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Create exciting school projects or cool promotional materials in seconds. This Azalea logo can be used in endless contexts. Just a sample of use: photo wallpaper, email advertising, brochures, newspaper Ads, postcard, headlines and mugs. Creating logos can not be any easier?
Our Azalea logos can be used for whatever you need. You can use them at work, home or school projects. Just a small example of usage: banner Ads, wedding invitations, magnets, calendars, classified ads, image processing and spreadsheets. All logos can be easily customized to your needs!
Our Azalea logos can be used for whatever you need. You can use them at work, home or school projects. Just a small example of usage: photo books, classified ads, flyers, photo wallpaper, film editing, place cards and post labels. Creating logos can not be any easier?
Our Azalea logos can be used for whatever you need. You can use them at work, home or school projects. Just a small example of usage: email advertising, brochures, booklets, party invitations, banners, campaigns and T-shirts. You can choose from hundreds of various logos!
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{
"redpajama_set_name": "RedPajamaC4"
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A Shafrir 1 és Shafrir 2 kis hatótávolságú, infravörös önirányítású légiharc-rakéták, melyeket Izraelben fejlesztettek ki az 1960-as években. A Shafrir 1 rakéta tervezését 1959-ben kezdték, és kis számban állították rendszerbe az Izraeli Légierő Mirage III-as vadászbombázó repülőgépein. A rakéta nem váltotta be a hozzá fűzött reményeket, ezért továbbfejlesztésével létrehozták a Shafrir 2 rakétát. Ez már megfelelt a légierő elvárásainak, ezért nagy mennyiségben rendszeresítették, és a jom kippuri háborúban tömegesen vetették be, 176 rakétaindításból 89 ellenséges repülőgép lelövését elérve (ez jobb, 50,7%-os találati arányt jelentett, mint a szintén bevetett amerikai AIM–9D és G rakétákkal elért arány). A rakéta továbbfejlesztésével hozták létre a Python 3 légiharc-rakétát.
Külső hivatkozások
Shafrir 1 – Az Israeli-Weapons.com cikke
Shafrir 2 – Az Israeli-Weapons.com cikke
Légiharc-rakéták
Izraelben kifejlesztett fegyverek
en:Shafrir
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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{"url":"https:\/\/www.rocketryforum.com\/threads\/anyone-recognize-this-bullpuppy.139787\/","text":"# Anyone recognize this Bullpuppy?\n\n### Help Support The Rocketry Forum:\n\n#### PokerJones\n\nI was rummaging through the DARS stash about a month ago picking up some CRs and saw this dusty PML Bullpuppy sitting in the corner, hard to pass it up for $10. I plan on rebuilding this back to it's original military livery but was curious if anyone recognized this rocket? Most likely belonged to a North Texas flyer at one time. Last edited: #### PokerJones ##### Well-Known Member Step one is to remove the 6\" long brass launch lug adorning the airframe. I used a dremel cutoff wheel to carefully cut through the epoxy fillet in line with the body tube to avoid any damage. I'll attach new rail buttons before final painting. Next was to sand down the entire airframe\/fins with 120 grit with the palm sander removing most of the original paint followed up with 220 grit. I did a little hand sanding in between the fins and along the fillets. The phenolic airframe is in pretty good shape with no major problems, one small gap between the airframe and boat tail was filled with bondo glazing putty. The fillets are still strong and smooth so I am just going to lightly sand with some 320 and then shoot the first coat of primer to see how it looks. #### qquake2k ##### Captain Low-N-Slow Nice find! 4\" airframe? What size motor mount? #### PokerJones ##### Well-Known Member Nice find! 4\" airframe? What size motor mount? This is the 3\" Bullpuppy with a 38mm MM. Not much room for motor retention with the boat tail but I will figure something out for that. #### rharshberger ##### Well-Known Member This is the 3\" Bullpuppy with a 38mm MM. Not much room for motor retention with the boat tail but I will figure something out for that. Even better is its the phenolic version, sooo much better than QT imo. #### PokerJones ##### Well-Known Member As usual I hit this with a quick coat of primer and every blemish on the rocket came shining through. After all of the rockets I have finished I know to expect this but it still always seems to surprise me. Sanded down that coat and applied another, the seam between the BT and boat tail is a little rougher than I had thought and will take some work. All in all it's going rather quickly. I ordered and received the decals from Mark@stickershock last week so all I have left is to spray this white, add the rail buttons, add a tracker bay to the NC and call it done. More pics when I finish up. Last edited: #### markkoelsch ##### Well-Known Member This is the 3\" Bullpuppy with a 38mm MM. Not much room for motor retention with the boat tail but I will figure something out for that. Friction fit with masking tape, and a wrap of aluminum tape over the boat tail and motor. Problem solved. #### PokerJones ##### Well-Known Member Friction fit with masking tape, and a wrap of aluminum tape over the boat tail and motor. Problem solved. Yep, thats my standard recipe for these tight boat tails and MD's. I've never had a lost motor when friction fitting. #### Incongruent ##### Well-Known Member Friction fit with masking tape, and a wrap of aluminum tape over the boat tail and motor. Problem solved. I have found through my \"experimenting\" (setting things on fire in the backyard) that the adhesive can separate from the aluminium foil when it gets hot enough. Does this happen? Or does the end of the motor not get hot enough\/different tape with different properties. #### dhbarr ##### Amateur Professional If the back end of your casing is getting that hot, you've got other problems. #### Incongruent ##### Well-Known Member If the back end of your casing is getting that hot, you've got other problems. Okay, thanks. Having not leapt off the high power diving board yet, I wouldn't know. Back to the original discussion. #### PokerJones ##### Well-Known Member Rocket$10\nPaint & Primer $10 Decals$28\n\n\\$48 and it looks like a Bullpup again!\n\nNice to have another rocket that I can fly on small 38mm H's and I's (easier on the wallet) at the local launches.\n\n#### qquake2k\n\n##### Captain Low-N-Slow\nThat turned out really nice!\n\n#### AfterBurners\n\n##### Well-Known Member\nIt looks like a PML kit. Did it have a piston ejection system?\n\n#### PokerJones\n\n##### Well-Known Member\nIt looks like a PML kit. Did it have a piston ejection system?\nIt is a PML kit, 3\" Bullpuppy with the phenolic BT. The piston is long gone but the red 1\" nylon dog leash remains. I cut it down to just above the edge of the BT and sewed a loop in it, good enough for now.\n\n#### AfterBurners\n\n##### Well-Known Member\nIt is a PML kit, 3\" Bullpuppy with the phenolic BT. The piston is long gone but the red 1\" nylon dog leash remains. I cut it down to just above the edge of the BT and sewed a loop in it, good enough for now.\nI have this same kit which is ready for paint. Just waiting for a nice day, but its their \"Q\" tube that I got. Great looking kit.","date":"2021-03-03 02:43:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.18275199830532074, \"perplexity\": 6894.90115178603}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178365186.46\/warc\/CC-MAIN-20210303012222-20210303042222-00195.warc.gz\"}"}
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\section{Introduction}
This work is a collaboration with the company JayWay in Halmstad.
In order to enter the office today, employees need a tag-key.
Those who do not have a tag-key (for example visitors or couriers),
or if employees forget their tag,
they must ring the doorbell.
Then, someone has to open the door manually,
usually those at the desks located closer to the entrance.
This does not only consume time and focus of the workday for
these staff members but also disturbs others with desks
located nearby.
Accordingly, the goal of this work is to develop a system that uses face recognition to control the lock system of the entrance door for employees,
and speech-to-text conversion to notify the appropriate staff member that a visitor or delivery is waiting outside.
The project will help the company and its staff in two ways.
First, it will simplify the process of entering the office for employees.
Second, it will also reduce the disturbance when a visitor wants to enter the office
or when an employee forgets their tag-key.
The work of this paper will be limited to the fundamentals
of the access control system.
In terms of hardware, the fundamentals
are a small computer, display, camera, mic and speaker (Figure~\ref{fig:hardware}).
The requirements set for this project are:
\begin{itemize}
\item To control the access to the office with the help of a small computer and a camera.
\item To identify employees by means of a picture that will be compared against images in the personnel database.
\item To unlock the entrance door when an employee is granted access.
\item To use speech-to-text to handle visitors and guests by identifying the employee that should handle the visit.
\item To notify the employee that a visitor is waiting at the door.
\end{itemize}
A user interface has been
developed as well to handle user interaction via the display (Figure~\ref{fig:gui_main_menu}).
The display will show a graphical user interface (GUI) for the
users to interact with and present different outputs.
\begin{figure}[htb]
\centering
\includegraphics[width=0.48\textwidth]{hardware_parts.png}
\caption{Hardware parts used in the project.}
\label{fig:hardware}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.35\textwidth]{gui_main_menu.png}
\caption{Main menu on the GUI.}
\label{fig:gui_main_menu}
\end{figure}
\subsection{Related Works}
Today there are different technologies for facial recognition,
including 2D, 2D-3D and 3D facial recognition \cite{ABATE20071885}.
The accuracy will depend on several factors including
light,
pose,
occlusion,
or expression changes \cite{SINGH2018536}.
Face recognition technologies are in extensive use today
due to social networks and the massive availability of cameras
in many personal devices.
In such contexts, cloud computing moves the computing power and data storage burden to the cloud, bringing the possibility of biometric applications to a range of devices that otherwise would not have sufficient capacity to perform the operations locally \cite{6822213}.
Solutions have been proposed for example for face recognition \cite{6844616,6920723},
iris \cite{Kesava14},
voice \cite{6138538},
or keystroke \cite{Xi}.
Cloud speech-to-text solutions are also in increasing use by millions of person in applications such as Siri, Google, Amazon Transcribe, Alexa etc.
Speech-to-text solutions provide the capability to translate verbal language to text, enabling a more natural human-machine interaction, given that voice is the most common method for people to communicate \cite{Das}.
Notifying an employee can be done in several ways, e.g. via SMS, E-mail or Slack. Slack is a tool that can be used for communication within a group, a whole office, etc.\footnote{https://slack.com/} Different channels can be created as well for different projects. The company involved in this work uses Slack to communicate within their offices. A bot can be created in the slack workspace that can be used to send messages to inform the employees regarding if there is a guest outside waiting for them, if there is a delivery at the door, etc. This bot will be referred to as Slack-bot.
Both Amazon and Google offers services to send E-mail with respective API, but only Amazon offers the API to send an SMS. However these services are pay-per-use.
Biometric systems can be attacked via spoofing attacks \cite{[Jain11a]}.
These are different from regular IT attacks, in which system channels or modules
are hijacked.
In spoofing attacks, a fake biometric characteristic is presented to the sensor
with the purpose to fool the system and pretend to be somebody else.
Besides the apparent vulnerability that this may entail,
such type of attacks does not require any knowledge or access to the inner of the system,
and they can be carried out without any engineering skills.
In the case of face systems, for example, a picture, a video, or a 3D mask of a genuine used can be held in front of the camera \cite{6990726}.
Unfortunately, today it is straightforward to get a picture of a person directly or from the internet, even inadvertently.
For this reason, it is of utmost importance to implement measures to counteract
spoofing attacks.
\begin{figure}[htb]
\centering
\includegraphics[width=0.4\textwidth]{flow_system.png}
\caption{Overview of the whole system.}
\label{fig:flow_system}
\end{figure}
\section{System Design}
The project is divided into two phases, the experimentation
phase and the implementation phase. The experimentation
phase aims to find the most suitable setup for this project.
It lays the foundation of how the system will be implemented.
The implementation phase seeks to describe how the software
and hardware are integrated to form the intended system.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.6\textwidth]{flow_employee.png}
\caption{Flowcharts of employee function (top) and GUI (bottom).}
\label{fig:flow_employee}
\end{figure*}
\subsection{Software Components}
Python v3 was chosen as the programming language. We also considered Java, but Python offered more APIs and libraries.
Overall, Python offers a simplicity when it comes to libraries and APIs for calculations and recognitions, plus it is a well-known programming language used for image and data analysis \cite{8665702}.
Currently, the company has its information on each employee on Kperson, which is their custom-built website where they keep all the data within the company. Kperson is deployed on Amazon. The information has the form of a json string, and it contains all the necessary strings such as name, last name, id, company, link to images etc. This information can be used for face recognition, which saves the effort of creating a new database.
For face recognition, we employed Amazon Rekognition \cite{AmazonRek} and Amazon Simple Storage Service (Amazon S3), both from Amazon Web Services (AWS).
Amazon Rekognition is a cloud-based software providing computer vision capabilities, while Amazon S3 provides object storage.
The reason for choosing AWS was that it provided the most important services that were needed. Another reason is also that the company had its data on Amazon.
We also considered the use of Google Cloud Platform (GCP), a
platform which provided similar services to what AWS did.
The reasons why GCP was not chosen were because it would have demanded the transfer of employee data (already at Amazon) to GCP.
Moreover, GCP did not provide a face recognition service, only face detection.
All the data for each employee was fetched from Kperson, such like their full name and images.
Speech-to-text is used for the guest function, where the guest has to speak the name of the employee they want to meet with. Both Amazon Transcribe (AT) \cite{AmazonSpeechText} and Google's Cloud Speech-to-Text (GCST) \cite{GoogleSpeechText} services were considered.
We have chosen Google's service, since in our tests, Amazon's engine took 30 to 60 seconds to do a translation.
This may be because AT first record a file, send it to the cloud and then translate it into a string, while GCST takes the input directly and translate it into a string much faster.
Another reason to choose Google's service is that Amazon does not support Swedish language (the working language of the company involved).
The chosen method for notifying a employee in the guest handling is Slack. Amazon offered services for both E-mail and SMS, but both costs depending on the amount of messages sent, while Slack is free. Therefore, to reduce the costs of the project and because Slack is already used as a primary communication tool within the company, Slack was chosen for the notifications. To be able to send notifications, a slack-bot was created.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.6\textwidth]{flow_guest.png}
\caption{Flowcharts of guest function (top) and GUI (bottom).}
\label{fig:flow_guest}
\end{figure*}
\subsection{Hardware Components}
The hardware-tools needed for the project are the following: two Raspberry Pi, a camera for the face recognition, a microphone for the speech-to-text services, a speaker, a touch screen to show the GUI, access to a database with the face of the company's employees, and finally the lock system, which should be connected to a Raspberry Pi and therefore be able to be controlled by code.
There were two hardware candidates suitable for the project, either a Raspberry Pi or an Arduino. One of the most primary programming languages hosted on the Raspberry Pi is Python, while Arduino cannot run a code written in Python, but it is possible for it to communicate with a device using Python. To be able to use the cloud services, a Raspberry Pi can easily be connected to internet while a Arduino needs a external hardware. Overall the Raspberry Pi and Arduino are very similar but the Raspberry Pi are more powerful than the Arduino and is better on handling multiple tasks, making it more suitable for this project. Therefore Raspberry Pi (Model: Raspberry Pi 3 Model B+) was selected. The Raspberry Pi also has other devices that are specially adapted for it, which was used for the project and stated below.
Two devices of Raspberry Pi was used just because of security reasons (Figure~\ref{fig:flow_system}). One will be placed outside the office with a touch screen, camera, microphone and a speaker. These components only handle the capturing of the image and sending the image to the cloud storage. The second Raspberry Pi will be placed inside the office and handle the critical tasks of face recognition and controlling the lock system (this is the reason why this Raspberry Pi is placed inside the office). Both of the Raspberry Pi's are running on the operating system called Raspbian.
The Raspberry Pi-to-Raspberry Pi communication was done using transmission control protocol (TCP).
The Raspberry Pi device that was placed indoors was the server and it had two clients connected to it. One client was the other Raspberry Pi, that will be mounted outside the entrance door and the other one was the slack-bot.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.6\textwidth]{flow_delivery.png}
\caption{Flowcharts of delivery function (top) and GUI (bottom).}
\label{fig:flow_delivery}
\end{figure*}
A 7 inch LCD-touch display (Model: RASPBERRYPI-DISPLAY) was attached to the device to show the GUI. It is a multi-touch capacitive touch screen and supports up to 10 fingers at the same time. The resolution on the screen is 800$\times$480 and has a update frequency on 60hz. The display needed a Raspberry Pi as power supply with at least 2.5A, and it was connected with a ribbon cable to a display-port on the Raspberry Pi.
A Raspberry Pi Camera Module V2 (Model: RPI 8MP CAMERA BOARD) specially made to fit a Raspberry Pi was used to capture a image. It has a Sony IMX219 8-megapixel sensor and is capable to take images up to 3280$\times$2464. It was connected through a 15cm ribbon cable to a CSI-port on the Raspberry Pi. There are many third party libraries created for it and can therefore be used in Python with the Picamera Python library \cite{Picamera}.
As external soundcard,
we employed the Plexgear USB-sound card USC-100.
As plug-in microphone, we used the Hama Mini Mikrofon Notebook Silver.
It is a small microphone with a frequency range between 30Hz and 16 kHz, a sensitivity on -62 dB. and the impedance is 1kΩ.
The Roxcore Crossbeat Portable Bluetooth speaker (Model: Roxcore Crossbeat) was used as speaker.
All the hardware parts used in the project are shown in Figure~\ref{fig:hardware}, except the second Raspberry Pi which is attached to the backside of the display.
The company has its office on the second floor in a building, and the entrance is in the stairwell, therefore the devices was not exposed to rain, wind, sunlight etc. However there were some small echos in the stairwell, but that was not considered as loud enough to disturb the speech-to-text process. The lighting in the building was good enough for capturing a good picture, and there was no window near the entrance where the sunlight could cause a bad picture.
\subsection{Working Modes and GUI}
The system is divided in three functions for $i$) employees, $ii$) guests and $iii$) deliveries. The employee function has two authentication steps, the face recognition and a random generated code that needs to be confirmed to protect against spoofing.
The guest function includes the speech-to-text service to state an employee's name that the guest wants to meet, and the employee is then notified.
The delivery function informs the specific persons in the office that are responsible for the deliveries by sending a notification.
The touch screen displays a GUI for the user and is built with the Tkinter library in Python.
The main menu on the GUI shows the company logo and three buttons, one for each function.
\subsubsection{Employee Function}
The flowcharts of the employee function and GUI are shown in Figure~\ref{fig:flow_employee}.
If it is an employee, the system is going to ask the person visually through the GUI to stand in front of the camera to take a picture. When the picture has been captured, it is sent to the S3 bucket and deleted after the comparison.
The image is encrypted on the client-side before it is uploaded to the cloud. The encryption algorithm used is the XOR cipher. XOR cipher is an easy used symmetric encryption algorithm that can be used for encrypting and decrypting images by changing the pictures byte arrays with a key, and this encryption algorithm is hard to crack by using the brute force method \cite{NATSHEH2016175}.
The image is decrypted before the comparison, and then compared against the collections using the Amazon Face Recognition API. To improve protection against spoofing attacks, a two factor authentication is implemented, by combination of face recognition and a code. When the comparison is done, it returns the similarity score. If the similarity score is above a predefined threshold, the system generates a random 4 digit code and sends it to the employee as a private message on Slack. This prevents unauthorized persons to enter the office. Even if the face recognition system recognizes the person on a picture, the person holding the picture would not be able to enter since the code is sent to the employee on the picture. The code needs to be entered on the keypad of the GUI.
The system allows 3 unsuccessful attempts before it returns to the main menu, and then the process needs to be done again. Every new try, it generates a new random code to prevent a brute force attack. If the code is correct, the server unlocks the door, and sends a message to the GUI, which then displays a ``Welcome to the office'' message and the employee name.
If the similarity score is below the threshold, the person is denied access, and the GUI will go back to the main menu.
The lock system for the main door installed at the company works in a way such a voltage of 12V is constantly applied to the lock circuit to keep it closed. To activate (unlock) the door, a pulse signal of 0V has to be applied, and when the signal goes from 0V to 12V again, the door will unlock during 5 seconds.
In our prototype, this is implemented with the General Purpose Input/Output (GPIO) pins available at the Raspberry Pi.
For security, unlocking of the door is handled by the Raspberry Pi
placed inside the office.
\subsubsection{Guest Function}
The flowcharts of the guest function and GUI are shown in Figure~\ref{fig:flow_guest}.
If the guest option is chosen, the person is asked visually through the GUI to pronounce the name of the employee that wishes to meet. The system records the input with the microphone and sends it to Google's Speech-to-Text service.
Since it can be difficult to use speech-to-text on names, a string comparison algorithm is used to compare the input with all names in the database. If the most similar string has
a similarity score over 80, the system sends a private message to the employee directly on Slack with the information that a guest is waiting outside the entrance door. The guest also gets information on the GUI that the employee have been notified.
If the score is below 30, it then means that the similarity score is too low and the system will ask the user to try again. If the score is between these values, the guest needs to confirm that it is the correct person that they want to meet, with help of ``Yes'' and ``No'' buttons which will be displayed on the GUI.
If the system returns the wrong employee name and the guest presses ``No'', the guest will be asked again who wishes to meet.
\subsubsection{Delivery Function}
The flowcharts of the delivery function and GUI are shown in Figure~\ref{fig:flow_delivery}.
If the delivery option is chosen, a notification is sent to the Slack channel that informs the employee in charge of deliveries that there is a delivery at the door. They will need to go to the door and open it manually since the delivery often requires a signature.
The GUI will show that the employee has been notified.
\subsection{Data Protection and Maintenance}
Regarding General Data Protection Regulation (GDPR), the GUI employed allows to fulfill the requirement of the GDPR. According to GDPR, all kind of information that can directly be linked to a person, can be counted as personal data, even pictures, drawings and movies if they show a person. An audio file with a person's voice can also be considered as personal data \cite{Datainspektionen1,Datainspektionen2}. To avoid that the camera takes pictures of every person that stands in front of it, a button on the GUI for the employees is used, so it has to be activated on purpose. When a picture is taken, it is encrypted before it is sent for comparison, and then decrypted when it is going to be compared to images in the database. No pictures or audio are saved locally on the Raspberry Pi.
An strategy to keep images of the database updated is also employed. Only face images with a similarity score greater than a threshold (higher than 99.5\%) are used to update the template of the employee in the database. This also means that someone who is not an employee will have very difficult to obtain a sufficient score to have his/her picture saved, and the picture will be deleted immediately after the comparison.
The 10 most recent pictures of an employee are kept in the database. If additional images have to be stored, then the oldest images will be overwritten.
\begin{figure}[htb]
\centering
\includegraphics[width=0.48\textwidth]{scores_histogram.png}
\caption{Results of face recognition experiments.}
\label{fig:scores_histogram}
\end{figure}
\section{Experiments and Results}
The proposed prototype was tested to evaluate its
usability and accuracy. For this purpose,
the system was used by several staff members of the company.
\subsection{Face Recognition}
A total of 40 employees were used for the experiments.
To test the performance of the face solution employed, 20 employees were asked to carry out 10 separate genuine tests each, leading to 200 genuine trials.
Also, 5 unauthorized attempts were simulated against each of the 40 employees of the database, leading to 200 impostor trials.
The similarity score was saved for each recognition trial, with the
distributions of genuine and impostor scores plotted in Figure~\ref{fig:scores_histogram}. %
From our tests, Amazon Rekognition provided perfect separation between genuine and impostors. The smallest genuine score is 94.25\%,
while the highest impostor score is 73.1\%.
The system can be therefore considered reliable to allow employees to access the office, and to deny access to unauthorized people, at least at the scale evaluated here and in the imaging environment where the company operates.
During the genuine trials,
we also measured the time taken by the different necessary functions: capturing a picture, comparing it with the database, and introducing the random code. The whole operation covers from when the user presses the button in the GUI until the door is opened.
Average results are given in Figure~\ref{fig:average_time}.
Approximately 22\% of the time is consumed by taking the picture, 51\% by the authentication process carried out in the Amazon cloud, and 27\% by the introduction of the PIN sent to the employee. Half of the time is spent on comparing the image to the database, while the rest is divided between the other two processes in approximately equal parts.
Overall, the process takes 20.3 seconds in average.
\subsection{Speech-to-Text}
The language on the speech-to-text service was set to Swedish, and the tests intend to show how well the service works depending on if it is a native or a non-native Swedish speaker who is pronouncing. Each employee name has been spoken by the testers to see how many attempts was needed to match with the right employee.
The results are shown in Figure~\ref{fig:stt}. As it can be shown, Swedish speakers managed to match the correct employee in the first try in the majority of cases (average amount of tries equal to 37/33= 1.12). On the other hand, non-Swedish speakers had to try more than once with approximately 27\% of the employees (9 out of 33 IDs), leading to an average amount of tries equal to 50/33=1.51.
\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{average_time.png}
\caption{Average time for the employee function in the system.}
\label{fig:average_time}
\end{figure}
\section{Discussion}
The goal of this work is to develop a system that can control the main entrance of an office by using face recognition and speech-to-text.
It makes use of two Raspberry Pi with a camera, a microphone, and a speaker.
Face recognition and speech-to-text conversion are done with the
cloud-based solutions provided by Amazon Web Services \cite{AmazonRek} and Google Speech-to-Text \cite{GoogleSpeechText}, respectively,
The system is complemented with a touch screen display, and
a graphical user interface (GUI) which presents the detected classes
to the user.
Disturbances are caused when a person inside the office needs to interrupt the work to open the door for an employee, a guest, or to receive delivery.
Face recognition minimizes the disturbances since no tag-keys are needed, and employees can enter the office even if they forget it, without anyone having to open the door manually for them.
The guest handling using speech-to-text also minimizes the interference, since it only notifies the employee in question which the guest declares in the system. The employee will receive a private message on Slack with the notification, so others at the office will not be disturbed. The delivery function also has its advantage, as it only sends a delivery notice on Slack to those who are responsible for receiving the deliveries.
GDPR is not an issue \cite{Datainspektionen1,Datainspektionen2} since the camera or the microphone are not recording continuously, but only when the corresponding function is activated in the GUI. In addition, none of the captured images or audio is saved, and the data is deleted immediately after it is being used for the intended purpose.
Another security measure is that data is encrypted before it is sent to the cloud.
\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{stt.png}
\caption{Speech-to-text accuracy by swedish speakers (top) and non-Swedish speakers (bottom).}
\label{fig:stt}
\end{figure}
The face recognition system works with very good accuracy in the environment of this work, as shown in the tests.
Spoof recognition is currently not supported in Amazon Rekognition, and the service does not include encryption/decryption of images either.
Using a two steps authentication process (face recognition combined with a confirmation code) makes the system more secure and provides protection against spoofing.
To avoid brute force attacks, a new random code is created every time, and it is only valid for three attempts.
A downside is that it is an extra step that takes about 5 extra seconds on average, which we will seek to overcome by including spoofing detection mechanisms \cite{6990726}.
The average time spent by an employee until the door unlocks is of 20.3 seconds, which may be perceived as high, although it provides a secure and accurate method for access control.
The speech-to-text is set to Swedish and works well for Swedish-speaking persons.
For a multicultural company, it can be more difficult to use the speech-to-text service.
A problem can also occur if two or more employees has the same first and last name. The system will return the first occurrence in the database, which might not be the correct person the guest want to visit.
Our system is implemented in a relatively quiet environment, but
it can be hard to listen and translate in setups with
a lot of background noise or if several people speak at the same time.
Our prototype works well in an office environment, but if it is connected to a slower internet connection, the face recognition and speech-to-text will take longer.
Also, the system is cloud based, which means that it always requires internet connection to operate.
There are also advantages in using a GUI. It allows to easily show information for the person using the system, and also to instruct the person on how to enter the office. Based on the feedback obtained, it is more comfortable to visualize the information for both the employee and the visitor.
\section*{Acknowledgment}
Authors K. H.-D. and F. A.-F. thank the Swedish Research Council for funding their research.
Authors also acknowledge the CAISR program of the Swedish Knowledge Foundation.
\bibliographystyle{IEEEtran}
\section{Introduction}
This work is a collaboration with the company JayWay in Halmstad.
In order to enter the office today, employees need a tag-key.
Those who do not have a tag-key (for example visitors or couriers),
or if employees forget their tag,
they must ring the doorbell.
Then, someone has to open the door manually,
usually those at the desks located closer to the entrance.
This does not only consume time and focus of the workday for
these staff members but also disturbs others with desks
located nearby.
Accordingly, the goal of this work is to develop a system that uses face recognition to control the lock system of the entrance door for employees,
and speech-to-text conversion to notify the appropriate staff member that a visitor or delivery is waiting outside.
The project will help the company and its staff in two ways.
First, it will simplify the process of entering the office for employees.
Second, it will also reduce the disturbance when a visitor wants to enter the office
or when an employee forgets their tag-key.
The work of this paper will be limited to the fundamentals
of the access control system.
In terms of hardware, the fundamentals
are a small computer, display, camera, mic and speaker (Figure~\ref{fig:hardware}).
The requirements set for this project are:
\begin{itemize}
\item To control the access to the office with the help of a small computer and a camera.
\item To identify employees by means of a picture that will be compared against images in the personnel database.
\item To unlock the entrance door when an employee is granted access.
\item To use speech-to-text to handle visitors and guests by identifying the employee that should handle the visit.
\item To notify the employee that a visitor is waiting at the door.
\end{itemize}
A user interface has been
developed as well to handle user interaction via the display (Figure~\ref{fig:gui_main_menu}).
The display will show a graphical user interface (GUI) for the
users to interact with and present different outputs.
\begin{figure}[htb]
\centering
\includegraphics[width=0.48\textwidth]{hardware_parts.png}
\caption{Hardware parts used in the project.}
\label{fig:hardware}
\end{figure}
\begin{figure}[htb]
\centering
\includegraphics[width=0.35\textwidth]{gui_main_menu.png}
\caption{Main menu on the GUI.}
\label{fig:gui_main_menu}
\end{figure}
\subsection{Related Works}
Today there are different technologies for facial recognition,
including 2D, 2D-3D and 3D facial recognition \cite{ABATE20071885}.
The accuracy will depend on several factors including
light,
pose,
occlusion,
or expression changes \cite{SINGH2018536}.
Face recognition technologies are in extensive use today
due to social networks and the massive availability of cameras
in many personal devices.
In such contexts, cloud computing moves the computing power and data storage burden to the cloud, bringing the possibility of biometric applications to a range of devices that otherwise would not have sufficient capacity to perform the operations locally \cite{6822213}.
Solutions have been proposed for example for face recognition \cite{6844616,6920723},
iris \cite{Kesava14},
voice \cite{6138538},
or keystroke \cite{Xi}.
Cloud speech-to-text solutions are also in increasing use by millions of person in applications such as Siri, Google, Amazon Transcribe, Alexa etc.
Speech-to-text solutions provide the capability to translate verbal language to text, enabling a more natural human-machine interaction, given that voice is the most common method for people to communicate \cite{Das}.
Notifying an employee can be done in several ways, e.g. via SMS, E-mail or Slack. Slack is a tool that can be used for communication within a group, a whole office, etc.\footnote{https://slack.com/} Different channels can be created as well for different projects. The company involved in this work uses Slack to communicate within their offices. A bot can be created in the slack workspace that can be used to send messages to inform the employees regarding if there is a guest outside waiting for them, if there is a delivery at the door, etc. This bot will be referred to as Slack-bot.
Both Amazon and Google offers services to send E-mail with respective API, but only Amazon offers the API to send an SMS. However these services are pay-per-use.
Biometric systems can be attacked via spoofing attacks \cite{[Jain11a]}.
These are different from regular IT attacks, in which system channels or modules
are hijacked.
In spoofing attacks, a fake biometric characteristic is presented to the sensor
with the purpose to fool the system and pretend to be somebody else.
Besides the apparent vulnerability that this may entail,
such type of attacks does not require any knowledge or access to the inner of the system,
and they can be carried out without any engineering skills.
In the case of face systems, for example, a picture, a video, or a 3D mask of a genuine used can be held in front of the camera \cite{6990726}.
Unfortunately, today it is straightforward to get a picture of a person directly or from the internet, even inadvertently.
For this reason, it is of utmost importance to implement measures to counteract
spoofing attacks.
\begin{figure}[htb]
\centering
\includegraphics[width=0.4\textwidth]{flow_system.png}
\caption{Overview of the whole system.}
\label{fig:flow_system}
\end{figure}
\section{System Design}
The project is divided into two phases, the experimentation
phase and the implementation phase. The experimentation
phase aims to find the most suitable setup for this project.
It lays the foundation of how the system will be implemented.
The implementation phase seeks to describe how the software
and hardware are integrated to form the intended system.
\begin{figure*}[htb]
\centering
\includegraphics[width=0.6\textwidth]{flow_employee.png}
\caption{Flowcharts of employee function (top) and GUI (bottom).}
\label{fig:flow_employee}
\end{figure*}
\subsection{Software Components}
Python v3 was chosen as the programming language. We also considered Java, but Python offered more APIs and libraries.
Overall, Python offers a simplicity when it comes to libraries and APIs for calculations and recognitions, plus it is a well-known programming language used for image and data analysis \cite{8665702}.
Currently, the company has its information on each employee on Kperson, which is their custom-built website where they keep all the data within the company. Kperson is deployed on Amazon. The information has the form of a json string, and it contains all the necessary strings such as name, last name, id, company, link to images etc. This information can be used for face recognition, which saves the effort of creating a new database.
For face recognition, we employed Amazon Rekognition \cite{AmazonRek} and Amazon Simple Storage Service (Amazon S3), both from Amazon Web Services (AWS).
Amazon Rekognition is a cloud-based software providing computer vision capabilities, while Amazon S3 provides object storage.
The reason for choosing AWS was that it provided the most important services that were needed. Another reason is also that the company had its data on Amazon.
We also considered the use of Google Cloud Platform (GCP), a
platform which provided similar services to what AWS did.
The reasons why GCP was not chosen were because it would have demanded the transfer of employee data (already at Amazon) to GCP.
Moreover, GCP did not provide a face recognition service, only face detection.
All the data for each employee was fetched from Kperson, such like their full name and images.
Speech-to-text is used for the guest function, where the guest has to speak the name of the employee they want to meet with. Both Amazon Transcribe (AT) \cite{AmazonSpeechText} and Google's Cloud Speech-to-Text (GCST) \cite{GoogleSpeechText} services were considered.
We have chosen Google's service, since in our tests, Amazon's engine took 30 to 60 seconds to do a translation.
This may be because AT first record a file, send it to the cloud and then translate it into a string, while GCST takes the input directly and translate it into a string much faster.
Another reason to choose Google's service is that Amazon does not support Swedish language (the working language of the company involved).
The chosen method for notifying a employee in the guest handling is Slack. Amazon offered services for both E-mail and SMS, but both costs depending on the amount of messages sent, while Slack is free. Therefore, to reduce the costs of the project and because Slack is already used as a primary communication tool within the company, Slack was chosen for the notifications. To be able to send notifications, a slack-bot was created.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.6\textwidth]{flow_guest.png}
\caption{Flowcharts of guest function (top) and GUI (bottom).}
\label{fig:flow_guest}
\end{figure*}
\subsection{Hardware Components}
The hardware-tools needed for the project are the following: two Raspberry Pi, a camera for the face recognition, a microphone for the speech-to-text services, a speaker, a touch screen to show the GUI, access to a database with the face of the company's employees, and finally the lock system, which should be connected to a Raspberry Pi and therefore be able to be controlled by code.
There were two hardware candidates suitable for the project, either a Raspberry Pi or an Arduino. One of the most primary programming languages hosted on the Raspberry Pi is Python, while Arduino cannot run a code written in Python, but it is possible for it to communicate with a device using Python. To be able to use the cloud services, a Raspberry Pi can easily be connected to internet while a Arduino needs a external hardware. Overall the Raspberry Pi and Arduino are very similar but the Raspberry Pi are more powerful than the Arduino and is better on handling multiple tasks, making it more suitable for this project. Therefore Raspberry Pi (Model: Raspberry Pi 3 Model B+) was selected. The Raspberry Pi also has other devices that are specially adapted for it, which was used for the project and stated below.
Two devices of Raspberry Pi was used just because of security reasons (Figure~\ref{fig:flow_system}). One will be placed outside the office with a touch screen, camera, microphone and a speaker. These components only handle the capturing of the image and sending the image to the cloud storage. The second Raspberry Pi will be placed inside the office and handle the critical tasks of face recognition and controlling the lock system (this is the reason why this Raspberry Pi is placed inside the office). Both of the Raspberry Pi's are running on the operating system called Raspbian.
The Raspberry Pi-to-Raspberry Pi communication was done using transmission control protocol (TCP).
The Raspberry Pi device that was placed indoors was the server and it had two clients connected to it. One client was the other Raspberry Pi, that will be mounted outside the entrance door and the other one was the slack-bot.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.6\textwidth]{flow_delivery.png}
\caption{Flowcharts of delivery function (top) and GUI (bottom).}
\label{fig:flow_delivery}
\end{figure*}
A 7 inch LCD-touch display (Model: RASPBERRYPI-DISPLAY) was attached to the device to show the GUI. It is a multi-touch capacitive touch screen and supports up to 10 fingers at the same time. The resolution on the screen is 800$\times$480 and has a update frequency on 60hz. The display needed a Raspberry Pi as power supply with at least 2.5A, and it was connected with a ribbon cable to a display-port on the Raspberry Pi.
A Raspberry Pi Camera Module V2 (Model: RPI 8MP CAMERA BOARD) specially made to fit a Raspberry Pi was used to capture a image. It has a Sony IMX219 8-megapixel sensor and is capable to take images up to 3280$\times$2464. It was connected through a 15cm ribbon cable to a CSI-port on the Raspberry Pi. There are many third party libraries created for it and can therefore be used in Python with the Picamera Python library \cite{Picamera}.
As external soundcard,
we employed the Plexgear USB-sound card USC-100.
As plug-in microphone, we used the Hama Mini Mikrofon Notebook Silver.
It is a small microphone with a frequency range between 30Hz and 16 kHz, a sensitivity on -62 dB. and the impedance is 1kΩ.
The Roxcore Crossbeat Portable Bluetooth speaker (Model: Roxcore Crossbeat) was used as speaker.
All the hardware parts used in the project are shown in Figure~\ref{fig:hardware}, except the second Raspberry Pi which is attached to the backside of the display.
The company has its office on the second floor in a building, and the entrance is in the stairwell, therefore the devices was not exposed to rain, wind, sunlight etc. However there were some small echos in the stairwell, but that was not considered as loud enough to disturb the speech-to-text process. The lighting in the building was good enough for capturing a good picture, and there was no window near the entrance where the sunlight could cause a bad picture.
\subsection{Working Modes and GUI}
The system is divided in three functions for $i$) employees, $ii$) guests and $iii$) deliveries. The employee function has two authentication steps, the face recognition and a random generated code that needs to be confirmed to protect against spoofing.
The guest function includes the speech-to-text service to state an employee's name that the guest wants to meet, and the employee is then notified.
The delivery function informs the specific persons in the office that are responsible for the deliveries by sending a notification.
The touch screen displays a GUI for the user and is built with the Tkinter library in Python.
The main menu on the GUI shows the company logo and three buttons, one for each function.
\subsubsection{Employee Function}
The flowcharts of the employee function and GUI are shown in Figure~\ref{fig:flow_employee}.
If it is an employee, the system is going to ask the person visually through the GUI to stand in front of the camera to take a picture. When the picture has been captured, it is sent to the S3 bucket and deleted after the comparison.
The image is encrypted on the client-side before it is uploaded to the cloud. The encryption algorithm used is the XOR cipher. XOR cipher is an easy used symmetric encryption algorithm that can be used for encrypting and decrypting images by changing the pictures byte arrays with a key, and this encryption algorithm is hard to crack by using the brute force method \cite{NATSHEH2016175}.
The image is decrypted before the comparison, and then compared against the collections using the Amazon Face Recognition API. To improve protection against spoofing attacks, a two factor authentication is implemented, by combination of face recognition and a code. When the comparison is done, it returns the similarity score. If the similarity score is above a predefined threshold, the system generates a random 4 digit code and sends it to the employee as a private message on Slack. This prevents unauthorized persons to enter the office. Even if the face recognition system recognizes the person on a picture, the person holding the picture would not be able to enter since the code is sent to the employee on the picture. The code needs to be entered on the keypad of the GUI.
The system allows 3 unsuccessful attempts before it returns to the main menu, and then the process needs to be done again. Every new try, it generates a new random code to prevent a brute force attack. If the code is correct, the server unlocks the door, and sends a message to the GUI, which then displays a ``Welcome to the office'' message and the employee name.
If the similarity score is below the threshold, the person is denied access, and the GUI will go back to the main menu.
The lock system for the main door installed at the company works in a way such a voltage of 12V is constantly applied to the lock circuit to keep it closed. To activate (unlock) the door, a pulse signal of 0V has to be applied, and when the signal goes from 0V to 12V again, the door will unlock during 5 seconds.
In our prototype, this is implemented with the General Purpose Input/Output (GPIO) pins available at the Raspberry Pi.
For security, unlocking of the door is handled by the Raspberry Pi
placed inside the office.
\subsubsection{Guest Function}
The flowcharts of the guest function and GUI are shown in Figure~\ref{fig:flow_guest}.
If the guest option is chosen, the person is asked visually through the GUI to pronounce the name of the employee that wishes to meet. The system records the input with the microphone and sends it to Google's Speech-to-Text service.
Since it can be difficult to use speech-to-text on names, a string comparison algorithm is used to compare the input with all names in the database. If the most similar string has
a similarity score over 80, the system sends a private message to the employee directly on Slack with the information that a guest is waiting outside the entrance door. The guest also gets information on the GUI that the employee have been notified.
If the score is below 30, it then means that the similarity score is too low and the system will ask the user to try again. If the score is between these values, the guest needs to confirm that it is the correct person that they want to meet, with help of ``Yes'' and ``No'' buttons which will be displayed on the GUI.
If the system returns the wrong employee name and the guest presses ``No'', the guest will be asked again who wishes to meet.
\subsubsection{Delivery Function}
The flowcharts of the delivery function and GUI are shown in Figure~\ref{fig:flow_delivery}.
If the delivery option is chosen, a notification is sent to the Slack channel that informs the employee in charge of deliveries that there is a delivery at the door. They will need to go to the door and open it manually since the delivery often requires a signature.
The GUI will show that the employee has been notified.
\subsection{Data Protection and Maintenance}
Regarding General Data Protection Regulation (GDPR), the GUI employed allows to fulfill the requirement of the GDPR. According to GDPR, all kind of information that can directly be linked to a person, can be counted as personal data, even pictures, drawings and movies if they show a person. An audio file with a person's voice can also be considered as personal data \cite{Datainspektionen1,Datainspektionen2}. To avoid that the camera takes pictures of every person that stands in front of it, a button on the GUI for the employees is used, so it has to be activated on purpose. When a picture is taken, it is encrypted before it is sent for comparison, and then decrypted when it is going to be compared to images in the database. No pictures or audio are saved locally on the Raspberry Pi.
An strategy to keep images of the database updated is also employed. Only face images with a similarity score greater than a threshold (higher than 99.5\%) are used to update the template of the employee in the database. This also means that someone who is not an employee will have very difficult to obtain a sufficient score to have his/her picture saved, and the picture will be deleted immediately after the comparison.
The 10 most recent pictures of an employee are kept in the database. If additional images have to be stored, then the oldest images will be overwritten.
\begin{figure}[htb]
\centering
\includegraphics[width=0.48\textwidth]{scores_histogram.png}
\caption{Results of face recognition experiments.}
\label{fig:scores_histogram}
\end{figure}
\section{Experiments and Results}
The proposed prototype was tested to evaluate its
usability and accuracy. For this purpose,
the system was used by several staff members of the company.
\subsection{Face Recognition}
A total of 40 employees were used for the experiments.
To test the performance of the face solution employed, 20 employees were asked to carry out 10 separate genuine tests each, leading to 200 genuine trials.
Also, 5 unauthorized attempts were simulated against each of the 40 employees of the database, leading to 200 impostor trials.
The similarity score was saved for each recognition trial, with the
distributions of genuine and impostor scores plotted in Figure~\ref{fig:scores_histogram}. %
From our tests, Amazon Rekognition provided perfect separation between genuine and impostors. The smallest genuine score is 94.25\%,
while the highest impostor score is 73.1\%.
The system can be therefore considered reliable to allow employees to access the office, and to deny access to unauthorized people, at least at the scale evaluated here and in the imaging environment where the company operates.
During the genuine trials,
we also measured the time taken by the different necessary functions: capturing a picture, comparing it with the database, and introducing the random code. The whole operation covers from when the user presses the button in the GUI until the door is opened.
Average results are given in Figure~\ref{fig:average_time}.
Approximately 22\% of the time is consumed by taking the picture, 51\% by the authentication process carried out in the Amazon cloud, and 27\% by the introduction of the PIN sent to the employee. Half of the time is spent on comparing the image to the database, while the rest is divided between the other two processes in approximately equal parts.
Overall, the process takes 20.3 seconds in average.
\subsection{Speech-to-Text}
The language on the speech-to-text service was set to Swedish, and the tests intend to show how well the service works depending on if it is a native or a non-native Swedish speaker who is pronouncing. Each employee name has been spoken by the testers to see how many attempts was needed to match with the right employee.
The results are shown in Figure~\ref{fig:stt}. As it can be shown, Swedish speakers managed to match the correct employee in the first try in the majority of cases (average amount of tries equal to 37/33= 1.12). On the other hand, non-Swedish speakers had to try more than once with approximately 27\% of the employees (9 out of 33 IDs), leading to an average amount of tries equal to 50/33=1.51.
\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{average_time.png}
\caption{Average time for the employee function in the system.}
\label{fig:average_time}
\end{figure}
\section{Discussion}
The goal of this work is to develop a system that can control the main entrance of an office by using face recognition and speech-to-text.
It makes use of two Raspberry Pi with a camera, a microphone, and a speaker.
Face recognition and speech-to-text conversion are done with the
cloud-based solutions provided by Amazon Web Services \cite{AmazonRek} and Google Speech-to-Text \cite{GoogleSpeechText}, respectively,
The system is complemented with a touch screen display, and
a graphical user interface (GUI) which presents the detected classes
to the user.
Disturbances are caused when a person inside the office needs to interrupt the work to open the door for an employee, a guest, or to receive delivery.
Face recognition minimizes the disturbances since no tag-keys are needed, and employees can enter the office even if they forget it, without anyone having to open the door manually for them.
The guest handling using speech-to-text also minimizes the interference, since it only notifies the employee in question which the guest declares in the system. The employee will receive a private message on Slack with the notification, so others at the office will not be disturbed. The delivery function also has its advantage, as it only sends a delivery notice on Slack to those who are responsible for receiving the deliveries.
GDPR is not an issue \cite{Datainspektionen1,Datainspektionen2} since the camera or the microphone are not recording continuously, but only when the corresponding function is activated in the GUI. In addition, none of the captured images or audio is saved, and the data is deleted immediately after it is being used for the intended purpose.
Another security measure is that data is encrypted before it is sent to the cloud.
\begin{figure}[t!]
\centering
\includegraphics[width=0.48\textwidth]{stt.png}
\caption{Speech-to-text accuracy by swedish speakers (top) and non-Swedish speakers (bottom).}
\label{fig:stt}
\end{figure}
The face recognition system works with very good accuracy in the environment of this work, as shown in the tests.
Spoof recognition is currently not supported in Amazon Rekognition, and the service does not include encryption/decryption of images either.
Using a two steps authentication process (face recognition combined with a confirmation code) makes the system more secure and provides protection against spoofing.
To avoid brute force attacks, a new random code is created every time, and it is only valid for three attempts.
A downside is that it is an extra step that takes about 5 extra seconds on average, which we will seek to overcome by including spoofing detection mechanisms \cite{6990726}.
The average time spent by an employee until the door unlocks is of 20.3 seconds, which may be perceived as high, although it provides a secure and accurate method for access control.
The speech-to-text is set to Swedish and works well for Swedish-speaking persons.
For a multicultural company, it can be more difficult to use the speech-to-text service.
A problem can also occur if two or more employees has the same first and last name. The system will return the first occurrence in the database, which might not be the correct person the guest want to visit.
Our system is implemented in a relatively quiet environment, but
it can be hard to listen and translate in setups with
a lot of background noise or if several people speak at the same time.
Our prototype works well in an office environment, but if it is connected to a slower internet connection, the face recognition and speech-to-text will take longer.
Also, the system is cloud based, which means that it always requires internet connection to operate.
There are also advantages in using a GUI. It allows to easily show information for the person using the system, and also to instruct the person on how to enter the office. Based on the feedback obtained, it is more comfortable to visualize the information for both the employee and the visitor.
\section*{Acknowledgment}
Authors K. H.-D. and F. A.-F. thank the Swedish Research Council for funding their research.
Authors also acknowledge the CAISR program of the Swedish Knowledge Foundation.
\bibliographystyle{IEEEtran}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 410
|
module Memstat
module Proc
class Smaps < Base
FIELDS = %w[size rss pss shared_clean shared_dirty private_clean private_dirty swap]
attr_accessor *FIELDS
attr_accessor :lines, :items
def initialize(options = {})
super
@path ||= "/proc/#{@pid}/smaps"
FIELDS.each do |field|
send("#{field}=", 0)
end
run
end
def run
@lines = File.readlines(@path).map(&:strip)
@items = []
item = nil
@lines.each.with_index do |line, index|
case line
when /[0-9a-f]+:[0-9a-f]+\s+/
item = Item.new
@items << item
item.parse_first_line(line)
when /\w+:\s+/
item.parse_field_line(line)
else
raise Error.new("invalid format at line #{index + 1}: #{line}")
end
end
@items.each do |item|
FIELDS.each do |field|
send "#{field}=", (send(field) + item.send(field))
end
end
end
def print
@print ||= begin
lines = []
lines << "#{"Process:".ljust(20)} #{@pid || '[unspecified]'}"
lines << "#{"Command Line:".ljust(20)} #{command || '[unspecified]'}"
lines << "Memory Summary:"
FIELDS.each do |field|
lines << " #{field.ljust(20)} #{number_with_delimiter(send(field)/1024).rjust(12)} kB"
end
lines.join("\n")
end
end
def command
return unless pid?
commandline = File.read("/proc/#{@pid}/cmdline").split("\0")
if commandline.first =~ /java$/ then
loop { break if commandline.shift == "-jar" }
return "[java] #{commandline.shift}"
end
return commandline.join(' ')
end
def number_with_delimiter(n)
n.to_s.gsub(/(\d)(?=\d{3}+$)/, '\\1,')
end
#
# Memstat::Proc::Smaps::Item
#
class Item
attr_accessor *FIELDS
attr_reader :address_start
attr_reader :address_end
attr_reader :perms
attr_reader :offset
attr_reader :device_major
attr_reader :device_minor
attr_reader :inode
attr_reader :region
def initialize
FIELDS.each do |field|
send("#{field}=", 0)
end
end
def parse_first_line(line)
parts = line.strip.split
@address_start, @address_end = parts[0].split('-')
@perms = parts[1]
@offset = parts[2]
@device_major, @device_minor = parts[3].split(':')
@inode = parts[4]
@region = parts[5] || 'anonymous'
end
def parse_field_line(line)
parts = line.strip.split
field = parts[0].downcase.sub(':','')
return if field == 'vmflags'
value = Integer(parts[1]) * 1024
send("#{field}=", value) if respond_to? "#{field}="
end
end
end
end
end
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,224
|
Dartmouth Police are telling Fun 107 that there is a herd of cows on the loose in town.
Dartmouth Police and Dartmouth Animal Control are reporting that they are "monitoring a herd of rogue cows" in the area of Route 6.
They are not sure who owns the cows yet, but the herd was last seen near Joe's Used Cycles on Route 6.
Police are concerned that the cows could interfere with the morning commute on Route 6 and are urging drivers to use caution in that area.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,125
|
{"url":"http:\/\/clay6.com\/qa\/28827\/a-wire-of-length-ell-is-bent-to-form-a-circular-coil-of-some-turns-a-curren","text":"Browse Questions\n\n# A wire of length $\\ell$ is bent to form a circular coil of some turns. A current $i$ flows through the coil and is placed in a uniform magnetic field $B$. The maximum torque on the coil can be\n\n$\\begin {array} {1 1} (a)\\;\\large\\frac{iB\\ell^2}{4 \\pi} & \\quad (b)\\;\\large\\frac{iB\\ell^2}{2 \\pi} \\\\ (c)\\;\\large\\frac{iB\\ell^2}{ \\pi} & \\quad (d)\\;\\large\\frac{2iB\\ell^2}{4 \\pi} \\end {array}$\n\nLet $n$ be the number of turns and $R$ be the radius of the coil\n$\\ell = 2\\pi Rn$\n$R = \\large\\frac{\\ell}{2 \\pi n}$\n$M = niA = \\large\\frac{i \\ell^ 2} {4 \\pi n}$\n$M$ is maximum when $n = 1$ so is the torque.\nTorque can be found by calculating its cross product with $B$\nAns : (a)","date":"2017-01-18 14:10:46","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9183989763259888, \"perplexity\": 207.59826242447923}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-04\/segments\/1484560280292.50\/warc\/CC-MAIN-20170116095120-00381-ip-10-171-10-70.ec2.internal.warc.gz\"}"}
| null | null |
The Duchy of Kalisz was a feudal district duchy in the Greater Poland, centered on the Kalisz Region. Its capital was Kalisz. The state was established in 1177, in the partition of the Duchy of Greater Poland, after the rebellion against Mieszko III. Duke Casimir II the Just of the Piast dynasty become its first ruler. It existed until 1279, when, it got united with duchies of Gniezno and Poznań, under the rule of Przemysł II, forming the Duchy of Greater Poland. It remained a fiefdom within the Duchy of Poland, until 1227, and after that, it become an independent state.
Citations
Notes
References
Bibliography
Andrzej Wędzki, Kalisz w państwie wczesnopiastowskim i w okresie rozbicia dzielnicowego. In: Władysław Rusiński (redactor), Dzieje Kalisza. Poznań: Wydawnictwo Poznańskie, 1977.
Józef Dobosz, Kazimierz II Sprawiedliwy
Bronisław Nowacki, Przemysł II
Jerzy Wyrozumski, Historia Polski do roku 1505
Former countries in Europe
Former monarchies of Europe
Duchies of Poland
Fiefdoms of Poland
History of Poland during the Piast dynasty
Kalisz
History of Greater Poland
12th-century establishments in Poland
13th-century disestablishments in Poland
States and territories established in 1177
States and territories disestablished in 1279
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 5,748
|
December 24, 1998 Columns & Opinion | Savage Love
By Dan Savage @fakedansavage
Hey, Faggot:
You ran a letter once from a woman who wanted to do her reluctant boyfriend with a dildo. I'm a man and I've always enjoyed this activity, but it strikes most women I date as rather odd. Aside from placing a personal ad, do you have any suggestions for meeting a compatible woman?
--Trying to Get My Role Reversed
Hey, TTGMRR:
It's understandable girlfriends find your desire for anal penetration odd. Straight boys who wanna take it up the butt are odd, in the sense that they're unusual, so you are odd. Finding herself in bed with a man who wants to be fucked is a unique experience for most straight women--and not all women are thrilled by it. If you're unwilling to go the personals route, I suggest you brace yourself for a lifetime of shocked reactions, tight-lipped refusals, and the occasional makes-it-all-worthwhile enthusiastic assent. But if you can learn to love the personals, they're a good way to meet like-minded pervs.
Don't take my word for it. Carol Queen, San Francisco-based sex guru and all-around swell gal, recently released a video on just this subject. Bend Over Boyfriend stars Carol and her oft-bent boyfriend, Robert, and together they guide breeder couples through the joys and triumphs of girl-on-boy butt fucking. To avoid the awkwardness of revealing perverse desires to a new partner, Carol suggests you advertise. "Personal ads will help him weed out the 'ewwwww!' types right away."
If you just can't do personal ads, "bi gals may be more open to such play than straight-and-narrow women--as might kinky women of any orientation, so he might keep an eye open for a BDSM support group," says Carol. "He'll obviously encounter women who will be into all kinds of play at an BDSM group, so he'll have to be pretty up-front about what he does (and doesn't) desire; fortunately, that environment supports and even demands that kind of disclosure.
"If he has a steady source of girlfriends who just aren't sure about this activity and need some info, he can show them Bend Over Boyfriend." Carol also suggests a field trip to the dildo/harness aisles at Good Vibrations in San Francisco, Toys in Babeland in Seattle or New York, Come As You Are in Toronto, or the sex-positive, women-owned sex shoppe in your area. "If they visit on a busy Saturday afternoon, chances are good they won't be the only boy-girl couple in that part of the store.
If the radical right knew how much of this is going on in America, their heads would blow up."
Bend Over Boyfriend is available from Good Vibrations for $34.95 plus shipping and handling. Call 800-BUY-VIBE to order yourself a copy. Do I even need to mention that Bend Over Boyfriend makes a thoughtful holiday gift?
I am a 55-year-old man. I was married for 6 years 20 years ago. Since then, I've been a virgin. When I was married I had sex every night. For some reason, looking for love and sex is like looking for a needle in a haystack. I am the king of masturbation and I hate it. I want to love and be loved. What can I do? --RC
Hey, RC:
What can I advise you to do that you probably haven't already tried? In the 20 years you've been looking for love and sex with no success, I can only assume you've done all the obvious things: perused the personal ads, cruised singles bars, volunteered for good causes, assessed your personal hygiene, asked friends to hook you up, attempted to pay for it.
If you've tried all that and none of it has worked, I can only conclude that something is very seriously wrong, something that can't be solved or salved in this format. Or ever, really.
I'm sorry if that sounds harsh, but there are men and women who for whatever reason--bad attitudes, wayward DNA, cruel circumstance--are destined to be alone. Maybe you're one of them. After 20 years of looking, perhaps the best advice I can give is the advice we advice professionals are never supposed to give: stop looking. Accept that it ain't gonna happen for you, that she isn't out there, and that no matter how long and hard you look, you ain't gonna find her. Reconcile yourself to being alone.
It's possible to be happy and single, but you'll have to disabuse yourself of a couple of myths in order to do so. These myths? "There's someone out there for everyone." "If you keep looking, someday you'll find him/her." How many people have to die alone and miserable before we stop jamming that someone-for-everyone crap down people's throats? The cold hard fact is that there isn't someone for everyone. Endlessly repeating that there is causes people for whom there isn't to waste their whole lives looking for someone who isn't out there.
To my mind, the stock advice given in response to questions like yours--keep looking, don't give up, she's out there somewhere--is far crueler than anything I'm telling you. False hopes dashed again and again can drive a person mad; cold hard facts, accepted and adjusted to, can relieve a person of much misery. So my advice for you is this: after 20 years, it's time to pack it in.
I was making out with my boyfriend--he's a violent kisser--and he was pulling on my tongue. That little membrane that connects the tongue to the bottom of the mouth snapped! It started to bleed all over the place, and it's pretty painful. My tongue is currently detached! I live with my parents, and they're in charge of all my medical business, as I'm a minor, and it would be somewhat humiliating for me if I presented them with my little problem. So I guess my question is, should I see a doctor, and have you ever heard of this before? --Speechless
Hey, S:
No, I haven't heard of this before and, yes, you should see a doctor. When you're bleeding all over the place, you're in pain, and your tongue is detached, seeing a doctor is probably a better idea than sending a letter to a sex-advice columnist and waiting weeks or months for a response. But since you wrote, and since we were going out to dinner anyway, I shared your letter with Dr. Barak Gaster, Savage Love's chief medical correspondent.
"The 'membrane' that Speechless is referring to is actually a thin layer of tissue called the frenulum that is so flimsy the tongue can do just fine without it. She's probably torn this tissue, and the resulting swelling may make it difficult for her to talk. But a tongue can't really 'detach,' since it's a bundle of muscles that extends far down the back of the throat, attaching to small bones near the Adam's apple. The only reason to see a doctor after an injury like this is if the bleeding, swelling, and pain continue to worsen, or if they don't start to get better within a day or two."
Send questions to Savage Love, Chicago Reader, 11 E. Illinois, Chicago 60611.
More Savage Love »
John Kass washing his hands of responsibility for last week's riot was a bridge too far
When to use 'come' and 'cum' in a sentence
The fight for the future
More by Dan Savage
My teenager is killing my freaky sex life
It's not normal for lesbian 'drama' to end in 911 calls
John Kass washing his hands of responsibility for last week's riot was a bridge too far 24
It's time to deplatform the right-wing Tribune columnist.
By John Greenfield | 01.11.21
A copy editor brings us to climax, linguistically speaking.
By Dan Savage | 01.13.21
The fight for the future 6
Congresswoman Mary Miller's apology is almost as bad as her original "Hitler was right" remark.
I pegged my boyfriend and now he wants to be 'the girl' 9
Kinky sex can be wonderful, but it won't fix your relationship.
"Can you come out your butt?" 18
And other questions from tomorrow's leaders
The gremlin and the EdD
She earned the title—still he was dissing her! Would he do the same to, say, Dr. Kissinger?
By Deanna Isaacs | 12.18.20
We should rename Chicago's shoreline highway DuSable Drive, but we don't have to drop LSD 57
Renaming the landmark for Chicago's Black founder would be a game-changer, but we can still also refer to Lake Shore Drive by its iconic appellation.
Democrats and ruling by fear 25
When politicians sell out to win, we all lose.
By Leonard C. Goodman | 12.18.20
CPS won't say why it suspended activist teacher Sarah Chambers 25
But Chambers has been an outspoken opponent of cuts to special ed funding.
Gay, middle-aged, and lonely as hell 14
Dan Savage advises a trio of late bloomers.
|
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| 1,112
|
{"url":"https:\/\/plainmath.net\/29182\/determine-whether-the-given-problem-is-an-equation-or-an-expression-if-it","text":"# Determine whether the given problem is an equation or an expression. If it is an\n\nDetermine whether the given problem is an equation or an expression. If it is an equation, then solve. If it is an expression, then simplify. $\\frac{2}{5}\\left(x+4\\right)+\\frac{1}{3}\\left(x-2\\right)$\n\nYou can still ask an expert for help\n\n\u2022 Questions are typically answered in as fast as 30 minutes\n\nSolve your problem for the price of one coffee\n\n\u2022 Math expert for every subject\n\u2022 Pay only if we can solve it\n\nrogreenhoxa8\nStep 1\nGiven:\n$\\frac{2}{5}\\left(x+4\\right)+\\frac{1}{3}\\left(x-2\\right)$\nStep 2\n$\\frac{2}{5}\\left(x+4\\right)+\\frac{1}{3}\\left(x-2\\right)$\ngiven is an algebraic expression\n$\\frac{2}{5}\\left(x+4\\right)+\\frac{1}{3}\\left(x-2\\right)$\n$=\\frac{2}{5}x+\\frac{8}{5}+\\frac{x}{3}-\\frac{2}{3}$\n$=\\frac{2x}{5}+\\frac{x}{3}+\\frac{8}{5}-\\frac{2}{3}$\n$=\\frac{6x+5x}{15}+\\frac{24-10}{15}$\n$=\\frac{11x+14}{15}$\n$=\\frac{11x}{15}+\\frac{14}{15}$","date":"2022-08-12 11:51:58","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 23, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8343248963356018, \"perplexity\": 1052.2063120341777}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-33\/segments\/1659882571692.3\/warc\/CC-MAIN-20220812105810-20220812135810-00139.warc.gz\"}"}
| null | null |
geology (Page 2)
Mayan Underworld Unveils a 10,000-Year-Old Story–Including a Violent Catastrophic Event
An archaeological expedition aimed at plumbing the secrets of Chichen Itza's underworld of the gods has uncovered the well-preserved remains of ancient humans and extinct animals that date back to the last ice age–far more than what the expedition's members bargained for when they set out to map and explore the sacred network of Mayan caves. The nature of some of the fossils found there also hinted at the occurrence of a "catastrophic event" that embedded some of the bones in the walls of the cave.
New Study Supports a Major Cometary Impact at the End of the Last Ice Age
The evidence pointing toward a major cometary impact that heralded the closure of the last ice age 13,000 years ago is steadily growing, with a new study from the University of Kansas offering more data that supports what is known as the Younger Dryas impact hypothesis.
"The hypothesis is that a large comet fragmented and the chunks impacted the Earth, causing this disaster," explains University of Kansas Emeritus Professor of Physics & Astronomy Adrian Melott. "A number of different chemical signatures–carbon dioxide, nitrate, ammonia and others–all seem to indicate that an astonishing 10 percent of the Earth's land surface, or about 10 million square kilometers [3,9 million square miles], was consumed by fires."
The Phenomenon of Mysterious Booms Continue Worldwide
The phenomenon of mysterious booms is continuing around the world, with the sound of unexplained explosions being reported from locales as diverse as Michigan, Lapland, St Ives, Swansea and Yorkshire.
Booms reported by residents of six counties in Alabama have been investigated by police, NASA, and the Birmingham Alabama National Weather Service, with the latter tweeting on November 14:
"Re: loud boom heard: we do not see anything indicating large fire/smoke on radar or satellite; nothing on USGS indicating an earthquake. We don't have an answer, and can only hypothesize with you. 1) sonic boom from aircraft; 2) meteorite w/ current Leonid shower?"
Australia Inhabited for at Least 65,000 Years, 20,000 Longer than Originally Assumed
On: July 21, 2017
The revised dating and uncovering of new artifacts from an archaeological site in Australia's Arnhem Land has prompted archaeologists to revise theories as to when the ancestors of present-day Aboriginies first settled in what we now call Australia, pushing that date back by 20,000 years to a point in time 65,000 years ago. This revised timeline also implies that modern humans may have begun the colonization of Asia much earlier than previously assumed.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 1,253
|
Q: Setting a cron job for a Meteor Method I have this piece of code:
Meteor.methods({
GetTickerInfo: function(){
Future = Npm.require('fibers/future');
var myFuture = new Future();
kraken.api('Ticker', {"pair": 'ETHXBT'}, function(error, data) {
if(error) {
console.log(error);
}
else {
console.log(data.result);
console.log(data.result.XETHXXBT.a);
myFuture.return(data.result);
}
});
console.log("EHEHEHEHEHHEEH");
console.log(myFuture.wait());
return myFuture.wait();
}
});
What it does it calls an API, gets some data back and when it's done it returns the data to the client so I can visualise in the graph. For now its a MANUAL click button on the client side which calls the method, does the job, and returns the data.
I would like to schedule a cron to do that. So every 5 sec make a API call and return the data back to the client (because there is where I visualise it). All the cron jobs are working with specific functions but I can't access the this function GetTickerInfo because it is defined and in the scope of Meteor.methods.
How can I call it be a cron job, but also leave the occasional Meteor Call from the client side when I want to manualy refresh in the given moment?
Can anyone show how would they implement this with for e.g. CRON package: percolatestudio/meteor-synced-cron
A: You have to be outside of the methods scope and I would personally do:
SyncedCron.add({
name: 'GetTickerInfo cron',
schedule: function(parser) {
return parser.text('every 5 seconds');
},
job: function() {
Meteor.call('GetTickerInfo');
}
});
SyncedCron.start()
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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| 4,534
|
<?php
namespace Drupal\jsonapi\EventSubscriber;
use Drupal\Core\Cache\CacheableMetadata;
use Drupal\Core\Render\RenderCacheInterface;
use Drupal\jsonapi\JsonApiResource\ResourceObject;
use Symfony\Component\EventDispatcher\EventSubscriberInterface;
use Symfony\Component\HttpKernel\Event\TerminateEvent;
use Symfony\Component\HttpKernel\KernelEvents;
/**
* Caches entity normalizations after the response has been sent.
*
* @internal
* @see \Drupal\jsonapi\Normalizer\ResourceObjectNormalizer::getNormalization()
* @todo Refactor once https://www.drupal.org/node/2551419 lands.
*/
class ResourceObjectNormalizationCacher implements EventSubscriberInterface {
/**
* Key for the base subset.
*
* The base subset contains the parts of the normalization that are always
* present. The presence or absence of these are not affected by the requested
* sparse field sets. This typically includes the resource type name, and the
* resource ID.
*/
const RESOURCE_CACHE_SUBSET_BASE = 'base';
/**
* Key for the fields subset.
*
* The fields subset contains the parts of the normalization that can appear
* in a normalization based on the selected field set. This subset is
* incrementally built across different requests for the same resource object.
* A given field is normalized and put into the cache whenever there is a
* cache miss for that field.
*/
const RESOURCE_CACHE_SUBSET_FIELDS = 'fields';
/**
* The render cache.
*
* @var \Drupal\Core\Render\RenderCacheInterface
*/
protected $renderCache;
/**
* The things to cache after the response has been sent.
*
* @var array
*/
protected $toCache = [];
/**
* Sets the render cache service.
*
* @param \Drupal\Core\Render\RenderCacheInterface $render_cache
* The render cache.
*/
public function setRenderCache(RenderCacheInterface $render_cache) {
$this->renderCache = $render_cache;
}
/**
* Reads an entity normalization from cache.
*
* The returned normalization may only be a partial normalization because it
* was previously normalized with a sparse fieldset.
*
* @param \Drupal\jsonapi\JsonApiResource\ResourceObject $object
* The resource object for which to generate a cache item.
*
* @return array|false
* The cached normalization parts, or FALSE if not yet cached.
*
* @see \Drupal\dynamic_page_cache\EventSubscriber\DynamicPageCacheSubscriber::renderArrayToResponse()
*/
public function get(ResourceObject $object) {
$cached = $this->renderCache->get(static::generateLookupRenderArray($object));
return $cached ? $cached['#data'] : FALSE;
}
/**
* Adds a normalization to be cached after the response has been sent.
*
* @param \Drupal\jsonapi\JsonApiResource\ResourceObject $object
* The resource object for which to generate a cache item.
* @param array $normalization_parts
* The normalization parts to cache.
*/
public function saveOnTerminate(ResourceObject $object, array $normalization_parts) {
assert(
array_keys($normalization_parts) === [
static::RESOURCE_CACHE_SUBSET_BASE,
static::RESOURCE_CACHE_SUBSET_FIELDS,
]
);
$resource_type = $object->getResourceType();
$key = $resource_type->getTypeName() . ':' . $object->getId();
$this->toCache[$key] = [$object, $normalization_parts];
}
/**
* Writes normalizations of entities to cache, if any were created.
*
* @param \Symfony\Component\HttpKernel\Event\TerminateEvent $event
* The Event to process.
*/
public function onTerminate(TerminateEvent $event) {
foreach ($this->toCache as $value) {
[$object, $normalization_parts] = $value;
$this->set($object, $normalization_parts);
}
}
/**
* Writes a normalization to cache.
*
* @param \Drupal\jsonapi\JsonApiResource\ResourceObject $object
* The resource object for which to generate a cache item.
* @param array $normalization_parts
* The normalization parts to cache.
*
* @see \Drupal\dynamic_page_cache\EventSubscriber\DynamicPageCacheSubscriber::responseToRenderArray()
* @todo Refactor/remove once https://www.drupal.org/node/2551419 lands.
*/
protected function set(ResourceObject $object, array $normalization_parts) {
$base = static::generateLookupRenderArray($object);
$data_as_render_array = $base + [
// The data we actually care about.
'#data' => $normalization_parts,
// Tell RenderCache to cache the #data property: the data we actually care
// about.
'#cache_properties' => ['#data'],
// These exist only to fulfill the requirements of the RenderCache, which
// is designed to work with render arrays only. We don't care about these.
'#markup' => '',
'#attached' => '',
];
// Merge the entity's cacheability metadata with that of the normalization
// parts, so that RenderCache can take care of cache redirects for us.
CacheableMetadata::createFromObject($object)
->merge(static::mergeCacheableDependencies($normalization_parts[static::RESOURCE_CACHE_SUBSET_BASE]))
->merge(static::mergeCacheableDependencies($normalization_parts[static::RESOURCE_CACHE_SUBSET_FIELDS]))
->applyTo($data_as_render_array);
$this->renderCache->set($data_as_render_array, $base);
}
/**
* Generates a lookup render array for a normalization.
*
* @param \Drupal\jsonapi\JsonApiResource\ResourceObject $object
* The resource object for which to generate a cache item.
*
* @return array
* A render array for use with the RenderCache service.
*
* @see \Drupal\dynamic_page_cache\EventSubscriber\DynamicPageCacheSubscriber::$dynamicPageCacheRedirectRenderArray
*/
protected static function generateLookupRenderArray(ResourceObject $object) {
return [
'#cache' => [
'keys' => [$object->getResourceType()->getTypeName(), $object->getId(), $object->getLanguage()->getId()],
'bin' => 'jsonapi_normalizations',
],
];
}
/**
* {@inheritdoc}
*/
public static function getSubscribedEvents() {
$events[KernelEvents::TERMINATE][] = ['onTerminate'];
return $events;
}
/**
* Determines the joint cacheability of all provided dependencies.
*
* @param \Drupal\Core\Cache\CacheableDependencyInterface|object[] $dependencies
* The dependencies.
*
* @return \Drupal\Core\Cache\CacheableMetadata
* The cacheability of all dependencies.
*
* @see \Drupal\Core\Cache\RefinableCacheableDependencyInterface::addCacheableDependency()
*/
protected static function mergeCacheableDependencies(array $dependencies) {
$merged_cacheability = new CacheableMetadata();
array_walk($dependencies, function ($dependency) use ($merged_cacheability) {
$merged_cacheability->addCacheableDependency($dependency);
});
return $merged_cacheability;
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 274
|
Another highlight of our recent Napa Valley Wine Experience 2013 was our tasting tour at Hall Vineyards Rutherford Estate. We first discovered Hall and their Napa Valley wines during our Napa Valley Wine Experience back in 2003. We've collected their signature label Napa Cabernet Sauvignon since. Along the way we picked up numerous vintage selections of their vineyard designated Sacrashe Vineyard Cabernet Sauvignon from the namesake vineyard.
The Hall Sacrashe Vineyard, pictured left, sits on the lower tier of the Vaca Mountain Range on the eastern Napa slope above the tony Auberge du Soleil Resort, Spa and Restaurant.
Our gallery of pictures of our Hall Rutherford visit are on our Napa Wine Experience 2013 - Hall page on www.unwindWine.com.
The Sacrashe vineyard designated label evolved to the Kathryn Hall Napa Valley Cabernet Sauvignon label with the 2006 vintage featured in our tasting journal back in March 18, 2010. This flagship label received numerous blockbuster ratings culminating in the 2009 release which was acclaimed the #2 wine in the world in Wine Spectator's annual Top 100. This was one of the highlights of the Wine Spectator's Grand Tour, Chicago, 2011.
We've had fun over the years giving the festive red labeled Hall wines to friends and family at Christmas, artfully decorated with a whimsical play on words, 'deck the Halls with boughs of holly..."
Hall has continued their development in Napa Valley with the opening of their Rutherford winery and tasting room located at the Sacrashe Vineyard. This has gone hand in hand with the proliferation of the number of labels on offer as well. They explain that the Rutherford crush facility is built to accommodate small lot production of crafted wines to complement the larger volume production capabilities of the Valley facility in St Helena.
The Rutherford winery features a stylish reception center with an outdoor patio and several interior tasting rooms with views overlooking the valley below. The center sits atop the winery production facility which connects to a network of caves dug into the mountain side below the famed Sacrashe Vineyard. The 14,000 square feet of caves were designed and built by hand by Friedrich Gruber of Gutenstein Austria. They are finished with handmade Austrian brick recovered from sites in and around Vienna.
Deep in the caves is the highlight of the facility, a magnificent cavernous tasting room with a massive table that can accommodate thirty. Over the table is a huge elegant crystal chandelier. The table offers picturesque views of the caves. Overlooking the room from the end wall is another large metal sculpture depicting the vines over a large table showcasing the selection of labels. To the rear of the room opposite the cave walk is a wine cellar housing some of the estates library wines.
Another remarkable feature of the cave facility is the displays of artwork from the Halls' art collection. The numerous works of metal and glass sculptures are displayed in elegant nooks built into the walls of the caves to showcase the collection that also includes a historic vintage large format bottle decanting contraption with a hand crank (shown below).
While not necessarily a collectable, we consider Sauvignon Blanc a mainstay of a comprehensive wine tasting with its clean crisp clear true representation of the white wine fruits. Hall is one of our favorites that we keep in our cellar for such occasions along with Duckhorn (featured in magnum at this years Open That Bottle Night 2013), David Arthur, and perhaps our favorite, Cliff Lede Napa Valley Sauvignon Blanc.
HALL "Kathryn Hall" Cabernet Sauvignon has become the flagship label replacing the historic vineyard designated Sacrashe label. It represents the most select fruit from the estate's vineyard blocks combined with select mountain fruit from vineyard partners. A blend of varietals results in a Bordeaux cuvée showcasing the dominant Sacrashe vineyard fruits.
Nearly opaque purple-black color, medium to full body, smooth and nicely polished, moderately complex with aromas of violets and cocoa and flavors of black berry and black raspberry and dark cherries,turning to hints of licorice, mocha and a subtle earthiness with moderate tannins on the finish.
One of the range of new labels extending the selection of the Hall brand, "Ellie's" is a tribute to Craig Hall's mother; an artist and teacher; who enlisted in World War II as soon as the U.S. Navy announced it would accept women. Ellie inspired Craig's interest in art collecting. In remembrance and further tribute to her, the label displays a pencil drawing by Ellie of an owl watching over a family vineyard.
Dark and full bodied, big fruit forward wine with layers complex layers of black fruits accented by tones of spicy clove and cassis with nicely integrated supple firm but nicely polished lingering tannins.
Interesting blend named after the town of Darwin, Australia, where Kathryn and Craig's small plane was required to make an emergency landing. Once safely on the ground, the Halls toasted their well-being with an Aussie Shiraz/Cabernet blend and pledged to craft a similar wine in the Napa Valley to commemorate the occasion. .
The 2009 HALL "Darwin" is an interesting blend of Syrah accented by Cabernet Sauvignon from the Hall Napa River Ranch estate. Dark, full bodied, forward black fruits accented by a layer of anise with tones of creosote and hints of tea and cedar.
More of our gallery of pictures of our Hall Rutherford visit are on our Napa Wine Experience 2013 - Hall page on www.unwindWine.com.
Oh my those cottages are adorable. I want to go now, looks fabulous. Cant wait to hear about the wine.
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"redpajama_set_name": "RedPajamaC4"
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I am no preacher, but, am a prime sinner and living in this world in HIS mercy. For a sinner like me too, when I read "The Holy Bible" KJV, the following verses were given in the context of "outward".
a thousand cubits round about.
had the oversight of the outward business of the house of God.
Est 6:4 And the king said, Who is in the court?
on the gallows that he had prepared for him.
thirty chambers were upon the pavement.
and the going up to it had eight steps.
Mat 23:27 Woe unto you, scribes and Pharisees, hypocrites!
2Co 10:7 Do ye look on things after the outward appearance?
that, as he is Christ's, even so are we Christ's.
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{
"redpajama_set_name": "RedPajamaC4"
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/*
ID: brandbe1
LANG: JAVA
TASK: gift1
*/
import java.util.*;
import java.io.*;
class gift1{
public static void main(String[] args) {
BufferedReader br = new BufferedReader(new FileReader(""));
PrintWriter pw = new PrintWriter(new BufferedWriter(new FileWriter("")));
StringTokenizer st = new StringTokenizer(br.readLine());
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
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Greta è un film del 2018 diretto da Neil Jordan.
Trama
Frances McCullen è una giovane cameriera che vive a New York City con la sua amica Erica. Frances è ancora sconvolta dalla morte di sua madre avvenuta un anno prima e mantiene una relazione tesa con suo padre Chris, un maniaco del lavoro. Una mattina Frances trova una borsetta su un treno della metropolitana; i documenti all'interno indicano che questa appartiene ad una certa Greta Hideg. Frances la va a trovare per restituirle la borsa e lei la invita a prendere una tazza di caffè.
Greta si presenta come una vedova francese, con una figlia che studia a Parigi. Frances inizia a trascorrere del tempo con Greta per tenerle compagnia, nonostante le obiezioni di Erica sul fatto che la loro amicizia è innaturale. Una sera, mentre cena da Greta, Frances trova un armadio pieno di borsette identiche a quella che aveva trovato sul treno. Attaccati alle borse ci sono dei post-it con nomi e numeri di telefono, compreso quello di Frances.
Frances, turbata dalla sua scoperta, decide di tagliare i legami con Greta. Questa, pensando morbosamente di potersi sostituire alla defunta madre della giovane, inizia a pedinarla, la chiama più volte al telefono e si presenta anche al ristorante dove lavora la ragazza, provocando una gran concitazione e finendo in ospedale; dimessa, Greta, la cui alterazione mentale è sempre più manifesta, prende ad inseguire anche Erica; a questo punto le due ragazze provano a richiedere un ordine restrittivo nei confronti della donna, ma viene loro detto che potrebbero volerci mesi per ottenerlo. Frances, in seguito, scopre quanto siano profonde le bugie di Greta: non solo è ungherese e non francese, ma sua figlia si è addirittura suicidata anni fa a causa del comportamento sadico della madre.
Frances è combattuta tra tornare a casa con suo padre o andare in vacanza con Erica. Quest'ultima le suggerisce di dire a Greta che se ne andrà via, ma in realtà dovrebbe restare a casa. La mattina dopo, Frances viene rapita dalla psicopatica, che la chiude in un baule di legno in una stanza segreta, poi usa il suo cellulare per mandare messaggi separati a Erica e al padre di Frances, facendo credere a quest'ultimo che la figlia sia in viaggio con la sua coinquilina. Allo stesso modo inganna anche Erica che pensa che l'amica sia da suo padre. Nel mentre Frances, ormai prigioniera della sua persecutrice, trova vestiti e documenti di altre giovani donne che Greta ha rapito in precedenza.
Erica e Chris alla fine si incontrano e scoprono che Frances non è con nessuno dei due. Con il passare del tempo Greta costringe Frances a imparare l'ungherese e a suonare il piano. Durante una lezione di cucina, Frances taglia un dito all'aguzzina e la fa perdere i sensi, cercando quindi di scappare: tutte le porte e le finestre della casa-prigione sono però sigillate. Allora corre nel seminterrato per cercare un'altra uscita e qui trova una delle precedenti vittime della folle, Samantha, ormai moribonda. Greta si avvicina di soppiatto a Frances e le avvolge una borsa intorno alla testa fino a farla svenire, riducendola di nuovo in suo potere.
Chris, sconvolto perché ha capito il pericolo che corre sua figlia, assume Cody, un investigatore privato, per trovare Frances e indagare su Greta. Cody scopre che quest'ultima era un'infermiera, licenziata per aver abusato di anestetici. Il detective va quindi ad incontrarla a casa sua. Frances, imbavagliata e legata, cerca di attirare la sua attenzione scuotendo il letto, ma Greta nasconde il rumore con la musica. Quando la psicopatica è fuori dalla stanza Cody scopre il vano nascosto dietro il pianoforte, dove è segregata la ragazza. Greta appare all'improvviso e gli affonda una siringa nel collo; l'uomo estrae la pistola e, mentre sta perdendo conoscenza a causa dell'iniezione, spara verso la donna, mancando il bersaglio. Greta si impossessa dell'arma e uccide l'investigatore.
Passa una quantità di tempo indeterminata e Greta, in cerca di una nuova preda, lascia un'altra borsetta in metropolitana. Un'altra ragazza si presenta a casa sua con la borsetta e lei la invita ad entrare preparando del caffè. Greta beve dalla sua tazza e inizia a sentirsi mancare. La ragazza si toglie la parrucca e si rivela essere Erica, che ha messo della droga nella bevanda. Erica dileggia con disprezzo l'ex infermiera dicendole che ha cercato a lungo la borsetta in metropolitana. Greta quindi sviene ed Erica trova Frances. Mentre cercano di scappare, la psicopatica, riprendendo conoscenza, emerge dall'ombra e afferra Frances prima di mancare di nuovo.
Erica e Frances rinchiudono Greta, priva di sensi, nella cassa dei giocattoli e usano una piccola riproduzione in metallo della Torre Eiffel per serrare il baule. Le ragazze lasciano la casa della pericolosa squilibrata per recarsi dalla polizia. Greta poco dopo prende a sbattere il coperchio della cassapanca e l'improvvisato chiavistello inizia a spostarsi.
Produzione
Le riprese del film sono iniziate nell'ottobre 2017 a Dublino e sono proseguite a Toronto e New York.
Promozione
Il trailer del film viene diffuso il 20 dicembre 2018.
Distribuzione
La pellicola è stata presentata al Toronto International Film Festival il 6 settembre 2018 e distribuita nelle sale cinematografiche statunitensi a partire dal 1º marzo 2019. In Italia la pellicola viene distribuita direttamente in home video dal 20 gennaio 2022.
Accoglienza
Critica
Sull'aggregatore Rotten Tomatoes il film riceve il 60% delle recensioni professionali positive con un voto medio di 5,66 su 10 basato su 257 critiche, mentre su Metacritic ottiene un punteggio di 54 su 100 basato su 42 critiche.
Note
Altri progetti
Collegamenti esterni
Film ambientati a New York
Film diretti da Neil Jordan
Film girati a New York
Film girati in Irlanda
Film girati in Canada
Film thriller psicologici
Film gialli
Film thriller drammatici
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"redpajama_set_name": "RedPajamaWikipedia"
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\section{Introduction}
\qquad In the framework of general relativity, weak fields and gravitational waves have been studied by many authors decades ago e.g.\cite{J.Weber},\cite{R.Wagoner},\cite{K.Thorne and S. Kovacs}. One of the fundamental problems in general relativity is the study of gravitational waves. The existence of gravitational waves in linear versions of the theory was already known in the early days of general relativity. First Einstein considered in a Minkowski spacetime with a metric $n_{\mu\nu}$ a small perturbation $\epsilon_{\mu\nu}$ such that the induced field $\alpha_{\mu\nu}=n_{\mu\nu}+\epsilon_{\mu\nu}$ with $\vert\epsilon_{\mu\nu}\vert\ll 1$ obeys by perturbation in linearized equations of motion. The linearized theory of gravity is an important theory because it can be utilized as a foundation for \textquotedblleft deriving\textquotedblright\thickspace General Relativity. By using the linearized field theory of gravitation some observable phenomena of our solar system and the universe can be detected
\cite{Farese},\cite{B.Sathyaprakash and B.Shutz} The weak field limit at a Finsler space-time has been studied in \cite{Calc},\cite{Stavrinos P.} and in the tangent bundle of a Finsler space by \cite{Pan.C.Stavrinos},\cite{V.Balan and P.C.Stavrinos},
\cite{N.Brinzei and S.Siparov}.
In our study we consider a pseudo- Finsler - Randers space-time of metric function \cite{G.Randers}
\begin{equation}
{\mathcal F}(x,y)=\sqrt{\alpha_{ij}(x)y^i y^j}+kA_i(x)y^i
\end{equation}
in the weak field limit where $\alpha_{ij}$ represents pseudo-Riemannian metric, $y^i=\frac{dx^i}{d\lambda}$(in applications $y^i$ represent velocity), $\lambda$ a parameter along the curve and $k$ a constant, $A_i$ represents the electromagnetic potential which is connected with the electromagnetic field by $F_{ij}=\frac{\partial A_j}{\partial x^i}-\frac{\partial A_i}{\partial x^j}$.
A Finsler-Randers space (FR) constitutes an important category of Finsler spaces from mathematical and physical perspective e.g.\cite{Bao D S-S Chern and Z Schen},\cite{S-K-S},\cite{Chang},\cite{Antonelli}.
In a FR space the condition of symmetry for the fundamental function ${\mathcal F}(x,y)$ is not satisfied ${\mathcal F}(x,y) \neq {\mathcal F}(x,-y)$. Causal considerations in pseudo-Finsler space-time that include only symmetries of fundamental functions \cite{Pfeifer C. and Wohlfarth M.} are very restricted and exclude by studying pseudo-FR spaces.\cite{Bao D S-S Chern and Z Schen},\cite{Antonelli},\cite{A.Bejancu and H.R. Farran}.
FR space can play a significant role in the theory of weak field and gravitational waves since ''a gravito - electromagnetic field'' is intrinsically included in its metric. Einstein's General Relativity shows indeed that gravito - magnetic field may be associated with mass currents \cite{L.Iorio C.Corda},\cite{Martens}. As well a gravito - magnetic force was postulated as an explanation for the anomalous precession of Mercury's perihelion \cite{Mashhoon}. In addition, previous works \cite{Bel1}, \cite{Bel2}, \cite{Matte} ,\cite{Penrose}, \cite{Pirani} showed how the electric and magnetic parts of the curvature tensors were related to the electric and magnetic parts of the gravitational field as well as with gravitatational waves.\cite{C.Tsagas}\\
This paper is organized as follows: we deal with the linearized metric form of a pseudo FR spacetime in relation to the deviation of geodesics and Raychaudhuri equation. In this approach we extend a previous consideration which was given in \cite{Calc}. By using Cartan and Berwald -like connections we get \textquotedblleft gravito - electromagnetic curvatures\textquotedblright\thickspace for this space. We derive the equations of motion and the Raychaudhuri equation in the framework of a tangent bundle of a $n$-dimensional manifold $M$. We also give the linearized connection coefficients as well as establishing the Lorentz equation of the weak field. In addition we attribute some physical interpretations in the geometrical concepts under consideration.
Moreover in the framework of the weak gravitational field of a generalized Finsler spacetime some applications for gravitational waves are given.
\section{Linearized field theory of Randers space-time}\qquad The behavior of particles in a gravitational and electromagnetic field is expected to indicate that the physical geometry in the direction of a geometrical unification is the Finsler geometry.
In a Finsler space the metric function ${\mathcal F}(x,y)$ can be considered as a potential function since the metric tensor (gravitational potential)
\begin{equation}\label{first}
g_{ij}(x,y)=\frac{1}{2}\frac{\partial^2 {\mathcal F}^2}{\partial y^i\partial y^j}
\end{equation}
is produced by this function. The metric of a Randers space is given by virtue of (\ref{first}) in the form \cite{Bao D S-S Chern and Z Schen},\cite{Diakogiannis}.
\begin{equation}\label{randers}
g_{ij}=\alpha_{ij}+\frac{2k}{\sigma}y^s\alpha_{s(i}A_{j)}+k^2A_iA_j+\frac{k}{\sigma}y^lA_lm_{ij}
\end{equation}
where $\sigma=\sqrt{\alpha_{ij}y^iy^j},$ $m_{ij}=\alpha_{ij}-\sigma^{-2}\alpha_{is}\alpha_{jl}y^sy^l$ and $A_{(ij)}=\frac{1}{2}\left( A_{ij}+A_{ji}\right).$ We observe that the presence of an electromagnetic field in a region of spacetime breaks the isotropy and the description of spacetime is given by two metrics, one of which has a pseudo-Riemann structure $a_{ij}(x)$ that corresponds to a motion of a particle with mass m in the gravitational field and the second is metric of a charged particle of mass m that corresponds to a Finsler space of metric $g_{ij}(x,v)$ which represents a dynamical field. Connection coefficients of pseudo FR space are produced by those metrics. In a FR space the second term $y^i b_i$ can represent a measure of a cosmological anisotropy,a magnetic field or a spin-velocity. This consideration is analogous to Rosen's (1940) in which at each point of space-time a Euclidean and pseudo-Riemannian metric corresponds in each point of space-time.(Bimetric theory)
Finsler spaces are endowed with Cartan, Berwald connections and other different types of connections. Cartan connection has very important properties (metric compatibility) for models are closely related to standard physics \cite{Vacaru S. Stavrinos P. Gabourov E. Gontsa D.},\cite{Vacaru S.}.
Berwald connection is not generally compatible with the ,metric structure on total space, since it has "weak" compatibility only on the h-space on the tangent bundle. However for the case of a standard model extension a Berwald structure can be used for Lorentz violation in relation to gravitational waves e.g. \cite{A.Kostelecky}
The explicit form of Randers connection of Berwald-like coefficients can be given in the form \cite{Asanov}
\begin{equation}\label{add}
L_{ij}^l=\alpha_{ij}^l+E_{ij}^l,
\end{equation}
where $\alpha_{ij}^l$ are the Riemannian Cristoffel symbols and $E_{ij}^l$ are given by
\begin{equation}\label{E}
E_{ij}^l=\frac{1}{2}\left(\alpha_{ij}y^kF_k^l+u_iF_j^l+u_jF^l_i\right)\alpha^{-1}-\frac{1}{2}u_iu_jy^kF^l_k\alpha^{-3}
\end{equation}
with $u_i=u_i(x,y)=\frac{\alpha_{ij}y^j}{\alpha(x,y)}\quad \alpha=\alpha(x,y)=\left(\alpha_{ij}(x)y^iy^j\right)^{1/2}.$ We note from (\ref{add}) and (\ref{E}) that the electromagnetic field enters in the connection coefficient of this space.
The geodesics of the Randers space are produced by the first variation of the action corresponding to the Lagrangian.
\begin{equation}
\frac{d}{d\lambda}\left(\frac{\partial{\mathcal F}}{\partial y^m}\right)-\frac{\partial{\mathcal F}}{\partial x^m}=0
\end{equation}
\begin{equation}\label{geo}
\frac{dy^m}{d\lambda}+L_{ij}^m(x,y)y^i y^j=0
\end{equation}
Because of (\ref{geo}) we get the well known Lorentz equation
\begin{equation} \label{lorentz}
\frac{d^2x^m}{d\lambda^2}+\alpha_{ij}^m\frac{d x^i}{d \lambda}\frac{d x^j}{d \lambda}+kF^m_j\frac{dx^j}{d\lambda}=0
\end{equation}
Eq.(\ref{lorentz}) represents the equation of motion of a charged particle in a gravitational and electromagnetic field, where $\lambda$ represents an affine parameter.
The equations of motion (\ref{lorentz}) under the perturbations $h_{ij}$ can be written in the form
\begin{equation}\label{lorentz2}
\frac{d^2x^m}{d\lambda^2}+\tilde{\alpha}_{ij}^m\frac{d x^i}{d \lambda}\frac{d x^j}{d \lambda}+kF^m_j\frac{dx^j}{d\lambda}=0
\end{equation}
where $\tilde{\alpha}_{ij}^m=\frac{1}{2}n^{ml}\left(\partial_i h_{jl}+\partial_j h_{im}-\partial_l h_{ij} \right)$ are the connection coefficients of the weak metric $\tilde\alpha_{ij}.$ The Eq. (\ref{lorentz2}) is useful for studying gravitational waves in a pseudo-FR space.
The curvature tensor of a Randers space can be produced by using Berwald-like connection coefficients in analogous to the form \cite{Asanov} for a weak field
\begin{equation}\label{curv}
H^i_{hjk}=R^i_{hjk}+E^i_{hjk}
\end{equation}
where $R^i_{hjk}$ is the Riemannian curvature tensor and $E^i_{hjk}$ is given by
\begin{equation}\label{ecurv}
\begin{split}
E^i_{hjk} & =\frac{1}{2}Q_{[jk]}\left( F^i_h F_{jk}+\alpha_{hk}F^m_j F^i_m-F_{hj}F^i_k\right)\\
& +Q_{[jk]}\left[(u_h\nabla_k F^i_j+y^m\alpha_{hj}\nabla_k F^i_m+u_j\nabla_k F^i_h)\alpha^{-1}-y^m u_h u_j\alpha^{-3}\nabla_k F^i_m\right]
\end{split}
\end{equation}
\\ with $\alpha=\alpha(x,y)=(\alpha_{ij}y^iy^j) ^{1/2}.$\\
Relation (\ref{curv}) is rewritten
\begin{equation}
H^i_{hjk}(x,y)=\Lambda^i_{hjk}(x)+Q_{[jk]}\left(u_h \nabla_k F^i_j +\cdots -\nabla_k F^i_m\right)
\end{equation}
with $\Lambda^i_{hjk}=R^i_{hjk}+\frac{1}{2}Q_{[jk]}\left( F^i_h F_{jk}+\cdots -F_{hj}F^i_k\right)$ and $Q_{[jk]}=\frac{1}{2}\left(Q_{kj}-Q_{jk}\right)$
Applying the condtion
\begin{equation}\label{condition}
Q_{[jk]}\left(u_h\nabla_k F^i_j+\dot{x}^m\alpha_{hj}\nabla_k F^i_m+u_j\nabla_k F^i_h\right)\alpha^{-1}-\dot{x}^m u_h u_j\alpha^{-3}\nabla_k F^i_m=0
\end{equation}
in (\ref{curv}) we get the Lagrangian of the classical gravitational and electromagnetic fields
\begin{equation}
\Lambda=\Lambda_{hji}^i\alpha^{hj}=R+kF_{mn}F^{mn}, \quad k \thickspace\text{constant}.
\end{equation}
The variation of action's integral
\begin{equation}
\delta I=\delta\int{\Lambda}\sqrt{\vert \det \alpha_{ij} \vert}d^4x
\end{equation}
leads to the weak field equations
\begin{equation}
\left(R_{mn}-\frac{1}{2}\tilde\alpha_{mn}R\right)+k\left(F_{nr}F^r_m-\frac{1}{4} \tilde{\alpha}_{mn}F_{rs}F^{rs}\right)=0
\end{equation}
which are Einstein - Maxwell field equations of the \textquotedblleft gravito - electromagnetic\textquotedblright\thickspace of the Randers space for the vacuum under the condition (\ref{condition}). These equations have the same form as in the Riemannian ansatz in the presence of an electromagnetic field.
From a physical point of view the curvature $H^i_{hjk}$ can be considered as a\textit{\textquotedblleft gravito - electromagnetic curvature \textquotedblright\thickspace } of the space. We can say that it involves a gravito - electromagnetic \textquotedblleft current\textquotedblright\thickspace source. The metric of a weak gravitational field can be decomposed into the flat Minkowski metric plus a small perturbation
\begin{equation}
\tilde\alpha_{ij}=n_{ij}+h_{ij},\quad \vert h_{ij}\vert\ll 1.
\end{equation} Under a linearized approach of the gravitational field, the Randers metric function can be written in the form of a first approximation of the Riemannian metric $\alpha_{ij}$
\begin{equation}
{\mathcal F}(x,y)=\sqrt{\left(n_{ij}+h_{ij}(x,v)\right)v^i v^j}+k A_iv^i
\end{equation}
where $v^i=dx^i/d\tau$ is the 4-velocity of the particle, $n_{ij}=diag(1,-1,-1,-1)$ is the Minkowski metric, $\vert h_{ij}\vert\ll 1$ represents small perturbations to the flat spacetime metric and $k$ is a constant. The linearized form of the metric tensor, as introduced by (\ref{randers}), becomes
\begin{equation}\label{lin}
g_{ij}=n_{ij}+h_{ij}+\frac{2k}{\acute{\sigma}}v^s n_{s(i}A_{j)}+k^2A_iA_j+\frac{k}{\acute{\sigma}}v^lA_l\theta_{ij}
\end{equation}
where $\acute{\sigma}=\sqrt{n_{ij}v^iv^j},\thickspace\theta_{ij}=n_{ij}-\acute{\sigma}^{-2}n_{si}n_{jl}v^s v^l$ and $a_{(ij)}=\frac{1}{2}\left(a_{ij}+a_{ji}\right).$
Considering in (\ref{lin}) the case where $v^s=(1,0,0,0)$ we get the Finslerian potential $g_{00}$ of the Randers space for a test material point in the static case
\begin{equation}
g_{00}=1+h_{00}+k^2\phi^2+2k\phi
\end{equation}
with $\phi=A_0.$ In the case of $\phi=0$ we get $g_{00}=1+h_{00}.$ This relation is useful in order to derive the Riemannian or Newtonian limit from the equation of motion in a Randers space. In this case the equation of motion has the form
\begin{equation}
\dot{v}^l+L^l_{00}v^0 v^0=\dot{v}^l+L^l_{00}=0
\end{equation}
with
\begin{equation}
L^\mu_{00}=-\frac{1}{2}n^{\mu\lambda}\partial_\lambda h_{00}.
\end{equation}
The full interpretation of $L^l_{ij}$ is given by (\ref{add}).
The Finslerian potential $g_{00}$ takes the value 1 for the values of electromagnetic vector potential
\begin{equation}
\phi_{1,2}=\left(-1\pm\left(1-h_{00}\right)^{1/2}\right)(k)^{-1}.
\end{equation}
The Cristoffel symbols and the curvature tensor of the linearized Randers space will take the following form. By using (\ref{add}) and (\ref{curv}) we get
\begin{equation}\label{linadd}
\tilde{L}^i_{lj}=\tilde{a}^i_{lj}+h^i_{lj}
\end{equation}
\begin{equation}\label{lincurv}
\tilde{H}^i_{ljk}=\tilde{R}^i_{ljk}+h^i_{ljk}
\end{equation}
where
\begin{equation}\label{epsilon}
\tilde{R}^i_{ljk}=\frac{1}{2}n^{is}\left(\partial^2_{[kl}h_{sj]}-\partial^2_{[js}h_{lk]}\right)
\end{equation}
$\tilde{a}^i_{jk}$ are the linearized Riemannian Christoffel symbols and the curvature tensor. From (\ref{linadd}) and (\ref{lincurv}) the rest terms will be given in the form
\begin{equation}
h^m_{ij}=\frac{1}{2}\left(n_{ij}v^k F^m_k+u_i F^m_j+u_j F^m_i\right)n^{-1}-\frac{1}{2}u_i u_j v^kF^m_kn^{-3}
\end{equation}
\begin{equation}\label{h}
\begin{split}
h^i_{hjk} & =\frac{1}{2}Q_{[jk]}\left(F^i_h F_{jk}+n_{hk}F^m_j F^i_m-F_{hj} F^i_k\right)\\
& +Q_{[jk]}\left(u_h\partial_k F^i_j + v^m n_{hj}\partial_k F^i_m + u_j\partial_k F^i_h\right)n^{-1}-v^m u_h u_j n^{-3}\partial_k F^i_m
\end{split}
\end{equation}
\\
By using Cartan covariant differentiation in a Randers space we can express the third curvature tensor of Cartan $\bar{R}^i_{jkl}$ in the form \cite{Yasuda-Shimada}
\begin{equation}
\begin{split}
\bar{R}^i_{jhk} &=R^i_{jhk}+D^i_{jh\vert k}-D^i_{jk\vert h}+D^i_{hk}D^m_{jk}-D^i_{mk}D^m_{jh}\\
&+C^i_{jm}(R^m_{0hk}+D^m_{0h\vert k}-D^m_{0k\vert h}+D^m_{sh}D^s_{0k}-D^m_{sk}D^s_{0h})
\end{split}
\end{equation}
where $R^i_{jhk}$ is the Riemannian curvature, the symbol $\vert$ represents the Cartan covariant derivative and $D^i_{jk}$ is the difference tensor of the Finslerian gravitational field given by
\begin{equation}
\begin{split}
D^i_{jk}=&k\ell^i A_{(j,k)}+\frac{1}{2}k\left(\omega^i_j A_{0,k}+\omega^i_k A_{0,j} -\omega_{jk} A_{0,s} g^{is}\right)-C^i_{jk}\\
&+g^{is}k\left(A_{[s,j]}\ell_k+A_{[s,k]}\ell_j\right)+\frac{k}{\tau}\alpha^{mt}\big[g^{is}(A_{([t,s])}+A_{[t,0]}p_s\tau)C_{jkm}\\
&-C^i_{km}(A_{[t,j]}+A_{[t,0]}p_j\tau)-C^i_{jm}(A_{[t,k]}+A_{[t,0]}p_k\tau)\big]
\end{split}
\end{equation}
where $\omega_{ij}(x,y)=\tau\left(a_{ij} -k^2A_iA_j\right), \tau={\mathcal F}/a^{1/2}$ and $\ell^i=y^i/{\mathcal F}.$ In a linearised form the Cartan curvature tensor is expressed by
\begin{equation}
\tilde{K}^i_{jhk} = \tilde{R}^i_{jhk}+\tilde{D}^i_{jh\vert k}-\tilde{D}^i_{jk\vert h}+\tilde{C}^i_{jm}\left(\tilde{R}^m_{0hk}+\tilde{D}^m_{0h\vert k}-\tilde{D}^m_{0k\vert h}\right)
\end{equation}
where $\tilde{R}^i_{jhk}$ is given by (\ref{epsilon}) and $\tilde{D}^i_{jh}, \tilde{C}^i_{jm}$ represent the weak difference tensor and the weak Cartan connection coefficients, $\tilde{C}_{ijk}=\frac{1}{2}\frac{\partial\tilde{g}_{ij}}{\partial y_k} $.\\In (31) we have ignored terms $\tilde{D}^._{..} \tilde{D}^._{..}$ because of the condition $\vert h_{ij}\vert\ll 1$. The Ricci tensor of the weak \textit{"gravito - electromagnetic"} field is given by
\begin{equation}\label{ricci}
\tilde{K}_{ij}=\tilde{K}^s_{ijs}.
\end{equation}
For a perfect fluid moving in a Randers space with Cartan curvature $K^i_{jhk}$ Einstein's equations can be given in the form of weak field
\begin{equation}
\tilde{K}_{il}=k\left(T_{il}(x,V(x))-\frac{1}{2}T^\kappa_\kappa g_{il}\right)
\end{equation}
with $\tilde{K}_{il}=\tilde{R}_{il}+E_{il}$, where $\tilde{K}_{il}$ is the Ricci tensor, $\tilde{R}_{il}$ is the Riemannian one, $E_{il}$ is the contraction of $E^i_{jkl}$ by (\ref{curv}) and $T_{il}$ the energy - momentum tensor of FR space. Randers type space-time in cosmological considerations for a weak anisotropic field $u^a$ with $\ ||u^a|| \ll 1$ has been studied in \cite{S-K-S}.
\section{Weak Deviation of geodesics.Raychaudhuri equation}
\qquad The deviation of geodesics play an important role in General Relativity and Gravitation. In the Finslerian space-time it has been studied from mathematical and physical point of view \cite{GS.Asanov and P.C.Stavrinos},\cite{P.Stavrinos},\cite{S.Rutz},\cite{V.Balan and Stavrinos P.}. In a pseudo-Randers space the deviation of geodesics can be expressed by using \textit{"gravito-electromagnetic" curvature} (\ref{curv})
\begin{equation}\label{deva}
\frac{\delta^2 \xi^i}{\delta\lambda^2}+\bar{H}^i_{jhk}(x,v)\xi^j v^h v^k=0
\end{equation}
where $\frac{\delta\xi^i}{\delta\lambda}=\xi^i_{\vert h}v^h$ and $\xi^i_{\vert h}=\frac{\partial\xi^i}{\partial x^h}+\bar{R}^i_{hk}(x,v)\xi^k$, with $\lambda$ affine parameter. $\bar{R}^i_{hk}$ are the Cartan connection coefficients, $\xi^i$ represents the deviation vector and $v^k$ the tangent vectors of a geodesic surface included in the Randers spacetime. We note from (\ref{deva}) that the deviation equation has two terms. The first term corresponds to the gravitational deviation, that will be observed if there is no electromagnetic field and it is associated with the $R^i_{hjk}$ part of the curvature tensor. The other term corresponds to a mixed geometrical and electromagnetic deviation and is associated with the $E^i_{hjk}$ part of the curvature tensor. The second term of deviation is connected to the force that two freely falling charged particles would exert to each other. Studying this case in a Riemannian spacetime has the consequence that the force does not necessarily result as a natural geometric effect as it does in a Finsler spacetime. If the $E^i_{hjk}$ vanishes then the deviation equation is reduced to the well known one of the Riemannian spacetime, namely
\begin{equation}
\frac{\delta^2 z^i}{\delta u^2}+R^i_{jkl}z^j v^k v^l=0
\end{equation}
\qquad Physically this means that we have two freely falling particles in the tidal field $R^i_{hjk}$ of spacetime. In the case where $R^i_{hjk}$ vanishes, we infer that the first term of the Randers metric corresponds to a Minkowski metric and the Finsler - Randers space becomes v - locally Minkowski \cite{R.Miron and M.Anastasiei}.
\begin{equation}
\mathcal {F}(x,v)=\sqrt{n_{\mu\nu} v^\mu v^\nu}+k A_i(x)v^i
\end{equation}
The only force that influences the two charged particles is due to the presence of the charged electromagnetic field. The deviation equation takes the form
\begin{equation}\label{devb}
\frac{\delta^2 z^i}{\delta u^2}+E^i_{jkl}z^j v^k v^l=0
\end{equation}
where $E^i_{jkl}$ is given by (\ref{ecurv}).
In this case the geometrical properties of the field are characterized by a homogeneous and anisotropic space. The metric fundamental tensor depends only on the velocities, which produce the anisotropic properties of the curved Finsler spacetime. Consequently there exists a frame of reference, where $\bar{R}^i_{hjk}$ vanishes. Under these circumstances the geodesic coordinates can be introduced for particles moving along these geodesics.
In a cosmological consideration the formula(36) can be given by $\mathcal {F}(x,V)=\sqrt{n_{\mu\nu} V^\mu V^\nu}+k W_i(x)V^i$ where $V=\tilde{H} d$ represents cosmological velocity depending on the cosmological Hubble parameter $\tilde{H}$ which is defined in the anisotropic Randers space-time with $\tilde{H} = \sqrt{H^2+Hz_{t}}$ cf. [13], $W_i$ represents an anisotropic field, $Z_t$ the variation of anisotropy and $d$ the distance. Such a consideration can be provided by a Finslerian osculating geometrical framework.If this field $W_i$ comes from by a curl the geodesics of this model are Riemannian. Gravitational waves in locally anisotropic spaces generate polarization patterns of the cosmic microwave background.
It is well known that the gravitational waves are connected to the deviation of geodesics. In order to study the weak field limit of a Randers space related to the deviation of the charged particles it is necessary to take into account relations (\ref{lincurv}) and (\ref{deva}). This is reasonable since in order to detect a gravitational wave at least two particles are needed.\\
Thus the deviation of geodesics of the weak Randers space is written in the form
\begin{equation}
\frac{d^2z^i}{d\tau^2}+\tilde{h}^i_{ljm}\frac{dz^j}{d\tau}\frac{dx^l}{d\tau}\frac{dx^m}{d\tau}=0
\end{equation}
\begin{equation}\label{devc}
\frac{d^2z^i}{d\tau^2}+\left(\epsilon^i_{ljm}+h^i_{ljm}+\right) \frac{dx^l}{d\tau}\frac{dz^j}{d\tau}\frac{dx^m}{d\tau}=0
\end{equation}
If we consider our test particles to be moving slowly then we can express the 4-velocities as a unit vector in the time direction. Hence we write
\begin{equation}\label{vec}
\frac{dx^i}{d\tau}=(1,0,0,0).
\end{equation}
In order to compute the Riemannian tensor in a first approximation we get from (\ref{devc})
\begin{equation}
\epsilon^i_{0j0}=\frac{1}{2}n^{ik}\left(\epsilon_{jk,00}-\epsilon_{0j,k0}-\epsilon_{0k,0j}+\epsilon_{00,kj}\right)
\end{equation}
In our case $\epsilon_{i0}=0$ and the Riemannian tensor takes the form
\begin{equation}\label{e}
\epsilon^i_{0j0}=\frac{1}
{2}\epsilon^i_{j,00}.
\end{equation}
The second term of (\ref{devc}) $h^i_{ljm}$ because of (\ref{vec}) and (\ref{h}) becomes
\begin{equation}\label{h2}
h^i_{0j0}=2F^i_0F^0_{j}+u_j\partial_0F^i_0.
\end{equation}
Furthermore (\ref{devc}) will take the form because of (\ref{e}) and (\ref{h2})
\begin{equation}\label{devd}
\frac{\partial^2z^i}{\partial \tau^2}+\left(\frac{1}{2}\frac{\partial^2\epsilon^i_j}{\partial \tau^2}+2F^i_0F^0_{j}+u_j\frac{\partial F^i_0}{\partial \tau}\right)\frac{\partial z^j}{\partial \tau}=0.
\end{equation}
The equation (\ref{devd}) coincides with the corresponding equation for a weak field limit of the Riemannian case, which is given in its full form by
\begin{equation}\label{deve}
\frac{d^2n^\mu}{ds^2}+R^\mu_{\nu\kappa\lambda}n^\kappa v^\nu v^\lambda=\Phi^\mu
\end{equation}
with
\begin{equation}
\Phi^\mu=k\left(\frac{dF^\mu_\kappa}{ds}v^\kappa+F^\mu_\nu F^\nu_\kappa v^\kappa\right),\quad k:\text{constant.}
\end{equation}
In general $\Phi^\mu$ represents a non-gravitational force, for instance a spring.
The difference between (\ref{deve}) and (\ref{deva}) is that in (\ref{deve}) the electromagnetic field has been added {\it ad hoc}. This means that the term $\Phi^\mu$ plays the role of the interaction external force between two nearby charged masses, moving in non-geodesical paths. In the equation (\ref{deva}) of the Randers space the electromagnetic field is incorporated in the geometry. In this approach the two charged masses move in geodesics of the Finsler space and their relative acceleration is determined by the curvature of the gravitational and electromagnetic fields, which is produced by the energy momentum tensor. Randers-type spaces best express a profound relation between physics and geometry.
An extension of the geodesic deviation equations constitutes the Raychadhuri equation. Raychadhuri equation is of important significance in Relativity theory and Cosmology because of its connection with singularities e.g. \cite{Raychaudhuri.A},\cite{S.Hawking-Ellis},\cite{A.Kouretsis and C.Tsagas}. In Finsler-Randers space-time this equation has been studied in a previous paper \cite{P.C.Stavrinos}. Its form in the weak Finslerian limit is given by
\begin{equation}\label{ray}
\frac{d\tilde{\theta}}{d\tau}=-\frac{1}{3}\tilde{\theta}^2-\tilde{\sigma}_{ik} \tilde{\sigma}^{ik}+\tilde{\omega}_{ik}\tilde{\omega}^{ik}-K_{i\ell}V^i V^\ell +\dot{V}^{i}_{;i}
\end{equation}
where $K_{il}$ represents the weak Cartan tensor,$\tilde{\theta}, \tilde{\omega_{ik}},\tilde{\sigma}_{ik}$ are the expansion,vorticity and the shear are defined by the
following forms:
\begin{subequations}\label{tso}
\begin{align}
& \widetilde{\theta}
=\Lambda_{ij}h^{ij}=V^{i}_{|i}-C^{i}_{im}\dot{V}^m\\
&\tilde{\omega}_{ik}=\Lambda_{[ik]}+\dot{V}_i V_k -\dot{V}_k V_i \\
&\tilde{\sigma}_{ik}=\Lambda_{(ik)}-\frac{1}{3}\tilde{\theta}
h_{ik}-2C_{ikm}V^m -\dot{V}_iV_k-\dot{V}_kV_i
\end{align}\end{subequations}
where $\Lambda_{(ik)}=V_{i;k}$ is the covariant derivative of the oscullating Riemannian space $V^i$ a unit vector $V^i V_{i} = 1$ In the case of geodesics the last term $\dot{V}^{i}_{;i}$ vanishes in (\ref{ray}) and (\ref{tso}). The introduction of Cartan tensor in (48) assigns an anisotropic structure for the Raychaudhuri equation. The linearized Raychaudhuri equation in a Randers spacetime in a first approach is expressed without vorticity and shear by
\begin{equation}\label{ray2}
\frac{d\tilde{\theta}}{d\tau}=-\frac{1}{3}\tilde{\theta}^2- \tilde{K}_{i\ell}V^i V^\ell
\end{equation}
In the case that $\dot{\tilde{\theta}}=0, \tilde{\sigma_{ij}}=0,\tilde{\omega}_{ij}=constant$, from
$\eqref{ray}$ the tidal field $K_{il}V^iV^l$ is due to the vorticity $\tilde{\omega}$ which plays the role of vacuum energy (cosmological
constant). It is analogous to a centrifugal field of the Newtonian theory. It counterbalances the tidal field.
\paragraph{\emph{Remark}} The fundamental sense of photon surfaces and their geometry has been defined and developed in [41],[42] for a timelike surface in a spherical symmetric space with determined properties.
In a pseudo-Finsler space-time $M$ with spherically symmetric metric [35] in which $\tilde{\sigma}_{ij}=0$ and $\tilde{\omega}_{ij}=0$, we can analogously consider a Finslerian photon surface $S$, where $S$ represents a timelike surface of $M$. Here the Raychaudhuri equation takes the form
\begin{equation}
\frac{d\theta_{(2)}}{d\tau} = -\frac{1}{2} \theta^2_{(2)} - K_{il}^{(3)} X^i X^l,
\end{equation}
where $\theta_{(2)}$ denotes the expansion of a vector field $X$ in the surface $S$, $\tau$ the affine parameter and $K_{il}^{(3)}$ the Ricci tensor of Cartan curvature.
On a physical viewpoint the anisotropic Cartan tensor is introduced in the geometry of spacetime because of a primordial vector field in the Finsler- Randers spacetime. Such a case has been studied in \cite{S-K-S} where the linearized Raychaudhuri equation stands
\begin{equation}
\frac{1}{3}\tilde{\theta}^2\tilde{q}=4\pi G(\mu+3P)\frac{H}{\tilde{H}}+f(\alpha,\dot\alpha,\ddot\alpha,z_t)
\end{equation}
where $\tilde{q}, \mu, p, \alpha, z_t$ represent the deceleration parameter, the density of matter, the pressure, the scale factor and the variation of anisotropy, which is connected to Cartan connection component $z_t=C_{000,0}$.
The Raychaudhuri equations can also be derived in the framework of a tangent bundle TM of a $n$-dimensional manifold by using of d-connection and the Ricci-identities. In this case we consider the d-curvature $R^{i}_{jkl}$ and the Ricci identities for a tangent horizontal vector field $X = X^H= X^i \delta/\delta x^i$ along a congruence of geodesics on TM [37]. So we have the relation
\begin{equation}
X^i_{|kl} - X^i_{|lk} = R^i_{jkl}X^j - T^h_{kl}X^i_{|h} - R^a_{kl}X^i|_a,
\end{equation}
or
\begin{equation}
X^l X^i_{|kl} = X^i_{|lk} X^l + R^i_{jkl}X^jX^l - T^h_{kl}X^i_{|h}X^l - R^a_{kl}X^i|_aX^l.
\end{equation}
Because of geodesics the relation $(X^lX^i_{|l})_{|k} =0$ is valid so we have
\begin{equation}
X^l X^i_{|kl} = -X^l X_{|k}X^i_{|l} + R^i_{jkl} X^j X^l - T^h_{kl} X^i_{|h} X^l -R^a_{kl} X^i|_a X^l.
\end{equation}
Taking the trace of the previous equation we have
\begin{equation}
X^l\tilde{h}^k_i X^i_{|kl} = - X^l_{|k} X^i_{|l} \tilde{h}^k_i + R^i_{jkl} X^j X^l \tilde{h}^k_i -T^h_{kl} X^i_{|h} X^l \tilde{h}^k_i - R^a_{kl} X^i|_a X^l \tilde{h}^k_i. \label{eq:55}
\end{equation}
We decompose the $h$-covariant derivative with kinematical terms
\begin{equation}\label{eq:149}
X^i_{|l} = \frac{1}{n-1} \tilde{\Theta}\tilde{h}^i_l + \tilde{\sigma}^i_l + \tilde{\omega}^i_l,
\end{equation}
where $\tilde{\Theta}$, $\tilde{\sigma}$, $\tilde{\omega}$ represent the expansion, shear and vorticity for the extended congruence of geodesics on TM, which are defined as
\begin{align}
\tilde{\Theta} &= X^i_{|l} \tilde{h}^l_i,\label{eq:150}\\
\tilde{\sigma}_{il} &= X_{i|l}+X_{l|i} - \frac{1}{3} \tilde{\Theta} \tilde{h}_{il},\\
\tilde{\omega}_{il} &= X_{i|l} - X_{l|i},
\end{align}
with
\begin{equation}
\tilde{h}_{il} = g_{il} - X_iX_l
\end{equation}
the projection operator, $h_{il}$ gives us $\tilde{h}_{il} X^i =0$ for a normalized $X_i$.
In the above mentioned relations, \eqref{eq:149}, \eqref{eq:150}, we used
\begin{equation}
\tilde{h}^i_l = g^{ik}\tilde{h}_{lk}.
\end{equation}
We finally get from \eqref{eq:55}
\begin{multline}\label{eq:Raychadhuri}
X^l \tilde{\Theta}_{|l} = \frac{d\tilde{\Theta}}{d\tau} = R_{kl}X^k X^l - T^h_{li} X^i_{|h} X^l - R^a_{li}X^i|_a X^l- X^l_{|k} X^k_{|l} = R_{il} X^i X^l\\
- T^h_{li} \left(\frac{1}{n-1} \tilde{\Theta}\tilde{h}^i_h + \tilde{\sigma}^i_h + \tilde{\omega}^i_h \right)X^l - R^a_{li} \left( \frac{1}{n-1} \Theta \tilde{h}^i_a + \sigma^i_a + \omega^i_a \right)X^l - \frac{1}{n-1} \tilde{\Theta} - \sigma^l_k \sigma^k_l - \omega^l_k\omega^k_l.
\end{multline}
where we put
\begin{displaymath}
X^i|_a = \frac{1}{n-1} \tilde{\Theta}\tilde{h}^i_a + \tilde{\sigma}^i_a + \tilde{\omega}^i_a,
\end{displaymath}
with $\tilde{h}^i_a = h^b_a \delta^i_c \delta^c_b$, $\sigma^i_a = \sigma^b_a \delta^i_c \delta^c_b$, $\omega^i_a = \omega^b_a \delta^i_c \delta^c_b$ and $\delta^i_c$ represent the generalized Kronecker symbols connecting with h-bases and v-bases. The equation \eqref{eq:Raychadhuri} is the Raychadhuri equation for the horizontal space of the tangent bundle.
\begin{equation}
R^a_{li} = \frac{\partial N^a_l}{\partial x^i} - \frac{\partial N^a_i}{\partial x^l}
\end{equation}
represents the curvature of non-linear connection.
By using of d-curvature $S^a_{bcd}$, the Ricci identities are written
\begin{equation}
X^a \vert_{bc} - X^a \vert_{cb} = S^a_{dbc} X^d - S^d_{bc} X^a \vert_{d},
\end{equation}
where the vector field $X=X^a \frac{\partial}{\partial y^a}$ belongs to the vertical space $S^a_{bc} = C^a_{bc} - C^a_{cb}$ represent the torsion and $C^a_{bc}$ the d-connection coefficients of the vertical space. In analogy to the consideration of Finslerian fluids cf.[43] we can get the Raychaudhuri equations. For the definitions of the decomposition of vertical covariant derivative of vertical geodesics expansion, shear and rotation we use the relations
\begin{equation}
X^a \vert_{b} = \frac{1}{3} \Theta h^a_{b} + \sigma^a_{b} + \omega^a_{b}
\end{equation}
where the expansion $\Theta$ is given by
\begin{equation}
\Theta = X^a \vert_{b} h^b_{a} = X^a \vert_{a},
\end{equation}
$h_{ab}(x,y)$ represents the v-metric on TM which is connected with the h-metric
$g_{ij}(x,y)=\delta^a_{i} \delta^b_{j} h_{ab} (x,y) \sigma_{ab}$ and $\omega_{ab}$. We define the shear and rotation by
\begin{align}
\sigma_{ab} &=X_a \vert_{b} +X_b \vert_{a} - \frac{1}{3} \Theta h_{ab},\\
\omega_{ab} &=X_{a} \vert_{b} - X_{b} \vert_{a}.
\end{align}
Because of the above mentioned relations one obtains an expression for the Raychaudhuri equation in the vertical space in the forms
\begin{multline}
X^c \Theta \vert_{c} = \frac{d\Theta}{d\tau}= S_{dc}X^d X^c - S^d_{ca} X^a \vert_{d} X^c - X^c \vert _{b} X^b \vert _{c}\\
=-\frac{1}{n-1} \Theta^2 - \sigma^a_{b} \sigma^b_{a} - \omega^a_{b} \omega^b_{a} + S_{dc} X^d X^c - S^b_{ca} X^c \left(\frac{1}{n-1} \Theta h^a_{b} + \sigma^a_{b} + \omega^a_{b} \right)
\end{multline}
\\
The Raychaudhuri equation can also be derived on the tangent bundle of a Finsler-Randers space-time as well as for its weak field limit by considering the analogous curvature in the rel.~\eqref{eq:Raychadhuri}.
\\
\\
\\
{\bf Applications:}\\
\\
{\bf 1.}\quad In the $(x^0,x^1,x^2,x^3)$ coordinates of an inertial frame generalized Finsler metrics that are very close to the flat metric can be written
\begin{equation}\label{weak}
g_{\mu\nu}(x,y)=\eta_{\mu\nu}+h_{\mu\nu}(x,y)
\end{equation}
where $y=\frac{dx}{dt}$ and $h_{\mu\nu}(x,y)$ are small anisotropic perturbations to the flat spacetime. These metric perturbations describe a gravitational wave. The line - element for a plane gravitational wave spacetime can be expressed in the form
\begin{equation}\label{li}
ds^2=dt^2-\left(1+f[(x_0,x_3),y]dx^2_1\right)-\left(1-f[(x_0,x_3),y]dx^2_2\right)+dx_3^2
\end{equation}
where the function $\delta_{\mu\nu}f(x,y)=h_{\mu\nu}(x,y),$ with
\begin{equation}
\left\vert f[(x^0-x^3),y]\right\vert\ll 1.
\end{equation}
If the wave has a definite frequency $\omega$, amplitude $\alpha$ and phase $\delta$ we can write
\begin{equation}
f[(x^0-x^3),y]=\alpha\sin [\omega(x^0-x^3)+\delta]\thickspace y.
\end{equation}
In the case of the weak field limit of a Finslerian or generalized Finslerian space-time the vacuum field equation holds analogous form to Newton's gravity and General Relativity.
We specify a gauge under a coordinate transformation $x^\mu\rightarrow \bar{x}^\mu=x^\mu+\frac{1}{2}Q^\mu_{\;\; \alpha \beta}x^\alpha x^\beta$ where $Q^\mu_{\;\; \alpha \beta}\sim \tilde{L}^\mu_{\;\;\alpha\beta}\sim \epsilon$, $\tilde{L}^\mu_{\;\;\alpha\beta}$ represent the linearized Christoffel symbols given by (25). We choose a Lorentz gauge $\eta^{\mu\nu} \tilde{L}^\rho_{\;\;\mu\nu}=0$.
In the weak field limit the connection coefficients $h^\rho_{\;\;\mu\nu}$ must be zero, so we have $\tilde{L}^\rho_{\;\;\mu\nu}=\tilde{a}^\rho_{\;\;\mu\nu}$ which leads to the equivalent Lorentz gauge
\begin{equation}
\partial_\mu h^{\mu}_{\;\lambda}(x,y)-\frac{1}{2}\partial_\lambda h=0.
\end{equation}
By using the linearized Einstein equations for the vacuum (27) with $\tilde{R}_{\mu\nu}=0$ we get the equivalent form of the wave equation in the new system of coordinates
\begin{equation}
\Box \bar{h}_{\mu\nu}(x,y)=0.
\label{gWEQ}
\end{equation}
A solution of (\ref{gWEQ}) is a gravitational wave
\begin{equation}
\bar{h}_{\mu\nu}(x,y)=\exp (ik_\sigma(y)x^\sigma)C_{\mu\nu}
\label{GWslt}
\end{equation}
with $C_{\mu\nu}=C_{\nu\mu}$ a constant polarization tensor and the wave vector $k_\mu (y)$ is a function of $y$ because of anisotropic metric (71).
By inserting (\ref{GWslt}) to (\ref{gWEQ}) we derive
\begin{equation}
\eta^{\mu\nu}k_{\mu}(y)k_{\nu}(y)=k^2=0
\end{equation}
with $k_\mu=(\omega,k_1,k_2,k_3)$. The plane wave (\ref{GWslt}) is therefore a solution to the linearized equations if the wave vector is null.
A linearized form of the curvature $\tilde{H}^i_{\;\;ljk}$ (relation (26)) can be produced by the linearized coefficients $\tilde{L}^i_{\;\; jk}$. In order to get gravitational waves the harmonic gauge condition is chosen as
\begin{equation}
\frac{\partial \tilde{h}^\mu_{\;\; \nu}}{\partial y^\mu}=0
\end{equation}
and traceless $\tilde{h}^\mu_{\;\; \nu}=0$.
The traceless and transverse (TT) gauge condition for the equation of motion is given by
\begin{equation}
\Box h^{TT}_{\;\;\mu\nu}(x,y)=0.
\end{equation}
A particular useful set of solutions to this wave equation are the plane waves
\begin{equation}
h^{TT}_{\;\;\mu\nu}(x,y)=C'_{\mu\nu}\exp (ik_a(y)x^a)
\end{equation}
with $C'_{\mu\nu}$ a constant symmetric tensor. The plane wave $ h^{TT}_{\;\;\mu\nu}$ is a solution to the linearized wave equation if the wave vector is null $k^a(y)k_a(y)=0$.
The geodesic deviation equation with the curvature $\tilde{H}^i_{\;\; jkl}$, for two particles with $\upsilon^\mu=(1,0,0,0)$ 4-velocity and separation vector $\zeta^{x^2}$ in the $x^2$-direction implies
\begin{eqnarray}
\frac{\partial ^2 \zeta ^{x^2}}{\partial t^2}=\frac{1}{2}\frac{\partial ^2}{\partial t^2} h^{TT}_{\;\; x^2x^2} \\
\frac{\partial ^2 \zeta ^{x^1}}{\partial t^2}=\frac{1}{2}\frac{\partial ^2}{\partial t^2} h^{TT}_{\;\; x^1x^2}
\end{eqnarray}
These equations are important to describe a polarization of the gravitational wave in $y=(x^1,x^2)$ direction in the framework of the anisotropic metric (71).
\\
\\
{\bf 2.} In the previous application the coordinates $x^i, \thickspace i=0,1,2,3$ are independent of time. Distances between test masses in Euclidean plane can be calculated from $X(t), Y(t)$ coordinates by using a Randers metric. We can put the coordinates in the form
\begin{subequations}
\begin{align}
X &=(1+\frac{1}{2}\alpha\sin\omega t)\thickspace x^1\\
Y &=(1-\frac{1}{2}\alpha\sin\omega t)\thickspace x^2\\
\end{align}
\end{subequations}
then
\begin{subequations}
\begin{align}
\dot{X}&=\frac{dX}{dt}=\frac{1}{2}\alpha\omega\cos\omega t\thickspace x^1\\
\dot{Y}&=\frac{dY}{dt}=-\frac{1}{2}\alpha\omega\cos\omega t\thickspace x^2
\end{align}
\end{subequations}
A plane gravitational wave of the form (\ref{li}) with $f[(x^0-x^3),y^3]=\alpha \sin[\omega (x^0-x^3)]\thickspace y^3,$ and $y^3=(0,0,1,0)$ propagates in the $y^3$ direction. Some test masses in the $x-y$ plane are as rest in a circle about a central test mass. After the gravitational wave passes in time t the circle is squeezed in the Y-direction and expanded in the X-direction, therefore the circle is transformed to an elliptic shape.
We define the indicartix curve $I_p$ to be a circle with center 0 and radius $\sqrt{1+f(x,y)}\thickspace OP$ where $P(x,y)$ is an arbitary point and $f(x,y)$ is a positive valued function. We can apply the Randers metric in order to calculate the distances between the masses \cite{Antonelli}. Therefore we have
\begin{equation}
{\mathcal L}(x,y)=\frac{\lambda+\sqrt{f(x,y)\rho^2+\lambda^2}}{f(x,y)}
\end{equation}
where
\begin{equation}
\rho^2=\frac{\dot{X}^2+\dot{Y}^2}{X^2+Y^2}\quad \lambda=\frac{X\dot{X}+Y\dot{Y}}{X^2+Y^2}
\end{equation}
\\
{\bf Conclusions}
\\
\\
We studied the behaviour of particles moving in a gravitational and electromagnetic field with the physical geometry of a Finsler--Randers (FR) space. Cartan and Berwald connections are applied for studying a linearized version of a weak field limit in F-R spaces.
In virtue of curvature tensors of the space of considerations some physical characterizations and interpretations in the sense of a \guillemotleft gravito--electromagnetic curvature \guillemotright are given. Such a concept could play a role in the bending of light geodesics and gravitational lensing in a region of locally anisotropic space-time.
In paragraph 3 the Raychaudhuri equations are extended and they were derived in the framework of a tangent bundle. This consideration can give an additional interest to a string theory.
Finally, some applications of Randers metric for gravitational waves are presented.
\\
\\
{\bf Acknowledgement}
\\
\\
The author is grateful to the University of Athens (Special Accounts for Research Grants) for the support to this work.
\newpage
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\section{Introduction}\label{secintro}
We refer to Slater's text \cite{Sl} for an introduction to
hypergeometric series, and to Gasper and Rahman's text \cite{GR}
for an introduction to basic hypergeometric series, whose notations we follow.
Throughout, we assume $|q|<1$ and $|z|<1$.
In \cite{A}, George Andrews proved the following two theorems:
\begin{theorem}\label{qwatson}
\begin{equation}
{}_4\phi_3\!\left[\begin{matrix}a,b,c^{\frac 12},-c^{\frac 12}\\
(abq)^{\frac 12},-(abq)^{\frac 12},c
\end{matrix}\,;q,q\right]=a^{\frac n2}
\frac{(aq,bq,cq/a,cq/b;q^2)_\infty}
{(q,abq,cq,cq/ab;q^2)_\infty},
\end{equation}
where $b=q^{-n}$ and $n$ is a nonnegative integer.
\end{theorem}
\begin{theorem}\label{qwhipple}
\begin{equation}\label{qwhippleeq}
{}_4\phi_3\!\left[\begin{matrix}a,q/a,c^{\frac 12},-c^{\frac 12}\\
-q,e,cq/e\end{matrix}\,;q,q\right]=q^{\binom{n+1}2}
\frac{(ea,eq/a,caq/e,cq^2/ae;q^2)_\infty}
{(e,cq/e;q)_\infty},
\end{equation}
where $a=q^{-n}$ and $n$ is a nonnegative integer.
\end{theorem}
By a standard polynomial argument \eqref{qwhippleeq} also holds
when $a$ is a complex variable but $c=q^{-2n}$ with
$n$ being a nonnegative integer. (This is the case we will
make use of.)
Theorems~\ref{qwatson} and \ref{qwhipple} are $q$-analogues of Watson's
and of Whipple's $_3F_2$ summation theorems, listed as
Equations (III.23) and (III.24) in \cite[p.~245]{Sl}, respectively.
\section{Two product formulas for basic hypergeometric
functions}
We now have the following two product formulas which
are derived using Theorems~\ref{qwatson} and \ref{qwhipple}.
The first one in Theorem~3 was already given earlier by
Jain and Srivastava~\cite[Equation~(4.9)]{JS}
(as Slobodan Damjanovi\'c has kindly pointed out to the author,
after seeing an earlier version of this note),
who established the result by specializing a
general reduction formula for double basic hypergeometric series.
The second formula in Theorem~4 appears to be new.
\begin{theorem}\label{qwatsonprod}
\begin{equation}
{}_2\phi_1\!\left[\begin{matrix}a,-a\\
a^2\end{matrix}\,;q,z\right]
{}_2\phi_1\!\left[\begin{matrix}b,-b\\
b^2\end{matrix}\,;q,-z\right]
={}_4\phi_3\!\left[\begin{matrix}ab,-ab,abq,-abq\\
a^2q,b^2q,a^2b^2\end{matrix};q^2,z^2\right].
\end{equation}
\end{theorem}
\begin{theorem}\label{qwhippleprod}
\begin{subequations}\label{qwhippleprodid}
\begin{align}
{}_2\phi_1\!\left[\begin{matrix}a,q/a\\
-q\end{matrix}\,;q,z\right]
{}_2\phi_1\!\left[\begin{matrix}b,q/b\\
-q\end{matrix}\,;q,-z\right]
=\sum_{j=0}^\infty\frac{(q^{2-j}/ab,aq^{1-j}/b;q^2)_j}
{(q^2;q^2)_j}q^{\binom j2}(bz)^j&\\
={}_4\phi_3\!\left[\begin{matrix}ab,q^2/ab,aq/b,bq/a\\
-q^2,q,-q\end{matrix}\,;q^2,z^2\right]
\qquad\qquad\qquad\qquad\qquad\quad&\notag\\[.1em]
{}-\frac{(a-b)(1-q/ab)}{1-q^2}\,z\,
{}_4\phi_3\!\left[\begin{matrix}abq,q^3/ab,aq^2/b,bq^2/a\\
-q^2,q^3,-q^3\end{matrix}\,;q^2,z^2\right]&.
\end{align}
\end{subequations}
\end{theorem}
\begin{proof}[Sketch of proofs.]
To prove Theorem~\ref{qwatsonprod}, compare coefficients
of $z^n$. The resulting identity is equivalent to Theorem~\ref{qwatson}.
The proof of Theorem~\ref{qwhippleprod} is similar.
Comparison of coefficients of $z^n$ gives an identity which is
equivalent to Theorem~\ref{qwhipple} (where in the latter theorem
the restriction $a=q^{-n}$ is replaced by $c=q^{-2n}$, as mentioned).
The second identity in Equation \eqref{qwhippleprodid} follows from
splitting the sum over $j$ into two parts depending on the parity of $j$.
(This is motivated by the particular numerator factors in the $j$-th summand.)
The technical details -- elementary manipulation of
$q$-shifted factorials -- are routine and thus omitted.
\end{proof}
Theorem~\ref{qwatsonprod} is a $q$-analogue of Bailey's
formula in \cite[p.~246, Equation~(2.11)]{B}:
\begin{equation}\label{qwatsonprodido}
{}_1F_1\!\left[\begin{matrix}a\\
2a\end{matrix}\,;z\right]
{}_1F_1\!\left[\begin{matrix}b\\
2b\end{matrix}\,;-z\right]
={}_2F_3\!\left[\begin{matrix}\frac 12(a+b),\frac 12(a+b+1)\\
a+\frac 12,b+\frac 12,a+b\end{matrix};\frac 14z\right].
\end{equation}
To obtain \eqref{qwatsonprodido} from Theorem~\ref{qwatsonprod},
replace $(a,b,z)$ by $(q^a,q^b,(1-q)z/2)$, and let $q\to 1$.
Similarly, Theorem~\ref{qwhippleprod} is a $q$-analogue of Bailey's
formula in \cite[p.~245, Equation~(2.08)]{B}:
\begin{align}\label{qwhippleprodido}
{}_2F_0\!&\left[\begin{matrix}a,1-a\\
-\end{matrix}\,;z\right]
{}_2F_0\!\left[\begin{matrix}b,1-b\\
-\end{matrix}\,;-z\right]\notag\\[.2em]
&{}={}_4F_1\!\left[\begin{matrix}\frac 12(1+a-b),\frac 12(1-a+b),
\frac 12(a+b),\frac 12(2-a-b)\\[.1em]
\frac 12\end{matrix}\,;4z^2\right]\notag\\[.1em]
&\qquad{}-(a-b)(a+b-1)\,z\notag\\
&\qquad\times{}_4F_1\!\left[\begin{matrix}\frac 12(2+a-b),\frac 12(2-a+b),
\frac 12(1+a+b),\frac 12(3-a-b)\\[.1em]
\frac 32\end{matrix}\,;4z^2\right].
\end{align}
To obtain \eqref{qwhippleprodido} from Theorem~\ref{qwhippleprod},
replace $(a,b,z)$ by $(q^a,q^b,2z/(1-q))$ and let $q\to 1$.
\section{Related results in the literature}
A different product formula for basic hypergeometric functions
was established by Srivastava \cite[Eq.~(21)]{S1} (see also
\cite[Eq.~(3.13)]{S2}):
\begin{equation}
{}_2\phi_1\!\left[\begin{matrix}a,b\\
-ab\end{matrix}\,;q,z\right]
{}_2\phi_1\!\left[\begin{matrix}a,b\\
-ab\end{matrix}\,;q,-z\right]
={}_4\phi_3\!\left[\begin{matrix}a^2,b^2,ab,abq\\
a^2b^2,-ab,-abq\end{matrix};q^2,z^2\right].
\end{equation}
This formula is a $q$-extension of Bailey's formula in
\cite[p.~245, Equation~(2.08)]{B}
(or, equivalently, of an identity recorded by
Ramanujan~\cite[Ch.\ 13, Entry 24]{R}).
Finally, we mention that in 1941 F.H.~Jackson~\cite{J}
had derived the identity
\begin{equation}
{}_2\phi_1\!\left[\begin{matrix}a^2,b^2\\
a^2b^2q\end{matrix}\,;q^2,z\right]
{}_2\phi_1\!\left[\begin{matrix}a^2,b^2\\
a^2b^2q\end{matrix}\,;q^2,qz\right]
={}_4\phi_3\!\left[\begin{matrix}a^2,b^2,ab,-ab\\
a^2b^2,abq^{\frac 12},-abq^{\frac 12}\end{matrix};q,z\right],
\end{equation}
which is a $q$-analogue of Clausen's formula of 1828,
\begin{equation}
\left({}_2F_1\!\left[\begin{matrix}a,b\\
a+b+\frac 12\end{matrix}\,;z\right]\right)^2
={}_3F_2\!\left[\begin{matrix}2a,2b,a+b\\
2a+2b,a+b+\frac 12\end{matrix};z\right].
\end{equation}
Another $q$-analogue of Clausen's formula
was delivered by Gasper in \cite{G}.
While it has the advantage that it expresses a square of a
basic hypergeometric series as a basic hypergeometric
series, it only holds provided the series terminate:
\begin{equation}\label{qClausen}
\left({}_4\phi_3\!\left[\begin{matrix}a,b,aby,ab/y\\
abq^{\frac 12},-abq^{\frac 12},-ab\end{matrix}\,;q,q\right]\right)^2
={}_5\phi_4\!\left[\begin{matrix}a^2,b^2,ab,aby,ab/y\\
a^2b^2,abq^{\frac 12},-abq^{\frac 12},-ab\end{matrix};q,q\right].
\end{equation}
See \cite[Sec.~8.8]{GR} for a nonterminating extension
of \eqref{qClausen} and related identities.
\section*{Acknowledgement}
I would like to thank George Gasper for his interest and for
informing me of the papers \cite{S1,S2} by Srivastava.
I am especially indebted to Slobodan Damjanovi\'c for pointing out that
Theorem~3 was already given by Jain and Srivastava~\cite[Equation~(4.9)]{JS}.
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Q: Formatting floating point numbers in C#: decimal separator always visible How do I get 12. from 12 value with standard .NET string formatting? I've tested 0.### and it does not work.
double number = 12;
string.Format("{0:0.###}", number); // returns "12" not "12."
double number = 12.345;
string.Format("{0:0.###}", number); // returns "12.345"
Currently I've resolved with string manipulation but is it somewhat possible with standard string.Format()?
Thanks.
A: I think you can first check if the double is actually and integer, and if yes, use a simple string.Format("{0}.", number):
double number = 12;
if (number % 1 == 0)
Console.Write(string.Format("{0}.", number));
C# demo
A: double number = 12;
string.Format((number % 1 == 0) ? "{0}." : "{0}", number);
Gives 12.
double number = 12.345;
string.Format((number % 1 == 0) ? "{0}." : "{0}", number);
Gives 12.345
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{"url":"https:\/\/zbmath.org\/?q=an:07142672","text":"# zbMATH \u2014 the first resource for mathematics\n\nTime4sys2imi: a tool to formalize real-time system models under uncertainty. (English) Zbl\u00a007142672\nHierons, Robert Mark (ed.) et al., Theoretical aspects of computing \u2013 ICTAC 2019. 16th international colloquium, Hammamet, Tunisia, October 31 \u2013 November 4, 2019. Proceedings. Cham: Springer (ISBN 978-3-030-32504-6\/pbk; 978-3-030-32505-3\/ebook). Lecture Notes in Computer Science 11884, 113-123 (2019).\nSummary: Time4sys is a formalism developed by Thales Group, realizing a graphical specification for real-time systems. However, this formalism does not allow to perform formal analyses for real-time systems. So a translation of this tool to a formalism equipped with a formal semantics is needed. We present here Time4sys2imi, a tool translating Time4sys models into parametric timed automata in the input language of IMITATOR. This translation allows not only to check the schedulability of real-time systems, but also to infer some timing constraints (deadlines, offsets $$\\ldots)$$ guaranteeing schedulability. We successfully applied Time4sys2imi to various examples.\nFor the entire collection see [Zbl 1425.68012].\n##### MSC:\n 68Qxx Theory of computing\n##### Software:\nCheddar; IMITATOR; Time4sys2imi; Uppaal\nFull Text:","date":"2021-06-13 06:19:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.5507484078407288, \"perplexity\": 3646.5207066917555}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-25\/segments\/1623487600396.21\/warc\/CC-MAIN-20210613041713-20210613071713-00420.warc.gz\"}"}
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{"url":"https:\/\/damnmodz.com\/q0d1tgmi\/79fe92-radius-of-incircle-of-right-angled-triangle","text":"## 26 Jan radius of incircle of right angled triangle\n\nThen, by CPCTC (congruent parts of congruent triangles are congruent) and the transitive property of congruence, IX\u203e\u2245IY\u203e\u2245IZ\u203e.\\overline{IX} \\cong \\overline{IY} \\cong \\overline{IZ}.IX\u2245IY\u2245IZ. Let r be the radius of the incircle of triangle ABC on the unit sphere S. If all the angles in triangle ABC are right angles, what is the exact value of cos r? Also, the incenter is the center of the incircle inscribed in the triangle. The formula above can be simplified with Heron's Formula, yielding The radius of an incircle of a right triangle (the inradius) with legs and hypotenuse is. The product of the incircle radius and the circumcircle radius of a triangle with sides , , and is: 189,#298(d) r R = a b c 2 ( a + b + c ) . Now we prove the statements discovered in the introduction. Hence, the incenter is located at point I.I.I. Solution First, let us calculate the measure of the second leg the right-angled triangle which \u2026 The center of the incircle will be the intersection of the angle bisectors shown . Prentice Hall. \\left[ ABC\\right] = \\sqrt{rr_1r_2r_3}.[ABC]=rr1\u200br2\u200br3\u200b\u200b. Then it follows that AY+BW+CX=sAY + BW + CX = sAY+BW+CX=s, but BW=BXBW = BXBW=BX, so, AY+BX+CX=sAY+a=sAY=s\u2212a,\\begin{aligned} AI &= r\\mathrm{cosec} \\left({\\frac{1}{2}A}\\right) \\\\\\\\ for integer values of the incircle radius you need a pythagorean triple with the (subset of) pythagorean triples generated from the shortest side being an odd number 3, 4, 5 has an incircle radius, r = 1 5, 12, 13 has r = 2 (property for shapes where the area value = perimeter value, 'equable') 7, 24, 25 has r = 3 9, 40, 41 has r = 4 etc. In order to prove these statements and to explore further, we establish some notation. Given the P, B and H are the perpendicular, base and hypotenuse respectively of a right angled triangle. The side opposite the right angle is called the hypotenuse (side c in the figure). Circumradius: The circumradius( R ) of a triangle is the radius of the circumscribed circle (having center as O) of that triangle. Right Triangle Equations. Prove that the radius r of the circle which touches the sides of the triangle is given by r=a+b-c\/2. Also, the incenter is the center of the incircle inscribed in the triangle. As sides 5, 12 & 13 form a Pythagoras triplet, which means 5 2 +12 2 = 13 2, this is a right angled triangle. BX1=BZ1=s\u2212c,CY1=CX1=s\u2212b,AY1=AZ1=s.BX_1 = BZ_1 = s-c,\\quad CY_1 = CX_1 = s-b,\\quad AY_1 = AZ_1 = s.BX1\u200b=BZ1\u200b=s\u2212c,CY1\u200b=CX1\u200b=s\u2212b,AY1\u200b=AZ1\u200b=s. Approach: Formula for calculating the inradius of a right angled triangle can be given as r = ( P + B \u2013 H ) \/ 2. Note that these notations cycle for all three ways to extend two sides (A1,B2,C3). Radius can be found as: where, S, area of triangle, can be found using Hero's formula, p - half of perimeter. Also, the incenter is the center of the incircle inscribed in the triangle. Therefore, the radii. Finally, place point WWW on AB\u203e\\overline{AB}AB such that CW\u203e\\overline{CW}CW passes through point I.I.I. A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). And in the last video, we started to explore some of the properties of points that are on angle bisectors. Since IX\u203e\u2245IY\u203e\u2245IZ\u203e,\\overline{IX} \\cong \\overline{IY} \\cong \\overline{IZ},IX\u2245IY\u2245IZ, there exists a circle centered at III that passes through X,X,X, Y,Y,Y, and Z.Z.Z. Question 2: Find the circumradius of the triangle \u2026 The incircle is the inscribed circle of the triangle that touches all three sides. Click hereto get an answer to your question \ufe0f In the given figure, ABC is right triangle, right - angled at B such that BC = 6 cm and AB = 8 cm. The center of the incircle is called the triangle's incenter.. An excircle or escribed circle of the triangle is a circle lying outside the triangle, tangent to one of its sides and tangent to the extensions of the other two. AB = 8 cm. asked Mar 19, 2020 in Circles by ShasiRaj ( 62.4k points) circles New user? Forgot password? PO = 2 cm. Click hereto get an answer to your question \ufe0f In the given figure, ABC is right triangle, right - angled at B such that BC = 6 cm and AB = 8 cm. https:\/\/brilliant.org\/wiki\/incircles-and-excircles\/. \\frac{1}{r} &= \\frac{1}{r_1} + \\frac{1}{r_2} + \\frac{1}{r_3}\\\\\\\\ r &= \\sqrt{\\frac{(s-a)(s-b)(s-c)}{s}} Now \u25b3CIX\\triangle CIX\u25b3CIX and \u25b3CIY\\triangle CIY\u25b3CIY have the following congruences: Thus, by HL (hypotenuse-leg theorem), \u25b3CIX\u2245\u25b3CIY.\\triangle CIX \\cong \\triangle CIY.\u25b3CIX\u2245\u25b3CIY. This is the same situation as Thales Theorem , where the diameter subtends a right angle to any point on a circle's circumference. Click hereto get an answer to your question \ufe0f In a right triangle ABC , right - angled at B, BC = 12 cm and AB = 5 cm . Using Pythagoras theorem we get AC\u00b2 = AB\u00b2 + BC\u00b2 = 100 AB, BC and CA are tangents to the circle at P, N and M. \u2234 OP = ON = OM = r (radius of the circle) By Pythagoras theorem, CA 2 = AB 2 + \u2026 Find the sides of the triangle. Recommended: Please try your approach on {IDE} first, before moving on to the solution. Find the radius of its incircle. Solving for angle inscribed circle radius: Inputs: length of side a (a) length of side b (b) Conversions: length of side a (a) = 0 = 0. length of side b (b) = 0 = 0. The center of the incircle is called the triangle's incenter. If we extend two of the sides of the triangle, we can get a similar configuration. Now we prove the statements discovered in the introduction. Therefore, all sides will be equal. b\u2212cr1+c\u2212ar2+a\u2212br3.\\frac {b-c}{r_{1}} + \\frac {c-a}{r_{2}} + \\frac{a-b}{r_{3}}.r1\u200bb\u2212c\u200b+r2\u200bc\u2212a\u200b+r3\u200ba\u2212b\u200b. First we prove two similar theorems related to lengths. Geometry calculator for solving the inscribed circle radius of a right triangle given the length of sides a, b and c. Right Triangle Equations Formulas Calculator - Inscribed Circle Radius Geometry AJ Design If a b c are sides of a triangle where c is the hypotenuse prove that the radius r of the circle which touches the sides of the triangle is given by r=a+b-c\/2 Sign up, Existing user? Right Triangle: One angle is equal to 90 degrees. (A1, B2, C3).(A1,B2,C3). Right triangle or right-angled triangle is a triangle in which one angle is a right angle (that is, a 90-degree angle). ))), 1r=1r1+1r2+1r3r1+r2+r3\u2212r=4Rs2=r1r2+r2r3+r3r1.\\begin{aligned} Inradius The inradius( r ) of a regular triangle( ABC ) is the radius of the incircle (having center as l), which is the largest circle that will fit inside the triangle. To find the area of a circle inside a right angled triangle, we have the formula to find the radius of the right angled triangle, r = ( P + B \u2013 H ) \/ 2. For right triangles In the case of a right triangle , the hypotenuse is a diameter of the circumcircle, and its center is exactly at the midpoint of the hypotenuse. Hence the area of the incircle will be PI * ( (P + B \u2013 H) \/ 2)2. Find the area of the triangle. \u2039 Derivation of Formula for Radius of Circumcircle up Derivation of Heron's \/ Hero's Formula for Area of Triangle \u203a Log in or register to post comments 54292 reads Precalculus Mathematics. We have found out that, BP = 2 cm. Area of a circle is given by the formula, Area = \u03c0*r 2 Another triangle calculator, which determines radius of incircle Well, having radius you can find out everything else about circle. The inradius rrr is the radius of the incircle. Then place point XXX on BC\u203e\\overline{BC}BC such that IX\u203e\u22a5BC\u203e,\\overline{IX} \\perp \\overline{BC},IX\u22a5BC, place point YYY on AC\u203e\\overline{AC}AC such that IY\u203e\u22a5AC\u203e,\\overline{IY} \\perp \\overline{AC},IY\u22a5AC, and place point ZZZ on AB\u203e\\overline{AB}AB such that IZ\u203e\u22a5AB\u203e.\\overline{IZ} \\perp \\overline{AB}.IZ\u22a5AB. The three angle bisectors all meet at one point. Given \u25b3ABC,\\triangle ABC,\u25b3ABC, place point UUU on BC\u203e\\overline{BC}BC such that AU\u203e\\overline{AU}AU bisects \u2220A,\\angle A,\u2220A, and place point VVV on AC\u203e\\overline{AC}AC such that BV\u203e\\overline{BV}BV bisects \u2220B.\\angle B.\u2220B. Contact: [email\u00a0protected] \u25a1_\\square\u25a1\u200b. In a triangle A B C ABC A B C, the angle bisectors of the three angles are concurrent at the incenter I I I. Consider a circle incscrbed in a triangle \u0394ABC with centre O and radius r, the tangent function of one half of an angle of a triangle is equal to the ratio of the radius r over the sum of two sides adjacent to the angle. The argument is very similar for the other two results, so it is left to the reader. Now we prove the statements discovered in the introduction. Since the triangle's three sides are all tangents to the inscribed circle, the distances from the circle's center to the three sides are all equal to the circle's radius. AY + BX + CX &= s \\\\ In this situation, the circle is called an inscribed circle, and its center is called the inner center, or incenter. These are very useful when dealing with problems involving the inradius and the exradii. 30, 24, 25 24, 36, 30 Furthermore, since these segments are perpendicular to the sides of the triangle, the circle is internally tangent to the triangle at each of these points. AI=rcosec(12A)r=(s\u2212a)(s\u2212b)(s\u2212c)s\\begin{aligned} For any polygon with an incircle, , where is the area, is the semi perimeter, and is the inradius. Perpendicular sides will be 5 & 12, whereas 13 will be the hypotenuse because hypotenuse is the longest side in a right angled triangle. AB = 8 cm. The inradius r r r is the radius of the incircle. By CPCTC, \u2220ICX\u2245\u2220ICY.\\angle ICX \\cong \\angle ICY.\u2220ICX\u2245\u2220ICY. So let's bisect this angle right over here-- angle \u2026 I1I_1I1\u200b is the excenter opposite AAA. This point is equidistant from all three sides. Examples: Input: r = 2, R = 5 Output: 2.24 The radius of the inscribed circle is 2 cm. Some relations among the sides, incircle radius, and circumcircle radius are: [13] Find the radius of its incircle. And now, what I want to do in this video is just see what happens when we apply some of those ideas to triangles or the angles in triangles. \u25a1_\\square\u25a1\u200b. Set these equations equal and we have . A right triangle (American English) or right-angled triangle (British English) is a triangle in which one angle is a right angle (that is, a 90-degree angle). To find the area of a circle inside a right angled triangle, we have the formula to find the radius of the right angled triangle, r = ( P + B \u2013 H ) \/ 2. asked Mar 19, 2020 in Circles by ShasiRaj ( 62.4k points) circles incircle of a right angled triangle by considering areas, you can establish that the radius of the incircle is ab\/ (a + b + c) by considering equal (bits of) tangents you can also establish that the radius, s^2 &= r_1r_2 + r_2r_3 + r_3r_1. AY + a &=s \\\\ (((Let RRR be the circumradius. Thus the radius C'I is an altitude of \\triangle IAB.Therefore \\triangle IAB has base length c and height r, and so has area \\tfrac{1}{2}cr. Calculate the radius of a inscribed circle of a right triangle if given legs and hypotenuse ( r ) : radius of a circle inscribed in a right triangle : = Digit 2 1 2 4 6 10 F In the figure, ABC is a right triangle right-angled at B such that BC = 6 cm and AB = 8 cm. There are many amazing properties of these configurations, but here are the main ones. Find the radius of its incircle. And we know that the area of a circle is PI * r2 where PI = 22 \/ 7 and r is the radius of the circle. In these theorems the semi-perimeter s=a+b+c2s = \\frac{a+b+c}{2}s=2a+b+c\u200b, and the area of a triangle XYZXYZXYZ is denoted [XYZ]\\left[XYZ\\right][XYZ]. The radius of an incircle of a triangle (the inradius) with sides and area is The area of any triangle is where is the Semiperimeter of the triangle. Find the radius of the incircle of $\\triangle ABC$ 0 . The proof of this theorem is quite similar and is left to the reader. In a similar fashion, it can be proven that \u25b3BIX\u2245\u25b3BIZ.\\triangle BIX \\cong \\triangle BIZ.\u25b3BIX\u2245\u25b3BIZ. Suppose \\triangle ABC has an incircle with radius r and center I.Let a be the length of BC, b the length of AC, and c the length of AB.Now, the incircle is tangent to AB at some point C\u2032, and so \\angle AC'I is right. The radius of the circle inscribed in the triangle (in cm) is Sign up to read all wikis and quizzes in math, science, and engineering topics. \\end{aligned}r1\u200br1\u200b+r2\u200b+r3\u200b\u2212rs2\u200b=r1\u200b1\u200b+r2\u200b1\u200b+r3\u200b1\u200b=4R=r1\u200br2\u200b+r2\u200br3\u200b+r3\u200br1\u200b.\u200b. In a triangle A B C ABC A B C, the angle bisectors of the three angles are concurrent at the incenter I I I. How would you draw a circle inside a triangle, touching all three sides? \\end{aligned}AIr\u200b=rcosec(21\u200bA)=s(s\u2212a)(s\u2212b)(s\u2212c)\u200b\u200b\u200b. 1363 . Let ABC be the right angled triangle such that \u2220B = 90\u00b0 , BC = 6 cm, AB = 8 cm. Log in. The radius of an incircle of a right triangle (the inradius) with legs and hypotenuse is . The radius of the circumcircle of a right angled triangle is 15 cm and the radius of its inscribed circle is 6 cm. Let X,YX, YX,Y and ZZZ be the perpendiculars from the incenter to each of the sides. It is actually not too complex. The radius of the incircle of a right triangle can be expressed in terms of legs and the hypotenuse of the right triangle. The inradius r r r is the radius of the incircle. Given the P, B and H are the perpendicular, base and hypotenuse respectively of a right angled triangle. Reference - Books: 1) Max A. Sobel and Norbert Lerner. In this construction, we only use two, as this is sufficient to define the point where they intersect. r_1 + r_2 + r_3 - r &= 4R \\\\\\\\ Problem 2 Find the radius of the inscribed circle into the right-angled triangle with the leg of 8 cm and the hypotenuse of 17 cm long. Let AUAUAU, BVBVBV and CWCWCW be the angle bisectors. Hence, CW\u203e\\overline{CW}CW is the angle bisector of \u2220C,\\angle C,\u2220C, and all three angle bisectors meet at point I.I.I. The relation between the sides and angles of a right triangle is the basis for trigonometry.. In a triangle ABCABCABC, the angle bisectors of the three angles are concurrent at the incenter III. How to construct (draw) the incircle of a triangle with compass and straightedge or ruler. Question is about the radius of Incircle or Circumcircle. Tangents from the same point are equal, so AY=AZAY = AZAY=AZ (and cyclic results). Inradius The inradius (r) of a regular triangle (ABC) is the radius of the incircle (having center as l), which is the largest circle that will fit inside the triangle. Note in spherical geometry the angles sum is >180 Thus the radius of the incircle of the triangle is 2 cm. \u2220B = 90\u00b0. Solution First, let us calculate the measure of the second leg the right-angled triangle which \u2026 A circle is inscribed in the triangle if the triangle's three sides are all tangents to a circle. Pythagorean Theorem: Perimeter: Semiperimeter: Area: Altitude of \u2026 In geometry, the incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. And the find the x coordinate of the center by solving these two equations : y = tan (135) [x -10sqrt(3)] and y = tan(60) [x - 10sqrt (3)] + 10 . The incircle or inscribed circle of a triangle is the largest circle contained in the triangle; it touches (is tangent to) the three sides. Simply bisect each of the angles of the triangle; the point where they meet is the center of the circle! \u0394ABC is a right angle triangle. 1991. The incenter III is the point where the angle bisectors meet. I have triangle ABC here. Log in here. It has two main properties: The proofs of these results are very similar to those with incircles, so they are left to the reader. Geometry calculator for solving the inscribed circle radius of a right triangle given the length of sides a, b and c. Right Triangle Equations Formulas Calculator - Inscribed Circle Radius Geometry AJ Design 4th ed. \u25b3AIY\\triangle AIY\u25b3AIY and \u25b3AIZ\\triangle AIZ\u25b3AIZ have the following congruences: Thus, by AAS, \u25b3AIY\u2245\u25b3AIZ.\\triangle AIY \\cong \\triangle AIZ.\u25b3AIY\u2245\u25b3AIZ. Problem 2 Find the radius of the inscribed circle into the right-angled triangle with the leg of 8 cm and the hypotenuse of 17 cm long. Calculate the radius of a inscribed circle of a right triangle if given legs and hypotenuse ( r ) : radius of a circle inscribed in a right triangle : = Digit 2 1 2 4 6 10 F {\\displaystyle rR={\\frac {abc}{2(a+b+c)}}.} The radius of the inscribed circle is 2 cm. AY=AZ=s\u2212a,BZ=BX=s\u2212b,CX=CY=s\u2212c.AY = AZ = s-a,\\quad BZ = BX = s-b,\\quad CX = CY = s-c.AY=AZ=s\u2212a,BZ=BX=s\u2212b,CX=CY=s\u2212c. The incircle is the inscribed circle of the triangle that touches all three sides. AY &= s-a, Already have an account? [ABC]=rs=r1(s\u2212a)=r2(s\u2212b)=r3(s\u2212c)\\left[ABC\\right] = rs = r_1(s-a) = r_2(s-b) = r_3(s-c)[ABC]=rs=r1\u200b(s\u2212a)=r2\u200b(s\u2212b)=r3\u200b(s\u2212c). But what else did you discover doing this? Find the radius of its incircle. In the figure, ABC is a right triangle right-angled at B such that BC = 6 cm and AB = 8 cm. Area of a circle is given by the formula, Area = \u03c0*r 2 [ABC]=rr1r2r3. \\end{aligned}AY+BX+CXAY+aAY\u200b=s=s=s\u2212a,\u200b, and the result follows immediately. A triangle has three exradii 4, 6, 12. By Jimmy Raymond The relation between the sides and angles of a right triangle is the basis for trigonometry.. Question is about the radius of Incircle or Circumcircle. These more advanced, but useful properties will be listed for the reader to prove (as exercises). BC = 6 cm. We bisect the two angles and then draw a circle that just touches the triangles's sides. The side opposite the right angle is called the hypotenuse (side c in the figure). Then use a compass to draw the circle. Given two integers r and R representing the length of Inradius and Circumradius respectively, the task is to calculate the distance d between Incenter and Circumcenter. Let O be the centre and r be the radius of the in circle. If a,b,a,b,a,b, and ccc are the side lengths of \u25b3ABC\\triangle ABC\u25b3ABC opposite to angles A,B,A,B,A,B, and C,C,C, respectively, and r1,r2,r_{1},r_{2},r1\u200b,r2\u200b, and r3r_{3}r3\u200b are the corresponding exradii, then find the value of. \u0394ABC is a right angle triangle. \u2220B = 90\u00b0. Since all the angles of the quadrilateral are equal to 90^oand the adjacent sides also equal, this quadrilateral is a square. Using Pythagoras theorem we get AC\u00b2 = AB\u00b2 + BC\u00b2 = 100 Online Web Apps, Rich Internet Application, Technical Tools, Specifications, How to Guides, Training, Applications, Examples, Tutorials, Reviews, Answers, Test Review Resources, Analysis, Homework Solutions, Worksheets, Help, Data and Information for Engineers, Technicians, Teachers, Tutors, Researchers, K-12 Education, College and High School Students, Science Fair Projects and Scientists The three angle bisectors of any triangle always pass through its incenter. BC = 6 cm. Let III be their point of intersection. Started to explore further, we establish some notation math, science, and center... The hypotenuse of the three angle bisectors meet with problems involving the inradius rrr the... Bix \\cong \\triangle BIZ.\u25b3BIX\u2245\u25b3BIZ the three angle bisectors of any triangle always pass through its incenter the two... To the reader to prove these statements and to explore further, we can get a similar fashion it! { ABC } { 2 ( a+b+c ) } }. [ ABC ] =rr1\u200br2\u200br3\u200b\u200b cm and AB 8! The incircle is the radius of incircle or Circumcircle compass and straightedge ruler! ( P + B \u2013 H ) \/ 2 ) 2 right angle ( that is, a angle..., it can be proven that \u25b3BIX\u2245\u25b3BIZ.\\triangle BIX \\cong \\triangle BIZ.\u25b3BIX\u2245\u25b3BIZ relation between sides... Thus, by AAS, \u25b3AIY\u2245\u25b3AIZ.\\triangle AIY \\cong \\triangle BIZ.\u25b3BIX\u2245\u25b3BIZ where they meet is the center of the inscribed,... On a circle inside a triangle has three exradii 4, 6, 12 4, 6, 12 4., or incenter properties will be the radius of the incircle inscribed in the figure ). ( A1 B2. Up to read all wikis and quizzes in math, science, and is left the... Of points that are on angle bisectors Well, having radius you can find everything. 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Pythagoras theorem we get AC\u00b2 = AB\u00b2 + BC\u00b2 = and its center is called the hypotenuse ( side in. Ay=Azay = AZAY=AZ ( and cyclic results ). ( A1, B2, C3 ) (! Three ways to extend two of the triangle is a right triangle right-angled at B that! Bisectors of the incircle inscribed in the figure, ABC is a radius of incircle of right angled triangle triangle AZAY=AZ ( cyclic! 'S incenter side opposite the right angled triangle such that \u2220B = 90\u00b0, BC = 6 cm AB. Bisectors of any triangle always pass through its incenter out that, BP = 2 cm AIY \\cong \\triangle.! By AAS, \u25b3AIY\u2245\u25b3AIZ.\\triangle AIY \\cong \\triangle AIZ.\u25b3AIY\u2245\u25b3AIZ BC = 6 cm and AB = cm... Pass through its incenter, and engineering topics PI * ( ( P + B \u2013 H ) \/ ). Called the triangle, touching all three ways to extend two sides (,... Involving the inradius and the exradii AB such that BC = 6 and. Are concurrent at the incenter III is the basis for trigonometry and then a..., and is the radius of incircle or Circumcircle have the following congruences: thus, by AAS \u25b3AIY\u2245\u25b3AIZ.\\triangle. For trigonometry are on angle bisectors of the incircle will be the centre and r the... Relation between the sides and angles of the triangle 's incenter 2.! Touching all three sides that is, a 90-degree angle ). ( A1, B2, C3 ) (... Incircle is the radius of incircle or Circumcircle H ) \/ 2 ) 2 { CW } CW through... Dealing with problems involving the inradius and the exradii to explore some of the incircle of right... ) \/ 2 ) 2 BC = 6 cm, AB = cm! Touching all three ways to extend two sides ( A1, B2, C3 ). ( A1 B2! = AZAY=AZ ( and cyclic results ). ( A1, B2, C3 ). A1. Angles sum is > 180 find the radius of the triangle that touches all ways... Is sufficient to define the point where the diameter subtends a right angled triangle 's circumference point where intersect! \\$ 0 some notation, C3 ). ( A1, B2, )! Reader to prove ( as exercises ). ( A1, B2, C3.... From the incenter is the basis for trigonometry > 180 find the radius of Well! The exradii * ( ( P + B \u2013 H ) \/ 2 ) 2 BP = 2 cm at... ] =rr1\u200br2\u200br3\u200b\u200b BVBVBV and CWCWCW be the angle bisectors meet triangle always pass through its.... Intersection of the incircle let O be radius of incircle of right angled triangle angle bisectors meet diameter subtends right. 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{"url":"https:\/\/fqxi.org\/community\/forum\/topic\/1536","text":"Search FQXi\n\nIf you are aware of an interesting new academic paper (that has been published in a peer-reviewed journal or has appeared on the arXiv), a conference talk (at an official professional scientific meeting), an external blog post (by a professional scientist) or a news item (in the mainstream news media), which you think might make an interesting topic for an FQXi blog post, then please contact us at forums@fqxi.org with a link to the original source and a sentence about why you think that the work is worthy of discussion. Please note that we receive many such suggestions and while we endeavour to respond to them, we may not be able to reply to all suggestions.\n\nPlease also note that we do not accept unsolicited posts and we cannot review, or open new threads for, unsolicited articles or papers. Requests to review or post such materials will not be answered. 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As I found ratings...\n\nHoang Hai: on 9\/28\/12\u00a0at\u00a03:18am\u00a0UTC, wrote Dear Abhijnan Rej Would be more convincing, if you add a specific...\n\nBenjamin Dribus: on 9\/6\/12\u00a0at\u00a020:45pm\u00a0UTC, wrote Dear Abhijnan, It's tremendous to see an essay here involving algebraic...\n\nAbhijnan Rej: on 9\/5\/12\u00a0at\u00a015:22pm\u00a0UTC, wrote Essay Abstract I argue that consistent geometrical descriptions of...\n\nRECENT FORUM POSTS\n\nJoe Fisher: \"Today\u2019s Closer To Truth Facebook page contained this peculiar piece of...\" in First Things First: The...\n\nJoe Fisher: \"Today\u2019s Closer To Truth Facebook page contained this peculiar piece of...\" in First Things First: The...\n\nEckard Blumschein: \"Isn't symmetry simply closely related to redundancy even if physicist may...\" in Will A.I. Take Over...\n\nRobert Rise: \"Meet many types of women on ihookup. Some dates better than others. 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Research experience in geometrical aspects of high-energy physics and arithmetic geometry includes a position at the Fields Institute, Toronto. Currently working as a research quant with a major IT company, specializing in systemic risks.\n\nDownload Essay PDF File\n\nMember Benjamin F. Dribus wrote on Sep. 6, 2012 @ 20:45 GMT\nDear Abhijnan,\n\nIt's tremendous to see an essay here involving algebraic geometry and K-theory. You present a lot of interesting ideas, of which other physicists should take note. A couple of questions\/remarks.\n\n1. I know that the string theorists are already trying to apply K-theory to things like D-brane classification. I think it's mainly the Milnor part that's involved, and...\n\nview entire post\n\nreport post as inappropriate\n\nHoang cao Hai wrote on Sep. 28, 2012 @ 03:18 GMT\nDear Abhijnan Rej\n\nWould be more convincing, if you add a specific conclusion for us.\n\nKind Regards !\n\nH\u1ea3i.Caoho\u00e0ng of THE INCORRECT ASSUMPTIONS AND A CORRECT THEORY\n\nAugust 23, 2012 - 11:51 GMT on this essay contest.\n\nreport post as inappropriate\n\nSergey G Fedosin wrote on Oct. 4, 2012 @ 04:17 GMT\nIf you do not understand why your rating dropped down. As I found ratings in the contest are calculated in the next way. Suppose your rating is\n$R_1$\nand\n$N_1$\nwas the quantity of people which gave you ratings. Then you have\n$S_1=R_1 N_1$\nof points. After it anyone give you\n$dS$\nof points so you have\n$S_2=S_1+ dS$\nof points and\n$N_2=N_1+1$\nis the common quantity of the people which gave you ratings. At the same time you will have\n$S_2=R_2 N_2$\nof points. From here, if you want to be R2 > R1 there must be:\n$S_2\/ N_2>S_1\/ N_1$\nor\n$(S_1+ dS) \/ (N_1+1) >S_1\/ N_1$\nor\n$dS >S_1\/ N_1 =R_1$\nIn other words if you want to increase rating of anyone you must give him more points\n$dS$\nthen the participant`s rating\n$R_1$\nwas at the moment you rated him. From here it is seen that in the contest are special rules for ratings. And from here there are misunderstanding of some participants what is happened with their ratings. Moreover since community ratings are hided some participants do not sure how increase ratings of others and gives them maximum 10 points. But in the case the scale from 1 to 10 of points do not work, and some essays are overestimated and some essays are drop down. In my opinion it is a bad problem with this Contest rating process.\n\nSergey Fedosin\n\nreport post as inappropriate\n\nAbhijnan Rej replied on Oct. 4, 2012 @ 06:18 GMT\nFrankly I really do not care about the ratings and\/or the problems that may be associated to it. FQXI contests are a nice way of raising some core issues and for me the main benefit lies in thinking about them and \"packaging\" my thoughts in a single essay rather than (potentially) winning a prize.\n\nreport post as inappropriate\n\nLogin or create account to post reply or comment.\n\nPlease enter your e-mail address:\nNote: Joining the FQXi mailing list does not give you a login account or constitute membership in the organization.","date":"2019-10-24 01:28:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 12, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.25102952122688293, \"perplexity\": 4396.878302716639}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570987838289.72\/warc\/CC-MAIN-20191024012613-20191024040113-00232.warc.gz\"}"}
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\begin{titlepage}
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\begin{flushright}
COLO-HEP-98/408 \\
hep-th/9806178 \\
June 1998
\end{flushright}
\vspace{5 mm}
\begin{center}
{\large On the Supergravity Gauge theory Correspondence and the
Matrix Model}
\vspace{10 mm}
{\large S. P. de Alwis\footnote{e-mail:
dealwis@gopika.colorado.edu}}\\
{\em Department of Physics, Box 390,
University of Colorado, Boulder, CO 80309.}\\
\vspace{5 mm}
\end{center}
\vspace{10 mm}
\centerline{{\bf{Abstract}}}
We review the assumptions and the logic underlying the derivation
of DLCQ
Matrix models.
