new sales. In practice, though, the buyer’s valuation of a product will change over time and this raises the concern of *perishability* of sales data.
In order to capture this point, we consider a richer model with varying coefficients: $$\begin{aligned}
\label{eq:VCmodel}
v_t(x_t) = \<x_t, \th_t\> + z_t\,.\end{aligned}$$ Model parameters $\theta_t$ may change over time and as a result, valuation of a product depends on both the product feature vector and the time index.
We study a dynamic pricing problem, where at each time period $t$, the firm has a product to sell and after observing the product feature vector $x_t$, posts a price $p_t$. If the buyer’s valuation is above the posted price, $v_t(x_t) \ge p_t$, a sale occurs and the firm collects a revenue of $p_t$. If the posted price exceeds the buyer’s valuation, $p_t > v_t(x_t)$, no sale occurs. Note that at each step, the firm has access to the previous feedbacks (sale/no sale) from the buyer and can use this information in setting the current price.
In this paper, we will analyze the varying-coefficient model and answer two fundamental questions:
> First, what is the value of knowing the sequence of model parameters $\theta_t$; in other words, what is the expected revenue loss (regret) compared to the clairvoyant policy that knows the parameters of the valuation model in advance? Second, what is a good pricing policy?
The answer to the first question intrinsically depends on the temporal variability in the sequence $\th_t$. If this variation is very large, then there is not much that can be learnt from previous feedback on the buyer’s behavior and the problem turns into a random price experimentation. On the other hand, if all of the parameters $\th_t$ are the same, then this feedback information can be used to learn the model parameters, which in turn helps in setting the future prices. In this case, an algorithm that performs a good balance between price exploration and best-guess pricing (exploitation) can lead to a small regret. In this work, we study this trade-off through a projected stochastic gradient descent algorithm and investigate the effect of variations of the sequence of $\th_t$ on the regret bounds.
Feature-based models have recently attracted interest in dynamic pricing. [@amin2014repeated] studied a similar model to (without the noise terms $z_t$), where the features $x_t$ are drawn from an unknown i.i.d distribution. A pricing strategy was proposed based on stochastic gradient descent, which results in a regret of the form $O(T^{2/3} \sqrt{\log T})$. This work also studied the problem of dynamic incentive compatibility in repeated posted-price auctions. Subsequently, [@cohen2016feature] studied model , wherein the feature vectors $x_t$ are chosen antagonistically by nature and not sampled i.i.d. This work proposes a pricing policy based on the ellipsoid method from convex optimization [@boyd2004convex] with a regret bound of $O(d^2 \log(T/d))$, under a low-noise setting. More accurately, the regret scales as $O(d^2 \log(\min\{T /d, 1/\delta\}) + d\delta T)$, where $\delta$ measures the noise magnitude: in case of bounded noise, $\delta$ represents the uniform bound on noise and in case of gaussian noise with variance $\sigma^2$, it is defined as $\delta = 2\sigma \sqrt{\log(T)}$. In [@lobel2016multidimensional], the regret bound of this policy was improved to $O(d \log T)$, under the noiseless setting. In [@JavanmardNazerzadeh], authors study and highlight the role of the structure of demand curve in dynamic pricing. They introduce model , and assume that the feature vectors $x_t$ are drawn i.i.d. from an unknown distribution. Further, motivated by real-world applications, it is assumed that the parameter vector $\th$ is sparse in the sense that only a few of its entries are nonzero. A regularized log-likelihood approach is taken to get an improved regret bound of order $s_0 (\log (d)+\log(T))$. We add to this body of work by considering feature-based models for valuation of products whose parameters vary over time.
Time-varying demand environments have also been studied recently by [@Keskin-TVC]. Explicitly, they consider a firm that sells one type of product to customers that arrive over a time horizon. After setting price $p_t$, the firm observes demand $D_t$ given by $D_t = \alpha_t + \beta_t p_t+\epsilon_t$, where $\alpha_t, \beta_t \in \reals$ are the unknown parameters of the demand model and $\epsilon_t$ are the unobserved demand shocks (noise). By contrast, in this work we consider different products, each characterized by a high-dimensional feature vector. Further, the seller only receives a binary feedback (sale/no sale) of the customer’s behavior at each step, rather than observing the customer’s valuation.
Organization of paper and our main contributions
------------------------------------------------
The remainder of this paper is structured as follows. In Section \[sec:model\], we formally define the model and formulate the problem. Technical assumptions and the notion of regret will be discussed in this section. We next propose a pricing policy based on projected stochastic gradient descent (PSGD) applied to the log-likelihood function. At each time period $t$, it returns an estimate $\hth_t$. The price $p_t$ is then set to the optimal price as if $\hth_t$ was the actual parameter $\th_t$. We next analyze the regret of our PSGD algorithm. Let $\delta_t = \|\th_{t+1}-\th_t\|$ be the variation in model parameters at time period $t$. In Section \[sec:regret\], we consider the setting where the product feature vectors $x_t$ are chosen antagonistically by nature and show that the regret of PSGD algorithm is of order $O(\sqrt{T} + \sum_{t=1}^T \sqrt{t} \delta_t)$. Interestingly, this bound is independent of the dimension $d$, which is a desirable property of our policy for high-dimensional applications. We next, in Section \[stochastic\], consider a stochastic features model, where the feature vectors $x_t$ are drawn independently from an unknown distribution (cf. Assumption \[SMF\]). Under this setting, we show that the regret of PSGD is of order $O(d^2 \log T + \sum_{t=1}^T t\delta_t/d)$. Note that setting $\delta_t=0$ corresponds to model and our PSGD pricing obtains a logarithmic regret in $T$. Section \[sec:thm\] is devoted to the proof of main theorems and the main lemmas are proved in Section \[sec:lem\]. Finally, proof of several technical steps are deferred to Appendices.
Related literature
------------------
Our works is at the intersection of dynamic pricing, online optimization and high-dimensional statistics. In the following, we briefly discuss the work most related to ours from these contexts.
[**Feature-based dynamic pricing.**]{} Recent papers on dynamic pricing consider models with features/covariates, motivated in part by new advances in big data technology that allow firms to collect large amount of fine-grained information. In the introduction, we discussed the work [@amin2014repeated; @JavanmardNazerzadeh; @cohen2016feature] which are closely related to our setting. Another recent work on feature-based dynamic pricing is [@qiang2016dynamic]. In this work, authors consider a model where the seller observes the demand entirely, rather than a binary feedback as in our setting. A greedy iterative least squares (GILS) algorithm is proposed that at each time period estimates the demand as a linear function of price by applying least squares to the set of prior prices and realized demands. The work underscores the role of feature-based approaches and show that they create enough price dispersion to achieve a regret of $O(\log (T))$. This is closely related to the work of [@den2013simultaneously] and [@keskin2014dynamic] in dynamic pricing (without demand covariates) that demonstrate the GILS is suboptimal and propose methods to integrate forced price-dispersion with GILS to achieve optimal regret.
[**Online optimization.**]{} This field offers a variety of tools for sequential prediction, where an agent measures its predictive performance according to a series of convex functions. Specifically, there is a sequence of a priori unknown reward functions $f_1, f_2, f_3, \dotsc$ and an agent must make a sequence of decisions: at each time period $t$, he selects a point $z_t$ and a loss $f_t(z_t)$ is incurred. Note that the function $f_t$ is not known to agent at step $t$, but he has access to all previous functions $f_1, \dotsc, f_{t-1}$. First order methods, like online gradient descent (OGD) or online mirror descent (OMD) only use the gradient of previous function at the selected points, i.e., $\partial f_t(z_t)$. The notion of regret here is defined by comparing the agent with the best fixed comparator [@shalev2011online].
[@Hall-DGD] proposed dynamic mirror descent that is capable of adapting adapts to a possibly non-stationary environment. In contrast to OMD [@beck2003mirror; @shalev2011online], the notion of regret is defined more generally with respect to the best comparator “sequence".
It is worth noting that the general framework of online learning does not directly apply to our problem. To see this, we define the the loss $f_t$ to be the negative of the revenue obtained in time period $t$, i.e., $f_t = -p_t \ind(p_t\ge v_t)$. Then 1) the loss functions are not convex; 2) the (first order information) of previous loss functions depend on the corresponding valuations $v_1, \dotsc, v_{t-1}$ which are never revealed to the seller. That said, we borrow some of the techniques from online optimization in proving our results. (See proof of Lemma \[lem:PE\].)
[**High-dimensional statistics.**]{} Among the work in this area, perhaps the most related one to our setting is the problem of 1-bit compressed sensing [@plan2013one; @plan2013robust; @ai2014one; @bhaskar20151]. In this problem, a set of linear measurements are taken from an unknown vector and the goal is to recover this vector having access to the sign of these measurements (1-bit information). This is related to the dynamic pricing problem on model , as the seller observes 1-bit feedback (sale/no sale from previous time periods). However, there are a few important differences between these two problem that are worth noting: 1) In dynamic pricing, the crux of the matter is the decisions (prices) made by the firm. Of course this task entails learning the model parameters and therefore the firm gets into the realm of exploration (learning) and exploitation (earning revenue). By contrast, 1-bit compressed sensing is only a learning task; 2) In dynamic pricing, the prices are set based on the previous (sale/no sale) feedbacks. Therefore, the feedbacks are inherently correlated and this makes the learning task challenging. However, in 1-bit compressed sensing it is assumed that the measurements (and therefore the observed signs ) are independent; 3) The majority of work on 1-bit compressed sensing consider an offline setting, while in the dynamic pricing, decision are made in an online manner.
Model {#sec:model}
=====
We consider a pricing problem faced by a firm that sells products in a sequential manner. At each time period $t=1,2,\cdots,T$ the firm has a product to sell and the product is represented by an [*observable*]{} vector of features (covariates) $x_t \in \cX \subseteq \reals^d$. The length of the time horizon, denoted by $T$, is *unknown* the to the firm and the set $\cX$ is bounded.
The product at time $t$ has a market value $v_t = v_t (x_t)$, depending on both $t$ and $x_t$, which is [*unobservable*]{}. At each period $t$, the firm (seller) posts a price $p_t$. If $p_t\le v_t$, a sale occurs, and the firm collects revenue $p_t$. If the price is set higher than the market value, $p_t>v_t$, no sale occurs and no revenue is generated. The goal of the firm is to design a pricing policy that maximizes the collected revenue.
We assume that the market value of a product is a linear function of its covariates, namely $$\begin{aligned}
\label{eq:model}
v_t(x_t) = \<\th_t, x_t\>+z_t\,.\end{aligned}$$ Here, $\theta_t$ and $x_t$ are $d$-dimensional and $\{z_t\}_{t\ge 1}$ are idiosyncratic shocks, referred to as noise, which are drawn independently and identically from a zero-mean distribution over $\reals$. We denote its cumulative distribution function by $F$, and the corresponding density by $f(x) = F'(x)$. Note that the noise can account for the features that are not measured. We refer to [@keskin2014dynamic; @debBoerZwart2014; @qiang2016dynamic] for a similar notion of demand shocks.
The sequence of parameters $\bth= (\th_1,\th_2,\dotsc)$ is *unknown* to the firm and they may vary across time. This paper focuses on arbitrary sequences $\bth$ and propose an efficient algorithm whose regret scale gracefully in time and the temporal variability in the sequences of $\th_t$. The regret is measured with respect to the clairvoyant policy that knows the sequence $\bth$ in advance. We will formally define the regret in Section \[sec:Benchmark\].
We let $y_t$ be the response variable that indicates whether a sale has occurred at period $t$: $$\begin{aligned}
y_t = \begin{cases}
+1&\text{ if }v_t \ge p_t\,,\\
-1&\text{ if }v_t <p_t\,.
\end{cases}\end{aligned}$$ Note that the above model for $y_t$ can be represented as the following probabilistic model: $$\begin{aligned}
\label{eq:prob-model}
y_t = \begin{cases}
+1&\text{ with probability }\, 1- F\left(p_t-\<\th_t,x_t\>\right)\,,\\
-1&\text{ with probability }\, F\left(p_t- \<\th_t, x_t\>\right)
\end{cases}\end{aligned}$$
Technical assumptions and notations
-----------------------------------
For a vector $v$, we write $\|v\|_p$ for the standard $\ell_p$ norm of a vector $v$, i.e., $\|v\|_p = (\sum_i |v_i|^p)^{1/p}$. Whenever the subscript $p$ is not mentioned it is deemed as the $\ell_2$ norm. For a matrix $A$, $\|A\|$ denotes its $\ell_2$ operator norm. For two vectors $a, b$, we use the notation $\<a,b\>$ to refer to their inner product.
To simplify the presentation, we assume that $\|x_t\| \le 1$, for all $x_t\in \cX$, and $\|\th_t\|\le \l1u$ for a known constant $\l1u$. We denote by $\Theta$ the $d$-dimensional $\ell_2$ ball of radius $\l1u$ (In fact, we can take $\Theta$ to be any convex set that contains parameters $\th_t$. The size of $\Theta$ effects our regret bounds up to a constant factor.)
We also make the following assumption on the distribution of noise $F$.
\[ass1\] The function $F(v)$ is strictly increasing. Further, $F(v)$ and $1-F(v)$ are log-concave in $v$.
Log-concavity is a widely-used assumption in the economics literature [@bagnoli2005log]. Note that if the density $f$ is symmetric and the distribution $F$ is log-concave, then $1-F$ is also log-concave. Assumption \[ass1\] is satisfied by several common probability distributions including normal, uniform, Laplace, exponential, and logistic. Note that the cumulative distribution function of all log-concave densities is also log-concave [@boyd2004convex].
We use the standard big-$O$ notation. In particular $f(n) = O(g(n))$ if there exists a constant $C > 0$ such that $|f(n)| \le Cg(n)$ for all $n$ large enough. We also use $\reals_{\ge 0}$ to refer to the set of non-negative real-valued numbers.
Benchmark policy and regret minimization {#sec:Benchmark}
----------------------------------------
For a pricing policy, we measures its performance via the notion of regret, which is the expected revenue loss compared to an oracle that knows the sequence of model parameters in advance (but not the realizations of $\{z_t\}_{t\ge 1}$).We first characterize this benchmark policy.
Using Eq. , the expected revenue from a posted price $p$ is equal to $p\times\prob(v_t\ge p)= p(1-F(p-\th_t\cdot x_t))$. First order condition for the optimal price $p^*(x_t,\th_t)$ reads $$\begin{aligned}
\label{eq:opt-p}
\popt(x_t,\th_t) = \frac{1-F\left(\popt(x_t,\th_t)-\<\th_t, x_t\>\right)}{f\left(\popt(x_t,\th_t)-\<\th_t, x_t\>\right)}\,.\end{aligned}$$ To lighten the notation, we drop the arguments $x_t$, $\th_t$ and denote by $p^*_t$ the optimal price at time $t$.
We next recall the *virtual valuation* function, commonly used in mechanism design [@Myerson81]: $$\varphi(v)\equiv v - \frac{1-F(v)}{f(v)}\,.$$ Writing Eq. in terms of function $\varphi$, we get $$\<\th_t, x_t\> + \varphi\left(\popt_t-\<\th_t, x_t\>\right)=0\,.$$ In order to solve for $p^*_t$, we define the pricing function $g$ as follows: $$\begin{aligned}
\label{eq:g}
g(v) \equiv v + \varphi^{-1}(-v)\,.\end{aligned}$$ By Assumption \[ass1\], $\varphi$ is injective and hence $g$ is well-defined. Further, it is easy to verify that $g$ is non-negative. Using the definition of $g$ and rearranging the terms we obtain $$\begin{aligned}
\label{eq:popt}
\popt_t = g(\<\th_t, x_t\>)\,.\end{aligned}$$
The performance metric we use in this paper is the worst-case regret with respect to a clairvoyant policy that knows the sequence $\bth$ in advance. Formally, for a policy $\pi$ to be the seller’s policy that sets price $p_t$ at period $t$, the worst-case regret is defined over $T$ periods is defined as: $$\begin{aligned}
\label{eq:Regret_def}
\Reg^\pi(T) \equiv \sup\, \left\{\Delta^{\pi}_{\bth,\bx}:\, \th_t\in \Theta, \, x_t \in \cX \right\}\,,\end{aligned}$$ where for $T\ge 1$, $\bth= (\th_1,\dotsc, \th_T)$ and ${\bx} = (x_1, x_2, \dotsc, x_T)$, $$\begin{aligned}
\Delta^\pi_{\bth,\bx}(T) = \E_{\bth,\bx} \left[\sum_{t=1}^T \bigg(p^*_t \ind(v_t \ge p^*_t) - p_t \ind(v_t \ge p_t) \bigg)\right]\,.\end{aligned}$$ Here the expectation $\E_{\bth,\bx}$ is with respect to the distributions of idiosyncratic noise, $z_t$. Note that $v_t$, $p_t$, and $p^*_t$ depend on $\bth$ and $\bx$.
Pricing policy {#sec:pricing_alg}
==============
function $g$, set $\Theta$, covariate vectors $\{x_t\}_{t\in \naturals}$
prices $\{p_t\}_{t\in \naturals}$
$p_1 \leftarrow 0$ and initialize $\hth_1 \in \Theta$
Set $\hth_{t+1}$ according to the following rule: $$\begin{aligned}
\label{eq:ML}
\hth_{t+1} = \Pi_\Theta(\hth_t - \eta_t \nabla\ell_t(\hth_t)) \end{aligned}$$ with $$\begin{aligned}
\label{eq:lt}
\ell_t(\th) = - \ind(y_t =1) \log (1-F(p_t - \<x_t, \th\> )) - \ind(y_t =-1) \log (F(p_t - \<x_t, \th\> )) \end{aligned}$$ Set price $p_{t+1}$ as $$\begin{aligned}
\label{eq:price}
p_{t+1} \leftarrow g(\<x_{t+1}, \hth_{t+1}\>) \end{aligned}$$
Our dynamic pricing policy consists of a projected gradient descent algorithm to predict parameters $\hth_t$. With each new product, it computes the negative gradient of the loss and shirts its prediction in that direction. The result is projected onto set $\Theta$ to produce the next prediction. The policy then sets the prices as $p_t = g(\<x_t,\hth_t\>)$. Note that by Eq. , $p_t$ is the optimal price if $\hth_t$ was the true parameter $\th_t$. Also, by log-concavity assumption on $F$ and $1-F$, the function $\ell_t(\th)$ is convex.
In projected gradient descent, the sequence of step sizes $\{\eta_t\}_{t\ge 1}$ is an arbitrary sequence of non-increasing values. In Sections \[sec:regret\] and \[stochastic\], we analyze the regret of our pricing policy and provide guidelines for choosing step sizes.
Regret analysis {#sec:regret}
---------------
We first define a few useful quantities that appear in our regret bounds. Define $$\begin{aligned}
M &\equiv& \l1u + \varphi^{-1}(0)\,,\label{eq:M}\\
u_{M} &\equiv& \sup_{|x|\le M} \left\{\max\Big\{ - \dx \log F(x) , -\dx \log(1-F(x)) \Big\}\right\}\,,\label{eq:uM}\\
\ell_{M} &\equiv& \inf_{|x|\le M} \left\{\min\Big\{ -\ddx \log F(x) , -\ddx\log(1-F(x)) \Big\}\right\}\,,\label{eq:lM}\end{aligned}$$ where the derivatives are with respect to $x$. We note that $M$ is an upper-bound on the maximum price offered and also, by the log-concavity property of $F$ and $1-F$, we have $$u_M = \max\Big\{-\dx\log F(-M) , -\dx\log(1-F(M)) \Big\}\,.$$ Further, by log-concavity property of $F$ and $1-F$, we have $\ell_M > 0$.
We also let $B = \max_{v} f(v)$ and $B' = \max_{v} f'(v)$, respectively denote the maximum value of the density function $f$ and the its derivative $f'$.
The following theorem bounds the regret of our PSGD policy.
\[thm1\] Consider model for the product market values and let Assumption \[ass1\] hold. Set $M = 2\l1u+\varphi^{-1}(0)$, with $\varphi$ being the virtual valuation function w.r.t distribution $F$. Then, the regret of PSGD pricing policy using a non-increasing sequence of step sizes $\{\eta_t\}_{t\ge1}$ is bounded as follows: $$\begin{aligned}
\label{thm:regret1}
\Reg(T) \le \frac{2(2B+MB')}{\ell_M} \max\bigg\{\frac{16}{\ell_M} \log T,
\,\frac{2\l1u^2}{\eta_{T+1}} + \frac{u_M^2}{2} \sum_{t=1}^T \eta_t + 2\l1u \sum_{t=1}^T \frac{\delta_t}{\eta_t} \bigg\}+\frac{M}{T}\,,\end{aligned}$$ where $\delta_t \equiv \|\th_{t+1}-\th_t\|$.
In particular, if $\eta_t \propto \frac{1}{\sqrt{t}}$, then there exists a constant $C = C(B,M,W,\ell_M, u_M) >0$, independent of $T$, such that $$\begin{aligned}
\Reg(T) \le C\Big(\sqrt{T} + \sum_{t=1}^T \sqrt{t} {\delta_t}\Big)\,.\end{aligned}$$
At the core of our regret analysis (proof of Theorem \[thm1\]) is the following Lemma that provides a prediction error bound for the customer’s valuations.
\[lem:PE\] Consider model for the product market values and let Assumption \[ass1\] hold. Set $M = 2\l1u+\varphi^{-1}(0)$, with $\varphi$ being the virtual valuation function w.r.t distribution $F$. Let $\{\hth_t\}_{t\ge1}$ be generated by PSGC pricing policy, using a non-increasing positive series $\eta_{t+1}\le \eta_t$. Then, with probability at least $1-\frac{1}{T^2}$ the following holds true: $$\begin{aligned}
\sum_{t=1}^T\<x_t,\th_t-\hth_t\>^2
&\le \frac{4}{\ell_M} \max\bigg\{\frac{16}{\ell_M} \log T, \nonumber\\
&\quad \quad \quad\quad\quad\;\; \frac{2\l1u^2}{\eta_1} + \sum_{t=1}^T \Big(\frac{1}{2\eta_{t+1}} - \frac{1}{2\eta_t} \Big) \|\th_{t+1}-\hth_{t+1}\|^2
+\frac{u_M^2}{2} \sum_{t=1}^T \eta_t + 2\l1u \sum_{t=1}^T \frac{\delta_t}{\eta_t} \bigg\}\,,\label{eq:PE}\end{aligned}$$ where $u_M, \ell_M$ are given by Equations , , respectively.
Lemma \[lem:PE\] is presented in a form that can also be used in proving our next results under the stochastic features model. For proving Theorem \[thm1\], we simplify bound as follows. Given that $\th_{t+1},\hth_{t+1}\in \Theta$, we have $\|\th_{t+1}-\hth_{t+1}\|\le 2\l1u$. Using the non-increasing property of sequence $\eta_t$, we write $$\begin{aligned}
\frac{2\l1u^2}{\eta_1} + \sum_{t=1}^T \left(\frac{1}{2\eta_{t+1}} - \frac{1}{2\eta_{t}}\right) \|\th_{t+1}-\hth_{t+1}\|^2 \le \frac{2\l1u^2}{\eta_1} + \sum_{t=1}^T \left(\frac{2\l1u^2}{\eta_{t+1}} - \frac{2\l1u^2}{\eta_{t}}\right)
\le \frac{2\l1u^2}{\eta_{T+1}}\,.\end{aligned}$$ Therefore, bound simplifies to: $$\begin{aligned}
\sum_{t=1}^T\<x_t,\th_t-\hth_t\>^2
\le \frac{4}{\ell_M} \max\bigg\{\frac{16}{\ell_M} \log T,
\frac{2\l1u^2}{\eta_{T+1}}
+\frac{u_M^2}{2} \sum_{t=1}^T \eta_t + 2\l1u \sum_{t=1}^T \frac{\delta_t}{\eta_t} \bigg\}\,,\label{eq:PE1}\end{aligned}$$ The regret bound is derived by relating regret at each time period to the prediction error at that time. We refer to Section \[sec:thm\] for the proof of Theorem \[thm1\].
The regret bound does not depend on the dimension $d$, which makes our pricing policy desirable for high-dimensional applications. Also, note that the temporal variation $\delta_t$ appears in our bound with coefficient $\sqrt{t}$. Therefore, variations at later times are more impactful on the regret of PSGD pricing policy. This is expected because at later times, the pricing policy is more relied on the accumulated information about the valuation model and an abrupt change in the model parameters can make this information worthless. On the other side, temporal changes at the beginning steps are not that effective since the policy is still experimenting different prices to learn the customer’s behavior.
While the regret bound is dimension-free, the computational complexity of PSGD pricing policy scales with dimension $d$. Specifically, the complexity of each step is $O(d)$. To see this, we note that the gradient $\nabla \ell_t(\th)$ can be computed in $O(d)$ by Equations and . Projection onto set $\Theta$ ($\ell_2$ projection) is also $O(d)$.
Stochastic features model {#stochastic}
=========================
In Theorem \[thm1\], we showed that our PSGD pricing policy achieves regret of order $O(\sqrt{T}+\sum_{t=1}^T \sqrt{t} \delta_t)$. Let us point out that in Theorem \[thm1\] the arrivals (feature vectors $x_t$) are modeled as adversarial. In this section, we assume that features $x_t$ are independent and identically distributed according to a probability distribution on $\reals^d$. Under such stochastic model, we show that the regret of PSGD pricing scales at most of order $O(d^2\log T+\sum_{t=1}^T t\delta_t/d)$.
We proceed by formally defining the stochastic features model.
\[SMF\] (*Stochastic features model*). Feature vectors $x_t$ are generated independently according to a probability distribution $\prob_{\bx}$, with a bounded support in $\reals^d$. We denote by $\Sigma$ the covariance matrix of distribution $\prob_{\bx}$ and assume that $\Sigma$ has bounded eigenvalues. Specifically, there exist constants $C_{\min}$ and $C_{\max}$ such that for every eigenvalue $\sigma$ of $\Sigma$, we have $0<\frac{1}{d}C_{\min} \le \sigma < \frac{1}{d}C_{\max}$.
Without loss of generality and to simplify the presentation, we assume that $\prob_{\bx}$ is supported on the unit $\ell_2$ ball in $\reals^d$. The rationale behind the above assumption on the scaling of eigenvalues is that $\Tr(\Sigma) = \E(\|x_t\|^2)\le 1$. Therefore, the assumption above on the eigenvalues of $\Sigma$ states that all the eigenvalues are of the same order.
Under the stochastic features model, we define the notion of worst-case regret as follows. For a policy $\pi$ be the seller’s policy that sets price $p_t$ at period $t$, the $T$-period regret is defined as: $$\begin{aligned}
\label{regret-2}
\Reg^\pi(T) \equiv \sup\, \left\{\Delta^\pi_{\bth,\prob_{\bx}}:\, \th_t\in \Theta, \, \prob_{\bx} \in Q \right\}\,,\end{aligned}$$ where $Q$ denotes the set of probability distribution supported on $\ell_2$ unit ball satisfying Assumption \[SMF\] (bounded eigenvalues). Further, for $T\ge 1$, $\bth= (\th_1,\dotsc, \th_T)$ and probability measure $\prob_{\bx}$, we define $$\begin{aligned}
\Delta^\pi_{\bth,\prob_{\bx}}(T) = \E_{\bth,\prob_{\bx}} \left[\sum_{t=1}^T \bigg(p^*_t \ind(v_t \ge p^*_t) - p_t \ind(v_t \ge p_t) \bigg)\right]\,.\end{aligned}$$ where the expectation is with respect to the distributions of idiosyncratic noise, $z_t$, and $\prob_{\bx}$, the distribution of feature vectors. Note the subtle difference with definition , in that the worst case is computed over $Q$ rather than $\cX$.
We propose a similar PSGD pricing policy for this setting, with a specific choice of the step sizes. Ideally, we want to set $\eta_t = 6/(\ell_M C t)$, where $C$ is an arbitrary fixed constant such that $0<C<\sigma_{\min}$, with $\sigma_{\min}$ being the minimum eigenvalue of population covariance $\Sigma$. Of course, $\Sigma$ is unknown and therefore we proceed as follows. We let $Q_t = (1/t)\sum_{\ell=1}^t x_\ell x_\ell^\sT$ be the empirical covariance based on the first $t$ features. Denote by $\sigma_t$ the minimum eigenvalue of $Q_t$. We then use the sequence $\sigma_t$, and set the step size $\eta_t$ as $$\eta_t = \frac{1}{\lambda_t \cdot t}\,,\quad \quad \lambda_t = \frac{\ell_M}{6} \left\{\frac{1}{t}\Big(1+\sum_{\ell=1}^t \sigma_\ell\Big) \right\}\,.$$ Description of the PSGD pricing policy is given in Table above.
Logarithmic regret bound
------------------------
function $g$, set $\Theta$, covariate vectors $\{x_t\}_{t\in \naturals}$
prices $\{p_t\}_{t\in \naturals}$
$p_1 \leftarrow 0$ and initialize $\hth_1 \in \Theta$ $Q_1 \leftarrow x_1x_1^\sT$
Define $\sig_t$ as the minimum eigenvalue of $Q_t$. Set $$\label{lambdat}
\lambda_t=
\frac{\ell_M}{6t}(1+\sum_{\ell=1}^t \sig_\ell)\,.$$ Set $$\begin{aligned}
\label{lambdat2}
\eta_t = \frac{1}{\lambda_t \cdot t}\end{aligned}$$ Set $\hth_{t+1}$ according to the following rule: $$\begin{aligned}
\label{eq:ML}
\hth_{t+1} = \Pi_\Theta(\hth_t - \eta_t \nabla\ell_t(\hth_t)) \end{aligned}$$ with $$\begin{aligned}
\label{eq:lt}
\ell_t(\th) = - \ind(y_t =1) \log (1-F(p_t - \<x_t,\th\> )) - \ind(y_t =-1) \log (F(p_t - \<x_t,\th\> )) \end{aligned}$$ $Q_{t+1}\leftarrow (\frac{t}{t+1}) Q_t + (\frac{1}{t+1}) x_{t+1}x_{t+1}^\sT$ Set price $p_{t+1}$ as $$\begin{aligned}
\label{eq:price}
p_{t+1} \leftarrow g( \<x_{t+1},\hth_{t+1} \>) \end{aligned}$$
The following theorem bounds the regret of our dynamics pricing policy.
\[thm:log-regret\] Consider model for the product market values and suppose Assumption \[ass1\] holds. Let $M = 2\l1u+\varphi^{-1}(0)$, with $\varphi$ being the virtual valuation function w.r.t distribution $F$. Under the stochastic features model (Assumption \[SMF\]), the regret of PSGD pricing policy is bounded as follows: $$\begin{aligned}
\label{eq:log-term}
\Reg(T) \le C_1 d^2 \log T + C_2\sum_{t=1}^T \frac{t}{d}\delta_t\,,\end{aligned}$$ where $\delta_t \equiv \|\th_{t+1}-\th_t\|$ and $C_1, C_2$ are constants that depend on $C_{\max}, C_{\min}, u_M, \ell_M, M, B, \l1u$ but are independent of dimension $d$.
Proof of Theorem \[thm:log-regret\] relies on the following lemma that is analogous to Lemma \[lem:PE\] and establishes a prediction error bound for the customer’s valuations.
\[lem:PE-random\] Consider model for the product market values and the stochastic features model (Assumption \[SMF\]). Suppose that Assumption \[ass1\] holds and set $M = 2\l1u+\varphi^{-1}(0)$, with $\varphi$ being the virtual valuation function w.r.t distribution $F$. Let $\{\hth_t\}_{t\ge 1}$ be generated by PSGD pricing policy. Then, $$\begin{aligned}
{C_{\min}}\sum_{t=1}^T \E(\|\th_t-\hth_t\|^2) \le&
\left[\frac{128}{\ell_M^2} + \frac{24u_M^2}{\ell_M^2} \left(\tilde{c}+ \frac{4}{C_{\min} d} \right) \right]\cdot d^3 \log T\\
& +8\l1u^2d \left(\frac{1}{T} +\frac{12}{\ell_M^2} + \frac{1}{c_2 d}\right) +{4\l1u}\sum_{t=1}^T t\delta_t\,.\end{aligned}$$ Here $\sigma_{\min}$ denotes the minimum eigenvalue of covariance $\Sigma$. (See Assumption \[SMF\].)
A lower bound on regret
-----------------------
In this section, we provide a theoretical lower bound on the minimum achievable regret of any pricing policy under the stochastic features model. Prior to that, we need to adopt a few notations.
For a given time horizon $T$ and a sequence of valuations parameters $\bth = (\th_1, \dotsc, \th_T)$, let $$\begin{aligned}
\label{V:T}
V_{\bth}(T) \equiv \sum_{t=1}^T t \|\th_{t+1}-\th_t\|\,. \end{aligned}$$ We also define, for $\nu\in[1/2,2]$, $$\begin{aligned}
\cV(T,B,\nu) \equiv \{\bth:\, \th_t\in \Theta,\, V_{\bth}(T)\le B d T^\nu \}\,.\end{aligned}$$ By assuming $\bth\in \cV(T,B,\nu)$ for all $T$, we are assuming that nature has a finite temporal variation budget to use in changing the valuation parameters throughout the time horizon. Of course, different variation metrics can be considered such as total variation $\sum_{t=1}^T\delta_t$ or the maximum temporal variation $\sup_{1\le t \le T} \delta_t$ and the performance of a pricing policy can be studied under different variation budget constraints. The specific choice of is putting higher weights at later variations in the sequence $\bth$ and is reasonable for applications where one expects the buyer’s preferences (valuation parameters) become stable over time. Note that designing favorable pricing policy for applications with gradual changes in buyer’s preferences is more challenging than that for environments with bursty changes. This might look counterintuitive at first glance because at any time, the accumulated information about valuations can become useless by an abrupt change in the valuation model. However, as noticed and analyzed in [@Keskin-TVC], this is not that case because, intuitively, gradual changes can be undetectable and lead to significant revenue loss, while for bursty changes, the policy can be designed in a way to detect the changes and reset its estimate of the valuation model after each change to avoid large estimation error and revenue loss. For a pricing policy $\pi$, consider the $T$-period regret, defined as $$\begin{aligned}
\Reg^\pi(T,B,\nu) \equiv \max\Big\{\Delta^{\pi}_{\bth,\prob_{\bx}}(T): \, \bth\in \cV(T,B,\nu),\, \prob_{\bx} \in Q \Big\} \end{aligned}$$ where we recall that $$\begin{aligned}
\label{Rtth}
\Delta^{\pi}_{\bth,\prob_{\bx}}(T) \equiv \sum_{t=1}^T \E_{\bth,\prob_{\bx}}\bigg(p^*_t \ind(v_t \ge p^*_t) - p_t \ind(v_t \ge p_t) \bigg)\,.\end{aligned}$$ Note that this is the same regret notion defined in , where we just make the variation budget constraint explicit in the notation.
Rephrasing the statement of Theorem \[thm:log-regret\], for PSGD pricing policy we have $\Reg^\pi(T,B,\nu) \le C_1d^2 \log T + C_2 B T^\nu$. We next provide a lower bound on the regret of any pricing policy. Indeed this lower bound applies to a powerful clairvoyant who fully observes the market values after the price is either accepted or rejected.
\[thm:LB-reg\] Consider linear model where the market values $v_t(x_t)$, $1\le t \le T$, are fully observed. We further assume that market value noises are generated as $z_t \sim \normal(0, \sigma^2)$. There exists a constant $c$, depending on $\sigma$, $C_{\max}$, such that $\Reg^\pi(T,B,\nu) \ge c\min\Big(\left({B^2 dT^{2\nu-1}} \right)^{1/3}, T/d\Big)$, for any pricing policy $\pi$ and time horizon $T$.
The high-level intuition behind this result is that the nature can change the valuation parameters in a gradual manner such that the seller should pay a revenue loss in order to detect the changes and learn the new valuation parameter after a change. To be more specific, we divide the time horizon into cycles of length $N$ periods, where $N$ is of order $(T^{4-2\nu}/d)^{1/3}$ and consider a setting where the value of $\th_t$ can change to one of two options $\th^0$, $\th^1$, only in the first period of a cycle. We choose the parameter change $\delta = \|\th^1-\th^0\|$ of order $\sqrt{d/N}$ to ensure that $(i)$ no policy can identify the change without incurring a revenue loss of order $N\delta^2/d$ $(ii)$ The variation metric $V_{\bth}(T)$ remains below the allowable limit of $Bd T^\nu$. Therefore, the total regret over $T$ periods works out at $T\delta^2/d$. In particular, for proving point $(i)$ we quantify the likelihood of valuations under the probability measures corresponding to $\th^0$ and $\th^1$, using Kullback-Leibler divergence. We use Pinsker inequality form probability theory and hypothesis testing results from information theory to show that there is a significant probability of not detecting the (potential) change, which consequently yields a revenue loss of order $N\delta^2/d$, over each cycle.
We refer to Section \[proof:LB-reg\] for the proof of Theorem \[thm:LB-reg\].
Numerical experiments {#sec:numerical}
=====================
We numerically study the performance of our PSGD pricing policy on synthetic data. In our experiments, we set $\l1u = 5$ and set $\theta_1 = (\l1u/2) (Z/\|Z\|)$, with $Z\sim \normal(0,\id_d)$ a multivariate normal variable. We then generate a sequence of parameters $\theta_t$ as follows: $$\theta_{t+1} = \th_t + r_t\,,$$ where $r_t = t^{-b} (\tZ/\|\tZ\|)$, with $\tZ\sim \normal(0,\id_d)$. Note that $\delta_t = \|\theta_{t+1}-\theta_t\| = \|r_t\| = t^{-b}$.
Next, at each time $t$, product covariates $x_t$ are independently sampled from a Gaussian distribution $\normal(0,\id_d)$ and normalized so that $\|x_t\| = 1$. Further, the market shocks are generated as $z_t \sim \normal(0,\sigma^2)$, with $\sigma = 1$. We run the PSGD pricing policy for stochastic features model.
[**Results.**]{} Figure \[fig:regret\_b\] compares the cumulative regret (averaged over 80 trials) of the PSGD policy, for $b = 0.5, 1, 2$, on the aforementioned synthetic data for $T = 50,000$ steps. The shaded region around each curve correspond to the $95\%$ confidence interval across the $80$ trials. As expected, increase in $b$ results in larger temporal variations and larger regret.
(-120,0)[$T$]{} (-240,95)
To better understand the behavior of regret for different values of $b$, we plotted the regret bounds in various scales in Figure \[fig:b\]. For $b= 0.5$, we have $\Reg(T)\sim T^{2/3}$, and for $b = 1, 2$, we have $\Reg~\sim \log(T)$. Comparing with Theorem \[thm:log-regret\], we see that the empirical regret in case of $b=0.5$, $1$, is smaller than the upper bound given by Equation , order-wise. However, it is worth noting that bound given in Theorem \[thm:log-regret\] applies to any adversarial choice of temporal variations $r_t$, while in our experiments we generated these terms independently at random.
Extension to nonlinear model {#sec:extension}
============================
Throughout the paper, we exclusively focused on linear models for buyer’s valuation with varying coefficients. In order to generalize our results to nonlinear models, we consider a setting where the market value of a product with feature vector $x_t$ is given by $$\begin{aligned}
v_t(x_t) = \psi (\<x_t,\th_t\>+z_t)\,.\end{aligned}$$ This model is often referred to as generalized linear model and captures nonlinear dependencies on features to some extent. We assume that the link function $\psi:\reals\mapsto \reals$ is a general log-concave function and is strictly increasing.
We next compute the pricing function. Since $\psi$ is strictly increasing, the expected revenue at a price $p$ amounts to $p\left(1-F\left(\psi^{-1}(p)-\<x_t,\th_t\>\right)\right)$. First order condition for the optimal price $p^*_t(x_t)$ reads as $$\begin{aligned}
\label{f6}
\psi'(\psi^{-1}(p_t^*)) = \frac{pf\left(\psi^{-1}(p_t^*)-\<x_t,\th_t\>\right)}{1-F\left(\psi^{-1}(p_t^*)-\<x_t,\th_t\>\right)}\,.\end{aligned}$$ Define $\lambda(v) = f(v)/(1-F(v))$ the hazard rate function for distribution $F$, and let $\tp = \psi^{-1}(p)$. Writing in terms of $\lambda$ function, we get $$\begin{aligned}
\label{f7}
\<x_t,\th_t\> = \tp^*_t - \lambda^{-1}\left(\frac{\psi'(\tp^*_t)}{\psi(\tp^*_t)}\right)\,.\end{aligned}$$ For real-valued $v$, define $$\begin{aligned}
\label{gpsi}
g_\psi^{-1}(v) \equiv v - \lambda^{-1} \left(\frac{\psi'(v)}{\psi(v)}\right)\,.\end{aligned}$$ Note that by log-concavity of $1-F$, the hazard function $\lambda$ is increasing. Also, by log-concavity of $\psi$, the term $\frac{\de}{\de v}\log\psi(v) = \psi'(v)/\psi(v)$ is decreasing. Putting these together, we obtain that $ - \lambda^{-1} ({\psi'(v)}/{\psi(v)})$ is increasing. Therefore, the right-hand side of is strictly increasing and the function $g_\psi$ is well-defined. Invoking Equation , we derive the optimal price as $$\begin{aligned}
p^*_t = \psi \left(g_\psi(\<x_t,\th_t\>)\right)\,.\end{aligned}$$ As noted before, since $\psi$ is increasing, at each period $t$, a sale happens if $z_t\ge \psi^{-1}(p_t) - \<x_t,\th_t\>$. Hence, the log-likelihood function reads as $$\begin{aligned}
\ell_t(\th) = - \ind(y_t =1) \log \left(1-F\left(\psi^{-1}(p_t) - \<x_t, \th\> \right)\right) - \ind(y_t =-1) \log \left(F\left(\psi^{-1}(p_t) - \<x_t, \th\> \right)\right)\,. \end{aligned}$$ In PSGD pricing policy, we run gradient step with this log-likelihood function and then set price $p_{t+1}$ at next step as $p_{t+1} = \psi \left(g_\psi(\<x_{t+1},\th_{t+1}\>)\right)$.
The results on the regret of PSGD pricing policy carries over to the generalized linear model as well. The analysis goes along the same lines and is omitted.
Proof of main theorems {#sec:thm}
======================
Proof of Theorem \[thm1\]
-------------------------
\[lem:M\] Set $M = 2\l1u+\varphi^{-1}(0)$, and for $\th\in \Theta$ define $u_t(\th) = p_t - \<x_t, \th\>$, where $p_t = g(\<x_t,\hth_t\>)$ is the posted price at time $t$. Then $|u_t(\th)|\le M$ for all $t\ge 1$.
Define function $h(;u)$ from $\reals_{\ge 0}$ to $\reals_{\ge 0}$ as $$h(p;u) = p(1-F(p-u))$$ This is the expected revenue at price $p$ when the noiseless valuation is $u$, i.e., $\<x_t,\th_t\> = u$. We let $$\begin{aligned}
\label{Rt}
R_t \equiv p_t^* \ind(v_t\ge p_t^*) - p_t\ind(v_t\ge p_t)\end{aligned}$$ be the regret incurred at time $t$, and define $\cF_t$ as the history up to time $t$ (Formally, $\cF_t$ is the $\sigma$-algebra generated by market noise $\{z_\ell\}_{\ell=1}^{t}$.) Then, $$\begin{aligned}
\E(R_t|\cF_{t-1}) = p_t^* \prob(v_t\ge p_t^*) - p_t \prob(v_t\ge p_t) = h(p^*_t;\<x_t,\th_t\>) - h(p_t;\<x_t,\hth_t\>)\,.\end{aligned}$$ The optimal price $p_t^*$ is the maximizer of $h(p;\<x_t,\th_t\>)$ and thus $h'(p_t^*;\<x_t,\th_t\>) = 0$. By Taylor expansion of function $h$, there exists a value $p$ between $p_t$ and $p^*_t$, such that, $$\begin{aligned}
\label{h-taylor}
h(p_t;\<x_t,\th_t\>) - h(p^*_t;\<x_t,\th_t\>)= \frac{1}{2} h''(p;\<x_t,\th_t\>) (p_t-p^*_t)^2\,.\end{aligned}$$
We next show that $|h''(p;\<x_t,\th_t\>)|\le C$ with $C = 2B+MB'$. Recall that $B = \max_v f(v)$ and $B'= \max_v f'(v)$. To see this, we write $$\begin{aligned}
\label{h"}
|h''(p;\<x_t,\th_t\>)| = \Big|2f(p-\<x_t,\th_t\>)+pf'(p-\<x_t,\th_t\>)\Big| \le 2B+MB'\,.\end{aligned}$$ Putting Equations , , and using the $1$-Lipschitz property of price function $g$, we conclude: $$\begin{aligned}
\label{eq:chain}
\E[R_t|\cF_{t-1}] &= h(p^*_t;\<x_t,\th_t\>) - h(p_t;\<x_t,\hth_t\>)\le \frac{2B+MB'}{2}(p_t-p^*_t)^2\nonumber\\
&= \frac{2B+MB'}{2}\Big(g(\<x_t,\hth_t\>)-g(\<x_t,\th_t\>)\Big)^2 \le \frac{2B+MB'}{2} \<x_t,\th_t-\hth_t\>^2\end{aligned}$$ To ease the presentation, define the shorthand $$A(T) \equiv \frac{4}{\ell_M} \max\bigg\{\frac{16}{\ell_M} \log T,
\,\frac{2\l1u^2}{\eta_{T+1}} + \frac{u_M^2}{2} \sum_{t=1}^T \eta_t + 2\l1u \sum_{t=1}^T \frac{\delta_t}{\eta_t} \bigg\}\,.$$ We further let $\cG$ be the probabilistic event that $\sum_{t=1}^T \<x_t,\th_t-\hth_t\>^2\le A(T)$. Employing Lemma \[lem:PE\] and using the fact that $\|\th_{t+1}-\hth_{t+1}\|^2\le 4\l1u^2$, we obtain that $\prob(\cG) \ge 1-\frac{1}{T^2}$.
We continue by bounding $E(R_t)$ as follows: $$\begin{aligned}
\E[R_t] &= \E[\E[R_t|\cF_{t-1}]] = \E\Big[\E[R_t|\cF_{t-1}]\cdot \Big(\ind(\cG)+\ind(\cG^c)\Big)\Big] \\
&= \frac{2B+MB'}{2} \E\Big[\<x_t,\th_t-\hth_t\>^2 \cdot \ind(\cG)\Big] + M\prob(\cG^c)\,.\end{aligned}$$ Consequently, $$\begin{aligned}
\Reg(T) \le \sum_{t=1}^T \E[R_t] &\le \frac{2B+MB'}{2}\E\Big[\sum_{t=1}^T \<x_t,\th_t-\hth_t\>^2 \cdot \ind(\cG)\Big] + M T\, \prob(\cG^c)
\le \frac{2B+MB'}{2} A(T) + \frac{M}{T}\,.\end{aligned}$$ The proof is complete.
Proof of Theorem \[thm:log-regret\]
-----------------------------------
Proof of Theorem \[thm:log-regret\] follows along the same lines as proof of Theorem \[thm1\]. Let $\ctF_t$ be the $\sigma$-algebra generated by market noises $\{z_\ell\}_{\ell=1}^t$ and feature vectors $\{x_{\ell}\}_{\ell=1}^{t}$. Further, let $\cF_t$ be the $\sigma$-algebra generated by $\ctF_t \cup\{x_{t+1}\}$. For term $R_t$ defined by and following the chain of inequalities as in , $$\begin{aligned}
\E[R_t|\cF_{t-1}] \le \frac{2B+MB'}{2}\, \<x_t,\th_t-\hth_t\>^2 \,.
\end{aligned}$$ For brevity in notation, let $\bar{B} = (2B+MB')/2$. Since, $\cF_{t}\supseteq \ctF_t$, by iterated law of iteration, $$\begin{aligned}
\E(R_t|\ctF_{t-1}) = \E(\E(R_t|\cF_{t-1})|\ctF_{t-1}) \le \bar{B} \<\th_t-\hth_t, \Sigma (\th_t-\hth_t)\> \le \frac{1}{d} \bar{B}C_{\max} \|\th_t-\hth_t\|^2
\end{aligned}$$ Applying Lemma \[lem:PE-random\], we get $$\begin{aligned}
\Reg(T) &\le \sum_{t=1}^T \E[R_t] \le \frac{1}{d} \bar{B}C_{\max} \sum_{t=1}^T \E(\|\th_t-\hth_t\|^2)\\
&\le \bar{B} \frac{C_{\max}}{C_{\min}}\left[\frac{128}{\ell_M^2} + \frac{24u_M^2}{\ell_M^2} \left(\tilde{c}+ \frac{4}{C_{\min} d} \right) \right]\cdot d^2 \log T\\
& +8\l1u^2\bar{B} \frac{C_{\max}}{C_{\min}} \left(\frac{1}{T} +\frac{12}{\ell_M^2} + \frac{1}{c_2 d}\right) + \bar{B} \frac{C_{\max}}{C_{\min}}\Big(\frac{4\l1u}{d}\Big) \sum_{t=1}^T t\delta_t\,.
\end{aligned}$$ The result follows by taking $$\begin{aligned}
C_1 &= \bar{B} \frac{C_{\max}}{C_{\min}} \left[{8\l1u^2}\left(\frac{1}{T} +\frac{12}{\ell_M^2} + \frac{1}{c_2 d}\right)
+ \frac{128}{\ell_M^2} + \frac{24u_M^2}{\ell_M^2} \left(\tilde{c}+ \frac{4}{C_{\min} d} \right) \right]\,,\\
C_2&= 4\l1u \bar{B} \frac{C_{\max}}{C_{\min}} \,.
\end{aligned}$$
Proof of Theorem \[thm:LB-reg\] {#proof:LB-reg}
-------------------------------
The proof methodology is similar to the proof of [@Keskin-TVC Theorem 1].
We first propose a setting for constructing the sequence of valuation parameters $\bth= (\th_1,\dotsc,\th_T)$. Divide the time horizon into cycles of length $N = \lceil m_0 T^{(4-2\nu)/3}\rceil$, where $m_0 = (\frac{\sigma^2}{C_{\max}B^2d})^{1/3}$. Consider a setting wherein the noise markets are generated as $z_t\sim\normal(0,\sigma^2)$ and the value of $\th_t$ can change only in the first period of a cycle, taking one of the two values $\{\th^0,\th^1\}$. Here, $\th^0, \th^1\in \reals^d$ are two arbitrary vectors such that $\|\th^0-\th^1\| =\delta$, with $\delta = \min(\sigma\sqrt{d/(C_{\max}N)},\sqrt{c_2})$. Note that for this sequence of $\bth$, we have $$\begin{aligned}
V_{\bth}(T) \le \sum_{k=1}^{\lceil T/N\rceil} (kN) \delta \le\frac{T^2}{N} \delta \le Bd T^\nu\end{aligned}$$ We consider a clairvoyant who fully observes the market values $v_t(x_t)$. Focus on a single cycle and let $\prob_0^\pi$ (resp. $\prob_1^\pi$) denote the probability distribution of valuations $(v_1, v_2, \dotsc, v_N)$ when all the parameters $\th_t $ are equal to $\th^0$ (resp. $\th^1$), for $1\le t \le N$. The KL divergence between $\prob_0^\pi$ and $\prob_1^\pi$ amounts to $$\begin{aligned}
\KL(\prob_0^\pi,\prob_1^\pi) \equiv \E_0^\pi \log \left(\frac{\prod_{t=1}^N \phi\left(\dfrac{v_t-\<x_t,\th_0\>}{\sigma}\right)}{\prod_{t=1}^N \phi\left(\dfrac{v_t-\<x_t,\th_1\>}{\sigma}\right)} \right)\end{aligned}$$ where $\E_0^\pi$ denotes expectation w.r.t $\prob_0^\pi$ and $\phi(s) = 1/(\sqrt{2\pi})e^{-s^2/2}$ is the standard Gaussian density. After simple algebraic manipulation, we obtain $$\begin{aligned}
\KL(\prob_0^\pi,\prob_1^\pi) &= -\frac{1}{2\sigma^2} \E^\pi_0 \bigg\{\sum_{t=1}^N (2z_t - \<x_t,\th^1-\th^0\>)\<x_t,\th^1-\th^0\> \bigg\}\\
&= \frac{1}{2\sigma^2} \sum_{t=1}^N \E^\pi_0 (\<x_t,\th^1-\th^0\>^2) \le \frac{1}{2\sigma^2 d} \sum_{t=1}^N C_{\max} \|\th^1-\th^0\|^2\\
&= \frac{1}{2\sigma^2}C_{\max} \frac{\delta^2 N}{d}\,.\end{aligned}$$ We next relate the expected regret to the KL divergence between $\prob_0^\pi$ and $\prob_1^\pi$.
\[lem:Rt-LB\] Let $R_t$ be the regret incurred at time $t$, defined as $R_t \equiv p_t^* \ind(v_t\ge p_t^*) - p_t\ind(v_t\ge p_t)$. Then, there exist constants $c_1, c_2$ depending on $\sigma$, $\l1u$, and $C_{\min}$, such that $$\begin{aligned}
\E(R_t) \ge \frac{c_1}{d} \E\Big\{ \min\left(\|\hth_t-\th_t\|_2^2,c_2\right)\Big\}\,.\end{aligned}$$
Proof of Lemma \[lem:Rt-LB\] goes along the proof of [@JavanmardNazerzadeh Equation (55)] and is omitted.
By applying Lemma \[lem:Rt-LB\], we have
$$\begin{aligned}
\Delta^\pi_{\bth,\prob_{\bx}}(N) = \sum_{t=1}^N \E_{\bth}(R_t) \ge \frac{c_1}{d} \sum_{t=1}^N \E\Big\{ \min\left(\|\hth_t-\th_t\|_2^2,c_2\right)\Big\}\,.\end{aligned}$$
For brevity in notations, for the sequence $\bth = (\th_1,\dotsc, \th_N)$, we define $\dis_a(\bth) = c_1\sum_{t=1}^N \min(\|\th_t-\th^a\|_2^2,c_2)$, for $a=1,2$. Define two sets $J_a$, for $a=1,2$ as follows: $$\begin{aligned}
J_a = \left\{\bth =(\th_1,\dotsc, \th_N):\, \th_i\in\reals^d\,,\, \dis_a(\bth) < \frac{1}{4} N \delta^2\right\}\,.\end{aligned}$$ Then,
$$\begin{aligned}
\max\left(\Delta^\pi_{0,\prob_{\bx}}(N),\Delta^\pi_{1,\prob_{\bx}}(N)\right) &\ge \frac{1}{d} \max\Big(\E_0^\pi (\dis_0(\bth)), \E^\pi_1(\dis_1(\bth)) \Big) \\
&\ge \frac{N}{4d} \delta^2 \max\Big(\prob^\pi_0(\bth\notin J_0), \prob^\pi_1(\bth\notin J_1) \Big)\\
&\stackrel{(a)}{\ge} \frac{N}{4d} \delta^2 \max\Big(\prob^\pi_0(\bth\notin J_0), \prob^\pi_1(\bth\in J_0) \Big)\\
&\ge \frac{N}{8d} \delta^2 \Big(\prob^\pi_0(\bth\notin J_0) + \prob^\pi_1(\bth\in J_0) \Big)\\
&\ge \frac{N}{8d} \delta^2 \Big(1- \prob^\pi_0(\bth\in J_0) + \prob^\pi_1(\bth\in J_0) \Big)\\
&\ge \frac{N}{8d} \delta^2 \Big(1- \sqrt{\frac{1}{2} \KL(\prob^\pi_0,\prob^\pi_1)} \Big) \quad \text{(By Pinsker inequality)}\\
&\ge \frac{N}{8d} \delta^2 \left(1- \frac{1}{2\sigma} \delta \sqrt{C_{\max} \frac{N}{d}} \right) \ge \frac{N\delta^2}{16d} \,.\end{aligned}$$
Here $(a)$ holds because $\bth\in J_0$ implies $\bth\notin J_1$. Otherwise, $\dis_0(\bth)< N\delta^2/4$ and $\dis_1(\bth)< N\delta^2/4$. Using the inequality $\min(a + b, c) \le \min(a, c) + \min(b, c)$ for $a, b, c \ge 0$, and applying triangle inequality, we get $$\begin{aligned}
N \min(\|\th^0-\th^1\|^2,c_2) \le 2\dis_0(\bth) + 2\dis_1(\bth) < N\delta^2\,,\end{aligned}$$ which is a contradiction because $\delta^2 = \|\th^0-\th^1\|^2 \le c_2$. Therefore, we conclude that $$\begin{aligned}
\Reg^\pi(T,B,\nu) &\ge \Big\lfloor \frac{T}{N} \Big\rfloor \max\left(\Delta^\pi_{0,\prob_{\bx}}(N),\Delta^\pi_{1,\prob_{\bx}}(N)\right) \nonumber\\
&\ge \frac{T\delta^2}{16d} = \frac{T}{16} \min\Big(\frac{\sigma^2}{C_{\max}N}, \frac{c_2}{d}\Big)\nonumber\\
&= \frac{1}{16} \min\left\{\Big(\frac{\sigma^2}{C_{\max}} \Big)^{2/3} ({B^2 dT^{2\nu-1}} )^{1/3}, \frac{c_2 T}{d}\right\}\,.\end{aligned}$$ The result follows.
Proof of main lemmas {#sec:lem}
====================
Proof of Lemma \[lem:PE\]
-------------------------
We prove Lemma \[lem:PE\] by developing an upper bound and a lower bound for the quantity $\sum_{t=1}^T \ell_t(\hth_t) - \sum_{t=1}^T \ell_t(\th_t)$. The result follows by combining these two bounds.
\[lem:DUB\] Suppose $\{\th_t\}_{t\ge 1}$ is an arbitrary sequence in $\Theta$, and $\|\th\|\le \l1u$ for all $\th\in \Theta$. Set $M = 2\l1u+\varphi^{-1}(0)$, with $\varphi$ being the virtual valuation function w.r.t distribution $F$. Further, let $\{\hth_t\}_{t\ge1}$ be generated by PSGD policy using a non-increasing positive series $\eta_{t+1}\le \eta_t$. Then $$\begin{aligned}
\sum_{t=1}^T \ell_t(\hth_t) -\sum_{t=1}^T \ell_t(\th_t) \le &\frac{2\l1u^2}{\eta_{1}} + \sum_{t=1}^T \Big(\frac{1}{2\eta_{t+1}}-\frac{1}{2\eta_t}\Big)\|\th_{t+1}-\hth_{t+1}\|^2 \nonumber\\
&+\frac{u_M^2}{2} \sum_{t=1}^T \eta_t + 2\l1u \sum_{t=1}^T \frac{\delta_t }{\eta_t} -\frac{\ell_M}{2}\sum_{t=1}^T\<x_t,\th_t-\hth_t\>^2\,,\label{DUB}\end{aligned}$$ where $\delta_t \equiv \|\th_{t+1}-\th_t\|$ and we recall $u_M$ from Equation .
The proof of Lemma \[lem:DUB\] uses similar ideas to the regret bounds established in [@Hall-DGD], but uses the log-concavity of $F$ and $1-F$ and also definition of $u_M$ and $\ell_M$ as per Equations and to get a more refined bound including quadratic terms $\<x_t,\hth_t-\th_t\>^2$. We refer to Appendix \[app:DUB\] for the proof of Lemma \[lem:DUB\]. Our next Lemma provides a probabilistic lower bound on $\sum_{t=1}^T \ell_t(\hth_t) - \sum_{t=1}^T \ell_t(\th_t)$.
\[lem:DLB\] Consider model for the product market values and suppose Assumption \[ass1\] holds. Let $\{\hth_t\}_{t\ge 1}$ be an arbitrary sequence in $\Theta$. Then with probability at least $1-\frac{1}{T^2}$ the following holds true $$\begin{aligned}
\label{DLB}
\sum_{t=1}^T \ell_t(\hth_t) - \sum_{t=1}^T\ell_t(\th_t) \ge -2\sqrt{\log T} \Big\{\sum_{t=1}^T \<x_t,\th_t-\hth_t\>^2\Big\}^{1/2} \,.\end{aligned}$$
Proof of Lemma \[lem:DLB\] is given in Appendix \[app:DLB\]. It uses convexity of $\ell_t(\hth)$ and an application of a concentration bound on martingale difference sequences.
Combining Equations and we obtain that with probability at least $1-\frac{1}{T^2}$ the following holds true $$\begin{aligned}
-2\sqrt{\log T} \Big\{\sum_{t=1}^T \<x_t,\th_t-\hth_t\>^2\Big\}^{1/2} \le& \frac{2\l1u^2}{\eta_1} + \sum_{t=1}^T \Big(\frac{1}{2\eta_{t+1}} - \frac{1}{2\eta_t} \Big) \|\th_{t+1}-\hth_{t+1}\|^2 \nonumber\\
&+\frac{u_M^2}{2} \sum_{t=1}^T \eta_t + 2\l1u \sum_{t=1}^T \frac{\delta_t }{\eta_t} -\frac{\ell_M}{2}\sum_{t=1}^T\<x_t,\th_t-\hth_t\>^2\end{aligned}$$ Rearranging the terms, we get $$\begin{aligned}
\frac{\ell_M}{2}&\sum_{t=1}^T\<x_t,\th_t-\hth_t\>^2 -2\sqrt{\log T} \Big\{\sum_{t=1}^T \<x_t,\th_t-\hth_t\>^2\Big\}^{1/2} \nonumber\\
&\le \frac{2\l1u^2}{\eta_1} + \sum_{t=1}^T \Big(\frac{1}{2\eta_{t+1}} - \frac{1}{2\eta_t} \Big) \|\th_{t+1}-\hth_{t+1}\|^2
+\frac{u_M^2}{2} \sum_{t=1}^T \eta_t + 2\l1u \sum_{t=1}^T \frac{\delta_t }{\eta_t}\label{f4}\end{aligned}$$ Define $A\equiv \sum_{t=1}^T\<x_t,\th_t-\hth_t\>^2 $ and denote by $B$ the right-hand side of Equation .
Writing in terms of $A$ and $B$, we have $$\begin{aligned}
\label{f5}
A - \frac{4}{\ell_M} \sqrt{A \log T} \le \frac{2B}{\ell_M}\,.
\end{aligned}$$ We next upper bound $A$ as follows. Consider two cases:
[**Case 1:**]{} Assume that $$\sqrt{A\log T} \le \frac{\ell_M}{8}A\,.$$ Using this in Equation , we get $A\le 4B/\ell_M$.
[**Case 2:**]{} Assume that $$\sqrt{A\log T} > \frac{\ell_M}{8}A\,.$$ Then, $A< (64/\ell_M^2)\log T$.
Combining the above two cases, we obtain $$A\le \frac{4}{\ell_M}\max\Big(\frac{16}{\ell_M}\log T, {B}\Big)\,.$$
Substituting for $A$ and $B$, we have $$\begin{aligned}
\sum_{t=1}^T\<x_t,\th_t-\hth_t\>^2 &\le \frac{4}{\ell_M}\max\bigg\{\frac{16}{\ell_M} \log T, \\
&\quad \quad \quad \quad\quad\; \;\frac{2\l1u^2}{\eta_1} + \sum_{t=1}^T \Big(\frac{1}{2\eta_{t+1}} - \frac{1}{2\eta_t} \Big) \|\th_{t+1}-\hth_{t+1}\|^2
+\frac{u_M^2}{2} \sum_{t=1}^T \eta_t + 2\l1u \sum_{t=1}^T \frac{\delta_t }{\eta_t} \bigg\}\end{aligned}$$ The proof is complete.
Proof of Lemma \[lem:PE-random\] {#proof:PE-random}
--------------------------------
\[pro:eigmin\] Let $\sig_t$ denote the minimum eigenvalue of $Q_t \equiv (1/t)\sum_{\ell=1}^t x_\ell x_\ell^\sT$. Further, let $\sigma_{\min}$ be the minimum eigenvalue of $\Sigma$, where $\Sigma$ is the population covariance of feature vectors as in Assumption \[SMF\]. Then, there exist constants $c_1,c_2>0$, such that $$\begin{aligned}
\forall t\ge c_1d:\,\, \prob\Big(\frac{1}{2} \sigma_{\min} \le \sig_t\le \frac{3}{2}\sigma_{\min}\Big) \ge 1- 2e^{-c_2 t/d}\,. \label{eq:prob}\end{aligned}$$ Further, $\sig_t\le 1$, for all $t\ge 1$.
Let $\cF_t$ be the $\sigma$ algebra generated by market shocks $\{z_\ell\}_{\ell=1}^t$ and features $\{x_\ell\}_{\ell=1}^t$. We further define $D_t = \<x_t,\hth_t-\th_t\>^2 - \|\Sigma^{1/2}(\hth_t-\th_t)\|^2$. Note that $\hth_t$ is $\cF_{t-1}$ measurable and $x_t$ is independent of $\cF_{t-1}$, which implies $\E(D_t|\cF_{t-1}) = 0$. Hence, $\E(D_t) = 0$ by iterated law of expectation and therefore $\sum_{t=1}^T \E(D_t) = 0$. Equivalently, $$\begin{aligned}
\label{eq:PredictionB1}
\E\left[\sum_{t=1}^T \<x_t,\hth_t-\th_t\>^2 \right] = \sum_{t=1}^T \E\left[\|\Sigma^{1/2}(\hth_t-\th_t)\|^2 \right] \ge \sigma_{\min}\, \E\left[\sum_{t=1}^T \|\hth_t-\th_t\|^2\right]\end{aligned}$$ Define $\cG_T$ the event that bound holds true. Then, $$\begin{aligned}
\E\left[\sum_{t=1}^T \<x_t,\hth_t-\th_t\>^2 \right] &= \E\left[\sum_{t=1}^T \<x_t,\hth_t-\th_t\>^2 \cdot(\ind_\cG+\ind_{\cG^c}) \right]\nonumber\\
&\le \E\left[\sum_{t=1}^T \<x_t,\hth_t-\th_t\>^2 \cdot\ind_\cG \right] + 4\l1u^2 T\, \prob(\cG^c)\nonumber\\
&\le \E\left[\sum_{t=1}^T \<x_t,\hth_t-\th_t\>^2 \cdot\ind_\cG \right] + \frac{4\l1u^2}{T}\,.
\label{eq:term0}
\end{aligned}$$ Further, using inequality $\max(a,b) \le |a|+|b|$, we get $$\begin{aligned}
\E\left[\sum_{t=1}^T \<x_t,\hth_t-\th_t\>^2 \cdot\ind_\cG \right] \le \frac{4}{\ell_M}\bigg\{&
\frac{16}{\ell_M}\log T + \frac{12\l1u^2}{\ell_M} + \frac{1}{2} \sum_{t=1}^T \E\bigg[\Big((t+1) \lambda_{t+1}-t \lambda_t \Big) \cdot \|\th_{t+1}-\hth_{t+1}\|^2\bigg]\nonumber\\
& + \frac{u_M^2}{2}\sum_{t=1}^T\E\left[\frac{1}{t\lambda_t}\right]+2\l1u \sum_{t=1}^T \E[t\lambda_t] \delta_t\bigg\}\,.\label{eq:PredictionB2}\end{aligned}$$ We next bound the terms on the right-hand side individually. $$\begin{aligned}
\sum_{t=1}^T &\E\bigg[\Big((t+1) \lambda_{t+1}-t \lambda_t \Big) \cdot \|\th_{t+1}-\hth_{t+1}\|^2\bigg]
\le \frac{\ell_M}{6}\sum_{t=1}^T \E\bigg[\sig_{t+1} \cdot \|\th_{t+1}-\hth_{t+1}\|^2\bigg]\nonumber\\
&\le\frac{\ell_M}{6} \sum_{t=1}^T \E\bigg[\sig_{t+1} \|\th_{t+1}-\hth_{t+1}\|^2\, \ind(\sig_{t+1}<3\sigma_{\min}/2)\bigg]+ \frac{\ell_M}{6} \sum_{t=1}^T \E\bigg[\sig_{t+1} \, \|\th_{t+1}-\hth_{t+1}\|^2\, \ind(\sig_{t+1} > 3\sigma_{\min}/2)\bigg] \nonumber\\
&\le \frac{\ell_M}{4}\sigma_{\min}\sum_{t=1}^T \E\Big(\|\th_{t+1}-\hth_{t+1}\|^2\Big) + \sum_{t=1}^T 2\ell_M\l1u^2 e^{-c_2 t/d}\nonumber\\
&\le \frac{\ell_M}{4} \sigma_{\min}\sum_{t=1}^T \E\Big(\|\th_{t+1}-\hth_{t+1}\|^2\Big) + \frac{2\ell_M}{c_2d} \l1u^2 \,,\label{term1}\end{aligned}$$ where in the last inequality, we used $\prob(\sig_{t+1} > 3\sigma_{\min}/2) \le 2e^{-c_2d t}$, $\sigma_t\le 1$ and $\|\hth_t-\th_t\|\le 2\l1u$, according to Proposition \[pro:eigmin\].
The next term on the right-hand side of is bounded in the following proposition.
\[pro:lambda\_reciprocal\] Using rule for $\lambda_t$, we have $$\begin{aligned}
\E\left[\frac{1}{t\lambda_t}\right]\le \frac{6}{\ell_M}\left(\tilde{c} d^2\log T+\frac{4d}{C_{\min}} \log T\right)\,,\label{term2}\end{aligned}$$ where $\tilde{c} = \max(c_1,1/c_2)$ and constants $c_1$ and $c_2$ are defined in Proposition \[pro:eigmin\] .
Finally, for the last term, we note that $Q_t$ is rank deficient for $t\le d$ and hence $\sigma_t = 0$, for $1\le t\le d$. Further, the minimum eigenvalue of a matrix is a concave function over PSD matrices. By Jensen inequality, we have $$\begin{aligned}
\E (\lambda_t) &= \frac{\ell_M}{6t} (1+\sum_{\ell=1}^t \E(\sigma_\ell)) = \frac{\ell_M}{6t} \Big(1+\sum_{\ell=d+1}^t \E(\sigma_\ell)\Big)\nonumber \\
&\le \frac{\ell_M}{6t} \Big(1+\sum_{\ell=d+1}^t \sigma_{\min}\Big) \le \frac{\ell_M}{6t} \Big(1+\frac{t-d}{d}\Big) = \frac{\ell_M}{6d}\,.\end{aligned}$$ In the last inequality, we used the fact that $\Tr(\Sigma) = \E(\|x_t\|^2) = 1$, and thus $\sigma_{\min}\le 1/d$. Hence, $$\begin{aligned}
\sum_{t=1}^T \E[t\lambda_t] \delta_t \le \frac{\ell_M}{6d} \sum_{t=1}^T t\delta_t\,, \label{term3}\end{aligned}$$
Using Equations , , to bound the right-hand side of , we get $$\begin{aligned}
\E\left[\sum_{t=1}^T \<x_t,\hth_t-\th_t\>^2 \cdot\ind_\cG \right] \le&
\left[\frac{64}{\ell_M^2} + \frac{12u_M^2}{\ell_M^2} \left(\tilde{c}+ \frac{4}{C_{\min} d} \right) \right]\cdot d^2 \log T\nonumber\\
& +\frac{48\l1u^2}{\ell_M^2} + \frac{4\l1u^2 }{c_2 d} +\frac{2\l1u}{d} \sum_{t=1}^T t\delta_t + \frac{\sigma_{\min}}{2}\sum_{t=1}^T \E(\|\th_t-\hth_t\|^2)\,. \label{term4}\end{aligned}$$ Combining bounds , and , we obtain $$\begin{aligned}
\frac{\sigma_{\min}}{2}\sum_{t=1}^T \E(\|\th_t-\hth_t\|^2) \le&
\left[\frac{64}{\ell_M^2} + \frac{12u_M^2}{\ell_M^2} \left(\tilde{c}+ \frac{4}{C_{\min} d} \right) \right]\cdot d^2 \log T\\
& + 4\l1u^2 \left(\frac{1}{T} +\frac{12}{\ell_M^2} + \frac{1}{c_2 d} \right) + \frac{2\l1u}{d} \sum_{t=1}^T t\delta_t \,.\end{aligned}$$ The result follows by recalling that $\sigma_{\min}\ge C_{\min}/d$ as stated by Assumption \[SMF\].
Acknowledgements {#acknowledgements .unnumbered}
================
The author was partially supported by a Google Faculty Research Award.
Proof of Lemma \[lem:M\] {#app:M}
========================
We first state some properties of the the virtual valuation function $\varphi$ and the price function $g$, given by Equation .
\[propo:M\] If $1-F$ is log-concave, then the virtual valuation function $\varphi$ is strictly monotone increasing and the price function $g$ satisfies $0<g'(v)<1$, for all values of $v\in \reals$.
We refer to [@JavanmardNazerzadeh] (Lemmas 1 and 2 in Appendix A therein) for a proof of Proposition \[propo:M\].
For $\th\in \Theta$ we have $\|\th\|\le \l1u$ and hence $|\<x_t,\th\>|\le \|x_t\| \|\th\| \le \l1u$ for all $t$. Applying Proposition \[propo:M\] (1-Lipschitz property of $g$), $$p_t = g(\<x_t,\th_t\>) \le g(0) + |\<x_t,\th_t\>| \le \varphi^{-1}(0) + \l1u\,.$$ Therefore, $$\begin{aligned}
\label{eq:utB}
|u_t(\th)|\le |p_t| + |\<x_t,\th\>|\le \varphi^{-1}(0) + 2\l1u\,.\end{aligned}$$
Proof of Lemma \[lem:DUB\] {#app:DUB}
==========================
We note that the update rule can be recast as $\hth_{t+1} = \arg\min_{\th\in \Theta} \cC_t(\th)$, where $$\cC_t(\th) = \eta_t\<\nabla \ell_t(\hth_t),\th\>+\frac{1}{2}\|\th-\hth_t\|^2\,.$$ By convexity of $\cC_t$ and optimality of $\hth_{t+1}$, we have $\<\th-\hth_{t+1},\nabla \cC_t(\hth_{t+1})\>\ge 0$ for all $\th\in \Theta$. Setting $\th=\th_t$, $$\begin{aligned}
\<\th_t-\hth_{t+1}, \eta_t \nabla \ell_t(\hth_t) + \hth_{t+1}-\hth_{t}\> \ge 0\,.\label{eq:E1}\end{aligned}$$ Expanding $\ell_t(\th)$ around $\hth_t$, we have $$\begin{aligned}
\label{eq:E2}
\ell_t(\hth_t)-\ell(\th_t) = \<\nabla\ell_t(\hth_t), \hth_t-\th_t\> -\frac{1}{2}\<\th_t-\hth_t, \nabla^2\ell_t(\tth) (\th_t-\hth_t)\> \,,
$$ for some $\tilth$ on the line segment between $\tth_t$ and $\hth_t$. Recalling , the gradient and the hessian of $\ell_t$ read as $$\begin{aligned}
\label{eq:nabla-nabla2}
\nabla \ell_t(\th) = \mu_t(\th) x_t\,, \quad \nabla^2 \ell_t(\th) = \eta_t(\th) x_tx_t^\sT\,,\end{aligned}$$ with, $$\begin{aligned}
\mu_t(\th) &=& -\frac{{f}(u_t(\th))}{F(u_t(\th))}\ind(y_t = -1) + \frac{{f}(u_t(\th))}{1-F(u_t(\th))} \ind(y_t = +1) \nonumber\\
&=& -\dx \log F(u_t(\th)) \ind(y_t = -1) - \dx \log(1-F(u_t(\th))) \ind(y_t = +1)\label{eq:mu}\end{aligned}$$ $$\begin{aligned}
\eta_t(\th) &=& \bigg(\frac{f(u_t(\th))^2}{F(u_t(\th))^2} - \frac{{f'}(u_t(\th))}{F(u_t(\th))} \bigg)\ind(y_t = -1) +\bigg(\frac{f(u_t(\th))^2}{(1-F(u_t(\th)))^2}+ \frac{{f'}(u_t(\th))}{1-F(u_t(\th))} \bigg)\ind(y_t = +1)\nonumber\\
&=& -\ddx \log F(u_t(\th)) \ind(y_t = -1) - \ddx \log (1-F(u_t(\th))) \ind(y_t = +1)\,. \label{eq:eta}\end{aligned}$$ Here, $u_t(\th) = p_t-\<x_t,\th\>$, and $\dx \log F(x)$ and $\ddx \log F(x)$ represent first and second derivative w.r.t $x$, respectively. In addition, using Equation $$\begin{aligned}
\label{eq:utB}
|u_t(\th)| \le \varphi^{-1}(0) + 2\l1u = M\,, \quad \forall \th\in \Theta\,. \end{aligned}$$ Hence, invoking the definition of $\ell_M$, as per Equation , we get that $\eta_t(\th) \ge \ell_M$ and hence $\nabla^2\ell_t(\tilth) \succeq \ell_M x_t x_t^\sT$.
Continuing from Equation , we get $$\begin{aligned}
\label{eq:E3}
\ell_t(\hth_t)-\ell(\th_t) &\le \<\nabla\ell_t(\hth_t), \hth_t-\th_t\> -\frac{\ell_M}{2}\<x_t,\th_t-\hth_t\>^2 \nonumber\\
&= \<\nabla\ell_t(\hth_t), \hth_{t+1}-\th_t\> + \<\nabla\ell_t(\hth_t), \hth_t-\hth_{t+1}\> -\frac{\ell_M}{2}\<x_t,\th_t-\hth_t\>^2\nonumber\\
&\le \frac{1}{\eta_t} \<\th_t-\hth_{t+1},\hth_{t+1}-\hth_t\>+ \<\nabla\ell_t(\hth_t), \hth_t-\hth_{t+1}\> -\frac{\ell_M}{2}\<x_t,\th_t-\hth_t\>^2\nonumber\\
&= \frac{1}{2\eta_t} \Big\{\|\th_t-\hth_t\|^2-\|\th_t-\hth_{t+1}\|^2-\|\hth_{t+1} - \hth_t\|^2 \Big\} \nonumber\\
&\quad \,\,+ \<\nabla\ell_t(\hth_t), \hth_t-\hth_{t+1}\> -\frac{\ell_M}{2}\<x_t,\th_t-\hth_t\>^2\nonumber\\
&=\frac{1}{2\eta_t}\Big\{\|\th_t-\hth_t\|^2-\|\th_{t+1}-\hth_{t+1}\|^2\Big\} + \frac{1}{2\eta_t}\Big\{\|\th_{t+1}-\hth_{t+1}\|^2 - \|\th_t-\hth_{t+1}\|^2 \Big\} \nonumber\\
&\quad\,\, -\frac{1}{2\eta_t} \|\hth_{t+1} - \hth_t\|^2 + \<\nabla\ell_t(\hth_t), \hth_t-\hth_{t+1}\> -\frac{\ell_M}{2}\<x_t,\th_t-\hth_t\>^2\end{aligned}$$ We next note that the second term above can be bounded as $$\begin{aligned}
\label{eq:E4}
\frac{1}{2\eta_t} \Big\{\|\th_{t+1}-\hth_{t+1}\|^2 - \|\th_t-\hth_{t+1}\|^2 \Big\} = \frac{1}{\eta_t}\<\th_{t+1}-\hth_{t+1}, \th_{t+1}-\th_t\> \le \frac{2}{\eta_t} \l1u \delta_t\,,\end{aligned}$$ because $\th_{t+1}, \hth_{t+1}\in \Theta$ and hence $\|\th_{t+1}-\hth_{t+1}\|\le 2\l1u$ by triangle inequality.
Further, $$\begin{aligned}
\label{eq:E5}
\<\nabla\ell_t(\hth_t), \hth_t-\hth_{t+1}\> &\le \frac{1}{2\eta_t} \|\hth_{t+1} - \hth_t\|^2 + \frac{\eta_t}{2}\|\nabla \ell_t(\hth_t)\|^2 \nonumber\\
&\le \frac{1}{2\eta_t} \|\hth_{t+1} - \hth_t\|^2 + \frac{\eta_t}{2} |\mu(\hth_t)|^2 \|x_t\|^2 \le \frac{1}{2\eta_t} \|\hth_{t+1} - \hth_t\|^2 + \frac{\eta_t}{2} u_M^2\,,\end{aligned}$$ where we used the inequality $2ab\le a^2+b^2$ and the characterization of gradient . Note that by , $|u_t(\hth)|\le M$ and by definition , $|\mu_t(\hth_t)|\le u_M$. Plugging in bounds from and in Equation , we arrive at $$\begin{aligned}
\ell_t(\hth_t)-\ell(\th_t) \le \frac{1}{2\eta_t}\Big\{\|\th_t-\hth_t\|^2-\|\th_{t+1}-\hth_{t+1}\|^2\Big\} + \frac{2}{\eta_t} \l1u \delta_t + \frac{\eta_t}{2}u_M^2 -\frac{\ell_M}{2}\<x_t,\th_t-\hth_t\>^2\end{aligned}$$ We use the shorthand $D_t = \frac{1}{2} \|\th_t-\hth_t\|^2$. The result follows by summing the above bound over time: $$\begin{aligned}
\sum_{t=1}^T \ell_t(\hth_t) - \sum_{t=1}^T \ell_t(\th_t) =& \sum_{t=1}^T \Big(\frac{D_t}{\eta_t} - \frac{D_{t+1}}{\eta_{t+1}} \Big) + \sum_{t=1}^T D_{t+1}\Big(\frac{1}{\eta_{t+1}}-\frac{1}{\eta_t} \Big) \\
&+\frac{u_M^2}{2} \sum_{t=1}^T \eta_t + 2\l1u \sum_{t=1}^T \frac{\delta_t }{\eta_t} -\frac{\ell_M}{2}\sum_{t=1}^T\<x_t,\th_t-\hth_t\>^2\,.\end{aligned}$$ The proof is concluded because $D_1\le 2\l1u^2$ as $\hth_1,\th_1\in \Theta$; therefore $$\begin{aligned}
\sum_{t=1}^T \Big(\frac{D_t}{\eta_t} - \frac{D_{t+1}}{\eta_{t+1}} \Big) = \frac{D_1}{\eta_1}-\frac{D_{T+1}}{\eta_{T+1}} \le \frac{D_1}{\eta_1} \le \frac{2\l1u^2}{\eta_1}\,.\end{aligned}$$
Proof of Lemma \[lem:DLB\] {#app:DLB}
==========================
By convexity of $\ell_t(\th)$, we have $$\begin{aligned}
\label{Eq6}
\ell_t(\th_t) - \ell_t(\hth_t) \le \<\nabla \ell_t(\th_t),\hth_t-\th_t\> = \mu_t(\th_t) \<x_t,\th_t-\hth_t\>\,.\end{aligned}$$ We denote $D_t = \mu_t(\th_t) \<x_t,\th_t-\hth_t\>$ and let $\cF_t$ be the $\sigma$-algebra generated by $\{z_t\}_{t=1}^T$. Since $\hth_t$ is $\cF_{t-1}$ measurable, we have $$\begin{aligned}
\E(D_t|\cF_{t-1}) = \E(\mu_t(\th_t)|\cF_{t-1}) \<x_t,\th_t-\hth_t\> = 0\,,\end{aligned}$$ where $\E(\mu_t(\th_t)|\cF_{t-1}) = 0$ follows readily from Equation . Therefore, $D(T) \equiv \sum_{t=1}^T D_t$ is a martingale adapted to the filtration $\cF_{t}$.
We next bound $\E[e^{\lambda D_t}|\cF_{t-1}]$ for any $\lambda\in \reals$. Conditional on $\cF_{t-1}$, we have $|D_t|\le \beta_t$, with $\beta_t \equiv u_M|\<x_t,\th_t-\hth_t\>|$. Since $e^{\lambda z}$ is convex, $$\begin{aligned}
\E[e^{\lambda D_t}|\cF_{t-1}] &\le \E\bigg[\frac{\beta_t-D_t}{2\beta_t} e^{-\lambda \beta_t} +\frac{\beta_t+D_t}{2\beta_t} e^{\lambda \beta_t}\bigg| \cF_{t-1} \bigg]\nonumber\\
&=\E\bigg[\frac{e^{-\lambda \beta_t}+e^{\lambda \beta_t}}{2} \bigg] + \E[D_t|\cF_{t-1}] \bigg(\frac{e^{-\lambda \beta_t}+e^{\lambda \beta_t}}{2\beta_t} \bigg) = \cosh(\lambda \beta_t) \le e^{\lambda^2 \beta_t^2/2}\,.\end{aligned}$$ We are now ready to apply the following Bernstein-type concentration bound for martingale difference sequences, whose proof is given in Appendix \[app:MD\] for the reader’s convenience.
\[propo:MD\] Consider a martingale difference sequence $D_t$ adapted to a filtration $\cF_t$, such that for any $\lambda\ge 0$, $\E[e^{\lambda D_t}|\cF_{t-1}] \le e^{\lambda^2\sigma_t^2/2}$. Then, for $D(T) = \sum_{t=1}^T D_t$, the following holds true: $$\begin{aligned}
\prob(D(T)\ge \xi) \le e^{-{\xi^2}/({2\sum_{t=1}^T \sigma_t^2})}\,.\end{aligned}$$
Combining Equation and the result of Proposition \[propo:MD\] we obtain $$\begin{aligned}
\prob\bigg(\sum_{t=1}^T \ell_t(\hth_t) - \sum_{t=1}^T\ell_t(\th_t) \le- 2\sqrt{\log T} \Big\{\sum_{t=1}^T \<x_t,\th_t-\hth_t\>^2\Big\}^{1/2} \bigg) \le \frac{1}{T^2}\,.\end{aligned}$$ The result follows.
Proof of Proposition \[propo:MD\] {#app:MD}
=================================
We follow the standard approach of controlling the moment generating function of $D(T)$.Conditioning on $\cF_{t-1}$ and applying iterated expectation yields $$\begin{aligned}
\E[e^{\lambda D(T)} ] = \E\bigg[e^{\lambda \sum_{t=1}^{T-1} D_t} \cdot \E[e^{\lambda D_T}|\cF_{T-1}] \bigg] \le \E\bigg[e^{\lambda \sum_{t=1}^{T-1} D_t} \bigg] e^{\lambda^2 \sigma_T^2/2}\,.\end{aligned}$$ Iterating this procedure gives the bound $\E[e^{\lambda \sum_{t=1}^T D_t}]\le e^{\lambda^2 \sum_{t=1}^T \sigma_t^2/2}$, for all $\lambda \ge 0$.
Now by applying the exponential Markov inequality we get $$\begin{aligned}
\prob(D(T)\ge \xi) = \prob(e^{\lambda D(T)} \ge e^{\lambda \xi})\le e^{-\lambda \xi} \E[e^{\lambda \sum_{t=1}^T D_t}] \le e^{-\lambda \xi} e^{\lambda^2 (\sum_{t=1}^T \sigma_t^2)/2}\,.\end{aligned}$$ Choosing $\lambda = \xi/(\sum_{t=1}^T \sigma_t^2)$ gives the desired result.
Proof of Proposition \[pro:eigmin\]
===================================
We prove the result in a more general case, namely when the features are independent random vectors with bounded subgaussian norms.
For a random variable $z$, its subgaussian norm, denoted by $\|z\|_{\psi_2}$ is defined as $$\begin{aligned}
\|z\|_{\psi_2} = \sup_{p\ge 1} \;p^{-1/2} (\E|z|^p)^{1/p}\,.\end{aligned}$$ Further, for a random vector $z$ its subgaussian norm is defined as $$\begin{aligned}
\|z\|_{\psi_2} = \sup_{\|u\|\ge 1} \; \|\<z,u\>\|_{\psi_2}\,.\end{aligned}$$
We next recall the following result from [@vershynin] about random matrices with independent rows.
\[versh\] Suppose $x_\ell\in \reals^d$ are independent random vectors generated from a distribution with covariance $\Sigma$ and their subgaussian norms are bounded by $K$. Further, let $Q_t = (1/t) \sum_{\ell=1}^t x_\ell x_\ell^\sT$. Then for every $s\ge 0$, the following inequality holds with probability at least $1-2\exp(-cs^2)$: $$\begin{aligned}
\Big\|Q_t-\Sigma \Big\| \le \max(\delta,\delta^2) \quad \quad \quad \text{ where }\delta = C\sqrt{\frac{d}{t}} + \frac{s}{\sqrt{t}}\,. \end{aligned}$$ Here $C$ and $c>0$ are constants that depend solely on $K$.
We next show that the feature vectors in our problem have bounded subgaussian norm. Given that $\|x_\ell\|\le 1$, for $\|u\|\le 1$, we have $$\begin{aligned}
\|\<x_\ell,u\>\|_{\psi_2} =\sup_{p\ge 1}\; p^{-1/2} (\E|\<x_\ell,u\>|^p)^{1/p}\le \sup_{p\ge 1}\; p^{-1/2} (\E[\|x_\ell\|\|u\|]^p)^{1/p}\le 1\,.\end{aligned}$$ Applying Proposition with $K=1$, there exist constants $c_1,c_2$ (depending on $C_{\min}$), such that for $t\ge c_1d^2$, we have $$\begin{aligned}
\label{pert}
\|Q_t -\Sigma\| \le \frac{1}{2d} C_{\min} \le \frac{1}{2} \sigma_{\min}\,,\end{aligned}$$ with probability at least $1-2e^{-c_2 t/d}$. Weyl’s inequality then implies that $|\sig_t-\sigma_{\min}|\le \sigma_{\min}/2$. Also note that for $t\ge 1$, $$\sigma_t\le \|Q_t\| \le \frac{1}{t}\sum_{\ell=1}^t \|x_\ell x_\ell^\sT\| = \frac{1}{t}\sum_{\ell=1}^t \|x_\ell\|^2 = 1\,.$$ The proof is complete.
Proof of Lemma \[pro:lambda\_reciprocal\] {#app:lambda_reciprocal}
=========================================
The way we set $\lambda_t$ (see Equation ), we have $$\frac{1}{t\lambda_t} = \left(\frac{6}{\ell_M}\right) \frac{1}{1+\sig_1+\sig_2+\dotsc+\sig_t}$$
Clearly, for $t\ge 1$, $1/(t\lambda_t)\le 6/\ell_M$. Let $t_0 = \tilde{c} d^2 \log T$, with $\tilde{c}= \max(c_1,1/c_2)$. For $T\ge t_0$, define the event $\cE_T$ as follows $$\begin{aligned}
\cE_T = \{\sigma_t \ge \sigma_{\min}/2, \, \text{for } t_0\le t\le T\}\,.\end{aligned}$$ By applying Proposition \[pro:eigmin\] and union bounding over $t$, we get $$\begin{aligned}
\prob(\cE_T) \ge 1 - \sum_{t = t_0}^T 2e^{-c_2 t/d} \ge 1 - \frac{2d}{c_2} e^{-c_2t_0/d} \end{aligned}$$ Therefore, $$\begin{aligned}
\sum_{t=t_0}^T \E\left[\frac{1}{t\lambda}\right] &\le \E\left[\left(\sum_{t=t_0}^T\frac{1}{t\lambda}\right) \ind(\cE_T)\right] + \frac{6T}{\ell_M} \prob(\cE_T^c)\nonumber\\
& = \frac{6}{\ell_M}\E\left[\left(\sum_{t=t_0}^T\frac{1}{1+\sigma_1+\dotsc+\sigma_t}\right)\cdot \ind(\cE_T)\right] + \frac{6T}{\ell_M} \prob(\cE_T^c)\nonumber\\
&\le \frac{6}{\ell_M}\left(\sum_{t=1}^T \frac{1}{1+\frac{t}{2}\sigma_{\min}} +\frac{2d}{c_2} T^{1-c_2\tilde{c}d}\right)\nonumber\\
&\le \frac{12}{\ell_M} \left(\frac{1}{\sigma_{\min}} \log T + \frac{d}{c_2} T^{1-d} \right) \le \frac{24d}{\ell_M C_{\min}} \log T\,.\end{aligned}$$ For $t\ge 1$, we use the bound $1/(t\lambda_t)\le 6/\ell_M$. Hence, $$\begin{aligned}
\sum_{t=1}^T \E\left[\frac{1}{t\lambda}\right] \le \frac{6}{\ell_M}\left(t_0+\frac{4d}{C_{\min}} \log T\right) \le \frac{6}{\ell_M}\left(\tilde{c} d^2\log T+\frac{4d}{C_{\min}} \log T\right)\end{aligned}$$
The proof is complete.
|
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abstract: 'Wireless network topologies change over time and maintaining routes requires frequent updates. Updates are costly in terms of consuming throughput available for data transmission, which is precious in wireless networks. In this paper, we ask whether there exist low-overhead schemes that produce low-stretch routes. This is studied by using the underlying geometric properties of the connectivity graph in wireless networks.'
author:
-
bibliography:
- 'doubling.bib'
title: Hierarchical Routing over Dynamic Wireless Networks
---
distributed routing algorithms; wireless networks; geometric random graphs; competitive analysis; mobility
|
---
abstract: 'In the discretization of differential problems on complex geometrical domains, discretization methods based on polygonal and polyhedral elements are powerful tools. Adaptive mesh refinement for such kind of problems is very useful as well and states new issues, here tackled, concerning good quality mesh elements and reliability of the simulations. In this paper we numerically investigate optimality with respect to the number of degrees of freedom of the numerical solutions obtained by the different refinement strategies proposed. A geometrically complex geophysical problem is used as test problem for several general purpose and problem dependent refinement strategies.'
address:
- |
Dipartimento di Scienze Matematiche, Politecnico di Torino\
Corso Duca degli Abruzzi 24, Torino, 10129, Italy
- Member of the INdAM research group GNCS
author:
- Stefano Berrone
- Andrea Borio
- 'Alessandro D’Auria'
bibliography:
- 'scico2mp.bib'
- 'dfn.bib'
- 'articolo.bib'
- 'VEMbibl.bib'
title: 'Refinement strategies for polygonal meshes applied to adaptive VEM discretization.'
---
=1
Mesh adaptivity ,Polygonal mesh refinement ,Virtual Element Method ,Discrete Fracture Network flow simulations ,Simulations in complex geometries ,A posteriori error estimates
65N30 ,65N50 ,68U20 ,86-08 ,86A05
|
---
abstract: |
The boundary problem for graphs like Pascal’s but with general multiplicities of edges is related to a ‘backward’ problem of moments of the Hausdorff type.\
**Keywords.** Boundary problem, ‘small’ moment problem, Markov chains, asymptotics of combinatorial numbers.
author:
- 'Alexander Gnedin[^1] and Jim Pitman[^2]'
date: '\'
title: Moment problems and boundaries of number triangles
---
The extreme boundary
====================
Let $T_n:=\{(n,0),(n,1),\ldots,(n,n)\}$ and $T:=\cup_{n=0}^\infty T_{n}$. We endow $T$ with the structure of a directed graph in which every node $(n,k)$ has two outgoing edges $(n,k)\to (n+1,k)$ and $(n,k)\to (n+1,k+1)$ with [*multiplicities*]{} $\ell_{nk}$ and $r_{nk}$ (respectively), where $\{\ell_{nk};~ (n,k)\in T\}$ and $\{r_{nk};~ (n,k)\in T\}$ are given triangular arrays with (strictly) positive entries. A classical example is the Pascal graph with unit multiplicities $\ell_{nk}=r_{nk}=1$.
Let $\cal V$ be the set of nonnegative solutions $V=\{V_{nk};~ (n,k)\in T\}$ to the backward recursion $$\label{Vrec}
V_{nk}=\ell_{nk} V_{n+1,k}+ r_{nk} V_{n+1,k},~~~~~~~(n,k)\in T$$ with normalisation $V_{00}=1$. The set $\cal V$ is convex and compact in the product topology of functions on $T$. By some general theory in Dynkin (1978) the [*extreme boundary*]{} ${\rm ext} {\cal V}$, comprised of indecomposable elements of $\cal V$, is a Borel set. Moreover, $\cal V$ is a Choquet simplex, meaning that each $V\in {\cal V}$ has a unique representation as convex combination $$\label{VU}
V=\int_{{\rm ext}{\cal V}} U\,\mu({\rm d}U)$$ with some probability measure $\mu$ supported by ${\rm ext} {\cal V}$. The [*boundary problem*]{} for the graph $T$ is to find some explicit description of the set of extremes, meaning, if possible, a simple parametrisation of ${\rm ext} {\cal V}$ along with the kernel that is implicit in (\[VU\]).
The recursion (\[Vrec\]) for the Pascal graph appeared in the work of Hausdorff on summation methods (1921, p. 78) and the ‘small’ problem of moments on $[0,1]$. In this case the bivariate array $V$ is completely determined by $V_{\bullet,0}$ according to the rule $V_{\bullet+k,k}=\nabla^k (V_{\bullet,0})$, where $\bullet$ stands for the variable $n$, and $\nabla^k$ is the $k$th iterate of the difference operator $\nabla (U_\bullet):=U_\bullet-U_{\bullet+1}$. The condition $V\geq 0$ means that $V_{\bullet, 0}$ is completely monotone, hence by Hausdorff’s theorem $V_{\bullet, 0}$ is a sequence of moments $$V_{n0}=\int_{[0,1]} x^n \mu({\rm d}x)$$ of some probability measure $\mu$. That is to say, the set of extremes ${\rm ext}{\cal V}$ can be identified with the unit interval, and the extremes have the form $V_{nk}(x)=x^{n-k}(1-x)^k$ for $x\in [0,1]$; in particular, $V_{n0}=x^n$.
For general multiplicities the recursion (\[Vrec\]) is equivalent to $$V_{\bullet, k}=\nabla_k(\cdots(\nabla_1(V_{\bullet,0}))\cdots),$$ where $\nabla_k(U_{\bullet})=(U_{\bullet}-\ell_{\bullet,k}U_{\bullet+1})/r_{\bullet, k}$ is a generalised difference operator. By analogy with Hausdorff’s criterion, the question about positivity of the generalised iterated differences of $V_{\bullet,0}$ may be regarded as a ‘backward’ problem of moments. A direct problem of moments of the Hausdorff type appears when we determine the $V_{n0}$’s for extreme solutions as functions on the boundary, and consider the integral representation of the generic $V_{n0}$ in the form (\[VU\]).
A bivariate array $V\in {\cal V}$ could be also computed by suitable differencing the diagonal sequence $(V_{nn})$, but this leads to the same type of the moment problem by virtue of the transposition of $T$ which exchanges the multiplicities $\ell_{nk}$’s with $r_{n,n-k}$’s.
A special feature of $T$, as compared with more complicated graphs like Young’s lattice (see Kerov (2003), Borodin and Olshanski (2000)), is a natural total order on the extreme boundary. In this note we extend the argument of Gnedin and Pitman (2006) to show that the total order allows ${\rm ext}{\cal V}$ to be embedded into $[0,1]$. We shall also survey the connection of the boundary problem with asymptotic properties of some classical arrays of combinatorial numbers.
Markov chain approach
=====================
The [*weight*]{} of a path in $T$ joining the root $(0,0)$ and some other node $(n,k)$ is defined as the product of multiplicities of edges along the path (for instance the weight of $(0,0)\to(1,0)\to(2,1)$ is $\ell_{00}r_{10}$). The [*dimension*]{} $D_{nk}$ of $(n,k)\in T$ is defined to be the sum of weights of all paths from $(0,0)$ to $(n,k)$. The dimensions are computable from the forward recursion $$\label{Drec}
D_{nk}=r_{n-1,k-1} D_{n-1,k-1}+ \ell_{n-1,k} D_{n-1,k},~~~~~~~(n,k)\in T,$$ (where the first term in the right-hand side is absent for $k=0$ and the second term is absent for $k=n$), with the initial condition $D_{00}=1$. The [*number triangle*]{} associated with $T$ is the array $\{D_{nk};~ (n,k)\in T\}$.
Each $V\in {\cal V}$ determines the law ${\mathbb P}_V$ of a inhomogeneous Markov chain $K_\bullet$ whose backward transition probabilities for $0\leq k\leq n,~n>0$ are $$\label{backtr}
{\mathbb P}_V(K_{n-1}=j\,|\, K_n=k)= {D_{n-1,j}\over D_{nk}}(\ell_{n-1,j}\delta_{jk}+r_{n-1,j}\delta_{j,k-1}),$$ and whose distribution at time $n$ is ${\mathbb P}_V(K_n=k)=D_{nk} V_{nk}.$ It is important that the probabilities (\[backtr\]) are determined solely by the multiplicities of edges and do not depend on $V$. Hence $\cal V$ is in essence a class of distributions for Markov chains on $T$ with given backward transition probabilities.
For each fixed integer $\nu$ and $0\leq \varkappa\leq \nu$ let $V^{\nu\varkappa}$ be the function on $T$ which satisfies the recursion (\[Vrec\]) for $n< \nu$, satisfies $V^{\nu\varkappa}_{\nu k}=\delta_{k\varkappa}$, and equals $0$ on $\cup_{n>\nu}T_n$. Such $V^{\nu \varkappa}$ determines the probability law of a finite Markov chain $(K_0,\ldots,K_\nu)$ conditioned on $K_\nu=\varkappa$.
We define the [*sequential boundary*]{} $\partial_\uparrow{\cal V}$ to be the set of elements of ${\cal V}$ representable as limits $V=\lim_{\nu\to\infty} V^{\nu,\varkappa(\nu)}$ taken along infinite paths $\{\varkappa(\nu);~\nu=0,1,\ldots\}$ in $T$. The sequential boundary $\partial_\uparrow{\cal V}$ may be smaller than the set of all accumulation points for $\{V^{\nu\varkappa}; (\nu,\varkappa)\in T\}$ (the [*Martin boundary*]{}), but it is large enough to cover ${\rm ext}{\cal V}$, as is seen from the following lemma, which is a variation on the theme of sufficiency (see Diaconis and Freedman (1984)).
If $V\in {\rm ext}{\cal V}$ then the random functions $$\label{defVX}
V^{\nu,K_\nu}:=\sum_{\varkappa=0}^{\nu} 1_{\{K_\nu= \varkappa\}} V^{\nu\varkappa}$$ satisfy $V^{\nu,K_\nu}\to V$, as $\nu\to\infty$, ${\mathbb P}_V$-almost surely.
Let ${\cal F}_\nu$ be the sigma-algebra generated by $K_{\nu}, K_{\nu+1}, \ldots$, and ${\cal F}_\infty=\cap {\cal F}_\nu$. Let ${\mathbb P}_V$ correspond to some extreme $V$. Choose any $(n,k)$ and consider random variables $$V^{\nu,K_\nu}_{nk}={\mathbb P}_V(K_n=(n,k)\,|\, K_\nu)/D_{nk}={\mathbb P}_V(K_n=(n,k)\,|\, {\cal F}_\nu)/D_{nk},~~~\nu\geq n$$ where the first equality follows from the definition (\[defVX\]), and the second equality is a consequence of the Markov property. Applying Doob’s reversed martingale convergence theorem to the conditional expectations given ${\cal F}_1\supset {\cal F}_2\supset \ldots$ we obtain $$V^{\nu, K_\nu}_{nk}\to {\mathbb P}_V(K_n=(n,k)\,|\, {\cal F}_\infty)/D_{nk}~~~~~{\mathbb P}_V{\rm-a.s.}$$ The assumption $V\in{\rm ext} {\cal V}$ implies that $\cal F_\infty$ is trivial, hence $${\mathbb P}_V(K_n=(n,k)\,|\, {\cal F}_\infty)={\mathbb P}_V(K_n=(n,k))=V_{nk}D_{nk}.$$
0.3cm
Thus ${\rm ext}{\cal V}\subset\partial_\uparrow{\cal V}$ (in general the inclusion is strict). To state this conclusion in analytical terms, define the weight of a path in $T$ connecting two nodes $(n,k)$ and $(\nu,\varkappa)$ as the product of multiplicities along the path, and define the [*extended dimension*]{} $D^{\nu\varkappa}_{nk}$ as the sum of weights over all such paths (so that $D^{\nu\varkappa}_{00}=D_{\nu\varkappa}$). We then have a fundamental relation $$\label{reldim}
V_{nk}^{\nu\varkappa}={D_{nk}^{\nu\varkappa}\over D^{\nu\varkappa}},$$ which connects the boundary problem with asymptotic properties of $T$. Specifically, the convergence of $V^{\nu,\varkappa(\nu)}$ amounts to the convergence of these ratios for all $(n,k)\in T$ along the path (in fact, it is enough to focus on $V_{\bullet,0}$).
Order
=====
A special feature of $T$ which yields the order is that the only possible increments of the variable $k$ along any path are $0$ and $1$. The next lemma appeared in Gnedin and Pitman (2006) with a different proof.
\[L2\] For $\nu> n$ fixed, $V_{n0}^{\nu\varkappa}$ is nonincreasing in $\varkappa$.
Choose $0\leq\varkappa<\varkappa'\leq \nu$ and consider two Markov chains $K_\bullet, K_\bullet'$ which run in reverse time $n=\nu,\nu-1,\ldots,0$ according to (\[backtr\]) and start with $K_\nu=\varkappa, K_\nu'=\varkappa'$. Suppose the chain $K_\bullet'$ jumps independently of $K_\bullet$ as long as they are in distinct states, and suppose that $K_\bullet'$ is coupled with $K_\bullet$ at some random time $0\leq \tau<\nu$ when the states become the same. In the reverse time, only transitions $k\to k,~ k\to k-1$ for $k>0$ and $0\to 0$ are possible, hence we always have $K_n'\geq K_n$. Therefore the event $K'_n=0$ occurs exactly when $K_n=0$ and $\tau\geq n$, which implies $${\mathbb P}(K_n=0\,|\, K_\nu=\varkappa)\geq {\mathbb P}(K_n=0\,|\, K_\nu=\varkappa').$$
0.3cm
A minor modification of the above argument shows that if $K_\nu$ under ${\mathbb P}_V$ is strictly stochastically smaller than $K_\nu$ under some other ${\mathbb P}_{V'}$, then the same relation holds true for every $n\leq \nu$.
We focus now on $V_{10}$. Suppose $V\in \partial_\uparrow{\cal V}$ is induced, via (\[reldim\]), by some infinite path $\{\varkappa(\nu);~\nu=0,1,\ldots\}$, and $V'\in \partial_\uparrow{\cal V}$ is induced by some other path $\{\varkappa'(\nu);~\nu=0,1,\ldots\}$. If $\varkappa(\nu)=\varkappa'(\nu)$ for infinitely many $\nu$ then, of course, $V=V'$. If $\varkappa(\nu)<\varkappa'(\nu)$ for infinitely many $\nu$ and $\varkappa(\nu)>\varkappa'(\nu)$ for infinitely many $\nu$ then by Lemma \[L2\] we have $V_{\bullet, 0}=V_{\bullet, 0}'$ and $V=V'$. Thus $V\neq V'$ can only occur if the same strict inequality holds for all sufficiently large $\nu$. To be definite, let $\varkappa(\nu)< \varkappa'(\nu)$ for all large enough $\nu$, but then $V\neq V'$ implies that $K_n$ under ${\mathbb P}_V$ is strictly stochastically smaller than $K_n$ under ${\mathbb P}_{V'}$ for all $n>0$, in particular this holds for $n=1$ which means that $V_{10}>V_{10}'$. We see that for $V,V'\in \partial_\uparrow {\cal V}$, the inequality $V_{10}>V_{10}'$ holds if and only if $K_n$ under ${\mathbb P}_V$ is strictly stochastically smaller than $K_n$ under ${\mathbb P}_{V'}$ for all $n>0$. This defines a strict order $\lhd$ on $\partial_\uparrow {\cal V}$.
The sequential boundary $\partial_\uparrow {\cal V}$ is compact.
Suppose $V^j\in \partial_\uparrow {\cal V}$ $(j=1,2,\ldots)$ is a sequence converging to some $V\in{\cal V}$. We know that $\cal V$ is a metrisable compactum with some distance function $\rm dist$. Passing to a subsequence we can restrict consideration to the case of increasing or decreasing sequence, so to be definite assume that $V^{j+1}\lhd V^j$ for $j=1,2,\ldots$ Choosing some path $\{\varkappa^j(\nu);~ \nu=0,1,\ldots\}$ which induces $V^j$, the ordering implies that $\varkappa^j(\nu)\to\infty$ as $\nu\to\infty$ and $\varkappa^j(\nu)<\varkappa^{j+1}(\nu)$ for all large enough $\nu$. As $\nu$ varies, define inductively in $j$ a function $\varkappa(\nu)$ which coincides for some $\nu$ with $\varkappa^j(\nu)$. Specifically, $\varkappa(\nu)=\varkappa^j(\nu)$ until $\varkappa^{j+1}(\nu)<\varkappa^j(\nu)$ starts to hold along with ${\rm dist}(V^{\nu,\varkappa_j},V)<1/j$ and ${\rm dist}(V^{\nu,\varkappa_{j+1}},V)<1/j$, then let $\varkappa(\nu)$ decrement by $1$ until it becomes equal to $\varkappa^{j+1}(\nu)$. This defines an infinite path in $T$, for which one can use monotonicity to show that $V^{\nu,\varkappa(\nu)}\to V$.
Recalling that $\ell_{00}V_{10}+r_{00}V_{11}=1$ we obtain:
The function $V\mapsto \ell_{00}V_{10}$ is an ordered homeomorphism of the sequential boundary $\partial_{\uparrow}{\cal V}$ with order $\lhd$ into $[0,1]$ with order $>$.
Two extreme cases $\ell_{00}V_{01}=0$ and $\ell_{00}V_{01}=1$ correspond to trivial Markov chains $K_\bullet=0$ and $K_\bullet=\bullet$, respectively.
Discrete or continuous?
=======================
In the situation covered by the following lemma, setting $\varkappa(\nu)=m$ (for large $\nu$) for $m=1,2,\ldots$ is the only way to induce nontrivial limits. Then ${\rm ext} {\cal V}$ is discrete and coincides with the sequential boundary.
[(Gnedin and Pitman (2006))]{}\[disc\] Suppose for $m=0,1,\ldots$ there are solutions $V(m) \in {\cal V}$ such that $V_{nm} (m)\,D_{nm}\to 1$ as $n\to\infty$, then each $V(m)$ is extreme and satisfies $K_n\to m$ ${\mathbb P}_{V(m)}$-[a.s.]{}. If also $V_{10}(m)\to 0$ as $m\to\infty$ then $V(m)$ converges to the trivial solution $V(\infty)$ with $K_\bullet=\bullet$ ${\mathbb P}_{V(\infty})$-[a.s.]{}, and in this case ${\rm ext}{\cal V}=\partial_\uparrow{\cal V}=\{V(0), V(1),\ldots, V(\infty)\}$.
In some cases the limits can be obtained by setting $\varkappa(\nu)\sim s\, c(\nu)$ with suitable scaling $c(\nu)\to\infty$ and $s\geq 0$. Under conditions in the next lemma, ${\rm ext}{\cal V}$ coincides with $\partial_\uparrow{\cal V}$ and is homeomorphic to $[0,1]$. The scaling determines the order of growth of $K_\bullet$ under ${\mathbb P}_V$’s.
\[cont\][(Gnedin and Pitman (2006))]{} Suppose there is a sequence of positive constants $\{c(\nu); \nu=0,1,\ldots\}$ with $c(\nu)\to\infty$, and for each $s\in [0,\infty]$ there is a solution $V(s)\in {\cal V}$ which satisfies $K_\nu/c(\nu)\to s$ ${\mathbb P}_{V(s)}$-[a.s.]{}. Suppose the mapping $s\mapsto V(s)$ is a continuous injection from $[0,\infty]$ to ${\cal V}$ with $0$ and $\infty$ corresponding to the trivial solutions. Then a path $\{\varkappa(\nu);\,\nu=0,1,\ldots\}$ induces a limit if and only if $\varkappa(\nu)/c(\nu)\to s$ for some $s\in [0,\infty]$, in which case the limit is $V(s)$. Moreover, ${\rm ext}{\cal V}=\partial{\cal V}_\uparrow=\{V(s), s\in [0,\infty]\}$.
Minor variations of the above two situations are obtained by transposing multiplicities $\ell_{nk}\leftrightarrow r_{n,n-k}$. Still, this does not exhaust all possibilities. See Kerov (2003) (Section 1.3, Theorem 2) for examples of boundaries with both discrete and continuous components.
Number triangles
================
[**The Pascal triangle.**]{} For the Pascal graph the dimensions are $D_{nk}={n\choose k}$ and $D^{\nu\varkappa}_{nk}={\nu-n\choose \varkappa-k}$. The ratios $V^{\nu,\varkappa(\nu)}_{nk}={\nu-n\choose \varkappa(\nu)-k}/{\nu\choose \varkappa(\nu)}$ converge iff $\varkappa(\nu)/\nu\to x\in [0,1]$, in which case the limit is $V_{n0}(x)=x^n$. This identification of extremes is equivalent to de Finetti’s theorem (see Aldous (2003)), since $V\in {\cal V}$ determines the law of some infinite sequence of exchangeable Bernoulli trials. A closely related type of moment problem with a monotonicity constraint have been discussed recently in Gnedin and Pitman (2007).
[**The $q$-Pascal triangle.**]{} This graph has multiplicities $\ell_{nk}=1, r_{nk}=q^{n-k}$, $(n,k)\in T$, and may be seen as a parametric deformation of the Pascal graph. The extreme boundary was found in Kerov (2003) by an algebraic method and justified by Olshanski (2001) by the analysis of (\[reldim\]). The dimensions are expressible through $q$-binomial coefficients as $$D_{nk}={n\choose k}_q,~~~~~D^{\nu,\varkappa}_{nk}=q^{(\varkappa-k)(n-k)}{\nu-n\choose \varkappa-k}_q{\bigg /}{\nu\choose \varkappa}_q.$$ Suppose first that $0<q<1$. Lemma \[disc\] is applicable, and all nontrivial extremes are given by $$V_{nk}(m)= {q^{(m-k)(n-k)}(1-q)\cdots (1-q^m)\over(1-q)\cdots(1-q^{m-k})}~ 1_{\{0\leq k\leq m\}}, ~~~~m=1,2,\ldots$$ In particular, $V_{n0}(m)=q^{mn}$, $(m=0,1,\ldots,\infty$).
The function $V\mapsto \ell_{00}V_{10}$ identifies the extreme boundary with $\{q^m,~m=0,1,\ldots,\infty\}$. The decomposition (\[VU\]) into extremes corresponds to a version of Hausdorff’s moment problem on $[0,1]$ with kernel $x^n$, but subject to the constraint that the measure is to be supported by a geometric progression. That is to say, a sequence $V_{\bullet,0}$ with $V_{00}=1$ is representable as a mixture $$V_{n0}=\sum_{m\in \{0,1,\ldots,\infty\}} p_m q^{mn}$$ with some probability distribution $\{p_m;~ m=0,1,\ldots,\infty\}$ if and only if $V_{\bullet, k}=\nabla_k(\cdots(\nabla_1(V_{\bullet, 0})\cdots)\geq 0$ for all $k\geq 0$, where $\nabla_k(U_\bullet)= (U_\bullet -U_{\bullet+1})/q^{\bullet-k}$.
In the case $q>1$ the extreme boundary is $\{1-q^{-m},~m=0,1,\ldots,\infty\}$ (this case is reducible to $q<1$ by transposition of $T$ and replacing $q$ by $q^{-1}$). The only accumulation point of ${\rm ext}{\cal V}$ for $q<1$ is $0$ and for $q>1$ is $1$. A phase transition occurs at $q=1$, when the extreme boundary is continuous.
[**Stirling triangles.**]{} Let $r_{nk}=1$ and $\ell_{nk}=(n+1)-\alpha(k+1)$ for $-\infty<\alpha<1$. For $\alpha=-\infty$ take $\ell_{nk}=k+1$. The dimension is $D_{nk}=\left[\!\! \begin{array}{c} n+1\\k+1 \end{array}\!\!\right]_\alpha$. The notation stands for the generalised Stirling numbers defined as connection coefficients in $$(t)_{n\uparrow}=\sum_{k=1}^n \left[\!\! \begin{array}{c} n\\k \end{array}\!\!\right]_\alpha
\alpha^n(t/\alpha)_{n\uparrow},$$ (where $\uparrow$ denotes the rising factorial), with the convention that these are the Stirling numbers of the second kind for $\alpha=-\infty$. For $\alpha=0$ these are the signless Stirling numbers of the first kind.
For $-\infty\leq\alpha<0$ the extreme boundary is discrete, with $$V_{n0}(m)={1\over (m|\alpha|+1)_n}~~~{\rm for~~}-\infty<\alpha<0, ~~~~V_{n0}(m)={1\over m^n}~~~{\rm for~~}\alpha=-\infty.$$ These kernels underly a moment problem for measures on the set $\{0,1,\ldots,\infty\}$.
A phase transition occurs at $\alpha=0$. Lemma \[cont\] applies with $\varkappa(\nu)\sim s\,\log n$, the extreme boundary is continuous and the kernel is $$V_{n0}(s) ={1\over (s+1)_{n\uparrow}}~~~~~s\in[0,\infty].$$ This case is closely related to random permutations, records and Ewens’ sampling formula (see Arratia et al (2003)).
In the case $0<\alpha<1$ we should take $\varkappa(\nu)\sim s\, n^\alpha$ to generate the boundary, see Gnedin and Pitman (2006) for formulas for $V_{n0}(s)$ (to adjust the notation in Gnedin and Pitman (2006) to the present setting, one should replace $(n,k)$ by $(n+1,k+1)$). This family of solutions is related to Poisson-Kingman partitions, see Gnedin and Pitman (2006) and references therein.
Several results and (still open) conjectures about boundaries of more general Stirling graphs, with multiplicities of the form $\ell_{nk}=b_n+a_k, ~r_{nk}=1$, are given in Kerov (2003).
[**The Eulerian triangle.**]{} For multiplicities $\ell_{nk}=k+1, ~r_{nk}=n-k+1$ the dimension is the Eulerian number $\langle {n+1\atop k}\rangle$ (that counts permutations with a given number of descents). The boundary problem was solved in Gnedin and Olshanski (2006). The extreme solutions are given by $$V_{nk}(m)={1\over (n+1)!} \prod_{i=-k}^{n-k} \left( 1+{i\over m}\right)$$ with $m\in {\mathbb Z}\cup\{\infty\}$. Note that the range of $\ell_{00}V_{10}(m)=(m+1)/(2m)$ is symmetric about $1/2$, with $1/2$ being the only accumulation point. The symmetry of the boundary stems in this case from the invariance of multiplicities under transposition.
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[^1]: Utrecht University; e-mail gnedin@math.uu.nl
[^2]: University of California, Berkeley; e-mail pitman@stat.Berkeley.EDU
|
---
author:
- 'D. Bettoni'
- 'G. Galletta'
- 'S. García-Burillo'
- 'A. Rodríguez-Franco'
date: 'Received 23 February 2201/Accepted 7 May 2001'
title: 'The gas content of peculiar galaxies: counterrotators and polar rings[^1] [^2] '
---
Introduction {#intro}
============
The existence of kinematically decoupled disks of gas and/or stars with anti-parallel spins has been reported for a significant number of galaxies [see @gall96 for a review]. The phenomenon of counterrotation may be seen in the ionized gas [@bettoni84] but in almost half of the reported cases it is found in pure [*stellar*]{} disks [see @rubin for a review], being in some cases accompanied by gas counterrotation. Evidence of kinematical decoupling for the cold gas, either atomic or molecular, is also present in a high percentage of counterrotating galaxies [@oosterloo; @braun; @casoli; @vandriel; @sage2; @santi; @santi2].
Different scenarios have been proposed to explain the counterrotation present in elliptical and disk galaxies. Most of them invoke the capture of matter which comes from outside the acceptor galaxies. The various models examine different masses and time-scales involved in the accretion process. An external origin is also invoked to explain the existence of polar ring galaxies, where gaseous disks or rings are seen to rotate almost perpendicularly with respect to the main stellar body of the system [@whitmore]. However, the link between polar rings and counterrotators remains unclear. Alternatively, it has been suggested that a primordial mechanism, invoking a dissipationless cosmological collapse perturbed by tidal fields, could explain the formation of counterrotating galaxies [@harsoula].
Within the accretion scenario, the morphology of the acceptor system and its dynamic evolution would depend on several factors: the nature of the accreted matter (gas and/or stars), the ratio between the accreted mass and that of the acceptor galaxy and, finally, the accretion speed. Whereas the collision between equally massive galaxies may lead to a merging like the ‘Antennae’, ending up as a giant elliptical galaxy with counterrotation [@barnes], the disruption of the acceptor´s disk could be avoided by progressive infall of gas whose spin is anticorrelated with the main stellar body [@quinn; @thakar; @voglis]. However, the accretion of a gas-rich satellite may heat the stellar disk [@thakar]. Observations show that lenticulars and spiral galaxies hosting counterrotation do not necessarily present disrupted stellar disks. Their stellar kinematics appear globally regular [@oosterloo] and the stellar disks show hardly any sign of thickening [@n4550]. The end product of the accretion process at the present epoch seems to have reached, in most of the known studied cases, an equilibrium configuration for the stellar component. Either the time-scales to reach equilibrium are short enough or, alternatively, accretion caused no traumatic changes in the kinematics of the stars.
The gas, however, is expected to reflect the consequences of the accretion process more violently than the stars. If gas was accreted by a disk galaxy with a non negligible amount of interstellar gas, a strong interaction between the accreted and the primary gas is likely. The existence of violent cloud-cloud collisions (the relative velocity between the interacting clouds would be: v$\sim$2v$_{rot}$) and the highly dissipative nature of the encounters might lead to the onset of large-scale shocks. These might convert the atomic gas into molecular gas [@braine; @sage1; @young] and eventually induce starbursts [@wang; @read; @santi2]. If the described scenario holds, one would expect that the content of molecular gas would be higher in counterrotating galaxies than in a comparison sample of non-interacting galaxies of the same Hubble type. On the other hand, if the origin of counterrotation is primordial, or alternatively, if large-scale shocks are not efficient or short-lived, the H$_2$ content should be similar for counterrotating and normal galaxies.
It is also unclear whether polar rings and counterrotators represent different steps in the process of mass accretion. A comparison of their H$_2$ content could reveal if there is an evolutionary link between the two families of accretors.
This paper represents a first step in answering some of the above-mentioned questions by studying the global gas content in a sample of counterrotators. In this work we estimates the content of molecular, atomic and hot gas for a sample of 58 galaxies of different morphological Hubble types and of different types of counterrotation. Molecular gas masses are derived from $^{12}$CO(J=1–0) observations made with the 15m SEST radiotelescope on 10 galaxies with counterrotation and 1 polar ring (section 2). Results from these new observations are described in Section 4. The published data for 48 objects where there are indications of counterrotation in the gas and/or stars have been compiled and added to this sample(see Section 5). The H$_2$ content of counterrotating galaxies will be studied relative to a comparison sample of normal galaxies that was built up from the literature, as described in section 6. The gas content of counterrotators has also been compared with those of polar ring galaxies, using available data from the literature (see below for detailed references). A similar comparative study has been done with the HI content (from various sources), with the warm dust mass (derived from IRAS data) and with the amount of hot gas, the dominant gas component in early type objects (values derived from X-ray data taken by ROSAT and by EINSTEIN).
The SEST sample
===============
The 11 galaxies in our sample observed using the SEST telescope were selected from the list of objects published by @gall96. The galaxies are described individually in section \[results\], together with the main results inferred from this CO study. The relevant parameters of the systems, such as the diameters (D$_{25}$: the de-projected linear diameter corresponding to the blue isophote at 25 [ ]{}), absolute B magnitudes (M$_B$), distances ($d$) and morphological types are shown in columns 3–6 of Table 1. These data have been extracted from the recently up-dated Lyon-Meudon database LEDA [@leda].
The conversion factor between the integrated intensities of $^{12}$CO(J=1–0) and the H$_2$ column densities was taken from @strong, i.e.:
$$\chi=N(H_2)/I_{10}=2.3\times10^{20} mol/Kkms^{-1}.$$
The total mass of molecular hydrogen (in ) under the 45beam of SEST for the observed galaxies was obtained from N(H$_2$) using: $$M(H_2)=7.8\, 10^{-16}\times N(H_2)\times d^{2} ({\hbox{M$_\odot$}})$$ where $d$ is the galaxy distance in Mpc and N(H$_2$) the measured column density in molcm$^{-2}$. The total molecular gas content under the beam (M$_{mol}$) is derived by multiplying M(H$_2$) by 1.36 in order to include the Helium mass fraction. Although a variation of the $\chi$ conversion factor cannot be excluded among the observed galaxies, we will take the above derived values as a good estimate of the molecular gas masses.
HI masses (M$_{HI}$) have been taken from various sources or calculated from the m$_{21}$ parameter of LEDA [@leda], assuming: $$M_{HI} = 2.35\ 10^5\ 10^{-0.4(m_{21c}-17.4)}\ d^2$$ with $d$ being the distance in Mpc. Hereafter, M$_{gas}$ is used for the total mass of cold gas, i.e., M$_{gas}$=M$_{mol}$+M$_{HI}$.
The mass values of X-ray emitting gas (M$_X$) have been derived from ROSAT data [@beuing] and from EINSTEIN data [@fabbiano; @burstein], assuming: $$M_{X} = 10^{-24}\ L_x^{0.5}\ L_B^{1.2}$$ Note that this formula [@roberts] is generally valid for early-type galaxies.
Finally, warm dust masses (M$_{dust}$) have been calculated from IR fluxes (S$_{60}$ and S$_{100}$) published by @knapp and from LEDA raw data, kindly furnished by G. Paturel, assuming: $$M_d = 4.78\
10^{-3}\ S_{100}\ d^2\ (exp(144.06/T_d)-1)$$ where dust temperature $T_d=49*(S_{60}/S_{100})^{0.4}$ and d is the distance in Mpc. A more accurate calculation giving the total dust mass, including the coldest component not detected by IRAS, is beyond the scope of this work.
All distance-dependent values available from the literature have been re-scaled to the adopted distances from LEDA, explicitly listed in Table 1.
Observations {#obs}
============
Emission in the $J$=1–0 and $J$=2–1 transitions of $^{12}$CO among the sample of 11 galaxies was searched for using the Swedish-ESO Submillimeter Telescope (SEST) at La Silla, equipped with the dual channel IRAM 115/230 GHz receivers which allow for simultaneous observations. There were three different observation runs: November 6th-11th 1998, May 28th–31st 1999 and May7th-11th 2000. Beam sizes were 45 and 23 at 115 GHz and 230 GHz, respectively. Unless explicitly stated, the temperature scale used throughout the paper is antenna temperature, corrected from atmospheric losses and rear spillover (T$_a^*$). When deriving line ratios, it is assumed that main beam efficiencies $\eta_{beam}$(115 GHz)=0.70 and $\eta_{beam}$(230 GHz)=0.50, in order to refer temperatures to the main beam brightness scale. Spectrometers cover bandwidths of 995 MHz (1290 ) and 543 MHz (1410) for the J=2–1 and J=1–0 lines respectively. Dual-beam switching was used, with a beam throw of 12 to produce a flat baseline. Typical system temperatures ranged from 190-430 K. Pointing and focus were checked every 2-3 hours, using several SiO maser sources located near the target galaxy. The RMS accuracy of the pointing model was typically $\le$2, assuring an absolute positional accuracy better than 5.
Except for one case (NGC3497), we made only single point maps centered on the nuclei of the galaxies. Individual scans at each position were coadded to get total integration times ranging from 1h to 7h. Spectra were Hanning-smoothed to a velocity resolution of 30 (for both lines) with the exception of narrow lines for which a higher resolution was kept (see below). Linear baselines were fitted and subtracted from the smoothed spectra using the GILDAS software package.
-2cm
Results for the SEST sample {#results}
===========================
This section presents the main results of the CO observations for the 11 galaxies of the SEST sample, preceded by a short description of the systems.
ESO263-48
---------
This galaxy, also denoted Anon 1029-45, has been included in a list of dust-lane ellipticals by @hawarden. It has a prominent and strongly warped dust lane going across the major axis up to r=30 (5.2 kpc). Due to its prominent dust lane which gives the system the appearance of an edge-on disk, it has often been classified as S0, although it has all the properties (photometric profile, luminosity and size) of giant ellipticals. Furthermore, long exposure plates of the galaxy show no signature of a stellar disk.
The stellar kinematics have been studied by @bertola1, who derived a maximum rotational velocity of 210 reached at r=20 (3.4 kpc) and a velocity dispersion of 260 . Ionized gas in counterrotation, with spin velocities of $\sim$ 250 at r=7 (1.2 kpc), has been detected along the major axis [@bertola2]. There are no HI observations available for ESO263-48; the galaxy also remains undetected in the ROSAT survey. However, this elliptical is particularly rich in dust; from the IRAS fluxes we estimate M$_{dust} \sim$3.810$^{5}$. A continuum radio emission has been detected at 1.4 and 4.9 GHz, which extends perpendicularly to the dust lane [@bertola1].
The J=2–1 and J=1–0 CO spectra of Fig.1 show $\sim$800 –wide emission profiles centered at v=2810 (hereafter taken as the CO-based systemic velocity; v$_{sys}$). The J=1–0 spectrum is asymmetrical with respect to v$_{sys}$ as it can be fitted by three gaussian-like components. Emission of the two extreme velocity components at v=2517 and v=3170 can be explained by the presence of an unresolved H$_2$ disk with a rotation speed of v$_{rot}$=325 , reached within r$=$2 kpc (the upper limit on r is set by the J=2–1 beam). The value of v$_{rot}$ derived from CO is significantly larger than that which is inferred for the ionized gas at the same radius ($\sim$250 ). This discrepancy may indicate that the H$_2$ disk seen in projection extends farther out. The asymmetry in the $^{12}$CO(J=1–0) spectrum is caused by the existence of a strong component at V=2980 , redshifted by 170 with respect to v$_{sys}$, and having no blue-shifted counterpart. The latter can come from an asymmetrical distribution in the H$_2$ disk or, alternatively, be the signature of a warp in the molecular gas disk (as suggested by the distorted dust lane).
The molecular gas content under the J=1–0 beam can be derived within a radius r=22 (4 kpc) (close to the maximum extent of the dust lane feature). It was calculated to be M$_{mol}$=810$^{8}$ , a high value for an elliptical galaxy.
IC1459
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The absolute magnitude and the intrinsic diameter of this galaxy (see Table 1) are both characteristic of a giant elliptical. IC1459 has a massive counterrotating stellar core (M$\sim$10$^{10}$ , according to @franx) and its outer stellar isophotes are twisted, an indication that the stellar body is triaxial. This galaxy shows dust absorption in the central 10 [@sparks] and faint pseudo-arms in the outer part (r$\sim$3.5 @malin). The counterrotating core, which hosts a compact radio source [@franx], has a radius of $\sim$2 (200 pc) and a projected spin velocity of 170$\pm$20 . On dynamical grounds, the core can be described as a disk, rather than as an ellipsoid. In contrast, the outer directly-rotating stellar body has a slower rotation figure; it reaches $\sim$45$\pm$8 at r=40 (4 kpc). The galaxy is crossed by a disk of ionized gas, whose emission is evenly detected up to r=35 (3.5 kpc); the ionized gas rotates in the same direction than the outer stellar body, but at a higher speed (350 ) [@franx]. Therefore counterrotation in this galaxy seems confined to the inner core and affects only the stars. X-ray emission peaks in the galaxy nucleus [@roberts], an indication of central activity.
@walsh report a negative detection of HI emission, the upper limit on M$_{HI}$ being $\sim$10$^7$ . Our $^{12}$CO(J=1–0) spectrum also shows a negative detection for molecular gas emission. In contrast, the integrated $^{12}$CO(J=2–1) spectrum shows clearly a narrow emission line of FWHM$\sim$35 , centered at 1782 (see Fig.1). The velocity centroid of CO is redshifted by $\sim$100 with respect to the galaxy systemic velocity (v$_{sys}$=1691 ). This discrepancy of velocity centroids and the narrowness of the $^{12}$CO(J=2–1) line both indicate that emission cannot come from a rotating molecular gas disk in equilibrium that could be associated with the ionized disk or, alternatively, with the counterrotating core. In the discarded scenario of a molecular gas disk, the CO line should be 3–5 times wider than is actually observed, considering the size of our beam. Instead, the CO profile may come from a Giant Molecular Association (GMA). Furthermore, the derived upper limit on the mass of molecular gas, M$_{mol}<$2 10$^7$ , is noticeably low. Additional support for the interpretation outlined above comes from the high resolution deconvolved V-band images of the dust distribution, obtained with the HST. The morphology of the dust lane source in the inner 4 of the galaxy is very irregular and indicates non equilibrium motions [@forbes]. Finally, the estimated 3$\sigma$ upper limit on the (J=2–1)/(J=1–0) ratio ($>$1.3) suggests that the CO emission might be partly optically thin.
Similar molecular gas components have been found in other early-type galaxies, such as NGC404 [@wiklind], classified as a gas accreting elliptical with a minor axis dust lane [@bg]. This GMA may be a residual of one of the galaxies involved in the passed merger that is supposed to be at the origin of IC1459 [@franx].
IC2006
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The size and the luminosity of IC 2006 put this galaxy among the dwarf ellipticals. Counterrotation is present in a outer ring of atomic gas, which is aligned with the apparent major axis of the galaxy. At the radius of the HI ring, the galaxy luminosity decreases to B$\sim$27 [ ]{}[@schweizer]. Schweizer and collaborators describe the HI distribution as a 2 kpc wide circular ring of radius $\sim$11 kpc, inclined at 37. With the adopted parameters, the rotation speed of the HI gas would be 200 at this distance. HI gas remains undetected inside the ring. In contrast, faint emission from ionized gas is detected within 2.5 kpc of the nucleus, characterized by a velocity gradient which is smaller and also inverted with respect to that of the stars (from -70 (NE) to 50 (SW), relative to v$_{sys}$=1385 ). The counterrotating ionized gas disk is highly turbulent, with a measured velocity dispersion of 190 [@schweizer].
The upper limit set by ROSAT observations [@beuing] indicates that, contrary to IC1459, IC2006 has no relevant quantity of hot gas (see Table 1). Based on the optical photometry and the kinematics of the outer HI ring, @schweizer derive the presence of a dark halo which contains about twice the mass of the luminous stellar body. Our single-point $^{12}$CO(J=1–0) map sampled the galaxy nucleus up to r=3.5 kpc. This region includes the ionized gas disk but excludes the outer HI ring. CO emission was not detected; the latter implies an upper limit for the central (r$<$3.5 kpc) molecular gas content of M$_{mol}<$1.4 10$^7$ .
NGC1216
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This galaxy belongs to the Hickson compact group N23. It is an almost edge-on disk galaxy classified as S0-a in the LEDA database. Counterrotation in the ionized gas is detected from optical emission lines [@rubin2]. The gaseous disk extends up to a radius of 2 kpc, with an observed maximum rotational speed of 75 ; this velocity is significantly lower than that of the stars at the same radius ($\sim$175 ). The latter may be an indication that the gaseous disk is either warped or tilted with respect to the galaxy plane. The optical images of this galaxy show no signature of dust absorption. @williams report a negative detection for this galaxy in the 21cm line, which gives an upper limit of M$_{HI}$$<$510$^{7}$. IRAS, ROSAT and, finally, our CO observations report negative detections for NGC1216. The latter give an upper limit of M$_{mol}<$310$^8$ for the molecular gas content, within a region of radius r=7 kpc.
NGC2217
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NGC2217 is a barred spiral (SBa) seen almost face-on. It has an outer stellar+gaseous ring of radius $\sim$8 kpc and an inner oval ring of radius $\sim$4 kpc, which encircles the bar.
@bettoni90 have studied the distribution and kinematics of the ionized gas, confined to the innermost 20 ($\sim$2 kpc) of the galaxy. The gas distribution suggests the presence of a two-arm spiral structure, whereas the kinematics are characterised by counterrotation with respect to the stars, inside r$\sim$10 ($\sim$1 kpc). However, a detailed analysis of the data by the authors shows that the gas counterrotation is not real, and it may be better accounted for if one assumes the gas to be in a warped disk seen in projection. The ionized gas inside r=1 kpc would lie in a series of polar rings almost at 90with respect to both the bar and the stellar disk. In the outer region (r=1-2 kpc) the plane of the gas rings would have settled towards the disk of the stars, and changed its inclination by nearly 90 degrees. The latter explains why the gas and the stars rotate in the same direction for r$>$1 kpc. We have classified the system as a polar ring galaxy.
The apparent counterrotation of ionized gas inside r=1 kpc produces the largest extention of radial velocities along the bar minor axis: from v=1450 to 1800 . The v$_{sys}$ derived from the gas and the star kinematics agree within the margin of errors, being close to 1640 . @bettoni90 fit a rotation curve to their data, inferring de-projected rotational velocities of 125 and 150 for the stars and the gas, respectively, at a radius r=1 kpc.
HI observations reported by @h82 show a double-horned emission profile centered at v$_{sys}$=1615 , close to the value found by @bettoni90 and with a total width at zero power of $\sim$300 . The re-scaled HI mass is M$_{HI}$=2.710$^{9}$. As there is no high-resolution HI map of NGC 2217, the location of the HI gas is uncertain.
The spectra in Fig.1 show the detection of molecular gas emission for the two lines of $^{12}$CO. The two profiles differ significantly, however. Whereas the J=1–0 line is centered at v=1720 (i.e., redshifted 100 with respect to v$_{sys}$), the J=2–1 line peaks at v$\sim$v$_{sys}$ (as defined above). Although the low spatial resolution of these observations tells us little on the precise location of molecular clouds, the reported asymmetry of the J=1–0 profile (which samples the disk up to r=2 kpc) suggests that the distribution of H$_2$ gas in NGC2217 is highly asymmetrical and/or that the kinematics of molecular clouds might depart from circular motions. Most noticeably, the integrated HI profile also shows a pronounced asymmetry. The FWHM of the two CO lines are close to the values found in HI and in optical emission lines ($\sim$250-300 ).
We derived a molecular gas content of M$_{mol}\sim$9 10$^{7}$up to r=2 kpc.
NGC3497
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NGC3497 is a major-axis dust lane elliptical known by different names (NGC3525=NGC3528=IC2624). The ringed dusty disk shown in the B-R color maps of @eb85 seems to have the same extent as the stellar disk (diameter$\sim$70). As with all major-axis dust-lane ellipticals, e.g. ESO 263-48 in this paper, the dust signature is interpreted to be the result of an accretion episode. The galaxy has a fainter galaxy at 2 (named NPM1G -19.0362) and a companion with similar redshift at 5 (NGC 3529=IC2625).
Stellar and gas rotation curves have been derived by @bertola1, who measured a radial velocity difference of $\sim$240$\pm$30 between the western and eastern sides of the major axis (on the NE side stars are receding). The measured systemic velocity is v$_{sys}$=3672$\pm$25 . No X-ray, IR or HI data are available in the literature.
The emission of both lines of $^{12}$CO were observed in three positions located along the major axis of the disk: the (0,0) offset centered on the galaxy nucleus and two off-centered positions at r=$\pm$22 . The J=2–1 and J=1–0 spectra shown in Fig.1 reveal the presence of molecular gas in the central region (up to r=5 kpc) and also the detection of the J=1–0 line of $^{12}$CO in the NE offset. In the SW position, however, CO emission was not detected. The $^{12}$CO(J=1–0) central spectrum is fitted well by a single gaussian profile, centered at 3774 and with FWHM=600 ($\sim$1000 at zero power); therefore, it is 100redshifted with respect to v$_{sys}$. In contrast, the J=2–1 profile shows three velocity components at v=3497 , v=3693 (close to the optically determined v$_{sys}$) and v=3497 . The velocity asymmetry in the central J=1–0 spectrum may indicate that H$_2$ distribution is slightly asymmetrical in the disk within r=5 kpc.
A comparison between the radial velocity measured on the CO spectrum ($\sim$3570 ) and the stellar velocities observed in the NE side of the major axis (redshifted with respect to v$_{sys}$), indicates that molecular gas is counterrotating with respect to the stars.
The molecular gas mass within the central r=5 kpc, derived from the $^{12}$CO(J=1–0) integrated intensity, would be M$_{mol}$=1.410$^{9}$. The amount of molecular gas detected in the NE offset is M$_{mol}$=2.710$^{8}$.
NGC4772
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@haynes considers NGC4772 as a case of apparent counterrotation of the ionized gas versus the stars. The kinematical decoupling of the nuclear ionized gas (r$<$5=0.3 kpc), observed along both the minor and the major axes, has been interpreted as the signature of a misaligned embedded gas bar, rather than as evidence of counterrotation. However, this Sa galaxy shares many features with other prototypical counterrotators. Mimicking NGC3626, the HI content of NGC4772 is distributed in two separate rings, probably non coplanar. As is the case for NGC 3626, the central region of NGC4772 is HI-poor. Furthermore, the deep optical photometry of the galaxy reveals the presence of a round, low surface brightness disk in the outer part, reminiscent of a similar feature reported by @buta in the Sab counterrotator NGC 7217.
The J=1–0 spectrum of $^{12}$CO (Fig.1) reveals the presence of molecular gas (inside r=1.5 kpc). The line profile, centered at $\sim$1120 and with FWHM$\sim$300 , is slightly redshifted with respect to the optically determined v$_{sys}$=1040 (the same as derived from HI).
Emission in the J=2–1 line is undetected, however. The molecular gas mass within the central r=1.5 kpc, derived from the $^{12}$CO(J=1–0) integrated intensity, would be M$_{mol}$=5.410$^{7}$M$_{\sun}$.
NGC5854
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NGC5854 is an early spiral (Sa) characterized by a low gas content and the absence of current star formation. @haynes have studied the stellar kinematics, using H$\alpha$ and MgIb optical absorption lines, and the kinematics of ionized gas, using the N\[II\] and O\[III\] optical emission lines. These data reveal the existence of a counterrotating gas disk extending up to r$\sim$7 (0.8 kpc), with a total velocity range of 120 . The stellar velocities measured at r$\sim$40 (4.5 kpc) reach $\pm$160 . Although HI content is low, @magri detected a signal in the nucleus. The HI profile is centered at 1663 , close to the optically determined value for v$_{sys}$=1669$\pm$30 [@fouque]. The narrowness of the HI spectrum (FWHM$\sim$100 , namely, less than the measured stellar velocity spread) suggests a close association of the HI component with the counterrotating ionized gas (see discussion in @haynes).
Although weak, the $^{12}$CO(J=1–0) spectrum shows the existence of H$_2$ gas within r=2.3 kpc. There is a hint of a double-horned profile, with two velocity components at v=1539 (with FWHM$\sim$100 ) and v=1930 (with FWHM$\sim$170 ), equidistant from v=1735 . The CO spectrum in the J=2–1 line confirms the detection of molecular gas. Not surprisingly, the J=2–1 and J=1–0 profiles differ. The J=2–1 line shows hints of two velocity components, although with a smaller velocity spread (v=1630 and 1740 ) and is centered at v$\sim$1690 , in reasonable agreement with HI. However the larger velocity spread of the J=1–0 spectrum would suggest that, compared to the counterrotating ionized gas core, the H$_2$ disk may extend farther out. The molecular gas mass within the central r=2.3 kpc was derived from the $^{12}$CO(J=1–0) integrated intensity giving; M$_{mol}$=1.610$^{8}$M$_{\sun}$.
NGC5898
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NGC5898 was studied by @bettoni84 and @bertola88, who discovered the first case of ionized gas counterrotation in a dust lane elliptical in this galaxy. Their data, which extended out to $\sim$10 (1.4 kpc), have recently been completed by @caon who analysed the stellar and the gas kinematics farther out (up to r$\sim$35 (4.8 kpc)). The new data along the major axis show the existence of a stellar core of radius r$\sim$5(0.7 kpc) which counterrotates with respect to the outer stellar body. The ionized gas counterrotates with respect to the inner stellar core, and it therefore corotates with the outer stars. In contrast, gas is seen to counterrotate at all radii along the minor axis. This might indicate that angular momentum vectors of the ionized gas and the stars are certainly misaligned, but not antiparallel. The observations of this galaxy at X-ray and IR wavelengths show an upper limit for hot gas of $\sim$310$^8$ and a moderate quantity of dust, of a few 10$^4$ .
The J=1–0 spectrum of $^{12}$CO (Fig.1) shows a tentative detection of molecular gas inside r=3.1 kpc. The line profile, centered at $\sim$2020 and with FWHM$\sim$190 , is slightly blueshifted with respect to the optically determined value of v$_{sys} (\sim$2100 , derived from @bertola88). The observed asymmetry might arise if the H$_2$ gas was associated with the ionized disk, which shows a marked extension towards the SW (where ionized gas radial velocities are blueshifted). In this scenario molecular gas would also be counterrotating.
The $^{12}$CO emission is undetected in the J=2–1 line, however. We have calculated the molecular gas mass within the central r=3.1 kpc, using the $^{12}$CO(J=1–0) integrated intensity, giving M$_{mol}$=10$^{8}$M$_{\sun}$.
NGC7007
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The optical images of NGC 7007 show an elliptical-like body encircled by an off-centered bow-shaped dust lane on the eastern side. @dettmar discovered, in this galaxy, the signature of a counterrotating ionized gas disk by comparing the spectrograms of gas emission (NII $\lambda$6853) and stellar absorption lines (H$_{\alpha}$). Spectra taken later [@bettoni] allowed a detailed analysis of the stellar and the gas kinematics, characterised by maximum rotational velocities of $\pm$150 and $\pm$175 respectively, reached at r=10. The central velocity dispersion for stars is 150 , whereas gas lines have instrumental width ($<$100 ). The galaxy contains a source of infrared emission detected by IRAS, and at X-ray wavelengths the published work reports an upper limit (see Table 1).
Our J=2–1 and J=1–0 spectra both show a weak narrow line of $\sim$30 FWHM, centered at $\sim$2850 , close to the optical redshift of 2924 $\pm$ 66 reported in RC3 [@rc3]. These results are, however, at odds with that quoted by @dacosta (3053 $\pm$20kms$^{-1}$), who estimated v$_{sys}$ as a weighted average between gas emission and stellar absorption data. If the @dacosta value is more accurate, as indicated by the additional spectra of @bettoni, the reported difference between the CO peak and the optical systemic velocity could be explained with an asymmetry in the distribution of cold gas. The derived molecular gas mass, $\sim$ 610$^7$ , could be well accounted for if the emission observed came from a few Giant Molecular Associations (GMAs) in the center of the galaxy, as observed in IC1459. The narrowness of lines in both transitions supports this scenario.
NGC 7079
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NGC 7079 is a weakly barred SB0 galaxy, a member of an interacting pair. @bettoni97 detected a counterrotating disk-like structure of ionized gas which extends up to a radius of r=2 kpc. The radial velocities for the gas span from 2600 to 2800 . The stellar kinematics is typical of an undisturbed disk. The measured radial velocities (up to r=4.5 kpc) range from v$\sim$2500 to v$\sim$2900 and give a v$_{sys}$=2680 and a central velocity dispersion of 150 . No X-ray emission has been detected from this galaxy; the IRAS satellite detected infrared emission.
The J=2–1 and J=1–0 lines of CO are detected in this galaxy, showing the presence of molecular gas (see Fig.1). The line profiles, $\sim$170 wide at zero power, are centered on the galaxy systemic velocity, derived from optical data. The linewidths of both CO lines agree satisfactorily with the velocity range measured for the counterrotating ionized gas. In contrast, the CO widths are much smaller than the velocity interval measured in the stellar lines. This may indicate that H$_2$ gas is confined to the inner portion of the galaxy and that it shares the same kinematics as the counterrotating ionized gas. The inferred molecular gas content is M$_{mol}\sim$1.210$^{8}$up to r=2 kpc.
The enlarged sample of accreting galaxies
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The newly acquired data described above would nevertheless be insufficent to extrapolate estimates on the global gas content to all counterrotators. In order to improve the SEST sample on statistical grounds we added the published data from those counterrotators [@gall96; @kannappan] with an estimate from any of the different gas tracers: M$_{mol}$, M$_{HI}$, M$_{dust}$ or M$_X$. Masses are derived with the same assumptions used for the SEST sample. The properties of the enlarged sample are in first part of Table 1, together with the list of references relevant for this compilation. This new sample allows a complete study of the global gas content of counterrotators, using different gas tracers in a statistically significant sample of 58 objects.
In order to compare counterrotators and polar rings, the available data on a sample of 46 polar ring galaxies have also been compiled(see references in Table 1). Data include 36 polar ring lenticulars and spirals (@rubin3 [@whitmore] and this work), and 10 polar ring ellipticals, known in the literature as ellipticals with minor-axis dust-lanes, such as NGC 5128 [@bg]. Polar ring ellipticals have traditionally been classified as S0s due to the presence of a dark ring or disk in optical pictures. However, the luminosity profiles of these galaxies do not follow an exponential law, typical of stellar disks, but rather a r$^{1/4}$ law, characteristic of spheroidal systems. We have therefore re-classified these galaxies as ellipticals, whenever they appear as S0s in catalogs. Not all polar ring lenticulars and spirals present in @whitmore’s catalogue were finally included in our list. We only selected those systems where the perpendicularity between the ring and the stellar body is clearly visible in the images, discarding all systems appearing doubtful in our inspection of the catalogue. Altogether the sample of accreting systems includes 104 objects.
Building up a comparison sample
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The main aim of this paper is to study the molecular gas content of accreting galaxies (counterrotators and polar rings) and compare it with the [*average*]{} value for [*normal*]{} non-interacting galaxies as a function of the Hubble type. The first non-obvious task is the definition of a comparison sample of [*normal*]{} galaxies. The sample should contain a statistically significant number of objects. This requirement is critical for early-type galaxies, as the majority of accreting systems are of types earlier than Sa (morphological type code t=1). Moreover, the sample should avoid the inclusion of abnormal objects, suspected to be interacting and/or merging galaxies, e.g those reported in Arp’s catalogs.
In the past, two different research groups have built up comparison samples in order to study the variation of the gas content of galaxies along the Hubble sequence: @bregman and @casoli. @bregman derived the content of molecular gas, HI, X-ray emitting gas and dust, working on a sample of 467 early-type galaxies, ranging from pure ellipticals (E, t=–5) to early spirals (Sa, t=1). In their analysis they favoured the use of the total blue luminosity (L$_B$) of galaxies as the necessary normalisation factor, i.e., the inferred numbers being M$_{mol}$/L$_B$, M$_{HI}$/L$_B$, M$_{dust}$/L$_B$ and M$_X$/L$_B$. They concluded, first, that the gas content of elliptical galaxies is dominated by the hot phase(M$_{X}>$M$_{mol}$+M$_{HI}$) and secondly, that M$_{mol}$/L$_B$ and M$_{HI}$/L$_B$ both show a strong positive gradient from E to Sa-type systems. However, the number of early type galaxies detected either in H$_2$ or in HI gas is scarce: for H$_2$, 1 E-type detected with 11 upper limits (hereafter UL) and 6 SOs detected with 18 UL. Poor statistics cast some doubt on their conclusions.
@casoli discussed the molecular and atomic gas content for a set of 582 objects, mainly disk galaxies, normalizing the gas masses by D$_{25}^2$. For comparison, numbers for H$_2$ are: 3 E detected and 7 SOs detected (with 2 UL). Results from @casoli are noticeably at odds with that of @bregman: the sharp increase of gas content from t=–5 to t=1 reported by @bregman (1–2 orders of magnitude) is not confirmed using @casoli data. This discrepancy is illustrated in Figure \[normal\] where the mean values are represented as a function of t for the two samples. To reconcile these discrepant trends one must assume an unrealistic decrease by 2 orders of magnitude in L$_B$/D$_{25}^2$, going from E to Sa systems. Results from both samples for types t$<$1 are however dubious, considering the poor statistics in this range. Moreover, the two samples include a non-negligible percentage of interacting galaxies (estimated to be close to $\sim$20$\%$ in both samples). Interacting galaxies should be discarded when putting together a comparison sample of [*normal*]{} non-interacting objects.
These limitations were the motivation to build up a new comparison sample of normal galaxies. Paper II by @paper2 will discuss extensively the details of this compilation. The over-all numbers give a grand total of 1773 normal galaxies selected from a processed sample of 3800 objects. @paper2 purposedly excluded from the selected sample those galaxies belonging to the interacting or disturbed categories (most of them appearing in @arp, @vv, and @am catalogues). Galaxies listed in the @veron catalogue of AGN systems have also been excluded because in some cases their peculiar activity has been attributed to gas accretion. We have taken from @paper2 the normalized values M/L$_B$ for the molecular, atomic and X-ray emitting gas, as well as for the warm dust content inferred from IRAS. The global statistics for detections (and UL) are: 247 in H$_2$ (113 UL), 774 in HI (149 UL), 196 in X-rays (661 UL) and 861 in IR (555 UL). This sample improves the statistics for early type galaxies compared to previous works, the numbers for H$_2$ being: 10 E-type detected (plus 18 UL) and 10 lenticulars detected (plus 17 UL). Note that the galaxies used to build the mean values for the different ISM tracers are not always the same; however, the majority of galaxies in our sample (1135) have detections or upper limits in at least two wavebands.
We have applied a survival analysis method to the different ensembles of M/L$_B$ data. This analysis tool takes properly into account both detections and UL in order to derive representative averages. The mean values are derived and plotted under the label ‘normal galaxies’, and are binned according to the morphological type code (with $\Delta$t=1). Most noticeably, UL lower significantly the estimated mean M/L$_B$ values for normal galaxies of early types (see @paper2). The derivation of mean values of molecular gas content from @casoli used survival analysis also. For HI data, all the galaxies of their sample were detected and so no survival analysis needs be applied. Also, the analysis of @bregman takes into account the different detection rates of the various morphological types, but using different non-parametric tests, based on rank.
At first sight, the comparison between the mean values derived from these three different studies [@bregman; @casoli; @paper2] shows that Log(M$_{HI}$/D$_{25}^2$) and Log(M$_{mol}$/D$_{25}^2$) of @paper2 are intermediate between @casoli and @bregman values. Although the cold gas content increases by a factor $\sim$10 from E to Sa-types, this gradient is less steep than that reported by @bregman (see Fig. \[normal\]).
To identify any potential bias in the galaxy samples compared in this work (normal galaxies, counterrotators and polar rings), we have analysed the statistical distribution of the following intrinsic properties: M$_B$, D$_{25}$ and FIR flux (given by m$_{FIR}$). Kolgomorov-Smirnov tests applied to these quantities indicate that the distributions of M$_B$, D$_{25}$ and m$_{FIR}$ are not significantly different for the 3 samples, at a confidence level better than 95$\%$.
The ISM of gas accretors
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We discuss in this section the results obtained from the comparison of the gas/dust content of gas-accreting and normal galaxies. Mean values of log M/L were obtained for each morphological type, using a $\Delta$t=1 code binning. We studied the deviations from the reference values issued from the survival analysis method applied to normal galaxies (see above). The statistical significance of any difference found between the samples was evaluated by a Student t-test applied to the mean of the values of Log M/L, binned according to morphological type. Fig.\[HI\]–\[X\] illustrate this comparison, whose results can be summarised as follows:
- Log(M$_{HI}$/L$_B$) and Log(M$_{mol}$)/L$_B$) show a large dispersion for accretors of types t$<$0 (Figures \[HI\] and \[mol\]). This result holds for counterrotators and polar rings. The values of the gas content for normal galaxies are also highly dispersed for t$<$0.
- Polar ring spirals and lenticulars have a HI content $\sim$1 order of magnitude higher than normal galaxies (Fig.\[HI\]). The reported difference between the samples is established with a 99$\%$ statistical significance. In contrast, there is no significant difference between normal galaxies and polar ring ellipticals regarding the HI content. This result, previously found and discussed by @richter94 and @huchtmeier97 is still valid when Log(M$_{HI}$/D$^2$), instead of Log(M$_{HI}$/L$_B$), is used as the gas content estimator. Therefore, this tendency cannot be attributed to any colour bias unexpectedly affecting PRs.
On the other hand, the HI content of counterrotating galaxies shows no significant departure from the expected normal value (Fig. \[HI\]).
- The molecular gas content of polar ring galaxies lies above [*normal*]{} values for all galaxy types (Fig. \[mol\]). On average, it is calculated that M$_{mol}$/L$_B$ is $\sim$1 order of magnitude higher in polar rings; this result, which agrees satisfactorily with the conclusions of @gall97, is established with a 98$\%$ statistical significance. Therefore, the global content of cold gas (M$_{gas}$) in S0/S-polar rings lies $>$1 order of magnitude above standard values with a 99$\%$ certainty (see Fig. \[gas\]). On the other hand, the M(mol)/M$_{HI}$ ratio in polar rings stays close to normal values for all types (Fig. \[mol\_HI\]).
The molecular gas content of counterrotating galaxies is marginally lower than normal for types t$<$0 (Fig. \[mol\]). This deficiency is not firmly established, its statistical significance being low (78$\%$). In contrast, counterrotating galaxies reach normal H$_2$ masses for t$>$0. Contrary to polar rings, the global content of cold gas in counterrotators, given by Log(M$_{gas}$/L$_B$), shows no relevant departures from normal values for all types (Fig. \[gas\]).
However, the HI phase seems to dominate in early type counterrotators (t$<$0). We tentatively identify an increase by $\sim$1 order of magnitude in the M(H$_2$)/M$_{HI}$ ratio from t=–6 to t=6, a factor significantly larger than that observed in normal galaxies (Fig. \[mol\_HI\]). The latter indicates that some mechanism favouring the transformation of HI into H$_2$ might be at work for counterrotators on the right-side of the Hubble sequence (see below).
- As derived from Log(M$_{dust}$)/L$_B$, polar ring galaxies have a dust content $\sim$0.5-1 order of magnitude higher than normal. This result is established with a 99$\%$ statistical significance for spirals and lenticulars, whereas it is only at a 90$\%$ certainty level for ellipticals. In contrast, galaxies with counterrotation have a warm dust content not significantly different to normal galaxies for all types (Fig. \[dust\]).
- One fifth of counterrotators and nearly half of the polar ring ellipticals have been detected in X-rays. On average, the estimated masses of hot gas (see Table 1) are slightly lower than the ones of normal galaxies. Moreover, the slope fitted to the 17 detections in the L$_X$–L$_B$ diagram ($\sim$ 1.7, Fig. \[X\]) lies between the prototypical value for emission mainly due to hot diffuse gas (L$_{gas}$, with slope $\sim$2), and the one for emission being dominated by discrete sources(L$_{discr}$, with $\sim$1; see [@ciotti] for details). Furthermore, L$_{X}$ values for accretors are below the emission level predicted by cooling flows models, assuming the recent SN I rate equal to 0.18 (i.e., 1.8 10$^{-3}$ SN I events per year per unit of 10$^{10}$ L$_B/L_{\sun}$) [@cappellaro]. We can conclude that the observed normalised X-ray luminosity of accretors needs no huge starburst event as an explanation.
- A source of uncertainty for M$_{mol}$ comes from the undersampling of some of the sources. Apart from NGC3497, only single point maps centered in the galaxy nuclei are available from the SEST observations. Therefore, the derived M$_{mol}$ should be taken as a lower limit for the SEST sample, as well as for some of the galaxies in the enlarged sample of accretors for which no complete maps are available. Although it is difficult to evaluate accurately the bias introduced on M$_{mol}$, we are confident to have on average $\sim$70$\%$ of the total molecular gas masses under the SEST beam for spirals. This estimate is based on the comparison between the mean ratios of D$_{beam}$/D$_{25}$ ($\sim$0.4, where D$_{beam}$=22$\arcsec$) for the SEST galaxies and the predicted ratios of D$_{CO}$/D$_{25}\sim$0.5 for typical spiral galaxies. The values of D$_{CO}$, defined as the diameter of the canonical distribution of CO in Virgo spirals which contains 70$\%$ of the total CO flux (@kenney88), indicate that molecular gas is highly concentrated in the inner optical disks. Therefore, the reported differences (a factor of $\sim$10!) in the molecular gas content between polar rings and counterrotators can hardly be attributed to a systematic undersampling of counterrotators, considering also that undersampling in CO maps affects polar ring galaxies to a comparable extent. Furthermore, results obtained using HI as a tracer of cold gas (much less affected by undersampling), confirm a similar trend as that shown by CO: polar rings have a gas content significantly higher than counterrotators.
Discussion and conclusions
==========================
Gas and stars along polar orbits can be explained as the result of the acquisition of cold infalling gas by an accreting galaxy. The accreted gas can smear out into a ring after a few orbital periods. The fate of this ring will depend on its orientation, relative to the mass distribution of the accretor, and most importantly, on the mass/self-gravity of the gas. A polar ring may form after a high angle impact with gas which has a spin perpendicular to the equatorial plane of the accretor, remaining in an equilibrium configuration for several Hubble times. In contrast, counterrotating galaxies may form after a low angle impact with a gas disk which has a spin antiparallel to that of the accreting system.
In the most general case, however, the impact angle is intermediate between polar and planar. In this case the orbits of gas clouds will experience differential precession in the non-spherical potential of the galaxy, characterized by a quadrupole component [@steiman82]. The latter applies for disk galaxies (axisymmetric) and ellipticals (triaxials). @sparke86 and @arnaboldi94 have studied in detail the dynamics of self-gravitating annuli of matter inclined to the principal axes of axisymmetric and triaxial potentials. If the strength of gas self-gravity is negligible, the inclined ring may rapidly settle towards the equatorial plane, appearing either as a [*co*]{}- or as a [*counter*]{}-rotating disk. In contrast, if the gas ring is [*heavy*]{}, the self-coupling can stabilise the ring for several Hubble times. In an intermediate case, several subrings (near polar or close to the equator), characterized by different precessing rates, can coexist in a single galaxy (e.g. NGC 660).
Studying the morphology of polar rings, @vangorkom and @sage1 have found that their ages may vary from 400 Myr to $\ge$ 4 Gyr, if the smooth rings are the oldest. Some polar rings or inclined rings could be the result of recent acquisitions, whereas others appear to be evolved systems. However, twisted polar rings are uncommon and some have had time to form stars. This requires the existence of a stabilising mechanism. The observations of atomic and molecular gas show that the quantity of gas mass present in polar rings is sufficient to stabilise them through self-gravity (@vangorkom [@sage1; @gall97], and [*this work*]{}). We derived a global content of cold gas (M$_{gas}$) in polar rings which is 1–2 orders of magnitude higher than in normal galaxies.
In contrast to polar rings, the derived content of cold gas in counterrotators is close to normal. Although counterrotators and polar rings probably share a common origin, the estimated gas masses confirm that [*light*]{} gas rings may have evolved faster. If the mass of gas originally accreted is not sufficient to stabilise the ring through self-gravity, the ring settles toward the equatorial plane in less than a Hubble time. In this case, the merger relic could be a counterrotator. Once the gas disk has settled to the plane with an antiparallel spin it can interact with the gas of the primary disk. Since the two components have opposite rotating directions, there can be large-scale shocks and angular momentum annihilation when they come into contact. Near this transition region the transformation of atomic into molecular gas could be enhanced, especially if the primordial gas content is high, i.e., for late-type accretors. Confirming these expectations, the measured M(H$_2$)/M$_{HI}$ ratio seems larger in counterrotators than in normal galaxies for types t$>$0.
In the course of this process, a starburst might be triggered in the circumnuclear molecular gas disk [@santi2]. The time-scale for gas infall could be extremely short, being close to the free-fall time, i.e., $\sim$10$^{7-8}$Myr. The mass of gas involved in the starburst episode however is kept low enough (10$^{8-9}$M$_{\sun}$) for a typical counterrotating galaxy. In polar rings, although the cold gas content is larger than the one in normal galaxies, star formation in the dynamically stable ring proceeds calmly. Confirming this scenario it was found that counterrotators, polar rings and normal galaxies have a similar content of hot gas, according to their normalized X-ray luminosities. Also, the normalized L$_{FIR}$ lies within the typical boundaries of aged (T$\sim$1Gy) mild starbursts, far from the values characteristic of massive mergers (see @read).
Acknowledgments
We would like to thank dr. G. Paturel for kindly making available to the authors the FIR raw data of LEDA database and to Dr. F. Ochsenbein for the changes made to the Vizier’s query form after our request. Thanks to the referee’s comments for useful suggestions on the statistical analysis. This research made use of Vizier service [@vizier] and of NASA’s Astrophysics Data System Abstract Service, mirrored in CDS of Strasbourg. SGB and ARF thank financial support from the Spanish CICYT under grant number PB96-0104 and CICYT-PNIE under grant number 1FD1997-1442. GG has made use of funds from University of Padova (Fondi 60%-2000).
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[^1]: Based on observations collected at SEST telescope, European Southern Observatory, La Silla, Chile.
[^2]: Table 1 is only available in electronic form.
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abstract: 'The evidence for a quantum phase transition under the superconducting dome in the high-$T_c$ cuprates has been controversial. We report low temperature normal state thermopower(S) measurements in electron-doped Pr$_{2-x}$Ce$_{x}$CuO$_{4-\delta}$ as a function of doping (*x* from 0.11 to 0.19). We find that at 2 K both S and S/T increase dramatically from *x*=0.11 to 0.16 and then saturate in the overdoped region. This behavior has a remarkable similarity to previous Hall effect results in Pr$_{2-x}$Ce$_{x}$CuO$_{4-\delta}$. Our results are further evidence for an antiferromagnetic to paramagnetic quantum phase transition in electron-doped cuprates near *x*=0.16.'
author:
- Pengcheng Li$^1$
- 'K. Behnia$^2$'
- 'R. L. Greene$^1$'
title: 'Evidence for a quantum phase transition in electron-doped Pr$_{2-x}$Ce$_{x}$CuO$_{4-\delta}$ from thermopower measurements'
---
The existence of a quantum phase transition at a doping under the superconducting dome in high-$T_c$ superconductors is still controversial. Evidence for a quantum critical point has been given for hole-doped cuprates[@Tallon; @Loram; @Ando] but the T=0 normal state is difficult to access because of the large critical field(H$_{c2}$). Electron-doped cuprates have a relatively low H$_{c2}$ and several studies have suggested that a quantum phase transition exists in those cuprates. Electrical transport[@Yoram] on electron-doped Pr$_{2-x}$Ce$_{x}$CuO$_{4-\delta}$(PCCO) shows a dramatic change of Hall coefficient around doping $x_c$=0.16, which indicates a Fermi surface rearrangement at this critical doping. Optical conductivity experiments[@Zimmers] revealed that a density-wave-like gap exists at finite temperatures below the critical doping $x_c$ and vanishes when $x\geq x_c$. Neutron scattering experiments[@Kang] on Nd$_{2-x}$Ce$_{x}$CuO$_{4-\delta}$(NCCO) found antiferromagnetism as the ground state below the critical doping while no long range magnetic order was observed above $x_c$. Other suggestive evidence[@Fournier] comes from the observation of a low temperature normal state insulator to metal crossover as a function of doping, and the disappearance of negative spin magnetoresistance at a critical doping[@Yoramupturn]. All these experiments strongly suggest that an antiferromagnetic(AFM) to paramagnetic quantum phase transition(QPT) occurs under the superconducting dome in the electron-doped cuprates.
The quantum phase transition in electron-doped cuprates is believed to be associated with a spin density wave(SDW) induced Fermi surface reconstruction[@Lin; @Zimmers]. Angle resolved photoemission spectroscopy(ARPES) experiments[@Armitage] on NCCO reveal a small electron-like pocket at$(\pi, 0)$ in the underdoped region and both electron- and hole-like Fermi pockets near optimal doping. This interesting feature is thought to arise as a result of the SDW instability that fractures the conduction band into two different parts[@Lin]. If one continues to increase the doping(above $x_c$), the weakening of the spin density wave leads to a large hole-like Fermi pocket centered at $(\pi, \pi)$ in the overdoped region[@Lin; @Matsui].
Nevertheless, the presence of a quantum critical point(QCP) under the superconducting dome in electron-doped cuprates is still quite controversial[@Greven]. Other experimental probes of the critical region are needed. In this paper, we present a systematic study of the magnetic field driven normal state thermopower on PCCO films. We find a doping dependence similar to that seen in the low temperature normal state Hall effect measurements[@Yoram]. From a simple free electron model comparison of these two quantities, we find a strikingly similar behavior of the effective number of carriers. This strongly suggests that a quantum phase transition takes place near x=0.16 in PCCO.
High quality PCCO films with thickness about 3000Å were fabricated by pulsed laser deposition on SrTiO$_3$ substrates (10$\times$5 mm$^2$). Detailed information can be found in our previous papers[@Peng; @Maiser]. The films were characterized by AC susceptibility, resistivity measurements and Rutherford Back Scattering(RBS).
High resolution thermopower is measured using a steady state method by switching the temperature gradient to cancel the Nernst effect and other possible background contributions. The sample is mounted between two thermally insulated copper blocks. The temperature gradient is built up by applying power to heaters on each block and the gradient direction is switched by turning on or off the heaters. The temperature gradient is monitored by two Lakeshore Cernox bare chip thermometers. Thermopower data is taken when the gradient is stable and averaged for many times to reduce the systematic error. The voltage leads are phosphor bronze, which has a small thermopower even at high field[@Wangyy]. The thermopower contribution from the wire is calibrated against YBa$_2$Cu$_3$O$_7$(T$_{c}$=92 K) for T$<$90 K and Pb film for T$>$90 K, and is subtracted out to get the absolute thermopower of the PCCO sample.
We measured the zero field and in field resistivity of all the doped PCCO films. The results are similar to our previous report[@Yoram]. A 9 T magnetic field(H$\parallel$c) is enough to suppress the superconductivity for all the dopings. This enables us to investigate the low temperature normal state properties in PCCO. A low temperature resistivity upturn is seen for doping below *x*=0.16, which suggests a possible insulator to metal crossover as a function of doping[@Fournier].
Thermopower is measured on the PCCO films doped from *x*=0.11 to 0.19. In zero field, a sharp superconducting transition is clearly seen in the thermopower. In the inset of Fig. \[fig1\], we show the thermopower S of *x*=0.16(T$_c$=16.5 K) as a function of temperature. Our high resolution thermopower setup enables us to observe small changes of signal. When the sample goes to the superconducting state, S=0, a small change $\triangle$S=0.5 $\mu$V/K is easily detectable, which indicates a better sensitivity than our previous one-heater-two-thermometer setup[@Budhani]. We also show the Hall coefficient R$_H$ as a function of temperature for the same film in the graph. A sign change of both S and R$_H$ is observed at the same temperature.
In the main panel of Fig. \[fig1\], we show the zero field thermopower for all the superconducting films. A clear superconducting transition is seen in these films. The normal state S(T$>$T$_c$) is negative in the underdoped region. It becomes positive in the overdoped region at low temperature(to be shown later). The magnitude of S in the underdoped region is large as expected for a system with less charge carrier density while it is much smaller in the overdoped region. Previous zero field thermopower measurements on NCCO crystals[@Wang] are qualitatively similar to our data.
When a 9 T magnetic field is applied along the c-axis, the superconducting films are driven to the normal state for T$<$T$_c$. As seen from the inset of Fig. \[fig1\], when the superconductivity is destroyed, the normal state thermopower is obtained. In Fig. \[fig2\], we show the normal state thermopower for all the films. The low temperature(T$<$15 K) normal state thermopower is shown in the inset. We showed in Fig. \[fig1\] that for *x*=0.16 the thermopower changes from negative to positive for T$<$30 K, in good agreement with the Hall effect measurements[@Yoram]. For the overdoped films *x*=0.17 and 0.18, we observe similar behavior with a sign change occurring below 45 K and 60 K respectively. However, the thermopower is always positive for *x*=0.19. Similar to the the Hall effect, the thermopower for *x*$\geq$0.16 is nearly same for T$<$10 K, as shown in the inset of Fig. \[fig2\]. The dramatic change of the thermopower at low temperature from *x*=0.15 to the overdoped region suggests a sudden Fermi surface rearrangement around the critical doping *x*=0.16.
In the Boltzmann picture, thermopower and electrical conductivity are related through the expression[@Ashcroft]: $$\label{1}
S=\frac{-\pi^{2}k_{B}^{2}T}{3e}\frac{\partial{ln\sigma(\epsilon)}}{\partial{\epsilon}}|_{\epsilon=E_{F}}$$ In the simple case of a free electron gas, this yields: $S/T=
\frac{-\pi^{2}k_{B}^{2}}{3e}\frac{N(\epsilon_F)}{n}$ (N($\epsilon_F$) is the density of states at the Fermi energy and $n$ is the total number of charge carriers). However, in real metals, the energy-dependence of the scattering time at the Fermi level, $(\frac{\partial\ln\tau(\epsilon)}{\partial\epsilon})_{\epsilon=\epsilon_{f}}$, also affects the thermopower. In the zero-temperature limit, it has been shown that this term also becomes proportional to $\frac{N(\epsilon_F)}{n}$ when the impurity scattering dominates[@Miyake]. In electron-doped cuprates, there is strong evidence[@Yoram] for impurity scattering at low temperatures. The residual resistivity is about 50 $\mu\Omega$-cm for an optimally-doped film, which is quite large compared to clean metals, and the temperature dependence of the resistivity becomes almost constant below 20 K. This is all suggestive of strong impurity scattering. The scattering most likely comes from Ce and oxygen disorder and one would expect a similar disorder at all dopings, although this is hidden by the anomalous (and unexplained) resistivity upturn for the lower dopings. Therefore, we expect that the thermopower is proportional to N(E$_F$)/n will be a valid approximation for our electron-doped PCCO films. This theory thus provides a solid theoretical basis for an experimental observation: in a wide variety of correlated metals, there is an experimental correlation between the magnitude of thermopower and specific heat in the zero-temperature limit[@Behnia].
Let us examine our data with this picture in mind. Fig. \[fig3\](a) presents S/T as a function of temperature below 40 K for all the doped films. As seen in the figure, there is a dramatic difference between the underdoped and the overdoped films. For underdoped, S/T displays a strong temperature dependence below 20 K, which is reminiscent of the low temperature upturn in resistivity and Hall effect[@Fournier; @Yoram]. One possible explanation for this feature would be charge localization [@Fournier3]. If all, or some of, the itinerant carriers localize at very low temperatures, then the decrease in conductivity is expected to be concomitant with an increase in the entropy per itinerant carrier (which is the quantity roughly measured by S/T). We find this to be qualitatively true as shown in Fig. \[fig4\], which displays S/T and conductivity for *x*=0.11 in a semilog plot. Below 10 K, both quantities are linear functions of $\log$T. Note that for the resistivity, it has been shown[@Fournier] that the logarithmic divergence saturates below 1 K. Therefore, further thermopower measurements below 2 K would be very useful.
In contrast to the underdoped films, the temperature dependence of S/T in the overdoped region is weaker and there is clearly a finite S/T even at zero temperature. Taking the magnitude of S/T at 2 K as our reference, we can examine the doping dependence of the ratio $\frac{N(\epsilon_F)}{n}$ for itinerant carriers at this temperature. Fig. \[fig3\](b) presents the doping dependence of S/T at 2 K. A strong doping dependence for $x\leq$0.16, a sharp kink around *x*=0.16 and a saturation in the overdoped region are visible. The dramatic change of S/T at low temperatures from the underdoped to overdoped regions is similar to the Hall effect[@Yoram] at 0.35 K, in which a sharp kink was observed around *x*=0.16. Both S/T and R$_H$ change from negative in the underdoped region to a saturated positive value above *x*=0.16.
The similarity of the doping dependence of S/T and R$_H$ implies a common physical origin. To explore the relation between S/T and R$_H$, let us assume a simple free electron model, where thermopower displays a very simple correlation with the electronic specific heat, $C_{el}= \frac{\pi^{2}k_{B}^{2}T}{3}N(\epsilon_F)$. Following the analysis of Ref.20, a dimensionless quantity $$\label{2}
q=\frac{S}{T}\frac{N_{Av}e}{\gamma}$$ can be defined($N_{Av}$ is Avogadro’s number and $\gamma=C_{el}/T$), which is equal to $N_{Av}/n$. For a simple metal, R$_H=V/ne$ ($V$ is the total volume). If we define $$\label{3}
q'=R_He/V_m$$ where $V_m$ is unit cell volume, then $q'$ is also equal to $N_{Av}/n$. By this simple argument, we can compare S and R$_H$ directly. Because we do not have data for $\gamma$ except at optimal doping, we assume it does not change much with doping. With the $\gamma$ value($4mJ/K^2mole$)[@Hamza] for *x*=0.15 and S/T and R$_H$ at 2 K, we can plot both $q$ and $q'$ together, as shown in Fig. \[fig5\]. We find a remarkable similarity in the doping dependence of these two dimensionless quantities, both in trend and in magnitude. Note that no dramatic changes in either $q$ or $q'$ are observed near *x*=0.13, where it is claimed that AFM long range order vanishes[@Greven] from recent neutron scattering measurements. We should mention that assuming a constant $\gamma$ as a function of doping in our range of investigation (*x*=0.11 to 0.19) is, of course, subject to caution due to a lack of experimental data. However, it has been found[@Hamza] that the specific heat coefficient $\gamma$ is the same for an as-grown crystal and a superconducting Pr$_{1.85}$Ce$_{0.15}$CuO$_4$ crystal. Neutron scattering studies have shown that an as-grown *x*=0.15 crystal is equivalent to an annealed Pr$_{1.88}$Ce$_{0.12}$CuO$_4$ crystal[@Greven2]. This strongly suggests that $\gamma$ will not change much with Ce doping at least in the critical range around optimal doping. Therefore, no significant change in the doping dependence of $q$ due to this correction is expected.
We believe that the saturation of S/T in the overdoped region is a result of the Fermi surface rearrangement due to the vanishing of antiferromagnetism above a critical doping. To our knowledge, there is no theoretical prediction for the doping dependence of the thermopower in an antiferromagnetic quantum critical system. Although the temperature dependence of thermopower near zero temperature is given by Paul *et al*.[@Paul] for such a system near critical doping, we are not yet able to access the very low temperature region(T$<$2 K) to test these predictions in PCCO. Nevertheless, an amazing agreement between thermopower and Hall effect measurements is shown in our simple free electron model. This model is certainly oversimplified since there is strong evidence for two types of carriers near optimal doping[@Jiang; @Fournier2; @Gollnik]. But, much of this transport data[@Jiang; @Fournier2; @Gollnik] implies that one type of carrier dominates at low temperature. Thus a simple model may be reasonable. However, to better understand this striking result a more detailed theoretical analysis will be needed.
Interestingly, the number $q$ in overdoped PCCO is close to 1. It was shown that when $q$ is close to unity, a Fermi liquid behavior is found in many strongly correlated materials[@Behnia]. This suggests that overdoped PCCO is more like a Fermi liquid metal than underdoped PCCO. When x is above the critical doping *x*=0.16, $q$ and $q'$ are close to $1/(1-x)$, which suggests that the hole-like Fermi surface is recovered in accordance with local density approximation band calculations and the Luttinger theorem.
In summary, we performed high resolution measurements to investigate the low temperature normal state thermopower(S) of electron-doped cuprates Pr$_{2-x}$Ce$_{x}$CuO$_{4-\delta}$(PCCO). We find a strong correlation between S/T and the Hall coefficient (R$_H$) at 2 K as a function of doping. Using a simple free electron model, which relates thermopower to the electronic specific heat, we conclude that our observations support the view that a quantum phase transition occurs near *x*=0.16 in the PCCO system.
This work is supported by NSF Grant DMR-0352735. We thank Drs. Andy Millis and Victor Yakovenko for fruitful discussions.
Y. Ando *et al*., Phys. Rev. Lett.**92**, 247004 (2004)and references therein. J. L. Tallon and J. W. Loram Physica C **349**, 53 (2001). G.Q. Zheng *et al*., Phys. Rev. Lett. **94**, 047006 (2005). Y. Dagan *et al*., Phys. Rev. Lett.**92**, 167001 (2004). A. Zimmers *et al*., Europhys.. Lett.**70**, 225 (2005). H. Kang *et al*., Nature(London) **423**, 522 (2003). P. Fournier *et al*., Phys. Rev. Lett.**81**, 4720 (1998). Y. Dagan *et al*., Phys. Rev. Lett.**94**, 057005 (2005). J. Lin and A. J. Millis, Phys. Rev. B **72**, 214506 (2005). N. P.Armitage,*et al*.,Phys. Rev. Lett. **88**, 257001 (2002). H. Matsui *et al*.,Phys. Rev. Lett. **94**, 047005 (2005); H. Matsui, *et al.*, Phys. Rev. Lett. **95**, 017003 (2005).
Recent neutron scattering experiments on $Nd_{2-x}Ce_xCuO_4$ argue that the QCP is at x=0.13 and that the superconductivity and AFM do not coexist, E.M. Motoyama *et al*., cond-mat/0609386. J. L. Peng *et al*., Phys. Rev. B **55** R6145 (1997). E. Maiser *et al*.,Physica(Amsterdam) **297C**, 15(1998). Y. Wang *et al*., Nature(London) **423**, 425 (2003). R. C. Budhani *et al*., Phys. Rev. B **65**, 100517(R) (2002). C. H. Wang *et al*., Phys. Rev. B **72**,132506 (2005). N. W. Ashcroft and N. D. Mermin, *Solid State Physics*, Saunders College Publishing (1976).
K. Miyake and H. Kohno, J. Phys. Soc. Jpn, **74**, 254 (2005). K. Behnia, D. Jaccard and J. Flouquet, J. Phys.:Condens. Matter **16**, 5187 (2004). P. Fournier *et al*., Phys. Rev. B **62**, R11993 (2000). H. Balci and R. L. Greene, Phys. Rev. B **70** 140508(R) (2004). P. K. Mang *et al.*, Phys. Rev. Lett. *93*, 027002 (2004). I. Paul and G. Kotliar, Phys. Rev. B **64**, 184414 (2001). W. Jiang *et al*., Phys. Rev. Lett.**73**, 1291 (1994). P. Fournier *et al*., Phys. Rev. B **56**, 14149(1997). F. Gollnik and M. Naito, Phys. Rev. B **58**, 11734(1998).
|
---
abstract: 'Recent research on discourse relations has found that they are cued not only by discourse markers (DMs) but also by other textual signals and that signaling information is indicative of genres. While several corpora exist with discourse relation signaling information such as the Penn Discourse Treebank (PDTB, @PrasadEtAl2008) and the Rhetorical Structure Theory Signalling Corpus (RST-SC, @das2018rst), they both annotate the Wall Street Journal (WSJ) section of the Penn Treebank (PTB, @MarcusSantoriniMarcinkiewicz1993), which is limited to the news domain. Thus, this paper adapts the signal identification and anchoring scheme [@liu2019discourse] to three more genres, examines the distribution of signaling devices across relations and genres, and provides a taxonomy of indicative signals found in this dataset.'
author:
- |
Yang Liu\
Department of Linguistics\
Georgetown University\
[yl879@georgetown.edu]{}
bibliography:
- 'naaclhlt2019.bib'
title: |
Beyond The Wall Street Journal:\
Anchoring and Comparing Discourse Signals across Genres
---
Introduction {#intro}
============
Sentences do not exist in isolation, and the meaning of a text or a conversation is not merely the sum of all the sentences involved: an informative text contains sentences whose meanings are relevant to each other rather than a random sequence of utterances. Moreover, some of the information in texts is not included in any one sentence but in their arrangement. Therefore, a high-level analysis of discourse and document structures is required in order to facilitate effective communication, which could benefit both linguistic research and NLP applications. For instance, an automatic discourse parser that successfully captures how sentences are connected in texts could serve tasks such as information extraction and text summarization.
A discourse is delineated in terms of relevance between textual elements. One of the ways to categorize such relevance is through *coherence*, which refers to semantic or pragmatic linkages that hold between larger textual units such as <span style="font-variant:small-caps;">Cause, Contrast</span>, and <span style="font-variant:small-caps;">Elaboration</span> etc. Moreover, there are certain linguistic devices that systematically signal certain discourse relations: some are generic signals across the board while others are indicative of particular relations in certain contexts. Consider the following example from the Georgetown University Multilayer (GUM) corpus [@zeldes2017gum],[^1] in which the two textual units connected by the DM *but* form a <span style="font-variant:small-caps;">Contrast</span> relation, meaning that the contents of the two textual units are comparable yet not identical.
. Related cross-cultural studies have resulted in insufficient statistical power, ***but*** interesting trends (e.g., Nedwick, 2014 ). \[academic\_implicature\] \[implicature\]
However, the coordinating conjunction *but* is also a frequent signal of another two relations that can express adversativity: <span style="font-variant:small-caps;">Concession</span> and <span style="font-variant:small-caps;">Antithesis</span>. <span style="font-variant:small-caps;">Concession</span> means that the writer acknowledges the claim presented in one textual unit but still claims the proposition presented in the other discourse unit while <span style="font-variant:small-caps;">Antithesis</span> dismisses the former claim in order to establish or reinforce the latter. In spite of the differences in their pragmatic functions, these three relations can all be frequently signaled by the coordinating conjunction *but*: symmetrical <span style="font-variant:small-caps;">Contrast</span> as in \[implicature\], <span style="font-variant:small-caps;">Concession</span> as in \[implicature\_concession\], and <span style="font-variant:small-caps;">Antithesis</span> as in \[chomsky\_antithesis\]. It is clear that *but* is a generic signal here as it does not indicate strong associations with the relations it signals.
. This was a very difficult decision, ***but*** one that was made with the American public in mind. \[news\_nasa\] \[implicature\_concession\]
. NATO had never rescinded it, ***but*** they had and started some remilitarization. \[interview\_chomsky\] \[chomsky\_antithesis\]
As suggested by @taboada2003rhetorical, some discourse signals are indicative of certain genres: they presented how to characterize appointment-scheduling dialogues using their rhetorical and thematic patterns as linguistic evidence and suggested that the rhetorical and the thematic analysis of their data can be interpreted functionally as indicative of this type of task-oriented conversation. Furthermore, the study of the classification of discourse signals can serve as valuable evidence to investigate their role in discourse as well as the relations they signal.
One limitation of the RST Signalling Corpus is that no information about the location of signaling devices was provided. As a result, @liu2019discourse presented an annotation effort to anchor discourse signals for both elementary and complex units on a small set of documents in RST-SC (see Section \[anchoring\] for details). The present study addresses methodological limitations in the annotation process as well as annotating more data in more genres in order to investigate the distribution of signals across relations and genres and to provide both quantitative and qualitative analyses on signal tokens.
Background
==========
**Rhetorical Structure Theory** (RST, @mann1988rhetorical) is a well-known theoretical framework that extensively investigates discourse relations and is adopted by @das2017signalling and the present study. RST is a functional theory of text organization that identifies hierarchical structure in text. The original goals of RST were discourse analysis and proposing a model for text generation; however, due to its popularity, it has been applied to several other areas such as theoretical linguistics, psycholinguistics, and computational linguistics [@taboada2006applications].
RST identifies hierarchical structure and nuclearity in text, which categorizes relations into two structural types: <span style="font-variant:small-caps;">Nucleus-Satellite</span> and <span style="font-variant:small-caps;">Multinuclear</span>. The <span style="font-variant:small-caps;">Nucleus-Satellite</span> structure reflects a hypotactic relation whereas the <span style="font-variant:small-caps;">Multinuclear</span> structure is a paratactic relation [@taboada2013annotation]. The inventory of relations used in the RST framework varies widely, and therefore the number of relations in an RST taxonomy is not fixed. The original set of relations defined by @mann1988rhetorical included 23 relations. Moreover, RST identifies textual units as Elementary Discourse Units (EDUs), which are non-overlapping, contiguous spans of text that relate to other EDUs [@zeldes2017gum]. EDUs can also form hierarchical groups known as complex discourse units.
Relation Signaling {#rs}
------------------
When it comes to relation signaling, the first question to ask is what a signal is. In general, signals are the means by which humans identify the realization of discourse relations. The most typical signal type is DMs (e.g. ‘although’) as they provide explicit and direct linking information between clauses and sentences. As mentioned in Section \[intro\], the lexicalized discourse relation annotations in PDTB have led to the discovery of a wide range of expressions called <span style="font-variant:small-caps;">Alternative Lexicalizations (*AltLex*)</span> [@prasad2010realization]. RST-SC provides a hierarchical taxonomy of discourse signals beyond DMs (see Figure \[taxonomy\] for an illustration, reproduced from @das2017signalling [p.752].
Intuitively, DMs are the most obvious linguistic means of signaling discourse relations, and therefore extensive research has been done on DMs. Nevertheless, focusing merely on DMs is inadequate as they can only account for a small number of relations in discourse. To be specific, @das2017signalling reported that among all the 19,847 signaled relations (92.74%) in RST-SC (i.e. 385 documents and all 21,400 annotated relations), relations exclusively signaled by DMs only account for 10.65% whereas 74.54% of the relations are exclusively signaled by other signals, corresponding to the types they proposed.
![Taxonomy of Signals in RST-SC (Fragment).[]{data-label="taxonomy"}](taxonomy.png){width="75mm"}
The Signal Anchoring Mechanism {#anchoring}
------------------------------
As mentioned in Section \[intro\], RST-SC does not provide information about the location of discourse signals. Thus, @liu2019discourse presented an annotation effort to anchor signal tokens in the text, with six categories being annotated. Their results showed that with 11 documents and 4,732 tokens, 923 instances of signal types/subtypes were anchored in the text, which accounted for over 92% of discourse signals, with the signal type *semantic* representing the most cases (41.7% of signaling anchors) whereas discourse relations anchored by DMs were only about 8.5% of anchor tokens in this study, unveiling the value of signal identification and anchoring.
Neural Modeling for Signal Detection
------------------------------------
@Zeldes2018 trained a Recurrent Neural Network (RNN) model for the task of relation classification, and then latent associations in the network were inspected to detect signals. It is relatively easy to capture DMs such as ‘then’ or a relative pronoun ‘which’ signaling an <span style="font-variant:small-caps;">Elaboration</span>. The challenge is to figure out what features the network needs to know about beyond just word forms such as meaningful repetitions and variable syntactic constructions. With the human annotated data from the current project, it is hoped that more insights into these aspects can help us engineer meaningful features in order to build a more informative computational model.
Methodology
===========
**Corpus**. The main goal of this project is to anchor and compare discourse signals across genres, which makes the Georgetown University Multilayer (GUM) corpus the optimal candidate, in that it consists of eight genres including interviews, news stories, travel guides, how-to guides, academic papers, biographies, fiction, and forum discussions. Each document is annotated with different annotation layers including but not limited to dependency (`dep`), coreference (`ref`), and rhetorical structures (`rst`). For the purpose of this study, the `rst` layer is used as it includes annotation on discourse relations, and signaling information will be anchored to it in order to produce a new layer of annotation. However, it is worth noting that other annotation layers are great resources to delve into discourse signals on other levels.
![A Visualization of How Strongly Each Genre Signals in the GUM Corpus.[]{data-label="gurt"}](signal_strengths.png){width="80mm"}
Moreover, due to time limitations and the fact that this is the first attempt to apply the taxonomy of signals and the annotation scheme to other genres outside RST-DT’s newswire texts, four out of eight genres in the GUM corpus were selected: *academic*, *how-to guides*, *interviews*, and *news*, which include a collection of 12 documents annotated for discourse relations. The rationale for choosing these genres is that according to @Zeldes2018’s neural approach to discourse signal prediction on the GUM corpus, how-to guides and academic articles in the GUM corpus signal most strongly, with interviews and news articles slightly below the average and fiction and reddit texts the least signaled, as shown in Figure \[gurt\] (reproduced from @zeldes2018neural [p.19]). It is believed that the selection of these four genres is a good starting point of the topic under discussion.
**Annotation Tool**. One of the reasons that caused low inter-annotator agreement (IAA) in @liu2019discourse is the inefficient and error prone annotation tools they used: no designated tools were available for the signal anchoring task at the time. We therefore developed a better tool tailored to the purpose of the annotation task. It is built over an interface offering full RST editing capabilities called rstWeb [@Zeldes2016] and provides mechanisms for viewing and editing signals [@Gessler2019].
**Annotation Reliability**. In order to evaluate the reliability of the scheme, a revised inter-annotator agreement study was conducted using the same metric and with the new interface on three documents from RST-SC, containing 506 tokens with just over 90 signals. Specifically, agreement is measured based on token spans. That is, for each token, whether the two annotators agree it is signaled or not. The results demonstrate an improvement in Kappa, 0.77 as opposed to the previous Kappa 0.52 in @liu2019discourse.
**Taxonomy of Discourse Signals**. The most crucial task in signaling annotation is the selection of signal types. The taxonomy of discourse signals used in this project is adapted from that of @das2017signalling, with additional types and subtypes to better suit other genres. Two new types and four new subtypes of the existing types are proposed: the two new types are *Visual* and *Textual* in which the subtype of the former is *Image* and the subtypes of the latter are *Title, Date,* and *Attribution*. The three new subtypes are *Modality* under the type *Morphological* and *Academic article layout, Interview layout* and *Instructional text layout* under the type *Genre*.
**Signal Anchoring Example**. Semantic features have several subtypes, with *lexical chain* being the most common one. Lexical chains are annotated for words with the same lemma or words or phrases that are semantically related. Another characteristic of lexical chains is that words or phrases annotated as lexical chains are open to different syntactic categories. For instance, the following example shows that the relation <span style="font-variant:small-caps;">Restatement</span> is signaled by a *lexical chain* item corresponding to the phrase *a lot of* in the nucleus span and *quantity* in the satellite span respectively.
. \[They compensate for this by creating the impression that they have ***a lot of*** friends –\]~N~ \[they have a ‘***quantity***, not quality’ mentality.\]~S~ \[whow\_arrogant\] \[lc\]
Results & Analysis
==================
This pilot study annotated 12 documents with 11,145 tokens across four different genres selected from the GUM corpus. *Academic articles, how-to guides*, and *news* are written texts while *interview* is spoken language. Generally speaking, all 20 relations used in the GUM corpus are signaled and anchored. However, this does not mean that all occurrences of these relations are signaled and anchored. There are several signaled but unanchored relations, as shown in Table \[unanchored\]. In particular, the 5 unsignaled instances of the relation <span style="font-variant:small-caps;">Joint</span> result from the design of the annotation scheme (see Section \[scheme\] for details). Additionally, the unanchored signal types and subtypes are usually associated with high-level discourse relations and usually correspond to genre features such as *interview layout* in interviews where the conversation is constructed as a question-answer scheme and thus rarely anchored to tokens.
With regard to the distribution of the signal types found in these 12 documents, the 16 distinct signal types amounted to 1263 signal instances, as shown in Table \[overviewresults\]. There are only 204 instances of DMs out of all 1263 annotated signal instances (16.15%) as opposed to 1059 instances (83.85%) of other signal types. In RST-SC, DM accounts for 13.34% of the annotated signal instances as opposed to 81.36%[^2] of other signal types [@das2017signalling]. The last column in Table \[overviewresults\] shows how the distribution of each signal type found in this dataset compares to RST-SC. The reason why the last column does not sum to 100% is that not all the signal types found in RST-SC are present in this study such as the combined signal type *Graphical + syntactic*. And since *Textual* and *Visual* are first proposed in this study, no results can be found in RST-SC, and the category *Unsure* used in RST-SC is excluded from this project.
Distribution of Signals across Relations {#distribution-rels}
----------------------------------------
Table \[relation\_based\] provides the distribution of discourse signals regarding the relations they signal. The first column lists all the relations used in the GUM corpus. The second column shows the number of signal instances associated with each relation. The third and fourth columns list the most signaled and anchored type and subtype respectively.
The results show a very strong dichotomy of relations signaled by DMs and semantic-related signals: while DMs are the most frequent signals for five of the relations – <span style="font-variant:small-caps;">Condition, Concession, Antithesis, Cause</span>, and <span style="font-variant:small-caps;">Circumstance</span>, the rest of the relations are all most frequently signaled by the type *Semantic* or *Lexical*, which, broadly speaking, are all associated with open-class words as opposed to functional words or phrases. Furthermore, the type *Lexical* and its subtype *indicative word* seem to be indicative of <span style="font-variant:small-caps;">Justify</span> and <span style="font-variant:small-caps;">Evaluation</span>. This makes sense due to the nature of the relations, which requires writers’ or speakers’ opinions or inclinations for the subject under discussion, which are usually expressed through positive or negative adjectives (e.g. *serious, outstanding, disappointed*) and other syntactic categories such as nouns/noun phrases (e.g. *legacy, excitement, an unending war*) and verb phrases (e.g. *make sure, stand for*). Likewise, words like *Tips, Steps*, and *Warnings* are indicative items to address communicative needs, which is specific to a genre, in this case, the how-to guides. It is also worth pointing out that <span style="font-variant:small-caps;">Evaluation</span> is the only discourse relation that is not signaled by any DMs in this dataset.
** **
---------------------------------------------------------------- ---- -------
<span style="font-variant:small-caps;">**Preparation**</span> 28 22.2
<span style="font-variant:small-caps;">**Solutionhood**</span> 11 32.35
<span style="font-variant:small-caps;">Joint</span> 5 1.92
<span style="font-variant:small-caps;">Background</span> 3 2.68
<span style="font-variant:small-caps;">Cause</span> 1 4
<span style="font-variant:small-caps;">Evidence</span> 1 4.2
<span style="font-variant:small-caps;">Motivation</span> 1 4.76
: Distribution of Unanchored Relations.[]{data-label="unanchored"}
[width=,center]{}
***signal\_type*** ***frequency*** ****** ******
------------------------ ----------------- -------- --------
Semantic 563 44.58 24.8
DM 204 16.15 13.34
Lexical 156 12.35 3.89
Reference 71 5.62 2.00
Semantic + syntactic 51 4.04 7.36
Graphical 46 3.64 3.46
Syntactic 44 3.48 29.77
Genre 30 2.38 3.22
Morphological 26 2.06 1.07
Syntactic + semantic 25 1.98 1.40
Textual 24 1.90 N/A
Numerical 8 0.63 0.09
Visual 7 0.55 N/A
Reference + syntactic 3 0.24 1.86
Lexical + syntactic 3 0.24 0.41
Syntactic + positional 2 0.16 0.23
Total 1263 100.00 92.9
: Distribution of Signal Types and its Comparison to RST-SC.[]{data-label="overviewresults"}
[width=,center]{}
** **
-------------------------------------------------------------- ----- ---------------- ------------------------
<span style="font-variant:small-caps;">Joint</span> 260 Semantic (147) lexical chain (96)
<span style="font-variant:small-caps;">Elaboration</span> 243 Semantic (140) lexical chain (96)
<span style="font-variant:small-caps;">Preparation</span> 129 Semantic (54) lexical chain (30)
<span style="font-variant:small-caps;">Background</span> 112 Semantic (62) lexical chain (42)
<span style="font-variant:small-caps;">Contrast</span> 68 Semantic (39) lexical chain (31)
<span style="font-variant:small-caps;">Restatement</span> 60 Semantic (34) lexical chain (28)
<span style="font-variant:small-caps;">**Concession**</span> 49 DM (23) DM (23)
<span style="font-variant:small-caps;">**Justify**</span> 49 Lexical (25) indicative word (23)
<span style="font-variant:small-caps;">**Evaluation**</span> 42 Lexical (31) indicative word (31)
<span style="font-variant:small-caps;">Solutionhood</span> 34 Semantic (12) lexical chain (5)
<span style="font-variant:small-caps;">**Condition**</span> 31 DM (25) DM (25)
<span style="font-variant:small-caps;">Antithesis</span> 31 DM (12) DM (12)
<span style="font-variant:small-caps;">Sequence</span> 26 Semantic (7) lexical chain (6)
<span style="font-variant:small-caps;">Cause</span> 25 DM (12) DM (12)
<span style="font-variant:small-caps;">Evidence</span> 24 Semantic (8) lexical chain (7)
<span style="font-variant:small-caps;">Result</span> 21 Semantic (8) lexical chain (7)
<span style="font-variant:small-caps;">Motivation</span> 21 Semantic (8) lexical chain (7)
<span style="font-variant:small-caps;">**Purpose**</span> 21 Syntactic (9) infinitival clause (7)
<span style="font-variant:small-caps;">Circumstance</span> 20 DM (11) DM (11)
: Distribution of Most Common Signals across Relations. []{data-label="relation_based"}
Even though some relations are frequently signaled by DMs such as <span style="font-variant:small-caps;">Condition</span> and <span style="font-variant:small-caps;">Antithesis</span>, most of the signals are highly lexicalized and indicative of the relations they indicate. For instance, signal tokens associated with the relation <span style="font-variant:small-caps;">Restatement</span> tend to be the repetition or paraphrase of the token(s). Likewise, most of the tokens associated with <span style="font-variant:small-caps;">Evaluation</span> are strong positive or negative expressions. As for <span style="font-variant:small-caps;">Sequence</span>, in addition to the indicative tokens such as *First & Second* and temporal expressions such as *later*, an indicative word pair such as *stop & update* can also suggest sequential relationship. More interestingly, world knowledge such as the order of the presidents of the United States (e.g. that Bush served as the president of the United States before Obama) is also a indicative signal for <span style="font-variant:small-caps;">Sequence</span>.
***relations***
------------------------------------------------------------ ----------------------------------------------------------------------------
<span style="font-variant:small-caps;">Joint</span> ; (16), **and** (15), also (10), *Professor Eastman* (3), he (3), they (2)
<span style="font-variant:small-caps;">Elaboration</span>
<span style="font-variant:small-caps;">Preparation</span> : (6), How to (2), Know (2), Steps (2), Getting (2)
<span style="font-variant:small-caps;">Background</span> Therefore, Indeed, build on, previous, *Bob McDonnell*, Looking back
<span style="font-variant:small-caps;">Contrast</span>
<span style="font-variant:small-caps;">Restatement</span>
<span style="font-variant:small-caps;">Concession</span>
<span style="font-variant:small-caps;">Justify</span>
<span style="font-variant:small-caps;">Evaluation</span>
<span style="font-variant:small-caps;">Solutionhood</span>
<span style="font-variant:small-caps;">Condition</span>
<span style="font-variant:small-caps;">Antithesis</span>
<span style="font-variant:small-caps;">Sequence</span>
<span style="font-variant:small-caps;">Cause</span>
<span style="font-variant:small-caps;">Evidence</span>
<span style="font-variant:small-caps;">Result</span>
<span style="font-variant:small-caps;">Motivation</span>
<span style="font-variant:small-caps;">Purpose</span>
<span style="font-variant:small-caps;">Circumstance</span>
------------------ --
***academic***
**
***news***
***interviews***
------------------ --
Another way of seeing these signals is to examine their associated tokens in texts, regardless of the signal types and subtypes. Table \[anchoredtokens\] lists some representative, generic/ambiguous (in boldface), and coincidental (in italics) tokens that correspond to the relations they signal. Each item is delimited by a comma; the *&* symbol between tokens in one item means that this signal consists of a word pair in respective spans. The number in the parentheses is the count of that item attested in this project; if no number is indicated, then that token span only occurs once. The selection of these single-occurrence items is random in order to better reflect the relevance in contexts. For instance, lexical items like *Professor Eastman* in <span style="font-variant:small-caps;">Joint</span>, *NASA* in <span style="font-variant:small-caps;">Elaboration</span>, *Bob McDonnell* in <span style="font-variant:small-caps;">Background</span>, and *NATO* in <span style="font-variant:small-caps;">Restatement</span> appear to be coincidental because they are the topics or subjects being discussed in the articles. These results are parallel to the findings in @Zeldes2018 [p.180], which employed a frequency-based approach to show the most distinctive lexemes for some relations in GUM.
Distribution of Signals across Genres {#distribution-genres}
-------------------------------------
Table \[genre\_based\] shows the distribution of the signaled relations in different genres. Specifically, the number preceding the vertical line is the number of signals indicating the relation and the percentage following the vertical line is the corresponding proportional frequency. The label <span style="font-variant:small-caps;">N/A</span> suggests that no such relation is present in the sample from that genre.