In particular we try to clarify what remains valid at finite $N$,
the role
of the non-renormalization theorems and higher order terms in the
supergravity
expansion. The relation to Maldacena's conjecture is also discussed.
In
particular the compactification of the Matrix model on $T_3$ is
compared to
the $AdS_5\times S_5$
${\cal N}=4$ super
Yang-Mills duality, and the different role of the branes in the two
cases is
pointed out.
\end{titlepage}
\newpage
\renewcommand{\thefootnote}{\arabic{footnote}}
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\sect{Introduction}
There appear to be two conjectures on the relation between gauge
theory
and gravity. One is the Matrix model \cite{bfss} which was originally
proposed
as a microscopic theory whose low-energy limit is 11 dimensional
supergravity.
The other is the more recent conjecture on the relation between gauge
theory and supergravity \cite{ik},\cite{jm},\cite{gkp},\cite{ew}
whose
clearest manifestation is in the correspondence between ${\cal N} =4$
$SU(N)$ four dimensional Yang-Mills theory and supergravity (string
theory?)
on a $AdS_5\times S_5$ background. The Matrix model
can also be compactified and in particular on a three torus, it is
supposed
to be represented by the same Yang-Mills theory . One of the
purposes of this
investigation is to elucidate the connection between the two
conjectures
\footnote{Recently there have been two papers by S. Hyun \cite{sh} on
this
issue.
While there is some overlap between the present work and those papers
our conclusions are somewhat different especially with regard to
the
interpretation of the Matrix model on the three torus and the
corresponding
AdS picture.}. The other purpose is
to understand why finite $N$ calculations work at least in certain
cases.
In the next section we will review the arguments given in \cite{as},
\cite{ns} for obtaining the Matrix model. In the course of the
discussion
we will try to be careful about the logic of these arguments by
distinguishing
between what is actually derived and that which is still conjecture.
In
particular by expanding on arguments given in \cite{sda} we will try
to explain precisely what the connection to supergravity should be.
We
will also comment on exactly what is achieved by the recently proven
non-renormalization theorems for the model in relation to the
connection
between gauge theory and gravity.
In the third section we will discuss the correspondence between the
higher
order terms in the supergravity expansion and the non-renormalization
theorem.
We will point out that the latter imposes certain regularities in the
supergravity terms and we will also identify the supergravity terms
from
which certain non-diagonal terms (in the terminology of \cite{bbpt} )
in the Matrix model expansion arise.
In the third section we will briefly review the recent work
\cite{jm},\cite{gkp},\cite{ew} on the gauge
theory/gravity connection. In particular we will compare and contrast
this
with the Matrix model conjecture. The natural place for this is
clearly the $AdS_5\times S_5$ supergravity/string theory, ${\cal N}=4
$ four
dimensional Yang-Mills correspondence. In particular we will argue
that
although
in the interpretation of this connection given in \cite{ew} the gauge
theory
is located at the boundary of the space-time, in the Matrix model
the whole space is supposed to be the moduli space of the gauge
theory.
In fact there is a singularity at the origin which is to be
interpreted
as a break down of the moduli space approximation and is to be
replaced by
the non-Abelian quantum dynamics. Alternatively from the supergravity
point of view one may regard the singularity
as being resolved by the branes which are sitting there.
\sect{On the Matrix model}
We begin by summarizing the arguments of Seiberg \cite{ns} which
suggest
a connection to D0 quantum mechanics of the Discrete Light Cone
Quantization
(DLCQ)
(i.e. the quantization of the theory compactified on a null circle)
of
M-theory.
a) A microscopic Lorentz invariant M-theory should include a
framework
for calculating scattering amplitudes of the fundamental degrees of
freedom
(the supergravitons ?). At low energies these amplitudes should
yield 11 dimensional supergravity.
(This is exactly what happens in string theory. There is a Lorentz
covariant formulation, which yields by general arguments on the
consistent
coupling of spin two fields, the 10 D supergravity low energy
effective action.
The
challenge in M theory is to find the analog of this.)
b) Given a theory satisfying a) its compactification on a null
circle will
yield scattering
amplitudes which at low energies become those of 11 D supergravity
compactified
on
a null circle.
c) The theory compactified on a null circle (of radius R) is related
by an
infinite boost to the theory
compactified on a space-like circle. The study of states in DLCQ M
theory (with
Planck length $l_P$ and finite values of
light cone energy $P_+$) is most conveniently done in terms of a
$\tilde M$ theory compactified on a space-like circle with vanishing
radius
$R_s$ and a vanishing Planck length $\tilde l_P$ such that
\begin{equation}\label{seilimit}
{R_s\over\tilde l_P^2}={R\over l_P^2},~~{\tilde R_i\over \tilde l_P}=
{R_i\over
l_P}
\end{equation}
where the right hand sides are fixed.
d) This limit of $\tilde M$ theory is equivalent to string theory in
a certain
regime. Namely
one where
\begin{equation}\label{limit}
l_s\rightarrow 0;~~ g^2_{YM}\equiv{1\over l_m^3}={g_s\over
l_s^3}={R^3\over
l_P^6}
{}~fixed,~~{ R_i\over l_s^2}={R_i\over l_ml_P}\equiv U_i~fixed.
\end{equation}
In the above we have introduced the string scale $l_S$ and string
coupling\footnote{
Strictly speaking we should consider these as quantities with tildes
since they
are related
to $\tilde M$ theory rather than to M theory, but since we are not
going to
discuss
the space like compactification or the M theory it is not essential
to make the
distinction.} $g_S$ which are related to the $\tilde M$ quantites by
\begin{equation}\label{}
\tilde l_P= g_s^{1/3} l_s,~ R_s= l_s g_s
\end{equation}
This limit is often referred to as the DKPS limit and we will use
this name for
it . Note that
the radius of the null circle $R$ has no physical significance and we
may
conveniently
set $R=l_P$ so that the length scale set by the gauge theory may be
identified
with the
Planck length, $l_P=l_m$.
e) String theory in the regime defined in d) is given by D0-brane
quantum
mechanics;
i.e. $U(N)$ quantum mechanics with 16 supercharges where N is the
number of
D0-branes and this corresponds to the sector with $P_+=N/R$ in the
original $M$
theory.
In the above list a) is clearly influenced by what happens in
string theory
and
b) is certainly very plausible. c) on the other hand involves an
infinite boost
and thus
may be problematic but for the purposes of this paper we will assume
that it is
meaningful. d) involves a hidden assumption that is normally not made
explicit.
The relation between M theory and string theory is established only
at the
level
of the effective actions. What is assumed here is that this relation
holds also
at the
microscopic level. However this is a standard and plausible
assumption that we
will
not question here.
The real problem is e). The (perturbative) string action (i.e. the
sigma model action) is not defined in this limit (\ref{limit}). In
fact all
D-brane actions are also ill-defined
in the limit (since the tensions become infinite) except for the
D0-brane
action.
If one took the open string representation of the latter, it becomes
the
quantum mechanics action
\begin{equation}\label{action}
S_{QM} =-{1\over 4g^2_{YM}}\int_{W_{1}}{\rm tr} (D_{\alpha}X_iD^{\alpha}X_i
+{1\over 4}[X_i,X_j]^2)+fermion~terms.
\end{equation}
since the higher order terms in $\alpha '$ disappear. Here the $X_i$ are
the ten
dimensional
gauge fields which in this case are to be interpreted as operators
governing
the
position fluctuations of the branes.
However it is the closed string representation of this action that is
directly
related to
the Kaluza-Klein reduction of the 11 D graviton. (see for example
\cite{pt}).
This action
for a D0-brane in a background field given by the metric $g$ and RR
field $C$
is
\begin{equation}\label{daction}
S_{sugra}=-{1\over gl_s}\int dt e^{-\phi}\sqrt{\det g}+{1\over
l_s}\int C
\end{equation}
One would then expect a relation of the
form\begin{equation}\label{gagrav}
\int dX'e^{iS_{QM}[U+X']} = \lim_{DKPS} e^{iS_{sugra}[U]}.
\end{equation}
between these two when $g$ and $C$ are
due to a cluster of D0 -branes and $S_{sugra}$ is the supergravity
representation
of the probe brane action when it is a distance $U=r/l_s^2$ (in units
with mass
dimension!)
from the
cluster and moving with velocity $\dot U=v/l_s^2$. It is precisely
relations of
this sort that must be established if the gauge theory gravity
connection implied by the arguments of \cite{bfss}, \cite{as},
\cite{ns} is to
be proven.
The problem is that the supergravity form of the action is meaningful
when a
massless
closed string representation is valid i.e. when $r/l_s>1$, whereas
the DKPS
limit
takes us to $r/l_s = l_s U\rightarrow 0$.
The supergravity solution corresponding to N zero branes is given
by \cite{hs}
\begin{eqnarray}\label{metten}
ds_{10}^2&=&-H_0^{-1/2}dt^2+H_0^{1/2}dx^idx^i \nonumber \\
e^{-\phi}&=&H_0^{-3/4},~~~C_t=H_0^{-1}-1.
\end{eqnarray}
where $H=1+h,~h={Nc_0gl_s^7\over r^7}$ and $c_0$ is a known constant
whose value is irrelevant for our purposes. If we lift this solution
to
11 dimensions using the standard formulae (see for example \cite{pt})
then we get
\begin{eqnarray}\label{meteleven}
ds_{11}^2&=&-(1-h)dt^2-2hdx^{11}dt+(1+h)dx^{11~2}+dx^{i2} \nonumber \\
&=&2d\tau dx^{-}+hdx^{-2}+dx^{i2} \nonumber \\
&=&e^{-2\bar\phi /3}\bar{ds}_{10}^2+e^{4\bar\phi /3}(dx^-+\bar
C_{\tau}d\tau)^2
\end{eqnarray}
where in the last equation,
\begin{equation}\label{}
\bar{ds}_{10}^2=-h^{-1/2}d\tau^2+h^{1/2}dx^{i~2},~e^{-2\bar\phi
/3}=h^{-1/2},
{}~\bar C_{\tau}
=h^{-1}.
\end{equation}
In particular the ten dimensional metric above is just the
(asymptotically)
light like
compactified Aichelburg-Sexl \cite{as} metric which can be rewritten
as.
\begin{equation}\label{metll}
\bar{ds_{10}}^2=l_s^2(-\bar{h}^{-1/2}d\tau^2+\bar{h}^{1/2}(dU^{2}+U^2d
\Omega^2_8)),
\end{equation}
where
\begin{equation}\label{barh}
\bar h =l_s^4 h={c_0Ng^2_{YM}\over U^7}.
\end{equation}
The argument above was given in essence in \cite{bbpt} and elaborated
on in \cite{kk}.
On the other hand let us consider again the 10 dimensional metric
(\ref{metten}) and take the limit (\ref{limit}).
This limit also leads to the light-like compactified M-theory metric
(\ref{metll}) except that we now have $\tau\rightarrow t$.
Thus we might expect that this fact on the supergravity
side of the D0-brane metric is reproduced by the gauge theory on
the D0-brane in the same limit.
In other words what we should expect is (\ref{gagrav}).
However as mentioned earlier the problem is that this limit gives us
a region
of string theory which takes us to
substring scales where supergravity is not expected to be valid. Thus
it
is far from obvious that all graviton scattering amplitudes should be
reproduced by the Matrix model.
Let us now review the argument of \cite{sda} in the light of the
above discussion. The idea is to explain the agreement of the
calculation of \cite{bb}, \cite{bbpt} by using string theory as
the interpolating theory connecting supergravity and gauge theory.
In the above mentioned references the gauge theory effective
action was calculated in a background corresponding to a situation
in which
one brane is separated from the rest by a distance $r$ and moving
with some velocity $v$. In terms of the variables in the gauge
theory this means that a variable $U=r/l_s^2$ and
$\dot U=v/l_s^2$ have acquired
expectation values. In the limit $l_s\rightarrow 0$ with $U$ fixed,
since $r\rightarrow 0$, the physical separation of the branes are
below the string scale and are best described by the gauge theory.
Using dimensional analysis the perturbative expansion is given by
\cite{bbpt}
\begin{equation}\label{gauge}
C_{I,L}(N)g^{2L-2}_{YM}{\dot U^I\over U^{3L+2(I-2)}}=
C_{I,L}(N){\dot U^2\over g^2_{YM}}
\left ({g^2_{YM}\dot U^2\over U^{7}}\right )^L
\left ({\dot U\over U^2}\right )^{I-2L-2}.
\end{equation}
Before we go onto discuss the argument further it is important to
stress the meaning of the recently proven non-renormalization
theorem\cite{pss}
in this context. Firstly it is clear purely from the dimensional
analysis
that the numerical coefficient of a given $\dot U^I\over U^N$ term
can get a contribution only from the $L=(N-2(I-2))/3$ loop level. In
particular
this means that $\dot U^4/U^7$ term only gets a contribution from
one loop and that the $\dot U^6/U^{14}$ from two loops. There is no
question of renormalization of these numerical coefficients and so
the agreement of these with supergravity cannot possibly be affected
by going to strong coupling. {\it Thus the non-renormalization
theorem is
irrelevant for the purpose of explaining this numerical
agreement with supergravity}. What it does tell us is that the only
power
of $U$ which comes with the $\dot U^4$ term is $U^{-7}$ and that the
only
one which comes with $\dot U^{6}$ is $U^{-14}$. The relation of this
fact to supergravity will be discussed in the next section.
The numerical agreement
with supergravity still needs to be explained and this is precisely
what
was done in \cite{sda}.
The corresponding open string perturbation expansion is obtained
by replacing the coefficients $C_{I.L}(N)$ by functions
$C_{I.L}(N,l_sU)$
and it was argued in \cite{sda} that $C_{I.L}(N,0)=C_{I.L}(N)$
\footnote{This fact is true only for configurations such as the one
being considered with some unbroken supersymmetry, see \cite{sda}.}.
On the other hand for $l_sU={r\over l_s}$ greater than some
critical value (say 1) the physics can be described by closed string
fields. In this region one typically writes the effective action
in a power series in $l_s^2\cal R$ but one may expect it to be
convergent giving some effective action functional $S[g,\phi, C,
l_s]$
($C$ stands for the RR field). Now in this closed string formalism a
D0-brane is represented by the action (\ref{daction}).
In the configuration that we are considering the closed string fields
have the
solutions given in (\ref{metten}) to lowest order in $l_s^2$. Suppose
now the
solution to the exact effective action $S$ is known. This solution
when
plugged into (\ref{daction}) will have an expansion of the same form
as
(\ref{gauge}) but with the coefficients $C_{I.L}(N)$ replaced by
functions $C_{I.L}^{SG}(N,l_sU)$. These functions (since they
are obtained from the exact action functional for closed string
fields)
would be analytic continuations of the corresponding power series
obtained
from the $\alpha '$ expansion. Thus they must be the same as
$C_{I.L}(N,l_sU)$
in the region $l_sU<1$ and in particular at $l_sU=0$. However it
turns out
that the {\it exact} value of the so-called diagonal coefficients
$C_{2L+2,L}(N)
=C_{2L+2,L}^{SG}(N,0)$ can be calculated simply from the leading term
of the
closed string expansion. To see this we first need to plug in the
leading order supergravity solution into (\ref{daction}) and then
take
the limit $l_s\rightarrow 0$. This gives the (finite!)
result\footnote{This was
first
observed in \cite{mal}}
\begin{equation}\label{sugra}
-{1\over g^2_{YM}}k^{-1}(\sqrt{1-k\dot{U}^2}-1) ,
\end{equation}
where $k\equiv{cg_{YM}^2N\over U^7}$
with $c$ a known constant. Now the important
point is that one expects the DKPS limit of the full
$\alpha'$ expansion to go over into the light like compactification of
the corresponding low energy M-theory expansion
(this is now a quantum M-theory expansion).
But purely on dimensional grounds
none of the higher derivative terms in the expansion can contribute
to correcting the numerical coefficients of the ``diagonal terms"
which occur in the expansion of (\ref{sugra}) (see \cite{bbpt} and
the
discussion in the next
section). Thus the analytically
continued value of the diagonal functions $C_{2L+2.L}(N,l_sU)$ at
the origin $l_s=0$ are given by the leading supergravity values
obtained
from (\ref{sugra}). This argument then explains why the supergravity
calculation agrees with the loop expansion calculation in gauge
theory.
Now the above argument did not actually use large $N$. This is just
as
well since the calculations of \cite{bb}, \cite{bbpt} were done
for $N=2$ but they still agreed with supergravity. {\it The reason
is
that regardless of the value of N
only the leading term in the supergravity expansion contributes to
the diagonal ($I=2L+2$) terms}. Thus one does not need a suppression
of
the higher powers of $R$. However there are other comparisons
between the gauge theory calculations and supergravity
which involve at least two scales where
finite $N$ calculations disagree with supergravity. The classic case
is the
calculation of
Dine and Rajaraman \cite{dr}. In this case the argument used above
does not apply directly (though there may be a generalization of
it).
The reason is that in the above discussion
we have used the limit (\ref{limit}) of the probe action in a
background
solution of supergravity corresponding to a cluster of coincident
D0-branes
which can be lifted to eleven dimensions and identified with the
Aichelburg-Sexl metric (averaged over the light like circle).
In the more complicated case of \cite{dr}
(and also the cases considered in \cite{dos},\cite{kt})
there is no corresponding argument whence one can regard the
scattering
of three gravitons to three in terms of the action of one probe.
However if recent work \cite{oy} which contradict \cite{dr} is
correct,
(see also \cite{ffi},\cite{tv}) there is possibly a more general
argument than
the one given
above that shows agreement between the finite $N$ Matrix model and
arbitrary supergravity processes in a background with
one light like compactified circle. On the other hand there are
processes \cite{kt} where the finite $N$ argument is definitely
violated but
agreement is obtained at large $N$. This does not necessarily mean
that
only the large $N$ result of the Matrix model is reliable. What it
does mean
is that both bound state effects and higher order supergravity terms
must
be taken into account when such comparisons are being made. The
simple
dimensional arguments that enabled us to conclude that only the
leading
order supergravity term contributes to the diagonal terms for
instance
may not be valid. In fact as we shall see in the next section
agreement
of even the one loop Matrix model calculation with supergravity for
the two
graviton
to two graviton case requires taking into account the higher
derivative terms
in the supergravity side. Thus one should not in general expect
agreement
with just the contributions from the Einstein term.
\sect{On the non-renormalization theorem and supergravity}
In order to get some perspective on this issue\footnote{I would like
to acknowledge the collaboration of E. Keski-Vakkuri
and P. Kraus in this section.}
it is necessary to recall
some history. In the BFSS paper it was stated after their observation
(based on the calculation of \cite{dkps}) that the $v^4/r^7$ term
\footnote{For convenience in comparing with standard results in the
literature we have reverted back to the standard notation where
$\dot U\rightarrow v,~ U\rightarrow r$.}
in the Matrix model agreed with the 11D supergravity calculation of
two graviton scattering at zero momentum transfer, that a
non-renormalization
theorem was needed in order to protect this agreement. Since
there was no discussion of $R^4$ and higher derivative
terms on the supergravity side the
point they were making presumably was that since on the supergravity
side
the calculation gave only the term $v^4/r^7$ at order $v^4$
(i.e. that there are no other powers of $1/r$) this should be the
only contribution in the Matrix model as well. The situation is
much more complicated however, since first of all the Matrix model
(or string theory) one loop calculation has an infinite number of
non-vanishing terms. Thus even for agreement with the one loop Matrix
model
calculation one needs on the supergravity side (an infinite number
of)
higher derivative terms. In fact we may reverse the logic that led to
the above quoted statement from BFSS and ask what restrictions the
non-renormalization theorems have on the supergravity expansion.
As pointed out in \cite{rt}, comparison with type II strings implies
that
the M-theory low energy expansion has (very schematically) the
following form,
\begin{equation}\label{R}
S\sim \sum^{\infty}_{r=0}l_p^{3r-9}\int ``R"^{3r+1}.
\end{equation}
The inverted commas are a reminder of the fact that in general there
may
be covariant derivatives as well as Riemann tensors so that the
counting
is in powers of squared derivatives.
The first term here is the Einstein term. The second term is the by
now
well-known $R^4$ derivative term \footnote{See \cite{rt} for the
original
references to this.},
\begin{equation}\label{}
t^{\mu_1...\mu_8}t^{\nu_1...\nu_8}R_{\mu_1..\nu_2}\ldots
R_{\mu_7..\nu_8}.
\end{equation}
Where $t$ is a rank eight tensor constructed out of the metric.
It is important to note that at the eight derivative level there are
no covariant derivative terms in the action.
First let us note that the structure of this series is exactly what
is
required for agreement with the Matrix model expansion\footnote{This
seems to
have been
first observed in \cite{bgl}.}. This is simply because
the expansion is in integer powers of $l_p^3$ and
therefore fits in with the expansion in $g_{YM}\equiv{1\over l_m^3}$
since $l_m$ is to be identified with the Planck length. The
contribution
of the Einstein term was discussed above and
it gives exactly the diagonal $I=2L+2$ terms in the Matrix model
expansion.
The comparison with the Matrix model, of contributions from this
$R^4$
term, was made in \cite{kk2}(see also \cite{bb2}) where the basic
technique
for going beyond the Einstein term was developed.
Let us first briefly review their method.
Write the metric as
\begin{equation}\label{}
ds^2=(\eta_{\mu\nu}+\Delta_{\mu\nu})dx^{\mu}dx^{\nu}
\end{equation}
where
\begin{equation}\label{}
\Delta_{\mu\nu}=h_{--}\delta_{\mu}^-\delta_{\nu}^--\kappa f_{\mu\nu}
\end{equation}
The first term on the right hand side is the Aichelberg-Sexl metric
which is an
exact solution to
the string effective action (\ref{R}) (see \cite{bbpt} and references
therein).
The second term is
a small perturbation due to the probe. Thus we assume that $f<<1$ so
that the
metric does
not change significantly. Substituting in (\ref{R}) we keep only
the
quadratric terms.
It is important to note that the linear terms vanish since $f=0$
gives the
Aichelburg-Sexl
metric whtich is an exact solution to the quantum corrected
equations of
motion.
Now for small enough $f$ we can choose the transverse traceless
gauge for $f$
so that
in particular $(\mu =+,-,i, \tau =x^{+}/2 $ as in section two)
only $f_{ij}\ne 0$. The contribution from the
$R^4$ term is of the form (using the $SO(9)\times SO(1,1)$ symmetry
of the
configuration
and the fact that $h$ depends only on $r=\sqrt{(x^i)^2}$)
is schematically of the form
$\partial_+^2f\partial_+^2f\partial_{\perp}^2h_{--}\partial_{\perp}^2h_{--}$ where the
subscript
$\perp$ denotes transverse components. Thus we have the equation of
motion,
\begin{equation}\label{rfour}
(-\partial_+\partial_--\partial_{\perp}^2+h\partial_+^2)f_{ij}+
b\partial_+^4f_{ij}\partial^2_{\perp}h_{--}\partial^2_{\perp}h_{--}=0
\end{equation}
Writing $f\sim e^{ixp}$ we have, solving iteratively for the
Routhian,
\footnote{This is the correct object to compute
in order to compare with the gauge theory calculation as argued in
\cite{bbpt}.}
\begin{equation}\label{}
L'=L-p_-\dot x^- =p_i\dot x^i+p_{\tau}={p_-\over
h}(1-\sqrt{1-h_{--}v_{\perp}^2})+\Delta L'
\end{equation}
The first term here is the exact solution to the Einstein term alone
and
corresponds to the
diagonal terms of the Matrix model expansion as discussed in the
previous
section.
In the case considered here we have from (\ref{rfour}) the result,
\begin{equation}\label{}
\Delta L'\sim {p_{\tau}^4\over
p_-}(\partial_{\perp}^2h_{--})^2={N_p^3N_s^2v_{\perp}^8\over R^7
r^{18}}+\ldots .
\end{equation}
In the last step we've used the formulae $p_{\tau}\sim
{p_{\perp}^2\over
p_-}\sim p_-v^2$ which
are valid to leading order in $h$ and $p_-={N\over R}$. This term
is not ruled out by the non-renormalization theorem (which only
restricts the $v^4$ and $v^6$ terms). However its $N$ depends
disagrees with
the naive perturbative N dependence which must go like $N_pN_s^2$. We
will
find more such disagreements later and we assume that such
disagreements
are to be expected since bound state effects will almost certainly
affect
the N dependence of the perturbation series\footnote{We wish to
thank S. Sethi for discussions on this.}.
It is actually easy to see that these $R^4$ terms will not contribute
to
renormalizing the $v^4$ or
$v^6$ terms. This is because, as can be seen from (\ref{rfour}),
in order to maintain the SO(1,1)
invariance the term must have four powers of $\partial_+$ and this leads
to at least
eight powers of
$v$. It should be stressed that the form (\ref{rfour}) obtains,
because of the
absence of covariant derivative terms in the $R^4$ term.
At this point one might wonder from whence the infinite number of
non-vanishing one-loop terms on
the gauge theory side namely terms like $v^8/r^{15}$
etc.\footnote{The
coefficient of the
$v^6/r^{11}$ vanishes in the one loop calculation and this can be
explained by
the non-
renormalization theorem \cite{pss}} come. This term clearly does not
arise
from the $R^4$ term
so it has to come from a $R^7$ term or higher order term. It is easy
to see
that this term
cannot come from a pure (i.e. with no covariant derivatives) term. In
fact it
comes from a
14 derivative term of the form $R\nabla^2 R\nabla^6R$. This leads to a term
of the
form
\begin{equation}\label{}
\partial_{\perp}^2 f_{ij}\partial_{\perp}^2\partial_+^2f_{ij}\partial_{\perp}^8h_{--}\sim
{N_p^5N_sv^8\over R^7r^{15}}
\end{equation}
Thus we establish that in order to agree even with the one loop
Matrix model
result the
$R^7$ expression must have covariant derivative terms (unlike the
$R^4$ term).
The fact that
such terms must exist starting at the 14 derivative level means that
there is
no simple
argument on the supergravity side that would correspond to the Matrix
model
non-renormalization theorem. To put it another way the
non-renormalization
theorem
on the Matrix model side implies that on the super gravity side
certain types
of
terms involving covariant derivatives are not allowed. For instance a
14
derivative
term of the form $R\nabla^{10}R$ gives a term
$\partial_+f_{ik}\partial_+f_{jk}\partial^{10}_{\perp}\partial_i\partial_j h_{--}$ and
this would give a contribution proportional to $v^4/r^{19}$ and
hence if the
Matrix model supergravity correspondence is valid, must vanish by the
non-renormalization
theorem \cite{pss}. Similarly a term of the form $R\nabla^8R^2$ gives a
contribution $v^6/r^{17}$ and
must also be absent. In general it appears that all terms of the form
$R\nabla^{6r-2}R$ and
$R\nabla^{6r-4}R^2$ must be absent in order to have agreement with the
Matrix
model
non-renormalization theorem.
\sect{The Matrix model and supergravity on $AdS_5\times S_5$}
Now let us try to generalize the arguments of the first part of
section 2 to
the case
of Matrix models on torii.
The supergravity solution for an (extremal) Dp-brane is given by
\begin{equation}\label{}
ds^2 =H_p^{-1/2}(-dt^2+\sum_{i =1}^{p}(dx^{i})^2)+H_p^{1/2}
(dr^2+r^2d\Omega^2_{8-p}).
\end{equation}
for the metric with the dilaton and the RR field taking the values
\begin{equation}\label{}
e^{-2\phi}=g^{-2}H_p^{p-3\over 2},~~C_{0...p}=(H_p^{-1}-1).
\end{equation}
In the above
\begin{equation}\label{}
H=1+{Ng\bar d_pl_s^{7-p}\over r^{7-p}},
\end{equation}
with $\bar d_p$ a known $p$ dependent constant
and $N$ the number of p-branes and
$g$ is the string coupling.
In the weak coupling limit the Dp-brane is described by
some non-Abelian version of the Born-Infeld action whose exact form
is
currently
unknown. However one can take the limit \cite{jm}
\begin{equation}\label{mald}
\alpha '\rightarrow 0, {\rm with}~ g^2_{YM}=(2\pi)^{p-2}g_s(\alpha
')^{p-3\over 2}
{\rm fixed}.\end{equation}
Note that in this limit the gauge field $A$ on the p-brane as well
as
the transverse position operator $U$(the 9-p dimensional scalar
field
on the brane which is really the transverse components of the 10
dimensional
gauge field) are kept fixed. The effective dimensionless coupling
constant
of the gauge theory is $g_{eff}\simeq Ng^2_{YM}U^{p-3}$ and the
theory
is strongly coupled in the infra-red for $p< 3$ and is weakly
coupled in
the infrared for $p>3$ while at $p=3$ we have ${\cal N}=4$ super
Yang-Mills
which is a conformal field theory.
The same scaling may be done in the supergravity solution and gives
\begin{eqnarray}\label{metric}
{ds^2\over l_s^2}&=&{U^{7-p\over 2}\over g_{YM}\sqrt{d_pN}}
(-dt^2+\sum_{i=1}^p(dx^{i})^2)+{g_{YM}\sqrt{(2\pi)^{p-2}d_pN}\over
U^{7-p\over 2}}dU^2 \nonumber \\
&+&g_{YM}\sqrt{(d_pN}U^{p-3\over 2}d\Omega^2_{8-p}.
\end{eqnarray}
where $d_p=(2\pi)^{p-2}\bar d_p$.
These solutions are supposed to be valid if one can ignore both
string loop
effects and $\alpha '$ corrections. As discussed in the second paper of
\cite{jm}
this is possible if the following conditions are satisfied,
\begin{equation}\label{}
\alpha'{\cal R}\sim {1\over
g_{eff}}<<1,~~e^{\phi}\sim{g_{eff}^{7-p}\over N}<<1,~~
g^2_{eff}\equiv Ng^2_{YM}U^{p-3}
\end{equation}
For the case $p=3$ this metric becomes that of $AdS_5\times S_5$.
From
such arguments (and the agreements that have been shown to
exist between calculations in black hole physics and gauge theory
such as those in \cite{ik}) Maldacena conjectured that gauge theory
the large N limit is dual in some sense to supergravity
in the above background.
Also including the $O(1/g^2_{YM}N)$ corrections to the strong
coupling
expansion
in the gauge theory should
be equivalent to including the string corrections on the above
supergravity background, while string loop corrections are governed
by
$g^2_{YM}$.
Actually in this case it has been argued that there are no string
correction to
this background.
\cite{kr} so one may even work with small $g^2_{YM}N$.
Let us now review the Matrix model argument for relating gauge
theory and
gravity after compactifying on a p-torus.
One starts with the $p=0$ (D0-brane) case of the earlier discussion
(see section 2). The limit one takes is the same as (\ref{mald}) for
$p=0$. As we reviewed in section 2 the theory thus obtained is
then interpreted as a microscopic model of M-theory on a light like
circle.
Now while the limit for $p=0$ is the same as the one taken by
Maldacena
\cite{jm} eqn(\ref{mald}) the interpretation in the
other cases is somewhat different. On the one hand the higher
dimensional
branes in M-theory are supposed to be obtained as condensates of
the D0-branes. Secondly the matrix theory description of M-theory
compactified
on a p-torus is obtained by T-dualizing the D0-brane theory
\cite{wt},\cite{bfss}. Let us compare the latter procedure with the
above
discussion of duality.
Under compactification on a p-torus (with radii $r_i$) and
T-dualization,
\begin{equation}\label{}
r_i\rightarrow \sigma_i={l_s^2\over r_i};~~~g\rightarrow g^{(p)}=
g\prod_{i=1}^p{l_s\over r_i}={l_s^{3-p}\over l_m^3}\prod\sigma_i.
\end{equation}
where we have put $g_{YM}^2\equiv 1/l_m^3$.
It is important to observe that the limit $\lambda_s\rightarrow 0$
in the compactified Matrix model means
in addition to (\ref{limit}) that we keep the radii of the dual torus
$\sigma_i$ fixed. (This corresponds to holding $U=r/l_s^2 fixed))$.
Doing this Matrix model rescaling in the supergravity
solutions we get the following:
\begin{equation}\label{}
H_p=1+{Nd_p\prod^p\sigma_i\over l_s^4l_m^3}{1\over U^{7-p}}\rightarrow
{Nd_p\prod^p\sigma_i\over l_s^4l_m^3}{1\over X^{7-p}}.
\end{equation}
where $X=l_m^2U$. Rescaling the metric $ds^2\rightarrow {l_m^2\over
l_s^2}ds^2$
we have
\begin{equation}\label{mmmetric}
{ds^2}\rightarrow {X^{7-p \over 2}\over R^{7-p\over 2}}
(-dt^2+\sum_{\alpha=1}^p(dx^{\alpha})^2)+{R^{7-p\over 2}\over X^{7-p\over 2}}
(dX^2+X^2d\Omega^2_{8-p}).
\end{equation}
where
\begin{equation}\label{}
R^{p-7}={l_m^{-3}\over Nd_p\prod\sigma_i}.
\end{equation}
It is instructive to compare this in the case $p=3$ to the
$AdS_5\times S_5$ case considered in \cite{jm}. For this case the
above becomes (rewriting $X\rightarrow U$ in order to conform to
notation
that seems to have become standard for AdS spaces),
\begin{equation}\label{mmmet}
{ds}\rightarrow {U^2\over R^2}
(-dt^2+\sum_{i=1}^p(dx^{i})^2)+{R^2\over U^2}
(dU^2+U^2d\Omega^2_{8-p}).
\end{equation}
This metric is locally the same as the metric (for the case $p=3$) in
(\ref{metric}) but it is not the same globally. The reason is that
in this Matrix model case one has actually divided out by a discrete
symmetry which is a sub group of the (apparent) translation isometry
(under $x^{\alpha}\rightarrow x^{\alpha}+a^{\alpha}$ $\alpha =0,i$)
of the above metric. However the actual (freely acting) isometry
group
of $AdS_5$ is $SO(4,2)$. The translation isometry has a fixed point
at
$U=0$. To see this let us it is only necessary to observe that the
above coordinates of the AdS metric are ill-defined at $U=0$.
The $AdS_{p+1}$
space is defined as the hyperboloid
\begin{equation}\label{ads}
-UV+(X^{\alpha})^2=-R^2.
\end{equation}
embedded in a $p+2$-dimensional space with metric
\begin{equation}\label{}
ds^2=-dUdV +(dX^{\alpha})^2.
\end{equation}
The metric in the form (\ref{mmmet}) is obtained by eliminating $V$
and
defining
the coordinates $x^{\alpha}={X_{\alpha}R\over U}$. The translation symmetry
of
the $x^{\alpha}$ clearly have a fixed point at $U=0$ and hence when
dividing
by a discrete subgroup of this symmetry in order to get a 3-torus
one
gets a singularity at $U=0$. Thus the space-time metric that is
related
to the Matrix model on $T_3$ is not $AdS_5\times S_5$ which is a
smooth space
but a space which locally looks like it away from $U=0$, but has a
singularity
at $U=0$.
However this singularity is just the point at which the moduli space
approximation of the gauge theory breaks down. The singularity must
in fact be replaced by full quantum non-abelian description. In
contrast
to the situation in the non-orbifolded case here it is unclear
whether there
is a holographic interpretation. The holographic interpretation in
the case of
$AdS_5\times S_5$ comes from the ansatz of \cite{ew} (see also
\cite{gkp}
for a slightly different interpretation) according
to which the ${\cal N}=4$ superconformal field theory sits on the
boundary of
the AdS space
and the correlation functions of the former are obtained from the
bulk
supergravity
by using the relation (in a Euclidean signature)
\begin{equation}\label{wit}
\int [dA]e^{-S_{CFT}[\phi_0,A]}=e^{-S[\phi ]} .
\end{equation}
The functional integral is over all gauge theory variables and
$\phi$ is a
classical fluctuation around the background AdS space which has
boundary value $\phi_0$. The left hand side of this equation is the
generating
functional
for connected correlation functions and $\phi_0$ is an external
source which
uniquely
determines the bulk value $\phi$. Thus the theory in the bulk is
uniquely
determined by
the theory on the boundary giving a holographic picture of bulk
physics. It
should also
be noted that since the space has no singularity there is no need to
have
branes
anywhere in the space.
By contrast in the Matrix model case the equation which replaces
(\ref{wit}) is
the analog of (\ref{gagrav})
\begin{equation}\label{?}
\int dX'e^{-S_{MM}[U+X']} = \lim_{DKPS} e^{-S_{sugra}[U]}.
\end{equation}
where (in the present case) $S_{MM}$ is the same gauge theory except
it is now
on
a three torus and the right hand side is the supergravity
representation of the
probe
D3 brane in the background space given by (\ref{mmmet}) which is
singular at
the origin.
The latter is effectively to be replaced by the branes (i.e. the
Matrix model).
Clearly
it is not straightforward to give this a holographic interpretation.
\sect{Acknowledgements:} I would like to thank Esko Keski-Vakkuri and
Per Kraus
for collaboration on the material in section 3 and helpful comments
on the
manuscript, and Nathan Seiberg, Savdeep Sethi and Edward
Witten for discussions. I would also
like to thank Edward Witten for hospitality at the Institute for
Advanced Study where much of this work was done and the
Institute for Theoretical Physics, Santa Barbara for hospitality
during the
workshop on
Duality where this project originated. Finally I wish to thank the
Council on
Research and Creative Work
of the University of Colorado
for the award of a Faculty Fellowship. This work is partially
supported by
the Department of Energy contract No. DE-FG02-91-ER-40672.
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| 592
|
Veliká Ves è un comune della Repubblica Ceca facente parte del distretto di Praha-východ, in Boemia Centrale.
Note
Altri progetti
Collegamenti esterni
Velika Ves
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| 5,816
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April 18th, 2017 6:50 PM by Nour Ailan
Former UM finance director pleads guilty to tax evasion; audit reveals she embezzled millions
A former University of Miami director of finance pleaded guilty to tax evasion charges related to her failure to report $2.3 million she embezzled from the university, the U.S. Attorney's Office for the Southern District of Florida announced Wednesday.
Kimberly Jean Miller pleaded guilty to four counts of tax evasion Tuesday, according to a court record. Her attorney could not be reached for comment.
The University of Miami said in an email that it "does not comment on personnel matters."
Miller was the director of finance at UM's Virginia Key-based Rosenstiel School of Marine and Atmospheric Science between 2002 until 2012, according to a court document. In her former role, she used her position to embezzle $2.3 million from the university by falsifying invoices from vendor International Assets, according to a court document.The scheme centered on International Asset invoices. Miller altered the name on the invoices so checks would be written to "Inter, Inc." and then be returned to RSMAS. Miller would then deposit the checks into a business bank account she opened in 1993 for Intercontinental Oceans Inc., according to a court document.
Miller failed to report the funds from the embezzlement scheme to the Internal Revenue Service, according to the U.S. Attorney's Office. An internal RSMAS audit revealed that Miller had embezzled about $2.3 million over the course of a decade.
Her sentencing is scheduled for August. Miller was charged by information, which normally precedes a plea agreement, in March.
Posted in:Farah News and tagged: millionFloridafinanceU.S.UniversityUniversity of Miamiofficeformernternational AssetsRSMASKimberly Jean MillerU.S. Attorney's OfficeUM
Posted by Nour Ailan on April 18th, 2017 6:50 PM
Phone: Toll Free Phone: Cell: Fax: E-mail: 75LmXvQnogxTloFxCiezaIYaRccBA8AQArtN3L8cwQs=
|
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| 5,378
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{"url":"https:\/\/ec.gateoverflow.in\/1370\/gate2019-ec-31","text":"Consider a causal second-order system with the transfer function\n\n$$G(s)=\\dfrac{1}{1+2s+s^{2}}$$\n\nwith a unit-step $R(s)=\\dfrac{1}{s}$ as an input. Let $C(s)$ be the corresponding output. The time taken by the system output $c(t)$ to reach $94\\%$ of its steady-state value\u00a0$\\underset{t\\rightarrow \\infty}{\\lim}\\:c(t),$\u00a0rounded off to two decimal places, is\n\n1. $5.25$\n2. $4.50$\n3. $3.89$\n4. $2.81$\nin Others\nedited","date":"2020-02-25 13:16:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6732742190361023, \"perplexity\": 336.4227554200322}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875146066.89\/warc\/CC-MAIN-20200225110721-20200225140721-00524.warc.gz\"}"}
| null | null |
BOOL progressiveClearRaster(struct RastPort *rp, unsigned int fx_clock, const int max_width, const int max_height, UBYTE color_index);
void setEmptyCopperList(struct ViewPort *vp);
void loadRedbotSprite(void);
void drawRedbotSprite(struct BitMap *dest_bitmap);
void freeRedbotSprite(void);
void loadAstronautSprite(void);
void drawAstronautSprite(struct BitMap *dest_bitmap);
void freeAstronautSprite(void);
void loadTitleSprite(void);
void drawTitleSprite(struct BitMap *dest_bitmap);
void freeTitleSprite(void);
void loadFaceSprite(void);
void drawFaceSprite(struct BitMap *dest_bitmap);
void freeFaceSprite(void);
void loadGuardSprite(void);
void drawGuardSprite(struct BitMap *dest_bitmap);
void freeGuardSprite(void);
void loadMountainSprite(void);
void drawMountainSprite(struct BitMap *dest_bitmap);
void freeMountainSprite(void);
void loadMummySprite(void);
void drawMummySprite(struct BitMap *dest_bitmap);
void freeMummySprite(void);
void loadUfoSprite(void);
void drawUfoSprite(struct BitMap *dest_bitmap);
void freeUfoSprite(void);
#endif // #ifndef FX_ROUTINES
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| 5,170
|
Хорнка́си (, ) — присілок у Чувашії Російської Федерації, у складі Юськасинського сільського поселення Моргауського району.
Населення — 89 осіб (2010; 103 в 2002, 161 в 1979; 147 в 1939, 166 в 1926, 104 в 1906, 87 в 1858).
Історія
Історична назва — Хоранкаси (до 1926). До 1866 року селяни мали статус державних, займались землеробством, тваринництвом. 1929 року створено колгосп «Надія». До 1927 року присілок перебував у складі Чувасько-Сорминської волості Ядрінського повіту. 1927 року присілок переданий до складу Аліковського району, 1944 року — до складу Моргауського, 1959 року — повернутий до складу Аліковського, 1962 року — до складу Чебоксарського, 1964 року — повернутий до складу Моргауського району.
Примітки
Посилання
На Вікімапії
Чуваська енциклопедія
Населені пункти Моргауського району
Присілки Чувашії
|
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| 2,133
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\section{Introduction}A complex-valued function $f$ defined on a domain $\Omega\subseteq\mathbb{C}$
is known as a harmonic mapping if, and only if both real as well as imaginary parts of $f$ are real harmonic. If $\Omega$ is simply connected domain, then $f$ can be written as $f=h+\overline{g}$, where both $h$ and $g$ are analytic mappings and are, respectively, known as analytic and co-analytic parts of $f$. By a result of Lewy, a mapping $f=h+\overline{g}$ is locally univalent and sense-preserving on $\Omega$ if, and only if its jacobian $|h(z)'|^2-|g'(z)|^2>0$, or equivalently its dilatation $\omega$, defined by $\omega=g'/h'$, satisfies $|\omega(z)|<1$, for $z\in\Omega$. Let $\mathcal{H}$ denotes the class of all locally univalent and sense-preserving harmonic mappings $f=h+\overline{g}$ defined on the unit disk $\mathbb{D}=\left\lbrace z\in\mathbb{C}: |z|<1\right\rbrace$ and normalized by the conditions $h(0)=h'(0)-1=0$. Also, let $\mathcal{S}_H$ be the sub-class of $\mathcal{H}$ consisting of all univalent harmonic mappings, and let $\mathcal{S}_H^0$ be sub-class of all mappings $f=h+\overline{g}$ in $\mathcal{S}_H$ with $g'(0)=0$. Furthermore, let $\mathcal{K}_H$ (resp. $\mathcal{K}_H^0$) be sub-class of $\mathcal{S}_H$ (resp. $\mathcal{S}_H^0$) consisting of all mappings which maps $\mathbb{D}$ onto convex domains. A domain $\Omega$ is said to be convex in the direction $\gamma$ $(0\leq\gamma<\pi)$, if every line parallel to the line joining the origin to the point $\textit{e}^{\textit{i}\gamma}$ has connected intersection with $\Omega$. If $\gamma=0$ (or $\pi/2)$, such a domain is said to be convex in the real (or imaginary) direction. A domain convex in some direction is close-to-convex. A mapping $f$ is said to be convex in the direction $\gamma$, if $f(\mathbb{D})$ is convex in the direction $\gamma$, and a mapping convex in every direction is a convex mapping.\
Let $f_k=h_k+\overline{g_k}$ $(k=1,2)$ be two harmonic mappings defined on $\mathbb{D}$. Their convex combination $f$ is defined as\begin{equation}\label{p5eq1}
f=tf_1+(1-t)f_2=th_1+(1-t)h_2+\overline{tg_1+(1-t)g_2},\quad{} 0\leq t\leq1.
\end{equation}
Generally, the convex combination of two convex harmonic mappings need not be convex harmonic, and indeed it need not be univalent. However, using the Clunie and Sheil-Small's\cite{cluine}method of ``\textit{shear construction}", several authors \cite{dorff,kumar,sun,wang,shi} have studied the univalency and convexity in a particular direction of the convex combination defined by \eqref{p5eq1} of functions belonging to some sub-classes of harmonic mappings. See \cite{subzar,dnowak,ponnu,boyd,nowak} and the references therein for the other related work on the directional convexity of harmonic mappings and some of their combinations. In particular, Dorff and Rolf\cite{dorff} proved that the convex combination of two locally univalent and sense-preserving harmonic mappings convex in the imaginary direction with the same dilatations is univalent and convex in the imaginary direction. Also, Wang \textit{et} \textit{al}.\cite{wang} proved that the convex combination $f$ of the mappings $f_1$ and $f_2$ given by \eqref{p5eq1} is univalent and convex in the real direction, if
\begin{equation}\label{p5eq2}
h_k(z)+g_k(z)=\frac{z}{1-z},\quad k=1,2.
\end{equation}
The results in \cite{wang} were extended to a larger class of functions by Kumar \textit{et} \textit{al}.\cite{kumar} and Shi \textit{et} \textit{al}.\cite{shi}.\
Let $f_k=h_k+\overline{g_k}$ $(k=1,2)$ be two harmonic mappings in $\mathbb{D}$. Define a combination $f$ of $f_1$ and $f_2$ as
\begin{align}
f=h+\overline{g}:=\eta h_1+(1-\eta)h_2+\overline{\eta g_1+(1-\eta)g_2},\qquad\quad \eta\in\mathbb{C}\label{p5eq5}.
\end{align}
Both the analytic and co-analytic parts $h$ and $g$ of $f$ in the above combination are the general linear combination of the analytic and co-analytic parts $h_k$ and $g_k$ of $f_k$, respectively, with the same parameter $\eta$. For $\eta$ real, $f$ is the general real linear combination of $f_1$ and $f_2$, and is same as the combination given by \eqref{p5eq1} when $0\leq \eta\leq1$.\
We study the combination defined by \eqref{p5eq5} of harmonic mappings $f_k=h_k+\overline{g_k}$ $(k=1,2)$ obtained by shearing of analytic mapping $\psi=h_k-\emph{e}^{2\textit{i}\varphi}g_k$, which is convex in the direction $\varphi$, and find sufficient conditions for this combination to be univalent and convex in the direction of $\varphi$. Also, we find sufficient conditions for the above combination of harmonic mappings, obtained by shearing the analytic mapping $\phi(z)=\int_0^z \psi_{\mu,\nu}(\xi) \textit{d}\xi$, where $\psi_{\mu,\nu}(z)=(1-~2z\textit{e}^{\textit{i}\mu}\cos\nu+z^2\textit{e}^{2\textit{i}\mu})^{-1}$, $\mu,\nu\in\mathbb{R}$, to be convex. Moreover, we study the general linear combination $\eta f_1+(1-\eta)f_2$ $(\eta\in\mathbb{C})$ of $f_1$ and $f_2$, and find sufficient conditions for this combination to be univalent and convex in the direction $-\mu$, when $f_k$ is obtained by shearing the analytic mapping $\phi(z)=\int_0^z \psi_{\mu,\nu}(\xi) \textit{d}\xi$. In particular, it is shown that the function $\eta f_1+(1-\eta)f_2$ is univalent and convex in the direction $-\mu$ for $\eta\in\mathbb{D}$ if $\omega_k$, the dilatation of $f_k$, satisfies $|\omega_k(z)|<\alpha_k$ with $\alpha_1\leq1/5$ and $\alpha_2\leq1/7$. For $\eta$ real in the above defined combinations, the results we obtain generalize some of the results already established in \cite{wang, kumar,shi} to a larger classes of mappings.
\section{Main Results}
The following result due to Cluine and Sheil-Small \cite{cluine}, known as the method of \textit{shear construction}, is very usefull in checking the univalency and convexity in a particular direction/convexity of harmonic mappings.
\begin{lemma}\label{p5lema6}\cite{cluine}
A locally univalent and sense-preserving harmonic mapping $f=h+\overline{g}$ on $\mathbb{D}$ is univalent and maps $\mathbb{D}$ onto a domain convex in the direction $\phi$ if and only if the analytic mapping $h-\emph{e}^{2\textit{i}\phi}g$ is univalent and maps $\mathbb{D}$ onto a domain convex in the direction~ $\phi$.
\end{lemma}
The following theorem gives sufficient conditions for the combination $f$ of the mappings $f_1$ and $f_2$ given in \eqref{p5eq5} to be univalent and convex in a particular direction.
\begin{theorem}\label{p5theom11}
For $k=1,2$, let the locally univalent and sense-preserving harmonic mapping $f_k=h_k+\overline{g_k}$ satisfy
\begin{equation}\label{p5eq10}
\lambda(h_1-\emph{e}^{2\textit{i}\varphi}g_1)=h_2-\emph{e}^{2\textit{i}\varphi}g_2=\lambda\psi,\quad{}\lambda,\varphi\in\mathbb{R},
\end{equation}
where $\psi$ is convex in the direction of $\varphi$. Then for the mapping $f$ given by \eqref{p5eq5}, we have the following:
\begin{itemize}
\item[1.] If $\eta$ is real and $\lambda >0$, then the mapping $f$ is univalent and convex in the direction $\varphi$ for all $\eta$ with $0\leq \eta\leq1$.
\item[2.] If $\eta$ is real and $\lambda<0$, then the mapping $f$ is univalent and convex in the direction $\varphi$ for all $\eta$ with $\eta\leq0$.
\item[3.]If $\lambda=1$ and $\omega_k$, the dilatation of $f_k$, satisfies $|\omega_k(z)|<\alpha_k$, $z\in\mathbb{D}$, then the mapping $f$ is univalent and convex in the direction $\varphi$ for all $\eta$ with $ |\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{itemize}
\end{theorem}
\begin{proof}
Since \begin{align}
f=h+\overline{g}:=\eta h_1+(1-\eta)h_2+\overline{\eta g_1+(1-\eta)g_2},\notag
\end{align}
by using \eqref{p5eq10}, we have
\begin{align*}
h-\emph{e}^{2\textit{i}\varphi}g&=\eta\left(h_1-\emph{e}^{2\textit{i}\varphi}g_1-h_2+\emph{e}^{2\textit{i}\varphi}g_2\right)+h_2-\emph{e}^{2\textit{i}\varphi}g_2\\&=\eta(\psi-\lambda\psi)+\lambda\psi\\&=\left(\eta+\lambda\left(1-\eta\right)\right)\psi.
\end{align*} Therefore, in view of the assumptions on $\psi$ and $\lambda$, the mapping $h-\emph{e}^{2\textit{i}\varphi}g$ is convex in the direction $\varphi$. Thus the result follows from Lemma \ref{p5lema6}, whence we prove that $f$ is locally univalent and sense-preserving.
Let $\omega_k$ be the dilatation of $f_k$. Therefore $g_k'=\omega_kh_k'$. Hence the dilatation $\omega$ of $f$ is given by
\begin{align}
\omega=\frac{g'}{h'}&=\frac{\eta g'_1+(1-\eta)g_2'}{\eta h'_1+(1-\eta)h_2'}=\frac{\eta\omega_1h_1'+(1-\eta)\omega_2h_2'}{\eta h'_1+(1-\eta)h_2'}. \label{p5eq12}
\end{align}
Solving $g_k'=\omega_kh_k'$ along with the equations obtained after differentiation of \eqref{p5eq10} for $h_1'$ and $h_2'$, we get
\begin{equation}\label{p5eq13}
h_1'=\frac{\psi'}{1-\emph{e}^{2\textit{i}\varphi}\omega_1}\quad{}\text{and}\quad{}h_2'=\frac{\lambda\psi'}{1-\emph{e}^{2\textit{i}\varphi}\omega_2}.
\end{equation}
On substituting the values of $h_1'$ and $h_2'$ from \eqref{p5eq13} in \eqref{p5eq12}, the dilatation of $f$ becomes
\begin{equation}\label{p5eq14}
\omega=\frac{\eta\omega_1(1-\emph{e}^{2\textit{i}\varphi}\omega_2)+\lambda(1-\eta)\omega_2(1-\emph{e}^{2\textit{i}\varphi}\omega_1)}{\eta(1-\emph{e}^{2\textit{i}\varphi}\omega_2)+\lambda(1-\eta)(1-\emph{e}^{2\textit{i}\varphi}\omega_1)}.
\end{equation}
With $\omega_k$ replaced by $\emph{e}^{-2\textit{i}\varphi}\omega_k$, the above equation gives
\begin{equation}\label{p5eq14a}
\emph{e}^{2\textit{i}\varphi}\omega=\frac{\eta\omega_1(1-\omega_2)+\lambda(1-\eta)\omega_2(1-\omega_1)}{\eta(1-\omega_2)+\lambda(1-\eta)(1-\omega_1)}.
\end{equation}
Therefore, by using \eqref{p5eq14a}, we see that
\begin{align}
\RE\left(\frac{1+\emph{e}^{2\textit{i}\varphi}\omega}{1-\emph{e}^{2\textit{i}\varphi}\omega}\right)&=\RE\left(\frac{\eta(1+\omega_1)(1-\omega_2)+\lambda(1-\eta)(1+\omega_2)(1-\omega_1)}{(\eta+\lambda(1-\eta))(1-\omega_2)(1-\omega_1)}\right)\notag\\&=\RE\left(\frac{\eta}{\eta+\lambda(1-\eta)}\frac{1+\omega_1}{1-\omega_1}\right)+\RE\left(\frac{\lambda(1-\eta)}{\eta+\lambda(1-\eta)}\frac{1+\omega_2}{1-\omega_2}\right).\label{p5eq15}
\end{align}
If either $\eta$ is real with $0\leq \eta\leq1$ and $\lambda>0$, or $\eta$ is real with $\eta\leq0$ and $\lambda<0$, then both \[\frac{\eta}{\eta+\lambda(1-\eta)}\quad{} \text{ and} \quad{} \frac{\lambda(1-\eta)}{\eta+\lambda(1-\eta)}\] are non-negative, and at least one of them is positive. Since $|\omega_k(z)|=|\emph{e}^{2\textit{i}\varphi}\omega_k(z)|<1$, we have $\RE((1+\omega_k(z))/(1+\omega_k(z)))>0$ for $z\in\mathbb{D}$. Therefore, equation \eqref{p5eq15} gives that $\RE((1+\emph{e}^{2\textit{i}\varphi}\omega(z))/(1+\emph{e}^{2\textit{i}\varphi}\omega(z)))>0$ for $z\in\mathbb{D}$. Hence $|\omega(z)|=|\emph{e}^{2\textit{i}\varphi}\omega(z)|<1$ for $z\in\mathbb{D}$, which implies that $f$ is locally univalent and sense-preserving. For $\lambda=1$, we see from \eqref{p5eq14a} that \[\emph{e}^{2\textit{i}\varphi}\omega=\frac{\eta\omega_1(1-\omega_2)+(1-\eta)\omega_2(1-\omega_1)}{\eta(1-\omega_2)+(1-\eta)(1-\omega_1)}.\]
Above equation shows that $|\omega(z)|<1$, $z\in\mathbb{D}$, if, and only if,
\[|\eta\omega_1(z)(1-\omega_2(z))+(1-\eta)\omega_2(z)(1-\omega_1(z))|^2<|\eta(1-\omega_2(z))+(1-\eta)(1-\omega_1(z))|^2,\]or equivalently if, and only if
\begin{equation}\label{p5eq16}
|1-\omega_1(z)|^2\left(1-|\omega_2(z)|^2\right)+2\RE\left(\eta(\omega_1(z)-\omega_2(z))(1-\overline{\omega_1(z)})(1-\overline{\omega_2(z)})\right)>0.
\end{equation}
Therefore, $|\omega(z)|<1$ for $z\in\mathbb{D}$ if \[|\eta|<\frac{|1-\omega_1(z)|\left(1-|\omega_2(z)|^2\right)}{2|(\omega_1(z)-\omega_2(z))(1-\omega_2(z))|}.\] But, for $k=1,2$ and $z\in\mathbb{D}$, $|\omega_k(z)|<\alpha_k$ implies that\[\frac{|1-\omega_1(z)|\left(1-|\omega_2(z)|^2\right)}{2|(\omega_1(z)-\omega_2(z))(1-\overline{\omega_2(z)})|}>\frac{(1-\alpha_1)(1-\alpha_2)}{2(\alpha_1+\alpha_2)}.\] Hence, $|\omega(z)|<1$ for $z\in\mathbb{D}$, for all $\eta$ with $|\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{proof}
\begin{remark}\label{remak1} The result in part $3$ is sharp in the sense that it doesn't holds good for all the values of $\eta$ in any disk of radius greater than $(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$. Put $\omega_1(z)=\alpha z$ and $\omega_2(z)=-\alpha z$ and take $\eta$ to be negative real number in the left-hand side of inequality \eqref{p5eq16}, and then by letting $z\rightarrow1$ through the real values, we see that the inequality \eqref{p5eq16} doesn't hold good for $\eta<-(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{remark}
Since the mapping $\phi(z):=\int_0^z\psi_{\mu,\nu}(\xi)d\xi$ is convex (convexity of $\phi$ is easily seen by observing that $\RE\left(1+z\phi''(z)/\phi'(z)\right)>0$ for $\mathbb{D}$), Theorem \ref{p5theom11} gives the following result.
\begin{corollary}\label{p5corl21}
For $k=1,2$ and $\mu,\nu,\varphi\in\mathbb{R}$, let the mapping $f_k=h_k+\overline{g_k}\in\mathcal{H}$ satisfy
\[h_k(z)-\emph{e}^{2\textit{i}\varphi}g_k(z)=\phi(z),\quad{}z\in\mathbb{D}.\] Then for the mapping $f$ given by \eqref{p5eq5}, we have the following:
\begin{itemize}
\item[1.] If $\eta$ is real, then the mapping $f\in\mathcal{S}_H^0$ and convex in the direction $\varphi$ for all $\eta$ with $0\leq \eta\leq1$.
\item[2.]If $\omega_k$, the dilatation of the mapping $f_k$ satisfies $|\omega_k(z)|<\alpha_k$, $z\in\mathbb{D}$, then the mapping $f\in\mathcal{S}_H^0$ and convex in the direction $\varphi$ for all $\eta$ with $ |\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{itemize}
\end{corollary}
However, if we take $\varphi=\pi/2+\mu$ in the above corollary, we get $f\in\mathcal{K}_H^0$. In fact, in such a case, we have a more generalized result, see Theorem \ref{p5theom22}. For any non-negative integer $n$, define the differential operator $\mathcal{D}^n:A \longrightarrow A$ on the family $A$ of all analytic mappings $f$ as: $
\mathcal{D}^0f(z)=f(z)$ and $\mathcal{D}^nf(z)=z(\mathcal{D}^{n-1}f)'(z)$ for $n\geq1$.
For the harmonic mapping $f=h+\overline{g}$, define $\mathcal{D}^nf:=\mathcal{D}^nh+\overline{\mathcal{D}^ng}$. In order to prove our next result, we use the following straight forward generalization of Sheil-Small's \cite{sheil} result on the relation between the starlike and convex harmonic mappings.
\begin{theorem}\label{p5theom21b}
If $f = h + \overline{g}$ is a starlike harmonic mapping, and $H$ and $G$ are the analytic mappings defined by
\[\mathcal{D}^nH=h,\quad \mathcal{D}^nG=(-1)^ng,\quad H(0) = H'(0)-1=G(0) = 0,\]
then the mapping $F = H + \overline{G}\in\mathcal{K}_H$.
\end{theorem}
\begin{theorem}\label{p5theom22}
For $k=1,2$ and $\mu,\nu\in\mathbb{R}$, let the mapping $f_k=h_k+\overline{g_k}\in\mathcal{H}$ be such that, the mapping $\mathcal{D}^{n-1}f_k$ is locally univalent, sense-preserving, and
\begin{equation}\label{p5eq23}
\frac{(h_k(z)+\emph{e}^{2\textit{i}\mu}(-1)^{n-1}g_k(z)}{z}=\frac{1}{z}\int_0^{z_n=z}\left(\dots\frac{1}{z_1}\int_0^{z_1} \psi_{\mu,\nu}(\xi)\textit{d}\xi\dots\right)\textit{d}z_{n-1}.
\end{equation}
Then for the mapping $f$ given by \eqref{p5eq5}, we have the following:
\begin{itemize}
\item[1.] If $\eta$ is real, then the mapping $f\in\mathcal{K}_H^0$ for all $\eta$ with $0\leq \eta\leq1$.
\item[2.]If $\omega_k$, the dilatation of $\mathcal{D}^{n-1}f_k$, satisfies $|\omega_k(z)|<\alpha_k$ for $z\in\mathbb{D}$, then the mapping $f\in\mathcal{K}_H^0$ for all $\eta$ with $ |\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{itemize}
\end{theorem}
\begin{proof}
Since
\begin{align}
f=h+\overline{g}:=\eta h_1+(1-\eta)h_2+\overline{\eta g_1+(1-\eta)g_2},\label{p5eq23a}
\end{align}
we have
\begin{align*}
h(z)+\emph{e}^{2\textit{i}\mu}g(z)&=\eta\left(h_1(z)+\emph{e}^{2\textit{i}\mu}g_1(z)-h_2(z)-\emph{e}^{2\textit{i}\mu}g_2(z)\right)+h_2(z)+\emph{e}^{2\textit{i}\mu}g_2(z)\\&=h_2(z)+\emph{e}^{2\textit{i}\mu}g_2(z).
\end{align*}
Therefore, in view of \eqref{p5eq23}, we see that
\[\mathcal{D}^{n-1}h(z)+\emph{e}^{2\textit{i}\mu}(-1)^{n-1}\mathcal{D}^{n-1}g(z)=\int_0^z\psi_{\mu,\nu}(\xi)\textit{d}\xi,\]
or equivalently
\begin{equation}\label{p5eq24}
H(z)+\emph{e}^{2\textit{i}\mu}G(z)=\int_0^z\psi(\xi)\textit{d}\xi,
\end{equation}
where $H(z):=\mathcal{D}^{n-1}h(z)$ and $G(z):=(-1)^{n-1}\mathcal{D}^{n-1}g(z)$. In view of the assumptions on $\mathcal{D}^{n-1}f_k$, Theorem \ref{p5theom11} shows that the mapping $F:=H+\overline{G}$ is locally univalent and sense-preserving. We will show it is convex. To prove this, in view of Lemma \ref{p5lema6}, it suffices to show that the mapping $H-\textit{e}^{2\textit{i}\theta}G$ is convex in the direction $\theta$ for all $\theta$ ranging in an interval of length $\pi$. In other words, it is sufficient to show that the mapping $\textit{e}^{-\textit{i}(\mu+\theta)}(H-\textit{e}^{2\textit{i}\theta}G)$ is convex in the direction $-\mu$ for all $\theta$ such that $0\leq\mu+\theta<\pi$. Consider the case $0\leq\mu+\theta<\pi/2$. Since $f$ is locally univalent and sense-preserving, $|G'(z)/H'(z)|<1$, $z\in\mathbb{D}$, and hence $$\RE\left(\frac{H'(z)-\textit{e}^{-2\textit{i}\mu}G'(z)}{H'(z)+\textit{e}^{-2\textit{i}\mu}G'(z)}\right)>0,\quad{}z\in\mathbb{D}.$$Using above inequality, we have
\begin{align}
\RE\left(\frac{\textit{e}^{-\textit{i}(\mu+\theta)}(H-\textit{e}^{2\textit{i}\theta}G)'(z)}{H'(z)+\textit{e}^{-2\textit{i}\mu}G'(z)}\right)&=\RE\left(\frac{(\textit{e}^{-\textit{i}(\mu+\theta)}H'(z)-\textit{e}^{-2\textit{i}\mu}\textit{e}^{\textit{i}(\mu+\theta)}G'(z)}{H'(z)+\textit{e}^{-2\textit{i}\mu}G'(z)}\right)\notag\\&=\RE\left(\frac{H'(z)-\textit{e}^{-2\textit{i}\mu}G'(z)}{H'(z)+\textit{e}^{-2\textit{i}\mu}G'(z)}\cos(\mu+\theta)-\textit{i}\sin(\mu+\theta)\right)\notag\\&=\cos(\mu+\theta)\RE\left(\frac{H'(z)-\textit{e}^{-2\textit{i}\mu}G'(z)}{H'(z)+\textit{e}^{-2\textit{i}\mu}G'(z)}\right)>0, \quad\in\mathbb{D}.\label{p5eq24a}
\end{align}
Also, differentiation of equation \eqref{p5eq24} gives that $H'(z)+\textit{e}^{-2\textit{i}\mu}G'(z)=\psi_{\mu,\nu}(z)$. Therefore, in view of equation \eqref{p5eq24a}, Theorem \ref{p5theom7} shows that the mapping $\textit{e}^{-\textit{i}(\mu+\theta)}(H-~\textit{e}^{2\textit{i}\theta}G)$ is convex in the direction $-\mu$ for all $\theta$ such that $0\leq\mu+\theta<~\pi/2$. Similarly with $\gamma=\mu+\pi$, Theorem \ref{p5theom7} shows that the mapping $\textit{e}^{-\textit{i}(\mu+\theta)}(H-~\textit{e}^{2\textit{i}\theta}G)$ is convex in the direction $-\mu$ for all $\theta$ such that $\pi/2\leq\mu+\theta<\pi$. Thus $F$ is convex, and hence starlike. Also, equation \eqref{p5eq23a} shows that the normalization of $f_k$ implies the normalization of $f$. The result now follows by Theorem \ref{p5theom21b}.
\end{proof}
Above result is extended to $n$ mappings as.
\begin{corollary}\label{p5corl25}
For $k=1,2,\dots,n$ and $\mu,\nu\in\mathbb{R}$, let the normalized harmonic mapping $f_k=h_k+\overline{g_k}$ satisfy equation \eqref{p5eq23}, and let the mapping $\mathcal{D}^{n-1}f_k$ be locally univalent and sense-preserving. Then we have the following:
\begin{itemize}
\item[1.] If $\sum_{k=1}^n t_k=1$, $0\leq t_k\leq1$, then the mapping $f=\sum_{k=1}^n t_kf_k\in\mathcal{K}_H^0$.
\item[2.] Let $\omega_k$, the dilatation of the mapping $\mathcal{D}^{n-1}f_k$, satisfy $|\omega_k(z)|<\alpha_1$ for $k=1,2,\dots,n-1$ and $|\omega_n(z)|<\alpha_2$, $z\in\mathbb{D}$. Also, let $t_1,t_2,\dots,1-t_n$ are of same sign and $\sum_1^n t_k=1$. Then the mapping $f=\sum_{k=1}^n t_kf_k\in\mathcal{K}_H^0$ for all $t_n$ with $0\leq t_n\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{itemize}
\end{corollary}
\begin{proof}
Part 1 simply follows by the repeated application of part 1 in Theorem \ref{p5theom22}. However, for part 2, we see that
\begin{align*}
f&=t_1f_1+t_2f_2+\dots+t_{n}f_{n}\\
&=(1-t_n)\left(\frac{t_1}{1-t_n}f_1+\dots+\frac{t_{n-1}}{1-t_n}f_{n-1}\right)+t_nf_n\\&=:(1-t_n)F+t_nf_n.
\end{align*}
By the assumptions on $t_k$, we see that $t_1/(1-t_n),t_2
/(1-t_n)\dots, t_{n-1}/(1-t_n)$ are all positive and $\sum_{k=1}^{n-1}t_k/(1-t_n)=1$. Also, $|\omega_k(z)|<\alpha_1$, $z\in\mathbb{D}$, for $k=1,2,\dots,n-1$ . Therefore by \cite[Lemma 6]{sun}, the dilatation $\omega$ of the mapping \[\mathcal{D}^{n-1}F=\frac{t_1}{1-t_n}\mathcal{D}^{n-1}f_1+\dots+\frac{t_{n-1}}{1-t_n}\mathcal{D}^{n-1}f_{n-1}\] satisfies $
|\omega(z)|<\alpha_1$, $z\in\mathbb{D}$. Since $\sum_{k=1}^{n-1}t_k/(1-t_n)=1$, the mapping $F$ satisfies equation \eqref{p5eq23}, and the normalization of the mappings $f_k$ implies the normalization of the mapping $F$. Therefore the result follows from part 2 of Theorem \ref{p5theom22}.
\end{proof}
Observe that \[\int_0^z\psi_{0,0}(\xi)\textit{d}\xi=\int_0^z \frac{\textit{d}\xi}{1-2\xi+\xi^2}
=\frac{z}{1-z}\] and\[\int_0^z\psi_{\pi/2,\pi/2}(\xi)\textit{d}\xi=\int_0^z \frac{\textit{d}\xi}{1-\xi^2}=\frac{1}{2}\log\left(\frac{1+z}{1-z}\right).\]
Therefore, Theorem \ref{p5theom22} gives the following results of Sun \textit{et al.}.
\begin{corollary}\cite[Theorems 4, 5]{sun}
For $k=1,2$, let the mapping $f_k=h_k+\overline{g_k}\in\mathcal{H}$ satisfy
\[h_k(z)+g_k(z)=\frac{z}{1-z}\quad{}\text{or}\quad h_k(z)-g_k(z)=\frac{1}{2}\log\left(\frac{1+z}{1-z}\right), \quad{}z\in\mathbb{D}.\] Then, for $0\leq t\leq1$, the mapping $f=tf_1+(1-t)f_2\in\mathcal{K}_H^0$.
\end{corollary}
The following result of Royster and Zeigler \cite{royster} can be used to check the directional convexity of the harmonic mappings (arising from shearing of analytic mappings) through the method of \textit{shear construction}.
\begin{theorem}\label{p5theom7}\cite{royster}
Let $\phi$ be a non-constant analytic mapping in $\mathbb{D}$. Then $\phi$ maps $\mathbb{D}$ onto a domain convex in the direction $\gamma$ if, and only if, there are real numbers $\mu$ $(0\leq\mu<2\pi)$ and $\nu$ $(0\leq\nu<\pi)$, such that
\begin{equation}\label{p5eq8}
\RE\left\lbrace\textit{e}^{\textit{i}(\mu-\gamma)}(1-2z\textit{e}^{-\textit{i}\mu}\cos\nu+z^2\textit{e}^{-2\textit{i}\mu})\phi'(z) \right\rbrace\geq0,\quad z\in \mathbb{D}.
\end{equation}
\end{theorem}
Put $\gamma=\mu$ and then replace it by $-\mu$ in the above theorem, we see the non-constant analytic mapping $\phi$ is convex in the direction $-\mu$, if for some $\nu\in\mathbb{R}$, $\RE\left(\phi'(z)/\psi_{\mu,\nu}(z)\right)>0$. Using this argument, Theorem \ref{p5theom11} gives the following result.
\begin{theorem}\label{p5corl8a}
For $\mu,\nu\in\mathbb{R}$, let the locally univalent and sense-preserving mapping $f_k=h_k+\overline{g_k}$ $ (k=1,2)$ satisfy
\begin{equation}\label{p5eq8b}
h_k(z)-\emph{e}^{-2\textit{i}\mu}g_k(z)=\int_0^z \psi_{\mu,\nu}(\xi)p(\xi)d\xi, \quad{}z\in\mathbb{D},
\end{equation}
where $p$ is an analytic mapping with $\RE p(z)>0$ for$z\in\mathbb{D}$. Then for the mapping $f$ given by \eqref{p5eq5}, we have the following:
\begin{itemize}
\item[1.]If $\eta$ is real, then the mapping $f$ is univalent and convex in the direction $-\mu$ for all $\eta$ with $0\leq \eta\leq1$.
\item[2.]If $\lambda=1$ and $\omega_k$, the dilatation of $f_k$, satisfies $|\omega_k(z)|<\alpha_k$, $z\in\mathbb{D}$, then the mapping $f$ is univalent and convex in the direction $-\mu$ for all $\eta$ with $ |\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{itemize}
\end{theorem}
\begin{proof}
Since $\RE p(z)>0$, $z\in\mathbb{D}$, we have
\begin{align*}
\RE\bigg(\frac{1}{\psi_{\mu,\nu}(z)}\left(\int_0^z \psi_{\mu,\nu}(\xi)p(\xi)d\xi\right)'\bigg)&=\RE\bigg(\frac{1}{\psi_{\mu,\nu}(z)}\psi_{\mu,\nu}(z)p(z)\bigg)\\&=\RE p(z)>0,\quad\qquad z\in\mathbb{D}.
\end{align*}
Hence, by Theorem \ref{p5theom7}, the mapping $\int_0^z \psi_{\mu,\nu}(\xi)p(\xi)d\xi$ is convex in the direction $-\mu$. Therefore, in view of equation \eqref{p5eq8b}, Theorem \ref{p5theom11} follows the result.
\end{proof}
On varying the mapping $p$ in the above theorem, we get different results which in a special cases not only generalizes the already established results on convex combination of harmonic mappings (e.g see \cite{kumar, sun}), but on taking smaller values of $\alpha_1$ and $\alpha_2$, part $(2)$ shows that such results are extended to a wider range of the values of $\eta$ as well.
\begin{corollary}\label{p5corl8c}
For $k=1,2$, and $\mu,\nu_1,\nu_2\in\mathbb{R}$, let the locally univalent and sense-preserving mapping $f_k=h_k+\overline{g_k}$ ($k=1,2$) satisfy
\begin{equation}\label{p5eq8d}
h_k(z)+\emph{e}^{-2\textit{i}\mu}g_k(z)=A.\frac{z(1-z\textit{e}^{\textit{i}\mu}\cos\nu_1)}{1-z^2\textit{e}^{2\textit{i}\mu}}+B.\int_0^z\psi_{\mu,\nu_2}(\xi)d\xi, \quad{}z\in\mathbb{D},
\end{equation}
where $A,B\geq 0$ with $A+B>0$. Then for the mapping $f$ given by \eqref{p5eq5}, we have the following:
\begin{itemize}
\item[1.]If $\eta$ is real, then the mapping $f$ is univalent and convex in the direction $-(\mu+\pi/2)$ for all $\eta$ with $0\leq \eta\leq1$.
\item[2.]If $\lambda=1$ and $\omega_k$, the dilatation of $f_k$, satisfies $|\omega_k(z)|<\alpha_k$, $z\in\mathbb{D}$, then the mapping $f$ is univalent and convex in the direction $-(\mu+\pi/2)$ for all $\eta$ with $ |\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{itemize}
\end{corollary}
\begin{proof}
Let$$p(z)=A.\frac{1-2z\textit{e}^{\textit{i}\mu}\cos\nu_1+z^2\textit{e}^{2\textit{i}\mu}}{1-z^2\textit{e}^{2\textit{i}\mu}}+B.\frac{1-z^2\textit{e}^{2\textit{i}\mu}}{1-2z\textit{e}^{\textit{i}\mu}\cos\nu_2+z^2\textit{e}^{2\textit{i}\mu}}$$
Since, for any $\gamma$ real and $z\in\mathbb{D}$,
\begin{align}
\RE\left(\frac{1-z^2\textit{e}^{2\textit{i}\mu}}{1-2z\textit{e}^{\textit{i}\mu}\cos\gamma+z^2\textit{e}^{2\textit{i}\mu}}\right)\notag&=\frac{1-|z|^4-2\cos\gamma(1-|z|^2)\RE(\textit{e}^{\textit{i}\mu}z)}{|1-2z\textit{e}^{\textit{i}\mu}\cos\gamma+z^2\textit{e}^{2\textit{i}\mu}|^2}\notag\\&\geq\frac{(1-|z|^2)(1+|z|^2-2|\cos\gamma|\RE(\textit{e}^{\textit{i}\mu}z))}{|1-2z\textit{e}^{\textit{i}\mu}\cos\gamma+z^2\textit{e}^{2\textit{i}\mu}|^2}>0\notag,
\end{align}
we get $\RE p(z)>0$ for $z\in\mathbb{D}$.
Also, in view of equation \eqref{p5eq8d}, we see that
\begin{align*}
\int_0^z \psi_{\mu+\pi/2,0}(\xi)p(\xi)d\xi&= \int_0^z \frac{p(\xi)d\xi}{1-\xi^2\textit{e}^{2\textit{i}\mu}}\\&=\int_0^z \bigg(A.\frac{1-2\xi\textit{e}^{\textit{i}\mu}\cos\nu_1+\xi^2\textit{e}^{2\textit{i}\mu}}{(1-\xi^2\textit{e}^{2\textit{i}\mu})^2}+B.\psi_{\mu,\nu_2}\bigg)d\xi\\&=A.\frac{z(1-z\textit{e}^{\textit{i}\mu}\cos\nu_1)}{1-z^2\textit{e}^{2\textit{i}\mu}}+B.\int_0^z \psi_{\mu,\nu_2}d\xi\\&=h_k(z)+\emph{e}^{-2\textit{i}\mu}g_k(z).
\end{align*}
Therefore, the result follows by Theorem \ref{p5corl8a}.
\end{proof}
\begin{remark}
By taking $A=1$, $B=0$, $\mu=\pi$ and $\gamma_1=0$, and $A=1$, $B=0$, $\mu=\pi$ in the above corollary, we get Theorem 3 of Wang \textit{et al.} \cite{wang} and Theorem 2.1 of Kumar \textit{et al.} \cite{kumar}, respectively. Also, observe that
\[\int_0^z\psi_{\mu,\nu_2}(\xi)d\xi=-\frac{\textit{e}^{-\textit{i}\mu}}{2\textit{i}\sin\nu_2}\log\left(\frac{1-z\textit{e}^{\textit{i}(\mu+\nu_2)}}{1-z\textit{e}^{\textit{i}(\mu-\nu_2)}}\right).\]
for $\nu_2\neq n \pi$, $n\in\mathbb{N}$. Therefore by taking $\mu=\pi$ in the above corollary, we get a result of Sun \textit{et al.} \cite[p.371]{sun}.
\end{remark}
Taking
\[p(z)=A.\frac{1-2z\textit{e}^{\textit{i}\mu}\cos\gamma+z^2\textit{e}^{2\textit{i}\mu}}{1-z^2\textit{e}^{2\textit{i}\mu}}+B.\frac{1-z^2\textit{e}^{2\textit{i}\mu}}{1-2z\textit{e}^{\textit{i}\mu}\cos\gamma+z^2\textit{e}^{2\textit{i}\mu}}\]
in Theorem \ref{p5corl8a}, and proceeding as in Corollary \ref{p5corl8c}, we get the following result.
\begin{corollary}
For $k=1,2$, and $\mu,\nu\in\mathbb{R}$, let the locally univalent and sense-preserving harmonic mapping $f_k=h_k+\overline{g_k}$ ($k=1,2$) satisfy
\[h_k(z)-\emph{e}^{-2\textit{i}\mu}g_k(z)=A.\frac{1}{2\textit{e}^{\textit{i}\mu}}\log\left(\frac{1+z\textit{e}^{\textit{i}\mu}}{1-z\textit{e}^{\textit{i}\mu}}\right)+B. z\psi_{\mu,\nu}(z), \quad{}z\in\mathbb{D},\]
where $A,B\geq 0$ with $A+B>0$. Then for the mapping $f$ given by \eqref{p5eq5}, we have the following:
\begin{itemize}
\item[1.]If $\eta$ is real, then the mapping $f$ is univalent and convex in the direction $-(\mu+\pi/2)$ for all $\eta$ with $0\leq \eta\leq1$.
\item[2.]If $\lambda=1$ and $\omega_k$, the dilatation of $f_k$, satisfies $|\omega_k(z)|<\alpha_k$, $z\in\mathbb{D}$, then the mapping $f$ is univalent and convex in the direction $-(\mu+\pi/2)$ for all $\eta$ with $ |\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{itemize}
\end{corollary}
Lastly, we consider the general linear combination $\mathcal{F}$ of the mappings $f_1$ and $f_2$, and is defined as
\begin{align*}
\mathcal{F}=\eta f_1+(1-\eta)f_2=\eta h_1+(1-\eta)h_2+\overline{\overline{\eta}g_1+(1-\overline{\eta})g_2}=:h+\overline{g}\notag,\quad{}\eta\in\mathbb{C}\notag.
\end{align*}
Like the $f$ combination defined , above combination $\mathcal{F}$ too have both the analytic and co-analytic parts as general linear combination of the corresponding analytic and co-analytic parts of $f_1$ and $f_2$, respectively, except that the parameters of the combination are not same, but complex conjugates. If $\eta$ is real, the above combination is simply a special case of the combination $f$ of the mappings $f_1$ and $f_2$ defined by \eqref{p5eq5}. We find sufficient conditions for this combination of the locally univalent and sense-preserving harmonic mappings to be univalent and convex in a particular direction. First, we prove the following lemma.
\begin{lemma}\label{p5lema29p}
For $k=1,2$, let the mapping $f_k=h_k+\overline{g_k}\in\mathcal{H}$ satisfy
\begin{equation}\label{p5eq29q}
h_k(z)+\emph{e}^{-2\textit{i}\mu}g_k(z)=\int_0^z \psi_{\mu, \nu}(\xi)d\xi
\end{equation}
for some $\mu,\nu\in\mathbb{R}$. Then the mapping $\mathcal{F}=\eta f_1+(1-\eta)f_2$ is univalent and convex in the direction $-\mu$ if it is locally univalent and sense-preserving for all $\eta\in\mathbb{C}$ satisfying anyone of the following:
\begin{itemize}
\item[(i)] $0\leq \RE\eta\leq1$.
\item[(ii)] $-1\leq \RE\eta\leq1$ provided $\omega_k$, the dilatation of $f_k$, satisfies $|\omega_k(z)|<\alpha_k$ with $\alpha_1\leq1/5$ and $\alpha_2\leq1/7$.
\end{itemize}
\end{lemma}
\begin{proof}Let $\eta=|\eta|e^{i\theta}$.
Since \begin{align*}
\mathcal{F}=\eta f_1+(1-\eta)f_2=\eta h_1+(1-\eta)h_2+\overline{\overline{\eta}g_1+(1-\overline{\eta})g_2}:=h+\overline{g}\notag,
\end{align*}
we have
\begin{align*}
h-\emph{e}^{-2\textit{i}\mu}g&=|\eta|\left(\emph{e}^{\textit{i}\theta}(h_1-h_2)-\emph{e}^{-2\textit{i}\mu}\emph{e}^{-\textit{i}\theta}(g_1-g_2)\right)+h_2-\emph{e}^{-2\textit{i}\mu}g_2\\&=|\eta|\left(h_1-h_2)-\emph{e}^{-2\textit{i}\mu}(g_1-g_2)\right)\cos\theta-\textit{\textit{i}}\sin\theta\left(h_1-h_2)+\emph{e}^{-2\textit{i}\mu}(g_1-g_2)\right)\\&\quad+h_2-\emph{e}^{-2\textit{i}\mu}g_2\\&=|\eta|\cos\theta(h_1-\emph{e}^{-2\textit{i}\mu}g_1)+(1-|\eta|\cos\theta)(h_2-\emph{e}^{-2\textit{i}\mu}g_2).
\end{align*}
Therefore, in view of \eqref{p5eq29q}, we see that
\begin{equation}\label{p5eq29r}
\frac{h'-\emph{e}^{-2\textit{i}\mu}g'}{\psi_{\mu,\nu}}=\RE(\eta)\frac{h_1'-\emph{e}^{-2\textit{i}\mu}g_1'}{h_1'+\emph{e}^{-2\textit{i}\mu}g_1'}+(1-\RE\eta)\frac{h_2'-\emph{e}^{-2\textit{i}\mu}g_2'}{h_2'+\emph{e}^{-2\textit{i}\mu}g_2'}
\end{equation}
Since the mapping $f_k=h_k+\overline{g_k}$ $(k=1,2)$ is locally univalent and sense-preserving, $|g_k'(z)/h_k'(z)|<1$ for $z\in\mathbb{D}$, or equivalently
\[\RE\left(\frac{h_k'(z)-\emph{e}^{-2\textit{i}\mu}g_k'(z)}{h_k'(z)+\emph{e}^{-2\textit{i}\mu}g_k'(z)}\right)>0,\qquad\quad z\in\mathbb{D}.\]
Hence, above equation along with \eqref{p5eq29r} shows that
\[\RE\left(\frac{h(z)-\emph{e}^{-2\textit{i}\mu}g(z)}{\psi_{\mu,\nu}(z)}\right)>0,\qquad\quad z\in\mathbb{D},\] if $0\leq \RE\eta\leq1$. Therefore by Theorem \ref{p5theom7}, the function $h-\emph{e}^{-2\textit{i}\mu}g$ is convex in the direction $-\mu$. Thus, for all $\eta$ with $0\leq \RE\eta\leq1$, lemma \ref{p5lema6} shows that the mapping $\mathcal{F}$ is univalent and convex in the direction of $-\mu$ provided it is locally univalent and sense-preserving. This follows the result in case (i). Let $\omega_k$ be the dilatation of the mapping $f_k$. Therefore equation \eqref{p5eq29r} is equivalent to
\begin{equation}\label{p5eq29u}
\frac{h'-\emph{e}^{-2\textit{i}\mu}g'}{\psi_{\mu,\nu}}=\RE(\eta)\frac{1-\emph{e}^{-2\textit{i}\mu}\omega_1}{1+\emph{e}^{-2\textit{i}\mu}\omega_1}+(1-\RE\eta)\frac{1-\emph{e}^{-2\textit{i}\mu}\omega_2}{1+\emph{e}^{-2\textit{i}\mu}\omega_2}.
\end{equation}
Note that\[\RE\frac{1-z/5}{1+z/5}<\frac{3}{2}\quad\text{and}\quad\RE\frac{1-z/7}{1+z/7}>\frac{3}{4},\qquad\quad z\in\mathbb{D}.\]
Hence, if $|\omega_1(z)|<1/5$ and $|\omega_1(z)|<1/7$, then
for $-1\leq \RE\eta<0$, equation \eqref{p5eq29u} shows that
\begin{align*}
\RE\frac{h'(z)-\emph{e}^{-2\textit{i}\mu}g'(z)}{\psi_{\mu,\nu}(z)}>\frac{3}{2}\RE\eta+\frac{3}{4}(1-\RE\eta)&=\frac{3}{4}(1+\RE\eta)\geq0,\qquad\quad z\in\mathbb{D}.
\end{align*}
Above equation along with Theorem \ref{p5theom7} and Lemma \ref{p5lema29p} shows that, for all $\eta$ with $-1\leq\RE\eta<0$, the mapping $\mathcal{F}$ is univalent and convex in the direction of $-\mu$ provided it is locally univalent and sense-preserving. This along with the result in case (i) follows the result in case (ii).
\end{proof}
The problem in the above lemma is now to find out the vales of $\eta$ for which $\mathcal{F}$ is locally univalent and sense-preserving. In the next theorem, we prove this for the values of $\eta$ lying in the disk with origin as center and radius given in-terms of the bounds of the dilatations of $f_k$.
\begin{theorem}\label{p5theom29s}
For $k=1,2$, let the mapping $f_k=h_k+\overline{g_k}\in\mathcal{H}$ satisfy
\[
h_k(z)+\emph{e}^{-2\textit{i}\mu}g_k(z)=\int_0^z \psi_{\mu, \nu}(\xi)d\xi
\]
for some $\mu\in\mathbb{R}$. Let the dilatation $\omega_k$ of $f_k$ satisfies $|\omega_k|<\alpha_k$, $\alpha_k\in\mathbb{R}$. Then the mapping $\mathcal{F}=\eta f_1+(1-\eta)f_2$ $(\eta\in\mathbb{C})$ is locally univalent and sense-preserving for $|\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{theorem}
\begin{proof}
Since $\omega_k$ is the dilatation of the mapping $f_k$, following similarly as in Theorem \ref{p5theom11}, the dilatation $\omega$ of $\mathcal{F}$ is given by
\[
\omega=\frac{\eta\omega_1(1-\emph{e}^{-2\textit{i}\mu}\omega_2)+(1-t\emph{e}^{\textit{i}\theta}\eta)\omega_2(1-\eta\omega_1)}{\overline{\eta}(1-\emph{e}^{-2\textit{i}\mu}\omega_2)+(1-\overline{\eta})(1-\emph{e}^{-2\textit{i}\mu}\omega_1)}.
\]
With $\omega_k$ replaced by $\emph{e}^{2\textit{i}\mu}\omega_k$, the above equation gives
\[
\emph{e}^{-2\textit{i}\varphi}\omega=\frac{\eta\omega_1(1-\omega_2)+(1-\eta)\omega_2(1-\omega_1)}{\overline{\eta}(1-\omega_2)+(1-\overline{\eta})(1-\omega_1)}.
\]
Thus $|\omega(z)|<1$ for $z\in\mathbb{D}$ if, and only if
\[|\eta\omega_1(z)(1-\omega_2(z))+(1-\eta)\omega_2(z)(1-\omega_1(z))|^2<|\overline{\eta}(1-\omega_2(z))+(1-\overline{\eta})(1-\omega_1(z))|^2,\]or equivalently, if and only if
\[
|1-\omega_1(z)|^2\left(1-|\omega_2(z)|^2\right)+2\RE\left(\eta(\omega_1(z)-\omega_2(z))(1-\overline{\omega_1(z)})(\emph{e}^{-2\textit{i}\theta}-\overline{\omega_2(z)})\right)>0,
\]where $\theta$ is the argument of $\eta$.
Therefore $|\omega(z)|<1$, $z\in\mathbb{D}$, if \[|\eta|<\frac{|1-\omega_1(z)|\left(1-|\omega_2(z)|^2\right)}{2|(\omega_1(z)-\omega_2(z))(\emph{e}^{-2\textit{i}\theta}-\overline{\omega_2(z)})|}.\] But, for $k=1,2$ and $z\in\mathbb{D}$, $|\omega_k(z)|<\alpha_k$ implies that\[\frac{|1-\omega_1(z)|\left(1-|\omega_2(z)|^2\right)}{2|(\omega_1(z)-\omega_2(z))(\emph{e}^{-2\textit{i}\theta}-\overline{\omega_2(z)})|}>\frac{(1-\alpha_1)(1-\alpha_2)}{2(\alpha_1+\alpha_2)}.\] Hence $|\omega(z)|<1$, $z\in\mathbb{D}$, for all $\eta$ with $|\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{proof}
\begin{remark}
Clearly Remark \ref{remak1} shows that the result in the above theorem is sharp in the sense that it doesn't holds good for all the values of $\eta$ in any disk of radius greater than $(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$.
\end{remark}
Theorem \ref{p5theom29s} along with Lemma \ref{p5lema29p} gives the following result.
\begin{theorem}
For $k=1,2$, let the mapping $f_k=h_k+\overline{g_k}\in\mathcal{H}$ satisfy
\begin{equation}\label{p5eq29t}
h_k(z)+\emph{e}^{-2\textit{i}\mu}g_k(z)=\int_0^z \psi_{\mu, \nu}(\xi)d\xi
\end{equation}
for some $\mu,\nu\in\mathbb{R}$. Let, $\omega_k$, the dilatation of $f_k$ satisfies $|\omega_k|<\alpha_k$, $\alpha_k\in\mathbb{R}$. Then the mapping $\mathcal{F}=\eta f_1+(1-\eta)f_2$ $(\eta\in\mathbb{C})$ is univalent and convex in the direction $-\mu$ for $|\eta|\leq(1-\alpha_1)(1-\alpha_2)/(2(\alpha_1+\alpha_2))$ and satisfying anyone of the following:
\begin{itemize}
\item[(i)] $0\leq \RE\eta\leq1$.
\item[(ii)] $-1\leq \RE\eta\leq1$ provided $\omega_k$, the dilatation of $f_k$, satisfies $|\omega_k(z)|<\alpha_k$ with $\alpha_1\leq1/5$ and $\alpha_2\leq1/7$.
\end{itemize}
\end{theorem}
\begin{remark}
For $k=1,2$, if the mapping $f_k=h_k+\overline{g_k}\in\mathcal{H}$ satisfy \eqref{p5eq29t} and its dilatation $\omega_k$ satisfies $|\omega_k(z)|<\alpha_k$, with $\alpha_1\leq1/5$ and $\alpha_2\leq1/7$, then the mapping $\mathcal{F}=\eta f_1+(1-\eta)f_2$ is univalent and convex in the direction $-\mu$ for all $\eta\in\mathbb{D}$ .
\end{remark}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,127
|
Q: Three,js Trackball rotate only one object I am new to three.js and computer graphics. So I am facing quite simple problem. I want to rotate only cube and axes set static. How to achieve it, because now cube and axes are rotating? Maybe I have to use another type of rotation?
My code:
<html>
<body>
<script type="text/javascript" src="../libs/three.js"></script>
<script type="text/javascript" src="../libs/TrackBallControls.js"></script>
<script>
var camera, controls, secene, render;
init();
animate();
function init() {
camera = new THREE.PerspectiveCamera(45, window.innerWidth / window.innerHeight, 1, 1000);
camera.position.z = 500;
var debugaxis = function(axisLength){
//Shorten the vertex function
function v(x,y,z){
return new THREE.Vector3(x,y,z);
}
//Create axis (point1, point2, colour)
function createAxis(p1, p2, color){
var line, lineGeometry = new THREE.Geometry(),
lineMat = new THREE.LineBasicMaterial({color: color, lineWidth: 1});
lineGeometry.vertices.push(p1, p2);
line = new THREE.Line(lineGeometry, lineMat);
scene.add(line);
}
createAxis(v(-axisLength, 0, 0), v(axisLength, 0, 0), 0xFF0000);
createAxis(v(0, -axisLength, 0), v(0, axisLength, 0), 0x00FF00);
createAxis(v(0, 0, -axisLength), v(0, 0, axisLength), 0x0000FF);
};
controls = new THREE.TrackballControls(camera);
controls.addEventListener('change', render);
scene = new THREE.Scene();
debugaxis(200);
var geometry = new THREE.CubeGeometry(100,100,100);
var material = new THREE.MeshNormalMaterial({ wireFrame: false, transparent: true, opacity: 0.5});
var mesh = new THREE.Mesh( geometry, material);
scene.add(mesh);
for (var f = 0, fl = mesh.geometry.faces.length; f < fl; f++) {
var face = mesh.geometry.faces[f];
var centroid = new THREE.Vector3(0, 0, 0);
centroid.add(mesh.geometry.vertices[face.a]);
centroid.add(mesh.geometry.vertices[face.b]);
centroid.add(mesh.geometry.vertices[face.c]);
centroid.divideScalar(3);
var arrow = new THREE.ArrowHelper(
face.normal,
centroid,
25,
0x3333FF,
5,
5);
mesh.add(arrow);
}
renderer = new THREE.WebGLRenderer();
renderer.setSize(window.innerWidth, window.innerHeight)
document.body.appendChild(renderer.domElement);
}
function animate(){
requestAnimationFrame(animate);
controls.update();
}
function render(){
renderer.render(scene, camera);
}
</script>
</body>
</html>
A: I solved this problem by rotating only a cube. In this example i can spin cube with a mouse. With trackball animation it is impossible to achieve it, because you are rotating camera (as HenryHey said).
<!DOCTYPE html>
<html lang="en">
<head>
<title>three.js canvas - geometry - cube</title>
<meta charset="utf-8">
<meta name="viewport" content="width=device-width, user-scalable=no, minimum-scale=1.0, maximum-scale=1.0">
<style>
body {
font-family: Monospace;
background-color: #f0f0f0;
margin: 0px;
overflow: hidden;
}
</style>
</head>
<body>
<script src="../libs/three.js"></script>
<script src="../libs/Projector.js"></script>
<script src="../libs/CanvasRenderer.js"></script>
<script src="../libs/stats.js"></script>
<script>
var container, stats;
var camera, scene, renderer;
var cube;
var targetRotation = 0;
var targetRotationOnMouseDown = 0;
var mouseX = 0;
var mouseXOnMouseDown = 0;
var windowHalfX = window.innerWidth / 2;
var windowHalfY = window.innerHeight / 2;
init();
animate();
function init() {
container = document.createElement( 'div' );
document.body.appendChild( container );
var info = document.createElement( 'div' );
info.style.position = 'absolute';
info.style.top = '10px';
info.style.width = '100%';
info.style.textAlign = 'center';
info.innerHTML = 'Drag to spin the cube';
container.appendChild( info );
var debugaxis = function(axisLength){
//Shorten the vertex function
function v(x,y,z){
return new THREE.Vector3(x,y,z);
}
//Create axis (point1, point2, colour)
function createAxis(p1, p2, color){
var line, lineGeometry = new THREE.Geometry(),
lineMat = new THREE.LineBasicMaterial({color: color, lineWidth: 1});
lineGeometry.vertices.push(p1, p2);
line = new THREE.Line(lineGeometry, lineMat);
scene.add(line);
}
createAxis(v(-axisLength, 0, 0), v(axisLength, 0, 0), 0xFF0000);
createAxis(v(0, -axisLength, 0), v(0, axisLength, 0), 0x00FF00);
createAxis(v(0, 0, -axisLength), v(0, 0, axisLength), 0x0000FF);
};
camera = new THREE.PerspectiveCamera( 70, window.innerWidth / window.innerHeight, 0.1, 1000 );
camera.position.y = 100;
camera.position.z = 500;
camera.position.x = 100;
scene = new THREE.Scene();
debugaxis(200);
// Cube
var geometry = new THREE.CubeGeometry( 100, 100, 100 );
var material = new THREE.MeshNormalMaterial( { wireFrame: false, transparent: true, opacity: 0.5 } );
cube = new THREE.Mesh( geometry, material );
cube.position.y = 0;
cube.position.x = 0;
for (var f = 0, fl = cube.geometry.faces.length; f < fl; f++) {
var face = cube.geometry.faces[f];
var centroid = new THREE.Vector3(0, 0, 0);
centroid.add(cube.geometry.vertices[face.a]);
centroid.add(cube.geometry.vertices[face.b]);
centroid.add(cube.geometry.vertices[face.c]);
centroid.divideScalar(3);
var arrow = new THREE.ArrowHelper(
face.normal,
centroid,
30,
0x3333FF,
7,
7);
cube.add(arrow);
}
scene.add( cube );
// Plane
var geometry = new THREE.PlaneBufferGeometry( 200, 200 );
geometry.applyMatrix( new THREE.Matrix4().makeRotationX( - Math.PI / 2 ) );
var material = new THREE.MeshBasicMaterial( { color: 0xe0e0e0, overdraw: 0.5 } );
//plane = new THREE.Mesh( geometry, material );
//scene.add( plane );
renderer = new THREE.CanvasRenderer();
renderer.setClearColor( 0xf0f0f0 );
//renderer.setPixelRatio( window.devicePixelRatio );
renderer.setSize( window.innerWidth, window.innerHeight );
container.appendChild( renderer.domElement );
stats = new Stats();
stats.domElement.style.position = 'absolute';
stats.domElement.style.top = '0px';
container.appendChild( stats.domElement );
document.addEventListener( 'mousedown', onDocumentMouseDown, false );
document.addEventListener( 'touchstart', onDocumentTouchStart, false );
document.addEventListener( 'touchmove', onDocumentTouchMove, false );
//
window.addEventListener( 'resize', onWindowResize, false );
}
function onWindowResize() {
windowHalfX = window.innerWidth / 2;
windowHalfY = window.innerHeight / 2;
camera.aspect = window.innerWidth / window.innerHeight;
camera.updateProjectionMatrix();
renderer.setSize( window.innerWidth, window.innerHeight );
}
//
function onDocumentMouseDown( event ) {
event.preventDefault();
document.addEventListener( 'mousemove', onDocumentMouseMove, false );
document.addEventListener( 'mouseup', onDocumentMouseUp, false );
document.addEventListener( 'mouseout', onDocumentMouseOut, false );
mouseXOnMouseDown = event.clientX - windowHalfX;
targetRotationOnMouseDown = targetRotation;
}
function onDocumentMouseMove( event ) {
mouseX = event.clientX - windowHalfX;
targetRotation = targetRotationOnMouseDown + ( mouseX - mouseXOnMouseDown ) * 0.02;
}
function onDocumentMouseUp( event ) {
document.removeEventListener( 'mousemove', onDocumentMouseMove, false );
document.removeEventListener( 'mouseup', onDocumentMouseUp, false );
document.removeEventListener( 'mouseout', onDocumentMouseOut, false );
}
function onDocumentMouseOut( event ) {
document.removeEventListener( 'mousemove', onDocumentMouseMove, false );
document.removeEventListener( 'mouseup', onDocumentMouseUp, false );
document.removeEventListener( 'mouseout', onDocumentMouseOut, false );
}
function onDocumentTouchStart( event ) {
if ( event.touches.length === 1 ) {
event.preventDefault();
mouseXOnMouseDown = event.touches[ 0 ].pageX - windowHalfX;
targetRotationOnMouseDown = targetRotation;
}
}
function onDocumentTouchMove( event ) {
if ( event.touches.length === 1 ) {
event.preventDefault();
mouseX = event.touches[ 0 ].pageX - windowHalfX;
targetRotation = targetRotationOnMouseDown + ( mouseX - mouseXOnMouseDown ) * 0.05;
}
}
//
function animate() {
requestAnimationFrame( animate );
render();
stats.update();
}
function render() {
cube.rotation.y += ( targetRotation - cube.rotation.y ) * 0.05;
renderer.render( scene, camera );
}
</script>
</body>
</html>
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,291
|
package com.scientificsoft.iremote.platform.tools;
import java.io.InputStream;
import java.io.OutputStream;
import android.content.Context;
import com.scientificsoft.iremote.android.iremote.tools.ContextHolder;
public class File {
public static InputStream openFile(String filename) {
try {
return ContextHolder.instance().context().openFileInput(filename);
} catch ( Exception e ) {
return null;
}
}
public static OutputStream createFile(String filename) {
try {
return ContextHolder.instance().context().openFileOutput(filename, Context.MODE_PRIVATE);
} catch ( Exception e ) {
return null;
}
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,284
|
This project will continue work to develop new organometallic methodology to prepare C-C bonds in a stereoselective fashion, and applications of this methodology. In particular, the preparation of (2-alkenyl-3-pentene-1,5-diyl)iron complexes and their transformation into vinylcyclopropanes and divinylcyclopropanes will be targeted. The divinylcyclopropanes are likely to undergo rearrangement to cycloheptadienes. These reactions will be applied to the synthesis of beta-aminocyclopropanecarboxylic acids and frondosin A, an inhibitor of the binding of interleukin-8 to its receptor.
With this award, the Organic and Macromolecular Chemistry Program is supporting the research of Professor William A. Donaldson of the Department of Chemistry at Marquette University. Professor Donaldson's research efforts revolve around the development of organometallic methodology for the formation of cyclopropanes and cycloheptadienes, structural motifs present in a variety of biologically relevant molecules. Successful development of the methodology will have an impact on synthesis in the pharmaceutical and agricultural industries.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 9,476
|
Jean Messiha (; born Hossam Boutros Messiha, , 10 September 1970) is an Egyptian-born French economist and media personality, formerly a politician and senior civil servant. He was appointed Deputy Undersecretary of Management at the Ministry of Defence in 2014 before he joined the National Front (FN) in 2016, when he became spokesman of Horaces, a group of high-ranking civil servants and business executives who meet once a month to discuss the party platform. Messiha stood as a candidate in the 2017 legislative election in the 4th constituency of the Aisne department.
In 2020, he left the party to assume the presidency of the Apollon Institute, a far-right think tank. In 2022, Messiha joined Éric Zemmour's newly-founded Reconquête party and became its spokesman, although he left the party following the presidential election to return as president of the Apollon Institute.
Early life
Messiha was born Hossam Boutros Messiha in 1970 in Cairo, Egypt, to a family of Coptic Christians; his father was a diplomat. He lived in Bogotá, Colombia from the age of 3 to 7. At the age of 8, he arrived with his family in France, reportedly "not speaking a word of French". He then grew up in Mulhouse. In 1990, upon his naturalisation as a French citizen, he changed his first name to Jean.
Messiha graduated from Sciences Po, where one of his professors was Henri Guaino. He earned a PhD in Economics. His thesis was about the budgetary policies of the Maastricht Treaty and Amsterdam Treaty. He graduated from the École nationale d'administration in 2005.
Career
Messiha began his career as a high-ranking civil servant in 2005. He was appointed as Deputy Undersecretary of Management at the Ministry of Defence in 2014.
Messiha became an advisor to National Rally leader Marine Le Pen in 2014. In May 2016, he became the spokesman of the "Horaces", a group of high-ranking civil servants and business executives, supporting Marine Le Pen, who meet once a month and discuss the political platform of the National Rally. While the group announces more than 155 members, Messiha is the only one whose name has been publicly known so far. According to Dominique Albertini of Libération, Messiha's role within the National Rally is to represent "the drawing power of [the party] towards high-ranking civil servants".
Messiha has asserted his belief in Renaud Camus's Great Replacement conspiracy theory, whereby Christian populations are being "replaced" through non-European immigration, specifically from Muslim and African countries. On social media, he has expressed that Islam is at odds with France's republican system. He is also a critic of the European Union.
A candidate in the 2017 French legislative election to represent Aisne's 4th constituency in the National Assembly, Messiha was defeated in the second round by La République En Marche! candidate Marc Delatte, with 43.73% of valid votes against Delatte's 56.27%.
In November 2020, several news outlets reported that Messiha was going to leave the National Rally. This was later confirmed by Messiha, who announced his departure in an interview published in Valeurs actuelles. In March 2021, Jean Messiha made the headlines in affirming the existence of a "black privilege" during the 46th César Awards. In August 2021, Twitter has permanently suspended his account @jeanmessiha for multiple violations of hateful conduct policy.
In January 2022, Messiha joined Reconquête, the party Éric Zemmour founded in December 2021.
Personal life
Messiha became a naturalised French citizen at the age of 20, changing his first name to "Jean" in the process. He has described himself as a "naturalized ethnic Frenchman" () and "Arab outside, French inside". In February 2017, he was surprised to learn that, in spite of his naturalisation, he was still considered an immigrant by the national statistics bureau of France, Institut national de la statistique et des études économiques (INSEE).
References
1970 births
Living people
Egyptian emigrants to France
French people of Coptic descent
Politicians from Mulhouse
French critics of Islam
École nationale d'administration alumni
20th-century French economists
21st-century French economists
French civil servants
National Rally (France) politicians
Reconquête politicians
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 274
|
Asnières-sur-Seine je město v severozápadní části metropolitní oblasti Paříže ve Francii v departmentu Hauts-de-Seine a regionu Île-de-France. Od centra Paříže je vzdálené 7,9 km. Žije zde obyvatel.
Geografie
Asnières leží na pravém břehu Seiny. Dříve bylo spojeno s Gennevilliers.
Sousední obce: Clichy, Bois-Colombes, Gennevilliers a Courbevoie.
Vývoj počtu obyvatel
Počet obyvatel
Osobnosti města
Albert Grisar, belgický operní skladatel
Jean Lescure, spisovatel
Zajímavosti
Zajímavostí je, že na území Asnières vznikl roku 1899 první hřbitov zvířat na světě (Cimetière des chiens). Nejsou v něm pohřbívání jen psi, kočky či ptáci, ale též koně, lvi a opice.
Z pěveckého sboru města Asnières v 70. letech vznikla známá skupina Les Poppys. Původně se jmenovala Petits Chanteurs d'Asnières.
Partnerská města
Špandava
Reference
Související články
Seznam obcí v departementu Hauts-de-Seine
Externí odkazy
Města v Île-de-France
Obce v departementu Hauts-de-Seine
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 1,516
|
Apomys est un genre de rongeurs de la sous-famille des Murinés.
Liste des espèces
Selon :
Apomys abrae (Sanborn, 1952)
Apomys datae (Meyer, 1899)
Apomys gracilirostris Ruedas 1995
Apomys hylocoetes Mearns, 1905
Apomys insignis Mearns, 1905
Apomys littoralis (Sanborn, 1952)
Apomys microdon Hollister, 1913
Apomys musculus Miller, 1911
Apomys sacobianus Johnson, 1962
Selon
Apomys abrae (Sanborn, 1952)
Apomys brownorum (Heaney, et al. 2011)
Apomys datae (Meyer, 1899)
Apomys gracilirostris Ruedas 1995
Apomys hylocoetes Mearns, 1905
Apomys insignis Mearns, 1905
Apomys littoralis (Sanborn, 1952)
Apomys microdon Hollister, 1913
Apomys musculus Miller, 1911
Apomys sacobianus Johnson, 1962
Liens externes
Rongeur (nom scientifique)
Muridae
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 7,884
|
Q: Built Scrapy spider, but it isn't following links I wrote a simple spider to get links of hikes. It seems like it isn't looking at the URLs at all to scrape the site:
[scrapy] INFO: Crawled 0 pages (at 0 pages/min), scraped 0 items (at 0 items/min)
Here's my simple spider:
from scrapy.spiders import Spider
from scrapy.selector import Selector
from oregon_hikes_scrapper.items import HikeLinkItem
ENDPOINTS = [ 'from="%27%27Peter_Iredale%27%27&to=Bonney_Meadows-Hidden_Meadows_Trail_Junction', \
'from=Bonney_Meadows-Hidden_Meadow_Trail_Junction&to=Clatsop_Loop_Hike',
]
class OrHikeSpider(Spider):
name ='or_hikes'
allowed_domains = "oregonhikers.org"
start_url = [
"http://www.oregonhikers.org/field_guide/Special:AllPages&" + l for l in ENDPOINTS
]
def parse(self, response):
hikes = Selector.xpath('//*[@id="mw-content-text"]/table[2]/tbody/tr[1]/td[1]/div/a')
for hike in hikes:
item = HikeLinkItem()
item['hike'] = hike.xpath('@title').extract()
item['link'] = hike.xpath('@href').extract()
yield item
A: Syntax error:
start_urls instead of start_url
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 3,529
|
{"url":"https:\/\/manned.org\/rsync.1","text":"# rsync\n\nrsync(1) User Commands rsync(1)\n\nNAME\nrsync - a fast, versatile, remote (and local) file-copying tool\n\nSYNOPSIS\nLocal:\nrsync [OPTION...] SRC... [DEST]\n\nAccess via remote shell:\nPull:\nrsync [OPTION...] [USER@]HOST:SRC... [DEST]\nPush:\nrsync [OPTION...] SRC... [USER@]HOST:DEST\n\nAccess via rsync daemon:\nPull:\nrsync [OPTION...] [USER@]HOST::SRC... [DEST]\nrsync [OPTION...] rsync:\/\/[USER@]HOST[:PORT]\/SRC... [DEST]\nPush:\nrsync [OPTION...] SRC... [USER@]HOST::DEST\nrsync [OPTION...] SRC... rsync:\/\/[USER@]HOST[:PORT]\/DEST)\n\nUsages with just one SRC arg and no DEST arg will list the source files\n\nDESCRIPTION\nRsync is a fast and extraordinarily versatile file copying tool. It can\ncopy locally, to\/from another host over any remote shell, or to\/from a\nremote rsync daemon. It offers a large number of options that control\nevery aspect of its behavior and permit very flexible specification of\nthe set of files to be copied. It is famous for its delta-transfer\nalgorithm, which reduces the amount of data sent over the network by\nsending only the differences between the source files and the existing\nfiles in the destination. Rsync is widely used for backups and mirroring\nand as an improved copy command for everyday use.\n\nRsync finds files that need to be transferred using a \"quick check\"\nalgorithm (by default) that looks for files that have changed in size or\nin last-modified time. Any changes in the other preserved attributes (as\nrequested by options) are made on the destination file directly when the\nquick check indicates that the file's data does not need to be updated.\n\nSome of the additional features of rsync are:\n\no support for copying links, devices, owners, groups, and\npermissions\n\no exclude and exclude-from options similar to GNU tar\n\no a CVS exclude mode for ignoring the same files that CVS would\nignore\n\no can use any transparent remote shell, including ssh or rsh\n\no does not require super-user privileges\n\no pipelining of file transfers to minimize latency costs\n\no support for anonymous or authenticated rsync daemons (ideal for\nmirroring)\n\nGENERAL\nRsync copies files either to or from a remote host, or locally on the\ncurrent host (it does not support copying files between two remote\nhosts).\n\nThere are two different ways for rsync to contact a remote system: using\na remote-shell program as the transport (such as ssh or rsh) or\ncontacting an rsync daemon directly via TCP. The remote-shell transport\nis used whenever the source or destination path contains a single colon\n(:) separator after a host specification. Contacting an rsync daemon\ndirectly happens when the source or destination path contains a double\ncolon (::) separator after a host specification, OR when an rsync:\/\/ URL\nSHELL CONNECTION\" section for an exception to this latter rule).\n\nAs a special case, if a single source arg is specified without a\ndestination, the files are listed in an output format similar to \"ls -l\".\n\nAs expected, if neither the source or destination path specify a remote\n\nRsync refers to the local side as the client and the remote side as the\nserver. Don't confuse server with an rsync daemon. A daemon is always a\nserver, but a server can be either a daemon or a remote-shell spawned\nprocess.\n\nSETUP\nSee the file README.md for installation instructions.\n\nOnce installed, you can use rsync to any machine that you can access via\na remote shell (as well as some that you can access using the rsync\ndaemon-mode protocol). For remote transfers, a modern rsync uses ssh for\nits communications, but it may have been configured to use a different\nremote shell by default, such as rsh or remsh.\n\nYou can also specify any remote shell you like, either by using the -e\ncommand line option, or by setting the RSYNC_RSH environment variable.\n\nNote that rsync must be installed on both the source and destination\nmachines.\n\nUSAGE\nYou use rsync in the same way you use rcp. You must specify a source and\na destination, one of which may be remote.\n\nPerhaps the best way to explain the syntax is with some examples:\n\nrsync -t *.c foo:src\/\n\nThis would transfer all files matching the pattern *.c from the current\ndirectory to the directory src on the machine foo. If any of the files\nalready exist on the remote system then the rsync remote-update protocol\nis used to update the file by sending only the differences in the data.\nNote that the expansion of wildcards on the command-line (*.c) into a\nlist of files is handled by the shell before it runs rsync and not by\nrsync itself (exactly the same as all other Posix-style programs).\n\nrsync -avz foo:src\/bar \/data\/tmp\n\nThis would recursively transfer all files from the directory src\/bar on\nthe machine foo into the \/data\/tmp\/bar directory on the local machine.\nThe files are transferred in archive mode, which ensures that symbolic\nlinks, devices, attributes, permissions, ownerships, etc. are preserved\nin the transfer. Additionally, compression will be used to reduce the\nsize of data portions of the transfer.\n\nrsync -avz foo:src\/bar\/ \/data\/tmp\n\nA trailing slash on the source changes this behavior to avoid creating an\nadditional directory level at the destination. You can think of a\ntrailing \/ on a source as meaning \"copy the contents of this directory\"\nas opposed to \"copy the directory by name\", but in both cases the\nattributes of the containing directory are transferred to the containing\ndirectory on the destination. In other words, each of the following\ncommands copies the files in the same way, including their setting of the\nattributes of \/dest\/foo:\n\nrsync -av \/src\/foo \/dest\nrsync -av \/src\/foo\/ \/dest\/foo\n\nNote also that host and module references don't require a trailing slash\nto copy the contents of the default directory. For example, both of\nthese copy the remote directory's contents into \"\/dest\":\n\nrsync -av host: \/dest\nrsync -av host::module \/dest\n\nYou can also use rsync in local-only mode, where both the source and\ndestination don't have a ':' in the name. In this case it behaves like\nan improved copy command.\n\nFinally, you can list all the (listable) modules available from a\nparticular rsync daemon by leaving off the module name:\n\nrsync somehost.mydomain.com::\n\nSee the following section for more details.\n\nThe syntax for requesting multiple files from a remote host is done by\nspecifying additional remote-host args in the same style as the first, or\nwith the hostname omitted. For instance, all these work:\n\nrsync -av host:file1 :file2 host:file{3,4} \/dest\/\nrsync -av host::modname\/file{1,2} host::modname\/file3 \/dest\/\nrsync -av host::modname\/file1 ::modname\/file{3,4}\n\nOlder versions of rsync required using quoted spaces in the SRC, like\nthese examples:\n\nrsync -av host:'dir1\/file1 dir2\/file2' \/dest\nrsync host::'modname\/dir1\/file1 modname\/dir2\/file2' \/dest\n\nThis word-splitting still works (by default) in the latest rsync, but is\nnot as easy to use as the first method.\n\nIf you need to transfer a filename that contains whitespace, you can\neither specify the --protect-args (-s) option, or you'll need to escape\nthe whitespace in a way that the remote shell will understand. For\ninstance:\n\nrsync -av host:'file\\ name\\ with\\ spaces' \/dest\n\nCONNECTING TO AN RSYNC DAEMON\nIt is also possible to use rsync without a remote shell as the transport.\nIn this case you will directly connect to a remote rsync daemon,\ntypically using TCP port 873. (This obviously requires the daemon to be\nrunning on the remote system, so refer to the STARTING AN RSYNC DAEMON TO\nACCEPT CONNECTIONS section below for information on that.)\n\nUsing rsync in this way is the same as using it with a remote shell\nexcept that:\n\no you either use a double colon :: instead of a single colon to\nseparate the hostname from the path, or you use an rsync:\/\/ URL.\n\no the first word of the \"path\" is actually a module name.\n\no the remote daemon may print a message of the day when you connect.\n\no if you specify no path name on the remote daemon then the list of\naccessible paths on the daemon will be shown.\n\no if you specify no local destination then a listing of the\nspecified files on the remote daemon is provided.\n\no you must not specify the --rsh (-e) option (since that overrides\nthe daemon connection to use ssh -- see USING RSYNC-DAEMON\nFEATURES VIA A REMOTE-SHELL CONNECTION below).\n\nAn example that copies all the files in a remote module named \"src\":\n\nrsync -av host::src \/dest\n\nSome modules on the remote daemon may require authentication. If so, you\nwill receive a password prompt when you connect. You can avoid the\npassword you want to use or using the --password-file option. This may\nbe useful when scripting rsync.\n\nWARNING: On some systems environment variables are visible to all users.\nOn those systems using --password-file is recommended.\n\nYou may establish the connection via a web proxy by setting the\nenvironment variable RSYNC_PROXY to a hostname:port pair pointing to your\nweb proxy. Note that your web proxy's configuration must support proxy\nconnections to port 873.\n\nYou may also establish a daemon connection using a program as a proxy by\nsetting the environment variable RSYNC_CONNECT_PROG to the commands you\nwish to run in place of making a direct socket connection. The string\nmay contain the escape \"%H\" to represent the hostname specified in the\nrsync command (so use \"%%\" if you need a single \"%\" in your string). For\nexample:\n\nexport RSYNC_CONNECT_PROG='ssh proxyhost nc %H 873'\nrsync -av targethost1::module\/src\/ \/dest\/\nrsync -av rsync:\/\/targethost2\/module\/src\/ \/dest\/\n\nThe command specified above uses ssh to run nc (netcat) on a proxyhost,\nwhich forwards all data to port 873 (the rsync daemon) on the targethost\n(%H).\n\nNote also that if the RSYNC_SHELL environment variable is set, that\nprogram will be used to run the RSYNC_CONNECT_PROG command instead of\nusing the default shell of the system() call.\n\nUSING RSYNC-DAEMON FEATURES VIA A REMOTE-SHELL CONNECTION\nIt is sometimes useful to use various features of an rsync daemon (such\nas named modules) without actually allowing any new socket connections\ninto a system (other than what is already required to allow remote-shell\naccess). Rsync supports connecting to a host using a remote shell and\nthen spawning a single-use \"daemon\" server that expects to read its\nconfig file in the home dir of the remote user. This can be useful if\nyou want to encrypt a daemon-style transfer's data, but since the daemon\nis started up fresh by the remote user, you may not be able to use\nfeatures such as chroot or change the uid used by the daemon. (For\nanother way to encrypt a daemon transfer, consider using ssh to tunnel a\nlocal port to a remote machine and configure a normal rsync daemon on\nthat remote host to only allow connections from \"localhost\".)\n\nFrom the user's perspective, a daemon transfer via a remote-shell\nconnection uses nearly the same command-line syntax as a normal rsync-\ndaemon transfer, with the only exception being that you must explicitly\nset the remote shell program on the command-line with the --rsh=COMMAND\noption. (Setting the RSYNC_RSH in the environment will not turn on this\nfunctionality.) For example:\n\nrsync -av --rsh=ssh host::module \/dest\n\nIf you need to specify a different remote-shell user, keep in mind that\nthe user@ prefix in front of the host is specifying the rsync-user value\n(for a module that requires user-based authentication). This means that\nyou must give the '-l user' option to ssh when specifying the remote-\nshell, as in this example that uses the short version of the --rsh\noption:\n\nrsync -av -e \"ssh -l ssh-user\" rsync-user@host::module \/dest\n\nThe \"ssh-user\" will be used at the ssh level; the \"rsync-user\" will be\nused to log-in to the \"module\".\n\nSTARTING AN RSYNC DAEMON TO ACCEPT CONNECTIONS\nIn order to connect to an rsync daemon, the remote system needs to have a\ndaemon already running (or it needs to have configured something like\ninetd to spawn an rsync daemon for incoming connections on a particular\nport). For full information on how to start a daemon that will handling\nincoming socket connections, see the rsyncd.conf(5) man page -- that is\nthe config file for the daemon, and it contains the full details for how\nto run the daemon (including stand-alone and inetd configurations).\n\nIf you're using one of the remote-shell transports for the transfer,\nthere is no need to manually start an rsync daemon.\n\nSORTED TRANSFER ORDER\nRsync always sorts the specified filenames into its internal transfer\nlist. This handles the merging together of the contents of identically\nnamed directories, makes it easy to remove duplicate filenames, and may\nconfuse someone when the files are transferred in a different order than\nwhat was given on the command-line.\n\nIf you need a particular file to be transferred prior to another, either\nseparate the files into different rsync calls, or consider using --delay-\nupdates (which doesn't affect the sorted transfer order, but does make\nthe final file-updating phase happen much more rapidly).\n\nEXAMPLES\nHere are some examples of how I use rsync.\n\nTo backup my wife's home directory, which consists of large MS Word files\nand mail folders, I use a cron job that runs\n\nrsync -Cavz . arvidsjaur:backup\n\neach night over a PPP connection to a duplicate directory on my machine\n\"arvidsjaur\".\n\nTo synchronize my samba source trees I use the following Makefile\ntargets:\n\nget:\nrsync -avuzb --exclude '*~' samba:samba\/ .\nput:\nrsync -Cavuzb . samba:samba\/\nsync: get put\n\nThis allows me to sync with a CVS directory at the other end of the\nconnection. I then do CVS operations on the remote machine, which saves\na lot of time as the remote CVS protocol isn't very efficient.\n\nI mirror a directory between my \"old\" and \"new\" ftp sites with the\ncommand:\n\nrsync -az -e ssh --delete ~ftp\/pub\/samba nimbus:\"~ftp\/pub\/tridge\"\n\nThis is launched from cron every few hours.\n\nOPTION SUMMARY\nHere is a short summary of the options available in rsync. Please refer\nto the detailed description below for a complete description.\n\n--verbose, -v increase verbosity\n--info=FLAGS fine-grained informational verbosity\n--debug=FLAGS fine-grained debug verbosity\n--stderr=e|a|c change stderr output mode (default: errors)\n--quiet, -q suppress non-error messages\n--no-motd suppress daemon-mode MOTD\n--checksum, -c skip based on checksum, not mod-time & size\n--archive, -a archive mode; equals -rlptgoD (no -H,-A,-X)\n--no-OPTION turn off an implied OPTION (e.g. --no-D)\n--recursive, -r recurse into directories\n--relative, -R use relative path names\n--no-implied-dirs don't send implied dirs with --relative\n--backup, -b make backups (see --suffix & --backup-dir)\n--backup-dir=DIR make backups into hierarchy based in DIR\n--suffix=SUFFIX backup suffix (default ~ w\/o --backup-dir)\n--inplace update destination files in-place\n--append append data onto shorter files\n--append-verify --append w\/old data in file checksum\n--dirs, -d transfer directories without recursing\n--mkpath create the destination's path component\n--perms, -p preserve permissions\n--executability, -E preserve executability\n--chmod=CHMOD affect file and\/or directory permissions\n--acls, -A preserve ACLs (implies --perms)\n--xattrs, -X preserve extended attributes\n--owner, -o preserve owner (super-user only)\n--group, -g preserve group\n--devices preserve device files (super-user only)\n--specials preserve special files\n-D same as --devices --specials\n--times, -t preserve modification times\n--atimes, -U preserve access (use) times\n--open-noatime avoid changing the atime on opened files\n--crtimes, -N preserve create times (newness)\n--omit-dir-times, -O omit directories from --times\n--fake-super store\/recover privileged attrs using xattrs\n--sparse, -S turn sequences of nulls into sparse blocks\n--preallocate allocate dest files before writing them\n--write-devices write to devices as files (implies --inplace)\n--dry-run, -n perform a trial run with no changes made\n--whole-file, -W copy files whole (w\/o delta-xfer algorithm)\n--checksum-choice=STR choose the checksum algorithm (aka --cc)\n--one-file-system, -x don't cross filesystem boundaries\n--block-size=SIZE, -B force a fixed checksum block-size\n--rsh=COMMAND, -e specify the remote shell to use\n--rsync-path=PROGRAM specify the rsync to run on remote machine\n--existing skip creating new files on receiver\n--ignore-existing skip updating files that exist on receiver\n--remove-source-files sender removes synchronized files (non-dir)\n--del an alias for --delete-during\n--delete delete extraneous files from dest dirs\n--delete-before receiver deletes before xfer, not during\n--delete-during receiver deletes during the transfer\n--delete-delay find deletions during, delete after\n--delete-after receiver deletes after transfer, not during\n--delete-excluded also delete excluded files from dest dirs\n--ignore-missing-args ignore missing source args without error\n--delete-missing-args delete missing source args from destination\n--ignore-errors delete even if there are I\/O errors\n--force force deletion of dirs even if not empty\n--max-delete=NUM don't delete more than NUM files\n--max-size=SIZE don't transfer any file larger than SIZE\n--min-size=SIZE don't transfer any file smaller than SIZE\n--max-alloc=SIZE change a limit relating to memory alloc\n--partial keep partially transferred files\n--partial-dir=DIR put a partially transferred file into DIR\n--delay-updates put all updated files into place at end\n--prune-empty-dirs, -m prune empty directory chains from file-list\n--numeric-ids don't map uid\/gid values by user\/group name\n--groupmap=STRING custom groupname mapping\n--timeout=SECONDS set I\/O timeout in seconds\n--contimeout=SECONDS set daemon connection timeout in seconds\n--ignore-times, -I don't skip files that match size and time\n--size-only skip files that match in size\n--modify-window=NUM, -@ set the accuracy for mod-time comparisons\n--temp-dir=DIR, -T create temporary files in directory DIR\n--fuzzy, -y find similar file for basis if no dest file\n--compare-dest=DIR also compare destination files relative to DIR\n--copy-dest=DIR ... and include copies of unchanged files\n--compress, -z compress file data during the transfer\n--compress-choice=STR choose the compression algorithm (aka --zc)\n--compress-level=NUM explicitly set compression level (aka --zl)\n--skip-compress=LIST skip compressing files with suffix in LIST\n--cvs-exclude, -C auto-ignore files in the same way CVS does\n--filter=RULE, -f add a file-filtering RULE\n-F same as --filter='dir-merge \/.rsync-filter'\nrepeated: --filter='- .rsync-filter'\n--exclude=PATTERN exclude files matching PATTERN\n--exclude-from=FILE read exclude patterns from FILE\n--include=PATTERN don't exclude files matching PATTERN\n--include-from=FILE read include patterns from FILE\n--files-from=FILE read list of source-file names from FILE\n--from0, -0 all *-from\/filter files are delimited by 0s\n--protect-args, -s no space-splitting; wildcard chars only\n--copy-as=USER[:GROUP] specify user & optional group for the copy\n--port=PORT specify double-colon alternate port number\n--sockopts=OPTIONS specify custom TCP options\n--blocking-io use blocking I\/O for the remote shell\n--outbuf=N|L|B set out buffering to None, Line, or Block\n--stats give some file-transfer stats\n--8-bit-output, -8 leave high-bit chars unescaped in output\n--progress show progress during transfer\n-P same as --partial --progress\n--itemize-changes, -i output a change-summary for all updates\n--remote-option=OPT, -M send OPTION to the remote side only\n--out-format=FORMAT output updates using the specified FORMAT\n--log-file=FILE log what we're doing to the specified FILE\n--log-file-format=FMT log updates using the specified FMT\n--early-input=FILE use FILE for daemon's early exec input\n--list-only list the files instead of copying them\n--bwlimit=RATE limit socket I\/O bandwidth\n--stop-after=MINS Stop rsync after MINS minutes have elapsed\n--stop-at=y-m-dTh:m Stop rsync at the specified point in time\n--write-batch=FILE write a batched update to FILE\n--only-write-batch=FILE like --write-batch but w\/o updating dest\n--protocol=NUM force an older protocol version to be used\n--iconv=CONVERT_SPEC request charset conversion of filenames\n--checksum-seed=NUM set block\/file checksum seed (advanced)\n--ipv4, -4 prefer IPv4\n--ipv6, -6 prefer IPv6\n--version, -V print the version + other info and exit\n--help, -h (*) show this help (* -h is help only on its own)\n\nRsync can also be run as a daemon, in which case the following options\nare accepted:\n\n--daemon run as an rsync daemon\n--bwlimit=RATE limit socket I\/O bandwidth\n--config=FILE specify alternate rsyncd.conf file\n--dparam=OVERRIDE, -M override global daemon config parameter\n--no-detach do not detach from the parent\n--port=PORT listen on alternate port number\n--log-file=FILE override the \"log file\" setting\n--log-file-format=FMT override the \"log format\" setting\n--sockopts=OPTIONS specify custom TCP options\n--verbose, -v increase verbosity\n--ipv4, -4 prefer IPv4\n--ipv6, -6 prefer IPv6\n--help, -h show this help (when used with --daemon)\n\nOPTIONS\nRsync accepts both long (double-dash + word) and short (single-dash +\nletter) options. The full list of the available options are described\nbelow. If an option can be specified in more than one way, the choices\nare comma-separated. Some options only have a long variant, not a short.\nIf the option takes a parameter, the parameter is only listed after the\nlong variant, even though it must also be specified for the short. When\nspecifying a parameter, you can either use the form --option=param or\nreplace the '=' with whitespace. The parameter may need to be quoted in\nsome manner for it to survive the shell's command-line parsing. Keep in\nmind that a leading tilde (~) in a filename is substituted by your shell,\nso --option=~\/foo will not change the tilde into your home directory\n(remove the '=' for that).\n\n--help, -h (*)\nPrint a short help page describing the options available in rsync\nand exit. (*) The -h short option will only invoke --help when\nused without other options since it normally means --human-\n\n--version, -V\nPrint the rsync version plus other info and exit.\n\nThe output includes the default list of checksum algorithms, the\ndefault list of compression algorithms, a list of compiled-in\ncapabilities, a link to the rsync web site, and some\n\n--verbose, -v\nThis option increases the amount of information you are given\nduring the transfer. By default, rsync works silently. A single\n-v will give you information about what files are being\ntransferred and a brief summary at the end. Two -v options will\ngive you information on what files are being skipped and slightly\nmore information at the end. More than two -v options should only\nbe used if you are debugging rsync.\n\nIn a modern rsync, the -v option is equivalent to the setting of\ngroups of --info and --debug options. You can choose to use these\nnewer options in addition to, or in place of using --verbose, as\nany fine-grained settings override the implied settings of -v.\nBoth --info and --debug have a way to ask for help that tells you\nexactly what flags are set for each increase in verbosity.\n\nHowever, do keep in mind that a daemon's \"max verbosity\" setting\nwill limit how high of a level the various individual flags can be\nset on the daemon side. For instance, if the max is 2, then any\ninfo and\/or debug flag that is set to a higher value than what\nwould be set by -vv will be downgraded to the -vv level in the\ndaemon's logging.\n\n--info=FLAGS\nThis option lets you have fine-grained control over the\ninformation output you want to see. An individual flag name may\nbe followed by a level number, with 0 meaning to silence that\noutput, 1 being the default output level, and higher numbers\nincreasing the output of that flag (for those that support higher\nlevels). Use --info=help to see all the available flag names,\nwhat they output, and what flag names are added for each increase\nin the verbose level. Some examples:\n\nrsync -a --info=progress2 src\/ dest\/\nrsync -avv --info=stats2,misc1,flist0 src\/ dest\/\n\nNote that --info=name's output is affected by the --out-format and\n--itemize-changes (-i) options. See those options for more\ninformation on what is output and when.\n\nThis option was added to 3.1.0, so an older rsync on the server\nside might reject your attempts at fine-grained control (if one or\nmore flags needed to be send to the server and the server was too\nabove when dealing with a daemon.\n\n--debug=FLAGS\nThis option lets you have fine-grained control over the debug\noutput you want to see. An individual flag name may be followed\nby a level number, with 0 meaning to silence that output, 1 being\nthe default output level, and higher numbers increasing the output\nof that flag (for those that support higher levels). Use\n--debug=help to see all the available flag names, what they\noutput, and what flag names are added for each increase in the\nverbose level. Some examples:\n\nrsync -avvv --debug=none src\/ dest\/\nrsync -avA --del --debug=del2,acl src\/ dest\/\n\nNote that some debug messages will only be output when\n--stderr=all is specified, especially those pertaining to I\/O and\nbuffer debugging.\n\nBeginning in 3.2.0, this option is no longer auto-forwarded to the\nserver side in order to allow you to specify different debug\nvalues for each side of the transfer, as well as to specify a new\ndebug option that is only present in one of the rsync versions.\nIf you want to duplicate the same option on both sides, using\nbrace expansion is an easy way to save you some typing. This\nworks in zsh and bash:\n\nrsync -aiv {-M,}--debug=del2 src\/ dest\/\n\n--stderr=errors|all|client\nThis option controls which processes output to stderr and if info\nmessages are also changed to stderr. The mode strings can be\nabbreviated, so feel free to use a single letter value. The 3\npossible choices are:\n\no errors - (the default) causes all the rsync processes to\nsend an error directly to stderr, even if the process is on\nthe remote side of the transfer. Info messages are sent to\nthe client side via the protocol stream. If stderr is not\navailable (i.e. when directly connecting with a daemon via\na socket) errors fall back to being sent via the protocol\nstream.\n\no all - causes all rsync messages (info and error) to get\nwritten directly to stderr from all (possible) processes.\nThis causes stderr to become line-buffered (instead of raw)\nand eliminates the ability to divide up the info and error\nmessages by file handle. For those doing debugging or\nusing several levels of verbosity, this option can help to\navoid clogging up the transfer stream (which should prevent\nany chance of a deadlock bug hanging things up). It also\nenables the outputting of some I\/O related debug messages.\n\no client - causes all rsync messages to be sent to the client\nside via the protocol stream. One client process outputs\nall messages, with errors on stderr and info messages on\nstdout. This was the default in older rsync versions, but\ncan cause error delays when a lot of transfer data is ahead\nof the messages. If you're pushing files to an older\nrsync, you may want to use --stderr=all since that idiom\nhas been around for several releases.\n\nThis option was added in rsync 3.2.3. This version also began the\nforwarding of a non-default setting to the remote side, though\nrsync uses the backward-compatible options --msgs2stderr and --no-\nmsgs2stderr to represent the all and client settings,\nrespectively. A newer rsync will continue to accept these older\noption names to maintain compatibility.\n\n--quiet, -q\nThis option decreases the amount of information you are given\nduring the transfer, notably suppressing information messages from\nthe remote server. This option is useful when invoking rsync from\ncron.\n\n--no-motd\nThis option affects the information that is output by the client\nat the start of a daemon transfer. This suppresses the message-\nof-the-day (MOTD) text, but it also affects the list of modules\nthat the daemon sends in response to the \"rsync host::\" request\n(due to a limitation in the rsync protocol), so omit this option\nif you want to request the list of modules from the daemon.\n\n--ignore-times, -I\nNormally rsync will skip any files that are already the same size\nand have the same modification timestamp. This option turns off\nthis \"quick check\" behavior, causing all files to be updated.\n\n--size-only\nThis modifies rsync's \"quick check\" algorithm for finding files\nthat need to be transferred, changing it from the default of\ntransferring files with either a changed size or a changed last-\nmodified time to just looking for files that have changed in size.\nThis is useful when starting to use rsync after using another\nmirroring system which may not preserve timestamps exactly.\n\n--modify-window=NUM, -@\nWhen comparing two timestamps, rsync treats the timestamps as\nbeing equal if they differ by no more than the modify-window\nvalue. The default is 0, which matches just integer seconds. If\nyou specify a negative value (and the receiver is at least version\n3.1.3) then nanoseconds will also be taken into account.\nSpecifying 1 is useful for copies to\/from MS Windows FAT\nfilesystems, because FAT represents times with a 2-second\nresolution (allowing times to differ from the original by up to 1\nsecond).\n\nIf you want all your transfers to default to comparing\nnanoseconds, you can create a ~\/.popt file and put these lines in\nit:\n\nrsync alias -a -a@-1\nrsync alias -t -t@-1\n\nWith that as the default, you'd need to specify --modify-window=0\n(aka -@0) to override it and ignore nanoseconds, e.g. if you're\ncopying between ext3 and ext4, or if the receiving rsync is older\nthan 3.1.3.\n\n--checksum, -c\nThis changes the way rsync checks if the files have been changed\nand are in need of a transfer. Without this option, rsync uses a\n\"quick check\" that (by default) checks if each file's size and\ntime of last modification match between the sender and receiver.\nThis option changes this to compare a 128-bit checksum for each\nfile that has a matching size. Generating the checksums means\nthat both sides will expend a lot of disk I\/O reading all the data\nin the files in the transfer, so this can slow things down\nsignificantly (and this is prior to any reading that will be done\nto transfer changed files)\n\nThe sending side generates its checksums while it is doing the\nfile-system scan that builds the list of the available files. The\nreceiver generates its checksums when it is scanning for changed\nfiles, and will checksum any file that has the same size as the\ncorresponding sender's file: files with either a changed size or a\nchanged checksum are selected for transfer.\n\nNote that rsync always verifies that each transferred file was\ncorrectly reconstructed on the receiving side by checking a whole-\nfile checksum that is generated as the file is transferred, but\nthat automatic after-the-transfer verification has nothing to do\nwith this option's before-the-transfer \"Does this file need to be\nupdated?\" check.\n\nThe checksum used is auto-negotiated between the client and the\nserver, but can be overridden using either the --checksum-choice\n(--cc) option or an environment variable that is discussed in that\noption's section.\n\n--archive, -a\nThis is equivalent to -rlptgoD. It is a quick way of saying you\nwant recursion and want to preserve almost everything (with -H\nbeing a notable omission). The only exception to the above\nequivalence is when --files-from is specified, in which case -r is\nnot implied.\n\nNote that -a does not preserve hardlinks, because finding\nmultiply-linked files is expensive. You must separately specify\n-H.\n\n--no-OPTION\nYou may turn off one or more implied options by prefixing the\noption name with \"no-\". Not all options may be prefixed with a\n\"no-\": only options that are implied by other options (e.g. --no-\nD, --no-perms) or have different defaults in various circumstances\n(e.g. --no-whole-file, --no-blocking-io, --no-dirs). You may\nspecify either the short or the long option name after the \"no-\"\nprefix (e.g. --no-R is the same as --no-relative).\n\nFor example: if you want to use -a (--archive) but don't want -o\n(--owner), instead of converting -a into -rlptgD, you could\nspecify -a --no-o (or -a --no-owner).\n\nThe order of the options is important: if you specify --no-r -a,\nthe -r option would end up being turned on, the opposite of\n-a --no-r. Note also that the side-effects of the --files-from\noption are NOT positional, as it affects the default state of\nseveral options and slightly changes the meaning of -a (see the\n--files-from option for more details).\n\n--recursive, -r\n(-d).\n\nBeginning with rsync 3.0.0, the recursive algorithm used is now an\nincremental scan that uses much less memory than before and begins\nthe transfer after the scanning of the first few directories have\nbeen completed. This incremental scan only affects our recursion\nalgorithm, and does not change a non-recursive transfer. It is\nalso only possible when both ends of the transfer are at least\nversion 3.0.0.\n\nSome options require rsync to know the full file list, so these\noptions disable the incremental recursion mode. These include:\n--delete-before, --delete-after, --prune-empty-dirs, and --delay-\nupdates. Because of this, the default delete mode when you\nspecify --delete is now --delete-during when both ends of the\nconnection are at least 3.0.0 (use --del or --delete-during to\n--delete-delay option that is a better choice than using --delete-\nafter.\n\nIncremental recursion can be disabled using the --no-inc-recursive\noption or its shorter --no-i-r alias.\n\n--relative, -R\nUse relative paths. This means that the full path names specified\non the command line are sent to the server rather than just the\nlast parts of the filenames. This is particularly useful when you\nwant to send several different directories at the same time. For\nexample, if you used this command:\n\nrsync -av \/foo\/bar\/baz.c remote:\/tmp\/\n\nwould create a file named baz.c in \/tmp\/ on the remote machine.\n\nrsync -avR \/foo\/bar\/baz.c remote:\/tmp\/\n\nthen a file named \/tmp\/foo\/bar\/baz.c would be created on the\nremote machine, preserving its full path. These extra path\nelements are called \"implied directories\" (i.e. the \"foo\" and the\n\"foo\/bar\" directories in the above example).\n\nBeginning with rsync 3.0.0, rsync always sends these implied\ndirectories as real directories in the file list, even if a path\nelement is really a symlink on the sending side. This prevents\nsome really unexpected behaviors when copying the full path of a\nfile that you didn't realize had a symlink in its path. If you\nvia its path, and referent directory via its real path. If you're\ndealing with an older rsync on the sending side, you may need to\nuse the --no-implied-dirs option.\n\nIt is also possible to limit the amount of path information that\nis sent as implied directories for each path you specify. With a\nmodern rsync on the sending side (beginning with 2.6.7), you can\ninsert a dot and a slash into the source path, like this:\n\nrsync -avR \/foo\/.\/bar\/baz.c remote:\/tmp\/\n\nThat would create \/tmp\/bar\/baz.c on the remote machine. (Note that\nthe dot must be followed by a slash, so \"\/foo\/.\" would not be\nabbreviated.) For older rsync versions, you would need to use a\nchdir to limit the source path. For example, when pushing files:\n\n(cd \/foo; rsync -avR bar\/baz.c remote:\/tmp\/)\n\n(Note that the parens put the two commands into a sub-shell, so\nthat the \"cd\" command doesn't remain in effect for future\ncommands.) If you're pulling files from an older rsync, use this\nidiom (but only for a non-daemon transfer):\n\nrsync -avR --rsync-path=\"cd \/foo; rsync\" \\\nremote:bar\/baz.c \/tmp\/\n\n--no-implied-dirs\nThis option affects the default behavior of the --relative option.\nWhen it is specified, the attributes of the implied directories\nfrom the source names are not included in the transfer. This\nmeans that the corresponding path elements on the destination\nsystem are left unchanged if they exist, and any missing implied\ndirectories are created with default attributes. This even allows\nthese implied path elements to have big differences, such as being\na symlink to a directory on the receiving side.\n\nFor instance, if a command-line arg or a files-from entry told\nrsync to transfer the file \"path\/foo\/file\", the directories \"path\"\nand \"path\/foo\" are implied when --relative is used. If \"path\/foo\"\nis a symlink to \"bar\" on the destination system, the receiving\nrsync would ordinarily delete \"path\/foo\", recreate it as a\ndirectory, and receive the file into the new directory. With\n--no-implied-dirs, the receiving rsync updates \"path\/foo\/file\"\nusing the existing path elements, which means that the file ends\nup being created in \"path\/bar\". Another way to accomplish this\nalso affect symlinks to directories in the rest of the transfer).\n\nWhen pulling files from an rsync older than 3.0.0, you may need to\nuse this option if the sending side has a symlink in the path you\nrequest and you wish the implied directories to be transferred as\nnormal directories.\n\n--backup, -b\nWith this option, preexisting destination files are renamed as\neach file is transferred or deleted. You can control where the\nbackup file goes and what (if any) suffix gets appended using the\n--backup-dir and --suffix options.\n\nNote that if you don't specify --backup-dir, (1) the --omit-dir-\ntimes option will be forced on, and (2) if --delete is also in\neffect (without --delete-excluded), rsync will add a \"protect\"\nfilter-rule for the backup suffix to the end of all your existing\nexcludes (e.g. -f \"P *~\"). This will prevent previously backed-up\nfiles from being deleted. Note that if you are supplying your own\nfilter rules, you may need to manually insert your own\nexclude\/protect rule somewhere higher up in the list so that it\nhas a high enough priority to be effective (e.g., if your rules\nspecify a trailing inclusion\/exclusion of *, the auto-added rule\nwould never be reached).\n\n--backup-dir=DIR\nThis implies the --backup option, and tells rsync to store all\nbackups in the specified directory on the receiving side. This\ncan be used for incremental backups. You can additionally specify\na backup suffix using the --suffix option (otherwise the files\nbacked up in the specified directory will keep their original\nfilenames).\n\nNote that if you specify a relative path, the backup directory\nwill be relative to the destination directory, so you probably\nwant to specify either an absolute path or a path that starts with\n\"..\/\". If an rsync daemon is the receiver, the backup dir cannot\ngo outside the module's path hierarchy, so take extra care not to\ndelete it or copy into it.\n\n--suffix=SUFFIX\nThis option allows you to override the default backup suffix used\nwith the --backup (-b) option. The default suffix is a ~ if no\n--backup-dir was specified, otherwise it is an empty string.\n\n--update, -u\nThis forces rsync to skip any files which exist on the destination\nand have a modified time that is newer than the source file. (If\nan existing destination file has a modification time equal to the\nsource file's, it will be updated if the sizes are different.)\n\nNote that this does not affect the copying of dirs, symlinks, or\nother special files. Also, a difference of file format between\nthe sender and receiver is always considered to be important\nenough for an update, no matter what date is on the objects. In\nother words, if the source has a directory where the destination\nhas a file, the transfer would occur regardless of the timestamps.\n\nThis option is a transfer rule, not an exclude, so it doesn't\naffect the data that goes into the file-lists, and thus it doesn't\naffect deletions. It just limits the files that the receiver\nrequests to be transferred.\n\n--inplace\nThis option changes how rsync transfers a file when its data needs\nto be updated: instead of the default method of creating a new\ncopy of the file and moving it into place when it is complete,\nrsync instead writes the updated data directly to the destination\nfile.\n\nThis has several effects:\n\no Hard links are not broken. This means the new data will be\nvisible through other hard links to the destination file.\nMoreover, attempts to copy differing source files onto a\nmultiply-linked destination file will result in a \"tug of\nwar\" with the destination data changing back and forth.\n\no In-use binaries cannot be updated (either the OS will\nprevent this from happening, or binaries that attempt to\nswap-in their data will misbehave or crash).\n\no The file's data will be in an inconsistent state during the\ntransfer and will be left that way if the transfer is\ninterrupted or if an update fails.\n\no A file that rsync cannot write to cannot be updated. While\na super user can update any file, a normal user needs to be\ngranted write permission for the open of the file for\nwriting to be successful.\n\no The efficiency of rsync's delta-transfer algorithm may be\nreduced if some data in the destination file is overwritten\nbefore it can be copied to a position later in the file.\nThis does not apply if you use --backup, since rsync is\nsmart enough to use the backup file as the basis file for\nthe transfer.\n\nWARNING: you should not use this option to update files that are\nbeing accessed by others, so be careful when choosing to use this\nfor a copy.\n\nThis option is useful for transferring large files with block-\nbased changes or appended data, and also on systems that are disk\nbound, not network bound. It can also help keep a copy-on-write\nfilesystem snapshot from diverging the entire contents of a file\nthat only has minor changes.\n\nThe option implies --partial (since an interrupted transfer does\nnot delete the file), but conflicts with --partial-dir and\n--delay-updates. Prior to rsync 2.6.4 --inplace was also\n\n--append\nThis special copy mode only works to efficiently update files that\nare known to be growing larger where any existing content on the\nreceiving side is also known to be the same as the content on the\nsender. The use of --append can be dangerous if you aren't 100%\nsure that all the files in the transfer are shared, growing files.\nYou should thus use filter rules to ensure that you weed out any\nfiles that do not fit this criteria.\n\nRsync updates these growing file in-place without verifying any of\nthe existing content in the file (it only verifies the content\nthat it is appending). Rsync skips any files that exist on the\nreceiving side that are not shorter than the associated file on\nthe sending side (which means that new files are trasnferred).\n\nThis does not interfere with the updating of a file's non-content\nattributes (e.g. permissions, ownership, etc.) when the file does\nnot need to be transferred, nor does it affect the updating of any\ndirectories or non-regular files.\n\n--append-verify\nThis special copy mode works like --append except that all the\ndata in the file is included in the checksum verification (making\nit much less efficient but also potentially safer). This option\ncan be dangerous if you aren't 100% sure that all the files in the\ntransfer are shared, growing files. See the --append option for\nmore details.\n\nNote: prior to rsync 3.0.0, the --append option worked like\n--append-verify, so if you are interacting with an older rsync (or\nthe transfer is using a protocol prior to 30), specifying either\nappend option will initiate an --append-verify transfer.\n\n--dirs, -d\nTell the sending side to include any directories that are\nencountered. Unlike --recursive, a directory's contents are not\ncopied unless the directory name specified is \".\" or ends with a\ntrailing slash (e.g. \".\", \"dir\/.\", \"dir\/\", etc.). Without this\noption or the --recursive option, rsync will skip all directories\nit encounters (and output a message to that effect for each one).\nIf you specify both --dirs and --recursive, --recursive takes\nprecedence.\n\nThe --dirs option is implied by the --files-from option or the\n--list-only option (including an implied --list-only usage) if\n--recursive wasn't specified (so that directories are seen in the\nlisting). Specify --no-dirs (or --no-d) if you want to turn this\noff.\n\nThere is also a backward-compatibility helper option, --old-dirs\n(or --old-d) that tells rsync to use a hack of -r --exclude='\/*\/*'\nto get an older rsync to list a single directory without\nrecursing.\n\n--mkpath\nCreate a missing path component of the destination arg. This\nallows rsync to create multiple levels of missing destination dirs\nand to create a path in which to put a single renamed file. Keep\nin mind that you'll need to supply a trailing slash if you want\nthe entire destination path to be treated as a directory when\ncopying a single arg (making rsync behave the same way that it\nexisted).\n\nFor example, the following creates a copy of file foo as bar in\nthe sub\/dir directory, creating dirs \"sub\" and \"sub\/dir\" if either\ndo not yet exist:\n\nrsync -ai --mkpath foo sub\/dir\/bar\n\nIf you instead ran the following, it would have created file foo\nin the sub\/dir\/bar directory:\n\nrsync -ai --mkpath foo sub\/dir\/bar\/\n\ndestination.\n\nWhen symlinks are encountered, the item that they point to (the\nreferent) is copied, rather than the symlink. In older versions\nof rsync, this option also had the side-effect of telling the\ndirectories. In a modern rsync such as this one, you'll need to\nspecify --keep-dirlinks (-K) to get this extra behavior. The only\nexception is when sending files to an rsync that is too old to\nunderstand -K -- in that case, the -L option will still have the\nside-effect of -K on that older receiving rsync.\n\nThis tells rsync to copy the referent of symbolic links that point\noutside the copied tree. Absolute symlinks are also treated like\nordinary files, and so are any symlinks in the source path itself\nwhen --relative is used. This option has no additional effect if\n\nNote that the cut-off point is the top of the transfer, which is\nthe part of the path that rsync isn't mentioning in the verbose\noutput. If you copy \"\/src\/subdir\" to \"\/dest\/\" then the \"subdir\"\ndirectory is a name inside the transfer tree, not the top of the\ntransfer (which is \/src) so it is legal for created relative\nsymlinks to refer to other names inside the \/src and \/dest\ndirectories. If you instead copy \"\/src\/subdir\/\" (with a trailing\nslash) to \"\/dest\/subdir\" that would not allow symlinks to any\nfiles outside of \"subdir\".\n\nThis tells rsync to ignore any symbolic links which point outside\nthe copied tree. All absolute symlinks are also ignored. Using\nthis option in conjunction with --relative may give unexpected\nresults.\n\nThis option tells rsync to (1) modify all symlinks on the\nreceiving side in a way that makes them unusable but recoverable\n(see below), or (2) to unmunge symlinks on the sending side that\nhad been stored in a munged state. This is useful if you don't\nquite trust the source of the data to not try to slip in a symlink\nto a unexpected place.\n\nThe way rsync disables the use of symlinks is to prefix each one\nwith the string \"\/rsyncd-munged\/\". This prevents the links from\nbeing used as long as that directory does not exist. When this\noption is enabled, rsync will refuse to run if that path is a\ndirectory or a symlink to a directory.\n\nThe option only affects the client side of the transfer, so if you\nneed it to affect the server, specify it via --remote-option.\n(Note that in a local transfer, the client side is the sender.)\n\nThis option has no affect on a daemon, since the daemon configures\nsupport directory of the source code.\n\nThis option causes the sending side to treat a symlink to a\ndirectory as though it were a real directory. This is useful if\nyou don't want symlinks to non-directories to be affected, as they\n\nWithout this option, if the sending side has replaced a directory\nwith a symlink to a directory, the receiving side will delete\nanything that is in the way of the new symlink, including a\ndirectory hierarchy (as long as --force or --delete is in effect).\n\nside.\n\nsource. If you want to follow only a few specified symlinks, a\ntrick you can use is to pass them as additional source args with a\ntrailing slash, using --relative to make the paths match up right.\nFor example:\n\nrsync -r --relative src\/.\/ src\/.\/follow-me\/ dest\/\n\nThis works because rsync calls lstat(2) on the source arg as\ngiving rise to a directory in the file-list which overrides the\nsymlink found during the scan of \"src\/.\/\".\n\nThis option causes the receiving side to treat a symlink to a\ndirectory as though it were a real directory, but only if it\nmatches a real directory from the sender. Without this option,\ndirectory.\n\nFor example, suppose you transfer a directory \"foo\" that contains\na file \"file\", but \"foo\" is a symlink to directory \"bar\" on the\n\"foo\", recreates it as a directory, and receives the file into the\nsymlink and \"file\" ends up in \"bar\".\n\nOne note of caution: if you use --keep-dirlinks, you must trust\nall the symlinks in the copy! If it is possible for an untrusted\nuser to create their own symlink to any directory, the user could\nthen (on a subsequent copy) replace the symlink with a real\ndirectory and affect the content of whatever directory the symlink\nreferences. For backup copies, you are better off using something\nhierarchy.\n\nside.\n\nThis tells rsync to look for hard-linked files in the source and\nlink together the corresponding files on the destination. Without\nthis option, hard-linked files in the source are treated as though\nthey were separate files.\n\nThis option does NOT necessarily ensure that the pattern of hard\nlinks on the destination exactly matches that on the source.\nCases in which the destination may end up with extra hard links\ninclude the following:\n\no If the destination contains extraneous hard-links (more\nlinking than what is present in the source file list), the\ncopying algorithm will not break them explicitly. However,\nif one or more of the paths have content differences, the\nnormal file-update process will break those extra links\n(unless you are using the --inplace option).\n\no If you specify a --link-dest directory that contains hard\n--link-dest files can cause some paths in the destination\nassociations.\n\nNote that rsync can only detect hard links between files that are\ninside the transfer set. If rsync updates a file that has extra\nwill be broken. If you are tempted to use the --inplace option to\navoid this breakage, be very careful that you know how your files\nare being updated so that you are certain that no unintended\nchanges happen due to lingering hard links (and see the --inplace\noption for more caveats).\n\nIf incremental recursion is active (see --recursive), rsync may\ntransfer a missing hard-linked file before it finds that another\nlink for that contents exists elsewhere in the hierarchy. This\ndoes not affect the accuracy of the transfer (i.e. which files are\nhard-linked together), just its efficiency (i.e. copying the data\nfor a new, early copy of a hard-linked file that could have been\nfound later in the transfer in another member of the hard-linked\nset of files). One way to avoid this inefficiency is to disable\nincremental recursion using the --no-inc-recursive option.\n\n--perms, -p\nThis option causes the receiving rsync to set the destination\nthe --chmod option for a way to modify what rsync considers to be\nthe source permissions.)\n\nWhen this option is off, permissions are set as follows:\n\no Existing files (including updated files) retain their\nexisting permissions, though the --executability option\nmight change just the execute permission for the file.\n\no New files get their \"normal\" permission bits set to the\nsource file's permissions masked with the receiving\ndirectory's default permissions (either the receiving\nprocess's umask, or the permissions specified via the\ndestination directory's default ACL), and their special\npermission bits disabled except in the case where a new\ndirectory inherits a setgid bit from its parent directory.\n\nThus, when --perms and --executability are both disabled, rsync's\nbehavior is the same as that of other file-copy utilities, such as\ncp(1) and tar(1).\n\nIn summary: to give destination files (both old and new) the\nsource permissions, use --perms. To give new files the\ndestination-default permissions (while leaving existing files\nunchanged), make sure that the --perms option is off and use\n--chmod=ugo=rwX (which ensures that all non-masked bits get\nenabled). If you'd care to make this latter behavior easier to\ntype, you could define a popt alias for it, such as putting this\nline in the file ~\/.popt (the following defines the -Z option, and\nincludes --no-g to use the default group of the destination dir):\n\nrsync alias -Z --no-p --no-g --chmod=ugo=rwX\n\nYou could then use this new option in a command such as this one:\n\nrsync -avZ src\/ dest\/\n\n(Caveat: make sure that -a does not follow -Z, or it will re-\nenable the two --no-* options mentioned above.)\n\nThe preservation of the destination's setgid bit on newly-created\ndirectories when --perms is off was added in rsync 2.6.7. Older\nrsync versions erroneously preserved the three special permission\nbits for newly-created files when --perms was off, while\noverriding the destination's setgid bit setting on a newly-created\ndirectory. Default ACL observance was added to the ACL patch for\nrsync 2.6.7, so older (or non-ACL-enabled) rsyncs use the umask\neven if default ACLs are present. (Keep in mind that it is the\nversion of the receiving rsync that affects these behaviors.)\n\n--executability, -E\nThis option causes rsync to preserve the executability (or non-\nexecutability) of regular files when --perms is not enabled. A\nregular file is considered to be executable if at least one 'x' is\nturned on in its permissions. When an existing destination file's\nexecutability differs from that of the corresponding source file,\nrsync modifies the destination file's permissions as follows:\n\no To make a file non-executable, rsync turns off all its 'x'\npermissions.\n\no To make a file executable, rsync turns on each 'x'\npermission that has a corresponding 'r' permission enabled.\n\nIf --perms is enabled, this option is ignored.\n\n--acls, -A\nThis option causes rsync to update the destination ACLs to be the\nsame as the source ACLs. The option also implies --perms.\n\nThe source and destination systems must have compatible ACL\nentries for this option to work properly. See the --fake-super\noption for a way to backup and restore ACLs that are not\ncompatible.\n\n--xattrs, -X\nThis option causes rsync to update the destination extended\nattributes to be the same as the source ones.\n\nFor systems that support extended-attribute namespaces, a copy\nbeing done by a super-user copies all namespaces except system.*.\nA normal user only copies the user.* namespace. To be able to\nbackup and restore non-user namespaces as a normal user, see the\n--fake-super option.\n\nThe above name filtering can be overridden by using one or more\nfilter options with the x modifier. When you specify an xattr-\naffecting filter rule, rsync requires that you do your own\nsystem\/user filtering, as well as any additional filtering for\nwhat xattr names are copied and what names are allowed to be\ndeleted. For example, to skip the system namespace, you could\nspecify:\n\n--filter='-x system.*'\n\nTo skip all namespaces except the user namespace, you could\nspecify a negated-user match:\n\n--filter='-x! user.*'\n\nTo prevent any attributes from being deleted, you could specify a\nreceiver-only rule that excludes all names:\n\n--filter='-xr *'\n\nNote that the -X option does not copy rsync's special xattr values\n(e.g. those used by --fake-super) unless you repeat the option\n(e.g. -XX). This \"copy all xattrs\" mode cannot be used with\n--fake-super.\n\n--chmod=CHMOD\nThis option tells rsync to apply one or more comma-separated\n\"chmod\" modes to the permission of the files in the transfer. The\nresulting value is treated as though it were the permissions that\nthe sending side supplied for the file, which means that this\noption can seem to have no effect on existing files if --perms is\nnot enabled.\n\nIn addition to the normal parsing rules specified in the chmod(1)\nmanpage, you can specify an item that should only apply to a\ndirectory by prefixing it with a 'D', or specify an item that\nshould only apply to a file by prefixing it with a 'F'. For\nexample, the following will ensure that all directories get marked\nset-gid, that no files are other-writable, that both are user-\nwritable and group-writable, and that both have consistent\nexecutability across all bits:\n\n--chmod=Dg+s,ug+w,Fo-w,+X\n\nUsing octal mode numbers is also allowed:\n\n--chmod=D2775,F664\n\nIt is also legal to specify multiple --chmod options, as each\nadditional option is just appended to the list of changes to make.\n\nSee the --perms and --executability options for how the resulting\npermission value can be applied to the files in the transfer.\n\n--owner, -o\nThis option causes rsync to set the owner of the destination file\nto be the same as the source file, but only if the receiving rsync\nis being run as the super-user (see also the --super and --fake-\nsuper options). Without this option, the owner of new and\/or\ntransferred files are set to the invoking user on the receiving\nside.\n\nThe preservation of ownership will associate matching names by\ndefault, but may fall back to using the ID number in some\ndiscussion).\n\n--group, -g\nThis option causes rsync to set the group of the destination file\nto be the same as the source file. If the receiving program is\nnot running as the super-user (or if --no-super was specified),\nonly groups that the invoking user on the receiving side is a\nmember of will be preserved. Without this option, the group is\nset to the default group of the invoking user on the receiving\nside.\n\nThe preservation of group information will associate matching\nnames by default, but may fall back to using the ID number in some\ndiscussion).\n\n--devices\nThis option causes rsync to transfer character and block device\nfiles to the remote system to recreate these devices. This option\nhas no effect if the receiving rsync is not run as the super-user\n\n--specials\nThis option causes rsync to transfer special files such as named\nsockets and fifos.\n\n-D The -D option is equivalent to --devices --specials.\n\n--write-devices\nThis tells rsync to treat a device on the receiving side as a\nregular file, allowing the writing of file data into a device.\n\nThis option implies the --inplace option.\n\nBe careful using this, as you should know what devices are present\non the receiving side of the transfer, especially if running rsync\nas root.\n\nThis option is refused by an rsync daemon.\n\n--times, -t\nThis tells rsync to transfer modification times along with the\nfiles and update them on the remote system. Note that if this\noption is not used, the optimization that excludes files that have\nnot been modified cannot be effective; in other words, a missing\n-t or -a will cause the next transfer to behave as if it used -I,\ncausing all files to be updated (though rsync's delta-transfer\nalgorithm will make the update fairly efficient if the files\nhaven't actually changed, you're much better off using -t).\n\n--atimes, -U\nThis tells rsync to set the access (use) times of the destination\nfiles to the same value as the source files.\n\nIf repeated, it also sets the --open-noatime option, which can\nhelp you to make the sending and receiving systems have the same\naccess times on the transferred files without needing to run rsync\nan extra time after a file is transferred.\n\nNote that some older rsync versions (prior to 3.2.0) may have been\nbuilt with a pre-release --atimes patch that does not imply\n--open-noatime when this option is repeated.\n\n--open-noatime\nThis tells rsync to open files with the O_NOATIME flag (on systems\nthat support it) to avoid changing the access time of the files\nthat are being transferred. If your OS does not support the\nO_NOATIME flag then rsync will silently ignore this option. Note\nalso that some filesystems are mounted to avoid updating the atime\non read access even without the O_NOATIME flag being set.\n\n--crtimes, -N,\nThis tells rsync to set the create times (newness) of the\ndestination files to the same value as the source files.\n\n--omit-dir-times, -O\nThis tells rsync to omit directories when it is preserving\nmodification times (see --times). If NFS is sharing the\ndirectories on the receiving side, it is a good idea to use -O.\nThis option is inferred if you use --backup without --backup-dir.\n\nThis option also has the side-effect of avoiding early creation of\ndirectories in incremental recursion copies. The default --inc-\nrecursive copying normally does an early-create pass of all the\nsub-directories in a parent directory in order for it to be able\nto then set the modify time of the parent directory right away\n(without having to delay that until a bunch of recursive copying\nhas finished). This early-create idiom is not necessary if\ndirectory modify times are not being preserved, so it is skipped.\nSince early-create directories don't have accurate mode, mtime, or\nownership, the use of this option can help when someone wants to\navoid these partially-finished directories.\n\nThis tells rsync to omit symlinks when it is preserving\nmodification times (see --times).\n\n--super\nThis tells the receiving side to attempt super-user activities\neven if the receiving rsync wasn't run by the super-user. These\nactivities include: preserving users via the --owner option,\npreserving all groups (not just the current user's groups) via the\n--groups option, and copying devices via the --devices option.\nThis is useful for systems that allow such activities without\nbeing the super-user, and also for ensuring that you will get\nerrors if the receiving side isn't being run as the super-user.\nTo turn off super-user activities, the super-user can use --no-\nsuper.\n\n--fake-super\nWhen this option is enabled, rsync simulates super-user activities\nby saving\/restoring the privileged attributes via special extended\nattributes that are attached to each file (as needed). This\nincludes the file's owner and group (if it is not the default),\nthe file's device info (device & special files are created as\nempty text files), and any permission bits that we won't allow to\nbe set on the real file (e.g. the real file gets u-s,g-s,o-t for\nsafety) or that would limit the owner's access (since the real\nsuper-user can always access\/change a file, the files we create\ncan always be accessed\/changed by the creating user). This option\nalso handles ACLs (if --acls was specified) and non-user extended\nattributes (if --xattrs was specified).\n\nThis is a good way to backup data without using a super-user, and\nto store ACLs from incompatible systems.\n\nThe --fake-super option only affects the side where the option is\nused. To affect the remote side of a remote-shell connection, use\nthe --remote-option (-M) option:\n\nrsync -av -M--fake-super \/src\/ host:\/dest\/\n\nFor a local copy, this option affects both the source and the\ndestination. If you wish a local copy to enable this option just\nfor the destination files, specify -M--fake-super. If you wish a\nlocal copy to enable this option just for the source files,\ncombine --fake-super with -M--super.\n\nThis option is overridden by both --super and --no-super.\n\nfile.\n\n--sparse, -S\nTry to handle sparse files efficiently so they take up less space\non the destination. If combined with --inplace the file created\nmight not end up with sparse blocks with some combinations of\nkernel version and\/or filesystem type. If --whole-file is in\neffect (e.g. for a local copy) then it will always work because\nrsync truncates the file prior to writing out the updated version.\n\nNote that versions of rsync older than 3.1.3 will reject the\ncombination of --sparse and --inplace.\n\n--preallocate\nThis tells the receiver to allocate each destination file to its\neventual size before writing data to the file. Rsync will only\nuse the real filesystem-level preallocation support provided by\nLinux's fallocate(2) system call or Cygwin's posix_fallocate(3),\nnot the slow glibc implementation that writes a null byte into\neach block.\n\nWithout this option, larger files may not be entirely contiguous\non the filesystem, but with this option rsync will probably copy\nmore slowly. If the destination is not an extent-supporting\nfilesystem (such as ext4, xfs, NTFS, etc.), this option may have\nno positive effect at all.\n\nIf combined with --sparse, the file will only have sparse blocks\n(as opposed to allocated sequences of null bytes) if the kernel\nversion and filesystem type support creating holes in the\nallocated data.\n\n--dry-run, -n\nThis makes rsync perform a trial run that doesn't make any changes\n(and produces mostly the same output as a real run). It is most\ncommonly used in combination with the --verbose, -v and\/or\n--itemize-changes, -i options to see what an rsync command is\ngoing to do before one actually runs it.\n\nThe output of --itemize-changes is supposed to be exactly the same\non a dry run and a subsequent real run (barring intentional\ntrickery and system call failures); if it isn't, that's a bug.\nOther output should be mostly unchanged, but may differ in some\nareas. Notably, a dry run does not send the actual data for file\ntransfers, so --progress has no effect, the \"bytes sent\", \"bytes\nreceived\", \"literal data\", and \"matched data\" statistics are too\nsmall, and the \"speedup\" value is equivalent to a run where no\nfile transfers were needed.\n\n--whole-file, -W\nThis option disables rsync's delta-transfer algorithm, which\ncauses all transferred files to be sent whole. The transfer may\nbe faster if this option is used when the bandwidth between the\nsource and destination machines is higher than the bandwidth to\ndisk (especially when the \"disk\" is actually a networked\nfilesystem). This is the default when both the source and\ndestination are specified as local paths, but only if no batch-\nwriting option is in effect.\n\n--checksum-choice=STR, --cc=STR\nThis option overrides the checksum algorithms. If one algorithm\nname is specified, it is used for both the transfer checksums and\n(assuming --checksum is specified) the pre-transfer checksums. If\ntwo comma-separated names are supplied, the first name affects the\ntransfer checksums, and the second name affects the pre-transfer\nchecksums (-c).\n\nThe checksum options that you may be able to use are:\n\no auto (the default automatic choice)\n\no xxh128\n\no xxh3\n\no xxh64 (aka xxhash)\n\no md5\n\no md4\n\no none\n\nRun rsync --version to see the default checksum list compiled into\nyour version (which may differ from the list above).\n\nIf \"none\" is specified for the first (or only) name, the --whole-\nfile option is forced on and no checksum verification is performed\non the transferred data. If \"none\" is specified for the second\n(or only) name, the --checksum option cannot be used.\n\nThe \"auto\" option is the default, where rsync bases its algorithm\nchoice on a negotiation between the client and the server as\nfollows:\n\nWhen both sides of the transfer are at least 3.2.0, rsync chooses\nthe first algorithm in the client's list of choices that is also\nin the server's list of choices. If no common checksum choice is\nfound, rsync exits with an error. If the remote rsync is too old\nto support checksum negotiation, a value is chosen based on the\nprotocol version (which chooses between MD5 and various flavors of\nMD4 based on protocol age).\n\nThe default order can be customized by setting the environment\nvariable RSYNC_CHECKSUM_LIST to a space-separated list of\nacceptable checksum names. If the string contains a \"&\"\ncharacter, it is separated into the \"client string & server\nstring\", otherwise the same string applies to both. If the string\n(or string portion) contains no non-whitespace characters, the\ndefault checksum list is used. This method does not allow you to\nspecify the transfer checksum separately from the pre-transfer\nchecksum, and it discards \"auto\" and all unknown checksum names.\nA list with only invalid names results in a failed negotiation.\n\nThe use of the --checksum-choice option overrides this environment\nlist.\n\n--one-file-system, -x\nThis tells rsync to avoid crossing a filesystem boundary when\nrecursing. This does not limit the user's ability to specify\nitems to copy from multiple filesystems, just rsync's recursion\nthrough the hierarchy of each directory that the user specified,\nand also the analogous recursion on the receiving side during\ndeletion. Also keep in mind that rsync treats a \"bind\" mount to\nthe same device as being on the same filesystem.\n\nIf this option is repeated, rsync omits all mount-point\ndirectories from the copy. Otherwise, it includes an empty\ndirectory at each mount-point it encounters (using the attributes\nof the mounted directory because those of the underlying mount-\npoint directory are inaccessible).\n\nis treated like a mount-point. Symlinks to non-directories are\nunaffected by this option.\n\n--existing, --ignore-non-existing\nThis tells rsync to skip creating files (including directories)\nthat do not exist yet on the destination. If this option is\ncombined with the --ignore-existing option, no files will be\nupdated (which can be useful if all you want to do is delete\nextraneous files).\n\nThis option is a transfer rule, not an exclude, so it doesn't\naffect the data that goes into the file-lists, and thus it doesn't\naffect deletions. It just limits the files that the receiver\nrequests to be transferred.\n\n--ignore-existing\nThis tells rsync to skip updating files that already exist on the\ndestination (this does not ignore existing directories, or nothing\n\nThis option is a transfer rule, not an exclude, so it doesn't\naffect the data that goes into the file-lists, and thus it doesn't\naffect deletions. It just limits the files that the receiver\nrequests to be transferred.\n\nThis option can be useful for those doing backups using the\n--link-dest option when they need to continue a backup run that\ngot interrupted. Since a --link-dest run is copied into a new\ndirectory hierarchy (when it is used properly), using --ignore-\nexisting will ensure that the already-handled files don't get\ntweaked (which avoids a change in permissions on the hard-linked\nfiles). This does mean that this option is only looking at the\nexisting files in the destination hierarchy itself.\n\n--remove-source-files\nThis tells rsync to remove from the sending side the files\n(meaning non-directories) that are a part of the transfer and have\nbeen successfully duplicated on the receiving side.\n\nNote that you should only use this option on source files that are\nquiescent. If you are using this to move files that show up in a\nparticular directory over to another host, make sure that the\nfinished files get renamed into the source directory, not directly\nwritten into it, so that rsync can't possibly transfer a file that\nis not yet fully written. If you can't first write the files into\na different directory, you should use a naming idiom that lets\nrsync avoid transferring files that are not yet finished (e.g.\nname the file \"foo.new\" when it is written, rename it to \"foo\"\nwhen it is done, and then use the option --exclude='*.new' for the\nrsync transfer).\n\nStarting with 3.1.0, rsync will skip the sender-side removal (and\noutput an error) if the file's size or modify time has not stayed\nunchanged.\n\n--delete\nThis tells rsync to delete extraneous files from the receiving\nside (ones that aren't on the sending side), but only for the\ndirectories that are being synchronized. You must have asked\nrsync to send the whole directory (e.g. \"dir\" or \"dir\/\") without\nusing a wildcard for the directory's contents (e.g. \"dir\/*\") since\nthe wildcard is expanded by the shell and rsync thus gets a\nrequest to transfer individual files, not the files' parent\ndirectory. Files that are excluded from the transfer are also\nexcluded from being deleted unless you use the --delete-excluded\noption or mark the rules as only matching on the sending side (see\nthe include\/exclude modifiers in the FILTER RULES section).\n\nPrior to rsync 2.6.7, this option would have no effect unless\n--recursive was enabled. Beginning with 2.6.7, deletions will\nalso occur when --dirs (-d) is enabled, but only for directories\nwhose contents are being copied.\n\nThis option can be dangerous if used incorrectly! It is a very\ngood idea to first try a run using the --dry-run option (-n) to\nsee what files are going to be deleted.\n\nIf the sending side detects any I\/O errors, then the deletion of\nany files at the destination will be automatically disabled. This\nis to prevent temporary filesystem failures (such as NFS errors)\non the sending side from causing a massive deletion of files on\nthe destination. You can override this with the --ignore-errors\noption.\n\nThe --delete option may be combined with one of the --delete-WHEN\noptions without conflict, as well as --delete-excluded. However,\nif none of the --delete-WHEN options are specified, rsync will\nchoose the --delete-during algorithm when talking to rsync 3.0.0\nor newer, and the --delete-before algorithm when talking to an\n\n--delete-before\nRequest that the file-deletions on the receiving side be done\nbefore the transfer starts. See --delete (which is implied) for\nmore details on file-deletion.\n\nDeleting before the transfer is helpful if the filesystem is tight\nfor space and removing extraneous files would help to make the\ntransfer possible. However, it does introduce a delay before the\nstart of the transfer, and this delay might cause the transfer to\ntimeout (if --timeout was specified). It also forces rsync to use\nthe old, non-incremental recursion algorithm that requires rsync\nto scan all the files in the transfer into memory at once (see\n--recursive).\n\n--delete-during, --del\nRequest that the file-deletions on the receiving side be done\nincrementally as the transfer happens. The per-directory delete\nscan is done right before each directory is checked for updates,\nso it behaves like a more efficient --delete-before, including\ndoing the deletions prior to any per-directory filter files being\nupdated. This option was first added in rsync version 2.6.4. See\n--delete (which is implied) for more details on file-deletion.\n\n--delete-delay\nRequest that the file-deletions on the receiving side be computed\nduring the transfer (like --delete-during), and then removed after\nthe transfer completes. This is useful when combined with\n--delay-updates and\/or --fuzzy, and is more efficient than using\n--delete-after (but can behave differently, since --delete-after\ncomputes the deletions in a separate pass after all updates are\ndone). If the number of removed files overflows an internal\nbuffer, a temporary file will be created on the receiving side to\nhold the names (it is removed while open, so you shouldn't see it\nduring the transfer). If the creation of the temporary file\nfails, rsync will try to fall back to using --delete-after (which\nit cannot do if --recursive is doing an incremental scan). See\n--delete (which is implied) for more details on file-deletion.\n\n--delete-after\nRequest that the file-deletions on the receiving side be done\nafter the transfer has completed. This is useful if you are\nsending new per-directory merge files as a part of the transfer\nand you want their exclusions to take effect for the delete phase\nof the current transfer. It also forces rsync to use the old,\nnon-incremental recursion algorithm that requires rsync to scan\nall the files in the transfer into memory at once (see\n--recursive). See --delete (which is implied) for more details on\nfile-deletion.\n\n--delete-excluded\nIn addition to deleting the files on the receiving side that are\nnot on the sending side, this tells rsync to also delete any files\non the receiving side that are excluded (see --exclude). See the\nFILTER RULES section for a way to make individual exclusions\nbehave this way on the receiver, and for a way to protect files\nfrom --delete-excluded. See --delete (which is implied) for more\ndetails on file-deletion.\n\n--ignore-missing-args\nWhen rsync is first processing the explicitly requested source\nfiles (e.g. command-line arguments or --files-from entries), it\nis normally an error if the file cannot be found. This option\nsuppresses that error, and does not try to transfer the file.\nThis does not affect subsequent vanished-file errors if a file was\ninitially found to be present and later is no longer there.\n\n--delete-missing-args\nThis option takes the behavior of (the implied) --ignore-missing-\nargs option a step farther: each missing arg will become a\ndeletion request of the corresponding destination file on the\nreceiving side (should it exist). If the destination file is a\nnon-empty directory, it will only be successfully deleted if\n--force or --delete are in effect. Other than that, this option\nis independent of any other type of delete processing.\n\nThe missing source files are represented by special file-list\nentries which display as a \"*missing\" entry in the --list-only\noutput.\n\n--ignore-errors\nTells --delete to go ahead and delete files even when there are\nI\/O errors.\n\n--force\nThis option tells rsync to delete a non-empty directory when it is\nto be replaced by a non-directory. This is only relevant if\ndeletions are not active (see --delete for details).\n\nNote for older rsync versions: --force used to still be required\nwhen using --delete-after, and it used to be non-functional unless\nthe --recursive option was also enabled.\n\n--max-delete=NUM\nThis tells rsync not to delete more than NUM files or directories.\nIf that limit is exceeded, all further deletions are skipped\nthrough the end of the transfer. At the end, rsync outputs a\nwarning (including a count of the skipped deletions) and exits\nwith an error code of 25 (unless some more important error\ncondition also occurred).\n\nBeginning with version 3.0.0, you may specify --max-delete=0 to be\nwarned about any extraneous files in the destination without\nremoving any of them. Older clients interpreted this as\n\"unlimited\", so if you don't know what version the client is, you\ncan use the less obvious --max-delete=-1 as a backward-compatible\nway to specify that no deletions be allowed (though really old\nversions didn't warn when the limit was exceeded).\n\n--max-size=SIZE\nThis tells rsync to avoid transferring any file that is larger\nthan the specified SIZE. A numeric value can be suffixed with a\nstring to indicate the numeric units or left unqualified to\nspecify bytes. Feel free to use a fractional value along with the\nunits, such as --max-size=1.5m.\n\nThis option is a transfer rule, not an exclude, so it doesn't\naffect the data that goes into the file-lists, and thus it doesn't\naffect deletions. It just limits the files that the receiver\nrequests to be transferred.\n\nThe first letter of a units string can be B (bytes), K (kilo), M\n(mega), G (giga), T (tera), or P (peta). If the string is a\nsingle char or has \"ib\" added to it (e.g. \"G\" or \"GiB\") then the\nunits are multiples of 1024. If you use a two-letter suffix that\nends with a \"B\" (e.g. \"kb\") then you get units that are multiples\nof 1000. The string's letters can be any mix of upper and lower-\ncase that you want to use.\n\nFinally, if the string ends with either \"+1\" or \"-1\", it is offset\nby one byte in the indicated direction. The largest possible\nvalue is usually 8192P-1.\n\nExamples: --max-size=1.5mb-1 is 1499999 bytes, and --max-size=2g+1\nis 2147483649 bytes.\n\nNote that rsync versions prior to 3.1.0 did not allow --max-\nsize=0.\n\n--min-size=SIZE\nThis tells rsync to avoid transferring any file that is smaller\nthan the specified SIZE, which can help in not transferring small,\njunk files. See the --max-size option for a description of SIZE\nand other information.\n\nNote that rsync versions prior to 3.1.0 did not allow --min-\nsize=0.\n\n--max-alloc=SIZE\nBy default rsync limits an individual malloc\/realloc to about 1GB\nin size. For most people this limit works just fine and prevents\na protocol error causing rsync to request massive amounts of\nmemory. However, if you have many millions of files in a\ntransfer, a large amount of server memory, and you don't want to\nsplit up your transfer into multiple parts, you can increase the\nper-allocation limit to something larger and rsync will consume\nmore memory.\n\nKeep in mind that this is not a limit on the total size of\nallocated memory. It is a sanity-check value for each individual\nallocation.\n\nSee the --max-size option for a description of how SIZE can be\nspecified. The default suffix if none is given is bytes.\n\nBeginning in 3.2.3, a value of 0 specifies no limit.\n\nYou can set a default value using the environment variable\nRSYNC_MAX_ALLOC using the same SIZE values as supported by this\noption. If the remote rsync doesn't understand the --max-alloc\noption, you can override an environmental value by specifying\n--max-alloc=1g, which will make rsync avoid sending the option to\nthe remote side (because \"1G\" is the default).\n\n--block-size=SIZE, -B\nThis forces the block size used in rsync's delta-transfer\nalgorithm to a fixed value. It is normally selected based on the\nsize of each file being updated. See the technical report for\ndetails.\n\nBeginning in 3.2.3 the SIZE can be specified with a suffix as\ndetailed in the --max-size option. Older versions only accepted a\nbyte count.\n\n--rsh=COMMAND, -e\nThis option allows you to choose an alternative remote shell\nprogram to use for communication between the local and remote\ncopies of rsync. Typically, rsync is configured to use ssh by\ndefault, but you may prefer to use rsh on a local network.\n\nIf this option is used with [user@]host::module\/path, then the\nremote shell COMMAND will be used to run an rsync daemon on the\nremote host, and all data will be transmitted through that remote\nshell connection, rather than through a direct socket connection\nto a running rsync daemon on the remote host. See the section\n\"USING RSYNC-DAEMON FEATURES VIA A REMOTE-SHELL CONNECTION\" above.\n\nBeginning with rsync 3.2.0, the RSYNC_PORT environment variable\nwill be set when a daemon connection is being made via a remote-\nshell connection. It is set to 0 if the default daemon port is\nbeing assumed, or it is set to the value of the rsync port that\nwas specified via either the --port option or a non-empty port\nvalue in an rsync:\/\/ URL. This allows the script to discern if a\nnon-default port is being requested, allowing for things such as\nan SSL or stunnel helper script to connect to a default or\nalternate port.\n\nCommand-line arguments are permitted in COMMAND provided that\nCOMMAND is presented to rsync as a single argument. You must use\nspaces (not tabs or other whitespace) to separate the command and\nargs from each other, and you can use single- and\/or double-quotes\nto preserve spaces in an argument (but not backslashes). Note\nthat doubling a single-quote inside a single-quoted string gives\nyou a single-quote; likewise for double-quotes (though you need to\npay attention to which quotes your shell is parsing and which\nquotes rsync is parsing). Some examples:\n\n-e 'ssh -p 2234'\n-e 'ssh -o \"ProxyCommand nohup ssh firewall nc -w1 %h %p\"'\n\n(Note that ssh users can alternately customize site-specific\nconnect options in their .ssh\/config file.)\n\nYou can also choose the remote shell program using the RSYNC_RSH\nenvironment variable, which accepts the same range of values as\n-e.\n\noption.\n\n--rsync-path=PROGRAM\nUse this to specify what program is to be run on the remote\nmachine to start-up rsync. Often used when rsync is not in the\ndefault remote-shell's path (e.g. --rsync-\npath=\/usr\/local\/bin\/rsync). Note that PROGRAM is run with the\nhelp of a shell, so it can be any program, script, or command\nsequence you'd care to run, so long as it does not corrupt the\nstandard-in & standard-out that rsync is using to communicate.\n\nOne tricky example is to set a different default directory on the\nremote machine for use with the --relative option. For instance:\n\nrsync -avR --rsync-path=\"cd \/a\/b && rsync\" host:c\/d \/e\/\n\n--remote-option=OPTION, -M\nThis option is used for more advanced situations where you want\ncertain effects to be limited to one side of the transfer only.\nFor instance, if you want to pass --log-file=FILE and --fake-super\nto the remote system, specify it like this:\n\nrsync -av -M --log-file=foo -M--fake-super src\/ dest\/\n\nIf you want to have an option affect only the local side of a\ntransfer when it normally affects both sides, send its negation to\nthe remote side. Like this:\n\nrsync -av -x -M--no-x src\/ dest\/\n\nBe cautious using this, as it is possible to toggle an option that\nwill cause rsync to have a different idea about what data to\nexpect next over the socket, and that will make it fail in a\ncryptic fashion.\n\nNote that it is best to use a separate --remote-option for each\noption you want to pass. This makes your usage compatible with\nthe --protect-args option. If that option is off, any spaces in\nyour remote options will be split by the remote shell unless you\ntake steps to protect them.\n\nWhen performing a local transfer, the \"local\" side is the sender\nand the \"remote\" side is the receiver.\n\nNote some versions of the popt option-parsing library have a bug\nin them that prevents you from using an adjacent arg with an equal\nin it next to a short option letter (e.g. -M--log-file=\/tmp\/foo).\nIf this bug affects your version of popt, you can use the version\nof popt that is included with rsync.\n\n--cvs-exclude, -C\nThis is a useful shorthand for excluding a broad range of files\nthat you often don't want to transfer between systems. It uses a\nsimilar algorithm to CVS to determine if a file should be ignored.\n\nThe exclude list is initialized to exclude the following items\n(these initial items are marked as perishable -- see the FILTER\nRULES section):\n\nRCS SCCS CVS CVS.adm RCSLOG cvslog.* tags TAGS .make.state\n.nse_depinfo *~ #* .#* ,* _$* *$ *.old *.bak *.BAK *.orig\n*.rej .del-* *.a *.olb *.o *.obj *.so *.exe *.Z *.elc *.ln\ncore .svn\/ .git\/ .hg\/ .bzr\/\n\nthen, files listed in a $HOME\/.cvsignore are added to the list and any files listed in the CVSIGNORE environment variable (all cvsignore names are delimited by whitespace). Finally, any file is ignored if it is in the same directory as a .cvsignore file and matches one of the patterns listed therein. Unlike rsync's filter\/exclude files, these patterns are split on whitespace. See the cvs(1) manual for more information. If you're combining -C with your own --filter rules, you should note that these CVS excludes are appended at the end of your own rules, regardless of where the -C was placed on the command-line. This makes them a lower priority than any rules you specified explicitly. If you want to control where these CVS excludes get inserted into your filter rules, you should omit the -C as a command-line option and use a combination of --filter=:C and --filter=-C (either on your command-line or by putting the \":C\" and \"-C\" rules into a filter file with your other rules). The first option turns on the per-directory scanning for the .cvsignore file. The second option does a one-time import of the CVS excludes mentioned above. --filter=RULE, -f This option allows you to add rules to selectively exclude certain files from the list of files to be transferred. This is most useful in combination with a recursive transfer. You may use as many --filter options on the command line as you like to build up the list of files to exclude. If the filter contains whitespace, be sure to quote it so that the shell gives the rule to rsync as a single argument. The text below also mentions that you can use an underscore to replace the space that separates a rule from its arg. See the FILTER RULES section for detailed information on this option. -F The -F option is a shorthand for adding two --filter rules to your command. The first time it is used is a shorthand for this rule: --filter='dir-merge \/.rsync-filter' This tells rsync to look for per-directory .rsync-filter files that have been sprinkled through the hierarchy and use their rules to filter the files in the transfer. If -F is repeated, it is a shorthand for this rule: --filter='exclude .rsync-filter' This filters out the .rsync-filter files themselves from the transfer. See the FILTER RULES section for detailed information on how these options work. --exclude=PATTERN This option is a simplified form of the --filter option that defaults to an exclude rule and does not allow the full rule- parsing syntax of normal filter rules. See the FILTER RULES section for detailed information on this option. --exclude-from=FILE This option is related to the --exclude option, but it specifies a FILE that contains exclude patterns (one per line). Blank lines in the file and lines starting with ';' or '#' are ignored. If FILE is '-', the list will be read from standard input. --include=PATTERN This option is a simplified form of the --filter option that defaults to an include rule and does not allow the full rule- parsing syntax of normal filter rules. See the FILTER RULES section for detailed information on this option. --include-from=FILE This option is related to the --include option, but it specifies a FILE that contains include patterns (one per line). Blank lines in the file and lines starting with ';' or '#' are ignored. If FILE is '-', the list will be read from standard input. --files-from=FILE Using this option allows you to specify the exact list of files to transfer (as read from the specified FILE or '-' for standard input). It also tweaks the default behavior of rsync to make transferring just the specified files and directories easier: o The --relative (-R) option is implied, which preserves the path information that is specified for each item in the file (use --no-relative or --no-R if you want to turn that off). o The --dirs (-d) option is implied, which will create directories specified in the list on the destination rather than noisily skipping them (use --no-dirs or --no-d if you want to turn that off). o The --archive (-a) option's behavior does not imply --recursive (-r), so specify it explicitly, if you want it. o These side-effects change the default state of rsync, so the position of the --files-from option on the command-line has no bearing on how other options are parsed (e.g. -a works the same before or after --files-from, as does --no-R and all other options). The filenames that are read from the FILE are all relative to the source dir -- any leading slashes are removed and no \"..\" references are allowed to go higher than the source dir. For example, take this command: rsync -a --files-from=\/tmp\/foo \/usr remote:\/backup If \/tmp\/foo contains the string \"bin\" (or even \"\/bin\"), the \/usr\/bin directory will be created as \/backup\/bin on the remote host. If it contains \"bin\/\" (note the trailing slash), the immediate contents of the directory would also be sent (without needing to be explicitly mentioned in the file -- this began in version 2.6.4). In both cases, if the -r option was enabled, that dir's entire hierarchy would also be transferred (keep in mind that -r needs to be specified explicitly with --files-from, since it is not implied by -a). Also note that the effect of the (enabled by default) --relative option is to duplicate only the path info that is read from the file -- it does not force the duplication of the source-spec path (\/usr in this case). In addition, the --files-from file can be read from the remote host instead of the local host if you specify a \"host:\" in front of the file (the host must match one end of the transfer). As a short-cut, you can specify just a prefix of \":\" to mean \"use the remote end of the transfer\". For example: rsync -a --files-from=:\/path\/file-list src:\/ \/tmp\/copy This would copy all the files specified in the \/path\/file-list file that was located on the remote \"src\" host. If the --iconv and --protect-args options are specified and the --files-from filenames are being sent from one host to another, the filenames will be translated from the sending host's charset to the receiving host's charset. NOTE: sorting the list of files in the --files-from input helps rsync to be more efficient, as it will avoid re-visiting the path elements that are shared between adjacent entries. If the input is not sorted, some path elements (implied directories) may end up being scanned multiple times, and rsync will eventually unduplicate them after they get turned into file-list elements. --from0, -0 This tells rsync that the rules\/filenames it reads from a file are terminated by a null ('\\0') character, not a NL, CR, or CR+LF. This affects --exclude-from, --include-from, --files-from, and any merged files specified in a --filter rule. It does not affect --cvs-exclude (since all names read from a .cvsignore file are split on whitespace). --protect-args, -s This option sends all filenames and most options to the remote rsync without allowing the remote shell to interpret them. This means that spaces are not split in names, and any non-wildcard special characters are not translated (such as ~,$, ;, &, etc.).\nWildcards are expanded on the remote host by rsync (instead of the\nshell doing it).\n\nIf you use this option with --iconv, the args related to the\nremote side will also be translated from the local to the remote\ncharacter-set. The translation happens before wild-cards are\n\nYou may also control this option via the RSYNC_PROTECT_ARGS\nenvironment variable. If this variable has a non-zero value, this\noption will be enabled by default, otherwise it will be disabled\nby default. Either state is overridden by a manually specified\npositive or negative version of this option (note that --no-s and\n--no-protect-args are the negative versions). Since this option\nwas first introduced in 3.0.0, you'll need to make sure it's\ndisabled if you ever need to interact with a remote rsync that is\nolder than that.\n\nRsync can also be configured (at build time) to have this option\nenabled by default (with is overridden by both the environment and\nthe command-line). Run rsync --version to check if this is the\ncase, as it will display \"default protect-args\" or \"optional\nprotect-args\" depending on how it was compiled.\n\nThis option will eventually become a new default setting at some\nas-yet-undetermined point in the future.\n\n--copy-as=USER[:GROUP]\nThis option instructs rsync to use the USER and (if specified\nafter a colon) the GROUP for the copy operations. This only works\nif the user that is running rsync has the ability to change users.\nIf the group is not specified then the user's default groups are\nused.\n\nThis option can help to reduce the risk of an rsync being run as\nroot into or out of a directory that might have live changes\nhappening to it and you want to make sure that root-level read or\nwrite actions of system files are not possible. While you could\nalternatively run all of rsync as the specified user, sometimes\nyou need the root-level host-access credentials to be used, so\nthis allows rsync to drop root for the copying part of the\noperation after the remote-shell or daemon connection is\nestablished.\n\nThe option only affects one side of the transfer unless the\ntransfer is local, in which case it affects both sides. Use the\n--remote-option to affect the remote side, such as -M--copy-\nas=joe. For a local transfer, the lsh (or lsh.sh) support file\nprovides a local-shell helper script that can be used to allow a\n\"localhost:\" or \"lh:\" host-spec to be specified without needing to\nsetup any remote shells, allowing you to specify remote options\nthat affect the side of the transfer that is using the host-spec\n(and using hostname \"lh\" avoids the overriding of the remote\ndirectory to the user's home dir).\n\nFor example, the following rsync writes the local files as user\n\"joe\":\n\nsudo rsync -aiv --copy-as=joe host1:backups\/joe\/ \/home\/joe\/\n\nThis makes all files owned by user \"joe\", limits the groups to\nthose that are available to that user, and makes it impossible for\nthe joe user to do a timed exploit of the path to induce a change\nto a file that the joe user has no permissions to change.\n\nThe following command does a local copy into the \"dest\/\" dir as\nuser \"joe\" (assuming you've installed support\/lsh into a dir on\nyour $PATH): sudo rsync -aive lsh -M--copy-as=joe src\/ lh:dest\/ --temp-dir=DIR, -T This option instructs rsync to use DIR as a scratch directory when creating temporary copies of the files transferred on the receiving side. The default behavior is to create each temporary file in the same directory as the associated destination file. Beginning with rsync 3.1.1, the temp-file names inside the specified DIR will not be prefixed with an extra dot (though they will still have a random suffix added). This option is most often used when the receiving disk partition does not have enough free space to hold a copy of the largest file in the transfer. In this case (i.e. when the scratch directory is on a different disk partition), rsync will not be able to rename each received temporary file over the top of the associated destination file, but instead must copy it into place. Rsync does this by copying the file over the top of the destination file, which means that the destination file will contain truncated data during this copy. If this were not done this way (even if the destination file were first removed, the data locally copied to a temporary file in the destination directory, and then renamed into place) it would be possible for the old file to continue taking up disk space (if someone had it open), and thus there might not be enough room to fit the new version on the disk at the same time. If you are using this option for reasons other than a shortage of disk space, you may wish to combine it with the --delay-updates option, which will ensure that all copied files get put into subdirectories in the destination hierarchy, awaiting the end of the transfer. If you don't have enough room to duplicate all the arriving files on the destination partition, another way to tell rsync that you aren't overly concerned about disk space is to use the --partial-dir option with a relative path; because this tells rsync that it is OK to stash off a copy of a single file in a subdir in the destination hierarchy, rsync will use the partial- dir as a staging area to bring over the copied file, and then rename it into place from there. (Specifying a --partial-dir with an absolute path does not have this side-effect.) --fuzzy, -y This option tells rsync that it should look for a basis file for any destination file that is missing. The current algorithm looks in the same directory as the destination file for either a file that has an identical size and modified-time, or a similarly-named file. If found, rsync uses the fuzzy basis file to try to speed up the transfer. If the option is repeated, the fuzzy scan will also be done in any matching alternate destination directories that are specified via --compare-dest, --copy-dest, or --link-dest. Note that the use of the --delete option might get rid of any potential fuzzy-match files, so either use --delete-after or specify some filename exclusions if you need to prevent this. --compare-dest=DIR This option instructs rsync to use DIR on the destination machine as an additional hierarchy to compare destination files against doing transfers (if the files are missing in the destination directory). If a file is found in DIR that is identical to the sender's file, the file will NOT be transferred to the destination directory. This is useful for creating a sparse backup of just files that have changed from an earlier backup. This option is typically used to copy into an empty (or newly created) directory. Beginning in version 2.6.4, multiple --compare-dest directories may be provided, which will cause rsync to search the list in the order specified for an exact match. If a match is found that differs only in attributes, a local copy is made and the attributes updated. If a match is not found, a basis file from one of the DIRs will be selected to try to speed up the transfer. If DIR is a relative path, it is relative to the destination directory. See also --copy-dest and --link-dest. NOTE: beginning with version 3.1.0, rsync will remove a file from a non-empty destination hierarchy if an exact match is found in one of the compare-dest hierarchies (making the end result more closely match a fresh copy). --copy-dest=DIR This option behaves like --compare-dest, but rsync will also copy unchanged files found in DIR to the destination directory using a local copy. This is useful for doing transfers to a new destination while leaving existing files intact, and then doing a flash-cutover when all files have been successfully transferred. Multiple --copy-dest directories may be provided, which will cause rsync to search the list in the order specified for an unchanged file. If a match is not found, a basis file from one of the DIRs will be selected to try to speed up the transfer. If DIR is a relative path, it is relative to the destination directory. See also --compare-dest and --link-dest. --link-dest=DIR This option behaves like --copy-dest, but unchanged files are hard linked from DIR to the destination directory. The files must be identical in all preserved attributes (e.g. permissions, possibly ownership) in order for the files to be linked together. An example: rsync -av --link-dest=$PWD\/prior_dir host:src_dir\/ new_dir\/\n\nIf file's aren't linking, double-check their attributes. Also\ncheck if some attributes are getting forced outside of rsync's\ncontrol, such a mount option that squishes root to a single user,\nor mounts a removable drive with generic ownership (such as OS X's\n\"Ignore ownership on this volume\" option).\n\nBeginning in version 2.6.4, multiple --link-dest directories may\nbe provided, which will cause rsync to search the list in the\norder specified for an exact match (there is a limit of 20 such\ndirectories). If a match is found that differs only in\nattributes, a local copy is made and the attributes updated. If a\nmatch is not found, a basis file from one of the DIRs will be\nselected to try to speed up the transfer.\n\nThis option works best when copying into an empty destination\nhierarchy, as existing files may get their attributes tweaked, and\nthat can affect alternate destination files via hard-links. Also,\nitemizing of changes can get a bit muddled. Note that prior to\nversion 3.1.0, an alternate-directory exact match would never be\nfound (nor linked into the destination) when a destination file\n\nNote that if you combine this option with --ignore-times, rsync\nfiles together as a substitute for transferring the file, never as\nan additional check after the file is updated.\n\nIf DIR is a relative path, it is relative to the destination\n\nNote that rsync versions prior to 2.6.1 had a bug that could\nprevent --link-dest from working properly for a non-super-user\nwhen -o was specified (or implied by -a). You can work-around\nthis bug by avoiding the -o option when sending to an old rsync.\n\n--compress, -z\nWith this option, rsync compresses the file data as it is sent to\nthe destination machine, which reduces the amount of data being\ntransmitted -- something that is useful over a slow connection.\n\nRsync supports multiple compression methods and will choose one\nfor you unless you force the choice using the --compress-choice\n(--zc) option.\n\nRun rsync --version to see the default compress list compiled into\n\nWhen both sides of the transfer are at least 3.2.0, rsync chooses\nthe first algorithm in the client's list of choices that is also\nin the server's list of choices. If no common compress choice is\nfound, rsync exits with an error. If the remote rsync is too old\nto support checksum negotiation, its list is assumed to be \"zlib\".\n\nThe default order can be customized by setting the environment\nvariable RSYNC_COMPRESS_LIST to a space-separated list of\nacceptable compression names. If the string contains a \"&\"\ncharacter, it is separated into the \"client string & server\nstring\", otherwise the same string applies to both. If the string\n(or string portion) contains no non-whitespace characters, the\ndefault compress list is used. Any unknown compression names are\ndiscarded from the list, but a list with only invalid names\nresults in a failed negotiation.\n\nThere are some older rsync versions that were configured to reject\na -z option and require the use of -zz because their compression\nlibrary was not compatible with the default zlib compression\nmethod. You can usually ignore this weirdness unless the rsync\nserver complains and tells you to specify -zz.\n\nsuffixes that will be transferred with no (or minimal)\ncompression.\n\n--compress-choice=STR, --zc=STR\nThis option can be used to override the automatic negotiation of\nthe compression algorithm that occurs when --compress is used.\nThe option implies --compress unless \"none\" was specified, which\n\nThe compression options that you may be able to use are:\n\no zstd\n\no lz4\n\no zlibx\n\no zlib\n\no none\n\nRun rsync --version to see the default compress list compiled into\nyour version (which may differ from the list above).\n\nNote that if you see an error about an option named --old-compress\nor --new-compress, this is rsync trying to send the --compress-\nchoice=zlib or --compress-choice=zlibx option in a backward-\ncompatible manner that more rsync versions understand. This error\nindicates that the older rsync version on the server will not\nallow you to force the compression type.\n\nNote that the \"zlibx\" compression algorithm is just the \"zlib\"\nalgorithm with matched data excluded from the compression stream\n(to try to make it more compatible with an external zlib\nimplementation).\n\n--compress-level=NUM, --zl=NUM\nExplicitly set the compression level to use (see --compress, -z)\ninstead of letting it default. The --compress option is implied\nas long as the level chosen is not a \"don't compress\" level for\nthe compression algorithm that is in effect (e.g. zlib compression\ntreats level 0 as \"off\").\n\nThe level values vary depending on the checksum in effect.\nBecause rsync will negotiate a checksum choice by default (when\nthe remote rsync is new enough), it can be good to combine this\noption with a --compress-choice (--zc) option unless you're sure\nof the choice in effect. For example:\n\nrsync -aiv --zc=zstd --zl=22 host:src\/ dest\/\n\nFor zlib & zlibx compression the valid values are from 1 to 9 with\n6 being the default. Specifying 0 turns compression off, and\nspecifying -1 chooses the default of 6.\n\nFor zstd compression the valid values are from -131072 to 22 with\n3 being the default. Specifying 0 chooses the default of 3.\n\nFor lz4 compression there are no levels, so the value is always 0.\n\nIf you specify a too-large or too-small value, the number is\nsilently limited to a valid value. This allows you to specify\nsomething like --zl=999999999 and be assured that you'll end up\nwith the maximum compression level no matter what algorithm was\nchosen.\n\nIf you want to know the compression level that is in effect,\nspecify --debug=nstr to see the \"negotiated string\" results. This\nwill report something like \"Client compress: zstd (level 3)\"\n(along with the checksum choice in effect).\n\n--skip-compress=LIST\nOverride the list of file suffixes that will be compressed as\nlittle as possible. Rsync sets the compression level on a per-\nfile basis based on the file's suffix. If the compression\nalgorithm has an \"off\" level (such as zlib\/zlibx) then no\ncompression occurs for those files. Other algorithms that support\nchanging the streaming level on-the-fly will have the level\nminimized to reduces the CPU usage as much as possible for a\nmatching file. At this time, only zlib & zlibx compression\nsupport this changing of levels on a per-file basis.\n\nThe LIST should be one or more file suffixes (without the dot)\nseparated by slashes (\/). You may specify an empty string to\nindicate that no files should be skipped.\n\nSimple character-class matching is supported: each must consist of\na list of letters inside the square brackets (e.g. no special\nclasses, such as \"[:alpha:]\", are supported, and '-' has no\nspecial meaning).\n\nThe characters asterisk (*) and question-mark (?) have no special\nmeaning.\n\nHere's an example that specifies 6 suffixes to skip (since 1 of\nthe 5 rules matches 2 suffixes):\n\n--skip-compress=gz\/jpg\/mp[34]\/7z\/bz2\n\nThe default file suffixes in the skip-compress list in this\nversion of rsync are:\n\n3g2 3gp 7z aac ace apk avi bz2 deb dmg ear f4v flac flv gpg gz\niso jar jpeg jpg lrz lz lz4 lzma lzo m1a m1v m2a m2ts m2v m4a\nm4b m4p m4r m4v mka mkv mov mp1 mp2 mp3 mp4 mpa mpeg mpg mpv\nmts odb odf odg odi odm odp ods odt oga ogg ogm ogv ogx opus\notg oth otp ots ott oxt png qt rar rpm rz rzip spx squashfs\nsxc sxd sxg sxm sxw sz tbz tbz2 tgz tlz ts txz tzo vob war\nwebm webp xz z zip zst\n\nThis list will be replaced by your --skip-compress list in all but\none situation: a copy from a daemon rsync will add your skipped\nsuffixes to its list of non-compressing files (and its list may be\nconfigured to a different default).\n\n--numeric-ids\nWith this option rsync will transfer numeric group and user IDs\nrather than using user and group names and mapping them at both\nends.\n\nBy default rsync will use the username and groupname to determine\nwhat ownership to give files. The special uid 0 and the special\ngroup 0 are never mapped via user\/group names even if the\n--numeric-ids option is not specified.\n\nIf a user or group has no name on the source system or it has no\nmatch on the destination system, then the numeric ID from the\n\"use chroot\" setting in the rsyncd.conf manpage for information on\nhow the chroot setting affects rsync's ability to look up the\nnames of the users and groups and what you can do about it.\n\n--usermap=STRING, --groupmap=STRING\nThese options allow you to specify users and groups that should be\nmapped to other values by the receiving side. The STRING is one\nor more FROM:TO pairs of values separated by commas. Any matching\nFROM value from the sender is replaced with a TO value from the\nreceiver. You may specify usernames or user IDs for the FROM and\nTO values, and the FROM value may also be a wild-card string,\nwhich will be matched against the sender's names (wild-cards do\nNOT match against ID numbers, though see below for why a '*'\nmatches everything). You may instead specify a range of ID\nnumbers via an inclusive range: LOW-HIGH. For example:\n\nThe first match in the list is the one that is used. You should\nspecify all your user mappings using a single --usermap option,\nand\/or all your group mappings using a single --groupmap option.\n\nNote that the sender's name for the 0 user and group are not\ntransmitted to the receiver, so you should either match these\nvalues using a 0, or use the names in effect on the receiving side\n(typically \"root\"). All other FROM names match those in use on\nthe sending side. All TO names match those in use on the\nreceiving side.\n\nAny IDs that do not have a name on the sending side are treated as\nhaving an empty name for the purpose of matching. This allows\nthem to be matched via a \"*\" or using an empty name. For\ninstance:\n\n--usermap=:nobody --groupmap=*:nobody\n\nWhen the --numeric-ids option is used, the sender does not send\nany names, so all the IDs are treated as having an empty name.\nThis means that you will need to specify numeric FROM values if\nyou want to map these nameless IDs to different values.\n\nFor the --usermap option to have any effect, the -o (--owner)\noption must be used (or implied), and the receiver will need to be\nthe --groupmap option to have any effect, the -g (--groups) option\nmust be used (or implied), and the receiver will need to have\npermissions to set that group.\n\n(-s).\n\n--chown=USER:GROUP\nThis option forces all files to be owned by USER with group GROUP.\nThis is a simpler interface than using --usermap and --groupmap\ndirectly, but it is implemented using those options internally, so\nyou cannot mix them. If either the USER or GROUP is empty, no\nmapping for the omitted user\/group will occur. If GROUP is empty,\nthe trailing colon may be omitted, but if USER is empty, a leading\ncolon must be supplied.\n\nIf you specify \"--chown=foo:bar\", this is exactly the same as\nspecifying \"--usermap=*:foo --groupmap=*:bar\", only easier. If\n\n--timeout=SECONDS\nThis option allows you to set a maximum I\/O timeout in seconds.\nIf no data is transferred for the specified time then rsync will\nexit. The default is 0, which means no timeout.\n\n--contimeout=SECONDS\nThis option allows you to set the amount of time that rsync will\nwait for its connection to an rsync daemon to succeed. If the\ntimeout is reached, rsync exits with an error.\n\nBy default rsync will bind to the wildcard address when connecting\nto an rsync daemon. The --address option allows you to specify a\noption in the --daemon mode section.\n\n--port=PORT\nThis specifies an alternate TCP port number to use rather than the\ndefault of 873. This is only needed if you are using the double-\ncolon (::) syntax to connect with an rsync daemon (since the URL\nsyntax has a way to specify the port as a part of the URL). See\nalso this option in the --daemon mode section.\n\n--sockopts=OPTIONS\nThis option can provide endless fun for people who like to tune\ntheir systems to the utmost degree. You can set all sorts of\nsocket options which may make transfers faster (or slower!). Read\nthe man page for the setsockopt() system call for details on some\nof the options you may be able to set. By default no special\nsocket options are set. This only affects direct socket\nconnections to a remote rsync daemon.\n\nThis option also exists in the --daemon mode section.\n\n--blocking-io\nThis tells rsync to use blocking I\/O when launching a remote shell\ntransport. If the remote shell is either rsh or remsh, rsync\ndefaults to using blocking I\/O, otherwise it defaults to using\nnon-blocking I\/O. (Note that ssh prefers non-blocking I\/O.)\n\n--outbuf=MODE\nThis sets the output buffering mode. The mode can be None (aka\nUnbuffered), Line, or Block (aka Full). You may specify as little\nas a single letter for the mode, and use upper or lower case.\n\nThe main use of this option is to change Full buffering to Line\nbuffering when rsync's output is going to a file or pipe.\n\n--itemize-changes, -i\nRequests a simple itemized list of the changes that are being made\nto each file, including attribute changes. This is exactly the\nsame as specifying --out-format='%i %n%L'. If you repeat the\noption, unchanged files will also be output, but only if the\nreceiving rsync is at least version 2.6.7 (you can use -vv with\nolder versions of rsync, but that also turns on the output of\nother verbose messages).\n\nThe \"%i\" escape has a cryptic output that is 11 letters long. The\ngeneral format is like the string YXcstpoguax, where Y is replaced\nby the type of update being done, X is replaced by the file-type,\nand the other letters represent attributes that may be output if\nthey are being modified.\n\nThe update types that replace the Y are as follows:\n\no A < means that a file is being transferred to the remote\nhost (sent).\n\no A > means that a file is being transferred to the local\n\no A c means that a local change\/creation is occurring for the\nitem (such as the creation of a directory or the changing\n\no A h means that the item is a hard link to another item\n\no A . means that the item is not being updated (though it\nmight have attributes that are being modified).\n\no A * means that the rest of the itemized-output area\ncontains a message (e.g. \"deleting\").\n\nThe file-types that replace the X are: f for a file, a d for a\ndirectory, an L for a symlink, a D for a device, and a S for a\nspecial file (e.g. named sockets and fifos).\n\nThe other letters in the string indicate if some attributes of the\nfile have changed, as follows:\n\no \".\" - the attribute is unchanged.\n\no \"+\" - the file is newly created.\n\no \" \" - all the attributes are unchanged (all dots turn to\nspaces).\n\no \"?\" - the change is unknown (when the remote rsync is old).\n\no A letter indicates an attribute is being updated.\n\nThe attribute that is associated with each letter is as follows:\n\no A c means either that a regular file has a different\nchecksum (requires --checksum) or that a symlink, device,\nor special file has a changed value. Note that if you are\nsending files to an rsync prior to 3.0.1, this change flag\nwill be present only for checksum-differing regular files.\n\no A s means the size of a regular file is different and will\nbe updated by the file transfer.\n\no A t means the modification time is different and is being\nupdated to the sender's value (requires --times). An\nalternate value of T means that the modification time will\nbe set to the transfer time, which happens when a\nfile\/symlink\/device is updated without --times and when a\n(Note: when using an rsync 3.0.0 client, you might see the\ns flag combined with t instead of the proper T flag for\nthis time-setting failure.)\n\no A p means the permissions are different and are being\nupdated to the sender's value (requires --perms).\n\no An o means the owner is different and is being updated to\nthe sender's value (requires --owner and super-user\nprivileges).\n\no A g means the group is different and is being updated to\nthe sender's value (requires --group and the authority to\nset the group).\n\no A u|n|b indicates the following information: u means the\naccess (use) time is different and is being updated to the\nsender's value (requires --atimes); n means the create time\n(newness) is different and is being updated to the sender's\nvalue (requires --crtimes); b means that both the access\nand create times are being updated.\n\no The a means that the ACL information is being changed.\n\no The x means that the extended attribute information is\nbeing changed.\n\nOne other output is possible: when deleting files, the \"%i\" will\noutput the string \"*deleting\" for each item that is being removed\n(assuming that you are talking to a recent enough rsync that it\nlogs deletions instead of outputting them as a verbose message).\n\n--out-format=FORMAT\nThis allows you to specify exactly what the rsync client outputs\nto the user on a per-update basis. The format is a text string\ncontaining embedded single-character escape sequences prefixed\nwith a percent (%) character. A default format of \"%n%L\" is\nassumed if either --info=name or -v is specified (this tells you\njust the name of the file and, if the item is a link, where it\npoints). For a full list of the possible escape characters, see\nthe \"log format\" setting in the rsyncd.conf manpage.\n\nSpecifying the --out-format option implies the --info=name option,\nwhich will mention each file, dir, etc. that gets updated in a\nsignificant way (a transferred file, a recreated symlink\/device,\nor a touched directory). In addition, if the itemize-changes\nescape (%i) is included in the string (e.g. if the --itemize-\nchanges option was used), the logging of names increases to\nmention any item that is changed in any way (as long as the\nreceiving side is at least 2.6.4). See the --itemize-changes\noption for a description of the output of \"%i\".\n\nRsync will output the out-format string prior to a file's transfer\nunless one of the transfer-statistic escapes is requested, in\nwhich case the logging is done at the end of the file's transfer.\nWhen this late logging is in effect and --progress is also\nspecified, rsync will also output the name of the file being\ntransferred prior to its progress information (followed, of\ncourse, by the out-format output).\n\n--log-file=FILE\nThis option causes rsync to log what it is doing to a file. This\nis similar to the logging that a daemon does, but can be requested\nfor the client side and\/or the server side of a non-daemon\ntransfer. If specified as a client option, transfer logging will\nbe enabled with a default format of \"%i %n%L\". See the --log-\nfile-format option if you wish to override this.\n\nHere's a example command that requests the remote side to log what\nis happening:\n\nrsync -av --remote-option=--log-file=\/tmp\/rlog src\/ dest\/\n\nThis is very useful if you need to debug why a connection is\nclosing unexpectedly.\n\n--log-file-format=FORMAT\nThis allows you to specify exactly what per-update logging is put\ninto the file specified by the --log-file option (which must also\nbe specified for this option to have any effect). If you specify\nan empty string, updated files will not be mentioned in the log\nfile. For a list of the possible escape characters, see the\n\"log format\" setting in the rsyncd.conf manpage.\n\nThe default FORMAT used if --log-file is specified and this option\nis not is '%i %n%L'.\n\n--stats\nThis tells rsync to print a verbose set of statistics on the file\ntransfer, allowing you to tell how effective rsync's delta-\ntransfer algorithm is for your data. This option is equivalent to\n--info=stats2 if combined with 0 or 1 -v options, or --info=stats3\nif combined with 2 or more -v options.\n\nThe current statistics are as follows:\n\no Number of files is the count of all \"files\" (in the generic\nsense), which includes directories, symlinks, etc. The\ntotal count will be followed by a list of counts by\nfiletype (if the total is non-zero). For example: \"(reg:\n5, dir: 3, link: 2, dev: 1, special: 1)\" lists the totals\nfor regular files, directories, symlinks, devices, and\nspecial files. If any of value is 0, it is completely\nomitted from the list.\n\no Number of created files is the count of how many \"files\"\n(generic sense) were created (as opposed to updated). The\ntotal count will be followed by a list of counts by\nfiletype (if the total is non-zero).\n\no Number of deleted files is the count of how many \"files\"\n(generic sense) were created (as opposed to updated). The\ntotal count will be followed by a list of counts by\nfiletype (if the total is non-zero). Note that this line\nis only output if deletions are in effect, and only if\nprotocol 31 is being used (the default for rsync 3.1.x).\n\no Number of regular files transferred is the count of normal\nfiles that were updated via rsync's delta-transfer\nalgorithm, which does not include dirs, symlinks, etc.\nNote that rsync 3.1.0 added the word \"regular\" into this\n\no Total file size is the total sum of all file sizes in the\ntransfer. This does not count any size for directories or\nspecial files, but does include the size of symlinks.\n\no Total transferred file size is the total sum of all files\nsizes for just the transferred files.\n\no Literal data is how much unmatched file-update data we had\nto send to the receiver for it to recreate the updated\nfiles.\n\no Matched data is how much data the receiver got locally when\nrecreating the updated files.\n\no File list size is how big the file-list data was when the\nsender sent it to the receiver. This is smaller than the\nin-memory size for the file list due to some compressing of\nduplicated data when rsync sends the list.\n\no File list generation time is the number of seconds that the\nsender spent creating the file list. This requires a\nmodern rsync on the sending side for this to be present.\n\no File list transfer time is the number of seconds that the\nsender spent sending the file list to the receiver.\n\no Total bytes sent is the count of all the bytes that rsync\nsent from the client side to the server side.\n\no Total bytes received is the count of all non-message bytes\nthat rsync received by the client side from the server\nside. \"Non-message\" bytes means that we don't count the\nbytes for a verbose message that the server sent to us,\nwhich makes the stats more consistent.\n\n--8-bit-output, -8\nThis tells rsync to leave all high-bit characters unescaped in the\noutput instead of trying to test them to see if they're valid in\nthe current locale and escaping the invalid ones. All control\ncharacters (but never tabs) are always escaped, regardless of this\noption's setting.\n\nThe escape idiom that started in 2.6.7 is to output a literal\nbackslash (\\) and a hash (#), followed by exactly 3 octal digits.\nFor example, a newline would output as \"\\#012\". A literal\nbackslash that is in a filename is not escaped unless it is\nfollowed by a hash and 3 digits (0-9).\n\nOutput numbers in a more human-readable format. There are 3\npossible levels: (1) output numbers with a separator between each\nset of 3 digits (either a comma or a period, depending on if the\ndecimal point is represented by a period or a comma); (2) output\nnumbers in units of 1000 (with a character suffix for larger\nunits -- see below); (3) output numbers in units of 1024.\n\nThe default is human-readable level 1. Each -h option increases\nthe level by one. You can take the level down to 0 (to output\nnumbers as pure digits) by specifying the --no-human-readable\n(--no-h) option.\n\nThe unit letters that are appended in levels 2 and 3 are: K\n(kilo), M (mega), G (giga), T (tera), or P (peta). For example, a\n1234567-byte file would output as 1.23M in level-2 (assuming that\na period is your local decimal point).\n\nBackward compatibility note: versions of rsync prior to 3.1.0 do\nnot support human-readable level 1, and they default to level 0.\nThus, specifying one or two -h options will behave in a comparable\nmanner in old and new versions as long as you didn't specify a\n--no-h option prior to one or more -h options. See the --list-\nonly option for one difference.\n\n--partial\nBy default, rsync will delete any partially transferred file if\nthe transfer is interrupted. In some circumstances it is more\ndesirable to keep partially transferred files. Using the\n--partial option tells rsync to keep the partial file which should\nmake a subsequent transfer of the rest of the file much faster.\n\n--partial-dir=DIR\nA better way to keep partial files than the --partial option is to\nspecify a DIR that will be used to hold the partial data (instead\nof writing it out to the destination file). On the next transfer,\nrsync will use a file found in this dir as data to speed up the\nresumption of the transfer and then delete it after it has served\nits purpose.\n\nNote that if --whole-file is specified (or implied), any partial-\ndir file that is found for a file that is being updated will\nsimply be removed (since rsync is sending files without using\nrsync's delta-transfer algorithm).\n\nRsync will create the DIR if it is missing (just the last dir --\nnot the whole path). This makes it easy to use a relative path\n(such as \"--partial-dir=.rsync-partial\") to have rsync create the\npartial-directory in the destination file's directory when needed,\nand then remove it again when the partial file is deleted. Note\nthat the directory is only removed if it is a relative pathname,\nas it is expected that an absolute path is to a directory that is\nreserved for partial-dir work.\n\nIf the partial-dir value is not an absolute path, rsync will add\nan exclude rule at the end of all your existing excludes. This\nwill prevent the sending of any partial-dir files that may exist\non the sending side, and will also prevent the untimely deletion\nof partial-dir items on the receiving side. An example: the above\n--partial-dir option would add the equivalent of \"-f '-p .rsync-\npartial\/'\" at the end of any other filter rules.\n\nIf you are supplying your own exclude rules, you may need to add\nyour own exclude\/hide\/protect rule for the partial-dir because (1)\nthe auto-added rule may be ineffective at the end of your other\nrules, or (2) you may wish to override rsync's exclude choice.\nFor instance, if you want to make rsync clean-up any left-over\npartial-dirs that may be lying around, you should specify\n--delete-after and add a \"risk\" filter rule, e.g. -f 'R .rsync-\npartial\/'. (Avoid using --delete-before or --delete-during unless\nyou don't need rsync to use any of the left-over partial-dir data\nduring the current run.)\n\nIMPORTANT: the --partial-dir should not be writable by other users\nor it is a security risk. E.g. AVOID \"\/tmp\".\n\nYou can also set the partial-dir value the RSYNC_PARTIAL_DIR\nenvironment variable. Setting this in the environment does not\nforce --partial to be enabled, but rather it affects where partial\nfiles go when --partial is specified. For instance, instead of\nusing --partial-dir=.rsync-tmp along with --progress, you could\nset RSYNC_PARTIAL_DIR=.rsync-tmp in your environment and then just\nuse the -P option to turn on the use of the .rsync-tmp dir for\npartial transfers. The only times that the --partial option does\nnot look for this environment value are (1) when --inplace was\nspecified (since --inplace conflicts with --partial-dir), and (2)\nwhen --delay-updates was specified (see below).\n\nWhen a modern rsync resumes the transfer of a file in the partial-\ndir, that partial file is now updated in-place instead of creating\nyet another tmp-file copy (so it maxes out at dest + tmp instead\nof dest + partial + tmp). This requires both ends of the transfer\nto be at least version 3.2.0.\n\nFor the purposes of the daemon-config's \"refuse options\" setting,\n--partial-dir does not imply --partial. This is so that a refusal\nof the --partial option can be used to disallow the overwriting of\ndestination files with a partial transfer, while still allowing\nthe safer idiom provided by --partial-dir.\n\nThis option puts the temporary file from each updated file into a\nholding directory until the end of the transfer, at which time all\nthe files are renamed into place in rapid succession. This\nattempts to make the updating of the files a little more atomic.\nBy default the files are placed into a directory named .~tmp~ in\neach file's destination directory, but if you've specified the\n--partial-dir option, that directory will be used instead. See\nthe comments in the --partial-dir section for a discussion of how\nthis .~tmp~ dir will be excluded from the transfer, and what you\ncan do if you want rsync to cleanup old .~tmp~ dirs that might be\nlying around. Conflicts with --inplace and --append.\n\nThis option implies --no-inc-recursive since it needs the full\nfile list in memory in order to be able to iterate over it at the\nend.\n\nThis option uses more memory on the receiving side (one bit per\nfile transferred) and also requires enough free disk space on the\nreceiving side to hold an additional copy of all the updated\nfiles. Note also that you should not use an absolute path to\n--partial-dir unless (1) there is no chance of any of the files in\nthe transfer having the same name (since all the updated files\nwill be put into a single directory if the path is absolute) and\n(2) there are no mount points in the hierarchy (since the delayed\nupdates will fail if they can't be renamed into place).\n\nfor an update algorithm that is even more atomic (it uses --link-\ndest and a parallel hierarchy of files).\n\n--prune-empty-dirs, -m\nThis option tells the receiving rsync to get rid of empty\ndirectories from the file-list, including nested directories that\nhave no non-directory children. This is useful for avoiding the\ncreation of a bunch of useless directories when the sending rsync\nis recursively scanning a hierarchy of files using\ninclude\/exclude\/filter rules.\n\nNote that the use of transfer rules, such as the --min-size\noption, does not affect what goes into the file list, and thus\ndoes not leave directories empty, even if none of the files in a\ndirectory match the transfer rule.\n\nBecause the file-list is actually being pruned, this option also\naffects what directories get deleted when a delete is active.\nHowever, keep in mind that excluded files and directories can\nprevent existing items from being deleted due to an exclude both\nhiding source files and protecting destination files. See the\nperishable filter-rule option for how to avoid this.\n\nYou can prevent the pruning of certain empty directories from the\nfile-list by using a global \"protect\" filter. For instance, this\noption would ensure that the directory \"emptydir\" was kept in the\nfile-list:\n\n--filter 'protect emptydir\/'\n\nHere's an example that copies all .pdf files in a hierarchy, only\ncreating the necessary destination directories to hold the .pdf\nfiles, and ensures that any superfluous files and directories in\nthe destination are removed (note the hide filter of non-\ndirectories being used instead of an exclude):\n\nrsync -avm --del --include='*.pdf' -f 'hide,! *\/' src\/ dest\n\nIf you didn't want to remove superfluous destination files, the\nmore time-honored options of --include='*\/' --exclude='*' would\nwork fine in place of the hide-filter (if that is more natural to\nyou).\n\n--progress\nThis option tells rsync to print information showing the progress\nof the transfer. This gives a bored user something to watch.\nWith a modern rsync this is the same as specifying\n--info=flist2,name,progress, but any user-supplied settings for\nthose info flags takes precedence (e.g.\n\"--info=flist0 --progress\").\n\nWhile rsync is transferring a regular file, it updates a progress\nline that looks like this:\n\n782448 63% 110.64kB\/s 0:00:04\n\nIn this example, the receiver has reconstructed 782448 bytes or\n63% of the sender's file, which is being reconstructed at a rate\nof 110.64 kilobytes per second, and the transfer will finish in 4\nseconds if the current rate is maintained until the end.\n\nThese statistics can be misleading if rsync's delta-transfer\nalgorithm is in use. For example, if the sender's file consists\nof the basis file followed by additional data, the reported rate\nwill probably drop dramatically when the receiver gets to the\nliteral data, and the transfer will probably take much longer to\nfinish than the receiver estimated as it was finishing the matched\npart of the file.\n\nWhen the file transfer finishes, rsync replaces the progress line\nwith a summary line that looks like this:\n\n1,238,099 100% 146.38kB\/s 0:00:08 (xfr#5, to-chk=169\/396)\n\nIn this example, the file was 1,238,099 bytes long in total, the\naverage rate of transfer for the whole file was 146.38 kilobytes\nper second over the 8 seconds that it took to complete, it was the\n5th transfer of a regular file during the current rsync session,\nand there are 169 more files for the receiver to check (to see if\nthey are up-to-date or not) remaining out of the 396 total files\nin the file-list.\n\nIn an incremental recursion scan, rsync won't know the total\nnumber of files in the file-list until it reaches the ends of the\nscan, but since it starts to transfer files during the scan, it\nwill display a line with the text \"ir-chk\" (for incremental\nrecursion check) instead of \"to-chk\" until the point that it knows\nthe full size of the list, at which point it will switch to using\n\"to-chk\". Thus, seeing \"ir-chk\" lets you know that the total\ncount of files in the file list is still going to increase (and\neach time it does, the count of files left to check will increase\nby the number of the files added to the list).\n\n-P The -P option is equivalent to --partial --progress. Its purpose\nis to make it much easier to specify these two options for a long\ntransfer that may be interrupted.\n\nThere is also a --info=progress2 option that outputs statistics\nbased on the whole transfer, rather than individual files. Use\nthis flag without outputting a filename (e.g. avoid -v or specify\n--info=name0) if you want to see how the transfer is doing without\nscrolling the screen with a lot of names. (You don't need to\nspecify the --progress option in order to use --info=progress2.)\n\nFinally, you can get an instant progress report by sending rsync a\nsignal of either SIGINFO or SIGVTALRM. On BSD systems, a SIGINFO\nis generated by typing a Ctrl+T (Linux doesn't currently support a\nSIGINFO signal). When the client-side process receives one of\nthose signals, it sets a flag to output a single progress report\nwhich is output when the current file transfer finishes (so it may\ntake a little time if a big file is being handled when the signal\narrives). A filename is output (if needed) followed by the\n--info=progress2 format of progress info. If you don't know which\nof the 3 rsync processes is the client process, it's OK to signal\nall of them (since the non-client processes ignore the signal).\n\nCAUTION: sending SIGVTALRM to an older rsync (pre-3.2.0) will kill\nit.\n\nThis option allows you to provide a password for accessing an\nrsync daemon via a file or via standard input if FILE is -. The\nfile should contain just the password on the first line (all other\nlines are ignored). Rsync will exit with an error if FILE is\nworld readable or if a root-run rsync command finds a non-root-\nowned file.\n\nThis option does not supply a password to a remote shell transport\nsuch as ssh; to learn how to do that, consult the remote shell's\ndocumentation. When accessing an rsync daemon using a remote\nshell as the transport, this option only comes into effect after\nthe remote shell finishes its authentication (i.e. if you have\nalso specified a password in the daemon's config file).\n\n--early-input=FILE\nThis option allows rsync to send up to 5K of data to the \"early\nexec\" script on its stdin. One possible use of this data is to\ngive the script a secret that can be used to mount an encrypted\nfilesystem (which you should unmount in the the \"post-xfer exec\"\nscript).\n\nThe daemon must be at least version 3.2.1.\n\n--list-only\nThis option will cause the source files to be listed instead of\ntransferred. This option is inferred if there is a single source\narg and no destination specified, so its main uses are: (1) to\nturn a copy command that includes a destination arg into a file-\nlisting command, or (2) to be able to specify more than one source\narg (note: be sure to include the destination). Caution: keep in\nmind that a source arg with a wild-card is expanded by the shell\ninto multiple args, so it is never safe to try to list such an arg\nwithout using this option. For example:\n\nrsync -av --list-only foo* dest\/\n\nStarting with rsync 3.1.0, the sizes output by --list-only are\naffected by the --human-readable option. By default they will\ncontain digit separators, but higher levels of readability will\noutput the sizes with unit suffixes. Note also that the column\nwidth for the size output has increased from 11 to 14 characters\nfor all human-readable levels. Use --no-h if you want just digits\nin the sizes, and the old column width of 11 characters.\n\nCompatibility note: when requesting a remote listing of files from\nan rsync that is version 2.6.3 or older, you may encounter an\nerror if you ask for a non-recursive listing. This is because a\nfile listing implies the --dirs option w\/o --recursive, and older\nrsyncs don't have that option. To avoid this problem, either\nspecify the --no-dirs option (if you don't need to expand a\ndirectory's content), or turn on recursion and exclude the content\nof subdirectories: -r --exclude='\/*\/*'.\n\n--bwlimit=RATE\nThis option allows you to specify the maximum transfer rate for\nthe data sent over the socket, specified in units per second. The\nRATE value can be suffixed with a string to indicate a size\nmultiplier, and may be a fractional value (e.g. \"--bwlimit=1.5m\").\nIf no suffix is specified, the value will be assumed to be in\nunits of 1024 bytes (as if \"K\" or \"KiB\" had been appended). See\nthe --max-size option for a description of all the available\nsuffixes. A value of 0 specifies no limit.\n\nFor backward-compatibility reasons, the rate limit will be rounded\nto the nearest KiB unit, so no rate smaller than 1024 bytes per\nsecond is possible.\n\nRsync writes data over the socket in blocks, and this option both\nlimits the size of the blocks that rsync writes, and tries to keep\nthe average transfer rate at the requested limit. Some burstiness\nmay be seen where rsync writes out a block of data and then sleeps\nto bring the average rate into compliance.\n\nDue to the internal buffering of data, the --progress option may\nnot be an accurate reflection on how fast the data is being sent.\nThis is because some files can show up as being rapidly sent when\nthe data is quickly buffered, while other can show up as very slow\nwhen the flushing of the output buffer occurs. This may be fixed\nin a future version.\n\n--stop-after=MINS\nThis option tells rsync to stop copying when the specified number\nof minutes has elapsed.\n\nRsync also accepts an earlier version of this option: --time-\nlimit=MINS.\n\nFor maximal flexibility, rsync does not communicate this option to\nthe remote rsync since it is usually enough that one side of the\nconnection quits as specified. This allows the option's use even\nwhen only one side of the connection supports it. You can tell\nthe remote side about the time limit using --remote-option (-M),\nshould the need arise.\n\n--stop-at=y-m-dTh:m\nThis option tells rsync to stop copying when the specified point\nin time has been reached. The date & time can be fully specified\nin a numeric format of year-month-dayThour:minute (e.g.\n2000-12-31T23:59) in the local timezone. You may choose to\nseparate the date numbers using slashes instead of dashes.\n\nThe value can also be abbreviated in a variety of ways, such as\nspecifying a 2-digit year and\/or leaving off various values. In\nall cases, the value will be taken to be the next possible point\nin time where the supplied information matches. If the value\nspecifies the current time or a past time, rsync exits with an\nerror.\n\nFor example, \"1-30\" specifies the next January 30th (at midnight\nlocal time), \"14:00\" specifies the next 2 P.M., \"1\" specifies the\nnext 1st of the month at midnight, \"31\" specifies the next month\nwhere we can stop on its 31st day, and \":59\" specifies the next\n59th minute after the hour.\n\nFor maximal flexibility, rsync does not communicate this option to\nthe remote rsync since it is usually enough that one side of the\nconnection quits as specified. This allows the option's use even\nwhen only one side of the connection supports it. You can tell\nthe remote side about the time limit using --remote-option (-M),\nshould the need arise. Do keep in mind that the remote host may\nhave a different default timezone than your local host.\n\n--write-batch=FILE\nRecord a file that can later be applied to another identical\ndestination with --read-batch. See the \"BATCH MODE\" section for\ndetails, and also the --only-write-batch option.\n\nThis option overrides the negotiated checksum & compress lists and\nalways negotiates a choice based on old-school md5\/md4\/zlib\nchoices. If you want a more modern choice, use the --checksum-\nchoice (--cc) and\/or --compress-choice (--zc) options.\n\n--only-write-batch=FILE\ndestination system when creating the batch. This lets you\ntransport the changes to the destination system via some other\nmeans and then apply the changes via --read-batch.\n\nNote that you can feel free to write the batch directly to some\nportable media: if this media fills to capacity before the end of\nthe transfer, you can just apply that partial transfer to the\ndestination and repeat the whole process to get the rest of the\nchanges (as long as you don't mind a partially updated destination\nsystem while the multi-update cycle is happening).\n\nAlso note that you only save bandwidth when pushing changes to a\nremote system because this allows the batched data to be diverted\nfrom the sender into the batch file without having to flow over\nthe wire to the receiver (when pulling, the sender is remote, and\nthus can't write the batch).\n\nApply all of the changes stored in FILE, a file previously\ngenerated by --write-batch. If FILE is -, the batch data will be\nread from standard input. See the \"BATCH MODE\" section for\ndetails.\n\n--protocol=NUM\nForce an older protocol version to be used. This is useful for\ncreating a batch file that is compatible with an older version of\nrsync. For instance, if rsync 2.6.4 is being used with the\n--write-batch option, but rsync 2.6.3 is what will be used to run\nthe --read-batch option, you should use \"--protocol=28\" when\ncreating the batch file to force the older protocol version to be\nused in the batch file (assuming you can't upgrade the rsync on\n\n--iconv=CONVERT_SPEC\nRsync can convert filenames between character sets using this\noption. Using a CONVERT_SPEC of \".\" tells rsync to look up the\ndefault character-set via the locale setting. Alternately, you\ncan fully specify what conversion to do by giving a local and a\nremote charset separated by a comma in the order\n--iconv=LOCAL,REMOTE, e.g. --iconv=utf8,iso88591. This order\nensures that the option will stay the same whether you're pushing\nor pulling files. Finally, you can specify either --no-iconv or a\nCONVERT_SPEC of \"-\" to turn off any conversion. The default\nsetting of this option is site-specific, and can also be affected\nvia the RSYNC_ICONV environment variable.\n\nFor a list of what charset names your local iconv library\nsupports, you can run \"iconv --list\".\n\nIf you specify the --protect-args option (-s), rsync will\ntranslate the filenames you specify on the command-line that are\n\nNote that rsync does not do any conversion of names in filter\nfiles (including include\/exclude files). It is up to you to\nensure that you're specifying matching rules that can match on\nboth sides of the transfer. For instance, you can specify extra\ninclude\/exclude rules if there are filename differences on the two\nsides that need to be accounted for.\n\nWhen you pass an --iconv option to an rsync daemon that allows it,\nthe daemon uses the charset specified in its \"charset\"\nconfiguration parameter regardless of the remote charset you\nactually pass. Thus, you may feel free to specify just the local\ncharset for a daemon transfer (e.g. --iconv=utf8).\n\n--ipv4, -4 or --ipv6, -6\nTells rsync to prefer IPv4\/IPv6 when creating sockets or running\nssh. This affects sockets that rsync has direct control over,\nsuch as the outgoing socket when directly contacting an rsync\ndaemon, as well as the forwarding of the -4 or -6 option to ssh\nwhen rsync can deduce that ssh is being used as the remote shell.\nFor other remote shells you'll need to specify the\n\"--rsh SHELL -4\" option directly (or whatever ipv4\/ipv6 hint\noptions it uses).\n\nThese options also exist in the --daemon mode section.\n\nIf rsync was complied without support for IPv6, the --ipv6 option\nwill have no effect. The rsync --version output will contain\n\"no IPv6\" if is the case.\n\n--checksum-seed=NUM\nSet the checksum seed to the integer NUM. This 4 byte checksum\nseed is included in each block and MD4 file checksum calculation\n(the more modern MD5 file checksums don't use a seed). By default\nthe checksum seed is generated by the server and defaults to the\ncurrent time(). This option is used to set a specific checksum\nseed, which is useful for applications that want repeatable block\nchecksums, or in the case where the user wants a more random\nchecksum seed. Setting NUM to 0 causes rsync to use the default\nof time() for checksum seed.\n\nDAEMON OPTIONS\nThe options allowed when starting an rsync daemon are as follows:\n\n--daemon\nThis tells rsync that it is to run as a daemon. The daemon you\nstart running may be accessed using an rsync client using the\nhost::module or rsync:\/\/host\/module\/ syntax.\n\nIf standard input is a socket then rsync will assume that it is\nbeing run via inetd, otherwise it will detach from the current\nterminal and become a background daemon. The daemon will read the\nconfig file (rsyncd.conf) on each connect made by a client and\nrespond to requests accordingly. See the rsyncd.conf(5) man page\nfor more details.\n\nBy default rsync will bind to the wildcard address when run as a\ndaemon with the --daemon option. The --address option allows you\nto specify a specific IP address (or hostname) to bind to. This\nmakes virtual hosting possible in conjunction with the --config\nmanpage.\n\n--bwlimit=RATE\nThis option allows you to specify the maximum transfer rate for\nthe data the daemon sends over the socket. The client can still\nspecify a smaller --bwlimit value, but no larger value will be\nallowed. See the client version of this option (above) for some\nextra details.\n\n--config=FILE\nThis specifies an alternate config file than the default. This is\nonly relevant when --daemon is specified. The default is\n\/etc\/rsyncd.conf unless the daemon is running over a remote shell\nprogram and the remote user is not the super-user; in that case\nthe default is rsyncd.conf in the current directory (typically\n$HOME). --dparam=OVERRIDE, -M This option can be used to set a daemon-config parameter when starting up rsync in daemon mode. It is equivalent to adding the parameter at the end of the global settings prior to the first module's definition. The parameter names can be specified without spaces, if you so desire. For instance: rsync --daemon -M pidfile=\/path\/rsync.pid --no-detach When running as a daemon, this option instructs rsync to not detach itself and become a background process. This option is required when running as a service on Cygwin, and may also be useful when rsync is supervised by a program such as daemontools or AIX's System Resource Controller. --no-detach is also recommended when rsync is run under a debugger. This option has no effect if rsync is run from inetd or sshd. --port=PORT This specifies an alternate TCP port number for the daemon to listen on rather than the default of 873. See also the \"port\" global option in the rsyncd.conf manpage. --log-file=FILE This option tells the rsync daemon to use the given log-file name instead of using the \"log file\" setting in the config file. --log-file-format=FORMAT This option tells the rsync daemon to use the given FORMAT string instead of using the \"log format\" setting in the config file. It also enables \"transfer logging\" unless the string is empty, in which case transfer logging is turned off. --sockopts This overrides the socket options setting in the rsyncd.conf file and has the same syntax. --verbose, -v This option increases the amount of information the daemon logs during its startup phase. After the client connects, the daemon's verbosity level will be controlled by the options that the client used and the \"max verbosity\" setting in the module's config section. --ipv4, -4 or --ipv6, -6 Tells rsync to prefer IPv4\/IPv6 when creating the incoming sockets that the rsync daemon will use to listen for connections. One of these options may be required in older versions of Linux to work around an IPv6 bug in the kernel (if you see an \"address already in use\" error when nothing else is using the port, try specifying --ipv6 or --ipv4 when starting the daemon). These options also exist in the regular rsync options section. If rsync was complied without support for IPv6, the --ipv6 option will have no effect. The rsync --version output will contain \"no IPv6\" if is the case. --help, -h When specified after --daemon, print a short help page describing the options available for starting an rsync daemon. FILTER RULES The filter rules allow for flexible selection of which files to transfer (include) and which files to skip (exclude). The rules either directly specify include\/exclude patterns or they specify a way to acquire more include\/exclude patterns (e.g. to read them from a file). As the list of files\/directories to transfer is built, rsync checks each name to be transferred against the list of include\/exclude patterns in turn, and the first matching pattern is acted on: if it is an exclude pattern, then that file is skipped; if it is an include pattern then that filename is not skipped; if no matching pattern is found, then the filename is not skipped. Rsync builds an ordered list of filter rules as specified on the command- line. Filter rules have the following syntax: RULE [PATTERN_OR_FILENAME] RULE,MODIFIERS [PATTERN_OR_FILENAME] You have your choice of using either short or long RULE names, as described below. If you use a short-named rule, the ',' separating the RULE from the MODIFIERS is optional. The PATTERN or FILENAME that follows (when present) must come after either a single space or an underscore (_). Here are the available rule prefixes: exclude, '-' specifies an exclude pattern. include, '+' specifies an include pattern. merge, '.' specifies a merge-file to read for more rules. dir-merge, ':' specifies a per-directory merge-file. hide, 'H' specifies a pattern for hiding files from the transfer. show, 'S' files that match the pattern are not hidden. protect, 'P' specifies a pattern for protecting files from deletion. risk, 'R' files that match the pattern are not protected. clear, '!' clears the current include\/exclude list (takes no arg) When rules are being read from a file, empty lines are ignored, as are comment lines that start with a \"#\". Note that the --include & --exclude command-line options do not allow the full range of rule parsing as described above -- they only allow the specification of include \/ exclude patterns plus a \"!\" token to clear the list (and the normal comment parsing when rules are read from a file). If a pattern does not begin with \"- \" (dash, space) or \"+ \" (plus, space), then the rule will be interpreted as if \"+ \" (for an include option) or \"- \" (for an exclude option) were prefixed to the string. A --filter option, on the other hand, must always contain either a short or long rule name at the start of the rule. Note also that the --filter, --include, and --exclude options take one rule\/pattern each. To add multiple ones, you can repeat the options on the command-line, use the merge-file syntax of the --filter option, or the --include-from \/ --exclude-from options. INCLUDE\/EXCLUDE PATTERN RULES You can include and exclude files by specifying patterns using the \"+\", \"-\", etc. filter rules (as introduced in the FILTER RULES section above). The include\/exclude rules each specify a pattern that is matched against the names of the files that are going to be transferred. These patterns can take several forms: o if the pattern starts with a \/ then it is anchored to a particular spot in the hierarchy of files, otherwise it is matched against the end of the pathname. This is similar to a leading ^ in regular expressions. Thus \/foo would match a name of \"foo\" at either the \"root of the transfer\" (for a global rule) or in the merge-file's directory (for a per-directory rule). An unqualified foo would match a name of \"foo\" anywhere in the tree because the algorithm is applied recursively from the top down; it behaves as if each path component gets a turn at being the end of the filename. Even the unanchored \"sub\/foo\" would match at any point in the hierarchy where a \"foo\" was found within a directory named \"sub\". See the section on ANCHORING INCLUDE\/EXCLUDE PATTERNS for a full discussion of how to specify a pattern that matches at the root of the transfer. o if the pattern ends with a \/ then it will only match a directory, not a regular file, symlink, or device. o rsync chooses between doing a simple string match and wildcard matching by checking if the pattern contains one of these three wildcard characters: '*', '?', and '[' . o a '*' matches any path component, but it stops at slashes. o use '**' to match anything, including slashes. o a '?' matches any character except a slash (\/). o a '[' introduces a character class, such as [a-z] or [[:alpha:]]. o in a wildcard pattern, a backslash can be used to escape a wildcard character, but it is matched literally when no wildcards are present. This means that there is an extra level of backslash removal when a pattern contains wildcard characters compared to a pattern that has none. e.g. if you add a wildcard to \"foo\\bar\" (which matches the backslash) you would need to use \"foo\\\\bar*\" to avoid the \"\\b\" becoming just \"b\". o if the pattern contains a \/ (not counting a trailing \/) or a \"**\", then it is matched against the full pathname, including any leading directories. If the pattern doesn't contain a \/ or a \"**\", then it is matched only against the final component of the filename. (Remember that the algorithm is applied recursively so \"full filename\" can actually be any portion of a path from the starting directory on down.) o a trailing \"dir_name\/***\" will match both the directory (as if \"dir_name\/\" had been specified) and everything in the directory (as if \"dir_name\/**\" had been specified). This behavior was added in version 2.6.7. Note that, when using the --recursive (-r) option (which is implied by -a), every subdir component of every path is visited left to right, with each directory having a chance for exclusion before its content. In this way include\/exclude patterns are applied recursively to the pathname of each node in the filesystem's tree (those inside the transfer). The exclude patterns short-circuit the directory traversal stage as rsync finds the files to send. For instance, to include \"\/foo\/bar\/baz\", the directories \"\/foo\" and \"\/foo\/bar\" must not be excluded. Excluding one of those parent directories prevents the examination of its content, cutting off rsync's recursion into those paths and rendering the include for \"\/foo\/bar\/baz\" ineffectual (since rsync can't match something it never sees in the cut- off section of the directory hierarchy). The concept path exclusion is particularly important when using a trailing '*' rule. For instance, this won't work: + \/some\/path\/this-file-will-not-be-found + \/file-is-included - * This fails because the parent directory \"some\" is excluded by the '*' rule, so rsync never visits any of the files in the \"some\" or \"some\/path\" directories. One solution is to ask for all directories in the hierarchy to be included by using a single rule: \"+ *\/\" (put it somewhere before the \"- *\" rule), and perhaps use the --prune-empty-dirs option. Another solution is to add specific include rules for all the parent dirs that need to be visited. For instance, this set of rules works fine: + \/some\/ + \/some\/path\/ + \/some\/path\/this-file-is-found + \/file-also-included - * Here are some examples of exclude\/include matching: o \"- *.o\" would exclude all names matching *.o o \"- \/foo\" would exclude a file (or directory) named foo in the transfer-root directory o \"- foo\/\" would exclude any directory named foo o \"- \/foo\/*\/bar\" would exclude any file named bar which is at two levels below a directory named foo in the transfer-root directory o \"- \/foo\/**\/bar\" would exclude any file named bar two or more levels below a directory named foo in the transfer-root directory o The combination of \"+ *\/\", \"+ *.c\", and \"- *\" would include all directories and C source files but nothing else (see also the --prune-empty-dirs option) o The combination of \"+ foo\/\", \"+ foo\/bar.c\", and \"- *\" would include only the foo directory and foo\/bar.c (the foo directory must be explicitly included or it would be excluded by the \"*\") The following modifiers are accepted after a \"+\" or \"-\": o A \/ specifies that the include\/exclude rule should be matched against the absolute pathname of the current item. For example, \"-\/ \/etc\/passwd\" would exclude the passwd file any time the transfer was sending files from the \"\/etc\" directory, and \"-\/ subdir\/foo\" would always exclude \"foo\" when it is in a dir named \"subdir\", even if \"foo\" is at the root of the current transfer. o A ! specifies that the include\/exclude should take effect if the pattern fails to match. For instance, \"-! *\/\" would exclude all non-directories. o A C is used to indicate that all the global CVS-exclude rules should be inserted as excludes in place of the \"-C\". No arg should follow. o An s is used to indicate that the rule applies to the sending side. When a rule affects the sending side, it prevents files from being transferred. The default is for a rule to affect both sides unless --delete-excluded was specified, in which case default rules become sender-side only. See also the hide (H) and show (S) rules, which are an alternate way to specify sending-side includes\/excludes. o An r is used to indicate that the rule applies to the receiving side. When a rule affects the receiving side, it prevents files from being deleted. See the s modifier for more info. See also the protect (P) and risk (R) rules, which are an alternate way to specify receiver-side includes\/excludes. o A p indicates that a rule is perishable, meaning that it is ignored in directories that are being deleted. For instance, the -C option's default rules that exclude things like \"CVS\" and \"*.o\" are marked as perishable, and will not prevent a directory that was removed on the source from being deleted on the destination. o An x indicates that a rule affects xattr names in xattr copy\/delete operations (and is thus ignored when matching file\/dir names). If no xattr-matching rules are specified, a default xattr filtering rule is used (see the --xattrs option). MERGE-FILE FILTER RULES You can merge whole files into your filter rules by specifying either a merge (.) or a dir-merge (:) filter rule (as introduced in the FILTER RULES section above). There are two kinds of merged files -- single-instance ('.') and per- directory (':'). A single-instance merge file is read one time, and its rules are incorporated into the filter list in the place of the \".\" rule. For per-directory merge files, rsync will scan every directory that it traverses for the named file, merging its contents when the file exists into the current list of inherited rules. These per-directory rule files must be created on the sending side because it is the sending side that is being scanned for the available files to transfer. These rule files may also need to be transferred to the receiving side if you want them to affect what files don't get deleted (see PER-DIRECTORY RULES AND DELETE below). Some examples: merge \/etc\/rsync\/default.rules . \/etc\/rsync\/default.rules dir-merge .per-dir-filter dir-merge,n- .non-inherited-per-dir-excludes :n- .non-inherited-per-dir-excludes The following modifiers are accepted after a merge or dir-merge rule: o A - specifies that the file should consist of only exclude patterns, with no other rule-parsing except for in-file comments. o A + specifies that the file should consist of only include patterns, with no other rule-parsing except for in-file comments. o A C is a way to specify that the file should be read in a CVS- compatible manner. This turns on 'n', 'w', and '-', but also allows the list-clearing token (!) to be specified. If no filename is provided, \".cvsignore\" is assumed. o A e will exclude the merge-file name from the transfer; e.g. \"dir-merge,e .rules\" is like \"dir-merge .rules\" and \"- .rules\". o An n specifies that the rules are not inherited by subdirectories. o A w specifies that the rules are word-split on whitespace instead of the normal line-splitting. This also turns off comments. Note: the space that separates the prefix from the rule is treated specially, so \"- foo + bar\" is parsed as two rules (assuming that prefix-parsing wasn't also disabled). o You may also specify any of the modifiers for the \"+\" or \"-\" rules (above) in order to have the rules that are read in from the file default to having that modifier set (except for the ! modifier, which would not be useful). For instance, \"merge,-\/ .excl\" would treat the contents of .excl as absolute-path excludes, while \"dir- merge,s .filt\" and \":sC\" would each make all their per-directory rules apply only on the sending side. If the merge rule specifies sides to affect (via the s or r modifier or both), then the rules in the file must not specify sides (via a modifier or a rule prefix such as hide). Per-directory rules are inherited in all subdirectories of the directory where the merge-file was found unless the 'n' modifier was used. Each subdirectory's rules are prefixed to the inherited per-directory rules from its parents, which gives the newest rules a higher priority than the inherited rules. The entire set of dir-merge rules are grouped together in the spot where the merge-file was specified, so it is possible to override dir-merge rules via a rule that got specified earlier in the list of global rules. When the list-clearing rule (\"!\") is read from a per-directory file, it only clears the inherited rules for the current merge file. Another way to prevent a single rule from a dir-merge file from being inherited is to anchor it with a leading slash. Anchored rules in a per- directory merge-file are relative to the merge-file's directory, so a pattern \"\/foo\" would only match the file \"foo\" in the directory where the dir-merge filter file was found. Here's an example filter file which you'd specify via --filter=\". file\": merge \/home\/user\/.global-filter - *.gz dir-merge .rules + *.[ch] - *.o - foo* This will merge the contents of the \/home\/user\/.global-filter file at the start of the list and also turns the \".rules\" filename into a per- directory filter file. All rules read in prior to the start of the directory scan follow the global anchoring rules (i.e. a leading slash matches at the root of the transfer). If a per-directory merge-file is specified with a path that is a parent directory of the first transfer directory, rsync will scan all the parent dirs from that starting point to the transfer directory for the indicated per-directory file. For instance, here is a common filter (see -F): --filter=': \/.rsync-filter' That rule tells rsync to scan for the file .rsync-filter in all directories from the root down through the parent directory of the transfer prior to the start of the normal directory scan of the file in the directories that are sent as a part of the transfer. (Note: for an rsync daemon, the root is always the same as the module's \"path\".) Some examples of this pre-scanning for per-directory files: rsync -avF \/src\/path\/ \/dest\/dir rsync -av --filter=': ..\/..\/.rsync-filter' \/src\/path\/ \/dest\/dir rsync -av --filter=': .rsync-filter' \/src\/path\/ \/dest\/dir The first two commands above will look for \".rsync-filter\" in \"\/\" and \"\/src\" before the normal scan begins looking for the file in \"\/src\/path\" and its subdirectories. The last command avoids the parent-dir scan and only looks for the \".rsync-filter\" files in each directory that is a part of the transfer. If you want to include the contents of a \".cvsignore\" in your patterns, you should use the rule \":C\", which creates a dir-merge of the .cvsignore file, but parsed in a CVS-compatible manner. You can use this to affect where the --cvs-exclude (-C) option's inclusion of the per-directory .cvsignore file gets placed into your rules by putting the \":C\" wherever you like in your filter rules. Without this, rsync would add the dir- merge rule for the .cvsignore file at the end of all your other rules (giving it a lower priority than your command-line rules). For example: cat <<EOT | rsync -avC --filter='. -' a\/ b + foo.o :C - *.old EOT rsync -avC --include=foo.o -f :C --exclude='*.old' a\/ b Both of the above rsync commands are identical. Each one will merge all the per-directory .cvsignore rules in the middle of the list rather than at the end. This allows their dir-specific rules to supersede the rules that follow the :C instead of being subservient to all your rules. To affect the other CVS exclude rules (i.e. the default list of exclusions, the contents of$HOME\/.cvsignore, and the value of $CVSIGNORE) you should omit the -C command-line option and instead insert a \"-C\" rule into your filter rules; e.g. \"--filter=-C\". LIST-CLEARING FILTER RULE You can clear the current include\/exclude list by using the \"!\" filter rule (as introduced in the FILTER RULES section above). The \"current\" list is either the global list of rules (if the rule is encountered while parsing the filter options) or a set of per-directory rules (which are inherited in their own sub-list, so a subdirectory can use this to clear out the parent's rules). ANCHORING INCLUDE\/EXCLUDE PATTERNS As mentioned earlier, global include\/exclude patterns are anchored at the \"root of the transfer\" (as opposed to per-directory patterns, which are anchored at the merge-file's directory). If you think of the transfer as a subtree of names that are being sent from sender to receiver, the transfer-root is where the tree starts to be duplicated in the destination directory. This root governs where patterns that start with a \/ match. Because the matching is relative to the transfer-root, changing the trailing slash on a source path or changing your use of the --relative option affects the path you need to use in your matching (in addition to changing how much of the file tree is duplicated on the destination host). The following examples demonstrate this. Let's say that we want to match two source files, one with an absolute path of \"\/home\/me\/foo\/bar\", and one with a path of \"\/home\/you\/bar\/baz\". Here is how the various command choices differ for a 2-source transfer: Example cmd: rsync -a \/home\/me \/home\/you \/dest +\/- pattern: \/me\/foo\/bar +\/- pattern: \/you\/bar\/baz Target file: \/dest\/me\/foo\/bar Target file: \/dest\/you\/bar\/baz Example cmd: rsync -a \/home\/me\/ \/home\/you\/ \/dest +\/- pattern: \/foo\/bar (note missing \"me\") +\/- pattern: \/bar\/baz (note missing \"you\") Target file: \/dest\/foo\/bar Target file: \/dest\/bar\/baz Example cmd: rsync -a --relative \/home\/me\/ \/home\/you \/dest +\/- pattern: \/home\/me\/foo\/bar (note full path) +\/- pattern: \/home\/you\/bar\/baz (ditto) Target file: \/dest\/home\/me\/foo\/bar Target file: \/dest\/home\/you\/bar\/baz Example cmd: cd \/home; rsync -a --relative me\/foo you\/ \/dest +\/- pattern: \/me\/foo\/bar (starts at specified path) +\/- pattern: \/you\/bar\/baz (ditto) Target file: \/dest\/me\/foo\/bar Target file: \/dest\/you\/bar\/baz The easiest way to see what name you should filter is to just look at the output when using --verbose and put a \/ in front of the name (use the --dry-run option if you're not yet ready to copy any files). PER-DIRECTORY RULES AND DELETE Without a delete option, per-directory rules are only relevant on the sending side, so you can feel free to exclude the merge files themselves without affecting the transfer. To make this easy, the 'e' modifier adds this exclude for you, as seen in these two equivalent commands: rsync -av --filter=': .excl' --exclude=.excl host:src\/dir \/dest rsync -av --filter=':e .excl' host:src\/dir \/dest However, if you want to do a delete on the receiving side AND you want some files to be excluded from being deleted, you'll need to be sure that the receiving side knows what files to exclude. The easiest way is to include the per-directory merge files in the transfer and use --delete- after, because this ensures that the receiving side gets all the same exclude rules as the sending side before it tries to delete anything: rsync -avF --delete-after host:src\/dir \/dest However, if the merge files are not a part of the transfer, you'll need to either specify some global exclude rules (i.e. specified on the command line), or you'll need to maintain your own per-directory merge files on the receiving side. An example of the first is this (assume that the remote .rules files exclude themselves): rsync -av --filter=': .rules' --filter='. \/my\/extra.rules' --delete host:src\/dir \/dest In the above example the extra.rules file can affect both sides of the transfer, but (on the sending side) the rules are subservient to the rules merged from the .rules files because they were specified after the per-directory merge rule. In one final example, the remote side is excluding the .rsync-filter files from the transfer, but we want to use our own .rsync-filter files to control what gets deleted on the receiving side. To do this we must specifically exclude the per-directory merge files (so that they don't get deleted) and then put rules into the local files to control what else should not get deleted. Like one of these commands: rsync -av --filter=':e \/.rsync-filter' --delete \\ host:src\/dir \/dest rsync -avFF --delete host:src\/dir \/dest BATCH MODE Batch mode can be used to apply the same set of updates to many identical systems. Suppose one has a tree which is replicated on a number of hosts. Now suppose some changes have been made to this source tree and those changes need to be propagated to the other hosts. In order to do this using batch mode, rsync is run with the write-batch option to apply the changes made to the source tree to one of the destination trees. The write-batch option causes the rsync client to store in a \"batch file\" all the information needed to repeat this operation against other, identical destination trees. Generating the batch file once saves having to perform the file status, checksum, and data block generation more than once when updating multiple destination trees. Multicast transport protocols can be used to transfer the batch update files in parallel to many hosts at once, instead of sending the same data to every host individually. To apply the recorded changes to another destination tree, run rsync with the read-batch option, specifying the name of the same batch file, and the destination tree. Rsync updates the destination tree using the information stored in the batch file. For your convenience, a script file is also created when the write-batch option is used: it will be named the same as the batch file with \".sh\" appended. This script file contains a command-line suitable for updating a destination tree using the associated batch file. It can be executed using a Bourne (or Bourne-like) shell, optionally passing in an alternate destination tree pathname which is then used instead of the original destination path. This is useful when the destination tree path on the current host differs from the one used to create the batch file. Examples:$ rsync --write-batch=foo -a host:\/source\/dir\/ \/adest\/dir\/\n$scp foo* remote:$ ssh remote .\/foo.sh \/bdest\/dir\/\n\n$rsync --write-batch=foo -a \/source\/dir\/ \/adest\/dir\/$ ssh remote rsync --read-batch=- -a \/bdest\/dir\/ <foo\n\nIn these examples, rsync is used to update \/adest\/dir\/ from \/source\/dir\/\nand the information to repeat this operation is stored in \"foo\" and\n\"foo.sh\". The host \"remote\" is then updated with the batched data going\ninto the directory \/bdest\/dir. The differences between the two examples\nreveals some of the flexibility you have in how you deal with batches:\n\no The first example shows that the initial copy doesn't have to be\nlocal -- you can push or pull data to\/from a remote host using\neither the remote-shell syntax or rsync daemon syntax, as desired.\n\no The first example uses the created \"foo.sh\" file to get the right\nrsync options when running the read-batch command on the remote\nhost.\n\no The second example reads the batch data via standard input so that\nthe batch file doesn't need to be copied to the remote machine\nfirst. This example avoids the foo.sh script because it needed to\nuse a modified --read-batch option, but you could edit the script\nfile if you wished to make use of it (just be sure that no other\noption is trying to use standard input, such as the \"--exclude-\nfrom=-\" option).\n\nCaveats:\n\nThe read-batch option expects the destination tree that it is updating to\nbe identical to the destination tree that was used to create the batch\nupdate fileset. When a difference between the destination trees is\nencountered the update might be discarded with a warning (if the file\nappears to be up-to-date already) or the file-update may be attempted and\nthen, if the file fails to verify, the update discarded with an error.\nThis means that it should be safe to re-run a read-batch operation if the\ncommand got interrupted. If you wish to force the batched-update to\nalways be attempted regardless of the file's size and date, use the -I\noption (when reading the batch). If an error occurs, the destination\ntree will probably be in a partially updated state. In that case, rsync\ncan be used in its regular (non-batch) mode of operation to fix up the\ndestination tree.\n\nThe rsync version used on all destinations must be at least as new as the\none used to generate the batch file. Rsync will die with an error if the\nprotocol version in the batch file is too new for the batch-reading rsync\nto handle. See also the --protocol option for a way to have the creating\nrsync generate a batch file that an older rsync can understand. (Note\nthat batch files changed format in version 2.6.3, so mixing versions\nolder than that with newer versions will not work.)\n\nWhen reading a batch file, rsync will force the value of certain options\nto match the data in the batch file if you didn't set them to the same as\nthe batch-writing command. Other options can (and should) be changed.\nFor instance --write-batch changes to --read-batch, --files-from is\ndropped, and the --filter \/ --include \/ --exclude options are not needed\nunless one of the --delete options is specified.\n\nThe code that creates the BATCH.sh file transforms any\nfilter\/include\/exclude options into a single list that is appended as a\n\"here\" document to the shell script file. An advanced user can use this\nto modify the exclude list if a change in what gets deleted by --delete\nis desired. A normal user can ignore this detail and just use the shell\nscript as an easy way to run the appropriate --read-batch command for the\nbatched data.\n\nThe original batch mode in rsync was based on \"rsync+\", but the latest\nversion uses a new implementation.\n\nThree basic behaviors are possible when rsync encounters a symbolic link\nin the source directory.\n\nBy default, symbolic links are not transferred at all. A message\n\"skipping non-regular\" file is emitted for any symlinks that exist.\n\nIf --links is specified, then symlinks are recreated with the same target\non the destination. Note that --archive implies --links.\n\ntheir referent, rather than the symlink.\n\nRsync can also distinguish \"safe\" and \"unsafe\" symbolic links. An\nexample where this might be used is a web site mirror that wishes to\nensure that the rsync module that is copied does not include symbolic\nlinks to \/etc\/passwd in the public section of the site. Using --copy-\nunsafe-links will cause any links to be copied as the file they point to\nto have any effect.)\n\nwith \/), empty, or if they contain enough \"..\" components to ascend from\nthe directory being copied.\n\nHere's a summary of how the symlink options are interpreted. The list is\nin order of precedence, so if your combination of options isn't\nmentioned, use the first line that is a complete subset of your options:\n\nother options to affect).\n\nTurn all unsafe symlinks into files and duplicate all safe\n\nTurn all unsafe symlinks into files, noisily skip all safe\n\nDuplicate safe symlinks and skip unsafe ones.\n\nDIAGNOSTICS\nrsync occasionally produces error messages that may seem a little\ncryptic. The one that seems to cause the most confusion is \"protocol\nversion mismatch -- is your shell clean?\".\n\nThis message is usually caused by your startup scripts or remote shell\nfacility producing unwanted garbage on the stream that rsync is using for\nits transport. The way to diagnose this problem is to run your remote\nshell like this:\n\nssh remotehost \/bin\/true > out.dat\n\nthen look at out.dat. If everything is working correctly then out.dat\nshould be a zero length file. If you are getting the above error from\nrsync then you will probably find that out.dat contains some text or\ndata. Look at the contents and try to work out what is producing it.\nThe most common cause is incorrectly configured shell startup scripts\n(such as .cshrc or .profile) that contain output statements for non-\n\nIf you are having trouble debugging filter patterns, then try specifying\nthe -vv option. At this level of verbosity rsync will show why each\nindividual file is included or excluded.\n\nEXIT VALUES\n0 Success\n\n1 Syntax or usage error\n\n2 Protocol incompatibility\n\n3 Errors selecting input\/output files, dirs\n\n4 Requested action not supported: an attempt was made to manipulate\n64-bit files on a platform that cannot support them; or an option\nwas specified that is supported by the client and not by the\nserver.\n\n5 Error starting client-server protocol\n\n6 Daemon unable to append to log-file\n\n10 Error in socket I\/O\n\n11 Error in file I\/O\n\n12 Error in rsync protocol data stream\n\n13 Errors with program diagnostics\n\n14 Error in IPC code\n\n21 Some error returned by waitpid()\n\n22 Error allocating core memory buffers\n\n23 Partial transfer due to error\n\n24 Partial transfer due to vanished source files\n\n25 The --max-delete limit stopped deletions\n\n35 Timeout waiting for daemon connection\n\nENVIRONMENT VARIABLES\nCVSIGNORE\nThe CVSIGNORE environment variable supplements any ignore patterns\nin .cvsignore files. See the --cvs-exclude option for more\ndetails.\n\nRSYNC_ICONV\nSpecify a default --iconv setting using this environment variable.\n(First supported in 3.0.0.)\n\nRSYNC_PROTECT_ARGS\nSpecify a non-zero numeric value if you want the --protect-args\noption to be enabled by default, or a zero value to make sure that\nit is disabled by default. (First supported in 3.1.0.)\n\nRSYNC_RSH\nThe RSYNC_RSH environment variable allows you to override the\ndefault shell used as the transport for rsync. Command line\noptions are permitted after the command name, just as in the -e\noption.\n\nRSYNC_PROXY\nThe RSYNC_PROXY environment variable allows you to redirect your\nrsync client to use a web proxy when connecting to a rsync daemon.\nYou should set RSYNC_PROXY to a hostname:port pair.\n\nauthenticated rsync connections to an rsync daemon without user\nintervention. Note that this does not supply a password to a\nremote shell transport such as ssh; to learn how to do that,\nconsult the remote shell's documentation.\n\nUSER or LOGNAME\nThe USER or LOGNAME environment variables are used to determine\nthe default username sent to an rsync daemon. If neither is set,\n\nHOME The HOME environment variable is used to find the user's default\n.cvsignore file.\n\nFILES\n\/etc\/rsyncd.conf or rsyncd.conf\n\nrsync-ssl(1), rsyncd.conf(5)\n\nBUGS\ntimes are transferred as *nix time_t values\n\nWhen transferring to FAT filesystems rsync may re-sync unmodified files.\nSee the comments on the --modify-window option.\n\nfile permissions, devices, etc. are transferred as native numerical\nvalues\n\nPlease report bugs! See the web site at https:\/\/rsync.samba.org\/.\n\nVERSION\nThis man page is current for version 3.2.3 of rsync.\n\nINTERNAL OPTIONS\nThe options --server and --sender are used internally by rsync, and\nshould never be typed by a user under normal circumstances. Some\nawareness of these options may be needed in certain scenarios, such as\nwhen setting up a login that can only run an rsync command. For\ninstance, the support directory of the rsync distribution has an example\nscript named rrsync (for restricted rsync) that can be used with a\n\nCREDITS\nCOPYING for details.\n\nA web site is available at https:\/\/rsync.samba.org\/. The site includes\nan FAQ-O-Matic which may cover questions unanswered by this manual page.\n\nWe would be delighted to hear from you if you like this program. Please\ncontact the mailing-list at rsync@lists.samba.org.\n\nThis program uses the excellent zlib compression library written by Jean-\n\nTHANKS\nSpecial thanks go out to: John Van Essen, Matt McCutchen, Wesley W.\nTerpstra, David Dykstra, Jos Backus, Sebastian Krahmer, Martin Pool, and\n\nThanks also to Richard Brent, Brendan Mackay, Bill Waite, Stephen\nRothwell and David Bell. I've probably missed some people, my apologies\nif I have.\n\nAUTHOR\nrsync was originally written by Andrew Tridgell and Paul Mackerras. Many\npeople have later contributed to it. It is currently maintained by Wayne\nDavison.\n\nMailing lists for support and development are available at\nhttps:\/\/lists.samba.org\/.\n\nrsync 3.2.3 07 Aug 2020 rsync(1)","date":"2020-12-01 05:56:41","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.45685240626335144, \"perplexity\": 7192.679798341916}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-50\/segments\/1606141652107.52\/warc\/CC-MAIN-20201201043603-20201201073603-00506.warc.gz\"}"}
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\begin{center}
\huge{Department of Computer Science and Engineering}\\[0.5cm]
\normalsize
\textsc{National Institute of Technology Calicut}\\[2.0cm]
\emph{\LARGE Certificate}\\[2.5cm]
\end{center}
\normalsize This is to certify that this is a bonafide record of the project presented by the students whose names are given below during Monsoon semester 2013 in partial fulfilment of the requirements of the degree of Bachelor of Technology in Computer Science and Engineering.\\[1.0cm]
\begin{table}[h]
\centering
\begin{tabular}{lr}
Roll No & Names of Students \\ \\ \hline
\\
B100312CS & Sudev A C \\
B100229CS & Sharath Hari N \\
\end{tabular}
\end{table}
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\begin{flushright}
Dr K. Muralikrishnan\\
(Project Guide)\\[1.5cm]
Jayaraj P B\\
(Course Coordinator)\\
\end{flushright}
\begin{flushleft}
Date: 31/10/2013
\end{flushleft}
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{
"redpajama_set_name": "RedPajamaGithub"
}
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Denise Doring VanBuren was elected the 45th President General of the National Society Daughters of the American Revolution in June 2019, a three-year term.
DAR service
VanBuren joined the DAR through her Patriot Ancestors, father and son, Jacob and Marcus Plattner. She has been involved with the DAR in the City of Beacon, New York, then with the New York State organization before her role at a national level.
As Regent of the Beacon, New York, Melzingah Chapter from 1998 to 2001, VanBuren chaired the Executive Board and was responsible for the stewardship of the 1709 Madam Brett Homestead, the oldest building in Dutchess County. She was named New York State's Outstanding Chapter Regent in 1999. She led the Melzingah Chapter's efforts to erect a municipal bust in honor of George Washington in Beacon. In 2000, she led a hike to the top of Mount Beacon that involved more than 600 people rededicating Melzingah's 1900 monument to Revolutionary War soldiers.
She served in three State Chairmanships and as State Historian before serving as New York State Regent from 2010 to 2013. As State Regent, her theme was "Celebrate the Empire State. Excelsior!"
At the National Society, VanBuren served as Organizing Secretary General from 2013 to 2016 and First Vice President General from 2016 to 2019. She has been Editor in Chief of American Spirit and Daughters since 2004.
VanBuren was installed as the 45th DAR President General in 2019 during the 128th Continental Congress. She chose the theme "Rise and Shine for America" with the goals of "passionate purpose, increased membership and an improved public image."
Highlights from the VanBuren Administration:
First ever virtual Continental Congress, held digitally due to the COVID-19 pandemic.
Release of the DAR's Continuing Commitment to Equality, in which the DAR "reaffirm[ed] to the membership and the public alike that our organization condemns racism." and stated that, "Bias, prejudice and intolerance have no place in the DAR or America."
She is also a member of the Daughters of Union Veterans of the Civil War.
Historical organizations and books
VanBuren served five-terms as President of the Beacon Historical Society, and co-authored two books, Historic Beacon (1998) and Beacon Revisited (2003), to benefit that organization. She served two terms as President of the Dutchess County Historical Society, and two terms as President of the Exchange Club of Southern Dutchess.
Professional work
In 1993, VanBuren joined the media relations group at Central Hudson Gas & Electric Corporation in Poughkeepsie, NY. She became a Vice President in 2000, ultimately serving as Vice President of Public Relations from 1993 to her retirement from that organization effective January 2020.
Board service
VanBuren serves on the following Boards:
Mid-Hudson Regional Hospital
Hudson River Valley Institute at Marist College
Boscobel, Inc.
Dutchess Tourism, Inc. (past Chair)
Education
VanBuren graduated from St. Bonaventure University, and then went on to obtain an MBA from Mount St. Mary College.
References
Living people
Year of birth missing (living people)
Place of birth missing (living people)
Daughters of the American Revolution people
People from Beacon, New York
St. Bonaventure University alumni
Mount St. Mary's University (Los Angeles) alumni
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
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Zápalník je součástka bicího ústrojí ručních palných zbraní. Při výstřelu úderník narazí na zápalník a ten přenese úder na zápalku náboje.
Konstrukce zápalníku
Místo, kterým zápalník udeří do zápalky náboje se nazývá hrot zápalníku. Musí jít o malou plochu, ale zároveň hrot zápalníku nesmí zápalku prorazit. Úkolem hrotu je promáčnout dno zápalky ke kovadlince tak, aby došlo k zažehnutí. Pro náboje se středovým zápalem se obvykle používá půlkulatý hrot. Příkladem výjimky jsou pistole Glock. Pro náboj s okrajovým zápalem, například .22 Long Rifle se používá většinou hrot pravoúhlý.
Vzdálenost o kterou se hrot zápalníku vysune je důležitý údaj související s konstrukcí nábojů a zápalek. Následující tabulka ukazuje doporučené hodnoty pro lovecké zbraně podle předpisů v Rakousku.
Spojení zápalníku s úderníkem
U některých konstrukcí zbraní bývá zápalník spojen s úderníkem. V tomto případě jde o úderník s integrovaným zápalníkem (nebo jen úderník se zápalníkem). Tento princip je použit například u pistolí Glock. Používání pojmu úderník se zápalníkem nebývá v některých pramenech důsledné. Někteří autoři používají jen pojem zápalník a v jiných zdrojích je použit pro stejnou součástku pojem úderník. Zápalník jako samostatná součástka je použit například v pistoli CZ 75, nebo u útočné pušky Samopal vzor 58.
Identifikace zbraně
Hrot zápalníku zanechává při úderu na dno zápalky (respektive dno nábojnice u střeliva s okrajovým zápalem) stopu, podle které může být prováděna idetifikace zbraně ve které byla použita zkoumaná nábojnice. V principu jde o porovnání otisků mikro-nerovností vzniklých na povrchu hrotu zápalníku při výrobě.
Odkazy
Reference
Související články
Úderník
Externí odkazy
Palné zbraně
Lovecké zbraně
Součásti palných zbraní
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Back Off, Big Apple
Some ideas are so lowdown awful, criticism is not enough. One needs to seize the offending notion by the scruff of the neck, shove it into a corner, and stand over it and scold it: "Bad idea! Bad, bad idea!" The recent trademark application filed by New York City to monopolize the phrase "The World's Second Home" for a wide range of products -- from paperweights to baby bibs to temporary tattoos to fanny packs -- is such a notion, according to Lawrence J. Siskind.
By Lawrence J. Siskind | March 17, 2005 at 12:00 AM
Some ideas are so lowdown awful, criticism is not enough. One needs to seize the offending notion by the scruff of the neck, shove it into a corner, and stand over and scold it: "Bad idea! Bad, bad idea!" The recent trademark application filed by New York City to monopolize the phrase "The World's Second Home" for a wide range of products — from paperweights to baby bibs to temporary tattoos to fanny packs — is such a notion. It is not enough to discredit the idea. Someone really needs to give New York a time out. New York City wants to host the 2012 Olympics. To increase its chances, the city's attorneys have filed a trademark application to register "The World's Second Home" on an intent-to-use basis for scores of goods and services in eight different classes. The strategy, apparently, is to emphasize that New York is a city of immigrants, with constituents from all over the globe, and thus the logical locale for an international event such as the Olympics. There are obvious problems with the gambit, starting with the fact that New York is not the World's Second Home. Granted, with 36 percent of its population foreign born, New York is the second home for many, many people. But it lags behind Miami (with 51 percent of its population foreign born), Los Angeles (41 percent) and even Vancouver (37 percent). New Yorkers might respond that Miami is the second home mainly to Cuban-Americans, and Los Angeles the second home mainly to Mexican-Americans, while New York's second-homers come from many different places. But even in the category of second-home diversity, New York trails behind Toronto, which has been named by the United Nations as "the world's most ethnically diverse city" five times in a row. But these are minor matters. The main problem with New York's attempt to monopolize "The World's Second Home" is the violence it does to bedrock principles of trademark law. Since the dawn of recorded history — actually before the dawn — trademarks have served the function of distinguishing one party's products from those of another. Paintings on the walls of the Lascaux caves in southern France show bison bearing marks which, scholars say, indicate ownership. The cavemen wanted to distinguish their bison from those of their prehistoric rivals. Ancient Egyptians, Greeks and Chinese used seals and other markings to distinguish one party's pottery items from those of another. All these venerable trademark practitioners started with a product. Then they developed a mark to append to it. And so trademarks evolved through history. Antonio Stradivari crafted more than 1,100 violins and other musical instruments — and the "Stradivarius" mark emerged to identify his handiwork and models. Arthur Guinness first developed a remarkable stout in Dublin, and then attached the family name to distinguish it from his competitors' beers. But now comes New York City, reversing the process. First they developed the catchy if inaccurate slogan "The World's Second Home." Then they compiled the longest possible list of tchotchkes imaginable — including lawn signs, bean bag chairs, pencil sharpeners, key chains, sandals, rompers and knickers (yes, knickers) — upon which to attach the mark. Why? Obviously, not for the traditional purpose of trademark law: to distinguish New York City's products from those of other cities. Can anyone imagine a shopper searching the globe for knickers, until he finally finds a pair emblazoned with the "World's Second Home" slogan, then exclaiming: "At last! Thanks to this slogan, I know I've finally found a pair of genuine knickers manufactured by the good people of New York City! Just what I've been looking for!" No, the knickers could (and probably will) come from Bangladesh, for all the shopper will care. It's not the product New York will be selling, but the mark. New York wants to be known as the World's Second Home. The cheap products are just so many tiny, portable billboards to advertise that aspiration. Of course, New York City is not alone in the campaign to stand trademark law on its head. That noted New Yorker, Donald Trump, no sooner had a hit show under his belt than he applied to register "You're Fired" for games, playthings, pillows, alcoholic beverages, and men's cologne. Jessie Conners, a failed contestant on the show, showed her ability to learn from the master. After she was "fired," she promptly filed to register "You're Fired" for a line of clothing. Of course, no shopper is going to search for the mark "You're Fired" on toys, pillows, or drinks to gain assurance that the products truly are made by Donald Trump or Jessie Conners. Trump and Conners do not intend to use the slogan to promote their products. They intend to use the slogan to promote themselves. Aside from undermining the fundamentals of trademark law, what is wrong with this trend? Is it a crime for New York City to want to have the exclusive right to sell tote bags with "The World's Second Home" slogan? Well, yes it is. And not just a crime against good taste. It is a crime against freedom of speech. Other cities may vie with New York for the title of the World's Second Home. By the simple expedient of trademark registration practice, and the vagaries of the international classification system, New York City can effectively block its rivals from expressing that sentiment in just about any medium other than newsprint. Which is, of course, the whole point of the campaign. New York is not out to promote its tchotchke industry. That industry is already quite healthy, thank you. Rather, the campaign's aim is to send a message and to prevent others from sending a competing message. This inversion of trademark law leads to perverse incentives. Rival cities, denied the right to use the slogan "The World's Second Home" on their own chazzerei, may rush to reserve "World's Third Home" or "World's Fourth Home." Our trademark registration system allows anyone to file any number of "intent to use" applications. After the application is approved, the applicant has up to three years to actually use the mark on a product. Thus, a crafty city might apply for and so tie up for years the slogans "The World's [Third], [Fourth], [Fifth], [and so on, up to 99th] Home." I grew up in Marblehead, a small town north of Boston, where every schoolchild learns that his hometown is "The Birthplace of the American Navy." Our claim was supported by the fact that the men rowing Washington's army across the Delaware in Emanuel Leutze's famous painting were Marblehead fishermen, one and all. For some unfathomable reason, the smaller town of Beverly, a few miles farther north, also lays claim to the title, and motorists could not enter the community without passing a billboard welcoming them to this other putative "Birthplace of the American Navy." There was a fair amount of argument between the towns and petty vandalism of each others' signs. But on the whole, the battle was waged ethically. It never occurred to anyone, in that quainter and more innocent age, to file trademark applications to obtain exclusive rights to use the "Birthplace of the American Navy" slogan on electric light switch plates, tattoos, or bean bag chairs. We were fortunate. Since anyone can file an intent to use trademark application, Marblehead (and Beverly, too, for that matter) might have lost out to Omaha, Neb., for exclusive rights to the "Birthplace of the American Navy" slogan. Contributing writer Lawrence J. Siskind, of San Francisco's Harvey Siskind Jacobs, specializes in intellectual property law.
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Q: accessing httpcontext properties from System.Timers.Timer I have a problem when accessing httpcontext.current value at application_start method.
There is a lots of discussions about this topic.
But I want to share my business logic and I need some advice how to handle problem.
Let's look at this business logic together step by step
1-I want to design "custom object" which has "static global List" property and any user can add "LogObj" object to this list whereever actions occured.
public class MyLog
{
public static void List<LogObj> LogObjList {get;set;}
static MyLog()
{
LogObjList = new List<LogObj>();
}
}
2- If I have a "System.Timers.Timer" object which checks the "static global List" every X milliseconds and performs some action which defined in the code
public static init(){
System.Timers.Timer t = new System.Timers.Timer();
t.Elapsed += T_Elapsed;
t.Interval = 3000;
t.Start();
}
private void T_Elapsed(object sender, System.Timers.ElapsedEventArgs e)
{
//perform some code.
var s = HttpContext.Current.Session["test"];
var logObj = MyLog.LogObjList[0] + s;
//save as text file ...
}
3- If I start init() method in application_start event at global.asax I get this error "object reference ..." where the line of "..HttpContext.Current.Session" started.
So
If I do not want to access any httpcontext.current's properties I have no problem at this situation.
But If I need to access any properties of httpcontext.current at Timer_Elapsed event I have problem about it.
So I need your advice or alternative way to making my algorithm.
Thank you
|
{
"redpajama_set_name": "RedPajamaStackExchange"
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\section{Introduction}
\label{sec:intro}
\input{intro.tex}
\section{Preliminaries}
\label{sec:prelim}
\input{prelim.tex}
\section{An Erd{\"o}s-P{\'o}sa Type Theorem}
\label{sec:erdosposa}
\input{erdos.tex}
\section{Fixed-Parameter Tractability and Kernelization Complexity}
\label{sec:fpt-ker1}
\input{fpt-ker1.tex}
\section{An Improved FPT Algorithm and A Smaller Kernel}
\label{sec:fpt-ker2}
\input{fpt-ker2.tex}
\section{Concluding Remarks}
\label{sec:concl}
\input{concl.tex}
\subsection{An FPT Algorithm}
Consider an instance $\mathcal{I}=(T,k)$ of \textsc{ACT}. Let $n$ denote $|V(T)|$ and $m$ denote $|A(T)|$. Suppose $\mathcal{I}$ is a yes-instance and $\mathcal{C}$ is a set of $k$ arc-disjoint cycles in $T$. From Lemma \ref{lem:short-cycle}, we may assume that the total number of arcs that are in cycles in $\mathcal{C}$ is at most $(2k+1)k$. Using this observation, we proceed as follows. We color the arcs of $T$ uniformly at random from the color set $[\ell]$ where $\ell=2k^2+k$. Let $\chi:A(T) \rightarrow [\ell]$ denote this coloring.
\begin{proposition}[\cite{color-coding}]
\label{prop:colorful}
If $E$ is a subset of $A(T)$ of size $\ell$, then the probability that the arcs in $E$ are colored with pairwise distinct colors is at least $e^{-\ell}$.
\end{proposition}
Next, we define the notion of a colorful solution for our problem.
\begin{definition}{\bf (Colorful set of cycles)}
\label{def:colorful-sol}
A set $\mathcal{C}$ of arc-disjoint cycles in $T$ that satisfies the property that for any two (not necessarily distinct) cycles $C,C' \in \mathcal{C}$ and for any two distinct arcs $e \in A(C), e' \in A(C')$, $\chi(e) \neq \chi(e')$ holds is said to be a {\em colorful set of cycles}.
\end{definition}
Rephrasing Proposition \ref{prop:colorful} in the context of our problem, we have the following observation.
\begin{observation}
\label{obs:colorful-opt}
If $\mathcal{C}$ is a solution of $\mathcal{I}$ with the property that for each $C \in \mathcal{C}$, $|C| \leq 2k+1$, then $\mathcal{C}$ is a colorful set of cycles in $T$ with probability at least $e^{-\ell}$.
\end{observation}
Armed with the guarantee that a solution (if one exists) of $\mathcal{I}$ is colorful with sufficiently high probability, we focus on finding a colorful set of cycles in $T$.
\begin{lemma}
\label{lem:colorful-cycles}
If $T$ has a colorful set of $k$ cycles, then such a set can be obtained in $\ell! n^{\mathcal{O}(1)}$ time.
\end{lemma}
\begin{proof}
Consider a permutation $\sigma$ of $[\ell]$. For each $i \in [k]$, let $D^\sigma_i$ denote the subgraph of $T$ with $V(D^\sigma_i)=V(T)$ and $A(D^\sigma_i)=A(T) \cap \{ (u,v) \in A(T) \mid 2(i-1)k+i \leq \sigma(\chi((u,v))) \leq 2k+2(i-1)k+i\}$. That is, $A(D^\sigma_1)$ is the set of arcs of $T$ that are colored with the first $2k+1$ colors, $A(D^\sigma_2)$ is the set of arcs of $T$ that are colored with the next $2k+1$ colors and so on. For each $i \in [k]$, let $C^\sigma_i$ denote a cycle (if one exists) in $D^\sigma_i$. Let $\mathcal{C}_\sigma$ denote the set $\{C^\sigma_i \mid i \in [k]\}$. For each permutation $\sigma$ of $[\ell]$, we compute the corresponding set $\mathcal{C}_\sigma$. If $T$ has a colorful set of $k$ cycles, then $|\mathcal{C}_\pi|=k$ for some permutation $\pi$ of $[\ell]$. Therefore, by computing $\mathcal{C}_\pi$ for every permutation $\pi$ of $[\ell]$, we can obtain a colorful set of $k$ cycles in $T$ (if one exists).
\end{proof}
Using the standard technique of derandomization of color coding based algorithms \cite{color-coding,fpt-book,splitters}, we have the following result by taking $m=|A(T)|$.
\begin{proposition}[\cite{color-coding,fpt-book,splitters}]
\label{prop:perfect-family}
Given integers $m,\ell \geq 1$, there is a family $\mathcal{F}_{m,\ell}$ of coloring functions $\chi:A(T) \rightarrow [\ell]$ of size $e^{\ell} {\ell}^{\mathcal{O}(\log \ell)} \log m$ that can be constructed in $e^{\ell} {\ell}^{\mathcal{O}(\log \ell)} m \log m$ time satisfying the following property: for every set $E \subseteq A(T)$ of size $\ell$, there is a function $\chi \in \mathcal{F}_{m,\ell}$ such that $\chi(e) \neq \chi(e')$ for any two distinct arcs $e,e' \in E$.
\end{proposition}
Then, we have the following result.
\begin{theorem}
\label{thm:col-code-algo}
\textsc{ACT} can be solved in $\mathcal{O}^\star(2^{\mathcal{O}(k^2 \log k)})$ time.
\end{theorem}
\begin{proof}
Consider an instance $\mathcal{I}=(T,k)$ of \textsc{ACT}. Let $\ell=2k^2+k$. First, we compute the family $\mathcal{F}_{m,\ell}$ of $e^{\ell} {\ell}^{\mathcal{O}(\log \ell)} \log m$ coloring functions using Proposition \ref{prop:perfect-family} where $m$ is the number of arcs in $T$. Then, for each coloring function $\chi:A(T) \rightarrow [\ell]$ in $\mathcal{F}_{m,\ell}$, we determine if $T$ has a colorful set of $k$ cycles using Lemma \ref{lem:colorful-cycles}. Due to the properties of $\mathcal{F}_{m,\ell}$ guaranteed by Proposition \ref{prop:perfect-family}, it follows that $\mathcal{I}$ is a yes-instance if and only if $T$ has a set of $k$ cycles that is colorful with respect to at least one of the coloring functions. The overall running time is $\mathcal{O}^\star(2^{\mathcal{O}(k^2 \log k)})$.
\end{proof}
Observe that the running time of the algorithm to find a colorful set of $k$ cycles can be improved to $2^\ell n^{\mathcal{O}(1)}$ by employing a standard dynamic programming scheme. This will result in an $\mathcal{O}^\star(2^{\mathcal{O}(k^2)})$ time algorithm for \textsc{ACT}. However, we skip the details of the same as we will describe an $\mathcal{O}^\star(2^{\mathcal{O}(k \log k)})$ time algorithm for \textsc{ACT} in Section \ref{sec:fpt-ker2}.
\subsection{A Polynomial Kernel}
Now, we show that \textsc{ACT} admits a polynomial kernel. We use Theorem \ref{thm:lin-ep} to describe a quadratic vertex kernel.
\begin{theorem}
\label{thm:cp-quad-kernel}
\textsc{ACT} admits a kernel with $\mathcal{O}(k^2)$ vertices.
\end{theorem}
\begin{proof}
Let $(T,k)$ denote an instance of \textsc{ACT}. From Theorem \ref{thm:lin-ep}, we know that $T$ has either $k$ arc-disjoint triangles or a feedback arc set $F$ of size at most $6(k-1)$. In the former case, we return a trivial yes-instance of constant size as the kernel. In the latter case, $S=V(F)$ is a feedback vertex set of $T$ of size at most $12k$. Let $D$ denote the transitive tournament $T-S$ and $\delta$ denote its unique topological ordering. Observe that for each $v \in S$, the subtournament of $T$ induced by $V(D) \cup \{v\}$ is also transitive. If there is a cycle in $D \cup \{v\}$, then this cycle (which is also a cycle in $T$) has no arc from $F$ leading to a contradiction.
For each $v \in S$, let $R(v)$ be the set of first (with respect to $\delta$) $2k+1$ vertices in $N^{+}(v)$. Let $T'$ be the subtournament of $T$ induced by $S \cup \{R(v) \mid v \in S\}$. Clearly, $T'$ has $\mathcal{O}(k^2)$ vertices. We claim that $(T',k)$ is the required kernel of $(T,k)$. We need to show that $T$ has $k$ arc-disjoint cycles if and only if $T'$ has $k$ arc-disjoint cycles. The reverse direction of the claim holds trivially. Let us now prove the forward direction. Suppose $T$ has a set of $k$ arc-disjoint cycles. Among all such sets, let $\mathcal{C}$ be one that minimizes $\sum_{C \in \mathcal{C}} |V(C) \cap (V(T) \setminus V(T'))|$. Suppose there is a cycle $C$ in $\mathcal{C}$ that is not in $T'$. Then, there is a vertex $v_i \in V(C)$ that is not in $T'$. As argued earlier, any cycle in $T$ has at least two vertices from $S$. Let $x$ and $y$ be two such vertices in $C$ where $(x,v_1,\dots,v_i,\dots,v_q,y)$ is a path in $C$ from $x$ to $y$ with internal vertices from $V(D)$.
The subtournaments $\widehat{T}=D \cup \{x\}$ and $\widetilde{T}=D \cup \{y\}$ are transitive with unique topological orderings $\sigma$ and $\pi$, respectively. Observe that for all distinct $u,v \in V(D)$, $\pi(u)<\pi(v)$ if and only if $\sigma(u)<\sigma(v)$. As $(x,v_1,\dots,v_i,\dots,v_q)$ is a path in $\widehat{T}$, it follows that $\sigma(x)<\sigma(v_j)$ for each $j \in [q]$. Similarly, as $(v_1,\dots,v_i,\dots,v_q,y)$ is a path in $\widetilde{T}$, we have $\pi(y)>\pi(v_j)$ for each $j \in [q]$. As $v_i \notin V(T')$, it follows that $v_i \notin R(x)$ and $|R(x)| = 2k+1$. Then, there is at least one vertex $z$ in $R(x)$ such that the arcs $(x,z)$ and $(z,y)$ are not in any cycle in $\mathcal{C}$. Now, $\sigma(z)<\sigma(v_i)$ as $z,v_i \in N^{+}(x)$, $v_i \notin R(x)$ and $z \in R(x)$. Thus, we have $\pi(z)<\pi(v_i)$. As $\pi(v_i)<\pi(y)$, it follows that $(z,y) \in A(T)$ as $\pi(z)<\pi(y)$. Then, by replacing the path $(x,v_1,\dots,v_i,\dots,v_q,y)$ by $(x,z,y)$, we obtain another set $\mathcal{C}'$ of $k$ arc-disjoint cycles such that $\sum_{C \in \mathcal{C}} |V(C) \cap (V(T) \setminus V(T'))|>\sum_{C \in \mathcal{C}'} |V(C) \cap (V(T) \setminus V(T'))|$. However, this leads to a contradiction by the choice of $\mathcal{C}$.
\end{proof}
\subsection{A Linear Vertex Kernel}
We show that the linear kernelization described in \cite{jcss11} for \textsc{Feedback Arc Set in Tournaments} also leads to a linear kernelization for our problem. In order to describe the kernel, we need to state some terminology defined in \cite{jcss11}. Let $T$ be a tournament on $n$ vertices. First, we apply the following reduction rule.
\begin{reduction rule}
\label{rule1}
If a vertex $v$ is not in any cycle, then delete $v$ from $T$.
\end{reduction rule}
This rule is clearly safe as our goal is to find $k$ cycles and $v$ cannot be in any of them. To describe our next rule, we need to state some terminology and a lemma known from \cite{jcss11}. For an ordering $\sigma$ of $V(T)$, let $T_\sigma$ denote the tournament $T$ whose vertices are ordered according to $\sigma$. Clearly, $V(T_\sigma)=V(T)$ and $A(T_\sigma)=A(T)$ since $T$ and $T_\sigma$ denote the same tournament. An arc $(u,v) \in A(T_\sigma)$ is called a {\em back arc} if $\sigma(u)>\sigma(v)$ and it is called a {\em forward arc} otherwise. An {\em interval} is a consecutive set of vertices in $T_\sigma$.
\begin{lemma}[\cite{jcss11}]\footnote{Lemma \ref{lem:safe-part} is Lemma 3.9 of \cite{jcss11} that has been rephrased to avoid the use of several definitions and terminology introduced in \cite{jcss11}.}
\label{lem:safe-part}
Let $T_\sigma$ be an ordered tournament on which Reduction Rule \ref{rule1} is not applicable. Let $B$ denote the set of back arcs in $T_\sigma$ and $E$ denote the set of arcs in $T_\sigma$ with endpoints in different intervals. If $|V(T_\sigma)| \geq 2 |B|+1$, then there exists a partition $\mathcal{J}$ of $V(T_\sigma)$ into intervals with the following properties that can be computed in polynomial time.
\begin{itemize}
\item There is at least one arc $e=(u,v) \in A(T)$ with $e \in B \cap E$.
\item There are $|B \cap E|$ arc-disjoint cycles using only arcs in $E$.
\end{itemize}
\end{lemma}
Our reduction rule that is based on this lemma is as follows.
\begin{reduction rule}
\label{rule2}
Let $T_\sigma$ be an ordered tournament on which Reduction Rule \ref{rule1} is not applicable. Let $B$ denote the set of back arcs in $T_\sigma$ and $E$ denote the set of arcs in $T_\sigma$ with endpoints in different intervals. Let $\mathcal{J}$ be a partition of $V(T_\sigma)$ into intervals satisfying the properties specified in Lemma \ref{lem:safe-part}. Reverse all arcs in $B \cap E$ and decrease $k$ by $|B \cap E|$.
\end{reduction rule}
\begin{lemma}
Reduction Rule \ref{rule2} is safe.
\end{lemma}
\begin{proof}
Let $T'_\sigma$ be the tournament obtained from $T_\sigma$ by reversing all arcs in $B \cap E$. Suppose $T'_\sigma$ has $k-|B \cap E|$ arc-disjoint cycles. Then, it is guaranteed that each such cycle is completely contained in an interval. This is due to the fact that $T'_\sigma$ has no back arc with endpoints in different intervals. Indeed, if a cycle in $T'_\sigma$ uses a forward (back) arc with endpoints in different intervals, then it also uses a back (forward) arc with endpoints in different intervals. It follows that for each arc $(u,v) \in E$, neither $(u,v)$ nor $(v,u)$ is used in these $k-|B \cap E|$ cycles. Hence, these $k-|B \cap E|$ cycles in $T'_\sigma$ are also cycles in $T_\sigma$. Then, we can add a set of $|B \cap E|$ cycles obtained from the second property of Lemma \ref{lem:safe-part} to these $k-|B \cap E|$ cycles to get $k$ cycles in $T_\sigma$. Conversely, consider a set of $k$ cycles in $T_\sigma$. As argued earlier, we know that the number of cycles that have an arc that is in $E$ is at most $|B \cap E|$. The remaining cycles (at least $k-|B \cap E|$ of them) do not contain any arc that is in $E$, in particular, they do not contain any arc from $B \cap E$. Therefore, these cycles are also cycles in $T'_\sigma$.
\end{proof}
\begin{theorem}
\label{thm:cp-linear-kernel}
\textsc{ACT} admits a kernel with $\mathcal{O}(k)$ vertices.
\end{theorem}
\begin{proof}
Let $(T,k)$ denote the instance obtained from the input instance by applying Reduction Rule \ref{rule1} exhaustively. From Lemma \ref{thm:lin-ep}, we know that either $T$ has $k$ arc-disjoint triangles or has a feedback arc set of size at most $6(k-1)$ that can be obtained in polynomial time. In the first case, we return a trivial yes-instance of constant size as the kernel. In the second case, let $F$ be the feedback arc set of size at most $6(k-1)$ of $T$. Let $\sigma$ denote a topological ordering of the vertices of the directed acyclic graph $T-F$. As $V(T-F)=V(T)$, $\sigma$ is an ordering of $V(T)$ such that $T_\sigma$ has at most $6(k-1)$ back arcs. If $|V(T_\sigma)| \geq 12k-11$, then from Lemma \ref{lem:safe-part}, there is a partition of $V(T_\sigma)$ into intervals with the specified properties. Therefore, Reduction Rule \ref{rule2} is applicable (and the parameter drops by at least 1). When we obtain an instance where neither of the Reduction Rules \ref{rule1} and \ref{rule2} is applicable, it follows that the tournament in that instance has at most $12k$ vertices.
\end{proof}
\subsection{A Faster FPT Algorithm}
Here, we show that \textsc{ACT} can be solved in $\mathcal{O}^\star(2^{\mathcal{O}(k \log k)})$ time. The idea is to reduce the problem to the following \textsc{Arc-Disjoint Paths} problem in directed acyclic graphs.
\defparprob{Arc-Disjoint Paths}
{A digraph $D$ on $n$ vertices and $k$ ordered pairs $(s_1,t_1),\dots,(s_k,t_k)$ of vertices of $D$.}
{$k$}
{Do there exist arc-disjoint paths $P_1,\dots,P_k$ in $D$ such that $P_i$ is a path from $s_i$ to $t_i$ for each $i \in [k]$?}
On directed acyclic graphs, \textsc{Arc-Disjoint Paths} is known to be \NP-complete \cite{dag-edp-npc}, \W[1]-hard \cite{dag-edp} and solvable in $n^{\mathcal{O}(k)}$ time \cite{dag-edp-xp}. Despite its fixed-parameter intractability, we will show that we can use the $n^{\mathcal{O}(k)}$ algorithm to describe another (and faster) \FPT\ algorithm for \textsc{ACT}.
\begin{theorem}
\label{thm:reduc-cp-edp}
\textsc{ACT} can be solved in $\mathcal{O}^\star(2^{\mathcal{O}(k \log k)})$ time.
\end{theorem}
\begin{proof}
Consider an instance $(T,k)$ of \textsc{ACT}. Using Theorem \ref{thm:cp-linear-kernel}, we obtain a kernel $\mathcal{I}=(\widehat{T},\widehat{k})$ such that $\widehat{T}$ has $\mathcal{O}(k)$ vertices. Further, $\widehat{k} \leq k$. By definition, $(T,k)$ is a yes-instance if and only if $(\widehat{T},\widehat{k})$ is a yes-instance. Using Theorem \ref{thm:lin-ep}, we know that $\widehat{T}$ either contains $\widehat{k}$ arc-disjoint triangles or has a feedback arc set of size at most $6(\widehat{k}-1)$ that can be obtained in polynomial time. If Theorem \ref{thm:lin-ep} returns a set of $\widehat{k}$ arc-disjoint triangles in $\widehat{T}$, then we declare that $(T,k)$ is a yes-instance.
Otherwise, let $\widehat{F}$ be the feedback arc set of size at most $6(\widehat{k}-1)$ returned by Theorem \ref{thm:lin-ep}. Let $D$ denote the (acyclic) digraph obtained from $\widehat{T}$ by deleting $\widehat{F}$. Observe that $D$ has $\mathcal{O}(k)$ vertices. Suppose $\widehat{T}$ has a set $\mathcal{C}=\{C_1,\dots,C_{\widehat{k}}\}$ of $\widehat{k}$ arc-disjoint cycles. For each $C \in \mathcal{C}$, we know that $A(C) \cap \widehat{F} \neq \emptyset$ as $\widehat{F}$ is a feedback arc set of $\widehat{T}$. We can guess that subset $F$ of $\widehat{F}$ such that $F=\widehat{F} \cap A(\mathcal{C})$. Then, for each cycle $C_i \in \mathcal{C}$, we can guess the arcs $F_i$ from $F$ that it contains and also the order $\sigma_i$ in which they appear. This information is captured as a partition $\mathcal{F}$ of $F$ into $\widehat{k}$ sets, $F_1$ to $F_{\widehat{k}}$ and the set $\{\sigma_1,\dots,\sigma_{\widehat{k}}\}$ of permutations where $\sigma_i$ is a permutation of $F_i$ for each $i \in [\widehat{k}]$. Any cycle $C_i$ that has $F_i \subseteq F$ contains a $(v,x)$-path between every pair $(u,v)$, $(x,y)$ of consecutive arcs of $F_i$ with arcs from $A(D)$. That is, there is a path from $\head(\sigma_i^{-1}(j))$ and $\tail(\sigma_i^{-1}((j+1) \mod |F_i|))$ with arcs from $D$ for each $j \in [|F_i|]$. The total number of such paths in these $\widehat{k}$ cycles is $\mathcal{O}(|F|)$ and the arcs of these paths are contained in $D$ which is a (simple) directed acyclic graph.
The number of choices for $F$ is $2^{|\widehat{F}|}$ and the number of choices for a partition $\mathcal{F}=\{F_1,\dots,F_{\widehat{k}}\}$ of $F$ and a set $X=\{\sigma_1,\dots,\sigma_{\widehat{k}}\}$ of permutations is $2^{\mathcal{O}(|\widehat{F}| \log |\widehat{F}|)}$. Once such a choice is made, the problem of finding $\widehat{k}$ arc-disjoint cycles in $\widehat{T}$ reduces to the problem of finding $\widehat{k}$ arc-disjoint cycles $\mathcal{C}=\{C_1,\dots,C_{\widehat{k}}\}$ in $\widehat{T}$ such that for each $1 \leq i \leq \widehat{k}$ and for each $1 \leq j \leq |F_i|$, $C_i$ has a path $P_{ij}$ between $\head(\sigma_i^{-1}(j))$ and $\tail(\sigma_i^{-1}((j+1) \mod |F_i|))$ with arcs from $D=\widehat{T}-\widehat{F}$. This problem is essentially finding $r=\mathcal{O}(|\widehat{F}|)$ arc-disjoint paths in $D$ and can be solved in ${|V(D)|}^{\mathcal{O}(r)}$ time using the algorithm in \cite{dag-edp-xp}. Therefore, the overall running time of the algorithm is $\mathcal{O}^\star(2^{\mathcal{O}(k \log k)})$ as $|V(D)|=\mathcal{O}(k)$ and $r=\mathcal{O}(k)$.
\end{proof}
|
{
"redpajama_set_name": "RedPajamaArXiv"
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| 4,067
|
{"url":"http:\/\/math.stackexchange.com\/questions\/674615\/how-find-this-prod-n-2-infty-left1-frac1n6-right","text":"# How find this $\\prod_{n=2}^{\\infty}\\left(1-\\frac{1}{n^6}\\right)$\n\nHow find this $$\\prod_{n=2}^{\\infty}\\left(1-\\dfrac{1}{n^6}\\right)$$\n\nI think we can find this value have closed form $$\\prod_{n=2}^{\\infty}\\left(1-\\dfrac{1}{n^{2k}}\\right)$$ since $$1-\\dfrac{1}{n^6}=\\left(1-\\dfrac{1}{n^3}\\right)\\left(1+\\dfrac{1}{n^3}\\right)$$ so I think we must find this $$\\prod_{n=2}^{\\infty}\\left(1-\\dfrac{1}{n^3}\\right)$$ and $$\\prod_{n=2}^{\\infty}\\left(1+\\dfrac{1}{n^3}\\right)$$\n\nThank you\n\n-\nWhat have you tried? You might consider finding a common denominator and factoring. Or you could try taking log of the product. \u2013\u00a0SpamIAm Feb 13 '14 at 4:26\n\u2013\u00a0Mhenni Benghorbal Feb 13 '14 at 5:20\n\nAs Mhenni Benghorbal mentioned, there is a similar problem which has been treated yesterday and I shall use a similar approach to the one he proposed.\n\n$$\\prod_{n=2}^{m}\\left(1-\\dfrac{1}{n^3}\\right)=\\frac{\\cosh \\left(\\frac{\\sqrt{3} \\pi }{2}\\right) \\Gamma \\left(m-\\frac{i \\sqrt{3}}{2}+\\frac{3}{2}\\right) \\Gamma \\left(m+\\frac{i \\sqrt{3}}{2}+\\frac{3}{2}\\right)}{3 \\pi m^3 \\Gamma (m)^2}$$ and $$\\prod_{n=2}^{m}\\left(1+\\dfrac{1}{n^3}\\right)=\\frac{(m+1) \\cosh \\left(\\frac{\\sqrt{3} \\pi }{2}\\right) \\Gamma \\left(m-\\frac{i \\sqrt{3}}{2}+\\frac{1}{2}\\right) \\Gamma \\left(m+\\frac{i \\sqrt{3}}{2}+\\frac{1}{2}\\right)}{2 \\pi \\Gamma (m+1)^2}$$ So, the product, from $n=2$ to $n=m$, write $$\\frac{(m+1) \\cosh ^2\\left(\\frac{\\sqrt{3} \\pi }{2}\\right) \\Gamma \\left(m-\\frac{i \\sqrt{3}}{2}+\\frac{1}{2}\\right) \\Gamma \\left(m-\\frac{i \\sqrt{3}}{2}+\\frac{3}{2}\\right) \\Gamma \\left(m+\\frac{i \\sqrt{3}}{2}+\\frac{1}{2}\\right) \\Gamma \\left(m+\\frac{i \\sqrt{3}}{2}+\\frac{3}{2}\\right)}{6 \\pi ^2 m^5 \\Gamma (m)^4}$$ If $m$ goes to infinity, the limit is then $$\\frac{1+\\cosh \\left(\\sqrt{3} \\pi \\right)}{12 \\pi ^2} =\\dfrac{\\cosh^2\\left(\\pi\\dfrac{\\sqrt3}2\\right)}{6\\pi^2}$$ as shown by Lucian.\n\n-\nThe connection between the Gammas of the odd exponents and the sines of the even ones is the result of Euler's reflection formula. \u2013\u00a0Lucian Feb 13 '14 at 6:39\n@Lucian. Thank you for that since I learn a lot from you. I must say that I really like your answer. I thought that extending what was done on yesterday was also an interesting execrcise. Cheers. \u2013\u00a0Claude Leibovici Feb 13 '14 at 7:00\n\n$$P_6(x)=\\prod_{n=1}^\\infty\\bigg(1-\\frac{x^6}{\\pi^6n^6}\\bigg)=\\frac{\\sin x}{x^3}\\cdot\\frac{\\cosh\\big(x\\sqrt3\\big)-\\cos x}2\\quad$$\n\nSee Basel problem. For $x\\to\\pi$, use l'Hopital after first dividing through $1-\\dfrac{x^6}{\\pi^6}$ , in order to finally arrive at $\\dfrac{\\cosh^2\\left(\\pi\\dfrac{\\sqrt3}2\\right)}{6\\pi^2}$ . In general, $P_{2k}(x)=s_k\\displaystyle\\prod_{j=0}^{k-1}\\frac{\\sin\\Big((-1)^{j\/k}x\\Big)}x$ , where $s_k$ forms a cycle of length four, $(+,+,-,-)$, starting with $k=1$. And, of course, when $x\\to\\pi$, use l'Hopital's rule after first dividing through $1-\\bigg(\\dfrac x\\pi\\bigg)^{2k}$, just as before. Hope this helps.\n\n-","date":"2016-07-28 12:41:24","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9360881447792053, \"perplexity\": 300.1453318680615}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-30\/segments\/1469257828282.32\/warc\/CC-MAIN-20160723071028-00240-ip-10-185-27-174.ec2.internal.warc.gz\"}"}
| null | null |
Q: How to correct a KeyError or continue after it I am using this code to follow a twitter account and if a specific word is tweeted it prints the text and the tweet id then deletes the tweet. The code below works as in it finds the word 'hi' when tweeted, prints it and the id, and deletes the tweet. The problem is that immediately after it deletes the tweet I get the error- KeyError 'text' . and the script stops. I'm new to python and don't know much about dictionaries, exceptions, or KeyError. How do I make an exception to ignore this error and continue or create a dictionary and additional code so no error happens?
import time
from twython import TwythonStreamer
from twython import Twython
# Twitter application authentication
APP_KEY = ''
APP_SECRET = ''
OAUTH_TOKEN = ''
OAUTH_TOKEN_SECRET = ''
twitter = Twython(APP_KEY, APP_SECRET, OAUTH_TOKEN, OAUTH_TOKEN_SECRET)
# Setup callbacks from Twython Streamer
class TweetStreamer(TwythonStreamer):
def on_success (self, data):
if (data['text']) == ('hi'):
print (data['text'])
print (data['id'])
time.sleep(2)
twitter.destroy_status(id=data['id'])
print ('Tweet was deleted')
def on_error(self, status_code, data):
print (status_code)
# Create streamer
try:
stream = TweetStreamer(APP_KEY, APP_SECRET, OAUTH_TOKEN, OAUTH_TOKEN_SECRET)
stream.statuses.filter(follow = '')
except KeyboardInterrupt:
print ('Process manually stopped')
Full error:
Traceback (most recent call last):
File "C:\Users\MainUser\Desktop\Python Scripts\twitterstreamuser.py", line 35, in <module>
stream.statuses.filter(follow = '')
File "C:\Users\MainUser\AppData\Local\Programs\Python\Python35-32\lib\site-packages\twython\streaming\types.py", line 66, in filter
self.streamer._request(url, 'POST', params=params)
File "C:\Users\MainUser\AppData\Local\Programs\Python\Python35-32\lib\site-packages\twython\streaming\api.py", line 154, in _request
if self.on_success(data): # pragma: no cover
File "C:\Users\MainUser\Desktop\Python Scripts\twitterstreamuser.py", line 20, in on_success
if (data['text']) == ('hi'):
KeyError: 'text'
A: You try to get "text" from data
data["text"]
but data doesn't have key "text"
You can check if data has key "text"
if 'text' in data:
if data['text'] == 'hi':
print(data['text'])
or shorter
if 'text' in data and data['text'] == 'hi':
print(data['text'])
or use data.get("text") (or data.get("text", "default text")) to get "text" or "default text" (or None)
msg = data.get("text") # it gives data["text"] or `None`
if msg == 'hi':
print(msg)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 5,284
|
Whitening one's teeth is a dream come true for many—most especially if you have that disgusting yellowish tinge on your teeth. Getting that perfect set of teeth apt for the billboards and TV ads is such a joy to a person…however, not everyone can have that blissful set of teeth. In fact, most of the celebrities we see on TV and movies do not have those gleaming teeth prior to entering show business. They may have already undergone a lot of dental treatments to attain that pearly white effect on their teeth.
Predisposing diseases and treatments like chemotherapy and radiation therapy which removes the white appearance of the teeth enamel.
During dental laser whitening, the coating of the teeth, more specifically the enamel can start to thin out. This enamel is a protective covering of the teeth necessary to avoid sensitivity. As such, teeth sensitivity to hot and cold drinks and foods are common dental laser whitening side effects.
Tooth decay is one of the most dreaded dental laser whitening side effects because of a reason similar to what was mentioned above—the thinning of the teeth enamel. One of the purposes of our tooth enamel is to protect our teeth from bacteria that cause tooth decay. With a thin enamel, the protection from tooth decay also weakens.
Teeth whitening should be done with extreme care, especially when using a laser. Laser teeth whitening is done through radiation of the teeth using laser so if a certain group of teeth has been exposed to too much laser for quite a time longer than the other teeth, uneven whitening comes as common dental laser whitening side effects.
Undergoing dental laser whitening can also lead to gum problems since the laser is not only contained to the teeth. The gums can also be exposed to a laser which can cause sensitivity, soreness and even ulcerations of the gums and inner cheeks.
Undergoing teeth whitening should not be a hasty decision. Apart from the costs that you may incur, you also have to be aware of the dental laser whitening side effects that may come along with the procedure.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 1,746
|
Oscar Nomnations
89th Academy Awards, ABC, Academy Awards, All About Eve, Andrew Garield, Arrival, Barry Jenkins, Captain Fantastic, Casey Afleck, Damien Chazelle, Deadpool, Denis Villeneuve, Denzel Washington, Dev Patel, Elle, emma stone, Fantastic Beasts and Where to Find Them, Fences, Florence Foster Jenkins, Hacksaw Ridge, Hell or High Water, Hidden Figures, Isabelle Huppert, Jackie, Jeff Bridges, Kenneth Lonergan, Kubo and the Two Stirngs, La La Land, Lione, loving, Lucas Hedges, Mahershala Ali, Manchester By the Sea, Mel Gibson, Meryl Streep, Michael Shannon, Michelle Williams, Moana, Moonlight, Naomi Harris, Natalie Portman, Nichole Kidman, Nocturnal Animals, nominations, Octavis Spencer, oscars, Rogue One: A Star Wars Story, Ruth Negga, ryan gosling, Ryan Reynolds, Titanic, Viggo Mortensen, Viola Davis, Zootopia
La La Land Gets 14 Oscar Nominations
La La Land received 14 nominations for the 89th Academy Awards that were announced Tuesday morning. The film tied Titantic and All About Eve for the most nominations for any movie in history. The musical's nominations include best picture, stars Ryan Gosling and Emma Stone, director Damien Chazelle and two for best song.
The Academy Award nominations were announced Tuesday morning with La La Land received 14.
(Photo: Academy of Motion Picture Arts and Science)
Other standouts in this year's Oscars, which will be announced Feb. 26 at 8:30 p.m. on ABC, include Moonlight, which got nominated in eight categories, including best picture, director, supporting actor and supporting actress. Also up for eight awards is Arrival, including best picture, director, production design, cinematography and sound awards.