[width=,center]{}
***relations*** ***academic*** ***how-to guides*** ***news*** ***interview***
----------------------------------------------------------------- ------------------ --------------------- --------------------- ------------------
[ <span style="font-variant:small-caps;">Joint</span>]{} [ 65 | 23.13%]{} [ 76 | 18.67%]{} [ 65 | 25.39%]{} [ 54 | 16.77%]{}
[ <span style="font-variant:small-caps;">Elaboration</span>]{} [ 61 | 21.71%]{} [ 79 | 19.41%]{} [ 53 | 20.70%]{} [ 50 | 15.53%]{}
[ <span style="font-variant:small-caps;">Preparation</span>]{} [ 25 | 8.9%]{} [ 55 | 13.51]{} [ 15 | 5.86%]{} [ 34 | 10.56%]{}
[ <span style="font-variant:small-caps;">Background</span>]{} [ 33 | 11.74%]{} [ 24 | 5.9%]{} [ 28 | 10.94%]{} [ 27 | 8.39%]{}
[ <span style="font-variant:small-caps;">Contrast</span>]{} [ 17 | 6.05%]{} [ 21 | 5.16%]{} [ 19 | 7.42%]{} [ 11 | 3.42%]{}
[ <span style="font-variant:small-caps;">Restatement</span>]{} [ N/A]{} [ 20 | 4.91%]{} [ 11 | 4.3%]{} [ 29 | 9.01%]{}
[ <span style="font-variant:small-caps;">Concession</span>]{} [ 17 | 6.05%]{} [ 13 | 3.19%]{} [ 10 | 3.91%]{} [ 9 | 2.8%]{}
[ <span style="font-variant:small-caps;">Justify</span>]{} [ 1 | 0.36%]{} [ 11 | 2.7%]{} [ 15 | 5.86%]{} [ 22 | 6.83%]{}
[ <span style="font-variant:small-caps;">Evaluation</span>]{} [ 10 | 3.56%]{} [ 12 | 2.95%]{} [ 7 | 2.73%]{} [ 13 | 4.04%]{}
[ <span style="font-variant:small-caps;">Solutionhood</span>]{} [ 2 | 0.71%]{} [ 8 | 1.97%]{} [ N/A]{} [ 24 | 7.45%]{}
[ <span style="font-variant:small-caps;">Condition</span>]{} [ N/A]{} [ 25 | 6.14%]{} [ 3 | 1.17%]{} [ 3 | 0.93%]{}
[ <span style="font-variant:small-caps;">Antithesis</span>]{} [ 3 | 1.07%]{} [ 10 | 2.46%]{} [ 1 | 0.39%]{} [ 17 | 5.28%]{}
[ <span style="font-variant:small-caps;">Sequence</span>]{} [ 12 | 4.27%]{} [ 4 | 0.98]{} [ 5 | 1.95%]{} [ 5 | 1.55%]{}
[ <span style="font-variant:small-caps;">Cause</span>]{} [ 6 | 2.14%]{} [ 12 | 2.95]{} [ 6 | 2.34%]{} [ 1 | 0.31%]{}
[ <span style="font-variant:small-caps;">Evidence</span>]{} [ 10 | 3.56%]{} [ N/A]{} [ 5 | 1.95%]{} [ 9 | 2.8%]{}
[ <span style="font-variant:small-caps;">Result</span>]{} [ 3 | 1.07%]{} [ 6 | 1.47%]{} [ 6 | 2.34%]{} [ 6 | 1.86%]{}
[ <span style="font-variant:small-caps;">Motivation</span>]{} [ N/A]{} [ 21 | 5.16%]{} [ N/A]{} [ N/A]{}
[ <span style="font-variant:small-caps;">Purpose</span>]{} [ 14 | 4.98%]{} [ 5 | 1.23%]{} [ N/A]{} [ 2 | 0.62%]{}
[ <span style="font-variant:small-caps;">Circumstance</span>]{} [ 2 | 0.36%]{} [ 5 | 1.23%]{} [ 7 | 2.73%]{} [ 6 | 1.86%]{}
[ *Total*]{} [ 281 | 100%]{} [ **407** | 100%]{} [ **256** | 100%]{} [ 322 | 100%]{}
: Distribution of Signaled Relations across Genres.[]{data-label="genre_based"}
As can be seen from Table \[genre\_based\], how-to guides involve the most signals (i.e. 407 instances), followed by interviews, academic articles, and news. It is surprising to see that news articles selected from the GUM corpus are not as frequently signaled as they are in RST-SC, which could be attributed to two reasons. Firstly, the source data is different. The news articles from GUM are from Wikinews while the documents from RST-SC are Wall Street Journal articles. Secondly, RST-DT has finer-grained relations (i.e. 78 relations as opposed to the 20 relations used in GUM) and segmentation guidelines, thereby having more chances for signaled relations. Moreover, it is clear that <span style="font-variant:small-caps;">Joint</span> and <span style="font-variant:small-caps;">Elaboration</span> are the most frequently signaled relations in all four genres across the board, followed by <span style="font-variant:small-caps;">Preparation</span> in how-to guides and interviews or <span style="font-variant:small-caps;">Background</span> in academic articles and news, which is expected as these four relations all show high-level representations of discourse that involve more texts with more potential signals.
Table \[genre-tokens\] lists some signal tokens that are indicative of genre (in boldface) as well as generic and coincidental ones (in italics). The selection of these items follows the same criteria used in Section \[distribution-rels\]. Even though DMs *and* and *but* are present in all four genres, no associations can be established between these DMs and the genres they appear in. Moreover, as can be seen from Table \[genre-tokens\], graphical features such as semicolons, colons, dashes, and parentheses play an important role in relation signaling. Although these punctuation marks do not seem to be indicative of any genres, academic articles tend to use them more as opposed to other genres. Although some words or phrases are highly frequent, such as *discrimination* in academic articles, *arrogant people* in how-to guides, *IE6* in news, and *Sarvis* in interviews, they just seem to be coincidental as they happen to be the subjects or topics being discussed in the articles.
**Academic writing** is typically formal, making the annotation more straightforward. The results from this dataset suggest that academic articles contain signals with diverse categories. As shown in Table \[genre-tokens\], in addition to the typical DMs and some graphical features mentioned above, there are several lexical items that are very strong signals indicating the genre. For instance, the verb *hypothesized* and its synonym *posited* are indicative in that researchers and scholars tend to use them in their research papers to present their hypotheses. The phrase *based on* is frequently used to elaborate on the subject matter. Furthermore, Table \[genre-tokens\] also demonstrates that academic articles tend to use ordinal numbers such as *First* and *Second* to structure the text. Last but not least, the word *Albeit* indicating the relation <span style="font-variant:small-caps;">Concession</span> seems to be an indicative signal of academic writing due to the register it is associated with.
**How-to Guides** are the most signaled genre in this dataset. This is due to the fact that instructional texts are highly organized, and the cue phrases are usually obvious to identify. As shown in Table \[genre-tokens\], there are several indicative signal tokens such as the *wh*-word *How*, an essential element in instructional texts. Words like *Steps*, *Tips*, and *Warnings* are strongly associated with the genre due to its communicative needs. Another distinct feature of how-to guides is the use of imperative clauses, which correspond to verbs whose first letter is capitalized (e.g. *Know, Empty, Fasten, Wash*), as instructional texts are about giving instructions on accomplishing certain tasks and imperative clauses are good at conveying such information in a straightforward way.
**News articles**, like academic writing, are typically organized and structured. As briefly mentioned at the beginning of this section, news articles selected in this project are not as highly signaled as the news articles in RST-SC. In addition to the use of different source data, another reason is that RST-DT employs a finer-grained relation inventory and segmentation guidelines; as a result, certain information is lost. For instance, the relation <span style="font-variant:small-caps;">Attribution</span> is signaled 3,061 times out of 3070 occurrences (99.71%) in RST-SC, corresponding to the type *syntactic* and its subtype *reported speech*, which does not occur in this dataset. However, we do have some indicative signal tokens such as *market* and *the major source*.
**Interviews** are the most difficult genre to annotate in this project for two main reasons. Firstly, it is (partly) spoken language; as a result, they are not as organized as news or academic articles and harder to follow. Secondly, the layout of an interview is fundamentally different from the previous three written genres. For instance, the relation <span style="font-variant:small-caps;">Solutionhood</span> seems specific to interviews, and most of the signal instances remain unanchored (i.e. 11 instances), which is likely due to the fact that the question mark is ignored in the current annotation scheme. As can be seen from Table \[genre-tokens\], there are many *wh-*words such as *What* and *Why*. These can be used towards identifying interviews in that they formulate the question-answer scheme. Moreover, interviewers and interviewees are also important constituents of an interview, which explains the high frequencies of the two interviewees *Sarvis* and *Noam Chomsky* and the interviewer *Wikinews*. Another unique feature shown by the signals in this dataset is the use of spoken expressions such as *Well* and *So* when talking, which rarely appear in written texts.
Discussion
==========
Annotation Scheme {#scheme}
-----------------
For syntactic signals, one of the questions worth exploring is which of these are actually attributable to sequences of tokens, and which are not. For example, sequences of auxiliaries or constructions like imperative clauses might be identifiable, but more implicit and variable syntactic constructions are not such as ellipsis.
In addition, one of the objectives of the current project is to provide human annotated data in order to see how the results produced by machine learning techniques compare to humans’ judgments. In particular, we are interested in whether or not contemporary neural models have a chance to identify the constructions that humans use to recognize discourse relations in text based on individual sequences of word embeddings, a language modeling technique that converts words into vectors of real numbers that are used as the input representation to a neural network model based on the idea that words that appear in similar environments should be represented as close in vector space.
Another dilemma that generally came up during the discussion about signal anchoring was whether or not to mark the first constituent of a multinuclear relation. In Figure \[multinuclear\], four juxtaposed segments are linked together by the <span style="font-variant:small-caps;">Joint</span> relation, with associated signal tokens being highlighted. The first instance of <span style="font-variant:small-caps;">Joint</span> is left unsignaled/unmarked while the other instances of <span style="font-variant:small-caps;">Joint</span> are signaled. The rationale is that when presented with a parallelism, the reader only notices it from the second instance. As a result, signals are first looked for between the first two spans, and then between the second and the third. If there is no signal between the second and the third spans, then try to find signals in the first and the third spans. Because this is a multinuclear relation, transitivity does exist between spans. Moreover, the current approach is also supported by the fact that a multinuclear relation is often found in the structure like X, Y *and* Z, in which the discourse marker *and* is between the last two spans, and thus this *and* is only annotated for the relation between the last two spans but not between the first two spans. However, the problem with this approach is that the original source for the parallelism cannot be located.
![A Visualization of a Multinuclear Relation.[]{data-label="multinuclear"}](multinuclear.png){width="80mm"}
Distribution of Discourse Signals
---------------------------------
So far we have examined the distributions of signals across relations (Section \[distribution-rels\]) and genres (Section \[distribution-genres\]) respectively. Generally speaking, DMs are not only ambiguous but also inadequate as discourse signals; most signal tokens are open-class lexical items. More specifically, both perspectives have revealed the fact that some signals are highly indicative while others are generic or ambiguous. Thus, in order to obtain more valid discourse signals and parse discourse relations effectively, we need to develop models that take signals’ surrounding contexts into account to disambiguate these signals.
Based on the results found in this dataset regarding the indicative signals, they can be broadly categorized into three groups: ***register-related***, ***communicative-need related***, and ***semantics-related***. The first two are used to address genre specifications whereas the last one is used to address relation classification. Words like *Albeit* are more likely to appear in academic papers than other genres due to the register they are associated with; words like *Steps, Tips,* and *Warnings* are more likely to appear in instructional texts due to the communication effect they intend to achieve. Semantics-related signals play a crucial role in classifying relations as the semantic associations between tokens are less ambiguous cues, thereby supplementing the inadequacy of DMs.
Validity of Discourse Signals
-----------------------------
It is also worth pointing out that some tokens are frequent signals in several relations, which makes their use very ambiguous. For instance, the coordinating conjunction *and* appears in <span style="font-variant:small-caps;">Joint, Restatement, Sequence</span>, and <span style="font-variant:small-caps;">Result</span> in this dataset. Similarly, the subordinating conjunctions *since* and *because* serve as signals of <span style="font-variant:small-caps;">Justify</span>, <span style="font-variant:small-caps;">Cause</span>, and <span style="font-variant:small-caps;">Evidence</span> in these 12 documents. These ambiguities would pose difficulties to the validity of discourse signals. As pointed out by @Zeldes2018, a word like *and* is extremely ambiguous overall, since it appears very frequently in general, and is attested in all discourse functions. However, it is noted that some ‘and’s are more useful as signals than others: adnominal ‘and’ (example \[adnominal\]) is usually less interesting than intersentential ‘and’ (example \[intersentential\]) and sentence initial ‘and’ (example \[initial\]).
. The owners, \[William ***and*** Margie Hammack\], are luckier than any others.[^3] – <span style="font-variant:small-caps;">Elaboration-Additional</span> \[adnominal\]
. \[Germany alone had virtually destroyed Russia, twice,\]~n1~ \[***and*** Germany backed by a hostile military alliance, centered in the most phenomenal military power in history, that’s a real threat.\]~n2~ – <span style="font-variant:small-caps;">Joint</span> \[interview\_chomsky\] \[intersentential\]
. \[It arrests us.\]~N~ \[***And*** then you say you won’t commit a mistake, so you’ll commit new mistakes. It doesn’t matter.\]~S~ –<span style="font-variant:small-caps;">Antithesis</span> \[interview\_peres\] \[initial\]
Hence, it would be beneficial to develop computational models that score and rank signal words not just based on how proportionally often they occur with a relation, but also on how (un)ambiguous they are in contexts. In other words, if there are clues in the environment that can tell us to safely exclude some occurrences of a word, then those instances shouldn’t be taken into consideration in measuring its ‘signalyness’.
Conclusion
==========
The current study anchors discourse signals across several genres by adapting the hierarchical taxonomy of signals used in RST-SC. In this study, 12 documents with 11,145 tokens across four different genres selected from the GUM corpus are annotated for discourse signals. The taxonomy of signals used in this project is based on the one in RST-SC with additional types and subtypes proposed to better represent different genres. The results have shown that different relations and genres have their indicative signals in addition to generic ones, and the indicative signals can be characterized into three categories: register-related, communicative-need related, and semantics-related.
The current study is limited to the `rst` annotation layer in GUM; it is worth investigating the linguistic representation of these signals through other layers of annotation in GUM such as coreference and bridging, which could be very useful resources constructing theoretical models of discourse. In addition, the current project provides a qualitative analysis on the validity of discourse signals by looking at the annotated signal tokens across relations and genres respectively, which provides insights into the disambiguation of generic signals and paves the way for designing a more informative mechanism to quantitatively measure the validity of discourse signals.
[^1]: The square brackets at the end of each example contain the document ID from which this example is extracted. Each ID consists of its genre type and one keyword assigned by the annotator at the beginning of the annotation task.
[^2]: This result excludes the class *Unsure* used in RST-SC.
[^3]: This example is chosen from the RST-DT corpus [@CarlsonEtAl2003] for illustration due to the apposition. Note that the relation inventory also differs.
|
---
abstract: 'Supervised learning-based segmentation methods typically require a large number of annotated training data to generalize well at test time. In medical applications, curating such datasets is not a favourable option because acquiring a large number of annotated samples from experts is time-consuming and expensive. Consequently, numerous methods have been proposed in the literature for learning with limited annotated examples. Unfortunately, the proposed approaches in the literature have not yet yielded significant gains over random data augmentation for image segmentation, where random augmentations themselves do not yield high accuracy. In this work, we propose a novel task-driven data augmentation method for learning with limited labeled data where the synthetic data generator, is optimized for the segmentation task. The generator of the proposed method models intensity and shape variations using two sets of transformations, as additive intensity transformations and deformation fields. Both transformations are optimized using labeled as well as unlabeled examples in a semi-supervised framework. Our experiments on three medical datasets, namely cardiac, prostate and pancreas, show that the proposed approach significantly outperforms standard augmentation and semi-supervised approaches for image segmentation in the limited annotation setting. The code is made publicly available at https://github.com/krishnabits001/task$\_$driven$\_$data$\_$augmentation.'
author:
- '[^1]'
bibliography:
- 'references.bib'
title: 'Semi-supervised Task-driven Data Augmentation for Medical Image Segmentation'
---
Data augmentation, semi-supervised learning, machine learning, deep learning, medical image segmentation
Introduction {#sec:intro}
============
Accurate image segmentation is important for many clinical applications that rely on medical images. In the recent years, deep neural networks have been successful in yielding high segmentation performance at the expense of requiring large amount of annotated training data. Obtaining many annotated examples is difficult for medical images since getting clinical experts to annotate a large number of segmentation masks, which require per-pixel annotations, is an expensive and time-consuming process. Hence, it is not a preferable solution in clinical settings. At the heart of this issue lies the fundamental gap between generalization performance of humans and current Deep Learning (DL) methods. While humans can generalize well for image segmentation after observing very few examples, even one or two examples seem to suffice in some applications, this is not the case with the current DL algorithms.In this work, we focus on algorithmic approaches aiming to close the mentioned gap for medical image segmentation.
First we contemplate the question: why do we need large annotated data for the deep neural networks to obtain high segmentation accuracy? We hypothesize that for segmentation task, when trained with large number of annotated samples, the target relationship learned by the neural networks between images and segmentation masks is robust to the variations in shape and intensity characteristics of the target and surrounding structures. Algorithms trained on a small number of annotated samples may not be exposed to a sufficient amount of variations, and consequently, perform poorly on unseen test images that contain variations not observed during training. These variations in shape arise due to anatomical variations in the population and variations in intensity characteristics are due to differences in (i) tissue properties and its composition, and (ii) image acquisition and scanner protocols, especially in Magnetic Resonance Imaging (MRI). To address the need for a large number of annotated data to achieve high segmentation performance with DL methods, in this work we propose a task-driven and semi-supervised data augmentation method (shown in Fig. \[fig:gans\]). The method is based on learning generative models that can be used to sample deformation fields and additive intensity transformations. Segmentation cost is included during training of these models for the synthesis of image-label pairs, which incorporate the task-driven nature. Semi-supervised nature, on the other hand, is incorporated by including unlabeled data in the training through an adversarial term, which help generators synthesize diverse set of shape and intensity variations present in the population, even in scenarios where the number of labeled examples are extremely low. The proposed approach in essence aims to optimize the augmentation task and can be intuitively understood by drawing analogies with existing augmentation methods. For example, if we consider random elastic deformations proposed in [@simard2003best; @ronneberger2015u], the augmentation is based on a deformation model with a few number of parameters like Gaussian kernel size and width. These parameters can be seen as hyper-parameters and their values can be optimized by training separate segmentation networks for a number of combinations, and selecting the combination that yields the best performance on validation images. Our approach uses a much more flexible transformation model using networks and optimizes its weights using both labeled and unlabeled training images. Its only hyper-parameter is the number of training iterations which is determined based on performance on validation images. Thus, both the training and validation images play a crucial role. In our experiments, we evaluated the proposed approach using three different publicly available datasets of cardiac, prostate and pancreas. We present comparisons with existing alternatives as well as an ablation study empirically analyzing the proposed approach. A preliminary version of this work has been presented at the conference on Information Processing in Medical Imaging [@kcipmi19]. In this extended version we additionally:
- analyze quantitatively how each term of regularization loss, namely adversarial loss and large deviation loss components affects the performance gains obtained in the proposed method (Refer Results in Sec 5.A).
- investigate the benefit of optimizing the generator jointly with the segmentation network as compared to independent optimization of generator w.r.t segmentation network (Refer Results in Sec 5.B).
- examine the generality of the proposed method by evaluating it on two more datasets, namely prostate and pancreas.
- compare with a larger set of related methods including self-training with conditional random fields and image-level adversarial training.
Related work
------------
We broadly classify the relevant literature into two categories in light of the proposed method in this work:
**Data Augmentation**: Data Augmentation is a simple technique to enlarge the training set based on generating synthetic image-label pairs. The idea is to transform the images in such a way that the label remains the same, or the transformation is well-defined for both the image and the label. The popular approaches are random affine transformations [@cirecsan2011high], random elastic transformations [@simard2003best; @ronneberger2015u] and random contrast transformations [@hong2017convolutional; @perez2018data]. These methods are very simple to implement and empirically have been shown to reduce overfitting and improve performance on unseen examples. Several recent works trained Generative Adversarial Networks(GANs) [@goodfellow2014generative], using the existing labeled dataset to generate realistic image-label pairs. The idea has been applied to various analysis tasks [@salehinejad2018generalization; @frid2018synthetic; @guibas2017synthetic; @hou2017unsupervised; @wu2018conditional; @liu2018pixel; @sixt2018rendergan] including MR image segmentation [@costa2018end; @shin2018medical; @bowles2018gan]. MixUp [@zhang2017mixup] is different from the above approaches in the fashion that the generated data is not realistic in nature. Here, the additional data is obtained by linearly interpolating available images and corresponding labels, respectively. Despite the unrealistic nature, it seems to improve the performance of the neural networks on standard benchmark datasets [@zhang2017mixup] and also on medical image segmentation [@eaton2018improving] in limited annotation setting. All of the above mentioned methods have parameters that control the generation process. These are either set by experience or learned to generate realistic samples based on the available labeled examples, which itself often requires large number of training samples. None of the methods optimize the parameters with respect to the task performance nor leverage unlabeled data. The proposed approach optimizes the parameters of the generator to get the best segmentation performance and leverages unlabeled data during the optimization.
The closest work to ours was proposed in [@wang2018low]. In a meta-learning setup, the authors proposed a fully supervised approach that incorporated the classification performance in learning the generator so to generate augmented images that are optimal for the task. Although it is a few-shot learning method, we still need a large number of different classes samples to train the generator. Here, no modeling assumptions are considered in the generator setup, and the augmented images are synthesized directly. While we instead incorporate domain knowledge and model the generator to output deformation and intensity transformations to generate the synthetic images. Due to these assumptions of modeling transformations as well as leveraging the unlabeled data in the generation process, we do not need a large number of labeled examples during training. In the proposed method, we can readily obtain the synthetic mask using the generated transformation which cannot be obtained with the method [@wang2018low]. We show in the results (Section \[eff\_terms\_reg\] and Fig. \[fig:vary\_terms\]) that the semi-supervised framework of our method yields significant improvements over using only the labeled examples with task-specific loss as in [@wang2018low]. **Semi-supervised learning** (SSL) methods leverage unlabeled data to accompany limited annotated data during training. The underlying idea is to regularize training by leveraging the unlabeled data and avoid overfitting. We divide SSL works into 3 sub-categories: (i) self-training, (ii) adversarial training, and (iii) learned registration based approaches.
*Self-training* approaches [@yarowsky1995unsupervised; @li2005setred] are based on the concept of iteratively re-training an already-trained network on the estimated labels (pseudo-labels) of unlabeled data as ground truths. In [@bai2017semi] authors show an improvement in the segmentation performance on cardiac MRIs using a self-training approach along with a Conditional Random Field (CRF) model post-processing intermediate predictions before re-training. However, it has also been shown that the self-training approaches can suffer if the initial predictions are erroneous on the unlabeled data [@chapelle2009semi; @zhu2009introduction].
*Adversarial* training and GANs have been used in the semi-supervised setting both using the generator as the segmentation network and the discriminator in a regularization term [@zhang2017deep], and vice-versa [@souly2017semi]. In limited annotation setting, we compare and illustrate that the proposed method outperforms the earlier stated semi-supervised approaches.
Alternatively, in a concurrent work [@zhao2019data], the authors proposed a one-shot data augmentation approach that learns registration between unlabeled image and labeled image where two independent models are trained to learn spatial and appearance transformations for the registration. Later, both learned models are used to generate augmented image-label pairs which are used to train a segmentation network. As in the other augmentation works, the generator of this model is also not optimized to yield the best task performance. Furthermore, the approach relies on image registration, which, on one hand, is a difficult task for non-brain anatomy, and on the other hand, leads to task-irrelevant background structures substantially influence the augmentation process.
Lastly, with **weakly-supervised learning** the issue of expensive and time-consuming pixel-wise annotations is addressed using weaker labels during training such as scribbles [@can2018scribble] and image-wide labels [@andermatt2018pathology], which are different approaches that can be complementary to data-augmentation.
Methods
=======
Let $X_L$ be a set of training images and $Y_L$ be the set of corresponding ground truth segmentation labels, $S$ be a segmentation network and $w_s$ be its trainable parameters. In the supervised learning setting, a loss function $L_s \left(X_L,Y_L\right)$ is defined as the disagreement between the labels predicted by the network $S$ for the set of input images $X_L$ and ground truth labels $Y_L$. The objective is to minimize the loss function $L_s$ with respect to the parameters $w_s$ as can be stated as in Eq. \[eq:sup\_obj\]. $$\min_{w_{S}} L_S(X_L,Y_L)
\label{eq:sup_obj}$$
When data augmentation is used, the minimization becomes$$\min_{w_{S}} L_S(X_L \cup X_G, Y_L \cup Y_G)
\label{eq:sup_plus_aug_obj}$$ where $X_G$ and $Y_G$ denote the sets of generated images and corresponding labels, which can be generated using the transformations mentioned previously. In this minimization, the effective training set is formed of the augmented sets $X_L \cup X_G$ and $Y_L \cup Y_G$. When model-based transformations are used for generating the augmentation sets $X_G$ and $Y_G$, such as geometric or contrast transformations, each augmented image $x_G\in X_G$ and the corresponding label $y_G\in Y_G$ are created by a conditional generator function that takes as input an existing labeled pair and applies a random transformation with fixed parameters. We can represent this with the following notation, as in Eq. \[eq:notation\]. $$\begin{split}
& (x_G, y_G) = G\left((x_L, y_L), z; w_G\right),\\
& (x_L, y_L) \sim p\left(X_L,Y_L\right),\ z\sim p(z),
\label{eq:notation}
\end{split}$$ where $(x_L, y_L)$ are the image-label pair sampled from the set of labeled examples, $G(\cdot,\cdot;w_G)$ is the transformation function, $z$ is the random component of the transformation and $w_G$ are the parameters of the transformation. For instance, for the random elastic deformations proposed in [@ronneberger2015u], $G$ would be the deformation model, $w_G$ would include the grid spacing between anchor points and standard deviation of the distribution of displacement vectors at each anchor point, while $z$ would correspond to a random draw of displacement vectors. The same transformation would be applied to both $x_L$ and $y_L$ to generate the augmented pairs. As described in the introduction, in the model-based transformations, parameters are often pre-defined and their number is kept low.
When GANs are used for generation, a neural network is used as the generator as $(x_G, y_G) = G(z; w_G)$, and its parameters are determined by optimizing Eq. \[eq:gan\] as per the adversarial learning framework [@goodfellow2014generative], using an additional discriminator network $D$ with its own set of parameters $w_D$. (The GAN can also be a conditional image generator defined as $G(z, x_L; w_G)$ and the discussion still holds.) $$\begin{split}
\min_{w_G} \max_{w_D} \mathbb{E}_{x,y\sim p(X_L,Y_L)}[\log D(x, y; w_D)] +\\
\mathbb{E}_{z\sim p(z)}[\log(1 - D(x_G, y_G; w_D))],\ (x_G, y_G) = G(z; w_G)
\label{eq:gan}
\end{split}$$ The generator $G$ is input only with a random draw from the distribution $p(z)$ to output an image-label pair $(x_G,y_G)$. The labeled pairs are still used but during the training. The discriminator $D$ is optimized to distinguish between generated pairs from $G$ and real pairs $(x_L,y_L)$, while the generator $G$ is optimized to produce $(x_G,y_G)$ such that $D$ can not differentiate between generated and real pairs. This forces the generated set $(X_G,Y_G)$ to be as “realistic” as possible.
In both model-based and GAN-based approaches, generators would be pre-defined or trained in advanced without considering the task, and during training random draws would be sampled from the generator models to create $X_G$ and $Y_G$.
Semi-Supervised and Task-Driven Data Augmentation
-------------------------------------------------
In this work, we propose to generate augmented image-label pairs that are optimized for the segmentation task(Fig. \[fig:gans\]). We achieve this by optimizing the generator function, similar to the GAN-based approach, but with a crucial difference, we integrate the task loss and leverage unlabeled images in the process. The proposed model optimizes Eq. \[eq:sup\_plus\_aug\_task\_driven\_obj\] instead of Eq. \[eq:gan\]. $$\min_{w_G}\big(\min_{w_{S}} L_S\left(X_L\cup X_G, Y_L\cup Y_G\right) + L_{\text{reg}, G}(X_{UL})\big),
\label{eq:sup_plus_aug_task_driven_obj}$$ where $X_{UL}$ denotes a set of unlabeled images. Furthermore, we would like to be able to optimize the parameters with limited number of labeled examples. To this end, we integrate domain knowledge into the generation process, similar to the model-based approaches, but allowing a network parameterization of the transformation models to increase flexibility while remaining trainable. We define two conditional generators to model shape and intensity variations using *deformation fields generator* (non-affine spatial transformations) and *intensity fields generator*, respectively. Crucially, both models are constructed such that the segmentation mask $y_{G}$ corresponding to an augmented image $x_{G}$ is obtained by applying the same transformation to the input image mask $y_{L}$.
With this optimization, we want to incorporate two sets of ideas with the two loss-terms in Eq. \[eq:sup\_plus\_aug\_task\_driven\_obj\]. The first term ensures that the model generates set of augmented pairs $(X_G,Y_G)$ such that they are helpful for the minimization of segmentation loss, which is the *task-driven* nature of the approach. The second term is a regularization term built on adversarial loss and integrates a preference for larger transformations leveraging the unlabeled images in the generation process, which is the *semi-supervised* component of the method.
### Deformation Field Generator
The deformation field generator $G_V$ is trained to output a deformation field transformation. The conditional generator $G_V$, a network with parameters $w_{G_V}$, takes as input a labeled image $x_L$ and a $z$ vector randomly drawn from a unit Gaussian distribution to produce a dense per-pixel deformation field $\textbf{v} = G_V(x_L, z; w_{G_V})$. Later, the input image and its corresponding label (in one-hot encoding form) are warped using bi-linear interpolation based on the generated deformation field $\textbf{v}$ to produce the augmented image-label pair $x_{G_V}$ and $y_{G_V}$, respectively. The augmented image and label sets are denoted by $X_{G_V}$ and $Y_{G_V}$, and each sample pair is defined using the following: $x_{G_V}= \textbf{v} \circ x_L$, $y_{G_V}= \textbf{v} \circ y_L$, where $\circ$ denotes warping operation.
### Additive Intensity Field Generator
Similar to $G_V$, here the generator $G_I$ is trained to output an additive intensity mask transformation. The generator $G_I$, again a network with parameters $w_{G_I}$, takes as input a labeled image from $x_L$ and a $z$ vector randomly drawn from a unit Gaussian distribution to output an additive intensity mask $\Delta I = G_I(x_L, z; w_{G_I})$. Then $\Delta I$ is added to the input image $x_L$ to obtain the augmented image $x_{G_I}$, and its corresponding segmentation mask $y_{G_I}$ remains unchanged as the initial mask $y_L$. The augmented image-label set is denoted by $\{X_{G_I},Y_{G_I}\}$ and one sample pair is defined using the following: $x_{G_I}= x_L + \Delta I $, $y_{G_I}= y_L$.
[0.9]{} ![Data augmentation modules that generate augmented image-label pair with task-driven optimization defined in a semi-supervised framework. Here, the dotted-red line indicates the inclusion of segmentation loss for generator optimization.[]{data-label="fig:gans"}](images/geometric_gan.pdf "fig:"){width="\textwidth"}
[0.9]{} ![Data augmentation modules that generate augmented image-label pair with task-driven optimization defined in a semi-supervised framework. Here, the dotted-red line indicates the inclusion of segmentation loss for generator optimization.[]{data-label="fig:gans"}](images/intensity_gan.pdf "fig:"){width="\textwidth"}
### Regularization Loss
The regularization term $L_{\text{reg}}$ is defined as in Eq. \[eq:gan\_G\_C\] for both conditional generators. $$\label{eq:gan_G_C} L_{\text{reg}, G_C} = \lambda_{\text{adv}}L_{\text{adv}, G_C} + \lambda_{\text{LD}}L_{\text{LD}, G_C},\ \text{for }C={V,I}.$$ The first term is the following standard adversarial loss $$\label{eq:adv_G_C}
\begin{split}
{} & L_{\text{adv}, G_C} = \max_{w_{D_{C}}} \mathbb{E}_{x\sim p(X_{UL})}[\log D_{C}(x; w_{D_C})] + \\
& \mathbb{E}_{z\sim p(z),x_{L}\sim p(X_L)}[\log(1 - D_{C}(G_C(x_L, z; w_{G_C}); w_{D_C}))]
\end{split}$$ where $D_C$, $w_{D_C}$ denotes the discriminator network and its weights. This first term incorporates the semi-supervised nature of the model by including the set of unlabeled images $X_{UL}$ in the optimization. It allows samples in the unlabeled set that show different shape and intensity variations than the labeled examples guide the optimization of the generators. The second term in Eq. \[eq:gan\_G\_C\] is the *Large Deviation* (LD) term that embeds a preference for large transformations. The LD term prevents the generator from producing very small deformations and intensity fields, which would satisfy the adversarial loss as well as lead to lower segmentation loss in the cost given in Eq. \[eq:sup\_plus\_aug\_task\_driven\_obj\]. The definition of $L_{\text{LD},G_C}$ depends on the generator type. We define the two terms for the deformation and intensity field generators as $L_{\text{LD}, G_V} = -\|\textbf{v}\|_1$ and $L_{\text{LD}, G_I} = -\|\Delta_{I}\|_1$. The negative signs ensures that minimizing the LD term maximizes the $L_1$ norms. The LD term forces the generator to produce large transformations while the adversarial loss tries to constrain them. Optimizing for both terms yield augmented images that differ substantially from the input labeled images for both the generators.
Finally, the weighting terms $\lambda_{adv}$ and $\lambda_{\text{LD}}$ balances the effect of the two terms on the optimization.
### Optimization Sequence
Before the optimization of the conditional generators, the segmentation network ($S$) in the proposed setup is pre-trained on only the labeled data for a few epochs as per Eq. \[eq:sup\_obj\] and later optimized with both the labeled and generated data from the conditional generators as per Eq. \[eq:sup\_plus\_aug\_obj\]. This is done to ensure that $S$ produces reasonable segmentation masks for the labeled data when the optimization of the generators begins. Following the pre-training of $S$, both generative models for deformation and intensity fields are trained separately by minimizing the cost given in Eq. \[eq:sup\_plus\_aug\_task\_driven\_obj\] with corresponding regularization terms. Note that this minimization trains over $S$, the generators and the discriminators. This implies training of the set of networks $(S,G_V,D_V)$ and $(S,G_I,D_I)$ independently for deformation and intensity fields generation, respectively. The optimization sequence is run for a fixed number of iterations, and the segmentation network is evaluated on the validation images using the Dice’s similarity coefficient (DSC) at every iteration. Once the optimization is complete, we fix both the generators $G_V$ and $G_I$ with the parameters that yielded the best mean DSC over the validation images. Then, the segmentation network $S$ is optimized using Eq. \[eq:sup\_plus\_aug\_obj\] once again from a random initialization. The data used for this final training comprises of both the original labeled sets, $X_L$ and $Y_L$, and the augmented sets, $X_G$ and $Y_G$ sampled from the trained generators $G_V$ or $G_I$ or both (where we perform two back to back transformations, e.g. $G_I$ is applied over the deformed image obtained from $G_V$).
The validation images play a crucial role in the defined optimization as they determine the parameters of the generator model chosen to generate the augmented data that is later used for the independent optimization of the segmentation network.
Available training data: labeled set $(X_{L},Y_{L})$ and unlabeled set $(X_{UL})$.\
**Step 1**:\
(a) train deformation field generator $(G_{V})$ as per Eq. (5) using the available data.\
(b) train intensity field generator $(G_{I})$ as per Eq. (5) using the available data.\
**Step 2**:\
Post optimization, the generators are used to sample augmented image, label pairs conditioned on the labeled set.\
(a) The shape transformed image, label pairs $(X_{G_V}, Y_{G_V})$ are sampled using generator $(G_{V})$.\
(b) The intensity transformed image, label pairs $(X_{G_I}, Y_{G_I})$ are sampled using generator $(G_{I})$.\
(c) The image, label pairs that contain both the shape and intensity transformations (denoted by $(X_{G_{VI}}, Y_{G_{VI}})$) are obtained by inputting the sampled shape transformed image, label pairs $(X_{G_V}, Y_{G_V})$ from $(G_{V})$ through $(G_{I})$.\
**Step 3**:\
Train the segmentation network with all the available labeled and generated augmented data. The training set includes: original labeled data $(X_{L}, Y_{L})$, affine transformed data $(X_{Aff}, Y_{Aff})$, shape transformed data $(X_{G_V}, Y_{G_V})$, intensity transformed data $(X_{G_I}, Y_{G_I})$ and data with both shape and intensity transformations applied $(X_{G_{VI}}, Y_{G_{VI}})$.
Dataset and Network details
===========================
Dataset details
---------------
**Cardiac Dataset**\[sec:dataset\_acdc\]: This is a publicly available dataset hosted as part of MICCAI’17 ACDC challenge [@bernard2018deep] [^2]. It comprises of 100 subjects’ short-axis cardiac cine-MR images captured using 1.5T and 3T scanners. The in-plane resolution ranges from 0.70x0.70mm to 1.92x1.92mm and through-plane resolution ranges from 5mm to 10mm. The segmentation masks are provided for left ventricle (LV), myocardium (Myo) and right ventricle (RV) for both end-systole (ES) and end-diastole (ED) phases of each subject. This dataset is divided into 5 sub-groups (details in [@bernard2018deep] [^3]), each comprising of 20 subjects, respectively. For our experiments we only used the ES images.
**Prostate Dataset**: This is a public dataset made available as part of MICCAI’18 medical segmentation decathlon challenge [^4]. It comprises of 48 subjects T2 weighted MR scans of prostate. The in-plane resolution ranges from 0.60x0.60mm to 0.75x0.75mm and through-plane resolution ranges from 2.99mm to 4mm. Segmentation masks comprise of two adjoint regions: peripheral zone (PZ) and central gland (CG).
**Pancreas Dataset**: This dataset also is from medical decathlon MICCAI’18 challenge [^5]. It comprises of 282 subjects CT scans. Segmentation masks available comprise of labels with large (background), medium (pancreas) and small (tumor) structures. The in-plane resolution ranges from 0.6x0.6mm to 0.97x0.97mm and through-plane resolution ranges from 0.7mm to 7.5mm. In this work, we create two labels for segmentation task, where label 1 denotes the foreground which was created by merging the labels of pancreas and tumor, and label 0 denotes the background label.
Pre-processing
--------------
N4 [@tustison2010n4itk] bias correction was performed on the cardiac and prostate datasets. The below pre-processing was applied to all images of all the datasets. (i) intensity normalization: all the volumes were normalized using min-max normalization according to: $(x-x_2)/(x_{98}-x_2)$, where $x_2$ and $x_{98}$ denote the $2^{nd}$ and $98^{th}$ intensity percentiles of the 3D volume. (ii) re-sampling: all 2D image slices and their corresponding label maps were re-sampled to a fixed in-plane resolution $r$ using bi-linear and nearest-neighbour interpolation, respectively and then cropped or padded with zeros to a fixed size of $f_z$. The resolution $r$ and fixed size $f_z$ were chosen empirically for each dataset, and the values were: (a) cardiac dataset: $r = $1.367x1.367mm and $f_z = $224x224, (b) prostate dataset: $r = $0.6x0.6mm and $f_z = $224x224, (c) pancreas dataset: $r = $0.8x0.8mm and $f_z = $320x320.
Network Architecture
--------------------
There are 3 types of networks in the proposed method (see Fig. \[fig:gans\]): a segmentation network $S$, a discriminator network $D$ and a generator network $G$. We describe the architectures of these networks below. $G_V$ and $G_I$ use the same architecture except at the last layer, which are used to model the deformation field and the intensity mask, respectively.
**Generator**: Generator $G$ takes an image $x_L$ and a randomly drawn $z$ vector of dimension 100 as input. Both inputs are initially passed through 2 sub-networks namely $G_{subnet,X}$ and $G_{subnet,z}$. $G_{subnet,X}$ comprises of 2 convolutional layers and $G_{subnet,z}$ comprises of a fully-connected layer, followed by reshaping of output into down-sampled image dimensions. Then a set of bi-linear upsampling and convolutional layers are applied consecutively to output feature maps of image dimensions. The resulting outputs of both the sub-networks, which are of same dimensions, are then concatenated and passed through a common sub-network $G_{subnet,common}$, which consists of 4 convolutional layers where the last layer is different for $G_V$ and $G_I$. The final convolution for $G_V$ yields two feature maps that correspond to dense per-pixel deformation field $\textbf{v}$, while for $G_I$, it outputs a single feature map that corresponds to the intensity mask $\Delta I$. The final layer of $G_I$ uses $tanh$ activation to restrict the range of values in the intensity mask. All the convolutional layers except the final layers are followed by batch normalization layers and ReLU activation. All convolutional layers’ kernels are 3x3 except the final layers’ which are 1x1.
**Discriminator**: $D$ has an architecture similar to the DCGAN [@radford2015unsupervised], which comprises of five convolutional layers each with a kernel size of 5x5 and a stride of 2. The convolutions are followed by batch normalization layers and leaky ReLU activations with the leak value of the negative slope set to 0.2. After the convolutional layers, the output is reshaped and passed through three fully-connected layers where the final layer has an output size of 2 that predicts the probability of the input being real or fake.
**Segmentation Network**: We chose the architecture of segmentation network $S$ similar to U-Net [@ronneberger2015u]. It comprises of encoding and decoding paths. The encoder comprises of four convolution blocks where each block has two 3x3 convolutions followed by a 2x2 max-pool layer. The decoder comprises of four convolution blocks where each block comprises of the concatenation of features from the corresponding level of the encoder, followed by two 3x3 convolutions and bi-linear upsampling by a factor of 2. Except for the last layer, all the layers have batch normalization and ReLU activation.
Training Details
----------------
The segmentation loss ($L_S$) used is weighted cross entropy. We empirically set the weights of background pixels as 0.1 and foreground pixels as 0.9 since the number of pixels belonging to the foreground are fewer in quantity and are of primary interest for the segmentation task at hand. For the datasets with more than one foreground label, we divided the foreground weight of 0.9 equally among all the labels. With this rationale, the weights of the output labels are as follows: (a) 0.1 for background and 0.3 for each of the three foreground classes of the cardiac dataset, (b) 0.1 for background and 0.45 for each of the two foreground classes of the prostate dataset, and (c) 0.1 for background and 0.9 for one foreground class of the pancreas dataset.
We split the data into training pool, validation, unlabeled and test sets. We empirically set $\lambda_{adv}$ as 1 to match the magnitude of adversarial loss to the segmentation loss. To determine $\lambda_{LD}$ parameter, we randomly sampled one 3D volume from the training pool of cardiac dataset (the sampled volume is one possibility of all training combinations used in full analysis) and trained the network, and evaluated performance on the validation images for three values of $\lambda_{LD}$: $10^{-2},10^{-3},10^{-4}$. The value of $10^{-3}$ for $\lambda_{LD}$ yielded the best validation performance for this experiment. So, we used this set values ($\lambda_{adv}=1, \lambda_{LD}=10^{-3}$) for all future experiments on all datasets. Owing to the computationally expensive nature of the proposed method, we did not perform an exhaustive grid search on many combinations of weights of the loss terms ($\lambda_{adv}$, $\lambda_{LD}$) on the validation set. But if one has enough computational resources, one could do a grid search of these hyper-parameters for each dataset to potentially obtain higher performance gains.
The batch size and the total number of iterations are set as 20 and 10000, respectively, based on the evolution of the training curves. For all the networks, while training, the iteration where the model yields the best performance on the validation images is saved for the evaluation. AdamOptimizer [@kingma2014adam] is used for the optimization of all the networks with learning rate of $10^{-3}$ and default beta values ($\beta_1=0.9$, $\beta_2=0.999$).
Experiments {#sec:setup}
===========
We evaluated the proposed method on three datasets: cardiac, prostate and pancreas. For each dataset, we split the data into 4 sets: labeled training ($X_{L,total}$), unlabeled training ($X_{UL}$), test ($X_{ts}$) and validation ($X_{vl}$). The size of each set is denoted by $N$ followed by a subscript indicating the set. The validation set consists of two 3D volumes ($N_{vl}$=2) for all datasets. For the cardiac, prostate and pancreas datasets, the number of 3D volumes ($N_{UL}$, $N_{ts}$) for unlabeled and test sets are (25, 20), (20, 13) and (25, 20) respectively. $X_{UL}$, $X_{ts}$ and $X_{vl}$ sets are selected randomly a-priori and fixed for all experiments. The remaining 3D volumes constitute the training pool $X_{L,total}$. Note that the entire training pool is never utilized for training. Rather a small number of training images ($N_{L}$) is sampled from $X_{L,total}$ for each experiment. As the interest of this work is to analyze the performance in the limited annotation setting, we set $N_{L}=1$ or $3$. Each experiment is run five times with different 3D training volumes. Further, to account for the variations in the random initialization and convergence of the networks, each of the five experiments is run three times. Thus, overall, we have 15 runs for each experiment. Since the cardiac dataset has five sub-groups of images (see Sec. \[sec:dataset\_acdc\]), we ensure that each set contains an equal number of images from each sub-group, and, when $N_{L}=1$ is run five times, each time the 3D volume is selected from a different sub-group.
Segmentation performances of the following models were compared: **No data augmentation (no aug)**: $S$ is trained without any data augmentation.
**Affine data augmentation (Aff)**: $S$ is trained with data augmentation comprising of affine transformations such as: (a) scaling (random scale factor is chosen from a uniform distribution with min and max value as 0.9 and 1.1), (b) flipping along x-axis, (c) rotation (randomly a value is chosen between -15 and +15 degrees and another type of rotation that is multiple of 45 degrees (defined as 45 deg\*N where N is randomly chosen between 0 and 8)). For each slice in the batch, we apply one of the above random transformation 80% of the time and 20% of the time we use the image as is.
For all the subsequent data augmentation methods, MixUp and semi-supervised methods, we include the random affine data augmentations by default as described above. For training of $S$, half of each batch was composed random affine augmentation and the remaining half was chosen from the specific augmentation technique. **Random elastic deformations (RD)**: Random elastic augmented images are created as stated in [@ronneberger2015u], where a deformation field is created using a matrix of size 3x3x2. Each element of this matrix is sampled from a Gaussian distribution with a mean of 0 and standard deviation of 10 and is then re-sampled to image dimensions using bi-cubic interpolation. (Refer appendix for results with different sigma & kernel sizes)
**Random contrast and brightness fluctuations** [@hong2017convolutional; @perez2018data] **(RI)**: These augmented images are created with the help of contrast adjustment step ($x = (x - \Bar{x}) * c + \Bar{x}$) and brightness adjustment step ($x = x + b$), where c and b are sampled uniformly from \[0.8,1.2\] and \[-0.1,0.1\], respectively and $\Bar{x}$ denotes mean of 2D image. (Refer appendix for more combinations of c and b evaluated)
**Deformation field transformations (GD)**: The deformation field generator trained with the proposed method $G_V$ is used to generate the augmented data i.e., $X_{G_V}$.
**Intensity field transformations (GI)**: The intensity field generator trained with the proposed method $G_I$ is used to generate the augmented data i.e., $X_{G_I}$.
**Both deformation and intensity field transformations (GD+GI)**: Augmented data included both $X_{G_V}$ and $X_{G_I}$, obtained from the generators $G_V$ and $G_I$, respectively. We also generated additional images $X_{G_{VI}}$ which have both the deformation and intensity transformations. These are obtained by applying intensity transformation using generator $G_I$ on the deformation field transformed images $X_{G_V}$. The augmented data consists of all the images generated $\{X_{G_V},X_{G_I},X_{G_{VI}}\}$.
**MixUp** [@zhang2017mixup] **(Mixup)**: The augmentation sets $X_G$ and $Y_G$ consist of the linear combination of available labeled images using the Mixup formulation as stated in Eq. \[eq:mixup\_eqn\] [@zhang2017mixup]. $${x_{Gi}} = \lambda x_{Li} + (1-\lambda) x_{Lj},\hspace{0.2cm}
{y_{Gi}} = \lambda y_{Li} + (1-\lambda) y_{Lj}
\label{eq:mixup_eqn}$$ where $\lambda$ is sampled from beta distribution Beta$(\alpha,\alpha)$ with $\alpha \in (0,\infty)$ and $\lambda \in [0,1)$. The $\alpha$ value of 0.2 yielded the best results for the datasets considered. $\lambda$ controls the ratio of mixing of two image-label pairs $(x_{Li},y_{Li})$, $(x_{Lj},y_{Lj})$ which are randomly sampled from the labeled image set.
**Mixup over deformation and intensity field transformations (GD+GI+Mixup)**: These set of images are obtained by applying Mixup over all possible pairs of available images: original data ($X_L$), their affine transformations and the generated images using deformation and intensity field generators
**Adversarial Training (Adv\_tr)**: For comparison, we investigate previously proposed adversarial training methods with the discriminator trained to operate on: (i) the image level discrimination [@zhang2017deep] and (ii) the pixel level discrimination [@souly2017semi] in a semi-supervised (SSL) setting. **Self-training (Self\_tr)**: The self-training based method as proposed in [@bai2017semi] is evaluated on the datasets.
**Ablation Studies:** In addition to the aforementioned comparisons, we carried out additional ablation studies as described below. These studies were done only on the cardiac dataset owing to lack of computational resources, as each experiment for each ablation study and dataset requires 15 runs. **A. Effects of adversarial ($\lambda_{adv}$) and large deviation ($\lambda_{LD}$) loss terms of regularization loss on segmentation performance**: We investigate the effect of each term of regularization on the performance of the proposed method. Different values of $\lambda_{adv}$ and $\lambda_{LD}$ are considered to examine how much each term impacts the performance. The training case of $\lambda_{adv} = \lambda_{LD} = 0$ is similar to earlier work [@wang2018low]. **B. Independent optimization of the generator and the segmentation networks**: Here, we optimize both $G_V$ and $G_I$ without the segmentation loss similar to [@shin2018medical; @bowles2018gan]. Later, augmented data created from these optimized generators are used for the independent training of the segmentation network. This experiment reveals the value of the segmentation loss. **C. Varying the number of unlabeled data**: We used different number of unlabeled volumes in the training of the generators. The number of 3D volumes studied ($N_{UL}$) were: 5, 10, 20 and 25. **D. Varying the number of labeled 3D volumes used in training**: The number of 3D volumes considered ($N_{L}$) were: 1, 3, 5, 10, 15, 40. This experiment is done to study if any improvement in Dice score is obtained when a large number of annotated volumes are available for training.
**E. Different set of train, validation, test and unlabeled 3D volumes**: We randomly sample another training, validation, test, and unlabeled image sets from the cardiac dataset different from the earlier sets and re-run learning deformation and intensity field transformations for this new set for one 3D training volume case ($N_{L}=1$). This experiment is done to analyze if the proposed method overfits to a specific dataset split or generalizes for any split. **F. No validation images**: Lastly, we report the performance observed when we do not use any validation images in the training of the generators. In this case, the training is stopped after running the model for some predefined number of iterations and these model parameters are used for generating images for augmentation. **Evaluation**: Dice’s similarity coefficient (DSC) is used to evaluate the segmentation performance of each method. The performance reported is obtained on $N_{ts}$ number of test images for the structures of each dataset as stated earlier.
Results and Discussion {#sec:results}
======================
[0.9]{} ![Left Ventricle (Cardiac data)[]{data-label="fig:lv_res"}](images/seg_results/acdc/rv__tr1_all_together_data_aug_ssl_compare_box_plt_avg_over_15runs.png "fig:"){width="\textwidth"}
[0.9]{} ![Left Ventricle (Cardiac data)[]{data-label="fig:lv_res"}](images/seg_results/acdc/myo__tr1_all_together_data_aug_ssl_compare_box_plt_avg_over_15runs.png "fig:"){width="\textwidth"}
[0.9]{} ![Left Ventricle (Cardiac data)[]{data-label="fig:lv_res"}](images/seg_results/acdc/lv__tr1_all_together_data_aug_ssl_compare_box_plt_avg_over_15runs.png "fig:"){width="\textwidth"}
[0.9]{} ![The segmentation performance of the proposed augmentation method (GD+GI) and several relevant works for three datasets are presented using Dice score (DSC). The number of labelled 3d volumes ($N_{L}$) used for training on cardiac, prostate, and pancreas datasets is one, one, and three, respectively (mean DSC and standard deviation values are reported on top of each boxplot). $\ast,\spadesuit,\star$ denotes the statistical significance of $GD$ over $RD$, $GI$ over $RI$, and $GD+GI$ over best performing related work, respectively (Wilcoxon signed rank test with threshold p value of 0.05).[]{data-label="fig:all_seg_res"}](images/seg_results/prostate/pz__tr1_all_together_data_aug_ssl_compare_box_plt_avg_over_15runs.png "fig:"){width="\textwidth"}
[0.9]{} ![The segmentation performance of the proposed augmentation method (GD+GI) and several relevant works for three datasets are presented using Dice score (DSC). The number of labelled 3d volumes ($N_{L}$) used for training on cardiac, prostate, and pancreas datasets is one, one, and three, respectively (mean DSC and standard deviation values are reported on top of each boxplot). $\ast,\spadesuit,\star$ denotes the statistical significance of $GD$ over $RD$, $GI$ over $RI$, and $GD+GI$ over best performing related work, respectively (Wilcoxon signed rank test with threshold p value of 0.05).[]{data-label="fig:all_seg_res"}](images/seg_results/prostate/tz__tr1_all_together_data_aug_ssl_compare_box_plt_avg_over_15runs.png "fig:"){width="\textwidth"}
[0.9]{} ![The segmentation performance of the proposed augmentation method (GD+GI) and several relevant works for three datasets are presented using Dice score (DSC). The number of labelled 3d volumes ($N_{L}$) used for training on cardiac, prostate, and pancreas datasets is one, one, and three, respectively (mean DSC and standard deviation values are reported on top of each boxplot). $\ast,\spadesuit,\star$ denotes the statistical significance of $GD$ over $RD$, $GI$ over $RI$, and $GD+GI$ over best performing related work, respectively (Wilcoxon signed rank test with threshold p value of 0.05).[]{data-label="fig:all_seg_res"}](images/seg_results/pancreas/panc__tr3_all_together_data_aug_ssl_compare_box_plt_avg_over_15runs.png "fig:"){width="\textwidth"}
Figures ( \[fig:rv\_res\]-to-\[fig:lv\_res\]),( \[fig:pz\_res\]-to-\[fig:cz\_res\]) and \[fig:panc\_res\] present the quantitative results of the experiments on the cardiac, prostate and pancreas datasets, respectively. The mean DSC and standard deviation values over all the runs are reported on the top of each boxplot in these figures. We observe that the proposed method of augmentation (i.e.,GD+GI) outperforms the other data augmentation and semi-supervised learning methods considered here. For qualitative inspection, we present some examples of visual results in Fig. \[fig:seg\_results\]. In the rest of this section, we present further analysis of the experimental results.
As expected, the lowest performance was observed when no data augmentation is used. Employing affine augmentation alone provided a substantial boost in performance. Adding random elastic deformations or random intensity fluctuations on top of affine augmentation yielded further improvements. Using the proposed learned deformation fields(GD) for augmentation yielded higher performance compared to random elastic deformations(RD). This results suggest that the proposed approach to learn a deformation field generator, by optimizing the segmentation accuracy along with a regularization term that leverages unlabeled examples, provided augmented examples more useful for obtaining high segmentation performance than random deformations. Some samples of the generated deformed images are illustrated in the Fig. \[fig:gen\_geogan\_imgs\]. Surprisingly, we observed that the generated images did not always have realistic anatomical shapes. This is contrary to the popular belief, but generating realistic images may not be necessary nor optimal to obtain the best segmentation network.
Similar to the deformation case, the proposed additive intensity mask(GI) based augmentations performed better than random intensity fluctuations(RI). Here as well, the result suggest benefits of optimizing the intensity transformation generator using the proposed approach. Fig. \[fig:gen\_geogan\_imgs\] illustrates that the images generated from the learned intensity transformation generator are not necessarily realistic.
Since both the generators $G_V$ and $G_I$ are modeled to encapsulate different characteristics of the entire population, using both the augmentations is expected to produce higher performance gains over using only one of the augmentation separately. In our experiments, we indeed observed a substantial improvement in DSC when both are used together. To our surprise, we observed that the Mixup augmentation yielded substantial performance gain over the affine transformations and random elastic deformations. This improvement was despite the augmented images being unrealistic. We attribute the performance improvement to the fact that Mixup creates soft probability maps for augmented images, which has been hypothesized to assist the optimization by providing additional information for training samples [@hinton2015distilling]. Applying Mixup over the augmented data obtained from the trained generators $G_V$ and $G_I$ yielded further marginal improvements, which suggests both approaches have complementary benefits. With self-training, we observed an improvement in DSC over the affine augmentations as illustrated in Fig. \[fig:all\_seg\_res\]. This has been well-documented in SSL literature [@yarowsky1995unsupervised; @li2005setred] that re-training the neural network with the estimated predictions of the unlabeled data can assist in improving the segmentation performance [@bai2017semi] in limited annotation setting. Although this yields some improvement over the affine augmentations, it did not outperform the proposed augmentations (GD+GI) except for the structure central gland of the prostate dataset. The semi-supervised adversarial training [@souly2017semi; @zhang2017deep] provided marginal performance gains over the baseline with affine augmentation (Results of [@souly2017semi] that uses image-level adversarial training are not reported as the GAN training did not converge to reasonable performance for the case of one labeled volume). This observation is not surprising as it has also been shown for other tasks in [@oliver2018realistic], SSL training may yield minimal performance gains when affine augmentations are included in the training.
In the Appendix, we provide additional analysis plots such as: (a) the performance improvement seen per test subject averaged over all the runs for each dataset in Fig. \[fig:dsc\_per\_each\_subj\]. We observed that for the majority of test subjects, the proposed method performs better or equal to random augmentations. (b) A sample of generated deformation and additive intensity fields obtained from the trained generators on the cardiac dataset are provided in Fig. \[fig:geo\_trans\_fields\] and Fig. \[fig:int\_trans\_fields\], respectively.
Despite getting performance improvements using the proposed method for the three datasets evaluated in this work, it is important to note that it is a computationally expensive method to deploy on any new dataset. This is because one would need to search for the optimal hyper-parameters ($\lambda_{adv}$, $\lambda_{LD}$) for the loss terms.
**Ablation studies:**\
**A. Effects of adversarial ($\lambda_{adv}$) and large deviation ($\lambda_{LD}$) loss terms of regularization loss on segmentation performance**\[eff\_terms\_reg\]: In this experiment, we analyze the effect of each term of the regularization loss on the performance by varying the values of $\lambda_{adv}$ and $\lambda_{LD}$ in the training of the generators $G_V$ and $G_I$. Fig. \[fig:vary\_terms\] presents the quantitative results of the analysis on the cardiac dataset for GD+GI augmentations. We observed the least performance gain over baseline when we disabled the whole regularization with $\lambda_{adv} = \lambda_{LD} = 0$, the setup similar to the work in [@wang2018low]. This setup yielded performance similar to the case when both random deformations and intensity fluctuations were leveraged for augmentation. The performance gain we observe when we enable only the adversarial loss in the regularization, i.e. ($\lambda_{adv}=1,\lambda_{LD}=0$), can be attributed to enforcing the model to match the distribution of generated images to that of unlabeled images. This matching propels the generator to synthesize examples showing the diverse set of shape and intensity variations present in the unlabeled data. We observed a deterioration in performance when only large-deviation loss is enabled on deformation and intensity fields ($\lambda_{adv}=0,\lambda_{LD}=10^{-3}$). This setup encourages the generators to explicitly produce larger deformation and intensity fields without any control from the adversarial term. Transformations output from these generators yields very unrealistic samples, in the most extreme case moving all the foreground pixels out of the frame. Such synthetic examples, naturally, are not useful for the segmentation task.
We observed the highest performance boost when both terms were enabled($\lambda_{adv}=1,\lambda_{LD}=10^{-3}$). This suggests the value of the combination of the terms and their complementary behavior. The large-deviation loss, when used in addition to the adversarial loss, prevents the network from simply replicating the training data with generating very small transformations. Instead, it compels the generator to produce larger fields as long as they satisfy discriminator’s objective. Effectively, producing augmented image-label pairs that are very different from the labeled image-label pairs. Adversarial loss on the other hand, contains the effects of the large-deviation loss by not allowing models to generate extreme transformations.
[1.0]{}
[1.0]{}
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**B. Independent optimization of the generator and the segmentation networks**\[disjoint\_opti\]: One of the biggest claims of the proposed method is the benefit of using the segmentation cost function for learning the generators. This leads to a joint optimization of the segmentation network’s parameters along with the generator’s. Here, we examine the impact on segmentation performance when the generators are optimized without the segmentation loss, only using the regularization term with adversarial and large-deviation terms. Results shown in Table \[tab:non\_joint\_opti\] show that when the segmentation loss is included in the learning of the generators, i.e., joint optimization leads to much higher DSC values than independent optimization. Thus, empirically proving our point that task-driven optimization is better than the traditional independent optimization, which was the approach taken in earlier works [@shin2018medical; @bowles2018gan] to generate the augmented data independent of the down-stream task.
[|m[1.6cm]{}|m[0.77cm]{}|m[0.77cm]{}|m[0.77cm]{}|m[0.77cm]{}|m[0.77cm]{}|m[0.77cm]{}|]{} methods of& &\
optimization & RV & Myo & LV & RV & Myo & LV\
independent & 0.527 & 0.553 & 0.719 & 0.793 & 0.797 & 0.909\
&(0.266) &(0.218) &(0.23) &(0.187) &(0.097) &(0.09)\
joint & 0.651 & 0.710 & 0.834 & 0.832 & 0.823 & 0.922\
&(0.23) &(0.157) &(0.171) &(0.148) &(0.076) &(0.072)\
**C. Varying number of unlabeled images**\[vary\_no\_unl\]: We investigate how varying the number of unlabeled 3D volumes for the training of the proposed method can influence the segmentation performance. This experiment is illustrated in Fig. \[fig:vary\_unlabeled\_data\] for the cardiac dataset. We observe that the improvements in segmentation accuracy do not change significantly while varying the number of unlabeled 3D volumes used. **D. Varying number of labeled images**\[vary\_no\_lab\]: Here, we investigated how the performance gap between affine, random, and proposed augmentations varies as we increase the number of training volumes involved in the training. We observe that the performance gap between the augmentation approaches reduces as we increase the number of labeled training volumes as shown in Fig. \[fig:vary\_labeled\_data\]. This is expected since as the labeled examples increase, the network sees larger number cases and gains robustness to variations present in these images. For the case of 40 3D training volumes, we see that the performance of affine augmentations is almost similar to the proposed augmentations. One striking observation is that with the proposed model, segmentation accuracy using 10 labeled examples is similar to using 40 examples using random augmentations. **E. Different set of train, validation, test, and unlabeled 3D volumes**\[vary\_val\_set\]: Lastly, we show that the results hold for any randomly chosen dataset split, and does not overfit to a specific set of validation images as illustrated in Table \[tab:diff\_tr\_tst\_sets\].
Methods RV Myo LV
--------- --------------- --------------- ---------------
RD+RI 0.506 (0.213) 0.647 (0.159) 0.819 (0.149)
GD+GI 0.683 (0.212) 0.693 (0.139) 0.842 (0.136)
: Mean Dice scores with standard deviations for the cardiac dataset for a different set of train, validation, test, and unlabeled volumes for $N_L=1$ (see \[vary\_val\_set\]-E).[]{data-label="tab:diff_tr_tst_sets"}
**F. No validation images**\[no\_val\_set\]: Additionally, we also present results for the case when no validation images were used in the training in Fig. \[fig:no\_val\_imgs\] in the appendix. Here, the chosen model parameters are obtained after training the generators for predefined number of iterations. Surprisingly, we observe that the performance obtained without validation images does not vary much w.r.t the performance obtained using validation images.
\(a) (b) (c) (d) (e) (f) (g)\
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Input Image $|$ Images generated by $G_V$ $\rightarrow$\
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Input Image $|$ Images generated by $G_I$ $\rightarrow$
Conclusion
==========
In the clinical setting, deployment of successful deep learning algorithms for medical image analysis is limited due to the difficulty of assembling large-scale annotated datasets. In this work, we proposed a semi-supervised task-driven data augmentation method to tackle the issue of obtaining robust segmentation in limited data setting for training. To achieve this we proposed two novel contributions: (i) task-driven based optimization where the generation of the augmentation data is optimal for the segmentation performance, and (ii) semi-supervised nature is induced by using the unlabeled data in the generative modeling setup, where we design two conditional generative models to output transformations that capture two factors of variations: shape and intensity characteristics present in the population. Using three publicly available datasets, we demonstrated the proposed method for segmenting the cardiac, prostate and pancreas using limited annotated examples, reporting substantial performance gains over existing methods. Surprisingly, the augmented images generated via the proposed task-driven approach were not necessarily realistic yet yielded improved segmentation performance, questioning the validity of the assumption that generating realistic examples is the optimal way.
Acknowledgements
================
The presented work is partially funded by Swiss Data Science Center (DeepMicroIA) and the Clinical Research Priority Program Grant on Artificial Intelligence in Oncological Imaging Network from University of Zurich. We thank Nvidia for their GPU donation.
Appendix
========
\
\[fig:rd\_vary\_sig\_ks\]
\
\[fig:rd\_vary\_cont\_brit\]
\
\[fig:no\_val\_imgs\]
\
(a) Cardiac dataset (mean dice per test subject)\
\
(b) Prostate dataset (mean dice per test subject)\
\
(c) Pancreas dataset (mean dice per test subject)\
\[fig:dsc\_per\_each\_subj\]
\(a) image ($X$) (b) deformation field (c) transformed\
over image ($\textbf{v}$) image ($X \circ \textbf{v}$)\
\
\
\
\[fig:geo\_trans\_fields\]
\(a) image ($X$) (b) additive intensity (c) transformed\
field ($\Delta I$) image ($X + \Delta I$)\
\
\
\
\[fig:int\_trans\_fields\]
[^1]: KC, NK, CFB and EK are with the Computer Vision Laboratory, ETH Zurich, Switzerland. Email: {krishna.chaitanya}@vision.ee.ethz.ch. AB and OD are with the University Hospital of Zurich, Switzerland.
[^2]: https://www.creatis.insa-lyon.fr/Challenge/acdc
[^3]: https://www.creatis.insa-lyon.fr/Challenge/acdc
[^4]: http://medicaldecathlon.com/index.html
[^5]: http://medicaldecathlon.com/index.html
|
---
abstract: 'We consider hybrid inflation in the braneworld scenario. In particular, we consider inflation in global supersymmetry with the D-terms in the scalar potential for the inflaton field to be the dominant ones (D-term inflation). We find that D-term dominated inflation can naturally accomodate all requirements of the successful hybrid inflationary model also in the framework of D-brane cosmology with global supersymmetry. The reheating temperature after inflation can be high enough ($\sim 10^{10} \: GeV$) for successful thermal leptogenesis.'
---
**D-term inflation in D-brane cosmology**
G.Panotopoulos$^1$
$^1$ Department of Physics, University of Crete,
0.2 cm
Heraklion, Crete, Hellas
0.2cm
email: [*panotop@physics.uoc.gr*]{}
Introduction
============
Recently there has been considerable interest in higher dimensional cosmological models. In those models our four-dimensional world lives on a three-dimensional extended object (brane) which is embedded in a higher dimensionl space (bulk). The models of this kind are string-inspired ones, as it is known that in Type I string theory [@pol] there are two sectors, the open and the closed ones, and that the theory contains extended objects, called D-branes, where open strings can end. The fields in the closed sector (including gravity) can propagate in the bulk, whereas the fields in the open sector are confined to the brane. In such string-inspired scenarios the extra dimensions need not be small [@dvali] and in fact they can even be non-compact [@rs]. It is important to note that in the context of extra dimensions and the braneworld idea one discovers a generalized Friedmann equation, which is different from the usual Friedmann equation in conventional cosmology. This means that the rate of expansion of the universe in this novel cosmology is altered and accordingly the physics in the early universe can be different from what we know already. So it would be very interesting to study the cosmological implications of these new ideas about extra dimensions and braneworlds. Perhaps the best laboratory for such a study is inflation [@linde], which has become the standard paradigm in the Big-Bang cosmology and which is in favour after the recent discovery from WMAP satellite (see e.g [@wmap]) that the universe is almost flat. It is known that there is not a theory for inflation yet. All we have is a big collection of inflationary models. The single-field models for inflation, such as ’new’ [@linde1] or ’chaotic’ [@linde2], are characterized by the disadvantage that they require ’tiny’ coupling constants in order to reproduce the observational data. This difficulty was overcome by Linde who proposed, in the context of non-supersymmetric GUTS, the hybrid inflationary scenario [@linde3]. It turns out that one can consider hybrid inflation in supersymmetric theories (for a review on supersymmetry and supergravity see [@nilles]) too. In fact, inflation looks more natural in supersymmetric theories rather in non-supersymmetric ones [@riotto]. In a supersymmetric theory, the tree-level potential is the sum of an F-term and a D-term. These two terms have rather different properties and in all inflationary models only one of them dominates [@sakel]. The case of F-term inflation (where F-terms dominate) was considered for the first time in [@wands], while the case of D-term inflation (where D-terms dominate) was considered in [@halyo]. In fact, if one considers supergravity then D-term inflation looks more promising, since it avoids the problem associated with the inflaton mass [@halyo]. F-term inflation in braneworld was studied in [@chafik]. In the present note we discuss the implications of D-term inflation.
Before proceeding our discussion, let us specify our setup. The braneworld modelthat we shall consider is the supersymmetric version of the RS II model (see e.g[@bagger]). However,the cosmological solution of this extended model is the same as that in the non-supersymmetric model, since Einstein’s equations belong to the bosonic part. The only sourse in the bulk is a five-dimensional cosmological constant. There is matter confined to the brane and during inflation, which is the cosmological era we shall be interested in, this matter is dominated by ascalar field, called the inflaton field $\phi$.
The paper consists of six sections of which this introduction is the first. We present D-term inflation in the second section and brane cosmology in the third. Our results for the inflationary dynamics on the brane are discussed in the fourth section. We discuss reheating after inflation in the fifth section and finally we conclude in the fifth section.
D-term inflation
================
In this section we explain what D-term inflation is, following essentially [@riotto]. Inflation, by definition, breaks global supersymmetry since it requires a non-zero cosmological constant $V$ (false vacuum energy of the inflaton). For a D-term spontaneous breaking of supersymmetry a term linear in the auxiliary field $D$ is needed (Fayet-Iliopoulos mechanism [@fayet]). If the theory contains an abelian $U(1)$ gauge symmetry (anomalous or not), the Fayet-Iliopoulos D-term $$\xi \int d^4\theta V=\xi D$$ where $V$ is the vector superfield, is supersymmetric and gauge invariant and therefore allowed by the symmetries. We remark that an anomalous $U(1)$ symmetry is usually present in string theories and the anomaly is cancelled by the Green-Schwarz mechanism. However, here we will consider a non-anomalous $U(1)$ gauge symmetry. In the context of global supersymmetry, D-term inflation is derived from the superpotential $$W=\lambda \Phi \Phi_{+} \Phi_{-}$$ where $\Phi, \Phi_{-}, \Phi_{+}$ are three chiral superfields and $\lambda$ is the superpotential coupling. Under the $U(1)$ gauge symmetry the three chiral superfields have sharges $Q_{\Phi}=0, Q_{\Phi_{+}}=+1$ and $Q_{\Phi_{-}}=-1$, respectively. The superpotential given above leads to the following expression for the scalar potential $$V(\phi_{+}, \phi_{-}, |\phi|)=\lambda^2 (|\phi|^2 (|\phi_{+}|^2+|\phi_{-}|^2)+|\phi_{+} \phi_{-}|^2)+\frac{g^2}{2} (|\phi_{+}|^2-|\phi_{-}|^2+\xi)^2$$ where $\phi$ is the scalar component of the superfield $\Phi$, $\phi_{\pm}$ are the scalar components of the superfields $\Phi_{\pm}$, $g$ is the gauge coupling of the $U(1)$ symmetry and $\xi$ is a Fayet-Iliopoulos term, chosen to be positive. The global minimum is supersymmetry conserving, but the gauge group $U(1)$ is spontaneously broken $$<\phi>=<\phi_{+}>=0, \; <\phi_{-}>=\sqrt{\xi}$$ However, if we minimize the potential, for fixed values of $\phi$, with respect to other fields, we find that for $\phi>\phi_{c}=\frac{g}{\lambda} \sqrt{\xi}$, the minimum is at $\phi_{+}=\phi_{-}=0$. Thus, for $\phi>\phi_{c}$ and $\phi_{+}=\phi_{-}=0$ the tree-level potential has a vanishing curvature in the $\phi$ direction and large positive curvature in the remaining two directions $m_{\pm}^2=\lambda^2 |\phi|^2 \pm g^2 \xi$.
For arbitrary large $\phi$ the tree-level value of the potential remains constant and equal to $V_{0}=(g^2/2) \xi^2$, thus $\phi$ plays naturally the role of an inflaton field. Along the inlationary trajectory the F-term vanishes and the universe is dominated by the D-term, which splits the masses in the $\Phi_{+}$ and $\Phi_{-}$ superfields, resulting to the one-loop effective potential for the inflaton field. The radiative corrections are given by the Coleman-Weinberg formula [@coleman] $$\Delta V_{1-loop}=\frac{1}{64 \pi} \sum_{i} (-1)^{F_{i}} m_{i}^4 ln\frac{m_{i}^2}{\Lambda^2}$$ where $\Lambda$ stands for a renormalization scale which does not affect physical quantities and the sum extends over all helicity states $i$, with fermion number $F_{i}$ and mass squared $m_{i}^2$. The radiative corrections given by the above formula lead to the following effective potential for D-term inflation $$\label{eq:1}
V(\phi)=\frac{g^2 \xi^2}{2} \left (1+\frac{g^2}{16 \pi^2} ln\frac{|\phi|^2 \lambda^2}{\Lambda^2} \right )$$ The end of inflation is determined either by the failure of the slow-roll conditions or when $\phi$ approaches $\phi_{c}$.
Effective gravitational equations on the brane
==============================================
Here we review the basic equations of brane cosmology. We work essentially in the context of Randall-Sundrum II model [@rs]. In the bulk there is just a cosmological constant $\Lambda_{5}$, whereas on the brane there is matter with energy-momentum tensor $\tau_{\mu \nu}$. Also, the brane has a tension $T$. The five dimensional Planck mass is denoted by $M_{5}$. If Einstein’s equations hold in the five dimensional bulk, then it has been shown in [@shir] that the effective four-dimensionl Einstein’s equations induced on the brane can be written as $$G_{\mu \nu}+\Lambda_{4} g_{\mu \nu}=\frac{8 \pi}{M_{p}^2} \tau_{\mu \nu}+(\frac{8 \pi}{M_{5}^3})^2 \pi_{\mu \nu}-E_{\mu \nu}$$ where $g_{\mu \nu}$ is the induced metric on the brane, $\pi_{\mu \nu}=\frac{1}{12} \: \tau \: \tau_{\mu \nu}+\frac{1}{8} \: g_{\mu \nu} \: \tau_{\alpha \beta} \: \tau^{\alpha \beta}-\frac{1}{4} \: \tau_{\mu \alpha} \: \tau_{\nu}^{\alpha}-\frac{1}{24} \: \tau^2 \: g_{\mu \nu}$, $\Lambda_{4}$ is the effective four-dimensional cosmological constant, $M_{p}$ is the usual four-dimensional Planck mass and $E_{\mu \nu} \equiv C_{\beta \rho \sigma} ^\alpha \: n_{\alpha} \: n^{\rho} \: g_{\mu} ^{\beta} \: g_{\nu} ^{\sigma}$ is a projection of the five-dimensional Weyl tensor $C_{\alpha \beta \rho \sigma}$, where $n^{\alpha}$ is the unit vector normal to the brane. The tensors $\pi_{\mu \nu}$ and $E_{\mu \nu}$ describe the influense of the bulk in brane dynamics. The five-dimensional quantities are related to the corresponding four-dimensional ones through the relations $$M_{p}=\sqrt{\frac{3}{4 \pi}} \frac{M_{5}^3}{\sqrt{T}}$$ and $$\Lambda_{4}=\frac{4 \pi}{M_{5}^3} \left( \Lambda_{5}+\frac{4 \pi T^2}{3 M_{5}^3} \right )$$ In a cosmological model in which the induced metric on the brane $g_{\mu \nu}$ has the form of a spatially flat Friedmann-Robertson-Walker model, with scale factor $a(t)$, the Friedmann-like equation on the brane has the generalized form [@binetry] $$H^2=\frac{\Lambda_{4}}{3}+\frac{8 \pi}{3 M_{p}^2} \rho+(\frac{4 \pi}{3 M_{5}^3})^2 \rho^2+\frac{C}{a^4}$$ where $C$ is an integration constant arising from $E_{\mu \nu}$. The cosmological constant term and the term linear in $\rho$ are familiar from the four-dimensional convensional cosmology. The extra terms, i.e the “dark radiation” term and the term quadratic in $\rho$, are there because of the presense of the extra dimension. Adopting the Randall-Sundrum fine-tuning $$\Lambda_{5}=-\frac{4 \pi T^2}{3 M_{5}^3}$$ the four-dimensional cosmological constant vanishes. Furthermore, the term with the integration constant $C$ will be rapidly diluted during inflation and can be ignored. So the generalized Friedmann equation takes the final form $$H^2=\frac{8 \pi}{3 M_{p}^2} \rho \left( 1+\frac{\rho}{2 T} \right )$$ We notice that in the low density regime $\rho \ll T$ we recover the usual Friedmann equation. However, in the high energy regime $\rho \gg T$ the unity can be neglected and then the Friedmann-like equation becomes $$H^2=\frac{4 \pi \rho^2}{3 T M_{4}^2}$$ Note that in this regime the Hubble parameter is linear in $\rho$, while in conventional cosmology it goes with the square root of $\rho$.
Inflationary dynamics on the brane
==================================
As already mentioned, we will consider the case in which the energy momentum on the brane is dominated by a scalar field $\phi$ confined on the brane with a self-interaction potential $V(\phi)$ given in (\[eq:1\]). The field $\phi$ is a function of time only, as dictated by the isotropy and homogeneity of the observed four-dimensional universe. A homogeneous scalar field behaves like a perfect fluid with pressure $p=(1/2) \dot{\phi}^2-V$ and energy density $\rho=(1/2) \dot{\phi}^2+V$. There is no energy exchange between the brane and the bulk, so the energy-momentum tensor $T_{\mu \nu}$ of the scalar field is conserved, that is $\nabla ^ \nu T_{\mu \nu}=0$. This is equivalent to the continuity equation for the pressure $p$ and the energy density $\rho$ $$\dot{\rho}+3 H (p+\rho)=0$$ where $H$ is the Hubble parameter $H=\dot{a}/a$. Therefore we get the equation of motion for the scalar field $\phi$, which is the following $$\ddot{\phi}+3 H \dot{\phi}+V'(\phi)=0$$ This is of course the Klein-Gordon equation for a scalar field in a Robertson-Walker background. The equation that governs the dynamics of the expansion of the universe is the Friedmann-like equation of the previous section. Inflation takes place in the early stages of the evolution of the universe, so in the Friedmann equation the extra term dominates and therefore the equation for the scale factor is $$H^2=\frac{4 \pi \rho^2}{3 T M_{p}^2}$$ In the slow-roll approximation the slope and the curvature of the potential must satisfy the two constraints $\epsilon \ll 1$ and $|\eta| \ll 1$, where $\epsilon$ and $\eta$ are the two slow-roll parameters which are defined by $$\epsilon \equiv -\frac{\dot{H}}{H^2}$$ $$\eta \equiv \frac{V''}{3 H^2}$$ In this approximation the equation of motion for the scalar field takes the form $$\dot{\phi} \simeq -\frac{V'}{3 H}$$ while the generalized Friedmann equation becomes $(V \gg \dot{\phi}^2)$ $$H^2 \simeq \frac{4 \pi V^2}{3 T M_{p}^2}$$ The number of e-folds during inflation is given by $$N \equiv ln \frac{a_{f}}{a_{i}} = \int _{t_{i}}^{t_{f}} \: H dt$$ For a strong enough inflation we take $N=60$. In the slow-roll approximation the number of e-folds and the slow-roll parameters are given by the formulae [@maartens] $$\epsilon \simeq \frac{M_{p}^2}{16 \pi} \: \left (\frac{V'}{V} \right )^2 \: \frac{4 T}{V}$$ $$\eta \simeq \frac{M_{p}^2}{8 \pi} \: \left (\frac{V''}{V} \right )^2 \: \frac{2 T}{V}$$ $$N \simeq -\frac{8 \pi}{M_{p}^2} \: \int _{\phi_{i}} ^ {\phi_{f}} \: \frac{V}{V'} \: \frac{V}{2 T} \: d \phi$$ The main cosmological constraint comes from the amplitude of the scalar perturbations which is given in this new context by [@maartens] $$A_{s}^2=\frac{512 \pi}{75 M_{p}^6} \: \frac{V^3}{V'^2} \: \left (\frac{V}{2 T} \right )^2$$ where the right-hand side is evaluated at the horizon-crossing when the comoving scale equals the Hubble radius during inflation. Finally, the spectral index for the scalar perturbations is given in terms of the slow-roll parameters $$n_{s}-1 \equiv \frac{d \: ln A_{s}^2}{d \: ln k}=2 \eta - 6 \epsilon$$ and the tensor-to-scalar ratio is given by $$\frac{A_{t}^2}{A_{s}^2}=\epsilon \: \frac{T}{V}$$ In what follows we will assume that $g \sim 0.5$ and that inflation ends at $\phi_{c}=(g/ \lambda) \: \sqrt{\xi}$. To make sure that the slow-roll conditions are satisfied we impose the constraint $$\frac{T M_{p}^2 \lambda^2}{16 \pi^3 g^2 \xi^3} \ll 1$$ Also, we have assumed that the potential $V$ is much larger than the brane tension $T$. Therefore another constraint to be satisfied is $$\frac{g^2 \xi^2}{4 T} \gg 1$$ Now that we have written all the necessary formulae, we can proceed to the presentation of our results. For arbitrary $\lambda$ it is not possible to satisfy both the datum from COBE that $A_{s}=2 \cdot 10^{-5}$ and the slow-roll conditions. For this to happen the superpotential coupling $\lambda$ has to be smaller or equal to $0.0185$ (approximately). Then, for a given value for $\lambda$, the brane tension cannot become arbitrarily large because in that case the constraint that the potential should be much larger than the brane tension is not satisfied. We find the following upper bound for the brane tension $T$ $$T \leq 5 \cdot 10^{55} \: GeV^4$$ Now that we have set upper bounds for $T$ and $\lambda$ so that our constraints and the data from COBE are satisfied, we can compute the spectral index $n_{s}$ and the tensor-to-scalar ratio $r$. For example, for the values $T=5 \cdot 10^{55} \: GeV^4$ and $\lambda=0.0185$ we find $$n_{s}=0.99, \; r=2 \cdot 10^{-4}$$ A detailed analysis shows that for a particular value for $\lambda$ (below the upper bound of course) the spectral index does not depend on $T$ and is always very close to 1. As $\lambda$ becomes smaller and smaller the spectral index slightly increases and gets even closer to 1. Also, in all cases the tensor perturbations are negligible. Finally, we find that for the maximum value for the brane tension $\sqrt{\xi} \sim 10^{14} \: GeV$, whereas $\sqrt{\xi}$ becomes smaller as $T$ decreases. We note that according to our analysis $\lambda$ a priori can take arbitrarily small values. However, this would be unnatural and for that reason we do not consider values for $\lambda$ much smaller than $10^{-3}$. In that case we find that the values of the inflaton remain safely below Planck mass and therefore global supersymmetry is a good approximation.
Reheating
=========
Finally, let us turn to the discussion of reheating after inflation and to the computation of the reheating temperature $T_R$. After slow-roll the inflaton decays with a decay rate $\Gamma$ and the decay products quickly thermalize. This is the way the universe re-enters the radiation era of standard Big-Bang cosmology. The reheating temperature $T_R$ is related to two more cosmological topics, namely the gravitino problem [@khlopov] and the baryogenesis through leptogenesis. In gravity mediated SUSY breaking models and for an interesting range of the gravitino mass, $m_{3/2} \sim 0.1-1 TeV$, if the gravitino is unstable it has a long lifetime and decays after the BBN. The decay products destroy light elements produced by the BBN and hence the primordial abundance of the gravitino is constrained from above to keep the success of the BBN. This leads to an upper bound on the reheating temperature $T_R$ after inflation, since the abundance of the gravitino is proportional to $T_R$. A detailed analysis derived a stringent upper bound $T_R \leqslant 10^6-10^7 \: GeV$ when gravitino has hadronic modes [@ellis]. On the other hand, primordial lepton asymetry is converted to baryon asymmetry [@yanagida] in the early universe through the “sphaleron” effects of the electroweak gauge theory [@rubakov]. This baryogenesis through leptogenesis requires a lower bound on the reheating temperature. Leptogenesis can be thermal or non-thermal. For a thermal leptogenesis $T_{R} \geqslant 2 \cdot 10^9 \: GeV$ [@giudice], whereas for non-thermal leptogenesis $T_{R} \geqslant 10^6 \: GeV$ [@asaka]. It seems that it is impossible to satisfy both constaints for the reheating temperature coming from leptogenesis and the gravitino problem. However, the authors of [@gravitino] have showed that in the brane world scenario, that we discuss here, it is possible to solve the gravitino problem allowing for the reheating temperature to be as high as $10^{10} \: GeV$. According to ref. [@gravitino] the gravitino abundance is proportional not to the reheating temperature, as is the case in conventional cosmology, but to a transition temperature $T_t$ between high temperatures ($T_R$) and low ones (today’s temperature $T_0$). That way the requirement for not over-production of gravitini leads to an upper bound for this transition temperature and not for the reheating temperature, which can be as high as a satisfactory leptogenesis requires.
The reheating temperature is given by the formula $$T_R=\left (\frac{90 T \Gamma^2 m_p^2}{2 \pi^3 g^2 \xi^2 g_{eff}^2} \right )^{1/4}$$ where $g_{eff}$ is the effective number of degrees of freedom at the reheating temperature and for the MSSM is $g_{eff}=\frac{915}{4}$. Assuming that the inflaton $\phi$ decays to the lighest of the three heavy right handed neutrinos $\psi$ $$\phi \; \rightarrow \; \psi + \psi$$ the decay rate of the inflaton is [@sakel] $$\Gamma=\frac{m_{infl}}{8 \pi} \left ( \frac{M_1}{\sqrt{\xi}} \right )^2$$ where $m_{infl}$ is the inflaton mass, $M_1$ is the smallest of the three neutrino mass eigenvalues and $m_{infl} > 2 M_{1}$. The mass of the inflaton is given in terms of the coupling constant $g$ and the Fayet-Iliopoulos parameter $\xi$ by $$m_{infl} = \sqrt{2} \: g \: \sqrt{\xi}$$ If the value of the mass of the lightest right handed neutrino is $M_{1}=10^{10} \: GeV$, which is a representative value, then the reheating temperature $T_{R}$ turns out to be of the order of the right handed neutrino mass. In fact the reheating temperature is a little bit larger than the neutrino mass. So we see that the reheating temperature is of the right order of magnitude for thermal leptogenesis. When the right handed neutrino mass increases (remaining though smaller than $m_{infl}/2$), the reheating temperature increases too and is always of the order of the neutrino mass but a little bit larger (see Figure 1). For a given value of $M_{1}$ and of the superpotential coupling $\lambda$ the reheating temperature does not change varying the tension of the brane $T$. Finally, for a given $M_1$, when $\lambda$ increases then $T_{R}$ decreases, but only slighty so as to remain of the order of magnitude of the mass $M_{1}$ (see Figure 2).
Conclusions
===========
To summarize, we have reexamined supersymmetric D-term dominated hybrid inflation in brane cosmology. We have found that we can reproduce the observational data provided that each of the brane tension, five-dimensional Planck mass and the superpotential coupling does not exceed a particular value. For a given value for the superpotential coupling, when the brane tension takes the maximum allowed value then the scale of inflation $\sqrt{\xi}$ is of the order of $\sim 10^{14} \: GeV$. This value of the inflationary scale is lower than the (supersymmetric) GUT scale, but close to it. Also, we have found that for natural values of the superpotential coupling $\lambda$ the inflaton field cannot take large values and stays well below the four-dimensional Planck mass, consistent with the global supersymmetry approximation adopted here. Furthermore, we have seen that our results are compatible with the corresponding results in the standard four-dimensional cosmology. This means that the advantages of the hybrid model are naturally preserved in the framework of brane cosmology. Finally, our study shows that the reheating temperature after inflation can naturally be of order $10^{10} \: GeV$ (or larger) allowing for a successful thermal leptogenesis.
Acknowlegements {#acknowlegements .unnumbered}
===============
The author would like to thank T.N.Tomaras for valuable comments on the manuscript and I.Antoniadis and G.Kofinas for useful discussions. The author would like also to thank CERN theory division (where part of this work was completed) for its warmest hospitality. Work supported by the Greek Ministry of education research program “Heraklitos” and by the EU grant MRTN-CT-2004-512194.
[999]{}
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|
---
abstract: |
The long time behavior of a couple of interacting asymmetric exclusion processes of opposite velocities is investigated in one space dimension. We do not allow two particles at the same site, and a collision effect (exchange) takes place when particles of opposite velocities meet at neighboring sites. There are two conserved quantities, and the model admits hyperbolic (Euler) scaling; the hydrodynamic limit results in the classical Leroux system of conservation laws, *even beyond the appearence of shocks*. Actually, we prove convergence to the set of entropy solutions, the question of uniqueness is left open. To control rapid oscillations of Lax entropies via logarithmic Sobolev inequality estimates, the symmetric part of the process is speeded up in a suitable way, thus a slowly vanishing viscosity is obtained at the macroscopic level. Following [@fritz1; @fritz2], the stochastic version of Tartar–Murat theory of compensated compactness is extended to two-component stochastic models.\
[Key words:]{} hydrodynamic limit, hyperbolic scaling, systems of conservation laws, compensated compactness\
[AMS 2001 subject classification: 60K35, 82C24]{}
author:
- |
[József Fritz Bálint Tóth]{}\
Institute of Mathematics\
Budapest University of Technology and Economics
title: 'Derivation of the Leroux system as the hydrodynamic limit of a two-component lattice gas'
---
Introduction {#section:intro}
============
The main purpose of this paper is to derive a couple of Euler equations (hyperbolic conservation laws) in a regime of shocks. While the case of smooth macroscopic solutions is quite well understood, see [@yau] and [@ovy], serious difficulties emerge when the existence of classical solutions breaks down. A general method to handle attractive systems has been elaborated in [@rezakhanlou], see also [@fritz1] and [@kipnislandim] for further references. Hyperbolic models with two conservation laws, however, can not be attractive in the usual sense because the phase space is not ordered in a natural way. We have to extend some advanced methods of PDE theory of hyperbolic conservation laws to stochastic (microscopic) systems. Lax entropy and compensated compactness are the main key words here, see [@lax1], [@lax2], [@murat], [@tartar1], [@tartar2], [@diperna] for the first ideas, and the textbook [@serre] for a systematic treatment. The project has been initiated in [@fritz1], a full exposition of techniques in the case of a one-component asymmetric Ginzburg–Landau model is presented in [@fritz2]. Here we investigate the simplest possible, but nontrivial two-component lattice gas with collisions, further models are to be discussed in a forthcoming paper [@fritztoth2]. Since the underlying PDE theory is restricted to one space dimension, we also have to be satisfied with such models. The proof is based on a strict control of entropy pairs at the microscopic level as prescribed by P. Lax, L. Tartar and F. Murat for approximate solutions to hyperbolic conservation laws. A Lax entropy is macroscopically conserved along classical solutions, but the microscopic system can not have any extra conservation law, thus we are facing with rapidly oscillating quantities. These oscillations are to be controlled by means of logarithmic Sobolev inequality estimates, and effective bounds are obtainable only if the symmetric part of the microscopic evolution is strong enough. That is why the *microscopic viscosity* of the model goes to infinity, i.e. the model is changed when we rescale it. Of course, the *macroscopic viscosity* vanishes in the limit and thus the effect of speeding up the symmetric part of the microscopic infinitesimal generator is not seen in the hydrodynamic limit.
Unfortunately, compensated compactness yields only *existence* of weak solutions, the Lax entropy condition is not sufficient for weak *uniqueness* in the case of two component systems. That is why we can prove convergence of the conserved fields to the set of entropy solutions only, we do not know whether this set consists of a single trajectory specified by its initial data. Let us remark that [@quastelyau] has the same difficulty concerning the derivation of the incompressible Navier–Stokes equation in 3 space dimensions. The Oleinik type conditions of weak uniqueness are out of reach of our methods because they require a one sided uniform Lipschitz continuity of the Riemann invariants of the macroscopic system, see [@bressan] for most recent results of PDE theory in this direction. It is certainly not easy to get such bounds at the microscopic level.
The paper is organized as follows. The microscopic model and the macroscopic equations are introduced in the next two sections. The main result and its conditions are formulated in Section 4. Proofs are presented in Section 5, while some technical details are postponed to the Appendix.
Microscopic model {#section:model}
=================
State space, conserved quantities, infinitesimal generator {#subs:statespace}
----------------------------------------------------------
We consider a pair of coupled asymmetric exclusion processes on the discrete torus, particles move with an average speed $+1$ and $-1$, respectively. Since we allow at most one particle per site, the individual state space consists of three elements. There is another effect in the interaction, something like a collision: if two particles of opposite velocities meet at neighboring sites, then they are also exchanged after some exponential holding times. We can associate velocities $\pm1$ to particles according to their categories, thus particle number and momentum are the natural conserved quantities; the numbers of $+1$ and $-1$ particles could have been another choice.
Throughout this paper we denote by $\Tn$ the discrete torus $\Z/n\Z$, $n\in\N$, and by $\T$ the continuous torus $\R/\Z$. The local spin space is $S=\{-1,0,1\}$. The state space of the interacting particle system of size $n$ is $$\Omn:=S^{\Tn}.$$ Configurations will generally be denoted as $$\uo:=(\omega_j)_{j\in \Tn}\in\Omn,$$ We need to separate the symmetric (reversible) part of the dynamics. This will be speeded up sufficiently in order to enhance convergence to local equilibrium also at a mesoscopic scale. The phenomenon of compensated compactness is materialized at this scale in the hydrodynamic limiting procedure. So (somewhat artificially) we consider separately the asymmetric and symmetric parts of the rate functions $r: S\times S\rightarrow \R_+$, respectively, $s:S\times S\rightarrow \R_+$. The dynamics of the system consists of elementary jumps exchanging nearest neighbor spins: $(\omega_j,\omega_{j+1}) \rightarrow (\omega'_j,\omega'_{j+1}) =
(\omega_{j+1},\omega_{j})$, performed with rate $ \lambda r(\omega_j,\omega_{j+1}) + \kappa s(\omega_j,\omega_{j+1})$, where $\lambda, \kappa > 0$ are fixed speed-up factors, depending on the size of the system in the limiting procedure.
The rate functions are chosen as follows: $$\begin{array}{rr}
r(1,-1)=0, & \qquad r(-1,1)=2,
\\[5pt]
r(0,-1)=0, & \qquad r(-1,0)=1,
\\[5pt]
r(1,0)=0, & \qquad r(0,1) =1,
\end{array}$$ that is the rate of collisions is twice as large as that of simple jumps, and $$r(\omega_j,\omega_{j+1})=\omega^-_j(1-\omega^-_{j+1})
+\omega^+_{j+1}(1-\omega^+_j)\,,$$ where $\omega^+_j:=\ind_{\{\omega_j=1\}}\,,$ $\omega^-_j:=\ind_{\{\omega_j=-1\}}$ and $\ind_A$ denotes the indicator of a set $A\,.$ The rates of the symmetric component are simply $$s(\omega_j,\omega_{j+1}) = \ind_{\{\omega_j\not=\omega_{j+1}\}}\,.$$ The rates $r$ define a *totally asymmetric* dynamics, while the rates $s$ define a *symmetric* one. The infinitesimal generators defined by these rates are: && f() := \_[j]{} r(\_j,\_[j+1]{}) (f( \_[j,j+1]{} )-f())\
&& f() = \_[j]{} s(\_j,\_[j+1]{}) (f(\_[j,j+1]{} )-f()), where $\Theta_{i,j}$ is the spin-exchange operator, ( \_[i,j]{} )\_[k]{} = {
[lcl]{} \_[j]{} & & k=i\
\_[i]{} & & k=j\
\_[k]{} & & k=i,j.
. Recall that periodic boundary conditions are assumed in the definition of $\Ln$ and $\Kn$.
To get exactly the familiar Leroux system as the limit, the two conserved quantities, $\eta$ and $\xi$ should be chosen as \_j = (\_j) := 1-[|[\_j]{}|]{} \_j = (\_j):=\_j. The microscopic dynamics of the model has been defined so that $\sum_j\xi_j$ and $\sum_j\eta_j$ are conserved, we shall see that there is no room for other (independent) hidden conserved observables. In terms of the conservative quantities we have $$\begin{aligned}
\label{eq:rates}
r(\omega_j,\omega_{j+1})&=\frac{1}{4}(1-\eta_j-\xi_j)
(1+\eta_{j+1}+\xi_{j+1})\\&+\frac{1}{4}(1+\eta_j-\xi_j)
(1-\eta_{j+1}+\xi_{j+1})\,.\notag\end{aligned}$$
The rate functions are so chosen that the product measures \_[,u]{}()= \_[j]{}\_[,u]{}(\_j), with one-dimensional marginals $$\pi_{\rho,u}(0)=\rho, \quad
\pi_{\rho,u}(\pm1)=\frac{1-\rho\pm u}{2}.$$ are stationary in time. We shall call these Gibbs measures. The parameters take values from the set $${\cal D}:=
\{(\rho,u)\in[0,1]\times[-1,1]\,:\,\rho+{\left|{u}\right|}\le1\},$$ and the uniform $\pin:=\pin_{1/3,0}$ will serve as a reference measure. Due to conservations, the stationary measures $\pin_{\rho,u}$ are not ergodic. Expectation with respect to the measures $\pin_{\rho,u}$ will be denoted by $\expect_{\rho,u}(\cdot)$. In particular, given a local observable $\ups_i:=\ups(\omega_{i-m},\dots,\omega_{i+m})$ with $m$ fixed, its equilibrium expectation will be denoted as (,u):= \_[,u]{}(\_i).
The system of microscopic size $n$ will be driven by the infinitesimal generator = n+ n\^2, where $\sigma=\sigma(n)$ is the *macroscopic viscosity*, the factor $n\sigma(n)$ can be interpreted as the *microscopic viscosity*. A priori we require that $\sigma(n)\ll1$ as $n\to\infty$. A very important restriction, $\sqrt{n}\sigma(n)\gg1$ will be imposed on $\sigma(n)$, see condition (\[cond:viscosity\]) in subsection \[subs:result\].
Let $\mun_0$ be a probability distribution on $\Omn\,,$ which is the initial distribution of the microscopic system of size $n$, and denote \_t:=\_0e\^[t]{} the distribution of the system at (macroscopic) time $t$. The Markov process on the state space $\Omn$ driven by the infinitesimal generator $\Gn$, started with initial distribution $\mun_0$ will be denoted by $\Xn_t$.
Fluxes {#subs:fluxes}
------
Elementary computations show that the infinitesimal generators $\Ln$ and $\Kn$ act on the conserved quantities as follows, see .
[rll]{} \_i= & - (\_[i]{},\_[i+1]{}) + (\_[i-1]{},\_[i]{}) & =: -\_[i]{} + \_[i-1]{},\
\_i= & - (\_[i]{},\_[i+1]{}) + (\_[i-1]{},\_[i]{}) & =: -\_[i]{} + \_[i-1]{},\
\_i= & - \^s(\_[i]{},\_[i+1]{}) + \^s(\_[i-1]{},\_[i]{}) & =: -\^s\_[i]{} + \^s\_[i-1]{},\
\_i= & - \^s(\_[i]{},\_[i+1]{}) + \^s(\_[i-1]{},\_[i]{}) & =: -\^s\_[i]{} + \^s\_[i-1]{},
where \[eq:phipsidef\]
[rrl]{} \_j &=& r(\_i,\_[i+1]{})(\_i-\_[i+1]{})\
&=& 12 {\_[j]{}\_[j+1]{} +\_[j+1]{}\_[j]{}} + 12 {\_[j]{}-\_[j+1]{}}\
\_j &=& r(\_i,\_[i+1]{})(\_i-\_[i+1]{})\
&=& 12 { \_[j]{} + \_[j+1]{}-2 + 2\_[j]{}\_[j+1]{}}+12 { \_[j+1]{}\_[j]{} - \_[j]{}\_[j+1]{} } + {\_[j]{} - \_[j+1]{}} ,\
\^s\_j &=& \_[j]{}-\_[j+1]{},\
\^s\_j &=& \_[j]{}-\_[j+1]{}.
Note that the microscopic fluxes of the conserved observables induced by the symmetric rates $s(\omega_{j},\omega_{j+1})$ are (discrete) gradients of the corresponding conserved variables.
It is easy to compute the macroscopic fluxes: \[eq:macrofluxes\]
[l]{} (,u) := \_[,u]{}(\_j) = u\
(,u) := \_[,u]{}(\_j) = + u\^2
Leroux’s equation – a short survey {#section:leroux}
==================================
Having the macroscopic fluxes (\[eq:macrofluxes\]) computed, the Euler equations of the system considered are expected to be \[eq:leroux\] {
[l]{} + (u)=0\
u + (+ u\^2)=0.
. with given initial data \[eq:ic\] u(0,x)=u\_0(x), (0,x)=\_0(x). This is exactly Leroux’s equation well known in the PDE literature, see [@serre]. In the present section we shortly review the main facts about this PDE. The first striking fact is that such equations may have classical solutions only for some special initial data, in general shocks are developed in a finite time. Therefore solutions should be understood in a weak (distributional) sense, and there are many weak solutions for the same initial values.
The following vectorial notations sometimes make our formulas more compact: := (
[c]{}\
u
), := (
[c]{}\
), := (
[cc]{} &
), \^2:= (
[cc]{} &\
&
) We shall use alternatively, at convenience, the compact vectorial and the explicit notation.
Lax entropy pairs {#subs:lent}
-----------------
In the case of classical solutions (\[eq:leroux\]) can be written as $\partial_t\vu+D(\vu)\partial_x\vu=0,$ where D(,u):=(,u) = (
[cc]{} u &\
1 & 2 u
) is the matrix of the linearized system. The eigenvalues of $D$ are just && = (,u) := u+ 12 { + u },\
&& = (,u) := u- 12 { -u}. This means that (\[eq:leroux\]) is *strictly hyperbolic* in the domain {(,u): 0, u, (,u)=(0,0)}, with marginal degeneracy (i.e. coincidence of the two characteristic speeds, $\lambda=\mu$) at the point $(\rho,u)=(0,0)$.
*Lax entropy/flux pairs* $\bigl(S(\vu),F(\vu)\bigr)$ are solutions of the linear hyperbolic system $\nabla F(\vu)=\nabla S(\vu)\cdot\nabla\vPhi(\vu)\,,$ that is $\partial_tS(\vu)+\partial_xF(\vu)=0$ along classical solutions. This means that an entropy $S$ is a conserved observable. In our particular case this reads \[entropia\] {
[ll]{} F’\_=&uS’\_+S’\_u,\
F’\_u=&S’\_+2uS’\_u.
. or, written as a second order linear equation for $S$: \[entropia2\] S”\_+u S”\_[u]{} -S”\_[uu]{}=0. This equation is known to have many convex solutions, see [@lax1]. We call an entropy/flux pair *convex* if the map $(\rho,u)\mapsto S(\rho,u)$ is convex. In particular, a globally convex Lax entropy/flux pair defined on the whole half plane $\R_+\times\R$ is $$S(\rho,u):=\rho\log\rho +\frac{u^2}{2},
\quad
F(\rho,u):=u\rho+u\rho\log\rho+\frac{2u^3}{3}\,.$$ Weak solutions of (\[entropia\]) are called *generalized entropy/flux pairs*. Riemann’s method of solving second order linear hyperbolic PDEs in two variables (see Chapter 4 of [@john]) and compactness of $\dom$ imply that generalized entropy/flux pairs can be approximated pointwise by twice differentiable entropy/flux pairs.
An *entropy solution* of the Cauchy problem (\[eq:leroux\]), (\[eq:ic\]) is a measurable function $[0,T]\times\T\ni(t,x)\mapsto \vu(t,x)\in\R_+\times\R$ which for any convex entropy/flux pair $(S,F),$ and any nonnegative test function $\varphi:[0,T]\times\T\to\R$ with support in $[0,T)\times\T$ satisfies && \_0\^t\_( (t,x) S((t,x)) + (t,x) F((t,x)) ) dxdt\
\[eq:wlax\] && + \_(0,x)S((0,x)) dx 0 Note that $S(\rho,u)=\pm\rho, F(\rho,u)=\pm\rho u$, respectively, $S(\rho,u)=\pm u, F(\rho,u)=\pm(\rho + u^2)$ are entropy/flux pairs, thus entropy solutions are (a special class of) weak solutions. Entropy solutions of the Cauchy problem (\[eq:leroux\]), (\[eq:ic\]) form a (strongly) closed subset of the Lebesgue space $L^p([0,T]\times\T, \,dt\,dx)=:L^p_{t,x}$ for any $p\in[1,\infty)$.
Young measures, measure valued entropy solutions {#subs:young}
------------------------------------------------
A Young measure on $([0,T]\times\T)\times{\cal D}$ is $\nu=\nu(t,x;d\vv)$, where\
(1) for any $(t,x)\in [0,T]\times\T$ fixed, $\nu(t,x;d\vv)$ is a probability measure on ${\cal D}$, and,\
(2) for any $A\subset{\cal D}$ fixed the map $(t,x)\mapsto\nu(t,x;A)$ is measurable.\
Given a probability measure $\nu$ on $\R_+\times\R$, we shall use the notation $$\langle\nu\,,\,f\rangle:=
\int_{\cal D} f(\vv)\,\nu(d\vv).$$ The set of Young measures will be denoted by ${\cal Y}$. A sequence $\nu^{n}\in{\cal Y}$ *converges vaguely* to $\nu\in{\cal Y}$, denoted $\nu^n\vto\nu$, if for any $f\in C([0,T]\times\T\times{\cal D})$ \_[n]{} \_0\^T\_\^n(t,x),f(t,x,)dtdx = \_0\^T\_(t,x),f(t,x,)dtdx, or, equivalently, if for any test function $\varphi\in C([0,T]\times\T)$ and any $g\in C({\cal D})$ \_[n]{} \_0\^T\_(t,x) \^n(t,x),gdtdx = \_0\^T\_(t,x) (t,x),gdtdx. The set ${\cal Y}$ of Young measures will be endowed with the vague topology induced by this notion of convergence. ${\cal Y}$ endowed with the vague topology is metrizable, separable and compact. We also consider (without explicitly denoting this) the Borel structure on $\cal Y$, induced by the vague topology.
We say that the Young measure $\nu(t,x;d\vv)$ is *Dirac-type* if there exists a measurable function $\vu:[0,T]\times\T\to{\cal D}$ such that for almost all $(t,x)\in[0,T]\times\T$, $\nu(t,x;d\vv)=\delta_{\vu(t,x)}(d\vv)$. We denote the subset of Dirac-type Young measures by ${\cal U}\subset{\cal Y}$. It is a fact (see Chapter 9 of [@serre]) that $${\cal Y}
=
\overline{\text{co}({\cal Y})}
=
\overline{\text{co}({\cal U})}
=
\overline{{\cal U}},$$ where ‘co’ stands for convex hull and closure is meant according to the vague topology.
We say that the Young measure $\nu(t,x;d\vv)$ is a *measure valued entropy solution* of the Cauchy problem (\[eq:leroux\]), (\[eq:ic\]) iff for any convex entropy/flux pair $(S,F)$ and any positive test function $\varphi:[0,T]\times\T\to\R_+$ with support in $[0,T)\times\T$, \_0\^T\_( (t,x) (t,x),S + (t,x) (t,x),F) dxdt &&\
\[eq:mlax\] + \_(0,x) (0,x),S dx 0 && holds true. Measure valued entropy solutions of the Cauchy problem (\[eq:leroux\]), (\[eq:ic\]) form a (vaguely) *closed* subset of $\cal Y$.
Clearly, if $\vu:[0,T]\times\T\to{\cal D}$ is an entropy solution of the Cauchy problem (\[eq:leroux\]), (\[eq:ic\]) in the sense of (\[eq:wlax\]), then the Dirac-type Young measure $\nu(t,x;d\vv):=\delta_{\vu(t,x)}(d\vv)$ is a measure valued entropy solution in the sense of (\[eq:mlax\]). The convergence of subsequences of approximate solutions to measure solutions is almost immediate by vague compactness, the crucial issue is to show the Dirac property of measure valued entropy solutions. This is the aim of the theory of compensated compactness.
Tartar factorization {#subs:tarfac}
--------------------
A probability measure $\nu(d\rho,du)$ on $\R^2$ satisfies the *Tartar factorization* property with respect to a couple $(S_i,F_i)\,,$ $i=1,2$ of entropy/flux pairs if $$\label{tarfac}
\langle\nu,S_1F_2-S_2F_1\rangle=\langle\nu,S_1\rangle
\langle\nu,F_2\rangle-\langle\nu,S_2\rangle\langle\nu,F_1\rangle\,.$$ Dirac measures certainly posses this property, and in some cases, there is a converse statement, too. The following one-parameter families of entropy/flux pairs play an essential role in the forthcoming argument: \[parent\]
[ll]{} S\_a(,u):=+au-a\^2, & F\_a(,u):=(a+u)S\_a(,u),\
\_a(,u):=|+au-a\^2|, & \_a(,u):=(a+u)\_a(,u),
where the parameter, $a\in\R\,.$ The case of $(S_a,F_a)$ is obvious because it is a linear function of the basic conserved observables and their fluxes.
The pair $(\Bar{S}_a,\Bar{F}_a)$ satisfies in the generalized (weak) sense. This is due to the facts that the line of non-differentiability, $\rho+au-a^2=0$, is just a characteristic line of the PDE (\[entropia\]), and $(\Bar{S}_a,\Bar{F}_a)$ coincides with $(\pm S_a,\pm F_a)$ on the domains $D_{\pm}:=\{\pm(\rho+au-a^2)>0\}$.
\[lemma:dirac\] Suppose that a compactly supported probability measure, $\nu$ on $\R^2$ satisfies for any two entropy/flux pairs of type . Then $\nu$ is concentrated to a single point, i.e. it is a Dirac mass.
This is Exercise 9.1 in [@serre], where detailed instructions are also added. For Reader’s convenience we reproduce the easy proof. Suppose first that $S_a=\rho+au-a^2=0$ $\nu$-a.s. for some $a\in\R\,,$ then $\langle\nu,\rho\rangle+a\langle\nu,u\rangle=a^2\,;$ let $a_1$ and $a_2$ denote the roots of this equation. Since $S_{a_1}(\rho,u)=0$ implies $S_{a_2}(\rho,u)=0,$ $u=a_1+a_2$ and $\rho=-a_1a_2$ $\nu$-a.s. Therefore we may, and do assume that $$g(a):=\frac{\langle\nu,\Bar{F}_a\rangle}
{\langle\nu,\Bar{S}_a\rangle}
=a+\frac{\langle\nu,u|\rho+au-a^2|\rangle}
{\langle\nu,|\rho+au-a^2|\rangle}$$ is well defined for all $a\in\R\,.$ It is plain that $g(a)-a$ is continuous, bounded, and $g(a)-a\to\langle\nu,u\rangle$ as $a\to\pm\infty.$ Applying to $(S_a,F_a)$ and $(\Bar{S}_a,\Bar{F}_a)$ we get $g(a)-a=\langle\nu,uS_a\rangle/\langle\nu,S_a\rangle\,.$ On the other hand, from for $(S_a,F_a)$ and $(S_b,F_b)$ we get $$\label{ba}
(b-a)\left(\langle\nu,S_aS_b\rangle
-\langle\nu,S_a\rangle\langle\nu,S_b\rangle\right)
=\langle\nu,S_a\rangle\langle\nu,uS_b\rangle
-\langle\nu,S_b\rangle\langle\nu,uS_a\rangle\,,$$ thus dividing by $(b-a)\langle\nu,S_a\rangle\langle\nu,S_b\rangle\,,$ and letting $b\to a$ we see that $g$ is differentiable, and $g'(a)\geq1,$ consequently $g(a)=a+\langle\nu,u\rangle$ for all $a\in\R\,.$ This means that $$\langle\nu,\rho u\rangle+a\langle\nu,u^2\rangle
-a^2\langle\nu,u\rangle=\langle\nu,\rho\rangle
\langle\nu,u\rangle+a\langle\nu,u\rangle^2 -a^2\langle\nu,u\rangle$$ for all $a\in\R\,,$ whence $\nu(u^2)=\nu^2(u)\,.$ Substitute now $u=\langle\nu,u\rangle$ back into . Since $b-a\neq\langle\nu,u\rangle$ may be assumed, we have $$\langle\nu,S_aS_b\rangle
=\langle\nu,S_a\rangle\langle\nu,S_b\rangle\,,$$ consequently $\langle\nu,\rho^2\rangle=\langle\nu,\rho\rangle^2\,.$
This lemma establishes that measure solutions satisfying Tartar’s factorization property are, in fact, weak solutions.
The hydrodynamic limit under Eulerian scaling {#section:hdl}
=============================================
Block averages {#subs:blocks}
--------------
We choose a *mesoscopic* block size $l=l(n)$. A priori 1 l(n) n, but more serious restrictions will be imposed, see condition (\[cond:blocksize\]) in subsection \[subs:result\]. and define the *block averages* of local observables in the following way:\
We fix once for ever a weight function $a:\R\to\R_+$. It is assumed that:\
(1) $x\mapsto a(x)$ has support in the compact interval $[-1,1]$,\
(2) it has total weight $\int a(x)\,dx=1$,\
(3) it is even: $a(-x)=a(x)$, and\
(4) it is twice continuously differentiable.
Given a local variable $\ups_i$ its block average *at macroscopic space* $x$ is defined as \[eq:blocks\] [\^[n]{}]{}(x) = [\^[n]{}]{}(,x) := \_ja()\_j. Note that, since $l=l(n)$, we do not denote explicitly dependence of the block average on the mesoscopic block size $l$.
We shall use the handy (but slightly abused) notation [\^[n]{}]{}(t,x):=[\^[n]{}]{}(\_t,x). This is the empirical block average process of the local observable $\ups_i$.
In accordance with the compact vectorial notation introduced at the beginning of Section \[section:leroux\] we shall denote \_j := (
[c]{} \_j\
\_j
), \_j := (
[c]{} \_j\
\_j
), [\^[n]{}]{}(x) := (
[c]{} [\^[n]{}]{}(x)\
[\^[n]{}]{}(x)
), [\^[n]{}]{}(x) := (
[c]{} [\^[n]{}]{}(x)\
[\^[n]{}]{}(x)
), and so on.
Let ${\widehat{\vxi}^{n}}(t,x)$ be the sequence of empirical block average processes of the conserved quantities, as defined above, regarded as elements of $L^1_{t,x}:=L^1([0,T]\times\T)$. We denote by $\P^n$ the distribution of these in $L^1_{t,x}$: \[eq:l1distr\] ¶\^n(A) := (\^nA), where $A\in L^1_{t,x}$ is (strongly) measurable. Tightness and weak convergence of the sequence of probability measures $\P^n$ will be meant according to the norm (strong) topology of $L^1_{t,x}$. Weak convergence of a subsequence $\P^{n'}$ will be denoted $\P^{n'}\wto\P$.
Further on, we denote by $\nu^n$ the sequence of Dirac-type random Young measures concentrated on the trajectories of the empirical averages ${\widehat{\vxi}^{n}}(t,x)$ and by $\Q^n$ their distributions on ${\cal Y}$: \[eq:young\] \^n(t,x;d) := \_[[\^[n]{}]{}(t,x)]{}(d), \^n(A) := (\^nA), where $A\in{\cal Y}$ is (vaguely) measurable. Due to vague compactness of $\cal Y$, the sequence of probability measures $\Q^n$ is automatically tight. Weak convergence of a subsequence $\Q^{n'}$ will be meant according to the vague topology of $\cal Y$ and will be denoted $\Q^n\wvto\Q$. In this case we shall also say that the subsequence of random Young measures $\nu^{n'}$ (distributed according to $\Q^{n'}$) *converges vaguely in distribution* to the random Young measure $\nu$ (distributed according to $\Q$), also denoted $\nu^n\wvto\nu$.
Main result {#subs:result}
-----------
All results are valid under the following conditions
(A) \[cond:viscosity\] The macroscopic viscosity $\sigma=\sigma(n)$ satisfies \[eq:viscosity\] n\^[-1/2]{}1.
(B) \[cond:blocksize\] The mesoscopic block size $l=l(n)$ is chosen so that \[eq:blocksize\] n\^[2/3]{}\^[1/3]{} l n
(C) \[cond:iniprofile\] The initial density profiles converge weakly in probability (or, equivalently in any $L^p$, $1\le p<\infty$). That is: for any test function $\vvarphi:\T\to\R\times\R$ \[eq:iniprofile\] \_[n]{}( [|[ \_(x)([\^[n]{}]{}(0,x)-\_0(x)) dx ]{}|]{} ) =0.
Our main result is the following
\[thm:main\] Conditions (\[cond:viscosity\]), (\[cond:blocksize\]), and (\[cond:iniprofile\]) are in force. The sequence of probability measures $\P^n$ on $L^1_{t,x}$, defined in (\[eq:l1distr\]) is tight (according to the norm topology of $L^1_{t,x}$). Moreover, if $\P^{n'}$ is a subsequence which converges weakly (according to the norm topology of $L^1_{t,x}$), $\P^{n'}\wto\P$, then the limit probability measure $\P$ is concentrated on the entropy solutions of the Cauchy problem (\[eq:leroux\]), (\[eq:ic\]).
[**Remark:**]{} Assuming *uniqueness* of the entropy solution $\vu(t,x)$ of the Cauchy problem (\[eq:leroux\]), (\[eq:ic\]), we could conclude that $${\widehat{\vxi}^{n}}\buildrel{L^1_{t,x}}\over{\longrightarrow}\vu,
\qquad
\text{ in probability.}$$
Proof {#section:proof}
=====
Outline of proof {#subs:notations}
----------------
We broke up the proof into several subsections according to what we think to be a logical and transparent structure.
In subsection \[subs:apriori\] we state the precise *quantitative form* of the convergence to local equilibrium: the logarithmic Sobolev ineqaulity valid for our model and Varadhan’s large deviation bound on space-time averages of block variables. As main consequence of these we obtain our a priori estimates: the so-called *one-block estimate* and a version of the so-called *two-block estimate*, formulated for spatial derivatives of the empirical block averages. These estimates are of course the main probabilistic ingredients of the further arguments. The proof of these estimates is postponed to the Appendix of the paper.
In subsection \[subs:basic\] we write down an identity which turns out to be the stochastic approximation of the PDE (\[eq:leroux\]). Various error terms are defined here which will be estimated in the forthcoming subsections.
In subsection \[subs:bounds\] we introduce the relevant *Sobolev norms* and by using the previously proved a priori estimates we prove the necessary upper bounds on the apropriate Sobolev norms of the error terms.
In subsection \[subs:mvsol\] we show that choosing a subsequence of the random Young measures (\[eq:young\]) which converges vaguely in distribution, the limit (random) Young measure is almost surely measure valued entropy solution of the Cauchy problem (\[eq:leroux\]), (\[eq:ic\]).
Subsection \[subs:coco\] contains the stochastic version of the method of *compensated compactness*. It is further broken up into two sub-subsections as follows. In sub-subsection \[subsubs:murat\] we preent the stochastic version of Murat’s Lemma: we prove that for any smooth Lax entropy/flux pair the entropy production process is tight in the Sobolev space $H^{-1}_{t,x}$. In sub-subsection \[subsubs:tartar\] we apply (an almost sure version of) Tartar’s Div-Curl Lemma leading to the desired almost sure factorization property of the limiting random Young measures. Finally, as main consequence of Tartar’s Lemma, we conclude that choosing any subsequence of the random Young measures (\[eq:young\]) which converges vaguely in distribution, the limit (random) Young measure is almost surely of Dirac type.
The results of subsection \[subs:mvsol\] and sub-subsection \[subsubs:tartar\] imply the Theorem. The concluding steps are presented in subsection \[subs:end\].
Local equilibrium and a priori bounds {#subs:apriori}
-------------------------------------
The hydrodynamic limit relies on macroscopically fast convergence to (local) equilibrium in blocks of mesoscopic size $l$. Fix the block size $l$ and $(N,Z)\in\N\times\Z$ with the restriction $N+{\left|{Z}\right|}\le l$ and denote \^l\_[N,Z]{} &:=& { \^l: \_[j=1]{}\^l\_j=N, \_[j=1]{}\^l\_j=Z },\
\_[N,Z]{}() &:=& \_[,u]{}(| \_[j=1]{}\^l\_j=N, \_[j=1]{}\^l\_j=Z), and, for $f:\Omega^l_{N,Z}\to\R$ K\^l\_[N,Z]{}f() &:=& \_[j=1]{}\^[l-1]{} (f(\_[j,j+1]{})-f()),\
D\^l\_[N,Z]{}(f) &:=& 12 \_[j=1]{}\^[l-1]{} \^l\_[N,Z]{} ( (f(\_[j,j+1]{})-f())\^2 ). In plain words: $\Omega^l_{N,Z}$ is the hyperplane of configurations $\uome\in\Omega^l$ with fixed values of the conserved quantities, $\pi^l_{N,Z}$ is the *microcanonical distribution* on this hyperplane, $K^l_{N,Z}$ is the symmetric infinitesimal generator restricted to the hyperplane $\Omega^l_{N,Z}$, and finally $D^l_{N,Z}$ is the Dirichlet form associated to $K^l_{N,Z}$. Note, that $K^l_{N,Z}$ is defined with free boundary conditions. Expectations with respect to the measures $\pi^l_{N,Z}$ are denoted by $\expect^l_{N,Z}\big(\cdot\big)$. The convergence to local equilibrium is *quantitatively controlled* by the following uniform logarithmic Sobolev estimate:
\[lemma:lsi\] There exists a finite constant $\aleph$ such that for any $l\in\N$, $(N,Z)\in\N\times\Z$ with $N+{\left|{Z}\right|}\le l$ and any $h:\Omega^l_{N,Z}\to\R_+$ with $\expect^l_{N,Z}(h)=1$ the following bound holds: \[eq:lsi\] \^l\_[N,Z]{}(hh) l\^2 D\^l\_[N,Z]{} ().
[**Remark:**]{} In [@yau2] (see also [@leeyau]) the similar statement is proved (inter alia) for symmetric simple exclusion process. That proof can be easily adapted to our case. Instead of stirring configurations of two colours we have stirring of configurations of three colours. No really new ideas are involved. For sake of completeness however, we sketch the proof in subsection \[subs:lsiproof\] of the Appendix.
The following large deviation bound goes back to Varadhan [@varadhan]. See also the monographs [@kipnislandim] and [@fritz1].
\[lemma:varadhan\] Let $l\le n$, ${\cal V}:S^l\to \R_+$ and denote ${\cal V}_j(\uome) :=
{\cal V}(\omega_{j},\dots,\omega_{j+l-1})$. Then for any $\beta>0$ \[eq:varadhan\] \_[j]{} \_0\^T \_[\_s]{}( [V]{}\_j ) ds C + \_[ N, Z ]{} \^[l]{}\_[N,Z]{} ({ })
[**Remarks:**]{} (1) Assuming only uniform bound of order $l^{-2}$ on the spectral gap of $K^l_{N,Z}$ (rather than the stronger logarithmic Sobolev inequality (\[eq:lsi\])) and using Rayleigh-Schrödinger perturbation (see Appendix 3 of [@kipnislandim]) we would get && -17mm \_[j]{} \_0\^T \_[\_s]{}( [V]{}\_j ) ds\
&& C + T\_( + ), which wouldn’t be sufficient for our needs.\
(2) The proof of the bound (\[eq:varadhan\]) explicitly relies on the logarithmic Sobolev inequality (\[eq:lsi\]). It appears in [@yau3] and it is reproduced in several places, see e.g. [@fritz1; @fritz2]. We do not repeat it here.
The main probabilistic ingredients of our proof are the following two consequences of Lemma \[lemma:varadhan\]. These are variants of the celebrated *one block estimate*, respectively, *two blocks estimate* of Varadhan and co-authors.
\[propo:apriori\] Assume conditions (\[cond:viscosity\]) and (\[cond:blocksize\]). Given a local variable $\ups_j$ there exists a constant $C$ (depending only on $\ups_j$) such that the following bounds hold: \[eq:obrepl\] -10mm ( \_0\^T\_ [|[ [\^[n]{}]{}(s,x) - ([\^[n]{}]{}(s,x)) ]{}|]{}\^2 dxdt ) && C\
\[eq:gradbound\] -10mm ( \_0\^T\_ [|[ (s,x) ]{}|]{}\^2 dxdt ) && C \^[-1]{}
The proof of Proposition \[propo:apriori\] is postponed to subsection \[subs:aprioriproof\] in the Appendix. It relies on the large deviation bound (\[eq:varadhan\]) and an elementary probability lemma stated in subsection \[subs:epl\] of the Appendix.
We shall refer to (\[eq:obrepl\]) as the *block replacement bound* and to (\[eq:gradbound\]) as the *gradient bound*.
The basic identity {#subs:basic}
------------------
Given a smooth function $f:\dom \to \R$ we write f([\^[n]{}]{}(t,x)) = f([\^[n]{}]{}(t,x)) + M\^n\_f(t,x), where the process $t\mapsto M^n_f(t,x)$ is a martingale. Here and in the future $\pt f({\widehat{\vxi}^{n}}(t,x))$ and $\pt M^n_f(t,x)$ are meant as *distributions* in their time variable.
In this order we compute the action of the infinitesimal generator $\Gn=n\Ln + n^2\sigma\Kn$ on $f({\widehat{\vxi}^{n}}(x))$. First we compute the asymmetric part: \[eq:lf\] nf([\^[n]{}]{}(x)) &=& - f([\^[n]{}]{}(x))[\^[n]{}]{}(x) + A\^[1,n]{}\_f(x) where \[eq:a1\] A\^[1,n]{}\_f(x) = A\^[1,n]{}\_f(,x) := n\_[j]{} r(\_j,\_[j+1]{})4.5cm\
{ f( [\^[n]{}]{}(x) - ( a()-a() ) ( \_j-\_[j+1]{}) ) - f([\^[n]{}]{}(x))\
+ a’() f([\^[n]{}]{}(x)) (\_j-\_[j+1]{}) }. See formula (\[eq:phipsidef\]) for the definition of $\vphi$. $A^{1,n}_f$ is a numerical error term which will be easy to estimate.
Next, the symmetric part: \[eq:kf\] n\^2f([\^[n]{}]{}(x)) &=& f([\^[n]{}]{}(x))\^2[\^[n]{}]{}(x) + A\^[2,n]{}\_f(x) where \[eq:a2\] A\^[2,n]{}\_f(x) = A\^[2,n]{}\_f(,x) :=\
n\^2\_[j]{} { f( [\^[n]{}]{}(x) - ( a()-a() ) ( \_j-\_[j+1]{}) )\
- f([\^[n]{}]{}(x)) + a”() f([\^[n]{}]{}(x)) \_j }. This is another numerical error term easy to estimate.
Hence our *basic identity* \[eq:basic\] && -5mm f([\^[n]{}]{}(t,x)) + f([\^[n]{}]{}(t,x)) ([\^[n]{}]{}(t,x)) [\^[n]{}]{}(t,x) =\
&& \_[i=1]{}\^2 ( A\^[i,n]{}\_f(t,x) + B\^[i,n]{}\_f(t,x) + C\^[i,n]{}\_f(t,x) ) + M\^[n]{}\_f(t,x). The various terms on the right hand side are \[eq:b1\] && -8mm B\^[1,n]{}\_f(x) = B\^[1,n]{}\_f(,x) := { f([\^[n]{}]{}(x)) ( ([\^[n]{}]{}(x))-[\^[n]{}]{}(x) ) }\
\[eq:b2\] && -8mm B\^[2,n]{}\_f(x) = B\^[2,n]{}\_f(,x) := \^2f([\^[n]{}]{}(x)) = { f([\^[n]{}]{}(x)) [\^[n]{}]{}(x) }\
&& -8mm C\^[1,n]{}\_f(x) = C\^[1,n]{}\_f(,x) := - ([\^[n]{}]{}(x))\^\^2f([\^[n]{}]{}(x)) ( ([\^[n]{}]{}(x))-[\^[n]{}]{}(x) )\
\[eq:c1\]\
\[eq:c2\] && -8mm C\^[2,n]{}\_f(x) = C\^[2,n]{}\_f(,x) := - ([\^[n]{}]{}(x))\^\^2f([\^[n]{}]{}(x)) ([\^[n]{}]{}(x)) and && A\^[i,n]{}\_f(t,x):=A\^[i,n]{}\_f(\_t,x),\
&& B\^[i,n]{}\_f(t,x):=B\^[i,n]{}\_f(\_t,x),\
&& C\^[i,n]{}\_f(t,x):=C\^[i,n]{}\_f(\_t,x). In the present paper we shall apply the basic identity (\[eq:basic\]) only for Lax entropies $f(\vu)=S(\vu)$. In this special case the left hand side gets the form of a conservation law: \[eq:lentbasic\] && -5mm S([\^[n]{}]{}(t,x)) + F([\^[n]{}]{}(t,x)) =\
&& \_[i=1]{}\^2 ( A\^[i,n]{}\_S(t,x) + B\^[i,n]{}\_S(t,x) + C\^[i,n]{}\_S(t,x) ) + M\^[n]{}\_S(t,x),
Bounds {#subs:bounds}
------
We fix $T<\infty$ and use the $L^p$ norms \_[L\^[p]{}\_[t,x]{}]{}\^p := \_0\^T\_ [|[g(t,x)]{}|]{}\^pdxdt and the Sobolev norms $${\left\Vert{g}\right\Vert}_{W^{-1,p}_{t,x}}
:=
\sup\big\{
\int_0^T \int_{\T}
\varphi(t,x) g(t,x) \,dx\,dt\,:\,
{\left\Vert{\pt\varphi}\right\Vert}_{L^{q}_{t,x}}^q +
{\left\Vert{\px\varphi}\right\Vert}_{L^{q}_{t,x}}^q \le1
\big\}$$ where $p^{-1}+q^{-1}=1$ and $\varphi:[0,T]\times\T\to \R$ is a test function. We use the standard notation $W_{t,x}^{-1,2}=:H_{t,x}^{-1}$.
[**Remark on notation:**]{} The numerical error terms $A^{i,n}_f(t,x)$, $i=1,2$, will be estimated in $L^{\infty}_{t,x}$ norm. In these estimates only Taylor expansion bounds are used, no probabilistic argument is involved. The more sophisticated terms $B^{i,n}_f(t,x)$, $i=1,2$, respectively, $C^{i,n}_f(t,x)$, $i=1,2$, will be estimated in $H^{-1}_{t,x}$, respectively, $L^{1}_{t,x}$ norms. The martingale derivative $\pt M^n_f(t,x)$ will be estimated in $H^{-1}_{t,x}$ norm.
By straightforward numerical estimates (which do not rely on any probabilistic arguments) we obtain
\[lemma:bounds1\] Assume conditions (\[cond:viscosity\]) and (\[cond:blocksize\]). Let $f:\dom\to\R$ be a twice continuously differentiable function with bounded derivatives. Then almost surely \_[L\^\_[t,x]{}]{} = o(1)\_[L\^\_[t,x]{}]{} = o(1) as $n\to\infty$.
Indeed, using nothing more than Taylor expansion and boundedness of the local variables we readily obtain \[eq:a1bound\] && \_[x]{}\_ [|[A\^[1,n]{}\_f(,x)]{}|]{} C =o(1)\
\[eq:a2bound\] && \_[x]{}\_ [|[A\^[2,n]{}\_f(,x)]{}|]{} C =o(1). We omit the tedious but otherwise straightforward details.
Applying Proposition \[propo:apriori\] we obtain the following more sophisticated bounds
\[lemma:bounds2\] Assume conditions (\[cond:viscosity\]) and (\[cond:blocksize\]). Let $f:\dom\to\R$ be a twice continuously differentiable function with bounded derivatives. The following asymptotics hold, as $n\to\infty$: (i) && (\_[H\^[-1]{}\_[t,x]{}]{} ) =o(1)\
(ii) && (\_[H\^[-1]{}\_[t,x]{}]{} ) =o(1)\
(iii) && (\_[L\^[1]{}\_[t,x]{}]{} ) =o(1)\
(iv) && (\_[L\^[1]{}\_[t,x]{}]{} ) =(1)
\
$(i)$ We use the block replacement bound (\[eq:obrepl\]): && -7mm ( [|[ \_0\^T\_ v(t,x)B\^[1,n]{}\_f(t,x) dxdt ]{}|]{} )\
&& = ( [|[ \_0\^T\_ v(t,x) f([\^[n]{}]{}(t,x)) ( ([\^[n]{}]{}(t,x))-[\^[n]{}]{}(t,x) ) dxdt ]{}|]{} )\
&& \_[|[f()]{}|]{} \_[L\^2\_[t,x]{}]{} ( \_0\^T\_ [|[ ([\^[n]{}]{}(t,x))-[\^[n]{}]{}(t,x) ]{}|]{}\^2 dxdt )\^[1/2]{}\
&& C \_[L\^2\_[t,x]{}]{} . $(ii)$ We use the gradient bound (\[eq:gradbound\]): && -12mm ( [|[ \_0\^T\_ v(t,x)B\^[2,n]{}\_f(t,x) dxdt ]{}|]{} )\
&& = ( [|[ \_0\^T\_ v(t,x) f([\^[n]{}]{}(t,x)) ( (t,x) ) dxdt ]{}|]{} )\
&& \_[|[f()]{}|]{} \_[L\^2\_[t,x]{}]{} ( \_0\^T\_ [|[ (t,x) ]{}|]{}\^2 dxdt )\^[1/2]{}\
&& C \_[L\^2\_[t,x]{}]{} \^[1/2]{}. $(iii)$ We use both, the block replacement bound (\[eq:obrepl\]) and the gradient bound (\[eq:gradbound\]): && -10mm ( \_0\^T\_ [|[ C\^[1,n]{}\_f(t,x) ]{}|]{} dxdt )\
&& \_[|[\^2f()]{}|]{} ( \_0\^T\_ [|[ [\^[n]{}]{}(s,x) - ([\^[n]{}]{}(s,x)) ]{}|]{}\^2 dxdt )\^[1/2]{}\
&& ( \_0\^T\_ [|[ (s,x) ]{}|]{}\^2 dxdt )\^[1/2]{}\
&& C . $(iv)$ We use again the gradient bound (\[eq:gradbound\]): && -18mm ( \_0\^T\_ [|[ C\^[2,n]{}\_f(t,x) ]{}|]{} dxdt )\
&& \_[|[\^2f()]{}|]{} \_0\^T\_ [|[ (s,x) ]{}|]{}\^2 dxdt\
&& C.
\[lemma:bounds3\] Assume conditions (\[cond:viscosity\]) and (\[cond:blocksize\]). Let $f:\dom\to\R$ be a twice continuously differentiable function with bounded derivatives. There exists a constant $C$ (depending only on $f$) such that the folowing asymptotics holds as $n\to\infty$: (\_[H\^[-1]{}\_[t,x]{}]{} ) =o(1)
Since \^2\_[H\^[-1]{}\_[t,x]{}]{} \^2\_[L\^[2]{}\_[t,x]{}]{}, we have to bound the expectation of the right hand side. ( \_0\^T\_( M\^[n]{}\_f(t,x) )\^2 dxdt ) = ( \_0\^T\_M\^[n]{}\_f(t,x) dxdt ), where $t\mapsto \langle M^{n}_f(t,x) \rangle$ is the conditional variance process of the martingale $M^{n}_f(t,x)$: \[eq:condcov\] M\^[n]{}\_f(t,x) &=& n ( f\^2([\^[n]{}]{}(t,x)) - 2f([\^[n]{}]{}(t,x))f([\^[n]{}]{}(t,x)) )\
&& + n\^2( f\^2([\^[n]{}]{}(t,x)) - 2f([\^[n]{}]{}(t,x))f([\^[n]{}]{}(t,x)) ). Using the expressions (\[eq:lf\]) and (\[eq:kf\]) we obtain M\^[n]{}\_f(t,x) &=& A\^[1,n]{}\_[f\^2]{}(t,x)-2f([\^[n]{}]{}(t,x))A\^[1,n]{}\_f(t,x)\
&& + A\^[2,n]{}\_[f\^2]{}(t,x)-2f([\^[n]{}]{}(t,x))A\^[2,n]{}\_f(t,x). Hence, by the bounds (\[eq:a1bound\]) and (\[eq:a2bound\]) (which apply as well of course to the function $f^2$), we obtain $$\sup_{t\in[0,T]}
\sup_{x\in\T}
\,
\langle
M^{n}_f(t,x)
\rangle
\le
C\frac{n^2\sigma}{l^3}
=o(1),$$ which proves the lemma.
Convergence to measure valued entropy solutions {#subs:mvsol}
-----------------------------------------------
\[prop:mvsol\] Conditions (\[cond:viscosity\]), (\[cond:blocksize\]), and (\[cond:iniprofile\]) are in force. Let $\Q^{n'}$ be a subsequence of the probability distributions defined in (\[eq:young\]), which converges weakly in the vague sense: $\Q^{n'}\wvto\Q$. Then the probability measure $\Q$ is concentrated on the measure valued entropy solutions of the Cauchy problem (\[eq:leroux\]), (\[eq:ic\]).
Due to separability of $C([0,T]\times\T)$ it is sufficient to prove that for any convex Lax entropy/flux pair $(S,F)$ and any positive test function $\varphi:[0,T]\times\T\to\R_+$, (\[eq:mlax\]) holds $\Q$-almost-surely. So we fix $(S,F)$ and $\varphi$, and denote the real random variable X\^n &:=& - \_0\^T\_(t,x) ( S([\^[n]{}]{}(t,x)) + F([\^[n]{}]{}(t,x)) ) dxdt\
&=& \_0\^T\_( (t,x) \^n(t,x), S + (t,x) \^n(t,x), F ) dxdt\
&& 3.9cm + \_(0,x) \^n(0,x), Sdx. The right hand side is a continuous function of $\nu^n$, so from the asumption $\Q^n\wvto\Q$ it follows that \[eq:xtox\] X\^nX, where X &:=& \_0\^T\_( (t,x) (t,x),S + (t,x) (t,x),F ) dxdt\
&& 3.9cm + \_\_[D]{} (0,x;d) (0,x)S() dx. and $\nu$ is distributed according to $\Q$.
We apply the basic identity (\[eq:basic\]) speciafied for $f(\vu)=S(\vu)$, that is identity (\[eq:lentbasic\]). It follows that \[eq:decompo\] X\^n= Y\^n+Z\^n where Y\^n &:=& \_0\^T\_(t,x) C\^[2,n]{}\_S(t,x) dxdt\
&=& \_0\^T\_(t,x) ([\^[n]{}]{}(t,x))\^\^2 S([\^[n]{}]{}(t,x)) ([\^[n]{}]{}(t,x)) and Z\^n := \_0\^T\_(t,x) ( \_[i=1]{}\^2 ( A\^[i,n]{}\_S + B\^[i,n]{}\_S ) + C\^[1,n]{}\_S + M\^n\_S ) (t,x) dxdt. Due to convexity of $S$ and positivity of $\varphi$ we have \[eq:ypositive\] Y\^n0, On the other hand, from Lemmas \[lemma:bounds1\], \[lemma:bounds2\], \[lemma:bounds3\] we conclude that \[eq:ztozero\] \_[n]{} ([|[Z\^n]{}|]{}) =0. Finally, from (\[eq:xtox\]), (\[eq:decompo\]), (\[eq:ypositive\]) and (\[eq:ztozero\]) the statement of the Proposition follows.
Compensated compactness {#subs:coco}
-----------------------
### Murat’s lemma {#subsubs:murat}
\[lemma:murat\] Assume conditions (\[cond:viscosity\]) and (\[cond:blocksize\]). Given a twice continuously differentiable Lax entropy/flux pair $(S,F)$, the sequence X\^[n]{}(t,x):= S(\^[n]{}(t,x)) + F(\^[n]{}(t,x)) is tight in $H^{-1}_{t,x}$.
Note that $ X^{n}(t,x)$ is exactly the left hand side of the basic identity (\[eq:lentbasic\]) and racall that this expression (in particular $\pt S({\widehat{\vxi}^{n}}(t,x))$) is a random distribution in its $t$ variable.
By definition and a priori boundedness of the domain $\dom$, there exists a constant $C<\infty$ such that \[eq:murat1\] ( \_[W\^[-1,]{}\_[t,x]{}]{}C ) =1. We decompose \[eq:decomp\] X\^[n]{}(t,x) = Y\^[n]{}(t,x)+Z\^[n]{}(t,x), where && Y\^[n]{}(t,x) := B\^[1,n]{}\_S(t,x) + B\^[2,n]{}\_S(t,x) + M\^[n]{}\_S(t,x),\
&& Z\^[n]{}(t,x) := A\^[1,n]{}\_S(t,x) + A\^[2,n]{}\_S(t,x) + C\^[1,n]{}\_S(t,x) + C\^[2,n]{}\_S(t,x). For the definitions of the terms $A^{i,n}_S$, $B^{i,n}_S$, $C^{i,n}_S$, $i=1,2$, see (\[eq:a1\]), (\[eq:a2\]) and (\[eq:b1\])–(\[eq:c2\]).
From Lemmas \[lemma:bounds1\], \[lemma:bounds2\] and \[lemma:bounds3\] it follows that \[eq:murat2\] ( \_[H\^[-1]{}\_[t,x]{}]{} ) 0, and \[eq:murat3\] ( \_[L\^[1]{}\_[t,x]{}]{} ) C. Further on, from (\[eq:murat2\]), respectively, (\[eq:murat3\]) it follows that for any $\vareps>0$ one can find a *compact* subset $K_\vareps$ of $H^{-1}_{t,x}$ and a *bounded* subset $L_\vareps$ of $L^1_{t,x}$ such that \[eq:smallprob\] (Y\^nK\_)</2, (Z\^nL\_)</2. On the other hand, Murat’s lemma (see [@murat] or Chapter 9 of [@serre]) says that $$M_\vareps:=
\big(K_\vareps + L_\vareps\big)\cap
\{X\in H^{-1}_{t,x}: {\left\Vert{X}\right\Vert}_{ W^{-1,\infty}_{t,x}}\le C\}$$ is compact in $H_{t,x}^{-1}$. From (\[eq:murat1\]), (\[eq:decomp\]) and (\[eq:smallprob\]) it follows that $$\prob\Big(X^n\notin M_\vareps\Big)<\vareps,$$ uniformly in $n$, which proves the lemma.
### Tartar’s lemma and its consequence {#subsubs:tartar}
\[lemma:tartar\] Assume conditions (\[cond:viscosity\]) and (\[cond:blocksize\]). Let $\Q^{n'}$ be a subsequence of the probability measures on $\cal Y$ defined in (\[eq:young\]), which converges weakly in the vague sense: $\Q^{n'}\wvto\Q$. Then $\Q$ is concentrated on the (vaguely closed) subset of Young measures satisfying . That is, $\Q$-a.s. for any two generalized Lax entropy/flux pairs $(S_1,F_1)$ and $(S_2,F_2)$ and any test function $\varphi:[0,T]\times\T\to\R$, \[eq:tartar\] && -6mm \_0\^T\_(t,x) (t,x) , S\_1F\_2-S\_2F\_1 dxdt =\
&& -3mm \_0\^T\_(t,x) ( (t,x) , S\_1(t,x) , F\_2- (t,x) , S\_2(t,x) , F\_1) dxdt.
First we prove (\[eq:tartar\]) for twice continuously differentiable entropy/flux pairs. Due to separability of $C([0,T]\times\T)$ it is sufficient to prove that for any two twice continuously differentiable Lax entropy/flux pairs $(S_1,F_1)$ and $(S_2,F_2)$ and any test function $\varphi:[0,T]\times\T\to\R$, (\[eq:tartar\]) holds $\Q$-almost-surely. So we fix $(S_1,F_1)$, $(S_2,F_2)$ and $\varphi$. Note that X\^n\_j(t,x) &:=& S\_j([\^[n]{}]{}(t,x)) + F\_j([\^[n]{}]{}(t,x))\
&=& \^n(t,x),S\_j+ \^n(t,x),F\_j$j=1,2$.
Due to Skorohod’s representation theorem (see Theorem 1.8 of [@ethierkurtz]) and Lemma \[lemma:murat\] we can realize the random Young measures $\nu^n(t,x;d\vv)$ and $\nu(t,x;d\vv)$ *jointly* on an enlarged probablity space $(\Xi,{\cal A}, \prob)$ so that *$\prob$-almost-surely* \^[n’]{} , {X\^[n’]{}\_j:[n’]{}, j=1,2 } H\^[-1]{}\_[t,x]{}. So, applying Tartar’s Div-Curl Lemma (see [@tartar1], [@tartar2], or Chapter 9 of [@serre]) we conclude that (in this realization) almost surely the factorization (\[eq:tartar\]) holds true.
Since $\dom$ is compact, from Riemann’s method of solving the linear hyperbolic PDE (\[entropia2\]) (see Chapter 4 of [@john]) it follows that generalized entropy/flux pairs are approximated pointwise by smooth ones. Thus the Tartar factorization (\[eq:tartar\]) extends from smooth to generalized entropy/flux pairs. Hence the lemma.
The main consequence of Lemma \[lemma:tartar\] is the following
\[prop:diperna\] Assume conditions (\[cond:viscosity\]) and (\[cond:blocksize\]). Let $\Q^{n'}$ be a subsequence of the probability measures on $\cal Y$ defined in (\[eq:young\]), which converges weakly in the vague sense: $\Q^{n'}\wvto\Q$. Then the probability measure $\Q$ is concentrated on a set of Dirac-type Young measures, that is $\Q({\cal U})=1$.
In view of Lemma \[lemma:tartar\] this is a direct consequence of Lemma \[lemma:dirac\].
[**Remark:**]{} This is the only point where we exploit the very special features of the PDE (\[eq:leroux\]). Note that the proof of Lemma \[lemma:dirac\] relies on elementary explicit computations. In case of general $2\times2$ hyperbolic systems of conservation laws, instead of these explicit computations we should refer to DiPerna’s arguments from [@diperna], possibly further complicated by the existence of singular (non-hyperbolic) points isolated at the boundary of the domain $\cal D$. More general results will be presented in the forthcoming paper [@fritztoth2].
End of proof {#subs:end}
------------
From Propositions \[prop:mvsol\] and \[prop:diperna\] it follows that from any subsequence $n'$ one can extract a sub-subsequence $n''$ such that $\Q^{n''}\wvto\Q$ and $\Q$ is concentrated on the set of Dirac-type measure valued entropy solutions of the Cauchy problem. From now on we denote simply by $n$ this sub-subsequence. Referring again to Skorohod’s Representation Theorem we realize the Dirac-type random Young measures $\nu^{n}_{t,x}(d\vv):=\delta_{{\widehat{\vxi}^{n}}(t,x)}(d\vv)$ and $\nu_{t,x}(d\vv):=\delta_{\vu(t,x)}(d\vv)$ jointly on an enlarged probability space $(\Xi,{\cal A}, \prob)$, so that $\nu^{n}\vto\nu$ almost surely and $(t,x)\mapsto\vu(t,x)$ is almost surely entropy solution of the Cauchy problem. From basic functional analytic considerations (see e.g. Chapter 9 of [@serre]) it follows that, in case that the limit Youg measure is also Dirac-type, the vague convergence $\nu^{n}\vto\nu$ implies strong (i.e. norm) convergence of the underlying functions, \[eq:l1cvg\] [\^[n]{}]{}L\^1\_[t,x]{}. So, we have realized jointly on the probability space $(\Xi,{\cal A}, \prob)$ the empirical block average processes ${\widehat{\vxi}^{n}}(t,x)$ and the random function $\vu(t,x)$ so that the latter one is almost surely entropy solution of the Cauchy problem, and (\[eq:l1cvg\]) almost surely holds true. This proves the theorem.
Appendix {#section:app}
========
The logarithmic Sobolev inequality for random stirring of $r$ colours on the linear graph $\{1,2,\dots,l\}$ {#subs:lsiproof}
-----------------------------------------------------------------------------------------------------------
Let $r\ge2$ be a fixed intger. For $l\in\N$ we consider $r$-tuples of integers $N=(N_1,\dots,N_r)$ such that \[eq:ncond\] && N\_0, =1,…,r N\_1+…+N\_r=l,\
&& \^l\_N := { {1,…,r}\^l: \_[j=1]{}\^l\_[{\_j=}]{}=N\_, =1,…,r }. Let $\pi^l_N$ denote the uniform probability measure on $\Omega^l_N$: $$\pi^l_N(\uome)=
\frac{N_1!\cdots N_r!}{l!},
\qquad
\uome\in\Omega^l_N.$$ The one dimensional marginals of $\pi^l_N$ are $$\pi^{l,1}_N(\alpha)
=
\frac{N_\alpha}{l}.$$ The random element of $\Omega^l_N$ distributed according to $\pi^l_N$ will be denoted $\uzet=(\zeta_1,\zeta_2,\dots,\zeta_l)$. Expectation with respect to $\pi^l_N$, respectively, $\pi^{l,1}_N$ will be denoted by $\expect^l_N\big(\cdots\big)$, respectively, $\expect^{l,1}_N\big(\cdots\big)$. Conditional expectation, given the first coordinate $\zeta_1$ will be denoted $\expect^l_N\big(\cdots\big|\zeta_1\big)$. Note that $$\expect^{l}_{N}\big(f(\uzet)\big|\zeta_1=\alpha \big)
=
\expect^{l-1}_{N^{\alpha}}\big(f(\alpha,\zeta_2,\dots,\zeta_l)\big)$$ where $\expect^{l-1}_{N^{\alpha}}\big(\cdots\big)$ stands for expectation with respect to $(\zeta_2,\dots,\zeta_l)$ distributed according to $\pi^{l-1}_{N^{\alpha}}$ and, given $N=(N_1,\dots,N_\alpha,\dots,N_r)$ with $N_\alpha\ge1$, $N^\alpha:=
(N_1,\dots,N_\alpha-1,\dots,N_r)$.
Given a probability density $h$ over $(\Omega^l_N,\pi^l_N)$, its entropy is H\^l\_N(h) := \^l\_N(h()h()).
Further on, for $i,j\in\{1,\dots,l\}$ let $\Theta_{i,j}:\Omega^l_N\to\Omega^l_N$ be the spin exchange operator $$\left(\Theta_{i,j}\uome\right)_k=
\left\{
\begin{array}{ll}
\omega_j\quad&\text{ if } k=i,
\\
\omega_i\quad&\text{ if } k=j,
\\
\omega_k\quad&\text{ if } k\not=i,j,
\end{array}
\right..$$ For $f:\Omega^l_N\to\R$ we define the Dirichlet form and the conditional Dirichlet form, given $\zeta_1$ D\^l\_N(f) &:=& 12 \_[i=1]{}\^[l-1]{} \^l\_N ( (f(\_[i,i+1]{})-f())\^2 ),\
D\^l\_N(f|\_1) &:=& 12 \_[i=1]{}\^[l-1]{} \^l\_N ( (f(\_[i,i+1]{})-f())\^2 | \_1 )\
&=& D\^[l-1]{}\_[N\^[\_1]{}]{}(f(\_1,)).
The logarithmic Sobolev inequality is formulated in the following
There exist a finite constant $\aleph$ such that for any number of colours $r$, any block size $l\in \N$, any distribution of colours $N=(N_1,\dots,N_r)$ satisfying (\[eq:ncond\]) and any probability density $h$ over $(\Omega^l_N, \pi^l_N)$, the following inequality holds: \[eq:lsigen\] H\^l\_N(h) l\^2 D\^l\_N(h).
[**Remark:**]{} The proof follows [@yau2] (see also [@leeyau]). Due to exchangeability of the measures $\pi^l_N$ some steps are considearbly simpler than there.
We shall prove the Proposition by induction on $l$. Denote $$W(l):=
\sup_{N}\sup_{h}
\frac{H^l_N\big(h\big)}{D^l_N\big(\sqrt h\big)}.$$ The following identity is straightforward H\^l\_N(h) &=& \^[l,1]{}\_N ( \^l\_N(h() | \_1 ) \^[l]{}\_[N]{} ( h\_1() h\_1() | \_1 ) )\
\[eq:entropydecomp\] && + \^[l,1]{}\_N ( \^l\_N(h() |\_1) \^l\_N(h() |\_1) ), where in the first term of the right hand side $$h_1(\uzet)
:=
\frac
{h(\uzet)}
{\expect^l_N \big( h(\uzet) \big| \zeta_1 \big)}.$$
First we bound the first term on the right hand side of (\[eq:entropydecomp\]). By the induction hypothesis && -2cm \^[l,1]{}\_N ( \^l\_N(h() | \_1 ) \^[l]{}\_[N]{} ( h\_1() h\_1() | \_1 ) )\
&=& \^[l,1]{}\_N ( \^l\_N(h() | \_1 ) \^[l-1]{}\_[N\^[\_1]{}]{} ( h\_1() h\_1() ) )\
&& W(l-1) \^[l,1]{}\_N ( \^l\_N(h() | \_1 ) D\^[l-1]{}\_[N\^[\_1]{}]{} ( ) )\
&=& W(l-1) \^[l,1]{}\_N ( D\^[l-1]{}\_[N\^[\_1]{}]{} ( ) )\
\[eq:firstterm\] && W(l-1) D\^[l]{}\_[N]{} ( ).
Next we turn to the second term on the right hand side of (\[eq:entropydecomp\]). In order to simplify notation in the next argument we denote \[eq:notation\] \_:=, q\_(j):= \^l\_N(h() \_[{\_j=}]{} ). It is straightforward that for any $K<\infty$ there exists a finite constant $C=C(K)$ such that for any $v\in[0,K]$ vv (v-1) + C (v -1 )\^2 and, furthermore, the constant $C$ can be chosen so that for any $v>K$ vv C v\^[3/2]{}. Hence, with the notation introduced in (\[eq:notation\]), we get the following upper bound for the second term on the right hand side of (\[eq:entropydecomp\]) \[eq:upper1\] && -6mm \^[l,1]{}\_N ( \^l\_N(h() |\_1) \^l\_N(h() |\_1) ) = \_[=1]{}\^r \_\
&& C \_[=1]{}\^r \_ { ( - 1)\^[ 2]{} \_[{K}]{} + ()\^[ 3/2]{} \_[{> K}]{} }. We use the straightforward inequality $$\sum_{\alpha=1}^r
\varrho_\alpha
\left(
\frac{q_\alpha(1)}{\varrho_\alpha}-1
\right)
\ind_{\{\frac{q_\alpha(1)}{\varrho_\alpha}\le K\}}
\le 0.$$ We choose $K$ sufficiently large in order that Lemma 4.1 of [@yau2] can be applied to $\{1,2,\dots,l\}\ni j\mapsto \sqrt{q_\alpha(j)/\varrho_\alpha}$. Thus we obtain the upper bound ( - 1)\^[ 2]{} \_[{K}]{} + ()\^[ 3/2]{} \_[{> K}]{} &&\
\[eq:upper2\] C’ l \_[j=1]{}\^[l-1]{} ( - )\^[ 2]{}. && Putting together (\[eq:upper1\]) and (\[eq:upper2\]) and returning to the explicit notation we obtain the following upper bound for the second term on the right hand side of (\[eq:entropydecomp\]): && -7mm \^[l,1]{}\_N ( \^l\_N(h() |\_1) \^l\_N(h() |\_1) )\
&& C” l \_[j=1]{}\^[l-1]{} \_[=1]{}\^r ( - )\^[ 2]{}\
&& = C” l \_[j=1]{}\^[l-1]{} \_[=1]{}\^r ( - )\^[ 2]{}\
&& = C” l \_[j=1]{}\^[l-1]{} ( - )\^[ 2]{}\
\[eq:secondterm\] && C” l \_[j=1]{}\^[l-1]{} \^l\_N ( ( - )\^2 ) = C” l D\^l\_N(h). In the second step we used *exchangeability* of the canonical measures $\pi^l_N$. In the last inequality we note that the map $$\R_+\times\R_+\ni(x,y)\mapsto\left(\sqrt x-\sqrt y\right)^2$$ is *convex* and we use Jensen’s inequality.
From (\[eq:entropydecomp\]), (\[eq:firstterm\]) and (\[eq:secondterm\]) eventually we obtain $$W(l)\le W(l-1) + C'' l,$$ which yields (\[eq:lsigen\]).
An elementary probability lemma {#subs:epl}
-------------------------------
The contents of the present subsection, in paericular Lemma \[lemma:epl\] and its Corollary \[corollary:epl\] are borrowed form [@tothvalko2]. For their proofs see that paper.
Let $(\Omega,\pi)$ be a finite probability space and $\omega_i$, $i\in\Z$ i.i.d. $\Omega$-valued random variables with distribution $\pi$. Further on let
[ll]{} :\^d, & \_i:=(\_i),\
:\^[m]{}, & \_i:=(\_[i]{}…,\_[i+m-1]{}).
For $\vx\in\text{co}(\text{Ran}(\vxi))$ denote $$\Ups(\vx):=
\frac
{\expect_\pi\big(\ups_1\exp\{\sum_{i=1}^{m}\vlam\cdot\vxi_i\}\}\big)}
{\expect_\pi\big(\exp\{\vlam\cdot\vxi_1\}\}\big)^{m}},$$ where $\text{co}(\cdot)$ stands for ‘convex hull’ and $\vlam\in\R^d$ is chosen so that $$\frac
{\expect_\pi\big(\vxi_1\exp\{\vlam\cdot\vxi_1\}\}\big)}
{\expect_\pi\big(\exp\{\vlam\cdot\vxi_1\}\}\big)}
=
\vx.$$ For $l\in\N$ we denote *plain* block averages by $$\overline{\vxi}_l:=\frac1l\sum_{j=1}^l\vxi_j.$$ Finally, let $b:[0,1]\to\R$ be a fixed smooth function and denote $$M(b):=\int_0^1 b(s)\,ds.$$ We also define the block averages *weighted by $b$* as $$\langle b\,,\,\vxi\rangle_l
:=
\frac1l\sum_{j=0}^l
b(j/l)\vxi_j,
\quad
\langle b\,,\,\ups\rangle_l
:=
\frac1l\sum_{j=0}^l
b(j/l)\ups_j,$$ The following lemma relies on elementary probability arguments:
\[lemma:epl\] There exists a constant $C<\infty$, depending only on $m$, on the joint distribution of $(\ups_i,\vxi_i)$ and on the function $b$, such that the following bounds hold uniformly in $l\in\N$ and $\vx\in({\rm{Ran}}(\vxi)+\dots+{\rm{Ran}}(\vxi))/l$:\
(i) If $M(b)=0$, then \[eq:epl11\] ( { b,\_l } | \_l=) {C(\^2+/l)}. (ii) If $M(b)=1$ then \[eq:epl12\] ( { ( [b,]{}\_l - (b,\_l) ) } | \_l=) {C(\^2+/l)}.
The proof of this lemma appears in [@tothvalko2].
\[corollary:epl\] There exists a $\gamma_0>0$, depending only on $m$, on the joint distribution of $(\ups_i,\vxi_i)$ and on the function $b$, such that the following bounds hold uniformly in $l\in\N$ and $\vx\in({\rm{Ran}}(\vxi)+\dots+{\rm{Ran}}(\vxi))/l$:\
(i) If $M(b)=0$, then \[eq:epl1\] ( { \_0 l b,\_l\^2 } | \_l=) 2. (ii) If $M(b)=1$ then \[eq:epl2\] ( { \_0 l ( [b,]{}\_l - (b,\_l) )\^2 } | \_l=) 2.
The bounds (\[eq:epl1\]) and (\[eq:epl2\]) follow from (\[eq:epl11\]), respectively, (\[eq:epl12\]) by exponential Gaussian averaging.
Proof of the a priori bounds (Proposition \[propo:apriori\]) {#subs:aprioriproof}
------------------------------------------------------------
### Proof of the block replacement bound (\[eq:obrepl\])
We note first that by simple numerical approximation (no probability bounds involved) [|[ \_ [|[ [\^[n]{}]{}(x) - ([\^[n]{}]{}(x)) ]{}|]{}\^2 dx - 1n\_[j=1]{}\^n [|[ [\^[n]{}]{}(j/n) - ([\^[n]{}]{}(j/n)) ]{}|]{}\^2 ]{}|]{} &&\
Cl\^[-2]{} = o(). && We apply Lemma \[lemma:varadhan\] with $${\cal V}_j=
{\left|{
{\widehat{\ups}^{n}}(j/n)
-
\Ups({\widehat{\vxi}^{n}}(j/n))
}\right|}^2.$$ We use the bound (\[eq:epl2\]) of Lemma \[corollary:epl\] with the function $b=a$ of (\[eq:blocks\]). Note that $\gamma=\gamma_0l$ can be chosen in (\[eq:varadhan\]). This yields the bound (\[eq:obrepl\]).
### Proof of the gradient bound (\[eq:gradbound\])
Again, we start with numerical approximation: [|[ \_ [|[ (x) ]{}|]{}\^2 dx - 1n\_[j=1]{}\^n [|[ (j/n) ]{}|]{}\^2 ]{}|]{} C = o(\^[-1]{}). We apply Lemma \[lemma:varadhan\] with $${\cal V}_j=
{\left|{
\px{\widehat{\ups}^{n}}(j/n)
}\right|}^2.$$ We use now the bound (\[eq:epl1\]) of Lemma \[corollary:epl\] with the function $b=a'$, where $a$ is the weighting function from (\[eq:blocks\]). The same choice $\gamma=\gamma_0l$ applies. This will yield the bound (\[eq:gradbound\]).
[**Acknowledgement:**]{} We thank the kind hospitality of Institut Henri Poincaré where part of this work was done. We also acknowledge the financial support of the Hungarian Science Foundation (OTKA), grants T26176 and T037685.
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|
---
author:
- 'M. García-Marín'
- 'L. Colina'
- 'S. Arribas'
- 'A. Monreal-Ibero'
date: 'Received ; accepted'
subtitle: 'I. The Data[^1]'
title: 'Integral field optical spectroscopy of a representative sample of ULIRGs:'
---
[Ultraluminous infrared galaxies are among the brightest objects in the local Universe. They are powered by strong star formation and/or an AGN. They are also likely to be the progenitors of elliptical galaxies. The study of the structure and kinematics of samples of local ULIRGs is necessary to understand the physical processes that these galaxies undergo, and their implications for our understanding of similar types of galaxies at high redshift. ]{} [The goal of the project is to analyze the structure, dust distribution, ionization state, and kinematics of a representative sample of 22 ULIRGs. The galaxies in the sample undergo different merger phases (they are evenly divided between pre- and post-coalescence systems) and ionization stages (27% H[ii]{}, 32% LINER, 18% Seyfert, and 23% mixed classifications) over a wide infrared luminosity range (11.8$\leq$L$_{IR}$/L$_{\odot}$$\leq$12.6), with some galaxies of low-luminosity. The main aims of this paper are to present the sample and discuss the structure of the stellar and ionized gas components. ]{} [Our study relies on the use of integral field optical spectroscopy data obtained with the INTEGRAL instrument at the William Herschel Telescope. ]{} [The structure of the ionized gas as traced by different emission lines has been studied and compared with that of the stellar continuum. We find structural variations between the gaseous and the stellar components, with offsets in the emission peaks positions of up to about 8 kpc. Young star formation (as traced by the H$\alpha$ emission) is present in all regions of the galaxies. However, for 64% of ULIRGs in an early interaction phase, the young star formation peak does not coincide with the stellar maxima. In contrast, galaxies undergoing advanced mergers have a H$\alpha$ peak that is located in their nuclear regions. In three of the studied ULIRGs, hard ionizing photons traced by the \[O[iii]{}\]$\lambda$5007 line excite extra-nuclear nebulae out to distances of about 7 kpc. These regions do not show bright stellar emission, but are rather dominated by nebular emission. These galaxies have nuclei classified as Seyfert in the literature. Approximately 40% of the pre-coalescence ULIRGs exhibit shifts between the peaks of their red continuum and that local to the \[O[i]{}\]$\lambda$6300 line. However, some of these peaks are associated with the secondary stellar nucleus of the system. In contrast, the emission in post-coalescence ULIRGs is concentrated towards the nuclei. These results imply that evolution caused by a merger is ocurring in the ionized gas structure of ULIRGs.]{}
Introduction
============
Ultraluminous Infrared Galaxies (ULIRGs, 10$^{12}$L$_{\odot}$$\le$ L$_{bol}$$\sim$ L$_{IR}$$[8-1000\mu m]$ $\le$ 10$^{13}$L$_{\odot}$) are a key galaxy class characterized by the emission of the bulk of their energy in the infrared. They are as luminous as QSOs and, in the local Universe, twice as numerous (see reviews by Sanders & Mirabel 1996; Lonsdale et al.2006). The existence of galaxies with a strong infrared excess was reported by Low & Kleinmann (1968) and Rieke & Low (1972), but the study of ULIRGs as a galaxy class started with their detection in large numbers by the IRAS satellite (e.g., Soifer et al.1984; Sanders et al.1988). Subsequently, optical long-slit spectroscopy was used to investigate their main energy source (e.g., Veilleux et al.1995). It has been generally accepted that ULIRGs are powered mainly by intense star formation, whereas the presence and relative importance of an AGN was a matter of debate (Genzel et al.1998; Veilleux, Kim & Sanders 1999; Risaliti et al. 2006; Farrah et al. 2007). However, IR studies suggest that the contribution of AGN to the bolometric luminosity of the system may be relevant in 15–20% of cases (see Nardini et al.2008 and Risaliti et al.2006).
ULIRGs are characterized by interaction/merger processes that strongly drive their dynamics. The merger triggers the starburst activity (i.e.,the ultra-luminous phase), is responsible for the complex tidally-dominated morphologies (e.g., Borne et al.2000; Bushouse et al.2002; Farrah et al.2001), dominates the ionized gas kinematics (e.g., Colina, Arribas & Monreal 2005), and possibly transforms spiral galaxies into intermediate mass ellipticals (e.g., Genzel et al.2001; Tacconi et al.2002; Dasyra et al.2006).
In a cosmological context, the importance of ULIRGs is significant because they appear to be the low-[*z*]{} analogs to at least part of the population of star-forming luminous galaxies giving rise to the far-IR background. These are the sub-mm galaxies (Smail, Ivison & Blain 1997) and the ULIRGs detected by *Spitzer* (Pérez-González et al.2005). It has been shown that, in contrast to the situation in the local Universe, the importance of the contribution of ULIRGs and the less luminous LIRGs (Luminous Infrared Galaxies, 10$^{11}$L$_{\odot}$$\le$ L$_{bol}$$\sim$ L$_{IR}$$[8-1000\mu m]$ $\le$ 10$^{12}$L$_{\odot}$) to the IR extragalactic light increases with redshift (Lagache, Puget & Dole 2005; Le Floc’h et al.2004; Lonsdale et al.2004; Yan et al.2004; Caputi et al.2006a, 2006b). They are the main contributors to the comoving star density of the Universe at z$>$1.0 (Elbaz et al.2002; Le Floc’h et al.2005; Pérez-González et al.2005; Caputi et al.2007). Generally, these high redshift studies do not focus on the detailed characterization of individual objects, but rather on their general integrated properties. To be able to study spatially resolved information, it is however important to understand their internal structure, dust distribution, ionization structure, and kinematics. This information is limited at high redshifts, and studies have used integral field spectrographs to analyze these high redshift objects in detail (e.g., Genzel et al2006; Förster-Schreiber et al.2006, 2009).
In the local Universe, where the spatial resolution allows us to study the internal galaxy structure in greater detail, comprehensive studies of local ULIRGs have been completed using imaging and long-slit spectroscopy in the optical and infrared (e.g., Veilleux et al.1995; Genzel et al.1998; Kim et al.1998; Scoville et al.2000; Farrah et al.2001; Bushouse et al.2002; Imanishi et al.2007). In some cases, the combination of these images with optical two-dimensional spectral information such as that provided by integral field spectroscopy (IFS) has been used to carry out comprehensive studies of individual or a few ULIRGs (e.g., Arribas, Colina & Borne 1999; Monreal-Ibero 2004; Colina, Arribas & Monreal-Ibero 2005).
To obtain statistical knowledge of the nature of ULIRGs, we have started a program to study, on the kpc scale, the dust structure, the two-dimensional ionization statem, and the ionized gas kinematics of a representative sample of local ULIRGs, based on optical IFS data obtained with the instrument INTEGRAL. A similar study but for a representative sample of LIRGs is being conducted using VIMOS at the VLT (Arribas et al.2008) and PMAS on the Calar Alto 3.5 m (Alonso-Herrero et al.2009). All these studies are part of the same project, which investigates the two-dimensional extinction, ionization, and kinematic kpc-scale structure of a representative sample of low$-z$ LIRGs and ULIRGs, along with its implications for their high-redshift analogs. For instance, the IFS data presented in this paper sample regions similar to those that will be studied by the instruments MIRI (Mid IR Instrument) and NIRSpec (Near Infrared Spectrograph), due to be launched onboard of the *James Webb Space Telescope*, at redshifts above 1.
In this paper, we combine data from previously published works (see Colina, Arribas & Monreal 2005; García-Marín et al.2006 and references therein) and new data of 13 unpublished galaxies. The paper is organized as follows. We describe the sample selection in Sect.2; the observations and data reduction are outlined in Sect.3, while the data analysis is presented in Sect.4. The morphology of the continuum stellar emission and ionized gas is presented in Sect.5. Finally, the summary is given in Sect.6. The second paper of the series will present the extinction of the sample. The third paper will focus on the ionization state, whereas the fourth will present the kinematics. Throughout the paper, we use $\Omega_{\Lambda}$=0.7, $\Omega_{M}$=0.3, and H$_{0}$=70 km s$^{-1}$ Mpc$^{-1}$.
Sample selection and properties
===============================
The present sample of low-*z* ULIRGs was selected using the following criteria: (1) to sample the IR luminosity range (Fig. \[fig0\]); (2) to cover all types of nuclear activity, that is different excitation mechanisms such as H- (i.e., star formation activity), LINER- (i.e., superwinds, shocks), and Seyfert-like (i.e., presence of an AGN); (3) to optimize the linear scales by selecting low-*z* galaxies; and (4) to span different phases of the interaction process.
We selected our sample from those Sanders et al.(1988; 1995), Melnick & Mirabel (1990), Leech et al. (1994), Kim et al. (1995), Lawrence et al. (1996), and Clements et al. (1996). Our final selection of 22 northern hemisphere ULIRGs (33 individual galaxies) is shown in Fig. \[galaxias1WFPC2\], and its main characteristics are presented in Table\[TableSample\]. In the following paragraphs, we describe the selection criteria in detail.
The range of luminosities covered by the sample is 11.8$\le$log(L$_{IR}$/L$_{\odot}$) $\le$12.6, including two systems classified as LIRGs, but close enough to the low IR luminosity range of the ULIRG class. The luminosity distribution of the galaxies in our sample is similar to that of the complete flux-limited[^2] IRAS Revised Bright Galaxy Sample (RBGS; Sanders et al.2003), which covers the entire sky surveyed by IRAS at Galactic latitudes $|$b$|$$>$5$^{\circ}$ (Fig. \[fig0\]).
![Luminosity distribution of the IRAS Revised BGS (Sanders et al.2003, orange dashed histogram) compared with our sample of ULIRGs (black dashed histogram). The luminosities of the BGS galaxies have been re-calculated to consistently compare both datasets. The luminosity range is 12.0$\le$log(L$_{IR}$/L$_{\odot}$) $\le$12.6 with bins of 0.2. The two LIRGs of the sample have not been included in the graphic.[]{data-label="fig0"}](HistogramaConSanders.jpg){width="\columnwidth"}
All the excitation mechanisms are well represented (see Table\[TableSample\]): 27%, 32%, and 18%of the ULIRGs are classified as H[ii]{}, LINER, and Seyfert, respectively. The remaining 23%represent objects with unclear or mixed classifications. Using the 1 Jy sample, a flux-limited sample of ULIRGs identified from the IRAS Faint Source Catalog, Kim et al. (1998) derived the following distribution for ULIRGs: 28%for H[ii]{}, 38%for LINER, and 34%for Seyfert. The H[ii]{} and LINER fractions compare well with ours, whereas the Seyfert fraction is almost double. This difference is mainly caused by the wider luminosity covered by Kim et al. (1998; their sample goes up to L$_{IR}$=12.8L$_\odot$, whereas ours stops at 12.6L$_\odot$), which results in a higher percentage of AGN (Veilleux et al.1999).
The galaxies are located at distances of between about 40 and 900 Mpc, with an angular sampling that ranges from 0.2 to 3.1 kpc arcsec$^{-1}$ (see individual values in Table\[TableSample\]). The median redshift is 0.11, which compares well with 0.14, the median value of the 1 Jy sample (Kim & Sanders 1998).
Finally, the ULIRGs in our sample exhibit a variety of morphologies, from interacting disks with a projected nuclear distance up to 36 kpc, to close nuclei or individual nucleus surrounded by a common stellar envelope. There are detailed classification schemes based on the interaction stage and morphological features that lead to the existence of many classes and subclasses (see e.g., Surace 1998; Veilleux et al.2002). We, however, chose to separate our sample of ULIRGs into two broad categories using the projected nuclear distance as a discriminator: (1) *Pre-coalescence galaxies:* those with a projected nuclear separation larger than 1.5 kpc. (2) *Post-coalescence galaxies:* those with a projected nuclear separation smaller or similar than 1.5 kpc. This accurate distance calculation was possible because of the high spatial resolution of the *HST* images (typical scale 01 per pixel). The selected distance evenly distributes the galaxies into the two categories, and is appropriate for separating ULIRGs into early and late merger phases. In addition to this, theoretical models suggest that by the time the two nuclei have reached a separation of $\lesssim$ 1 kpc, the stellar system has basically achieved equilibrium, although their nuclei can still be separate structures (e.g., Mihos 1999; Mihos & Hernquist 1996; Bendo & Barnes 2000; Naab et al.2006). However, it is important to clearly state that the use of projected nuclear distances may lead to some misclassification because galaxies identified as post-coalescence galaxies could well be in a pre-coalescence state.
Considering all these aspects, we conclude that our sample is essentially representative of the ULIRG class.
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IRAS Name R.A.(J2000) DEC(J2000) Spectral Class *z* D$_{\rm{L}}$[^3] ${\log(L_{\rm{IR}})}$[^4] Scale Morphology[^5]
(hh:mm:ss) (deg:mm:ss) (Optical) (Mpc) (L$_{\odot}$) (kpc arcsec$^{-1}$)
IRAS 13156+0435 13:18:07.6 +04:18:59.4 H[ii]{}/LINER (G) 0.113 525 12.13 2.058 IP at 36.0 kpc
IRAS 13342+3932 13:36:23.3 +39:16:41.9 Sey1 (E) 0.179 866 12.51 3.026 IP at 25.1 kpc
IRAS 18580+6527 18:58:06.3 +65:31:32.2 H[ii]{}/Sey2 (G) 0.176 850 12.26 2.986 IP at 15.0 kpc
IRAS 16007+3743 16:02:36.9 +37:34:39.7 LINER (H) 0.185 898 12.11 3.100 IP at 14.2 kpc
IRAS 06268+3509 06:30:12.6 +35:07:51.9 H[ii]{} (C)[^6] 0.169 813 12.51[^7] 2.895 IP at 9.1 kpc
IRAS 08572+3915 09:00:25.4 +39:03:54.4 LINER (E)[^8] 0.058 259 12.17 1.130 IP at 6.1 kpc
IRAS 14348-1447 14:37:38.4 -15:00:22.8 LINER (E) 0.083 378 12.39 1.556 IP at 5.5 kpc
Arp 299[^9] 11:28:30.4 +58:34:10.0 H[ii]{}/Sey2 (D)[^10] 0.010 43 11.81 0.205 IP at 5.0 kpc
IRAS 13469+5833 13:48:40.3 +58:18:50.0 H[ii]{} (I)[^11] 0.158 755 12.31 2.726 IP at 4.5 kpc
IRAS 12112+0305 12:13:42.9 +02:48:29.0 LINER (E) 0.073 330 12.37 1.395 IP at 4.0 kpc
Mrk 463 13:56:02.8 +18:22:17.2 Sey1/Sey2 (F, G)[^12] 0.050 222 11.81 0.984 DN at 3.8 kpc
IRAS 06487+2208 06:51:45.7 +22:04:27.0 H[ii]{} (C) 0.144 682 12.57[^13] 2.522 DN at 1.5 kpc
IRAS 11087+5351 11:11:36.4 +53:35:02.0 Sey1 (G) 0.143 677 12.13 2.507 DN at 1.5 kpc
Mrk 273 13:44:42.0 +55:53:12.1 LINER/Sey2 (B, E)[^14] 0.038 167 12.18 0.749 DN at 0.7 kpc
Arp 220 15:34:57.1 +23:30:11.5 LINER (E) 0.018 78 12.20 0.368 DN at 0.4 kpc
IRAS 17208-0014 17:23:21.9 -00:17:00.9 LINER (A)[^15] 0.043 190 12.43 0.844 SN
IRAS 15250+3609 15:26:59.4 +35:58:37.5 LINER (H)[^16] 0.055 245 12.09 1.072 SN
IRAS 12490-1009 12:51:40.7 -10:25:26.1 H[ii]{}/LINER (G) 0.101 465 12.07 1.854 SN
IRAS 14060+2919 14:08:18.9 +29:04:46.9 H[ii]{} (E) 0.117 545 12.18 2.113 SN
IRASF 09427+1929 09:45:32.4 +19:15:34.9 H[ii]{} (H) 0.149 708 12.10 2.600 SN
IRAS 15206+3342 15:22:38.0 +33:31:35.9 H[ii]{} (E) 0.124 580 12.27 2.232 SN
Mrk 231 12:56:14.2 +56:52:25.2 Sey1 (E) 0.042 186 12.57 0.832 SN
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Observations and data reduction
===============================
IFS observations were obtained with INTEGRAL, a fiber-based optical integral field system (Arribas et al.1998) connected to the Wide Field Fibre Optic Spectrograph (WYFFOS; Bingham et al.1994) and mounted on the 4.2 m William Herschel Telescope. The observations were carried out during a number of observing runs between 1998 and 2004. We used three INTEGRAL configurations, namely the so-called standard bundles 1 (SB1), 2 (SB2), and 3 (SB3). The fiber diameters are 045, 09, and 27 for SB1, SB2, and SB3, respectively, resulting in the field of view (FoV) given in Table\[INTEGRAL-bundles\]. The three bundles have a similar configuration in the focal plane. The majority of the fibers form a rectangular area centered on the object, whereas a subset of fibers that form an outer ring of 45in radius is simultaneously used for measuring the sky (see configuration details in Table \[INTEGRAL-bundles\]). A remarkable capability of the instrument INTEGRAL is its flexibility in the bundle change, which allows us to select the most convenient instrument configuration depending on the seeing conditions. For the majority ($\sim$75%) of our cases, the SB2 bundle was the preferred one: the fiber size (09) is similar to the typical seeing of La Palma, and in general an entire ULIRG fits in the FoV (160$\times$123).
The spectra were taken with a 600 lines mm$^{-1}$ grating, which provides an effective spectral resolution (FWHM) of 6.0, 6.0, and 9.8 Åfor the SB1, SB2, and SB3 bundles, respectively[^17]. The covered spectral range of interest was $\lambda\lambda$4500-7000 Å(rest-frame). The total integration time, fiber bundle used on each galaxy, and individual comments on some observations are presented in Table \[datos\].
The data reduction was performed within the IRAF[^18] environment, followed the standard procedures applied to this type of data (see Arribas et al.1997 and references therein), and can be summarized as follows. After subtracting an averaged BIAS frame, we proceeded to define the fibers apertures. This step involves the identification of information coming from each fiber, to preserve the full spatial and spectral information provided by IFS. For this reduction step, we used a sky flat image, which clearly indicates the position of spectra from the fibers along the detector because of its high signal over the entire wavelength range. The location of the INTEGRAL+WYFFOS system at the Nasmyth focus of the telescope ensures its stability throughout a given night (del Burgo 2000), thus the information obtained for the sky flat can be used in all the scientific images observed during the same night. Once the fibers have been identified, we proceed to subtract the stray light and cross-talk contamination. Both signals create a background that has a diluting effect on the spectra, affecting in particular spectra from low surface brightness regions. The spurious light outside the defined apertures was analyzed, modeled two-dimensionally, and extracted from every image. The light transmitted by each fiber onto the detector was afterwards measured, added along the spatial direction, and subtracted to obtain a final one dimensional spectrum for every single fiber.
The one dimensional spectra were individually wavelength calibrated using a well-characterized lamp arc image. We checked the wavelength calibration using well identified sky lines, and obtained standard deviations of between 0.08 and 0.18 Å.
The next step in the data reduction involves the flat-field correction. By combining the information provided by lamp and sky flats, which accounted for pixel-to-pixel and fiber-to-fiber variations, respectively, we derived a response image, which was applied to all science data. Finally, the sky contribution to the spectra was defined and subtracted using the combined information of the outer ring of fibers present in every bundle. We carefully checked that the fibers covering the sky were not contaminated by any contribution of the galaxy.
For the flux calibration, several spectrophotometric standard stars[^19] were observed (BD+28 4211, Feige 34, BD +3326 42, and GD 153) using the same instrumental configuration and data reduction procedures as for the galaxies. Once the data reduction was completed, we combined all the individual exposures (a minimum of three), improving the S/N and rejecting cosmic rays. Finally, when needed, a correction factor accounting for the percentage of flux not integrated by the central fiber (i.e., the fiber where the calibration star is centered) was used. Using the four spectrophotometric standard stars during six different observing runs, the average uncertainty obtained in the flux calibration of the galaxies was in the range 10–15%. For further details about the flux calibration technique, we refer to Monreal-Ibero et al. (2007).
We also retrieved broad-band imaging data for our galaxies from the *HST* archive. The images were taken with the Wide Field and Planetary Camera (WFPC2) using the F814W filter, and are available for the entire sample. This filter is equivalent to the ground-based Johnson-Cousins I (Origlia & Leitherer 2000). The images were calibrated on the fly, with the highest quality available reference files at the time of retrieval[^20].
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IRAS Name INTEGRAL INTEGRAL Date Exp. time Comment on IFU data
(Bundle) (FoV arcsec$^{2}$) month/year (s)
IRAS 13156+0435 SB2, SB3 32.9$\times$25.3, 69.1$\times$60.5 03/02 18000[^21] \[O[iii]{}\] non detected in N comp.
IRAS 13342+3932 SB3 101.7$\times$89.0 03/02 6$\times$1500
IRAS 18580+6527 SB2 47.8$\times$36.7 04/01 7$\times$1500
IRAS 16007+3743 SB2 49.6$\times$38.1 04/01 6$\times$1500
IRAS 06268+3509 SB2 46.3$\times$35.6 03/02 6$\times$1500
IRAS 08572+3915[^22] SB2 18.0$\times$13.9 04/99 9$\times$1800
IRAS 14348$-$1447$^{a}$ SB2 1.3$\times$1.0 04/98 4$\times$1800
Arp 299 [^23] SB2, SB3 3.3$\times$2.5, 6.9$\times$6.0 01/04 14400[^24]
IRAS 13469+5833 SB1 21.3$\times$17.4 03/02 5$\times$1800 Sky line to H$\alpha$+\[N[ii]{}\] complex
Not enough S/N
IRAS 12112+0305$^{a}$[^25] SB2 27.9$\times$19.5 04/98 5$\times$1800, 4$\times$1500
Mrk 463 SB2 15.7$\times$12.1 04/01 5$\times$900
IRAS 06487+2208 SB1 19.7$\times$16.1 03/02 5$\times$1500
IRAS 11087+5351 SB1 19.5$\times$16.0 03/02 6$\times$1500
Mrk 273$^{a}$ SB2 12.0$\times$9.2 04/98 3$\times$1500
Arp 220$^{a}$ SB2, SB3 5.9$\times$4.5, 12.4$\times$10.8 05/00 48000[^26] \[O[iii]{}\] not detected, H$\beta$ not covered
IRAS 17208$-$0014$^{a}$ SB2 13.5$\times$10.4 04/99 4$\times$1950
IRAS 15250+3609$^{a}$ SB2 17.1$\times$13.2 04/98 5$\times$1800
IRAS 12490$-$1009 SB2 29.7$\times$22.8 03/02 4$\times$1500
IRAS 14060+2919 SB2 1.9$\times$1.4 03/02 4$\times$1800
IRAS 09427+1929 SB3 87.4$\times$76.4 03/02 2320[^27] Short inhomogeneous exposures
IRAS 15206+3342$^{a}$ SB2 35.7$\times$27.4 04/99 4$\times$1800
Mrk 231 SB1 6.5$\times$5.3 03/02 4$\times$900 H$\beta$ not covered
Broad emission lines
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\[Table\_log\]
Data analysis
=============
After data reduction, every galaxy has a set of spectra, each of them associated with a particular galaxy region observed by an individual fiber. The nuclear spectra of the systems under study are shown in Fig.\[espectros1\]. Depending on the instrument set-up, the angular scale of the spectra is 045, 09, or 27 for the SB1, SB2, and SB3 bundles, respectively. In this context, nuclear spectra are the ones corresponding to the fiber closest to the continuum peak.
Almost all galaxies exhibit nuclear emission, the most important optical emission lines including those that are both strong (H$\beta$, \[O[iii]{}\]$\lambda\lambda$4959, 5007, H$\alpha$+\[N[ii]{}\]$\lambda\lambda$6548, 6484, and \[S[ii]{}\]$\lambda\lambda$6716, 6731) and weak (\[O[i]{}\]$\lambda$6300). Some galaxies also exhibit other weak emission lines (\[N[i]{}\]$\lambda$5199, He[i]{}$\lambda$5876, and He[i]{}$\lambda$6678), as well as interstellar absorption lines (Na[i]{}$\lambda\lambda$5890, 5896).
To obtain maps of the most relevant optical lines tracing the ionization state, we first fitted the lines to Gaussian functions using the `DIPSO` package (Howarth & Murray 1988), inside the STARLINK environment[^28]. It is well known that, under certain conditions, such as line blending, the fitting algorithms may derive different solutions. We therefore decided to apply some restrictions to constrain our results more reliably. The H$\beta$ and \[O[i]{}\]$\lambda$6300 lines were individually fitted with a single component, with no particular constraint. We note that no H$\beta$ in absorption was detected, and hence no correction was applied. The \[O[iii]{}\]$\lambda\lambda$4959, 5007, and H$\alpha$+\[N[ii]{}\] complexes were fitted with two and three Gaussians, respectively, which were assumed to have identical kinematics. We applied these conditions based on the assumption that these line complexes trace identically the ionized gas kinematics. Additionally, we fixed the line intensity ratios of the \[O[iii]{}\] and \[N[ii]{}\] lines according to atomic parameters. For the \[S[ii]{}\]$\lambda\lambda$6716, 6731 lines, we assume only that they share the same kinematics, since their line ratio variations can be used to trace the electron density.
The spectral maps shown in this paper have all been derived using the values obtained with a single Gaussian component fitting for each line. In few cases, we detected additional kinematical components in the line profiles, which may be indicative of the presence of winds, superwinds, and AGN flows. Some regions of a few galaxies are also consistent with the presence of a broad component in the hydrogen recombination lines. These particular cases will not be addressed here but in dedicated papers about the kinematics and excitation conditions, because the purpose of the present work is to provide a general overview of the data.
The mappings tracing the stellar component were derived using a continuum emission line-free filter with a rectangular bandwidth of about 150 Å for each galaxy (filters being centered approximately at 4700 Å and 6150 Å restframe).
Finally, the derivation of all these maps involves the ordering of the fibers according to their astronomical position (RA, Dec). To that aim, we used the free software IDA (García-Lorenzo et al.2002), a tool specifically designed for the INTEGRAL IFU. To preserve the instrumental spatial resolution, the pixel size was equivalent to the fiber size used during the observations (see Figs.\[maps1\] - \[maps28\]). The HST images and the spectral maps were aligned by comparing with the peaks and external envelope structure of the red continuum map.
Observed morphologies of the stellar and ionized gas components
===============================================================
General trends
--------------
We present and discuss the differences between the structure of the stellar and ionized gas components for our sample of ULIRGs.
The high resolution HST images provide evidence for very complex stellar structures, with bright nuclear regions, dust structures, and knots of star formation. In the pre-coalescence systems, the large scale structure is characterized by merger-induced structures, i.e., bridges, tails, and plumes connecting the individual galaxies. The original spiral arms are disrupted and form kpc-size tidal tails with, in some cases, chains of star-forming regions (Monreal-Ibero et al.2007). For the post-coalescence galaxies, the large scale structure traces an outer common envelope, which resembles an elliptical structure (García-Marín2007), in addition to late merger features such as disordered nuclear regions and tails.
In spite of the different angular sampling, the INTEGRAL-based continuum images of the galaxies obtained at wavelengths blueward of H$\alpha$ and H$\beta$ recover the stellar structure observed in the high angular resolution *HST* images. This occurs even in the low-surface brightness regions, reassuring us that all the observed stellar and ionized gas structures are real. Only small differences between the INTEGRAL red and blue continua are detected (e.g., Arp 299, IRAS 08572+3915, IRAS 12112+0305), and they can probably be explained by extinction effects.
The general morphology of the ionized gas low surface brightness regions is similar to that of the stellar component, although in about 25% of the galaxies structural variations such us differences in shape or the presence of external clumps are associated with gas in high excitation states (Arp 299, Mrk 463, Mrk 273, IRAS 18580+6527, and IRAS 11087+5351). These differences are mainly traced by the \[O[iii]{}\]$\lambda$5007. In the pre-coalescence systems, one of the optical nuclei is usually a weak line emitter, with a surface brightness similar to that of the interface between the galaxies.
The largest structural variations between the stellar and ionized gas components are observed in the high surface brightness regions (i.e., the main body of the galaxies), and are detected as differences in the location of the emission peaks (see the distance between continuum and emission lines peaks for individual ULIRGs in Table\[Separacion\]). In what follows, we consider a significant shift as those equal to or larger than 1 kpc (within the errors given in Table\[Separacion\]).
There is also evidence of substructures in the different ionization states of the gas. These variations may be due to mechanisms such as star formation, AGNs, and shocks caused by tidal effects. To trace this more reliably, we have selected respectively the strong hydrogen recombination line H$\alpha$, the \[O[iii]{}\]$\lambda$5007 line, which traces hard ionizing photons, and \[O[i]{}\]$\lambda$6300, which is considered to be a sensitive shock tracer. Other effects, such as extinction and differences in stellar populations and in metallicities, may also play a role, but the detailed analysis of the excitation mechanisms in ULIRGs is beyond the scope of the present paper. In the following, we present the main structural characteristics of the pre- and post-coalescence systems.
Some ULIRGs of the sample were not included for a variety of reasons, such as the lack of \[O[iii]{}\]$\lambda$5007 detection (Arp 220), the low S/N of their data precluding the detection of emission lines (e.g., IRAS 13469+5833), poor spatial resolution (e.g., IRAS 13342+3932), and severe AGN contamination of the host galaxy light that makes it impossible to distinguish the ionized gas component (Mrk 231, see Appendix). The analysis presented here was carried out for 18 ULIRGs (i.e., 20 INTEGRAL pointings, see Table\[Table\_log\]). This includes two individual pointings for the spatially separated galaxies that form the systems Arp 299 and IRAS 13156+0435. On average, the linear scale corresponding to our angular resolution is about 1.8 kpc. Therefore, all the results presented here should be considered valid on this scale.
Pre-coalescence ULIRGs
----------------------
The ionized gas traced by the H$\alpha$ maps in the pre-coalescence systems exhibit emission on scales of about 5-10 kpc measured from the stellar nuclei (see Figs, \[maps1\] - \[maps16\]). Their intensity peak are located at different positions: in the brightest stellar nuclei (e.g., IRAS 18580+6527), in the main body/overlapping region between the galaxies (e.g., NGC 3690), in one of the tails (e.g., IRAS 16007+3743), in an extra-nuclear stellar knot (e.g., IRAS 12112+0305), or in the secondary nucleus (e.g.Mrk 463). In about 65% of the galaxies studied, there are significant offsets between the H$\alpha$ and the stellar continuum peaks, with projected separations of between 1-8 kpc. Extra-nuclear star-forming regions, mainly located in tidal tails, are also detected as secondary H$\alpha$ peaks (IRAS 16007+3743, IRAS 08572+3915, IRAS 14348-1447, and IRAS 12112+0305). These additional light peaks are identified as Tidal Dwarf Galaxies (TDGs), candidates (Monreal-Ibero et al.2007). The H$\beta$ structure is rather similar to that of H$\alpha$, but it is intrinsically weaker and more affected by extinction, especially in the galaxy nuclei.
The overall morphology, relative brightness of the nuclei, and the peak distribution of the brightest forbidden lines \[N[ii]{}\]$\lambda$6484, and \[S[ii]{}\]$\lambda\lambda$6716, 6731 are in good agreement with that of H$\alpha$. Nonetheless, there are some cases where small scale differences in their peak positions are found (IRAS 18580+6527 and IRAS 12112+0305), and one galaxy where their peaks are located 8.4 kpc away from the one of H$\alpha$ (IRAS 16007+3743). Given the wavelength proximity of these emission lines, it is unlikely that these differences are produced by any extinction effects. They are probably produced instead by the presence of different ionization mechanisms associated with tidal shocks.
The results for the highly excited line \[O[iii]{}\]$\lambda$5007 are an interesting case study. As for H$\alpha$, in about 50% of the pre-coalescence systems, there are significant shifts between the stellar continuum and the nebular emission peaks. In comparison with other emission-line maps, we find that the \[O[iii]{}\]$\lambda$5007 peak coincides with that of H$\alpha$ in 70% of the studied cases. In some galaxies, there are regions that appear to be dominated by local ionization sources. For instance, in IRAS 08572+3915 and IRAS 12112+0305, the \[O[iii]{}\]$\lambda$5007 line has its maxima in regions identified as TDG candidates in the stellar continuum images. These TDG candidates are also secondary H$\alpha$ emitters. These differences may be caused by the effects of extinction on \[O[iii]{}\]$\lambda$5007, which are more important in the nuclei than in these external regions. In the case of IC 694, the peak of the \[O[iii]{}\]$\lambda$5007 coincides with a region of the galaxy believed to be one of the original spiral arms disrupted by the merger. It is therefore also bright in the stellar component and in H$\alpha$.
The galaxy IRAS 18580+6527 also deserves special mention, since the extra-nuclear local maxima located at about 10 kpc from the eastern nucleus (also bright in H$\alpha$, see Fig.\[maps5\], $\Delta$$\alpha$$\sim$-4, $\Delta$$\delta$$\sim$-3) is not associated with any particular stellar mass concentration, but is rather dominated by nebular emission. Interestingly enough, this galaxy is classified as a Seyfert. As explained below, similar extra-nuclear highly excited nebulae are detected in the post-coalescence ULIRGs Mrk 273 and IRAS 11087+5351.
Approximately 40% of the pre-coalescence ULIRGs present shifts of up to 6 kpc between the peaks of the red continuum and the \[O[i]{}\]$\lambda$6300 line. However, in some cases such as IRAS 06268+3509 and Mrk 463, the line maxima is associated with the stellar secondary nucleus of the system. This would mean that in general the \[O[i]{}\]$\lambda$6300 activity is mainly related to the nuclei.
Post-coalescence ULIRGs
-----------------------
For the morphologically evolved ULIRGs, the comparison between the stellar and ionized gas components provides different results from above. The most important one is that all the emission tends to be concentrated in the nuclear region. The previously reported structural differences between the stellar and ionized gas are not so common for this type of ULIRGs. The spectral maps indicate that in all cases (see Figs. \[maps17\] - \[maps28\]) the H$\alpha$ peak coincides with the peak of the stellar emission (i.e. within the central kpc), indicating a nuclear concentration of the excitation sources. The H$\alpha$ overall structure generally coincides with that of the red continuum. One galaxy (IRAS 15250+3609) contains a TDG candidate that is also an H$\alpha$ emitter (Monreal-Ibero 2007). For Mrk 273, Arp 220, and IRAS 11087+5351, there are also secondary extra-nuclear structures that do not correlate spatially with the stellar distribution. Likewise for the pre-coalescence systems, the H$\alpha$ map generally agree with those of H$\beta$, although the latter is more affected by extinction that can cause their maxima position not to be coincident.
The behavior of the \[N[ii]{}\]$\lambda$6484, and \[S[ii]{}\]$\lambda\lambda$6716, 6731 follows that of H$\alpha$, and only small scale variations are measured in IRAS 15250+3609.
As in the pre-coalescence systems, there are two galaxies (IRAS 11087+5351 and Mrk 273) that exhibit a peak in the \[O[iii]{}\]$\lambda$5007 emission within the extra-nuclear regions (7.2 and 5.1 kpc, respectively). These regions are not dominated by local ionization sources, because there is no particular stellar mass concentration there. Interestingly, both galaxies have a Seyfert classification (see Table \[TableSample\]). Secondary H$\alpha$ structures are also detected in these extra-nuclear regions.
The \[O[i]{}\]$\lambda$6300 emission line maps also follow the nuclear concentration trend exhibited by the post-coalescence systems, with only the maxima position of IRAS 11087+5351 being shifted with respect to that of the continuum and almost coinciding with the \[O[iii]{}\]$\lambda$5007 peak.
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Galaxy Nuc Sep \[O\] H$\alpha$ \[O\] H$\beta$ \[S\] \[N\] Comment
(kpc) (kpc) (kpc) (kpc) (kpc) (kpc) (kpc)
IRAS 13156+0435N 36.0 1.8$\pm$0.9 1.8$\pm$0.9 2.0$\pm$0.9 1.8$\pm$0.9 1.8$\pm$0.9\* 1.8$\pm$0.9
IRAS 13156+0435S 36.0 1.8$\pm$0.9 1.8$\pm$0.9 1.8$\pm$0.9 1.8$\pm$0.9 1.8$\pm$0.9 1.8$\pm$0.9
IRAS 18580+6527 15.0 0.0$\pm$1.2 2.7$\pm$1.2 2.7$\pm$1.2 2.7$\pm$1.2 0.0$\pm$1.2 2.7$\pm$1.2
IRAS 16007+3743 14.2 3.9$\pm$1.4 8.4$\pm$1.4 8.4$\pm$1.4 8.4$\pm$1.4 0.0$\pm$1.4 0.0$\pm$1.4
IRAS 06268+3509 9.1 5.8$\pm$1.3 2.6$\pm$1.3 2.6$\pm$1.3 3.7$\pm$1.3 2.6$\pm$1.3 2.6$\pm$1.3
IRAS 08572+3915 6.1 5.9$\pm$0.5 5.9$\pm$0.5 5.1$\pm$0.5 1.0$\pm$0.5 5.9$\pm$0.5 5.9$\pm$0.5
IRAS 14348$-$1447 5.5 0.0$\pm$0.7 0.0$\pm$0.7 0.0$\pm$0.7 0.0$\pm$0.7 0.0$\pm$0.7 0.0$\pm$0.7
Arp 299/NGC 3690 5.0 0.4$\pm$0.1 1.3$\pm$0.1 1.2$\pm$0.1 1.2$\pm$0.1 1.2$\pm$0.1 1.3$\pm$0.1
Arp 299/IC 694 5.0 0.8$\pm$0.1 0.9$\pm$0.1 1.3$\pm$0.1 1.3$\pm$0.1 0.9$\pm$0.1 0.9$\pm$0.1
IRAS 12112+0305 4.0 1.3$\pm$0.6 2.0$\pm$0.6 1.2$\pm$0.6 2.5$\pm$0.6 1.2$\pm$0.6 1.2$\pm$0.6
Mrk 463 3.8 3.5$\pm$0.4 3.5$\pm$0.4 3.5$\pm$0.4 3.5$\pm$0.4 3.5$\pm$0.4 3.5$\pm$0.4
IRAS 06487+2208 1.5 1.1$\pm$0.6 1.1$\pm$0.6 1.1$\pm$0.6 0.0$\pm$0.6 0.0$\pm$0.6 1.1$\pm$0.6
IRAS 11087+5351 1.5 6.6$\pm$0.6 1.6$\pm$0.6 7.2$\pm$0.6 7.2$\pm$0.6 1.6$\pm$0.6 1.6$\pm$0.6
Mrk 273 0.7 0.7$\pm$0.3 0.7$\pm$0.3 5.1$\pm$0.3 5.1$\pm$0.3 0.7$\pm$0.3 0.7$\pm$0.3
Arp220 0.4 0.0$\pm$0.2 0.5$\pm$0.2 N/A N/A 0.3$\pm$0.2 0.3$\pm$0.2
IRAS 12490$-$1009 0.0 1.7$\pm$0.8 1.7$\pm$0.8 0.0$\pm$0.8 0.0$\pm$0.8 1.7$\pm$0.8 1.7$\pm$0.8
IRAS 14060+2919 0.0 1.9$\pm$0.9 1.9$\pm$0.9 1.9$\pm$0.9 1.9$\pm$0.9 1.9$\pm$0.9 1.9$\pm$0.9
IRAS 15206+3342 0.0 0.0$\pm$1.0 0.0$\pm$1.0 0.0$\pm$1.0 0.0$\pm$1.0 0.0$\pm$1.0 0.0$\pm$1.0
IRAS 15250+3609 0.0 0.0$\pm$0.5 1.4$\pm$0.5 1.4$\pm$0.5 1.4$\pm$0.5 0.0$\pm$0.5 0.0$\pm$0.5
IRAS 17208$-$0014 0.0 0.0$\pm$0.4 0.0$\pm$0.4 1.1$\pm$0.4 1.1$\pm$0.4 0.0$\pm$0.4 0.0$\pm$0.4
------------------- --------- ------------- ------------- ------------- ------------- --------------- ------------- ---------
Summary
=======
This is the first paper in a series to study in detail the two-dimensional morphology, dust distribution, excitation processes, and kinematics of a representative sample of 22 local ULIRGs. To achieve this, we have used optical IFS data combined with high resolution *HST* images. In this paper, we have described the sample selection and the observations, and presented the morphological properties of the continuum and ionized gas components. The ULIRGs were selected to cover a wider range of the IR luminosity distribution, different merger phases, and to exhibit a variety of nuclear activities (H[ii]{}, LINER, Seyfert). Using the projected nuclear distance as a criteria, we have adopted a simple classification scheme that divides the sample into pre- and post-coalescence galaxies, and identifies the different behaviors at different stages of the merger. The main results of this paper can be summarized as:
- Despite the different resolution, the structure of the INTEGRAL red continua is consistent with that of the HST F814W band. The red and blue continua may exhibit slight differences caused by the effect of the extinction. In contrast, the structure of the warm ionized gas can be significantly decoupled from that of the stellar continuum. These variations are due to the different spatial distribution of the ionization sources present in the galaxy structure, and to the dust distribution.
- We have compared the stellar and ionized gas structures of ULIRGs undergoing different merger phases (pre- and post-coalescence), and found structural variations between the gaseous and the stellar components, with offsets in the emission peak positions of up to about 8 kpc in the pre-coalescence systems. For 64% of ULIRGs in an early interaction phase, the H$\alpha$ peak does not coincide with the stellar maxima. In contrast, galaxies undergoing advanced mergers have their H$\alpha$ peak located in the nuclear regions.
- Four pre- and one post-coalescence ULIRGs appear to contain TDGs candidates, which we identify with H$\alpha$ emission associated with stellar emission knots.
- The ionization structure traced by different emission lines has been also studied. We detected variations in lines with similar rest-frame, meaning that the differences are unlikely to be explained by the presence of dust. This is more common in the pre-coalescence ULIRGs, and it is probably a consequence of different excitation mechanisms. Differences in metallicity and stellar populations may also be playing a role.
- The \[O[iii]{}\] line traces highly excited extra-nuclear clouds in both pre- and post-coalescence systems, with no relevant stellar counterpart. These regions are also secondary H$\alpha$ emitters. All these galaxies have a Seyfert nuclei, classified using optical emission lines ratios.
- The peak of the \[O[i]{}\] line is shifted with respect to the stellar one, but in general its emission is nuclear.
- This analysis infers that there is an evolutionary trend in the ionized gas behavior. In the pre-coalescence ULIRGs, the structural differences are remarkable, and the extra-nuclear regions, out to distances of about 8 kpc, play an important role and contribute to the excitation level. In contrast, the post-coalescence ULIRGs tend to have their activity concentrated in the nucleus. The only exception to this are the extra-nuclear nebulae traced by the \[O[iii]{}\]. We have detected them in both galaxy types, and they appear to be related to the presence of a Seyfert nucleus.
We would like to thank Almudena Alonso-Herrero for useful comments and discussions. This paper uses the plotting package `jmaplot`, developed by Jesús Maíz-Apellániz. http:$//$dae45.iaa.csic.es:8080$/$$\sim$jmaiz/software. This research has made use of the NASA/IPAC Extragalactic Database (NED) which is operated by the Jet Propulsion Laboratory, California Institute of Technology, under contract with the National Aeronautics and Space Administration.
This work has been supported by the Spanish Ministry of Education and Science, under grant BES-2003-0852, project AYA2002-01055. MG-M is supported by the German federal department for education and research (BMBF) under the project numbers: 50OS0502 & 50OS0801. AMI is supported by the Spanish Ministry of Science and Innovation (MICINN) under the program “Specialization in International Organisms” ref. ES2006-0003.
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Stellar component and ionized gas maps for the sample of ULIRGs:
================================================================
In this Appendix, the maps of interest for all the galaxies of the sample, as obtained with the INTEGRAL system, are presented. The emission-line free stellar continuum, along with the most relevant optical-emission line (H$\beta$, \[O[iii]{}\]$\lambda$5007, \[O[i]{}\]$\lambda$6300, H$\alpha$+\[N[ii]{}\]$\lambda\lambda$6548, 6584, \[S[ii]{}\]$\lambda\lambda$)6716, 6731) maps are shown. Complementary optical (0.8 $\mu$m) *HST* images are presented too. The galaxies are sorted by decreasing nuclear separation.
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[^1]: Based on observations with the William Herschel Telescope operated on the island of La Palma by the ING in the Spanish Observatorio del Roque de los Muchachos of the Instituto de Astrofísica de Canarias. Based also on observations with the NASA-ESA Hubble Space Telescope, obtained at the Space Telescope and Science Institute, which is operated by the Association of Universities for Research in Astronomy, Inc. under NASA contract number NAS5-26555.
[^2]: threshold criteria S$_{60\mu \rm{m}}$$>$5.4 Jy, referred to the IRAS band centered in 60$\mu$m
[^3]: For the luminosity distances, the Wright (2006) cosmology calculator has been used.