Receiving six nominations each were Hacksaw Ridge, Lion and Manchester by the Sea. Mel Gibson was nominated for his direction of Hacksaw Ridge after years as a Hollywood pariah, but he faces stiff competition. Fences and Hell or High Water each received four nods.
Fantastic Beasts and Where to Find Them and Rogue One: A Star Wars Story were snubbed in the nominations with each getting only two. The former got nods in the costume design and production design categories, while the later was nominated in sound mixing and visual effects. Despite getting nominations for best comedy/musical and best actor for Ryan Reynolds, Deadpool was shut out of the Academy Award nominations completely, even with its Oscar campaign and much love from the critics.
Below is the list of nominations:
Actor in a Leading Role
Casey Affleck, Manchester by the Sea
Andrew Garfield, Hacksaw Ridge
Ryan Gosling, La La Land
Viggo Mortensen, Captain Fantastic
Denzel Washington, Fences
Actress in a Leading Role
Isabelle Huppert, Elle
Ruth Negga, Loving
Natalie Portman, Jackie
Emma Stone, La La Land
Meryl Streep, Florence Foster Jenkins
Actor in a Supporting Role
Mahershala Ali, Moonlight
Jeff Bridges, Hell or High Water
Lucas Hedges, Manchester by the Sea
Dev Patel, Lion
Michael Shannon, Nocturnal Animals
Actress in a Supporting Role
Viola Davis, Fences
Naomie Harris, Moonlight
Nichole Kidman, Lion
Octavia Spencer, Hidden Figures
Michelle Williams, Manchester by the Sea
Animated Feature Film
My Life As a Zucchini
The Red Turtle
Denis Villeneuve, Arrival
Mel Gibson, Hacksaw Ridge
Damien Chazelle, La La Land
Kenneth Lonergan, Manchester by the Sea
Barry Jenkins, Moonlight
Writing (Adapted Screenplay)
Eric Heisserer, Arrival
August Wilson, Fences
Allison Schroeder and Theodore Melfi, Hidden Figures
Luke Davies, Lion
Barry Jenkins, screenplay; Tarell Alvin McCraney, story, Moonlight
Writing (Original Screenplay)
Taylor Sheridan, Hell or High Water
Yorgos Lanthimos and Efthimis Filippou, The Lobster
Mike Mills, 20th Century Women
Documentary (Feature)
Fire at Sea
Life, Animated
Documentary (Short Subject)
Joes's Violin
Watani: My Homeland
The White Helments
Foreign Language Film
Land of Mine, Denmark
A Man Called Ove, Sweden
The Salesman, Iran
Tanna, Austrailia
Toni Erdmann, Germany
Short Film (Animated)
Blind Vaysha
Pear Cider and Cigarettes
Short Film (Live Action)
Ennemis Interieurs
La Femme et le TGV
Silent Nights
Joe Walker, Arrival
John Gilbert, Hacksaw Ridge
Jake Roberts, Hell or High Water
Tom Cross, La La Land
Nat Sanders and Joi McMillon, Moonlight
Cinematograpny
Bradford Young, Arrival
Linus Sandgren, La La Land
Greig Fraser, Lion
James Laxton, Moonlight
Rodrigo Prieto, Silence
Joanna Johnston, Allied
Colleen Atwood, Fantastic Beasts and Where to Find Them
Consolata Boyle, Florence Foster Jenkins
Madeline Fontaine, Jackie
Mary Zophres, La La Land
Makeup and Hairstyling
Eva von Bahr and Love Larson, A Man Called Ove
Joel Harlow and Richard Alonzo, Star Trek Beyond
Alessandro Bertolazzi, Giorgio Gregorini and Christopher Nelson, Suicide Squad
Music (Original Score)
Mica Levi, Jackie
Justin Hurwitz, La La Land
Dustin O'Halloran and Hauschka, Lion
Nicholas Britell, Moonlight
Thomas Newman, Passengers
Music (Original Song)
Audition (The Fools Who Dream) from La La Land
Can't Stop the Feeling from Trolls
City of Stars from La La Land
The Empty Chair from Jim: The James Foley Story
How Far I'll Go from Moana
Production Design: Patrice Vermette; Set Decoration: Paul Hotte, Arrival
Production Design: Stuart Craig; Set Decoration: Anna Pinnock, Fantastic Beasts and Where to Find Them
Production Design: Jess Gonchor; Set Decoration: Nancy Haigh, Hail! Ceasar
Production Design: David Wasco; Set Decoration: Sandy Reynolds-Wasco, La La Land
Production Design: Guy Hendrix Dyas; Set Decoration: Gene Serdena, Passengers
Sylvain Bellemare, Arrival
Wylie Stateman and Renée Tondelli, Deepwater Horizon
Robert Mackenzie and Andy Wright, Hacksaw Ridge
Ai-Ling Lee and Mildred Iatrou Morgan, La La Land
Alan Robert Murray and Bub Asman, Sully
Bernard Gariépy Strobl and Claude La Haye, Arrival
Kevin O'Connell, Andy Wright, Robert Mackenzie and Peter Grace, Hacksaw Ridge
Andy Nelson, Ai-Ling Lee and Steve A. Morrow, La La Land
David Parker, Christopher Scarabosio and Stuart Wilson, Rogue One: A Star Wars Story
Greg P. Russell, Gary Summers, Jeffrey J. Haboush and Mac Ruth, 13 Hours: The Secret Soldiers of Benghazi
Craig Hammack, Jason Snell, Jason Billington and Burt Dalton, Deepwater Horizon
Stephane Ceretti, Richard Bluff, Vincent Cirelli and Paul Corbould, Doctor Strange
Robert Legato, Adam Valdez, Andrew R. Jones and Dan Lemmon, The Jungle Book
Steve Emerson, Oliver Jones, Brian McLean and Brad Schiff, Kubo and the Two Strings
John Knoll, Mohen Leo, Hal Hickel and Neil Corbould, Rogue One: A Star Wars Story
Editor-in-Chief Mark Heckathorn is a journalist, movie buff and foodie. He oversees DC on Heels editorial operations as well as strategic planning and staff development. Reach him with story ideas or suggestions at dcoheditor (at) gmail (dot) com.
More posts by the Author »
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 7,226
|
package com.hyrax.microservice.project.rest.api.domain.response.wrapper;
import com.hyrax.microservice.project.rest.api.domain.response.TeamResponse;
import lombok.Builder;
import lombok.Data;
import java.util.List;
@Data
@Builder
public class TeamResponseWrapper {
private final List<TeamResponse> teamResponses;
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 733
|
DIABOLIQUE (Blu-Ray) - #35
I first saw Henri-Georges Clouzot's 1955 thriller Diabolique by accident. It must have been 1995 or thereabouts, back when I was editing a comic book called Triple-X . The creators, Arnold and Jacob Pander, kept referencing a movie called Diabolique as being an inspiration for the Eurotrash look of their futuristic revolutionaries. Intrigued, I went out and rented a copy of the film. When I was done, I was kind of confused. What did a black-and-white shocker about two women murdering the man who done them wrong have to do with hooded tricksters and underground newspapers?
Well, nothing, as it turns out. I had rented the wrong movie. They meant Danger: Diabolik , which my neighborhood Hollywood Video didn't have. Lucky for me, though, because I might not have otherwise found this Diabolique, or when I had, I might not have been able to go into it as a blank slate. It's a movie that is full of surprises, delivering both emotional suckerpunches and truly gruesome cinematic scares. If you haven't seen it and know nothing more about it, I kind of want to dissuade you from reading further. Just go get it. You'll thank me later.
Diabolique is one hell of a genre picture, one that plays with the twisty conventions of a Hitchcock movie, wringing all of the horror out of the scandalous scenario it possibly can, while also building a movie full of pro-women proclamations and enough double-crosses to compete with the best film noir. The director's wife Vera Clouzot stars in Diabolique as Christina Delassalle, a language teacher whose dowery subsidizes the school run by her no-good husband Michel (Paul Meurisse). Michel is cheating on Christina with another teacher, the cynical blonde Nicole (Simone Signoret). The two women, the wife and the mistress, work side by side, giving them opportunity to compare notes. Both have plenty of reason to hate Michel. At the start of Diabolique, Nicole has to wear dark sunglasses because Michel has been smacking her around and she has a black eye. Oh, and did I mention that Christina is sickly with a heart condition? Yeah, if Michel survives this movie, he's got one hell of a career in politics ahead of him.
And surviving this movie will be his challenge. The ladies are looking to take the opportunity of a three-day weekend to enact a murder plot and take care of Michel once and for all. That's right, the sisters are going to do it for themselves, setting aside their differences to take care of the one thing that plagues them both. Their scheme involves poisoning and drowning--the rest, well, keep it schtum. The ladies play to type, with Christina being nervous and increasingly unhinged and Nicole being more calculating and determined. Both are excellent and they play off each other well, particularly as Christina's nerves start to affect Nicole. Simone Signoret, we know, is good at being in charge, but she's also excellent at falling apart.
On the same track, Paul Meurisse plays a real convincing creep. He is sneering and snide and just nasty enough that he never really crosses over into caricature; Michel ends up being an honest-to-goodness villain. You'll be rooting for him to get what's coming his way. Diabolique's after-plot involves transporting the body for discovery in a place far from where the ladies have established their alibi--which is when things go weird. Clouzot employs every trick in the book to give his gals a fright. We even tread into some territory straight out of ghost stories. Whose face is that in the window in the class picture? Could it be...?
Diabolique is as tightly wound a film as you're likely to find. It's a remarkable thing watching all the pieces move into place, like observing the interior of a well-designed clock. Gears turn, springs stretch and contract, every element does its part to ring the hour. Clouzot shot on what looks like real sets, choosing interiors with lots of doors and intersecting corridors. He also puts silence to use, letting moments linger, the ambient noise of the surroundings adding to the suspense. The director never tips his hand to what is really going on, there is no winking to the audience, it all comes across with a very straight face. Just try not to get sucked in. Go on, try!
Diabolique was an early entry in the Criterion Collection, and the thirteen-year-old DVD was more than ready for an upgrade. The new high-definition transfer on the Blu-Ray is flat-out gorgeous. The overall image quality is crystal clear, with lovely renderings of black, white, and all the grays in between. Digital noise and scratches have been erased, making for a spotless viewing experience. No distractions here, you can just get caught up in the story, no tsk tsking at DVD glitches.
Amongst the trio of new extras on this disc, which consist mostly of critical commentary by admirers of the film, is an introduction to Diabolique by director Serge Bromberg. Bromberg is the director of the full-length documentary Henri-Georges Clouzot's Inferno, which I was also lucky enough to review on Blu-Ray recently. Though Bromberg's film doesn't dig too deep into Clouzot's earlier filmography, one can easily see how obsession and an attention to detail led to the influential director making such exacting motion pictures. L'enfer, the unfinished film unearthed in Henri-Georges Clouzot's Inferno, was a movie that tried to bring jealousy to life. It starred Romy Schneider as the object of these delusions, and it's a shame no one ever cast her and Simone Signoret to play daughter and mother. I can't be the only one to think they look alike!
Special note should be made of the new packaging of Diabolique. Criterion contracted David Plunkett to provide the awesome illustrations for the cover of the case and also the cover of the interior booklet. The images cross simulated woodcut techniques with classic poster design to create pulpy illos that would be at home on any spinner rack selling dime novels. Check out more from Plunkett and Spur Design at their website.
This disc was provided by the Criterion Collection for purposes of review.
Screen captures are from the DVD, not the Blu-Ray.
Posted by Jamie S. Rich at 11:40 PM
Labels: blu-ray, clouzot, hitchcock
THE GREAT DICTATOR (Blu-Ray) - #565
SIDELINE: LIAR'S KISS by Eric Skillman and Jhomar ...
PALE FLOWER (Blu-Ray) - #564
SOMETHING WILD - #563
SMILES OF A SUMMER NIGHT (Blu-Ray) - #237
KES (Blu-Ray) - #561
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
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Aureliana angustifolia är en potatisväxtart som beskrevs av R. C. Almeida-lafeta. Aureliana angustifolia ingår i släktet Aureliana och familjen potatisväxter. Inga underarter finns listade i Catalogue of Life.
Källor
Potatisväxter
angustifolia
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{
"redpajama_set_name": "RedPajamaWikipedia"
}
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\section{Introduction}
The worldwide network of company ownership provides crucial information
for the systemic analysis of the world
economy~\cite{schweitzer_networks_2009,farmer_complex_2012}. A complete
understanding of its properties and how they are formed has a wide range
of potential applications, including assessment and evasion of systemic
risk~\cite{battiston_debtrank_2012}, collusion and antitrust regulation
\cite{gulati_strategicnetworks_2000,gilo_collusion_2006}, market
monitoring~\cite{diamond_delegated_1984,chirinko_germanbanks_2006}, and
strategic investment~\cite{teece_cooperation_1992}. Recently, Vitali et
al~\cite{vitali_network_2011} inferred the network structure of global
corporate control, using the Orbis 2007 marketing
database~\footnote{\url{http://www.bvdinfo.com/products/company-information/international/orbis}}.
Analyzing its structure, they found a tightly connected ``core'' made of
a small number of large companies (mostly financial institutions) which
control a significant part of the global economy. A central question
which arises is what is the dominant mechanism behind this
centralization of control. The answer is not obvious, since the decision
of firms to buy other firms can be driven by diverse goals: Banks act as
financial intermediaries doing monitoring for uninformed investors
\cite{diamond_delegated_1984,chirinko_germanbanks_2006}, managers can
improve their power by buying other firms instead of paying dividends
\cite{jensen_takeovers_1986}, speculation on stock prices as well as
dividend earnings can be a significant source of
revenue~\cite{modiglia_cost_1958,
porta_dividend_2000,jensen_takeovers_1986}, and companies can have
strategic advantages, e.g.\ due to knowledge
sharing~\cite{teece_cooperation_1992,hamel_strategic_1991,dyer_relational_1998}.
Another possible hypothesis for control centralization is that managers
collude to form influential alliances: Indeed, agents (e.g. board
members) often work for different firms in central
positions~\cite{battiston_boards_2004}. Although all these factors are
likely to play a role, we here investigate a different hypothesis,
namely that a centralized structure may arise spontaneously, as a result
of a simple ``richt-get-richer'' dynamics~\cite{simon_richer_1955},
without any explicit underlying strategy from the part of the
companies. We consider a simple adaptive feedback
mechanism~\cite{gross_adaptive_2008}, which incorporates the indirect
control that companies have on other companies they own, which in turn
increases their buying power. The higher buying power can then be used
to buy portions of more important companies, or a larger number of less
important ones, which further increases their relative control, and
progressively marginalizes smaller companies. We show that this simple
dynamical ingredient suffices to reproduce many of the qualitative
features observed in the real data~\cite{vitali_network_2011}, including
the emergence of a core-periphery structure and the relative portion of
control exerted by the dominating core. Although this does not preclude
the possibility that companies may take advantage and further
consolidate their privileged positions in the network, it does suggest
that deliberate strategizing may not be the dominating factor which
leads to global centralization.
\section{Model description}\label{sec:model}
We consider a network of $N$ companies, where a directed edge between
two nodes $j\to i$ means company $j$ owns a portion of company $i$. The
relative amount of $i$ which $j$ owns is given by the matrix $w_{ij}$
(i.e. the ownership shares), such that $\sum_j w_{ij} = 1$. We note that
it is possible for self-loops to exist, i.e. a company can in principle
buy its own shares. In the following, we describe a model with two main
mechanisms: 1. The evolution of the relative control of companies, given
a static network;
2. The evolution of the network topology via adaptive rewiring of the
edges.
\subsection{Evolution of control}
Here we assume that if $j$ owns $i$, it exerts some influence on $i$ in
a manner which is proportional to $w_{ij}$. If we let $v_j$ describe the
relative amount of control a company $j$ has on other companies, we can
write
\begin{equation}\label{eq:v}
v_j = 1-\alpha + \alpha \sum_i A_{ij} w_{ij} v_i,
\end{equation}
where $A_{ij}$ is the adjacency matrix, the parameter $\alpha$
determines the propagation of control and $1-\alpha$ is an intrinsic
amount of independence between companies~\footnote{Eq.~\ref{eq:v} can be
seen as a weighted version of the Katz centrality
index~\cite{katz_new_1953}, which is one of many ways of measuring the
relative centrality of nodes in a directed network, such as
PageRank~\cite{page_PageRank_1999} and
HITS~\cite{kleinberg_authoritative_1999}. It converges for $0\leq\alpha<
1$ and we enforce normalization with $\sum_i v_i=N$.}. We further assume
that the control value $v_j$ directly affects other features such as
profit margins, and overall market influence, such that the buying power
of companies with large $v_j$ is also increased. This means that the
ownership of a company $i$ is distributed among the owners $j$,
proportionally to their control $v_j$, i.e.
\begin{equation}\label{eq:w}
w_{ij} = \frac{A_{ij} v_j}{\sum_l A_{il} v_l},
\end{equation}
(see Fig.~\ref{fig:control}).
These equations are assumed to evolve in a faster time scale, such that
equilibrium is reached before the topology changes, as described in the
next section.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.35\columnwidth]{graph_control.pdf}
\includegraphics[width=.35\columnwidth]{graph_weights.pdf}
\caption{Illustration of the control of firms including indirect
control (left) and the ownership being proportional to the control
(right), as described in the text.}
\label{fig:control}
\end{center}
\end{figure}
\subsection{Evolution of the network topology}
Companies may decide to buy or sell shares of a given company at a given
time. The actual mechanisms regulating these decisions are in general
complicated and largely unknown, since they may involve speculation,
actual market value, and other factors, which we do not attempt to model
in detail here. Instead, we describe these changes probabilistically,
where an edge may be deleted or inserted randomly in the network, and
such moves may be accepted or rejected depending on how much it changes
the control of the nodes involved. For simplicity, we force the total
amount of edges in the network to be kept constant, such that a random
edge deletion is always accompanied by a random edge insertion. Such
``moves'' may be rejected or accepted, based on the change they bring to
the $v_j$ values of the companies involved. If we let $m$ be the company
which buys new shares of company $l$, and $j$ which sells shares of
company $i$, the probability that the move is accepted is
\begin{equation}\label{eq:rewire}
p = \min\left(1,e^{\beta(\tilde{w}_{lm} v_l - w_{ij} v_i)}\right),
\end{equation}
where $w_{ij}$ is computed before the move and $\tilde{w}_{lm}$
afterwards, and the parameter $\beta$ determines the capacity companies
have to foresee the advantage of the move, such that for $\beta=0$ all
random moves are accepted, and for $\beta\to\infty$ they are only
accepted if the net gain is positive (see Fig.~\ref{fig:adaptiv}). Note
that in Eq.~\ref{eq:rewire} it is implied that companies with larger
control will tend to buy more than companies with smaller control, which
is well justified by our assumption that control is correlated with
profit and wealth.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.35\columnwidth]{graph_adaptiv_0.pdf}
\includegraphics[width=.35\columnwidth]{graph_adaptiv_1.pdf}
\caption{Illustration of the adaptive process, before the rewiring
(left) and afterwards (right), as described in the text.}
\label{fig:adaptiv}
\end{center}
\end{figure}
The overall dynamics is composed by performing many rewiring steps as
described above, until an equilibrium is reached, i.e. the observed
network properties do not change any longer. In order to preserve a
separation of time scales between the control and rewiring dynamics, we
performed a sufficiently large number of iterations of Eqs.~\ref{eq:v}
and~\ref{eq:w} before each attempted edge move.
\section{Centralization of control}\label{sec:concentration}
A typical outcome of the dynamics can be seen in
Fig.~\ref{fig:condensation} for a network with $N=3\times10^4$ nodes and
average degree $\left<k\right>=2$, after an equilibration time of about
$6\times10^{9}$ steps. In contrast to the case with $\beta=0$, which
results in a fully random graph, for a sufficiently high value of
$\beta$ the distribution of firm ownerships (i.e. the out-degree of the
nodes) becomes very skewed, with a bimodal form. We can divide the most
powerful companies into a broad range which owns shares from $10$ to
about $150$ other companies, and a separate group with
$k_{\text{out}}>150$. The correlation matrix of this network shows that
these high-degree nodes are connected strongly among themselves, and own
a large portion of the remaining companies (see
Fig.~\ref{fig:condensation}). This corresponds to a highly connected
``core'' of about 45 nodes with $\left<k_{\text{sub}}\right>\approx
39.8$, which is highlighted in red in Fig.~\ref{fig:condensation}c and
can be seen separately in Fig.~\ref{fig:condensation}d. The distribution
of in-degree (not shown) is bimodal as well with highest values for the
inner core. With values up to $k_{\rm in}=50$, the highest in-degree
(number of owners) is considerably below the highest out-degree (number
of firms owned at once).
\begin{figure*}[Htb!]
\begin{center}
\begin{minipage}{0.3\textwidth}\centering
\includegraphics[width=\textwidth]{pic_new_dens_k_k2_a0,5_N30000.pdf}
(a)
\end{minipage}
\begin{minipage}{0.3\textwidth}\centering
\includegraphics[width=\textwidth]{pic_new_deg_corr_k2_a0,5_b10_n30000.pdf}
(b)
\end{minipage}
\begin{minipage}{0.24\textwidth}\centering
\includegraphics[width=1\textwidth]{blob.png}
(c)
\end{minipage}
\begin{minipage}{0.14\textwidth}\centering
\includegraphics[width=\textwidth]{blob_core.png}
(d)
\end{minipage}
\caption{(a) Degree distribution of the resulting network for
$\left<k\right>=2$, a control propagation value of $\alpha=0.5$,
$N=30000$ and different values of prior knowledge $\beta$;
(b) Degree correlation matrix for $\beta=10$,
showing the resulting core-periphery structure; (c) Graph layout of
the whole network, with red nodes representing a chosen fraction of
the most highly connected core, and blue ones the periphery; (d)
Subgraph of the most powerful companies with $v_i > 20$ (about
100). The node colors and sizes correspond to the $v_i$ values.}
\label{fig:condensation}
\end{center}
\end{figure*}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=.49\columnwidth]{pic_new_dens_control_k2_a0,5_N30000.pdf}
\includegraphics[width=.49\columnwidth]{pic_new_control_fraction_a0,5_N30000.pdf}
\caption{Left: Distribution of inherited control $v_i - (1-\alpha)$ for
$\alpha = 0.5$ and different values of $\beta$; Right: Relative fraction
of control as a function of fraction of most powerful companies.}
\label{fig:condensation_v}
\end{center}
\end{figure}
Similarly to the out-degree, the distribution of control values $v_i$ is
also bimodal for larger values of $\beta$, as can be seen in
Fig.~\ref{fig:condensation_v}, and is strongly correlated with the
out-degree values. The total fraction of companies controlled by the
most powerful ones is very large, as shown on the right panel of
Fig.~\ref{fig:condensation_v}. For instance, we see that a fraction of
around $0.15\%$ of the central core controls about $57\%$ of all
companies. The companies with intermediary values of control (and
out-degree) also possess a significant part of the global control, e.g.
around $.85\%$ of the most powerful control an additional $25\%$ of the
network. It is important to emphasize the difference between these two
classes of companies for two reasons: Firstly the inner core inherits
control from intermediate companies without the need to gather up all
the minor companies. In fact the ownership links going out from the
inner core (about $10^4$) is enough to cover the direct control of only
a third of all companies, while the effective control is more than a
half. Secondly, the fraction of intermediary companies increases for
larger networks. For a network with $N=3\times10^5$, the inner core
includes a fraction of only $0.04\%$, controlling an effective $41\%$ of
the total companies. Nonetheless, all the most powerful companies
together account for around $1\%$ of the network and $82\%$ of the total
control; values which do not change considerably with system size.
Let us compare the results presented so far with empirical data
presented in \cite{vitali_network_2011}. For different reasons, this
comparison can only be qualitative. First of all, the empirical data
includes economic agents with different functions (shareholders,
transnational companies and participated companies) out of different
sectors (eg. financial and real economy), while we consider identical
agents. Secondly, we force every company to be owned 100\%, while the
empirical data neglects restrained shares and diversified holdings.
Thirdly, the control analysis in \cite{vitali_network_2011} is done
somewhat differently: All the $600,508$ economic agents were considered
for the topological characterization, while many companies (80\% of all
agents there) were neglected for the control analysis. In the empirical
data, a strongly connected component of $1,318$ companies controls more
than a half of all companies arranged in the out component. This
concentration is compatible with the core-periphery structure presented
in Fig.~\ref{fig:condensation}, however the empirical data does not show
a distinct bimodal structure. Nonetheless, there are highly connected
substructures in the core, e.g. a structure with 22 highly connected
financial companies ($\left<k_{\rm
sub}\right>\approx 12$) was highlighted in
\cite{battiston_debtrank_2012}. The control concentration in the
empirical data was reported as a fraction of $0.5\%$ which controls
$80\%$ of the network. This is similar to the results of our model (see
Fig.~\ref{fig:condensation_v} on the right). There are, however,
features that our model does not reproduce, the most important of which
being the out-degree distribution of the network, which
in~\cite{vitali_network_2011} is very broad, and displays no discernible
scales, where in our case it is either bimodal or Poisson-like. One
possible explanation for this discrepancy is that we have focused on
equilibrium steady-state configurations of the dynamics, whereas the
real economy is surely far away from such an equilibrium. A more precise
model would need to incorporate such transient dynamics in a more
realistic way. Nevertheless, the general tendency of the control to be
concentrated on relatively few companies is evident in such equilibrium
states, and features very prominently in the empirical data as well.
\subsection{Transition to centralization}
To investigate the transition from homogeneous no centralized networks
with increasing $\beta$, we measured the inverse participation ratio
$I=\left[\frac{1}{TN}\sum_{ti} v_i(t)^2 \right]^{-1}$ with the time $t$
summing over a sufficiently long time window of length $T$ after
equilibration. Since $\frac{1}{N}\leq I\leq 1$, we expect $I=1$ in the
perfectly homogeneous case where $v_i=1$ for all nodes, and
$I=\frac{1}{N}$ if only one node has $v_i > 0$, and the control is
maximally concentrated. As can be seen in
Fig.~\ref{fig:condensation_trans}, we observe a smooth transition from
very homogeneous companies connected in fully random manner for
$\beta=0$, to a pronounced concentration of control for increased
$\beta$, for which the aforementioned core-periphery is observed. The
transition becomes more abrupt when either the average degree
$\left<k\right>$ is increased or the parameter $\alpha$ (which
determines the fraction of inherited control) is decreased.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.49\columnwidth]{pic_new_concentration_over_beta_k2___N10000.pdf}
\includegraphics[width=0.49\columnwidth]{pic_new_concentration_over_beta_kvar_a0,5_N10000.pdf}
\caption{Inverse participation ratio $I=\left[\frac{1}{TN}\sum_{ti}
v_i(t)^2 \right]^{-1}$ as a function of $\beta$, for a network with
$N=10^4$, and for (left) $\left<k\right>=2$ and different values of
$\alpha$ and (right) $\alpha=0.5$ and different values of
$\left<k\right>$.}
\label{fig:condensation_trans}
\end{center}
\end{figure}
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.49\columnwidth]{pic_new_dens_k_k2_b10_N30000.pdf}
\includegraphics[width=0.49\columnwidth]{pic_new_dens_control_k2_b10_N30000.pdf}
\caption{Distribution of out degrees (left) and inherited control
$v_i-(1-\alpha)$ (right) for $\beta=10$, $\left<k\right>=2$ and $N=30000$
as in Fig.~\ref{fig:condensation} and \ref{fig:condensation_v},
but for different values of $\alpha$.}
\label{fig:condensation_dist}
\end{center}
\end{figure}
Centralization of control can emerge in different ways depending on the
parameters $\alpha$ and $\beta$. In Fig.~\ref{fig:condensation_dist}, it
is shown that different values of $\alpha$ for a high value of
$\beta=10$ can lead to a detached controlling core ($\alpha=0.2$) or to
broadly distributed control values ($\alpha=0.8$). With smaller values
of $\alpha$, indirect control is suppressed and companies can gain power
only by owning large numbers of marginal companies. E.g.:\ for
$\alpha=0.2$, this leads to a highly connected core of $41$ companies
having $\left<k_{\rm sub}\right>\approx 18.2$, the rest of the companies
have very little influence. For larger values of $\alpha$, indirect
control has a larger effect, which leads to a hierarchical network where
companies with small numbers of owned firms $k_{\rm out}$ may
nevertheless inherit large control values $v_i$. The case with
$\alpha=0.5$ and $\beta=10$ shown in Figs.~\ref{fig:condensation} and
\ref{fig:condensation_v} exhibits a mixture of these two scenarios. The
transition to a centralized core also occurs when increasing $\beta$ and
keeping $\alpha$ constant (see right panel in
Fig.~\ref{fig:condensation_trans}).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.41\columnwidth]{lattice.pdf}
\includegraphics[width=0.57\columnwidth]{pic_new_dens_control_tau0_k2_N30000.pdf}
\caption{Left: Graph layout of a $10\times 10$ lattice with
$\alpha=0.9$. The vertex sizes and colors correspond to the $v_i$
values, and the edge thickness to the $w_{ij}$ values.
Right:
Distribution of inherited control $v_i-(1-\alpha)$ for static poisson graphs
having $\left<k\right>=2$ and $N=30\,000$, with different values of
$\alpha$ (for $\alpha=0.5$ and $\alpha=0.8$ shifted).
The dashed line is a power law with exponent $-1$.}
\label{fig:lattice}
\end{center}
\end{figure}
One interesting aspect of the centralization of control as we have
formulated is that it is not entirely dependent on the adaptive
dynamics, and occurs also to some extent on graphs which are
static. Simply solving Eqs.~\ref{eq:v} and~\ref{eq:w} will lead to a
non-trivial distribution of control values $v_i$ which depend on the (in
this case fixed) network topology and the control inheritance parameter
$\alpha$. In Fig.~\ref{fig:lattice} is shown on the left the control
values obtained for a square $2D$ lattice with periodic boundary
conditions, and bidirectional edges. What is observed is a spontaneous
symmetry breaking, where despite the topological equivalence shared
between all nodes, a hierarchy of control is formed, which is not unique
and will vary between each realization of the dynamics. A similar
behavior is also observed for fully random graphs, as shown on the right
of Fig.~\ref{fig:lattice}, where the distribution of control values
becomes increasingly broader for larger values of $\alpha$,
asymptotically approaching a power-law $\rho(v) \sim v^{-1}$ for
$\alpha\to1$. This behavior is similar to a phase transition at
$\alpha=1$, where at this point Eq.~\ref{eq:v} no longer converges to a
solution.
\section{Conclusion}
We have tested the hypothesis that a rich-get-richer process using a
simple, adaptive dynamics is capable of explaining the phenomenon of
concentration of control observed in the empirical network of company
ownership~\cite{vitali_network_2011}. The process we proposed
incorporates the indirect control that companies have on other companies
they own, which increases their buying power in a feedback fashion, and
allows them to gain even more control. In our model, the system
spontaneously organizes into a steady-state comprised of a well-defined
core-periphery structure, which reproduces many qualitative observations
in the real data presented in \cite{vitali_network_2011}, such as the
relative portion of control exerted by the dominating companies. Our
model shows that this kind of centralized structure can emerge without
it being an explicit goal of the companies involved. Instead, it can
emerge simply as a result of individual decisions based on local
knowledge only, with the effect that powerful companies can increase
their relative advantage even further.
It is interesting to compare our model to other agent based models
featuring agents competing for centrality. The emergence of
hierarchical, centralized states with interesting patterns of global
order was reported for agents creating links according to game theory
\cite{holme_centrality_2006,lee_multiadaptive_2011,do_patterns_2010} as
well as for very simple effective rules of rewiring according to
measured centrality
\cite{koenig_centrality_2011,bardoscia_climbing_2013}. The latter is
combined with phase transitions according to the noise in the rewiring
process. The stylized model of a society studied in
\cite{bardoscia_climbing_2013} shows a hierarchical structure, if the
individuals have a preference for social status. The intuitive emergence
of hierarchy is associated with shrinking mobility of single agents
within the hierarchy. This effect is present in our model as well and
deserves further investigation.
Our results may shed light on certain antitrust regulation strategies.
As we found that a simple mechanism without collusion suffices for
control centralization, any regulation which is targeted to diminish
such activities may prove fruitless. Instead, targeting the
self-organizing features which lead to such concentration, such as
e.g. limitations on the indirect control of shareholders representing
other companies, may appear more promising.
\bibliographystyle{apsrev4-1}
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{"url":"https:\/\/codegolf.stackexchange.com\/questions\/23790\/point-free-look-and-say-sequence\/23795","text":"# Point-free look and say sequence [duplicate]\n\nYou are too make a program that take an integer as the input and outputs the first what ever that number was of the look and say sequence.\n\nFor example:\n\n$.\/LAS 8 [1,11,21,1211,111221,312211,13112221,1113213211] The exact way you output the list is unimportant, as long as users can distinctly see the different numbers of the sequence. Here is the catch though. You can't use any sort of user-defined variable. For example: 1. No variables, including scoped variables. 2. When you have functions, they can not have a name. (Exception, if your language requires a main function or similar to work, you may have that function.) 3. When you have functions, they can not have named arguments. Also, you may not use a library with specific capabilities relating to the look and say sequence, and you can not access the network, or provide your program with any files (although it can generate and use its own.) This is code golf, so shortest code in characters wins! \u2022 What is \"EXTREME POINT FREENESS\"? Mar 12 '14 at 3:44 \u2022 @Quincunx I had to look it up: stackoverflow.com\/questions\/944446\/\u2026 Mar 12 '14 at 5:37 \u2022 Can you explain this rule: When you have functions, they can not have named arguments.? Mar 12 '14 at 7:35 \u2022 @ n\u0334\u030b\u0316h\u0337\u0303\u0349a\u0337\u033f\u032dh\u0338\u0305\u0321t\u0335\u0344\u0328d\u0337\u0340\u0330h\u0337\u0302\u0333 In several languages (like the J language or stack\/based languages like forth or postscript), functions don't have argument; they apply to some external context (a stack or arguments coming from an external scope). Mar 12 '14 at 11:54 \u2022 VTC as unclear. What is a \"user defined variable\"? For example, Perl has many variables which may be used, but are defined (initially) by the interpreter itself. It also has a default variable which is referenced by many functions implicitly. Dec 22 '20 at 3:20 ## 15 Answers ## GolfScript (31 chars) ~[]\\{[1\\{.2$={;\\)}1if\\}*].n@}*;\n\n\nAdapted from my answer to a previous look-and-say question. This one has a less onerous restriction for functional languages, which allows saving 5 chars, but because most answers to the previous question can't be adapted (it's a crazily onerous restriction for non-functional languages) I don't think it makes sense to close it as a dupe.\n\nimport Data.List\nimport Control.Applicative\nimport Data.Function\nmain= readLn >>= print .(flip take (map read $fix ((\"1\":). map (concat .(map ((++)<$>(show . length)<*>((:[]). head))). group))::[Integer]))\n\n\nIt works by using the group function to group them into groups of equal things. Then it uses applicatives with functions to build a function that simultaneously reads the length, and appends it to with one the elements. It uses a fix and a map to create a recursive definition (point-free.) And there ya go.\n\n# J (42 chars)\n\nPoint-free (also called tacit) programming is natural in J.\n\n,@:((#,{.);.1~(1,}.~:}:))&.>^:(<((<1)\"_))\n\n\nThat's a function, to use it you write the code, a space, and the input number. For example,\n\n ,@:((#,{.);.1~(1,}.~:}:))&.>^:(<((<1)\"_)) 8\n\u250c\u2500\u252c\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u252c\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2510\n\u25021\u25021 1\u25022 1\u25021 2 1 1\u25021 1 1 2 2 1\u25023 1 2 2 1 1\u25021 3 1 1 2 2 2 1\u25021 1 1 3 2 1 3 2 1 1\u2502\n\u2514\u2500\u2534\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2534\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2500\u2518\n\n\nNotice the pretty boxes in the output.\n\nAddendum: Here are a couple of \"cheats\" I was too bashful to use at first, but now that I've seen other use them first...\n\n\u2022 Here's a 36 char version with a different \"calling convention\": replace 8 with the number of terms you want.\n\n,@:((#,{.);.1~(1,}.~:}:))&.>^:(<8)<1\n\n\u2022 And if having extra zeroes in the output is OK, here's a 32 char version:\n\n,@:((#,{.);.1~(1,}.~:}:))^:(<8)1\n\n\n# Bash and coreutils, 111 73 chars\n\neval echo 1\\|yes 'tee -a o|fold -1|uniq -c|(tr -dc 0-9;echo)|'|sed $1q: uniq -c is doing the heavy lifting to produce the next number in the sequence. yes, sed and eval create the necessary number of repeats of the processing pipeline. The rest is just formatting. Output is placed in a file called o.: $ .\/looksay.sh 8\nubuntu@ubuntu:~$cat o 1 11 21 1211 111221 312211 13112221 1113213211$\n\n\n## GolfScript, 36 chars\n\n~([1]\\{.[0\\{.2$=!{0\\.}*;\\)\\}\/](;}*] Variables are pretty rarely used in GolfScript, and this task certainly doesn't need them. Input is on stdin, output to stdout. For example, the input 8 gives the output: [[1] [1 1] [2 1] [1 2 1 1] [1 1 1 2 2 1] [3 1 2 2 1 1] [1 3 1 1 2 2 2 1] [1 1 1 3 2 1 3 2 1 1]] I may write a detailed explanation of this code later, but at least you can easily tell that it uses no variables by the fact that it doesn't include the variable assignment operator : anywhere. # Haskell, 118 chars (80 without imports) import Data.List import Control.Monad main=readLn>>=print.flip take(iterate(ap((++).show.length)(take 1)<=<group)\"1\") # Mathematica, 65 chars FromDigits\/@NestList[Flatten@Reverse[Tally\/@Split@#,3]&,{1},#-1]& ### Example: FromDigits\/@NestList[Flatten@Reverse[Tally\/@Split@#,3]&,{1},#-1]&[8] {1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211} ## J, 37 characters 1([:,((1,2&(~:\/\\))(#,{.);.1]))@[&0~i. Based on my answer to the Pea Pattern question. There may be some potential for shortening here. Usage is as for the other J answer: 1([:,((1,2&(~:\/\\))(#,{.);.1]))@[&0~i. 7 1 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 2 1 0 0 0 0 0 0 1 2 1 1 0 0 0 0 1 1 1 2 2 1 0 0 3 1 2 2 1 1 0 0 1 3 1 1 2 2 2 1 It also has the extra zeroes problem my pea pattern answer had. \u2022 Ah, there's more than one previous question, and more answers from that one can be copied to this one without any tweaks at all than from the question I found. I'm almost convinced to vote to close as dupe. Mar 12 '14 at 12:32 \u2022 @PeterTaylor The Pea pattern one is slightly different in that you have to sort the numbers in the previous line before creating the next. Mar 12 '14 at 12:35 # Perl 6: 63 53 characters say (1,*.subst(\/(\\d)$0*\/,{.chars~.[0]},:g)...*)[^get]\n\n\nCreate a lazy list of the Look and Say sequence (1,*.subst(\/(\\d)$0*\/,{.chars~.[0]},:g)...*), and then get as many elements as specified by the user ([^get], which is an array subscript and means [0..(get-1)]), and say them all. The lazy list works by first taking 1, then to generate each successive number, it takes the last one it found and substitutes all the sequences of the same digit, as matched by \/(\\d)$0*\/, and replaces them with {how many}+{what digit}, or .chars~.[0].\n\nThe only variables in this code are $0, the first capture of the match, and the implicit, topical $_ variable that bare .methods call, and neither of these are user-defined.\n\n# Dyalog APL, 35 characters\n\n(\u22a2,\u2282\u2218\u220a\u2218((\u2262,\u2283)\u00a8\u2283\u2282\u23682\u2262\/0,\u2283)\u2218\u233d)\u2363(\u2395-1)\u22a21\n\n\u2395 is evaluated input. In the link I've replaced it with 8, as tryapl.org does not allow user input.\n\nNo named variables (a\u21901), no named functions (f\u2190{}), no arguments (\u237a, \u2375).\n\nOnly composition of functions:\n\n\u2022 monadic operators\u2014each:f\u00a8, reduce:f\/, commute:f\u2368\n\u2022 dyadic operators\u2014power:f\u2363n, compose:f\u2218g\n\u2022 forks\u2014(f g h)B \u2190\u2192 (f B)g(h B); A(f g h)B \u2190\u2192 (A f B)g(A h B)\n\u2022 atops\u2014(f g)B \u2190\u2192 f(g B); A(f g)B \u2190\u2192 f(A g B)\n\u2022 4-trains (fork-atops)\u2014(f g h k) \u2190\u2192 (f (g h k))\n\nPrimitive functions used:\n\n\u2022 right:A\u22a2B \u2190\u2192 B\n\u2022 reverse:\u233dB\n\u2022 first:\u2283B\n\u2022 concatenate:A,B\n\u2022 not match:A\u2262B, count:\u2262B\n\u2022 enclose:\u2282B, partition:A\u2282B\n\u2022 flatten:\u220aB\n\nIn tryapl.org, if you remove the trailing \u22a21, which is the argument to this massive composed thing, you can see a diagram of how it's parsed:\n\n \u2363\n\u250c\u2500\u2534\u2500\u2510\n\u250c\u2500\u253c\u2500\u2510 7\n\u22a2 , \u2218\n\u250c\u2534\u2510\n\u2218 \u233d\n\u250c\u2500\u2500\u2534\u2500\u2500\u2500\u2510\n\u2218 \u250c\u2500\u2534\u2500\u2510\n\u250c\u2534\u2510 \u00a8 \u250c\u2500\u253c\u2500\u2500\u2500\u2510\n\u2282 \u220a \u250c\u2500\u2518 \u2283 \u2368 \u250c\u2500\u253c\u2500\u2500\u2500\u2510\n\u250c\u2500\u253c\u2500\u2510 \u250c\u2500\u2518 2 \/ \u250c\u2500\u253c\u2500\u2510\n\u2262 , \u2283 \u2282 \u250c\u2500\u2518 0 , \u2283\n\u2262\n\n\n# GolfScript, 57 43 chars\n\nMy own approach. Ended up longer than the existing one sadly =(.\n\n~[1 9]{.);p[{...1<^0=?.@(\\@(>.,(}do 0=]}@*;\n\n\nSample output for stdin of 8:\n\n[1]\n[1 1]\n[2 1]\n[1 2 1 1]\n[1 1 1 2 2 1]\n[3 1 2 2 1 1]\n[1 3 1 1 2 2 2 1]\n[1 1 1 3 2 1 3 2 1 1]\n\n\nAlternate version w\/out the 9 sentinel, yet it's longer at 47 characters. I suspect it has more potential:\n\n~[1]{.p[{...1<^.{0=?.@(\\@(>1}{;,\\0=0}if}do]}@*;\n\n\n# Scala 178\n\n(0 to Console.in.readLine.toInt).map(i=>Function.chain(List.fill[String=>String](i)(y=>(('0',0,\"\")\/:(y+\" \")){case((a,b,c),d)=>if(d==a)(a,b+1,c)else(d,1,c+b+a)}._3.drop(2)))(\"1\"))\n\n\u2022 I'm pretty sure that the i in i=> is a variable. Nov 14 '14 at 17:14\n\n# Husk, 11 bytes\n\n\u2191\u00a1o\u1e41\u00a7eL\u2190g;1\n\n\nTry it online!\n\n1-indexed.\n\n## J 66 (with I\/O)\n\n\".@(_5}&',@((#,&:\":{.);.1~1&(0})&(~:_1|.]))^:(<X)\":1')@{.&.stdin''\n\n\nwithout IO, scores 43:\n\nNB. change the 8 for the number of numbers you'd want\n,@((#,&:\":{.);.1~1&(0})&(~:_1|.]))^:(<8)'1'\n\n\nFunny question to pose yourself, when is the first 9 to show up?\n\n\u2022 Never look at the integer sequence page. Mar 12 '14 at 20:04\n\u2022 Ok, I see. Then ... why so? Mar 12 '14 at 21:59\n\u2022 voices.yahoo.com\/\u2026 Mar 12 '14 at 22:10\n\u2022 Nice trick in the IO version of replacing the X by the input in a string and then calling eval!\n\u2013\u00a0Omar\nMar 13 '14 at 13:24\n\u2022 As for the funny question: isn't it pretty clear you only ever have 1, 2 & 3? I mean to get a 4 or higher, in the previous step you'd need four consecutive equal digits, xaaaay, but that can't happen since you'd be saying a further step earlier you saw \"x a's, a a's\" or \"a a's, a a's\".\n\u2013\u00a0Omar\nMar 13 '14 at 13:41\n\n# Jelly, 11 10 bytes\n\n1\u0152rUF\u018a\u2019}\u00a1)\n\n\nTry it online!\n\nJelly doesn't have variables.\n\n## Explanation\n\n1\u0152rUF\u018a\u2019}\u00a1)\n) For each i in 1..n:\n1 Begin with 1\n\u2019}\u00a1 i - 1 times\n\u018a (\n\u0152r Run-length encode\nU Reverse each\nF Flatten\n\u018a )\n`","date":"2021-09-20 21:10:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.38751542568206787, \"perplexity\": 5986.472653670114}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780057091.31\/warc\/CC-MAIN-20210920191528-20210920221528-00138.warc.gz\"}"}
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\section*{Methods}
We have taken the case of equal electron and hole densities $(n_e=n_h=n)$ and equal effective masses $(m^\star=m_e^\star=m_h^\star)$. For zero temperature, the mean field Eqs.\ (3)-(4) in the main text for the momentum-dependent gap function $\Delta^{\gamma}_{\mathbf{k}}\equiv \Delta^{\gamma}_{\mathbf{k}}(T=0)$ and chemical potentials $\mu^e$ and $\mu^h$ reduce to coupled equations for $\Delta^{\gamma}_{\mathbf{k}}$ and the average chemical potential, $\mu=(\mu_e+\mu_h)/2$,
\begin{eqnarray}
\Delta^{\gamma}_{\mathbf{k}}&=& -\sum_{{\mathbf{k}'}\gamma'} \frac{1}{2}V^{eh}_{\mathbf{k}-\mathbf{k}'} \frac{\Delta^{\gamma'}_{\mathbf{k}'}} {2E^{\gamma'}_{\mathbf{k}'}}
\label{Delta-eqn} \\
n &=& 2g_v\sum_{\mathbf{k}} (v^{\gamma=1}_{\mathbf{k}})^2\ .
\label{n-eh}
\end{eqnarray}
$\xi^{\gamma}_{\mathbf{k}}=\gamma \hbar^2k^2/(2m_r^\star)-\mu$, where $m_r^\star$ is the reduced effective mass of the electron-hole pairs. The screened zero-$T$ electron-hole Coulomb attraction $V^{eh}_{\mathbf{k}-\mathbf{k}'}$ depends on the momentum exchanged in the scattering process, thus making the gap momentum-dependent. The extra factor of $\frac{1}{2}$ in Eq.\ (6) comes from the graphene geometrical form factor $F^{\gamma\gamma'}_{\mathbf{k}\mathbf{k}'}$. In the phase space region of interest to us, coupling is strong and the $s$-wave harmonic contribution to $F^{\gamma\gamma'}_{\mathbf{k}\mathbf{k}'}$ dominates, leading to the factor $\frac{1}{2}$.
The numerical sums are transformed into continuous integrals and then numerically evaluated using the Gauss-Legendre method. We numerically solved the $T=0$ non-linear system of integral equations, Eqs.\ (6) and (7), iteratively using a multivariable Newton method. The convergence ratio requirement for the variables was set at $10^{-6}$. Retaining only the s-wave harmonic of the form factor, results in a gap function $\Delta_{\mathbf{k}}$ that depends only on the modulus of the momentum ${\mathbf{k}}$.
Concerning the ultraviolet behavior of the mean field equations, the integral over the momentum is well-defined, thanks to the exponential decay in momentum space of the bare Coulomb interaction $v_{\mathbf{q}}\mathrm{e}^{-qD_{eh}}$ between electrons and holes in the bilayers separated by a dielectric barrier of thickness $D_{eh}$. In our calculations we fixed the upper momentum cutoff for the integrations as $k_{c}=5\times\max(1/D_{eh},k_F)$. Numerical stability was checked by increasing $k_c$ and increasing number of integration points. We confirmed that the results obtained by the Newton method for $\mu>0$ agreed with a solution of the mean field equations independently obtained by a direct recursive method.\\
\section*{Suppression of screening: polarization bubble}
In the superfluid state at low temperature, the static limit of the polarization bubble in the diagrammatic (RPA-like) resummation is suppressed at small momenta $q$. Reference 13 in the main text discusses this effect in the limiting case of no dielectric barrier, $D_{eh}= 0$, for energy bands with linear dispersion. The polarization bubble is responsible for renormalizing the bare Coulomb interaction. A small-$q$ suppression of the screening allows strong unscreened electron-hole pairing peaked at small-$q$ to occur, and this strong pairing can lead to a large gap in the excitation spectrum.
We performed numerical calculations for the polarization bubbles for the case of quadratic energy bands. In the superfluid state the polarization bubble is given by the sum of the normal and anomalous bubbles (Eq.\ 4 in the main text and supplement Fig.\ 4).
\begin{figure}
\includegraphics[angle=0,width=0.45\textwidth] {Fig.1.supp.eps}
\caption{$\Pi_0^{(n)}(q)$ and $\Pi_0^{(a)}(q)$ with two normal and two anomalous Green functions, respectively.}
\label{polbubbles}
\end{figure}
In RPA the polarization bubbles are constructed with the normal and
anomalous Green's functions of BCS theory.
Assuming the Green's functions are diagonal in the band indices, the normal and anomalous Green's functions are given by,
\begin{eqnarray}
{\cal{G}}^{\gamma\gamma}(\mathbf{k},\mathrm{i}\epsilon_n)
&=&\frac{(u_{k}^{\gamma})^2}{\mathrm{i}\epsilon_n-E_{k}^{\gamma}}
+\frac{(v_{k}^{\gamma})^2}{\mathrm{i}\epsilon_n+E_{k}^{\gamma}}\\
{\cal{F}}^{\gamma\gamma}(\mathbf{k},\mathrm{i}\epsilon_n)
&=&\frac{u_{k}^{\gamma}v_{k}^{\gamma}}{\mathrm{i}\epsilon_n-E_{k}^{\gamma}}
-\frac{u_{k}^{\gamma}v_{k}^{\gamma}}{\mathrm{i}\epsilon_n+E_{k}^{\gamma}}
\ ,
\label{GandF}
\end{eqnarray}
respectively, where $u_{k}^{\gamma}$ and $v_{k}^{\gamma}$ are the Bogoliubov factors
\begin{eqnarray}
(u_{k}^{\gamma})^2&=&\frac{1}{2}\left(1+\frac{\xi_k^\gamma}{E_k^\gamma}\right)\nonumber\\
(v_{k}^{\gamma})^2&=&\frac{1}{2}\left(1-\frac{\xi_k^\gamma}{E_k^\gamma}\right)\ .
\label{uandv}
\end{eqnarray}
The summation
of the fermionic Matsubara frequencies $\epsilon_n$ of the particle-hole loop can then be
done analytically, and taking the static limit the expressions for the
bubbles are given by,
\begin{eqnarray}
\Pi_0^{(n)}(q,T)\!\!&=&\!\! -2g_v\sum_{\mathbf{k}\gamma\gamma'}
\frac{1}{2}\frac{(u_{k}^{\gamma})^2(v_{{\mathbf{k}}-{\mathbf q}}^{\gamma'})^2
+(v_{k}^{\gamma})^2 (v_{{\mathbf{k}}-{\mathbf q}}^{\gamma'})^2} {E_{\mathbf{k}}^{\gamma}+E_{{\mathbf{k}}-{\mathbf q}}^{\gamma'}}\nonumber\\
\label{pi-n} \\
\Pi_0^{(a)}(q,T)\!\!&=&\!\! 2g_v\sum_{\mathbf{k}\gamma\gamma'}
\frac{1}{2}\frac{2u_{\mathbf{k}}^{\gamma} v_{\mathbf{k}}^{\gamma}
u_{{\mathbf{k}}-{\mathbf q}}^{\gamma'} v_{{\mathbf{k}}-{\mathbf q}}^{\gamma'}}{E_{\mathbf{k}}^{\gamma} +E_{{\mathbf{k}}-{\mathbf q}}^{\gamma'}}\ .
\label{pi-a}
\end{eqnarray}
When the chemical potential $\mu$ is positive, the contributions
of the lower band $(\gamma$ or $\gamma'=-1)$ to the total
polarization bubbles are small because the corresponding $k$-states are filled and far in energy from $\mu$. Hence they contribute little to the particle-hole processes. We considered only the $\gamma=\gamma'=1$ contribution and, as in the gap equation, only the s-wave harmonic of the
geometrical form factor of graphene $F^{\gamma\gamma'}_{\mathbf{k},\mathbf{k}-\mathbf{q}}$.
Physically, the effect of screening is suppressed because of the opening of the energy gap $\Delta$ at the Fermi surface. This exponentially suppresses particle-hole processes with energies less than the gap energy $\Delta$ and it is precisely these low-energy processes that are needed to screen the long-range Coulomb interaction. Once a gap appears in the excitation spectrum, screening exactly vanishes in the $q=0$ limit. From a diagrammatic point of view, in addition to a suppression of the bubble with diagonal Green functions at low momentum, there is an additional canceling contribution from the anomalous bubble with off-diagonal Green functions. At $q=0$ the cancelation is exact for any non-zero $\Delta$.
\end{document}
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| 6,425
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Ballota és un gènere compost per unes 35 espècies d'angiospermes incloses en la família de les lamiàcies. Aquest gènere és natiu de regions temperades d'Europa, del nord d'Àfrica i de l'oest d'Àsia, amb la major diversitat en la regió Mediterrània.
Les espècies de Ballota han sigut usades com aliment per a larves d'alguns Lepidòpters inclòs Coleophora: C. ballotella, C. lineolea i C. ochripennella.
Espècies seleccionades
Ballota acetabulosa
Ballota frutescens
Ballota hirsuta
Ballota nigra - Malrubí Negre
Ballota pseudodictamnus
El gènere es tanca afegint el gènere Marrubium, algunes espècies del qual havien format part d'aquest gènere en el passat.
Lamieae
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{"url":"https:\/\/socratic.org\/questions\/how-do-you-change-fractions-into-decimals","text":"# How do you change fractions into decimals?\n\n$17 \\div i \\mathrm{de} 6 = 2.8333 \\ldots \\implies \\frac{17}{6} = 2.8333 \\ldots$","date":"2021-05-14 20:47:30","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 1, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3619314134120941, \"perplexity\": 9097.99620188317}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-21\/segments\/1620243991207.44\/warc\/CC-MAIN-20210514183414-20210514213414-00470.warc.gz\"}"}
| null | null |
Art - Three in a Row
Artist: Espen Gangvik (1993).
This decoration on the wall of the Department of Petroleum Engineering is more or less part of the architecture. The sculptures, or rather the installation, are made of laminated wooden plates coated with steel. With its faceted zigzag forms, the design is dynamic. The three sections are mounted on the façade on different levels, the uppermost being 18 metres above ground.
© Copyright - National and international copyright laws protect the works of art presented at this website. The artworks may not be reproduced or made public in any way, analogue or digital, without permission from the right holders / BONO. Please contact BONO (Norwegian Visual Artists Copyright Society) in order to obtain a license. http://www.bono.no
Text descriptions of art made before the year 2000 are taken from the book 'Skulpturguiden for Trondheim' by Anne Grønli and Grethe Britt Fredriksen. Text descriptions of art made after the year 2000 are written by Per Christiansen.
|
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\section{Introduction}
\IEEEPARstart{D}{ata} processing techniques are often based on the \emph{manifold assumption}: Meaningful data (signals) lie near a low dimensional manifold, although their apparent dimension is much larger \cite{Carlsson2009,Peyre2009} \cite[Section 5.11.3]{Goodfellow2016} \cite[Section 9.3]{Elad2010}. This fact has classically been exploited in two different ways.
On the one hand, for decades if not centuries, scientists have handcrafted \emph{analytical models}. This amounts to come up with a mathematical description of the manifold, based on domain knowledge and careful observation of the phenomena of interest. This approach reaches its limits for complex phenomena that are difficult to model with a reasonable number of parameters, in which case simplifying assumptions have to be made, hindering model relevance.
On the other hand, thanks to the advent of modern computers, \emph{machine learning} techniques have emerged and led to tremendous successes in various domains \cite{Lecun2015,Goodfellow2016}. One of their main feature is to avoid any explicit mathematical description of the manifold at hand, which is taken into account via a large amount of training data sampling it. Such an approach is particularly successful in application domains for which building analytical models is difficult, since it is much more flexible. However, flexibility comes at the price of computationally heavy learning and difficulties to inject a priori knowledge on the phenomena at hand.
Recently, a promising approach meant to combine the advantages of the two aforementioned approaches has been proposed under the name of \emph{deep unfolding} (also known as deep unrolling). It amounts to unfold iterative algorithms initially based on analytical models so as to express them as neural networks that can be optimized \cite{Gregor2010,Hershey2014,Kamilov2016}. This has the advantage of adding flexibility to algorithms based on classical models, and amounts to constrain the search space of neural networks by using domain knowledge. Moreover, this leads to inference algorithms of controlled complexity (see \cite{Monga2019} and references therein for a complete survey).
Channel estimation is of paramount importance for communication systems in order to optimize the data rate/energy consumption tradeoff.
In modern systems, the possibly large number of transmit/receive antennas and subcarriers makes this task difficult. For example, it has been recently proposed to use massive multiple input multiple output (massive MIMO) wireless systems \cite{Rusek2013,Larsson2014,Lu2014} with a large number of antennas in the millimeter-wave band \cite{Rappaport2013, Swindlehurst2014}, where a large bandwidth can be exploited. In that case the channel comprises hundreds or even thousands of complex numbers, whose estimation is a very challenging signal processing problem \cite{Heath2016}. Fortunately, despite the high dimension of the channel, realistic channels are often well approximated by only a few dominant propagation paths \cite{Samimi2016} (typically less than ten). Such channels are said \emph{sparse}.
For massive MIMO channel estimation, it is thus customary to use an analytical model based on the physics of propagation in order to ease the problem. This amounts to parameterize a manifold by physical parameters such as the directions, delays and gains of the dominant propagation paths, the dimension of the manifold being equal to the number of real parameters considered in the model. Physical channel models allow injecting strong a priori knowledge based on solid principles \cite{Sayeed2002,Bajwa2010,Lemagoarou2018}, but necessarily make simplifying assumptions (e.g. the plane wave assumption \cite{Lemagoarou2019b}) and require knowing exactly the system configuration (positions of the antennas, gains, etc.). Such requirements being unrealistic in practice, the massive MIMO channel estimation task could perfectly benefit from the flexibility offered by the deep unfolding approach.
\noindent{\bf Contributions.}
In this paper, we introduce {\sf mpNet}, an unfolded neural network specifically designed for massive MIMO channel estimation. The unfolded algorithm is matching pursuit \cite{Mallat1993}, which is a greedy computationally efficient algorithm taking advantage of the massive MIMO channel sparsity. The network takes as input the least squares channel estimates, with the objective to denoise them. The weights of {\sf mpNet} are initialized using an imperfect physical model. The main concerns while designing the method were to make it:
\begin{itemize}
\item {\bf Unsupervised}: no need for clean channels to train {\sf mpNet}, the method uses the least squares channel estimates as training data and a reconstruction cost function. It is thus trained as an autoencoder \cite{Rumelhart1986}.
\item {\bf Online}: no need for a separate offline training phase, {\sf mpNet} is initialized with an imperfect physical model and trained incrementally using online gradient descent \cite{Bottou1998}.
\item {\bf SNR adaptive}: no need for several networks trained at different SNRs, {\sf mpNet} automatically adapts its depth to incoming data.
\item {\bf Computationally efficient}: backpropagation through {\sf mpNet} is cheaper than classical channel estimation using a greedy sparse recovery algorithm (which corresponds to the forward pass of {\sf mpNet}).
\end{itemize}
To the best of the authors' knowledge, the proposed method is the only one meeting all these requirements. Such a method is particularly suited to imperfectly known or non-calibrated systems. Indeed, starting from an imperfect physical channel model, our method allows a base station to automatically correct its channel estimation algorithm based on incoming data.
Note that this paper is partly based on a previously published work \cite{Lemagoarou2020}. This paper is however much more exhaustive, since the method is exposed in a more comprehensive way, SNR adaptivity is totally new, and a much more extensive set of experiments is performed in order to empirically assess the method.
\noindent{\bf Related work.} Machine learning holds promise for wireless communications (see \cite{Oshea2017,Wang2017,Qin2019} for exhaustive surveys). More specifically, since the physics of propagation provides pretty accurate analytical models, model-driven machine learning approaches \cite{He2019} seem particularly suited. The method we propose can be seen as an instance of the model-driven approach.
In the context of massive MIMO channel estimation, it has recently been proposed to use adaptive data representations using dictionary learning techniques \cite{Ding2018}. However, classical dictionary learning employing algorithms such as K-SVD \cite{Aharon2006}, as proposed in \cite{Ding2018}, is very computationally heavy, and thus not suited to online learning, as opposed to the {\sf mpNet} approach.
Deep unfolding has also been considered by communications researchers (see \cite{Balatsoukas2019} for a survey). It has mainly been used for symbol detection, unfolding projected gradient descent \cite{Samuel2017} or approximate massage passing \cite{He2018a} algorithms.
Regarding MIMO channel estimation, it has been proposed in \cite{He2018} to unfold a sparse recovery algorithm named denoising-based approximate message passing (DAMP) \cite{Metzler2016}. However, the method is directly adapted from image processing \cite{Metzler2017} and does not make use of a physical channel model as initialization as we propose here. A recent work also proposes to use deep unfolding of the DAMP algorithm for MIMO channel estimation \cite{Wei2019}, using a physical model to optimize the shrinkage functions used by DAMP. However, these previously proposed methods all require collecting a database of clean channels and an offline training phase, due to their intrinsic supervised nature. This may hinder their practical applicability. Moreover, the proposed unfolded neural networks are much more computationally complex than {\sf mpNet}, and do not comprise an automatic way to adapt to the SNR.
Finally, deep learning has also been applied to channel estimation in an orthogonal frequency-division multiplexing (OFDM) context, which is mathematically very close to the one studied here. In \cite{Gao2018}, a neural network is used as a post-treatment of the least squares channel estimates in order to denoise them, and included in a joint channel estimation and detection framework. In \cite{Soltani2019}, the noisy time-frequency response of the channel is viewed as an image and is denoised using classical denoising neural networks. Once again, these approaches are supervised, and are of high complexity compared to classical methods \cite{Vanlier2020}.
\noindent{\bf Organization.} The remaining of the paper is organized as follows. First, section~\ref{sec:problem_formulation} introduces the problem at hand and describes the physical model on which {\sf mpNet} is based. Then, the motivations behind the proposed solution is presented in section~\ref{sec:impact}. Section~\ref{sec:prop_approach} introduces in details {\sf mpNet}: the deep unfolding based strategy we propose for MIMO channel estimation. In section~\ref{sec:experiments}, different experiments are conducted in order to assess and validate the potential of our approach. Finally, section~\ref{sec:conclusion} discusses the contributions and concludes the paper.
\section{Problem formulation}
\label{sec:problem_formulation}
\subsection{System settings}
We consider in this paper a massive MIMO system, also known as multi-user MIMO (MU-MIMO) system \cite{Rusek2013,Larsson2014,Lu2014}, in which a base station equipped with $N$ antennas communicates with $K$ single antenna users ($K<N$). Let us consider for ease of presentation a transmission on a single subcarrier, even though everything presented in the paper can obviously be generalized to the multicarrier case. The system operates in time division duplex (TDD) mode, so that channel reciprocity holds and the channel is estimated in the uplink: each user sends a pilot sequence $\mathbf{p}_k \in \mathbb{C}^{T}$ of duration $T$ (orthogonal to the sequences of the other users, $\mathbf{p}_k^H\mathbf{p}_l = \delta_{kl}$) for the base station to estimate the channel. The received signal is thus expressed
\begin{equation}
\mathbf{R}=\sum_{k=1}^K\mathbf{h}_k\mathbf{p}_k^H +\mathbf{N},
\end{equation}
where $\mathbf{N}\in \mathbb{C}^{N \times T}$ is Gaussian noise.
After correlating the received signal with the pilot sequences, and assuming no pilot contamination from adjacent cells for simplicity, the base station gets noisy measurements of the channels of all users, taking the form
\begin{equation}
\mathbf{x}_l \triangleq \mathbf{R}\mathbf{p}_l =\underbrace{\sum\nolimits_{k=1}^K\mathbf{h}_k\mathbf{p}_k^H\mathbf{p}_l}_{\mathbf{h}_l} + \underbrace{\mathbf{N}\mathbf{p}_l}_{\mathbf{n}_l},
\end{equation}
for $l=1,\dots,K$, with $\mathbf{n}_l \sim \mathcal{CN}(0,\sigma^2\mathbf{Id}),\, \forall l$.
In order to simplify notations in the remaining parts of the paper, we drop the user index and denote such measurements with the canonical expression
\begin{equation}
\mathbf{x} = \mathbf{h} + \mathbf{n},
\label{eq:observations}
\end{equation}
where $\mathbf{h}$ is the channel of the considered user and $\mathbf{n}$ is noise, with $\mathbf{n} \sim \mathcal{CN}(0,\sigma^2\mathbf{Id})$. Such a dropping of the user index does not harm the description of our approach, since it treats the channels of all users indifferently. Note that $\mathbf{x}$ is already an unbiased estimator of the channel, obtained by solving a least squares estimation problem, so that we call it the least squares (LS) estimator in the sequel. Its performance can be assessed by the input signal-to-noise ratio
$$
\text{SNR}_{\text{in}} \triangleq \frac{\left\Vert \mathbf{h} \right\Vert_2^2}{N\sigma^2}.
$$
However, one can get better channel estimates using a physical model which allows to denoise the least squares estimate, as is explained in the next subsection.
\subsection{Physical model}
Let us denote $\{g_1,\dots,g_N\}$ the complex gains of the base stations antennas and $\{\overrightarrow{a_1},\dots,\overrightarrow{a_N}\}$ their positions with respect to the centroid of the antenna array. Then, under the plane wave assumption and assuming omnidirectional antennas (isotropic radiation patterns), the channel resulting from a single propagation path with direction of arrival (DoA) $\overrightarrow{u}$ is proportional to the \emph{steering vector}
$$
\mathbf{e}(\overrightarrow{u}) \triangleq (
g_1\mathrm{e}^{-\mathrm{j}\frac{2\pi}{\lambda}\overrightarrow{a_1}.\overrightarrow{u}}, \dots, g_N\mathrm{e}^{-\mathrm{j}\frac{2\pi}{\lambda}\overrightarrow{a_N}.\overrightarrow{u}})^T
$$
which reads $\mathbf{h} = \beta \mathbf{e}(\overrightarrow{u}),$
with $\beta \in \mathbb{C}$. In that case, a sensible estimation strategy \cite{Sayeed2002,Bajwa2010,Lemagoarou2018} is to build a dictionary of steering vectors corresponding to $A$ potential DoAs: $\mathbf{E} \triangleq \begin{pmatrix}
\mathbf{e}(\overrightarrow{u_1}),\dots,\mathbf{e}(\overrightarrow{u_A})
\end{pmatrix}
$ and to compute a channel estimate with the procedure
\begin{align}
\begin{split}
&\overrightarrow{v}=\text{argmax}_{\overrightarrow{u_i}} \,\,\, |\mathbf{e}(\overrightarrow{u_i})^H\mathbf{x}|, \\
&\hat{\mathbf{h}} = \mathbf{e}(\overrightarrow{v})\mathbf{e}(\overrightarrow{v})^H\mathbf{x}.
\end{split}
\label{eq:estim_strat}
\end{align}
The first step of this procedure amounts to find the column of the dictionary the most correlated with the observation (least square channel estimate) to estimate the DoA and the second step amounts to project the observation on the corresponding steering vector. The SNR at the output of this procedure reads
$$\text{SNR}_\text{out} \triangleq \frac{\Vert \mathbf{h} \Vert_2^2}{\mathbb{E}\big[\Vert\mathbf{h}-\hat{\mathbf{h}}\Vert_2^2\big]},$$
and we have at best $\text{SNR}_\text{out} =N\text{SNR}_\text{in}$ (neglecting the discretization error), if the selected steering vector is collinear to the actual channel. This is a direct consequence of the Cauchy-Schwarz inequality, and is intuitively explained by the fact that from the $N$ complex dimensions of $\mathbf{x}$ corrupted by noise, only one is kept when projecting on the best steering vector, so that the effective noise variance is divided by $N$. The potential gain of using such a physical model can be huge, especially for massive MIMO systems in which the number of antennas $N$ is large.
Moreover, this strategy can be generalized to estimate sparse multipath channels of the form
\begin{equation}
\mathbf{h} = \sum\nolimits_{p=1}^P\beta_p \mathbf{e}(\overrightarrow{u_p}),
\label{eq:multipath_channel}
\end{equation}
by iterating the procedure \eqref{eq:estim_strat} until some predefined stopping criterion is met. This leads to greedy sparse recovery algorithms such as matching pursuit (MP) \cite{Mallat1993} or orthogonal matching pursuit (OMP) \cite{Tropp2010}. Since the method proposed in this paper is based on the unfolding of the matching pursuit algorithm, a high level overview of it applied to channel estimation, with dictionary $\mathbf{E}$ and input $\mathbf{x}$ is given in algorithm~\ref{alg:mp}.
\begin{algorithm}[htb]
\caption{Matching pursuit \cite{Mallat1993} (high level overview)}
\begin{algorithmic}[1]
\REQUIRE Dictionary $\mathbf{E}$, input $\mathbf{x}$ (noisy channel)
\STATE $\mathbf{r} \leftarrow \mathbf{x}$
\WHILE{Stopping criterion not met}
\STATE Find the most correlated atom: $s\leftarrow\underset{i}{\text{argmax}} \,\,\, |\mathbf{e}_i^H\mathbf{r}|$
\STATE Update the residual: $\mathbf{r} \leftarrow \mathbf{r} - \mathbf{e}_s\mathbf{e}_s^H\mathbf{r}$
\ENDWHILE
\ENSURE $\hat{\mathbf{h}} \leftarrow \mathbf{x} - \mathbf{r}$ (denoised channel)
\end{algorithmic}
\label{alg:mp}
\end{algorithm}
\section{Motivation: imperfect models}
\label{sec:impact}
Basing an estimation strategy on a physical model, as suggested in the previous section, requires knowing precisely the physical parameters of the system (in particular the positions and gains of the antennas) in order to build an appropriate dictionary. Then, even in the case of perfect system knowledge, some simplifying hypotheses (such as the plane wave assumption considered in the previous section) have to be made in order to keep the model mathematically tractable. Consequently, every model, regardless of its sophistication, is necessarily imperfect. Such a situation is well-known and summarized by the aphorism \emph{``All models are wrong''} \cite{Box1976}.
In the context of MIMO channel estimation, what is the impact of an imperfect knowledge of the physical parameters and/or of the invalidity of some hypotheses? In order to address this question, let us perform a simple experiment.
Consider an antenna array of $N=64$ antennas at the base station, whose known \emph{nominal} configuration is a uniform linear array (ULA) of unit gain antennas separated by half-wavelengths and aligned with the $x$-axis. This nominal configuration corresponds to gains and positions $\{\tilde{g}_i,\tilde{\overrightarrow{a_i}}\}_{i=1}^N$. Now, suppose the knowledge of the system configuration is imperfect, meaning that the unknown \emph{true} configuration of the system is given by the gains and positions $\{g_i,\overrightarrow{a_i}\}_{i=1}^N$, with
\begin{align}
\begin{split}
&g_i = \tilde{g}_i + n_{g,i}, \, n_{g,i} \sim \mathcal{CN}(0,\sigma_g^2),\\
&\overrightarrow{a_i} = \tilde{\overrightarrow{a_i}} + \lambda\mathbf{n}_{p,i}, \, \mathbf{n}_{p,i} = {\small\begin{pmatrix} e_{p,i}, &0, & 0 \end{pmatrix}^T}, e_{p,i} \sim \mathcal{N}(0,\sigma_p^2).
\end{split}
\label{eq:imperfection}
\end{align}
This way, $\sigma_g$ (resp. $\sigma_p$) quantifies the uncertainty about the antenna gains (resp. spacings). Moreover, let
$$
\tilde{\mathbf{e}}(\overrightarrow{u}) \triangleq (
\tilde{g}_1\mathrm{e}^{-\mathrm{j}\frac{2\pi}{\lambda}\tilde{\overrightarrow{a_1}}.\overrightarrow{u}},\dots,
\tilde{g}_N\mathrm{e}^{-\mathrm{j}\frac{2\pi}{\lambda}\tilde{\overrightarrow{a_N}}.\overrightarrow{u}}
)^T
$$
be the nominal steering vector and $\tilde{\mathbf{E}} \triangleq \begin{pmatrix}
\tilde{\mathbf{e}}(\overrightarrow{u_1}),\dots,\tilde{\mathbf{e}}(\overrightarrow{u_A})
\end{pmatrix}$ be a dictionary of nominal steering vectors. The experiment consists in comparing the estimation strategy of \eqref{eq:estim_strat} using the true (perfect but unknown) dictionary $\mathbf{E}$ with the exact same strategy using the nominal (imperfect but known) dictionary $\tilde{\mathbf{E}}$. To do so, we generate measurements according to \eqref{eq:observations} with channels of the form $\mathbf{h} = \mathbf{e}(\overrightarrow{u})$ where $\overrightarrow{u}$ corresponds to azimuth angles chosen uniformly at random, and $\text{SNR}_{\text{in}}$ is set to $10\,\text{dB}$. Then, the dictionaries $\mathbf{E}$ and $\tilde{\mathbf{E}}$ are built by choosing $A=32N$ directions corresponding to evenly spaced azimuth angles. Let $\hat{\mathbf{h}}_{\mathbf{E}}$ be the estimate obtained using $\mathbf{E}$ in \eqref{eq:estim_strat}, and $\hat{\mathbf{h}}_{\tilde{\mathbf{E}}}$ the estimate obtained using $\tilde{\mathbf{E}}$. The SNR loss (performance decrease) caused by using the imperfect but known dictionary $\tilde{\mathbf{E}}$ instead of the perfect but unknown dictionary $\mathbf{E}$ is measured by the quantity
$$
\frac{\Vert \hat{\mathbf{h}}_{\tilde{\mathbf{E}}} - \mathbf{h} \Vert_2^2}{\Vert \hat{\mathbf{h}}_\mathbf{E} - \mathbf{h} \Vert_2^2}.
$$
Results in terms of SNR loss, in average over $10$ antenna array realizations and $1000$ channel realizations per antenna array realization are shown on figure~\ref{fig:imperfect_model}. From the figure, it is obvious that even a relatively small uncertainty about the system configuration can cause a great SNR loss. For example, an uncertainty of $0.03\lambda$ on the antenna spacings and of $0.09$ on the antenna gains leads to an SNR loss of more than $10\,\text{dB}$, which means that the mean squared error undergoes a more than tenfold increase.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.9\columnwidth]{model_imperfection.pdf}
\caption{SNR loss in decibels (dB) due to imperfect knowledge of the system.}
\label{fig:imperfect_model}
\end{figure}
This short experiment simply highlights the fact that using imperfect models can severely harm MIMO channel estimation performance. Note that we took here as an example of imperfection the uncertainty about antenna positions and gains, but many other sources of imperfection will impact real world practical systems. The main contribution of this paper is to propose a method that takes into account and corrects to some extent imperfect physical models using machine learning, and more specifically deep unfolding.
\section{Proposed approach: mpNet}
\label{sec:prop_approach}
Let us now propose a strategy based on deep unfolding allowing to correct a channel estimation algorithm based on an imperfect physical model incrementally, via online learning.
\subsection{Unfolding matching pursuit}
\label{ssec:unfolding_MP}
The estimation strategy summarized in algorithm~\ref{alg:mp} can be unfolded as a neural network taking the observation $\mathbf{x}$ as input and outputting a channel estimate $\hat{\mathbf{h}}$. Indeed, the first step in the while loop amounts to perform a linear transformation (multiplying the input by the matrix $\mathbf{E}^H$) followed by a nonlinear one (finding the inner product of maximum amplitude and setting all the others to zero) and the second step corresponds to a linear transformation (multiplying by the matrix $\mathbf{E}$). Such a strategy is parameterized by the dictionary of steering vectors $\mathbf{E}$. In the case where the optimal dictionary $\mathbf{E}$ is unknown (or imperfectly known), we propose to learn the dictionary matrix used in \eqref{eq:estim_strat} directly on data via backpropagation \cite{Rumelhart1986}.
\noindent{\bf Neural network structure.} This is done by considering the dictionary matrix as weights of the neural network we introduce, called {\sf mpNet}, whose forward pass is given in algorithm~\ref{alg:forwardmpnet}. The notation $\text{HT}_1$ refers to the hard thresholding operator which keeps only the entry of greatest modulus of its input and sets all the others to zero. The parameters of this neural network are the weights $\mathbf{W} \in \mathbb{C}^{N \times A}$, where $A$ is an hyperparameter denoting the number of considered atoms in the dictionary. Note that complex weights and inputs are handled classically by stacking the real and imaginary parts for vectors and using the real representation for matrices. The forward pass of {\sf mpNet} can be seen as a sequence of $K$ iterations whose schematic description is shown on figure~\ref{fig:mpnetK}. The stopping criterion determining the number $K$ of replications of the aforementioned structure, which corresponds to the number of estimated paths, is studied in section~\ref{ssec:stopping_criterion}, with the objective to make the depth of {\sf mpNet} adaptive to the SNR.
\begin{algorithm}[htb]
\caption{Forward pass of {\sf mpNet}}
\begin{algorithmic}[1]
\REQUIRE Weight matrix $\mathbf{W}\in \mathbb{C}^{N\times A}$, input $\mathbf{x}$
\STATE $\mathbf{r} \leftarrow \mathbf{x}$
\WHILE{Stopping criterion not met}
\STATE $\mathbf{r} \leftarrow \mathbf{r} - \mathbf{W}\text{HT}_1(\mathbf{W}^H\mathbf{r})$
\ENDWHILE
\ENSURE $\texttt{FW}(\mathbf{W},\mathbf{x}) \leftarrow \mathbf{x} - \mathbf{r}$
\end{algorithmic}
\label{alg:forwardmpnet}
\end{algorithm}
\begin{figure}[htbp]
\includegraphics[width=\columnwidth]{schema_K.pdf
\caption{One layer of {\sf mpNet}.}
\label{fig:mpnetK}
\end{figure}
\subsection{Training mpNet}
\label{ssec:training_mpnet}
The method we propose to jointly estimate channels while simultaneously correcting an imperfect physical model is summarized in algorithm~\ref{alg:learningmpnet}. Note that {\sf mpNet} is fed with normalized inputs of the form $\frac{\mathbf{x}}{\Vert \mathbf{x}\Vert_2}$, since we noticed it improved its performance. The training strategy amounts to initialize the weights of {\sf mpNet} with a dictionary of nominal steering vectors $\tilde{\mathbf{E}}$ and then to perform a minibatch gradient descent \cite{Bottou2010} on the weights $\mathbf{W}$ to minimize the risk
\begin{equation}
R \triangleq \mathbb{E} \left[\frac{1}{2} \frac{\Vert \mathbf{x} - \hat{\mathbf{h}} \Vert_2^2}{\Vert \mathbf{x} \Vert_2^2}\right].
\end{equation}
Denoting $\mathcal{B}$ the current minibatch of size $B$, the expectation involved in the risk is approximated by computing an average over the minibatch observations, leading to the cost function
\begin{equation}
C \triangleq \frac{1}{2B}\sum_{\mathbf{x} \in \mathcal{B}} \frac{\Vert \mathbf{x} - \hat{\mathbf{h}} \Vert_2^2}{\Vert \mathbf{x} \Vert_2^2}.
\label{eq:cost}
\end{equation}
Note that this cost function evolves with time, since all minibatches are made of different observations, thus allowing real-time adaptation of {\sf mpNet} to changes in the channel distribution.
The network is trained to minimize the average discrepancy between its inputs and outputs, exactly as a classical autoencoder. It operates \emph{online}, on streaming observations $\mathbf{x}_t,\, t=1,\dots,\infty$ of the form \eqref{eq:observations} acquired over time (coming from all users indifferently). Note that, as opposed to the classical unfolding strategies \cite{Gregor2010,Hershey2014,Kamilov2016}, the proposed method is totally \emph{unsupervised}, meaning that it requires only noisy channel observations and no database of clean channels to run. Note that in all the experiments performed in this letter, we use minibatches of $200$ observations and the Adam optimization algorithm \cite{Kingma2014} with an exponentially decreasing learning rate starting at $0.001$ and being multiplied by $0.9$ every $200$ minibatchs. By abuse of notation, we denote $\texttt{Adam}(\mathbf{W},\frac{\partial C}{\partial \mathbf{W}},\alpha)$ the update of the weights $\mathbf{W}$ by the Adam algorithm on cost function $C$ using the learning rate $\alpha$.
\begin{algorithm}[htb]
\caption{Online training of {\sf mpNet}}
\begin{algorithmic}[1]
\REQUIRE Nominal dictionary $\tilde{\mathbf{E}}\in \mathbb{C}^{N\times A}$, minibatch size $B$, learning rate $\alpha$
\STATE Initialize the weights: $\mathbf{W}\leftarrow \tilde{\mathbf{E}}$
\STATE Initialize the cost function: $C\leftarrow 0$
\FOR{$t=1,\dots,\infty$}
\STATE Get observation $\mathbf{x}_t$ following \eqref{eq:observations}
\STATE Estimate the channel (forward pass):\newline $\hat{\mathbf{h}}_t \leftarrow \Vert \mathbf{x}_t \Vert_2\texttt{FW}\left(\mathbf{W},\frac{\mathbf{x}_t}{\Vert \mathbf{x}_t \Vert_2}\right)$
\STATE Increment the cost function:\newline $C \leftarrow C + \frac{1}{2B\Vert \mathbf{x}_t \Vert_2^2}\Vert \mathbf{x}_t - \hat{\mathbf{h}}_t \Vert_2^2$
\IF{$t \mod B = 0$}
\STATE Update the weights (backward pass):\newline $\mathbf{W} \leftarrow \texttt{Adam}(\mathbf{W},\frac{\partial C}{\partial \mathbf{W}},\alpha)$
\STATE Reset the cost function: $C\leftarrow 0$
\ENDIF
\ENDFOR
\end{algorithmic}
\label{alg:learningmpnet}
\end{algorithm}
\begin{figure*}[t]
\begin{subfigure}[b]{0.333\textwidth}
\includegraphics[width=\columnwidth]{left_mp}
\end{subfigure}
\begin{subfigure}[b]{0.333\textwidth}
\includegraphics[width=\columnwidth]{center_mp}
\end{subfigure}
\begin{subfigure}[b]{0.333\textwidth}
\includegraphics[width=\columnwidth]{right_mp}
\end{subfigure}
\caption{Channel estimation performance on synthetic realistic channels for various SNRs and model imperfections.}
\label{fig:multi}
\end{figure*}
\noindent{\bf Computational complexity.} Let us denote $K$ the number of times line~$3$ of algorithm~\ref{alg:forwardmpnet} is executed, which corresponds to the number of estimated channel paths. This number depends on the chosen stopping criterion, which is studied in section~\ref{ssec:stopping_criterion}. The forward pass of {\sf mpNet} costs $\mathcal{O}(KNA)$ arithmetic operations, its complexity being dominated by the multiplication of the input by the matrix $\mathbf{W}^H$ (first block of figure~\ref{fig:mpnetK}).
The backpropagation step costs only $\mathcal{O}(KN)$ arithmetic operations ($A$ times less), since the error flows through only one columns of the weight matrix $\mathbf{W}$ at each step, due to the hard thresholding operation done during the forward pass. This short complexity analysis means that jointly estimating the channel (forward pass) and learning the model (backward pass) is done at a cost that is overall the same order as the one of simply estimating the channel with a greedy algorithm (MP or OMP), without adapting the model at all to data (which corresponds to computing only the forward pass). This very light computational cost makes the method particularly well suited to online learning, as opposed to previously proposed channel estimation strategies based on deep unfolding \cite{He2018,Wei2019,Vanlier2020}.
\subsection{Choosing the stopping criterion}
\label{ssec:stopping_criterion}
{\sf mpNet} is the unfolded version of the matching pursuit algorithm, and its number $K$ of layers corresponds to the number of iterations of the said algorithm. It also represents the number of estimated channel paths. In fact, $K$ is nothing else but an hyperparameter that needs to be optimized. But how can we determine this number appropriately?
In the preliminary version of this study \cite{Lemagoarou2020}, this was done by testing different values of $K$ and choosing the one yielding the best results in terms of relative error in average over channel observations, by cross-validation. In that case, the number of estimated paths was the same for every channel observation. For practical systems, this strategy is suboptimal. Indeed, the number of estimated paths should ideally depend on the SNR: the higher the SNR the more paths can be estimated reliably. The SNR depending on the distance separating the users and the base station, the path loss and gains of the different propagation paths, the depth of {\sf mpNet} should be allowed to vary in order to estimate a number of path adapted to each channel observation.
In order to do so, let us take advantage of the adaptive stopping criteria proposed for greedy sparse recovery algorithms. In \cite{Cai2011,Wu2012}, the authors show that OMP with the stopping criterion
\begin{equation}\label{eq:sc_old}
\mathtt{SC_1}: \,\|\mathbf{r}\|_2^2 \leq \tilde{\sigma}^2 (N+2\sqrt{N\log N}),
\end{equation}
with $\tilde{\sigma}^2\triangleq\frac{\sigma^2}{\|\mathbf{x}\|_2^2}$ is optimal in a support recovery sense. Moreover, for a small number of iterations and an incoherent enough dictionary, MP and OMP give very close results. Hence, we propose to use $\mathtt{SC_1}$ as stopping criterion for {\sf mpNet}. Implementing this stopping criterion requires knowing the noise variance $\sigma^2$ or at least having an estimate $\hat{\sigma}^2$. Fortunately, the noise variance can be estimated quite reliably in MIMO systems \cite{Das2012}. In the sequel, we assume that a perfect noise variance estimate is available ($\hat{\sigma}^2 = \sigma^2$).
Note that practical channels $\mathbf{h}$ and learned weights $\mathbf{W}$ do not follow exactly the generative model used in \cite{Cai2011,Wu2012} to derive the optimal stopping criterion, so that $\mathtt{SC_1}$ may lose its optimality in the case studied here. For this reason, we propose also to use the simpler and more intuitive stopping criterion
\begin{equation}\label{eq:sc_new}
\mathtt{SC_2}: \,\|\mathbf{r}\|_2^2 \leq \tilde{\sigma}^2 N.
\end{equation}
From a neural network perspective, using the stopping criteria $\mathtt{SC_1}$ and $\mathtt{SC_2}$ means training a neural network whose depth is adaptive and dynamically adjusted during learning and inference. The structure of figure~\ref{fig:mpnetK} is indeed replicated until the used stopping criterion is met.
The two stopping criteria $\mathtt{SC_1}$ and $\mathtt{SC_2}$ are empirically compared on realistic synthetic channels in section~\ref{sec:experiments}.
\section{Experimental validation}
\label{sec:experiments}
Let us now assess {\sf mpNet} on realistic synthetic channels. To do so, we consider the SSCM channel model \cite{Samimi2016} in order to generate non-line-of-sight (NLOS) channels at $28\,\text{GHz}$ (see \cite[table~IV]{Samimi2016}) corresponding to all users. We consider the same setting as in section~\ref{sec:impact}, namely a base station equipped with an ULA of $64$ antennas, with an half-wavelength nominal spacing and unit nominal gains used to build the imperfect nominal dictionary $\tilde{\mathbf{E}}$ (with $A=8N$) which serves as an initialization for {\sf mpNet}. The actual antenna arrays are generated the same way as in section~\ref{sec:impact}, using \eqref{eq:imperfection}, and are kept fixed for the whole experiment.
\subsection{Fixed SNR}
\label{xsec:fixed_SNR}
For the first set of experiments, we consider two model imperfections: $\sigma_p = 0.05,\,\sigma_g = 0.15$ (small uncertainty) and $\sigma_p = 0.1,\,\sigma_g = 0.3$ (large uncertainty) to build the unknown ideal dictionary $\mathbf{E}$. The input SNR is fixed during the experiment and takes the values $\{5,10\}\,\text{dB}$.
We compare the performance of various configurations of {\sf mpNet}, namely the version with a fixed number of iterations $K$, with $K$ set to $6$ for the SNR of $5\,\text{dB}$ and to $8$ for the SNR of $10\,\text{dB}$ (determined by cross validation) and the versions with an adaptive stopping criterion, using criteria $\mathtt{SC_1}$ and $\mathtt{SC_2}$ described in section~\ref{ssec:stopping_criterion}.
In addition, the proposed method is compared to the least squares estimator and to the OMP algorithm using the stopping criterion $\mathtt{SC_2}$, with either the imperfect nominal dictionary $\tilde{\mathbf{E}}$ or the unknown ideal dictionary $\mathbf{E}$. Finally, in order to show the interest of the imperfect model initialization, we compare the proposed method to {\sf mpNet} using the well-known Xavier initialization \cite{Glorot2010}.
\noindent{\bf Results.} The results of this experiment are shown on figure~\ref{fig:multi} as a function of the number of channels of the form~\eqref{eq:observations} seen by the base station over time. The performance measure is the relative mean squared error ($\text{rMSE} = \Vert \hat{\mathbf{h}} - \mathbf{h} \Vert_2^2/\Vert \mathbf{h} \Vert_2^2$) averaged over minibatches of $200$ channels.
Several comments are in order:
\begin{enumerate}[leftmargin=*]
\item The imperfect model is shown to be well corrected by {\sf mpNet}s, the green and the two blue curves being very close to the red one (ideal unknown dictionary) after a certain amount of time. This is true both for a small uncertainty and for a large one and at all tested SNRs. Note that using the nominal dictionary (initialization of {\sf mpNet}) may be even worse than the least squares method, showing the interest of correcting the model, since {\sf mpNet} always ends up outperforming the least squares, thanks to the learning process.
\item Comparing the leftmost and center figures, it is interesting to notice that learning is faster and the attained performance is better with a large SNR (the green and the two blue curves get closer to the red one faster), which can be explained by the better quality of data used to train the model.
\item Comparing the leftmost and rightmost figures, it is apparent that a smaller uncertainty, which means a better initialization since the nominal dictionary is closer to the ideal unknown dictionary, leads to a faster convergence, but obviously also to a smaller improvement.
\item Looking at the purple curve on all figures, it is apparent that initialization matters. Indeed, the random initialization performs much worse than the initialization with the nominal dictionary and takes longer to converge.
\item The green and the two blue curves are all close to each other. In terms of performance, the light blue curve (corresponding to the use of $\mathtt{SC_2}$ as a stopping criterion) always leads to the best performance, followed by the green curve (corresponding to a fixed depth) and finally the dark blue curve (corresponding to the use of $\mathtt{SC_1}$). The central figure shows that with a lower SNR, the gap between the dark blue and the rest of the {\sf mpNet} curves is more pronounced. To further understand the difference between $\mathtt{SC_1}$ and $\mathtt{SC_2}$, we show on figures~\ref{fig:K_histo_SC1} and \ref{fig:K_histo_SC2} histograms of the selected depths for the experimental settings corresponding to the leftmost figure in figure~\ref{fig:multi}. We see that, for $\mathtt{SC_1}$, the distribution is centered at $K=5$ which is 3 iterations behind the network with the best results at a fixed depth. However, for $\mathtt{SC_2}$, the distribution is centered at $K=8$ which is the optimal value for a fixed depth. Moreover, the $\mathtt{SC_2}$ version outperforming the fixed depth one is a result of its adaptability to each channel observation.
\end{enumerate}
These conclusions are very promising and highlight the applicability of the proposed method.
\begin{figure}[htp]
\centering
\includegraphics[clip,width=0.85\columnwidth]{old_sc_histo}%
\caption{Histograms of depths selected by {\sf mpNet} when equipped with $\mathtt{SC_1}$.}
\label{fig:K_histo_SC1}
\end{figure}
\begin{figure}[htp]
\centering
\includegraphics[clip,width=0.85\columnwidth]{new_sc_histo}%
\caption{Histograms of depths selected by {\sf mpNet} when equipped with $\mathtt{SC_2}$.}
\label{fig:K_histo_SC2}
\end{figure}
\subsection{Varying SNR}
In practical scenarios, the SNR is not fixed, and rather depends on the distance between the transmitter and the receiver as well as propagation conditions. This variability has to be taken into account for the learning algorithm to be efficiently optimized. Using $\mathtt{SC_2}$, which takes into account both the noise level and the signal intensity, suggests that our model is capable of automatically adapting to the SNR. To verify this claim, we consider the same SSCM channel model described in the previous section, but this time, the distance between the transmitter and the receiver is sampled from a uniform distribution ranging from $60\,\text{m}$ to $200\,\text{m}$. Our model is then compared to its fixed depth versions for a selected set of values of $K$ ($\{3,6,8,14\}$), as well as to the other previously presented estimation methods. The results are shown in figure~\ref{fig:variable_snr}. While all versions of {\sf mpNet} are capable of learning overtime, the adaptive one is clearly outperforming the others throughout the whole experiment, starting once again at the same error level as OMP equipped with the nominal dictionary (the maroon curve). Furthermore, compared to the precedent experiment at a fixed SNR, the gap between the fixed depth and the adaptive version is more pronounced. The performance of {\sf mpNet} with $\mathtt{SC_2}$ is indeed very close in terms of performance in this more realistic scenario to what is achievable with a perfect knowledge of the physical parameters (here represented by the red curve).
Finally, figure~\ref{fig:snr_histo} shows how the SNR of the generated channel observations is distributed. Note that we choose not to consider observations with a low SNR ($<1\,\text{dB}$) as the model struggles to learn from very noisy instances. In practice, this could be achieved by setting a threshold on the intensity of received signals as a way to filter out useless observations. This leads to a truncated normal distribution of the SNR centered at $10\,\text{dB}$.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.9\columnwidth]{snr_histo}
\caption{SNR distribution over 5000 generated channel observations.}
\label{fig:snr_histo}
\end{figure}
\begin{figure}[htbp]
\includegraphics[width=\columnwidth]{variable_snr}
\caption{Channel estimation performance on synthetic realistic channels for a varying SNR.}
\label{fig:variable_snr}
\end{figure}
\subsection{Anomaly detection and recovery}
\begin{figure*}[t]
\begin{subfigure}[b]{0.333\textwidth}
\includegraphics[width=\columnwidth]{break_low}
\end{subfigure}
\begin{subfigure}[b]{0.333\textwidth}
\includegraphics[width=\columnwidth]{break_medium}
\end{subfigure}
\begin{subfigure}[b]{0.333\textwidth}
\includegraphics[width=\columnwidth]{break_high}
\end{subfigure}
\caption{Adaptation to antenna damage. The horizontal bar at the middle marks the moment at which the break happens.}
\label{fig:break}
\end{figure*}
One of the benefits of online learning is the continuous adaptation of the model to incoming data. If this data is disturbed, the impact would be observed on the cost function used by the system. Indeed, the distribution of the new data would be different from the one on which the system has learned so far. The speed at which the error increases would be proportional to the rate at which the distribution of the new data shifts from its original state. Furthermore, this increase in the error is simultaneously compensated by the training that is done on the network. This behavior may prove useful for detecting and recovering from anomalies. Anomalies could occur for many reasons in a massive MIMO system, including:
\begin{itemize}
\item Bent or broken antennas due to natural causes
\item Improperly adjusted antennas after a technical intervention
\item Disoriented array due to wind
\end{itemize}
In this section, we propose to test the ability of our model to detect and adapt to various types of anomalies.
\noindent{\bf Out of order antennas.} Let us first simulate antennas that go out of order, in a base station equipped with 64 antennas. This is done by setting some of the antenna gains to zero at a certain point of time during training. In this case, we consider 3 scenarios: 10\%, 30\% and 50\% of broken antennas (chosen uniformly at random). Similarly to previous experiments, the channels are generated following the SSCM model at a variable SNR and the adaptive version of {\sf mpNet} with $\mathtt{SC_2}$ is compared to the other estimation methods. Figure~\ref{fig:break} shows the results. It can be seen that the training starts as usual with our model slowly approaching the performance of OMP with the optimal dictionary. The anomaly can be observed at the middle of training when the number of seen channels reaches ${\sim} 100000$ channels. All the estimation methods see their error jump, with the exception of LS where no change is observed. The amount by which the error increases is proportional to the number of broken antennas. The error starts decreasing again for {\sf mpNet} as the training resumes, but naturally stays the same for OMP based methods that do not correct the dictionary they use. By the end of the experiment, and depending on the number of broken antennas, {\sf mpNet} can completely recover from the damage and its error reaches once again the level it successfully attained right before the anomaly.
The LS estimation method does not depend on any physical parameter, which explains the stable error level it displayed throughout the experiment. OMP methods, however, see their error increase because the physical parameters they are based on become less precise and thus induce a bigger error. Those methods are hence capable of detecting the exact moment where the damage happens but are incapable of adapting. On the other hand, and for the same reason, {\sf mpNet} is also capable of detecting the anomaly but rapidly adapts its parameters (dictionary). In practice, the detection could be implemented via a simple threshold.
Note that the anomaly recovery only concerns the channel estimation performance. Indeed, a base station with broken antennas will always be less efficient than a fully functional one, especially in terms of channel capacity. In summary, the model does the best it can on the channel estimation task with the available means.
\begin{figure*}[htbp]
\begin{subfigure}[b]{0.333\textwidth}
\includegraphics[width=\columnwidth]{aging_low}
\end{subfigure}
\begin{subfigure}[b]{0.333\textwidth}
\includegraphics[width=\columnwidth]{aging_medium}
\end{subfigure}
\begin{subfigure}[b]{0.333\textwidth}
\includegraphics[width=\columnwidth]{aging_high}
\end{subfigure}
\caption{Adaptation to antenna aging. The horizontal bars mark the different moments at which the aging happens.}
\label{fig:aging}
\end{figure*}
\noindent{\bf Aging antenna array.} The second type of anomaly we consider is antenna aging. It corresponds to antenna gains slowly shifting away from their initial values. Considering antenna aging, using the initially ideal dictionary will lead to an increase of the error over time since the physical parameters are less and less precisely known. To simulate this phenomenon, we consider the same settings as for the precedent experiment. We then iteratively add noise to the antenna gains over the course of training. This is done for 10 iterations. Antenna gains at iteration $t$ are thus expressed as
\begin{equation}
g_{i,t} = g_{i,t-1} + n_{i}, \, n_{i} \sim \mathcal{CN}(0,\sigma_a^2),
\end{equation}
where $\sigma_a$ could be seen as a measure of the severity (speed) of aging. We consider 3 levels of aging: $\sigma_a=0.05$ (mild), $\sigma_a=0.1$ (medium) and $\sigma_a=0.2$ (severe). Again, channels are generated following the SSCM model at a variable SNR and the adaptive version of {\sf mpNet} with $\mathtt{SC_2}$ is compared to the other estimation methods. Results are shown on figure~\ref{fig:aging}. We observe that both OMP based estimations see their error progressively increase at a rate proportional to the severity of aging, but performance worsens more rapidly when starting with an ideal dictionary. On the other hand, the neural network continues to learn and the impact of aging is barely noticeable. {\sf mpNet} online learning compensates for the error induced by the continuous change of physical parameters. Finally, no noticeable change is observed on the LS estimation method.
\subsection{From ULA to UPA}
\begin{figure}[b]
\includegraphics[width=\columnwidth]{UPA}
\caption{Channel estimation performance on synthetic realistic channels for a base station equipped with a UPA.}
\label{fig:UPA}
\end{figure}
The physical model on which is based {\sf mpNet} is structure-agnostic, meaning that it is meant to work with any antenna array structure. This suggests that our model, which was initially tested on ULAs, is capable of working with any structure as well. To verify this claim, we propose to adapt it to uniform planar arrays (UPAs). A change in the way steering vectors are generated is required to take into account the rotational symmetry that was verified for ULAs and that no longer holds for UPAs. Therefore, the dictionary of steering vectors has to be built from DoAs sampled from the whole 3D half space, instead of a half-plane in the case of ULAs.
We conducted an experiment similar to the ones described in section~\ref{sec:experiments}. We consider an UPA consisting of a square grid of $8\times8$ antennas ($N=64$) separated by half-wave lengths and placed on the $xz$-plane . Channels are generated at a fixed SNR of $10\,\text{dB}$. To take into account the additional dimension of UPAs, ideal antenna positions are this time given by
\begin{equation}
\overrightarrow{a_i} = \tilde{\overrightarrow{a_i}} + \lambda\mathbf{n}_{p,i}, \, \mathbf{n}_{p,i} = {\small\begin{pmatrix} e_{x,p,i}, &0, & e_{z,p,i} \end{pmatrix}^T},
\end{equation}
with $e_{x,p,i},e_{z,p,i} \sim \mathcal{N}(0,\sigma_p^2)$.
We compare the adaptive version of {\sf mpNet} equipped with $\mathtt{SC_2}$ to other estimation methods. The results are shown on figure~\ref{fig:UPA}. Once again, the model successfully learns overtime reducing its estimation error while the other classical methods maintain their performances.
This small experiment can be seen as a sanity check to show that, indeed, the model is capable of accepting any antenna structure with minimal changes in the implementation.
\section{Conclusion and perspectives}
\label{sec:conclusion}
In this paper, we introduced {\sf mpNet}: a neural network allowing adding flexibility to physical models used for MIMO channel estimation. It is based on the deep unfolding strategy that views a classical algorithm (matching pursuit in this case) as a neural network, whose parameters can be trained. The proposed method was shown to correct incrementally (via online learning) an imperfect or imperfectly known physical model in order to make channel estimation as efficient as if the unknown ideal model were known. It is trained in an unsupervised manner as an autoencoder, and {\sf mpNet} can be seen as a denoiser for channel observations. Training {\sf mpNet} thus does not necessitate a database of clean channels, nor an offline training phase, which makes it particularly attractive for practical systems and unalike previously proposed methods.
We have shown that initializing the network with a dictionary of imperfect steering vectors (as opposed to using a random initialization) improves performance considerably. In the experimental part of the paper, we simulated the model imperfection by introducing uncertainties on the antenna gains and positions, but it is important to highlight the fact that the method could in principle correct many other model imperfections (such as uncertainties about the antenna diagrams, couplings between neighboring antennas, etc.).
Moreover, we introduced a stopping criterion, inspired by previous work on the OMP algorithm, to dynamically select the optimal depth of {\sf mpNet}. This was shown to be particularly convenient when working on observations with a varying SNR level, which more accurately resembles real world channel observations. Evaluated on realistic synthetic data, this approach showed great results compared to other methods.
In addition, online learning enabled us to exploit the observed change in data distribution following an anomaly occurrence to detect and recover from it. We simulated two types of anomalies: antenna damage and aging. In both cases, our model was capable of efficiently recovering from the decrease in performance overtime.
Finally, we proved that our model is capable of adapting to any antenna structure with no apparent drop in performance or change in behavior. In particular, we showed that a simple change in the steering vector generation process was required to adapt the model, previously based on a ULA structure, to a UPA structure.
In future work, we could explore the unfolding of more sophisticated sparse recovery algorithms (such as iterative soft thresholding \cite{Daubechies2004} or approximate message passing \cite{Donoho2009}) using the same strategy. In addition, other stopping criteria could also be integrated to the model and tested. Finally, the deep unfolding of the matching pursuit algorithm initialized with a dictionary based on an imperfect model is by no means limited to the MIMO channel estimation task and could be exploited for other tasks, as long as an initial model is available.
\bibliographystyle{unsrt}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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\section{Introduction}
Finding a unified field theory of gauge bosons and gravitons has been a holy grail of physics since the days of Einstein, Heisenberg and Pauli.
While Yang-Mills theory and Einstein gravity share common features such as local symmetries, their actions
appear to be very different. In particular, their perturbative quantization in weakly interacting situations
strongly differ: while the Feynman diagrammatic expansion for gravity is notoriously involved and not renormalizable,
the Yang-Mills case is under good control and the basis of high precision predictions for scattering experiments at ever increasing orders in perturbation theory.
This is why the surprising construction of Bern-Carrasco-Johansson \cite{Bern:2008qj,Bern:2010ue} for the integrand of quantum gravitational
scattering amplitudes in terms of a double copy of Yang-Mills ones has been highly inspirational and points to a surprisingly
direct and intimate connection between these two fundamental theories of nature, see \cite{Bern:2019prr} for a recent review.
Concretely, the double copy construction arranges the building blocks of gluon scattering amplitudes in terms of kinematic numerators,
color factors and scalar propagators in such a fashion, that the kinematic numerators obey identities akin to
the Jacobi identity constitutional for the color factors resulting from the color gauge symmetry. Replacing the color factors by the
thereby identified kinematic numerators of the gluon amplitudes then
yields the integrands of scattering amplitudes in axion-dilaton-gravity (or $\mathcal{N}=0$ supergravity, i.e.~the low energy limit
of the bosonic string). Formal proofs of the double copy have been provided at tree-level
using a variety of methods.
This ``color-kinematic duality'' has been extended to a large class of theories and the question of which (gravitational) theories
admit a double copy is an interesting and in general open one.
At the same time the double copy presents a very efficient tool to construct amplitude integrands in (super)gravity to very high loop orders, see \cite{Bern:2019prr} for an account. Yet, its deeper nature, in particular the nature of the `kinematic algebra', remains ill understood and is a subject of intense research in the modern amplitude program, see \cite{Dixon:1996wi,*Elvang:2013cua,*Henn:2014yza} for reviews.
The double copy relation for quantum scattering amplitudes leads to a natural challenge for classical general relativity. Namely, is there a classical double
copy transforming solutions of Yang-Mills (or Maxwell's) theory to gravity (with a dilaton) as well? In fact a number of such constructions
has been provided \cite{Monteiro:2014cda,*Luna:2015paa,*Luna:2016hge,*Carrillo-Gonzalez:2017iyj,*Guevara:2020xjx,*Monteiro:2020plf,*White:2020sfn,*Chacon:2021wbr,*Godazgar:2021iae}, most prominently perhaps for the Kerr-Newman solution of a spinning black-hole \cite{Arkani-Hamed:2019ymq}.
Yet, from the quantum origin of the color-kinematic
duality it should be clear that the \emph{perturbative} nature of the classical solution, i.e.~the systematic expansion about a
flat Minkowski background (known as the post-Minkowskian expansion in general relativity), should be central for the existence of a classical double copy prescription. Fascinatingly, this is also the domain relevant for analytic gravitational wave physics, describing the inspiral (or scattering) of two massive bodies (black-holes, neutron stars or stars). The emitted gravitational radiation in that two-body process being detectable in present and future gravitational wave detectors (without the dilaton, and for the bound system in the post-Newtonian expansion, which is a combined weak field and slow motion expansion).
The approach of constructing perturbative classical solutions via the double copy was pioneered
in \cite{Goldberger:2016iau}. Here the scattering of two point particles via dilaton and graviton interactions was constructed via a double
copy of the perturbative solutions of the Yang-Mills equations coupled to point particles carrying color charge. This was further extended and clarified to the next-to-next-to leading order (NNLO) by Shen in \cite{Shen:2018ebu}, spin effects and extensions were studied in \cite{Goldberger:2017vcg,Goldberger:2017ogt,Goldberger:2019xef}
and the analogue problem in bi-adjoint scalar field theory was considered in \cite{Goldberger:2017frp,Bastianelli:2021rbt}. All these approaches operate at the level
of the equations of motion, i.e.~one perturbatively solves both the field and particle equations of motion.
An alternative route was taken in the works \cite{Plefka:2018dpa,Plefka:2019hmz}, involving the present authors, where
a path integral based approach was taken. Here, starting from the actions describing the coupling of massive, charged particles to Yang-Mills or dilaton-gravity the force mediating fields (gluons, dilatons and graviton) were integrated out yielding
an effective action for the point particles, thereby taking the classical $\hbar\to 0$ limit.
It was shown at LO and NLO that the resulting effective action could be obtained by a suitably generalized double copy
prescription \cite{Plefka:2018dpa} taking inspiration from the amplitudes approach. Concretely, the need for a trivalent graph structure was artificially introduced via delta functions on the worldline for higher valence worldline-bulk field vertices.
Yet, this double copy prescription was shown to break down for the effective action at the NNLO \cite{Plefka:2019hmz}.
It was speculated in \cite{Plefka:2019hmz} that the reason for this breakdown lies in the attempt of double copying a gauge-variant and off-shell quantity -- the effective action -- which is at tension with the on-shell nature of the scattering amplitude double copy.
Returning to the realm of applying quantum field theory based techniques to the post-Minkowskian perturbative gravitational
scattering problem, an approach termed worldline quantum field theory (WQFT) was put forward recently \cite{Mogull:2020sak,Jakobsen:2021zvh} that explains the relation between the two presently common approaches employing the
classical limit of scattering amplitudes, see
e.g.~\cite{Kosower:2018adc,delaCruz:2020bbn,Bern:2019crd,Damour:2017zjx,*DiVecchia:2021bdo,*Bjerrum-Bohr:2018xdl,*Cheung:2018wkq,*Bern:2019nnu,*Bjerrum-Bohr:2021din,*Herrmann:2021tct,*Brandhuber:2021eyq,*Bern:2021dqo,*Cristofoli:2021vyo}, and the PM effective field theory (EFT) approach,
see e.g.~\cite{Porto:2016pyg,*Levi:2018nxp,*Kalin:2020mvi,*Kalin:2020fhe,*Dlapa:2021npj}.
In essence it starts out from a \emph{first quantized} description of the matter field's (scalar, spinor, vector) propagator
in a gravitational background \`a la Feynman-Schwinger and demonstrates how this leads in a classical limit to the
effective field theory description. Yet, the WQFT formalism not only is of conceptual relevance (introducing an
approximate supersymmetry in the description of spin for black holes and neutron stars \cite{Jakobsen:2021zvh}),
it also provides a very efficient tool to quickly arrive at the classical observables of the scattering process without
the need to go through an iterative solution of the equations of motion in the standard EFT approach or to deal with the subtleties of the classical
limit in the amplitudes based approach. As such the deflection, the spin-kick or the explicit gravitational Bremsstrahlung waveform as well as the eikonal being the generating function of these have been established at NLO \cite{Jakobsen:2021smu,Jakobsen:2021lvp,Jakobsen:2021zvh}.
In this work we therefore address the conceptually important question whether a double copy prescription exists for the
WQFT as well? On the face of it this is to be expected, as the WQFT may be thought of as a (partially) first order form
of the scattering amplitude problem. For this we first settle for the relevant worldline quantum field theories in section
\ref{sec2}. This includes in particular the case of the worldline coupled bi-adjoint scalar field theory which we indeed require
in order to separate the kinematic numerators from the propagator terms in the Yang-Mills case - a prerequisite for the
double copy construction. In section \ref{sec2} we establish the relevant Feynman rules for all three WQFTs, coupling to
bi-adjoint scalars, Yang-Mills and dilaton-gravity, as well as point out the relevance of the eikonal or free energy of the
WQFT as generating function for the observable of the particle's deflections. In section \ref{Se:doublecopy} we develop the
classical double copy prescription for WQFTs and construct the eikonal at NLO level. Section \ref{sec3} is devoted to the lift
of our results for the (three-particle) eikonal to the radiative waveform and in section \ref{sec4} we detail the relation
of our WQFT double copy to the established one for the quantum scattering amplitudes. After concluding we collect our conventions
in the appendix.
\section{Worldline quantum field theories}
\label{sec2}
We wish to apply the worldline quantum field theory formalism \cite{Mogull:2020sak} to massive point particles coupled to bi-adjoint scalar field theory (BS), Yang-Mills theory (YM) and dilaton-gravity (DG).
Compactly, the actions for the three theories may be written as
\begin{align}
\label{eq:actionWQFT}
S^{\mathrm{WQFT}} = S^{\mathrm{BS/YM/DG}} + \sum_{i} S^\mathrm{cc/pc/pm}_i,
\end{align}
where $S^{\mathrm{BS/YM/DG}}$ is the respective field theory action and
$S^\mathrm{cc/pc/pm}_i$ the respective $i$'th particle worldline action.
Note that multiple worldlines are included in eq.\eqref{eq:actionWQFT} in order to allow for interactions.
We will now focus on the worldline actions $S^\mathrm{cc/pc/pm}_i$ and delegate the rather well known field theory actions into the Appendix \ref{Ap:convention}.
The action of a massive point charge coupled to a non-abelian gauge field $A_{\mu}^{a}$ is \cite{Bastianelli:2013pta, Bastianelli:2015iba}
\begin{align} \label{eq:actionPC}
S^\mathrm{pc} = \!-\! \int \! \mathrm{d}\tau
\left( \frac{m}{2} \left(e^{-1} \dot{x}^2 \!+\! e \right)
\!-\! i \Psi^\dagger \dot{\Psi}
\!-\! g \dot{x}^\mu A_\mu^a C^a\right),
\end{align}
where $e(\tau)$ is the einbein,
and the dot over a symbol denotes a derivative with respect to $\tau$.
The ``color wave function'' $\Psi_{\alpha}(\tau)$ is an auxiliary field carrying the color degrees of freedom of the particle,
the $\alpha,\beta,\ldots=1,\ldots, d_{R}$ are indices of the $d_{R}$ dimensional representation of the gauge group,
and $C^a(\tau) = \Psi^{\dagger\alpha} (T^a)_{\alpha}{}^{\beta} \Psi_{\beta}$ is the associated color charge
that determines the coupling to the gauge field $A^{a}_\mu(x)$. We shall take the generators $(T^a)_{\alpha}{}^{\beta}$ to be in the fundamental of $SU(N)$ such that $d_{R}=N$ and the adjoint indices $a,b,\ldots=1,\ldots N^{2}-1$.
This action is invariant under the reparametrization of $\tau$.
The kinetic term can be transformed into the more familiar form $- m\int \mathrm{d}\tau \sqrt{\dot{x}^2}$
by solving the algebraic equations of motion for the einbein $e(\tau)$ and reinserting the solution into the action.
However, for convenience we will fix $e(\tau)=1$ such that $ \dot{x}^2=1 $ and $\tau$ is then the proper time.
Similarly, the action of a worldline minimally coupled to dilaton-gravity reads
\begin{align}
\label{eq:Spmdef}
S^{\mathrm{pm}} =
- \frac{m}{2} \int \mathrm{d} \tau
\left( e^{-1} e^{2\kappa \varphi} g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu
+ e \right),
\end{align}
where $\varphi(x)$ is the dilaton and $e(\tau)$ is again the einbein.
The coupling constant is $\kappa = \sqrt{32 \pi G}$, where $G$ is Newton's constant.
Again, upon integrating out $e(\tau)$ we arrive at the more common form of the action
$- m \int \mathrm{d} \tau e^{\kappa \varphi} \sqrt{g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu }$.
We gauge fix $e(\tau)=1$ so that $e^{2\kappa \varphi} g_{\mu\nu} \dot{x}^\mu \dot{x}^\nu = 1$.
In the weak gravitational field limit we expand
\begin{align} \label{eq:perturbG}
g_{\mu\nu} = \eta_{\mu\nu} + \kappa\, h_{\mu\nu},
\end{align}
where $\eta_{\mu\nu}$ is the flat-space Minkowskian metric.
Because the dilaton appears as an exponent in the action, the interaction terms become cumbersome in perturbation theory of both the worldline and field theory actions.
To simplify the calculation, we adopt the gravity gauge-fixing term and field redefinitions of $\{\varphi, h_{\mu\nu}\}$ introduced in \cite{Plefka:2018dpa} eqs.~(51)-(56).
This will decouple the worldline from $\varphi$ up to quadratic order, as well as recast the cubic
self-interaction of $h_{\mu\nu}$ into a simpler form.
The redefined field theory action can be found in \eqn{eqA:dilgravfinal} of Appendix \ref{Ap:convention}.
The worldline action in terms of the redefined fields then reads
\begin{align}
S^{\mathrm{pm}} =
- \frac{m}{2} \int \mathrm{d} \tau
\Big(& \dot{x}^2 + \kappa h_{\mu\nu} \dot{x}^\mu \dot{x}^\nu \nonumber \\
&+ \frac{\kappa^2}{2} h_{\mu\rho}h_{\nu}^{\ \rho} \dot{x}^\mu \dot{x}^\nu
\Big)
+ \mathcal{O}(\kappa^3).
\end{align}
The indices are lowered or raised by the Minkowskian metric.
Finally, let us introduce the massive point particle coupling to a bi-adjoint scalar field theory.
Here, we use the bi-adjoint scalar theory to identify the double copy kernel as introduced in \cite{Shen:2018ebu}.
A point particle interacting with a bi-adjoint scalar field (WBS) is described by \cite{Shen:2018ebu,Bastianelli:2021rbt}
\begin{multline} \label{eq:Sccaction}
S^\mathrm{cc} =
- \int \! \mathrm{d}\tau
\Big( \frac{m}{2} \left(e^{-1} \dot{x}^2 + e \right)
- i \Psi^\dagger \dot{\Psi}
- i \tilde{\Psi}^\dagger \dot{\tilde{\Psi}} \\
-e \frac{y}{m} \phi_{a \tilde{a}} C^{a} \tilde{C}^{\tilde{a}} \Big),
\end{multline}
where $y$ is the coupling constant and $\phi_{a \tilde{a}}(x)$ is the bi-adjoint scalar field carrying two distinct color indices $a$
and $\tilde a$ related to the color and dual-color gauge groups respectively.
$\Psi_{\alpha}(\tau)$ and $\tilde{\Psi}_{\tilde\alpha}(\tau)$ are the color and dual color wave functions.
The corresponding charges are defined in a similar way as in $S^{\text{pc}}$ of \eqn{eq:actionPC}:
$C^a = \Psi^{\dagger} T^a \Psi$ and
$\tilde{C}^{\tilde{a}} = \tilde{\Psi}^{\dagger} \tilde{T}^{\tilde{a}} \tilde{\Psi}$.
Note that setting $e(\tau) = 1$ in this case will enforce the constraint $\dot{x}^2 + \frac{2y}{m^2} \phi_{a \tilde{a}} C^a \tilde{C}^{\tilde{a}} =1$.
In the worldline quantum field theory (WQFT) formalism, describing the scattering of two particles, we expand the coordinate $x^{\mu}(\tau)$ along a straight line trajectory background
\begin{align}
x^\mu(\tau) &= b^\mu + v^\mu \tau + z^\mu(\tau),
\end{align}
with $v^2=1$ and the fluctuation $z^{\mu}$. Not that the straight line background solves the equations of motion in the field free scenario(s) $\phi_{a\tilde{a}}=A_{\mu}^{a}=h_{\mu\nu}=\varphi=0$.
We take $b \cdot v = 0$ which may always be achieved upon shifting $\tau$.
As explained in \cite{Mogull:2020sak}, the physical meanings of $b^\mu$ and $v^\mu$ depend on the type of the worldline propagator
(advanced/retarded/time symmetric).
As in this work our main concern for the double copy construction is the integrand the $i\epsilon$ description of the propagators is of no direct concern.
Likewise, we decompose the color wave function in the background
\begin{align}
\Psi(\tau) &= \psi + \varPsi(\tau),
\end{align}
where $\psi = \Psi(-\infty)=\text{const}$ is the initial condition, and $\varPsi(\tau)$ is the fluctuation that will be quantized.
Consequently, the color charge is
\begin{align}
C^a = c^a + \psi^\dagger T^a \varPsi + \varPsi^\dagger T^a \psi + \varPsi^\dagger T^a \varPsi,
\end{align}
where we have defined the background color charge
$c^a = \psi^\dagger T^a \psi$.
A similar decomposition applies to the dual color wave function $\tilde{\Psi}(\tau)$, and all respective dual quantities are denoted with a tilde.
In WQFT, the physical observables are computed as the expectation values of the corresponding operators.
We will integrate out the BS/YM/DG fields $\phi_{a\tilde{a}}; A_{\mu}^{a}; h_{\mu\nu},\varphi$ as well as all fluctuations of worldline degrees of freedom $z(\tau), \varPsi(\tau), \tilde{\varPsi}(\tau)$ in the path integral, so the results only depend on the background fields $b, v, \psi$.
In the path integral, the expectation value of an operator $\mathcal{O}$ is expressed as
\begin{align}\label{eq:expectation}
\langle \mathcal{O} \rangle = \frac{1}{\mathcal{Z_\mathrm{WQFT}}} \!\int D[\Phi] \!
\prod_i D[z_i] \left(D[\varPsi_i, \tilde{\varPsi}_i] \right)
\mathcal{O}\, e^{i S^\mathrm{WQFT}},
\end{align}
where $\Phi \in \{ \phi_{a \tilde{a}}, A_\mu^a, h_{\mu\nu}, \varphi \}$ denotes the bosonic fields in the respective theories.
$\mathcal{Z_\mathrm{WQFT}}$ is the partition function,
\begin{align}
\mathcal{Z}_\mathrm{WQFT} = \int D[\Phi] \!
\prod_i D[z_i] \left(D[\varPsi_i, \tilde{\varPsi}_i] \right)
e^{i S^\mathrm{WQFT}}.
\end{align}
In the binary case ($i=1,2$) $\mathcal{Z_\mathrm{WQFT}}$ may be identified with the exponentiated eikonal phase $\chi$.
The momentum deflection of a particle $\Delta p_i^\mu$ can be calculated by taking the derivative of the eikonal with respect to $b_i^\mu$.
Here we claim that this relation holds for an arbitrary number of worldlines.
We will provide a simple proof in the path integral formalism.
Let us first consider the derivative of $\ln \mathcal{Z_\mathrm{WQFT}}$ with respect to $b_i^\mu$,
\begin{align} \label{eq:logZtob}
i \frac{\partial \ln \mathcal{Z}_\mathrm{WQFT}}{\partial b_i^\mu} =
\left\langle
- \frac{ \partial S^\mathrm{WQFT}} {\partial b_i^\mu} \right\rangle
=- \int_{-\infty}^{+\infty}\!\! \mathrm{d} \tau
\left\langle \frac{ \partial L^\mathrm{pp}} {\partial x_i^\mu} \right\rangle,
\end{align}
where in the last step we exploit the fact that in the full action $b_i^\mu$ only appears as the $\tau$-independent background of $x_i^\mu(\tau)$ in the point particle action $S^{\mathrm{pp}} = \int \mathrm{d} \tau L^{\mathrm{pp}}$, where $L^{\mathrm{pp}}$ is the Lagrangian.
As the expectation value of the equation of motion for $x(\tau)$ vanishes, we can rewrite eq.\eqref{eq:logZtob} as
\begin{align}
i \frac{\partial \ln \mathcal{Z}_\mathrm{WQFT}}{\partial b_i^\mu}
= -\int_{-\infty}^{+\infty}\!\! \mathrm{d} \tau
\left\langle \frac{\mathrm{d} }{\mathrm{d} \tau} \frac{\partial L^\mathrm{pp}} {\partial \dot{x}_i^\mu} \right\rangle
= \left. \left\langle p^{\mathrm{can}}_{i, \mu} \right\rangle \right|_{-\infty}^{+\infty},
\end{align}
where $p^{\mathrm{can}}_{i, \mu} = -\partial L^\mathrm{pp}/{\partial \dot{x}_i^\mu} $ is the canonical momentum conjugated to $x^\mu$.
Since we are studying a scattering process, in past and future infinity we may assume that the point particles are so far separated that the interaction terms vanish.
In this case $p^{\mathrm{can}}_{i, \mu}$ reduces to the kinematic momentum
$m_i \dot{x}_i^\mu$, so we have
\begin{align}
m_i \Delta \dot{x}_i^\mu
=
i \frac{\partial \ln \mathcal{Z}_\mathrm{WQFT}}{\partial b_{i,\mu}}.
\end{align}
Therefore in this letter, we define the generalized eikonal phase for more than two worldlines as,
\begin{align}
\chi = -i \ln \mathcal{Z}_\mathrm{WQFT}.
\end{align}
In section \ref{Se:doublecopy}, we will perform a double copy for the eikonal to next-to-leading order.
Since we will mostly work in momentum space, it will be useful to express the worldline fluctuations as
\begin{equation}
\begin{split}
\label{eq:wlFourier}
z^\mu(\tau) &= \int_{\omega} e^{-i \omega \tau} z^{\mu}(\omega), \\
\varPsi(\tau) &= \int_{\omega} e^{-i \omega\tau} \varPsi(\omega), \\
\varPsi^\dagger(\tau) &= \int_{\omega} e^{-i \omega \tau} \varPsi^{\dagger}(-\omega) \, .
\end{split}
\end{equation}
The dual color wave function $\tilde{\varPsi}$ in momentum space is defined in the same way as $\varPsi$.
For convenience we use the integral shorthands
$\int_\omega := \int \frac{\mathrm{d}\omega}{2\pi}$,
$\int_k := \int \frac{\mathrm{d}^4 k}{(2\pi)^4}$ as well as $\delta\!\!\!{}^-\!(\omega) := 2\pi \delta(\omega)$ and
$\delta\!\!\!{}^-\!^{(4)} (k^\mu) := (2\pi)^4 \delta^{(4)} (k^\mu)$.
When evaluated on the worldline, the generic field $\Phi$ may be expanded as
\begin{widetext}
\begin{align}
\label{eq:decompose}
\Phi(x(\tau))
&=\int_{k} e^{i k \cdot(b+v \tau+z(\tau))} \Phi(-k)
= \sum_{n=0}^{\infty} \frac{i^{n}}{n !} \int_{k} e^{i k \cdot(b+v \tau)}(k \cdot z(\tau))^{n} \Phi(-k) \nonumber \\
&=\int_{k} e^{i k \cdot b} \Phi(-k)
\left( e^{i k \cdot v\tau}
+ i \int_\omega e^{i (k \cdot v + \omega) \tau} k \cdot z(-\omega)
\right) + \mathcal{O}(z^2).
\end{align}
\end{widetext}
We take the expansion only to linear order in $z^\mu$ since this is the highest term we need in this letter.
A complete expression of $h_{\mu\nu}$ to all orders in $z$ may be found in \cite{Mogull:2020sak}.
Next we extract the Feynman rules from the worldline actions.
The worldline propagators are the same in all three theories,
\begin{align}
\label{eq:propZ}
\begin{tikzpicture}[baseline={(0,0)}]
\begin{feynman}
\vertex[sdot, label=180:$z^\mu$] at (-1,0) (v1) {};
\vertex[sdot, label=0:$z^\nu$] at (0,0) (v2) {};
\diagram*{
(v1) -- [ultra thick, momentum={[arrow distance=5pt]\( \omega \)}] (v2)
};
\end{feynman}
\end{tikzpicture}
&= -\frac{i}{m} \frac{\eta^{\mu \nu}}{\omega^2} \\
\label{eq:propPsi}
\begin{tikzpicture}[baseline={(0,0)}]
\begin{feynman}
\vertex[sdot, label=180:$\varPsi^\dagger$] at (-1,0) (v1) {};
\vertex[sdot, label=0:$\varPsi$] at (0,0) (v2) {};
\diagram*{
(v1) -- [wavefunc, momentum={[arrow distance=5pt]\(\color{black} \omega \)}] (v2)
};
\end{feynman}
\end{tikzpicture}
&= \frac{i}{\omega}\, .
\end{align}
The propagator of the dual field $\tilde{\varPsi}$ is identical to the one for $\varPsi$.
Let us now begin with the analysis of the Yang-Mills coupled WQFT.
With \eqref{eq:wlFourier} and \eqref{eq:decompose} we can expand the interaction term of $S^{\text{pc}}$ from eq.~\eqn{eq:actionPC} as
\begin{align}
\label{eq:WYMinteraction}
S^{\mathrm{pc}}_{\mathrm{int}}
=& g \int \mathrm{d} \tau \, \dot{x}^\mu(\tau) \cdot A^a(x(\tau)) \, C^a(\tau) \\
=& g \int_k e^{i k \cdot b} v\cdot A^a(-k) \delta\!\!\!{}^-\!(k\cdot v) c^a \nonumber\\
&+ g \int_{k,\omega} e^{i k \cdot b} A_\mu^a(-k) \delta\!\!\!{}^-\!(k\cdot v + \omega) \nonumber\\
&\quad\times \Big[ i \big( \omega z^\mu(-\omega) + v^\mu k\cdot z(-\omega) \big) c^a \nonumber\\
&\quad\quad + v^\mu (\psi^\dagger T^a \varPsi(-\omega)
\!+\! \varPsi^\dagger(\omega) T^a \psi ) \Big]\!+\! \mathcal{O} \left( (z, \varPsi)^2 \right)
\nonumber
\end{align}
where we keep the interaction to linear order in worldline fluctuations.
The Feynman rules of the worldline-gluon vertices can be directly read off from \eqref{eq:WYMinteraction},
\begin{align}
\label{eq:vertex0z1gYM}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex (v1) at (-1,0) {};
\vertex (v2) at (1,0) {};
\vertex[sdot] at (0,0) (v0) {};
\vertex (k) at (0,-1.3) {$A_\mu^a$};
\diagram*{
(v1) -- [scalar] (v0) --[scalar] (v2),
(v0) -- [opacity=0, momentum'={\(k\)} ] (k),
(v0) -- [gluon] (k),
};
\end{feynman}
\end{tikzpicture}
&= i g e^{i k \cdot b} \delta\!\!\!{}^-\!(k\cdot v) v^\mu c^a \\
\begin{tikzpicture}[baseline={(0,-0.8)}]
\begin{feynman}
\vertex at (-1,0) (v1) {};
\vertex at (1.2,0) (v2) {$z^\rho$};
\vertex[sdot] at (0,0) (v0) {};
\vertex at (0,-1.3) (k) {$A_\mu^a$};
\diagram*{
(v1) -- [scalar] (v0) -- [ultra thick, momentum={[arrow distance=5pt]\( \omega \)}] (v2),
(v0) -- [opacity=0, momentum'={\(k\)} ] (k),
(v0) -- [gluon] (k),
};
\end{feynman}
\end{tikzpicture}
&
\begin{aligned}
= -& g e^{i k \cdot b} \delta\!\!\!{}^-\!(k\! \cdot \! v + \omega) \\
&~ \times ( \omega \eta^{\mu\rho} + v^\mu k^\rho ) c^a
\end{aligned} \\
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex at (-1,0) (v1) {};
\vertex at (1.2,0) (v2) {$\varPsi^\dagger$};
\vertex[sdot] at (0,0) (v0) {};
\vertex at (0,-1.3) (k) {$A_\mu^a$};
\diagram*{
(v1) -- [scalar] (v0) -- [wavefunc,arrow size=1.1pt, momentum={[arrow distance=5pt]\( \color{black} \omega \)}] (v2),
(v0) -- [opacity=0, momentum'={\(k\)} ] (k),
(v0) -- [gluon] (k),
};
\end{feynman}
\end{tikzpicture}
&= i g e^{i k \cdot b} \delta\!\!\!{}^-\!(k\cdot v + \omega) v^\mu (T^a \psi) \\
\label{eq:vertex1psiin1gYM}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex at (-1,0) (v1) {$\varPsi$};
\vertex at (1.2,0) (v2) {};
\vertex[sdot] at (0,0) (v0) {};
\vertex at (0,-1.3) (k) {$A_\mu^a$};
\diagram*{
(v1) -- [wavefunc, momentum={[arrow distance=5pt]\( \color{black} \omega \)}] (v0) -- [scalar] (v2),
(v0) -- [opacity=0, momentum={\(k\)} ] (k),
(v0) -- [gluon] (k),
};
\end{feynman}
\end{tikzpicture}
&= i g e^{i k \cdot b} \delta\!\!\!{}^-\!(k\cdot v - \omega) v^\mu (\psi^\dagger T^a).
\end{align}
Turning to the bi-adjoint scalar coupled WQFT, we can expand the worldline-scalar coupling of \eqn{eq:Sccaction} in the same way,
\begin{align}
\label{eq:WBSinteraction}
S^{\mathrm{cc}}_{\mathrm{int}} =& \frac{y}{m} \int \mathrm{d} \tau \phi^{a \tilde{a}}(x(\tau)) C^a(\tau) C^{\tilde{a}}(\tau) \\
=& \frac{y}{m} \int_k e^{i k \cdot b} \phi^{a \tilde{a}}(-k) \delta\!\!\!{}^-\!(k\cdot v) c^a c^{\tilde{a}} \nonumber\\
&+ \frac{y}{m} \int_{k,\omega} e^{i k \cdot b} \phi^{a \tilde{a}}(-k) \delta\!\!\!{}^-\!(k\cdot v + \omega)
\Big[ i k\cdot z(-\omega) c^a c^{\tilde{a}} \nonumber\\
&\quad + \left( \psi^\dagger T^a \varPsi(-\omega)
+ {\varPsi}^\dagger(\omega) T^{{a }} {\psi } \right) c^{\tilde{a}}\nonumber\\
&\quad + c^{{a}}\! \left(\tilde{\psi}^\dagger \tilde{T}^{\tilde{a}} \tilde{\varPsi}(-\omega)
+ \tilde{\varPsi}^\dagger(\omega) \tilde{T}^{\tilde{a }} \tilde{\psi } \!\right)
\!\Big] \!+\! \mathcal{O} \left( (z, \varPsi)^2 \right) .\nonumber
\end{align}
Again, we keep only the terms that we need in this work.
From the interaction \eqref{eq:WBSinteraction} we extract the Feynman rules
\begin{align} \label{eq:vertex0z1gBS}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex (v1) at (-1,0);
\vertex (v2) at (1,0);
\vertex[sdot] at (0,0) (v0) {};
\vertex (k) at (0,-1.3) {$\phi^{ab}$};
\diagram*{
(v1) -- [scalar] (v0) --[scalar] (v2),
(v0) -- [opacity=0, momentum'={\(k\)} ] (k),
(v0) -- [photon] (k),
};
\end{feynman}
\end{tikzpicture}
=&
\frac{iy}{m} e^{i k \cdot b} \delta\!\!\!{}^-\!(k\cdot v) c^a c^{\tilde{a}} \\
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex at (-1,0) (v1) {};
\vertex at (1.2,0) (v2) {$z^\rho$};
\vertex[sdot] at (0,0) (v0) {};
\vertex at (0,-1.3) (k) {$\phi^{ab}$};
\diagram*{
(v1) -- [scalar] (v0) -- [ultra thick, momentum={[arrow distance=5pt]\( \omega \)}] (v2),
(v0) -- [opacity=0, momentum'={\(k\)} ] (k),
(v0) -- [photon] (k),
};
\end{feynman}
\end{tikzpicture}
=& - \frac{y}{m} e^{i k \cdot b} \delta\!\!\!{}^-\!(k\cdot v + \omega) k^\rho c^a c^{\tilde{a}}
\end{align}
\begin{align}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex at (-1,0) (v1) {};
\vertex at (1.2,0) (v2) {$\varPsi^\dagger$};
\vertex[sdot] at (0,0) (v0) {};
\vertex at (0,-1.3) (k) {$\phi^{ab}$};
\diagram*{
(v1) -- [scalar] (v0) -- [wavefunc,arrow size=1.1pt, momentum={[arrow distance=5pt]\( \color{black} \omega \)}] (v2),
(v0) -- [opacity=0, momentum'={\(k\)} ] (k),
(v0) -- [photon] (k),
};
\end{feynman}
\end{tikzpicture}
&
= \frac{iy}{m} e^{i k \cdot b} \delta\!\!\!{}^-\!(k\cdot v + \omega)
(T^a \psi) c^{\tilde{a}}
\label{eq:vertex1psiout1gBS} \\
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex at (-1,0) (v1) {$\varPsi$};
\vertex at (1.2,0) (v2) {};
\vertex[sdot] at (0,0) (v0) {};
\vertex at (0,-1.3) (k) {$\phi^{ab}$};
\diagram*{
(v1) -- [wavefunc, momentum={[arrow distance=5pt]\( \color{black} \omega \)}] (v0) -- [scalar] (v2),
(v0) -- [opacity=0, momentum={\(k\)} ] (k),
(v0) -- [photon] (k),
};
\end{feynman}
\end{tikzpicture}
&
\!\!= \frac{i y}{m} e^{i k \cdot b} \delta\!\!\!{}^-\!(k\cdot v - \omega)
(\psi^\dagger T^a ) c^{\tilde{a}}.
\label{eq:vertex1psiin1gBS}
\end{align}
For vertices that involves the dual wave function, we simply use \eqref{eq:vertex1psiout1gBS} or \eqref{eq:vertex1psiin1gBS} and change ${\Psi}$ to $\tilde{\Psi}$.
In the dilaton-gravity coupled WQFT, the interaction term is remarkably simplified by the field redefinitions of $\{\varphi, h_{\mu\nu} \}$.
In the end the linear order in $h_{\mu\nu}$ is no different than the interaction term of a point mass in pure gravity, which is given in \cite{Mogull:2020sak} to all orders in $z(\omega)$.
Here we provide the first terms we need in this letter,
\begin{widetext}
\begin{align}
S^{\mathrm{pm}}_{\mathrm{int}} = &
-\frac{m \kappa}{2} \int_{k} e^{i k \cdot b} \delta(k \cdot v) h_{\mu \nu}(-k) v^{\mu} v^{\nu}
-i \frac{m \kappa}{2} \int_{k, \omega} e^{i k \cdot b} \delta(k \cdot v+\omega) h_{\mu \nu}(-k) z^{\rho}(-\omega)\left(2 \omega v^{(\mu} \delta_{\rho}^{\nu)}+v^{\mu} v^{\nu} k_{\rho}\right) \nonumber\\
& -\frac{m \kappa^2}{4} \int_{k_1,k_2}
e^{(k_1+k_2) \cdot b}\delta\!\!\!{}^-\!((k_1+k_2) \! \cdot \! v)
h_{\mu\rho}(-k_1) h_{\nu}^{\ \rho}(-k_2) v^\mu v^\nu
+ \mathcal{O}(k^3, z^2),
\end{align}
\end{widetext}
from which we obtain the Feynman rules,
\begin{align}
\label{eq:vertex0z1hDG}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex (v1) at (-1,0);
\vertex (v2) at (1,0);
\vertex[sdot] at (0,0) (v0) {};
\vertex (k) at (0,-1.3) {$h_{\mu\nu}$};
\diagram*{
(v1) -- [scalar] (v0) --[scalar] (v2),
(v0) -- [opacity=0, momentum'={\(k\)} ] (k),
(v0) -- [graviton] (k),
};
\end{feynman}
\end{tikzpicture}
=& -i \frac{m \kappa}{2} e^{i k \cdot b} \delta\!\!\!{}^-\!(k \cdot v) v^{\mu} v^{\nu} \\
\begin{tikzpicture}[baseline={(0,-0.8)}]
\begin{feynman}
\vertex at (-1,0) (v1) {};
\vertex at (1.2,0) (v2) {$z^\rho$};
\vertex[sdot] at (0,0) (v0) {};
\vertex at (0,-1.3) (k) {$h_{\mu\nu}$};
\diagram*{
(v1) -- [scalar] (v0) -- [ultra thick, momentum={[arrow distance=5pt]\( \omega \)}] (v2),
(v0) -- [opacity=0, momentum'={\(k\)} ] (k),
(v0) -- [graviton] (k),
};
\end{feynman}
\end{tikzpicture}
&
\begin{aligned}
=& \frac{m \kappa}{2} e^{i k \cdot b} \delta\!\!\!{}^-\!(k \cdot v+\omega) \\
& \qquad \left(2 \omega v^{(\mu} \delta_{\rho}^{\nu)}+v^{\mu} v^{\nu} k_{\rho}\right)
\end{aligned} \\
\label{eq:vertex2z1hDG}
\begin{tikzpicture}[baseline={(0,-0.8)}]
\begin{feynman}
\vertex at (-1,0) (v1) {};
\vertex at (1.2,0) (v2) {};
\vertex[sdot] at (0,0) (v0) {};
\vertex at ($(v0)!1!-40:(0,-1.4)$) (k1) {$h_{\mu\nu}$};
\vertex at ($(v0)!0.05!-40:(0,-1.4)$) (f1) {};
\vertex at ($(v0)!1!40:(0,-1.4)$) (k2) {$h_{\rho\sigma}$};
\vertex at ($(v0)!0.05!40:(0,-1.4)$) (f2) {};
\diagram*{
(v1) -- [scalar] (v0) -- [scalar] (v2),
(f1) -- [opacity=0, momentum'={[arrow distance=5pt, label distance=-6pt, arrow shorten=0.25] $k_1\ \ $} ] (k1),
(v0) -- [graviton] (k1),
(f2) -- [opacity=0, momentum={[compact arrow]$\ \ k_2$} ] (k2),
(v0) -- [graviton] (k2),
};
\end{feynman}
\end{tikzpicture}
&
\begin{aligned}
= - &\frac{m \kappa^2}{2} \int_{k_1,k_2}
e^{i(k_1+k_2) \cdot b} \\
&\delta\!\!\!{}^-\! \left((k_1 \!+\!k_2) \! \cdot \! v\right) v^{(\mu} \eta^{\nu)(\rho} v^{\sigma)},
\end{aligned}
\end{align}
where the parenthesis of Lorentz indices denotes symmetrization with unit weight, e.g.~$v_1^{(\mu} v_2^{\nu)} = \frac{1}{2}(v_1^{\mu} v_2^{\nu} + v_1^{\nu} v_2^{\mu})$.
\section{Classical double copy}
\label{Se:doublecopy}
One of the main challenges of constructing the double copy in the classical limit of quantum field theories
is that the locality structure is concealed.
This is rooted in the classical limit of the massive scalar propagator \cite{Mogull:2020sak},
which contains both double and single propagators as we can see in WQFT from \eqref{eq:propZ} and \eqref{eq:propPsi}.
Following \cite{Shen:2018ebu} we tackle this difficulty by using the bi-adjoint scalar theory to identify the correct locality structure, i.e.~disentangle the kinematical numerators from the propagator terms.
Another important strategy to establish the classical double copy is to consider more than two worldlines even if we are ultimately interested only in two-body interactions.
This is to avoid the situation where some color factors in the two-body situation are vanishing but the corresponding numerators do not, which under the double copy map may yield non-zero contributions. This may be evaded if we use as many worldlines as worldline-field interactions occur.
Specifically, we will consider an $(n+2)$-body system at $\mathrm{N^{n}LO}$.
In the WQFT formalism, this is equivalent to taking into account only tree diagrams.
To retrieve the binary system from this, we need to sum all possible ways of fusing the $(n+2)$ worldlines into $2$ worldlines.
In summary, our double copy relation of the eikonal phase reads
\begin{subequations} \label{eq:dcEikonal}
\begin{align}
\label{eq:dcEikonalBS}
\chi^{\mathrm{BS}}_n =& -y^{2n} \int {\mathrm{d} \mu_{1,2,...,(n+1)}} \sum_{i,j} C_i K_{ij} \tilde{C}_j, \\
\label{eq:dcEikonalYM}
\chi^{\mathrm{YM}}_n =& -(ig)^{2n} \int {\mathrm{d} \mu_{1,2,...,(n+1)}} \sum_{i,j} C_i K_{ij} {N}_j, \\
\label{eq:dcEikonalDG}
\chi^{\mathrm{DG}}_n =& -\left( \frac{\kappa}{2} \right)^{2n} \int {\mathrm{d} \mu_{1,2,...,(n+1)}}\sum_{i,j} N_i K_{ij} N_j,
\end{align}
\end{subequations}
where $C_i, \tilde{C}_j$ denotes the color and dual color factors, $N_j$ are the numerators, and $K_{ij}$ are the so-called double copy kernels that encodes the locality structure.
For further convenience, we have also defined the integral measure
\begin{align}
\mathrm{d} \mu_{1,2,...,n} = \prod_{i=1}^{n} \left( \frac{\mathrm{d}^4 k_i}{(2\pi)^4} e^{i k_i \cdot b_i}
\delta\!\!\!{}^-\!\left(k_i \! \cdot \! p_i \right) \right)\delta\!\!\!{}^-\!^{(4)} \bigg(\sum_{i=1}^n k_i^\mu \bigg),
\end{align}
where $k_i$ is the total outgoing momentum of bosonic fields $\Phi(x)$ attached to a worldline.
Note that we have defined the momentum of the massive particle as
\begin{align}
p_i^\mu := m_i v_i^\mu,
\quad \text{so that} \quad
\delta\!\!\!{}^-\!(k_i \! \cdot \! p_i) = \frac{\delta\!\!\!{}^-\!(k_i \! \cdot \! v_i)}{m_i}.
\end{align}
Hereafter we will always express the numerator $N_j$ in terms of the momentum $p_i^\mu$ which is necessary in order to balance the mass dimension under the double copy.
The kinematic numerators $N_i$ are arranged to satisfy the same algebraic equations as the color factors $C_i$,
\begin{align}
&C_i + C_j + C_k = 0
\quad \Rightarrow \quad
N_i + N_j + N_k = 0.
\end{align}
It is worth mentioning that we have the color-kinematic duality already at quartic order in the coupling constant.
\subsection{Eikonal at leading order (LO)}
The locality structure at leading order is trivial, so we do not need to employ the bi-adjoint scalar theory in order to
double copy YM color charged particles to DG ones.
In Yang-Mills coupled WQFT (WYM) the eikonal phase at this order involves only one diagram.
Using the Feynman rules \eqref{eq:vertex0z1gYM} and the gluon propagator \eqref{eq:propA}, we have
\begin{align} \label{eq:eikonalYMLO}
i \chi^{\mathrm{YM}}_1
=&
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex (v1) at (-1,0) {$1$};
\vertex (v2) at (0.8,0);
\vertex[sdot] at (0,0) (v0) {};
\vertex[sdot] (w0) at (0,-1) {};
\vertex (w1) at (-1,-1) {$2$};
\vertex (w2) at (0.8,-1);
\diagram*{
(v1) -- [scalar] (v0) -- [scalar] (v2),
(v0) -- [opacity=0, momentum'={\(k_1\)} ] (w0),
(v0) -- [gluon] (w0),
(w1) -- [scalar] (w0) -- [scalar] (w2)
};
\end{feynman}
\end{tikzpicture}
= ig^2 \int \mathrm{d} \mu_{1,2}
\frac{(p_1 \! \cdot \! p_2) (c_1 \! \cdot \! c_2) }{k_1^2}
\end{align}
where we have massaged the formula to fit the form as \eqref{eq:dcEikonalYM}.
We can identify the color factor, the numerator and the double copy kernel as
\begin{align}
C = (c_1 \! \cdot \! c_2), \quad N = (p_1 \! \cdot \! p_2), \quad K = \frac{1}{k_1^2}.
\end{align}
In worldline coupled dilaton-gravity (WDG), thanks to the decoupling of $\varphi$ from the worldline, we have also only one diagram mediated by $h_{\mu\nu}$. With \eqref{eq:vertex0z1hDG} and the graviton propagator \eqref{eq:proph}, we obtain
\begin{align}
i\chi^{\mathrm{DG}}_1 =
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex (v1) at (-1,0) {$1$};
\vertex (v2) at (0.8,0);
\vertex[sdot] at (0,0) (v0) {};
\vertex[sdot] (w0) at (0,-1) {};
\vertex (w1) at (-1,-1) {$2$};
\vertex (w2) at (0.8,-1);
\diagram*{
(v1) -- [scalar] (v0) -- [scalar] (v2),
(v0) -- [opacity=0, momentum'={\(k_1\)} ] (w0),
(v0) -- [graviton] (w0),
(w1) -- [scalar] (w0) -- [scalar] (w2)
};
\end{feynman}
\end{tikzpicture}
= \frac{-i \kappa^2}{4} \int {\mathrm{d} \mu_{1,2}}
\frac{(p_1 \cdot p_2)^2}{k_1^2}.
\end{align}
Hence at the leading order the eikonal of Yang-Mills and dilaton gravtiy obviously possess a double copy relation \eqref{eq:dcEikonal}.
\subsection{Eikonal at Next-to-Leading order (NLO)}
As explained before, at next-to-leading order, to avoid the vanishing of some contributions in worldline coupled bi-adjoint scalar theory (WBS) and Yang-Mills coupled WQFT theory, we will consider three worldlines.
At this order the locality structure is non-trivial.
As we will see, the double copy kernel is off-diagonal.
Therefore, we will first consider the bi-adjoint scalar theory to identify the kernel.
The Feynman diagrams in WBS can be calculated using the Feynman rules \eqref{eq:vertex0z1gBS}-\eqref{eq:vertex1psiin1gBS} and the three-point vertex of $\phi_{a\tilde a}$ \eqref{eq:vertex3phi},
\begin{widetext}
\begin{align}
\label{eq:BSNLOzprop}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [label=180:$1$] at (0,0) (a0) ;
\vertex [sdot, right=.6cm of a0] (a1) {};
\vertex [sdot, right=.7cm of a1] (a2) {};
\vertex [right=.6cm of a2] (a3);
\vertex [below=1 of a1] (i1);
\vertex [sdot] at ($(a1)!1!-30:(i1)$) (b1) {};
\vertex [left=.4cm of b1] (i2);
\vertex [label=180:2] at ($(b1)!1!-30:(i2)$) (b0);
\vertex [right=.4cm of b1] (i3);
\vertex at ($(b1)!1!-30:(i3)$) (b2);
\vertex [below=1 of a2] (j1);
\vertex [sdot] at ($(a2)!1!30:(j1)$) (c1) {};
\vertex [left=.4cm of c1] (j2);
\vertex [label=180:3] at ($(c1)!1!30:(j2)$) (c0);
\vertex [right=.4cm of c1] (j3);
\vertex at ($(c1)!1!30:(j3)$) (c2);
\diagram*{
(a0) --[scalar] (a1) --[ultra thick] (a2) --[scalar] (a3),
(b0) --[scalar] (b1) --[scalar] (b2),
(c0) --[scalar] (c1) --[scalar] (c2),
(a1) --[photon, rmomentum={[compact arrow]$\ \ k_2$} ] (b1),
(a2) --[photon, rmomentum={[compact arrow]$\ \ k_3$} ] (c1),
};
\end{feynman}
\end{tikzpicture}
=& {-iy^4}\int \frac{\mathrm{d} \mu_{1,2,3}}{k_2^2 k_3^2}
\frac{k_2 \cdot k_3}{(k_2 \! \cdot \! p_1)^2}
(c_1\! \cdot \! c_2) (c_1\! \cdot \! c_3)
(\tilde{c}_1\! \cdot \! \tilde{c}_2) (\tilde{c}_1\! \cdot \! \tilde{c}_2)
\\
\label{eq:BSNLOinprop}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [label=180:$1$] at (0,0) (a0);
\vertex [sdot, right=.6cm of a0] (a1) {};
\vertex [sdot, right=.7cm of a1] (a2) {};
\vertex [right=.6cm of a2] (a3);
\vertex [below=1 of a1] (i1);
\vertex [sdot] at ($(a1)!1!-30:(i1)$) (b1) {};
\vertex [left=.4cm of b1] (i2);
\vertex [label=180:$2$] at ($(b1)!1!-30:(i2)$) (b0);
\vertex [right=.4cm of b1] (i3);
\vertex at ($(b1)!1!-30:(i3)$) (b2);
\vertex [below=1 of a2] (j1);
\vertex [sdot] at ($(a2)!1!30:(j1)$) (c1) {};
\vertex [left=.4cm of c1] (j2);
\vertex [label=180:$3$] at ($(c1)!1!30:(j2)$) (c0);
\vertex [right=.4cm of c1] (j3);
\vertex at ($(c1)!1!30:(j3)$) (c2);
\diagram*{
(a0) --[scalar] (a1) --[wavefunc] (a2) --[scalar] (a3),
(b0) --[scalar] (b1) --[scalar] (b2),
(c0) --[scalar] (c1) --[scalar] (c2),
(a1) -- [photon, rmomentum={[compact arrow]$\ \ k_2$}] (b1),
(a2) -- [photon, rmomentum={[compact arrow]$\ \ k_3$}] (c1),
};
\end{feynman}
\end{tikzpicture}
=& {-iy^4} \int \frac{\mathrm{d} \mu_{1,2,3}}{k_2^2 k_3^2}
\frac{1}{k_2\! \cdot \! p_1}
\big( (c_1^{ba} c_2^a c_3^b) (\tilde{c}_1\! \cdot \! \tilde{c}_2) (\tilde{c}_1\! \cdot \! \tilde{c}_3)
+ (c_1\! \cdot \! c_2) (c_1\! \cdot \! c_3) (\tilde{c}_1^{\tilde{b}\tilde{a}} \tilde{c}_2^{\tilde{a}} \tilde{c}_3^{\tilde{b}}) \big)
\\
\label{eq:BSNLOoutprop}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [label=180:$1$] at (0,0) (a0);
\vertex [sdot, right=.6cm of a0] (a1) {};
\vertex [sdot, right=.7cm of a1] (a2) {};
\vertex [right=.6cm of a2] (a3);
\vertex [below=1 of a1] (i1);
\vertex [sdot] at ($(a1)!1!-30:(i1)$) (b1) {};
\vertex [left=.4cm of b1] (i2);
\vertex [label=180:$2$] at ($(b1)!1!-30:(i2)$) (b0);
\vertex [right=.4cm of b1] (i3);
\vertex at ($(b1)!1!-30:(i3)$) (b2);
\vertex [below=1 of a2] (j1);
\vertex [sdot] at ($(a2)!1!30:(j1)$) (c1) {};
\vertex [left=.4cm of c1] (j2);
\vertex [label=180:$3$] at ($(c1)!1!30:(j2)$) (c0);
\vertex [right=.4cm of c1] (j3);
\vertex at ($(c1)!1!30:(j3)$) (c2);
\vertex at ($(a2)!0.2!0:(b1)$) (k2) {};
\vertex at ($(a1)!0.2!(c1)$) (k3) {};
\diagram*{
(a0) --[scalar] (a1) --[wavefunc] (a2) --[scalar] (a3),
(b0) --[scalar] (b1) --[scalar] (b2),
(c0) --[scalar] (c1) --[scalar] (c2),
(a1) -- [photon] (c1),
(k3) -- [opacity=0, rmomentum={[compact arrow]$\ \ k_3$}] (c1),
(a2) -- [photon] (b1),
(k2) -- [opacity=0, rmomentum={[compact arrow]$\ \ k_2$}] (b1),
};
\end{feynman}
\end{tikzpicture}
=& {-iy^4} \int \frac{\mathrm{d} \mu_{1,2,3}}{k_2^2 k_3^2}
\frac{-1}{k_2\! \cdot \! p_1}
\big( (c_1^{ab} c_2^a c_3^b) (\tilde{c}_1\! \cdot \! \tilde{c}_2) (\tilde{c}_1\! \cdot \! \tilde{c}_3)
+ (c_1\! \cdot \! c_2) (c_1\! \cdot \! c_3) (\tilde{c}_1^{\tilde{a}\tilde{b}} \tilde{c}_2^{\tilde{a}} \tilde{c}_3^{\tilde{b}}) \big)
\\
\label{eq:BSNLO3phi}
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [sdot] (g1) {};
\vertex [below=0.5cm of g1] (g2);
\vertex [sdot, above=1cm of g1] (a0) {};
\vertex [sdot, left=0.87cm of g2] (b0) {};
\vertex [sdot, right=0.87cm of g2] (c0) {};
\vertex [left=0.6cm of a0, label=180:$1$] (a1);
\vertex [right=0.6cm of a0] (a2);
\vertex [above=0.52cm of b0] (i1);
\vertex [left=0.3cm of i1, label=180:$2$] (b1);
\vertex [below=0.52cm of b0] (i2);
\vertex [right=0.3cm of i2] (b2);
\vertex [below=0.52cm of c0] (i3);
\vertex [left=0.3cm of i3, label=180:$3$] (c1);
\vertex [above=0.52cm of c0] (i4);
\vertex [right=0.3cm of i4] (c2);
\diagram*{
(a1) -- [scalar] (a0) -- [scalar] (a2),
(b1) -- [scalar] (b0) -- [scalar] (b2),
(c1) -- [scalar] (c0) -- [scalar] (c2),
(g1) -- [photon, rmomentum={[arrow distance=5pt, label distance=-4pt, arrow shorten=0.2]$k_1$}] (a0),
(g1) -- [photon, rmomentum={[arrow distance=5pt, label distance=-4pt, arrow shorten=0.2]$k_2$}] (b0),
(g1) -- [photon, rmomentum={[arrow distance=5pt, label distance=-4pt, arrow shorten=0.2]$k_3$}] (c0),
};
\end{feynman}
\end{tikzpicture}
=& {-iy^4} \int \frac{\mathrm{d} \mu_{1,2,3}}{k_1^2 k_2^2 k_3^2}
\ 2 f^{abc} c_1^a c_2^b c_3^c
\tilde{f}^{\tilde{a}\tilde{b}\tilde{c}} \tilde{c}_1^{\tilde{a}} \tilde{c}_2^{\tilde{b}} \tilde{c}_3^{\tilde{c}}
\end{align}
\end{widetext}
where for compactness we have defined
\begin{align}
c^{ab} := \left(\psi^\dagger T^a T^b \psi \right)\, , \quad
\tilde {c}^{\tilde a\tilde b} := \left(\tilde\psi^\dagger \tilde T^{\tilde a} \tilde T^{\tilde b} \tilde\psi \right)\, .
\end{align}
Note that in \eqref{eq:BSNLOinprop} and \eqref{eq:BSNLOoutprop}, the propagator with an arrow denotes either the color or dual color wave function, and we have added up their contributions.
We stress that the factors $c^{ab}$ are absent in the equation of motion, so they will not explicitly appear in the classical solutions \cite{Wong:1970fu}.
In fact, summing up \eqref{eq:BSNLOinprop} and \eqref{eq:BSNLOoutprop} we can remove $c^{ab}$ by
\begin{align}\label{eq:Jacobic}
c^{ab} - c^{ba} = f^{abc} c^{c}\, ,
\end{align}
and similarly for the dual-color sector.
However, these factors turn out to be critical for the double copy:
because of them we find classical numerators that satisfy color-kinematics duality at this order.
From \eqref{eq:BSNLOzprop} - \eqref{eq:BSNLOoutprop} we can identify 3 (dual-)color factors,
\begin{align}
\label{eq:eikonalNLOColor}
C_i^{\mathrm{(123)}} =& \big\{ (c_1\! \cdot \! c_2) (c_1\! \cdot \! c_3) \, ,\, (c_1^{ab} c_2^a c_3^b) \, ,\, (c_1^{ba} c_2^a c_3^b) \big\} \\
\label{eq:eikonalNLOdColor}
\tilde{C}_i^{\mathrm{(123)}} =& \big\{ (\tilde{c}_1\! \cdot \! \tilde{c}_2) (\tilde{c}_1\! \cdot \! \tilde{c}_3)\, , \,
(\tilde{c}_1^{\tilde{a}\tilde{b}} \tilde{c}_2^{\tilde{a}} \tilde{c}_3^{\tilde{b}}) \, , \,
(\tilde{c}_1^{\tilde{b}\tilde{a}} \tilde{c}_2^{\tilde{a}} \tilde{c}_3^{\tilde{b}}) \big\} \, .
\end{align}
Note that here we only consider diagrams with worldline propagators of particle $1$. There are also contributions involving propagators of $2$ and $3$, which can be gained simply by relabeling $(123)$ in \eqref{eq:BSNLOzprop}-\eqref{eq:BSNLOoutprop} and give us another 6 color factors.
Together with the single (dual)-color factor emerging from \eqref{eq:BSNLO3phi}
\begin{align}
\label{eq:eikonalNLOColor0}
C_i^{\mathrm{(0)}} = f^{abc} c_1^a c_2^b c_3^c,
\qquad
\tilde{C}_i^{\mathrm{(0)}} = \tilde{f}^{\tilde{a}\tilde{b}\tilde{c}} \tilde{c}_1^{\tilde{a}} \tilde{c}_2^{\tilde{b}} \tilde{c}_3^{\tilde{c}},
\end{align}
we see that the double copy kernel $K_{ij}$ is 10-dimensional.
Fortunately, $K_{ij}$ is block-diagonal. The block that corresponds to the three-dimensional space \eqref{eq:eikonalNLOColor} is
\begin{align}
\label{eq:eikonalNLOKernel}
K_{ij}^{\mathrm{(123)}} =& \frac{1}{k_2^2 k_3^2} \left(
\begin{array}{cccc}
\frac{k_2 \cdot k_3}{(k_2 \cdot p_1)^2} & \frac{-1}{k_2\cdot p_1} & \frac{1}{k_2\cdot p_1} \\
\frac{-1}{k_2\cdot p_1} & 0 & 0 \\
\frac{1}{k_2\cdot p_1} & 0 & 0 \\
\end{array}
\right).
\end{align}
and analogously for the color-dual \eqref{eq:eikonalNLOdColor}.
By permutations of $(123)$ we may obtain other blocks.
The last block coupling to the structure constant is extracted from \eqref{eq:BSNLO3phi} and is $1$-dimensional, we have
\begin{gather}
\label{eq:eikonalNLOKernel0}
K_{ij}^{\mathrm{(0)}} = \frac{2}{k_1^2 k_2^2 k_3^2}.
\end{gather}
We now proceed to consider the Yang-Mills coupled WQFT (WYM) theory.
The Feynman diagrams are very similar to those of WBS theory.
With the WYM Feynman rules \eqref{eq:vertex0z1gYM} - \eqref{eq:vertex1psiin1gYM}, we may compute the contributions to the eikonal phase
\begin{widetext}
\begin{align}
\label{eq:YMNLOzprop}
&\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [label=180:$1$] at (0,0) (a0) ;
\vertex [sdot, right=.6cm of a0] (a1) {};
\vertex [sdot, right=.7cm of a1] (a2) {};
\vertex [right=.6cm of a2] (a3);
\vertex [below=1 of a1] (i1);
\vertex [sdot] at ($(a1)!1!-30:(i1)$) (b1) {};
\vertex [left=.4cm of b1] (i2);
\vertex [label=180:2] at ($(b1)!1!-30:(i2)$) (b0);
\vertex [right=.4cm of b1] (i3);
\vertex at ($(b1)!1!-30:(i3)$) (b2);
\vertex [below=1 of a2] (j1);
\vertex [sdot] at ($(a2)!1!30:(j1)$) (c1) {};
\vertex [left=.4cm of c1] (j2);
\vertex [label=180:3] at ($(c1)!1!30:(j2)$) (c0);
\vertex [right=.4cm of c1] (j3);
\vertex at ($(c1)!1!30:(j3)$) (c2);
\diagram*{
(a0) --[scalar] (a1) --[ultra thick] (a2) --[scalar] (a3),
(b0) --[scalar] (b1) --[scalar] (b2),
(c0) --[scalar] (c1) --[scalar] (c2),
(a1) --[gluon, rmomentum={[compact arrow]$\ \ k_2$} ] (b1),
(a2) --[gluon, rmomentum={[compact arrow]$\ \ k_3$} ] (c1),
};
\end{feynman}
\end{tikzpicture}
= {-ig^4} \int \frac{\mathrm{d} \mu_{1,2,3}}{k_2^2 k_3^2}
(c_1\! \cdot \! c_2) (c_1\! \cdot \! c_3)
\left( \frac{k_2\! \cdot \! k_3}{(k_2\! \cdot \! p_1)^2} n_0
+ \frac{ 1 }{k_2 \! \cdot \! p_1} n_1
\right) \\
\label{eq:YMNLOinprop}
&\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [label=180:$1$] at (0,0) (a0) ;
\vertex [sdot, right=.6cm of a0] (a1) {};
\vertex [sdot, right=.7cm of a1] (a2) {};
\vertex [right=.6cm of a2] (a3);
\vertex [below=1 of a1] (i1);
\vertex [sdot] at ($(a1)!1!-30:(i1)$) (b1) {};
\vertex [left=.4cm of b1] (i2);
\vertex [label=180:2] at ($(b1)!1!-30:(i2)$) (b0);
\vertex [right=.4cm of b1] (i3);
\vertex at ($(b1)!1!-30:(i3)$) (b2);
\vertex [below=1 of a2] (j1);
\vertex [sdot] at ($(a2)!1!30:(j1)$) (c1) {};
\vertex [left=.4cm of c1] (j2);
\vertex [label=180:3] at ($(c1)!1!30:(j2)$) (c0);
\vertex [right=.4cm of c1] (j3);
\vertex at ($(c1)!1!30:(j3)$) (c2);
\diagram*{
(a0) --[scalar] (a1) --[wavefunc] (a2) --[scalar] (a3),
(b0) --[scalar] (b1) --[scalar] (b2),
(c0) --[scalar] (c1) --[scalar] (c2),
(a1) --[gluon, rmomentum={[compact arrow]$\ \ k_2$} ] (b1),
(a2) --[gluon, rmomentum={[compact arrow]$\ \ k_3$} ] (c1),
};
\end{feynman}
\end{tikzpicture}
= {-ig^4} \int \frac{\mathrm{d} \mu_{1,2,3}}{k_2^2 k_3^2}
\frac{(c_1^{ba} c_2^a c_3^b) }{k_2\! \cdot \! p_1} n_0 \\
\label{eq:YMNLOoutprop}
&\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [label=180:$1$] at (0,0) (a0);
\vertex [sdot, right=.6cm of a0] (a1) {};
\vertex [sdot, right=.7cm of a1] (a2) {};
\vertex [right=.6cm of a2] (a3);
\vertex [below=1 of a1] (i1);
\vertex [sdot] at ($(a1)!1!-30:(i1)$) (b1) {};
\vertex [left=.4cm of b1] (i2);
\vertex [label=180:$2$] at ($(b1)!1!-30:(i2)$) (b0);
\vertex [right=.4cm of b1] (i3);
\vertex at ($(b1)!1!-30:(i3)$) (b2);
\vertex [below=1 of a2] (j1);
\vertex [sdot] at ($(a2)!1!30:(j1)$) (c1) {};
\vertex [left=.4cm of c1] (j2);
\vertex [label=180:$3$] at ($(c1)!1!30:(j2)$) (c0);
\vertex [right=.4cm of c1] (j3);
\vertex at ($(c1)!1!30:(j3)$) (c2);
\vertex at ($(a2)!0.2!0:(b1)$) (k2) {};
\vertex at ($(a1)!0.2!(c1)$) (k3) {};
\diagram*{
(a0) --[scalar] (a1) --[wavefunc] (a2) --[scalar] (a3),
(b0) --[scalar] (b1) --[scalar] (b2),
(c0) --[scalar] (c1) --[scalar] (c2),
(a1) -- [gluon] (c1),
(k3) -- [opacity=0, rmomentum={[compact arrow]$\ \ k_3$}] (c1),
(a2) -- [gluon] (b1),
(k2) -- [opacity=0, rmomentum={[compact arrow]$\ \ k_2$}] (b1),
};
\end{feynman}
\end{tikzpicture}
= {-ig^4} \int \frac{\mathrm{d} \mu_{1,2,3}}{k_2^2 k_3^2}
\frac{-(c_1^{ab} c_2^a c_3^b) }{k_2\! \cdot \! p_1} n_0 \\
\label{eq:YMNLO3g}
& \begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [sdot] (g1) {};
\vertex [below=0.5cm of g1] (g2);
\vertex [sdot, above=1cm of g1] (a0) {};
\vertex [sdot, left=0.87cm of g2] (b0) {};
\vertex [sdot, right=0.87cm of g2] (c0) {};
\vertex [left=0.6cm of a0, label=180:$1$] (a1);
\vertex [right=0.6cm of a0] (a2);
\vertex [above=0.52cm of b0] (i1);
\vertex [left=0.3cm of i1, label=180:$2$] (b1);
\vertex [below=0.52cm of b0] (i2);
\vertex [right=0.3cm of i2] (b2);
\vertex [below=0.52cm of c0] (i3);
\vertex [left=0.3cm of i3, label=180:$3$] (c1);
\vertex [above=0.52cm of c0] (i4);
\vertex [right=0.3cm of i4] (c2);
\diagram*{
(a1) -- [scalar] (a0) -- [scalar] (a2),
(b1) -- [scalar] (b0) -- [scalar] (b2),
(c1) -- [scalar] (c0) -- [scalar] (c2),
(g1) -- [gluon, rmomentum={[arrow distance=5pt, label distance=-4pt, arrow shorten=0.2]$k_1$}] (a0),
(g1) -- [gluon, rmomentum={[arrow distance=5pt, label distance=-4pt, arrow shorten=0.2]$k_2$}] (b0),
(g1) -- [gluon, rmomentum={[arrow distance=5pt, label distance=-4pt, arrow shorten=0.2]$k_3$}] (c0),
};
\end{feynman}
\end{tikzpicture}
= {-ig^4} \int \mathrm{d} \mu_{1,2,3}
\frac{2 f^{abc} c_1^a c_2^b c_3^c} {k_1^2 k_2^2 k_3^2} \left( -n_1 \right)
\end{align}
\end{widetext}
where we have defined
\begin{align}
&n_0 = p_1\! \cdot \! p_2\, p_1\! \cdot \! p_3 \\
&n_1 = k_2 \! \cdot \! p_3\, p_1 \! \cdot \! p_2 - k_3 \! \cdot \! p_2\, p_1 \! \cdot \! p_3 - k_2 \! \cdot \! p_1\, p_2 \! \cdot \! p_3.
\end{align}
Based on the color factors identified in \eqref{eq:eikonalNLOColor}, \eqref{eq:eikonalNLOColor0} and the double copy kernel \eqref{eq:eikonalNLOKernel}, \eqref{eq:eikonalNLOKernel0}, we are led to organize the numerators as
\begin{align}
\label{eq:N123atNLO}
N_j^{\mathrm{(123)}} =& \left\{ n_0 \, ,\, \frac{-n_1}{2} \,, \, \frac{n_1}{2} \right\} \\
\label{eq:N0atNLO}
N_j^{\mathrm{(0)}} =& \, -n_1 \, ,
\end{align}
so that the WYM eikonal may be decomposed in the form of \eqref{eq:dcEikonalYM},
\begin{align}
\chi_2 =
-g^4 &\int \mathrm{d} \mu_{1,2,3}
\sum_{i,j} \Big(
C_i^{\mathrm{(0)}} K_{ij}^{\mathrm{(0)}} N_j^{\mathrm{(0)}} \nonumber\\
&+ \big(C_i^{\mathrm{(123)}} K_{ij}^{\mathrm{(123)}} N_j^{\mathrm{(123)}} + \text{cyclic} \big)
\Big).
\end{align}
Fortunately, this decomposition automatically satisfies the color-kinematics duality
\begin{align}
c_1^{ab} c_2^a c_3^b - c_1^{ba} c_2^a c_3^b =& f^{abc} c_1^a c_2^b c_3^c \\
\frac{-n_1}{2} - \frac{n_1}{2} =& -n_1.
\end{align}
Note that the decomposition of $N_j^{(123)}$ is not unique due to the Jacobi relation \eqn{eq:Jacobic}\footnote{For example, we could also have $N_j^{\mathrm{(123)}} = \left( n_0 \quad {0} \quad {n_1} \right)$. Color-kinematic duality still holds and the double copy gives the correct gravitational result. We have chosen to write $N_j^{(123)}$ in a symmetric form.}.
In principal, we are now prepared to execute the double copy as proposed in \eqref{eq:dcEikonalDG} to get the eikonal phase in the worldline coupled dilaton gravity theory (WDG).
In order to check the validity of our double copy prescription,
we directly compute the eikonal in WDG theory with \eqref{eq:vertex0z1hDG} - \eqref{eq:vertex2z1hDG} we find for the graphs
not involving bulk graviton interactions
\begin{widetext}
\begin{align}
\label{eq:DGNLO1z}
\begin{tikzpicture}[baseline={(0,-0.7)}]
\begin{feynman}
\vertex [label=180:$1$] at (0,0) (a0) ;
\vertex [sdot, right=.6cm of a0] (a1) {};
\vertex [sdot, right=.7cm of a1] (a2) {};
\vertex [right=.6cm of a2] (a3);
\vertex [below=1 of a1] (i1);
\vertex [sdot] at ($(a1)!1!-30:(i1)$) (b1) {};
\vertex [left=.4cm of b1] (i2);
\vertex [label=180:2] at ($(b1)!1!-30:(i2)$) (b0);
\vertex [right=.4cm of b1] (i3);
\vertex at ($(b1)!1!-30:(i3)$) (b2);
\vertex [below=1 of a2] (j1);
\vertex [sdot] at ($(a2)!1!30:(j1)$) (c1) {};
\vertex [left=.4cm of c1] (j2);
\vertex [label=180:3] at ($(c1)!1!30:(j2)$) (c0);
\vertex [right=.4cm of c1] (j3);
\vertex at ($(c1)!1!30:(j3)$) (c2);
\diagram*{
(a0) --[scalar] (a1) --[ultra thick] (a2) --[scalar] (a3),
(b0) --[scalar] (b1) --[scalar] (b2),
(c0) --[scalar] (c1) --[scalar] (c2),
(a1) --[graviton] (b1),
(a1) --[opacity=0, rmomentum={[compact arrow]$\ \ k_2$} ] (b1),
(a2) --[graviton] (c1),
(a2) --[opacity=0, rmomentum={[compact arrow]$\ \ k_3$} ] (c1),
};
\end{feynman}
\end{tikzpicture}
&
\begin{aligned}
= \frac{-i\kappa^4}{16} \int& \frac{\mathrm{d} \mu_{1,2,3}} {k_2^2 k_3^2 (k_2\! \cdot \! p_1)^2}
\big( (k_2\! \cdot \! k_3) (p_1\! \cdot \! p_2)^2 (p_1\! \cdot \! p_3)^2
-4 (k_2\! \cdot \! p_1)^2 (p_1\! \cdot \! p_2) (p_1\! \cdot \! p_3) (p_2\! \cdot \! p_3) \\
& \quad
-2 (k_3\! \cdot \! p_2) (k_2 \! \cdot \! p_1) (p_1\! \cdot \! p_2) (p_1\! \cdot \! p_3)^2
+2 (k_2\! \cdot \! p_3) (k_2 \! \cdot \! p_1) (p_1\! \cdot \! p_3) (p_1\! \cdot \! p_2)^2
\big)
\end{aligned}\\
\label{eq:DGNLO0z}
\begin{tikzpicture}[baseline={(0,-0.7)}]
\begin{feynman}
\vertex [label=180:$1$] at (0,0) (a0) ;
\vertex [sdot, right=.8cm of a0] (a1) {};
\vertex [right=.8cm of a1] (a3);
\vertex [below=1 of a1] (i1);
\vertex [sdot] at ($(a1)!1!-40:(i1)$) (b1) {};
\vertex [left=.4cm of b1] (i2);
\vertex [label=180:2] at ($(b1)!1!-40:(i2)$) (b0);
\vertex [right=.4cm of b1] (i3);
\vertex at ($(b1)!1!-40:(i3)$) (b2);
\vertex [sdot] at ($(a1)!1!40:(i1)$) (c1) {};
\vertex [left=.4cm of c1] (j2);
\vertex at ($(c1)!1!40:(j2)$) (c0);
\vertex [right=.4cm of c1] (j3);
\vertex [label=0:3] at ($(c1)!1!40:(j3)$) (c2);
\diagram*{
(a0) --[scalar] (a1) --[scalar] (a3),
(b0) --[scalar] (b1) --[scalar] (b2),
(c0) --[scalar] (c1) --[scalar] (c2),
(a1) --[graviton] (b1),
(a1) --[opacity=0, rmomentum'={[compact arrow]$k_2\ \ $} ] (b1),
(a1) --[graviton] (c1),
(a1) --[opacity=0, rmomentum={[compact arrow]$\ \ k_3$} ] (c1),
};
\end{feynman}
\end{tikzpicture}
&= \frac{-i\kappa^4}{16} \int \frac{\mathrm{d} \mu_{1,2,3}}{k_2^2 k_3^2}
\big(2 (p_1\! \cdot \! p_2) (p_1\! \cdot \! p_3) (p_2\! \cdot \! p_3) \big).
\end{align}
\end{widetext}
Summing up the two diagrams \eqref{eq:DGNLO1z} and \eqref{eq:DGNLO0z}, we can check that the result can be written as
\begin{align}
\label{eq:dcNLO1}
& \frac{-i\kappa^4}{16} \int \frac{\mathrm{d} \mu_{1,2,3}}{k_2^2 k_3^2}
\left( \frac{k_2\! \cdot \! k_3 n_0^2}{(k_2\! \cdot \! p_1)^2}
+ \frac{2 n_0 n_1}{k_2\! \cdot \! p_1} \right) \\
=& \frac{-i\kappa^4}{16} \int {\mathrm{d} \mu_{1,2,3}}
\sum_{i,j} N_i^{\mathrm{(123)}} K_{ij}^{\mathrm{(123)}} N_j^{\mathrm{(123)}}. \nonumber
\end{align}
In the last line we have arranged the result to the form of \eqref{eq:dcEikonalDG} with the double copy kernel $K_{ij}^{\mathrm{(123)}}$ and the numerator $N_i^{\mathrm{(123)}}$ defined in \eqref{eq:eikonalNLOKernel} and \eqref{eq:N123atNLO} respectively.
Turning to the bulk graviton interaction graphs
thanks to the field redefinition of $\{\varphi, h_{\mu\nu}\}$, the three-graviton vertex \eqref{eq:vertex3h} is directly proportional to the square of three-gluon vertex, so we can easily compute the last diagram which is manifestly a double-copy of the WYM one
\begin{align}
\label{eq:dcNLO2}
&\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [sdot] (g1) {};
\vertex [below=0.5cm of g1] (g2);
\vertex [sdot, above=1cm of g1] (a0) {};
\vertex [sdot, left=0.87cm of g2] (b0) {};
\vertex [sdot, right=0.87cm of g2] (c0) {};
\vertex [left=0.6cm of a0, label=180:$1$] (a1);
\vertex [right=0.6cm of a0] (a2);
\vertex [above=0.52cm of b0] (i1);
\vertex [left=0.3cm of i1, label=180:$2$] (b1);
\vertex [below=0.52cm of b0] (i2);
\vertex [right=0.3cm of i2] (b2);
\vertex [below=0.52cm of c0] (i3);
\vertex [left=0.3cm of i3, label=180:$3$] (c1);
\vertex [above=0.52cm of c0] (i4);
\vertex [right=0.3cm of i4] (c2);
\diagram*{
(a1) -- [scalar] (a0) -- [scalar] (a2),
(b1) -- [scalar] (b0) -- [scalar] (b2),
(c1) -- [scalar] (c0) -- [scalar] (c2),
(g1) -- [graviton] (a0),
(g1) -- [graviton] (b0),
(g1) -- [graviton] (c0),
(g1) -- [opacity=0, rmomentum={[arrow distance=5pt, label distance=-4pt, arrow shorten=0.2]$k_1$}] (a0),
(g1) -- [opacity=0, rmomentum={[arrow distance=5pt, label distance=-4pt, arrow shorten=0.2]$k_2$}] (b0),
(g1) -- [opacity=0, rmomentum={[arrow distance=5pt, label distance=-4pt, arrow shorten=0.2]$k_3$}] (c0),
};
\end{feynman}
\end{tikzpicture}
= \frac{-i\kappa^4}{16} \int \mathrm{d} \mu_{1,2,3} \frac{2 n_1^2}{k_1^2 k_2^2 k_3^2}.
\end{align}
From \eqref{eq:dcNLO1} and \eqref{eq:dcNLO2} we therefore conclude that the double copy of the WYM eikonal coincides with the one of WDG also at the next-to-leading order ($\mathcal{O}(\kappa^4)$).
\section{Radiative double copy}
\label{sec3}
In this letter we are mainly considering the conservative sector of the WQFT, however, with a slight modification we can generalize the eikonal double copy \eqref{eq:dcEikonal} to classical radiation.
In WQFT, the $\Phi$ field radiation is computed as \cite{Jakobsen:2021smu,Jakobsen:2021lvp}
\begin{align}
-i k^2 \left. \langle \Phi(k) \rangle \right|_{k^2=0}
\end{align}
For $\Phi \in \{A_\mu^a, h_{\mu\nu} \}$, we also need to contract it with the polarizations $\{\epsilon^\mu, \epsilon^{\mu\nu}\}$ respectively.
We take the gluon radiation as an example.
Loosely speaking, the radiation at order $\mathcal{O}(g^{2n-1})$ can be obtained from the eikonal phase at $\mathcal{O}(g^{2n})$ by cutting off one worldline.
Diagrammatically, the gluon radiation of a binary source at leading order can be gained from \eqref{eq:YMNLOzprop}-\eqref{eq:YMNLO3g} by cutting the propagator $k_3$ and identifying $k_3$ with the momentum of the radiated gluon.
The on-shell condition $k_3 \cdot \epsilon = 0$ plays the same role as the $\delta\!\!\!{}^-\!(k_3 \cdot p_3)$ in the measure of the eikonal phase.
This ensures that the gluon radiation can be decomposed into $C_i K_{ij} N_j$, with $C_i$ attained from \eqref{eq:eikonalNLOColor} and \eqref{eq:BSNLO3phi} by striping off $c_3$, $N_i$ from \eqref{eq:N123atNLO} and \eqref{eq:N0atNLO} by replacing $p_3^\mu$ by $\epsilon^\mu$ and $K_{ij}$ being identical to \eqref{eq:eikonalNLOKernel}.
From the same approach we can also get the gravitational radiation and decompose it as $N_i K_{ij} N_j$.
Therefore we conclude that the double copy construction works for radiation, too.
We note that this is equivalent to the approach considered by Shen \cite{Shen:2018ebu} and Goldberger and Ridgway \cite{Goldberger:2016iau} where the radiation is calculated by solving the equations of motion.
\section{From amplitude to eikonal}
\label{sec4}
The expectation values in WQFT are directly linked to the classical limit of S-matrix element.
Consequently, we can expect that the classical double copy of WQFT discussed is also closely related to the double copy at the level of the scattering amplitude.
In this section, we will consider scalar QCD, i.e.~massive scalar fields coupled to Yang-Mills whose double copy has been studied in \cite{Plefka:2019wyg}.
We claim that the classical limit of the scattering amplitude of $n$ distinct scalar pairs corresponds to the WYM eikonal phase at $\mathcal{O} \left(g^{2(n-1)} \right)$ and show the connection explicitly at $\mathcal{O} \left(g^{4} \right)$.
Moreover, we will demonstrate that the double copy of the eikonal phase is the classical limit of the BCJ double copy of the scattering amplitude.
The exponentiated eikonal phase is directly related to the classical limit of scattering amplitude\cite{Amati:1987wq, Amati:1990xe},
\begin{align}
(1+\Delta_{q}) e^{i \chi} -1 = \sum_{n=2} \frac{1}{2^n} \int {\mathrm{d} \mu_{1,2,...,n}} \lim_{\hbar \to 0}\mathcal{A} (n \to n)
\end{align}
where $\mathcal{A} (n \to n)$ denotes an amplitude of $n$ pairs of distinct massive scalars,
and $\chi$ is the total eikonal phase, which scales as $\hbar^{-1}$ and receives contributions from all higher-loop amplitudes.
The introduction of the ``quantum remainder'' $\Delta_{q}$ (scaling as $\hbar^{n\geq 0}$) is needed for consistency \cite{Heissenberg:2021tzo}.
Here, we only care about tree diagrams, therefore we have
\begin{align}
\label{eq:AtoEikonal}
\chi_{n-1} = \frac{-i}{2^n} \int {\mathrm{d} \mu_{1,2,...,n}} \lim_{\hbar \to 0}\mathcal{A}^{\mathrm{tree}} (n \to n).
\end{align}
The correspondence at the $2\to 2$ level is rather trivial, so we will focus on the $3\to 3$ case.
The leading order 6-scalar amplitude in SQCD is \cite{Plefka:2019wyg}\footnote{We have converted the result of \cite{Plefka:2019wyg} to follow our conventions.}
\begin{align} \label{eq:6scalars}
\mathcal{A}^{\mathrm{tree}} (3& \to 3) =
\begin{tikzpicture}[baseline={(current bounding box.center)}]
\begin{feynman}
\vertex [small blob] (c) {};
\vertex [left=0.9cm of c, label=180:\(\hat{p}_2\!+\!\frac{k_2}{2}\text{, } l\)] (l2);
\vertex [label=180:$\hat{p}_1\!+\!\frac{k_1}{2}\text{, } j$] at ($(c)!1!-60:(l2)$) (l1);
\vertex [label=180:$\hat{p}_3\!+\!\frac{k_3}{2}\text{, } n$] at ($(c)!1!60:(l2)$) (l3);
\vertex [right=0.9cm of c, label=0:$k\text{, }\hat{p}_2\!-\!\frac{k_2}{2}$] (r2);
\vertex [label=0:$i\text{, }\hat{p}_1\!-\!\frac{k_1}{2}$] at ($(c)!1!60:(r2)$) (r1);
\vertex [label=0:$m\text{, }\hat{p}_3\!-\!\frac{k_3}{2}$] at ($(c)!1!-60:(r2)$) (r3);
\diagram*{
(l1) -- [fermion, arrow size=1.1pt, red] (c) -- [fermion, arrow size=1.1pt, red] (r1),
(l2) -- [fermion, arrow size=1.1pt, blue] (c) -- [fermion, arrow size=1.1pt, blue] (r2),
(l3) -- [fermion, arrow size=1.1pt, green] (c) -- [fermion, arrow size=1.1pt, green] (r3),
};
\end{feynman}
\end{tikzpicture}
\nonumber \\
=& {8\hat{c}^{(0)} \hat{n}^{(0)} \over k_1^2 k_2^2 k_3^2}
+ \bigg[ \frac{8}{k_2^2 k_3^2} \bigg(
\frac{\hat{c}^{(123)} \hat{n}^{(123)}}{2 \hat{p}_1\! \cdot \! k_2 - k_2 \! \cdot \! k_3} \nonumber\\
& + \frac{\hat{c}^{(132)} \hat{n}^{(132)}}{2 \hat{p}_1\! \cdot \! k_3 - k_3 \! \cdot \! k_2}
\bigg) + \text{cyclic} \bigg],
\end{align}
where we have introduced $\hat{p}_{i}$ as the average of the in- and outgoing momentum of particle $i$ which is orthogonal
to its momentum transfer $\hat{p}_i \cdot k_i =0$.
The color factors are
\begin{align}
\hat{c}^{(0)} &= f^{abc} T^a_{ij} T^b_{kl} T^c_{mn} \nonumber \\
\hat{c}^{(123)} &= (T^b T^a)_{ij} T^a_{kl} T^b_{mn} \\
\hat{c}^{(132)} &= (T^a T^b)_{ij} T^a_{kl} T^b_{mn}.\nonumber
\end{align}
where $i,j,...,l$ denote the color indices of the massive scalars.
The corresponding numerators are
\begin{align}
\hat{n}^{(0)} =& {-ig^4} \hat{p}_{1,\mu} \hat{p}_{2,\nu} \hat{p}_{3,\rho} V^{\mu\nu\rho}_{123} \\
\label{eq:numerator123}
\hat{n}^{(123)}\! =& \frac{-ig^4}{2} \Big( 4\hat{p}_1\! \cdot \! \hat{p}_2\, \hat{p}_1\! \cdot \! \hat{p}_3
+ 2\hat{p}_1\! \cdot \! \hat{p}_3 \, k_1\! \cdot \! \hat{p}_2
- 2 \hat{p}_1\! \cdot \! \hat{p}_2 \, k_1\! \cdot \! \hat{p}_3
\nonumber \\ &
- 2 \hat{p}_1 \! \cdot \! k_2 \, \hat{p}_2 \! \cdot \! \hat{p}_3
- k_1 \! \cdot \! \hat{p}_2 \, k_1\! \cdot \! \hat{p}_3
+ k_2 \! \cdot \! k_3 \, \hat{p}_2\! \cdot \! \hat{p}_3 \Big) \\
\label{eq:numerator132}
\hat{n}^{(132)}\! =& \frac{-ig^4}{2} \Big( 4\hat{p}_1\! \cdot \! \hat{p}_2\, \hat{p}_1\! \cdot \! \hat{p}_3
+ 2\hat{p}_1\! \cdot \! \hat{p}_2 \, k_1\! \cdot \! \hat{p}_3
- 2 \hat{p}_1\! \cdot \! \hat{p}_3 \, k_1\! \cdot \! \hat{p}_2
\nonumber \\ &
- 2 \hat{p}_1 \! \cdot \! k_3 \, \hat{p}_2 \! \cdot \! \hat{p}_3
- k_1 \! \cdot \! \hat{p}_2 \, k_1\! \cdot \! \hat{p}_3
+ k_2 \! \cdot \! k_3 \, \hat{p}_2\! \cdot \! \hat{p}_3 \Big),
\end{align}
which has been brought into a form to satisfy color-kinematic duality $\hat{n}^{(132)} - \hat{n}^{(123)} = \hat{n}^{(0)}$.
In the classical limit we take small momentum transfers $k_i \to \hbar k_i$, and consider the expansion in small $\hbar$ following \cite{Kosower:2018adc}.
In \eqref{eq:numerator123} and \eqref{eq:numerator132}, we have already sorted the terms in powers of $k_i$.
We identify the momentum as $\hat{p}_i = p_i$, although since $\hat{p}_i ^2 \neq m_i^2$, we need to change the definition of $p_i$ to $p_i=\hat{m}_i v_i$ with $\hat{m}_i^2 = \hat{p}_i ^2$.\footnote{This is related to the fact that we use Feynman propagators in the amplitudes, for the details, see \cite{Mogull:2020sak}.}
At this order, the redefinition will not change the WQFT result.
The massive propagators will become
\begin{align}
\frac{1}{2 \hat{p}_1\! \cdot \! k_2 - k_2 \! \cdot \! k_3} \to
\frac{1}{\hbar} \frac{1}{2 p_1\! \cdot \! k_2} + \frac{k_2 \! \cdot \! k_3}{4 (p_1\! \cdot \! k_2)^2}
+ \mathcal{O}(\hbar).
\end{align}
Performing the classical limit of the Yang-Mills amplitude, we also need to consider the classical limit of the color factors, which was recently investigated by de la Cruz et al. \cite{delaCruz:2020bbn}.
Built on their insight, we propose the classical limit of the color factors to be
\begin{align}
T^a_{ij} \rightarrow&\ c^a \\
\label{eq:classicalTT}
(T^aT^b)_{ij} \rightarrow&\ c^a c^b + \hbar\, c^{ab}\\
f^{abc} \rightarrow&\ \hbar f^{abc}.
\end{align}
Note that the sub-leading term in \eqref{eq:classicalTT} guarantees that the Jacobi identity holds in the classical limit.
It is now straightforward to compute the the classical limit of the amplitude \eqref{eq:6scalars} and extract the eikonal using \eqref{eq:AtoEikonal}. Keeping only the leading order terms in the classical $\hbar \to 0$ limit, we have
\begin{align}
{\hat{c}^{(0)} \hat{n}^{(0)} \over k_1^2 k_2^2 k_3^2}
\to&\ C_i^{(0)} K_{ij}^{(0)} N_j^{(0)} \\
\frac{1}{ k_2^2 k_3^2 } \bigg(
\frac{\hat{c}^{(123)} \hat{n}^{(123)}}{ 2 \hat{p}_1\! \cdot \! k_2 - k_2 \! \cdot \! k_3 } +& \frac{\hat{c}^{(132)} \hat{n}^{(132)}}{ 2 \hat{p}_1\! \cdot \! k_3 - k_2 \! \cdot \! k_3 } \bigg)
\nonumber\\
\to& \ C_i^{(123)} K_{ij}^{(123)} N_j^{(123)}.
\end{align}
We therefore recover the eikonal phase of SQCD from the WQFT, which directly operates at the classical level.
We can double copy the SQCD amplitude \eqref{eq:6scalars} by replacing the color factors by the numerators.
This results in an amplitude of massive scalars coupled to gravity and the dilaton.
We can then likewise consider the classical limit of this gravitational amplitude,
\begin{align}
{\hat{n}^{(0)} \hat{n}^{(0)} \over k_1^2 k_2^2 k_3^2}
\to&\ N_i^{(0)} K_{ij}^{(0)} N_j^{(0)} \\
\frac{1}{ k_2^2 k_3^2 } \bigg(
\frac{\hat{n}^{(123)} \hat{n}^{(123)}}{ 2 \hat{p}_1\! \cdot \! k_2 - k_2 \! \cdot \! k_3 } +& \frac{\hat{n}^{(132)} \hat{n}^{(132)}}{ 2 \hat{p}_1\! \cdot \! k_3 - k_2 \! \cdot \! k_3 } \bigg)
\nonumber\\
\to& \ N_i^{(123)} K_{ij}^{(123)} N_j^{(123)},
\end{align}
which coincides with our calculation in WDG.
We have therefore verified that the classical double copy of the world line quantum field theory is in full agreement with the quantum double copy of amplitudes at LO and NLO.
Note that the double copy of SQCD contains self-interactions of massive scalars \cite{Plefka:2019wyg}, however, these are short-range interactions and do not contribute to the classical theory. So we don't need to introduce additional terms in WDG and the double copy automatically works out.
\section{Conclusions}
In this work, we have extended the WQFT formalism to massive, charged point particles coupled to the bi-adjoint scalar field, Yang-Mills and dilaton-gravity theories.
We proposed a classical double copy prescription for the eikonal phases, alias free energies, in these theories and explicitly
verified the validity of the double copy up to quartic order (NLO) in the coupling constants. This entails the double copy relation
for the particle's deflection (or scattering angle) as the derivative of the eikonal with respect to the impact parameter.
With minor modifications our double copy prescription also applies to classical radiation emitted in the scattering process.
As a technical tool it was necessary to increase the number of worldlines of scattered particles with the order of
perturbation theory, i.e.~an $(n+2)$-body system for the N${}^{n}$LO order as well as to consider the eikonal of WBS in parallel
to establish the double copy kernels.
In fact, we expect all expectation values in WYM and WDG to feature the double copy relation as they are directly related to the quantum scattering amplitudes.
To illustrate the connection, we compared our eikonal phase to the classical limit of the corresponding eikonal emerging from the massive scalar six-point amplitude finding agreement.
These insights give us the expectation that the classical double copy for the WQFT will prevail to NNLO and higher. This
would cure the breakdown observed in \cite{Plefka:2019hmz} of a double copy prescription for the off-shell effective action of the
particle's worldline coordinates $x^{\mu}(\tau)$. The essential difference of our approach to the off-shell effective action of \cite{Plefka:2018dpa,Plefka:2019hmz} is that in WQFT \emph{both} the force
mediating fields and the fluctuations on the worldline are integrated out in the path integral.
The most important application of WQFT will be to Einstein gravity. This, however, continues to be a challenge for the
double copy prescription as it suffers from pollution of the dilaton and in principle even the Kalb-Ramond two-form.
Many approaches have been explored to remove the dilaton from the double copy construction
\cite{Luna:2017dtq,Johansson:2014zca,Bern:2019crd,Carrasco:2021bmu}.
Since the WQFT provides a simple way to extract classical quantities, it might be easier to project out dilaton
at the classical level. It will be interesting to explore this in future work.
An obvious extension of the classical double copy is to incorporate spin, some attempts in this direction are \cite{Goldberger:2017ogt,Li:2018qap, Goldberger:2019xef}.
The recently discovered hidden supersymmetry in the worldline description of spinning compact bodies \cite{Jakobsen:2021zvh}
should be applicable to the Yang-Mills case as well. In particular the limitation of coupling higher-spin worldline theories
to gravity might be overcome upon using the double copy. Concretely, the construction of the $\sqrt{\text{Kerr}}$ solution
\cite{Arkani-Hamed:2019ymq} in the language of the WQFT would be an interesting starting point.
\acknowledgments
This project has received funding from the European Union's Horizon 2020 research
and innovation program under the Marie Sklodowska-Curie grant agreement No. 764850 ``SAGEX''.
JP thanks the Max-Planck-Institut f\"ur Physik (Werner-Heisenberg Institut) for hospitality.
Some of our figures were produced with the help of TikZ-Feynman \cite{Ellis:2016jkw}.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 1,803
|
{"url":"https:\/\/chat.stackexchange.com\/transcript\/3740\/2018\/3\/25","text":"10:18 AM\n0\n\nAs recently pointed on this thread, we are going to get rid of functions. Today I noticed that we also have partial-functions. I feel that this tag can go just as well as part of the process.\n\n6 hours later\u2026\n4:28 PM\n0\n\n2017 Outstanding Tag Management: From 2016 tag management Decision needs to be made on [tag:map-projections]. What to do about graded algebraic structure tags Pluralize [tag:division-ring] Semicontinuity Proposal to rename [tag:generalizedeigenvector] Discussion on [tag:probability], [tag:expec...\n\n4 hours later\u2026\n8:42 PM\n1\n\nTheorem: Suppose $(X,d)$ is a metric space and $f:[0,1] \\rightarrow X$ is a path in $X$ with no-zero finite length $L$. Then, there exists a path $g:[0,1] \\rightarrow X$ from $f(0)$ to $f(1)$ that has the same image as $f$ and satisfies $lth_t(g) = tL ~\\forall ~t \\in [0,1]$. In particular $g$ is ...\n\nIs sufficiently different from the existing tag?\n1\n\nTwo discrete random variables $X$ and $Y$, whose values are positive integers, have the joint probability mass function: $$p(x,y) = 2^{-x-y}$$ I need to determine the marginal probability mass functions, which I believe to be defined as $p(x) = \\sum p(x,y)$ for $y$ and $p(y) = \\sum p(x,y)$ for ...\n\n1\n\nIs there any situation in that joint probability $p(x,y)$ equals to marginal probability $p(x)$? What is the interpretation of this situation?\n\n2\n\nWhere does the term \"marginal\" in \"marginal probability\" or \"marginal distribution\" come from?\n\n2\n\nLet $X$ and $Y$ be random variables with joint probability mass function $f(x,y) = k \\cdot \\dfrac {2^{x+y}}{x!y!}$, for $x, y \\in \\{ 0, 1, 2, \\cdots \\}$ and for a positive constant $k$. How can I derive the marginal probability mass function of $X$? How do I evaluate $k$? Are $X$ and $Y$ ind...\n\n1\n\nHi , I tried to resolve this question by taking the marginal pdf of $Y$ , then find it's expected value . But apparently this is not correct, they used the pdf of $f(x,y)$ in the solution and performed double integral to find $E(Y)$, can someone explain me why? Thanks\n\n1\n\nIn the book of Haskell Programming by C. Allen, at page 39, it is given the following lambda expression $$(\ud835\udf06\ud835\udc65\ud835\udc66.\ud835\udc65\ud835\udc65\ud835\udc66)(\ud835\udf06\ud835\udc65.\ud835\udc65\ud835\udc66)(\ud835\udf06\ud835\udc65.\ud835\udc65\ud835\udc67)$$ According to me, this equals to by applying the left two expression as an input for the rightmost expression $$(\ud835\udf06\ud835\udc65.\ud835\udc65\ud835\udc66)(\ud835\udf06\ud835\udc65.\ud835\udc65\ud835\udc66)(\ud835\udf06\ud835\udc65.\ud835\udc65\ud835\udc67)... The tag also has tag-excerpt: \"For questions involving the notion of the Radon-Nikodym derivative or the Radon-Nikodym theorem. Use this tag along with (probability-theory) or (measure-theory).\" 1 I am curious about the following problem: Let B_t be a standard Brownian motion on (\\Omega, \\mathcal F, \\mathcal F_t, \\mathbb P_a), where the filtration is generated by B_t. On a finite interval [0,T] we define X_t as the one solving the SED$$\\mathrm dX_t=\\mu_a\\,\\mathrm dt+\\sigma\\,\\,\\...\n\n7\n\nThe problem: Let $T >0$, and let $(\\Omega, \\mathscr F, \\{ \\mathscr F_t \\}_{t \\in [0,T]}, \\mathbb P)$ be a filtered probability space where $\\mathscr F_t = \\mathscr F_t^W$ where $W = \\{W_t\\}_{t \\in [0,T]}$ is standard $\\mathbb P$-Brownian motion. Let $X = \\{X_t\\}_{t \\in [0,T]}$ be a stoch...\n\n2\n\nLet $X$ be a random variable on a probability space $(\\Omega,\\mathscr F, P)$. Define a new probability measure $$\\tilde P(A) = E[1_A X]$$ for all $A\\in\\mathscr F$. Let $\\tilde E$ be expectation taken with respect to the new measure $\\tilde{P}$. Suppose now that $Y$ is also a random variable $(\\... 0 How do I prove the following? I don't know where to start. If$X$is a random variable with$E^{\\mathbb P}[X] = \\mathbb P(X>0)=1$and$ \\mathbb Q$is the probability measure defined by$ \\mathbb Q(A)=E^{\\mathbb Q}[X1_A] $then$E^{\\mathbb Q}[Y]=E^{\\mathbb P}[XY]$2 Let$\\alpha$and$\\beta$be equivalent probability measures on$(\\Omega, \\mathcal{F})$, with Radon-Nikodym density of$\\alpha$wrt$\\beta$is$\\eta$, i.e., for all$A \\in \\mathcal{F}, \\beta(A) = \\int_A\\eta d\\alpha$. Let$\\mathcal{G}$be a sub-$\\sigma$-field of$\\mathcal{F}$. Show that$\\eta$i... 1 i've come across this problem in Petersen's \"Ergodic Theory\": Let$(X,\\mathcal{B},T,\\mu)$be an ergodic dynamical system. Let$\\nu\\ll\\mu$be a measure un$(X,\\mathcal{B})$such that$\\nu T^{-1}\\ll\\nu$. Show that$\\nu=\\nu T^{-1}$and that$\\nu$is a constant multiple of$\\mu$. I've tried solving... 1 Consider the Black-Scholes Model where we have the following risky asset$dS_t = \\mu S_t dt + \\sigma S_t dW_t, t\\in[0,T] t\u22650 ,S_0 = s >0 $where$\\mu,\\sigma$are positive constants and a risk-free asset$dB_t = rB_tdt , B_0 = 1 (W_t)_{t\u22650}$denotes a standard brownian motion with its ... 1 Is there any situation in that joint probability$p(x,y)$equals to marginal probability$p(x)$? What is the interpretation of this situation? 2 Where does the term \"marginal\" in \"marginal probability\" or \"marginal distribution\" come from? 1 Hi , I tried to resolve this question by taking the marginal pdf of$Y$, then find it's expected value . But apparently this is not correct, they used the pdf of$f(x,y)$in the solution and performed double integral to find$E(Y)$, can someone explain me why? Thanks 3 I am reading Mac Lane's Categories for the Working Mathematician. He mentioned that the usual completion of metric space is universal for the evident forgetful functor (from complete metric spaces to metric spaces). (p.57) I am not sure what is this forgetful functor 'forgetting'. I think this... 2 I have read in many category theory textbooks the term \"forgetful functor\". But no has ever given me a precise definition of this term. I want a rigorous definition, not merely an answer that basically says, \"I know it when I see it.\". Has someone developed a perfectly rigorous definition of forg... 1 Let$K$be a closed convex cone. Then$K$is solid if and only if$K$is reproducing. Hint: If$K$is a convex cone then$K-K$has a nonempty interior.$K-K$is the minimal subspace containing$K$. Definitions: A set$K$in a Euclidean space$V$is a convex cone if for every$x, y \\i...\n\n2\n\nI studying metrization and in different parts I encountered different formulations of theorems, for example in the Nagata\u2013Smirnov metrization theorem I found: A topological space $X$ is metrizable if and only if it is $T_3$ and Hausdorff and has a $\\sigma$-locally finite base, and other: ...\n\n0\n\nI have read about the following example from Muller: $(M) \\begin{cases} x' = f(t,x) \\\\[1mm] x(0) = 0 \\end{cases}$ where $f: \\mathbb{R}\\times\\mathbb{R} \\rightarrow \\mathbb{R}$ is the function: \\$f(t,x) = \\begin{cases} 0 & t \\leq 0, x \\in \\mathbb{R} \\\\ 2t & t>0,x < 0 \\\\ 2t - \\frac{4x}{t} & t >...","date":"2020-03-30 23:53:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9882912039756775, \"perplexity\": 329.869065102221}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-16\/segments\/1585370497309.31\/warc\/CC-MAIN-20200330212722-20200331002722-00140.warc.gz\"}"}
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{"url":"http:\/\/mathhelpforum.com\/calculus\/41708-particle-curvature-question.html","text":"1. ## particle\/curvature question\n\nHi guys can anyone help me out with this?\n\nA moving particle at time t \u2208 [0, 10] (seconds) has position vector in metres from the origin (0, 0, 0) given by the vector function r(t) = (10 \u2212 t)i + (t^2\n\u2212 10t)j + sin tk.\ni. Describe the path of the particle, as seen from above (the positive k-direction), and also describe it in three dimensions.\nii. Find the curvature of the path, at t = 2\u03c0 \u2248 6.28 seconds.\niii. Find the angle between the path (at start and end-points) and the k-direction.\n\nThanks\n\n2. Originally Posted by katie\nHi guys can anyone help me out with this?\n\nA moving particle at time t \u2208 [0, 10] (seconds) has position vector in metres from the origin (0, 0, 0) given by the vector function r(t) = (10 \u2212 t)i + (t^2\n\u2212 10t)j + sin tk.\ni. Describe the path of the particle, as seen from above (the positive k-direction), and also describe it in three dimensions.\n\nMr F says:\n\nx = 10 - t => t = 10 - x .... (1)\n\ny = t^2 - 10t .... (2)\n\nSubstitute (1) into (2).\n\nii. Find the curvature of the path, at t = 2\u03c0 \u2248 6.28 seconds.\n\nMr F says: Substitute into the formula.\n\niii. Find the angle between the path (at start and end-points) and the k-direction.\n\nMr F says: Use the dot product.\n\nThanks\n..\n\n3. when you say substitute into the formula is that just into the r(t) formula?\n\nand the dot product, i am not sure what the other vector is to be using the dot product with?\n\nthanks\n\n4. Originally Posted by jimmy\nwhen you say substitute into the formula is that just into the r(t) formula?\n\nand the dot product, i am not sure what the other vector is to be using the dot product with?\n\nthanks\nThe curvature formula.\n\nDirection of motion given by dr\/dt. Evaluate at t = 0 and t = 10. Consider the dot product of each with k ....","date":"2017-06-24 17:50:20","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8026615381240845, \"perplexity\": 899.9696600991394}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-26\/segments\/1498128320270.12\/warc\/CC-MAIN-20170624170517-20170624190517-00521.warc.gz\"}"}
| null | null |
Gracie Fields
Dame Gracie Fields, DBE (born Grace Stansfield; 9 January 1898 – 27 September 1979) was an English actress, singer and comedienne and star of both cinema and music hall.[1][2] She spent the later part of her life on the isle of Capri, Italy. Fields was made a Commander of the Order of the British Empire (CBE) for Services to Entertainment in 1938, and in 1979, seven months before her death, she was invested a Dame by Queen Elizabeth II.[3]
Fields on Capri (Allan Warren, 1973)
Grace Stansfield
(1898-01-09)9 January 1898
Rochdale, Lancashire, England
27 September 1979(1979-09-27) (aged 81)
La Canzone Del Mare, Capri, Italy
Archie Pitt
Monty Banks
Boris Alperovici (d. 1983)
Life and workEdit
Fields was born Grace Stansfield, over a fish and chip shop owned by her grandmother, Sarah Bamford, in Molesworth Street, Rochdale, Lancashire. She made her first stage appearance as a child in 1905, joining children's repertory theatre groups such as Haley's Garden of Girls and the Nine Dainty Dots. Her two sisters, Edith and Betty, and brother, Tommy, all went on to appear on stage, but Gracie was the most successful. Her professional debut in variety took place at the Rochdale Hippodrome theatre in 1910 and she soon gave up her job in the local cotton mill, where she was a half-timer, spending half a week in the mill and the other half at school.
Fields met the comedian and impresario Archie Pitt and they began working together. Pitt gave Fields champagne on her 18th birthday, and wrote in an autograph book to her that he would make her a star. Pitt began to manage her career and they began a relationship; they married in 1923 at Clapham Register Office. Their first revue was called Yes I Think So in 1915 and the two continued to tour Britain together until 1924. That year they appeared in the revue Mr Tower of London, with other shows By Request, It's A Bargain and The Show's The Thing, during the following years.
Pitt was the brother of Bert Aza, founder of the Aza agency, which was responsible for many entertainers of the day including the actor and comedian Stanley Holloway, who was introduced to Aza by Fields. Fields and Holloway first worked together on her film Sing As We Go in 1934 and the two remained close friends for the rest of their lives.[4]
FameEdit
Fields came to major public notice in Mr Tower of London, which appeared in London's West End.[5] Her career accelerated from this point with legitimate dramatic performances and the beginning of a recording career.
At one point, Fields was playing three shows a night in London's West End. She appeared in the Pitt production SOS with Gerald Du Maurier, a legitimate production staged at the St James's Theatre.
Fields' most famous song, which became her theme, "Sally", was worked into the title of her first film, Sally in Our Alley (1931), which was a major box office hit. She went on to make several films initially in Britain and later in the United States (for which she was paid a record fee of £200,000 for four films). Regardless, she never enjoyed performing without a live audience, and found the process of film-making boring. She tried to opt out of filming, before director Monty Banks persuaded her otherwise, landing her the lucrative Hollywood deal. Fields demanded that the four films be filmed in Britain and not Hollywood, and this was the case.
The final few lines of the song "Sally", which Fields sang at every performance from 1931 onwards, were written by her husband's mistress, Annie Lipman. Fields claimed in later life that she wanted to "Drown blasted Sally with Walter with the aspidistra on top!", a reference to two other of her well-known songs, "Walter, Walter", and "It's the Biggest Aspidistra in the World".[6]
The famous opera star Luisa Tetrazzini heard her singing an aria and asked her to sing in grand opera. Gracie decided to stay "where I knew I belonged."[7]
Charity workEdit
In the 1930s her popularity peaked and she was given many honours: the Officer of the Venerable Order of St. John (for charity work), the Commander of the Order of the British Empire (CBE) (for services to entertainment) in 1938 and the Freedom of the Borough of Rochdale in 1937.
She donated her house in The Bishops Avenue, north London (which she had not much cared for and which she had shared with her husband Archie Pitt and his mistress) to an orphanage after the marriage broke down. In 1939, she became seriously ill with cervical cancer.[8] The public sent over 250,000 goodwill messages and she retired to her villa on Capri. After she recovered, she recorded a very special 78rpm record simply called Gracie's Thanks, in which she thanks the public for the many cards and letters she received while in hospital. During World War II, she paid for all servicemen/women to travel free on public transport within the boundaries of Rochdale.
Fields also helped Rochdale F.C. in the 1930s when they were struggling to pay fees and buy sports equipment.
In 1933 she set up the Gracie Fields Children's Home and Orphanage at Peacehaven, Sussex, for children of those in the theatre profession who could not look after their children. She kept this until 1967, when the home was no longer needed. This was near her own home in Peacehaven, and Fields often visited, with the children all calling her 'Aunty Grace'.[9]
World War IIEdit
Fields, accompanied by an RAF orchestra, entertains airmen at their 1939 Christmas party
Fields shares a joke with troops in a village near Valenciennes, France, April 1940
In 1939, Fields suffered a breakdown and went to Capri to recuperate.[10] World War II was declared while she was recovering in Capri, and Fields – still very ill after her cancer surgery – threw herself into her work and signed up for the Entertainments National Service Association (ENSA) headed by her old film producer, Basil Dean. Fields travelled to France to entertain the troops in the midst of air-raids, performing on the backs of open lorries and in war-torn areas. She was the first artist to play behind enemy lines in Berlin.
Following her divorce from Archie Pitt, she married Italian-born film director Monty Banks in March 1940. However, because Banks remained an Italian citizen and would have been interned in the United Kingdom after Italy declared war in 1940, she went with him to North America, possibly at the suggestion of Winston Churchill who told her to "Make American Dollars, not British Pounds",[8] which she did, in aid of the Navy League and the Spitfire Fund. She and Banks moved to their home in Santa Monica, California. Fields occasionally returned to Britain, performing in factories and army camps around the country. After their initial argument, Parliament offered her an official apology.
Although she continued to spend much of her time entertaining troops and otherwise supporting the war effort outside Britain, this led to a fall-off in her popularity at home. She performed many times for Allied troops, travelling as far as New Guinea, where she received an enthusiastic response from Australian personnel.[11] In late 1945 she toured the South Pacific Islands.
Past World War IIEdit
After the war, Fields continued her career less actively. She began performing in Britain again in 1948 headlining the London Palladium over Eartha Kitt who was also on the bill. The BBC gave her her own radio show in 1947 dubbed Our Gracie's Working Party in which 12 towns were visited by Fields, and a live show of music and entertainment was broadcast weekly with Fields compering and performing, and local talents also on the bill. This tour commenced in Rochdale. Like so many BBC shows at the time, this show transferred to Radio Luxembourg in 1950, sponsored by Wisk soap powder. Billy Ternent and his Orchestra accompanied her.
In 1951, Fields took part in the cabaret which closed the Festival of Britain celebrations[12]. She proved popular once more[citation needed], though never regaining the status she enjoyed in the 1930s. She continued recording, but made no more films, moving more towards light classical music as popular tastes changed, often adopting a religious theme. She continued into the new medium of LP records, and recorded new takes of her old favourite songs, as well as new and recent tracks to 'liven things up a bit'.
Monty Banks died on 8 January 1950[13] of a heart attack while travelling on the Orient Express. On 18 February 1952 in Capri, Fields married Boris Alperovici, a Romanian radio repairman.[14] She claimed that he was the love of her life, and that she couldn't wait to propose to him. She proposed on Christmas Day in front of friends and family. They married at the Church of St. Stefano on Capri in a quiet ceremony before honeymooning in Rome.
She lived on her beloved Isle of Capri for the remainder of her life, at her home La Canzone Del Mare, a swimming and restaurant complex which Fields' home overlooked. It was favoured by many Hollywood stars during the 1950s, with regular guests including Richard Burton, Elizabeth Taylor, Greta Garbo and Noël Coward.
She began to work less, but still toured the UK under the management of Harold Fielding, manager of top artists of the day such as Tommy Steele and Max Bygraves. Her UK tours proved popular, and in the mid-1960s she performed farewell tours in Australia, Canada and America – the last performance was recorded and released years later.
In 1956, Fields was the first actress to portray the title character in Miss Marple[15] in a US TV production of Agatha Christie's A Murder is Announced.[16] The production featured Jessica Tandy and Roger Moore, and predates the Margaret Rutherford films by some five years. She also starred in television productions of A Tale of Two Cities, The Old Lady Shows Her Medals – for which she won a TV Award – and Mrs 'Arris Goes to Paris, which was remade years later with Angela Lansbury as Mrs Harris, a charwoman in search of a fur coat (or a Christian Dior gown in Lansbury's case.)
In 1957, her single "Around the World" peaked at No.8 in the UK Singles Chart, with her recording of "Little Donkey" reaching No.20 in November 1959.[17] The sheet music for the song was the UK's best-seller for seven weeks.[18]
She was the subject of This Is Your Life in 1960 when she was surprised by Eamonn Andrews at the BBC Television Theatre.[19]
Fields regularly performed in TV appearances, being the first entertainer to perform on Val Parnell's Sunday Night at the London Palladium.[20] Fields had two Christmas TV specials in 1960 and 1961, singing her old favourites and new songs in front of a studio audience. 1971 saw A Gift For Gracie, another TV special presented by Fields and Bruce Forsyth. This followed on from her popularity on Stars on Sunday, a religious programme on Britain's ITV, in which well-known performers sang hymns or read extracts from the Bible. Fields was the most requested artist on the show.
In 1968, Fields headlined a two-week Christmas stint at the West Riding of Yorkshire's prestigious Batley Variety Club. "I was born over a fish and chip shop – I never thought I'd be singing in one!" claimed Fields during the performance recorded by the BBC.[21]
In 1975, her album The Golden Years reached No. 48 in the UK Albums Chart.[17]
In 1978, she opened the Gracie Fields Theatre, next to Oulder Hill Community School in her native Rochdale, performing a concert there recorded by the BBC to open the show. Fields appeared in ten Royal Variety Performances from 1928 onwards, her last being in 1978 at the age of 80 when she appeared as a surprise guest in the finale, in which she appeared and sang her theme song, "Sally".[5]
Her final TV appearance came in January 1979 when she appeared in a special octogenarian edition of The Merv Griffin Show in America, in which she sang the song she popularised in America, "The Biggest Aspidistra in the World".[22] Fields was notified by her confidante John Taylor while she was in America that she had the Queen's invitation to become a Dame Commander of the Order of the British Empire, to which she replied: "Yes I'll accept, yes I can kneel – but I might need help getting back up, and yes I'll attend – as long as they don't call Boris 'Buttons'."
Fields' health declined in July 1979, when she contracted pneumonia after performing an open-air concert on the Royal Yacht which was docked in Capri's harbour.[citation needed] After a spell in hospital, she seemed to be recovering, but died on 27 September 1979.[23] The press reported she died holding her husband's hand, but in reality he was at their Anacapri home at the time, while Gracie was home with the housekeeper, Irena. She is buried in Capri's Protestant Cemetery[23] in a white marble tomb. Her coffin was carried by staff from her restaurant. Her husband Boris died on 3 July 1983.[24]
Honours and popular cultureEdit
Fields was appointed Commander of the Order of the British Empire in 1938. In February 1979, she was invested as a Dame Commander of the Order of the British Empire[25] seven months before her death at her home on Capri, aged 81.
Gracie Fields was mentioned in the 1987 film Wish You Were Here, the 1996 film Intimate Relations, and the 2006 film The History Boys.
On 3 October 2009 the final train to run on the Oldham Loop before it closed to be converted to a tramway, a Class 156, was named in her honour.[26]
Fields was granted the Freedom of Rochdale.[27] The local theatre in Rochdale, the Gracie Fields Theatre, was opened by her in 1978.[28]
In September 2016, a statue of Fields was unveiled outside Rochdale Town Hall which is the first statue of a woman to be erected for over a century in Greater Manchester.[29]
Gracie Fields was the Mystery Guest on the May 1, 1955 airing of What's My Line? After Bennett Cerf asked about one of her songs, Dorothy Kilgallen correctly guessed it was her.
Notable songsEdit
Gracie Fields 1937
"We're All living at the Cloisters", You didn't want me when you had me
"Sally", The Kerry Dance
"Sing As We Go"
"Thing-Ummy-Bob (That's Gonna Win The War)"
"The Biggest Aspidistra in the World", Three Green bonnets
"I Took my Harp to a Party", The Trek
"Pedro the Fisherman"
"Only a Glass of Champagne", Speak softly love
"Angels Guard Thee", Around the world
"Nuns' Chorus", Little Donkey
"Now Is the Hour" The Carefree heart
"The Isle of Capri", The woodpecker song
"Walter, Walter, Lead Me to the Altar", Young at heart
"Christopher Robin is Saying His Prayers", Far Away
"If I Had a Talking Picture of You", Home
"Wish Me Luck as You Wave Me Goodbye", the Holy City
"When I Grow Too Old to Dream"
"If I Knew You Were Comin' I'd've Baked a Cake"
"The Twelfth of Never"
"Those Were The Days" (performed live at The Batley Variety Club in 1968)
"Singin' in the bathtub"
"Stop and shop at the Co-op shop"
"I never cried so much in all my life"
Gracie Fields in Stage Door Canteen (1943)
1931 Sally in Our Alley Sally Winch
1932 Looking on the Bright Side Gracie
1933 This Week of Grace Grace Milroy
1934 Love, Life and Laughter Nellie Gwynn
Sing As We Go Gracie Platt
1935 Look Up and Laugh Gracie Pearson
1936 Queen of Hearts Grace Perkins
1937 The Show Goes On Sally Scowcroft
1938 We're Going to Be Rich Kit Dobson
Young and Beautiful (short) Herself
Keep Smiling Gracie Gray
1939 Shipyard Sally Sally Fitzgerald
1943 Stage Door Canteen Alice Chalice
Holy Matrimony Herself
1945 Molly and Me Molly Barry
Paris Underground Emmeline Quayle
Box office rankingEdit
For a number of years, British film exhibitors voted her among the top ten stars in Britain at the box office via an annual poll in the Motion Picture Herald.
1936 – 1st (3rd most popular star over all)[30]
1937 – 1st (3rd overall)[31]
1938 – 2nd[32]
1939 - 2nd[33]
1940 - 3rd[34]
1941 - 8th[35]
^ "Gracie Fields".
^ "Gracie Fields - Biography, Movie Highlights and Photos - AllMovie". AllMovie.
^ "BFI Screenonline: Fields, Gracie (1898-1979) Biography". www.screenonline.org.uk.
^ Richard Anthony Baker (20 April 2010). "Obituaries / Jack Beckitt". The Stage. Retrieved 12 May 2012.
^ a b "FIELDS, Gracie (1898-1979) - English Heritage". www.english-heritage.org.uk.
^ Helen Smith Bevington (1983). The Journey is Everything: A Journal of the Seventies. Durham, NC: Duke University Press. p. 158. ISBN 0-8223-0553-4.
^ Rochdale Observer, Saturday 29 September 1979
^ a b David Bret (27 August 2009). "Truth about Gracie's life in exile". Daily Express. Retrieved 9 February 2018.
^ "Famous Residents", Peacehaven Council
^ The Afro American, newspaper, 2 December 1939, page 6 – (article) Twelve Sing Way Back to America by William N. Jones – "Bricktop has been back in America several weeks while Adelaide Hall has been singing for the soldiers. Miss Hall, whose popularity with the British Tommy's ranks with that of Gracie Fields, may remain in England as Miss Fields has recently suffered a breakdown."(retrieved 14 October 2015): https://news.google.com/newspapers?nid=2211&dat=19391118&id=5k9AAAAAIBAJ&sjid=RgMGAAAAIBAJ&pg=1335,3800200&hl=en
^ "Home - Australian War Memorial" (PDF). awm.gov.au. Retrieved 9 March 2015.
^ 'End To Festival', The Times(London), 1 October 1951, p.4:"The crowd...rose to great heights of enthusiasm when Miss Gracie Fields came to the microphone...and...joined in singing at her behest more heartily than probably anyone else could have persuaded them to do."
^ "Death Of Gracie Fields Husband" (scan). The Morning Bulletin (№ 27622). Rockhampton, Queensland. 10 January 1950. p. 1. Retrieved 9 March 2015 – via Trove-National Library of Australia.
^ Gavaghan, Julian (17 February 2014). "On This Day: British star Gracie Fields marries Romanian repairman in Italy". uk.news.yahoo.com. Retrieved 9 March 2015. Gracie Fields married a Romanian radio repairman [Boris Alperovici] in a quiet ceremony in Italy on this day [February 18] in 1952.
^ "First actress to portray Miss Marple on TV". www.guinnessworldrecords.com. Guinness World Records Limited. Archived from the original on 16 February 2015. Retrieved 16 February 2015. The earliest actress to portray Miss Marple was the UK's Gracie Fields in the NBC (USA) Goodyear TV Playhouse: A Murder is Announced in 1956.
^ "A Murder Is Announced (Full Cast & Crew)". Goodyear Playhouse. Internet Movie Database (IMDb). 1956. Retrieved 9 March 2015.
^ a b Roberts, David (2006). British Hit Singles & Albums (19th ed.). London: Guinness World Records Limited. p. 199. ISBN 1-904994-10-5.
^ "Around The World". UK Sheet Music Charts. onlineweb.com. 25 May 1957. Retrieved 9 March 2015. [failed verification]
^ "Gracie Fields". www.bigredbook.info.
^ marcus, laurence. "SUNDAY NIGHT AT THE LONDON PALLADIUM - A TELEVISION HEAVEN REVIEW". www.televisionheaven.co.uk.
^ "The variety club that made Batley the 'Las Vegas of the North'". www.yorkshirepost.co.uk.
^ "The Merv Griffin Show". TVGuide.com.
^ a b "Gracie Fields (1899–1979)". findagrave.com. Retrieved 9 March 2015.
^ "Boris Alperovici (? – 1983)". findagrave.com. Retrieved 9 March 2015.
^ "Supplement to the London Gazette". Her Majesty's Stationery Office. Retrieved 27 November 2009.
^ "Train named after Rochdale star". Rochdale Online. Retrieved 15 October 2009.
^ "Alderman Crowder grants the Freedom of Rochdale to Gracie Fields". YouTube. 8 September 2010. Retrieved 12 May 2012.
^ "Gracie Fields Theatre, Rochdale". Graciefieldstheatre.com. 27 September 1979. Retrieved 19 February 2015.
^ Halliday, Josh (19 September 2016). "'Our Gracie' comes home: Rochdale salutes Gracie Fields with statue". the Guardian. Retrieved 10 June 2018.
^ "Star names at the Box Office-British Preferences" (scan). The Mercury. Vol. CXLVI (№ 20, 731). Hobart, Tasmania. 10 April 1937. p. 5. Retrieved 27 April 2012 – via Trove-National Library of Australia.
^ 'Sir Cedric Hardwicke and Sara Allgood Will Star in Next Subscription Play at the National, 3 Jan.: 'Shadow and Substance,' Abbey Theater Hit, Fifth Guild-American Society Offering; 'Night Must Fall' Voted 1937's Best; Shirley Temple Again Biggest Box-Office Name; Mitzi Writes a Letter.', The Washington Post 20 December 1937: 14.
^ "FORMBY IS POPULAR ACTOR" (scan). The Mercury. Vol. CL (№ 21, 295). Hobart, Tasmania. 25 February 1939. p. 5. Retrieved 27 April 2012 – via Trove-National Library of Australia.
^ "Motion Picture Herald". archive.org.
Fields, Gracie (1960). Sing As We Go. London: Frederick Muller Limited. pp. 1–228.
Lassandro, Sebastian (2019). Pride of Our Alley, vol 1 and 2. Bear Manor Media.
Gracie Fields: The Authorised Biography (1995) by David Bret
"Gracie Fields" by Jeffrey Richards in the Oxford Dictionary of National Biography
Cullen, Frank; Hackman, Florence; McNeilly, Donald (2007), "Gracie Fields", in . (ed.), Vaudeville, old & new: an encyclopedia of variety performers in America, Vol.1, New York: Routledge Press, pp. 380–383, ISBN 978-0-415-93853-2, retrieved 2 September 2010 – via Google Books
"Gracie Fields: English comedienne mugs and sings", Life: 124, 126, 21 December 1942, retrieved 2 September 2010 – via Google Books
Hunter, Jefferson (2010), English Filming, English Writing, Bloomington, Indiana: Indiana University Press, ISBN 978-0-253-35443-3, retrieved 2 September 2010 – via Google Books – Paperback ISBN 978-0-253-22177-3
Johnston, Sheila M.F (2001), "London Little Theatre Era: Gracie Fields", in . (ed.), Let's go to the Grand!: 100 years of entertainment at London's Grand Theatre, Toronto, Canada: Natural Heritage/Natural History Inc, pp. 95–96, ISBN 1-896219-75-6, retrieved 2 September 2010 – via Google Books
Joyce, Patrick (1994), Visions of the people: industrial England and the question of class 1848–1914, Cambridge, New York: Press Syndicate of the University of Cambridge, pp. 215–219, 318–320, ISBN 0-521-37152-X, retrieved 2 September 2010 – via Google Books (First published 1991)
Landy, Marcia (2001), "The extraordinary ordinariness of Gracie Fields: the anatomy of a British film star", in Babington, Bruce (ed.), British Stars and Stardom: From Alma Taylor to Sean Connery, Manchester, UK: Manchester University Press, pp. 56–67, ISBN 0-7190-5840-6, retrieved 2 September 2010 – via Google Books – Paperback ISBN 0-7190-5841-4
Scannell, Paddy (1996), Radio, television, and modern life: a phenomenological approach, Oxford, UK & Cambridge, Massachusetts: Blackwell, pp. 65, 72, ISBN 0-631-19874-1, retrieved 2 September 2010 – via Google Books – Paperback ISBN 0-631-19875-X. Digitalised 2002
Slide, Anthony (1985), "Sing As We Go", in . (ed.), Fifty classic British films, 1932–1982: a pictorial record, General Publishing Co., Canada; and Constable & Co, UK, pp. 16–18, ISBN 0-486-24860-7, retrieved 2 September 2010 – via Google Books
Shafer, Stephen C (1997), British popular films, 1929–1939: the cinema of reassurance, London: Routledge, ISBN 0-415-00282-6, retrieved 2 September 2010 – via Google Books
Short, Ernest Henry; Compton Rickett, Arthur (1970), Ring up the curtain: being a pageant of English entertainment covering half a century, Freeport, N.Y: Books for Libraries Press, ISBN 0-8369-5299-5, retrieved 2 September 2010 – via Google Books (First published 1938)
Wikimedia Commons has media related to Gracie Fields.
Gracie Fields on IMDb
Gracie Fields at the BFI's Screenonline
The Official Dame Gracie Fields website
Gracie Fields at Turner Classic Movies
Gracie Fields: A Biography by Joan Moules
Gracie Fields' appearance on This Is Your Life
Photographs and literature
Nine digitally restored Gracie Fields recordings
Gracie Fields and Thomas Thompson
Gracie Fields interview on Parkinson, 05/11/1977
"The Gracie Fields Show". RadioEchoes. 1944.
Retrieved from "https://en.wikipedia.org/w/index.php?title=Gracie_Fields&oldid=905511128"
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Q: Transport Tracker to Track buses on Google Map in MVC Web Application I am developing an MVC based application in which I have to track the buses on google map like Uber Cab tracking system.
Can someone please suggest me the proper approach how can I achieve this?
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{"url":"https:\/\/www.groundai.com\/project\/rational-neural-networks4263\/","text":"Rational neural networks\n\n# Rational neural networks\n\n## Abstract\n\nWe consider neural networks with rational activation functions. The choice of the nonlinear activation function in deep learning architectures is crucial and heavily impacts the performance of a neural network. We establish optimal bounds in terms of network complexity and prove that rational neural networks approximate smooth functions more efficiently than ReLU networks with exponentially smaller depth. The flexibility and smoothness of rational activation functions make them an attractive alternative to ReLU, as we demonstrate with numerical experiments.\n\n## 1 Introduction\n\nDeep learning has become an important topic across many domains of science due to its recent success in image recognition, speech recognition, and drug discovery\u00a0Hinton et al. (2012); Krizhevsky et al. (2012); LeCun et al. (2015); Ma et al. (2015). Deep learning techniques are based on neural networks, which contain a certain number of layers to perform several mathematical transformations on the input. A nonlinear transformation of the input determines the output of each layer in the neural network: , where is a matrix called the weight matrix, is a bias vector, and is a nonlinear function called the activation function (also called activation unit). The computational cost of training a neural network depends on the total number of nodes (size) and the number of layers (depth). A key question in designing deep learning architectures is the choice of the activation function to reduce the number of trainable parameters of the network while keeping the same approximation power\u00a0Goodfellow et al. (2016).\n\nWhile smooth activation functions such as sigmoid, logistic, or hyperbolic tangent are widely used, they suffer from the \u201cvanishing gradient problem\u201d\u00a0Bengio et al. (1994) because their derivatives are zero for large inputs. Neural networks based on polynomial activation functions are an alternative\u00a0Cheng et al. (2018); Daws Jr. and Webster (2019); Goyal et al. (2019); Guarnieri et al. (1999); Ma and Khorasani (2005); Vecci et al. (1998), but can be numerically unstable due to large gradients for large inputs\u00a0Bengio et al. (1994). Moreover, polynomials do not approximate non-smooth functions efficiently\u00a0Trefethen (2013), which can lead to optimization issues in classification problems. A popular choice of activation function is the Rectified Linear Unit (ReLU) defined as \u00a0Jarrett et al. (2009); Nair and Hinton (2010). It has numerous advantages, such as being fast to evaluate and zero for many inputs\u00a0Glorot et al. (2011). Many theoretical studies characterize and understand the expressivity of shallow and deep ReLU neural networks from the perspective of approximation theory\u00a0DeVore et al. (1989); Liang and Srikant (2016); Mhaskar (1996); Telgarsky (2016); Yarotsky (2017).\n\nReLU networks also suffer from drawbacks, which are most evident during training. The main disadvantage is that the gradient of ReLU is zero for negative real numbers. Therefore, its derivative is zero if the activation function is saturated\u00a0Maas et al. (2013). To tackle these issues, several adaptations to ReLU have been proposed such as Leaky ReLU\u00a0Maas et al. (2013), Exponential Linear Unit (ELU)\u00a0Clevert et al. (2015), Parametric Linear Unit (PReLU)\u00a0He et al. (2015), and Scaled Exponential Linear Unit (SELU)\u00a0Klambauer et al. (2017). These modifications outperform ReLU in image classification applications, and some of these activation functions have trainable parameters, which are learned by gradient descent at the same time as the weights and biases of the network. To obtain significant benefits for image classification and partial differential equation (PDE) solvers, one can perform an exhaustive search over trainable activation functions constructed from standard units\u00a0Jagtap et al. (2020); Ramachandran et al. (2017). However, most of the \u201cexotic\u201d activation functions in the literature are motivated by empirical results and are not supported by theoretical statements on their potentially improved approximation power over ReLU.\n\nIn this work, we study rational neural networks, which are neural networks with activation functions that are trainable rational functions. In\u00a0Section 3, we provide theoretical statements quantifying the advantages of rational neural networks over ReLU networks. In particular, we remark that a composition of low-degree rational functions has a good approximation power but a relatively small number of trainable parameters. Therefore, we show that rational neural networks require fewer nodes and exponentially smaller depth than ReLU networks to approximate smooth functions to within a certain accuracy. This improved approximation power has practical consequences for large neural networks, given that a deep neural network is computationally expensive to train due to expensive gradient evaluations and slower convergence. The experiments conducted in\u00a0Section 4 demonstrate the potential applications of these rational networks for solving PDEs and Generative Adversarial Networks (GANs).1 The practical implementation of rational networks is straightforward in the TensorFlow framework and consists of replacing the activation functions by trainable rational functions. Finally, we highlight the main benefits of rational networks: the fast approximation of functions, the trainability of the activation parameters, and the smoothness of the activation function.\n\n## 2 Rational neural networks\n\nWe consider neural networks whose activation functions consist of rational functions with trainable coefficients and , i.e., functions of the form:\n\n F(x)=P(x)Q(x)=\u2211rPi=0aixi\u2211rQj=0bjxj,aP\u22600,bQ\u22600, (1)\n\nwhere and are the polynomial degrees of the numerator and denominator, respectively. We say that is of type and degree .\n\nThe use of rational functions in deep learning is motivated by the theoretical work of Telgarsky, who proved error bounds on the approximation of ReLU neural networks by high-degree rational functions and vice versa\u00a0Telgarsky (2017). On the practical side, neural networks based on rational activation functions are considered by Molina et al.\u00a0Molina et al. (2019), who defined a safe Pad\u00e9 Activation Unit (PAU) as\n\n F(x)=\u2211rPi=0aixi1+|\u2211rQj=1bjxj|.\n\nThe denominator is selected so that does not have poles located on the real axis. PAU networks can learn new activation functions and are competitive with state-of-the-art neural networks for image classification. However, this choice results in a non-smooth activation function and makes the gradient expensive to evaluate during training. In a closely related work, Chen et al.\u00a0Chen et al. (2018) propose high-degree rational activation functions in a neural network, which have benefits in terms of approximation power. However, this choice can significantly increase the number of parameters in the network, causing the training stage to be computationally expensive.\n\nIn this paper, we use low-degree rational functions as activation functions, which are then composed together by the neural network to build high-degree rational functions. In this way, we can leverage the approximation power of high-degree rational functions without making training expensive. We highlight the approximation power of rational networks and provide optimal error bounds to demonstrate that rational neural networks theoretically outperform ReLU networks. Motivated by our theoretical results, we consider rational activation functions of type , i.e., and . This type appears naturally in the theoretical analysis due to the composition property of Zolotarev sign functions (see Section 3.1): the degree of the overall rational function represented by the rational neural network is a whopping , while the number of trainable parameters only grows linearly with respect to the depth of the network. Moreover, a superdiagonal type allows the rational activation function to behave like a nonconstant linear function at , unlike a diagonal type, e.g.,\u00a0, or the ReLU function. A low-degree activation function keeps the number of trainable parameters small, while the implicit composition in a neural network gives us the approximation power of high-degree rationals. This choice is also motivated empirically, and we do not claim that the type is the best choice for all situations as the configurations may depend on the application (see Figure 6 of the Supplementary Material). Our experiments on the approximation of smooth functions and GANs suggest that rational neural networks are an attractive alternative to ReLU networks (see Section 4). We observe that a good initialization, motivated by the theory of rational functions, prevents rational neural networks from having arbitrarily large values.\n\n## 3 Theoretical results on rational neural networks\n\nHere, we demonstrate the theoretical benefit of using neural networks based on rational activation functions due to their superiority over ReLU in approximating functions. We derive optimal bounds in terms of the total number of trainable parameters (also called size) needed by rational networks to approximate ReLU networks as well as functions in the Sobolev space . Throughout this paper, we take to be a small parameter with . We show that an -approximation on the domain of a ReLU network by a rational neural network must have the following size (indicated in brackets):\n\n Rational\u00a0[\u03a9(log(log(1\/\u03f5)))]\u2264ReLU\u2264% Rational\u00a0[O(log(log(1\/\u03f5)))], (2)\n\nwhere the constants only depend on the size and depth of the ReLU network. Here, the upper bound means that all ReLU networks can be approximated to within by a rational network of size . The lower bound means that there is a ReLU network that cannot be -approximated by a rational network of size less than , for some constant . In comparison, the size needed by a ReLU network to approximate a rational neural network within the tolerance of is given by the following inequalities:\n\n ReLU\u00a0[\u03a9(log(1\/\u03f5))]\u2264Rational\u2264ReLU\u00a0[O(log(1\/\u03f5))3], (3)\n\nwhere the constants only depend on the size and depth of the rational neural network. This means that all rational networks can be approximated to within by a ReLU network of size , while there is a rational network that cannot be -approximated by a ReLU network of size less than . A comparison between\u00a0(2) and\u00a0(3) suggests that rational networks could be more resourceful than ReLU. A key difference between rational networks and neural networks with polynomial activation functions is that polynomials perform poorly on non-smooth functions such as ReLU, with an algebraic convergence of \u00a0Trefethen (2013) rather than the (root-)exponential convergence with rationals (see Figure 1\u00a0(left)).\n\n### 3.1 Approximation of ReLU networks by rational neural networks\n\nTelgarsky showed that neural networks and rational functions can approximate each other in the sense that there exists a rational function of degree2 that is -close to a ReLU network\u00a0Telgarsky (2017), where is a small number. To prove this statement, Telgarsky used a rational function constructed with Newman polynomials\u00a0Newman (1964) to obtain a rational approximation to the ReLU function that converges with square-root exponential accuracy. That is, Telgarsky needed a rational function of degree to achieve a tolerance of . A degree rational function can be represented with coefficients, i.e., and in\u00a0Equation 1. Therefore, the rational approximation to a ReLU network constructed by Telgarsky requires at least parameters. In contrast, for any rational function, Telgarsky showed that there exists a ReLU network of size that is an -approximation on .\n\nOur key observation is that by composing low-degree rational functions together, we can approximate a ReLU network much more efficiently in terms of the size (rather than the degree) of the rational network. Our theoretical work is based on a family of rationals called Zolotarev sign functions, which are the best rational approximation on , with , to the sign function\u00a0Achieser (2013); Petrushev and Popov (2011), defined as\n\n sign(x)=\u23a7\u23a8\u23a9\u22121,x<0,0,x=0,1,x>0.\n\nA composition of Zolotarev sign functions of type has type but can be represented with parameters instead of . This property enables the construction of a rational approximation to ReLU using compositions of low-degree Zolotarev sign functions with parameters in Lemma .\n\n###### Lemma\n\nLet . There exists a rational network of size such that\n\n \u2225R\u2212ReLU\u2225\u221e:=maxx\u2208[\u22121,1]|R(x)\u2212ReLU(x)|\u2264\u03f5.\n\nMoreover, no rational network of size smaller than can achieve this.\n\nThe proof of Lemma (see Supplementary Material) shows that the given bound is optimal in the sense that a rational network requires at least parameters to approximate the ReLU function on to within the tolerance . The convergence of the Zolotarev sign functions to the ReLU function is much faster, with respect to the number of parameters, than the rational constructed with Newman polynomials (see\u00a0Figure 1 (left)).\n\nThe converse of Lemma , which is a consequence of a theorem proved by Telgarsky\u00a0(Telgarsky, 2017, Theorem\u00a01.1), shows that any rational function can be approximated by a ReLU network of size at most .\n\n###### Lemma\n\nLet . If is a rational function, then there exists a ReLU network of size such that .\n\nTo demonstrate the improved approximation power of rational neural networks over ReLU networks ( versus ), it is known that a ReLU networks that approximates , which is rational, to within on must be of size at least \u00a0(Liang and Srikant, 2016, Theorem\u00a011).\n\nWe can now state our main theorem based on\u00a0Lemmas and\u00a03.1. Theorem provides bounds on the approximation power of ReLU networks by rational neural networks and vice versa. We regard\u00a0Theorem as an analogue of\u00a0(Telgarsky, 2017, Theorem\u00a01.1) for our Zolotarev sign functions, where we are counting the number of training parameters instead of the degree of the rational functions. In particular, our rational networks have high degrees but can be represented with few parameters due to compositions, making training more computationally efficient. While Telgarsky required a rational function with parameters to approximate a ReLU network with fewer than nodes in each of layers to within a tolerance of , we construct a rational network that only has size .\n\n###### Theorem\n\nLet and let denote the vector 1-norm. The following two statements hold:\n\n1. Let be a rational network with layers and at most nodes per layer, where each node computes and is a rational function with Lipschitz constant (, , and are possibly distinct across nodes). Suppose further that and . Then, there exists a ReLU network of size\n\n O(kMlog(MLM\/\u03f5)3)\n\nsuch that .\n\n2. Let be a ReLU network with layers and at most nodes per layer, where each node computes and the pair (possibly distinct across nodes) satisfies . Then, there exists a rational network of size\n\n O(kMlog(log(M\/\u03f5)))\n\nsuch that .\n\nTheorem highlights the improved approximation power of rational neural networks over ReLU networks. ReLU networks of size are required to approximate rational networks while rational networks of size only are sufficient to approximate ReLU networks.\n\n### 3.2 Approximation of functions by rational networks\n\nA popular question is the required size and depth of deep neural networks to approximate smooth functions\u00a0Liang and Srikant (2016); Montanelli et al. (2019); Yarotsky (2017). In this section, we consider the approximation theory of rational networks. In particular, we consider the approximation of functions in the Sobolev space , where is the regularity of the functions and . The norm of a function is defined as\n\n \u2225f\u2225Wn,\u221e([0,1]d)=max|n|\u2264nesssupx\u2208[0,1]d|Dnf(x)|,\n\nwhere is the multi-index , and is the corresponding weak derivative of . In this section, we consider the approximation of functions from\n\n Fd,n:={f\u2208Wn,\u221e([0,1]d),\u2225f\u2225Wn,\u221e([0,1]d)\u22641}.\n\nBy the Sobolev embedding theorem\u00a0Brezis (2010), this space contains the functions in , which is the class of functions whose first derivatives are Lipschitz continuous. Yarotsky derived upper bounds on the size of neural networks with piecewise linear activation functions needed to approximate functions in \u00a0(Yarotsky, 2017, Theorem\u00a01). In particular, he constructed an -approximation to functions in with a ReLU network of size at most and depth smaller than . The term is introduced by a local Taylor approximation, while the term is the size of the ReLU network needed to approximate monomials, i.e., for , in the Taylor series expansion.\n\nWe now present an analogue of Yarotsky\u2019s theorem for a rational neural network.\n\n###### Theorem\n\nLet , , , and . There exists a rational neural network of size\n\n O(\u03f5\u2212d\/nlog(log(1\/\u03f5)))\n\nand maximum depth such that .\n\nThe proof of Theorem consists of approximating by a local Taylor expansion. One needs to approximate the piecewise linear functions and monomials arising in the Taylor expansion by rational networks using Lemma and Proposition (see Supplementary Material). The main distinction between Yarotsky\u2019s argument and the proof of\u00a0Theorem is that monomials can be represented by rational neural networks with a size that does not depend on the accuracy of . In contrast, ReLU networks require parameters. Meanwhile, while ReLU neural networks can exactly approximate piecewise linear functions with a constant number of parameters, rational networks can approximate them with a size of a most (see\u00a0Lemma ). That is, rational neural networks approximate piecewise linear functions much faster than ReLU networks approximate polynomials. This allows the existence of a rational network approximation to with exponentially smaller depth () than the ReLU networks constructed by Yarotsky.\n\nA theorem proved by DeVore et al.\u00a0DeVore et al. (1989) gives a lower bound of on the number of parameters needed by a neural network to express any function in with an error , under the assumption that the weights are chosen continuously. Comparing and , we find that rational neural networks require exponentially fewer nodes than ReLU networks with respect to the optimal bound of to approximate functions in .\n\n## 4 Experiments using rational neural networks\n\nIn this section, we consider neural networks with trainable rational activation functions of type . We select the type based on empirical performance; roughly, a low-degree (but higher than ) rational function is ideal for generating high-degree rational functions by composition, with a small number of parameters. The rational activation units can be easily implemented in the open-source TensorFlow library\u00a0Abadi et al. (2016) by using the polyval and divide commands for function evaluations. The coefficients of the numerators and denominators of the rational activation functions are trainable parameters, determined at the same time as the weights and biases of the neural network by backpropagation and a gradient descent optimization algorithm.\n\nOne crucial question is the initialization of the coefficients of the rational activation functions\u00a0Chen et al. (2018); Molina et al. (2019). A badly initialized rational function might contain poles on the real axis, leading to exploding values, or converge to a local minimum in the optimization process. Our experiments, supported by the empirical results of Molina et al.\u00a0Molina et al. (2019), show that initializing each rational function with the best rational approximation to the ReLU function (as described in Lemma ) produces good performance. The underlying idea is to initialize rational networks near a network with ReLU activation functions, widely used for deep learning. Then, the adaptivity of the rational functions allows for further improvements during the training phase. We represent the initial rational function used in our experiments in Figure 1 (right). The coefficients of this function are obtained by using the minimax command, available in the Chebfun software\u00a0Driscoll et al. (2014); Filip et al. (2018) for numerically computing rational approximations (see Table 1 in the Supplementary Material).\n\nIn the following experiments, we use a single rational activation function of type at each layer, instead of different functions at each node to reduce the number of trainable parameters and the computational training expense.\n\n### 4.1 Approximation of functions\n\nRaissi, Perdikaris, and Karniadakis\u00a0Raissi (2018); Raissi et al. (2019) introduce a framework called deep hidden physics models for discovering nonlinear partial differential equations (PDEs) from observations. This technique requires to solving the following interpolation problem: given the observation data at the spatio-temporal points , find a neural network (called the identification network), that minimizes the loss function\n\n L=1NN\u2211i=1|N(xi,ti)\u2212ui|2. (4)\n\nThis technique has successfully discovered hidden models in fluid mechanics\u00a0Raissi et al. (2020), solid mechanics\u00a0Haghighat et al. (2020), and nonlinear partial differential equations such as the Korteweg\u2013de Vries (KdV) equation\u00a0Raissi et al. (2019). Raissi et al. use an identification network, consisting of layers and nodes per layer, to interpolate samples from a solution to the KdV equation. Moreover, they observe that networks based on smooth activation functions, such as the hyperbolic tangent () or the sinusoid (), outperform ReLU neural networks\u00a0Raissi (2018); Raissi et al. (2019). However, the performance of these smooth activation functions highly depends on the application.\n\nMoreover, these functions might not be adapted to approximate non-smooth or highly oscillatory solutions. Recently, Jagtap, Kawaguchi, and Karnidakis\u00a0Jagtap et al. (2020) proposed and analyzed different adaptive activation functions to approximate smooth and discontinuous functions with physics-informed neural networks. More specifically, they use an adaptive version of classical activation functions such as sigmoid, hyperbolic tangent, ReLU, and Leaky ReLU. The choice of these trainable activation functions introduces another parameter in the design of the neural network architecture, which may not be ideal for use for a black-box data-driven PDE solver.\n\nWe illustrate that rational neural networks can address the issues mentioned above due to their adaptivity and approximation power (see Section 3). Similarly to Raissi\u00a0Raissi (2018), we use a solution to the KdV equation:\n\n ut=\u2212uux\u2212uxxx,u(x,0)=\u2212sin(\u03c0x\/20),\n\nas training data for the identification network (see the left panel of Figure 2). We train and compare four neural networks, which contain ReLU, sinusoid, rational, and polynomial activation functions, respectively.3 The mean squared error (MSE) of the neural networks on the validation set throughout the training phase is reported in the right panel of Figure 2. We observe that the rational neural network outperforms the sinusoid network, despite having the same asymptotic convergence rate. The network with polynomial activation functions (chosen to be of degree 3 in this example) is harder to train than the rational network, as shown by the non-smooth validation loss (see the right panel of Figure 2). We highlight that rational neural networks are never much bigger in terms of trainable parameters than ReLU networks since the increase is only linear with respect to the number of layers. Here, the ReLU network has parameters (consisting of weights and biases), while the rational network has . The ReLU, sinusoid, rational, and polynomial networks achieve the following mean square errors after epochs:\n\n MSE(uReLU)=1.9\u00d710\u22124, MSE(usin)=3.3\u00d710\u22126, MSE(urat)=1.2\u00d710\u22127, MSE(upoly)=3.6\u00d710\u22125.\n\nThe absolute approximation errors between the different neural networks and the exact solution to the KdV equation is illustrated in Figure 5 of the Supplementary Material. The rational neural network is approximatively five times more accurate than the sinusoid network used by Raissi and twenty times more accurate than the ReLU network. Moreover, the approximation errors made by the ReLU network are not uniformly distributed in space and time and located in specific regions, indicating that a network with non-smooth activation functions is not appropriate to resolve smooth solutions to PDEs.\n\nGenerative adversarial networks (GANs) are used to generate fake examples from an existing dataset\u00a0Goodfellow et al. (2014). They usually consist of two networks: a generator to produce fake samples and a discriminator to evaluate the samples of the generator with the training dataset. Radford et al.\u00a0Radford et al. (2015) describe deep convolutional generative adversarial networks (DCGANs) to build good image representations using convolutional architectures. They evaluate their model on the MNIST and Imagenet image datasets\u00a0Deng et al. (2009); LeCun et al. (1998). This section highlights the simplicity of using rational activation functions in existing neural network architectures by training an Auxiliary Classifier GAN (ACGAN)\u00a0Odena et al. (2017) on the MNIST dataset. In particular, the neural network4, denoted by ReLU network in this section, consists of convolutional generator and discriminator networks with ReLU and Leaky ReLU\u00a0Maas et al. (2013) activation units (respectively) and is used as a reference GAN. As in the experiment described in Section 4.1, we replace the activation units of the generative and discriminator networks by a rational function with trainable coefficients (see\u00a0Figure 1). We initialize the activation functions in the training phase with the best rational function that approximates the ReLU function on .\n\nWe show images of digits from the first five classes generated by a ReLU and rational GANs at different epochs of the training in Figure 3 (the samples are generated randomly and are not manually selected). We observe that a rational network can generate realistic images with a broader range of features than the ReLU network, as illustrated by the presence of bold numbers at the epoch 20 in the bottom panel of Figure 3. However, the digits one generated by the rational network are identical, suggesting that the rational GAN suffers from mode collapse. It should be noted that generative adversarial networks are notoriously tricky to train\u00a0Goodfellow et al. (2016). The hyper-parameters of the reference model are intensively tuned for a piecewise linear activation function (as shown by the use of Leaky ReLU in the discriminator network). Moreover, many stabilization methods have been proposed to resolve the mode collapse and non-convergence issues in training, such as Wasserstein GAN\u00a0Arjovsky et al. (2017), Unrolled Generative Adversarial Networks\u00a0Metz et al. (2016), and batch normalization\u00a0Ioffe and Szegedy (2015). These techniques could be explored and combined with rational networks to address the mode collapse issue observed in this experiment.\n\n## 5 Conclusions\n\nWe have investigated rational neural networks, which are neural networks with smooth trainable activation functions based on rational functions. Theoretical statements demonstrate the improved approximation power of rational networks in comparison with ReLU networks. In practice, it seems beneficial to select the activation function as very low-degree rationals, making training more computationally efficient. We emphasize that it is simple to implement rational networks in existing deep learning architectures, such as TensorFlow, together with the ability to have trainable activation functions.\n\nThere are many future research directions exploring the potential applications of rational networks in fields such as image classification, time series forecasting, and generative adversarial networks. These applications already employ nonstandard activation functions to overcome various drawbacks of ReLU. Another exciting and promising field is the numerical solution and data-driven discovery of partial differential equations with deep learning. We believe that popular techniques such as physics-informed neural networks\u00a0Raissi et al. (2019) could benefit from rational neural networks to improve the robustness and performances of PDE solvers, both from a theoretical and practical viewpoint.\n\nNeural networks have applications in diverse fields such as facial recognition, credit-card fraud, speech recognition, and medical diagnosis. There is a growing understanding of the approximation power of neural networks, which is adding theoretical justification to their use in societal applications. We are particularly interested in the future applicability of rational neural networks in discovering and solving of partial differential equations (PDEs). Neural networks, in particular rational neural networks, have the potential to revolutionize fields where PDE models derived by mechanistic principles are lacking.\n\n{ack}\n\nThe authors thank the National Institute of Informatics (Japan) for funding a research visit, during which this project was initiated. We thank Gilbert Strang for making us aware of Telgarsky\u2019s paper\u00a0Telgarsky (2017). We also thank Matthew Colbrook and Nick Trefethen for their suggestions on the paper. This work is supported by the EPSRC Centre For Doctoral Training in Industrially Focused Mathematical Modelling (EP\/L015803\/1) in collaboration with Simula Research Laboratory. The work of the third author is supported by the National Science Foundation grant no. 1818757.\n\n## Appendix A Supplementary Material\n\n### a.1 Deferred proofs of Section 3.1\n\nWe first show that a rational function can approximate the absolute value function on with square-root exponential convergence.\n\n###### Lemma\n\nFor any integer , we have\n\n minr\u2208Rk,kmaxx\u2208[\u22121,1]||x|\u2212xr(x)|\u22644e\u2212\u03c0\u221ak\/2,\n\nwhere is the space of rational functions of type at most . Thus, is a rational approximant to of type at most .\n\nMoreover, if for some and integers , then can be written as , where .\n\n###### Proof\n\nLet be a real number and consider the sign function on the domain , i.e.,\n\n sign(x)={\u22121,x\u2208[\u22121,\u2212\u2113],+1,x\u2208[\u2113,1].\n\nBy\u00a0[4, Equation\u00a0(33)], we find that for any ,\n\n minr\u2208Rk,kmaxx\u2208[\u22121,\u2212\u2113]\u222a[\u2113,1]|sign(x)\u2212r(x)|\u22644[exp(\u03c022log(4\/\u2113))]\u2212k.\n\nLet be the rational function of type that attains the minimum\u00a0[4, Equation\u00a0(12)]. We refer to such as the Zolotarev sign function. It is given by\n\n r(x)=Mx\u220f\u230a(k\u22121)\/2\u230bj=1x2+c2j\u220f\u230ak\/2\u230bj=1x2+c2j\u22121,cj=\u21132sn2(jK(\u03ba)\/k;\u03ba)1\u2212sn2(jK(\u03ba)\/k;\u03ba).\n\nHere, is a real constant selected so that equioscillates on , , is the first Jacobian elliptic function, and is the complete elliptic integral of the first kind. Since we have the following inequality,\n\n maxx\u2208[\u22121,\u2212\u2113]\u222a[\u2113,1]||x|\u2212xr(x)| =maxx\u2208[\u22121,\u2212\u2113]\u222a[\u2113,1]\u2223\u2223x\u22c5sign(x)\u2212xr(x)\u2223\u2223 \u2264maxx\u2208[\u22121,\u2212\u2113]\u222a[\u2113,1]\u2223\u2223sign(x)\u2212r(x)\u2223\u2223.\n\nThe last inequality follows because on . Moreover, since for (see\u00a0[4, Equation\u00a0(12)]) we have\n\n maxx\u2208[\u2212\u2113,\u2113]||x|\u2212xr(x)|\u2264maxx\u2208[\u2212\u2113,\u2113]|x|\u2264\u2113.\n\nTherefore,\n\n maxx\u2208[\u22121,1]||x|\u2212xr(x)|\u2264max\u23a7\u23a8\u23a9\u2113,4[exp(\u03c022log(4\/\u2113))]\u2212k\u23ab\u23ac\u23ad.\n\nNow, select to minimize this upper bound. One finds that and the result follows immediately.\n\nFor the final claim, let be the Zolotarev sign function of type on , with . By definition, is the best rational approximation of degree to the sign function on . We know from\u00a0[32, 44] that there exist Zolotarev sign functions , where each is of type , such that\n\n r(x):=Zk(x;\u2113)=Rp(\u22ef(R2(R1(x))\u22ef). (5)\n\nThe proof of Lemma is a direct consequence of the previous lemma and the properties of Zolotarev sign functions.\n\n###### Proof (Proof of Lemma )\n\nLet , , , and be the Zolotarev sign function of type . Again from\u00a0[32, 44], we see that there exist Zolotarev sign functions of type such that their composition equals , i.e.,\n\n r(x):=Z3k(x;\u2113)=Rk(\u22ef(R2(R1(x))\u22ef). (6)\n\nFollowing the proof of Lemma , we have the inequality\n\n maxx\u2208[\u22121,1]||x|\u2212xr(x)|\u22644e\u2212\u03c0\u221a3k\/2, (7)\n\nwhere we chose . Now, we take\n\n k=\u2308ln(2\/\u03c02)+2ln(ln(4\/\u03f5))ln(3)\u2309, (8)\n\nso that the right-hand side of\u00a0Equation 7 is bounded by . Finally, we use the identity\n\n ReLU(x)=|x|+x2,x\u2208R,\n\nto define a rational approximation to the ReLU function on the interval as\n\n ~r(x)=12(xr(x)1+\u03f5+x).\n\nTherefore, we have the following inequalities for ,\n\n |ReLU(x)\u2212~r(x)| =12\u2223\u2223\u2223|x|\u2212xr(x)1+\u03f5\u2223\u2223\u2223\u226412(1+\u03f5)(||x|\u2212xr(x)|+\u03f5|x|) \u2264\u03f51+\u03f5\u2264\u03f5.\n\nThen, is a composition of rational functions of type and can be represented using at most coefficients (see\u00a0Equation 5). Moreover, using\u00a0Equation 8, we see that , which means that is representable by a rational network of size . Finally, for .\n\nThe upper bound on the complexity of the neural network obtained in Lemma is optimal, as proved by Vyacheslavov\u00a0[59].\n\n###### Theorem (Vyacheslavov)\n\nThe following inequalities hold:\n\n C1e\u2212\u03c0\u221ak\u2264maxx\u2208[\u22121,1]||x|\u2212rk(x)|\u2264C2e\u2212\u03c0\u221ak,k\u22650, (9)\n\nwhere is the best rational approximation to in from . Here, are constants that are independent of .\n\nWe first deduce the following corollary, giving lower and upper bounds on the optimal rational approximation to the ReLU function.\n\n###### Corollary\n\nThe following inequalities hold:\n\n C12e\u2212\u03c0\u221ak\u2264\u2225ReLU\u2212rk\u2225\u221e\u2264C22e\u2212\u03c0\u221ak,k\u22650, (10)\n\nwhere is the best rational approximation to ReLU on in and are constants given by Theorem .\n\n###### Proof\n\nLet be an integer and let be any rational function of degree . Now, define . Since , we have\n\n \u2225ReLU\u2212rk\u2225\u221e=maxx\u2208[\u22121,1]\u2223\u2223\u222312(r% abs(x)+x)\u221212(|x|+x)\u2223\u2223\u2223=maxx\u2208[\u22121,1]12\u2223\u2223rabs(x)\u2212|x|\u2223\u2223\u226512C1e\u2212\u03c0\u221ak,\n\nwhere the inequality is from\u00a0Theorem . Now, let be the best rational approximation to on . Now, define . We find that\n\n \u2225ReLU\u2212rReLU\u2225\u221e=maxx\u2208[\u22121,1]\u2223\u2223\u222312(|x|+x)\u221212(rk(x)+x)\u2223\u2223\u2223=maxx\u2208[\u22121,1]12||x|\u2212rk(x)|\u226412C2e\u2212\u03c0\u221ak,\n\nwhich proves that the best approximation to ReLU satisfies the upper bound.\n\nWe now show that a rational neural network must be at least in size (total number of nodes) to approximate the ReLU function to within .\n\n###### Proposition\n\nLet . A rational neural network that approximates the ReLU function on to within has size of at least .\n\n###### Proof\n\nLet be a rational neural network with nodes at each of its layers, and assume that its activation functions are rational functions of type at most . Let be the maximum of the degrees of the activation functions of . Such a network has size . Note that itself is a rational function of degree , where from additions and compositions of rational functions we have . If is an -approximation to the ReLU function on , we know by\u00a0Corollary that\n\n C12e\u2212\u03c0\u221ad\u2265\u03f5,d\u2265(1\u03c0ln(C12\u03f5))2. (11)\n\nThe statement follows by minimizing the size of , i.e., subject to\n\n dMrM\u220fi=1ki\u2265(1\u03c0ln(C12\u03f5))2.\n\nThat is,\n\n M\u2211i=1ln(ki)+Mln(dr)\u22652ln(ln(C12\u03f5))\u22122ln(\u03c0). (12)\n\nWe introduce a Lagrange multiplier and define the Lagrangian of this optimization problem as\n\n L(k1,\u2026,kM,\u03bb)=M\u2211i=1ki+\u03bb[2ln(ln(C12\u03f5))\u22122ln(\u03c0)\u2212M\u2211i=1ln(ki)\u2212Mln(dr)].\n\nOne finds using the Karush\u2013Kuhn\u2013Tucker conditions\u00a0[31] that . Then, using Equation 12, we find that satisfies\n\n ln(\u03bb)\u22652M[ln(ln(C12\u03f5))\u2212ln(\u03c0)]\u2212ln(dr)=:ln(\u03bb\u2217). (13)\n\nTherefore, the rational network with layers that approximates the ReLU function to within on has a size of at least , where is given by\u00a0Equation 13 and depends on . We now minimize with respect to the number of layers . We remark that minimizing is equivalent of minimizing , where\n\n ln(s(M))=ln(M)+ln(\u03bb\u2217)=ln(M)+2M[ln(ln(C12\u03f5))\u2212ln(\u03c0)]\u2212ln(dr).\n\nOne finds that one should take and . The result follows.\n\nWe now show that ReLU neural networks can approximate rational functions.\n\n###### Proof (Proof of Lemma )\n\nLet and be a rational function. Take , which is still a rational function. Without loss of generality, we can assume that is an irreducible rational function (otherwise cancel factors till it is irreducible). Since is a rational, it can be written as with . Moreover, we know that for so we can assume that for (it is either positive or negative by continuity). Since is continuous on , there is an integer such that for . Furthermore, we find that for because and for . By\u00a0[56, Theorem\u00a01.1], there exists a ReLU network of size such that\n\n maxx\u2208[0,1]\u2223\u2223\u2223f(x)\u2212p(x)q(x)\u2223\u2223\u2223\u2264\u03f52.\n\nWe now define a scaled ReLU network such that for . Therefore, for all ,\n\n \u2223\u2223~f(x)\u2212~R(x)\u2223\u2223=\u2223\u2223\u2223f(x)1+\u03f5\/2\u2212p(x)q(x)\u2223\u2223\u2223\u226411+\u03f5\/2(\u2223\u2223\u2223f(x)\u2212p(x)q(x)\u2223\u2223\u2223+\u03f52\u2223\u2223\u2223p(x)q(x)\u2223\u2223\u2223)\u2264\u03f5.\n\nTherefore, is a ReLU neural network of size that is an -approximation to on .\n\nWe can now prove Theorem that shows how rational neural networks can approximate ReLU networks and vice versa. The structure of the proof closely follows\u00a0[56, Lemma\u00a01.3].\n\n###### Proof (Proof of Theorem )\n\nThe statement of\u00a0Theorem comes in two parts, and we prove them separately. 1.\u00a0\u00a0Consider the subnetwork of the rational network , consisting of the layers of up to the th layer for some . Let denote the ReLU network obtained by replacing each rational function in by a ReLU network approximation at a given tolerance for and , such that for (see Lemma\u00a03.1). Let be the output of the rational network at layer and node for . Now, approximate node in the st layer by a ReLU network with tolerance (see Lemma\u00a03.1). The approximation error between the rational and the approximating ReLU network at layer and node satisfies\n\n Ei,J+1 =|fri,J+1(a\u22a4i,J+1HReLU(x)+bi,J+1)\u2212ri,J+1(a\u22a4i,J+1H(x)+bi,J+1)| \u2264|fri,J+1(a\u22a4i,J+1HReLU(x)+bi,J+1)\u2212ri,J+1(a\u22a4i,J+1HReLU(x)+bi,J+1)|\ue152\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue151\ue150\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue153(1) +|ri,J+1(a\u22a4i,J+1HReLU(x)+bi,J+1)\u2212ri,J+1(a\u22a4i,J+1H(x)+bi,J+1)|\ue152\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue151\ue150\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue154\ue153(2).\n\nThe first term is bounded by\n\n (1)\u2264maxx\u2208[\u22121,1]\u2223\u2223ri,J+1(x)\u2212fri,J+1\u2223\u2223\u2264\u03f5J+1,\n\nsince by assumption. The second term is bounded as the Lipschitz constant of is at most . That is,\n\n (2)\u2264L\u2225ai,J+1\u22251maxx\u2208[\u22121,1]d\u2225\u2225HReLU(x)\u2212H(x)\u2225\u2225\u221e\u2264Lmaxx\u2208[\u22121,1]d\u2225\u2225HReLU(x)\u2212H(x)\u2225\u2225\u221e,\n\nwhere we used the fact that and for . We find that we have the following set of inequalities:\n\n max1\u2264i\u2264kj+1Ei,j+1\u2264Lmax1\u2264i\u2264kjEi,j+\u03f5j+1,1\u2264i\u2264kj,1\u2264j\u2264J+1,\n\nwith . If we select , then we find that . When , the ReLU network approximates the original rational network, , and the ReLU network has size\n\n O(kM\u2211j=1log(MLj\u2212M\u03f5)3).\n\nwhere we used the fact that for . This can be simplified a little since\n\n M\u2211j=1log(MLj\u2212M\u03f5)3=M\u2211j=1(log(MLM\/\u03f5)+jlog(1\/L))3=O(Mlog(MLM\/\u03f5)3).\n\n2.\u00a0\u00a0Telgarsky proved in\u00a0[56, Lemma\u00a01.3] that if is a neural network obtained by replacing all the ReLU activation functions in by rational functions for , which satisfies and for , then\n\n maxx\u2208[\u22121,1]d|f(","date":"2021-02-27 10:34:13","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8841518759727478, \"perplexity\": 479.30834156744965}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-10\/segments\/1614178358798.23\/warc\/CC-MAIN-20210227084805-20210227114805-00228.warc.gz\"}"}
| null | null |
Q: How do I properly combine these SELECTS (MySQL)? I want to get all my posts data to show it on my webpage with this SELECT. I have my table posts that contains most of the posts (+ replies) data and a table social that tracks who views and likes it(each like is a new row).
Normally I can get the username, post time, content... but I'm struggling to get the number of views the post gets, the number of likes, and the number of replies in the same SELECT. My base SELECT looks like this:
SELECT posts.username, posts.time, cat.cat_name,
posts.title, posts.content, posts.reply,
posts.user_file, posts.audio, social.id,
social.views, social.likes
FROM posts
LEFT JOIN user on posts.user_id = user.id
LEFT JOIN cat ON posts.cat_id = cat.id
LEFT JOIN social ON posts.id = social.post_id
If I wanted to get the number of comments per post I would use
(if the value inside reply is 0 it's a post if it's a reply it contains the post id it's referring to):
SELECT COUNT(*) AS `comments` FROM posts GROUP BY reply
/* this returns an error: SQL Error (1242): Subquery returns more than 1 row */
And I would get the number of likes and views like this:
SELECT MAX(social.views) AS views FROM social GROUP BY post_id
SELECT social.likes FROM social WHERE social.id = (SELECT MAX(social.id) FROM social GROUP BY post_id
But if I use it together in the earlier SELECT it just fills every row with the same number. Example:
... posts.audio, social.id, (SELECT MAX(social.views) AS views FROM social GROUP BY post_id) FROM posts ...
This just fills every row even if it shouldn't have views with 25 (correct value for 1 specific row but wrong for everything else).
What would be a proper way of making a bigger SELECT like this?
Not sure if it matters but I am using it with a MySQL module in NodeJS.
A: Try this way
SELECT posts.username, posts.time, cat.cat_name,
posts.title, posts.content, posts.reply,
posts.user_file, posts.audio,
COUNT(social.views), COUNT(social.likes)
FROM posts
LEFT JOIN user on posts.user_id = user.id
LEFT JOIN cat ON posts.cat_id = cat.id
LEFT JOIN social ON posts.id = social.post_id
GROUP BY
posts.username, posts.time, cat.cat_name,
posts.title, posts.content, posts.reply,
posts.user_file, posts.audio
A: This is the closest I've gotten to SELECTing everything in one query but I still don't have a way of counting the number of comments:
SELECT posts.username, posts.time, cat.cat_name,
posts.title, posts.content, posts.reply,
posts.user_file, posts.audio,
social.views, social.likes
FROM posts
LEFT JOIN user on posts.user_id = user.id
LEFT JOIN cat ON posts.cat_id = cat.id
LEFT JOIN social ON posts.id = social.post_id
WHERE social.likes IN (SELECT social.likes FROM social
WHERE social.id IN (SELECT MAX(social.id)
FROM social GROUP BY post_id))
GROUP BY social.post_id
HAVING posts.reply = 0
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,802
|
Wycombe Wanderers FC is een Engelse voetbalclub uit High Wycombe, Buckinghamshire.
Geschiedenis
De club werd opgericht in 1887 door een groep meubelmakers. In 1895 nam de club het stadion Loakes Park in gebruik en een jaar later sloten ze zich aan bij de Southern League maar daar hadden ze weinig succes omdat er veel profclubs waren. In 1908 verhuisde Wycombe naar de Suburban League en na de Eerste Wereldoorlog naar de Spartan League. Na hier kampioen geworden te zijn voegde de club zich bij de Isthmian League in 1921. Hoewel de club in 1931 de FA Amateur Cup won zou het tot 1956 duren vooraleer de eerste van 8 Isthmian League titels gewonnen werd.
In 1985 promoveerde de club naar de Football Conference en degradeerde na één seizoen, maar kon ook weer na één seizoen terugkeren. Vijf jaar later werd het Loaks Park stadion omgeruild voor het huidige Adams Park en werd Martin O'Neill trainer-coach. Met hem zou de club twee keer de FA Throphy winnen en uiteindelijk promoveren naar de Football League.
De intrede in de Football League was een groot succes en Wycombe promoveerde meteen door naar de 3de klasse. Daar werd promotie naar de 2de klasse net gemist en in 1995 verliet O'Neill de club. De volgende seizoenen werd de club een middenmoter, maar kon wel standhouden in de 3de klasse. In het seizoen 2000/01 haalde de club de halve finale van de FA Cup waar uiteindelijk Liverpool FC te sterk was. In 2004 degradeerde de club na tien seizoenen uit de 3de klasse.
Het eerste seizoen in de nieuwe League Two was middelmatig met de tiende plaats in de eindrangschikking, maar in 2005/06 liep het vlotjes en de eerste 21 wedstrijden van de competitie bleef de club ongeslagen. Tijdens de tweede helft van het seizoen kreeg de club met tegenslagen te kampen. Op 14 januari 2006, toen de club speelde tegen Notts County, overleed speler Mark Philo, de 21-jarige middenvelder die de dag ervoor een zwaar auto-ongeluk had gehad.
Hierna verloor de club zes keer op rij en maakte zo geen kans meer op automatische promotie en werd 6de, in de play-offs was Cheltenham Town te sterk. In 2009 promoveerde de club na een derde plaats. In 2010 degradeerde de club alweer terug naar de Football League Two. In het seizoen 2010/11 eindigde Wycombe Wanderers op de derde plaats, waardoor het weer promoveerde naar de League One, waar het na één seizoen direct weer degradeerde. In het seizoen 2013/14 ontsnapte Wycombe ternauwernood aan degradatie naar de Football Conference door een iets beter doelsaldo (+3) dan Bristol Rovers. Na het spelen van de promotie play-offs na het seizoen 2017/18 promoveerde Wycombe Wanderers weer terug naar de League One.
Op 13 juli 2020 won Wycombe de promotiefinale naar de tweede divisie in het Wembley Stadium tegen Oxford United FC met 2-1. Het was de eerste keer in de 133-jaar clubgeschiedenis van Wycombe Wanderers dat de club promoveerde naar de Championship.
Leagues
1896-97 – Sloot zich aan bij Southern League Division Two
1908-09 – Sloot zich aan bij de Great Western Suburban League
1919-20 - Sloot zich aan bij Spartan League
1921-22 - Sloot zich aan bij Isthmian League na twee opeenvolgende Spartan League-titels
1930-31 - FA Amateur Cup-winnaar
1953-54 – Werd net geen vicekampioen door slechter doelsaldo
1955-56 - Isthmian League-kampioen
1956-57 - Isthmian League-kampioen; FA Amateur Cup-runner-up
1957-58 - Isthmian League-runner-up
1959-60 - Isthmian League-runner-up
1969-70 - Isthmian League-runner-up
1970-71 - Isthmian League-kampioen
1971-72 - Isthmian League-kampioen
1973-74 - Isthmian League-kampioen
1974-75 - Isthmian League-kampioen (op basis van doelsaldo)
1975-76 - Isthmian League-runner-up
1976-77 - Isthmian League-runner-up
1981-82 - Halvefinalist FA Trophy
1982-83 - Isthmian League-kampioen
1985-86 – Promoveerde naar Alliance Premier League, degradeerde na één seizoen
1986-87 - Isthmian League-kampioen (8ste keer)
1987-88 – Terug naar Conference (ex-Alliance Premier League)
1990-91 - FA Trophy-winnaar
1991-92 - Conference-runner-up (miste titel en promotie naar Football League op basis van doelsaldo)
1992-93 - Conference-kampioen, FA Trophy-winnaar, promotie naar Football League Division Three
1993-94 - Promotie Division Two na play-offs (finale - Wycombe Wanderers 4, Preston North End 2 in Wembley)
2000-01 - Halvefinalist FA Cup
2003-04 – Degradeerde naar Division Three, die nu "League Two" werd
2008-09 - Promotie naar League One
2009-10 - Degradatie naar League Two
2010-11 - Promotie naar League One
2019-20 - Promotie naar Championship
Erelijst
FA Trophy
1991, 1993
Eindklasseringen vanaf 1987/88
Bekende (oud-)spelers
Externe links
Officiële website
chairboys.co.uk
smbu.co.uk
wsab.co.uk
gasroom.co.uk
Engelse voetbalclub
Sport in South East England
Buckinghamshire
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 9,665
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require('./src/util/util.spec');
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 7,974
|
\section{Derivation of the FDR}
For completeness, here we add a proof of the work FDR for the driven quantum dot. The results follow from the general considerations of Refs.~\citealp{Mandal2016,Miller2019work}, but provide a more explicit derivation for the specific setting considered.
Recall that we consider a quantum dot with a two-fold degenerate energy level $E(t)$ that is driven for a time $t \in [0,\tau]$, which is in contact with a thermal fermionic reservoir.
It is convenient to express the quantities of interest as $x_s \equiv x(s\tau)$ with $s\in (0,1)$. In terms of the renormalised variables, the equation of motion reads:
\begin{align}
\frac{1}{\tau} \dot{p}_s = \Gamma_s (p^{\rm eq}_s -p_s)
\label{difeqapp}
\end{align}
with $\dot{q}_s={\rm d} q_s/{\rm d} s$, and where we introduced $\Gamma_s=\Gamma_0 (1+b E_s)$ and $p^{\rm eq}_s=1/(1+2e^{\beta(E_s)})$.
The stochastic variable $n_s \in \{0,1\}$ determines whether the dot is occupied or not at time $s\tau$. Hence, $\mathcal{E}[n_s]=p_s$ where $\mathcal{E}[.]$ denotes averaging over all trajectories. The work extracted in a given trajectory reads:
$W=\int_0^\tau {\rm d} s \hspace{1mm} n_s \dot{e}_s$. The average work is given by
\begin{align}
\langle W \rangle=
\mathcal{E}[ W ] =\int_0^1 {\rm d} s \hspace{1mm} \mathcal{E}[n_s] \dot{E}_s =\int_0^1 {\rm d} s \hspace{1mm} p_s \dot{E}_s.
\label{AvWapp}
\end{align}
On the other hand, the work fluctuations read
\begin{align}
\sigma^2_W=\mathcal{E}[ W^2 ]-(\mathcal{E}[W ])^2
\label{workFluctapp}
\end{align}
with
$\mathcal{E}[ W^2 ] =\int_0^1 {\rm d} s \hspace{1mm}\int_0^1 {\rm d} t \hspace{1mm} \mathcal{E}[n_s n_t] \dot{E}_s \dot{E}_t$.
For $t\geq s$, $\mathcal{E}[n_s n_t]$ is given by
$\mathcal{E}[n_s n_t]=p_s p_{t|s}$, where $p_{t|s}$ is the conditional probability of having $n_t=1$ given that $n_s =1 $ at time $s<t$ (this ensures that the product $n_t n_s$ is non-zero).
Hence we have
$\mathcal{E}[W^2 ] =2 \int_0^\tau {\rm d} t \hspace{1mm}\int_0^t {\rm d} s \hspace{1mm} p_s p_{t|s} \dot{E}_s \dot{E}_t$, and also:
\begin{align}
\sigma_W^2 = 2 \int_0^\tau {\rm d} t \hspace{1mm}\int_0^t {\rm d} s \hspace{1mm} p_s (p_{t|s} - p_t) \dot{E}_s \dot{E}_t
\label{ex2}
\end{align}
for the work fluctuations.
In principle, both $ \langle W \rangle$ and $\sigma^2_W$ can be computed exactly by solving the differential equation~\eqref{difeqapp}. We obtain the formal solutions:
\begin{align}
p_s = p_0 e^{-\tau \int_0^s \Gamma_y \hspace{1mm}{\rm d}y}+ \tau \int_0^s e^{-\tau \int_x^s \Gamma_y \hspace{1mm}{\rm d}y} \Gamma_x p^{\rm eq}_x \hspace{1mm}{\rm d}x.
\end{align}
and
\begin{align}
p_{t|s} = e^{-\tau \int_s^t \Gamma_y \hspace{1mm}{\rm d}y}+ \tau \int_s^t e^{-\tau \int_x^t \Gamma_y \hspace{1mm}{\rm d}y} \Gamma_x p^{\rm eq}_x \hspace{1mm}{\rm d}x.
\label{formSolptsApp}
\end{align}
In order to obtain the FDR, we are interested in obtaining $ \langle W \rangle$ and $\sigma^2_W$ analytically in the slow driving regime, i.e., at leading order in $1/\tau$. We proceed differently for each term. For $ \langle W \rangle$, it is convenient to solve the differential equation \eqref{difeqapp}
perturbatively for $1/\tau$ by introducing the expansion:
$p_s=p_s^{\rm (0)}+ \frac{1}{ \tau}p_s^{\rm (1)}+...$ \cite{cavinaSlowDynamicsThermodynamics2017} (see also Refs. \citealp{Campisi2012geometric,Deffner2013,Ludovico2016,cavinaSlowDynamicsThermodynamics2017,Abiuso2020entropy}).
At zeroth order ($1/\tau =0$), we simply find the equilibrium solution $p_s^{(0)}=p_s^{\rm eq}$. At first order, we have $p_s^{(1)}= - \dot{p}_s^{(0)} /\Gamma_s$, and hence:
\begin{align}
p_s&=p_s^{\rm eq}- \frac{1}{\tau \Gamma_s} \dot{p}_s^{\rm eq}+\mathcal{O}\left(\frac{1}{\tau \Gamma_0} \right)^2
\nonumber\\
&=p_s^{\rm eq}+ \frac{\beta \dot{E}_s}{\tau \Gamma_s} (1-p_s^{\rm eq})p_s^{\rm eq}+\mathcal{O}\left(\frac{1}{\tau \Gamma_0} \right)^2
\label{pertSolp}
\end{align}
where we used
$p^{\rm eq}_s=1/(1+2e^{\beta(E_s)})$ in the second line. Plugging this solution into \eqref{AvWapp}, we obtain:
\begin{align}
\langle W \rangle = \Delta F + \frac{\beta }{\tau} \int_0^1 {\rm d}s \hspace{1mm} (1-p_s^{\rm eq})p_s^{\rm eq} \frac{\dot{E}_s^2}{ \Gamma_s} +\mathcal{O}\left(\frac{1}{\tau \Gamma_0} \right)^2
\end{align}
with $\Delta F = \int_0^1 {\rm d} s \hspace{1mm} p_s^{\rm eq} \dot{E}_s$. Obtaining $\sigma_W^2$ at leading order in $1/\tau$ is more subtle, and we have to proceed differently. The underlying reason is that the perturbative solution \eqref{pertSolp} disregards the memory of initial condition (which decays exponentially with time). This is well justified for $p_s$ away from $s=0$, but not for $p_{t|s}$ where the regime $t\approx s$ is the most relevant one. Instead, we can expand directly the formal solution \eqref{formSolptsApp} around $s=t$, obtaining at leading order:
\begin{align}
p_{t|s} &= e^{-\tau \Gamma_t (t-s)}+\tau \int_s^t e^{-\tau \Gamma_t (t-x)} \Gamma_t p_t^{\rm eq} {\rm d}x +\mathcal{O}\left((t-s)e^{-\tau \Gamma_0 (t-s)}\right)
\nonumber\\
&= p_t^{\rm eq} +e^{-\tau \Gamma_t (t-s)} (1-p_t^{\rm eq})+ \mathcal{O}\left((t-s)e^{-\tau \Gamma_0 (t-s)}\right)
\end{align}
Plugging this expression into \eqref{ex2}, and consistently expanding around $s=t$, we obtain:
\begin{align}
\sigma_W^2 &= 2 \int_0^\tau {\rm d} t \hspace{1mm} p_t^{\rm eq} (1-p_t^{\rm eq}) \dot{E}_t^2 \int_0^t \left[ e^{-\tau \Gamma_t (t-s)}+ \mathcal{O} \left((t-s)e^{-\tau \Gamma_0 (t-s)}\right) \right]{\rm d} s
§\nonumber\\
&= \frac{2}{\tau} \int_0^\tau {\rm d} t \hspace{1mm} p_t^{\rm eq} (1-p_t^{\rm eq}) \frac{\dot{E}_t^2}{\Gamma_t} \hspace{1mm} +\mathcal{O}\left(\frac{1}{\tau \Gamma_0} \right)^2.
\end{align}
This provides the desired result:
\begin{align}
W_{\rm diss} = \frac{\beta}{2} \sigma_W^2 +\mathcal{O}\left(\frac{1}{\tau \Gamma_0} \right)^2.
\end{align}
\end{document}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 7,851
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Experience life through the eyes of seven-year-old Agatha, an insomniac who finds herself conflicted between friendship with the animals in her mother's butcher shop and her unwavering love for eating meat. Embark on a satirical narrative point-and-click adventure as you go on an unpredictable journey through Agatha Knife's quirky and hand-crafted world. Aiming to make the animals unafraid of their inevitable fate in the butcher shop where Agatha works, you'll head into the whimsical Psychotic Universe looking for answers.
Upon her discovery of religion, Agatha quickly realizes it could be used as a valuable tool to win over the animals, quash their fears, and make them unafraid of death. Join Agatha as she undertakes the challenge of creating a new religion of her own, Carnivorism, and works to convince the animals that the sacrifice of their flesh is the secret to eternal happiness.
Taking place in the same world as Mango Protocol's first title, MechaNika, Agatha Knife is riddled with dark humor, geek references, and challenging puzzles. Utilizing humor and wit to boldly explore controversial issues, players will find themselves immersed in Agatha's struggles while experiencing the world through the eyes of a naive child.
Agatha Knife was released on April 27th this year. The game didn't get much visibility but so far it has been very well received by press and users, achieving a 98% of positive reviews on Steam. They have enjoyed the weird and fun story of the 7 yo butcher creating her own religion, the cute and colorful art style of this Psychotic Universe, and the lighthearted and catchy melodies of the original soundtrack.
Finalist in Best Music Awrad in Gameboss 2017. The other nominees were two amazing games: Aragami and Candle (winner).
Finalist in Best Music and Audio Design Award in Gamelab 2017. The winner of this award was the outstanding Rime.
Finalist in Diversity Award in TIGA Awards 2017. Some of the other nominees were Horizon Zero Dawn, Herald, and Women in Games (winner).
Winner of the Excellence in Narrative Award in BICFest 2017 (YAY!!!). Old Man's Journey and our very good friends from Appnormals Team with STAY were also finalists.
We use pop culture and geek references in Agatha Knife, including movies, music, anime and video games, as a narrative device (and because reasons).
We review the art pipeline we worked with during the development of our second Psychotic Adventure, Agatha Knife.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 6,516
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Q: Spring + Hibernate with Hazelcast as 2nd level cache I have a spring project (spring core 3.1.2) configured with Hibernate (hibernate core 4.2.8) and i want to set up Hazelcast as a 2nd level cache. I want to have the cache distributed in P2P, embedded cluster mode (each application instance runs a hazelcast instance on the same machine).
This is my current sessionFactory configuration.
<bean id="sessionFactory"
class="org.springframework.orm.hibernate4.LocalSessionFactoryBean" scope="singleton">
<property name="dataSource" ref="dataSource" />
<property name="packagesToScan" value="com.myProject.beans" />
<property name="hibernateProperties">
<props>
<prop key="hibernate.database">ORACLE</prop>
<prop key="hibernate.show_sql">false</prop>
<prop key="hibernate.dialect">org.hibernate.dialect.Oracle10gDialect</prop>
<!--enable batch operations-->
<prop key="hibernate.jdbc.batch_size">20</prop>
<prop key="hibernate.order_inserts">true</prop>
<prop key="hibernate.order_updates">true</prop>
<!-- 2nd level cache configuration-->
<prop key="hibernate.cache.use_second_level_cache">true</prop>
<prop key="hibernate.cache.region.factory_class">com.hazelcast.hibernate.HazelcastLocalCacheRegionFactory</prop>
<prop key="hibernate.cache.use_query_cache">false</prop>
</props>
</property>
</bean>
This configuration seems to work on my local machine as i ran a small test that checks for a 2nd level cache hit.
The question is:
What other configurations must I make in order to have the cache distributed among the instances. How are the different machines going to "know about each other"?
Also, is there a way to create a test scenario that checks that the cache is indeed distributed among a couple of machines?(ex: launch 2 jvms) An example would be greatly appreciated.
Any other tips or warnings about this configuration are welcome.
Disclaimer: Its the first time I use Hazelcast.
My Hazelcast version is 3.5.4
Thank you!
A: You don't need to do anymore config to form a cluster.
Just start another instance of your app and two hazelcast instances should see each other and form a cluster.
By default hazelcast members uses multicast to find each other, of course you can change this behaviour by adding a custom hazelcast.xml to your project.
Here is a detailed example of Spring-Hibernate-Hazelcast integration.
https://github.com/hazelcast/hazelcast-code-samples/tree/master/hazelcast-integration/spring-hibernate-2ndlevel-cache
applicationContext-hazelcast.xml is the file to modify if you want to play with Hazelcast config.
For example, in this sample project port-auto-increment is set to false
that means Hazelcast won't start if the specified port is already occupied. (5701 by default)
Just set this property to true and start another Application instance, you should see that the cache is distributed.
Please don't forget to comment out last line of Application class before starting first instance to keep processes alive
Start first instance like below;
public static void main(String[] args) {
InitializeDB.start();
ApplicationContext context = new ClassPathXmlApplicationContext("applicationContext.xml");
DistributedMapDemonstrator distributedMapDemonstrator = context.getBean(DistributedMapDemonstrator.class);
distributedMapDemonstrator.demonstrate();
//Hazelcast.shutdownAll(); Keep instances alive to see form a cluster
}
And second one like below;
public static void main(String[] args) {
//InitializeDB.start(); DB will be initialized already by the first instance
ApplicationContext context = new ClassPathXmlApplicationContext("applicationContext.xml");
DistributedMapDemonstrator distributedMapDemonstrator = context.getBean(DistributedMapDemonstrator.class);
distributedMapDemonstrator.demonstrate();
//Hazelcast.shutdownAll(); Keep instances alive to see form a cluster
}
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 2,447
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My husband and I are struggling to pay off our student loans. We have a three year old daughter whom we want to have a home for but we can't seem to pay off the loans and save for a house. We live with family and it just seems like a long road to pay off debt. I just am seeking prayers to help remain focused on God and His plan for our lives. He hasn't forgotten me; but I struggle with remembering that He loves me and my family.
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 4,029
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Gastroenterology Research and Practice
Gastroenterology Research and Practice/
AbstractIntroductionMethodsResultsDiscussionData AvailabilityConflicts of InterestAuthors' ContributionsReferencesCopyrightRelated Articles
Volume 2018 | Article ID 4195968 | https://doi.org/10.1155/2018/4195968
Sustained Clinical Efficacy and Mucosal Healing of Thiopurine Maintenance Treatment in Ulcerative Colitis: A Real-Life Study
Daniela Pugliese,1Annalisa Aratari,2Stefano Festa,2Pietro Manuel Ferraro,3Rita Monterubbianesi,4Luisa Guidi,1Maria Lia Scribano,4Claudio Papi,2and Alessandro Armuzzi1
Academic Editor: Vicent Hernández
Received12 Mar 2018
Accepted19 Aug 2018
Published03 Oct 2018
Background and Aims. Thiopurines are commonly used for treating ulcerative colitis (UC), despite the fact that controlled evidence supporting their efficacy is limited. The aim of this study was to evaluate the long-term outcome of thiopurines as maintenance therapy in a large cohort of UC patients. Methods. All UC patients receiving thiopurine monotherapy at three tertiary IBD centers from 1995 to 2015 were identified. The primary endpoint was steroid-free clinical remission. Secondary endpoints were mucosal healing (MH), defined as Mayo endoscopic subscore 0, long-term safety, and predictors of sustained clinical remission. Results. We identified 192 patients, contributing a total of 747 person-years of follow-up (median follow-up 36 months, range 1–210 months). Steroid dependency was the most common indication for thiopurine treatment (58%). Steroid-free remission occurred in 45.3% of patients; 36.3% stopped thiopurines because of treatment failure and 18.2% for adverse events or intolerance. The cumulative probability of maintaining steroid-free remission while on thiopurine treatment was 87%, 76%, 67.6%, and 53.4% at 12, 24, 36, and 60 months, respectively. MH occurred in 57.9% of patients after a median of 18 months (range 5–96). No independent predictors of sustained clinical remission could be identified. Conclusions. Thiopurines represent an effective and safe long-term maintenance therapy for UC patients.
Ulcerative colitis (UC) is an inflammatory bowel disease (IBD) needing chronic maintenance therapies in order to prevent symptom relapses and disease progression [1]. Aminosalicylates are the first-line medical option for remission maintenance in the long term for mild to moderate disease [2, 3]. Nevertheless, after a moderate-to-severe disease flare requiring systemic corticosteroids, up to 20% of patients need to escalate therapies because of the development of steroid-dependency and approximately 15% because of steroid-refractoriness [4]. Thiopurines, azathioprine (AZA), and 6-mercaptopurine (6MP) have been considered the reference maintenance treatment for patients with steroid-dependent and steroid-refractory moderate-to-severe UC for many years and are recommended as the first line immunosuppressive therapy by major guidelines [1, 5].
Controlled data supporting the efficacy of thiopurines in UC are limited and are not as robust as in Crohn's disease (CD) [6, 7]. Few old randomized controlled trials (RCTs) addressing AZA and 6MP for the treatment of UC have relevant methodological limitations such as small sample size, inadequate thiopurine dose, heterogeneity of patient populations, limited follow-up, and not well-defined endpoints [8–14]. Despite these limitations, a systematic review and meta-analysis addressing the use of thiopurines in UC concluded that AZA and 6MP are more effective than placebo for the prevention of relapse in UC, with a number needed to treat (NNT) of 5 and an absolute risk reduction (ARR) of 23% compared to placebo [15]. Moreover, the efficacy of thiopurines in UC is supported by several uncontrolled observational studies: a mean efficacy of 65% and 75% for remission induction and maintenance, respectively, has been reported [15]. However, study designs, patients' characteristics, length of follow-up, and endpoints considered are very heterogeneous across studies, making robust conclusions very challenging. Furthermore, mucosal healing (MH) in UC has been poorly investigated with thiopurines, despite the fact that MH has recently emerged as a therapeutic goal in the management of IBDs, both for clinical trials and clinical practice [16].
The aim of this study is to evaluate the long-term effectiveness of thiopurines for maintaining clinical and endoscopic remissions in a large cohort of UC patients in a real-life setting and to explore possible predictors of sustained effectiveness.
2. Patients and Methods
This is an open-label retrospective study of consecutive UC patients treated with thiopurines at three IBD referral centers in Rome, Italy (Presidio Columbus, Fondazione Policlinico Universitario A. Gemelli IRCCS Università Cattolica del Sacro Cuore; S. Filippo Neri Hospital; and San Camillo Forlanini Hospital). Eligible patients included men and women older than 18 years with an established diagnosis of UC, who received maintenance treatment with thiopurine monotherapy from 1995 to 2015. Patients receiving thiopurine monotherapy after a course of anti–tumour necrosis factor (TNF) alpha treatment or after rescue therapy with cyclosporine for severe steroid refractory UC were excluded.
A shared common database was used to collect demographic and clinical data. The following variables were recorded: age at diagnosis, gender, disease duration, disease extent, endoscopic activity, smoking habit, indication for thiopurine treatment, type of thiopurines used (AZA or 6MP), and concomitant medications during induction and maintenance phases. The indications for thiopurine therapy were classified as the following: (1) steroid dependence, (2) maintenance therapy after a severe acute attack responsive to intravenous (iv) steroids, and (3) maintenance therapy for patients with mild to moderate disease with frequent relapses despite optimized treatment with aminosalicylates. Steroid dependency was defined according to the Italian Group for the Study of IBD (IG-IBD) guidelines [5] or as need of at least two steroid courses in the previous year. Patients with two or more clinical relapses in the last year despite appropriate oral and rectal aminosalicylates were considered having frequent relapses. Disease extent was defined according to the Montreal classification [17]; endoscopic activity was evaluated according to the Mayo endoscopic subscore [18]. Baseline endoscopy had to be performed within 3 months before starting thiopurines; follow-up endoscopies were scheduled at variable time points according to clinical judgment. MH was defined as Mayo endoscopic subscore of 0 and assessed for patient achieving sustained steroid-free clinical remission [18]. At the last follow-up visit, data regarding disease activity and whether patients were still on thiopurine maintenance were recorded. The reasons for discontinuation of thiopurines were classified as (1) sustained steroid-free clinical remission; (2) thiopurine failure, defined as clinical relapse requiring therapeutic escalation with corticosteroids and/or biologics or need for colectomy; and (3) intolerance or adverse events (AEs).
The primary endpoint was steroid-free clinical remission, defined as no diarrhea, no haematochezia, and no need of steroids, anti-TNF alpha agents, or surgery during maintenance therapy with thiopurines. Secondary endpoints were the occurrence rate of MH in patients in steroid-free remission and long-term safety. Finally, potential clinical predictors of steroid-free clinical remission and mucosal healing were analysed.
Data were described using means with standard deviation (SD) and medians with range for continuous data and percentages for discrete data. Cumulative probabilities of continuing thiopurine treatment while in remission and cumulative probability of colectomy in a patient who failed thiopurines were estimated by the Kaplan-Meier method. Associations between clinical variables and treatment efficacy (both for steroid-free remission and mucosal healing) were analysed with logistic regression analysis and expressed as odds ratio (OR) and 95% confidence intervals (95% CI). The following covariates were considered: gender, age, disease duration, disease extension, smoking habit, indication for thiopurine therapy, and concomitant aminosalicylate treatment. A two-tailed value < 0.05 was regarded as statistically significant. StatsDirect statistical tools (copyright 1990–2001) were used for all calculations.
3.1. Baseline Patients' Characteristics
One hundred and ninety-two UC patients (88 male and 104 female) receiving thiopurines as maintenance treatment were enrolled. The demographic and clinical characteristics of patients are summarised in Table 1. Median age at diagnosis was 36 years (range 16–69 years), and the median disease duration was 3.3 years (range 0–31 years). One hundred and seventeen patients (60%) had extensive colitis, and 75 patients (40%) had left-sided disease. Most patients were nonsmokers or former smokers (88%). Steroid dependency was the most common indication for thiopurine treatment (111 of 192 patients, 58%); 36 of 192 patients (19%) received thiopurines following a severe acute attack responsive to intravenous steroids, and 45 of 192 patients (23%) received thiopurines because of frequent clinical relapses despite optimized treatment with aminosalicylates.
Demographic and clinical characteristics of patients.
At baseline, 148 of 192 patients (77%) were concomitantly treated with corticosteroids. More than 90% of patients received concomitant aminosalicylate maintenance.
AZA was the preferred thiopurine compared to 6MP (90% vs. 10%). All patients received thiopurines at the standard dose of 2.0–2.5 mg/kg for AZA and of 1.0–1.5 mg/kg for 6MP. For both drugs, 50 mg/day was the initial dose progressively increased to the standard dose; dose adjustment was performed during treatment according to clinical judgment. Thiopurine metabolite monitoring, as well as thiopurine methyltransferase (TPMT) activity, was not performed because it is not routinely available in clinical practice in Italy.
Endoscopic data at baseline were available for 175 of 192 patients (91.1%): 91 patients (52%) had moderate endoscopic activity classified as Mayo endoscopic subscore = 2, and 68 patients (39%) had severe endoscopic activity classified as Mayo endoscopic subscore = 3.
3.2. Outcomes
The median follow-up while on thiopurine maintenance was 36 months (range 1–210 months). Participants contributed a total of 747 person-years of follow-up. Overall, 87 of 192 patients (45.3%) achieved steroid-free clinical remission within a median follow-up of 39 months (range 1–210 months). Conversely, 105 of 192 patients (54.6%) withdrew from thiopurines because of treatment failure (, 36.3%) or occurrence of AEs or intolerance (, 18.2%) (Figure 1).
Percentage of patients achieving steroid-free clinical remission and treatment discontinuation for failure and adverse events.
Treatment failure occurred after a median follow-up of 36 months (range 3–173 months), while most patients who discontinued thiopurines for intolerance withdrew the drug within the first year (59%).
The cumulative probability of maintaining steroid-free remission while on thiopurine treatment was 87%, 76%, 67.6%, and 53.4% at 12, 24, 36, and 60 months, respectively (Figure 2). Among the 87 patients who achieved steroid-free remission, 65 (73.8%) were still on thiopurine therapy at the end of the follow-up, while 22 (25%) were discontinued because of sustained remission after a median length of thiopurine treatment of 39 months (range 14–128 months). Among the 70 patients who were considered treatment failures, 57 (81.4%) received at least one course of systemic corticosteroids, 59 patients (84.2%) escalated to anti-TNF alpha agents, and 15 (21.4%) ultimately required colectomy. The cumulative probability of a course free of colectomy within 5 years after thiopurine failure was 90%, 84.4%, 82.0%, and 67.6% at 12, 24, 36, and 60 months, respectively (Figure 3).
Cumulative probability of maintaining steroid-free remission while on thiopurine maintenance in the entire population.
Cumulative probability of a course free of colectomy after thiopurine discontinuation for treatment failure.
As far as MH is concerned, data are available for a subgroup of 69 of 87 responders, whose baseline and follow-up endoscopy data were available. Follow-up endoscopies were performed after a median time of 18 months (range 5–96 months) after starting thiopurines, according to clinical judgment. Endoscopic activity, expressed as Mayo endoscopic subscore, at baseline and during follow-up is shown in Figure 4. Overall, 40 of 69 patients (57.9%) achieved complete MH while on thiopurine maintenance (Mayo endoscopic subscore = 0).
Endoscopic activity according to the Mayo endoscopic subscore at baseline and during follow-up in the subgroup of 69 of 87 responders who underwent endoscopy both before starting thiopurines and during follow-up. Mucosal healing (Mayo endoscopic subscore = 0) was achieved in 58% of patients.
A logistic regression analysis was performed to explore possible clinical predictors of treatment success. None of the clinical variables included in the model was associated with the probability of steroid-free remission (Table 2) or mucosal healing (data not shown).
Logistic regression analysis of predictors of steroid-free clinical remission.
3.3. Safety
A total of 45 patients experienced at least one AE related to thiopurine exposure. Overall, 35 patients discontinued thiopurines because of AEs or intolerance. The description and frequency of all AE events in our cohort are reported in Table 3. Gastrointestinal intolerance (including nausea and vomiting) occurred in 13 patients (29%). In 3 patients, switch to 6MP was attempted without success. Thirteen patients (29%) experienced leukopenia (a white blood cell count < 3000/mm), and among them, 10 needed drug discontinuation. Elevation of serum transaminases (more than 2–3 times the upper limit of normal) was recorded in 6 patients (13%), and 5 patients were consequently discontinued. No one developed chronic liver disease. In 5 patients (11%), elevation of serum pancreatic enzymes occurred, but only two patients (4%) developed acute pancreatitis requiring hospital admission. Infections were recorded in 14 patients (31%), but only three of them (2 cases of Listeria monocytogenes infection and 1 case of Cytomegalovirus colitis) were considered severe and required hospitalization. Two patients (4%) developed malignancies (1 anal cancer and 1 gastric cancer).
Adverse events related to thiopurine exposure.
Although thiopurines are widely used as a maintenance treatment in UC and are considered at least as effective as in CD patients [19], controversy still exists regarding their efficacy in maintaining remission in the long term [15]. Evidence-based data supporting the efficacy of AZA and 6MP in UC are limited, and the main evidence comes from observational studies, mainly retrospective. Observational studies report substantial variability in effectiveness of thiopurines in UC, ranging from 40% to 70% [20–26]. However, significant heterogeneity across studies, methodological limitations, small sample size, variable length of follow-up, and different endpoint definitions highlight the uncertainty of the available data.
Our study focuses on the long-term outcome of thiopurine treatment in UC patients in a real-life setting. Although the main limitation of our study is its retrospective design, the large number of patients included and the consistent length of follow-up (760 person-years) are the main strengths. Moreover, we report data addressing MH in a large subgroup of patients, and this represents a peculiarity of our study because thiopurine-induced MH has not been extensively studied and it is usually not assessed in most observational studies [24–26]. Another strength of our study is the strict definition of steroid-free clinical remission, our primary endpoint, that is, the absence of diarrhea and blood in stools, without need of any escalation of therapy, including steroids, anti-TNF alpha agents, or surgery. As previously reported, stool frequency and rectal bleeding alone provide reasonable estimates of disease activity as well as the Mayo scoring system, commonly used in RCTs [27].
MH has been strictly defined as a Mayo endoscopic subscore = 0. Although in several RCTs and cohort experiences MH is usually defined as a Mayo endoscopic subscore ≤ 1 [28], recent observations suggest that there is an improved long-term outcome in patients achieving complete MH (Mayo subscore = 0) compared to patients achieving partial MH (Mayo subscore = 1) [29]. Finally, a survival regression model has been performed to explore possible predictors of sustained efficacy of thiopurines.
Overall results show that approximately 45% of UC patients receiving thiopurines achieve steroid-free remission, 37% fail to respond to thiopurines and need escalation therapy, and less than 20% discontinue the drug because of AEs or intolerance.
Our results suggest a favourable profile of thiopurines in UC in terms of long-term efficacy and safety and are comparable to data recently reported by Sood et al. In their cohort of 255 UC patients, after a median follow-up of 30 months, 60.4% achieved remission, approximately 20% required escalation of therapy, and 30% experienced AEs resulting in thiopurine discontinuation [26]. Other smaller observational studies report comparable results [24, 25]. The probability of achieving steroid-free clinical remission is unpredictable; logistic regression analysis failed to identify any clinical predictor of treatment success confirming previous observations [26]. However, it is interesting to note that early introduction of thiopurines, within the first year after diagnosis, was associated with a reduced probability of achieving steroid-free remission although the data is not statistically significant. We can speculate that patients who require early introduction of thiopurines have a more severe disease onset and a more aggressive early clinical course leading to a worse outcome.
Data concerning endoscopic remission are not available in recently published large series [26]. We have studied the occurrence rate of MH in a subgroup of patients who achieved steroid-free remission and who underwent colonoscopy at baseline and after a median of 12 months (range 1–132) after starting thiopurines. Complete endoscopic remission, defined as a Mayo endoscopic subscore = 0, was observed in more than 50% of patients. In recent years, targeting MH is an emerging therapeutic endpoint in the management of UC [16, 30]. MH has been associated to a more favourable outcome in terms of reduction of clinical relapse, steroid needs, hospitalizations, colorectal cancer, and surgery [31]. Although it is commonly accepted that thiopurines are able to induce MH, this effect is slow, the occurrence rate of MH in thiopurine-treated UC has not been systematically investigated, and few data are available. In a recent multicenter retrospective French study on 80 UC patients receiving thiopurine monotherapy, MH (defined as a Mayo endoscopic subscore ≤ 1 and Ulcerative Colitis Endoscopic Index of Severity (UCEIS) < 2) was observed in 43.7% after a mean follow-up of 38 ± 31 months after thiopurine introduction [32]. These findings are similar to our observations.
AEs requiring withdrawal from therapy occurred in 18.2% of patients, a figure similar to that reported in other observational studies [19–24]. However, in other cohort studies, some of which include both CD and UC patients, the occurrence rate of AEs leading to thiopurine discontinuation may be as high as 25–40% [23, 26, 33–35]. In our study, the most common causes of AZA cessation were gastrointestinal symptoms, despite the fact that a slow dose escalation approach was adopted in most patients. A switch to 6MP was attempted in a minority of patients. Myelotoxicity and hepatotoxicity requiring drug discontinuation occurred in about 5% and 3% of patients, respectively. We have no data on TPMT activity and serum thiopurine metabolite concentrations: monitoring metabolites is not a routine practice in Italy, and this approach is not available in most of the hospitals.
In conclusion, in our real-life experience on a large cohort of UC patients, thiopurines are effective for maintaining long-term steroid-free clinical remission and for inducing MH. No predictors of long-term benefit could be identified. Less than 20% of patients discontinue the drug because of AEs or intolerance supporting a favourable benefit/risk profile of thiopurines in UC.
The general dataset is available upon request writing to the corresponding author, Daniela Pugliese.
Daniela Pugliese received lecture fees from Takeda and AbbVie. Annalisa Aratari, Pietro Manuel Ferraro, and Rita Monterubbianesi have none to declare. Stefano Festa received lecture fees and consulting from Takeda, Ferring, Sofar, and Alfa-Wasserman. Luisa Guidi received lecture fees and consulting from AbbVie, MSD, Mundipharma, Takeda, and Zambon. Maria Lia Scribano received lecture fees and consulting from AbbVie, Takeda, Pfizer, and Janssen. Claudio Papi received consultancy and educational projects from Takeda, MSD, AbbVie, Sofar, Chiesi, and Alfa-Wasserman. Alessandro Armuzzi received consulting and lecture fees from AbbVie, AstraZeneca, Chiesi, Ferring, Hospira, Lilly, MSD, Mundipharma, Otsuka, Sofar, Takeda, and Zambon and a grant for research from MSD.
DP, LG, AA (Aratari), RM, MLS, and SF identified eligible patients and collected data; CP and AA (Armuzzi) designed the study design and revised the final manuscript; PMF performed statistical analysis; DP and SF wrote the manuscript.
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Copyright © 2018 Daniela Pugliese et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 6,587
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oneid.jwts
==========
.. automodule:: oneid.jwts
:members: make_jwt, verify_jwt, make_jws, extend_jws_signatures,
remove_jws_signatures, get_jws_key_ids, verify_jws, get_jws_headers
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 3,795
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Plan B will pit college football unbeatens BYU, Coastal Carolina
The 14th-ranked Chanticleers were scheduled to face No. 25 Liberty on Saturday in an unexpected Top 25 matchup that persuaded ESPN to send "College GameDay" to Conway, S.C., for the first time.
But the coronavirus had crept into the Liberty program. Coastal Carolina needed a Plan B — as in BYU. Head coach Kalani Sitake's eighth-ranked Cougars have been primed to pounce on short notice if the opportunity arose to bolster their chances to reach a major bowl.
"Last week I said: 'Kalani, it's kind of like when there's a married couple about ready to have a baby and you have your bag packed by the door. That's how it's going to be maybe,'" BYU athletic director Tom Holmoe said. "You might have to pick up your bag and go."
With ESPN and the Sun Belt Conference's assistance, Coastal Carolina and BYU finalized a deal Thursday morning to play a game about 56 hours later.
"Once you're in that situation this year, you know we're all kind of playing by a different book," Hogue said. "We obviously have a lot invested in this weekend, so we wanted to start exploring what opportunities might be out there."
Head coach Kalani Sitake and eighth-ranked BYU were prepared to go even before they had an opponent to play.
Jeff Swinger / Associated Press
Kurt Dargis, director of college football for ESPN, said he got a call from Sun Belt Conference officials Wednesday afternoon, informing him of the potential problem with what the network had turned into a showcase game.
ESPN owns the television rights for both the Sun Belt and BYU, and has been helping the independent Cougars rebuild a schedule that fell apart when Power 5 conferences decided to play mostly league games during the pandemic.
Dargis called Holmoe and asked if BYU was interested. Holmoe said he needed to run it by Sitake, but it took the AD only about 45 minutes to get back with the news: If Liberty could not play, BYU would.
"I'm grateful to our coaches," Holmoe said. "They've watched a lot of film this week."
Briefly: Still buzzing over its win over Oregon, Oregon State will visit Utah on Saturday without quarterback Tristan Gebbia (hamstring), who threw for 263 yards and a TD and ran for another score against the Ducks. ... Israel Tucker ran for 161 yards and two TDs for Louisiana Tech, which played for the first time since Halloween and won 42-31 at North Texas.
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Why Does Octopath Traveler feel so repetitive and boring? and what could devs learn from it?
by Gabriel Gutierrez in September 17, 2021 September 18, 2021
Nintendo Switch PC
OT is probably one of the most beloved games that wink at the 16-bit era that many gamers grew up with, bringing a nostalgic feel that few games manage to touch. But now without its good share of issues.
Octopath Traveler is NOT A BAD GAME AT ALL… The graphics are stunningly beautiful. The music, especially the boss themes, is an absolute blast. The battle mechanics, a great update to the classic 16-bit era.
However, unlike other games, I started to notice the exact same critique: "it's stunning… but boring". And personally, it's a sentiment I absolutely share. This game was so promising but… it lacked something and I grew bored of it.
Driven by curiosity, I made several polls about which JRPG game you WOULD NOT recommend among 4 games from the Switch (Octopath, Dragon Quest XI, Xenoblade Chronicles, and Ys VII) and Octopath was consistently voted as the least recommended (to many fans' surprise, for a total of 382 votes), almost DOUBLING the number of votes from the 2nd less liked the game, which was Xenoblade Chronicles – Definitive Edition (with 200 votes).
Their reason? Again, "It's boring!", "I couldn't bother to finish it", "the storyline is bland".
As an aspiring Game Developer, I found this case intriguing. It's the first time I've seen such a successful game get the same type of critique popping up in a constant way that I decided to dive and search which aspects made the game so boring after a while for a lot of players.
After playing other JRPG's, OT's flaws were immediately evident to me, and I hope this list will serve as a footprint of how NOT to make a JRPG:
1. LINEAR DUNGEONS WITH A COMPLETE LACK OF 'DYNAMICS'
In the games of the 16-bit era, it was common that RPG's dungeons had some linearity to them, an aspect that OT emulates pretty well… while being incredibly boring after a while. Something that didn't really didn't happen in older games like Final Fantasy 6, and the reason is that OT lacks something that other games had that I call "Dungeon Dynamics", which is the way you complete a dungeon.
In Octopath Traveler you just go from "entrance" to "boss spot" like if you went through a loooooong corridor with little deviations to get some chests. And that's it. EVERY SINGLE DUNGEON IS PLAYED THE SAME, making this aspect a very linear, boring, and unengaging experience for a lot of players, where the only difference of one dungeon from another is a slightly different layout, different visuals, and the monsters.
LESSON:
If you take a game like Final Fantasy 9 you will notice the exact opposite of this issue. NO DUNGEON IS PLAYED THE SAME. Every single one has its quirks beyond the visuals: The Ice Cave where Vivi can melt down some walls, piquing your curiosity to see what could be behind that frozen wall. In Fossil Roo you have to pull switches to change the flow of water to either reach hidden chests or to be able to go forward. In FF6 you have to search and push some buttons in the Magitek factory to keep going. In few words, every single dungeon has its very own and unique dynamics in form of mini-puzzles that are easy enough to solve and keep you entertained, making the experiences very different despite having the same basic structure.
2. SAME STORY STRUCTURE
After getting through the 1st chapter of each character you would have already noticed something that makes you feel like it has become repetitive. The story structure, which goes like this:
1) Go to Town A.
2) Story development. Expect some "plot twist".
3) Go to the dungeon that is just beside the town.
4) Go through the linear dungeon to find the boss.
5) Defeat the boss.
6) More story with some "plot twist".
7) Go to Town B. And repeat ad nauseam… 32 TIMES!
It's a structure that has NO VARIATIONS and adds another layer to the whole experience that makes it feel very mechanic, predictable, and repetitive.
It's pretty evident why this structure was made, though: To save time since the team had to write 8 different stories, so they couldn't really give themselves the luxury of writing stories that had "arcs" that lasted longer than the others, thus, sacrificing a lot of spontaneity that a proper story development has.
3. LACK OF MEANINGFUL CHARACTER INTERACTION
Another CRITICAL aspect of a likable story is not only how well fleshed out the characters are, but how they actually interact with each other and watch how their actions impact other characters that make them evolve in interesting ways. But in OT that is nonexistent too.
OT's defenders argue that "they are 8 different stories, they were meant to develop independently." And they are kinda right in theory, but in practice, the game falls into a very contradicting issue.
What could possibly make Olberic help a complete stranger break into a mansion and steal? Why would Primrose help an acolyte complete a pilgrimage that is entirely alien to her goals?
The potential for a very deep, intriguing, interesting, and even unpredictable development is there to take… only to sit on the sides with all that lost potential for a story that could have evolved into epic proportions.
The setting was INCREDIBLY promising: 8 stories that would slowly connect, but that actually never happens UNTIL THE NEAR END, when you have already completed like 95% of the game! Sure, they have some chit chat between chapters that are optional to see, but they are superficial and provide almost nothing to the development of the story, which makes them a chore to read after a while due to how irrelevant they are to the story and their evident disconnection from one to another.
While it's possible to make the stories connect, it's impossible to do so without sacrificing the player's freedom of how s/he wants the character's stories to develop since some linearity will be inevitable. But giving the player COMPLETE freedom will result in disconnected stories that feel bland.
4. OVERFOCUS ON THE BREAK SYSTEM
OT's Break System is an innovative system but at the cost of making the player too overfocused on it.
One major flaw is that if the player doesn't take advantage of the Break System, the game severely punishes him by dragging out the battles because the damage done is too low.
Another issue is that if you happen to run into a dungeon full of monsters with X weakness and none of your characters have attacks of said element or weapon, prepare for a LOOOOONG battle, an issue that is even more accentuated with the Boss battles that can feel like hours if your team happens to lack the skills or weapons needed.
This sole issue is enough to make the player focus so damn much on the Break System that it leaves little room for other strategic elements because the player feels rewarded to break the enemies as fast as possible or else face a dragged-on battle, making other choices less relevant.
This means that once you get a complete understanding of which attacks to use, almost half the battles will be finished in your 1st turn without ever giving the monsters a chance to act by using the SAME attacks and skills in the SAME ORDER, something I have never experienced before so constantly and that strips a lot of the challenge to the game, making a lot of the journey easy and too linear
I believe the reason for this overfocus was because the reward was too high and the alternative was a dragged and long battle. If you want a player to use a certain battle mechanic, make the reward good enough, but not to the point of making other options irrelevant. Personally? I would've made the Break System like this:
1) If a monster is weak to lightning, fire attacks still deal the same damage instead of less.
2) Instead of knocking the monster and lose a turn, it would only make it weaker to all attacks. That way most battles would've been more challenging.
5. LACK OF ENGAGING SIDE QUESTS.
While it's understandable that not everyone likes side quests (or even mini-games), they are also a staple in every good JRPG. OT's side quests? They are barely there in the figurative sense.
In OT they were so damn obscure, with barely any tips that after a while you would completely forget about them and even not care due to how little details you would get from them.
In (again) FF9 most side quests are easy to remember because they had little and constant reminders as you progressed in the game in very subtle ways without the need of a log.
When exploring you would find coins that are part of the Stellazio coin collection. Every mog you met that served as a save point was also part of the Mognet letter delivery quest. Every new marsh you arrived at reminds you of Quina's frog gathering. And every Chocobo track seen on the world map was also a reminder of the friendly monsters' battles.
6. IS THE STORY REALLY THAT BAD AND BORING?
Let's face it, story quality is one of the most subjective aspects of any videogame, but one thing is undeniable: the story quality is one of the most valued aspects in any JRPG given how they are strongly story-driven games.
Again, in a small poll, Storyline had more than double the votes as the most valued aspect in a JRPG, followed by battle mechanics.
Octopath Traveler's storylines are not bad, they are just average at best, and barely memorable due to how the structure had to be made. So, if you are a player who loves a very well fleshed-out character development, good plot twists, and an intriguing story, for most critics (myself included), this game will fall short for you.
But again, it really depends on your standards and how much you value this aspect that may be a breaking point into buying this game or just avoid it.
QUICK RECommendations: Other retro rpgs you might want to check instead
SUIKODEN II
Probably the best-underrated JRPG ever made.
Also, it's the inspiration behind Eiyuden Chronicle. CHECK IT ON AMAZON
From the creators of the Dragon Quest series. This is
Square as its finest. CHECK IT ON AMAZON
Level-Grind is real. But seriously many FF veterans
regards FFVI as the best game in the series. CHECK IT ON AMAZON
Lunar 2: Eternal Blue
Retro JRPG with great voice acting and fully animated
hand-drawn cutscenes. CHECK IT AMAZON
Tales of Phantasia
The OG in the Tales franchise is also hands down 1 of the
The best in the series definitely a must-try. CHECK IT ON AMAZON
OT is undeniably stunning with its visuals which is one of the best among the "retro" trending that really itches that tick for a 16-bit look and a return to that glorious SNES era. But the issues like repetitive gameplay, repetitive dungeons, lack of puzzles, lack of meaningful character interaction, and bland storylines leave A LOT to be desired from this game.
Sure, a lot of players won't mind a linear dungeon, or repetitive gameplay, or a bland storyline as long as other aspects of the game make up for it… but all of them together is a sure recipe to make A LOT of players incredibly bored of your game, and even drop it midway.
OT feels like that REALLY BEAUTIFUL girl (or boy, if you're a gamer girl 😉 ) that you would LOVE to have a girlfriend and happens to like you back… only to find out after a couple of dates that she's boring, with a severe lack of topics to talk about and with little in common. You really want to love her due to how stunning and cute she looks, but… the spark is just not there.
Gabriel Gutierrez
Gabriel is an aspiring Game Developer, currently in the works for a Final Fantasy Tactics inspired game.
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by RJ Go in January 30, 2021
This is really on point and I love how brutally honest the opinions here. Great read!
by michael in August 12, 2021
this is very well written and I agree with a lot of this. The random encounters and bench members not leveling up certainly didn't help. But the battle system isn't really "New" just because it's "different" than other ones out there. You are just hitting the monster with what it's weak to. That's a no brainer that's in every game. That is not strategy, especially when you have done it 100 times already. Turn based JRPGS often make up for this lack of meaningful choice and skill by having the strategy be on the progression side, with character development, equipment, etc, but guess what is extremely barebones in octopath traveler? Yep. In games like FF7 and trails you have materia choices which let you play around with your characters' stats and what they can do, and most modern games let you avoid enemies because, if you are sick of battles, you don't want to frickin battle anymore!! Action RPGs demand skill - ideally - while Strategy RPGs have meaningful choices. Turn based RPGs often lack both of those. Trails makes up for it a lot by including timing in there. When to use an S break, etc. I think there is an opportunity to use positioning more in JRPG because in an actual battle, historically it's arguably the most important thing other than force. And force is brainless. There is also opportunities to have more of a trade off. Things like I can use this move that is powerful but costs HP that will endanger my life. Or I can use MP to cast Explosion!!! but then I won't have MP to heal. Sure there are plenty of systems that have that potential but none of them really force you to make choices. Usually what you need to do is pretty obvious. Often the only battles that are challenging are ones with status effects or instant KOs and that's just because they take control away from you.
by RJ Go in August 13, 2021
I agree Michael. While the retro graphics and music are definitely superb I just couldn't get myself to finish the game. I feel they forced the game to have those gimmicks but the gameplay just fell apart for me.
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Chris Chatham
Mr. Chatham has represented major stars, producers, directors, writers, financiers, independent production companies, broadcasters, and major corporations. He acts as general counsel to key players in the entertainment industry and provides advice regarding production matters, management strength, deal structuring, new media business, employment matters and other strategic planning. Mr. Chatham is a recognized Hollywood dealmaker in consumer brands backed by celebrity influencers or industry icons. Mr. Chatham has also litigated civil matters in state and in federal courts, as well as mediations and arbitrations before JAMS and AAA, involving a myriad of complex litigation matters including claims for breach of contract, negligence, fraud, wrongful termination, shareholder disputes, trademark and copyright infringement, and other general business disputes and tortious conduct.
Before his legal career, Mr. Chatham worked on Wall Street as a corporate bond trader at Barclays (formally Lehman Brothers, Inc.). Mr. Chatham was a Registered Representative of the National Association of Securities Dealers and helped coordinate many high profile debt offerings for S&P 500 companies, which included facilitating and trading bonds in the corporate and high yield markets. Mr. Chatham obtained his undergraduate degree at the University of Virginia where he received a tennis scholarship.
Mr. Chatham has been profiled in a number of publications, including Variety's Legal Impact Report, The Hollywood Reporter, Los Angeles Times, Los Angeles Magazine, Variety's Dealmakers Report, Super Lawyer Magazine and has been a legal analyst for nationally syndicated television shows. Mr. Chatham is admitted to practice in California before U.S. Court of Appeals, 9th Circuit; Northern District of California; and Central District of California. He is also admitted to practice in Hawaii and the District of Columbia. Mr. Chatham is a member of the Beverly Hills Bar Association, the Los Angeles County Bar Association, the Hawaii Bar Association and the D.C. Bar Association.
Christopher Chatham
LEGAL DISCLAIMER AND NOTICE:
By viewing this website, you agree that no attorney client relationship is intended or created. The content within this web site is informational and general in nature only. The information contained herein may not reflect the most current legal developments, is not intended to constitute legal advice and should not be used for this purpose. Visitors to this website should not act on any of the information contained herein without first obtaining qualified legal advice.
CHRIS CHATHAM SELECTED AS A SUPER LAWYER FOR 2021
CHRIS CHATHAM HAS BEEN RECOGNIZED IN VARIETY MAGAZINE'S 2020 LEGAL IMPACT REPORT AS ONE OF THE TOP ENTERTAINMENT ATTORNEYS FOR THE 7TH CONSECUTIVE YEAR
CHRIS CHATHAM SELECTED AS ONE OF VARIETY MAGAZINE'S TOP DEALMAKERS FOR 7TH CONSECUTIVE YEAR
© 2021 Chatham Law Group.
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#EV and #Batteries Snapshot (TSXV: $NBM.V) (OTCQB: $NBMFF) (NYSE: $F) (NYSE: $GM) (NYSE: $QS) (NASDAQ: $ENVX) @neo_battery @Ford @GM @QuantumScapeCo @Enovix3D
Global #Environmental Concerns Drive New Era for #EV's and #Batteries
Point Roberts WA, Delta, BC – November 8, 2021 - Investorideas.com, a leading investor news resource covering EV and battery stocks releases a special report on the growing EV market as both vehicle and battery manufacturers jockey for position, racing to both improve performance and production capacity, all in the name of "environmentalism," featuring Vancouver-based NEO Battery Materials Ltd. (TSXV: NBM) (OTCQB: NBMFF). NEO intends to become a silicon anode active materials supplier to the electric vehicle industry.
Read this news, featuring NBM in full at https://www.investorideas.com/news/2021/renewable-energy/11082EV-Batteries-NBM.asp
The global electric vehicle market size is expected to reach USD 917.70 Billion in 2028 and register a revenue CAGR of 20.6% over the forecast period, according to a recent report by Reports and Data. Supportive government policies and regulations, rising environmental concerns, decreasing prices of batteries, and advancements in charging technologies are some key factors expected to drive market revenue growth.
For battery materials, the silicon anode market is facing an accelerated uprising due to the material's potential to store up to ten times more lithium-ions compared to current graphite anodes used in EV batteries. In a report by The Korea Economic Daily, the industry is expected to experience a growth of a 70% CAGR until 2025 to a market size of USD 2.6 Billion to 3.4 Billion, capturing 15% of the total anode market compared to the current 3%.
NEO Battery Materials Ltd. (TSXV: NBM) (OTCQB: NBMFF) recently announced that over the past three months the Company has successfully completed the Silicon (Si) Anode Production Capacity Upscaling Project.
From the initial production rate of several grams per hour for manufacturing silicon anode materials at the lab-scale, NEO's engineering team has accomplished to expand the rate to a level of several kilograms per hour. This is a result of improving productivity by more than 1,000-fold, and the success of the Project at this level has given stronger validation for the 120-ton semi-commercial plant that is scheduled to be commissioned by the end of next year. In addition to increasing the throughput rate (production speed) from this Project, NEO has reduced the amount of solvents used in the one-pot synthesis by more than 50%, thereby significantly lowering the processing cost of the Si anode material.
From the news: Dr. J.H. Park, Director and Chief Scientific Advisor commented, "NEO Battery's Nanocoated Silicon Anode that is based on low-cost Metallurgical-Grade Si (Metal-Si) utilizes NEO's optimized milling technique and nanocoating technologies simultaneously, which innovatively improve the poor cycling performance and life of Metal-Si. By aiming to implement a continuous process, NEO is attempting to shift the paradigm for production methods and efficiencies of existing battery anode and cathode materials."
Mr. Spencer Huh, President and CEO, added, "As NEO understands the need to fast-track into mass production, we are pleased to announce the accomplishment of the Upscaling Project. The Company is at the forefront of developing unique Si anode lines through the low-cost manufacturing process, and we are customizing solutions for various downstream users to optimize the products for high-power electric vehicle lithium-ion battery applications."
Establishment of NEO Battery Materials Korea Co., Ltd.:
From the news: As of the week of November 1, 2021, the South Korean subsidiary, NEO Battery Materials Korea Co., Ltd., ("NBMK") has been established and has been registered as a foreign-invested corporation. NBMK will provide the flexibility to operate and to finalize and contract the semi-commercial site location. Through NBMK, the Company will seek to create relationships with the Korean provincial governments to apply for grants and to expand business opportunities in the lithium-ion battery supply chain.
Ford Motor Company (NYSE:F) and its financing subsidiary, Ford Motor Credit Company, recently introduced the North America auto industry's first sustainable financing framework, focusing on and paying for ambitious plans in vehicle electrification and other environmental and social areas.
From the news: Separately, Ford also announced a cash tender offer to repurchase up to $5 billion of the company's higher-cost debt. Actions such as the debt tender offer and the issuance of 0% convertible notes earlier this year, together with anticipated broader access to capital from the new sustainable financing framework, are consistent with Ford's objectives to further strengthen its balance sheet and financial flexibility and return its credit ratings to investment grade.