[^4]: L$_{IR} $(8-1000 $\mu$m) was derived following Sanders & Mirabel 1996 ($L(8-1000\mu m)=4\pi {D_{L}}^{2}F_{IR}[L_{\odot}]$, with $F_{FIR}$=$1.26\times 10^{-14}\cdot(2.58f_{60}+f_{100})[Wm^{-2}]$. The quantities $f_{12}$, $f_{25}$, $f_{60}$, and $f_{100}$ are the *IRAS* flux densities in Jy at 12, 25, 60 and 100 $\mu$m.). The IRAS fluxes were obtained from the IRAS Faint Galaxy Sample (Moshir et al.1993, Vizier on line catalogue II/156A).
[^5]: IP means interacting pair, DN double nucleus and SN single nucleus. The distances in kpc represent the projected nuclear distance as measured in the F814W HST image.
[^6]: \(C) Darling & Giovanelli 2006
[^7]: IRAS fluxes obtained from the IRAS point Source Catalogue.
[^8]: \(E) Kim, Veilleux & Sanders 1998.
[^9]: Arp 299 is the system formed by NGC 3690 (to the east) and IC 694 (to the west).
[^10]: \(D) García-Marín et al.2006.
[^11]: Veilleux et al.1999.
[^12]: \(F) Miller & Goodrich 1990; (G) García-Marín 2007.
[^13]: IRAS fluxes obtained from the IRAS point Source Catalogue.
[^14]: \(B) Colina, Arribas & Borne 1999.
[^15]: \(A) Arribas & Colina 2003.
[^16]: \(H) Veilleux et al.1995.
[^17]: These values correspond to the old camera mounted on WYFFOS, which was used for the present observations. In August 2004, a new camera was commissioned for the instrument. See more details in http://www.iac.es/proyecto/integral.
[^18]: IRAF software is distributed by the National Optical Astronomy Observatory (NOAO), which is operated by the Association of Universities for Research in Astronomy (AURA), Inc., in cooperation with the National Science Foundation.
[^19]: These stars are part of the Hubble Space Telescope spectrophotometric standards.
[^20]: More information about data process can be found at www.stsci.edu/hst/wfpc2/analysis/analysis.html
[^21]: 3$\times$1500 s for the South component with the SB2 bundle; 3$\times$1800 for the North component with the SB2 bundle, 5$\times$1500 s for the entire system with the SB3 bundle.
[^22]: Previously published galaxy. See Colina, Arribas & Monreal-Ibero 2005, and references therein.
[^23]: García-Marín et al. 2006.
[^24]: 3$\times$1200 s for each individual galaxy of the system, NGC 3690 and IC 694 observed with the SB2 bundle. 3$\times$1800 s for the entire system as observed with the SB3 bundle.
[^25]: For IRAS 12112+0305 we present the results obtained by combining two different pointings of the SB2 INTEGRAL bundle.
[^26]: 6$\times$1500 s for the nuclear region observed with the SB2 bundle, 26$\times$1500 s for the nucleus and the extended nebula as observed with the SB3 bundle.
[^27]: 1500 s, 520 s, 300 s.
[^28]: See http://www.starlink.rl.ac.uk/.
|
---
abstract: 'Metallic transition-metal oxides undergo a metal-to-insulator transition (MIT) as the film thickness decreases across a critical thickness of several monolayers (MLs), but its driving mechanism remains controversial. We have studied the thickness-dependent MIT of the ferromagnetic metal La$_{0.6}$Sr$_{0.4}$MnO$_3$ by x-ray absorption spectroscopy and x-ray magnetic circular dichroism. As the film thickness was decreased across the critical thickness of the MIT (6-8 ML), a gradual decrease of the ferromagnetic signals and a concomitant increase of paramagnetic signals were observed, while the Mn valence abruptly decreased towards Mn$^{3+}$. These observations suggest that the ferromagnetic phase gradually and most likely inhomogeneously turns into the paramagnetic phase and both phases abruptly become insulating at the critical thickness.'
author:
- 'G. Shibata'
- 'K. Yoshimatsu'
- 'E. Sakai'
- 'V. R. Singh'
- 'V. K. Verma'
- 'K. Ishigami'
- 'T. Harano'
- 'T. Kadono'
- 'Y. Takeda'
- 'T. Okane'
- 'Y. Saitoh'
- 'H. Yamagami'
- 'A. Sawa'
- 'H. Kumigashira'
- 'M. Oshima'
- 'T. Koide'
- 'A. Fujimori'
bibliography:
- 'ref.bib'
- 'MyNotes.bib'
title: 'Thickness-dependent ferromagnetic metal to paramagnetic insulator transition in La$_{0.6}$Sr$_{0.4}$MnO$_3$ thin films studied by x-ray magnetic circular dichroism'
---
\[Intro\]Introduction
=====================
The physical properties of 3$d$ transition-metal oxides (TMOs) are usually controlled by the bandwidth and/or band filling of the 3$d$ bands [@MITrev]. Recently, attempts have also been made to control them by the thickness of thin film samples, namely, by dimensionality [@HongLSMO; @HuijbenLSMO; @YoshiLSMO; @LSMO110_PRL12; @XiaSRO; @ToyotaSRO; @YoshiSVO10; @YoshiSVO11; @LNOthickdep]. In many metallic oxide thin films, including ferromagnetic ones such as La$_{1-x}$Sr$_{x}$MnO$_3$ (LSMO) [@HongLSMO; @HuijbenLSMO; @YoshiLSMO; @LSMO110_PRL12] and SrRuO$_3$ (SRO) [@XiaSRO; @ToyotaSRO], and paramagnetic ones such as SrVO$_3$ (SVO) [@YoshiSVO10; @YoshiSVO11] and LaNiO${_3}$ [@LNOthickdep], the resistivity increases when the film thickness is decreased, and metal-to-insulator transitions (MITs) occur at a critical thickness of several monolayers (MLs). Evidence for the MITs was found by transport measurements [@HongLSMO; @HuijbenLSMO; @XiaSRO; @LNOthickdep] and photoemission spectroscopy (PES) [@YoshiLSMO; @ToyotaSRO; @YoshiSVO10; @YoshiSVO11]. According to the PES studies of TMO thin films, the density of states at the Fermi level ($E_F$) disappears below 4 ML (SRO [@ToyotaSRO], SVO [@YoshiSVO10]) to 8 ML (LSMO [@YoshiLSMO]), resulting in a large insulating gap (of order $\sim$ 1 eV) at $E_F$. In order to explain such MITs, transport theories for conventional metal thin films [@FSeq1; @FSeq2], which take into account only surface roughness, predict extremely small critical thicknesses, and one has to invoke strong electron-electron scattering, namely, strong electron correlation. For example, the thickness-dependent MIT of SVO thin films has been considered as a bandwidth-controlled Mott transition caused by the decreased number of nearest-neighbor V atoms [@YoshiSVO10]. Furthermore, the decrease of the film thickness leads to the lowering of spatial dimension and symmetry, and the increase of interfacial effects, resulting in the changes of the electric and magnetic properties. In LSMO [@HuijbenLSMO; @YoshiLSMO] as well as in SRO [@XiaSRO], not only metallic conduction but also ferromagnetism disappears simultaneously. For LSMO thin films, where the number of electrons is not an integer, the simple bandwidth-controlled Mott transition mechanism alone is not sufficient to explain the MIT and one has also to take into account the effect of disorder or local lattice distortion to induce the localization of charge carriers. Therefore, it is important to consider mechanisms such as charge ordering and/or the splitting of the $d$ bands due to the lower structural symmetry at the surface and interface. Such mechanisms are usually accompanied by changes in the magnetic properties. Thus, it is strongly desired to probe both the electronic states and magnetic properties on the microscopic level at the same time as a function of film thickness. For this purpose, we have investigated the electric and magnetic properties of LSMO ($x=0.4$) thin films as functions of film thickness using x-ray magnetic circular dichroism (XMCD). XMCD in core-level x-ray absorption spectroscopy (XAS) is a powerful tool to obtain information about the magnetism of specific elements, together with information about the valence states, and is especially suitable for the study of thin films and nanostructures, because it can probe the intrinsic magnetism without contribution from the substrate and other extrinsic effects.
\[Exp\]Experiment
=================
LSMO thin films with various thicknesses were fabricated on the TiO$_2$-terminated (001) surface of SrTiO$_3$ (STO) substrates by the laser molecular beam epitaxy (laser-MBE) method. The LSMO thickness ranged from 2 ML to 15 ML. After the deposition, the films were capped with 1 ML of La$_{0.6}$Sr$_{0.4}$TiO$_3$ (LSTO) and then 2 ML of STO \[Fig. \[XASall\](a)\]. The LSTO layer was inserted to keep the local environment of the topmost MnO$_2$ layer the same as that of the deeper MnO$_2$ layers [@YoshiLSMO], namely, each MnO$_2$ layer is sandwiched by La$_{0.6}$Sr$_{0.4}$O (LSO) layers. All the fabrication conditions were identical to those of Ref. . The surface morphology of the multilayers was checked by atomic force microscopy, and atomically flat step-and-terrace structures were clearly observed in all the films studied. Four-circle x-ray diffraction measurements confirmed the coherent growth of the films. Prior to measurements, the samples were annealed in 1 atm of O$_2$ at 400 $^{\circ}\text{C}$ for 45 minutes in order to eliminate oxygen vacancies. The XAS and XMCD measurements were performed at polarization-variable undulator beamlines BL-16A of the Photon Factory (PF) and at BL23SU of SPring-8 [@BL23SU]. The maximum magnetic field was $\mu_{0} H_{\text{ext}} = 3\ \text{T}$ at PF and $\mu_{0} H_{\text{ext}} = 8\ \text{T}$ at SPring-8. At both beamlines, the magnetic field was applied perpendicular to the film surface. Photons were incident normal to the sample surface and their helicity was reversed to measure XMCD. The measurements were performed at $T=20$ K. All the spectra were taken in the total electron yield mode. Most of the spectra shown in the figures were taken at PF, while data taken at high magnetic fields ($\mu_{0} H_{\text{ext}} > 3\ \text{T}$) were measured at SPring-8.
\[Res\]Results and Discussion
=============================
Figures \[XASall\](b) and \[XASall\](b)(c) show the XAS and XMCD spectra of all the LSMO samples. The spectral line shapes of XMCD \[Fig. \[XASall\](c)\] are similar to those of bulk LSMO [@KoideLSMO]. Reflecting the thickness-dependent magnetic properties of LSMO thin films [@HongLSMO; @HuijbenLSMO; @YoshiLSMO], the XMCD intensity decreased with decreasing thickness. Using the XMCD sum rules [@orbsum; @spinsum], we have estimated the spin ($M_{\text{spin}}$) and orbital ($M_{\text{orb}}$) magnetic moments of the Mn ions . The ratio $|M_{\text{orb}}/M_{\text{spin}}|$ was found to be less than $\sim 1/100$ for all the LSMO thicknesses; therefore $M_{\text{spin}}$ mainly contributes to the magnetic moment of Mn. Figure \[MHcurves\] shows thus estimated magnetization curves ($M \equiv M_\text{spin} + M_\text{orb} \sim M_\text{spin}$) of the LSMO thin films with various thicknesses. In Fig. \[MHcurves\](a), $M$’s are plotted as functions of applied magnetic field $H_\text{ext}$. In Fig. \[MHcurves\](b) the same data are replotted as functions of magnetic field corrected for the demagnetizing field $H_\text{demag}$ perpendicular to the film as $H = H_{\text{ext}}+H_\text{demag} = H_{\text{ext}}-M$. After the demagnetization-field correction, the saturation field is reduced to $\mu_{0} H \sim 0.3$ - $0.8\ \text{T}$, but is still an order of magnitude larger than the saturation field when the external magnetic field is applied parallel to the film and the demagnetizing field is absent ($\mu_{0} H \sim 10^{-2}\ \text{T}$) [@Ishii_APL05]. Therefore, we conclude that the LSMO thin films have in-plane easy magnetization axes due to magnetocrystalline anisotropy, which probably originates from the tensile strain from the STO substrate [@Konishi_JPSJ; @LSMOLD_Natcom12] (note that $a = 0.387\ \text{nm}$ for bulk LSMO while $a = 0.3905\ \text{nm}$ for STO).
![(Color online) XAS and XMCD spectra of the La$_{0.6}$Sr$_{0.4}$MnO$_3$ (LSMO) thin films for various thicknesses. (a) Schematic drawing of the thin film samples consisting of LSMO, SrTiO$_3$ (STO), and La$_{0.6}$Sr$_{0.4}$TiO$_3$ (LSTO) layers. (b) Mn $2p$ XAS spectra. (c) Mn $2p$ XMCD spectra. All the spectra were taken at $T=20$ K with an external magnetic field $\mu_{0} H_\text{ext} = 3$ T. In panel (b), Mn $2p$ XAS spectra of LaMnO$_{3}$ (Mn$^{3+}$) [@Mn23XAS] and SrMnO$_{3}$ (Mn$^{4+}$) [@SMOXAS] are also shown as references. []{data-label="XASall"}](Fig1_ver11.eps){width="8cm"}
Next, we decompose the magnetization curves into two components as shown in the inset of Fig. \[MHcurves\](b): the ferromagnetic component which saturates below $\mu_{0} H \sim 1\ \text{T}$ and the paramagnetic component which increases linearly with $H$ up to the highest magnetic fields . Thus the intercept of the magnetization curve gives the ferromagnetic moment $M_{\text{ferro}}$ and the slope of the magnetization curve gives paramagnetic susceptibility $\chi_{\text{para}}$ of the Mn ions. We confirmed the linear increase of $M$ as a function of $H$ up to higher magnetic fields ($3\ \text{T} < \mu_{0} H_{\text{ext}} \leq 8\ \text{T}$) for several samples (not shown), which justifies the separation of the magnetization curves into the ferromagnetic and paramagnetic components. In Fig. \[Ndep\](a), the $M_{\text{ferro}}$ and $\chi_{\text{para}}$ values thus obtained are plotted as functions of film thickness. With decreasing film thickness, $M_{\text{ferro}}$ decreases and $\chi_{\text{para}}$ increases, indicating a gradual transition from the ferromagnetic state to the paramagnetic state. The measured $\chi_{\text{para}}$ is, however, somewhat larger than the one predicted by the Curie law of the Mn$^{3+}$-Mn$^{4+}$ mixed valence state in the entire thickness range [^1]. This indicates that the system exhibits a ferromagnetic-to-paramagnetic phase separation and that even in the paramagnetic state there are ferromagnetic correlations between the Mn local moments.
![(Color online) Magnetic field dependence of the magnetization ($M$) of LSMO thin films estimated by XMCD. (a) $M$ plotted against the external field $H_{\text{ext}}$. (b) $M$ plotted against the magnetic field corrected for the demagnetizing field (see text). Inset shows the decomposition of $M$ into the ferromagnetic ($M_{\text{ferro}}$) and paramagnetic ($M_{\text{para}}$) components. []{data-label="MHcurves"}](Fig2_ver17.eps){width="8cm"}
![(Color online) Thickness dependencies of the magnetic and electronic properties of the LSMO thin films. (a) Thickness dependence of the ferromagnetic moment per Mn ($M_{\text{ferro}}$) and the paramagnetic susceptibility ($\chi_{\text{para}}$) per Mn of the LSMO thin films, estimated from the magnetization curves in Fig. \[MHcurves\]. The paramagnetic susceptibility simulated using the Curie law ($\chi_{\text{para}}^{\text{Curie}}$) is shown by a dashed curve, indicating an enhancement of the paramagnetic signals. The $M_{\text{ferro}}$ and $\chi_{\text{para}}$ estimated from the magnetization measurements up to $\mu_{0} H_{\text{ext}} \leq 8\ \text{T}$ are shown by open symbols. (b) Peak position of the Mn L$_3$ edge estimated from the XAS spectra and the valence band maximum (VBM) position relative to the Fermi level ($E_{F}$) measured by PES [@YoshiLSMO]. []{data-label="Ndep"}](Fig3_ver17.eps){width="8cm"}
We note that while the thickness-dependent MIT occurs rather abruptly according to PES [@YoshiLSMO], Fig. \[Ndep\](a) shows a *gradual* decrease and increase of the ferromagnetic and paramagnetic components, respectively, with decreasing film thickness. Indeed, the thickness dependence of the valence-band maximum (VBM) measured by PES [@YoshiLSMO], as shown in Fig. \[Ndep\](b), indicates an abrupt opening of the energy gap below the critical thickness. This means that the paramagnetic component already exists slightly above the critical thickness of the MIT and the ferromagnetic component persists below it as an ferromagnetic insulating (FM-I) phase. Therefore, the present results suggest that a paramagnetic metallic (PM-M) or paramagnetic insulating (PM-I) phase starts to appear in the ferromagnetic metallic (FM-M) phase from slightly above the critical thickness, and that the FM-I and the PM-I phases coexist below the critical thickness of MIT.
In order to obtain the information about changes in the electronic structure across the MIT, we examine the thickness dependence of the line shapes and the energy positions of the XAS and XMCD spectra. Comparing the experimental XAS spectra with those of LaMnO$_3$ (Mn$^{3+}$) [@Mn23XAS] and SrMnO$_3$ (Mn$^{4+}$) [@SMOXAS] \[Fig. \[XASall\](b)\], the intensities of structures a and c, which originate from Mn$^{3+}$, become stronger when the LSMO thickness is reduced. In addition, as shown in Figs. \[XASall\](b) and \[Ndep\](b), the peak positions of the Mn L$_3$ and L$_2$ edges are abruptly shifted to lower energies by $\sim 0.2\ \text{eV}$ between 8 ML and 6 ML, where the thickness-dependent MIT occurs [@YoshiLSMO]. Similar peak shifts are also observed in the XMCD spectra, as shown in Fig. \[XASall\](c). In the reference XAS spectra in Fig. \[XASall\](b), both the L$_3$ and L$_2$ edges are located at lower photon energies for Mn$^{3+}$ than for Mn$^{4+}$. From these spectral changes, we conclude that the effective hole concentration decreases as the LSMO thickness decreases, and that it suddenly drops at the critical thickness of MIT. Considering that bulk LSMO enters the FM-I phase in the low hole concentration region $0.09 \lesssim x \lesssim 0.16$ [@Urushi], the observed valence shift towards Mn$^{3+}$ in the thin films is certainly related with the FM-M to FM-I transition with decreasing thickness.
The observed valence change towards Mn$^{3+}$ with decreasing film thickness may be partly explained by the presence of the LSTO layer in the cap. As mentioned in Sec. \[Exp\] a 1-ML-thick LSTO layer is inserted between the LSMO film and the STO cap layer in our samples \[Fig. \[XASall\](a)\]. The (LSO)$^{0.6+}$ layer in LSTO acts as an electron donor. Because the work function of STO is smaller than that of LSMO, the electrons supplied by the LSO layer are doped into the LSMO side rather than the STO side [@Kumi_Ti4+]. Thus the average Mn valence is shifted towards the $3+$ side from the nominal valence $3.4+$. This effect is more significant in the thinner films because the number of the doped electrons per monolayer is larger. However, the amount of electron charges is not enough to explain the observed valence change. There is also a possibility that some oxygen vacancies may exist in the LSMO films and/or STO substrates, which leads to additional electron doping. We note that such a valence change towards Mn$^{3+}$ with decreasing film thickness has also been reported in previous studies [@JSLee_reverse; @FelipXRD].
The coincidence of the MIT and the abrupt valence change at 6-8 ML implies some connection between the MIT and the valence change. One possible scenario is that the MIT induces changes in the charge distribution in the LSMO film, leading to the apparent decrease of the hole concentration over the entire LSMO film. When the film is thick and metallic, the free electric charges (holes) will be distributed at the top and bottom interfaces of LSMO, so that there is no potential gradient inside the film. When the film becomes thinner and insulating, the holes are distributed over the entire LSMO layer. If the holes at the bottom interface are distributed in a more extended region, it may cause the abrupt Mn valence change observed by XMCD (the probing depth of which is 3-5 nm [@TEYdepth] or 8-12 ML). In order to clarify the relationship between the observed valence change and the MIT, further experiments would be required, especially on the depth profiles of the electronic states and magnetism in the LSMO thin films.
\[Summ\]Summary
===============
We have performed XAS and XMCD studies of LSMO thin films with varying thickness in order to investigate the origin of the thickness-dependent MIT and the concomitant loss of ferromagnetism. With decreasing film thickness, a gradual decrease of the ferromagnetic component and an increase of the paramagnetic component were observed. The experimental paramagnetic susceptibility was larger than the Curie law, indicating that spin correlations between Mn atoms are ferromagnetic. The Mn valence was found to approach $\text{Mn}^{3+}$ below the critical thickness of the MIT. The ferromagnetic-to-paramagnetic transition occurred gradually as a function of thickness, whereas the MIT and the valence change towards $\text{Mn}^{3+}$ took place abruptly. These results can be understood within the picture of mixed phases: the films above the critical thickness as a mixture of the FM-M and PM-I or PM-M phases while the films below the critical thickness as a mixture of the FM-I and PM-I phases. The mechanism of the valence change has to be investigated as a future theoretical problem.
This work was supported by a Grant-in-Aid for Scientific Research from the JSPS (No. S22224005) and the Quantum Beam Technology Development Program from the JST. The experiment was done under the approval of the Photon Factory Program Advisory Committee (Proposal No. 2010G187 and No. 2010S2-001) and under the Shared Use Program of JAEA Facilities (Proposal No. 2011A3840/BL23SU). G.S. acknowledges support from Advanced Leading Graduate Course for Photon Science (ALPS) at the University of Tokyo and the JSPS Research Fellowships for Young Scientists (Project No. 26.11615).
[^1]: We have compared the magnitude of $\chi_{\text{para}}$ with that of non-interacting Mn$^{3+}$/Mn$^{4+}$ local moments calculated for the Curie paramagnetic state, namely, $$\begin{aligned}
\chi_{\text{para}}^{\text{Curie}} & = \frac{S(S+1)g^2\mu_B}{3k_BT}\left( 1 - \frac{M_{\text{ferro}}}{M_0} \right),\end{aligned}$$ where $g \simeq 2.0$ is the $g$-factor, $M_0 = 3.6 \mu_B/\text{Mn}$ is the saturation magnetization of LSMO, and $1 - M_{\text{ferro}}/M_0$ is the number ratio of the paramagnetic Mn atoms.
|
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abstract: 'We show the existence of rainbow perfect matchings in $\mu n$-bounded edge colourings of Dirac bipartite graphs, for a sufficiently small $\mu>0$. As an application of our results, we obtain several results on the existence of rainbow $k$-factors in Dirac graphs and rainbow spanning subgraphs of bounded maximum degree on graphs with large minimum degree.'
author:
- 'Matthew Coulson[^1] and Guillem Perarnau[^2]'
title: Rainbow matchings in Dirac bipartite graphs
---
[^1]: School of Mathematics, University of Birmingham, UK. Email: mjc685@bham.ac.uk.
[^2]: School of Mathematics, University of Birmingham, UK. Email: g.perarnau@bham.ac.uk.
|
---
abstract: 'Motivated by recent works of Sen [@Sen:2002nu; @Sen:2002in] and Gibbons [@Gibbons:2002md], we study the evolution of a flat and homogeneous universe dominated by tachyon matter. In particular, we analyse the necessary conditions for inflation in the early roll of a single tachyon field.'
author:
- |
Malcolm Fairbairn[^1]\
Michel H.G. Tytgat[^2]\
\
Service de Physique Théorique, CP225\
Université Libre de Bruxelles\
Bld du Triomphe, 1050 Brussels, Belgium
date: 'April 9, 2002'
title: 'Inflation from a Tachyon Fluid ?'
---
Introduction
============
Despite numerous efforts, it seems difficult to reconcile string theory with the highly successfull paradigm of inflation. In this brief note, following Gibbons [@Gibbons:2002md], we investigate whether some form of Sen’s tachyonic matter [@Sen:2002nu; @Sen:2002in] might provide the necessary ingredients for a phase of inflationary expansion in the early universe.
According to Sen [@Sen:2002in], (see also [@Gibbons:2001hf]), a rolling tachyon condensate in either bosonic or supersymmetric string theory can be described by a fluid which in the homogeneous limit has energy density $$\rho = {V(T)\over\sqrt{1 - \dot T^2}}$$ and pressure \[eos\] p = - V(T) - (T) (1 - T\^2) with $T$ the tachyon field and $V(T)$ the tachyon potential. These expressions are obtained from the tachyon matter effective lagrangian \[lag\] [L]{} = - V(T) . Given the generic properties of $V(T)$ for $T\geq 0$, a most remarkable feature of Sen’s equation of state (\[eos\]) is that tachyon matter interpolates smoothly between p = - w = -1 for $\dot T=0$ and p = 0 w = 0 as $\dot T$ reaches its limiting value $\dot T=1$. As already emphasized by Gibbons [@Gibbons:2002md], if the tachyon condensate starts to roll down the potential with small intial $\dot T$, a universe dominated by this new form of matter will smoothly evolve from a phase of accelerated expansion to a phase dominated by a non-relativistic fluid. It is tempting to speculate that the latter could contribute to some new form of dark matter. However, the topic of this paper is whether or not the tachyon condensate could be relevant for inflation. (For related speculations, see for instance [@speculations][@anupam].)
The shape of the tachyon condensate effective potential depends on the system under consideration. In bosonic string theory for instance, this potential has a maximum $V=V_0$ at $T=0$, where $V_0$ is the tension of some unstable bosonic D-brane, a local minimum with $V=0$, generically at $T \rightarrow + \infty$, corresponding to a metastable closed bosonic string vacuum, and a runaway behaviour for negative $T$. An exact classical potential ([*[i.e.]{}*]{} exact to all orders in $\alpha^\prime$, but only tree level in $g_s$) encompassing these properties has been computed [@Kutasov:2000qp], \[potential\] V(T) = V\_0(1+ [T/T\_0]{}). Note that the curvature at the top of the potential (\[potential\]) is $
{d^2V/ dT^2} = - {V_0/T_0^2}
$ . As the tachyon field has dimension $[T] = E^{-1}$, if $\dot T \ll 1$, from Eq.(\[lag\]) we see that it is natural to rescale $T$ by $\sqrt{V_0} T \equiv \phi$. At $V=V_0$, the mass of the canonically normalized $\phi$ is then $M_\phi^2 =
-1/T_0^2$. In [@Kutasov:2000qp], $T_0 \sim l_s$ and $V_0$ is the tension of a bosonic Dp-brane, $V_0 \sim 1/g_s
l_s^{p+1}$. We shall see that these values of $V_0$ and $T_0$ seem marginally incompatible with inflation: the potential is simply too steep. However, the values for $V_0$ and $T_0$ required to obtain slow-roll inflation are within an order of magnitude of these values.
Tachyon matter cosmology
========================
As shown by Gibbons, for a Roberston-Walker tachyon matter dominated universe, the Friedman equation takes the standard form H\^2 = [\^2 3]{} = [\^2 3]{}[V]{} with $\kappa^2= 8 \pi G= 8 \pi /M_{pl}^2$ and where we have assumed spatial flatness[^3] and have chosen to put the cosmological constant to zero. Entropy conservation gives as usual = - 3 H (+ p) the latter being equivalent to the equation of motion for the tachyon field $T$, \[eom\] [V T1 - T\^2]{} + 3 H VT + V\^= 0 where $V^\prime = dV/dT$. We would like to use these equations to determine the slow-roll conditions for inflation. The first condition to be satisfied is that the expansion is accelerating (see for instance [@liddle] or [@Lyth:1998xn]) $$\begin{aligned}
{\ddot a\over a} \equiv H^2 + \dot H &=& - {\kappa^2 \over 6}(\rho + 3 p) \;>\; 0\nonumber\\
&=& \frac{\kappa^2 }{3}{V\over \sqrt{1 - \dot T^2}}\left (1 - {3\over 2} \dot T^2\right)\;>\; 0\end{aligned}$$ which requires that[^4] \[inflation\] T\^2 < [23]{}. In order to have a sufficiently long period of inflation, the tachyon field should start rolling with small initial $\dot T$. To relate the condition (\[inflation\]) to the shape of the potential, we have to calculate the slow-roll conditons. Following a standard procedure [@liddle], we can express the evolution of the universe as a function of $T$ rather than time. Using H = - [\^2 2]{} [V T\^2]{} and because of the monoticity of $T$ with respect to time, we can rewrite this equation as H\^(T) = - [\^2 2]{} [V T]{} where the prime denotes derivation with respect to $T$. Taking the square of the Friedmann equation we get \[HJ\] [H\^]{}\^2 - [94]{} H\^4(T) + [\^4 4]{} V(T)\^2 = 0. This first order differential equation is the analog for tachyon matter of the Hamilton-Jacobi form of the Friedmann equation with a single inflaton field [@Salopek:1990jq]. It expresses the fact that $H^\prime$ is negligible as long as H\^2 V. Solving (\[HJ\]) for $H(T)$, we can then get $\dot T$ as a function of $T$ = 1 - ([\^2 V3 H\^2]{})\^2. Using the potential (\[potential\]) one can solve (\[HJ\]) numerically. We have chosen the field $T=T_i$ to be slightly displaced from the maximum of the potential with initial $\dot T_i=0$. We also assume $T_i>0$ for obvious reasons. The amount of inflation obtained depends upon the variable $T/T_0$ and on a dimensionless parameter $X_0$ which characterizes the flatness of the potential close to its peak X\_0\^2 = \^2 T\_0\^2 V\_0. In Figure \[dotT2\] we plot $\dot T^2$ for different values of $X_0$.
The pertinent feature of this figure is that for increasing $X_0$, inflation ends at increasing values of $T$. This is equivalent to requiring that as the rolling of the field commences, the $\ddot T$ term is negligible in the equation of motion for the tachyon field (\[eom\]), so that 3 H V T - V\^. \[slow\] Another quantity of interest is the number of e-folds during the inflationary phase, N(T) = \_t\^[t\_[end]{}]{} H dt = - \_T\^[T\_[end]{}]{} [H\^2 VV\^]{} dT This is shown in Figure \[efolds\]. To successfully use inflation to solve the horizon problem and bring us a suitably flat spectrum of perturbations we require $N~\geq~50-60$ e-folds of inflation whilst the field is rolling. As always, there is some uncertainty in the number of e-folds related to the choice of initial conditions. However, if we assume that $\dot T^2$ is small initially and that $T/T_0$ starts at less than about $0.1$ then the total number of e-folds becomes insensitive to the exact initial conditions. We choose the value of $X_0$ so as to accommodate enough e-folds and the normalization to the COBE spectrum, which we shall describe in the next section. For the time being, we simply note that inflation lasts longer if $X_0$ is larger, as shown in Figure \[dotT2\].
Once inflation is over the expansion of the tachyon-matter dominated universe slows rapidly which precludes the possibility of the modes generated during the slow roll regime being pushed outside our present day horizon.
Slow-rolling tachyon matter
===========================
The numerical analysis shows that inflation requires that $X_0 \gsim 3$. Once this is satisfied the field $T$ evolves in a framework analogous to the slow-roll approximation for scalar fields. In order for the conditions H\^2 V and H V T - V\^to hold, the following inequalities must be satisfied: \[epsilon\] 1 and \[eta\] - [V”\^2 V\^2]{} 1. These are just the usual definitions of the slow-roll parameters $\epsilon$ and $\eta$. Their form is different from the usual expressions because we are taking derivatives with respect to $T$. Near $V_0$, for small $\dot T$, one can use the canonically normalized field $\phi = \sqrt{V_0} T$ which brings (\[epsilon\]) and (\[eta\]) to their usual form [@liddle]. As we assume that the tachyon is rolling from the top of the potential, generically $\epsilon \ll \eta$ for small $T/T_0$, so that the second condition is the most stringent one. Using the potential (\[potential\]) to be specific, (\[eta\]) becomes \^2 V\_0 T\_0\^2 X\_0\^2 1 \[condition1\] a condition which is consistent with the numerical analysis of the preceding section.
Finally we must estimate the size of the fluctuations and compare this estimate with the magnitude of the density pertubations observed by COBE. Assuming that the slow-roll approximation can be made to hold over a significant range of cosmic scales, the spectral index $n(k)$ is taken to be scale independent with n 1 - 2 1. \[spectrum\] We also follow the standard procedure to estimate the density perturbations ||T where . We then use the slow roll condition ($\ref{slow}$) to eliminate $V^{\prime}$ leading to the expression || and we obtain the expression for the density perturbation during inflation when $T/T_0$ is of order one || which correponds to roughly 60 efolds before the end of inflation for $5\lsim X_0 \lsim 7$. Since $\delta\rho/\rho \approx 2\times 10^{-5}$ we see that \^[3]{} V\_0 T\_0 610\^[-4]{}. \[condition2\] In order to calculate the magnitude of our perturbations, we assumed that $\dot{T}$ is negligible and that we are close to the peak of the potential so that we can substitute $T$ for the canonical scalar field $\sqrt{V_0}\phi$. However, without a full derivation of the perturbations starting with the Lagrangian ($\ref{lag}$) we cannot say at what point our approximations break down, and what kind of behaviour replaces the normal lore at that point.
Discussion
==========
We have not mentioned the precise origin of the tachyon field in question, but given the string theory origin of Sen’s equation of state, it is tempting to write the parameters $V_0$ and $T_0$ in terms of the string length $l_s$ and the open string coupling constant $g_s$, V\_0=,T\_0=\_0 l\_s. where $v_0$ and $\tau_0$ are dimensionless parameters such that $V_0/v_0$ is the tension of a D3-brane and $\tau_0 l_s$ is the inverse tachyon mass[@strings]. The gravitational coupling in 4 dimensions is given in terms of the stringy parameters by $$\kappa^2\equiv 8 \pi G_N =\pi g_s^2 l_s^2\left(\frac{l_s}{R}\right)^{6}.$$ Here $R$ is the compactification radius of the compact 6 dimensional manifold, taken here to be a 6-torus.[^5] For the $D=4$ effective theory to be applicable one usually requires that $ R \gg l_s$ since $R=l_s$ denotes the self T-dual point where the mass spectrum of KK and winding modes become degenerate [@strings]. The volume of the compact space therefore must satisfy the inequality $V > (2\pi R)^6$.
In order for the gravitational waves at the end of inflation to be compatible with CMB observtions, there is a further condition $\cite{linde}$ $$\frac{H_{end}}{M_{pl}}\le 3.6\times 10^{-5}$$ which together with the requirement for enough e-folds ($\ref{condition1}$) and the magnitude of the perturbations as given by equation ($\ref{condition2}$) leads to three conditions which can be written as $$\begin{aligned}
\left(\frac{l_s}{R}\right)^{12}v_0 g_s^3 &\lsim& 1.4\times 10^{-6} \;\;\; \mbox{\rm (no grav. waves)}\label{ngw}\\
\left(\frac{l_s}{R}\right)^{9}v_0 \tau_0 g_s^2 &\sim& 2.7\times 10^{-2}\;\;\; \mbox{\rm(COBE normalization)}\\
\left(\frac{l_s}{R}\right)^{6}v_0 \tau_0^2 g_s &\gsim& 7.1\times 10^{2} \;\;\;\mbox{\rm (inflationary cond.)}\label{ic}\end{aligned}$$ These conditions cannot all be satisfied simultaneously with the values one would choose as a first guess, $v_0 \sim \tau_0 \sim 1$, for any $R \gsim l_s$.[^6]
For $R \sim l_s$ and insisting on $v_0 \sim 1$, (\[ngw\]) gives $g_s \lsim 10^{-2}$ while (\[ic\]) imposes $\tau_0 \gsim 10^{2}$, corresponding to a rather light tachyon mass $m_T \lsim 10^{-2}/l_s$. This is phenomenologically the simplest solution, but is at odds with expectations from string theory. (See however [@Kim:2002rv].)
If one insists on a tachyon mass $\sim 1/l_s$, the only alternative without entering the strong coupling regime is to increase the energy density, for instance by increasing the number of branes, in the spirit of [@anupam] where the potential arises from the combined tachyonic potential of a number of brane/anti-brane systems. The problem here is that in order for each individual brane to act in the way described by Sen’s effective action the typical seperation between branes within the compact directions would have to be larger than $l_s$.[^7] A miminal, although [*ad hoc*]{}, setting is to take $v_0$ to be equal to the number of string length size volumes within the compact space, i.e. $$v_0=\left(\frac{2\pi R}{l_s}\right)^6$$ Then it is possible to fulfill the above requirements with $g_s\sim 2\times 10^{-2}$, $R/l_s\sim 10$, corresponding to a very large number of branes ($\sim 10^{11}$). This initial condition is quite baroque and at first sight rather unattractive. However we take note that such a large number is not foreign to speculations on the dS/CFT correspondence [@Strominger:2001pn], which suggests that the number of light degrees of freedom of the putative Euclidean CFT dual to the inflationary phase of the universe should be very large $c \sim 10^8$ [@Larsen:2002et].
One last issue we would like to comment on is that of reheating at the end of inflation. In this paper our approach has been rather phenomenological. We took Sen’s equation of state together with a string-theory motivated effective potential. Both ingredients rest on quite severe approximations, but are supposed to capture at least some of the physics of tachyon condensation. Remarkable features are that 1) the minimum of the tachyon potential is at $T \rightarrow \infty$ and that 2) the energy of the brane stays confined to its hyperplane in the form of a pressureless fluid. As far as we understand, the first feature is supposed to be generic. As has been recently emphasized by Kovman and Linde [@linde], this is quite worrisome for the purpose of reheating the universe at the end of inflation. They also note that this is potentially a problem for [*all*]{} string-inspired models of inflation based on the annihilation or decay of branes.
From a phenomenological point of view the simplest resolution would be to have a potential that vanishes at finite $T$ so that reheating can proceed through oscillations of the tachyon fluid around its minimum. Either way, this raises the issue of the decay of the tachyon fluid. The standard lore in the string-community is that unstable D-branes should ultimately decay into closed string modes. To make this process manifest one should take into account $g_s$ corrections to the tachyon effective action, a program which has not been completed yet. We believe that a related issue will be to understand the precise nature of the tachyon fluid at the minimum of the potential. It is expected to be stable only in the limit $g_s \rightarrow 0$[@Sen] and should in principle be a good zeroth order approximation to the problem of tachyon matter decay.
An intruiging characteristic of Sen’s effective action is that at the minimum of the potential, the tachyon fluid equations of motions reduced to, in Hamiltonian form and in flat space-time, [@Sen:2002an] $$\begin{aligned}
1 &=& (\dot T)^2 - (\partial_i T)^2\\
\dot \Pi &=& \partial_i \left(\Pi {\partial_i T\over \dot T}\right) \end{aligned}$$ where $\Pi$ is the momentum conjugate to $\dot T$. These are precisely the equation of motion of a relativistic pressureless fluid of partons of mass unity, with velocity field $\vec v = \nabla T/\dot T \equiv \vec k/\omega(k)$ and energy density $\Pi(x,t)$. Solutions of these equations of motions are quite trivial to work out. A time-independent but otherwise arbitrary configuration of the energy-density $\Pi(x)$ corresponds in the parton picture to a bunch of massive partons [*at rest*]{}. However, generic solutions have caustic singularities and the free parton picture has probably a limited range. It makes nevertheless clear why there are no propagating tachyon waves at the minimum of the potential, simply because there are no collective excitations in a pressureless fluid.
In the limit of $g_s\rightarrow 0$, the tachyon fluid is stable and behaves as some new form of dark matter. The problem with this is that the universe would never become radiation dominated and for tachyon matter to be relevant in cosmology requires some fine-tuning [@Shiu:2002qe; @linde]. For finite $g_s$ however tachyon matter should be unstable and decay. If we are willing to take the parton picture seriously, it is plausible that their decay could solve the issue of reheating of the universe at the end of tachyon rolling, however, not being string theorists, we have no idea how this question could be addressed. As an illustration we simply parametrize the decay rate as $$\Gamma \sim \frac{g_s^n}{l_s} \sim \left(\frac{l_s}{R}\right)^3 g_s^{n+1}M_{pl}$$ The reheat temperature is then (presumably over-) estimated to be T\_[RH]{}\~ \~()\^g\_s\^M\_[pl]{} which for generic powers of $n$ is high enough for most cosmological puposes. Obviously it would be interesting, both for string theory and for phenomenological applications to develop further insights on this issue.
Conclusions
===========
In this paper we have investigated the possibility of using Sen’s equation of state to see if we can obtain viable cosmological inflation. We have described the conditions which the potential and the tachyon mass must fulfil in order to provide enough e-folds of inflation and density perturbations of the correct magnitude. Our approach has been essentially phenomenological. In particular, the conditions for slow-roll are essentially independent of the precise shape of the potential and whether it vanishes at finite or infinite $T$. One attractive feature of Sen’s equation of state is that it give an explicit realization of so-called k-inflation [@Armendariz-Picon:1999rj]. A related interesting property is that the exit from inflation is automatic for generic string-inspired effective potentials due to the limiting value $\dot T= 1$ for homogeneous configurations. As already emphasized numerous times a straightforward matching with string theory is problematic but perhaps not altogether impossible. An important issue, both for cosmological applications and, we presume, for string theory is to understand the ultimate fate of the tachyon matter at the bottom of the potential.
Acknowledgements {#acknowledgements .unnumbered}
================
We thank Laurent Houart, Samuel Leach and Ashoke Sen for useful conversations or comments.
[99]{}
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J. Polchinski, “String Theory.” [*Cambridge, UK: Univ. Pr. (1998)*]{}. C. V. Johnson, “D-brane primer,” arXiv:hep-th/0007170. H. s. Kim, “Supergravity approach to tachyon potential in brane-antibrane systems,” arXiv:hep-th/0204191. P. Horava, “Type IIA D-branes, K-theory, and matrix theory,” Adv. Theor. Math. Phys. [**2**]{} (1999) 1373 \[arXiv:hep-th/9812135\].
A. Strominger, “The dS/CFT correspondence,” JHEP [**0110**]{} (2001) 034 \[arXiv:hep-th/0106113\]; “Inflation and the dS/CFT correspondence,” JHEP [**0111**]{} (2001) 049 \[arXiv:hep-th/0110087\]; M. Gutperle and A. Strominger, “Spacelike branes,” JHEP [**0204**]{} (2002) 018 \[arXiv:hep-th/0202210\]. F. Larsen, J. P. van der Schaar and R. G. Leigh, “de Sitter holography and the cosmic microwave background,” JHEP [**0204**]{} (2002) 047 \[arXiv:hep-th/0202127\]; E. Halyo, “Holographic inflation,” arXiv:hep-th/0203235. L. Kofman and A. Linde, “Problems with tachyon inflation,” arXiv:hep-th/0205121.
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[^1]: mfairbai@ulb.ac.be
[^2]: mtytgat@ulb.ac.be
[^3]: As tachyon matter turns into a non-relativistic fluid for large $T$, a closed Universe will eventually recollapse. We assume flatness for simplicity. Alternatively, the recollapse can be supposed to be in the far future of our model universe.
[^4]: This condition for inflation is in contrast to that obtained for a normal scalar field, $\dot \phi^2 < V(\phi)$.
[^5]: Like in most higher dimensional scenarios, we assume that some unknown mechanism freezes the moduli associated to the extra dimensions. Also, for simplicity we assume that we can neglect the evolution of other fields, like the dilaton.
[^6]: In the first version of this paper we overlooked a factor of $g_s$ in the expression of the Planck mass which led us to underestimate strong gravity effects.
[^7]: If the branes are coincident, there are extra tachyonic degrees of freedom which all together transform either in the adjoint of the gauge group that lives on a stack of non-BPS D-brane or in the bi-fundamental for $Dp-\bar Dp$ pairs [@Horava:1998jy].
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abstract: 'We report high angular resolution (3$\arcsec$) Submillimeter Array (SMA) observations of the molecular cloud associated with the Ultra-Compact HII region G5.89-0.39. Imaged dust continuum emission at 870$\mu$m reveals significant linear polarization. The position angles (PAs) of the polarization vary enormously but smoothly in a region of 2$\times$10$^{4}$ AU. Based on the distribution of the PAs and the associated structures, the polarized emission can be separated roughly into two components. The component “x” is associated with a well defined dust ridge at 870 $\mu$m, and is likely tracing a compressed B field. The component “o” is located at the periphery of the dust ridge and is probably from the original B field associated with a pre-existing extended structure. The global B field morphology in G5.89, as inferred from the PAs, is clearly disturbed by the expansion of the HII region and the molecular outflows. Using the Chandrasekhar-Fermi method, we estimate from the smoothness of the field structures that the B field strength in the plane of sky can be no more than 2$-$3 mG. We then compare the energy densities in the radiation, the B field, and the mechanical motions as deduced from the C$^{17}$O 3-2 line emission. We conclude that the B field structures are already overwhelmed and dominated by the radiation, outflows, and turbulence from the newly formed massive stars.'
author:
- 'Ya-Wen Tang'
- 'Paul T. P. Ho'
- Josep Miquel Girart
- Ramprasad Rao
- Patrick Koch
- 'Shih-Ping Lai'
title: 'Evolution of Magnetic Fields in High Mass Star Formation: SMA dust polarization image of the UCHII region G5.89-0.39'
---
Introduction
============
One of the main puzzles in the study of star formation is the low star formation efficiency in molecular clouds. Since molecular clouds are known to be cold, the thermal pressure is small. Hence, if there are no other supporting forces against gravity, the free-fall time scale will be short and the star formation rate will be much higher than what is observed. Magnetic (B) fields have been suggested to play the primary role in providing a supporting force to slow down the collapsing process (see the reviews by Shu et al. (1999) and Mouschovias & Ciolek (1999)). In these models, the B field is strong enough and has an orderly structure in the molecular cloud. The B field lines, which are anchored to the ionized particles, will then be dragged in along the direction of accretion, only when the ambipolar diffusion process allows the neutral component to slip pass the ionized component. In the standard low-mass star formation model (Galli & Shu 1993; Fiedler & Mouschovias 1993), an hourglass-like B field morphology is expected with an accreting disk near the center of the pinched field. Alternatively, turbulence has also been suggested as a viable source of support against contraction (see reviews by Mac Low & Klessen (2004) and Elmegreen & Scalo (2004)). The relative importance of B field and turbulence continues to be a hot topic as the two methods of support will lead to different scenarios for the star formation process.
Compared with the low mass stars, the formation process of high mass stars is really poorly understood. High mass star forming regions, because of their rarity, are usually at larger distances and are always located in dense and massive regions, because they are typically formed in a group. Hence, both poor resolution and complexity have hampered past observational studies. Furthermore, the environments of high mass star forming regions are very different from the low mass case because of higher radiation intensity, higher temperature, and stronger gravitational fields. Will the B fields in massive star forming sites have a similar morphology to the low mass cases?
Polarized emission from dust grains can be used to study the B field in dense regions, because the dust grains are not spherical in shape. They are thought to be aligned with their minor axes parallel to the B field in most of the cases, even if the alignment is not magnetic (Lazarian 2007). Due to the differences in the emitted light perpendicular and parallel to the direction of alignment, the observed thermal dust emission will be polarized, the direction of polarization is then perpendicular to the B field. Although the alignment mechanism of the dust grains has been a difficult topic for decades (see review by Lazarian 2007), the radiation torques seem to be a promising mechanism to align the dust grains with the B field (e.g., Draine & Weingartner 1996; Lazarian & Hoang 2007). However, other processes such as mechanical alignments by outflows can also be important.
Polarized dust emission has been detected successfully at arcsecond scales. The best example might be the low mass star forming region NGC 1333 IRAS 4A (Girart, Rao & Marrone 2006), which reveals the classic predicted hourglass B field morphology. Results on the massive star formation regions, such as W51 e1/e2 cores (Lai et al. 2001), NGC2024 FIR5 (Lai et al. 2002), DR21 (OH) (Lai et al. 2003), G30.79 FIR 10 (Cortes et al. 2006) and G34.4+0.23 MM (Cortes et al. 2008), typically show an organized and smooth B field morphology. However, this could be due to a lack of spatial resolution. Indeed, for the nearby high mass cases such as Orion KL (Rao et al. 1998) and NGC2071IR (Cortes, Crutcher, & Matthews 2006), abrupt changes of the polarization direction on small physical scales have been seen, which may suggest mechanical alignments by outflows as proposed by these authors. Whether high mass star forming regions will all show complicated B field structures on small scales remains to be examined.
In this study, we report on one of the first SMA measurements of dust polarization for a high mass star forming region, G5.89-0.39 (hereafter, G5.89). The linearly polarized thermal dust emission is used to map the B field at $\sim$3$\arcsec$ resolution, and the C$^{17}$O 3-2 line is used to study the structure and kinematics of the dense molecular cloud. The description of the source, the observations and the data analysis, the results, and the discussion are in Sec. 2, 3, 4 and 5, respectively. The conclusions and summary are in Sec. 6.
Source Description
==================
G5.89 is a shell-like ultracompact HII (UCHII) region (Wood et al. 1989) at a distance of 2 kpc (Acord et al. 1998). The UCHII region is 0.01 pc in size, and its dynamical age is 600 years, estimated from the expansion velocity (Acord et al. 1998). Observations of the K$_{s}$ and $L'$ magnitudes and color by Feldt et al. (2003) suggest that G5.89 contains an O5 V star.
Just as in other cases of massive stars, G5.89 contains most likely a cluster of stars. The detections of associated H$_{2}$O masers (Hofner & Churchwell 1996), OH masers (Stark et al. 2007; Fish et al. 2005) and class I CH$_{3}$OH masers (Kurtz et al. 2004), suggest that multiple stars have formed in this region. Furthermore, the morphology of the detected molecular outflows also suggest the presence of multiple driving sources, because different orientations are observed in different tracers. In CO 1-0, the large scale outflow is almost in the east-west direction (Harvey & Forveille 1988; Watson et al. 2007). In C$^{34}$S and the OH masers, the outflow is in the north-south direction (Cesaroni et al. 1991; Zijlstra et al. 1990). In SiO 5$-$4, the outflow is at a position angle (PA) of 28$\degr$ (Sollins et al. 2004). In the CO 3-2 line, the outflows (Hunter et al. 2008) are in the north-south direction and at the PA of 131$\degr$, and the latter one is associated with the Br$\gamma$ outflow (Puga et al. 2006). In addition, the detected 870 $\mu$m emission has also been resolved into multiple peaks (labelled in Fig. 1(a); Hunter et al. 2008). The different masers, the multiple outflows, and the multiple dust peaks, are all consistent with the formation of a cluster of young stars.
G5.89 should be expected to have a substantial impact on its environment. In terms of the total energy in outflows in this region, G5.89 is definitely one of the most powerful groups of outflows ever detected (Churchwell 1997).
Observation and Data Analysis
=============================
The observations were carried out on July 27, 2006 and September 10, 2006 using the Submillimeter Array (Ho, Moran & Lo (2004))[^1] in the compact configuration, with 7 of the 8 antennas available for both tracks. The projected lengths of the baselines ranged from 6.5 to 70 k$\lambda$ ($\lambda\approx$870$\mu$m). Therefore, our observational results are insensitive to structures larger than 39$\arcsec$. The SMA receivers are intrinsically linearly polarized and only one polarization is available at the current time. Thus, quarter-wave plates (see Marrone & Rao 2008) were installed in order to convert the linear polarization (LP) to circular polarization (CP). The quarter-wave plates were rotated by 90$ \degr$ on a 5 minutes cycle using a Walsh function to switch between 16 steps in order to sample all the 4 Stokes parameters. The integration time spent on the source in each step was approximately 15 seconds. The overhead required in switching between the different states was approximately 5 seconds. In each cycle all four cross-correlations (LL, LR, RL, and RR) were each calculated 4 times. The data were then averaged over this complete cycle in order to obtain quasi simultaneous dual polarization visibilities. We assume that the smearing due to the change of the polarization angles on this time scale is negligible.
The local oscillator frequency was tuned to 341.482 GHz. With a 2 GHz bandwidth in each sideband we were able to cover the frequency range from 345.5 to 347.5 GHz and from 335.5 to 337.5 GHz in the upper and lower sideband, respectively. The correlator was set to a uniform frequency resolution of 0.65 MHz ($\sim$ 0.7 km s$^{-1}$) for both sidebands. While our main emphasis was to map the polarized continuum emission from the dust, we were also able to detect a number of molecular lines simultaneously. These results will be published separately.
Generally, the conversions of the LP to CP of the receivers are not perfect. This non-ideal characteristic of the receiver will cause an unpolarized source to appear polarized, which is known as instrumental polarization or leakage. Nevertheless, these leakage terms (see Sault, Hamaker, & Bregman 1996) can be calibrated by observing a strong linearly polarized quasar. In this study, the leakage and bandpass were calibrated by observing 3c279 for the first track and 3c454.3 for the second track. Both sources were observed for 2 hours while they were transiting in order to get the best coverage of parallactic angles. The leakage terms are frequency dependent, $\sim$1% and $\sim$3% for the upper and lower sideband before the calibration, respectively. After calibration, the leakage is less than 0.5$\%$ in both sidebands. Besides the calibration for the polarization leakage, the amplitudes and phases were calibrated by observing the quasars 1626-298 and 1924-292 every 18 minutes. These two gain calibrators in both tracks were used because of the availabilities of the calibrators during the observations. Finally, the absolute flux scale was calibrated using Callisto.
The data were calibrated and analyzed using the MIRIAD package (Sault, Teuben, & Wright 1995). After the standard gain calibration, self-calibration was also performed by selecting the visibilities of G5.89 with uv distances longer than 30 k$\lambda$. As a result, the sidelobes and the noise level of the Stokes $I$ image were reduced by a factor of 2. In order to get the images from the measured visibilities, the task INVERT in the MIRIAD package was used. The Stokes $Q$ and $U$ maps are crucial for the derivation of the polarization segments. We used the dirty maps of $Q$ and $U$ to derive the polarization to avoid a possible bias introduced from the CLEAN process. The Stokes $I$ map shown in this paper is after CLEAN.
The Stokes $I$, $Q$ and $U$ images of the continuum were constructed with natural weighting in order to get a better S/N ratio for the polarization. The final synthesized beam is $3\arcsec.0 \times1\arcsec.9$ with the natural weighting. The C$^{17}$O images are presented with a robust weighting 0.5 in order to get a higher angular resolution, and the synthesized beam is 2.8$\arcsec \times$1.8$\arcsec$ with PA of 13$\degr$. The noise levels of the $I$, $Q$ and $U$ images are $\sim$ 30, 5 and 5 mJy Beam$^{-1}$, respectively. Note that the noise level of the Stokes $I$ image is much larger than the ones in the Stokes $Q$ and $U$ images. The large noise level of the $I$ image is most likely due to the extended structure, which can not be recovered with our limited and incomplete uv sampling. The strength (I$_{p}$) and percentage ($P$) of the linearly polarized emission is calculated from: $I_{p}^{2} = Q^2 +U^2-\sigma_{Q,U}^{2}$ and $P$ = I$_{p}$/I, respectively. The term $\sigma_{Q,U}$ is the noise level of the Stokes $Q$ and $U$ images, and it is the bias correction due to the positive measure of I$_{p}$. The noise of I$_{p}$ ($\sigma_{I_p}$) is thus 5 mJy Beam$^{-1}$. The presented polarization is derived using the task IMPOL in the MIRIAD package, where the bias correction of $\sigma_{I_p}$ is included.
Results
=======
In this section, we present the observational results of the dust continuum and the dust polarization at 870$\mu$m, and the C$^{17}$O 3-2 emission line. No polarization was detected in the CO 3-2 emission line.
Continuum Emission
------------------
The total continuum emission at 870 $\mu$m, shown in Fig. 1(a), is resolved with a total integrated flux density 12.6$\pm$1.3 Jy. In general, the morphology of the continuum emission at 870$\mu$m is similar to the emission at 1.3 mm by Sollins et al. (2004). However, the 870$\mu$m emission peaks at $\sim$ 1$\arcsec$ west of the position of the O5 star, which is offset toward the north-west by $\sim$1$\arcsec$.7 from the peak of the 1.3 mm continuum emission. Because there is still a significant contribution from the free-free emission to the continuum at 870 $\mu$m and at 1.3mm, the differences between the 870 $\mu$m and 1.3 mm maps most likely result from the increasing contribution from the dust emission as compared to the free-free emission at shorter wavelengths. Due to the importance of a correct dust continuum image in the derivation of the polarization, we describe here how the free-free continuum was estimated and removed from the 870$\mu$m total continuum emission.
### Removing the free-free emission
The free-free continuum at 2cm (shown in color scale in Fig.1 (a) and (b)) was imaged from the VLA archival database observed on August 7, 1986. The VLA synthesized beam of the 2cm free-free image is 0$ \arcsec.92\times$0$\arcsec$.45 with natural weighting of the uv data. Since the free-free shell is expanding at a rate of 2.5 mas year$^{-1}$ (Acord et al. 1998), at a distance of 2 kpc, this expansion motion over the intervening 20 years is negligible within the synthesized beam of our SMA observation.
The contribution from the free-free continuum was removed by the following steps. Firstly, we adopted a spectral index $\alpha=-$0.154 calculated in Hunter et al. (2008) for the free-free continuum emission between 2cm to 870$\mu$m. The resulting estimated free-free continuum strength at 870$\mu$m was 4.9 Jy. Secondly, we further assumed that the morphology of the free-free continuum at 870 $\mu$m and at 2cm were identical. We then smoothed the VLA 2cm image to the SMA resolution and scaled the total flux density to 4.9 Jy. Finally, we subtracted this image from the total continuum at 870 $\mu$m. The resultant 870 $\mu$m dust continuum image is shown in Fig. 1(b). The total flux density of the dust continuum is therefore 7.7$\pm$0.8 Jy.
### Dust continuum: mass and morphology
The corresponding gas mass (M$_{gas}$) was calculated from the flux density of the dust continuum at 870 $\mu$m following Lis et al. (1998):
$$\label{mh2}
M_{gas} = \frac{2\lambda^{3} Ra \rho d^{2}}{3hcQ(\lambda)J(\lambda,T_d)}S(\lambda)$$
Here, we assumed a gas-to-dust mass ratio $R$ of 100, a grain radius $a$ of 0.1 $\mu$m, a mean grain mass density $\rho$ of 3 g cm$^{-3}$, a distance to the source $d$ of 2 kpc, a dust temperature $T_{d}$ of 44 K, an observed flux density S$(\lambda)$ of 7.7 Jy, the Planck factor $J(\lambda, T_{d})=[exp(hc/\lambda k
T_{d})-1]^{-1}$. $h$, $c$ and $k$ are the Planck constant, the speed of light and the Boltzmann constant, respectively. The grain emissivity $Q$($\lambda$) was estimated to be $1.5\times 10^{-5}$ after assuming $Q(350\mu m)$ of $7.5\times 10^{-4}$ and $\beta$ of 2 (cold dust component), and using the relation $Q(\lambda)=Q(350\mu m)(350\mu m/\lambda)^{\beta}$ (Hunter et al. 2000). As suggested in the same paper, the dust emission can be modeled by two temperature components, with the emission dominated by the colder component at T$_{d}$ $\sim$ 44 K. We adopted this value for T$_{d}$, and therefore, the mass given here refers only to the cold component and is an underestimate of the total mass. The derived gas mass of the dust core M$_{gas}$ is $\sim$ 300 M$_{\sun}$, with a number density $n_{H_{2}}=$ 5.3$\times$10$^{6}$ cm$^{-3}$ averaged over the emission region. The sizescale along the line of sight is assumed to be 0.13 pc, which is the diameter of the circle with the equivalent emission area.
The dust emission presented in Fig. 1(b) has an extension toward the northeast, east and southwest and has a steep roll off on the northwestern edge of the ridge. In the higher angular resolution (0.8$\arcsec$) observation at the same wavelength by Hunter et al. (2008), the dust core is resolved into 5 peaks, where the two strongest peaks align in the north-south direction to the west of the O5 star. The dust continuum emission associated with SMA-N, SMA-1 and SMA-2 is called *sharp dust ridge* hereafter because of its strong emission and its morphology. There is no peak detected at the position of the O5 star. It is likely that the O5 star is located in a dust-free cavity, as proposed by Feldt et al. (1999) and Hunter et al. (2008).
Dust polarization
-----------------
We first compare the dust polarization derived from the 870 $\mu$m total continuum (Fig. 1(c)) and from the 870 $\mu$m dust continuum (Fig. 1(d)). In both cases the derived polarization is at the same location with the same PAs. The only difference of the polarization in Fig. 1(c) and 1(d) is that the percentage of polarization near the HII region is increased in Fig. 1(d). This is because of the fact that the free-free continuum is not polarized, and the $Q$ and $U$ components are not affected by the free-free continuum subtraction. Therefore, the expected polarization percentage will increase when the free-free continuum is removed from the 870 $\mu$m continuum. The total detected polarized intensity I$_{p}$ is 59 mJy. All the polarization shown in the figures besides Fig. 1(c) is calculated from the derived dust continuum image. The off-set positions, percentages and PAs of the polarization segments are listed in Table 1.
### Morphology of the detected polarization
The polarized emission is not uniformly distributed. Detected polarization at 2$\sigma_{I_p}$ are shown as blue segments and detections above 3$\sigma_{I_p}$ are shown by red segments. Most of the polarized emission is located in the northern half of the dust core close to the HII region and appears as 4 patches, mostly with $\sigma_{I_p} \geq 3$ (Fig. 2(a) in color scale). There is a sharp gap where no polarization is detected extending from the NE to the SW across the O star. The southern half of the dust core is free of polarization, except for a few positions at the edge of the dust core. However, the polarization in the south half of the dust core is at 2 to 3$\sigma_{I_p}$ level only. We will focus our discussions on the more significant detections in the core of the cloud.
We separate the polarized emission into two groups. We are guided principally by the fact that one group is associated with the periphery of the total dust emission, while the other group tracks the strongest parts of the total dust emission. The polarized patches to the east of the O star and to the west of the Br$\gamma$ outflow source have similar PAs of $\sim 50
\degr$ (Fig. 2(b)). These polarization segments are located at the fainter edges of the higher resolution 870 $\mu$m dust continuum image (Fig. 2(c); Hunter et al. 2008) and at the less steep part of the 3$\arcsec$ resolution image (this paper). This may suggest that this polarization originates from a more extended overall structure, rather than from the detected condensations. Therefore, these polarization segments are suggested to be the component “o” (defined in the next section). The rest of the polarization in the northern part is all next to the sharp gap where no polarization is detected. Most of the polarization is on the 870$\mu$m *sharp dust ridge* observed with 0.8$\arcsec$ resolution, except for the ones at the NE and SW ends where the polarization patches stretch toward the extended structure. At these NE and SW ends the polarization is probably the sum of the extended and the condensed structures. These polarization segments are suggested to belong to the component “x”.
The 0.8$\arcsec$ resolution observations show that there is a hole in the southern part of the detected dust continuum. This hole is not resolved with the 3$\arcsec$ synthesized beam of our map. That may explain why polarization is not detected at this position. Here, and also for the dust ridge sharply defined with 0.8$\arcsec$ resolution, the dust polarization is sensitive to the underlying structures and can help to identify unresolved features which are smaller than our resolution.
### Distribution of the polarization segments
The detected PAs vary enormously over the entire map, ranging from $-60\degr$ to 61$\degr$ (Fig. 3(a)). Nevertheless, they vary smoothly along the dust ridge and show organized patches. We have roughly separated the polarized emission into two different components according to their locations (as discussed in Sec. 4.2.1) and their PAs. The “o” component is probably from an extended structure with PAs ranging from 33$\degr$ to 61 $\degr$. The mean PA weighted with the observational uncertainties of component “o” is 49$\pm$3$\degr$, with a standard deviation of 11$\degr$. The “x” component associated with the *sharp dust ridge* has PAs ranging from $-60\degr$ to 4$\degr$. Its weighted mean PA is $-$24$\pm$1$\degr$, with a standard deviation of 18$\degr$. If the polarization were not separated into two components, the weighted mean PA is $-$9$\degr$ with a standard deviation of 39$\degr$.
The relation between the percentage of polarization and the intensity is shown in Fig. 3(b). The percentage of polarization decreases towards the denser regions, which has already been seen for other star formation sites, such as the ones listed in Sec. 1. This is possibly due to a decreasing alignment efficiency in high density regions, because the radiation torques are relatively ineffective (Lazarian & Hoang 2007). It can also be due to the geometrical effects, such as differences in the viewing angles (Gonçalves et al. 2005), or due to the results from averaging over a more complicated underlying field morphology.
C$^{17}$O 3-2 emission line
---------------------------
In order to trace the physical environments and the gas kinematics in G5.89, we choose to use the C$^{17}$O 3-2 emission line because of its relatively simple chemistry. The critical density of C$^{17}$O 3-2 is $\sim$ 10$^{5}$ (cm$^{-3}$), assuming a cross-section of 10$^{-16}$ (cm$^{-2}$) and a velocity of 1 km s$^{-1}$, and therefore, it will trace both the relative lower (n$_{H_2}$ $\sim$10$^{5}$ (cm$^{-3}$)) and higher (n$_{H_2}$$\sim$ 10$^{6}$ (cm$^{-3}$)) density regions. Although its critical density is much smaller than the estimated gas density of 5.3$\times$10$^{6}$ (cm$^{-3}$) from the dust continuum, it is apparently tracing the same regions as the dust continuum because of the similar morphology of the integrated intensity image, shown in the next section. We therefore assume that the kinematics traced by C$^{17}$O represents the bulk majority of the molecular cloud and that it is well correlated with the dust continuum.
### Morphology of C$^{17}$O 3-2 emission
The emission of the C$^{17}$O 3-2 line covers a large velocity range, from $-$7 to 28 km s$^{-1}$, as shown in the channel maps in Fig. 4. The majority of the gas traced by the C$^{17}$O 3-2 line is relatively quiescent and has a morphology similar to the 870$\mu$m dust continuum emission. Besides the components which trace the dust continuum, an arc feature is seen in the south-east corner of the panel covering 10 to 15 km s$^{-1}$. There is no associated 870$\mu$m dust continuum detected at this location, probably due to the low total column density or mass of this feature. Another feature seen in the more quiescent gas is the clump extending towards the south of the dust core (see the panel covering 6 to 10 km s$^{-1}$ in Fig. 4). This clump has a similar morphology as seen in the 870 $\mu$m dust continuum where no polarization has been found. At the higher velocity ends, i.e. from $-$7 to $-$3 km s$^{-1}$ and from 23 to 28 km s$^{-1}$, the emission appears at the 870$\mu$m dust ridge. This suggests that at the *sharp dust ridge*, there are high velocity components besides the majority of quiescent material. Furthermore, the brightest HII features appear correlated with the strongest C$^{17}$O emission, especially at low velocities (v$_{lsr}=$ 6 to 15 km s$^{-1}$), which may point toward an interaction between the molecular gas and the HII region.
The total integrated intensity (0th moment) image (Fig. 5(*upper-panel*)) of the C$^{17}$O 3-2 emission line shows a similar morphology as the 870 $\mu$m dust continuum. The morphology of the C$^{17}$O gas to the west of the O star is similar to the dense dust ridge, i.e. there is an extension from north to south. The steep roll off of the dust continuum in the north-west and an extension from NE to the west of the O star are also seen in C$^{17}$O. Besides these similar features to the dust continuum, a strong C$^{17}$O peak is found at position A, where no dust continuum peak is detected. This feature A likely does not have much mass, and we will not discuss its properties further in this paper.
### Total gas mass from C$^{17}$O 3-2 line
The total gas mass M$_{gas}$ in this region can be derived from the C$^{17}$O 3-2 line. This provides a complementary estimate, which is independent from the mass derived from the dust continuum in Eq. 1. Assuming that the observed C$^{17}$O 3-2 line is optically thin and in local thermal equilibrium (LTE), the mean column density $N_{C^{17}O}$ is calculated following the standard derivation of radiative transfer (see Rohlfs & Wilson 2004): $$\label{1}
N_{C^{17}O} = 1.3 \times 10^{13} \times
\frac{T_{R3-2}\triangle V}{D(n,T_{k})}$$
Here, the T$_{R3-2}\triangle$V term is the mean flux density of the entire emission region in K km s$^{-1}$. The D parameter depends on the number density $n$ and the kinetic temperature T$_{k}$ and is given by: $$\label{1}
D(n,T_{k})=f_{2}[J_{\nu}(T_{ex})-J_{\nu}(T_{bk})][1-exp(-16.597/T_{ex})],$$
where f$_2$ is the population fraction of C$^{17}$O molecules in the J$=$2 state. T$_{ex}$ and T$_{bk}$ are the excitation and background temperatures, respectively. The adopted value of D is 1.5 from the LVG calculation by Choi, Evans II $\&$ Jaffe (1993). In their calculation, this D value is correct within a factor of 2 for 10 $<$ T$_{k}$$<$ 200 K in the LTE condition. The total gas mass M$_{gas}$ is given by:
$$\label{2}
M_{gas} = \mu m_{H_{2}} d^2 \Omega \frac{N_{C^{17}O}}{X_{C^{17}O}}$$
$\mu$ is 1.3, which is a correction factor for elements heavier than hydrogen. m$_{H_2}$ is the mass of a hydrogen molecule. $d$ and $\Omega$ are the distance to the source and the solid angle of the emission, respectively. The C$^{17}$O abundance $X_{C^{17}O}$ is assumed to be [5 $\times$ 10$^{-8}$]{} (Frerking & Langer 1982; Kramer et al. 1999). The derived mean $N_{C^{17}O}$ is [2$\times$10$^{16}$ cm$^{-2}$]{}. The mean gas number density $n_{H_{2}}$ is [1.6$\times$10$^{6}$ $cm^{-3}$]{}, assuming the size of the molecular cloud is 0.13 pc along the line of sight, which is the diameter of the circle with the equivalent emission area. The derived M$_{gas}$ from the C$^{17}$O 3-2 emission is $\sim$100 M$_{\sun}$.
The gas mass calculated using the C$^{17}$O 3-2 line is a factor of 3 smaller than the value derived from the dust continuum (300 M$_{\sun}$). This difference has also been seen in the C$^{17}$O survey towards the UCHII regions by Hofner et al. (2000). Their M$_{gas}$ estimated from the measurement of the C$^{17}$O emission tends to be a factor of 2 smaller than the measurement from the dust continuum. The uncertainty of the estimate here possibly results from the assumptions of the dust emissivity, the gas to dust ratio, the abundance of the C$^{17}$O, and from the possibility that C$^{17}$O might not be entirely optically thin.
Discussion
==========
We discuss the possible reasons of the non-detected polarization in the CO 3-2 line in the next paragraph. In order to interpret our results, we have also analyzed the kinematics of the molecular cloud in G5.89 using the C$^{17}$O 3-2 1st and 2nd moment images, the position velocity (PV) diagrams, and the spectra at various positions. The strength of the B field inferred from the dust polarization is calculated using the Chandrasekhar-Fermi method. A possible scenario of the dust polarization is discussed based on the calculation of the mass to flux ratio and the energy density.
CO 3-2 polarization
-------------------
Under the presence of the B field, the molecular lines can be linearly polarized if the molecules are immersed in an anisotropic radiation field and the rate of radiative transitions is at least comparable with the rate of collisional transitions. This effect is called the Goldreich-Kylafis (G-K) effect (Goldreich & Kylafis (1981); Kylafis (1983)). The G-K effect provides a viable way to probe the B field structure of the molecular cores, because the polarization direction is either parallel or perpendicular to the B field. The degree of the polarization depends on several factors: the degree of anisotropy; the ratio of the collision rate to the radiative rate; the optical depth of the line; and the angle between the line of sight, the B field, and the axis of symmetry of the velocity field. In general, the maximum polarization occurs when the line optical depth is $\sim$ 1 (Deguchi & Watson 1984). Although the predicted polarization can be as high as 10%$-$20%, the G-K effect is only detected in a limited number of star formation sites: the molecular outflow as traced by the CO molecular lines with BIMA in the source NGC 1333 IRAS 4A (Girart & Crutcher 1999), and the outer low-density envelope in G34.4+0.23 MM (Cortes et al. 2008), G30.79 FIR 10 (Cortes & Crutcher 2006) and DR 21(OH) (Lai et al. 2003). High resolution observations are required to separate regions with different physical conditions.
We have checked the polarization in the molecular lines. No detection in the CO 3-2 and other emission lines was found. The molecular outflows as seen in the CO 3-2 and SiO 8-7 emission lines will be shown in Tang et al. (in prep.). We briefly discuss the possible reasons for the lack of polarization in the molecular lines here.
One possible reason is the high optical depth ($\tau$) of the CO 3-2 line. It has been shown by Goldreich and Kylafis (1981) that the percentage of polarization depends on the value of $\tau$, decreasing rapidly as the line becomes optically thick. When corrected for multi-level populations, Deguchi & Watson (1984) suggested that the percentage of polarization decreases further by about a factor of 2. In G5.89, $\tau$ of the CO 3$-$2 line is $\sim$10 at v$_{lsr}$ = 25 km s$^{-1}$ (Choi et al. 1995), which is the channel where the emission is strongest in our SMA observation. Note that this emission does not peak at the systematic velocity ($v_{sys}$) of 9.4 km s$^{-1}$, which is most likely due to the missing extended structure which our observation cannot reconstruct. We then estimate that the expected percentage of polarization will be about 1.5$\%$, or a polarized flux density of 0.5 Jy Beam$^{-1}$ for the CO 3-2 line, which is below our sensitivity.
Besides an optimum $\tau$, the anisotropic physical conditions, such as the velocity gradient and the density of the molecular cloud, are needed to produce a polarized component from the spectral line. The fraction and direction of polarization will also change as a function of $\tau$ if there are external radiation sources nearby (Cortes et al. 2005). Here, we are not able to distinguish between these possible reasons.
The kinematics traced by C$^{17}$O 3-2 emission line
----------------------------------------------------
As shown in Sec. 4.3.1, high velocity components of the molecular gas are traced by the C$^{17}$O 3-2 emission near the HII region. Here, we examine the kinematics in G5.89.
The intensity weighted velocity (1st moment) image provides the information on the line-of-sight motion (mean velocity). The molecular cloud is red-shifted with respect to the v$_{sys}$ of 9.4 (km s$^{-1}$) in the NW (position $B$) and SE (position $D$) of the O5 star (middle panel of Fig. 5). Next to the south of the O5 star, a blue-shifted clump with respect to 9.4 km s$^{-1}$ is detected. The molecular cloud in G5.89 has significant variations in mean velocity within a radius of 5$\arcsec$ around the O5 star.
To further investigate the relative motions, the total velocity dispersion $\delta v_{total}$ (2nd moment) image is also presented (Fig. 5 (*lower-panel*)). $\delta v_{total}$ is related to the spectral linewidth at full-width half maximum (FWHM) for a Gaussian line profile: FWHM $=$ 2.355$\delta v_{total}$. Around the HII region in G5.89, $\delta v_{total}$ has a maximum of $\sim$ 6 km s$^{-1}$ (FWHM $\sim$ 14 km s$^{-1}$) near the O5 star and decreases in the regions away from the O5 star. In terms of mean velocity and velocity dispersion, the molecular gas near the HII region is clearly disturbed. Besides the feature near the HII region, the velocity dispersion along the *sharp dust ridge* is larger and has a correspondent extension (NE$-$SW), which suggests that the molecular cloud along the *sharp dust ridge* is more turbulent (see also Sec. 4.3.1). This enhanced turbulent motion supports our separation of the polarized emission into component “o” and “x” in Sec. 4.2.2. These two polarized components are most likely tracing different physical environments.
The PV plots cut at various PAs at the position of the O5 star and cut along the extension in the NE and SW direction on the 2nd moment image (white segments on the lower panel of Fig. 5) are shown in Fig. 6. The strongest emission is at $v_{sys}$ with an extension of 18$\arcsec$, which suggests that the majority of the gas is quiescent. Besides the quiescent gas, a ring-like structure, indicated as red-dashed ellipses in Fig. 6, can be seen clearly, especially at the PA of 60$\degr$ to 100$\degr$. Both an infalling motion (e.g. Ho & Young 1996) and an expansion can produce a ring-like structure in the PV plots. In an infalling motion, the expected free-fall velocity is $\sim$ 5 km s$^{-1}$ for a central mass of 50 M$_{\sun}$ at a distance of 2$\arcsec$ from the central star. This is smaller than the value measured in the ring-like structure in G5.89. This C$^{17}$O 3-2 ring-like structure in the PV plots is therefore more likely tracing the expansion along with the HII region because of its high velocity ($\pm$10 km s$^{-1}$) and its dimension (2$\arcsec$ in radius). However, the ring structure is not complete. This may be because the material surrounding the HII region is not homogeneously distributed, or the HII region is not completely surrounded by the molecular gas.
Besides the expansion motion along with the HII region, there are higher velocity components extending up to 30 km s$^{-1}$ (red-shifted) and $-$5 km s$^{-1}$ (blue-shifted) (Fig. 6). The high velocity structure extending from the position of 2.5$\arcsec$ to the velocity of $\sim$30 km s$^{-1}$ is clearly seen in the PV cuts at PA of 0$\degr$ to 40$\degr$ (indicated as cyan arcs in Fig. 6). These high velocity components are probably due to the sweeping motion of the molecular outflows in G5.89, because there is no other likely energy source which can move the material to such a high velocity. From the PV plots at the position of the O5 star at various PAs and the PV plot cut along the *sharp dust ridge*, we conclude that the molecular cloud is most likely both expanding along with the HII region and being swept-up by the molecular outflows, all in addition to the bulk of the quiescent gas.
The examination of the spectra at various positions also helps to analyze the kinematics in G5.89. The spectra (Fig. 7) near the HII ring (positions $C$, $D$, $F$, $G$ and $H$) have broad line-widths. Furthermore, the spectra are not Gaussian-like, or with distinct components at high velocities ($\pm$ 10 km s$^{-1}$). At position $F$ and $H$, both spectra show a strong peak at v$_{sys}$. The high velocity wing at the position $F$ is red-shifted, and it is blue-shifted at position H. This is consistent with the NS molecular outflow. The molecular gas near the positions $E$ and $I$ is more quiescent because of its narrow linewidth. The spectrum taken at position $I$ has a FWHM of 4 km s$^{-1}$ and a peak intensity at $\sim$ 7 km s$^{-1}$. Comparing with the spectra at other positions, the cloud around the position $I$ is relatively quiescent and unaffected by the HII region or the outflows. This cloud in the south near the position $I$ may be a more independent component which is further separated along the line of sight. We conclude that the C$^{17}$O 3-2 spectra demonstrate that the kinematics and morphology have been strongly affected by the expansion of the HII region. The nearly circular structure in the PV plots, and in the channel maps near the systemic velocity, as well as the spectra, suggest that a significant part of the mass has been pushed by the HII region. An impact from the molecular outflow can also be seen in the PV plots and spectra.
Estimate of the B field strength
--------------------------------
The B field strength projected in the plane of sky (B$_{\bot}$) can be estimated by means of the Chandrasekhar-Fermi (CF) method (Chandrasekhar & Fermi 1953; Falceta-Gonçalves, Lazarian, & Kowal 2008). In general, the CF method can be applied to both dust and line polarization measurements. We apply the CF method only to the dust continuum polarization, because there is no line polarization detected in this paper. Although the dust grains can also be mechanically aligned, the existing observational evidence in NGC 1333 IRAS4A (Girart et al. 2006) demonstrates that the dust grains can align with the B field in the low mass star formation regions. Here, we assume that the dust grains also align with the B field in G5.89.
The strength of B$_{\bot}$ can be calculated from: $$\label{1}
B_{\bot} = Q \sqrt{4\pi\bar{\rho}}\frac{\delta v_{los, A}}{\delta\phi}
= 63 \sqrt{n_{H_2}}\frac{\delta v_{los, A}}{\delta \phi}$$
Here, B$_{\bot}$ is in the unit of mG. The term Q is a dimensionless parameter smaller than 1. Q is $\sim$0.5 (Ostriker, Stone & Gammie 2001), depending on the inhomogeneities within the cloud, the anisotropies of the velocity perturbations, the observational resolution and the differential averaging along the line of sight. The term $\bar{\rho}$ is the mean density. $\delta \phi$ is the dispersion of the polarization angles in units of degree. $\delta
v_{los, A}$ is the velocity dispersion along the line of sight in units of km s$^{-1}$, which is associated with the Alfvénic motion. $n_{H{2}}$ is the number density of H$_{2}$ molecules in units of 10$^{7}$ cm$^{-3}$. It has been shown numerically that the CF method is a good approximation for $\delta \phi <
25^{\degr}$ (Ostriker, Stone, & Gammie (2001)).
$\delta v_{los, A}$ is estimated from $\delta v_{total}$ in the 2nd moment image (lower panel in Fig. 5). $\delta v_{total}$ contains the information of the dispersions caused by the Alfvénic turbulent motion ($\delta \textit{v}_{los, A}$) and the dispersions caused by the HII expansion and outflow motions ($\delta \textit{v}_{bulk}$). The relation of these three components is:
$$\label{1}
\delta v_{total} \sim \sqrt{\delta v_{los, A}^{2}+\delta v_{bulk}^{2}}$$
Here, we neglect the minor contributions from the thermal Doppler motions. The measured $\delta v_{total}$ at the positions of detected polarization are listed in Table 1. $\delta v_{total}$ is in the range of 1 to 6 km s$^{-1}$. However, the molecular gas near the HII region is clearly disturbed by both the HII expansion and the molecular outflows (see Sec. 4.3.1 and 5.2). Therefore, the detected $\delta v_{total}$ at these positions is dominated by the bulk motion. Since $\delta v_{total}$ in the relatively quiescent regions is more likely tracing the Alfvénic motion only, we adopt the minimum value $\delta v_{total}$ of 1 km s$^{-1}$ at these positions for $\delta v_{los, A}$ in order to derive B$_{\bot}$.
The term $n_{H_{2}}$ is $\sim$3$\times$10$^{6}$ (cm$^{-3}$), estimated from the averaged $n_{H_{2}}$ from the 870$\mu$m dust continuum and the C$^{17}$O 3-2 line emission (Sec. 4.1.2 and Sec. 4.3.2). $\delta\phi$ in Eq. 4 can be extracted from the observed standard deviation of the PAs $\delta \phi_{obs}$. $\delta
\phi_{obs}$ contains both the observational uncertainty $\sigma_{\phi,obs}$ and $\delta \phi$. The relation is: $\delta\phi_{obs}^{2}$ = $\delta\phi^{2} + \sigma_{\phi,obs}^{2}$. Since the polarization in G5.89 results probably from two different systems (discussed in Sec. 4.2.1 and 4.2.2), it is more reasonable to separate these two groups when deriving $\delta\phi$. The derived $\sigma_{\phi,obs}$, $\delta\phi_{obs}$ and $\delta\phi$ are 3$\degr$ and 11$\degr$ and $\sim$11$\degr$ for component “o”, and 2$\degr$, 18$\degr$ and $\sim$18$\degr$ for component “x”, respectively. By using Eq. (4), the derived B$_{\bot}$ is 3mG and 2mG for component “o” and “x”, respectively.
The estimated B$_{\bot}$ is highly uncertain. Due to the bulk motions, it is difficult to extract the $\delta v_{los, A}$ component from the observed $\delta v_{total}$. The uncertainty introduced from $\delta v_{los, A}$ is within a factor of 6. Of course, the grouping of “o” and “x” components of the polarization, as motivated in Sec. 4.2.1, 4.2.2 and 5.2, is not a unique interpretation. If $\delta \phi$ is calculated without grouping, a more complex model of the larger scale B field morphology is needed to calculate the deviation due to the Alfvénic motion. More observations with sufficient uv coverage are required to establish such a model. Based on the standard deviation $\delta \phi$ of 39$\degr$ from the detected polarization without subtracting the larger scale B field and without grouping, the calculated lower limit of B$_{\bot}$ is $\sim$1mG. Therefore, the estimated B$_{\bot}$ from the grouping of component “o” and “x” seems reasonable. The value is comparable to the ones estimated via the CF method in other massive star formation regions with an angular resolution of a few arcseconds: $\sim$ 1mG in DR 21(OH) (Lai et al. 2003) and $\sim$1.7 mG in G30.79 FIR 10 (Cortes & Crutcher 2006). Moreover, B$_{\bot}$ is similar to B$_{\parallel}$ measured from the Zeeman pairs of the OH masers by Stark et al. (2007), ranging from $-$2 to 2 mG. Although B$_{\parallel}$ measured from OH masers is most likely tracing special physical conditions, such as shocks or dense regions, it is the only direct measurement of B$_{\parallel}$ in G5.89, and hence, is of interest to compare. Assuming B$_{\bot}$ and B$_{\parallel}$ have the same strengths of 2mG, the total B field strength in G5.89 is $\sim$ 3 mG.
Collapsing cloud or not?
------------------------
The mass to flux ratio $\lambda$, a crucial parameter for the magnetic support/ambipolar diffusion model, can be calculated from: $\lambda$ = 7.6 $\times$ 10$^{-21}$ $\frac{N_{H_{2}}}{B}$ (Mouschovias & Spitzer 1976; Nakano & Nakamura 1978). $N_{H_2}$ is in cm$^{-2}$. $B$ is in $\mu$G. In the case of $\lambda$ $<$ 1, the cloud is in a subcritical stage and magnetically supported. In the case of $\lambda$ $>$ 1, the cloud is in a collapsing stage.
Since there is no observation of the B field strength as a function of position in the entire cloud, we assume that the B field is uniform with the strength of 3 mG in the entire cloud when $\lambda$ is calculated. For consistency, when comparing with the kinetic pressure in the next section, $N_{H_{2}}$ is derived from the C$^{17}$O 3-2 emission (section 4.3.2). The derived $\lambda$ in G5.89 is $>$1 in most parts of the molecular cloud, as shown in the upper panel of Fig. 8. If the statistical geometrical correction factor of $\frac{1}{3}$ is considered (Crutcher 2004), the $\lambda_{corr}$ in the *sharp dust ridge* is still close to 1, whereas at the positions of the component “o” and the outer part, it is much smaller than 1. This suggests that G5.89 is probably in a supercritical phase near the HII region and in a subcritical phase in the outer part of the dust core.
This conclusion is based on the assumption that the B field in the entire cloud is uniform with a strength of 3 mG. This assumption seems to be crucial at first glance. However, the derived $\lambda$ increases from 0.1 to 2.5 toward the UCHII region, which is due to the high contrast of the column density across the cloud. Unless the actual B field strength differs by a factor of 25 across the region and compensates for the contrast in the column density, such a variation of $\lambda$ in G5.89 is indeed possible.
Compressed field?
-----------------
The coincident location of the detected polarization of component “x” and the *sharp dust ridge* is quite interesting. One possible scenario is that the B field lines are compressed by the shock front, i.e. HII expansion. In a magnetized large molecular cloud with a B field traced by a component “o”, and with a shock sent out from the east of the narrow dust ridge, we expect a rapid change of the polarization PA. This is similar to the results in magnetohydrodynamic simulations by Krumholz et al. (2007). Because of our limited angular resolution, polarization with a large dispersion in the PAs over a small physical scale will be averaged out within the synthesized beam. In our result, in fact, there is a gap where polarized emission is not detected right next to the *sharp dust ridge*, and a series of OH masers are detected in this gap. Note that the OH masers are most likely from the shock front. From the discussion in Sec. 5.2, evidence for the molecular cloud expanding with the HII region is found in the molecular gas traced by the C$^{17}$O 3-2 emission. The 870$\mu$m *sharp dust ridge* can be explained by the swept-up material along with the molecular gas from the HII expansion. In this scenario, the component “o” is tracing the B field in the pre-shock region, while the “x” component is tracing the compressed field.
However, the swept-up flux density (summation of the flux density of SMA-N, SMA-1 and SMA-2 reported in Hunter et al. 2008) is $\sim$ 20% of the total detected flux density in this paper. This requires a huge amount of energy to sweep up the material with this mass. Is the radiation pressure (P$_{rad}$) from the central star large enough to overcome the kinetic pressure (P$_{kin}$) and the B field pressure (P$_{B}$)? Here we compare these pressure terms.
P$_{rad}$ can be calculated from the luminosity in G5.89 following the equation: P$_{rad}$ $=$ $\frac{L}{cA}$, where $L$, $c$ and A are the luminosity, speed of light and the area, respectively. Since the G5.89 region is dense, most of the radiation is absorbed and redistributed into the surrounding material. The total far infrared luminosity of G5.89 is 3$\times$10$^{5}$L$_{\sun}$ (Emerson, Jennings, & Moorwood 1973) and the radius of the HII region at 2cm is $\sim$ 2$\arcsec$ (4000 AU). The energy density and hence, the radiation pressure (P$_{rad}$) in the sphere with a radius of 2$\arcsec$ is 8.5$\times$10$^{-7}$ (dyne cm$^{-2}$).
P$_{kin}$ is calculated by using the 0th moment ($MOM0$) and 2nd moment ($MOM2$) images of the C$^{17}$O 3-2 line: $$\label{1}
P_{kin} = \frac{1}{2} \rho \delta v_{total}^2 = 3.4 \times10^{-9} \times (MOM0) \times
(MOM2)^2$$
where P$_{kin}$ is in dyne cm$^{-2}$, $\rho$ is the gas density in g cm$^{-3}$ and $\delta v_{total}$ is the velocity dispersion in cm s$^{-1}$, $MOM0$ is in units of K km s$^{-1}$, and $MOM2$ is in units of km s$^{-1}$. $\rho$ is calculated following Sec. 4.3.2, with the size of 0.13 pc for the molecular cloud along the line of sight. The derived P$_{kin}$ image is shown in the middle and lower panels of Fig. 8. P$_{kin}$ is in the range of $\sim$ 1$\times$10$^{-9}$ to 1.4$\times$10$^{-6}$ dyne cm$^{-2}$. P$_{kin}$ is calculated under the assumption that the length along the line of sight is uniform in G5.89, which is the main bias in the calculation. The estimated total B field strength is 3 mG, thus the B-field pressure (P$_B$) is 3.6$\times$10$^{-7}$ (dyne cm$^{-2}$). Although the upper limit of P$_{kin}$ is 1.6 times larger than P$_{rad}$ at a radius of 2$\arcsec$, any variation of the structure in the direction along the line of sight in G5.89 - which is most likely the case - will affect the estimated P$_{kin}$. Nevertheless, P$_{rad}$ is at the same order as P$_{kin}$ and P$_{B}$ at the radius of 2$\arcsec$ around the O5 star. Therefore, in terms of pressure, the radiation from the central star is likely sufficient to sweep up the material and compress the B field lines along the narrow dust ridge. $\lambda_{corr}$ close to 1 near the UCHII region suggests that the B fields play a minor role as compared with the gravity.
The B field direction traced by component “x” is parallel to the major axis of this sharp dust ridge, which is also seen in some other star formation sites such as Cepheus A (Curran $\&$ Chrysostomou 2007) and DR 21(OH) (Lai et al. 2003). However, in most of the cases, the detected B field direction is parallel to the minor axis of the dust ridge, e.g. W51 e1/e2 cores (Lai et al. 2001) and G34.4+0.23 MM (Cortes et al. 2008), which agrees with the ambipolar diffusion model. A possible explanation is that the polarization is from the swept-up material, which interacts with the original dense filament. Thus, the polarization here may represent the swept-up field lines. This scenario is supported by the energy density and also the morphology of the field lines in the case of G5.89.
Comparison with Other Star Formation Sites
------------------------------------------
The detected B field structure in the G5.89 region is more complicated than the B fields in other massive star formation sites detected so far with interferometers. Both the compressed field structure and the more organized larger scale B field are detected in G5.89.
The B field lines vary smoothly in the cores at earlier star formation stages, such as the W51 e1/e2 cores (Lai et al. 2001), G34.4+0.23 MM (Cortes et al. 2008) and NGC 2024 FIR 5 (Lai et al. 2002). These cores are still in a collapsing stage (Ho et al. 1996 ; Ramesh et al. 1997; Mezger et al. 2001). Among these observations, the B fields inferred from the dust polarization show an organized structure over the scale of 10$^{5}$ AU (15$\arcsec$) at a distance of 7 kpc in the W51 e1/e2 cores and also on the scale of 10$^{5}$ AU (35$\arcsec$) at a distance of 3.9 kpc in G34.4+0.23 MM. In contrast, the B fields in NGC 2024 FIR 5, which is closer at a distance of 415 pc, show an hourglass morphology on a scale of 4$\times$10$^{3}$ AU (9$\arcsec$). Such small scale structures would not be resolved in the current data of W51 e1/e2 core and G34.4+0.23 MM due to the resolution effect. Compared to these sources, G5.89 is more complicated with polarization structures on both small (4$\times$10$^{3}$ AU) and large (2$\times$10$^{4}$ AU) scales. However, higher angular resolution polarization images of the cores in the earlier stages are necessary in order to compare the B field morphology with the later stages in the massive star formation process. At this moment, we cannot conclude at which stage the B field structures become more complex.
Currently, the best observational evidence supporting the theoretical accretion model is the polarization observation of the source NGC 1333 IRAS 4A (Girart, Rao & Marrone 2006) carried out with the SMA. The NGC 1333 IRAS 4A is a low mass star formation site, at a distance of $\sim$300 pc. The detected pinched B-field structure is at a scale of 2400 AU (8$\arcsec$). If NGC 1333 IRAS 4A were at a distance of 2kpc, we could barely resolve it at our resolution of 3$\arcsec$. Higher angular resolution polarization measurements are required to resolve the underlying structure in G5.89.
Conclusions and Summary
=======================
High angular resolution (3$\arcsec$) studies at 870$\mu$m have been made of the magnetic (B) field structures, the dust continuum structures, and the kinematics of the molecular cloud around the Ultra-Compact HII region G5.89-0.39. The goal is to analyze the role of the B field in the massive star forming process. Here is the summary of our results:
1. The gas mass (M$_{gas}$) is estimated from the dust continuum and from the C$^{17}$O 3$-$2 emission line. The continuum emission at 870 $\mu$m is detected with its total flux density of 12.6$\pm$1.3 Jy. After removing the free-free emission from the detected continuum, the flux density of the 870 $\mu$m dust continuum is 7.7 Jy, which corresponds to M$_{gas}$ $\sim$300 M$_{\sun}$. M$_{gas}$ derived from the detected C$^{17}$O 3-2 emission line is $\sim$100 M$_{\sun}$, which is 3 times smaller than the value derived from the dust continuum. The discrepancy of M$_{gas}$ derived from the dust continuum and the C$^{17}$O emission line is also seen in other UCHII regions, e.g. Hofner et al. (2000). The lower values measured from C$^{17}$O could be due to optical depth effects or abundance problems.
2. The linearly polarized 870 $\mu$m dust continuum emission is detected and resolved. The dust polarization is not uniformly distributed in the entire dust core. Most of the polarized emission is located around the HII ring, and there is no polarization detected in the southern half of the dust core except at the very southern edges. The position angles (PAs) of the polarization vary enormously but smoothly in a region of 2$\times$10$^{4}$ AU (10$\arcsec$), ranging from $-$60$\degr$ to 61$\degr$. Furthermore, the polarized emission is from organized patches, and the distribution of the PAs can be separated into two groups. We suggest that the polarization in G5.89 traces two different components. The polarization group “x”, with its PAs ranging from $-$60$\degr$ to $-$4$\degr$, is located at the 870$\mu$m *sharp dust ridge*. In contrast, the group “o”, with its PAs ranging from 33$\degr$ to 61$\degr$, is at the periphery of the *sharp dust ridge*. The inferred B field direction from group “x” is parallel to the major axis of the 870$\mu$m dust ridge. One possible interpretation of the polarization in group “x” is that it may represent swept-up B field lines, while the group “o” traces more extended structures. In the G5.89 region, both the large scale B field (group “o”) and the compressed B field (group “x”) are detected.
3. By using the Chandrasekhar-Fermi method, the estimated strength of B$_{\bot}$ from component “o” and from component “x” is in between 2 to 3 mG, which is comparable to the Zeeman splitting measurements of B$_{\parallel}$ from the OH masers, ranging from $-$2 to 2 mG by Stark et al. (2007). The derived lower limit of B$_{\bot}$ from the detected polarization without grouping and without modeling larger scale B field is $\sim$1mG. Assuming that B$_{\bot}$ and B$_{\parallel}$ have the same strengths of 2 mG in the entire cloud, the derived $\lambda$ increases from 0.1 to 2.5 toward the UCHII region, which is due to the high contrast of the column density across the cloud. Unless the actual B field strength differs by a factor of 25 across the region and compensates for the contrast in the column density, such a variation of $\lambda$ in G5.89 is suggested. The corrected mass to flux ratio ($\lambda_{corr}$) is closer to 1 near the HII region and is much smaller than 1 in the outer parts of the dust core. G5.89 is therefore most likely in a supercritical phase near the HII region.
4. The kinematics of the molecular gas is analyzed using the C$^{17}$O 3-2 emission line. From the analysis of the channel maps, the position velocity plots and spectra, the molecular gas in the G5.89 region is expanding along with the HII region, and it is also possibly swept-up by the molecular outflows. Assuming the size along the line of sight is uniform in G5.89, P$_{kin}$ is in the range of $\sim$ 1$\times$10$^{-9}$ to 1.4$\times$10$^{-6}$ dyne cm$^{-2}$. The calculated radiation pressure (P$_{rad}$) at a radius of 2$\arcsec$ and the B field pressure (P$_B$) with a field strength of 3mG are 8.5$\times$10$^{-7}$ and 3.6$\times$10$^{-7}$ dyne cm$^{-2}$, respectively. Although the upper limit of P$_{kin}$ is 1.6 times larger than P$_{rad}$ at a radius of 2$\arcsec$, any variation of the structure in the direction along the line of sight in G5.89 - which is most likely the case - will affect the estimated P$_{kin}$. Nevertheless, P$_{rad}$ is on the same order as P$_{kin}$ and P$_{B}$ at the radius of 2$\arcsec$ around the O5 star. The scenario that the matter and B field in the 870$\mu$m *sharp dust ridge* have been swept-up is supported in terms of the available pressure.
G5.89 is in a more evolved stage as compared with the corresponding structures of other sources in the collapsing phase. The morphologies of the B field in the earlier stages of the evolution show systematic or smoothly varying structures, e.g. on the scale of 10$^{5}$ AU for W51 e1/e2 and G34.4+0.23 MM, and on the scale of 4$\times$10$^{3}$ AU for NGC 2024 FIR 5. With the high resolution and high sensitivity SMA data, we find that the B field morphology in G5.89 is more complicated, being clearly disturbed by the expansion of the HII region and the molecular outflows. The large scale B field structure on the scale of 2$\times$10$^{4}$ AU in G5.89 can still be traced with dust polarization. From the analysis of the C$^{17}$O 3-2 kinematics and the comparison of the available energy density (pressure), we propose that the B fields have been swept up and compressed. Hence, the role of the B field evolves with the formation of the massive star. The ensuing luminosity, pressure and outflows overwhelm the existing B field structure.
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![*(a)* The SMA 870 $\mu$m total continuum image (contours) overlayed on the VLA 2cm continuum (free-free continuum; color scale). The white contours represent the continuum emission strength at 3, 5, 10, 15, 20, 25 ... 60 and 65 $\sigma$ levels, and the black contours in the center represent 70, 80, 90, 100 and 110 $\sigma$ levels, where [$1\sigma$ is 30 mJy Beam$^{-1}$]{}. The star marks the O star detected by Feldt et al. (2003). The asterisk marks the origin of the Br$\gamma$ outflow detected by Puga et al. (2006). The SMA and VLA synthesized beams are shown as white and black ellipses at the lower-left corner, respectively. The white “+” mark the positions of the sub-mm peaks identified in Hunter et al. (2008). *(b)* The same as in (a), with the white contours representing the SMA 870 $\mu$m dust continuum (after the subtraction of the free-free continuum). The contours start from and step in $3\sigma$, where [$1\sigma$ is 30 mJy Beam$^{-1}$]{}. The color wedge on the upper-right edge represents the strength of the 2cm free-free continuum in the units of Jy Beam $^{-1}$. The red and blue arrows indicate the axes of the molecular outflows. The outflows in the N-S and NW-SW direction in the west of O5 star are identified in Hunter et al. (2008). The 3rd outflow in the east of O5 star is identified in Tang et al. (in prep.). *(c)* The polarization (red segments) derived by using image (a). The length of the red segment represents the percentage of the polarized intensity. The 870 $\mu$m continuum is shown both in white contours with the steps as in Fig. (a) and in the color scale. *(d)* The polarization (red segments) derived by using image (b). The 870 $\mu$m dust continuum is again shown both in white contours with the steps as in Fig. (b) and in the color scale. The color wedge at the lower-right edge shows the strength of the dust continuum in the units of Jy Beam$^{-1}$. In *(c)* and *(d)*, the polarization plotted is above 3$\sigma_{I_p}$. []{data-label=""}](f1.eps)
![(a) The 870$\mu$m dust continuum (white and grey contours) overlaid on the polarized intensity (I$_{P}$) image (color scale). The contours plotted are the same as in Fig. 1(b). The color wedge shows the strength of polarization intensity in units of mJy Beam$^{-1}$. The smallest white open circles and plus signs mark the positions of the Zeeman pairs of the OH maser (Stark et al. 2007) with different polarimetries. The other symbols are the same as in Fig. 1. The larger solid white circles mark component “o”, defined in Sec. 4.2.2. (b) The polarization (red and blue segments) overlaid on the 870 $\mu$m dust continuum (black contours) and the 2cm free-free continuum emission (color scale). In red and blue segments are polarization segments above 3$\sigma_{I_p}$ and between 2 to 3$\sigma_{I_p}$, respectively. The contours, star, asterisk, circles and plus signs are all the same as in (a). The color wedge shows the strength of the 2cm free-free emission in units of Jy Beam$^{-1}$. The ellipses in the lower-left corner are the synthesized beams of this paper, shown in black, and of the 2cm free-free continuum image, shown in white. (c) The inferred B field (red segments) overlaid on the 870 $\mu$m dust continuum (blue contours) in this paper and in Hunter et al. (2008) (grey scale). The ellipses in the lower-left corner are the synthesized beams of this work, shown in black, and of Hunter et al. (2008), shown in white. The white crosses mark the sub-mm sources detected by Hunter et al. (2008). The triangles mark the positions of the H$_{2}$ knots identified in Puga et al. (2006).[]{data-label="pol"}](f2_a "fig:") ![(a) The 870$\mu$m dust continuum (white and grey contours) overlaid on the polarized intensity (I$_{P}$) image (color scale). The contours plotted are the same as in Fig. 1(b). The color wedge shows the strength of polarization intensity in units of mJy Beam$^{-1}$. The smallest white open circles and plus signs mark the positions of the Zeeman pairs of the OH maser (Stark et al. 2007) with different polarimetries. The other symbols are the same as in Fig. 1. The larger solid white circles mark component “o”, defined in Sec. 4.2.2. (b) The polarization (red and blue segments) overlaid on the 870 $\mu$m dust continuum (black contours) and the 2cm free-free continuum emission (color scale). In red and blue segments are polarization segments above 3$\sigma_{I_p}$ and between 2 to 3$\sigma_{I_p}$, respectively. The contours, star, asterisk, circles and plus signs are all the same as in (a). The color wedge shows the strength of the 2cm free-free emission in units of Jy Beam$^{-1}$. The ellipses in the lower-left corner are the synthesized beams of this paper, shown in black, and of the 2cm free-free continuum image, shown in white. (c) The inferred B field (red segments) overlaid on the 870 $\mu$m dust continuum (blue contours) in this paper and in Hunter et al. (2008) (grey scale). The ellipses in the lower-left corner are the synthesized beams of this work, shown in black, and of Hunter et al. (2008), shown in white. The white crosses mark the sub-mm sources detected by Hunter et al. (2008). The triangles mark the positions of the H$_{2}$ knots identified in Puga et al. (2006).[]{data-label="pol"}](f2_b "fig:")
![(a) Distribution of the PAs (defined in the range $-$90$\degr$ to 90$\degr$) of the polarization in G5.89. (b) Total intensity (I) versus percentage (%) of the polarized flux density. In both (a) and (b) panels, the statistics are from the detected polarization segments above 3 $\sigma_{I_p}$ confidence level. The “cross” represents the component “x”, which is associated with the sharp dust ridge. The “circle” represents the component “o”, which is associated with the extended structure.[]{data-label="vec_dist"}](f3.ps)
[^1]: The Submillimeter Array is a joint project between the Smithsonian Astrophysical Observatory and the Academia Sinica Institute of Astronomy and Astrophysics and is funded by the Smithsonian Institution and the Academia Sinica.
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---
abstract: 'We introduce the stochastic band structure, a method giving the dispersion relation for waves propagating in periodic media or along waveguides, and subject to material loss or radiation damping. Instead of considering an explicit or implicit functional relation between frequency $\omega$ and wavenumber $k$, as is usually done, we consider a mapping of the resolvent set in the dispersion space $(\omega, k)$. Bands appear as as the trace of Lorentzian responses containing local information on propagation loss both in time and space domains. For illustration purposes, the method is applied to a lossy sonic crystal, a radiating surface phononic crystal, and a radiating optical waveguide. The stochastic band structure can be obtained for any system described by a time-harmonic wave equation.'
author:
- Vincent Laude
- 'Maria E. Korotyaeva'
title: Stochastic band structure for waves propagating in periodic media or along waveguides
---
Introduction
============
The dispersion relation is essential information to describe wave propagation [@brillouinBOOK2003]. This is especially true for structures that support wave propagation, such as waveguides and artificial crystals, including photonic [@yablonovitchPRL1987; @johnPRL1987; @luNP2014] and phononic [@kushwahaPRL1993; @maldovanN2013; @husseinAMR2014] crystals. The dispersion relation gives the possible propagation modes and relates the angular frequency $\omega$ with the wavevector $\bm{k}$. In the case of periodic media and crystals, it is termed the band structure. It is very often obtained by looking for the eigenvalues and eigenfunctions of a matrix, in the case of finite-dimensional problems. Finding the eigenvalues and eigenfunctions of a finite-size matrix is indeed nowadays a well mastered numerical problem [@golubJCAM2000]. Solvers are routinely used to obtain them, at least in the case of lossless waveguides and crystals. Indeed, in the absence of loss, one often obtains a self-adjoint propagation operator, i.e. an operator satisfying Hermitian symmetry. In this case, the Hilbert-Schmidt theorem tells us that the spectrum lies on the real line [@renardyBOOK2006]. If the operator is furthermore compact, eigenvalues are discrete and isolated (in case they are degenerate, the corresponding eigenfunctions are orthogonal). Physically, this situation is generally implied when one plots the dispersion relation as a graph $\omega(\bm{k})$. Both frequency and wavevector are real quantities and the dispersion relation is composed of distinct bands that can be numbered. Note that compactness here refers to the domain of definition being finite and the operator being bounded.
![ Application of the stochastic band structure method to a two-dimensional sonic crystal of rigid cylinders in air, with material loss taken into account. (a) The finite-size computation domain is here a unit-cell of the crystal. (b) A spatially random excitation is applied and the response (c) is obtained. From the response, the stochastic band structure (d) can be plotted. For further details see Section \[sec3.1\] and Fig. \[fig2\]. []{data-label="fig1"}](fig1b){width="65mm"}
The situation becomes more obscure when propagation loss or some infinite dimension enters the picture. In the case of loss, the self-adjoint property is lost. In the case of an infinite domain of definition, the operator is no more compact. In the latter case, there exist computational techniques to approximate the problem on a finite domain, such as the perfectly matched layer (PML) [@berengerJCP1994]. As a result, however, self-adjointness is generally lost. Fortunately, a theorem similar to the Hilbert-Schmidt theorem still holds for compact operators in finite dimensional Hilbert spaces, but eigenvalues are complex [@renardyBOOK2006]. As a result, the bands composing the dispersion relation are not real functions anymore. Different physical examples where this situation happens will be considered in the following of this paper: a lossy sonic crystal [@moiseyenkoPRB2011], such as the one depicted in Figure \[fig1\], a phononic crystal of pillars sitting on a semi-infinite substrate [@khelifPRB2010], and an open optical waveguide. Specifically, the goal of this paper is to obtain a generalized representation of the dispersion relation that applies to these difficult cases. We term the method the stochastic band structure, because it relies on an analysis of the response of the propagation medium to a random excitation having a definite frequency and wavevector. Fig. \[fig1\](b) shows a typical random excitation applied to a unit-cell of a crystal. The idea of considering the dispersion relation as a graph of discrete bands $\omega(\bm{k})$ is replaced by a response function $E(\omega, \bm{k})$ similar to a density of states. The response itself is obtained from the solution to the forced problem, as shown in Fig. \[fig1\](c). Infinite domains are transformed to a compact domain using a PML technique implemented as a complex coordinate transform [@hugoninJOSAA2005]. As we show in Section 2, the stochastic response reveals the spectrum as the complement of the resolvent set of the operator and gives essential information on the amount of propagation loss at a particular dispersion point. Furthermore, the response can be made almost independent of the particular realization of the random excitation. In Section 3, we describe the application of the method to the three different situations mentioned above. In Section 4, and before concluding the paper, we discuss the merits of the stochastic band structure method and its relations to previous methods.
Theory
======
Let us first recall some results from spectral theory [@renardyBOOK2006]. We place ourselves in an appropriate functional space, usually a Hilbert space. For a bounded linear operator $A$, $R(\lambda)=(A - \lambda I)^{-1}$ is by definition the resolvant operator. the resolvent set is the set of all complex numbers $\lambda$ such that $R(\lambda)$ exists and is bounded. The spectrum is defined as the complement of the resolvent set in the complex plane. Every eigenvalue of the operator $A$ belongs to the spectrum. Note that this definition is more general than the usual definition of eigenvalues of matrices of finite-dimensional spaces. It also avoids the difficulties of defining the spectrum through a singular equation by considering instead its non-singular complement.
According to the Hilbert-Schmidt theorem, the spectrum of a self-adjoint operator lies on the real line and is in general a spectral combination of a point spectrum of discrete eigenvalues and a continuous spectrum. For compact self-adjoint operators, eigenvalues are discrete and isolated. Eigenvalues $\lambda_n$ and eigenfunctions $\bm{e}_n$ satisfy $A \bm{e}_n = \lambda_n \bm{e}_n$. The eigenfunction expansion theorem tells us that any function can be written $\bm{f} = \sum_n f_n \bm{e}_n$ with the eigenfunctions $\bm{e}_n$ forming an orthonormal basis. The notation $f_n = \langle \bm{f}, \bm{e}_n \rangle$ is for the scalar product in the functional space. As we indicated in the introduction, in case the operator is not self-adjoint, a similar theorem holds for compact operators in finite dimensional Hilbert spaces, providing the eigenvalues are considered complex. Note that they can further be algebraically degenerated, but we will not consider this complication in this paper. This assumption is equivalent to assuming the operator is isomorphic to a diagonalizable matrix.
Linear wave equations, including those for acoustic, elastic, and optical waves, can generally be written for time-harmonic waves as $$\begin{aligned}
(K(\bm{k}) - \omega^2 M) \bm{u}(\omega, \bm{k}) &= \bm{f}(\omega, \bm{k})
\label{eq1}\end{aligned}$$ where $K$ is a stiffness operator and $M$ a mass operator, $\bm{u}(\omega, \bm{k})$ is a function describing the solution in reciprocal space, and $\bm{f}(\omega, \bm{k})$ is a forcing term at a particular frequency and wavevector. Equation is obtained from a Fourier transform of the original wave equation over the time coordinate. The stiffness operator $K$ is a differential operator of the space coordinates and can depend on an imposed wavevector $\bm{k}$, for waveguide and artificial crystal problems. We can then define the resolvent operator as $$\begin{aligned}
R(\lambda) &= (M^{-1} K - \lambda I)^{-1}
\label{eq2}\end{aligned}$$ with $\lambda=\omega^2$. Note that the inversion operation of the mass operator should be understood as symbolic and is introduced for convenience; in practice there is no need to invert a matrix or operator. The solution to Eq. is formally $$\begin{aligned}
\bm{u} &= R(\lambda) M^{-1} \bm{f} .
\label{eq3}\end{aligned}$$ Introducing the eigenvalues $\lambda_n$ and eigenfunctions $\bm{e}_n$ of $M^{-1} K$, we obtain from the eigenfunction expansion theorem that $$\begin{aligned}
u_n &= (\lambda_n - \lambda)^{-1} g_n
\label{eq4}\end{aligned}$$ with $\bm{g} = M^{-1} \bm{f}$. Overall, the solution is $$\begin{aligned}
\bm{u} &= \sum_n (\lambda_n - \lambda)^{-1} g_n \bm{e}_n .
\label{eq5}\end{aligned}$$ This equation expresses the well known fact that the solution to a linear equation is a linear combination of eigenfunctions. The coefficients of the combination are complex Lorentzian functions or poles, centered on the eigenvalues, and are also proportional to the projection of the excitation on each eigenfunction. When $\lambda \approx \lambda_n$, we have $\bm{u} \approx (\lambda_n - \lambda)^{-1} g_n \bm{e}_n$, i.e. the solution approaches in the limit the particular eigenfunction. This observation leads to a practical way to obtain every eigenvalue and eigenfunction: one can explore all values of $\lambda$, i.e. the resolvent set; since the eigenvalues are isolated each pole can be isolated and identified. Of course, such a procedure would be very lengthy compared to existing eigenvalue solvers. If we forget the idea of obtaining exactly all eigenvalues, but only wish to obtain a view of the landscape of the resolvent set with a given – and limited – resolution, then the method can be useful, as we illustrate in the following section.
Eq. can be viewed as the dynamical equation obtained from the Euler-Lagrange principle with a Laplacian combining potential elastic energy, kinetic energy, and the work of the forcing term. A Hamiltonian operator can then be defined as $$\begin{aligned}
H &= \frac{1}{2}(K + \omega^2 M)\end{aligned}$$ and we evaluate the response as the self-energy of the solution, or $$\begin{aligned}
E &= \langle H \bm{u}, \bm{u} \rangle .\end{aligned}$$ Using the eigenfunction expansion for the solution, we have $$\begin{aligned}
E(\lambda) &= \sum_{n,n'} \langle H \bm{e}_n, \bm{e}_{n'} \rangle g_n (\lambda - \lambda_n)^{-1} g^*_{n'} (\lambda^* - \lambda_{n'}^*)^{-1}\end{aligned}$$ which is real and positive by construction, if $K$ and $M$ are real operators. The response is an expansion over complex poles centered on the eigenvalues. Close to an eigenvalue, i.e. when $\lambda \approx \lambda_n$, $E(\lambda) \approx \langle H \bm{e}_n, \bm{e}_{n} \rangle |g_n|^2 |\lambda - \lambda_n|^{-2}$. The response has then locally a – possibly damped – Lorentzian shape.
In order not to miss any of the poles, it is needed that $g_n = \langle \bm{g}, \bm{e}_n \rangle \neq 0$ for any $n$. In practice, we can consider a spatially random excitation. If the number of degrees of freedom is large, then the probability that any $g_n = 0$ is very small. The result can further be made almost independent of the exact value of $g_n$ by considering the log-derivative of the response $$\begin{aligned}
\frac{\partial}{\partial \lambda} \log E = \frac{1}{E} \frac{\partial E}{\partial \lambda} .\end{aligned}$$ Indeed, writing $\lambda - \lambda_n = \alpha + \imath \beta$, with both $\alpha$ and $\beta$ real, we have locally $$\begin{aligned}
\frac{\partial}{\partial \alpha} \log E &\approx -2 \alpha (\alpha^2 + \beta^2)^{-1}
\label{eq10}\end{aligned}$$ which is a real Lorentzian function. The response is thus practically independent of the excitation. Of course, Eq. is only valid close to eigenvalue $\lambda_n$ that is separated from the other eigenvalues. In practice, it also requires that the eigenvalues are sufficiently isolated compared to the analyzing resolution.
As a whole, the stochastic response $E(\omega, \bm{k})$ contains information on the bands in the form of a continuous map in the dispersion space $(\omega, \bm{k})$. Each band leaves a trace which is locally a Lorentzian function. This Lorentzian function gives both the real part and the imaginary part of the local eigenvalue; in particular the width of the response informs on propagation damping in both time and space.
Results {#sec3}
=======
In this section we consider three different examples of the application of the stochastic band structure method. In each case, a direct eigenvalue analysis would have been difficult, as we argue.
Bloch waves in a lossy sonic crystal {#sec3.1}
------------------------------------
Let us consider a two-dimensional sonic crystal composed of steel cylinders in air, as depicted in Figure \[fig1\](a). The structure is an infinite periodic repetition of a square primitive unit-cell of length $a$, periodic along axes $x_1$ and $x_2$. The steel cylinders are assumed infinitely long in the $x_3$ direction. Time-harmonic waves satisfy the linear acoustic equation for pressure $p$ in air $$\begin{aligned}
- \nabla \cdot \left( \frac{1}{\rho} \nabla p \right) - \omega^2 \frac{p}{B} &= f
\label{eq11}\end{aligned}$$ where the mass density $\rho(\bm{r})$ and the elastic modulus $B(\bm{r})$ are functions of position and $f(\bm{r})$ is an applied forcing term. These functions are discontinuous at the interface between matrix and inclusion. The band structure of a sonic crystal gives the dispersion relation of its Bloch waves. Bloch waves are in this case of the form $p(\bm{r},t)=\bar{p}(\bm{r})\exp(\imath(\omega t - \bm{k}\cdot\bm{r}))$, with $\bar{p}(\bm{r})$ the periodic part of the solution. Because of the very large acoustic impedance mismatch between steel and air, the boundary condition on the cylinders can be approximated by $\frac{\partial p}{\partial n}=0$ (the normal acceleration vanishes on the boundary). Appendix \[app1\] summarizes how to transform Eq. to an integral equation, using the finite element method (FEM), that is then easily cast into a linear system and to an eigenvalue system in case the applied force vanishes.
![ A two-dimensional square-lattice sonic crystal of rigid cylinders in air, with $d/a=0.8$. (a) The band structure for the lossless crystal is obtained by solving for frequency as a function of wavevector, $\omega(\bm{k})$, along the path M-$\Gamma$-X-M of the first Brillouin zone. (b) The stochastic excitation is applied in air as a random periodic field. (c-d) The normalized stochastic response and its log-derivative for small loss ($\mu / B a=10^{-9}$ ) show mostly undamped Lorentzian functions following the bands in (a). (e-f) The normalized stochastic response and its log-derivative for larger loss ($\mu / B a=10^{-7}$ ) show how damping distributes along each band.[]{data-label="fig2"}](fig2){width="85mm"}
In the absence of loss, the eigenvalue problem can be solved readily and the band structure is usually presented as reduced frequency, $\omega a/2\pi$, as a function of reduced wavenumber, $k a/2 \pi$. The result is shown in Fig. \[fig2\](a). As the problem is self-adjoint and the unit-cell is closed (compact), eigenfrequencies are discrete and isolated.
Let us now consider material loss. Specifically, loss is generally frequency dependent and is often modeled by a complex elastic modulus of the form $B' = B + \jmath \omega \mu$, with $\mu$ the viscosity [@auldBOOK1973]. Bloch’s theorem remains valid for coefficients that depend on frequency, so we can still use the same formulation as before. The eigenvalue problem, however, becomes nonlinear. An alternative is to consider the complex band structure, i.e. to solve for $k$ as a function of $\omega$ [@laudePRB2009; @romeroAPL2010]. In this way, the spatial damping of time-harmonic waves can be obtained as a function of frequency.
In order to apply the stochastic band structure method, we apply a stochastic Bloch-Floquet excitation, $f(\bm{r}; \bm{k},\omega)=\bar{f}(\bm{r})\exp(\imath(\omega t - \bm{k}\cdot\bm{r}))$, with stochastic periodic part, to the acoustic equation. We then explore the resolvant set as a function of $k$ and $\omega$, and obtain the stochastic band structures in Fig. \[fig2\](c-f). For illustration purposes, we have considered two arbitrary amounts of viscous damping, either $\mu / B a=10^{-9}$ or $\mu / B a=10^{-7}$ . With these values, and for the highest reduced frequency in the band structure, the ratio $\omega \mu/B$ of the imaginary part to the real part of the elastic modulus is at most $4\times 10^{-6}$ and $4\times 10^{-4}$, respectively. For the smaller value of viscous damping, the stochastic band structure is almost not affected and every band in the lossless band structure is visible with about the same intensity. For the larger value of viscous damping, bands are increasingly damped with increasing frequency.
Frequency-dependent loss can be estimated by looking at a cross-section of the stochastic band structure at constant $k$. For instance, Fig. \[fig3\] shows this cross-section at the X point of the first Brillouin zone. A superposition of damped Lorentzian functions is clearly observed. Such vertical cross-sections reveal the amount of temporal damping for every eigenvalue. reciprocally, considering horizontal cross-sections would reveal spatial damping of the same eigenvalues. The stochastic band structure thus contains the information of the complex band structure, with the imaginary part of the wavenumber replaced by the width of the Lorentzian response for each band. At the same time, it also contains information on temporal damping that is absent of the complex band structure.
![Cross-section of the stochastic band structure of Fig. \[fig2\], taken at the X point of the first Brillouin zone, without and with loss ($\mu / B a=10^{-7}$ ). Temporal damping affecting the different Bloch waves is apparent.[]{data-label="fig3"}](cross_loss){width="75mm"}
Surface Bloch waves of a phononic crystal of pillars {#sec3.2}
----------------------------------------------------
Surface elastic wave propagation on the surface of a phononic crystal has attracted a lot of attention, partly with regards to applications to surface acoustic wave technology [@olssonRST2009; @benchabaneAPL2015], to micro-electro-mechanical systems (MEMS) [@mohammadiAPL2009; @hsuAPL2011], and to thermal transport [@hopkinsNL2010; @maldovanN2013], but also from a fundamental point of view. If initial questions were related to the generalization of the definition of Rayleigh surface waves to periodic media [@tanakaPRB1998], it was soon realized that surface phononic crystals must as well support the propagation of radiating guided waves [@wuPRB2004; @laudePRE2005]. These waves can be described as surface excitations that are coupled with radiation modes of the substrate supporting the crystal. Radiation modes exist in the region of the dispersion diagram called the sound cone. By definition, the boundary of the sound cone is obtained by looking for the slowest bulk wave propagating in a given direction and along the surface of the substrate, with the direction of wave propagation being measured by the Poynting vector [@laudeAPL2011]. This procedure leads in general to an anisotropic but non dispersive velocity surface $v_{sc}(\bm{k}/|\bm{k}|)$ whose projection on the band structure looks like a cone.
The elastodynamic equation is $$\begin{aligned}
- T_{i j,j} - \rho \omega^2 u_{i} &= f_i
\label{eq12}\end{aligned}$$ where $\bm{u}$ is the displacement vector and $T_{i j}$ is the stress tensor. $f_i$ are body forces and the constitutive relation of elasticity (Hooke’s law) is $$\begin{aligned}
T_{i j} &= c_{i j k l} S_{k l} ,
\label{eq13}\end{aligned}$$ with $c_{i j k l}$ the elastic tensor and $S_{i j} = \frac{1}{2} \left( u_{i,j} + u_{j,i} \right)$ the strain tensor.
In the general case of surface waves, and in contrast to bulk waves, there does not exist an eigenvalue equation giving the band structure. Surface waves are instead found by looking for the zeros of a determinant of the boundary conditions, or any equivalent secular equation [@wuPRB2004; @laudePRE2005]. This procedure, however, is strictly speaking limited to lossless surface waves, whose dispersion lies outside the sound cone. Leaky guided surface waves, whose dispersion lies inside the sound cone, have been obtained by looking for minima of the boundary condition determinant [@laudePRE2005]. A difficulty is that radiation modes of the substrate have to be selected according to a partial wave selection rule. In the general case of a finite depth phononic crystal sitting on a semi-infinite substrate, this procedure is cumbersome and the usual approach has been to consider only purely guided waves, i.e. non-radiative surface waves lying outside the sound cone [@khelifPRB2010; @assouarAPL2011; @yudistiraAPL2012]. An immediate drawback is that the non-radiative band structure is defined in a quite restricted sense and neglects the interaction of surface waves with bulk waves radiated in the substrate.
![A two-dimensional square-lattice phononic crystal of silicon pillars on a silicon substrate. (a) The unit-cell is arbitrarily cut at a depth $h+w$ in the substrate, with $w$ the depth of a perfectly matched layer (PML) introduced to approximate radiation inside the semi-infinite substrate. The height of the pillar is $h_1/a=1$ and the diameter is $d/a=0.5$. (b) The classical band structure, generated by solving an eigenvalue problem with damping in the PML set off, gives the dispersion of non-radiative surface waves only outside the sound cone highlighted in gray. (c-d) The normalized stochastic response and its log-derivative map the stochastic band structure throughout the $(\omega, k)$ dispersion plane.[]{data-label="fig4"}](fig4){width="85mm"}
As an example, let us consider the phononic crystal of pillars of Fig. \[fig4\]. A unit-cell of the crystal shown in Fig. \[fig4\](a) consists of a cylindrical pillar of the same material as the substrate, silicon. The crystal has a square lattice with lattice constant $a$. For obvious practical reasons, the unit-cell has to be limited to a certain depth. Fig. \[fig4\](b) shows the band structure for surface guided waves computed according to the method in Ref. [@khelifPRB2010]. The overlayed sound cone indicates the bands that are removed from consideration; those bands are actually strongly dependent on the substrate thickness and most of them are obviously spurious Lamb waves.
In order to apply the stochastic band structure method, the semi-infinite radiation medium has to be replaced a finite region. A solution could be to couple the solution in a finite crystal layer with an homogeneous radiation medium. This is however applicable only in specific cases for which the Green’s function is known explicitly, such as isotropic infinite media. Instead, we approximate numerically the semi-infinite substrate by an elastic PML, as summarized in Appendix \[app2\]. A Bloch-Floquet stochastic body force is applied in the layer region.
The stochastic response and its log-derivative are shown in Fig. \[fig4\](c-d). Below the sound cone, the stochastic band structure is very similar to the non-radiative band structure, as expected. Inside the sound cone, however, spurious bands are removed and radiation damping associated with each band can be easily evaluated. In particular, the avoided crossings appearing at the intersection of local resonances of the pillars with propagating surface waves become clearly apparent. In the classical band structure, they were damaged by interference with the spurious Lamb waves.
Leaky guided waves in the light cone {#sec3.3}
------------------------------------
Waveguides are structures with one invariance axis that are able to confine propagating waves around a central core. They have many applications in engineering, starting with the optical fiber. There are different guidance mechanisms, including guidance provided by a photonic or a phononic band gap, but the simplest guidance mechanism is total internal reflection. For simplicity, we will consider optical waves in the remaining of this section. As depicted in Fig. \[fig5\](a), waves can be guided in a ’slow’ core surrounded by a ’fast’ cladding, providing the dispersion of the guided waves lies in between the light lines of the core and the cladding [@marcuseBOOK1991]. When this condition is met, the dispersion point $(\omega, k)$ is inside the light cone of the core but outside the light cone of the cladding. The light field is then sinusoidal in the core and evanescent in the cladding, implying confinement around the core as light propagates along the axis of the waveguide.
![ Optical guidance along a free-standing silicon microwire in air ($d=1$ ). (a) Axis $x_3$ is an invariance axis along which the wavenumber $k$ is counted. (b) The dispersion diagram for waves guided by total internal reflection is divided in three different regions: guided waves can be either fully evanescent, guided inside the slow core, or coupled to the radiation modes of air. (c) The computation domain for the stochastic band structure is divided between core, cladding, and a perfectly matched layer approximating radiation to infinity. (d) The stochastic band structure maps the dispersion relation throughout the $(\omega, k)$ dispersion plane, especially informing on radiation damping inside the light cone for air. []{data-label="fig5"}](fig5){width="80mm"}
Now there are two more regions of dispersion space in Fig. \[fig5\](a). In the doubly-evanescent region, there are no guided solutions in optical dielectric waveguides [@marcuseBOOK1991] but interface guided waves may exist in other types of systems, such as the Stoneley wave of elastic media [@auldBOOK1973]. In the radiation region, however, there exist radiation guided waves that can be regarded as a combination of waves propagating in the core with radiation modes of the infinite cladding. Many different methods have been proposed in order to obtain radiation guided waves and their approximation with leaky waves; see for instance Ref. [@huAOP2009]. In the following we illustrate that the stochastic band structure method yields a direct mapping of dispersion and an estimate for propagation losses due to radiation.
Let us consider a free-standing silicon microwire in air, as depicted in Fig. \[fig5\](b). Maxwell’s equations in dielectric media lead to the following vector wave equation for the magnetic field vector $\bm{H}$ $$\begin{aligned}
\nabla \times \left( \frac{1}{\epsilon} \nabla \times \bm{H} \right) - \frac{\omega^2}{c^2} \bm{H} &= \bm{f} ,
\label{eq14}\end{aligned}$$ with $\epsilon$ the relative dielectric constant and $c$ the speed of light in a vacuum. Because of invariance of the structure along axis $x_3$, solutions can be written as $\bm{\bar{H}}(x_1,x_2) \exp(\imath(\omega t - k x_3))$. Hence the unknown becomes the modal shape $\bm{\bar{H}}(x_1,x_2)$ defined in two-dimensional transverse space. As a result, it is enough the represent the solution on a two-dimensional mesh. Appendix \[app3\] gives a hybrid-mode variational formulation of guided wave optics, from which an equation system such as Eq. can be obtained, as well as the expression of a PML to terminate the computation domain.
![ Optical guidance along a free-standing silicon microwire in air ($d=1$ ). (a) The log-derivative of the stochastic response is shown for $kd/(2\pi)=0.7$. (b) The first damped Lorentzian resonance appearing in the light cone is fitted to the model of Eq. . (c) The magnetic field distribution at the maximum of the stochastic response in (b) approximates a particular radiation guided mode. (d-e) Same as (b-c) for the second resonance. (f-g) Same as (b-c) for the third resonance. []{data-label="fig6"}](fig6){width="80mm"}
Considering guided wave stochastic excitation inside the fiber core, we obtain the stochastic band structure of Fig. \[fig5\](c). In the guidance region below the light cone for air, the usual sequence of guided modes leaves Lorentzian traces, with higher order guided modes appearing as the frequency is increased. In the radiation region, i.e. inside the light cone for air, the stochastic band structure shows that higher order guided modes are continuously connected to transverse resonances of the microwire, with a cut-off frequency at $k=0$. Bands in the light cone for air are affected by radiation damping, which can be estimated from the damped Lorentzian functions. A vertical cross-section of the stochastic band structure at $kd/(2\pi)=0.7$ is presented in Fig. \[fig6\]. For frequencies under the light line the bands for guided modes are undamped. Above the light line, however, all Lorentzian functions acquire a certain level of damping that can be attributed to coupling with radiation modes. Each of them can be fitted individually to the model of Eq. , from which the quality factor can estimated as $Q \approx \beta/\Re(\lambda_n)$. Three examples of the fitting procedure are given in Fig. \[fig6\], together with the magnetic field distributions obtained at each maxima of the stochastic response.
Discussion and conclusions
==========================
As a summary, we have introduced the stochastic band structure as a generic method to obtain a mapping of the dispersion relation for wave equations. Instead of relying on an eigenvalue problem or an equivalent root-finding method, we apply a stochastic excitation in a unit cell, with given wavenumber and frequency, and observe the response to this excitation. Plotting the stochastic response yields a mapping of the resolvant set, of which the spectrum is the singular complement. Close to an eigenvalue, the solution to the forced problem is proportional to the eigenfunction and the response has a simple damped Lorentzian shape, containing information on damping in both time and space. As we have illustrated, the method (i) can be applied to frequency-dependent material loss – it would work for $k$-dependent loss too –, (ii) takes into account radiation in an infinite or a semi-infinite medium, by combining it with a perfectly matched layer or other numerical technique, (iii) and reveals radiating guided waves and resonant modes of vibration.
Compared to eigenvalue-based methods, we don’t need to assume the existence of an explicit $\omega(k)$ or $k(\omega)$ functional dispersion relation. Instead, an implicit response $E(\omega, k)$ is obtained that is similar to a local density of states. The full complex $(\omega, k)$ dispersion space can be explored if desired, though we have only considered its real restriction in this paper. In the case of material loss, it would be interesting to compare further the stochastic band structure with the complex band structure [@davanccoOE2007; @fietzOE2011; @laudePRB2009; @husseinPRB2009; @moiseyenkoPRB2011].
The numerical efficiency of the stochastic band structure is rather poor, because the forced problem has to be solved for each dispersion point, and hence a great number of times in most practical problems. It thus should not be used to replace an eigenvalue computation when the latter is possible. The method is also best suited to small unit cells, as those used for artificial crystal and waveguide problems.
As we have shown, the result appears to be independent to a large extent from the precise random realization of the driving force, i.e. it is almost deterministic even though the generating mechanism is stochastic. Clearly, this is possible because the stochastic excitation only appears as the right-hand-side of a linear equation; the singularities of the propagation operator filter out the solution to deliver only the eigenfunctions as maxima of the response. By analogy with thermal fluctuations generating acoustic phonons, as observed for instance via Brillouin light scattering [@carlottiAP2018] or from visualization of the vibration modes of nanomechanical resonators from thermal noise [@bunchS2007; @tsioutsiosNL2017], the response has fluctuations but there is little doubt that all eigenmodes are excited for each noise realization.
We note that photonic band structures have been obtained via finite-difference time-domain (FDTD) computations [@chanPRB1995]. The FDTD method was extended to metal-dielectric [@baidaPRB2006] and phononic band structures as well [@tanakaPRB2000; @sigalasJAP2000]. It works by applying a spatially random excitation, with a temporal excitation, to a finite computation domain terminated by periodic boundary conditions. By computing a Fourier transform of the solution as a function of time, a response similar to the one we have introduced is obtained. A difference is that we work in the frequency - wavenumber domain directly, so the frequency resolution can be arbitrarily high, whereas it takes an increasingly long computation time with FDTD. As a result, the stochastic band structure is adapted to resonant structures and very low group velocities. Furthermore, the computation error is not growing as a function of time, as with FDTD, but is instead increasing with frequency, because the mesh captures less and less of the wave details that are of the order of the wavelength.
Finally, we suggest that the stochastic band structure can be computed for any medium supporting wave propagation as described by a time-harmonic wave equation or Helmholtz equation, which includes pressure waves in fluids, water waves, elastic waves in solids, electromagnetic waves described with Maxwell’s equations – including plasmons,– or structures described with Schrödinger’s equation.
Acknowledgments {#acknowledgments .unnumbered}
===============
This work was supported by the Agence Nationale de la Recherche through the Labex ACTION program (grant No. ANR-11-LABX-0001-01). MK was supported by a grant from Région de Bourgogne Franche-Comté.
Mathematical models
===================
In this appendix, we summarize the mathematical models that were used to generate the results in Section \[sec3\].
Sonic crystal {#app1}
-------------
A weak form of the linear acoustic equation is [@ihlenburgBOOOK2006] $$\begin{aligned}
\int_\Omega \nabla q^* \cdot \left( \frac{1}{\rho} \nabla p \right)
- \omega^2 \int_\Omega q^* \frac{1}{B} p &= \int_\Omega q^* f\end{aligned}$$ with $q$ test functions taken in the same functional space as the solution $p$. $\Omega$ is the domain of definition, i.e. a unit cell of the sonic crystal. Assuming a Bloch wave form for all field quantities, – i.e. $p(\bm{r},t)=\bar{p}(\bm{r})\exp(\imath(\omega t - \bm{k}\cdot\bm{r}))$, $q(\bm{r},t)=\bar{q}(\bm{r})\exp(\imath(\omega t - \bm{k}\cdot\bm{r}))$, $f(\bm{r},t)=\bar{f}(\bm{r})\exp(\imath(\omega t - \bm{k}\cdot\bm{r}))$, – we obtain a weak form for the periodic parts as [@laudeBOOK2015] $$\begin{aligned}
\int_\Omega (\nabla \bar{q} - \imath \bm{k} \bar{q})^* \cdot \left( \frac{1}{\rho} (\nabla \bar{p} - \imath \bm{k} \bar{p}) \right) \nonumber \\
- \omega^2 \int_\Omega \bar{q}^* \frac{1}{B} \bar{p} = \int_\Omega \bar{q}^* \bar{f}\end{aligned}$$ with periodic boundary conditions applied on pairs of external boundaries. When the applied force vanishes, this is an eigenvalue equation giving $\lambda=\omega^2$ as a function of $\bm{k}$. When the applied forcing term is non zero, this is an equation system of the form of Eq. . The corresponding total energy of the solution is computed as $$\begin{aligned}
\langle H p, p \rangle &= \frac{1}{2} \int_\Omega \nabla p^* \cdot \left( \frac{1}{\rho} \nabla p \right) + \frac{1}{2} \; \omega^2 \int_\Omega p^* \frac{1}{B} p\end{aligned}$$
Surface phononic crystal {#app2}
------------------------
Since the stress and strain tensors are symmetric, we can employ the contracted notation for symmetric pairs of indices [@auldBOOK1973]: the contracted indices $I=(ij)$ and $J=(kl)$ run from 1 to 6 according to the rule $1=(11)$, $2=(22)$, $3=(33)$, $4=(23)$, $5=(13)$, and $6=(12)$. With the definitions $T_I=T_{ij}$, and $S_J=S_{kl}$ for $I=1,2,3$ and $S_J=2S_{kl}$ for $I=4,5,6$, Hooke’s law can be written $T_I = c_{IJ} S_J$. Considering test functions $\bm{v}$ taken in the same functional space as the solution $\bm{u}$, a weak form of the elastodynamic equation for Bloch waves is [@husseinPRS2009; @laudeBOOK2015] $$\begin{aligned}
\int_\Omega S_I(\bm{v})^* c_{IJ} S_J(\bm{u}) - \omega^2 \int_\Omega \bar{\bm{v}}^* \cdot \rho \bar{\bm{u}} = \int_\Omega \bar{\bm{v}}^* \cdot \bar{\bm{f}}
\label{eqA4}\end{aligned}$$ with $$\begin{aligned}
S_1(\bm{u}) &= \frac{\partial \bar{u}_1}{\partial x_1} -\imath k_1 \bar{u}_1 , \\
S_2(\bm{u}) &= \frac{\partial \bar{u}_2}{\partial x_2} -\imath k_2 \bar{u}_2 , \\
S_3(\bm{u}) &= \frac{\partial \bar{u}_3}{\partial x_3} -\imath k_3 \bar{u}_3 , \\
S_4(\bm{u}) &= \frac{\partial \bar{u}_3}{\partial x_2} + \frac{\partial \bar{u}_2}{\partial x_3} -\imath (k_3 \bar{u}_2 + k_2 \bar{u}_3) , \\
S_5(\bm{u}) &= \frac{\partial \bar{u}_3}{\partial x_1} + \frac{\partial \bar{u}_1}{\partial x_3} -\imath (k_3 \bar{u}_1 + k_1 \bar{u}_3) , \\
S_6(\bm{u}) &= \frac{\partial \bar{u}_2}{\partial x_1} + \frac{\partial \bar{u}_1}{\partial x_2} -\imath (k_2 \bar{u}_1 + k_1 \bar{u}_2) .\end{aligned}$$ Eq. is used to solve an eigensystem in case the forcing term vanishes or to obtain an equation system of the form of Eq. for the stochastic band structure. The total energy of the solution is computed as $$\begin{aligned}
\langle H \bm{u}, \bm{u} \rangle &= \frac{1}{2} \int_\Omega S_I(\bm{u})^* c_{IJ} S_J(\bm{u}) \nonumber \\
&+ \frac{1}{2} \omega^2 \int_\Omega \bm{u}^* \cdot \rho \bm{u}\end{aligned}$$
The perfectly matched layer (PML) is next introduced to transform the infinite problem into a finite problem. The idea is to seek a solution to the dynamical equations by using a coordinate transform from a complex infinite space, that admits evanescent waves as eigenfunctions instead of plane waves, to the real finite space [@hugoninJOSAA2005; @zschiedrichJCAM2006]. Given coordinates $\bm{x}$ of real space, we introduce coordinates $\bm{y}$ of complex space via a transform $y_i=y_i(\bm{x})$. Upon introducing the Jacobian matrix $$J_{ij}=\frac{\partial y_i}{\partial x_j}$$ we can rewrite the integrals of the variational formulation. In an integral, the integration element changes proportionally to $\det(J)$. Consider a function $u(\bm{x})=\tilde{u}(\bm{y})$. The gradient of the displacement vector transforms as $$\nabla \tilde{u} = \frac{\partial \tilde{u}}{\partial y_i} = \frac{\partial x_j}{\partial y_i} \frac{\partial u}{\partial x_j} = J^{-t} \nabla u.$$ The inverse Jacobian has elements $J^{-1}_{ij} = \frac{\partial x_i}{\partial y_j}$ and $(...)^t$ denotes the transposition operator.
In the case of the elastodynamic equation, the weak form becomes [@laudeBOOK2015] $$\begin{aligned}
\int_\Omega S_I(\bm{v})^* c_{IJ} S_J(\bm{u}) \det(J) &- \omega^2 \int_\Omega \bar{\bm{v}}^* \cdot \rho \bar{\bm{u}} \det(J) \nonumber \\
&= \int_\Omega \bar{\bm{v}}^* \cdot \bar{\bm{f}}\end{aligned}$$ with the modified definition of the strains $$\begin{aligned}
S_{1}(\bm{u}) &= J^{-1}_{m,1} u_{1,m}, \\
S_{2}(\bm{u}) &= J^{-1}_{m,2} u_{2,m}, \\
S_{3}(\bm{u}) &= J^{-1}_{m,3} u_{3,m}, \\
S_{4}(\bm{u}) &= J^{-1}_{m,2} u_{3,m} + J^{-1}_{m,3} u_{2,m}, \\
S_{5}(\bm{u}) &= J^{-1}_{m,1} u_{3,m} + J^{-1}_{m,3} u_{1,m}, \\
S_{6}(\bm{u}) &= J^{-1}_{m,1} u_{2,m} + J^{-1}_{m,2} u_{1,m},\end{aligned}$$ where summation over $m$ is implied and $u_{i,m}=\frac{\partial \bar{u}_i}{\partial x_m} -\imath k_m \bar{u}_i$.
In practice we use the following coordinate transform for the SAW problem, for $x_3<-h$, $$\begin{aligned}
y_3 &= x_3 + \frac{i}{\omega} \int_{-h}^{x_3} \sigma(s) \mathrm{d}s\end{aligned}$$ with $\sigma(s) = \beta |s + h| / w^2$, where $w$ is the PML width and $\beta$ is a numerical coefficient whose value is tuned to optimize absorption. In Fig. \[fig4\], the values $h=2a$ and $w=a$ were used.
Dielectric optical waveguide {#app3}
----------------------------
A weak form of the guided-wave optical equation is [@jinBOOK2002] $$\begin{aligned}
\int_\Omega \frac{1}{\epsilon} \left( {\textrm{rot}(\bar{\bm{H}}')} {\textrm{rot}(\bar{\bm{H}})} + {\textrm{div}(\bar{\bm{H}}')} {\textrm{div}(\bar{\bm{H}})} + k^2 \bar{\bm{H}}' \cdot \bar{\bm{H}} \right) \nonumber \\
+ \int_{\delta\Omega} \bar{H}'_n {\textrm{div}(\bar{\bm{H}})} \left[ \frac{1}{\epsilon} \right]
- \frac{\omega^2}{c^2} \int_\Omega \bar{\bm{H}}' \cdot \bar{\bm{H}} = \int_\Omega \bar{\bm{H}}' \cdot \bar{\bm{f}} .
\label{eqA22}\end{aligned}$$ This expression is obtained by keeping as unknowns only the first two components of $\bm{H}$, i.e. $\bm{H} = (H_1, H_2)$, since the third component is set by the auxiliary Maxwell equation $\nabla \cdot \bm{H} = 0$. Here we use the transverse divergence ${\textrm{div}(\bar{\bm{H}})} = \bar{H}_{1,1} + \bar{H}_{2,2}$ and transverse rotational ${\textrm{rot}(\bar{\bm{H}})} = \bar{H}_{2,1} - \bar{H}_{1,2}$, $\bar{H}'_n$ is the normal component of $\bar{\bm{H}}'$ at the boundary $\delta\Omega$, and $\left[ \frac{1}{\epsilon} \right]$ denotes the jump of the permittivity. Note that the boundary integral appears because on the non continuity of the electric field at the interface between different dielectric media. In Fig. \[fig5\], the boundary $\delta\Omega$ is the interface between silicon and air.
The total energy of the solution is computed as $$\begin{aligned}
E = & \frac{1}{2} \int_\Omega \frac{1}{\epsilon} \left( {\textrm{rot}(\bar{\bm{H}})} {\textrm{rot}(\bar{\bm{H}})} \right. \nonumber \\
& \left. + {\textrm{div}(\bar{\bm{H}})} {\textrm{div}(\bar{\bm{H}})} + k^2 \bar{\bm{H}} \cdot \bar{\bm{H}} \right) \nonumber \\
& + \frac{1}{2} \int_{\delta\Omega} \bar{H}_n {\textrm{div}(\bar{\bm{H}})} \left[ \frac{1}{\epsilon} \right] \nonumber \\
& + \frac{\omega^2}{2c^2} \int_\Omega \bar{\bm{H}} \cdot \bar{\bm{H}} .
\label{eqA23}\end{aligned}$$
Again, Eq. leads to an equation system of the form of Eq. . A PML is constructed as in the elastic case from a complex coordinate transformation, now along both axes $x_1$ and $x_2$. The integrands of the first and the third integrals in Eq. are multiplied with $\det(J)$, while the definition of the transverse divergence and rotational become $$\begin{aligned}
{\textrm{div}(\bar{\bm{H}})} &= J^{-1}_{m,1} \bar{H}_{1,m} + J^{-1}_{m,2} \bar{H}_{2,m}, \\
{\textrm{rot}(\bar{\bm{H}})} &= J^{-1}_{m,1} \bar{H}_{2,m} - J^{-1}_{m,2} \bar{H}_{1,m},\end{aligned}$$ where summation over $m$ is implied.
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---
abstract: 'The KLOE experiment at the DA$\Phi$NE $\phi$-factory has performed a new precise measurement of the pion form factor using Initial State Radiation events, with photons emitted at small polar angle. Results based on an integrated luminosity of 240 pb$^{-1}$ and extraction of the $\pi\pi$ contribution to $a_\mu$ in the mass range $[0.35,0.95]$ GeV$^2$ are presented, the systematic uncertainty is reduced with respect to the published KLOE result.'
address: 'Università degli Studi e Sezione INFN “Roma TRE”, Via della Vasca Navale 84, 00146 Roma, Italy'
author:
- 'KLOE Collaboration[^1] presented by Federico Nguyen (*e-mail address*: nguyen@fis.uniroma3.it)'
title: 'A precise new KLOE measurement of $F_\pi$ with ISR events and determination of $\pi\pi$ contribution to $a_\mu$ for $[0.35,0.95]$ GeV$^2$'
---
Introduction {#sec:1}
============
The anomalous magnetic moment of the muon has recently been measured to an accuracy of 0.54 ppm [@Bennett:2006fi]. The main source of uncertainty in the value predicted [@Jegerlehner:2007xe] in the Standard Model is given by the hadronic contribution, $a_\mu^{hlo}$, to the lowest order. This quantity is estimated with a dispersion integral of the hadronic cross section measurements.
In particular, the pion form factor, $F_\pi$, defined via $\sigma_{\pi\pi}\equiv\sigma_{e^+ e^-\to\pi^+\pi^-}\propto
s^{-1}\beta^3_\pi(s) |F_\pi(s)|^2$, accounts for $\sim70\%$ of the central value and for $\sim60\%$ of the uncertainty in $a_\mu^{hlo}$.
The KLOE experiment already published [@Aloisio:2004bu] a measurement of $F_\pi$ with the method described below, using an integrated luminosity of 140 pb$^{-1}$, taken in 2001, henceforth referred to as KLOE05.
Measurement of $\sigma(e^+e^-\to\pi^+\pi^-\gamma)$ at DA$\Phi$NE {#sec:2}
================================================================
![Fiducial volume for the small angle photon (narrow cones) and for the the pion tracks (wide cones).[]{data-label="fig:1"}](pfdetector_new2.eps "fig:"){width="14pc" height="15.6pc"}-0.5cm
DA$\Phi$NE is an $e^+ e^-$ collider running at $\sqrt{s}\simeq M_\phi$, the $\phi$ meson mass, which has provided an integrated luminosity of about 2.5 fb$^{-1}$ to the KLOE experiment up to year 2006. In addition, about 250 pb$^{-1}$ of data have been collected at $\sqrt{s}\simeq 1$ GeV, in 2006. Present results are based on 240 pb$^{-1}$ of data taken in 2002. The KLOE detector consists of a drift chamber [@Adinolfi:2002uk] with excellent momentum resolution ($\sigma_p/p\sim 0.4\%$ for tracks with polar angle larger than $45^\circ$) and an electromagnetic calorimeter [@Adinolfi:2002zx] with good energy ($\sigma_E/E\sim 5.7\%/\sqrt{E~[\mathrm{GeV}]}$) and precise time ($\sigma_t\sim 54~\mathrm{ps}/\sqrt{E~[\mathrm{GeV}]}\oplus
100~\mathrm{ps}$) resolution.
At DA$\Phi$NE, we measure the differential spectrum of the $\pi^+\pi^-$ invariant mass, $M_{\pi\pi}$, from Initial State Radiation (ISR) events, $e^+ e^-\to\pi^+\pi^-\gamma$, and extract the total cross section $\sigma_{\pi\pi}\equiv\sigma_{e^+ e^-\to\pi^+\pi^-}$ using the following formula [@Binner:1999bt]: $$M_{\pi\pi}^2~ \frac{{{\rm d}}\sigma_{\pi\pi\gamma}}
{{{\rm d}}M_{\pi\pi}^2} = \sigma_{\pi\pi}
(M_{\pi\pi}^2)~ H(M_{\pi\pi}^2)~,
\label{eq:1}$$ where $H$ is the radiator function. This formula neglects Final State Radiation (FSR) terms.
The cross section for ISR photons has a divergence in the forward angle (relative to the beam direction), such that it dominates over FSR photon production. The fiducial volume – shown in Fig. \[fig:1\] – is based on the following criteria:
- two tracks with opposite charge within the polar angle range $50^\circ<\theta<130^\circ$;
- small angle photon, $\theta_\gamma<15^\circ\,
(\theta_\gamma>165^\circ)$, the photon is not explicitly detected and its direction is reconstructed from the track momenta in the $e^+e^-$ center of mass system, $\vec{p}_\gamma=-(\vec{p}_{\pi^+} +\vec{p}_{\pi^-})$.
The above criteria result in events with good reconstructed tracks and enhance the probability of having an ISR photon. Furthermore,
- FSR at the Leading Order is reduced to the $0.3\%$ level;
- the contamination from the resonant process $e^+e^-\to
\phi\to\pi^+\pi^-\pi^0$ – where at least one of photons coming from the $\pi^0$ is lost – is reduced to the level of $\sim5\%$.
Discrimination of $\pi^+\pi^-\gamma$
![Signal and background distributions in the $m_{trk}$-$M^2_{\pi\pi}$ plane; the selected area is shown.[]{data-label="fig:2"}](trackmass_qq.eps "fig:"){width="16.5pc" height="15.5pc"}-0.5cm
from $e^+ e^-\to e^+ e^-\gamma$ events is done via particle identification [@knote:192] based on the time of flight, on the shape and the energy of the clusters associated to the tracks. In particular, electrons deposit most of their energy in the first planes of the calorimeter while minimum ionizing muons and pions release uniformly the same energy in each plane. An event is selected if at least one of the two tracks has not being identified as an electron. Fig. \[fig:2\] shows that contaminations from the processes $e^+e^-\to\mu^+\mu^-\gamma$ and $\phi\to\pi^+\pi^-\pi^0$ are rejected by cuts on the track mass variable, $m_{trk}$, defined by the four-momentum conservation, assuming a final state consisting of two particles with the same mass and one photon
Improvements with respect to the published analysis {#sec:3}
===================================================
The analysis of data taken since 2002 benefits from cleaner and more stable running conditions of DA$\Phi$NE, resulting in less machine background and improved event filters than KLOE05. In particular, the following changes are implemented:
- a new trigger level was added at the end of 2001 to eliminate the 30% loss from pions penetrating to the outer calorimeter plane and thus were misidentified as cosmic rays events. For the 2002 data, this inefficiency has decreased down to 0.2%, as evaluated from a control sample;
- the offline background filter, which contributed the largest experimental systematic uncertainty to the published work [@Aloisio:2004bu], has been improved. The filter efficiency increased from 95% to 98.5%, with negligible systematic uncertainty;
- the vertex requirement on the two tracks – used in KLOE05 – is not applied, therefore eliminating the systematic uncertainty from this source.
The absolute normalization of the data sample is measured using large angle Bhabha scattering events, $55^\circ<\theta<125^\circ$.
![Differential cross section in the $\pi\pi$ invariant mass for the process $e^+e^-\to\pi^+\pi^-\gamma$, from an integrated luminosity of 240 pb$^{-1}$.[]{data-label="fig:3"}](diffxsecppg.eps "fig:"){width="17.5pc" height="16.5pc"}-0.5cm
The integrated luminosity, $\mathcal{L}$, is obtained [@Ambrosino:2006te] from the observed number of events, divided by the effective cross section evaluated from the Monte Carlo generator `Babayaga` [@Carloni; @Calame:2000pz], including QED radiative corrections with the parton shower algorithm, inserted in the code simulating the KLOE detector. An updated version of the generator, `Babayaga@NLO` [@Balossini:2006wc], decreased the predicted cross section by 0.7%, while the theoretical relative uncertainty improved from 0.5% to 0.1%. The experimental relative uncertainty on $\mathcal{L}$ is 0.3%.
Evaluation of $F_\pi$ and $a_\mu^{\pi\pi}$ {#sec:4}
==========================================
The $\pi\pi\gamma$ differential cross section is obtained from the observed spectrum, $N_{obs}$, after subtracting the residual background events, $N_{bkg}$, and correcting for the selection efficiency, $\varepsilon_{sel}(M_{\pi\pi}^2)$, and the luminosity: $$\frac{{{\rm d}}\sigma_{\pi\pi\gamma}}
{{{\rm d}}M_{\pi\pi}^2} = \frac{N_{obs}-N_{bkg}}
{\Delta M_{\pi\pi}^2}\, \frac{1}{\varepsilon_{sel}(M_{\pi\pi}^2)~ \mathcal{L}}~ ,
\label{eq:2}$$ Fig. \[fig:3\] shows the differential cross section from the selected events.
After unfolding, with the inversion of the resolution matrix obtained from Monte Carlo,
[|l|c|c|]{}\
& KLOE05 & KLOE08\
offline filter & 0.6 & negligible\
background & 0.3 & 0.6\
$m_{trk}$ cuts & 0.2 & 0.2\
$\pi$/e ID & 0.1 & 0.1\
vertex & 0.3 & not used\
tracking & 0.3 & 0.3\
trigger & 0.3 & 0.1\
acceptance & 0.3 & 0.1\
FSR & 0.3 & 0.3\
luminosity & 0.6 & 0.3\
$H$ function eq.(\[eq:1\]) & 0.5 & 0.5\
VP & 0.2 & 0.1\
total & 1.3 & 1.0\
for events with both an initial and a final photon, the differential cross section is corrected using `Phokhara` for shifting them from $M_{\pi\pi}$ to the virtual photon mass, $M_{\gamma^*}$. Then, it is divided by the radiator function (`Phokhara` setting the pion form factor $F_\pi=1$) to obtain the measured total cross section $\sigma_{\pi\pi(\gamma)}(M_{\gamma^*})$, of eq.(\[eq:1\]).
The pion form factor is evaluated subtracting the FSR term, $\eta_{FSR}$ [@Schwinger:1989ix], $$\sigma_{\pi\pi(\gamma)}~=~ \frac{\pi}{3}~
\frac{\alpha_{em}^2\,\beta_\pi^3}{M_{\gamma^*}^2}~
|F_\pi|^2 \left(1+\eta_{FSR}\right)~.$$ The cross section for the $a_\mu^{\pi\pi}$ dispersion integral – inclusive of FSR – is obtained after removing vacuum polarization, VP, effects [@Jegerlehner:2006ju], $$\sigma_{\pi\pi(\gamma)}^{bare}=\sigma_{\pi\pi(\gamma)}
\left[\frac{\alpha_{em}(0)}{\alpha_{em}(M_{\gamma^*})}\right]^2~.$$ Table \[tab:1\] shows the list of relative systematic uncertainties in the evaluation of $a_\mu^{\pi\pi}$ in the mass range \[0.35,0.95\] GeV$^2$, for KLOE05 and for the analysis of this new data set, KLOE08.
Results {#sec:5}
=======
[|l|c|]{}\
published 05 & $388.7~\pm~0.8_{stat}~\pm~4.9_{sys}$\
updated 05 & $384.4~\pm~0.8_{stat}~\pm~4.6_{sys}$\
new data 08 & $389.2~\pm~0.6_{stat}~\pm~3.9_{sys}$\
\
CMD-2 [@Ignatov:2008] & $361.5~\pm~1.7_{stat}~\pm~2.9_{sys}$\
SND [@Ignatov:2008] & $361.0~\pm~2.0_{stat}~\pm~4.7_{sys}$\
KLOE08 & $358.0~\pm~0.6_{stat}~\pm~3.4_{sys}$\
The published analysis, updated for the new Bhabha cross section and for a bias in the trigger correction [@Ambrosino:2007vj], is compared with KLOE08, and also with the results obtained by the VEPP–2M experiments [@Ignatov:2008], in the mass range $M_{\pi\pi}\in[630,958]{\mathrm{\ MeV}}$. Table \[tab:2\] shows the good agreement amongst KLOE results, and also with the published
![Comparison of the pion form factor measured from CMD-2, SND and KLOE, where for this latter statistical errors (light band) and summed statistical and systematic errors (dark band) are shown.[]{data-label="fig:4"}](diffcomplbls.eps "fig:"){width="18pc" height="17pc"}-0.5cm
CMD-2 and SND values. They agree with KLOE08 within one standard deviation.
The band of Fig. \[fig:4\] shows the KLOE08 pion form factor smoothed – accounting for both statistical and systematic errors – and normalized to fix the 0 in the ordinate scale. CMD-2 and SND data points are interpolated and compared to this band, in the same panel.
Conclusions and outlook
=======================
We obtained the $\pi\pi$ contribution to $a_\mu$ in the mass range $M_{\pi\pi}^2\in[0.35,0.95]{\mathrm{\ GeV}}^2$ integrating the $\pi\pi\gamma$ differential cross section for the ISR events $e^+ e^-\to\pi^+\pi^-\gamma$, with photon emission at small angle:
1. KLOE08 confirms KLOE05, but with more accuracy;
2. KLOE08 is in agreement within one standard deviation with SND and CMD-2 values in the mass range $M_{\pi\pi}\in[630,958]{\mathrm{\ MeV}}$ [@Ignatov:2008].
Thus, $a_\mu^{\pi\pi}$ is measured to an accuracy of 0.1%. Independent analyses are in progress:
- measure $\sigma_{\pi\pi(\gamma)}$ using detected photons emitted at large angle, which would improve the knowledge of the FSR interference effects from KLOE $f_0(980)$ measurements [@Ambrosino:2005wk; @Ambrosino:2006hb];
- measure the pion form factor directly from the ratio, bin-by-bin, of $\pi^+\pi^-\gamma$ to $\mu^+\mu^-\gamma$ spectra [@Muller:2006bk] (see Fig. \[fig:2\] for the selection of $\mu\mu\gamma$ events);
- extract the pion form factor from data taken at $\sqrt{s}=1$ GeV, off the $\phi$ resonance, where $\pi^+\pi^-\pi^0$ background is negligible.
[99]{} G. W. Bennett [*et al.*]{} \[Muon G-2 Collaboration\], Phys. Rev. D [**73**]{} (2006) 072003 F. Jegerlehner, “Essentials of the Muon g-2”, arXiv:hep-ph/0703125 A. Aloisio [*et al.*]{} \[KLOE Collaboration\], Phys. Lett. B [**606**]{} (2005) 12 A. Denig [*et al.*]{} \[KLOE Collaboration\], KLOE Note 192, July 2004, [`www.lnf.infn.it/kloe/pub/knote/kn192.ps`]{} M. Adinolfi [*et al.*]{}, \[KLOE Collaboration\] Nucl. Instrum. Meth. A [**488**]{} (2002) 51 M. Adinolfi [*et al.*]{}, \[KLOE Collaboration\] Nucl. Instrum. Meth. A [**482**]{} (2002) 364 S. Binner, J. H. Kühn and K. Melnikov, Phys. Lett. B [**459**]{} (1999) 279 G. Rodrigo, H. Czyż, J. H. Kühn and M. Szopa, Eur. Phys. J. C [**24**]{} (2002) 71 H. Czyż, A. Grzelinska, J. H. Kühn and G. Rodrigo, Eur. Phys. J. C [**33**]{} (2004) 333 H. Czyż and E. Nowak-Kubat, Phys. Lett. B [**634**]{} (2006) 493 F. Ambrosino [*et al.*]{} \[KLOE Collaboration\], Eur. Phys. J. C [**47**]{} (2006) 589 C. M. Carloni Calame [*et al.*]{}, Nucl. Phys. B [**584**]{} (2000) 459 G. Balossini [*et al.*]{}, Nucl. Phys. B [**758**]{} (2006) 227 F. Jegerlehner, Nucl. Phys. Proc. Suppl. [**162**]{} (2006) 22 J. S. Schwinger, “Particles, Sources, and Fields. VOL. 3”, [*Redwood City, USA: ADDISON-WESLEY (1989) 318 P. (Advanced Book Classics Series)*]{} F. Ambrosino [*et al.*]{} \[KLOE Collaboration\], arXiv:0707.4078 \[hep-ex\] F. Ignatov, these proceedings F. Ambrosino [*et al.*]{} \[KLOE Collaboration\], Phys. Lett. B [**634**]{} (2006) 148 F. Ambrosino [*et al.*]{} \[KLOE Collaboration\], Eur. Phys. J. C [**49**]{} (2007) 473 S. E. Müller and F. Nguyen [*et al.*]{} \[KLOE Collaboration\], Nucl. Phys. Proc. Suppl. [**162**]{} (2006) 90
[^1]: F. Ambrosino, A. Antonelli, M. Antonelli, F. Archilli, C. Bacci, P. Beltrame, G. Bencivenni, S. Bertolucci, C. Bini, C. Bloise, S. Bocchetta, F. Bossi, P. Branchini, P. Campana, G. Capon, T. Capussela, F. Ceradini, F. Cesario, S. Chi, G. Chiefari, P. Ciambrone, F. Crucianelli, E. De Lucia, A. De Santis, P. De Simone, G. De Zorzi, A. Denig, A. Di Domenico, C. Di Donato, B. Di Micco, A. Doria, M. Dreucci, G. Felici, A. Ferrari, M. L. Ferrer, S. Fiore, C. Forti, P. Franzini, C. Gatti, P. Gauzzi, S. Giovannella, E. Gorini, E. Graziani, W. Kluge, V. Kulikov, F. Lacava, G. Lanfranchi, J. Lee-Franzini, D. Leone, M. Martemianov, M. Martini, P. Massarotti, W. Mei, S. Meola, S. Miscetti, M. Moulson, S. Müller, F. Murtas, M. Napolitano, F. Nguyen, M. Palutan, E. Pasqualucci, A. Passeri, V. Patera, F. Perfetto, M. Primavera, P. Santangelo, G. Saracino, B. Sciascia, A. Sciubba, A. Sibidanov, T. Spadaro, M. Testa, L. Tortora, P. Valente, G. Venanzoni, R. Versaci, G. Xu.
|
---
author:
- 'François Gay-Balmaz$^{1}$ and Hiroaki Yoshimura$^{2}$'
title: 'A free energy Lagrangian variational formulation of the Navier-Stokes-Fourier system'
---
\[theorem\][Remark]{}
Introduction
============
The dynamics of a viscous heat conducting fluid is governed by the Navier-Stokes-Fourier equations given by a system of PDEs describing the balance of fluid momentum, the balance of mass, and the conservation of energy. The latter can be equivalently formulated in terms of the entropy or the temperature. It is well-known that in absence of the irreversible processes of viscosity and heat conduction, these equations, as well as the general equations of reversible continuum mechanics, arise from the Hamilton principle applied to the Lagrangian trajectory of fluid particles.
In [@GBYo2016a; @GBYo2016b], we proposed a systematic extension of Hamilton’s principle to include irreversible processes by introducing the concept of thermodynamic displacements and making use of a generalization of the Lagrange-d’Alembert principle with nonlinear nonholonomic constraints. This approach covers both discrete and continuum systems and naturally involves the entropy as an independent variable. For a concrete use in applications, it is often more practical to use the temperature rather than the entropy as the independent variable. Temperature is indeed a much easier measurable quantity and the phenomenological coefficients (such as heat conductivity or viscosity) are naturally expressed in terms of the temperature rather than the entropy. In this case, the variational formulation must be expressed in terms of the free energy.
In this paper, we present a variational formulation for the Navier-Stokes-Fourier system based on a [*free energy Lagrangian*]{}, which complements the approach developed in [@GBYo2016b]. The variational derivation is first expressed in the material (or Lagrangian) description, from which the spatial (or Eulerian) description is deduced. The variational formulation follows from an infinite dimensional extension of the free energy Lagrangian variational formulation for nonequilibrium thermodynamics of discrete systems. It has a systematic structure which relies on the concepts of variational and phenomenological constraints.
Discrete systems and a free energy Lagrangian {#section_2}
=============================================
In this section we review the variational formulation for nonequilibrium thermodynamics of discrete (i.e., finite dimensional) systems developed in [@GBYo2016a]. The formulation is first given in terms of “classical Lagrangians", i.e., Lagrangians expressed in terms of the internal energy of the system. This naturally implies the use of the entropy $S$ as an independent variable in the variational formulation. Then, we present a variational formulation based on a *free energy Lagrangian* that allows the treatment of the temperature $T$ rather than the entropy as the independent variable.
Variational formulation of nonequilibrium thermodynamics of simple systems
--------------------------------------------------------------------------
We shall present the variational formulation by first considering *simple thermodynamic systems* before going into the general setting of the discrete systems. We follow the systematic treatment of thermodynamic systems presented in [@StSc1974], to which we also refer for the precise statement of the two laws of thermodynamics.
**Simple discrete systems.** A *discrete thermodynamic system* $ \boldsymbol{\Sigma} $ is a collection $ \boldsymbol{\Sigma} = \cup_{A=1}^N$ of a finite number of interacting simple thermodynamic systems $ \boldsymbol{\Sigma} _A $. By definition, a *simple thermodynamic system* is a macroscopic system for which one (scalar) thermal variable and a finite set of mechanical variables are sufficient to describe entirely the state of the system. From the second law of thermodynamics (e.g., [@StSc1974]), we can always choose the entropy $S$ as a thermal variable. A typical example of such a simple system is the one-cylinder problem. We refer to [@Gr1999] for a systematic treatment of this system via Stueckelberg’s approach.
**Variational formulation.** We now quickly review from [@GBYo2016a] the variational formulation of nonequilibrium thermodynamics for the particular case of simple closed systems.
Let $Q$ be the configuration manifold associated to the mechanical variables of the simple system. We denote by $TQ$ the tangent bundle to $Q$ and use the classical local notation $(q, v) \in TQ$ for the elements in the tangent bundle. Our approach is of course completely intrinsic and does not depend on the choice of coordinates on $Q$. The Lagrangian of a simple thermodynamic system is a function $$L: TQ \times \mathbb{R} \rightarrow \mathbb{R} , \quad (q, v, S) \mapsto L(q, v, S),$$ where $S \in\mathbb{R}$ is the entropy. We assume that the system involves exterior and friction forces given by fiber preserving maps $F^{\rm ext}, F^{\rm fr}:TQ\times \mathbb{R} \rightarrow T^* Q$, i.e., such that $F^{\rm fr}(q, v, S)\in T^*_qQ$, similarly for $F^{\rm ext}$, where $T^*Q$ is the cotangent bundle to $Q$. Finally we assume that the system is subject to an external heat power supply $P^{\rm ext}_H(t)$.
We say that a curve $(q(t),S(t)) \in Q \times \mathbb{R}$, $t \in [t _1 , t _2 ] \subset \mathbb{R}$ is a [*solution of the variational formulation of nonequilibrium thermodynamics*]{} if it satisfies the variational condition $$\label{LdA_thermo_simple}
\delta \int_{t _1 }^{ t _2}L(q , \dot q , S)dt +\int_{t_1}^{t_2}\left\langle F^{\rm ext}(q, \dot q, S), \delta q\right\rangle dt =0, \quad\;\;\; \textsc{\small Variational Condition}$$ for all variations $ \delta q(t) $ and $\delta S(t)$ subject to the constraint $$\label{CV_simple}
\frac{\partial L}{\partial S}(q, \dot q, S)\delta S= \left\langle F^{\rm fr}(q , \dot q , S),\delta q \right\rangle,\qquad\qquad\, \textsc{\small Variational Constraint}$$ with $ \delta q(t_1)=\delta (t_2)=0$, and also if it satisfies the phenomenological constraint $$\label{CK_simple}
\frac{\partial L}{\partial S}(q, \dot q, S)\dot S = \left\langle F^{\rm fr}(q, \dot q, S) , \dot q \right\rangle- P^{\rm ext}_H, \quad \textsc{\small Phenomenological Constraint}$$ where $ \dot q=\frac{dq}{dt}$ and $\dot S=\frac{dS}{dt}$.
From this variational formulation, we deduce the system of evolution equations for the simple thermodynamic system as $$\label{simple_systems}
\left\{
\begin{array}{l}
\displaystyle\vspace{0.2cm}\frac{d}{dt}\frac{\partial L}{\partial \dot q}- \frac{\partial L}{\partial q}= F^{\rm fr}(q, \dot q, S)+F^{\rm ext}(q, \dot q, S),\\
\displaystyle\frac{\partial L}{\partial S}\dot S= \left\langle F^{\rm fr}(q, \dot q, S), \dot q \right\rangle - P^{\rm ext}_H.
\end{array} \right.$$
\[PC\_VS\_VC\][ The explicit expression of the constraint involves phenomenological laws for the friction force $F^{\rm fr}$, this is why we refer to it as a *phenomenological constraint*. The associated constraint is called a *variational constraint* since it is a condition on the variations to be used in . Note that the constraint is nonlinear and also that one passes from the variational constraint to the phenomenological constraint by formally replacing the variations $ \delta q$, $\delta S$ by the time derivatives $ \dot q$, $\dot S$. Such a systematic correspondence between the phenomenological and variational constraints still holds for the general discrete systems, as we shall recall below. We refer to [@GBYo2017a] for the relation with other variational formulations used in nonholonomic mechanics with linear or nonlinear constraints. See also §\[3\_2\] below. ]{}
For the case of adiabatically closed systems (i.e., $P^{\rm ext}_H=0$), the evolution equations can be geometrically formulated in terms of Dirac structures induced from the phenomenological constraint and from the canonical symplectic form on $T ^\ast Q$ or on $T ^\ast (Q\times \mathbb{R} )$, see [@GBYo2017c].
In absence of the entropy variable $S$, this variational formulation recovers Hamilton’s variational principle in classical mechanics, where becomes the Euler-Lagrange equations.
Variational formulation of nonequilibrium thermodynamics of discrete systems {#2_2}
----------------------------------------------------------------------------
**Discrete systems.** We now consider the case of a discrete system $ \boldsymbol{\Sigma} = \cup_{A=1}^N \boldsymbol{\Sigma} _A$, composed of interconnecting simple systems $ \boldsymbol{\Sigma} _A$, $A=1,...,N$ that can exchange heat and mechanical power, and interact with *external heat sources* $ \boldsymbol{\Sigma} _R$, $R=1,...,M$. We follow the description of discrete systems given in [@StSc1974] and [@Gr1997].
By definition, a [*heat source*]{} is a simple system $ \boldsymbol{\Sigma} _R$ uniquely defined by a single variable $S_R$. Its energy is thus given by $U_R=U_R(S_R)$, the temperature is $T^R:= \frac{\partial U_R}{\partial S_R}$, and $ \frac{d}{dt} U_R=T^R\dot S_R=P^{R \rightarrow \boldsymbol{\Sigma} }_H$, where $P_H^{R \rightarrow \boldsymbol{\Sigma} }$ is the heat power flow due to the heat exchange with $ \boldsymbol{\Sigma} $.
The state of the discrete system $ \boldsymbol{\Sigma} $ is described by geometric variables $q \in Q_{ \boldsymbol{\Sigma} } $ and entropy variables $S_A$, $A=1,...,N$. Note that the entropy $S_A$ has the index $A$ since it is associated to the simple system $ \boldsymbol{\Sigma} _A$. The geometric variables, however, are not indexed by $A$ since in general they are associated to several systems $ \boldsymbol{\Sigma} _A$ that can interact with. The Lagrangian of a discrete system is thus a function $$\label{Lagrangian_S_discrete}
L:T Q_{ \boldsymbol{\Sigma} } \times \mathbb{R}^N \rightarrow \mathbb{R} , \quad (q, \dot q, S_1,...,S_N) \mapsto L(q, \dot q, S_1,...,S_N).$$
As before, the power supplied from the exterior is due to that by external forces and by transfer of heat. For simplicity, we ignore internal and external matter exchanges in this section. Hence, in particular, we consider the case in which the system is closed. The external force reads $F^{\rm ext}:=\sum_{A=1}^N F^{{\rm ext} \rightarrow A}$, where $F^{{\rm ext} \rightarrow A}$ is the external force acting on the system $\boldsymbol{\Sigma} _A$. The external heat power associated to heat transfer is $P^{\rm ext}_H =\sum_R \big(\sum_{A=1}^NP_H^{R \rightarrow A}\big)=\sum_{A=1}^NP_H^{{\rm ext} \rightarrow A}$, where $P_H^{R \rightarrow A}$ denotes the power of heat transfer between the external heat source $\boldsymbol{\Sigma} _R$ and the system $ \boldsymbol{\Sigma} _A$. The friction force associated to system $\boldsymbol{\Sigma} _A$ is $F^{{\rm fr}(A)}:TQ_{ \boldsymbol{\Sigma} } \times \mathbb{R}^N \rightarrow T^* Q_ {\boldsymbol{\Sigma} }$ with $F^{\rm fr}:=\sum_{A=1}^N F^{{\rm fr}(A)}$. The internal heat power exchange between $ {\boldsymbol{\Sigma} } _A$ and $ {\boldsymbol{\Sigma}} _B$ can be described by $$P_H^{B \rightarrow A}= \kappa _{AB}(q , S ^A , S ^B)(T^B-T^A),$$ where $ \kappa _{AB}= \kappa _{BA}\geq 0$ are the heat transfer phenomenological coefficients.
A typical, and historically relevant, example of a discrete (non-simple) system is the [*adiabatic piston*]{}. We refer to [@Gr1999] for a systematic treatment of the adiabatic piston from Stueckelberg’s approach.
**Variational formulation.** Our variational formulation is based on the introduction of new variables, called *thermodynamic displacements*, that allow a systematic inclusion of all the irreversible processes involved in the system. In our case, since we only consider the irreversible processes of the mechanical friction and heat conduction, we just need to introduce (in addition to the mechanical displacement $q$) the *thermal displacements*$^{1}$, $ \Gamma ^A $, $A=1,...,N$ such that $\dot{\Gamma}^{A}=T^{A}$, where $ \Gamma ^A $ are monotonically increasing real functions of time $t$ and hence the temperatures $T^{A}$ of $ \boldsymbol{\Sigma} _A$ take positive real values, i.e., $(T^{1},...,T^{N}) \in \mathbb{R} _{+}^{N}$.
Each of these variables is accompanied with its dual variable $ \Sigma _A$ whose time rate of change is associated to the entropy production of the simple system $A$. The meaning of the variable $\Sigma_A$ and its distinction with the entropy variable $S_A$ may be clarified in the context of continuum systems, as will be seen in §\[3\].
We say that a curve $\left(q(t), S _A (t), \Gamma ^A (t) , \Sigma _A (t) \right) \in Q _{ \boldsymbol{\Sigma} }\times \mathbb{R}^{3N} $, $t \in [t _1 , t _2 ] \subset \mathbb{R}$ is *solution of the variational formulation of nonequilibrium thermodynamics* if it satisfies the *variational condition* $$\label{LdA_thermo_discrete_systems}
\delta \int_{ t _1 }^{ t _2 }\Big[ L(q , \dot q , S _1 , ... S _N )+ \sum_{A=1}^N(S_A - \Sigma_A)\dot{\Gamma}^A \Big] dt +\int_{ t _1 }^{ t _2 } \left\langle F^{\rm ext}, \delta q \right\rangle dt =0,$$ for all variations $\delta q (t), \delta \Gamma^A (t) , \delta \Sigma_A (t)$ subject to the *variational constraint* $$\label{Virtual_Constraints_Systems}
\begin{split}
\frac{\partial L}{\partial S_A } \delta \Sigma _A
= \left\langle F^{{\rm fr}(A)}, \delta q \right\rangle-\sum_{B=1}^N \kappa _{AB}&(\delta \Gamma ^B-\delta \Gamma ^A ), \;\;\textrm{$($no sum on $A)$}
\end{split}$$ with $\delta q( t _i )=0$ and $ \delta \Gamma ( t _i )=0$, for $i=1,2$, and also if it satisfies the nonlinear *phenomenological constraint* $$\label{Kinematic_Constraints_Systems}
\begin{split}
\frac{\partial L}{\partial S _A } \dot{\Sigma}_A= \left\langle F^{{\rm fr}(A)}, \dot q \right\rangle - \sum_{B=1}^N \kappa _{AB}&(\dot \Gamma ^B-\dot \Gamma ^A ) - P_H^{\rm ext}. \\
\end{split}$$ From this variational formulation, we deduce the system of evolution equations for the discrete thermodynamic system as $$\label{discrete_systems}
\left\{
\begin{array}{l}
\displaystyle\vspace{0.2cm}\frac{d}{dt}\frac{\partial L}{\partial \dot q}- \frac{\partial L}{\partial q}= \sum_{A=1}^N F^{{\rm fr}(A)}+ F^{\rm ext},\\
\displaystyle\frac{\partial L}{\partial S_A}\dot S_A= \left\langle F^{{\rm fr}(A)}, \dot q \right\rangle +\sum_{B=1}^N \kappa _{AB} \left( \frac{\partial L}{\partial S_B}-\frac{\partial L}{\partial S_A}\right) - P^{\rm ext}_H, \;\; A=1,...,N.
\end{array} \right.$$ We refer to [@GBYo2016a] for a complete treatment of discrete systems. In a similar way with the situation of simple thermodynamic systems, one passes from the variational constraint to the phenomenological constraint by formally replacing the $ \delta $-variations $ \delta q, \delta \Sigma _A,\delta \Gamma _A$ by the time derivatives $\dot q, \dot \Sigma _A, \dot\Gamma _A$ (see Remark \[PC\_VS\_VC\]). This is possible thanks to the introduction of the thermodynamic displacements $\Gamma _A$.
Formulation based on the free energy and heat equations {#2_3}
-------------------------------------------------------
We now present the variational formulation for discrete thermodynamic systems based on a free energy Lagrangian, in which one makes use of temperature $T$ rather than entropy $S$ as an independent variable. We start with the case of a simple discrete thermodynamic system.
**Simple systems.** Given the Lagrangian $L:TQ \times \mathbb{R} \rightarrow \mathbb{R} $ of a discrete simple system, the associated *free energy Lagrangian* $ \mathcal{L} :TQ \times \mathbb{R}_+ \rightarrow \mathbb{R}$ is defined (see [@GBYo2016a]) by $$\mathcal{L}(q,\dot q,T):= L(q,\dot q, S(q, \dot q, T))+TS(q, \dot q, T),$$ where we assumed that the function $S \in \mathbb{R} \mapsto \frac{\partial L}{\partial S}(q,\dot q, S) \in \mathbb{R} _+ $ is a diffeomorphism for all $(q, \dot q) \in TQ$ and where the function $S(q,\dot q,T)$ is defined by the condition $ -\frac{\partial L}{\partial S}(q,\dot q, S)=T$, for all $(q,\dot q, S) \in TQ \times \mathbb{R}$.
In most physical examples, the partial derivative $ \frac{\partial L}{\partial S}$ does not depend on $\dot q$. In this case the Lagrangian has the standard form $L(q, \dot q, S)=L_{\rm mech}(q, \dot q)- U(q,S)$, where $L_{\rm mech}: TQ \rightarrow \mathbb{R} $ is a Lagrangian of the mechanical part of the simple system and $U:Q \times \mathbb{R} \rightarrow \mathbb{R}$ is an internal energy. The associated free energy Lagrangian is $ \mathcal{L} (q, \dot q, T)= L_{\rm mech}(q, \dot q)- \mathscr{F}(q,T)$, where $\mathscr{F}:Q\times \mathbb{R} _+\rightarrow \mathbb{R} $ is the free energy associated to $U:Q\times\mathbb{R} \rightarrow \mathbb{R} $, where the relation between temperature and entropy may be understood in the dual context of Lagrangians$^{2}$.
The friction and external forces $F^{\rm fr}, F^{\rm ext}: T Q \times \mathbb{R} _+\rightarrow T^*Q$ are now expressed in terms of the temperature rather than the entropy. As before, the variational formulation needs the introduction of the variables $ \Gamma $ and $ \Sigma $.
Recall that $T=\dot{\Gamma}$. We say that a curve $(q(t), \Gamma (t), \Sigma (t)) \in Q \times \mathbb{R}^{2}$, $t \in [t _1 , t _2 ] \subset \mathbb{R}$ is a [*solution of the free energy variational formulation of nonequilibrium thermodynamics*]{} if it satisfies the *variational condition* $$\label{LdA_thermo_discrete}
\delta \int_{t _1 }^{ t _2} \left( \mathcal{L} (q , \dot q , \dot\Gamma )- \Sigma \dot\Gamma \right) dt +\int_{t_1}^{t_2}\left\langle F^{\rm ext}(q, \dot q, T), \delta q\right\rangle dt =0,$$ for all variations $ \delta q(t), \delta \Gamma(t)$ and $\delta \Sigma(t)$ subject to the *variational constraint* $$\label{CV_discrete}
\dot\Gamma \delta \Sigma = - \left\langle F^{\rm fr}(q , \dot q , T),\delta q \right\rangle,$$ with $ \delta q(t_1)=\delta\Gamma (t_2)=0$, and also if it satisfies the nonlinear nonholonomic *phenomenological constraint* $$\label{CK_discrete}
\dot\Gamma \dot \Sigma = - \left\langle F^{\rm fr}(q, \dot q, T) , \dot q \right\rangle + P^{\rm ext}_H.$$ From this variational formulation, we get the system of evolution equations for the simple system in terms of the free energy as $$\label{discrete_systems_free}
\left\{
\begin{array}{l}
\displaystyle\vspace{0.2cm}\frac{d}{dt}\frac{\partial \mathcal{L} }{\partial \dot q}- \frac{\partial \mathcal{L} }{\partial q}= F^{\rm fr}(q, \dot q, T)+F^{\rm ext}(q, \dot q, T),\\
\displaystyle T \frac{d}{dt} \frac{\partial \mathcal{L} }{\partial T}= - \left\langle F^{\rm fr}(q, \dot q, T), \dot q \right\rangle + P^{\rm ext}_H.
\end{array} \right.$$ This is an appropriate form to derive the heat equation of the simple system. Since the Lagrangian has the form $ \mathcal{L} (q, \dot q, T)= L_{\rm mech}(q, \dot q)- \mathscr{F}(q,T)$, the second equation in becomes $$T \left( \frac{\partial ^2 \mathscr{F}}{\partial T ^2 }(q,T)\dot T+ \frac{\partial ^2 \mathscr{F}}{\partial q \partial T}(q,T)\dot q \right) =\left\langle F^{\rm fr}(q, \dot q, T), \dot q \right\rangle - P^{\rm ext}_H,$$ from which we obtain the *heat equation* $$c_v(q,T) \dot T=T \frac{\partial ^2 \mathscr{F}}{\partial q \partial T}(q,T)\dot q - \left\langle F^{\rm fr}(q, \dot q, T), \dot q \right\rangle+ P^{\rm ext}_H ,$$ where $c_v(q,T)= - T \frac{\partial ^2\mathscr{F}}{\partial T ^2 }(q,T)$ is the specific heat.
**Discrete systems.** Consider a discrete system with configuration space $Q_{ \boldsymbol{\Sigma} }$ and Lagrangian $L: T Q_{ \boldsymbol{\Sigma} } \times \mathbb{R} ^N \rightarrow \mathbb{R} $. The corresponding free energy Lagrangian is defined by generalizing the above definition as $$\begin{aligned}
\mathcal{L} (q, \dot q, T^1,...,T^N):&= L \left( q, \dot q, S_1(q, \dot q, T^1,...,T^N),..., S_N(q, \dot q, T^1,...,T^N)\right) \\
&\qquad\qquad \qquad + \sum_{A=1}^N T^A S_A(q, \dot q, T^1,...,T^N),\end{aligned}$$ where we assumed the function $(S_1,.., S_N) \in \mathbb{R} ^N \mapsto \left( \frac{\partial L}{\partial S_1}, ..., \frac{\partial L}{\partial S_N} \right) \in ( \mathbb{R}_+ ) ^N $ is a diffeomorphism for all $(q, \dot q) \in TQ_{ \boldsymbol{\Sigma} }$ and where the functions $S_A(q,\dot q, T^1,...,T^N)$, $A=1,...,N$ are defined from the conditions $ - \frac{\partial L}{\partial S_A}(q, \dot q, S_1,...,S_N)=T^A$, for all $A=1,...,N$ and for all $q, \dot q, S_1,...,S_N$.
Recall $T^{A}=\dot{\Gamma}^{A}$. We say that a curve $\left(q(t), \Gamma ^A (t) , \Sigma _A (t) \right) \in Q_{ \boldsymbol{\Sigma} } \times \mathbb{R} ^{2N} $, $t \in [t _1 , t _2 ]$ is a *solution of the free energy variational formulation of nonequilibrium thermodynamics* if it satisfies the *variational condition* $$\label{LdA_thermo_discrete_systems-new}
\begin{aligned}
\delta \int_{ t _1 }^{ t _2 }\Big( \mathcal{L}(q , \dot q , \dot \Gamma ^{1},...,\dot \Gamma ^{N} )&- \sum_{A=1}^N \Sigma _A\dot\Gamma ^A \Big) dt +\int_{ t _1 }^{ t _2 }\left\langle F^{\rm ext}, \delta q \right\rangle dt =0,
\end{aligned}$$ for all variations $ \delta q , \delta \Gamma^A , \delta \Sigma_A$ subject to the *variational constraint* $$\label{Virtual_Constraints_Systems-new}
\dot \Gamma ^A \delta \Sigma _A = - \left\langle F^{{\rm fr}(A)}(...), \delta q \right\rangle+ \sum_{B=1}^N \kappa _{AB}(\delta \Gamma ^B-\delta \Gamma ^A ),\;\; \textrm{$($no sum on $A)$}$$ with $ \delta q( t _i )=0$ and $ \delta \Gamma^A ( t _i )=0$ for $i=1,2$, and if it satisfies the *phenomenological constraint* $$\label{Kinematic_Constraints_Systems-new}
\dot \Gamma ^A \dot{\Sigma}_A = -\left\langle F^{{\rm fr}(A)}(...), \dot q \right\rangle + \sum_{B=1}^N \kappa _{AB}(\dot \Gamma ^B- \dot \Gamma ^A ) + P_H^{{\rm ext} \rightarrow A}.$$ From this variational formulation, we deduce the system of evolution equations for the discrete system in terms of the free energy as
[$$\label{al_thermo_mech_equations_discrete_systems2}
\left\{
\begin{array}{l}
\displaystyle\vspace{0.1cm}\frac{d}{dt} \frac{\partial \mathcal{L}}{\partial \dot q }- \frac{\partial \mathcal{L}}{\partial q }= \sum_{A=1}^N F^{{\rm fr}(A)}+F^{\rm ext},\\[2mm]
\displaystyle T ^A \frac{d}{dt}\frac{\partial \mathcal{L}}{\partial T^{A}} = - \left\langle F^{{\rm fr}(A)} , \dot q \right\rangle +\sum_{B=1}^N \kappa _{AB}( T^{B}-T ^A )+P_H^{{\rm ext} \rightarrow A}, \quad A=1,...,N.
\end{array}
\right.$$]{}
For a free energy Lagrangian of the form $ \mathcal{L} (q, \dot q, T_1,...,T_N)= L_{\rm mech}(q, \dot q)- \sum_{A=1}^NF ^A (q, T_A)$, the *heat equation* is computed from the second equation of as $$c_{v,A} (q,T ^A) \dot T ^A=T^A \frac{\partial ^2 F^A}{\partial q \partial T^A}(q,T^A)\dot q - \left\langle F^{\rm fr}, \dot q \right\rangle +\sum_{B=1}^N \kappa _{AB}( T^{B}-T ^A ) + P^{\rm ext}_H ,$$ where $c_{v, A}(q,T^A )= - T^A \frac{\partial ^2 F^A}{\partial (T ^A)^2 }(q,T^A)$ is the specific heat of system $ \boldsymbol{\Sigma} _A$.
The Navier-Stokes-Fourier equations {#3}
===================================
We shall now systematically extend to the continuum setting the previous free energy variational formulations by focalising on the case of a heat conducting viscous fluid. We refer to [@GBYo2016b] for the corresponding formulation in terms of the entropy.
The variational formulation of the Navier-Stokes-Fourier equation is first formulated in the Lagrangian (or material) representation, because it is in this representation that the variational formulation is deduced from that of discrete systems described in §\[2\_3\]. All the equations are intrinsically written in a differential-geometric form.
Configuration space and Lagrangians
-----------------------------------
We assume that the domain occupied by the fluid is a smooth compact manifold with smooth boundary $ \partial\mathcal{D}$. The configuration space is $Q= \operatorname{Diff}_0( \mathcal{D} )$, the group of all diffeomorphisms$^{3}$ of $ \mathcal{D} $ that keep the boundary $ \partial \mathcal{D} $ pointwise fixed. This corresponds to no-slip boundary conditions. This choice of the configuration space aims to describe only strong solutions of the partial differential equation. We assume that the manifold $ \mathcal{D} $ is endowed with a Riemannian metric $g$.
Given a curve $ \varphi _t$ of diffeomorphisms, starting at the identity at $t=0$, we denote by $ x= \varphi _t(X)= \varphi (t,X) \in \mathcal{D}$ the current position of a fluid particle which at time $t=0$ is at $X \in \mathcal{D} $. The *mass density* $\varrho (t,X)$ and the *entropy density* $S(t,X)$ in the Lagrangian (or material) description are respectively related to the corresponding quantities $ \rho (t,x)$ and $ s(t,x)$ in the Eulerian (or spatial) description as $$\label{material_rho_S}
\varrho (t,X)= \rho (t, \varphi _t(X)) J _{\varphi _t}(X) \quad\text{and}\quad S(t,X)= s (t, \varphi_t (X)) J_{ \varphi _t}(X),$$ were $ J _{\varphi_t }$ denotes the Jacobian of $ \varphi _t$ relative to the Riemannian metric $g$, i.e., $ \varphi_t^\ast \mu _g = J_{\varphi_t} \mu _g $, with $ \mu _g $ the Riemannian volume form.
From the conservation of the total mass, we have $ \varrho (t, X)= \varrho _{\rm ref}(X)$, i.e., the mass density in the material description is time independent. It therefore appears as a parameter in the Lagrangian function and in the variational formulation. This is not the case for the material entropy $S(t,X)$, which is a dynamic field with corresponding variations $ \delta S$ that must be taken into account in the variational formulation.
**The Lagrangian.** In a similar way to the case of discrete systems in , the Lagrangian in the material description is a map $$\label{Lagrangian}
L_{ \varrho _{\rm ref}}: T \operatorname{Diff}_0( \mathcal{D} ) \times \mathcal{F} ( \mathcal{D} )\rightarrow \mathbb{R}, \quad ( \varphi , \dot \varphi , S) \mapsto L_{ \varrho _{\rm ref}}( \varphi , \dot \varphi , S),$$ where $T \operatorname{Diff}_0( \mathcal{D} )$ is the tangent bundle to $ \operatorname{Diff}_0( \mathcal{D} )$ and $ \mathcal{F} ( \mathcal{D} )$ is a space of real valued functions on $ \mathcal{D} $ with a given high enough regularity, so that all the formulas used below are valid. The index notation in $L_{\rm \varrho _{\rm ref}}$ is used to recall that $L$ depends parametrically on $ \varrho _{\rm ref}$. By $( \varphi , \dot \varphi )$ we denote an arbitrary vector in the tangent space $T_ \varphi \operatorname{Diff}_0( \mathcal{D} )$. We choose to follow here the traditional coordinate notation$^{4}$
(i.e., of the type $L(q, \dot q)$) used in classical mechanics and in §\[section\_2\], although our point of view is completely intrinsic. Recall that the tangent space to $\operatorname{Diff}_0( \mathcal{D} )$ at $ \varphi $ is given by $T_ \varphi \operatorname{Diff}_0(\mathcal{D} )= \{ \mathbf{V} _\varphi : \mathcal{D} \rightarrow T \mathcal{D} \mid \mathbf{V} _ \varphi (X) \in T_{\varphi (X)} \mathcal{D},\;\; \mathbf{V} _\varphi |_ {\partial \mathcal{D}} =0 \}$, where the map $ \mathbf{V} _\varphi : \mathcal{D} \rightarrow T \mathcal{D}$ has the same regularity with $ \varphi $.
Consider a gas with a given state equation $ \varepsilon = \varepsilon (\rho, s)$ where $ \varepsilon $ is the internal energy density, and the Lagrangian is given by
[$$\label{Lagrangian_NSF}
\begin{aligned}
L_{ \varrho _{\rm ref}}( \varphi , \dot\varphi , S)&\!=\! \int_ \mathcal{D} \frac{1}{2}\varrho _{\rm ref}( X ) | \dot\varphi ( X )| ^2 _g \mu _g(X) \!-\! \int_ \mathcal{D} \varepsilon \left( \frac{ \varrho _{\rm ref}(X)}{J_\varphi (X)} , \frac{ S(X)}{J_\varphi (X)} \right) J_\varphi (X) \mu _g(X)\\
&= \int_ \mathcal{D} \mathfrak{L} (\varphi (X) , \dot \varphi (X),T _X\varphi , \varrho _{\rm ref}(X),S(X)) \mu _g (X),
\end{aligned}$$]{}
where $T _ X\varphi :T_ X \mathcal{D} \rightarrow T_ { \varphi (X)}\mathcal{D} $ is the tangent map to $ \varphi $. The first term of $L_{ \varrho _{\rm ref}}$ represents the total kinetic energy of the gas, computed with the help of the Riemannian metric $g$, and the second term represents the total internal energy. The second term is deduced from $\varepsilon ( \rho , s)$ by using the relations . Both terms are written here in terms of material quantities. In the second line we defined the Lagrangian density $\mathfrak{L} (\varphi ,\dot \varphi , T \varphi , \varrho _{\rm ref},S)\mu _g$ as the integrand of the Lagrangian $L$. The *material temperature* is given by $$\mathfrak{T}= - \frac{\partial \mathfrak{L} }{\partial S}= \frac{\partial \varepsilon }{\partial s}( \rho , s) \circ \varphi = T \circ\varphi ,$$ where $T$ is the *Eulerian temperature*.
**The free energy Lagrangian.** Generally, given a Lagrangian $L_{\varrho _{\rm ref}}:T \operatorname{Diff}_0( \mathcal{D} ) \times \mathcal{F} ( \mathcal{D} )\rightarrow \mathbb{R}$ with Lagrangian density $ \mathfrak{L} (\varphi ,\dot \varphi , T \varphi , \varrho _{\rm ref},S)$, we define the associated *free energy Lagrangian* $\mathcal{L} _{\varrho _{\rm ref}}: T \operatorname{Diff}_0( \mathcal{D} ) \times \mathcal{F} _+ ( \mathcal{D} )\rightarrow \mathbb{R}$ as $$\label{FEL}
\begin{aligned}
&\mathcal{L}_{\varrho _{\rm ref}} ( \varphi , \dot\varphi , \mathfrak{T} ):=\int_ \mathcal{D} \mathscr{L}(\varphi (X), \dot\varphi (X),T_X\varphi , \varrho _{\rm ref}(X),\mathfrak{T} (X)) \mu _g,
\end{aligned}$$ where the free energy Lagrangian density $\mathscr{L}$ is defined by $$\label{definition_mathscrL}
\mathscr{L}( \alpha ,\mathfrak{T} ):= \mathfrak{L} (\alpha , S( \alpha , \mathfrak{T} )) + \mathfrak{T} S(\alpha ,\mathfrak{T} ).$$ In we used the abbreviation $ \alpha = (\varphi , \dot\varphi ,T \varphi , \varrho _{\rm ref})$, we assumed that the function $ S \in\mathbb{R} \mapsto \frac{\partial\mathfrak{L} }{\partial S}( \alpha , S) \in \mathbb{R} _+ $ is invertible for all $ \alpha $, and we defined the function $ S( \alpha , \mathfrak{T} )$ by the condition $ \frac{\partial \mathcal{L} }{\partial S}(\alpha , S( \alpha , \mathfrak{T} ) )= \mathfrak{T} $, for all $ \alpha $. In , $\mathcal{F} _+ ( \mathcal{D} )$ denotes a set of strictly positive functions on $ \mathbb{R} $.
For the case of the Lagrangian , we obtain
[$$\label{FE_Lagrangian_NSF}
\begin{aligned}
\mathcal{L} _{\varrho _{\rm ref}}( \varphi , \dot\varphi , & \mathfrak{T} )=: \int_ \mathcal{D} \mathscr{L} \big( \varphi (X),\dot \varphi (X), T _X\varphi , \varrho _{\rm ref}( X ), \mathfrak{T} (X)\big)\mu _g(X)\\
&= \int_ \mathcal{D} \frac{1}{2}\varrho _{\rm ref}( X ) | \dot\varphi ( X )| ^2 _g \mu _g(X) - \int_ \mathcal{D} \psi \left( \frac{ \varrho _{\rm ref}(X)}{J_\varphi (X)} , \mathfrak{T} (X)\right) J_\varphi (X) \mu _g(X),
\end{aligned}$$]{}
where $ \psi ( \rho , T)$ is the (Helmholtz) free energy density associated to the internal energy density $ \varepsilon ( \rho , s)$. The free energy density is given in material representation as $$\label{FE_material}
\Psi ( \varphi , T_X \varphi , \varrho _{\rm ref}, \mathfrak{T} ):= \psi \left( \frac{ \varrho _{\rm ref} }{J_\varphi } , \mathfrak{T} \right) J_\varphi.$$
For example, for the case of a perfect gas, the internal energy density and the free energy density are respectively given by $$\begin{aligned}
\varepsilon ( \rho , s)&= \varepsilon _0 e^ {\frac{1}{C_v} \left( \frac{s}{ \rho }-\frac{s_0}{\rho _0} \right) }\left( \frac{ \rho }{\rho _0}\right) ^{C_p/C_v},\\
\psi ( \rho ,T)&=\rho T \left(\frac{\psi _0 }{\rho _0 T_0} + R _{\rm gas} \ln \left( \frac{ \rho }{\rho _0} \right) -C_v \ln \left( \frac{T}{T_0}\right) \right),\end{aligned}$$ where the constant $C_v$ is the specific heat coefficient at constant volume, the constant $C_p$ is the specific heat at constant pressure, $R_{\rm gas}= C_p-C_v$ is the gas constant, and $ \rho _0, s_0, T_0, \varepsilon _0, \psi _0$ are given constant reference values verifying $ \varepsilon _0= \varepsilon ( \rho _0, s_0)$ and $ \psi _0= \psi ( \rho _0, T_0)$.
Variational formulation in material representation {#3_2}
--------------------------------------------------
It is well-known that in absence of irreversible processes the equations of continuum mechanics in material representation arise from Hamilton’s principle, see, e.g., [@MaHu1983 Ch.5]. Before presenting the extension of this variational formulation to the irreversible case, we shall briefly review Hamilton’s principle as it applies to the Lagrangian . In this case, the resulting system becomes the *perfect adiabatic compressible fluid*.
**Hamilton’s principle in the reversible case.** In absence of irreversible processes, the entropy is conserved so that we have $S(t, X)= S_{\rm ref}(X)$ in material representation. The equations of motion thus follow from Hamilton’s principle applied to the Lagrangian $L_{\varrho _{\rm ref}}$ in , in which the variable $S=S_{\rm ref}$ is understood as a fixed parameter field in the same way as $\varrho _{\rm ref}$. Writing $L_{\varrho _{\rm ref}, S_{\rm ref}}:T \operatorname{Diff}_0( \mathcal{D} ) \rightarrow \mathbb{R}$, with $L_{\varrho _{\rm ref}, S_{\rm ref}}( \varphi , \dot \varphi ):= L_{\varrho _{\rm ref}}(\varphi , \dot\varphi , S_{\rm ref})$, Hamilton’s principle is $$\label{HP_easy}
\delta \int_ {t_1}^{t_2} L_{\varrho _{\rm ref}, S_{\rm ref}}( \varphi , \dot\varphi )dt=0 ,$$ with respect to variations $ \delta \varphi $ such that $\delta \varphi |_{ \partial\mathcal{D} }=0$ and $ \delta \varphi (t _i )=0$, $i=1,2$. The stationarity condition yields the Euler-Lagrange equations for $L_{\varrho _{\rm ref}, S_{\rm ref}}$ in the form $$\label{EL_fluids}
\varrho _{\rm ref}\frac{D}{Dt} \mathbf{V} = \operatorname{DIV} \mathbf{P} ^{\rm cons},$$ where $ \mathbf{V} $ is the material velocity and $ \mathbf{P} ^{\rm cons}$ is the first Piola-Kirchhoff stress tensor.
We now describe in details the geometric objects involved in this equation. The material velocity is defined by $ \mathbf{V}_t(X):= \frac{\partial }{\partial t} \varphi _ t(X) \in T_{ \varphi_t(X)}\mathcal{D} $. At each time $t$, the material velocity $ \mathbf{V} _t$ is a vector field on $ \mathcal{D} $ along $ \varphi_t$. The first Piola-Kirchhoff tensor is defined by $$\label{Pcons1}
\mathbf{P} ^{\rm cons}= - \left( \frac{\partial \mathfrak{L} }{\partial T_{X} \varphi } \right) ^\sharp,$$ where $ \sharp$ denotes the index rising operator relative to the Riemannian metric $g$. At each time $t$, the first Piola-Kirchhoff tensor $ \mathbf{P} ^{\rm cons}(t,\_\,)$ is a two-point tensor along $ \varphi _t$. More precisely, for all $X \in \mathcal{D} $, we have $$\mathbf{P} ^{\rm cons}(t,X): T_ x ^\ast \mathcal{D} \times T_ X ^\ast \mathcal{D} \rightarrow \mathbb{R} , \quad \text{where} \quad x= \varphi _t(X).$$ On the left hand side of , $D \mathbf{V} /Dt$ denotes the covariant time derivative of the vector field $ \mathbf{V} _t$ along $ \varphi_t $, relative to the Riemannian metric $g$. This covariant derivative yields again a vector field along $ \varphi _t$. On the right hand side of , $ \operatorname{DIV} \mathbf{P} ^{\rm cons}$ denotes the divergence${}^4$ of the two-point tensor $ \mathbf{P} ^{\rm cons}$ along $ \varphi _t $, relative to the Riemannian metric $g$. It yields a vector field on $ \mathcal{D} $ along $ \varphi _t $. We refer to [@MaHu1983] for a detailed account of two-point tensors along diffeomorphisms and their covariant derivatives.
A direct computation shows that for the Lagrangian , the first Piola-Kirchhoff tensor leads to the pressure in material representation, i.e., $$\label{P_cons_NSF}
\mathbf{P} ^{\rm cons}(X)( \alpha _x, \beta _X)= - p ( \rho , s) J _\varphi \,g\big( T_x\varphi^{-1} ( \alpha _x), \beta _X\big), \quad p( \rho , s)= \frac{\partial \varepsilon}{\partial \rho }\rho + \frac{\partial \varepsilon }{\partial s}s - \varepsilon,$$ where $x= \varphi _t(X)$.
The system , with $ \mathbf{P} ^{\rm cons}$ given in , is the material description of the Euler equations for a compressible perfect adiabatic fluid, which is usually written in spatial representation as $$\label{viscoelastic_spat}
\left\{
\begin{array}{l}
\vspace{0.2cm}\rho ( \partial _t \mathbf{v} + \mathbf{v} \cdot \nabla \mathbf{v} )=- \operatorname{grad} p,\\[1mm]
\partial _t \rho + \operatorname{div} ( \rho \mathbf{v} )=0, \quad \partial _t s+ \operatorname{div}(s \mathbf{v} )=0,
\end{array}
\right.$$ where $ \nabla $ is the Levi-Civita covariant derivative associated to $g$ and the operators $ \operatorname{grad}$ and $ \operatorname{div}$ are both associated to $g$. A systematic approach to derive the variational principle in spatial representation from Hamilton’s principle in material representation, is provided by the Euler-Poincaré reduction theory on Lie groups, see [@HoMaRa1998].