From the news: "Winning businesses are financially healthy and lead in sustainability – it's not a choice, they rely on each other," said John Lawler, Ford's CFO. "We're again putting our money where our mouth is, prioritizing and allocating capital to environmental and social initiatives that are good for people, good for the planet, and good for Ford."
The announcement was made on the fifth anniversary of the Paris Climate Agreement, as Ford executives joined world leaders, environmental advocates and other forward-looking companies at the United Nations Climate Change Conference (COP26) in Glasgow, Scotland.
From the news: Among other expected benefits, initiatives outlined in Ford's sustainable financing framework are intended to help the company become carbon neutral no later than 2050, in line with its commitment to the Paris Agreement. Ford was one of the first full-line U.S. automakers to pledge to reduce greenhouse gas emissions from its vehicles, operations and supply chain in alignment with goals of the accord. This pledge is backed by science-based interim targets the automaker intends to achieve by 2035.
From the news: The potential positive environmental and social influence of projects described in Ford's sustainable financing framework earned an "advanced" rating – the highest possible – from Vigeo Eiris. Vigeo Eiris, an arm of Moody's Corp., makes independent assessments of organizations' goals and performance against environmental, social and governance matters.
Guided by aggressive environmental and social goals, a significant portion of related financing will go toward accelerating Ford's leadership in electric vehicles. Objectives include expanding EV technology and charging infrastructure to remove obstacles to adoption and improve the customer experience, and EV and battery manufacturing to reduce emissions.
Last month, General Motors Co. (NYSE: GM) provided a detailed roadmap of how the company plans to double its annual revenue and expand margins to 12 to 14 percent by 2030, as a result of GM's transformation into a growth company driven by EVs, connected services and new businesses.
"GM has changed the world before and we're doing it again," said GM Chair and CEO, Mary Barra. "We have multiple drivers of long-term growth and I've never been more confident or excited about the opportunities ahead."
From the news: GM concluded the first of two days of investor meetings by sharing its growth plans. Leaders – many of whom recently joined GM from other companies – detailed how GM's compelling hardware and software platforms will combine to create growth, expand margins, add customers and diversify revenues.
"GM is unlocking a secular growth story that is changing the trajectory of our business," said Paul Jacobson, Executive Vice President and Chief Financial Officer. "Simply stated, we are at an inflection point in which we expect revenue to double by 2030 while also expanding our margins. We will achieve this by growing our core business of designing, building, and selling world-class ICE, electric and autonomous vehicles, growing software and services with high margins and entering and commercializing new businesses."
From the news: With most automakers switching to EV's, there is a great demand for high performance batteries which battery makers are working diligently to meet.
QuantumScape Corporation (NYSE: QS), a leader in the development of next-generation solid-state lithium-metal batteries for use in electric vehicles, is also in the process of upscaling performance and production having recently announced the release of an independent third-party laboratory testing report on the performance of its solid-state lithium-metal battery cells.
QuantumScape's single-layer cells were tested by Mobile Power Solutions, an independent battery lab, and met automotive-relevant conditions: over 800 cycles at 25 °C, 1C (one hour) charge/discharge rates, 100% depth of discharge and under 3.4 atmospheres of pressure. We believe that the results from the tests, covering a group of three single-layer cells, are consistent with those initially reported by QuantumScape in its December 2020 Battery Showcase presentation.
"We are happy that these independent test results substantially replicate the cycling performance we reported at our December 2020 Battery Showcase," said Jagdeep Singh, CEO and co-founder of QuantumScape. "With the publication of this report, we will continue to focus on our product roadmap goals and delivering cells to our customers."
Just last month, Enovix Corporation (NASDAQ: ENVX) (NASDAQ: ENVXW), a leader in the design and manufacture of next generation 3D Silicon™ Lithium-ion batteries, announced it had achieved a major milestone—manufacturing battery cells from its first automated factory in Fremont, Calif. Additionally, the company announced it designed, fabricated and released pre-production quantities of a new cell design for Augmented Reality (AR) glasses for a top-tier consumer electronics company.
From the news: "This is a major accomplishment for Enovix and I'm incredibly proud of our team," said Harrold Rust, Co-founder, President and Chief Executive Officer of Enovix. "Manufacturing the first cell off of our automated line is proof that our machine set is ready for production. It's the culmination of years of long hours, dedication and hard work from our world-class team and it's further proof that we are on track to meet our goal of not only delivering a battery with up to 110% greater energy density, but also we're on target for commercial production in Q1 2022 and first product revenue in Q2 2022."
From the news: The first cell off the line is a manufacturing achievement that requires more than 25 machines to work in concert. The Enovix factory is state-of-the-art since it uses both established lithium-ion battery manufacturing equipment, including electrode fabrication and the majority of battery packaging and formation, as well as the Company's proprietary roll-to-stack cell assembly, a precise, high-speed replacement for conventional lithium-ion wound cell assembly. This enables its roll-to-stack production tools to "drop in" to existing lithium-ion battery manufacturing lines and increase watt-hour capacity.
From the news: Battery capacity is an important factor in the ever-evolving consumer electronics space. It is increasingly important to support compute-intensive applications for high-end wearables, mobile phones and laptop/tablet platforms. Increased computing capability supported with high battery capacity is necessary for the large-scale adoption of wearable devices, such as AR glasses. This form factor has significantly less available volume to house batteries that can provide enough energy to run compute-intensive platforms. As such, a step-change increase in battery energy density is essential to enable products that will appeal to mass market audiences.
While this boom in the EV space is great for automakers and battery manufacturers alike, the question still remains - which of these batteries will be able to meet the demands of large scale EV production and will the environmental benefits of an EV world outweigh the costs of new vehicle and battery production (mining precious metals, energy needed for battery production, testing, waste, etc.)? As of right now, the future looks bright, but as with all great changes in technology, only time will tell.
Investorideas.com publishes breaking stock news, third party stock research, guest posts and original articles and podcasts in leading stock sectors. Learn about investing in stocks and get investor ideas in cannabis, crypto, AI and IoT, mining, sports biotech, water, renewable energy, gaming and more. Investor Idea's original branded content includes podcasts and columns : Crypto Corner , Play by Play sports and stock news , Investor Ideas Potcasts Cannabis News and Stocks on the Move podcast , Cleantech and Climate Change , Exploring Mining , Betting on Gaming Stocks Podcast and the AI Eye Podcast.
Disclaimer/Disclosure: Our site does not make recommendations for purchases or sale of stocks, services or products. Nothing on our sites should be construed as an offer or solicitation to buy or sell products or securities. All investing involves risk and possible losses. This site is currently compensated for news publication and distribution, social media and marketing, content creation and more. Disclosure is posted for each compensated news release, content published /created if required but otherwise the news was not compensated for and was published for the sole interest of our readers and followers. Disclosure: this news article featuring NBM.V is a paid for news release on Investorideas.com – (two thousand) More disclaimer info: https://www.investorideas.com/About/Disclaimer.asp Learn more about publishing your news release and our other news services on the Investorideas.com newswire https://www.investorideas.com/News-Upload/ and tickertagstocknews.com Global investors must adhere to regulations of each country. Please read Investorideas.com privacy policy: https://www.investorideas.com/About/Private_Policy.asp
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}
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#region License
//
// Copyright Tony Beveridge 2015. All rights reserved.
// MIT license applies.
//
#endregion
using System.Collections.Generic;
namespace FSM {
public interface IContainer {
TType Resolve<TType>(string name = null) where TType : class;
IEnumerable<TType> ResolveAll<TType>() where TType : class;
}
}
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{
"redpajama_set_name": "RedPajamaGithub"
}
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Q: Laravel 5.2: Error sending mail using GMail I'm trying to send an email from localhost with Laravel 5.2 (using PHP 7.0.8 and XAMPP), but I get this error:
Swift_TransportException in StreamBuffer.php line 269:
Connection could not be established with host smtp.gmail.com [ #0]
The values of my loaded mail config looks like this (using Config::get("mail")):
"driver" => "smtp"
"host" => "smtp.gmail.com"
"port" => "465"
"from" => array:2 [
"address" => "***@gmail.com"
"name" => "MY_NAME"
]
"encryption" => "ssl"
"username" => "***@gmail.com"
"password" => "***"
"sendmail" => "C:\xampp\sendmail\sendmail.exe -bs"
I've googled the last few days around 10 hours, but did not find a solution which is working. I've also tried to change "port" => "587" and "encryption" => "tls", but this did not work either. The error is:
ErrorException in StreamBuffer.php line 95:
stream_socket_enable_crypto(): SSL operation failed with code 1. OpenSSL Error messages:
error:14090086:SSL routines:ssl3_get_server_certificate:certificate verify failed
In my Google Account I've already allowed "less secure apps". Is there anything I can try?
btw: with the PHP mail function it is possible to send emails using my gmail account.
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On the first day after Winter Break, a group of employees representing Boulder Valley School District's classrooms, schools and departments gathered to help identify the Long-Term Outcomes and Strategic Themes that will be the foundation for BVSD's new strategic plan.
"It is so exciting to be here today to begin our new strategic planning process," said BVSD Assistant Superintendent of Instructional Services and Equity Sam Messier.
The day began with an analysis of data about the district, as well as stakeholder feedback collected during Superintendent Rob Anderson's first 100 days.
"We are looking at our data to see what are the patterns and trends and going through a process to address how we can improve our goals for the school district," said Lafayette Elementary Principal Stephanie Jackman.
The goal of the Strategic Plan is to move the district forward.
"We need to be making sure that we do not get stuck. We must continue to move forward and make sure that we are really focusing on our students and their learning and preparing them for jobs that aren't even created yet today," said BVSD Director of IT Project Management Susan Oasheim.
Superintendent Rob Anderson hopes BVSD's strategic plan will be similarly inspiring to the students, teachers, staff, families and community members of BVSD.
"[This work] is really going to give us an opportunity to create a foundation for this district to make it the absolute best school district in the country," said Anderson.
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Q: Expression for the bound on the error of a Poisson approximation? I'm hoping that somebody can help clarify a question I had regarding the Poisson approximation and its applications. My textbook presents the following theorem, which I'm having trouble making sense of. My assumption is that the first capital $N$ is meant to be a lowercase $n$, and that all subsequent $N$'s represent a discrete random variable with a binomial distribution:
Theorem 2.5. Consider independent events $A_i$, $i=1,2,...,n$, with probabilities $p_i=P(A_i)$. Let $N$ be the number of events that occur, let $\lambda=p_1+···+p_n$, and let $Z$ have a Poisson distribution with parameter $\lambda$. Then, for any set of integers $B$,$$\left\vert P(N\in B)-P(Z\in B)\right\vert\leq\sum_{i=1}^n p_i^2\tag{2.14}$$ We can simplify the right-hand side by noting $$\sum_{i=1}^n p_i^2\leq \max_i p_i \sum_{i=1}^n p_i=\lambda\max_ip_i$$ This says that if all the $p_i$ are small then the distribution of $N$ is close to a Poisson with parameter $\lambda$. Taking $B=\{k\}$, we see that the individual probabilities $P(N=k)$ are close to $P(Z=k)$, but this result says more. The probabilities of events suchas $P(3\leq N\leq 8)$ are close to $P(3\leq N\leq 8)$ and we have an explicit bound on the error.
The text then refers back to an example comparing the exact probability of obtaining exactly one double $6$ in twelve rolls of a pair of dice to the corresponding Poisson approximation:
Suppose we roll two dice $12$ times and we let $D$ be the number of times a double $6$ appears. Here, $n=12$ and $p=1/36$, so $np=1/3$. We now compare $P(D=k)$ with the Poisson approximation for $k=1$.
$$k=1 \text{ exact answer:}\,\,\,\,\,\,P(D=1)=\left(1-\frac{1}{36}\right)^{12} =0.7132$$ $$\text{Poisson approximation:}\,\,\,\,\,\,P(D=1)=e^{-1/3}=0.7165$$
For a concrete situation, consider [the example above], where $n=12$ and all the $p_i=1/36$. In this case the error bound is $$\sum_{i=1}^{12} p_i^2 =12\left(\frac{1}{36}\right) ^2=\frac{1}{108}=0.00926$$ while the error for the approximation for $k=1$ is $0.0057$.
From this context, my thought is that the error bound they allude to is specifically that for the Poisson approximation to the binomial distribution, and not the Poisson approximation to some other type of distribution. Can somebody with a more complete understanding of the Poisson distribution (and its relationship to the binomial) confirm or refute this assertion? I'd also be curious to know where the proof of this theorem comes from, as my text doesn't offer any obvious reference.
Citation: Elementary Probability for Applications, Rick Durrett.
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« Back to Recent Articles
Transportation Commissioner Dan Saltzman and Director Leah Treat meet with the PBOT Bureau and Budget Advisory Committee in January 2017. Photo: Portland Bureau of Transportation
(Sept. 19, 2017) We are thrilled to unveil the PBOT Bureau of Transportation's Bureau and Budget Advisory Committee (BBAC) for fiscal year 2017-2018. BBAC members will inform the bureau's annual transportation budget; review program priorities and infrastructure project lists; and provide input on the strategy and direction for incorporating equity into PBOT's work and engaging communities that PBOT has traditionally underserved.
All City of Portland bureaus are required to have Budget Advisory Committees. These committees provide residents the opportunity to provide important input into the budget priorities of the individual bureaus. In 2015, Director Leah Treat expanded PBOT's own Budget Advisory Committee into the Bureau and Budget Advisory Committee in an effort to offer a more robust avenue for public input year around.
"PBOT is committed to hearing from a diverse cross section of Portlanders so that we better understand the people we serve and their transportation needs. Our public advisory groups are key venues for us to seek public input and reimagine the future of Portland's transportation system," said Director Treat. "Our policies and programs have become stronger because of the BBAC's engagement and we plan to continue on that path this year. The 2017-2018 committee members represent both the veterans of the transportation world and the next generation of transportation leaders. The diverse mix of experience, backgrounds, areas of expertise, geographic and community affiliations will be a great resource for our upcoming budget and policy discussions."
The 2017-2018 BBAC will include 5 new members and 13 returning members. BBAC members are appointed by PBOT Director Leah Treat for a one year term and may sit on the committee up to five years. The group includes representatives from PBOT's modal committees (Bicycle Advisory Committee, Pedestrian Advisory Committee and Portland Freight Committee) and representatives from PBOT employee unions. The new members were selected via a competitive application process over the summer. 2017-2018 BBAC member bios are available below. For more information about the BBAC and to follow committee business throughout the year, please visit the committee website.
September Meeting Notice
The first meeting of the Bureau and Budget Advisory Committee is scheduled for Thursday, September 21, 2017 from 4:00 – 6:00pm at the Portland Building (8th floor, Hawthorne Conference room). The meeting is open to the public. The committee meets monthly on the third Thursday of each month from September 2017 to June 2018. Meeting materials will be posted online after each meeting.
New BBAC Members
Patricia Montgomery
Patricia Montgomery has worked in the transportation industry for over 30 years. She has taken on various roles within that industry that includes previous ownership of a medical transportation company, COO of New Rose City Cab Company and Bantu Enterprises, and over 20 experience as a taxi driver in the field. Her vast experience includes Private for Hire Transportation Board Voting Member, and numerous transportation committees. She currently is Co-Chair of the Elliott Neighborhood Association, and Junior Warden of St. Philip the Deacon Episcopal Church. With the changing transportation industry Patricia has joined this committee to provide direct insight and knowledge from her experience.
Kevin Vandermore
Kevin Vandemore is an experienced Certified Public Accountant who specializes in enhancing organizational value by providing risk based and objective assurance, advice and insight. His career includes work in public accounting and private industry, and he has experience working with a diverse set of companies from smaller entities to large public corporations. He earned a Masters of Science in Financial Analysis from Portland State University where he took the university's motto to heart – let knowledge serve the city. In addition to volunteering and being active in the community he enjoys mountaineering, traveling and riding his bike.
Maria Hernandez
Maria Hernandez Segoviano was born and raised in La Cruz de Aguilar, Guanajuato, México. She moved to Woodburn, Oregon at the age of 12, with her family, and has since considered Woodburn home. After acquiring a bachelor's of science degree from Willamette University in Political Science and minoring in Sociology and Latin American Studies, Maria went on to do a Public Affairs Fellowship with CORO Northern California. Most recently, Maria was the Deputy Campaign Manager working to elect State Representative Teresa Alonso Leon, the first Indigenous Latina to represent a district in the state of Oregon. Presently, Maria is the Advocacy Coordinator for OPAL Environmental Justice Oregon. Maria leads OPAL's statewide community-connecting, providing our partners access to solidarity networks and opportunities to build local power. She also serves on the Oregon Latino Agenda for Action Board.
Frannie Knight
Frannie Knight is a senior at Valor Christian School International located in Beaverton, Oregon. She wants to pursue a Bachelor's degree in Environmental Science and Political Science and then either go to law school or get a graduate degree in Environmental Engineering. In high school, Frannie participates in numerous science competitions and Mock Trials. She enjoys reading, going to school, and being with friends and family.
Molly Baer Kramer
Molly has worked in nonprofit fundraising and administration in Portland since 2001, primarily for conservation organizations. Most recently she served as Deputy Director for the Oregon League of Conservation Voters. She is currently a consultant and holds a DPhil in history from the University of Oxford.
Molly's key focus as a member of PBOT's BBAC will be to represent the interests of disabled communities. She is able-bodied but her long-term partner is paralyzed and uses a wheelchair. She will utilize her connections with disability-rights organizations, including the Portland Commission on Disability, to ensure that the needs of the disabled are understood and integrated into PBOT's planning and budgeting.
Returning BBAC Members
Arlene Kimura
East Portland supporter/activist since 1992. Arlene initially became involved through the neighborhood system with land use planning, transportation issues, including urban trails, and environmental concerns. As East Portland has changed, Kimura has also become interested in health and economic development opportunities.
Elaine O'Keefe
Pedestrian Advisory Committee Representative
Elaine O'Keefe worked in local government for more than two decades. Including over a decade with Portland Fire and Rescue. Currently, she is a board member of the Sellwood-Moreland Improvement League (SMILE), a member of the SMILE Transportation Committee, and a member of the Portland Pedestrian Advisory Committee.
Heather Bowman
Heather Bowman is a partner with the law firm Bodyfelt Mount where her litigation practice includes employment discrimination and professional liability defense. Bowman's practice includes engagement in civil rights issues and other volunteer work includes examining equity issues in legal practice. She uses all forms of transportation, and particularly appreciates transportation cycling.
Heather McCarey
Bicycle Advisory Committee Representative
Heather McCarey has a master's degree in City and Regional Planning from Georgia Tech and works with Transportation Management Associations in urban, suburban, and park settings. McCarey is currently the Executive Director of Explore Washington Park, one of the first Transportation Management Associations in the nation created to address transportation issues both to and throughout a city park.
Kaliska Day
Kaliska Day, is a native Oregonian and an Alaska Native of the Tligint/Haida Tribe. With a degree in Construction Management from Arizona State University, Day has multi-year experience in the construction management sector, including serving as a construction management consultant for various public works agencies in California and Oregon.
Meesa Long
A resident of Southeast Portland, Meesa Long is a Reading Specialist in an East County Middle School and is also passionate about serving her community and neighborhood. In her work with transportation issues in Portland, Long's main goal has been to increase safe pedestrian travel for children and families within under-served neighborhoods, and to think outside the box to create positive and equitable transportation improvements within the city.
Momoko Saunders
BBAC Co-Chair
Momoko Saunders is a software engineer and resident of East Portland. She is on the board of the non-profit Bike Farm, which she co-founded in 2007. Momoko is also an active volunteer for App Camp 4 Girls and board member of Portland Society.
Pia Welch
Freight Advisory Committee Representative
Pia Welch began her career with Flying Tigers in California which was later acquired by FedEx Express. She has since worked for FedEx for close to three decades. Welch has served as President of Portland Air Cargo Association, Board Member American Association of University Women, and member and Vice Chair of the Portland Freight Committee. She is currently the Chair of the Freight Committee. She has been involved in city projects including; The Comprehensive Plan, Airport Way Project and various sub-committee groups when topics required more in-depth study.
Ruthanne Bennett
PTE Local 17 Representative
An civil engineer with PBOT, Ruthanne Bennett represents PTE Local 17/COPPEA Chapter. She has been a union member for 20 years and a COPPEA Steward for five years. She has consistently advocated for transportation priorities, including supporting the Fix Our Streets package and the COPPEA Value Capture program. She was instrumental in creating the COPPEA Value Capture program, which is an innovative program to encourage and fund the construction of safe street infrastructure during development projects. In addition to her B.S. in Civil Engineering she has a B.S. and M.S. in Mathematics from Portland State University.
Ryan Hashagen
Ryan Hashagen is a volunteer with Better Block PDX. A Professional Tricyclist, he has founded and run several tricycle based businesses in Canada & the U.S. Hashagen won the Cargo Messenger World Championship in 2003 & 2004 in Seattle & Edmonton. He enjoys working to connect, collaborate, and facilitate tactical urbanism projects with a wide range of organizations, businesses, and agencies.
Samuel Gollah
Sam Gollah has over a decade of experience in entitlement processing, including land use and permit compliance as a public and private planner throughout the Willamette Valley. Gollah has also provided zoning and equity consulting services for the City of Portland's Comprehensive Plan update (2035). He currently serves as a member of the City of Portland's Transportation Expert Group (TEG).
Thomas Karwaki
Thomas Karwaki chairs the University Park Neighborhood Association, an organization with over 9,000 members and that includes the University of Portland. Karwaki coordinates land use, public safety, emergency response, communication and public relations efforts of the UPNA.
Tony Lamb
Tony is a graduate of Portland State University's Community Development program with a focus on community empowerment, economic development and the creation of a livable community for all without displacement. He is currently pursuing a graduate degree at the Portland State University Master of Urban and Regional Planning (MURP) program.
Tony has served on numerous social justice and economic development initiatives including the following: Social Justice and Civic Leadership Cohort with the Urban League of Portland, East Portland Action Plan Economic Development Subcommittee, PBOT Transportation Expert Group, Multnomah County Digital Inclusion member, Steering Committee for McLennan County Reintegration Round Table, City of Waco Poverty Reduction Committee and Open-Table Anti-Poverty Program International Tech Committee.
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Q: Is it possible to separate minibuffer and echo area in Emacs? I'm curious, whether it is possible to separate echo area and minibuffer, so two different places (lines, panes, frames) are used for output of messages and input of commands.
As it is said in Hide Emacs echo area during inactivity , it is impossible to get rid of echo area completely, but some proposals are:
*
*move minibuffer to a dedicated window;
*filter messages in echo area - for example, print only keystrokes in echo area and use *Messages* buffer for other messages.
What options do i have? Is it possible in theory to separate echo area and minibuffer? Would it theoretically require rewriting C source code and recompiling Emacs? Please post any thoughts and ideas.
A: Based on the manual and a look at the C code, I believe the answer is "no".
M-: (info "(elisp) Echo Area Customization") RET says:
The variable `max-mini-window-height', which specifies the maximum height for resizing minibuffer windows, also applies to the echo area (which is really a special use of the minibuffer window; *note Minibuffer Misc::).
The Minibuffer Misc link doesn't discuss that specific point further, but if the echo area explicitly uses the minibuffer window then you'll not be able to separate them.
Edit:
For confirmation, if you look at the source for the C function message3_nolog() in xdisp.c, it obtains the frame for the selected frame's minibuffer, selects that, and then passes through to echo_area_display() which uses the currently-selected frame's minibuffer window as the echo area window.
(Emacs 24.0.95)
So the "mini window" used for the minibuffer and echo area is indeed one and the same, just as the manual states.
The only possibility I can think of is to try to find a way of automatically copying echo area messages to some other window, but as this is all happening in C code, in functions not exposed to elisp, I suspect that's not possible either.
Edit 2:
Would it theoretically require rewriting C source code and recompiling Emacs?
If you need genuine separation, then yes, I believe that is the case.
If the copying approach was sufficient, you might be able to manage that purely in elisp by advising all of the functions which can result in messages being written to the echo area. You could start reading here to see what that might entail:
M-: (info "(elisp) The Echo Area") RET
(but if you are really desperate to implement this, I would suggest your time would be better spent working in C and providing a patch which would allow such a separation to be made, because I'm a little doubtful all that advice would be robust in the long term.)
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\section{Introduction}
Conf\/iguration spaces endowed with some algebraic structures are of interest in various areas
of mathematical physics. As a rule, Hamiltonian systems def\/ined on their cotangent bundles
have certain mathematically and physically interesting features, especially when their
Hamiltonians are somehow suited to the mentioned algebraic structures, e.g., are invariant
under their automorphism groups or subgroups. The best known example is the theory of
Hamiltonian systems on the cotangent bundles of Lie groups or their group spaces (or even more
general homogeneous spaces) where by the group space we mean the homogeneous space with
trivial isotropy groups, i.e., groups which ``forgot'' about having the distinguished neutral
element. The special attention in applications is paid to Hamiltonians invariant under left or
right translations or under both of them. The examples are the rigid bodies, incompressible
ideal f\/luids \cite{Arn78}, af\/f\/inely-rigid bodies (see for example \cite{all04,all05} and
references therein), etc.
Usually in physics one deals with linear groups, i.e., groups faithfully realizable by f\/inite
matrices. The only relatively known exceptions are $\overline{{\rm GL}(n,\mathbb{R})}$ and
$\overline{{\rm SL}(n,\mathbb{R})}$, i.e., the covering groups of ${\rm GL}(n,\mathbb{R})$ and
${\rm SL}(n,\mathbb{R})$ respectively. However, in spite of various attempts of F.~Hehl, Y.~Ne'eman and others (see for example~\cite{HKH,HN}), their physical applicability is as yet rather
doubtful and questionable. So, one of the best known examples of Hamiltonian systems on
algebraic structures are (usually invariant) ones on the cotangent bundles of matrix groups or
more gene\-ral\-ly some matrix manifolds. From the purely algebraic point of view, such
conf\/iguration spaces consist of second-order (and non-degenerate) tensors in some linear
spaces. Geometrically they represent linear transformations. Some questions appear here in a
natural way. Namely, it is a rule that all second-order tensors, i.e., not only mixed ones,
are of particular importance in physics. Twice covariant or contravariant tensors represent
various scalar products, e.g., metric tensors, electromagnetic f\/ields, gauge f\/ields, etc. In a
purely analytical sense all second-order tensors are matrices. Obviously, due to the
dif\/ference in the transformation rules, all they are geometrically completely dif\/ferent
objects. Nevertheless the natural question arises as to the existence of geometrically and
physically interesting Hamiltonian systems on the cotangent bund\-les of manifolds of
second-order tensors of other type than linear transformations. We mean here f\/irst of all the
manifolds of scalar products, both real-symmetric and complex-sesquilinear-hermitian. Also the
twice covariant and contravariant tensors without any special symmetries may be interesting.
One of our motivations has to do with certain ideas concerning nonlinear quantum mecha\-nics.
Various ways towards nonlinearity in quantum case were presented, e.g., in the review papers~\cite{Sv_rev,Sv}, from those motivated by paradoxes of the
quantum measurement, the interplay of unitary evolution and reduction,
etc., to certain ideas based on geometry like, for instance, the Doebner--Goldin nonlinearity \cite{DG,DGN,G}. However in
this article we are motivated by another idea. Namely, it is well known that the unitary
evolution of a quantum system, described by the Schr\"{o}dinger equation, may be interpreted
as a Hamiltonian system on Hilbert space. The most convenient way to visualize this is to
start from f\/inite-dimensional, i.e., ``$n$-level'', quantum systems~\mbox{($n<\infty$)}. The scalar
product is then f\/ixed once for all and is an absolute element of the system. The true ``degrees
of freedom'' are represented only by the vector of the underlying Hilbert space ``wave
functions''. And here some natural analogy appears with the situation in Special Relativity vs.\
General Relativity:
\begin{itemize}\itemsep=0pt
\item In specially-relativistic theories the metric tensor is f\/ixed once for all as an
absolute object, whereas all physical f\/ields are ``f\/lexible'' and satisfy dif\/ferential equations
as a rule derivable from the variational principle. The f\/ixed metric tensor is then used as a
``glue'' to contract tensor indices in order to build the scalar density of weight one dependent
algebraically on f\/ields and their f\/irst-order derivatives.
\item In generally-relativistic theories the metric tensor becomes f\/lexible as well, it is
included to degrees of freedom and satisf\/ies dif\/ferential equations together with the other
``physical'' f\/ields. Moreover it becomes itself the physical f\/ield, in this case the
gravitational one.
\end{itemize}
One can wonder whether one should not follow a similar pattern in quantum mechanics. Just to
make the scalar product ``f\/lexible'' and dynamically coupled to the $\psi$-object, i.e., to the
``wave function''. But, as mentioned, the scalar product is a twice covariant tensor. And so we
return to the idea of Hamiltonian systems on manifolds of scalar products or more general
twice covariant or twice contravariant tensors. And the point is that such manifolds carry
some natural Riemannian, pseudo-Riemannian or hermitian metric structures (almost canonical)
which are essentially non-Euclidean, i.e., describe some curved geometries on manifolds of
scalar products. Because of this the coef\/f\/icients at their derivatives in Lagrangians (as
quadratic forms of those velocities) are irreducibly non-constant. The resulting
Euler--Lagrange equations for them, and therefore also for the systems ``wave function $+$
f\/lexible scalar products'', are essentially nonlinear. This is the non-perturbative
nonlinearity, i.e., it cannot be interpreted as an artif\/icial extra correction to some basic
linear background. So, physically this is one of natural candidates for the ef\/fective and
geometrically interpretable nonlinearity in quantum mechanics, perhaps somehow explaining the
conf\/lict between unitary evolution and reduction, which exists essentially due to the
linearity of the standard quantum mechanics.
Beside the above-mentioned physical motivations, one should also stress that such Hamiltonian
models are interesting in themselves from the purely geometric point of view. They are somehow
similar to the (pseudo-)Riemannian metric structures on semisimple Lie groups, in particular
to the Killing tensor. Nevertheless their algebraic and geometric structure is dif\/ferent. As
to our knowledge, such Riemannian geometries have not been yet studied in mathematics. One has
the feeling that being so canonical as Killing metrics on groups they may have some
interesting geometric properties and are worth to be investigated.
\section{General problem}
Let us take a set of $n$ elements and some function $\psi$ def\/ined on it, i.e.,
\begin{gather*
\mathcal{N}=\{1,\ldots,n\}\in \mathbb{N},\qquad \psi:\mathcal{N}\rightarrow\mathbb{C}.
\end{gather*}
Then we can def\/ine the ``wave function'' of the $n$-level quantum system as a following
$n$-vector
\begin{gather*
\psi=\left[
\begin{array}{c}
\psi^{1} \\
\vdots \\
\psi^{n}
\end{array}
\right],\qquad \psi^{a}=\psi(a)\in\mathbb{C}.
\end{gather*}
Let $H$ be a unitary space with the scalar product
\begin{gather*
G:H\times H\rightarrow\mathbb{C},
\end{gather*}
which is a sesquilinear hermitian form. Then such an $H$ will be our $n$-dimen\-sional ``Hilbert''
space ($\mathbb{C}^{n}$).
So, let us consider the general Lagrangian
\begin{gather}
L=\alpha_{1}iG_{\bar{a}b}\big(\overline{\psi}{}^{\bar{a}}\dot{\psi}^{b}-
\dot{\overline{\psi}}{}^{\bar{a}}\psi^{b}\big)+
\alpha_{2}G_{\bar{a}b}\dot{\overline{\psi}}{}^{\bar{a}}
\dot{\psi}^{b}+
\big[\alpha_{4}G_{\bar{a}b}+\alpha_{5} H_{\bar{a}b}\big]
\overline{\psi}{}^{\bar{a}}\psi^{b}\nonumber\\
\phantom{L=}{} +\alpha_{3}\big[G^{b\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{b}\big]
\dot{G}_{\bar{a}b}+
\Omega[\psi,G]^{d\bar{c}b\bar{a}}
\dot{G}_{\bar{a}b}\dot{G}_{\bar{c}d}-\mathcal{V}\left(\psi,G\right),\label{eq8}
\end{gather}
where
\begin{gather*}
\Omega[\psi,G]^{d\bar{c}b\bar{a}}=
\alpha_{6}\big[G^{d\bar{a}}+\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{d}
\big]\big[G^{b\bar{c}}+\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{b}
\big]+
\alpha_{7}\big[G^{b\bar{a}}+\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{b}
\big]\big[G^{d\bar{c}}+\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{d}
\big]\nonumber\\
\phantom{\Omega[\psi,G]^{d\bar{c}b\bar{a}}=}{} +
\alpha_{8}\overline{\psi}{}^{\bar{a}}\psi^{b}
\overline{\psi}{}^{\bar{c}}\psi^{d},\qquad
\Omega[\psi,G]^{d\bar{c}b\bar{a}}=\Omega[\psi,G]^{b\bar{a}d\bar{c}}
\end{gather*}
and the potential $\mathcal{V}$ can be taken, for instance, in the following quartic form
\begin{gather*
\mathcal{V}\left(\psi,G\right)=\varkappa\big(
G_{\bar{a}b}\overline{\psi}{}^{\bar{a}}\psi^{b}\big)^{2}.
\end{gather*}
The f\/irst and second terms in (\ref{eq8}) (those with
$\alpha_{1}$ and $\alpha_{2}$) describe the free
evolution of wave function $\psi$ while $G$ is f\/ixed.
The Lagrangian for trivial part of the linear dynamics
(those with~$\alpha_{4}$) can be also taken in the more general form
$f\left(G_{\bar{a}b}\overline{\psi}{}^{\bar{a}}\psi^{b}\right)$,
where $f:\mathbb{R}\rightarrow\mathbb{R}$.
The term with~$\alpha_{5}$ corresponds to the Schr\"{o}dinger
dynamics while $G$ is f\/ixed and then
\begin{gather*
H^{a}{}_{b}=G^{a\bar{c}}H_{\bar{c}b}
\end{gather*}
is the usual Hamilton operator. If we properly choose the
constants $\alpha_{1}$ and $\alpha_{5}$, then we obtain precisely the Schr\"{o}dinger equation.
The dynamics of the scalar product $G$ is described by the terms linear and quadratic in the time derivative of $G$.
In the above formulae $\overline{\psi}{}^{\bar{a}}=\overline{\psi^{a}}$ denotes the usual
complex conjugation and $\alpha_{i}$, $i=\overline{1,9}$, and $\varkappa$ are some constants.
Then applying the variational procedure we obtain the equations of motion as follows
\begin{gather*}
\frac{\delta L}{\delta \overline{\psi}{}^{\bar{a}}}=
\alpha_{2}G_{\bar{a}b}\ddot{\psi}^{b}+\big(\alpha_{2}\dot{G}_{\bar{a}b}
-2\alpha_{1}iG_{\bar{a}b}\big)\dot{\psi}^{b}-
2\alpha_{8}\dot{G}_{\bar{a}b}\psi^{b}\dot{G}_{\bar{c}d}
\overline{\psi}{}^{\bar{c}}\psi^{d}
\nonumber\\
\phantom{{\delta \overline{\psi}{}^{\bar{a}}}=}{} -2\alpha_{9}\big(\alpha_{6}\dot{G}_{\bar{a}d}\dot{G}_{\bar{c}b}+
\alpha_{7}\dot{G}_{\bar{a}b}\dot{G}_{\bar{c}d}\big)
\psi^{b}\big(G^{d\bar{c}}+\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{d}\big)
\nonumber\\
\phantom{{\delta \overline{\psi}{}^{\bar{a}}}=}{}
+\big[\big(2\varkappa G_{\bar{c}d}\overline{\psi}{}^{\bar{c}}\psi^{d}-
\alpha_{4}\big)G_{\bar{a}b}-\alpha_{5} H_{\bar{a}b}-
\big[\alpha_{3}\alpha_{9}+\alpha_{1}i\big]\dot{G}_{\bar{a}b}
\big]\psi^{b}=
\end{gather*}
and
\begin{gather}
\frac{\delta L}{\delta G_{\bar{a}b}}=
2\Omega[\psi,G]^{b\bar{a}d\bar{c}}\ddot{G}_{\bar{c}d}+
2\dot{\Omega}[\psi,G]^{b\bar{a}d\bar{c}}\dot{G}_{\bar{c}d}+
\big(2\varkappa G_{\bar{c}d}\overline{\psi}{}^{\bar{c}}\psi^{d}-
\alpha_{4}\big)\overline{\psi}{}^{\bar{a}}\psi^{b}\nonumber\\
\phantom{{\delta G_{\bar{a}b}}=}{} +
2G^{d\bar{a}}\big[\alpha_{6}G^{b\bar{e}}\big(G^{f\bar{c}}+
\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{f}\big)+
\alpha_{7}G^{b\bar{c}}\big(G^{f\bar{e}}+
\alpha_{9}\overline{\psi}{}^{\bar{e}}\psi^{f}\big)
\big]\dot{G}_{\bar{c}d}\dot{G}_{\bar{e}f}\nonumber\\
\phantom{{\delta G_{\bar{a}b}}=}{}-
\alpha_{2}\dot{\overline{\psi}}{}^{\bar{a}}\dot{\psi}^{b}+
\big[\alpha_{3}\alpha_{9}+\alpha_{1}i\big]
\dot{\overline{\psi}}{}^{\bar{a}}\psi^{b}+
\big[\alpha_{3}\alpha_{9}-\alpha_{1}i\big]
\overline{\psi}{}^{\bar{a}}\dot{\psi}^{b}=0,\label{eq20}
\end{gather}
where
\begin{gather*}
\dot{\Omega}[\psi,G]^{b\bar{a}d\bar{c}} =
\alpha_{8}\big(\dot{\overline{\psi}}{}^{\bar{a}}\psi^{b}
\overline{\psi}{}^{\bar{c}}\psi^{d}+\overline{\psi}{}^{\bar{a}}\dot{\psi}^{b}
\overline{\psi}{}^{\bar{c}}\psi^{d}+\overline{\psi}{}^{\bar{a}}\psi^{b}
\dot{\overline{\psi}}{}^{\bar{c}}\psi^{d}+\overline{\psi}{}^{\bar{a}}\psi^{b}
\overline{\psi}{}^{\bar{c}}\dot{\psi}^{d}\big)
\nonumber\\
\phantom{\dot{\Omega}[\psi,G]^{b\bar{a}d\bar{c}}=}{} +
\alpha_{6}\alpha_{9}\big(\big[\dot{\overline{\psi}}{}^{\bar{a}}\psi^{d}+
\overline{\psi}{}^{\bar{a}}\dot{\psi}^{d}\big]\big[G^{b\bar{c}}+
\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{b}\big]+
\big[\dot{\overline{\psi}}{}^{\bar{c}}\psi^{b}+
\overline{\psi}{}^{\bar{c}}\dot{\psi}^{b}\big]\big[G^{d\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{d}\big]\big)
\nonumber\\
\phantom{\dot{\Omega}[\psi,G]^{b\bar{a}d\bar{c}}=}{} +
\alpha_{7}\alpha_{9}\big(\big[\dot{\overline{\psi}}{}^{\bar{a}}\psi^{b}+
\overline{\psi}{}^{\bar{a}}\dot{\psi}^{b}\big]\big[G^{d\bar{c}}+
\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{d}\big]+
\big[\dot{\overline{\psi}}{}^{\bar{c}}\psi^{d}+
\overline{\psi}{}^{\bar{c}}\dot{\psi}^{d}\big]\big[G^{b\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{b}\big]\big)
\nonumber\\
\phantom{\dot{\Omega}[\psi,G]^{b\bar{a}d\bar{c}}=}{} -
\alpha_{6}\big[G^{d\bar{e}}G^{f\bar{a}}\big(G^{b\bar{c}}+
\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{b}\big)+
G^{b\bar{e}}G^{f\bar{c}}\big(G^{d\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{d}\big)\big]\dot{G}_{\bar{e}f}
\nonumber\\
\phantom{\dot{\Omega}[\psi,G]^{b\bar{a}d\bar{c}}=}{} -
\alpha_{7}\big[G^{b\bar{e}}G^{f\bar{a}}\big(G^{d\bar{c}}+
\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{d}\big)+
G^{d\bar{e}}G^{f\bar{c}}\big(G^{b\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{b}\big)\big]\dot{G}_{\bar{e}f}
\end{gather*}
\section{Towards the canonical formalism}
The Legendre transformations leads us to the following canonical variables
\begin{gather}\label{eq01}
\pi_{b}=\frac{\partial L}{\partial \dot{\psi}^{b}}=
\alpha_{2}G_{\bar{a}b}\dot{\overline{\psi}}{}^{\bar{a}}+
\alpha_{1}iG_{\bar{a}b}\overline{\psi}{}^{\bar{a}},\qquad
\overline{\pi}_{\bar{a}}=
\frac{\partial L}{\partial \dot{\overline{\psi}}{}^{\bar{a}}}=
\alpha_{2}G_{\bar{a}b}\dot{\psi}{}^{b}-
\alpha_{1}iG_{\bar{a}b}\psi^{b},
\\
\label{eq02}
\pi^{\bar{a}b}=\frac{\partial L}{\partial \dot{G}_{\bar{a}b}}=
\alpha_{3}\big[G^{b\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{b}\big]+
2\Omega[\psi,G]^{b\bar{a}d\bar{c}}\dot{G}_{\bar{c}d}.
\end{gather}
The energy of our $n$-level Hamiltonian system is as follows
\begin{gather*}
E=\dot{\overline{\psi}}{}^{\bar{a}}\frac{\partial L}{\partial
\dot{\overline{\psi}}{}^{\bar{a}}}+\dot{\psi}^{b}\frac{\partial L}{\partial\dot{\psi}^{b}}+
\dot{G}_{\bar{a}b}\frac{\partial L}{\partial \dot{G}_{\bar{a}b}}-L\nonumber\\
\phantom{E}{} =
\alpha_{2}G_{\bar{a}b}\dot{\overline{\psi}}{}^{\bar{a}}
\dot{\psi}^{b}-
\left(\alpha_{4}
G_{\bar{a}b}+\alpha_{5}
H_{\bar{a}b}\right)\overline{\psi}{}^{\bar{a}}\psi^{b}
+\Omega[\psi,G]^{\bar{a}b\bar{c}d}
\dot{G}_{\bar{a}b}\dot{G}_{\bar{c}d}
+\varkappa\big(
G_{\bar{a}b}\overline{\psi}{}^{\bar{a}}\psi^{b}\big)^{2}
\end{gather*}
Inverting the expressions (\ref{eq01}), (\ref{eq02}) we obtain that
\begin{gather*
\dot{\overline{\psi}}{}^{\bar{a}}=
\frac{1}{\alpha_{2}}G^{b\bar{a}}\pi_{b}-
\frac{\alpha_{1}}{\alpha_{2}}i\overline{\psi}{}^{\bar{a}},\qquad
\dot{\psi}{}^{b}=\frac{1}{\alpha_{2}}G^{b\bar{a}}\overline{\pi}_{\bar{a}}+
\frac{\alpha_{1}}{\alpha_{2}}i\psi^{b},
\\
\dot{G}_{\bar{a}b}=
\frac{1}{2}\Omega[\psi,G]^{-1}_{\bar{a}b\bar{c}d}
\big(\pi^{\bar{c}d}-\alpha_{3}
\big[G^{d\bar{c}}+
\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{d}\big]\big),
\end{gather*}
where
\begin{gather*
\Omega[\psi,G]^{-1}_{\bar{a}b\bar{c}d}=
\Lambda[\psi,G]^{-1}_{\bar{a}b\bar{c}d}-
\frac{\alpha_{8}}{1+\alpha_{8}\theta_{2}[\psi,G]}
\Lambda[\psi,G]^{-1}_{\bar{a}b\bar{e}f}
\overline{\psi}{}^{\bar{e}}\psi^{f}
\Lambda[\psi,G]^{-1}_{\bar{c}d\bar{g}h}
\overline{\psi}{}^{\bar{g}}\psi^{h},\\
\Lambda[\psi,G]^{-1}_{\bar{a}b\bar{c}d}
=\frac{1}{\alpha_{6}}\lambda[\psi,G]^{-1}_{\bar{a}d}
\lambda[\psi,G]^{-1}_{\bar{c}b}-\frac{\alpha_{7}}{\alpha_{6}\left(
\alpha_{6}+n\alpha_{7}\right)}
\lambda[\psi,G]^{-1}_{\bar{a}b}\lambda[\psi,G]^{-1}_{\bar{c}d},\\
\lambda[\psi,G]^{-1}_{\bar{a}b}=
G_{\bar{a}b}-\frac{\alpha_{9}}{1+\alpha_{9}\theta_{1}[\psi,G]}
G_{\bar{a}d}G_{\bar{c}b}\overline{\psi}{}^{\bar{c}}\psi^{d},\\
\theta_{2}[\psi,G]=\Lambda[\psi,G]^{-1}_{\bar{a}b\bar{c}d}
\overline{\psi}{}^{\bar{a}}\psi^{b}
\overline{\psi}{}^{\bar{c}}\psi^{d}=\frac{
\alpha_{6}+\left(n-1\right)\alpha_{7}}{
\alpha_{6}\left(\alpha_{6}+n\alpha_{7}\right)}
\left(\frac{\theta_{1}[\psi,G]}{1+\alpha_{9}\theta_{1}[\psi,G]}\right)^{2},\\
\theta_{1}[\psi,G]= G_{\bar{a}b}\overline{\psi}{}^{\bar{a}}\psi^{b},
\end{gather*}
and then the Hamiltonian has the following form
\begin{gather*}
H=\frac{1}{\alpha_{2}}G^{b\bar{a}}\overline{\pi}_{\bar{a}}\pi_{b}
+\frac{\alpha_{1}}{\alpha_{2}}i\big(\psi^{b}\pi_{\psi b}-
\overline{\psi}{}^{\bar{a}}\overline{\pi}_{\bar{a}}\big)
-\left[\left(\alpha_{4}-\frac{\alpha^{2}_{1}}{\alpha_{2}}\right)G_{\bar{a}b}
+\alpha_{5}H_{\bar{a}b}\right]
\overline{\psi}{}^{\bar{a}}\psi^{b}\nonumber\\
\phantom{H=}{} +\frac{1}{4}\Omega[\psi,G]^{-1}_{\bar{a}b\bar{c}d}\pi^{\bar{a}b}\pi^{\bar{c}d}
-\frac{\alpha_{3}}{2}\Omega[\psi,G]^{-1}_{\bar{a}b\bar{c}d}\big[G^{b\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{b}\big]\pi^{\bar{c}d}\nonumber\\
\phantom{H=}{} +\frac{\alpha^{2}_{3}}{4}
\Omega[\psi,G]^{-1}_{\bar{a}b\bar{c}d}\big[G^{b\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{b}\big]
\big[G^{d\bar{c}}+
\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{d}\big]
+\varkappa\big(
G_{\bar{a}b}\overline{\psi}{}^{\bar{a}}\psi^{b}\big)^{2}
\end{gather*}
\section{Special cases}
\subsection[Pure dynamics for $G$]{Pure dynamics for $\boldsymbol{G}$}
First of all, if we consider the pure dynamics of scalar product $G$ while the wave function
$\psi$ is f\/ixed, then from (\ref{eq20}) we obtain the following equations of motion
\begin{gather}
\Omega[\psi,G]^{b\bar{a}d\bar{c}}\ddot{G}_{\bar{c}d}=
\left(\frac{\alpha_{4}}{2}-\varkappa \theta_{1}[\psi,G]\right)
\overline{\psi}{}^{\bar{a}}\psi^{b}+
\alpha_{7}\theta_{3}[\psi,G]
\big(G^{b\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{b}\big)
\nonumber\\
\phantom{\Omega[\psi,G]^{b\bar{a}d\bar{c}}\ddot{G}_{\bar{c}d}=}{} +
\alpha_{6}\dot{G}_{\bar{c}d}\dot{G}_{\bar{e}f}
\big(\gamma[\psi,G]^{b\bar{e}f\bar{c}d\bar{a}}
+\gamma[\psi,G]^{f\bar{a}d\bar{e}b\bar{c}}
-\gamma[\psi,G]^{b\bar{e}d\bar{a}f\bar{c}}\big),\label{eq25b}
\end{gather}
where
\begin{gather*
\theta_{3}[\psi,G]=
G^{d\bar{e}}G^{f\bar{c}}\dot{G}_{\bar{c}d}\dot{G}_{\bar{e}f},\qquad
\gamma[\psi,G]^{f\bar{e}d\bar{c}b\bar{a}}=
G^{f\bar{e}}G^{d\bar{c}}\big(G^{b\bar{a}}+
\alpha_{9}\overline{\psi}{}^{\bar{a}}\psi^{b}\big).
\end{gather*}
If we additionally suppose that $\alpha_{4}=\alpha_{8}=\alpha_{9}=\varkappa=0$, then (\ref{eq25b}) simplif\/ies signif\/icantly
\begin{gather*
\big(\alpha_{6}G^{b\bar{c}}G^{d\bar{a}}+
\alpha_{7}G^{b\bar{a}}G^{d\bar{c}}\big)
\big(\ddot{G}_{\bar{c}d}-
\dot{G}_{\bar{c}f}G^{f\bar{e}}\dot{G}_{\bar{e}d}\big)=0.
\end{gather*}
Hence, the pure dynamics of the scalar product is described by the following equations
\begin{gather}\label{eq25}
\ddot{G}_{\bar{a}b}-\dot{G}_{\bar{a}d}G^{d\bar{c}}\dot{G}_{\bar{c}b}=0.
\end{gather}
Let us now demand that $\dot{G}G^{-1}$ is equal to some constant value $E$, i.e., $\dot{G}=EG$, then
\begin{gather*
\ddot{G}=E\dot{G}=E^{2}G
\end{gather*}
and
\begin{gather*
\dot{G}G^{-1}\dot{G}=EGG^{-1}EG=E^{2}G,
\end{gather*}
therefore our equations of motion (\ref{eq25}) are fulf\/illed automatically and the solution is
as follows
\begin{gather*
G(t)_{\bar{a}b}=\left(\exp(Et)\right)^{\bar{c}}{}_{\bar{a}}G_{0}{}_{\bar{c}b}.
\end{gather*}
Similarly if we demand that $G^{-1}\dot{G}$ is equal to some other constant $E^{\prime}$,
i.e., $\dot{G}=GE^{\prime}$,
\begin{gather*
\ddot{G}=\dot{G}E^{\prime 2}=GE^{\prime 2},
\\
\dot{G}G^{-1}\dot{G}=GE^{\prime}G^{-1}GE^{\prime}=GE^{\prime 2},
\end{gather*}
then the equations of motion are also fulf\/illed and the solution is as follows
\begin{gather*
G(t)_{\bar{a}b}=G_{0}{}_{\bar{a}d}\left(\exp(E^{\prime}t)\right)^{d}{}_{b}.
\end{gather*}
The connection between these two dif\/ferent constants $E$ and $E^{\prime}$ is written below
\begin{gather*
\dot{G}(0)=\dot{G}_{0}=G_{0}E^{\prime}=EG_{0}.
\end{gather*}
\subsection[Usual and first-order modified Schr\"{o}dinger equations]{Usual and f\/irst-order modif\/ied Schr\"{o}dinger equations}
The second interesting special case is obtained when we suppose that the scalar product $G$ is
f\/ixed, i.e., the equations of motion are as follows
\begin{gather}\label{eq33}
\alpha_{2}\ddot{\psi}^{a}-2\alpha_{1}i\dot{\psi}^{a}+
\left(2\varkappa\theta_{1}\left[\psi,G\right]-
\alpha_{4}\right)\psi^{a}-\alpha_{5}H^{a}{}_{b}\psi^{b}=0.
\end{gather}
Then if we also take all constants of model to be equal to $0$ except of the following ones
\begin{gather*
\alpha_{1}=\frac{\hbar}{2},\qquad \alpha_{5}=-1,
\end{gather*}
we end up with the well-known usual Schr\"{o}dinger equation
\begin{gather*
i\hbar\dot{\psi}^{a}=H^{a}{}_{b}\psi^{b}.
\end{gather*}
Its f\/irst-order modif\/ied version is obtained when we suppose that $G$ is a dynamical variable
and~$\alpha_{2}$ is equal to $0$, i.e.,
\begin{gather}
i\hbar\dot{\psi}^{a} = H^{a}{}_{b}\psi^{b}-\left[\frac{i\hbar}{2}+\alpha_{3}\alpha_{9}\right]
G^{a\bar{c}}\dot{G}_{\bar{c}b}\psi^{b}+
\left(2\varkappa\theta_{1}\left[\psi,G\right]-
\alpha_{4}\right)\psi^{a}
\nonumber\\
\phantom{i\hbar\dot{\psi}^{a} =}{}-
2\alpha_{8}G^{a\bar{c}}\dot{G}_{\bar{c}b}\psi^{b}\dot{G}_{\bar{e}d}
\overline{\psi}{}^{\bar{e}}\psi^{d} -
2\alpha_{9}G^{a\bar{c}}\big(\alpha_{6}\dot{G}_{\bar{c}d}\dot{G}_{\bar{e}b}+
\alpha_{7}\dot{G}_{\bar{c}b}\dot{G}_{\bar{e}d}\big)
\psi^{b}\big(G^{d\bar{e}}+
\alpha_{9}\overline{\psi}{}^{\bar{e}}\psi^{d}\big),\!\!\!\label{eq37}\\
2\Omega[\psi,G]^{b\bar{a}d\bar{c}}\ddot{G}_{\bar{c}d}
=\left[\frac{i\hbar}{2}-\alpha_{3}\alpha_{9}\right]
\overline{\psi}{}^{\bar{a}}\dot{\psi}^{b}-
\left[\frac{i\hbar}{2}+\alpha_{3}\alpha_{9}\right]
\dot{\overline{\psi}}{}^{\bar{a}}\psi^{b}\nonumber\\
\phantom{2\Omega[\psi,G]^{b\bar{a}d\bar{c}}\ddot{G}_{\bar{c}d}=}{}-
2G^{d\bar{a}}\big[\alpha_{6}G^{b\bar{e}}\big(G^{f\bar{c}}+
\alpha_{9}\overline{\psi}{}^{\bar{c}}\psi^{f}\big)+
\alpha_{7}G^{b\bar{c}}\big(G^{f\bar{e}}+
\alpha_{9}\overline{\psi}{}^{\bar{e}}\psi^{f}\big)
\big]\dot{G}_{\bar{c}d}\dot{G}_{\bar{e}f}\nonumber\\
\phantom{2\Omega[\psi,G]^{b\bar{a}d\bar{c}}\ddot{G}_{\bar{c}d}=}{}-
\left(2\varkappa\theta_{1}\left[\psi,G\right]-
\alpha_{4}\right)\overline{\psi}{}^{\bar{a}}\psi^{b}-
2\dot{\Omega}[\psi,G]^{b\bar{a}d\bar{c}}\dot{G}_{\bar{c}d}.\nonumber
\end{gather}
We can rewrite (\ref{eq37}) in the following form
\begin{gather*
i\hbar\dot{\psi}^{a}=H_{\rm ef\/f}{}^{a}{}_{b}\psi^{b},
\end{gather*}
where the ef\/fective Hamilton operator is given as follows:
\begin{gather*}
H_{\rm ef\/f}{}^{a}{}_{b}
=
H^{a}{}_{b}-\left[\frac{i\hbar}{2}+\alpha_{3}\alpha_{9}\right]
G^{a\bar{c}}\dot{G}_{\bar{c}b}+
\left(2\varkappa\theta_{1}\left[\psi,G\right]-
\alpha_{4}\right)\delta^{a}{}_{b}-
2\alpha_{8}G^{a\bar{c}}\dot{G}_{\bar{c}b}\dot{G}_{\bar{e}d}
\overline{\psi}{}^{\bar{e}}\psi^{d}\nonumber\\
\phantom{H_{\rm ef\/f}{}^{a}{}_{b}=}{} -
2\alpha_{9}G^{a\bar{c}}\big(\alpha_{6}\dot{G}_{\bar{c}d}\dot{G}_{\bar{e}b}+
\alpha_{7}\dot{G}_{\bar{c}b}\dot{G}_{\bar{e}d}\big)\big(G^{d\bar{e}}+
\alpha_{9}\overline{\psi}{}^{\bar{e}}\psi^{d}\big)
\end{gather*}
\subsection[Second-order modified Schr\"{o}dinger equation]{Second-order modif\/ied Schr\"{o}dinger equation}
The idea of introducing the second time derivative of the wave function into the usual Schr\"{o}dinger equation as a correction term is not completely new and has been already discussed in the literature. The similar problems were studied a long time ago by A.~Barut and more recently have been re-investigated by V.V.~Dvoeglazov, S.~Kruglov, J.P.~Vigier and others (see, e.g.,~\cite{VVD,SIK} and references therein; the authors of this article are grateful to one of the referee for pointing them to above-mentioned references). Among others there is also an interesting article where the authors used the analogy between the Schr\"{o}dinger and Fourier equations~\cite{JMK}.
The quantum Fourier equation which describes the heat (mass) dif\/fusion on the atomic level has
the following form
\begin{gather*
\frac{\partial T}{\partial t}=\frac{\hbar}{m}\nabla^{2}T.
\end{gather*}
If we make the substitutions $t\rightarrow it/2$ and $T\rightarrow \psi$, then we end up with
the free Schr\"{o}dinger equation
\begin{gather*
i\hbar\frac{\partial \psi}{\partial t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi.
\end{gather*}
The complete Schr\"{o}dinger equation with the potential term $V$ after the reverse
substitutions $t\rightarrow -2it$ and $\psi\rightarrow T$ gives us the parabolic quantum
Fokker--Planck equation, which describes the quantum heat transport for $\triangle t>\tau$, where $\tau=\hbar/m\alpha^{2}c^{2}\sim 10^{-17}$ sec and $c\tau\sim 1$~nm, i.e.,
\begin{gather*
\frac{\partial T}{\partial t}=\frac{\hbar}{m}\nabla^{2}T-\frac{2V}{\hbar}T.
\end{gather*}
For ultrashort time processes when $\triangle t<\tau$ one obtains the generalized quantum
hyperbolic heat transport equation
\begin{gather*
\tau\frac{\partial^{2} T}{\partial t^{2}}+\frac{\partial T}{\partial
t}=\frac{\hbar}{m}\nabla^{2}T-\frac{2V}{\hbar}T
\end{gather*}
(its structure and solutions for ultrashort thermal processes were investigated in
\cite{JMK_mono}) which leads us to the second-order modif\/ied Schr\"{o}dinger equation
\begin{gather}\label{eq56}
2\tau\hbar\frac{\partial^{2}\psi}{\partial t^{2}}+i\hbar\frac{\partial \psi}{\partial
t}=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi+V\psi
\end{gather}
in which the additional term describes the interaction of electrons with surrounding
space-time f\/illed with virtual positron-electron pairs. It is easy to see that (\ref{eq56}) is
analogous to (\ref{eq33}) if we suppose that
\begin{gather*
\alpha_{1}=\frac{\hbar}{2},\qquad \alpha_{2}=-2\tau\hbar,\qquad \alpha_{4}=0,\qquad
\alpha_{5}=-1,\qquad \varkappa=0.
\end{gather*}
\section[Conservation laws and GL$(n,\mathbb{C})$-invariance]{Conservation laws and GL$\boldsymbol{(n,\mathbb{C})}$-invariance}
So, if we investigate the invariance of our general Lagrangian (\ref{eq8})
under the group GL$(n,\mathbb{C})$ and consider some one-parameter group of transformations
\begin{gather*
\left\{\exp\left(A\tau\right):\tau\in\mathbb{R}\right\},\qquad A\in {\rm L}(n,\mathbb{C}),
\end{gather*}
then the inf\/initesimal transformations rules for $\psi$ and $G$ are as follows
\begin{gather*}
\psi^{a} \mapsto L^{a}{}_{b}\psi^{b},\qquad
G^{a\bar{c}} \mapsto L^{a}{}_{b}\overline{L}{}^{\bar{c}}{}_{\bar{e}}G^{b\bar{e}},
\qquad
G_{\bar{a}b} \mapsto G_{\bar{c}d}\overline{L^{-1}}{}^{\bar{c}}{}_{\bar{a}}L^{-1d}{}_{b},
\end{gather*}
where
\begin{gather*
L^{a}{}_{b}=\delta^{a}{}_{b}+\epsilon A^{a}{}_{b},\qquad L^{-1a}{}_{b}\approx
\delta^{a}{}_{b}-\epsilon A^{a}{}_{b},\qquad \epsilon\approx 0.
\end{gather*}
So leaving only the f\/irst-order terms with respect to $\epsilon$ we obtain that the variations
of $\psi$ and $G$ are as follows
\begin{gather*}
\delta\psi^{a}=\epsilon A^{a}{}_{b}\psi^{b},\qquad \delta\overline{\psi}{}^{\bar{a}}=\epsilon
\overline{A}{}^{\bar{a}}{}_{\bar{c}}\overline{\psi}{}^{\bar{c}},\\
\delta G^{a\bar{c}}=\epsilon\big(A^{a}{}_{b}G^{b\bar{c}}+
\overline{A}{}^{\bar{c}}{}_{\bar{e}}G^{a\bar{e}}\big),\qquad \delta
G_{\bar{a}b}=-\epsilon\big(G_{\bar{c}b}\overline{A}{}^{\bar{c}}{}_{\bar{a}}+
G_{\bar{a}d}A^{d}{}_{b}\big)
\end{gather*}
then
\begin{gather}\label{eq44}
\frac{1}{\epsilon}\left(\frac{\partial L}{\partial \dot{\overline{\psi}}{}^{\bar{a}}}\delta
\overline{\psi}{}^{\bar{a}}+\frac{\partial L}{\partial
\dot{\psi}^{b}}\delta\psi^{b}\right)=G_{\bar{a}b}
\big(\alpha_{2}\dot{\overline{\psi}}{}^{\bar{a}}+
\alpha_{1}i\overline{\psi}{}^{\bar{a}}\big)A^{b}{}_{d}\psi^{d}
+G_{\bar{a}b}\big(\alpha_{2}\dot{\psi}^{b}-\alpha_{1}i\psi^{b}\big)
\overline{A}{}^{\bar{a}}{}_{\bar{c}}\overline{\psi}{}^{\bar{c}}
\end{gather}
and
\begin{gather}
\frac{1}{\epsilon}\frac{\partial L}{\partial\dot{G}_{\bar{a}b}}\delta
G_{\bar{a}b}=-\big[\alpha_{3}\big(\delta^{b}{}_{f}+
\alpha_{9}G_{\bar{a}f}\overline{\psi}{}^{\bar{a}}\psi^{b}\big)
+2\Omega[\psi,G]^{b\bar{a}d\bar{c}}
G_{\bar{a}f}\dot{G}_{\bar{c}d}\big]A^{f}{}_{b}\nonumber\\
\hphantom{\frac{1}{\epsilon}\frac{\partial L}{\partial\dot{G}_{\bar{a}b}}\delta
G_{\bar{a}b}=}{}
-\big[\alpha_{3}\big(\delta^{\bar{a}}{}_{\bar{e}}+
\alpha_{9}G_{\bar{e}b}\overline{\psi}{}^{\bar{a}}\psi^{b}\big)
+2\Omega [\psi,G ]^{b\bar{a}d\bar{c}}
G_{\bar{e}b}\dot{G}_{\bar{c}d}\big]\overline{A}{}^{\bar{e}}{}_{\bar{a}}.\label{eq45}
\end{gather}
If we consider some f\/ixed scalar product $G_{0}$ and take the $G_{0}$-hermitian $A$'s, then
\begin{gather*
A^{a}{}_{b}=G_{0}{}^{a\bar{c}}\widetilde{A}_{\bar{c}b},\qquad
\overline{A}^{\bar{a}}{}_{\bar{c}}=\widetilde{A}_{\bar{c}b}G_{0}^{b\bar{a}},\qquad \widetilde{A}{}^{\dag}=\widetilde{A},
\end{gather*}
and therefore the expressions (\ref{eq44}) and (\ref{eq45}) are written together in the matrix form as follows
\begin{gather*
\mathcal{J}\left(A\right)={\rm Tr}\big(V\widetilde{A}\big),
\end{gather*}
where the hermitian tensor $V$ describing the system of conserved physical quantities is given as follows
\begin{gather*}
V=\alpha_{2}\big(\psi\dot{\psi}^{\dag}GG^{-1}_{0}+G^{-1}_{0}G\dot{\psi}\psi^{\dag}\big)
+\big(\alpha_{1}i-\alpha_{3}\alpha_{9}\big)\psi\psi^{\dag}GG^{-1}_{0}\nonumber\\
\phantom{V=}{} -
\big(\alpha_{1}i+\alpha_{3}\alpha_{9}\big)G^{-1}_{0}G\psi\psi^{\dag}
-2\alpha_{3}G^{-1}_{0}
-2\big(G^{-1}_{0}G\omega [\psi,G ]+\omega [\psi,G ] GG^{-1}_{0}\big)
\end{gather*}
where
\begin{gather*
\omega\left[\psi,G\right]^{b\bar{a}}=
\Omega\left[\psi,G\right]^{b\bar{a}d\bar{c}}\dot{G}_{\bar{c}d}.
\end{gather*}
Similarly for the $G_{0}$-antihermitian $A$'s, i.e., when $\widetilde{A}^{\dag}=-\widetilde{A}$, we
obtain another hermitian tensor~$W$ as a conserved value
\begin{gather*
\mathcal{J}\left(A\right)={\rm Tr}\big(iW\widetilde{A}\big),
\end{gather*}
where
\begin{gather*}
iW = \alpha_{2}\big(\psi\dot{\psi}^{\dag}GG^{-1}_{0}-
G^{-1}_{0}G\dot{\psi}\psi^{\dag}\big)
+ (\alpha_{1}i-\alpha_{3}\alpha_{9} )\psi\psi^{\dag}GG^{-1}_{0}
\nonumber\\
\phantom{iW =}{} +
(\alpha_{1}i+\alpha_{3}\alpha_{9} )G^{-1}_{0}G\psi\psi^{\dag}
+2\big(G^{-1}_{0}G\omega [\psi,G ]-\omega [\psi,G ] GG^{-1}_{0}\big)
\end{gather*}
\section{Final remarks}
This is a very preliminary, simplif\/ied f\/inite-level model. It is still not clear whether it is consistent with the usual statistical interpretation of quantum mechanics. This model is thought on as a~step towards discussing the wave equations obtained by combining the f\/irst and second time derivatives. There are some indications that such a combination might be reasonable. Within a rather dif\/ferent context (motivated by the idea of conformal invariance) we studied such a~problem in \cite{GF,KGD} where the wave equations with the superposition of Dirac and d'Alembert operators were considered.
\subsection*{Acknowledgements}
This paper contains results obtained within the framework of the research project 501 018 32/1992 f\/inanced from the Scientif\/ic Research Support Fund
in 2007--2010. The authors are greatly indebted to the Ministry of Science and Higher Education for this f\/inancial support.
The authors are also very grateful to the referees for their valuable remarks and comments concerning this article.
\pdfbookmark[1]{References}{ref}
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 8,216
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package com.cognifide.slice.mapper.strategy.impl;
import java.lang.reflect.Field;
import com.cognifide.slice.mapper.annotation.IgnoreProperty;
import com.cognifide.slice.mapper.strategy.MapperStrategy;
/**
* MapperStrategy which allows all fields of a class except for the fields annotated by {@link IgnoreProperty}
* to be mapped.
*/
public class AllFieldMapperStrategy implements MapperStrategy {
@Override
public boolean shouldFieldBeMapped(Field field) {
return !field.isAnnotationPresent(IgnoreProperty.class);
}
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 1,277
|
Michał Sita
Michał Sita /
Củ Chi /
History of Poland /
Zoological archive /
Iraq /
Scenes of no importance /
Power station /
Karabakh /
Wielkopolska /
Fertile grounds to the East of first hills of Smaller Caucasus never belonged to Armenians. Similarly to other parts of Nagorno Karabakh Republic, plains like this one, along Askeran – Martakert road, were inhabited by Azeris. Although today Armenians are in control calling it a 'buffer zone', Azeris describe it with the term 'occupied territories'. In 1993, after a series of military gains over Azerbaijan's army, Armenians did not hold their men at the foothills where they comprised an ethnic majority, but moved forward. They cleared a stripe of land of its Muslim inhabitants, forcing them to flee and leave their homes, fields and cities behind. An area several kilometers wide, over a hundred kilometers long, is comprised of ruins of homes, villages, and monstrous ghost towns. It is this moonlike landscape where young men spend their compulsory military service pointing guns at their Azeri counterparts. For twenty years this situation has been stable, with a gun firing on one side or another form time to time.
Karabakh security forces try to keep these landscapes along Azerbaijan border out of reach of visitors, not only because of the proximity of unpredictable front line, but also because the lands conquered by Armenians give clear idea on how recent is the history of ethnic cleansing committed in Karabakh at a scale that is hard to imagine. They remind of a fact that Azeris have very valid reasons to try, one day, to recapture ruins of their cities; Agdam and Fizuli.
In Fizuli, Armin explains, what was valuable was everything that could be extracted and sold. The first came home appliances, after these; windows, doors and pipes. Today, when Fizuli infrastructure has been reduced to rocks, these rocks are used by construction companies. Armin works in one of them, collecting, transporting, and utilizing stones at construction sites across Karabakh.
At the same place, after an unsuccessful hare hunt, several men from neighboring Togh village party by the banks of defunct hydropower plant reservoir, surrounded by remains of Fizuli city. A small airport serving as regional transport hub before the war, and Soviet monuments, complete the scene. One of the men explains in broken Russian; 'what you see are terrains we bravely grabbed from Azerbaijan…' corrected by a second man; 'not grabbed, but conquered, or recouped in retaliation for Azeri atrocities, committed at Armenians on incomparable scale. The point is that these are lands we deserved and earned, and that we will defend'.
Crowds of drunk fathers roam the streets of Hadrut, celebrating an oath marking the beginning of compulsory military service of their sons. For two years they will be going grey while their kids will sit in trenches pointing guns at Azeri side. In one of improvised cafes, serving usually as barber shop or butchery, family parties in a joyless manner. Toast bidding starts, and since Armenian toasts are elaborate, they give room to share views on Azeris, Turks, their mothers, sisters and anyone willing to treat them as human beings. Paranoid politics talks, uncles' advice on military adventures make the atmosphere dense enough to prevent Armenian hospitality from tolerating an outsider any more. In another similar place a family has much less of an idea how to bear the situation, sitting silent, waiting for an evening to come that will allow them to return to Yerevan, or anywhere in Armenia in other cases. Those who could not afford hiring a driver stay in cheap hotels, sustained in Hadrut thanks to proximity of the military base, largest and the most important in Southern Karabakh.
Arthur's mother works in a grocery store earning less that 150 $. Prices in that store, due to an isolation of the region and large distance lorries have to travel, are higher than in Armenia. That means it's impossible to earn a living doing a simple job, a hairdresser, a butcher, or a cook. Every usual occupation makes sense only if there is a member of the family abroad, supporting everyday expenses back home. Reasonable salaries are in the army only.
Arthur picks me up from the store, shares homemade vine and introduces me to his friends; a corporal, a contract soldier, policeman, intelligence analyst. All of them after compulsory two years of service decided to stay in the forces. They unanimously agree, 9 out of ten of their friends do the same. There is no family in Chartar without at least one member in the military.
Karabakh army consists of approximately 25.000 men, 18% of the Republic population. If you add police, security service and remaining uniformed forces, the picture of a remote, militarized region drawn by young soldiers – seems reliable. Only in the army, in their opinion, one can afford to stay home with their family instead of emigrating to Russia.
Thermal springs in remote Northern parts of the Republic are a common tourist destination, especially in winter, when huge mosquitoes are not making it impossible to dive into a hot pool. Here, I was sure, I will meet people looking for a few hours rest, like myself, the ones like farm workers, emigrants to Russia, spending their holiday here. A little further to the West, in another area, in hot water I sit among sad special forces making a break in their border patrol, two policemen and a chief of local security services observing the whole scene from behind a car. I'm not allowed to photograph, since the view of military in a tub or a police car next to a grill could be considered a security breach. At that moment I am already well aware of strategic significance of raw material transfer line in the form of home gas pipeline, of military equipment disguised as an old Lada or military personnel embodied in a father saying goodbye to his son, all of which cannot be depicted. It is allowed to spend the night in there, but with caution. Azeri saboteurs can be encountered, just like the ones last year. They asked, sniffed, looked for something. And you, yourself? Are you sure you've not just visited Baku? That you're rally not Turkish? That you don't greet your mother As-Salaam Alaikum? Maybe you'll show your documents anyway?
I selected these scenes of daily life in Nagorno Karabakh Republic because I believe they illustrate precisley the uncompromising routine of daily life in this area, in a moment preceding a large scale outbreak of violence that took place in early April 2016, several weeks after this documentary was completed.
Mountainous region of Nagorno Karabakh has been ruled autonomously since 1994, when Armenians won de facto independence for these areas in a bitter war against Azerbaijan. Armenian support has sustained the breakaway republic economically and militarily ever since. An unsettled conflict is considered ready to blow any moment, like it happened this year.
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
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| 4,084
|
И́рис по́здний, или каса́тик поздний () — вид многолетних травянистых растений рода семейства .
Ботаническое описание
Луковичное многолетнее растение со стеблем высотой 40-60 см.
Луковицы яйцевидные, закутаны в основания листьев.
Нижние листья 30-60×2,6 мм, верхние короче и шире.
Цветков 2-3, голубовато-фиолетовых, с желтоватой серединой.
Плод — коробочка. Семена полукруглые, сжатые, желтоватые.
Цветёт во второй половине лета, позже других ирисов подрода Xiphium, которые цветут весной.
Распространение и экология
Эндемик гор юго-восточной Испании. Встречается в сосняках и на сухих лугах.
Таксономия
Вид Ирис поздний входит в род семейства порядка .
Примечания
Литература
Ссылки
Ирисовые
Эндемики Испании
Флора Европы
|
{
"redpajama_set_name": "RedPajamaWikipedia"
}
| 8,222
|
{"url":"http:\/\/math.soimeme.org\/~arunram\/Notes\/091207ftcProbSet.xhtml","text":"Problem Set - The Fundamental Theorem of Calculus\n\n## Problem Set - The Fundamental Theorem of Calculus\n\n What does $\u222b a b f x d x$ mean? How does one usually calculate Give an example which shows that this method does not always work. Why doesn't it? Give an example which shows that $\u222b a b f x d x$ is not always the true area under $f\\left(x\\right)$ between $a$ and $b$ even if $f\\left(x\\right)$ is contunuous between $a$ and $b$. What is the Fundamental Theorem of Calculus? Let $f\\left(x\\right)$ be a function which is continuous and let $A\\left(x\\right)$ be the area under $f\\left(x\\right)$ from $a$ to $x$. Compute the derivative of $A\\left(x\\right)$ by using limits. Why is the Fundamental Theorem of Calculus true? Explain carefully and thoroughly. Give an example which illustrates the Fundamental Theorem of Calculus. In order to do this, compute an area by summing up the areas of tiny boxes and then show that applying the Fundamental Theorem of Calculus gives the same result.","date":"2019-01-20 19:31:54","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 14, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8837946653366089, \"perplexity\": 52.8917916005724}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-04\/segments\/1547583730728.68\/warc\/CC-MAIN-20190120184253-20190120210253-00212.warc.gz\"}"}
| null | null |
\chapter{Elements of neutrino physics}
\label{App:StandMod}
\setlength{\epigraphwidth}{0.45\textwidth}
\epigraph{The only thing I'm not good at is modesty. Because I'm great at it.}{Gina Linetti, \emph{Brooklyn Nine-Nine} [S05E17]}
{
\hypersetup{linkcolor=black}
\minitoc
}
In this appendix, we summarize some useful results regarding the description of neutrinos and their interactions. We first discuss the Standard Model case, before giving the parameters that are commonly used to describe massive neutrino mixings.
\section{Neutrinos in the Standard Model of particle physics}
The Standard Model (SM) is a gauge theory based on the local symmetry group $\mathrm{SU(3)}_C \times \mathrm{SU(2)}_L \times \mathrm{U(1)}_Y$, where the subscripts $C$, $L$, $Y$ denote respectively colour, left-handed chirality and hypercharge. The interaction of neutrinos is determined by the electroweak part of the SM, based on the gauge group $\mathrm{SU(2)}_L \times \mathrm{U(1)}_Y$.
\subsection{Tools for the Clifford algebra}
We refer the reader to, e.g.~\cite{PeskinSchroeder,SchwartzQFT,Srednicki} for a detailed introduction to quantum field theory. Here, we simply recall the useful relations involving the $\gamma$ matrices, which as a reminder are defined such that
\begin{equation}
\label{eq:Clifford}
\{ \gamma^\mu, \gamma^\nu \} = 2 \eta^{\mu \nu} \, ,
\end{equation}
where $\eta^{\mu \nu}$ is the Minkowski metric (here taken with signature (+---), for consistency with the FLRW metric).
The left and right chiral parts of a Dirac spinor $\psi = \psi_L + \psi_R$ are obtained from the projectors
\begin{equation}
P_L \equiv \frac{1-\gamma^5}{2} \qquad , \qquad P_R \equiv \frac{1+ \gamma^5}{2} \, ,
\end{equation}
with the fifth gamma-matrix $\gamma^5 \equiv {\mathrm i} \gamma^0 \gamma^1 \gamma^2 \gamma^3$.
The useful identities read:
\begin{equation*}
\label{eq:trace_identities}
\boxed{ \begin{aligned}
\eta^{\mu \nu} \eta_{\mu \nu} &= 4 \\
\mathrm{tr} \left[ \gamma^\mu \gamma^\nu P_{L,R} \right] &= 2 \eta^{\mu \nu} \\
\mathrm{tr}\left[\gamma^\sigma \gamma^\mu \gamma^\lambda \gamma^\nu P_{L,R} \right] &= 2 \left(\eta^{\sigma \mu} \eta^{\lambda \nu} - \eta^{\sigma \lambda} \eta^{\mu \nu} + \eta^{\sigma \nu} \eta^{\mu \lambda} \right) \pm 2 i \epsilon^{\sigma \mu \lambda \nu} \\
\epsilon^{\mu \nu \rho \sigma} \epsilon_{\mu \nu \tau \lambda} &= - 2 ({\delta^\rho}_\tau {\delta^\sigma}_\lambda - {\delta^\rho}_\lambda {\delta^\sigma}_\tau) \\
\gamma^\mu \gamma^\nu \gamma^\rho \gamma_\mu &= 4 \eta^{\nu \rho} \\
\gamma^\mu \gamma^\nu \gamma^\rho \gamma^\sigma \gamma_\mu &= - 2 \gamma^\sigma \gamma^\rho \gamma^\nu
\end{aligned}}
\end{equation*}
Note that any specific choice of matrices satisfying the fundamental anticommutation relations~\eqref{eq:Clifford} constitutes a \emph{representation} of the $\gamma$ matrices. Two are often used:
\begin{itemize}
\item the Dirac basis, in which
\[ \gamma^0_D = \begin{pmatrix} \mathbb{1} & 0 \\ 0 & \mathbb{1} \end{pmatrix} \, , \quad \gamma^i_D = \begin{pmatrix} 0 & \sigma^i \\ - \sigma^i & 0 \end{pmatrix} \, , \quad \gamma^5_D = \begin{pmatrix} 0 & \mathbb{1} \\ \mathbb{1} & 0 \end{pmatrix} \, , \]
with $\sigma^i$ the Pauli matrices,
\item the Weyl (chiral) basis, for which one choice is \cite{PeskinSchroeder}
\[ \gamma^0_C = \begin{pmatrix} 0 & \mathbb{1} \\ \mathbb{1} & 0 \end{pmatrix} \, , \quad \gamma^i_C = \begin{pmatrix} 0 & \sigma^i \\ - \sigma^i & 0 \end{pmatrix} \, , \quad \gamma^5_C = \begin{pmatrix} - \mathbb{1} & 0 \\ 0 & \mathbb{1} \end{pmatrix} \, . \]
This is the choice consistent\footnote{By consistent, we mean "which makes clear": we can always define $\psi_{L,R}$ based on the actions of the left and right projectors on $\psi$, but they do not separate into two Weyl spinors.} with the decomposition of the 4-component spinor field $\psi(x)$ into the left-handed and right-handed two-component Weyl spinors:
\[ \psi_C = \begin{pmatrix} \psi_L \\ \psi_R \end{pmatrix} \, , \quad P_L \psi_C = \frac{1 - \gamma^5_C}{2} = \begin{pmatrix} \mathbb{1} & 0 \\ 0 & 0 \end{pmatrix} \psi_C = \psi_L \, . \]
\end{itemize}
\subsection{Neutrino interactions and Fermi theory}
\label{subsec:Hamiltonians}
The relevant two-body interactions correspond to Standard Model interactions involving neutrinos and antineutrinos. In the early universe, they interact through weak processes with electrons, positrons (also muons and antimuons) and other (anti)neutrinos. Therefore, we must take as interaction Hamiltonian \eqref{eq:defHint} the useful part of the SM Hamiltonian of weak interactions, that is given by
\begin{equation}
\label{eq:defHsm}
\hat{H}_{\mathrm{int}} = \hat{H}_{CC} + \hat{H}_{NC}^{\mathrm{mat}} + \hat{H}_{NC}^{\nu \nu} \, ,
\end{equation}
where we separate three contributions:
\begin{itemize}
\item the charged-current hamiltonian,
\begin{multline}
\label{eq:hcc_app}
\hat{H}_{CC} = 2 \sqrt{2} G_F m_W^2 \int{\ddp{1} \ddp{2} \ddp{3} \ddp{4}} \ (2\pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\ \times [\overline{\psi}_{\nu_e}(\vec{p}_1)\gamma_\mu P_L\psi_e(\vec{p}_4)] W^{\mu \nu}(\Delta) [\overline{\psi}_e(\vec{p}_2) \gamma_\nu P_L \psi_{\nu_e}(\vec{p}_3)] \, ,
\end{multline}
with $\psi(\vec{p}) = \sum_{h} \left[ \hat{a}(\vec{p},h) u^h(\vec{p})+ \hat{b}^\dagger(-\vec{p},h) v^h(-\vec{p}) \right]$ the Fourier transform of the quantum fields, and the gauge boson propagator
\begin{equation}
\label{eq:app_propagator}
W^{\mu \nu}(\Delta) = \frac{\eta^{\mu \nu} - \frac{\Delta^\mu \Delta^\nu}{m_W^2}}{m_W^2 - \Delta^2} \simeq \frac{\eta^{\mu \nu}}{m_W^2} + \frac{1}{m_W^2}\left(\frac{\Delta^2 \eta^{\mu \nu}}{m_W^2} - \frac{\Delta^\mu \Delta^\nu}{m_W^2}\right) \, .
\end{equation}
The lowest order in this expansion is the usual 4-Fermi effective theory. The momentum transfer is $\Delta = p_1-p_4$ for a $t$-channel ($\nu_e-e^-$ scattering), and $\Delta= p_1 + p_2$ for the $s$-channel ($\nu_e-e^+$).
\item the neutral-current interactions with the matter background (electrons and positrons, we would write the same term for $\mu^\pm$),
\begin{multline}
\label{eq:hncmat}
\hat{H}_{NC}^{\mathrm{mat}} = 2 \sqrt{2} G_F m_Z^2 \sum_{\alpha} \int{\ddp{1} \ddp{2} \ddp{3} \ddp{4}} \ (2\pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\ \times [\overline{\psi}_{\nu_\alpha}(\vec{p}_1)\gamma_\mu P_L\psi_{\nu_\alpha}(\vec{p}_3)] Z^{\mu \nu}(\Delta) [\overline{\psi}_e(\vec{p}_2) \gamma_\nu (g_L P_L + g_R P_R) \psi_e(\vec{p}_4)] \, ,
\end{multline}
where $Z^{\mu \nu}$ is identical to $W^{\mu \nu}$ with the replacement $m_W \to m_Z$. The neutral-current couplings are $g_L = -1/2 + \sin^2{\theta_W}$ and $g_R = \sin^2{\theta_W}$, where $\sin^2{\theta_W} \simeq 0.231$ is the weak-mixing angle.
\item the self-interactions of neutrinos,\footnote{To understand the different prefactor from $\hat{H}_{NC}^{\mathrm{mat}}$, start from the general neutral-current Hamiltonian:
\[
\hat{H}_{NC} = 2 \sqrt{2} G_F m_Z^2 \sum_{f,f'} \int{\cdots \ \left[\overline{\psi}_{f} \gamma_\mu (g_L^f P_L + g_R^f P_R) \psi_f\right]Z^{\mu \nu}(\Delta)\left[\overline{\psi}_{f'}\gamma_\nu (g_L^{f'} P_L + g_R^{f'} P_R) \psi_{f'}\right]} \]
Now the multiplicity of each term and the use of $g_L^\nu = 1/2$, $g_R^\nu=0$ lead to the Hamiltonians above.
}
\begin{multline}
\label{eq:hncself}
\hat{H}_{NC}^{\nu \nu} = \frac{G_F}{\sqrt{2}} m_Z^2 \sum_{\alpha, \beta} \int{\ddp{1} \ddp{2} \ddp{3} \ddp{4}} \ (2\pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\ \times [\overline{\psi}_{\nu_\alpha}(\vec{p}_1)\gamma_\mu P_L\psi_{\nu_\alpha}(\vec{p}_3)] Z^{\mu \nu}(\Delta) [\overline{\psi}_{\nu_\beta}(\vec{p}_2) \gamma_\nu P_L \psi_{\nu_\beta}(\vec{p}_4)] \, .
\end{multline}
\end{itemize}
In order to calculate the collision term, we will restrict to the low-energy 4-Fermi theory (cf. section~\ref{app:collision_term}), while the next-to-leading order must be used to obtain the relevant mean-field terms in the Quantum Kinetic Equations.
\section{Neutrino masses and mixings}
\label{subsec:Values_Mixing}
\paragraph{PMNS mixing matrix}
For all numerical calculations in this work, we employ the standard parameterization of the PMNS matrix which reads \cite{Gariazzo_2019,GiuntiKim,PDG}
\begin{equation}
\label{eq:PMNS}
U = R_{23} R_{13} R_{12} = \begin{pmatrix}
c_{12} c_{13} & s_{12} c_{13} & s_{13} \\
- s_{12}c_{23} - c_{12}s_{23}s_{13} & c_{12} c_{23} - s_{12}s_{23}s_{13} & s_{23} c_{13} \\
s_{12}s_{23} - c_{12}c_{23}s_{13} & -c_{12}s_{23} - s_{12}c_{23}s_{13} & c_{23} c_{13}
\end{pmatrix} \, ,
\end{equation}
with $c_{ij} = \cos{\theta_{ij}}$, $s_{ij}=\sin{\theta_{ij}}$ and $\theta_{ij}$ the mixing angles. $R_{ij}$ is the real rotation matrix of angle $\theta_{ij}$ in the $i$-$j$ plane, namely, $(R_{ij})^i_i = (R_{ij})^j_j = c_{ij}$, $(R_{ij})^k_k = 1$ where $k \neq i,j$, $(R_{ij})^i_j = - (R_{ij})^j_i = s_{ij}$ and the other components are zero. Note that we do not introduce here a CP-violating phase, postponing its treatment to specific sections of this thesis, namely~\ref{subsec:Decoupling_CP} and~\ref{SecDiracPhase}. We use the most recent values from the Particle Data Group \cite{PDG}:
\begin{align}
\left(\frac{\Delta m_{21}^2}{10^{-5} \, \mathrm{eV}^2},\frac{\Delta m_{31}^2}{10^{-3} \, \mathrm{eV}^2},s_{12}^2,s_{23}^2,s_{13}^2\right)_\mathrm{ NH} &= \left(7.53, 2.53, 0.307, 0.546, 0.0220 \right) \, , \label{ValuesStandard} \\
\left(\frac{\Delta m_{21}^2}{10^{-5} \, \mathrm{eV}^2},\frac{\Delta m_{31}^2}{10^{-3} \, \mathrm{eV}^2},s_{12}^2,s_{23}^2,s_{13}^2\right)_\mathrm{ IH} &= \left(7.53, -2.46, 0.307, 0.539, 0.0220 \right) \, ,
\end{align}
where $\Delta m_{ij}^2 \equiv m_i^2 - m_j^2$ is the difference of the squared masses of the mass eigenstates $i$ and $j$. The associated values of the mixing angles are $\theta_{12} = 0.587$, $\theta_{13}=0.149$ and $\theta_{23}=0.831$ in normal ordering, the only different value is $\theta_{23} = 0.824$ in inverted ordering.
For completeness, we also give the most recent values of the physical constants used \cite{PDG}: the Fermi constant $G_F = 1.1663787 \times 10^{-5} \, \mathrm{GeV^{-2}}$ and the gravitational constant $\mathcal{G} = 6.70883 \times 10^{-39} \, \mathrm{GeV^{-2}}$.
Note that we could also use the values from the global fit of neutrino oscillation data~\cite{deSalas_Mixing}, a choice made in~\cite{Bennett2021}. The results we get for the standard value $N_{\mathrm{eff}}$ are identical at the level of a few $10^{-6}$.
\paragraph{Parameterizations of $\bm{\mathrm{SU(2)}}$ and $\bm{\mathrm{SO(3)}}$} In chapter~\ref{chap:Asymmetry}, we study the case of two-flavour mixing, for which a vector representation of density matrices is possible. This allows for a more intuitive representation of their time evolution, namely in terms of precession. Let us precise here some definitions concerning the useful matrix groups $\mathrm{SU(2)}$ and $\mathrm{SO(3)}$.
Any matrix of $\mathrm{SU(2)}$ can be expressed in terms of Euler angles as
\begin{equation}
\mathcal{R}_2(\alpha,\beta,\gamma) = \mathcal{R}_\mathrm{z}(\alpha)\cdot
\mathcal{R}_\mathrm{y}(\beta) \cdot \mathcal{R}_\mathrm{z}(\gamma) = \begin{pmatrix}
\mathrm{e}^{-{\mathrm i}(\alpha+\gamma)/2} \cos(\beta/2) & -\mathrm{e}^{-{\mathrm i}(\alpha-\gamma)/2}\sin(\beta/2) \\
\mathrm{e}^{{\mathrm i}(\alpha-\gamma)/2} \sin(\beta/2)& \mathrm{e}^{{\mathrm i}(\alpha+\gamma)/2}\cos(\beta/2)
\end{pmatrix}
\end{equation}
with $\mathcal{R}_i(\theta) \equiv \exp(- {\mathrm i} \theta \sigma_i/2)$. Similarly a $\mathrm{SO(3)}$ matrix is also expressed with Euler angles as
\begin{equation}
R_3(\alpha,\beta,\gamma) = R_\mathrm{z}(\alpha)\cdot R_\mathrm{y}(\beta) \cdot R_\mathrm{z}(\gamma)
\end{equation}
where $R_j(\theta) \equiv \exp(-{\mathrm i} \theta \mathcal{J}^j)$ and
$(\mathcal{J}^i)_{jk} = -{\mathrm i} \epsilon_{ijk}$. Both sets of matrices are related since they share the same Lie algebra, thanks to
\begin{equation}
\label{eq:SU2SO3}
\mathcal{R}_2\cdot \sigma_i \cdot \mathcal{R}_2^\dagger = \sigma_j\left(R_3\right)^j_{\,\,\,i}
\,,
\end{equation}
where it is implied that the Euler angles defining $\mathcal{R}_2$ and $R_3$ are the same. Therefore, the conjugation of the traceless part of a two-neutrino density matrix by an element $\mathcal{R}_2$ of $\mathrm{SU(2)}$ is equivalent to the associated rotation $R_3$ applied on its vector representation defined in equation~\eqref{MatrixToVector}.
Note that the PMNS matrix is thus defined as
\begin{equation}
\label{eq:PMNS_bis}
U = R_\mathrm{x}(-\theta_{23})R_\mathrm{y}(\theta_{13}) R_\mathrm{z}(-\theta_{12}) \,.
\end{equation}
\end{document}
\chapter{On the BBGKY formalism}
\label{App:BBGKY}
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In this Appendix, we give some technical details on the BBGKY formalism and how it applies to (anti)neutrinos in the early Universe, in addition to the elements introduced in chapter~\ref{chap:QKE}. First, we explicit the components of the density matrix and the interaction potential, discussing some different presentations existing in the literature. Then, we explain how antiparticles can be included without much effort in the framework we presented in chapter~\ref{chap:QKE}.
\section{Further details on the formalism}
\label{app:BBGKY_Antisym}
The BBGKY hierarchy is often used in nuclear physics~\cite{Cassing1990, Reinhard1994, Lac04,Lac14}, which corresponds to a slightly different physical context than neutrinos and antineutrinos in the early Universe. In particular, we have chosen to use systematically a second-quantized approach (compared to the first-quantized formalism sometimes used), and we give in this section the equivalence between the various formalisms.
\subsection{Components of the density matrix}
\label{app:vrho_components}
Let's prove that the components of the $s$-body operator defined in \eqref{eq:defrhos} are given by \eqref{eq:defrhos2}. We roughly follow \cite{Simenel}, Eqs.~(12)--(15). For simplicity, we derive the result for $\hat{\varrho}^{(1)}$ only, but it can be readily generalized.
We need the closure relation which reads
\begin{equation}
\label{eq:closure_relation}
\hat{\mathbb{1}}_N = \frac{1}{N!} \sum_{k_1 \cdots k_N}{\ket{k_1 \cdots k_N} \bra{k_1 \cdots k_N}} \, ,
\end{equation}
following~\cite{Simenel}, Eq.~(A28). Moreover, the trace of a $N-$particle operator $\hat{A}$ reads
\begin{equation}
\label{eq:def_trace}
\mathrm{Tr} \, \hat{A} = \sum_{k_1 \cdots k_N} \frac{1}{\sqrt{N!}} \bra{k_1 \cdots k_N} \hat{A} \ket{k_1 \cdots k_N} \frac{1}{\sqrt{N!}} \, ,
\end{equation}
consistently with (A22) and (C3) in~\cite{Simenel}. The \emph{partial traces} are given by:
\begin{equation}
\label{eq:def_partial_trace}
\left. \mathrm{Tr}_{s+1 \dots N}(\hat{A}) \right \rvert^{i_1 \cdots i_s}_{j_i \cdots j_s} = \sum_{k_{s+1}\cdots k_N} \frac{1}{\sqrt{N!}} \bra{i_1 \cdots i_s k_{s+1} \cdots k_N} \hat{A} \ket{j_1 \cdots j_s k_{s+1} \cdots k_N} \frac{1}{\sqrt{N!}} \, .
\end{equation}
Note that we have evidently $\mathrm{Tr}_{1 \dots N} \hat{A} = \mathrm{Tr} \, \hat{A}$.
Starting from~\eqref{eq:defrho}, and inserting twice the closure relation~\eqref{eq:closure_relation},
\begin{align}
\langle \Psi | \hat{a}_j^\dagger \hat{a}_i | \Psi \rangle
&= \langle \Psi | \frac{1}{N!}\sum_{k_1 \cdots k_N}{|k_1 \cdots k_N \rangle \langle k_1 \cdots k_N |} \, \hat{a}_j^\dagger \hat{a}_i \, \frac{1}{N!}\sum_{k'_1 \cdots k'_N}{|k'_1 \cdots k'_N \rangle \langle k'_1 \cdots k'_N |} \Psi \rangle \nonumber
\\ &= \frac{1}{(N!)^2} \sum_{\substack{k_1 \cdots k_N \\ k'_1 \cdots k'_N}}{\langle k'_1 \cdots k'_N | \Psi \rangle \langle \Psi | k_1 \cdots k_N \rangle \times \langle k_1 \cdots k_N | \hat{a}_j^\dagger \hat{a}_i |k'_1 \cdots k'_N \rangle} \nonumber
\\ &= \frac{N^2}{(N!)^2} \sum_{\substack{k_2 \cdots k_N \\ k'_2 \cdots k'_N}}{\langle i \, k'_2 \cdots k'_N | \Psi \rangle \langle \Psi | j \, k_2 \cdots k_N \rangle \times \langle k_2 \cdots k_N | k'_2 \cdots k'_N \rangle} \nonumber
\\ \intertext{Indeed, $\sum_{k'_2 \cdots k'_N}{\langle i \, k'_2 \cdots k'_N | \Psi \rangle \hat{a}_i |k'_1 \cdots k'_N \rangle} = N \langle i \, k'_2 \cdots k'_N | \Psi \rangle | k'_2 \cdots k'_N \rangle$ since, due to antisymmetry, only one of the $k'_p$ can be equal to $i$, and the minus signs appearing when anticommuting $\hat{a}_i$ and $\hat{a}^\dagger_{k'_1}\cdots \hat{a}^\dagger_{k'_p}$ are compensated when $k'_p = i$ is moved at the beginning of the bra $\bra{i k'_2 \cdots k'_N}$.} &= \frac{1}{[(N-1)!]^2} \sum_{\substack{k_2 \cdots k_N \\ k'_2 \cdots k'_N}}{\langle i \, k'_2 \cdots k'_N | \Psi \rangle \langle \Psi | j \, k_2 \cdots k_N \rangle \times (N-1)! \cdot \delta_{k_2 k'_2}\cdots\delta_{k_N k'_N}} \nonumber
\\ &= N \sum_{k_2 \cdots k_N}{\frac{1}{\sqrt{N!}}\langle i \, k_2 \cdots k_N | \Psi \rangle \langle \Psi | j \, k_2 \cdots k_N \rangle \frac{1}{\sqrt{N!}}} \nonumber
\\ &= N \left. \mathrm{Tr}_{2 \dots N} \hat{D} \right |_{ij} \, ,
\end{align}
where we use the definition~\eqref{eq:def_partial_trace} of partial traces. We thus recover the expression~\eqref{eq:defrhos}.
\subsection{"Labelled particles" notation}
It is very common in the literature~\cite{Lac04,Simenel,Lac14} to adopt a notation that overlooks --- temporarily --- the fact that the particles are indistinguishable. In other words, one "labels" particles and introduces tensor product states. For instance, for two particles, the ket $\ket{1:i,2:j}$ allow to define the antisymmetrized version (valid for fermions):\footnote{The ket $\ket{ij}$ is, in second quantization, directly defined from the vacuum via $\ket{ij} \equiv \hat{a}^\dagger_i \hat{a}^\dagger_j \ket{0}$, from which the antisymmetry properties follow.}
\begin{equation}
\ket{ij} \equiv \frac{1}{\sqrt{2}} \left(\ket{1:i,2:j} - \ket{1:j,2:i}\right) \, .
\end{equation}
\paragraph{Consistency of definitions} The interaction matrix elements are then usually defined as $\tilde{v}^{ik}_{jl} = v^{ik}_{jl} - v^{ik}_{lj}$, where \[v^{ik}_{jl} \equiv \bra{1:i,2:k}\hat{H}_{\mathrm{int}}\ket{1:j,2:l} \, . \] This definition is consistent with~\eqref{eq:defvint}. Indeed, one has:
\begin{align}
\bra{ik}\hat{H}_{\mathrm{int}}\ket{jl} &= \frac{1}{\sqrt{2}} \left( \bra{1:i,2:k} - \bra{1:k,2:i}\right) \hat{H}_{\mathrm{int}} \left( \ket{1:j, 2:l} - \ket{1:l,2:j} \right) \frac{1}{\sqrt{2}} \nonumber \\
&= \frac12 \bra{1:i,2:k} \hat{H}_{\mathrm{int}} \left( \ket{1:j, 2:l} - \ket{1:l,2:j} \right) \nonumber \\
&\qquad \qquad \qquad \qquad + \frac12 \underbrace{\bra{1:k, 2:i} \hat{H}_{\mathrm{int}} \left( \ket{1:l, 2:j} - \ket{1:j,2:l} \right)}_{\text{first term with } 1 \leftrightarrow 2} \nonumber \\
&= \bra{1:i,2:k} \hat{H}_{\mathrm{int}} \left( \ket{1:j, 2:l} - \ket{1:l,2:j} \right) \nonumber \\
&= v^{ik}_{jl} - v^{ik}_{lj} \label{eq:check_defHint}
\end{align}
We used the fact that nothing changes if we rename $1 \leftrightarrow 2$ (i.e. $v^{ik}_{jl} = v^{ki}_{lj}$).
We prefer to use solely the definition~\eqref{eq:defvint} which uses only creation/annihilation operators and no nonphysical labeling. We will however refer to it in this Appendix when necessary, to connect our equations with the corresponding ones in the literature.
\paragraph{Two-body operators} In this new "language", the interaction Hamiltonian is not written directly in second quantization, but in terms of two-body operators:
\begin{equation}
\hat{H}_\mathrm{int} = \frac12 \sum_{q \neq q'}{\hat{V}(q,q')} = \sum_{q<q'}{\hat{V}(q,q')} \, ,
\end{equation}
where the indices $q$ and $q'$ label particles, and noting that $\hat{V}(q,q') = \hat{V}(q',q)$. The matrix elements of $\hat{V}$ read
\[v^{ik}_{jl} \equiv \bra{1:i,2:k}\hat{V}(1,2) \ket{1:j,2:l} \, ,\]
hence the same definition is valid with $\hat{H}_\mathrm{int}$, i.e. $v^{ik}_{jl} = \bra{1:i,2:k}\hat{H}_{\mathrm{int}}\ket{1:j,2:l}$. Note that this overlooks the fact that $\hat{H}_\mathrm{int}$ is a $N$-body operator and $\hat{V}(1,2)$ a 2-body one, such that they don't act on the same spaces.
Finally, we emphasize that the expression~\eqref{eq:defHint}, which involves annihilation and creation operators, is valid for any number of particles, hence in the entire Fock space~\cite{CohenIII}.
\paragraph{Traces and notation of the BBGKY hierarchy} Finally, let us discuss the expression of traces in this "labelled particles" notation --- an important point as traces appear in all equations of the BBGKY hierarchy.
An important relation is (we prove its consistency with the above definitions later, and admit it for now):
\begin{equation}
\label{eq:tr_v_rho}
\mathrm{Tr}_2 \left(\hat{V}(1,2) \hat{\varrho}^{(12)}\right)^{\textcolor{firebrick}{i}}_{\textcolor{firebrick}{j}} \equiv \sum_{k,l,r}{v^{\textcolor{firebrick}{i}k}_{rl} \varrho^{rl}_{\textcolor{firebrick}{j}k}} \, .
\end{equation}
The summations over $l$, $r$ come from the product operation, and the sum over $k$ is due to the trace. Given the antisymmetry properties of $\hat{\varrho}^{(12)}$, namely $\varrho^{rl}_{jk} = - \varrho^{lr}_{jk}$, we get (with a relabeling of the indices $l$, $r$ for one term):
\begin{align}
\mathrm{Tr}_2 \left(\hat{V}(1,2) \hat{\varrho}^{(12)}\right)^i_j &=\frac12 \sum_{k,l,r}{\left(v^{ik}_{rl} - v^{ik}_{lr}\right) \varrho^{rl}_{jk}} \nonumber \\
&= \frac12 \sum_{k,l,r}{\tilde{v}^{ik}_{rl} \varrho^{rl}_{jk}} \nonumber \\
&= \frac12 \mathrm{Tr}_2 \left( \hat{\tilde{v}}^{(12)} \hat{\varrho}^{(12)}\right)^i_j
\end{align}
This relations explain why the BBGKY hierarchy \emph{looks} a priori different depending on the references, cf.~for instance Eq.~(88) in~\cite{Simenel}, Eq.~(3) in~\cite{Lac04} or Eq.~(9) in~\cite{Volpe_2013}.
Let us now prove~\eqref{eq:tr_v_rho}. From~\eqref{eq:def_trace}, we have
\begin{align*}
\mathrm{Tr}_2 \left(\hat{V}(1,2) \hat{\varrho}^{(12)}\right)^i_j &=\frac{1}{2!} \sum_{k}{\bra{ik} \hat{V}(1,2) \hat{\varrho}^{(12)} \ket{jk}} \\
&= \frac{1}{2!} \frac{1}{2!} \sum_{k,l,r} \underbrace{\bra{ik} \hat{V}(1,2) \ket{rl}}_{\displaystyle v^{ik}_{rl} - v^{ik}_{lr}} \underbrace{\bra{rl} \hat{\varrho}^{(12)} \ket{jk}}_{\displaystyle \varrho^{rl}_{jk} - \varrho^{rl}_{kj}} \, , \\
\end{align*}
where the matrix elements are consistent with~\eqref{eq:check_defHint}. With the antisymmetry property of $\hat{\varrho}^{(12)}$, we have
\begin{align*}
\mathrm{Tr}_2 \left(\hat{V}(1,2) \hat{\varrho}^{(12)}\right)^i_j &= \frac{1}{2!} \frac{1}{2!} \sum_{k,l,r}{(v^{ik}_{rl}-v^{ik}_{lr}) \times 2 \varrho^{rl}_{jk}} \\
&= \sum_{k,l,r}{v^{ik}_{rl} \varrho^{rl}_{jk}}
\end{align*}
thanks to the relabeling $l \leftrightarrow r$ in one term.
\section{Quantum kinetic equations with antiparticles}
\label{app:antiparticles}
\subsection{QKE for $\bm{\bar{\varrho}}$}
We present in this section the inclusion of antiparticles to the BBGKY formalism.
\paragraph{Generalized definitions}
One must adapt the definitions \eqref{eq:defrhos2} and \eqref{eq:defHint} to include the annihilation and creation operators $\hat{b},\hat{b}^\dagger$. Throughout this appendix, we will emphasize the indices which are associated to antiparticles with a barred notation $(\bar{\imath},\bar{\jmath})$. Therefore, with capital indices $I$ being either $i$ or ${\bar{\imath}}$, we have:
\begin{align}
\label{eq:defrho_anti}
\varrho^{I_1 \cdots I_s}_{J_1 \cdots J_s} &\equiv \langle \hat{c}_{J_s}^\dagger \cdots \hat{c}_{J_1}^\dagger \hat{c}_{I_1} \cdots \hat{c}_{I_s} \rangle \, , \\
\hat{H}_0 &= \sum_{I,J}{t^{I}_{J} \, \hat{c}^\dagger_I \hat{c}_J} \, , \\
\hat{H}_\mathrm{int} &= \frac14 \sum_{I,J,K,L}{\tilde{v}^{IK}_{JL} \, \hat{c}^\dagger_I \hat{c}^\dagger_K \hat{c}_L \hat{c}_J} \, , \label{eq:vint_anti}
\end{align}
where $\hat{c}_I = \hat{a}_i$ or $\hat{b}_{\bar{\imath}}$ depending on the index $I$ labelling a particle or an antiparticle.
The evolution equations \eqref{eq:hierarchy} and \eqref{eq:eqvrho} are naturally extended to the antiparticle case thanks to the global indices. The downside of this strategy is that the transformation law of tensors is now implicit: since $\hat{a}$ transforms like $\hat{b}^\dagger$ under a unitary transformation $\psi^a = \mathcal{U}^a_i \psi^i$, the behaviour of upper and lower indices is inverted whenever they label an antiparticle degree of freedom, for instance:
\begin{equation}
\label{eq:transfo_unit}
t^{i}_{j} = {\mathcal{U}^\dagger}^{i}_{a} \, t^{a}_{b} \, \mathcal{U}^{b}_{j} \qquad ; \qquad t^{{\bar{\imath}}}_{{\bar{\jmath}}} = \mathcal{U}^{a}_{i} \, t^{\bar{a}}_{\bar{b}} \, {\mathcal{U}^\dagger}^{j}_b \, .
\end{equation}
Since we assume an isotropic medium, there are no "abnormal" or "pairing" densities \cite{Volpe_2013,SerreauVolpe,Volpe_2015} such as $\langle \hat{b} \hat{a} \rangle$, which ensures the separation of the two-body density matrix between the neutrino density matrix (for which we keep the notation $\varrho$) and the antineutrino one $\bar{\varrho}$. In order for $\bar{\varrho}$ to have the same transformation properties as $\varrho$, we need to take a transposed convention for its components:
\begin{equation}
\bar{\varrho}^{\bar{\imath}}_{\bar{\jmath}} = \varrho^{\{J=\bar{\jmath}\}}_{\{I=\bar{\imath}\}} = \langle \hat{c}^\dagger_{\bar{\imath}} \hat{c}_{\bar{\jmath}} \rangle = \langle \hat{b}^\dagger_i \hat{b}_j \rangle \, .
\end{equation}
One could further take transposed conventions for the antiparticle indices in $t$ and $\tilde{v}$, which would ensure a clear correspondence between index position and transformation law --- contrary to \eqref{eq:transfo_unit}. For instance, $\bar{t}^{\bar{\imath}}_{\bar{\jmath}} \equiv t^{\bar{\jmath}}_{\bar{\imath}}$ transforms as $t^i_j$. However, in order to keep a unique expression for the mean-field potential or the collision term, we stick to the general definitions above. For instance, we have:
\begin{equation}
\label{eq:Gamma_full}
\Gamma^{i}_{j} = \sum_{K,L}{\tilde{v}^{iK}_{jL} \varrho^{L}_{K}} = \sum_{k,l}{\tilde{v}^{ik}_{jl} \varrho^{l}_{k}} + \sum_{\bar{k},\bar{l}}{\tilde{v}^{i \bar{k}}_{j \bar{l}} \bar{\varrho}^{\bar{k}}_{\bar{l}}} \, .
\end{equation}
Since the annihilation and creation operators do not appear naturally in normal order in the Hamiltonian \eqref{eq:defHsm}, recasting it in the form~\eqref{eq:vint_anti} leads to extra minus signs in $\tilde{v}$ involving antiparticles (cf.~table~\ref{Table:MatrixElements}).
These conventions being settled, we can include the full set of interaction matrix elements and compute all relevant contributions to the neutrino QKEs \eqref{eq:QKE_rho}. In the following, we derive the QKE for $\bar{\varrho}$.
\paragraph{QKE for antineutrinos}
Thanks to our conventions, the evolution equation for the antineutrino density matrix $\bar{\varrho}$ is similarly obtained within the BBGKY formalism,
with some differences compared to the neutrino case. First and foremost, the evolution equation for $\bar{\varrho}^{\bar{\imath}}_{\bar{\jmath}}$ correspond in the general formalism to the equation for $\varrho^{\bar{\jmath}}_{\bar{\imath}}$:
\begin{equation}
\label{eq:drhobjbi}
{\mathrm i} \frac{\mathrm{d} \bar{\varrho}^{{\bar{\imath}}}_{{\bar{\jmath}}}}{\mathrm{d} t} = {\mathrm i} \frac{\mathrm{d} \varrho^{{\bar{\jmath}}}_{{\bar{\imath}}}}{\mathrm{d} t} = \left( \left[t^{{\bar{\jmath}}}_{K} + \Gamma^{{\bar{\jmath}}}_{K}\right] \varrho^{K}_{{\bar{\imath}}} - \varrho^{{\bar{\jmath}}}_{K} \left[t^{K}_{{\bar{\imath}}} + \Gamma^{K}_{{\bar{\imath}}}\right] \right) + {\mathrm i} \, \hat{\mathcal{C}}^{\bar{\jmath}}_{\bar{\imath}} \, ,
\end{equation}
showing that taking the commutator with a transposed convention leads to a minus sign. Moreover,
\begin{itemize}
\item we express the kinetic terms $t^{\bar{\jmath}}_{\bar{\imath}}$, starting from the mass basis:
\begin{equation}
t^{\bar{\jmath}}_{\bar{\imath}} = U^a_j \left. \frac{\mathbb{M}^2}{2p}\right|^{\bar{a}}_{\bar{b}} {U^\dagger}^i_b = {U^\dagger}^i_b \left. \frac{\mathbb{M}^2}{2p}\right|^b_a U^a_j = t^i_j \, ;
\end{equation}
\item $\tilde{v}^{\bar{\jmath} k}_{\bar{\imath} l}$ is the coefficient in front of $\hat{b}^\dagger_j \hat{a}^\dagger_k \hat{a}_l \hat{b}_i$, so it will have the same expression (apart from the interchange of $u$ and $v$ spinors for neutrinos, which leaves the result identical) as the coefficient in front of $\hat{a}_j \hat{a}^\dagger_k \hat{a}_l \hat{a}^\dagger_i = - \hat{a}^\dagger_i \hat{a}^\dagger_k \hat{a}_l \hat{a}_j$, that is $- \tilde{v}^{ik}_{jl}$. Therefore, $\Gamma^{\bar{\jmath}}_{\bar{\imath}} = - \Gamma^{i}_{j}$.
\end{itemize}
Including these two results in~\eqref{eq:drhobjbi} show that, compared to the neutrino case, the vacuum term gets a minus sign (from the reversed commutator), but not the mean-field. Formally,
\begin{equation}
{\mathrm i} \frac{\mathrm{d} \bar{\varrho}^i_j}{\mathrm{d} t} = \left[- \hat{t} + \hat{\Gamma}, \hat{\bar{\varrho}}\right]^{i}_{j} + {\mathrm i} \, \hat{\mathcal{C}}^{\bar{\jmath}}_{\bar{\imath}} \, .
\end{equation}
Two additional remarks:
\begin{itemize}
\item $s$ and $t$ channels are inverted when the particle $1$ is an antineutrino ($2$ and $4$ left unchanged). For instance, the scattering between $\bar{\nu}_e$ and $e^-$ is a $s-$channel (exchanged momentum $\Delta = p_1+p_2$), contrary to the scattering between $\nu_e$ and $e^-$ ($\Delta = p_1 - p_2$). This changes the sign of $\Delta^2$, leading to another minus sign for $\Gamma$ at order $1/m_{W,Z}^2$;
\item the collision integral $\bar{\mathcal{C}}$ is obtained from $\mathcal{C}$ through the replacements $\varrho \leftrightarrow \bar{\varrho}$ and $g_L \leftrightarrow g_R$, as detailed in~\ref{app:collision_term}.
\end{itemize}
Considering all these remarks, we obtained the QKE for $\bar{\varrho}$ \eqref{eq:QKE_rhobar}.
\subsection{Particle/antiparticle symmetry and consistency of the QKEs}
\label{subsec:QKE_consistency}
If there is no asymmetry, we expect that $\varrho = \bar{\varrho}^T$ where $^T$ represents the transposed of the matrix. Indeed, we recall that the definitions of $\varrho$ and $\bar{\varrho}$ are transposed: $\varrho^\alpha_\beta \propto \langle \hat{a}^\dagger_{\nu_\beta} \hat{a}_{\nu_\alpha} \rangle$ and $\bar{\varrho}^\alpha_\beta \propto \langle \hat{b}^\dagger_{\nu_\alpha} \hat{b}_{\nu_\beta} \rangle$. Note that the density matrices being Hermitian, $\bar{\varrho}^T = \bar{\varrho}^*$, but talking about transposition will be more convenient here (essentially because of the commutators in the QKEs).
A consistency check of the QKEs consists in verifying that, assuming $\varrho = \bar{\varrho}^T$, equations~\eqref{eq:QKE_rho} and~\eqref{eq:QKE_rhobar} are equivalent. To do so, we transpose~\eqref{eq:QKE_rhobar}. We assume that there is no CP-phase in the PMNS matrix, as its effect would naturally be to break this equivalence.
\subsubsection{Mean-field consistency}
The key relation is $[A,B]^T = (AB-BA)^T = - [A^T,B^T]$. Then, assuming $\varrho = \bar{\varrho}^T$,
\begin{align*}
\left(U \frac{\mathbb{M}^2}{2p}U^\dagger\right)^T &= U^* \frac{\mathbb{M}^2}{2p} U^T = U\frac{\mathbb{M}^2}{2p} U^\dagger &\text{(the absence of CP phase is crucial),} \\
(\mathbb{N}_\nu - \mathbb{N}_{\bar{\nu}})^T &= \mathbb{N}_{\bar{\nu}} - \mathbb{N}_\nu = - (\mathbb{N}_\nu - \mathbb{N}_{\bar{\nu}}) \, , \\
(\mathbb{E}_\mathrm{lep} + \mathbb{P}_\mathrm{lep})^T &= \mathbb{E}_\mathrm{lep} + \mathbb{P}_\mathrm{lep} &\text{because these matrices are diagonal,} \\
(\mathbb{E}_\nu + \mathbb{E}_{\bar{\nu}})^T &= \mathbb{E}_{\bar{\nu}} + \mathbb{E}_\nu \, .
\end{align*}
Therefore,
\begin{align*}
{\mathrm i} \left[ \frac{\partial}{\partial t} - H p \frac{\partial}{\partial p}\right] \bar{\varrho}^T &= \Big[ \left(U \frac{\mathbb{M}^2}{2p}U^\dagger\right)^T, \bar{\varrho}^T \Big] - \sqrt{2} G_F \Big[ (\mathbb{N}_\nu - \mathbb{N}_{\bar{\nu}})^T, \bar{\varrho}^T \Big] \\
&\qquad \qquad \qquad + 2 \sqrt{2} G_F p \Big[ \frac{(\mathbb{E}_\text{lep} + \mathbb{P}_\text{lep})^T}{m_W^2} + \frac43 \frac{(\mathbb{E}_{\nu} + \mathbb{E}_{\bar{\nu}})^T}{m_Z^2},\bar{\varrho}^T \Big ] + {\mathrm i} \bar{\mathcal{I}}^T \\
&= \Big[ U \frac{\mathbb{M}^2}{2p}U^\dagger , \varrho \Big] + \sqrt{2} G_F \Big[(\mathbb{N}_\nu - \mathbb{N}_{\bar{\nu}}, \varrho \Big] \\
&\qquad \qquad \qquad + 2 \sqrt{2} G_F p \Big[ \frac{\mathbb{E}_\text{lep} + \mathbb{P}_\text{lep}}{m_W^2} + \frac43 \frac{\mathbb{E}_{\nu} + \mathbb{E}_{\bar{\nu}}}{m_Z^2},\varrho \Big ] + {\mathrm i} \bar{\mathcal{I}}^T \\
\end{align*}
This coincides with~\eqref{eq:QKE_rho} if $\bar{\mathcal{I}}^T=\mathcal{I}$, which we check now.
\subsubsection{Collision term consistency}
Let us show that $\bar{\mathcal{I}}^T = \mathcal{I}$ if there is no asymmetry (i.e.~if we assume that $\bm{{\varrho = \bar{\varrho}^T}}$). We will systematically observe that the processes in $\mathcal{I}$ correspond to the ones in $\bar{\mathcal{I}}^T$ where particles are exchanged with their associated antiparticles. Let us show it on a few processes:
\paragraph*{Annihilation into charged leptons} The process $\nu + \bar{\nu} \leftrightarrow e^- + e^+$ indeed coincides between both collision integrals:
\begin{align*}
\bar{\mathcal{I}}^T_{[\bar{\nu} \nu \to e^- e^+]} &\propto \int{\ddp{2} \cdots \, 4 (p_1\cdot p_3)(p_2 \cdot p_4) \times \underbrace{f_3 \bar{f}_4}_{f_3 f_4} \left[ G_L (\mathbb{1} -\varrho_2)G_L(\mathbb{1} -\bar{\varrho}_1) + \mathrm{h.c.}\right]^T} + \cdots \\
&= \int{\ddp{2} \cdots \, 4 (p_1\cdot p_3)(p_2 \cdot p_4) \times f_3 f_4 \left[ (\mathbb{1} -\bar{\varrho}_1^T) G_L (\mathbb{1} -\varrho_2^T)G_L + \mathrm{h.c.}\right]} + \cdots \\
&= \int{\ddp{2} \cdots\, 4 (p_1\cdot p_4)(p_2 \cdot p_3) \times f_4 f_3 \left[ (\mathbb{1} -\varrho_1) G_L (\mathbb{1} -\bar{\varrho}_2)G_L + \mathrm{h.c.}\right]} + \cdots \\
&= \mathcal{I}_{[\nu \bar{\nu} \to e^- e^+]}
\end{align*}
We recognize the second term of the statistical factor \eqref{eq:F_ann_cl}.
\paragraph*{Scattering with charged leptons} The process $\nu + e^+ \to \nu + e^+$ in $\mathcal{I}$ is "exchanged" with $\bar{\nu} + e^- \to \bar{\nu} + e^-$ in $\bar{\mathcal{I}}$. Indeed, we have:
\begin{align*}
F_\mathrm{sc}^{LL}(\bar{\nu}^{(1)},e^{(2)},\bar{\nu}^{(3)},e^{(4)})^T &= f_4(\mathbb{1} -f_2)\left[G_L \bar{\varrho}_3 G_L (\mathbb{1} -\bar{\varrho}_1) + \mathrm{h.c.} \right]^T - \{\mathrm{loss}\} \\
&= \bar{f}_4(\mathbb{1} -\bar{f}_2)\left[(\mathbb{1} - \varrho_1) G_L \varrho_3 G_L + \mathrm{h.c}. \right]^T - \{\mathrm{loss}\} \\
&= F_\mathrm{sc}^{LL}(\nu^{(1)},\bar{e}^{(2)},\nu^{(3)},\bar{e}^{(4)})
\end{align*}
and the prefactor of these statistical factors is identical in both collision integrals, namely $(p_1 \cdot p_4)(p_2 \cdot p_3)$.
\paragraph*{(Anti)neutrino scattering} The correspondence is now between $\nu + \nu \to \nu + \nu$ and $\bar{\nu} + \bar{\nu} \to \bar{\nu} + \bar{\nu}$. Indeed, let's compare the statistical factors \eqref{eq:F_sc_nn} and \eqref{eq:F_sc_bnbn}. Since
\[ \left[ (\mathbb{1} - \bar{\varrho}_1) \bar{\varrho}_3 (\mathbb{1} - \bar{\varrho}_2) \bar{\varrho}_4 \right]^T = \varrho_4 (\mathbb{1} -\varrho_2) \varrho_3 (\mathbb{1} -\varrho_1) \, , \]
we do have $\mathcal{I}_{[\nu \nu \to \nu \nu]} = \bar{\mathcal{I}}^T_{[\bar{\nu}\bar{\nu}\to\bar{\nu}\bar{\nu}]}$.
This proves the consistency of the QKEs. If one adopts the opposite point of view, this shows that if $\varrho = \bar{\varrho}^*$ initially, then this symmetry is preserved along the evolution.
\chapter{Computing the terms of the QKEs}
\setlength{\epigraphwidth}{0.3\textwidth}
\epigraph{The Game is On!}{Sherlock Holmes, \emph{Sherlock}}
{
\hypersetup{linkcolor=black}
\minitoc
}
We provide here the derivation of the terms of the QKE that were not discussed in chapter~\ref{chap:QKE}, that is the neutral-current mean-field potentia, the collision integral with charged leptons, the antineutrino collision integral. We also detail the dimensional reduction of the collision integral.
\section{Interaction matrix elements and mean-field potential}
\label{app:matrix_el_MF}
We have detailed the example of the calculation of the interaction matrix elements $\tilde{v}$ for charged-current processes in the chapter~\ref{chap:QKE}. Let us show briefly how the other contributions are computed and how they lead to the different terms in the full mean-field potential~\eqref{eq:Gamma_potential}. For brevity, we only treat the interactions in the four-Fermi approximation, but the contributions at order $\Delta^2/m_{W,Z}^2$ can be computed following the same procedure as in section~\ref{subsec:weak_matrix_el_QKE}. Moreover, once the matrix element is known for the interaction with a given particle, the result for the interaction for the associated antiparticle is obtained exactly as in section~\ref{subsec:weak_matrix_el_QKE}.
\subsection{Charged-current mean-field}
The mean-field potential due to the forward coherent scattering with $e^\pm$ is discussed in chapter~\ref{chap:QKE}.
\subsection{Neutral-current mean-field}
For neutral-current processes, the procedure is exactly similar to the charged-current case, with the Hamiltonian
\begin{subequations}
\label{eq:hnc}
\begin{align}
\hat{H}_{NC} &= \frac{G_F}{\sqrt{2}} \sum_{f,f'} \int{{\mathrm{d}^3 \vec{x} \ \left[\bar{\psi}_{f} \gamma^\mu (g_V^f -g_A^{f} \gamma^5) \psi_f\right]\left[\bar{\psi}_{f'}\gamma_\mu (g_V^{f'} - g_A^{f'} \gamma^5) \psi_{f'}\right]}} \, , \\
&= 2 \sqrt{2} G_F \sum_{f,f'} \int{{\mathrm{d}^3 \vec{x} \ \left[\bar{\psi}_{f} \gamma^\mu (g_L^f P_L + g_R^{f} P_R) \psi_f\right]\left[\bar{\psi}_{f'}\gamma_\mu (g_L^{f'} P_L + g_R^{f'} P_R) \psi_{f'}\right]}} \, ,
\end{align}
\end{subequations}
where the different couplings are related via (remember that $P_L = (1-\gamma^5)/2$ and $P_R = (1+\gamma^5)/2$)
\begin{equation}
g_L = \frac{g_V + g_A}{2} \quad , \quad g_R = \frac{g_V - g_A}{2} \, .
\end{equation}
We split the Hamiltonian in the contributions involving neutrinos and the matter background and only (anti)neutrinos, \eqref{eq:hncmat} and~\eqref{eq:hncself}.
\subsubsection{Matter background}
We can follow the exact same steps as in the charged-current case. In the following we deal with particles, but as show before considering antiparticles basically only require to add a minus sign.
We find:
\begin{multline}
\label{eq:vNCmat}
\tilde{v}^{\nu_\alpha(1) f(2)}_{\nu_\alpha(3) f(4)} = \frac{G_F}{\sqrt{2}} \ (2 \pi)^3 \delta^{(3)}(\vec{p}_1+\vec{p}_2- \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_\alpha}^{h_1}(\vec{p}_1) \gamma^\mu (1 - \gamma^5) u_{\nu_\alpha}^{h_3}(\vec{p}_3)] \ [\bar{u}_{f}^{h_2}(\vec{p}_2) \gamma_\mu (g_V^{f} - g_A^{f} \gamma^5) u_{f}^{h_f}(\vec{p}_4)] \, .
\end{multline}
Assuming a homogeneous and unpolarized background for the fermion $f$, we can use the trace technology on the $u_f$ spinor product:
\begin{equation}
\sum_{h_f}{[\bar{u}_{f}^{h_f}(\vec{p}) \gamma_\mu (g_V^f - g_A^f \gamma^5) u_{f}^{h_f}(\vec{p})]} = \mathrm{tr} [(\gamma^\alpha p_\alpha + m_f) \gamma_\mu (g_V^f - g_A^f \gamma^5)] = g_V^f p_\alpha \mathrm{tr} [\gamma^\alpha \gamma_\mu] = 4 g_V^f p_\mu \, ,
\end{equation}
which leads to, writing $f_f(p)$ the distribution function of the fermion $f$,
\begin{align}
\Gamma^{\nu_\alpha(\vec{p}_1,-)}_{\nu_\alpha(\vec{p}_3,-)} \underset{\text{NC, $f$}}{=} &\frac{G_F}{\sqrt{2}} \times (2 \pi)^3 \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times \int{\frac{\mathrm{d}^3\vec{p}}{(2 \pi)^3 2 E_p} 4 \, g_V^f \, p_\mu \times f_f(p)} \nonumber \\
&\qquad \qquad \qquad \qquad \qquad \qquad \times \underbrace{[\bar{u}_{\nu_\alpha}^{(-)}(\vec{p}_1) \gamma^\mu (1 - \gamma^5) u_{\nu_\alpha}^{(-)}(\vec{p}_3)]}_{4 {p_1}^\mu} \nonumber \\
= \ \ &\frac{G_F}{\sqrt{2}} \times (2 \pi)^3 \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times 4 p_1 \times g_V^f \times 2 \int{\frac{\mathrm{d}^3\vec{p}}{(2 \pi)^3} f_f(p)} \nonumber \\
= \ \ &(2 \pi)^3 2 p_1 \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times \sqrt{2} G_F g_V^f n_f \nonumber\\
= \ \ &\sqrt{2} G_F g_V^f n_f \ \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, .
\end{align}
We assumed that the fermions were spin-1/2 fermions, with 2 helicity states.
Therefore, in the early universe, including for completeness protons and neutrons, the full mean-field potential due to neutral-current interactions with matter reads
\begin{equation}
\label{eq:Gamma_NC_temp}
\Gamma^{\nu_\alpha(\vec{p}_1,-)}_{\nu_\alpha(\vec{p}_3,-)} \underset{\text{NC, mat}}{=} \sqrt{2} G_F \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, \left[g_V^e (n_e - \bar{n}_e) + g_V^p n_p + g_V^n n_n \right]
\end{equation}
This term is the same for all three neutrino flavors, hence it is proportional to $\mathbb{1}$ and does not contribute to the mean-field dynamics.\footnote{This is true only if there are no sterile species. Otherwise the mean-field matrix would be, noting $\Gamma_\text{NC}$ the value~\eqref{eq:Gamma_NC_temp}, $\mathrm{diag}(\Gamma_\text{NC},\Gamma_\text{NC},\Gamma_\text{NC},0,\cdots,0)$ with as many zero diagonal entries as there are sterile neutrino species.} Moreover, using $g_V^p = - g_V^e$, $g_V^n = - 1/2$ and the electric neutrality of the Universe $n_e - \bar{n}_e - n_p = 0$, we can write the final result:
\begin{equation}
\label{eq:Gamma_NC_mat_final}
\boxed{\Gamma^{\nu_\alpha(\vec{p}_1,-)}_{\nu_\alpha(\vec{p}_3,-)} \underset{\text{NC, mat}}{=} - \frac{G_F}{\sqrt{2}} \, \delta_{\alpha \beta} \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, n_n} \, .
\end{equation}
\subsubsection{Neutrino self-interactions}
To derive $\tilde{v}^{\nu_\alpha \nu_\beta}_{\nu_\alpha \nu_\beta}$, we rewrite the Hamiltonian \eqref{eq:hncself} in the form~\eqref{eq:defHint}. For convenience, we distinguish the diagonal and non-diagonal terms, and restrict to Fermi order.
\paragraph{Non-diagonal terms}
The Hamiltonian contains the terms
\begin{multline}
\hat{H}_{NC}^{\nu \nu} \supset \frac{G_F}{4 \sqrt{2}} \sum_{h_1 \dots} \int{[\mathrm{d}^3 \vec{p}_2 \,] \cdots} \ (2 \pi)^3 \delta^{(3)}(\vec{p}_1+\vec{p}_2-\vec{p}_3-\vec{p}_4) \times [\bar{u}_{\nu_\alpha}^{h_1}(\vec{p}_1) \gamma^\mu (1 - \gamma^5) u_{\nu_\alpha}^{h_4}(\vec{p}_4))] \\ \times [\bar{u}_{\nu_\beta}^{h_2}(\vec{p}_2) \gamma_\mu (1 - \gamma^5) u_{\nu_\beta}^{h_3}(\vec{p}_3)]
\times \hat{a}^\dagger_{\nu_\alpha}(\vec{p}_1,h_1) \hat{a}_{\nu_\alpha}(\vec{p}_4,h_4) \hat{a}^\dagger_{\nu_\beta}(\vec{p}_2,h_2) \hat{a}_{\nu_\beta}(\vec{p}_3,h_3)
\end{multline}
where all the possible contributions in normal ordering, so as to be in the form~\eqref{eq:defHint}, read
\begin{multline}
\label{eq:factor14offdiag}
\hat{a}^\dagger_{\nu_\alpha}(1) \hat{a}_{\nu_\alpha}(4) \hat{a}^\dagger_{\nu_\beta}(2) \hat{a}_{\nu_\beta}(3) = \frac14 \Big(- \hat{a}^\dagger_{\nu_\alpha}(1) \hat{a}^\dagger_{\nu_\beta}(2) \hat{a}_{\nu_\alpha}(4) \hat{a}_{\nu_\beta}(3) + \hat{a}^\dagger_{\nu_\alpha}(1) \hat{a}^\dagger_{\nu_\beta}(2) \hat{a}_{\nu_\beta}(3) \hat{a}_{\nu_\alpha}(4) \\ + \hat{a}^\dagger_{\nu_\beta}(2) \hat{a}^\dagger_{\nu_\alpha}(1) \hat{a}_{\nu_\alpha}(4) \hat{a}_{\nu_\beta}(3) - \hat{a}^\dagger_{\nu_\beta}(2) \hat{a}^\dagger_{\nu_\alpha}(1) \hat{a}_{\nu_\beta}(3) \hat{a}_{\nu_\alpha(4)}\Big)
\end{multline}
This term appears \emph{twice} in $\hat{H}_{NC}^{\nu \nu}$, therefore we can identify
\begin{multline}
\label{eq:vNCnunu}
\tilde{v}^{\nu_\alpha(1) \nu_\beta(2)}_{\nu_\beta(3) \nu_\alpha(4)} = \frac{G_F}{2\sqrt{2}} \ (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_\alpha}^{h_1}(\vec{p}_1) \gamma^\mu (1 - \gamma^5) u_{\nu_\beta}{h_3}(\vec{p}_3)] \ [\bar{u}_{\nu_\beta}^{h_2}(\vec{p}_2) \gamma_\mu (1 - \gamma^5) u_{\nu_\alpha}^{h_4}(\vec{p}_4)]
\end{multline}
Note that we used a Fierz identity to get this result (cancellation of two minus signs). To get the mean-field, we follow the standard procedure,
\begin{equation}
\Gamma^{\nu_\alpha(\vec{p}_1,h_1)}_{\nu_\beta(\vec{p}_3,h_3)} \underset{\text{NC, $\nu$}}{=} \sum_{h_2,h_4} \int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_4]} \ \tilde{v}^{\nu_\alpha(1) \nu_\beta(2)}_{\nu_\beta(3) \nu_\alpha(4)} \times \underbrace{\varrho^{\nu_\alpha(\vec{p}_4,h_4)}_{\nu_\beta(\vec{p}_2,h_2)}}_{(2\pi)^3 \, \delta_{h_2 h_4} \, 2p_2 \, \delta^{(3)}(\vec{p}_2 - \vec{p}_4) \varrho^\alpha_\beta(p_2)} \, .
\end{equation}
Using trace technology, we get the result:
\begin{align}
\Gamma^{\nu_\alpha(\vec{p}_1,-)}_{\nu_\beta(\vec{p}_3,-)} &\underset{\text{NC, $\nu$}}{=} \frac{G_F}{2 \sqrt{2}} \times (2 \pi)^3 \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times \int{\frac{\mathrm{d}^3\vec{p}_2}{(2 \pi)^3 2 p_2} 16 p_2^\mu p_{1,\mu} \times \varrho^\alpha_\beta(p_2)} \nonumber \\
&\ = \sqrt{2} G_F \times (2 \pi)^3 \, 2 p_1 \, \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times \int{\frac{\mathrm{d}^3\vec{p}_2}{(2 \pi)^3} \, \varrho^\alpha_\beta(p_2)} \nonumber \\
&\ = \sqrt{2} G_F \, \left. \mathbb{N}_\nu \right\rvert^\alpha_\beta \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, .
\end{align}
where we have eliminated angular integrals involving $\vec{p}_2 \cdot \vec{p}_1$, which vanish thanks to isotropy.
There is yet another term to consider: the propagation $\nu_\alpha \to \nu_\beta$ in a background of antineutrinos, namely, the matrix elements $\bra{\nu_\alpha \bar{\nu}_\alpha} \hat{H}_{NC}^{\nu \nu} \ket{\nu_\beta \bar{\nu}_\beta}$. It is equivalently the coefficient in front of $\hat{a}^\dagger_{\nu_\alpha}(1) \hat{b}^\dagger_{\nu_\alpha}(2) \hat{b}_{\nu_\beta}(4) \hat{a}_{\nu_\beta}(3)$ in the expansion of the Hamiltonian, namely,
\begin{multline}
\tilde{v}^{\nu_\alpha(1) \bar{\nu}_\beta(2)}_{\nu_\beta(3) \bar{\nu}_\alpha(4)} = \frac{G_F}{2\sqrt{2}} \ (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_\alpha}^{h_1}(\vec{p}_1) \gamma^\mu (1 - \gamma^5)v_{\nu_\alpha}^{h_4}(\vec{p}_4)] \ [\bar{v}_{\nu_\beta}^{h_2}(\vec{p}_2) \gamma_\mu (1 - \gamma^5) u_{\nu_\beta}^{h_3}(\vec{p}_3)]
\end{multline}
We then multiply this term by
\begin{equation}
\langle \hat{b}^\dagger_{\nu_\alpha}(\vec{p}_4,h_4) \hat{b}_{\nu_\beta}(\vec{p}_2, h_2) \rangle = (2 \pi)^3 \delta_{h_2 h_4} 2 p_2 \delta^{(3)}(\vec{p}_2 - \vec{p}_4) \bar{\varrho}^{\alpha}_{\beta}(p_2) \, .
\end{equation}
Note that it is $\bar{\varrho}$, and not $\bar{\varrho}^*$ that appears, thanks to the transposed definition of $\bar{\varrho}$ compared to $\varrho$. Using again a Fierz identity, we get an extra minus sign such that
\begin{align}
\Gamma^{\nu_\alpha(\vec{p}_1,-)}_{\nu_\beta(\vec{p}_3,-)} &\underset{\text{NC, $\bar{\nu}$}}{=} - \frac{G_F}{2 \sqrt{2}} \times (2 \pi)^3 \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times \int{\frac{\mathrm{d}^3\vec{p}_2}{(2 \pi)^3 2 p_2} 16 p_2^\mu p_{1,\mu} \times \bar{\varrho}^\alpha_\beta(p_2)} \nonumber \\
&\ = - \sqrt{2} G_F \times (2 \pi)^3 \, 2 p_1 \, \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times \int{\frac{\mathrm{d}^3\vec{p}_2}{(2 \pi)^3} \, \bar{\varrho}^\alpha_\beta(p_2)} \nonumber \\
&\ = - \sqrt{2} G_F \, \left. \mathbb{N}_{\bar{\nu}} \right\rvert^\alpha_\beta \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, .
\end{align}
\paragraph{Diagonal terms} In order to compute $\Gamma^{\nu_\alpha}_{\nu_\alpha}$, two different interaction matrix elements are needed:
\begin{itemize}
\item $\tilde{v}^{\nu_\alpha \nu_\beta}_{\nu_\alpha \nu_\beta}$ with $\alpha \neq \beta$, which are given in~\ref{eq:vNCnunu},
\item $\tilde{v}^{\nu_\alpha \nu_\alpha}_{\nu_\alpha \nu_\alpha}$, which are twice as great as the former matrix elements. Indeed, they consider to only \emph{one} term in the $\hat{H}_{NC}^{\nu \nu}$, but since they only involve one species, there is no rewriting~\eqref{eq:factor14offdiag}, which leads to an extra factor of \emph{four} --- all in all, there is an extra factor of 2.\footnote{In terms of Feynman diagrams, this corresponds to the possibility of coherent forward scattering with a $t-$ and a $u-$ channel (or $s-$channel for the interaction with $\bar{\nu}_\alpha$), contrary to the off-diagonal case where only one channel is available.}
\end{itemize}
This allows to write the matrix elements gathered in Table~\ref{Table:MatrixElements}.
Finally, the diagonal part of the self-interaction mean-field reads (we do not detail the interaction with antiparticles, which as usual gives an extra minus sign)
\begin{align}
\Gamma^{\nu_\alpha(\vec{p}_1,-)}_{\nu_\alpha(\vec{p}_3,-)} &\underset{\text{NC}}{=} \sqrt{2} G_F \times (2 \pi)^3 \, 2 p_1 \, \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \sum_{\beta} \int{\frac{\mathrm{d}^3\vec{p}_2}{(2 \pi)^3} \left(\varrho^{\beta}_\beta(p_2) - \bar{\varrho}^\beta_\beta(p_2)\right) \times [1 + \delta_{\alpha \beta}]} \nonumber \\
&\, = \sqrt{2} G_F \, \left(\mathbb{N} - \mathbb{N}_{\bar{\nu}}\right)^\alpha_\alpha \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} + \sqrt{2} G_F \, \mathrm{Tr} \left(\mathbb{N} - \mathbb{N}_{\bar{\nu}}\right) \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, .
\end{align}
The second term being flavour-independent, it does not contribute to flavour evolution (in other words, it is a contribution $\propto \mathbb{1}$ to the mean-field, hence its contribution vanishes inside the commutators).
\section{Collision term}
\label{app:collision_term}
We follow the same procedure as in section~\ref{subsec:collision_integral} to compute the remaining contributions to the collision term, that is the parts involving charged leptons. We then present the calculation of the antineutrino collision integral.
\subsection{Neutrino-electron scattering}
\label{subsec:Nu_e_scatt}
We use the general interaction matrix element~\eqref{eq:vtilde_nue_full}, such that our expression will be valid even for non-standard interactions (which are not considered in our numerical calculations):
\begin{multline}
\label{eq:vtilde_nu_e_app}
\tilde{v}^{\nu_\alpha(1) e(2)}_{\nu_\beta(3) e(4)} = 2 \sqrt{2} G_F \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_\alpha}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_\beta}^{h_3} (\vec{p}_3)] \ [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu (G_L^{\alpha \beta} P_L + G_R^{\alpha \beta} P_R ) u_{e}^{h_4} (\vec{p}_4)] \, .
\end{multline}
Let us compute the contribution to the scattering kernel for which $2=e^-,3=\nu_\gamma,3'=\nu_\delta,1'=\nu_\sigma$. We have:
\begin{align*}
\tilde{v}^{\nu_\alpha(1) e(2)}_{\nu_\gamma(3)e(4)} \times \tilde{v}^{\nu_\delta(3') e(4')}_{\nu_\sigma(1') e(2')} &= 8 G_F^2 (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) (2 \pi)^3 \delta^{(3)}(\vec{p}_{\underline{1}}+\vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\ &\qquad \qquad \sum_{h_2,h_3,h_4} [\bar{u}_{\nu_\alpha}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_\gamma}^{h_3} (\vec{p}_3)] [\bar{u}_{\nu_\delta}^{h_3} (\vec{p}_3) \gamma^\nu P_L u_{\nu_\sigma}^{h_1} (\vec{p}_1)] \\ \times [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu &(G_L^{\alpha \gamma} P_L + G_R^{\alpha \gamma} P_R ) u_{e}^{h_4} (\vec{p}_4)][\bar{u}_{e}^{h_4} (\vec{p}_4)\gamma_\nu (G_L^{\delta \sigma} P_L + G_R^{\delta \sigma} P_R ) u_{e}^{h_2} (\vec{p}_2)] \\
&= 8 G_F^2 \, (2\pi)^6 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \delta^{(3)}(\vec{p}_1-\vec{p}_{\underline{1}}) \\
&\times \mathrm{tr}\left[\gamma^\rho p_{1\rho} \gamma^\mu P_L \gamma^\lambda p_{3 \lambda} \gamma^\nu P_L \right] \\ &\times \mathrm{tr}\left[(\gamma_\tau p_2^\tau + m_e) \gamma_\mu (G_L^{\alpha \gamma} P_L + G_R^{\alpha \gamma} P_R ) (\gamma_\eta p_4^\eta + m_e) \gamma_\nu (G_L^{\delta \sigma} P_L + G_R^{\delta \sigma} P_R ) \right]
\end{align*}
The first trace reads (we use $\{P_L,\gamma^\mu\} = 0$):
\[p_{1\rho} p_{3 \lambda} \mathrm{tr}\left[\gamma^\rho \gamma^\mu P_L \gamma^\lambda \gamma^\nu P_L \right] = 2 \left[ p_1^\mu p_3^\nu + p_1^\nu p_3^\mu - (p_1 \cdot p_3) g^{\mu \nu} \right] + 2 {\mathrm i} p_{1\rho} p_{3 \lambda} \epsilon^{\rho \mu \lambda \nu} \, . \]
In the second, only even powers of $m_e$ survive:
\begin{align*}
\mathrm{tr}&\left[(\gamma_\tau p_2^\tau + m_e) \gamma_\mu (G_L^{\alpha \gamma} P_L + G_R^{\alpha \gamma} P_R ) (\gamma_\eta p_4^\eta + m_e) \gamma_\nu (G_L^{\delta \sigma} P_L + G_R^{\delta \sigma} P_R ) \right] \\
&= p_2^\tau p_4^\eta \mathrm{tr} \left[ \gamma_\tau \gamma_\mu \gamma_\eta \gamma_\nu (G_L^{\alpha \gamma} G_L^{\delta \sigma} P_L + G_R^{\alpha \gamma} G_R^{\delta \sigma} P_R) \right] + m_e^2 \mathrm{tr} \left[\gamma_\mu \gamma_\nu (G_R^{\alpha \gamma} G_L^{\delta \sigma} P_L + G_L^{\alpha \gamma} G_R^{\delta \sigma} P_R) \right]
\\ &= 2 \left[ p_{2\mu} p_{4\nu} + p_{2\nu} p_{4\mu} - (p_2 \cdot p_4) g_{\mu \nu} \right] (G_L^{\alpha \gamma} G_L^{\delta \sigma} + G_R^{\alpha \gamma} G_R^{\delta \sigma}) \\
&\qquad \qquad + 2 {\mathrm i} p_2^\tau p_4^\eta \epsilon_{\tau \mu \eta \nu} (G_L^{\alpha \gamma} G_L^{\delta \sigma} - G_R^{\alpha \gamma} G_R^{\delta \sigma}) + 2 m_e^2 g_{\mu \nu} (G_R^{\alpha \gamma} G_L^{\delta \sigma} + G_L^{\alpha \gamma} G_R^{\delta \sigma})
\end{align*}
The product of both terms reads:
\begin{itemize}
\item product of imaginary parts
\begin{multline*}
-4 p_{1\rho} p_{3 \lambda} p_2^\tau p_4^\eta (G_L^{\alpha \gamma} G_L^{\delta \sigma} - G_R^{\alpha \gamma} G_R^{\delta \sigma}) \epsilon^{\rho \mu \lambda \nu} \epsilon_{\tau \mu \eta \nu} \\
= 8 (G_L^{\alpha \gamma} G_L^{\delta \sigma} - G_R^{\alpha \gamma} G_R^{\delta \sigma}) \left[ (p_1 \cdot p_2) (p_3 \cdot p_4) - (p_1 \cdot p_4)(p_2 \cdot p_3) \right] \, ,
\end{multline*}
\item product of real parts, $\mathcal{O}(m_e^0)$
\begin{multline*}4 (G_L^{\alpha \gamma} G_L^{\delta \sigma} + G_R^{\alpha \gamma} G_R^{\delta \sigma}) \left[ p_1^\mu p_3^\nu + p_1^\nu p_3^\mu - (p_1 \cdot p_3) g^{\mu \nu} \right] \left[ p_{2\mu} p_{4\nu} + p_{2\nu} p_{4\mu} - (p_2 \cdot p_4) g_{\mu \nu} \right] \\
= 8 (G_L^{\alpha \gamma} G_L^{\delta \sigma} + G_R^{\alpha \gamma} G_R^{\delta \sigma}) \left[(p_1 \cdot p_2)(p_3 \cdot p_4) + (p_1 \cdot p_4)(p_2 \cdot p_3) \right] \, ,
\end{multline*}
\item product of real parts, $\mathcal{O}(m_e^2)$
\begin{multline*} 4 (G_R^{\alpha \gamma} G_L^{\delta \sigma} + G_L^{\alpha \gamma} G_R^{\delta \sigma}) \left[ p_1^\mu p_3^\nu + p_1^\nu p_3^\mu - (p_1 \cdot p_3) g^{\mu \nu} \right] m_e^2 g_{\mu \nu} \\
= - 8 (G_R^{\alpha \gamma} G_L^{\delta \sigma} + G_L^{\alpha \gamma} G_R^{\delta \sigma}) (p_1 \cdot p_3) m_e^2 \, .
\end{multline*}
\end{itemize}
Therefore,
\begin{align*}
\tilde{v}^{\nu_\alpha(1) e(2)}_{\nu_\gamma(3)e(4)} \times \tilde{v}^{\nu_\delta(3') e(4')}_{\nu_\sigma(1') e(2')} = 2^5 G_F^2 \, &(2\pi)^6 \, \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \, \delta^{(3)}(\vec{p}_1-\vec{p}_{\underline{1}}) \\
\qquad \times \Big[ &4 (p_1 \cdot p_2)(p_3 \cdot p_4) (G_L^{\alpha \gamma} G_L^{\delta \sigma} )+ 4 (p_1 \cdot p_4)(p_2 \cdot p_3) (G_R^{\alpha \gamma} G_R^{\delta \sigma}) \\
- \, &2 (p_1 \cdot p_3) m_e^2 (G_R^{\alpha \gamma} G_L^{\delta \sigma} + G_L^{\alpha \gamma} G_R^{\delta \sigma}) \Big]
\end{align*}
All $\tilde{v}\tilde{v}^*$ products in~\eqref{eq:C11} are equal to this, the only difference being the indices $\alpha,\gamma,\delta,\sigma$ which must respect the matrix structure. This expression is in perfect agreement with, for instance, Eq.~(2.10) of~\cite{Relic2016_revisited}. Schematically,
\[\underbrace{\tilde{v}^{\nu_\alpha e}_{\nu_\gamma e}}_{\to G_{\alpha \gamma}^a} \varrho^\gamma_\delta (3)f_e^{(4)} \underbrace{\tilde{v}^{\nu_\delta e}_{\nu_\sigma e}}_{\to G_{\delta \sigma}^b} (\mathbb{1} -\varrho^{(1)})^{\sigma}_{\beta} (1 -f_e^{(2)}) = f_e^{(4)}(1-f_e^{(2)}) \left[G^a \cdot \varrho^{(3)}\cdot G^b \cdot (\mathbb{1} -\varrho^{(1)})\right]^\alpha_\beta \, . \]
Remember that we get the same contribution exchanging $3$ and $4$. Therefore :
\begin{equation}
\label{eq:C_sc_cl}
\begin{aligned}
\mathcal{C}^{[\nu e^- \to \nu e^-]} = &(2 \pi)^3 \delta^{(3)}(\vec{p}_1-\vec{p}_{\underline{1}}) \frac{2^5 G_F^2}{2}\int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_3] [\mathrm{d}^3 \vec{p}_4] (2 \pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)} \\
&\Big[ 4 (p_1 \cdot p_2)(p_3 \cdot p_4) F_\mathrm{sc}^{LL}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) \\
&+ 4 (p_1 \cdot p_4)(p_2 \cdot p_3) F_\mathrm{sc}^{RR}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) \\
&- 2 (p_1 \cdot p_3) m_e^2 \left(F_\mathrm{sc}^{LR}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) + F_\mathrm{sc}^{RL}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) \right) \Big] \, .
\end{aligned}
\end{equation}
\noindent The statistical factors read:
\begin{multline}
\label{eq:F_sc_cl}
F_\mathrm{sc}^{AB}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) = f_4 (1-f_2) \left [ G^A \varrho_3 G^B (\mathbb{1} -\varrho_1) + (\mathbb{1} - \varrho_1) G^B \varrho_3 G^A \right] \\ - (1-f_4)f_2 \left [ G^A (\mathbb{1} -\varrho_3) G^B \varrho_1 + \varrho_1 G^B (\mathbb{1} -\varrho_3) G^A \right] \, ,
\end{multline}
with the compact notations $f_i =f_e^{(i)} = f_e(p_i)$ and $\varrho_i = \varrho^{(i)} = \varrho(p_i)$.
\vspace{0.5cm}
The other scattering amplitudes can all be obtained via "crossing symmetry" methods, as will be shown in the next two sections.
\subsection{Neutrino-positron scattering}
Now note that the relevant matrix elements are (see Table~\ref{Table:MatrixElements})
\begin{multline}
\tilde{v}^{\nu_\alpha(1) \bar{e}(2)}_{ \nu_\beta(3) \bar{e}(4)} = - 2 \sqrt{2} G_F \ (2 \pi)^3 \, \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_\alpha}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_\beta}^{h_3} (\vec{p}_3)] \ [\bar{v}_{e}^{h_4} (\vec{p}_4)\gamma_\mu (G_L^{\alpha \beta} P_L + G_R^{\alpha \beta} P_R ) v_{e}^{h_2} (\vec{p}_2)]
\end{multline}
Therefore, the matrix structure is exactly identical, but we have to interchange in the matrix elements $p_2 \leftrightarrow p_4$.
All in all, the scattering with charged leptons reads:
\begin{equation}
\label{eq:C_sc_cl_full}
\begin{aligned}
\mathcal{C}^{[\nu e^\pm \to \nu e^\pm]} = &(2 \pi)^3 \delta^{(3)}(\vec{p}_1-\vec{p}_{\underline{1}}) \frac{2^5 G_F^2}{2}\int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_3] [\mathrm{d}^3 \vec{p}_4] (2 \pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)} \\
&\Big[ 4 (p_1 \cdot p_2)(p_3 \cdot p_4) \left(F_\mathrm{sc}^{LL}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) + F_\mathrm{sc}^{RR}(\nu^{(1)},\bar{e}^{(2)},\nu^{(3)},\bar{e}^{(4)})\right) \\
&+ 4 (p_1 \cdot p_4)(p_2 \cdot p_3) \left(F_\mathrm{sc}^{RR}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) + F_\mathrm{sc}^{LL}(\nu^{(1)},\bar{e}^{(2)},\nu^{(3)},\bar{e}^{(4)}) \right) \\
&- 2 (p_1 \cdot p_3) m_e^2 \left(F_\mathrm{sc}^{LR}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) + F_\mathrm{sc}^{LR}(\nu^{(1)},\bar{e}^{(2)},\nu^{(3)},\bar{e}^{(4)}) + \{L \leftrightarrow R \}\right) \Big] \, .
\end{aligned}
\end{equation}
\subsection{Neutrino-antineutrino annihilation}
Now, the relevant matrix element is:
\begin{multline}
\tilde{v}^{\nu_\alpha(1) \bar{\nu}_\beta(2)}_{ e(3) \bar{e}(4)} = - 2 \sqrt{2} G_F \ (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_\alpha}^{h_1} (\vec{p}_1) \gamma^\mu P_L v_{\nu_\beta}^{h_2} (\vec{p}_2)] \ [\bar{v}_{e}^{h_4} (\vec{p}_4)\gamma_\mu (G_L^{\alpha \beta} P_L + G_R^{\alpha \beta} P_R ) u_{e}^{h_3} (\vec{p}_3)] \, .
\end{multline}
\noindent Two remarks must be made:
\begin{itemize}
\item Thanks to the transposed definition of $\bar{\varrho}$, the statistical factor keeps a simple expression not involving extra transpositions. Indeed, we have:
\[\underbrace{\tilde{v}^{\nu_\alpha \bar{\nu}_\gamma}_{e \bar{e}}}_{\to G_{\alpha \gamma}^a} f_e^{(3)} \bar{f}_e^{(4)} \underbrace{\tilde{v}^{e \bar{e}}_{\nu_\delta \bar{\nu}_\sigma}}_{\to G_{\sigma \delta}^b} (\mathbb{1}-\varrho^{(1)})^\delta_\beta \underbrace{(\mathbb{1}-\varrho^{(2)})^{\bar{\sigma}}_{\bar{\gamma}}}_{=(\mathbb{1}-\bar{\varrho}^{(2)})^\gamma_\sigma} = f_e^{(3)}\bar{f}_e^{(4)} \left[G^a \cdot (\mathbb{1}-\bar{\varrho}^{(2)})\cdot G^b \cdot (\mathbb{1}-\varrho^{(1)})\right]^\alpha_\beta \, .\]
\item As can be seen with the associated Feynman diagrams, the amplitudes are obtained from $\nu + e^-$ scattering through:
\[p_1 \to p_1 \ ; \ p_2 \to - p_4 \ ; \ p_3 \to - p_2 \ ; \ p_4 \to p_3 \, . \]
If we stay in our formalism, the product of $\tilde{v}^{\nu \bar{\nu}}_{e \bar{e}} \times \tilde{v}^{e \bar{e}}_{\nu \bar{\nu}}$ involves the combinations $u_e^{h_3}(\vec{p}_3) \bar{u}_e^{h_3}(\vec{p}_3)$ and $v_e^{h_4}(\vec{p}_4) \bar{v}_e^{h_4}(\vec{p}_4)$. Therefore the $m_e^2$ term reverses sign [product $(\gamma_\rho p_3^\rho + m_e)(\gamma_\omega p_4^\omega - m_e)$].
\end{itemize}
This leads to the following statistical factor:
\begin{multline}
\label{eq:F_ann_cl}
F_\mathrm{ann}^{AB}(\nu^{(1)},\bar{\nu}^{(2)},e^{(3)},\bar{e}^{(4)}) = f_3 \bar{f}_4 \left[ G^A (1-\bar{\varrho}_2) G^B (1-\varrho_1) + (1- \varrho_1) G^B (1-\bar{\varrho}_2) G^A \right] \\ - (1-f_3)(1-\bar{f}_4) \left[ G^A \bar{\varrho}_2 G^B \varrho_1 + \varrho_1 G^B \bar{\varrho}_2 G^A \right] \, ,
\end{multline}
and the collision term contribution
\begin{equation}
\label{eq:C_ann_cl}
\begin{aligned}
\mathcal{C}^{[\nu \bar{\nu} \to e^- e^+]} = &(2 \pi)^3 \delta^{(3)}(\vec{p}_1-\vec{p}_{\underline{1}}) \frac{2^5 G_F^2}{2}\int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_3] [\mathrm{d}^3 \vec{p}_4] (2 \pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)} \\
&\Big[ 4 (p_1 \cdot p_4)(p_2 \cdot p_3) F_\mathrm{ann}^{LL}(\nu^{(1)},\bar{\nu}^{(2)},e^{(3)},\bar{e}^{(4)}) \\
&+ 4 (p_1 \cdot p_3)(p_2 \cdot p_4) F_\mathrm{ann}^{RR}(\nu^{(1)},\bar{\nu}^{(2)},e^{(3)},\bar{e}^{(4)}) \\
&+ 2 (p_1 \cdot p_2) m_e^2 \left(F_\mathrm{ann}^{LR}(\nu^{(1)},\bar{\nu}^{(2)},e^{(3)},\bar{e}^{(4)}) + F_\mathrm{ann}^{RL}(\nu^{(1)},\bar{\nu}^{(2)},e^{(3)},\bar{e}^{(4)}) \right) \Big] \, .
\end{aligned}
\end{equation}
\subsection{Antineutrino collision term}
The antineutrino collision term $\bar{\mathcal{C}}$ for $\bar{\varrho}$ can be deduced from $\mathcal{C}$ based on the transformations $\varrho \leftrightarrow \bar{\varrho}$ and $L \leftrightarrow R$. Let us prove it explicitly.
As explained in section~\ref{app:collision_term}, the evolution equation for $\bar{\varrho}^\alpha_\beta$ is obtained similarly to the one for $\varrho^\beta_\alpha$. In other words, $\bar{\mathcal{C}}^\alpha_\beta$ is calculated with the general formula~\eqref{eq:C11}, where we take $i_1 = \bar{\nu}_\beta(\vec{p}_1)$ and $i_1' = \bar{\nu}_\alpha(\vec{p}_1)$.
\subsubsection{Antineutrino - charged lepton scattering}
The relevant matrix element is now (note the minus sign with respect to~\eqref{eq:vtilde_nu_e_app}, as explained in section~\ref{app:antiparticles}):
\begin{multline}
\tilde{v}^{\bar{\nu}_\beta(1) e(2)}_{\bar{\nu}_\alpha(3) e(4)} = - 2 \sqrt{2} G_F \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{v}_{\nu_\alpha}^{h_3} (\vec{p}_3) \gamma^\mu P_L v_{\nu_\beta}^{h_1} (\vec{p}_1)] \ [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu (G_L^{\alpha \beta} P_L + G_R^{\alpha \beta} P_R ) u_{e}^{h_4} (\vec{p}_4)] \, .
\end{multline}
When inserting it in the collision term formula~\eqref{eq:C11}, the only differences with respect to the derivation of subsection~\ref{subsec:Nu_e_scatt} are the transposition of the $G$ matrices --- which actually corresponds to the exchange of the first and second lines in~\eqref{eq:C11} ---, and most importantly the exchange $p_1 \leftrightarrow p_3$. Therefore, the collision term reads
\begin{equation}
\label{eq:Cbar_sc_cl}
\begin{aligned}
\bar{\mathcal{C}}^{[\bar{\nu} e^- \to \bar{\nu} e^-]} = &(2 \pi)^3 \delta^{(3)}(\vec{p}_1-\vec{p}_{\underline{1}}) \frac{2^5 G_F^2}{2}\int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_3] [\mathrm{d}^3 \vec{p}_4] (2 \pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)} \\
&\Big[ 4 (p_1 \cdot p_4)(p_2 \cdot p_3) F_\mathrm{sc}^{LL}(\bar{\nu}^{(1)},e^{(2)},\bar{\nu}^{(3)},e^{(4)}) \\
&+ 4 (p_1 \cdot p_2)(p_3 \cdot p_4) F_\mathrm{sc}^{RR}(\bar{\nu}^{(1)},e^{(2)},\bar{\nu}^{(3)},e^{(4)}) \\
&- 2 (p_1 \cdot p_3) m_e^2 \left(F_\mathrm{sc}^{LR}(\bar{\nu}^{(1)},e^{(2)},\bar{\nu}^{(3)},e^{(4)}) + F_\mathrm{sc}^{RL}(\bar{\nu}^{(1)},e^{(2)},\bar{\nu}^{(3)},e^{(4)}) \right) \Big] \, ,
\end{aligned}
\end{equation}
which is indeed the neutrino collision term~\eqref{eq:C_sc_cl}, with the replacement $\varrho \to \bar{\varrho}$ and $L \leftrightarrow R$. The statistical factor is given by~\eqref{eq:F_sc_cl} with antineutrino density matrices, that is
\begin{multline}
\label{eq:F_sc_bar_cl}
F_\mathrm{sc}^{AB}(\bar{\nu}^{(1)},e^{(2)},\bar{\nu}^{(3)},e^{(4)}) = f_4 (1-f_2) \left [ G^A \bar{\varrho}_3 G^B (\mathbb{1} -\bar{\varrho}_1) + (\mathbb{1} - \bar{\varrho}_1) G^B \bar{\varrho}_3 G^A \right] \\ - (1-f_4)f_2 \left [ G^A (\mathbb{1} -\bar{\varrho}_3) G^B \bar{\varrho}_1 + \bar{\varrho}_1 G^B (\mathbb{1} -\bar{\varrho}_3) G^A \right] \, .
\end{multline}
To avoid any confusion with the possible transpositions, let us justify one term of the statistical factor. The first term in~\eqref{eq:C11} reads
\[\underbrace{\tilde{v}^{\bar{\nu}_\beta e}_{\bar{\nu}_\gamma e}}_{\to G_{\gamma \beta}^A} \overbrace{\varrho^{\bar{\gamma}}_{\bar{\delta}}(3)}^{\bar{\varrho}^\delta_\gamma(3)} f_e^{(4)} \underbrace{\tilde{v}^{\bar{\nu}_\delta e}_{\bar{\nu}_\sigma e}}_{\to G_{\sigma \delta}^B} \overbrace{(\mathbb{1} -\varrho^{(1)})^{\bar{\sigma}}_{\bar{\alpha}}}^{(\mathbb{1} - \bar{\varrho}^{(1)})^\alpha_\sigma} (1 -f_e^{(2)}) = f_e^{(4)}(1-f_e^{(2)}) \left[(\mathbb{1} -\bar{\varrho}^{(1)}) \cdot G^B \cdot \bar{\varrho}^{(3)} \cdot G^A \right]^\alpha_\beta \, , \]
which is the second term in~\eqref{eq:F_sc_bar_cl}.
The scattering with positrons is treated in the same fashion.