[It is important to observe that while the entropy variable $S_{\rm ref}(X)$ is seen as a fixed parameter in the material description of the reversible case, this is not true for the temperature as it clearly follows from its definition $ \mathfrak{T} = - \frac{\partial \mathfrak{L} }{\partial S}(\varphi , \dot\varphi ,T \varphi , \varrho _{\rm ref}, S_{\rm ref})$. Therefore, when working with the corresponding free-energy Lagrangian $ \mathcal{L}_{ \varrho _{\rm ref}} ( \varphi , \dot\varphi , \mathfrak{T} )$ in , one cannot avoid considering the variations $ \delta \mathfrak{T} $ in the variational principle, even in the reversible case. We will consider this situation later, as a particular case of our variational formulation for the irreversible case.]{}
\[Material\_G\][The general geometric formulation of continuum mechanics needs a more general setting than the one presented above, namely, one has to consider the field $ \varphi $ as an embedding from a reference manifold $ \mathcal{B}$ into an ambient manifold $ \mathcal{S} $. Each of these manifolds is endowed with a Riemannian metric, typically denoted by $ G$ on $ \mathcal{B} $ and by $g$ on $ \mathcal{S} $. This is the appropriate setting to study the material and spatial symmetries in nonlinear continuum mechanics, see [@MaHu1983], [@SiMaKr1988], [@GBMaRa2012]. In the present situation, since we are studying the special case of a fluid in a fixed domain, we have $ \mathcal{B} = \mathcal{S} =\mathcal{D}$, so we can make the choice $g=G$.]{}
**Navier-Stokes-Fourier equations in material representation.** The processes of viscosity and heat conduction are described by the inclusion of the corresponding thermodynamic fluxes given, in the material description, by a friction Piola-Kirchhoff tensor $ \mathbf{P} ^{\rm fr} (t, X)$ and an entropy flux density $ \mathbf{J} _S(t, X)$, where $ \mathbf{P} ^{\rm fr}$ is a two-point tensor along $ \varphi $ and $ \mathbf{J} _S$ is a vector field on $ \mathcal{D} $. We will recall their usual phenomenological expressions later in the Eulerian description.
By complete analogy with the case of discrete systems developed earlier, our formulation needs the notion of *thermal displacements* ([@GrNa1991]) in material representation, i.e., a variable $ \Gamma (t, X)$ such that $$\frac{d}{dt} \Gamma (t, X)=\mathfrak{T} (t, X).$$ This is a particular case of *thermodynamic displacement variables* introduced in [@GBYo2016a; @GBYo2016b]. In addition to these internal irreversible processes, we assume that the fluid is heated from the exterior, which is represented by a heat power supply density $ \varrho _{\rm ref}(X)R(t,X)$ in material representation. Recall that $ \mathscr{L}$ denotes the free energy Lagrangian density.
By complete analogy with the variational formulation given in – for discrete systems, we consider the variational formulation for a curve $( \varphi (t), \Gamma (t), \Sigma (t)) \in \operatorname{Diff}_0( \mathcal{D} ) \times \mathcal{F} ( \mathcal{D} ) \times \mathcal{F} ( \mathcal{D} ) $ as follows: $$\label{VC_NSF_material}
\delta \int_{t_1}^{t_2} \!\!\int_ \mathcal{D} \left[ \mathscr{L} \big( \varphi , \dot \varphi , T\varphi , \varrho _{\rm ref}, \dot \Gamma\big)- \Sigma \dot \Gamma\right] \mu _g dt=0, \quad\;\;\; \textsc{\small Variational Condition}$$ for all variations $\delta\varphi$, $ \delta \Gamma $, $ \delta \Sigma $ subject to the constraint $$\label{CV_NSF_material}
\dot\Gamma \delta \Sigma = (\mathbf{P} ^{\rm fr})^\flat : \nabla \delta \varphi -\mathbf{J} _S \cdot \mathbf{d} \delta \Gamma, \qquad\, \textsc{\small Variational Constraint}$$ with $ \delta \varphi (t_i)=0$ and $ \delta \Gamma (t_i)=0$, for $i=1,2$, where the curve $( \varphi (t), \Gamma (t), \Sigma (t))$ satisfies the constraint $$\label{KC_NSF_material}
\dot\Gamma \dot \Sigma = (\mathbf{P} ^{\rm fr}) ^\flat : \nabla \dot \varphi -\mathbf{J} _S \cdot \mathbf{d} \dot\Gamma+ \varrho _{\rm ref}R, \qquad\, \textsc{\small Phenomenological Constraint}.$$
In the constraints and , $ \delta \varphi , \dot\varphi : \mathcal{D} \rightarrow T \mathcal{D} $ are vector fields on $ \mathcal{D} $ covering $ \varphi $. The objects $ \nabla \delta \varphi $ and $ \nabla \dot\varphi$ are two-point tensor fields covering $ \varphi $, obtained by taking the Levi-Civita covariant derivative associated to $g$. The notation $``:"$ means the full contraction of the two-point tensor fields along $ \varphi $. The flat operator on $ \mathbf{P} ^{\rm cons}$ applies to the spatial index.
Note the analogy between the three conditions -- for the variational formulation of discrete systems and the three conditions -- above. As before, the constraint is nonlinear and one can pass from the variational constraint to the phenomenological constraint by formally replacing the variations $ \delta \varphi $, $\delta \Gamma $, $ \delta \Sigma $ by the time derivatives $ \dot\varphi $, $\dot \Gamma $, $\dot\Sigma $. This variational formulation does not impose any restriction concerning the dependance of the thermodynamic fluxes on the variables $ \varphi , \dot \varphi , \varrho _{\rm ref}, \Gamma , \dot \Gamma $ and on their spatial derivatives.
Since $ \delta \varphi |_{ \partial \mathcal{D} }=0$ and $ \delta \varphi (t _i )=0$, $ \delta \Gamma (t _i )=0$, $i=1,2$, by applying the variational condition , we get $$\int_{t _1 }^{ t _2 } \!\!\int_ \mathcal{D} \left[ \left( \frac{\partial^ \nabla \mathscr{L} }{\partial \varphi }- \operatorname{DIV} \frac{\partial \mathscr{L} }{\partial T_X \varphi }- \frac{D}{Dt}\frac{\partial \mathscr{L} }{\partial \dot \varphi }\right) \delta \varphi + \left( \dot \Sigma - \frac{d}{dt}\frac{\partial \mathscr{L}}{\partial\dot \Gamma } \right) \delta \Gamma - \dot \Gamma \delta \Sigma \right] \mu _g dt=0.$$ Here $ \frac{\partial^ \nabla \mathscr{L} }{\partial \varphi }$ is the partial derivative of $\mathscr{L} $ relative to the spatial coordinate $x= \varphi (X)$. Such a partial derivative needs to be defined with respect to a Riemannian metric, here $g$, this is why we insert the exponent $ \nabla $. For the Lagrangian in , we have $ \frac{\partial^ \nabla \mathscr{L} }{\partial \varphi }=0$. The other partial derivatives of $ \mathscr{L}$ do not need the use of a Riemannian metric. The operator $ D/Dt$ and $ \operatorname{DIV}$ are associated to $g$, as explained earlier. Using the variational constraint and again $ \delta \varphi |_{ \partial \mathcal{D} }=0$ and $ \delta \varphi (t _i )=0$, $ \delta \Gamma (t _i )=0$, $i=1,2$, and collecting the terms associated to the variations $ \delta \varphi $ and $ \delta \Gamma $, we obtain the system $$\begin{aligned}
\delta \varphi : \quad &\;\;\rho _{\rm ref} \frac{D \mathbf{V} }{Dt}= \operatorname{DIV} \left( \mathbf{P} ^{\rm cons}+\mathbf{P} ^{\rm fr} \right) , \label{delta_phi} \\
\delta \Gamma : \quad &\;\; \frac{d}{dt}\frac{\partial \Psi }{\partial \mathfrak{T} } = \operatorname{DIV}\mathbf{J} _S -\dot \Sigma\quad\text{and}\quad \mathbf{J} _S \cdot \mathbf{n} ^\flat =0 \;\; \text{on $ \partial\mathcal{D} $}, \label{delta_Gamma}\end{aligned}$$ where $ \mathbf{n} $ is the outward pointing unit normal vector field along the boundary $ \partial \mathcal{D} $ and $ \mathbf{n} ^\flat $ is the associated one-form which is obtained by applying the index lowering operator $ \flat $ associated to $g$. The second equation in arises from the freeness of the variations $ \delta \Gamma $ at the boundary. In the conservative Piola-Kirchhoff stress tensor $ \mathbf{P} ^{\rm cons}$ is defined from the free energy Lagrangian density $ \mathscr{L}$ (or the free energy density $\Psi $) as $$\mathbf{P} ^{\rm cons}= - \left( \frac{\partial \mathscr{L} }{\partial T_{X} \varphi } \right) ^\sharp=\left( \frac{\partial \Psi }{\partial T_{X} \varphi } \right) ^\sharp.$$ Here we note that $ \mathbf{P} ^{\rm cons}$ is defined in terms of $ \mathscr{L}$ whereas in it is defined in terms of $\mathfrak{L}$. From the general definition of the free energy Lagrangian density in , we see that these two definitions of $ \mathbf{P} ^{\rm cons}$ coincide, one being expressed in terms of the entropy , and the other with respect to the temperature. For the free energy Lagrangian , we can compute the partial derivative with respect to $ T_ X \varphi $ to get $$\mathbf{P} ^{\rm cons}( \alpha _x, \beta _X)= - p ( \rho , T) J_ \varphi \,g\big( T_x\varphi^{-1} ( \alpha _x), \beta _X\big), \quad p( \rho , T)= \frac{\partial \psi }{\partial \rho }\rho - \psi ,$$ where $x= \varphi_{t} (X)$. This expression coincides with although it is here expressed in terms of the temperature.
Using the relation in the phenomenological constraint yields $$\mathfrak{T} \left( -\frac{d}{dt}\frac{\partial \Psi }{\partial \mathfrak{T} } + \operatorname{DIV} \mathbf{J} _S\right) = ( \mathbf{P} ^{\rm fr})^\flat: \nabla \dot \varphi - \mathbf{J} _S \cdot \mathbf{d} \mathfrak{T} + \varrho _{\rm ref}R,$$ where we recall that $\mathfrak{T} := \dot \Gamma $ is the temperature in material representation.
These results are summarized as follows.
The Navier-Stokes-Fourier equations in material representation, given by $$\label{summary_NSF}
\left\{
\begin{array}{l}
\vspace{0.2cm}\displaystyle\varrho _{\rm ref} \frac{D \mathbf{V} }{Dt}= \operatorname{DIV}( \mathbf{P} ^{\rm cons}+ \mathbf{P} ^{\rm fr}),\\[2mm]
\displaystyle\mathfrak{T} \left( -\frac{d}{dt}\frac{\partial \Psi }{\partial \mathfrak{T} }+ \operatorname{DIV} \mathbf{J} _S \right) = ( \mathbf{P} ^{\rm fr}) ^\flat : \nabla \mathbf{V} - \mathbf{J} _S \cdot \mathbf{d} \mathfrak{T} + \varrho _{\rm ref}R,
\end{array}
\right.$$ with boundary conditions $\mathbf{V} |_{ \partial \mathcal{D} }=0$ and $\mathbf{J} _S \cdot \mathbf{n} ^\flat |_{ \partial \mathcal{D} }= 0 $, follow from the variational condition together with the variational and phenomenological constraints and .
When applied to a general free energy Lagrangian density $ \mathscr{L}$, this theorem yields the general system $$\label{general_L}
\left\{
\begin{array}{l}
\vspace{0.2cm}\displaystyle \frac{D }{Dt} \frac{\partial \mathscr{L}}{\partial \dot \varphi } - \frac{\partial ^{\nabla\!\!}\mathscr{L}}{\partial \varphi } = \operatorname{DIV} \left( - \frac{\partial \mathscr{L}}{\partial T_ X \varphi } + (\mathbf{P} ^{\rm fr} ) ^\flat \right) ,\\[2mm]
\displaystyle\mathfrak{T} \left( \frac{d}{dt} \frac{\partial \mathscr{L}}{\partial \mathfrak{T}} + \operatorname{DIV} \mathbf{J} _S \right) = ( \mathbf{P} ^{\rm fr}) ^\flat : \nabla \mathbf{V} - \mathbf{J} _S \cdot \mathbf{d} \mathfrak{T} + \varrho _{\rm ref}R,
\end{array}
\right.$$ with boundary conditions $\mathbf{V} |_{ \partial \mathcal{D} }=0$ and $\mathbf{J} _S \cdot \mathbf{n} ^\flat |_{ \partial \mathcal{D} }= 0 $. This system is the continuum version of system and describes the equations of motion for a continuum theory with free energy Lagrangian density $\mathscr{L}$ and subject to the irreversible processes of viscosity and heat conduction.
**Geometric structure associated to the variational formulation.** We now comment on the general geometric setting underlying the formulation – and its relation with the variational formulation in nonholonomic mechanics. Let us consider formally the manifold $ \mathcal{Q} := \operatorname{Diff}_0( \mathcal{D} ) \times \mathcal{F} ( \mathcal{D} ) \times \mathcal{F} ( \mathcal{D} )$ and denote an element in $\mathcal{Q}$ by $q:= (\varphi , \Gamma , \Sigma)$. We consider the vector bundle $T \mathcal{Q} \times _\mathcal{Q} T \mathcal{Q} \rightarrow \mathcal{Q} $ whose vector fiber at $q \in \mathcal{Q} $ is given by the vector space $T_q \mathcal{Q} \times T_q \mathcal{Q} $. For convenience, we shall write an element in this fiber by using the local$^{5}$ notation $( q, \dot q, \delta q) \in T_q \mathcal{Q} \times T_q \mathcal{Q}$.
Geometrically, the variational constraint defines a subset $C_V \subset T \mathcal{Q} \times _ \mathcal{Q} T \mathcal{Q}$ as follows: $( q, \dot q, \delta q) \in C_V$ $\Leftrightarrow$ $( q, \dot q, \delta q)$ satisfies , where $( q, \dot q, \delta q)= (\varphi , \Gamma , \Sigma,\dot \varphi , \dot\Gamma , \dot\Sigma, \delta \varphi , \delta \Gamma , \delta \Sigma)$. This variational constraint satisfies the following property: for each $(q, \dot q) \in T \mathcal{Q} $, the set $C_V(q, \dot q)$ defined by $C_V(q, \dot q):= C_V \cap \{(q, \dot q)\} \times T_q \mathcal{Q}$ is a vector space.
The phenomenological constraint on $(q, \dot q)= (\varphi , \Gamma , \Sigma,\dot \varphi , \dot\Gamma , \dot\Sigma)$ geometrically defines a subset $C_K \subset T \mathcal{Q}$ of the tangent bundle to $\mathcal{Q} $. For the case of adiabatically closed systems (i.e., $ \varrho _{\rm ref} R=0$), the subset $C_K$ is obtained from the variational constraint $C_V$ via the following general construction $$\label{CKCV}
C_K:=\{(q, \dot q) \in T \mathcal{Q} \mid (q, \dot q)\in C_V(q, \dot q)\}.$$ Constraints $C_K$ and $C_V$ are related as in , which we refer to as *constraints of the thermodynamic type*, see [@GBYo2017c].
In terms of the above constraint sets $C_V$ and $C_K$, the variational formulation – of the Navier-Stokes-Fourier system can thus be written as follows:
A curve $q(t) = ( \varphi (t), \Gamma (t), \Sigma (t))\in \mathcal{Q} $, $t \in [t _1 , t _2 ]$ satisfies the Navier-Stokes-Fourier system if and only if it satisfies the variational condition $$\label{VC_general}
\delta \int_{t_1}^{t_2} \mathsf{L}(q, \dot q) dt=0,$$ for all variations $$\label{CV_general}
\delta q \in C_V(q, \dot q)$$ with $ \delta q(t _1 )= \delta q(t _2)=0$ and where the curve $q(t)$ satisfies $$\label{CK_general}
\dot q(t) \in C_K.$$ The Lagrangian $\mathsf{L}$ in denotes the full expression under the time integral in , namely, $$\mathsf{L}(q,\dot{q})=\int_ \mathcal{D} \left[ \mathscr{L} \big( \varphi , \dot \varphi , T\varphi , \varrho _{\rm ref}, \dot \Gamma\big)- \Sigma \dot \Gamma\right] \mu _g.$$
From a mathematical point of view, the variational formulation of – is a [*nonlinear*]{} (and infinite dimensional) extension of the Lagrange-d’Alembert principle used for the treatment of nonholonomic mechanical systems with *linear* constraints, see e.g., [@Bl2003]. Such linear constraints are given by a distribution $ \Delta \subset T \mathcal{Q} $ on $ \mathcal{Q} $. In this linear case, we have $C_K= \Delta \subset T \mathcal{Q} $ and the variational constraint is $C_V= T \mathcal{Q} \times _Q \Delta $.
For the case of *nonlinear* constraints $C _K\subset T \mathcal{Q} $ on velocities in *mechanics*, which are called [*kinematic constraints*]{}, a generalization of the Lagrange-d’Alembert principle has been considered in [@Ch1934], see also [@Ap1911], [@Pi1983]. In Chetaev’s approach, the variational constraint $C_{V}$ is derived from the kinematic constraint $C_K$. However, it has been pointed out in [@Ma1998] that this principle does not always lead to the correct equations of motion for mechanical systems and in general one has to consider the kinematic and variational constraints as independent notions. A general geometric variational approach for nonholonomic systems with nonlinear and (possibly) higher order kinematic and variational constraints has been described in [@CeIbdLdD2004]. This setting generalizes both the Lagrange-d’Alembert and Chetaev approaches. It is important to point out that for these generalizations, including Chetaev’s approach, energy may not be conserved along the solution of the equations of motion. The variational formulation – falls into the general setting described in [@CeIbdLdD2004], extended here to the infinite dimensional setting. In the special case of constraints of the thermodynamic type, i.e., related through , the energy is conserved, see [@GBYo2017c], consistently with the fact that in such a situation the system is isolated.
Variational formulation in spatial representation
-------------------------------------------------
We shall now develop the spatial (or Eulerian) representation of the variational formulation --. The spatial fields associated to $\dot \varphi $, $ \varrho _{\rm ref}$, $\Gamma $, $ \Sigma $, $ \mathfrak{T} $, are denoted by $\mathbf{v}$, $ \rho $, $s$, $ \sigma $, $ \gamma $, $T$. The spatial and Lagrangian fields are related as follows $$\label{def_Eulerian_variables}
\begin{array}{lll}
\vspace{0.2cm}&\mathbf{v}= \dot \varphi\circ\varphi ^{-1}, &\rho =(\varrho _{\rm ref}\circ \varphi^{-1} )J_ \varphi ^{-1},\\
\vspace{0.2cm}&s = (S\circ \varphi^{-1} )J_ \varphi ^{-1}, \qquad & \sigma = (\Sigma \circ \varphi^{-1} )J_ \varphi ^{-1},\\
\vspace{0.2cm}&\gamma = \Gamma \circ \varphi^{-1} , & T=\mathfrak{T} \circ \varphi^{-1}.
\end{array}$$ The Eulerian quantities associated to $ \mathbf{J} _S$, $ \mathbf{P} ^{\rm fr}$, $R$ are denoted by $ \mathbf{j} _s$, $ \boldsymbol{\sigma} ^{\rm ref}$, $r$, and are defined as $$\label{def_Eulerian_flux}
\begin{aligned}
&\mathbf{j}_s= (T \varphi \circ \mathbf{J} _S \circ \varphi^{-1} )J_\varphi ^{-1}, \qquad r= R \circ \varphi^{-1},\\
&\boldsymbol{\sigma} ^{\rm fr}(x)( \alpha _x, \beta _x)= J_ \varphi ^{-1}\mathbf{P} ^{\rm fr}( \varphi ^{-1} (x) )( \alpha _x, T ^\ast _X \varphi ( \beta _x)),
\end{aligned}$$ for all $x \in \mathcal{D} $ and for all $ \alpha _x , \beta _x \in T^* _x \mathcal{D} $.
From the expression of the free energy Lagrangian , we deduce its spatial representation as $$\label{Lagrangian_NSF_Euler}
\ell( \mathbf{v} , \rho , T)=\int_ \mathcal{D} \Lambda( \mathbf{v}, \rho , T) \mu _g= \int_ \mathcal{D} \frac{1}{2} \rho | \mathbf{v}|_g ^2\mu _g-\int_ \mathcal{D} \psi (\rho , T) \mu _g.$$ Proceeding similarly as in [@GBYo2016b] we use the relations and to rewrite the variational formulation -- in spatial representation for a curve $( \mathbf{v} (t), \rho (t), \gamma (t), \sigma (t))$ as follows: $$\label{VC_NSF_Eulerian}
\delta \int_{t_1}^{t_2} \!\!\int_ \mathcal{D} \left[ \Lambda \big(\mathbf{v} , \rho , D_t\gamma \big)- \sigma D_t \gamma \right] \mu _g dt=0, \quad\;\;\; \textsc{Variational Condition}$$ with respect to variations $$\label{variations}
\delta \mathbf{v} = \partial _t \boldsymbol{\zeta} +[ \mathbf{v} , \boldsymbol{\zeta} ], \quad \delta \rho =- \operatorname{div}( \rho \boldsymbol{\zeta} ), \quad \delta \gamma , \quad \text{and} \quad \delta \sigma,$$ subject to the constraint $$\label{CV_NSF_Eulerian}
D_t\gamma \bar D_ \delta \sigma = ( \boldsymbol{\sigma} ^{\rm fr})^\flat : \nabla \boldsymbol{ \zeta } -\mathbf{j} _S \cdot \mathbf{d} D_ \delta \gamma, \qquad\, \textsc{Variational Constraint}$$ with $ \boldsymbol{\zeta} (t_i)=0$ and $ \delta \gamma (t_i)=0$, for $i=1,2$, where the curve $( \mathbf{v} (t), \rho (t), \gamma (t), \sigma (t))$ satisfies the constraint $$\label{KC_NSF_Eulerian}
D_t\gamma \bar D_t\sigma = ( \boldsymbol{\sigma} ^{\rm fr}) ^\flat : \nabla \mathbf{v} -\mathbf{j} _S \cdot \mathbf{d} D_t \gamma+ \rho r, \;\;\, \textsc{Phenomenological Constraint}.$$
The first two expressions in are obtained by taking the variations with respect $ \varphi$, $ \mathbf{v} $, and $ \rho $, of the first two relations in and by defining the vector field $ \boldsymbol{\zeta} := \delta \varphi \circ\varphi^{-1} $. These formulas can be directly justified by employing the Euler-Poincaré reduction theory on Lie groups, for instance, see [@HoMaRa1998].
In , , and , we introduced the notations $$\begin{array}{lll}
\vspace{0.2cm}&D_tf:= \partial _t f+ \mathbf{v}\cdot \mathbf{d} f, & \qquad \bar D_t f := \partial _t f + \operatorname{div}( f \mathbf{v} ),\\
\vspace{0.2cm}&D_ \delta f:= \delta f+ \boldsymbol{\zeta} \cdot \mathbf{d} f, &\qquad \bar D_ \delta f := \delta f + \operatorname{div}( f \boldsymbol{\zeta} ),
\end{array}$$ for the Lagrangian time derivatives and variations of scalar fields and density fields.
By applying , using the expression for the variations $ \delta \mathbf{v} $ and $ \delta \rho $, and $ \mathbf{v} |_{ \partial \mathcal{D} }=0$, $ \delta \gamma (t _i )=0$, $i=1,2$, we find the condition $$\begin{aligned}
&\int_{t_1}^{t_2} \!\!\int_ \mathcal{D} \left[ \left( \frac{\partial \Lambda }{\partial \mathbf{v} }+ \left( \frac{\delta \Lambda}{\delta T} - \sigma \right) \mathbf{d} \gamma\right) \cdot (\partial _t \boldsymbol{\zeta} + [\mathbf{v} , \boldsymbol{\zeta}] )-\frac{\partial \Lambda}{\partial \rho } \operatorname{div}( \rho \boldsymbol{\zeta} ) \right.\\
&\left. \qquad\qquad\qquad \qquad\qquad \qquad \qquad \qquad - \bar D_t\left( \frac{\delta \Lambda}{\delta T} - \sigma \right) \delta \gamma -\delta \sigma D_t\gamma \right] \mu _g\, dt=0.\end{aligned}$$ Using the variational constraint , collecting the terms proportional to $ \boldsymbol{\zeta} $ and $ \delta \gamma $, and using $\boldsymbol{\zeta} |_{ \partial \mathcal{D} }=0$, $ \mathbf{v} |_{ \partial \mathcal{D} }=0$, $ \boldsymbol{\zeta} (t _i )=0$, $i=1,2$, we obtain the three conditions $$\begin{aligned}
\boldsymbol{\zeta} : \quad &( \partial _t + \pounds _ \mathbf{v} ) \left( \frac{\partial \Lambda}{\partial \mathbf{v} }+ \left( \frac{\delta \Lambda}{\delta T} - \sigma \right) \mathbf{d} \gamma\right)= \rho \,\mathbf{d} \frac{\partial \Lambda}{\partial \rho }- \sigma \mathbf{d} (D_t \gamma )+\operatorname{div} \boldsymbol{\sigma} ^{\rm fr} - (\operatorname{div} \mathbf{j} _s )\,\mathbf{d} \gamma, \\
\delta \gamma : \quad & \bar D_t\left( \frac{\delta \Lambda}{\delta T} - \sigma \right) = - \operatorname{div} \mathbf{j} _s \qquad\text{and}\qquad \mathbf{j} _s\cdot \mathbf{n} ^\flat =0 \;\; \text{on $ \partial\mathcal{D} $} ,\end{aligned}$$ where we introduced the Lie derivative notation $\pounds _ \mathbf{v} \mathbf{m}: = \mathbf{v} \cdot \nabla \mathbf{m} + \nabla \mathbf{v} ^{\mathsf{T}} \cdot \mathbf{m} + \mathbf{m}\operatorname{div} \mathbf{v} $ for a one-form density $ \mathbf{m}$ along a vector field $ \mathbf{v} $. Further computations and the phenomenological constraint finally yield the system $$\label{system_Eulerian}
\left\{
\begin{array}{l}
\vspace{0.2cm}\displaystyle
( \partial _t + \pounds _ \mathbf{v} ) \frac{\partial \Lambda}{\partial \mathbf{v} }= \rho \,\mathbf{d} \frac{\partial \Lambda }{\partial \rho }- \frac{\partial \Lambda}{\partial T } \mathbf{d} T+ \operatorname{div} \boldsymbol{\sigma} ^{\rm fr}\\
\vspace{0.2cm}\displaystyle T \left( \bar D_t \frac{\delta \Lambda}{\delta T} + \operatorname{div} \mathbf{j} _s \right) = (\boldsymbol{\sigma} ^{\rm fr} ) ^\flat : \nabla \mathbf{v} - \mathbf{j} _s \cdot \mathbf{d} T+ \rho r \\
\bar D_t \rho =0,
\end{array}
\right.$$ whose last equation, the mass conservation equation, follows from the definition of $ \rho $ in terms of $\varrho _{\rm ref} $. These are the general equations of motion, in free energy Lagrangian form, for fluid dynamics subject to the irreversible processes of viscosity and heat conduction. By specifying this system to the Lagrangian , one immediately gets the Navier-Stokes-Fourier system in the form $$\label{NSF_Eulerian}
\left\{
\begin{array}{l}
\vspace{0.2cm}\displaystyle
\rho ( \partial_t \mathbf{v} +\mathbf{v} \cdot \nabla \mathbf{v} )= - \operatorname{grad} p +\operatorname{div} \boldsymbol{\sigma} ^{\rm fr}\\
\vspace{0.2cm}\displaystyle T\bar D_t \frac{\partial \psi }{\partial T} = \operatorname{div} (T\mathbf{j} _s ) -(\boldsymbol{\sigma} ^{\rm fr} ) ^\flat: \nabla \mathbf{v} - \rho r\\
\bar D_t \rho =0,
\end{array}
\right.$$ where $p= \rho \frac{\partial \psi}{\partial \rho }- \psi $. The heat equation can be rewritten as $$T \left( \frac{\partial s }{\partial T }D_tT + \frac{\partial p}{\partial T} \operatorname{div} \mathbf{v} \right) = (\boldsymbol{\sigma} ^{\rm fr} ) ^\flat: \nabla \mathbf{v} - \operatorname{div}( T \mathbf{j} _s ) + \rho r ,$$ where the partial derivatives are taken at constant mass density and constant temperature. In terms of usual coefficients$^{6}$ (the specific heat at constant volume $C_v$, the speed of sound $c_s ^2 $ and the diabatic temperature gradient $ \Gamma $, which may all depend on $ \rho $ and $T$) it reads $$\rho C_v \left( D_tT+ \rho c_s ^2 \Gamma \operatorname{div} \mathbf{v}\right) = (\boldsymbol{\sigma} ^{\rm fr} ) ^\flat : \nabla \mathbf{v} - \operatorname{div}( T \mathbf{j} _s ) + \rho r.$$ This heat equation is valid for any state equations. In the case of the perfect gas, it simplifies since $C_v$ is a constant and $ \rho ^2 c_s ^2 C_v\Gamma = p$. These results are summarized as follows.
The Navier-Stokes-Fourier equations in spatial representation, given by $$\label{summary_NSF_Eulerian}
\left\{
\begin{array}{l}
\vspace{0.2cm}\displaystyle\rho ( \partial_t \mathbf{v} +\mathbf{v} \cdot \nabla \mathbf{v} )= -\mathbf{d} p +\operatorname{div} \boldsymbol{\sigma} ^{\rm fr},\\[2mm]
\displaystyle\rho C_v \left( D_tT+ \rho c_s ^2 \Gamma \operatorname{div} \mathbf{v}\right) = \boldsymbol{\sigma} ^{\rm fr} : \nabla \mathbf{v} - \operatorname{div}( T \mathbf{j} _s ) + \rho r,
\end{array}
\right.$$ with boundary conditions $\mathbf{v} |_{ \partial \mathcal{D} }=0$ and $\mathbf{j} _s \cdot \mathbf{n} ^\flat |_{ \partial \mathcal{D} }= 0 $, follow from the variational condition with the variational and phenomenological constraints , .
[In order to close the system , it is necessary to provide phenomenological expressions of the thermodynamic fluxes in terms of the thermodynamic affinities, compatible with the second law of thermodynamics. In our case, the thermodynamic fluxes are $ \boldsymbol{\sigma} ^{\rm fr}$ and $ \mathbf{j} _s$ and we have the well-known relations $$\label{friction_stress_NSF}
\boldsymbol{\sigma} ^{\rm fr}=2 \mu (\operatorname{Def} \mathbf{v})^{\sharp }+ \left( \zeta - \frac{2}{3}\mu \right)(\operatorname{div} \mathbf{v} ) g^\sharp\quad\text{and}\quad T\mathbf{j} _s^{\flat}= - \kappa \mathbf{d} T \;\; \text{(Fourier law)},$$ where $ \operatorname{Def} \mathbf{v} = \frac{1}{2} (\nabla \mathbf{v} + \nabla \mathbf{v} ^\mathsf{T})$, $ \mu \geq 0 $ is the first coefficient of viscosity (shear viscosity), $ \zeta \geq 0$ is the second coefficient of viscosity (bulk viscosity), and $ \kappa \geq 0$ is the thermal conductivity. Generally, these coefficients depend on $ \rho $ and $T$. ]{}
**Constraints in infinite dimensions.** The variational formulation for the Navier-Stokes-Fourier system developed in this paper is based on *nonlinear* and *infinite dimensional* generalizations of the Lagrange-d’Alembert principle of nonholonomic mechanics. In the present case, the infinite dimensional constraint is *not* associated to a mechanical constraint, it is the expression of the entropy production of the system. For infinite dimensional constrained *mechanical* systems, variational formulations have been used for example in [@GBPu2012], [@GBPu2015a] and [@GBPu2014], [@GBPu2015b] to derive and study geometrically exact models for elastic strands with rolling contact and for flexible fluid-conducting tubes. An infinite dimensional generalization of the Lagrange-d’Alembert principle was proposed in [@GBYo2016a] to treat the case of 2$^{nd}$ order Rivlin-Ericksen fluids in the context of nonholonomic systems. We refer to [@ShBKZeBl2015] for a treatment of infinite dimensional constrained mechanical systems via Hamel’s formalism.
**Conclusion and future direction.** In this paper, we presented a Lagrangian variational formulation for the Navier-Stokes-Fourier system based on the free energy. This formulation is developed in a systematic way from the free energy variational formulation of the thermodynamics of discrete systems described in §\[2\_3\]. The approach presented in this paper complements that made in [@GBYo2016b] as it uses the temperature, rather than the entropy, as an independent variable. The proposed free energy variational formulation is also well-adapted to include additional irreversible processes such as diffusion and chemical reactions treated in [@GBYo2016b]. It can also be extended to cover the case of moist atmospheric thermodynamics following [@FGB2017].
**Acknowledgements.** F.G.B. is partially supported by the ANR project GEOMFLUID, ANR-14-CE23-0002-01; H.Y. is partially supported by JSPS Grant-in-Aid for Scientific Research (26400408, 16KT0024, 24224004) and the MEXT “Top Global University Project”.
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---
abstract: 'We show that adiabatic evolution of a low-dimensional lattice of quantum spins with a spectral gap can be simulated efficiently. In particular, we show that as long as the spectral gap $\Delta E$ between the ground state and the first excited state is any constant independent of $n$, the total number of spins, then the ground-state expectation values of local operators, such as correlation functions, can be computed using polynomial space and time resources. Our results also imply that the local ground-state properties of any two spin models in the same quantum phase can be efficiently obtained from each other. A consequence of these results is that adiabatic quantum algorithms can be simulated efficiently if the spectral gap doesn’t scale with $n$. The simulation method we describe takes place in the Heisenberg picture and does not make use of the finitely correlated state/matrix product state formalism.'
author:
- 'Tobias J. Osborne'
title: Simulating adiabatic evolution of gapped spin systems
---
Introduction
============
The low-temperature physics of lattices of interacting quantum spins is typically very complex. The computational cost of even *approximating* basic properties, such as the ground-state energy eigenvalue, of these systems is prohibitive. Indeed, for 2D lattices of interacting spins, the task of computing an approximation to the ground-state energy eigenvalue correct to within some polynomial confidence interval is fantastically difficult — this problem is complete for the complexity class [QMA]{}, which is the quantum version of the complexity class [NP]{} [@oliveira:2005a; @kempe:2004a; @kitaev:2002a].
It might therefore seem that the computational task of approximating the low-temperature behaviour of interacting quantum spins is entirely hopeless. However, for physically realistic models, this is not the case in practice. Many algorithms have been developed which appear to provide efficient approximations to a wide variety of local properties of physically realistic systems, such as correlators, at low temperature. Perhaps the most successful of these methods has been the family of algorithms based on the density matrix renormalisation group (DMRG) (See [@schollwoeck:2005a] and references therein for a review of the DMRG and description of extensions.)
The DMRG is a remarkably flexible and adaptable algorithm, admitting a slew of generalisations. Applications include: simulating dynamics [@vidal:2003a; @vidal:2003b], dissipative systems [@verstraete:2004b; @zwolak:2004a], disordered systems [@paredes:2005a], and higher dimensional lattices [@verstraete:2004a]. At least part of the flexibility of the DMRG is due to the fact that it is equivalent to a variational minimisation over the space of *finitely correlated states* (FCS) [@fannes:1992a]. Hence, the methodology of the DMRG can be adapted to any situation where the principle object of study, be it an eigenstate or propagator, can be approximated using a FCS vector on Hilbert space. An alternative to methods based on variations over FCS has been recently proposed which appears to offer spectacular computational speedups over the DMRG and relatives [@vidal:2005a].
In practice it appears that the DMRG and related algorithms can efficiently obtain arbitrarily accurate approximations to the local ground-state properties of a $1$D collection of interacting quantum spins. However, at the current time, there is no satisfactory understanding of the *correctness* (i.e. will the DMRG *always* return a faithful approximation to the ground state and not some other eigenstate) and the *complexity* (i.e., assuming correctness, how much computational resources are required to obtain a good approximation to a ground state) of the DMRG.
The correctness of the DMRG is far from obvious. This is because the ground-state approximation obtained by the DMRG *cannot* be certified; the DMRG only returns an approximate ground-state eigenvector and cannot guarantee that this vector is close to the true ground state. It is therefore extremely desirable to determine *a priori* the class of systems for which the DMRG and relatives provably return faithful approximations to the ground state. The complexity [@endnote40] of the DMRG is also difficult to ascertain. Assuming we could prove correctness of the DMRG for a class of realistic physical systems, the actual complexity of the DMRG depends subtly on many detailed properties of the system, such as geometric entropy of the ground state, and nonconvexity of the objective function which is minimised.
Recently this situation is changing [@hastings:2006a; @verstraete:2005a; @osborne:2005d]. In [@hastings:2005b] an analysis of the resource scaling of a DMRG-like algorithm to obtain approximations to the ground states of $1$D gapped local models was undertaken. This paper provides the first general subexponential estimate for the time and space resource requirements of any *provably correct* method to compute approximations to the ground states of gapped models; it was found that if the model is gapped then resources scaling as $n^{c\log n}$, with $c$ some constant, are sufficient to obtain and store a computational representation of the ground state of a gapped local model [@endnote41]. In [@verstraete:2005a] it was shown that the ground state of some [@endnote42] *critical* $1$D spin models can be *stored* efficiently. Unfortunately, there is currently no theoretical argument which implies that these approximations to the ground states can be obtained efficiently. Indeed, the results of this paper imply that if such approximations are obtained via adiabatic continuation then *exponential* computational resources may be required to obtain them. (Note, however, that we can say nothing about the other methods to obtain such FCS approximations.) Finally, in [@osborne:2005d] it was shown that an approximation to the propagator for a $1$D lattice of quantum spins can be obtained and stored (as a FCS vector) using polynomial resources in $n$ and the error $\epsilon$ and exponential resources in the time $|t|$. (It is straightforward to extend the argument of [@osborne:2005d] to show an analogous result in $2$D.)
There is at least one solid reason why we believe that DMRG-like methods ought to provide a computationally efficient recipe to compute approximations to the ground states of gapped systems. Namely, we know that the ground-state correlation functions for any gapped system are *clustering* or rapidly decaying with separation [@hastings:2004a; @hastings:2004b; @nachtergaele:2005a]. This result, which is the natural analogue of Fredenhagen’s proof [@fredenhagen:1985a] of clustering for relativistic quantum field theories, is especially impressive given that it applies to an extremely wide class of quantum lattice systems in low dimensions. As a consequence of clustering results we conclude that gapped spin systems are essentially free — an intuition which is persuasively backed up by classical renormalisation-group style argumentation — and thus can be modelled as noninteracting *effective* spins, which can be simulated easily.
Another way of arriving at this conclusion is to think of correlations as roughly “measuring” the degree of *quantum* correlations in the ground state. Since the amount of quantum correlations in a quantum state limits the extent to which a state can be approximated by a FCS [@verstraete:2005a], we are strongly encouraged to think that the clustering results may actually imply that DMRG-like algorithms may converge rapidly for at least some realistic gapped systems.
Unfortunately, knowing that the correlations decay is not enough information to infer that the eigenstates are finitely correlated. To understand this simply consider a *generic quantum state* [@hayden:2004b] which is a quantum state chosen uniformly from Haar measure induced on state-space. A generic quantum state exhibits rapidly decaying correlations (indeed, all $m$-point correlation functions are essentially zero for $m < \frac{n}{2}$) yet such a state is extremely entangled and cannot be efficiently represented as a finitely correlated state. Nevertheless, it might be argued that the results of [@hastings:2004a; @hastings:2004b; @nachtergaele:2005a] avoid this counterexample because they prove something stronger, namely *exponential clustering*, which says that the reduced density operator $\rho_{AB}$ of the ground state for two arbitrarily large separated regions $A$ and $B$ is indistinguishable from a product $\rho_A\otimes\rho_B$ when it is used to compute expectations for *product* observables $M_A
M_B$. Interestingly, a naive attempt to exploit this exponential clustering runs into problems. The reason is that there exist highly entangled states $\sigma_{AB}$, called *data-hiding states*, which exhibit precisely these properties [@hayden:2004c]. Thus, to prove that the ground state of a gapped local hamiltonian is well-approximated by finitely correlated state with polynomial resources we appear to need more information than that given by correlation functions.
Despite some recent progress a solution to the fundamental problem, namely, to prove correctness of *any* algorithm which obtains approximations to local ground-state properties for gapped $1$D models and to further provide a *polynomial* theoretical worst-case estimate on the resource requirements such an algorithm, still seems far away. Let us summarise the various approaches to finding approximations to the ground state of a spin model and the theoretical obstructions encountered in each of these approaches.
There are at least four ways to obtain an approximation to the ground state of a quantum system: (i) variation over a class of ansatz ground states; (ii) simulation of the thermalisation process via imaginary time evolution or similar; (iii) approximation of the convex set of reduced density operators of translation-invariant quantum states; and (iv) adiabatic continuation from the ground states of classical spin models. The DMRG is an example of the first method, namely it is a variation over the class of FCS with fixed auxiliary dimension. Unfortunately this variation is, in general, nonconvex and it has been recently discovered [@eisert:2006a] that hard instances for a closely related variation problem can be constructed. Thus it seems likely that the DMRG is not correct in general. The second approach, namely imaginary time evolution, suffers from the shortcoming that an initial guess $|\Omega'\rangle$ for the ground state $|\Omega\rangle$ which has nontrivial overlap with the actual ground state is required. If such an initial guess is unavailable then the storage requirements of the imaginary time evolution approach could be, in the worst case, exponential [@endnote43]. It seems plausible that obtaining such a guess could be as hard as solving the original problem. The third method requires an exponentially good characterisation of the convex set of reduced density operators of translation-invariant quantum states in order to obtain $O(1)$ estimates for local operators. The final method, which is the focus of this paper, suffers from the limitation that it is not known if the ground state of an arbitrary gapped spin model can be obtained via adiabatic continuation from a classical model without encountering a quantum phase transition. However, it has been recently proved [@yarotsky:2004a; @yarotsky:2005a; @yarotsky:2005b] that in the *neighbourhood* of a classical spin model adiabatic continuation *will* work. Thus, using this approach, we are able to provide the first polynomial estimates on the resource requirements of a correct method to obtain a representation of the ground state of at least a subclass of gapped models.
There is an intimate connection between simulating adiabatic continuation for quantum lattice models and simulating quantum computations [@aharonov:2004a; @kempe:2004a]. Namely, if adiabatic evolution for an arbitrary 2D lattice model with a gap that scales as an inverse polynomial of the system size could be simulated efficiently on a classical computer then [@endnote44] [BQP]{}$\subseteq$[P]{}, thus obviating the need to design and engineer a quantum computer in the first place! Naturally, our results are nowhere near strong enough to show the complexity class inclusion [BQP]{}$\subseteq$[P]{}, but they do have implications for error correction methods for adiabatic quantum algorithms.
A complete theory of quantum error correction for adiabatic quantum algorithms [@farhi:2001a] is still being developed. For example, for general thermalisation decoherence, we really have no idea how to calculate a fault-tolerance threshold for adiabatic quantum algorithms (see [@nielsen:2000a] and [@preskillnotes] for a discussion of quantum error correction and fault tolerance.). Presumably a general quantum error-correcting code for a quantum adiabatic algorithm would involve encoding the adiabatic evolution in a larger system such that the minimum spectral gap encountered along the evolution was larger [@endnote45]. This would mean it would cost the environment more energy/unit time to induce a transition from the ground state during the evolution (an “error”). It is natural to assume that the gap could be boosted to a large constant, independent of the number $n$ of spins, with a polynomial increase in size. Our results show that if this were possible then we could simulate adiabatic quantum algorithms efficiently on a classical computer! Thus, conditioned on the strict complexity class containment [P]{} $\subset$ [BQP]{}, we obtain a bound on how large the gap could be boosted by encoding for adiabatic quantum algorithms.
The method we develop in this paper is very closely related to the method studied in [@wen:2005a]. In [@wen:2005a] the authors investigate the evolution of local operators under a quasi-adiabatic change in a local hamiltonian. As long as the hamiltonian has a spectral gap throughout the evolution, it was found that local operators remained local and thus it was possible to say that local gauge invariance remains when two hamiltonians are in the same phase. Our task is similar: we wish to understand the expectation values of local operators in the ground state of a system that has *undergone* adiabatic evolution. We wish to show that the computation of such expectation values can be done efficiently on a classical computer as long as the smallest gap encountered during the adiabatic evolution is $O(1)$. While this calculation can be treated using quasi-adiabatic evolution and the methods developed in [@wen:2005a] to study such evolutions, we prefer to study *exact* adiabatic evolution. We do this primarily in anticipation of the application of these results to studying entropy-area laws for systems in the same phase.
We provide an efficient computational method to compute the expectation values of local operators in the ground states of hamiltonians undergoing exact adiabatic evolution, a method which works equally well for hamiltonians with spatially varying interactions. Our method does not make use of the FCS formalism. Rather, we develop our simulation method in the *Heisenberg picture*, where locality is manifest. Indeed, if we were to make use of state representations in the Schrödinger picture, i.e. the 2D FCS formalism (PEPS), we would be unable to apply our results because even if we could construct PEPS approximations to the adiabatically continued ground state it is currently unknown how to efficiently extract expectation values of local operators from the PEPS representation. We sidestep this issue by providing a ground-state *certificate* in the form of a specification of a local hamiltonian which can be efficiently numerically simulated in the Heisenberg picture to extract local expectation values.
The outline of this paper is as follows. We begin in §\[sec:form\] by introducing the class of local hamiltonians we consider and stating the problem we wish to solve. In §\[sec:efflocal\] we then show how adiabatic evolution for quantum lattices of spins can be described by unitary dynamics of an *effective* local hamiltonian. We use this effective dynamics in §\[sec:effsim\] to construct an approximate local dynamics which can then be used to efficiently extract local properties of the adiabatically continued ground state. We conclude with some discussion of our results in §\[sec:disc\]. We detail some simple properties of compactly supported $C^\infty$ functions in Appendix \[app:cutoff\].
Formulation {#sec:form}
===========
In this section we introduce the Hilbert space and operator algebras for the systems we consider. We define what we mean by strictly local and approximately local hamiltonians. Finally, we specify the computational task that will occupy us for the rest of this paper.
We consider quantum systems defined on a set of vertices $V$ with a finite dimensional Hilbert space $\mathcal{H}_x$ attached to each vertex $x\in V$. We always assume that $V$ is finite. (There are some minor theoretical obstructions which currently preclude a simple extension of our results to infinite lattices; we’ll discuss this in a further paper.) For $X\subset V$, the Hilbert space associated to $X$ is the tensor product $\mathcal{H}_X=\bigotimes_{x\in X}\mathcal{H}_x$, and the algebra of observables on $X$ is denoted by $\mathcal{A}_X=\mathcal{B}(\mathcal{H}_X)$, where $\mathcal{B}(\mathcal{H}_X)$ denotes the $C^*$-algebra of bounded operators on $\mathcal{H}_X$ with norm $$\|A\| = \sup_{|\psi\rangle \in \mathcal{S}(\mathcal{H}_X)}
\|A|\psi\rangle\|,$$ and $\mathcal{S}(\mathcal{H}_X)$ is the state space for $\mathcal{H}_X$. We assume that $V$ is equipped with a metric $d$. In the most common cases $V$ is the vertex set of a graph, and the metric is given by the graph distance, $d(x,y)$, which is the length of the shortest path of edges connecting $x$ and $y$ in the graph. Finally, by tensoring with the unit operators on $Y\setminus X$, we consider $\mathcal{A}_X$ as a subalgebra of $\mathcal{A}_Y$, whenever $X\subset Y$.
We will, for the sake of clarity, introduce and describe our results for a collection of $n$ distinguishable spin-$\frac{1}{2}$ particles. Thus, the Hilbert space $\mathcal{H}$ for our system is given by $\mathcal{H} = \bigotimes_{j=0}^{n-1} \mathbb{C}^2$. We now fix the metric for our vertex set $V$ to be that of a low-dimensional periodic lattice $L$ of $n=m^\eta$ vertices, where $m\in\mathbb{N}$ and $\eta$ is the dimension. Because the case $\eta=2$ is the only really nontrivial case that interests us, we fix $\eta=2$ from now on. We refer to vertices as *sites* and identify each site $v$ with its coordinates $\mathbf{j} = (j_x,
j_y)$. Because the lattice is periodic we identify coordinates: $(j_x=m) \equiv (j_x = 0)$ and $(j_y=m) \equiv (j_y = 0)$. It is entirely straightforward to generalise our results to higher-dimensional lattices, higher dimensional spins, and to more general lattices.
We consider a distinguished basis, the *standard product basis*, for $\mathcal{H}_{V}$ given by $|\mathbf{z}\rangle =
\bigotimes_{{j_x}=0}^{m-1}\bigotimes_{{j_y}=0}^{m-1}
|z_{(j_x,j_y)}\rangle$, $z_{\mathbf{j}}\in \mathbb{Z}/2\mathbb{Z}$. We’ll also have occasion to refer to a certain orthonormal basis for $\mathcal{A}_{V}$: we denote by $ \sigma^{\boldsymbol{\alpha}} =
\bigotimes_{{j_x}=0}^{m-1}\bigotimes_{{j_y}=0}^{m-1}
\sigma_{(j_x,j_y)}^{\alpha_{(j_x,j_y)}}$, $\alpha_{\mathbf{j}}\in
\mathbb{Z}/4\mathbb{Z}$, the *standard operator basis*, where $
\sigma^{0} = \left(\begin{smallmatrix} 1 & 0 \\
0 & 1
\end{smallmatrix}\right)$, $\sigma^{1} = \left(\begin{smallmatrix} 0 & 1 \\ 1 & 0
\end{smallmatrix}\right)$, $\sigma^{2} = \left(\begin{smallmatrix} 0 & -i \\ i & 0
\end{smallmatrix}\right)$, and $\sigma^{3} = \left(\begin{smallmatrix} 1 & 0 \\ 0 & -1
\end{smallmatrix}\right)$, are the Pauli sigma matrices.
We define the *support* ${\operatorname{supp}}(M)\subset V$ of an operator $M\in\mathcal{A}_V$ to be the smallest subset $\Lambda \subset V$ such that $M\in\mathcal{A}_{\Lambda}$, i.e., the smallest subset upon which $M$ acts nontrivially. Let $M\subset L$ and $N\subset L$. We define the *sumset* $M+N\subset L$ of $M$ and $N$ by $M+N =
\{\mathbf{x}+\mathbf{y}\,|\, \mathbf{x}\in M, \mathbf{y}\in N\}$ where the addition operation $\mathbf{x}+\mathbf{y}$ is inherited from the standard addition on $L\equiv(\mathbb{Z}/m\mathbb{Z})\times
(\mathbb{Z}/m\mathbb{Z})$. This operation is the natural generalisation of the convolution operation on the real numbers to the finite group $L$. (It is fairly straightforward to generalise these operations to more general graphs.)
We now introduce the family $H(s)$ of parameter-dependent hamiltonians we are going to focus on. To define our family we’ll initially fix some parameter-dependent interaction term $h(s)\in\mathcal{A}_V$ which has bounded norm [@endnote46]: $\|h(s)\| \le O(1)$. We think of $h(s)$ as being “centred” on site $\mathbf{0}$, i.e. we demand that $\mathbf{0}\in{\operatorname{supp}}(h(s))$. Our family $H(s)$ of quantum systems is then defined by $$\label{eq:tihamdef}
H(s) = \sum_{\mathbf{j}\in L}
\mathcal{T}_y^{j_y}(\mathcal{T}_x^{j_x}(h(s))) = \sum_{\mathbf{j}\in
L} h_{\mathbf{j}}(s),$$ where $\mathcal{T}_x$ (respectively, $\mathcal{T}_y$) is the unit translation operator which translates the subsystems one site across in the $x$ (respectively, $y$) direction, eg., $$\mathcal{T}_x\left(\bigotimes_{{j_x}=0}^{m-1}\bigotimes_{{j_y}=0}^{m-1}
\sigma_{(j_x,j_y)}^{\alpha_{(j_x,j_y)}}\right) =
\bigotimes_{{j_x}=0}^{m-1}\bigotimes_{{j_y}=0}^{m-1}
\sigma_{(j_x+1,j_y)}^{\alpha_{(j_x,j_y)}},$$ and $h_{\mathbf{j}}(s) =
\mathcal{T}_y^{j_y}(\mathcal{T}_x^{j_x}(h(s)))$. While the hamiltonian $H(s)$ generated by this construction is translation-invariant, none of our subsequent calculations depend on this fact in any serious way. Hence the results of this paper apply equally to hamiltonians with spatially varying interactions.
We are going to make three simplifying assumptions about our hamiltonian $H(s)$. The first is that $H(s)$ is assumed to be *strictly local* which means that $|{\operatorname{supp}}(h(s))|$ is an $O(1)$ constant. The second assumption we make is that the interaction $h(s)$ that generates $H(s)$ can be written as $h(s) = h_0 + s h'$, where $h_0$ and $h'$ are two operators with $O(1)$ norm. The final assumption is that the ground state is unique and the spectral gap $\Delta E(s)$ between the ground- and first-excited states for $H(s)$ satisfies the inequality $\Delta E(s) \ge \Delta$, $\forall
s\in[0,1]$, where $\Delta E(s)$ is an $O(1)$ constant. Note that the first two assumptions can be lifted with a little extra work, however, the assumption that the gap $\Delta E(s)$ is an $O(1)$ constant cannot be relaxed: the simulation algorithm we present scales exponentially with $\Delta E(s)$.
We will also have occasion to discuss *approximately local* hamiltonians. Such hamiltonians are obtained in the same way as in (\[eq:tihamdef\]), that is, we fix some initial interaction term $k(s)$ which we then average over translates to generate our hamiltonian $K(s)$. In this case, however, the initial interaction term is allowed to have support equal to all of $V$. The only constraint we make is that $k(s)$ must *decay rapidly* which means that $k(s)$ can be written as a sum: $$k(s) = \sum_{\alpha=0}^{m-1} k_{\alpha}(s)$$ where ${\operatorname{supp}}(k_{\alpha}(s)) = \Lambda_\alpha$, and $\Lambda_\alpha$ consists of all the sites within a distance $\alpha$ of site $0$, i.e., $\Lambda_\alpha = \{\mathbf{j}\, | \, d(\mathbf{0},\mathbf{j})
\le \alpha \}$. As a result, $k_\alpha(s)$ is an operator with a support (or “radius”) consisting of $\alpha$ sites centred on site $0$. The rapid decay condition is then that $$\|k_{\alpha}(s)\| \le f(\alpha), \quad 0 \le \alpha < m.$$ where $f(\alpha)$ is some rapidly decreasing function of $\alpha$.
We say that a hamiltonian $K(s)$ constructed from the interaction $k(s)$ has *rapid decay*. We write the final hamiltonian resulting from this construction as $$\label{eq:ksdef}
K(s) = \sum_{\mathbf{j}\in L}\sum_{\alpha = 0}^{m-1}
k_{\mathbf{j},\alpha}(s),$$ where $k_{\mathbf{j},\alpha}(s) =
\mathcal{T}_y^{j_y}(\mathcal{T}_x^{j_x}(k_{\alpha}(s)))$.
Finally, we set out the problem we aim to solve. We suppose $H(s)$ is a strictly local parameter-dependent hamiltonian for a $2$D lattice of the form (\[eq:tihamdef\]), with interaction $h(s)$ having $O(1)$ norm and $O(1)$ support. We assume that the ground state $|\Omega(s)\rangle$ is unique and, further, that the spectral gap $\Delta E(s)$ between the ground state and first excited state satisfies $\Delta E(s) \ge \Delta$, $\forall s\in [0, 1]$, where $\Delta$ is a constant independent of $n$. Finally, we suppose that expectation values of arbitrary local operators $A\in\mathcal{A}_L$, with $O(1)$ support, in the initial ground state $|\Omega(0)\rangle$ can be computed efficiently, i.e., $\omega_0(A) = \langle \Omega(0)|
A |\Omega(0)\rangle$ can be computed efficiently for all $A\in\mathcal{A}_L$. This would be the case when, for example, $H(0)$ is any regular classical hamiltonian, that is, $[h_{\mathbf{j}}(s), h_{\mathbf{k}}(s)]=0$, $\forall \mathbf{j},
\mathbf{k} \in L$. Alternatively, this occurs when $H(s)$ has a ground state which is exactly a $2$D finitely correlated state. (When $H(s)$ has spatially varying interactions we must require that the ground state of $H(0)$ is a *known* product state. We need to do this in order to avoid the constructions of [@barahona:1982], which show that computing the ground state of a disordered classical systems is at least [NP]{}-hard.)
Our approximation problem is therefore the following. First fix some error $\epsilon$. Then our problem is to find an efficient computational method to compute, for any local operator $A$ with bounded support [@endnote47], uniform approximants $\omega'_{s}(A)$ to the exact expectation values $\omega_s(A) =
\langle \Omega(s)|A|\Omega(s)\rangle$. That is, our problem is to efficiently compute $\omega'_{s}(A)$ so that $|\omega'_{s}(A) -
\omega_s(A)|<\epsilon$ for all $s\in[0,1]$ and for all bounded local operators $A$ with bounded support.
The constraint that the observables whose expectation values are to be simulated must have bounded support stems from the condition that in the large-$n$ limit such operators should be elements of the quasi-local algebra $\mathcal{A}_L$. We lose no generality in this assumption when applying it to the simulation of quantum algorithms because the answer that the algorithm computes should be encoded in the ground state in such a way that it can be read out from the expectation value of a local operator. It is also worth noting that any correlation function involving a bounded number of subsystems satisfies our definition of having bounded support.
Before we end this section we introduce some notation for approximations. Because we have occasion to refer to functions for which only *bounds* on growth, derivatives, etc. are known it is convenient to adopt the following notation. If we have two quantities $A$ and $B$ then we use the notation $A\lesssim B$ to denote the estimate $A\le CB$ for some constant $C$ which only depends on unimportant quantities. In almost all the cases we consider the only important quantity is $n$, the total number of spins. Thus, unless we indicate otherwise, $A\lesssim B$ means that $A\le CB$ for some $C$ independent of $n$. Because we’ll be interested in the consequences of allowing the minimum gap $\Delta$ to depend on $n$ we’ll explicitly retain any dependence on $\Delta$ in our calculations.
Effective local dynamics for exact adiabatic evolution {#sec:efflocal}
======================================================
In this section we study exact adiabatic evolution for quantum spin systems. We show that if there is a gap throughout the evolution then the exact adiabatic evolution is equivalent to unitary dynamics generated by an approximately local hamiltonian.
We consider adiabatic quantum evolution generated by $H(s)$ as $s$ is varied adiabatically from $s=0$ to $s=1$. Thus we would like to understand the ground state $|\Omega(s)\rangle$ of $H(s)$. We do this by setting up a differential equation for $|\Omega(s)\rangle$: $$\label{eq:pformula}
\frac{d}{ds} |\Omega(s)\rangle = P'(s)|\Omega(s)\rangle,$$ where $P'(s) = \frac{d}{ds}(|\Omega(s)\rangle\langle\Omega(s)|)$ and we’ve set phases [@endnote48] so that $\langle
\Omega'(s)|\Omega(s)\rangle = 0$. Because $P'(s)$ is not antihermitian the dynamics generated by this equation are not unitary.
There are at least two ways to set up differential equations for $|\Omega(s)\rangle$ which *do* generate unitary dynamics. The first is via *exact adiabatic evolution* (see [@avron:1987a; @avron:1993a] for a rigourous discussion of rather general results about exact adiabatic evolution): $$\label{eq:gpformula}
\frac{d}{ds} |\Omega(s)\rangle = -[P(s), P'(s)]|\Omega(s)\rangle.$$ Because of the gap condition on $H(s)$, the “hamiltonian” $[P(s),P'(s)]$ for this dynamics is given by first-order stationary perturbation theory: $$\begin{gathered}
[P(s), P'(s)] = |\Omega(s)\rangle \langle \Omega(s)| \frac{\partial
H(s)}{\partial s} \frac{\mathbb{I}}{\Omega(s)\mathbb{I}-H(s)} - \\
\frac{\mathbb{I}}{\Omega(s)\mathbb{I}-H(s)} \frac{\partial
H(s)}{\partial s} |\Omega(s)\rangle \langle \Omega(s)|,\end{gathered}$$ where $\Omega(s)$ is the ground-state energy of $H(s)$, and we define $\frac{\mathbb{I}}{\Omega(s)\mathbb{I}-H(s)}$ via the Moore-Penrose inverse: $\frac{\mathbb{I}}{\Omega(s)\mathbb{I}-H(s)}|\Omega(s)\rangle =0$.
The other way, which we call *effectively local exact adiabatic evolution*, is obtained by rewriting $P'(s)$. We exploit the fact that $H(s)$ has a spectral gap to find $$\label{eq:gsproj}
P(s) = \int_{-\infty}^{\infty} \chi_{\gamma}(t)
e^{-it\Omega(s)}e^{itH(s)}dt,$$ where $\chi_{\gamma}(t)$ is an even real function whose fourier transform $\widehat{\chi}_\gamma$ is $C^\infty$, has compact support in $[-\gamma, \gamma]$, and is normalised so that $\widehat{\chi}_\gamma(0) =1$. (See Appendix \[app:cutoff\] for a description of $C^\infty$ cutoff functions and their properties.) We must set $\gamma < \Delta$ to ensure that only the ground state appears on the RHS of (\[eq:gsproj\]). The formula (\[eq:gsproj\]) for $P(s)$ may be verified by writing $e^{itH}$ in its eigenbasis and exploiting the $L_2$ unitarity of the fourier transform.
We next use the Duhamel formula $$\frac{d}{ds}e^{itH(s)} = i\int_0^t e^{iuH(s)}\frac{\partial
H(s)}{\partial s} e^{i(t-u)H(s)} du,$$ to rewrite (\[eq:pformula\]):
$$\frac{d}{ds} |\Omega(s)\rangle = -i \frac{d\Omega(s)}{ds}
\int_{-\infty}^{\infty} t\chi_\gamma(t) dt|\Omega(s)\rangle +
i\int_{-\infty}^{\infty}
\chi_{\gamma}(t)e^{-it\Omega(s)}\left(\int_0^t
\tau_u^{H(s)}\left(\frac{\partial H(s)}{\partial s}\right) du\right)
e^{itH(s)}dt|\Omega(s)\rangle,$$
where $\tau_u^{H(s)}(M) = e^{iuH(s)}Me^{-iuH(s)}$. Using the fact that $\chi_\gamma(t)$ is an even function of $t$ and cancelling phases we obtain $$\frac{d}{ds} |\Omega(s)\rangle = i\int_{-\infty}^{\infty}
\chi_{\gamma}(t)\left(\int_0^t \tau_u^{H(s)}\left(\frac{\partial
H(s)}{\partial s}\right) du\right)
dt|\Omega(s)\rangle.$$ By integrating this expression for $\frac{d}{ds} |\Omega(s)\rangle$ in the energy eigenbasis of $H(s)$ and using the assumed gap structure one can find that this expression is equivalent to the usual expression obtained from first-order perturbation theory: $$\frac{d}{ds} |\Omega(s)\rangle =
\frac{\mathbb{I}}{\Omega(s)\mathbb{I} - H(s)}\frac{\partial
H(s)}{\partial s} |\Omega(s)\rangle.$$
Thanks to our assumed form of $H(s) = H_0 + s H'$, with $H' =
\sum_{\mathbf{j}\in L} h'_{\mathbf{j}} = \sum_{\mathbf{j}\in L}
\mathcal{T}_y^{j_y}(\mathcal{T}_x^{j_x}(h'))$, we notice that $\frac{\partial H(s)}{\partial s} = \sum_{\mathbf{j}\in L}
h'_{\mathbf{j}}$, and we write $$\label{eq:adqevol}
\frac{d}{ds} |\Omega(s)\rangle = i\sum_{\mathbf{j}\in
L}\mathcal{F}_s(h_{\mathbf{j}}') |\Omega(s)\rangle,$$ with initial condition that $|\Omega(0)\rangle$ is the ground state of $H(0)$ and where $\mathcal{F}_s(M) = \int_{-\infty}^{\infty}
\chi_{\gamma}(t) \left(\int_0^t \tau_u^{H(s)}(M)\, du \right) dt$.
The equation (\[eq:adqevol\]) tells us that $|\Omega(s)\rangle$ can be obtained from $|\Omega(0)\rangle$ by *unitary* dynamics according to the time-dependent hermitian hamiltonian $K(s) =
\sum_{\mathbf{j}\in L}\mathcal{F}_s(h_{\mathbf{j}}') =
\sum_{\mathbf{j}\in L} k_\mathbf{j}(s)$, where we write $k_\mathbf{j}(s)=\mathcal{F}_s(h_{\mathbf{j}}')$. We also write $k(s) = \mathcal{F}_s(h')$ for the interaction term $k(s)$ which generates $K(s)$. Furthermore, we claim that $K(s)$ is *approximately local* for all $s\in[0,1]$.
The way to see that $K(s)$ is approximately local is to use the standard Lieb-Robinson bound [@lieb:1972a; @hastings:2004a; @nachtergaele:2005a; @hastings:2005b]. The Lieb-Robinson bound reads $$\label{eq:liebrobinson}
\|[\tau_{t}^{H(s)}(A), B]\| \le |Y| e^{-v d(x,Y)}(e^{\kappa |t|}-1),$$ for any two norm-$1$ operators $A\in\mathcal{A}_x$ and $B\in
\mathcal{A}_{Y}$, with $\{x\}\cap Y = \emptyset$ which are initially separated by a distance $d(x,Y)$. The constants $v$ and $\kappa$ are independent of $n$ and depend only on $\|h(s)\|$, which is an $O(1)$ constant.
What we do is define $$k_{0}(s) = \mathcal{F}_s^{H_{\Lambda_0}(s)}(h')$$ and $$k_{\alpha}(s) = \mathcal{F}_s^{H_{\Lambda_\alpha}(s)}(h') -
\mathcal{F}_s^{H_{\Lambda_{\alpha-1}}(s)}(h'), \quad 0 < \alpha < m,$$ where we define $$\mathcal{F}_s^{H_{\Lambda_\alpha}(s)}(M) = \int_{-\infty}^{\infty}
\chi_{\gamma}(t) \left(\int_0^t \tau_u^{H_{\Lambda_\alpha}(s)}(M)
\,du \right)dt,$$ with $$H_{\Lambda_\alpha}(s) = \sum_{\mathbf{j}\in\Lambda_{\alpha}}
h_{\mathbf{j}}(s),$$ where $\Lambda_\alpha = \{\mathbf{j}\, | \, d(\mathbf{0},\mathbf{j})
\le \alpha \}$. Obviously $k_{\alpha}(s)$ has support ${\operatorname{supp}}(k_{\alpha}(s)) = \Lambda_\alpha+{\operatorname{supp}}(h')$. Also note that $k(s) = \sum_{\alpha=0}^{m-1} k_{\alpha}(s)$ (recall that $m$ is the diameter of the lattice).
We now show how the Lieb-Robinson bound provides an estimate on the decay of $\|k_\alpha(s)\|$. Firstly, we rewrite the Lieb-Robinson bound (\[eq:liebrobinson\]) so that it is more useful:
$$\label{eq:liebrobinson2}
\begin{split}
\|\tau_t^{H_{\Lambda_\alpha}}(A) -
\tau_t^{H_{\Lambda_{\alpha-1}}}(A)\| &= \left\|\int_{0}^t ds\,
\frac{d}{d{t'}} (\tau_{t'}^{H_{\Lambda_{\alpha-1}}}(\tau_{t-t'}^{H_{\Lambda_{\alpha}}}(A))) \right\| \\
&= \left\| \int_{0}^t d{t'}\,
\tau_{t'}^{H_{\Lambda_{\alpha-1}}}([H_{\Lambda_\alpha}-H_{\Lambda_{\alpha-1}}, \tau_{t-{t'}}^{H_{\Lambda_\alpha}}(A)]) \right\| \\
&\le \int_0^{|t|} d{t'}\,
\|[H_{\Lambda_\alpha}-H_{\Lambda_{\alpha-1}},
\tau_{{t'}}^{H_{\Lambda_\alpha}}(A)]\| \\
&\le 2\int_0^{|t|} d{t'}\,
\|[H_{\Lambda_\alpha}-H_{\Lambda_{\alpha-1}}]\| e^{-v
\alpha+\kappa|t'|} \\
&\lesssim \alpha e^{\kappa|t| - v\alpha},
\end{split}$$
where we used the fundamental theorem of calculus to get the first line, the triangle inequality and unitary invariance of the norm to get the third line, we substituted the Lieb-Robinson bound (\[eq:liebrobinson\]) in fourth line, and we integrated the bound to get the fourth line. The $\alpha$ term in the fourth line comes from the fact that the operator $H_{\Lambda_\alpha}-H_{\Lambda_{\alpha-1}}$ consists of $\alpha$ terms (the number of terms crossing the boundary). The Lieb-Robinson bound, in this form, says that the evolution of $A$ with respect to $H_{\Lambda_\alpha}$ is almost the same as that for $H_{\Lambda_{\alpha-1}}$, i.e., the boundary effects are unimportant for short times.
Now consider
$$\label{eq:kbound1}
\begin{split}
\|k_\alpha(s)\| &= \left\|\int_{-\infty}^{\infty} \chi_{\gamma}(t)
\left(\int_0^t \left(\tau_u^{H_{\Lambda_\alpha}(s)}(h') -
\tau_u^{H_{\Lambda_{\alpha-1}}(s)}(h') \right)du \right)dt\right\| \\
&\le 2\int_{0}^{\infty} |\chi_{\gamma}(t)| \left(\int_0^t
\left\|\tau_u^{H_{\Lambda_\alpha}(s)}(h') -
\tau_u^{H_{\Lambda_{\alpha-1}}(s)}(h') \right\|du\right) dt \\
&\le 2\int_{0}^{\infty} |\chi_{\gamma}(t)| \left(\int_0^t
\min\{2\|h'\|, c\alpha e^{\kappa |u| - v\alpha}\}du\right) dt \\
&\lesssim \alpha\int_{0}^{c\alpha}|\chi_\gamma(t)|e^{\kappa |t| -
v\alpha}dt + \int_{c\alpha}^\infty |\chi_\gamma(t)||t|dt \\
&\lesssim \alpha\int_{0}^{c\alpha} e^{\kappa t - v\alpha} dt +
\int_{c\alpha}^\infty \frac{1}{\gamma^l |t|^{l-1}}dt, \quad \forall l > 1 \\
&\lesssim \alpha e^{(\kappa c - v)\alpha} +
\frac{1}{c\gamma^l\alpha^{l-1}}, \quad \forall l > 1,
\end{split}$$
where to get the second line we applied the triangle inequality, to get the third line we applied the Lieb-Robinson bound in the form (\[eq:liebrobinson2\]) with $\alpha =
{\operatorname{diam}}(\Lambda_{\alpha})+\text{const.}$ (we’ve dropped the dependance of the interactions $H_{\Lambda_\alpha}(s)$ on the parameter $s$ because for these inequalities the evolution is independent of the parameter $s$), in the third line we’ve broken the integral into two pieces and applied the different regimes of the Lieb-Robinson bound separately with $c$ some constant [@endnote49] to be chosen later, and in the final line we applied the decay estimates on $\chi_\gamma(t)$ (see Appendix \[app:cutoff\] for a derivation of these estimates). Thus, by choosing $c < v/\kappa$ we see that $\|k_\alpha(s)\|$ is decaying faster than the inverse of any polynomial in $\alpha$ for $\alpha \gtrsim 1/\gamma$, i.e., for $\alpha
> c/\Delta$, where $c$ is some constant. In this way we see that exact adiabatic evolution can be thought of as unitary dynamics according to the paramater-dependent hamiltonian $K(s)$ which is approximately local with respect to the metric $d$ on the lattice. For an illustration of the interactions of $K(s)$ see Figure \[fig:hint\].
Efficient simulation of adiabatic evolution {#sec:effsim}
===========================================
In this section we apply a Lieb-Robinson bound to show that dynamics according to effectively local exact adiabatic evolution keep local operators approximately local and hence show that expectation values of local operators in adiabatically evolved ground states can be computed efficiently.
Recall that we can write the ground state $|\Omega(s)\rangle$ by integrating (\[eq:adqevol\]) as $$|\Omega(s)\rangle = \mathcal{U}(s;0)|\Omega(0)\rangle,$$ where $$\mathcal{U}(s;0) = \mathcal{T}e^{i\int^s_0 K(s') ds'},$$ and $\mathcal{T}$ denotes the time-ordering operation. Our objective is to uniformly approximate $$\omega_s(A) = \langle \Omega(0)|\mathcal{U}^\dag(s;0) A
\mathcal{U}(s;0)|\Omega(0)\rangle,$$ for all $s\in[0,1]$. The way we do this is to show that the operator $A(s) \equiv \mathcal{U}^\dag(s;0) A \mathcal{U}(s;0)$ remains approximately local for all $s\in[0,1]$ and use the assumed fact that $\omega_0(B)$ can be computed efficiently for all local operators $B$. For simplicity we assume that the operator $A$ is located at the origin and has support $|{\operatorname{supp}}(A)| = 1$. It is easy to extend the results of this section to apply to operators with bounded support on disconnected regions, such as correlators.
We now study the locality of $A(s)$. What we do is first show that $A(s)$ can be uniformly approximated in operator norm by the series of approximants $$A_{\alpha}(s) \equiv \mathcal{V}^\dag_{\Lambda_\alpha}(s;0) A
\mathcal{V}_{\Lambda_\alpha}(s;0),$$ where $\mathcal{V}_{\Lambda_\alpha}(s;0)$ satisfies the differential equation $$\frac{d}{ds}\mathcal{V}_{\Lambda_\alpha}(s;0) = i\sum_{\mathbf{j}\in
\Lambda_\alpha}\mathcal{F}_s(h_{\mathbf{j}}')\mathcal{V}_{\Lambda_\alpha}(s;0)
= iK_{\Lambda_\alpha}(s)\mathcal{V}_{\Lambda_\alpha}(s;0),$$ with $\mathcal{V}_{\Lambda_\alpha}(0;0) = \mathbb{I}$ and where $K_{\Lambda_\alpha}(s) = \sum_{\mathbf{j}\in
\Lambda_\alpha}\mathcal{F}_s(h_{\mathbf{j}}')$ and $\Lambda_\alpha =
\{\mathbf{j}\, | \, d(\mathbf{0},\mathbf{j}) \le \alpha \}$. In words: the approximation $A_{\alpha}(s)$ is that operator obtained by evolving $A$ with respect only to those interaction terms in $K(s)$ whose centres are within a distance $\alpha$ of $A$. Naturally this means that $A_{m-1}(s)=A(s)$. We use a Lieb-Robinson bound to show that $\|A(s)-A_{\alpha}(s)\|$ is rapidly decaying.
To show this we prove $\|\tau_{s;0}^{K(s)}(A) -
\tau_{s;0}^{K_{\Lambda_\alpha}(s)}(A)\|$ is small for $|s| \le 1$ and large constant $\alpha$ where $\tau_{s;s'}^{K(s)}(M) =
\mathcal{U}^\dag(s;s')M\mathcal{U}(s;s')$. To make this expression easier to deal with, and to more explicitly relate it to group-velocity bounds, we rewrite it:
$$\label{eq:hamapprox}
\begin{split}
\|\tau_{s;0}^{K(s)}(A) - \tau_{s;0}^{K_{\Lambda_\alpha}(s)}(A)\| &=
\left\|\int_{0}^s ds'\,
\frac{d}{ds'} (\tau_{s';0}^{K_{\Lambda_\alpha}(s')}(\tau_{s;s'}^{K(s)}(A))) \right\| \\
&= \left\| \int_{0}^s ds'\,
\tau_{s';0}^{K_{\Lambda_\alpha}(s')}([K_{\Lambda_\alpha^c}(s'), \tau_{s;s'}^{K(s)}(A)]) \right\| \\
&\le \int_0^{|s|} ds'\, \|[K_{{\Lambda_\alpha}^c}(s'),
\tau_{s;s'}^{K(s)}(A)]\|,
\end{split}$$
where $K_{{\Lambda_\alpha}^c}(s) = \sum_{\mathbf{j}\in L\setminus
{\Lambda_\alpha}} k_{\mathbf{j}}(s)$.
We now apply a general Lieb-Robinson bound recently proved in [@hastings:2005b]. In order to apply the Lieb-Robinson bound of [@hastings:2005b] we need to establish that our hamiltonian $K(s)$ satisfies the conditions of Assumption 2.2 of [@hastings:2005b], which in our case reads $$\sum_{\alpha=0}^{m-1} \|k_\alpha\|
(1+2\alpha(\alpha+1))^2(2+2\alpha)^\eta \le s_1,$$ where $\eta$ is a positive constant and $s_1$ is some constant. We need to ensure that the sum on the left evaluates to a constant instead of diverging. The only flexibility we have is to choose a decay estimate for $\|k_\alpha\|$ which is strong enough to overwhelm the polynomial in $\alpha$ it is multiplied by. The highest power of $\alpha$ appearing in this sum is $\alpha^{4+\eta}$. Thus we use the decay estimate (\[eq:kbound1\]) and choose $l\ge 7+\eta$, and so we find that this constant $s_1$ equates to $$s_1 = \sigma(l)/\gamma^{l},$$ where $$\sigma(l) = c_l\sum_{\alpha=1}^{m-1}
\frac{(1+2\alpha(\alpha+1))^2(2+2\alpha)^\eta}{\alpha^{l-1}},$$ and $c_l$ is the constant arising from the estimate (\[eq:kbound1\]), and $l$ is any chosen power $l\ge7+\eta$. (The proof of the general Lieb-Robinson bound described in [@hastings:2005b] is easily extended to cover parameter-dependent hamiltonians such as $K(s)$.) This reads $$\|[\tau_{s}^{K(s)}(A), B]\| \le
\frac{\rho(l)|Y|\left(e^{\frac{\sigma(l)}{\gamma^{l}}|s|}-1\right)}{(1+d(x,Y))^l},
\quad \forall l>7+\eta,$$ for any two norm-$1$ operators $A\in\mathcal{A}_x$ and $B\in
\mathcal{A}_{Y}$, with $\{x\}\cap Y = \emptyset$ which are initially separated by a distance $d(x,Y)$ and $\rho(l)$ is a constant which depends only on $l$. The constant $v$ is independent of $n$ and depends only on $\|h(s)\|$. We isolate the dependence of this bound on the minimum gap $\gamma$ by defining $$g(\gamma, l) =
\rho(l)\left(e^{\frac{\sigma(l)}{\gamma^{l}}|s|}-1\right).$$ Note that we are going to systematically redefine this function in our subsequent derivations to absorb extra constants and occurrences of $\gamma$. With this bound we first obtain an upper bound on $\|[\tau_{s,s'}^{K(s)}(A), k_{\mathbf{j},\alpha}(s')]\|$ (recall that the operators $k_{\mathbf{j},\alpha}(s)$ are defined via Eq. (\[eq:ksdef\])): $$\|[A(s), k_{\mathbf{j},\alpha}(s)]\| \le
\begin{cases} \frac{g(\gamma,l)(1+2\alpha(\alpha+1))\|k_{\alpha}(s)\|}{(1+\delta-\alpha)^l}, \quad \alpha < \delta \\ 2\|A\| \|k_{\alpha}(s)\|, \quad \alpha \ge \delta, \end{cases}$$ where $\delta = d(\mathbf{0}, \mathbf{j})$ and $(1+2\alpha(\alpha+1))= |\Lambda_\alpha(\mathbf{j})| = |\{
\mathbf{x}\,|\, d(\mathbf{j}, \mathbf{x}) \le \alpha\}|$. We use the estimate (\[eq:kbound1\]) and redefine $g(\gamma,l)$ to find the upper bound $$\|[A(s), k_{\mathbf{j},\alpha}(s)]\| \le
\begin{cases} \frac{g(\gamma,l)(1+2\alpha(\alpha+1))}{\alpha^{l+2}(1+\delta-\alpha)^l}, \quad \alpha < \delta \\ 2\|A\| \|k_{\alpha}(s)\|, \quad \alpha \ge \delta. \end{cases}$$ We next find the minimum of the denominator $\alpha^{l+2}(1+\delta-\alpha)^l$ on the interval $1\le\alpha\le
\delta$, which is $\delta^l$, and redefine $g(\gamma,l)$ to arrive at the final upper bound $$\label{eq:kjabound}
\|[A(s), k_{\mathbf{j},\alpha}(s)]\| \le
\begin{cases} \frac{g(\gamma,l)}{\delta^l}, \quad \alpha < \delta \\ \frac{2c_l\|A\|}{\gamma^{l+1}\alpha^l}, \quad \alpha \ge \delta. \end{cases}$$
Thus, by choosing the centre $\mathbf{j}$ far enough away from the centre $\mathbf{0}$ of $A(s)$ we find the behaviour $$\|[A(s), k_{\mathbf{j},\alpha}(s)]\| \lesssim \frac{g(\gamma,l)}{
d(\mathbf{0},\mathbf{j})^l}, \quad \forall l>1,$$ i.e., the quantity $\|[A(s), k_{\mathbf{j},\alpha}(s)]\|$ decays faster than any polynomial in $d(\mathbf{0},\mathbf{j})$.
We next use our upper bound (\[eq:kjabound\]) to obtain an upper bound on $\|[A(s), k_{\mathbf{j}}(s)]\|$: $$\label{eq:1stadecay}
\begin{split}
\|[A(s), k_{\mathbf{j}}(s)]\| &\le \sum_{\alpha=0}^{m-1} \|[A(s),
k_{\mathbf{j},\alpha}(s)]\| \\
&\le \frac{g(\gamma,l)}{\delta^{l-1}} + \sum_{\alpha=\delta}^{m-1}
\frac{2c_{l+1}\|A\|}{\gamma^{l+1}\alpha^l} \\
&\le \frac{g(\gamma,l)}{\delta^{l-1}},
\end{split}$$ where we’ve redefined $g(\gamma,l)$ in the last line.
Now we use the decay estimate (\[eq:1stadecay\]) in (\[eq:hamapprox\]) to provide an upper bound for $\|A(s)-A_{\alpha}(s)\|$: $$\label{eq:adecay1}
\begin{split}
\|A(s)-A_{\alpha}(s)\| &\le \sum_{\mathbf{j} \in L\setminus
\Lambda_\alpha}\int_{0}^{1} ds \|[A(s), k_{\mathbf{j}}(s)]\| \\
&\le \sum_{\delta = \alpha}^{m-1} \frac{(1+2\delta(\delta+1))
g(\gamma,l)}{\delta^{l-1}} \\
&\le \frac{g(\gamma,l)}{\alpha^{l-4}},
\end{split}$$ where we’ve redefined $g(\gamma,l)$.
So, as long as $\alpha$ is chosen to be so large that it overwhelms the $O(1)$ constant $g(\gamma,l)$ we find that $\|A(s)-A_{\alpha}(s)\|$ can be made to decay faster than any polynomial in $\alpha$, and hence, can be made as small as desired. Thus there exists some constant $\alpha$ such that $\|A(s)-A_{\alpha}(s)\| < \epsilon$. Note that, because $k(s)$ has support throughout $L$, $A_\alpha(s)$ has support throughout $L$.
In order to provide a simulation method to compute approximations to ground-state expectation values $\omega_s(A)$ we need to show that $A_\alpha(s)$ can be approximated by an operator with support only on a constant number of sites around ${\operatorname{supp}}(A)=\mathbf{0}$. The way we do this is to show that $A_\alpha(s)$ is operator-norm close to $$\widetilde{A}_{\alpha,\beta}(s) =
\widetilde{\mathcal{V}}^\dag_{\Lambda_{\alpha,\beta}}(s) A
\widetilde{\mathcal{V}}_{\Lambda_{\alpha,\beta}}(s),$$ where $\widetilde{\mathcal{V}}_{\Lambda_{\alpha,\beta}}$ satisfies the differential equation $$\begin{gathered}
\label{eq:alphabetav}
\frac{d}{ds}\widetilde{\mathcal{V}}_{\Lambda_{\alpha,\beta}}(s) = \\
i\sum_{\mathbf{j}\in
\Lambda_{\alpha}}\widetilde{\mathcal{F}}_{\mathbf{j},s}^{H_{\Lambda_\beta(\mathbf{j})}}(h_{\mathbf{j}}')\widetilde{\mathcal{V}}_{\Lambda_{\alpha,\beta}}(s)
=
i\widetilde{K}_{\Lambda_{\alpha,\beta}}(s)\widetilde{\mathcal{V}}_{\Lambda_{\alpha,\beta}}(s),\end{gathered}$$ and $$\widetilde{\mathcal{F}}_{\mathbf{j},s}^{H_{\Lambda_\beta(\mathbf{j})}}(h_{\mathbf{j}}')
= \int_{-\infty}^{\infty} \chi_{\gamma}(t) \left(\int_0^t
\tau_u^{H_{\Lambda_\beta(\mathbf{j})}(s)}(h_{\mathbf{j}}') du
\right)dt,$$ with $\Lambda_\beta(\mathbf{j}) = \{ \mathbf{x}\,|\, d(\mathbf{j},
\mathbf{x}) \le \beta\}$.
To show that $\widetilde{A}_{\alpha,\beta}(s)$ is close to $A_{\alpha}(s)$ we first exploit the general inequality $$\|\mathcal{V}_{\Lambda_\alpha}(s)-\widetilde{\mathcal{V}}_{\Lambda_{\alpha,\beta}}(s)\|
\le \int_{0}^{|s|}
\|K_{{\Lambda_\alpha}}(s')-\widetilde{K}_{\Lambda_{\alpha,\beta}}(s')\|
ds'$$ which is proved, for example, by exploiting the Lie-Trotter expansion, and then upper-bound the right-hand side using the triangle inequality by $$\begin{gathered}
\int_{0}^{|s|}
\|K_{{\Lambda_\alpha}}(s')-\widetilde{K}_{\Lambda_{\alpha,\beta}}(s')\|
ds' \le \\ \sum_{\mathbf{j}\in \Lambda_\alpha} \int _{0}^{|s|}
\|k_{\mathbf{j}}(s')-\widetilde{k}_{\mathbf{j},\beta}(s')\| ds',\end{gathered}$$ where $\widetilde{k}_{\mathbf{j},\beta}(s) =
\widetilde{\mathcal{F}}_{\mathbf{j},s}^{H_{\Lambda_\beta(\mathbf{j})}}(h_{\mathbf{j}}')$. We can upper-bound the integral on the right-hand side by using an argument identical to the one used to show (\[eq:kbound1\]). We thus obtain $$\label{eq:aalphadecay}
\begin{split}
\sum_{\mathbf{j}\in \Lambda_\alpha} \int _{0}^{|s|}
\|k_{\mathbf{j}}(s')-\widetilde{k}_{\mathbf{j}, \beta}(s')\| ds'
&\lesssim \sum_{\mathbf{j}\in \Lambda_\alpha}
\frac{1}{\gamma^l\beta^{l-1}} \\
&\lesssim \frac{\alpha^2}{\gamma^l\beta^{l-1}},
\end{split}$$ where $l$ is any power, and we’ve used the fact that the number of sites in $\Lambda_\alpha$ is given by $1+2\alpha(\alpha+1)$. By choosing $\beta \gtrsim \alpha$ we find that $\mathcal{V}_{\Lambda_\alpha}(s)$ can be made as close as desired to $\widetilde{\mathcal{V}}_{\Lambda_{\alpha, \beta}}(s)$.
To obtain closeness of our final approximation $\widetilde{A}_{\alpha,\beta}(s)$ to $A(s)$ we use the triangle inequality $$\begin{split}
\|A(s)-\widetilde{A}_{\alpha,\beta}(s)\| &\le \|A(s)-A_\alpha(s)\| +
\|A_\alpha(s)-\widetilde{A}_{\alpha,\beta}(s)\| \\
&\le \frac{g(\gamma, l)}{\alpha^{l}} +
\frac{\alpha^2}{\gamma^{l'}\beta^{l'-1}},
\end{split}$$ where we’ve used the upper bound (\[eq:adecay1\]) with an adjusted value of $l$ and we’ve also used (\[eq:aalphadecay\]) with an appropriate choice of power $l'$. We therefore find that it is sufficient, for a given constant $\epsilon$ to choose large (but $O(1)$) $\alpha$ and $\beta$ so that $$\label{eq:finalaestimate}
\|A(s)-\widetilde{A}_\alpha(s)\| \le \epsilon.$$
The actual values of $\alpha$ and $\beta$ required to reduce the error (\[eq:finalaestimate\]) to below $\epsilon$ scales better than linearly with $w=\max(g(\gamma,l), 1/\gamma)$, where $\gamma$ is a constant multiplied by the minimum energy $\Delta E$ encountered along the adiabatic path. Thus the support of the final approximation $\widetilde{A}_{\alpha,\beta}(s)$ is given, in the worst case, by ${\operatorname{supp}}(\widetilde{A}_{\alpha,\beta}(s)) \lesssim w$. Note that $w$ depends, via $g(\gamma,l)$, exponentially on $1/\gamma$, i.e., the inverse energy gap.
Because the final approximation $\widetilde{A}_{\alpha,\beta}(s)$ can be computed via integrating (\[eq:alphabetav\]), and by noticing that this integration can be performed by restricting our attention to the finite-dimensional subalgebra $\mathcal{A}_{W}$, where $W={\operatorname{supp}}(\widetilde{A}_{\alpha,\beta}(s))$, we see that $\widetilde{A}_{\alpha,\beta}(s)$ can be computed using resources which scale as $2^{cw}$, with $c$ some constant.
Discussion {#sec:disc}
==========
In this paper we have shown how to efficiently calculate the ground-state expectation values of local operators with constant support for gapped adiabatically evolving spin systems. In order to provide our simulation method we reduced the problem to showing that under exact adiabatic evolution the expectation value of a local operator can be computed from the expectation value of an approximately local operator in the unevolved ground state. Given this observation we then argued that if it is easy to compute expectation values of local operators in the original ground state then one could approximate the desired expectation values arbitrarily well by using time and space resources that scale with the inverse gap.
Our approach has several shortcomings. The first is that the scaling of the simulation resources with the error $\epsilon$ scales faster than $2^{1/\epsilon}$. This means that if the expectation value of an operator which is a sum of many local operators is desired then our simulation method may require superpolynomial resources. For example, if the expectation value of the *total* magnetisation $M = \sum_{\mathbf{j}\in L} \sigma_j^z$ (as opposed to the more traditional *average* magnetisation $m = M/n$) is required to some accuracy $\epsilon$ then our simulation method will require superpolynomial resources. This is not entirely unexpected, after all, in the thermodynamic limit such operators are unbounded and cannot be approximated at all. Another manifestation of this shortcoming is that if the expectation values of the local operators are required to an accuracy which scales as $\epsilon < 1/n$ then our method may require superpolynomial resources. These problems do not manifest themselves for the applications we have in mind. Namely, when applied to the calculation of average properties of two states in the same quantum phase we only require accuracy to some small constant $\epsilon$ which doesn’t scale with the system size, and when applied to simulating adiabatic quantum algorithms we only need $\epsilon$ to scale as a constant in order to read out the answer of the algorithm.
The second shortcoming of our method is that, by the current method, we are unable to directly approximate the scaling of the geometric entropy [@endnote50] $S_\Lambda$ with $\Lambda$. The reason for this is that our current method approximates $\rho_\Lambda(s)$ by calculating approximations to all the expectation values of a basis of operators for $\mathcal{A}_\Lambda$. Because we are computing *approximations* to expectation values we end up computing only an *approximation* $\widetilde{\rho}_{\Lambda}(s)$ to $\rho_\Lambda(s)$. The best continuity result available for the von Neumann entropy is Fannes inequality (see, for example, [@nielsen:2000a]) for a derivation) which implies that the error in the approximation $\widetilde{S}_\Lambda$ calculated from $\widetilde{\rho}_{\Lambda}(s)$ grows larger as $\Lambda$ increases. We’ll describe an approach to this problem using exact adiabatic evolution in a future paper.
The principle characteristic of our approach is that approximations are made in the *Heisenberg picture*. What we mean here is that instead of approximating the evolved quantum state of the spin system in operator norm we instead compute approximations to the evolved local operators. We should expect this strategy to be successful because the locality of the interactions in the hamiltonian doesn’t manifest itself in the Schr[ö]{}dinger picture but, thanks to the Lieb-Robinson bound, it is precisely clear what locality implies for local operators in the Heisenberg picture. Because in the thermodynamic limit we are only able to physically access local operators (such as average magnetisation and correlators) this approach doesn’t lead to any loss of generality over computations carried out in the Schr[ö]{}dinger picture.
It is possible that our analysis acutally applies to all gapped spin models. This is because it is possible that any gapped spin model is adiabatically connected [@endnote51] to a classical spin model with trivial ground state. Classical renormalisation-group style argumentation certainly seems to back this statement up: after all, we know that the RG fixed points are either trivial (classical) or quantum critical points. However, there is as yet no rigourous general proof of this statement for quantum spin systems.
We would like to suggest that the following description of the space of local (translation-invariant) spin models is correct. Firstly, in this space there are many distinguished points, classical spin systems, where the ground state can be calculated trivially. Around each of these points is a small region in hamiltonian space of hamiltonians which are provably adiabatically connected to the classical spin model points [@yarotsky:2004a; @yarotsky:2005a; @yarotsky:2005b]. In these regions we have shown that the local ground-state properties can be determined efficiently. Outside these small regions there are other regions which may or may not be adiabatically connected to the classical spin model points where the hamiltonians are gapped. In these regions it is known that the local ground-state properties can be calculated using subexponential resources [@hastings:2005b]. On the boundaries between the quantum phases there are quantum critical walls. For these points, in $1$D, it is known that an approximation to the ground state as a finitely correlated state can be stored using polynomial space [@verstraete:2005a]. It is not known if these approximations can be obtained efficiently. This picture is summarised in Figure \[fig:phases\].
Acknowledgments {#acknowledgments .unnumbered}
---------------
I would like to thank Jens Eisert, Matthew Hastings, Jiannis Pachos, Tony Short, Barbara Terhal, David DiVincenzo, and Andreas Winter for helpful correspondence, comments, and discussions.
Properties of smooth cutoff functions {#app:cutoff}
=====================================
In this Appendix we briefly review the properties of compactly supported $C^\infty$ cutoff functions.
Of fundamental utility in our derivations is a class of functions known as *compactly supported $C^\infty$ bump functions*. These functions are defined so that their fourier transform $\widehat{\chi}_\gamma(\omega)$ is compactly supported on the interval $[-\gamma, \gamma]$, and equal to $1$ on the middle third of the interval. Such functions satisfy the following derivative bounds $$\label{eq:chiderbound}
\frac{d^j\widehat{\chi}_\gamma(\omega)}{d\omega^j} \lesssim
\gamma^{-j},$$ for all $j$ with the implicit constant depending on $j$. This is just about the best estimate possible given Taylor’s theorem with remainder and the constraints that $\widehat{\chi}_\gamma(\omega)$ is equal to $1$ at $\omega = 0$ and $\widehat{\chi}_\gamma(\omega)$ is compactly supported.
The function $\chi_\gamma(t)$ has support throughout $\mathbb{R}$ but it is decaying rapidly. To see this consider $$\chi_\gamma(t) = -\frac{1}{2\pi}\int_{-\infty}^{\infty} \frac{1}{it}
e^{-it\omega}\frac{d}{d\omega}\widehat{\chi}_\gamma(\omega) d\omega$$ which comes from integrating by parts. Continuing is this fashion allows us to arrive at $$\chi_\gamma(t) = \frac{1}{2\pi}\int_{-\infty}^{\infty}
\left(-\frac{1}{it}\right)^j
e^{-it\omega}\frac{d^j}{d\omega^j}\widehat{\chi}_\gamma(\omega)
d\omega$$ Since $\widehat{\chi}_\gamma(\omega)$ has all its derivatives bounded, according to (\[eq:chiderbound\]), and using the compact support of $\widehat{\chi}_\gamma(\omega)$ we find $$\begin{split}
|\chi_\gamma(t)| &\lesssim \left|\int_{-\gamma}^{\gamma}
\left(\frac{1}{it}\right)^j e^{-it\omega} \gamma^{-j} d\omega\right|
\\
&\lesssim \int_{0}^{\gamma} \frac{1}{|\gamma t|^j} d\omega \\
&\lesssim \frac{1}{\gamma^{j-1}|t|^j},
\end{split}$$ for all $j\in \mathbb{N}$. Thus we find that $\chi_{\gamma}(t)$ decays to $0$ faster than the inverse of any polynomial in $t$ with characteristic “width” $1/\gamma$. The existence and construction of such functions is discussed, for example, in [@vaaler:1981a; @vaaler:1985a].
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---
abstract: '\[introduction\] In this paper, we propose a novel approach of word-level Indic script identification using only character-level data in training stage. Our method uses a multi-modal deep network which takes both offline and online modality of the data as input in order to explore the information from both the modalities jointly for script identification task. We take handwritten data in either modality as input and the opposite modality is generated through intermodality conversion. Thereafter, we feed this offline-online modality pair to our network. Hence, along with the advantage of utilizing information from both the modalities, the proposed framework can work for both offline and online script identification which alleviates the need for designing two separate script identification modules for individual modality. We also propose a novel conditional multi-modal fusion scheme to combine the information from offline and online modality which takes into account the original modality of the data being fed to our network and thus it combines adaptively. An exhaustive experimental study has been done on a data set including English(Roman) and 6 other official Indic scripts. Our proposed scheme outperforms traditional classifiers along with handcrafted features and deep learning based methods. Experiment results show that using only character level training data can achieve competitive performance against traditional training using word level data.'
address:
- 'Institute for Media Innovation, Nanyang Technological University, Singapore.'
- 'Centre for Vision, Speech and Signal Processing, University of Surrey, England, United Kingdom.'
- 'Department of ECE, Institute of Engineering & Management, Kolkata, India.'
- 'Department of EE, Institute of Engineering & Management, Kolkata, India.'
- 'Department of Electrical Engineering, Jadavpur University.'
- 'Department of CSE, Indian Institute of Technology, Roorkee, India.'
- 'Computer Vision and Pattern Recognition Unit, Indian Statistical Institute, Kolkata, India.'
author:
- Ayan Kumar Bhunia
- Subham Mukherjee
- Aneeshan Sain
- Ankan Kumar Bhunia
- Partha Pratim Roy
- Umapada Pal
bibliography:
- 'mybibfile.bib'
title: 'Indic Handwritten Script Identification using Offline-Online multi-modal Deep Network'
---
`Handwritten Script Identification, Deep Neural Network, multi-modal Learning, Offline and Online Handwriting, Character Level Training.`
Introduction
============
Script identification [@Sahare2017; @singh2015offline] is a key step of Optical Character Recognition (OCR) in multi-script documents. Script can be defined as a writing system consisting of a set of specific symbols and graphical shapes. Each script features specific attributes that distinguish it from other ones. Script identification is of utmost importance to understand handwritten documents automatically. Identification of handwritten script has gained prime importance in document image processing community; one of the reasons being global digitization of several handwritten scriptures and books. The basis of such script identification is the unique spatial relation among the strokes of a particular script, which helps in distinguishing the scripts from one another.
In multilingual [@singh2017handwritten] environment, script recognition gains significant importance, since every handwritten text recognition system is language specific. In a country like India, where there are 12 official scripts[^1], handwriting recognition becomes complicated. Hence, a robust script identification system is necessary to automate the process of text recognition [@ghosh2010script; @ubul2017script]. The critical challenge encountered in such a task is that the handwritten text suffers from inherent challenges due to free flow nature of handwriting, unlike machine generated text which have fairly uniform structure. The variation in writing style among the individuals and complex shapes of characters are some of the major hindrances for handwritten script identification. Figure \[fig:fig1\] shows some handwritten text documents written in different scripts.
Most of the existing literature for script identification/ recognition focuses on extracting linguistic and statistical features at the word or the line level [@ghosh2010script; @ubul2017script; @chanda2009word; @rajput2010handwritten]. On the contrary, the proposed framework is based on the hypothesis that the character set of a particular script alone contains distinctive features for the purpose of script recognition at word level. To extract the feature we employ both offline (image representation) as well as online (stroke order representation) modalities of handwritten text simultaneously, to utilize their combined potential for script identification task. To ensure the same, we have designed a deep neural architecture which can be trained in an end to end manner. To our knowledge, the proposed method is the first work which introduces the idea of combining offline and online modality in a single deep network to maximize the script identification task. Along with this multi-modal deep network, we also stress on the models trained using character-level data only and test the script identification task for both character and word level data.
Only character level data in training stage provides us several advantages. Firstly, it is easier to collect large amount of character level data in lesser time, which saves both time and effort while preparing the dataset. Secondly, there are lesser number of unique characters compared to word level data, since several combinations of character sequence are possible for word level data. Thirdly, character level training is much lighter than that of word level data, thus making it ideal for employing in hand-held computing devices, e.g., mobile devices. It can be noted that character level data possesses a significant amount of information about the script which can be utilized to achieve word level script-identification performance.
The proposed deep network utilizes the information from both offline and online modalities jointly to leverage the respective advantages of both representations simultaneously. Online handwritten data [@namboodiri2004online] comprises of a 1-D signal in the form of sequence of points, whereas offline data comprises of images. The reason behind combining online modality is that it holds both spatial and temporal information of the character. On the contrary, the offline word images do not contain the temporal information. From the literature, it is noted that due to availability of dynamic information such as temporal order of the points, the performance of online system is comparatively better than the offline one [@hamdani2009combining]. However, offline data contains pixel level spatial information which partly complements the temporal and spatial information of online data.
Recently, the advent of convolutional deep architectures[@gomez2017improving] has made it possible to improve the performance of traditional feature-based approaches significantly. We draw our inspiration of the multi-modal framework from popular deep learning frameworks in RGB-D data [@xu2017multi; @han2017cnns; @wang2015mmss; @asif2017multi] where it aims to utilize both RGB image and depth modality in a single deep network to get the advantage from both modalities simultaneously. Our proposed method is designed to handle both online and offline data in a single network rather than requiring different models. It eliminates the necessity of two different models for offline and online handwritten data individually. Our proposed framework first converts the online data into its offline form (and vice versa), thereafter feeding the data represented in two different modalities to the network, for learning the combined vector space for script identification.
In addition to this, we have proposed a novel conditional fusion scheme for script identification. The primary motivation is to include complementary information of offline and online modalities. For instance, if the original data is from online (offline) modality we convert it to its offline (online) equivalent, thereafter feeding both them into our deep network. During the multi-modal fusion, we combine features from both the modalities adaptively, by imposing the condition that the original data is from online modality. Note that, the data comes in single modality, i.e. both modalities of that particular data are not present. Thus, we perform inter modality conversion to generate the other one. Use of both modalities simultaneously provides us with two major advantages. Firstly, it provides better script identification performance; and secondly, this makes it possible to design a single system which can work for both offline as well as online handwritten data. The underlying idea here is that our proposed conditional multi-modal fusion mechanism would encourage the deep network to combine both the modalities considering their individual contributions.
The main contributions of the paper are as follows. Firstly, we propose a deep multi-modal framework for script identification which uses online and offline modalities of character level data to exploit their shared information. We thus have a single framework which can handle both modalities by converting the data from the present modality to the other modality and learns a joint embedding to utilize the information from both the modalities for better performance. Here, we also develop a novel conditional multi-modal fusion scheme for effective combination of both the modalities. In addition to that, it avoids the need of requiring two different models to handle data in two different modalities. The trained model is used to identify the script for both online and offline character as well as word level data. To the best of our knowledge, this is the first attempt to develop a single framework which works in parallel for both online and offline data for handwritten script identification. Secondly, our proposed method enjoys the advantage of using light weight training model for script identification since we are using character level data for training which has fewer combinations compared to word level data. Thirdly, we have done an exhaustive experiments using a number of different scripts from different modalities, online and offline, to justify the feasibility and competitive performance with other existing baseline methods.
The rest of the paper is organized in the following manner. In Section 2 we discuss related work developed for script identification, stroke recovery task and popular deep multi-modal framework for different computer vision problems. In Section 3, the proposed framework for script identification has been described. In Section 4, we elaborate the experimental setup and discuss about the results of the various experiments conducted to justify the significance and efficiency of our proposed method. Finally, conclusions and future directions are given in Section 5.
Related Work
============
**Handwritten script identification:** Handwritten script identification is an important task in developing a multilingual handwriting recognition system where more than one script might be present. A comprehensive survey on script identification has been presented in [[@ghosh2010script; @ubul2017script]]{}. Various works have been reported on printed document script identification in [@chanda2009word]. However, script identification in handwritten documents is much more difficult compared to that in printed documents, due to varying handwriting styles of different individuals. A popularly explored approach in handwritten script identification is to extract linguistic and statistical feature followed by Support Vector Machine (SVM) as the classifier. In [@hiremath2010script], such solution for word-level offline classification is proposed for Thai-Roman script classification. A technique for script identification in torn documents is proposed by Chanda et al. [@chanda2009word], in which Roman and Indic scripts are considered to evaluate the performance. They worked with rotation invariant Zernike features and the rotation dependent gradient feature, using PCA-based methods to predict orientation and then apply an SVM classifier at the character level. The results are calculated for the word level using majority voting at the character level, followed by prediction at the document level in a similar fashion. Pal et al. [@pal2007handwritten] proposed a modified quadratic classifier using directional feature for recognition of off-line handwritten numerals of six popular Indic scripts. The bounding box of the numeric characters are divided into smaller blocks to capture the local information and directional feature is extracted at two levels, one from the original images and other one from the down sampled version of it through Gaussian Filtering. Moalla et al. proposed methods [@moalla2002extraction; @moalla2004extraction] to separate out the Arabic text from the documents containing both Arabic and English words. In [@ferrer2014multiple], Ferrer et al. has proposed a method for script identification in offline word images using word information index which estimates the amount of information included in a word. Different classifiers are trained using words with similar amount of information. During testing, the appropriate classifier is chosen based on the word information index of the query keyword. Regional local feature is studied in the work [@dhandra2007morphological]. In [@rajput2010handwritten], Rajput et al. proposed a method based upon the features extracted using Discrete Cosine Transform(DCT) and Wavelets along with KNN classifier to identify seven major scripts, namely, Devanagari, Gujarati, Gurumukhi, Kannada, Malayalam, Tamil, and Telugu at block level. Also, different textures feature have been explored for script identification task in various works [@hiremath2010script; @hangarge2010offline; @pal2012handwriting].
Neural network based solutions are also popular for script identification as discussed in [@ghosh2010script]. In one of the earliest works, neural nets were employed for script identification in postal automation systems[@roy2005system; @roy2005neural]. In [@sankaran2012recognition], BLSTM is used for printed Devanagari Script recognition which uses five different features, namely, (a) the lower profile, (b) the upper profile, (c) the ink-background transitions, (d) the number of black pixels, and (e) the span of the foreground pixels. These features are fed to a Bi-RNN architecture using Connectionist Temporal Classification objective function which provides an improvement of more than 20% Word Error Rate (WER) compared to the best available OCR system during its publication year. In [@ul2015sequence], a 1D-LSTM architecture, with one hidden layer is used for script identification at the text-line level to learn binary script models, and the reported identification accuracy for English-Greek scripts is 98.19%.
Recently, Singh et al. [@singh2015word] proposed a word level script identification approach for handwritten images where they designed a set of 82 features using a combination of elliptical and polygonal approximation techniques. Authors considered a total of 7000 handwritten text words from six different Indic scripts - Bangla, Devanagarai, Gurumukhi, Malayalam, Oriya, Telugu and Roman script. They reported a maximum accuracy of 95.35% using Multi-Layer Perceptron(MLP) classifier. Handwritten numeral script identification has been proposed by Obaidullah et al. [@obaidullah2015numeral]. Obaidullah et al. [@obaidullah2018handwritten] have done a comprehensive survey on handwritten indic script identification. Extreme Learning Machine(ELM) has been used for handwritten script identification in [@obaidullah2018extreme]. The work in [@obaidullah2019automatic] analyzed a particular question, which level of document (word, line, page) provides optimum performance for handwritten script identification. Authors concluded that line-level data provides most consistent performance among al, followed by page, block and word-level. Combination of different handcrafted feature extractors [@obaidullah2018automatic], like directional stroke image component fractal dimension, structural and visual appearance, interpolation and Gabor energy based texture features, are used for line level indic-handwritten script identification.
Along with these, a few deep learning based approaches [@shi2016script; @gomez2017improving; @mei2016scene] for script identification in scene images have appeared in the literature recently. However, the potential of deep neural network for handwritten script identification has not been explored completely. In [@shi2016script], local deep features are extracted using a pretrained CNN model and discriminative clustering is carried out to obtain the mid level representation by learning a set of discriminative patterns from extracted local features. Following this, the deep features and the mid-level representations are jointly optimized in a deep network and their proposed model is termed as Discriminative Convolutional Neural Network(DisCNN). Gomez et al.[@gomez2017improving] used the ensembles conjoined networks in order to learn from the stroke patches along with their relative importance.
**Online Trajectory Retrieval:** Restoring of temporal order from offline handwriting has been worked since long [@doermann1995recovery; @boccignone1993recovering]. Stroke and sub-stroke properties were utilized and authors provided a a taxonomy of local, regional and global temporal clues which were found to be beneficial for stroke recovery problem. In [@elbaati2009temporal], Elbaati et al. proposed an approach to recover the stroke by segmenting the offline word image into strokes and labeling all the edges as successive parts of the strokes. Then, a Genetic Algorithm is applied to optimize these strokes and produce the best possible stroke order. An application of the above method is used in [@hamdani2009combining] which combines offline and online data for Hidden Markov Model(HMM) based Arabic handwriting recognition. The offline features and temporal stroke order from online data are complementary in nature, which in combination improve the recognition accuracy of the framework. In [@kato2000recovery], Kato et al. proposed a stroke recovery technique which works for single stroke characters. This system labels each edge of the word image and connects them based on a predefined algorithm without any learning method. Very recently, Bhunia et al. [@kumarbhunia2018handwriting] proposed a deep-learning based solution using traditional sequence to sequence learning architecture for handwriting trajectory recovery.
**Deep multi-modal learning:** multi-modal learning [@ngiam2011multi-modal; @srivastava2012multi-modal] is a very popular concept in computer vision community in order to combine information from more than one sources. However, in document image analysis, more specifically for handwriting recognition task, there are hardly any application of multi-modal framework. In recent time, due to advancement of deep learning technology different modalities are combined for better accuracy in different problems like scene understanding, RGB-D object detection etc. In the task of Image Captioning [@karpathy2015deep] and Visual Question Answering [@ilievski2017multi-modal], the language feature and image feature are combined using deep neural network architecture. In RGB-D data, both image and depth modalities are explored for various tasks like Object Recognition[@wang2015mmss], Scene classification[@zhu2016discriminative], Object Detection [@xu2017multi]. Given a vast literature for multi-modal learning, we attempt to use a deep multi-modal framework to explore the joint information from both offline and online handwritten data for script identification task.
Proposed Framework
==================
The proposed approach can be divided into two steps. In the first step, we extract the data from original modality to its equivalent opposite modality. In the next step, the data from both the modalities are considered simultaneously as input to deep neural network to combine their information. Note that only character level data is used to train the network. During testing, it can identify the script for both character and word level data from both modalities. In our work, we design Convolutional-LSTM architecture where Convolutional Network intends to extract more robust sequential feature from the data and LSTM module captures the contextual information of the sequence for better performance. An overview of our proposed multi-modal framework is given in Figure \[fig:fig3\]. In this section, we describe our different modules of our framework serially.
Inter-Modality Conversion
-------------------------
Inter modality conversion is a key step in our framework. Since our proposed deep network takes the offline-online pair as the input, a modality-conversion is required in order to get the opposite modality from the input data.
### Offline to Online Conversion:
Offline to online conversion of handwritten data has been performed using [@nel2005estimating]. First, the skeleton image of the handwritten text is extracted through a thinning process to extract a parametric curve. In order to perform a comparison between a static image skeleton and a dynamic exemplar, translation is performed until the centroids of the two become aligned and following this, it is converted into a static image. The authors have thickened the static image skeleton and also the image obtained from the dynamic exemplar to a line width of approximately five pixels. After this, matching between the two images is carried out followed by trajectory extraction. The method that has been used for determining the pen trajectory from a static and a normalized image by making use of a Hidden Markov Model. For the problem of stroke order recovery, the sequence of states of the HMM are used to describe the sequence of pen positions as the image is produced. The HMM model is built from the skeleton of the static image. The dynamic exemplar is matched to the static image by making use of the Hidden Markov Model. An example of offline to online conversion is shown in Figure \[fig:fig9\].
### Online to Offline Conversion:
Conversion of online data to offline is a trivial one. Online data consists of consecutive co-ordinate points representing the flow of writing. To convert the online data to its offline equivalent, we first define an empty image matrix based on the difference of maximum and minimum $(x,y)$ coordinate values. Thereafter, we mark those pixel positions of the empty image matrix based on the online coordinate points and join consecutive points serially. This process generates the skeleton image of the handwritten data. Following this, a morphological thickening operation is performed in order to make the equivalent offline word image similar to the real offline word image data. Online to offline conversion is shown in Figure \[fig:fig8\]
Network Architecture for Online modality
----------------------------------------
Online modality of handwritten data consists of sequential $(x,y)$ co-ordinate points representing the flow of writing. One of the naive ways could be to feed these sequential $(x,y)$ co-ordinate points to a LSTM module and get the state of the last time step as the final feature representation of the online data. However, in this proposed approach, we have designed a Convolutional LSTM architecture for the online stream of our network. The key idea of using CNN is to achieve a certain extent of shift, scale and distortion in-variance. For a 2D image, the local connectivity of CNN learns the correlation among neighbouring pixels. The objective of using CNN on the sequential co-ordinate points before feeding it to the LSTM module is to learn the correlation among the neighbouring co-ordinate points. In addition to that, it intends to achieve a certain degree of distortion or shift invariance which may arise due to free flow of writing of different individuals, and during capturing of co-ordinate points by sensors. Hence, convolutional network will convert the $(x,y)$ co-ordinate points into a high dimensional space by incorporating the spatial correlation among nearby sequential points in order to make it less variant to different individuals’ handwriting and distortions.
For every online data sample, there are N sequential points represented by two values, $ X $ co-ordinate value and $ Y $- co-ordinate value for each point. However, in order to formulate the 1-D convolution over these points, we consider these points as 1-dimensional signal with 2 channels, i.e. $D \in \mathbb{R}^{N\times 1 \times 2}$. We employ a 1-D convolution with filter $W \in \mathbb{R}^{M\times Q \times 1\times P}$ ($M$ filters with dimension of $ {Q\times 1\times P}$). We restrict the length of the filter $Q$ to 5. $P$ is the number of channels or feature maps in the input. For example, $P = 2$ for the first convolution layer since the input is 1-dimensional 2 channel having a length N. The output $G \in \mathbb{R}^{N\times 1 \times M}$ of 1D-convolution is given by
$${G = Conv_{1D}(D;W)}
\centering$$
1-D convolution operation follows the same rule as that of 2D convolution with a small difference, which is that the filter used here is of size $L \times 1$ and it strides over only one direction(here time direction). A graphical illustration of 1D-convolution over the online coordinate points is shown in Figure \[fig:fig7\]. In order to introduce the invariance against free flow writing of different individuals and noisy acquiring of data from sensors, we have added one maxpooling operation along the time direction after the second convolution layers. The window size of maxpooling operation is $2\times 1$, hence it reduces the number of data points to half.
However, note that, online data represents the flow of writing with time. Hence, it is observed that more than one maxpooling operation reduces the performance. Inspired from the network of [@engelmann2017exploring], we have used global maxpooling [@lin2013network] operation to obtain the global feature vector which contains the holistic information of all the co-ordinate points. This global feature vector is concatenated with the feature map of every data point and passes through to last convolution layer. This convolution layer intends to combine each point wise feature with a global feature adaptively. In our online stream, we have used six convolution layers with the number of filters being 32, 64, 128, 256, 256, and 512 respectively. Hence, we obtain an output tensor of size $\mathbb{R}^{(N/2) \times 1 \times 512}$, which thereafter is to be fed to the LSTM module. Following this, we create a custom ‘Map-to-Sequence’ layer as the bridge between convolutional layers and recurrent layers as mentioned in [@shi2016end]. This ‘Map-to-Sequence’ layer converts the 3-dimensional tensor to a 2-dimensional time distributed feature representation of size $\mathbb{R}^{(N/2) \times 512}$ for the online data points.
Network Architecture for offline modality
-----------------------------------------
The online data was already time distributed, containing successive co-ordinate points which represent the flow of writing. In contrast to this, the offline data does not have any time information. Hence, in order to feed this to a LSTM module, we need to convert the offline word image into sequential feature representation. In handwriting recognition, one of the popular approaches is to use sliding based [@roy2016hmm; @BHUNIA201812] feature extraction. However, the handcrafted features has its own limitation. To solve this problem, we have used a convolutional neural network[@shi2016end] in order to extract the sequential feature from offline images which thereafter is fed to the LSTM module. There may exist several cases where character from one script may resemble the character of a different script. For those instances, it will be beneficial to look at the contexts of those ambiguous characters. Hence, we employ a Convolutional-LSTM architecture[@shi2016end] for the script identification task where convolutional network is used to convert the offline image into its sequential deep convolutional feature representation. Then the LSTM module is used to capture the contextual information with in a sequence. It can take input images of arbitrary width which is one of the major requirements of our framework since we are training the network using character level data and predict the result for both character and word level data as well. Word level data usually has much longer width and there is large variation in the length depending on the number of characters present in that word. Resizing the width to a fixed size is not a good choice since it distorts the word image and it may eliminate some good script specific information. However, it is needed to scale all the images to a fixed height to feed them in the network keeping the aspect ratio same.
Generation of each feature vector of a feature sequence is done in a left-to-right manner on the feature maps, taken column wise. This denotes that the the concatenation of the i-th columns of all the maps gives rise to the i-th feature vector. As per the architecture, the width of each such column is maintained at one pixel. Features are translation invariant due to the fact that layers of convolution, max pooling and element wise activation function operate in local neighborhood. Hence, each column of the feature maps actually maps to a specific area of the original image. Such regions are found in the same sequence to their corresponding columns on the feature maps from left to right. Each vector in the feature sequence can thus be regarded as a local image descriptor. Figure \[fig:fig6\] graphically shows the process of feature sequence generation using convolutional architecture [@shi2016end]. Our convolutional architecture for feature sequence extraction is composed of seven convolutional layer. The major change we have done in our network is to include a global maxpooling [@lin2013network] operation to get a global feature vector for the sequence which is concatenated with every left-to-right feature map and are fed to one last convolutional layer before converting it to final feature sequence using ‘Map to Sequence’ operation. The primary objective of including the global feature in our method is that besides gathering the local information, we can also consider the holistic representation of the entire image.
Conditional Multi-modal Fusion
------------------------------
After obtaining the time distributed feature sequence from both offline and online stream of every data sample using 1-D and 2-D Convolutional Network, we feed those sequential features to two different LSTM (Long Short Term Memory)modules for each modality. Traditional RNN suffers from the problem of vanishing and exploding gradient. To overcome these drawbacks, a different type of RNN is used known as LSTM (Long Short Term Memory). A memory cell along with three multiplicative gates constitute an LSTM. These gates are called input gate, forget gate and output gate. From the conceptual point of view, the past contents are stored in memory cells while the input and output cells are used to enable the cell to store contents for a long period of time. The forget gate is used to clear the memory in the cell. The main advantage of an LSTM is its ability to handle better long term dependency. The core idea of using LSTM module is to extract the feature from cell state of last time step after LSTM has seen the complete sequence of the offline or online feature sequence. This leads to the consideration of sequential relation among all feature vectors of a sequence. This is highly expected in a task like script identification where two different scripts might have a few related characters which possess a certain extent of similarity. However, using the sequential approach, we can avoid this confusion by considering global representation including the sequential relation among successive feature vectors of a sequence. For combination of offline and online information, we proposed a multi-modal conditional fusion method. Simple concatenation of the features of the modalities results redundant information decreasing the performance of the model. Also, it is necessary to take the most relevant information from the two modalities for correctly classifying the script. Thus, we used a novel fusion technique that dynamically assigns weights across the modalities representations. It learns the correlations between offline and online modality along with their adaptive contribution in fusion method.
Let the feature from the final time step of the LSTM network be $F_{online}$ and $F_{offline}$ with size $\mathbb{R}^{1\times K}$ for online and offline modality respectively. At first, we concatenate $F_{online}$ and $F_{offline}$ to obtain $F_{concat}$ of size $\mathbb{R}^{1\times 2K}$.
$${F_{concat}^{1\times2K} = Concat(F_{online}^{1\times K}, F_{offline}^{1\times K})}
\centering$$
The concatenated feature representation is conditioned on a 2 bit binary vector $Z$ representing the original modality of the input data. It is important to let the model know the actual form of the input data, whether it is online data or offline data. It allows the model to give the priority to the original modality adaptively in calculating the final feature representation. After feeding $Z$ to the concatenated representation we get a feature vector $F_{cond}$ of size $\mathbb{R}^{(1\times 2K)+2}$. Thereafter, we pass it through a fully connected layer($FC$) of weights $W \in \mathbb{R}^{(2K+2) \times K}$. The primary objective of such fully connected layer is to learn the correlation between the two modalities in order to assign their respective weightage accordingly. The output of this fully connected layer is $F_{fc}$ with size $\mathbb{R}^{1\times K}$. Finally, the sigmoid function outputs the weight parameters. Using equation \[eq\_Poff\] and \[eq\_Pon\] we get the weights $P_{Offline}^{1\times K}$ and $P_{Offline}^{1\times K}$. These weights are element-wise multiplied with their corresponding modality representations. The final feature vector is obtained by adding these two feature representation. Then, a fully connected layer is used which has the same number of neuron as the number of classes. A softmax layer outputs the probability distribution of the script over the classes. The conditional fusion is carried out by the following operations.
$${F_{fc}^{1\times K} = FC(F_{cond}^{1\times 2K+2} ; W_{fc,1}^{2K+2 \times K}) }
\centering$$
$$\label{eq_Poff}
{ P_{Offline}^{1\times K} = Sigmoid(F_{fc}^{1\times K}) }
\centering$$
$$\label{eq_Pon}
{P_{Online}^{1\times K} = 1 - P_{Offline}^{1\times K} }
\centering$$
$${{F_{offline,weighted}^{1\times K}} = F_{offline}^{1\times K} \odot P_{offline}^{1\times K}}
\centering$$
$${{F_{online,weighted}^{1\times K}} = F_{online}^{1\times K} \odot P_{online}^{1\times K}}
\centering$$
$${ {F_{Fusion, Final}^{1\times K}} = {F_{offline,weighted}^{1\times K}} + {F_{online,weighted}^{1\times K}} }
\centering$$
Implementation Details {#impl}
----------------------
Our proposed multi-modal network uses character level data during training and it predicts the script identification result for both character and word level data. Let $(X_{i},Y_{i})$ be the online-offline modality pair for a particular sample data which is to be fed as the input to our network. It is to be noted that only one of the modalities, i.e. either $X_{i}$ or $Y_{i}$ is present originally, and we convert the other one using the method mentioned in section 3.1. $(L_{i}, Z_{i})$ are the two given labels to train the network in a supervised manner. Here, $L_{i}$ is the corresponding script label and $Z_{i}$ is the original modality from which the original data sample was fetched. $Z_{i}$ is the extra supervision we use to impose a condition during multi-modal fusion. During training, $L_{i}$ is used to calculate the cross-entropy loss for classification to train the network through back-propagation. During testing, the network predicts the identified script $L_{i}$ as output. However, $Z_{i}$ is present during both training and testing for conditional fusion of two different modalities.
The network architecture of our framework is shown pictorially in Figure \[fig:fig4\]. We have included one global average pooling operation in both offline and online stream networks in order to capture the holistic information about the data sample. The architecture for offline stream consists of 7 convolutional layers and 4 maxpooling layers. For the 3rd and 4th maxpooling layers, the filter size was fixed at 2x1, in order to get the feature maps with larger width, thus generating a larger feature sequence, which was found to be beneficial for capturing the spatial dependency among characters of words. We normalize every offline data to a height of 32 keeping the aspect ratio same. In order to feed the character level offline data in batches during training, we resize every character to a size of 32x32. However, we can feed offline images of arbitrary width to keep the aspect ratio constant during testing, but only one at a time. This is expected because word images(during testing) usually have much longer width, i.e. much higher aspect ratio compared to single characters. On the contrary, for online data it has no such font size limitation, since it is already time distributed. The architecture of online stream consists of 6 convolutional layer and one maxpooling operation. In order to accelerate the training process, we have added two batch normalization layers in both of our offline and online stream networks. Next, we have used two layers LSTM network with 512 hidden LSTM units. Hence, K equals to 512 in our framework. The final feature for offline and online modality is extracted from the cell state of the last time step of two LSTM modules, after which multi-modal conditional fusion is carried out.
We have implemented our complete system using Python and Tensorflow framework in 2.50 GHz Intel(R) Xeon(R) CPU, 32GB RAM and an NVIDIA Titan-X GPU. The weights of the model are initialized according to the Xavier initializer. All convolution and fully connected layers use Rectified Linear Units (ReLU). The training is carried out using Stochastic Gradient Descent algorithm with a momentum of 0.9 and learning rate 0.01. The network converges after 30K iterations with a batch size of 32. The learning rate is multiplied by 0.1 when the validation error stops decreasing for enough number of iterations. The weight decay regularization parameter is set to $5\times 10^{4}$.
Experiments
===========
In this section, we report the performance of our script identification framework. We first introduce the datasets used for our study and then present the detailed script identification performance along with different baseline methods, error analysis and discussions.
Datasets
--------
As per our findings, there exists no such standard datasets for handwritten script recognition. Here, we have collected various publicly available word and character recognition datasets [@BHUNIA201812; @roy2016hmm] of different scripts to prepare our required database for script recognition. In our experiments, we have considered a total of 7 scripts for the performance evaluation, namely, Devanagari, Bangla, Odia, Gurumukhi, Tamil, Telugu and English. Among these scripts, Devanagari, Bangla, Gurumukhi and Odia are descended [@ghosh2010script] from the common ancestor script in the Brahmi script family. There exist a good extent of similarity between Bangla and Devanagari, Devanagari and Gurumukhi as mentioned in [@BHUNIA201812]. Similarly, Tamil and Telugu are two south Indian scripts. On the other side, English is a global language which is a medium of communication for different parts of the world. Most of the documents are bi-script which contains one of the regional languages with English. Hence, our selection of the scripts for performance evaluation is based on the intention to make the task of script identification more difficult. Table \[tab1\] gives the detail of our dataset used for script recognition. All the experiments for script recognition have been done in a 10 fold cross validation mode with 7:2:1 training, validation and testing. By this, 70% data of dataset was used for training, 10% data for validation and 20% data for testing.
------------ -------- --------- -------- ---------
Online Offline Online Offline
Bangla 10,509 10,897 10,589 10,687
Devanagari 10,897 11,021 10,847 10,645
Gurumukhi 9,897 9,789 9,569 9,657
Odia 9,457 9,789 9,147 9,234
Tamil 9,456 9,476 9,874 9,476
Telugu 10,486 10,789 10,687 10,694
English 10,489 10,879 11,023 11,458
------------ -------- --------- -------- ---------
: Details of our script identification dataset[]{data-label="tab1"}
Different Baselines
-------------------
As mentioned earlier, there is no such earlier framework for handwritten script identification using multi-modal deep network which can perform for both offline and online data simultaneously using a single model. However, one of the naive approaches is to convert the data from either modality to its equivalent opposite modality, and feed the required modality to the framework based on the modality it has been trained on. For instance, if a framework has been trained using offline handwritten data, and we have online handwritten data for testing. In this case, the naive approach is to convert online data to its offline equivalent and test using the model trained from offline handwritten data. However, there is a major limitation of using such naive approach. Although, handwritten data can be converted between two modalities, the data distribution of converted data is not similar to the real one, thus limiting the performance. In order to justify the superiority of our method, we evaluated both in-modality and cross-modality performance for all the baseline methods. Another contribution of our framework is that our framework can be trained with light weight character level data and can achieve performance as that of traditional way of training a script identification model using word level data. Hence, we report the performance of our framework for both character level and word level data for training. To perform a fair comparison between word level and character level data for training, we use the nearly equal number of word and character level data for training. In Table \[tab1\], we have mentioned the number of data for word level and character level data from offline and online modalities are present; and it shows the number of sample is nearly with in a same range. For every experiment, cross modality(training from online data and testing on offline data or vice-versa) or cross level(i.e. training from character level data and testing on word level data or vice-versa), we use 7 fold data of a particular level or modality for training, and testing and validation have been done on the data of other modality or level with 2 and 1 fold each, respectively.
To compare our proposed framework, we have defined a few base line methods based on deep neural network architecture. We have justified the limitations of every baseline with respect to our proposed framework. All the base line methods are defined for single modality data in order to justify the improvement in performance, we achieve due to our multi-modal framework. Also, our proposed novel multi-modal fusion method is compared with different traditional multi-modal fusion methods in section 4.3. The base line methods using different traditional classifiers and hand-crafted features are detailed in section 4.5 separately. For the first two baseline methods, we have just sliced the offline and online stream of our multi-modal network into two different baselines for online and offline data respectively. The performance comparison with these two baseline justifies the necessity of designing a multi-modal deep framework.
#### DL\_1:
For this baseline, we use 1D-convolutional-LSTM network for online handwritten script identification. This is the same configuration as used for our online stream network and is only trained from online data. The softmax classification layer is used at the output of last time step. Although the performance is competitive in case of online data, the major limitation is, the cross-modality performance is restricted since the network is unaware of the data distribution of offline data.
#### DL\_2:
For this baseline, we use 2D-convolutional-LSTM network for offline handwritten script identification identification following the same architecture as mentioned for offline stream of our network. This has been trained only from the offline images. Here also, the main limitation is that it does not perform well for cross-modal data.
#### DL\_3:
One of the important contributions in our online stream network is the use of 1D-convolutional network over the online coordinate points in order to learn the structural correlation of among neighboring pixel points. Hence, we define our third base line in order to justify the improvement, we achieve due to use of this 1D-convolutional network. Here, we directly feed the coordinate points into the LSTM network for script identification and evaluate the identification performance. The architecture and setup for different hyper parameters are kept same as mentioned in section \[impl\].
Our framework is trained from data in paired modality, where the original modality is present from both offline and online modality with equal distribution in the training data. We have evaluated the performance our model for both real online and offline data individually. We found that our model generalizes well for both online and offline data with no significant change in the accuracy. The results are reported in Table 2. However, we have evaluated the performance using only one modality as the source of data for training. It has been observed that the performance of script identification decreases in this case compared to training the network using both the modalities as the source. Hence, we conclude that our proposed architecture generalizes well for both the modalities when it is trained from both online and offline modality. A comparative study has been shown in Fig. \[fig:fig11\]. Interestingly, note that, model trained from only online-data performs poorly than offline in case of last three combinations (char-word, word-char and word-word), though difference is marginal. The reasoning behind such observation could be (a) while we redraw offline version from online data, we connect the successive points, followed by a morphological thickening operation. Following Figure \[fig:fig8\] and \[fig:fig14\], there exist some noticeable difference between redrawn and original offline images, mainly in terms of non-uniform thickness of the stroke or random jittering in the shape of strokes in case of real offline images, compared to uniform and smooth stroke width of redrawn sample. Owing to this domain gap, the trained model from redrawn offline image (for offline branch) could not generalize well for original offline image of testing set. (b) Although online modality is claimed to perform better for handwriting recognition, this may not be always true for script recognition in Indic script. Due to complex nature of indic script’s characters, presence of matra, and compound character, stroke sequence of a word could be written in different ways, therefore it could be challenging if some unknown writing stroke sequence appears in testing data. The confusion matrix for both character and word level training is shown in Figure \[fig:fig12\].
[|c|c|c|c|c|]{} Method &
----------
Training
Data
----------
: Script identification performance using different baseline schemes[]{data-label="my-label"}
&
---------
Testing
Data
---------
: Script identification performance using different baseline schemes[]{data-label="my-label"}
&
------------------
With in modality
Accuracy
------------------
: Script identification performance using different baseline schemes[]{data-label="my-label"}
&
----------------
Cross modality
Accuracy
----------------
: Script identification performance using different baseline schemes[]{data-label="my-label"}
\
& Character & Character & 95.61 & 90.94\
& Character & Word & 94.84 & 89.84\
& Word & Character & 92.13 & 87.67\
& Word & Word & 95.14 & 90.12\
& Character & Character & 95.47 & 90.96\
& Character & Word & 94.59 & 90.04\
& Word & Character & 91.83 & 87.23\
& Word & Word & 95.31 & 90.54\
& Character & Character & 92.48 & 87.47\
& Character & Word & 91.43 & 86.64\
& Word & Character & 89.32 & 85.14\
& Word & Word & 92.12 & 87.34\
[|c|c|c|c|c|]{}
------------
Training
Data Level
------------
: Script identification performance using our proposed framework[]{data-label="tab:Acc"}
&
------------
Testing
Data Level
------------
: Script identification performance using our proposed framework[]{data-label="tab:Acc"}
&
-----------------
Online Modality
Accuracy
-----------------
: Script identification performance using our proposed framework[]{data-label="tab:Acc"}
&
------------------
Offline Modality
Accuracy
------------------
: Script identification performance using our proposed framework[]{data-label="tab:Acc"}
&
----------
Average
Accuracy
----------
: Script identification performance using our proposed framework[]{data-label="tab:Acc"}
\
Character & Character & 98.59 & 98.48 & 98.55\
Character & Word & 97.75 & 97.81 & 97.79\
Word & Character & 94.89 & 94.91 & 94.92\
Word & Word & 98.11 & 98.23 & 98.17\
Comparative study with different multi-modal fusion methods
-----------------------------------------------------------
One of most popular application of multi-modal fusion approaches is Visual Question Answering[@ilievski2017multimodal], where image and language feature representation are combined. Here, we also combine the offline and online feature representations for script identification using a conditional multi-modal fusion method. We compare our multi-modal fusion method with different traditional fusion methods popular in the literature. Lets denote the feature representation of offline and online modality as $F_{1}$ and $F_{2}$, both of which is of dimension $\mathbb{R}^{1\times K}$.The traditional approaches used in our comparison are as follows: Firstly, One of the approaches is to concatenate these two features $F_{1}$ and $F_{2}$ and feed it to a fully connected and a softmax layer for classification. Secondly, $F_{1}$ and $F_{2}$ can be added or multiplied element wise followed by a fully connected and a softmax layer for classification. Thirdly, we have considered the outer product between $F_{1}$ and $F_{2}$ and bilinear pooling [@lin2015bilinear] followed by a fully connected and a softmax layer for final classification. Fourthly, we use multi-modal Compact Bilinear Pooling with pooling dimension 4K for multi-modal fusion. More details about multi-modal Compact Bilinear(MCB) Pooling can be found in [@gao2016compact]. We have also evaluated the performance of our multi-modal fusion method with out using conditional fusion. The results for different multi-modal fusion strategies are reported in Table \[fusion\].
[|c|c|c|c|c|]{} &\
Method & Char-Char & Char-Word & Word-Char & Word-Word\
Elementwise Sum & 96.12 & 95.61 & 92.87 & 95.94\
Concatenation & 96.81 & 96.34 & 93.37 & 96.64\
Elementwise Product & 96.17 & 95.57 & 93.01 & 96.05\
--------------------------
Outer Product + Bilinear
Pooling
--------------------------
: Comparative study with different multi-modal fusion strategy[]{data-label="fusion"}
& 96.94 & 96.59 & 93.61 & 96.83\
MCB Pooling & 97.14 & 96.72 & 93.87 & 97.03\
------------------------
Proposed Fusion Method
(Without Condition)
------------------------
: Comparative study with different multi-modal fusion strategy[]{data-label="fusion"}
& 97.89 & 97.08 & 94.27 & 97.49\
------------------------
Proposed Fusion Method
(With Condition)
------------------------
: Comparative study with different multi-modal fusion strategy[]{data-label="fusion"}
& 98.55 & 97.79 & 94.92 & 98.17\
Comparative study with different traditional classifiers
--------------------------------------------------------
In this paper, we have considered convolutional layers stacked with LSTMs for due to their superior performance reported in recent literature. It is a deep multi-modal network which can be trained in a end to end manner using back propagation and it takes input for both offline and online modality of the data. In recent literature, Deep Neural Network has achieved a great superiority over the traditional classifiers and handcrafted features. Most of the previous methods for handwritten script identification are based on this traditional classifiers and handcrafted features. However, since there is no standard dataset, we can not compare our deep learning based system with those methods. Hence, we employed some baseline methods using some popular traditional classifiers and popular handcrafted features. We name these methods as Traditional Approaches(TA) and report the results in our 7 handwritten scripts dataset. We show the performance using both character level and word level data as training. Also, we report the cross modality script identification accuracy. However, training has been done from single modality, i.e. no multi-modal combination, for these traditional methods. The different baseline approaches considered in our experiments are described in the following subsections.
### Performance on Offline Data
For offline word images, we have used PHOG (Pyramidal Histogram of Oriented Gradient) and LBP (Local Binary Pattern) for feature extraction. PHOG is gradient based feature which has been utilized in sliding window based handwriting recognition task [@roy2016hmm; @BHUNIA201812]. LBP is a texture descriptor which has been further used in different tasks like, facial expression recognition [@moore2011local], handwritten word spotting in historical documents [@dey2016local] etc.
#### TA\_1:
For this baseline, we have used two popular state-of the art traditional classifiers [@fernandez2014we], SVM(Support Vector Machine) and Random Forest, for the script identification task along with PHOG and LBP features, respectively. Both SVM and Random Forest have been studied in various classification problems extensively in both Computer Vision and Document Image Analysis communities. Here, we have extracted the handcrafted feature using PHOG and LBP respectively, and evaluate the classification performance using SVM and Random Forest classifiers, respectively. The results are reported in Table \[tab5\] as TA\_1. It is to be noted that using this baseline, it is not possible to predict the script at word level using character level training data, since SVMs or Random Forests are not good in handling sequential data. Hence, the results are reported accordingly.
#### TA\_2:
In this baseline, we have explored the sequence-based script identification approach using two traditional sequential classifiers HMM and HCRF, respectively. For sequential feature extraction from word or character level images, a sliding window moves from left to right of the image and PHOG or LBP feature is extracted from each sliding window, which denotes the feature at single time step. This type of sliding window based classification approach has been used in music score writer identification [@roy2017hmm]. As we are dealing with sequence based classification approach in this baseline, it is possible to use character level data for training in order to predict the script at word level. The results are reported in Table \[tab5\] as TA\_2.
#### TA\_3:
Here, we use the same sliding window based feature extraction approach as used in TA\_2, the only difference is that we use 2 layers LSTM as the classifier. Performance of different handcrafted features with LSTM has been studied in [@chherawala2016feature] for offline handwriting recognition. Hence, we employ a similar framework for the script identification task, and name it as TA\_3. Results of each corresponding experiment has been reported in Table \[tab6\]
[|c|c|c|c|c|c|c|]{} Method & Classifier & Feature &
------------
Training
Data Level
------------
: Comparative study with different traditional baseline methods[]{data-label="tab5"}
&
------------
Testing
Data Level
------------
: Comparative study with different traditional baseline methods[]{data-label="tab5"}
&
---------------
With in
Modality Acc.
---------------
: Comparative study with different traditional baseline methods[]{data-label="tab5"}
&
---------------
Cross
Modality Acc.
---------------
: Comparative study with different traditional baseline methods[]{data-label="tab5"}
\
& & & Character & Character & 90.78 & 85.48\
& & & Word & Character & 87.29 & 83.12\
& & & Word & Word & 89.44 & 84.69\
& & & Character & Character & 91.48 & 85.78\
& & & Word & Character & 88.32 & 83.67\
& & & Word & Word & 90.49 & 85.14\
& & & Character & Character & 90.87 & 85.94\
& & & Word & Character & 87.21 & 83.64\
& & & Word & Word & 89.51 & 84.47\
& & & Character & Character & 91.54 & 86.71\
& & & Word & Character & 88.57 & 84.39\
& & & Word & Word & 90.42 & 85.49\
& & & Character & Character & 91.48 & 86.65\
& & & Character & Word & 90.53 & 85.17\
& & & Word & Character & 88.51 & 84.74\
& & & Word & Word & 91.11 & 86.39\
& & & Character & Character & 91.94 & 86.46\
& & & Character & Word & 90.91 & 85.37\
& & & Word & Character & 89.14 & 84.69\
& & & Word & Word & 91.81 & 86.44\
& & & Character & Character & 91.34 & 86.46\
& & & Character & Word & 90.37 & 85.71\
& & & Word & Character & 88.97 & 84.72\
& & & Word & Word & 91.23 & 86.82\
& & & Character & Character & 91.87 & 86.55\
& & & Character & Word & 90.84 & 85.67\
& & & Word & Character & 89.34 & 85.41\
& & & Word & Word & 91.97 & 86.91\
[|c|c|c|c|c|c|c|]{} Method & Classifier & Feature &
------------
Training
Data Level
------------
: Comparative study with different traditional baseline methods[]{data-label="tab6"}
&
------------
Testing
Data Level
------------
: Comparative study with different traditional baseline methods[]{data-label="tab6"}
&
---------------
With-in
Modality Acc.
---------------
: Comparative study with different traditional baseline methods[]{data-label="tab6"}
&
--------------
Cross
Modality Acc
--------------
: Comparative study with different traditional baseline methods[]{data-label="tab6"}
\
& & & Character & Character & 91.64 & 86.47\
& & & Character & Word & 90.54 & 85.69\
& & & Word & Character & 88.87 & 84.03\
& & & Word & Word & 91.63 & 86.23\
& & & Character & Character & 92.11 & 87.39\
& & & Character & Word & 91.01 & 86.46\
& & & Word & Character & 89.31 & 84.97\
& & & Word & Word & 91.82 & 86.89\
& & & Character & Character & 92.34 & 84.44\
& & & Character & Word & 91.47 & 83.69\
& & & Word & Character & 88.94 & 81.41\
& & & Word & Word & 92.03 & 83.87\
& & & Character & Character & 92.31 & 84.47\
& & & Character & Word & 91.37 & 83.41\
& & & Word & Character & 88.67 & 81.67\
& & & Word & Word & 92.08 & 83.69\
& & & Character & Character & 92.54 & 84.89\
& & & Character & Word & 91.67 & 83.85\
& & & Word & Character & 89.05 & 81.09\
& & & Word & Word & 92.17 & 84.12\
### Performance on Online Data
Online data contains the successive coordinate points representing the trajectory of pen’s movement. For online handwriting recognition, one of the most popular handcrafted feature descriptor is NPEN++ [@jaeger2001online]. We have considered this feature in our traditional baseline method for handwritten online script identification. Here, five different features namely, Curliness (CR), Writing direction (WD), Linearity (LR), Slope (SP), and Curvature (CV) are extracted from the sequence of coordinate points and are concatenated to form the final feature descriptor. More details about these features can be found in [@jaeger2001online].
#### TA\_4:
In this baseline, we have used the NPEN++ feature along with sequential classifier HMM and HCRF for each case. HMM and NPEN++ feature have been used in online handwriting recognition in [@jaeger2001online]. Hence, it is reasonable to evaluate the performance of these combinations for online handwritten script identification task. The results are reported in Table \[tab6\] as TA\_4.
#### TA\_5:
In this framework, we directly feed the online sequential coordinate point in a LSTM network for script identification. This type of approach has been addressed in the literature for handwriting recognition in [@graves2008unconstrained]. We report the result for this approach in Table \[tab6\].
Performance on Public PHDIndic Dataset
--------------------------------------
Recently, authors in [@obaidullah2018phdindic_11] introduced a new dataset for handwritten Indic script identification, which consists of 1000 words each from 11 different scripts, namely Bengali, Devanagari, Gujarati, Gurumukhi, Kannada, Malayalam, Odia, Roman, Tamil, Telegu, Urdu. Some word level samples are shown in Figure \[fig:phdindic\]. Note that PHDIndic is completely a word-level offline dataset and no online dataset is available for all the 11 scripts. Please also note that our framework is designed with a motivation to work both for offline and online data parallelly. In order to justify the contribution of our multi-modal framework, we need such a dataset which has both offline and online data for all the scripts, so that we can evaluate within modality and cross modality performance that would explain the contribution of different design choices in our network architecture. Nevertheless, we have evaluated the performance on offline word-level data from PHDIndic dataset in order to validate our framework on a publicly available dataset. Following [@ukil2018deep], we use 80% images from each script as training and rest 20% for testing. For the online stream, we feed online-converted data from offline images. We achieved 94.74% accuracy on the testing set. As shown in Figure \[fig:fig11\], that our framework generalizes well while we train it from both real offline-online data. Therefore, had we been able to train the model also including real online data for all the 11 scripts, the accuracy is supposed to get improved. Moreover, our method is competitive with other recently introduced frameworks [@obaidullah2018phdindic_11; @ukil2019improved]. The foremost thing here is to noted that all these frameworks [@obaidullah2018phdindic_11; @ukil2019improved] have been designed explicitly for offline data which has no way to handle online data equally; however our framework can handle both offline and online data parallely with nearly equal efficiency (see Table \[tab:Acc\]).
Error Analysis
--------------
Inter modality conversion is one of the main crucial steps in our framework, since our proposed architecture takes offline-online modality pair of a data sample as the input. Although the conversion of online to offline modality is a trivial and simple one, the recovery of stroke information form offline data is a bit challenging due to free flow nature of handwriting by different individuals. For this task, we have adopted the method in [@nel2005estimating]. However, during our experimental analysis, we have observed that the offline to online conversion algorithm fails to recover proper stroke sequences in some specific cases. Figure \[fig:fig10\] shows some examples, where the the our adopted algorithm fails to recover the expected stroke sequence in some specific cases. One of the main reasons of this problem is skeletonization error that appears as unwanted skeleton branches with incorrect angles due to uneven thickness of the handwritten data and surface noise. It is also observed from the second image of Figure \[fig:fig10\] that the algorithm tends to miss strokes at the junction points due to the presence of the Matra which is very common feature among Indic scripts. This problem can be solved to some extent by Matra removal as proposed by [@roy2016hmm].
Conclusions and Future Work
===========================
In this paper we have proposed a new method for script identification which has provided us with satisfactory results. Handwritten text present in either modality, online or offline has been received and the absent modality has been recreated using inter modality conversion. After such recreation, both modalities had been fed in pair into a deep neural network. This designed neural network uses both sets of information from both modalities to employ multi-modal fusion thus combining the features adaptively. Two significant achievements obtained are the designing of one single training model which encompasses the training of both modalities of data. Secondly, the features of two modalities are combined to produce more accurate results.
As evident from the results, our method performs better than almost every other state-of-the art method for handwritten script identification. A few drawbacks include incomplete conversion of modality to the other, but the conditions existent for such cases are rare. The offline to online conversion can be handled using a deep end-to-end network [@kumarbhunia2018handwriting] and can be included as a sub-module in our architecture. Our future work would include fine tuning our proposed deep network architecture and including more Indic scripts to increase the scope of application of our method. One of the most promising future research directions would be to design a single deep model for offline and online handwriting recognition in a single deep neural network through exploring information from both the modalities.
[^1]: <https://en.wikipedia.org/wiki/Languages_of_India>, accessed on 20/02/2018
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---
address: |
Department of Mathematics and Statistics\
The University of North Carolina at Greensboro
author:
-
bibliography:
- 'pram.bib'
title: 'High-dimensional robust approximated $M$-estimators for mean regression with asymmetric data'
---
Introduction {#sec1}
============
Asymmetry along with heteroscedasticity or contamination often occurs with the growth of data dimensionality. In high-dimensional settings, particularly when random errors follow irregular distributions such as asymmetry and heteroscedasticity, simultaneous mean estimation and variable selection are still of interest in many applications. For example, in economics where asymmetric data is prevalent, it is of interest to study how mean GDP is affected by many features. Another example can be found in RNA-seq data analysis, the highly skewed nature and mean-variance dependency of RNA-Seq data may pose difficulties on building prognostic gene signatures. Although certain transformations of RNA-seq data have been studied for improving detection of important genes [@zwiener2014RNA-seq], we may be still interested in building a mean regression model on the original data.
In this paper, we are interested in high-dimensional mean regression that is robust to the following irregular settings: (a) the data are not symmetric due to the skewness of random errors ([@fan2017estimation]); (b) the data are heteroscedastic ([@daye2012high], [@wang2012quantile]); and (c) the data are contaminated in both response and a large number of variables ([@rousseeuw2005robust]). However, above irregular settings are often overlooked for high-dimensional data analysis, especially for the theoretical development.
Despite the extensive work on penalized robust M-estimator in high-dimensional regression (e.g. @huber1964robust, @lambert2011robust, @gao2010asymptotic, @wang2013l1, @loh2017statistical), most of them either do not estimate the conditional mean regression function or require the error distribution to be symmetric and/or homogeneous. To tackle this problem, @fan2017estimation proposed a so-called RA-Lasso estimator, in which they waived the symmetry requirement by using the Huber loss with a diverging parameter in order to reduce the bias when the error distribution is asymmetric. @fan2017estimation obtained nice asymptotic properties of the RA-Lasso estimator, and proved its estimation consistency at the minimax rate enjoyed by LS-Lasso.
However, the Huber loss approximation used in the RA-Lasso dose not downweight the very large residual due to its non-decreasing $\Psi$-function. [@shevlyakov2008redescending] showed that M-estimators given by non-decreasing $\Psi$-function do not possess finite variance sensitivity, meaning the asymptotic variance can be largely affected if the assume model is only approximately true. In that paper, the authors proposed to consider re-descending M-estimators with $\Psi$-function re-descending to zero to address this problem. They further showed that re-descending M-estimator can be designed by maximizing the minimum variance sensitivity under a global minimax criterion. For instance, the Smith’s estimator and Tukey’s biweight estimator are the optimal M-estimator with minimax variance sensitivity for a class of densities with a bounded variance and a bounded fourth moment, respectively [@shevlyakov2008redescending]. Therefore it is tempting to also include re-descending $M$-estimator in the study of complex high-dimensional settings.
For decades both the theoretical and computational result in penalized re-descending M-estimator in high-dimensional settings have been very limited, due to the non-convexity of loss functions. Recently @loh2017statistical established a form of local statistical consistency for the high-dimensional $M$-estimators allowing both the loss and penalty functions to be non-convex. However, this study does not address the problem of asymmetry and heteroscedasticity. Also, their numerical studies neglect settings for asymmetric data and lack of comparisons among different $M$-estimations.
In this paper, we consider high-dimensional linear regression in more general irregular settings: the data can be contaminated or include possible large outliers in both random errors and covariates, the random errors may lack of symmetry and homogeneity. In particular, we investigate both statistical and computational properties of high-dimensional mean regression in the penalized $M$-estimator framework with diverging robustness parameters. This framework allows both the loss function and the penalty to be non-convex. Our perspective is different from @loh2017statistical since all loss functions considered in our study converge to a quadratic loss when the corresponding robustness parameter diverges. To be more specific, we proposed a class of Penalized Robust Approximated quadratic $M$-estimators (PRAM) to address all irregular settings in (a-c) mentioned above. Inspired by @fan2017estimation, PRAM uses a family of loss functions with a diverging parameter $\alpha$ to control the robustness as well as the discrepancy to the quadratic loss. By controlling the divergent rate of $\alpha$, PRAM estimators are able to reduce the bias induced by asymmetric error distribution and meanwhile preserve the robustness to approximate the mean estimators. Additionally, we extend the PRAM to a more general setting by relaxing the sub-Gaussian assumption on covariates.
Our theoretical contributions in this paper include the investigation of statistical properties for a class of PRAM estimators with only weak assumptions on both random errors and covariates. In particular, We first introduce sufficient conditions under which a PRAM estimator has local estimation consistency at the same rate as the minimax rate enjoyed by the LS-Lasso. We then show that the PRAM estimator actually equals the local oracle solution with the correct support if an appropriate non-convex penalty is used. Based on this oracle result we further establish the asymptotic normality of the PRAM estimators. As we will see, with the devise of diverging parameters in the loss functions, our theoretical result is applicable for a wide class of PRAM estimators which are robust to general irregular settings, when the dimensionality of data grows with the sample size at an almost exponential rate.
Computationally, we also implement the PRAM estimator through a two-step optimization procedure and investigate the performance of six PRAM estimators generated from three types of loss function approximation (the Huber loss, Tukey’s biweight loss and Cauchy loss) combined with two types of penalty functions (the Lasso and MCP penalties). While our numerical results demonstrate satisfactory finite sample performance of the PRAM estimators under general irregular settings, it suggests that in practice, when the data are heavy-tailed or contaminated, a well-behaved PRAM estimator can be chosen by considering a re-descending loss function approximation and a concave penalty, using the RA-Lasso as an initial.
#### Related Works:
we end this section by highlighting a few things on how our work is different from some recent related work:
- As introduced earlier, the RA-Lasso proposed by [@fan2017estimation] waives the symmetry requirement by allowing the parameter of Huber loss to diverge. The idea is that by controlling the divergent rate of the parameter, while preserving certain robustness, the Huber loss becomes ‘closer’ to the $\ell_2$ loss and thus potentially reduces the bias when the error distribution is asymmetric. Our work in this paper relax the convexity restriction of loss functions and answer the question on how in general a loss function with strong robustness should converge to the $\ell_2$ loss to achieve the estimation consistency at the minimax rate. While [@fan2017estimation] focuses exclusively on the Lasso penalty, our framework also allows concave penalties and therefore inherits certain oracle property under some conditions. Furthermore, we relax the sub-Gaussian assumption on covariates in [@fan2017estimation] by incorporating weight functions in the extension of PRAM estimators.
- [@loh2017statistical] also establishes a form of local statistical consistency for high-dimensional non-convex M-estimators. However, we address the problem of asymmetry and heteroscedasticity. In particular, our proposed framework is more general: we consider the empirical loss function $\mL_{\alpha, n}$ satisfying $\lim_{\alpha \to \infty} E[\nabla \mL_{\alpha, n}(\bb^*)] =\b0$, where $\bb^*$ is the true parameter vector and $\alpha$ is the diverging parameter. In contrast, [@loh2017statistical] requires the condition $E[\nabla \mL_{\alpha, n}(\bb^*)] =\b0$ for each $\alpha >0$, which may not hold with the lack of homogeneity and symmetry in general. Additionally, [@loh2017statistical] does not suggest which estimators to be considered in real applications. We further investigate this problem by comparing different PRAM estimators in numerical studies.
The remainder of our paper is organized as follow. In Section 2, we introduce the basic setup regarding PRAM estimators and corresponding generalizations. In Section 3, we establish the local estimation consistency for the PRAM estimators under sufficient conditions. For non-convex regularized PRAM estimators, we also present our statistical theory concerning the selection consistency and the asymptotic normality of PRAM estimators. We discuss the implementation of PRAM estimators including both the computational algorithm and the tuning parameter selection in Section 4. In section 5, we conduct some simulation studies to demonstrate the performance of the PRAM estimators under different settings. We also apply those PRAM estimators for NCI-60 data analysis and illustrates all results in Section 6. Section 7 concludes and summarizes the paper. All technical proofs are relegated to the Appendix.
#### Notation:
We use bold symbols to denote matrices or vectors. For a matrix or a vector $\bv$, we write $\bv^T$ to denote its transpose. We write $\|\cdot\|_1$ and $\|\cdot\|_2$ to denote the $L_1$ norm and the $L_2$ norm of a vector, respectively. For a function $g: \RR^p \mapsto \RR$, we write $\nabla g$ to denote a gradient of the function. We write $u_+$ to denote $\max(u,0)$ for any $u \in \RR$.
The PRAM method {#sec2}
===============
Model settings {#sec2-1}
--------------
Consider an ultra high-dimensional linear regression model $$\label{linear model}
y_i=\bx_i^T\bb^*+\epsilon_i,$$ where $\bx_i=(x_{i1},\cdots,x_{ip})^T$ for $i=1,\cdots,n$ are independent and identically distributed (i.i.d) $p$-dimensional covariate vectors such that $E(\bx_i)=\bf 0$, $\{\epsilon_i\}_{i=1}^n$ are independent errors such that $E(\epsilon_i \mid \bx_i)=0$ and thus we allow the conditional heteroschedasticity. Note $\bb^*=(\beta^*_1,\cdots,\beta^*_p)^T\in\RR^p$ is an $s$-sparse conditional mean coefficient vector (only include $s$ nonzero elements) and $p \gg n$.
Our model settings permit the existence of all the following irregular settings on both $\epsilon_i$s and $\bx_i$s: (a) asymmetry of $\epsilon_i$; (b) heteroscedasty of $\epsilon_i$ and $\epsilon_i$ may depend on $\bx_i$; (c) data contamination of $\epsilon_i$ and $\bx_i$.
We are interested in penalized mean regression estimators such that $$\label{eq:estimator}
\hat{\bb}\in {\operatornamewithlimits{argmin}}_{\|\bb\|_1\le R} \left\{ \mL_{\alpha,n}(\bb)+\rho_\lambda(\bb)\right\},$$ where $\mL_{\alpha, n}$ is the empirical loss function and $\rho_\lambda$ is a penalty function which encourages the sparsity in the solution. Here $\alpha >0$ is a parameter controlling the robustness, which is allowed to diverge. As mentioned in Section \[sec1\], we consider the loss function $\mL_{\alpha, n}$ satisfying $$\label{population condition}
\lim_{\alpha \to \infty} E[\nabla \mL_{\alpha, n}(\bb^*)] =\b0.$$ This condition in (\[population condition\]) relaxes the condition, $E[\nabla \mL_{\alpha, n}(\bb^*)] =\b0$ for each $\alpha >0$, required in [@loh2017statistical], which may be invalid with the lack of homogeneity and symmetry. The condition (\[population condition\]) permits the random error to be heterogeneous and/or asymmetric, as long as $E[\nabla \mL_{\alpha, n}(\bb^*)]$ converges to $\b0$ with diverging $\alpha$.
We also include the side condition $\|\bb\|_1 \le R$ in the penalized optimization problem in (\[eq:estimator\]), in order to guarantee the existence of local/global optima, for the case where the loss function or the regularizer may be non-convex. We also require $\|\bb^*\|_1 \le R$ so that $\bb^*$ is feasible in (\[eq:estimator\]). In real applications, we can choose $R$ to be a sufficiently large number.
Penalty functions {#sec2-2}
-----------------
Since the coefficients vector $\bb^*$ is assumed to be $s$-sparse in the high-dimensional linear regression model in , we only consider penalties which generate sparse solutions. In particular, we require the penalty function $\rho_{\lambda}$ in to satisfy following properties listed in Assumption 1.
\[as:penalty\] The penalty function is coordinate-separable such that $\rho_\lambda(\bb)=\sum_{j=1}^{p} \rho_\lambda(\beta_j)$ for some scalar function $\rho_\lambda: \RR \mapsto \RR$. In addition,
1. the function $t \mapsto \rho_\lambda(t)$ is symmetric around zero and $\rho_\lambda(0)=0$;
2. the function $t \mapsto \rho_\lambda(t)$ is non-decreasing on $\RR^+$;
3. the function $t \mapsto \frac{\rho_\lambda(t)}{t}$ is non-increasing on $\RR^+$;
4. the function $t \mapsto \rho_\lambda(t)$ is differentiable for $t \ne 0$;
5. $ \lim_{t \to 0^+} \rho_\lambda'(t) = \lambda$;
6. there exists $\mu > 0$ such that the function $t \mapsto \rho_{\lambda} (t) + \frac{\mu}{2}t^2$ is convex;
7. there exists $\delta \in (0, \infty)$ such that $\rho_{\lambda}'(t)=0$ for all $t \ge \delta \lambda$.
Those properties in Assumption 1 are related to the penalty functions studied in @loh2013regularized and @loh2017statistical, where $\rho_{\lambda}$ is said to be $\mu$-amenable if $\rho_{\lambda}$ satisfies conditions (i)-(vi) for $\mu$ defined in (vi). If $\rho_{\lambda}$ also satisfies condition (vii), we say that $\rho_{\lambda}$ is $(\mu, \delta)$-amenable. Some popular choices of amenable penalty functions include Lasso [@tibshirani1996regression], SCAD [@fan2001variable], and MCP [@zhang2010nearly] given as follows:
- The [**Lasso**]{} penalty, $\rho_{\lambda}(t)=\lambda |t|$, is $0$-amenable but not $(0,\delta)$-amenable for any $\delta <\infty$.\
- The [**SCAD**]{} penalty, $$\rho_{\lambda}(t)=\begin{cases}
\lambda |t| \quad & {\rm for~} |t| \le \lambda,\\
-\frac{t^2-2a\lambda|t|+\lambda^2}{2(a-1)} \quad & {\rm for~} \lambda < |t| \le a\lambda,\\
\frac{(a+1)\lambda^2}{2} \quad & {\rm for~}|t| > a\lambda,
\end{cases}$$ where $a>2$ is a fixed parameter. The SCAD penalty is also $(\mu, \delta)$-amenable with $\mu=\frac{1}{a-1}$ and $\delta = a$.\
- The [**MCP**]{} penalty, $$\rho_{\lambda}(t)={\rm sign}(t) \lambda \int_{0}^{|t|}\left(1-\frac{z}{\lambda b}\right)_+ dz,$$ where $b>0$ is a fixed parameter. The MCP penalty is also $(\mu, \delta)$-amenable with $\mu=\frac{1}{b}$ and $\delta=b$.
It has been shown that the folded concave penalty, such as SCAD or MCP, possesses better variable selection properties than the convex penalty like the Lasso.
Loss functions {#sec2-3}
--------------
From the linear model setting in Section \[sec2-1\], we know $E(y_i | \bx_i)=\bx_i^T\bb^*$. We are interested in finding a well-behaved mean-regression estimator of $\bb^*$. Since we consider a general setting discussed in Section \[sec2-1\], we wish to study the empirical loss function $\mL_{\alpha,n}$ that are robust to outliers and/or heavy-tailed distribution. Let $l_{\alpha}: \RR \mapsto \RR $ denote a residual function, or a loss function, defined on each observation pair ($\bx_i, y_i$). The corresponding empirical loss function for (\[eq:estimator\]) is then given by $$\label{eq:M-estimator}
\mL_{\alpha,n}(\bb) = \frac{1}{n} \sum_{i=1}^{n}l_{\alpha}(y_i - \bx_i^T\bb).$$ With a well chosen non-quadratic function $l_{\alpha}$, the penalized mean regression estimators from (\[eq:estimator\]) can be robust to outliers or heavy-tailed distribution in the additive noise term $\epsilon_i$. However, it may generate bias to the conditional mean when the conditional distribution of $\epsilon_i$ is not symmetric.
To reduce such bias induced by the non-quadratic loss, we consider a family of loss function with flexible robustness and diverging parameters satisfying (\[population condition\]) to approximate the traditional quadratic loss. In particular, we require the following approximation: $$\label{eq:approximate loss}
\textbf{Approximation Equation:} \lim_{\alpha \to \infty} l_{\alpha}(u)=\frac{1}{2} u^2, \quad \forall u \in \RR.$$ The empirical loss function satisfy (\[eq:approximate loss\]) is called a robust approximated quadratic loss function. The following approximations take the Huber loss, Tukey’s biweight loss and Cauchy loss to robustly approximate the quadratic loss functions:
- [**Huber Approximation**]{} $$l_\alpha(u)=\begin{cases}
\frac{u^2}{2} & \text{if } |u| \le \alpha, \\
\alpha |u| - \frac{\alpha^2}{2} & \text{if } |u| \ge \alpha.
\end{cases}$$
- [**Tukey’s biweight Approximation**]{} $$l_\alpha(u)=\begin{cases}
\frac{\alpha^2}{6}(1-(1-\frac{u^2}{\alpha^2})^3) & \text{if } |u| \le \alpha, \\
\frac{\alpha^2}{6} & \text{if } |u| \ge \alpha.
\end{cases}$$
- [**Cauchy Approximation**]{} $$l_\alpha(u)=\frac{\alpha^2}{2}\log(1+\frac{u^2}{\alpha^2}).$$
It is straight forward to verify that all above three loss functions satisfy equation (\[eq:approximate loss\]). In addition, the Tukey’s biweight loss and Cauchy loss produce re-descending $M$-estimators. In the robust regression literature, we call an $M$-estimator re-descending if there exists $u_0>0$ such that $|l_{\alpha}'(u)|=0$ or decrease to 0 smoothly, for all $|u| \ge u_0$. In that case, large residuals can be downweighted. See more discussions in [@muller2004redescending] and [@shevlyakov2008redescending].
PRAM estimators and the extensions {#sec2-4}
----------------------------------
A class of PRAM estimators takes the form: $$\label{eq:PRAM-estimator}
\hat{\bb} \in {\operatornamewithlimits{argmin}}_{\|\bb\|_1\le R} \left\{\frac{1}{n} \sum_{i=1}^{n}l_{\alpha}(y_i-\bx_i^T \bb)+\rho_\lambda(\bb)\right\},$$ where the penalty function $\rho_\lambda$ satisfies Assumption \[as:penalty\], the loss function $l_{\alpha}$ is a scalar function satisfying equation and $\alpha >0$ is a robustness parameter which is allowed to diverge.
Whereas a PRAM estimator in equation (\[eq:PRAM-estimator\]) takes into account the contamination or heavy-tailed distribution in asymmetric additive error, a single outlier in $\bx_i$ may still cause the corresponding estimator to perform arbitrarily badly. We downweight large values of $\bx_i$ and extend the class of PRAM estimators to $$\label{eq:GPRAM-estimator}
\hat{\bb} \in {\operatornamewithlimits{argmin}}_{\|\bb\|_1\le R} \left\{
\frac{1}{n} \sum_{i=1}^{n} \frac{w(\bx_i)}{v(\bx_i)}l_{\alpha}((y_i-\bx_i^T \bb)v(\bx_i))+\rho_\lambda(\bb)\right
\},$$ where $w,v$ are weight functions mapping from $\RR^p$ to $\RR^+$. When $w \equiv v\equiv 1$, (\[eq:GPRAM-estimator\]) is reduced to the PRAM class defined in (\[eq:PRAM-estimator\]). A few options for choosing the weight functions can be found in [@mallows1975some], [@hill1977robust], [@merrill1971bad]. Such a downweighting strategy was also adopted in [@loh2017statistical].
For the rest of the paper, we specify the PRAM estimator with the Huber approximation, Tukey’s biweight approximation and Cauchy approximation as the HA-type, TA-type and CA-type PRAM estimator, respectively. In particular, we also specify a PRAM estimator using a re-descending loss function approximation (e.g. Tukey’s biweight approximation and Cauchy approximation) a re-descending PRAM estimator. Additionally, we classify a PRAM estimator with the Lasso penalty and MCP penalty as the Lasso-type and MCP-type PRAM estimator correspondingly.
Statistical Properties {#sec3}
======================
Estimation Consistency {#sec3-1}
----------------------
As in (\[eq:GPRAM-estimator\]), we consider a class of PRAM estimators with the loss function in a general setting, $$\label{eq:gloss}
\mL_{\alpha,n}(\bb)=\frac{1}{n} \sum_{i=1}^{n} \frac{w(\bx_i)}{v(\bx_i)}l_{\alpha}((y_i-\bx_i^T \bb)v(\bx_i)).$$ To obtain the estimation consistency, we make the following additional assumptions on $l_\alpha$.
\[as:loss\] $l_\alpha: \RR \mapsto \RR$ is a scalar function for $\alpha > 0$ with the existence of the first derivative $l_\alpha'$ everywhere and the second derivative $l_\alpha''$ almost everywhere. In addition,
1. there exists a constant $0 < k_1 < \infty$ such that $|l_\alpha'(u)| \le k_1 \alpha$ for all $ u \in \RR$;
2. for all $\alpha >0$, $l_{\alpha}'(0)=0$ and $l_{\alpha}'$ is Lipschitz such that $|l_{\alpha}'(x) - l_{\alpha}'(y)| \le k_2|x-y|$ for all $x, y \in \RR$ and some $0 <k_2 <\infty$;
3. for some $k \ge 2$, there exists a constant $d_1>0$ such that $|1-l_{\alpha}''(u)| \le d_1|u|^k\alpha^{-k}$ for almost all $|u| \le \alpha$.
Note that Assumption \[as:loss\](i) indicates that the magnitude of $l_\alpha'$ is bounded from above at the same rate of $\alpha$ so that the PRAM estimator can achieve robustness. Assumption \[as:loss\](ii) implies $|l_\alpha'(u)| \le k_2|u|$ for all $u \in \RR$ and $|l_{\alpha}''(u)| \le k_2 $ for almost every $u \in \RR$. In particular, the loss functions we study in this paper actually satisfy Assumption \[as:loss\](ii) with $k_2=1$, showing that $l_{\alpha}$ is bounded by the quadratic loss function $u^2/2$ for any $\alpha$. Assumption \[as:loss\](iii) indicates that for almost all $u \in \RR$, $l''_\alpha$ converges point-wisely to 1 with at least the order of $\alpha^{-k}$ for $k\ge 2$.
The above assumptions cover a wide range of loss functions, including the Huber loss, Hampel loss, Tukey’s biweight loss and Cauchy loss.
By some simple math, we can show that $\lim_{\alpha \to \infty} l'_{\alpha}(u)= u$ for all $u\in \RR$ based on Assumption \[as:loss\]. Suppose in addition that $l_\alpha(0)=0$, we can further obtain the approximation equation (\[eq:approximate loss\]), indicating that Assumption 2 alone gives sufficient conditions for $l_{\alpha}$ to approximate the quadratic loss.
By dominated convergence theorem, we have $$\begin{split}
\lim_{\alpha \to \infty} E[\nabla \mL_{\alpha, n}(\bb^*)] &= \lim_{\alpha \to \infty} E[w(\bx_i)\bx_il_{\alpha}'(\epsilon_i v(\bx_i))] \\
& = E[w(\bx_i)\bx_i (\epsilon_i v(\bx_i))] = E[w(\bx_i)\bx_iE(\epsilon_i\mid \bx_i) v(\bx_i))]=\b0 .
\end{split}$$ So under Assumption \[as:loss\], we have $\lim_{\alpha \to \infty} E[\nabla \mL_{\alpha, n}(\bb^*)]=\b0$ and thus it allows the random error to be heterogeneous and/or asymmetric.
We now make some weak assumptions on both random error $\epsilon$ and covariate vector $\bx$ for the investigation of the approximation error.
\[as:x\] For $w(\bx)$ and $v(\bx)$ given in , the random error $\epsilon$ with $E[\epsilon \mid \bx]=0$ and covariate vector $\bx$ with $E[\bx]=\b0$ satisfy:
1. $E[E(|\epsilon|^k \mid \bx)v(\bx)^k]^2 \le M_{k} < \infty$, for $k \ge 2$ in Assumption 2(iii);
2. $\sup_{\|u\|_2=1} E[v(\bx)\bx^Tu]^{2k}=q_k < \infty$, for $k \ge 2$ in Assumption 2(iii);
3. $0<k_l<\lambda_{min}(E[w(\bx)v(\bx)\bx\bx^T])$ and $\lambda_{max}(E[w(\bx)^2\bx\bx^T]) < k_u$;
4. for any $\pmb \nu \in \RR^p$, $w(\bx)\bx^T\pmb \nu$ is sub-Gaussian with parameter at most $k_0^2\|\pmb \nu\|_2^2$.
Note that condition (i) requires only the existence of second conditional moment of $\epsilon$, indicating that this condition is independent of the distribution of $\epsilon$ itself and can hold for heavy-tailed or skewed distribution. If $w(\bx) \equiv v(\bx)\equiv 1$, the conditions(ii) and (iv) hold when $\bx_i^T\pmb \nu$ is sub-Gaussian for any $\pmb \nu \in \RR^p$. In this case, Assumption \[as:x\] becomes conditions (C1-C3) in @fan2017estimation. If covariate $\bx$ is contaminated or heavy-tailed distributed, conditions(ii)-(iv) nonetheless holds with some proper choices of $w(\bx)$ and $v(\bx)$ (e.g. $w(\bx)\bx^T \pmb \nu$ is bounded for any $\pmb \nu \in \RR^p$), which potentially relaxes the sub-Gaussian assumption on $\bx$.
Let $\bb_{\alpha}^*$ be a local non-penalized population minimizer under the PRAM loss, $$\label{eq:beta_alpha_star}
\bb_{\alpha}^* \in {\operatornamewithlimits{argmin}}_{\|\bb - \bb^*\|_2 \le R_0} \left \{ E \left[\frac{w(\bx)}{v(\bx)}l_{\alpha}((y-\bx^T\bb)v(\bx))\right] \right \},$$ for some $0 < R_0< \infty$. Note that $\bb_{\alpha}^*$ is a local minimizer of (\[eq:beta\_alpha\_star\]) within a neighborhood of $\bb^*$. If the regularization parameter $\lambda$ in equation (\[eq:GPRAM-estimator\]) converges to 0 sufficiently fast, then $\hat{\bb}$ is a natural unpenalized $M$-estimator of $\bb_{\alpha}^*$ for any $\alpha>0$. Whereas $\bb_{\alpha}^*$ differs from $\bb^*$ in general, $\bb_{\alpha}^*$ is expected to converge to $\bb^*$ when $\alpha\to \infty$, due to the approximation equation (\[eq:approximate loss\]) for PRAM. The rate of the approximation error $\|\bb_{\alpha}^* - \bb^*\|_2$ is established in Theorem 1.
\[Tm:1\] Under the Assumption \[as:loss\] and \[as:x\], there exists a universal positive constant $C_1$, such that $\|\bb_{\alpha}^*-\bb^*\|_2 \le 2^k C_1k_l^{-1}\sqrt{k_u}(\sqrt{M_k}+R_0^k\sqrt{q_k}){\alpha}^{1-k}$. Here $k$, $k_l$, $k_u$, $M_k$, $q_k$ appear in Assumption \[as:loss\], \[as:x\] and $R_0$ appears in (\[eq:beta\_alpha\_star\]).
Theorem \[Tm:1\] gives an upper bound of the approximation error between the true parameter vector and the non-penalized PRAM population minimizer. The approximation error vanishes when $\alpha\to \infty$. It vanishes faster if a higher moment of $\epsilon | \bx$ exists. In fact, Theorem 1 demonstrates that the approximation of the loss function $l_\alpha$ to the quadratic loss helps to reduce the bias induced by the asymmetry on $\epsilon$. If we let $l_{\alpha}$ in equation (\[eq:gloss\]) be the Huber loss and $w(\bx) \equiv v(\bx)\equiv 1$, Theorem 1 gives the upper bound of the approximation error studied in @fan2017estimation.
In order to obtain the estimation consistency for the PRAM estimator in (\[eq:GPRAM-estimator\]), we also require the loss function $\mL_{\alpha,n}$ to satisfy the following uniform Restricted Strong Convexity (RSC) condition.
\[as:RSC\] There exist $\gamma$, $\tau$, $\alpha_0>0$ and a radius $r>0$ such that for all $\alpha \ge \alpha_0$, the loss function $\mL_{\alpha,n}$ in satisfies $$\label{eq:URSC}
\langle \nabla \mL_{\alpha,n}(\bb_1)- \nabla \mL_{\alpha,n}(\bb_2), \bb_1-\bb_2 \rangle \ge \gamma \|\bb_1-\bb_2\|_2^2 - \tau \frac{\log p}{n}\|\bb_1 - \bb_2\|_1^2,$$ where $\bb_j \in \RR^p$ such that $\|\bb_j-\bb^*\|_2\le r$ for $j=1, 2$.
Note that the uniform RSC assumption is only imposed on $\mL_{\alpha,n}$ inside the ball of radius $r$ centered at $\bb^*$. Thus the loss function used for robust regression can be wildly nonconvex while it is away from the origin. The radius $r$ essentially specifies a local ball centered around $\bb^*$ in which stationary points of the PRAM estimator are well-behaved.
In [@loh2013regularized] and [@loh2017statistical], the RSC condition were imposed on a specific loss function. Although Assumption \[as:RSC\] requires that the RSC condition is satisfied uniformly over a family of loss functions generated from a range of $\alpha$, this assumption is in fact not stronger: Assumption \[as:RSC\] holds naturally if there exists $\alpha_0>0$ such that $\mL_{\alpha_0,n}$ satisfies Assumption 2 and inequality (\[eq:URSC\]) for some $\gamma, \tau >0$. We further establish the uniform RSC condition in Appendix.
We present our main estimation consistency result on the PRAM estimator in the following Theorem \[Tm: est\].
\[Tm: est\] Suppose the random error and covariates satisfy Assumption \[as:x\] and $\mL_{\alpha,n}$ in satisfies Assumption \[as:loss\]. Then we have the following results.
1. If $\max\{(\frac{2d}{R_0})^{\frac{1}{k-1}},C_2 (\frac{n}{\log p})^{\frac{1}{2(k-1)}}\} \le \alpha \le C_3\sqrt{\frac{n}{\log p}}$, then with probability greater than $1-2\exp(-C_4\log p)$, $\mL_{\alpha,n}$ satisfies $$\label{eq:Ln-bound}
\|\nabla \mL_{\alpha,n}(\bb^*)\|_{\infty} \le C_5\sqrt{\frac{\log p}{n}}.$$
2. Suppose $\mL_{\alpha,n}$ also meets the uniform RSC condition in Assumption \[as:RSC\]. Suppose $\rho_\lambda$ is $\mu$-amenable with $\frac{3}{4}\mu < \gamma$ in Assumption \[as:penalty\]. Let $\hat{\bb}$ be a local PRAM estimator in the uniform RSC region. Then for $R \ge \|\bb^*\|_1$, $\lambda \ge \max\{ 4 \|\nabla \mL_{\alpha,n}(\bb^*)\|_\infty, 8\tau R \frac{\log p}{n}\}$ and $n \ge C_0r^{-2}k\log p$, $\hat{\bb}$ exists and satisfies the bounds $$\|\hat{\bb}-\bb^*\|_2 \le \frac{24\lambda\sqrt{s}}{4\gamma-3\mu} \text{ and } \|\hat{\bb}-\bb^*\|_1 \le \frac{96\lambda s}{4\gamma-3\mu}.$$
The statistical consistency result of Theorem \[Tm: est\] holds even when the random errors lack of symmetry and homogeneity, and the regressors lack of sub-Gaussian assumption. It shows that with high probability one can choose $\lambda=\mathcal{O}\left (\sqrt{\frac{\log p}{n}} \right)$ such that $\|\hat{\bb}-\bb^*\|_2=\mathcal{O}_p \left (\sqrt{\frac{s\log p}{n}} \right)$ and $\|\hat{\bb}-\bb^*\|_1=\mathcal{O}_p \left (\sqrt{\frac{s^2 \log p}{n}} \right)$. Hence, it guarantees that when the parameter $\alpha$ diverges at a certain rate, a local PRAM estimator within the local region of radius $r$ is statistically consistent at the minimax rate enjoyed by the LS-Lasso. The rate range of $\alpha$ stated in Theorem \[Tm: est\](i) in fact reveals that in the presence of asymmetric and heavy-tailed/contaminated data, $\alpha$ should diverge faster enough, for example, faster than $\bo \left ((\frac{n}{\log p})^{\frac{1}{2(k-1)}} \right )$, to reduce the bias sufficiently but meanwhile not too fast, for instance, slower than $\bo \left ((\frac{n}{\log p})^\frac{1}{2} \right)$, in order to preserve certain robustness of a PRAM estimator. The existence of a higher moment of $\epsilon|\bx$ (a larger $k$) actually allows $\alpha$ to diverge at a lower rate.
The proof of Theorem \[Tm: est\] in Appendix reveals that the estimation consistency result also holds for the local stationary points in program (\[eq:estimator\]). Here $\tilde{\bb}$ is a stationary point of the optimization in (\[eq:estimator\]) if
$$\langle \nabla \mL_{\alpha, n}(\tilde{\bb}) + \nabla \rho_{\lambda}(\tilde{\bb}), \bb - \tilde{\bb} \rangle \ge 0,$$
for all feasible $\bb$ in a neighbour of $\tilde{\bb}$. Note that stationary points include both the interior local maxima as well as all local and global minima. Hence Theorem \[Tm: est\] guarantees that all stationary points within the ball of radius $r$ centered at $\bb^*$ have local statistical consistency at the minimax rate enjoyed by the LS-Lasso.
Oracle Properties
-----------------
In this section, we establish the oracle properties for the PRAM estimators in program (\[eq:GPRAM-estimator\]). We first define the local oracle estimator as $$\label{eq:oracle}
\hat{\bb}_{S}^{\mathcal{O}}= {\operatornamewithlimits{argmin}}_{\bb \in \RR^S:\|\bb - \bb^*\|_2 \le r} \left\{ \mL_{\alpha, n} (\bb)
\right \},$$ where we set $S=\{j: \beta^*_j \ne 0 \}$. Let $\beta^*_{\min}= \min_{j \in S} |\beta^*_j|$ denote a minimum signal strength on $\bb^*$. Our oracle result shows that when the penalty $\rho_\lambda$ is $(\mu, \delta)$-amenable and the assumptions stated earlier are satisfied, those stationary points of the PRAM estimator in program (\[eq:GPRAM-estimator\]) within the local neighborhood of $\bb^*$ are actually unique and agree with the oracle estimator (\[eq:oracle\]), as stated in the following theorem.
\[Tm: oracle\] Suppose the penalty $\rho_{\lambda}$ is $(\mu, \delta)$-amenable and conditions in Theorem \[Tm: est\] hold. Suppose in addition that $v(\bx)\bx_j$ is sub-Gaussian for all $j=1,\cdots,p$, $\|\bb^*\|_1 \le \frac{R}{2}$ for some $R>\frac{192\lambda s}{4\gamma-3\mu}$, $\beta^*_{\min} \ge C_6\sqrt{\frac{\log p}{ns}} + \delta \lambda$, and $n \ge C_{01} s \log p$ for a sufficiently large constant $C_{01}$. Suppose $\alpha$ satisfies $C_{22} \left (\frac{ns^2}{\log p} \right) ^{\frac{1}{2(k-1)}} \le \alpha \le C_3 \sqrt{\frac{n}{\log p}}$ and $s^2 = \bo \left ((\frac{n}{\log p})^{k-2} \right)$. Let $\tilde{\bb}$ be a stationary point of program (\[eq:GPRAM-estimator\]) in the uniform RSC region. Then with probability at least $1-
C_8\exp(-C_{41} \frac{\log p}{s^2})$, $\tilde{\bb}$ satisfies $ supp (\tilde{\bb})\subseteq S$ and $\tilde{\bb}_S = \hat{\bb}_{S}^{\mathcal{O}}$ .
Two most often considered $(\mu, \delta)$-amenable penalties are SCAD and MCP, as introduced in Section \[sec2-2\]. Since the Lasso penalty is not $(\mu, \delta)$-amenable, the Lasso-type PRAM estimator does not have the oracle properties. In Theorem \[Tm: oracle\], the lower bound rate of $\alpha$ is higher than the one in Theorem \[Tm: est\], with a ratio $\bo \left (s^{\frac{1}{k-1}} \right)$. Thus to have the oracle properties, $s$ cannot grow with $n$ too fast. In particular, $ s =\bo \left ((\frac{n}{\log p})^{\frac{k-2}{2}} \right)$ for $k \ge 2$. Note that the feasibility condition $\|\bb^*\| \le \frac{R}{2}$ instead of $R$ in Theorem 2, is for the technical proof. It means that (\[eq:GPRAM-estimator\]) is optimized in a larger neighborhood of $\bb^*$ in order to cover $(\hat{\bb}^\bo_S, \b0_{S^c})$ such that $\|\hat{\bb}^\bo_S - \bb^*\|_2 < r$.
The condition $s^2 = \bo \left ((\frac{n}{\log p})^{k-2} \right)$ shows that, if the number of non-zero parameters $s$ is finite, Theorem \[Tm: oracle\] requires only the existence of second moment of $\epsilon|\bx$ $(k=2)$; if we also allow $s$ to grow with sample size $n$, the oracle result holds when at least the third moment of $\epsilon|\bx$ exists $(k \ge 3)$.
Since $ \hat{\bb}_{S}^{\mathcal{O}}$ is essentially an $s$-dimensional M-estimator, to analyze the asymptotic behavior of $\tilde{\bb}$ and $\hat{\bb}_{S}$, Theorem \[Tm: oracle\] allows us to apply previous results in the literature concerning the asymptotic distribution of low-dimensional M-estimators. In particular, [@he2000parameters] established the asymptotic normality for a fairly general class of convex M-estimators where $p$ is allowed to grow with $n$. Although the loss function we considered may be highly nonconvex, the restricted program in (\[eq:oracle\]) can still be convex under the uniform RSC condition. Hence by applying our Theorem \[Tm: oracle\] and the standard results for M-estimators with a diverging number of parameters in [@he2000parameters], we can obtain the following theorem concerning the asymptotic normality of any stationary point of the program (\[eq:GPRAM-estimator\]). For the sake of simplicity, we only provide the result under $w(\bx) \equiv v(\bx)\equiv 1$. The result of a weighted PRAM can be derived accordingly.
\[Tm: normality\] Suppose conditions in Theorem \[Tm: oracle\] hold and the loss function $\mL_{\alpha, n}$ given in (\[eq:gloss\]) is twice differentiable within the $\ell_2$-ball of radius r around $\bb^*$. Suppose for all $\alpha >0$, $l_{\alpha}''$ is Lipschitz such that $|l_{\alpha}''(x) - l_{\alpha}''(y)| \le k_3|x-y|$ for all $x, y \in \RR$ and some $0< k_3 <\infty$. Suppose in addition that $\alpha > (2C_9/k_l)^{1/k}$ and $\alpha ^{1-k} = o(n^{-1/2})$. Let $\tilde{\bb}$ be a stationary point of program (\[eq:GPRAM-estimator\]) in the uniform RSC region. If $\frac{s \log^3 s}{n} \rightarrow 0$, then $\|\tilde{\bb} - \bb^*\|_2 = \bo_p \left (\sqrt{\frac{s}{n}} \right)$. If $\frac{s^2 \log s}{n} \rightarrow 0$, then for any $\bv \in \RR^p$, we have $$\frac{\sqrt{n}}{\sigma_{\bv}} \cdot \bv^T(\tilde{\bb} - \bb^*) \xrightarrow{d} N(0,1),$$ where $$\sigma_{\bv}^2 = \bv^T_S E[(\nabla^2\mL_{\alpha, n}(\bb^*))_{SS}]^{-1} Var(l_\alpha'(\epsilon_i)(\bx_i)_S) E[(\nabla^2\mL_{\alpha, n}(\bb^*))_{SS}]^{-1} \bv_S.$$
The condition $\alpha ^{1-k} = o(n^{-1/2})$ indicates that $\alpha$ should diverge at least faster than $n^{\frac{1}{2(k-1)}}$, in addition to the rate stated in Theorem \[Tm: oracle\]. Together with the result in Theorem \[Tm:1\], it means that the approximation error $\|\bb_\alpha^* -\bb^*\|_2$ should vanish at a rate of $o(n^{-1/2})$, in order to obtain the asymptotic normality properties. Note that the condition $\alpha > (2C_9/k_l)^{1/k}$ is required to guarantee the invertibility of matrix $E[(\nabla^2\mL_{\alpha, n}(\bb^*))_{SS}]$.
To further understand the condition $\alpha ^{1-k} = o(n^{-1/2})$, we take $\alpha = \bo \left (\sqrt{\frac{n}{\log p}} \right)$ as an example, the fastest divergent rate indicated in Theorem \[Tm: oracle\]. Then the condition requires $ \frac{\log p}{n}\cdot n^{\frac{1}{k-1}} \rightarrow 0$. Thus $\frac{1}{k-1} < 1$ and then $k>2$. Therefore the asymptotic normality result holds only when at least the third moment of $\epsilon|\bx$ exists. In particular, when $k=3$, we obtain $n^{-\frac{1}{2}} \log p \rightarrow 0$.
Implementation of the PRAM estimators {#sec:4}
=====================================
Note that the optimization in (\[eq:estimator\]) may not be a convex optimization problem since we allow both loss function $\mL_{\alpha,n}$ and $\rho_\lambda$ to be non-convex. To obtain the corresponding stationary point, we use the composite gradient descend algorithm [@nesterov2013gradient]. Denoting $q_\lambda(\bb) = \lambda\|\bb\|_1-\rho_\lambda(\bb)$ and $\bar{L}_{\alpha,n}(\bb)=\mL_{\alpha,n}(\bb)-q_\lambda(\bb)$, we can rewrite the program as $$\hat{\bb} \in {\operatornamewithlimits{argmin}}_{\|\bb\|_1 \le R} \left\{ \bar{L}_{\alpha,n}(\bb) + \lambda\|\bb\|\right\}.$$ Then the composition gradient iteration is given by $$\label{eq:interate}
\bb^{t+1} \in {\operatornamewithlimits{argmin}}_{\|\bb\|_1 \le R} \left\{ \frac{1}{2} \| \bb - (\bb^t - \eta \nabla \bar{L}_{\alpha,n}(\bb^t))\|_2^2 + \eta\lambda \|\bb\|_1 \right\},$$ where $\eta >0$ is the step size for the update and can be determined by the backtracking line search method described in [@nesterov2013gradient]. A simple calculation shows that the iteration in (\[eq:interate\]) takes the form $$\bb^{t+1} = S_{\eta\lambda } \left(\bb^t - \eta\nabla \bar{L}_{\alpha,n}(\bb^t)\right),$$ where $S_{\eta \lambda}(\cdot)$ is the soft-thresholding operator defined as $$[ S_{\eta\lambda }(\bb)]_j = {\rm sign}(\beta_j) \left(|\beta_j| - \eta\lambda\right)_+.$$ We further adopt the two-step procedure discussed in @loh2017statistical to guarantee the convergence to a stationary point for the non-convex optimization problem:
1. Run the composite gradient descent using the convex Huber loss function with the convex Lasso penalty to get an initial PRAM estimator.
2. Run the composite gradient descent on the desired high-dimensional PRAM estimator using the initial PRAM estimator from Step 1.
For tuning parameters selection, the optimal values of $\alpha$ and $\lambda$ are chosen by a two-dimensional grid search using the cross-validation. In Particular, the searching grid is formed by partitioning a rectangle uniformly in the scale of ($\alpha, \log(\lambda)$). The optimal values are found by the combination that minimizes the cross-validated trimmed mean squared prediction error.
Simulation Studies {#sec:simulation}
==================
In this section, we assess the performance of the PRAM estimators by considering different types of loss and penalty functions through various models. The simulation setting is similar to the one in @fan2017estimation. The data is generated from the following model $$y_i=\bx_i^T\bb^*+\epsilon_i.$$ We choose the true regression coefficient vector as $\bb^*=(\textbf{3}_5^T, \textbf{2}_5^T, \textbf{1.5}_5^T, \textbf{0}_{p-15}^T)^T$, where the first 15 elements consist of 5 numbers of $3$, $2$, $1.5$ receptively and the rest are 0. In all simulation settings, we let $\textit{n=100}$ and $\textit{p=500}$.
\[ex:1\] [**(Homogeneous case)**]{} The covariates vector $\bx_i$s are generated from a multivariate normal distribution with mean $\bf 0$ and covariance $\bI_p$ independently. The random errors $\epsilon_i=e_i-E[e_i]$, where $e_i$ are generated independently from the following 5 scenarios:
1. $N(0,4)$: Normal with mean 0 and variance 4;
2. $\sqrt{2}t_3$: $\sqrt{2}$ times the $t$-distribution with degrees of freedom $3$;
3. MixN: Equal mixture of Normal distributions N(-1, 4) and N(8, 1);
4. LogNormal: Log-normal distribution such that $ e_i = \exp(1.3z_i)$, where $z_i\sim N(0,1)$.
5. Weibull: Weibull distribution with the shape parameter $ 0.3$ and the scale parameter $0.15$.
We consider three types of loss functions equipped with diverging parameters (the Huber loss, Tukey’s biweight loss and Cauchy loss) and two types of penalty functions (the Lasso and MCP penalties). Thus it produces 6 different PRAM estimators: HA-Lasso, TA-Lasso, CA-Lasso, HA-MCP, TA-MCP and CA-MCP. Note the HA-Lasso becomes the RA-Lasso estimator in @fan2017estimation, where the HA-Lasso has been demonstrated to perform better than the Lasso and R-Lasso, especially when the errors were asymmetric and heavy-tailed (LogNormal and Weibull). Thus in our simulation we skip those comparisons and only evaluate the performance of all those 6 PRAM estimators. Their performances on both mean estimation and variable selection under the five scenarios were reported by the following five measurements:
1. $L_2$ error, which is defined as $\|\hat{\bb} - \bb^*\|_2$.
2. $L_1$ error, which is defined as $\|\hat{\bb} - \bb^*\|_1$.
3. Model size (MS), the average number of selected covariates.
4. False positives rate (FPR), the percent of selected but unimportant covariates: $$\label{eq:FPR}
FPR=\frac{|\hat{S} \bigcap S^c|}{|S^c|} \times 100\%.$$
5. False negatives rate (FNR), the percent of non-selected but important covariates: $$FNR=\frac{|\hat{S}^c \bigcap S|}{|S|} \times 100\%.$$
Here $\hat{S}=\{j: \hat{\beta}_j \ne 0 \}$ and $S=\{j: \beta^*_j \ne 0 \}$. The model considered in Example \[ex:1\] is homogeneous, in which the error distribution is independent of covariate $\bx$. We also assess the performance of PRAM estimators under heteroscedastic model in the next example.
\[ex:2\] [**(Heteroscedastic case)**]{} We generate the data from $$y_i=\bx_i^T\bb^*+c^{-1}(\bx_i^T \bb^*)^2 \epsilon_i,$$ where the constant $c=\sqrt{3}\|\bb^*\|_2^2$ makes $E[c^{-1}(\bx_i^T \bb^*)^2]^2=1$. We also consider $\bx_i \sim N(\b0, \bI_p)$ and generate the random error $\epsilon$ from the same five scenarios described in Example \[ex:1\].
Finally, we design a simulation setting to evaluate the performance of the generalized PRAM estimators under weaker distribution assumptions on the covariates.
\[ex:3\] [**(Non-Gaussian $\bx$ case)**]{} Similar to Example \[ex:1\], except that the covariate $\bx$ in $20\%$ of observations are first generated from independent chi-square variables with 10 degrees of freedom, and then recentered to have mean zero.
For all three examples described above, we run 100 simulations for each scenario. In Example \[ex:3\], we consider the generalized PRAM estimators with $v(\bx) \equiv 1$ and $w(\bx)=\min\left\{1,\frac{4} {\|\bx\|_{\infty}}\right\}$. For all six PRAM estimators, tuning parameters $\lambda$ and $\alpha$ are chosen optimally by 10-fold cross-validation, with $\alpha$ ranges in ($0.1\sqrt{\frac{n}{\log p}}, 10\sqrt{\frac{n}{\log p}}$) and $\lambda$ ranges in ($0.01\sqrt{\frac{\log p}{n}}, 2.5\sqrt{\frac{\log p}{n}}$). These ranges are motivated from Theorem \[Tm: est\]. The mean values out of $100$ iterations (with standard errors in parentheses) are reported in Table \[tb:homo\], \[tb:heter\], \[tb:contaminatex\], respectively.
We have two findings based on results in Table \[tb:homo\] and \[tb:heter\]. Firstly, all the MCP-type PRAM estimators largely outperform the Lasso-type estimators in all the measurements, rendering satisfactory finite sample performances under different settings. This is consistent with the oracle property of the PRAM estimators using a proper non-convex penalty stated in Theorem \[Tm: oracle\]. Secondly, for estimators with the same penalty, although all estimators perform comparably for light-tailed settings ($N(0,4)$ and MixN), the TA-type and CA-type PRAM estimators outperform the HA-type estimators using the same penalty in heavy-tailed settings ($\sqrt{2}t_3$, LogNormal and Weibull). This is actually not surprising due to the following two facts: (1) re-descending M-estimators can achieve the minimax variance sensitivity under certain global minimax criterion [@shevlyakov2008redescending]; (2) the HA-Lasso estimation is used as the initial in the optimization process of TA-type and CA-type PRAM estimators. Note that the error terms $c^{-1}(\bx_i^T \bb^*)^2 \epsilon_i$ in the heteroscedastic model have the same variance as those in the homogeneous model, however, their distribution possess heavier tails. Hence in the heteroscedastic model, except for a few errors being far away on tail, most of the others get even closer to the center. This fact explains why the performances in Table \[tb:heter\] are consistently better than those in Table \[tb:homo\].
In Example \[ex:3\], we only report results from the MCP-type PRAM estimators, since they have been shown to perform better than the Lasso-type estimators. In the homogeneous model with non-Gaussian covariates, Table \[tb:contaminatex\] clearly indicates that the PRAM estimators with well chosen $w(\bx)$ perform better in all cases than those PRAM with $w(\bx)=1$. In addition, among those three weighted PRAM estimators, the weighted TA-MCP (WTA-MCP) and the weighted CA-MCP (WCA-MCP) again show advantages over the weighted HA-MCP (WHA-MCP) when the errors are heavy-tailed, which is consistent with the findings obtained in Example \[ex:1\] and \[ex:2\]. In conclusion, the PRAM estimator with a folded concave penalty (e.g. MCP penalty) render promising performances in different settings, which is consistent with our theoretical results. Our simulation study also shed some lights on how to implement robust high-dimensional M-estimators for real applications: when the data are strongly heavy-tailed or contaminated, regardless of asymmetry and/or heteroschedasticity, a re-descending PRAM estimator with a concave penalty yields better performance than a convex PRAM estimator in practice.
Real Data Example
=================
In this section, we use the NCI-60 data, a gene expression data set collected from Affymetrix HG-U133A chip, to illustrate the performance of the PRAM estimators evaluated in Section [\[sec:simulation\]]{}. The NCI-60 data consists of data from 60 human cancer cell lines and can be downloaded via the web application CellMiner (http://discover.nci.nih.gov/cellminer/). The study is to predict the protein expression on the KRT18 antibody from other gene expression levels. The expression levels of the protein [*keratin*]{} 18 is known to be persistently expressed in carcinomas ([@oshima1996oncogenic]). After removing the missing data, there are $n=59$ samples with $21,944$ genes in the dataset. One can refer [@shankavaram2007transcript] for more details.
We perform some pre-screenings and keep only $p_1$ genes with largest variations and then choose $p_2$ genes out of them which are most correlated with the response variable. Here the final dataset is obtained by choosing $p_1=2000$ and $p_2=500$, yielding $n=59$ and $p=500$ for PRAM data analysis. Similar to our simulation studies, we then apply $6$ PRAM estimators to select important genes, with tuning parameters $\alpha$ and $\lambda$ chosen from the $10$-fold cross validation. Since the TA-type and CA-type PRAM estimators perform similarly, we will only report results from four methods: HA-Lasso, CA-Lasso, HA-MCP and CA-MCP.
The number of selected genes from four PRAM methods are 27 (HA-Lasso), 31 (CA-Lasso), 12 (HA-MCP), 5 (CA-MCP), respectively. HA-Lasso and CA-Lasso that selected 27 and 31 genes respectively could potentially result in over selection since the total sample size is only 59. Figure \[fig:resid-box\](a) and Figure \[fig:resid-box\](b) show that the residual distributions generated from HA-MCP and CA-MCP both had a longer tail on the left side. It indicates that PRAM estimators with non-convex penalties can be resistant to the data contamination or data’s irregularity due to the flexible robustness and nice variable selection property.
For the sake of simplicity, we only report those selected genes and corresponding coefficient estimation by HA-MCP and CA-MCP in Table \[tb:vs\]. According to our analysis, genes KRT8, NRN1 and GPX3 are selected by all four methods. It is not surprising for gene KRT8 since it has the largest correlation with the response variable and has a long history of being paired with KRT18 in cancer studies for cell death and survival, cellular growth and proliferation, organismal injury and abnormalities, and so on [@li2016silac; @Walker2007cancer]. Gene NRN1 was investigated to be involved in melanoma migration, attachment independent growth, and vascular mimicry [@bosserhoff2017neurotrophin]. Recent studies showed that gene GPX3 plays as a tumor suppressor in lung cancer cell line [@an2018gpx3] and its down-regulation is related to pathogenesis of melanoma [@chen2016hypermethylation]. Notice that gene ATP2A3 is also singled out by both HA-MCP and CA-MCP. This gene encodes the enzyme involved in calcium sequestration associated with muscular excitation and contraction, and was shown to act an important role in resveratrol anticancer activity in breast cancer cells [@izquierdo2017atp2a3]. In addition, Table \[tb:vs\] indicates that gene GPNMB is only selected by CA-MCP. The GPNMB expression was found to be associated with reduction in disease-free and overall survival in breast cancer and its over-expression had been identified in numerous cancers [@maric2013glycoprotein]. Therefore, both genes (ATP2A3 and GPNMB) deserve further study in genetics research.
To further evaluate the prediction performance of those PRAM estimators, we randomly choose 6 observations as the test set and applied four methods to the rest patients to get the coefficients estimation, then compute the prediction error on the test set. We repeat the random splitting 100 times and the boxplots of the Relative Mean Squared Prediction Error (RPE) with respect to HA-Lasso are shown in Figure \[fig:resid-box\](c). A method with $\text{RPE}<1$ indicates a bettern performance than HA-Lasso. It is clearly seen from Figure \[fig:resid-box\](c) that the MCP-type PRAM estimators have better predictions than those from the Lasso-type estimators, even though they select much smaller number of variables. In addition, Figure \[fig:resid-box\](c) together with Table \[tb:vs\] show that a re-descending PRAM estimator with a non-convex penalty (e.g. CA-MCP) is more likely to give a more parsimonious model with better prediction performance, which is consistent with the findings from our simulation studies.
Discussion
==========
The irregular settings including data asymmetry, heteroscedasticity and data contamination often exist due to the data high-dimensionality. It is very important to address these irregular settings both theoretically and numerically in high-dimensional data analysis. In this paper we have proposed a class of PRAM estimators for robust high-dimensional mean regression. The key feature of the PRAM estimators is using a family of loss functions with flexible robustness and diverging parameters to approximate the mean function produced from the traditional quadratic loss. This approximation process can reduce the bias generated by data’s irregularity in high-dimensional mean regression. The proposed framework is very general and it covers a wide range of loss functions and penalty functions, allowing both functions to be non-convex.
Theoretically, we establish statistical properties of PRAM in ultra high-dimensional settings when $p$ grows with $n$ at an almost exponential rate. In particular, we show its local estimation consistency at the minimax rate enjoyed by the LS-Lasso and further establish the oracle properties of the PRAM estimators, including both selection consistency and asymptotic normality, when an amenable penalty is used. The theoretical result is applicable for general irregular settings, including the data are contaminated by heavy-tailed distribution and/or outliers in the random errors and covariates, the random errors lack of symmetry and/or homogeneity.
One fundamental difference between our proposed PRAM estimator and the common penalized M-estimator is that we require $\lim_{\alpha \to \infty} E[\nabla \mL_{\alpha, n}(\bb^*)] =\b0$ instead of $E[\nabla \mL_{\alpha, n}(\bb^*)] =\b0$ for every $\alpha >0$. To establish the estimation consistency and the oracle properties, the divergent rate of $\alpha$ plays a crucial role. In the presence of asymmetric and heavy-tailed/contaminated data, the PRAM estimators will either not be able to reduce the bias sufficiently (when $\alpha$ diverges too slowly) or lose robustness (when $\alpha$ diverges too fast). The divergent rate of $\alpha$ stated in Theorem \[Tm: est\] and Theorem \[Tm: oracle\] actually shows us how $\alpha$ should diverge with $n$, in order to obtain a robust sparse PRAM estimator in high-dimensional mean regression under general irregular settings.
Additionally, our numerical studies show satisfactory finite sample performance of the PRAM estimators under irregular settings, which is consistent with our theoretical findings. Among all the possible choices of PRAM estimators, our numerical results also suggest to implement a re-descending PRAM estimator with a concave penalty such as TA-MCP and CA-MCP, using the HA-Lasso as the initial estimator, when the data are strongly heavy-tailed or contaminated.
Our research in this paper provides a systematic study of penalized M-estimation in high-dimensional regression data analysis. We hope this study shed some lights in future directions of research, including devising similar theoretical guarantees for estimators with grouping structures in the covariates, or study of high-dimensional models with varying coefficients under general irregular settings.
Appendix 1 {#appendix-1 .unnumbered}
==========
Establishing the uniform RSC condition {#establishing-the-uniform-rsc-condition .unnumbered}
--------------------------------------
Let $\varepsilon_T= E\left[ P\left(|\epsilon_i| \ge \frac{T}{2}|\bx \right)\right]$ be the expected tail probability. Below we establish some sufficient conditions where an unweighted $\mL_{\alpha, n}$ ($w(\bx) \equiv v(\bx)\equiv 1$) satisfies the uniform RSC condition in Assumptions \[as:RSC\] with high probability. The uniform RSC condition for weighted loss can be established accordingly.
\[Tm: RSC\] Suppose $\mL_{\alpha,n}$ satisfies Assumption \[as:loss\] and the covariate $\bx$ satisfies Assumption \[as:x\]. If s $ n \ge C_{10}s\log p$, then with probability at least $1-C_{11}\exp(-C_{12}\log p)$, the loss function $\mL_{\alpha,n}$ satisfies the Uniform RSC condition in Assumption \[as:RSC\] with $$\gamma=\frac{k_l}{32}, \quad \quad \tau=\frac{C_{13}(3+ 2k_2)^2k_0^2T_0^2}{2r^2} \quad \text{and} \quad \alpha_0=\max\{ (2d_1)^{\frac{1}{k}},1\}\cdot T_0,$$ where $T_0 >0$ is a sufficiently large constant that satisfies $$\label{ineq:RSC}
C_{14}k_0^2\left(\sqrt{\varepsilon_{T_0}}+\exp\left(-\frac{C_{15}T_0^2}{k_0^2r^2}\right) \right) < \frac{k_l}{2+4k_2}.$$
Theorem \[Tm: RSC\] guarantees that the loss function $\mL_{\alpha, n}$ satisfies the uniform RSC condition with probability converging to 1. Note that the left hand side of inequality (\[ineq:RSC\]) is monotonically decreasing on $T_0$, meaning that inequality (\[ineq:RSC\]) is always satisfied for a sufficiently large $T_0$. In addition, while keeping inequality (\[ineq:RSC\]) satisfied, a larger $T_0$ (thus larger $\alpha_0$) actually allows a larger radius $r$ of local ball around $\bb^*$ and a more contaminated distribution of $\epsilon$. Theorem $\ref{Tm: RSC}$ implies that the Huber loss, Hampel loss, Tukey’s biweight loss and Cauchy loss satisfy Assumption \[as:RSC\] with high probability.
Appendix 2 {#appendix-2 .unnumbered}
==========
Proofs {#proof .unnumbered}
------
[**Proof of Theorem \[Tm:1\]**]{}\
Let $l(x)=\frac{1}{2}x^2$. Observe that $$\begin{split}
E[\nabla \frac{w(\bx)}{v(\bx)}l((y-\bx^T\bb^*)v(\bx))]&=E[w(\bx)v(\bx)(y-\bx^T\bb^*)(-\bx)]\\
&=E[w(\bx)v(\bx)\epsilon(-\bx)]\\
&=E[E[\epsilon|\bx]w(\bx)v(\bx)(-\bx)]\\
&=\b0,
\end{split}$$ where the last equality follows from $E[\epsilon|\bx]=0$. Hence $\bb^*$ is the minimizer of $E[\frac{w(\bx)}{v(\bx)}l((y-\bx^T\bb)v(\bx))]$. Then it follows from Assumption \[as:x\](iii) that $$\label{eq:bound1}
\begin{split}
& E[\frac{w(\bx)}{v(\bx)}l((y-\bx^T\bb_{\alpha}^*)v(\bx)) -\frac{w(\bx)}{v(\bx)}l((y-\bx^T\bb^*)v(\bx))]\\
&=E\{w(\bx)v(\bx)[l(y-\bx^T\bb_{\alpha}^*)-l(y-\bx^T\bb^*)]\}\\
&=\frac{1}{2}(\bb_{\alpha}^* - \bb^*)^TE[w(\bx)v(\bx)\bx\bx^T](\bb_{\alpha}^* - \bb^*) \ge \frac{1}{2}k_l \|\bb_{\alpha}^* - \bb^*\|_2^2
\end{split}$$ Let $g_{\alpha}(x)=l(x)- l_{\alpha}(x)$. Since $\bb_{\alpha}^*$ is the minimizer of $E [\frac{w(\bx)}{v(\bx)}l_{\alpha}((y-\bx^T\bb)v(\bx))]$ within a neighbour of $\bb^*$, we have $$\label{ineq: exp_l_upper}
\begin{split}
& E[\frac{w(\bx)}{v(\bx)}l((y-\bx^T\bb_{\alpha}^*)v(\bx)) - \frac{w(\bx)}{v(\bx)}l((y-\bx^T\bb^*)v(\bx))] \\
=& E\{\frac{w(\bx)}{v(\bx)}[l((y-\bx^T\bb_{\alpha}^*)v(\bx))-l_{\alpha}((y-\bx^T\bb_{\alpha}^*)v(\bx))]\} + \\
&E\{\frac{w(\bx)}{v(\bx)}[l_{\alpha}((y-\bx^T\bb_{\alpha}^*)v(\bx))- l_{\alpha}((y-\bx^T\bb^*)v(\bx))]\}+\\
& E\{\frac{w(\bx)}{v(\bx)}[l_{\alpha}((y-\bx^T\bb^*)v(\bx))-l((y-\bx^T\bb^*)v(\bx))]\\
\le & E[\frac{w(\bx)}{v(\bx)}g_{\alpha}((y-\bx^T\bb_{\alpha}^*)v(\bx))]-E[\frac{w(\bx)}{v(\bx)}g_{\alpha}((y-\bx^T\bb^*)v(\bx))]
\end{split}$$
It follows from mean value theorem that $$\label{ineq:ga_diff}
\begin{split}
&E[\frac{w(\bx)}{v(\bx)}g_{\alpha}((y-\bx^T\bb_{\alpha}^*)v(\bx))-\frac{w(\bx)}{v(\bx)}g_{\alpha}((y-\bx^T\bb^*)v(\bx))]\\
=& E[w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*) (z-l_{\alpha}'(z))] \\
\le & E[|w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)|| z-l_{\alpha}'(z)|]
\end{split}$$ where $z=(y-\bx^T\tilde{\bb})v(\bx)$ and $\tilde{\bb}$ is a vector lying between $\bb^*$ and $\bb_{\alpha}^*$. Notice $l'_{\alpha}(0)=0$ in Assumption \[as:loss\](ii). By taking integral on each side of inequality in Assumption \[as:loss\](iii), we have $$\label{ineq:l1bound}
|u-l_{\alpha}'(u)| \le \frac{d_1}{k+1}|u|^{k+1}\alpha^{-k},$$ for all $|u| \le \alpha$. Observe that $$\label{eq:I1+I2}
\begin{split}
E[|z-l_{\alpha}'(z)||\bx] =& E[|z-l_{\alpha}'(z)| \mathbbm{1}(|z| \le \alpha)|\bx] + E[|z-l_{\alpha}'(z)| \mathbbm{1}(|z| > \alpha)|\bx]\\
=& I_1 + I_2.
\end{split}$$ From (\[ineq:l1bound\]) we have $$\label{eq:I1}
\begin{split}
I_1 =& E[|z-l_{\alpha}'(z)| \mathbbm{1}(|z| \le \alpha)|\bx]\\
\le & \frac{d_1\alpha^{-k}}{k+1} E[ |z|^{k+1}\mathbbm{1}(|z| \le \alpha)|\bx] \\
\le & \frac{d_1\alpha^{-k}}{k+1} E[ \frac{\alpha}{|z|}|z|^{k+1}|\bx] \\
= & \frac{d_1\alpha^{1-k}}{k+1} E[|z|^{k}|\bx].
\end{split}$$ Also observe that $$\label{eq:I2}
\begin{split}
I_2 = & E[|z-l_{\alpha}'(z)| \mathbbm{1}(|z| > \alpha)|\bx]\\
\le & E[|z| \mathbbm{1}(|z| > \alpha)|\bx] + E[|l_{\alpha}'(z)| \mathbbm{1}(|z| > \alpha)|\bx] \\
< & \frac{1}{\alpha^{k-1}}E[|z|^{k}|\bx] + k_1\alpha E[\mathbbm{1}(|z| > \alpha)|\bx] \\
= & \alpha^{1-k}E[|z|^{k}|\bx] + k_1 \alpha^{1-k}E[|z|^{k}|\bx] \\
= & (1+k_1)\alpha^{1-k}E[|z|^{k}|\bx],
\end{split}$$ where the second inequality follows from Assumption \[as:loss\](i). Combining (\[eq:I1+I2\]), (\[eq:I1\]) and (\[eq:I2\]), we obtain $$\label{eq: z-l'(z)1}
E[|z-l_{\alpha}'(z)||\bx] \le (\frac{d_1}{k+1} +1 + k_1)\alpha^{1-k}E[|z|^{k}|\bx] = C_1 \alpha^{1-k}E[|z|^{k}|\bx]$$ where $C_1 = \frac{d_1}{k+1} +1 + k_1$ and $k$ is the constant that stated in Assumption \[as:loss\](iii), Assumption \[as:x\](i) and 3(ii).
Combining inequalities (\[ineq: exp\_l\_upper\]), (\[ineq:ga\_diff\]) and (\[eq: z-l’(z)1\]), we obtain $$\label{eq: bound2}
\begin{split}
& E[\frac{w(\bx)}{v(\bx)}l((y-\bx^T\bb_{\alpha}^*)v(\bx)) - \frac{w(\bx)}{v(\bx)}l((y-\bx^T\bb^*)v(\bx))]\\\le& C_1\alpha^{1-k}E\{|y-\bx^T\tilde{\bb}|^{k}v(\bx)^k|w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)|\}\\
=&C_1\alpha^{1-k}E\{|\epsilon+\bx^T(\bb^*-\tilde{\bb})|^{k}v(\bx)^k|w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)|\}\\
\le & C_1(2/\alpha)^{k-1}\{ E[|\epsilon|^{k}v(\bx)^k|w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)|] +\\ &E[|\bx^T(\bb^*-\tilde{\bb})|^{k}v(\bx)^k|w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)|]\},
\end{split}$$ where the last inequality follows from Minkowski inequality. Note that $$\label{ineq:left}
\begin{split}
E[|\epsilon|^{k}v(\bx)^k|w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)|]=&E[E(|\epsilon|^{k}|\bx)v(\bx)^k|w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)|]\\
\le & \{E[E(|\epsilon|^{k}|\bx)v(\bx)^k]^2\}^{\frac{1}{2}} \{E[w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)]^2\}^{\frac{1}{2}}\\
\le &\sqrt{M_kk_u}\|\bb_{\alpha}^*-\bb^*\|_2,
\end{split}$$ where the first inequality follows from Hölder inequality and the last inequality follows from Assumption \[as:x\](i) and (iii). Observe that, $$\label{ineq:right}
\begin{split}
E[|\bx^T(\bb^*-\tilde{\bb})|^{k}v(\bx)^k|w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)|] \le &
\{E[v(\bx)\bx^T(\bb^*-\tilde{\bb})]^{2k}\}^{\frac{1}{2}} \{E[w(\bx)\bx^T(\bb_{\alpha}^*-\bb^*)]^2\}^{\frac{1}{2}}\\
\le & R_0^k\sqrt{q_{k}k_u}\|\bb_{\alpha}^*-\bb^*\|_2,
\end{split}$$ where $R_0$ is defined in (\[eq:beta\_alpha\_star\]) and the last inequality follows from Assumption \[as:x\](ii) and \[as:x\](iii). By inequalities (\[eq:bound1\]), (\[eq: bound2\]), (\[ineq:left\]), (\[ineq:right\]) we have $$\|\bb_{\alpha}^*-\bb^*\|_2 \le 2^k C_1k_l^{-1}\sqrt{k_u}(\sqrt{M_k}+R_0^k\sqrt{q_k}){\alpha}^{1-k}.$$ $\Box$\
[**Proof of Theorem \[Tm: est\]**]{}\
The gradient of $\mL_{\alpha,n}$ is $$\begin{split}
\nabla \mL_{\alpha,n}(\bb^*)=&-\frac{1}{n} \sum_{i=1}^{n} w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))\bx_i.
\end{split}$$ Recall $\bb_{\alpha}^*$ is the minimizer of $E [\frac{w(\bx)}{v(\bx)}l_{\alpha}((y-\bx^T\bb)v(\bx))]$ within a neighbour of $\bb^*$ defined in (\[eq:beta\_alpha\_star\]). When $\alpha \ge (\frac{2d}{R_0})^\frac{1}{k-1}$ where $d=2^k C_1k_l^{-1}\sqrt{k_u}(\sqrt{M_k}+R_0^k\sqrt{q_k})$, we have $\|\bb_{\alpha}^*-\bb^*\|_2 \le \frac{R_0}{2} < R_0$ under the result of Theorem \[Tm:1\]. Hence $\bb_{\alpha}^*$ is an interior point of program (\[eq:beta\_alpha\_star\]). Then we have $E[w(\bx)l_{\alpha}'((y-\bx^T\bb_{\alpha}^*)v(\bx))\bx]=\b0$. Observe that $$\label{eq:meanBound}
\begin{split}
E[w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}] = & E[w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}] - E[w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb_{\alpha}^*)v(\bx))x_{ij}]\\
\le & k_2E[|v(\bx_i)\bx_i^T(\bb_{\alpha}^*-\bb^*)||w(\bx_i)x_{ij}|]\\
\le & k_2\{E|v(\bx_i)\bx_i^T(\bb_{\alpha}^*-\bb^*)|^2\}^{\frac{1}{2}}\{E|w(\bx_i)x_{ij}|^2\}^{\frac{1}{2}}\\
\le & k_2\sqrt{q_1}\|\bb_{\alpha}^*-\bb^*\|_2\sqrt{k_0^2+d_2^2}\\
\le & d_3 \alpha^{1-k},
\end{split}$$ where $\max_{1 \le j \le p}|E[w(\bx_i)\bx_{ij}]| < d_2 <\infty$ and $d_3=2^kk_2\sqrt{q_1(k_0^2+d_2^2)k_u}C_1k_l^{-1}(\sqrt{M_k}+2^kR_0^k\sqrt{q_k})$. Note that the first inequality is from Assumption \[as:loss\](ii) and the third inequality follows from Assumption \[as:x\](ii) and (iv). And the last inequality is from Theorem \[Tm:1\].
Let $\mu_j = E[w(\bx_i)\bx_{ij}]$, $j=1,2,\dots, p$. Then we have $$\label{eq:boundwxm}
\begin{split}
E|w(\bx_i)\bx_{ij}|^m=& E|w(\bx_i)\bx_{ij}-\mu_j+\mu_j|^m\\
\le & E[2^{m-1}(|w(\bx_i)\bx_{ij}-\mu_j|^m+|\mu_j|^m)]\\
\le & 2^{m-1} [E|w(\bx_i)\bx_{ij}-\mu_j|^m + d_2^m]\\
\le & 2^{m-1} [m(\sqrt{2})^{m}k_0^m \Gamma(\frac{m}{2}) + d_2^m],
\end{split}$$ where the last inequality follows from Assumption \[as:x\](iv), by which $w(\bx_i)x_{ij}$ is sub-Gaussian hence for $m > 0$([@rivasplata2012subgaussian]) $$\label{eq:gaussian}
E|w(\bx_i)x_{ij}-\mu_j|^m \le m(\sqrt{2})^{m}k_0^m\Gamma(\frac{m}{2}).$$ Next we bound the $E[w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}]^m$ from the above. For $m \ge 2$, by Assumption \[as:loss\] and \[as:x\](i) we have $$\label{ineq:el1m}
\begin{split}
E|w(\bx_i)l_{\alpha}'(\epsilon_iv(\bx_i))x_{ij}|^m \le& E[(k_1\alpha)^{m-2}(k_2\epsilon_iv(\bx_i))^2|w(\bx_i)x_{ij}|^m]\\
\le & k_1^{m-2}\alpha^{m-2}k_2^2E[(\epsilon_iv(\bx_i))^2|w(\bx_i)x_{ij}|^m]\\
\le & k_1^{m-2}\alpha^{m-2}k_2^2\{ E[E(\epsilon_i^2|\bx_i)v(\bx_i)^2]^2\}^{1/2}\{E[(w(\bx_i)x_{ij})^{m}]^2 \}^{1/2}\\
\le & k_1^{m-2}\alpha^{m-2}k_2^2\sqrt{M_2}\{E[(w(\bx_i)x_{ij})^{m}]^2 \}^{1/2}.
\end{split}$$ By taking $m=2$ in (\[ineq:el1m\]), we have $$\label{ineq:psi_squred}
\begin{split}
E[w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}]^2 \le &k_2^2\sqrt{M_2}\{E[(w(\bx_i)x_{ij})^{2}]^2 \}^{1/2}\\
\le & k_2^2\sqrt{M_2}(128k_0^4 + 8d_2^4)^\frac{1}{2}\\
\le & d_4,
\end{split}$$ where $d_4=\sqrt{2}k_2^2\sqrt{M_2}(8k_0^2+2d_2^2) $ and the second inequality follows from (\[eq:boundwxm\]).
For $m \ge 3$, by replacing $m$ by $2m$ in (\[eq:boundwxm\]), we obtain $$\label{ineq: wx2mbound}
\begin{split}
\{E|w(\bx_i)\bx_{ij}|^{2m} \}^{\frac{1}{2}} \le & \{2^{2m-1} (2m)2^{m}k_0^{2m} \Gamma(m) + 2^{2m-1} d_2^{2m}\}^{\frac{1}{2}}\\
\le& 2^{\frac{3m}{2}} k_0^m\sqrt{m!} + 2^{m-\frac{1}{2}}d_2^m\\
=& (2^{\frac{3m}{2}}k_0^m \frac{2}{\sqrt{m!}} + \frac{2^{m+\frac{1}{2}}d_2^m}{m!})\frac{m!}{2}\\
\le & (2^{\frac{3m}{2}}k_0^m + 2^{m-1}d_2^m)\frac{m!}{2}\\
= & [(2^{\frac{3}{2}}k_0)^{m-2} \cdot (2^{\frac{3}{2}}k_0)^2 + (2d_2)^{m-2} \cdot 2d_2^2]\frac{m!}{2} \\
\le& \frac{m!}{2}(2^{\frac{3}{2}}k_0 + 2d_2)^{m-2}(8k_0^2 + 2d_2^2).
\end{split}$$
Combining inequality (\[ineq:el1m\]) and (\[ineq: wx2mbound\]), we have $$\begin{split}
E|w(\bx_i)l_{\alpha}'(\epsilon_iv(\bx_i))x_{ij}|^m \le & k_1^{m-2}\alpha^{m-2}k_2^2\sqrt{M_2}[\frac{m!}{2}(2^{\frac{3}{2}}k_0 + 2d_2)^{m-2}(8k_0^2 + 2d_2^2)]\\
< &\frac{m!}{2}(4(k_0+d_2)k_1\alpha)^{m-2}(k_2^2\sqrt{M_2}(8k_0^2+2d_2^2))\\
< &\frac{m!}{2}(4(k_0+d_2)k_1\alpha)^{m-2}d_4,
\end{split}$$ By Bernstein inequality (Proposition 2.9 of @massart2007concentration) we have $$\begin{array}{ll}
&P\left(|\frac{1}{n}\sum_{i=1}^{n}w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}-\frac{1}{n}\sum_{i=1}^{n}E[w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}]|\right.\\
&\quad\quad\quad\ge \left. \sqrt{\frac{2d_4t}{n}}+\frac{4(k_0+d_2)k_1\alpha t}{n} \right)\\
&\le 2\exp(-t).
\end{array}$$ It implies that $$\begin{array}{ll}
&P\left((|\frac{1}{n}\sum_{i=1}^{n}w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}|\right.\\
&\quad\quad\quad\ge \left.\sqrt{\frac{2d_4t}{n}}+\frac{4(k_0+d_2)k_1 \alpha t}{n}+|\frac{1}{n}\sum_{i=1}^{n}E[w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}]|\right)\\
&\le 2\exp(-t).
\end{array}$$ By the bound in (\[eq:meanBound\]), $$\label{eq:concentration}
\begin{array}{ll}
P(|\frac{1}{n}\sum_{i=1}^{n}w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}|
\ge \sqrt{\frac{2d_4t}{n}}+\frac{4(k_0+d_2)k_1\alpha t}{n}+d_3\alpha^{1-k}) \le 2\exp(-t).
\end{array}$$ Let $k_{\lambda}$ be a constant such that $2C^2d_4 < k_{\lambda}^2 $ and $ k_{\lambda} ^{\frac{k-2}{k-1}} \le \frac{C(C-8)d_4}{16(8d_3d_5^{k-2})^{\frac{1}{k-1}}(k_0+d_2)k_1}$, $C$ is a sufficiently large constant to guarantee such $k_{\lambda}$ exists and $d_5$ be an universal constant such that $\sqrt{\frac{\log p}{n}} \le d_5$. Let $\lambda_n=k_{\lambda}\sqrt{\frac{\log p}{n}}$ and $t=\frac{\lambda_n^2 n}{2C^2d_4}$. Then $$\label{ineq:lambdan_1}
\sqrt{\frac{2d_4t}{n}} = \frac{\lambda_n}{C}.$$ Consider $\alpha$ that satisfies $$\label{eq:alpha}
\left (\frac{8d_3}{\lambda_n} \right )^{\frac{1}{k-1}} \le \alpha \le \frac{C(C-8)d_4}{16(k_0+d_2)k_1\lambda_n}.$$ Note that together with $\lambda_n=k_{\lambda}\sqrt{\frac{\log p}{n}}$ we obtain $C_2 (\frac{n}{\log p})^{\frac{1}{2(k-1)}} \le \alpha \le C_3\sqrt{\frac{n}{\log p}}$, where $C_2=(\frac{8d_3}{k_\lambda})^\frac{1}{k-1}$ and $C_3=\frac{C(C-8)d_4}{16(k_0+d_2)k_1k_\lambda}$. By $\alpha \ge \left( \frac{8d_3}{\lambda_n} \right)^{\frac{1}{k-1}}$ we have $$\label{ineq:lambdan_2}
d_3\alpha^{1-k} \le \frac{\lambda_n}{8}.$$ By $\alpha \le \frac{C(C-8)d_4}{16(k_0+d_2)k_1\lambda_n}$ we have $$\frac{4(k_0+d_2)k_1\alpha t}{n} \le \frac{C(C-8)d_4t}{4 n\lambda_n } = \frac{C(C-8)d_4}{4n\lambda_n} \cdot \frac{\lambda_n^2 n}{2C^2d_4} =\frac{\lambda_n(C-8)}{8C}$$ Together with (\[ineq:lambdan\_1\]) and (\[ineq:lambdan\_2\]), we obtain $$\sqrt{\frac{2d_4t}{n}}+\frac{4(k_0+d_2)k_1\alpha t}{n}+d_3\alpha^{1-k} \le \frac{\lambda_n}{4}.$$ Hence by (\[eq:concentration\]), it gives $$\label{ineq: 1d1}
P\left (|\frac{1}{n}\sum_{i=1}^{n}w(\bx_i)l_{\alpha}'((y_i-\bx_i^T\bb^*)v(\bx_i))x_{ij}|
\ge \frac{\lambda_n}{4} \right ) \le 2\exp\left (-\frac{n\lambda_n^2}{2C^2d_4}\right ).$$ It then follows from union inequality that $$\label{ineq:pd1}
P\left (\|\nabla \mL_{\alpha,n}(\bb^*)\|_{\infty}
\ge C_5\sqrt{\frac{\log p}{n}} \right) \le 2\exp\left (-\frac{n\lambda_n^2}{2C^2d_4}+\log p \right ) \le 2 \exp(-C_4 \log p ),$$ where $C_4=\frac{k_{\lambda}^2}{2C^2d_4}-1$ and $C_5=\frac{k_\lambda}{4}$. Note that $C_4>0$ by $2C^2d_4<k_{\lambda}^2$. This complete the proof for equation (\[eq:Ln-bound\]). And the rest of the result follows immediately from the Theorem 1 in Loh(2017).
By side conditions $\|\bb^*\|_1\le R$ and $\|\hat{\bb}\|_1 \le R$ introduced in (\[eq:GPRAM-estimator\]), we have $\|\hat{\bb} - \bb^*\|_2 \le 2R$. Thus if $\mL_{\alpha,n}$ satisfies the uniform RSC condition with some $r \ge 2R$, which by Theorem \[Tm: RSC\] is achievable with high probability for a sufficiently large $\alpha$, then $\hat{\bb}$ satisfies $\|\hat{\bb}-\bb^*\|_2 \le r$ and thus a well-behaved PRAM estimator $\hat{\bb}$ in Theorem \[Tm: est\](ii) is attainable.
$\Box$\
To prove Theorem \[Tm: oracle\], we need the following result adopted directly from the Lemma 1 in [@loh2017statistical].
\[lemma:restrict-cov\] Suppose $\mL_{\alpha, n}$ satisfies the local RSC condition (\[as:RSC\]) and $n \ge \frac{2\tau}{\gamma} s \log p$. Then $\mL_{\alpha, n}$ is strongly convex over the region $S_r = \{ \bb \in \RR^p: supp(\bb) \subseteq S, \|\bb - \bb^*\|_2 \le r\}$.
Proof. The proof is similar to the proof of Lemma 1 in [@loh2017statistical].
$\Box$\
[**Proof of Theorem \[Tm: oracle\]**]{}\
The proof is an adaptation of the arguments of Theorem 2 in the paper [@loh2017statistical]. We follow the three steps of the primal-dual witness (PDW) construction described in that paper:
1. Optimize the restricted program $$\label{eq:restircted}
\hat{\bb}_S \in {\operatornamewithlimits{argmin}}_{\bb \in \bb^S: \|\bb\|_1 \le R} \left\{ \mL_{\alpha, n}(\bb) + \rho_{\lambda}(\bb)
\right \},$$ and establish that $\|\hat{\bb}_S\|_1 < R$.
2. Recall $q_\lambda(\bb) = \lambda\|\bb\|_1-\rho_\lambda(\bb)$ defined in Section \[sec:4\]. Define $\hat{\bz}_S \in \partial \|\hat{\bb}_S\|_1$, and choose $\hat{ \bz}= (\hat{\bz}_S, \hat{\bz}_{S^c})$ to satisfy the zero-subgradient condition $$\label{eq:zero-subgradient}
\nabla \mL_{\alpha, n}(\hat{\bb}) - \nabla q_{\lambda}(\hat{\bb})+\lambda \hat{\bz} = \b0,$$ where $\hat{\bb} = (\hat{\bb}_S, \pmb 0_{S^c})$. Show that $\hat{\bb}_S = \hat{\bb}^\bo_S$ and establish strict dual feasibility: $\|\hat{\bz}_{S^c}\|_\infty < 1$.
3. Verify via second order conditions that $\hat{\bb}$ is a local minimum of the program (\[eq:GPRAM-estimator\]) and conclude that all stationary points $\tilde{\bb}$ satisfying $\|\tilde{\bb} - \bb^*\|_2 \le r$ are supported on S and agree with $\hat{\bb}^\bo$.
**Proof of Step (i)** : By applying Theorem \[Tm: est\] to the restricted program (\[eq:restircted\]), we have $$\|\hat{\bb}_S-\bb^*_S\|_1 \le \frac{96\lambda s}{4\gamma-3\mu},$$ and thus $$\|\hat{\bb}_S\|_1 \le \|\bb^*\|_1 + \|\hat{\bb}_S-\bb^*_S\|_1 \le \frac{R}{2} + \frac{96\lambda s}{4\gamma-3\mu} < R,$$ under the assumption of the theorem. This complete step (i) of the PDW construction.
$\Box$\
To prove step (ii), we need the following Lemma \[Lemma: restrict-bound\] and \[Lemma: restricted-strict-convex\]:
\[Lemma: restrict-bound\] Under the conditions of Theorem \[Tm: oracle\], we have the bound $$\|\hat{\bb}^\bo_S-\bb^*_S\|_2 \le C_6 \sqrt \frac{\log p}{ns}$$ and $\hat{\bb}_S = \hat{\bb}^\bo_S$ with probability at least $1-2 \exp(-C_{41} \frac{\log p}{s^2})$.
*Proof.* Recall $\hat{\bb}^\bo = (\hat{\bb}^\bo_S, \pmb 0_{S^c})$. By the optimality of the oracle estimator in (\[eq:oracle\]), we have $$\label{ineq:oracle}
\mL_{\alpha, n} (\hat{\bb}^\bo) \le \mL_{\alpha, n}(\bb^*).$$ When $n \ge \frac{2\tau}{\gamma}s\log p$, by Lemma \[lemma:restrict-cov\], $\mL_{\alpha, n}(\bb)$ is strongly convex over restricted region $S_r=\{\|\bb - \bb^*\|_2 \le r\}$ . Hence, $$\mL_{\alpha, n}(\bb^*) + \langle \nabla \mL_{\alpha, n}(\bb^*), \hat{\bb}^\bo - \bb^* \rangle + \frac{\gamma}{4} \|\hat{\bb}^\bo - \bb^*\|_2^2 \le \mL_{\alpha, n}(\hat{\bb}^\bo).$$ Together with inequality (\[ineq:oracle\]) we obtain $$\begin{array}{ll}
\frac{\gamma}{4} \|\hat{\bb}^\bo - \bb^*\|_2^2 &\le \langle \nabla \mL_{\alpha, n}(\bb^*), \bb^* - \hat{\bb}^\bo \rangle \le \| \nabla (\mL_{\alpha, n}(\bb^*))_S\|_\infty \cdot \|\hat{\bb}^\bo - \bb^*\|_1\\
& \le \sqrt{s} \| \nabla (\mL_{\alpha, n}(\bb^*))_S\|_\infty \cdot \|\hat{\bb}^\bo - \bb^*\|_2,
\end{array}$$ implying that $$\label{ineq:res-l2-bound}
\|\hat{\bb}^\bo - \bb^*\|_2 \le \frac{4\sqrt{s}}{\gamma}\| \nabla (\mL_{\alpha, n}(\bb^*))_S\|_\infty.$$ Following the similar argument of equations (\[eq:alpha\]) , (\[ineq: 1d1\]) and (\[ineq:pd1\]) in Theorem 2, we have $$P(\|\nabla (\mL_{\alpha, n}(\bb^*_S))\|_\infty) \ge \frac{\lambda_n}{4}) \le 2\exp(-\frac{n\lambda_n^2}{2C^2d_4} + \log s),$$ for $C_{21} \lambda_n^{-\frac{1}{k-1}} \le \alpha \le C_{31}\lambda_n^{-1}$. Let $\lambda_n = C_{51} \sqrt{\frac{\log p}{ns^2}}$, we obtain $$\label{ineq:res-tm1}
\|(\nabla \mL_{\alpha, n}(\bb^*))_S\|_\infty = \|\nabla (\mL_{\alpha, n}(\bb^*_S))\|_\infty \le \frac{1}{4}C_{51} \sqrt{\frac{\log p}{ns^2}}$$ with probability at least $1-2 \exp(-C_{41} \frac{\log p}{s^2})$, where we require $s^2\log s =\bo (\log p)$. Then $\alpha$ satisifies $$\label{ineq: alpha_range2}
C_{22} (\frac{ns^2}{\log p})^{\frac{1}{2(k-1)}} \le \alpha \le C_{32}\sqrt{\frac{ns^2}{\log p}}.$$ Combining inequality (\[ineq:res-l2-bound\]) and (\[ineq:res-tm1\]), we obtain $$\label{ineq: res-inf-bound}
\|\hat{\bb}^\bo - \bb^*\|_2 \le C_6\sqrt{\frac{\log p}{ns}}$$ as desired, where $C_6=C_{51}/\gamma$.
Next we show $\hat{\bb}_S = \hat{\bb}^\bo_S$. When $n > \frac{C_6^2}{r^2} \frac{\log p}{s}$, we have $\|\hat{\bb}^\bo_S - \bb^*_S\|_2 < r$ and thus $\hat{\bb}^\bo_S$ is an interior point of the oracle program in (\[eq:oracle\]), implying $$\label{eq:oracle-gradient}
\nabla \mL_{\alpha, n}(\hat{\bb}^\bo_S) =\b0.$$ By assumption that $\beta^*_{\min} \ge C_6\sqrt{\frac{\log p}{ns}} + \delta \lambda$ and inequality (\[ineq: res-inf-bound\]), we have $$\begin{array}{ll}
|\hat{\beta}^\bo_j| \ge |\beta^*_j| - |\hat{\beta}^\bo_j - \beta^*_j| &\ge \beta^*_{\min} - \|\hat{\bb}^\bo_S-\bb^*_S\|_\infty \\
& \ge ( C_6\sqrt{\frac{\log p}{ns}} + \delta \lambda) -C_6\sqrt{\frac{ \log p}{ns}} \\
& = \delta \lambda.
\end{array}$$ for all $j \in S$. Together with the assumption that $\rho_{\lambda}$ is $(\mu, \delta)$-amenable, that is, Assumption \[as:penalty\](vii), we have $$\label{eq:oracle-penalty}
\nabla q_{\lambda}(\hat{\bb}^\bo_S) = \lambda \text{sign}(\hat{\bb}^\bo_S) = \lambda \hat{\bz}^\bo_S,$$ where $\hat{\bz}^\bo_S \in \partial \|\hat{\bb}^\bo_S\|_1$. Combining equation (\[eq:oracle-gradient\]) and (\[eq:oracle-penalty\]), we obtain $$\label{eq:oracle-zero-subgradient}
\nabla \mL_{\alpha, n}(\hat{\bb}^\bo_S) - \nabla q_{\lambda}(\hat{\bb}^\bo_S) + \lambda \hat{\bz}^\bo_S = \b0.$$ Hence $\hat{\bb}^\bo_S$ satisifes the zero-subgradient condition on the restricted program (\[eq:restircted\]). By step (i) $\hat{\bb}_S$ is an interior point of the program (\[eq:restircted\]), then it must also satisfy the zero-subgradient condition on the restricted program. Using the strict convexity from Lemma \[Lemma: restricted-strict-convex\], we obtain $\hat{\bb}_S = \hat{\bb}^\bo_S$.
$\Box$\
The following lemma guarantees that the program in (\[eq:restircted\]) is strictly convex:
\[Lemma: restricted-strict-convex\] Suppose $\mL_{\alpha, n}$ satisfies the uniform RSC condition (\[as:RSC\]) and $\rho_{\lambda}$ is $\mu$-amenable. Suppose in addition the sampel size satisifies $n > \frac{2\tau}{\gamma - \mu}s \log p$, then the restricted program in (\[eq:restircted\]) is strictly convex.
We omit the proof since it is similar to the proof of Lemma 2 in [@loh2017support].
$\Box$\
**Proof of step (ii)** : We rewrite the zero-subgradient condition (\[eq:zero-subgradient\]) as $$\left (\nabla \mL_{\alpha, n}(\hat{\bb}) - \nabla \mL_{\alpha, n}(\bb^*) \right ) + \left (\nabla \mL_{\alpha, n}(\bb^*)- \nabla q_{\lambda}(\hat{\bb}) \right )+\lambda \hat{\bz} = \b0.$$ Let $\Hat{Q}$ be a $p \times p$ matrix $\hat{Q}= \int_0^1 \nabla^2 \mL_{\alpha, n} \blp \bb^* + t(\hat{\bb} - \bb^*) \brp dt$. By the zero-subgradient condition and the fundamental theorem of calculas, we have $$\hat{Q}(\hat{\bb} - \bb^*) + \left ( \nabla \mL_{\alpha, n}(\bb^*) - \nabla q_{\lambda}(\hat{\bb}) \right ) + \lambda \hat{\bz}=\b0.$$ And its block form is $$\label{eq: block-zero-subgradient}
\left[ {\begin{array}{*{20}c}
\hat{Q}_{SS} & \hat{Q}_{SS^c} \\
\hat{Q}_{S^c S} & \hat{Q}_{S^c S^c} \\
\end{array} } \right]
\left[ {\begin{array}{*{20}c}
\hat{\bb}_S - \bb^*_S \\
\b0 \\
\end{array} } \right]
+ \left[ {\begin{array}{*{20}c}
\nabla \mL_{\alpha, n}(\bb^*)_S - \nabla q_{\lambda}(\hat{\bb}_S) \\
\nabla \mL_{\alpha, n}(\bb^*)_{S^c} - \nabla q_{\lambda}(\hat{\bb}_{S^c}) \\
\end{array} } \right] + \lambda \left[ {\begin{array}{*{20}c}
\hat{\bz}_{S} \\
\hat{\bz}_{S^c} \\
\end{array} } \right] = \b0.$$
The selection property implies $\nabla q_{\lambda}(\hat{\bb}_{S^c})=\b0$. Plugging this result into equation (\[eq: block-zero-subgradient\]) and performing some algebra, we conclude that $$\hat{\bz}_{S^c} = \frac{1}{\lambda} \left \{ \hat{Q}_{S^cS}(\bb^*_S - \hat{\bb}_S) - (\nabla \mL_{\alpha, n}(\bb^*))_{S^c} \right \}.$$ Therefore, $$\label{ineq:pd-feasible}
\begin{array}{ll}
\|\hat{\bz}_{S^c}\|_\infty & \le \frac{1}{\lambda}\|\hat{Q}_{S^c S}(\hat{\bb}_S - \bb^*_S)\|_\infty + \frac{1}{\lambda} \|\nabla \mL_{\alpha, n}(\bb^*))_{S^c}\|_\infty\\
& \le \frac{1}{\lambda} \left \{ \max_{j \in S^c} \| e^T_j \hat{Q}_{S^c S} (\hat{\bb}_S - \bb^*_S)\|_2 \right \} + \frac{1}{\lambda} \|\nabla \mL_{\alpha, n}(\bb^*))_{S^c}\|_\infty\\
& \le \frac{1}{\lambda} \left \{\max_{j \in S^c} \| e^T_j \hat{Q}_{S^c S}\|_2 \right \} \| (\hat{\bb}_S - \bb^*_S)\|_2 + \frac{1}{\lambda} \|\nabla \mL_{\alpha, n}(\bb^*))_{S^c}\|_\infty.
\end{array}$$ Observe that $$\begin{array}{ll}
[(e_j^T\hat{Q}_{S^cS})_m]^2 & \le [\frac{1}{n}\sum_{i=1}^n w(\bx_i)\bx_{ij} v(\bx_i) \bx_{im}\int_0^1 l''((y_i - \bx_i^T \bb^* - t(\bx_i\hat{\bb} - \bx_i\bb^*))v(\bx_i)){\mathop{}\!\mathrm{d}}t ]^2\\
& \le k_2^2 [\frac{1}{n} \sum_{i=1}^n w(\bx_i)\bx_{ij} \cdot v(\bx_i) \bx_{im} ]^2,\\
\end{array}$$ for all $j \in S^c$ and $m \in S$, where the second inequality follows from Assumption \[as:loss\](ii). By condition of Theorem \[Tm: oracle\], the variables $w(\bx_i)\bx_{ij}$ and $v(\bx_i) \bx_{im}$ are both sub-Gaussian. Using standard concentration results for i.i.d sums of products of sub-Gaussian variables, we have $$P([(e_j^T\hat{Q}_{S^cS})_m]^2 \le c_1) \ge 1-c_2\exp(-c_3n).$$ It then follows from union inequality that $$\label{ineq:maxQ}
P( \max_{j \in S^c} \| e^T_j \hat{Q}_{S^c S}\|_2 \le \sqrt{c_1s}) \ge 1 - c_2\exp(-c_3n + \log(s(p-s))) \ge 1-c_2\exp(-\frac{c_3}{2}n),$$ where $n \ge \frac{2}{c_3} \log \left(s(p-s) \right)$. By Lemma \[Lemma: restrict-bound\] we obtain $$\label{ineq:pd-feasible2}
\|\hat{\bb}_S - \bb^*_S\|_2 \le C_6\sqrt{\frac{\log p}{ns}}.$$ Furthermore, Theorem \[Tm: est\] gives $$\label{ineq:pd-feasible3}
\|\nabla \mL_{\alpha, n}(\bb^*))_{S^c}\|_\infty \le \|\nabla \mL_{\alpha, n}(\bb^*))\|_\infty \le C_5\sqrt{\frac{\log p}{n}},$$ Combining inequality (\[ineq:pd-feasible\]), (\[ineq:maxQ\]), (\[ineq:pd-feasible2\]) and (\[ineq:pd-feasible3\]), we have $$\|\hat{\bz}_{S^c}\|_\infty \le \frac{1}{\lambda}C_7\sqrt{\frac{\log p}{n}}.$$ with probability at least $1-C_8 \exp(-C_{41} \frac{\log p}{s^2})$. Note that $\alpha$ is required to satisfy both ranges in Theorem \[Tm: est\] and (\[ineq: alpha\_range2\]). Combing these two ranges we have $$C_{22}(\frac{ns^2}{\log p})^{\frac{1}{2(k-1)}} \le \alpha \le C_3 \sqrt{\frac{n}{\log p}},$$ where $s^2 = \bo \left ((\frac{n}{\log p})^{k-2} \right)$. In paticular, for $\lambda > C_7\sqrt{\frac{\log p}{n}}$, we conclude at last that the strict dual feasibility condition $ \|\hat{\bz}_{S^c}\|_\infty < 1$ holds, completing step (ii) of the PDW construction.
**Proof of step (iii)** : Since the proof for this step is almost identical to the proof in Step (iii) of Theorem 2 in [@loh2017statistical], except for the slightly different notation, we refer the reader to the arguments provided in that paper.
$\Box$\
To prove Theorem \[Tm: normality\], we need to generalized the asymptotic normality results for lower dimensional non-penalized M-estimator from [@he2000parameters] to the following Lemma:
\[Lemma:Shao2000\] Suppose $\bz_1, \bz_2, \dots, \bz_n \in \RR^{p}$ are independent observations from probability distribution $F_{i,\bb}$, $i=1, 2, \dots, n$, with a common parameter $\bb \in \RR^s$. And $s$ may increase with the sample size $n$. Suppose $\mL_n(\bb) = \frac{1}{n}\sum_{i=1}^{n} \rho(\bz_i, \bb)$ is convex in $\bb$ in a neighborhood of $\bb^*$ and has a unique local minimizer $\hat{\bb}$. Define $\psi(\bz_i,\bb)= \frac{\partial}{\partial \bb} \rho(\bz_i, \bb) $ and $\eta_i(\tilde{\bb}, \bb) = \psi(\bz_i, \tilde{\bb}) - \psi(\bz_i, \bb) -E\psi(\bz_i, \tilde{\bb}) + E\psi(\bz_i, \bb) $ and $B_s = \{ \bnu \in \RR^s: \|\bnu\|_2=1\}$. Suppose $\bb^* \in \RR^s$ such that $$\label{eq: true_beta}
\|\sum_{i=1}^n \psi(\bz_i, \bb^*)\|_2 = \bo_p((ns)^{1/2}).$$ Assume the following conditions are satisfied:
1. $\|\sum_{i=1}^n \psi(\bz_i, \hat{\bb})\|_2 = o_p(n^{1/2})$.
2. There exist $C$ and $r \in (0,2]$ such that $\max_{i \le n} E_{\bb} \sup_{\tilde{\bb}:\|\tilde{\bb}-\bb\|_2 \le d} \|\eta_i(\tilde{\bb}, \bb)\|_2^2 \le n^C d^r$, for $0 <d \le 1$.
3. There exists a sequence of $s$ by $s$ matrices $\bD_n$ with $\lim \inf_{n \rightarrow \infty} \lambda_{min}(\bD_n) > 0$ such that for any $K>0$ and uniformly in $\bnu \in B_s$, $$\sup_{\|\bb-\bb^*\|_2 \le K(s/n)^{1/2}} |\bnu^T\sum_{i=1}^{n} E_{\bb^*}(\psi(\bz_i, \bb) - \psi(\bz_i, \bb^*)) - n \bnu^T\bD_n(\bb - \bb^*)|=o((ns)^{1/2}).$$
4. $\sup_{\tilde{\bb}: \|\tilde{\bb} - \bb\|_2 \le K(s/n)^{1/2}} \sum_{i=1}^{n} E_{\bb} |\bnu^T \eta_i(\tilde{\bb}, \bb)|^2 =\bo(A(n,s))$ for any $\bb \in \RR^s$, $\bnu \in B_s$ and $K>0$.
5. $\sup_{\bnu \in S_s}\sup_{\tilde{\bb}: \|\tilde{\bb} - \bb\|_2 \le K(s/n)^{1/2}} \sum_{i=1}^{n} (\bnu^T \eta_i(\tilde{\bb}, \bb))^2 =\bo_p(A(n,s))$ for any $\bb \in \RR^s$ and $K>0$.
If $A(n,s)=o(n/\log n)$, we have $$\|\hat{\bb} - \bb^*\|_2^2=\bo_p(s/n).$$ Furthermore, if $A(n,s)=o(n/(s \log n))$, then for any unit vector $\bnu \in \RR^s$, we have $$\hat{\bb} - \bb^* = - n^{-1} \sum_{i=1}^n \bD_n^{-1} \psi(\bz_i, \bb^*) +\br_n,$$ with $\|\br_n\|_2 = o_p(n^{-1/2})$.
*Proof.* Our proof is similar to the proof of Theorem 1 and 2 in [@he2000parameters]. Note that in that paper, $\bb^*$ is defined to be the solution of $\sum_{i=1}^n E_{\bb} \psi(x_i, \bb) = \b0$, in addition to the condition in equation (\[eq: true\_beta\]). However, a careful inspection of the proofs in that paper reveals that the results still holds if we only require $\bb^*$ to satisfied equation (\[eq: true\_beta\]).
$\Box$\
[**Proof of Theorem \[Tm: normality\]**]{}\
We then apply the result to the oracle estimator $\Hat{\bb}_S^{\bo}$ defined in equation (\[eq:oracle\]) with $w(\bx) \equiv v(\bx)\equiv 1$. Although Lemma \[Lemma:Shao2000\] requires $\mL_n$ to be convex, a throughout examination of the proofs in [@he2000parameters] shows that the results still hold if we restrict our attention to a subset of $\RR^p$ on which $\mL_n$ is convex and $\hat{\bb}$ is the unique minimizer. Since $\Hat{\bb}_S^{\bo}$ is $s$-dimensional vector without sparsity, we denote $\bx_i$, $\bb$ and $\bb^*$ all as $s$-dimensional vectors for the rest of our proof. We also denote $\bb_{\alpha}^*$ as $(\bb_{\alpha}^*)_S$. Let $\bz_i=(\bx_i, y_i)$ and we rewrite $\rho(\bz_i, \bb)$ as $l_\alpha(y_i - \bx_i^T\bb)$, with $\mL_{\alpha, n}$ taking the place of $\mL_{n}$. Then $\psi(\bz_i, \bb)= -l_\alpha'(y_i - \bx_i^T\bb)\bx_i$.
We start with verifying equation (\[eq: true\_beta\]), which can be rewrited as $$\label{eq: true_beta2}
\|\sum_{i=1}^n l_\alpha'(\epsilon_i)\bx_i\|_2 = \bo_p((ns)^{1/2}).$$ Observe that for any $\bnu \in B_s$, $$\begin{array}{ll}
P(|\sum_{i=1}^n \bnu^Tl_\alpha'(\epsilon_i)\bx_i - \sum_{i=1}^n E[\bnu^Tl_\alpha'(\epsilon_i)\bx_i]|>t) &\le nVar(\bnu^Tl_\alpha'(\epsilon_i)\bx_i)t^{-2} \\
&\le nE|\bnu^Tl_\alpha'(\epsilon_i)\bx_i|^2t^{-2} \\
&\le n E\|l_\alpha'(\epsilon_i)\bx_i\|_2^2t^{-2}\\
& \le nsd_4t^{-2},
\end{array}$$ where the last inequality follows from inequality (\[ineq:psi\_squred\]). We then have $$\label{ineq:Opbound}
P(|\sum_{i=1}^n \bnu^Tl_\alpha'(\epsilon_i)\bx_i|>t + \sum_{i=1}^n |E [\bnu^Tl_\alpha'(\epsilon_i)\bx_i]|)\le nsd_4t^{-2}.$$ Observe that $$\label{ineq: bound E-norm}
\begin{array}{ll}
|E [\bnu^Tl_\alpha'(\epsilon_i)\bx_i]| &= |E[l_\alpha'(y_i - \bx_i^T\bb^*)\bnu^T\bx_i ]|\\
&=|E[l_\alpha'(y_i - \bx_i^T\bb^*)\bnu^T\bx_i ]-E[l_{\alpha}'(y_i-\bx_i^T\bb_{\alpha}^*)\bnu^T\bx_i ]|\\
& \le k_2E[|\bx_i^T(\bb_{\alpha}^*-\bb^*)||\bnu^T\bx_i|]\\
& \le k_2\{E|\bx_i^T(\bb_{\alpha}^*-\bb^*)|^2\}^{\frac{1}{2}}\{E|\bnu^T\bx_i|^2\}^{\frac{1}{2}}\\
& \le k_0^2k_2\|\bb_{\alpha}^*-\bb^*\|_2,
\end{array}$$ where the first and last inequalities follow from Assumption \[as:loss\](ii) and Assumption \[as:x\](iv) respectively. Together with the results in Theorem \[Tm:1\] and condition $\alpha ^{1-k} = o(n^{-1/2})$, we obtain $$\label{eq:E-norm-bound}
E[\bnu^Tl_\alpha'(\epsilon_i)\bx_i]= o(n^{-1/2}).$$ Thus by inequality (\[ineq:Opbound\]) and (\[eq:E-norm-bound\]) we have $\sum_{i=1}^n \bnu^Tl_\alpha'(\epsilon_i)\bx_i = \bo_p((ns)^{1/2})$ for any $\bnu \in B_s$. It then implies equation (\[eq: true\_beta2\]). Next we verify the conditions (i)-(v). Since the $\mL_{\alpha,n}$ is differentiable, the left hand side of condition (i) is 0 and thus it is satisfied. By definition of $\eta_i$, we have $$\eta_i(\tilde{\bb}, \bb) = l_\alpha'(y_i-\bx_i^T\bb)\bx_i - l_\alpha'(y_i - \bx_i^T\tilde{\bb})\bx_i - E l_\alpha'( y_i - \bx_i^T\bb)\bx_i +
E l_\alpha'(y_i - \bx_i^T\tilde{\bb})\bx_i
.$$ Observe that $$\begin{array}{ll}
\| \eta_i(\tilde{\bb}, \bb)\|_2 &\le \|l_\alpha'(y_i - \bx_i^T\tilde{\bb})\bx_i - l_\alpha'(y_i-\bx_i^T\bb)\bx_i\|_2 + \| E l_\alpha'(y_i - \bx_i^T\tilde{\bb})\bx_i -
E l_\alpha'(y_i-\bx_i^T\bb)\bx_i\|_2\\
& \le k_2|\bx_i^T(\tilde{\bb} - \bb)|\cdot\|\bx_i\|_2 + k_2\|E \bx_i^T(\tilde{\bb} - \bb)\bx_i\|_2 \\
& \le k_2\|\tilde{\bb} - \bb\|_2 \|\bx_i\|_2^2 + k_2E\|\bx_i^T(\tilde{\bb} - \bb)\bx_i\|_2\\
& \le k_2\|\tilde{\bb} - \bb\|_2 \|\bx_i\|_2^2 + k_2\|\tilde{\bb} - \bb\|_2 E \|\bx_i\|_2^2,
\end{array}$$ where the second and third inequality follow from Assumption \[as:loss\](ii) and Jensen’s inequality, respectively. We then obtain $$\max_{i \le n} E_{\bb} \sup_{\tilde{\bb}:\|\tilde{\bb}-\bb\|_2 \le d} \|\eta_i(\tilde{\bb}, \bb)\|_2^2 \le \max_{i \le n} 4 k_2^2d^2 E \|\bx_i\|_2^4.$$ Since Assumption \[as:x\](iv) implies $E \|\bx_i\|_2^4 =\bo (s^2)$ for $i=1,\cdots,n$, condition(ii) holds with $r=2$ and if $s=\bo(n^{r_1})$ for $r_1>0$.
Similarly, for any $\bnu \in B_s$, we have $$\begin{array}{ll}
\bnu^T \eta_i(\tilde{\bb}, \bb) &\le |l_\alpha'( y_i - \bx_i^T\tilde{\bb}) -l_\alpha'(y_i-\bx_i^T\bb)||\bnu^T\bx_i| + E [|l_\alpha'( y_i - \bx_i^T\tilde{\bb}) -l_\alpha'(y_i-\bx_i^T\bb)||\bnu^T\bx_i|]\\
& \le k_2|\bx_i^T(\tilde{\bb} - \bb)||\bnu^T \bx_i| + k_2E[|\bx_i^T(\tilde{\bb} - \bb)||\bnu^T \bx_i|]\\
& \le k_2\|\tilde{\bb} - \bb\|_2 |\tilde{\bnu}^T\bx_i|| \bnu^T \bx_i | + k_2\|\tilde{\bb} - \bb\|_2 \{E|\tilde{\bnu}^T\bx_i|^2\}^{1/2} E\{| \bnu^T \bx_i |^2\}^{1/2}\\
& \le k_2\|\tilde{\bb} - \bb\|_2(|\tilde{\bnu}^T\bx_i|| \bnu^T \bx_i | + k_0^2),
\end{array}$$ where $\tilde{\bnu}=(\tilde{\bb} - \bb)/\|\tilde{\bb} - \bb\|_2$. The second and last inequalities follow from Assumption \[as:loss\](ii) and Assumption \[as:x\](iv) respectively. It then gives $$| \bnu^T \eta_i(\tilde{\bb}, \bb)|^2 \le k_2^2\|\tilde{\bb} - \bb\|_2^2(|\tilde{\bnu}^T\bx_i|^2| \bnu^T \bx_i |^2 + 2k_0^2|\tilde{\bnu}^T\bx_i|| \bnu^T \bx_i |+k_0^4).$$ Together with Assumption \[as:x\](iv), we obtain $$E| \bnu^T \eta_i(\tilde{\bb}, \bb)|^2 = \bo(\|\tilde{\bb} - \bb\|_2^2)$$ and $$| \bnu^T \eta_i(\tilde{\bb}, \bb)|^2 = \bo_p(\|\tilde{\bb} - \bb\|_2^2).$$ Hence condition (iv) and (v) hold with $A(n,s)=s$.
Finally we show condition (iii). Let $\bD_{\alpha, n} = E[\nabla^2 \mL_{\alpha, n}(\bb^*)]$ and thus it is an $s$ by $s$ matrix. Observe that $$\begin{array}{ll}
E[l_{\alpha}''(\epsilon_i)|\bx_i] & = E[l''_\alpha (\epsilon_i)\mathbbm{1}(|\epsilon_i| \le \alpha)|\bx_i ] + E[l''_\alpha (\epsilon_i)\mathbbm{1}(|\epsilon_i| > \alpha)|\bx_i] \\
& \ge E[(1-d_1|\epsilon_i|^k\alpha^{-k}) \mathbbm{1}(|\epsilon_i| \le \alpha)|\bx_i ] + E[l''_\alpha (\epsilon_i)\mathbbm{1}(|\epsilon_i| > \alpha)|\bx_i]\\
& \ge P(|\epsilon_i| \le \alpha |\bx_i) - d_1\alpha^{-k}E[|\epsilon_i|^k|\bx_i]- k_2\alpha^{-k}E[|\epsilon_i|^k|\bx_i]
\\
& \ge 1-\alpha^{-k}E[|\epsilon_i|^k|\bx_i] - d_1\alpha^{-k}E[|\epsilon_i|^k|\bx_i]- k_2\alpha^{-k}E[|\epsilon_i|^k|\bx_i]\\
&= 1- (d_1+k_2+1)\alpha^{-k}E[|\epsilon_i|^k|\bx_i],
\end{array}$$ where the first and second inequalities follow from Assumption \[as:loss\](iii) and (ii), respectively. Thus for any $\bnu \in B_s$, we have $$\begin{array}{ll}
\bnu^T \bD_{\alpha, n} \bnu & = E[l_{\alpha}''(\epsilon_i) \bnu^T\bx_i \bx_i^T \bnu]\\
& \ge E[(1- (d_1+k_2+1)\alpha^{-k} E[|\epsilon_i|^k|\bx_i])\bnu^T\bx_i \bx_i^T \bnu]
\\
& = \bnu^T E[\bx_i \bx_i^T] \bnu - (d_1+k_2+1) \alpha^{-k} E[E(|\epsilon_i|^k|\bx_i) (\bnu\bx_i)^2 ] \\
& \ge k_l - (d_1+k_2+1) \alpha^{-k} \{ E[E(|\epsilon_i|^k|\bx_i)]^2 \}^{1/2} \{E[(\bnu\bx_i)^4]\}^{1/2} \\
& \ge k_l - C_9\alpha^{-k},
\end{array}$$ where second inequality follows from Assumption \[as:x\](i) and $C_9$ is a constant that only depends on $k_0$, $k_2$, $d_1$, $M_k$. Hence if $\alpha > (2C_9/k_l)^{1/k}$, we have $\lambda_{min}( \bD_{\alpha, n}) >k_l/2$. It then implies $\lim \inf_{n \rightarrow \infty} \lambda_{min}(\bD_{\alpha, n}) > 0$. Observe that $$\begin{array}{ll}
&|\bnu^T\sum_{i=1}^{n} E_{\bb^*}(\psi(\bx_i, \bb) - \psi(\bx_i, \bb^*)) - n \bnu^T\bD_{\alpha, n}(\bb - \bb^*)|\\
= & |\bnu^T\sum_{i=1}^{n} E_{\bb^*}\{(l'_\alpha( y_i - \bx_i^T\bb^*)\bx_i - l'_\alpha(y_i - \bx_i^T\bb)\bx_i - l''_\alpha(y_i - \bx_i^T\bb^*)\bx_i\bx_i^T(\bb - \bb^*)\}|\\
= & |\bnu^T\sum_{i=1}^{n} E_{\bb^*}\{(l''_\alpha(y_i - \bx_i^T\tilde{\bb})\bx_i^T(\bb - \bb^*)\bx_i - l''_\alpha( y_i - \bx_i^T\bb^*)\bx_i\bx_i^T(\bb - \bb^*)\}|\\
\le & |\bnu^T\sum_{i=1}^{n} E_{\bb^*}\{(k_3|\bx_i^T(\tilde{\bb} -\bb^*)| |\bx_i^T(\bb - \bb^*)\bx_i| \}| \\
\le & k_3\|\bb - \bb^*\|_2^2\sum_{i=1}^{n} E_{\bb^*}\{|\bx_i^T\tilde{\bnu}|^2 |\bx_i^T\bnu| \},
\end{array}$$ where $\tilde{\bb}$ is a vector lying between $\bb$ and $\bb^*$ and $\tilde{\bnu}=(\tilde{\bb} - \bb)/\|\tilde{\bb} - \bb\|_2$. Note that the second equation follows from mean value theorem and the first inequality follows from the condition that $l_{\alpha}''$ is Lipschitz. By Assumption \[as:x\] (iv) we have $\sum_{i=1}^{n} E_{\bb^*}\{|\bx_i^T\tilde{\bnu}|^2 \bx_i^T\bnu| \} = \bo(n)$. Hence condition (iii) holds if $s/n \rightarrow 0$.
We conclude that the desired results hold for the oracle estimator $\hat{\bb}^\bo_S$. In particular, we have $$\label{eq:est-error-sum}
\begin{split}
\hat{\bb}^\bo_S - \bb^* = & n^{-1} \sum_{i=1}^n \bD_{\alpha, n}^{-1} l_\alpha'(\epsilon_i)\bx_i + \br_n\\
= & n^{-1} \sum_{i=1}^n \{\bD_{\alpha, n}^{-1} l_\alpha'(\epsilon_i)\bx_i - E[\bD_{\alpha, n}^{-1} l_\alpha'(\epsilon_i)\bx_i] \} + E[\bD_{\alpha, n}^{-1} l_\alpha'(\epsilon_i)\bx_i]+ \br_n,
\end{split}$$ with $\|\br_n\|_2 = o_p(n^{-1/2})$. Observe that $$\label{eq:mu-bound}
\begin{array}{ll}
\| E[\bD_{\alpha, n}^{-1}l_\alpha'(\epsilon_i)\bx_i] \|_2 &= \| \bD_{\alpha, n}^{-1}E[l_\alpha'(\epsilon_i)\bx_i] \|_2\\
&= \|\bD_{\alpha, n}^{-1} \tilde{\bnu}\|_2\|E[l_\alpha'(\epsilon_i)\bx_i]\|_2\\
& \le [\lambda_{\min}(\bD_{\alpha, n})]^{-1} \|E[l_\alpha'(\epsilon_i)\bx_i]\|_2 \\
&= o(n^{-1/2}),
\end{array}$$ where the last equality follows from equation (\[eq:E-norm-bound\]). By equations (\[eq:est-error-sum\]) and (\[eq:mu-bound\]), we obtain $$\label{eq: asymptotic}
\frac{\sqrt{n}}{\sigma_\bnu} \cdot \bnu^T(\hat{\bb}^\bo_S - \bb^*) \xrightarrow{d} N(0,1),$$ where $\sigma_\bnu^2 = \bnu^T \bD_{\alpha, n}^{-1} Var(l_\alpha'(\epsilon_i)\bx_i) \bD_{\alpha, n}^{-1} \bnu$. By Theorem \[Tm: oracle\], the asymptotic result in (\[eq: asymptotic\]) is also applicable for the stationary point $\tilde{\bb}$.
$\Box$\
To prove Theorem \[Tm: RSC\], we need the following result:
\[Lemma: RSC\_loh\] Suppose covariate $\bx$ satisfies Assumption \[as:x\](iv) and $l''_\alpha(u)$ satisfies Assumption \[as:loss\](ii). For any fixed $\alpha>0$, suppose the bound $ C_{14}k_0^2 \left(\sqrt{\varepsilon_{T}}+\exp\left(-\frac{C_{15}T^2}{k_0^2r^2}\right)\right) < \frac{\gamma_{\alpha, T}}{\gamma_{\alpha, T} + k_2} \cdot \frac{k_l}{2}$ holds, where $ \gamma_{\alpha, T} = \min_{|u| \le T}l''_{\alpha}(u)>0$. Suppose in addition that the sample size satisfies $n \ge C_{10}s \log p$. With probability at least $1-C_{11}\exp(-C_{12}\log p)$, the loss function $\mL_{\alpha,n}$ satisfies $$\label{eq:RSC}
\langle \nabla \mL_{\alpha,n}(\bb_1)- \nabla \mL_{\alpha,n}(\bb_2), \bb_1-\bb_2 \rangle \ge \gamma_{\alpha} \|\bb_1-\bb_2\|_2^2 - \tau_{\alpha} \frac{\log p}{n}\|\bb_1 - \bb_2\|_1^2,$$ where $\bb_j \in \RR^p$ such that $\|\bb_j-\bb^*\|_2\le r$ for $j=1, 2$ with $$\label{eq:RSC_coeff}
\gamma_{\alpha}=\frac{\gamma_{\alpha, T}k_l}{16} \quad \text{and} \quad \tau_{\alpha}=\frac{C_{13}(\gamma_{\alpha, T} + k_2)^2k_0^2T^2}{r^2}.$$ Here the constants $C_{10}, C_{11}, C_{12}, C_{13}, C_{14}, C_{15}$ do not depend on $\alpha$.
Proof. The proof is similar to the proof of Proposition 2 in [@loh2017statistical]. Note that in that paper, it assumes $\bx_i {\perp\mkern-9.5mu\perp}\epsilon_i$. However, a careful inspection of the proofs reveals that the result stills holds if we allow $\epsilon_i$ to depend on $\bx_i$. We refer the reader to the arguments provided in that paper.\
$\Box$\
[**Proof of Theorem \[Tm: RSC\]**]{}\
Recall $\gamma_{\alpha, T} = \min_{|u| \le T}l''_{\alpha}(u)$. By Assumption \[as:loss\](iii) , $\alpha \ge \alpha_0$ and $\alpha_0=\max\{ (2d_1)^{\frac{1}{k}},1\}\cdot T_0$ we have
$$\label{eq:lb_gamma}
\gamma_{\alpha, T_0} = \min_{|u| \le T_0} l_{\alpha}(u)
\ge \min_{|u| \le T_0} (1-d_1|u|^{k}\alpha^{-k}) \ge 1-d_1|T_0|^{k}\alpha_0^{-k} \ge \frac{1}{2}.$$
And $$\label{eq:ub_gamma}
\gamma_{\alpha, T_0} = \min_{|u| \le T_0} l_{\alpha}(u)
\le \min_{|u| \le T_0} (1+d_1|u|^{k}\alpha^{-k}) \le 1+d_1|T_0|^{k}\alpha_0^{-k} \le \frac{3}{2}.$$ By equation (\[eq:lb\_gamma\]), we obtain $$\frac{\gamma_{\alpha, T_0}}{\gamma_{\alpha, T_0} + k_2} \cdot \frac{k_l}{2} \ge \frac{\frac{1}{2}}{\frac{1}{2} + k_2} \cdot \frac{k_l}{2} \ge \frac{k_l}{2+4k_2}.$$ Together with condition $C_{14}k_0^2 \left(\sqrt{\varepsilon_{T_0}}+\exp \left(-\frac{C_{15}T_0^2}{k_0^2r^2} \right) \right) < \frac{k_l}{2+4k_2}$, we have $$\label{eq:RSC_condition}
c_{14}k_0^2 \left(\sqrt{\varepsilon_{T_0}}+\exp \left(-\frac{c_{15}T_0^2}{k_0^2r^2} \right) \right) < \frac{\gamma_{\alpha, T_0}}{\gamma_{\alpha, T_0} + k_2} \cdot \frac{k_l}{2}.$$ By equation (\[eq:lb\_gamma\]), (\[eq:ub\_gamma\]), (\[eq:RSC\_condition\]) and Lemma \[Lemma: RSC\_loh\] we complete the proof.
$\Box$
|
---
abstract: |
Visual localization under large changes in scale is an important capability in many robotic mapping applications, such as localizing at low altitudes in maps built at high altitudes, or performing loop closure over long distances. Existing approaches, however, are robust only up to about a 3x difference in scale between map and query images.
We propose a novel combination of deep-learning-based object features and state-of-the-art SIFT point-features that yields improved robustness to scale change. This technique is training-free and class-agnostic, and in principle can be deployed in any environment out-of-the-box. We evaluate the proposed technique on the KITTI Odometry benchmark and on a novel dataset of outdoor images exhibiting changes in visual scale of $7\times$ and greater, which we have released to the public. Our technique consistently outperforms localization using either SIFT features or the proposed object features alone, achieving both greater accuracy and much lower failure rates under large changes in scale.
author:
- 'Andrew Holliday$^{1}$ and Gregory Dudek$^{2}$[^1]'
bibliography:
- 'IEEEabrv.bib'
- 'mybib.bib'
title: '**Scale-Robust Localization Using General Object Landmarks** '
---
INTRODUCTION
============
In this work, we attempt to address the problem of performing metric localization in a known environment under extreme changes in visual scale. Our localization approach is based on the identification of objects in the environment, and their use as landmarks. By “objects" we here mean physical entities which are distinct from their surroundings and have some consistent physical properties of structure and appearance.
Many robotic applications involve repeated traversals of a known environment over time. In such applications, it is usually beneficial to first construct a map of the environment, which can then be used by a robot to navigate the environment in subsequent missions. Surveying the environment from a very high altitude allows complete geographic coverage of the environment to be obtained by shorter, and thus more efficient, paths by the surveyor. At the same time, a robot that makes use of this high-altitude map to localize may have mission parameters requiring it to operate at a much lower altitude.
\
Far Image
Near Image
\
\
SIFT
res3d, $64 \times 64$ input
\
One such scenario is that of performing visual surveys of benthic environments, such as coral reefs, as in Johnson-Roberson et al. [@underwaterMappingExample]. A fast-moving surface vehicle may be used to rapidly map a large area of a reef. This map may then be used by a slower-moving, but more maneuverable, autonomous underwater vehicle (AUV) such as the Aqua robot [@aquaRobot], to navigate the reef while capturing imagery very close to the sea floor. Another relevant scenario is that of a robot performing loop closure over long distances as part of . Loop closure, the recognition of a previously-viewed location when viewing it a second time, is key to accurate , and the overall accuracy of techniques could be considerably improved if loop closure could be conducted across major changes in scale and perspective.
In scenarios such as the above, a robot must deal with the change in visual scale between two perspectives, which may be $5\times$ or even greater. In some scenarios, such as in benthic environments, other factors may also intrude, such as colour-shifting due to the optical properties of water, and image noise due to particulate suspended in the water. Identifying scenes across such large changes in scale is very challenging for modern visual localization techniques. Even the most scale-robust techniques, such as , can only localize reliably under scale factors less than about $3\times$.
We hypothesize that the hierarchical features computed by the intermediate layers of a [@deepTextbook] may prove robust to changes in scale, due to their high degree of abstraction. We propose a technique for performing metric localization across significant changes in scale by identifying and describing non-semantic objects in a way that allows them to be associated between scenes. We show that these associations can be used to guide the matching of features between images in a way that improves the robustness of matching to scale changes, allowing accurate localization under visual scale factors of 3 and greater. The proposed system does not require any environment-specific training, and in principle can be deployed out-of-the-box in arbitrary environments. The objects used by our system are defined functionally, in terms of their utility as scale-invariant landmarks, and are not limited to semantically-meaningful object categories.
We specifically consider the problem of localizing between pairs of images known to contain the same scene at different visual scale. A solution to this problem is an essential component of a system that can perform full global localization across large scale changes, and in certain cases - such as the low-vs-high-altitude case described above - could suffice on its own for global localization. We demonstrate the approach both on standard localization benchmarks and on a novel dataset of image pairs from urban scenes exhibiting major scale changes.
RELATED WORK
============
Visual localization refers to the problem of determining a robot’s pose using images from one or more cameras, with reference to a map or set of previously-seen images. This may be done with some prior on the robot’s position, or with no such prior, called global localization [@dudek2010computational]. Visual odometry is a form of non-global localization, while global localization is closely related to loop closure; both of these are important components of , and there is a large body of literature exploring both problems. Prominent early work includes Leonard et al. [@leonard1991mobile] and Mackenzie et al. [@mackenzie1994precise], and Fox et al. [@fox1999markov].
Many traditional visual approaches to these problems, and particularly global localization, have been based on the recognition of whole-image descriptors of particular scenes, such as GIST features [@gist]. Successful instances include SeqSLAM [@seqslam], which uses a heavily downsampled version of the input image as a descriptor, and LSD-SLAM [@lsdslam], which performs direct image alignment for loop closure, as well as Hansen et al. [@seqslamvar1], Cadena et al., [@robustPlaceRecogWithSequences], Liu et al. [@zhangWholeImageComparison] and Naseer et al. [@seqslamvar2]. Because whole-image descriptors encode the geometric relationships of features in the 2D image plane, images of the same scene from different perspectives can have very different descriptors, making such methods very sensitive to changes in perspective and scale.
Another common approach is to discretize point-feature descriptors and build bag-of-words histograms of the input images. FAB-MAP [@fabmap], ORB-SLAM [@orbslam], and the system of Ho et al. [@seqslamvar3] perform variants of this for loop closure, starting from SURF [@surf], ORB [@orb], and [@sift] features, respectively. While suitable for place-recognition tasks, such approaches alone are not appropriate for global localization, because spatial information about the visual words is not contained in the histogram. Hence, state-of-the-art systems such as ORB-SLAM and LSD-SLAM rely on visual odometry for pose estimation. Their visual odometry techniques are limited in robustness to changes in scale, perspective, and appearance, and so rely on successive estimations from closely-spaced frames.
Other global localization approaches attempt to recognize particular landmarks in an image, and use those to produce a metric estimate of the robot’s pose. SLAM++ of Salas-Moreno et al. [@salas2013slam++] performs by recognizing landmarks from a database of 3D object models. Linegar et al. [@landmarks24Hour] and Li et al. [@highLevelFeaturesUnderwater] both train a bank of support vector machines (SVMs) to detect specific landmarks in a known environment, one SVM per landmark. More recently, [@semanticSlam] made use of a Deformable Parts Model (DPM) [@deformablePartsModel] to detect objects for use as loop-closure landmarks in their system. All of these approaches require a pre-existing database of either object types or specific objects to operate. These databases can be costly to construct, and these systems will fail in environments in which not enough landmarks belonging to the database are present.
Some work has explored the use of for localization. PoseNet [@posenet] is a that learns a mapping from images in an environment to metric camera poses, but it can only operate on the environment on which it was trained. In S[ü]{}nderhauf et al. [@convWholeImagePlaceRecog], the intermediate activations of a trained for image classification were used as whole-image descriptors for place recognition, a non-metric form of global localization. In a similar fashion, Vysotska et al. [@vysotska2016lazy] use whole-image descriptors from a in a SeqSLAM-like framework. Subsequent work of S[ü]{}nderhauf et al. [@convNetLandmarks] refined this approach by using the same descriptor for object proposals within an image instead of the whole image. Cascianelli et al. [@cascianelli2017robust] and Panphattarasap et al. [@panphattarasap2016visual] both expand on this technique. These works consider only place recognition, however, and do not attempt to deal with the more challenging problem of full global localization (which necessitates returning a pose estimate). Schmidt et al. [@denseDeepPointDescriptors] and Simo-Serra et al. [@sparseDeepPointDescriptors] have both explored the idea of learning point-feature descriptors with a , which could replace classical point features in a bag-of-words model.
When exploring robustness to perspective change, all of these works only consider positional variations of at most a few meters, when the scenes exhibit within-image scale variations of tens or hundreds of meters, and when the reference or training datasets consisted of images taken over traversals of environments ranging from hundreds to thousands of meters. As a result, little significant change in scale exists between map images and query images in these experiments. To the best of our knowledge, ours is the first to attempt to combine deep object-like features and point features into a single, unified representation of landmarks. This synthesis provides superior metric localization to either technique in isolation, particularly under significant ($3\times$ and greater) changes in scale.
PROPOSED SYSTEM {#sec:system}
===============
The first stage of our metric localization pipeline consists in detecting objects in a pair of images, computing convolutional descriptors for them, and matching these descriptors between images. Our approach here closely follows that used for image-retrieval by S[ü]{}nderhauf et al. [@convWholeImagePlaceRecog]; we differ in using Selective Search (SS), as proposed by Uijlings et al. [@selectiveSearch], to propose object regions, and in our use of a more recent architecture.
To extract objects from an image, Selective Search object proposals are first extracted from the image, and filtered to remove objects with bounding boxes less than 200 pixels in size and with aspect ratio greater than 3 or less than 1/3. The image regions defined by each surviving SS bounding box are then extracted from the image, rescaled to a fixed size via bilinear interpolation, and run through a . We use a ResNet-50 architecture trained on the ImageNet image-classification dataset, as described in He et al. [@resnet]. Experiments were run using six different layers of the network as feature descriptors, and with inputs to the network of four different resolutions. The network layers and resolutions are listed in Table \[table:resNetSizes\].
Having extracted objects and their descriptors from a pair of images, we perform brute-force matching of the objects between the images. Following [@convNetLandmarks], we take the match of each object descriptor $\mathbf{u}$ in image $i$ to be the descriptor $\mathbf{v}$ in image $j$ that has the smallest cosine distance from $\mathbf{u}$, defined as $t_{cos. err.}(\mathbf{u}, \mathbf{v}) = 1 - \frac{\mathbf{u} \cdot \mathbf{v}}{||\mathbf{u}||_2 \cdot ||\mathbf{v}||_2}$. Matches are validated by cross-checking; a match $(\mathbf{u}, \mathbf{v})$ is only considered valid if $\mathbf{u}$ is the most similar object to $\mathbf{v}$ in image $i$ and $\mathbf{v}$ is the most similar object to $\mathbf{u}$ in image $j$.
Input res. pool1 res2c res3d res4f res5c pool5
------------------ ------- ------- --------- ------- ---------- -------
$224 \times 224$ 201k 803k 401k 200k **100k** 2k
$128 \times 128$ 66k 262k 131k 65k 131k 8k
$64 \times 64$ 16k 66k **32k** 66k **131k** 8k
$32 \times 32$ 4k 16k 32k 66k 131k 8k
: The sizes, as a number of floating-point values, of the output layers of ResNet-50 at different input resolutions. Values in bold indicate layer-resolution pairs which provided the best results in any of our experiments.
\[table:resNetSizes\]
Once object matches are found, we extract features from both images, using 3 octave layers, an initial Gaussian with $\sigma=1.6$, an edge threshold of 10, and a contrast threshold of 0.04. For each pair of matched objects, we match features that lie inside the corresponding bounding boxes to one another. features are matched via their Euclidean distance, and cross-checking is again used to filter out bad matches. By limiting the space over which we search for matches to matched object regions, we hypothesize that the scope for error in matching will be significantly reduced, and thus the accuracy of the resulting metric pose estimates will be increased. As a baseline against which to compare our results, experiments were also run using alone, with no objects, and objects alone, without features - this last is essentially a naïve application of the place-recognition system of S[ü]{}nderhauf [@convNetLandmarks] to metric localization. In these baseline experiments, matching was performed in the same way, but the search for matches was conducted over all features in both images. When object proposals alone were used, they were matched in the same manner described above, and their bounding box centers were used as match points.
The resulting set of match points are used to produce a metric pose estimate. Depending on the experiment, we compute either a homography $H$ or an essential matrix $E$ [@Hartley2004]. In either case, the calculation of $H$ or $E$ from point correspondences is done via a RANSAC algorithm with an inlier threshold of 6, measured in pixel units.
![A simplified illustration of our object-detection architecture.[]{data-label="fig:ssResNetArch"}](Images/SS_and_ResNet.pdf){width="8.5cm"}
KITTI EXPERIMENTS {#sec:kitti}
=================
Experimental Setup
------------------
To evaluate the robustness of our proposed method to changes in scale, we conducted experiments on the KITTI Odometry benchmark dataset [@kitti]. This dataset consists of data sequences from a variety of sensors, including colour stereo imagery captured at a 15Hz frame rate, taken from a sensor rig mounted on a car as it drives along twenty-two distinct routes in the daytime. Eleven of these sequences contain precise ground truth poses for each camera frame taken on each trajectory. These trajectories were used to evaluate the proposed method.
Our evaluation consisted of first subsampling each sequence by taking every fifth frame, to make the size of the overall dataset more manageable and increase the level of scale change present between adjacent frames in the sequence. A set of image pairs was generated for each subsampled sequence by taking each frame $i$ in the sequence and pairing $i$ with the 10 subsequent frames, $i + j\ \forall j \in [1, 10]$. Each successive value of $j$ gave an image pair $(i, i+j)$ with a greater degree of visual scale change, as shown in Fig. \[fig:kittiScalePairSamples\].
We finally filtered out any frame pairs whose gaze directions differed by more than $45^\circ$ in any axis, in order to consider only pairs that actually look at the same scene (in practice, only the yaw differs significantly in KITTI). In total, 40,748 image pairs were used in our evaluation. For each image pair, the images from the left colour camera (designated camera 2 in KITTI) were used for localization. An example set of images is shown in Fig. \[fig:kittiScalePairSamples\].
![Sample frame from sequence 00. The frames below it are separated by $j = 1, 5, 10$ in the subsampled sequence, respectively. This gives an indication of the range of visual scale changes observed over all the pairs of the dataset.[]{data-label="fig:kittiScalePairSamples"}](Images/kitti/image_sep_small.png){width="6cm"}
To estimate a transform between an image pair, a set of point matches was produced between the two images according to each of the three methods we compare, as described in section \[sec:system\]. In each case, these point matches were used to estimate an essential matrix $E$, from which was derived a pose estimate $(\mathbf{q}_e, \mathbf{t}_e)$ via a standard method of applying SVD and cheirality checking [@Hartley2004]. $(\mathbf{q}_e, \mathbf{t}_e)$ describes the transform between the two frames. To assess the quality of the estimate, we used two error metrics. The first was the relative positional error, as defined in Eq. \[eqn:relDist\]:
$$\label{eqn:relDist}
t_{err} = \frac{||\mathbf{t}_g - \mathbf{t}_e||_2}{||\mathbf{t}_g||_2 + ||\mathbf{t}_e||_2}$$
where $t_g$ is the ground-truth translation between the two frames and $t_e$ is the estimated translation. We normalize the vector from the estimated pose to the true pose to remove any correlation of that vector’s length with the magnitude of the true translation. Values of $t_{err}$ range from 0 to 1.
The second error metric was the rotational error, which following [@rotationMetrics] is defined in Eq. \[eqn:rotDist\]:
$$\label{eqn:rotDist}
r_{err} = 1 - |\mathbf{q}_g \cdot \mathbf{q}_e|$$
Where $\mathbf{q}_g$ and $\mathbf{q}_e$ are quaternions representing the ground-truth and estimated gaze directions, respectively. For some image pairs, no pose could be estimated, due to insufficient or inconsistent point matches. We refer to this as localization failure, and for both $t_{err}$ and $r_{err}$ we substitute a value of 1, the maximum possible error under each metric, in these failure cases.
A preliminary evaluation was carried out over the space of CNN input resolutions and output layers by running them on the first 1000 image pairs from the first subsampled sequence (sequence 00). We found that using an input resolution of $224 \times 224$ and the res5c feature layer as output gave both the highest accuracy and lowest localization failure rate. This configuration was used for all object-landmark experiments on KITTI that we describe below.
Results {#subsec:kittiResults}
-------
All metrics were plotted against the ground-truth translational distance, $||\mathbf{t}_g||_2$, between the frames in the image pairs. To make these plots readable, we grouped image pairs by their frame-separation $j$, and plotted the mean error of each group against its mean ground-truth distance, in Fig. \[fig:kittiPosErr\] (for $t_{err}$) and Fig. \[fig:kittiRotErr\] (for $r_{err}$). A logarithmic curve was fitted against each, as we expected that performance would initially worsen rapidly with distance, then level off. We also display the failure rate of each group versus the group’s mean distance in Fig. \[fig:kittiFailureRates\].
--------------------------------------------------------------------------------------
Method $t_{err}$ $r_{err}$ failure count
----------------- ---------------------------------------- ----------- ---------------
SIFT only 0.680 0.149 1854
Objects only 0.744 0.232 7146
Proposed method **[0.641]{} & **[0.086]{} & **[785]{}\
******
--------------------------------------------------------------------------------------
: []{data-label="table:kittiPerformance"}
![This plot shows the mean normalized positional error ($t_{err}$) versus the mean inter-camera distance of each group of image pairs. Image pairs are grouped by the number of frames separating them in the sequence, from 1 to 10. $t_{err}$ is a unitless error metric that ranges from 0 (best) to 1 (worst). The improvement of the proposed method over SIFT is small but consistent.[]{data-label="fig:kittiPosErr"}](Images/kitti/All_relative_T_error.png){width="7.5cm"}
![The mean rotational error ($r_{err}$) versus the mean inter-camera distance of each group of image pairs. ($r_{err}$), like ($t_{err}$), is a unitless error metric that ranges from 0 (best) to 1 (worst). This metric shows a more significant improvement over SIFT in the proposed method.[]{data-label="fig:kittiRotErr"}](Images/kitti/All_rotational_error.png){width="7.5cm"}
![The fraction of the pairs in each image-pair group for which localization failure occurred, versus the mean inter-camera distance of each group. In all groups, our proposed method has far fewer failures than either SIFT or object features alone.[]{data-label="fig:kittiFailureRates"}](Images/kitti/All_failure_rates.png){width="7.5cm"}
The overall performance of each method across all pairs is provided in Table \[table:kittiPerformance\]. This table shows that our proposed method improves on SIFT under each metric: a small improvement of 6% in $t_{err}$, and more significant improvements of 43% in $r_{err}$, and 58% in failure rate, overall. Meanwhile, the objects-only method performs significantly worse than both our method and SIFT on all metrics and at all pair distances.
Fig. \[fig:kittiRotErr\] shows that on $r_{err}$ the improvement of our method over SIFT is negligible at $j=1$, but grows significantly and consistently with the distance between frames. In Fig. \[fig:kittiPosErr\] meanwhile, we see that on $t_{err}$ the improvement grows at first, and is greater than 0.05 for most of the intermediate gaps, but shrinks again at the largest gaps. Fig. \[fig:kittiFailureRates\] shows similar behaviour in the localization failure rate - it is lowest for all methods at the largest gaps.
From visual inspection of these extreme image pairs, this improvement at high $j$ appears to be caused by sections where the vehicle drives down a long, straight road for some distance. In these cases, the visual scale of objects visible near the end of the road will show little change over even a gap of $j = 10$, making localization relatively easy. Unlike more winding roads, such long, straight sections will not have any high-$j$ pairs removed due to the images being on either side of a bend in the road, meaning that the high-$j$ groups will contain disproportionately many pairs from these straight sections.
MONTREAL IMAGE PAIR EXPERIMENTS
===============================
Experimental Setup
------------------
To test the effectiveness of the proposed system in a real-world scenario, a set of 31 image pairs were taken across eleven scenes surrounding the Montreal campus. Scenes were chosen to contain a roughly-centred object of approximately uniform depth in the scene, so that a uniform change in image scale could be achieved by taking images at various distances from the object. This ensures that successful matches must be made under one change in scale, and makes the relationship between the images amenable to description by a homography. The image pairs exhibit changes in scale ranging from factors of about 1.5 to about 7, with the exception of one image pair showing scale change of about 15 in a prominent foreground object. All images were taken using the rear-facing camera of a Samsung Galaxy S3 phone, and were downsampled to $1200\times900$ pixels via bilinear interpolation for all experiments. Each image pair was hand-annotated with a set of 10 point correspondences, distributed approximately evenly over the nearer image in each pair. We have made this dataset publicly available [^2].
The proposed system was used to compute point matches between each image pair, and from these point matches, a homography $H$ was computed as described in section \[sec:system\]. $H$ was used to calculate the total symmetric transfer error (STE) for the image pair $e$ over the ground truth points:
$$\textup{STE} = \sum^N_i||p^i_{far} - H p^i_{near}||_2 + ||p^i_{near} - H^{-1}p^i_{far}||_2$$
Whenever no $H$ could be found for an image pair by some method, its error on that image pair was set to the maximum STE we observed for any attempted method, $\textup{STE}_\textup{max} = 15,833,861,380.8$. The plain STE ranges over many orders of magnitude on this dataset, so we present the results using the logarithmic STE, making the results easier to interpret.
The same set of parameters were run over this dataset as in the KITTI experiments - our system at six network layers and four input resolutions, plus alone and objects alone, for comparison. However, the results from objects alone were substantially worse at all configurations than those of either SIFT or the proposed method, similar to what we observed in section \[sec:kitti\]. For the sake of brevity, we ignore the objects-only results in the discussion and figures below.
Results {#subsec:realWorldResults}
-------
Table \[table:realErrorByLayer\] shows the performance of each feature layer and each input resolution over the whole Montreal dataset, and shows the results from using features alone as well. As this table shows, the total error using just features is significantly greater than that of the best-performing input resolution for each feature layer. Also, the average error of the intermediate layers res2c, res3d, and res4f, are all very comparable. It is interesting to note that in this experiment, more intermediate layers are favoured, while the KITTI experiments favoured the highest resolution and the second-deepest layer of the network. This may arise from the difference in the native resolution of the images - KITTI’s image resolutions vary from sequence to sequence, but are all close to $1230\times370$.
Fig. \[fig:topThreeLayersPerPair\] show the error of each of the three best-performing configurations, as well as the -only approach, on each of the image pairs in the dataset, plotted versus the median scale change over all pairs of ground-truth matches $(p^i, p^j)$ in each image. The scale change between matches $(p^i, p^j)$ is defined as: $\textup{scale change}_{i,j} = \frac{||p^i_{near} - p^j_{near}||_2}{||p^i_{far} - p^j_{far}||_2}$. The lines of best fit for each method further emphasize the improvement of our system over features at all scale factors up to 6. The best-fit lines for all of the top-three configurations of our system overlap almost perfectly, although there is a fair degree of variance in their performances on individual examples.
The use of homographies to relate the image pairs allows us to visually inspect the quality of the estimated $H$, by using $H$ to map all pixels in the farther image to their estimated locations in the nearer image. Visual inspection of these mappings for the 31 image pairs confirm that those configurations with lower logarithmic STEs tend to have more correct-looking mappings, although all configurations of our system with mean logarithmic STE $< 10$ produce comparable mappings for most pairs, and on some pairs, higher-error configurations such as res4f with $64 \times 64$-pixel inputs produce a subjectively better mapping than the lowest-error configuration. Fig. \[fig:nearTrottier2\] and Fig. \[fig:nearTrotter1\] display some example homography mappings.
Input res. pool1 res2c res3d res4f res5c pool5
------------------ -------- -------- ----------- -------- -------- --------
$224 \times 224$ 11.102 13.057 9.631 10.277 10.537 10.011
$128 \times 128$ 10.389 11.716 10.231 9.458 9.930 9.505
$64 \times 64$ 10.921 9.381 **9.339** 9.777 10.234 9.667
$32 \times 32$ 10.134 10.162 10.473 9.658 10.301 10.607
SIFT 10.654
: A table showing the logarithmic STE of each configuration of the system, averaged over all image pairs. The best-performing feature overall is res3d with a $64 \times 64$ input size, followed closely by res2c with $64 \times 64$ inputs and res4f with $128 \times 128$ inputs. The mean log. STE of features alone is presented as well, for comparison.[]{data-label="table:realErrorByLayer"}
![The error of features alone, and the three best-performing configurations of our system, on each image pair in the dataset, plotted versus the median scale change exhibited in the image pair, along with a line of best fit for each method.[]{data-label="fig:topThreeLayersPerPair"}](Images/realWorld/errorsVsScale.png){width="9cm"}
\
Far Image
Near Image
SIFT
\
\
res3d, $64 \times 64$ input
res4f, $128 \times 128$ input
res5c, $128 \times 128$ input
\
CONCLUSIONS
===========
One strength of our proposed system is that it requires no domain-specific training, making use only of a pre-trained . However, as future work we wish to explore the possibility of training a with the specific objective of producing a scale- and perspective-invariant object descriptor, as doing so may result in more accurate matching of objects. We also wish to explore the possibility that including matches from multiple layers of the network in the localization process could improve the system’s accuracy.
The most natural extension of this work, however, is to extend it to the full global-localization problem, where the system must localize within a large map or database of images with no prior on the position, and must moreover do so across major scale changes. Depending on the scenario, this may require combining our localization method with a similarly scale-robust place-recognition system.
We have shown that by combining deep learning with classical methods, we can perform accurate localization across major changes in scale. Our system uses a pre-trained deep network to describe arbitrary objects and correctly match them between images for use as navigation landmarks. Restricting feature matching to matched object regions substantially improves the robustness of matching both to changes in image noise and to changes in scale. Despite much prior work on place recognition and localization using both classical methods and deep learning, our result sets a new benchmark for metric localization performance across significant scale changes.
[^1]: $^{1}$Andrew Holliday and Gregory Dudek are with the Center for Intelligent Machines, McGill University, 845 Sherbrooke St, Montreal, Canada [ahollid@cim.mcgill.ca, dudek@cim.mcgill.ca]{}
[^2]: <http://www.cim.mcgill.ca/~mrl/montreal_scale_pairs/>
|
---
abstract: |
Building upon earlier observations which demonstrate substantial star-to-star differences in the carbon abundances of M13 subgiants, we present new Keck LRIS spectra reaching more that 1.5 mag below the M13 main-sequence turn-off (to V $\approx$ 20). Our analysis reveals a distribution of C abundances similar to that found among the subgiants, implying little change in the compositions of the M13 stars at least through the main-sequence turn-off. We presume these differences to be the result of some process operating early in the cluster history.
Additional spectra of previously studied bright M13 giants have been obtained with the Hale 5-m. A comparison of C abundances derived using the present methods and those from the literature yield a mean difference of 0.03$\pm$0.14 dex for four stars in common with @1996AJ....112.1511S and 0.14$\pm$0.07 dex for stars also observed by @1981ApJS...47....1S (if one extreme case is removed). We conclude that the lower surface C abundances of these luminous giants as compared to the subgiants and main-sequence stars are likely the result of mixing rather than a difference in our abundance scales.
NH band strengths have also been measured for a handful of the most luminous M13 turn-off stars. While molecular band formation in such stars is weak, significant star-to-star NH band strength differences are present. Moreover, for the stars with both C and N measurements, differences between stars in these two elements appear to be anticorrelated.
Finally, the most recent C and N abundances for main-sequence, main-sequence turn-off, and subgiant stars in 47 Tuc, M71, M5, and the present M13 data are compared.
author:
- 'Michael M. Briley, Judith G. Cohen, Peter B. Stetson'
title: 'The Chemical Inhomogeneity of Faint M13 Stars: C and N Abundances '
---
INTRODUCTION
============
It has been known since the early 1970’s that the otherwise indistinguishable members of any given Galactic globular cluster (GC) exhibit significant star-to-star variations in surface abundances of certain light elements (most notably C and N, as well as O, and often Na, Al, and Mg)[^1]. However, while the abundance patterns commonly observed point to an origin in proton capture nucleosynthesis [@1990SvAL...16..275D; @1995PASP..107.1177L; @1996ApJ...464L..79C], identification of the specific reaction site(s) and a full theoretical description of the abundance modifying process(es) remain uncertain. As has been pointed out in numerous reviews [see @1987PASP...99...67S; @1994PASP..106..553K; @1998IAUS..189..193D] two possibilities exist:
First, the present day cluster stars may have modified their own surface compositions through some mixing process not included in standard models (i.e., an “in situ” scenario). By far the most promising candidate site in this regard is the region above the H-burning shell after first dredge-up in evolving cluster giants, where conditions for partial CN and possibly ON-cycle reactions exist [see @1979ApJ...229..624S for one of the earliest treatments]. Subsequent circulation of this material into the stellar envelope via meridional currents or turbulent diffusion [for example @2003ApJ...593..509D] will result in decreasing C abundances and increasing N with evolutionary state as has been observed along the red giant branches (RGB) of several metal-poor clusters [see @1982ApJS...49..207C; @1983ApJ...266..144T; @1990ApJ...359..307B for classic examples]. Moreover, the operation of such a mechanism can at least qualitatively explain the O and Mg versus Na and Al anticorrelations found among the most luminous red giants in several clusters [e.g. @1998AJ....115.1500K and references therein].
Common to all models of this process is the prohibition of “extra mixing” by the molecular weight gradient left behind by the inward excursion of the convective envelope during first dredge-up. Only after the molecular weight discontinuity has been destroyed by the outward moving H-burning shell, an event marked by the RGB luminosity function (LF) bump, is mixing expected to take place . This theoretical prediction appears to be borne out by observations of decreased Li abundances following the LF bump in NGC 6752 and similar drops in [$^{12}$C$/^{13}$C]{} seen by @2003ApJ...585L..45S in NGC 6528 and M4.
However, this cannot be the entire picture. As early as 1978, it was noted by @1978ApJ...223L.117H that the subgiant branch (SGB) and likely the main-sequence (MS) stars of 47 Tuc also possessed star-to-star differences in CH and CN band strengths. This has been most recently followed in 47 Tuc to $\approx 2.5$ mag below the MS turn-off (MSTO) by @2003AJ....125..197H. An analysis of their observed CN and CH band strengths yields factors of 10 variations in N anticorrelated with factors of 3 differences in C [@Briley-2004], matching those found among the more evolved members. Such CN and CH (N and C) variations have also been shown to exist among the MS, MSTO, or SGB stars of NGC 6752, M71, and M5 [@1991ApJ...381..160S; @1999AJ....117.2434C; @2002AJ....123.2525C] Moreover, star-to-star variations in Na, Al, O, and Mg, similar to those found among the luminous cluster stars, have been identified among the SGB and MSTO stars of 47 Tuc [@1996Natur.383..604B], NGC 6752 , M71 [@2002AJ....123.3277R], and M5 [@2003AJ....125..224R]. Although the various correlations and anticorrelations among these elements suggest the presence of material exposed to proton-capture reactions, such stars lie well below the LF bump and, particularly in the case of the MS stars, no mechanism is known for circulating significant quantities of CN(O) nucleosynthesized material to their surfaces.
Thus the second possible origin of the GC abundance variations - they have been set in place before RGB ascent and are due to the operation of some mechanism early in the cluster history (sometimes referred to as a “primordial” scenario). A number of possibilities exist as are discussed extensively by @1998MNRAS.298..601C, including: that the proto-cluster gas was inhomogeneous in these elements (a true primordial origin), that there was an extended period of star formation of sufficient duration to allow some low-mass stars to form with material ejected from more massive already-evolved cluster asymptotic giant branch (AGB) stars, or that the present day cluster stars have accreted AGB ejecta onto their surfaces after their formation. The appeal of AGB stars as sites of the proton capture nucelosynthesis lies in their ability to modify the cluster gas in light elements [including C, N, O, Al, Na, and Mg - see @2001ApJ...550L..65V] while not altering the abundances of heavy elements.
As the reader has likely noted, observational evidence exists for both mixing and early contamination scenarios, which has led many investigators to conclude that the compositions of the cluster stars we observe today are not the result of one or the other scenario exclusively, but rather both. Unfortunately, this leads to difficulties in disentangling the contributions of each process among the more luminous cluster stars - a problem that can only be reconciled by exploring the compositions of a cluster’s stars to the MSTO and below. Clearly, abundance trends found among a cluster’s MS stars reflect the original makeup of the bright giants, while deviations from this “baseline” composition are likely the result of mixing. This was recently demonstrated in the case of M13 by @2002ApJ...579L..17B (hereafter BCS02) - that a large spread in C abundances exists among the SGB stars of M13, which presumably reflects star-to-star variations in C abundances set early in the cluster history. However, the SGB C abundances also appear larger than those found by other investigators among the more luminous M13 stars, implying the operation of a mixing mechanism on the RGB which has reduced surface C abundances.
In the present paper, we return to M13 and extend our sample more than two magnitudes fainter to include MS stars. In addition, we have also obtained spectra of M13 bright giants observed in earlier studies to verify our abundance scale. Our results confirm those of BCS02 - that a primordial spread in the distribution of light elements exists in M13 which has further been modified during RGB ascent. Measurements of the 3360Å NH bands also were obtained for a handful of the more luminous stars in our sample. N abundances calculated from these bands suggest a C versus N anticorrelation at the level of the MSTO.
OBSERVATIONS
============
THE FAINT STAR SAMPLE
---------------------
The initial sample of stars in M13 was aimed to produce subgiants at the base of the RGB. It consisted of those stars from the photometric database [described by @1998PASP..110..533S; @2000PASP..112..925S] located more than 150 arcsec from the center of M13 (to avoid crowding) with $16.9<V<17.35$ and with $0.86 < (V-I) < 0.96$ mag. A slitmask with 0.7 arcsec wide slitlets, narrower than normal to enhance the spectral resolution and minimize contributions from adjacent stars in these crowded fields, was designed using JGC’s software from this sample and used in May 2001 with LRIS [@1995PASP..107..375O] at Keck. For this slitmask, as for all those used for the M13 stars, the red side of LRIS was set to include the NaD lines and H$\alpha$. We used the highest possible dispersion, 0.64Å/pixel (29 [km s$^{-1}$]{}/pixel) or 1.9Å/spectral resolution element there, to facilitate radial velocity confirmation of cluster membership. Given that the radial velocity of M13 is –246 [km s$^{-1}$]{}, distinguishing field stars from cluster members is then straightforward.
The blue side of LRIS [@LRIS] was used with the 600 line/mm grism blazed at 5000 Å. The detector for LRIS-B at that time was a 2048x2048 CCD not optimized for UV response. The spectra covered the range from $\sim$3400 to 5000 Å, thus including the strongest CN band at 3885Å and the G band of CH at 4300Å, with a resolution of $\sim$4 Å (1.0 Å/pixel). Two additional slitmasks were defined from this sample and used in May 2002 during less than ideal weather conditions for 6 exposures of 4800 sec each. The spectra were dithered by moving the stars along the length of the slitlets by 2 arcsec between exposures. These spectra are part of those presented in BCS02.
Because of the crowded fields, in addition to the intended stars some slitlets contained additional stars bright enough to provide suitable spectra, and these were utilized as well. As might be expected from the luminosity function, most of the secondary sample consists of stars at or just below the main-sequence turnoff. Hence subtraction of sequential exposures was not possible, and they were reduced individually using Figaro [@figaro] scripts, then the 1D spectra for each object were summed.
Based on the serendipitous main sequence stars found in the 2002 observations (see the plots in BCS02), we decided to try to reach main sequence stars well below the turnoff in M13, sufficiently faint to be cool enough to have detectable CH bands. The criteria used to define the sample from the photometric database were 19.3 $<$ I $<$ 19.7, V-I within 0.06 mag of the main sequence of M13, taken as 1.26+0.28(I-19.4), and located more than 200 arcsec from the center of M13. A single slitmask with 0.8 arcsec wide slitlets was designed and used at Keck with LRIS June 26, 2003. The blue spectra cover the full range from the atmospheric cutoff to 5000 Å, with 1.0 Å/pixel and a spectral resolution of $\sim$ 4 Å. Four exposures totalling 4200 sec were obtained. The new very high quantum efficiency detector for LRIS-B consisting of two 2kx4k Marconi CCDs, with 15 $\mu$ pixels and a readout noise of 4.0 e$^-$, was completed and installed into LRIS in June 2002, and so was available for these observations. The high UV throughput of LRIS-B with this new sensitive detector for the first time enabled us to reach the NH bands in the brighter of these stars with some precision. The locations of the faint program stars on the M13 color-magnitude diagram (CMD) are shown in Figure \[fig\_1\].
THE BRIGHT STAR SAMPLE
----------------------
There are published surveys [@1981ApJS...47....1S; @1996AJ....112.1511S] in which CH indices have been used to determine [\[C/Fe\]]{} values for large samples of the highest luminosity giants in M13. However, our Keck/LRIS sample of low luminosity stars in M13 has no overlap with these earlier works. To ensure that the merger of our data for faint stars in M13 with these published datasets for CH band strengths in M13 giants is valid, we need to verify the consistency of the different measurements of the CH indices and resulting abundances. To demonstrate this, we obtained new blue spectra of a small sample of bright giants with published CH band strengths from earlier studies, and remeasured their CH indices with the same procedures used for the lower luminosity M13 stars of our main sample (as described below). These spectra were taken in April and May, 2003 at the Hale Telescope on Palomar Mountain during observing runs intended primarily for other projects. The blue channel of the Double Spectrograph [@1982PASP...94..586O] was used with a 1200 line/mm grating and a Loral 512x2788 15$\mu$ pixel CCD, giving 0.55 Å/pixel with a spectral resolution of 1.9 Å for a 1 arcsec slit.
ANALYSIS
========
THE FAINT STAR SAMPLE
---------------------
Our analysis essentially repeats that of BCS02 and is fully described in @2001AJ....122..242B (hereafter BC01) and the reader is referred to these works for details. To summarize: strengths of the 4350Å CH (G) bands of our program stars were measured via the I(CH) index of @1999AJ....117.2428C [@1999AJ....117.2434C] - an index which compares the flux removed by the G-band to the adjacent continuum on both sides. The resulting indices, calculated using bandpasses corrected for the radial velocity of M13, are plotted for the sample of faint stars as a function of I magnitude in Figure \[fig\_2\]. The decrease in CH band strengths near I $\approx$ 18 is due to the higher temperatures of the MSTO stars (as pointed out by BC01). However, among the fainter MS stars in the sample (near I $\approx$ 19.5), the surface temperatures have dropped by roughly 300K and again a large and significant scatter in CH band strengths is apparent. The one sigma error bars plotted for the present sample have been determined entirely from Poisson statistics in the molecular-band and continuum spectral windows.
In a similar manner, the strength of absorption by the 3350Å NH band was measured in spectra of the more luminous members of the Keck MS/MSTO sample using the double sided logarithmic $s_{NH}$ index as defined in @1993PASP..105.1260B. The resulting indices (and one sigma Poisson error bars) are also plotted in Figure \[fig\_2\]. This marks the first time NH bands have been observed among such faint stars in a globular cluster. Spectra of two MSTO stars exhibiting differing NH band strengths, and two MS stars with differing CH band strengths are shown in Figure \[fig\_3\].
In order to relate the observed indices to the underlying [\[C/Fe\]]{}, we employ a series of synthetic spectra based on MARCS model atmospheres. Our models are those used in BC01 and BCS02 and based on the 16 Gyr $[Fe/H]$ = $-$1.48 O-enhanced isochrone grid of @2001ApJ...556..322B. The locations of the model points on the M13 I, V$-$I CMD are shown in Figure \[fig\_1\] assuming $(m-M)_V$ = 14.43 and a reddening of E(B$-$V) = 0.02 as in BC01 and BCS02.
From each model and a given set of C/N/O abundances, synthetic spectra were computed using the SSG program [@1994MNRAS.268..771B and references therein] and the line list of @1995AJ....110.3035T at a step size of 0.02Å (see BC01 for further details) assuming the average heavy element compositions of @1993AJ....106.1490K [@1997AJ....113..279K]. The result was then convolved with a Gaussian kernel to match the resolution of the observed spectra and I(CH) and $s_{NH}$ indices were measured. The model indices for I(CH) are illustrated in Figure \[fig\_2\] for four C abundances (as in BCS02): [\[C/Fe\]]{} = $-$0.85 and [\[C/Fe\]]{} = $-$1.1, which roughly match the observed compositions of M13’s CN-weak and strong bright giants respectively [see @1996AJ....112.1511S], and [\[C/Fe\]]{} = 0.0 and [\[C/Fe\]]{} = $-$0.5. Also plotted in Figure \[fig\_2\] are $s_{NH}$ indices for a variety of [\[N/Fe\]]{} values. Note that among these relatively warm MS/MSTO stars, there is little sensitivity in the CH (NH) band strengths to changes in N, O (C, O) abundances (as opposed to the cool giants where molecular equilibrium must be considered, particularly with regard to O). As a check of this, Table \[tbl-1\] shows the sensitivity of I(CH) and $s_{NH}$ to such changes in a cool MS model (T$_{eff}$=5601, log g=4.66, corresponding to an M13 MS star with $I$=19.60).
Following BCS02, we have applied the method of @1990ApJ...359..307B to convert the observed indices to corresponding C and N abundances: the model isoabundance curves were interpolated to the $M_I$ of each program star, and the observed index converted into the corresponding abundance based on the synthetic index at that $M_I$. Resulting C and N abundances are plotted in Figures \[fig\_4\] and \[fig\_5\]. Note that the large error bars which accompany the stars of Figure \[fig\_4\] near I=19 and the stars of low [\[C/Fe\]]{} ($\approx -1$) are due to the overall weakness of the CH bands — small errors in I(CH) therefore result in large changes in [\[C/Fe\]]{}. Likewise, a similar situation exists among the MSTO stars with measured NH band strengths.
THE BRIGHT STAR SAMPLE
----------------------
As with the faint stars, the I(CH) index was measured from the spectra of the six bright M13 giants. For each star this value was compared to synthetic indices generated from model atmospheres whose stellar parameters were taken from the high resolution analyses of @1993AJ....106.1490K [@1997AJ....113..279K], and @1996AJ....112..545P, including their heavy element and \[O/Fe\] abundances. Where available, [\[N/Fe\]]{} values from @1996AJ....112.1511S were also used. For two stars (K188 and III-7), N abundances were not available from the literature, and a value of +1.0 was assumed. For star III-7, an \[O/Fe\] of 0.0 was used. The model parameters and the resulting [\[C/Fe\]]{} abundance which matched the observed I(CH) indices are listed in Table \[tbl-2\] along with the C abundances from @1996AJ....112.1511S and @1981ApJS...47....1S.
For the four bright giants in common with @1996AJ....112.1511S, we find an average offset of 0.03($\pm$ 0.14) dex in [\[C/Fe\]]{} (present $-$ Smith). We therefore consider our C abundances to be essentially on the same scale, as might be expected considering the similar analysis tools used. The difference between our results and those of @1981ApJS...47....1S are somewhat larger: 0.25($\pm$0.23). However, almost half of this offset is driven by the result for II-76. Excluding this star reduces the average difference to 0.14($\pm$0.07). Note that II-76 has both a high \[O/Fe\] and a lower [\[N/Fe\]]{} abundance as might be expected from a star with a lesser amount of CN(O)-cycle material in its atmosphere . The source of this discrepancy is likely the cooler model used for II-76 by Suntzeff (T$_{eff}$ = 4220K versus the 4350K used here), as well as the lower O abundance (\[O/Fe\] = 0.0) and higher microturbulent velocity (2.5 [km s$^{-1}$]{}). Repeating our analysis with the values used by Suntzeff reduces our resulting [\[C/Fe\]]{} by 0.32 dex to $-$0.96. The luminous stars of Suntzeff plotted in Figure \[fig\_4\] have therefore been shifted by 0.14 in [\[C/Fe\]]{} to place them on our abundance scale.
Given the use of the same modeling codes, line lists, and CH indices throughout our analysis, we presume the resulting C abundances from both the faint and bright star samples, as well as those of @1996AJ....112.1511S and @1981ApJS...47....1S (with the appropriate shift), to be on the same abundance scale. Any systematic differences due to different telescope/spectrograph systems will be minimized by the use of the I(CH) index which uses continuum bands both blueward and redward of the CH feature to remove slope differences due to variations in instrumental response.
RESULTS
=======
There are several points to be made about the present results, which are given in tabular form in Tables \[tbl-A1\] and \[tbl-A2\] of Appendix A. First, as can be seen in Figure \[fig\_2\], significant differences in [\[C/Fe\]]{} exist among stars at least 1.5 mag fainter than the MSTO in M13. This corresponds to a mass of approximately 0.66 $M_\odot$ using the isochrone of Figure \[fig\_1\]. Among these old MS stars, CN(O)-cycle reactions are entirely confined to the central core [see for example Figures 4 and 5 of @2002ApJ...568..979R] and as has been pointed out by numerous investigators, MS stars such as these are not thought to possess a mechanism that connects their surface with regions of energy generation (namely the core). Indeed, should such mixing take place, the subsequent paths of the stars in the CMD would be radically altered by the infusion of fresh H into the core [e.g. @1988PASP..100..314V]. One must conclude the source of the observed differences in [\[C/Fe\]]{} is likely not the stars themselves. Moreover, the values of [\[C/Fe\]]{} among the MS stars are consistent with those found by BCS02 among the M13 SGB stars (see Figure \[fig\_4\]) and imply little change in composition has occurred from the MS to at least the base of the SGB.
Figure \[fig\_4\] also includes the [\[C/Fe\]]{} values of @1996AJ....112.1511S and @1981ApJS...47....1S (shifted upwards by 0.14 dex). As discussed in BCS02, there appears to be a marked decline in [\[C/Fe\]]{} towards higher luminosities among the M13 giants. Clearly an evolutionary change such as this can be best interpreted as the result of a mixing process bringing up C-depleted material from a region in which at least CN-cycle reactions are operating (see BCS02 for a more detailed discussion). Also shown in Figure \[fig\_4\] is the location of the LF bump in M13 [from @1998MNRAS.293..434P] — the event which marks the destruction of the molecular weight gradient thought to inhibit deep mixing. Unfortunately, the luminosity at which the onset of C depletion begins is uncertain due to the gap in the available data (from 15 $<$ V $<$ 17). However, since an extrapolation of the trend in giant-branch \[C/Fe\] abundance faintward intersects the magnitude of the LF bump at the average abundance of the fainter stars, it is reasonable to infer that the abundance decline begins near that event; neither a significant decrease nor a significant increase in carbon abundance with a subsequent recovery to the original value hidden within the gap in our data is reasonably to be expected.
The [\[N/Fe\]]{} values determined from the NH band strengths of the MSTO stars are plotted in Figure \[fig\_5\]. Although the error bars are admittedly larger than one would like owing to the weaknesses of the CH and NH bands among the warmer MSTO stars, a general anticorrelation between [\[C/Fe\]]{} and [\[N/Fe\]]{} is suggested. Note that these abundances do not suffer from the inherent tendency towards C/N anticorrelations of analyses based on CH and CN band strengths. Of course an overall C/N anticorrelation is known to be present among the evolved M13 stars and the values for the bright RGB stars of @1996AJ....112.1511S are also shown in Figure \[fig\_5\]. Immediately apparent is the shift of the RGB stars towards lower [\[C/Fe\]]{}, as is expected from Figure \[fig\_4\]. If C-poor/N-rich material is indeed being circulated into the stellar envelopes during RGB ascent, the lack of near solar [\[N/Fe\]]{} RGB stars is also explained (although the error bars on the two lower [\[N/Fe\]]{} MSTO stars severely limit the weight which can be placed on this statement). That higher N abundances do not appear to be found among the RGB stars under these circumstances is perhaps not a surprise if these stars are already leaving the MSTO with large [\[N/Fe\]]{} overabundances: an M13 MSTO star with [\[C/Fe\]]{} = $-$0.4 and [\[N/Fe\]]{} = 1.0 which undergoes a mixing episode reducing [\[C/Fe\]]{} to $-$1.2 will experience a rise in [\[N/Fe\]]{} of only 0.05 dex — in essence, the N abundances are already so large, the addition of freshly minted N via the CN-cycle results in only a small fractional change in [\[N/Fe\]]{}. Thus, while the error bars in Figure \[fig\_5\] are large, we can at least claim it is not inconsistent with the assertion that we are seeing substantial star-to-star variations in C (and N) set early in the cluster history, which are further being modified by mixing during RGB ascent. The possibility of also mixing ON-cycle material to the surface is more difficult to assess because of the large N variations among the MSTO stars. In the example above, an additional reduction in \[O/Fe\] from +0.45 to $-0.35$ would increase [\[N/Fe\]]{} by 0.46 dex — and among the bright giants, even larger O depletions (as much as \[O/Fe\] = $-0.7$ to $-0.8$) have been noted. Starting with an even larger N overabundance of +1.4 reduces the change in \[N/Fe\] to +0.25. However, at least from the small sample of Figure \[fig\_5\], it appears that none of the bright RGB stars possess larger N abundances than their MSTO counterparts, which in turn suggests the envelopes of at least the initially N-rich stars may not be cycled through a region of ON-cycle reactions while on the RGB. Clearly knowledge of the O abundances of the M13 MSTO stars would help settle this question.
A similar result was noted in the more metal-poor clusters M92 and M15 by @1982ApJS...49..207C and @1983ApJ...266..144T (respectively) — that substantial N overabundances are present from the SGB to AGB that are not necessarily correlated with C abundances. Indeed, an analogous situation can been seen in the present results and those of @2002AJ....123.2525C for M5 (see Figures \[fig\_5\] and \[fig\_6\]): the “higher” [\[C/Fe\]]{} MSTO stars (at $\approx -0.4$) span almost a dex in [\[N/Fe\]]{}. It is clear that if we are to ascribe the same mechanism to the origin of the SGB/MSTO inhomogeneities in these clusters, it must be operating at the MSTO or earlier.
DISCUSSION
==========
That significant and correlated star-to-star differences in C and N, as well as O, Na, Al, and Mg have been found among the SGB, MSTO, and MS stars of several clusters (see references above), implies the operation of some process external to the present stars, presumably having taken place early in the cluster history. The discussion of @1998MNRAS.298..601C includes a comprehensive look at various possibilities. It is worth while however, to revisit a few of the more critical constraints on any theory of the origin of the abundance variations.
First, whatever mechanism has altered the light-element compositions of the cluster stars has left the heavy elements essentially untouched, at least to the limits of our ability to determine them — the analysis of M5 by @2003AJ....125..224R is an excellent example. This alone would seem to exclude the possibility of the light-element variations arising from the merger of two distinct proto-cluster clouds (as has been pointed out by numerous authors).
Second, these abundance variations appear to be almost ubiquitous among the population of Galactic globular clusters. To highlight this, we have plotted in Figure \[fig\_6\] the [\[C/Fe\]]{} and [\[N/Fe\]]{} values for the present sample of M13 MSTO stars, the 47 Tuc MS stars of @Briley-2004, the M5 SGB stars of @2002AJ....123.2525C, and the MSTO stars of M71 from BC01. Note that BC01 did not directly extract C and N abundances from their observed indices — we have converted them here following the procedure outlined in @2002AJ....123.2525C and using the indices and models presented in BC01; the values are given in Appendix A, Table \[tbl-A3\].
Third, the elements which are observed to vary are associated with proton capture nucleosynthesis under conditions of CN and ON-cycling. The source/site must process these CNO-group elements and return this material to the cluster to be incorporated into the present population of low mass stars either before, during, or after their formation.
A popular model which fits these constraints is the incorporation of ejecta from intermediate mass (3-6$M_\odot$) AGB stars undergoing hot bottom burning and third dredge-up [see @2001ApJ...550L..65V], although difficulties such as the establishment of an O-Na anti-correlation remain [see for example @2003ApJ...590L..99D]. However, as is discussed in @2001AJ....122.2561B and BCS02, the quantities of material required to produce the observed star-to-star differences among the low luminosity stars (most notably extreme C depletions), which are clearly not diluted as the convective envelopes deepen during RGB ascent, rules out any sort of simple accretion model. Indeed, for the present M13 stars, roughly 70% of a C-poor MS star’s total mass must be captured ejecta if the accreted matter is completely free of C (see BCS02). It is of course unclear how such an enormous amount of material can be returned to the cluster without appealing to a shallow initial mass function [see @2001AJ....122.2561B], nor how the present stars can sweep up the necessary mass of ejecta . We note in Figure \[fig\_6\] that the depletions in C do appear smaller in the more metal-rich clusters M71 and 47 Tuc in accord with the prediction the of AGB ejecta models of @2001ApJ...550L..65V. Yet at the same time, if one presumes the highest [\[C/Fe\]]{} SGB/MSTO stars in M13 and M5 to represent the original (accretion free) C abundance of the cluster stars, they are still some 0.4 dex more C-poor than their 47 Tuc/M71 counterparts, implying either truly primordial (i.e., pre-accretion) differences in at least C or that nearly all the present stars in M13 and M5 have undergone at least some accretion of C-poor material. However, the spread in [\[N/Fe\]]{} is essentially identical among all four clusters. Clearly, knowledge of the patterns of \[O/Fe\] and \[Na/Fe\] among the present stars would help constrain the AGB ejecta theories.
An interesting counterpoint to this model is the scenario suggested by @1982ApJS...49..207C and @1983ApJ...266..144T to explain similar results among M92 and M15 SGB stars — that the stars of these clusters were inhomogeneously “polluted” by an injection of raw C from intermediate mass AGB stars which is subsequently converted into N in the present stars before SGB evolution thereby explaining both the C deficiencies and large N enhancements as well as star-to-star differences in (C+N). This has the additional advantage of requiring considerably more modest composition modifications (a factor of 4 or so in C from star to star), which in turn lowers the mass of captured ejecta required. However, to explain the large C depletions already in place by the MSTO, significant processing of the envelope through a region of CN-cycling must have taken place while the stars occupied the MS. One then returns to the difficulty of mixing in such stars discussed above.
Another site of the proton-capture reactions has recently been suggested by @Li-Burstein03, who note that the high mass (250-300 $M_\odot$) zero metallicity models of @2001ApJ...550..372F tend to mix He and He-burning products into their H-burning shells during the later stages of He-burning. This fresh C, N, O is partially processed into N while at the same time, the stars expand into red supergiants. If mass loss also occurs at this point, the cluster could be seeded with freshly produced C/O-poor, N-rich material. Such a scenario is presented within the context of the cluster formation history of — that the GCs formed from primordial material (zero-metal) that was subsequently enriched by the supernovae of massive stars before low mass stars could form. However, the problem remains that the production/seeding and mixing of the heavy-elements must be decoupled from that of the light-elements in order to explain the remarkable homogeneity of Fe, Ti, Ca, etc. within the GCs. In the context of GC formation in a well mixed supershell [e.g. @1991ApJ...376..115B] this is difficult to explain if the CNO-modified material is ejected prior to the driving supernovae and subsequent supershell expansion/mixing.
The entire Keck/LRIS user community owes a huge debt to Jerry Nelson, Gerry Smith, Bev Oke, and many other people who have worked to make the Keck Telescope and LRIS a reality. We are grateful to the W. M. Keck Foundation, and particularly its late president, Howard Keck, for the vision to fund the construction of the W. M. Keck Observatory. We also wish to express our thanks to Roger Bell whose SSG code was instrumental in this project and the anonymous referee for their suggestions. Partial support was provided by the National Science Foundation under grant AST-0098489 to MMB and grant AST-0205951 to JGC and by the F. John Barlow professorship and UW Oshkosh Faculty Development Program (MMB).
TABLES OF OBSERVED INDICES AND RESULTING ABUNDANCES
===================================================
[cccccccc]{} 41211\_2349 & 17.30 & 17.10 & 0.20 & -0.010 & -0.024 & - & -\
41217\_2535 & 17.63 & 16.88 & 0.75 & 0.104 & 0.228 & -0.49 & 1.09\
41132\_2535 & 18.20 & 17.61 & 0.59 & 0.013 & 0.060 & - & 1.37\
41185\_2646 & 18.52 & 17.98 & 0.54 & 0.008 & 0.031 & - & 1.24\
41165\_2813 & 18.64 & 18.10 & 0.54 & 0.034 & -0.026 & -0.25 & -0.14\
41204\_2622 & 18.89 & 18.34 & 0.55 & 0.034 & 0.004 & -0.39 & 0.81\
41302\_2212 & 18.90 & 18.32 & 0.58 & 0.057 & - & 0.30 & -\
41228\_2301 & 19.05 & 18.53 & 0.52 & 0.022 & 0.054 & - & 1.20\
41211\_2513 & 19.06 & 18.49 & 0.57 & 0.032 & -0.022 & -0.50 & 0.02\
41157\_2535 & 19.07 & 18.49 & 0.58 & 0.028 & 0.069 & -0.77 & 1.32\
41170\_2333 & 19.07 & 18.50 & 0.57 & 0.024 & 0.094 & - & 1.46\
41191\_2527 & 19.38 & 18.77 & 0.61 & 0.030 & - & -0.91 & -\
41274\_2133 & 19.41 & 18.73 & 0.68 & 0.000 & - & - & -\
41265\_2254 & 19.55 & 18.94 & 0.61 & 0.061 & - & -0.12 & -\
41197\_2648 & 19.58 & 18.97 & 0.61 & 0.038 & 0.175 & -0.66 & 1.47\
41171\_2802 & 19.92 & 19.33 & 0.59 & 0.043 & - & -0.93 & -\
41173\_2418 & 19.97 & 19.30 & 0.67 & 0.097 & - & -0.16 & -\
41188\_2516 & 19.98 & 19.31 & 0.67 & 0.089 & - & -0.24 & -\
41107\_2643 & 20.00 & 19.33 & 0.67 & 0.071 & - & -0.44 & -\
41334\_2223 & 20.00 & 19.35 & 0.65 & 0.070 & - & -0.47 & -\
41243\_2227 & 20.01 & 19.39 & 0.62 & 0.043 & - & -1.02 & -\
41122\_2707 & 20.03 & 19.36 & 0.67 & 0.100 & - & -0.21 & -\
41227\_2356 & 20.04 & 19.30 & 0.74 & 0.102 & - & -0.12 & -\
41120\_2625 & 20.05 & 19.36 & 0.69 & 0.032 & - & -1.47 & -\
41264\_2258 & 20.05 & 19.31 & 0.74 & 0.070 & - & -0.43 & -\
41296\_2221 & 20.05 & 19.40 & 0.65 & 0.113 & - & -0.15 & -\
41135\_2723 & 20.06 & 19.38 & 0.68 & 0.101 & - & -0.22 & -\
41277\_2355 & 20.11 & 19.35 & 0.76 & 0.049 & - & -0.78 & -\
41198\_2646 & 20.13 & 19.42 & 0.71 & 0.075 & - & -0.49 & -\
41219\_2530 & 20.15 & 19.48 & 0.67 & 0.037 & - & -1.41 & -\
41167\_2650 & 20.19 & 19.50 & 0.69 & 0.115 & - & -0.25 & -\
41276\_2132 & 20.21 & 19.60 & 0.61 & 0.077 & - & -0.60 & -\
41202\_2321 & 20.23 & 19.55 & 0.68 & 0.056 & - & -0.87 & -\
41210\_2353 & 20.23 & 19.56 & 0.67 & 0.127 & - & -0.23 & -\
41130\_2540 & 20.24 & 19.54 & 0.70 & 0.077 & - & -0.56 & -\
41340\_2151 & 20.24 & 19.55 & 0.69 & 0.104 & - & -0.38 & -\
41130\_2641 & 20.25 & 19.54 & 0.71 & 0.130 & - & -0.19 & -\
41131\_2639 & 20.26 & 19.50 & 0.76 & 0.086 & - & -0.47 & -\
41207\_2619 & 20.26 & 19.56 & 0.70 & 0.113 & - & -0.33 & -\
41243\_2229 & 20.27 & 19.56 & 0.71 & 0.096 & - & -0.45 & -\
41279\_2407 & 20.27 & 19.60 & 0.67 & 0.095 & - & -0.49 & -\
41191\_2530 & 20.28 & 19.54 & 0.74 & 0.113 & - & -0.31 & -\
41208\_2557 & 20.31 & 19.60 & 0.71 & 0.087 & - & -0.53 & -\
41255\_2229 & 20.32 & 19.55 & 0.77 & 0.103 & - & -0.39 & -\
41278\_2232 & 20.32 & 19.64 & 0.68 & 0.097 & - & -0.50 & -\
41301\_2213 & 20.32 & 19.65 & 0.67 & 0.125 & - & -0.35 & -\
41315\_2143 & 20.33 & 19.62 & 0.71 & 0.135 & - & -0.25 & -\
41270\_2213 & 20.35 & 19.60 & 0.75 & 0.105 & - & -0.43 & -\
41166\_2612 & 20.38 & 19.65 & 0.73 & 0.141 & - & -0.24 & -\
41183\_2653 & 20.40 & 19.66 & 0.74 & 0.118 & - & -0.41 & -\
41255\_2307 & 20.46 & 19.66 & 0.80 & 0.053 & - & -1.15 & -
[cccccc]{} 41230\_2604 & 16.83 & 16.00 & 0.83 & 0.166 & -0.34\
41244\_2423 & 16.88 & 16.05 & 0.83 & 0.182 & -0.25\
41224\_2734 & 16.92 & 16.11 & 0.81 & 0.154 & -0.40\
41299\_2630 & 16.95 & 16.12 & 0.83 & 0.212 & -0.07\
41213\_2642 & 16.99 & 16.20 & 0.79 & 0.098 & -0.72\
41259\_2821 & 17.03 & 16.23 & 0.80 & 0.182 & -0.22\
41210\_2830 & 17.07 & 16.30 & 0.77 & 0.168 & -0.29\
41320\_2941 & 17.07 & 16.31 & 0.76 & 0.118 & -0.56\
41207\_2719 & 17.09 & 16.31 & 0.78 & 0.136 & -0.47\
41296\_2957 & 17.11 & 16.34 & 0.77 & 0.201 & -0.10\
41249\_2549 & 17.30 & 16.46 & 0.84 & 0.104 & -0.63\
41210\_2834 & 17.05 & 16.48 & 0.57 & 0.128 & -0.48\
41301\_2440 & 17.32 & 16.49 & 0.83 & 0.208 & -0.03\
41188\_2619 & 17.34 & 16.57 & 0.77 & 0.081 & -0.84\
41260\_3026 & 17.35 & 16.60 & 0.75 & 0.112 & -0.55\
41212\_2744 & 17.39 & 16.60 & 0.79 & 0.165 & -0.24\
41256\_2801 & 17.43 & 16.66 & 0.77 & 0.082 & -0.79\
41260\_2850 & 17.43 & 16.69 & 0.74 & 0.127 & -0.43\
41340\_2401 & 17.54 & 16.72 & 0.82 & 0.072 & -0.92\
41252\_2524 & 17.53 & 16.73 & 0.80 & 0.149 & -0.28\
41282\_2908 & 17.49 & 16.76 & 0.73 & 0.094 & -0.62\
41284\_2922 & 17.54 & 16.81 & 0.73 & 0.113 & -0.47\
41217\_2534 & 17.63 & 16.88 & 0.75 & 0.106 & -0.47\
41260\_2459 & 17.69 & 16.88 & 0.81 & 0.116 & -0.40\
41256\_3005 & 17.67 & 16.94 & 0.73 & 0.023 & -\
41249\_2548 & 18.09 & 17.03 & 1.06 & -0.032 & -\
42103\_2722 & - & 17.03 & - & 0.091 & -\
41253\_2521 & 17.79 & 17.07 & 0.72 & 0.068 & -\
42104\_2748 & - & 17.15 & - & 0.067 & -\
42108\_2808 & - & 17.18 & - & 0.039 & -\
42071\_2457 & - & 17.20 & - & 0.051 & -\
42055\_2321 & - & 17.23 & - & 0.053 & -\
42097\_2610 & - & 17.24 & - & 0.042 & -\
42088\_2635 & - & 17.33 & - & 0.046 & -\
42077\_2623 & - & 17.40 & - & 0.033 & -\
41112\_2621 & 18.05 & 17.44 & 0.61 & 0.012 & -\
41284\_2930 & 18.02 & 17.47 & 0.55 & 0.031 & -\
42062\_2223 & - & 17.51 & - & 0.030 & -\
42095\_2656 & - & 17.52 & - & 0.101 & -\
42104\_2854 & - & 17.55 & - & 0.009 & -\
42068\_2553 & - & 17.60 & - & 0.017 & -\
42040\_2211 & - & 17.63 & - & 0.048 & -\
42073\_2606 & - & 17.68 & - & 0.016 & -\
42028\_2442 & - & 17.71 & - & 0.006 & -\
42033\_2200 & - & 17.72 & - & 0.015 & -\
42072\_2737 & 18.32 & 17.74 & 0.58 & 0.035 & -\
42071\_2508 & - & 17.74 & - & 0.010 & -\
42068\_2541 & - & 17.76 & - & 0.011 & -\
42129\_2833 & - & 17.78 & - & 0.039 & -\
42096\_2708 & - & 17.79 & - & 0.047 & -\
42071\_2515 & - & 17.92 & - & -0.002 & -\
41281\_2911 & 18.15 & 17.96 & 0.19 & 0.001 & -\
42051\_2422 & - & 17.96 & - & 0.018 & -\
42062\_2405 & - & 17.96 & - & 0.044 & -\
42064\_2355 & - & 17.96 & - & 0.044 & -\
42078\_2253 & - & 17.99 & - & 0.014 & -\
41213\_2651 & 18.54 & 18.00 & 0.54 & 0.020 & -\
42060\_2650 & - & 18.04 & - & 0.013 & -\
42034\_2250 & - & 18.04 & - & 0.043 & -\
42065\_2305 & - & 18.05 & - & 0.029 & -\
42029\_2445 & - & 18.07 & - & 0.013 & -\
42050\_2430 & - & 18.10 & - & 0.028 & -\
42058\_2452 & - & 18.17 & - & -0.002 & -\
41099\_2615 & 18.75 & 18.19 & 0.56 & -0.006 & -\
42064\_2349 & - & 18.21 & - & -0.009 & -\
41341\_2405 & 18.83 & 18.22 & 0.61 & 0.004 & -\
42035\_2345 & - & 18.24 & - & 0.012 & -\
42099\_2820 & - & 18.36 & - & 0.035 & -\
41136\_2455 & 19.01 & 18.42 & 0.59 & 0.028 & -\
41081\_2630 & 19.05 & 18.46 & 0.59 & 0.039 & -\
41157\_2535 & 19.07 & 18.49 & 0.58 & 0.024 & -\
41228\_2301 & 19.05 & 18.53 & 0.52 & 0.020 & -\
41121\_2548 & 19.12 & 18.54 & 0.58 & 0.009 & -\
41102\_2643 & 19.14 & 18.54 & 0.60 & 0.008 & -\
41096\_2617 & 19.14 & 18.56 & 0.58 & 0.130 & -\
41173\_2525 & 19.15 & 18.56 & 0.59 & 0.040 & -\
41150\_2415 & 19.16 & 18.56 & 0.60 & 0.021 & -\
41172\_2423 & 19.17 & 18.58 & 0.59 & 0.046 & -\
41216\_2412 & 19.18 & 18.61 & 0.57 & 0.034 & -\
41116\_2614 & 19.21 & 18.62 & 0.59 & 0.023 & -\
41280\_2145 & 19.21 & 18.62 & 0.59 & 0.034 & -\
41139\_2533 & 19.21 & 18.62 & 0.59 & 0.041 & -\
41317\_2148 & 19.21 & 18.66 & 0.55 & 0.038 & -\
41256\_2223 & 19.22 & 18.66 & 0.56 & 0.040 & -\
41204\_2446 & 19.27 & 18.66 & 0.61 & 0.054 & -\
41279\_2334 & 19.27 & 18.67 & 0.60 & 0.044 & -\
41200\_2315 & 19.28 & 18.68 & 0.60 & 0.060 & -\
41263\_2211 & 19.28 & 18.69 & 0.59 & 0.021 & -\
41236\_2215 & 19.37 & 18.77 & 0.60 & 0.019 & -\
41255\_2324 & 19.39 & 18.79 & 0.60 & 0.023 & -\
41174\_2358 & 19.39 & 18.79 & 0.60 & 0.030 & -\
41134\_2417 & 19.41 & 18.80 & 0.61 & 0.017 & -\
41231\_2331 & 19.42 & 18.81 & 0.61 & 0.048 & -\
41094\_2540 & 19.43 & 18.82 & 0.61 & 0.050 & -\
41136\_2550 & 19.45 & 18.85 & 0.60 & -0.005 & -\
41310\_2206 & 19.65 & 18.88 & - & 0.030 & -\
41290\_2218 & 19.49 & 18.89 & 0.60 & 0.016 & -\
41219\_2401 & 19.50 & 18.96 & 0.54 & 0.010 & -\
41161\_2532 & 19.80 & 19.18 & 0.62 & 0.029 & -\
41175\_2421 & 19.93 & 19.25 & 0.68 & 0.085 & -\
41286\_2327 & 20.13 & 19.40 & 0.73 & 0.090 & -\
41140\_2549 & 20.23 & 19.52 & 0.71 & 0.093 & -\
41266\_2209 & 20.43 & 19.71 & 0.72 & 0.093 & -
[ccccccc]{} C51228\_3737 & 17.00 & 1.38 & 0.138 & 0.131 & -0.17 & 0.32\
C51265\_3739 & 17.01 & 1.31 & 0.122 & 0.141 & -0.25 & 0.49\
C51314\_3755 & 17.01 & 1.34 & 0.096 & 0.340 & -0.39 & 1.41\
C51385\_4166 & 17.01 & 1.40 & 0.092 & 0.378 & -0.40 & 1.54\
C51312\_3634 & 17.03 & 1.35 & 0.156 & 0.388 & 0.04 & 1.19\
C51418\_4158 & 17.03 & 1.36 & 0.091 & 0.347 & -0.39 & 1.48\
C51419\_3870 & 17.03 & 1.38 & 0.112 & 0.333 & -0.24 & 1.29\
C51291\_3655 & 17.05 & 1.33 & 0.125 & 0.296 & -0.13 & 1.11\
C51417\_3943 & 17.05 & 1.41 & 0.146 & 0.124 & -0.03 & 0.25\
C51396\_4020 & 17.11 & 1.33 & 0.140 & 0.327 & 0.04 & 1.15\
C51285\_3749 & 17.12 & 1.29 & 0.120 & 0.142 & -0.10 & 0.55\
C51260\_4161 & 17.13 & 1.40 & 0.096 & 0.351 & -0.20 & 1.49\
C51266\_3848 & 17.13 & 1.36 & 0.090 & 0.280 & -0.28 & 1.34\
C51254\_3957 & 17.14 & 1.42 & 0.143 & 0.143 & 0.06 & 0.43\
C51413\_4033 & 17.14 & 1.27 & 0.092 & 0.236 & -0.27 & 1.18\
C51352\_4055 & 17.15 & 1.31 & 0.127 & 0.115 & -0.02 & 0.30\
C51424\_3823 & 17.15 & 1.31 & 0.086 & 0.279 & -0.29 & 1.38\
C51400\_3529 & 17.16 & 1.31 & 0.118 & 0.105 & -0.08 & 0.26\
C51250\_3763 & 17.17 & 1.39 & 0.136 & 0.107 & 0.05 & 0.18\
C51373\_3631 & 17.17 & 1.31 & 0.128 & 0.102 & -0.01 & 0.17\
C51368\_4074 & 17.18 & 1.31 & 0.077 & 0.253 & -0.37 & 1.39\
C51306\_3738 & 17.19 & 1.32 & 0.124 & 0.113 & -0.01 & 0.31\
C51277\_3950 & 17.20 & 1.34 & 0.094 & 0.255 & -0.20 & 1.25\
C51270\_3931 & 17.21 & 1.40 & 0.082 & 0.285 & -0.27 & 1.43\
C51386\_3659 & 17.21 & 1.28 & 0.123 & 0.103 & -0.01 & 0.21\
C51416\_3834 & 17.22 & 1.31 & 0.118 & 0.124 & -0.03 & 0.45\
C51266\_4149 & 17.24 & 1.34 & 0.092 & 0.273 & -0.18 & 1.32\
C51346\_4124 & 17.24 & 1.39 & 0.133 & 0.340 & 0.10 & 1.26\
C51267\_4025 & 17.25 & 1.35 & 0.127 & 0.121 & 0.04 & 0.38\
C51378\_3975 & 17.25 & 1.34 & 0.124 & 0.106 & 0.02 & 0.25\
C51404\_3918 & 17.25 & 1.35 & 0.111 & 0.109 & -0.07 & 0.35\
C51308\_3765 & 17.26 & 1.27 & 0.099 & 0.283 & -0.12 & 1.31\
C51316\_3960 & 17.26 & 1.29 & 0.071 & 0.276 & -0.37 & 1.53\
C51377\_3737 & 17.26 & 1.25 & 0.105 & 0.122 & -0.10 & 0.51\
C51430\_3648 & 17.29 & 1.34 & 0.083 & 0.234 & -0.25 & 1.27\
C51385\_3962 & 17.30 & 1.24 & 0.107 & 0.113 & - & -\
C51290\_3644 & 17.33 & 1.36 & 0.113 & 0.109 & -0.03 & 0.36\
C51224\_4027 & 17.35 & 1.31 & 0.098 & 0.161 & -0.12 & 0.84\
C51279\_3957 & 17.35 & 1.36 & 0.120 & 0.139 & 0.02 & 0.57\
C51287\_4050 & 17.35 & 1.38 & 0.116 & 0.135 & 0.00 & 0.55\
C51261\_4172 & 17.36 & 1.32 & 0.121 & 0.108 & 0.03 & 0.30\
C51281\_3638 & 17.37 & 1.28 & 0.111 & 0.113 & -0.04 & 0.41\
C51331\_4042 & 17.38 & 1.28 & 0.076 & 0.229 & -0.30 & 1.34\
C51252\_4126 & 17.39 & 1.39 & 0.098 & 0.134 & -0.12 & 0.68\
C51279\_3842 & 17.39 & 1.28 & 0.124 & 0.082 & 0.05 & -0.14\
C51310\_3849 & 17.39 & 1.29 & 0.109 & 0.140 & -0.04 & 0.64\
C51334\_3732 & 17.40 & 1.29 & 0.115 & 0.112 & 0.00 & 0.37\
C51262\_3874 & 17.42 & 1.33 & 0.114 & 0.103 & -0.01 & 0.29\
C51377\_3766 & 17.42 & 1.29 & 0.080 & 0.231 & -0.25 & 1.31\
C51243\_4217 & 17.43 & 1.35 & 0.072 & 0.211 & -0.34 & 1.31\
C51412\_4143 & 17.43 & 1.27 & 0.089 & 0.170 & - & -\
C51287\_3658 & 17.44 & 1.30 & 0.114 & 0.099 & -0.01 & 0.23\
C51315\_4161 & 17.44 & 1.30 & 0.134 & 0.458 & 0.22 & 1.60\
C51324\_3542 & 17.45 & 1.38 & 0.068 & 0.236 & -0.38 & 1.45\
C51405\_3749 & 17.45 & 1.24 & 0.072 & 0.234 & -0.31 & 1.39\
C51252\_3923 & 17.46 & 1.29 & 0.081 & 0.253 & -0.23 & 1.38\
C51235\_3931 & 17.47 & 1.34 & 0.106 & 0.106 & -0.06 & 0.37\
C51244\_3757 & 17.47 & 1.35 & 0.116 & 0.126 & 0.01 & 0.50\
C51296\_3969 & 17.48 & 1.25 & 0.113 & 0.086 & -0.02 & 0.02\
C51279\_4119 & 17.49 & 1.36 & 0.123 & 0.105 & 0.06 & 0.25\
C51289\_3768 & 17.49 & 1.23 & 0.139 & 0.261 & 0.18 & 1.01\
C51416\_4137 & 17.51 & 1.39 & 0.061 & 0.207 & -0.48 & 1.44\
C51229\_3628 & 17.52 & 1.28 & 0.082 & 0.240 & -0.23 & 1.33\
C51391\_3723 & 17.52 & 1.26 & 0.132 & 0.163 & 0.12 & 0.65\
C51393\_4120 & 17.52 & 1.27 & 0.115 & 0.135 & 0.00 & 0.58\
C51288\_4040 & 17.53 & 1.30 & 0.130 & 0.140 & 0.11 & 0.52\
C51292\_3964 & 17.54 & 1.42 & 0.107 & 0.117 & -0.05 & 0.48\
C51398\_3758 & 17.54 & 1.24 & 0.078 & 0.226 & -0.27 & 1.31\
C51336\_3569 & 17.55 & 1.23 & 0.107 & 0.110 & -0.06 & 0.41\
C51338\_3624 & 17.55 & 1.28 & 0.107 & 0.137 & -0.05 & 0.64\
C51402\_3627 & 17.55 & 1.30 & 0.118 & 0.101 & 0.02 & 0.23\
C51409\_4045 & 17.56 & 1.28 & 0.108 & 0.105 & -0.05 & 0.35\
C51275\_3873 & 17.57 & 1.32 & 0.073 & 0.227 & -0.33 & 1.37\
C51299\_4161 & 17.57 & 1.28 & 0.118 & 0.084 & 0.02 & -0.05\
C51307\_3821 & 17.59 & 1.28 & 0.082 & 0.232 & -0.24 & 1.30\
C51419\_3843 & 17.59 & 1.29 & 0.074 & 0.239 & -0.32 & 1.41
UPDATE ON ANOMALOUS STARS PREVIOUSLY OBSERVED IN M5
===================================================
In @2002AJ....123.2525C, we studied the CH bands in a large sample of stars in M5. Even taking into account the substantial star-to-star variation seen among the CH band strengths of the stars in our sample, we denoted six of these stars as anomalous. Since that time, we have checked the data for these stars yet again. We have found that two of the six stars were misidentified. C18206\_0533 with V=18.42 is actually C18188\_0733, with BVI = 17.71, 17.03, 16.17. With this correction, as compared to the bulk of our M5 sample [see Figures 8 and 9 of @2002AJ....123.2525C] the star has normal CH for its [$T_{eff}$]{}, although its uvCN is still anomalously strong, but not as much as previously. Also, star C18211\_0559 (V=18.06) is actually C18191\_0559, with BVI = 18.27, 17.57, 16.74. Its CN is now reasonable for its corrected $V$ mag, but its CH index is still unexpectedly strong.
In our earlier paper, we presented low accuracy radial velocities from the LRIS spectra at H$\alpha$ which suggested that 4 of the 6 stars classified as anomalous are radial velocity members of M5. Such data was not available for one star, while the radial velocity of C18211\_0559 (now identified as C18191\_0559) was 25 km/sec higher than that of the cluster.
To verify the membership of C18191\_0559 in M5, we obtained low SNR spectra with HIRES [@1994SPIE.2198..362V] for it and for a second star from the LRIS sample. A single 1200 sec exposure for each was made on May 1, 2002, a night with considerable clouds. The HIRES slit for one of these two also included a second M5 star. The heliocentric radial velocities for these three stars derived from the NaD lines are presented in Table \[tbl-B1\]. The radial velocity for M5 found by @2002AJ....123.3277R from an extensive high dispersion analysis of stars over a wide range in luminosity is +55.0 km/sec, so we conclude all three of these stars are members.
[cccc]{} C18225\_0537 & 17.07 & +58.2 & Anomalous star in Cohen et al.\
C18191\_0554 & 17.12 & +63.1 & in LRIS sample, but not anomalous\
C18191\_0558 & 17.57 & +58.5 & Anomalous in Cohen et al. as C18211\_0559
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[ccccc]{}
-0.50 & 0.0 & +0.40 & 0.075 & 0.044\
-0.50 & 1.0 & +0.40 & 0.074 & 0.232\
-0.50 & 0.0 & 0.00 & 0.079 & 0.048\
-1.00 & 0.0 & +0.40 & 0.042 & 0.044
[lcccccccccc]{}
IV-25/L-954 & 0.166 & 4000 & 0.15 & 2.25 & 12.09 & $-$0.90 & +1.22 & $-$1.31 & $-$1.36 & -\
II-67/L-70 & 0.165 & 3950 & 0.20 & 2.10 & 12.12 & $-$0.79 & +1.33 & $-$1.32 & $-$1.34 & -\
II-76/L-96 & 0.200 & 4350 & 1.15 & 1.85 & 12.52 & +0.46 & +0.59 & $-$0.62 & $-$0.82 & $-$1.2\
III-18/L-77 & 0.156 & 4350 & 1.15 & 1.85 & 12.77 & $-$0.18 & +1.10 & $-$1.11 & $-$0.97 & $-$1.3\
K188/A1 & 0.238 & 4550 & 1.50 & 1.80 & 13.39 & +0.45 & +1.00 & $-$0.32 & - & $-$0.5\
III-7/L-114 & 0.173 & 4600 & 1.65 & 2.00 & 13.45 & 0.00 & +1.00 & $-$0.83 & - & -0.9\
[^1]: Note we are excluding $\omega$ Cen and M22, both of which appear to have experienced some degree of self-enrichment.
|
---
author:
- Ali Sharif Razavian
- Hossein Azizpour
- |
\
Atsuto Maki
- Josephine Sullivan
- Carl Henrik Ek
- Stefan Carlsson
bibliography:
- 'egbib.bib'
title: Persistent Evidence of Local Image Properties in Generic ConvNets
---
Acknowledgment {#acknowledgment .unnumbered}
==============
We would like to gratefully acknowledge the support of NVIDIA for the donation of multiple GPU cards for this research.
|
---
abstract: 'Similar to Weyl fermions, a recently discovered topological fermion “triple point" can be generated from the splitting of Dirac fermion while the system has inversion symmetry (IS) breaking or time reversal symmetry (TRS) breaking. Inducing triple points in IS breaking symmorphic systems have been well studied, whereas in TRS breaking symmorphic systems have not yet. In this work, we extend the theory of searching triple points to all symmorphic magnetic systems. We list all $k$ paths of all symmorphic systems which allow the existence of triple points. With this systematic study, we also find out that the coexistence of Dirac points and triple points is symmetrically allowed in some particular symmetric systems. Our works will not only be helpful for searching triple points but also extend the knowledge of such a new topological fermion.'
author:
- 'Chi-Ho Cheung'
- 'R. C. Xiao'
- 'Ming-Chien Hsu'
- 'Huei-Ru Fuh'
- 'Yeu-Chung Lin'
- 'Ching-Ray Chang'
title: '**Systematic analysis for triple points in all magnetic symmorphic systems and symmetry-allowed coexistence of Dirac points and triple points**'
---
I. Introduction
===============
Over the past few decades, topology has been emerging in condensed matter physics. The development started from quantum Hall effect[@in1-1; @in1-2] which is the quantum-mechanical version of Hall effect. The second stage of development is quantum anomalous Hall effect[@in2-1; @in2-2; @in2-3; @in2-4] which is a quantum Hall effect without external magnetic field. The third stage of development is quantum spin Hall effect[@in3-1; @in3-2; @in3-3; @in3-4; @in3-5; @in3-6] which is a quantum Hall effect without breaking time reversal symmetry (TRS). Analogous to quantum spin Hall effect which pumps spin, there are topological crystalline insulators[@in4-1; @in4-2; @in4-3; @in4-4] which can pump the eigenvalues of mirror symmetry. All these four topological phenomena are insulating in bulk band, but have topologically protected surface state which is conducting.
Besides looking for topological phenomena in bulk insulating materials, scientists also look for topological phenomenon in bulk metallic materials. Recently, topological metals such as Dirac semimetal[@in5-1; @in5-2; @in5-3; @in5-4], Weyl semimetal[@in6-1; @in6-4; @in6-3; @in6-2] and triple point semimetal[@in7-1; @in7-2; @in7-3; @in7-4; @in7-5; @in7-6; @in7-7; @in7-8; @in7-9; @in7-10; @in7-11; @in7-12] have been discovered. These topological metals have topologically protected surface state just like those quantum Hall effects. No matter metallic in bulk or insulating in bulk, as long as their surface state are topologically protected, they can be promising candidates for electronic devices or even spintronic devices. Thus they can be valuable for industrial applications. On the other hand, topological metal provides a different playground and relatively lower price to search for those elementary particles described by relativistic quantum field theory. Since topological metal is valuable for both academic research and industrial applications, it has drawn a lot of attention in recent years.
One of the topological metals hosting a quasiparticle analogue of an elementary particle is Dirac semimetal. The earliest found Dirac semimetal is $Na_{3}Bi$[@in5-3]. $Na_{3}Bi$ has both IS and TRS, thus all bands at every $k$ points in the Brillouin zone are at least doubly degenerate. While any doubly degenerate band linearly crossing over another doubly degenerate band at a $k$ point, a four-fold degenerate Dirac point is formed. Such a Dirac point can be an analogue of the Dirac fermion which described by relativistic quantum field theory in high energy physics.
In high energy physics, breaking TRS or IS can cause Dirac fermion splitting into Weyl fermions. In condensed matter physics, bands can be non-degenerate while system does not have TRS and IS. When a non-degenerate band linearly crossing over another non-degenerate band at a $k$ point, a two-fold degenerate Weyl point is formed. Such a Weyl point can be an analogue of Weyl fermion in high energy physics too.
However, in condensed matter physics, fermions in crystal are constrained by magnetic space group symmetries rather than by Lorentz invariance. This gives rise to the uncertainty that doubly degenerate bands may or may not split while TRS or IS is broken. In this paper, we will discuss a new fermion-triple point which has no counterparts in high-energy physics and can be formed by a non-degenerate band linearly crossing over another doubly degenerate band at a $k$ point. In general, the formations of triple points can be caused by nonsymmorphic or symmorphic magnetic space groups symmetries, but as we emphasize in the title, we only discuss those triple points which caused by symmorphic magnetic space group symmetries.
If Dirac fermions in condensed matter must has TRS and IS just like the Dirac fermions in high energy physics, then it cannot coexist with triple point which cannot exist in a system with both TRS and IS. However, recent research shows that Dirac fermions in condensed matter can exist in a system without TRS $\cdot$ IS[@in8-1; @in8-2]. This gives rise to the possibility of finding several systems which have two k paths with two different symmetry groups, one allows the existence of Dirac points while another one allows the existence of triple points.
We organize this paper as follows. In section II, we review the condition of forming triple points by discussing a special case. In section III, we generalize this condition to all magnetic point groups and list all possible k paths of all possible symmorphic systems which allow the existence of triple points. In section IV, we point out that the coexistence of Dirac points and triple points is symmetrically allowed in some particular symmetric systems. In section V, we summarize the contributions of this paper.
II. The condition of forming triple points
==========================================
Similar to Weyl fermions in high energy physics, triple points in condensed matter physics can be split from Dirac fermions while TRS or IS of the system is broken. It is well known that Dirac fermions can exist in a system which has both TRS and $D_{6h}$ point group symmetry. In this section, we are going to use this system as an example to show how triple points split from Dirac fermions and point out the necessary condition of forming triple points.
$D_{6h}$ point group includes $C_{3z}$, $C_{2z}$, $M_{x}$ and IS. Since the system has TRS and IS, all bands have spin degeneracy at any $k$ point. As Dirac fermions are a crossing point of two 2-fold degenerate bands, Dirac fermion is a point of 4-fold degeneracy.
If all bands have spin degeneracy at any $k$ point, triple point cannot be formed (triple point is a point of 3-fold degeneracy). Thus TRS or IS must be broken to induce triple points. However, at a high symmetry $k$ point/path/plane, TRS and IS are not the only symmetries which protect the degeneracy of bands. Therefore, considering other crystal symmetries is needed.
To be more specific, we assume the irreducible representations of the bands which form the Dirac points are $\overline{E}_{1g/u}$ and $\overline{E}_{3g/u}$. With the irreducible representations, the matrix forms of the symmetry operators are as follows: $$\label{eq:1}
\begin{split}
\begin{array}{lcl}
for \ basis:
\overline{E}_{1g/u} \\
\ \ \ \ \ TRO \ \ \ \ \ \ \ \ \ \ \ IS \ \ \ \ \ \ \ \ \ M_{x} \ \ \ \ \ \ C_{2z} \ \ \ \ \ \ \ \ \ C_{3z} \\
{\begin{matrix}
\pm\begin{bmatrix}
0 & -1 \\
1 & 0
\end{bmatrix}K & \pm\begin{bmatrix}
1 & 0 \\
0 & 1
\end{bmatrix} & \pm\begin{bmatrix}
0 & i \\
i & 0
\end{bmatrix} & \begin{bmatrix}
i & 0 \\
0 & -i
\end{bmatrix} & \begin{bmatrix}
e^{\frac{i\pi}{3}} & 0 \\
0 & e^{\frac{-i\pi}{3}}
\end{bmatrix}
\end{matrix}}, \\
\\
for \ basis:
\overline{E}_{3g/u} \\
\ \ \ \ \ TRO \ \ \ \ \ \ \ \ \ \ \ \ \ \ IS \ \ \ \ \ \ \ \ \ \ \ M_{x} \ \ \ \ \ \ C_{2z} \ \ \ \ \ \ \ \ C_{3z} \\
{\begin{matrix}
\pm\begin{bmatrix}
0 & 1 \\
-1 & 0
\end{bmatrix}K & \mp\begin{bmatrix}
-1 & 0 \\
0 & -1
\end{bmatrix} & \mp\begin{bmatrix}
0 & i \\
i & 0
\end{bmatrix} & \begin{bmatrix}
-i & 0 \\
0 & i
\end{bmatrix} & \begin{bmatrix}
e^{i\pi} & 0 \\
0 & e^{-i\pi}
\end{bmatrix}
\end{matrix}},
\end{array}
\end{split}
$$ where $TRO$ is the operator of TRS and $K$ is complex conjugate operator.
If a symmetry operator S (S can be a unitary or an anti-unitary operator) acts on a $k_{h}$ vector, such that $Sk_{h}=k_{h}+nG$, where G is any reciprocal lattice vector and n is any integer number, then all such symmetry operators form the little group of $k_{h}$.
Firstly, we only consider the unitary subgroup of the little group of $k_{h}$. Hamiltonian $H(k_{h})$ has to commute with all the symmetry operators of the unitary subgroup of the little group of $k_{h}$. If any symmetry operators of this unitary subgroup does not commute with each other in a subspace of the Hilbert space, then $H(k_{h})$ has to be degenerate in this subspace, otherwise $H(k_{h})$ cannot commute with all the symmetry operators of the unitary subgroup simultaneously.
Furthermore, those anti-unitary symmetry operators of the little group of $k_{h}$ could cause extra degeneracy. In symmorphic system, in order to consider all the symmetry operators of the little group of $k_{h}$, we have to treat the little group as a magnetic point group rather than to treat it as the original point group, no matter the system does or does not have any magnetic moment. If the system does not have any magnetic moment, then it has TRS. Thus the symmetry group of the system is one of the grey groups of the 122 magnetic point groups. The little group of $k_{h}$ of this system is a subgroup of the grey group. Therefore, the little group of $k_{h}$ of a paramagnetic system could be any magnetic point group. All 122 magnetic point groups can be classified into three types: 32 ordinary point groups, 32 “grey" point groups and 58 “black and white" magnetic point groups. The degeneracies of the ordinary point groups have been discussed hereinabove. The extra degeneracies that caused by the TRS of any grey point groups are known to be Kramers degeneracy which has been well discussed too[@con1]. The extra degeneracies which caused by those anti-unitary symmetry operators of any black and white magnetic point groups are discussed in the Appendix of this paper.
In the system with $D_{6h}$ and TRS, any $k$ point on $\Gamma-Z$ axis-$k_{z}$ is invariant under $C_{3z}$, $C_{2z}$ rotation or $M_{x}$ reflection or (TRO$\cdot$IS) operation, thus the symmetry group of $\Gamma-Z$ axis is $6/m'mm$ which is a black and white magnetic point group. According to Eq. \[eq:1\], $C_{2z}$ does not commute with $M_{x}$ in both $\overline{E}_{1g/u}$ and $\overline{E}_{3g/u}$. Furthermore, according to Table. \[tab:S1\], the anti-unitary operators in $6/m'mm$ do not cause any extra degeneracy. Thus, $\overline{E}_{1g/u}$ and $\overline{E}_{3g/u}$ are both 2-fold degeneracy along the $k_{z}$ path. Besides, $\overline{E}_{1g/u}$ and $\overline{E}_{3g/u}$ are two different irreducible representations in $k_{z}$ path, so any coupling between these two representations (bands) are forbidden. Hence, there will be no gap opening when these two bands come across each other at $k_{z}$ path. Therefore, under such symmetry condition, a linear crossing between two 2-fold degenerate bands is allowed and so is the 4-fold degenerate Dirac point.
The symmetry condition of allowing the existence of Dirac points can be streamlined and generalized as follows: Dirac points can exist at a $k$ path whose symmetry group has two or more than two 2-dimensional double group irreducible representations.
If the TRS of the system is broken, the symmetry group of $k_{z}$ path is reduced from $6/m'mm$ to $C_{6v}$. Since the 2-fold degeneracy of $\overline{E}_{1g/u}$ and of $\overline{E}_{3g/u}$ remain being protected by $C_{2z}$ and $M_{x}$, the Dirac points on $k_{z}$ path do not split just because of TRS breaking. If $M_{x}$ symmetry is chosen for further symmetry breaking, all symmetry operators of the little group of $k_{z}$ path commute with each other. Both $\overline{E}_{1g/u}$ and $\overline{E}_{3g/u}$ will be split. If we want to induce triple points, breaking $M_{x}$ symmetry is not an option. If $C_{2z}$ is chosen for the further symmetry breaking, the symmetry group of the little group of $k_{z}$ path becomes $C_{3v}$. All symmetry operators in $\overline{E}_{3g/u}$ commute with each other, the symmetry operators in $\overline{E}_{1g/u}$ do not commute with each other. Thus $\overline{E}_{1g/u}$ remains to be a 2-fold degeneracy whereas $\overline{E}_{3g/u}$ splits into two non-degenerate bands. On $k_{z}$ path, since $C_{3z}$ symmetry can prevent any coupling between $\overline{E}_{1g/u}$ and $\overline{E}_{3g/u}$, these representations still belong to different irreducible representations. Therefore, the crossing point will not be gapped. Thus each Dirac point will split into two triple points when the $C_{2z}$ and TRS are broken. The variations of band structures and of system symmetry are shown in Fig. \[Fig: 3\].
![The schematic figure of the symmetry reduction processes of the system. After the system symmetry has been reduced from $D_{6h}+TRS(D_{6h})$ to $D_{6h}$, both $\overline{E}_{1g/u}$ and $\overline{E}_{3g/u}$ do not split. When the symmetry of the system becomes $D_{3d}$, $\overline{E}_{1g/u}$ remains to be a 2-fold degeneracy whereas $\overline{E}_{3g/u}$ splits into two non-degenerate bands. The Dirac points are marked in black circles and the triple points are marked in red circles. []{data-label="Fig: 3"}](Fig/fig3){width="8.5cm"}
This physical phenomenon will be further clarified if we use the $k \cdot P $ expansion and method of invariants to calculate the Hamiltonian around $\Gamma$ point for $k_{z}$ path: $$\label{eq:2}
\begin{array}{lcl}
H_{D_{3d}}(k_{z})=\varepsilon_{0}(k_{z})+ \\
\begin{medsize}\begin{pmatrix}
C_{0}-C_{1}k^{2}_{z} & 0 & 0 & 0 \\
0 & C_{0}-C_{1}k^{2}_{z} & 0 & 0 \\
0 & 0 & -C_{0}+C_{1}k^{2}_{z} & D \\
0 & 0 & D & -C_{0}+C_{1}k^{2}_{z} \\
\end{pmatrix}\end{medsize},
\end{array}$$ this is the Hamiltonian for the system with $D_{3d}$ symmetry and without TRS; the expansion is only up to the first order of $k$ for off-diagonal matrix elements and up to the second order of $k$ for diagonal matrix elements; $\varepsilon_{0}(k_{z})=A_{0}+A_{1}k_{z}^{2}$. $C_{0}$ and $C_{1}$ are real positive $k$ independent coefficients. $A_{0}$, $A_{1}$ and $D$ are real $k$ independent coefficients.
When the $C_{3z}$ operator acts on the effective Hamiltonian $C_{3z}H_{D_{3d}}(k)C^{-1}_{3z}$, according to the $C_{3z}$ symmetry operator of Eq. \[eq:1\], a phase factor $e^{\frac{2i\pi}{3}}$ or $-e^{\frac{i\pi}{3}}$ will be generated on the matrix elements $H_{12}$, $H_{21}$ and the matrix elements of off-diagonal block. Therefore these matrix elements must have $k_{+}$ or $k_{-}$ ($k_{\pm}=k_{x}\pm ik_{y}$) factor to match the $C_{3z}$ symmetry condition $C_{3z}H_{D_{3d}}(k)C^{-1}_{3z}=H_{D_{3d}}(C^{-1}_{3z}k)$. If only considering the Hamiltonian of $\Gamma-Z$ axis, all these matrix elements become zero. This explains hereinabove mentioned: on $k_{z}$ path, $C_{3z}$ symmetry can prevent any coupling between $\overline{E}_{1g/u}$ and $\overline{E}_{3g/u}$, such that these representations belong to different irreducible representations. Furthermore breaking $C_{2z}$ symmetry and TRS induces $D(\tau_{0}\sigma_{x}-\tau_{z}\sigma_{x})/2$ ($\tau$ is the Hilbert space of combining $\overline{E}_{1g/u}$ and $\overline{E}_{3g/u}$; $\sigma$ is the Hilbert space within $\overline{E}_{1g/u}$ or within $\overline{E}_{3g/u}$). This term splits $\overline{E}_{3g/u}$ to two 1-dimensional irreducible representations. Thus the 4-fold degenerate Dirac point splits into two 3-fold degenerate triple points.
Base on the above analysis, the symmetry condition of allowing the existence of triple points can be streamlined as follows: triple points only can exist at a k path whose symmetry group contains both 1-dimensional and 2-dimensional double group irreducible representations.
III. Triple points in all symmorphic systems
============================================
In this section, we are going to find out all possible $k$ paths of all possible symmorphic systems which allow the existence of triple points.
In symmorphic system, symmetry group of any k points is one of the magnetic point groups. So the first step is to check those 122 types magnetic point group and list all magnetic point groups which can match the symmetry condition of allowing the existence of triple points.
Among 32 types of ordinary point groups, there are 3 types, namely $C_{3v}$, $D_{3}$ and $D_{3d}$, of point groups containing both 1-dimensional and 2-dimensional double group irreducible representations. In all the Brillouin zone of the 14 types Bravais lattice, only 6 types of $k$ path contain $C_{3}$ symmetry (there is no $k$ plane contains $C_{3}$ symmetry). These 6 types of $k$ path are $\Lambda$ and $P$ of trigonal; $\Gamma-Z$ and $K-H$ of hexagonal; $1 1 1$ direction of cubic $P$, cubic $F$ and cubic $I$; F of cubic $I$ (since there is no unified $k$ path symbol, the Brillouin zones are demonstrated in Fig. \[Fig: 1\] to define the 6 symbols which are used for marking the 6 types of $k$ path). These 6 types of $k$ path only can contain $C_{3v}$, none of them can contain $D_{3}$ or $D_{3d}$. Therefore, among the ordinary point groups only $C_{3v}$ can match the symmetry condition of allowing the existence of triple points.
![The Brillouin zone for (a) trigonal with $a > \sqrt{2} c$, (b) trigonal with $a < \sqrt{2} c$, (c) hexagonal, (d) cubic $P$, (e) cubic $F$ and (f) cubic $I$. All the $k$ paths, which contain $C_{3}$ symmetry, are marked in green. Other Brillouin zones, which cannot contain $C_{3}$ symmetry, are omitted. []{data-label="Fig: 1"}](Fig/kpath){width="5.5cm"}
Since operating TRO on $k$ is to change the sign of $k$, only $k$ points, not $k$ paths, allow grey point group to be their symmetry group. Thus, if the symmetry group of $k$ is a grey point group, triple point cannot exist on this $k$.
All black and white magnetic point groups contain a set of unitary operators which form a unitary subgroup (one of the ordinary point group), and this unitary subgroup has a set of double group irreducible representations. The rest of the operators of the black and white magnetic point group are anti-unitary operators. These anti-unitary operators cannot generate any double group irreducible representation, but they can further degenerate the ordinary double group irreducible representations (as shown in Appendix). The further degeneracies could allow the existence of triple points, whereas the ordinary double group irreducible representations forbid that. Or contrarily, the further degeneracies forbid the existence of triple points, whereas the ordinary double group irreducible representations allow that.
Among all the 58 types of black and white magnetic point groups, 17 types have further degeneracies (as shown in Table. \[tab:S1\]). Among the 17 types, only $-6'$ allows the existence of triple points, whereas the double group irreducible representations of its unitary subgroup forbid that. All the other 16 types forbid the existence of triple points due to: having the element-TRS $\cdot$ IS or having no 1-dimensional double group irreducible representation or no $k$ path belonging to the black and white magnetic point group. Among the 16 types, $-3'm$ is the one that forbids the existence of triple points, whereas the double group irreducible representations of its unitary subgroup (its unitary subgroup is $C_{3v}$) allow that. For later discussion, it would be important to notice that $-3'm$ has the element of $-1'$.
Next step is to look for $k$ paths, in all symmorphic systems, which allow existence of triple points. We can directly look for these $k$ paths in all the symmorphic magnetic space groups. However, there are too many symmorphic magnetic space groups. Thus we choose to analyze magnetic point group of the symmorphic magnetic space group first. In this way a large number of unqualified magnetic space groups can be excluded. Then we contrast those qualified magnetic point group systems with their magnetic space group symbols.
As mentioned above, there are two kinds of $k$ paths allowing the existence of triple points: the first kind is the $k$ paths which belong to $-6'$ black and white magnetic point group; the second kind is the $k$ paths whose unitary subgroup belongs to $C_{3v}$ while the $k$ paths do not have $-1'$ symmetry. We will search these two kinds of $k$ paths separately.
Firstly, we search the $k$ paths which belong to $-6'$. If the Brillouin zone of a system contains the $-6'$ $k$ path, the crystal symmetry (directions of magnetic moments are not counted for crystal symmetry) of the system must contains $C_{3h}$ symmetry. According to the subgroup decomposition of the 32 point groups in Ref. [@con1], the crystal symmetry of the system contains $C_{3h}$ symmetry only if the crystal symmetry of the system is $D_{6h}$ or $D_{3h}$ or $C_{6h}$ or $C_{3h}$. The Bravais lattice of these four kinds of crystal symmetry is Hexagonal. In the Brillouin zone of Hexagonal Bravais lattice, only $\Gamma-Z$ can contain all the symmetry elements of $-6'$. Thus we only need to determine whether the $\Gamma-Z$ of all magnetic point group systems of $D_{6h}$, $D_{3h}$, $C_{6h}$ and $C_{3h}$ belong to $-6'$. The results are shown at Table. \[tab:1\].
--------------------------------------- ---------------------------
Label of magnetic point group systems Does $\Gamma-Z$ belong to
$-6'$?
($C_{6h}$) 6/m no
6/m1’ no
6’/m’ yes
6’/m no
6/m’ no
($D_{3h}$) -6m2 no
-6m21’ no
-6m’2’ no
-6’m2’ no
-6’m’2 yes
($D_{6h}$) 6/mmm no
6/mmm1’ no
6/m’mm no
6/mm’m’ no
6/m’m’m’ no
6’/mmm’ no
6’/m’mm’ no
($C_{3h}$) -6 no
-61’ yes
-6’ yes
--------------------------------------- ---------------------------
: The list for searching first kind of $k$ paths. The first column is the label of magnetic point group systems. Answer of “Does $\Gamma-Z$ of the system belong to $-6'$?" is given in second column.
\[tab:1\]
Secondly, we search the $k$ paths whose unitary subgroup belongs to $C_{3v}$ while the $k$ paths do not have $-1'$ symmetry. If the Brillouin zone of a system contains such $k$ path, the symmetry group of the system must contain $C_{3v}$ symmetry. According to the subgroup decomposition of the 32 point groups in Ref. [@con1], the symmetry group of the system contains $C_{3v}$ symmetry only if the crystal symmetry of the system is one of the seven symmetries, namely $C_{3v}$, $T_{d}$, $O_{h}$, $C_{6v}$, $D_{3d}$, $D_{3h}$ and $D_{6h}$. Furthermore, as (TRO $\cdot$ IS) acting on any $k$ is equal to $k$, any $k$ point in the Brillouin zone contains (TRO $\cdot$ IS) symmetry if and only if the system contains (TRO $\cdot$ IS) symmetry. Thus we can use two symmetry conditions to filter most of the magnetic point groups which belong to the seven crystal symmetry: 1. the system must contains $C_{3v}$ symmetry; 2. the system must not contains (TRO $\cdot$ IS) symmetry. Besides, the $k$ paths which contain $C_{3v}$ symmetry must contain $C_{3}$ symmetry. Hence, as mentioned above, the $k$ path whose unitary subgroup is $C_{3v}$ symmetry must be one of the following: $\Lambda$ and $P$ of trigonal; $\Gamma-Z$ and $K-H$ of hexagonal; $1 1 1$ direction of cubic $P$, cubic $F$ and cubic $I$; F of cubic $I$. All we need to do is to determine whether the unitary subgroup, of these 6 types of $k$ paths in the Brillouin zone of filtered magnetic point group systems, is $C_{3v}$ symmetry. The determining processes and results are given in Table. \[tab:2\]. Notice that for the Brillouin zone of trigonal Bravais lattice, the $k$ path-$P$ appears only if lattice constants fulfill the condition of $a > \sqrt{2} c$. Sometimes, the $k$ path-$K-H$ of Hexagonal Bravais lattice is not contained in the mirror plane of $C_{3v}$ symmetry, such that $K-H$ does not contain $C_{3v}$ symmetry. Hence, we need the magnetic space group to determine whether the existence of triple points is allowed on this $k$ path. Thus, some answers in the fifth column of Table. \[tab:2\] are “MSG is needed". Combining Table. \[tab:1\] and Table. \[tab:2\], then contrasting with magnetic space group, we can get all $k$ paths of all symmorphic systems which allow the existence of triple points (as shown in Table. \[tab:3\]).
symmetry of the system contains $C_{3v}$ or not? contains $-1'$ or not? Brillouin zone $k$ path the existence of triple points
------------------------ -------------------------------------------------- ---------------- ------------ --------------------------------
($C_{3v}$) 3m contains $C_{3v}$, but no $-1'$ trigonal $\Lambda$ yes
P yes
Hexagonal $\Gamma-Z$ yes
$K-H$ MSG is needed
3m1’ contains $C_{3v}$, but no $-1'$ trigonal $\Lambda$ yes
P yes
Hexagonal $\Gamma-Z$ yes
$K-H$ MSG is needed
3m’ no $C_{3v}$
($T_{d}$) -43m contains $C_{3v}$, but no $-1'$ cubic P 1 1 1 yes
cubic F 1 1 1 yes
cubic I 1 1 1 yes
F yes
-43m1’ contains $C_{3v}$, but no $-1'$ cubic P 1 1 1 yes
cubic F 1 1 1 yes
cubic I 1 1 1 yes
F yes
-4’3m’ no $C_{3v}$
($O_{h}$) m-3m contains $C_{3v}$, but no $-1'$ cubic P 1 1 1 yes
cubic F 1 1 1 yes
cubic I 1 1 1 yes
F yes
m-3m1’ contains $-1'$
m’-3’m contains $-1'$
m-3m’ no $C_{3v}$
m’-3’m’ contains $-1'$
($C_{6v}$) 6mm contains $C_{3v}$, but no $-1'$ Hexagonal $\Gamma-Z$ no
$K-H$ yes
6mm1’ contains $C_{3v}$, but no $-1'$ Hexagonal $\Gamma-Z$ no
$K-H$ yes
6’mm’ contains $C_{3v}$, but no $-1'$ Hexagonal $\Gamma-Z$ yes
$K-H$ MSG is needed
6m’m’ no $C_{3v}$
($D_{3d}$) -3m contains $C_{3v}$, but no $-1'$ trigonal $\Lambda$ yes
P yes
Hexagonal $\Gamma-Z$ yes
$K-H$ MSG is needed
-3m1’ contains $-1'$
-3’m contains $-1'$
-3’m’ no $C_{3v}$
-3m’ no $C_{3v}$
($D_{3h}$) -6m2 contains $C_{3v}$, but no $-1'$ Hexagonal $\Gamma-Z$ yes
$K-H$ MSG is needed
-6m21’ contains $C_{3v}$, but no $-1'$ Hexagonal $\Gamma-Z$ yes
$K-H$ MSG is needed
-6m’2’ no $C_{3v}$
-6’m2’ contains $C_{3v}$, but no $-1'$ Hexagonal $\Gamma-Z$ yes
$K-H$ MSG is needed
-6’m’2 no $C_{3v}$
($D_{6h}$) 6/mmm contains $C_{3v}$, but no $-1'$ Hexagonal $\Gamma-Z$ no
$K-H$ yes
6/mmm1’ contains $-1'$
6/m’mm contains $-1'$
6/mm’m’ no $C_{3v}$
6/m’m’m’ contains $-1'$
6’/mmm’ contains $-1'$
6’/m’mm’ contains $C_{3v}$, but no $-1'$ Hexagonal $\Gamma-Z$ yes
$K-H$ MSG is needed
: The list for searching second kind of $k$ paths. The first column is the label of magnetic point group systems. The second column shows whether the system be filtered out by the symmetry condition of containing $C_{3v}$ while does not contain $-1'$. The third column is the class of Brillouin zone. The fourth column is the label of $k$ path. The fifth column shows that whether the $k$ path allows the existence of triple points. Sometimes, the $k$ path-$K-H$ of Hexagonal Bravais lattice is not contained in the mirror plane of $C_{3v}$ symmetry and consequently $K-H$ does not contain $C_{3v}$ symmetry. Hence, we need the magnetic space group to determine whether the existence of triple points is allowed on this $k$ path. Thus, some answers in the fifth column are “MSG is needed". For the Brillouin zone of trigonal Bravais lattice, the $k$ path-$P$ appears only if lattice constants fulfill the condition of $a > \sqrt{2} c$.
\[tab:2\]
--------------------- ------------ ----------------------
Label of magnetic $k$ path the kind of $k$ path
point group systems
P3m1 $\Gamma-Z$ first kind
P3m11’ $\Gamma-Z$ first kind
P31m $\Gamma-Z$ first kind
$K-H$ first kind
P31m1’ $\Gamma-Z$ first kind
$K-H$ first kind
R3m $\Lambda$ first kind
P first kind
R3m1’ $\Lambda$ first kind
P first kind
P-31m $\Gamma-Z$ first kind
$K-H$ first kind
P-3m1 $\Gamma-Z$ first kind
R-3m $\Lambda$ first kind
P first kind
P-61’ $\Gamma-Z$ second kind
P-6’ $\Gamma-Z$ second kind
P6’/m’ $\Gamma-Z$ second kind
P6mm $K-H$ first kind
P6mm1’ $K-H$ first kind
P6’m’m $\Gamma-Z$ first kind
P6’mm’ $\Gamma-Z$ first kind
$K-H$ first kind
P-6m2 $\Gamma-Z$ first kind
P-6m21’ $\Gamma-Z$ first kind
P-6’m’2 $\Gamma-Z$ second kind
P-6’m2’ $\Gamma-Z$ first kind
P-62m $\Gamma-Z$ first kind
$K-H$ first kind
P-62m1’ $\Gamma-Z$ first kind
$K-H$ first kind
P-6’2’m $\Gamma-Z$ first kind
$K-H$ first kind
P-6’2m’ $\Gamma-Z$ second kind
P6/mmm $K-H$ first kind
P6’/m’m’m $\Gamma-Z$ first kind
P6’/m’mm’ $\Gamma-Z$ first kind
$K-H$ first kind
P-43m 1 1 1 first kind
P-43m1’ 1 1 1 first kind
F-43m 1 1 1 first kind
F-43m1’ 1 1 1 first kind
I-43m 1 1 1 first kind
F first kind
I-43m1’ 1 1 1 first kind
F first kind
Pm3m 1 1 1 first kind
Fm3m 1 1 1 first kind
Im3m 1 1 1 first kind
F first kind
--------------------- ------------ ----------------------
: The list of all $k$ paths of all symmorphic systems which allow the existence of triple points. The first column is the label of magnetic point group systems which allow the existence of triple points. The second column are the $k$ paths of the system which allow the existence of triple points. The third column is the kind of the $k$ path. The first kind of $k$ paths is from Table. \[tab:1\] and the second kind is from Table. \[tab:2\].
\[tab:3\]
IV. The coexistence of Dirac point and triple point
===================================================
The degeneracy of a Dirac fermion in high energy physics is protected by TRS and IS. However, the degeneracy of a Dirac fermion in condensed matter can be preserved in a $k$ path whose symmetry group has two or more than two 2-dimensional double group irreducible representations (as mentioned in section. II). That means Dirac fermion can exist in a system which does not contain (TRO $\cdot$ IS) symmetry. This gives rise to the possibility of the coexistence of Dirac fermion and odd-fold degenerate fermion. We know that triple points can exist in a $k$ path whose symmetry group contains both 1-dimensional and 2-dimensional double group irreducible representations (as mentioned in section. II too). Combining the condition of existence of Dirac fermion with the condition of existence of triple point, we find that there are several systems which allow the coexistence of Dirac points and triple points. $T$ and $\Delta$ of $Pm3m$, $\Delta$ of $Fm3m$, $\Delta$ of $Im3m$, $\Gamma-Z$ of $P6/mmm$, $\Gamma-Z$ of $P6mm$ and $\Gamma-Z$ of $P6mm{1}'$ allow the existence of Dirac points while the $k$ paths of these several systems, which allow the existence of triple points, are given in Table. \[tab:3\]. The Brillouin zone of these several systems are shown in Fig. \[Fig: 2\].
![The Brillouin zone of the systems which allow the coexistence of Dirac points and triple points. The Brillouin zone of (a) P6/mmm, P6mm and P6mm1’. The Brillouin zone of (b) Pm3m, (c) Fm3m and (d) Im3m. The $k$ paths, which allow the existence of triple points, are marked in green. The $k$ paths, which allow the existence of Dirac points, are marked in blue. []{data-label="Fig: 2"}](Fig/kpath2){width="5.5cm"}
Then there are two questions surfaced:
\(1) Is the condition of existence of Weyl points similar to the condition of existence of triple points? More specifically, can Weyl points exist in a $k$ path whose symmetry group contains two 1-dimensional double group irreducible representations?
\(2) Can the coexistence of Dirac points, triple points and Weyl points be allowed in some symmetric systems?
To answer question (1), we analyze two systems: The first system has $D_{3d}$ point group symmetry with hexagonal Bravais lattice. The symmetry group of $\Gamma-Z$ in its Brillouin zone is $C_{3v}$. $C_{3v}$ contains two 1-dimensional double group irreducible representations. These two representations have different mirror symmetry eigenvalues ($i$ and $-i$) which means they are two different representations in the mirror plane of $C_{3v}$ symmetry. Thus the crossing point in $\Gamma-Z$ must extend onto the mirror planes of $C_{3v}$ symmetry to form a Weyl nodal line other than discrete points. This means, in this case, Weyl points are not able to exist in a $k$ path whose symmetry group contains two 1-dimensional double group irreducible representations.
The second system has $C_{3i}$ point group symmetry with hexagonal Bravais lattice. The symmetry group of $\Gamma-Z$ is $C_{3}$. $C_{3}$ contains two 1-dimensional double group irreducible representations. This system can be viewed as the first system to which a uniform magnetic field-$B_{z}$, in parallel with principle axis, is applied. The $B_{z}$ breaks the mirror symmetry of $C_{3v}$ and the Weyl nodal line is gap opening. Furthermore, the band structure along $k_{z}$ path can be described by Eq. \[eq:2\] plus several $B_{z}$ induced terms: $$\label{eq:3}
\begin{array}{lcl}
H_{C_{3i}}(k_{z})=\varepsilon_{0}(k_{z})+ \\
\begin{medsize}\begin{pmatrix}
M(k_{z})+\beta_{1} & 0 & 0 & 0 \\
0 & M(k_{z})+\beta_{2} & 0 & 0 \\
0 & 0 & -M(k_{z})+\beta_{3} & D+\beta_{5} \\
0 & 0 & D\beta_{5}^{*} & -M(k_{z})+\beta_{4} \\
\end{pmatrix}\end{medsize},
\end{array}$$ where $\beta_{1}, \ \beta_{2}, \ \beta_{3}$ and $\beta_{4}$ are real while $\beta_{5}$ is complex; $M(k_{z})$ is $C_{0}-C_{1}k^{2}_{z}$. Solving the eigenvalues of Eq. \[eq:3\], we find that the 2-dimensional double group irreducible representation-$\overline{E}_{1g/u}$ in Eq. \[eq:2\] splits into two 1-dimensional double group irreducible representations. That means each triple point in Eq. \[eq:2\] splits into two crossing points. Using $k \cdot P$ expansion and method of invariants to calculate the Hamiltonian around these crossing points can prove that these crossing points are Weyl points[@coex1]. Therefore, the two 1-dimensional double group irreducible representations of $\Gamma-Z$ in this system allow the existence of Weyl points.
According to the above analysis for the two systems, sometimes Weyl points can exist in a $k$ path whose symmetry group contains two 1-dimensional double group irreducible representations and sometimes Weyl points cannot. Thus, the condition of existence of Weyl points should not be as stated in question (1).
For answering question (2):
The $\Gamma-Z$ of the two above mentioned systems does not allow the coexistence of Dirac points, triple points and Weyl points, but recent research shows that triple point and Weyl point are able to coexist with each other[@in7-1].
The symmetry requirement of Weyl points is much lower than that of triple points. Weyl points are robust against the perturbations of most of the symmetry breaking. Weyl points even can exist at the $k$ points whose symmetry group belongs to $C_{1}$[@coex2]. Such Weyl points in bulk of 3-dimensional systems only need periodic symmetry in the absence of (TRO $\cdot$ IS).
Therefore, in view of symmetry analysis hereinabove, the coexistence of Dirac points, triple points and Weyl points is allowed in some systems. If that happens, it will cause chaos when the systems are defined to be Dirac semimetal or triple point topological metal or Weyl semimetal. There is a viewpoint that triple point topological metal is an intermediate phase between Dirac and Weyl semimetal[@in7-4]. It can be true, but there are overlapping among these three phases and the boundary of the phases is still obscure so far. Such a state of chaos is raised by using symmetry to classify these three phases. In condensed matter systems, the symmetry group of a system is unique, but, in the Brillouin zone of a system, there are many different $k$ paths/planes belonging to different symmetry groups. Usually, the existences of Dirac points or triple points in a $k$ path depend on the symmetry group of the $k$ path other than only depend on the symmetry group of the system. As a result, the coexistence of Dirac points, triple points and Weyl points is symmetry-allowed.
V. Conclusion
=============
In high energy physics, breaking TRS or breaking IS but keep Lorentz invariance, one Dirac fermion will split into two Weyl fermions. In condensed matter systems, breaking TRS or IS, Dirac point can remain intact or split into triple points or split into Weyl points. One Dirac point can split into two triple points, two triple points can split into four Weyl points. Existence of triple point between Dirac phase and Weyl phase was well hidden in the past. The recent discovery of triple point drives researchers into study of unknown characteristics of triple point. Therefore, we extend the theory of searching triple points to all symmorphic magnetic systems, and list all $k$ paths of all symmorphic systems which allow the existence of triple points. Our systematic study is helpful for searching triple points in various systems. Besides, we also find out that the coexistence of Dirac points and triple points is symmetrically allowed in some particular symmetric systems. Our works will not only be helpful for searching triple points but also extend the knowledge of such a new topological fermion.
**ACKNOWLEDGMENTS**
This work is supported by the Ministry of Science and Technology of Taiwan under Grant No. MOST 104-2112-M-002-007-MY3.
APPENDIX: Corepresentations of black and white magnetic point group
===================================================================
All black and white point groups can be express as follows: $$\label{eq:A1}
M=H+TRO(G-H),
$$ where $M$ is black and white point group, $H$ is the unitary subgroup of $M$ and $G$ is one of the ordinary point groups.
Here, we denote element of $H$ by $U$, and element of $TRO(G-H)$ by $V$. We suppose that $\Delta$ is a unitary irreducible representation of $H$ (the dimension of $\Delta$ can be greater than 1), therefore: $$\label{eq:A2}
U\left\langle \varphi \right|=\left\langle \varphi \right|\Delta(U).$$ Now we introduce a basis $\left| \phi \right>$ which is produced by operating $V$ on $\left| \phi \right>$: $$\label{eq:A3}
V\left\langle \varphi \right|=\left\langle \phi \right|.$$ From Eq. \[eq:A2\], Eq. \[eq:A3\] and $V^{-1}UV$ is belong to $H$, we can get: $$\label{eq:A4}
\begin{array}{lcl}
U\left\langle \phi \right|=UV\left\langle \varphi \right| \\
\ \ \ \ \ \ \ =V(V^{-1}UV)\left\langle \varphi \right| \\
\ \ \ \ \ \ \ =V\left\langle \varphi \right|\Delta(V^{-1}UV) \\
\ \ \ \ \ \ \ =\left\langle \phi \right|\Delta^{*}(V^{-1}UV),
\end{array}$$ complex conjugate is denoted by asterisk. Let $$\label{eq:A5}
\left\langle \zeta \right|=\left\langle \varphi, \phi \right|.$$ From Eq. \[eq:A2\], Eq. \[eq:A4\] and Eq. \[eq:A5\], we have: $$\label{eq:A6}
U\left\langle \zeta \right|=\left\langle \zeta \right|D(U),$$ where $$\label{eq:A7}
D(U)=\begin{medsize}\begin{pmatrix}
\Delta(U) & 0 \\
0 & \Delta^{*}(V^{-1}UV) \\
\end{pmatrix}\end{medsize},$$ for all $U$ that belong to $H$. Since $\Delta^{*}(V^{-1}UV)$ is also a representation of $H$[@ap1], the anti-unitary operators of $M$ do not create any extra irreducible representation. Anti-unitary operators only cause the irreducible representations of $H$ degenerate with each other or degenerate with itself, but usually anti-unitary operators do not cause any extra degeneracy.
If $\Delta(U)$ and $\Delta^{*}(V^{-1}UV)$ are equivalent, then there exists an unitary operator P such that: $$\label{eq:A8}
\Delta(U)=P\Delta^{*}(V^{-1}UV)P^{-1},$$ for all $U$ belonging to $H$.
If $$\label{eq:A9}
PP^{*}=\Delta(V^{2}),$$ then anti-unitary operators do not cause any extra degeneracy, and we call it case(1).
If $$\label{eq:A10}
PP^{*}=-\Delta(V^{2}),$$ then anti-unitary operators cause the irreducible representation $\Delta$ degenerate with itself, and we call it case(2).
If $\Delta(U)$ and $\Delta^{*}(V^{-1}UV)$ are not equivalent, then they are degenerate with each other, and we call it case(3).
Now we can classify all the representations of all black and white point groups. All those black and white point groups contain representations belonging to case(2) or case(3) are listed in Table. \[tab:S1\].
$M$ $H$ irreducible representation of $H$ case
------- ---------------- ------------------------------------------------------------------------------------------------------------------------------------------------------------ ------
-1’ 1 ($C_{1}$) $\overline{A}$ 2
2’/m m ($C_{1h}$) ${}^{1}\overline{E}$, ${}^{2}\overline{E}$ 3
2/m’ 2 ($C_{2}$) ${}^{1}\overline{E}$, ${}^{2}\overline{E}$ 3
4’ 2 ($C_{2}$) ${}^{1}\overline{E}$, ${}^{2}\overline{E}$ 3
-4’ 2 ($C_{2}$) ${}^{1}\overline{E}$, ${}^{2}\overline{E}$ 3
4’/m 2/m ($C_{2h}$) ${}^{1}\overline{E}_{g}$, ${}^{2}\overline{E}_{g}$, ${}^{1}\overline{E}_{u}$, ${}^{2}\overline{E}_{u}$ 3
4/m’ 4 ($C_{4}$) ${}^{1}\overline{E}_{2}$, ${}^{2}\overline{E}_{1}$, ${}^{1}\overline{E}_{1}$, ${}^{2}\overline{E}_{2}$ 3
4’/m’ -4 ($S_{4}$) ${}^{1}\overline{E}_{2}$, ${}^{2}\overline{E}_{1}$, ${}^{1}\overline{E}_{1}$, ${}^{2}\overline{E}_{2}$ 3
-3’ 3 ($C_{3}$) $\overline{A}$ 2
${}^{1}\overline{E}$, ${}^{2}\overline{E}$ 3
-3’m 3m ($C_{3v}$) ${}^{1}\overline{E}$, ${}^{2}\overline{E}$ 3
$\overline{E}_{1}$ 1
-3’m’ 32 ($D_{3}$) ${}^{1}\overline{E}$, ${}^{2}\overline{E}$ 3
$\overline{E}_{1}$ 1
6’ 3 ($C_{3}$) ${}^{1}\overline{E}$, ${}^{2}\overline{E}$ 3
$\overline{A}$ 1
-6’ 3 ($C_{3}$) ${}^{1}\overline{E}$, ${}^{2}\overline{E}$ 3
$\overline{A}$ 1
6’/m -6 ($C_{3h}$) ${}^{1}\overline{E}_{1}$, ${}^{2}\overline{E}_{3}$, ${}^{1}\overline{E}_{2}$, ${}^{2}\overline{E}_{2}$, ${}^{2}\overline{E}_{1}$, ${}^{1}\overline{E}_{3}$ 3
6/m’ 6 ($C_{6}$) ${}^{1}\overline{E}_{1}$, ${}^{2}\overline{E}_{3}$, ${}^{1}\overline{E}_{2}$, ${}^{2}\overline{E}_{2}$, ${}^{2}\overline{E}_{1}$, ${}^{1}\overline{E}_{3}$ 3
6’/m’ -3 ($C_{3i}$) ${}^{1}\overline{E}_{g}$, ${}^{2}\overline{E}_{g}$, ${}^{1}\overline{E}_{u}$, ${}^{2}\overline{E}_{u}$ 3
$\overline{A}_{g}$, $\overline{A}_{u}$ 1
m’3 23 ($T$) ${}^{1}\overline{F}$, ${}^{2}\overline{F}$ 3
$\overline{E}$ 1
: The list for all the black and white magnetic point groups which have representations belonging to case(2) or case(3). The first column is the label of $M$. The second column is the label of $H$. The third column is the label of irreducible representation of $H$. The fourth column is the classification of the irreducible representation of $H$.
\[tab:S1\]
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abstract: 'A general real-space multigrid algorithm MIKA (Multigrid Instead of the K-spAce) for the self-consistent solution of the Kohn-Sham equations appearing in the state-of-the-art electronic-structure calculations is described. The most important part of the method is the multigrid solver for the Schrödinger equation. Our choice is the Rayleigh quotient multigrid method (RQMG), which applies directly to the minimization of the Rayleigh quotient on the finest level. Very coarse correction grids can be used, because there is in principle no need to represent the states on the coarse levels. The RQMG method is generalized for the simultaneous solution of all the states of the system using a penalty functional to keep the states orthogonal. Special care has been taken to optimize the iterations towards the self-consistency and to run the code in parallel computer architectures. The scheme has been implemented in multiple geometries. We show examples from electronic structure calculations employing nonlocal pseudopotentials and/or the jellium model. The RQMG solver is also applied for the calculation of positron states in solids.'
address: 'Laboratory of Physics, Helsinki University of Technology, P.O. Box 1100, FIN-02015 HUT, FINLAND'
author:
- 'T. Torsti, M. Heiskanen, M. J. Puska, and R. M. Nieminen'
title: 'MIKA: a multigrid-based program package for electronic structure calculations'
---
=10000
[2]{}
Introduction {#sec:introduction}
============
The goal of computational materials science and also that of modeling of nanoscale man-made structures is to calculate from first principles the various chemical and/or physical properties. This requires the solution of the electronic (and ionic) structures of the system in question. The density-functional theory (DFT) [@kohn98] makes a huge step towards this goal by casting the untractable problem of many interacting electrons to that of noninteracting particles under the influence of an effective potential. However, in order to apply DFT in practice one has to resort to approximations for electron exchange and correlation such as the local-density approximation (LDA) or the generalized-gradient approximation (GGA). Moreover, in the case of systems consisting of hundreds or more atoms it is still a challenge to solve numerically efficiently for the ensuing Kohn-Sham equations.
We have developed a real-space multigrid method called MIKA (Multigrid Instead of the K-spAce) for the numerical solution of the Kohn-Sham equations [@mgarticle1]. In real-space methods[@beckrev; @arias; @waghmare], the values of the wave-functions and potentials are presented using three-dimensional point grids, and the partial differential equations are discretized using finite differences. Multigrid methods[@brandt1; @beckrev] overcome the critical slowing-down (CSD) phenomenon occuring with basic real-space relaxation methods. Several approaches employing the multigrid idea have appeared during recent years[@briggs; @ancilotto; @fattebert2; @wang1].
From the different multigrid methods available for the solution of the Schrödinger equation, we have picked up the Rayleigh Quotient Multigrid (RQMG) method introduced by Mandel and McCormick [@McCormick]. This approach differs from full-approximation-storage[@brandt2; @beck1; @wang1; @costiner] (FAS) methods, as well as from those methods[@briggs], where the eigenproblem is linearized.
In the RQMG method the coarse grid relaxation passes are performed so that the Rayleigh quotient calculated on the [*fine*]{} grid will be minimized. In this way there is no requirement for the solution to be well represented on a coarse grid and the coarse grid representation problem is avoided. Mandel and McCormick[@McCormick] introduced the method for the solution of the eigenpair corresponding to the lowest eigenvalue. We have generalized it to the simultaneous solution of a desired number of lowest eigenenergy states by developing a scheme which keeps the eigenstates separated by the use of a penalty functional[@mgarticle1].
Numerical Methods {#sec:methods}
=================
In our RQMG application the coarse grid relaxations are performed by the so-called coordinate relaxation method. It solves the discretized eigenproblem $$H u = \lambda B u$$ by minimizing the Rayleigh quotient $$\label{Ray}
\frac{\langle u\arrowvert H\arrowvert u\rangle}
{\langle u\arrowvert B\arrowvert u\rangle}.$$ Above, $H$ and $B$ are matrix operators chosen so that the Schrödinger equation discretized on a real-space point grid with spacing $h$ is satisfied to a chosen order $O(h^n)$. In Eq. (\[Ray\]) $u$ is a vector containing the wave function values at the grid points. In the relaxation method, the current estimate $u$ is replaced by $ u' = u + \alpha d$, where the search vector $d$ is simply chosen to be unity in one grid point and to vanish in all other points, and $\alpha$ is chosen to minimize the Rayleigh quotient. This leads to a simple [^1] quadratic equation for $\alpha$. A complete coordinate relaxation pass is then obtained by performing the minimization at each point in turn and these passes can be repeated until the lowest state is found with desired accuracy.
Naturally, also the coordinate relaxation suffers from CSD because of the use of local information only in updating $u$ in a certain point. In order to avoid it one applies the multigrid idea. In the multigrid scheme by Mandel and McCormick[@McCormick] the crucial point is that [*coarse*]{} grid coordinate relaxation passes are performed so that the Rayleigh quotient calculated on the [*fine*]{} grid will be minimized. In this way there is no requirement for the solution to be well represented on a coarse grid. In practice, a coarse grid search substitutes the fine grid solution by $$\label{rqmgchgeq}
u_f' = u_f + \alpha I_c^f e_c,$$ where the subscripts $f$ and $c$ stand for the fine and coarse grids, respectively, and $I_c^f$ a prolongation operator interpolating the coarse grid vector to the fine grid. The Rayleigh quotient to be minimized is then $$\begin{aligned}
\label{rqmgeq}
& \frac{\langle u_f + \alpha I_c^f d_c \arrowvert H_f \arrowvert
u_f + \alpha I_c^f d_c \rangle}
{\langle u_f + \alpha I_c^f d_c \arrowvert B_f \arrowvert
u_f + \alpha I_c^f d_c \rangle} = \qquad \qquad \qquad \qquad \nonumber \\
&\qquad \qquad \qquad
\frac{ \langle u_f \arrowvert H_f u_f \rangle
+ 2\alpha \langle I_f^c H_f u_f \arrowvert d_c \rangle
+ \alpha^2 \langle d_c \arrowvert H_c d_c \rangle
}
{ \langle u_f \arrowvert B_f u_f \rangle
+ 2\alpha \langle I_f^c B_f u_f \arrowvert d_c \rangle
+ \alpha^2 \langle d_c \arrowvert B_c d_c \rangle
}.\end{aligned}$$ The second form is obtained by relating the coarse grid operators, $H_c$ and $B_c$, with the fine grid ones, $H_f$ and $B_f$, by the Galerkin condition $$\label{galerkincond}
H_c = I_f^c H_f I_c^f; \quad
B_c = I_f^c B_f I_c^f; \quad
I_f^c = \left(I_c^f\right)^T.$$ The key point to note is that when $H_f u_f$ and $B_f u_f$ are provided from the fine grid to the coarse grid, the remaining integrals can be calculated on the coarse grid itself. Thus one really applies coordinate relaxation on the coarse grids to minimize the *fine level* Rayleigh quotient. This is a major departure from the earlier methods, which to some extent rely on the ability to represent the solution of some coarse grid equation on the coarse grid itself. Here, on the other hand, one can calculate the *exact* change in the Rayleigh quotient due to *any* coarse grid change, no matter how coarse the grid itself is. There is no equation whose solution would have to be representable.
In the MIKA package we have generalized the RQMG method to the simultaneous solution of several mutually orthogonal eigenpairs. The separation of the different states is divided into two or three subtasks. First, in order to make the coarse grid relaxations converge towards the desired state we apply a penalty functional scheme. Given the current approximations for the $k$ lowest eigenfunctions, the next lowest, $(k+1)$’th state is updated by minimizing the functional $$\label{rqmgneq}
\frac{\langle u_{k+1}\arrowvert H\arrowvert u_{k+1}\rangle}
{\langle u_{k+1}\arrowvert B\arrowvert u_{k+1}\rangle}
+ \sum\limits_{i=1}^{k}
q_i \frac{\left|\langle u_i | u_{k+1}\rangle\right|^2}
{\langle u_i | u_i\rangle \cdot
\langle u_{k+1} | u_{k+1}\rangle}.$$ The minimization of this functional is equivalent to imposing the orthonormality constraints against the lower $k$ states, when $q_i \rightarrow \infty$. By increasing the shifts $q_i$ any desired accuracy can be obtained, but in order to obtain a computationally efficient algorithm a reasonable finite value should be used, for example $$q_i = (\lambda_{k+1}-\lambda_i) + {\rm Q},$$ where $Q$ is a sufficiently large positive constant. In our test calculations $Q$ is of the order of $Q=0.5\ldots 2$ Ha.
The substitution (\[rqmgchgeq\]) is introduced in the functional (\[rqmgneq\]) and the minimization with respect to $\alpha$ leads again to a quadratic equation. This time the coefficients contain terms due to the penalty part.
While the penalty functional keeps the states separated on the coarse levels, we apply a simple relaxation method (Gauss-Seidel) on the finest level. The Gauss-Seidel method converges to the nearest eigenvalue, so ideally no additional orthogonalizations would be needed. In practice, however, we use Gramm-Schmidt orthogonalizations and subspace rotations[@mgarticle1]. However, the number of fine grid orthogonalizations remains quite plausible, for example, in comparison with the conjugate gradient search of eigenpairs employing only the finest grid [@seitsonen].
The Kohn-Sham equations have to be solved self-consistently, [*i.e.*]{} the wave functions solved from the single-particle equation determine via the density (solution of the Poisson equation and the calculation of the exchange-correlation potential) the effective potential for which they should again be solved. To approach this self-consistency requires an optimized strategy so that numerical accuracy of the wave functions and the potential increase in balance, enabling the most efficient convergence [@wang1; @waghmare]. Our strategy in MIKA for self-consistency iterations is illustrated in Fig. \[fig:strategy\]. The Poisson equation for the Coulomb potential is solved also by the multigrid method.
Examples {#sec:results}
========
We have demonstrated[@mgarticle1] the performance of the MIKA scheme in calculating the electronic structures of small molecules and solid-state systems described by pseudopotentials. As a typical application Fig. \[fig:deep\_state\] shows the electron density of the so-called deep state localized at a neutral, ideal vacancy in bulk Si. It was shown, that the accuracy of 1 meV for the total energy was reached after three or four V-cycles, and that the amount of cpu-time needed was of the same order as when applying state-of-the-art plane-wave codes. We obtained an average convergence rate of approximately one decade per self-consistency iteration. This is of the same order as those reported by Wang and Beck [@wang1] in their FAS scheme or by Kresse and Furthmüller [@kresse2] in their plane-wave scheme employing self-consistency iterations. The convergence rate of one decade per self-consistency iteration is better than that obtained by Ancilotto [*et al.*]{} [@ancilotto] in the FMG scheme and much better than the rate reached in the linearized multigrid scheme by Briggs [*et al.*]{} [@briggs].
We have applied the RQMG method also for the calculation of positron states in solids. Fig. \[fig:positrons\] shows how the delocalized positron state in the perfect $\alpha$-quartz is trapped in to a Si-vacancy. Positron states are a particularly simple case for our method, because only the lowest energy wave function needs to be calculated in a given potential, so that no orthogonalizations or penalty functionals are needed. Moreover, in a simple scheme an electron density calculated without the influence of the positron can be used as the starting point [@pos_rev]. However, even for the positron states the superior performance of the multigrid method in comparison with straightforward relaxation schemes is evident. For example, we have calculated the positron state in the Si-vacancy in bulk Si using a supercell containing 1727 atoms. The solution of the positron wave-function using the RQMG-method took less than a minute of cpu-time on a typical work station. To put this in proper context, J. E. Pask [*et al*]{} [@Pask] report a similar calculation, based on the finite element method but without multigrid acceleration, for a supercell containing 4096 Cu atoms. The result converged within 1 ps took ’ just 14.3 hr ’ of CPU time.
We have also applied the MIKA scheme in two-dimensional problems for quantum dots employing the current-spin-density functional theory (CSDFT), see Ref. [@Henri]. Moreover, we have implemented the RQMG-method in cylindrical coordinates enabling very efficient and accurate calculations for atomic chains, or systems which can be described using axisymmetric jellium models. Fig. \[fig:cylinder\] shows a selected wavefunction of a system where a chain of four carbon atoms is sandwhiched between two planar jellium leads.
Summary and outlook {#sec:conclusions}
===================
In the MIKA program package the RQMG method introduced by Mandel and McCormick [@McCormick] is generalized for the simultaneous solution of a desired number of lowest eigenenergy states. The approach can be viewed to belong to a third group of multigrid methods, in addition to FAS and techniques where the eigenproblem is linearized. In principle, one can use arbitrarily coarse grids in RQMG, whereas in the other multigrid methods one has to be able to represent all the states also on the coarsest grid.
We are convinced that our method will compete with the standard plane-wave methods for electronic structure calculations. However, some straightforward programming is still required. Implementation of the Hellmann-Feynman forces, required for the optimization of the ionic structures is under way.
During the RQMG V-cycle, the states are all relaxed simultaneously and independently of each other. A parallelization over states would therefore be natural to implement on a shared memory architecture. We have parallelized the MIKA codes over k-points, and over real-space domains. The domain decomposition is the appropriate method for distributed memory parallel computers.
We acknowledge the contributions by Henri Saarikoski, Paula Havu, Esa Räsänen, Tero Hakala, and Sampsa Riikonen in sharing their experience of the use of the MIKA package in different applications and preparing the figures \[fig:positrons\] (T.H.) and \[fig:cylinder\] (P.H.). T.T. acknowledges financial support by the Vilho, Yrjö and Kalle Väisälä foxundation. This research has been supported by the Academy of Finland through its Centre of Excellence Programme (2000 - 2005).
W. Kohn, Rev. Mod. Phys. [**71**]{}, 1253 (1998). M. Heiskanen, T. Torsti, M. J. Puska, and R. M. Nieminen, Phys. Rev. B [**63**]{}, 245106 (2001). T. L. Beck, Rev. Mod. Phys. [**72**]{}, 1041 (2000). T. A. Arias, Rev. Mod. Phys. [**71**]{}, 267 (1999). U. V. Waghmare, H. Kim, I. J. Park, N. Modine, P. Maragakis, and E. Kaxiras, Comp. Phys. Comm. 137, 341 (2001). A. Brandt, Math. Comp. [**31**]{}, 333 (1977). E. L. Briggs and D. J. Sullivan and J. Bernholc, Phys. Rev. B, [**52**]{}, 5471 (1995); [*ibid.*]{}, [**54**]{}, 14362 (1996). F. Ancilotto, P. Blandin, and F. Toigo, Phys. Rev. B, [**59**]{}, 7868 (1999). J.-L. Fattebert, J. Comput. Phys. [**149**]{}, 75 (1999). J. Wang and T. L. Beck J. Chem. Phys. [**112**]{}, 9223 (2000). J. Mandel and S. McCormick, J. Comput. Phys. [**80**]{} (1989) 442. A. Brandt, S. F. McCormick and J. W. Ruge, SIAM J. Sci. Stat. Comput. [**4**]{}, 244 (1983). T. L. Beck and K. A. Iyer and M. P. Merrick, Int. J. Quantum Chem. [**61**]{}, 341 (1997). S. Costiner and S. Ta’asan, Phys. Rev. E, [**52**]{}, 1181 (1995). A. P. Seitsonen and M. J. Puska and R. M. Nieminen, Phys. Rev. B, [**51**]{}, 14057 (1995). G. Kresse and J. Furthmüller, Comput. Mat. Sci. [**6**]{}, 15 (1996). M. J. Puska and R. M. Nieminen, Rev. Mod. Phys. [**66**]{}, 841 (1994). J.E. Pask, B.M. Klein, P.A. Sterne and C.Y. Fong, Comp. Phys. Comm. [**135**]{}, 1, (2001). H. Saarikoski, M. J. Puska, and R. M. Nieminen, in this conference proceedings.
[^1]: For the sake of simplicity, the wave-functions, and thus $\alpha$, are here assumed real. We have implemented the complex case as well.