\subsubsection{Antineutrino - neutrino annihilation}
In this case, the appropriate exchange is $p_1 \leftrightarrow p_2$. The results then read
\begin{multline}
F_\mathrm{ann}^{AB}(\bar{\nu}^{(1)},\nu^{(2)},e^{(3)},\bar{e}^{(4)}) = f_3 \bar{f}_4 \left[ G^A (1-\varrho_2) G^B (1-\bar{\varrho}_1) + (1- \bar{\varrho}_1) G^B (1-\varrho_2) G^A \right] \\ - (1-f_3)(1-\bar{f}_4) \left[ G^A \varrho_2 G^B \bar{\varrho}_1 + \bar{\varrho}_1 G^B \varrho_2 G^A \right] \, ,
\end{multline}
and
\begin{equation}
\begin{aligned}
\bar{\mathcal{C}}^{[\bar{\nu} \nu \to e^- e^+]} = &(2 \pi)^3 \delta^{(3)}(\vec{p}_1-\vec{p}_{\underline{1}}) \frac{2^5 G_F^2}{2}\int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_3] [\mathrm{d}^3 \vec{p}_4] (2 \pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)} \\
&\Big[ 4 (p_1 \cdot p_3)(p_2 \cdot p_4) F_\mathrm{ann}^{LL}(\bar{\nu}^{(1)},\nu^{(2)},e^{(3)},e^{(4)}) \\
&+ 4 (p_1 \cdot p_4)(p_2 \cdot p_3) F_\mathrm{ann}^{RR}(\bar{\nu}^{(1)},\nu^{(2)}, e^{(3)},e^{(4)}) \\
&+ 2 (p_1 \cdot p_2) m_e^2 \left(F_\mathrm{ann}^{LR}(\bar{\nu}^{(1)},\nu^{(2)},e^{(3)},e^{(4)}) + F_\mathrm{ann}^{RL}(\bar{\nu}^{(1)},\nu^{(2)},e^{(3)},e^{(4)}) \right) \Big] \, .
\end{aligned}
\end{equation}
Once again, they correspond to Eqs.~\eqref{eq:F_ann_cl} and~\eqref{eq:C_ann_cl} with the changes $\varrho \leftrightarrow \bar{\varrho}$ and $L \leftrightarrow R$.
\subsubsection{Antineutrino self-interactions}
\paragraph{Antineutrino-antineutrino scattering}
We can compare the relevant matrix element:
\begin{multline}
\tilde{v}^{\bar{\nu}_\beta(1) \bar{\nu}_\alpha(2)}_{ \bar{\nu}_\beta(3) \bar{\nu}_\alpha(4)} = (1 + \delta_{\alpha \beta}) \times \sqrt{2} G_F \ (2 \pi)^3 \, \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{v}_{\nu_\beta}^{h_3} (\vec{p}_3) \gamma^\mu P_L v_{\nu_\beta}^{h_1} (\vec{p}_1)] \ [\bar{v}_{\nu_\alpha}^{h_4} (\vec{p}_4)\gamma_\mu P_L v_{\nu_\alpha}^{h_2} (\vec{p}_2)]
\end{multline}
with the one for neutrino-neutrino scattering given in Table~\ref{Table:MatrixElements}. Apart from the replacement of $u$ spinors by $v$ spinors, the appropriate exchanges are $1 \leftrightarrow 3$ and $2 \leftrightarrow 4$, which leaves the scattering amplitude unchanged compared to neutrino-neutrino scattering (see the calculation in section~\ref{subsec:collision_integral}), i.e. it will still be $(p_1 \cdot p_2)(p_3 \cdot p_4)$.
Concerning the statistical factor, we have for instance:
\[\tilde{v}^{\bar{\nu}_\beta \bar{\nu}_\gamma}_{\bar{\nu}_\gamma \bar{\nu}_\beta} \varrho^{\bar{\gamma}}_{\bar{\sigma}}(3) \varrho^{\bar{\beta}}_{\bar{\delta}}(4) \tilde{v}^{\bar{\nu}_\sigma \bar{\nu}_\delta}_{\bar{\nu}_\sigma \bar{\nu}_\delta} (\mathbb{1} -\varrho^{(1)})^{\bar{\sigma}}_{\bar{\alpha}} (\mathbb{1} -\varrho^{(2)})^{\bar{\delta}}_{\bar{\gamma}} \propto \left[(\mathbb{1} -\bar{\varrho}^{(1)}) \cdot \bar{\varrho}^{(3)} \cdot (\mathbb{1} - \bar{\varrho}^{(2)}) \cdot \bar{\varrho}^{(4)} \right]^\alpha_\beta \, , \]
such that the full statistical factor is identical to~\eqref{eq:F_sc_nn} with the replacement $\varrho \to \bar{\varrho}$, that is
\begin{multline}
\label{eq:F_sc_bnbn}
F_\mathrm{sc}(\bar{\nu}^{(1)},\bar{\nu}^{(2)},\bar{\nu}^{(3)},\bar{\nu}^{(4)}) = \left[ \bar{\varrho}_4 (\mathbb{1} - \bar{\varrho}_4) + \mathrm{Tr}(\cdots) \right] \bar{\varrho}_3 (\mathbb{1} -\bar{\varrho}_1) + (\mathbb{1} - \bar{\varrho}_1) \bar{\varrho}_3 \left[ (\mathbb{1} - \bar{\varrho}_2) \bar{\varrho}_4 + \mathrm{Tr}(\cdots)\right] \\
- \left[ (\mathbb{1} - \bar{\varrho}_4) \bar{\varrho}_2 + \mathrm{Tr}(\cdots)\right] (\mathbb{1} -\bar{\varrho}_3) \bar{\varrho}_1 - \bar{\varrho}_1 (\mathbb{1} -\bar{\varrho}_3) \left[\bar{\varrho}_2(\mathbb{1} -\bar{\varrho}_4) + \mathrm{Tr}(\cdots)\right] \, .
\end{multline}
\paragraph{Antineutrino-neutrino scattering/annihilation} Now the appropriate exchange from the neutrino-antineutrino matrix elements is $p_1 \leftrightarrow p_2$ and $p_3 \leftrightarrow p_4$, such that the prefactor $(p_1 \cdot p_2)(p_3 \cdot p_4)$ is left invariant.
The expressions for the statistical factors are Eqs.~\eqref{eq:F_sc_nbn} and~\eqref{eq:F_ann_nn} with $\varrho \leftrightarrow \bar{\varrho}$:
\begin{multline}
\label{eq:F_sc_bnn}
F_\mathrm{sc}(\bar{\nu}^{(1)},{\nu}^{(2)},\bar{\nu}^{(3)},{\nu}^{(4)}) = \left[ (\mathbb{1} - {\varrho}_2) {\varrho}_4 + \mathrm{Tr}(\cdots) \right] \bar{\varrho}_3 (\mathbb{1} -\bar{\varrho}_1) + (\mathbb{1} - \bar{\varrho}_1) \bar{\varrho}_3 \left[ {\varrho}_4 (\mathbb{1} - {\varrho}_2) + \mathrm{Tr}(\cdots)\right] \\
- \left[ {\varrho}_2 (\mathbb{1} -{\varrho}_4) + \mathrm{Tr}(\cdots) \right] (\mathbb{1} -\bar{\varrho}_3) \bar{\varrho}_1 - \bar{\varrho}_1 (\mathbb{1} -\bar{\varrho}_3) \left[ (\mathbb{1} -{\varrho}_4) {\varrho}_2 + \mathrm{Tr}(\cdots)\right] \, ,
\end{multline}
\begin{multline}
\label{eq:F_ann_bnbn}
F_\mathrm{ann}(\bar{\nu}^{(1)},{\nu}^{(2)},\bar{\nu}^{(3)},{\nu}^{(4)}) = \left[ \bar{\varrho}_3 {\varrho}_4 + \mathrm{Tr}(\cdots) \right] (\mathbb{1} -{\varrho}_2) (\mathbb{1} -\bar{\varrho}_1) + (\mathbb{1} - \bar{\varrho}_1) (\mathbb{1} -{\varrho}_2) \left[ {\varrho}_4 \bar{\varrho}_3 + \mathrm{Tr}(\cdots)\right] \\
- \left[ (\mathbb{1} -\bar{\varrho}_3) (\mathbb{1} -{\varrho}_4) + \mathrm{Tr}(\cdots) \right] {\varrho}_2 \bar{\varrho}_1 - \bar{\varrho}_1 {\varrho}_2 \left[ (\mathbb{1} -{\varrho}_4) (\mathbb{1} -\bar{\varrho}_3) + \mathrm{Tr}(\cdots)\right] \, .
\end{multline}
\vspace{0.5cm}
\noindent Finally, the full self-interaction antineutrino collision term reads
\begin{equation}
\begin{aligned}
\bar{\mathcal{C}}^{[\bar{\nu} \bar{\nu}]} = &(2 \pi)^3 \delta^{(3)}(\vec{p}_1-\vec{p}_{\underline{1}}) \frac{2^5 G_F^2}{2}\int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_3] [\mathrm{d}^3 \vec{p}_4] (2 \pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)} \\
&\Big[ 4 (p_1 \cdot p_2)(p_3 \cdot p_4) F_\mathrm{sc}(\bar{\nu}^{(1)},\bar{\nu}^{(2)},\bar{\nu}^{(3)},\bar{\nu}^{(4)}) \\
&+ (p_1 \cdot p_4)(p_2 \cdot p_3)\left( F_\mathrm{sc}(\bar{\nu}^{(1)},\nu^{(2)}, \bar{\nu}^{(3)}, \nu^{(4)} + F_\mathrm{ann}(\bar{\nu}^{(1)},\nu^{(2)}, \bar{\nu}^{(3)}, \nu^{(4)}) \right) \Big] \, .
\end{aligned}
\end{equation}
\section{Reduction of the collision integral}
\label{app:reduc_collision_integral}
For completeness, we detail the reduction of the collision integral from nine to two dimensions, following~\cite{Dolgov_NuPhB1997}.
\subsection{Method}
The collision integral for each reaction reads generally, as shown in the calculations of the previous sections (recall that $\mathcal{C} = (2 \pi)^3 \, 2 E_1\, \delta^{(3)}(\vec{p}_1 - \vec{p}_{\underline{1}}) \mathcal{I}[\varrho]$):
\begin{equation}
\label{eq:general_I_app}
\mathcal{I}= \frac{1}{2 E_1} \int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_3] [\mathrm{d}^3 \vec{p}_4] \, (2 \pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)} \times S \langle \abs{\mathcal{M}}^2\rangle \times F[\varrho] \, ,
\end{equation}
with $F[\varrho]$ the statistical factor, $\langle \abs{\mathcal{M}}^2 \rangle$ the reaction matrix element and $S$ the symmetrization factor.\footnote{The symmetrization factor appears as part of the usual Feynman rules in diagrammatic Quantum Field Theory. Within the BBGKY formalism, these numerical prefactors arise from the combination of the interaction matrix elements $\tilde{v}$ — which is absolutely equivalent.} As explained in section~\ref{subsec:reduced_equations_QKE}, the key trick then consists in using the integral representation of the Dirac delta function:
\[ \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) = \int{\frac{\mathrm{d}^3 \vec{\lambda}}{(2 \pi)^3} \, e^{{\mathrm i} \vec{\lambda} \cdot (\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4)}} \, , \]
and decompose the entire collision integral with spherical coordinates. The "$\vec{e}_\mathrm{z}$ unit vector" for $\vec{\lambda}$ is aligned with $\vec{p}_1$, while $\vec{\lambda}$ is the "$\vec{e}_\mathrm{z}$ unit vector" for $\vec{p}_{i \geq 2}$, that is,
\[ \cos{\theta_\lambda} \equiv \frac{\vec{p}_1 \cdot \vec{\lambda}}{p_1 \lambda} \qquad ; \qquad \cos{\theta_i} \equiv \frac{\vec{p}_i \cdot \vec{\lambda}}{p_i \lambda} \ \ \text{for } i = 2,3,4 \, , \]
the associated azimuthal angles $\varphi_\lambda, \varphi_{i \geq 2}$ being defined as usual. Recalling that $[\mathrm{d}^3 \vec{p}] = \mathrm{d}^3 \vec{p}/(2 \pi)^3 2 E = p^2 \mathrm{d}{p} \mathrm{d}{\Omega} / (2 \pi)^3 2 E$, with $\mathrm{d}{\Omega} = \sin{\theta} \mathrm{d}{\theta} \mathrm{d}{\varphi}$ for the solid angles, we rewrite~\eqref{eq:general_I_app}
\begin{align*}
\mathcal{I} &= \frac{1}{2^4 E_1} \frac{1}{(2 \pi)^8} \int{\lambda^2 \mathrm{d}{\lambda} \mathrm{d}{\Omega_\lambda}}\prod_{i=2}^{4}\frac{p_i^2 \mathrm{d}{p_i} \mathrm{d}{\Omega_i}}{E_i} \, e^{{\mathrm i} \vec{\lambda}\cdot(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4)} \\
&\qquad \qquad \qquad \times \delta(E_1 + E_2 - E_3 - E_4) \, S \langle \abs{\mathcal{M}}^2\rangle \, F[\varrho] \\
&= \frac{1}{2^6 \pi^3 E_1 p_1} \int{\prod_{i=2}^{4}{\frac{p_i \mathrm{d}{p_i}}{E_i}} \, \delta(E_1 + E_2 - E_3 - E_4) \, F[\varrho] \times D(p_1, p_2, p_3, p_4)} \, ,
\end{align*}
where we defined
\begin{multline}
\label{eq:definition_Dfunction}
D(p_1,p_2,p_3,p_4) \equiv \frac{p_1 p_2 p_3 p_4}{2^6 \pi^5} \int_{0}^{\infty}{\lambda^2 \mathrm{d}{\lambda}} \int{e^{{\mathrm i} \vec{p}_1 \cdot \vec{\lambda}} \mathrm{d}{\Omega_\lambda}} \int{e^{{\mathrm i} \vec{p}_2 \cdot \vec{\lambda}} \mathrm{d}{\Omega_2}} \\ \int{e^{-{\mathrm i} \vec{p}_3 \cdot \vec{\lambda}} \mathrm{d}{\Omega_3}} \int{e^{-{\mathrm i} \vec{p}_4 \cdot \vec{\lambda}} \mathrm{d}{\Omega_4}} \ S \langle \abs{\mathcal{M}}^2\rangle \, .
\end{multline}
It is finally the particular form of the matrix elements $S \langle \abs{\mathcal{M}}^2\rangle$ which allows for further simplifications. Indeed, in the Fermi approximation of weak interactions, there are only two kinds of matrix elements (see equations~\eqref{eq:C_nnscatt}, \eqref{eq:C_sc_cl_full}, \eqref{eq:C_ann_cl} and similarly for antineutrinos):
\begin{equation}
\label{eq:app_matrix_el}
\begin{aligned}
K (q_1 \cdot q_2)(q_3 \cdot q_4) &= K (E_1 E_2 - \vec{q}_1 \cdot \vec{q}_2)(E_3 E_4 - \vec{q}_3 \cdot \vec{q}_4) \\
\text{and} \qquad \qquad K' m_e^2 (q_3 \cdot q_4) &= K' m_e^2 (E_3 E_4 - \vec{q}_3 \cdot \vec{q}_4) \, ,
\end{aligned}
\end{equation}
where each $q_i$ corresponds to one of the $p_i$. The scalar products appearing in the matrix elements are thus explicitly
\[ \vec{q}_i \cdot \vec{q}_j = q_i q_j (\sin{\theta_i} \sin{\theta_j} \cos(\varphi_i - \varphi_j) + \cos{\theta_i} \cos{\theta_j}) \, , \]
but the first term vanishes after the $\varphi$ integration thanks to the isotropy of the system.\footnote{Simply stated, it comes from $\int_{0}^{2 \pi}{\cos{\varphi} \mathrm{d}{\varphi}} = 0$.} With the second term, the $\varphi$ integrations give factors of $(2 \pi)$, which leads to two possible kinds of integrals appearing in $D$ depending on the matrix element:
\begin{subequations}
\label{eq:plus_ou_moins}
\begin{align}
\int{e^{\pm {\mathrm i} \lambda q \cos{\theta}} \sin{\theta} \, \mathrm{d}{\theta}} &= \frac{2}{\lambda q} \sin{(\lambda q)} \, , \\
\int{e^{\pm {\mathrm i} \lambda q \cos{\theta}} \cos{\theta} \sin{\theta} \, \mathrm{d}{\theta}} &= \mp \frac{2 {\mathrm i}}{\lambda q} \left[\cos{(\lambda q)} - \frac{\sin{(\lambda q)}}{\lambda q}\right] \, .
\end{align}
\end{subequations}
With this intermediate result, we can perform all integrations except the $\lambda$ one, and obtain the expressions for the $D-$functions depending on the matrix element gathered in Table~\ref{tab:integral_D_functions}, where we define $D_1$, $D_2$ and $D_3$.
\renewcommand{\arraystretch}{0.1}
\begin{table}[!htb]
\centering
\begin{tabular}{|M{2.1cm} |M{12cm} |}
\hline
\vspace{0.15cm} $S \langle \abs{\mathcal{M}}^2\rangle$ \vspace{0.15cm} & $D(q_1,q_2,q_3,q_4)$ \\ \hline \hline
\[1\] & \[D_1 = \frac{4}{\pi} \int_{0}^{\infty}{\frac{\mathrm{d} \lambda}{\lambda^2} \sin{(\lambda q_1)} \sin{(\lambda q_2)} \sin{(\lambda q_3)} \sin{(\lambda q_4)}}\] \\
\[- \vec{q}_3 \cdot \vec{q}_4\] & \begin{multline*} D_2(3,4) = \frac{4 q_3 q_4}{\pi} \int_{0}^{\infty}\frac{\mathrm{d} \lambda}{\lambda^2} \sin{(\lambda q_1)} \sin{(\lambda q_2)} \\ \times \left[\cos{(\lambda q_3)} - \frac{\sin{(\lambda q_3)}}{\lambda q_3}\right] \left[\cos{(\lambda q_4)} - \frac{\sin{(\lambda q_4)}}{\lambda q_3}\right] \end{multline*} \\
\[(\vec{q}_1 \cdot \vec{q}_2)(\vec{q}_3 \cdot \vec{q}_4)\] & \[D_3 = \frac{4 q_1 q_2 q_3 q_4}{\pi} \int_{0}^{\infty}\frac{\mathrm{d} \lambda}{\lambda^2} \left[\cos{(\lambda q_1)} - \frac{\sin{(\lambda q_1)}}{\lambda q_1}\right] \! \cdots \! \left[\cos{(\lambda q_4)} - \frac{\sin{(\lambda q_4)}}{\lambda q_3}\right]\] \\ \hline
\end{tabular}
\caption[Integral expression of the $D-$functions]{Integral expression of the $D-$functions. It is important to emphasize that if the two arguments of $D_2$ do not correspond to both incoming or both outgoing particles, it changes sign — see equation~\eqref{eq:plus_ou_moins}.
\label{tab:integral_D_functions}}
\end{table}
\renewcommand{\arraystretch}{1}
When inserting the matrix elements~\eqref{eq:app_matrix_el} in the definition~\eqref{eq:definition_Dfunction}, we thus get\footnote{There is a typo in equation (A.14) of~\cite{Dolgov_NuPhB1997}.}
\begin{equation}
\begin{aligned}
D &= K \left[ E_1 E_2 E_3 E_4 D_1 + D_3 + E_1 E_2 D_2(3,4) + E_3 E_4 D_2(1,2) \right] \\
\text{and} \qquad \quad D &= K' m_e^2 \left[E_3 E_4 D_1 + D_2(3,4) \right] \, .
\end{aligned}
\end{equation}
\subsection{Expressions of the $D-$functions}
As shown in the expressions of Table~\ref{tab:integral_D_functions}, $D_1$ and $D_3$ are symmetric with respect to permutations of any variables, while $D_2$ is symmetric under the permutations $1 \leftrightarrow 2$ and $3 \leftrightarrow 4$. Therefore, we give in the following the expressions in the case $q_1 > q_2$ and $q_3 > q_4$, without loss of generality.
Distinguishing the following four physical cases, the integrals of Table~\ref{tab:integral_D_functions} can be computed and simplified using, e.g., \emph{Mathematica}.
\paragraph{Case (a)} $q_1 + q_2 > q_3 + q_4$, $q_1 + q_4 > q_2 + q_3$ and $q_1 \leq q_2 + q_3 + q_4$
\begin{subequations}
\begin{align}
D_1 &= \frac12 (q_2 + q_3 + q_4 - q_1) \, , \\
D_2 &= \frac{1}{12} \left( (q_1 - q_2)^3 + 2 (q_3^3 + q_4^3) - 3 (q_1 - q_2)(q_3^2 + q_4^2) \right) \, , \\
D_3 &= \frac{1}{60} \left(q_1^5 - 5 q_1^3 q_2^2 + 5 q_1^2 q_2^3 - q_2^5 - 5 q_1^3 q_3^2 + 5 q_2^3 q_3^2 + 5 q_1^2 q_3^3 + 5 q_2^2 q_3^3 \right. \nonumber \\
&\qquad \quad \left. - q_3^5 - 5 q_1^3q_4^2 + 5 q_2^3 q_4^2 + 5 q_3^3 q_4^2 + 5 q_1^2 q_3^3 + 5 q_2^2 q_4^3 + 5 q_3^2 q_4^3 - q_4^5 \right) \, .
\end{align}
\end{subequations}
Having $q_1 > q_2 + q_3 + q_4$ would be unphysical, and yields $D_1 = D_2 = D_3 = 0$.
\paragraph{Case (b)} $q_1 + q_2 > q_3 + q_4$ and $q_1 + q_4 < q_2 + q_3$
\begin{subequations}
\begin{align}
D_1 &= q_4 \, , \\
D_2 &= \frac13 q_4^3 \, , \\
D_3 &= \frac{1}{30} q_4^3 \left(5 q_1^2 + 5 q_2^2 + 5 q_3^2 - q_4^2 \right) \, .
\end{align}
\end{subequations}
\paragraph{Case (c)} $q_1 + q_2 < q_3 + q_4$, $q_1 + q_4 < q_2 + q_3$ and $q_3 \leq q_1 + q_2 + q_4$
\begin{subequations}
\begin{align}
D_1 &= \frac12 (q_1 + q_2 + q_3 - q_4) \, , \\
D_2 &= \frac{1}{12} \left( - (q_1 + q_2)^3 - 2 q_3^3 + 2 q_4^3 + 3 (q_1 + q_2)(q_3^2 + q_4^2) \right) \, , \\
D_3 &= \frac{1}{60} \left(- q_1^5 + 5 q_1^3 q_2^2 + 5 q_1^2 q_2^3 - q_2^5 + 5 q_1^3 q_3^2 + 5 q_2^3 q_3^2 - 5 q_1^2 q_3^3 - 5 q_2^2 q_3^3 \right. \nonumber \\
&\qquad \quad \left. + q_3^5 + 5 q_1^3 q_4^2 + 5 q_2^3 q_4^2 - 5 q_3^3 q_4^2 + 5 q_1^2 q_4^3 + 5 q_2^2 q_4^3 + 5 q_3^2 q_4^3 - q_4^5 \right) \, .
\end{align}
\end{subequations}
The expression for $D_3$ corresponds to case (a) with the exchanges $q_1 \leftrightarrow q_3$ and $q_2 \leftrightarrow q_4$. The case $q_3 > q_1 + q_2 + q_4$ would be unphysical, and yields $D_1 = D_2 = D_3 = 0$.
\paragraph{Case (d)} $q_1 + q_2 < q_3 + q_4$ and $q_1 + q_4 > q_2 + q_3$
\begin{subequations}
\begin{align}
D_1 &= q_2 \, , \\
D_2 &= \frac16 q_2 \left(3 q_3^2 + 3 q_4^2 - 3 q_1^2 - q_2^2 \right) \, , \\
D_3 &= \frac{1}{30} q_2^3 \left(5 q_1^2 + 5 q_3^2 + 5 q_4^2 - q_2^2 \right) \, .
\end{align}
\end{subequations}
\end{document}
\chapter[The numerical code \texttt{NEVO}][The numerical code NEVO]{The numerical code \texttt{NEVO}}
\label{App:Numerics}
\setlength{\epigraphwidth}{0.53\textwidth}
\epigraph{Is this an instrument of communication or torture?}{Lady Violet Crawley, \emph{Downton Abbey} [S02E05]}
{
\hypersetup{linkcolor=black}
\minitoc
}
During this PhD, we have developed the numerical code \texttt{NEVO} (\texttt{N}eutrino \texttt{EVO}lver) to follow neutrino evolution in the early Universe. We presented its main features in chapter~\ref{chap:Decoupling}, and give some additional information in this Appendix. First, we review in more details how the degrees of freedom in density matrices are serialized. We then provide an extensive description of our method of calculation of the Jacobian of the differential system of equations, in particular how it is extended from the {ATAO}-$\mathcal{V}\,$ (chapter~\ref{chap:Decoupling}) to the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme (chapter~\ref{chap:Asymmetry}). Indeed, we show that this implementation allows to gain an order of magnitude in computation time, whether we consider the standard calculation or the asymmetric case.
\section{Serialization of density matrices in flavour space}
Since the spectra of density matrices are sampled on a grid of $N$ comoving momenta, the $\{y_n\}$, the variables which need to be solved for are the $\varrho_{\alpha\beta}(y_n)$, and $\bar{\varrho}_{\alpha\beta}(y_n)$. In the discretized numerical resolution, integrals are replaced by a quadrature method, that is by a weighted sum on the $y_n$, the weights depending on the chosen grid points.
In order to alleviate the explanations, we will ignore the presence of antineutrinos and we shall consider that the variables are just the $\varrho_{\alpha\beta}(y_n)$, for which we use the short notation $\varrho_{\alpha\beta,n}$. In the {ATAO}-$(\Hself\pm\mathcal{V})\,$ method, this is clearly wrong since an asymmetry matrix $\mathcal{A}_{\alpha\beta}$ requires that neutrinos and antineutrinos should have different distributions and one must always evolve both neutrinos and antineutrinos --- but that does not change the arguments presented here.
For each $n$, we serialize the matrix components. This requires to define a basis ${P^a}$ for hermitian matrices with $N_\nu^2$ elements. These are divided into $N_\nu(N_\nu+1)/2$ basis matrices for the real components and $N_\nu(N_\nu-1)/2$ for the imaginary components. For instance when $N_\nu=2$ the matrices are
\begin{equation}
P^1 = \begin{pmatrix}
1&0\\
0&0\end{pmatrix} \, , \quad
P^2 = \begin{pmatrix}
0&0\\
0&1\end{pmatrix} \, , \quad
P^3 = \begin{pmatrix}
0&1\\
1&0\end{pmatrix} \, , \quad
P^4 = \begin{pmatrix}
0& - {\mathrm i}\\
{\mathrm i}&0\end{pmatrix}\,.
\end{equation}
An inner product between two hermitian matrices is $(A,B) \equiv \mathrm{Tr}(A \cdot B^\dagger)$, hence the norms of the basis matrices are
\begin{equation}
\norm{P^a}^2 = \sum_{\alpha\beta} |{P^a}_{\alpha\beta}| ^2= \mathrm{Tr}(P^a \cdot P^{a\dagger})\,.
\end{equation}
Any density matrix is decomposed on serialized components as
\begin{equation}
\varrho_{\alpha \beta,n} \equiv \varrho_{a,n} {P^a}_{\alpha\beta}\,.
\end{equation}
The serialized components are also related to the components in the matter basis through
\begin{equation}\label{DefPaij}
\widetilde \varrho_{ij,n} = \varrho_{a,n} {P^{a}}_{ij}(n)\quad \text{with} \quad {P^{a}}_{ij}(n) \equiv [U^\dagger(n) \cdot P^a \cdot U(n)]_{ij} \, ,
\end{equation}
where $U(n)$ stands for $U_{\mathcal{H}}(y_n)$, $\mathcal{H}$ being the appropriate Hamiltonian (it will depend on the numerical scheme chosen). Note that we use the same notation $P^a$ in the matter ($P^a_{ij}$) and flavour ($P^a_{\alpha \beta}$) bases, the difference being identified through the indices. Conversely the serialized components $\varrho_{a,n}$ are obtained from
\begin{equation}\label{Convertrhoarhoi}
\varrho_{a,n} = \varrho_{\alpha\beta,n} {P^{\alpha\beta}}_{a}(n)=\frac{\widetilde{\varrho}_{ij,n} {P^{a}}_{ji}(n)}{\norm{P^a}^2 }\quad \text{with} \quad {P^{\alpha\beta}}_{a}(n) \equiv \frac{ {P^{a\star}}_{\alpha\beta}(n)}{\norm{P^a}^2 } =\frac{ {P^{a}}_{\beta \alpha}(n) }{\norm{P^a}^2 }\,.
\end{equation}
In any ATAO scheme, we are only interested in the diagonal components of the matter basis, since by construction all off-diagonal components vanish, hence we define $\widetilde \varrho_{i,n} \equiv \widetilde \varrho_{ii,n}$ and obtain the following relations between the serialized (flavour) basis and the (diagonal) matter components
\begin{equation}
\begin{aligned}
\widetilde \varrho_{i,n} &= \varrho_{a,n} T^{a}_i(n)\quad &&\text{with}\quad T^{a}_i(n) \equiv {P^{a}}_{ii}(n) \, , \\
\varrho_{a,n}&= \widetilde \varrho_{i,n} T^i_{a}(n) \quad &&\text{with}\quad T^i_{a}(n) \equiv \frac{1}{\norm{P^a}^2} T^{a}_i(n) \,.
\end{aligned}
\end{equation}
Since the $U(n)$ depend both on $x$ and on $y_n$, the $T^a_i(n)$ and $T^i_a(n)$ also depend on these variables, that is for each time step they must be computed for all points of the momentum grid.
\section{Direct computation of the Jacobian}
A key feature of the code \texttt{NEVO} is the calculation of the Jacobian of the system of differential equations, which is performed directly instead of relying on the extremely time-consuming default finite difference method.
Throughout this section we use a prime to denote a derivative with respect to $x$, and we stress again that we do not mention antineutrinos for the sake of clarity, but all developments must be carried out taking them into account.
\subsection{QKE scheme}
We must solve for the evolution of $z$ and of the flavour space serialized variables $\varrho_{a,n}$. Equation~\eqref{D1Froustey2020} dictates the evolution of $z$. The evolution of the $\varrho_{a,n}$ is governed by~\eqref{eq:QKE_compact_asym}
\begin{equation}\label{eq:QKE_compactserialized}
\varrho_{a,n}' = M_{a,n}^c \varrho_{c,n} + \mathcal{K}_{a,n} \, ,
\end{equation}
with
\begin{equation}
M_{a,n}^c \equiv (\mathcal{V}_{b,n} + \mathcal{J}_b) {C^{bc}}_a\quad\text{and}\quad {C^{bc}}_a \equiv -{\mathrm i}[P^b,P^c]_{\alpha \beta} {P^{\alpha\beta}}_a\,.
\end{equation}
The first term in \eqref{eq:QKE_compactserialized} comes from mean-field effects and the second term from collisions.
The associated Jacobian has the general structure
\begin{equation}\label{JacobNumQKE}
\begin{pmatrix}
0&0\\ \\
\displaystyle \left(\frac{\partial M_{a,n}^c }{\partial z} \varrho_{c,n} \right)&
\displaystyle \left(\frac{\partial M_{a,n}^c }{\partial \varrho_{b,m}} \varrho_{c,n} + M_{a,n}^b \delta^n_m \right)
\end{pmatrix}+
\begin{pmatrix}
\displaystyle \frac{\partial z'}{\partial z}& \displaystyle \frac{\partial z'}{\partial \varrho_{b,m}}\\ \\
\displaystyle \frac{\partial {\mathcal{K}}_{a,n}}{\partial z}& \displaystyle \frac{\partial {\mathcal{K}}_{a,n}}{\partial \varrho_{b,m}}
\end{pmatrix}
\end{equation}
where, as before, the first matrix is due to mean-field effects, and the second to collisions. The contributions from the mean-field effects require to calculate
\begin{equation}
\frac{\partial M_{a,n}^c }{\partial z} =\frac{\partial \mathcal{V}_{b,n}}{\partial z}{C^{bc}}_a\,, \qquad
\frac{\partial M_{a,n}^c }{\partial \varrho_{b,m}} = \frac{\partial \mathcal{J}_d}{\partial \varrho_{b,m}} {C^{cd}}_a\,,
\end{equation}
and the quantity $\partial \mathcal{J}_d/\partial \varrho_{b,m}$ is read on the integral definition \eqref{DefJ}. The computation of the Jacobian associated with mean-field effects is at most $\mathcal{O}(N^2)$ when self-interactions are taken into account, and only $\mathcal{O}(N)$ when they are ignored or when there is no asymmetry. Let us now review the complexity of the remaining terms.
\begin{itemize}
\item $\partial \mathcal{K}_{a,n}/ \partial \varrho_{b,m}$ is the time-consuming part. Since the complexity for computing the collision term is $\mathcal{O}(N^3)$, using a finite difference method would scale as $\mathcal{O}(N^4)$. A method to reduce the complexity to $\mathcal{O}(N_\nu^2 N^3)$ is detailed in \cite{Froustey2020}, hence considerably speeding the numerical resolution.
\item $\partial \mathcal{K}_{a,n}/\partial z$ is just the collision term where the contribution coming from the distributions of electrons/positrons is varied with respect to $z$. Hence it has the same $\mathcal{O}(N^3)$ complexity as the collision term.
\item $\partial z'/\partial z$ can be obtained from equation~\eqref{D1Froustey2020}. In practice we simply use a finite difference method.
\item $\partial z'/\partial \varrho_{b,m}$ is obtained from the chain rule as
\begin{equation}\label{dzpdrho}
\frac{\partial z'}{\partial \varrho_{b,m}} = \frac{\partial z'}{\partial \mathcal{K}_{a,n}} \frac{\partial \mathcal{K}_{a,n}}{\partial \varrho_{b,m}}
\end{equation}
and we only need the variation $\partial z'/\partial \mathcal{K}_{a,n}$ which is easily read on equation~\eqref{D1Froustey2020}. Indeed, since only the trace of the collision term sources $z'$, the only serialized components $a$ leading to a non-vanishing $\partial z'/\partial \mathcal{K}_{a,n}$ are those for which $\mathrm{Tr}(P^a) \neq 0$.
\end{itemize}
\subsection{{ATAO}-$\mathcal{V}\,$ scheme}
In the {ATAO}-$\mathcal{V}\,$ scheme we integrate $z$ with \eqref{D1Froustey2020}, and the diagonal components $\widetilde \varrho_i$ with \eqref{BasicATAOH}, that is
\begin{equation}\label{QKEcompact}
\widetilde{\varrho}'_{i,n}= \widetilde{\mathcal{K}}_{i,n}\,.
\end{equation}
Note that this is a very compact notation which hides the fact that what is known in general are the $\mathcal{K}_{\alpha \beta}(y_n)$ which depend on the $\varrho_{\alpha\beta}(y_n)$. Hence at each step one must transform the matter basis components $\widetilde \varrho_{i,n}$ to the flavour basis, compute the collision terms, and convert back into the matter basis, and keep only the diagonal terms. The relation between the diagonal matter basis components and the (flavour basis) serialized components reads
\begin{equation}
\label{eq:relation_K_Ktilde}
\widetilde{\mathcal{K}}_{i,n} = \mathcal{K}_{a,n} T^a_i(n).
\end{equation}
The general form of the Jacobian is then
\begin{equation}\label{JacobNumATAOH}
\begin{pmatrix}
\displaystyle \frac{\partial z'}{\partial z}& \displaystyle \frac{\partial z'}{\partial \widetilde\varrho_{j,m}}\\ \\
\displaystyle \frac{\partial \widetilde{\mathcal{K}}_{i,n}}{\partial z}& \displaystyle \frac{\partial \widetilde{\mathcal{K}}_{i,n}}{\partial \widetilde \varrho_{j,m}}
\end{pmatrix}
\end{equation}
since in this scheme there are no mean-field effects to solve for, as they are hidden in the evolution of the matter basis. Again a finite difference method to compute $\partial \widetilde{\mathcal{K}}_{i,n}/\partial \widetilde \varrho_{j,m}$ would be of complexity $\mathcal{O}(N^4)$, but using the method detailed in \cite{Froustey2020} it is reduced to a complexity $\mathcal{O}(N_\nu N^3)$. This is even slightly faster (reduced by a factor $N_\nu$) than for computing $\partial \mathcal{K}_{a,n}/ \partial \varrho_{b,m}$ because there are only $N_\nu$ diagonal matter components instead of $N_\nu^2$ flavour components. Both Jacobian blocks are related thanks to
\begin{equation}\label{RelateJacob}
\frac{\partial \widetilde{\mathcal{K}}_{i,n}}{\partial \widetilde\varrho_{j,m}} = T^{a}_i(n) \frac{\partial {\mathcal{K}}_{a,n}}{\partial \varrho_{b,m}} T_{b}^j(m) \, .
\end{equation}
Let us review the other blocks in \eqref{JacobNumATAOH}. First we use the chain rule
\begin{equation}\label{dzpdrhoi}
\frac{\partial z'}{\partial \widetilde \varrho_{j,m}} = \frac{\partial z'}{\partial \widetilde{\mathcal{K}}_{i,n}} \frac{\partial \widetilde{\mathcal{K}}_{i,n}}{\partial \widetilde\varrho_{j,m}}\,,
\end{equation}
where $\partial z' / \partial \widetilde{\mathcal{K}}_{i,n} $ is easily read from equation~\eqref{D1Froustey2020} since only the trace of the collision term sources $z'$. And finally $\partial \widetilde{\mathcal{K}}_{i,n} / \partial z $ is similar to the computation of a collision term, but with the contribution from the $e^\pm$ distribution varied upon $z$.
\subsection{{ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme}
The general {ATAO}-$(\Hself\pm\mathcal{V})\,$ equation \eqref{BasicATAOJH}, when written explicitly in matter basis components and using the previous notation, is also of the form \eqref{QKEcompact}. However in the {ATAO}-$(\Hself\pm\mathcal{V})\,$ we also solve at the same time the evolution of $\mathcal{A}_{\alpha\beta}$ given by \eqref{dJdx}. Note that it depends on the full collision term $\mathcal{K}_{\alpha \beta}(y_n)$ (or the $\mathcal{K}_{a,n}$ in serialized basis) and not just on the diagonal components in the matter basis $\widetilde{\mathcal{K}}_{i,n}$, contrary to the evolution of the $\widetilde \varrho_i$ in \eqref{QKEcompact}.
Since we supplement $z$ and the $\widetilde\varrho_{i,n}$ with the $N_\nu^2$ variables $\mathcal{A}_a$, this extends the size of the Jacobian. We now show how the new blocks in the Jacobian can be computed, and that this preserves the $\mathcal{O}(N^3)$ complexity. The general form of the Jacobian is
\begin{equation}\label{JacobNum}
\begin{pmatrix}
\displaystyle \frac{\partial z'}{\partial z}& \displaystyle \frac{\partial z'}{\partial \widetilde\varrho_{j,m}}& \displaystyle \frac{\partial z'}{ \partial \mathcal{A}_b}\\ \\
\displaystyle \frac{\partial \widetilde{\mathcal{K}}_{i,n}}{\partial z}& \displaystyle \frac{\partial \widetilde{\mathcal{K}}_{i,n}}{\partial \widetilde \varrho_{j,m}}& \displaystyle\frac{\partial \widetilde{\mathcal{K}}_{i,n}}{\partial \mathcal{A}_b}\\ \\
\displaystyle \frac{\partial \mathcal{A}'_a}{\partial z}& \displaystyle \frac{\partial \mathcal{A}'_a}{\partial \widetilde\varrho_{j,m}}& \displaystyle \frac{\partial \mathcal{A}'_a}{\partial \mathcal{A}_b}
\end{pmatrix} \, ,
\end{equation}
and only the blocks in the right column or the bottom line are specific to the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme.
As detailed hereafter, in order to compute these new blocks we shall need the computation of $\partial \mathcal{K}_{a,n}/\partial \varrho_{b,m}$, which is what is needed when computing the Jacobian in the QKE method. The block $\partial \widetilde{\mathcal{K}}_{i,n}/\partial \widetilde \varrho_{j,m}$ is then deduced through \eqref{RelateJacob}.
We also need to know how the $U(n)$ vary when the components of $\mathcal{A}$ are varied. Let us define the set of anti-hermitian matrices
\begin{equation}\label{DefWan}
W^{a,n} \equiv \frac{\partial U(n)}{\partial \mathcal{A}_a} \cdot U^\dagger(n) = - U(n) \cdot \frac{\partial U^\dagger(n)}{\partial \mathcal{A}_a}\,.
\end{equation}
which allow to know how the flavour components vary when $\mathcal{A}$ varies, for fixed matter basis components. They are obtained thanks to
\begin{equation}\label{CrypticW}
\left(W^{a,n}\right)_{ij}= \frac{\sqrt{2} G_F}{(xH)}\left(\frac{m_e}{x}\right)^3 \frac{\left[U^\dagger(n) \cdot P^a \cdot U(n)\right]_{ij}}{(\mathcal{V}_{j,n} + \mathcal{J}_{j,n} -\mathcal{V}_{i,n} - \mathcal{J}_{i,n})}\quad \text{for}\quad i\neq j\,,
\end{equation}
where the $\left(W^{a,n}\right)_{ij}$ are the components of $W^{a,n}$ in the matter basis, that is they are defined as $\left[U^\dagger(n)\cdot W^{a,n} \cdot U(n)\right]_{ij}$\,. The $(\mathcal{V}+\mathcal{J})_{j,n}$ are the diagonal components of $\mathcal{V}+\mathcal{J}$ in the matter basis, which are by definition its eigenvalues. The $W^{a,n}$ are then found by transforming~\eqref{CrypticW} to the flavour basis with $U(n)$. Using their definition~\eqref{DefWan}, one then finds
\begin{equation}\label{drhoadAb}
\frac{\partial \varrho_{a,n}}{\partial \mathcal{A}_b} =\varrho_{c,n} [W^{b,n}, P^c]_{\alpha \beta} {P^{\alpha \beta}}_a(n)\,.
\end{equation}
We now have all the tools to compute the blocks in the Jacobian \eqref{JacobNum} that are specific to the presence of $\mathcal{A}$.
\begin{itemize}
\item $\partial \widetilde{\mathcal{K}}_{i,n}/\partial \mathcal{A}_b$
This is deduced from~\eqref{eq:relation_K_Ktilde}. Using the Leibniz rule, we deduce
\begin{equation}\label{dKidAb}
\frac{\partial \widetilde{\mathcal{K}}_{i,n}}{\partial \mathcal{A}_b} = \frac{\partial \mathcal{K}_{a,n}}{\partial \mathcal{A}_b}T^a_i(n) + \mathcal{K}_{a,n} \frac{\partial T^a_i(n)}{\partial \mathcal{A}_b}\quad \text{with} \quad \frac{\partial T^a_i(n)}{\partial \mathcal{A}_b} = - \left(U^\dagger(n) \cdot [ W^{b,n}, P^a ]\cdot U(n)\right)_{ii} \, .
\end{equation}
\item $\partial z'/\partial A_b$
We only need to apply the chain rule using equation \eqref{dKidAb} since
\begin{equation}
\frac{\partial z'}{\partial \mathcal{A}_b} = \frac{\partial z'}{\partial \widetilde{\mathcal{K}}_{i,n}} \frac{\partial \widetilde{\mathcal{K}}_{i,n}}{\partial \mathcal{A}_b}\,,
\end{equation}
with $\partial z'/\partial \widetilde{\mathcal{K}}_{i,n}$ already needed for equation \eqref{dzpdrhoi}.
\item $\partial \mathcal{A}'_a/\partial z$
This is similar to the treatment of equation~\eqref{dJdx} with the replacement $\mathcal{V} \to \partial \mathcal{V}/\partial z$ and $\mathcal{K} \to \partial \mathcal{K}/\partial z$.
\item $\partial \mathcal{A}'_a/\partial \widetilde \varrho_{j,n}$
Let us define the following derivative :
\begin{equation}\label{TotalStrangeDer}
\frac{\partial \mathcal{A}'_a}{\partial \varrho_{b,n}} \equiv \left(\left. \frac{\partial \mathcal{A}'_a}{\partial \varrho_{b,n}}\right|_\mathrm{mf} + \frac{\partial \mathcal{A}'_a}{\partial \mathcal{K}_{c,m}} \frac{\partial \mathcal{K}_{c,m}}{\partial \varrho_{b,n}}\right)\,,
\end{equation}
where it is understood that the first term corresponds to the mean-field term, that is the first term on the rhs of equation~\eqref{dJdx}, and the second term is the indirect contribution via the dependence of the collision term. Since the integrals appearing in equation~\eqref{dJdx} are computed with a quadrature method, that is a sum on the grid of comoving momenta with appropriate weights, these derivatives select only one term in these sums (the one corresponding to the comoving momentum $y_n$). We then immediately get from equation~\eqref{Convertrhoarhoi} and the chain rule
\begin{equation}
\frac{\partial \mathcal{A}'_a}{\partial \widetilde\varrho_{j,n}} = \frac{\partial \mathcal{A}'_a}{\partial \varrho_{b,n}}T^j_b(n)\,.
\end{equation}
\item $\partial \mathcal{A}'_a/\partial \mathcal{A}_b$
Finally, using again the derivative \eqref{TotalStrangeDer}, we find a simple expression for the last block
\begin{equation}\label{dApdA}
\frac{\partial \mathcal{A}'_a}{\partial \mathcal{A}_b} = \frac{\partial \mathcal{A}'_a}{\partial \varrho_{c,n}} \frac{\partial \varrho_{c,n}}{\partial \mathcal{A}_b} \,.
\end{equation}
\end{itemize}
In practice, pairs of indices like ${i,n}$ or ${a,n}$ are also serialized (e.g. with $I = n N_\nu + i$ and $A = n N_\nu^2 +a$), such that all products with implicit summations in this section appear as matrix multiplications when implemented in the code.
\vspace{0.3cm}
To summarize we need to compute the Jacobian as in the QKE method, which gives $\partial \mathcal{K}_{a,n}/\partial \varrho_{b,m}$. It corresponds to the variation of all flavour components in the collision term with respect to variations in all flavour components of the density matrices. We also need to compute the $T^a_i(n)$ from equation~\eqref{DefPaij}, and the $W^{a,n}$ from equation~\eqref{CrypticW}. We then deduce from equation~\eqref{drhoadAb} the $\partial \varrho_{a,n}/\partial \mathcal{A}_b$. Finally, knowing $\partial z'/\partial \widetilde{\mathcal{K}}_i(n)$, $\left.\partial \mathcal{A}'_a/\partial \varrho_{b,n}\right|_\mathrm{mf}$ and $\partial \mathcal{A}'_a/\partial \mathcal{K}_{c,m}$ from the equations governing the evolution of $z$ and $\mathcal{A}_{\alpha\beta}$, we can compute the five new blocks of the Jacobian as described from equations~\eqref{dKidAb} to~\eqref{dApdA}. The step which is the most time-consuming is the first one, that is the computation of $\partial \mathcal{K}_{a,n}/\partial \varrho_{b,m}$, whose complexity is $\mathcal{O}(N^3)$. Since this is already the longest step in the direct computation of the Jacobian in the QKE scheme, we deduce that for large $N$ the direct computation of the Jacobian in the {ATAO}-$(\Hself\pm\mathcal{V})\,$ method takes roughly the same time as the direct computation of the Jacobian in the QKE method.
\end{document}
\chapter[Primordial neutrino asymmetry evolution][Primordial neutrino asymmetry evolution]{Primordial neutrino asymmetry evolution}
\label{chap:Asymmetry}
\setlength{\epigraphwidth}{0.5\textwidth}
\epigraph{It's supposed to be a little asymmetrical. Apparently a small flaw somehow improves it.}{Sheldon Cooper, \emph{The Big Bang Theory} [S11E24]}
{
\hypersetup{linkcolor=black}
\minitoc
}
\boxabstract{The material of this chapter was published in~\cite{Froustey2021}.}
The specific features of the cosmic neutrino background and its effects on primordial nucleosynthesis have been studied in the previous chapters, assuming systematically that neutrinos and antineutrinos kept the same distributions. This absence of asymmetry --- or rather, the extremely small asymmetry --- between matter and antimatter is measured for baryons via the parameter $\eta \sim 10^{-9}$~\cite{Fields:2019pfx}. Since, by charge conservation, $n_{e^-} - n_{e^+} = n_p$ during BBN ($\mathrm{H}$ being by far the most numerous species), there can be no sizable asymmetry in the charged lepton sector. However, thanks to their electric neutrality, no such constraint exists for neutrinos.
In thermal and chemical equilibrium, the chemical potential of a given neutrino flavour $\alpha$ and the corresponding antineutrino chemical potential are related through $\mu_\alpha =- \bar \mu_\alpha $. The initial asymmetry in a given flavour $\alpha$, defined as the difference between the neutrino and antineutrino comoving densities, is related to $\mu_\alpha$, which is not constrained \emph{a priori} for the reason explained above. One thus hopes to constrain them from their impact on cosmological observables. More specifically, to take into account the effects of momenta redshifting due to cosmological expansion, we aim at constraining the degeneracy parameters $\xi_\alpha \equiv \mu_\alpha/T_{\mathrm{cm}}$ which are conserved by expansion. There is first an effect of $\xi_e$ on the neutron/proton freeze-out, which affects Big-Bang Nucleosynthesis (BBN). Indeed, the detailed balance relation~\eqref{eq:nse} is shifted to
\begin{equation}
\label{eq:nse_xie}
\frac{n_n}{n_p} = \left. \frac{n_n}{n_p}\right\rvert_{\xi_e = 0} \times e^{- \xi_e} \, .
\end{equation}
Also the total energy density of a given neutrino flavour and its corresponding antineutrino is supplemented by a term $\propto \xi_\alpha^2$, leading to a modification of $N_{\mathrm{eff}}$, and this has an impact on cosmological expansion which affects both BBN and the cosmic microwave background (CMB) anisotropies. Hence, assuming a full equilibration of neutrino asymmetries with a common $\xi$, a constraint can be obtained from BBN alone~\cite{Simha:2008mt,Fields:2019pfx}, from CMB alone~\cite{Oldengott:2017tzj,Planck18}, or using a combination of both~\cite{Pitrou_2018PhysRept} to give
\begin{equation}
\label{eq:xi_BBN_CMB}
\xi = 0.001 \pm 0.016\,.
\end{equation}
If standard baryogenesis models involving sphalerons suggest that $\xi$ should be of the order of the baryon asymmetry $\eta = n_b/n_\gamma \simeq 6.1\times 10^{-10}$~\cite{Neutrino_Cosmology,Davidson_Leptogenesis}, other proposed models like~\cite{McDonald:1999in,March-Russell:1999hpw,Gu:2010dg} manage to combine a large lepton asymmetry with the value of $\eta$. Therefore potentially "high" values of $\xi$ are not forbidden and \eqref{eq:xi_BBN_CMB} motivates why we focus in this chapter on degeneracy parameters in the range $[10^{-3},10^{-1}]$.
The total asymmetry, that is the sum over each flavour asymmetry, is preserved by the physical processes at play. However, individual asymmetries can evolve towards the average, in which case we can talk about "\emph{flavour equilibration}". The goal of this chapter is to review the physics of this equilibration, that is the evolution of the degeneracy parameters, accounting for all relevant physical effects at play during neutrino decoupling. To that end, it is necessary to solve the Quantum Kinetic Equations (QKEs) that dictate the evolution of (anti)neutrino density matrices, taking into account vacuum oscillations, mean-field effects with leptons and neutrinos (the latter being referred to as self-interaction mean-field), and collision processes.
First, self-interactions have a crucial effect in delaying equilibration as they are responsible for the so-called \emph{synchronous oscillations}~\cite{Pastor:2001iu,Dolgov_NuPhB2002,Abazajian2002,Wong2002}, and we find that in general there is also a second regime with \emph{quasi-synchronous oscillations} having much larger frequencies. Furthermore we find that the complete form of the neutrino collision term must be used, including the full matrix structure of both reactions among neutrinos and with electrons/positrons.
Unless the chemical potential differences are very small, there is always a period when self-interactions dominate over the lepton mean-field contribution and the vacuum Hamiltonian contribution. One of the dramatic consequences is that solving the exact evolution of neutrino number densities involves very short time scales compared to the cosmological time scale, which implies that it is numerically very difficult to treat them exactly. So far, the main approach when considering non-vanishing degeneracies consisted in using a damping approximation for the collision term~\cite{Bell98,Dolgov_NuPhB2002,Pastor:2008ti,Gava:2010kz,Gava_corr,Mangano:2010ei,Johns:2016enc,Barenboim:2016shh}, either for all its components or only for its off-diagonal components. Indeed the computation of the collision term is the time-consuming step with a $\mathcal{O}(N^3)$ complexity, where $N$ is the number of points used to sample the neutrino spectra. We use none of these approximations, and we find from the structure of the full collision term that it cannot efficiently damp all types of synchronous oscillations, a feature that is lost when relying on damping approximations.
We have shown in chapter~\ref{chap:Decoupling} that the numerical resolution could be considerably improved, altering only subdominantly the precision of results, by using an approximate scheme which consisted in averaging over neutrino oscillations in the adiabatically evolving matter basis. In this chapter, we extend this method with initial degeneracies, that is taking into account the effect of the self-interaction mean-field. In section~\ref{SecTheory} we summarize the formalism used to describe the evolution of neutrino and antineutrino density matrices in the context of non-zero asymmetries, and in section~\ref{SecResScheme} we detail the various numerical schemes we developed, notably the extension of the ATAO scheme when considering self-interactions. Restricting to oscillations with only two neutrinos in section~\ref{Sec2Neutrinos}, we derive analytic expressions for synchronous and quasi-synchronous oscillations. Two physically motivated cases with two-neutrino flavours are then investigated in details in section~\ref{SecRelevant2Neutrinos}. They allow to understand the evolution of neutrino asymmetry in the general case with three neutrinos, which is presented in section~\ref{Sec:3neutrinos}, along with an assessment of the dependence on the main mixing parameters (mass ordering, mixing angles, Dirac phase). Finally we discuss the main differences with existing results in the literature in section~\ref{SecDiscussion}.
\section{Neutrino evolution in the primordial Universe with degeneracies}\label{SecTheory}
In order to determine neutrino evolution in the early Universe, one must solve a set of quantum kinetic equations in the expanding Universe, involving both neutrino oscillations and collisions. We present in this section all the variables relevant to this problem, with a particular emphasis on the physical quantities related to the presence of a neutrino/antineutrino asymmetry.
\subsection{QKE with a non-zero neutrino/antineutrino asymmetry}
Let us recall the general form of the QKE~\eqref{eq:QKE_fullfinal}, valid in the early Universe:
\begin{multline}
\frac{\partial \varrho(x,y_1)}{\partial x} = - \frac{{\mathrm i}}{xH} \left(\frac{x}{m_e}\right) \left[ U \frac{\mathbb{M}^2}{2y_1}U^\dagger, \varrho \right] + {\mathrm i} \frac{2 \sqrt{2} G_F}{xH} y_1 \left(\frac{m_e}{x}\right)^5 \left[ \frac{\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}}{m_W^2} ,\varrho \right ] \\
- {\mathrm i} \frac{\sqrt{2} G_F}{x H} \left(\frac{m_e}{x}\right)^3 \left[ \overline{\mathbb{N}}_\nu - \overline{\mathbb{N}}_{\bar{\nu}}, \varrho \right] + {\mathrm i} \frac{8 \sqrt{2} G_F}{3 x H} y_1 \left( \frac{m_e}{x}\right)^5 \left[\frac{\bar{\mathbb{E}}_\nu + \bar{\mathbb{E}}_{\bar{\nu}}}{m_Z^2}, \varrho\right] + \frac{1}{xH} \mathcal{I} \, ,
\end{multline}
We will neglect the symmetric term proportional to (anti)neutrino energy densities, as it is always negligible compared to the same term proportional to charged lepton energy densities. Indeed, for initial Fermi-Dirac distributions at the same temperature and without degeneracies, this term is purely proportional to the identity matrix, hence it does not contribute to the dynamics of density matrices, which was the reason we discarded this term in chapter~\ref{chap:Decoupling}. Considering initial degeneracies, we have $\bar{\rho}_{\nu_\alpha}+\bar{\rho}_{\bar{\nu}_\alpha} \propto \xi_\alpha^2$, therefore this contribution is typically smaller than $\bar{\rho}_\mathrm{lep}$ (for relativistic leptons) by a factor which is of the order of the $\xi_\alpha^2$ differences.
We rewrite the QKE for $\varrho$, along with the equation for $\bar{\varrho}$, in a more concise way:
\begin{subequations}
\label{eq:QKE_compact_asym}
\begin{align}
\frac{\partial \varrho}{\partial x} &= - {\mathrm i} [\mathcal{V} +
\mathcal{J},\varrho] + \mathcal{K} \, ,\\
\frac{\partial \bar{\varrho}}{\partial x} &= + {\mathrm i} [\mathcal{V} -\mathcal{J},\bar{\varrho}] + \overline{\mathcal{K}} \, ,
\end{align}
\end{subequations}
with $\mathcal{V} = \mathcal{H}_0 + \mathcal{H}_\mathrm{lep}$, where the different Hamiltonians have been defined in~\eqref{eq:Hvac} and~\eqref{eq:Hlep}. For convenience, we recall here the expressions:
\begin{equation}
\label{eq:Hamil_general}
\mathcal{H}_0 \equiv \frac{1}{xH} \left(\frac{x}{m_e}\right) U \frac{\mathbb{M}^2}{2 y_1} U^\dagger \quad , \qquad
\mathcal{H}_\mathrm{lep} \equiv - \frac{1}{xH} \left(\frac{m_e}{x}\right)^5 2 \sqrt{2} G_F y_1 \frac{\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}}{m_W^2} \, .
\end{equation}
It proves convenient to separate the $y-$dependence in the Hamiltonian --- see section \ref{Sec2Neutrinos}. Hence we define
\begin{equation}
\label{eq:Hamil_y}
\mathcal{H}_0 \equiv \underline{\mathcal{H}}_0 / y \, , \qquad \mathcal{H}_\mathrm{lep} \equiv \underline{\mathcal{H}}_\mathrm{lep} y \, .
\end{equation}
Contrary to the standard case studied in chapter~\ref{chap:Decoupling}, the self-interaction Hamiltonian $\mathcal{J}$ must be included when considering neutrino asymmetries. We introduce the notation
\begin{equation}
\label{DefJ}
\mathcal{J} = \frac{1}{xH} \left(\frac{m_e}{x}\right)^3 \sqrt{2} G_F \mathcal{A} \ , \ \ \text{where} \quad \mathcal{A} \equiv (\overline{\mathbb{N}}_\nu - \overline{\mathbb{N}}_{\bar{\nu}}) = \int (\varrho-\bar{\varrho})\mathcal{D}y\,.
\end{equation}
Note that $\mathcal{A}$, referred to as the "(integrated) neutrino asymmetry", is simply proportional to the lepton number matrix $\eta_\nu \equiv \mathcal{A}/n_\gamma = \pi^2/[2 \zeta(3) z^3] \times \mathcal{A}$. We introduced the convenient notation $\mathcal{D}y \equiv y^2 \mathrm{d}{y}/(2 \pi^2)$.
The Hubble rate $H$ is given by the Friedmann equation, that we recall here to highlight its dependence on $x$,
\begin{equation}
\label{eq:scaling_Hubble_x}
H = \frac{m_e}{m_\mathrm{Pl}} \times \frac{m_e}{x^2} \times \sqrt{\frac{\bar{\rho}}{3}} \qquad \text{where} \quad \bar{\rho} =\bar{\rho}_\gamma + \bar{\rho}_{\nu, \bar{\nu}} + \bar{\rho}_{e^\pm} + \bar{\rho}_{\mu^\pm} \, ,
\end{equation}
where we stress again that the "barred" energy densities are the comoving ones, differing by a factor $(m_e/x)^4$ from the physical ones. $m_\mathrm{Pl} \simeq 2.435 \times 10^{18} \, \mathrm{GeV}$ is the reduced Planck mass.
\paragraph{Mixing parameters} We use the standard parameterization of the PMNS matrix~\eqref{eq:PMNS_bis} and the values~\eqref{ValuesStandard}, unless otherwise specified. We do not take into account the CP phase except in the dedicated subsection~\ref{SecDiracPhase}.
\paragraph{Dirac or Majorana neutrinos} It could be natural to believe that an asymmetry between neutrinos and antineutrinos is not possible if neutrinos are Majorana particles\footnote{For instance, the argument was raised in~\cite{Bernstein1982} before being corrected in~\cite{Langacker1982a,Langacker1982b} --- the original mistake was acknowledged in~\cite{Bernstein1984}.}. Indeed, in that case "neutrinos are their own antiparticles", which should lead necessarily to $\mathbb{N}_\nu = \mathbb{N}_{\bar{\nu}}$. But this overlooks the fact that there are \emph{helicity} degrees of freedom to take into account. Due to the Majorana condition $\nu = \nu^C$ (where $^C$ denote charge conjugation), there are twice as many degrees of freedom for Dirac neutrinos (left-handed\footnote{Left-handed (resp. right-handed) referring to a negative (resp. positive) \emph{helicity}.} and right-handed neutrinos, left-handed and right-handed antineutrinos), while there are only left-handed and right-handed neutrinos in the Majorana case. It is then common to refer to right-handed neutrinos in the Majorana case as "antineutrinos". However, all these states are not in thermal equilibrium, since helicity-flip rates are suppressed by a factor $\mathcal{O}(m_\nu^2/E_\nu^2) \ll 1$ compared to helicity-conserving reactions.
In summary, a positive asymmetry is interpreted:
\begin{itemize}
\item for Majorana neutrinos, as an excess of left-handed over right-handed neutrinos,
\item for Dirac neutrinos, as an excess of left-handed neutrinos over right-handed antineutrinos.
\end{itemize}
The counting of degrees of freedom with details about $CPT$ transformations is explained in~\cite{GiuntiKim}, Section~6.2.2. The thermal population of the \emph{a priori} two extra degrees of freedom in the Dirac case is discussed in~\cite{LesgourguesPastor}: the "wrong-helicity" states cannot be populated except if they had a mass at the $\mathrm{keV}$ scale~\cite{Dolgov_2002PhysRep}, which is excluded for active neutrinos.
\subsubsection{Evolution of the plasma temperature}
For large temperatures, that is before we start the numerical resolution ($x<x_\mathrm{init}$), the evolution of the comoving plasma temperature is estimated assuming that neutrino spectra are thermal with the same temperature ($z_{\nu}=z$). Afterwards, the evolution of $z$ is computed using the full (anti)neutrino density matrices and the exact form for collisions between neutrinos and electrons/positrons. Including QED corrections~\cite{Heckler_PhRvD1994,Mangano2002,Bennett2020}, we use
\begin{equation}\label{D1Froustey2020}
\frac{\mathrm{d} z}{\mathrm{d} x} = \frac{\displaystyle \frac{x}{z}J(x/z) - {S}_\nu + G_1(x/z)}{ \displaystyle \frac{x^2}{z^2}J(x/z) + Y(x/z) + \frac{1}{4}\sum_\alpha Y_{\nu}(\xi_\alpha/z) + \frac{2 \pi^2}{15} + G_2(x/z)} \, ,
\end{equation}
where we defined
\begin{equation}
\begin{aligned}
& S_\nu = 0\,, \quad Y_{\nu}(\zeta_\alpha) \equiv \frac{1}{\pi^2} \int_{0}^{\infty}{\mathrm{d} \omega \, \omega^3 (\omega-\zeta_\alpha) \frac{\exp{(\omega-\zeta_\alpha)}}{[\exp{(\omega-\zeta_\alpha)}+1]^2}}\quad &\text{for}\quad x \leq x_\mathrm{init}\,, \\
&Y_\nu = 0 \,,\quad S_\nu \equiv \frac{1}{4 \pi^2 z^3} \int_{0}^{\infty}{\mathrm{d} y \, y^3 \left( \mathrm{Tr} [\mathcal{K}] + \mathrm{Tr} [\overline{\mathcal{K}}] \right)} \ \quad &\text{for}\quad x > x_\mathrm{init}\,,
\end{aligned}
\end{equation}
the functions $J,\, Y, \, G_1, \, G_2$ having been introduced in~\eqref{eq:zQED}. The sum on $\alpha$ in the denominator of \eqref{D1Froustey2020} runs on $2 N_\nu$ elements, being all neutrinos and antineutrinos species.
The starting condition $z_\mathrm{init}$ at $x_\mathrm{init}$ is found by solving the differential equation~\eqref{D1Froustey2020}, with the initial condition $z=1$ at $x=0$.\footnote{In principle $z$ increases at each species annihilation, and in particular we should consider $\mu^\pm$ annihilations since these leptons appear in the mean-field effects. This choice of initial conditions is however consistent with neglecting the interactions with $\mu^\pm$ in the collision term, which therefore do not reheat the plasma of neutrinos, photons and $e^\pm$.} When there are no neutrino degeneracies, it matches the condition found by all coupled species entropy conservation. However, it gives a slightly different $z_\mathrm{init}$ in the presence of initial degeneracies since entropy conservation is then violated.
\subsection{Neutrino asymmetry matrix $\mathcal{A}$}
\label{subsec:Anti}
Long before neutrino decoupling, that is for temperatures much larger than $2\,\mathrm{MeV}$, neutrinos and antineutrinos are maintained at kinetic and chemical equilibrium, thus generally following Fermi-Dirac (FD) distributions with a chemical potential
\[g(T_\nu, \mu, p) \equiv \left[e^{(p - \mu)/T_\nu}+1\right]^{-1} \, .\]
Introducing the reduced variables $z_\nu = T_\nu/T_{\mathrm{cm}}$ and $\xi = \mu/T_{\mathrm{cm}}$, we rewrite this FD distribution
\[g(z_\nu, \xi, y) = \left[e^{(y - \xi)/z_\nu}+1\right]^{-1} \, .\]
In most of the temperature range of interest, since electrons and positrons have not annihilated and all species are coupled, we can consider\footnote{We only take $z_\nu = 1$ for the analytical discussion in order to simplify the presentation. In the numerical resolution, the spectra evolve following the QKEs and $e^\pm$ annihilations increase the neutrino temperatures.} $z_\nu = 1$. We thus define $g(\xi, y) \equiv g(1, \xi, y)$, and the initial conditions read
\begin{equation}\label{rhoinit}
\varrho_\mathrm{init} = \mathrm{diag}\left[g(\xi_\alpha,y)\right] \, , \quad\bar{\varrho}_\mathrm{init} = \mathrm{diag}\left[g(-\xi_\alpha,y)\right]\,.
\end{equation}
\paragraph{Useful properties of Fermi-Dirac spectra} In this chapter, we will often use the following relations:\footnote{They respectively intervene in the calculation of the asymmetry, the sum of energy densities, the leading and next-to-leading orders of the asymmetry oscillation frequency.}
\begin{subequations}
\label{IntegralsFD}
\begin{align}
\int [g(\xi,y) - g(-\xi,y)] \mathcal{D} y &= \frac{\xi}{6} + \frac{\xi^3}{6\pi^2}\label{Intg1}\\
\int y \, [g(\xi,y) + g(-\xi,y)] \mathcal{D} y &= \frac{7 \pi^2}{120}+ \frac{\xi^2}{4} + \frac{\xi^4}{8\pi^2}\label{Intg2}\\
\int y^{-1} \, [g(\xi,y) + g(-\xi,y)] \mathcal{D} y &= \frac{1}{12} + \frac{\xi^2}{4\pi^2}\label{Intg3}\\
\int y^{-2} \, [g(\xi,y) - g(-\xi,y)] \mathcal{D} y &= \frac{\xi}{2 \pi^2} \label{Intg4}
\end{align}
\end{subequations}
From equation~\eqref{Intg1}, the asymmetry matrix is initially
\begin{equation}
\label{eq:init_Anti}
\mathcal{A}_\mathrm{init} = \frac{1}{6}\mathrm{diag}\left[\xi_\alpha+\frac{\xi_\alpha^3}{\pi^2}\right] \, .
\end{equation}
For reasons detailed in section~\ref{SecResScheme}, we also introduce the evolution equation for $\mathcal{A}$. It is obtained by combining the QKE \eqref{eq:QKE_compact_asym} with the definition~\eqref{DefJ},
\begin{equation}\label{dJdx}
\frac{\mathrm{d} \mathcal{A}}{\mathrm{d} x} = -{\mathrm i} \int \left[\mathcal{V},
\varrho + \bar{\varrho}\right] \mathcal{D}y+ \int \left(\mathcal{K} -
\overline{\mathcal{K}}\right) \mathcal{D}y\,.
\end{equation}
In principle there is no need to solve this equation for $\mathcal{A}$ because it is a simple consequence of the definition~\eqref{DefJ} with equations~\eqref{eq:QKE_compact_asym}. However, some approximate resolution schemes promote $\mathcal{A}$ to an independent variable, thus requiring this additional equation to ensure the overall consistency.
\subsection{MSW transitions}
Schematically, the lepton mean-field term scales as $T_{\mathrm{cm}}^5$, whereas the vacuum oscillation Hamiltonian scales as $1/T_{\mathrm{cm}}$ (discarding the common $1/xH$ scaling). Hence there is always a Mikheev-Smirnov-Wolfenstein (MSW) transition~\cite{MSW_W,MSW_MS} from lepton mean-field domination to vacuum domination, which can be resonant or not depending on the mixing angles and the mass ordering. There are two differences with the MSW transition in stars. First, the lepton mean-field term in stellar environments is $\sqrt{2}G_F n_{e^-}$, but it is cancelled here by the positron contribution $-\sqrt{2}G_F n_{e^+}$ since the electron/positron asymmetry is negligible. Hence, in the cosmological case the dominant lepton mean-field contribution is given by~\eqref{eq:Hamil_general}. Second, the role of the electron density profile crossed by emitted neutrinos in a star is now played by the thermal evolution of the Universe. In the cosmological context, there are three transitions which are illustrated in Figure~\ref{fig:ODG_QKE}.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/ODG_QKE.pdf}
\caption[Orders of magnitude of the different rates involved in the QKE]{\label{fig:ODG_QKE} Orders of magnitude of the different rates involved in the QKE, for $y=y_\mathrm{eff}=3.15$ (this averaged value will be justified in section~\ref{FreqSyncOsc}). $\mathcal{J}$ is plotted with $\mathcal{A}$ given by \eqref{eq:init_Anti} and $\xi = 0.01$. The oscillation frequencies, set by the Hamiltonian eigenvalues, are very large compared to the collision rate and its variation (see section~\ref{sec:ATAO} for this discussion). As the temperature decreases, the dominant contribution in the Hamiltonian changes from $\mathcal{J}$ to $\mathcal{V}$, $\mathcal{V}$ itself being dominated first by $\mathcal{H}_\mathrm{lep}$ and then by $\mathcal{H}_0$. We estimate the magnitude of the collision rate as in Figure 1 of~\cite{Mirizzi2012}.}
\end{figure}
\begin{enumerate}
\item Since $m_\mu/m_e \simeq 207$, the first MSW transition, that we call the muon-driven MSW transition, occurs when the $\mu^\pm$ mean-field effects become of the same order as the vacuum Hamiltonian associated with the large mass gap $\Delta m_{31}^2$ (or equivalently $\Delta m_{32}^2$), and this occurs around $T_\mathrm{MSW}^{(\mu)} \simeq 12\,\mathrm{MeV}$ (see section~\ref{SecDescriptionMuonTransition}), when muons are not relativistic.
\item When the $e^\pm$ mean-field effects also become of the same order as the vacuum Hamiltonian associated with the large mass gap $\Delta m_{31}^2$, we encounter the first electron-driven MSW transition around $T_\mathrm{MSW}^{(e), 1} \simeq 5\,\mathrm{MeV}$ (see section~\ref{SecElectronMSW}).
\item Finally, when the same mean-field term becomes of the same order as the vacuum Hamiltonian associated with the small mass gap $\Delta m_{21}^2$, we reach the second electron-driven MSW transition around $T_\mathrm{MSW}^{(e), 2} \simeq 2.8\,\mathrm{MeV}$ (see section~\ref{SecElectronMSW}).
\end{enumerate}
The presence of a neutrino asymmetry modifies this picture because self-interaction mean-field effects (abbreviated as \emph{self-interactions} when it is clear that we do not refer to collisions between (anti)neutrinos) scale as $T_{\mathrm{cm}}^3$ and the traceless part of the neutrino asymmetry is proportional to $|\xi_\alpha-\xi_\beta|$. Unless the degeneracy differences are very small, there is always a period when self-interactions dominate over the lepton mean-field contribution until they become smaller than the vacuum contribution (see Figure~\ref{fig:ODG_QKE}). At the beginning of this period of self-interaction mean-field domination we can encounter a Matter Neutrino Resonance (MNR)~\cite{Malkus:2014iqa,Johns:2016enc}, when lepton mean-field effects become smaller than self-interaction effects. However in that early phase all matrix densities and all mean-field contributions (save the negligible vacuum one), are diagonal in flavour space, therefore no conversion can occur. Conversely, describing the end of the self-interaction domination, when the vacuum Hamiltonian takes over the self-interaction effects, is rather complicated owing to the physics of synchronous oscillations which takes place, and which depends on the lepton-driven MSW transitions. One of the goals of this chapter is precisely to revisit the physics of these oscillations and their consequences for the equilibration of asymmetries.
Finally, note that cases where degeneracies are so small that self-interactions are at most of the order of the vacuum or lepton mean-field contributions around the MSW transition, lead to rather different physical effects since this condition is largely dependent on the magnitude of neutrino momenta. These low degeneracy regimes have been investigated in~\cite{Johns:2016enc}, but we will not explore such small values, motivated by the fact that BBN constraints are of the order of $10^{-2}$ on $\xi_e$, see equation~\eqref{eq:xi_BBN_CMB}.
\section{Resolution schemes}\label{SecResScheme}
The QKEs~\eqref{eq:QKE_compact_asym} are challenging to solve for various reasons, the main one being the coexistence of multiple time scales: the different terms in the Hamiltonian correspond to different oscillation frequencies, that need to be compared to the collision rate---the latter being in addition particularly computationally expensive. The orders of magnitude of the different terms involved in the QKE~\eqref{eq:QKE_compact_asym} are shown on Figure~\ref{fig:ODG_QKE}.
\paragraph{ATAO approximation}
The separation of these time scales allows for the use of effective resolution schemes.
In general, for a given Hamiltonian $\mathcal{H}$ governing the evolution of a density matrix $\varrho$, i.e., if $\partial_x \varrho = - {\mathrm i} [\mathcal{H},\varrho]$, the eigenvalues of $\mathcal{H}$ give the oscillation frequencies of $\varrho$. More precisely, noting $U_\mathcal{H}$ the unitary matrix which diagonalizes $\mathcal{H}$ (that is $\mathcal{H} = U_\mathcal{H} D_\mathcal{H} U_\mathcal{H}^\dagger$ with $D_\mathcal{H}$ diagonal), the density matrix in the "$\mathcal{H}$-basis" is $U_\mathcal{H}^\dagger \varrho U_\mathcal{H}$. The off-diagonal components of this matrix have oscillatory phases equal to the differences of the diagonal components of $D_\mathcal{H}$.
If $U_\mathcal{H}$ evolves slowly enough\footnote{As explained in section~\ref{sec:ATAO}, $U_\mathcal{H}$ must evolve slowly compared to the inverse oscillation frequency, that is schematically $\abs{(U_\mathcal{H}^\dagger \partial_x U_\mathcal{H})^i_j} \ll \abs{(D_\mathcal{H})^i_i - (D_\mathcal{H})^j_j}$, which corresponds roughly to comparing the Hubble rate to the oscillation frequencies.}, the oscillation frequencies are so large that the off-diagonal components of $U_\mathcal{H}^\dagger \varrho U_\mathcal{H}$ are averaged out. Therefore, transforming back to the flavour basis, we define the averaged matrix $\langle \varrho \rangle_\mathcal{H} $ by
\begin{equation}
\label{DefAverage}
\langle\varrho \rangle_\mathcal{H} \equiv U_\mathcal{H} \reallywidetilde{\left( U^\dagger_\mathcal{H} \varrho U_\mathcal{H} \right)} U_\mathcal{H}^\dagger \, .
\end{equation}
The wide overtilde notation means that we keep only the diagonal part---thus neglecting the fast off-diagonal oscillatory evolution which averages to zero. This procedure requires that the diagonalizing basis changes slowly relative to the oscillations, which is a standard case of adiabatic approximation. Since oscillations are averaged throughout the adiabatic evolution of the Hamiltonian, the \emph{adiabatic transfer of averaged oscillations} (ATAO) consists in the approximation
\begin{equation}
\label{eq:def_ATAO}
\varrho \simeq \langle \varrho \rangle_\mathcal{H} \quad \text{i.e.} \quad \varrho = U_\mathcal{H} \widetilde{\varrho}_\mathcal{H} U_\mathcal{H}^\dagger \, ,
\end{equation}
with $\widetilde{\varrho}_\mathcal{H}$ diagonal. When including collisions, we account for their effects on time scales much larger than the one set by $\mathcal{H}$, which leads to the evolution equations
\begin{equation}\label{ATAOrhodot}
\partial_x \widetilde{\varrho}_\mathcal{H} = U^\dagger_\mathcal{H} \langle\mathcal{K}\rangle_\mathcal{H} U_\mathcal{H} = \widetilde{\mathcal{K}}_\mathcal{H}\,.
\end{equation}
Note that the collision term depends on $\varrho$, which is evaluated with the approximation~\eqref{eq:def_ATAO}.
Such a situation is encountered by neutrinos in the early universe: the results of Figure~\ref{fig:ODG_QKE} show that the Hamiltonian governing the evolution of $\varrho$ is progressively dominated, as the temperature decreases, by the self-potential (and the lepton mean-field), then by the vacuum contribution, and we now detail the associated approximation schemes.
\subsection{{ATAO}-$\mathcal{V}\,$}
If the self-potential can be ignored (for instance if we consider a case without neutrino asymmetries), the fast scale is set by the Hamiltonian $\mathcal{V}$ and we will call this situation the {ATAO}-$\mathcal{V}\,$ approximation, which was used in chapter~\ref{chap:Decoupling}. As previously explained, we thus approximate\footnote{Concerning $\bar{\varrho}$, it is equivalent to average it around $\pm \mathcal{V}$, hence our choice to use $\mathcal{V}$ for both $\varrho$ and $\bar{\varrho}$.} $\varrho \simeq \langle \varrho \rangle_{\mathcal{V}}$ and $\bar{\varrho} \simeq \langle \bar{\varrho} \rangle_{\mathcal{V}}$, such that
\begin{equation}\label{rtildetorhoATAOH}
\varrho = U_{\mathcal{V}} \, \widetilde{\varrho}_{\mathcal{V}} \,U^\dagger_{ \mathcal{V}}\,,\qquad \bar{\varrho} = U_{\mathcal{V}}\, \widetilde{\bar{\varrho}}_{\mathcal{V}} \,U^\dagger_{\mathcal{V}}\,,
\end{equation}
with $\widetilde{\varrho}_{\mathcal{V}}$ and $\widetilde{\bar{\varrho}}_{\mathcal{V}}$ being diagonal. Therefore, it is convenient to solve for the $N_\nu$ diagonal components of these variables instead of the $N_\nu^2$ variables of the density matrices in flavour basis (which, in this approximation, are not independent). The evolution equation~\eqref{ATAOrhodot} leads to
\begin{equation}\label{BasicATAOH}
\partial_x \widetilde{\varrho}_\mathcal{V} = \widetilde{\mathcal{K}}_\mathcal{V}[\varrho,\bar{\varrho}]\,,\qquad \partial_x \widetilde{\bar{\varrho}}_\mathcal{V} = \widetilde{\overline{\mathcal{K}}}_\mathcal{V}[\varrho,\bar{\varrho}]\,.
\end{equation}
Since the collision term depends on $\varrho,\bar{\varrho}$, this means that the evolved variables $\widetilde{\varrho}_\mathcal{V}$ and $\widetilde{\bar{\varrho}}_\mathcal{V}$ are transformed to the flavour basis with \eqref{rtildetorhoATAOH}, so as to evaluate the collision term whose values in flavour space are eventually transformed back into the matter basis. We then keep only their diagonal components through
\begin{equation}
\widetilde{\mathcal{K}}_\mathcal{V} \equiv \reallywidetilde{\left( U^\dagger_\mathcal{V} \mathcal{K} U_\mathcal{V} \right)}\,,\qquad \widetilde{\overline{\mathcal{K}}}_\mathcal{V} \equiv \reallywidetilde{\left( U^\dagger_\mathcal{V} \overline{\mathcal{K}} U_\mathcal{V} \right)}\,.
\end{equation}
Actually, $\mathcal{V}$ depends on both $x$ and $y$, and so does $U_\mathcal{V}$. Hence this averaging scheme is momentum-dependent, which is a central feature to understand the evolution of density matrices. When lepton mean-field effects can be ignored, then the $y$ dependence is the same for all momenta (a $1/y$ prefactor in $\mathcal{V} \simeq \mathcal{H}_0$) and the unitary matrices $U_{\mathcal{V}}$ do not depend on $y$ anymore since they all reduce to the PMNS matrix.
\subsection{{ATAO}-$(\Hself\pm\mathcal{V})\,$}
When neutrino asymmetries cannot be ignored, we see on Figure~\ref{fig:ODG_QKE} that there is a range of temperatures for which $\mathcal{J}$ must necessarily be included in the Hamiltonian. As can be seen in the QKEs~\eqref{eq:QKE_compact_asym}, the Hamiltonian for $\varrho$ is then $\mathcal{J} + \mathcal{V}$ while it is $\mathcal{J} - \mathcal{V}$ for $\bar{\varrho}$. Therefore, the {ATAO}-$(\Hself\pm\mathcal{V})\,$ approximation reads $\varrho \simeq \langle\varrho \rangle_{\mathcal{J}+\mathcal{V}}$ and $\bar{\varrho} \simeq \langle\bar{\varrho} \rangle_{\mathcal{J}-\mathcal{V}}$, such that
\begin{equation}
\varrho = U_{\mathcal{J} + \mathcal{V}} \, \widetilde{\varrho}_{\mathcal{J} + \mathcal{V}} \,U^\dagger_{\mathcal{J} + \mathcal{V}}\,,\qquad \bar{\varrho} = U_{\mathcal{J} - \mathcal{V}}\, \widetilde{\bar{\varrho}}_{\mathcal{J} - \mathcal{V}} \,U^\dagger_{\mathcal{J} - \mathcal{V}}\, ,
\end{equation}
where $\widetilde{\varrho}_{\mathcal{J} + \mathcal{V}}$ and $\widetilde{\bar{\varrho}}_{\mathcal{J} - \mathcal{V}}$ are diagonal. We solve the evolution of $\varrho, \bar{\varrho}$ on timescales much larger than the one set by $\mathcal{J} \pm \mathcal{V}$, on which oscillations are averaged, hence the evolution equation is given by~\eqref{ATAOrhodot}
\begin{equation}\label{BasicATAOJH}
\partial_x \widetilde{\varrho}_{\mathcal{J} +\mathcal{V}} = \widetilde{\mathcal{K}}_{\mathcal{J} +\mathcal{V}}[\varrho,\bar{\varrho}]\,,\qquad \partial_x \widetilde{\bar{\varrho}}_{\mathcal{J} - \mathcal{V}} = \widetilde{\overline{\mathcal{K}}}_{\mathcal{J} - \mathcal{V}}[\varrho,\bar{\varrho}]\,.
\end{equation}
The method is similar to the {ATAO}-$\mathcal{V}\,$ case, but we need to handle the fact that the Hamiltonian itself depends on $\varrho$, through the self-potential $\mathcal{J}$. In order to compute it at each time step, we would need to keep track of the $N_\nu^2$ entries of each density matrix in the flavour basis. A better possibility, which we choose, consists in promoting $\mathcal{J}$ (actually, $\mathcal{A}$) to be an independent variable with its own evolution equation~\eqref{dJdx}. Equation~\eqref{DefJ} is then only used to set the initial value of $\mathcal{J}$ from the initial conditions on $\varrho$, $\bar{\varrho}$. In doing so, we go from $2 \times N\times N_\nu^2$ to $2 \times N \times N_\nu + N_\nu^2$ variables, with $N$ the number of momentum nodes (cf.~section~\ref{subsec:numerics}). We stress that the evolution of $\mathcal{A}$ depends on the full collision terms in flavour space, and not just on the diagonal components in the matter basis $\widetilde{\mathcal{K}}_{\mathcal{J} \pm\mathcal{V}}$, as is the case for $\widetilde{\varrho}_{\mathcal{J} +\mathcal{V}}$ and $\widetilde{\bar{\varrho}}_{\mathcal{J} -\mathcal{V}}$.
\paragraph{High temperatures: {ATAO}-$\Hself\,$} It is clear from Figure~\ref{fig:ODG_QKE} that at large temperatures, $\mathcal{J}$ largely dominates $\mathcal{V} \simeq \mathcal{H}_\mathrm{lep}$ (except for very small $\xi$ that lie outside the range of values we span here). That is why one could consider an even simpler {ATAO}-$\Hself\,$ approximation, where $\mathcal{J} \pm \mathcal{V}$ is replaced by $\mathcal{J}$. In that case, the changes of basis for $\varrho$ and $\bar{\varrho}$ are achieved with the same matrix $U_\mathcal{J}$. In section~\ref{FreqSyncOsc}, we show that this "leading order" Hamiltonian leads to theoretical estimates of synchronous oscillations frequencies in agreement with the existing literature, while using the full {ATAO}-$(\Hself\pm\mathcal{V})\,$ allows to get an important correction which is responsible for quasi-synchronous oscillations.
The weight of $\mathcal{V}$ in the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme becomes more important when the temperature decreases (i.e., $x$ increases), since $\mathcal{J} \propto x^{-2}$ and $\mathcal{H}_0 \propto x^2$.
\subsection{QKE}
The QKE method is not an approximation scheme, but consists instead in solving exactly the neutrino and antineutrino evolutions, that is equations~\eqref{eq:QKE_compact_asym}. However
these equations are very stiff at early times given that all terms except the vacuum one increase for
large temperatures. Therefore, integration times are typically much longer, in addition to the fact that we need to keep track of the $N_\nu^2$ entries of each density matrix in the flavour basis, contrary to the $N_\nu$ diagonal ones in the matter basis when using an ATAO framework.
\subsection{Numerical methods}
\label{subsec:numerics}
The general method used to solve for the time evolution of density matrices is described in section~\ref{sec:numeric_Dec}. The neutrino spectra are sampled on a grid and we have several possible choices for the spacing of the reduced momenta $y$ in this grid. We found that in the context of asymmetry equilibration, a linear spacing is much more adequate than the Gauss-Laguerre quadrature. All numerical results presented in this chapter are performed with an extension of the code \texttt{NEVO}, using a linear grid with $N=40$ points, the minimum and maximum momenta being chosen as described in section~\ref{sec:numeric_Dec}. We start the numerical resolution at $T_{\mathrm{cm}} = 20\,\mathrm{MeV}$, the final temperature depending on the particular configuration investigated. For initial conditions, we set $z_\mathrm{init}$ using that photons, $e^\pm$ and neutrinos are fully thermalized with a common temperature, see equation~\eqref{D1Froustey2020}. In the case of vanishing degeneracies, this determines $z_\mathrm{init}-1 \simeq 7.42 \times 10^{-6}$.
In the general QKE method, the only difference in the code is the contribution of commutators of the type $[\mathcal{A}, \varrho]$ and $[\mathcal{A}, \bar{\varrho}]$ in \eqref{eq:QKE_compact_asym}. However, when using the {ATAO}-$(\Hself\pm\mathcal{V})\,$ method, one needs to add $N_\nu^2$ variables corresponding to the degrees of freedom of $\mathcal{A}$ whose evolution is determined by \eqref{dJdx}.
When equations are stiff, we must rely on implicit methods that
require the computation of the Jacobian of the system of differential
equations. The default method consists in using a finite difference
estimation. The complexity of the calculation of the collision term is $\mathcal{O}(N^3)$ since for each momentum one must compute on a two-dimensional integral \cite{Dolgov_NuPhB1997}. Hence with finite differences the complexity for the Jacobian is $\mathcal{O}(N^4)$. However we can provide its explicit form to the solver and it reduces its evaluation to $\mathcal{O}(N^3)$. This method was used in chapter~\ref{chap:Decoupling} in both the QKE and the {ATAO}-$\mathcal{V}\,$ schemes.
This powerful numerical technique can be extended to the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme, and the essential steps are described in appendix~\ref{App:Numerics}.
Since we only add $N_\nu^2$ variables, the complexity remains $\mathcal{O}(N^3)$. All in all, we found that the code was at least ten times faster with the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme, and even more at low temperatures where the fast oscillations (see next section) slow even more the QKE algorithm.
\section{Synchronous oscillations with two neutrinos}\label{Sec2Neutrinos}
The presence (and domination) of the self-interaction mean-field in the QKEs radically changes the phenomenology of neutrino evolution. This non-linear term notably leads to oscillations of all momentum-modes at a common frequency, a phenomenon named \emph{synchronous oscillations}, studied both numerically \cite{Pastor:2001iu,Dolgov_NuPhB2002,Mangano:2010ei} and analytically \cite{Abazajian2002,Wong2002}. In this section, we extend this theoretical work in the framework of the ATAO approximations we developed: this allows to explicitly calculate the next-to-leading order contribution to the oscillation frequency that was not considered in previous works, and that we check numerically in the next section.
We restrict to a two-flavour case, which allows to easily perform the following calculations thanks to the vector representation of $2 \times 2$ Hermitian matrices. We do not specify yet the values of the mixing parameters, as they will be set for different physical setups in section~\ref{SecRelevant2Neutrinos}.
Let us thus consider in this section the vacuum Hamiltonian of the form
\begin{equation}
\label{eq:Hvac_2nu}
\mathcal{H}_0 = \frac{1}{xH} \left( \frac{x}{m_e} \right) U \begin{pmatrix} 0 & 0 \\ 0 & \Delta m^2/2y
\end{pmatrix} U^\dagger \quad \text{with} \quad U = \begin{pmatrix} \cos{\theta} & \sin{\theta} \\
- \sin{\theta} & \cos{\theta} \end{pmatrix} \, ,
\end{equation}
along with the lepton mean-field contribution of the type
\begin{equation}
\label{eq:Hlep_2nu}
\mathcal{H}_\mathrm{lep} = - \frac{1}{xH} \left(\frac{m_e}{x}\right)^5 \frac{2 \sqrt{2} G_F y}{m_W^2}
\begin{pmatrix}
\bar{\rho}_{l^\pm} + \bar{P}_{l^\pm} & 0 \\
0 & 0
\end{pmatrix} \, .
\end{equation}
In order to maintain a similar expansion history as in the case of three neutrinos, we add one fully decoupled thermalised neutrino flavour to the energy content of the Universe when studying the case of only two neutrino oscillations.
\subsection{Transformation to vectors}
It is customary to rephrase the density matrix evolution as an evolution for vectors using the relation between a Hermitian $2\times2$ matrix $P$, and a vector of $\mathbb{R}^3$ $\vec{P}$
\begin{equation}\label{MatrixToVector}
P = \frac{1}{2}P^0 \mathbb{1} +\frac{1}{2} \vec{P} \cdot \vec{\sigma} \, ,
\end{equation}
where $\vec{\sigma} = \left(\sigma_\mathrm{x}, \sigma_\mathrm{y}, \sigma_z \right)$ is the "vector" of Pauli matrices. Commutators of matrices are then handled using $[\sigma_i,\sigma_j] = 2 {\mathrm i}
\epsilon_{ijk} \sigma_k$ as we obtain
\begin{equation}
-{\mathrm i} [P, Q] = \frac{1}{2} \left( \vec{P} \wedge \vec{Q} \right) \cdot \vec{\sigma} \,.
\end{equation}
The evolution of the neutrino and antineutrino density matrices in vector notations\footnote{For consistency, we write the "vector part" of the two-neutrino density matrix $\vec{\varrho}$, while it is common in the literature to call this the \emph{polarization vector} $\vec{P}$ \cite{SiglRaffelt,Dolgov_NuPhB2002,Johns:2016enc}.} are immediately obtained to be
\begin{equation}
\label{eq:QKE_2nu}
\partial_x \vec{\varrho} = \left(\vec{\mathcal{V}} + \vec{\mathcal{J}}\right) \wedge \vec{\varrho} +
\vec{\mathcal{K}}\,,\qquad
\partial_x \vec{\bar{\varrho}} = \left( -\vec{\mathcal{V}} +
\vec{\mathcal{J}}\right) \wedge \vec{\bar{\varrho}} + \vec{\overline{\mathcal{K}}}\,,
\end{equation}
which we must supplement by
\begin{equation}
\partial_x \varrho^0 = \mathcal{K}^0\,,\qquad \partial_x \bar{\varrho}^0 = \overline{\mathcal{K}}^0\,,
\end{equation}
to account for the evolution of the trace part of density matrices.
In the QKE~\eqref{eq:QKE_2nu}, the vector form of the Hamiltonian $\vec{\mathcal{V}}= \vec{\mathcal{H}}_0 + \vec{\mathcal{H}}_\mathrm{lep}$ is the sum of the vacuum contribution obtained from \eqref{eq:Hvac_2nu}
\begin{equation}
\label{eq:Hvec}
\vec{\mathcal{H}}_0 = \frac{1}{xH} \left(\frac{x}{m_e}\right) \frac{\Delta m^2}{2 y} \begin{pmatrix} \sin(2 \theta) \\ 0 \\ - \cos(2 \theta) \end{pmatrix} \, ,
\end{equation}
and the lepton mean-field one, derived from \eqref{eq:Hlep_2nu},
\begin{equation}
\label{eq:Hlepvec}
\vec{\mathcal{H}}_\mathrm{lep} = - \frac{1}{xH} \left(\frac{m_e}{x}\right)^5 \frac{2 \sqrt{2} G_F y}{m_W^2} \begin{pmatrix} 0 \\ 0 \\ \bar{\rho}_{l^\pm} + \bar{P}_{l^\pm} \end{pmatrix} \, .
\end{equation}
Finally, the asymmetry vector evolves as
\begin{equation}\label{ddxvecJ}
\frac{\mathrm{d} \vec{\mathcal{A}}}{\mathrm{d} x} = \int{\left(\vec{\mathcal{V}} \wedge [\vec{\varrho } + \vec{\bar{\varrho}}] \right) \mathcal{D}y} + \int{\left(\vec{\mathcal{K}} - \vec{\overline{\mathcal{K}}}\right) \mathcal{D} y}\,.
\end{equation}
This vector formalism allows for a more visual representation of the ATAO schemes. Averaging $\varrho$ with respect to an Hamiltonian $\mathcal{H}$ corresponds to \emph{projecting $\vec{\varrho}$ onto $\vec{\mathcal{H}}$}. To see this, we first note that the restriction to the diagonal part of an Hermitian two-by-two matrix corresponds to a projection along $\vec{e}_\mathrm{z}$ in vector notation. Hence when applying the averaging definition~\eqref{DefAverage}, the first step is the rotation which aligns $\vec{\mathcal{H}}$ with $\vec{e}_\mathrm{z}$, then the diagonal part restriction selects only the $\mathrm{z}$-component of this rotation $\vec{\varrho}_\mathcal{H}$, and finally it is rotated back into the initial frame. As a result one has, in the case of two neutrinos,
\begin{equation}
\overrightarrow{\langle \varrho \rangle}_{\mathcal{H}} = (\vec{\varrho} \cdot
\hat{\mathcal{H}}) \hat{\mathcal{H}} \, ,
\end{equation}
where $\hat{\mathcal{H}}$ is the unit vector in the direction of $\vec{\mathcal{H}}$. Since the equations of motion~\eqref{eq:QKE_2nu} correspond to instantaneous precessions set by $\vec{\mathcal{H}}$ (up to the collision term), the averaging procedure corresponds to projecting along that precession vector, i.e. removing the fast rotating part that is orthogonal to it.
\subsection{Frequency of synchronous oscillations}
\label{FreqSyncOsc}
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/th23_mu1_0p01_NoCollSyncOsc.pdf}
\caption[Synchronous oscillations in a two-neutrino $\nu_\mu - \nu_\tau$ case]{\label{fig:SyncOsc} Synchronous oscillations in a two-neutrino $\nu_\mu - \nu_\tau$ case with $\Delta m^2 = 2.45\times10^{-3} \, \mathrm{eV}^2$, $\theta=0.831$, without collisions. In blue $\varrho_\mathrm{z} = \varrho^{\mu}_{\mu} - \varrho^{\tau}_{\tau}$ and in orange $\bar{\varrho}_\mathrm{z}$. The initial degeneracy parameters are $\xi_\mu = 0.01$ and $\xi_\tau = 0$.}
\end{figure}
In some setups where the non-linear self-potential term in the QKEs dominates, such as dense neutrino gases or the early universe (for not too small asymmetries), it has been shown in references~\cite{Samuel:1993uw,Kostelecky:1993yt,Kostelecky:1993dm,Kostelecky:1993ys,Pastor:2001iu,Dolgov_NuPhB2002} that neutrinos develop so-called momentum-independent \emph{synchronous oscillations}, with all $y-$modes being "locked" on the asymmetry vector $\vec{\mathcal{A}}$. This is shown on Figure~\ref{fig:SyncOsc}, where the physical parameters are the same as in the upcoming section~\ref{SecMuonMSW}.
To understand this phenomenon and make quantitative predictions regarding the behaviour of the system of neutrinos and antineutrinos in different setups, we will first ignore the effect of collisions. The initial density matrices are given by equations~\eqref{rhoinit}.
Hence the initial vector components are $\varrho_\mathrm{z}(y) = g(\xi_1,y) - g(\xi_2,y)$ and $\bar{\varrho}_\mathrm{z}(y) = g(-\xi_1,y) - g(-\xi_2,y)$, and $\varrho_{\mathrm{x}, \mathrm{y}}(y) = \bar{\varrho}_{\mathrm{x}, \mathrm{y}}(y) = 0$. Since we neglect collisions in this section, it is clear from~\eqref{eq:QKE_2nu} that the norms of $\vec{\varrho}$ and $\vec{\bar{\varrho}}$ are conserved. The adiabatic evolution of these vectors thus consists in a rotation so as to follow the direction of their Hamiltonian. Therefore, in the {ATAO}-$(\Hself\pm\mathcal{V})\,$ approximation, we can write the density matrix vectors
\begin{equation}
\label{eq:vrho_ATAOJV}
\begin{aligned}
\vec{{\varrho}} &= \abs{g(\xi_1,y)- g(\xi_2,y)} \widehat{\mathcal{J}+\mathcal{V}} \, , \\
\vec{{\bar{\varrho}}} &= - \abs{g(-\xi_1,y)- g(-\xi_2,y)} \widehat{\mathcal{J}-\mathcal{V}} \, ,
\end{aligned}
\end{equation}
where $\widehat{\mathcal{J}+\mathcal{V}}$ is the unit vector in the direction of $\vec{\mathcal{J}}+\vec{\mathcal{V}}$. One must remember that at the initial temperatures we consider ($T_{\mathrm{cm}} \sim 20 \, \mathrm{MeV}$), the Hamiltonian is largely dominated by $\mathcal{J}$. The {ATAO}-$\Hself\,$ approximation then corresponds to discarding $\mathcal{V}$ in the above expressions, and this will give the leading order behaviour of the asymmetry.
\subsubsection{Evolution of the asymmetry vector}
\paragraph{Leading order} Let us then focus on this high temperature region first, when the misalignment between $\vec{\varrho}$ and $\vec{\bar{\varrho}}$ is negligible, i.e.,
\begin{equation}
\vec{\varrho} = \abs{g(\xi_1,y)- g(\xi_2,y)} \widehat{\mathcal{J}} \, , \qquad
\vec{{\bar{\varrho}}} = - \abs{g(-\xi_1,y)- g(-\xi_2,y)} \widehat{\mathcal{J}} \, .
\end{equation}
Hence, the asymmetry vector is obtained from~\eqref{DefJ} and~\eqref{IntegralsFD} and reads
\begin{equation}
\label{JtohatJ}
\vec{\mathcal{A}} = \frac{1}{6}\left\lvert \xi_1-\xi_2 \right\rvert \left(1+\frac{\xi_1^2+\xi_2^2+\xi_1\xi_2}{\pi^2}\right)\widehat{\mathcal{A}} \, ,
\end{equation}
with the unit vector definition $\widehat{\mathcal{A}} = \widehat{\mathcal{J}}$, which is equal initially to $\mathrm{sgn}(\xi_1 - \xi_2) \vec{e}_\mathrm{z}$. We can use the expressions of $\vec{\varrho}, \, \vec{\bar{\varrho}}$ to explicitly compute the $y-$integral appearing in~\eqref{ddxvecJ}. It is then particularly convenient to use the quantities~\eqref{eq:Hamil_y} which isolate the momentum dependence of the Hamiltonian. Therefore, using the integrals given in \eqref{IntegralsFD}, we can rewrite~\eqref{ddxvecJ} as
\begin{equation}
\label{eq:derivA_leading}
\frac{\mathrm{d} \vec{\mathcal{A}} }{\mathrm{d} x}= F(\xi_1,\xi_2)
\vec{\underline{\mathcal{V}}}_\mathrm{eff}\wedge \vec{\mathcal{A}}\qquad \text{where} \qquad
\vec{\underline{\mathcal{V}}}_\mathrm{eff} \equiv \left(\vec{\underline{\mathcal{H}}}_0 + y^2_\mathrm{eff}
\vec{\underline{\mathcal{H}}}_\mathrm{lep} \right) \, ,
\end{equation}
where we defined the slowness factor
\begin{equation}
\label{eq:slowness}
F(\xi_1,\xi_2) \equiv \frac{3}{2} \frac{\xi_1+\xi_2}{\pi^2 +\xi_1^2 +
\xi_2^2+\xi_1 \xi_2}\,,
\end{equation}
in agreement with \cite{Abazajian2002,Wong2002}. The typical "average" momentum is
\begin{equation}
y_\mathrm{eff} \equiv \pi\sqrt{1+\frac{\xi_1^2+\xi^2_2 }{2\pi^2}} \simeq \pi \, ,
\end{equation}
in agreement with equation~(33) in \cite{Wong2002} or equation~(2.19) in \cite{Abazajian2002} (derived in the particular case $\xi_1 = 0$).
This standard result allows to recover the key features of synchronous oscillations. The evolution of all $y-$modes is locked on the evolution of $\vec{\mathcal{A}}$, which precesses around the effective Hamiltonian computed for $y = y_\mathrm{eff}$. However, the oscillation frequency is greatly reduced compared to standard oscillations set by the Hamiltonian $\mathcal{V}(y_\mathrm{eff})$, since for small degeneracies $F \propto (\xi_1 + \xi_2) \ll 1$. Let us provide a numerical evaluation. After muons and antimuons have annihilated (their remaining asymmetry is completely negligible here), and before electrons and positrons did so, that is in the range $200 \,\mathrm{MeV}\ge T_{\mathrm{cm}} \ge 0.5 \,\mathrm{MeV}$, the Hubble parameter~\eqref{eq:scaling_Hubble_x} reads
\begin{equation}\label{Htox2}
H \simeq \frac{m_e}{M_\mathrm{Pl}} \times \frac{m_e}{x^2} \sqrt{\frac{\pi^2}{45} \times \left[1 +
(N_\nu+2)\frac{7}{8}\right]}\simeq \frac{m_e}{x^2}\times 2.278 \cdot 10^{-22}\,,
\end{equation}
where in the last step we have taken $N_\nu=3$. When entering the correct numbers and approximating the slowness
factor by its lowest order in the $\xi_\alpha$, that is $F(\xi_1,\xi_2)\simeq 3(\xi_1+\xi_2)/(2\pi^2)$, we estimate the precession frequency to be
\begin{equation}\label{EvaluationOmega}
\Omega(x) \simeq 1.28\times 10^6 \,x^2\, \abs{\xi_1+\xi_2} \, \frac{\Delta m^2}{10^{-3}\,\mathrm{eV}^2} \, .
\end{equation}
Initially, all unit vectors are aligned $\widehat{\mathcal{A}} \parallel \hat{\mathcal{V}} \simeq \hat{\mathcal{H}}_\mathrm{lep} \parallel \vec{e}_\mathrm{z}$. Then, as the temperature decreases, $\mathcal{J}$ dominates less compared to $\mathcal{V}$ and the vectors $\vec{\varrho}$ and $\vec{\bar{\varrho}}$ become aligned with different directions (namely, $\widehat{\mathcal{J} + \mathcal{V}}$ and $\widehat{\mathcal{J} - \mathcal{V}}$), leading to~\eqref{eq:vrho_ATAOJV}.
\paragraph{Next-to-leading order} Let us therefore now account for the effect of $\mathcal{V}$, in that $\varrho$ and $\bar{\varrho}$ do not get projected on the exact same directions. We will assume that $\lvert \vec{\mathcal{V}} \rvert \ll \lvert \vec{\mathcal{J}} \rvert$, such that we can perform an expansion of the unit vector
\begin{equation}
\label{ExpandSH}
\widehat{\mathcal{J}+\mathcal{V}} \simeq \widehat{\mathcal{J}} + \frac{\vec{\mathcal{V}} }{ |\vec{\mathcal{J}}| } - \left(\frac{\vec{\mathcal{V}}\cdot \vec{\mathcal{J}}}{|\vec{\mathcal{J}}|^2}\right) \widehat{\mathcal{J}}+\cdots
\end{equation}
For $\widehat{\mathcal{J}-\mathcal{V}} $ the expression is identical up to $\vec{\mathcal{V}} \to -\vec{\mathcal{V}}$. This expansion gives, in the {ATAO}-$(\Hself\pm\mathcal{V})\,$ approximation, the next-to-leading order (NLO) terms that were not explicited in previous works (cf., for instance, equation~(25) in~\cite{Wong2002}). Including this expansion in equation~\eqref{ddxvecJ}, we can once again recast the evolution of the asymmetry as a precession equation
\begin{equation}
\label{eq:precession}
\frac{\mathrm{d} \vec{\mathcal{A}}}{\mathrm{d} x} = \vec{\Omega} \wedge \vec{\mathcal{A}} \, ,
\end{equation}
where the oscillation frequency (in $x$ variable) reads, retaining only the vacuum contribution $\mathcal{V} = \mathcal{H}_0$ for simplicity,\footnote{This is not an oversimplification. Indeed, as long as $\mathcal{H}_\mathrm{lep}$ dominates over $\mathcal{H}_0$, all vectors are aligned along $\vec{e}_\mathrm{z}$ and no precession takes place.}
\begin{equation}
\label{eq:Omega_JH}
\vec\Omega = F(\xi_1, \xi_2 ) \vec{\underline{\mathcal{H}}}_0 \left[1
-\frac{12}{\sqrt{2} G_F}\left(\frac{x}{m_e}\right)^3 \frac{x H \vec{\underline{\mathcal{H}}}_0 \cdot
\widehat{\mathcal{A}}}{|\xi_1-\xi_2|(\xi_1 + \xi_2)} \right]\,.
\end{equation}
The second term between brackets is the NLO term, which accounts for the difference between purely synchronous and \emph{quasi}-synchronous oscillations, given that its origin is rooted in the orientation differences between $\vec{\varrho}$ and $\vec{\bar{\varrho}}$. Note that we also took the lowest order contribution $\lvert \vec{\mathcal{A}} \rvert \simeq \abs{\xi_1 - \xi_2}/6$, valid for small degeneracy parameters (we can use the leading order expression~\eqref{JtohatJ} since the evolution of $\vec{\mathcal{A}}$ is a precession, hence its norm is unchanged). The expression~\eqref{eq:Omega_JH} leaves a priori the possibility of divide-by-zero if $\xi_1 = \pm \, \xi_2$. We discuss these special cases at the end of this section.
To estimate the precession frequency, we consider as before that initially $\widehat{\mathcal{A}} = \mathrm{sgn}(\xi_1 - \xi_2) \vec{e}_z$ and assume the transition between $\mathcal{H}_\mathrm{lep}$ and $\mathcal{H}_0$ to be abrupt enough such that we can estimate, using \eqref{eq:Hvac_2nu}, $x H \vec{\underline{\mathcal{H}}}_0 \cdot \widehat{\mathcal{A}} = - \mathrm{sgn}(\xi_1 - \xi_2) (x/m_e) (\Delta m^2 / 2) \cos(2\theta)$. Therefore we get
\begin{equation}
\label{eq:Omega_norm}
\lvert \vec\Omega \rvert = \frac{\abs{F(\xi_1, \xi_2 ) \Delta m^2}}{2 m_e H} \times \left\lvert 1
+ \left(\frac{x}{x_\mathrm{tr}}\right)^4 \frac{\mathrm{sgn}(\Delta m^2 \cos(2 \theta))}{\xi_1^2 - \xi_2^2} \right \rvert \,,
\end{equation}
where we defined
\begin{equation}
x_\mathrm{tr} \equiv m_e \left(\frac{\sqrt{2} G_F}{6 |\Delta m^2
\cos(2\theta)|}\right)^{1/4}\simeq 3.7 \left(\frac{10^{-3}\,\mathrm{eV}^2}{|\Delta m^2
\cos(2\theta)|}\right)^{1/4}\,.
\end{equation}
Given the scaling $H \propto x^{-2}$ recalled in~\eqref{eq:scaling_Hubble_x}, the frequency of synchronous oscillations keeps increasing as the Universe expands, first as $\Omega \propto x^2$ (leading order) and then as $\Omega \propto x^6$ (NLO domination).
This second behaviour is a novel result of this thesis. Although one might expect that the effect of $\mathcal{V}$ would be completely subdominant compared to $\mathcal{J}$, the particular form of the equation of motion changes this picture. Keeping the dominant term in the expansion~\eqref{ExpandSH} (i.e., $\widehat{\mathcal{J}}$) leads in the evolution equation~\eqref{ddxvecJ} to an integral symmetric under $\xi_1 \to - \xi_1$ and similarly for $\xi_2$. Therefore, the associated contribution is proportional to $\xi_1^2 - \xi_2^2$, which after dividing by the norm of $\vec{\mathcal{A}}$ accounts for the precession frequency $\propto \xi_1 + \xi_2$ obtained in \eqref{eq:slowness}. However, the first order correction in~\eqref{ExpandSH} is odd with respect to $\vec{\mathcal{V}}$, hence an antisymmetric integral with respect to $\xi_{1,2} \to - \xi_{1,2}$. The corresponding result is $\propto \xi_1 - \xi_2$, which is enhanced compared to the leading order term.
The transition from leading to next-to-leading order is then found to be around
\begin{equation}
\label{eq:xNLO}
x_\mathrm{NLO} \equiv x_\mathrm{tr}|\xi_1^2-\xi_2^2|^{1/4}\,.
\end{equation}
Note that depending on the sign of $\Delta m^2 \cos(2\theta)/(\xi_1^2-\xi_2^2)$ the frequency can go through zero, which means that $\vec{\mathcal{A}}$ can precess in one direction, slow down, and then precess in the opposite direction with a frequency increasing as $\propto x^6$.
\paragraph{Particular cases} While the previous calculation seemed fairly general, there are two specific cases that deserve to be discussed.
\begin{itemize}
\item \emph{Equal asymmetries ---} if $\xi_1 = \xi_2$ the asymmetry vector $\vec{\mathcal{A}}$ is strictly zero and the previous formalism is inadequate (namely, \eqref{eq:derivA_leading} cannot be obtained anymore since initially $\vec{\varrho} = \vec{\bar{\varrho}} = \vec{0}$).
\item \emph{Equal but opposite asymmetries ---} if $\xi_1 = - \xi_2$, the leading order term in~\eqref{eq:Omega_JH} vanishes (since $F(\xi_1,\xi_2) \propto \xi_1 + \xi_2$), but not the next-to-leading order contribution, a special case investigated at the end of section~\ref{SecMuonMSW}.
\end{itemize}
\paragraph{Summary: evolution of $\vec{\mathcal{A}}$} Initially, at high temperatures (typically $T_{\mathrm{cm}} \sim 20 \, \mathrm{MeV}$), the lepton term dominates over the vacuum one and $\vec{\mathcal{A}} \propto \vec{\mathcal{H}}_{\mathrm{lep}} \parallel \vec{e}_z$. All vectors are aligned, and this situation does not change until the MSW transition between $\mathcal{H}_\mathrm{lep}$-domination to $\mathcal{H}_0$-domination. If this transition is slow enough compared to the precession frequency, then $\vec{\mathcal{A}}$ keeps following $\vec{\underline{\mathcal{V}}}_\mathrm{eff}$ and ends up aligned with $\vec{\underline{\mathcal{H}}}_0$. This corresponds to an \emph{adiabatic} evolution of the asymmetry vector itself. Conversely, if the transition is too abrupt (that is much shorter than the precession timescale), $\vec{\mathcal{A}}$ gets brutally misaligned with $\vec{\mathcal{H}}_0$ and oscillations develop. Let us stress that the evolution of $\varrho$ and $\bar{\varrho}$ is in general adiabatic, but it is the evolution of the vector that they track, namely $\vec{\mathcal{J}}$, which can be non-adiabatic depending on the value of the slowness factor~\eqref{eq:slowness}.
If oscillations do develop, initially at the frequency~\eqref{EvaluationOmega}, the increasing influence of $\mathcal{V}$ compared to $\mathcal{J}$ leads to a new behaviour: beyond $x_\mathrm{NLO}$ given by~\eqref{eq:xNLO}, the frequency increases faster and $\Omega \propto x^6$. These features are illustrated in section~\ref{SecRelevant2Neutrinos}. Note that the calculation of the NLO assumes $\lvert \vec{\mathcal{V}} \rvert \ll \lvert \vec{\mathcal{A}} \rvert$, but at some point the vacuum term becomes dominant over the self-potential one (cf.~Figure~\ref{fig:ODG_QKE}) and the {ATAO}-$(\Hself\pm\mathcal{V})\,$ regime breaks down. This is discussed in more detail in section~\ref{SecTransitionToATAOV}.
\subsubsection{Adiabaticity parameter}
In the case with only two neutrinos, let us consider the evolution of the asymmetry vector $\vec{\mathcal{A}}$ without collisions and at leading order, which is dictated by equation~\eqref{eq:derivA_leading}. If the transition from a mean-field dominated to a vacuum dominated Hamiltonian, that is the MSW transition, is slow enough, then $\vec{\mathcal{A}}$ evolves adiabatically and follows $\vec{\underline{\mathcal{V}}}_\mathrm{eff} $. In order to assess the degree of (non-)adiabaticity, we thus need to quantify the speed at which the transition takes place. Let us first define a tipping angle $\beta$, illustrated in Figure~\ref{FigAngles}, by
\begin{equation}
\cos(2\beta) = -\hat{\underline{\mathcal{V}}}_\mathrm{eff} \cdot \vec{e}_\mathrm{z} \quad \text{since }\quad \hat{\underline{\mathcal{V}}}_\mathrm{eff}(x \ll x_\mathrm{MSW})= -\vec{e}_\mathrm{z}\,.
\end{equation}
\begin{figure}[!ht]
\centering
\includegraphics{figs/Schemas_adiab.pdf}
\caption[Parameterization of the adiabaticity of the evolution of $\vec{\mathcal{A}}$]{\label{FigAngles} Definition of the tipping angle $\beta$ on the left. The condition $\beta = \theta/2$ shown on the right corresponds to our definition of the MSW transition.}
\end{figure}
Initially the tipping angle vanishes, and long after the transition it reaches $\theta$. We define the location of the MSW transition $x_\mathrm{MSW}$ as the moment when the tipping angle takes half of its final value, $\beta_\mathrm{MSW} = \theta/2$, which corresponds to $|\vec{\underline{\mathcal{H}}}_0| = y^2_\mathrm{eff}|\vec{\underline{\mathcal{H}}}_\mathrm{lep}|$ (as can be checked by trigonometric manipulations). Let us then define an adiabaticity parameter as
\begin{equation}\label{Defgammatr1}
\gamma \equiv \frac{|\vec{\Omega}|}{\partial_x (2 \beta)}\, ,
\end{equation}
with $\vec{\Omega} = F(\xi_1,\xi_2)\vec{\underline{\mathcal{V}}}_\mathrm{eff}$, whose value is estimated from\footnote{The next-to-leading order contribution to $\vec{\Omega}$ is not relevant here as we focus on cases where $x_\mathrm{MSW} \ll x_\mathrm{NLO}$.} \eqref{eq:derivA_leading}. A large $\gamma$ corresponds to a rate of change of the Hamiltonian direction ($2 \partial_x \beta$) much smaller than the instantaneous precession frequency ($|\vec{\Omega}|$), that is to a very adiabatic evolution. We find
\begin{equation}
\gamma^{-1} = -\frac{(\underline{\mathcal{H}}_0^\perp)^2 y_\mathrm{eff}^2}{\abs{F(\xi_1,\xi_2)}|\vec{\underline{\mathcal{V}}}_\mathrm{eff}|^3} \partial_x\left(\frac{|\vec{\underline{\mathcal{H}}}_\mathrm{lep}|}{\underline{\mathcal{H}}_0^\perp}\right)\,,\quad \text{where}\quad \underline{\mathcal{H}}_0^\perp \equiv |\vec{\underline{\mathcal{H}}}_0| \sin(2 \theta)\,.
\end{equation}
We have used $\partial_x (2\beta) = -\sin^2(2 \beta)y^2_\mathrm{eff}\partial_x\left(|\vec{\underline{\mathcal{H}}}_\mathrm{lep}|/\underline{\mathcal{H}}_0^\perp \right)$ and $\sin(2 \beta) = \underline{\mathcal{H}}_0^\perp/|\vec{\underline{\mathcal{V}}}_\mathrm{eff}|$.
In order to assess the adiabaticity of the transition, we must estimate how large $\gamma$ is at the transition. Since $|\vec{\underline{\mathcal{H}}}_\mathrm{lep}|/ \underline{\mathcal{H}}_0^\perp \propto 1/x^6$, and using that at the transition we have $|\vec{\underline{\mathcal{V}}}_\mathrm{eff}| = 2 |\vec{\underline{\mathcal{H}}}_0| \cos \theta$ (see the geometry on the right plot of Figure~\ref{FigAngles}), the value of the adiabaticity parameter at the transition reduces to
\begin{equation}\label{Defgammatr2}
\gamma_\mathrm{MSW} = \left.\abs{F(\xi_1,\xi_2)}\frac{2}{3} \frac{x |\vec{\underline{\mathcal{H}}}_0| \cos^2 \theta}{\sin\theta}\right|_{x=x_\mathrm{MSW}}\,.
\end{equation}
Let us define the degree of non-adiabaticity through
\begin{equation}
P_\mathrm{n.a} \equiv \frac{1}{2}\left(1 \mp \widehat{\mathcal{A}} \cdot \widehat{\underline{\mathcal{V}}}_\mathrm{eff}\right)\,,
\end{equation}
with a $-$ sign (resp.~$+$ sign) if initially $\vec{\mathcal{A}}$ is aligned (resp.~anti-aligned) with $\vec{\Hamil}_{\mathrm{eff}}$ (i.e., $- \vec{e}_\mathrm{z}$). Its asymptotic value $P^\infty_\mathrm{n.a}$ when $x \gg x_\mathrm{MSW}$ estimates the misalignment of the final asymmetry vector due to lack of adiabaticity. Indeed if the transition is perfectly adiabatic, $\vec{\mathcal{A}}$ keeps tracking $\vec{\underline{\mathcal{V}}}_\mathrm{eff}$ and we always have $P_\mathrm{n.a}=0$. In general, the degree of non-adiabaticity needs not be much larger than unity to lead to a small $P^\infty_\mathrm{n.a}$, that is to a very non-adiabatic transition---see for instance the Landau-Zener approximation~\eqref{LZ}.
We note from equation~\eqref{Defgammatr2} that the adiabaticity parameter is of order $x \abs{F(\xi_1,\xi_2)}|\vec{\underline{\mathcal{H}}}_0|$ evaluated at the transition, modulated by a geometric factor $(2/3) \cos^2 \theta/\sin \theta$. A transition is resonant when at some point the tipping angle goes through $2\beta = \pi/2$, that is through $\vec{\underline{\mathcal{V}}}_\mathrm{eff}$ having no component along $\vec{e}_\mathrm{z}$. Hence, a very non-resonant transition corresponds to $\theta \ll 1$, and in that case the geometric factor is enhanced by $1/\sin(\theta)$. It is less likely to encounter a small adiabaticity parameter because the tipping angle is small, and $|\vec{\Omega}|$ at the transition is larger than its final value (since $|\vec{\underline{\mathcal{V}}}_\mathrm{eff}|$ keeps decreasing). Conversely for a very resonant transition, $\pi/2-\theta \ll 1$, and the geometric factor is reduced by $\cos^2 \theta $ which is small, that is leads to a smaller adiabaticity parameter. This is partly because of the large tipping angle, but also because $|\vec{\Omega}|$ at the transition is much smaller than its final value (recall that at the transition $|\vec{\underline{\mathcal{V}}}_\mathrm{eff}| = 2 |\vec{\underline{\mathcal{H}}}_0| \cos \theta$).
In the very resonant configuration ($\pi/2-\theta \ll 1$), the adiabaticity parameter for the dynamics of $\vec{\mathcal{A}}$ takes the simpler form
\begin{equation}\label{EqVeryResonant}
\gamma^{(\pi/2-\theta \ll 1)}_\mathrm{MSW} = \abs{F(\xi_1,\xi_2)}\underline{\mathcal{H}}_0^\perp \Delta x_\mathrm{MSW} \, , \ \ \text{with}\qquad \frac{1}{\Delta x_\mathrm{MSW}} \equiv y_\mathrm{eff}^2\left.\partial_x \left(\frac{|\vec{\underline{\mathcal{H}}}_\mathrm{lep}|}{\underline{\mathcal{H}}_0^\perp}\right) \right|_{x=x_\mathrm{MSW}}\,.
\end{equation}
\paragraph{Landau-Zener formula} The Landau-Zener~\cite{Landau1932,Zener1932,Haxton:1986bc,Abazajian:2001nj,Johns:2016enc,Wittig} formula is an approximation of the degree of non-adiabaticity in this very resonant situation, using the approximation that the diagonal components of $\mathcal{V}$ are linear in $x$ and that off-diagonal ones are constant, which reads
\begin{equation}\label{LZ}
P^\infty_\mathrm{n.a} \simeq \exp(-\pi \gamma_\mathrm{MSW}/2)\,.
\end{equation}
Note that the factors $F(\xi_1,\xi_2)$ and $y_\mathrm{eff}^2$ in \eqref{EqVeryResonant} are specific to the fact that we consider the evolution of $\vec{\mathcal{A}}$. If we had considered the evolution of $\vec{\varrho}$ given by equation~\eqref{eq:QKE_2nu} without self-interactions nor collisions, that is $\partial_x \vec{\varrho} = \vec{\mathcal{V}} \wedge \vec{\varrho} $, we would have obtained with a similar analysis (all quantities are written here for a given momentum $y$) the usual expression for the Landau-Zener adiabatic parameter
\begin{equation}\label{GammaLZ}
\gamma^{(\pi/2-\theta \ll 1)}_\mathrm{MSW} =\mathcal{H}_0^\perp \Delta x_\mathrm{MSW}\, , \quad \text{with} \qquad \frac{1}{\Delta x_\mathrm{MSW}} \equiv \left.\partial_x \left(\frac{|\vec{\mathcal{H}}_{\mathrm{lep}}|}{\mathcal{H}_0^\perp}\right)\right|_{x=x_\mathrm{MSW}}
\end{equation}
where $\mathcal{H}_0^\perp = \mathcal{H}_0 \sin(2 \theta)$. It is similar to equation~(9b) of \cite{Haxton:1986bc} and equation~(7.8) of \cite{Abazajian:2001nj},
the only difference being that $\mathcal{H}_0^\perp$ is considered constant when dealing with solar neutrinos, whereas in the cosmological context it scales as $\propto x$, hence explaining why the expression \eqref{GammaLZ} for the transition width, $\Delta x_\mathrm{MSW}$, takes into account this evolution. However, note that this necessary modification for the expression of the adiabaticity parameter is absent from equation (28) of \cite{Johns:2016enc}, although this impacts only marginally the estimation of adiabaticity.
\section{Relevant two-neutrino cases for the primordial Universe}\label{SecRelevant2Neutrinos}
The previous results derived with only two neutrinos can shed some light on the physics at play in the standard case with three neutrinos and a general PMNS matrix. After the muon-driven MSW transition and before the electron-driven one, the oscillations only take place in the $\nu_\mu-\nu_\tau$ subspace since the unitary matrix $U_{\mathrm{eff}}$ that diagonalizes $\mathcal{V}$ is approximately
\begin{equation}\label{UtoU}
U_{\mathrm{eff}} = R_{23}(\theta_{23}^{\mathrm{eff}}) = \begin{pmatrix}
1 & 0 & 0\\
0 & \cos \theta_{23}^{\mathrm{eff}} &\sin \theta_{23}^{\mathrm{eff}} \\
0 & -\sin \theta_{23}^{\mathrm{eff}} & \cos \theta_{23}^{\mathrm{eff}}
\end{pmatrix} \,,
\end{equation}
this form being rigorously valid in the limit $m_\mu/m_e \to \infty$. Expanding in the ratio $\epsilon = \Delta m_{21}^2 / \Delta m_{32}^2$, we find
\begin{equation}\label{th23th23}
\tan(2 \theta_{23}^{\mathrm{eff}}) = \tan(2 \theta_{23}) - \epsilon
\frac{ \sin (\theta_{13}) \sin(2 \theta_{12})}{\cos^2 (\theta_{13}) \cos^2(2 \theta_{23})} + \mathcal{O}(\epsilon^2)\,.
\end{equation}
Given the values~\eqref{ValuesStandard} we find $\theta_{23}^{\mathrm{eff}} \simeq \theta_{23}$ with a difference of order $0.25\,\%$. Hence, we investigate the case $\Delta m^2 = \Delta m_{32}^2$ and $\theta=\theta_{23}$ in section~\ref{SecMuonMSW} to study the evolution of density matrices after the muon-driven transition.
Unfortunately, the system is not so easily reduced to a two-neutrino system when it comes to the description of the subsequent electron-driven MSW transitions. For simplicity, we choose to consider a fictitious configuration where $\theta_{13}=\theta_{23}=0$ such that oscillations take only place in the $\nu_e-\nu_\mu$ subspace, with the electrons/positrons being the relevant contribution to the lepton mean-field effects \eqref{eq:Hlep_2nu}. This configuration is detailed in section~\ref{SecElectronMSW}. For numerical applications we consider $\Delta m^2 = \Delta m_{21}^2$ and $\theta=\theta_{12}$. Although it is an ideal setup, it will provide important insight for the full three-neutrino case in section~\ref{Sec:3neutrinos}.
\subsection{Muon-driven MSW transition}\label{SecMuonMSW}
Let us consider a muon-driven MSW transition with $\theta = \theta_{23} \simeq 0.831$ and $\Delta m^2 = \Delta m_{32}^2 \simeq 2.453 \times 10^{-3} \, \mathrm{eV}^2$ for numerics \cite{PDG}. We restrict to normal ordering for simplicity, and do not include collisions yet. This means that electrons and positrons are absent from this description, except for their contribution to the energy density and thus the Hubble parameter~\eqref{eq:scaling_Hubble_x}.
\subsubsection{Description of the transition}\label{SecDescriptionMuonTransition}
As outlined before, synchronous oscillations of the neutrino ensemble can develop when the MSW transition occurs, provided this transition is abrupt enough for $\vec{\mathcal{A}}$ to get suddenly misaligned with $\vec{\Hamil}_{\mathrm{eff}}$ and precess around it. Let us first estimate the location of this transition. The energy density of muons/antimuons drops very rapidly once they become non-relativistic. In this limit, we get
\begin{equation}
\rho_{\mu^\pm}+ P_{\mu^\pm} = 4 \sigma^{5/2}
\left(\frac{x}{2\pi}\right)^{3/2} \mathrm{e}^{-\sigma x} \times \frac{m_e^4}{x^3}
\end{equation}
with $\sigma = m_\mu/m_e\simeq 206.77$. Approximating $y_{\mathrm{eff}} \simeq \pi$, we find that the vacuum term is equal in magnitude to the lepton term --- which is our definition of the transition, --- for $x_\mathrm{MSW}\simeq 0.043$, that is for $T_{\mathrm{cm}} \simeq 12\,\mathrm{MeV}$. Given the exponential drop $\exp(-\sigma x)$ of muons energy density, one can note that $x_\mathrm{MSW}$ is very mildly sensitive to the value of $\Delta m^2$.
The adiabaticity parameter given by \eqref{Defgammatr2} is then
\begin{equation}
\label{eq:gammaMSW_muon}
\gamma_\mathrm{MSW} \simeq 100 \times \abs{\xi_1 + \xi_2} \,.
\end{equation}
For $\xi_1+\xi_2$ of a few percent or smaller, we find $\gamma_\mathrm{MSW}<1$ and the transition is abrupt, that is the evolution of $\vec\mathcal{A}$ during the transition is very non-adiabatic. Hence we expect that as the direction of the effective Hamiltonian moves away from the vertical axis, $\vec{\mathcal{A}}$ will develop oscillations at the frequency $\Omega$. For much larger $\xi_1+\xi_2$ such that $\gamma_\mathrm{MSW} > 1$, and considering the Landau-Zener estimation for the degree of adiabaticity~\eqref{LZ}, $\vec{\mathcal{A}}$ should tend to follow adiabatically the transition to the vacuum Hamiltonian.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/ATAOs_th23_mu1_NoColl.pdf} \\
\includegraphics[]{figs/ATAOs_th23_mu2_NoColl.pdf}
\caption[Evolution of the flavour asymmetries for a $\nu_\mu - \nu_\tau$ system without collisions]{\label{fig:2nu_mu-tau} Evolution of the flavour asymmetries for a two-neutrino $\nu_\mu$ (green) - $\nu_\tau$ (red) system without collisions, with $\Delta m^2 = 2.45 \times 10^{-3} \, \mathrm{eV}^2$ and $\theta = 0.831$. We compare different numerical schemes: in solid line QKE, in dots {ATAO}-$(\Hself\pm\mathcal{V})\,$ (hidden behind QKE), in dot-dashes {ATAO}-$\Hself\,$, and in dashes {ATAO}-$\mathcal{V}\,$. The initial degeneracy parameters are on the first row $\xi_2 =0$ and $\xi_1=0.1,0.01,0.001$ from left to right ; on the second row $\xi_1 =0$ and $\xi_2=0.1,0.01,0.001$. On the $\mathrm{y}$-axis label, $\xi$ stands for the non-zero initial $\xi_i$.}
\end{figure}
The evolution of asymmetry is illustrated in Figure~\ref{fig:2nu_mu-tau}. It is clear that the evolution with the self-interaction mean-field is completely different from the evolution where this has been ignored and which corresponds to the {ATAO}-$\mathcal{V}\,$ line: no synchronous oscillations take place in this scheme. These oscillations, in agreement with our adiabaticity estimate, do develop significantly for initial degeneracies smaller than one percent. On the contrary, for $\xi_1 + \xi_2 = 0.1$, the transition is quasi-adiabatic and $\vec{\mathcal{A}}$ follows the direction set by $\vec{\Hamil}_{\mathrm{eff}}$ with oscillations of much smaller amplitude compared to the smaller $\xi$ cases (right plots). Furthermore, it appears that at small degeneracies, the {ATAO}-$\Hself\,$ results differ from the more accurate {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme, the latter matching perfectly the QKE method. The difference between {ATAO}-$\Hself\,$ and {ATAO}-$(\Hself\pm\mathcal{V})\,$ can be understood by considering the NLO contribution to the precession frequency: this extra contribution explains the "wrong" frequency in the {ATAO}-$\Hself\,$ case (see for instance the bottom right plot on Figure~\ref{fig:2nu_mu-tau}), or even the wrong qualitative behaviour of the asymmetry (top right plot).
\paragraph{Synchronous oscillation frequency}
To estimate the frequency from our runs we compute $(\partial_x \vec{\mathcal{A}} \wedge \vec{\mathcal{A}})/|\vec{\mathcal{A}}|^2$ which gives, given the precession equation~\eqref{eq:precession}, the projection of the rotation vector $\vec{\Omega}$ orthogonally to $\vec{\mathcal{A}}$. If the MSW transition is abrupt, the precession takes place around $\vec{\mathcal{H}}_0$ with an angle $2 \theta$, hence the former quantity should be equal to $\lvert \sin(2 \theta) \vec{\Omega} \rvert$. Both frequencies are shown on Figure~\ref{fig:Omega_2nu_mu-tau}. We clearly see the transition from the regime $\Omega \propto x^2$ to
$\Omega \propto x^6$, that is the transition to the NLO regime, and in particular how the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme fits the QKE results, while (as expected by construction) the {ATAO}-$\Hself\,$ scheme completely misses this change of regime. In the region $x_\mathrm{MSW} < x \ll x_\mathrm{NLO}$, $\mathcal{J}$ largely dominates over $\mathcal{V}$ and all three schemes coincide.
Also, since $\theta=0.831 > \pi/4$, $\cos(2\theta)<0$ and according to~\eqref{eq:Omega_norm} the frequency can go through zero for normal ordering ($\Delta m^2>0$) with $|\xi_1|>|\xi_2|$ or inverted ordering ($\Delta m^2<0$) with $|\xi_2|>|\xi_1|$. That is the case in the top right plot of Figure~\ref{fig:2nu_mu-tau} ($\xi_1 = 0.001$, $\xi_2 = 0$ and normal ordering): we observe a back and forth motion of $\mathcal{A}$ for\footnote{The value predicted using \eqref{eq:xNLO} is slightly different from the one obtained numerically, because \eqref{eq:xNLO} assumes zero adiabaticity, such that the angle between $\vec{\mathcal{A}}$ and $\vec{\mathcal{H}}_0$ is exactly $2 \theta$, whereas in reality $\vec{\mathcal{A}}$ partially follows the direction of $\vec{\Omega}$ during the MSW transition (see the bottom right plot on Figure~\ref{fig:Coll_e-mu} for a similar behaviour in an electron-driven transition).} $T_\mathrm{NLO} \simeq 2.8 \, \mathrm{MeV}$, which corresponds to $\Omega = 0$ as visible on Figure~\ref{fig:Omega_2nu_mu-tau}, left plot. This transition between two frequency regimes with a change of rotation direction is a feature also seen in Figures 9 and 10 of \cite{Johns:2016enc}.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/th23_mu1_0p001_NoCollOmega.pdf}
\includegraphics[]{figs/th23_mu2_0p001_NoCollOmega.pdf}
\caption[Frequency of synchronous oscillations for a $\nu_\mu - \nu_\tau$ system]{\label{fig:Omega_2nu_mu-tau} Frequency of synchronous oscillations in the case $\xi_1=0.001$, $\xi_2=0$
(\emph{left}) and the case $\xi_1=0$, $\xi_2=0.001$ (\emph{right}). We consider a $\nu_\mu - \nu_\tau$ system with $\Delta m^2 = 2.45 \times 10^{-3}\,\mathrm{eV}^2$ and $\theta = 0.831$. The vertical red line is the location of the MSW
transition. The dashed black line is $|\Omega \sin(2\theta)|$, that is the analytic approximation~\eqref{eq:Omega_norm}. The coloured lines correspond to $\lvert (\partial_x \vec{\mathcal{A}} \wedge
\vec{\mathcal{A}}) \rvert /|\vec{\mathcal{A}}|^2$, in the QKE method (blue), the
{ATAO}-$\Hself\,$ scheme in orange and the {ATAO}-$(\Hself\pm\mathcal{V})\,$ in green (hidden behind QKE).}
\end{figure}
If we consider smaller degeneracies, such that
\begin{equation}
|\xi_1^2-\xi_2^2| < \frac{x^4_\mathrm{MSW}}{x^4_\mathrm{tr}} \simeq
1.8\times 10^{-8}\left(\frac{|\Delta m^2
\cos(2\theta)|}{10^{-3}\,\mathrm{eV}^2}\right)
\end{equation}
then the NLO contribution to the precession frequency will dominate already when the MSW transition occurs. The adiabaticity parameter $\gamma_\mathrm{MSW}$ must be rescaled by multiplying it by the factor in square brackets in equation~\eqref{eq:Omega_JH}, and we get approximately
\begin{equation}
\gamma_\mathrm{MSW} =\frac{4.1 \times 10^{-7}}{\abs{\xi_1 - \xi_2}} \, ,
\end{equation}
assuming the NLO contribution does dominate in~\eqref{eq:Omega_JH}, which amounts to multiplying~\eqref{eq:gammaMSW_muon} by $(x_\mathrm{MSW}/x_\mathrm{tr})^4 /\abs{\xi_1^2 - \xi_2^2}$. Therefore, if we satisfy the condition $|\xi_1-\xi_2| \gg 4.1 \times10^{-7}$, the transition is still abrupt and oscillations do develop. For even smaller degeneracies, there is no clear region where $|\vec{\mathcal{J}}| \gg |\vec{\mathcal{V}}|$, and the subsequent phenomenology can only be captured by a full QKE resolution as in~\cite{Johns:2016enc}.
\paragraph{Beginning of oscillations} Provided the MSW transition is non-adiabatic, oscillations of $\vec{\mathcal{A}}$ appear, driving each individual mode. However, one can see on Figure~\ref{fig:2nu_mu-tau} that depending on the value of $(\xi_1, \xi_2)$, the apparent "start" of these oscillations looks shifted while $x_\mathrm{MSW}$ is the same. We can estimate how oscillations develop, which provides an additional check of our analytical developments.
The asymmetry evolves with a frequency $\Omega(x)$, therefore the phase of the oscillations is at any time given by
\begin{equation}
\frac{\mathrm{d} \Phi}{\mathrm{d} x} = \Omega(x) \qquad \text{hence} \qquad \Phi(x) = \frac{1}{3} \Omega(x) x \, ,
\end{equation}
where we used the fact that $\Omega \propto x^2$, keeping only the leading order contribution~\eqref{EvaluationOmega}. Half a period of oscillation is reached when $\Phi(x_\pi) = \pi$, which happens for
\begin{equation}
\label{eq:x_startosc}
x_\pi = \left( \frac{1}{1.28 \times 10^6} \times \frac{10^{-3} \, \mathrm{eV}^2}{\abs{\Delta m^2}} \times \frac{1}{\abs{\xi_1 + \xi_2}} \times 3 \pi \right)^{1/3} \simeq 0.067 \, ,
\end{equation}
for $\xi_1 = 0.01$ and $\xi_2=0$, which agrees with Figure~\ref{fig:2nu_mu-tau}, top middle plot.
\subsubsection{Particular cases}\label{SecParticular}
In this subsection, we use the $\nu_\mu - \nu_\tau$ framework to discuss the particular cases of equal and equal but opposite asymmetries, for which the calculations of section~\ref{FreqSyncOsc} are no longer valid.
If $\xi_1 = \xi_2$, the vector parts of $\varrho(y)$, $\bar{\varrho}(y)$ and $\mathcal{A}$ are all equal to zero, and will therefore remain so. The self-potential term cancels in the QKE and the {ATAO}-$\mathcal{V}\,$ scheme describes accurately the neutrino evolution.
The case $\xi_1 = - \xi_2$ would correspond to a vanishing total lepton number density, while each flavour could display large asymmetries. This would result in a possibly significant contribution to the total energy density, hence the interest for this particular case. It was shown in \cite{Pastor:2001iu,Dolgov_NuPhB2002} that in this scenario synchronous oscillations are hampered as long as $\mathcal{J}$ dominates. This is in perfect agreement with our theoretical analysis: at leading order, as $F(\xi, -\xi) = 0$ the first term in \eqref{eq:Omega_JH} vanishes. However, our calculation of the NLO contribution shows that oscillations can still take place, but directly with a frequency $\propto x^6$.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/th23_0p001m0p001_NoCollxi1mxi2.pdf}
\caption[Equal but opposite asymmetries $\xi_1 = -\xi_2 = \xi = 0.001$]{\label{fig:xi1mxi2} Equal but opposite asymmetries $\xi_1 = -\xi_2 = \xi = 0.001$. We clearly see that, if we discarded the next-to-leading order contribution in~\eqref{eq:vrho_ATAOJV} ({ATAO}-$\Hself\,$ curve, in dashed-dots), oscillations would be switched off. However, the lowest order term in the frequency expression is now $\propto x^6$ and gives rise to quasi-synchronous oscillations beyond $x\simeq 0.2$, see~equation~\eqref{eq:x_xi1mxi2}.}
\end{figure}
To check this prediction, we plot the evolution of asymmetries for $\xi_1 = -\xi_2 = \xi = 0.001$ on Figure~\ref{fig:xi1mxi2}. In the {ATAO}-$\Hself\,$ scheme (which ignores the NLO contribution), oscillations never appear, contrary to the actual QKE evolution, correctly captured by the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme. The onset of synchronous oscillations is delayed compared for instance to the right plots of Figure~\ref{fig:2nu_mu-tau}, and we can estimate the location of this starting point exactly as in the previous section, the only difference being that we use the NLO part of~\eqref{eq:Omega_norm} $\Omega \propto x^6$. We find that the location of the first half-oscillation is
\begin{equation}
\label{eq:x_xi1mxi2}
x_\pi = \left( \frac{8 \pi ^2}{3} \times 2.278 \times 10^{-22} \times \frac{m_e^2}{\abs{\Delta m^2}} \times x_\mathrm{tr}^4 \times \xi \times 7 \pi \right)^{1/7} \simeq 0.20 \, ,
\end{equation}
for $\xi = 0.001$, in excellent agreement with Figure~\ref{fig:xi1mxi2}.
\subsection{Electron-driven MSW transition}\label{SecElectronMSW}
We now consider an electron/positron driven transition in the (fictitious) $\nu_e - \nu_\mu$ subspace, with the mixing angle $\theta = \theta_{12} \simeq 0.587$ and the small mass gap $\Delta m^2 = \Delta m_{21}^2 \simeq 7.53 \times 10^{-5} \, \mathrm{eV}^2$ for numerics. The difference with the previous case comes from the fact that the MSW transition now takes place when electrons are still relativistic. Moreover, we show in the following section how the collision term is very different from the one in the $\nu_\mu - \nu_\tau$ subspace.
The relativistic limit is sufficient to estimate the location of the MSW transition, therefore we use (we take the comoving plasma temperature $z=1$ for simplicity, which is justified since $e^\pm$ annihilations are just beginning at this stage):
\begin{equation}
\rho_{e^\pm}+P_{e^\pm} = \frac{7 \pi^2}{45} (m_e/x)^4\, ,
\end{equation}
to deduce that the transition takes place for
\begin{equation}
x_\mathrm{MSW} = \left(\frac{m_e^6 G_F}{m_W^2 \abs{\Delta
m^2}}\frac{28\sqrt{2}\pi^2 y^2_\mathrm{eff}}{45}\right)^{1/6}=0.118\,\left(\frac{10^{-3} \, \mathrm{eV}^2}{\abs{\Delta m^2}}\right)^{1/6}\,.
\end{equation}
For the numerical values chosen, we find $x_\mathrm{MSW} \simeq 0.18$, that is $T_{\mathrm{cm}} \simeq 2.8\,\mathrm{MeV}$\footnote{Note also that the first electron-driven transition associated with the large mass gap should be around $T_{\mathrm{cm}} = 5\,\mathrm{MeV}$ by application of this estimate with $\Delta m^2 = \Delta m_{31}^2$.}. We estimate the adiabaticity of this transition with~\eqref{Defgammatr2}, and find
\begin{equation}
\gamma_\mathrm{MSW} \simeq 485 \times \abs{\xi_1 + \xi_2} \, .
\end{equation}
The larger prefactor compared to the estimate~\eqref{eq:gammaMSW_muon} in the muon-driven case makes the transition adiabatic up to smaller degeneracies. This is in agreement with the results of Figure~\ref{fig:2nu_e-mu}: for instance, the transition is much more adiabatic (small amplitude of synchronous oscillations) for $\xi_1 + \xi_2 = 0.01$ compared to Figure~\ref{fig:2nu_mu-tau}.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/ATAOs_th12_mu1_NoColl.pdf} \\
\includegraphics[]{figs/ATAOs_th12_mu2_NoColl.pdf}
\caption[Evolution of the flavour asymmetries for a $\nu_e - \nu_\mu$ system without collisions]{\label{fig:2nu_e-mu}Evolution of the flavour asymmetries for a two-neutrino $\nu_e$ (blue) - $\nu_\mu$ (green) system without collisions, with $\Delta m^2 = 7.53 \times 10^{-5} \, \mathrm{eV}^2$ and $\theta = 0.587$. We compare different numerical schemes: in solid line QKE, in dots {ATAO}-$(\Hself\pm\mathcal{V})\,$ (hidden behind QKE), in dot-dashes {ATAO}-$\Hself\,$, and in dashes {ATAO}-$\mathcal{V}\,$. The initial degeneracy parameters are on the first row $\xi_2 =0$ and $\xi_1=0.1,0.01,0.001$ from left to right ; on the second row $\xi_1 =0$ and $\xi_2=0.1,0.01,0.001$.}
\end{figure}
The frequency regimes outlined in section~\ref{FreqSyncOsc} are once again observed on Figure~\ref{fig:Omega_2nu_e-mu}, where we plot the quantities $|\Omega \sin(2\theta)|$ and $\lvert (\partial_x \vec{\mathcal{A}} \wedge \vec{\mathcal{A}}) \rvert /|\vec{\mathcal{A}}|^2$. We see the transition from $\Omega \propto x^2$ to $\Omega \propto x^6$ and the possible cancellation of the frequency at this transition. Contrary to the case studied in section~\ref{SecMuonMSW}, it happens now in normal ordering for $\abs{\xi_2}>\abs{\xi_1}$ because $\cos(2 \theta) > 0$.
\begin{figure}[!ht]
\centering
\includegraphics{figs/th12_mu1_0p001_NoCollOmega.pdf}
\includegraphics{figs/th12_mu2_0p001_NoCollOmega.pdf}
\caption[Frequency of synchronous oscillations for a $\nu_e - \nu_\mu$ system]{\label{fig:Omega_2nu_e-mu} Same plot as Figure~\ref{fig:Omega_2nu_mu-tau} for the $\nu_e - \nu_\mu$ system with mixing parameters $\theta=0.587$ and $\Delta m^2 = 7.53\times10^{-5}\,\mathrm{eV}^2$.}
\end{figure}
\subsection{Effect of collisions}\label{SecCollisions}
In the previous sections, we systematically discarded the collision term in the QKEs in order to focus on the synchronous oscillation phenomenon and how approximate numerical schemes (namely, the {ATAO}-$(\Hself\pm\mathcal{V})\,$ procedure) accurately capture the physics at play.
Taking into account the scattering and annihilation processes is nevertheless crucial for a precision calculation, not only since these processes will determine neutrino decoupling and partial reheating \cite{Dolgov_NuPhB1997,Esposito_NuPhB2000,Mangano2002,Mangano2005,Relic2016_revisited,Grohs2015,Grohs:2016cuu,Froustey2019,Froustey2020,Bennett2021}, but also because they can reduce flavour asymmetry differences. This second effect was notably shown in references~\cite{Dolgov_NuPhB2002,Johns:2016enc}, but these works used approximate expressions for the collision term (so-called damping approximation). We aim at showing the effect of the exact collision term, whose expression was derived for instance in~\cite{SiglRaffelt,BlaschkeCirigliano} and in chapter~\ref{chap:QKE} of this manuscript.
For the following discussion, we only recall that the collision term $\mathcal{K}[\varrho,\bar{\varrho}]$ is an integral whose matrix structure is determined by the statistical factors associated to two-body reactions $(1) + (2) \to (3) + (4)$. They read typically (we write $\varrho_i = \varrho(y_i)$ for particle $i$):
\begin{equation}
\label{eq:stat_fact1}
\left[\varrho_4 (1- \varrho_2) + \mathrm{Tr}\left(\varrho_4 (1-\varrho_2) \right) \right] \varrho_3 (1- \varrho_1) - \{ \text{loss} \} + \mathrm{h.c.} \, ,
\end{equation}
for neutrino elastic scattering (this term corresponds to the process $\nu^{(1)} + \nu^{(2)} \to \nu^{(3)} + \nu^{(4)}$) and
\begin{equation}
\label{eq:stat_fact2}
f_4^{(e)} (1- f_2^{(e)})G^{L/R} \varrho_3 G^{L/R} (1- \varrho_1) - \{ \text{loss} \} + \mathrm{h.c.} \, ,
\end{equation}
for reactions with electrons and positrons (this particular terms stands for a scattering $\nu^{(1)} + e^{(2)} \to \nu^{(3)} + e^{(4)}$). In the above expressions, the loss part corresponds to the exchange $\left\{ \varrho_i \leftrightarrow (1 - \varrho_i) \right\}$ for all distributions, and $\mathrm{h.c.}$ stands for "hermitian conjugate". The coupling matrices $G^L$ and $G^R$ are diagonal in flavour space, and read in the full three-neutrino framework
\begin{equation}
\label{eq:coupling_matrices}
G^L = \begin{pmatrix} g_L +1 & 0 & 0 \\ 0 & g_L & 0 \\ 0 & 0 & g_L \end{pmatrix} \, , \quad
G^R = \begin{pmatrix} g_R & 0 & 0 \\ 0 & g_R & 0 \\ 0 & 0 & g_R \end{pmatrix} \, ,
\end{equation}
with $g_L = -\frac12 + \sin^2{\theta_W}$, $g_R = \sin^2{\theta_W}$ where $\theta_W$ is Weinberg's angle. In the $ee$ entry of $G^L$, the extra factor of $1$ accounts for the charged currents between $e^\pm$ and $\nu_e$. Since we do not consider collisions with other charged leptons (due to their negligible density in the range of temperatures of interest), this is the only additional factor in $G^L$. Therefore the collision terms satisfy the general property
\begin{equation}\label{MagicKFundamental}
\mathcal{K}[U_\mathrm{s} \varrho U^\dagger_\mathrm{s}, U_\mathrm{s} \bar{\varrho} U^\dagger_\mathrm{s}] = U_\mathrm{s}\mathcal{K}[\varrho,\bar{\varrho}] U^\dagger_\mathrm{s}\,,\qquad \overline{\mathcal{K}}[U_\mathrm{s} \varrho U^\dagger_\mathrm{s}, U_\mathrm{s} \bar{\varrho} U^\dagger_\mathrm{s}] = U_\mathrm{s}\overline{\mathcal{K}}[\varrho,\bar{\varrho}] U^\dagger_\mathrm{s}
\end{equation}
for constant unitary matrices of the type
\begin{equation}\label{UtoU2}
U_\mathrm{s} = \begin{pmatrix}
1 & 0 \\
0 & \mathcal{U}
\end{pmatrix} \,,\qquad \mathcal{U} \in \mathrm{U}(2) \,.
\end{equation}
In general, the collision term $\mathcal{K}[\varrho, \bar{\varrho}]$, being made of statistical factors like \eqref{eq:stat_fact1} and \eqref{eq:stat_fact2}, tends to make the density matrices in flavour basis diagonal, with entries being Fermi-Dirac distributions --- or $\varrho$ and $\bar{\varrho}$ must be obtained from conjugation of such matrices with a unitary matrix of the type~\eqref{UtoU2}. The degeneracies are not constrained by processes like $\nu_\alpha + \nu_\beta \to \nu_\alpha + \nu_\beta$ or $\nu_\alpha + \bar{\nu}_\alpha \to \nu_\beta + \bar{\nu}_\beta$. The only constraint is due to the processes $\nu_\alpha + \bar{\nu}_\alpha \to e^- + e^+$, which impose $\xi_\alpha = - \bar{\xi}_\alpha$ at equilibrium. Therefore, if collisions are strong enough, the density matrices are pushed towards
\begin{equation}
\label{eq:vrho_coll0}
\varrho \sim \mathrm{diag}[ g(\xi_\alpha,y) ]\,, \qquad \bar{\varrho} \sim \mathrm{diag}[ g(-\xi_\alpha,y) ]\, ,
\end{equation}
where $\sim$ stands for the possible conjugation by a matrix of the form~\eqref{UtoU2}.
\paragraph{Muon-driven transition} In the framework of section~\ref{SecMuonMSW}, we considered a two-neutrino case with only $\nu_\mu$ and $\nu_\tau$. When focusing on the $\nu_\mu-\nu_\tau$ subspace of \eqref{eq:coupling_matrices}, the $G^L$ and $G^R$ matrices are proportional to the identity matrix.
Initially, the collision term vanishes since $\varrho$ and $\bar{\varrho}$ are in the form~\eqref{eq:vrho_coll0}. What may come as a surprise is the fact that it keeps vanishing even though $\varrho$ and $\bar{\varrho}$ evolve. Indeed, at high temperature the {ATAO}-$\Hself\,$ scheme is valid and both $\varrho$ and $\bar{\varrho}$ are diagonalized by the same matrix $U_{\mathcal{J}}$, that is furthermore identical for all momenta $y$. This means that we have\footnote{For clarity, we omit the subscript $\mathcal{J}$ for the matter density matrix $\widetilde{\varrho}_{\mathcal{J}}$. More generally in this section, $\widetilde{\varrho}$ will be the diagonal density matrix, whether the Hamiltonian is $\mathcal{J}$, $\mathcal{J} + \mathcal{V}$, \dots} $\varrho = U_{\mathcal{J}} \widetilde{\varrho} U_{\mathcal{J}}^\dagger$ (and similarly for $\bar{\varrho}$), so from the restriction of the general property~\eqref{MagicKFundamental} we deduce the relations
\begin{equation}\label{MagicKUJ}
\mathcal{K}[\varrho,\bar{\varrho}] = U_{\mathcal{J}}
\mathcal{K}\left[\widetilde\varrho,\widetilde\bar{\varrho}\right] U^\dagger_{\mathcal{J}} \ , \qquad
\overline{\mathcal{K}}(\varrho,\bar{\varrho}) = U_{\mathcal{J}} \bar {\mathcal{K}}\left[\widetilde\varrho,\widetilde\bar{\varrho}\right] U^\dagger_{\mathcal{J}} \, .
\end{equation}
Thanks to this peculiar "factorization", the collision term keeps vanishing as long as $\widetilde{\varrho}$ and $\widetilde{\bar{\varrho}}$ are diagonal matrices (which they are by definition) of Fermi-Dirac distributions. This is much less restrictive, and remains satisfied as long as $\mathcal{J} \gg \mathcal{V}$ since $\widetilde{\mathcal{K}} = 0$ leads to $\partial_x \widetilde{\varrho} = 0$, hence the collision term keeps vanishing, and so on. With or without collisions, the evolution of $\varrho, \bar{\varrho}$ is purely due to the change of direction of $\vec{\mathcal{A}}$, which oscillates more or less around $\vec{\mathcal{V}}$ depending on the adiabaticity of the MSW transition.
Note that the previous argument is only exact for the part of $\mathcal{K}$ corresponding to neutrino self-interactions. It extends to the scattering with electrons/positrons as long as all particles share the same temperature. But even beyond this, when $e^\pm$ annihilations populate the neutrinos, they do so in creating pairs of neutrinos/antineutrinos, so the collision term acts to maintain thermal distributions, but not to equilibrate asymmetries.
All in all, the asymmetries are not affected at all by the collision term as long as the {ATAO}-$\Hself\,$ scheme is a good description of neutrino evolution. However, we have shown that below $\sim 10 \, \mathrm{MeV}$ the refined {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme is necessary to capture the physics. The very fact that $\varrho$ and $\bar{\varrho}$ are not diagonalized with the same unitary matrix (either $U_{\mathcal{J}+\mathcal{V}}$ or $U_{\mathcal{J}-\mathcal{V}}$), and furthermore the $y$-dependence of these matrices, means that we lose the property \eqref{MagicKUJ}, that is
\begin{equation}\label{MagicKUJLost}
\mathcal{K}[\varrho,\bar{\varrho}] \neq U_{\mathcal{J}+\mathcal{V}}(y)
\mathcal{K}[\widetilde\varrho,\widetilde\bar{\varrho}] U^\dagger_{\mathcal{J}+\mathcal{V}}(y) \ , \qquad
\overline{\mathcal{K}}[\varrho,\bar{\varrho}] \neq U_{\mathcal{J}-\mathcal{V}}(y) \bar {\mathcal{K}}[\widetilde\varrho,\widetilde\bar{\varrho}] U^\dagger_{\mathcal{J}-\mathcal{V}}(y) \,.
\end{equation}
When this non-equality is not meaningless --- it is necessarily suppressed by a factor $|
\vec{\mathcal{V}}|/|\vec{\mathcal{\mathcal{J}}}| \propto x^4 /|\xi_1-\xi_2|$ --- the collision term starts to have a mild effect, which is even smaller for large $\xi_\alpha$ differences. Therefore, only for rather small $\xi_\alpha$ differences can a slight equilibration effect due to collisions be expected. However, since the frequency $\Omega$ of synchronous oscillations is then smaller, the actual start of oscillations is delayed until a moment when collisions are inefficient. This is why we expect collisions to have a negligible effect throughout the evolution in this $\nu_\mu - \nu_\tau$ system. We check this on Figure~\ref{fig:Collisions_mu}, left plot, where the evolution is indistinguishable from the one without collisions (Figure~\ref{fig:2nu_mu-tau}, top right plot). On the right plot, we artificially multiplied the collision term by $1000$, and we do observe in that case the damping of quasi-synchronous oscillations when $\abs{\mathcal{V}} \sim \abs{\mathcal{J}}$, which corresponds to a reduction of asymmetry differences between the two flavours.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/th23_mu1_0p001_testcoll.pdf}
\includegraphics[]{figs/th23_mu1_0p001_testcollx1000.pdf}
\caption[Effect of collisions on the muon-driven transition for $\xi_1=0.001$, $\xi_2=0$]{\label{fig:Collisions_mu} Effect of collisions on the muon-driven transition for $\xi_1=0.001$, $\xi_2=0$, with the collision term set to its actual value (\emph{left}) and artificially multiplied by $1000$ (\emph{right}). We only plot the result of two numerical schemes: {ATAO}-$\mathcal{V}\,$ (dashes) and {ATAO}-$(\Hself\pm\mathcal{V})\,$ (solid, equivalent to QKE).}
\end{figure}
In principle, taking into account scattering and annihilations with muons and antimuons invalidates this picture since the charged currents with the $\nu_\mu$ and $\bar \nu_\mu$ make $G^L$ non-proportional to the identity, and the general property \eqref{MagicKUJ} would be lost. However their density is so suppressed in this range of temperature that we were able to check that the previous results are not affected.
Note also that the behaviour is very different if we ignore the self-interaction mean-field and rely on the pure {ATAO}-$\mathcal{V}\,$ scheme. In that case, the system is made of pure mass states of $\mathcal{H}_0$ after the MSW transition (since each $\varrho$ is diagonal in the mass basis). However, given the $y$-dependence of $\mathcal{V}$, this transition does not happen at the same time for all momenta. Thus there cannot be a property of the type~\eqref{MagicKUJ} with $U_{\mathcal{J}} \to U_{\mathcal{V}}$ because there is no unique $\mathcal{V}$, and furthermore $U_{\mathcal{V}}(y)$ depends on $y$ which prevents its factorization out of the collision integral. Therefore the collision term will tend to restore diagonality in flavour space (that is reduce flavour coherence), and this can only be compatible with pure mass states (a requirement of the {ATAO}-$\mathcal{V}\,$ approximation) when all flavours have reached the same distributions, that is when the asymmetry matrix $\mathcal{A}$ is proportional to the identity and thus $\vec{\mathcal{A}}=\vec{0}$. In other words, the collision term is strongly incompatible with the evolution of asymmetries dictated by the {ATAO}-$\mathcal{V}\,$ scheme, and thus damps them. We observe this behaviour on Figure~\ref{fig:Collisions_mu}: $\mathcal{A}_\mathrm{z} \to 0$ in the presence of collisions, which was not the case on Figure~\ref{fig:2nu_mu-tau}.
To conclude, if we ignore the self-interaction mean-field, the collision term efficiently damps the asymmetry differences, because the unitary adiabatic evolution is not the same for density matrices at various momenta. When including the additional self-interaction potential, as long as it dominates the vacuum and lepton mean-fields, the density matrices at various momenta evolve adiabatically with the common
unitary matrix $U_{\mathcal{J}}$ and this preserves the initial absence of
effect of the collision term. It is only when {ATAO}-$\Hself\,$ is insufficient and one has to rely on {ATAO}-$(\Hself\pm\mathcal{V})\,$ that one starts to see the effect of the unitary evolution differing between momenta, but also between neutrinos and antineutrinos. This allows the collision term to damp the asymmetry vector. But this comes with a very large delay and the collision term is only able to barely damp $\mathcal{A}_\mathrm{z}$.
\paragraph{Electron-driven transition} In the framework of section~\ref{SecElectronMSW}, the difference with the $\nu_\mu - \nu_\tau$ case is the fact that $G^L$ is no longer proportional to the identity: $G^L = \mathrm{diag}(g_L +1, g_L)$. Once oscillations develop and $U_\mathcal{J} \neq \mathbb{I}$, there is no property like~\eqref{MagicKUJ}. In other words, the matrix $G^L$ sets the direction $\vec{e}_\mathrm{z}$ towards which the collision term now unavoidably attracts $\vec{\varrho}$ and $\vec{\bar{\varrho}}$ (whereas before, $\mathcal{K}$ was blind to any global rotation of axes). Therefore, the collision term tends to erase flavour coherence much more efficiently: it damps oscillations but does not necessarily allow to fully reach a state where the two neutrino flavours have identical distributions, because the collision term becomes too weak at temperatures below the MSW transition.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/th12_mu1_0p001_coll.pdf}
\includegraphics[]{figs/th12_mu1_0p001Ay.pdf}
\includegraphics[]{figs/th12_mu1_0p001Anorm.pdf}
\includegraphics[]{figs/th12_mu1_0p001cosAH0.pdf}
\caption[Effect of collisions on the evolution of the $\nu_e - \nu_\mu$ system for $\xi_1 = 0.001, \, \xi_2 = 0$]{\label{fig:Coll_e-mu} Effect of collisions on the evolution of the $\nu_e - \nu_\mu$ system, with $(\xi_1 = 0.001, \, \xi_2 = 0)$. The dashed lines correspond to the {ATAO}-$\mathcal{V}\,$ scheme (no self-interactions in the mean-field), and the solid lines to the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme (equivalent to the full QKE resolution). \emph{Top left plot:} evolution of electron and muon flavour asymmetries. \emph{Top right plot:} $\mathrm{y}-$component of the asymmetry vector $\vec{\mathcal{A}}$. \emph{Bottom left plot:} norm of $\vec{\mathcal{A}}$. \emph{Bottom right plot:} angle between $\vec{\mathcal{A}}$ and the final precession direction $\vec{\mathcal{H}}_0$.}
\end{figure}
The top left plot of Figure~\ref{fig:Coll_e-mu} is equivalent to the top right plot of Figure~\ref{fig:2nu_e-mu} ($\xi_1 = 0.001$, $\xi_2 = 0$), but including collisions. As expected, the asymmetry is damped by $\mathcal{K}$ in both the {ATAO}-$\mathcal{V}\,$ and {ATAO}-$(\Hself\pm\mathcal{V})\,$ schemes, and the evolution looks quite similar, suggesting that one could be satisfied with the simpler {ATAO}-$\mathcal{V}\,$ resolution scheme. However, this misses some important physical features, as the other plots on Figure~\ref{fig:Coll_e-mu} show. First, if one neglects the self-potential there is no precession of $\vec{\mathcal{A}}$ around $\vec{\mathcal{V}}$, but simply the alignment of all $\vec{\varrho}, \, \vec{\bar{\varrho}}$ with $\vec{\mathcal{V}}$ which evolves from $\mathcal{H}_\mathrm{lep}$ domination to $\mathcal{H}_0$ domination. This is clearly seen on the top right plot of Figure~\ref{fig:Coll_e-mu} (dashed lines): the $\mathrm{y}$-component of $\vec{\mathcal{A}}$ is constantly equal to zero, which is expected as $\widehat{\mathcal{A}}$ evolves from $\vec{e}_\mathrm{z}$ to $\hat{\mathcal{H}}_0$ that lies in the $(\mathrm{x}-\mathrm{z})$ plane. In the correct {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme however, synchronous oscillations do develop when collisions are discarded. When one takes them into account, $\mathcal{A}_\mathrm{y}$ does not take sizeable values but starts an oscillation (that is strongly damped by $\mathcal{K}$), see the insert plot.