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---
abstract: '[**Abstract: **]{} Several conceptual aspects of quantum gravity are studied on the example of the homogeneous isotropic LQC model. In particular: $(i)$ The proper time of the co-moving observers is showed to be a quantum operator [and]{} a quantum spacetime metric tensor operator is derived. $(ii)$ Solutions of the quantum scalar constraint for two different choices of the lapse function are compared and contrasted. In particular it is shown that in case of model with masless scalar field and cosmological constant $\Lambda$ the physical Hilbert spaces constructed for two choices of lapse are the same for $\Lambda<0$ while they are significantly different for $\Lambda>0$. $(iii)$ The mechanism of the singularity avoidance is analyzed via detailed studies of an energy density operator, whose essential spectrum was shown to be an interval $[0,\rhoc]$, where $\rhoc\approx 0.41\rho_{\Pl}$. $(iv)$ The relation between the kinematical and the physical quantum geometry is discussed on the level of relation between observables.'
author:
- 'Wojciech Kamiński${}^{1}$'
- 'Jerzy Lewandowski${}^{1}$'
- 'Tomasz Paw[ł]{}owski${}^{2,1,3,4}$'
title: Physical time and other conceptual issues of QG on the example of LQC
---
$\hphantom{.}$
IGC-08/9-1
Introduction: the issues raised in the paper
============================================
Loop Quantum Cosmology [@lqc1; @lqc2] is a family of symmetry reduced models [built via methods]{} of Loop Quantum Gravity [@lqg]. It serves [both]{} as a testing ground for the quantization frameworks used in Quantum Gravity [@lqc2; @bahr; @kls-obs] and also a shortcut way to derive some physical predictions. One of the most surprising predictions it provides is the modification of the dynamics at near-Planck energy densities leading to the replacement of the classical Big Bang by a quantum Big Bounce. Although the most solid and robust results were obtained for isotropic cosmological models [@aps; @aps-imp; @apsv-spher; @skl-spher; @frw-hyper; @sLQC; @kl-sadj; @eff], there is an ongoing research (with various stages of rigour) treating homogeneous but anisotropic [@B1] or even inhomogeneous models [@eff-als; @spher; @gm-letter]. In this paper we are concerned with some conceptual aspects of quantum gravity and study them on the example of the homogeneous isotropic LQC model. They are: existence of a quantum spacetime metric tensor operator, definition of a solution to the quantum Einstein constraints, mechanism of singularity avoidance [and]{} the role of the kinematical quantum geometry for the properties of the physical quantum geometry.
Before going to the technical details of the LQC model used in this work, we [will present an]{} outline [of]{} our studies [(in Sec. \[sec:pre-qsp\] through \[sec:out-zero\])]{}. Next, in Sec. \[sec:LQC\] we will introduce the necessary technical details of the LQC model tested in this work, which is the model of isotropic, homogeneous spacetime interacting with a homogeneous scalar field introduced by Ashtekar, Pawlowski and Singh [@aps-imp]. Most of our results apply also (either directly or can be generalized) to the so called [solvable]{} LQC (sLQC) model [@sLQC].
A quantum relativistic time, a quantum spacetime {#sec:pre-qsp}
------------------------------------------------
One of the expectations upon the theory of quantum geometry is that it should provide a spacetime metric as a quantum operator $$\widehat{\rd s^2}\ =\ \widehat{g_{\alpha\beta}}\rd x^\alpha \rd x^\beta\ .$$
In the canonical formulation of the Einstein gravity, a general classical spacetime metric is written in the form $$\label{ds2gen}
\rd s^2\ =\ -(N^2 - N^a N^b q_{ab}) \rd t^2
+ N^b q_{ab} (\rd t\rd x^a + \rd x^a\rd t)
+ q_{ab}\rd x^a\rd x^b \ .$$ In the gauge choice free approach, the lapse and shift functions $N$ and $N^a$ respectively, are just non-dynamical [gauge parameters]{}. Therefore they should pass unchanged to the quantum theory, allowing in turn to write the metric tensor in the form
$$\widehat{\rd s^2}\ =\ -(N^2 - N^a N^b \hat{q}_{ab}) \rd t^2
+ N^b \hat{q}_{ab}(t,x) (\rd t\rd x^a + \rd x^a\rd t)
+ \hat{q}_{ab}(t,x)\rd x^a\rd x^b \ ,$$
where the un-hatted functions are [independent of]{} $\hat{q}$. As a consequence, even in the quantum theory, the $g_{tt}$ metric component commutes with all the other quantum metric components at any given instant $t$.
However, since Einstein’s gravity is a theory with constraints, the physical Hilbert space differs from the kinematical one, and only the Dirac observables can give rise to physical quantum observables. Therefore, the spacetime metric should be first reexpressed in terms of [them]{}. A quite well understood class [of]{} the Dirac observables are the [*partial observables*]{} developed recently by Rovelli, Dittrich and Thiemann [@rel-obs]. A partial observable is constructed out of a kinematical observable and a family of *clock functions* – functions defined on the classical phase space providing parametrizations of dynamical trajectories. One of possible choices of such clock function $T$ is a (coupled to the gravitational field) Kl[e]{}in-Gordon massless scalar field (which is exactly the choice made in the APS model of the quantum FRW spacetime [@aps-imp]). Upon that choice one can write the metric tensor (\[ds2gen\]) as, $$\rd s^2\ =\ g_{T T} \rd T^2
+ N'^b q'_{ab} (\rd T\rd x^a + \rd x^a\rd T) + q'_{ab}\rd x^a\rd x^b \ .$$ The function $g_{TT}$ is of the form $$\label{eq:N'}
g_{T T} \ =\ -\left( \frac{N^2 - N^a N^b q_{ab}}{N\frac{\pi_T}
{\sqrt{ {\rm det}q}}+N^aT_a } \right)$$ where $\tilde{\pi}_T$ is the momentum canonically conjugate to $T$, and the second equality follows from the canonical equations $$\frac{\partial T}{\partial t}\ =\
\{\,T,\,\int \frac{N}{2}(\frac{\tilde{\pi}_T^2}{\sqrt{\det q}}
+ q^{ab}\sqrt{{\rm det}q}T_{,a}T_{,b})\ +\ \int N^a\tilde{\pi}_T T_{,a}\} \ .$$
From it follows immediately that, since all terms on its righthand side are dynamical quantities, so is the function $g_{T T}$. Thus in quantum theory one should consider a Dirac observable corresponding to it. Whereas on the kinematical Hilbert space the operators ${\hat{\pi}_T}$ and $\hat{q}_{ab}$ commute, the corresponding partial observables do not, therefore the quantum counterpart of the righthand side of is not uniquely defined. This problem can be seen at the classical level already. Namely, if we denote by ${\cal O}_{\tilde{\pi}_T}$ and ${\cal
O}_{q_{ab}}$ the corresponding Dirac observables (we suppress the clock functions and other parameters needed to determine the observable), then their Poisson bracket *does not vanish*. Indeed, [(see [@GiesThiem-algiv] for details)]{} $$\{ {\cal O}_{\tilde{\pi}_T},\,{\cal O}_{q_{ab}}\}\ =\
{\cal O}_{\{{\tilde{\pi}_T,q_{ab}}\}^{\rm D}}$$ where ${\cal O}_{F(f)} = F({\cal O}_f)$ and $\{\cdot,\cdot\}^D$ is the Dirac bracket. Furthermore, one can show by inspection [(using eq. (2.18) of [@GiesThiem-algiv])]{}, that $$\{{\tilde{\pi}_T,q_{ab}}\}^{\rm D}\ \not=\ 0 \ .$$ In consequence:
(i) a quantum counterpart of $g_{TT}$ is an operator which does not commute with $\hat{q}_{ab}$ even at the same instant of time,
(ii) there is no unique definition of $\hat{g}_{TT}$ because of the ordering problem.
In this paper, we point out the issue and propose a definition of the quantum space-time metric tensor in the APS quantum FRW model, where the expression for the lapse function reduces (due to homogeneity) to $$\label{N'}
g_{TT}\ =\ \frac{\sqrt{\det q}}{\tilde{\pi}_T} \ .$$
[The physical meaning of the quantum geometry operators]{}
----------------------------------------------------------
[The quantum geometry operators are defined in the kinematical Hilbert space. They are build, briefly speaking, out of the 3-metric tensor.]{} The question regards the role and the properties of the quantum geometry operators in the physical Hilbert space. Considered operators can be defined by using the [relational]{} observables of Rovelli-Dittrich-Thiemann. On the one hand, they form in this case the same Poisson algebra as the kinematical ones. Also in simple examples ($\Lambda=0$) their quantum algebra is equivalent to the algebra of the kinematical quantum geometry operators. On the other hand, in the case of $\Lambda>0$ there are many differences between the kinematical and physical quantum geometry. We discuss them is Sec. \[sec:phys-oper\].
[Dependence of solutions to the constraint on lapse]{}
------------------------------------------------------
In a canonical approach to quantum gravity one has to define subsequently
- a quantum scalar constraint operator
- [the constraint condition, that is the mechanism via which the constraint operator selects the physical Hilbert space]{}
- a physical Hilbert space of solutions, [which involves in particular specification of the scalar product on it.]{}
In our case
the quantum constraint operator has the form $$\label{eq:Corig}
{{\hat{C}_{{{\rm tot}}}}}(N)\
=\ N(\frac{1}{2}{{\hat{\pi}_T}^2}\widehat{\sqrt{\det q}^{\ -1}}\ +\ {{\widehat{C_{{{\rm gr}}}}}})
\ ,$$ where $N$ is the lapse. One choice is to take lapse to be a number.
On the other hand, taking into account (\[N’\]) and the quantum nature of the lapse one is lead to [the constraint]{} operator $$\label{eq:Ctheta}
{{\hat{C}_{{{\rm tot}}}}}(N')\ =\
\frac{1}{2}{{\hat{\pi}_T}}\ +\ {\hat{\pi}_T}^{-1}\left[\widehat{{\sqrt{\det
q}\,}^{-1}}\right]^{-1}{{\widehat{C_{{{\rm gr}}}}}}\ ,$$ [suitably symmetrised in the second term]{}[^1]
Given either one of the constraints, [one can turn to the second step and define the corresponding constraint condition. It reads: take the spectral decomposition defined by the operator and allow only elements of the Hilbert space corresponding to the zero eigenvalue.]{}
At this point we make a suprising discovery:
- On the one hand, the first operator (\[eq:Corig\]) has a unique self-adjoint extension for arbitrary cosmological constant; we point it out in Section \[sec:C-noneq\] and give a mathematical argument.
- On the other hand, the second operator (\[eq:Ctheta\]) has inequivalent self-adjoint extensions if $\Lambda>0$. [^2]
[In other words, the second constraint operator does not define a constraint condition uniquely, because the spectral decomposition depends of a self-adjoint extension. Hence, solutions to that quantum scalar constraint depend on some additional choice which has to be made. [This apparent discrepancy forces us to ask a question:]{} What is a relation between the uniquely defined Hilbert space of solutions of the constraint (\[eq:Corig\]) and the extension dependent Hilbert spaces of solutions to the second constraint (\[eq:Ctheta\])?]{}
[To address it]{} [we explain in detail the difference in the properties of the operators and, respectively, in Sec. \[sec:C-noneq\]. Also, we briefly discuss the relation between Hilbert spaces of the solutions to those different constraints (the detailed analysis will be presented in [@ga]).]{}
[Big-Bounce and the energy density operator]{}
----------------------------------------------
[Within cosmological model specified at the end of Section \[sec:pre-qsp\] the equality satisfied by the lapse function (now given by ) can be also written in the following way $$N'^2\rd T^2\ =\ 2\rho^{-1} \rd T^2$$ where $$\rho\ =\ \frac{1}{2}\frac{\tilde{\pi}_T^2}{\det q}\ =\ T_{\mu\nu}n^\mu n^\nu$$ is the *energy density of the scalar field with* respect to the class of observers comoving with the universe.]{}
[The quantum energy density operator and its spectrum is another subject discussed in this paper on its own. The operator is used in the APS model as the measure of the avoidance of the singularity. At the early stages of LQC it was believed that the singularity avoidance is a kinematical effect implied by the non-singular way the metric determinant inverse shows up in the expression of the energy density. Indeed, the LQG motivated quantization of that expression has (up to factor ordering ambiguity) the form $$\hat{\rho}\ =\ \frac{1}{2}{\hat{\pi}_T}^2 \widehat{\det q^{-1}} \ ,$$ where the operator $\widehat{\det q^{-1}}$ is bounded, and actually annihilates the vector annihilated by $\widehat{\det q}$. A stronger result takes place in the APS model. Namely, the expectation value of the energy density $\langle\hat{\rho}\rangle(T)$ evolving with the time $T$ approaches certain universal value (of the order of Planck energy density $\rhoPl$)[[^3]]{} $$\langle\hat{\rho}\rangle(T)\ \leq\ \rhoc\
\approx\ 0.82\rhoPl$$ from below, and bounces back. Here we show, that the essential spectrum of $\hat{\rho}$ is $${\Sp{}_{\ess}}(\hat{\rho})\ =\ [0,\rhoc] \ .$$ There may still exist discrete spectrum elements bigger then $\rhoc$, however, the corresponding eigenfunctions are focused near the zero volume and therefore their contribution to semiclassical states focused at large [scalar field momentum]{} (and so at large volumes) is extremely small.]{}
[The role of the zero volume state]{} {#sec:out-zero}
-------------------------------------
[A technical subtlety concerning the constraint operators above, is that in the APS model the zero volume state $|0\rangle \in
{\cal H}_{\rm gr}$ is at the same time annihilated by the inverse-volume operator $$\widehat{\sqrt{\det q}^{-1}}|0\rangle\ =\ 0 \ .$$ This leads to [an impression of]{} incompleteness in a definition of the operator ${{\hat{C}_{{{\rm tot}}}}}(N')$ in $({{\Hil_{{{\rm gr}}}}},\,(\cdot|\widehat{\sqrt{\det q}^{-1}}\cdot))$ present even after the modification of the scalar product which removes that zero volume state. The solution to that [subtlety]{} is hidden in the results published in the literature [@aps-imp; @apsv-spher; @negL], but it [has never been]{} spelled out. We will present the details in Sec. \[sec:zero\] showing in particular some constraint [being]{} induced in ${{\Hil_{{{\rm gr}}}}}$ by the scalar constraint operator. The presence of this constraint allowed to define [rigorously]{} the evolution operator in [@aps-imp] and following works.]{} [The discussed structure allows in particular to immediately extend the results of [@kl-sadj] to superselection sector containing the $|v=0\rangle$ state.]{}
The elements of the LQC FRW {#sec:LQC}
===========================
In LQC, like in the other cosmological models, one restricts the Einstein’s theory to the space of the space-time metrics and other fields having a given symmetry. Here we consider the case of Friedman-Robertson-Walker (FRW) models corresponding to homogeneous and isotropic spacetimes. In this section we briefly introduce the quantum description of these models within LQC framework. For shortness we will introduce only those elements of the LQC models which will be relevant for our studies. For more detailed description of the quantization procedure the reader is referred to [@abl] and [@aps-imp].
On the classical level the spacetime is described by the product manifold $\mathbb{R}\times\Sigma$ and a metric tensor $$\rd s^2\ =\ -N^2 \rd t^2\ +\ a^2\fidq_{ab}\rd x^a\rd x^b$$ where $\fidq$ is a fixed, auxiliary, homogeneous, isotropic metric tensor on $\Sigma$, and $N$ is a homogeneous lapse function. The metric is coupled with a scalar field $T$ homogeneous on $\Sigma$. These properties boil down to conditions
\[eq:afhom\]$$\begin{aligned}
a(t,x)\ &=\ a(t) \ , &
T(t,x)\ &=\ T(t) \ , &
N(t,x)\ &=\ N(t) \ .
\tag{\ref{eq:afhom}}\end{aligned}$$
The diffeomorphism constraints are trivially satisfied, hence the only Einstein constraint is the scalar constraint. It takes the following form $${{\hat{C}_{{{\rm tot}}}}}(N) \ =\ N({{\widehat{C_{{{\rm gr}}}}}}+ \frac{1}{2}\frac{\tilde{\pi}_T^2}{|V|}) \ ,$$ where one fixes a finite region ([“cell”]{}) ${\Sigma}_0\subset\Sigma$ to integrate (if $\Sigma$ is compact a natural choice is $\Sigma_0=\Sigma$)
\[eq:basic-int\]$$\begin{aligned}
{{\pi_T}}\ &:=\ \int_{\Sigma_0}{\tilde{\pi}}_T \ , &
|V|\ &:=\ \int_{\Sigma_0} a^3\sqrt{{\rm det}q^{(0)}} \ , &
C_{\rm gr}\ &=\ \int_{\Sigma_0}\tilde{C}_{\rm gr}
\tag{\ref{eq:basic-int}}\end{aligned}$$
and $\tilde{C}_{\rm gr}$ is the Hamiltonian density of the gravitational field. One also introduces the oriented volume function ranging from $-\infty$ to $\infty$, namely $$V\ =\ \pm|V| \ ,$$ with the sign depending on the orientation in $\Sigma_0$ of the triad with respect to a fixed fiducial orientation of $\Sigma$. The kinematical Hilbert space and the quantum operators of the scalar field $T$ and its conjugate momentum ${{\pi_T}}$ are
$$\begin{aligned}
{\cal H}_{\rm sc}\ &=\ L^2({{\mathbb R}}) \ , \\
\hat{T}\psi(T)\ =\ T\psi(T) \ ,\qquad &{}\qquad
{\hat{\pi}_T}\psi(T)\ =\ -i\hbar\partial_T\psi(T) \ .\end{aligned}$$
The kinematical Hilbert space and the basic quantum operators for the gravitational field in the APS and sLQC model are,
$$\begin{aligned}
{{\Hil_{{{\rm gr}}}}}\ &=\ \overline{{{\rm Span}}(|v\rangle\ :\ v\in{{\mathbb R}})} \ , &
\langle v|v'\rangle\ &=\ \delta_{v,v'} \ , \\
\hat{V}|v\rangle\ &=\ \left(\frac{8\pi\gamma}{6}\right)^{\frac{3}{2}}
\frac{3\sqrt{3\sqrt{3}}}{2\sqrt{2}}\,v\,\lPl^3|v\rangle\
=:\ V_o\, v|v\rangle \ , &
\hat{h}_{\nu}|v\rangle\ &=\ |v+\nu\rangle \ ,\end{aligned}$$
where the operator $\hat{h}_{\nu}$ is a shift operator – a component of an operator corresponding to the classical holonomy function involving $d a/dt$.
The kinematical Hilbert space of the system is the tensor product ${{\Hil_{{{\rm sc}}}}}\otimes{{\Hil_{{{\rm gr}}}}}$. Every element $\psi\in {{\Hil_{{{\rm sc}}}}}\otimes{{\Hil_{{{\rm gr}}}}}$ is thought of as a function of the variables $T$ and $v$, and its values will be denoted by $\psi(T,v)$.
The quantum scalar constraint is considered in the following form $$\label{c}
\left(\frac{1}{2}{\hat{\pi}_T}^2\otimes 1\ +\ 1\otimes
\widehat{|V|^{-1}}^{-1}\widehat{C_{\rm gr}}\right)\,\psi(T,v)\
=\ 0 \ ,$$ where:
- $\widehat{|V|^{-1}}=V_o^{-1}B(\hat{V})$ is a result of a quantization of the classical $1/|V|$, with $B$ being a function. In the orthodox LQC it descends from the LQG definition of the orthonormal coframe expressed by commutators of various powers of the volume operator. For the studies performed in this article the exact form of $B$ does not matter. What is important are the following properties (true for both APS LQC and sLQC)
(i) $B(v) = B(-v)$,
(ii) [for non zero $v$]{} is finite and nonvanishing, and
(iii) for large $|v|$, $B(v) \simeq \frac{1}{|v|}$.
More specific assumptions will be made whenever necessary. Particular form of $B$ in models considered here is, respectively,
\[eq:B-form\]$$\begin{aligned}
B_{\rm sLQC}(v)\ &=\ \frac{1}{|v|} \ , &
B_{\rm APS}(v)\ &=\ \frac{27}{8}|v|
\big||v+1|^{\frac{1}{3}}-{|v-1|}^{\frac{1}{3}}\big|^3
\ .
\tag{\ref{eq:B-form}}\end{aligned}$$
- the operator $\widehat{C_{\rm gr}}$ has the form $$\label{cgr}
\widehat{C_{\rm gr}}\ =\ i(\hat{h}_{2} -
\hat{h}_{-2})A(\hat{V})i(\hat{h}_{2}-\hat{h}_{-2})
- \Lambda\hat{V} + W_k(\hat{V})$$ with $\Lambda$ being the cosmological constant, and $A$, $W_k$ being suitable symmetric functions, the second one depending on the type of the local symmetry group ($k=-1,0,1$). The assumption about $A$ we will refer to is the behavior $A(v)\sim |v|$ for large $|v|$ true in LQC as well as in the sLQC. In these two particular cases the form of $A$ reads
\[eq:A-form\]$$\begin{aligned}
A_{\rm APS}\ &=\ A_o\tilde{A}\ =\ A_0|v|\big||v+1|-|v-1|\big|
\ , & & \\
A_{\rm sLQC}\ &=\ 2A_o |v| \ , &
A_o\ &:=\ \frac{9\sqrt{3}\lPl}{32\sqrt{\pi}\gamma^{\frac{3}{2}}G}
\ .
\end{aligned}$$
The physical states are solutions to the quantum constraint, according to the APS model, thought of as maps $$\label{c'}
\mathbb{R} \ni T\ \mapsto\ \psi_T \in {{\Hil_{{{\rm gr}},B}}}\ ,$$ where the space ${\cal H}_{\rm gr, B}$ [(referred to further as an *auxiliary* space)]{} is defined by the same ${\rm
Span}(|v\rangle\! :\ v\in{{\mathbb R}})$ as before, however endowed by APS with the scalar product $$(\cdot|\cdot)_B\ =\ \langle \cdot |B(\hat{V})\cdot\rangle \ .$$ That definition of the new scalar product is suited to make the [*evolution operator*]{} $$\label{eq:Theta}
\hat{\Theta}\ :=\ - (B(\hat{V}))^{-1}{{\widehat{C_{{{\rm gr}}}}}}$$ symmetric, however the definition of this operator [in the form it is presented above,]{} needs to be completed. [Such precise definition, which was used in [@aps-imp; @apsv-spher; @negL], is discussed in Appendix \[sec:zero\].]{} Now, each solution to the scalar constraint takes the form $$\label{decomp}
\psi\ =\ \psi^{-}\ +\ \psi^+ \ ,$$ where $\psi^{\pm}$ satisfies, respectively, $$\label{cpm}
{\hat{\pi}_T}\psi^{\pm}_T(v) \ =\ \pm
\sqrt{2V_o\hat{\Theta}}\, \psi^{\pm}_T(v) \ ,$$ where each solution $\psi$ of (\[cpm\]) takes values $\psi_T$ in the part of the Hilbert space corresponding to the non-negative part of the spectrum of the operator $\hat{\Theta}$, and the square root is defined on that subspace. We will be assuming throughout this paper that this decomposition is unique, [which]{} is generically true[^4]. A non-unique case is considered in [@posL1]. Given two solutions $\psi$ and $\psi'$ of the quantum scalar constraint, APS define the following scalar product $$\label{scprphys}
(\psi|\psi')_{\rm phys}\ :=\ (\psi^+_T|\psi^+_T)_B\ +\
(\psi^-_T|\psi^-_T)_B$$ where the RHS is independent of $T$. Denote the resulting Hilbert space by ${{\Hil_{{{\rm phys}}}}}$.
A physical observable ${\hat{\pi}_T}$ is $$\label{pi}
{\hat{\pi}_T}\psi^\pm\ :=\ \pm
\sqrt{2V_o\hat{\Theta}}\, \psi^\pm \ ,$$ The volume operator $\hat{V}$ defined in the kinematical Hilbert space gives rise to the physical observable ${\hat{\cal
O}}_{V}(T_0)$ (modulo the discussion in Sec. \[sec:phys-oper\] below) determined by a number $T_0$ (the “instant of time”) and defined by the following [expression]{} $$({\hat{\cal O}}_{V}(T_0)\psi)_{T_0}\ =\ \hat{V}\psi_{T_0} \ .$$ In consequence it can be thought of as an operator in QM acting at an instant $T_0$ on a state evolving in the Schroedinger picture.
The final Hilbert space is selected as the irreducible subspace of all the quantum observables we choose. Classically, the system can be described completely by scalar field momentum ${{\pi_T}}$ and the volume of the fixed cell $V$. The first one commutes with the constraint, whereas the second defines the Dirac observables via the relational observables construction. Therefore, APS assume that the sufficient set of quantum operators to describe every quantum state consists of the following operators:
\[eq:obses\]$$\begin{aligned}
{\hat{\pi}_T}\psi^{\pm}(T,v)\
&:=\ (\id\times\sqrt{2V_o\hat{\Theta}})\,\psi^{\pm}(T,v) \ , \\
|\hat{V}|_{T_0}\psi^{\pm}(T,v)\
&:=\ e^{\pm i(T-T_0)\sqrt{2V_o\hat{\Theta}}}\,\hat{V}\,\psi^{\pm}(T_0,v) \ .\end{aligned}$$
[Furthermore]{}, there are subspaces $\Hil^\pm_{{{{\rm phys}}},\epsilon}$ preserved by the action of all the quantum observables, labeled by arbitrary $\epsilon\in [0,4)$ and a sign ‘$+$’ or ‘$-$’ corresponding to the decomposition (\[decomp\]). The subspace ${\cal H}^+_{{\rm
phys},\epsilon}$ (${\cal H}^-_{{\rm phys},\epsilon}$) is the space of solutions (\[c’\]) to (\[cpm\]) with $\pm=+$ ($\pm=-$) which take values in the following subspace of ${\cal H}_{{\rm gr},B}$ $$\label{Hpmepsilon}
{\cal H}_{\epsilon}\ =\ \overline{{{\rm Span}}(|v\rangle\ :\
v=\epsilon+4n,\ \ n\in{{\mathbb Z}})}$$ In the special cases of $\epsilon=0,\,2$, the subspace ${\cal
H}^\pm_{\epsilon}$ admits the action of the orientation changing operator $$(P\psi)_T(v)\ \mapsto\ \psi_T(-v) \ .$$ which commutes with $T$. In that case APS restrict the Hilbert space ${\cal H}^\pm_{\epsilon}$ further, to the subspace of the even functions.
The operator $\hat{\Theta}$ is well defined in every subspace $\Hil^\pm_{\epsilon}$ (in the domain ${{{\rm Span}}(|v\rangle\ :\
v=\epsilon+4n,\ \ n\in{{\mathbb Z}})}$) such that $\epsilon\not=0$, however for $\epsilon=0$ its definition is not [*a priori*]{} obvious and needs explanation. We provide it in [Appendix \[sec:zero\]]{} as well as our definition of the evolution operator $\hat{\Theta}$ in the $B(0)=0$ case.
[**Remarks.**]{}
- For the remaining values of the parameter $\epsilon$ we have $P(\Hil^\pm_{\epsilon})=\Hil^\pm_{4-\epsilon}$. Then, APS construct the space of the even functions spanned by elements of $\Hil^\pm_{\epsilon}$ and $P(\Hil^\pm_{\epsilon})$. In these cases construction reduces (is unitarily equivalent) to the single $\Hil^\pm_{\epsilon}$.
- We will often ignore the reducibility and consider the whole Hilbert space $\Hil_{{{\rm phys}}}$.
The quantum geometry operators in the physical theory {#sec:phys-oper}
=====================================================
In the Dirac program one of the most common techniques of constructing observables on ${{\Hil_{{{\rm phys}}}}}$ is an appropriate pull-back onto it of kinematical ones. However unless the quantity measured by given observable is a constant of motion such direct pull-back will not correspond to any physically interesting property of the system. Therefore in such cases one tries to construct operators measuring kinematical quantity “at a given time” (example of which is the operator $|\hat{v}|_{\phi}$ defined in [@aps-imp]). Technically this corresponds to the pull-back of kinematical observable to an auxiliary Hilbert space, the image of mapping . In this section we address (in context of LQC) the question of how the original properties of kinematical operators transfer to physical spaces. We will see that even the volume operator, seemingly under a perfect control, may surprisingly change much more, than it is expected in the LQC literature.
[Let us start our analysis in the context of an APS LQC, where $B(0)=0$. There the auxiliary Hilbert space is a space $\Hil_{{{\rm gr}},B}$ equipped with]{} the modified scalar product $\langle\cdot|B(\hat{V})\cdot\rangle$. The quantum volume operator is unchanged by this modification, and is still well defined and essentially self-adjoint in the domain ${{\rm Span}}(|v\rangle\ :\ v\in\mathbb{R})$. However a general operator $\hat{g}$ defined in that domain in ${\cal H}_{\rm gr}$ should be redefined such that modulo the ordering it corresponds to the same classical kinematical observable, but has the correct properties with respect to the ${}^\dagger$ operation. An example of such redefinition is replacing $\hat{g}$ defined in ${\cal H}_{\rm gr}$ by $$B(\hat{V})^{-1/2}\hat{g}B(\hat{V})^{1/2}$$ defined in ${\cal H}_{{\rm gr},B}$. In fact, this transformation coincides with the pull back of $\hat{g}$ by the unitary map used in the previous section, [which is]{}
\[eq:tr2\]$$\begin{aligned}
\Hil_{{{\rm gr}},B}\ &\rightarrow\ {{\Hil_{{{\rm gr}}}}}\ , &
\psi\ &\mapsto\ \sqrt{B(\hat{V})}\psi \ ,
\tag{\ref{eq:tr2}}\end{aligned}$$
and the inverse image of the domain ${{\rm Span}}(|v\rangle\ :\ v\in\mathbb{R})$ is ${{\rm Span}}(|v\rangle\ :\ 0\not= v\in\mathbb{R})$. As we mentioned, the volume operator $\hat{V}$ is not affected by that transformation (modulo the small restriction of the domain [consisting in]{} disappearing of the zero volume eigenvector $|0\rangle$.)
In the sLQC case, on the other hand, [the auxiliary space is directly]{} ${{\Hil_{{{\rm gr}}}}}$, so [the analog of the transformation is just an identity]{}.
[The transformation presented above does not however solve all the problems. To see that]{} let us go back to the construction of the physical Hilbert space. [To shorten the explanation we introduce the common notation denoting]{} by $({\cal H},(\cdot|\cdot))$ the Hilbert space $({\cal
H}_{{\rm gr},B},\langle\cdot|B\cdot\rangle)$ in the APS model case, or $({\cal H}_{{\rm gr}},\langle\cdot|\cdot\rangle)$ in the sLQC case. [Then each element of ${{\Hil_{{{\rm phys}}}}}$ is represented by a mapping]{} $${{\mathbb R}}\ni T\mapsto\psi_T\in{\cal H}$$ where the ${\cal H}$ valued functions $\psi$ satisfy the equation $$\label{thec}
{\frac{\partial^2}{\partial T^2}}\psi_T\ =\
-2V_o\hat{\Theta}\psi_T \ .$$ Choosing any instant of $T=T_0$, we have two maps from the space of the solutions to $\Hil$, $$\label{themap}
\psi\mapsto \psi^{\pm}_{T_0}\in {\cal H} \ ,$$ [corresponding, respectively, to the positive and negative frequency solutions.]{} Let us fix one of them (that is either ‘$+$’ or ‘$-$’). [Now]{}, the important observation is, that if the operator $\hat{\Theta}$ is not positive [(which happens for example in the case $\Lambda>0$)]{}, then the image of the map (\[themap\]) is not the entire Hilbert space ${\cal H}$. Indeed, a physical solution should satisfy at every instant $T_0$, $$\label{eq:positive}
(\psi_{T_0}\,|\hat{\Theta}\psi_{T_0})\ \ge\ 0 \ .$$ Assuming that the operator $\hat{\Theta}$ is self-adjoint, we can identify the image of the map with the subspace $\Hil_{\hat{\Theta}\geq0}$ of $\Hil$ corresponding to the non-negative part of the spectrum of $\hat{\Theta}$.
Let us now consider an example of the operator $\hat{g}$, the volume $\hat{g}\ =\ \hat{V}$. For the pullback of the operator at any $T$ to ${\cal H}_{\rm phys}$ to be well defined, the answer to the following two questions should be affirmative:
(i) \[it:q1\] Is any dense subset of $\Hil_{\hat{\Theta}\geq 0}$ contained in the (maximally extended) domain of the operator $\hat{V}$?
(ii) \[it:q2\] Is the space $\Hil_{\hat{\Theta}\geq 0}$ preserved by the volume operator $\hat{V}$?
When $\Lambda >0$, the answer to the question is likely to be negative. In particular, we do know that the eigenvectors of the evolutions operator $\hat{\Theta}$ are not in the domain of the volume operator. [A heuristic reason for that can be seen at the classical level already, when the trajectories reach infinite volume for finite $T$. To avoid this problem one has to ”compactify” the volume, that is to consider, instead of $\hat{V}$, an operator $f(\hat{V})$, where $f$ is a bounded (but monotonic) function.]{}
The most likely answer to the question is also “no”. [This means that]{}, given a solution $\psi$ of (\[thec\]) at an instant $T_0$ (taking values in the subspace $\Hil_{\hat{\Theta}\geq
0}$), in general there is no solution $\psi'$ of (\[thec\]) such that $$f(\hat{V})\psi_{T_0}\ =\ \psi'_{T_0} \ .$$ [To overcome this problem]{} one can employ the fact, that the sesquilinear form $(\cdot|f(\hat{V})\cdot)_B$ defined by $f(\hat{V})$ can be restricted to any subspace and define an operator therein. This is equivalent to using the orthogonal projection $$\label{eq:pos-proj}
P_{\Theta\geq 0}:\Hil_{{\rm gr},B}\rightarrow \Hil_{\Theta\geq 0}$$ and replacing he operator $f(\hat{V})$ by $$f(\hat{V})'\ :=\ P_{\Theta\geq 0}f(\hat{V})P_{\Theta\geq 0}.$$ The final operator $f(\hat{V})'$ is a well defined observable, in a sense that the answer to both and is affirmative.
To summarize, in the case when $\Theta$ is not positive definite the straightforward pull-back of the kinematical volume operator does not define correct physical observable. To define it correctly one has to implement additional modifications, like the ones presented above. [However, one]{} should be aware of the likelyhood of change of the commutation relations between projected operators, as in general for a projection operator $P$ we have $$[PAP,PBP]\ \not=\ P[A,B]P \ .$$
Finally, [let us]{} consider a relation of the physical volume operator with the original, kinematical one in the APS model.
When the operator $\hat{\Theta}$ is positive, the map $$\label{iso}
\Hil^\pm_{{{\rm phys}}}\ni\psi\mapsto \psi_{T_0}\in
\Hil_{{{\rm gr}},B}$$ is unitary. It pulls back the operator $\hat{V}$ to the observable operator $\hat{{\cal O}}_{V}(T_0)$. Hence, the spectrum of the resulting physical operator observable $\hat{{\cal O}}_{V}(T_0)$ coincides with the spectrum of the restriction of $\hat{V}$ and it is independent of $T_0$.
[The]{} situation changes if the operator $\hat{\Theta}$ is non-definite. The physical operator is now the pullback by (\[iso\]) of the modified operator $P_{\Theta\geq 0}f(\hat{V})P_{\Theta\geq 0}$ which is just a different operator than $f(\hat{V})$. In consequence their spectra may differ considerably.
The spacetime metric tensor operator from LQC {#sec:qmetric}
=============================================
Having the LQC FRW model at our disposal, we can implement our consideration from Sec. \[sec:pre-qsp\] concerning a quantum spacetime metric operator. For this model the classical spacetime metric tensor is $$\rd s^2\ =\ -\frac{V^2}{{{\pi_T}}^2}\rd T^2\ +\
\frac{{V}^{\frac{2}{3}}}{\int_{{\cal U}_0}\sqrt{{\rm det}\fidq}}
\,{\fidq_{ab}\rd x^a\rd x^b} \ .$$ Applying the discussion of Sec. \[sec:pre-qsp\] regarding lapse function we observe that the quantum operator corresponding to $\rd
s^2$ should have the form $$-\widehat{{\left(\frac{V^2}{{{\pi_T}}^2}\right)}}\rd T^2\ +\
\frac{{\hat{V}}^{\frac{2}{3}}}{\int_{{\cal U}_0}\sqrt{{\rm
det}\fidq}}
\,{\fidq_{ab}\rd x^a\rd x^b}$$ where ${\widehat{\left(\frac{V^2}{{{\pi_T}}^2}\right)}}$ stands for a quantum operator corresponding to the classical ${\frac{V^2}{{{\pi_T}}^2}}$. However, the operators $\hat{V}$ and ${\hat{\pi}_T}$ do not commute in ${\cal H}_{\rm
phys}$ (see (\[pi\])). Therefore there are two possibilities:
(i) the time part of the space time metric is only a semiclassical notion, and does not exists as a uniquely defined quantum operator, or
(ii) physics chooses one specific way of defining that operator, however we do not have sufficient information to guess that choice.
Remarkably, however, quantum test fields interacting with this quantum spacetime propagate in the unique way independent of that ambiguity [@qft-qsp]. Thus, the possible physical answer to that issue may be that the quantum metric is defined uniquely only through matter propagating on it.
The spacetime metric tensor, if it exists, can be used to calculate the geometric time of an interval $((T_1,x^a),(T_2,x^a))$ in a state (\[c’\]). It is given by the following formula $$\tau_{T_2,\,T_1}\ =\ \int_{T_1}^{T_2} {\sqrt{(\,\psi_T\,
|\,\widehat{\left(\frac{V^2}{{{\pi_T}}^2}\right)} \psi_{T}\,)}\rd T.}$$
Classically, the time component of the metric tensor can be expressed by the energy density ${\rho}$ of the homogeneous scalar field. The relation reads $${\frac{V^2}{{{\pi_T}}^2}}\rd T^2\ =\ {2}{\rho}^{-1}\rd T^2$$ Assuming that the relation holds on the quantum level, we have $$\label{eq:rho-appear}
{\widehat{\left(\frac{V^2}{{{\pi_T}}^2}\right)}}\rd T^2\ =\
{2}\hat{\rho}^{-1}\rd T^2.$$ However, we still have the similar ordering freedom in the definition of $\hat{\rho}$ operator [(see section \[sec:rho\])]{}.
Non-equivalence of the constraints $\widehat{C(1)}$ and $\widehat{C(|V|)}$ {#sec:C-noneq}
==========================================================================
In the previous sections, following the APS approach, we considered the scalar constraint in the form (\[c\]), that is $$\label{c1}
\widehat{C(|V|)}\ =\ \left(\frac{1}{2}{\hat{\pi}_T}^2\otimes 1\ -\
1\otimes V_o\hat{\Theta}\right)$$ defined in the Hilbert space ${\cal H}_{\rm sc}\otimes{\cal H}_{{\rm
gr},B}$ (see Sec. \[sec:LQC\], App. \[sec:zero\]).
This quantum constraint corresponds to the classical scalar constraint $C(N_1)$ given by the choice of the lapse function $$\label{N1}
N_1\ =\ |V| \ .$$
On the other hand one can choose different lapse, [in particular]{} $$\label{N2}
N_2\ =\ 1 \ ,$$ [which is more natural from the point of view of full LQG.]{} The corresponding quantum constraint operator is [of the form]{} $$\label{BC}
\widehat{C(1)}\ =\ ( \frac{1}{2}{\hat{\pi}_T}^2\otimes
|\widehat{V^{-1}}|
+ 1\otimes \widehat{C_{\rm gr}})$$ and it is defined right in the kinematical Hilbert space ${{\Hil_{{{\rm sc}}}}}\otimes{{\Hil_{{{\rm gr}}}}}$.
Given the quantum scalar constraint operator in either of the forms, the general construction (via the method of group averaging [@gave]) of the physical Hilbert space uses its spectral decomposition. The solutions are distributions defined on the spectrum and supported at the point $\lambda=0$. In the case of the operator (\[c1\]) the construction boils down to the APS construction outlined in Sec. \[sec:LQC\] [@aps]. [In this section we address the question]{} whether the second choice of the lapse function leads to the same result.
[To find an answer to this question we have to compare the spectral properties of constraints and . In the first case they are encoded in properties of the family of operators $\Theta_{{{\pi_T}}} := \frac{1}{2}{{\pi_T}}^2-V_o\hat{\Theta}$, (with ${{\pi_T}}\in
\mathbb{R}$) depending in turn on the spectral structure of $\hat{\Theta}$ in ${\cal H}_{{\rm gr},B}$. In the second case, on the other hand, the constraint inherits its properties from the family $\widehat{C_{{{\rm gr}},{{\pi_T}}}} :=
\frac{1}{2}{{\pi_T}}^2 |\widehat{V^{-1}}|+{{{\widehat{C_{{{\rm gr}}}}}}}$ (with ${{\pi_T}}\in\mathbb{R}$) defined in ${{\Hil_{{{\rm gr}}}}}$. In both cases the domain of considered families is ${\rm Span}(|v\rangle\ :\ v\in \mathbb{R})$]{}
Let us first turn our attention to the case of constraint . As discussed above its properties (in particular self-adjointness) are inherited from $\hat{\Theta}$, which has been recently extensively investigated. In particular
(i) for
- $\Lambda <0,\, k=-1,0,1$,
- $\Lambda=0,\, k=0, k=1$
- $\Lambda > \Lambda_{\crit},\, k=-1,0,1$,
- $\Lambda = \Lambda_{\crit},\, k=0$
(where $\Lambda_{\crit}:=8\pi G\rhoc$) the operator $\Theta$ is essentially self-adjoint [@kl-sadj; @sa], whereas
(ii) for $\Lambda\in (0,\Lambda_{\crit}), k=0$ [*it has inequivalent self-adjoint extensions* (see, for detailed analysis, [@posL1; @posL2] and also [@sa] for a summary, all currently in preparation)]{}.
[In the latter case]{} each self-adjoint extension of the operator $\hat{\Theta}$ defines via the APS construction a distinct quantum theory. The unitary non-equivalence of the different extensions follows from the difference between the corresponding discrete spectra.
That non-uniqueness in the self-adjoint extensions is related to the properties of the classical system: the evolution of the FRW spacetime reaches the end (the infinite physical time of the observers expanding with the universe [also corresponding to an infinite volume]{}) in a finite value of the scalar field $T$ used as a time variable. [In consequence to continue evolution in $T$ one has to specify boundary conditions at $|v|=\infty$.]{}
Surprisingly, those properties of the constraint operator (\[c1\]) in the case $\Lambda>0$, are in contrast with the properties of the constraint operator (\[BC\]) corresponding to the choice of the lapse function $N_2 = 1$, namely:
\[thm:o1\] [The operator $\widehat{C_{\rm gr}}$ is essentially self adjoint for arbitrary value of the cosmological constant $\Lambda$ and for arbitrary case $k=-1,0,1$.]{}
The technical reason [for this]{} is the following general fact [[@jacobi]]{}:
\[thm:l1\] In the Hilbert space $$\overline{{{\rm Span}}(|v\rangle\ :\ v\in
\mathbb{R})}\ \ \ \langle v|v'\rangle\ =\ \delta_{v,v'}$$ consider an operator $$(h_{2}-h_{-2})\tilde{A}(\hat{V})(h_{2}-h_{-2}) + W(\hat{V})$$ defined in the domain ${{{\rm Span}}(|v\rangle\,:\,v\in\mathbb{R})}$. That operator is essentially self-adjoint for every function $W$ and every nowhere-vanishing function $\tilde{A}$ such that $$\label{sum1/A}
\sum_{n\in\mathbb{Z}_{+}}\frac{1}{|\tilde{A}(\epsilon+4n)|}\ =\ \infty\quad
{\rm and}\quad
\sum_{n\in\mathbb{Z}_{-}}\frac{1}{|\tilde{A}(\epsilon+4n)|}\ =\ \infty \ .$$
Note, that the result holds for $\tilde{A}=A$ due to the [asymptotic behavior]{} $A(v)\propto |v|$ for $|v|\rightarrow \infty$. [On the other hand it does not hold for constraint (\[c1\]) for]{} considering it amounts to replacing the function $A$ by a function $\breve{A}\propto
|v|^2$ [(see )]{} which does not satisfy the condition (\[sum1/A\]).
The remaining [(not covered by Obs.\[thm:o1\])]{} term in $$\frac{1}{2}{{\pi_T}}^2 \widehat{V^{-1}}\ =\ \frac{1}{2V_o}{{\pi_T}}^2 B(\hat{V})$$ is bounded (in the APS case) and does not spoil the essential-self adjointness while added to $\widehat{C_{\rm gr}}$. [As the consequence,]{} self-adjoint extensions of the constraint operator is uniquely defined.
In summary, we have considered two operators (\[c1\]) and (\[BC\]) of the quantum scalar constraint corresponding to the classical constraint $C(N)$ with two different choices of a lapse function, namely: $N=N_1$ (\[N1\]) and $N=N_2$ (\[N2\]), respectively. [Provided,]{} the cosmological constant $\Lambda<0$, each of the operators $\widehat{C(|V|)}$ and $\widehat{C(1)}$ has a uniquely defined self-adjoint extension. However, if the value of the cosmological constant is positive $\Lambda_{\rm
cr}>\Lambda>0$, then the quantum scalar constraint operator $\widehat{C(|V|)}$ depends on a choice of a self adjoint extension of the operator $\hat{\Theta}$. Each choice determines a [(potentially) distinct]{} physical model. The operator $\widehat{C(1)}$ on the other hand, is essentially self-adjoint for every value of $\Lambda$.
How do those results fit together? What is the comparison between the sets of solutions to the quantum scalar constraint defined by using the operator $\widehat{C(|V|)}$ as opposed to those defined by using the operator $\widehat{C(1)}$?
It turns out, that in every case with $\Lambda<0$, the solutions to the quantum scalar constraint $\widehat{C(|V|)}$ coincide with the solutions to the quantum scalar constraint $\widehat{C(1)}$ and the physical model is independent of which constraint operator we use to construct it.
Let us turn now to the [positive]{} $\Lambda>0$ case. [One can ask:]{} what are the physical solutions obtained from the spectral decomposition of the operator $\widehat{C(1)}$. To answer it, let us restrict (for simplicity) the space ${{\Hil_{{{\rm gr}}}}}$ to the subspace $$\Hil_{o}^{\rm ev}\ :=\ {{\rm Span}}(|v\rangle + |-v\rangle\,:\,0\not=
v\in\mathbb{R} ) \ .$$ That subspace is preserved by the operator ${{{\widehat{C_{{{\rm gr}}}}}}}$, and actually, is exactly the subspace promoted to the physical Hilbert space in [@aps], which makes our restriction justified. A physical solution $\psi$ obtained by the spectral decomposition of $\widehat{C(1)}$ restricted to $\Hil_{o}^{\rm ev}$ is a family $$[0,\pi)\ni a\mapsto \psi^{(a)}$$ of solutions to the constraints $$\widehat{C(|V|)}^{(a)}\psi^{(a)}\ =\ 0 \ ,$$ where the $a$ labels the self-adjoint extensions of the constraint operator, and $\widehat{C(|V|)}^{(a)}$ stands for the corresponding extension. The physical scalar product derived from the spectral decomposition of $\widehat{C(1)}$ is $$(\psi|\psi')\ =\
\int_{0}^\pi \rd a {(\psi^{(a)}|\psi'^{(a)})}_{\rm phys} \ ,$$ where ${(\psi^{(a)}|\psi'^{(a)})}_{{{\rm phys}}}$ is the physical scalar product between the states in the APS model.
In conclusion, the physical Hilbert space constructed directly from the constraint “contains” [*all*]{} the solutions given by [*all the self adjoint extensions*]{} of the operator [$\Theta$]{}. The reason [for]{} the quotation mark is that the solutions to are not normalizable solutions of . The detailed construction of the Hilbert space of solutions to the constraint $\widehat{C(1)}$ will be presented in [@ga].
The scalar field energy density operators {#sec:rho}
=========================================
[Consider now the quantum scalar field energy density operator $\hat{\rho}$ introduced in . For the flat isotropic FRW systems considered so far in LQC [@aps-imp; @negL] the analysis of states semiclassical at late times has shown that the expectation values of $\hat{\rho}$ for such states are always bounded from above by a fundamental value $\rho_c$. This result was next extended in context of sLQC (for $k=0$, $\Lambda=0$) to full physical Hilbert space [@sLQC]. In this section we address the issue of the boundedness of energy density both in solvable and APS formulation (see Sect. \[sec:LQC\]) of LQC from a slightly different perspective, namely by analysing the spectrum of the operator $\hat{\rho}$.]{}
[Here,]{} for the convenience, we will work with a slightly different representation of the physical states defined by the unitary embedding
\[eq:transf\]$$\begin{aligned}
{\cal H}_{{\rm gr},B}\ &\rightarrow {\cal H}_{\rm gr} &
\psi\ &\mapsto\ B^{-\frac{1}{2}}\psi \ .
\tag{\ref{eq:transf}}\end{aligned}$$
The transformation does not affect the representation used in the sLQC case in Appendix \[sec:zero\] because that representation is defined directly in ${{\Hil_{{{\rm gr}}}}}$. [In consequence $\Theta$ is still well defined.]{}
An energy density operator should be given by a suitable symmetrization ‘$\vdots$’ of the following operator product $$\hat{\rho}= -\vdots\frac{\rhoc}{8}
B(v)^{\frac{1}{2}}(h_{-2}-h_{2})\tilde{A}(v)
(h_{-2}-h_{2})B(v)^{\frac{1}{2}}\vdots\,$$ where $\rhoc:=\sqrt{3}/(16\pi^2\gamma^2G^2\hbar)$ is the critical energy density defined in [@aps-imp].
We will start our discussion with the simplest possible case and increase the level of complexity. To begin with let us consider the $\Lambda=0=k$ case (see (\[cgr\])), and $\epsilon\not=0$. Also, as the functions $A$, $B$ we take the simplest ones – corresponding to sLQC
\[eq:AB-sLQC\]$$\begin{aligned}
A\ &=\ A_{\rm sLQC}\ =\ 2A_0|v| \ , &
B\ &=\ B_{\rm sLQC}\ =\ \frac{1}{|v|} \ .
\tag{\ref{eq:AB-sLQC}}\end{aligned}$$
Defining the ordering “all the functions in between the holonomy operators”, we get the following operator $$\label{rhoideal}
\hat{\rho}\ =\ -{\frac{\rhoc}{4}(h_2-h_{-2})^2} \ .$$ The spectrum of this operator is $[0,\,\rhoc]$. Notice, that $\rhoc$ is exactly the maximal value achieved by $\langle\hat{\rho}\rangle(T)$ during the Big-Bounce.
Now, consider the case
\[eq:AB-APS\]$$\begin{aligned}
A\ &=\ A_{\rm APS} \ , & B\ &=\ B_{\rm APS} \ .
\tag{\ref{eq:AB-APS}}\end{aligned}$$
Here we can set the same ordering as in , $$\label{eq:rho-ideal}
\hat{\rho}_1\ =\ -{\frac{\rhoc}{8}}
(h_{-2}-h_{2})B(v)\tilde{A}(v)(h_{-2}-h_{2}) \ ,$$ although there is a multitude of other possibilities available, like for example $$\label{eq:rho-used}
\hat{\rho}_2\ =\ -\frac{\rhoc}{8}
B(v)^{\frac{1}{2}}(h_{-2}-h_{2})\tilde{A}(v)
(h_{-2}-h_{2})B(v)^{\frac{1}{2}} \ .$$ In either case the resulting operator is plus a compact operator. Therefore the essential spectrum is still $[0,\,\rhoc]$, however there maybe a non-empty point spectrum above $\rhoc$ (with possible accumulation point $\rhoc$). The presence and structure of the point spectrum depends on particular ordering chosen. However, if the operator $\hat{\rho}$ is not explicitly bounded from above by $\rhoc$, this opens the possibility that the energy density expectation value can a-priori exceed $\rhoc$ for some physically interesting states. We address this issue now.
By a calculation similar to that in [@sa], one can show that the asymptotic behaviour of the solution to equation $$\label{eq:eig-eq}
\rho^{\dagger}_2\psi\ =\ \lambda\psi \ ,$$ in dual space is the following ($\alpha,\alpha'$ are some coefficients depending only on $\lambda$) $$\label{eq:limits}
\psi(v)\ =\ \begin{cases}
\alpha e^{i\gamma v}+\alpha'e^{-i\gamma v}+\bar{O}(v^{-1})
\ , & \lambda<\rhoc \ , \\
\alpha e^{\gamma v}(1+\bar{O}(v^{-1}))\ {\rm
or}\ \alpha'e^{-\gamma v}(1+\bar{O}(v^{-1}))
\ , & \lambda>\rhoc \ ,
\end{cases}$$ where $\bar{O}(v^{-n})$ is a [*bounded*]{} rest term decaying like $v^{-n}$ for large $|v|$, and $\gamma$ satisfies
$$\begin{aligned}
\label{eq:gamma1}
\rhoc\sin^2 \gamma\ &=\ \lambda \ , \quad \lambda<\rhoc \ ,\\
\label{eq:gamma2}
\rhoc\sinh^2 \gamma\ &=\ \lambda \ , \quad \lambda>\rhoc \ .\end{aligned}$$
This asymptotic behavior is valid for $\lambda\not=\rhoc$ and gives exponential decay of eigenfunctions with eigenvalues $\lambda>\rhoc$.
It is also worth noting, that (at least for orderings given by (\[eq:rho-ideal\], \[eq:rho-used\])) there are no normalizable solutions to with $\lambda=0$, therefore $\hat{\rho}$ is invertible.
At this moment we cannot exclude possibilities of existing eigenfunctions of $\hat{\rho}$ with $\lambda>\rhoc$. Indeed, the numerical check performed for $\hat{\rho}$ given by (via methods used for analysis of the spectrum of $\Theta$ operator in [@apsv-spher; @negL]) revealed the existence of such eigenfunctions. An example of one of them is shown on Fig. \[fig:norm-eig\]. Nevertheless we showed that any eigenfunction of $\lambda>\rhoc$ decays exponentially in $v$, with the exponent growing logarithmically with $(\lambda-\rhoc)$ (see ). On the other hand the numerical simulations show that for large energies eigenfunctions of $\Theta$ are supported away from small values of $v$. In consequence their scalar product with eigenvalues of $\hat{\rho}$ under consideration is very small. This fact explains why the influence of the latter cannot be observed.
![A normalizable eigenfunction [$\psi_\lambda$]{} of the $\hat{\rho}$ operator defined via corresponding to eigenvalue $\lambda>\rhoc$. An example presented here has eigenvalue very close to $\rhoc$, namely $(\lambda-\rhoc)\approx 7.7\cdot 10^{-7}\rhoc$.[]{data-label="fig:norm-eig"}](discr-eig.eps){width="80.00000%"}
An important difficulty emerges in the case of $\Lambda>0$. Then, the scalar field energy density operator is not any longer non-negative in $\Hil_{\rm gr}$, because it is of the form $$\hat{\rho}= -\vdots\frac{\rhoc}{8}
B(v)^{\frac{1}{2}}(h_{-2}-h_{2})\tilde{A}(v)
(h_{-2}-h_{2})B(v)^{\frac{1}{2}}\vdots\ -\
\Lambda \frac{1}{V_o}\hat{V}\hat{B} \ ,$$ [and the essential part of its spectrum is shifted with respect to $\Lambda=0$ case by $-8\pi G\Lambda$.]{} However, the solutions of the quantum constraint (\[eq:c-gen\]) take values in the subspace of $\Hil_{\rm gr}$ corresponding to the non-negative part of the spectrum of $\Theta$. The question (we do not know the answer to) is whether or not $\hat{\rho}$ restricted to that physical subspace becomes non-negative. Since the answer probably depends on the choice of the density operator, the non-negativity is a condition of the ordering in a definition of $\hat{\rho}$ consistent with the definition of $\Theta$.
Concluding remarks
==================
The problem which we leave open is whether or not the quantum spacetime metric tensor operator should be uniquely defined in QG. [The proposal of such construction was made in Section \[sec:qmetric\] however it suffers the factor ordering ambiguity.]{} [At this point there are two aspects of that issue which are worth commenting.]{}
[First,]{} the ordering ambiguity is restricted by the group averaging techniques. Since the starting point for [that procedure]{} is the kinematical Hilbert space, the operator ${\hat{\pi}_T}$ commutes with the geometry operators. Then the ambiguity is restricted [just]{} to a symmetrization of the product [${\hat{\pi}_T}\delta(\hat{T}-T_0)$]{}. We [provide an extended explanation]{} of that point in [@ga].
[Second,]{} [it is possible to]{} derive the propagation equation for a quantum test field on the quantum geometry background [@qft-qsp]. Not surprisingly, the result involves the quantum metric tensor components. Remarkably, the expression is uniquely defined, whether the quantum metric tensor [operator itself exists]{} or not. Thus, the possible physical solution of that issue may be that the quantum metric is defined uniquely only through matter propagating on it.
Another unsolved issue is how to understand the space of solutions to the quantum scalar constraint corresponding to the lapse function $N=1$. [The properties of each of the individual quantum constraint operators $\hat{C}(V)$ and, respectively, $\hat{C}(1)$ are familiar from the Schroedinger quantum mechanics. In the first case the classical trajectories are not complete in the evolution parameter, namely infinite volume is achieved in finite time. That is usually an indication that a self-adjoint extension is likely to be not unique. In the second case, the infinite volume is achieved only in infinite time. (The classical trajectories are incomplete at the zero volume as well, but in LQC this does not cause any evolution unbiguity.) Therefore there is the analogy with the quantum mechanics. The difference, and a new ambiguity is, that in gravity we can have two characteristics in a single theory depending on choice of the evolution parameter. The details of the construction of the solutions to the constraint operator $\hat{C}(1)$ from the solutions of the variuos extensions of the constraint operator $\hat{C}(V)$]{} will be presented in [@ga]. Calculation of the partial observables [might]{} bring even more surprises.
Finally, in this work we have considered a simplest LQC model. However the full Quantum Gravity can be given a similar structure if [it is formulated]{} according to the Brown-Kuchar model [@BK; @GiesThiem-algiv]. Therefore many results discussed in this paper is likely to admit generalizations to the [full]{} QG.
Acknowledgments {#acknowledgments .unnumbered}
===============
We would like to thank Abhay Ashtekar and Guillermo Mena-Marugán for extensive discussions and helpful comments. We also profited from discussions with Martin Bojowald, Alex Corichi, Bianca Dittrich, Marcin Domaga[ł]{}a, Kristina Giesel, Carlo Rovelli, Parampreet Singh, [Ł]{}ukasz Szulc and Thomas Thiemann. We are grateful to the referee for his important comments. The work was partially supported by the Polish Ministerstwo Nauki i Szkolnictwa Wyzszego grant 1 P03B 075 29 and grant 182/N-QGG/2008/0, by 2007-2010 research project N202 n081 32/1844 , the National Science Foundation (NSF) grant PHY-0456913 and by the Foundation for Polish Science grant “Master”. TP acknowledges financial aid provided by the I3P framework of CSIC and the European Social Fund, and the funds of Polish Academy of Sciences (PAN).
Evolution operator in LQC: rigorous definition {#sec:zero}
==============================================
In this appendix we present the completion of the definition of the symmetric evolution operator $\hat{\Theta}$ introduced in (\[eq:Theta\]), taking special care of the difficulties related with either the vanishing $$B(0)\ =\ 0\label{B00}$$ (in the case of $B=B_{\rm APS}$) or divergence $$B(0)\ =\ \infty\label{B0infty}$$ (in the $B=B_{\rm sLQC}$ case).
Let us begin with (\[B00\]). The operator $\hat{\Theta}$ has been well defined in Sec \[sec:LQC\] in every subspace $\Hil_{\epsilon}\subset{{\Hil_{{{\rm gr}},B}}}$ and ${\epsilon\in (0,4)}$ through formula (\[eq:Theta\]) already. The remaining case is $\epsilon=0$ (this problem is solved in [@aps-imp] however it is not spelled out).
We start with a more suitable form of the constraint operator, namely we consider the solutions to the equation $$-\frac{1}{2}\widehat{V^{-1}}\frac{\partial^2}{\partial
T^2}\psi_T\ =\ \hat{C}_{\rm gr}\psi_T,\label{eq:constr}$$ where $\psi_T\in {\cal H}_{\rm gr}$ and the action of ${{\widehat{C_{{{\rm gr}}}}}}$ can be written (following (\[cgr\])) as $$\label{eq:Cg}
[\hat{C}_{\rm gr}\psi_T](v)\ =\
C^+(v)\psi_T(v+4)+C^o(v)\psi_T(v)+C^-(v)\psi_T(v-4) \ ,$$ with $C^{o,\pm}(v)$ being real functions, of which $C^{o}(v)<0$.
Taking the scalar product of the left and the right hand sides respectively with the vector $|0\rangle\in{{\Hil_{{{\rm gr}}}}}$ we find $$\langle 0|{{\widehat{C_{{{\rm gr}}}}}}\psi_T\rangle \ =\ 0 \ .$$ This is a condition that has to be satisfied by $\psi_T\in{{\Hil_{{{\rm gr}}}}}$ at every value of $T$. The meaning of this observation is, that the functions $T\mapsto \psi_T\in{{\Hil_{{{\rm gr}}}}}$ which satisfy the constraint equation (\[eq:constr\]) in fact take values only in the subspace $\Hil_{\langle 0|{{\widehat{C_{{{\rm gr}}}}}}\cdot\rangle=0}$ defined by the constraint $$\langle 0|{{\widehat{C_{{{\rm gr}}}}}}\psi\rangle \ =\ 0 \ .$$ However the subspace is not preserved by ${{\widehat{C_{{{\rm gr}}}}}}$. It (the intersection of the domain ${{\rm Span}}(|v\rangle\ :\ v\in\mathbb{R})\cap
\Hil_{\langle 0|{{\widehat{C_{{{\rm gr}}}}}}\cdot\rangle=0}$) is mapped into another subspace $\Hil_{\langle 0|\cdot\rangle=0}$ defined by the constraint $$\langle 0|\psi\rangle\ =\ 0 \ .$$
On the other hand, the orthogonal projection $${{\Hil_{{{\rm gr}}}}}\ \rightarrow\ \Hil_{\langle 0|\cdot\rangle=0}$$ maps isometrically $$\Hil_{\langle 0|\hat{C}_{\rm gr}\cdot\rangle=0}\ \rightarrow\
\Hil_{\langle 0|\cdot\rangle=0} \ .$$ This isomorphism can be used to push forward the operator ${{\widehat{C_{{{\rm gr}}}}}}$ to $\Hil_{\langle 0|\cdot\rangle=0}$. An action of the resulting operator (preserving $\Hil_{\langle 0|\cdot\rangle=0}$) is given by $$\label{eq:c-gen}
[\hat{\tilde{C}}_{\rm gr}\psi](v)\ =\ \begin{cases}
C^+(v)\psi(v+4) + C^o(v)\psi(v)
+ C^-(v)\psi(v-4) \ , & v\not\in\{-4,0,4\} \ , \\
\left\{ \begin{split}
&C^+(4)\psi(8) + C^o(4)\psi(4) \\
&- C^-(4)\left[\frac{C^+(0)}{C^o(0)}\psi(4)
+ \frac{C^-(0)}{C^o(0)}\psi(-4)\right]
\end{split} \right\} \ , & v=4 \ , \\
\qquad\qquad\qquad\qquad\quad 0 \ , & v=0 \ , \\
\left\{ \begin{split}
&C^-(-4)\psi(-8) + C^o(-4)\psi(-4) \\
&- C^+(-4)\left[\frac{C^-(0)}{C^o(0)}\psi(-4)
+ \frac{C^+(0)}{C^o(0)}\psi(4)\right]
\end{split} \right\} \ , & v=-4 \ , \\
\end{cases} \ .$$
It is worth to be stressed, that the Hilbert space isomorphism is not unitary in the kinematical Hilbert product, but it becomes unitary, after one endows the Hilbert space ${\cal H}_{\rm gr}$ with the $\langle\cdot|B\cdot\rangle$ product. Finally, the APS constraint is imposed on functions $T\mapsto\psi_T\in {\cal H}_{\langle
0|\cdot\rangle=0}={\cal H}_{{\rm gr},B}$ and it is $$\label{eq:evo-gen}
\partial^2_{T}\psi_T\ =\
2V_oB(\hat{V})^{-1}\hat{\tilde{C}}_{\rm gr}{\psi}_T\
=: \ -2V_o\hat{\Theta}\psi_T\ .$$ This extends the definition of the operator $\hat{\Theta}$ given in the previous section to the subspace ${\cal H}_{\rm \epsilon=0}$ in the sub-domain $${{\rm Span}}(|n\rangle\ \, : \, 0\not= n\in 4\mathbb{Z})\
\subset {\cal H}_{\rm \epsilon=0}.$$ The operator $\hat{\Theta}$ is defined in the Hilbert space ${\cal H}_{{\rm gr},B}$ in the domain ${{\rm Span}}(|v\rangle\,:\,\mathbb{R})$ (however the zero volume vector $|0\rangle$ has zero norm in this space). It is a symmetric operator which may have inequivalent self-adjoint extensions (see section \[sec:C-noneq\]), and one of them has to be chosen to make the quantum constraint equation well defined.
Exactly that method was used in the APS papers to study the physical solutions which take values in the subspace ${\cal H}_{\epsilon=0}$ (solutions preserved by the reflection $P|v\rangle=|-v\rangle$).
Let us turn now to the (\[B0infty\]) case. Now the problem is in introducing the scalar product $\langle\cdot|B\cdot\rangle$. However instead, we can modify the procedure of going from (\[eq:constr\]) to an analog of (\[eq:evo-gen\]) defining the operator $$\hat{\Theta}\ :=\ - B(\hat{V})^{-1/2} {{\widehat{C_{{{\rm gr}}}}}}B(\hat{V})^{-1/2} \ ,$$ which is well defined and symmetric (in the domain ${\rm
Span}(|v\rangle\ :\ v\in\mathbb{R})$) with respect to the original kinematical inner product of ${{\Hil_{{{\rm gr}}}}}$.
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[^1]: [For the sake of generality, we distinguish between $\widehat{{\sqrt{\det
q}\,}^{-1}}$ and $\widehat{{\sqrt{\det q}\,}}^{-1} $. This distinction takes place if one wants to derive the APS model by the group averaging method. However our results apply also to the sLQC in which there is no distinction of this type.]{}
[^2]: [This property (and its consequences) will be presented in detail in [@posL1; @posL2] currently in preparation.]{}
[^3]: Throughout of this paper we use the value of $\rhoc$ derived in [@aps-imp]. However recently it was shown [@entropy] that due to subtleties in constructing the loop of minimal area in LQC the so called area gap (lowest nonzero area eigenvalue) is twice bigger than the one used in [@aps-imp]. In consequence the value of $\rhoc$ (depending on it) is twice smaller and equals approximately $0.41\rhoPl$.
[^4]: That is as long, as $0$ is not an eigenvalue of $\hat{\Theta}$.
|
---
abstract: 'In this article, we study the rational cohomology rings of Voisin’s punctual Hilbert schemes ${X^{[n]}{_{\vphantom{[n]}} }}$ associated to a symplectic compact fourfold $X$. We prove that these rings can be universally constructed from $H\ee{_{\vphantom{[n]}} }(X,\Q)$ and $c_{1}{^{\vphantom{[n]}} }(X)$, and that Ruan’s crepant resolution conjecture holds if $c_{1}{^{\vphantom{[n]}} }(X)$ is a torsion class. Next, we prove that for any almost-complex compact fourfold $X$, the complex cobordism class of ${X^{[n]}{_{\vphantom{[n]}} }}$ depends only on the cobordism class of $X$.'
address: |
Institut de Mathématiques de Jussieu, UMR 7586\
Case 247\
Université Pierre et Marie Curie\
4, place Jussieu\
F-75252 Paris Cedex 05\
France
author:
- Julien Grivaux
title: 'Topological properties of punctual Hilbert schemes of almost-complex fourfolds (II)'
---
Introduction
============
The punctual Hilbert schemes of a smooth quasi-projective surface have been intensively studied in the past twenty years, starting with the works of Göttsche, Nakajima, Grojnowski, Lehn and others (see e.g. [@SchHilGo2]). These Hilbert schemes present a strong geometric interest because, among many other properties, they are smooth crepant resolutions of the symmetric products of the surfaces. If $X$ is a $K3$ surface, ${X^{[n]}{_{\vphantom{[n]}} }}$ are hyperkähler varieties by a result of Beauville [@SchHilBe] and Ruan’s conjecture predicts that for all positive integer $n$, $H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\C\bigr)$ is isomorphic as a ring to the orbifold cohomology ring of ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$ defined by Chen and Ruan [@SchHilCR], [@SchHilALR]. This is obtained by putting together results of Lehn-Sorger on the Hilbert scheme side [@SchHilLeSoBis] with the computations done in [@SchHilFG] and [@SchHilUr] on the orbifold side. The isomorphisms between $H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\C\bigr)$ and $H\ee_{\textrm{CR}}\bigl( {S^{\mkern 1 mu n\!}_{\vphantom{[}} X},\C\bigr)$ made it clear that the cohomology rings of punctual Hilbert schemes of a $K3$ surface depend only on the deformation class of the complex structure on $X$ in the space of almost-complex ones, since Chen-Ruan cohomology is a purely almost-complex theory. This is more generally the case for arbitrary projective surfaces: the description of the cohomology ring of Hilbert schemes done in [@SchHilLe] and [@SchHilLQWMA] shows that $H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ depends only on the ring $H\ee{_{\vphantom{[n]}} }(X,\Q)$ and on the first Chern class of $X$ in $H^{2}{_{\vphantom{[n]}} }(X,\Q)$. In the same spirit, it is proved in [@SchHilEGL] that the complex cobordism class of ${X^{[n]}{_{\vphantom{[n]}} }}$ depends only on the complex cobordism class of $X$. These facts received an explanation by a recent construction of Voisin [@SchHilVo1]: for any almost-complex compact fourfold $X$ and any positive integer $n$, it is possible to construct a punctual Hilbert scheme ${X^{[n]}{_{\vphantom{[n]}} }}$ which is a stable almost-complex differentiable manifold of real dimension $4n$. Besides, ${X^{[n]}{_{\vphantom{[n]}} }}$ is symplectic if $X$ is symplectic [@SchHilVo2].
In Nakajima’s theory [@SchHilNa], the correspondence action of the incidence schemes on the cohomology groups of Hilbert schemes is the main ingredient to understand their additive structure via representation theory. This approach has been carried out for the almost-complex case in [@SchHilGri]. Our first aim in this paper is now to study the multiplicative structure of the rational cohomology rings of almost-complex Hilbert schemes, generalizing the work of Lehn [@SchHilLe] and Li-Qin-Wang [@SchHilLQWMA]. We obtain a satisfactory result in the symplectic case:
\[Ch4suite2ThIntroUn\] If $(X,\omega )$ is a symplectic compact four-manifold, the rings $H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ can be constructed by universal formulae from the ring $H\ee{_{\vphantom{[n]}} }(X,\Q)$ and the first Chern class of $X$ in $H^{2}{_{\vphantom{[n]}} }(X,\Q)$.
When $X$ is projective, Lehn uses in an essential way algebraic curves on $X$ via their symmetric products imbedded in the Hilbert schemes to determine some excess intersection terms arising in correspondences computations. This argument fails a priori as soon as $X$ is not algebraic, since there is no pseudo-holomorphic curve on $X$ for a generic almost-complex structure on it. However, Donaldson’s theorem on symplectic submanifolds [@SchHilDo] allows to produce many paseudo-holomorphic curves for arbitrary small perturbations of a fixed almost-complex structure. As a corollary of an effective version of Theorem \[Ch4suite2ThIntroUn\], we obtain the cohomological crepant resolution conjecture for symplectic fourfolds with torsion first Chern class:
\[Ch4suite2ThIntroDeux\] Let $(X,\omega )$ be a symplectic four-manifold with zero first Chern class in $H^{2}{_{\vphantom{[n]}} }(X,\Q)$. Then for any positive integer $n$, $H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\C\bigr)$ is isomorphic as a ring to $H\ee_{\textrm{CR}}\bigl( {S^{\mkern 1 mu n\!}_{\vphantom{[}} X},\C\bigr)$.
This result provides many examples where Ruan’s conjecture holds in the symplectic category.
The second part of this article deals with the cobordism classes of almost-complex Hilbert schemes. The main result is the following:
\[Ch4suite2ThIntroTrois\] Let $(X,J)$ be an almost-complex compact four-manifold. For any positive integer $n$, the complex cobordism class of ${X^{[n]}{_{\vphantom{[n]}} }}$ given by its stable almost-complex structure depends only on the complex cobordism class of $X$.
The proof of [@SchHilEGL] for projective surfaces is rather delicate to adapt here because coherent sheaves, contrary to algebraic cycles or vector bundles, have no equivalent in almost-complex geometry. However, it turns out that the classical proof can be carried out in a relative context on spaces fibered in smooth analytic sets over a singular differentiable basis.
**Acknowledgement.** I want to thank my advisor Claire Voisin whose work on the almost-complex punctual Hilbert scheme is at the origin of this article. Her deep knowledge of the subject and her numerous advices have been most valuable to me. I also wish to thank her for her kindness and her patience.
The almost-complex Hilbert scheme
=================================
In this section, we recall definitions and fundamental properties of almost-complex Hilbert schemes for the reader’s convenience. More details can be found in [@SchHilGri], [@SchHilVo1] and [@SchHilVo2].
Definitions and basic properties
--------------------------------
Let $(X,J)$ be an almost-complex compact fourfold. For all $n\in\N\ee{_{\vphantom{[n]}} }$, we introduce the *incidence set* $$Z_{n}{^{\vphantom{[n]}} }=\bigl\{({{\underline{\vphantom{!}\vphantom{y}x}}},p)\ \textrm{in}\ {S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X\ \textrm{such that}\ p\in X\bigr\}\cdot$$ Let us fix a Riemannian metric $g$ on $X$, and $\varepsilon $ sufficiently small.
\[Ch4suite2DefUn\] Let $\bbb$ be the set of pairs $(W,{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }})$ such that
1. $W$ is a [neighbourhood]{} of $Z_{n}{^{\vphantom{[n]}} }$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$.
2. ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$ is a relative integrable structure on the fibers of $\apl{\operatorname{pr}_{1}{^{\vphantom{[n]}} }}{W}{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}$ depending smoothly on the parameter ${{\underline{\vphantom{!}\vphantom{y}x}}}$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$.
3. $\sup_{{{{\underline{\vphantom{!}\vphantom{y}x}}}\in{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}},\ {p\in W_{x}{^{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }\bigl|\bigl|{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{{\underline{\vphantom{!}\vphantom{y}x}}}}}(p)-J_{{{\underline{\vphantom{!}\vphantom{y}x}}}}{^{\vphantom{[n]}} }(p)\bigr|\bigr|_{g}{^{\vphantom{[n]}} }<\varepsilon $.
For $\varepsilon $ small enough, $\bbb_{\varepsilon }{^{\vphantom{[n]}} }$ is nonempty and weakly contractible.
\[Ch4suite2DefDeux\] Let $(W,{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }})$ be a relative integrable structure in $\bbb$. The *topological Hilbert scheme* $X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}$ is the subset of $W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ defined by $
X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}=\bigl\{({{\underline{\vphantom{!}\vphantom{y}x}}},\xi)\ \textrm{in}\ W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\ \textrm{such that}\ HC(\xi )={{\underline{\vphantom{!}\vphantom{y}x}}}\, \bigr\}
$, where $\apl{HC}{W^{[n]}_{{{\underline{\vphantom{!}\vphantom{y}x}}}}}{S^{n}{_{\vphantom{[n]}} }W_{{{\underline{\vphantom{!}\vphantom{y}x}}}}{^{\vphantom{[n]}} }}$ is the usual Hilbert-Chow morphism.
To obtain a differentiable structure on $X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}$, Voisin uses relative integrable structures in a contractible subset $\bbb'$ of $\bbb$ satisfying some additional geometric conditions. Her main result is the following:
[@SchHilVo1], [@SchHilVo2]\[Ch4suite2ThUn\] Let ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$ be a relative integrable structure in $\bbb'$. Then
1. $X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}$ has a natural differentiable structure. Furthermore, if $J'{\vphantom{J}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }$ is another relative integrable structure in $\bbb'$, $X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}$ and $X^{[n]}_{J'{_{\vphantom{[n]}} }{\vphantom{J}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$ are canonically isomorphic modulo diffeomorphisms isotopic to the identity.
2. There is a canonical Hilbert-Chow map $\apl{HC}{X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}}{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}$ whose fibers are homeomorphic to the fibers of the usual Hilbert-Chow morphism for projective surfaces.
3. $X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}$ can be endowed with a stable almost-complex structure.
4. If $X$ is symplectic and ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$ is compatible with the symplectic structure, $X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}$ is also symplectic.
For arbitrary relative integrable structures, this theorem has the following topological form:
[@SchHilGri]\[Ch4suite2ThDeux\] ${^{\vphantom{[n]}} }$
1. Let ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$ be a relative integrable structure in $\bbb$. Then $X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}$ is a topological manifold.
2. If $\bigl\{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{t}}\bigr\}_{t\in B(0,r)\suq\R^{d}{_{\vphantom{[n]}} }}{^{\vphantom{[n]}} }$ is a smooth path in $\bbb$, then the associated relative topological Hilbert scheme $\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\bigl\{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{t}}\bigr\}_{t\in B(0,r)}{^{\vphantom{[n]}} }\bigr)$ over $B(0,r)$ is a topological fibration.
3. For any ${{\underline{\vphantom{!}\vphantom{y}x}}}$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$ and any integrable structure in a [neighbourhood]{} $U_{{{\underline{\vphantom{!}\vphantom{y}x}}}}{^{\vphantom{[n]}} }$ of ${{\underline{\vphantom{!}\vphantom{y}x}}}$ sufficiently close to $J$, the Hilbert-Chow morphism $\apl{HC}{{X^{[n]}{_{\vphantom{[n]}} }}}{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}$ is locally isomorphic over a [neighbourhood]{} of ${{\underline{\vphantom{!}\vphantom{y}x}}}$ to the classical Hilbert-Chow morphism $\apl{HC}{U^{[n]}_{{{\underline{\vphantom{!}\vphantom{y}x}}}}}{S^{n}{_{\vphantom{[n]}} }U_{{{\underline{\vphantom{!}\vphantom{y}x}}}}{^{\vphantom{[n]}} }}$.
Point (ii) implies that for any couple $\bigl( {J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }},J'{\vphantom{J}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\bigr)$ of relative integrable structures in $\bbb$, $X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}$ and $X^{[n]}_{J'{\vphantom{J}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$ are homeomorphic and the isotopy class of this homeomorphism is canonical. Therefore, there exist canonical rings $H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ and $K\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr)$ such that for every relative integrable structure ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$ in $\bbb$, $H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ is canonically isomorphic to $H\ee{_{\vphantom{[n]}} }\bigl( X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}},\Q\bigr)$ and $K\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr)$ to $ K\bigl( X^{[n]}_{{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}}\bigr)$.
Point (iii) implies that Göttsche’s classical formula for the Betti numbers of punctual Hilbert schemes also holds in the almost-complex case (see [@SchHilGriTh]).
Incidence varieties and Nakajima operators
------------------------------------------
If $n$, $m$ $\in\N\ee{_{\vphantom{[n]}} }$, let $$Z_{n{\ensuremath{\times}}m}{^{\vphantom{[n]}} }=\bigl\{({{\underline{\vphantom{!}\vphantom{y}x}}},\,{{\underline{\vphantom{!}\vphantom{y}y}}},\,p)\ \textrm{in}\ {S^{\mkern 1 mu n\!}_{\vphantom{[}} X}\tim{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X\ \textrm{such that}\ p\in{{\underline{\vphantom{!}\vphantom{y}x}}}\cup{{\underline{\vphantom{!}\vphantom{y}y}}}\bigr\}\cdot$$ Relative integrable structures in a [neighbourhood]{} of $Z_{n{\ensuremath{\times}}m}{^{\vphantom{[n]}} }$ will be denoted by ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m}}$.
\[Ch4suite2DefTrois\] ${^{\vphantom{[n]}} }$
1. If $\bigl(W_{1}{^{\vphantom{[n]}} }, J^{1,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m}\bigr)$ and $\bigl(W_{2}{^{\vphantom{[n]}} }, J^{2,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m}\bigr)$ are two relative integrable structures in [neighbourhood]{}s of $Z_{n{\ensuremath{\times}}m}{^{\vphantom{[n]}} }$, the *product Hilbert scheme* $\bigl( X^{[n]\tim[m]}{_{\vphantom{[n]}} },J^{1,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m},J^{2,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m}\bigr)$ is defined by $$\begin{aligned}
\bigl( X^{[n]\tim[m]}{_{\vphantom{[n]}} },J^{1,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m},J^{2,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m}\bigr)=\bigl\{
(\xi ,\,\xi ',\,{{\underline{\vphantom{!}\vphantom{y}x}}},\,{{\underline{\vphantom{!}\vphantom{y}y}}})\ \textrm{in}\ W^{[n]}_{1,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}&\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}S^{m}{_{\vphantom{[n]}} }X}{^{\vphantom{[n]}} }W^{[m]}_{2,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\\& \textrm{such that}\
HC(\xi )={{\underline{\vphantom{!}\vphantom{y}x}}}\ \textrm{and}\ HC(\xi ')={{\underline{\vphantom{!}\vphantom{y}y}}}\bigr\}\cdot\end{aligned}$$
2. If $m>n$ and $\bigl( \ti{W},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n\tim(m-n)}}\bigr)$ is a relative integrable structure in a [neighbourhood]{} of $Z_{n\tim(m-n)}$, the *incidence variety* is defined by $$\begin{aligned}
\bigl({X^{[m,n]}{_{\vphantom{[n]}} }},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m}}\bigr)=\bigl\{
(\xi ,\,\xi ',\,&{{\underline{\vphantom{!}\vphantom{y}x}}},\,{{\underline{\vphantom{!}\vphantom{y}y}}})\ \textrm{in}\ \ti{W}^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}S^{m}{_{\vphantom{[n]}} }X}{^{\vphantom{[n]}} }\ti{W}^{[m]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\\ &\textrm{such that}\ \xi \subset\xi ',\ HC(\xi )={{\underline{\vphantom{!}\vphantom{y}x}}}\ \textrm{and}\ HC(\xi ')={{\underline{\vphantom{!}\vphantom{y}x}}}\cup{{\underline{\vphantom{!}\vphantom{y}y}}}\bigr\}\cdot\end{aligned}$$
As it is the case for topological Hilbert schemes, the product Hilbert scheme and the incidence varieties are uniquely defined up to homeomorphisms isotopic to the identity.
Let $\bigl( W,{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{(}n{\ensuremath{\times}}m}}\bigr)$, $\bigl( \ti{W},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n\tim(m-n)}}\bigr)$, $\bigl( W',{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{(}n}}\bigr)$ and $\bigl( W'',{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{(}m}}\bigr)$ be relative integrable structures in [neighbourhood]{}s of $Z_{\vphantom{(}n{\ensuremath{\times}}m}$, $Z_{n\tim(m-n)}$, $Z_{\vphantom{(}n}$ and $Z_{\vphantom{(}m}$. We introduce the following set of compatibility conditions of relative analytic spaces (see Section \[n\]):
1. $W\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}S^{m}{_{\vphantom{[n]}} }X}{^{\vphantom{[n]}} }({S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}S^{m-n}{_{\vphantom{[n]}} }X)\suq\ti{W}$, where the base change map is $\flgdba{({{\underline{\vphantom{!}\vphantom{y}x}}},{{\underline{\vphantom{!}\vphantom{y}y}}})}{({{\underline{\vphantom{!}\vphantom{y}x}}},{{\underline{\vphantom{!}\vphantom{y}x}}}\cup{{\underline{\vphantom{!}\vphantom{y}y}}}).}$
2. $W'\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }({S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}S^{m-n}{_{\vphantom{[n]}} }X)\suq\ti{W}$, where the base change map is the first projection.
3. $W''\tim_{S^{m}{_{\vphantom{[n]}} }X}{^{\vphantom{[n]}} }({S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}S^{m-n}{_{\vphantom{[n]}} }X)\suq\ti{W}$, where the base change map is $\flgdba{({{\underline{\vphantom{!}\vphantom{y}x}}},{{\underline{\vphantom{!}\vphantom{y}y}}})}{{{\underline{\vphantom{!}\vphantom{y}x}}}\cup{{\underline{\vphantom{!}\vphantom{y}y}}}.}$
– If (1) holds, there is a natural embedding of $\bigl( {X^{[m,n]}{_{\vphantom{[n]}} }},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n\tim(m-n)}}\bigr)$ into $\bigl( X^{[n]\tim[m]}{_{\vphantom{[n]}} },{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m}},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}m}}\bigr).$
– If (2) holds, there is a natural morphism $\lambda $ from $\bigl( {X^{[m,n]}{_{\vphantom{[n]}} }},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n\tim(m-n)}}\bigr)$ to $\bigl( {X^{[n]}{_{\vphantom{[n]}} }},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{(}n}}\bigr)$.
– If (3) holds, there is a canonical morphism $\nu $ from $\bigl( {X^{[m,n]}{_{\vphantom{[n]}} }},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n\tim(m-n)}}\bigr)$ to $\bigl( X^{[m]}{_{\vphantom{[n]}} },{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{(}m}}\bigr)$.
Unfortumately, conditions (2) and (3) cannot hold at the same time. However, since $X^{[n]\tim[m]}{_{\vphantom{[n]}} }$ is canonically homeomorphic up to isotopy to ${X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{[m]}{_{\vphantom{[n]}} }$, one still has morphisms from ${X^{[m,n]}{_{\vphantom{[n]}} }}$ to ${X^{[n]}{_{\vphantom{[n]}} }}$ and $X^{[m]}{_{\vphantom{[n]}} }$ whose homotopy classes are canonical. They will still be denoted by $\lambda $ and $\nu $.
The incidence varieties ${X^{[m,n]}{_{\vphantom{[n]}} }}$ are locally homeomorphic to the integrable model $U^{[m,n]}{_{\vphantom{[n]}} }$ where $U$ is an open set of $\C^{2}$. This allows to put a stratification on ${X^{[m,n]}{_{\vphantom{[n]}} }}$. In this way, ${X^{[m,n]}{_{\vphantom{[n]}} }}$ is a stratified topological space locally modelled over an analytic space, so it has a fundamental homology class.
The construction by Nakajima and Grojnowski of representations of the Heisenberg super-algebra of $H\ee{_{\vphantom{[n]}} }(X,\Q)$ into $\HHH:=\oplus_{n\in\N}{^{\vphantom{[n]}} }H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ via correspondence actions of incidence varieties done in [@SchHilNa] and [@SchHilGr] also holds in the almost-complex setting:
[@SchHilGri]\[Ch4suite2ThTrois\] If $(X,J)$ is an almost-complex compact fourfold, Nakajima operators $\mathfrak{q}_{i}{^{\vphantom{[n]}} }(\alpha )$, $i\in\Z$, $\alpha \in H\ee{_{\vphantom{[n]}} }(X,\Q)$, can be constructed. They depend only on the deformation class of $J$ and satisfy the Heisenberg commutation relations*:* $$\forall i,j\in\Z,\ \forall\alpha ,\beta\in H\ee{_{\vphantom{[n]}} }(X,\Q),\quad
\bigl[ \mathfrak{q}_{i}{^{\vphantom{[n]}} }(\alpha ),\mathfrak{q}_{j}{^{\vphantom{[n]}} }(\beta )\bigr]=i\,\delta _{i+j,0}{^{\vphantom{[n]}} }\,\Bigl( \int_{X}\alpha \beta\Bigr)\operatorname{id}_{\HHH}{^{\vphantom{[n]}} }.$$ Furthermore, these operators induce an irreducible representation of $\hh\bigl( H\ee{_{\vphantom{[n]}} }(X,\Q)\bigr)$ in $\HHH$ with highest weight vector $1$.
The boundary operator
=====================
The aim of this section is to carry out for symplectic fourforlds Lehn’s computation of the boundary operator done in [@SchHilLe]. The first part of Lehn’s argument can be adapted to the almost-complex case as in the proof of Nakajima relations done in [@SchHilGri]. This is explained in Section \[len\]. The result of Section \[SecRef\], based on Donaldson’s symplectic Kodaira’s theorem allows to carry out in section \[exce\] the second part of Lehn’s argument when $X$ is symplectic.
Lehn’s formula in the almost-complex case {#len}
-----------------------------------------
Let $(X,J)$ be an almost-complex compact fourfold and let $\T$ be the trivial complex line bundle on it. If ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$ is a relative structure in a [neighbourhood]{} of $Z_{n}{^{\vphantom{[n]}} }$, there is a relative tautological bundle $\T^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ on $W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ whose restriction to the fibers $\bigl( W^{[n]}_{{{\underline{\vphantom{!}\vphantom{y}x}}}},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{{\underline{\vphantom{!}\vphantom{y}x}}}}}\bigr)$ are the usual tautological bundles $\bigl( \oo{^{\vphantom{[n]}} }_{W^{[n]}_{{{\underline{\vphantom{!}\vphantom{y}x}}}}}\bigr)^{[n]}{_{\vphantom{[n]}} }$. If ${\T^{[n]}{_{\vphantom{[n]}} }}=\T^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{}_{\vert{X^{[n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }$, ${\T^{[n]}{_{\vphantom{[n]}} }}$ satisfies the following properties (see [@SchHilGri]):
1. The class of ${\T^{[n]}{_{\vphantom{[n]}} }}$ in $K\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr)$ is independent of ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$.
2. $-2c_{1}{^{\vphantom{[n]}} }\bigl( {\T^{[n]}{_{\vphantom{[n]}} }}\bigr)$ is Poincaré dual with $\Q$-coefficients to $\bigl[ \partial{X^{[n]}{_{\vphantom{[n]}} }}\bigr]$, where $$\partial{X^{[n]}{_{\vphantom{[n]}} }}=\bigl\{(\xi ,{{\underline{\vphantom{!}\vphantom{y}x}}})\ \textrm{in}\ {X^{[n]}{_{\vphantom{[n]}} }}\ \textrm{such that there exists}\ p\ \textrm{in}\ {{\underline{\vphantom{!}\vphantom{y}x}}}\ \textrm{with}\ \textrm{length}_{p}(\xi )\ge 2 \bigr\}$$ is the so-called *boundary of* ${X^{[n]}{_{\vphantom{[n]}} }}$.
3. If $\apl{\lambda }{{X^{[n+1,n]}{_{\vphantom{[n]}} }}}{{X^{[n]}{_{\vphantom{[n]}} }}}$ and $\apl{\nu }{{X^{[n+1,n]}{_{\vphantom{[n]}} }}}{{X^{[n+1]}{_{\vphantom{[n]}} }}}$ are the usual homotopy classes, $D$ is the exceptional divisor in ${X^{[n+1,n]}{_{\vphantom{[n]}} }}$ and $F$ is the complex line bundle on ${X^{[n+1,n]}{_{\vphantom{[n]}} }}$ whose first Chern class is Poincare dual to $-[D]$, then $\lambda \ee{_{\vphantom{[n]}} }\,\T^{[n+1]}{_{\vphantom{[n]}} }-\nu \ee{_{\vphantom{[n]}} }\,{\T^{[n]}{_{\vphantom{[n]}} }}=F$ in $K\bigl( {X^{[n+1,n]}{_{\vphantom{[n]}} }}\bigr)$.
We recall Lehn’s definition of the boundary operator:
\[ah\] Let $\HHH=\bigoplus_{n\ge 0}{^{\vphantom{[n]}} }H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$.
1. The *boundary operator* $\apl{\mathfrak{d}}{\HHH}{\HHH}$ is defined by $\mathfrak{d}\bigl[ (\alpha _{n}{^{\vphantom{[n]}} })_{n\ge 0}{^{\vphantom{[n]}} }\bigr]=\bigl( c_{1}{^{\vphantom{[n]}} }\bigl( {\T^{[n]}{_{\vphantom{[n]}} }}\bigr)\cup\alpha _{n}{^{\vphantom{[n]}} }\bigr)_{n\ge 0}{^{\vphantom{[n]}} }$.
2. If $A$ is an endomorphism of $\HHH$, *the derivative* $A'$ of $A$ is defined by the formula $$A'=[\mathfrak{d},A]=\mathfrak{d}\circ A-A\circ\mathfrak{d}.$$
We now state the following result:
\[uu\] Let $(X,J)$ be an almost-complex compact fourfold. There exist classes $\bigl( e_{n}{^{\vphantom{[n]}} }\bigr)_{n\ge 0}{^{\vphantom{[n]}} }$ in $H^{2}{_{\vphantom{[n]}} }(X,\Q)$ such that $$\forall n,m\in\Z,\ \forall \alpha ,\beta \in H\ee{_{\vphantom{[n]}} }(X,\Q),\quad
\bigl[ \mathfrak{q}'_{n}(\alpha ),\mathfrak{q}_{m}{^{\vphantom{[n]}} }(\beta )\bigr]=-nm\,\mathfrak{q}_{n+m}{^{\vphantom{[n]}} }(\alpha \beta )+\delta _{n+m,\,0}{^{\vphantom{[n]}} }\Bigl( \int_{X}e_{|n|}{^{\vphantom{[n]}} }\alpha \beta \Bigr)\operatorname{id}_{\HHH}{^{\vphantom{[n]}} }.$$
Let $n,\,m$ be positive integers such that $m\ge n$, ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n\tim(m-n)}}$ a relative integrable structure in a [neighbourhood]{} of $Z_{n\tim(m-n)}{^{\vphantom{[n]}} }$, and let $I^{[m,\,n]}_{\,\T,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ be the complex bundle on $W^{[m,\,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ whose restriction to the fibers $W^{[m,\,n]}_{{{\underline{\vphantom{!}\vphantom{y}x}}},\,{{\underline{\vphantom{!}\vphantom{y}y}}}}$ are the kernels of the natural surjective morphisms from $\bigl( \oo_{W^{[m,\,n]}_{{{\underline{\vphantom{!}\vphantom{y}x}}},\,{{\underline{\vphantom{!}\vphantom{y}y}}}}}\bigr)^{[m]}{_{\vphantom{[n]}} }$ to $\bigl( \oo_{W^{[m,\,n]}_{{{\underline{\vphantom{!}\vphantom{y}x}}},\,{{\underline{\vphantom{!}\vphantom{y}y}}}}}\bigr)^{[n]}{_{\vphantom{[n]}} }$. We put $I_{\,\T}^{[m,\,n]}=I_{\,\T,{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[m,\,n]}{}_{\vert{X^{[m,\,n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }$. Then $I_{\,\T}^{[m,\,n]}=\nu \ee{_{\vphantom{[n]}} }\,\T^{[n+1]}{_{\vphantom{[n]}} }-\lambda \ee{_{\vphantom{[n]}} }\,{\T^{[n]}{_{\vphantom{[n]}} }}$ in $K\bigl( {X^{[m,\,n]}{_{\vphantom{[n]}} }}\bigr)$, the proof being similar to [@SchHilGri Proposition 3.5]. This allows to prove that correspondences behave well under derivations:
\[hum\] Let $u$ be a class in $H_{*}{^{\vphantom{[n]}} }\bigl( {X^{[m,\,n]}{_{\vphantom{[n]}} }},\Q\bigr)$ and $\apl{u_{*}{^{\vphantom{[n]}} }}{H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)}{H\ee{_{\vphantom{[n]}} }\bigl( X^{[m]}{_{\vphantom{[n]}} },\Q\bigr)}$ be the associated correspondence map.Then $\bigl( u_{*}{^{\vphantom{[n]}} }\bigr)'{_{\vphantom{[n]}} }=\bigl[ u\cap c_{1}{^{\vphantom{[n]}} }\bigl( I_{\,\T}^{[m,\,n]}\bigr)\bigr]_{*}{^{\vphantom{[n]}} }$.
Let $\apl{\lambda }{{X^{[m,\,n]}{_{\vphantom{[n]}} }}}{{X^{[n]}{_{\vphantom{[n]}} }}}$ and $\apl{\nu }{{X^{[m,\,n]}{_{\vphantom{[n]}} }}}{{X^{[m]}{_{\vphantom{[n]}} }}}$ be the usual maps. The identity $$\nu \ee{_{\vphantom{[n]}} }\T^{[m]}{_{\vphantom{[n]}} }-\lambda \ee{_{\vphantom{[n]}} }{\T^{[n]}{_{\vphantom{[n]}} }}=I_{\,\T}^{[m,\,n]}$$ is satisfied in $K\bigl( {X^{[m,\,n]}{_{\vphantom{[n]}} }}\bigr)$. Thus $$\begin{aligned}
\bigl( u_{*}{^{\vphantom{[n]}} }\bigr)'\tau &=c_{1}{^{\vphantom{[n]}} }\bigl( \T^{[m]}{_{\vphantom{[n]}} }\bigr)\cup u_{*}{^{\vphantom{[n]}} }\tau-u_{*}{^{\vphantom{[n]}} }\bigl( c_{1}{^{\vphantom{[n]}} }\bigl(\T^{[n]}{_{\vphantom{[n]}} }\bigr)
\cup \tau \bigr) \\
&=PD^{-1}\Bigl[ \bigl( \nu _{*}{^{\vphantom{[n]}} }(u\cap\lambda \ee{_{\vphantom{[n]}} }\tau )\bigr)\cap c_{1}{^{\vphantom{[n]}} }\bigl( \T^{[m]}{_{\vphantom{[n]}} }\bigr)-\nu _{*}{^{\vphantom{[n]}} }\bigl( u\cap\lambda \ee{_{\vphantom{[n]}} }\bigl( c_{1}{^{\vphantom{[n]}} }\bigl( \T^{[n]{_{\vphantom{[n]}} }}\bigr)\cup\tau \bigr)\bigr)\Bigr]\\
&=PD^{-1}\Bigl[ \nu _{*}{^{\vphantom{[n]}} }\Bigl( u\cap \bigl[\bigl( \nu \ee{_{\vphantom{[n]}} }c_{1}{^{\vphantom{[n]}} }\bigl( \T^{[m]}{_{\vphantom{[n]}} }\bigr)-\lambda \ee{_{\vphantom{[n]}} }c_{1}{^{\vphantom{[n]}} }\bigl( \T^{[n]}{_{\vphantom{[n]}} }\bigr)\bigr)\cup\lambda \ee{_{\vphantom{[n]}} }\tau \bigr]\Bigr)\Bigr]\\
&=PD^{-1}\, \nu _{*}{^{\vphantom{[n]}} }\Bigl( \bigl[ u\cap c_{1}{^{\vphantom{[n]}} }\bigl(I_{\T}^{[m,\,n]} \bigr)\bigr]\cap \lambda \ee{_{\vphantom{[n]}} }\tau \Bigr).\end{aligned}$$
By this lemma, to prove the theorem it suffices to compute the commutator of two correspondences. We adapt Lehn’s proof exactly as in [@SchHilGri] for the Nakajima relations. This yields (see [@SchHilGriTh] for a detailed exposition):
$_{{\ensuremath{\displaystyle}}*}$ For all $m,n$ in $\Z$ with $m\neq n$, $\bigl[ \mathfrak{q}'_{n}(\alpha ),\mathfrak{q}_{m}{^{\vphantom{[n]}} }(\beta )\bigr]=\mu _{n,\,m}{^{\vphantom{[n]}} }\,\mathfrak{q}_{n+m}{^{\vphantom{[n]}} }(\alpha \beta )$, where $\mu _{n,\,m}{^{\vphantom{[n]}} }\in\Z$.
$_{{\ensuremath{\displaystyle}}*}$ For all $n$ in $\Z$, $\bigl[ \mathfrak{q}'_{n}(\alpha ),\mathfrak{q}_{-n}{^{\vphantom{[n]}} }(\beta )\bigr]=\bigl( \int_{X}e_{|n|}{^{\vphantom{[n]}} }\,\alpha \beta\bigr)\operatorname{id}_{\HHH}{^{\vphantom{[n]}} }$, where the classes $e_{|n|}{^{\vphantom{[n]}} }$ belong to $ H^{2}{_{\vphantom{[n]}} }(X,\Q)$.
The terms $\mu _{n,\,m}{^{\vphantom{[n]}} }$ and $e_{n}{^{\vphantom{[n]}} }$ are *the excess contributions*. The multiplicity $\mu _{n,\,m}{^{\vphantom{[n]}} }$ can computed locally on $X$, so that Lehn’s proof applies and gives $\mu _{n,\,m}{^{\vphantom{[n]}} }=-nm$. However, this is not the case for the class $e_{n}{^{\vphantom{[n]}} }$ which involves the *global* geometry of $X$.
\[aas\] When $\alpha $ runs through a basis of $H\ee{_{\vphantom{[n]}} }(X,\Q)$, the operators $\mathfrak{d}$ and $\mathfrak{q}_{1}{^{\vphantom{[n]}} }(\alpha )$ generate $\HHH$ from the vector $1$.
This is a consequence of the relations $\bigl[ q'_{1}(\alpha ),q_{m}{^{\vphantom{[n]}} }(1)\bigr]=-m\,q_{m+1}{^{\vphantom{[n]}} }(\alpha )$, $m\in\N$.
Holomorphic curves in symplectic fourfolds {#SecRef}
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Until now, we have only considered integrable structures in small open sets of $(X,J)$. To compute the excess classes $e_{n}{^{\vphantom{[n]}} }$, we will construct pseudo-holomorphic curves in $X$ for perturbed almost-complex structures. To do so we use the following theorem of Donaldson [@SchHilDo], which is a symplectic version of Kodaira’s imbedding theorem:
[@SchHilDo]\[ThDo\] Let $(V,\omega )$ be a symplectic manifold of dimension $2n$ such that $\omega $ is an integral class. If $h$ is a lift of $\omega $ in $H^{2}{_{\vphantom{[n]}} }(V,\Z)$, then, for $k\gg 0$, the classes $PD(kh)$ in $ H_{2n-2}{^{\vphantom{[n]}} }(V,\Z)$ can be realized by homology classes of symplectic submanifolds. More precisely, if $J$ is an almost-complex structure on $V$ compatible with $\omega $, it is possible to write $PD(kh)=\bigl[ S_{k}{^{\vphantom{[n]}} }\bigr]$ for $k$ large enough, where $S_{k}{^{\vphantom{[n]}} }$ is a $J_{k}{^{\vphantom{[n]}} }$-holomorphic codimension $2$ submanifold in $V$ and $||J_{k}{^{\vphantom{[n]}} }-J||_{C^{0}{_{\vphantom{[n]}} }}{^{\vphantom{[n]}} }\le C/\sqrt{k}$.
We apply this theorem to our situation:
\[cor\] Let $(X,\omega )$ be a symplectic compact fourfold and $J$ an adapted almost-complex structure on $X$. Then there exist almost-complex structures $\bigl( J_{i}{^{\vphantom{[n]}} }\bigr)_{1\le i\le N}{^{\vphantom{[n]}} }$ arbitrary close to $J$ and $J_{i}{^{\vphantom{[n]}} }$-holomorphic curves $\bigl( C_{i}{^{\vphantom{[n]}} }\bigr)_{1\le i\le N}{^{\vphantom{[n]}} }$ such that*:*
1. For all $i$, $J_{i}{^{\vphantom{[n]}} }$ is integrable in a [neighbourhood]{} of $C_{i}{^{\vphantom{[n]}} }$.
2. The classes $\bigl[ C_{i}{^{\vphantom{[n]}} }\bigr]$ span $H_{2}{^{\vphantom{[n]}} }(X,\Q)$.
We pick $\alpha _{1}{^{\vphantom{[n]}} },\dots,\alpha _{N}{^{\vphantom{[n]}} }$ in $H^{2}{_{\vphantom{[n]}} }(X,\R)$ such that the $\omega +\alpha _{i}{^{\vphantom{[n]}} }$’s are symplectic forms in $H^{2}{_{\vphantom{[n]}} }(X,\Q)$ which generate $H^{2}{_{\vphantom{[n]}} }(X,\Q)$. There exist almost-complex structures $\bigl( \ti{J}_{i}{^{\vphantom{[n]}} }\bigr)_{1\le i\le N}{^{\vphantom{[n]}} }$ adapted to $(\omega +\alpha _{i}{^{\vphantom{[n]}} })_{1\le i\le N}{^{\vphantom{[n]}} }$ arbitrary close to $J$ if the $\alpha _{i}{^{\vphantom{[n]}} }$’s are small enough. For suitable sufficiently large values of $m$, the symplectic forms $m(\omega +\alpha _{i}{^{\vphantom{[n]}} })$ are integral and Donaldson’s theorem applies. We obtain curves $(C_{i})_{1\le i\le N}{^{\vphantom{[n]}} }$ where $J_{i}{^{\vphantom{[n]}} }$ is arbitrary close to $\ti{J}_{i}{^{\vphantom{[n]}} }$ and $\bigl[ C_{i}{^{\vphantom{[n]}} }\bigr]=PD\bigl( k_{i}{^{\vphantom{[n]}} }(\omega +\alpha _{i}{^{\vphantom{[n]}} })\bigr)$ in $H_{2}{^{\vphantom{[n]}} }(X,\Q)$. Let $U_{i }{^{\vphantom{[n]}} }$ be a small [neighbourhood]{} of the zero section of $N_{C_{i}{^{\vphantom{[n]}} }/X}$. We identify $U_{i}{^{\vphantom{[n]}} }$ with a [neighbourhood]{} of $C_{i}{^{\vphantom{[n]}} }$ in X. Since $\dim C_{i}{^{\vphantom{[n]}} }=2$, $J_{i}{^{\vphantom{[n]}} }{}_{\vert \, C_{i}}$ is integrable. Since $C_{i}{^{\vphantom{[n]}} }$ is $J_{i}{^{\vphantom{[n]}} }$-holomorphic, $N_{C_{i}{^{\vphantom{[n]}} }/X}$ is naturally a complex vector bundle over the complex curve $C_{i}{^{\vphantom{[n]}} }$, so that we can put a holomorphic structure on it. This gives an integrable structure $J'_{i}$ on $U_{i}{^{\vphantom{[n]}} }$ such that $J'_{i}{}_{\vert \, C_{i}}=J_{i}{^{\vphantom{[n]}} }{}_{\vert \, C_{i}}$. We glue together $J_{i}{^{\vphantom{[n]}} }$ and $J'_{i}$ in an annulus around $C_{i}{^{\vphantom{[n]}} }$. The resulting almost-complex structure can be chosen arbitrary close to $J_{i}{^{\vphantom{[n]}} }$ if $U_{i}{^{\vphantom{[n]}} }$ is small enough.
Computation of the excess term in the symplectic case {#exce}
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Our aim in this section is to prove the following theorem:
\[qqq\] If $(X,\omega )$ is a symplectic compact fourfold and $J$ is any almost-complex structure compatible with $\omega $, the excess contributions $e_{n}{^{\vphantom{[n]}} }$ of Theorem \[uu\] are given by $$\forall n\in\Z,\quad
e_{n}{^{\vphantom{[n]}} }=\frac{1}{2}\,n^{2}(|n|-1)c_{1}{^{\vphantom{[n]}} }(X).$$ This means that for all $n,m$ in $\Z$ and for all $\alpha ,\beta $ in $ H^{2}{_{\vphantom{[n]}} }(X,\Q)$, $$\bigl[ \mathfrak{q}'_{n}(\alpha ),\mathfrak{q}_{m}{^{\vphantom{[n]}} }(\beta )\bigr]=-nm\Bigl\{\mathfrak{q}_{n+m}{^{\vphantom{[n]}} }(\alpha \beta )-\dfrac{|n|-1}{2}\,\delta _{n+m,0}{^{\vphantom{[n]}} }\Bigl( \int_{X}c_{1}{^{\vphantom{[n]}} }(X)\alpha \beta \Bigr)\operatorname{id}_{\HHH}{^{\vphantom{[n]}} }\Bigr\}\cdot$$
If $X$ is a smooth projective surface, this theorem is due to Lehn ([@SchHilLe], Th. 3.10). First we sketch his argument, then we adapt it in the symplectic case.
In this proof the notation $[Z]$ will be used to denote the cohomological cycle class of a cycle $Z$, and its the homology class as it was previously the case.
If $X$ is a smooth projective surface and $C$ is a smooth algebraic curve on $X$, results of Grojnowski and Nakajima ([@SchHilGr], [@SchHilNabis]) describe explicitely the class $\bigl[ C^{[n]}{_{\vphantom{[n]}} }\bigr]$ in $H^{2n}{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ in terms of the classes $\mathfrak{q}_{i_{1}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }\bigl( [C]\bigr)\dots \mathfrak{q}_{i_{N}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }\bigl( [C]\bigr)\,.\,1$, where $i_{1}{^{\vphantom{[n]}} },\dots,i_{N}{^{\vphantom{[n]}} }$ are positive integers of total sum $n$.
Let $X_{0}^{[n]}$ be the set of schemes in ${X^{[n]}{_{\vphantom{[n]}} }}$ whose support is a single point. If $\partial C^{[n]}{_{\vphantom{[n]}} }=C^{[n]}{_{\vphantom{[n]}} }\cap\partial{X^{[n]}{_{\vphantom{[n]}} }}$, the term $I=\ds\int_{{X^{[n]}{_{\vphantom{[n]}} }}}\bigl[ X_{0}^{[n]}\bigr]\,.\,\bigl[ \partial C^{[n]}{_{\vphantom{[n]}} }\bigr]$ can be computed in two different ways:
\(i) The integral $I$ is equal to $\mathfrak{q}_{-n}{^{\vphantom{[n]}} }(1)\,\bigl( \bigl[ \partial C^{[n]}{_{\vphantom{[n]}} }\bigr]\bigr)$. Since $C^{[n]}{_{\vphantom{[n]}} }$ and $\partial {X^{[n]}{_{\vphantom{[n]}} }}$ intersect generically transversally, $\bigl[ \partial C^{[n]}{_{\vphantom{[n]}} }\bigr]=\bigl[ \partial{X^{[n]}{_{\vphantom{[n]}} }}\bigr]\,.\,\bigl[ C^{[n]}{_{\vphantom{[n]}} }\bigr]=-2\,c_{1}{^{\vphantom{[n]}} }(\T^{n}{_{\vphantom{[n]}} })\,.\,\bigl[ C^{[n]}{_{\vphantom{[n]}} }\bigr]=-2\,\mathfrak{d}\bigl[C^{[n]}{_{\vphantom{[n]}} }\bigr]$. Therefore, $I$ is a linear combination of terms of the form $\mathfrak{q}_{-n}{^{\vphantom{[n]}} }(1)\,{^{\vphantom{[n]}} }\mathfrak{q}_{i_{1}{^{\vphantom{[n]}} }}\bigl( [C]\bigr)\dots \mathfrak{q}'_{i_{k}{^{\vphantom{[n]}} }}\bigl( [C]\bigr)\dots \mathfrak{q}_{i_{N}{^{\vphantom{[n]}} }}\bigl( [C]\bigr)\,.\,1$, where $i_{1}{^{\vphantom{[n]}} },\dots, i_{N}{^{\vphantom{[n]}} }$ are positive integers of total sum $n$. These terms vanish except for:
1. $N=1$, $i_{1}{^{\vphantom{[n]}} }=n$. Then $\mathfrak{q}_{-n}{^{\vphantom{[n]}} }(1)\,\mathfrak{q}'_{n}([C])\,.\,1=\ds-\int_{X}e_{n}{^{\vphantom{[n]}} }\,.\,[C]$.
2. $N=2$, $i_{1}{^{\vphantom{[n]}} }+i_{2}{^{\vphantom{[n]}} }=n$. Then $\mathfrak{q}_{-n}{^{\vphantom{[n]}} }(1)\,\mathfrak{q}_{k}([C])\,\mathfrak{q}'_{n-k}([C])\,.\,1=0$ and $$\mathfrak{q}_{-n}{^{\vphantom{[n]}} }(1)\,\mathfrak{q}'_{k}([C])\,\mathfrak{q}_{n-k}{^{\vphantom{[n]}} }([C])\,.\,1=-nk\,\mathfrak{q}_{k-n}{^{\vphantom{[n]}} }([C])\,\mathfrak{q}_{n-k}{^{\vphantom{[n]}} }([C])\,.\,1=nk(n-k)\,[C]^{2}{_{\vphantom{[n]}} }.$$
This computation gives $I=\dfrac{1}{n}\ds\int_{X}e_{n}\,.\,[C]+\binom{n}{2}\,[C]^{2}{_{\vphantom{[n]}} }$.
\(ii) The cycle $C^{[n]}{_{\vphantom{[n]}} }$ intersect transversally $X_{0}^{[n]}$ in its smooth locus and $C^{[n]}{_{\vphantom{[n]}} }\cap X_{0}^{[n]}=C_{0}^{[n]}\simeq C$. Therefore $
I=\int_{{X^{[n]}{_{\vphantom{[n]}} }}}\bigl[ X_{0}^{[n]}\bigr]\,.\,\bigl[ C^{[n]}{_{\vphantom{[n]}} }\bigr]\,.\,\bigl[ \partial{X^{[n]}{_{\vphantom{[n]}} }}\bigr]=\deg_{\,C}{^{\vphantom{[n]}} }\bigl[ \oo_{{X^{[n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }\bigl( \partial{X^{[n]}{_{\vphantom{[n]}} }}\bigr)\bigr]=\deg
_{\,C}{^{\vphantom{[n]}} }\bigl[ \oo_{C^{[n]}{_{\vphantom{[n]}} }}{^{\vphantom{[n]}} }\bigl( \partial C^{[n]}{_{\vphantom{[n]}} }\bigr)\bigr]$, which is $-n(n-1)\,\deg_{\,C}{^{\vphantom{[n]}} }K_{X}{^{\vphantom{[n]}} }$ by direct computation.
In the algebraic case, the excess terms $e_{n}{^{\vphantom{[n]}} }$ lie in the Neron-Severi group of $X$ so that it is enough to prove that for every smooth algebraic curve $C$, $\ds\int_{X}\Bigl[ e_{n}{^{\vphantom{[n]}} }-\dfrac{1}{2}\,n^{2}{_{\vphantom{[n]}} }(n-1)\,c_{1}{^{\vphantom{[n]}} }(X)\Bigr]\,.\,[C]=0$. This is proved by comparison of the two expressions obtained for $I$.
Let us now suppose that $(X,\omega )$ is a symplectic compact fourfold and that $J$ is an adapted almost-complex structure on $X$. If $\gamma \in H\ee{_{\vphantom{[n]}} }(X)$ is a class of even degree, we define the vertex operators $\bigl( S_{m}{^{\vphantom{[n]}} }(\gamma )\bigr)_{m\ge 0}{^{\vphantom{[n]}} }$ by the formula $\sum_{m\ge 0}S_{m}{^{\vphantom{[n]}} }(\gamma )\, t^{m}{_{\vphantom{[n]}} }=\exp\left(\sum_{n>0}\frac{(-1)^{n-1}}{n}\, \mathfrak{q}_{n}^{\vphantom{A}}(\gamma )\, t^{n}{_{\vphantom{[n]}} }\right)$. Since $\gamma $ is of even degree, the operators $\bigl( \mathfrak{q}_{i}{^{\vphantom{[n]}} }(\gamma )\bigr)_{i>0}{^{\vphantom{[n]}} }$ commute in the usual sense, and the definition of $S_{m}{^{\vphantom{[n]}} }(\gamma )$ makes sense.
\[sans\] *([@SchHilGr], [@SchHilNabis] in the integrable case).* Let $\ti{J}$ be an almost-complex structure close to $J$, and let $C$ be a $\ti{J}$-holomorphic curve. Suppose that $\ti{J}$ is integrable in a [neighbourhood]{} of $C$. Then $\bigl[ C^{[n]}{_{\vphantom{[n]}} }\bigr]=S_{n}{^{\vphantom{[n]}} }\bigl( [C]\bigr)\, .\, 1$.
Since the Nakajima operators are invariant by deformation of the almost-complex structure, we can make the computations with almost-complex structures equal to $\ti{J}$ when all the points are near $C$. Let $U$ be a small [neighbourhood]{} of $C$ such that $\ti{J}$ is integrable in $U$. Then, for any positive integers $n$ and $m$ such that $m>n$, the Hilbert schemes ${X^{[m]}{_{\vphantom{[n]}} }}$, ${X^{[n]}{_{\vphantom{[n]}} }}$ and the incidence variety ${X^{[m\!,\,n]}{_{\vphantom{[n]}} }}$ are the usual integrable ones over $S^{m}{_{\vphantom{[n]}} }U$, $S^{n}{_{\vphantom{[n]}} }U$ and $S^{n}{_{\vphantom{[n]}} }U{\ensuremath{\times}}S^{n-m}{_{\vphantom{[n]}} }U$ respectively. Since Lemma \[sans\] holds in $H^{2n}_{c}\bigl( U^{[n]}{_{\vphantom{[n]}} },\Q\bigr)$, we are done.
If $(C,\ti{J})$ satisfies the hypotheses of Lemma \[sans\], Lehn’s computations recalled above apply verbatim and give $\ds\int_{X}\Bigl[ e_{n}{^{\vphantom{[n]}} }-\dfrac{1}{2}\,n^{2}{_{\vphantom{[n]}} }(|n|-1)\,c_{1}{^{\vphantom{[n]}} }(X)\Bigr]\,.\,[C]=0$. By Corollary \[cor\], $H^{2}{_{\vphantom{[n]}} }(X,\Q)$ is spanned by cohomology classes of such holomorphic curves. This gives the result.
The derivative of the Nakajima operators can be explicitely computed in terms of the Virasoro operators $\mathfrak{L}_{n}{^{\vphantom{[n]}} }(\alpha )$ defined in [@SchHilLe Section 3.1]:
\[www\] If $(X,\omega )$ is a symplectic compact fourfold and $J$ is a compatible almost-complex structure, then for all $n$ in $\Z$, $\mathfrak{q}'_{n}(\alpha )=n\, \mathfrak{L}_{n}{^{\vphantom{[n]}} }(\alpha )-\frac{1}{2}\,n(|n|-1)\mathfrak{q}_{n}^{\vphantom{A}}\bigl( c_{1}{^{\vphantom{[n]}} }(X)\,\alpha \bigr)$.
For the proof, see [@SchHilLe p. 180].
The ring structure of $H^{*}\bigl(X^{[n]},\Q\bigr)$
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Geometric tautological Chern characters
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Let $(X,J)$ be an almost-complex compact fourfold, $
\apl{\lambda }{{X^{[n+1]}{_{\vphantom{[n]}} }}}{{X^{[n]}{_{\vphantom{[n]}} }},}\ \apl{\nu }{{X^{[n+1,n]}{_{\vphantom{[n]}} }}}{X^{[n+1]}{_{\vphantom{[n]}} },}\ \apl{\rho }{{X^{[n+1,n]}{_{\vphantom{[n]}} }}}{X}
$ the three associated maps, which are canonical up to homotopy, and $D{\ensuremath{\subseteq}}{X^{[n+1,n]}{_{\vphantom{[n]}} }}$ the exceptional divisor.
If $E$ is a complex vector bundle on $X$, it is possible to associate to $E$ a sequence of tautological vector bundles $\bigl( E^{[n]}\bigr)_{n>0}{^{\vphantom{[n]}} }$ on ${X^{[n]}{_{\vphantom{[n]}} }}$. These tautological bundles are constructed in [@SchHilGri] using relative holomorphic structures on $E$, and their classes in are shown to be independent of these auxiliary structures. This defines tautological morphisms from $K(X)$ to $K\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr)$.
Let $F$ be the complex line bundle on ${X^{[n+1,n]}{_{\vphantom{[n]}} }}$ such that $c_{1}{^{\vphantom{[n]}} }(F)$ is Poincaré dual to $-[D]$. Then (see [@SchHilGri 3.2]), $\nu \ee{_{\vphantom{[n]}} }E^{[n+1]}{_{\vphantom{[n]}} }=\lambda {\ensuremath{^{\, *}}}E^{[n]}{_{\vphantom{[n]}} }+\rho \ee{_{\vphantom{[n]}} }E{\ensuremath{\otimes}}$F in $K\bigl( {X^{[n+1,n]}{_{\vphantom{[n]}} }}\bigr)$. This gives the relation $\nu \ee{_{\vphantom{[n]}} }\bigl( \operatorname{ch}\bigl( E^{[n+1]}{_{\vphantom{[n]}} }\bigr)\bigr)=\lambda \ee{_{\vphantom{[n]}} }\bigl( \operatorname{ch}\bigl( E^{[n]}{_{\vphantom{[n]}} }\bigr)\bigr)+\rho \ee{_{\vphantom{[n]}} }\operatorname{ch}(E)\,.\,c_{1}{^{\vphantom{[n]}} }(F)$ in $H^{\textrm{even}}{_{\vphantom{[n]}} }\bigl( X^{[n+1,n]}{_{\vphantom{[n]}} },\Q\bigr)$.
\[Ch4SuiteLemUn\] For every class $\alpha $ in $H^{\emph{even}}{_{\vphantom{[n]}} }(X,\Q)$ and every $n$ in $\N\ee{_{\vphantom{[n]}} }$, there exists a unique class $G(\alpha ,n)$ in $H^{\emph{even}}{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ such that $G(\alpha ,1)=\alpha $, and for all $n$ in $\N\ee{_{\vphantom{[n]}} }$, $$\nu \ee{_{\vphantom{[n]}} }G(\alpha ,n+1)-\lambda \ee{_{\vphantom{[n]}} }G(\alpha ,n)=\rho \ee{_{\vphantom{[n]}} }\alpha \,.\,c_{1}(F).$$
By the degeneration of the Atiyah-Hirzebruch spectral sequence from to cohomology with , we have isomorphisms $\apliso{\operatorname{ch}}{K\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr){\ensuremath{\otimes}}_{\Z}{^{\vphantom{[n]}} }\Q}{H^{\textrm{even}}{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr).}$ Therefore, we can define the classes $G(\alpha ,n)$ in $H^{\textrm{even}}\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ as follows: if $E$ is the unique class in $K(X){\ensuremath{\otimes}}_{\Z}{^{\vphantom{[n]}} }\Q$ such that $\operatorname{ch}(E)=\alpha $, then $G(\alpha ,n)=\operatorname{ch}\bigl( E^{[n]}{_{\vphantom{[n]}} }\bigr)$. Furthermore, $G(\alpha ,n)$ is unique since $\nu _{*}{^{\vphantom{[n]}} }\nu \ee{_{\vphantom{[n]}} }=\frac{1}{n+1}\operatorname{id}$.
Virtual tautological Chern characters {#decadix}
-------------------------------------
This section is somehow technical and can be omitted if we suppose that $b_{1}{^{\vphantom{[n]}} }(X)=0$, that is if $H^{\textrm{odd}}{_{\vphantom{[n]}} }(X,\Q)=0$. The purpose here is to extend Lemma \[Ch4SuiteLemUn\] to odd-dimensional cohomology classes. We adapt the method originally developed in the projective case by Li, Qin and Wang in [@SchHilLQWMA].
\[Ch4SuitePropUn\] For every class $\alpha $ in $H\ee{_{\vphantom{[n]}} }(X,\Q)$ and every $n$ in $\N\ee{_{\vphantom{[n]}} }$, there exists a unique class $G(\alpha ,n)$ in $H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ such that $G(\alpha ,1)=\alpha $ and for all $n\in\N\ee{_{\vphantom{[n]}} }$, $$\nu \ee{_{\vphantom{[n]}} }G(\alpha ,n+1)-\lambda \ee{_{\vphantom{[n]}} }G(\alpha ,n)=\rho \ee{_{\vphantom{[n]}} }\alpha \,.\,c_{1}(F).$$
\[Ch4SuiteRemUn\] If $X$ is projective and ${Y_{n}{^{\vphantom{[n]}} }}\suq{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X$ is the incidence locus, it is proved in [@SchHilLQWMA] that $G(\alpha ,n)=\operatorname{pr}_{1}\ee\bigl[ \operatorname{ch}(\oo_{{Y_{n}{^{\vphantom{[n]}} }}}{^{\vphantom{[n]}} })\,.\,\operatorname{pr}_{2}\ee\alpha \,.\,\operatorname{pr}_{2}\ee\,\operatorname{td}(X)\bigr]$.
We use the relative trick and the machinery of relative coherent sheaves developed in the appendix. Since these methods are going to be explained thoroughly in the second part of the paper with similar computations (e.g. in Section \[qwe\]), we skip some lengthy details.
If ${\ensuremath{\mathfrak{X} }}$ is a [relative smooth analytic space]{} and ${\ensuremath{\mathfrak{Z} }}$ a relative analytic subspace of ${\ensuremath{\mathfrak{X} }}$ (see Definition \[DefUnAppenCh4\]), we will denote $\lim\limits_{\genfrac{}{}{0pt}{}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}}}{^{\vphantom{[n]}} }H\ee_{{\ensuremath{\mathfrak{X}' }}\cap{\ensuremath{\mathfrak{Z} }}}({\ensuremath{\mathfrak{X}' }},\Q)$ by ${H\raisebox{-1.3ex}{$\mkern-20mu\leftarrow$}{^{\vphantom{[n]}} }{}}_{{\ensuremath{\mathfrak{Z} }}}\ee({\ensuremath{\mathfrak{X} }},\Q)$, where in the projective limit ${\ensuremath{\mathfrak{X}' }}$ runs through all open relatively compact relative analytic subspaces of ${\ensuremath{\mathfrak{X} }}$.
If $W$ is a [neighbourhood]{} of $Z_{n}{^{\vphantom{[n]}} }$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$, ${\ensuremath{\mathfrak{Y} }}_{n}{^{\vphantom{[n]}} }$ is the relative incidence locus and ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$ is the relative incidence [sheaf]{} on $W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W$ (see Definition \[a\] and Definition \[b\]), the topological class of ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$ lies in $\lim\limits_{\genfrac{}{}{0pt}{}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}}}{^{\vphantom{[n]}} }K_{{\ensuremath{\mathfrak{X}' }}\cap{\ensuremath{\mathfrak{Y} }}_{n}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }({\ensuremath{\mathfrak{X}' }})$. Let $\apl{\pi }{W}{X}$, $\apl{p}{W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W}{W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}}$ be the natural projections. We define the following cohomology classes:
1. $\mu ^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}$ is the Chern character of ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$ in ${H\raisebox{-1.3ex}{$\mkern-20mu\leftarrow$}{^{\vphantom{[n]}} }{}}\ee_{{\ensuremath{\mathfrak{Y} }}_{n}{^{\vphantom{[n]}} }}\bigl( W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W,\Q\bigr)$.
2. $G(\alpha ,n)^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }=p_{*}{^{\vphantom{[n]}} }\bigl[ \mu _{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\,.\,\pi \ee{_{\vphantom{[n]}} }\alpha \,.\,\pi \ee{_{\vphantom{[n]}} }\operatorname{td}(X)\bigr]$ in ${H\raisebox{-1.3ex}{$\mkern-20mu\leftarrow$}{^{\vphantom{[n]}} }{}}\ee{_{\vphantom{[n]}} }\bigl( W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},\Q\bigr)$.
3. $G(\alpha,n)=G(\alpha ,n)_{\vert{X^{[n]}{_{\vphantom{[n]}} }}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$.
We take the notations introduced at the beginning of Section \[Comparaison\], so that $W$ will be from now on a [neighbourhood]{} of $Z_{n{\ensuremath{\times}}1}$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$. Let $\apl{\pi }{W}{X,}$ $\apl{p}{W^{[n+1,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W}{W^{[n+1,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}$ and $\apl{q}{W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W}{W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}}$ be the natural projections. We define new cohomology classes:
1. $\ti{\mu }_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}=\operatorname{ch}\bigl( \ti{\oo}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}\bigr)$ in ${H\raisebox{-1.3ex}{$\mkern-20mu\leftarrow$}{^{\vphantom{[n]}} }{}}\ee_{\,\ti{\ensuremath{\mathfrak{Y} }}_{n}{^{\vphantom{[n]}} }}\bigl( W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W,\Q\bigr)$, where $\ti{{\ensuremath{\mathfrak{Y} }}}_{n}{^{\vphantom{[n]}} }=\operatorname{supp}\bigl( \ti\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}\bigr)$.
2. $\ti{G}(\alpha ,n)^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }=q_{*}\bigl[ \ti{\mu }_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\,.\,\pi \ee{_{\vphantom{[n]}} }\alpha \,.\,\pi \ee{_{\vphantom{[n]}} }\operatorname{td}(X)\bigr]$.
Then $\psi _{W}\ee\, \ti{G}
(\alpha ,n+1)^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }-\phi \ee_{W}\ti{G}(\alpha ,n)^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }=p_{*}{^{\vphantom{[n]}} }\bigl[ (\psi _{W}\ee\, \ti{\mu }^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n+1}-\phi \, \ee_{W}\, \ti{\mu }_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}})\,.\,\pi \ee{_{\vphantom{[n]}} }\alpha \,.\,\pi \ee{_{\vphantom{[n]}} }\operatorname{td}(X)\bigr]$. Lemma \[f\] shows that this quantity is equal to $p_{*}\bigl[ p\ee{_{\vphantom{[n]}} }\operatorname{ch}{(\LL)}\,.\,\rho \ee_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},\,W}\operatorname{ch}\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}}\bigr)\,.\,\pi \ee{_{\vphantom{[n]}} }\alpha \,.\,\pi \ee{_{\vphantom{[n]}} }\operatorname{td}(X)\bigr]$, which is equal to $\operatorname{ch}(\LL)\,\rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\ee\, r_{*}{^{\vphantom{[n]}} }\bigl[ \operatorname{ch}\bigl({\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}}\bigr)\,.\,\pi \ee{_{\vphantom{[n]}} }\alpha \,.\,\pi \ee{_{\vphantom{[n]}} }\operatorname{td}(X)\bigr]$ via the second diagram of the proof of Proposition \[SansLabBis\]. Using the same argument as in proof of Lemma \[EtToc\] (iv), $$\bigl[ \psi _{W}\ee\,\ti{G}(\alpha ,n+1)^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }-\phi _{W}\ee\,\ti{G}(\alpha ,n)^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\bigr]_{\vert{X^{[n+1,n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }=\operatorname{ch}(\LL)_{\vert{X^{[n+1,n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }\,\rho \ee{_{\vphantom{[n]}} }\bigl[ \operatorname{pr}_{1*}{^{\vphantom{[n]}} }\operatorname{ch}\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\Delta _{X}{^{\vphantom{[n]}} }}\bigr)\,.\,\alpha \,.\,\operatorname{td}(X)\bigr]$$ which is equal, by the Grothendieck-Riemann-Roch theorem of [@SchHilAtHi] applied to the diagonal injection, to $c_{1}{^{\vphantom{[n]}} }(F)\,.\,\rho \ee{_{\vphantom{[n]}} }(\alpha )$. To conclude the proof, it suffices to show that $$\psi \ee_{W}\,\ti{G}(\alpha ,n+1)^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }{}{^{\vphantom{[n]}} }_{\vert{X^{[n+1,n]}{_{\vphantom{[n]}} }}}=\nu \ee{_{\vphantom{[n]}} }\,G(\alpha ,n+1)\quad\textrm{and}\quad\phi \ee_{W}\,\ti{G}(\alpha ,n)^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }{}{^{\vphantom{[n]}} }_{\vert{X^{[n+1,n]}{_{\vphantom{[n]}} }}}=\lambda \ee{_{\vphantom{[n]}} }G(\alpha ,n).$$ This is performed exactly as in Lemma \[EtToc\], (i)–(iii).
The ring structure and the crepant resolution conjecture
--------------------------------------------------------
In this section, $X$ will be a symplectic compact fourfold endowed with a compatible almost-complex structure.
We introduce operators acting on $\HHH=\bop\nolimits_{n\in\N}{^{\vphantom{[n]}} }H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ by cup-product with the components of the virtual tautological Chern characters constructed in Section \[decadix\].
\[5\] Let $\alpha \in {H\ee{_{\vphantom{[n]}} }}(X,\Q)$ be a homogeneous cohomology class. Then
1. $G_{i}{^{\vphantom{[n]}} }(\alpha ,n)$ denotes the $(|\alpha |+2i)$-th component of $G(\alpha ,n)$.
2. $\mathfrak{S}_{i}{^{\vphantom{[n]}} }(\alpha )$ denotes the operator $\HHH$ which acts on ${H\ee{_{\vphantom{[n]}} }}({X^{[n]}{_{\vphantom{[n]}} }},\Q)$ by cup product with $G_{i}{^{\vphantom{[n]}} }(\alpha ,n)$.
Then the following result, originally proved by Lehn for geometric tautological Chern characters and generalized by Li, Qin and Wang for the virtual ones, is still valid:
\[UN\] For all $\alpha ,\beta $ in ${H\ee{_{\vphantom{[n]}} }}(X,\Q)$ and for all $k$ in $\N$, $\bigl[ \mathfrak{S}_{k}{^{\vphantom{[n]}} }(\alpha ),\mathfrak{q}_{1}{^{\vphantom{[n]}} }(\beta )\bigr]=\frac{1}{k!}\, \mathfrak{q}_{1}^{(k)}(\alpha \beta )$.
This is a consequence of Lemma \[hum\] and Proposition \[Ch4SuitePropUn\] (see [@SchHilLe Theorem 4.2]).
We can now state some significant results on the cohomology rings of Hilbert schemes of symplectic fourfolds. These results are known in the integrable case and are formal consequences of the various relations between $\mathfrak{q}_{n}^{\vphantom{A}}(\alpha )$, $\mathfrak{d}$, $\mathfrak{L}_{n}{^{\vphantom{[n]}} }(\alpha )$ and $\mathfrak{S}_{i}{^{\vphantom{[n]}} }(\alpha )$ (see e.g. Theorem 2.1 in [@SchHilLQW2003]), even though there is a lot of nontrivial combinatorics involved in the proofs. Thus the following results are formal consequences of Theorem \[qqq\], Corollary \[www\] and Proposition \[UN\].
\[6\] If $0\le i<n$ and $\alpha $ runs through a fixed basis of ${H\ee{_{\vphantom{[n]}} }}(X,\Q)$, the classes $G_{i}{^{\vphantom{[n]}} }(\alpha ,n)$ generate the ring ${H\ee{_{\vphantom{[n]}} }}\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$.
This result was initially proved in [@SchHilLQWMA] using vertex algebras. For other proofs, see [@SchHilLQW2001] and [@SchHilLQW2003].
\[asd\] For every integer $n$, the ring $H\ee{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ can be built by universal formulae from the ring $H\ee{_{\vphantom{[n]}} }(X,\Q)$ and the first Chern class of $X$ in $H^{2}{_{\vphantom{[n]}} }(X,\Q)$.
For the proof as well as an effective statement, see [@SchHilLQW2003].
There is a geometrical approach to the ring structure of ${H\ee{_{\vphantom{[n]}} }}\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ through orbifold cohomology. If $J$ is an adapted almost-complex structure on $X$, ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$ is an almost-complex Gorenstein orbifold. We can therefore consider the associated Chen-Ruan (or *orbifold*) cohomology ring $H\ee_{\textrm{CR}}\bigl( {S^{\mkern 1 mu n\!}_{\vphantom{[}} X},\Q\bigr)$ which is $\Z$-graded and depends only on the deformation class of $J$ (see [@SchHilCR], [@SchHilALR], [@SchHilFG]).
After works by Lehn-Sorger and Li-Qin-Wang, Qin and Wang developed a set of axioms which characterize $H^{*}_{\textrm{CR}}\bigl( {S^{\mkern 1 mu n\!}_{\vphantom{[}} X},\Q\bigr)$ as a ring (see [@SchHilALR]):
[@SchHilQW]\[QuinWang\] Let $A$ be a graded unitary ring and $(X,J)$ an almost-complex compact fourfold. We suppose that
1. $A$ is an irreducible $\hh\bigl( {H\ee{_{\vphantom{[n]}} }}(X,\C)\bigr)$-module and $1$ is a highest weight vector.
2. For all $\alpha $ in ${H\ee{_{\vphantom{[n]}} }}(X,\C)$ and for all $i$ $\N$, there exist classes $O_{i}{^{\vphantom{[n]}} }(\alpha ,n)\in A^{|\alpha |+2i}{_{\vphantom{[n]}} }$ such that if $\mathfrak{D}_{i}{^{\vphantom{[n]}} }(\alpha )$ is the operator of product by $\bop_{n}{^{\vphantom{[n]}} }O_{i}{^{\vphantom{[n]}} }(\alpha ,n)$ and $\mathfrak{D}_{1}{^{\vphantom{[n]}} }(1)=\mathfrak{d}$ is the derivation,
1. $\forall \alpha ,\beta \in{H\ee{_{\vphantom{[n]}} }}(X,\C)$, $\forall k\in\N$,$\bigl[ \mathfrak{D}_{k}{^{\vphantom{[n]}} }(\alpha ),\mathfrak{q}_{1}{^{\vphantom{[n]}} }(\beta )\bigr]=\mathfrak{q}_{1}^{(k)}(\alpha \beta )$.
2. If $\delta _{X}{^{\vphantom{[n]}} }$ is the class in $H\ee{_{\vphantom{[n]}} }(X,\Q)^{{\ensuremath{\otimes}}3}{_{\vphantom{[n]}} }$ mapped by the Künneth isomorphism to the cycle class of the diagonal in $X^{3}{_{\vphantom{[n]}} }$, $\ds\sum_{l_{1}{^{\vphantom{[n]}} }+l_{2}{^{\vphantom{[n]}} }+l_{3}{^{\vphantom{[n]}} }=0}:\mathfrak{q}_{l_{1}{^{\vphantom{[n]}} }} \mathfrak{q}_{l_{2}{^{\vphantom{[n]}} }} \mathfrak{q}_{l_{3}{^{\vphantom{[n]}} }}\!\!:(\delta _{X}{^{\vphantom{[n]}} })=-6\, \mathfrak{d}$.
Then $A$ is isomorphic as a ring to $H\ee_{\emph{CR}}\bigl( {S^{\mkern 1 mu n\!}_{\vphantom{[}} X},\C\bigr)$.
In $(2)$, we used the physicists’ normal ordering convention $$:\mathfrak{q}_{l_{1}{^{\vphantom{[n]}} }} \mathfrak{q}_{l_{2}{^{\vphantom{[n]}} }} \mathfrak{q}_{l_{3}{^{\vphantom{[n]}} }}\!\!:\,=
\mathfrak{q}_{l_{1}'}{^{\vphantom{[n]}} }\mathfrak{q}_{l_{2}'}{^{\vphantom{[n]}} }\mathfrak{q}_{l_{3}'}{^{\vphantom{[n]}} },\ \textrm{where}\
\{l_{1}{^{\vphantom{[n]}} },l_{2}{^{\vphantom{[n]}} },l_{3}{^{\vphantom{[n]}} }\}=\{l_{1}',l_{2}',l_{3}'\}\ \textrm{and}\ l'_{1}\le l_{2}'\le l_{3}'.$$ We apply this theorem to prove Ruan’s conjecture for the symmetric products of a symplectic fourfold with torsion first Chern class.
\[TROIS\] Let $(X,\omega )$ be a symplectic compact fourfold with vanishing first Chern class in $H^{2}{_{\vphantom{[n]}} }(X,\Q)$. Then Ruan’s crepant conjecture holds for ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$, i.e. the rings ${H\ee{_{\vphantom{[n]}} }}\bigl( {X^{[n]}{_{\vphantom{[n]}} }},\Q\bigr)$ and $H\ee_{\emph{CR}}\bigl( {S^{\mkern 1 mu n\!}_{\vphantom{[}} X},\Q\bigr)$ are isomorphic.
Let $O_{k}{^{\vphantom{[n]}} }(\alpha ,n)=k!\, \mathfrak{S}_{k}{^{\vphantom{[n]}} }(\alpha ,n)$. Then (1) is exactly Proposition \[UN\]. The relation (2) is a formal consequence of the Nakajima relations and of the formulae $\bigl[ \mathfrak{q}_{n}'(\alpha ),\mathfrak{q}_{m}^{\vphantom{A}}(\beta )\bigr]=-nm\, \mathfrak{q}_{n+m}^{\vphantom{A}}(\alpha \beta )$, $\mathfrak{q}'_{n}(\alpha )=n\, \mathfrak{L}_{n}{^{\vphantom{[n]}} }(\alpha )$.
The cobordism class of $X^{[n]}$ {#SecCinq}
================================
In this section, $(X,J)$ is an almost-complex compact fourfold, and no symplectic hypotheses are required. The almost-complex Hilbert schemes ${X^{[n]}{_{\vphantom{[n]}} }}$ are endowed with a stable almost complex structure (see [@SchHilVo1]), hence define almost-complex cobordism classes. By a fundamental result of Novikov [@SchHilNo] and Milnor [@SchHilMi], the almost-complex cobordism class of ${X^{[n]}{_{\vphantom{[n]}} }}$ is completely determined by the Chern numbers $\ds\int_{{X^{[n]}{_{\vphantom{[n]}} }}}P\bigl[ c_{1}{^{\vphantom{[n]}} }({X^{[n]}{_{\vphantom{[n]}} }}),\dots,c_{2n}{^{\vphantom{[n]}} }({X^{[n]}{_{\vphantom{[n]}} }})\bigr]$ where $P$ runs through all polynomials $P$ in $\Q\bigl[ T_{1}{^{\vphantom{[n]}} },\dots,T_{2n}{^{\vphantom{[n]}} }\bigr]$ of weighted degree $4n$, each variable $T_{k}{^{\vphantom{[n]}} }$ having degree $2k$. We intend to prove the following result:
\[QUATRE\] The almost-complex cobordism class of ${X^{[n]}{_{\vphantom{[n]}} }}$ depends only on the almost-complex cobordism class of $X$.
This means that if $P$ is a weighted polynomial in $\Q\bigl[ T_{1}{^{\vphantom{[n]}} },\dots,T_{2n}{^{\vphantom{[n]}} }\bigr]$ of degree $4n$, there exists a weighted polynomial $\ti{P}\bigl[ T_{1}{^{\vphantom{[n]}} },T_{2}{^{\vphantom{[n]}} }\bigr]$ of degree $4$, depending only on $P$ and $n$, such that $$\int_{{X^{[n]}{_{\vphantom{[n]}} }}}P\bigl[ c_{1}{^{\vphantom{[n]}} }({X^{[n]}{_{\vphantom{[n]}} }}),\dots,c_{2n}{^{\vphantom{[n]}} }({X^{[n]}{_{\vphantom{[n]}} }})\bigr]=\int_{X}\ti{P}\bigl[ c_{1}{^{\vphantom{[n]}} }(X),c_{2}{^{\vphantom{[n]}} }(X)\bigr].$$ This result has been proved by Ellinsgrud, Göttsche and Lehn in [@SchHilEGL] when $X$ is projective. Let us describe briefly the main steps of the argument and the tools we need.
Let $J_{\vphantom{n}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ be a relative integrable structure in a [neighbourhood]{} $W$ of $Z_{n}{^{\vphantom{[n]}} }$. Then the relative Hilbert scheme $\bigl( W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}\bigr)$ is fibered in smooth analytic spaces over ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$, so that we can consider its relative tangent bundle $T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\,W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}$ which is a continuous complex vector bundle on $W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$. It will turn out that the class $\bigl[ T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\,W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}\bigr]_{\vert{X^{[n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }$ in $K({X^{[n]}{_{\vphantom{[n]}} }})$ is exactly the class of the tangent bundle $T{X^{[n]}{_{\vphantom{[n]}} }}$, where ${X^{[n]}{_{\vphantom{[n]}} }}$ is endowed with the differentiable and stable almost-complex structures constructed by Voisin in [@SchHilVo1]. Therefore, the class of $T{X^{[n]}{_{\vphantom{[n]}} }}$ can be understood via the tangent bundle of classical Hilbert schemes.
The main idea in [@SchHilEGL] is to relate $X^{[n+1]}{_{\vphantom{[n]}} }$ and ${X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X$ via the smooth incidence Hilbert scheme $X^{[n+1,n]}{_{\vphantom{[n]}} }$. The essential step in their argument is the explicit comparison in $K\bigl( X^{[n+1,n]}{_{\vphantom{[n]}} }\bigr)$ of the classes of $T{X^{[n]}{_{\vphantom{[n]}} }}$ and $T X^{[n+1]}{_{\vphantom{[n]}} }$. This is carried out using the explicit description of $T{X^{[n]}{_{\vphantom{[n]}} }}$ as $\operatorname{pr}_{1*}{^{\vphantom{[n]}} }{\ensuremath{\mathcal{H}}}om\bigl( \jj_{n}{^{\vphantom{[n]}} },\oo_{n}{^{\vphantom{[n]}} }\bigr)$, where $\jj_{n}{^{\vphantom{[n]}} }$ is the ideal sheaf of the incidence locus ${Y_{n}{^{\vphantom{[n]}} }}\suq{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X$ and $\oo_{n}{^{\vphantom{[n]}} }$ is the structure sheaf of ${Y_{n}{^{\vphantom{[n]}} }}$. So it appears necessary to consider coherent sheaves and not only locally free ones.
In the almost-complex setting, the coherent sheaves have no equivalent. Instead of working directly on the almost-complex Hilbert schemes and the associated incidence varieties, we use the corresponding relative objects, which can be considered as homotopically equivalent to the original ones, but possess a much stronger structure: they are fibered in analytic spaces over a singular basis. Each fiber consists of the initial object (Hilbert scheme, incidence variety,...) associated to an open set of $X$ with an integrable structure on it. The almost-complex Hilbert scheme ${X^{[n]}{_{\vphantom{[n]}} }}$ will be for instance replaced by the fibration $W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ over ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$ associated to a relative integrable complex structure ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }}$: the corresponding fibers are the integrable Hilbert schemes $\bigl( W_{{{\underline{\vphantom{!}\vphantom{y}x}}}}^{[n]},J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{{\underline{\vphantom{!}\vphantom{y}x}}}}\bigr)_{{{\underline{\vphantom{!}\vphantom{y}x}}}\in{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }$.
In the appendix (Section \[AppendCh4\]), we develop a general formalism for relative coherent sheaves on spaces ${\ensuremath{\mathfrak{X} }}/B$ fibered in smooth analytic sets over a differentiable basis $B$ with quotient singularities, such as $W^{[n]}_{\operatorname{\vphantom{y}rel}}$. These [relative smooth analytic space]{}s carry a sheaf $\oo_{{\ensuremath{\mathfrak{X} }}}^{\, \operatorname{\vphantom{y}rel}}$ which is the sheaf of smooth functions holomorphic in the fibers (see Definition \[DefUnAppenCh4\]).
Intuitively, a relatively coherent sheaf $\ff$ on a [relative smooth analytic space]{} ${\ensuremath{\mathfrak{X} }}$ over $B$ is a family of coherent sheaves $\bigl( \ff_{b}{^{\vphantom{[n]}} }\bigr)_{b\in B}{^{\vphantom{[n]}} }$ on $\bigl( {\ensuremath{\mathfrak{X} }}_{b}{^{\vphantom{[n]}} }\bigr)_{b\in B}{^{\vphantom{[n]}} }$ varying smoothly with $b$ and locally trivially on ${\ensuremath{\mathfrak{X} }}$. If we take relative holomorphic coordinates on ${\ensuremath{\mathfrak{X} }}$, the local model for ${\ensuremath{\mathfrak{X} }}$ is $Z{\ensuremath{\times}}V$, where $Z$ is a smooth analytic set and $V$ is an open subset of the base $B$. Then the local model for a relatively coherent sheaf on $Z{\ensuremath{\times}}V$ is $\operatorname{pr}_{1}^{-1}\g{\ensuremath{\otimes}}_{\operatorname{pr}_{1}^{-1}\oo_{Z}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}$, where $\g$ is a coherent analytic sheaf on $Z$ (see Definition \[DefQuatreAppendCh4\]).
The usual operations on coherent sheaves (such as internal $\textrm{Hom}$, tensor product, dual, pull-back, push-forward and the associated derived operations) can be performed on relatively coherent sheaves for smooth morphisms holomorphic in the fibers satisfying some triviality conditions (see Definitions \[DefDeuxAppenCh4\] and \[DefSixAppendCh4\]). For the push-forward, we will only have to deal with the situation where the map is finite on the support of the [sheaf]{}, which is technically much simpler than the general case.
In this context, it is possible to construct a relative version of the usual analytic $K$-theory for coherent [sheaves]{} (see Definitions \[DefNeufAppendCh4\] and \[DefUnBisAppendCh4\]) as well as associated operations.
This being done, it will be necessary to consider relatively coherent sheaves as elements in topological . This is achieved by the following proposition:
\[SansLab\] If $\ff$ is a relatively coherent sheaf on a relative smooth analytic space ${\ensuremath{\mathfrak{X} }}$ over $B$ and ${\ensuremath{\mathfrak{X} }}'$ is relatively compact in ${\ensuremath{\mathfrak{X} }}$, then $\ff^{\, \infty }:=\ff{\ensuremath{\otimes}}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}}{^{\vphantom{[n]}} }{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{{\ensuremath{\mathfrak{X} }}}$ admits a resolution on ${\ensuremath{\mathfrak{X} }}'$ by complex vector bundles $\bigl( E_{i}{^{\vphantom{[n]}} }\bigr)_{1\le i\le N}{^{\vphantom{[n]}} }$. Besides, the element $[\ff^{\, \infty }{_{\vphantom{[n]}} }]:=\sum_{i=1}^{N}(-1)^{i-1}[E_{i}{^{\vphantom{[n]}} }]$ in $K({\ensuremath{\mathfrak{X} }}')$ is independent of $E_{\bullet}{^{\vphantom{[n]}} }$.
According to this proposition, it is possible to associate to any relatively coherent [sheaf]{} $\ff$ on a [relative smooth analytic space]{} ${\ensuremath{\mathfrak{X} }}$ a *topological class* $[\ff^{\,\infty }{_{\vphantom{[n]}} }]$ in $\ds\lim_{\genfrac{}{}{0pt}{2}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}\subset\subset {\ensuremath{\mathfrak{X} }}}}K({\ensuremath{\mathfrak{X}' }})$ and therefore Chern classes in $\ds\lim_{\genfrac{}{}{0pt}{2}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}\subset\subset {\ensuremath{\mathfrak{X} }}}}{H}\ee{_{\vphantom{[n]}} }({\ensuremath{\mathfrak{X}' }},\Z)$. Besides, the class $[\ff^{\,\infty }{_{\vphantom{[n]}} }]$ depends only on the relative class of $\ff$ in ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})$ by Proposition \[Compl\] (i).
This device enables us to carry out the proof of [@SchHilEGL] in a relative context.
The relative incidence [sheaf]{}
--------------------------------
In this section, we introduce the relative incidence sheaf ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$ and we compute its Chern classes. We will use the following notations.
1. $W$ is a small [neighbourhood]{} of the incidence locus $Z_{n}{^{\vphantom{[n]}} }$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$.
2. ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$ is a relative integrable structure on $W$ parametrized by ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$.
\[a\] The *relative incidence locus* ${\ensuremath{\mathfrak{Y}_{n} }}$ is the relative singular analytic subspace of $W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W$ defined by $${\ensuremath{\mathfrak{Y}_{n} }}=\bigl\{(\xi,\,w,\,{{\underline{\vphantom{!}\vphantom{y}x}}})\ \textrm{in}\ W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W\ \textrm{such that}\ w\in \operatorname{supp}(\xi)\bigr\}\cdot$$ The fibers $\bigl( {\ensuremath{\mathfrak{Y} }}_{n,\,{{\underline{\vphantom{!}\vphantom{y}x}}}}{^{\vphantom{[n]}} }\bigr)_{{{\underline{\vphantom{!}\vphantom{y}x}}}\in{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }$ are the usual incidence loci in $W_{{{\underline{\vphantom{!}\vphantom{y}x}}}}^{[n]}{\ensuremath{\times}}W_{{{\underline{\vphantom{!}\vphantom{y}x}}}}{^{\vphantom{[n]}} }$.
\[b\] The *relative incidence [sheaf]{}* ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$ is the relatively coherent [sheaf]{} ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{Y}_{n} }}}}$ on $W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W$. The associated ideal [sheaf]{}${\jj^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{ {\ensuremath{\mathfrak{Y}_{n} }}}}$ will be denoted by ${\jj^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$.
By Proposition \[SansLab\], ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$ defines a topological class in $\ds\lim_{\genfrac{}{}{0pt}{2}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}\subset\subset W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W}}K({\ensuremath{\mathfrak{X}' }})$ and therefore by restriction a class in $K\bigl( {X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X\bigr)$ which is independent of $J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}$ by Proposition \[Compl\] (ii).
\[c\] The classes $\mu _{i,\,n}{^{\vphantom{[n]}} }$ in $H^{2i}{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X,\Q\bigr)$ are defined by $\mu _{i,\,n}{^{\vphantom{[n]}} }=c_{i}{^{\vphantom{[n]}} }\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)_{\vert {X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }$.
– Let ${\ensuremath{\widetilde{W}}}$ be a small [neighbourhood]{} of the incidence locus $Z_{n,\,1}{^{\vphantom{[n]}} }$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$ and ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n,1}}$ a relative integrable structure parametrized by ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$. For $({{\underline{\vphantom{!}\vphantom{y}x}}},p)\in {S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$, ${\ti{W}}^{[n+1,n]}_{{{\underline{\vphantom{!}\vphantom{y}x}}},\,p}$ is smooth ([@SchHilCh], [@SchHilTi]). Therefore, ${\ti{W}}^{[n+1,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ is a relative smooth analytic space over ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$.
– Let $D^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }$ be the relative exceptional divisor in ${\ti{W}}^{[n+1,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ defined by $$D^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }=\bigl\{(\xi,\,w,\,{{\underline{\vphantom{!}\vphantom{y}x}}},\,p)\ \textrm{in}\ {\ti{W}}^{[n+1,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\ \textrm{such that}\ w\in\operatorname{supp}(\xi )\bigr\}\cdot$$ Then $D^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }$ is a relative singular analytic divisor of ${\ti{W}}^{[n+1,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$, and we get an associated relative holomorphic line bundle $\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\bigl( -D^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\bigr)$ on ${\ti{W}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,n]}$. Let $\apl{\rho }{X^{[n+1,n]}{_{\vphantom{[n]}} }}{X}$ be the residual morphism and $\apl{\sigma }{X^{[n+1,n]}{_{\vphantom{[n]}} }}{{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X}$ be the morphism $(\lambda ,\rho )$.
\[l\] We define a class $l$ in $H^{2}\bigl(X^{[n+1,n]}{_{\vphantom{[n]}} },\Q\bigr)$ by $l=c_{1}{^{\vphantom{[n]}} }\bigl[ {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{}}(-D_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} })\bigr]_{\vert X^{[n+1,n]}{_{\vphantom{[n]}} }}{^{\vphantom{[n]}} }$.
The class $l$ determines the Chern classes of the relative incidence [sheaf]{} in the following way:
\[d\] For all $i$, $n$ in $\N\ee$, $\mu _{i,\,n}{^{\vphantom{[n]}} }=\sigma _{*}{^{\vphantom{[n]}} }(l^{i})$.
We recall well known facts from the classical theory (see [@SchHilDa], [@SchHilCh], [@SchHilTi]). If $X$ is a quasi-projective surface and $Y_{n}{^{\vphantom{[n]}} }$ is the incidence locus in ${X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X$, then:
1. $\jj_{{Y_{n}{^{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }$ admits a locally free resolution of length $2$.
2. ${X^{[n+1,n]}{_{\vphantom{[n]}} }}\simeq\P\bigl( \jj_{{Y_{n}{^{\vphantom{[n]}} }}}\bigr)$ and ${X^{[n+1,n]}{_{\vphantom{[n]}} }}$ is smooth.
3. If $\sutrgd{{\ensuremath{\mathcal{A}}}}{{\ensuremath{\mathcal{B}}}}{\jj_{{Y_{n}{^{\vphantom{[n]}} }}}}$ is a resolution of $\jj_{{Y_{n}{^{\vphantom{[n]}} }}}$, $\apl{\pi }{\P({\ensuremath{\mathcal{B}}})}{{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X}$ is the projection and $s$ is the associated section of $\pi \ee{_{\vphantom{[n]}} }{\ensuremath{\mathcal{A}}}\ee{_{\vphantom{[n]}} }(1)$, then $s$ is transverse to the zero section.
Property (iii) follows from (ii) since the vanishing locus of $s$ (with its schematic structure) is isomorphic to $\P\bigl(\jj_{{Y_{n}{^{\vphantom{[n]}} }}}\bigr)$. Note that we use here Grothendieck’s convention for projective bundles.
We adapt now these properties to the relative setting as follows. First we can suppose that $W$, ${\ti{W}}$, ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$ and ${J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}1}}$ satisfy the compatibility condition: $W\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }\bigl( {S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X\bigr)\suq{\ti{W}}$ as [relative smooth analytic space]{}s. Let $\ti{{\ensuremath{\mathfrak{Y} }}}_{n}{^{\vphantom{[n]}} }$ be the relative singular analytic subspace of ${\ti{W}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]\tim[1]}$ defined by $$\ti{{\ensuremath{\mathfrak{Y} }}}_{n}{^{\vphantom{[n]}} }\!=\!\bigl\{(\xi,w,{{\underline{\vphantom{!}\vphantom{y}x}}},p)\ \textrm{in}\ {\ti{W}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]\tim[1]}\ \textrm{such that}\ w\in\operatorname{supp}{\xi }\bigr\}\cdot$$ If we consider the following diagram $$\xymatrix{
W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W\ar[d]&\bigl( W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W\bigr)\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }\bigl( {S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X\bigr)\,\ar[d]\ar[l]\ar@{^{(}->}[r]&{\ti{W}}^{[n]\tim[1]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\ar[d]\\
{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}&{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X\ar[l]_-{\operatorname{pr}_{1}{^{\vphantom{[n]}} }}\ar@{=}[r]&{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X
}$$ ${X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X$ embeds into the three relative smooth analytic spaces in a compatible way with the two morphisms on the first line. Therefore $\mu _{i,n}{^{\vphantom{[n]}} }=c_{i}{^{\vphantom{[n]}} }\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\ti{{\ensuremath{\mathfrak{Y} }}}_{n}}}\bigr)_{\vert{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }$. We will denote the two relative smooth analytic spaces ${\ti{W}}^{[n+1,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ and ${\ti{W}}^{[n]\tim[1]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ by ${\ensuremath{\mathfrak{X} }}$ and ${\ensuremath{\mathfrak{X}' }}$ respectively.
Let us take a family of charts $\bigl\{\apliso{\phi_{i}{^{\vphantom{[n]}} }}{\Omega _{i}{^{\vphantom{[n]}} }{\ensuremath{\times}}V_{i}{^{\vphantom{[n]}} }}{{\ti{W}}_{i}{^{\vphantom{[n]}} }}\bigr\}$ on ${\ti{W}}$, where the $\Omega _{i}{^{\vphantom{[n]}} }$’s are (possibly non connected) open sets in $\C^{2}$. They provide relative holomorphic charts $\phi _{i}^{[n+1,n]}$ and $\phi _{i}^{[n]\tim[1]}$ of ${\ensuremath{\mathfrak{X} }}$ and ${\ensuremath{\mathfrak{X}' }}$. For each index $i$, we pick a locally free resolution $\sutrgd{{\ensuremath{\mathcal{A}}}_{i}{^{\vphantom{[n]}} }}{{\ensuremath{\mathcal{B}}}_{i}{^{\vphantom{[n]}} }}{\jj_{Y_{n,i}}{^{\vphantom{[n]}} }}$ of length $2$ of $\jj_{Y_{n,i}}{^{\vphantom{[n]}} }$, where $Y_{n,i}{^{\vphantom{[n]}} }$ is the incidence locus in $\Omega _{i}^{[n]}{\ensuremath{\times}}\Omega _{i}{^{\vphantom{[n]}} }$. By the very construction of global smooth resolutions for relatively coherent [sheaves]{} (Proposition \[NouvProp\] (i)), there exists a global resolution $\sutrgd{{\ensuremath{\mathcal{A}}}}{{\ensuremath{\mathcal{B}}}}{\jj_{\ti{{\ensuremath{\mathfrak{Y} }}}_{n}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},\infty }}$ of length $2$ of $J_{\ti{{\ensuremath{\mathfrak{Y} }}}_{n}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},\infty }{\ensuremath{\otimes}}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}}{^{\vphantom{[n]}} }{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{{\ensuremath{\mathfrak{X}' }}}$ such that for all $i$ $\sutrgd{{\ensuremath{\mathcal{A}}}_{i}^{\,\infty }}{{\ensuremath{\mathcal{B}}}_{i}^{\,\infty }}{\jj_{\ti{{\ensuremath{\mathfrak{Y} }}}_{n}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},\infty }}$ is a sub-resolution of the global one. Let $\apl{\pi }{\P(\bbb)}{{\ensuremath{\mathfrak{X}' }}}$ be the projective bundle of $\bbb$ and $s$ the section of $\pi \ee{_{\vphantom{[n]}} }{\ensuremath{\mathcal{A}}}\ee{_{\vphantom{[n]}} }{\ensuremath{\otimes}}_{{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\P(\bbb)}}{^{\vphantom{[n]}} }{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\P(\bbb)}(1)$ obtained by the morphism $\sutr{\pi \ee{_{\vphantom{[n]}} }{\ensuremath{\mathcal{A}}}}{\pi \ee{_{\vphantom{[n]}} }{\ensuremath{\mathcal{B}}}}{{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\P(\bbb)}(1)}$. Then we have the following result:
\[e\] ${^{\vphantom{[n]}} }$
1. The vanishing locus $\textrm{Z}(s)$ of $s$ is canonically isomorphic to ${\ensuremath{\mathfrak{X} }}$.
2. After performing base change from ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$ to $X^{n}{_{\vphantom{[n]}} }{\ensuremath{\times}}X$, the section $s$ is transverse to the zero section.
3. If $j$ is the embedding of ${\ensuremath{\mathfrak{X} }}$ into $\P(\bbb)$, then ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}(-D_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }){\ensuremath{\otimes}}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}}{^{\vphantom{[n]}} }{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{{\ensuremath{\mathfrak{X} }}}\simeq j\ee{_{\vphantom{[n]}} }\, {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\P(\bbb)}(1)$.
\(i) We argue locally on ${\ensuremath{\mathfrak{X}' }}$. In relative coordinates $(z,v)$, the local resolutions of $\jj^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},\infty }_{\ti{\ensuremath{\mathfrak{Y} }}_{n}}$ is of the form $\xymatrix@C=17pt{0\ar[r]&\T^{r}{_{\vphantom{[n]}} }\ar[r]^-{M}&\T^{r+1}{_{\vphantom{[n]}} },}$ where $\T={\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{{\ensuremath{\mathfrak{X} }}}$ and $M$ is a $(r+1){\ensuremath{\times}}r$ matrix with holomorphic coefficients in the variable $z$. We can locally split the injection of ${\ensuremath{\mathcal{A}}}^{\infty }_{i}$ in ${\ensuremath{\mathcal{A}}}$. The global resolution of $\jj^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},\infty }_{\ti{\ensuremath{\mathfrak{Y} }}_{n}}$ is therefore locally isomorphic to $\xymatrix@C=17pt{0\ar[r]
&\T^{r}{_{\vphantom{[n]}} }\oplus\T^{m}{_{\vphantom{[n]}} }\ar[r]^-{\phi }&\T^{r+1}{_{\vphantom{[n]}} }\oplus\T^{m}{_{\vphantom{[n]}} }}$ where $\phi =
\begin{pmatrix}
M&0\\0&\operatorname{id}\end{pmatrix}\cdot
$ Over a small open set of ${\ensuremath{\mathfrak{X}' }}$, $\P(\bbb)$ is the trivial projective bundle with fiber $\P^{r+m}{_{\vphantom{[n]}} }(\C)\ee{_{\vphantom{[n]}} }$ and $s$ is given in coordinates by $s(u,z,v)(\alpha ,\beta )=u\bigl( M(z)\alpha ,\beta \bigr)$, where $u\in\P^{r+m}{_{\vphantom{[n]}} }(\C)\ee{_{\vphantom{[n]}} }$ and $(\alpha ,\beta )\in\C^{r+m}$ are the coordinates in the fibers of $\pi \ee{_{\vphantom{[n]}} }\aaa$. Therefore, $$\textrm{Z}(s)=\bigl\{(u,z,v)\ \textrm{such that}\ u_{\vert\C^{m}}{^{\vphantom{[n]}} }=0\ \textrm{and for all}\ \alpha \in \C^{r}{_{\vphantom{[n]}} },\ u\bigl( M(z)\alpha \bigr)=0\bigr\}\cdot$$ This implies that if we consider the local embedding $\xymatrix{\P({\ensuremath{\mathcal{B}}}_{i}^{\infty} )\,\ar@^{{(}->}[r]&\P(\bbb)}$ over ${\ensuremath{\mathfrak{X}' }}$ given by the splitting of the local resolution in the global one, then $\textrm{Z}(s)$ lies in $\P(\bbb_{i}^{\infty })$. It can be seen straightforwardly that the embedding of $\textrm{Z}(s)$ into $\P(\bbb_{i}^{\infty })$ is independant of the splitting. Let $\apl{\ti\pi }{\P(\bbb_{i}{^{\vphantom{[n]}} })}{\Omega _{i}^{[n]}\tim\Omega _{i}{^{\vphantom{[n]}} }}$ be the projective bundle of $\bbb_{i}{^{\vphantom{[n]}} }$ and $\ti{s}$ the section of $\ti\pi \ee{_{\vphantom{[n]}} }\aaa_{i}\ee(1)$ given by the morphism $\sutr{\ti{\pi }\ee{_{\vphantom{[n]}} }\aaa_{i}{^{\vphantom{[n]}} }}{\ti{\pi }\ee{_{\vphantom{[n]}} }\bbb_{i}{^{\vphantom{[n]}} }}{\oo_{\P(\bbb_{i}{^{\vphantom{[n]}} })}(1).}$ Then $\textrm{Z}(\ti{s})=\Omega _{i}^{[n+1,n]}$ and $\ti{s}$ is transverse to the zero section. This proves that over $\Omega _{i}^{[n]}\tim\Omega _{i}{^{\vphantom{[n]}} }{\ensuremath{\times}}V$, $\textrm{Z}(s)=\textrm{Z}(\ti{s}){\ensuremath{\times}}V=\Omega _{i}^{[n+1,n]}{\ensuremath{\times}}V$ so that $\textrm{Z}(s)$ is abstractly isomorphic over ${\ti{W}}_{i}^{[n]\tim[1]}$ to ${\ti{W}}_{i}^{[n+1,n]}$.
\(ii) In local coordinates, if for any $u\in\P^{r+m}{_{\vphantom{[n]}} }(\C)\ee{_{\vphantom{[n]}} }$ we define $u_{1}{^{\vphantom{[n]}} }=u_{\vert\C^{r+1}}{^{\vphantom{[n]}} }$ and $u_{2}=u_{\vert\C^{m}}{^{\vphantom{[n]}} }$, then $s$ is given near its zero locus by $\apl{s(u,z,v)}{(\alpha ,\beta )}{\bigl( \ti{s}(u_{1}{^{\vphantom{[n]}} },z)(\alpha ),u_{2}{^{\vphantom{[n]}} }(\beta )\bigr)}$. After base change, the variable $v$ lies in the smooth manifold $X^{n}{_{\vphantom{[n]}} }{\ensuremath{\times}}X$ and $s$ is clearly transverse to the zero section since $\ti{s}$ is.
\(iii) We have a chain of morphisms $\xymatrix{{\ti{W}}_{i}^{[n+1,n]}\,\ar@{^{(}->}[r]&\P(\bbb_{i}^{\,\infty })\,\ar@{^{(}.>}[r]&\P(\bbb)_{\vert {\ti{W}}_{i}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }}$ where the last morphism is only defined locally on $\P(\bbb^{\infty }_{i})$. To conclude, notice that ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\P(\bbb_{i}^{\infty })}}(1)$ is canonically isomorphic to ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigl( -D_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\bigr)_{\vert{\ti{W}}_{i}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }$ and that the restriction of ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\P(\bbb)_{\vert{\ti{W}}_{i}}}$ to $\P(\bbb_{i}^{\,\infty })$ is canonically isomorphic to ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\P(\bbb_{i}^{\,\infty })}(1)$.
Since $\P(\bbb)=\bigl[\P(\bbb)\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }(X^{n}{_{\vphantom{[n]}} }{\ensuremath{\times}}X)\bigr]/{\ensuremath{\mathfrak{S}}}_{n}{^{\vphantom{[n]}} }$, Lemma \[e\] (ii) implies that the homology class of the vanishing locus of $s$ in $H_{8n+8}{^{\vphantom{[n]}} }(\P(\bbb),\Q)$ is Poincaré dual to the top Chern class of $\pi \ee{_{\vphantom{[n]}} }\aaa\ee{_{\vphantom{[n]}} }{\ensuremath{\otimes}}_{{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\P(\bbb)}}{^{\vphantom{[n]}} }{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\P(\bbb)}(1)$. Let us consider the following diagram: $$\xymatrix{
\P(\bbb)\ar[d]_-{\pi }&{\ensuremath{\mathfrak{X} }}\supseteq D_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }\ar[l]_(.54){j}\ar[dl]^{\sigma _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}\\
{\ensuremath{\mathfrak{X}' }}}$$ If $\varepsilon =c_{1}{^{\vphantom{[n]}} }\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\P(\bbb)}(1)\bigr)\in H^{2}{_{\vphantom{[n]}} }(\P(\bbb),\Q)$, then by Lemma \[e\] (iii), $l=j\ee{_{\vphantom{[n]}} }\varepsilon _{\vert X^{[n+1,n]}{_{\vphantom{[n]}} }}{^{\vphantom{[n]}} }$. Now $$\begin{aligned}
\sigma _{\!{\ensuremath{\, \operatorname{\vphantom{y}rel}}}*}{^{\vphantom{[n]}} }(j\ee{_{\vphantom{[n]}} }\varepsilon ^{i}{_{\vphantom{[n]}} })&=\pi _{*}{^{\vphantom{[n]}} }([{\ensuremath{\mathfrak{X} }}]\,.\,\varepsilon ^{i}{_{\vphantom{[n]}} })=\sum_{k=0}^{d}c_{k}{^{\vphantom{[n]}} }(\aaa\ee{_{\vphantom{[n]}} })\,\pi _{*}{^{\vphantom{[n]}} }(\varepsilon ^{d+i-k}{_{\vphantom{[n]}} })=\sum_{k=0}^{d}c_{k}{^{\vphantom{[n]}} }(\aaa\ee{_{\vphantom{[n]}} })\,s_{i-k}{^{\vphantom{[n]}} }(\bbb\ee{_{\vphantom{[n]}} })\\
&=c_{i}{^{\vphantom{[n]}} }(\aaa\ee{_{\vphantom{[n]}} }-\bbb\ee{_{\vphantom{[n]}} })=(-1)^{i}c_{i}\bigl( \jj_{\ti{{\ensuremath{\mathfrak{Y} }}}_{n}{^{\vphantom{[n]}} }}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},\,\infty }\bigr)=(-1)^{i}c_{i}{^{\vphantom{[n]}} }\bigl( \oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},\,\infty }_{\ti{{\ensuremath{\mathfrak{Y} }}}_{n}{^{\vphantom{[n]}} }}\bigr)\qquad\textrm{since}\ i\ge 1.\end{aligned}$$ We have used here the Gysin morphism $\sigma _{\!{\ensuremath{\, \operatorname{\vphantom{y}rel}}}*}{^{\vphantom{[n]}} }$ with , which is possible since ${\ensuremath{\mathfrak{X} }}$ and ${\ensuremath{\mathfrak{X}' }}$ are rationally smooth. To conclude, we consider the diagram $$\xymatrix@C=40pt@R=20pt{
{\ensuremath{\mathfrak{X} }}\ar[d]_-{\sigma _{\!{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}&\,X^{[n+1,n]}{_{\vphantom{[n]}} }\ar[d]^-{\sigma }\ar@{_{(}->}[l]\\
{\ensuremath{\mathfrak{X}' }}&\,{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X\ar@{_{(}->}[l]
}$$ and get: $$\sigma _{*}{^{\vphantom{[n]}} }\, l^{i}=\sigma _{*}{^{\vphantom{[n]}} }(j\ee{_{\vphantom{[n]}} }\varepsilon ^{i}{_{\vphantom{[n]}} })_{\vert {X^{[n+1,n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }=\bigl[ \sigma _{\!{\ensuremath{\, \operatorname{\vphantom{y}rel}}}*}{^{\vphantom{[n]}} }(j\ee{_{\vphantom{[n]}} }\varepsilon ^{i}{_{\vphantom{[n]}} })\bigr]_{\vert{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }=(-1)^{i}c_{i}\bigl( \oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},\infty }_{\ti{{\ensuremath{\mathfrak{Y} }}}_{n}{^{\vphantom{[n]}} }}\bigr)_{\vert{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }=(-1)^{i}\mu _{i,n}{^{\vphantom{[n]}} }.$$
Computation of $TX^{[n]}$ in {#m}
-----------------------------
Let $W$ be a [neighbourhood]{} of the incidence locus $Z_{n}{^{\vphantom{[n]}} }$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$ and $J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}$ a relative integrable structure on it.
\[g\] We define ${\kappa }_{n\vphantom{1}}{^{\vphantom{[n]}} }$ as the class of the locally [sheaf]{} $T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ in ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}\bigl( W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\bigr)$.
The topological class of $\kappa _{n\vphantom{1}}{^{\vphantom{[n]}} }$ can be described as follows:
\[k\] The restriction to ${X^{[n]}{_{\vphantom{[n]}} }}$ of the topological class associated to $\kappa _{n\vphantom{1}}{^{\vphantom{[n]}} }$ is the class of the complex vector bundle $T{X^{[n]}{_{\vphantom{[n]}} }}$ in $K({X^{[n]}{_{\vphantom{[n]}} }})$.
If $J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{!}n}$ satisfies the conditions $(\mathscr{C})$ listed in [@SchHilVo1] page 711, then $X^{[n]}_{J_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}$ is smooth. Besides, the construction of the almost-complex structure done in [@SchHilVo1] shows that $T{X^{[n]}{_{\vphantom{[n]}} }}$ and $T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{}_{\vert {X^{[n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }$ have the same class in $K({X^{[n]}{_{\vphantom{[n]}} }})$. An arbitrary $J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{!}n}$ can be joined by a smooth path $\bigl\{J_{n,\,t}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\bigr\}_{t\in[0,1]}{^{\vphantom{[n]}} }$ to another relative integrable structure satisfying the conditions $(\mathscr{C})$. By rigidity of the topological , the class of $T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\bigl[ W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n,\,t}\bigr]_{\vert {X^{[n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }$ in $K\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr)$ is independent of $t$. This yields the result.
\[h\] For an arbitrary $J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{!}n}$, $X^{[n]}_{J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}$ is only a topological manifold (see [@SchHilGri]). Therefore, the advantage of using $T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ instead of $TX^{[n]}_{J_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}$ is that this complex vector bundle is defined for *any* relative integrable complex structure.
\[CinqNeuf\] In $K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\bigl( W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\bigr)$, the following identity holds: $$\kappa _{n\vphantom{1}}{^{\vphantom{[n]}} }=p_{*}{^{\vphantom{[n]}} }\bigl( \oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}+\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}-\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, .\, \oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\bigr).$$
We have $TW^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}=p_{*}{^{\vphantom{[n]}} }\,{\ensuremath{\mathcal{H}om}}\bigl( \jj_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)$. Since $\operatorname{supp}\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)$ has relative codimension $2$ in $W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W$, ${{\ensuremath{\mathcal{E}}}xt}^{i}{_{\vphantom{[n]}} }\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{n}}}\bigr)=0$ for $i<2$. Besides, $\jj_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ locally admits a free resolution of length $2$. Hence we get the following equalities in ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Y} }}_{n}{^{\vphantom{[n]}} }}}\bigl( W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }W\bigr)$: $$\begin{aligned}
\jj^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}_{n}\,.\,{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}&={\ensuremath{\mathcal{H}om}}\bigl( \jj_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)-{{\ensuremath{\mathcal{E}}}xt}^{1}{_{\vphantom{[n]}} }\bigl( \jj_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)+{{\ensuremath{\mathcal{E}}}xt}^{2}{_{\vphantom{[n]}} }\bigl( \jj_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)\\
&={\ensuremath{\mathcal{H}om}}\bigl( \jj_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)-{{\ensuremath{\mathcal{E}}}xt}^{2}{_{\vphantom{[n]}} }\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)\\
&={\ensuremath{\mathcal{H}om}}\bigl( \jj_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)-{{\ensuremath{\mathcal{E}}}xt}^{2}{_{\vphantom{[n]}} }\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{n}}}\bigr)\\
&={\ensuremath{\mathcal{H}om}}\bigl( \jj_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)-\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\end{aligned}$$ so that $p_{*}{^{\vphantom{[n]}} }{\ensuremath{\mathcal{H}om}}\bigl( \jj_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)=p_{*}{^{\vphantom{[n]}} }\bigl[\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{n}}}-\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\bigr)\,.\,{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}+\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee} \bigr]$
Comparison of $TX^{[n]}$ and $TX^{[n+1]}$ via the incidence variety $X^{[n+1,n]}$ {#Comparaison}
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Let ${W}$ be a [neighbourhood]{} of $Z_{n{\ensuremath{\times}}1}{^{\vphantom{[n]}} }$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X{\ensuremath{\times}}X$ and $J_{n{\ensuremath{\times}}1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ be a relative integrable complex structure on ${W}$. Let us consider the following morphisms over ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$:
1. $\apl{\rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}{{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}}{{W}}$ is the residual map.
2. $\apl{j_{\operatorname{\vphantom{y}rel}}{^{\vphantom{[n]}} }=\bigl( \operatorname{id},\rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }\bigr)}{{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}}{{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }{W}.}$
3. $p$ is the first projection $\xymatrix@C=17pt{
{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }{W}\ar[r]&{{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}}
.}$
4. If $\apl{f}{{\ensuremath{\mathfrak{X} }}}{{\ensuremath{\mathfrak{X}' }}}$ is a morphism over ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$, we define $$\apl{f_{{W}}{^{\vphantom{[n]}} }:=\mkern -2 mu f\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }\operatorname{id}_{W}{^{\vphantom{[n]}} }}{{\ensuremath{\mathfrak{X} }}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }{W}}{{\ensuremath{\mathfrak{X} }}'\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }{W}.}$$
5. $\xymatrix{{\psi }:{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\ar[r]&{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1]}}$ and $\xymatrix{{\phi }:{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\ar[r]&{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}}$ are the canonical projection maps.
6. $\xymatrix{{\sigma_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}=\bigl( {\phi },\rho_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }\bigr):W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\ar[r]&{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{W}.}$
We introduce the following relatively coherent sheaves:
1. $\ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ and $\ti{\oo}_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ are the relative incidence structure sheaves on ${W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W$ and ${W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W$.
2. $\LL=\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }(-D_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} })$, where $D_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }\suq{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}$ is the relative exceptional divisor.
3. $\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }$ is the relative diagonal in ${W}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }{W}$ and $\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}$ is the associated structure sheaf.
Then we have the following properties which are immediate consequences of the same results in the integrable case:
\[f\] ${^{\vphantom{[n]}} }$
1. On ${W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }{W}$ we have $j_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}*}{^{\vphantom{[n]}} }{\ensuremath{\mathcal{L}}}=p\ee{_{\vphantom{[n]}} }\LL{\ensuremath{\otimes}}\rho \ee_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\oo_{\Delta _{\operatorname{\vphantom{y}rel}}}^{\operatorname{\vphantom{y}rel}}$.\[EqRef\]
2. We have an exact sequence on ${W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }{W}$*:* $$\sutrgdpt{j_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}*}\LL{^{\vphantom{[n]}} }}{{\psi }_{W}\ee\, \ti\oo_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}{{\phi }_{W}\ee\,{\ensuremath{\widetilde{\oo}}}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}{.}$$
3. $\rho \ee_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}=\rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$,${{\psi }_{W}\ee\,{\ensuremath{\widetilde{\oo}}}_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}={{\psi }_{W}{\ensuremath{{\ensuremath{^{\, !}}}}}\,{\ensuremath{\widetilde{\oo}}}_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}$and ${{\phi }_{W}\ee\,{\ensuremath{\widetilde{\oo}}}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}={{\phi }_{W}{\ensuremath{{\ensuremath{^{\, !}}}}}\,{\ensuremath{\widetilde{\oo}}}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}.}$
We suppose that there exist a [neighbourhood]{} ${\ti{W}}$ of $Z_{n+1}{^{\vphantom{[n]}} }$ in $S^{n+1}{_{\vphantom{[n]}} }X{\ensuremath{\times}}X$ and a relative integrable complex structure $J_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ on it such that $W={\ti{W}}\tim_{S^{n+1}{_{\vphantom{[n]}} }X}{^{\vphantom{[n]}} }({S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X)$. This means that for all $({{\underline{\vphantom{!}\vphantom{y}x}}},p)$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$, $W_{{{\underline{\vphantom{!}\vphantom{y}x}}},\,p}{^{\vphantom{[n]}} }={\ti{W}}_{{{\underline{\vphantom{!}\vphantom{y}x}}}\, \cup\, p}{^{\vphantom{[n]}} }$ and $J_{n{\ensuremath{\times}}1,\, {{\underline{\vphantom{!}\vphantom{y}x}}},\,p}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}=J_{n+ 1,\, {{\underline{\vphantom{!}\vphantom{y}x}}}\, \cup\, p}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$. Then we get two weak morphisms $$\apl{\ti\psi }{{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}}{{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1]}}\ \textrm{and}\quad\apl{\ti\psi_{W}{^{\vphantom{[n]}} }}{{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }{W}}{{W}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1]}\tim_{S^{n+1}{_{\vphantom{[n]}} }X}{^{\vphantom{[n]}} }{W}}$$ (see Definition \[DefDeuxAppenCh4\] (ii)) obtained by composing $\psi $ and $\psi _{W}{^{\vphantom{[n]}} }$ with the base change map from $S^{n+1}{_{\vphantom{[n]}} }X$ to ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$ given by $\flgdba{({{\underline{\vphantom{!}\vphantom{y}x}}},p)}{\,{{\underline{\vphantom{!}\vphantom{y}x}}}\,\cup\,p}$. Then ${\ti{\psi }_{W}\ee\,{\ensuremath{\widetilde{\oo}}}_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}={{\psi }_{W}\ee\,\oo_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}$. Let $\ti{\kappa }_{n\vphantom{1}}{^{\vphantom{[n]}} }$ be the class of $T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }{W}^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ in $K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\bigl( {W}^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\bigr)$.
\[SansLabBis\] In $K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\bigl( {\ti{W}}^{[n+1,\, n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\bigr)$ we have $$\begin{aligned}
\ti\psi {\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\kappa _{n+1}{^{\vphantom{[n]}} }={\phi }{\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\ti{\kappa } _{n\vphantom{1}}{^{\vphantom{[n]}} }+\LL+\LL^{\vee}{_{\vphantom{[n]}} }.\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}
K_{W}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}-&\rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}
\bigl( \oo_{W}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}-T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }W+K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}_{W}\bigr)\\&-\LL\, .\, {\sigma_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\pe} {_{\vphantom{[n]}} }\ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}-\LL^{\vee}{_{\vphantom{[n]}} }.\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}K_{W}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\, .\, {\sigma_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\pee} {_{\vphantom{[n]}} }\ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}.\end{aligned}$$
Let $\apl{\ti{p}}{{\ti{W}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1]}\tim_{S^{\, n+1}{_{\vphantom{[n]}} }X}{^{\vphantom{[n]}} }{\ti{W}}}{{\ti{W}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1]}}$ be the projection on the first factor. By Proposition \[CinqNeuf\], $\ti\psi {\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\kappa _{n+1}{^{\vphantom{[n]}} }=\ti\psi {\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }{\ensuremath{\widetilde{p}}}_{*}{^{\vphantom{[n]}} }\bigl( \oo_{n+1 }^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}+\oo_{n+1 }^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}-\oo_{n+1 }^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, .\, \oo_{n+1 }^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\bigr)$. Let us consider the cartesian diagram $$\xymatrix@C=45pt{W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}
\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W\ar[r]^(.53){\ti\psi _{W}{^{\vphantom{[n]}} }}\ar[d]_{{p}}&{\ti{W}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1]}\tim_{S^{\, n+1}{_{\vphantom{[n]}} }X}{^{\vphantom{[n]}} }{\ti{W}}\ar[d]^{{\ensuremath{\widetilde{p}}}}\\
W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\ar[r]^{\ti\psi }&{\ti{W}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1]}
}$$ where the first column lies over ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X $ and the second one over $S^{\, n+1}{_{\vphantom{[n]}} }X$. Since ${\ensuremath{\widetilde{p}}}$ is finite on $\operatorname{supp}\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n+1}}\bigr)$, we obtain by Proposition \[PropUnBisAppendCh4\] (iv) and Lemma \[f\] (i) and (ii) the following relations in ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}\bigl( W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,n]}\bigr)$: $$\begin{aligned}
\ti\psi {\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\kappa _{n+1}{^{\vphantom{[n]}} }&={p}_{*}{^{\vphantom{[n]}} }\ti\psi {\ensuremath{{\ensuremath{^{\, !}}}}}_{W}\bigl( \oo_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}+\oo_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}-\oo_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, .\, \oo_{n+1}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\bigr)
={p}_{*}{^{\vphantom{[n]}} }{\phi}{\ensuremath{{\ensuremath{^{\, !}}}}}_{W}\bigl( \ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}+\ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}-\ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, .\, \ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\bigr)\\
&\hphantom{=}+{p}_{*}{^{\vphantom{[n]}} }\Bigl[{p}{\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\LL\, .\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\, \oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}+
{p}{\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\LL^{\vee}{_{\vphantom{[n]}} }.\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\, \oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}_{\Delta_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}-
{p}{\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\bigl( \LL\, .\, \LL^{\vee}{_{\vphantom{[n]}} }\bigr)\, .\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\bigl( \oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}.\, \oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}\bigr) \\
&\hphantom{=+\ti{p}_{*}{^{\vphantom{[n]}} }\Bigl[{p}{\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\LL\, .\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\, \oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}}-{p}{\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\LL\, .\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}\, .\, {\phi }_{W}{\ensuremath{{\ensuremath{^{\, !}}}}}\, \ti\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}-{p}{\ensuremath{{\ensuremath{^{\, !}}}}}_{{_{\vphantom{[n]}} }}\LL^{\vee}.\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\, \oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}\, .\,\, {\phi }_{W}{\ensuremath{{\ensuremath{^{\, !}}}}}\, \ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}
\Bigr].\end{aligned}$$
If $\apl{q}{W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W}{W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}$ is the projection on the first factor, then by Proposition \[CinqNeuf\], $\ti{\kappa }_{n\vphantom{1}}{^{\vphantom{[n]}} }=q_{*}{^{\vphantom{[n]}} }\bigl( \ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}+\ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}-\ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, .\, \ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\bigr)$ so that, since $q$ is finite on $\operatorname{supp}\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n}}\bigr)$, Proposition \[PropUnBisAppendCh4\] (iv) yields: ${\phi }{\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\ti{\kappa } _{n\vphantom{1}}{^{\vphantom{[n]}} }={p}_{*}{^{\vphantom{[n]}} }{\phi }{\ensuremath{{\ensuremath{^{\, !}}}}}_{W}\bigl( \ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}+\ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}-\ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, .\, \ti\oo_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\bigr)$. We also consider the diagram $$\xymatrix@C=45pt{W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}
\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W\ar[r]^(.53){\rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}{^{\vphantom{[n]}} }}\ar[d]_{{p}}&W\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W\ar[d]^{r}\\
W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\ar[r]^{\rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}&W
}$$where all the terms are over ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$. Since $r$ is injective on $\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }$, by Proposition \[PropUnBisAppendCh4\] (iv) again, $$\begin{aligned}
\ti\psi {\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\kappa _{n+1}{^{\vphantom{[n]}} }&={\phi }{\ensuremath{{\ensuremath{^{\, !}}}}}\ti{\kappa}_{n\vphantom{1}}{^{\vphantom{[n]}} }+\LL\, .\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}r_{*}{^{\vphantom{[n]}} }\oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}+\LL^{\vee}{_{\vphantom{[n]}} }.\, \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}r_{*}{^{\vphantom{[n]}} }\oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}-\rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}r_{*}{^{\vphantom{[n]}} }\bigl(\oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\,.\, \oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee} \bigr)\\
&-\LL\, .\, {p}_{*}{^{\vphantom{[n]}} }\bigl( j_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}*}{^{\vphantom{[n]}} }\oo_{W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, .\, {\phi }_{W}{\ensuremath{{\ensuremath{^{\, !}}}}}\, \ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}\bigr)
-\LL^{\vee}{_{\vphantom{[n]}} }.\, {p}_{*}{^{\vphantom{[n]}} }\Bigl[ \bigl( \rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}{\ensuremath{{\ensuremath{^{\, !}}}}}\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}\bigr)^{\vee}{_{\vphantom{[n]}} }\, .\, {\phi }{\ensuremath{{\ensuremath{^{\, !}}}}}_{W}\, \ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\Bigr].\end{aligned}$$ Now $r_{*}{^{\vphantom{[n]}} }\, \oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}=\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{W}$, $r_{*}{^{\vphantom{[n]}} }\, \oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}=K_{W}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}$ and if $\apl{\delta }{W}{W\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W}$ is the diagonal injection, $\rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\, \oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}=\rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\, \delta _{*}{^{\vphantom{[n]}} }\, K_{W}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}=j_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}*}{^{\vphantom{[n]}} }\rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, K_{W}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}\vee}$, thanks to Proposition \[PropUnBisAppendCh4\] (v) and to the diagram $$\xymatrix@C=45pt{
W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\ar[r]^(.4){j_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}\ar[d]^{\rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}&W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W\ar[d]^{\rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}{^{\vphantom{[n]}} }}\\
W\ar[r]^(0.4){\delta }&W\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W}$$
Cohomological computations {#qwe}
--------------------------
We are now going to consider the identity we obtained in Section \[Comparaison\] from a cohomological point of view. Recall that the classes $\mu _{i,n}{^{\vphantom{[n]}} }$ in $H^{2i}{_{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X,\Q\bigr)$ are defined by $\mu _{i,n}{^{\vphantom{[n]}} }=c_{i}{^{\vphantom{[n]}} }\bigl(\oo_{n}^{\, \operatorname{\vphantom{y}rel}}\bigr)_{\vert{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }$. Let us consider the following maps: $$\xymatrix@R=8pt{
&X&\\
&&\\
&{X^{[n+1,\, n]}{_{\vphantom{[n]}} }}\ar[uu]_-{\rho }\ar[dl]_-{\lambda }\ar[dr]^-{\nu }\ar[dd]^-{\sigma }\\
{X^{[n]}{_{\vphantom{[n]}} }}&&{X^{[n+1]}{_{\vphantom{[n]}} }}\\
&{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X&
}$$
\[EtToc\] ${^{\vphantom{[n]}} }$
1. $c_{i}{^{\vphantom{[n]}} }\bigl( {\phi }{\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\ti{\kappa}_{n\vphantom{1}}{^{\vphantom{[n]}} }\bigr)=\lambda \ee{_{\vphantom{[n]}} }c_{i}{^{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr)$.
2. $c_{i}{^{\vphantom{[n]}} }\bigl( {\phi }_{W}{\ensuremath{{\ensuremath{^{\, !}}}}}\, \ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\bigr)=(\lambda ,\operatorname{id})\ee{_{\vphantom{[n]}} }\mu _{i,n}{^{\vphantom{[n]}} }$.
3. $c_{i}{^{\vphantom{[n]}} }\bigl( {\sigma }\pee_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, \ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\bigr)_{\vert \, {X^{[n+1,\, n]}{_{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }=\sigma \ee{_{\vphantom{[n]}} }\mu _{i,n }{^{\vphantom{[n]}} }$.
4. $c_{i}{^{\vphantom{[n]}} }\bigl( \rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\ee\, T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }W\bigr)_{\vert\, {X^{[n+1,\, n]}{_{\vphantom{[n]}} }}}=\rho \ee{_{\vphantom{[n]}} }c_{i}{^{\vphantom{[n]}} }(X)$.
5. $c_{i}{^{\vphantom{[n]}} }\bigl( \rho {\ensuremath{{\ensuremath{^{\, !}}}}}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}},W}\, \oo_{\Delta _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{^{\vphantom{[n]}} }}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\bigr)_{\vert\, {X^{[n+1,\, n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }=(\rho ,\operatorname{id}){\ensuremath{^{\, *}}}c_{i}{^{\vphantom{[n]}} }\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\Delta _{X}{^{\vphantom{[n]}} }}\bigr)$.
By Lemma \[Compl\] (ii), the classes ${\phi }{\ensuremath{{\ensuremath{^{\, !}}}}}{_{\vphantom{[n]}} }\ti{\kappa}_{n\vphantom{1}}{^{\vphantom{[n]}} }$, $ {\phi }_{W}{\ensuremath{{\ensuremath{^{\, !}}}}}\, \ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ and ${\sigma }\pee_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\, \ti{\oo}_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ in topological are independent of the relative integrable structure $J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n{\ensuremath{\times}}1}$. Therefore, we can suppose that there exist a [neighbourhood]{} $\breve{W}$ of $Z_{n}{^{\vphantom{[n]}} }$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$ and a relative integrable structure $J_{n}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ on it such that $\breve{W}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }({S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X){\ensuremath{\subseteq}}W$. This means that for $({{\underline{\vphantom{!}\vphantom{y}x}}},p)$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$, $\breve{W}_{{{\underline{\vphantom{!}\vphantom{y}x}}}}{^{\vphantom{[n]}} }{\ensuremath{\subseteq}}W_{{{\underline{\vphantom{!}\vphantom{y}x}}},\,p}{^{\vphantom{[n]}} }$ and $J_{n{\ensuremath{\times}}1,\,{{\underline{\vphantom{!}\vphantom{y}x}}},\,p}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{}_{\vert \breve{W}_{{{\underline{\vphantom{!}\vphantom{y}x}}}}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }=J^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{n,\,{{\underline{\vphantom{!}\vphantom{y}x}}}}$. Then (i), (ii) and (iii) are straightforward.
For (iv), let $\widehat{W}=W\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}{^{\vphantom{[n]}} }X$, where the base change morphism is given by the diagonal injection of $X$ in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$. We consider the following diagram: $$\xymatrix@C=45pt{W_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}^{[n+1,\, n]}\ar[r]^-{\rho _{\operatorname{\vphantom{y}rel}}{^{\vphantom{[n]}} }}&W&\,\widehat{W}\ar@{_{(}->}[l]\\
{X^{[n+1,\, n]}{_{\vphantom{[n]}} }}\ar@{^{(}->}[u]\ar[r]_-{\rho }&X{^{\vphantom{[n]}} }\ar@{^{(}->}[u]\ar@{=}[r]&X{^{\vphantom{[n]}} }\ar@{^{(}->}[u]
}$$ Then, $c_{i}{^{\vphantom{[n]}} }\bigl( \rho _{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\ee\, T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }W\bigr)_{\vert\, {X^{[n+1,\, n]}{_{\vphantom{[n]}} }}}=\rho \ee{_{\vphantom{[n]}} }c_{i}{^{\vphantom{[n]}} }\bigl( T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\widehat{W}\bigr)_{\vert X}{^{\vphantom{[n]}} }$. Since $\widehat{W}$ is a [neighbourhood]{} of $\Delta _{X}{^{\vphantom{[n]}} }$ in $X{\ensuremath{\times}}X$, $T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\widehat{W}_{\vert X}{^{\vphantom{[n]}} }\simeq TX$, so that $c_{i}{^{\vphantom{[n]}} }\bigl( T^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }\widehat{W}\bigr)_{\vert X}{^{\vphantom{[n]}} }=c_{i}{^{\vphantom{[n]}} }(X)$.
For (v), we extend the structure $J_{n{\ensuremath{\times}}1}^{\, \operatorname{\vphantom{y}rel}}$ to a relative integrable structure $J_{n{\ensuremath{\times}}1{\ensuremath{\times}}1}^{\, \operatorname{\vphantom{y}rel}}$ in a [neighbourhood]{} ${\ensuremath{\overline{W}}}$ of $Z_{n{\ensuremath{\times}}1{\ensuremath{\times}}1}{^{\vphantom{[n]}} }$ such that for any $({{\underline{\vphantom{!}\vphantom{y}x}}},p,q)\in {S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X{\ensuremath{\times}}X $ near the incidence locus $p=q$, the equality $J_{n{\ensuremath{\times}}1{\ensuremath{\times}}1,\,{{\underline{\vphantom{!}\vphantom{y}x}}},\,p,\,q}^{\, \operatorname{\vphantom{y}rel}}=J_{n{\ensuremath{\times}}1,\,{{\underline{\vphantom{!}\vphantom{y}x}}},\,p}^{\, \operatorname{\vphantom{y}rel}}$ holds. This means that $W\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }({S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X{\ensuremath{\times}}X)\suq{\ensuremath{\overline{W}}}$. We define $\widehat{W}$ by $\widehat{W}={\ensuremath{\overline{W}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X{\ensuremath{\times}}X}{^{\vphantom{[n]}} }(X{\ensuremath{\times}}X)$, the base change morphism from ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X{\ensuremath{\times}}X$ to $X{\ensuremath{\times}}X$ being given by $\flgdba{(x,x')}{(\delta (x), x,x')}$, where $\apl{\delta }{X}{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}}$ is the diagonal injection. Then we consider the diagram: $$\xymatrix@C=50pt@R=25pt{
W^{[n+1,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W\ar[r]^-{\rho _{\operatorname{\vphantom{y}rel},\,W}{^{\vphantom{[n]}} }}\ar@{^{(}->}[d]&W\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X}{^{\vphantom{[n]}} }W\ar@{^{(}->}[d]\\
{\ensuremath{\overline{W}}}^{[n+1,n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X{\ensuremath{\times}}X}{\ensuremath{\overline{W}}}\ar[r]^-{\rho _{\operatorname{\vphantom{y}rel},\,{\ensuremath{\overline{W}}}}{^{\vphantom{[n]}} }}&{\ensuremath{\overline{W}}}\tim_{{S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X{\ensuremath{\times}}X}{^{\vphantom{[n]}} }{\ensuremath{\overline{W}}}&\,\widehat{W}\tim_{X{\ensuremath{\times}}X}{^{\vphantom{[n]}} }\widehat{W}\ar@{_{(}->}[l]\\
{X^{[n+1,n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X{^{\vphantom{[n]}} }\ar[r]_-{(\rho ,\,\operatorname{id})}\ar@{^{(}->}[u]&X{\ensuremath{\times}}X{^{\vphantom{[n]}} }\ar@{^{(}->}[u]&X{\ensuremath{\times}}X{^{\vphantom{[n]}} }\ar@{=}[l]\ar@{^{(}->}[u]_(.45){\iota}
}$$ If $\Delta _{\widehat{W}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ is the relative diagonal in $\widehat{W}\tim_{X{\ensuremath{\times}}X}{^{\vphantom{[n]}} }\widehat{W}$, then $\iota \ee{_{\vphantom{[n]}} }\bigl[ {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\Delta _{\widehat{W}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}}\bigr]=\bigl[ {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\Delta _{X}{^{\vphantom{[n]}} }}\bigr]$ in $K(X{\ensuremath{\times}}X)$. This gives the result.
Let $d_{i}{^{\vphantom{[n]}} }$ be the Chern class of ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{\Delta _{X}{^{\vphantom{[n]}} }}$ in $H^{2i}{_{\vphantom{[n]}} }(X{\ensuremath{\times}}X,\Q)$. By [@SchHilAtHi], $d_{i}$ is a universal polynomial in $c_{1}{^{\vphantom{[n]}} }(X)$ and $c_{2}{^{\vphantom{[n]}} }(X)$ with rational coefficients.
\[LaDer\] ${^{\vphantom{[n]}} }$
*(i)* For all $i,n$ in $\N\ee{_{\vphantom{[n]}} }$, $(\nu ,\operatorname{id})\ee{_{\vphantom{[n]}} }\mu _{i,n+1}{^{\vphantom{[n]}} }-(\lambda ,\operatorname{id})\ee{_{\vphantom{[n]}} }\mu _{i,n}{^{\vphantom{[n]}} }=\ds\sum_{k=0}^{i}\operatorname{pr}\ee_{1}\, l^{\, k}{_{\vphantom{[n]}} }\, .\, (\rho ,\operatorname{id})\ee{_{\vphantom{[n]}} }\, d_{i-k}$, where $\mu _{i,n}{^{\vphantom{[n]}} }$ and $l$ are defined in \[c\] and \[l\].
*(ii)* For all $i,n$ in $\N\ee{_{\vphantom{[n]}} }$, $\nu \ee{_{\vphantom{[n]}} }c_{i}{^{\vphantom{[n]}} }\bigl( {X^{[n+1]}{_{\vphantom{[n]}} }}\bigr)-\lambda \ee{_{\vphantom{[n]}} }c_{i}{^{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr)$ is a universal polynomial in the classes $l$, $\rho \ee{_{\vphantom{[n]}} }c_{i}{^{\vphantom{[n]}} }(X)$ and $\sigma \ee{_{\vphantom{[n]}} }\mu _{i,n}{^{\vphantom{[n]}} }$.
This is a consequence of Lemma \[EtToc\], Lemma \[f\] (i) and (ii), and of Proposition \[SansLabBis\].
We are now going to perform the induction step.
\[11\] If $n,m$ are positive integers, let $P$ be a polynomial in $
\operatorname{pr}_{0}{\ensuremath{^{\, *}}}c_{i}{^{\vphantom{[n]}} }\bigl( {X^{[n+1]}{_{\vphantom{[n]}} }}\bigr)$, $\operatorname{pr}\ee_{0k}\, \mu _{i,n+1}{^{\vphantom{[n]}} }$, $\operatorname{pr}\ee_{kl}\, d_{i}{^{\vphantom{[n]}} }$, $\operatorname{pr}_{k}{\ensuremath{^{\, *}}}c_{i{^{\vphantom{[n]}} }}(X)$ $ (1\le k,l\le m)$, which are cohomology classes on ${X^{[n+1]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{\vphantom{[}m}{_{\vphantom{[n]}} }.
$ Then there exists a polynomial $\ti{P}$ depending only on $P$, in the classes analogously defined on ${X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{\vphantom{[}m+1}{_{\vphantom{[n]}} }$, such that $\ds\int_{{X^{[n+1]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{\vphantom{[}m}{_{\vphantom{[n]}} }}P=\int_{{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{\vphantom{[}m+1}{_{\vphantom{[n]}} }}\ti{P}$.
We consider the incidence diagram
(-3.5,-3)(10,1) (100,10)[${X^{[n+1,n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{m}{_{\vphantom{[n]}} }$]{} (15,-40)[${X^{[n+1]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{m}{_{\vphantom{[n]}} }$]{} (185,-40)[$\bigl( {X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X\bigr){\ensuremath{\times}}X^{m}{_{\vphantom{[n]}} }$]{} (62,3)(25,-30) (137,3)(174,-30) (35,-7)[$\scriptstyle{(\nu ,\operatorname{id})}$]{} (165,-7)[$\scriptstyle{(\sigma ,\operatorname{id})}$]{}
Since $(\nu ,\operatorname{id})$ and $(\sigma ,\operatorname{id})$ are generically finite of degrees $n+1$ and $1$, $$\ds\int _{{X^{[n+1]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{m}{_{\vphantom{[n]}} }}P=\frac{1}{n+1}\int_{{X^{[n]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{m+1}{_{\vphantom{[n]}} }}(\sigma ,\operatorname{id})_{*}{^{\vphantom{[n]}} }(\nu ,\operatorname{id})\ee{_{\vphantom{[n]}} }P.$$ By Proposition \[LaDer\] (ii), $(\nu ,\operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{0}\ee\, c_{i}{^{\vphantom{[n]}} }\bigl( {X^{[n+1]}{_{\vphantom{[n]}} }}\bigr)-(\sigma, \operatorname{id})\ee\, pr_{0}\ee\, c_{i}{^{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr)$ is a polynomial in the classes $pr_{0}\ee\, l$, $(\sigma ,\operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{1}{\ensuremath{^{\, *}}}\, c_{i}{^{\vphantom{[n]}} }(X)$ and $(\sigma,\operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{01}\ee\, \mu _{i,n}{^{\vphantom{[n]}} }$.
By Proposition \[d\], $(\sigma ,\operatorname{id})_{*}{^{\vphantom{[n]}} }\, l^{\, i}=(-1)^{i}{_{\vphantom{[n]}} }\, pr_{01}\ee\, \mu _{i,n}{^{\vphantom{[n]}} }$. Thus, $(\sigma ,\operatorname{id})_{*}{^{\vphantom{[n]}} }\, (\nu ,\operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{0}\ee\, c_{i}{^{\vphantom{[n]}} }\bigl( {X^{[n+1]}{_{\vphantom{[n]}} }}\bigr)$ is a polynomial in $pr_{01}\ee\, \mu _{i,n}{^{\vphantom{[n]}} }$ and $pr_{1}\ee\, c_{i}{^{\vphantom{[n]}} }(X)$.
By Proposition \[LaDer\] (i), $(\nu ,\operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{0k}\ee\, \mu _{i,n+1}{^{\vphantom{[n]}} }-(\sigma ,\operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{0,k+1}\ee\, \mu _{i,n}{^{\vphantom{[n]}} }$ is a polynomial in the classes $pr_{0}\ee\, l$ and $(\sigma \operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{1k}\ee\, d_{i}{^{\vphantom{[n]}} }$. Then we can use Proposition \[d\] again.
To conclude, we use the relations $(\nu,\operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{kl}{^{\vphantom{[n]}} }\, d_{i}{^{\vphantom{[n]}} }=(\sigma ,\operatorname{id})\ee{_{\vphantom{[n]}} }pr_{k+1,\, l+1}{^{\vphantom{[n]}} }\, d_{i}{^{\vphantom{[n]}} }$ and $(\nu ,\operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{k}\ee\, c_{i}{^{\vphantom{[n]}} }(X)=(\sigma ,\operatorname{id})\ee{_{\vphantom{[n]}} }\, pr_{k+1}\ee\, c_{i}{^{\vphantom{[n]}} }(X)$.
We can now finish the proof of Theorem \[QUATRE\]. We write $$\int_{{X^{[n]}{_{\vphantom{[n]}} }}}P\Bigl( c_{1}{^{\vphantom{[n]}} }\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr),\dots,c_{2n}\bigl( {X^{[n]}{_{\vphantom{[n]}} }}\bigr)\Bigr)=\int_{{X^{[n-1]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X}\ti{P}_{1}{^{\vphantom{[n]}} }=\int_{{X^{[n-2]}{_{\vphantom{[n]}} }}{\ensuremath{\times}}X^{2}{_{\vphantom{[n]}} }}\ti{P}_{2}{^{\vphantom{[n]}} }=\dots=\int_{X^{n}{_{\vphantom{[n]}} }}\ti{P}$$ where $\ti{P}$ is a polynomial in the classes $pr_{k}\ee\, c_{i}{^{\vphantom{[n]}} }(X)$ and $pr_{kl}\ee\, d_{i}{^{\vphantom{[n]}} }$. Since $d_{i}{^{\vphantom{[n]}} }$ is a polynomial in $c_{1}{^{\vphantom{[n]}} }(X)$ and $c_{2}(X)$, we are done.
$\square$
Appendix {#AppendCh4}
========
Our aim in this appendix is to develop a general framework for [sheaves]{} on spaces which are fibered in smooth analytic spaces over a differentiable orbifold. This formalism is somehow heavy but necessary to carry out the computations of [@SchHilEGL] in a relative setting.
Relative analytic spaces {#n}
------------------------
Let $M$ be a smooth manifold and let $G$ be a finite subgroup of $\textrm{Diff}(M)$ (in fact, any effective differentiable orbifold would do, but no such generality is required here). Let $B=M/G$. If $\apl{p}{M}{B}$ is the projection, $B$ will be endowed with the [sheaf]{} of rings ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{B}:=p_{*}{^{\vphantom{[n]}} }\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{M}\bigr)^{G}{_{\vphantom{[n]}} }\!\!.$
\[DefUnAppenCh4\] ${^{\vphantom{[n]}} }$
1. A *[relative smooth analytic space]{} over $B$* is a ringed topological space $\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$ endowed with a surjective morphism $\apl{\pi }{\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)}{\bigl( B,{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{B}\bigr)}$ such that $\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$ is locally on ${\ensuremath{\mathfrak{X} }}$ isomorphic over $B$ to a ringed topological space of the type $\bigl( Z{\ensuremath{\times}}V,{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}\bigr)$, where $V$ is an open subset of $B$, $Z$ is a smooth analytic space and ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}$ is the [sheaf]{} of differentiable functions on $Z{\ensuremath{\times}}V$ which are holomorphic on the slices $Z\tim\{v\}$, $v\in V$.
2. Let $\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$ be a [relative smooth analytic space]{} over $B$ and let ${\ensuremath{\mathfrak{Z} }}$ be a subset of ${\ensuremath{\mathfrak{X} }}$. We say that ${\ensuremath{\mathfrak{Z} }}$ is *a relative analytic subspace of $B$* if for all $x\in{\ensuremath{\mathfrak{X} }}$ and for all relative holomorphic chart $\apliso{\phi }{Z{\ensuremath{\times}}V}{\,U_{x}{^{\vphantom{[n]}} }}$, $\phi ^{-1}\bigl( {\ensuremath{\mathfrak{Z} }}_{\vert U_{x}}{^{\vphantom{[n]}} }\bigr)=Z'{\ensuremath{\times}}V$, where $Z'$ is a (possibly singular) analytic subset of $Z$.
\[RemUnAppendCh4\] ${^{\vphantom{[n]}} }$
– If $\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$ is a [relative smooth analytic space]{} over $B$, the fibers $\bigl({\ensuremath{\mathfrak{X} }}_{b}{^{\vphantom{[n]}} }\bigr)_{b\in B}{^{\vphantom{[n]}} }$ are smooth analytic sets, but they do not form in general a fibration over $B$, since the projection map $\pi $ is not proper. However, they locally vary in a trivial way on ${\ensuremath{\mathfrak{X} }}$.
– The main example of a relative smooth analytic space we can keep in mind is $W^{[n]}_{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}$ over ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$ where $W$ is a [neighbourhood]{} of the incidence set in ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}{\ensuremath{\times}}X$, endowed with a relative integrable structure parametrized by ${S^{\mkern 1 mu n\!}_{\vphantom{[}} X}$.
We now introduce morphisms between relative smooth analytic spaces.
\[DefDeuxAppenCh4\] ${^{\vphantom{[n]}} }$
1. Let ${\ensuremath{\mathfrak{X} }}$ and ${\ensuremath{\mathfrak{X}' }}$ be two [relative smooth analytic space]{}s over $B$ and $\apl{f}{{\ensuremath{\mathfrak{X} }}}{{\ensuremath{\mathfrak{X}' }}}$ be a continuous map over $B$. We say that $f$ is a *morphism* if for all $x\in {\ensuremath{\mathfrak{X} }}$ we can find trivializations of ${\ensuremath{\mathfrak{X} }}$ and ${\ensuremath{\mathfrak{X}' }}$ around $x$ and $f(x)$ in which $\apl{f}{Z{\ensuremath{\times}}V}{Z'{\ensuremath{\times}}V}$ has the form $\flgdba{(z,v)}{(g(z),v)}$, where $\apl{g}{Z}{Z'}$ is holomorphic.
2. Let ${\ensuremath{\mathfrak{X} }}$ and ${\ensuremath{\mathfrak{X}' }}$ be two [relative smooth analytic space]{}s over $B$ and $B'$ and $\apl{f}{{\ensuremath{\mathfrak{X} }}}{{\ensuremath{\mathfrak{X}' }}}$ be a continuous map. We say that $f$ is a *weak morphism* if there exist a smooth map $\apl{u}{B}{B'}$ and a morphism $\apl{\ti{f}}{{\ensuremath{\mathfrak{X} }}}{{\ensuremath{\mathfrak{X}' }}\tim_{B'}{^{\vphantom{[n]}} }B}$ over $B$ such that $f$ is obtained by composing $\ti{f}$ with the base change morphism $\apl{u}{{\ensuremath{\mathfrak{X}' }}\tim_{B'}{^{\vphantom{[n]}} }B}{{\ensuremath{\mathfrak{X}' }}.}$
\[RemDeuxAppendCh4\] ${^{\vphantom{[n]}} }$
– If ${\ensuremath{\mathfrak{X} }}$ and ${\ensuremath{\mathfrak{X}' }}$ are two [relative smooth analytic space]{}s over $B$ and $B'$, a continuous map $\apl{f}{{\ensuremath{\mathfrak{X} }}}{{\ensuremath{\mathfrak{X}' }}}$ over a smooth map $\apl{u}{B}{B'}$ which is holomorphic in the fibers is not a weak morphism in general.
– Weak morphisms can be characterized by their expression in suitable charts as it is the case for morphims. Indeed, $\apl{f}{{\ensuremath{\mathfrak{X} }}}{{\ensuremath{\mathfrak{X}' }}}$ is a weak morphism if for all $x\in{\ensuremath{\mathfrak{X} }}$ there exist trivializations of ${\ensuremath{\mathfrak{X} }}$ and ${\ensuremath{\mathfrak{X}' }}$ around $x$ and $f(x)$ in which $\apl{f}{Z{\ensuremath{\times}}V}{Z'{\ensuremath{\times}}V'}$ has the form $\flgdba{(z,v)}(\!g(z),u(v))$, where $\apl{g}{Z}{Z'}$ is holomorphic and $\apl{u}{V}{V'}$ is smooth.
– The term “morphism” has to be carefully understood because morphisms cannot *a priori* be composed in this context.
Relatively coherent sheaves {#q}
---------------------------
We introduce now the main object of our study.
\[DefQuatreAppendCh4\] Let ${\ensuremath{\mathfrak{X} }}$ be a [relative smooth analytic space]{} over $B$. A [sheaf]{} $\ff$ of ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}$-modules will be *[relatively coherent]{} * if for all $x\in{\ensuremath{\mathfrak{X} }}$, if $\apliso{\phi }{Z{\ensuremath{\times}}V}{\,U_{x}{^{\vphantom{[n]}} }}$ is a trivialization of ${\ensuremath{\mathfrak{X} }}$ in a [neighbourhood]{} $U_{x}{^{\vphantom{[n]}} }$ of $x$, there exists a coherent [sheaf]{} ${{\ensuremath{\overline{\ff}}}}$ on $Z$ such that $$\phi ^{-1}\ff_{\vert U_{x}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }\simeq {\ensuremath{\operatorname{pr}_{1}^{-1}}}{{\ensuremath{\overline{\ff}}}}{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}}{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}$$ as [sheaves]{} of ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}$-modules.
\[RemQuatreAppenCh4\] ${^{\vphantom{[n]}} }$
– If ${\ensuremath{\mathfrak{X} }}$ is a [relative smooth analytic space]{} over $B$ and ${\ensuremath{\mathfrak{Z} }}$ is a relative analytic subspace of ${\ensuremath{\mathfrak{X} }}$, then ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{Z} }}}}$ is a relatively coherent [sheaf]{} on ${\ensuremath{\mathfrak{X} }}$.
– A [relatively coherent]{} [sheaf]{} on ${\ensuremath{\mathfrak{X} }}$ can also be defined by glueing conditions: it is given by a family of relative holomorphic charts $\bigl\{\apliso{\phi _{i}}{Z_{i}{^{\vphantom{[n]}} }{\ensuremath{\times}}V_{i}{^{\vphantom{[n]}} }}{U_{i}{^{\vphantom{[n]}} }}\bigr\}_{i}{^{\vphantom{[n]}} }$ with transition functions $\xymatrix@C=17pt{\phi _{ji}:Z_{ij}{^{\vphantom{[n]}} }{\ensuremath{\times}}V_{ij}{^{\vphantom{[n]}} }\ar[r]^-{\sim}& Z_{ji}{^{\vphantom{[n]}} }{\ensuremath{\times}}V_{ji}{^{\vphantom{[n]}} },}$ and a family of coherent [sheaves]{} $\bigl\{{{\ensuremath{\overline{\ff}}}}_{i}{^{\vphantom{[n]}} }\bigr\}_{i}{^{\vphantom{[n]}} }$ on $\{Z_{i}{^{\vphantom{[n]}} }\}$ together with isomorphisms $$\phi _{ji}^{-1}\Bigl[\bigl(\,{{\ensuremath{\overline{\ff}}}}_{j}{^{\vphantom{[n]}} }{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}\oo_{Z_{j}}}{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z_{j}{^{\vphantom{[n]}} }{\ensuremath{\times}}V_{j}{^{\vphantom{[n]}} }}}\bigr)_{\vert Z_{ji}{^{\vphantom{[n]}} }{\ensuremath{\times}}U_{ji}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }\,\Bigr]\simeq\bigl(\,{{\ensuremath{\overline{\ff}}}}_{i{^{\vphantom{[n]}} }}{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}\oo_{Z_{i}}}{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z_{i}{^{\vphantom{[n]}} }{\ensuremath{\times}}V_{i}{^{\vphantom{[n]}} }}}\bigr)_{\vert Z_{ij}{^{\vphantom{[n]}} }{\ensuremath{\times}}U_{ij}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }$$ of [sheaves]{} of ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z_{ij}{^{\vphantom{[n]}} }{\ensuremath{\times}}V_{ij}{^{\vphantom{[n]}} }}}$-modules satisfying the usual cocycle condition.
– Let $\ff$ be a [relatively coherent]{} [sheaf]{} on ${\ensuremath{\mathfrak{X} }}$ given by a family of [sheaves]{} $\{{{\ensuremath{\overline{\ff}}}}_{i}{^{\vphantom{[n]}} }\}_{i\in I}{^{\vphantom{[n]}} }$. Then, for any $b\in B$, if $J=\{i\in I\ \textrm{such that}\ b\in V_{i}{^{\vphantom{[n]}} }\}$, the [sheaves]{} $\{{{\ensuremath{\overline{\ff}}}}_{i}{^{\vphantom{[n]}} }\}_{i\in J}{^{\vphantom{[n]}} }$ on $\{Z_{i}{^{\vphantom{[n]}} }{\ensuremath{\times}}b\}_{i\in J}{^{\vphantom{[n]}} }$ patch together into a coherent [sheaf]{} on ${\ensuremath{\mathfrak{X} }}_{b}{^{\vphantom{[n]}} }$, which we will denote by $\ff_{b}{^{\vphantom{[n]}} }$.
The glueing procedure allows to perform standard operations for coherent [sheaves]{} in the relative setting.
\[DefSixAppendCh4\] ${^{\vphantom{[n]}} }$
1. Let $\ff$ and $\g$ be [relatively coherent]{} [sheaves]{} on ${\ensuremath{\mathfrak{X} }}$ given by two families $\{{{\ensuremath{\overline{\ff}}}}_{i}{^{\vphantom{[n]}} }\}$ and $\{{{\ensuremath{\overline{\g}}}}_{i}{^{\vphantom{[n]}} }\}$. The [sheaves]{} ${{\ensuremath{\mathcal{E}}}xt}^{\,k}{_{\vphantom{[n]}} }(\ff,\g)$ and ${\ttt\! or}^{k}{_{\vphantom{[n]}} }(\ff,\g)$ are defined by the families $\bigl\{{{\ensuremath{\mathcal{E}}}xt}^{\,k}_{\oo_{Z_{i}}}({{\ensuremath{\overline{\ff}}}}_{i}{^{\vphantom{[n]}} },{{\ensuremath{\overline{\g}}}}_{i}{^{\vphantom{[n]}} })\bigr\}$ and $\bigl\{{\ttt\! or}^{k}_{\oo_{Z_{i}}}({{\ensuremath{\overline{\ff}}}}_{i}{^{\vphantom{[n]}} },{{\ensuremath{\overline{\g}}}}_{i}{^{\vphantom{[n]}} })\bigr\}.$ We put ${\ensuremath{\mathcal{H}}}om(\ff,\g)={{\ensuremath{\mathcal{E}}}xt}^{\,0}{_{\vphantom{[n]}} }(\ff,\g)$ and $\ff{\ensuremath{\otimes}}\g={\ttt\! or}^{0}{_{\vphantom{[n]}} }(\ff,\g)$.
2. Let $\apl{f}{\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)}{\bigl( {\ensuremath{\mathfrak{X}' }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}\bigr)}$ be a weak morphism between relative smooth analytic spaces. Let $\{\phi _{i}{^{\vphantom{[n]}} }\}_{i\in I}{^{\vphantom{[n]}} }$ and $\{\phi '_{i}\}_{j\in J}{^{\vphantom{[n]}} }$ be two families of charts such that:
1. $I{\ensuremath{\subseteq}}J$ and $f(U_{i}{^{\vphantom{[n]}} }){\ensuremath{\subseteq}}U_{i}'$.
2. In the charts $\phi _{i}{^{\vphantom{[n]}} }$ and $\phi _{i}'$, $f$ has the form $\flgdba{(z,v)}{(g_{i}{^{\vphantom{[n]}} }(z),u_{i}{^{\vphantom{[n]}} }(v)).}$
If $\g$ is a [relatively coherent]{} [sheaf]{} on ${\ensuremath{\mathfrak{X}' }}$ given by a family $\{{{\ensuremath{\overline{\g}}}}_{j}{^{\vphantom{[n]}} }\}_{j\in J}{^{\vphantom{[n]}} }$, the [sheaves]{} ${\ttt\! or}^{k}(\g,f)$ are defined by the families $\Bigl\{{\ttt\! or}^{k}_{g^{-1}_{i}\oo_{Z_{i}'}}\bigl(g_{i}^{-1}{{\ensuremath{\overline{\g}}}}_{i}{^{\vphantom{[n]}} }, \oo_{Z_{i}{^{\vphantom{[n]}} }}\bigr)\Bigr\}_{i\in I}{^{\vphantom{[n]}} }$. We put $f\ee{_{\vphantom{[n]}} }\g={\ttt\! or}^{0{_{\vphantom{[n]}} }}(\g,f)$.
3. Let $\apl{f}{\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)}{\bigl( {\ensuremath{\mathfrak{X}' }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}\bigr)}$ be a morphism of [relative smooth analytic space]{}s over $B$ and $\ff$ a [relatively coherent]{} [sheaf]{} on ${\ensuremath{\mathfrak{X} }}$ such that for all $b\in B$, $f_{b}{^{\vphantom{[n]}} }$ is finite on $\operatorname{supp}(\ff_{b}{^{\vphantom{[n]}} })$. Let $\{\phi _{i}{^{\vphantom{[n]}} },U_{i}{^{\vphantom{[n]}} }\}_{i\in I}{^{\vphantom{[n]}} }$ and $\{\phi '_{j},U'_{j}\}_{j\in J}{^{\vphantom{[n]}} }$ be two families of charts such that
1. $J{\ensuremath{\subseteq}}I$ and $f^{-1}(U'_{j})\cap\operatorname{supp}(\ff){\ensuremath{\subseteq}}U_{j}$.
2. If $W_{j}{^{\vphantom{[n]}} }$ is a [neighbourhood]{} of $f^{-1}(U'_{j})\cap\operatorname{supp}(\ff)$, $\apl{f}{W_{j}{^{\vphantom{[n]}} }}{U_{j}{^{\vphantom{[n]}} }}$ has the form $\flgdba{(z,w)}{(g_{j}(z),v)}$.
If $\ff$ is given by a family $\{\ff_{i{^{\vphantom{[n]}} }}\}_{i\in I}{^{\vphantom{[n]}} }$, the [sheaf]{} $f_{*}{^{\vphantom{[n]}} }\ff$ is defined by the family $\{g_{j*}{^{\vphantom{[n]}} }\ff_{j}{^{\vphantom{[n]}} }\}_{j\in I}{^{\vphantom{[n]}} }$.
\[RemSixAppenCh4\] ${^{\vphantom{[n]}} }$
– For all $b$ in $ B$, $$\begin{aligned}
{{\ensuremath{\mathcal{E}}}xt}^{k}{_{\vphantom{[n]}} }(\ff,\g)_{b}{^{\vphantom{[n]}} }&={{\ensuremath{\mathcal{E}}}xt}^{k}_{\oo_{{\ensuremath{\mathfrak{X} }}_{b}}}(\ff_{b}{^{\vphantom{[n]}} },\g_{b}{^{\vphantom{[n]}} }),&
{\ttt\! or}^{k}{_{\vphantom{[n]}} }(\ff,\g)_{b}{^{\vphantom{[n]}} }&={\ttt\! or}^{k}_{\oo_{{\ensuremath{\mathfrak{X} }}_{b}}}(\ff_{b}{^{\vphantom{[n]}} },\g_{b}{^{\vphantom{[n]}} }),\\
{\ttt\! or}^{k}{_{\vphantom{[n]}} }(\g,f)_{b}{^{\vphantom{[n]}} }&={\ttt\! or}^{k}_{f^{-1}_{b}\oo_{{\ensuremath{\mathfrak{X}' }}_{u(b)}}}\bigl( f^{-1}_{b}\g_{u(b)}{^{\vphantom{[n]}} },\oo_{{\ensuremath{\mathfrak{X} }}_{b}}\bigr)&
\bigl( f_{*}{^{\vphantom{[n]}} }\ff\bigr)_{b}{^{\vphantom{[n]}} }&=\bigl( f_{b}{^{\vphantom{[n]}} }\bigr)_{*}{^{\vphantom{[n]}} }\ff_{b}.\end{aligned}$$
– The finiteness hypothesis will be verified for the computations of Sections \[m\] and \[Comparaison\]. Therefore, we need not construct direct images in full generality.
– It is straightforward that ${\ensuremath{\mathcal{H}}}om (\ff,\g)={\ensuremath{\mathcal{H}}}om_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}}{^{\vphantom{[n]}} }(\ff,\g),$$\ff{\ensuremath{\otimes}}\g=\ff{\ensuremath{\otimes}}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}}{^{\vphantom{[n]}} }\g,$$
f\ee{_{\vphantom{[n]}} }\g=f^{-1}\g{\ensuremath{\otimes}}_{f^{-1}{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}}{^{\vphantom{[n]}} }{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}$ and that $f_{*}{^{\vphantom{[n]}} }\ff$ is the usual direct image of $\ff$ by $f$ via the morphism $\flgd{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}}{f_{*}{^{\vphantom{[n]}} }{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}}$. The associated derived results are also true and are consequences of Lemma \[LemUnAppenCh4\], (i). Thanks to the finiteness hypothesis, there are no higher direct images $R^{i}{_{\vphantom{[n]}} }f_{*}{^{\vphantom{[n]}} }$.
Analytic for relatively coherent sheaves {#p}
----------------------------------------
We are now going to define morphisms of [relatively coherent]{} sheaves. The natural idea is to consider the [relatively coherent]{} [sheaves]{} on $\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$ as a full subcategory of $\textrm{Mod}\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$. It is not appropriate because this category would be non-abelian. Before giving the definition, we start with a preliminary flatness lemma which will be essential in the sequel.
\[LemUnAppenCh4\] Let $U$ be an open set of $\R^{n}{_{\vphantom{[n]}} }$, $G$ a finite group acting smoothly on $U$, $V=U/G$ and let $Z$ be a smooth analytic set. Then
1. ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}$ is flat over ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}$,
2. ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Z{\ensuremath{\times}}V}$ is flat over ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}$.
\(i) Let $\apl{\delta }{U}{V}$ be the projection and $\mm$ be a [sheaf]{} of ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}$-modules. Then $$(\delta ,\operatorname{id})^{-1}\bigl( \mm{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}}{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Z{\ensuremath{\times}}V}\bigr)=
(\delta ,\operatorname{id})^{-1}\mm{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}}\bigl({\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Z{\ensuremath{\times}}U}\bigr)^{G}{_{\vphantom{[n]}} }\simeq\bigl[(\delta ,\operatorname{id})^{-1}\mm{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}}{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Z{\ensuremath{\times}}U}\bigr]^{G}{_{\vphantom{[n]}} }.$$ Since the functor $\flgdba{\ff}{\ff^{G}{_{\vphantom{[n]}} }}$ from $\textrm{Mod}_{G}{^{\vphantom{[n]}} }\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Z{\ensuremath{\times}}U}\bigr)$ to $\textrm{Mod}\bigl[ \bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Z{\ensuremath{\times}}U}\bigr)^{G}{_{\vphantom{[n]}} }\,\bigr]$ is exact, it suffices to prove that ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Z{\ensuremath{\times}}U}$ is smooth over ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}$. If $Y$ is a real-analytic manifold, we will denote by ${\ensuremath{\mathcal{C}^{\mkern 2 mu\omega \vphantom{_{p}}}}}_{Y}$ the [sheaf]{} of real-analytic functions on $Y$. Then, ${\ensuremath{\mathcal{C}^{\mkern 2 mu\omega \vphantom{_{p}}}}}_{Z}$ is flat over ${\oo_{Z}{^{\vphantom{[n]}} }}$, ${\ensuremath{\mathcal{C}^{\mkern 2 mu\omega \vphantom{_{p}}}}}_{Z{\ensuremath{\times}}U}$ is flat over ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\ensuremath{\mathcal{C}^{\mkern 2 mu\omega \vphantom{_{p}}}}}_{Z}$ and ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Z{\ensuremath{\times}}U}$ is flat over ${\ensuremath{\mathcal{C}^{\mkern 2 mu\omega \vphantom{_{p}}}}}_{Z{\ensuremath{\times}}U}$ by [@SchHilMa Th 2].
\(ii) As in (i), it suffices to prove that ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}U}}$ is flat over ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}$. Let $k=\lfloor (n+1)/2\rfloor$. Then $U{\ensuremath{\times}}\R^{k}{_{\vphantom{[n]}} }$ can be seen as an open subset $\ti{U}$ in $\C^{(n+k)/2}{_{\vphantom{[n]}} }$. By [@SchHilMa Th 2 bis], ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z\tim\ti{U}}}$ is flat over $\oo_{Z\tim\ti{U}}{^{\vphantom{[n]}} }$ and $\oo_{Z\tim\ti{U}}{^{\vphantom{[n]}} }$ is flat over ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}$. Thus ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z\tim\ti{Y}}} $ is flat over ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}$. If $\apl{q}{Z\tim\ti{U}}{Z{\ensuremath{\times}}U}$ is the projection, $q^{-1}{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}U}}$ is a direct factor of ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z\tim\ti{U}}}$ (as ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}$-modules), so that ${\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}U}}$ is flat over ${\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}$.
This being done, the definition of a morphism of [relatively coherent]{} [sheaves]{} runs as follows:
\[DefNeufAppendCh4\] Let $\ff$ and $\g$ be [relatively coherent]{} [sheaves]{} on a [relative smooth analytic space]{} $\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$. A morphism $u\in\textrm{Hom}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}}{^{\vphantom{[n]}} }(\ff,\g)$ will be said to be *strict* if for every $x\in{\ensuremath{\mathfrak{X} }}$ and any trivialization $\apliso{\phi }{Z{\ensuremath{\times}}V}{\,U_{x}{^{\vphantom{[n]}} }}$ in a [neighbourhood]{} of $x$, there exist two isomorphisms $\phi ^{-1}\ff_{\vert U_{x}{^{\vphantom{[n]}} }}\simeq {\ensuremath{\operatorname{pr}_{1}^{-1}}}{{\ensuremath{\overline{\ff}}}}{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}}{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}$ and $\phi ^{-1}\g_{\vert U_{x}{^{\vphantom{[n]}} }}\simeq {\ensuremath{\operatorname{pr}_{1}^{-1}}}{{\ensuremath{\overline{\g}}}}{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}}{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}$, where ${{\ensuremath{\overline{\ff}}}}$ and ${{\ensuremath{\overline{\g}}}}$ are coherent on $Z$, and $v$ in $ \textrm{Hom}_{{\oo_{Z}{^{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }({{\ensuremath{\overline{\ff}}}},{{\ensuremath{\overline{\g}}}})$, such that the following diagram commutes: $$\xymatrix@C=60pt{
\phi ^{-1}\ff\ar[r]^-{\phi ^{-1}u}\ar[d]^-{\sim}&\phi ^{-1}\g\ar[d]^-{\sim}\\
{\ensuremath{\operatorname{pr}_{1}^{-1}}}{{\ensuremath{\overline{\ff}}}}{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}\ar[r]^-{v{\ensuremath{\otimes}}\operatorname{id}}&{\ensuremath{\operatorname{pr}_{1}^{-1}}}{{\ensuremath{\overline{\g}}}}{\ensuremath{\otimes}}_{{\ensuremath{\operatorname{pr}_{1}^{-1}}}{\oo_{Z}{^{\vphantom{[n]}} }}}{^{\vphantom{[n]}} }{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z{\ensuremath{\times}}V}}
}$$
\[RemNeufAppendCh4\] Let $\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$ be a [relative smooth analytic space]{}. Lemma \[LemUnAppenCh4\] (i) implies that the category $\operatorname{Coh}^{\operatorname{\vphantom{y}rel}}({\ensuremath{\mathfrak{X} }})$ of [relatively coherent]{} [sheaves]{} on ${\ensuremath{\mathfrak{X} }}$ with strict morphisms is an abelian subcategory of $\textrm{Mod}\bigl( {\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$.
\[DefUnBisAppendCh4\] Let $\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$ be a [relative smooth analytic space]{}.
1. We define the *relative analytic $K$-theory of ${\ensuremath{\mathfrak{X} }}$* by $${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})=\lim_{\genfrac{}{}{0pt}{2}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}}}K\bigl( \textrm{Coh}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }({\ensuremath{\mathfrak{X}' }})\bigr),$$ where ${\ensuremath{\mathfrak{X}' }}$ runs through relatively compact open analytic subsets in ${\ensuremath{\mathfrak{X} }}$.
2. In the same way, if ${\ensuremath{\mathfrak{Z} }}$ is a relative sub-analytic space of ${\ensuremath{\mathfrak{X} }}$, the relative analytic $K$-theory of ${\ensuremath{\mathfrak{X} }}$ with suppoort in ${\ensuremath{\mathfrak{Z} }}$ is defined as $${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z} }}}}({\ensuremath{\mathfrak{X} }})=\lim_{\genfrac{}{}{0pt}{2}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}}}K\bigl( \textrm{Coh}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{Z} }}\cap{\ensuremath{\mathfrak{X}' }}}({\ensuremath{\mathfrak{X}' }})\bigr),$$ where $\textrm{Coh}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{Z} }}\cap{\ensuremath{\mathfrak{X}' }}}({\ensuremath{\mathfrak{X}' }})$ is the abelian category of relatively coherent sheaves on ${\ensuremath{\mathfrak{X}' }}$ supported in ${\ensuremath{\mathfrak{Z} }}$ and ${\ensuremath{\mathfrak{X}' }}$ runs through relatively compact open analytic subsets in ${\ensuremath{\mathfrak{X} }}$.
As for coherent [sheaves]{}, we can define usual operations on relative analytic $K$-theory. Here is a list of these operations:
1. *The product.* A product from ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }}){\ensuremath{\otimes}}_{\Z}{^{\vphantom{[n]}} }{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})$ to ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})$ is defined by $$\ff\, .\, \g=\sum_{k\ge 0}(-1)^{k}{\ttt\! or}^{k}(\ff,\g).$$ Similarly, if ${\ensuremath{\mathfrak{Z} }}$ is a relative analytic subspace of ${\ensuremath{\mathfrak{X} }}$, a product with support from ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }}){\ensuremath{\otimes}}_{\Z}{^{\vphantom{[n]}} }{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z} }}}}({\ensuremath{\mathfrak{X} }})$ to ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z} }}}}({\ensuremath{\mathfrak{X} }})$ is given by the same formula.
2. *The dual morphism.* It is an involution of ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})$ given by $$\ff^{\vee}{_{\vphantom{[n]}} }=\sum_{k\ge 0}(-1)^{k}{{\ensuremath{\mathcal{E}}}xt}^{k}\bigl( \ff,{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr).$$
3. *The pull-back.* If $\apl{f}{\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)}{\bigl( {\ensuremath{\mathfrak{X}' }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}\bigr)}$ is a weak morphism, the pull-back map $\apl{f\pe{_{\vphantom{[n]}} }}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X}' }})}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})}$ is defined by $$f\pe\g=\sum_{k\ge 0}(-1)^{k}{_{\vphantom{[n]}} }{\ttt\! or}^{k}(\g,f).$$ If ${\ensuremath{\mathfrak{Z}' }}$ is a relative analytic subspace of ${\ensuremath{\mathfrak{X}' }}$, we also have a pull-back morphism with supports $\apl{f\pe{_{\vphantom{[n]}} }}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z}' }}}}({\ensuremath{\mathfrak{X}' }})}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}f^{-1}({\ensuremath{\mathfrak{Z}' }})}}({\ensuremath{\mathfrak{X}' }})}$ given by the same formula.
4. *The Gysin map.* If $\apl{f}{\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)}{\bigl( {\ensuremath{\mathfrak{X}' }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}\bigr)}$ is a morphism and ${\ensuremath{\mathfrak{Z} }}$ is a relative analytic subspace of ${\ensuremath{\mathfrak{X} }}$ such that for every $b$ in $B$, $f_{b}{^{\vphantom{[n]}} }{}_{\vert {\ensuremath{\mathfrak{Z} }}_{b}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }$ is finite, the Gysin morphism $\apl{f_{*}{^{\vphantom{[n]}} }}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z} }}}}({\ensuremath{\mathfrak{X} }})}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X}' }})}$ is induced by the exact functor $\apl{f_{*}{^{\vphantom{[n]}} }}{\textrm{Coh}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{Z} }}}({\ensuremath{\mathfrak{X} }})}{\textrm{Coh}({\ensuremath{\mathfrak{X}' }}).}$
We now list all the properties we need relating the operations introduced above.
\[PropUnBisAppendCh4\] ${^{\vphantom{[n]}} }$
1. If $\bigl( {\ensuremath{\mathfrak{X} }},{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}\bigr)$ is a [relative smooth analytic space]{}, ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})$ is a unitary ring. Furthermore, if ${\ensuremath{\mathfrak{Z} }}$ is a relative analytic subspace of ${\ensuremath{\mathfrak{X} }}$, ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z} }}}}({\ensuremath{\mathfrak{X} }})$ is a module over ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})$.
2. The pull-back morphism in relative $K$-theory is contravariant and the Gysin map is covariant.
3. The projection formula holds. More precisely, if $\apl{f}{ {\ensuremath{\mathfrak{X} }}}{ {\ensuremath{\mathfrak{X}' }}}$ is a morphism, ${\ensuremath{\mathfrak{Z} }}$ a relative analytic subspace of ${\ensuremath{\mathfrak{X} }}$ such that $f_{\vert {\ensuremath{\mathfrak{Z} }}}{^{\vphantom{[n]}} }$ is finite, $\ff$ a [relatively coherent]{} [sheaf]{} on ${\ensuremath{\mathfrak{X} }}$ supported in ${\ensuremath{\mathfrak{Z} }}$ and $\g$ a [relatively coherent]{} [sheaf]{} on ${\ensuremath{\mathfrak{X}' }}$, then $f_{*}{^{\vphantom{[n]}} }\bigl( \ff\,.\,f\pe{_{\vphantom{[n]}} }\g\bigr)=f_{*}{^{\vphantom{[n]}} }\ff\,.\,\g$.
4. Let ${\ensuremath{\mathfrak{X} }}$ be a relative smooth analytic space over $B$, ${\ensuremath{\mathfrak{X}' }}$ and $\Delta $ be two [relative smooth analytic space]{}s over $B'$ and $\apl{f}{ {\ensuremath{\mathfrak{X} }}}{ {\ensuremath{\mathfrak{X}' }}}$ a weak morphism; $f$ induces a weak morphism $\apl{f_{\Delta }{^{\vphantom{[n]}} }}{{\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }\bigl( \Delta{\ensuremath{\times}}_{B'}{^{\vphantom{[n]}} }B\bigr)}{{\ensuremath{\mathfrak{X}' }}\tim_{B}{^{\vphantom{[n]}} }\Delta.}$ Let ${\ensuremath{\mathfrak{Z}' }}$ be a relative analytic subspace of ${\ensuremath{\mathfrak{X}' }}\tim_{B}{^{\vphantom{[n]}} }\Delta $ such that the projection $\apl{q}{{\ensuremath{\mathfrak{X}' }}\tim_{B}\Delta }{{\ensuremath{\mathfrak{X}' }}}$ is finite on ${\ensuremath{\mathfrak{Z}' }}$, and let ${\ensuremath{\mathfrak{Z} }}=f^{-1}_{\Delta }({\ensuremath{\mathfrak{Z}' }})$. We consider the diagram $$\xymatrix@C=40pt{
{\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }\bigl( \Delta{\ensuremath{\times}}_{B'}{^{\vphantom{[n]}} }B\bigr) \ar[r]^-{f_{\Delta }{^{\vphantom{[n]}} }}\ar[d]_-{p}&{\ensuremath{\mathfrak{X}' }}\tim_{B}{^{\vphantom{[n]}} }\Delta \ar[d]^-{q}\\
{\ensuremath{\mathfrak{X} }}\ar[r]_-{f}&{\ensuremath{\mathfrak{X}' }}}$$ as well as the following pull-back and push-forward operations: $$\begin{aligned}
\hspace*{12mm}\apl{f\pe{_{\vphantom{[n]}} }&}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X}' }})}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})}&\apl{\!\!f_{\Delta }\pe&}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z}' }}}}({\ensuremath{\mathfrak{X}' }}\tim_{B}{^{\vphantom{[n]}} }\Delta )}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z} }}}}\bigl[ {\ensuremath{\mathfrak{X} }}\!\tim_{B}{^{\vphantom{[n]}} }\bigl( \Delta{\ensuremath{\times}}_{B'}{^{\vphantom{[n]}} }B\bigr)\bigr]}\\
\hspace*{12mm}
\apl{p_{*}{^{\vphantom{[n]}} }&}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z} }}}}\bigl[ {\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }\bigl( \Delta{\ensuremath{\times}}_{B'}{^{\vphantom{[n]}} }B\bigr)\bigr] }{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})}&\apl{q_{*}{^{\vphantom{[n]}} }&}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{Z}' }}}}({\ensuremath{\mathfrak{X}' }}\tim_{B}{^{\vphantom{[n]}} }\Delta )}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X}' }})}\end{aligned}$$ Then, for every $\g$ in $\operatorname{Coh}_{{\ensuremath{\mathfrak{Z}' }}}{^{\vphantom{[n]}} }({\ensuremath{\mathfrak{X}' }}\tim_{B}{^{\vphantom{[n]}} }\Delta )$ we have $
f\pe{_{\vphantom{[n]}} }q_{*}{^{\vphantom{[n]}} }\g=p_{*}f\pe_{\Delta }\g$ in $ {K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})$.
5. Let ${\ensuremath{\mathfrak{X} }}$, $\Delta $ be two relative smooth analytic spaces over $B$, ${\ensuremath{\mathfrak{X}' }}$ a relative smooth analytic subspace of $\Delta $ and $\apl{f}{{\ensuremath{\mathfrak{X} }}}{{\ensuremath{\mathfrak{X}' }}}$ a morphism. Consider the cartesian diagram $$\xymatrix@C=40pt{
{\ensuremath{\mathfrak{X} }}\ar[r]^-{(\operatorname{id},i\circ f)}\ar[d]_-{f}&{\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }\Delta \ar[d]^-{f_{\Delta }{^{\vphantom{[n]}} }}\\
{\ensuremath{\mathfrak{X}' }}\ar[r]_-{(\operatorname{id},i)}&{\ensuremath{\mathfrak{X}' }}\tim_{B}{^{\vphantom{[n]}} }\Delta
}$$ as well as the following pull-back and push-forward operations: $$\begin{aligned}
\hspace{2mm}\apl{&f\pe{_{\vphantom{[n]}} }}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X}' }})}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})}&&\apl{f_{\Delta }\pe}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{X}' }}}}({\ensuremath{\mathfrak{X}' }}\tim_{B}{^{\vphantom{[n]}} }\Delta )}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{X} }}}}({\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }\Delta )}\\
\hspace{2mm}\apl{&(\operatorname{id},i)_{*}{^{\vphantom{[n]}} }}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X}' }})}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{X}' }}}}({\ensuremath{\mathfrak{X}' }}{\ensuremath{\times}}_{B}{^{\vphantom{[n]}} }\Delta )}&&\apl{(\operatorname{id},i\circ f)_{*}{^{\vphantom{[n]}} }}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})}{{K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{X} }}}}({\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }\Delta )}\end{aligned}$$ Then for every [relatively coherent]{} [sheaf]{} $\g$ on ${\ensuremath{\mathfrak{X}' }}$, we have $f_{\Delta }\pe(\operatorname{id},i)_{*}{^{\vphantom{[n]}} }\g=(\operatorname{id},i\circ f)_{*}{^{\vphantom{[n]}} }f\pe{_{\vphantom{[n]}} }\g$ in ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}{\ensuremath{\mathfrak{X} }}}}({\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }\Delta )$.
\(i) If $\ff$, $\g$ and $\hh$ are relatively coherent sheaves on ${\ensuremath{\mathfrak{X} }}$, for each ${\ensuremath{\mathfrak{X}' }}$ open and relatively compact in ${\ensuremath{\mathfrak{X} }}$, we have a spectral sequence such that $$\begin{cases}
E^{\,p,\,q}_{2}&\!\!\!\!={\ttt\! or}^{\,p}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}}\bigl( {\ttt\! or}^{q}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}}(\ff,\g),\hh\bigr)\\
E^{\,p,\,q}_{\infty }&\!\!\!\!=\operatorname{Gr}_{p}\,{^{\vphantom{[n]}} }{\ttt\! or}^{\,p+q}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}}(\ff,\g,\hh).
\end{cases}$$ and $E^{\,p,\,q}_{2}=0$ except for finitely many couples $(p,q)$. Furthermore, by Lemma \[LemUnAppenCh4\] (i), the [sheaves]{} $E_{2}^{\,p,\,q}$ are relatively coherent on ${\ensuremath{\mathfrak{X}' }}$ and the morphisms $d_{2}^{\,p,\,q}$ are strict. Therefore, for all $r\ge 2$, the [sheaves]{} $E_{r}^{\,p,\,q}$ are relatively coherent and the morphisms $d_{2}^{\,p,\,q}$ are strict so that $$\sum_{p,\,q\ge 0}(-1)^{p+q}E^{\,p,\,q}_{2}=\sum_{n\ge 0}(-1)^{n}\,{\ttt\! or}^{\,n}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X}' }}}}}(\ff,\g,\hh)$$ in ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X}' }})$. This gives the result.
\(ii) The proof is similar to the proof of (i), using spectral sequences associated to the composition of two functors.
The proofs of (iii), (iv) and (v) are done in the same way. We will prove (iv).
\(iv) Let $x\in{\ensuremath{\mathfrak{X} }}$. We take charts $U_{x}{^{\vphantom{[n]}} }\simeq Z{\ensuremath{\times}}V$ and $U_{f(x)}{^{\vphantom{[n]}} }\simeq Z'{\ensuremath{\times}}V'$ such that $\apl{f}{U_{x}{^{\vphantom{[n]}} }}{U_{f(x)}}$ has the form $\flgdba{(z,v)}{(g(z),u(v))}$, where $\apl{g}{Z}{Z'}$ is holomorphic and $\apl{u}{V}{V'}$ is smooth. Let $\delta _{_{1}}{^{\vphantom{[n]}} },\dots,\delta _{_{N}}{^{\vphantom{[n]}} }\in \Delta $ such that $q^{-1}(f(x))\cap{\ensuremath{\mathfrak{Z} }}=\cup_{i=1}^{N}\bigl( f(x),\delta _{_{i}}{^{\vphantom{[n]}} }\bigr)$. We take a chart $U_{\delta _{_{1}},\dots,\,\delta _{_{N}}}{^{\vphantom{[n]}} }\simeq Y{\ensuremath{\times}}V'$ in a [neighbourhood]{} of the $\delta _{_{i}}{^{\vphantom{[n]}} }$’s. Then the diagram looks locally on ${\ensuremath{\mathfrak{X}' }}$ as follows: $$\xymatrix@C=40pt@R=30pt{
Z{\ensuremath{\times}}Y{\ensuremath{\times}}V\ar[r]^-{(g,\,\operatorname{id},\,u)}\ar[d]_-{\operatorname{pr}_{13}{^{\vphantom{[n]}} }}&Z'{\ensuremath{\times}}Y{\ensuremath{\times}}V'\ar[d]^(.55){\operatorname{pr}_{13}}\\
Z{\ensuremath{\times}}V\ar[r]_-{(g,\,u)}&Z'{\ensuremath{\times}}V'}$$ Furthermore, ${\ensuremath{\mathfrak{Z} }}={\ensuremath{\overline{{\ensuremath{\mathfrak{Z} }}}}}{\ensuremath{\times}}V$ where ${\ensuremath{\overline{{\ensuremath{\mathfrak{Z} }}}}}$ is an analytic subset of $Z{\ensuremath{\times}}Y$ and $\operatorname{pr}_{_{1}}{^{\vphantom{[n]}} }{}_{\!\!\vert{\ensuremath{\overline{{\ensuremath{\mathfrak{Z} }}}}}}{^{\vphantom{[n]}} }$ is finite. Then for any analytic [sheaf]{} ${\ensuremath{\overline{\g}}}$ on $Z'{\ensuremath{\times}}Y$, we have $$(g,u)\ee{_{\vphantom{[n]}} }\operatorname{pr}_{_{13}*}{^{\vphantom{[n]}} }\bigl( \operatorname{pr}_{_{12}}^{-1}{\ensuremath{\overline{\g}}}{\ensuremath{\otimes}}_{\operatorname{pr}_{12}^{-1}\oo_{Z'{\ensuremath{\times}}Y}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }\!\!{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z'{\ensuremath{\times}}Y{\ensuremath{\times}}V'}}\bigr)\simeq
\operatorname{pr}_{_{13}*}{^{\vphantom{[n]}} }\,(g,\operatorname{id},u)\ee{_{\vphantom{[n]}} }\bigl( \operatorname{pr}_{_{12}}^{-1}{\ensuremath{\overline{\g}}}{\ensuremath{\otimes}}_{pr_{12}^{-1}\oo_{Z'{\ensuremath{\times}}Y}{^{\vphantom{[n]}} }}{^{\vphantom{[n]}} }\!\!{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{Z'{\ensuremath{\times}}Y{\ensuremath{\times}}V'}}\bigr).$$ Taking the derivative with respect to ${\ensuremath{\overline{\g}}}$ and using Lemma \[LemUnAppenCh4\] (i), we obtain the result.
Topological classes
-------------------
In Sections \[q\] and \[p\], we have constructed a theory for relative coherent sheaves as well as associated operations. It remains to obtain cohomological informations about these objects. To do so, we will construct global smooth resolutions for relatively coherent [sheaves]{}. We start with a general result about smooth resolutions.
\[NouvProp\] Let $M$ be a smooth manifold, $G$ a finite group of $\emph{Diff}(M)$ and $Y=M/_{{\ensuremath{\displaystyle}}G}$. Let $\hh$ be a sheaf of ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}$-modules which admits a finite free resolution in a [neighbourhood]{} of any point $y\in Y$. Then
1. $\hh$ admits a finite locally free resolution in a [neighbourhood]{} of any compact set of $Y$.
2. Two resolutions of $\hh$ in a [neighbourhood]{} of a compact set are sub-resolutions of a third one.
We will use several times the following lemma:
\[Utile\] Let $Y=M/G$ and let $\sutrgd{\ff}{\g}{\hh}$ be an exact sequence of ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}$-modules on an open set $U{\ensuremath{\subseteq}}Y$ such that $\hh$ is locally free. Then this exact sequence globally splits on $U$.
It is sufficient to prove that $\textrm{Ext}^{1}_{{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}}(U,\hh,\ff)=0$. The ${\ensuremath{\mathcal{E}}}xt\Longrightarrow \textrm{Ext}$ spectral sequence satisfies $$\begin{cases}
E_{2}^{p,q}=H^{p}{_{\vphantom{[n]}} }\Bigl[ U,{\ensuremath{\mathcal{E}}}xt^{q}_{{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}}\bigl( \hh,\ff\bigr)\Bigr]\\
E_{\infty }^{p,q}=\textrm{Gr}^{p}{_{\vphantom{[n]}} }\, \, \textrm{Ext}^{p+q}_{{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}}(U,\hh,\ff).
\end{cases}$$ Since $\hh$ is locally free, ${\ensuremath{\mathcal{E}}}xt^{q}(\hh,\ff)=0$ for $q>0$ and since ${\ensuremath{\mathcal{H}}}om_{{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}}{^{\vphantom{[n]}} }(\hh,\ff) $ is a fine sheaf, $E^{p,0}_{2}=0$ for $p\ge 1$. Therefore all the terms $E_{2}^{p,q}$ vanish except $E_{2}^{0,0}$. This implies $\emph{Ext}^{\, 1}_{{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}}(U,\hh,\ff)=0$.
\(i) Let $K{\ensuremath{\subseteq}}Y$ be a compact. We choose a finite covering $\bigl( U_{i}{^{\vphantom{[n]}} }\bigr)_{1\le i\le d}{^{\vphantom{[n]}} }$ of $K$ and open sets $\bigl( V_{i}{^{\vphantom{[n]}} }\bigr)_{1\le i\le d}{^{\vphantom{[n]}} }$ such that ${\ensuremath{\overline{U{^{\vphantom{[n]}} }}}}_{i}{^{\vphantom{[n]}} }{\ensuremath{\subseteq}}V_{i}{^{\vphantom{[n]}} }$ and $\hh_{\, \vert V_{i}{^{\vphantom{[n]}} }} $ admits a finite free resolution. Using smooth cut-off functions, we obtain for each $i$ a complex of sheaves $$\xymatrix{0\ar[r]&\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}\bigr)^{n_{iN}{^{\vphantom{[n]}} }}{_{\vphantom{[n]}} }\ar[r]&\cdots\cdots\ar[r]&\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}\bigr)^{n_{i1}{^{\vphantom{[n]}} }}{_{\vphantom{[n]}} }\ar[r]^(.6){\pi _{i}{^{\vphantom{[n]}} }}&\hh\ar[r]&0}$$ which is exact in $U_{i}{^{\vphantom{[n]}} }$. If $E=\bop_{i=1}^{d}\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}\bigr)^{n_{i1}{^{\vphantom{[n]}} }}{_{\vphantom{[n]}} }$ and $\apl{\pi :=\bop_{i=1}^{d}\pi _{i}{^{\vphantom{[n]}} }}{E}{\hh}$, the morphism $\pi $ is surjective. Let $\nn_{i}{^{\vphantom{[n]}} }=\ker \pi _{i}{^{\vphantom{[n]}} }$ and $\nn=\ker \pi $. We have an exact sequence: $$\xymatrix{0\ar[r]&\nn_{i}{^{\vphantom{[n]}} }\ar[r]&\nn_{\vert\, U_{i}{^{\vphantom{[n]}} }}\ar[r]&\bop_{j\not=i}\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}\bigr)^{n_{j1}{^{\vphantom{[n]}} }}{_{\vphantom{[n]}} }\ar[r]&0.}$$ Thus $\nn_{\vert\, U_{i}{^{\vphantom{[n]}} }}$ is locally isomorphic to $ \nn_{i}{^{\vphantom{[n]}} }\oplus\bigl( {\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{Y}\bigr)^{\sum_{j\not = i}n_{j1}{^{\vphantom{[n]}} }}{_{\vphantom{[n]}} }$. Furthermore $\nn_{i}{^{\vphantom{[n]}} }$ admits a finite free resolution of length $N-1$. Thus $\nn$ admits a finite free resolution of length at most $N-1$ in a [neighbourhood]{} of every point in $K$ and we can start the argument again. After at most $N$ steps, the kernel will be locally free.
\(ii) Let $\bigl( E_{i}{^{\vphantom{[n]}} }\bigr)_{1\le i\le N}{^{\vphantom{[n]}} }$ and $\bigl( F_{i}{^{\vphantom{[n]}} }\bigr)_{1\le i\le N}{^{\vphantom{[n]}} }$ be two finite locally free resolutions in a [neighbourhood]{} of $K$. Suppose that we have constructed $\bigl( G_{i}{^{\vphantom{[n]}} }\bigr)_{1\le i\le k}{^{\vphantom{[n]}} }$ and injections $\xymatrix{
E_{_{\bullet}}{^{\vphantom{[n]}} }\ar@{^{(}->}[r]& G_{_{\bullet}}{^{\vphantom{[n]}} }}$ and $\xymatrix{
F_{_{\bullet}}\ar@{^{(}->}[r]& G_{_{\bullet}}{^{\vphantom{[n]}} }}$ Let $Q_{k}{^{\vphantom{[n]}} }=G_{k}{^{\vphantom{[n]}} }/E_{k}{^{\vphantom{[n]}} }$ and $R_{k}{^{\vphantom{[n]}} }=G_{k}{^{\vphantom{[n]}} }/F_{k}{^{\vphantom{[n]}} }$. The sheaves $Q_{1}{^{\vphantom{[n]}} },\dots,Q_{k}{^{\vphantom{[n]}} },R_{1}{^{\vphantom{[n]}} },\dots,R_{k}{^{\vphantom{[n]}} }$ are locally free. Let $N_{k}{^{\vphantom{[n]}} }=\ker\bigl(\!\! \xymatrix@C=12pt{E_{k}{^{\vphantom{[n]}} }\ar[r]&E_{k-1}{^{\vphantom{[n]}} }}\!\!\bigr)$, $N'_{k}=\ker\bigl(\!\! \xymatrix@C=12pt{F_{k}{^{\vphantom{[n]}} }\ar[r]&F_{k-1}{^{\vphantom{[n]}} }}\!\!\bigr)$, $N''_{k}=\ker\bigl(\!\! \xymatrix@C=12pt{G_{k}{^{\vphantom{[n]}} }\ar[r]&G_{k-1}{^{\vphantom{[n]}} }}\!\!\bigr)$, $\ti{Q}_{k}{^{\vphantom{[n]}} }=\ker\bigl(\!\! \xymatrix@C=12pt{Q_{k}{^{\vphantom{[n]}} }\ar[r]&Q_{k-1}{^{\vphantom{[n]}} }}\!\!\bigr)$ and $\ti{R}_{k}{^{\vphantom{[n]}} }=\ker\bigl(\!\! \xymatrix@C=12pt{R_{k}{^{\vphantom{[n]}} }\ar[r]&R_{k-1}{^{\vphantom{[n]}} }}\!\!\bigr)$. Then $\ti{Q}_{k}{^{\vphantom{[n]}} }$ and $\ti{R}_{k}{^{\vphantom{[n]}} }$ are locally free. We have two exact sequences $\sutrgd{N_{k}{^{\vphantom{[n]}} }}{N''_{k}}{\ti{Q}_{k}{^{\vphantom{[n]}} }}$ and $\sutrgdpt{N'_{k}}{N''_{k}}{\ti{R}_{k}{^{\vphantom{[n]}} }}{.}$ By Lemma \[Utile\], $N''_{k}\simeq N_{k}{^{\vphantom{[n]}} }\oplus\ti{Q}_{k}{^{\vphantom{[n]}} }\simeq N'_{k}\oplus \ti{R}_{k}{^{\vphantom{[n]}} }$, and we define $G_{k+1}{^{\vphantom{[n]}} }=\bigl( E_{k+1}{^{\vphantom{[n]}} }\oplus\ti{Q}_{k}{^{\vphantom{[n]}} }\bigr)\oplus\bigl( F_{k+1}{^{\vphantom{[n]}} }\oplus\ti{R}_{k}{^{\vphantom{[n]}} }\bigr)$. We put $G_{N+1}{^{\vphantom{[n]}} }=N''_{N}$ to end the resolution $G_{_{\bullet}}{^{\vphantom{[n]}} }$.
We apply now this result in our context. Let $\ff$ be a relatively coherent [sheaf]{} on a [relative smooth analytic space]{} ${\ensuremath{\mathfrak{X} }}$ over $B$, where $B=M/G$. Then ${\ensuremath{\mathfrak{X} }}=\bigl( {\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }M\bigr)/G$ and ${\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }M$ is smooth. Furthermore, by Lemma \[LemUnAppenCh4\] (ii), the [sheaf]{} $\ff^{\infty }{_{\vphantom{[n]}} }:=\ff{\ensuremath{\otimes}}_{{\oo^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{{\ensuremath{\mathfrak{X} }}}}}{^{\vphantom{[n]}} }{\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{{\ensuremath{\mathfrak{X} }}}$ locally admits finite resolutions. By Proposition \[NouvProp\] (i), $\ff^{\infty }{_{\vphantom{[n]}} }$ admits a globally locally resolution $E_{_{\bullet}}{^{\vphantom{[n]}} }$ on any relatively compact open analytic subset ${\ensuremath{\mathfrak{X}' }}$ of ${\ensuremath{\mathfrak{X} }}$. Besides, Proposition \[NouvProp\] (ii) implies that the element $\sum_{i=1}^{N}(-1)^{i-1}\bigl[ E_{i}{^{\vphantom{[n]}} }\bigr]$ in $K({\ensuremath{\mathfrak{X}' }})$ is independent of the chosen resolution $E_{_{\bullet}}{^{\vphantom{[n]}} }$.
In conclusion, we can associate to each relatively coherent sheaf $\ff$ on ${\ensuremath{\mathfrak{X} }}$ a topological class $\bigl[ \ff^{\infty }{_{\vphantom{[n]}} }\bigr]$ in $\lim\limits_{\genfrac{}{}{0pt}{2}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}}}K({\ensuremath{\mathfrak{X}' }})$, where ${\ensuremath{\mathfrak{X}' }}$ runs through all open relatively compact analytic subspaces of ${\ensuremath{\mathfrak{X} }}$. We now state properties of this topological class:
\[Compl\] Let ${\ensuremath{\mathfrak{X} }}$ be a [relative smooth analytic space]{} over $B$.
1. The topological class map from $\operatorname{Coh}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}{_{\vphantom{[n]}} }({\ensuremath{\mathfrak{X} }})$ to $\lim\limits_{\genfrac{}{}{0pt}{2}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}}}K({\ensuremath{\mathfrak{X}' }})$ factors through ${K^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}_{\vphantom{f}}}({\ensuremath{\mathfrak{X} }})$.
2. Let $\ff$ be a relatively coherent [sheaf]{} on ${\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }(B\tim[0,1])$ and for all $t\in[0,1]$, let $\apl{i_{t}{^{\vphantom{[n]}} }}{{\ensuremath{\mathfrak{X} }}\tim_{B}{^{\vphantom{[n]}} }(B\tim\{t\})}{{\ensuremath{\mathfrak{X} }}\tim_{B\tim\{t\}}{^{\vphantom{[n]}} }(B\tim[0,1])}$ be the associated base change morphism. Then the class $\bigl[ i_{t}\ee\ff^{\infty }{_{\vphantom{[n]}} }\bigr]$ in $\lim\limits_{\genfrac{}{}{0pt}{2}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}}}K({\ensuremath{\mathfrak{X}' }})$ is independent of $t$.
\(i) We must show that if $\sutrgd{\ff}{\g}{\hh}$ is a strict exact sequence of relatively coherent sheaves, then $[\ff^{\, \infty }{_{\vphantom{[n]}} }]-[\g^{\, \infty }{_{\vphantom{[n]}} }]+[\hh^{\, \infty }{_{\vphantom{[n]}} }]=0$. This sequence is locally isomorphic to $$\sutrgdpt{pr^{-1}_{1}\, {\ensuremath{\overline{\ff{^{\vphantom{[n]}} }}}}{\ensuremath{\otimes}}_{pr^{-1}_{1}\oo_{Z}{^{\vphantom{[n]}} }}\oo_{Z{\ensuremath{\times}}V}^{\, \operatorname{\vphantom{y}rel}}}{pr^{-1}_{1}\, {\ensuremath{\overline{\g{^{\vphantom{[n]}} }}}}{\ensuremath{\otimes}}_{pr^{-1}_{1}\oo_{Z}{^{\vphantom{[n]}} }}\oo_{Z{\ensuremath{\times}}V}^{\, \operatorname{\vphantom{y}rel}}}{pr^{-1}_{1}\, {\ensuremath{\overline{\hh{^{\vphantom{[n]}} }}}}{\ensuremath{\otimes}}_{pr^{-1}_{1}\oo_{Z}{^{\vphantom{[n]}} }}\oo_{Z{\ensuremath{\times}}V}^{\, \operatorname{\vphantom{y}rel}}}{,}$$ and is obtained by extension of the structure sheaves from an exact sequence of coherent analytic sheaves $\sutrgd{{\ensuremath{\overline{\ff{^{\vphantom{[n]}} }}}}}{{\ensuremath{\overline{\g{^{\vphantom{[n]}} }}}}}{{\ensuremath{\overline{\hh{^{\vphantom{[n]}} }}}}}$ on $Z$. We can construct locally free resolutions $E_{\, {\ensuremath{\overline{\ff{^{\vphantom{[n]}} }}}},\bullet }{^{\vphantom{[n]}} }$, $E_{\, {\ensuremath{\overline{\g{^{\vphantom{[n]}} }}}}, \bullet }{^{\vphantom{[n]}} }$, $E_{\, {\ensuremath{\overline{\hh{^{\vphantom{[n]}} }}}}, \bullet }{^{\vphantom{[n]}} }$ of ${\ensuremath{\overline{\ff{^{\vphantom{[n]}} }}}}$, ${\ensuremath{\overline{\g{^{\vphantom{[n]}} }}}}$, ${\ensuremath{\overline{\hh{^{\vphantom{[n]}} }}}}$ related by an exact sequence $
\sutrgdpt{E_{\, {\ensuremath{\overline{\ff{^{\vphantom{[n]}} }}}},\bullet }{^{\vphantom{[n]}} }}{E_{\, {\ensuremath{\overline{\g{^{\vphantom{[n]}} }}}}, \bullet }{^{\vphantom{[n]}} }}{E_{\, {\ensuremath{\overline{\hh{^{\vphantom{[n]}} }}}}, \bullet }{^{\vphantom{[n]}} }}{.}
$ Using cut-off functions again, we patch these exact sequences together step by step and obtain resolutions $E_{\ff^{\, \infty },\bullet }{^{\vphantom{[n]}} }$, $E_{\g^{\, \infty },\bullet }{^{\vphantom{[n]}} }$, $E_{\hh^{\, \infty },\bullet }{^{\vphantom{[n]}} }$ of $\ff^{\, \infty }{_{\vphantom{[n]}} }$, $\g^{\, \infty }{_{\vphantom{[n]}} }$, $\hh^{\, \infty }{_{\vphantom{[n]}} }$ on ${\ensuremath{\mathfrak{X} }}'$ related by an exact sequence $$\sutrgdpt{E_{\ff^{\, \infty },\bullet }{^{\vphantom{[n]}} }}{E_{\g^{\, \infty },\bullet }{^{\vphantom{[n]}} }}{E_{\hh^{\, \infty },\bullet }{^{\vphantom{[n]}} }}{.}$$
\(ii) Let ${\ensuremath{\mathfrak{X}' }}$ be an open relatively compact analytic subset of ${\ensuremath{\mathfrak{X} }}$ and $E_{_{\bullet}}{^{\vphantom{[n]}} }$ be a resolution of $\ff^{\infty }{_{\vphantom{[n]}} }$ on ${\ensuremath{\mathfrak{X}' }}$. The class $\alpha (t)=\sum_{i=1}^{N}(-1)^{i-1}\bigl[ i_{t}\ee\,E_{i}{^{\vphantom{[n]}} }\bigr]$ in $K({\ensuremath{\mathfrak{X}' }})$ is independent of $t$. Since $\ff^{\infty }{_{\vphantom{[n]}} }$ is flat over ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{B\tim[0,1]}$, it is also flat over ${\ensuremath{\mathcal{C}^{\mkern 1 mu\infty\vphantom{_{p}}}}}_{[0,1]}$, so that $\alpha (t)=\bigl[ i_{t}\ee\ff^{\infty }{_{\vphantom{[n]}} }\bigr]$ in $K({\ensuremath{\mathfrak{X}' }})$.
If ${\ensuremath{\mathfrak{Z} }}$ is a relative analytic subspace of ${\ensuremath{\mathfrak{X} }}$, there is also a topological class map with support from $\textrm{Coh}_{{\ensuremath{\mathfrak{Z} }}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}({\ensuremath{\mathfrak{X} }})$ to $\lim\limits_{\genfrac{}{}{0pt}{2}{\longleftarrow}{{\ensuremath{\mathfrak{X}' }}}}K_{{\ensuremath{\mathfrak{Z} }}\cap{\ensuremath{\mathfrak{X}' }}}{^{\vphantom{[n]}} }\,({\ensuremath{\mathfrak{X}' }})$ which factors through $K_{{\ensuremath{\mathfrak{Z} }}}^{{\ensuremath{\, \operatorname{\vphantom{y}rel}}}}({\ensuremath{\mathfrak{X} }})$.
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|
---
author:
- 'F. Castelli'
- 'R.L. Kurucz'
title: New energy levels from stellar spectra
---
[The spectra of B-type and early A-type stars show numerous unidentified lines in the whole optical range, especially in the 5100-5400Å interval. Because transitions to high energy levels should be observed in this region, we used semiempirical predicted wavelengths and gf-values of to identify unknown lines. ]{} [Semiempirical line data for computed by Kurucz are used to synthesize the spectrum of the slow-rotating, -overabundant CP star HR6000. ]{} [ We determined a total of 109 new 4f levels for with energies ranging from 122324cm$^{-1}$ to 128110cm$^{-1}$. They belong to the subconfigurations 3d$^{6}$($^{3}$P)4f (10 levels), 3d$^{6}$($^{3}$H)4f (36levels), 3d$^{6}$($^{3}$F)4f (37levels), and 3d$^{6}$($^{3}$G)4f (26levels). We also found 14 even levels from 4d (3levels), 5d (7levels), and 6d (4levels) configurations. The new levels have allowed us to identify more than the 50% of previously unidentified lines of HR6000 in the wavelength region 3800-8000Å. Tables listing the new energy levels are given in the paper; tables listing the spectral lines with $\log\,gf$$\ge$$-$1.5 that are transitions to the 4f energy levels are given in the Online Material. These new levels produce 18000 lines throughout the spectrum from the ultraviolet to the infrared. ]{}
Introduction
============
In a previous paper (Castelli, Kurucz & Hubrig, 2009) (Paper I) we have determined 21 new 3d$^{6}$($^{3}$H)4f high energy levels of on the basis of predicted energy levels, computed $\log\,gf$ values for , and unidentified lines in UVES high resolution, high signal-to-noise spectra of HR6000 and 46Aql. Both stars are iron overabundant CP stars and have rotational velocity v[*sini*]{} of the order of 1.5kms$^{-1}$ and 1.0kms$^{-1}$, respectively.
In this paper we continue the effort to determine new high-energy levels of . We used the same spectra and models for HR6000 that we adopted in PaperI, together with line lists which include transitions between observed-observed, observed-predicted, and predicted-predicted energy levels. In this paper we increase the number of the new energy levels from the 21 listed in PaperI, to a total of 109 energy levels, which belong to the subconfigutations: 3d$^{6}$($^{3}$P)4f (10 levels), 3d$^{6}$($^{3}$H)4f (36 levels), 3d$^{6}$($^{3}$F)4f (37 levels), and 3d$^{6}$($^{3}$G)4f (26 levels), and 14 levels from the even configurations 4d (3levels), 5d (7levels), and 6d (4levels). The new levels have allowed us to identify more than the 50% of the previously unidentified lines in the wavelength region 3800-8000Å of HR6000 (Castelli & Hubrig, 2007). The method that we adopted to determine the new energy levels is the same as described in PaperI. It is recalled here in Sect.3. The comparison of the observed spectrum of HR6000 with the synthetic spectrum which includes the new lines is available on the Castelli web site[^1].
The star HR6000
===============
According to PaperI, the CP star HR6000 (HD144667) has an estimated rotational velocity of 1.5kmsec$^{-1}$. The model stellar parameters for an individual abundance ATLAS12 (Kurucz 2005) model are [$T_{\rm eff}$]{}=13450K, [$\log\,g$]{}=4.3. In addition to the large iron overabundance \[$+$0.9\], overabundances of (\[$+$4.6\]), ($>$\[$+$1.5\]), (\[$+$0.55\]), (\[$+$0.2\]), (\[$+$1.5\]), Y (\[$+$1.2\]), and Hg (\[$+$2.7\]) were observed. This peculiar chemical composition, together with the underabundances of He, C, N, O, Al, Mg, Si, S, Cl, Sc, V, Co, Ni, and Sr gives rise to an optical line spectrum very rich in lines, with transitions involving upper energy levels close to the ionization limit (Johansson 2009). Also numerous and lines are observable in the spectrum.
The method
==========
To determine the new energy levels we used high-resolution UVES spectra of HR6000 (see PaperI), the corresponding synthetic spectrum, and the list of the computed transitions with predicted values for levels with no experimentally available energies. Predicted energy levels and $\log\,gf$ values were computed by Kurucz with his version of the Cowan (1981) code (Kurucz 2009). The calculation included 46 even configurations d$^{7}$, d$^{6}$4s$-$9s, d$^{6}$4d$-$9d, d$^{6}$5g$-$9g, d$^{6}$7i-9i, d$^{6}$9l, d$^{5}$4s$^{2}$, d$^{5}$4s5s$-$9s, d$^{5}$4s4d$-$9d, d$^{5}$4s5g-9g, d$^{5}$4s7i$-$9i, d$^{5}$4s9l, d$^{4}$4s$^{2}$4d, and d$^{5}$4p$^{2}$ with 19771 levels least-squares fitted to 418 known levels. The 39 odd configurations included d$^{6}$4p$-$9p, d$^{6}$4f$-$9f, d$^{6}$6h$-$9h, d$^{6}$8k$-$9k, d$^{5}$4s4p$-$9p, d$^{5}$4s4f$-$9f, d$^{5}$4s6h$-$9h, d$^{5}$4s8k-9k, d$^{4}$4s$^{2}$4p$-$5p, and d$^{4}$4s$^{2}$4f with 19652 levels least-squares fitted to 596 known levels. The calculations were done in LS coupling with all configuration interactions included, with scaled Hartree-Fock starting guesses, and with Hartree-Fock transition integrals. A total of 7080169 lines were saved from the transition array of which 102833 lines are between known levels and have good wavelengths. The computed line list was sorted into tables of all the strong lines connected to every predicted level. When a given predicted level gives rise to at least two lines having $\log\,gf$$\ge$$-$1.0, we selected one of these transitions and searched in the spectrum for those unidentified lines which have wavelength within $\pm$50Å and residual flux within about $\pm$ 5% of those of the selected predicted line. From the observed wavelength of one of these unidentified lines and from the known energy of the lower or upper level of the predicted transition, we derived a possible energy for the predicted level. If most of transitions obtained with this energy correspond to lines observed in the spectrum, we kept the tentative energy value as a real value, otherwise we repeated the procedure using another line taken from the unidentified ones, and continued the searching until we found that energy for which most of the predicted lines correspond to the observed lines. Whenever one or more new levels were found, the whole semiempirical calculation was repeated to produce improved predicted wavelengths and $\log\,gf$-values. Because all configuration interactions were included, and because the mixing is exceptionally strong in the 4d and 5d configurations, every new level changed the predictions. Mixing between close levels can produce large uncertainties in the $\log\,gf$ values for lines that involve those levels.
This procedure is very successfull for levels which produce two or more transitions with $\log\,gf$$>$0.0, but becomes more and more difficult as the intensity of the predicted lines decreases. In fact, weak lines are usually blended with stronger components, so that the method may fail in these cases.
The new energy levels
=====================
The new energy levels of the 3d$^{6}$($^{3}$P)4f, 3d$^{6}$($^{3}$H)4f, 3d$^{6}$($^{3}$F)4f, and 3d$^{6}$($^{3}$G)4f subconfigurations and from the even configurations 3d$^{6}$4d, 3d$^{6}$5d, and 3d$^{6}$6d are listed in Tables1$-$5. Because the 3d$^{6}$4f states of tend to appear in pairs we have used the j$_{c}$\[K\]$_{j}$ notation of jK coupling for them, where [**j**]{}$_{c}$ is the total angular momentum of the core and [**K**]{}=[**J$_{c}$**]{}$+$[**l**]{} is the coupling of [**J$_{c}$**]{} with the orbital angular momentum [**l**]{} of the active electron. The level pairs correspond to the two separate values of the total angular momentum [**J**]{} obtained when the spin s=$\pm$1/2 of the active electron is added to [**K**]{}. The positive energies are those obtained by comparing observed and predicted line profiles, as described in Sect.3 and shown in Fig.2. The energies between parentheses in Tables1$-$4 are predicted values for which we have been not able to find the corresponding observed level. The reason for the failure is that either all the lines from the energy level are weak or, even if some of the transitions are predicted as moderately strong ($\log\,gf$$>$0.0), they are blended with other stronger components, so that their identification is uncertain. The columns with label “c$-$o” in Tables 1-5 show the difference between the predicted and observed energy levels.
The 4d even energy levels listed in Table5 give rise to some of the transitions listed in the Online Material. The strongest transitions related with the 5d, and 6d even energy levels occur in the 6000-8000Å region and in the 4000-5000Å region, respectively. The transitions to the odd energy levels are discussed in Sect.5
The observed energy levels, the least squares fits, the predicted energy levels, and the line lists can be found on the Kurucz web site[^2]. The observed levels come from the following sources: Johansson (1978), Sugar & Corliss (1985), Adam et al. (1987), Johansson & Baschek (1988), Johansson (1988, private communication), Rosberg & Johansson (1992), Castelli, Johansson & Hubrig (2008), Castelli, Kurucz, Hubrig (2009), and this work. The calculations on the web site are updated whenever there are improvements to the energy levels.
[crcrcrcrcrcrrrr]{} & & & & & & & & & & &\
& & & & & & & & & & &\
2\[5\] &11/2 & 122351.810 &$-$20.236\
&9/2 & 122324.142 &$-$18.980\
\
2\[4\] &9/2 &122355.116 &$-$6.685 & 1\[4\]& 9/2 &123629.520&$-$4.606\
&7/2 &122355.553 &$-$6.801 & & 7/2 &123637.833&$-$6.417\
\
2\[3\] &7/2 & 122351.488 &$-$18.489 & 1\[3\]& 7/2 &123615.875 &$-$2.642 & 0\[3\] & 7/2 &(124167.229) &\
&5/2 & (122353.541) & & & 5/2 &123649.493 &$-$5.687 & & 5/2 & 124157.060&$+$15.841\
\
2\[2\] &5/2 &(122342.921) & & 1\[2\] & 5/2 &(123637.063) &\
&3/2 &(122336.098) & & & 3/2 &(123646.360) &\
\
2\[1\] & 3/2 &(122358.405) &\
& 1/2 &(122332.608) &\
[crcrcrcrcrcrrrr]{} & & & & & & & & & & &\
& & & & & & & & & & &\
6\[9\] & 19/2 & 122954.180&$+$14.465\
& 17/2 & 122952.730&$+$20.251\
\
6\[8\] & 17/2 & 123007.910 &$+$26.752 & 5\[8\] & 17/2 & 123219.200&$-$10.198\
& 15/2 & 122910.920 &$-$16.531 & & 15/2 & 123193.090&$-$17.864\
\
6\[7\] & 15/2 & 123018.430 &$+$34.439 & 5\[7\]& 15/2 & 123238.440&$-$6.653 &4\[7\] & 15/2 & 123396.250&$-$33.027\
& 13/2 & 123015.400 &$+$40.333 & & 13/2 & 123168.680&$-$33.645 & & 13/2 & 123355.490&$-$36.436\
\
6\[6\] &13/2 & 122990.620 &$-$2.720 & 5\[6\]& 13/2 & 123249.650&$-$6.519 &4\[6\] &13/2 & 123414.730&$-$32.244\
&11/2 & 123037.430 &$+$26.878 & & 11/2 & 123270.340&$+$0.899 & &11/2 & 123427.119&$-$33.418\
\
6\[5\] &11/2 & 123002.288 &$+$33.455 & 5\[5\]& 11/2 & 123251.470&$-$1.320 &4\[5\] & 11/2 & 123441.100&$-$26.889\
&9/2 & 123026.350 &$+$18.587 & & 9/2 & 123269.378&$+$2.937 & &9/2 & 123435.468&$-$17.705\
\
6\[4\] &9/2 & 122988.215 &$+$30.836 & 5\[4\]& 9/2 & 123258.994&$-$1.556 & 4\[4\] & 9/2 & 123460.690&$-$26.898\
&7/2 & 122980.408 &$+$26.752 & & 7/2 & 123258.021&$-$1.362 & &7/2 & 123435.277&$-$16.103\
\
6\[3\] &7/2 & 122946.419 &$+$21.403 & 5\[3\] & 7/2 & 123235.165&$+$3.471 & 4\[3\] & 7/2 & 123451.449&$-$21.115\
&5/2 &(122967.896)& & & 5/2 & (123248.017)& & & 5/2 & 123430.181&$-$16.906\
\
& & & & 5\[2\] & 5/2 & 123211.159&$-$1.017 & 4\[2\] & 5/2 & (123401.927)\
& & & & & 3/2 & 123213.323&$-$12.585 & & 3/2 & (123384.857)&\
\
& & & & & & & &4\[1\] & 3/2& (123356.410)\
& & & & & & & & & 1/2 & (123343.705)\
[crcrcrcrcrcrrrr]{} & & & & & & & & & & & &\
& & & & & & & & & & &\
4\[7\] & 15/2 & 124421.468 &$+$12.238 &\
& 13/2 & 124436.436 &$+$36.895 &\
\
4\[6\] &13/2 & 124400.107 &$+$4.567 & 3\[6\]& 13/2 & 124661.274 &$+$15.827 &\
&11/2 & 124402.557 &$-$3.593 & & 11/2 & 124656.535 &$+$7.092 &\
\
4\[5\] &11/2 & 124388.840 &$+$3.174 & 3\[5\]& 11/2 & 124626.900 &$+$3.179 &2\[5\] & 11/2& 124803.873&$+$20.054\
&9/2 & 124385.706 &$+$2.938 & & 9/2 & 124636.116 &$+$3.120 & & 9/2 & 124809.727&$+$15.721\
\
4\[4\] &9/2 & 124401.939 &$+$4.674 & 3\[4\]& 9/2 & 124623.120 &$+$3.085 &2\[4\] & 9/2 & 124793.905 &$+$12.624\
&7/2 & 124385.010 &$+$0.698 & & 7/2 & 124620.914 &$+$7.289 & & 7/2 & 124783.748 &$+$15.272\
\
4\[3\] &7/2 & 124416.110&$+$13.187 & 3\[3\] & 7/2 & 124641.989 &$+$9.092 &2\[3\] & 7/2 &(124814.025)&\
&5/2 & 124403.474 &$+$1.243 & & 5/2 & 124653.022 &$-$8.651 & & 5/2 &(124808.178)\
\
4\[2\] &5/2 & 124434.563 &$+$23.142 & 3\[2\] & 5/2 &(124670.316)& & 2\[2\] & 5/2 &(124835.676) &\
&3/2 & 124460.410 &$-$11.802 & & 3/2 &(124678.325) & & & 3/2 &(124833.418) &\
\
4\[1\] &3/2 & (124487.989)& & 3\[1\] &3/2 &(124697.077) & & 2\[1\] & 3/2 &(124876.972)&\
&1/2 & (124484.721)& & &1/2 &(124708.453) & & & 1/2 &(124874.375)&\
\
& & & & 3\[0\] &1/2 & 124731.762 &$-$4.875\
[crcrcrcrcrcrrrr]{} & & & & & & & & & & & &\
& & & & & & & & & & &\
5\[8\] & 17/2 & 127507.241& $-$5.657\
& 15/2 & 127524.1227&$+$14.501\
\
5\[7\] & 15/2 & 127484.653 &$-$1.445 &4\[7\]& 15/2 & 127892.981&$+$4.313\
& 13/2 & 127515.235 &$+$2.816 & & 13/2 & 127895.260&$+$3.367\
\
5\[6\] &13/2 & 127489.429 &$-$4.823 &4\[6\]& 13/2 & 127875.000 &$+$2.236 & 3\[6\] &13/2 & 128110.214&$-$2.182\
&11/2 & 127489.977 &$-$0.294 & & 11/2 & 127880.436 &$+$1.216 & &11/2 &(128076.012)&\
\
5\[5\] &11/2 &127482.748&$+$3.147 & 4\[5\]& 11/2 & 127869.158 &$+$0.993 & 3\[5\] & 11/2& 128071.171&$-$10.517\
&9/2 & (127484.561)& & & 9/2 & 127855.952 &$-$16.898 & & 9/2 & 128055.658&$-$16.898\
\
5\[4\] &9/2 & 127485.362 &$-$15.194 & 4\[4\]& 9/2 & 127869.892 &$-$4.920 & 3\[4\] & 9/2 & 128062.710&$-$15.669\
&7/2 & 127485.699&$+$9.404 & & 7/2 & (127871.098)& & & 7/2 & 128066.823&$-$22.228\
\
5\[3\] &7/2 &(127476.624)& & 4\[3\] & 7/2 &(127877.776) & & 3\[3\] & 7/2 & (128047.849)\
&5/2 &127510.913 &$+$9.552 & & 5/2 & 127874.745&$+$5.549 & & 5/2 & 128063.103&$-$8.192\
\
5\[2\] &5/2 &(127499.343) & & 4\[2\] & 5/2 &(127868.807)& &3\[2\] & 5/2 & 128089.313&$+$10.032\
&3/2 &127487.681 &$-$0.341 & & 3/2 &(127895.930) & & & 3/2 & (128069.044) &\
\
& & & &4\[1\] &3/2 &(127876.787) & & 3\[1\] & 3/2 &(128099.051) &\
& & & & &1/2 &(127898.510) & & & 1/2 & (128099.237) &\
\
& & & & & & & &3\[0\] &1/2 &(128161.312) &\
The new lines
=============
The new lines in the 3800-8000Å region, produced by transitions to the subconfigurations ($^{3}$P)4f, ($^{3}$H)4f, ($^{3}$F)4f, and ($^{3}$G)4f, are shown in Tables6$-$9, respectively. Only lines with $\log\,gf$$\ge$$-$1.50 are listed, because lines with lower $\log\,gf$ values are not observable in this wavelength region of HR6000. The new lines are mostly concentrated in the 5100-5400Å interval. The upper energy levels (cols.1$-$4) were derived as described in Sect.3, the lower energy levels (cols.5$-$6) are those described in Sect.4, the calculated wavelength (col.7) is the Ritz wavelength in air, the $\log\,gf$ values (col.8) were computed by Kurucz, the observed wavelengths (col.9) are the wavelengths of lines well observable in the HR6000 spectrum. Most of them were listed as unidentified lines in Castelli & Hubrig (2007)[^3]. In the last column, comments derived from the comparison of the observed and computed spectra are added for most lines. In a few cases, both computed and observed stellar lines correspond to lines measured by Johansson in laboratory works (Johansson 1978; Castelli, Johansson, & Hubrig 2008). The notes “J78” and “lab” are added for these lines. When lines are computed weaker than the observed ones the disagreement can be due either to a too low $\log\,gf$ value or to some unknown component which increases the line intensity. When lines are computed much stronger than the observed ones, some problem with the energy levels or/and $\log\,gf$ computations is very probably present. When we observed a very good agreement between the observed and computed lines, either isolated or blends, we added the note “good agreement”.
[llrlrlrr]{} & & &\
& & &\
3d$^{6}$($^{3}$P)4d& $^{2}$F &7/2 & 103191.917 & $+$27.014\
3d$^{6}$($^{3}$P)4d& $^{2}$D &5/2 & 103597.402 & $-$5.701\
3d$^{6}$($^{3}$F)4d& $^{2}$F &7/2 & 105775.491 & $-$42.697\
\
3d$^{6}$($^{3}$H)5d& $^{4}$H &13/2 & 124208.725 & $+$47.495\
3d$^{6}$($^{3}$H)5d& $^{4}$G &11/2 & 124251.805 & $+$44.041\
3d$^{6}$($^{3}$H)5d& $^{4}$K &15/2 & 124297.017 & $-$5.220\
3d$^{6}$($^{3}$H)5d& $^{4}$I &15/2 & 124357.304 & $+$12.292\
3d$^{6}$($^{3}$H)5d& $^{4}$K &13/2 & 124415.353 & $-$14.256\
3d$^{6}$($^{3}$H)5d& $^{2}$I &11/2 & 124976.008 & $-$38.096\
3d$^{6}$($^{3}$F)5d& $^{4}$H &13/2 & 125732.991 & $+$9.243\
\
3d$^{6}$($^{5}$D)6d& $^{6}$D &5/2 & 113934.466 & $-$58.836\
3d$^{6}$($^{5}$D)6d& $^{4}$D &7/2 & 114009.934 & $-$3.477\
3d$^{6}$($^{5}$D)6d& $^{6}$G &7/2 & 114428.399 & $+$51.787\
3d$^{6}$($^{5}$D)6d& $^{6}$G &5/2 & 114619.007 & $+$22.415\
Figure1 shows the spectrum in the 5185-5196Åinterval, computed before and after the determination of the new energy levels. Figure2 compares the observed spectrum of HR6000 with the synthetic spectrum computed with the line list including the new lines. When the two figures are considered together, the improvement in the comparison between the observed and computed spectra is evident.
-0.2cm
-1.0cm
Conclusions
===========
Computed atomic data and stellar spectra observed at high resolution and high signal-to-noise ratio of the iron$-$overabundant, slow$-$rotating star HR6000 were used to extend laboratory studies on energy levels and line transitions. We identified as about 500 unidentified spectral lines in the 3800$-$8000Å region. A few of these lines were already identified as iron from laboratory analyses (Johansson 2007, private communication), but they were never classified. Because numerous other new lines are components of blends they contribute to improve the agreement between observed and computed spectra. On the other hand, there is a small number of new lines which are not observed in the spectrum. We believe that they are due to computational problems related with the mixing of the even energy levels rather than to incorrect energy values for the new 4f odd levels.
In spite of the large number of the new identified lines, several medium-strong lines and a conspicuous number of weak lines remain still unidentified in the spectral region we analyzed. If we examine the list of the lines which correspond to transitions from predicted energy levels, we can count about 4600 lines with $\log\,gf$$\ge$$-$1.0, where about 400 of them have $\log\,gf$$\ge$0.0. Because the transitions producing these lines occur between high-excitation energy levels that are not strongly populated, most of the lines are weak in a star like HR6000. This large number of weak predicted lines could explain the spectrum of HR6000 longward of about 5800Å. The spectrum looks like it is affected by a noise larger than that due to the instrumental effects. Castelli & Hubrig (2007) explained this “noise” with the presence of a T-Tauri star affecting the HR6000 spectrum. After this study, we prefer to state that the spectrum shows the presence of numerous weak lines from high-excitation levels, probably 4d, 5d, 6d $-$ 4f, 5f, 6f transitions, which still have to be identified. The hypothesis of the presence of the T-Tauri star affecting the HR6000 spectrum is an example of an incorrect conclusion that can be drawn owing to the use of incomplete line lists. We will extend this study of the spectrum to the near infrared region in the near future using CRIRES (CRyogenic high-resolution InfraRed Echelle Spectrograph) observations of HR6000 and 46Aql. The observations are scheduled in summer 2010 (ESO proposal 41380, P.I. S. Hubrig).
Adam, J., Baschek, B., Johansson, S., Nilsson, A. E., & Brage, T. 1987, , 312, 337
Bi[é]{}mont, E., Johansson, S., & Palmeri, P. 1997, , 55, 559
Castelli, F., & Hubrig, S. 2007, , 475, 1041
Castelli, F., Johansson, S., & Hubrig, S. 2008, Journal of Physics Conference Series, 130, 012003
Castelli, F., Kurucz, R., & Hubrig, S. 2009, , 508, 401 (Paper I)
Cowan, R. D. 1981, The Theory of Atomic Structure and Spectra (Berkeley: Univ. California Press)
Johansson, S. 1978, , 18, 217
Johansson, S. 2009, , T134, 014013
Johansson, S., & Baschek, B. 1988, Nuclear Instruments and Methods in Physics Research B, 31, 222
Kurucz, R. L. 2005, Memorie della Societa’ Astronomica Italiana, Supplementi, 8, 14
Kurucz, R. L. 2009, American Institute of Physics Conference Series, 1171, 43
Rosberg, M., & Johansson, S. 1992, , 45, 590
Sugar, J., & Corliss, C. 1985, J. Phys. Chem. Ref. Data, 14, Supp.2
[llllllllllll]{} & & & & &\
& & & & & & & &\
122351.810 & ($^{3}$P)4f & 2\[5\] & 11/2 & 103165.320 & ($^{3}$P)4d $^{4}$F$_{9/2}$ & 5210.550 & $+$0.795& 5210.55 & good agreement\
& & & & 103683.070 & ($^{5}$D)5d $^{4}$F$_{9/2}$ & 5355.059 & $+$0.164& 5355.06 & computed too strong\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5380.493 & $-$1.047& & at the noise level\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5698.178 & $-$0.539& & blend with a telluric line\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5733.913 & $-$0.635& 5733.90 & computed too weak\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6396.332 & $-$0.741& 6396.32 & computed too weak\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 7972.359 & $-$0.985& & at the noise level\
\
122324.142 & ($^{3}$P)4f & 2\[5\] & 9/2 & 103102.860 & ($^{3}$P)4d $^{4}$D$_{7/2}$ & 5201.118 & $-$0.056& & wrong,not observed\
& & & & 103191.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ & 5225.329 & $+$0.634& & blend, good agreement\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 5451.698 & $-$1.133& & blend, good agreement\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5488.097 & $-$0.362& & blend, good agreement\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5603.024 & $-$0.170& 5603.05 &\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5811.956 & $-$1.441& &blend,good agreement\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 6041.116 & $-$0.837&6041.1 & weak,good agreement\
\
122355.116 & ($^{3}$P)4f & 2\[4\] & 9/2 & 102394.718 & ($^{5}$D)6s $^{4}$D$_{7/2}$ & 5008.523 & $-$0.809& &weak, computed too strong\
& & & & 103102.860 & ($^{3}$P)4d $^{4}$D$_{7/2}$ & 5192.750 & $+$0.657& 5192.75& lab, good agreement\
& & & & 103165.320 & ($^{3}$P)4d $^{4}$F$_{9/2}$ & 5209.652 & $-$0.035& 5209.66& lab, good agreement\
& & & & 103191.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ & 5216.883 & $-$0.404& & blend\
& & & & 103683.070 & ($^{5}$D)5d $^{4}$F$_{9/2}$ & 5354.110 & $-$0.637& 5354.1 & weak\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5478.781 & $-$1.319& & at the continuum level\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5697.105 & $-$1.443& & at the continuum level\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 6413.457 & $-$1.407& & blend\
\
122355.550 & ($^{3}$P)4f & 2\[4\] & 7/2 & 102394.718 & ($^{5}$D)6s $^{4}$D$_{7/2}$ & 5008.414 & $-$1.258& & good agreement\
& & & & 102802.312 & ($^{5}$D)6s $^{4}$D$_{5/2}$ & 5112.818 & $-$0.959& 5112.82 & computed too weak\
& & & & 103002.670 & ($^{3}$P)4d $^{4}$D$_{5/2}$ & 5165.751 & $+$0.441& 5165.75 & lab, good agreement\
& & & & 103102.860 & ($^{3}$P)4d $^{4}$D$_{7/2}$ & 5192.633 & $+$0.155& 5192.62&lab, computed too weak\
& & & & 103165.320 & ($^{3}$P)4d $^{4}$F$_{9/2}$ & 5209.534 & $-$1.105& & blend, good agreement\
& & & & 103191.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ & 5216.765 & $-$0.764& &blend\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ & 6425.418 & $-$1.436& & at the continuum level\
\
122351.488 & ($^{3}$P)4f & 2\[3\] & 7/2 & 103102.860 & ($^{3}$P)4d $^{4}$D$_{7/2}$ & 5193.729 & $-$1.320& &blend\
& & & & 103191.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ & 5217.871 & $-$0.250& 5217.870 & lab\
& & & & 103597.402 & ($^{3}$P)4d $^{2}$D$_{5/2}$ & 5330.689 & $+$0.525& 5330.680 & lab\
& & & & 104023.910 & ($^{3}$H)4d $^{4}$G$_{5/2}$ & 5454.742 & $-$1.327& &at the continuum level\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5479.870 & $-$1.320& &at the continuum level\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5594.450 & $-$1.116& 5594.42 & computed too weak ?\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5622.022 & $-$0.573& 5622.02 & computed too weak ?\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5840.440 & $-$1.282& & at the continuum level\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 6689.941 & $-$0.330& 6689.91 &\
\
123629.520 & ($^{3}$P)4f & 1\[4\] & 9/2 & 103102.860 & ($^{3}$P)4d $^{4}$D$_{7/2}$ & 4870.353 & $-$1.402& & at the continuum level\
& & & & 104000.810 & ($^{5}$D)5d $^{6}$P$_{7/2}$ & 5093.159 & $-$0.981& &blend\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5121.112 & $+$0.327& 5121.1 & lab, good agreement\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5221.043 & $+$0.408& 5221.04 & lab, good agreement\
& & & & 104873.230 & ($^{5}$D)5d $^{4}$D$_{7/2}$ & 5330.062 & $-$1.183& & blend\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5364.564 & $-$0.118&5364.55& computed too strong\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5401.999 & $-$0.418& &blend\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5430.640 & $-$1.066&5430.64 & computed too weak\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5599.422 & $-$0.624& 5599.42 & good agreement\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 5928.743 & $-$0.677& 5928.72 & at the noise level\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 7426.139 & $-$1.173& &\
\
[lllllllllll]{} & & & & &\
& & & & & & & &\
123637.833 & ($^{3}$P)4f & 1\[4\] & 7/2 & 102802.312 & ($^{5}$D)6s $^{4}$D$_{5/2}$ & 4798.155 & $-$1.297& &at the continuum level\
& & & & 103002.670 & ($^{3}$P)4d $^{4}$D$_{5/2}$ & 4844.743 & $-$0.954& &computed too strong\
& & & & 103597.402 & ($^{3}$P)4d $^{2}$D$_{5/2}$ & 4988.521 & $-$0.339& 4988.51 & lab\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5118.932 & $-$0.819& 5118.95 & lab, computed too weak\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 5122.163 & $-$1.282& &\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5218.777 & $-$0.644& &blend\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5242.763 & $+$0.180& 5242.775 & lab\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5362.172 & $-$1.268& & at the continuum level\
& & & & 105127.770 & ($^{5}$D)5d $^{4}$D$_{5/2}$ & 5400.965 & $-$1.143& & at the continuum level\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5432.211 & $-$0.531& & wrong, not observed\
& & & & 105379.430 & ($^{3}$F)4d $^{4}$D$_{5/2}$ & 5475.409 & $-$0.552& 5475.42 & computed too strong\
& & & & 105711.730 & ($^{5}$D)5d $^{6}$S$_{5/2}$ & 5576.909 & $-$1.432& &at the continuum level\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5735.883 & $-$1.221& &at the continuum level\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ & 5936.184 & $-$1.317& & at the level of the noise\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 5960.996 & $-$0.565& 5961.00 &\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 6159.712 & $-$0.665& 6179.75 & blend ?\
& & & & 110428.280 & ($^{3}$G)4d $^{4}$F$_{5/2}$ & 7568.195 & $-$1.229& & no spectrum\
\
123615.875 & ($^{3}$P)4f & 1\[3\] & 7/2 & 103597.402 & ($^{3}$P)4d $^{2}$D$_{5/2}$ & 4993.993 & $-$1.435& &\
& & & & 104023.910 & ($^{3}$H)4d $^{4}$G$_{5/2}$ & 5102.711 & $-$0.526& 5102.7 & lab, good agreement\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5124.694 & $-$1.046& 5124.69 & good agreement\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 5127.932 & $-$0.244& &wrong, not obs\
& & & & 104209.610 & ($^{3}$H)4d $^{2}$F$_{5/2}$ & 5151.540 & $-$0.081 & 5151.52 & J78, lab, computed too weak\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5224.766 & $-$0.973& 5227.77 & good agreement\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5248.807 & $-$0.232& 5248.801 & computed too strong\
& & & & 105127.770 & ($^{5}$D)5d $^{4}$D$_{5/2}$ & 5407.380 & $-$1.391& 5407.37 & computed too weak\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5438.700 & $-$0.416& 5438.70 & computed too strong\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5743.118 & $-$0.454& 5743.10 & good agreement\
\
123649.493 & ($^{3}$P)4f & 1\[3\] & 5/2 & 104209.610 & ($^{3}$H)4d $^{2}$F$_{5/2}$ & 5142.631 & $-$1.288& & at the continuum level\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5239.559 & $-$1.150& 5239.56 & good agreement\
& & & & 104572.920 & ($^{3}$P)4d $^{4}$F$_{3/2}$ & 5240.573 & $+$0.071& 5240.587 & lab, good agreement\
& & & & 104588.710 & ($^{5}$D)5d $^{6}$D$_{3/2}$ & 5244.914 & $-$1.288& & blend\
& & & & 104839.998 & ($^{3}$P)4d $^{2}$D$_{3/2}$ & 5314.985 & $-$0.441& & blend,computed too strong\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5428.771 & $-$1.471& & blend\
& & & & 105317.440 & ($^{3}$P)4d $^{2}$P$_{3/2}$ & 5453.411 & $+$0.082& 5453.42 & lab, computed too strong\
& & & & 105518.140 & ($^{3}$H)4d $^{4}$F$_{3/2}$ & 5513.777 & $-$0.591& & wrong, not observed\
& & & & 106846.650 & ($^{3}$F)4d $^{4}$F$_{3/2}$ & 5949.725 & $-$1.358& & at the continuum level\
& & & & 107430.250 & ($^{3}$F)4d $^{2}$D$_{3/2}$ & 6163.810 & $-$0.253& & wrong, not observed\
& & & & 108105.900 & ($^{3}$F)4d $^{2}$P$_{3/2}$ & 6431.741 & $-$0.724& & blend\
\
124157.060 & ($^{3}$P)4f & 0\[3\] & 5/2 & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5103.788 & $-$1.191& 5103.8 & good agreement\
& & & & 104572.920 & ($^{3}$P)4d $^{4}$F$_{3/2}$ & 5104.750 & $+$0.094& 5104.75 & lab, good agreement\
& & & & 104588.710 & ($^{5}$D)5d $^{6}$D$_{3/2}$ & 5108.869 & $-$1.369& &\
& & & & 104839.998 & ($^{3}$P)4d $^{2}$D$_{3/2}$ & 5175.329 & $-$1.125& &blend\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5283.154 & $-$0.937& &blend\
& & & & 105317.440 & ($^{3}$P)4d $^{2}$P$_{3/2}$ & 5306.486 & $-$1.020& 5306.49 & computed too weak\
& & & & 105460.230 & ($^{3}$F)4d $^{4}$D$_{3/2}$ & 5347.013 & $-$0.482& 5347.05 & blend\
& & & & 105518.140 & ($^{3}$H)4d $^{4}$F$_{3/2}$ & 5363.626 & $+$0.082& 5363.61 &computed too strong\
& & & & 106846.650 & ($^{3}$F)4d $^{4}$F$_{3/2}$ & 5775.269 & $-$0.286& 5775.25 & good agreement\
& & & & 107430.250 & ($^{3}$F)4d $^{2}$D$_{3/2}$ & 5976.771 & $-$0.922& & blend\
& & & & 108105.900 & ($^{3}$F)4d $^{2}$P$_{3/2}$ & 6228.356 & $-$0.686& 6228.34 & good agreement\
& & & & 110609.540 & ($^{3}$G)4d $^{4}$F$_{3/2}$ & 7379.392 & $-$1.370& &at the continuum level\
[lllllllllll]{} & & & & &\
& & & & & & & &\
122954.180 & ($^{3}$H)4f & 6\[9\] & 19/2 & 103644.800 & ($^{3}$H)4d $^{4}$K$_{17/2}$ & 5177.388 & $+$1.169 & 5177.394 & J78, lab, good agreement\
\
122952.730 & ($^{3}$H)4f &6\[9\] & 17/2 & 103644.800 & ($^{3}$H)4d $^{4}$K$_{17/2}$ & 5177.777 & $-$0.930 & & blend\
& & & & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5194.384 & $+$0.798 & 5194.387 & lab, good agreement\
& & & & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 5241.181 & $+$0.558 & 5241.183 & J78, lab, good agreement\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5308.346 & $+$0.518 & 5308.350 & J78,lab, good agreement\
\
123007.910 & ($^{3}$H)4f & 6\[8\] & 17/2 & 103644.800 & ($^{3}$H)4d $^{4}$K$_{17/2}$ & 5163.021 & $+$0.498 & 5163.018 & J78,lab, good agreement\
& & & & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5179.534 & $+$0.534 & 5179.540 & J78, lab, good agreement\
& & & & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 5226.062 & $+$0.820 & 5226.070 & lab, good agreement\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5292.838 & $-$1.419 & &\
& & & & 108337.860 & ($^{3}$G)4d $^{4}$I$_{15/2}$ & 6814.729 & $-$1.183 & & at the noise level\
\
122910.920 & ($^{3}$H)4f & 6\[8\] & 15/2 & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5205.693 & $-$0.207 & 5205.70 & blend\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5239.942 & $+$0.015 & 5239.948 & J78, lab, computed too weak\
& & & & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 5252.695 & $-$0.107 & 5252.702 & lab, computed too weak\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5304.620 & $-$0.357 & 5304.60 & lab, computed too weak\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5320.157 & $+$0.082 & 5320.18 & lab, good agreement\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5376.136 & $+$0.132 & 5376.12 & lab, computed too weak\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5466.362 & $+$0.698 & 5466.38 & good agreement\
& & & & 108463.910 & ($^{3}$G)4d $^{4}$I$_{13/2}$ & 6919.939 & $-$0.887& & at the continuum level\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$ & 7009.596 & $-$1.436& 7009.6 ? & computed too weak ?\
& & & & 109049.600 & ($^{3}$G)4d $^{2}$I$_{13/2}$ & 7212.332 & $-$1.456& 7212.33 ? & computed too weak ?\
\
123018.430 & ($^{3}$H)4f & 6\[7\] & 15/2 & 103617.580 & ($^{3}$H)4d $^{4}$H$_{13/2}$ & 5152.978 & $+$0.761 & 5152.985 & lab, good agreement\
& & & & 103644.800 & ($^{3}$H)4d $^{4}$K$_{17/2}$ & 5160.218 & $-$0.354 & 5160.213 & lab, good agreement\
& & & & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5176.713 & $+$0.364 & 5176.722 & J78,lab, good agreement\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5210.580 & $-$1.104 & 5210.65 ? & computed too weak ?\
& & & & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 5223.190 & $+$0.447 & 5223.25 & blend, good agreement\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5274.530 & $-$1.138 & 5274.53 & good agreement\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5289.892 & $-$0.894 & 5289.899 & lab, good agreement\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5434.415 & $-$1.378 & & at the noise level\
& & & & 108337.860 & ($^{3}$G)4d $^{4}$I$_{15/2}$ & 6809.845 & $-$1.228 & & at the noise level\
\
123015.400 & ($^{3}$H)4f& 6\[7\]& 13/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 5149.230 & $+$0.424& 5149.243 & lab, good agreement\
& & & & 103617.580 & ($^{3}$H)4d $^{4}$H$_{13/2}$ & 5153.783 & $+$0.761& 5153.786 & lab, good agreement\
& & & & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5177.525 & $-$0.341& & blend\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5189.655 & $-$0.783& & blend, good agreement\
& & & & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 5224.017 & $-$0.132& 5224.025 & lab, good agreement\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5290.740 & $-$1.258& 5290.730 & computed too weak\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5477.945 & $-$1.275& 5477.95 & good agreement\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5568.910 & $-$1.164& 5568.92 & good agreement\
& & & & 105288.850 & ($^{3}$F)4d $^{4}$H$_{13/2}$ & 5639.690 & $-$1.357& & blend\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5891.220 & $-$1.302& & blend\
& & & & 108181.550 & ($^{3}$G)4d $^{4}$G$_{11/2}$ & 6739.478 & $-$1.459& & at the noise level\
\
122990.620 & ($^{3}$H)4f & 6\[6\] & 13/2 & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5184.178 & $-$0.976 & & blend\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5196.339 & $-$0.126& 5196.32 & computed too weak\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5218.143 & $-$0.028& 5218.149 & lab, good agreement\
& & & & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 5230.790 & $-$1.208& 5230.80 & good agreement\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5257.034 & $-$0.940& & blend\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5282.281 & $-$1.039& 5282.29 &blend,computed too weak\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5297.687 & $-$1.010& 5297.7 & blend\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5313.049 & $-$0.954& & blend\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5353.192 & $+$0.205& 5353.22 & blend, computed too strong\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5442.643 & $+$0.049& 5442.65 & J78, lab, good agreement\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5485.393 & $+$0.141& 5485.40& computed too strong\
[llllllllllll]{} & & & & &\
& & & & & & & &\
122990.620 & cont. & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5576.608 & $-$0.487& 5576.60& computed too strong\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5803.114 & $-$0.380& 5803.12& computed too weak\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5899.835 & $+$0.277& 5899.82 & good agreement\
& & & & 108630.429 & ($^{1}$I)5s e$^{2}$I$_{11/2}$ & 6961.775 & $-$1.168& &at the continuum level\
& & & & 109049.600 & ($^{3}$G)4d $^{2}$I$_{13/2}$ & 7171.100 & $-$1.477& & at the continuum level\
& & & & 109389.880 & ($^{3}$G)4d $^{2}$I$_{11/2}$ & 7350.516 & $-$1.297& 7350.49 ? & computed too weak ?\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 7512.581 & $-$0.706& & blend, computed too weak ?\
\
123037.430 & ($^{3}$H)4f & 6\[6\] & 11/2 & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5183.727 & $+$0.242 & 5183.713 &J78, lab, blend\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5189.016 & $-$0.187 & 5189.013 & lab\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5205.425 & $-$0.558 & 5205.427 & lab, blend\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5216.891 & $-$0.503 & & blend\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5269.248 & $-$0.797 & 5269.235 &\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5339.807 & $-$0.759 &\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5428.808 & $-$0.405 & 5428.80& lab\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5471.340 & $-$0.934 &\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5483.874 & $-$0.019 & 5483.85 & lab\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5516.963 & $-$0.234 & &wrong, not obs\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5562.084 & $-$1.223 &\
& & & & 105398.850 & ($^{3}$F)4d $^{4}$H$_{11/2}$ & 5667.818 & $-$1.176 & &\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5787.389 & $-$0.146 & 5787.35\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5883.582 & $+$0.287 & 5883.58 & J78\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5901.584 & $-$0.581 & & blend\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6204.452 & $-$1.391 &\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 7486.247 & $-$0.596& &\
\
123002.288 & ($^{3}$H)4f & 6\[5\] & 11/2 & 103165.320 & ($^{3}$P)4d $^{4}$F$_{9/2}$ & 5039.690 & $-$0.526&\
& & & & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 5152.712 & $+$0.662& 5152.70 & lab\
& & & & 103617.580 & ($^{3}$H)4d $^{4}$H$_{13/2}$ & 5157.271 & $+$0.380& & blend\
& & & & 103683.070 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5174.754 & $-$0.491& 5174.75& lab\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5193.192 & $-$0.719 & 5193.191& blend\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5198.501 & $-$1.338 &\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5481.886 & $-$1.256 &\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5494.468 & $-$0.835 &\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5527.686 & $-$1.221 & 5527.68& computed too weak\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5572.983 & $-$0.697 & 5572.98\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5895.778 & $-$1.407 &\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6140.765 & $-$0.940 &\
& & & & 108181.550 & ($^{3}$G)4d $^{4}$G$_{11/2}$ & 6745.444 & $-$1.310 &\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 7579.208 & $-$1.201 &\
\
123026.350 & $^{3}$H)4f & 6\[5\] & 9/2 & 103102.860 & ($^{3}$P)4d $^{4}$D$_{7/2}$ & 5017.801 & $-$1.092&\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5186.706 & $-$0.152& 5186.722& lab\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5192.002 & $+$0.073& 5192.010 & lab\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5219.909 & $-$0.488& & blend\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5284.389 & $-$0.355&\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5390.860 & $-$1.184&\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5487.209 & $+$0.186& 5487.21 & lab\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5520.339 & $-$0.063& &wrong, not observed\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5544.006 & $-$1.091&\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5791.103 & $-$0.522& 5791.05\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5887.421 & $-$0.109& 5887.42\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5905.446 & $-$0.710&\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6131.699 & $-$1.253&\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 6148.685 & $-$1.351&\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6208.722 & $-$0.916&\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 7492.464 & $-$1.002& &\
[llllllllllll]{} & & & & &\
& & & & & & & &\
122988.215 & ($^{3}$H)4f & 6\[4\] & 9/2 & 103165.320 & ($^{3}$P)4d $^{4}$F$_{9/2}$ & 5043.266 & $-$0.030&\
& & & & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 5156.450 & $+$0.529& 5156.45 & lab\
& & & & 103683.070 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5178.524 & $-$0.018& 5178.53 & lab\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5196.989 & $-$0.773&\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5202.306 & $-$0.787&\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5486.117 & $-$1.286 &\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5498.718 & $-$0.382& 5498.72\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5531.988 & $-$1.028&\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5577.356 & $-$0.785 & 5577.35\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5900.673 & $-$1.342&\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6146.075 & $-$0.412& 6146.08\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6223.461 & $-$1.178&\
& & & & 108181.550 & ($^{3}$G)4d $^{4}$G$_{11/2}$ & 6751.852 & $-$1.421 &\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 7587.298 & $-$0.695 &\
\
122980.408 & ($^{3}$H)4f & 6\[4\] & 7/2 & 103102.860 & ($^{3}$P)4d $^{4}$D$_{7/2}$ & 5029.399 & $-$0.735&\
& & & & 103165.320 & ($^{3}$P)4d $^{4}$F$_{9/2}$ & 5045.253 &$-$0.962&\
& & & & 103683.070 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5180.619 &$-$1.116& &\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5204.420 & $-$0.034& 5204.419\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5232.461 & $-$0.656& &\
& & & & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5245.466 & $-$1.235& &\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5297.253 & $+$0.049& 5297.26&\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5404.248 & $-$0.598&\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5501.081 & $-$0.147&\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5534.379 & $-$0.071&\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5558.167 & $-$0.731&\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5921.516 & $-$0.986&\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6149.026 & $-$0.728&\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 6166.108 & $-$1.069&\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6226.487 & $-$1.380&\
\
122946.419 & ($^{3}$H)4f & 6\[3\] & 7/2 & 103102.860 & ($^{3}$P)4d $^{4}$D$_{7/2}$ & 5038.014 & $-$1.413&\
& & & & 103165.320 & ($^{3}$P)4d $^{4}$F$_{9/2}$ & 5053.922 &$+$0.160&\
& & & & 103683.070 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5189.760 &$+$0.167& 5189.763 & lab.\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5213.645 & $-$0.746&\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5306.811 & $-$0.814&\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5511.388 & $-$0.043& 5511.40\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5619.156 & $-$1.229 &\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5636.890 & $-$1.411&\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5933.462 & $-$1.332&\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6161.908 & $-$0.227& 6161.90\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6239.696 & $-$0.856&\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 7611.442 & $-$0.504 &\
\
123219.200 & ($^{3}$H)4f & 5\[8\] & 17/2 & 103644.800 & ($^{3}$H)4d $^{4}$K$_{17/2}$ & 5107.290 & $-$0.983&\
& & & & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5123.448 & $+$0.347& 5123.45 & lab\
& & & & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 5168.969 & $+$0.064& & blend\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5234.285 & $+$0.991& 5234.283 & lab\
\
123193.090 & ($^{3}$H)4f & 5\[8\] &15/2 & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5130.313 & $-$0.507&\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5163.574 & $+$0.908& 5163.55 & lab\
& & & & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 5175.957 & $-$0.540& 5175.95 &\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5226.368 & $-$0.216& & blend\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5241.450 & $-$0.301& 5241.465& lab\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5295.776 & $-$0.452& 5295.773&\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5383.304 & $+$0.146& 5383.32 & blend\
[llllllllllll]{} & & & & &\
& & & & & & & &\
123238.440 & ($^{3}$H)4f& 5\[7\] & 15/2 & 103617.580 & ($^{3}$H)4d $^{4}$H$_{13/2}$ & 5095.196 & $-$0.836 & 5095.19\
& & & & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5118.401 & $-$0.254 & 5118.40 & lab\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5151.507 & $-$0.716 & & blend\
& & & & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 5163.831 & $-$0.599 & 5163.82 & lab\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5214.007 & $+$0.873 & 5214.99 & blend\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5229.017 & $-$0.045 & 5229.030 & lab\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5283.085 & $+$0.323 & 5283.093 & lab\
& & & & 105288.850 & ($^{3}$F)4d $^{4}$H$_{13/2}$ & 5569.611 & $-$1.005& & blend\
\
123168.680 & ($^{3}$H)4f & 5\[7\]& 13/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 5108.895 & $-$1.165\
& & & & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5136.747 & $-$1.256\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5148.687 & $+$0.010 & 5148.7 & lab\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5170.092 & $-$1.170\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5208.267 & $-$0.275 & 5208.268 & computed too weak\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5233.046 & $+$0.138 & 5233.041\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5263.242 & $-$0.600\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5302.633 & $-$0.581\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5390.389 & $+$0.010 & 5390.38 &computed too strong\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5432.319 & $+$0.495 & 5432.31 & lab\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5521.763 & $-$0.481 & 5521.78\
& & & & 105398.850 & ($^{3}$F)4d $^{4}$H$_{11/2}$ & 5625.954 & $-$1.425\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5743.747 & $-$0.321 & 5743.75 &computed too strong\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5838.483 & $-$0.311\
& & & & 108630.429 & ($^{1}$I)5s e$^{2}$I$_{11/2}$ & 6876.509 & $-$1.228\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 7413.385 & $-$0.848\
\
123249.650 & ($^{3}$H)4f & 5\[6\] & 13/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 5087.842 & $-$0.510 & 5087.85 & lab\
& & & & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5115.465 & $-$1.027\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5127.305 & $+$0.392 & 5127.32& lab, blend\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5148.533 & $+$0.357 & 5148.52& lab\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5186.389 & $+$0.210 & 5186.396 & lab\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5210.960 & $-$0.403 & 5210.964\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5225.953 & $-$0.742 & & blend\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5240.901 & $-$0.464 & 5240.911\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5279.957 & $-$0.647 & &blend\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5366.958 & $+$0.032 & 5366.95 & lab\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5497.178 &$-$1.156 &\
& & & & 105288.850 & ($^{3}$F)4d $^{4}$H$_{13/2}$ & 5566.135 &$-$1.260 &\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5717.150 &$-$0.553 & 5717.18 &\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5811.004 & $-$0.182 & 5811.00\
& & & & 109049.600 & ($^{3}$G)4d $^{2}$I$_{13/2}$ & 7040.287 & $-$1.496\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 7369.139 & $-$1.023\
\
123270.340 & ($^{3}$H)4f & 5\[6\] & 11/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 5082.491 &$-$0.827 & & blend\
& & & & 103683.070 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5103.934 &$-$1.365& &\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5121.871 &$+$0.373& 5121.89 & lab\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5127.035 &$-$0.542 & 5127.05\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5143.054 &$-$0.456 & 5143.05\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5154.246 &$+$0.127 & 5154.25 & lab\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5180.829 &$-$0.529 & 5180.84 & lab\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5205.347 &$-$0.844 & 5235.225\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5235.223 &$-$0.536\
& & & & 104192.480 & ($^{3}$H)4d $^{4}$I$_{9/2}$ & 5240.220 &$-$1.229\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5274.195 & $-$1.310\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5361.004 & $-$0.422 & 5361.00& lab\
[llllllllllll]{} & & & & &\
& & & & & & & &\
123270.340 & cont. & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5414.696 &$-$0.589 & 5414.7 & blend\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5446.953 & $-$0.182& 5446.95\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5490.931 &$-$1.162 &\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5518.678 & $-$0.927 & & wrong,not observed\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5710.394 &$-$0.287 & 5710.40\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5804.025 &$-$0.029 & 5804.02\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6041.291 &$-$1.018\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6116.045 &$-$1.092\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 7357.917 &$-$0.867\
\
123251.470 &($^{3}$H)4f &5\[5\] & 11/2 & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5126.827 & $-$0.236 & & blend\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5132.001 & $+$0.078 & 5132.0 & lab\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5159.265 & $+$0.007 & 5159.29 & lab, blend\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5185.899 & $+$0.058 & 5185.901 & lab\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5210.466 & $-$0.583\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5240.401 & $-$0.177 & 5240.405 & lab\
& & & & 104192.480 & ($^{3}$H)4d $^{4}$I$_{9/2}$ & 5245.408 & $-$1.139 & & blend\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5279.449 & $-$1.308\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5407.990 & $+$0.040 & 5407.99 &lab\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5420.234 & $-$1.131\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5452.558 & $-$0.967 & 5452.55\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5496.628 & $-$0.739 & 5496.62\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5524.433 & $-$1.032& &\
& & & & 105524.460 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5639.544 & $-$1.347 &\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5801.269 & $-$0.770 & & computed too strong\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5810.389 & $-$1.328\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5827.945 & $-$0.015& 5827.95& computed too weak\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6123.114 & $-$0.236\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 7336.744 & $-$1.064\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 7548.984 & $-$1.185\
\
123269.378 &($^{3}$H)4f &5\[5\] & 9/2 & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5122.123 & $-$1.173 & &blend\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5127.287 & $-$0.734 & &blend\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5154.501 & $+$0.418 & 5154.50 & lab\
& & & & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5167.121 & $-$0.470 & 5167.1 & computed too weak\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5181.086 & $-$0.545 & 5181.1 & blend, computed too weak\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 5183.700 & $-$0.079 & & blend\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 5184.458 & $-$0.485 & 5184.463 & computed too strong\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5217.365 & $-$1.017 &\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5235.486 & $-$0.560 &\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5402.756 & $-$0.812 &\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5414.977 & $-$0.955\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5470.281 & $-$1.409\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5509.211 & $-$0.290 & 5509.2\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5539.003 & $-$1.382\
& & & & 105524.460 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5633.853 & $-$1.381 &\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5795.246 & $-$0.974 &\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5821.868 & $-$0.325& 5821.88\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6041.643 &$-$1.431\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 6107.415 &$-$0.980\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6116.405 & $-$0.472& & blend\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 7327.115 & $-$1.238\
\
123258.994&($^{3}$H)4f &5\[4\] & 9/2 & 103165.320 & ($^{3}$P)4d $^{4}$F$_{9/2}$ & 4975.303 & $-$1.479\
& & & & 103191.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ & 4981.898 & $-$0.587\
& & & & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 5085.425 & $-$1.404\
& & & & 103683.070 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5106.894 & $-$0.960 & &\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5124.850 & $+$0.047 & 5124.82 & lab\
[llllllllllll]{} & & & & &\
& & & & & & & &\
123258.994& cont. & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5130.020 & $+$0.269 & 5130.0 & lab\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5157.263 & $-$0.663 & &blend\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5324.070 & $-$0.506 & & blend\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5418.025 & $-$0.657 & 5418.02 & lab\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5450.323 & $+$0.051 & 5450.30& wrong, computed too strong\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5494.356 & $-$1.301\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5512.367 & $-$0.848\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5522.138 & $-$0.450 & 5522.10 & computed too strong\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5539.264 & $-$1.434 &\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5714.098 & $-$0.740 & 5714.10\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5807.851 & $-$0.440 & 5807.85 & blend\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5825.392 & $-$0.814\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6045.483 & $-$0.970\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 6061.948 & $-$1.148\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 6111.293 & $-$1.488\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 6724.229 & $-$1.436\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 7364.069 & $-$1.370\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 7636.319 & $-$1.343\
\
123258.021 &($^{3}$H)4f &5\[4\] & 7/2& 102802.312 & ($^{5}$D)6s $^{4}$D$_{5/2}$ & 4887.246 & $-$1.497 & &blend\
& & & & 103002.670 & ($^{3}$P)4d $^{4}$D$_{5/2}$ & 4935.589 & $-$1.223& & blend\
& & & & 103102.860 & ($^{3}$P)4d $^{4}$D$_{7/2}$ & 4960.124 & $-$1.397 & &at the continuum level\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5130.276 & $-$0.633 & &blend\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5157.521 & $-$0.254 & & blend\
& & & & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5170.156 & $-$0.375 & &blend\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 5186.755 & $-$0.078 & &blend\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 5187.514 & $-$0.396 & 5187.52 &\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5220.459 & $-$1.202 & &computed too strong\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 5223.820 & $-$0.829 & &blend\
& & & & 104209.610 & ($^{3}$H)4d $^{2}$F$_{5/2}$ & 5248.321 & $-$0.898 & &blend\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5349.313 & $-$0.940 & &wrong, not observed\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5450.611 & $-$1.412 & &blend\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5473.683 & $-$0.926 & &blend\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5512.661 & $+$0.003 & 5512.65\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5542.490 & $-$1.205 & &blend\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5799.064 & $-$1.320 & &blend\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5825.721 & $-$0.559 & 5825.73\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 6099.124 & $-$1.189 & &blend\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 6111.655 & $-$0.698 & &blend\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6120.658 & $-$0.942 & &at the continuum level\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 7636.885 & $-$1.434 & & no spectrum\
\
123235.165 &($^{3}$H)4f &5\[3\] & 7/2& 103191.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ & 4987.820 & $-$0.173 &\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5136.300 & $-$0.037 & 5136.30\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5163.610 & $-$0.154 & & blend\
& & & & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5176.274 & $-$0.716 & 5176.25&\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 5192.913 & $-$0.799 & & blend\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 5193.673 & $-$0.887 & & blend\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5226.698 & $-$1.309 & &\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5330.834 & $-$0.226 & 5330.81 & computed too strong\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5425.030 & $-$0.825 & 5425.01 &\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5457.411 & $-$0.238 & 5457.40 &\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5519.618 & $-$1.438 &\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5529.415 & $-$0.668 & 5529.40& wrong, computed too strong\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5549.523 & $-$1.242 &\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5571.298 & $-$1.482 &\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 6054.160 & $-$1.224\
[llllllllllll]{} & & & & &\
& & & & & & & &\
123235.165 & cont. & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 6070.719 & $-$0.626 & 6070.71\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 7650.242 & $-$0.970\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 7893.688 & $-$1.448\
\
123211.159 &($^{3}$H)4f &5\[2\] & 5/2 & 103193.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ & 4993.801 &$-$0.145 & 4993.80 & computed too strong\
& & & & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5182.716 &$-$1.163 & 5182.707 & good agreement\
& & & & 103986.330 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 5200.159 &$-$1.442\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5337.666 &$-$0.236 & & blend\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5487.763 &$-$1.396 & & blend\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5526.943 &$-$0.560 & 5526.92 & computed too strong\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5578.762 &$-$1.365 & & at the level of the noise\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 6079.581 &$-$0.532 & 6709.60 & good agreement\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 6129.215 &$-$1.126 & & blend\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 7664.321 &$-$0.703 & & in telluric\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 7908.679 &$-$1.384 & & in telluric\
\
123213.323 &($^{3}$H)4f &5\[2\] & 3/2 & 102802.312 & ($^{5}$D)6s $^{4}$D$_{5/2}$ & 4897.949 &$-$1.090 & 4897.90 & at the level of the noise\
& & & & 103597.402 & ($^{3}$P)4d $^{2}$D$_{5/2}$ & 5096.480 &$-$1.325 & & at the level of the noise\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 5236.050 &$-$0.269 & 5236.046 & computed too strong\
& & & & 104209.610 & ($^{3}$H)4d $^{2}$F$_{5/2}$ & 5260.666 &$-$0.338 & 5260.682 & lab, good agreement\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5362.139 &$-$0.684 & & wrong, not observed\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5560.475 &$-$1.142\
& & & & 105414.180 & ($^{3}$F)4d $^{4}$G$_{5/2}$ & 5616.690 &$-$1.055 & & blend\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ & 6089.687 &$-$1.322 & &blend\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 6115.802 &$-$0.758 & 6115.80 & good agreement\
& & & & 110428.280 & ($^{3}$G)4d $^{4}$F$_{5/2}$ & 7819.490 &$-$1.269 & & at the continuum level\
\
123396.250 & ($^{3}$H)4f & 4\[7\] & 15/2 & 103706.530 & ($^{3}$H)4d $^{4}$K$_{15/2}$ & 5077.377 &$-$1.404\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5109.953 &$-$0.102 & 5109.95 & lab\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5171.443 &$+$0.259 & 5171.45 & lab\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5239.390 &$+$0.861 & 5239.394& J78\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5325.048 &$+$0.257 & 5325.05 & J78, lab\
\
123355.490 & ($^{3}$H)4f & 4\[7\] & 13/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 5060.583 & $-$1.409\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5099.623 &$-$0.221 & 5099.6 & lab\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5120.621 &$-$1.170 & 5120.62 & lab, computed too weak\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5158.067 &$+$0.788 & 5158.05 & J78, lab\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5182.370 &$+$0.034 & 5182.371 & lab\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 5197.198 &$-$1.475\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5250.606 &$-$0.778 & 5250.609 & computed too weak\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5336.635 &$-$0.215 & 5336.62\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5377.729 &$-$0.165 & 5377.71 & J78, lab, computed too weak\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5682.754 &$-$0.574 & 5682.75\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5775.473 & $-$0.674\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 7312.092 & $-$1.277\
\
123414.730 & ($^{3}$H)4f & 4\[6\] & 13/2 & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 5084.259 &$-$0.750 &\
& & & & 103832.050 & ($^{3}$H)4d $^{4}$K$_{13/2}$ & 5105.131 &$-$0.704\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5142.349 &$-$0.245 & 5142.35 & lab\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 5166.504 &$-$0.525 & & blend\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5195.934 &$+$0.922& 5195.942& lab\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5234.320 &$-$0.791 & & blend\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5319.812 &$-$1.134 &\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5360.646 &$-$0.638 & 5360.65 & computed too weak\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5447.727 &$-$1.416 &\
& & & & 105398.850 & ($^{3}$F)4d $^{4}$H$_{11/2}$ & 5549.118 &$-$1.185 &\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5755.774 & $-$1.242\
[llllllllllll]{} & & & & &\
& & & & & & & &\
123427.119 & ($^{3}$H)4f & 4\[6\] & 11/2 & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5086.139 &$-$0.441 & 5086.15\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5112.917 &$-$0.423 & & blend\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5139.074 &$+$0.124 & 5139.10\
& & & & 104192.480 & ($^{3}$H)4d $^{4}$I$_{9/2}$ & 5197.506 &$+$0.465 & 5197.56 & blend\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5230.927 &$-$1.051\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5316.307 &$-$1.253\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5357.088 &$+$0.165& 5357.10 & J78,lab\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5369.102 &$-$1.260\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5444.051 &$-$0.902\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5659.712 &$-$0.911\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5742.735 &$-$0.704& & computed too strong\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5751.672 & $-$1.454\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5768.874 & $-$0.115 & 5768.90 & J78, computed too weak\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 5984.595 & $-$1.089\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6057.941 & $-$0.358& 6057.92 & blend\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 7243.378 & $-$1.142\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 7450.174 & $-$1.329\
\
123441.100 & ($^{3}$H)4f &4\[5\] &11/2& 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5082.524 &$-$0.439 & 5082.51 & computed too strong\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5109.263 &$+$0.037 & 5109.29 & lab\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5135.383 &$-$1.089\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5188.822 &$+$0.224 & 5188.831 & lab\
& & & & 104192.480 & ($^{3}$H)4d $^{4}$I$_{9/2}$ & 5193.731 &$+$0.573 & 5193.74 & J78, lab\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 5227.103& $-$1.390 &\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5353.077 & $-$0.299& & blend\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5439.910 &$-$1.230\
& & & & 105524.460 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5579.854 &$-$1.306\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5738.126 &$-$1.011& & computed too strong, not obs\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5764.224 & $-$0.455 & 5764.20\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 5979.588 & $-$1.109\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6052.813 & $-$0.460& 6052.8\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 7236.043 & $-$1.361\
\
123435.468 & ($^{3}$H)4f &4\[5\] &9/2 & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5123.141 &$+$0.119 & 5123.190 &blend\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5136.869 &$-$0.836& & blend\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 5139.439 &$+$0.314 & &blend\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 5140.184 &$-$0.208& 5140.2 & lab\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5172.529 &$-$1.242&\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5190.340 &$-$1.319\
& & & & 104192.480 & ($^{3}$H)4d $^{4}$I$_{9/2}$ & 5195.251 &$+$0.450& 5195.26& lab\
& & & & 105589.670 & ($^{3}$F)4d $^{4}$H$_{7/2}$ & 5602.005 &$-$1.242\
\
123460.690 & ($^{3}$H)4f &4\[4\] &9/2 & 103191.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ &4932.321 & $-$1.442\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5077.467 &$-$0.602 & 5077.5 & lab\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5104.153 &$-$0.047 & 5104.15\
& & & & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5116.528 &$-$0.613& 5116.52\
& & & & 103973.780 & ($^{3}$H)4d $^{4}$K$_{11/2}$ & 5130.220 &$-$1.289&\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 5132.783 &$-$0.961 &\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 5133.527 &$-$0.989&\
& & & & 104174.27 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 5183.552 &$-$0.937\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5267.488 &$-$0.494 & 5267.47\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5347.468 &$-$0.307 & 5347.45& lab\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5359.439 &$-$1.442\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5413.610 &$-$0.234 & 5413.60 & lab\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5434.117 &$-$1.217\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5451.734 &$-$0.292 & 5451.72\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5480.906 &$-$0.700& &blend\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5502.146 &$-$0.769\
& & & & 105449.540 & ($^{5}$D)5d $^{4}$G$_{7/2}$ & 5550.575 &$-$1.270\
[llllllllllll]{} & & & & &\
& & & & & & & &\
123460.690& cont. & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5731.681 &$-$0.446& & wrong, not observed\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5757.720& $+$0.118 & 5757.72 & J78, computed too low\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 5972.589 & $-$0.946\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 5988.704 & $-$1.212\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 6036.859 & $-$0.912\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6045.643 & $-$0.124& 6045.65\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 7225.797 & $-$0.960\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 7431.576 & $-$1.109\
\
123435.277 & ($^{3}$H)4f &4\[4\] &7/2 & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5123.191 &$-$0.068 & & blend\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 5139.489 &$+$0.217 & 5139.45 & lab, blend\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 5140.234 &$-$0.435 & 5140.20 & blend\
& & & & 104023.910 & ($^{3}$H)4d $^{4}$G$_{5/2}$ & 5150.186 &$+$0.144 & 5150.15 & lab\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 5175.880 &$-$1.206 & & blend\
& & & & 104192.480 & ($^{3}$H)4d $^{4}$I$_{9/2}$ & 5195.303 &$-$0.325 & & blend\
& & & & 104209.610 & ($^{3}$H)4d $^{2}$F$_{5/2}$ & 5199.932 &$-$1.066 & 5199.95 & computed too weak\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5299.053 &$-$0.753 & & computed too strong\
& & & & 105414.180 & ($^{3}$F)4d $^{4}$G$_{5/2}$ & 5547.511 &$-$1.009 & & at the level of the noise\
& & & & 105589.670 & ($^{3}$F)4d $^{4}$H$_{7/2}$ & 5602.065 &$-$1.328 & & blend\
& & & & 105630.750 & ($^{5}$D)5d $^{4}$G$_{5/2}$ & 5614.990 &$-$1.423 & & at the continuum level\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 6237.560 &$-$1.471 & &at the continuum level\
\
123451.449 & ($^{3}$H)4f &4\[3\] &7/2 & 103191.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ & 4934.571 &$-$1.453 &\
& & & & 103597.402 & ($^{3}$P)4d $^{2}$D$_{5/2}$ & 5035.352 &$-$0.856 &\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 5079.851 &$-$1.218 &\
& & & & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 5106.563 &$-$0.583 & 5106.55\
& & & & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5118.949 &$-$1.061&\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 5135.220 &$-$0.335&\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 5135.964 &$-$1.420& 5135.95\
& & & & 104023.910 & ($^{3}$H)4d $^{4}$G$_{5/2}$ & 5145.899 &$-$0.764\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 5168.256 &$-$1.230\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 5171.550 &$-$1.408&\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5270.054 &$-$0.654 & &blend\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5294.515 &$-$1.314\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5416.320 &$-$0.276 & 5416.32 & lab\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5454.483 &$-$0.324 & 5454.50 & blend\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5483.684 & $-$0.695& &\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5504.945 &$-$0.792 & 5504.95\
& & & & 105449.540 & ($^{5}$D)5d $^{4}$G$_{7/2}$ & 5553.424 &$-$1.292\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5734.719 &$-$1.053&\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5760.786& $-$0.536 & 5760.78 & computed too weak\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 5992.021 & $-$1.212\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 6040.230 & $-$1.110\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 6049.023 & $-$0.751&\
\
123430.181 & ($^{3}$H)4f &4\[3\] &5/2 & 103597.402 & ($^{3}$P)4d $^{2}$D$_{5/2}$ & 5040.752 &$-$1.238 & &blend\
& & & & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 5124.529 &$-$0.535 & 5124.52\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 5140.836 &$-$0.648 & 5140.83\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 5141.582 &$-$0.884 & & blend\
& & & & 104023.910 & ($^{3}$H)4d $^{4}$G$_{5/2}$ & 5151.538 &$+$0.030 & 5151.52 & J78, lab\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 5177.246 &$-$0.906 & & blend\
& & & & 104209.610 & ($^{3}$H)4d $^{2}$F$_{5/2}$ & 5201.311 &$-$0.851 & & blend, wrong ?\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 5300.485 &$-$0.786 & & blend, computed too strong\
& & & & 104572.920 & ($^{3}$P)4d $^{4}$F$_{3/2}$ & 5301.522 &$-$0.742 & & wrong, not observed\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5422.568 &$-$1.395 & & at the continuum level\
& & & & 105317.440 & ($^{3}$P)4d $^{2}$P$_{3/2}$ & 5519.442 &$-$1.271 & 5519.43 & at the level of the noise\
[llllllllllll]{} & & & & &\
& & & & & & & &\
123430.181 & cont. & & & 105379.430 & ($^{3}$F)4d $^{4}$D$_{5/2}$ & 5538.397 &$-$1.442 & &at the level of the noise\
& & & & 105414.180 & ($^{3}$F)4d $^{4}$G$_{5/2}$ & 5549.080 &$-$0.905 & & blend\
& & & & 105630.750 & ($^{5}$D)5d $^{4}$G$_{5/2}$ & 5616.598 &$-$1.451 & & blend\
& & & & 106846.650 & ($^{3}$F)4d $^{4}$F$_{3/2}$ & 6028.409 &$-$1.085 & 6028.40 & at the level of the noise\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 6035.729 &$-$1.269 &\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 6239.544 &$-$1.446 &\
& & & & 110428.280 & ($^{3}$G)4d $^{4}$F$_{5/2}$ & 7689.067 &$-$1.409 &\
& & & & 110609.540 & ($^{3}$G)4d $^{4}$F$_{3/2}$ & 7797.776 &$-$1.406 &\
[llllllllllll]{} & & & & &\
& & & & & & & &\
124421.468 & ($^{3}$F)4f & 4\[7\] & 15/2 & 103617.580 & ($^{3}$H)4d $^{4}$H$_{13/2}$ & 4805.451 & $-$0.972& 4805.42&\
& & & & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 4910.993 & $-$1.090& & at the continuum level\
& & & & 104119.710 & ($^{3}$H)4d $^{2}$K$_{15/2}$ & 4924.307 & $-$1.174& & not obs\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 5049.309 & $-$1.258& 5049.3 & very weak\
& & & & 105288.847 & ($^{3}$F)4d $^{4}$H$_{13/2}$ & 5225.221 & $+$0.974& 5225.229 & lab, J78\
\
124436.436 & ($^{3}$F)4f & 4\[7\] & 13/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 4798.043 & $-$1.190& &at the continuum level\
& & & & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 4833.123 & $-$1.441& &\
& & & & 104315.370 & ($^{3}$H)4d $^{2}$K$_{13/2}$ & 4968.529 & $-$1.078& 4968.53 & very weak\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5082.213 & $-$1.265& &blend\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5160.416 & $-$0.003& 5160.4 & lab\
& & & & 105288.847 & ($^{3}$F)4d $^{4}$H$_{13/2}$ & 5221.136 & $-$0.831& & blend,weak component\
& & & & 105398.852 & ($^{3}$F)4d $^{4}$H$_{11/2}$ & 5251.306 & $+$0.664& 5251.321 & blend\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5353.789 & $+$0.076& 5353.80 &\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5436.006 & $-$0.154& 5436.12 &\
& & & & 108630.429 & ($^{1}$I)5s e$^{2}$I$_{11/2}$ & 6324.960 & $-$1.433& &at the continuum level\
\
124400.107 & ($^{3}$F)4f & 4\[6\] & 13/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 4806.424 & $-$0.542& 4806.4&\
& & & & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 4942.792 & $-$1.458& &very weak\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5091.616 & $-$0.517& 5091.6&\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5170.111 & $+$0.742& 5170.10& J78,lab, blended\
& & & & 105288.850 & ($^{3}$F)4d $^{4}$H$_{13/2}$ & 5231.062 & $+$0.278& 5231.067& lab\
& & & & 105398.850 & ($^{3}$F)4d $^{4}$H$_{11/2}$ & 5261.345 & $+$0.080& 5261.339& shifted ?\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5364.226 & $-$0.538& 5364.22&\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5446.766 & $-$0.314& 5446.75&blend\
\
124402.557 & ($^{3}$F)4f & 4\[6\] & 11/2 & 103683.070 & ($^{5}$D)5d $^{4}$F$_{9/2}$ & 4825.028 & $-$1.407&\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5090.983 & $-$1.256& &blend\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5101.830 & $-$1.382 & 5101.82&\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5130.460 & $+$0.158 &\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5169.456 & $-$0.871& &computed too strong\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5194.042 & $-$0.084 & 5194.047\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5209.193 & $-$0.494 & 5209.199\
& & & & 105398.852 & ($^{3}$F)4d $^{4}$H$_{11/2}$ & 5260.668 & $-$0.049 & 5260.682&\
& & & & 105524.461 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5295.671 & $-$1.274 & 5295.662& computed too weak\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5363.520 & $-$0.269 & 5363.51 &\
& & & & 106018.643 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5438.027 & $-$0.914 & &blend\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5446.039 & $-$0.626 & 5446.05 &\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5461.459 & $+$0.179 & 5461.48 &\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 5654.418 & $-$0.044 & & computed too strong\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5719.850 & $+$0.097 & 5719.85 & lab,J78\
& & & & 109925.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 6765.246 & $-$1.049 & &\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 6945.303 & $-$1.190 & &\
\
124388.840 & ($^{3}$F)4f & 4\[5\] & 11/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 4809.029 & $-$0.852& 4809.02&\
& & & & 103683.070 & ($^{5}$D)5d $^{4}$F$_{9/2}$ & 4828.222 & $-$0.829& &\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 4848.889 & $-$0.699& & weak, on the H$_{\beta}$ wing\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5094.540 & $-$0.517& 5094.55 & lab\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5105.404 & $+$0.158& 5105.4 &\
& & & & 104868.500 & ($^{5}$D)5d $^{6}$G$_{9/2}$ & 5121.435 & $-$0.968& 5121.45 & weak\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5134.072 & $-$0.161& & blend\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5173.126 & $+$0.425& 5173.12 & lab\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5197.747 & $-$0.166& 5197.756 &\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5212.916 & $-$0.199& &blend\
& & & & 105288.847 & ($^{3}$F)4d $^{4}$H$_{13/2}$ & 5234.147 & $-$0.630& &blend\
& & & & 105398.852 & ($^{3}$F)4d $^{4}$H$_{11/2}$ & 5264.468 & $-$0.717& 5264.45 &\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5450.112 & $-$1.282& & blend\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 5658.806 & $-$0.643& & blend\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5724.343 & $-$0.429& &blend, computed too strong\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 6858.267.& $-$0.903& &at the continuum level\
[lllllllllll]{} & & & & &\
& & & & & & & &\
124385.706 & ($^{3}$F)4f & 4\[5\] & 9/2 & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 4849.626 & $-$1.159 & &H$_{\beta}$ wing, not obs.\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 4900.742 & $-$1.404& & at the continuum level\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5106.222 & $-$0.305& &\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5155.371 & $-$0.195& 5155.37 &computed too strong\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5173.965 & $-$0.955& 5173.98 &computed too weak\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5189.933 & $-$0.112& &blend\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5198.594 & $-$0.154& 5198.596 &\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5213.769 & $-$0.389& 5213.78 &\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5216.634 & $-$1.420& &\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5235.599 & $-$0.769& & blend\
& & & & 105398.852 & ($^{3}$F)4d $^{4}$H$_{11/2}$ & 5265.337 & $-$0.986& 5265.323 &\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5371.899 & $+$0.199& 5371.90 &\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5443.015 & $-$1.240& &\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5466.492 & $-$0.492& 5466.49 &blend\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 5659.810 & $-$1.436& &blend\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 5674.279 & $-$1.037& 5674.30 &\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5717.492 & $-$1.080& &blend\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5725.370 & $-$0.147& 5725.35 &\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 7031.188 & $-$1.480& ¬ observed\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 7236.302 & $-$1.125& ¬ observed\
\
124401.939 & ($^{3}$F)4f & 4\[4\] & 9/2 & 103683.070 & ($^{5}$D)5d $^{4}$F$_{9/2}$ & 4825.170 & $-$0.851& &\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 4845.810 & $-$1.216& & on the H$_{\beta}$ wing\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5018.593 & $-$0.782& & blend FeII 5018.440\
& & & & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 5091.141 & $-$1.199& 5091.15 &\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5101.991 & $-$0.285 & & wrong,not observed\
& & & & 104868.500 & ($^{5}$D)5d $^{6}$G$_{9/2}$ & 5118.000 & $-$0.871& 5117.98 &\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5130.621 & $+$0.114& 5130.60 &lab\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5151.058 & $-$0.280& 5151.07 &lab\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5169.622 & $-$0.361& 5169.6 &\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5194.209 & $-$1.245& & blend FeIII\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5209.359 & $-$1.260& &\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5211.949 & $+$0.055& 5211.953 & lab\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5231.152 & $-$0.836& & blend\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5363.698 & $-$1.391& & blend\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5367.218 & $-$0.182& 5367.22 &\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5461.644 & $-$0.455& 5461.65&\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 5654.613 & $-$0.197& 5654.62&\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5712.189 & $-$1.361& &at the level of the noise\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 6852.110 & $-$0.955& &at the level of the noise\
\
124385.010 & ($^{3}$F)4f & 4\[4\] & 7/2 & 103191.917 & ($^{3}$P)4d $^{2}$F$_{7/2}$ & 4717.199 & $-$1.461& &\
& & & & 103597.402 & ($^{3}$P)4d $^{2}$D$_{5/2}$ & 4809.214 & $-$1.233& &\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5106.403 & $-$1.091& &\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5155.556 & $-$0.412& 5155.56 &\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5190.121 & $-$0.246& 5190.123 &\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5198.782 & $-$0.950& &blend\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5213.958 & $-$1.188& &blend\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5216.553 & $-$1.332& &blend\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5220.268 & $-$1.463& &\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5235.790 & $-$0.829& &blend\
& & & & 105775.836 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5372.100 & $+$0.165& 5372.10 &lab\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5466.700 & $-$1.095& &at the level of the noise\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5500.096 & $-$0.922& &blend\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 5674.503 & $-$1.298& 5674.50 &computed too weak\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ & 5684.004 & $-$0.895& &\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 5706.743 & $-$0.920& &\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5717.719 & $-$1.023& ¬ observed\
[lllllllllll]{} & & & & &\
& & & & & & & &\
124385.010 & cont. & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5725.598 & $-$0.824& 5725.60 &\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 5888.617 & $-$0.044& 5888.61 &\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 7236.667 & $-$1.221& &at the level of the noise\
\
124416.110 & ($^{3}$F)4f & 4\[3\] & 7/2 & 103683.070 & ($^{5}$D)5d $^{4}$F$_{9/2}$ & 4821.172 & $-$1.273&\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5015.025 & $-$0.607& 5015.02 &\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5098.304 & $-$0.623& &\
& & & & 104868.500 & ($^{5}$D)5d $^{6}$G$_{9/2}$ & 5114.290 & $-$1.355& &computed too strong\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5126.892 & $-$0.477& 5126.84 & lab, blend\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ &5147.300 & $+$0.051& 5147.25 & blend,lab\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ &5181.754 & $-$1.028& 5181.75 & computed too weak\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ &5190.388 & $-$1.077& & blend\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ &5205.515 & $-$1.184& & blend\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ &5208.101 & $+$0.031& 5208.99 &\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ &5227.276 & $-$1.201& & blend\
& & & & 105379.430 & ($^{3}$F)4d $^{4}$D$_{5/2}$ &5251.555 & $-$1.289& &at the continuum level\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ &5363.137 & $-$0.687& 5363.15 &\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ &5457.419 & $-$1.335& & blend\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ &5650.084 & $-$0.819& & blend\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ &5664.504 & $-$1.029& & at the level of the noise\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ &5673.972 & $-$0.486& 5673.93 &blend\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ &5877.850 & $-$1.281& & at the level of the noise\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ &6845.461 & $-$1.364& & not observed\
\
124403.474 & ($^{3}$F)4f & 4\[3\] & 5/2 & 103597.402 & ($^{3}$P)4d $^{2}$D$_{5/2}$ & 4804.946 & $-$1.146& 4804.93 & computed too weak\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5150.651 & $-$0.855& &\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5185.150 & $-$0.746& 5185.141& lab,blend\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5215.240 & $-$1.455& & blend\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5230.732 & $-$1.416& & blend\
& & & & 105317.440 & ($^{3}$P)4d $^{2}$P$_{3/2}$ & 5237.975 & $-$1.304& & blend\
& & & & 105460.230 & ($^{3}$F)4d $^{4}$D$_{3/2}$ & 5277.458 & $-$0.778& &wrong, not observed\
& & & & 105518.140 & ($^{3}$H)4d $^{4}$F$_{3/2}$ & 5293.641 & $-$1.294& 5293.627 & computed too low ?\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5366.775 & $-$0.450& 5366.78 &\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5494.515 & $-$0.721& 5494.51 &\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ & 5678.044 & $-$1.006& &computed too strong\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 5700.741 & $-$0.790& 5700.76 &\
& & & & 107065.900 & ($^{3}$F)4d $^{4}$P$_{3/2}$ & 5766.220 & $-$1.192& &at the level of the noise\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 5882.220 & $-$0.040& 5882.22 &\
& & & & 107430.250 & ($^{3}$F)4d $^{2}$D$_{3/2}$ & 5890.000 & $-$0.918& & blend NaI\
& & & & 108105.900 & ($^{3}$F)4d $^{2}$P$_{3/2}$ & 6134.185 & $-$0.702& 6134.2 &\
& & & & 110611.800 & ($^{3}$G)4d $^{2}$F$_{5/2}$ & 7248.754 & $-$1.434& & blend with telluric\
\
124434.563 & ($^{3}$F)4f & 4\[2\] & 5/2 & 103597.402 & ($^{3}$P)4d $^{2}$D$_{5/2}$ & 4797.777 & $-$1.440& &\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 4921.269 & $-$0.982& & blend\
& & & & 104209.610 & ($^{3}$H)4d $^{2}$F$_{5/2}$ & 4943.008 & $-$1.371& 4943.0 &\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 5010.387 & $-$0.817& 5010.4 &\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5142.414 & $-$0.113& 5142.42 & lab\
& & & & 105213.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5176.803 & $-$1.156& & blend\
& & & & 105127.770 & ($^{5}$D)5d $^{4}$D$_{5/2}$ & 5178.082 & $-$1.132& 5178.08 & computed too weak\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5203.100 & $-$0.191& 5203.10 &\
& & & & 105379.430 & ($^{3}$F)4d $^{4}$D$_{5/2}$ & 5246.469 & $-$0.830& & at the noise level, computed too strong\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5357.833 & $-$1.105& &\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5485.142 & $-$1.413& &\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 5658.587 & $-$1.147& & blend\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ & 5668.035 & $-$0.132& 5668.05 & computed too strong\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 5690.652 & $-$1.300& 5690.68 & computed too weak\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 5871.480 & $-$1.133& &\
\
[lllllllllll]{} & & & & &\
& & & & & & & &\
124460.410& ($^{3}$F)4f & 4\[2\] & 3/2 & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 4915.015 & $-$1.449& &\
& & & & 104189.380 & ($^{5}$D)5d $^{4}$P$_{3/2}$ & 4931.772 & $-$1.122& & wrong,not observed\
& & & & 105234.060 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5199.747 & $-$1.496& &\
& & & & 105317.440 & ($^{3}$P)4d $^{2}$P$_{3/2}$ & 5222.396 & $-$0.923& & blend\
& & & & 105379.430 & ($^{3}$F)4d $^{4}$D$_{5/2}$ & 5239.362 & $-$1.350& & blend\
& & & & 105460.230 & ($^{3}$F)4d $^{4}$D$_{3/2}$ & 5261.644 & $-$0.436& &wrong, not observed\
& & & & 105518.140 & ($^{3}$H)4d $^{4}$F$_{3/2}$ & 5277.730 & $-$1.098& & blend\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5477.375 & $-$1.153& & at the level of the noise\
& & & & 106846.650 & ($^{3}$F)4d $^{4}$F$_{3/2}$ & 5675.805 & $-$1.332& & at the level of the noise\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 5682.292 & $-$0.926& & at the level of the noise\
& & & & 107065.930 & ($^{3}$F)4d $^{4}$P$_{3/2}$ & 5747.356 & $-$0.824& & at the level of the noise\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 5862.580 & $-$0.452& 5862.58 & at the level of thec noise\
& & & & 107430.250 & ($^{3}$F)4d $^{2}$D$_{3/2}$ & 5870.308 & $-$0.663& 5870.30 & computed too weak\
& & & & 108105.900 & ($^{3}$F)4d $^{2}$P$_{3/2}$ & 6112.829 & $-$0.452& & EMISSION ?\
\
124661.274 & ($^{3}$F)4f& 3\[6\] &13/2 & 103751.660 & ($^{3}$H)4d $^{4}$H$_{11/2}$ & 4781.152 &$-$1.241 & 4781.15 & computed too weak\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$ & 5101.212 &$-$1.511 & 5101.2 & computed too weak\
& & & & 105398.852 & ($^{3}$F)4d $^{4}$H$_{11/2}$ & 5190.010 &$+$0.482 & 5190.012&\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5290.092 &$+$0.589 & 5290.094&\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5370.350 &$+$0.111 & 5370.3 & FeII,5270.284 main comp.\
\
124656.535 & ($^{3}$F)4f& 3\[6\] &11/2 & 103874.260 & ($^{3}$H)4d $^{4}$H$_{9/2}$ & 4810.449 &$-$1.268 & 4810.45 & weak\
& & & & 104192.480 & ($^{3}$H)4d $^{4}$I$_{9/2}$ & 4885.254 &$-$1.238 & & blend\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5126.398 &$-$0.847 & & very weak\
& & & & 105398.852 & ($^{3}$F)4d $^{4}$H$_{11/2}$& 5191.288 &$-$1.025 & & blend\
& & & & 105524.461 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5225.371 &$+$0.768 & 5225.364& lab + unid\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5291.420 &$-$1.047 & & very weak\
& & & & 106018.643 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5363.923 &$+$0.201 & 5363.92 & lab\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 5574.341 &$-$1.111 & 5574.25 &\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5637.925 &$-$0.160 & 5637.92 &\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 6650.935 &$-$1.387 & & blend\
\
124626.900 & ($^{3}$F)4f& 3\[5\] &11/2 & 103683.070 & ($^{5}$D)5d $^{4}$F$_{9/2}$ & 4773.341 &$-$1.317 & &\
& & & & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 4793.540 &$-$0.748 & 4793.55 &\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5044.081 &$-$0.396 & & wrong, not observed\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5072.063 &$-$0.515 & 5072.05 &\
& & & & 105063.550 & ($^{3}$F)4d $^{4}$G$_{11/2}$& 5110.175 &$-$1.355 & & blend\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5134.199 &$+$0.353 & 5134.20 & blend\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5149.000 &$-$0.004 & & blend\
& & & & 105398.852 & ($^{3}$F)4d $^{4}$H$_{11/2}$ &5199.288 &$-$0.178 & 5199.29 &\
& & & & 105524.461 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5233.477 &$-$0.662 & 5233.47 & computed too weak\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5299.732 &$-$0.158 &5299.717& lab\
& & & & 106018.643 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5372.464 &$-$0.223 & & blend\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 5380.285 &$-$0.656 &5380.29 &\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5395.335 &$+$0.054 & 5395.32 & computed too strong\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 5583.566 &$-$1.347 & &\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5647.362 &$-$0.074 & & blend\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 6748.062 &$-$1.222 & & at the level of the noise\
\
124636.116 & ($^{3}$F)4f& 3\[5\] &9/2 & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 4791.423 &$-$1.349 & &at the level of the noise\
& & & & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 4869.996 &$-$1.378 & &blend\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 4960.280 &$-$1.109 & 4960.28 &weak\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 5041.737 &$-$1.101 & &weak\
& & & & 104873.230 & ($^{5}$D)5d $^{4}$D$_{7/2}$ & 5058.579 &$-$1.461 & &weak\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5069.692 &$-$1.055 & &weak\
[lllllllllll]{} & & & & &\
& & & & & & & &\
124636.116 & cont. & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5089.646 &$-$0.797 & &weak\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5123.331 &$-$1.032 & &\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5131.770 &$-$0.298 & &blend\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5146.557 &$-$0.622 & &blend\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5149.085 &$+$0.286 & 5149.1 &lab\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5167.827.&$-$0.884 & 5167.82 &computed too weak\
& & & & 105398.852 & ($^{3}$F)4d $^{4}$H$_{11/2}$ &5196.797 &$-$1.467 & &at the level of the noise\
& & & & 105524.461 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5230.953 &$-$0.507 & 5230.959 &computed too weak\
& & & & 105589.670 & ($^{3}$F)4d $^{4}$H$_{7/2}$ & 5248.862 &$-$0.754 & 5248.801 &blend\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5297.144 &$-$1.481 & &weak\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5300.576 &$-$0.373 & &weak\
& & & & 106018.643 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5369.805 &$-$0.547 & 5369.81 &\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5392.652 &$-$0.592 & ¬ obs, wrong\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 5594.760 &$-$0.050 & ¬ obs, wrong\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5636.766 &$-$0.061 & 5636.78 &computed too weak\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5644.423 &$-$0.918 & &blend\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{7/2}$ & 6784.867 &$-$1.141 & &at the level of the noise\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 6909.500 &$-$1.099 & &at the level of the noise\
\
124623.120 & ($^{3}$F)4f& 3\[4\] &9/2 & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 4829.221 &$-$1.017 & 4829.25 &computed too weak\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 4843.700 &$-$1.308 & &computed too strong, not obs\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 4844.361 &$-$1.133 & &\
& & & & 104916.550 & ($^{3}$H)4d $^{4}$F$_{9/2}$ & 5073.036 &$-$1.028 & &\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5093.016 &$-$1.142 &5093.01 &weak\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5126.745 &$-$0.382 &5126.75 &lab, blend\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5135.196 &$-$0.318 & &blend\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5150.003 &$-$0.755 &5150.02 &\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5152.534 &$-$1.333 & &blend\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5171.301 &$+$0.425 &5171.305 &\
& & & & 105398.852 & ($^{3}$F)4d $^{4}$H$_{11/2}$& 5200.310 &$-$1.359 & &blend\
& & & & 105449.540 & ($^{5}$D)5d $^{4}$G$_{7/2}$ & 5214.058 &$-$0.628 & &blend\
& & & & 105524.461 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5234.513 &$-$0.157 & &blend\
& & & & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 5300.794 &$-$1.386 & &blend\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5304.231 &$-$0.076 & 5304.25 &blend\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5373.555 &$-$1.277 & &\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5396.435 &$-$0.899 & 5396.45 &computed too weak\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5640.900 &$-$0.389 & 5640.9 &computed too strong\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5648.568 &$-$0.369 & 5648.57 &blend\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 7114.048 &$-$1.243 & &\
\
124620.914 & ($^{3}$F)4f& 3\[4\] &7/2 & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 4829.735 &$-$1.435 & &\
& & & & 104023.910 & ($^{3}$H)4d $^{4}$G$_{5/2}$ & 4853.719 &$-$0.883 & &\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 4985.721 &$-$0.873 &4985.72 &weak\
& & & & 104993.860 & ($^{3}$F)4d $^{4}$D$_{7/2}$ & 5093.588 &$-$1.437 & &blend\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{5/2}$ & 5127.325 &$-$0.784 & &blend\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5135.778 &$-$1.386 & &weak\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5156.745 &$-$0.254 & &blend\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5171.891 &$+$0.011 &5171.9\
& & & & 105379.430 & ($^{3}$F)4d $^{4}$D$_{5/2}$ & 5195.658 &$-$0.478 &5195.661 &lab\
& & & & 105414.180 & ($^{3}$F)4d $^{4}$G$_{5/2}$ & 5205.058 &$-$0.783 & &blend\
& & & & 105449.540 & ($^{5}$D)5d $^{4}$G$_{7/2}$ & 5214.658 &$-$1.042 & &weak\
& & & & 105524.461 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5235.117 &$-$1.185 & &blend\
& & & & 105711.730 & ($^{5}$D)5d $^{6}$S$_{5/2}$ & 5286.964 &$-$0.934 & &blend\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5304.852 &$-$0.525 &5304.87 &blend\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5429.627 &$-$0.531 &5429.62 &computed too weak\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 5630.922 &$-$1.421 & &weak\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5641.602 &$-$0.724 &5641.61 &weak\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5649.272 &$-$1.404 & ¬ observed\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 5807.914 &$-$0.295 &5807.9 &blend\
[lllllllllll]{} & & & & &\
& & & & & & & &\
124641.989 & ($^{3}$F)4f& 3\[3\] &7/2 & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 4868.603 &$-$1.393 & &\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 4871.525 &$-$1.423 & &\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 4958.835 &$-$1.370 & &blend\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5121.789 &$-$0.828 & &\
& & & & 105155.090 & ($^{3}$F)4d $^{4}$G$_{9/2}$ & 5130.223 &$-$0.928 & &blend\
& & & & 105211.062 & ($^{5}$D)5d $^{4}$G$_{9/2}$ & 5145.002 &$-$1.290 & &at the level of the noise\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5147.528 &$-$0.014 & 5147.52 &\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5166.258 &$-$1.096 & &weak\
& & & & 105379.430 & ($^{3}$F)4d $^{4}$D$_{5/2}$ & 5189.973 &$-$0.210 & &blend\
& & & & 105414.180 & ($^{3}$F)4d $^{4}$G$_{5/2}$ & 5199.353 &$-$1.041 & &blend\
& & & & 105589.670 & ($^{3}$F)4d $^{4}$H$_{7/2}$ & 5247.244 &$-$0.996 & 5247.25 &weak\
& & & & 105711.730 & ($^{5}$D)5d $^{6}$S$_{5/2}$ & 5281.078 &$-$0.874 & ¬ observed\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5298.926 &$-$0.405 & &blend\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5390.945 &$-$1.384 & &blend\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5423.419 &$-$0.138 &5423.41 &lab\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 5592.922 &$-$0.422 & &wrong\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ & 5602.152 &$-$0.795 & &blend\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 5624.245 &$-$1.195 & &blend\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5634.900 &$-$0.588 &5634.9 &computed too weak\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5642.552 &$-$1.377 & &at the level of the noise\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 5800.811 &$-$0.993 & &at the level of the noise\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 6906.696 &$-$1.294 & &at the level of the noise\
& & & & 110611.800 & ($^{3}$G)4d $^{2}$F$_{5/2}$ & 7125.523 &$-$1.233 & &at the level of the noise\
\
124653.022 & ($^{3}$F)4f& 3\[3\] &5/2 & 104023.910 & ($^{3}$H)4d $^{4}$G$_{5/2}$ & 4846.164 &$-$1.115 & & weak\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 4977.751 &$-$0.819 & 4977.75&computed too weak\
& & & & 104839.998 & ($^{3}$P)4d $^{2}$D$_{3/2}$ & 5045.778 &$-$0.981 & 5045.79 &computed too weak\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5118.896 &$-$1.484 &\
& & & & 105234.237 & ($^{3}$H)4d $^{4}$F$_{5/2}$ & 5148.219 &$-$0.286 & &computed too strong\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5163.314 &$-$0.700 & 5163.29&weak\
& & & & 105317.440 & ($^{3}$P)4d $^{2}$P$_{3/2}$ & 5170.372 &$-$1.129 & &\
& & & & 105379.430 & ($^{3}$F)4d $^{4}$D$_{5/2}$ & 5187.002 &$-$0.628 & 5187.0&\
& & & & 105414.180 & ($^{3}$F)4d $^{4}$G$_{5/2}$ & 5196.371 &$-$0.956 & &blend\
& & & & 105460.230 & ($^{3}$F)4d $^{4}$D$_{3/2}$ & 5208.839 &$-$0.132 & 5208.862&lab, computed too strong\
& & & & 105711.730 & ($^{5}$D)5d $^{6}$S$_{5/2}$ & 5278.002 &$-$1.442 &\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5295.829 &$-$1.021 & &blend\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5420.175 &$-$0.824 &5420.2 &computed too weak\
& & & & 106846.650 & ($^{3}$F)4d $^{4}$F$_{3/2}$ & 5614.409 &$-$0.773 & &computed too strong\
& & & & 107065.930 & ($^{3}$F)4d $^{4}$P$_{3/2}$ & 5684.411 &$-$1.018 & &\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 5797.100 &$-$0.273 &5797.1&\
& & & & 107430.250 & ($^{3}$F)4d $^{2}$D$_{3/2}$ & 5804.657 &$-$0.981 & &at the level of the noise\
& & & & 108105.900 & ($^{3}$F)4d $^{2}$P$_{3/2}$ & 6041.674 &$-$0.519 & &\
\
124731.762 & ($^{3}$F)4f & 3\[0\] & 1/2 & 104189.380 & ($^{5}$D)5d $^{4}$P$_{3/2}$ & 4866.625 &$-$0.710 & & on the H$_{\beta}$ wing\
& & & & 104588.710 & ($^{5}$D)5d $^{6}$D$_{3/2}$ & 4963.106 &$-$1.473 & &\
& & & & 104736.460 & ($^{3}$P)4d $^{2}$P$_{1/2}$ & 4999.780 &$-$1.476 & &\
& & & & 105460.230 & ($^{3}$F)4d $^{4}$D$_{3/2}$ & 5187.556 &$-$1.137 & &\
& & & & 105477.920 & ($^{3}$F)4d $^{4}$D$_{1/2}$ & 5192.323 &$-$0.902 & & blend\
& & & & 105518.140 & ($^{3}$H)4d $^{4}$F$_{3/2}$ & 5203.192 &$-$0.854 & & blend\
& & & & 107065.930 & ($^{3}$F)4d $^{4}$P$_{3/2}$ & 5659.074 &$-$0.650 & 5659.05 & computed too weak\
& & & & 107176.100 & ($^{5}$D)5d $^{4}$P$_{1/2}$ & 5694.588 &$-$0.810 & 5694.59 & good agreement\
& & & & 107430.250 & ($^{3}$F)4d $^{2}$D$_{3/2}$ & 5778.239 &$-$0.939 & & blend\
& & & & 108105.900 & ($^{3}$F)4d $^{2}$P$_{3/2}$ & 6013.060 &$-$1.184 & &\
[lllllllllll]{} & & & & &\
& & & & & & & &\
124803.873 & ($^{3}$F)4f& 2\[5\] &11/2 & 103771.320 & ($^{3}$H)4d $^{4}$G$_{9/2}$ & 4753.206 &$-$1.359 & &\
& & & & 104807.210 & ($^{3}$H)4d $^{2}$G$_{9/2}$ & 4999.441 &$-$1.315 & &\
& & & & 105524.461 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5185.437 &$+$0.377 &5185.422 & lab\
& & & & 106018.643 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5321.852 &$+$0.731 &5321.83 & lab\
& & & & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 5344.292 &$-$1.008 &5344.28 &\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5591.464 &$-$0.173 & & computed too strong\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 6586.373 &$-$1.344 & & not observed\
\
124809.727 & ($^{3}$F)4f& 2\[5\] &9/2 & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 4786.078 &$-$1.434&\
& & & & 103983.510 & ($^{3}$G)5s $^{2}$G$_{7/2}$ & 4800.298 &$-$1.342 & &\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5121.860 &$-$1.107 & & blend\
& & & & 105449.540 & ($^{5}$D)5d $^{4}$G$_{7/2}$ & 5163.801 &$-$1.335 & & blend\
& & & & 105524.461 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5183.862 &$-$1.227 & & blend\
& & & & 105589.670 & ($^{3}$F)4d $^{4}$H$_{7/2}$ & 5201.450 &$+$0.802 &5201.444 & lab\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5252.229 &$-$1.121 & &\
& & & & 106018.643 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5320.193 &$-$0.866 & & blend\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 5540.925 &$-$1.367 & &\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5582.123 &$-$0.405 &5582.12\
\
124793.905 & ($^{3}$F)4f& 2\[4\] &9/2 & 103921.630 & ($^{3}$H)4d $^{4}$G$_{7/2}$ & 4789.706 &$-$1.174& 4789.7 & computed too weak\
& & & & 103986.330 & ($^{3}$H)4d $^{4}$H$_{7/2}$ & 4804.599 &$-$1.426 & &blend\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 4921.748 &$-$1.081 & &blend\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5082.234 &$-$0.341 & &blend\
& & & & 105220.600 & ($^{3}$H)4d $^{4}$F$_{7/2}$ & 5107.576 &$-$0.574 & &blend\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5126.016 &$+$0.065 & 5126.00 & lab.\
& & & & 105449.540 & ($^{5}$D)5d $^{4}$G$_{7/2}$ & 5168.025 &$-$1.175 & &good agreement\
& & & & 105524.460 & ($^{3}$F)4d $^{4}$H$_{9/2}$ & 5188.118 &$-$0.544 & 5188.12 & good agreement\
& & & & 105589.670 & ($^{3}$F)4d $^{4}$H$_{7/2}$ & 5205.735 &$-$0.340 & & blend\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 5256.599 &$-$0.442 &5256.599 & good agreement\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5324.675 &$-$0.131 &5234.68 & good agreementy=\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5587.059 &$+$0.466 & & blend\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 5594.582 &$-$1.114 & & blend\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{7/2}$ & 6712.979 &$-$1.436 & &\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 6834.961 &$-$1.262 & &\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 7028.628 &$-$1.389 & &\
\
124783.748 & ($^{3}$F)4f& 2\[4\] &7/2 & 104023.910 & ($^{3}$H)4d $^{4}$G$_{5/2}$ & 4815.647 &$-$0.780 & ¬ observed\
& & & & 104120.270 & ($^{5}$D)5d $^{6}$P$_{5/2}$ & 4838.105 &$-$1.439 & &\
& & & & 104209.610 & ($^{3}$H)4d $^{2}$F$_{5/2}$ & 4859.114 &$-$1.499 & &\
& & & & 104569.230 & ($^{3}$P)4d $^{4}$F$_{5/2}$ & 4945.559 &$-$1.176 & & weak\
& & & & 105123.000 & ($^{3}$H)4d $^{2}$G$_{7/2}$ & 5084.859 &$-$1.401 & &\
& & & & 105291.010 & ($^{3}$F)4d $^{4}$G$_{7/2}$ & 5128.687 &$-$0.876 & & blend\
& & & & 105414.180 & ($^{3}$F)4d $^{4}$G$_{5/2}$ & 5161.300 &$+$0.512 &5161.3 &lab, computed too strong\
& & & & 105589.670 & ($^{3}$F)4d $^{4}$H$_{7/2}$ & 5208.490 &$-$0.196 &5208.501 &\
& & & & 105630.750 & ($^{5}$D)5d $^{4}$G$_{5/2}$ & 5219.661 &$-$0.923 & & blend\
& & & & 106018.640 & ($^{3}$F)4d $^{2}$H$_{9/2}$ & 5327.557 &$-$1.482 & &\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 5382.029 &$-$0.281 &5382.12 &\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 5590.233 &$-$0.326 &5590.22 &\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 5753.486 &$-$0.930 & & at the level of the noise\
& & & & 110611.800 & ($^{3}$G)4d $^{2}$F$_{5/2}$ & 7054.248 &$-$1.377 & & at the level of the noise\
[llllllllllll]{} & & & & &\
& & & & & & & &\
127507.241 & ($^{3}$G)4f & 5\[8\] & 17/2 & 103878.370 & ($^{3}$H)4d $^{4}$I$_{15/2}$ & 4230.919 & $-$1.017& 4230.93 &\
& & & & 108337.860 & ($^{3}$G)4d $^{4}$I$_{15/2}$ & 5215.200 & $+$1.119& 5215.21 &\
\
127524.122 & ($^{3}$G)4f & 5\[8\] & 15/2 & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 4261.475 & $-$1.477& &\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 4365.238 & $-$1.210& &\
& & & & 108133.440 & ($^{3}$G)4d $^{4}$H$_{13/2}$ & 5155.680 & $-$0.971& &\
& & & & 108463.910 & ($^{3}$G)4d $^{4}$I$_{13/2}$ & 5245.071 & $+$0.889& 5245.073 &lab, J78\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$ & 5296.420 & $-$0.047& 5296.418 &\
& & & & 109049.600 & ($^{3}$G)4d $^{2}$I$_{13/2}$ & 5411.356 & $+$0.449& &blend\
\
127484.653 & ($^{3}$G)4f & 5\[7\] & 15/2 & 108133.440 & ($^{3}$G)4d $^{4}$H$_{13/2}$ & 5166.196 & $+$0.934& 5166.2 & lab\
& & & & 108337.860 & ($^{3}$G)4d $^{4}$I$_{15/2}$ & 5221.353 & $+$0.453& 5221.335 & lab\
& & & & 108463.910 & ($^{3}$G)4d $^{4}$I$_{13/2}$ & 5255.955 & $-$0.980& &\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$ & 5307.518 & $-$0.940& &\
& & & & 109049.600 & ($^{3}$G)4d $^{2}$I$_{13/2}$ & 5422.941 & $-$1.415& &\
\
127515.235 & ($^{3}$G)4f & 5\[7\] & 13/2 & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 4595.998 & $-$1.059& &\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 4656.457 & $-$0.284& &\
& & & & 108133.440 & ($^{3}$G)4d $^{4}$H$_{13/2}$ & 5158.044 & $-$0.684& &\
& & & & 108181.550 & ($^{3}$G)4d $^{4}$G$_{11/2}$ & 5170.879 & $-$0.639& &\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5226.670 & $+$0.474& 5226.686 & lab\
& & & & 108463.910 & ($^{3}$G)4d $^{4}$I$_{13/2}$ & 5247.518 & $+$0.157& 5247.536 & lab\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$ & 5298.915 & $-$1.299& &\
& & & & 108775.080 & ($^{3}$G)4d $^{4}$I$_{11/2}$ & 5334.651 & $-$0.859& &\
& & & & 109049.600 & ($^{3}$G)4d $^{2}$I$_{13/2}$ & 5413.960 & $-$0.246& &\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 5606.354 & $+$0.514& 5606.38 &\
\
127489.429 & ($^{3}$G)4f & 5\[6\] & 13/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 4184.848 & $-$1.133& &\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 4662.061 & $-$1.312& &\
& & & & 108133.440 & ($^{3}$G)4d $^{4}$H$_{13/2}$ & 5164.921 & $+$0.601& 5164.9 & lab\
& & & & 108181.550 & ($^{3}$G)4d $^{4}$G$_{11/2}$ & 5177.791 & $+$0.705& 5177.77& lab &\
& & & & 108337.860 & ($^{3}$G)4d $^{4}$I$_{15/2}$ & 5220.051 & $-$0.463& &\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5233.732 & $-$1.225& &\
& & & & 108463.910 & ($^{3}$G)4d $^{4}$I$_{13/2}$ & 5254.636 & $-$0.596& &\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$ & 5306.173 & $-$0.818& &\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 5614.479 & $-$0.728& &\
\
127489.977 & ($^{3}$G)4f & 5\[6\] & 11/2 & 103600.430 & ($^{3}$H)4d $^{4}$G$_{11/2}$ & 4184.752 & $-$1.422& &\
& & & & 103683.070 & ($^{5}$D)5d $^{4}$F$_{9/2}$ & 4199.279 & $-$1.301& &\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 4661.942 & $-$1.108& &\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 4813.800 & $-$0.314& 4813.8 &\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 4861.143 & $-$0.513& &\
& & & & 108133.440 & ($^{3}$G)4d $^{4}$H$_{13/2}$ & 5164.775 & $-$0.273& 5164.77\
& & & & 108181.550 & ($^{3}$G)4d $^{4}$G$_{11/2}$ & 5177.644 & $+$0.437& 5177.64& lab &\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5233.581 & $-$0.349& 5233.58&\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 5234.562 & $-$0.887& &\
& & & & 109049.600 & ($^{3}$G)4d $^{2}$I$_{13/2}$ & 5421.376 & $-$1.110& &\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 5596.053 & $-$0.050& &computed too strong\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 5614.306 & $-$0.230& &\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 5655.161 & $-$0.047& 5655.15 &\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5718.689 & $-$0.545& &\
\
127482.748 & ($^{3}$G)4f & 5\[5\] & 11/2 & 105763.270 & ($^{3}$F)4d $^{2}$H$_{11/2}$ & 4602.873 & $-$1.478& &\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 4663.514 & $-$0.736& &\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 4815.476 & $-$0.239& & computed too strong\
& & & & 108133.440 & ($^{3}$G)4d $^{4}$H$_{13/2}$ & 5166.704 & $-$0.401& & computed too strong\
& & & & 108181.550 & ($^{3}$G)4d $^{4}$G$_{11/2}$ & 5179.583 & $+$0.320& & blend\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5235.563 & $-$0.190& 5235.585 & blend\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 5236.545 & $+$0.191& & blend, computed too strong\
[lllllllllll]{} & & & & &\
& & & & & & & &\
127482.748 & cont. & & & 108463.910 & ($^{3}$G)4d $^{4}$I$_{13/2}$ & 5256.482 & $-$0.830& 5256.5 &\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$ & 5308.055 & $-$1.341& &\
& & & & 108775.080 & ($^{3}$G)4d $^{4}$I$_{11/2}$ & 5343.915 & $-$1.043& &\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 5598.319 & $-$0.100& 5598.32& computed too weak\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 5616.586 & $-$0.042& 5616.6 & computed too weak\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 5657.474 & $-$0.662& 5657.50& computed too weak\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5721.054 & $-$0.506& &\
\
127485.362 & ($^{3}$G)4f & 5\[4\] & 9/2 & 104107.950 & ($^{3}$P)4d $^{4}$F$_{7/2}$ & 4276.430 & $-$1.168& &\
& & & & 104481.590 & ($^{3}$H)4d $^{2}$F$_{7/2}$ & 4345.891 & $-$1.316& &\
& & & & 105775.491 & ($^{3}$F)4d $^{2}$F$_{7/2}$ & 4604.910 & $-$1.176& &\
& & & & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 4662.945 & $-$1.404& &\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 4814.870 & $-$0.945& &\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 4825.337 & $-$1.318& &\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 4862.235 & $-$0.425& &\
& & & & 108181.550 & ($^{3}$G)4d $^{4}$G$_{11/2}$ & 5178.882 & $-$0.635& &\
& & & & 108365.320 & ($^{3}$G)4d $^{4}$D$_{7/2}$ & 5228.658 & $-$0.224& & blend\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5234.846 & $-$0.695& 5234.80 &\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 5235.828 & $-$0.195& 5235.80 & blend\
& & & & 108537.610 & ($^{3}$G)4d $^{4}$G$_{7/2}$ & 5276.203 & $-$1.169& &\
& & & & 108577.560 & ($^{3}$G)4d $^{4}$H$_{9/2}$ & 5287.351 & $-$1.391& &\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 5597.499 & $+$0.251& 5597.50 & computed too strong\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 5615.762 & $-$0.466& 5615.75 &\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 5656.638 & $-$0.349& 5656.55 & blend\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{7/2}$ & 5685.455 & $-$0.333& 5685.45 &\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5720.199 & $-$0.468& 5720.20 &\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 5772.711 & $-$1.064& &\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5910.253 & $-$0.120& & blend H2O\
\
127485.699 & ($^{3}$G)4f & 5\[4\] & 7/2 & 103683.070 & ($^{5}$D)5d $^{4}$F$_{9/2}$ & 4200.033 & $-$1.226& &\
& & & &106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 4814.791 & $+$0.017& 4814.8 & computed too strong\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 4825.259 & $-$0.375& 4825.30 & blend\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 4856.472 & $-$1.384& &\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 4862.155 & $-$0.753& &\
& & & & 108365.320 & ($^{3}$G)4d $^{4}$D$_{7/2}$ & 5228.566 & $+$0.266& & blend\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 5235.735 & $-$0.618& & blend\
& & & & 108537.610 & ($^{3}$G)4d $^{4}$G$_{7/2}$ & 5276.109 & $-$0.999& &\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 5597.394 & $-$1.025& &\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 5656.530 & $+$0.034& 5656.55 &\
& & & & 110065.750 & ($^{3}$G)4d $^{2}$D$_{5/2}$ & 5738.953 & $-$1.494& &\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 5772.598 & $-$0.676& &\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5910.135 & $-$1.369& &\
\
127510.913 & ($^{3}$G)4f & 5\[3\] & 5/2 & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 4819.393 & $-$0.294& 4819.40 &\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 4850.531 & $-$1.345& &\
& & & & 108365.320 & ($^{3}$G)4d $^{4}$D$_{7/2}$ & 5221.680 & $+$0.447& 5221.68 & lab\
& & & & 108537.610 & ($^{3}$G)4d $^{4}$G$_{7/2}$ & 5269.097 & $-$0.794& 5369.12&\
& & & & 110065.750 & ($^{3}$G)4d $^{2}$D$_{5/2}$ & 5730.658 & $-$0.761& &\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 5764.206 & $-$0.654& & blend\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5901.339 & $-$1.193& &\
\
127487.681 & ($^{3}$G)4f & 5\[2\] & 3/2 & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 4848.090 & $-$0.945& &\
& & & & 108642.410 & ($^{3}$G)4d $^{4}$D$_{5/2}$ & 5304.895 & $-$0.425& 5304.89 & blend\
& & & & 110065.750 & ($^{3}$G)4d $^{2}$D$_{5/2}$ & 5738.300 & $-$0.104& 5738.30 &\
[lllllllllll]{} & & & & &\
& & & & & & & &\
127892.981 & ($^{3}$G)4f & 4\[7\] & 15/2 & 104064.670 & ($^{3}$H)4d $^{4}$I$_{13/2}$ & 4195.506 & $-$1.455& &\
& & & & 104622.300 & ($^{3}$H)4d $^{2}$I$_{13/2}$ & 4296.044 & $-$1.387& &\
& & & & 108133.440 & ($^{3}$G)4d $^{4}$H$_{13/2}$ & 5059.436 & $-$0.484& 5059.42 & lab\
& & & & 108463.910 & ($^{3}$G)4d $^{4}$I$_{13/2}$ & 5145.493 & $-$0.007& 5145.5 &\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$ & 5194.901 & $+$0.482& &blend\
& & & & 109049.600 & ($^{3}$G)4d $^{2}$I$_{13/2}$ & 5305.427 & $+$0.862& 5305.42 & lab\
\
127895.260 & ($^{3}$G)4f & 4\[7\] & 13/2 & 104174.270 & ($^{3}$H)4d $^{4}$I$_{11/2}$ & 4214.489 & $-$1.351& &\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5124.848 & $-$0.679& & blend\
& & & & 108630.429 & ($^{1}$I)5s e$^{2}$I$_{11/2}$ & 5189.361 & $-$0.144& 5189.371 & lab\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$ & 5194.286 & $-$1.434& &\
& & & & 108775.080 & ($^{3}$G)4d $^{4}$I$_{11/2}$ & 5228.621 & $+$0.896& 5228.635 & lab\
& & & & 109389.880 & ($^{3}$G)4d $^{2}$I$_{11/2}$ & 5402.332 & $+$0.099& 5402.32 & lab\
\
127875.000 & ($^{3}$G)4f & 4\[6\] & 13/2 & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 4579.713 & $-$0.754& &\
& & & & 108133.440 & ($^{3}$G)4d $^{4}$H$_{13/2}$ & 5064.044 & $-$1.045& &\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5130.176 & $+$0.662& 5130.18 & lab\
& & & & 108463.910 & ($^{3}$G)4d $^{4}$I$_{13/2}$ & 5150.259 & $-$0.700& &\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$& 5199.759 & $-$0.190& & blend\
& & & & 109049.600 & ($^{3}$G)4d ${2}$I$_{13/2}$ & 5310.495 & $+$0.113& 5310.5 & lab\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 5495.480 & $+$0.481& 5495.49& lab, J78\
\
127880.436 & ($^{3}$G)4f & 4\[6\] & 11/2 & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 4589.468 & $-$0.765& &\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5128.745 & $-$0.375& & blend\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 5129.687 & $-$1.085& &\
& & & & 108577.560 & ($^{3}$G)4d $^{4}$H$_{9/2}$ & 5179.133 & $+$0.652& 5179.14 & lab\
& & & & 108630.429 & ($^{1}$I)5s e$^{2}$I$_{11/2}$ & 5193.357 & $-$0.797& &\
& & & & 108775.080 & ($^{3}$G)4d $^{4}$I$_{11/2}$ & 5232.678 & $-$0.047& & blend\
& & & & 108929.040 & ($^{3}$G)4d $^{4}$I$_{9/2}$ & 5275.188 & $-$0.897& &\
& & & & 109389.880 & ($^{3}$G)4d $^{2}$I$_{11/2}$ & 5406.663 & $-$0.491& &\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 5476.359 & $-$0.333& 5476.38 &\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 5493.838 & $-$1.052& &\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 5532.952 & $-$0.700& &\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5593.749 & $+$0.039& 5593.85 &\
\
127869.158 & ($^{3}$G)4f & 4\[5\] & 11/2 & 106045.690 & ($^{3}$H)4d $^{2}$H$_{11/2}$ & 4580.939 & $-$1.153& &\
& & & & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 4727.483 & $-$0.893& &\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5131.714 & $+$0.220& 5131.7 & lab\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 5132.657 & $+$0.408& & blend\
& & & & 108577.560 & ($^{3}$G)4d $^{4}$H$_{9/2}$ & 5182.161 & $-$0.938& &\
& & & & 108648.695 & ($^{1}$I)5s e$^{2}$I$_{13/2}$ & 5201.340 & $-$1.171& &\
& & & & 108775.080 & ($^{3}$G)4d $^{4}$I$_{11/2}$ & 5235.768 & $-$0.234& &blend\
& & & & 108929.040 & ($^{3}$G)4d $^{4}$I$_{9/2}$ & 5278.329 & $-$1.413& &\
& & & & 109049.600 & ($^{3}$G)4d $^{2}$I$_{13/2}$ & 5312.143 & $-$0.846& &\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 5479.744 & $-$0.089& 5479.72 & lab\
& & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 5497.245 & $+$0.050& 5497.25 &\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 5536.408 & $-$0.555& 5536.40 &\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5597.281 & $-$0.105& 5597.30 &\
\
127855.952 & ($^{3}$G)4f & 4\[5\] & 9/2 & 106722.170 & ($^{3}$F)4d $^{4}$F$_{9/2}$ & 4730.437 & $-$0.906& &\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 4740.541 & $-$0.409& &\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 4770.664 & $-$1.118& &\
& & & & 108365.320 & ($^{3}$G)4d $^{4}$D$_{7/2}$ & 5129.241 & $-$0.301& 5129.25 &\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5135.195 & $-$0.409& & blend\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 5136.140 & $+$0.294& & blend\
& & & & 108577.560 & ($^{3}$G)4d $^{4}$H$_{9/2}$ & 5185.710 & $-$0.829& &\
& & & & 108709.450 & ($^{3}$G)4d $^{4}$H$_{7/2}$ & 5221.432 & $-$1.407& &\
& & & & 109625.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 5483.714 & $+$0.010& 5483.70&\
[lllllllllll]{} & & & & &\
& & & & & & & &\
127855.952 & cont. & & & 109683.280 & ($^{3}$G)4d $^{2}$H$_{11/2}$ & 5501.240 & $-$0.659& &\
& & & & 109811.920 & ($^{3}$G)4d $^{4}$F$_{9/2}$ & 5540.460 & $-$0.431& 5540.47 &\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{7/2}$ & 5568.103 & $-$0.216& 5568.10 &\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 5651.767 & $-$0.160& 5651.78 & computed too weak\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5783.541 & $-$0.854& &\
\
127869.892 & ($^{3}$G)4f & 4\[4\] & 9/2 & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 4591.690 & $-$1.043& & no soectrum\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 4767.493 & $-$1.141& & no spectrum\
& & & & 108365.320 & ($^{3}$G)4d $^{4}$D$_{7/2}$ & 5125.575 & $-$1.117& & weak\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 5132.464 & $-$0.690& & blend\
& & & & 108537.610 & ($^{3}$G)4d $^{4}$G$_{7/2}$ & 5171.255 & $+$0.332& 5171.25 & lab, J78\
& & & & 108577.560 & ($^{3}$G)4d $^{4}$H$_{9/2}$ & 5181.963 & $+$0.101& 5181.97 & lab\
& & & & 108709.450 & ($^{3}$G)4d $^{4}$H$_{7/2}$ & 5217.634 & $-$1.196& & weak\
& & & & 108775.080 & ($^{3}$G)4d $^{4}$I$_{11/2}$ & 5235.567 & $-$0.810& & blend\
& & & & 108929.040 & ($^{3}$G)4d $^{4}$I$_{9/2}$ & 5278.125 & $-$0.704& & blend\
& & & & 109389.880 & ($^{3}$G)4d $^{2}$I$_{11/2}$ & 5409.748 & $-$1.407& &blend\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{7/2}$ & 5563.783 & $-$0.269& 5563.79 &\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5597.051 & $+$0.023& 5597.05 &\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 5647.317 & $-$0.723& & blend\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5778.881 & $-$0.074& 5778.88 &\
\
127874.745 & ($^{3}$G)4f & 4\[3\] & 5/2 & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 4736.320 & $-$0.862& &no spectrum\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ & 4742.937 & $-$1.442& &no spectrum\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 4758.764 & $-$0.354& &no spectrum\
& & & & 107407.800 & ($^{3}$F)4d $^{2}$D$_{5/2}$ & 4884.563 & $-$1.137& &blend\
& & & & 108365.320 & ($^{3}$G)4d $^{4}$D$_{7/2}$ & 5124.300 & $-$0.351& 5124.3 &\
& & & & 108537.610 & ($^{3}$G)4d $^{4}$G$_{7/2}$ & 5169.957 & $-$0.493& 5169.95 &\
& & & & 108613.960 & ($^{3}$G)4d $^{4}$G$_{5/2}$ & 5190.451 & $-$1.336& &blend\
& & & & 108642.410 & ($^{3}$G)4d $^{4}$D$_{5/2}$ & 5198.129 & $-$0.577& 5198.12 &\
& & & & 108859.470 & ($^{3}$G)4d $^{4}$D$_{3/2}$ & 5257.467 & $-$1.074& &weak\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{7/2}$ & 5562.281 & $-$0.790& &weak\
& & & & 110065.750 & ($^{3}$G)4d $^{2}$D$_{5/2}$ & 5613.582 & $-$0.302& 5613.55 & blend\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 5645.769 & $-$0.897& & weak\
& & & & 110428.280 & ($^{3}$G)4d $^{4}$F$_{5/2}$ & 5730.231 & $-$0.236& & blend\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5777.260 & $-$0.288& 5777.73 & computed too weak\
& & & & 110611.800 & ($^{3}$G)4d $^{2}$F$_{5/2}$ & 5791.149 & $-$1.493& & blend\
\
128110.214 & ($^{3}$G)4f & 3\[6\] & 13/2 & 104765.450 & ($^{3}$H)4d $^{2}$I$_{11/2}$ & 4282.411 & $-$1.266& &blend\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ & 5068.991 & $-$0.821&5068.99 &\
& & & & 108630.429 & ($^{1}$I)5s e$^{2}$I$_{11/2}$ & 5132.097 & $-$0.929& &blend\
& & & & 108775.080 & ($^{3}$G)4d $^{4}$I$_{11/2}$ & 5170.492 & $+$0.154& 5170.5 & lab\
& & & & 109389.880 & ($^{3}$G)4d $^{2}$I$_{11/2}$ & 5340.300 & $+$0.922& 5340.30 & lab, J78\
\
128071.171 & ($^{3}$F)4f& 3\[5\] & 11/2 & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 4549.630 &$-$0.731 & & no spectrum\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 4727.539 &$-$0.926 & & no spectrum\
& & & & 108387.920 & ($^{3}$G)4d $^{4}$H$_{11/2}$ &5079.046 &$-$1.376 & & blend\
& & & & 108391.500 & ($^{3}$G)4d $^{4}$G$_{9/2}$ & 5079.970 &$-$1.401 & & at the continuum level\
& & & & 108577.560 & ($^{3}$G)4d $^{4}$H$_{9/2}$ & 5128.457 &$+$0.377 &5128.47 & lab\
& & & & 108775.080 & ($^{3}$G)4d $^{4}$I$_{11/2}$ & 5180.954 &$-$0.687 & & blend\
& & & & 108929.040 & ($^{3}$G)4d $^{4}$I$_{9/2}$ & 5222.625 &$-$0.245 & 5222.62 & computed too strong\
& & & & 109389.880 & ($^{3}$G)4d $^{2}$I$_{11/2}$ & 5351.461 &$+$0.043 &5351.47 &\
& & & & 106925.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 5419.731 &$-$0.013 & 5419.73 & lab\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5534.681 &$+$0.459 & 5534.68 &\
[lllllllllll]{} & & & & &\
& & & & & & & &\
128055.658 & ($^{3}$F)4f& 3\[5\] & 9/2 & 106097.520 & ($^{3}$H)4d $^{2}$H$_{9/2}$ & 4552.844 &$-$1.204 & & no spectrum\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 4696.069 &$-$0.812 & & no spectrum\
& & & & 106924.430 & ($^{3}$F)4d $^{2}$G$_{9/2}$ & 4731.009 &$-$1.380 & & no spectrum\
& & & & 108537.610 & ($^{3}$G)4d $^{4}$G$_{7/2}$ & 5122.036 &$+$0.148 & 5122.02 & lab\
& & & & 108577.560 & ($^{3}$G)4d $^{4}$H$_{9/2}$ & 5132.541 &$+$0.038 & 5132.55 & lab\
& & & & 108709.450 & ($^{3}$G)4d $^{4}$H$_{7/2}$ & 5167.532 &$-$0.521 & & blend\
& & & & 108775.080 & ($^{3}$G)4d $^{4}$I$_{11/2}$ & 5185.122 &$-$1.448 & 5185.141& blend\
& & & & 109389.880 & ($^{3}$G)4d $^{2}$I$_{11/2}$ & 5355.908 &$-$0.925 & 5355.9 & weak\
& & & & 106925.200 & ($^{3}$G)4d $^{2}$G$_{9/2}$ & 5424.293 &$-$0.649 & & blend\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{7/2}$ & 5506.850 &$+$0.159 & 5506.85 &\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5539.439 &$+$0.045 & 5539.41 &\
& & & & 110167.280 & ($^{3}$G)4d $^{4}$F$_{7/2}$ & 5588.670 &$-$0.697 & 5588.65 &\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5717.485 &$-$0.176 & 5717.50 &\
\
128062.710 & ($^{3}$F)4f& 3\[4\] & 9/2 & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 4724.054 &$-$1.276 & & no spectrum\
& & & & 108709.450 & ($^{3}$G)4d $^{4}$H$_{7/2}$ & 5165.649 &$+$0.734 & 5165.65 & lab\
& & & & 108929.040 & ($^{3}$G)4d $^{4}$I$_{9/2}$ & 5224.934 &$+$0.139 & 5224.938 &\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{7/2}$ & 5504.712 &$-$0.840 & & not observed\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5537.275 &$-$1.268 & &at the level of the noise\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5715.180 &$-$1.173 & &at the level of the noise\
\
128066.823 & ($^{3}$F)4f& 3\[4\] & 7/2 & 104023.910 & ($^{3}$H)4d $^{4}$G$_{5/2}$ & 4158.057 &$-$1.351 & & not observed, wrong\
& & & & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 4573.647 &$-$1.130 & & no spectrum\
& & & & 106767.210 & ($^{3}$F)4d $^{4}$F$_{7/2}$ & 4693.607 &$-$1.067 & & no spectrum\
& & & & 106900.370 & ($^{3}$F)4d $^{2}$G$_{7/2}$ & 4723.136 &$-$1.319 & & no spectrum\
& & & & 108537.610 & ($^{3}$G)4d $^{4}$G$_{7/2}$ & 5119.108 &$-$0.444 & & computed too strong\
& & & & 108577.560 & ($^{3}$G)4d $^{4}$H$_{9/2}$ & 5129.601 &$-$1.316 & & blend\
& & & & 108613.960 & ($^{3}$G)4d $^{4}$G$_{5/2}$ & 5139.200 &$+$0.196 & 5139.20 & lab\
& & & & 108709.450 & ($^{3}$G)4d $^{4}$H$_{7/2}$ & 5164.552 &$-$0.146 & 5164.52 & computed too weak\
& & & & 108929.040 & ($^{3}$G)4d $^{4}$I$_{9/2}$ & 5223.811 &$-$0.993 & & blend\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{7/2}$ & 5503.465 &$-$0.078 & &blend\
& & & & 110008.300 & ($^{3}$G)4d $^{2}$H$_{9/2}$ & 5536.014 &$-$0.751 & 5536.0 &\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5713.836 &$-$0.308 & 5713.8 &\
& & & & 110611.800 & ($^{3}$G)4d $^{2}$F$_{5/2}$ & 5727.421 &$-$0.043 & 5727.45 &\
\
128063.103 & ($^{3}$G)4f& 3\[3\] &5/2 & 106864.650 & ($^{3}$G)4d $^{4}$F$_{3/2}$ & 4712.005 &$-$0.481 & &no spectrum\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 4716.475 &$-$1.431 & &no spectrum\
& & & & 107430.250 & ($^{3}$F)4d $^{2}$D$_{3/2}$ & 4845.286 &$-$0.946 & &blend,computed too strong\
& & & & 108613.960 & ($^{3}$G)4d $^{4}$G$_{5/2}$ & 5140.183 &$+$0.037 &5140.19 &\
& & & & 108642.410 & ($^{3}$G)4d $^{4}$D$_{5/2}$ & 5147.713 &$-$0.412 &5147.71 & computed too weak\
& & & & 108709.450 & ($^{3}$G)4d $^{4}$H$_{7/2}$ & 5165.544 &$-$0.693 & & blend\
& & & & 108859.470 & ($^{3}$G)4d $^{4}$D$_{3/2}$ & 5205.898 &$-$0.225 & 5205.879 &\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{5/2}$ & 5504.593 &$-$1.414 & &at the continuum level\
& & & & 110428.280 & ($^{3}$G)4d $^{4}$F$_{5/2}$ & 5669.025 &$-$0.651 & 5669.03 &\
& & & & 110461.260 & ($^{3}$G)4d $^{2}$D$_{3/2}$ & 5679.647 &$-$1.133 & & at the level of the noise\
& & & & 110609.540 & ($^{3}$G)4d $^{4}$F$_{3/2}$ & 5727.900 &$-$0.186 & 5727.90 &\
& & & & 110611.800 & ($^{3}$G)4d $^{2}$F$_{5/2}$ & 5728.642 &$-$0.772 & &weak\
\
128089.313 & ($^{3}$G)4f& 3\[2\] &5/2 & 106208.560 & ($^{3}$F)4d $^{2}$F$_{5/2}$ & 4568.946 &$-$1.396 & &no spectrum\
& & & & 106747.210 & ($^{5}$D)5d $^{4}$F$_{7/2}$ & 4688.657 &$-$1.457 & &no spectrum\
& & & & 106796.660 & ($^{3}$F)4d $^{4}$P$_{5/2}$ & 4695.142 &$-$1.393 & &no spectrum\
& & & & 106866.760 & ($^{3}$F)4d $^{4}$F$_{5/2}$ & 4710.650 &$-$1.102 & &no spectrum\
& & & & 108537.610 & ($^{3}$G)4d $^{4}$G$_{7/2}$ & 5113.219 &$-$1.022 & &at the continuum level\
& & & & 108642.410 & ($^{3}$G)4d $^{4}$D$_{5/2}$ & 5140.775 &$-$0.580 & & blend\
& & & & 108859.470 & ($^{3}$G)4d $^{4}$D$_{3/2}$ & 5198.803 &$-$0.577 & &blend\
& & & & 109901.500 & ($^{3}$G)4d $^{2}$G$_{5/2}$ & 5496.660 &$-$0.747 & & blend\
& & & & 110428.280 & ($^{3}$G)4d $^{4}$F$_{5/2}$ & 5660.612 &$-$0.985 & &blend\
& & & & 110461.260 & ($^{3}$G)4d $^{2}$D$_{3/2}$ & 5671.202 &$-$0.429 & 5671.20 &\
& & & & 110570.300 & ($^{3}$G)4d $^{2}$F$_{7/2}$ & 5706.501 &$-$0.913 & &at the level of the noise\
& & & & 110611.800 & ($^{3}$G)4d $^{2}$F$_{5/2}$ & 5720.051 &$+$0.065 & 5720.05 &\
[^1]: http://wwwuser.oat.ts.astro.it/castelli/hr6000new/hr6000.html
[^2]: http://kurucz.harvard.edu/atoms/2601
[^3]: http://wwwuser.oat.ts.astro.it/castelli/hr6000/unidentified.txt
|
---
abstract: 'I analyse the transport of particles of arbitrary statistics (Bose, Fermi and fractional exclusion statistics) through one-dimensional (1D) channels. Observing that the particle, energy, entropy and heat fluxes through the 1D channel are similar to the particle, internal energy, entropy and heat capacity of a quantum gas in a two-dimensional (2D) flat box, respectively, I write analytical expressions for the fluxes at arbitrary temperatures. Using these expressions, I show that the heat and entropy fluxes are independent of statistics at any temperature, and not only in the low temperature limit, as it was previously known. From this perspective, the quanta of heat conductivity represents only the low temperature limit of the 1D channel heat conductance and is equal (up to a multiplicative constant equal to the Plank constant times the density of states at the Fermi energy) to the universal limit of the heat capacity of quantum gases. In the end I also give a microscopic proof for the universal temperature dependence of the entropy and heat fluxes through 1D channels.'
author:
- 'Dragoş-Victor Anghel'
title: 'Universal heat conductance of one-dimensional channels'
---
Rego and Kirczenow [@PhysRevLett.81.232.1998.Rego] and, independently, Angelescu, Cross, and Roukes [@SuperlatticesMicrostructures23.673.1998.Angelescu], proved theoretically that the phonons heat conductance, $\kappa$, of a quasi one-dimensional dielectric wire in the ballistic regime in the low temperature limit, is quantized in units of $$\kappa_0\equiv\frac{\pi^2k_B^2T}{3h}, \label{kappa_LT} $$ namely $\kappa = {{\mathcal N}}_c\kappa_0$, where ${{\mathcal N}}_c$ is the number of phonon channels available along the wire, $h$ is the Plank constant and $k_B$ is the Boltzmann constant and $T$ is the average temperature between the ends 1 and 2 of the wire, $T=(T_1+T_2)/2$, assuming that $|T_1-T_2|/T\ll 1$.
These results have been experimentally confirmed in Refs. [@Nature404.974.2000.Schwab; @Nature.444.187.2006.Meschke] and have been extended by Rego and Kirczenow in Ref. [@PhysRevB.59.13080.1999.Rego], where they showed that the same quantization rule applies to the heat conductance of particles of any statistics.
Let us consider a two-terminal transport experiment in which the reservoirs 1 and 2 are connected by a quasi 1D wire. The temperatures and chemical potentials in the two reservoirs will be denoted by $T_{i}$ and $\mu_{i}$, respectively ($i=1,2$). The particles in the system may be bosons, fermions, or may obey fractional exclusion statistics (FES) of parameter $\alpha$ [@PhysRevLett.67.937.1991.Haldane; @PhysRevLett.73.922.1994.Wu; @PhysRevLett.104.198901.2010.Anghel; @PhysRevLett.104.198902.2010.Wu; @EPL.87.60009.2009.Anghel]. Then the Landauer formula for the particle and heat fluxes between the two reservoirs are [@PhysRevB.59.13080.1999.Rego; @PhysRep.395.159.2004.Blencowe]
\[IUgen\] $$\begin{aligned}
I &=& \sum _{n=1}^{{{\mathcal N}}_c}\int_{0}^{\infty}\frac{dk}{2\pi}v_{n}(k)\left[\eta_{1}(k)-\eta_{2}(k)\right]\zeta_n(k)
\label{Igen} \\
\dot U &=& \sum _{n=1}^{{{\mathcal N}}_c}\int_{0}^{\infty}\frac{dk}{2\pi}\epsilon v_{n}(k)\left[\eta_{1}(k)-\eta_{2}(k)\right]\zeta_n(k)
\label{Ugen}\end{aligned}$$
where $v_{n}(k)$ is the group velocity of the particles of momentum $k$, $\eta_{i}(k)$ denotes the thermal particle population in the reservoir $i$ and, finally, $\zeta_n(k)$ is the particle transmission coefficient through the wire. The summation is taken over the 1D (available) channels along the wire.
Since the particle group velocity is $v_{n}(k)=\hbar^{-1}(d\epsilon(k)/dk)$, and assuming further that $\zeta_n(k)\equiv\zeta_n$ is independent of $k$, Eqs. (\[IUgen\]) get the simple form
\[IUgen1\] $$\begin{aligned}
I &=& \frac{\zeta_n}{h}\sum _{n=1}^{{{\mathcal N}}_c}\int_{\epsilon_n(0)}^{\infty}d\epsilon\left[\eta_{1}(\epsilon)-\eta_{2}(\epsilon)\right]
\label{Igen1} \\
\dot U &=& \frac{\zeta_n}{h}\sum _{n=1}^{{{\mathcal N}}_c}\int_{\epsilon_n(0)}^{\infty}d\epsilon\,\epsilon\left[\eta_{1}(\epsilon)-\eta_{2}(\epsilon)\right]
\label{Ugen1}\end{aligned}$$
where $\epsilon_n(0)$ is the lowest energy level in the channel $n$.
To calculate the heat conductivity, $\kappa\equiv\dot U/(T_1-T_2)$, one introduces in (\[IUgen1\]) the FES populations [@PhysRevB.59.13080.1999.Rego; @PhysRevLett.73.922.1994.Wu; @EPL.90.10006.2010.Anghel],
\[EqFES\] $$\begin{aligned}
&&\eta_{i}(\epsilon)\equiv\eta_\alpha(\epsilon,\mu,T) = \left[w_\alpha(\mu,T)+\alpha\right], \label{Eq_eta} \\
&&w_\alpha^{\alpha}(\mu,T)[1+w_\alpha(\mu,T)]^{1-\alpha}=\exp[\beta(\epsilon-\mu)], \label{Eq_w}\end{aligned}$$
where $\beta\equiv 1/(k_BT)$ ($\alpha=0$ and 1 correspond to bosons and fermions, respectively). By doing so, Rego and Kirczenow calculated the low temperature limit of $\kappa$ and observed that it is independent of $\alpha$ and therefore the quantization relation (\[kappa\_LT\]) holds for particles of any statistics [@PhysRevB.59.13080.1999.Rego; @PhysRep.395.159.2004.Blencowe]. Nevertheless, a physical understanding of this mathematical result is still missing [@PhysRep.395.159.2004.Blencowe].
In this letter I extend the results of Rego and Kirczenow by showing that the heat conductivity of a 1D channel is independent of the statistics of particles at any temperature and I will provide a microscopic explanation for this result. This is done by observing that there is a close similarity between the stationary heat and particle transport in 1D and equilibrium thermodynamics in 2D (or, in systems with constant single-particle density of states, DOS [@ProcCambrPhilos42.272.1946.Auluc; @PhysRev.135.A1515.1964.May; @PhysRevE.55.1518.1997.Lee; @PhysA304.421.2002.Lee; @JPA35.7255.2002.Anghel; @RJP.54.281.2009.Anghel; @JPA38.9405.2005.Anghel]). The quanta of heat conductance is then nothing but the low temperature limit of the 1D heat conductance. Moreover, I calculate the entropy current through the channel and I show that this is also independent of statistics at any temperature as long as the particle current is zero.
Let us focus on systems with only one channel, $n$. We split the particle and heat fluxes into two parts, one coming from the reservoir 1 and one from the reservoir 2, $I_n\equiv I_{n,1}(\mu_1,T_1)-I_{n,1}(\mu_2,T_2)$ and $\dot U_n\equiv\dot U_{n,1}(\mu_1,T_1)-\dot U_{n,1}(\mu_2,T_2)$, where
\[IUnsplit\] $$\begin{aligned}
I_{n,1}(\mu_n,T_n) &\equiv& \frac{\zeta_n}{h}\int_{\epsilon_n(0)}^{\infty}\eta_{\alpha}(\epsilon,\mu_n,T_n)d\epsilon \label{Insplit} \\
\dot U_{n,1}(\mu_n,T_n) &\equiv& \frac{\zeta_n}{h}\int_{\epsilon_n(0)}^{\infty}\epsilon\eta_{\alpha}(\epsilon,\mu_n,T_n)d\epsilon
\label{Unsplit}\end{aligned}$$
We observe that $I_{n,1}(\mu_n,T_n)$ and $\dot U_{n,1}(\mu_n,T_n)$ have exactly the same expressions as the particle number, $N_\alpha(\mu_n,T_n)$, and internal energy, $U_\alpha(\mu_n,T_n)$, respectively, of a FES system of parameter $\alpha$ in equilibrium at temperature $T$ and chemical potential $\mu$, which has a constant DOS, $\sigma=\zeta_n/h$. Therefore we can apply here directly the results we obtained for these latter systems in Ref. [@JPA35.7255.2002.Anghel]. To specify the notations, let us briefly review the results of Ref. [@JPA35.7255.2002.Anghel] which are of interest here.
If a FES system of parameter $\alpha$ and density of states $\sigma=\zeta_n/h$, in equilibrium at temperature $T$, contains $N_\alpha$ particles, then we define the statistics independent parameter $y_0$ by the relation
\[def\_N\_y0\_mu\] $$N_\alpha \equiv k_BT\sigma\log(1+y_0). \label{N}$$ Knowing $y_0$, one may calculate the chemical potential, $\mu$, from the equation $$(1+y_0)^{1-\alpha}/y_0 = e^{-\beta\mu} \label{defy0}$$ and from Eqs. (\[N\]) and (\[defy0\]) we observe that $$\label{mu}
\exp{[(\mu - \alpha N_\alpha/\sigma)/k_{\rm B} T]} = 1- \exp{[-N_\alpha/(\sigma k_{\rm B}T)]}.$$
If we identify the (generalized) Fermi energy as $\epsilon_{\rm F}\equiv\lim_{T\to 0}\mu = \alpha N_\alpha/\sigma$ we observe that $\mu-\epsilon_{\rm F}$ is also independent of $\alpha$, or, *vice-versa*, $\mu-\epsilon_{\rm F}$ determine uniquely $N_\alpha$, for any $\alpha$.
In these notations, the grandcanonical potential and internal energy, $\Omega_\alpha$ and $U_\alpha$ are [@JPA35.7255.2002.Anghel] $$\begin{aligned}
\label{echiv}
\Omega_\alpha = -U_\alpha &=& (k_{\rm B}T)^2 \sigma\left[\frac{1-\alpha}{2}
\log^2{(1+y_0)} + Li_2(-y_0)\right] \nonumber \\
&& = \frac{1-\alpha}{2}\frac{N_\alpha^2}{\sigma} + (k_{\rm B} T)^2 \sigma Li_2(-y_0) ,\end{aligned}$$ where $Li_2$ is the Euler’s dilogarithm, $Li_2(z) = \sum_{i=1}^{\infty}z^k/k^2$ [@Lewin:book]. The fact that, at constant $N_\alpha$, the temperature dependent part of $\Omega_\alpha$ and $U_\alpha$, i.e. $(k_{\rm B} T)^2\sigma Li_2(-y_0)$, is independent of $\alpha$ is an expression of the thermodynamic equivalence of quantum gases of the same, constant DOS [@ProcCambrPhilos42.272.1946.Auluc; @PhysRev.135.A1515.1964.May; @PhysRevE.55.1518.1997.Lee; @PhysA304.421.2002.Lee; @JPA35.7255.2002.Anghel].
From Eqs. (\[echiv\]) and (\[N\]), one can obtain the entropy and the heat capacity [@JPA35.7255.2002.Anghel], $$\begin{aligned}
S &=& -k_{\rm B}^2 T \sigma [2Li_2(-y_0) + \log{(1+y_0)}\log{y_0}]
\label{entropia} \\
C_{\rm V} &=& - \frac{N_\alpha^2}{T\sigma}\frac{1+y_0}{y_0}- 2k_{\rm B}^2
T\sigma Li_2(-y_0) , \label{cv} \end{aligned}$$ which are both independent of $\alpha$. Since, according to Eq. (\[N\]), $\lim_{T\to0}y_0=\infty$, using the asymptotic behavior of the dilogarithm, $Li_2(-y_0) \sim -[\pi^2/6 + \log^2(y_0)/2]$, one can recover in the low temperature limit the universal asymptotic expression for the heat capacity of the system, namely [@Stone:book] $$C_V\sim(\pi^2/3)k_B^2T\sigma. \label{cv_limT0}$$ Now we have all the ingredients and we can transcribe the formalism above into a formalism for the particle and heat transport along the wire. For this, we identify $I_{n,1}(\mu_n,T_n)$ with $N_\alpha(\mu_n,T_n)$, and we introduce $y_{i0}$ by
\[IUpoly\] $$\begin{aligned}
&&I_{n,1}(\mu_n,T_n) \equiv \frac{\zeta_nk_BT_n}{h}\log(1+y_{i0}), \label{In} \\
&& (1+y_{i0})^{1-\alpha}/y_{i0} = e^{-\beta_n\mu_n}, \label{yin} \end{aligned}$$ $$\exp{[(\mu - \alpha h I_{n,1}/\zeta_n)/k_{B} T_n]} = 1- \exp{[-hI_{n,1}/(\zeta_n k_{B}T_n)]}, \label{muI}$$ in analogy to Eqs. (\[def\_N\_y0\_mu\])
From the equations above and observing that $\dot U_{n,1}(\mu_n,T_n)$ has an expression similar to that of $U_\alpha(\mu_n,T_n)$, we obtain $$\begin{aligned}
\dot U_{n,1} &=& -\frac{1-\alpha}{2}\frac{hI_{n,i}^2}{\zeta_n} - (k_{\rm B} T_n)^2 \frac{\zeta_n}{h} Li_2(-y_{i0}) .\label{dotUechiv2} \end{aligned}$$
From Eqs. (\[IUpoly\]) we calculate
\[dIdTdmu\] $$\begin{aligned}
\frac{\partial I_{n,1}}{\partial T} &=& \frac{I_{n,1}}{T}-\frac{\zeta_n\mu}{hT}\frac{1-\exp\left[-\frac{hI_{n,1}}{\zeta_n k_BT}\right]}{\alpha+(1-\alpha)\exp\left[-\frac{hI_{n,1}}{\zeta_n k_BT}\right]} \nonumber \\
&=& \frac{\zeta_n k_B}{h}\log(1+y_{i0}) - \frac{\zeta_n\mu}{hT}\frac{y_{i0}}{1+\alpha y_{i0}}, \label{dIdT}\end{aligned}$$ $$\frac{\partial I_{n,1}}{\partial \mu} = \frac{\frac{\zeta_n}{h}\left[1-\exp\left(-\frac{hI_{n,1}}{\zeta_n k_BT}\right)\right]}{\alpha+(1-\alpha)\exp\left(-\frac{hI_{n,1}}{\zeta_n k_BT}\right)}
= \frac{\zeta_n y_{i0}}{h(1+\alpha y_{i0})}.
\label{dIdmu}$$
\[ddotUdTdmu\] $$\begin{aligned}
\frac{\partial \dot U_{n,1}}{\partial T} &=& -(1-\alpha)\frac{hI_{n,1}}{\zeta_n}\frac{\partial I_{n,1}}{\partial T} - 2k_{\rm B}^2 T_n \frac{\zeta_n}{h} Li_2(-y_{i0}) \nonumber \\
&& + \frac{\mu I_{n,1}(1+y_{i0})}{T(1+\alpha y_{i0})}, \label{ddotUdT}
\\
\frac{\partial \dot U_{n,1}}{\partial\mu} &=&-(1-\alpha)\frac{hI_{n,1}}{\zeta_n}\frac{\partial I_{n,1}}{\partial\mu} - 2k_{\rm B}^2 T_n \frac{\zeta_n}{h} Li_2(-y_{i0}) \nonumber \\
&& + \frac{I_{n,1}(1+y_{i0})}{1+\alpha y_{i0}}, \label{ddotUdmu}\end{aligned}$$
To calculate the low temperature approximations, we use the fact that $\lim_{T\to0}y_{i0}=\infty$ for any $\alpha$ and obtain
\[lowTexpr1\] $$\begin{aligned}
y_{i0,\alpha>0} &\stackrel{T\to0}{\sim}& e^{\frac{\mu}{\alpha k_BT}}+\frac{1-\alpha}{\alpha},\label{lowTy0} \\
I_{n,1,\alpha>0} &\stackrel{T\to0}{\sim}& \frac{\zeta_n\mu}{h\alpha} + \frac{(1-\alpha)\zeta_n k_BT}{\alpha h}e^{-\frac{\mu}{\alpha k_BT}} \label{lowTI} \\
\dot U_{n,1,\alpha>0} &\stackrel{T\to0}{\sim}& \frac{\zeta_n\mu^2}{2h\alpha} +\frac{\pi^2\zeta_n(k_BT)^2}{6h} +\frac{(1-\alpha)\zeta_nk_BT\mu}{\alpha h}
\nonumber \\
&& \times e^{-\frac{\mu}{\alpha k_BT}} \label{lowTdotU}\end{aligned}$$ for $\alpha>0$, whereas for $\alpha=0$ we have $$\begin{aligned}
y_{i0,\alpha=0} &\stackrel{T\to0}{\sim}& -\frac{k_BT}{\mu}-\frac{1}{2},\label{lowTy0a0} \\
I_{n,1,\alpha=0} &\stackrel{T\to0}{\sim}& -\frac{\zeta_nk_BT}{h}\left[\log\left(-\frac{\mu}{k_BT}\right)+\frac{\mu}{2k_BT}\right] \label{lowTIa0} \\
\dot U_{n,1,\alpha=0} &\stackrel{T\to0}{\sim}& \frac{\pi^2\zeta_n(k_BT)^2}{6h} -\frac{\zeta_nk_BT\mu}{h}\log\left[-\frac{\mu}{k_BT}\right] \label{lowTdotUa0}\end{aligned}$$
From Eqs. (\[lowTexpr1\]) we calculate the derivatives of $I_{n,1}$ and $\dot U_{n,1}$ at low temperatures:
\[lowTderivs\] $$\begin{aligned}
\frac{\partial I_{n,1,\alpha>0}}{\partial T} &\stackrel{T\to0}{\sim}& \frac{(1-\alpha)\zeta_nk_B}{\alpha h}\left[\frac{\mu}{\alpha k_BT}+1\right]e^{-\frac{\mu}{\alpha k_BT}} \label{lowTdIpdtT0} \\
\frac{\partial I_{n,1,\alpha>0}}{\partial\mu} &\stackrel{T\to0}{\sim}& \frac{\zeta_n}{h\alpha} - \frac{(1-\alpha)\zeta_n}{\alpha^{2} h}e^{-\frac{\mu}{\alpha k_BT}} \label{lowTdIpdmuT0} \\
\frac{\partial \dot U_{n,1,\alpha>0}}{\partial T} &\stackrel{T\to0}{\sim}& \frac{\pi^2\zeta_nk_B^2T}{3h} +\frac{(1-\alpha)\zeta_nk_BT\mu}{\alpha h} \nonumber \\
&& + \frac{(1-\alpha)\zeta_nk_B\mu}{\alpha h}\left[\frac{\mu}{\alpha k_BT}+1\right]e^{-\frac{\mu}{\alpha k_BT}} \label{lowTddotUpdtT0} \\
\frac{\partial \dot U_{n,1,\alpha>0}}{\partial \mu} &\stackrel{T\to0}{\sim}& \frac{\zeta_n\mu}{h\alpha} -\frac{(1-\alpha)\zeta_nk_BT}{\alpha h}\left[\frac{\mu}{\alpha k_BT}-1\right] \nonumber \\
&& \times e^{-\frac{\mu}{\alpha k_BT}} \label{lowTddotUpdmuT0}\\
\frac{\partial I_{n,1,\alpha=0}}{\partial T} &\stackrel{T\to0}{\sim}& \frac{\zeta_nk_B}{\alpha h}\left[-\log\left(-\frac{\mu}{\alpha k_BT}\right)+1\right] \label{lowTdIpdta0} \\
\frac{\partial I_{n,1,\alpha=0}}{\partial\mu} &\stackrel{T\to0}{\sim}& -\frac{\zeta_n k_BT}{h\mu}\left(1+\frac{\mu}{2 k_BT}\right) \label{lowTdIpdmuT0a0} \\
\frac{\partial \dot U_{n,1,\alpha=0}}{\partial T} &\stackrel{T\to0}{\sim}& \frac{\pi^2\zeta_nk_B^2T}{3h} +\frac{\zeta_nk_B^2T}{h}\frac{\mu}{k_BT} \nonumber \\
&& \times\left[-\log\left(-\frac{\mu}{k_BT}\right)-1\right] \label{lowTddotUpdta0} \\
\frac{\partial \dot U_{n,1,\alpha=0}}{\partial \mu} &\stackrel{T\to0}{\sim}& \frac{\zeta_nk_BT}{h}\left[-\log\left(-\frac{\mu}{k_BT}\right)-1\right] \label{lowTddotUpdmua0}
\end{aligned}$$
We observe that the results (\[lowTderivs\]) for $\alpha>0$ coincide in the lowest order approximation with the results (16)-(19) of Ref. [@PhysRevB.59.13080.1999.Rego], but they are different in general for $\alpha=0$. For example $\lim_{T\to0}\partial I_{n,1}/\partial T=\infty$ for $\alpha=0$, whereas for $\alpha>0$ $\lim_{T\to0}\partial I_{n,1}/\partial T=0$. The only result that remains the same in the lowest order approximation for both, $\alpha>0$ and $\alpha=0$, is $\partial\dot U_{n,1}/\partial T=\zeta_n\pi^2k_B^2T/(3h)$, which is the quanta of heat conductance if we set $\zeta_n=1$.
Nevertheless, in real heat conductance measurements the heat transport takes place without particle transport. Therefore the derivative $(\partial\dot U_{n,1}/\partial T)_\mu$ (taken at constant $\mu$) is not the quantity of interest for us. The quantity which represents the heat conductivity is actually $(d\dot{U}_{n,1}/dT)_{I_{n}=0}\equiv(d\dot{U}_{n,1}/dT)_{I_{n,1}}$, which is the derivative of $\dot U_{n,1}$ with respect to $T$, at constant $I_{n,1}(\mu,T)$. In these conditions the variations of $T$ and $\mu$ between the two reservoirs are related by $$\left.\frac{d\mu}{dT}\right|_{I_{n,1}} = -\left.\frac{\partial I_{n,1}}{\partial T}\right|_{\mu}\left(\left.\frac{\partial I_{n,1}}{\partial \mu}\right|_{T}\right)^{-1}. \label{dmudT_I}$$ Plugging Eq. (\[dmudT\_I\]) into the expression for $(d\dot U_{n,1}/dT)_{I_{n,1}}$, we obtain $$\kappa=\left.\frac{d\dot U_{n,1}}{dT}\right|_{I_{n,1}} = \frac{\partial\dot U_{n,1}}{\partial T}-\frac{\partial\dot U_{n,1}}{\partial \mu}\left.\frac{\partial I_{n,1}}{\partial T}\right|_{\mu}\left(\left.\frac{\partial I_{n,1}}{\partial \mu}\right|_{T}\right)^{-1} \label{kappa_gen}$$ which, if we replace $I_{n,1}$ by $N_\alpha$ and $\dot U_{n,1}$ by $U_\alpha$, becomes identical to the expression for the heat capacity, $C_V$ (\[cv\]). Therefore we can transcribe Eq. (\[cv\]) for the heat conductivity as $$\kappa_{n} = - \frac{I_{n,1}^2}{T\sigma}\frac{1+y_{i,0}}{y_{i,0}}-2k_{\rm B}^2T\sigma Li_2(-y_{i,0}) , \label{kappa_univ}$$ which is independent of $\alpha$ at any temperature. In the low temperature limit Eq. (\[kappa\_univ\]) becomes the universal asymptotic expression, $\kappa_0$ (\[kappa\_LT\]), which is equivalent to (\[cv\_limT0\]) from equilibrium thermodynamics.
The result (\[kappa\_LT\]) was obtained before [@PhysRevLett.81.232.1998.Rego; @SuperlatticesMicrostructures23.673.1998.Angelescu; @PhysRevB.59.13080.1999.Rego; @PhysRep.395.159.2004.Blencowe] without imposing the condition $dI_{n,1}=0$ due to the fact that the second term at the right hand side of Eq. (\[kappa\_gen\]) converges to zero at $T\to0$, as one can readily check from the asymptotic expressions (\[lowTderivs\]).
In Ref. [@PhysRep.395.159.2004.Blencowe] Blencowe analyzed also the entropy flux through the 1D channel in the low temperature limit and observed that it is independent of $\alpha$. Let’s analyze it here from the perspective of Ref. [@JPA35.7255.2002.Anghel] and prove that it is independent of $\alpha$ at any temperature.
The entropy flux from one of the reservoirs may be calculated in a way similar to the calculation of heat and particle fluxes: $$\begin{aligned}
\dot S &=& \frac{k_B\zeta_n}{h}\int_{0}^{\infty}\big\{[1+(1-\alpha)n(\epsilon)]\log[1+(1-\alpha)n(\epsilon)]\nonumber \\
&& -n(\epsilon)\log[n(\epsilon)]-[1-\alpha n(\epsilon)]\log[1-\alpha n(\epsilon)]\big\}d\epsilon \nonumber
\\
&=& -\frac{k_{\rm B}^2 T\zeta_n}{h} [2Li_2(-y_{i,0}) + \log{(1+y_{i,0})}\log{y_{i,0}}] \label{entropia_flux} ,\end{aligned}$$ and represents the flux of the number of configurations of particle populations, $\{n(\epsilon)\}$, from one reservoir to the other. But Eq. (\[entropia\_flux\]) is identical in form with Eq. (\[entropia\]) and therefore it is independent of $\alpha$ at any temperature.
To clarify the microscopic reason for which the 1D entropy and heat fluxes are independent of statistics, we use Eqs. (\[IUpoly\]) and the relation $Li_2(x)+Li_2[-x/(1-x)]=-\frac{1}{2}\log^2(1-x)$ [@Lewin:book] to write $$\begin{aligned}
\dot U_{n,1}(\mu_i,T_i) &=& \frac{\alpha hI_{n,1}^2}{2\zeta_n} + \frac{\zeta_n(k_BT_i)^2}{h}Li_2\left(\frac{y_{0,i}}{y_{0,i}+1}\right) \nonumber \\
&\equiv& \dot U_{0,n,1}(I_{n,1}) + \dot U_{B,n,1}(I_{n,1},T_i) \label{def_dotUB}\end{aligned}$$ where $\dot U_{0,n,1}$ is the energy flux at zero temperature and $\dot U_{B,n,1}$ is the excitation energy flux–in a Bose gas, $\dot U_{0,n,1}\equiv0$ and $\dot U_{n,1}\equiv\dot U_{B,n,1}$, hence the notation. As one can see directly from Eq. (\[def\_dotUB\]), $\dot U_{B,n,1}$ is independent of $\alpha$ at any temperature and therefore at zero net current, $I_{n,1}(\mu_1,T_1)-I_{n,1}(\mu_2,T_2)=0$, the heat flux is equal to the difference between the excitation energy fluxes, $$\dot U_n = \dot U_{B,n,1}(I_{n,1},T_2)-\dot U_{B,n,1}(I_{n,1},T_1), \label{dotUN_B}$$ which is independent of $\alpha$ at any $T$.
The statistics independence of $\dot U_{B,n,1}(I,T)$ at any $T$ may be interpreted microscopically also starting from the analogy with the equilibrium thermodynamics of systems of constant DOS. In Ref. [@JPA35.7255.2002.Anghel] it was proven that for systems of particles of different statistics, but the same, constant DOS and particle number, there is a one-to-one mapping between configurations of particle populations, $\{n(\epsilon)\}$, which have the same excitation energy. This implies that the canonical partition functions of such systems are independent of $\alpha$ and therefore all their canonical thermodynamics is independent of statistics, including the entropy and the heat capacity.
![The microscopic analysis of the statistics independence of the heat conductivity: there is a one-to-one correspondence between micro-configurations of particles of different statistics which have the same excitation energy, $\dot U_{B,n,1}$, and therefore carry the same heat-fluxes.[]{data-label="B_F_flux"}](B_F_flux.eps){width="6cm"}
The same argument can be transcribed for energy fluxes in 1D channels. If we have two gases, one of parameter $\alpha$ and another of parameter $\alpha'$, both gases carrying the same particle flux, $I_{n,1}$, through a 1D channel, then one can establish a one-to-one correspondence between configurations of particle populations in the two gases, with the same $\dot U_{B,n,1}$. Two such configurations, one of bosons and one of fermions, are shown in Fig. \[B\_F\_flux\]. This implies that both, the excitation energy flux and the entropy flux (which is determined by the flux of the number of configurations), are independent of statistics.
In conclusion I showed that the particle, energy, entropy and heat fluxes through a 1D channel are analogue to the particle number, internal energy, entropy and the heat capacity of a gas of constant density of states. Using this analogy, I wrote analytical expressions for all the fluxes and their derivatives with respect to the chemical potential and temperature, I calculated their asymptotic expressions in the limit $T\to0$, and I showed that the heat and entropy fluxes are independent of the statistics of the particles involved in the transport at any temperature, not only when $T\to0$, as it was known before. Using a construction I introduced in Ref. [@JPA35.7255.2002.Anghel], I showed what is the microscopic reason for the independence of statistics of the constituent particles for heat and entropy fluxes in 1D channels.
The financial support from the Romanian National Authority for Scientific Research grant PN 09370102, and the Romanian, IFIN-HH–JINR collaboration grants 4027-3-10/11 and N4006 is gratefully acknowledged.
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abstract: '$A$-polynomials were introduced by Meyn and play an important role in the iterative construction of high degree self-reciprocal irreducible polynomials over the field $\mathbb F_2$, since they constitute the starting point of the iteration. The exact number of $A$-polynomials of each degree was given by Niederreiter. Kyuregyan extended the construction of Meyn to arbitrary even finite fields. We relate the $A$-polynomials in this more general setting to inert places in a certain extension of elliptic function fields and obtain an explicit counting formula for their number. In particular, we are able to show that, except for an isolated exception, there exist A-polynomials of every degree.'
address:
- 'Alp Bassa. Boğaziçi University, Faculty of Arts and Sciences, Department of Mathematics, 34342 Bebek, İstanbul, Turkey, '
- 'Ricardo Menares. Pontificia Universidad Católica de Chile, Facultad de Matemáticas, Vicuña Mackenna 4860, Santiago, Chile.'
author:
- 'Alp Bassa $^\P$'
- 'Ricardo Menares $^\dag$'
title: Enumeration of a special class of irreducible polynomials in characteristic 2
---
The $Q$-transform plays a prominent role in the construction of (self-reciprocal) irreducible polynomials. Given a polynomial $f\in \mathbb F_q[T]$, its $Q$-transform is given by $$f^Q(T):=T^{\deg f}\cdot f\Bigl(T+\frac1T\Bigr).$$ Then, $f^Q$ is a self-reciprocal polynomial of degree $2\cdot\deg f$. Clearly for $f^Q$ to be irreducible a necessary condition is that $f$ is irreducible. In characteristic 2, a simple sufficient and necessary condition for the irreducibility of $f^Q$ in terms of the coefficients of $f$ was established by Meyn ([@meyn], Theorem 6). More surprising is the fact that it is even possible to devise criteria to ensure that irreducibility is preserved under repeated applications of the $Q$-transform, hence giving an infinite sequence of self-reciprocal irreducible polynomials of increasing degree. Starting with an irreducible polynomial $f_0\in \mathbb F_q[T]$, we iteratively define a sequence of polynomials $f_m \in \mathbb F_q[T]$ by $$f_{m+1}=f_m^Q, \quad m \geq 0.$$
In what follows, we set $q=2^r$ and describe the conditions mentioned above in characteristic 2. Let $f(T)=T^n+a_{n-1}T^{n-1}+\ldots+a_1T+a_0 \in \mathbb F_q[T]$ be a monic irreducible polynomial of degree $n$. We say that $f$ is an $A$-*polynomial* if $Tr_{\mathbb F_q/\mathbb F_2}(a_{n-1})=1$ and $Tr_{\mathbb F_q/\mathbb F_2}(a_1/a_0)=1$. Then, whenever $f_0$ is an $A$-polynomial, the polynomial $f_m(T)$ is irreducible of degree $n2^m$ for all $m$. This fact was proved by Meyn [@meyn] and Varshamov [@var], when $r=1$, and generalized latter by Kyuregyan [@recurrent] for general $r \geq 1$.
When $q$ is odd, the $Q$-transform behaves in a more subtle way and the results are less complete. In that setting, S. Cohen introduced the related $R$-transform, which leads to a comparable iterative construction of irreducible reciprocal polynomials [@SCohen]. For a comparison of both transforms in terms of Galois theory see [@GaloisTheory].
In this note, we provide a closed formula for the number of $A$-polynomials in $\mathbb F_{2^r}[T]$ of given degree. On the other hand, it is known that there are no $A$-polynomials of degree 3 in $\mathbb F_2[T]$. In all other cases, our formula allows us to deduce the existence of $A$-polynomials of each degree over every finite field of characteristic 2.
\[main\] We denote by $A_r(n)$ the number of $A$-polynomials in $\mathbb F_{2^r}[T]$ of degree $n$.
1. Write $n=2^k\cdot m$, ($m$ odd) and let $\alpha=\frac{-1+\sqrt{-7}}{2}$. Then, $$A_r(n)=\frac{1}{4n} \sum_{d|m}\mu\Bigl(\frac{ m}{d}\Bigr)\bigl(q^{2^kd}+1-\alpha^{r 2^kd}-{\overline{\alpha}}^{r 2^k d}\bigr).$$ Here, $\mu$ is the Möbius function. In particular, we have that $$\label{estimate}
\left|A_r(n)-\frac{q^n}{4n}\right| \leq \frac{\sigma_0(m)}{4n}(q^{n/3}+1+2\cdot 2^{rn/6}).$$ Here, $\sigma_0(m)$ is the number of positive divisors of $m$. Hence, $A_r(n) \sim \frac{q^n}{4n}$ as $ n \rightarrow \infty.$
2. Assume $(r,n)\neq (1,3)$. Then, $A_r(n)\geq 1$.
In order to prove Theorem \[main\], we exploit the correspondence between irreducible polynomials of each degree $n$ and degree $n$ places that are inert in a particular unramified extension of elliptic function fields. Then, we show that an exact count can be obtained using the corresponding $L$-polynomials. A very rough estimate then ensures the existence of $A$-polynomials of any degree $n\geq 7$ over all even finite fields, which can be used as the starting point of an iterative construction.
Theorem \[main\] is a generalization of a result of Niederreiter in [@niederreiter], treating the case $r=1$ (cf. Remark \[nied\]). His method requires an explicit evaluation of certain Kloosterman sums attached to additive characters, which is available only when the base field is small. By turning around Niederreiter’s reasoning, we obtain an archimedean evaluation of a certain weighted average of Kloosterman sums (cf. Proposition \[Kloos\] below).
For results and notation about algebraic function fields we refer the reader to [@sti]. In [@MoisioRanto] a related approach is applied to the problem of counting polynomials with prescribed coefficients.
Interpreting $A$-polynomials in terms of an extension of elliptic function fields
=================================================================================
Let $q=2^r$, and let $\mathbb F_q$ be the finite field of $q$ elements. Consider the rational function field $F=\mathbb F_q(x)$ and the extensions $E_1=F(y_1)$ and $E_2=F(y_2)$, with $y_1^2+y_1=x$ and $y_2^2+y_2=1/x$. Both $E_1$ and $E_2$ are again rational function fields.
Let $f(x)=x^n+a_{n-1}x^{n-1}+\ldots+a_1x+a_0 \in \mathbb F_q[x]$ be a monic irreducible polynomial of degree $n$, with $f(x)\neq x$. We denote by $P_f$ the place of $F$ of degree $n$ associated to $f$. Then,
1. $P_f$ is inert in $E_1/F$ if and only if $Tr_{\mathbb F_q/\mathbb F_2}(a_{n-1})=1$
2. $P_f$ is inert in $E_2/F$ if and only if $Tr_{\mathbb F_q/\mathbb F_2}(a_1/a_0)=1.$
In particular, $f$ is an $A$-polynomial if and only if $P_f$ is inert in both extensions $E_1/F$ and $E_2/F$.
First we prove i). If $c$ is a root of $f$ in $\overline{\mathbb F}_q$, then $c \neq 0$ and $a_{n-1}=Tr_{\mathbb F_{q^n}/\mathbb F_q}(c)$. Hence by the transitivity of the trace, the condition $Tr_{\mathbb F_q/\mathbb F_2}(a_{n-1})=1$ is equivalent to $Tr_{\mathbb F_{q^n}/\mathbb F_2}(c)\neq 0$. By Hilbert’s Theorem 90, this happens exactly if $c$ is not of the form $\gamma^2-\gamma$ for any $\gamma\in \mathbb F_{q^n}$. In turn, this happens if and only if $f(y_1^2+y_1) \in \mathbb F[y_1]$ is irreducible. The last condition is equivalent to $P_f$ being inert in $E_1/F$, thus proving i).
Part ii) follows form Part i) applied to $f^*(x)=x^nf(1/x)\quad \diamond$\
Consider the compositum $E'=E_1\cdot E_2$ over $F$. The extension $E'/F$ is Galois, with Galois group $\mathbb Z/2\mathbb Z\times \mathbb Z/2\mathbb Z$, with $E_1$ and $E_2$ corresponding to the subgroups $\mathbb Z/2\mathbb Z\times \{0\}$ and $\{0\}\times \mathbb Z/2\mathbb Z$, respectively. Let $E$ be the subfield corresponding to third subgroup $H$, the diagonal subgroup. Clearly $E=F(y)$ with $y=y_1+y_2$ satisfying $y^2+y=x+1/x$. In other words, $E$ is the function field of the elliptic curve over $\mathbb F_q$ with $j=1$.
\[connection\] Let $P'$ be a place of $E'$ above $P_f$. We denote by $G(P'|P_f)$ be the associated decomposition group in $E'/F$. Let $C_r(n)$ be the number if inert places of degree $n$ in the extension $E'/E$. Then,
1. $f$ is an $A$-polynomial if and only if $G(P'|P_f)=H$
2. We have that $C_r(n)=2A_r(n)$.
The polynomial $T$ corresponds to the zero $P_T$ of $x$. Since only the pole $P_\infty$ of $x$ ramifies in the extension $E_1/F$, the places $P_T$ and $P_\infty$ are the only places of $F$ ramified in the extension $E'/F$. Both places are ramified in $E/F$ and in each case the place of $E$ lying above them splits in the extension $E'/E$. The extension $E'/E$ is unramified. Using the Riemann-Hurwitz genus formula, we see that the genera of $E$ and $E'$ are both $1$. For a place $P_f\neq P_T, P_\infty$, let $Z(P_f)$ be the associated decomposition group in the extension $E'/F$. The place $P_f$ is inert in $E_1/F$ and $E_2/F$, if and only if $Z(P_f)=H$. Hence to each degree $n$ place $P_f$ of $F$ that is inert in $E_1/F$ and $E_2/F$ there correspond two places of $E$ of degree $n$ that are inert in $E'/E$. In particular, to every $A$-polynomial $f$ there correspond two places of $E$ of degree $\deg f$ that are inert in $E'/E$. $\quad \diamond$
Enumerating $A$-polynomials over arbitrary even finite fields
=============================================================
In this section we provide a proof of Theorem \[main\]. Following Proposition \[connection\], we need to count the number of inert places of $E$ and $E'$. We will achieve this task by means of their $L$-polynomials.
All function fields can be defined already over $\mathbb F_2$. Hence we consider the extension $\mathbb F_2(x,y)/\mathbb F_2(x)$ with $y^2+y=x+1/x$. Among the rational places of $\mathbb F_2(x)$, the pole and zero of $x$ are ramified, the zero of $x-1$ splits in $\mathbb F_2(x,y)/\mathbb F_2(x)$, giving a total of $4$ rational places of $\mathbb F_2(x,y)$. The elliptic function field hence has trace $-1$ and $L$-polynomial $$2t^2+t+1=(1-\alpha t)(1-{\overline{\alpha}} t) \text{ with } \alpha=\frac{-1+\sqrt{-7}}{2}.$$ The $L$-polynomial of the constant field extension $E=\mathbb F_q(x,y)$ with $q=2^r$ is hence $$L_E(t)=(1-\alpha^r t)(1-{\overline{\alpha}}^r t).$$
The $L$-polynomial $L_{E'}$ of $E'$ has to be divisible by $L_E$ and be also of degree $2$, since $g(E')=1$. Hence $L_{E'}=L_{E}$. The number of degree $n$ places for each of the function fields is given by (see [@sti Propositions 5.1.16 and 5.2.9]) $$B(n)=\frac{1}{n}\sum_{d|n}\mu\Bigl(\frac{n}{d}\Bigr) \bigl(q^d+1-\alpha^{r d}-{\overline{\alpha}}^{r d}\bigr).\label{Bn}$$ For even $n$ the $B(n)$ places of $E'$ of degree $n$ come from the $C_r({n/2})$ inert places of degree $n/2$ of $E$ and the $B(n)-C_r(n)$ splitting places of degree $n$ (we get $2$ degree $n$ places for each splitting place). For odd $n$ the $B(n)$ places of $E'$ of degree $n$ come only from the $B(n)-C_r(n)$ splitting places of $E$ of degree $n$. Hence we obtain $$\begin{aligned}
B(n)=&C_r({n/2})+2\cdot (B(n)-C_r(n))& \text{for}\ n\ \text{even}\\
B(n)=&2\cdot (B(n)-C_r(n))& \text{for}\ n\ \text{odd}.\\\end{aligned}$$ Writing $n=2^k\cdot m$ for an odd integer $m$, we obtain $$\label{Cn}
C_r(2^k\cdot m)=\sum_{i=1}^{k+1}\frac{1}{2^i}B({2^{k+1-i}\cdot m}).$$
Using Proposition \[connection\], ii) and equations and , we obtain the formula stated in the first part of Theorem \[main\]. The estimate is obtained by using that $|\alpha|=\sqrt{2}$ and the fact that any proper divisor $d |m$ satisfies $d \leq m/3$.
The function field $E$ has genus $1$. Hence using estimates for the number of higher degree places (see for instance[@sti Corollary 5.2.10]) $B(n)\geq 1$ for any $n$ with $q^{(n-1)/2}(q^{1/2}-1)\geq 3$. Hence for $r\geq 4$ we have $B(n)\geq 1$ for any $n$. For $r=1$, we need $n\geq 7$ and for $r=2$ or $3$ we need $n\geq 3$ to ensure $B(n)\geq 1$ using this estimate. A case by case analysis using the $L$-polynomial shows that in all cases except $(r,n)=(1,3),$ we have $B(n)\geq 1$. Since $E'$ has genus 1, using Proposition \[connection\], ii) and equation , we obtain the second assertion in Theorem \[main\].
\[nied\] Note that for $r=1$ we recover the main Theorem in [@niederreiter]. Indeed, use the elementary identity $$\alpha^t+\overline{\alpha}^t=\frac{1}{2^{t-1}}\sum_{j=0}^{[t/2]}\left(\begin{array}{c}
t\\
2j
\end{array}\right)(-1)^{t+j}7^j,$$ valid for all integers $t\geq 1$.
Averages of Kloosterman sums
============================
Let $q=2^r$ and let $\chi : \mathbb F_q \rightarrow \mathbb C^*$ be an additive character. For any integer $n\geq 1$, let $\chi^{(n)} :\mathbb F_{q^n} \rightarrow \mathbb C^*$ be the additive character defined by $$\chi^{(n)}(u)=\chi\circ Tr_{\mathbb F_{q^n}/\mathbb F_{2}}(u).$$
Let $$K( \chi^{(n)};a,b)=\sum_{\alpha \in \mathbb F_{q^n}^*} \chi^{(n)}(a\alpha + b\alpha^{-1})$$ be the Kloosterman sum attached to $\chi^{(n)}$.
\[Kloos\] Assume $\chi$ is a non trivial additive character. Then, $$\label{archimedean}
\frac{1}{q^2}\sum_{u \in \mathbb F_q}\chi(u)\sum_{a,b \in \mathbb F_q \atop a+b=u} K(\chi^{(n)};a,b) = \sum_{d|n}\mu \left( \frac{n}{d}\right) \cdot d \cdot A_r(d),$$ for all $n \geq 1$.
let
$$\begin{aligned}
R(n) =&\{ \alpha \in \mathbb F_{q^n}^* : Tr_{\mathbb F_{q^n}/\mathbb F_{2}}(\alpha)=Tr_{\mathbb F_{q^n}/\mathbb F_{2}}(\alpha^{-1})=1 \}.\\
R^*(n)=&\{\alpha\in R(n) : [\mathbb F_q(\alpha):\mathbb F_q]=n\}.\end{aligned}$$
We have that $$\label{obvio}
|R^*(n)|=n\cdot A_r(n), \quad |R(n)| = \sum_{d|n} |R^*(d)|.$$
Since $\chi$ is non trivial, for all $u \in \mathbb F_q$ we have that [@LN Corollary 5.31], $$\frac{1}{q} \sum_{a \in \mathbb F_q}\chi(ua) = \left\{ \begin{array}{cc}
1 & \textrm{ if } u=0\\
0 & \textrm{ otherwise.}
\end{array}\right.$$
Set $T^{(n)}:=Tr_{\mathbb F_{q^n}/\mathbb F_{2}}.$ We have that
$$\begin{aligned}
|R(n)| &= \sum_{\alpha \in \mathbb F_{q^n}^*} 1_{\{T^{(n)}(\alpha)=1\}}\cdot 1_{\{T^{(n)}(\alpha^{-1})=1\}}\\
&= \sum_{\alpha \in \mathbb F_{q^n}^*} \frac{1}{q^2} \sum_{a \in \mathbb F_q}\chi\left((T^{(n)}(\alpha)+1)a\right) \sum_{b \in \mathbb F_q}\chi\left((T^{(n)}(\alpha^{-1})+1)b\right)\\
&= \frac{1}{q^2} \sum_{\alpha \in \mathbb F_{q^n}^*} \sum_{a,b \in \mathbb F_q}\chi(a+b)\chi\left(T^{(n)}(a\alpha+b\alpha^{-1})\right)\\
&= \frac{1}{q^2} \sum_{u\in \mathbb F_q} \chi(u) \sum_{a,b \in \mathbb F_q \atop a+b=u} \sum_{\alpha \in \mathbb F_{q^n}^*}\chi^{(n)}(a\alpha+b\alpha^{-1})\\
&= \frac{1}{q^2}\sum_{u \in \mathbb F_q}\chi(u)\sum_{a,b \in \mathbb F_q \atop a+b=u} K(\chi^{(n)};a,b).\end{aligned}$$
We conclude by combining this equality with the relations . $\quad \diamond$
When $r=1$, Niederreiter finds in [@niederreiter] an explicit evaluation in archimedean terms of each individual Kloosterman sum in the LHS of . It seems difficult to obtain a similar individual evaluation for general $r$. Our method proceeds by interpreting the whole LHS in terms of the number of places of appropriate degree in a particular elliptic extension and then use the explicit knowledge of the corresponding $L$-polynomials to determine such quantities.
[BHMP]{}
Alp Bassa, Emmanuel Hallouin, Ricardo Menares, and Marc Perret. Galois theory and iterative construction of irreducible polynomials. .
Stephen D. Cohen. The explicit construction of irreducible polynomials over finite fields. , 2(2):169–174, 1992.
Mels K. Kyuregyan. Recurrent methods for constructing irreducible polynomials over [${\rm GF}(2)$]{}. , 8(1):52–68, 2002.
Rudolf Lidl and Harald Niederreiter. , volume 20 of [*Encyclopedia of Mathematics and its Applications*]{}. Cambridge University Press, Cambridge, second edition, 1997. With a foreword by P. M. Cohn.
Helmut Meyn. On the construction of irreducible self-reciprocal polynomials over finite fields. , 1(1):43–53, 1990.
Marko Moisio and Kalle Ranto. Elliptic curves and explicit enumeration of irreducible polynomials with two coefficients prescribed. , 14(3):798–815, 2008.
Harald Niederreiter. An enumeration formula for certain irreducible polynomials with an application to the construction of irreducible polynomials over the binary field. , 1(2):119–124, 1990.
Henning Stichtenoth. , volume 254 of [ *Graduate Texts in Mathematics*]{}. Springer-Verlag, Berlin, second edition, 2009.
R. R. Varshamov. A general method of synthesis for irreducible polynomials over [G]{}alois fields. , 275(5):1041–1044, 1984.
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abstract: 'As a continued work of [@MXK], we are concerned with the Timoshenko system in the case of non-equal wave speeds, which admits the dissipative structure of *regularity-loss*. Firstly, with the modification of a priori estimates in [@MXK], we construct global solutions to the Timoshenko system pertaining to data in the Besov space with the regularity $s=3/2$. Owing to the weaker dissipative mechanism, extra higher regularity than that for the global-in-time existence is usually imposed to obtain the optimal decay rates of classical solutions, so it is almost impossible to obtain the optimal decay rates in the critical space. To overcome the outstanding difficulty, we develop a new frequency-localization time-decay inequality, which captures the information related to the integrability at the high-frequency part. Furthermore, by the energy approach in terms of high-frequency and low-frequency decomposition, we show the optimal decay rate for Timoshenko system in critical Besov spaces, which improves previous works greatly.'
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**AMS subject classification.** 35L45; 35B40; 74F05\
**Key words and phrases.** Global existence; minimal decay regularity; critical Besov spaces; Timoshenko system
Introduction
============
In this work, we are concerned with the following Timoshenko system (see [@T1; @T2]), which is a set of two coupled wave equations of the form $$\begin{aligned}
\left\{\begin{array}{l}
\varphi_{tt}-(\varphi_x-\psi)_x=0,\\[2mm]
\psi_{tt}-\sigma(\psi_{x})_{x}-(\varphi_x-\psi)+\gamma \psi_t =0.
\end{array}\right. \label{R-E1}\end{aligned}$$ System (\[R-E1\]) describes the transverse vibrations of a beam. Here, $t\geq 0$ is the time variable, $x\in \mathbb{R}$ is the spacial variable which denotes the point on the center line of the beam, $\varphi(t,x)$ is the transversal displacement of the beam from an equilibrium state, and $\psi(t,x)$ is the rotation angle of the filament of the beam. The smooth function $\sigma(\eta)$ satisfies $\sigma'(\eta)>0$ for any $\eta\in\mathbb{R}$, and $\gamma$ is a positive constant. We focus on the Cauchy problem of (\[R-E1\]), so the initial data are supplemented as $$(\varphi, \varphi_{t}, \psi, \psi_{t})(x,0)
=(\varphi_{0}, \varphi_{1}, \psi_{0}, \psi_{1})(x).\label{R-E2}$$ Based on the change of variable introduced by Ide, Haramoto, and the third author [@IHK]: $$\begin{aligned}
\label{R-E3}
v=\varphi_x-\psi, \quad
u=\varphi_t, \quad
z=a\psi_x, \quad
y=\psi_t,\end{aligned}$$ with $a>0$ being the sound speed defined by $a^2=\sigma'(0)$, it is convenient to rewrite (\[R-E1\])-(\[R-E2\]) as a Cauchy problem for the first-order hyperbolic system of $U=(v,u,z,y)^{\top}$ $$\begin{aligned}
\label{R-E4}
\left\{\begin{array}{l}
U_t + A(U)U_x + LU =0,\\[2mm]
U(x, 0) = U_0(x)
\end{array}\right.\end{aligned}$$ with $U_{0}(x)=(v_{0}, u_{0}, z_{0}, y_{0})(x)$, where $v_0=\varphi_{0,x}-\psi_{0}$, $u_0=\varphi_1$, $z_0=a\psi_{0,x}$, $y_0=\psi_1$ and $$\begin{aligned}
A(U)=-\left(
\begin{array}{cccc}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & a \\
0 & 0 &\frac{\sigma'(z/a)}{a} & 0
\end{array}
\right),\ \ \
L=\left(
\begin{array}{cccc}
0 & 0 & 0 & 1 \\
0 & 0 & 0 & 0 \\
0 & 0 & 0 & 0 \\
-1 & 0 & 0 & \gamma
\end{array}
\right).\end{aligned}$$ Note that $A(U)$ is a real symmetrizable matrix due to $\sigma'(z/a)>0$, and the dissipative matrix $L$ is nonnegative definite but not symmetric. Such degenerate dissipation forces (\[R-E4\]) to go beyond the class of generally dissipative hyperbolic systems, so the recent global-in-time existence (see [@XK1]) for hyperbolic systems with symmetric dissipation can not be applied directly, which is the main motivation on studying the Timoshenko system (\[R-E1\]).
Let us review several known results on (\[R-E1\]). In a bounded domain, it is known that (\[R-E1\]) is exponentially stable if the damping term $\varphi_{t}$ is also present on the left-hand side of the first equation of (\[R-E3\]) (see, e.g., [@RFSC]). Soufyane [@S] showed that (\[R-E1\]) could not be exponentially stable by considering only the damping term of the form $\psi_{t}$, unless for the case of $a=1$ (equal wave speeds). A similar result was obtained by Rivera and Racke [@RR2] with an alternative proof. In addition, Rivera and Racke [@RR1] also investigated the Timoshenko system with the heat conduction, which is described by the classical Fourier law. In the whole space, the third author and his collaborators [@IHK] considered the corresponding linearized form of (\[R-E4\]): $$\label{R-E5}
\left\{\begin{array}{l}
v_t-u_x+y=0,\\[2mm]
u_t-v_x=0,\\[2mm]
z_t-ay_x=0,\\[2mm]
y_t-az_x-v+\gamma y=0,\\[2mm]
(v, u, z, y)(x, 0)
=(v_{0}, u_{0}, z_{0}, y_{0})(x),
\end{array}\right.$$ and showed that the dissipative structure could be characterized by $$\left\{\begin{array}{l}
{\rm Re}\,\lambda(i\xi)\leq -c\eta_1(\xi)
\qquad {\rm for} \quad a=1, \\[1mm]
{\rm Re}\,\lambda(i\xi)\leq -c\eta_2(\xi)
\qquad {\rm for} \quad a\neq 1,
\end{array}\right.\label{R-E6}$$ where $\lambda(i\xi)$ denotes the eigenvalues of the system (\[R-E5\]) in the Fourier space, $\eta_1(\xi)=\frac{\xi^2}{1+\xi^2}$, $\eta_2(\xi)=\frac{\xi^2}{(1+\xi^2)^2}$, and $c>0$ is some constant. Consequently, the following decay properties were established for $U=(v,u,z,y)^{\top}$ of (\[R-E5\]) (see [@IHK] for details): $$\label{R-E7}
\|\partial_x^k U(t)\|_{L^2}
\lesssim
(1+t)^{-\frac{1}{4}-\frac{k}{2}}\|U_0\|_{L^1}
+e^{-ct}\|\partial_x^kU_0\|_{L^2}$$ for $a=1$, and $$\label{R-E8}
\|\partial_x^k U(t)\|_{L^2}
\lesssim
(1+t)^{-\frac{1}{4}-\frac{k}{2}}\|U_0\|_{L^1}
+(1+t)^{-\frac{l}{2}}\|\partial_x^{k+l}U_0\|_{L^2}$$ for $a\neq 1$. Recently, under the additional assumption $\int_{\mathbb{R}}U_{0}dx=0$, Racke and Said-Houari [@RS] strengthened (\[R-E7\])-(\[R-E8\]) such that linearized solutions decay faster with a rate of $t^{-\gamma/2}$, by introducing the integral space $L^{1,\gamma}(\mathbb{R})$.
Clearly, the high frequency part of (\[R-E7\]) yields an exponential decay, whereas the corresponding part of (\[R-E8\]) is of the regularity-loss type, since $(1+t)^{-\ell/2}$ is created by assuming the additional $\ell$-th order regularity on the initial data. Consequently, extra higher regularity than that for global-in-time existence of classical solutions is imposed to obtain the optimal decay rates.
In [@IK], Ide and the third author performed the time-weighted approach to establish the global existence and asymptotic decay of solutions to the nonlinear problem (\[R-E4\]). To overcome the difficulty caused by the regularity-loss property, the spatially regularity $s\geq6$ was needed. Denote by $s_{c}$ the critical regularity for global existence of classical solutions. Actually, the local-in-time existence theory of Kato and Majda [@K; @M] implies that $s_{c}=2$ for the Timoshenko system (\[R-E4\]), actually, the extra regularity is used to take care of optimal decay estimates. Consequently, some natural questions follow. Is $s=6$ the minimal decay regularity for (\[R-E4\]) with the regularity-loss? If not, which index characterises the minimal decay regularity? This motivates the following general definition.
\[defn1.1\] If the optimal decay rate of $L^{1}(\mathbb{R}^n)$-$L^2(\mathbb{R}^n)$ type is achieved under the lowest regularity assumption, then the lowest index is called the minimal decay regularity index for dissipative systems of regularity-loss, which is labelled as $s_{D}$.
Recently, we are concerned with the global existence and large-time behavior for (\[R-E4\]) in spatially critical Besov spaces. To the best of our knowledge, there are few results available in this direction for the Timoshenko system, although the critical space has already been succeeded in the study of fluid dynamical equations, see [@AGZ; @D1; @H; @PZ] for Navier-Stokes equations, [@D2; @XW; @XY; @Z] for Euler equations and related models. In [@XK1; @XK2], under the assumptions of dissipative entropy and Shizuta-Kawashima condition, the first and third authors have already investigated generally dissipative systems, however, the Timoshenko system admits the non-symmetric dissipation and goes beyond the class. Hence, as a first step, we [@MXK] considered the easy case, that is, (\[R-E4\]) with the equal wave speed ($a=1$). By virtue of an elementary fact in Proposition \[prop2.3\] (also see [@XK1]) that indicates the relation between homogeneous and inhomogeneous Chemin-Lerner spaces, we first constructed global solutions pertaining to data in the Besov space $B^{3/2}_{2,1}(\mathbb{R})$. Furthermore, the optimal decay rates of solution and its derivatives are shown in the space $B^{3/2}_{2,1}(\mathbb{R})\cap \dot{B}^{-1/2}_{2,\infty}(\mathbb{R})$ by the frequency-localization Duhamel principle and energy approach in terms of high-frequency and low-frequency decomposition.
In the present paper, we hope to establish similar results for (\[R-E4\]) with non-equal wave speeds ($a\neq1$) that has weaker dissipative mechanism. If done, we shall improve two regularity indices for Timoshenko system with regularity-loss: $s_{c}=3/2$ for global-in-time existence and $s_{D}=3/2$ for the optimal decay estimate, which lead to reduce significantly the regularity requirements on the initial data in comparison with [@IK].
Before main results, let us explain new technical points for (\[R-E4\]) with $a\neq1$ and the strategy to get round the obstruction. Firstly, as in [@MXK], the degenerate non-symmetric damping enables us to capture the dissipation from contributions of $(y,v,u_{x},z_{x})$, however, there is an additional norm related to $u_{x}$ in the proof for the dissipation of $v$. Indeed, we need to carefully take care of the topological relation between $\|u_{x}\|_{\widetilde{L}^{2}_{T}(\dot{B}^{-1/2}_{2,1})}$ and $\|u_{x}\|_{\widetilde{L}^{2}_{T}(B^{-1/2}_{2,1})}$ as in Proposition \[prop2.3\]. To do this, we localize (\[R-E4\]) with inhomogeneous blocks rather than homogeneous blocks to obtain the dissipative estimate for $v$.
Secondly, due to the weaker mechanism of regularity-loss, it seems that there is no possibility to capture optimal decay rates in the critical space $B^{3/2}_{2,1}(\mathbb{R})$, since the polynomial decay at the high-frequency part comes from the fact that the initial data is imposed extra higher regularity (see (\[R-E8\])). To overcome the outstanding difficulty, there are new ingredients in comparison with the case of equal wave speeds in [@MXK]. Precisely, we develop a new frequency-localization time-decay inequality for the dissipative rate $\eta(\xi)=\frac{|\xi|^2}{(1+|\xi|^2)^2}$ in $\mathbb{R}^{n}$, see Proposition \[prop3.1\]. At the formal level, we see that the high-frequency part decays in time not only with algebraic rates of any order as long as the function is spatially regular enough, but also additional information related the $L^p$-integrability is available. Consequently, the high-frequency estimate in energy approaches can be divided into two parts, and on each part, different values of $p$ (for example, $p=1$ or $p=2)$ are chosen to get desired decay estimates, see Lemma \[lem5.1\]. Additionally, it should be worth noting that the energy approach is totally different from that in [@MXK], where the frequency-localization Duhamel principle was used. Here, we shall employ somewhat “the square formula of the Duhamel principle" based on the Littlewood-Paley pointwise estimate in Fourier space for the linear system with right-hand side, see (\[R-E36\])-(\[R-E37\]) for details.
Our main results focus on the Timoshenko system with non-equal wave speeds ($a\neq1$), which are stated as follows.
\[thm1.1\] Suppose that $U_{0}\in B^{3/2}_{2,1}(\mathbb{R})$. There exists a positive constant $\delta_{0}$ such that if $$\|U_{0}\|_{B^{3/2}_{2,1}(\mathbb{R})}\leq
\delta_{0},$$ then the Cauchy problem (\[R-E4\]) has a unique global classical solution $U\in \mathcal{C}^{1}(\mathbb{R}^{+}\times
\mathbb{R})$ satisfying $$U \in\widetilde{\mathcal{C}}(B^{3/2}_{2,1}(\mathbb{R}))\cap\widetilde{\mathcal{C}}^{1}(B^{1/2}_{2,1}(\mathbb{R}))$$ Moreover, the following energy inequality holds that $$\begin{aligned}
&&\|U\|_{\widetilde{L}^\infty(B^{3/2}_{2,1}(\mathbb{R}))}+\Big(\|y\|_{\widetilde{L}^2_{T}(B^{3/2}_{2,1})}+\|(v,z_{x})\|_{\widetilde{L}^2_{T}(B^{1/2}_{2,1})}
+\|u_{x}\|_{\widetilde{L}^2_{T}(B^{-1/2}_{2,1})}\Big)\nonumber\\&\leq& C_{0}\|U_{0}\|_{B^{3/2}_{2,1}(\mathbb{R})}, \label{R-E9}\end{aligned}$$ where $C_{0}>0$ is a constant.
Theorem \[thm1.1\] exhibits the optimal critical regularity ($s_{c}=3/2$) of global-in-time existence for (\[R-E4\]), which was proved by the revised energy estimates in comparison with [@MXK], along with the local-in-time existence result in Proposition \[prop4.1\]. Observe that there is 1-regularity-loss phenomenon for the dissipation rate of $(v,u_{x})$.
Furthermore, with the aid of the new frequency-localization time-decay inequality in Proposition \[prop3.1\], we can obtain the the optimal decay estimates by using the time-weighted energy approach in terms of high-frequency and low-frequency decomposition.
\[thm1.2\] Let $U(t,x)=(v,u,z,y)(t,x)$ be the global classical solution of Theorem \[thm1.1\]. Assume that the initial data satisfy $U_{0}\in B^{3/2}_{2,1}(\mathbb{R})\cap\dot{B}^{-1/2}_{2,\infty}(\mathbb{R})$. Set $I_{0}:=\|U_{0}\|_{B^{3/2}_{2,1}(\mathbb{R})\cap\dot{B}^{-1/2}_{2,\infty}(\mathbb{R})}$. If $I_{0}$ is sufficiently small, then the classical solution $U(t,x)$ of (\[R-E4\]) admits the optimal decay estimate $$\begin{aligned}
\|U\|_{L^2}\lesssim I_{0}(1+t)^{-\frac{1}{4}}. \label{R-E10}\end{aligned}$$
Note that the embedding $L^1(\mathbb{R})\hookrightarrow \dot{B}^{-1/2}_{2,\infty}(\mathbb{R})$ in Lemma \[lem2.3\], as an immediate byproduct of Theorem \[thm1.2\], the usual optimal decay estimate of $L^{1}(\mathbb{R})$-$L^{2}(\mathbb{R})$ type is available.
\[cor1.1\] Let $U(t,x)=(v,u,z,y)(t,x)$ be the global classical solutions of Theorem \[thm1.1\]. If further the initial data $U_{0}\in L^1(\mathbb{R})$ and $\widetilde{I}_{0}:=\|U_{0}\|_{B^{3/2}_{2,1}(\mathbb{R})\cap L^1(\mathbb{R})}$ is sufficiently small, then $$\begin{aligned}
\|U\|_{L^2}\lesssim \widetilde{I}_{0}(1+t)^{-\frac{1}{4}}. \label{R-E11}\end{aligned}$$
Let us mention that Theorem \[thm1.2\] and Corollary \[cor1.1\] exhibit the optimal decay rate in the Besov space with $s_{c}=3/2$, that is, $s_{D}=3/2$, which implies that the minimal decay regularity coincides with the the critical regularity for global solutions, and the extra higher regularity is not necessary. In addition, it is worth noting that the present work opens a door for the study of dissipative systems of regularity-loss type, which encourages us to develop frequency-localization time-decay inequalities for other dissipative rates and investigate systems with the regularity-loss mechanism.
Finally, we would like to mention other studies on the dissipative Timoshenko system with different effects, see, e.g., [@RBS; @RFSC] for frictional dissipation case, [@FR; @SAJM; @SK] for thermal dissipation case, and [@ABMR; @ARMSV; @LK; @LP] for memory-type dissipation case.
The rest of this paper unfolds as follows. In Sect.\[sec:2\], we present useful properties in Besov spaces, which will be used in the subsequence analysis. In Sect.\[sec:3\], we shall develop new time-decay inequality with using frequency-localization techniques. Sect.\[sec:4\] is devoted to construct the global-in-time existence of classical solutions to (\[R-E4\]). Furthermore, in Sect.\[sec:5\], we deduce the optimal decay estimate for (\[R-E4\]) by employing energy approaches in terms of high-frequency and low-frequency decomposition. In Appendix (Sect.\[sec:6\]), we present those definitions for Besov spaces and Chemin-Lerner spaces for the convenience of reader.
**Notations.** Throughout the paper, $f\lesssim g$ denotes $f\leq Cg$, where $C>0$ is a generic constant. $f\thickapprox g$ means $f\lesssim g$ and $g\lesssim f$. Denote by $\mathcal{C}([0,T],X)$ (resp., $\mathcal{C}^{1}([0,T],X)$) the space of continuous (resp., continuously differentiable) functions on $[0,T]$ with values in a Banach space $X$. Also, $\|(f,g,h)\|_{X}$ means $
\|f\|_{X}+\|g\|_{X}+\|h\|_{X}$, where $f,g,h\in X$.
Tools
=====
\[sec:2\] In this section, we only collect useful analysis properties in Besov spaces and Chemin-Lerner spaces in $\mathbb{R}^{n}(n\geq1)$. For convenience of reader, those definitions for Besov spaces and Chemin-Lerner spaces are given in the Appendix. Firstly, we give an improved Bernstein inequality (see, *e.g.*, [@W]), which allows the case of fractional derivatives.
\[lem2.1\] Let $0<R_{1}<R_{2}$ and $1\leq a\leq b\leq\infty$.
- If $\mathrm{Supp}\mathcal{F}f\subset \{\xi\in \mathbb{R}^{n}: |\xi|\leq
R_{1}\lambda\}$, then $$\begin{aligned}
\|\Lambda^{\alpha}f\|_{L^{b}}
\lesssim \lambda^{\alpha+n(\frac{1}{a}-\frac{1}{b})}\|f\|_{L^{a}}, \ \ \mbox{for any}\ \ \alpha\geq0;\end{aligned}$$
- If $\mathrm{Supp}\mathcal{F}f\subset \{\xi\in \mathbb{R}^{n}:
R_{1}\lambda\leq|\xi|\leq R_{2}\lambda\}$, then $$\begin{aligned}
\|\Lambda^{\alpha}f\|_{L^{a}}\approx\lambda^{\alpha}\|f\|_{L^{a}}, \ \ \mbox{for any}\ \ \alpha\in\mathbb{R}.\end{aligned}$$
Besov spaces obey various inclusion relations. Precisely,
\[lem2.2\] Let $s\in \mathbb{R}$ and $1\leq
p,r\leq\infty,$ then
- If $s>0$, then $B^{s}_{p,r}=L^{p}\cap \dot{B}^{s}_{p,r};$
- If $\tilde{s}\leq s$, then $B^{s}_{p,r}\hookrightarrow
B^{\tilde{s}}_{p,r}$. This inclusion relation is false for the homogeneous Besov spaces;
- If $1\leq r\leq\tilde{r}\leq\infty$, then $\dot{B}^{s}_{p,r}\hookrightarrow
\dot{B}^{s}_{p,\tilde{r}}$ and $B^{s}_{p,r}\hookrightarrow
B^{s}_{p,\tilde{r}};$
- If $1\leq p\leq\tilde{p}\leq\infty$, then $\dot{B}^{s}_{p,r}\hookrightarrow \dot{B}^{s-n(\frac{1}{p}-\frac{1}{\tilde{p}})}_{\tilde{p},r}
$ and $B^{s}_{p,r}\hookrightarrow
B^{s-n(\frac{1}{p}-\frac{1}{\tilde{p}})}_{\tilde{p},r}$;
- $\dot{B}^{n/p}_{p,1}\hookrightarrow\mathcal{C}_{0},\ \ B^{n/p}_{p,1}\hookrightarrow\mathcal{C}_{0}(1\leq p<\infty);$
where $\mathcal{C}_{0}$ is the space of continuous bounded functions which decay at infinity.
\[lem2.3\] Suppose that $\varrho>0$ and $1\leq p<2$. It holds that $$\begin{aligned}
\|f\|_{\dot{B}^{-\varrho}_{r,\infty}}\lesssim \|f\|_{L^{p}}\end{aligned}$$ with $1/p-1/r=\varrho/n$. In particular, this holds with $\varrho=n/2, r=2$ and $p=1$.
Moser-type product estimates are stated as follows, which plays an important role in the estimate of bilinear terms.
\[prop2.1\] Let $s>0$ and $1\leq
p,r\leq\infty$. Then $\dot{B}^{s}_{p,r}\cap L^{\infty}$ is an algebra and $$\|fg\|_{\dot{B}^{s}_{p,r}}\lesssim \|f\|_{L^{\infty}}\|g\|_{\dot{B}^{s}_{p,r}}+\|g\|_{L^{\infty}}\|f\|_{\dot{B}^{s}_{p,r}}.$$ Let $s_{1},s_{2}\leq n/p$ such that $s_{1}+s_{2}>n\max\{0,\frac{2}{p}-1\}. $ Then one has $$\|fg\|_{\dot{B}^{s_{1}+s_{2}-n/p}_{p,1}}\lesssim \|f\|_{\dot{B}^{s_{1}}_{p,1}}\|g\|_{\dot{B}^{s_{2}}_{p,1}}.$$
In the analysis of decay estimates, we also need the general form of Moser-type product estimates, which was shown by Yong in [@Z].
\[prop2.2\] Let $s>0$ and $1\leq p,r,p_{1},p_{2},p_{3},p_{4}\leq\infty$. Assume that $f\in L^{p_{1}}\cap \dot{B}^{s}_{p_{4},r}$ and $g\in L^{p_{3}}\cap
\dot{B}^{s}_{p_{2},r}$ with $$\frac{1}{p}=\frac{1}{p_{1}}+\frac{1}{p_{2}}=\frac{1}{p_{3}}+\frac{1}{p_{4}}.$$ Then it holds that $$\begin{aligned}
\|fg\|_{\dot{B}^{s}_{p,r}}\lesssim \|f\|_{L^{p_{1}}}\|g\|_{\dot{B}^{s}_{p_{2},r}}+\|g\|_{L^{p_{3}}}\|f\|_{\dot{B}^{s}_{p_{4},r}}.\end{aligned}$$
In [@XK1], the first and third authors established a key fact, which indicates the connection between homogeneous Chemin-Lerner spaces and inhomogeneous Chemin-Lerner spaces.
\[prop2.3\] Let $s\in \mathbb{R}$ and $1\leq \theta, p,r\leq\infty$.
- It holds that $$\begin{aligned}
L^{\theta}_{T}(L^{p})\cap
\widetilde{L}^{\theta}_{T}(\dot{B}^{s}_{p,r})\subset \widetilde{L}^{\theta}_{T}(B^{s}_{p,r});\end{aligned}$$
- Furthermore, as $s>0$ and $\theta\geq r$, it holds that $$\begin{aligned}
L^{\theta}_{T}(L^{p})\cap
\widetilde{L}^{\theta}_{T}(\dot{B}^{s}_{p,r})=\widetilde{L}^{\theta}_{T}(B^{s}_{p,r})\end{aligned}$$
for any $T>0$.
The property of continuity for product in $\widetilde{L}^{\theta}_{T}(B^{s}_{p,r})$ is similar to in the stationary case (Proposition \[prop2.1\]), whereas the time exponent $\theta$ behaves according to the Hölder inequality.
\[prop2.4\] The following inequality holds: $$\|fg\|_{\widetilde{L}^{\theta}_{T}(B^{s}_{p,r})}\lesssim
(\|f\|_{L^{\theta_{1}}_{T}(L^{\infty})}\|g\|_{\widetilde{L}^{\theta_{2}}_{T}(B^{s}_{p,r})}
+\|g\|_{L^{\theta_{3}}_{T}(L^{\infty})}\|f\|_{\widetilde{L}^{\theta_{4}}_{T}(B^{s}_{p,r})})$$ whenever $s>0, 1\leq p\leq\infty,
1\leq\theta,\theta_{1},\theta_{2},\theta_{3},\theta_{4}\leq\infty$ and $$\frac{1}{\theta}=\frac{1}{\theta_{1}}+\frac{1}{\theta_{2}}=\frac{1}{\theta_{3}}+\frac{1}{\theta_{4}}.$$ As a direct corollary, one has $$\|fg\|_{\widetilde{L}^{\theta}_{T}(B^{s}_{p,r})}
\lesssim
\|f\|_{\widetilde{L}^{\theta_{1}}_{T}(B^{s}_{p,r})}\|g\|_{\widetilde{L}^{\theta_{2}}_{T}(B^{s}_{p,r})}$$ whenever $s\geq n/p,
\frac{1}{\theta}=\frac{1}{\theta_{1}}+\frac{1}{\theta_{2}}.$
Finally, we state a continuity result for compositions (see [@Abidi]) to end this section.
\[prop2.5\] Let $s>0$, $1\leq p, r, \rho\leq \infty$, $F\in
W^{[s]+1,\infty}_{loc}(I;\mathbb{R})$ with $F(0)=0$, $T\in
(0,\infty]$ and $v\in \widetilde{L}^{\rho}_{T}(B^{s}_{p,r})\cap
L^{\infty}_{T}(L^{\infty}).$ Then $$\|F(v)\|_{\widetilde{L}^{\rho}_{T}(B^{s}_{p,r})}\lesssim
(1+\|v\|_{L^{\infty}_{T}(L^{\infty})})^{[s]+1}\|v\|_{\widetilde{L}^{\rho}_{T}(B^{s}_{p,r})}.$$
Frequency-localization time-decay inequality
============================================
\[sec:3\] In the recent decade, harmonic analysis tools, especially for techniques based on Littlewood-Paley decomposition and paradifferential calculus have proved to be very efficient in the study of partial differential equations. It is well-known that the frequency-localization operator $\dot{\Delta}_{q}f$ (or $\Delta_{q}f$ ) has a smoothing effect on the function $f$, even though $f$ is quite rough. Moreover, the $L^p$ norm of $\dot{\Delta}_{q}f$ can be preserved provided $f\in L^p(\mathbb{R}^{n})$. To the best of our knowledge, so far there are few efforts about the decay property related to the operator $\dot{\Delta}_{q}f$. Here, the difficulty of regularity-loss mechanism forces us to develop the frequency-localization time-decay inequality. Precisely,
\[prop3.1\] Set $\eta(\xi)=\frac{\mid\xi\mid^{2}}{(1+\mid\xi\mid^{2})^{2}}$. If $f\in \dot{B}^{\sigma+\ell}_{2,r}(\mathbb{R}^{n})\cap \dot{B}^{-s}_{2,\infty}(\mathbb{R}^{n})$ for $\sigma\in \mathbb{R}, s\in \mathbb{R}$ and $1\leq r\leq\infty$ such that $\sigma+s>0$, then it holds that $$\begin{aligned}
&&\Big\|2^{q\sigma}\|\widehat{\dot{\Delta}_{q}f}e^{-\eta(\xi)t}\|_{L^{2}}\Big\|_{l^{r}_{q}}\nonumber\\ &\lesssim & \underbrace{(1+t)^{-\frac{\sigma+s}{2}}\|f\|_{\dot{B}^{-s}_{2,\infty}}}_{Low-frequency\ Estimate}
+\underbrace{(1+t)^{-\frac{\ell}{2}+\frac{n}{2}(\frac{1}{p}-\frac{1}{2})}\|f\|_{\dot{B}^{\sigma+\ell}_{p,r}}}_{High-frequency\ Estimate}, \label{R-E12}\end{aligned}$$ for $\ell>n(\frac{1}{p}-\frac{1}{2})$ [^1] with $1\leq p\leq2$.
For clarity, the proof is separated into high-frequency and low-frequency parts.
\(1) If $q\geq 0$, then $|\xi|\sim2^{q}\geq1$, which leads to $$\begin{aligned}
\|\widehat{\dot{\Delta}_{q}f}e^{-\eta(\xi)t}\|_{L^2}&\leq&\|\widehat{\dot{\Delta}_{q}f}e^{-c_{0}t|\xi|^{-2}}\|_{L^2(|\xi|\geq 1)}\nonumber\\&=&
\Big\||\xi|^{\ell}|\widehat{\dot{\Delta}_{q}f}|\frac{e^{-c_{0}t|\xi|^{-2}}}{|\xi|^{\ell}}\Big\|_{L^2(|\xi|\geq 1)}\nonumber\\&\leq&
\||\xi|^{\ell}\widehat{\dot{\Delta}_{q}f}\|_{L^{p'}}\Big\|\frac{e^{-c_{0}t|\xi|^{-2}}}{|\xi|^{\ell}}\Big\|_{L^{s}(|\xi|\geq 1)}
\ \ \Big(\frac{1}{p'}+\frac{1}{s}=\frac{1}{2},\ p'\geq2 \Big)\nonumber\\&\leq&
2^{q\ell}\|\dot{\Delta}_{q}f\|_{L^{p}}\Big\|\frac{e^{-c_{1}t|\xi|^{-2}}}{|\xi|^{\ell}}\Big\|_{L^{s}(|\xi|\geq 1)}\ \ \Big(\frac{1}{p}+\frac{1}{p'}=1\Big), \label{R-E13}\end{aligned}$$ where $c_{1}>0$ and the Hausdorff-Young’s inequality was used in the last line. By performing the change of variable as in [@XMK], we arrive at $$\begin{aligned}
\Big\|\frac{e^{-c_{1}t|\xi|^{-2}}}{|\xi|^{\ell}}\Big\|_{L^{s}(|\xi|\geq 1)}\lesssim (1+t)^{-\frac{\ell}{2}+\frac{n}{2}(\frac{1}{p}-\frac{1}{2})} \label{R-E14}\end{aligned}$$ for $\ell>n(\frac{1}{p}-\frac{1}{2})$. Besides, it can be also bounded by $(1+t)^{-\frac{\ell}{2}}$ for $\ell\geq0$ if $p=2$. Then it follows from (\[R-E13\])-(\[R-E14\]) that $$\begin{aligned}
2^{q\sigma}\|\widehat{\dot{\Delta}_{q}f}e^{-\eta(\xi)t}\|_{L^2}\lesssim 2^{q(\sigma+\ell)}(1+t)^{-\frac{\ell}{2}+\frac{n}{2}(\frac{1}{p}-\frac{1}{2})}\|\dot{\Delta}_{q}f\|_{L^{p}}. \label{R-E15}\end{aligned}$$
\(2) If $q<0$, then $|\xi|\sim2^{q}<1$, which implies that $$\begin{aligned}
|\widehat{\dot{\Delta}_{q}f}|e^{-\eta(\xi)t}\leq |\widehat{\dot{\Delta}_{q}f}|e^{-c_{2}t|\xi|^2}\lesssim |\widehat{\dot{\Delta}_{q}f}|e^{-c_{2}(2^{q}\sqrt{t})^2} \label{R-E16}\end{aligned}$$ for $c_{2}>0$. Furthermore, we can obtain $$\begin{aligned}
2^{q\sigma}\|\widehat{\dot{\Delta}_{q}f}e^{-\eta(\xi)t}\|_{L^{2}}
\lesssim\|f\|_{\dot{B}^{-s}_{2,\infty}}(1+t)^{-\frac{\sigma+s}{2}}[(2^{q}\sqrt{t})^{\sigma+s}e^{-c_{2}(2^{q}\sqrt{t})^2}] \label{R-E17}\end{aligned}$$ for $\sigma\in \mathbb{R}, s\in \mathbb{R}$ such that $\sigma+s>0$. Note that $$\begin{aligned}
\Big\|(2^{q}\sqrt{t})^{\sigma+s}e^{-c_{2}(2^{q}\sqrt{t})^2}\Big\|_{l^{r}_{q}}\lesssim 1, \label{R-E18}\end{aligned}$$ for any $r\in [1,+\infty]$. Combining (\[R-E15\]),(\[R-E17\])-(\[R-E18\]), we conclude that $$\begin{aligned}
&&\Big\|2^{q\sigma}\|\widehat{\dot{\Delta}_{q}f}e^{-\eta(\xi)t}\|_{L^{2}}\Big\|_{l^{r}_{q}}\nonumber\\ &\lesssim & \|f\|_{\dot{B}^{-s}_{2,\infty}}(1+t)^{-\frac{\sigma+s}{2}}
+\|f\|_{\dot{B}^{\sigma+\ell}_{p,r}}(1+t)^{-\frac{\ell}{2}+\frac{n}{2}(\frac{1}{p}-\frac{1}{2})}, \label{R-E19}\end{aligned}$$ which is just the inequality (\[R-E12\]).
Global-in-time existence
========================
\[sec:4\] As shown by [@MXK], the recent local existence theory in [@XK1] for generally symmetric hyperbolic systems can be applied to (\[R-E4\]) directly.
\[prop4.1\] ([@MXK]) Assume that $U_{0}\in{B^{3/2}_{2,1}}$, then there exists a time $T_{0}>0$ (depending only on the initial data) such that
- (Existence): system (\[R-E4\]) has a unique solution $U(t,x)\in\mathcal{C}^{1}([0,T_{0}]\times \mathbb{R})$ satisfying $U\in\widetilde{\mathcal{C}}_{T_{0}}(B^{3/2}_{2,1})\cap
\widetilde{\mathcal{C}}^1_{T_{0}}(B^{1/2}_{2,1})$;
- (Blow-up criterion): if the maximal time $T^{*}(>T_{0})$ of existence of such a solution is finite, then $$\limsup_{t\rightarrow T^{*}}\|U(t,\cdot)\|_{B^{3/2}_{2,1}}=\infty$$ if and only if $$\int^{T^{*}}_{0}\|\nabla
U(t,\cdot)\|_{L^{\infty}}dt=\infty.$$
Furthermore, in order to show that classical solutions in Proposition \[prop3.1\] are globally defined, we need to construct a priori estimates according to the dissipative mechanism produced by the Tomoshenko system. For this purpose, define by $E(T)$ the energy functional and by $D(T)$ the corresponding dissipation functional: $$E(T):=\|U\|_{\widetilde{L}^\infty_{T}(B^{3/2}_{2,1})}$$ and $$D(T):=\|y\|_{\widetilde{L}^2_{T}(B^{3/2}_{2,1})}+\|(v,z_{x})\|_{\widetilde{L}^2_{T}(B^{1/2}_{2,1})}
+\|u_{x}\|_{\widetilde{L}^2_{T}(B^{-1/2}_{2,1})}$$ for any time $T>0$. Hence, we have the following
\[prop4.2\] Suppose $U\in\widetilde{\mathcal{C}}_{T}(B^{3/2}_{2,1})\cap
\widetilde{\mathcal{C}}^1_{T}(B^{1/2}_{2,1})$ is a solution of (\[R-E4\]) for $T>0$. There exists $\delta_{1}>0$ such that if $E(T)\leq\delta_{1}, $ then $$\begin{aligned}
&&E(T)+D(T)\lesssim \|U_{0}\|_{B^{3/2}_{2,1}}+\Big(\sqrt{E(T)}+E(T)\Big)D(T). \label{R-E20}\end{aligned}$$ Furthermore, it holds that $$\begin{aligned}
&&E(T)+D(T)\lesssim \|U_{0}\|_{B^{3/2}_{2,1}}. \label{R-E21}\end{aligned}$$
Actually, in the case of non-equal wave speeds ($a\neq1$), a priori estimates on the dissipations for $y,z_{x}$ and $u_{x}$ coincide with the case of equal wave speeds. For brevity, we present them as lemmas only, the interested reader is referred to [@MXK] for proofs.
(The dissipation for $y$)\[lem4.1\] If $U\in\widetilde{\mathcal{C}}_{T}(B^{3/2}_{2,1})\cap
\widetilde{\mathcal{C}}^1_{T}(B^{1/2}_{2,1})$ is a solution of (\[R-E4\]) for any $T>0$, then $$\begin{aligned}
&&E(T)+\|y\|_{\widetilde{L}^{2}_{T}(B^{3/2}_{2,1})}\lesssim \|U_{0}\|_{B^{3/2}_{2,1}}+\sqrt{E(T)}D(T). \label{R-E22}\end{aligned}$$
\[lem4.2\] (The dissipation for $z_{x}$) If $U\in\widetilde{\mathcal{C}}_{T}(B^{3/2}_{2,1})\cap
\widetilde{\mathcal{C}}^1_{T}(B^{1/2}_{2,1})$ is a solution of (\[R-E4\]) for any $T>0$, then $$\begin{aligned}
\|z_{x}\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})} &\lesssim & E(T)+\|U_{0}\|_{B^{3/2}_{2,1}}+\|y\|_{\widetilde{L}^{2}_{T}(B^{3/2}_{2,1})}\nonumber\\ &&+
\|v\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}+\sqrt{E(T)}D(T). \label{R-E23}\end{aligned}$$
\[lem4.3\] (The dissipation for $u_{x}$) If $U\in\widetilde{\mathcal{C}}_{T}(B^{3/2}_{2,1})\cap
\widetilde{\mathcal{C}}^1_{T}(B^{1/2}_{2,1})$ is a solution of (\[R-E4\]) for any $T>0$, then $$\begin{aligned}
\|u_{x}\|_{\widetilde{L}^{2}_{T}(B^{-1/2}_{2,1})}\lesssim E(T)+\|U_{0}\|_{B^{3/2}_{2,1}}+\|v\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}+\|y\|_{\widetilde{L}^{2}_{T}(B^{3/2}_{2,1})}. \label{R-E24}\end{aligned}$$
However, the calculation for the dissipation of $v$ is a little different. We would like to give the proof as follows.
(The dissipation for $v$)\[lem4.4\] If $U\in\widetilde{\mathcal{C}}_{T}(B^{3/2}_{2,1})\cap
\widetilde{\mathcal{C}}^1_{T}(B^{1/2}_{2,1})$ is a solution of (\[R-E4\]) for any $T>0$, then $$\begin{aligned}
\|v\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}&\lesssim& E(T)+\|U_{0}\|_{B^{3/2}_{2,1}}+\varepsilon\|u_{x}\|_{\widetilde{L}^{2}_{T}(B^{-1/2}_{2,1})}
\nonumber\\&&+(1+C_{\varepsilon})\|y\|_{\widetilde{L}^{2}_{T}(B^{3/2}_{2,1})}
+E(T)D(T) \label{R-E25}\end{aligned}$$ for $\varepsilon>0$, where $C_{\varepsilon}$ is a position constant dependent on $\varepsilon$.
It is convenient to rewrite the system (\[R-E4\]) as follows: $$\begin{aligned}
\label{R-E26}
\left\{\begin{array}{l}
v_t-u_x+y=0,\\[2mm]
u_t-v_x=0,\\[2mm]
z_t-ay_x=0,\\[2mm]
y_t-az_{x}-v+\gamma y=g(z)_{x},
\end{array}\right.\end{aligned}$$ where the smooth function $g(z)$ is defined by $$g(z)=\sigma(z/a)-\sigma(0)-\sigma'(0)z/a=O(z^2)$$ satisfying $g(0)=0$ and $g'(0)=0$.
Firstly, applying the inhomogeneous frequency-localization operator $\Delta_{q}(q\geq-1)$ to (\[R-E26\]) gives $$\begin{aligned}
\label{R-E27}
\left\{\begin{array}{l}
\Delta_{q}v_t-\Delta_{q}u_x+\Delta_{q}y=0,\\[2mm]
\Delta_{q}u_t-\Delta_{q}v_x=0,\\[2mm]
\Delta_{q}z_t-a\Delta_{q}y_x=0,\\ [2mm]
\Delta_{q}y_t-a\Delta_{q}z_{x}-\Delta_{q}v+\gamma \Delta_{q}y=\Delta_{q}g(z)_{x}.
\end{array}\right.\end{aligned}$$ Next, multiplying the first equation in (\[R-E27\]) by $-\Delta_{q}y$, the second one by $-a\Delta_{q}z$, the third one by $-a\Delta_{q}u$ and the fourth one by $-\Delta_{q}v$, respectively, then adding the resulting equalities, we have $$\begin{aligned}
&&-(\Delta_{q}v\Delta_{q}y+a\Delta_{q}u\Delta_{q}z)_{t} +(a\Delta_{q}v\Delta_{q}z+a^2\Delta_{q}u\Delta_{q}y)_{x}+|\Delta_{q}v|^2\nonumber\\ &=&|\Delta_{q}y|^2+
(a^2-1)\Delta_{q}y\Delta_{q}u_x+\gamma\Delta_{q}y\Delta_{q}v-\Delta_{q}g(z)_{x}\Delta_{q}v. \label{R-E28}\end{aligned}$$
Integrating the equality (\[R-E28\]) in $x\in \mathbb{R}$, with the aid of Cauchy-Schwarz inequality, we obtain $$\begin{aligned}
&&\frac{d}{dt}E_{1}[\Delta_{q}U]+\frac{1}{2}\|\Delta_{q}v\|^2_{L^2}\nonumber\\ &\lesssim& \|\Delta_{q}y\|^2_{L^2}+|a^2-1|\|\Delta_{q}y\|_{L^2}\|\Delta_{q}u_{x}\|_{L^2}
\nonumber\\ &&+\|\Delta_{q}g(z)_{x}\|_{L^2}\|\Delta_{q}v\|_{L^2}, \label{R-E29}\end{aligned}$$ where $$E_{1}[\Delta_{q}U]:=-\int_{\mathbb{R}}(\Delta_{q}v\Delta_{q}y+\Delta_{q}u\Delta_{q}z)dx.$$ By performing the integral with respect to $t\in [0,T]$, we are led to $$\begin{aligned}
&&\|\Delta_{q}v\|^2_{L^2_{t}(L^2)}
\nonumber\\ &\lesssim &
\|\Delta_{q}U\|^2_{L^\infty_{T}(L^2)}+\|\Delta_{q}U_{0}\|^2_{L^2}+\|\Delta_{q}y\|^2_{L^2_{T}(L^2)}\nonumber\\ &&
+\|\Delta_{q}y\|_{L^2_{T}(L^2)}\|\Delta_{q}u_{x}\|_{L^2_{T}(L^2)}+\|\Delta_{q}g(z)_{x}\|^2_{L^2_{T}(L^2)}, \label{R-E30}\end{aligned}$$ where we have noticed the case of $a\neq1$. Furthermore, Young’s inequality enables us to get $$\begin{aligned}
&&2^{\frac{q}{2}}\|\Delta_{q}v\|_{L^2_{T}(L^2)}\nonumber\\ &\lesssim& c_{q}\|U\|_{\widetilde{L}^{\infty}_{T}(B^{1/2}_{2,1})}+c_{q}\|U_{0}\|_{B^{1/2}_{2,1}}+\varepsilon c_{q}\|u_{x}\|_{\widetilde{L}^{2}_{T}(B^{-1/2}_{2,1})}
\nonumber\\ &&
+c_{q}(1+C_{\varepsilon})\|y\|_{\widetilde{L}^{2}_{T}(B^{3/2}_{2,1})}
+c_{q}\|g(z)_{x}\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}\label{R-E31}\end{aligned}$$ for $\varepsilon>0$, where $C_{\varepsilon}$ is a position constant dependent on $\varepsilon$ and each $\{c_{q}\}$ has a possibly different form in (\[R-E31\]), however, the bound $\|c_{q}\|_{\ell^{1}}\leq1$ is well satisfied.
Recalling the fact $g'(0)=0$, it follows from Propositions \[prop2.4\]-\[prop2.5\] that $$\begin{aligned}
\|g(z)_{x}\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}&=&\|g'(z)z_{x}\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}
\nonumber\\ &\lesssim&\|g'(z)-g'(0)\|_{\widetilde{L}^{\infty}_{T}(B^{1/2}_{2,1})}\|z_{x}\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}
\nonumber\\ &\lesssim&\|z\|_{\widetilde{L}^{\infty}_{T}(B^{1/2}_{2,1})}\|z_{x}\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}.\label{R-E32}\end{aligned}$$ Hence, together with (\[R-E31\])-(\[R-E32\]), by summing up on $q\geq-1$, we deduce that $$\begin{aligned}
&&\|v\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}\nonumber\\ &\lesssim& \|U\|_{\widetilde{L}^{\infty}_{T}(B^{1/2}_{2,1})}+\|U_{0}\|_{B^{1/2}_{2,1}}+\varepsilon \|u_{x}\|_{\widetilde{L}^{2}_{T}(B^{-1/2}_{2,1})}
\nonumber\\ &&
+(1+C_{\varepsilon})\|y\|_{\widetilde{L}^{2}_{T}(B^{3/2}_{2,1})}
+\|z\|_{\widetilde{L}^{\infty}_{T}(B^{1/2}_{2,1})}\|z_{x}\|_{\widetilde{L}^{2}_{T}(B^{1/2}_{2,1})}, \label{R-E33}\end{aligned}$$ which leads to the inequality (\[R-E25\]) immediately.
Having Lemmas \[lem4.1\]-\[lem4.4\], by taking sufficiently small $\varepsilon>0$, we can achieve the proof of Proposition \[prop4.2\]. For brevity, we feel free to skip the details. Furthermore, along with local existence result (Proposition \[prop4.1\]) and a priori estimate (Proposition \[prop4.2\]), Theorem \[thm1.1\] follows from the standard boot-strap argument directly, see [@MXK] for similar details.
Optimal decay rates
===================
\[sec:5\] Due to the better dissipative structure in the case of $a=1$ (see [@MXK]), we performed the Littlewood-Paley pointwise estimates for the linearized problem (\[R-E5\]) and develop decay properties in the framework of Besov spaces. Furthermore, with the help of the frequency-localization Duhamel principle, the optimal decay estimates of (\[R-E4\]) are shown by localized time-weighted energy approaches. For the case of $a\neq1$, if the standard Duhamel principle is used, we need to deal with the weak mechanism of regularity-loss in the price of extra higher regularity, so it is impossible to achieve $s_{D}=3/2$. Hence, we involve new observations. Actually, we perform “the square formula of the Duhamel principle" based on the Littlewood-Paley pointwise estimate in Fourier space for the linear system with right-hand side, see (\[R-E36\])-(\[R-E37\]). Furthermore, we proceed the optimal decay estimate for (\[R-E4\]) in terms of high-frequency and low-frequency decompositions, with the aid of the frequency-localization time-decay inequality developed in Sect.\[sec:3\].
To do this, we define the following energy functionals: $$\mathcal{N}(t)=\sup_{0\leq\tau \leq t}(1+\tau)^{\frac{1}{4}}\|U(\tau)\|_{L^{2}},\ \ \ \mathcal{D}(t)=\|z_{x}(\tau)\|_{L^2_{t}(\dot{B}^{1/2}_{2,1})}. \nonumber$$ The optimal decay estimate lies in a nonlinear time-weighted energy inequality, which is include in the following
\[lem5.1\] Let $U=(v,u,z,y)^{\top}$ be the global classical solutions in Theorem \[thm1.1\]. Additionally, if $U_{0}\in\dot{B}^{-1/2}_{2,\infty}$, then it holds that $$\begin{aligned}
\mathcal{N}(t)\lesssim \|U_{0}\|_{B^{3/2}_{2,1}\cap \dot{B}^{-1/2}_{2,\infty}}+\mathcal{N}(t)\mathcal{D}(t)+\mathcal{N}(t)^2. \label{R-E34}\end{aligned}$$
As in [@MK], perform the energy method in Fourier spaces to get $$\begin{aligned}
\frac{d}{dt}E[\hat{U}]+c_{3}\eta_{1}(\xi)|\hat{U}|^2\lesssim \xi^2|\hat{g}|^2, \label{R-E35}\end{aligned}$$ with $\eta_{1}(\xi)=\frac{\xi^2}{(1+\xi^2)^2}$, where $E[\hat{U}]\approx|\hat{U}|^2$. As a matter of fact, following from the derivation of (\[R-E35\]), we can obtain the corresponding Littlewood-Paley pointwise energy inequality $$\begin{aligned}
\frac{d}{dt}E[\widehat{\dot{\Delta}_{q}U}]+c_{3}\eta_{1}|\widehat{\dot{\Delta}_{q}U}|^2\lesssim \xi^2|\widehat{\dot{\Delta}_{q}g}|^2, \label{R-E36}\end{aligned}$$ where $E[\widehat{\dot{\Delta}_{q}U}]\approx|\widehat{\dot{\Delta}_{q}U}|^2$. Gronwall’s inequality implies that $$\begin{aligned}
|\widehat{\dot{\Delta}_{q}U}|^2\lesssim e^{-c_{3}\eta_{1}t}|\widehat{\dot{\Delta}_{q}U_{0}}|^2+\int_{0}^{t}e^{-c_{3}\eta_{1}(t-\tau)}\xi^2|\widehat{\dot{\Delta}_{q}g}|^2d\tau. \label{R-E37}\end{aligned}$$ It follows from Fubini and Plancherel theorems that $$\begin{aligned}
\|U\|^2_{L^2}&=&\sum_{q\in \mathbb{Z}}\|\dot{\Delta}_{q}U\|^{2}_{L^{2}}\nonumber\\&\lesssim&
\sum_{q\in \mathbb{Z}}\|\widehat{\dot{\Delta}_{q}U_{0}}e^{-\frac{1}{2}c_{3}\eta_{1}(\xi)t}\|^{2}_{L^{2}}\nonumber\\&&+\int^{t}_{0}
\sum_{q\in \mathbb{Z}}\||\xi|\widehat{\dot{\Delta}_{q}g}e^{-\frac{1}{2}c_{3}\eta_{1}(\xi)(t-\tau)}\|^{2}_{L^{2}}d\tau
\nonumber\\&\triangleq& I_{1}+I_{2}. \label{R-E38}\end{aligned}$$ For $I_{1}$, by taking $p=r=2, \sigma=0, s=1/2$ and $\ell=1$ in Proposition \[prop3.1\], we arrive at $$\begin{aligned}
I_{1}&=&\Big(\sum_{q<0}+\sum_{q\geq0}\Big)\Big(\cdot\cdot\cdot\Big)\nonumber
\\&\lesssim&\|U_{0}\|^{2}_{\dot{B}^{-1/2}_{2,\infty}}(1+t)^{-\frac{1}{2}}+\sum_{q\geq0}2^{2q}\|\dot{\Delta}_{q}U_{0}\|^{2}_{L^{2}}(1+t)^{-1}\nonumber
\\&\lesssim&\|U_{0}\|^{2}_{\dot{B}^{-1/2}_{2,\infty}}(1+t)^{-\frac{1}{2}}+\|U_{0}\|^{2}_{\dot{B}^{1}_{2,2}}(1+t)^{-1}\nonumber
\\&\lesssim&\|U_{0}\|^{2}_{\dot{B}^{-1/2}_{2,\infty}\cap B^{3/2}_{2,1}}(1+t)^{-\frac{1}{2}}. \label{R-E39}\end{aligned}$$
Next, we begin to bound the nonlinear term on the right-hand side of (\[R-E38\]), which is written as the sum of low-frequency and high-frequency $$\begin{aligned}
I_{2}=\int^{t}_{0}\Big(\sum_{q<0}+\sum_{q\geq0}\Big)\Big(\cdot\cdot\cdot\Big)\triangleq I_{2L}+I_{2H}. \label{R-E40}\end{aligned}$$ For $I_{2L}$, by taking $r=2, \sigma=1$ and $ s=1/2$ in Proposition \[prop3.1\], we have $$\begin{aligned}
I_{2L}&&\leq\int^{t}_{0}(1+t-\tau)^{-\frac{3}{2}}\|g(z)\|^{2}_{\dot{B}^{-1/2}_{2,\infty}}d\tau\nonumber\\&&
\lesssim\int^{t}_{0}(1+t-\tau)^{-\frac{3}{2}}\|g(z)\|^{2}_{L^{1}}d\tau\nonumber\\&&\lesssim
\int^{t}_{0}(1+t-\tau)^{-\frac{3}{2}}\|z(\tau)\|^4_{L^2}d\tau\nonumber\\&&\lesssim
\mathcal{N}^{4}(t)\int^{t}_{0}(1+t-\tau)^{-\frac{3}{2}}(1+t)^{-1}d\tau\nonumber\\&&\lesssim \mathcal{N}^{4}(t)(1+t)^{-1}, \label{R-E41}\end{aligned}$$ where we used the embedding $L^1(\mathbb{R})\hookrightarrow \dot{B}^{-1/2}_{2,\infty}(\mathbb{R})$ in Lemma \[lem2.3\] and the fact $g(z)=O(z^2)$. For the high-frequency part $I_{2H}$, more elaborate estimates are needed. For the purpose, we write $$\begin{aligned}
I_{2H}=\Big(\int^{t/2}_{0}+\int^{t}_{t/2}\Big)\Big(\cdot\cdot\cdot\Big)\triangleq I_{2H1}+I_{2H2}.\end{aligned}$$ For $I_{2H1}$, taking $p=r=2, \sigma=1$ and $\ell=1/2$ in Proposition \[prop3.1\] gives $$\begin{aligned}
I_{2H1}&=&\int_{0}^{t/2}\sum_{q\geq0}2^{3q}\|\dot{\Delta}_{q}g(z)\|^{2}_{L^{2}}(1+t-\tau)^{-\frac{1}{2}}d\tau\nonumber
\\&\leq&\int_{0}^{t/2}(1+t-\tau)^{-\frac{1}{2}}\|g(z)\|^2_{\dot{B}^{3/2}_{2,2}}d\tau. \label{R-E411}\end{aligned}$$ On the other hand, recalling $g(z)=O(z^2)$, Proposition \[prop2.1\] and Lemmas \[lem2.1\]-\[lem2.2\] enable us to get $$\begin{aligned}
\|g(z)\|_{\dot{B}^{3/2}_{2,2}}\lesssim\|g(z)\|_{\dot{B}^{3/2}_{2,1}}\lesssim \|z\|_{L^\infty}\|z_{x}\|_{\dot{B}^{1/2}_{2,1}}. \label{R-E412}\end{aligned}$$ Combine (\[R-E411\]) and (\[R-E412\]) to arrive at $$\begin{aligned}
I_{2H1}&\lesssim&\int_{0}^{t/2}(1+t-\tau)^{-\frac{1}{2}}\|z(\tau)\|^2_{L^\infty}\|z_{x}(\tau)\|^2_{\dot{B}^{1/2}_{2,1}}d\tau
\nonumber \\&\lesssim& \sup_{0\leq\tau\leq t/2}\Big\{(1+t-\tau)^{-\frac{1}{2}}\|z(\tau)\|^{2}_{L^{\infty}}\Big\}\int_{0}^{t/2}\|z_{x}(\tau)\|^2_{\dot{B}^{1/2}_{2,1}}d\tau
\nonumber \\&\lesssim& (1+t)^{-\frac{1}{2}}\|U_{0}\|^2_{B^{3/2}_{2,1}}\mathcal{D}^2(t)
\nonumber \\&\lesssim& (1+t)^{-\frac{1}{2}}\|U_{0}\|^{2}_{B^{3/2}_{2,1}}. \label{R-E42}\end{aligned}$$ For the last step of (\[R-E42\]), we would like to explain a little. It follows from Proposition \[prop2.3\] and Remark \[rem6.1\] that $$\begin{aligned}
\mathcal{D}(t)\lesssim \|z_{x}\|_{\widetilde{L}^2_{t}(\dot{B}^{1/2}_{2,1})}\lesssim \|z_{x}\|_{\widetilde{L}^2_{t}(B^{1/2}_{2,1})} \lesssim \|U_{0}\|_{B^{3/2}_{2,1}},\label{R-E499}\end{aligned}$$ where we used the energy inequality (\[R-E9\]) in Theorem \[thm1.1\]. By choosing $r=2, p=\sigma=1$ and $\ell=1/2$ in Proposition \[prop3.1\], $I_{2H2}$ is proceeded as $$\begin{aligned}
I_{2H2}&=&\int_{t/2}^{t}\sum_{q\geq0}2^{3q}\|\dot{\Delta}_{q}g(z)\|^{2}_{L^{1}}d\tau\nonumber \\&\leq& \int_{t/2}^{t}\|g(z)\|^2_{\dot{B}^{3/2}_{1,2}}d\tau. \label{R-E43}\end{aligned}$$ Thanks to $g(z)=O(z^2)$, it follows from Proposition \[prop2.2\] that $$\begin{aligned}
\|g(z)\|_{\dot{B}^{3/2}_{1,2}}\leq\|g(z)\|_{\dot{B}^{3/2}_{1,1}}\lesssim\|z\|_{L^{2}}\|z_{x}\|_{\dot{B}^{1/2}_{2,1}}. \label{R-E44}\end{aligned}$$ Together with (\[R-E43\])-(\[R-E44\]), we are led to $$\begin{aligned}
I_{2H2}&\lesssim& \mathcal{N}^{2}(t)\int_{t/2}^{t} (1+\tau)^{-\frac{1}{2}}\|z_{x}(\tau)\|^2_{\dot{B}^{1/2}_{2,1}}d\tau \nonumber
\\&\lesssim&\mathcal{N}^{2}(t) \sup_{t/2\leq\tau\leq t}(1+\tau)^{-\frac{1}{2}} \int_{t/2}^{t}\|z_{x}(\tau)\|^2_{\dot{B}^{1/2}_{2,1}}d\tau
\nonumber
\\&\lesssim& (1+t)^{-\frac{1}{2}} \mathcal{N}^{2}(t)\mathcal{D}^2(t). \label{R-E45}\end{aligned}$$ Combine (\[R-E42\]) and (\[R-E45\]) to get $$\begin{aligned}
I_{2H}&\lesssim& (1+t)^{-\frac{1}{2}}\|U_{0}\|^{2}_{B^{3/2}_{2,1}}+(1+t)^{-\frac{1}{2}} \mathcal{N}^{2}(t)\mathcal{D}^2(t). \label{R-E46}\end{aligned}$$ Therefore, it follows from (\[R-E41\]) and (\[R-E46\]) that $$\begin{aligned}
I_{2}&\lesssim&(1+t)^{-1}\mathcal{N}^{4}(t)+(1+t)^{-\frac{1}{2}}\|U_{0}\|^{2}_{B^{3/2}_{2,1}}\nonumber
\\&&
+(1+t)^{-\frac{1}{2}} \mathcal{N}^{2}(t)\mathcal{D}^2(t). \label{R-E47}\end{aligned}$$ Finally, noticing (\[R-E38\])-(\[R-E39\]) and (\[R-E47\]), we conclude that $$\begin{aligned}
\|U\|^2_{L^2}&\lesssim& (1+t)^{-\frac{1}{2}}\|U_{0}\|^{2}_{\dot{B}^{-1/2}_{2,\infty}\cap B^{3/2}_{2,1}}+(1+t)^{-\frac{1}{2}} \mathcal{N}^{2}(t)\mathcal{D}^2(t)\nonumber
\\&&+(1+t)^{-1}\mathcal{N}^{4}(t) \label{R-E48}\end{aligned}$$ which leads to (\[R-E34\]) directly.
**Proof of Theorem \[thm1.2\].** Note that (\[R-E499\]), we arrive at $$\begin{aligned}
\mathcal{D}(t)\lesssim\|U_{0}\|_{B^{3/2}_{2,1}}\lesssim \|U_{0}\|_{B^{3/2}_{2,1}\cap\dot{B}^{-1/2}_{2,\infty}}. \label{R-E50}\end{aligned}$$ Thus, if the norm $\|U_{0}\|_{B^{3/2}_{2,1}\cap\dot{B}^{-1/2}_{2,\infty}}$ is sufficiently small, then we have $$\begin{aligned}
\mathcal{N}(t)\lesssim \|U_{0}\|_{B^{3/2}_{2,1}\cap\dot{B}^{-1/2}_{2,\infty}}+\mathcal{N}(t)^{2} \label{R-E51}\end{aligned}$$ which implies that $\mathcal{N}(t)\lesssim \|U_{0}\|_{B^{3/2}_{2,1}\cap\dot{B}^{-1/2}_{2,\infty}}$, provided that $\|U_{0}\|_{B^{3/2}_{2,1}\cap\dot{B}^{-1/2}_{2,\infty}}$ is sufficiently small. Consequently, the desired decay estimate in Theorem \[thm1.2\] follows $$\begin{aligned}
\|U\|_{L^{2}}\lesssim \|U_{0}\|_{B^{3/2}_{2,1}\cap\dot{B}^{-1/2}_{2,\infty}}(1+t)^{-\frac{1}{4}}. \label{R-E52}\end{aligned}$$ Hence, the proof of Theorem \[thm1.2\] is complete eventually. $\square$
Appendix
========
\[sec:6\] For convenience of reader, in this section, we review the Littlewood–Paley decomposition and definitions for Besov spaces and Chemin-Lerner spaces in $\mathbb{R}^{n}(n\geq1)$, see [@BCD] for more details.
Let ($\varphi, \chi)$ is a couple of smooth functions valued in \[0,1\] such that $\varphi$ is supported in the shell $\textbf{C}(0,\frac{3}{4},\frac{8}{3})=\{\xi\in\mathbb{R}^{n}|\frac{3}{4}\leq|\xi|\leq\frac{8}{3}\}$, $\chi$ is supported in the ball $\textbf{B}(0,\frac{4}{3})=
\{\xi\in\mathbb{R}^{n}||\xi|\leq\frac{4}{3}\}$ satisfying $$\chi(\xi)+\sum_{q\in\mathbb{N}}\varphi(2^{-q}\xi)=1,\ \ \ \ q\in
\mathbb{N},\ \ \xi\in\mathbb{R}^{n}$$ and $$\sum_{k\in\mathbb{Z}}\varphi(2^{-k}\xi)=1,\ \ \ \ k\in \mathbb{Z},\
\ \xi\in\mathbb{R}^{n}\setminus\{0\}.$$ For $f\in\mathcal{S'}$(the set of temperate distributions which is the dual of the Schwarz class $\mathcal{S}$), define $$\Delta_{-1}f:=\chi(D)f=\mathcal{F}^{-1}(\chi(\xi)\mathcal{F}f),\
\Delta_{q}f:=0 \ \ \mbox{for}\ \ q\leq-2;$$ $$\Delta_{q}f:=\varphi(2^{-q}D)f=\mathcal{F}^{-1}(\varphi(2^{-q}|\xi|)\mathcal{F}f)\
\ \mbox{for}\ \ q\geq0;$$ $$\dot{\Delta}_{q}f:=\varphi(2^{-q}D)f=\mathcal{F}^{-1}(\varphi(2^{-q}|\xi|)\mathcal{F}f)\
\ \mbox{for}\ \ q\in\mathbb{Z},$$ where $\mathcal{F}f$, $\mathcal{F}^{-1}f$ represent the Fourier transform and the inverse Fourier transform on $f$, respectively. Observe that the operator $\dot{\Delta}_{q}$ coincides with $\Delta_{q}$ for $q\geq0$.
Denote by $\mathcal{S}'_{0}:=\mathcal{S}'/\mathcal{P}$ the tempered distributions modulo polynomials $\mathcal{P}$. We first give the definition of homogeneous Besov spaces.
\[defn6.1\] For $s\in \mathbb{R}$ and $1\leq p,r\leq\infty,$ the homogeneous Besov spaces $\dot{B}^{s}_{p,r}$ is defined by $$\dot{B}^{s}_{p,r}=\{f\in S'_{0}:\|f\|_{\dot{B}^{s}_{p,r}}<\infty\},$$ where $$\|f\|_{\dot{B}^{s}_{p,r}}
=\left\{\begin{array}{l}\Big(\sum_{q\in\mathbb{Z}}(2^{qs}\|\dot{\Delta}_{q}f\|_{L^p})^{r}\Big)^{1/r},\
\ r<\infty, \\ \sup_{q\in\mathbb{Z}}
2^{qs}\|\dot{\Delta}_{q}f\|_{L^p},\ \ r=\infty.\end{array}\right.$$
Similarly, the definition of inhomogeneous Besov spaces is stated as follows.
\[defn6.2\] For $s\in \mathbb{R}$ and $1\leq p,r\leq\infty,$ the inhomogeneous Besov spaces $B^{s}_{p,r}$ is defined by $$B^{s}_{p,r}=\{f\in S':\|f\|_{B^{s}_{p,r}}<\infty\},$$ where $$\|f\|_{B^{s}_{p,r}}
=\left\{\begin{array}{l}\Big(\sum_{q=-1}^{\infty}(2^{qs}\|\Delta_{q}f\|_{L^p})^{r}\Big)^{1/r},\
\ r<\infty, \\ \sup_{q\geq-1} 2^{qs}\|\Delta_{q}f\|_{L^p},\ \
r=\infty.\end{array}\right.$$
On the other hand, we also present the definition of Chemin-Lerner spaces first initialled by J.-Y. Chemin and N. Lerner [@CL], which are the refinement of the space-time mixed spaces $L^{\theta}_{T}(\dot{B}^{s}_{p,r})$ or $L^{\theta}_{T}(B^{s}_{p,r})$.
\[defn6.3\] For $T>0, s\in\mathbb{R}, 1\leq r,\theta\leq\infty$, the homogeneous mixed Chemin-Lerner spaces $\widetilde{L}^{\theta}_{T}(\dot{B}^{s}_{p,r})$ is defined by $$\widetilde{L}^{\theta}_{T}(\dot{B}^{s}_{p,r}):
=\{f\in
L^{\theta}(0,T;\mathcal{S}'_{0}):\|f\|_{\widetilde{L}^{\theta}_{T}(\dot{B}^{s}_{p,r})}<+\infty\},$$ where $$\|f\|_{\widetilde{L}^{\theta}_{T}(\dot{B}^{s}_{p,r})}:=\Big(\sum_{q\in\mathbb{Z}}(2^{qs}\|\dot{\Delta}_{q}f\|_{L^{\theta}_{T}(L^{p})})^{r}\Big)^{\frac{1}{r}}$$ with the usual convention if $r=\infty$.
\[defn6.4\] For $T>0, s\in\mathbb{R}, 1\leq r,\theta\leq\infty$, the inhomogeneous Chemin-Lerner spaces $\widetilde{L}^{\theta}_{T}(B^{s}_{p,r})$ is defined by $$\widetilde{L}^{\theta}_{T}(B^{s}_{p,r}):
=\{f\in
L^{\theta}(0,T;\mathcal{S}'):\|f\|_{\widetilde{L}^{\theta}_{T}(B^{s}_{p,r})}<+\infty\},$$ where $$\|f\|_{\widetilde{L}^{\theta}_{T}(B^{s}_{p,r})}:=\Big(\sum_{q\geq-1}(2^{qs}\|\Delta_{q}f\|_{L^{\theta}_{T}(L^{p})})^{r}\Big)^{\frac{1}{r}}$$ with the usual convention if $r=\infty$.
We further define $$\widetilde{\mathcal{C}}_{T}(B^{s}_{p,r}):=\widetilde{L}^{\infty}_{T}(B^{s}_{p,r})\cap\mathcal{C}([0,T],B^{s}_{p,r})$$ and $$\widetilde{\mathcal{C}}^1_{T}(B^{s}_{p,r}):=\{f\in\mathcal{C}^1([0,T],B^{s}_{p,r})|\partial_{t}f\in\widetilde{L}^{\infty}_{T}(B^{s}_{p,r})\},$$ where the index $T$ will be omitted when $T=+\infty$.
By Minkowski’s inequality, Chemin-Lerner spaces can be linked with the usual space-time mixed spaces $L^{\theta}_{T}(X)$ with $X=B^{s}_{p,r}$ or $\dot{B}^{s}_{p,r}$.
\[rem6.1\] It holds that $$\|f\|_{\widetilde{L}^{\theta}_{T}(X)}\leq\|f\|_{L^{\theta}_{T}(X)}\,\,\,
\mbox{if}\,\, r\geq\theta;\ \ \ \
\|f\|_{\widetilde{L}^{\theta}_{T}(X)}\geq\|f\|_{L^{\theta}_{T}(X)}\,\,\,
\mbox{if}\,\, r\leq\theta.$$
Acknowledgments {#acknowledgments .unnumbered}
===============
J. Xu is partially supported by the National Natural Science Foundation of China (11471158), the Program for New Century Excellent Talents in University (NCET-13-0857) and the Fundamental Research Funds for the Central Universities (NE2015005). The work is also partially supported by Grant-in-Aid for Scientific Researches (S) 25220702 and (A) 22244009.
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[^1]: Let us remark that $\ell\geq0$ in the case of $p=2$.
|
---
author:
- |
Daniel F. Litim\
Department of Physics and Astronomy, University of Sussex, Brighton, BN1 9QH, U.K.\
E-mail:
title: Fixed Points of Quantum Gravity and the Renormalisation Group
---
Introduction
============
It is commonly believed that an understanding of the dynamics of gravity and the structure of space-time at shortest distances requires an explicit quantum theory for gravity. The well-known fact that the perturbative quantisation program for gravity in four dimensions faces problems has raised the suspicion that a consistent formulation of the theory may require a radical deviation from the concepts of local quantum field theory, $e.g.$ string theory. It remains an interesting and open challenge to prove, or falsify, that a consistent quantum theory of gravity cannot be accommodated for within the otherwise very successful framework of local quantum field theories.
Some time ago Steven Weinberg added a new perspective to this problem by pointing out that a quantum theory of gravity in terms of the metric field may very well exist, and be renormalisable on a non-perturbative level, despite it’s notorious perturbative non-renormalisability [@Weinberg]. This scenario, since then known as “asymptotic safety”, necessitates an interacting ultraviolet fixed point for gravity under the renormalisation group (RG) [@Weinberg; @Litim:2006dx; @Niedermaier:2006ns; @NiedermaierReuter; @Percacci:2007sz]. If so, the high energy behaviour of gravity is governed by near-conformal scaling in the vicinity of the fixed point in a way which circumnavigates the virulent ultraviolet (UV) divergences encountered within standard perturbation theory. Indications in favour of an ultraviolet fixed point are based on renormalisation group studies in four and higher dimensions [@Reuter:1996cp; @Souma:1999at; @Lauscher:2001ya; @Lauscher:2002mb; @Reuter:2001ag; @Percacci:2002ie; @Litim:2003vp; @Bonanno:2004sy; @Fischer:2006fz; @Fischer:2006at; @DL06; @Codello:2007bd; @Machado:2007ea], dimensional reduction techniques [@Forgacs:2002hz; @Niedermaier:2002eq], renormalisation group studies in lower dimensions [@Weinberg; @Gastmans:1977ad; @Christensen:1978sc; @Kawai:1992fz; @Aida:1996zn], four-dimensional perturbation theory in higher derivative gravity [@Codello:2006in], large-$N$ expansions in the matter fields [@Percacci:2005wu], and lattice simulations [@Hamber:1999nu; @Hamber:2005vc; @Ambjorn:2004qm].
In this contribution, we review the key elements of the asymptotic safety scenario (Sec. \[AS\]) and introduce renormalisation group techniques (Sec. \[RG\]) which are at the root of fixed point searches in quantum gravity. The fixed point structure in four (Sec. \[FP\]) and higher dimensions (Sec. \[ED\]), and the phase diagram of gravity (Sec. \[PD\]) are discussed and evaluated in the light of the underlying approximations. Results are compared with recent lattice simulations (Sec. \[Lattice\]), and phenomenological implications are indicated (Sec. \[pheno\]). We close with some conclusions (Sec. \[conclusions\]).
Asymptotic Safety {#AS}
=================
We summarise the basic set of ideas and assumptions of asymptotic safety as first laid out in [@Weinberg] (see [@Litim:2006dx; @Niedermaier:2006ns; @NiedermaierReuter; @Percacci:2007sz] for reviews). The aim of the asymptotic safety scenario for gravity is to provide for a path-integral based framework in which the metric field is the carrier of the fundamental degrees of freedom, both in the classical and in the quantum regimes of the theory. This is similar in spirit to effective field theory approaches to quantum gravity [@Donoghue:1993eb]. There, a systematic study of quantum effects is possible without an explicit knowledge of the ultraviolet completion as long as the relevant energy scales are much lower than the ultraviolet cutoff $\Lambda$ of the effective theory, with $\Lambda$ of the order of the Planck scale (see [@Burgess:2003jk] for a recent review).
The asymptotic safety scenario goes one step further and assumes that the cutoff $\Lambda$ can in fact be removed, $\Lambda\to\infty$, and that the high-energy behaviour of gravity, in this limit, is characterised by an interacting fixed point. It is expected that the relevant field configurations dominating the gravitational path integral at high energies are predominantly “anti-screening” to allow for this limit to become feasible. If so, it is conceivable that a non-trivial high-energy fixed point of gravity may exist and should be visible within $e.g.$ renormalisation group or lattice implementations of the theory, analogous to the well-known perturbative high-energy fixed point of QCD. Then the high-energy behaviour of the relevant gravitational couplings is “asymptotically safe” and connected with the low-energy behaviour by finite renormalisation group flows. The existence of a fixed point together with finite renormalisation group trajectories provides for a definition of the theory at arbitrary energy scales.
The fixed point implies that the high-energy behaviour of gravity is characterised by universal scaling laws, dictated by the residual high-energy interactions. No a priori assumptions are made about which invariants are the relevant operators at the fixed point. In fact, although the low-energy physics is dominated by the Einstein-Hilbert action, it is expected that (a finite number of) further invariants will become relevant, in the renormalisation group sense, at the ultraviolet fixed point.[^1] Then, in order to connect the ultraviolet with the infrared physics along some renormalisation group trajectory, a finite number of initial parameters have to be fixed, ideally taken from experiment. In this light, classical general relativity would emerge as a “low-energy phenomenon” of a fundamental quantum field theory in the metric field.
We illustrate this scenario with a discussion of the renormalisation group equation for the gravitational coupling $G$, following [@Litim:2006dx] (see also [@Niedermaier:2006ns; @NiedermaierReuter]). Its canonical dimension is $[G]=2-d$ in $d$ dimensions and hence negative for $d>2$. It is commonly believed that a negative mass dimension for the relevant coupling is responsible for the perturbative non-renormalisability of the theory. We introduce the renormalised coupling as $G(\mu)=Z^{-1}_G(\mu)\, G$, and the dimensionless coupling as $g(\mu)=\mu^{d-2}\,G(\mu)$; the momentum scale $\mu$ denotes the renormalisation scale. The graviton wave function renormalisation factor $Z_G(\mu)$ is normalised as $Z_G(\mu_0)=1$ at $\mu=\mu_0$ with $G(\mu_0)$ given by Newton’s coupling constant $G_N=6.67428
\cdot 10^{-11}\s0{m^3}{{\rm kg}\,s^2}$. The graviton anomalous dimension $\eta$ related to $Z_G(\mu)$ is given by $\eta=-\mu\s0{{\rm d}}{{\rm d}\mu}
\ln Z_G$. Then the Callan-Symanzik equation for $g(\mu)$ reads $$\label{dg}
\beta_g\equiv\mu\frac{{\rm d}g(\mu)}{{\rm d}\mu}
=(d-2+\eta)g(\mu)\,.$$ Here we have assumed a fundamental action for gravity which is local in the metric field. In general, the graviton anomalous dimension $\eta(g,\cdots)$ is a function of all couplings of the theory including matter fields. The RG equation displays two qualitatively different types of fixed points. The non-interacting (gaussian) fixed point corresponds to $g_*=0$ which also entails $\eta=0$. In its vicinity with $g(\mu_0)\ll 1$, we have canonical scaling since $\beta_g=(d-2)g$, and $$\label{Gauss}
G(\mu) = G(\mu_0)$$ for all $\mu<\mu_0$. Consequently, the gaussian regime corresponds to the domain of classical general relativity. In turn, can display an interacting fixed point $g_*\neq 0$ in $d>2$ if the anomalous dimension takes the value $\eta(g_*,\cdots)=2-d$; the dots denoting further gravitational and matter couplings. Hence, the anomalous dimension precisely counter-balances the canonical dimension of Newton’s coupling $G$. This structure is at the root for the non-perturbative renormalisability of quantum gravity within a fixed point scenario.[^2] Consequently, at an interacting fixed point where $g_*\neq0$, the anomalous dimension implies the scaling $$\label{G-asym}
G(\mu)= \frac{g_*}{\mu^{d-2}}$$ for the dimensionful gravitational coupling. In the case of an ultraviolet fixed point $g_*\neq 0$ for large $\mu$, the dimensionful coupling $G$ becomes arbitrarily small in its vicinity. This is in marked contrast to . Hence, indicates that gravity weakens at the onset of fixed point scaling. Nevertheless, at the fixed point the theory remains non-trivially coupled because of $g_*\neq 0$. The weakness of the coupling in is a dimensional effect, and should be contrasted with $e.g.$ asymptotic freedom of QCD in four dimensions where the dimensionless non-abelian gauge coupling becomes weak because of a non-interacting ultraviolet fixed point. In turn, if corresponds to a non-trivial infrared fixed point for $\mu\to 0$, the dimensionful coupling $G(\mu)$ grows large. A strong coupling behaviour of this type would imply interesting long distance modifications of gravity.
As a final comment, we point out that asymptotically safe gravity is expected to become, in an essential way, two-dimensional at high energies. Heuristically, this can be seen from the dressed graviton propagator whose scalar part, neglecting the tensorial structure, scales as ${\cal G}(p^2)\sim
p^{-2(1-\eta/2)}$ in momentum space. Here we have evaluated the anomalous dimension at $\mu^2\approx p^2$. Then, for small $\eta$, we have the standard perturbative behaviour $\sim p^{-2}$. In turn, for large anomalous dimension $\eta\to 2-d$ in the vicinity of a fixed point the propagator is additionally suppressed $\sim (p^2)^{-d/2}$ possibly modulo logarithmic corrections. After Fourier transform to position space, this corresponds to a logarithmic behaviour for the propagator ${\cal G}(x,y)\sim \ln(|x-y|\mu)$, characteristic for bosonic fields in two-dimensional systems.
Renormalisation Group {#Implications}
=====================
\[RG\] Whether or not a non-trivial fixed point is realised in quantum gravity can be assessed once explicit renormalisation group equations for the scale-dependent gravitational couplings are available. To that end, we recall the set-up of Wilson’s (functional) renormalisation group (see [@Litim:1998yn; @Litim:1998nf; @Bagnuls:2000ae; @BTW; @JP; @Pawlowski:2005xe; @Gies:2006wv] for reviews), which is used below for the case of quantum gravity. Wilsonian flows are based on the notion of a cutoff effective action $\Gamma_k$, where the propagation of fields $\phi$ with momenta smaller than $k$ is suppressed. A Wilsonian cutoff is realised by adding $\Delta
S_k=\s012\int\varphi(-q)\,R_k(q)\,\varphi(q)$ within the Schwinger functional $$\label{Z}
\ln\, Z_k[J]=\ln \int [D\varphi]_{\rm ren.}\exp\left(-S[\varphi]
-\Delta S_k[\varphi]
+\int J\cdot \varphi
\right)\,$$ and the requirement that $R_k$ obeys (i) $R_k(q)\to 0$ for $k^2/q^2\to 0$, (ii) $R_k(q)> 0$ for $q^2/k^2\to 0$, and (iii) $R_k(q)\to\infty$ for $k\to\Lambda$ (for examples and plots of $R_k$, see [@Litim:2000ci]). Note that the Wilsonian momentum scale $k$ takes the role of the renormalisation group scale $\mu$ introduced in the previous section. Under infinitesimal changes $k\to k-\Delta k$, the Schwinger functional obeys $\partial_t\ln
Z_k=-\langle\partial_t\Delta S_k\rangle_J$; $t=\ln k$. We also introduce its Legendre transform, the scale-dependent effective action $\Gamma_k[\phi]=\sup_J\left(\int J\cdot \phi -\ln Z_k[J]\right)-\s012\int \phi
R_k\phi$, $\phi=\langle \varphi\rangle_J$. It obeys an exact functional differential equation introduced by Wetterich [@Wetterich:1992yh] \[ERG\] \_t \_k= ([\_k\^[(2)]{}+R\_k]{})\^[-1]{}\_t R\_k, which relates the change in $\Gamma_k$ with a one-loop type integral over the full field-dependent cutoff propagator. Here, the trace $\tr$ denotes an integration over all momenta and summation over all fields, and $\Gamma_k^{(2)}[\phi](p,q)\equiv \delta^2\Gamma_k/\delta\phi(p)\delta\phi(q)$. A number of comments are in order:
- [**Finiteness and interpolation property.**]{} By construction, the flow equation is well-defined and finite, and interpolates between an initial condition $\Gamma_\Lambda$ for $k\to \Lambda$ and the full effective action $\Gamma\equiv\Gamma_{k=0}$. This is illustrated in Fig. \[Vergleich\]. The endpoint is independent of the regularisation, whereas the trajectories $k\to \Gamma_k$ depend on it.
- [**Locality.**]{} The integrand of is peaked for field configurations with momentum squared $q^2\approx k^2$, and suppressed for large momenta \[due to condition (i) on $R_k$\] and for small momenta \[due to condition (ii)\]. Therefore, the flow equation is essentially local in momentum and field space [@Litim:2000ci; @Litim:2005us].
- [**Approximations.**]{} Systematic approximations for $\Gamma_k$ and $\partial_t\Gamma_k$ are required to integrate . These include (a) perturbation theory, (b) expansions in powers of the fields (vertex functions), (c) expansion in powers of derivative operators (derivative expansion), and (d) combinations thereof. The iterative structure of perturbation theory is fully reproduced to all orders, independently of $R_k$ [@Litim:2001ky; @Litim:2002xm]. The expansions (b) - (d) are genuinely non-perturbative and lead, via , to coupled flow equations for the coefficient functions. Convergence is then checked by extending the approximation to higher order.
- [**Stability.**]{} The stability and convergence of approximations is, additionally, controlled by $R_k$ [@Litim:2000ci; @Litim:2001up]. Here, powerful optimisation techniques are available to maximise the physics content and the reliability through well-adapted choices of $R_k$ [@Litim:2000ci; @Litim:2001up; @Litim:2005us; @Pawlowski:2003hq; @Pawlowski:2005xe]. These ideas have been explicitly tested in $e.g.$ scalar [@Litim:2002cf] and gauge theories [@Pawlowski:2003hq].
- [**Symmetries.**]{} Global or local (gauge/diffeomorphism) symmetries of the underlying theory can be expressed as Ward-Takahashi identities for $n$-point functions of $\Gamma$. Ward-Takahashi identities are maintained for all $k$ if the insertion $\Delta S_k$ is compatible with the symmetry. In general, this is not the case for non-linear symmetries such as in non-Abelian gauge theories or gravity. Then the requirements of gauge symmtry for $\Gamma$ are preserved by either (a) imposing modified Ward identities which ensure that standard Ward identities are obeyed in the the physical limit when $k\to 0$, or by (b) introducing background fields into the regulator $R_k$ and taking advantage of the background field method, or by (c) using gauge-covariant variables rather than the gauge fields or the metric field [@Branchina:2003ek]. For a discussion of benefits and shortcomings of these options see [@Litim:1998nf; @Pawlowski:2005xe]. For gravity, most implementations presently employ option (b) together with optimisation techniques to control the symmetry [@Fischer:2006fz; @Fischer:2006at].
- [**Integral representation.**]{} The physical theory described by $\Gamma$ can be defined without explicit reference to an underlying path integral representation, using only the (finite) initial condition $\Gamma_\Lambda$, and the (finite) flow equation
0.001
(1000,260) (50,250)[[a)]{}]{}(500,250)[[b)]{}]{}(220,220)[$S$]{} (680,220)[$\Gamma_*$]{} (210,130)[$k\partial_k\Gamma_{k}$]{} (690,130)[$k\partial_k\Gamma_{k}$]{} (180,-20)[$\Gamma_{0}\approx\Gamma$]{} (660,-20)[$\Gamma_{0}\approx\Gamma_{\rm EH}$]{} .05![\[Vergleich\] Wilsonian flows for scale-dependent effective actions $\Gamma_k$ in the space of all action functionals (schematically); arrows point towards smaller momentum scales and lower energies $k\to 0$. [a)]{} Flow connecting a fundamental classical action $S$ at high energies in the ultraviolet with the full quantum effective action $\Gamma$ at low energies in the infrared (“top-down”). [b)]{} Flow connecting the Einstein-Hilbert action at low energies with a fundamental fixed point action $\Gamma_*$ at high energies (“bottom-up”).](Flow.eps "fig:"){width=".42\hsize"}.05![\[Vergleich\] Wilsonian flows for scale-dependent effective actions $\Gamma_k$ in the space of all action functionals (schematically); arrows point towards smaller momentum scales and lower energies $k\to 0$. [a)]{} Flow connecting a fundamental classical action $S$ at high energies in the ultraviolet with the full quantum effective action $\Gamma$ at low energies in the infrared (“top-down”). [b)]{} Flow connecting the Einstein-Hilbert action at low energies with a fundamental fixed point action $\Gamma_*$ at high energies (“bottom-up”).](FlowGravity.eps "fig:"){width=".42\hsize"}
.5cm
$$\label{integral}
\Gamma=\Gamma_\Lambda+
\int_\Lambda^0
\s0{dk}{k}\s012\,
\tr \left({\Gamma_k^{(2)}+R_k}\right)^{-1}\partial_t R_k\,.$$
This provides an implicit regularisation of the path integral underlying . It should be compared with the standard representation for $\Gamma$ via a functional integro-differential equation $$\label{DSE}
e^{-\Gamma}
=\int [D\varphi]_{\rm ren.}\exp
\left(-S[\phi+\varphi]
+\int \0{\delta\Gamma[\phi]}{\delta\phi}\cdot\varphi\right)$$ which is at the basis of $e.g.$ the hierarchy of Dyson-Schwinger equations.
- [**Renormalisability.**]{} In renormalisable theories, the cutoff $\Lambda$ in can be removed, $\Lambda\to\infty$, and $\Gamma_\Lambda\to \Gamma_*$ remains well-defined for arbitrarily short distances. In perturbatively renormalisable theories, $\Gamma_*$ is given by the classical action $S$, such as in QCD. In this case, illustrated in Fig. \[Vergleich\]a), the high energy behaviour of the theory is simple, given mainly by the classical action, and the challenge consists in deriving the physics of the strongly coupled low energy limit. In perturbatively non-renormalisable theories such as quantum gravity, proving the existence (or non-existence) of a short distance limit $\Gamma_*$ is more difficult. For gravity, illustrated in Fig. \[Vergleich\]b), experiments indicate that the low energy theory is simple, mainly given by the Einstein Hilbert theory. The challenge consists in identifying a possible high energy fixed point action $\Gamma_*$, which upon integration matches with the known physics at low energies. In principle, any $\Gamma_*$ with the above properties qualifies as fundamental action for quantum gravity. In non-renormalisable theories the cutoff $\Lambda$ cannot be removed. Still, the flow equation allows to access the physics at all scales $k<\Lambda$ analogous to standard reasoning within effective field theory [@Burgess:2003jk].
- [**Link with Callan-Symanzik equation.**]{} The well-known Callan-Symanzik equation describes a flow $k\s0d{dk}$ driven by a mass insertion $\sim k^2\phi^2$. In , this corresponds to the choice $R_k(q^2)=k^2$, which does not fulfill condition (i). Consequently, the corresponding flow is no longer local in momentum space, and requires an additional UV regularisation. This highlights a crucial difference between the Callan-Symanzik equation and functional flows . In this light, the flow equation could be interpreted as a functional Callan-Symanzik equation with [momentum-dependent]{} mass term insertion [@Symanzik:1970rt].
Now we are in a position to implement these ideas for quantum gravity [@Reuter:1996cp]. A Wilsonian effective action for gravity $\Gamma_k$ should contain the Ricci scalar $R(g_{\mu\nu})$ with a running gravitational coupling $G_k$, a running cosmological constant $\Lambda_k$ (with canonical mass dimension $[\Lambda_k]=2$), possibly higher order interactions in the metric field such as powers, derivatives, or functions of $e.g.$ the Ricci scalar, the Ricci tensor, the Riemann tensor, and, possibly, non-local operators in the metric field. The effective action should also contain a standard gauge-fixing term $S_{\rm gf}$, a ghost term $S_{\rm gh}$ and matter interactions $S_{\rm matter}$. Altogether, $$\label{EHk}
\Gamma_k=
\int d^dx \sqrt{\det g_{\mu\nu}}\left[
\0{1}{16\pi G_k}\left(-R+2\Lambda_k \right)
+\cdots+S_{\rm gf}+S_{\rm gh}+ S_{\rm matter}\right]\,,$$ and explicit flow equations for the coefficient functions such as $G_k$, $\Lambda_k$ or vertex functions, are obtained by appropriate projections after inserting into . All couplings in become running couplings as functions of the momentum scale $k$. For $k$ much smaller than the $d$-dimensional Planck scale $M_*$, the gravitational sector is well approximated by the Einstein-Hilbert action with $G_k\approx G_{k=0}$, and similarily for the gravity-matter couplings. At $k\approx M_*$ and above, the RG running of gravitational couplings becomes important. This is the topic of the following sections.
A few technical comments are in order: To ensure gauge symmetry within this set-up, we take advantage of the background field formalism and add a non-propagating background field $\bar g_{\mu\nu}$ [@Reuter:1996cp; @Litim:1998nf; @Reuter:1993kw; @Litim:1998qi; @Freire:2000bq; @Litim:2002hj; @Pawlowski:2001df]. This way, the extended effective action $\Gamma_k[g_{\mu\nu},\bar g_{\mu\nu}]$ becomes gauge-invariant under the combined symmetry transformations of the physical and the background field. A second benefit of this is that the background field can be used to construct a covariant Laplacean $-\bar D^2$, or similar, to define a mode cutoff at momentum scale $k^2=-\bar D^2$. This implies that the mode cutoff $R_k$ will depend on the background fields. The background field is then eliminated from the final equations by identifying it with the physical mean field. This procedure, which dynamically readjusts the background field, implements the requirements of “background independence” for quantum gravity. For a detailed evaluation of Wilsonian background field flows, see [@Litim:2002hj]. Finally, we note that the operator traces $\tr$ in are evaluated using heat kernel techniques. Here, well-adapted choices for $R_k$ [@Litim:2000ci; @Litim:2001up] lead to substantial algebraic simplifications, and open a door for systematic fixed point searches, which we discuss next.
Fixed Points {#FP}
============
In this section, we discuss the main picture in a simple approximation which captures the salient features of an asymptotic safety scenario for gravity, and give an overview of extensions. We consider the Einstein-Hilbert theory with a cosmological constant term and employ a momentum cutoff $R_k$ with the tensorial structure of [@Lauscher:2001ya] and variants thereof, an optimised scalar cutoff $R_k(q^2)\sim(k^2-q^2)\theta(k^2-q^2)$ [@Litim:2000ci; @Litim:2001up], and a harmonic background field gauge with parameter $\alpha$ in a specific limit introduced in [@Litim:2003vp]. The ghost wave function renormalisation is set to $Z_{C,k}=1$, and the effective action is given by with $S_{\rm matter}=0$. In the domain of classical scaling $G_k$ and $\Lambda_k$ are approximately constant, and reduces to the conventional Einstein-Hilbert action in $d$ euclidean dimensions. The dimensionless renormalised gravitational and cosmological constants are $$\label{glk}
\begin{array}{l}
g=k^{d-2}\, G_k\, \equiv k^{d-2}\, Z^{-1}_{G}(k)\ \bar G\ ,
\quad\quad
\lambda=\,k^{-2}\, \Lambda_k\
\end{array}$$ where it is understood that $g$ and $\lambda$ depend on $k$. Then the coupled system of $\beta$-functions is $$\begin{aligned}
\partial_t\lambda\equiv\beta_{\lambda}(\lambda,g)&=&
-2\lambda +\frac{g}{2}\,d\,(d+2)\,(d-5)
-d(d+2)g\, \frac{(d-1)g
+\frac{1}{d-2}(1-4\frac{d-1}{d}\lambda)}{2g-\frac{1}{d-2}(1-2\lambda)^2}
\label{beta-l-inf}
\\
\label{beta-g-inf}
\partial_t g\equiv\beta_{ g}(\lambda,g)&=&
(d-2)g+\frac{2(d-2)(d+2)g^2}{2(d-2)g-(1-2\lambda)^2}\,.\end{aligned}$$ We have rescaled $g\to g/c_d$ with $c_d=\Gamma(\s0d2+2)(4\pi)^{d/2-1}$ to remove phase space factors. This does not alter the fixed point structure. The scaling $g\to g/(384\pi^2)$ reproduces the $4d$ classical force law in the non-relativistic limit [@Robinson:2006yd]. For the anomalous dimension, we find $$\begin{aligned}
\label{eta-inf}
\eta(\lambda, g;d)&=&
\0{(d+2)\, g}{ g- g_{\rm bound}(\lambda)}\,,
\quad\quad
g_{\rm bound}(\lambda;d)
=
\0{\left(1-2\lambda\right)^2}{2(d-2)}\,.\end{aligned}$$
$$\begin{array}{c|c|c|c}%|c}
&
\quad\quad\theta'\quad\quad&
\quad\quad \theta''\quad\quad &
\quad\quad {\rm ref.}\quad\quad
\\ \hline
\quad a)\quad&
1.1-2.3 &
2.5-7.0 &
\cite{Lauscher:2001ya}
\\
b)&
1.4-2.0 &
2.4-4.3 &
\cite{Litim:2003vp}
\\
c)&
1.5-1.7 &
3.0-3.2 &
\cite{Fischer:2006fz}
\\
\end{array}$$
The anomalous dimension vanishes for vanishing gravitational coupling, and for $d=\pm2$. At a non-trivial fixed point the vanishing of $\beta_g$ implies $\eta_*=2-d$, and reflects the fact that the gravitational coupling is dimensionless in two dimensions. At $g=g_{\rm bound}$, the anomalous dimension $\eta$ diverges. The full flow is finite (no poles) and well-defined for all $k$, as are the full $\beta$-functions derived from it. Therefore the curve $g=g_{\rm bound}(\lambda)$ limits the domain of validity of the approximation.
We first consider the case $\lambda=0$ and find two fixed points, the gaussian one at $g_*=0$ and a non-gaussian one at $g_*=1/(4d)<g_{\rm bound}(0)$, which are connected under the renormalisation group. The universal eigenvalue $\partial \beta_g/\partial g|_*=-\theta$ at the fixed point are $\theta=2-d$ at the gaussian, and \[theta0-NG\] =2d at the non-gaussian fixed point.
Next we allow for a non-vanishing dynamical cosmological constant term $\Lambda_k\neq 0$ in . The coupled system exhibits the gaussian fixed point $(\lambda_*,g_*)=(0,0)$ with eigenvalues $-2$ and $d-2$.
$$\begin{array}{c|c|c|c|c|c|c}
&
\quad i \quad
&\theta'
&\quad \theta''\quad
&\quad \theta_3\quad
&\quad \theta_4\quad
&\quad {\rm ref.}\quad
%&{\rm comments}
\\ \hline
\quad a)\quad & 2& \quad 2.1 - 3.4 \quad &\quad 3.1 - 4.3\quad
&\quad 8.4 - 28.8\quad &-
&
\cite{Lauscher:2002mb}
\\
b)&2 &1.4 &2.8 &25.6 &-
&
\cite{DL06}
\\
c)&
2 &1.7 &3.1 &3.5 & -
&
\cite{DL06}
\\
d)&
2 &1.4 &2.3 &26.9 & -
&
\cite{Codello:2007bd}
\\
e)&
3 &2.7 &2.3 &2.1 & -4.2
&
\cite{Codello:2007bd}
\\
f)&
4 &2.9 &2.5 &1.6 & -3.9
&
\cite{Codello:2007bd}
\\
g)&
5 &2.5 &2.7 &1.8 & -4.4
&
\cite{Codello:2007bd}
\\
h)&
6 &2.4 &2.4 &1.5 & -4.1
&
\cite{Codello:2007bd}
\\
\end{array}$$
Non-trivial fixed points of and are found as follows: For non-vanishing $\lambda\neq 0$, we find a non-trivially vanishing $\beta_g$ for $g=g_0(\lambda)$, with $g_0(\lambda)=\s01{4d}(1-2\lambda)^2$. Note that $g_0(\lambda)<g_{\rm bound}(\lambda)$ for all $d>2$. Evaluating for $g=g_0(\lambda)$, we find $\beta_\lambda(\lambda,g_0(\lambda))=
\s014(d-4)(d+1)(1-2\lambda)^2-2d\,\lambda+\s0d2\,. $ The first term vanishes in $d=4$ dimensions. Consequently, we find a unique ultraviolet fixed point $\lambda_*=\s014$ and $g_*=\s01{64}$. In $d> 4$, the vanishing of $\beta_\lambda$ leads to two branches of real fixed points with either $\lambda_*>\s012$ or $\s012>\lambda_*>0$. Only the second branch corresponds to an ultraviolet fixed point which is connected under the renormalisation group with the correct infrared behaviour. This can be seen as follows: At $\lambda=\s012$, we find $\eta=d+2>0$. On a non-gaussian fixed point, however, $\eta<0$. Furthermore, $g$ cannot change sign under the renormalisation group flow . Consequently, $\eta$ cannot change sign either. Hence, to connect a fixed point at $\lambda>\s012$ with the gaussian fixed point at $\lambda=0$, $\eta$ would have to change sign at least twice, which is impossible. Therefore, we have a unique physically relevant solution given by $$\label{FP-d}
\lambda_*=
\frac{d^2-d-4-\sqrt{2d(d^2-d-4)}}{2(d-4)(d+1)}\,,\quad\quad
g_*=
\0{(\sqrt{d^2-d-4}-\sqrt{2d})^2}{2(d-4)^2(d+1)^2}\,.$$ An interesting property of this system is that the scaling exponents $\theta_1$ and $\theta_2$ – the eigenvalues of the stability matrix $\partial\beta_i/\partial g_j$ $(g_1\equiv g,g_2\equiv\lambda)$ at the fixed point – are a complex conjugate pair, $\theta_{1,2}=\theta'\pm i\theta''$ with $\theta'=\s053$ and $\theta''=\s0{\sqrt{167}}{3}$ in four dimensions. The reason for this is that the stability matrix, albeit real, is not symmetric. Complex eigenvalues reflect that the interactions at the fixed point have modified the scaling behaviour of the underlying operators $\int\sqrt{\det g_{\mu\nu}}R$ and $\int\sqrt{\det g_{\mu\nu}}$. This pattern changes for lower and higher dimensions, where eigenvalues are real [@Litim:2006dx].
At this point it is important to check whether the fixed point structure and the scaling exponents depend on technical parameters such as the gauge fixing procedure or the momentum cutoff function $R_k$, see Tab. \[tEH-4d\] and \[tRn-4d\]. For the Einstein-Hilbert theory in $4d$, results are summarised in Tab. \[tEH-4d\]. The $\alpha$-dependence of the $\beta$-functions is fairly non-trivial, $e.g.$ [@Reuter:1996cp; @Lauscher:2001ya; @Litim:2003vp]. It is therefore noteworthy that scaling exponents only depend mildly on variations thereof. Furthermore, the $R_k$-dependence is smaller than the dependence on gauge fixing parameters. We conclude that the fixed point is fully stable and $R_k$-independent for all technical purposes, with the presently largest uncertainty arising through the gauge fixing sector. In Tab. \[tRn-4d\], we discuss the stability of the fixed point under extensions beyond the Einstein-Hilbert approximation, including higher powers of the Ricci scalar both in Feynman gauge [@Lauscher:2002mb; @DL06] and in Landau-DeWitt gauge [@Codello:2007bd]. Once more, the fixed point and the scaling exponents come out very stable. Furthermore, starting from the operator $\int \sqrt{\det
g_{\mu\nu}}R^3$ and higher, couplings become irrelevant with negative scaling exponents [@Codello:2007bd; @Machado:2007ea]. This is an important first indication for the set of relevant operators at the UV fixed point being finite. Finally, we mention that the stability of the fixed point under the addition of non-interacting matter fields has been confirmed in [@Percacci:2002ie].
$d$ 5 6 7 8 9 10
------------ ------------- ------------- ------------- ------------- ------------- ------------- --
$\theta'$ 2.69 – 3.11 4.26 – 4.78 6.43 – 6.89 8.19 – 9.34 10.5 – 12.1 13.1 – 15.2
$\theta''$ 4.54 – 5.16 6.52 – 7.46 8.43 – 9.46 10.3 – 11.4 12.1 – 13.2 13.9 – 15.0
$|\theta|$ 5.31 – 6.06 7.79 – 8.76 10.4 – 11.6 13.2 – 14.7 16.1 – 17.9 19.1 – 21.3
: The variation of scaling exponent with dimensionality, gauge fixing parameters (using either Feynman gauge, or harmonic background field gauge with $0\leq \alpha \leq 1$), and the regulator $R_k$; data from [@Fischer:2006fz; @Fischer:2006at]. The $R_k$-variation, covering various classes of cutoff functions, is on the level of a few percent and smaller than the variation with $\alpha$.[]{data-label="tEH-d"}
Extra Dimensions {#ED}
================
It is interesting to discuss fixed points of quantum gravity specifically in more than four dimensions. The motivation for this is that, first of all, the critical dimension of gravity – the dimension where the gravitational coupling has vanishing canonical mass dimension – is two. For any dimension above the critical one, the canonical dimension is negative. Hence, from a renormalisation group point of view, the four-dimensional theory is not special. Continuity in the dimension suggests that an ultraviolet fixed point, if it exists in four dimensions, should persist towards higher dimensions. More generally, one expects that the local structure of quantum fluctuations, and hence local renormalisation group properties of a quantum theory of gravity, are qualitatively similar for all dimensions above the critical one, modulo topological effects for specific dimensions. Secondly, the dynamics of the metric field depends on the dimensionality of space-time. In four dimensions and above, the metric field is fully dynamical. Hence, once more, we should expect similarities in the ultraviolet behaviour of gravity in four and higher dimensions. Interestingly, this pattern is realised in the results [@Litim:2003vp], see the analytical fixed point . An extended systematic search for fixed points in higher-dimensional gravity for general cutoff $R_k$ has been presented in [@Fischer:2006fz; @Fischer:2006at], also testing the stability of the result against variations of the gauge fixing parameter (see Tab. \[tEH-d\]). The variation with $R_k$, ammended by stability considerations, is smaller than the variation with $\alpha$. We conclude from the weak variation that the fixed point indeed persists in higher dimensions. Further studies including higher derivative operators confirm this picture [@DL06]. This structural stability also strengthens the results in the four-dimensional case, and supports the view introduced above. A phenomenological application of these findings in low-scale quantum gravity is discussed below (see Sec. \[pheno\]).
0.001
(700,480) (750,20)[$\lambda_{4d}$]{} (-70,410)[$g_{4d}$]{} ![ The phase diagram for the running gravitational coupling $g_{4d}$ and the cosmological constant $\lambda_{4d}$ in four dimensions. The Gaussian and the ultraviolet fixed point are indicated by dots (red). The separatrix connects the two fixed points (full green line). The full (red) line indicates the bound $g_{\rm bound}(\lambda)$ where $1/\eta=0$. Arrows indicate the direction of the RG flow with decreasing $k\to 0$.[]{data-label="PD4"}](Flowsd4a0Smallb1.eps "fig:"){width=".7\hsize"}
Phase Diagram {#PD}
=============
In this section, we discuss the main characteristics of the phase portrait of the Einstein-Hilbert theory [@Reuter:2001ag; @Litim:2003vp] (see Fig. \[PD4\]). Finiteness of the flow implies that the line $1/\eta=0$ cannot be crossed. Slowness of the flow implies that the line $\eta=0$ can neither be crossed (see Sec. \[FP\]). Thus, disconnected regions of renormalisation group trajectories are characterised by whether $
g$ is larger or smaller $ g_{\rm bound}$ and by the sign of $g$. Since $\eta$ changes sign only across the lines $\eta=0$ or $1/\eta=0$, we conclude that the graviton anomalous dimension has the same sign along any trajectory. In the physical domain which includes the ultraviolet and the infrared fixed point, the gravitational coupling is positive and the anomalous dimension negative. In turn, the cosmological constant may change sign on trajectories emmenating from the ultraviolet fixed point. Some trajectories terminate at the boundary $g_{\rm bound}(\lambda)$, linked to the present approximation. The two fixed points are connected by a separatrix. The rotation of the separatrix about the ultraviolet fixed point reflects the complex nature of the eigenvalues. At $k\approx M_{\rm Pl}$, the flow displays a crossover from ultraviolet dominated running to infrared dominated running. The non-vanishing cosmological constant modifies the flow mainly in the crossover region rather than in the ultraviolet. In the infrared limit, the separatrix leads to a vanishing cosmological constant $\Lambda_k=\lambda_k
k^2\to 0$ and is interpreted as a phase transition boundary between cosmologies with positive or negative cosmological constant at large distances. Trajectories in the vicinity of the separatrix lead to a positive cosmological constant at large scales and are, therefore, candidate trajectories for realistic cosmologies [@cosmology]. This picture agrees very well with numerical results for a sharp cut-off flow [@Reuter:2001ag], except for the location of the line $1/\eta=0$ which is non-universal. Similar phase diagrams are found in higher dimensions [@Fischer:2006fz; @Fischer:2006at].
Lattice {#Lattice}
=======
Lattice implementations for gravity in four dimensions have been put forward based on Regge calculus techniques [@Hamber:1999nu; @Hamber:2005vc] and causal dynamical triangulations [@Ambjorn:2004qm]. In the Regge calculus approach, a critical point which allows for a lattice continuum limit has been given in [@Hamber:1999nu] using the Einstein Hilbert action with fixed cosmological constant. A scaling exponent has been measured in the four-dimensional simulation based on varying Newton’s coupling to the critical point, with $\partial_g\beta_g|_*=-\s01\nu$. The result reads $\nu\approx
\s013$, and should be contrasted with the RG result $\nu=1/\theta=\s038$ [@Litim:2003vp] as discussed in Sec. \[FP\]. In the large-dimensional limit, geometrical considerations on the lattice lead to the estimate $\nu=\s01{d-1}$ [@Hamber:2005vc], a behaviour which is in qualitative agreement with the explicit RG fixed point result $\nu=\s01{2d}$ in the corresponding limit, see .
Within the causal dynamical triangulation approach, global aspects of quantum space-times have been assessed in [@Ambjorn:2004qm]. There, the effective dimensionality of space-time has been measured as a function of the length scale by evaluating the return probability of random walks on the triangulated manifolds. The key result is that the measured effective dimensionality displays a cross-over from $d\approx 4$ at large scales to $d\approx 2$ at small scales of the order of the Planck scale. This behaviour compares nicely with the cross-over of the graviton anomalous dimension $\eta$ under the renormalisation group (see Sec. \[AS\]), and with renormalisation group studies of the spectral dimension (see [@Niedermaier:2006ns; @NiedermaierReuter; @Lauscher:2005qz]). These findings corroborate the claim that asymptotically safe quantum gravity behaves, in an essential way, two-dimensional at short distances.
Phenomenology {#pheno}
=============
The phenomenology of a gravitational fixed point covers the physics of black holes [@blackholes], cosmology [@cosmology; @Bonanno:2007wg; @Bentivegna:2003rr], modified dispersion relations [@Girelli:2006sc], and the physics at particle colliders [@Litim:2007iu; @Hewett:2007st; @Koch:2007yt]. In this section we concentrate on the later within low-scale quantum gravity models [@add; @aadd]. There, gravity propagates in $d=4+n$ dimensional bulk whereas matter fields are confined to a four-dimensional brane. The four-dimensional Planck scale $M_{\rm Pl}\approx 10^{19}$ GeV is no longer fundamental as soon as the $n$ extra dimensions are compact with radius $\sim L$. Rather, the $d=4+n$-dimensional Planck mass $M_*$ sets the fundamental scale for gravity, leading to the relation $M^2_{\rm Pl}\sim M^2_{*} (M_*\,L)^{n}$ for the four-dimensional Planck scale. Consequently, $M_*$ can be significantly lower than $M_{\rm Pl}$ provided $1/L\ll M_*$. If $M_*$ is of the order of the electroweak scale, this scenario lifts the hierarchy problem of the standard model and opens the exciting possibility that particle colliders could establish experimental evidence for the quantisation of gravity [@grw; @tao; @virtual_kk].
(1000,20)
0.001![ The scale-dependence of the gravitational coupling in a scenario with large extra dimension of size $\sim L$ with fundamental Planck scale $M_*$ and $M_*L\gg 1$. The fixed point behaviour in the deep ultraviolet enforces a softening of gravitational coupling (see text). [a)]{} In the infrared (IR) regime where $|\eta|\ll 1$, the coupling $g=G_k\,k^{d-2}$ displays a crossover from 4-dimensional to $(4+n)$-dimensional classical scaling at $k\approx 1/L$. The slope ${\rm d}\ln g/{\rm d}\ln k\approx d-2$ measures the effective number of dimensions. At $k\approx M_*$, a classical-to-quantum crossover takes place from $|\eta|\ll1$ to $\eta\approx
2-d$ (schematically). [b)]{} Classical-to-quantum crossover at the respective Planck scale for $G_k$ and the anomalous dimensions $\eta$ from numerical integrations of the flow equation; $d=4+n$ dimensions with $n=1,\cdots,7$ from right to left.[]{data-label="RunningG"}](FlowExtra.eps "fig:"){width=".4\hsize"} -.325.5![ The scale-dependence of the gravitational coupling in a scenario with large extra dimension of size $\sim L$ with fundamental Planck scale $M_*$ and $M_*L\gg 1$. The fixed point behaviour in the deep ultraviolet enforces a softening of gravitational coupling (see text). [a)]{} In the infrared (IR) regime where $|\eta|\ll 1$, the coupling $g=G_k\,k^{d-2}$ displays a crossover from 4-dimensional to $(4+n)$-dimensional classical scaling at $k\approx 1/L$. The slope ${\rm d}\ln g/{\rm d}\ln k\approx d-2$ measures the effective number of dimensions. At $k\approx M_*$, a classical-to-quantum crossover takes place from $|\eta|\ll1$ to $\eta\approx
2-d$ (schematically). [b)]{} Classical-to-quantum crossover at the respective Planck scale for $G_k$ and the anomalous dimensions $\eta$ from numerical integrations of the flow equation; $d=4+n$ dimensions with $n=1,\cdots,7$ from right to left.[]{data-label="RunningG"}](p2.eps "fig:"){width=".49\hsize"} -.025.5![ The scale-dependence of the gravitational coupling in a scenario with large extra dimension of size $\sim L$ with fundamental Planck scale $M_*$ and $M_*L\gg 1$. The fixed point behaviour in the deep ultraviolet enforces a softening of gravitational coupling (see text). [a)]{} In the infrared (IR) regime where $|\eta|\ll 1$, the coupling $g=G_k\,k^{d-2}$ displays a crossover from 4-dimensional to $(4+n)$-dimensional classical scaling at $k\approx 1/L$. The slope ${\rm d}\ln g/{\rm d}\ln k\approx d-2$ measures the effective number of dimensions. At $k\approx M_*$, a classical-to-quantum crossover takes place from $|\eta|\ll1$ to $\eta\approx
2-d$ (schematically). [b)]{} Classical-to-quantum crossover at the respective Planck scale for $G_k$ and the anomalous dimensions $\eta$ from numerical integrations of the flow equation; $d=4+n$ dimensions with $n=1,\cdots,7$ from right to left.[]{data-label="RunningG"}](p1.eps "fig:"){width=".49\hsize"}
(1000,5) (880,320)[[$|\eta|$]{}]{} (865,265)[[$n=1$]{}]{} (665,340)[[$n=7$]{}]{} (880,150)[[$\displaystyle \frac{G_k}{G_0}$]{}]{} (130,390)[[a)]{} schematically]{} (660,390)[[b)]{} numerically]{} (20,340)[$\ln g$]{} (270,310)[UV (fixed point)]{} (155,200)[IR ($d=4+n$)]{} (20,80)[IR ($d=4$)]{} (410,45)[$\ln k$]{} (740,20)[$\ln k$]{} (230,20)[$\ln M_*$]{} (130,20)[$\ln 1/L$]{}
The renormalisation group running of the gravitational coupling in this scenario has been studied in [@Fischer:2006fz; @Fischer:2006at; @Litim:2007iu] and is summarised in Fig. \[RunningG\]. The main effects due to a fixed point at high energies set in at momentum scales $k\approx M_*$, where the gravitational coupling displays a cross-over from perturbative scaling $G(k)\approx$ const. to fixed point scaling $G(k)\approx g_*
k^{2-d}$. Therefore we expect that signatures of this cross-over should be visible in scattering processes at particle colliders as long as these are sensitive to momentum transfers of the order of $M_*$.
We illustrate this at the example of dilepton production through virtual gravitons at the Large Hadron Collider (LHC) [@Litim:2007iu]. To lowest order in canonical dimension, the dilepton production amplitude is generated through an effective dimension–8 operator in the effective action, involving four fermions and a graviton [@grw]. Tree–level graviton exchange is described by an amplitude ${A} = {S}\cdot {T}$, where ${T} = T_{\mu\nu}
T^{\mu\nu} - \frac{1}{2+n} T_\mu^\mu T_\nu^\nu$ is a function of the energy-momentum tensor, and $$\label{S}
{S}= \frac{2\pi^{n/2}}{\Gamma(n/2)} \;
\frac{1}{M_*^{4}} \;
\int_0^\infty \frac{d m}{M_*} \; \left(\frac{m}{M_*}\right)^{n-1}\,
{\cal G}(s,m)$$ is a function of the scalar part ${\cal G}(s,m)$ of the graviton propagator [@grw; @gps]. The integration over the Kaluza-Klein masses $m$, which we take as continuous, reflects that gravity propagates in the higher-dimensional bulk. If the graviton anomalous dimension is small, the propagator is well approximated by ${\cal G}(s,m)=(s+m^2)^{-1}$. This propagator is used within effective theory settings, and applicable if the relevant momentum transfer is $\ll M_*$. In this case, is ultraviolet divergent for $n\ge 2$ due to the Kaluza-Klein modes [@grw]. Regularisation by an UV cutoff leads to a power-law dependence of the amplitude ${S}\sim M_*^{-4}( {\Lambda}/{M_*} )^{n-2}$ on the cutoff $\Lambda$. In a fixed point scenario, the behaviour of $S$ is improved due to the non-trivial anomalous dimension $\eta$ of the graviton, $e.g.$ . Evaluating $\eta$ at momentum scale $k^2\approx s+m^2$, we are lead to the dressed propagator ${\cal
G}(s,m)\approx\frac{M_*^{n+2}}{(s+m^2)^{n/2+2}}$ in the vicinity of an UV fixed point. The central observation is that becomes finite even in the UV limit of the integration. An alternative matching has been adapted in [@Hewett:2007st; @Koch:2007yt], based on the substitution $G(k)\to
G(\sqrt{s})$ in , setting $G=M_*^{2-d}$. In that case, however, remains UV divergent due to the Kaluza-Klein modes. We conclude that the large anomalous dimension in asymptotically safe gravity provides for a finite dilepton production rate.
(1000,112) 0.001(80,250)[[a)]{} effective theory]{} (550,250)[[b)]{} renormalisation group]{} {width=".33\textwidth"} {width=".66\textwidth"}
-.3cm
-.3cm
In Fig. \[fig:discovery\] we show the discovery potential in the fundamental Planck scale at the LHC, and compare effective theory studies [@gps] with a gravitational fixed point [@Litim:2007iu]. In either case the minimal signal cross sections have been computed for which a $5\sigma$ excess can be observed, taking into account the leading standard model backgrounds and assuming statistical errors. This translates into a maximum reach $M_D$ for the fundamental Planck scale $M_*$. To estimate uncertainties in the RG set-up, we allow for a 10% variation in the scale where the transition towards fixed point scaling sets in. Consistency is checked by introducing an artificial cutoff $\Lambda$ on the partonic energy [@gps], setting the partonic signal cross section to zero for $E_{\rm parton}>\Lambda$. It is nicely seen that $M_D$ becomes independent of $\Lambda$ for $\Lambda\to\infty$ when fixed point scaling is taken into account.
Conclusions
===========
The asymptotic safety scenario offers a genuine path towards quantum gravity in which the metric field remains the fundamental carrier of the physics even in the quantum regime. We have reviewed the ideas behind this set-up in the light of recent advances based on renormalisation group and lattice studies. The stability of renormalisation group fixed points and scaling exponents detected in four- and higher-dimensional gravity is remarkable, strongly supporting this scenario. Furthermore, underlying expansions show good numerical convergence, and uncertainties which arise through approximations are moderate. If the fundamental Planck scale is as low as the electroweak scale, signs for the quantisation of gravity and asymptotic safety could even be observed in collider experiments. It is intriguing that key aspects of asymptotic safety are equally seen in lattice studies. It will be interesting to evaluate these links more deeply in the future. Finally, asymptotically safe gravity is a natural set-up which leads to classical general relativity as a “low energy phenomenon” of a fundamental quantum field theory in the metric field.
Acknowledgements {#acknowledgements .unnumbered}
================
I thank Peter Fischer and Tilman Plehn for collaboration on the topics discussed here, and the organisers for their invitation to a very stimulating workshop.
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[^1]: For [infrared]{} fixed points, universality considerations often simplify the task of identifying the set of relevant, marginal and irrelevant operators. This is not applicable for interacting [ultraviolet]{} fixed points.
[^2]: Integer values for anomalous dimensions are well-known from other gauge theories at criticality and away from their canonical dimension. In the $d$-dimensional $U(1)$+Higgs theory, the abelian charge $e^2$ has mass dimension $[e^2]=4-d$, with $\beta_{e^2}=(d-4+\eta)\,
e^2$. In three dimensions, a non-perturbative infrared fixed point at $e^2_*\neq 0$ leads to $\eta_*=1$ [@Bergerhoff:1995zq]. The fixed point belongs to the universality class of conventional superconductors with the charged scalar field describing the Cooper pair. The integer value $\eta_*
=1$ implies that the magnetic field penetration depth and the Cooper pair correlation length scale with the same universal exponent at the phase transition [@Bergerhoff:1995zq; @Herbut:1996ut]. In Yang-Mills theories above four dimensions, ultraviolet fixed points with $\eta=4-d$ and implications thereof have been discussed in [@Kazakov:2002jd; @Gies:2003ic; @Morris:2004mg].
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[Continuous-Time Discrete-Space Models for Animal Movement Data]{}
Ephraim M. Hanks\
Department of Statistics, Colorado State University, Fort Collins, CO, U.S.A.\
email: hanks@stat.colostate.edu
Mevin B. Hooten\
U. S. Geological Survey, Colorado Cooperative Fish and Wildlife Research Unit,\
Colorado State University, Fort Collins, CO, U.S.A.
Mat W. Alldredge\
Colorado Parks and Wildlife\
Fort Collins, CO, U.S.A.
**Abstract**
The processes influencing animal movement and resource selection are complex and varied. Past efforts to model changing behavior over time used Bayesian statistical models with variable parameter space, such as reversible-jump Markov chain Monte Carlo approaches, which are computationally demanding and inaccessible to many practitioners. We present a continuous-time discrete-space (CTDS) model of animal movement that can be fit using standard generalized linear modeling (GLM) methods. This CTDS approach allows for the joint modeling of location-based as well as directional drivers of movement. Changing behavior over time is modeled using a varying-coefficient framework which maintains the computational simplicity of a GLM approach, and variable selection is accomplished using a group lasso penalty. We apply our approach to a study of two mountain lions (*Puma concolor*) in Colorado, USA.
<span style="font-variant:small-caps;">Keywords</span>: [Agent-Based Model; Animal Movement; Mountain Lion; Multiple Imputation; Varying-Coefficient Model.]{}
Introduction
============
Animal telemetry data have been used extensively in recent years to study animal movement, space use, and resource selection [e.g., @Johnson2011; @Hanks2011PLoS; @Fieberg2010]. The ease with which telemetry data are being collected is increasing, leading to vast increases in the number of animals being monitored, as well as the temporal resolution at which telemetry locations are obtained [@Cagnacci2010]. This combination can result in huge amounts of telemetry data on a single animal population under study. Additionally, the processes driving animal movement are complex, varied, and changing over time. For example, animal behavior could be driven by the local environment [e.g., @Hooten2010b], by conspecifics or predator/prey interactions [e.g., @Merrill2010], by internal states and needs [e.g., @Nathan2008], or by memory [e.g., @VanMoorter2009]. The animal’s response to each of these drivers of movement is also likely to change drastically over time [e.g., @Nathan2008; @Hanks2011PLoS; @McClintock2012].
The large amount of telemetry data available, even for one animal, and the complex behavior displayed in animal movement results in a challenging situation for statistical modeling. There is no shortage of existing statistical models of animal movement; however, most of these models are computationally demanding, and most are inaccessible to the practitioner. For example, consider the agent-based model of animal movement of [@Hooten2010b]. The agent-based framework is highly flexible, allowing for location-based and directional drivers of movement, but is computationally expensive. Analyzing the movement path of one animal with hourly locations over the course of a week using the approach of [@Hooten2010b] can require computational time on the order of days using standard computing resources. The velocity-based framework for modeling animal movement of [@Hanks2011PLoS] allows for time-varying behavior through a changepoint model of response to drivers of movement, and is more computationally efficient than the approach of [@Hooten2010b], requiring computational time on the order of hours for a similar problem. Similarly, the mechanistic state-switching approach of [@McClintock2012] allows for time-varying behavior through a state-switching approach. These three approaches use Bayesian statistical models, and both [@Hanks2011PLoS] and [@McClintock2012] allow for time-varying behavior by letting the model parameter space vary, either through a reversible-jump Markov chain Monte Carlo approach [@Green1995] or the related birth-death Markov chain Monte Carlo approach [@Stephens2000]. These methods can be quite computationally demanding, require the user to tune the algorithm to ensure convergence, and can be inaccessible to many practitioners.
The agent-based model of [@Hooten2010b] assumes a representation of the animal’s movement path that is discrete in both space (grid cells) and time (fixed time intervals). The velocity-based movement model of [@Hanks2011PLoS] assumes a representation of the movement path that is continuous in space and discrete in time. The state-switching model of [@McClintock2012] assumes a representation of the movement path that is discrete in time and continuous in space.
In this paper, we present a continuous-time, discrete-space (CTDS) model for animal movement which allows for flexible modeling of an animal’s response to drivers of movement in a computationally efficient framework. Instead of a Bayesian approach, we adopt a likelihood-based approach for inference, and instead of a state-switching or change-point model for changing behavior over time, we adopt a time-varying coefficient model. We also allow for variable selection using a lasso penalty. This CTDS approach is highly computationally efficient, requiring only minutes or seconds to analyze movement paths that would require hours using the approach of [@Hanks2011PLoS] or days using the approach of [@Hooten2010b], allowing the analysis of longer movement paths and more complex behavior than has been previously possible. To make this CTDS approach for modeling animal movement and resource selection accessible to practitioners, code to implement this approach is available online (www.stat.colostate.edu/\~hanks) in the form of a package for the R statistical computing environment [@R] with worked examples.
In Section 2 *Preliminaries*, we describe the continuous-time continuous-space model of [@Johnson2008] which is used to make inference on the posterior predictive distribution of an animal’s continuous movement path, conditioned on observed telemetry locations. We then describe the method of multiple imputation [@Rubin1987] which we use to integrate over the uncertainty in the animal’s continuous movement path. In Section 3 *Continuous-Time Discrete-Space Movement Model*, we describe the CTDS model for animal movement, and show how inference can be made on parameters in this model using standard software for generalized linear models (GLMs). In Section 4 *Time-Varying Behavior and Variable Selection* we use a varying-coefficient approach to model changing behavior over time, and use a lasso penalty for variable selection. In Section 5 *Drivers of Animal Movement* we discuss modeling potential covariates in the CTDS framework. In Section 6 *Example: Mountain Lions in Colorado* we illustrate our approach through an analysis of mountain lion (*Puma concolor*) movement in Colorado, USA. Finally, in Section 7 *Discussion* we discuss possible extensions to the CTDS approach.
Preliminaries
=============
Continuous-Time Continuous-Space Movement Model
-----------------------------------------------
To model animal movement, we make use of the continuous time correlated random walk (CTCRW) model of [@Johnson2008] to characterize a distribution for the continuous path conditioned on observed telemetry data. Let $\mathbf{S}=\{\mathbf{s}(t),t=t_0,t_1,\ldots,t_T\}$ be a collection of time-referenced telemetry locations for an animal. If the animal’s location and velocity at an arbitrary time $t$ are $\mathbf{s}(t)$ and $\mathbf{v}(t)$, respectively, then the CTCRW model can be specified as follows, ignoring the multivariate notation for simplicity: $$v(t)=\psi_1+\frac{\psi_2 e^{-\psi_3 t}}{\sqrt{2\psi_3}}
\omega\left( e^{2\psi_3 t}\right)\ ,$$ $$s(t)=s(0)+\int_0^t v(u)du\ ,$$ where $\boldsymbol\psi=[\psi_1,\psi_2,\psi_3]$ control the movement and $\omega(t)$ is standard Brownian motion. This model can be discretized and formulated as a state-space model, which allows for efficient computation of discretized paths $\tilde{\mathbf{S}}$ at arbitrarily fine time intervals via the Kalman filter [@Johnson2008a]. If a Bayesian framework is used for inference on $\boldsymbol\psi$, [@Johnson2008Biom] shows how the posterior predictive distribution of the animal’s continuous path $\tilde{\mathbf{S}}$ can be approximated using importance sampling. We will refer to the posterior predictive path distribution as $[\tilde{\mathbf{S}}|\mathbf{S}]$, where the bracket notation ‘$[\cdot]$’ denotes a probability distribution.
Multiple Imputation
--------------------
Our general strategy is to construct a model conditioned on the continuous path $\tilde{\mathbf{S}}$, and then integrate over the uncertainty in the posterior predictive distribution $[\tilde{\mathbf{S}}|\mathbf{S}]$ [e.g., @Hooten2010b; @Hanks2011PLoS]. If we treat the unobserved continuous path $\tilde{\mathbf{S}}$ as missing data, then we can make inference on model parameters using multiple imputation [@Rubin1987]. We motivate multiple imputation as posterior predictive inference on the imputation distribution within a Bayesian framework. Our treatment is similar to that of [@Rubin1987] and [@Rubin1996].
If we desire posterior predictive inference $[\boldsymbol\theta|\mathbf{S}]$ concerning environmentally relevant movement parameters $\boldsymbol\theta$, conditioned on the telemetry data $\mathbf{S}$ and the posterior predictive path distribution $[\tilde{\mathbf{S}}|\mathbf{S}]$, then we can write: $$[\boldsymbol\theta|\mathbf{S}] =\int_{\tilde{\mathcal{S}}}[\boldsymbol\theta|\tilde{\mathbf{S}}]
[\tilde{\mathbf{S}}|\mathbf{S}]d\tilde{\mathbf{S}}.$$ In the multiple imputation literature, the posterior predictive path distribution $[\tilde{\mathbf{S}}|\mathbf{S}]$ is called the imputation distribution and specifies a statistical model for the missing data $\tilde{\mathbf{S}}$ conditioned on the observed data $\mathbf{S}$. We will use the CTCRW model of [@Johnson2008] as the imputation distribution $[\tilde{\mathbf{S}}|\mathbf{S}]$ in our CTDS approach to modeling animal movement. The CTCRW model is a mechanistic model of animal movement that has been successfully applied to studies of aquatic [@Johnson2008] and terrestrial [@Hooten2010b] animals, and can represent a wide range of behavior.
[@Hooten2010b] and [@Hanks2011PLoS] use composition sampling to obtain samples from the posterior predictive distribution $[\boldsymbol\theta|\mathbf{S}]$ in (2) by sampling iteratively from $[\boldsymbol\theta|\tilde{\mathbf{S}}]$ and $[\tilde{\mathbf{S}}|\mathbf{S}]$. Alternately, under the multiple imputation framework the posterior distribution $[\boldsymbol\theta|\mathbf{S}]$ is assumed to be asymptotically Gaussian. The posterior can then be approximated using only the posterior predictive mean and variance, which can be obtained using conditional mean and variance formulae:
$$\begin{aligned}
E(\boldsymbol\theta|\mathbf{S}) &\approx \int_{\Theta}
\boldsymbol\theta \int_{\tilde{\mathcal{S}}}
[\boldsymbol\theta|\tilde{\mathbf{S}}]
[\tilde{\mathbf{S}}|\mathbf{S}]
d\tilde{\mathbf{S}}d\boldsymbol\theta \nonumber \\
&= \int_{\tilde{\mathcal{S}}}\left(\int_{\Theta}
\boldsymbol\theta
[\boldsymbol\theta|\tilde{\mathbf{S}}]d\boldsymbol\theta \right)
[\tilde{\mathbf{S}}|\mathbf{S}]
d\tilde{\mathbf{S}} \nonumber \\
&=E_{\tilde{\mathbf{S}}|\mathbf{S}}\left(E(\boldsymbol\theta|\tilde{\mathbf{S}})\right)\end{aligned}$$
and likewise: $$\text{Var}(\boldsymbol\theta|\mathbf{S}) \approx E_{\tilde{\mathbf{S}}|\mathbf{S}}\left(
\text{Var}(\boldsymbol\theta|\tilde{\mathbf{S}})\right)+\text{Var}_{\tilde{\mathbf{S}}|\mathbf{S}}\left(
E(\boldsymbol\theta|\tilde{\mathbf{S}})\right).$$
As the posterior distribution $[\boldsymbol\theta|\tilde{\mathbf{S}}]$ converges asymptotically to the sampling distribution of the maximum likelihood estimate (MLE) of $\boldsymbol\theta$, we can approximate $[\boldsymbol\theta|\tilde{\mathbf{S}}]$ by obtaining the asymptotic sampling distribution of the MLE. This allows us to use standard maximum likelihood approaches for inference, which can be much more computationally efficient than their Bayesian counterparts for this class of models.
The multiple imputation estimate $\hat{\boldsymbol\theta}_{MI}$ is typically obtained by approximating the integrals in (3) and (4) using a finite sample from the imputation distribution. The procedure can be summarized as follows:
1. Draw $K$ different realizations (imputations) $\tilde{\mathbf{S}}^{(k)} \sim [\tilde{\mathbf{S}}|\mathbf{S}]$ from the imputation distribution.
2. For each realization, find the MLE $\hat{\boldsymbol\theta}^{(k)}$ and asymptotic variance $Var(\hat{\boldsymbol\theta}^{(k)})$ of the estimate based on the the full data: $(\tilde{\mathbf{S}})$.
3. Combine results from different imputations using finite sample versions of the conditional expectation (3) and variance (4) results: $$\hat{\boldsymbol\theta}_{MI} =\frac{1}{K} \sum_{k=1}^K
\hat{\boldsymbol\theta}^{(k)}$$ $$Var(\hat{\boldsymbol\theta}_{MI}) =\frac{1}{K} \sum_{k=1}^K
var(\hat{\boldsymbol\theta}^{(k)})+\frac{1}{K}\sum_{k=1}^K
\left(\hat{\boldsymbol\theta}^{(k)}-\bar{\hat{\boldsymbol\theta}}\right)^2.$$
Equations (5) and (6) are the commonly used combining rules [@Rubin1987] for multiple imputation estimators in the scalar case. When $\boldsymbol\theta$ is vector-valued, similar approaches can be used [@Rubin1987].
Continuous-Time Discrete-Space Movement Model
=============================================
Having described the multiple imputation framework, we now focus on specifying a model of animal response to drivers of movement $[(\tilde{\mathbf{S}},\mathbf{S})|\boldsymbol\theta]$ that is flexible and computationally efficient. In doing so, we will assume a discrete (e.g., gridded) model for space [e.g., @Hooten2010b], and model animal movement as a continuous-time random walk through the discrete, gridded space.
Let the the study area be defined as a graph $(\mathbf{G},\mathbf{A})$ of $M$ locations $\mathbf{G}=(G_1,G_2,\ldots,G_M)$ connected by “edges” $\boldsymbol\Lambda=\{\lambda_{ij}:i\sim j,i=1,\ldots,M\}$ where $i\sim
j$ means that the nodes $G_i$ and $G_j$ are directly connected. For example, in a gridded space each grid cell is a node and the edges connect each grid cell to its first-order neighbors (e.g., a rook’s neighborhood). In typical studies, the spatial resolution of the grid cells in $\mathbf{G}$ will be determined by the resolution at which environmental covariates that may drive animal movement and selection are available.
A path realization $\tilde{\mathbf{S}}$ from the CTCRW model is continuous in time and space (Figure 1). If we consider a discrete, gridded space $\mathbf{G}$, then the continuous-time, continuous-space path $\tilde{\mathbf{S}}$ is represented by a continuous-time, discrete-space path $(\mathbf{g},\boldsymbol\tau)$ consisting of a sequence of grid cells $\mathbf{g}=(G_{i_1},G_{i_2},\ldots,G_{i_T})$ transversed by the animal’s continuous-space path and the residence times $\boldsymbol\tau=(\tau_1,\tau_2,\ldots,\tau_T)$ in each grid cell.
{width="\textwidth"}
In practice, this transformation from continuous space to discrete space results in a compression of the data to a temporal scale that is relevant to the resolution of the environmental covariates that may be driving movement and selection. For example, if an animal is moving slow relative to the time it takes to traverse a grid cell in $\mathbf{G}$, then the quasi-continuous path $\tilde{\mathbf{S}}$ may contain a long sequence of locations within one grid cell. The discrete-space representation of the path represents this long sequence of locations as one observation (a grid cell $G_i{_t}$ and residence time $\tau_t$). This data compression is especially relevant for telemetry data, in which observation windows can span years or even decades for some animals.
Random Walk Model
-----------------
The discrete-space representation $(\mathbf{g},\boldsymbol\tau)$ of the movement path allows us to use standard discrete-space random walk models to make inference about possible drivers of movement. While we will relax this assumption later to account for temporal autocorrelation in movement behavior, we initially assume that the the $t$-th observation $(G_{i_t},\tau_t)$ in the sequence is independent of all other observations in the sequence. Under this assumption, the likelihood of the sequence of transitions $\{(G_{i_t}
\rightarrow G_{i_{t+1}},\tau_t),t=1,2,\ldots,T\}$ is just the product of the likelihoods of each individual observation. We will focus on modeling each transition $(G_{i_t} \rightarrow G_{i_{t+1}},\tau_t)$, and will drop the $t$ subscript in this section to simplify notation.
If an animal is in cell $G_i$ at time $t$, then define the Poisson rate of transition from cell $G_i$ to a neighboring cell $G_j$ as $$\lambda_{ij}(\boldsymbol\beta)=\exp\{\mathbf{x}_{ij}' \boldsymbol\beta\}$$ where $\mathbf{x}_{ij}$ is a vector containing covariates related to drivers of movement specific to cells $i$ and $j$, and $\boldsymbol\beta$ is a vector of parameters that define how each of the covariates in $\mathbf{x}_{ij}$ are correlated with animal movement. The total transition rate $\lambda_i$ from cell $i$ is the sum of the transition rates to all neighboring cells: $\lambda_i(\boldsymbol\beta)=\sum_{j \sim i} \lambda_{ij}(\boldsymbol\beta)$ and the time $\tau$ that the animal resides in cell $G_i$ is exponentially-distributed with rate parameter equal to the total transition rate $\lambda_i(\boldsymbol\beta)$: $$[\tau|\boldsymbol\beta]=\lambda_{i}(\boldsymbol\beta)
\exp \left\{-\tau \lambda_{i}(\boldsymbol\beta)\right\}.$$
When the animal transitions from cell $G_i$ to one of its neighbors, the probability of transitioning to cell $G_k$, an event we denote as $G_i \rightarrow G_k$, follows a multinomial distribution with probability proportional to the transition rate $\lambda_{ik}$ to cell $G_k$: $$=\frac{\lambda_{ik}(\boldsymbol\beta)}{\sum_{j\sim i} \lambda_{ij}(\boldsymbol\beta)}=\frac{\lambda_{ik}(\boldsymbol\beta)}{\lambda_{i}(\boldsymbol\beta)}.$$
If, as is commonly assumed in the study of Markov random walks, the residence time and eventual destination are independent events, then the likelihood of the observation $(G_i \rightarrow G_k, \tau)$ is the product of the likelihoods of its parts: $$\begin{aligned}
[G_i \rightarrow G_k,
\tau|\boldsymbol\beta]&=\frac{\lambda_{ik}(\boldsymbol\beta)}{\lambda_{i}(\boldsymbol\beta)}
\cdot \lambda_{i}(\boldsymbol\beta)
\exp \left\{-\tau \lambda_{i}(\boldsymbol\beta)\right\} \nonumber \\
&=\lambda_{ik}(\boldsymbol\beta)\exp \left\{-\tau \lambda_{i}(\boldsymbol\beta)\right\}.\end{aligned}$$
Latent Variable Representation
------------------------------
We now introduce a latent variable representation of the transition process that is equivalent to (10), but allows for inference within a standard generalized linear modeling framework. For each $j$ such that $i\sim
j$, define $z_j$ as $$z_j=\begin{cases}
1 &,\ G_i \rightarrow G_j\\
0 &,\ \text{o.w.}
\end{cases}$$ and let $$[z_j,\tau|\boldsymbol\beta]\propto\lambda_{ij}^{z_j} \exp
\left\{-\tau \lambda_{ij}(\boldsymbol\beta)\right\}$$ where $n_i$ is the number of neighbors of the $i$-th grid cell. Then the product of $[z_j,\tau|\boldsymbol\beta]$ over all $j$ such that $i\sim j$ is proportional to the likelihood (10) of the observed transition: $$\begin{aligned}
\prod_{j:i\sim j}
[z_j,\tau|\boldsymbol\beta]&\propto \prod_{j:i\sim j}\lambda_{ij}^{z_j} \exp \left\{-\tau \lambda_{ij}(\boldsymbol\beta)\right\}\\
&=\lambda_{ik}(\boldsymbol\beta)\exp \left\{-\tau
\lambda_{i}(\boldsymbol\beta)\right\}\ ,\ \text{where } G_i
\rightarrow G_k \\
&=[G_i \rightarrow G_k,
\tau|\boldsymbol\beta]\end{aligned}$$
The benefit of this latent variable representation is that the likelihood of $z_j,\tau|\boldsymbol\beta$ in (11) is equivalent to the kernel of the likelihood in a Poisson regression with the canonical log link, where $z_j$ are the observations and $\log(\tau)$ is an offset or exposure term. The likelihood of the entire continuous-time, discrete-space path $(\mathbf{g},\boldsymbol\tau)$ can be written (returning $t$ to the notation) as: $$[\mathbf{g},\boldsymbol\tau|\boldsymbol\beta] =[\mathbf{Z},\boldsymbol\tau|\boldsymbol\beta]
\propto \prod_{t=1}^T \prod_{i_t \sim j_t}
\left[\lambda_{i_t j_t}^{z_{j_t}}(\boldsymbol\beta) \exp\{-\tau_t \lambda_{i_t j_t}(\boldsymbol\beta)\}\right]$$ where $\mathbf{Z}=(\mathbf{z}_1,\ldots,\mathbf{z}_T)'$ is a vector containing the latent variables $\mathbf{z}_i=(z_{i_1},z_{i_2},\ldots,z_{i_K})'$ for each grid cell in the discrete-space path. Inference can be made on $\boldsymbol\beta$ in (12) using standard Poisson GLM approaches (e.g., maximum likelihood). This provides a computationally efficient framework for the statistical analysis of potential drivers of movement within the multiple imputation framework of Section 2.2. Multiple path realizations (imputations) can be drawn from $[\tilde{\mathbf{S}}|\mathbf{S}]$. Each continuous path $\tilde{\mathbf{S}}$ can then be transformed into a CTDS path $(\mathbf{g},\boldsymbol\tau)$, which can then be used to make inference on $\boldsymbol\beta$ using (12). The results from the multiple imputed paths can then be combined using (5) and (6), resulting in a multiple imputation estimate $\hat{\boldsymbol\beta}_{MI}$ and estimated variance $Var(\hat{\boldsymbol\beta}_{MI})$.
Time-Varying Behavior and Variable Selection
============================================
In this section we describe how covariate effects can be allowed to vary over time using a varying-coefficient model, and how variable selection can be accomplished through shrinkage estimation.
Changing Behavior Over Time
---------------------------
Animal behavior and response to drivers of movement can change significantly over time. These changes can be driven by external factors such as changing seasons [e.g., @Grovenburg2009] or predator/prey interactions [e.g., @Lima2002], or by internal factors such as internal energy levels [e.g., @Nathan2008]. The most common approach to modeling time-varying behavior in animal movement is state-space modeling, typically within a Bayesian framework [e.g., @Morales2004; @Jonsen2005; @Getz2008; @Nathan2008; @Forester2009; @Gurarie2009; @Merrill2010]. Often, the animal is assumed to exhibit a number of behavioral states, each characterized by a distinct pattern of movement or response to drivers of movement. The number of states can be either known and specified in advance by the researcher [e.g., @Morales2004; @Jonsen2005] or allowed to be random [e.g., @Hanks2011PLoS; @McClintock2012].
State-space models are an intuitive approach to modeling changing behavior over time, but there are limits to the complexity that can be modeled using this approach. Allowing the number of states to be unknown and random requires a Bayesian approach with a changing parameter space. This is typically implemented using reversible-jump MCMC methods [e.g., @Green1995; @McClintock2012; @Hanks2011PLoS], which are computationally expensive and can be difficult to tune. Our approach is to use a computationally efficient GLM (12) to analyze parameters related to drivers of animal movement. Instead of using the common state-space approach, we employ varying-coefficient models [e.g., @Hastie1993] to model time-varying behavior in animal movement.
For simplicity in notation, consider the case where there is only one covariate $x$ in the model (7) and no intercept term. The model for the Poisson transition rate (7) will typically contain an intercept term and multiple covariates $\{x\}$, and the varying-coefficient approach we present generalizes easily to this case. In a time-varying-coefficient model, we allow the parameter $\beta(t)$ to vary over time in a functional (continuous) fashion. The transition rate (7) then becomes: $$\lambda_{ij}(\beta(t))=\exp \left\{x_{ij}\beta(t)\right\}$$ where $t$ is the time of the observation and $x_{ij}$ is the value of the covariate related to the Poisson rate of moving from cell $i$ to cell $j$. The functional regressor $\beta(t)$ is typically modeled as a linear combination of $n_{spl}$ spline basis functions $\{\boldsymbol\phi_k(t),k=1,\ldots,n_{spl}\}$ : $$\beta(t)=\sum_{k=1}^{n_{spl}} \alpha_k \boldsymbol\phi_k(t).$$ B-spline basis functions are among the most-widely used choices for $\{\phi_k(t)\}$, and are appropriate in most cases. Fourier basis functions are commonly used for $\{\phi_k(t)\}$ when the behavior is thought to be periodic (e.g., diurnal variation in behavior).
Under this varying-coefficient specification, (7) can be rewritten as $$\begin{aligned}
\lambda_{ij}(\beta(t))&=\exp \left\{x_{ij}\beta(t)\right\} \nonumber \\
&=\exp \left\{x_{ij}\sum_{k=1}^{n_{spl}} \alpha_k
\boldsymbol\phi_k((t))\right\} \nonumber \\
&=\exp \left\{\boldsymbol\phi_{ij}' \boldsymbol\alpha\right\}\end{aligned}$$ where $\boldsymbol\alpha=(\alpha_1,\ldots,\alpha_{n_{spl}})'$ and $\boldsymbol\phi_{ij}=x_{ij}\cdot(\phi_1(t),\ldots,\phi_{n_{spl}}(t))'$. The result is that the varying-coefficient model can be written as a GLM with a modified design matrix. This specification provides a flexible framework for allowing the effect of a driver of movement ($x$) to vary over time that is computationally efficient and simple to implement using standard GLM software.
Shrinkage Estimation
--------------------
The model we have specified in (12) is likely to be overparameterized, especially if we utilize a varying-coefficient model (13). Animal movement behavior is complex, and a typical study could entail a large number of potential drivers of movement, but an animal’s response to each of those drivers of movement is likely to change over time, with only a few drivers being relevant at any one time. Under these assumptions, many of the parameters $\alpha_k$ in (13) are likely to be very small or zero. Multicollinearity is also a potential problem, as many potential drivers of movement could be correlated with each other.
We propose a shrinkage estimator of $\boldsymbol\alpha$ using a lasso penalty [@Tibshirani1996]. The typical maximum likelihood estimate of $\boldsymbol\alpha$ is obtained by maximizing the likelihood $[\mathbf{Z},\boldsymbol\tau|\boldsymbol\alpha]$ from (12), or equivalently by maximizing the log-likelihood $\log
[\mathbf{Z},\boldsymbol\tau|\boldsymbol\alpha]$. The lasso estimate is obtained by maximizing the penalized log-likelihood, where the penalty is the sum of the absolute values of the regression parameters $\{\alpha_k\}$: $$\hat{\boldsymbol\alpha}_{\text{lasso}}=\max_{\boldsymbol\alpha} \left\{
\log
[\mathbf{Z},\boldsymbol\tau|\boldsymbol\alpha] -\gamma \sum_{k=1}^K |\alpha_k| \right\}.$$ As the tuning parameter $\gamma$ increases, the absolute values of the regression parameters $\{\alpha_k\}$ are “shrunk” to zero, with the parameters that best describe the variation in the data being shrunk more slowly than parameters that do not. Cross-validation is typically used to set the tuning parameter $\gamma$ at a level that optimizes the model’s predictive power.
Shrinkage approaches such as the lasso are well-developed for GLMs, and computationally-efficient methods are available for fitting GLMs to data [e.g., @Friedman2010]. Recent work has also applied the lasso to multiple imputation estimators [e.g., @Chen2011]. The main challenge in applying the lasso to multiple imputation is that a parameter may be shrunk to zero in the analysis of one imputation but not in the analysis of another. The solution is to use a so-called group lasso [@Yuan2006], in which a group of parameters is constrained to either all equal zero or all be non-zero together.
The lasso can be seen as a constrained optimization problem, with $\boldsymbol\alpha_{\text{lasso}}$ minimizing the mean squared error subject to the constraint that $||\boldsymbol\alpha_{\text{lasso}}||_1
\leq \nu$, where $||\cdot||$ is the L-1 norm and the value of $\nu$ is determined by the tuning parameter $\gamma$. As the estimate $\boldsymbol\alpha_{\text{lasso}}$ typically falls on the boundary of the constrained space, conventional approaches for estimating the standard error of $\boldsymbol\alpha_{\text{lasso}}$ are unavailable. Bayesian versions of the lasso [@Park2008] and group lasso [@Raman2009] provide alternatives that allow for estimation of the uncertainty about the parameters $\boldsymbol\alpha_{\text{lasso}}$ through posterior analysis.
The Bayesian approach entails significantly more computational complexity, and may not be as accessible to practitioners. We instead focus on the likelihood-based stacked lasso estimate of [@Chen2011]. In this estimate, instead of computing the lasso estimate $\boldsymbol\alpha_{\text{lasso}}$ for each imputation individually, and then combining the results using (5) and (6), the imputed data from all estimates are “stacked” together and a group lasso estimate is obtained for the combined data. We note that this likelihood-based stacked lasso approach does not allow for the estimation of the variance of $\boldsymbol\alpha_{\text{lasso}}$. If uncertainty estimates are a priority, we recommend choosing a parsimonious selection of potential drivers of movement *a priori* that exhibit little multicollinearity and computing the traditional multiple imputation estimates $\hat{\boldsymbol\alpha}_{MI}$.
Drivers of Animal Movement
==========================
We now provide some examples showing how a range of hypothesized drivers of movement could be modeled within the CTDS framework. Following [@Hooten2010b], we consider two distinct categories for drivers of movement from cell $G_i$ to cell $G_j$: location-based drivers ($\{p_{ki},k=1,2,\ldots,K\}$) which are determined only by the characteristics of cell $G_i$, and directional drivers ($\{q_{lij},l=1,2,\ldots,L\}$) which vary with direction of movement. Under a time-varying coefficient model for each driver, the transition rate (7) from cell $G_i$ to cell $G_j$ is $$\lambda_{ij}\left(\beta(t)\right)=\exp
\left\{\beta_0(t)+\sum_{k=1}^{K}p_{ki}\beta_k(t)+\sum_{l=1}^{L}q_{lij}\beta_l(t)\right\}$$ where $\beta_0(t)$ is a time-varying intercept term, $\{\beta_k(t)\}$ are time-varying effects related to location-based drivers of movement, and $\{\beta_l(t)\}$ are time-varying effects related to directional drivers of movement. We consider both location-based and directional drivers in what follows.
Location-Based Drivers of Movement
----------------------------------
[@Hooten2010b] denote static, non-directional drivers of movement as location-based drivers of movement. Location-Based drivers of movement can be used to examine differences in animal movement rates that can be explained by the environment an animal resides in. In the CTDS context, location-based drivers would be covariates dependent only on the characteristics of the cell where the animal is currently located. Large positive (negative) values of the corresponding $\beta_k(t)$ would indicate that the animal tends to transition quickly (slowly) from a cell containing the cover type in question.
Directional Drivers of Movement
-------------------------------
In contrast to location-based drivers, which describe the effect that the local environment in which the animal resides has on movement rates, directional drivers of movement [@Brillinger2001; @Hooten2010b; @Hanks2011PLoS] capture directional selection by the individual.
A directional driver of movement is defined by a vector which points toward (or away) from something that is hypothesized to attract (or repel) the animal in question. Let $\mathbf{v}_l$ be the vector corresponding to the $l$-th directional driver of movement. In the CTDS model for animal movement, the animal can only transition from cell $G_i$ to one of its neighbors $G_j:j\sim i$. Let $\mathbf{w}_{ij}$ be a unit vector pointing from the center of cell $G_i$ in the direction of the center of cell $G_j$. Then the covariate $q_{lij}$ relating the $l$-th directional driver of movement to the transition rate from cell $G_i$ to cell $G_j$ is the dot (or inner) product of $\mathbf{v}_l$ and $\mathbf{w}_{ij}$: $$q_{lij}=\mathbf{v}_l'\mathbf{w}_{ij}.$$ Then $p_{lij}$ will be positive when $\mathbf{v}_l$ points nearly in the direction of cell $G_j$, negative when $\mathbf{v}_l$ points directly away from cell $G_j$, and zero if $\mathbf{v}_l$ is perpendicular to the direction from cell $G_i$ to cell $G_j$.
Examples
--------
We now provide multiple examples of drivers of movement to illustrate the range of effects that can be modeled using this framework.
### Overall Movement Rate
The intercept term $\beta_0(t)$ in (15) can be seen as a driver of movement in which $p_{0i}=1$ for every cell $G_i \in
\mathbf{G}$. This intercept term controls the animal’s overall rate of transition from any cell, and thus models the animal’s overall movement rate. Allowing the intercept parameter $\beta_0(t)$ to vary over time could reveal changes in activity levels over time. For example, we might expect $\beta_0(t)$ to be larger at night for nocturnal species and smaller during the day.
### Movement Response to Land Cover Types
Indicator variables could be used to examine how animal movement differs between different landscape cover types (e.g., forest vs. plains) by setting $p_{ki}=1$ for each cell $G_i$ that is classified as containing the $k$-th cover type. As in the case of the intercept, allowing the parameter $\beta_k(t)$ related to the $k$-th cover type to vary over time can reveal variation in an animal’s movement pattern through the cover type. For example, an animal may move quickly through open terrain during the day, but may move more slowly through the same terrain at night.
### Environmental Gradients
Animals may use environmental gradients for navigation. For example, a mule deer might move predominantly in the direction of increasing elevation during a spring migration [e.g., @Hooten2010b], or a seal might follow gradients in sea surface temperature to navigate toward land [e.g., @Hanks2011PLoS]. Such effects can be modeled by including a directional driver of movement in (15) defined by a gradient vector $\mathbf{v}_l$ which points from the center of cell $G_i$ in the direction of steepest increase in the covariate $x_l$. Positive values of $\beta_l$ indicate that the animal moves generally towards cells with higher values of $x_l$, while negative values of $\beta_l$ indicate that the animal moves generally towards cells with lower values of $x_l$.
### Activity Centers
Many animals exhibit movement patterns that are centered on a location in space. This central location may be temporary, such as a kill site for a predator [e.g., @Knopff2009], or more permanent, such as a den for a central place forager [e.g., @Hanks2011PLoS; @McClintock2012]. The relatively new class of spatial capture-recapture models [e.g., @Royle2008] model detection probability as a decreasing function of distance from a central location (e.g., the “center” of an animal’s home range). Movement around an activity center can be modeled in the CTDS framework by including a directional driver of movement in (15) defined by a vector $\mathbf{v}_l$ which points from the center of cell $G_i$ to the location of the activity center. Then a positive value for $\beta_l$ would indicate that the animal is generally drawn toward this activity center. If the activity center is considered to be temporary (such as a kill site for a predator), then a time-varying-coefficient model should be used. The variable selection obtained through the lasso estimate can indicate the range of time in which the animal’s movement is centered around the activity center. If the activity center is considered to be permanent through the duration of the study, a varying-coefficient model may not be needed.
Under the likelihood-based specification of the CTDS model for animal movement, it is necessary to specify the locations of all hypothesized activity centers beforehand. In Section 6, we show an example of the specification of hypothesized activity centers (potential kill sites for mountain lions) using the original telemetry data. If a Bayesian formulation of the CTDS model were used, then the location of hypothesized activity centers could be random, and inference could be made on their locations jointly with inference on the movement parameters, similar to what is done in spatial capture-recapture models [e.g., @Royle2008].
### Conspecific Interaction
An animal’s movement patterns can be greatly affected by interaction with conspecifics. For example, one animal could follow the trail left by another animal, two animals could avoid one another by changing course when they become close enough to sense the other animal, or a pair of animals could maintain proximity as they move together across the landscape. While there are many possible approaches to modeling such dependence in behavior, we choose to model each of these interactions through the inclusion of directional effects in the CTDS modeling framework. For example, a directional driver could be included in the movement model for one animal that is defined by a vector pointing to the current location of another animal to examine whether the animal being modeled is attracted to ($\beta_l>0$) or avoids ($\beta_l<0$) the conspecific.
### Directional Persistence
The CTCRW model of [@Johnson2008] is based on a correlated random walk model for velocity that allows for directional persistence in animal movement. So far, we have assumed that each discrete movement step in our CTDS model is independent, but this assumption is not met if the animal exhibits any directional persistence. To account for directional persistence in the CTDS approach, we use an autoregressive approach by including a directional driver of movement at each discrete movement step that is defined by a vector pointing in the direction of the previous move. For example, if the animal moved west in the previous discrete movement step, then the autoregressive vector for the next step points west as well. Positive values of the $\beta$ related to this directional driver of movement indicate that the animal is likely to maintain its direction of movement over time.
Spatial and Temporal Scale
--------------------------
The choice of scale for a study can greatly influence results [e.g., @Boyce2006]. When speaking of the scale of a study, one could look at the grain, or resolution, at which the process is modeled, or the extent (coverage) over which the process is modeled. The spatial and temporal extent of a study of animal movement are determined by the telemetry data and the posterior predictive path distribution $[\tilde{\mathbf{S}}|\mathbf{S}]$. However, when implementing the CTDS approach, the researcher must make three choices pertaining to the grain or resolution: (1) the temporal scale at which the CTCRW movement path of the animal is sampled, (2) the spatial scale of the grid over which the discrete-space movement will be modeled, and (3) the temporal scale of the varying coefficient model, which is determined by the number and resolution of spline knots in the spline basis expansion.
As the CTCRW model of [@Johnson2008] is a continuous-time model, we recommend sampling from the movement path at as fine an interval as is feasible. In practice, this will be limited by computational resources and the size of the study. The temporal resolution needs to be fine enough that realizations from the posterior predictive path distribution $[\tilde{\mathbf{S}}|\mathbf{S}]$ are quasi-continuous and adequately capture the residence time $\tau$ in each grid cell in the CTDS representation of the movement path.
The choice of spatial resolution of the raster grid on which the CTDS process occurs implicitly specifies a time scale at which the movement process is modeled. Coarser spatial resolution (larger grid cells) will correspond to longer residence times $\boldsymbol\tau$ in the CTDS mode. The spatial resolution should be chosen so that the time scale at which an animal transitions from one grid cell to another is a time scale at which the animal in question can make meaningful choices about movement and resource selection. The time scale implicit in the choice of spatial resolution can be examined by plotting a histogram of the residence times in the CTDS representation of the movement path.
If the lasso penalty is used, then it is common to choose a saturated spline basis expansion in the varying-coefficient model, where one spline knot is specified at each data point in time. We recommend specifying a spline basis expansion with knots at a similar temporal resolution to the temporal resolution of the telemetry data. The lasso penalization will shrink the overparameterized expansion to a more parsimonious model that best fits the data. While a finer temporal resolution could be used, the posterior predictive path distribution is unlikely to show changes in behavior at time scales smaller than the time scale of the original data. Using a coarser temporal resolution will force $\beta(t)$ to be smooth. This would imply that changes in animal behavior are gradual and occur at time scales larger than the time scale of the data.
Example: Mountain Lions in Colorado
===================================
We illustrate our CTDS random walk approach to modeling animal movement through a study of mountain lions (*Puma concolor*) in Colorado, USA. As part of a larger study, a female mountain lion, designated AF79, and her subadult cub, designated AM80, were fitted with global positioning system (GPS) collars set to transmit location data every 3 hours. We analyze the location data $\mathbf{S}$ from one week of location information for these two animals (Figure 2).
{width=".8\textwidth"}
{width=".8\textwidth"}
We fit the CTCRW model of [@Johnson2008] to both animals’ location data using the ‘crawl’ package [@crawl] in the R statistical computing environment [@R]. Ten imputations from the posterior distribution of the quasi-continuous path distribution $[\tilde{\mathbf{S}}|\mathbf{S}]$ were obtained at one-minute intervals. The result is a quasi-continuous path at extremely fine temporal resolution for each imputation.
For covariate data, we used a landcover map of the state of Colorado created by the Colorado Vegetation Classification Project (http://ndis.nrel.colostate.edu/coveg/), which is a joint project of the Bureau of Land Management and the Colorado Division of Wildlife. The landcover map contained gridded landcover information at 100m square resolution. Figure 3 shows a histogram of the residence times $\boldsymbol\tau$ in each grid cell in the CTDS representation of the movement path of AM80. The area traveled by the two animals in our study was predominantly forested. To assess how the animals’ movement differed when in terrain other than forest, we created an indicator covariate where all forested grid cells were assigned a value of zero, and all cells containing other cover types, including developed land, bare ground, grassland, and shrubby terrain, were assigned a value of one (Figure 2a). This covariate and an intercept covariate were used as location-based covariates in the CTDS model for both animals.
{width=".7\textwidth"}
For each animal, we created a set of potential kill sites (PKS) by examining the original GPS location data (Figure 2). [@Knopff2009] classified a location as a PKS if two or more GPS locations were found within 200m of the site within a six-day period. We added an additional restraint that at least one of the GPS locations be during nighttime hours (9pm to 6am) for the point to be classified a PKS. We then created a covariate raster layer containing the distance to the nearest PKS for each grid cell (Figure 2). A directional covariate defined by a vector pointing towards the nearest PKS was included in the CTDS model for both animals.
To examine how the movement path of the cub AM80 affected the movement path of the mother AF79, we included a directional covariate in the CTDS model for AF79 defined by a vector pointing from the mother’s location to the cub’s location at each time point. Similarly, we included a directional covariate in the CTDS model for AM80 defined by a vector pointing from the cub’s location to the mother’s location at each time point.
For each animal, we also included a directional covariate pointing in the direction of the most recent movement at each time point. This covariate measures the strength of correlation between moves, and thus the strength of the directional persistence shown by the animal’s discrete-space movement path. As we are assuming an underlying correlated movement model (the CTCRW model of [@Johnson2008]), we expect the CTDS movement to be correlated in time as well.
To allow for varying behavior over time, we used a varying-coefficient model for each covariate in the model. For all covariates except the directional covariate related to directional persistence, we used a B-spline basis expansion, with regularly spaced spline knots at 3-hour intervals. We fit the CTDS model for each path using the ‘glmnet’ R package [@Friedman2010], using a Lasso penalty, with tuning parameter chosen to minimize the average squared error of the fit in a 10-fold cross-validation.
{width="\textwidth"}
{width="\textwidth"}
Results
-------
The time-varying results for the location-based and directional drivers of movement for AF79 are shown in Figure 4, with the corresponding results for AM80 shown in Figure 5. As we used a lasso penalty, we can only obtain point estimates (confidence intervals are unavailable) of the time-varying effects $\{\beta(t)\}$. A comparison of the differences between the results for AF79 and AM80 yields some insight into how the movement patterns of these two animals differ.
The intercept effect measures the animal’s general movement rate over time. Figure 4(a) and Figure 5(a) show the time-varying deviation from the grand mean for each animal. For example, the male AM80 moved at a higher than average rate of speed near sunrise on Julian day 90, as evidenced by the positive $\beta(t)$ in Figure 5(a). The time-varying intercept for both lions tends to be higher during nighttime hours and crepuscular periods than during the day, indicating higher overall rates of movement during nocturnal hours.
The location-based response to non-forested land cover (Figure 4b and Figure 5b) is zero for much of the study window for both AF79 and AM80. Forested land cover makes up over 90% of the study area, thus the mountain lions encounter different land cover infrequently. On day 90, the male (AM80) moves through a patch of non-forested land cover at a relatively high rate of speed. On day 89, the female moves through a patch of non-forested land cover at a low rate of speed. Both animals encounter non-forested patches at other times during the observation window, but the $\beta(t)$ related to movement through non-forested terrain is near-zero during these times. This $\beta(t)$ has been shrunk to zero by the Lasso procedure during these times, indicating that during these times the animal’s movement is not greatly different in non-forested patches than it is in forested patches.
The male cub (AM80) has a fairly consistent positive response to the directional covariate pointing toward the nearest PKS from day 88 to day 90 (Figure 5c), indicating that much of the male’s movement during these three days resembles a random walk centered on an attractive central location (a PKS). During day 91, the negative response to the direction to the nearest PKS indicates that the animal is moving away from the nearest PKS. At the same time, the positive response to the directional covariate pointing towards AF79 (Figure 5d) indicates that AM80 is moving towards AF79 during this time period.
Positive values of the autoregressive parameter (Figure 4e and Figure 5e) indicate that the animals are in a state of correlated movement (directional persistence). The magnitude of the autoregressive parameter is greater in general for the female (Figure 4e) than for the male (Figure 5e), indicating that the female may have a greater tendency for directed movements than the male. The CTDS movement paths are based on the CTCRW model of animal movement, which results in correlated movement paths. The inclusion of this autoregressive parameter is meant to account for the correlation in the CTDS path representation of the underlying correlated CTCRW movement path.
Discussion
==========
While we have couched our CTDS approach in terms of modeling animal movement, we can also view this approach in terms of resource selection [e.g., @Manly2002]. [@Johnson2008] describe a general framework for the analysis of resource selection from telemetry data using a weighted distribution approach where an observed distribution of resource use is seen as a re-weighted version of a distribution of available resources, and the resource selection function (RSF) defines the preferential use of resources by the animal. [@Warton2010] and [@Aarts2012] describe a point-process approach to resource selection that can be fit using a Poisson GLM, similar to the CTDS model we describe here. In the context of [@Warton2010], the CTDS approach can be viewed as a resource-selection analysis with the available resources at any time defined as the neighboring grid cells. The transition rate (15) of the CTDS process to each neighboring cell contains information about the availability of each cell, as well as the RSF that defines preferential use of the resources in each cell.
It is notable that the entire analysis in Section 6 requires less than ten minutes using a computer with 4 GB of memory and a 1.67 GHz quad-core processor. This increase in computational efficiency relative to the approaches of [@Johnson2008], [@Hooten2010b], [@Hanks2011PLoS], and [@McClintock2012] allows for inference on complex behavior at finer temporal resolution than has been possible previously. To make our CTDS approach accessible to practitioners, we have created an R-package (‘ctds’) that contains R-code to fit the CTDS model using multiple imputation as described in Sections 2-4. A script file contained in the ‘ctds’ package allows for the re-analysis of the telemetry data of the two mountain lions analyzed in Section 6. This R-package can be downloaded from the first author’s website (http://www.stat.colostate.edu/\~hanks).
The CTDS approach to modeling animal movement is flexible, and can be extended using standard approaches for GLMs. For example, if population-level inference is desired, the movement paths from multiple animals could be analyzed jointly, with population-level parameters in the GLM being shared by all animals. Similarly, interaction terms could be included in the model by including multiplied covariates in the design matrix. Fitting movement models in a GLM framework allows for many natural extensions with little additional effort.
We have focused on building an individual model for animal movement that is intuitive and computationally-efficient. If population-level inference on data from multiple animals is desired, there are multiple potential approaches. The first is to analyze movement paths from multiple animals jointly, with population-level parameters in the GLM being shared by all animals and individual variation modeled using standard random effects approaches. However, this approach may not be straightforward to implement or interpret in the context of a varying-coefficient model. Each individual animal is likely to encounter and be influenced by different drivers of movement (e.g., local environmental factors or nearby conspecifics) at differing times throughout the observation window. In this situation, it may make little sense to examine a population-level response to a particular driver of movement at a particular time, as in a typical random-effects analysis.
Instead, one approach to a population-level analysis could be a post-hoc analysis of the time-varying response to covariates $\boldsymbol\beta(t)$. For example, [@Hanks2011PLoS] used a cluster analysis to examine different movement regimes shared across individuals in the population. Differences in the movement patterns exhibited by subgroups (e.g., male vs. female) were then examined by a comparison of the proportion of time spent by each subgroup in each of the movement regimes.
The use of directional drivers of movement has a long history. [@Brillinger2001] model animal movement as a continuous-time, continuous-space random walk where the drift term is the gradient of a “potential function” that defines an animal’s external drivers of movement. [@Tracey2005] use circular distributions to model how an animal moves in response to a vector pointing towards an object that may attract or repel the animal. [@Hanks2011PLoS] and [@McClintock2012] make extensive use of gradients to model directed movements, and movements about a central location. In our study of mountain lion movement data, we used directional drivers of movement to model conspecific interaction between a mother (AF79) and her cub (AM80). Interactions between predators and prey could also be modeled using directional covariates defined by vectors pointing between animals. Some movements based on memory could also be modeled using directional covariates. For example, a directional covariate defined by a vector pointing to the animal’s location one year prior could be used to model seasonal migratory behavior. The ability to model both location-based and directional drivers of movement make the CTDS framework a flexible and extensible framework for modeling complex behavior in animal movement.
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