The bottom plots of Figure~\ref{fig:Coll_e-mu} show respectively the norm of $\vec{\mathcal{A}}$ and its alignment with $\vec{\mathcal{H}}_0$. In the no-collision case (blue curves), the final angle between $\vec{\mathcal{A}}$ and $\vec{\mathcal{H}}_0$ is non-zero (bottom right plot), but different from its initial value due to the very small adiabaticity of the MSW transition: $\vec{\mathcal{A}}$ slightly rotates towards $\vec{\mathcal{V}}_\mathrm{eff}$, and precesses with an angle slightly different from $2 \theta$. Concerning its norm, $\lvert \vec{\mathcal{A}} \rvert$ is conserved without collisions.\footnote{The small "trough" in the {ATAO}-$\mathcal{V}\,$ case (dashed blue line) around $3 \, \mathrm{MeV}$ is not a numerical artefact. Since in that case each individual $\vec{\varrho}(y)$ changes its direction from $\vec{\mathcal{H}}_{\mathrm{lep}}(y)$ to $\vec{\mathcal{H}}_0(y)$ at different times (instead of being all locked on $\vec{\mathcal{A}}$), the norm of $\vec{\mathcal{A}}$ can only be compared at early and late times.} In contrast, we observe that including collisions (brown curves) $\vec{\mathcal{A}}$ gets aligned with $\vec{\mathcal{H}}_0$ (but in the opposite direction due to the value of $\theta$), while the asymmetry differences are damped---a result of the competition between precession (which sets the preferred direction $\hat{\mathcal{H}}_0$) and collisions (with the preferred direction $\vec{e}_\mathrm{z}$).
\section{Evolution with three flavours of neutrinos}\label{Sec:3neutrinos}
Having presented in the previous sections the salient features of two-neutrino evolution in the presence of flavour asymmetries, we can now turn to the full three-neutrino framework. Our goal is not to provide a thorough exploration of the parameter space, but instead to highlight the main physical characteristics of neutrino evolution with non-zero asymmetries.
\subsection{Method}
We have shown the accuracy of the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme that we can confidently use instead of a full QKE resolution, more computationally expensive. Therefore, the results are here obtained with this method and are compared with the {ATAO}-$\mathcal{V}\,$ scheme where we recall that the self-interactions are ignored in the mean-field, so as to highlight how the self-potential changes the dynamics.
However, it is impossible to integrate correctly the evolution at low temperatures. First, oscillations become too fast as their frequency grows as $x^6$ when the NLO dominates. Then, we reach the point where the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme starts to fail and oscillations must become gradually not synchronized. Eventually the system must converge to a state where fast oscillations disappear, that is an {ATAO}-$\mathcal{V}\,$ scheme. We chose to switch to an {ATAO}-$\mathcal{V}\,$ scheme at low temperature to effectively capture this transition from {ATAO}-$(\Hself\pm\mathcal{V})\,$ to {ATAO}-$\mathcal{V}\,$. In principle one should use the full QKE scheme to integrate numerically this phase, but for the same reasons it is numerically daunting. We chose to switch to the {ATAO}-$\mathcal{V}\,$ scheme around $2\,\mathrm{MeV}$, since the final MSW transition is over and collisions become rather inefficient (except for electron/positron annihilations). This method of instantaneous switching to the {ATAO}-$\mathcal{V}\,$ scheme necessarily misses some physics since it hides the complexity of the transition, but as we discuss in section~\ref{SecTransitionToATAOV} we expect that this does not significantly affect the results obtained for the evolution of asymmetry.
\subsection{Results with standard mixing parameters}
Given the numerous possibilities for the values of the initial degeneracy parameters, we chose to restrict to two types of initial conditions. First, we consider the case where electronic flavour neutrinos have the largest degeneracy ($\xi_e=\xi_\mathrm{max}$), with $\xi_\tau=\xi_e/10$. Second we consider the case where muonic neutrinos have the largest non vanishing initial degeneracy ($\xi_\mu=\xi_\mathrm{max}$), with $\xi_\tau=\xi_\mu/10$. We do not report results where $\xi_\tau$ is the largest, because it is qualitatively very similar to the case where $\xi_\mu$ is the largest potential, since oscillations develop in exactly the same way. In any case, the third initial degeneracy parameter is set to zero.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/standard_mu1_0p1ATAOs.pdf}
\includegraphics[]{figs/standard_mu1_0p005ATAOs.pdf}\\ \vspace{0.2cm}
\includegraphics[]{figs/standard_mu2_0p1ATAOs.pdf}
\includegraphics[]{figs/standard_mu2_0p005ATAOs.pdf}
\caption[Evolution of flavour asymmetries in the standard three-neutrino case, for various initial conditions]{\label{figStandard} Initial conditions are $\xi_\alpha=(0.1,0,0.01)$ (top left), $\xi_\alpha=(0.005,0,0.0005)$ (top right), $\xi_\alpha=(0,0.1,0.01)$ (bottom left) and $\xi_\alpha=(0,0.005,0.0005)$ (bottom right).
The solid lines are the {ATAO}-$(\Hself\pm\mathcal{V})\,$ schemes (extended into a simple {ATAO}-$\mathcal{V}\,$ below $2\,\mathrm{MeV}$), and the dashed lines are {ATAO}-$\mathcal{V}\,$ schemes throughout.}
\end{figure}
Results are depicted in Figure~\ref{figStandard} for both typically large ($\xi_\mathrm{max}=0.1$) and typically small ($\xi_\mathrm{max}=0.005$) potentials. We can observe how synchronous oscillations develop in the $\nu_\mu-\nu_\tau$ space after the muon-driven transition. Their amplitude is reduced for large initial potentials since the transition is then more adiabatic in that case, as detailed in section~\ref{SecMuonMSW}. In the case of small initial potentials, the transition from leading order to NLO oscillations is also clearly visible (right plots of Figure~\ref{figStandard}). Note that even though the general behaviour is that asymmetries tend to converge, this trend stops before equilibration is complete, and is in general less complete than in the case where the self-interaction mean-field is ignored. Furthermore in some cases, the ordering of final asymmetry is not the same as the ordering of initial ones.
If we consider cases with much larger initial asymmetries in the muonic and tauic neutrinos, typically such that $\xi_\mu+\xi_\tau \gg 0.1$, the muon-driven transition is very adiabatic, and oscillations do not develop at that transition as the asymmetry vector closely follows the evolution of $\mathcal{V}(y_\mathrm{eff})$. This is illustrated in the left plot of Figure~\ref{figComparison}.
We stress that in this section we always consider the full collision term, and even though we show the results for the evolution of the asymmetry (since it is the relevant quantity to discuss synchronized oscillations), neutrinos are also partially reheated by electron/positron annihilations, which preserve the asymmetry. In Figure~\ref{figrho} we show the energy density fractional difference with respect to the one of a completely decoupled neutrino species with vanishing chemical potential $\bar{\rho}_\nu^{(0)} = 7 \pi^2/240$. In general the final effective temperature and distortions of electronic (anti-)neutrinos, which have a direct effect on neutron/proton freeze-out and thus BBN, depend on initial degeneracy parameters. Hence the impact on BBN predictions is not straightforward and one should perform a full BBN analysis for each set of initial conditions~\cite{Grohs:2016cuu}, as was done for the standard case of vanishing potentials in chapter~\ref{chap:BBN}.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/standard_mu2_0p005_rhonu.pdf}
\includegraphics[]{figs/standard_mu2_0p005_barrhonu.pdf}
\caption[Fractional difference of the energy density with respect to a decoupled neutrino without chemical potential]{\label{figrho} Fractional difference of the energy density with respect to a decoupled neutrino without chemical potential. The solid lines are the {ATAO}-$(\Hself\pm\mathcal{V})\,$ schemes (extended into a simple {ATAO}-$\mathcal{V}\,$ below $2\,\mathrm{MeV}$), and the dashed lines are {ATAO}-$\mathcal{V}\,$ schemes throughout. The left plot is for neutrinos and the right plot for antineutrinos. Initial conditions are $\xi_\alpha=(0,0.005,0.0005)$.}
\end{figure}
It is beyond the scope of this chapter to perform a full exploration of parameters with all initial degeneracy parameters and all mixing parameters. However we aim here at highlighting how results are qualitatively modified when considering different mixing parameters.
\subsection{Dependence on neutrino mass ordering}
In Figure~\ref{figcomp4} we show the dependence on the neutrino mass ordering on two examples. The main difference is the resonant nature of the first electron-driven MSW transition at $T_\mathrm{MSW}^{(e),1} \simeq 5\,\mathrm{MeV}$ in IO. It leads to a much faster evolution, a feature also observed in Figure 1 of~\cite{Mangano:2011ip}. On the examples of Figure~\ref{figcomp4}, we see two consequences of this resonant transition: the ordering of degeneracy parameters can be modified, and collisions are much more efficient in damping the synchronous oscillations right after the transition.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/standard_mu1_0p1_inverted_mu1_0p1.pdf}
\includegraphics[]{figs/standard_mu2_0p1_inverted_mu2_0p1.pdf}
\caption[Comparison of normal and inverted ordering]{\label{figcomp4} The normal ordering case in solid lines, is compared to the inverted ordering case in dashed lines. Initial conditions are $\xi_\alpha=(0.1,0,0.01)$ on the left, and $\xi_\alpha=(0,0.1,0.01)$ on the right.}
\end{figure}
\subsection{Dependence on mixing angles}\label{sec:mixing_angles}
In the two-neutrino case we have shown that $\vec{\mathcal{H}}_0$ sets the precession direction of $\vec{\mathcal{A}}$; in other words, the values of the mixing angles are key parameters to determine the final asymmetry differences. Even though they are now better and better constrained~\cite{PDG}, let us explore qualitatively in this section their influence on the equilibration process.
To that purpose, we compare in Figure~\ref{figcompamgles} (upper plots) the standard case discussed above with modified setups. First, when $\theta_{13}=0$ (the rest being unchanged), we notice that equilibration is less efficient. Furthermore setting $\theta_{12} = \theta_{23} = \pi/8$, with $\theta_{13}$ at its standard value, the equilibration is also much less efficient as depicted in Figure~\ref{figcompamgles} (lower plots). Therefore, the general result that asymmetries mostly tend to equilibrate crucially depends on the values of the mixing angles. Typically, for small values of the mixing angles $\theta_{12}$ and $\theta_{23}$, that is far away from $\pi/4$, equilibration is less efficient, and a non-vanishing value for $\theta_{13}$ also significantly helps the equilibration process as highlighted in~\cite{Dolgov_NuPhB2002,Mangano:2010ei,Mangano:2011ip}. Also, using $\theta_{23} = \pi/8$ instead of the larger standard value~\eqref{ValuesStandard} increases the geometric factor $\cos^2 \theta/\sin \theta$ of the adiabatic parameter~\eqref{Defgammatr2} by a factor $\simeq 3.6$. Therefore, the muon-driven transition is much more adiabatic and the resulting oscillations are suppressed (see the discussion in section~\ref{SecDescriptionMuonTransition}), as can be checked on the bottom plots of Figure~\ref{figcompamgles}.
Evidently, even though we do not report it here, when all mixing angles vanish, equilibration entirely disappears. This highlights the importance of a consistent treatment of neutrino mixing when studying flavour equilibration in the early Universe.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/standard_mu1_0p1_Noth13_mu1_0p1.pdf}
\includegraphics[]{figs/standard_mu2_0p1_Noth13_mu2_0p1.pdf}\\ \vspace{0.2cm}
\includegraphics[]{figs/standard_mu1_0p1_AllPi8_mu1_0p1.pdf}
\includegraphics[]{figs/standard_mu2_0p1_AllPi8_mu2_0p1.pdf}
\caption[Evolution of asymmetries with different mixing angles]{\label{figcompamgles} Comparison of the standard case (solid lines) with a modified setup (in dashed lines). In the upper plots, the modification is $\theta_{13}=0$, and in the lower plots $\theta_{12}=\theta_{23}=\pi/8$, with everything else unchanged. Initial conditions are $\xi_\alpha=(0.1,0,0.01)$ on the left, and $\xi_\alpha=(0,0.1,0.01)$ on the right. At low temperatures, the green lines overlap on the top right subplot, and similarly for the red lines on the bottom left subplot.}
\end{figure}
\subsection{Dependence on the Dirac phase}\label{SecDiracPhase}
We now examine the effect of the Dirac CP-violating phase $\delta$, that we discarded up until now in the PMNS matrix. Unless stated otherwise, all quantities in this section are considered in the case $\delta \neq 0$. $\varrho^{(\delta=0)}$ and $\bar{\varrho}^{(\delta=0)}$ refer to the solutions with vanishing Dirac phase, and we shall show how the general case with a non-zero phase can be deduced from it. It has been shown in \cite{Balantekin:2007es,Gava:2008rp,Gava:2010kz,Gava_corr} (see also section~\ref{subsec:Decoupling_CP}) that the evolution with a non-vanishing Dirac phase can be obtained from a transformation of the result obtained with a vanishing phase. More precisely, defining
$\check{S} \equiv R_{23} S R_{23}^\dagger$ (cf.~notations in section~\ref{subsec:Values_Mixing}), we can define
\begin{equation}\label{CheckAll}
\check{\mathcal{H}}_\mathrm{lep} = \check{S}^\dagger \mathcal{H}_\mathrm{lep} \check{S} \, , \qquad \check{\mathcal{H}}_0 = \check{S}^\dagger \mathcal{H}_0 \check{S} \, , \qquad \check{\varrho} \equiv \check{S}^\dagger \varrho \check{S} \, , \qquad \check{\bar{\varrho}} \equiv \check{S}^\dagger \bar{\varrho} \check{S}\,,
\end{equation}
and similar transformations for the collision terms. Since $\check{S}$ is of the type~\eqref{UtoU2} (see equation~\eqref{eq:Umat_SS} below), we infer from the property~\eqref{MagicKFundamental} that
\begin{equation}\label{MagiccheckK}
\check{\mathcal{K}}(\varrho,\bar{\varrho}) = \mathcal{K}(\check{\varrho},\check{\bar{\varrho}}) \quad \text{and} \quad \check{\overline{\mathcal{K}}}(\varrho,\bar{\varrho}) = \overline{\mathcal{K}}(\check{\varrho},\check{\bar{\varrho}}) \, .
\end{equation}
Furthermore, given that
\begin{equation}\label{MagicUU}
\check{S}^\dagger U = U^{(\delta=0)}S^\dagger\,,
\end{equation}
and $[\mathbb{M}^2,S]=0$, we deduce that
\begin{equation}
\label{eq:equalityofH0}
\check{\mathcal{H}}_0 = \mathcal{H}_0^{(\delta=0)} \, .
\end{equation}
Therefore, the evolution of $\check{\varrho}$ (resp. of $\check{\bar{\varrho}}$) is the same as the evolution of $\varrho^{(\delta=0)}$ (resp. of $\bar{\varrho}^{(\delta=0)}$) when the replacements $\mathcal{H}_\mathrm{lep} \to \check{\mathcal{H}}_\mathrm{lep}$ and $\mathcal{J} \to \check{\mathcal{J}}$ have been performed, that is
\begin{equation}
\frac{\partial \check{\varrho} }{\partial x} = - {\mathrm i} [\mathcal{H}_0^{(\delta=0)}+\check{\mathcal{H}}_\mathrm{lep} +
\check{\mathcal{J}},\check{\varrho}] + \mathcal{K}(\check{\varrho},\check{\bar{\varrho}})\,,\qquad \frac{\partial \check{\bar{\varrho}} }{\partial x} = + {\mathrm i} [\mathcal{H}_0^{(\delta=0)}+\check{\mathcal{H}}_\mathrm{lep} -
\check{\mathcal{J}},\check{\bar{\varrho}}] + \overline{\mathcal{K}}(\check{\varrho},\check{\bar{\varrho}})\,.
\end{equation}
In the standard case, that is with vanishing initial chemical potentials, $\check{\varrho}$ and $\varrho$ have the same initial conditions. If we further neglect the mean-field effects of muons/antimuons, then $[\check{S},\mathcal{H}_\mathrm{lep}]=0$, hence $\check{\mathcal{H}}_\mathrm{lep} = \mathcal{H}_\mathrm{lep}$ and $\check{\varrho}=\varrho^{(\delta=0)}$ (likewise for antineutrinos) at all times, as shown in \cite{Gava:2010kz,Gava_corr} and section~\ref{subsec:Decoupling_CP}. From this property, we obtain $\varrho$ from $\varrho^{(\delta=0)}$ using the inverse transformation, that is we get $\varrho = \check{S}\varrho^{(\delta =0)} \check{S}^\dagger$, with a similar relation for antineutrinos. It is equivalent to saying that both results are
exactly equal in their respective mass basis.
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/standard_mu1_0p1_CP245_mu1_0p1.pdf}
\includegraphics[]{figs/standard_mu2_0p1_CP245_mu2_0p1.pdf}
\caption[Effect of $\delta \neq 0$ on the evolution of asymmetries]{\label{figCP1} Effect of $\delta \neq 0$ on the evolution of asymmetries. The solid lines correspond to the standard case with $\delta=0$ (that is $\varrho^{(\delta=0)}$ and $\bar{\varrho}^{(\delta=0)}$). The dashed lines are the case with $\delta =245^\circ$ (central value in the most recent constraints~\cite{PDG}). The dotted lines correspond to the standard case results on which the transformation $\check{S}\varrho^{(\delta=0)} \check{S}^\dagger$ (and similarly for antineutrinos) has been applied. Initial conditions are $\xi_\alpha=(0.1,0,0.01)$ on the left and $\xi_\alpha=(0,0.1,0.01)$ on the right. Initially, the dashed lines are hidden behind the solid ones (see text). In the final stages the dotted lines are hidden behind the dashed lines hence showing that asymptotically $\varrho\simeq \check{S} \varrho^{(\delta=0)} \check{S}^\dagger$.
}
\end{figure}
However in the presence of initial degeneracies, the initial conditions for $\varrho$ are not necessarily equal to those of $\check{\varrho}$. It is interesting to note that going from $\varrho$ to $\check{\varrho}$
amounts to a rotation in the vector description of the $\nu_\mu-\nu_\tau$
subspace. Indeed, both $S$ and $\check{S}$ are of the \eqref{UtoU2} type, and the associated $\mathcal{U}_S$ and $\mathcal{U}_{\check{S}}$ are expressed in terms of rotations as
\begin{equation}
\label{eq:Umat_SS}
\mathcal{U}_S = \mathrm{e}^{{\mathrm i} \delta/2} \mathcal{R}_\mathrm{z}(\delta) \ , \qquad
\mathcal{U}_{\check{S}} = \mathrm{e}^{{\mathrm i} \delta/2}
\mathcal{R}_\mathrm{y} (-2 \theta_{23}) \cdot
\mathcal{R}_\mathrm{z}(\delta) \cdot \mathcal{R}_\mathrm{y}^\dagger(-2 \theta_{23})\,.
\end{equation}
Forgetting the global $\mathrm{e}^{{\mathrm i} \delta/2}$ factor which plays no role, we can use the property~\eqref{eq:SU2SO3} to interpret $\mathcal{U}_{\check{S}} $ as a rotation, when using the vector representation~\eqref{MatrixToVector}. It corresponds to a rotation of angle $\delta$ around an axis whose direction is obtained from the rotation $\mathcal{R}_\mathrm{y} (-2 \theta_{23})$ of the $\mathrm{z}$-axis. However, using that $\theta_{23}^\mathrm{eff} \simeq \theta_{23}$ from \eqref{th23th23}, this axis is approximately the one subtended by the $\nu_\mu-\nu_\tau$ restriction of the vacuum Hamiltonian. This has interesting consequences.
\begin{itemize}
\item Before the electron-driven transitions, using~\eqref{CheckAll} with $U$ of the effective form~\eqref{UtoU}, and~\eqref{eq:Umat_SS} taking $\theta_{23}^\mathrm{eff} \simeq \theta_{23}$, we infer the approximate relations
\begin{equation}
\left(\check{\mathcal{H}}_0\right)_{\alpha\beta} \simeq \left(\mathcal{H}_0\right)_{\alpha\beta} \quad \xRightarrow[\text{using \eqref{eq:equalityofH0}}]{} \quad \left(\mathcal{H}_0\right)_{\alpha\beta} \simeq \left(\mathcal{H}^{(\delta=0)}_0\right)_{\alpha\beta}\quad \text{for}\quad \alpha,\beta \in\{\mu,\tau \}\,.
\end{equation}
Therefore at early times the evolution of $\varrho$ and $\varrho^{(\delta=0)}$ are nearly completely similar (likewise for antineutrinos) as can be checked by comparing the solid and dashed lines of Figure~\ref{figCP1} at high temperatures.
\item Since the asymmetry vector precesses around that precise direction after the muon-driven transition, this amounts to the fact that after the muon-driven MSW transition and before the electron-driven ones, the oscillations of $\check{\varrho}$ (similarly for $\check{\bar{\varrho}}$) are simply phase shifted with respect to the ones of $\varrho^{(\delta=0)}$ (resp. $\bar{\varrho}^{(\delta=0)}$), by an angle $\delta$, as can be checked by comparing the dashed and dotted lines on Figure~\ref{figCP1} (see also the left plot of Figure~\ref{figComparison}).
\end{itemize}
Later, when the electron-driven transitions occur, the $\delta$-phase difference between $\check{\varrho}$ and $\varrho^{(\delta=0)}$ (likewise for antineutrinos) can only have an extremely marginal
effect because the oscillation frequency keeps increasing, and the amount of damping incurred in the magnitude of (the traceless part of) $\mathcal{A}$ is essentially only sensitive to the amplitude and axes of oscillations. Oscillations keep accelerating until synchronous oscillations disappear as we reach an average set by $\mathcal{H}_0$, which is captured by the {ATAO}-$\mathcal{V}\,$ scheme. Eventually the initial dephasing is lost, and it is impossible to distinguish between the final values of $\check{\varrho}$ and $\varrho^{(\delta=0)}$ (likewise for antineutrinos), therefore we can relate the final results to the case without Dirac phase by
\begin{equation}\label{FinalOperation}
\varrho \simeq \check{S}\varrho^{(\delta =0)} \check{S}^\dagger\,,\qquad \bar{\varrho} \simeq \check{S}\bar{\varrho}^{(\delta =0)} \check{S}^\dagger\,.
\end{equation}
There are two differences with the standard case without initial chemical potentials. On the one hand, \eqref{FinalOperation} is an approximate result and not an equality, based on the following approximations:
\begin{enumerate}
\item $\theta_{23}^\mathrm{eff} \simeq \theta_{23}$, which is guaranteed from equation~\eqref{th23th23} by $|\Delta m_{21}^2/\Delta m_{31}^2| \ll 1$ ;
\item the muon-driven MSW transition takes place well before the first electron-driven transition ($T_\mathrm{MSW}^{(\mu)} \gg T_\mathrm{MSW}^{(e),1}$), which is the case since $m_e/m_\mu \ll 1$ ;
\item the amplitude and directions of oscillations are not meaningfully affected by the $\delta$-dephasing, and eventually this dephasing should not be observable as oscillations are averaged at large $x$.
\end{enumerate}
On the other hand, it is only valid at late times, whereas in the standard case it is valid at all times. It can be checked on Figure~\ref{figCP1} that it is very accurate, as the dashed lines ($\varrho$) and dotted lines ($\check{S}\varrho^{(\delta =0)} \check{S}^\dagger$) are nearly indistinguishable at late times, and are only $\delta$-dephased at early times if oscillations develop.
\noindent Finally the property \eqref{FinalOperation} allows to understand the physical effects of the Dirac phase.
\begin{itemize}
\item When converted into mass basis components, the property~\eqref{MagicUU} and the relation~\eqref{FinalOperation} imply the (approximate) relation $\widetilde{\varrho} \simeq S \widetilde{\varrho}^{(\delta=0)} S^\dagger$. This means that in the {ATAO}-$\mathcal{V}\,$ approximation which holds at late times, we have $\widetilde \varrho = \widetilde \varrho^{(\delta=0)}$ and likewise for antineutrinos given that off-diagonal components in the matter basis vanish --- see section~\ref{subsec:Decoupling_CP}. Hence the difference in the final state, when interpreted in the flavour basis, is (approximately) only due to the different mass bases depending on the value of $\delta$. Structure formation, being sensitive to mass bases spectra, is therefore not affected by the Dirac phase. In addition, the trace of density matrices is conserved by~\eqref{FinalOperation} such that $N_\mathrm{eff}$ is preserved, and cosmological expansion is not modified. We conclude that there is no sizeable gravitational signature of the Dirac phase.
\item Since $\check{S}$ is of the~\eqref{UtoU2} type, the transformation~\eqref{FinalOperation} affects the number densities of $\nu_\mu$ and $\nu_\tau$, that is $\varrho^{\mu}_{\mu}$ and $\varrho^{\tau}_{\tau}$, but the number density for $\nu_{e}$, that is $\varrho^{e}_{e}$, is left invariant (likewise for antineutrinos). Only the coherence between the $e$ states and the $\mu$ and $\tau$ states, that is the $\varrho^{e}_{\mu}$ and $\varrho^{e}_{\tau}$ components, is affected. Therefore, there is also no perceptible effect on BBN because the neutron/proton freeze-out is sensitive only to the spectrum of electronic (anti)neutrinos.
\end{itemize}
\subsection{Equal but opposite asymmetries}
As noted in section~\ref{FreqSyncOsc}, the case of equal but opposite asymmetries is special because the leading order of synchronous oscillations vanishes --- meaning that the asymmetry should remain locked in its original configuration, --- but not the next-to-leading order. The results obtained in three-neutrino cases are depicted in Figure~\ref{figEBO}. When $\xi_\mu+\xi_\tau=0$, we expect from the analysis of section~\ref{SecParticular} that oscillations in the $\nu_\mu-\nu_\tau$ space should start at $1.4 \,\mathrm{MeV}$, while we observe the first half-oscillation at $4\,\mathrm{MeV}$. These oscillations must therefore be triggered by the first electron-driven transition.
One can compare the left plot of Figure~\ref{figEBO} with Figure~9 of~\cite{Dolgov_NuPhB2002}: the damping of synchronous oscillations is considerably reduced in our calculation, which we attribute to the use of the exact collision term, instead of a damping approximation (see also the discussion in section~\ref{SecDiscussion}).
\begin{figure}[!ht]
\centering
\includegraphics[]{figs/EBO_0p1_comparATAOs.pdf}
\includegraphics[]{figs/EBO_12_0p1ATAOs.pdf}
\caption[Special case of equal but opposite non-vanishing degeneracy parameters]{\label{figEBO} Special case of equal but opposite non-vanishing degeneracy parameters. The initial conditions on the left are $\xi_\alpha=(0,-0.1,0.1)$, and $\xi_\alpha=(0.1,-0.1,0)$ on the right. \emph{Numerical schemes:} {ATAO}-$(\Hself\pm\mathcal{V})\,$ in solid lines, {ATAO}-$\mathcal{V}\,$ in dashed lines. On the left plot, we extended the {ATAO}-$(\Hself\pm\mathcal{V})\,$ integration until $1.3 \, \mathrm{MeV}$ to see further the damping of the oscillations.}
\end{figure}
If the opposite initial degeneracy parameters are the electronic and muonic (or tauic) ones, there is no substantial difference with the "standard" case (Figure~\ref{figStandard}). Indeed, the $\nu_\mu-\nu_\tau$ oscillations equilibrate partially (at least in the case of large $\xi_\tau$ taken in the Figure) the asymmetries in the $\nu_\mu - \nu_\tau$ subspace, and the common asymmetry is then not the opposite of $\xi_e$.
\section{Discussion}\label{SecDiscussion}
\subsection{Transition from {ATAO}-$(\Hself\pm\mathcal{V})\,$ to {ATAO}-$\mathcal{V}\,$}\label{SecTransitionToATAOV}
We have shown that the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme works very well as long as $|\mathcal{V}| \ll |\mathcal{J}|$. Let us discuss here in more details the end of the {ATAO}-$(\Hself\pm\mathcal{V})\,$ regime, in a simplified case with only two flavours so as to use the vector formalism. When $\lvert \vec{\mathcal{V}} \rvert$ and $\lvert \vec{\mathcal{J}} \rvert$ are of the same order, quasi-synchronous oscillations cease to exist. Therefore, in principle, only the QKE scheme can handle this regime. However, the individual $\vec{\varrho}(y)$ then evolve independently rather than collectively, so that extremely rapid precessions around $\vec{\mathcal{V}}$ should take place for all momenta, and given the $y$-dependence of $\vec{\mathcal{V}}$, they tend to have different frequencies.
Given that $|\vec{\mathcal{V}}| \propto (\Delta m^2/(2 y m_e H) \simeq (\Delta m^2/10^{-3}\mathrm{eV}^2)\times 8 \cdot 10^6 \times x^2/y$, the oscillating part in the spectrum is typically a trigonometric function whose phase $\phi \propto x^3/y$. This implies extremely fast oscillations in the spectrum (i.e. in the variable $y$) whenever $x \gg 1$, that is at cosmological times. It is expected that even a mild collision term can average them out. Even if this is not the case, we only aim at describing the average of this incredibly fast oscillating spectrum, since this is the only part that will survive any measurement or physical process. This means that after a transitory regime, the {ATAO}-$\mathcal{V}\,$ scheme must become a good approximation. Sadly, given the $\mathcal{O}(N^3)$ complexity for computing the collision term, it becomes numerically impossible to integrate this transitory regime. Furthermore the $y$-grid becomes necessarily too sparse to account for these spectral oscillations. Hence one must rely on a certain approximation to handle the transition from a period where the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme applies to a regime where {ATAO}-$\mathcal{V}\,$ is sufficient.
We chose to push the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme as far as possible, typically down to $2\,\mathrm{MeV}$ and then to switch immediately to a {ATAO}-$\mathcal{V}\,$ scheme. In doing so we necessarily miss some features of the transitory regime. If $\vec{\mathcal{A}}$ is already well aligned with $\vec{\Hamil}_{\mathrm{eff}}$, it is nonetheless expected to be a very good approximation. Hence it misses some physics essentially due to the oscillations in the $\nu_\mu-\nu_\tau$ space which have developed right after the muon-driven transition, and for which we have seen that the collision term is not very effective in dampening these oscillations towards $\vec{\Hamil}_{\mathrm{eff}}$. We can however estimate the nature of the error made. If we focus on the two-neutrino case describing the $\nu_\mu-\nu_\tau$ space after the muon-driven transition, the density matrices are in the {ATAO}-$\mathcal{V}\,$ scheme in the form given by \eqref{eq:vrho_ATAOJV}. If we neglect terms of order $\mathcal{O}(\xi^2)$, then $\abs{g(-\xi_1,y)- g(-\xi_2,y)} \simeq \abs{g(\xi_1,y)- g(\xi_2,y)} $, that is we have essentially $\vec{{\varrho}} \propto \widehat{\mathcal{V}+\mathcal{J}}$ and $\vec{{\bar{\varrho}}} \propto \widehat{\mathcal{V}-\mathcal{J}} $ with the same prefactor. When the ratio $|\mathcal{V}|/|\mathcal{J}|$ grows this tends to displace both $\vec{{\varrho}}$ and $\vec{{\bar{\varrho}}}$ in the same direction, namely, the projection of $\vec{\mathcal{V}}$ in a plane orthogonal to $\vec{\mathcal{J}}$ (in agreement with the next-to-leading order term of the expansion \eqref{ExpandSH} of $\widehat{\mathcal{J}}$). The net result is that around the end of validity of the {ATAO}-$(\Hself\pm\mathcal{V})\,$ regime, neutrinos and antineutrinos of one flavour are converted to neutrinos and antineutrinos of the other flavour, but in similar proportions for neutrinos and antineutrinos, hence preserving the asymmetry. Of course, since the ratio $|\mathcal{V}| / |\mathcal{J}|$ reaches unity earlier for small $y$, it is expected that this concerns more the small momenta $y$. Also, the more $\vec{\mathcal{V}}$ and $\vec{\mathcal{J}}$ are misaligned in the end of the {ATAO}-$(\Hself\pm\mathcal{V})\,$ regime, the more this phenomenon takes place. This is nicely seen in Figure~9 of \cite{Johns:2016enc}, a configuration solved numerically without collisions, where we observe that both the number of neutrinos and antineutrinos of a given flavour increase while the opposite takes place for the other flavour. By construction, our approach based on an instantaneous switching from {ATAO}-$(\Hself\pm\mathcal{V})\,$ to {ATAO}-$\mathcal{V}\,$ cannot capture this phenomenon.
Finding a method to handle this transitory regime when including the collision term is an upcoming numerical challenge for the computation of equilibration in the early universe.
\subsection{Comparison with the literature}
Our numerical results differ from the literature in several aspects, which we review here.
\paragraph{Collision term} It is clear that the muon-driven oscillations are less damped in our case, compared to the results for instance reported in \cite{Dolgov_NuPhB2002}. We explain this by the fact that the general collision term satisfies the "factorization" property \eqref{MagicKFundamental}, and thus \eqref{MagicKUJ} in the $\nu_\mu-\nu_\tau$ subspace.
This implies that oscillations developing in this subspace are only very mildly damped, as detailed in section~\ref{SecCollisions}. When relying on an approximate collision term, this property is lost. For computations where all entries of the collision term are based on a damping approximation, it is the repopulation term which fails to satisfy the property \eqref{MagicKFundamental}. On the other hand, for computations which use the full collision term for on-diagonal components, but a damping approximation for off-diagonal components (as is the case in \cite{Dolgov_NuPhB2002}), the property is lost precisely because not all components are computed in the same method and this introduces preferred directions in the collision term. In all cases, using an approximation for the collision term results in much more damping of the oscillations in the $\nu_\mu-\nu_\tau$ space compared to our results. We already mentioned the case where $\xi_\mu=-\xi_\tau$ of \cite{Dolgov_NuPhB2002} (Figure 9), compared to our results in Figure~\ref{figEBO}. One of the consequences is that it is not possible to consider that by $10\,\mathrm{MeV}$ we would have generically achieved $\xi_\mu=\xi_\tau$, as is assumed for instance in \cite{Pastor:2008ti,Mangano:2010ei,Castorina2012}.
\begin{figure}[!h]
\centering
\includegraphics[]{figs/standard_mu3_0p5_E_CP180_mu3_0p5_E.pdf}
\includegraphics[]{figs/Mangano2010_E_Mangano2010_Noth13_E.pdf}
\caption[Comparison with previous results from the literature]{\label{figComparison} \emph{Left plot:} $\xi_\alpha=(0,0,0.5)$, with $\delta=0$ in solid line and $\delta=\pi$ in dashed line. Dotted lines correspond to the case $\delta=0$ on which the transformation $\check{S}\varrho^{(\delta=0)} \check{S}^\dagger$ has been applied, and are hidden behind dashed lines at late times. It can be compared with Figure~3 of \cite{Gava:2010kz}. \emph{Right plot:} $\xi_\alpha = (1.0732,-0.833,-0.833)$, with $\theta_{13}=0.20$ in solid line and $\theta_{13}=0$ in dashed line. It corresponds to Figure~1 of \cite{Mangano:2010ei}, noting that the initial conditions are $\eta_\mu=\eta_\nu = -0.61$ instead of the stated values $\eta_\mu=\eta_\nu = -0.41$.}
\end{figure}
\paragraph{Large mixing angle} Note that in references \cite{Dolgov_NuPhB2002,Gava:2010kz}, the large mixing angle value $\theta_{23}=\pi/4$ was used, which adds extra properties. Since $\cos(2 \theta_{23})=0$ we cannot use~\eqref{th23th23} to estimate $\theta_{23}^\mathrm{eff}$ after the muon-driven transition. Fortunately, in that case we get exactly
\begin{equation}
\tan(2 \theta_{23}^\mathrm{eff}) =\frac{(\epsilon^{-1} + 1 )\cos^2 (\theta_{13}) +\sin^2 \theta_{12} \sin^2 \theta_{13} - \cos^2 \theta_{12}}{\sin(\theta_{13}) \sin(2 \theta_{12})} \,,
\end{equation}
hence we also find from $\epsilon = \Delta m_{21}^2 / \Delta m_{32}^2 \ll 1$ that $\theta_{23}^\mathrm{eff} \simeq \pi/4 $, and the vector representation of the vacuum Hamiltonian restricted to the $\nu_\mu-\nu_\tau$ space is approximately along the $\mathrm{x}$-axis. Therefore the precession direction corresponds to a state where the asymmetry is the same for $\nu_\mu$ and $\nu_\tau$. In that case, and given the extra damping incurred by the approximations in the collision term, it is expected that the equilibration of the degeneracy parameters $\xi_\mu$ and $\xi_\tau$ is very efficient right after the muon-driven transition. This is seen for instance in Figures (7-10) of \cite{Dolgov_NuPhB2002}, or Figure 3 of \cite{Gava:2010kz} whereas in its counterpart here (the left plot of \ref{figComparison}), $\xi_\mu \neq \xi_\tau$ after the muon-driven transition. Also this special choice of mixing angle explains why $\xi_\mu$ and $\xi_\tau$ remain equal on Figure 1 of \cite{Mangano:2010ei} or \cite{Mangano:2011ip}, whereas in the right plot of Figure~\ref{figComparison} they differ once the electron-driven MSW transitions are crossed, which results in a full equilibration being never achieved. Note that we find nonetheless the same influence of $\theta_{13}$, as discussed in section~\ref{sec:mixing_angles}.
\paragraph{CP phase} Furthermore our results about the effect of the Dirac phase differ with respect to \cite{Gava:2010kz,Gava_corr}. Although we confirm that the effect of the Dirac phase must be maximal when $\delta=\pi$ (strictly speaking there is no CP-violation in that specific case, as it is equivalent to $\delta=0$ and a change of sign in $\theta_{13}$), we find that it must necessarily be negligibly small given the structure of the equations (see section \ref{SecDiracPhase}) whereas it is found small but not negligible in \cite{Gava:2010kz}. To be specific, in Figure 3 of \cite{Gava:2010kz} there is a small effect of the Dirac phase, whereas in the left plot of Figure \ref{figComparison} no perceptible effect is found. Again these differences must find their origin in the differences for the treatment of the collision term, given that the property~\eqref{MagiccheckK} is not satisfied by an approximate repopulation term.
\section*{Concluding remarks}
The complexity of the physics of neutrino evolution in the early Universe considerably increases when including initial degeneracies, a problem studied analytically and numerically in the last two decades. The {ATAO}-$\mathcal{V}\,$ scheme, presented in chapter~\ref{chap:Asymmetry}, which relied on the adiabaticity of the evolution of the Hamiltonian governing the dynamics of $\varrho, \bar{\varrho}$ and the very fast scale of oscillations, was extended to the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme to account for non-vanishing chemical potentials.
A restriction to two-flavour systems showed the excellent accuracy of this method compared to a much longer QKE resolution, which we found to be at least ten times slower and even more when dealing with low temperatures. Even though our code can perform this "exact" QKE resolution, it is thus sufficient, notably if one wants to explore a wide range of parameters, to rely on the {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme. Thanks to the {ATAO}-$(\Hself\pm\mathcal{V})\,$ approximation, we recover the famous synchronous oscillations, but also predict and understand new results such as the existence of a phase of quasi-synchronous oscillations, that is an increased frequency regime ($\Omega(x) \propto x^2 \to x^6$) when the vacuum + mean-field Hamiltonian $\mathcal{V}$ contribution becomes substantial compared to $\mathcal{J}$. The (non-)adiabaticity of the evolution of $\mathcal{A}$ during the lepton-driven transitions --- which depends on the degeneracies via the slowness factor~\eqref{eq:slowness} --- also allows to understand its qualitative behaviour, namely the (non-)efficiency of its alignment towards the vacuum Hamiltonian. In addition to their frequency, we can thus also estimate the beginning and the amplitude of the synchronous oscillations which develop afterwards.
We have shown that it is crucial to rely on the exact form of the collision term to fully take into account the physics of these oscillations --- approximate expressions previously overdamped degeneracy differences and led to a too rapid flavour equilibration. The $\mathcal{O}(N^3)$ complexity of the full collision term is the price to pay. Therefore, we argue that it is crucial to rely on a direct computation of the Jacobian to avoid worsening the problem. The method developed in the zero degeneracy case is extended to the situation with initial chemical potentials. Although it requires many more steps to implement it, as summarized in appendix~\ref{App:Numerics}, it keeps the appealing $\mathcal{O}(N^3)$ complexity.
The {ATAO}-$(\Hself\pm\mathcal{V})\,$ scheme fails when the vacuum potential is of the order of the self-interaction potential. In principle, the transition from the {ATAO}-$(\Hself\pm\mathcal{V})\,$ to the {ATAO}-$\mathcal{V}\,$ regime should be solved numerically with the full QKE method, but in practice this is numerically daunting. Nevertheless, by switching directly at sufficiently low temperature from {ATAO}-$(\Hself\pm\mathcal{V})\,$ to {ATAO}-$\mathcal{V}\,$, we argued that errors incurred on the neutrino asymmetry should be minimized.
In the standard case with three neutrinos, we have highlighted the influence of the various mixing parameters. Notably, the Dirac phase is found to have no perceptible effect as it essentially changes only the phase of synchronous oscillations, and its residual effect is accurately captured by the transformation~\eqref{FinalOperation}. In general, degeneracies tend to equilibrate partially, and this is due to the fact that the mixing angles $\theta_{12}$ and $\theta_{23}$ are not so different from maximal mixing ($\pi/4$), but this statement strongly depends on the non-vanishing of $\theta_{13}$. Given the complexity of the physics involved during the evolution of density matrices, the degree of equilibration depends non-trivially on the values of the initial degeneracies (e.g. equilibration is far from being achieved in the left plot of Figure~\ref{figComparison}), and requires further systematic study.
\end{document}
\chapter{Consequences for Big Bang Nucleosynthesis}
\label{chap:BBN}
\setlength{\epigraphwidth}{0.45\textwidth}
\epigraph{I'm too young to die! And too old to eat off the kids' menu! What a stupid age I am!}{Jason Mendoza, \emph{The Good Place} [S02E02]}
{
\hypersetup{linkcolor=black}
\minitoc
}
\boxabstract{The material of this chapter was partly published in~\cite{Froustey2019} and~\cite{Froustey2020}.}
During the MeV era, electron-positron annihilations and the subsequent distortions of neutrino spectra throughout their decoupling leave some imprints that we dicussed in the previous chapter: an increased energy density parameterized by $N_{\mathrm{eff}}$, and some non-thermal distortions. However, we only focused on the lepton sector, completely disregarding the baryons that are also present in the early Universe. There are indeed in negligible number, as the baryon-to-photon ratio is estimated at $\eta = n_b/n_\gamma \simeq 6.1\times 10^{-10}$~\cite{Fields:2019pfx}. Yet, the cooling of the Universe allows for the formation of light nuclides during the so-called Big Bang nucleosynthesis (BBN), topic of this chapter. Due to the value of $\eta$, it is justified to take neutrino evolution as a background result and to study how it affects BBN.
We show in Table~\ref{Table:General_BBN} the latest experimental values of the primordial abundances, and the ones predicted numerically. BBN numerical codes have indeed been developed over the past decades since the pioneering work of Wagoner~\cite{Wagoner}, the most widely used being \texttt{PRIMAT}~\cite{Pitrou_2018PhysRept}, \texttt{AlterBBN}~\cite{AlterBBN,AlterBBNv2} or \texttt{PArthENoPE}~\cite{Parthenope,Parthenope_reloaded,Parthenope_revolutions}. In this chapter, the values presented are updated from the works~\cite{Froustey2019,Froustey2020} and obtained with \texttt{PRIMAT}. We recall the notations introduced in~\ref{subsec:intro_BBN}: we call $n_i$ the number density of isotope $i$ and $n_b$ is the baryon density, which allow to define the \emph{number} fraction of isotope $i$, $X_i \equiv n_i/n_b$. The \emph{mass} fraction is therefore $Y_i \equiv A_i X_i$, where $A_i$ is the nucleon number. It is customary to define\footnote{This notation originates from the old astrophysical practice to call the mass fraction of hydrogen $X$, $Y$ for helium and $Z$ for heavier elements ("metals"), while the index $\mathrm{p}$ stands for "primordial".} $Y_{\mathrm{p}} \equiv Y_{\He4}$ and $i/\mathrm{H} \equiv X_i/X_\mathrm{H}$.
\renewcommand{\arraystretch}{1.3}
\begin{table}[!htb]
\centering
\begin{tabular}{|l|M{2.8cm} M{2.2cm} M{2.9cm} M{2.3cm} |}
\hline
Abundances & $Y_{\mathrm{p}}$ & $\mathrm{D}/\mathrm{H} \, (\times 10^{-5})$ & $\He3/\mathrm{H} \, (\times 10^{-5})$ & ${}^7{\mathrm{Li}}/\mathrm{H} \,( \times 10^{-10})$ \\
\hline \hline
Observations & $0.2453 \pm 0.0034$ \cite{Aver2020} & $2.527 \pm 0.030$ \cite{Cooke2017} & $\leq 1.1 \pm 0.2$ \cite{Bania2002,Cooke2022} & $1.6 \pm 0.3$ \cite{Sbordone2010} \\ \hline
This work & $0.24721 \pm 0.00014$ & $2.438 \pm 0.037$ & $1.039 \pm 0.014$ & $5.505 \pm 0.220$\\ \hline
\end{tabular}
\caption[State-of-the-art data on light element abundances]{Light element abundances: latest observations and results from this work. $\He3$ stands for $\He3 + \mathrm{T}$, and ${}^7{\mathrm{Li}}$ stands for ${}^7{\mathrm{Li}} + {}^7{\mathrm{Be}}$ to account for slow radioactive decays. The uncertainties on the predicted values are obtained assuming that the baryon density is determined from CMB+BAO~\cite{Planck18}, and performing a Monte-Carlo method on the posterior of this baryon abundance, but also on the uncertainties of nuclear rates and the neutron lifetime, see~\cite{Pitrou_2018PhysRept,Pitrou2020}.
\label{Table:General_BBN}}
\end{table}
The broad agreement between predictions and observations, spanning about nine orders of magnitude from helium-4 to lithium-7, has been a decisive argument towards the validation of the hot Big Bang model. However, problems persist such as the famous \emph{lithium problem}, with no good explanation to this day~\cite{Fields2011,Fields2022}.
This chapter is a synthesis and update of results partly published in two papers. In~\cite{Froustey2019}, we studied the various ways in which incomplete neutrino decoupling affects BBN, without taking into account flavour oscillations. Yet we compared the results depending on the corrections that we included in the BBN code. In~\cite{Froustey2020}, all corrections were included while we focused on the difference between the oscillation and no-oscillation cases. In this chapter, we will always use the results with flavour mixing, but we will study the interplay between neutrino decoupling and BBN without weak rates corrections to keep the discussion simple, before giving the "full" results in section~\ref{subsec:full_BBN}.
Comparisons with respect to a fiducial cosmology, where neutrinos are artificially decoupled instantaneously prior to electron-positron annihilations, require the ability to map different homogeneous cosmologies. There is no unique way to perform this cosmology mapping, that is, to compute variations, similarly to the gauge freedom that exists when comparing a perturbed cosmology with a background cosmology. For instance, we can compare the fiducial instantaneous decoupling with the full neutrino decoupling physics, either using the same cosmological times or the same cosmological factors, or even the same plasma temperatures. The fact that there is no unique choice complicates the discussion of the physical effects at play, but the physical observables, e.g., the final BBN abundances, do not depend on it. We will systematically specify which variable is left constant (cosmic time, scale factor, or photon temperature) when comparing the true Universe to the fiducial one. Quantities written with a superscript $^{(0)}$ correspond to the fiducial (instantaneous decoupling) cosmology, and the variation of a quantity $\psi$ will be written as
\begin{equation*}
\delta \psi \equiv \frac{\Delta \psi}{\psi^{(0)}} \equiv \frac{\psi - \psi^{(0)}}{\psi^{(0)}} \, .
\end{equation*}
\section{Incomplete neutrino decoupling and BBN}
\label{sec:overview_BBN}
By modifying the expansion rate of the Universe and affecting the neutron/proton weak reaction rates, incomplete neutrino decoupling will slightly modify the BBN abundances of light elements \cite{Pitrou_2018PhysRept,Mangano2005,Grohs2015}.
To get a clear understanding of the physics at play, it is useful to recall the standard picture of BBN \cite{KolbTurner,PeterUzan}.
\begin{enumerate}
\item Neutrons and protons track their equilibrium abundances,
\begin{equation}
\label{eq:nse}
\left. \frac{n_n}{n_p}\right|_\mathrm{eq} = \exp{(-\Delta/T_\gamma)} \,,
\end{equation}
where $\Delta = m_n - m_p \simeq 1.293 \, \mathrm{MeV}$ is the difference of nucleon masses, until the so-called ``weak freeze-out," when the rates of $n \leftrightarrow p$ reactions drop below the expansion rate,
\begin{equation}
\label{eq:defofFO}
\lambda \equiv \left. \frac{\Lambda_{n \to p} + \Lambda_{p \to n}}{H}\right|_{T_{\mathrm{FO}}} \simeq 1\,.
\end{equation}
\item After the freeze-out, neutrons only undergo beta decay until the beginning of nucleosynthesis, and a good approximation is
\begin{equation}
\label{eq:modelFO}
X_n(T_{\mathrm{Nuc}}) = X_n(T_{\mathrm{FO}}) \times \exp{\left[- \frac{t_\mathrm{Nuc} - t_\mathrm{FO}}{\tau_n}\right]}\, ,
\end{equation}
where $\tau_n \simeq 879.4 \, \mathrm{s}$~\cite{PDG} is the neutron mean lifetime. The nucleosynthesis temperature is usually defined when the \emph{deuterium bottleneck} is overcome, with the criterion $n_D/n_b \sim 1$ \cite{PeterUzan,Neutrino_Cosmology}. It can also be associated with the maximum in the evolution of the deuterium abundance~\cite{bernstein1989}, which coincides with the drop in the density of neutrons (converted into heavier elements). We will adopt this definition, which is very close to the other criterion. Note that $t_\mathrm{Nuc} - t_\mathrm{FO} \simeq t_\mathrm{Nuc}$, since $t_\mathrm{FO} \ll t_\mathrm{Nuc}$. Indeed, we have numerically $T_{\mathrm{FO}} \simeq 0.67 \, \mathrm{MeV}$ and $t_\mathrm{FO} \simeq 1.7 \, \mathrm{s}$ for the freeze-out, and $T_{\mathrm{Nuc}} \simeq 73 \, \mathrm{keV}$ and $t_\mathrm{Nuc} \simeq 245 \, \mathrm{s}$ for the start of nucleosynthesis.
There is a caveat in this oversimplified description: as shown in~\cite{Grohs:2016vef}, the evolution of the neutron abundance is not ruled only by beta decay below $T_{\mathrm{FO}}$. On the contrary, one needs to take into account all weak interactions even for smaller temperatures to avoid making potentially large mistakes on $Y_{\mathrm{p}}$ (see for instance Figure~8 in~\cite{Grohs:2016vef}). However, the approximation leading to~\eqref{eq:modelFO} is eventually valid for large enough times (below $T \simeq 0.28 \, \mathrm{MeV}$ according to~\cite{Pitrou_2018PhysRept}): since $t_\mathrm{Nuc} \gg t_\mathrm{FO}$, using this "beta decay" model will provide good results for the upcoming semi-analytical analysis, namely the calculation of $\delta X_n^{[\Delta t]}$ (see equations~\eqref{eq:deltaxn} and~\eqref{eq:dxnclock} below).
\item Almost all free neutrons are then converted into $\He4$, leading to
\begin{equation}
Y_{\mathrm{p}} \simeq 2 X_n(T_{\mathrm{Nuc}})\, .
\end{equation}
\end{enumerate}
This indicates very precisely where incomplete neutrino decoupling will intervene. Weak rates, and thus the freeze-out temperature, are modified through the changes in the distribution functions (different temperatures and spectral distortions $\delta g_{\nu_e}$). But the changes in the energy density will also modify the relation $t(T_\gamma)$, leaving more or less time for neutron beta decay and light element production. This is the so-called \emph{clock effect}, originally discussed in~\cite{Dodelson_Turner_PhRvD1992,Fields_PhRvD1993}. In summary, the neutron fraction at the onset of nucleosynthesis is modified as
\begin{align}
\delta X_n^{[\mathrm{Nuc}]} \equiv \frac{\Delta X_n(T_{\mathrm{Nuc}})}{X_n^{(0)}(T_{\mathrm{Nuc}})}
&= \frac{\Delta X_n(T_{\mathrm{FO}})}{X_n^{(0)}(T_{\mathrm{FO}})} - \frac{\Delta t_\mathrm{Nuc}}{\tau_n} \nonumber \\
&\equiv \delta X_n^{[\mathrm{FO}]} + \delta X_n^{[\Delta t]} \, , \label{eq:deltaxn}
\end{align}
with $\Delta t_\mathrm{Nuc} \equiv t_\mathrm{Nuc} - t_\mathrm{Nuc}^{(0)}$ (we neglected the variation of $t_\mathrm{FO}$). For freeze-out ($\delta X_n^{[\mathrm{FO}]}$), it is a variation at constant $\lambda = 1$, which we take as our definition of freeze-out. $\delta X_n^{[\mathrm{Nuc}]}$ is the neutron abundance variation between the onset of nucleosynthesis in the ``actual" Universe and the one in the reference universe. Given our definition of $T_{\mathrm{Nuc}}$, the constant quantity here is $\mathrm{d} X_D/\mathrm{d} t = 0$.
Note that this model of freeze-out is quite similar to the instantaneous decoupling approximation for neutrinos, i.e., we condense a gradual process into a snapshot. Actually, in the range $4 \gtrsim \lambda \gtrsim 0.2$, there is a smooth transition between nuclear statistical equilibrium [Eq.~\eqref{eq:nse}] and pure beta decay. For the sake of argument, we keep the criterion $\lambda \simeq 1$, and we will point out the limits of this model in the following discussions when necessary.
\section{Detailed analysis with \texttt{PRIMAT}}
In order to check the qualitative predictions of section~\ref{sec:overview_BBN}, we incorporate the results of neutrino decoupling from chapter~\ref{chap:Decoupling} into the BBN code \texttt{PRIMAT} and investigate the associated modification of abundances.
\subsection{Overview of the BBN code}
\texttt{PRIMAT} is a \emph{Mathematica} code developed from the \emph{Fortran} code used for instance in~\cite{Coc2006,CocVangioni2010,Coc2015}, designed to compute as precisely as possible the primordial abundances by including the various corrections to the weak and nuclear reaction rates. It is presented in~\cite{Pitrou_2018PhysRept,Pitrou:2019nub}, and is broadly designed as follows:
\begin{itemize}
\item it solves the dynamics of the background, first obtaining the scale factor as a function of the temperature $a(T_\gamma)$ (either from entropy conservation in the approximation of instantaneous neutrino decoupling, or taking into account the entropy transfer --- see below), then $a(t)$ via Friedmann equation;
\item once the thermodynamics and cosmological expansion are known, the weak rates are computed on a grid of plasma temperatures so as to interpolate them. Different corrections can be included in order to determine these rates with great precision;
\item finally, it builds and solves the system of differential equations that accounts for the nuclear and weak reactions.
\end{itemize}
We already mentioned the two levels at which incomplete neutrino decoupling intervenes: changing the relation $t(T_\gamma)$ via the different energy density and Friedmann equation, and affecting the weak rates via the different electronic (anti)neutrino distribution functions. Let us thus focus on the weak interaction reactions and how they are implemented in \texttt{PRIMAT}.
\subsubsection{Weak interaction reactions}
The reactions which determine the neutron-to-proton ratio are:
\begin{subequations}
\label{eq:weakrates_BBN}
\begin{align}
n + \nu_e &\longleftrightarrow p + e^- \label{eq:weakrates_BBN_1} \\
n &\longleftrightarrow p + e^- + \bar{\nu}_e \label{eq:weakrates_BBN_2} \\
n + e^+ &\longleftrightarrow p + \bar{\nu}_e \label{eq:weakrates_BBN_3}
\end{align}
\end{subequations}
The neutron and proton densities evolve according to
\begin{equation*}
\dot{n}_n + 3 H n_n = - n_n \Lambda_{n \to p} + n_p \Lambda_{p \to n} \quad \text{and} \quad \dot{n}_p + 3 H n_p = - n_p \Lambda_{p \to n} + n_n \Lambda_{n \to p} \, ,
\end{equation*}
where the rates $\Lambda_{n \to p}$ correspond to~\eqref{eq:weakrates_BBN} from left to right, and conversely for $\Lambda_{p \to n}$. In the \emph{Born approximation} (also called \emph{infinite nucleon mass approximation}), the scattering matrix elements take a simple form and we can write schematically (the bar shows that we are at the Born approximation level)
\begin{equation}
\label{eq:weakrate_schematic}
\overline{\Lambda} = K \int_{0}^{\infty}{p^2 \mathrm{d}{p} \, E_\nu^2 \times \left[\textit{Stat. fact.} \right]} \, ,
\end{equation}
with $p$ the electron or positron momentum. The prefactor reads $K = 4 G_F^2 \abs{V_{ud}}^2 (1+3 g_A^2)/(2 \pi)^3$ with $V_{ud}$ the first entry of the Cabibbo-Kobayashi-Maskawa (CKM) matrix~\cite{Cabibbo,KobayashiMaskawa} and $g_A = 1.2753(13)$ the axial-vector constant\footnote{Several conventions exist regarding this constant. In~\cite{PDG}, the vector and axial weak coupling constants $c_V$ and $c_A$ are defined such that the matrix elements include the term $[\gamma_\mu (c_V + c_A \gamma_5)]$, while $c_A$ is often defined with an opposite sign. With this convention however, we have $g_A = - c_V/c_A > 0$.} of nucleons~\cite{PDG}. The neutrino energy must satisfy the conservation condition, which actually limits the domain of integration. The statistical factor part contains only the product of electron/positron and (anti)neutrino distribution functions (that is the entries $\varrho^{e}_e$ and $\bar{\varrho}^{e}_e$ of the density matrix when we take into account mixing), since we can neglect the baryon Pauli-blocking factors due to the very small baryon-to-photon ratio, and the other neutron or proton distribution function is integrated upon, giving the density which gets factored out.
Let us consider reaction~\eqref{eq:weakrates_BBN_1}. Energy conservation requires, in the infinite nucleon mass approximation, $E_\nu = m_p + E - m_n = E - \Delta$, with $E$ the electron energy. Since we must have $E_\nu > 0$, the integral~\eqref{eq:weakrate_schematic} is thus limited to the domain $E > \Delta$, that is $p > \sqrt{\Delta^2 - m_e^2}$. The statistical factor is
\begin{equation}
\label{eq:statfact_BBN_1}
[\textit{Stat. fact.}] = [1-f_e(E)]f_{\nu_e}(E - \Delta) = f_e(-E) \times \frac{1 + \delta g_{\nu_e}(E-\Delta)}{e^{(E-\Delta)/T_{\nu_e}}+1} \, ,
\end{equation}
where we used the functional property of equilibrium Fermi-Dirac spectra $1 - f_e(E) = f_e(-E)$, and the parameterization of neutrino spectral distortions~\eqref{eq:param_rho}. If we assume instantaneous neutrino decoupling, the neutrino distribution function reads $f_\nu^{(\mathrm{eq})}$ (i.e. $T_{\nu_e} = T_{\mathrm{cm}}$ and $\delta g_{\nu_e}=0$), and the reaction rate can be written:
\begin{equation}
\overline{\Lambda}_{n+\nu_e \to p + e^-} = K \int_{E > \Delta}{p^2 \mathrm{d}{p} \, (E - \Delta)^2 f_e(-E) f_\nu^{(\mathrm{eq})}(E - \Delta)} \, ,
\end{equation}
A similar procedure can be applied to reaction~\eqref{eq:weakrates_BBN_2}. Energy conservation requires $E_{\bar{\nu}} = \Delta - E > 0$ hence $E < \Delta$. The statistical factor reads
\begin{equation}
\label{eq:statfact_BBN_2}
[\textit{Stat. fact.}] = [1-f_e(E)][1-f_{\bar{\nu}_e}(\Delta-E) ]= f_e(-E) \times \left[1- \frac{1 + \delta g_{\bar{\nu}_e}(\Delta-E)}{e^{(\Delta-E)/T_{\bar{\nu}_e}}+1}\right] \, ,
\end{equation}
which can also be simplified in the instantaneous decoupling limit, using $1 - f_\nu^{(\mathrm{eq})}(\Delta-E) = f_\nu^{(\mathrm{eq})}(E-\Delta)$, such that
\begin{equation}
\overline{\Lambda}_{n \to p + e^- + \bar{\nu}_e} = K \int_{E < \Delta}{p^2 \mathrm{d}{p} \, (E-\Delta)^2 f_e(-E) f_\nu^{(\mathrm{eq})}(E-\Delta)} \, ,
\end{equation}
Finally, for the reaction~\eqref{eq:weakrates_BBN_3}, energy conservation gives $E_{\bar{\nu}} = E + \Delta$ which does not put any constraint on $p$. The statistical factor reads
\begin{equation}
\label{eq:statfact_BBN_3}
[\textit{Stat. fact.}] = f_e(E)[1-f_{\bar{\nu}_e}(E + \Delta) ]= f_e(E) \times \left[1- \frac{1 + \delta g_{\bar{\nu}_e}(E+\Delta)}{e^{(E+\Delta)/T_{\bar{\nu}_e}}+1}\right] \, ,
\end{equation}
which we simplify in the instantaneous decoupling limit as
\begin{equation}
\overline{\Lambda}_{n + e^+ \to p + \bar{\nu}_e} = K \int_{0}^{\infty}{p^2 \mathrm{d}{p} \, (-E - \Delta)^2 f_e(E) f_\nu^{(\mathrm{eq})}(-E - \Delta)} \, .
\end{equation}
We therefore introduce the functions:
\begin{equation}
E_\nu^\mp(E) \equiv E \mp \Delta \qquad \text{and} \qquad \chi_\pm(E) \equiv \left[E_\nu^\mp(E)\right]^2 f_e(-E) \, f_\nu^{(\mathrm{eq})}\left(E_\nu^\mp(E)\right) \, .
\end{equation}
We can then gather all the contributions in a single expression for the Born rates:
\begin{equation}
\label{eq:Lambda_nTOp}
\overline{\Lambda}_{n \to p} = K \int_{0}^{\infty}{p^2 \mathrm{d}{p} \left[\chi_+(E) + \chi_+(-E)\right]} \, .
\end{equation}
The first term in the integrand comes from the sum of $\overline{\Lambda}_{n+\nu_e \to p + e^-}$ and $\overline{\Lambda}_{n \to p + e^- + \bar{\nu}_e}$, and the second term is $\overline{\Lambda}_{n + e^+ \to p + \bar{\nu}_e}$. This expression coincides with Eq.~(77) in~\cite{Pitrou_2018PhysRept}.
The reaction rate for protons is obtained by replacing $\Delta \to - \Delta$, that is $\chi_+ \to \chi_-$, which reads for completeness
\begin{equation}
\label{eq:Lambda_pTOn}
\overline{\Lambda}_{p \to n} = K \int_{0}^{\infty}{p^2 \mathrm{d}{p} \left[\chi_-(E) + \chi_-(-E)\right]} \, .
\end{equation}
In addition to these Born rates, several corrections are implemented: radiative corrections at zero and finite temperature, finite nucleon mass corrections, weak magnetism... and incomplete neutrino decoupling. We now focus on this latter particular feature, the extensive derivation above showing directly where to modify the rates to include the results from chapter~\ref{chap:Decoupling}. Note that, unless stated otherwise, QED corrections to the plasma thermodynamics are included in the calculation.
\subsection{Implementation of incomplete neutrino decoupling in \texttt{PRIMAT}}
In the version of \texttt{PRIMAT} used in~\cite{Pitrou_2018PhysRept}, the lack of effective temperatures and spectral distortion values across the nucleosynthesis era required an approximate strategy to include incomplete neutrino decoupling. It consisted in neglecting spectral distortions $\delta g_{\nu} = 0$ while considering that all neutrinos shared the same temperature, that is an effective average temperature $\widehat{T}_\nu$ consistent with the energy transfer between the QED plasma and neutrinos. We can define it from the effective temperatures $T_{\nu_\alpha}$:
\begin{equation}
\label{eq:def_Taverage}
\bar{\rho}_{\nu} = \frac78 \frac{\pi^2}{30} \left(z_{\nu_e}^4 + z_{\nu_\mu}^4 + z_{\nu_\tau}^4\right) \equiv 3 \times \frac78 \frac{\pi^2}{30} \times \hat{z}_\nu^4 \qquad \text{with} \qquad \hat{z}_\nu = \frac{\widehat{T}_\nu}{T_{\mathrm{cm}}} \, .
\end{equation}
It is not necessary to have the individual values of $T_{\nu_\alpha}$ to compute $\widehat{T}_{\nu}$, as one can use another key quantity: the \emph{heating rate}~\cite{Parthenope}
\begin{equation}
\label{eq:Nheating}
\mathcal{N} \equiv \frac{1}{z^4} \left(x \frac{\mathrm{d} (\bar{\rho}_\nu + \bar{\rho}_{\bar{\nu}})}{\mathrm{d} x} \right)_{x=x(z)} = \frac{1}{z^4} \frac{1}{2 \pi^2 H} \int{\mathrm{d}{y} y^3 \, \mathrm{Tr}[\mathcal{I} + \bar{\mathcal{I}}]} \, .
\end{equation}
which is obtained from~\eqref{eq:rhonu_matrix} and the QKE~\eqref{eq:QKE_final}, noting that the trace of a commutator is zero. $T_\gamma^4 \mathcal{N}$ can be viewed as the volume heating rate of the neutrino bath in units of the Hubble rate~\cite{Pitrou_2018PhysRept}. The values of $\mathcal{N}$ were obtained from a fit given in \texttt{PArthENoPE} \cite{Parthenope} [Eqs.~(A23)--(A25)], computed by Pisanti \emph{et al.}~from the results of~\cite{Mangano2002,Mangano2005}.
Note that we can also compute $\mathcal{N}$ from the variation on the comoving photon temperature $z$, which is more convenient if we want to treat our numerical results a posteriori, since we keep the values of $z(x)$ across the decoupling era. Starting from~\eqref{eq:zQED}, we have:
\begin{equation}
\label{eq:compute_N}
\mathcal{N}(x) = \frac{2 x}{z} \left\{\frac{x}{z} J\left(\frac{x}{z}\right) + G_1\left(\frac{x}{z}\right) - \left[ \frac{x^2}{z^2} J\left(\frac{x}{z}\right) + Y\left(\frac{x}{z}\right) + \frac{2 \pi^2}{15} + G_2\left(\frac{x}{z}\right)\right] \times \frac{\mathrm{d} z}{\mathrm{d} x} \right\} \, .
\end{equation}
We plot on Figure~\ref{fig:Compare_N} the quantity $\mathcal{N}$ deduced from the neutrino decoupling results of chapter~\ref{chap:Decoupling}, and the fit provided in \texttt{PArthENoPE}.
\begin{figure}[!ht]
\centering
\includegraphics{figs/Compare_N.pdf}
\caption[Dimensionless heating rate from \texttt{NEVO} and \texttt{PArthENoPE}]{\label{fig:Compare_N} Comparison of the dimensionless heating rate $\mathcal{N}$ computed with the results from \texttt{NEVO} and equation~\eqref{eq:compute_N}, and the fit given in \texttt{PArthENoPE}~\cite{Parthenope}. For consistency with the results at the time, we did not include the $\mathcal{O}(e^3)$ finite-temperature corrections to the plasma thermodynamics in this calculation.}
\end{figure}
This approximate handling of incomplete neutrino decoupling in the earlier version of \texttt{PRIMAT} correctly captures the changes in the expansion rate (since~\eqref{eq:def_Taverage} shows that the energy density is well computed from $\widehat{T}_\nu$), but \emph{a priori} it handles the weak rates poorly: electron neutrinos are too cold ($T_{\nu_e} > \widehat{T}_\nu$), and their spectrum is not distorted. This should in principle have consequences for the neutron-to-proton ratio at freeze-out, and thus on the final abundances.
We modified \texttt{PRIMAT} to introduce the results from neutrino transport analysis. Since the useful variable in nucleosynthesis is the plasma temperature $T_\gamma$, all other quantities\footnote{The non-thermal distortions depend both on the momentum (sampling on a grid with points $y_i$) and time, hence on the plasma temperature $T_\gamma$.} ($x$, $T_{\nu_\alpha}$, $\delta g_{\nu_\alpha}(y_i)$) are interpolated. Depending on the options chosen, one can then use the ``real" effective neutrino temperatures or the average temperature for comparison with the previous approach (keeping the total energy density unchanged in each case).
\paragraph{Summary of neutrino decoupling results} We plot on Figure~\ref{fig:summary_nubbn} the evolution of different quantities that play a significant role for BBN, obtained through the numerical resolution presented in chapter~\ref{chap:Decoupling}. The evolution of the comoving temperatures has already been discussed in the previous chapter. The reheating of the different species is due to the entropy transfer from electrons and positrons, which is visualized by plotting the variation of their number density. For $T_\gamma \gg m_e$, electrons are relativistic and $\bar{n}_{e^\pm} \equiv (n_{e^-} + n_{e^+})\times (x/m_e)^3$ is constant, while for $T_\gamma \ll m_e$ the density drops to zero. The variation between those two constants corresponds to the annihilation period, which indeed starts around $T_\gamma \sim m_e$ and is over for $T_\gamma \sim 30 \, \mathrm{keV}$. At the beginning of this period, neutrinos progressively decouple and there is a heat transfer from the plasma, visualized through the dimensionless heating rate $\mathcal{N}$ defined in~\eqref{eq:Nheating}. The slight overlap between the two curves in the bottom panel of Fig.~\ref{fig:summary_nubbn} is the very reason why neutrinos are partly reheated.
Finally, we plot the evolution of $N_\mathrm{eff}$, from $3$ before the MeV age to its frozen value $3.044$. To do so, we define it such that we can compute its value across the decoupling era and not only long after decoupling, via
\begin{equation}
\label{eq:defNeff_bbn}
\rho_\nu + \rho_{\bar{\nu}} = \frac78 \left(\frac{T_{\mathrm{cm}}}{T_\gamma}\right)^4 N_{\mathrm{eff}} \times \rho_\gamma \quad \text{hence} \quad \boxed{N_{\mathrm{eff}} = 3 \left(\frac{\hat{z}_\nu z^{(0)}}{z}\right)^4} \, .
\end{equation}
In this expression, $z^{(0)}$ is the photon temperature in the instantaneous decoupling limit. All these quantities are taken as functions of either the comoving temperature $T_{\mathrm{cm}}$ or the photon temperature $T_\gamma$. Comparing with Fig.~5 in~\cite{Grohs2015}, we note that there is no ``plateau" before the freeze-out. This behavior can be considered as an artifact due to plotting $N_{\mathrm{eff}}$ as a function of $x=m_e/T_{\mathrm{cm}}$: the plateau is due to the difference between $T_{\mathrm{cm}}$ and $T_{\mathrm{cm}}^{(0)}$ for a given $T_\gamma$, and does not represent a meaningful physical effect (see also Fig.~7 in~\cite{Esposito_NuPhB2000}). In other words, the asymptotic values of $N_{\mathrm{eff}}$ are meaningful, while the intermediate ones depend on the reference chosen, which is less significant.
\begin{figure}[!ht]
\centering
\includegraphics{figs/SummaryPlots.pdf}
\caption[Evolution of relevant quantities for neutrino decoupling, as a function of $T_\gamma$]{\label{fig:summary_nubbn} Evolution of relevant quantities for neutrino decoupling, as a function of the plasma temperature. \emph{Top}: Comoving (effective) temperatures of the plasma and neutrinos. \emph{Middle}: Effective number of neutrinos, as defined in Eq.~\eqref{eq:defNeff_bbn}. \emph{Bottom}: Neutrino heating rate and variation of the comoving electron+positron density (derivative taken with respect to $z/x = T_\gamma/m_e$).}
\end{figure}
\subsubsection{Corrections to the Born rates} The weak rates are modified at the Born level by including the effective temperatures and distortions of electronic (anti)neutrinos in the rates~\eqref{eq:Lambda_nTOp} and~\eqref{eq:Lambda_pTOn}. First, we have the same expressions where the neutrino distribution functions are replaced in $\chi_{\pm}$
\[ f_{\nu}^{(\mathrm{eq})}(E_\nu^\mp,T_{\mathrm{cm}}) \to f_{FD}(E_\nu^\mp,T_{\nu_e}) \qquad \text{i.e.} \qquad \frac{1}{e^{E_\nu^\mp/T_{\mathrm{cm}}}+1} \to \frac{1}{e^{E_\nu^\mp/T_{\nu_e}}+1} \, . \]
Then, we derive the contribution from the non-thermal distortions starting from the expressions of the statistical factors.
\begin{itemize}
\item[(a)] The case of eq.~\eqref{eq:statfact_BBN_1} is the simplest since the neutrino is in the initial state, we just need to add the extra contribution
\[(E-\Delta)^2 f_e(-E) \frac{\delta g_{\nu_e}(E-\Delta)}{e^{(E-\Delta)/T_{\nu_e}}+1} = (E_\nu^-)^2 f_e(-E) \frac{\delta g_{\nu_e}(E_\nu^-)}{e^{E_\nu^-/T_{\nu_e}}+1} \, , \]
and note that $E_\nu^- = \abs{E_\nu^-} > 0$.
\item[(b)] For the reaction $n \to p + e^- + \bar{\nu}_e$, i.e. the statistical factor~\eqref{eq:statfact_BBN_2}, note that $\Delta - E = - E_\nu^-(E) = \abs{E_\nu^-(E)}$, such that the extra contribution reads
\[(E-\Delta)^2 f_e(-E) \times (-1) \times \frac{\delta g_{\bar{\nu}_e}(\Delta-E)}{e^{(\Delta-E)/T_{\bar{\nu}_e}}+1} = - (E_\nu^-)^2 f_e(-E) \frac{\delta g_{\bar{\nu}_e}(\abs{E_\nu^-})}{e^{\abs{E_\nu^-}/T_{\bar{\nu}_e}}+1} \, . \]
\item[(c)] Finally, for~\eqref{eq:statfact_BBN_3}, the extra contribution is
\[(E+\Delta)^2 f_e(E) \times (-1) \times \frac{\delta g_{\bar{\nu}_e}(E+\Delta)}{e^{(E+\Delta)/T_{\nu_e}}+1} = - (E_\nu^-)^2 f_e(E) \frac{\delta g_{\nu_e}(-E_\nu^-)}{e^{-E_\nu^-/T_{\nu_e}}+1} \, , \]
where $E_\nu^- = -E - \Delta$ is evaluated at $-E$. Since $E_\nu^- < 0$, we can replace $- E_\nu^- = \abs{E_\nu^-}$.
\end{itemize}
If, as assumed in standard neutrino decoupling, we consider that neutrinos and antineutrinos have the same distributions, we can gather the corrections under the common equation (the case of $p \to n$ rates is treated in the same way):
\begin{subequations}
\label{eq:deltagamma}
\begin{align}
\Delta \Lambda_{n \to p} &= K \int_{0}^{\infty}{p^2 \mathrm{d} p \left[\delta \chi_+(E) + \delta \chi_+(-E)\right]} \, , \label{subeq:np} \\
\Delta \Lambda_{p \to n} &= K \int_{0}^{\infty}{p^2 \mathrm{d} p \left[\delta \chi_-(E) + \delta \chi_-(-E)\right]} \, , \label{subeq:pn}
\end{align}
\end{subequations}
with
\begin{equation}
\delta \chi_{\pm} (E) = \abs{E_\nu^\mp(E)}^2 f_{e}(-E) \, \frac{\sign{(E_\nu^\mp)} \times \delta g_{\nu_e}(\abs{E_\nu^\mp})}{e^{\abs{E_\nu^\mp}/T_{\nu_e}}+1} \, .
\end{equation}
As evidenced above, the $\sign$ function accounts for the fact that $f_{\nu_e}(\abs{E_\nu^\mp})$ appears as part of a Pauli blocking factor if $E_\nu^\mp < 0$, i.e., the neutrino is in a final state.
\subsubsection{Analysis via different implementations}
We consider three different implementations:
\begin{itemize}
\item[\textit{(i)}] The earlier \texttt{PRIMAT} approach (no distortions and an average neutrino temperature), where $\widehat{T}_\nu$ is computed from the effective temperatures via~\eqref{eq:def_Taverage}, although we checked that computing it from the heating rate $\mathcal{N}$ deduced from \texttt{NEVO} gave the same results. We call this approach ``$\widehat{T}_\nu$" in Tables~\ref{Table:Corrections} and \ref{Table:Full_Corrections} and Figs.~\ref{fig:analyzevariations} and \ref{fig:analyzevariationsfull}.
\item[\textit{(ii)}] The weak rates including the real electron neutrino temperature, but still without spectral distortions. We call this approach ``$T_{\nu_e},$ no distortions."
\item[\textit{(iii)}] Full results from neutrino evolution. We call this approach ``$T_{\nu_e},$ with distortions."
\end{itemize}
Note that these three scenarios take place in identical cosmologies, with the \emph{same} energy density; using the proper $\nu_e$ temperature and including distortions only affect the weak rates.
\renewcommand{\arraystretch}{1.3}
\begin{table}[!htb]
\centering
\begin{tabular}{|l|cccc|}
\hline
& $Y_{\mathrm{p}}$ & $\mathrm{D}/\mathrm{H} \times 10^5$ & $\He3/\mathrm{H} \times 10^5$ & ${}^7{\mathrm{Li}}/\mathrm{H} \times 10^{10}$ \\
\hline \hline
Inst. decoupling, no QED & $0.24268$ & $2.4027$ & $1.0339$ & $5.4690$ \\ \hline
$\widehat{T}_\nu$ & $0.24281$ & $2.4122$ & $1.0353$ & $5.4470$ \\
$T_{\nu_e},$ no distortions & $0.24278$ & $2.4120$ & $1.0353$ & $5.4466$ \\
$T_{\nu_e},$ with distortions & $0.24278$ & $2.4125$ & $1.0354$ & $5.4479$ \\ \hline \hline
Inst. decoupling, with QED & $0.24268$ & $2.4052$ & $1.0343$ & $5.4620$ \\ \hline
$\widehat{T}_\nu$ & $0.24280$ & $2.4146$ & $1.0357$ & $5.4403$ \\
$T_{\nu_e},$ no distortions & $0.24278$ & $2.4145$ & $1.0356$ & $5.4399$ \\
$T_{\nu_e},$ with distortions & $0.24286$ & $2.4149$ & $1.0357$ & $5.4412$\\ \hline
\end{tabular}
\caption[Light element abundances at the Born approximation level]{Light element abundances, at the Born approximation level, for various implementations of neutrino-induced corrections. See section~\ref{subsec:full_BBN} for results with the full corrections derived in~\cite{Pitrou_2018PhysRept}. The number of digits is larger than the nominal uncertainty but is chosen here so as to show the variations.
\label{Table:Corrections}}
\end{table}
\renewcommand{\arraystretch}{1.2}
We show in Table~\ref{Table:Previous} the relative variations of abundances when taking into account incomplete neutrino decoupling, i.e., when going from the instantaneous decoupling to the "$T_{\nu_e}$, with distortions" row in Table~\ref{Table:Corrections}. We also compare these variations with previous results in the literature. We check that our results are in close agreement with Grohs \emph{et al.}~\cite{Grohs2015}, but with opposite signs of variation (except for ${}^4 \mathrm{He}$) compared to the results of Mangano \emph{et al.}~\cite{Mangano2005}. The extensive study in the next section sheds a new light on the different phenomena involved: our aim is to justify physically these results.
\begin{table}[!ht]
\centering
\begin{tabular}{|l|rrrr|}
\hline
Variation of abundances & \multicolumn{1}{c}{$\delta Y_{\mathrm{p}}$ } & \multicolumn{1}{c}{$\delta \left(\mathrm{D}/\mathrm{H}\right) $}& \multicolumn{1}{c}{$ \delta \left(\He3/\mathrm{H}\right) $} & \multicolumn{1}{c|}{$\delta \left({}^7{\mathrm{Li}}/\mathrm{H}\right) $ }\\
\hline \hline
\emph{No QED corrections} & & & & \\
Grohs \emph{et al.}~\cite{Grohs2015} & $4.636 \times 10^{-4}$ & $3.686 \times 10^{-3}$ & $1.209 \times 10^{-3}$ &$-3.916 \times 10^{-3}$ \\
This work & $7.707 \times 10^{-4}$ & $4.086 \times 10^{-3}$ & $1.369 \times 10^{-3}$ & $-3.855 \times 10^{-3}$ \\ \hline \hline
\emph{QED corrections included} & & & & \\
Naples group \cite{Mangano2005} & $7.05 \times 10^{-4}$ & $-2.8 \times 10^{-3}$ & $-1.1 \times 10^{-3}$ & $3.92 \times 10^{-3}$ \\
This work & $7.648 \times 10^{-4}$ & $4.043 \times 10^{-3}$ & $1.355 \times 10^{-3}$ & $-3.814 \times 10^{-3}$ \\ \hline
\end{tabular}
\caption[Comparison with previous results]{Comparison with previous results. Note that baseline values are different in the cases that do or do not include QED corrections (see Table~\ref{Table:Corrections}). The values given by the Naples group in~\cite{Mangano2005} are absolute variations, and we need the baseline values to compute relative variations; as these were not given, we use our own baseline values. Besides, our results with QED corrections include the $\mathcal{O}(e^3)$ corrections which were absent in~\cite{Mangano2005} ; however, we checked that the relative variations were the same if we restricted to $\mathcal{O}(e^2)$ corrections.
\label{Table:Previous}}
\end{table}
\subsection{Neutron fraction at the onset of nucleosynthesis}
We now review the physics that allows us to understand the numerical results of Table~\ref{Table:Corrections}. We first detail the physics affecting the helium abundance, which is directly related to the neutron fraction at the onset of nucleosynthesis, before turning to the production of other light elements, for which the clock effect dominates.
\subsubsection{Neutron/proton freeze-out}
Previous articles~\cite{Dodelson_Turner_PhRvD1992,Fields_PhRvD1993,Mangano2005} studied the variation of $n \leftrightarrow p$ rates due to incomplete neutrino decoupling at constant scale factor, claiming that the Hubble rate $H$ was left unchanged at a given $x$. This argument of constant total energy density, namely $\Delta (\rho_\nu + \rho_{\bar{\nu}}) = - \Delta \rho_\mathrm{em}$ with $\rho_{\mathrm{em}}$ the energy density of the QED plasma, requires $T_\gamma \simeq T_\nu$, as proven in the Appendix 3 of~\cite{Dodelson_Turner_PhRvD1992}. However, by looking at the top panel of Fig.~\ref{fig:summary_nubbn} it appears that at freeze-out $T_\gamma$ and $T_{\nu_{\alpha}}$ differ by $\sim 1 \, \%$, which is the typical order of magnitude of variations we are interested in. Moreover, the analysis of~\cite{Fields_PhRvD1993} used thermal-equivalent distortions of neutrinos spectra (i.e., only effective temperatures, no $\delta g_{\nu}$), calling for a more precise study making full use of our numerical results.
Due to the rich interplay of the processes involved, an analytical estimate of $\delta X_n^{[\mathrm{FO}]}$ is particularly challenging. Since our goal is to provide a satisfactory physical picture of the role of neutrinos in BBN, and thus in particular to check Eq.~\eqref{eq:deltaxn}, we perform a numerical evaluation.
Figure~\ref{fig:analyzevariations} shows the variation of $X_n$ and $T_{\nu_e,\gamma}$ for the different implementations of neutrino-induced corrections around the time of freeze-out. In each case, incomplete neutrino decoupling leads to a decrease of $X_n$ at freeze-out.
\begin{figure}[!ht]
\centering
\includegraphics{figs/AnalyzeVariationsQED_Mix_.pdf}
\caption[Neutron fraction and temperature variations for various implementations of neutrino-induced corrections]{\label{fig:analyzevariations} Neutron fraction (\emph{top}) and temperature (\emph{bottom}) variations with respect to instantaneous decoupling, in the different implementations of neutrino-induced corrections. These quantities are plotted as a function of the ratio of the total $n \leftrightarrow p$ reaction rate and the Hubble expansion rate, which is approximately equal to $1$ at weak freeze-out.}
\end{figure}
For each implementation of neutrino-induced corrections the evolution of the photon temperature $z(x)$ is the same; the difference lies in whether or not we include $z_{\nu_e}$ and $\delta g_{\nu_e}$. But the quantities in Fig.~\ref{fig:analyzevariations} are plotted with respect to $\lambda$, which is a different function of $x$ in each case. For instance, when including the real $\nu_e$ temperature, weak rates increase and freeze-out is delayed, leading to a smaller $T_\gamma(\lambda \simeq 1) \equiv T_{\mathrm{FO}}$: the orange curve is below the blue one in the bottom panel of Fig.~\ref{fig:analyzevariations}. Adding the distortions increases the rates even more, and slightly decreases $T_{\mathrm{FO}}$ (green curve). One would then expect a reduction of $X_n$, which would track its equilibrium value longer. While this is true for thermal corrections (orange curve below the blue one in the top panel of Fig.~\ref{fig:analyzevariations}), adding the distortions disrupts this picture.
\paragraph{Disruption of detailed balance} Indeed, the main effect of including neutrino spectral distortions is to alter the detailed balance relation $\overline{\Lambda}_{p \to n} = e^{-\Delta/T} \overline{\Lambda}_{n \to p}$. Let us parameterize this deviation from detailed balance as\footnote{Note that a similar deviation from detailed balance would arise if electronic neutrinos had a chemical potential ($\sigma_\nu$ would then be $\mu_{\nu_e}/T$). However, this is a coincidence: here, neutrinos and antineutrinos have the same distributions, but the existence of non-thermal distortions lead to a disruption of detailed balance that we parameterize, \emph{for convenience}, like a chemical potential-like deviation.}
\begin{equation}
\label{eq:detailedbalance}
\Lambda_{p \to n} = \exp{ \left(-\frac{\Delta}{T} + \sigma_\nu\right)} \Lambda_{n \to p} \, ,
\end{equation}
with $\sigma_\nu \ll 1$. Writing this in terms of the Born rates $\overline{\Lambda}$ (which satisfy the detailed balance equation), we get
\begin{equation}
\sigma_\nu = \frac{\Delta \Lambda_{p \to n}}{\overline{\Lambda}_{p \to n}} - \frac{\Delta \Lambda_{n \to p}}{\overline{\Lambda}_{n \to p}} \, ,
\end{equation}
leading to a change in the equilibrium neutron abundance,
\begin{equation}
\label{eq:dxnsigma}
\delta X_n^{(\mathrm{eq})} = (1 - X_n) \sigma_\nu \, ,
\end{equation}
since $X_n/(1-X_n) = n_n/n_p$ and $(n_n/n_p)_{\mathrm{eq}} = \Lambda_{p \to n}/\Lambda_{n \to p}$. Corrections to the Born rates are shown in Fig.~\ref{fig:detailedbalance}. Equations \eqref{eq:detailedbalance} and thus \eqref{eq:dxnsigma} are not absolutely valid for $\lambda \simeq 1$ because deviations from detailed balance start earlier, but we can nonetheless estimate from this plot that $\sigma_\nu(\lambda \simeq 1) \simeq 0.001$. With $X_n(\lambda \simeq 1) \simeq 0.2$, we find from Eq.~\eqref{eq:dxnsigma} that including the spectral distortions increases the neutron fraction at freeze-out by
\begin{equation}
\label{eq:shift_disto}
\delta X_n^{[\mathrm{FO}],\delta g_{\nu_e}} \lesssim 0.08 \, \% \, ,
\end{equation}
where the definition of this value corresponds to the shift from the orange curve to the green curve in the top panel of Fig.~\ref{fig:analyzevariations}:
\begin{equation}
\delta X_n^{[\mathrm{FO}],\delta g_{\nu_e}} \equiv \delta X_{n,[T_{\nu_e},\text{with dist.}]}^{[\mathrm{FO}]} - \delta X_{n,[T_{\nu_e},\text{no dist.}]}^{[\mathrm{FO}]} \, .
\end{equation}
The value \eqref{eq:shift_disto} is overestimated because at $\lambda =1$, the neutron-to-proton ratio has already deviated from nuclear statistical equilibrium. In fact, one can reasonably consider that the shift in $\delta X_n^{[\mathrm{FO}]}$ is due to the deviation from detailed balance at higher temperatures, when nuclear statistical equilibrium was actually verified (namely, for $\lambda \sim 4$). Indeed, using Eq.~\eqref{eq:dxnsigma} for $\lambda \sim 4$, we obtain the observed shift $\delta X_n^{[\mathrm{FO}],\delta g_{\nu_e}} = 0.02 \, \%$.
\begin{figure}[!ht]
\centering
\includegraphics{figs/VariationRates.pdf}
\caption[Relative corrective to $n \leftrightarrow p$ weak rates]{\label{fig:detailedbalance} Relative corrections to $n \leftrightarrow p$ weak rates, with $\Delta \Lambda_{n \leftrightarrow p}$ defined in Eq.~\eqref{eq:deltagamma}. To ensure detailed balance requirements, we enforce $T_{\nu_e} = T_\gamma$.}
\end{figure}
We conclude this detailed analysis of neutron/proton freeze-out by stating the obtained value for $\delta X_n^{[\mathrm{FO}]}$, which can be read from Fig.~\ref{fig:analyzevariations} at $\lambda \sim 1$ in the ``$T_{\nu_e},$ with distortions" case:
\begin{equation}
\label{eq:dxnfo}
\delta X_n^{[\mathrm{FO}]} \simeq + 0.014 \, \% \, .
\end{equation}
\subsubsection{Clock effect}
The \emph{clock effect} is due to the higher radiation energy density for a given plasma temperature, which reduces the time necessary to go from $T_{\mathrm{FO}}$ to $T_{\mathrm{Nuc}}$. This leads to less neutron beta decay, and thus a higher $X_n(T_{\mathrm{Nuc}})$ and consequently a higher $Y_{\mathrm{p}}$. To estimate this contribution we will make several assumptions, justified by observing Fig.~\ref{fig:summary_nubbn}. Since $t_\mathrm{Nuc} \sim 245 \, \mathrm{s} \gg t_\mathrm{FO}$, the freeze-out modification discussed previously will only result in a very small change in duration; indeed, we find numerically that $\Delta t_\mathrm{FO} \simeq 0.002 \ \mathrm{s}$. We also checked that $T_{\mathrm{Nuc}}$ is almost not modified ($\delta T_{\mathrm{Nuc}} \simeq - 0.01 \ \%$), which is expected since the onset of nucleosynthesis is essentially determined only by $T_\gamma$. Therefore, the clock effect is mainly described by the change of duration between $T_{\mathrm{FO}}^{(0)}$ and $T_{\mathrm{Nuc}} \simeq T_{\mathrm{Nuc}}^{(0)}$.
An additional assumption is made by observing the time scale in Fig.~\ref{fig:summary_nubbn}: most of the neutron beta decay takes place when neutrinos have decoupled and electrons and positrons have annihilated. We will thus consider that between the freeze-out and the beginning of nucleosynthesis, neutrinos are decoupled and $N_\mathrm{eff} \simeq N_\mathrm{eff}^\mathrm{fin}$ is constant.
Therefore, we can write $H \propto 1/2t$ as we are in the radiation era, cf.~\eqref{eq:hubble_radiation}. Using Friedmann equation $H^2 \propto \rho$, we get
\begin{equation}
\frac{\Delta t_\mathrm{Nuc}}{t_\mathrm{Nuc}^{(0)}} = - \frac12 \left. \frac{\Delta \rho}{\rho^{(0)}}\right|_{T_\gamma = T_{\mathrm{Nuc}}} = - \left. \frac{\Delta \rho_\nu}{\rho_\nu^{(0)}}\right|_{T_{\mathrm{Nuc}}} \times \frac{\rho_\nu^{(0)}}{\rho^{(0)}}\,.
\end{equation}
The factor $1/2$ disappears since the variation of the energy density is $\Delta \rho = \Delta \rho_\nu + \Delta \rho_{\bar{\nu}} = 2 \Delta \rho_\nu$ (we consider the standard case without asymmetry). This shift in the neutrino energy density is parameterized by $N_\mathrm{eff}$, while the ratio of instantaneously decoupled energy densities is, at $T_{\mathrm{Nuc}}$,
\begin{equation}
\frac{\rho_\nu^{(0)}}{\rho^{(0)}} = \frac{\rho_\nu^{(0)}}{\rho_\gamma^{(0)} + \rho_\nu^{(0)} + \rho_{\bar{\nu}}^{(0)}} = \frac{\frac78 \frac{\pi^2}{30} \times 3 \times \left(\frac{4}{11}\right)^{4/3} T_{\mathrm{Nuc}}^4}{2 \times \frac{\pi^2}{30} T_{\mathrm{Nuc}}^4 + 2 \times \frac78 \frac{\pi^2}{30} \times 3 \times \left(\frac{4}{11}\right)^{4/3} T_{\mathrm{Nuc}}^4} \simeq 0.203 \, .
\end{equation}
This gives
\begin{equation*}
\frac{\Delta t_\mathrm{Nuc}}{t_\mathrm{Nuc}^{(0)}} \simeq - 0.203 \times \frac{\Delta N_\mathrm{eff}}{3} \simeq -3.0\times 10^{-3} \, ,
\end{equation*}
but since we included the QED corrections to the plasma thermodynamics at all stages of the calculation, the reference value for $N_{\mathrm{eff}}$ is not $3$ but $N_{\mathrm{eff}}^{(0),\text{QED}} = 3.00965$, which leads to
\begin{equation}
\frac{\Delta t_\mathrm{Nuc}}{t_\mathrm{Nuc}^{(0)}} \simeq - 0.203 \times \frac{\Delta N_\mathrm{eff}}{N_{\mathrm{eff}}^{(0),\text{QED}}} \simeq -2.3\times 10^{-3} \, ,
\end{equation}
This estimate is actually in very good agreement with the numerical result
\begin{equation}
\left. \frac{\Delta t_\mathrm{Nuc}}{t_\mathrm{Nuc}^{(0)}}\right|_{\texttt{PRIMAT}} \simeq -2.1 \times 10^{-3} \, .
\end{equation}
Hence, the estimate for the clock effect contribution is, from Eq.~\eqref{eq:deltaxn},
\begin{equation}
\label{eq:dxnclock}
\delta X_n^{[\Delta t]} = - \frac{\Delta t_\mathrm{Nuc}}{t_\mathrm{Nuc}^{(0)}} \times \frac{t_\mathrm{Nuc}^{(0)}}{\tau_n} \simeq 0.064 \, \% \, ,
\end{equation}
where we recall that $t_\mathrm{Nuc}^{(0)} \simeq 245 \, \mathrm{s}$ and $\tau_n = 879.4 \, \mathrm{s}$.
\subsection{Primordial abundances}
The previous results allow to estimate the changes to the primordial abundances. We separate the discussion between the $\He4$ abundance, which is essentially set by the neutron fraction at freeze-out, and the other light element abundances, for which the clock effect affects the nuclear reactions.
\subsubsection{Helium abundance}
The previous study allows us to estimate the change in the $\He4$ abundance. Since most neutrons are converted into $\He4$, by combining Eqs.~\eqref{eq:dxnfo} and \eqref{eq:dxnclock} (``$T_{\nu_e},$ with distortions" case) we get
\begin{equation}
\delta Y_{\mathrm{p}} = \delta X_n^{[\mathrm{Nuc}]} = \delta X_n^{[\mathrm{FO}]} + \delta X_n^{[\Delta t]} \simeq 0.078 \, \% \, ,
\label{eq:deltaYp}
\end{equation}
which is in excellent agreement with the result in Table~\ref{Table:Previous}.
The different values of $Y_{\mathrm{p}}$ depending on the implementations (see Table~\ref{Table:Corrections}) are very well described by this explanation: since the energy density is always the same, $\delta X_n^{[\Delta t]}$ remains identical, while the varying $\delta X_n^{[\mathrm{FO}]}$ (Fig.~\ref{fig:analyzevariations}) controls $\delta Y_{\mathrm{p}}$.
\subsubsection{Other abundances}
We now focus on the other light elements produced during BBN, up to ${}^7{\mathrm{Be}}$. To understand the individual variations of abundances due to incomplete neutrino decoupling, in Table~\ref{Table:LightElements} we separate the final abundances of $\He3$, $\mathrm{T}$, ${}^7{\mathrm{Be}}$, and ${}^7{\mathrm{Li}}$.
\begin{table}[h]
\centering
\begin{tabular}{|l|rrrrr|}
\hline
& \multicolumn{1}{c}{$\mathrm{D}/\mathrm{H}$} & \multicolumn{1}{c}{${\He3}/\mathrm{H}$}& \multicolumn{1}{c}{$\mathrm{T}/\mathrm{H}$} & \multicolumn{1}{c}{${{}^7{\mathrm{Be}}}/\mathrm{H}$} & \multicolumn{1}{c|}{${{}^7{\mathrm{Li}}}/\mathrm{H} $} \\
\hline \hline
$(i/\mathrm{H})^{(0),\infty}$ & $2.41 \times 10^{-5}$ & $1.03 \times 10^{-5}$ & $7.69 \times 10^{-8}$ & $5.17 \times 10^{-10}$ & $2.76 \times 10^{-11}$ \\
$\Delta (i/\mathrm{H})^\infty$ & $9.7 \times 10^{-8}$ & $1.4 \times 10^{-8}$ & $3.3 \times 10^{-10}$ & $- 2.2 \times 10^{-12}$ & $1.2 \times 10^{-13}$ \\
$\delta (i/\mathrm{H})^\infty$ & $0.40 \, \% $ & $0.13 \, \%$ & $0.43 \, \%$ &$- 0.42 \, \%$ & $0.43 \, \%$ \\ \hline
\end{tabular}
\caption[Variation of primordial abundances due to incomplete neutrino decoupling, at the Born level]{Neutrino-induced corrections to the primordial production of light elements other than $\He4$. QED corrections to the plasma thermodynamics are included up to order $\mathcal{O}(e^3)$, and the weak rates are computed at the Born level.
\label{Table:LightElements}}
\end{table}
There are two contributions to the change in the final abundance of an element:
\begin{equation}
\delta (i/\mathrm{H})^{\infty} = \delta X_i^{\infty} - \delta X_\mathrm{H}^{\infty} \simeq \delta X_i^{[\Delta t]} + \delta X_n^{[\mathrm{Nuc}]} \, .
\label{eq:abund_light}
\end{equation}
The variation of the proton final abundance is directly related to $\delta X_n^{[\mathrm{Nuc}]}$ given in Eq.~\eqref{eq:deltaxn}, because an increase of $X_n^{[\mathrm{Nuc}]}$ corresponds to a higher neutron-to-proton ratio and/or less beta decay, and thus less protons. On the other hand, the variation of $X_i^{\infty}$ is entirely encapsulated in the clock effect contribution $\delta X_i^{[\Delta t]}$ (it does not depend on $X_n(T_{\mathrm{Nuc}})$ at first order, since all light elements except $\He4$ only appear at trace level). Indeed, nucleosynthesis consists in elements being produced/destroyed until the reaction rates (which depend only on $T_\gamma$) become too small compared to the Hubble rate~\cite{SmithBBN}. Because of incomplete neutrino decoupling, a given value of $T_\gamma$ is reached sooner and the nuclear reactions have had less time to be efficient. In other words, there is less time to produce or destroy the different elements.\footnote{This argument does not apply to $\He4$ since it is the most stable light element: for such small variations of the expansion rate, almost all neutrons still end up in $\He4$, so $Y_{\mathrm{p}}$ is only affected by $\delta X_n^{[\mathrm{Nuc}]}$.}
\begin{figure}[!ht]
\centering
\includegraphics{figs/evolution_abundances.pdf}
\caption[Evolution of light element abundances, rescaled by their frozen-out value]{\label{fig:evolutionabundances} Evolution of light element abundances computed with \texttt{PRIMAT}, including incomplete neutrino decoupling corrections at the Born approximation level. To compare the evolution for different elements, all abundances are rescaled by their frozen-out value.}
\end{figure}
We can thus understand the values of Table~\ref{Table:LightElements} by looking at the evolution of abundances at the end of nucleosynthesis, shown in Fig.~\ref{fig:evolutionabundances}. All elements except ${}^7{\mathrm{Be}}$ are mainly destroyed when the temperature drops below $T_{\mathrm{Nuc}}$. The very similar evolutions of $\mathrm{D}$, $\mathrm{T}$, and ${}^7{\mathrm{Li}}$ explain their similar values of $\delta X_i^\infty$: their destruction rates go to zero more quickly, resulting in a higher final abundance value. For ${}^7{\mathrm{Be}}$ it is the opposite: it is more efficiently produced than destroyed, and the clock effect reduces the possible amount formed (hence, the negative $\delta X_{{}^7{\mathrm{Be}}}^{\infty}$). Moreover, its evolution is even sharper than that of tritium, and thus we expect $\abs{\delta X_{{}^7{\mathrm{Be}}}^{\infty}} > \delta X_\mathrm{T}^{\infty}$. Finally, $\He3$ has much smaller variations, with a small amplitude of abundance reduction from $T_{\mathrm{Nuc}}$. This explains the comparatively small value of $\delta X_{\He3}^{\infty}$.
To recover the aggregated variations of Table~\ref{Table:Previous} (for $\He3$ and $\mathrm{T}$, and ${}^7{\mathrm{Be}}$ and ${}^7{\mathrm{Li}}$), one performs the weighted average of individual variations. Since $(\He3/\mathrm{H})^\infty \gg (\mathrm{T}/\mathrm{H})^\infty$, the contribution of $\He3$ dominates, and this argument can be immediately applied to ${}^7{\mathrm{Be}}$ and ${}^7{\mathrm{Li}}$.
\subsection{Precision nucleosynthesis}
\label{subsec:full_BBN}
Having thoroughly studied the physics at play by focusing on the Born approximation level, we can now present the results incorporating all weak rates corrections derived in~\cite{Pitrou_2018PhysRept}. These additional contributions (radiative corrections, finite nucleon mass, and weak magnetism) cannot in principle be added linearly, due to nonlinear feedback between them. Concerning incomplete neutrino decoupling, this means that we also include radiative corrections inside the spectral distortion part of the rates: we modify Eq.~\eqref{eq:deltagamma}, following Eqs.~(100) and (103) in~\cite{Pitrou_2018PhysRept}. Since the neutrino sector physics was already "accurate" (given that we used results with QED corrections and flavour mixing), the corrections to the weak rates are the last missing ingredient at the nucleosynthesis level to have the most precise predictions of primordial abundances.
The results, once again for the three implementations of neutrino-induced corrections, are given in Table~\ref{Table:Full_Corrections}.
\begin{table}[!htb]
\centering
\begin{tabular}{|l|cccc|}
\hline
& $Y_{\mathrm{p}}$ & $\mathrm{D}/\mathrm{H} \times 10^5$ & $\He3/\mathrm{H} \times 10^5$ & ${}^7{\mathrm{Li}}/\mathrm{H} \times 10^{10}$ \\
\hline \hline
Inst. decoupling, all corr. & $0.24711$ & $2.4291$ & $1.0379$ & $5.5270$ \\ \hline
$\widehat{T}_\nu$ & $0.24716$ & $2.4381$ & $1.0392$ & $5.5038$ \\
$T_{\nu_e},$ no distortions & $0.24713$ & $2.4380$ & $1.0392$ & $5.5033$ \\
$\bm{T_{\nu_e},}$\textbf{ with distortions} & $\mathbf{0.24721}$ & $\mathbf{2.4384}$ & $\mathbf{1.0393}$ & $\mathbf{5.5045}$\\ \hline
\end{tabular}
\caption[Light element abundances with all corrections included]{Light element abundances, including all weak rate corrections and QED corrections to plasma thermodynamics, for various implementations of neutrino-induced corrections. See Table~\ref{Table:Corrections} for results at the Born approximation level. The final row (in boldface) gives the prediction for primordial abundances in the most accurate framework ; the values were reported in Table~\ref{Table:General_BBN}.
\label{Table:Full_Corrections}}
\end{table}
Compared to the Born approximation level (Table~\ref{Table:Corrections}), the additional corrections result in higher final abundances, as discussed in~\cite{Pitrou_2018PhysRept}. Starting then from a baseline where all of these corrections are included except for incomplete neutrino decoupling, the shift in abundances due to neutrinos is slightly reduced by roughly $- \, 0.03 \, \%$; for instance $\delta Y_{\mathrm{p}} = + \, 0.043 \, \%$ instead of $+ \, 0.076 \, \%$. The other conclusions of the previous sections remain valid: the average temperature implementation is close to the complete one, we explain $Y_{\mathrm{p}}$ through $X_n(T_{\mathrm{Nuc}})$, and the clock effect sources the variations of light elements other than $\He4$.
Since the additional corrections like finite nucleon mass contributions only affect the weak rates and not the energy density, we expect that the only difference compared to the picture at the Born level will lie in $\delta X_n^{[\mathrm{FO}]}$, while $\sigma_\nu$ and $\delta X_i^{[\Delta t]}$ will remain unchanged. This is indeed what we observe in Fig.~\ref{fig:analyzevariationsfull}: the reduction of the neutron fraction at freeze-out due to incomplete neutrino decoupling is enhanced when including all weak rates corrections. Moreover, by comparing Figs.~\ref{fig:analyzevariationsfull} and \ref{fig:analyzevariations} we find
\begin{equation}
\delta X_{n, \mathrm{All}}^{[\mathrm{FO}]} - \delta X_{n, \mathrm{Born}}^{[\mathrm{FO}]} \simeq - 0.03 \, \% \, ,
\end{equation}
which, by inserting this difference into Eqs.~\eqref{eq:deltaYp} and \eqref{eq:abund_light}, explains the results of Table~\ref{Table:Full_Corrections}.
\begin{figure}[!ht]
\centering
\includegraphics{figs/AnalyzeVariationsFullRC_FM_ThRC_QED_Mix_.pdf}
\caption[Neutron fraction variation for different implementations of neutrino-induced corrections (all other weak rate corrections included)]{\label{fig:analyzevariationsfull} Neutron fraction around freeze-out, in the different implementations of neutrino-induced corrections. Compared to Fig.~\ref{fig:analyzevariations}, all weak rate corrections are included.}
\end{figure}
\section*{Concluding remarks} Neutrino decoupling is now included in the standard BBN codes, at least via the $N_{\mathrm{eff}}$ parameter~\cite{AlterBBN,AlterBBNv2} or through the $\mathcal{N}$ heating function~\cite{Parthenope,Parthenope_reloaded,Parthenope_revolutions}. Note that the very small difference between the "$\widehat{T}_\nu$" and "$T_{\nu_e}$, with distortions" (way below the experimental uncertainties) justifies that the $\mathcal{N}$ method can give satisfactory results. However, no change to the weak rates seems to be taken into account in \texttt{AlterBBN}.
Our detailed analysis has evidenced that including incomplete neutrino decoupling leads to an increase of helium-4, deuterium and helium-3 abundances, and a reduction of lithium-7 abundance, in agreement with~\cite{Grohs2015} but disagreeing with~\cite{Mangano2005}.
Can we be satisfied with an approximate treatment of neutrino decoupling in a BBN calculation? Looking at Tables~\ref{Table:General_BBN} and~\ref{Table:Previous}, one can see that the scale of the variations due to incomplete neutrino decoupling (compared to instantaneous decoupling) is below the experimental uncertainties on the primordial abundances --- except for deuterium where the precision has reached the percent level. A possible tension concerning the abundance of deuterium has been suggested~\cite{Pitrou2020}, which shows that an accurate treatment of neutrino decoupling is crucial for precision nucleosynthesis and to make conclusions regarding the cosmological model. Nevertheless, the very small variation of abundances \emph{between the various implementations} shows that a treatment that is not completely exact, for instance via the heating rate $\mathcal{N}$, can be satisfactory. But, given our results that can be provided on demand, and the existence of other neutrino decoupling public codes~\cite{Bennett2021}, one might as well be as precise as possible on this point.
\end{document}
\chapter*{Conclusion}
\addcontentsline{toc}{chapter}{\protect\numberline{}Conclusion}
\setlength{\epigraphwidth}{0.48\textwidth}
\epigraph{Mozart, Beethoven and Chopin never died. They simply became music.}{Dr. Robert Ford, \emph{Westworld} [S01E10]}
Cosmology has entered exciting times, with the launch of terrestrial and spatial telescopes that will push always further our understanding of the Universe, and maybe unveil in the coming decade some of the enduring mysteries that plague the $\Lambda$CDM model: the nature of dark matter and dark energy, the $H_0$ tension, etc. In this era of "precision cosmology", the physics of neutrinos is absolutely essential. Indeed, neutrinos intervene at all stages of cosmological expansion: relativistic in the early Universe, they decouple from the plasma of photons, electrons and positrons precisely when primordial nucleosynthesis begins --- the initial conditions of BBN being dependent on the neutrino distributions! In the late Universe, massive neutrinos can become non-relativistic and affect the formation of large-scale structure. Moreover, the neutrino sector is an immense room for new physics: sterile states, non-standard interactions, ...
The aim of this PhD was to achieve new levels of precision in the study of neutrino evolution in the early Universe, assuming no beyond-the-Standard-Model physics --- except for the crucial phenomenon of flavour oscillations.
We developed a new method to derive the QKE which drives neutrino evolution, namely an \emph{extended-BBGKY hierarchy}: the perturbative expansion of~\cite{SiglRaffelt} is replaced by a well-controlled hierarchy of (un)correlated contributions to the $1-$, $2-$, $\cdots$ $n-$body density matrix. This method had been used to derive the mean-field terms of the QKE in~\cite{Volpe_2013}, in the so-called Hartree-Fock approximation. We went beyond this approximation and included higher order correlations in the molecular chaos ansatz to obtain the collision term (that is all contributions from scattering and annihilations between $\nu$, $\bar{\nu}$ and with $e^-$, $e^+$) and thus the full QKE.
This allowed us to perform a calculation of neutrino decoupling with, for the first time, the \emph{full} collision term (and the aforementioned QED corrections). Thus, we set a new recommended value of the cosmological observable $N_{\mathrm{eff}}$: $N_{\mathrm{eff}} = 3.0440$ with a precision of a few $10^{-4}$. This precision is partly due to the experimental uncertainty on the physical parameters (notably the mixing angle $\theta_{12}$), but mostly the numerical variability depending on the settings of our algorithm. The previous calculations taking into account flavour oscillations, which led to the value $N_{\mathrm{eff}} \simeq 3.045$~\cite{Relic2016_revisited}, did not consider the full collision term: its off-diagonal components were evaluated with a damping approximation. Including this term without any approximation is a real numerical challenge, in particular due to its stiffness and because it scales as $\mathcal{O}(N^3)$ with $N$ the size of the momentum grid. We ensured a reasonable computation time through a major improvement, namely the direct calculation of the Jacobian of the differential system. Our result on $N_{\mathrm{eff}}$ was later confirmed by~\cite{Bennett2021}.
We also introduced an effective description of flavour oscillations that gives results indistinguishable from the ones obtained solving the exact equation. It also substantially reduces the computation time, another key improvement of our code. This approximation relies on the existence of a large separation of scales between the oscillation frequencies and the collision rate, which allows to average over these oscillations. In other terms, the density matrix always remains diagonal in the matter basis (the basis of eigenstates of the Hamiltonian taking into account vacuum and mean-field effects). We named this simplified description the \emph{Adiabatic Transfer of Averaged Oscillations} (ATAO) approximation. Moreover, we used this approximation to get a deeper understanding of some results like the (absence of) effects of the CP phase in standard neutrino decoupling.
Solving the QKE, we obtain the frozen-out distributions of (anti)neutrinos, which in turn give access to parameters like $N_{\mathrm{eff}}$ (cf.~above) or the neutrino energy density parameter today $\Omega_\nu$. Thus, we are in possession of the two parameters that set the various effects of incomplete neutrino decoupling on the earliest probe of the history of the Universe we dispose of --- BBN ---: the distribution of $\nu_e, \, \bar{\nu}_e$ and the energy density parametrized by $N_{\mathrm{eff}}$.
We have also assessed the changes in the primordial abundances of helium, deuterium and lithium due to incomplete neutrino decoupling. First, the light element abundances depend on the expansion rate of the Universe (hence on $N_{\mathrm{eff}}$, via the so-called \emph{clock effect}). Then, the neutron abundance at the beginning of BBN is among other things set by the neutron-to-proton ratio which varies if one changes the distributions of $\nu_e, \bar{\nu}_e$. We have studied in detail how those effects interplayed, comparing their relative contributions and providing analytical estimates when possible. This theoretical work was conducted hand-in-hand with a numerical study, combining our neutrino evolution code and the BBN code \texttt{PRIMAT}. In particular, we were able to resolve an existing discrepancy in the literature between~\cite{Grohs2015} and~\cite{Mangano2005} regarding the variation of deuterium abundance due to incomplete neutrino decoupling.
The presence of non-zero neutrino asymmetries, \emph{a priori} allowed, adds considerable complexity to the physics of neutrino evolution. Indeed, there is now an additional self-interaction mean-field term in the QKE, which dominates throughout a large part of the neutrino decoupling era for asymmetries $\mu/T \in [10^{-3},10^{-1}]$. In line with our work on the resolution of the QKE in the standard case with the full collision term, we extended our code to the asymmetric case. Moreover, we generalized the ATAO approximation to account for self-interactions, which make the Hamiltonian non-linear. This ATAO framework allowed us to analytically recover known results about the collective \emph{synchronous oscillations}, but also to discover that this regime is generally followed by \emph{quasi}-synchronous oscillations with larger frequencies. We have provided numerous analytical and numerical checks of this new result, in the simplified two-flavour case but also in the general three-flavour framework. We further explored the dependency of the final neutrino configuration on the mixing parameters, and notably showed that the CP-violating Dirac phase cannot substantially affect the final $N_{\mathrm{eff}}$ nor the final electronic (anti)neutrino spectrum, and thus should not affect cosmological observables.
\paragraph{Prospects} The coming years will be exciting on the theoretical and experimental levels~\cite{Snowmass_Abazajian}. Neutrino properties will be constrained by cosmology and laboratory searches, these complementary results allowing to build a complete picture of the neutrino sector. On the cosmological side, the PTOLEMY experiment proposal~\cite{Long_PTOLEMY,PTOLEMY2018} which aims at observing directly the C$\nu$B is particularly exciting, as it would provide the first direct observation of the physics of neutrino decoupling, instead of all the "secondary" ones (BBN, effects on CMB, etc.). However, there has been some recent debate on the possibility of using such a setup (based on the measure of the beta decay and absorption processes of tritium bound to graphene), as this binding would lead to fundamental quantum uncertainties on the spectrum of the emitted $e^-$, well above the required energy resolution to detect the C$\nu$B~\cite{Cheipesh:2021fmg,PTOLEMY2022}. On the theoretical side, the results obtained during this PhD pave the way to many more applications. First, in the early Universe, we can explore some new physics, at the edge of the SM or including new mechanisms. For example, it is very important to provide precise constraints on the effect of sterile neutrinos of given masses and mixtures on cosmological observables, as they are often proposed as solutions to anomalies in laboratory experiments. Second, our thorough description of the evolution of primordial asymmetries is only the first step towards establishing new refined limits on the chemical potentials of neutrinos. Finally, the tools we have developed are not \emph{a priori} limited to cosmology, and the study of astrophysical environments such as binary neutron star mergers or core-collapse supernovae, where anisotropies and collective behaviours provide very rich and complex physics, is a clear path forward for research.
\end{document}
\chapter{Neutrinos in cosmology: an overview}
\label{chap:IntroCosmo}
\setlength{\epigraphwidth}{0.63\textwidth}
\epigraph{Our whole universe was in a hot, dense state \\
Then nearly fourteen billion years ago expansion started [...]}{Barenaked Ladies, \emph{Big Bang Theory Theme}}
{
\hypersetup{linkcolor=black}
\minitoc
}
Neutrinos are probably the most exciting particles in the Standard Model: being neutral fermions, they could be Majorana particles, and we \emph{know} (see section~\ref{sec:Intro_massive_nu}) that at least two neutrino states are massive --- a feature not predicted by the Standard Model. If many particle physics experiments have been able to determine the properties of these particles with increasing accuracy in the last decades, there is another promising laboratory that can be used: the Universe itself.
Cosmology indeed provides complementary results, as the imprints left by neutrino evolution on cosmological observables are directly dependent on neutrino properties. In this introductory chapter, we present the main elements of neutrino physics that are relevant to study their evolution in the early Universe. This presentation, which is necessarily limited, only scratches the surface of many topics that are developed in, e.g.~\cite{Neutrino_Cosmology,LesgourguesPastor}.
\section{Elements of standard cosmology}
We describe in this section the key features of the Standard Model of cosmology, introducing the necessary notations and equations for the forthcoming sections and chapters. It is intended as a concise and oriented presentation of cosmology, and we refer to many excellent references such as~\cite{KolbTurner,PeterUzan,WeinbergCosmology,ModernCosmology} for a more complete presentation of cosmology.
The reader familiar with standard cosmology can skip this first section and jump to section~\ref{sec:Intro_Neutrinos}, dedicated to an overview of neutrinos in the early Universe.
\subsection{The homogeneous and isotropic universe}
The Standard Model of cosmology is based on two main assumptions~\cite{PeterUzan}:
\begin{itemize}
\item General Relativity is an adequate theory of gravitation ;
\item the \emph{cosmological principle}: on the largest scales, the Universe is spatially homogeneous and isotropic.
\end{itemize}
This second hypothesis must be understood as a statistical, averaged property on scales typically $\gtrsim 100 \, \mathrm{Mpc}$. Under these assumptions, it can be shown that space-time must be described by a Friedmann-Lemaître-Robertson-Walker (FLRW) geometry, that is with the metric (we use the same conventions as~\cite{KolbTurner,GiuntiKim}):
\begin{equation}
\mathrm{d}{s}^2 = g_{\mu \nu} \mathrm{d}{x^\mu} \mathrm{d}{x^\nu} \equiv \mathrm{d}{t}^2 - a^2(t) \left(\frac{\mathrm{d}{r}^2}{1-Kr^2} + r^2 \mathrm{d}{\theta}^2 + r^2 \sin{\theta}^2 \mathrm{d}{\varphi}^2 \right) \, ,
\end{equation}
where $(t,r,\theta,\varphi)$ are the coordinates, $K=-1,0,+1$ for spaces with negative, zero and positive curvature, and $a(t)$ is the \emph{scale factor}. Such a cosmology is entirely determined by the evolution of $a(t)$, which is given by Einstein theory of General Relativity. It unveils the relationship between the geometry of spacetime (through the metric $g_{\mu \nu}$) and its energy content (through the stress-energy tensor $T_{\mu \nu}$). Einstein field equations read
\begin{equation}
\label{eq:Einstein}
R_{\mu \nu} - \frac12 R g_{\mu \nu} = 8 \pi \mathcal{G} T_{\mu \nu} + \Lambda g_{\mu \nu} \, .
\end{equation}
In this equation $R_{\mu \nu}$ is the Ricci tensor and $R = g^{\mu \nu} R_{\mu \nu}$ the Ricci scalar, $\mathcal{G}$ the gravitational constant, and $\Lambda$ the \emph{cosmological constant}. Note that we use natural units in which $\hbar = c = k_B = 1$. We assume that basics of General Relativity are known to the reader, and refer for instance to~\cite{PeterUzan,Wald,Carroll_RG}.
\subsubsection{Dynamical equations}
Thanks to the cosmological principle, the Universe can be described as a collection of perfect fluids, for which the energy-momentum tensor reads
\begin{equation}
T^{\mu \nu} = (\rho + P) u^\mu u^\nu - P g^{\mu \nu} \, ,
\end{equation}
where $\rho$ is the energy density, $P$ is the pressure, and $u^\mu = \mathrm{d}{x^\mu}/\mathrm{d} s$ is the four-velocity of the fluid. In the comoving frame where the perfect fluid is at rest, $u^\mu = (1,0,0,0)$ and one has
\begin{equation}
{T^\mu}_\nu = \mathrm{diag}(\rho,-P,-P,-P) \, .
\end{equation}
We see then from~\eqref{eq:Einstein} that it is possible to interpret the cosmological constant as the energy density of the vacuum, through
\begin{equation}
\rho_\Lambda = \frac{\Lambda}{8 \pi \mathcal{G}} \qquad \text{and} \qquad P_\Lambda = - \rho_\Lambda \, .
\end{equation}
We detail below that this \emph{a priori} peculiar \emph{negative} pressure amounts to the fact that the cosmological constant corresponds to a constant energy density.
\noindent We can now obtain the equations governing the dynamics of expansion:\footnote{Although we only quote the results, let us give here the non-vanishing Christoffel symbols of the FLRW metric, which we write $\mathrm{d}{s}^2 = \mathrm{d}{t}^2 - a^2(t) \gamma_{ij} \mathrm{d}{x^i}\mathrm{d}{x^j}$:
\begin{equation*}
{\Gamma^{0}}_{ij} = \dot{a} a \gamma_{ij} \quad , \quad {\Gamma^{i}}_{0j} = \frac{\dot{a}}{a} {\delta^i}_j \quad , \quad {\Gamma^{i}}_{jk} = \frac12 \gamma^{il}\left(\partial_j \gamma_{kl} + \partial_k \gamma_{jl} - \partial_l \gamma{jk} \right) = ^{(3)}\!{\Gamma^i}_{jk} \, .
\end{equation*}
This allows to compute the non-zero components of the Ricci tensor,
\begin{equation*}
R_{00} = - 3 \frac{\ddot{a}}{a} \quad , \quad R_{ij} = \left(\frac{\ddot{a}}{a} + 2 H^2 + \frac{2 K}{a^2} \right) a^2 \gamma_{ij} \quad \text{and the Ricci scalar} \quad R = - 6 \left( \frac{\ddot{a}}{a} + H^2 + \frac{K}{a^2} \right) \, .
\end{equation*}
}
\begin{itemize}
\item from the $0-0$ component of~\eqref{eq:Einstein}, one gets the famous \emph{Friedmann equation}:
\begin{equation}
\label{eq:Friedmann}
\boxed{\left(\frac{\dot{a}}{a}\right)^2 \equiv H^2 = \frac{8 \pi \mathcal{G}}{3} \rho - \frac{K}{a^2} } \, ,
\end{equation}
where we defined the \emph{Hubble parameter} $H \equiv \dot{a}/a$. Introducing the Planck mass $M_\mathrm{Pl} \equiv \mathcal{G}^{-1/2} \simeq 1.22 \times 10^{19} \, \mathrm{GeV}$, we have
\[ H^2 = \frac{8 \pi}{3 M_\mathrm{Pl}^2} \rho - \frac{K}{a^2} \, . \]
We will also sometimes use the reduced Planck mass $m_\mathrm{Pl} \equiv M_\mathrm{Pl}/\sqrt{8 \pi}$. The \emph{critical density} is the energy density corresponding to a flat ($K=0$) Universe today, $\rho_\mathrm{crit} \equiv 3 H_0^2 / 8 \pi \mathcal{G}$. The energy density parameter is then defined as $\Omega = \rho/\rho_\mathrm{crit}$, the different values in the standard model of cosmology being given below.
\item from the $i-i$ component of~\eqref{eq:Einstein}, one gets
\begin{equation}
\label{eq:acceleration}
\frac{\ddot{a}}{a} = - \frac{4 \pi \mathcal{G}}{3}(\rho + 3 P) \, .
\end{equation}
\item the energy-momentum conservation $\nabla_{\mu} T^{\mu \nu} = 0$ ($\nabla_\mu$ being the covariant derivative) reduces to
\begin{equation}
\label{eq:eq_conservation}
\boxed{\dot{\rho} + 3 H (\rho + P) = 0} \, .
\end{equation}
\end{itemize}
These three equations are not independent, which is a consequence of Bianchi identities. The most often used equations are then~\eqref{eq:Friedmann} and~\eqref{eq:eq_conservation}.
\subsubsection{Some solutions}
\paragraph{Friedmann equation} Perfect fluids are characterized by their \emph{equation of state} $P = w \rho$, where $w$ is independent of time. With such a relation, we see from~\eqref{eq:eq_conservation} that the energy density evolves as $\rho \propto a^{-3(1+w)}$. We distinguish three examples of interest:
\begin{itemize}
\item \emph{radiation} ($w = 1/3$), for which $\rho \propto a^{-4}$,
\item pressureless \emph{matter} ($w = 0$), for which $\rho \propto a^{-3}$,
\item \emph{dark energy} ($w = -1$), for which $\rho = \mathrm{const}$.
\end{itemize}
\paragraph{The $\bm{\Lambda}$CDM model} In the standard model of cosmology, which accounts extremely well for an incredible variety of observations,\footnote{There are of course some tensions, like the $H_0$ tension between "early" and "late" measurements of the Hubble constant. We do not discuss such limits of the $\Lambda$CDM model in this introduction.} the constituents of the Universe are~\cite{PDG}:
\begin{itemize}
\item baryonic matter (the "usual" matter), which behaves as pressureless matter, amounting today to $\Omega_b^0 \simeq 0.049$,
\item photons which behave as radiation and amounting to $\Omega_r^0 \sim 10^{-4}$,
\item neutrinos, which are the only known particles in the Standard Model which were ultrarelativistic at early times (during Big Bang Nucleosynthesis and Cosmic Microwave Background formation) and thus behaving as radiation, and are non-relativistic today (at least for two eigenstates, cf.~section~\ref{subsec:Omeganu}). Their contribution $\Omega_\nu^0$ is thus split between $\Omega_r^0$ and the total matter part $\Omega_m^0$,
\item "dark" components, which account for the missing energy (the baryonic and photon components which come from the Standard Model of particle physics only represent $5 \, \%$ of the total energy budget):
\begin{itemize}
\item a cosmological constant $\Lambda$ corresponding to \emph{dark energy}, which dominates the energy density today with $\Omega_\Lambda^0 \simeq 0.685$,
\item \emph{cold dark matter}, necessarily non-baryonic and whose nature is still unknown today, which amounts for $\Omega_c^0 \simeq 0.265$.
\end{itemize}
\end{itemize}
These last two components are at the origin of the name "$\Lambda$CDM" of the model.
We plot on Figure~\ref{fig:rho_scalefactor} the evolution of the energy density for the different constituents of the Universe. Given the different scalings of $\rho$ with the scale factor, it appears that, although today the dark energy is dominating the Universe, this was not the case in the past. In the early Universe, we were in the so-called \emph{radiation-dominated} era, until the energy densities of radiation and matter became equal ("matter-radiation equality") at the scale factor $a_\mathrm{eq}$. We then entered the \emph{matter-dominated} era, until the recent period of accelerated expansion driven by the cosmological constant, the transition taking place at the scale factor $a_\Lambda$ at which the acceleration of the expansion was zero.\footnote{Indeed, one can see from~\eqref{eq:acceleration} that the dominance of matter makes the Universe decelerate, while dark energy drives an accelerated expansion. The transition thus corresponds to a vanishing acceleration.}
\paragraph{Scale factor and time} Assuming that the Universe is flat ($K=0$), we can solve Friedmann equation~\eqref{eq:Friedmann} for a fluid of equation of state $P = w \rho$:
\begin{equation}
\dot{a} \propto a^{-(1+3w)/2} \quad \implies \quad a(t) \propto t^{\frac{2}{3(1+w)}} \, .
\end{equation}
In particular, we find for radiation the important relationship between the Hubble rate and cosmic time:
\begin{equation}
\label{eq:hubble_radiation}
a(t) \propto \sqrt{t} \quad \text{hence} \quad H = \frac{1}{2 t} \, .
\end{equation}
\vspace{-0.5cm}
\begin{figure}[!h]
\centering
\includegraphics{figs/rho_scale_factor.pdf}
\caption[Energy density as a function of the scale factor for different constituents]{\label{fig:rho_scalefactor} Energy density as a function of the scale factor for nonrelativistic matter ($\propto a^{-3}$), radiation ($\propto a^{-4}$) and a cosmological constant ($= \mathrm{const.}$). At early times, the energy density of the Universe is dominated by the radiation component.}
\end{figure}
\subsubsection{Redshifting of momenta}
We end this section with a very useful result valid in FLRW spacetime: the so-called "redshifting" of physical momentum as the Universe expands.
Let us show that the physical linear momentum of a free-falling particle decreases as $1/a$ as the Universe expands, following~\cite{Neutrino_Cosmology,ModernCosmology}. We start with the geodesic equation
\begin{equation}
\label{eq:geodesic}
\frac{\mathrm{d}^2{x^\mu}}{\mathrm{d} \lambda^2} + \Gamma^{\mu}_{\alpha \beta} \frac{\mathrm{d}{x^\alpha}}{\mathrm{d} \lambda} \frac{\mathrm{d}{x^\beta}}{\mathrm{d} \lambda} = 0 \, ,
\end{equation}
where the use of the affine parameter $\lambda$ instead of the proper time $\tau$ allows to treat at the same time massive and massless particles. It is implicitly defined such that the four-momentum $P^\mu = (E, P^i)$ reads
\begin{equation}
P^\mu \equiv \frac{\mathrm{d}{x^\mu}}{\mathrm{d} \lambda} \, .
\end{equation}
Note that the $0-$component of this definition gives $\mathrm{d} / \mathrm{d}{\lambda} = E \mathrm{d} / \mathrm{d}{t}$. The $0-$component of the geodesic equation~\eqref{eq:geodesic} can thus be written
\begin{equation}
E \frac{\mathrm{d} E}{\mathrm{d} t} + \Gamma^{0}_{ij} P^i P^j = 0 \, ,
\end{equation}
where we use the fact that only the spatial components of $\Gamma^0_{\alpha \beta}$ are non-zero. Since $\Gamma^0_{ij} = - (\dot{a}/a) g_{ij} = \dot{a} a \gamma_{ij}$, we have
\begin{equation}
\label{eq:geodesic_temp}
E \frac{\mathrm{d} E}{\mathrm{d} t} +\dot{a} a \gamma_{ij} P^i P^j = 0 \, .
\end{equation}
The norm of the four-momentum is $m^2 = P^\mu P_\mu = E^2 - a^2 \gamma_{ij} P^i P^j$. It is customary to introduce the \emph{physical} linear momentum $p^i \equiv a(t) P^i$, which allows to rewrite the norm of the four-momentum $E^2 - \lvert \vec{p} \rvert^2 = m^2$ with $\lvert \vec{p} \rvert^2 \equiv \gamma_{ij} p^i p^j$ (as in flat spacetime). Deriving this relation gives (note that the calculation is also valid if $m=0$)
\begin{equation}
\label{eq:norm_4momentum}
E \frac{\mathrm{d} E}{\mathrm{d} t} - \lvert \vec{p} \rvert \frac{\mathrm{d} \lvert \vec{p} \rvert}{\mathrm{d} t} = 0 \, .
\end{equation}
Rewriting~\eqref{eq:geodesic_temp} with the physical momentum finally leads to
\begin{equation}
E \frac{\mathrm{d} E}{\mathrm{d} t} +\frac{\dot{a}}{a} \lvert \vec{p} \rvert^2 = 0 \quad \xRightarrow[\text{using \eqref{eq:norm_4momentum}}]{} \quad \frac{1}{\lvert \vec{p} \rvert} \frac{\mathrm{d} \lvert \vec{p} \rvert}{\mathrm{d} t} = - \frac{\dot{a}}{a} \, .
\end{equation}
Therefore, we have proven that
\begin{equation}
\label{eq:scaling_p}
\boxed{\lvert \vec{p} \rvert \propto a^{-1}} \, .
\end{equation}
\subsection{Equilibrium thermodynamics}
Having described gravity in the homogeneous Universe, we must now turn to the equations governing matter and radiation. The statistical properties of the particles filling the Universe are described by their \emph{distribution functions} $f(p,t)$, which give the number of particles of momentum $p$ at time $t$ (there is no dependence on space nor on the direction of $\vec{p}$ thanks to homogeneity and isotropy):
\begin{equation}
\mathrm{d} N(p,t) = f(p,t) \frac{4 \pi p^2 \mathrm{d}{p}}{(2\pi)^3} \, .
\end{equation}
The number density $n$, energy density $\rho$, and pressure $P$ of a dilute, weakly-interacting of a gas of particles with $g$ internal degrees of freedom (for example, $2$ for photons or charged leptons) and distribution function $f(p,t)$ read
\begin{subequations}
\label{eq:thermo_intro}
\begin{align}
n &= \frac{g}{2 \pi^2} \int_{0}^{\infty}{p^2 \mathrm{d}{p} f(p,t)} \, , \\
\rho &= \frac{g}{2 \pi^2} \int_{0}^{\infty}{E(p) p^2 \mathrm{d}{p} f(p,t)} \, , \\
P &= \frac{g}{2 \pi^2} \int_{0}^{\infty}{\frac{p^2}{3 E(p)} p^2 \mathrm{d}{p} f(p,t)} \, . \label{eq:pressure_general}
\end{align}
\end{subequations}
If the reaction rates of a particle species are high enough (cf.~the discussion on decoupling below), it will be maintained in \emph{kinetic} equilibrium, such that its distribution function reads
\begin{equation}
f(p) = \frac{1}{\displaystyle e^{(E-\mu)/T} \pm 1} \, ,
\end{equation}
with $\mu$ the chemical potential and $T$ the temperature of the species. The $+$ sign is for fermions (Fermi-Dirac (FD) distribution), the $-$ sign for bosons (Bose-Einstein (BE) distribution). If this particle species is in \emph{chemical} equilibrium, there is an additional constraint on the chemical potentials, namely, the reaction $a + b \leftrightarrow c + d$ implies $\mu_a + \mu_b = \mu_c + \mu_d$. If this reaction is an elastic scattering, this is trivial since the incoming and outgoing particles are identical.\footnote{That is why we should rather talk about chemical equilibrium \emph{with} a given system. In contrast, kinetic equilibrium is an "intrinsic" property: self-interactions can maintain equilibrium spectra even if there is no external species to be at equilibrium with.} An interesting case is particle/antiparticle annihilation: for instance, annihilations into pairs of photons will impose $\mu = - \bar{\mu}$ since $\mu_\gamma = 0$.
Let us give the explicit results for the different thermodynamic quantities in the relativistic limit $T \gg m$, for non-degenerate particles $T \gg \mu$:
\begin{equation}
\label{eq:thermo_quantities_relat}
n = \left\{ \begin{aligned} &g \frac{\zeta(3)}{\pi^2} T^3 &\text{(BE)} \\
&g \frac34 \frac{\zeta(3)}{\pi^2} T^3 &\text{(FD)} \end{aligned} \right.
\qquad , \qquad
\rho = \left\{ \begin{aligned} &g \frac{\pi^2}{30} T^4 &\text{(BE)} \\
&g \frac78 \frac{\pi^2}{30} T^4 &\text{(FD)} \end{aligned} \right.
\qquad , \qquad
P = \frac{\rho}{3} \, ,
\end{equation}
while in the non-relativistic limit ($m \gg T$), the results are the same for fermions and for bosons:
\begin{equation}
\label{eq:thermo_quantities_NR}
n = g \left(\frac{mT}{2 \pi}\right)^{3/2} e^{-(m-\mu)/T} \qquad , \qquad \rho = m n \qquad , \qquad P = n T \ll \rho \, .
\end{equation}
These results justify the equations of state $w=1/3$ for radiation and $w=0$ for non-relativistic matter.
\paragraph{Entropy} One can show, combining the conservation equation~\eqref{eq:eq_conservation} and the derivative of pressure~\eqref{eq:pressure_general} with respect to temperature (with an equilibrium distribution), that the \emph{entropy density}
\begin{equation}
\label{eq:def_entropy}
s \equiv \frac{\rho + P - \mu n}{T} \, ,
\end{equation}
satisfies
\begin{equation}
\label{eq:entropy_evo}
\mathrm{d}{(sa^3)} = - \frac{\mu}{T} \mathrm{d}{(na^3)} \, .
\end{equation}
Therefore, for non-degenerate matter ($\mu/T \ll 1$) or when it is neither destroyed nor created ($\mathrm{d}{(n a^3)} = 0$), the product $s a^3$ is constant.
Note that these relations can be deduced from standard equilibrium thermodynamics. Indeed, the Gibbs free energy (or free enthalpy) $G = U + PV - TS$ is a function of $(T,P,N)$ thanks to the property of Legendre transforms: starting from $U(S,V,N)$, the two Legendre transforms via $+PV$ and $-TS$ change the natural variables to $(T,P,N)$, and the fundamental thermodynamic identity becomes
\begin{equation}
\label{eq:thermo_identity}
\mathrm{d}{U} = T \mathrm{d}{S} - P \mathrm{d}{V} + \mu \mathrm{d}{N} \qquad \implies \qquad \mathrm{d}{G} = - S \mathrm{d}{T} + V \mathrm{d}{P} + \mu \mathrm{d}{N} \, .
\end{equation}
However, $G$ is an additive function, and its only additive variable is $N$, therefore $G(T,P,N) = \mu(T,P) N$. We thus deduce that the entropy is $S = (U + PV - \mu N)/T$, which gives~\eqref{eq:def_entropy} after dividing by the volume. Moreover, using the identity~\eqref{eq:thermo_identity} for a comoving volume $V \propto a^3$, we get $T \mathrm{d}{(s a^3)} = \mathrm{d}{(\rho a^3)} + P \mathrm{d}{a^3} - \mu \mathrm{d}{(n a^3)}$, but the first two terms on the right-hand side cancel according to the conservation equation~\eqref{eq:eq_conservation}.
In the following section, we discuss some consequences of the expansion of the Universe. The conservation of entropy plays an important role as it gives directly some information on the evolution of the temperature.
\subsection{Thermal history of the Universe}
In short, the history of the Universe in the hot Big Bang model is the history of its cooling as it expands. This cooling has several consequences, and we discuss here two crucial ones to understand neutrino evolution in the (early) Universe.
\subsubsection{Nonrelativistic transition and entropy transfer}
It is useful to write the entropy density of the plasma as a function of the photon temperature $T_\gamma$:
\begin{equation}
s_\text{pl} \equiv \frac{2 \pi^2}{45} g_s(T_\gamma) T_\gamma^3 \, .
\end{equation}
Let us assume that we are in a non-degenerate case ($\mu =0$ for all species), such that the entropy is conserved according to~\eqref{eq:entropy_evo}: $s_\text{pl} a^3 = \text{const}$. We thus obtain the very important result:
\begin{equation}
\label{eq:gs}
\boxed{T_\gamma \propto g_s^{-1/3} a^{-1}} \, .
\end{equation}
Whenever $g_{s}$ is constant, the temperature decreases as $a^{-1}$. However, it is possible that particles in the plasma become non-relativistic, in which case their entropy is exponentially suppressed (in other words, they do not contribute anymore to $g_s$) --- cf.~the integrals given in~\eqref{eq:thermo_quantities_relat} and~\eqref{eq:thermo_quantities_NR}. The conservation of $s_\text{pl} a^3$ then shows that entropy is \emph{transferred} to the other species. The mechanism behind this transfer is the displacement of the equilibrium of the reaction $X + \bar{X} \leftrightarrow \gamma + \gamma $ towards the right when the temperature gets below $m_X$, since the average energy of photons is then too small to create $X-\bar{X}$ pairs. This temperature threshold is precisely the one of the non-relativistic transition, which is why we will equivalently talk about non-relativistic transition and entropy transfer, or particle/antiparticle annihilation.
We show a concrete example in section~\ref{sec:Intro_Neutrinos} when we discuss electron/positron annihilations and the associated reheating of photons.
\subsubsection{Decoupling}
The second consequence of the cooling of the Universe is the \emph{decoupling} of species when their interaction rate becomes too small compared to the expansion rate. Below the decoupling temperature, they interact too little to remain in thermal contact with other species. As a rule of thumb, we say that decoupling occurs when\footnote{Another way to justify this is to say that the heat bath temperature varies as $T_\gamma \propto a^{-1}$ (we neglect a variation of $g_s$ for this argument), such that $\dot{T}_\gamma/T_\gamma = - H$. Therefore, the relation~\eqref{eq:condition_decoupling} corresponds to the moment when the interactions are not fast enough to adjust to the changing temperature.}
\begin{equation}
\label{eq:condition_decoupling}
\frac{\Gamma}{H} \sim 1 \, ,
\end{equation}
with $\Gamma = n \langle \sigma v \rangle$, where $n$ is the number density of target particles, $\sigma$ is the cross-section and $v \sim 1$ the relative velocity (in the ultrarelativistic case). The angle brackets denote thermal averaging. This expression shows why decoupling occurs when the temperature decreases: the interactions may become too weak, or the target density can be suppressed (this is the case after recombination and thus for photon decoupling at $T \sim 0.3 \, \mathrm{eV}$).
\paragraph{Temperature evolution of a decoupled species} Once they are decoupled from the plasma, particles are free-streaming and their distribution functions are \emph{frozen}: if we write with a subscript $_D$ the quantities at decoupling, we have
\begin{equation}
\label{eq:temp_decoup}
f(p,t) = f(p_D, t_D) = f\left(\frac{a(t)}{a_D}p,t_D\right) \, ,
\end{equation}
where we have used the scaling relation~\eqref{eq:scaling_p}. In general, we cannot define an effective temperature and an effective chemical potential,\footnote{Such quantities have to be \emph{effective}, since equilibrium is not maintained anymore by interactions. However this is not in contradiction with the fact that, in the ultra- and non-relativistic limits, equilibrium distributions are maintained.} except in the two following limits~\cite{GiuntiKim}.
\begin{itemize}
\item If the particles are ultrarelativistic (and non-degenerate) at decoupling (which is the case for massless particles), we have from~\eqref{eq:temp_decoup} and using $E=p$,
\begin{equation}
\label{eq:decouple_massless}
f(p,t) = \frac{1}{\displaystyle e^{\frac{a}{a_D}p/T_D} \pm 1} = \frac{1}{\displaystyle e^{p/T} \pm 1} \quad \text{with} \quad \boxed{T = T_D \frac{a_D}{a} \propto a^{-1}} \, .
\end{equation}
The particles keep a relativistic equilibrium distribution with an effective temperature scaling as $a^{-1}$. Note that even if, at some point, the particles become nonrelativistic ($m \sim T$), the spectrum keeps this shape.
\item If the particles are nonrelativistic at decoupling, we can simplify $E_D = \sqrt{p_D^2 + m^2} \simeq m + (p^2/2 m)$, hence,
\begin{align}
f(p,t) = \frac{1}{\displaystyle e^{(E_D - \mu_D)/T_D} \pm 1} &\simeq e^{(\mu_D - m)/T_D} e^{-p_D^2/(2mT_D)} \nonumber \\
&\equiv e^{(\mu -m)/T} e^{-p^2/(2mT)} \, ,
\end{align}
where, using once again the scaling relation $p = p_D a_D/a$, we define the effective temperature
\begin{equation}
\boxed{T = T_D \left(\frac{a_D}{a}\right)^2 \propto a^{-2}} \, ,
\end{equation}
and the effective chemical potential
\begin{equation}
\mu =m + (\mu_D -m)\frac{T}{T_D} = m + (\mu_D -m)\left(\frac{a_D}{a}\right)^2 \, .
\end{equation}
Even if at decoupling $\mu_D=0$, it cannot remain equal to zero later.
\end{itemize}
\subsubsection{A (very) brief thermal history of the Universe}
If the number of relativistic degrees of freedom $g_s$ is constant, \eqref{eq:gs} shows that the temperature of the plasma (what we usually call "the temperature of the Universe") decreases as $a^{-1}$. As the Universe cools down, equilibrium between species can no longer be maintained, and massive particles become non-relativistic. There are also very high-energy phenomena that we did not discuss such as the electroweak phase transition. A summary of the major events that are predicted by the standard model of cosmology is presented in Table~\ref{Table:chrono_universe}. The earliest experimental probe of this model is Big Bang Nucleosynthesis (BBN), which we discuss in section~\ref{subsec:intro_BBN}. The observation of the Cosmic Microwave Background (CMB), that is photons that decoupled from electrons 380 000 years after the Big Bang, has been another decisive argument towards the validation of this standard model.
\renewcommand{\arraystretch}{1.2}
\begin{table}[!htb]
\centering
\begin{tabular}{|M{4cm} |M{3cm} | M{6cm} |}
\hline
$\sim$ Age of the Universe & Temperature $(\mathrm{K})$ & Major event(s) \\
\hline \hline
$< 10^{-43} \, \mathrm{s} $& $> 10^{32}$ & ??? \\
$ 10^{-43} - 10^{-35} \, \mathrm{s} $& $ 10^{32} - 10^{28} $ & Period of inflation \\
$ 10^{-35} - 10^{-12} \, \mathrm{s} $& $ 10^{28} - 10^{16} $ & Generation of matter/antimatter asymmetry \\
$ 10^{-12} \, \mathrm{s}$ & $10^{16}$ & Electroweak phase transition \\
$ 10^{-4} \, \mathrm{s} $& $ 10^{12} $ & Quark-hadron transition, $\mu^+ \mu^-$ annihilation \\
$ \bm{1 - 10^2 \, \mathrm{s}} $ & $\bm{\sim 10^{10} \, (\sim 1 \, \mathrm{MeV})}$ & \textbf{Neutrino decoupling, $\bm{e^+ e^-}$ annihilation, BBN} \\
$10^5 \, \mathrm{yr}$ & $4000$ & Recombination, formation of the Cosmic Microwave Background \\
$2 \times 10^5 - 10^9 \, \mathrm{yr}$ & & Galaxy formation \\
$13.7 \times 10^9 \, \mathrm{yr} $ & $2.73$ & Today \\ \hline
\end{tabular}
\caption[(Very) short history of the Universe]{Major events occurring during the expansion of the Universe in the hot Big Bang model (adapted from~\cite{MohapatraPal}). Note that the values given for the epoch of inflation are particularly model-dependent.
\label{Table:chrono_universe}}
\end{table}
As emphasized in Table~\ref{Table:chrono_universe}, the period during which the temperature of the Universe was about $1 \, \mathrm{MeV}$ (the so-called \emph{"MeV era"}) is eventful. Indeed, we show in the next section that neutrinos decouple only shortly before electrons/positrons annihilate, suggesting a partial overlap between these two phenomena. Moreover, this era also marks the beginning of BBN, meaning that neutrino decoupling influences the primordial abundances.
\section{Neutrinos in the early Universe}
\label{sec:Intro_Neutrinos}
In this section, we give an introduction to the main events summarized in Table~\ref{Table:chrono_universe} that take place during the MeV age: neutrino decoupling, $e^\pm$ annihilations, and primordial nucleosynthesis. The results presented here are standard, and will be the basis of the work presented in chapters~\ref{chap:Decoupling} and~\ref{chap:BBN}.
\noindent During this period, the constituents of the Universe are:
\begin{itemize}
\item a QED plasma of electrons, positrons and photons, tightly coupled by QED interactions (note that heavier charged leptons, $\mu^\mp$ and $\tau^\mp$, have annihilated),
\item a bath of neutrinos and antineutrinos, coupled to the QED plasma via weak interactions with electrons and positrons,
\item baryons (initially neutrons and protons which combine later on to form light elements during BBN) in negligible abundance compared to the leptons, as shown by the latest measurement of the baryon-to-photon ratio $\eta = n_b/n_\gamma \simeq 6.1 \times 10^{-10}$~\cite{Fields:2019pfx}.
\end{itemize}
We emphasize that the calculations presented in this section do not take into account all the physics known to play a role at this epoch, as neutrino oscillations (see section~\ref{sec:Intro_massive_nu}) are discarded. Full calculations including these phenomena and providing an in-depth analysis of their effects are the subject of this thesis.
\subsection{Instantaneous neutrino decoupling}
To roughly estimate the temperature of neutrino decoupling via~\eqref{eq:condition_decoupling}, we need to compare $\Gamma$ and $H$. In the early universe, neutrinos are kept in equilibrium via weak interaction processes like $\nu + e^{-} \to e^{-} + \nu$. The cross-section of such processes scales as $\sigma \sim G_F^2 T^2$, the particle density (for ultrarelativistic electrons) as $n_e \sim T^3$ (see~\eqref{eq:thermo_quantities_relat} above) and the relative velocity is $v \sim 1$. Hence the interaction rate \[\Gamma = n_e \langle \sigma v \rangle \sim G_F^2 T^5 \, .\]
From Eq.~\eqref{eq:Friedmann}, the Hubble rate is, in the radiation era,
\begin{equation*}
H = \sqrt{\frac{8 \pi}{3 M_\mathrm{Pl}^2} \rho_\mathrm{rad}} = \sqrt{\frac{8 \pi}{3 M_\mathrm{Pl}^2} g_* \frac{\pi^2}{30} T^4} \sim \sqrt{g_*} \frac{T_\gamma^2}{M_\mathrm{Pl}} \, ,
\end{equation*}
where $g_{*}$ is the number of relativistic degrees of freedom, defined such that
\begin{equation}
\rho_\text{rad} = \frac{\pi^2}{30} g_*(T_\gamma) T_\gamma^4 \, .
\end{equation}
At this time, the relativistic species in the Universe are photons (with two helicity states), three species of neutrinos and antineutrinos (each with one helicity state), plus electrons and positrons (with two spin states), thus
\begin{equation}
\label{eq:gstar}
g_* = 2 + 3\times 2 \times \frac{7}{8} + 2 \times 2 \times \frac{7}{8} = \frac{43}{4} = 10.75 \, .
\end{equation}
The ratio of the interaction rate to the expansion rate is thus
\begin{equation}
\frac{\Gamma}{H} \sim \frac{G_F^2 T^5}{\sqrt{g_*} \, T^2/M_\mathrm{Pl}} \simeq \left(\frac{T}{1 \ \mathrm{MeV}}\right)^3 \, .
\end{equation}
Therefore, at $T_{\nu D} \sim 1 \, \mathrm{MeV}$, there are not enough collisions compared to the expansion of the Universe and neutrinos subsequently decouple from the electromagnetic plasma. Once it has decoupled, the fluid of neutrinos and antineutrinos is called the \emph{Cosmic Neutrino Background} (C$\nu$B): if it could be directly detected, it would give a snapshot of the Universe as it was a few dozens of seconds after the Big Bang, a tremendous jump in the past compared to the CMB, which was formed 380 000 years later...
In the \emph{instantaneous decoupling approximation}, one considers that at $T_{\nu D}$, all neutrinos suddenly decouple. From this moment onward, their distribution function remains a Fermi-Dirac distribution~\eqref{eq:decouple_massless}
\[f_\nu(p) = \frac{1}{e^{p/T_\nu} +1} \qquad ; \qquad T_\nu \propto a^{-1} \, .\]
In the electromagnetic plasma, the equilibrium of the reaction $e^+ + e^- \leftrightarrow \gamma + \gamma$ gets very displaced on the right when the temperature drops below $m_e \simeq 0.511 \ \mathrm{MeV}$. Electrons and positrons annihilate and their entropy is transferred to the gas of photons, whose temperature will thus decrease slower than $a^{-1}$.
Let us estimate the final ratio of the photon-to-neutrino temperature once the $e^+ e^-$ annihilation is over. Using~\eqref{eq:gs}, the conservation of the entropy of the electromagnetic plasma reads\footnote{We put a superscript $^{\mathrm{(pl)}}$ to highlight the fact that neutrinos do not interact with the plasma anymore, and are thus not counted in $g_s^{(\mathrm{pl})}$.}
\[g_{s}^{(\mathrm{pl})} \times T_\gamma^3 a^3 = \mathrm{cst} \implies g_{s}^{(\mathrm{pl})} \times \left(\frac{T_\gamma}{T_\nu}\right)^3 = \mathrm{cst} \, ,\]
since $T_\nu \propto a^{-1}$. Long before the annihilation of electrons and positrons (but after neutrino decoupling), $T_\gamma = T_\nu$ and the relativistic species are photons and $e^\pm$ pairs, so $g_{s}^{(\mathrm{pl})} = 2 + 2 \times 2 \times \frac78 = \frac{11}{2}$. After the annihilation, there are only photons left and $g_{s}^{(\mathrm{pl})} = 2$. Therefore, long after decoupling, the ratio of temperatures is
\begin{equation}
\label{eq:TgTnu}
\boxed{\frac{T_\gamma}{T_\nu} = \left(\frac{11}{4}\right)^{1/3} \simeq 1.40} \ .
\end{equation}
This is of course an approximate result, for various reasons. First, since the temperatures of neutrino decoupling and $e^+ e^-$ annihilation are close, neutrinos are not fully decoupled when the annihilation takes place. This leads to a small "reheating" of neutrinos compared to the instantaneous decoupling (ID) limit. Historically, this has been parameterized in the following way: should ID be true, then after $e^+ e^-$ annihilation the radiation energy density would read
\begin{equation}
\rho_\text{rad} = \left(1 + 3 \times \frac78 \left(\frac{4}{11}\right)^{4/3}\right) \rho_\gamma \, ,
\end{equation}
where we used the expressions~\eqref{eq:thermo_quantities_relat} and the ratio of temperatures~\eqref{eq:TgTnu}. The departure from this "ideal" picture is defined such that $\rho_\text{rad}$ reads
\begin{equation}
\label{eq:intro_def_Neff}
\rho_\text{rad} = \left(1 + N_{\mathrm{eff}} \times \frac78 \left(\frac{4}{11}\right)^{4/3}\right) \rho_\gamma \, ,
\end{equation}
where the parameter $N_{\mathrm{eff}}$ is called\footnote{The key is in "effective": there are still exactly 3 active neutrinos, but their energetic contribution is equivalent to the one of not exactly 3 instantaneously decoupled neutrinos. In addition, note that any beyond-the-Standard-Model relativistic species that would contribute to $\rho_\text{rad}$ is taken into account in $N_{\mathrm{eff}}$.} the \emph{effective number of neutrino species}. It represents the (non-integer) number of instantaneously decoupled neutrinos that would have the same energy density as the actual 3 active neutrinos. It is a convenient cosmological observable as it encapsulates all the information on the energy density during the radiation-dominated era. Due to the overlap between $e^+ e^-$ annihilation and neutrino decoupling, we expect $N_{\mathrm{eff}} > 3$. It should be noted that $N_{\mathrm{eff}}$ is one of the main cosmological quantities we will be interested in throughout this manuscript.
Another limitation to the instantaneous decoupling limit is the fact that the more energetic neutrinos will remain in thermal contact with the plasma longer than the low-energy ones, which should source \emph{spectral distortions} of the neutrino distribution functions. Finally, the different flavours of neutrinos do not have the same coupling with electrons/positrons. Indeed, electronic neutrinos can interact with $e^\pm$ via neutral \emph{and} charged current processes (i.e., exchanges of $Z$ and $W$ bosons), while the other flavours of neutrinos can only interact with the heat bath via neutral current processes. Assuming that neutrinos had Maxwell-Boltzmann distributions, Dolgov found~\cite{Dolgov_2002PhysRep}:
\begin{equation*}
T_{\nu_e D} \simeq 1.87 \ \mathrm{MeV} \quad ; \quad T_{\nu_{\mu,\tau} D} \simeq 3.12 \ \mathrm{MeV} \, .
\end{equation*}
These different arguments show that, in reality, neutrino decoupling is a much more complicated process than what is described in the instantaneous decoupling approximation. The true decoupling process is referred to as \emph{incomplete neutrino decoupling}, and describing it requires to solve the Boltzmann equations which drive the evolution of the neutrino distributions functions. The following subsection describes standard numerical results on this topic.
\subsection{Incomplete neutrino decoupling}
\label{subsec:intro_decoupling}
Previous works have tackled the problem of incomplete neutrino decoupling, with increasing precision and taking into account the distortions of the spectra:\footnote{A good summary of the existing literature as of 2015 can be found in~\cite{Grohs2015}.} \cite{Dolgov1992,Dodelson_Turner_PhRvD1992} approximated the distribution functions with Maxwell-Boltzmann statistics, an approximation overcome in~\cite{Hannestad_PhRvD1995,Dolgov_NuPhB1997,Dolgov_NuPhB1999,Esposito_NuPhB2000} which used various numerical methods. Corrections to the plasma thermodynamics (see section~\ref{subsec:QED}) were included in~\cite{Mangano2002,Mangano2005,Grohs2015,Relic2016_revisited,Gariazzo_2019}. We can also quote the recent approximate but accurate methods developed in~\cite{Escudero_2018,Escudero_2020}.
We present in this section the results of a typical calculation of incomplete neutrino decoupling which \emph{does not} take into account neutrino masses and mixings, features that we describe in the next section. The full calculation is done in chapter~\ref{chap:Decoupling} and requires a more complex formalism. Without flavour oscillations, calculating neutrino decoupling~\cite{Mangano2002,Grohs2015,Froustey2019} requires to solve the covariant Boltzmann kinetic equation which reads for neutrinos~\cite{ModernCosmology}
\begin{equation}
\label{eq:boltzmann_nu}
\left[\frac{\partial}{\partial t} - H p \frac{\partial}{\partial p}\right]f_{\nu_\alpha}(p,t) = C_{\nu_\alpha}[f_\nu,f_{e^\pm}] \, ,
\end{equation}
where $C_{\nu_\alpha}[f_j]$ is the collision term. This collision integral is dominated by two-body reactions $1+2 \to 3+4$ and is given by (the sum is over reactions)
\begin{multline}
\label{eq:collision_integral_intro}
C_{\nu_1} = \frac{1}{2E_1} \sum{\int{\frac{\mathrm{d}^3 p_2}{2 E_2 (2\pi)^3}\frac{\mathrm{d}^3 p_3}{2 E_3 (2\pi)^3}\frac{\mathrm{d}^3 p_4}{2 E_4 (2\pi)^3} \times (2\pi)^4 \delta^{(4)}(p_1+p_2-p_3-p_4)}} \\
\times S \langle \abs{\mathcal{M}}^2\rangle \times F[f^{(1)},f^{(2)},f^{(3)},f^{(4)}] \, ,
\end{multline}
where $S$ is the symmetrization factor, $\langle \abs{\mathcal{M}}^2\rangle$ the summed-squared matrix element (given for instance in~\cite{Grohs2015}), and \[F \equiv f^{(3)} f^{(4)}(1-f^{(1)})(1-f^{(2)}) - f^{(1)} f^{(2)} (1-f^{(3)})(1-f^{(4)}) \, ,\]
the notation $f_a^{(j)}$ meaning $f_a(p_j)$. The standard calculation assumes no neutrino asymmetry $f_{\nu_\alpha} = f_{\bar{\nu}_\alpha}$. Finally, the distribution functions are the same for $\nu_\mu$ and $\nu_\tau$, as at the energy scales of interest the muon and tau neutrinos have the same interactions (while the distribution of $\nu_e$ is different because of charged current interactions with the background medium). We are thus restricted to two unknown neutrino distributions, $f_{\nu_e}$ and $f_{\nu_\mu}$. In addition to the Boltzmann equation for neutrinos, the last necessary equation\footnote{All remaining species (electrons, positrons, photons) are kept at equilibrium by fast QED interactions, so their properties are summarized by a single parameter, the plasma temperature $T_\gamma$ --- whose evolution is thus given by energy conservation.} is the total energy conservation~\eqref{eq:eq_conservation}.
\paragraph{Comoving variables} We define the comoving temperature $T_{\mathrm{cm}} \propto a^{-1}$ \cite{Grohs2015}, which corresponds to the physical temperature of all species when they are strongly coupled, i.e. $T_\nu = T_\gamma = T_{\mathrm{cm}}$ when $T_{\mathrm{cm}} \gg 1 \, \mathrm{MeV}$, and is also the temperature of neutrinos at all times in the instantaneous decoupling approximation $T_\nu^\mathrm{ID} = T_{\mathrm{cm}}$. From this proxy for the scale factor, we define the comoving variables \cite{Esposito_NuPhB2000,Mangano2005}
\begin{equation}
\label{eq:comoving_variables}
x \equiv m_e/T_{\mathrm{cm}}\, , \qquad y \equiv p/T_{\mathrm{cm}}\, ,\quad \text{and} \quad z \equiv T_\gamma/T_{\mathrm{cm}}\, ,
\end{equation}
which are respectively the reduced scale factor, the comoving momentum, and the dimensionless photon temperature, such that $f(p,t)$ is now expressed $f(x,y)$. We also introduce the dimensionless thermodynamic quantities $\bar{\rho} \equiv (x/m_e)^4 \rho $ and $\bar{P} \equiv (x/m_e)^4 P$.
\paragraph{Neutrino spectral distortions due to $e^\pm$ annihilation} It is then possible to follow the evolution of neutrino distribution functions across the decoupling era. Since this is an out-of-equilibrium process, we cannot properly talk about neutrino "temperatures", although such quantities are quite convenient to give a global picture of the results. Note that for instance in Refs.~\cite{Dodelson_Turner_PhRvD1992,Hannestad_PhRvD1995,Dolgov_NuPhB1997}, an "effective temperature" is defined as
\begin{equation*}
T_\mathrm{eff}(p) = \frac{p}{\ln{[1/f_\nu(p) -1]}}
\end{equation*}
and comparing $T_\mathrm{eff}$ with $T_\gamma$ along the evolution would seem to be a good indicator of decoupling. However, such a temperature is just the temperature of the only Fermi-Dirac spectrum which takes the value $f_\nu(p)$ at $p$, so it doesn't give the "global" information one is looking for when defining a temperature. Moreover, we are interested in the parameter $N_{\mathrm{eff}}$ which depends on the energy density of neutrinos --- but $T_\mathrm{eff}$ is not a convenient parameter to compute the energy density.
Therefore, we rather introduce the following parameterization:
\begin{equation}
\label{eq:param_fnu}
f_{\nu_\alpha}(x,y) \equiv \frac{1}{e^{y/z_{\nu_\alpha}} + 1} \left[1 + \delta g_{\nu_\alpha}(x,y)\right] \, ,
\end{equation}
where the reduced effective temperature $z_{\nu_\alpha} \equiv T_{\nu_\alpha}/T_{\mathrm{cm}}$ is the reduced temperature of the Fermi-Dirac spectrum with zero chemical potential which has the same energy density as the real distribution:
\begin{equation}
\label{eq:def_znu_intro}
\bar{\rho}_{\nu_\alpha} \equiv \frac78 \frac{\pi^2}{30} z_{\nu_\alpha}^4 \,.
\end{equation}
Note that the effective distortions are constrained so that~\eqref{eq:def_znu_intro} holds:
\begin{equation}
\int_{0}^{\infty}{\mathrm{d}{y} y^3\frac{\delta g_{\nu_\alpha}}{e^{y/z_{\nu_\alpha}}+1}} = 0 \, .
\end{equation}
The evolution of the effective temperatures is shown on Figure~\ref{fig:TnuNoMix}. As expected, photons are less reheated by $e^+ e^-$ annihilation, hence the smaller final value of $T_\gamma$. Even though we draw two lines, $T_{\nu_\mu}$ and $T_{\nu_\tau}$ are exactly equal. The higher value of $T_{\nu_e}$ is due to the charged current processes: these additional interactions maintain thermal contact longer and make electronic (anti)neutrinos the main channel of entropy transfer between the QED plasma and the neutrino bath. The effective number of neutrino species is conveniently computed from the effective temperatures:
\begin{equation}
\label{eq:defNeff}
\rho_\nu + \rho_{\bar{\nu}} = N_{\mathrm{eff}} \times \frac78 \left(\frac{4}{11}\right)^{4/3} \rho_\gamma \ \iff \ N_{\mathrm{eff}} \equiv \left[ \frac{(11/4)^{1/3}}{z} \right]^4 \times \left(z_{\nu_e}^4 + z_{\nu_\mu}^4 + z_{\nu_\tau}^4 \right) \, ,
\end{equation}
where we assumed no asymmetry between neutrinos and antineutrinos ($z_{\nu_\alpha} = z_{\bar{\nu}_\alpha}$). In the general case, one simply has to replace $z_{\nu_\alpha}^4 \to (z_{\nu_\alpha}^4+z_{\bar{\nu}_\alpha}^4)/2$.
\begin{figure}[!h]
\centering
\includegraphics{figs/TalkTempPlots.pdf}
\caption[Temperature evolution during neutrino decoupling]{\label{fig:TnuNoMix} Evolution of the (effective) temperatures of the relativistic species (photons and neutrinos) across the decoupling era. \emph{Top panel:} Comoving photon temperature in the instantaneous decoupling approximation (dashed grey line) and taking into account incomplete neutrino decoupling (solid orange line). The asymptotic value of $T_\gamma^{\mathrm{ID}}/T_{\mathrm{cm}}$ is, as derived in~\eqref{eq:TgTnu}, $(11/4)^{1/3}\simeq 1.40102$. The neutrino effective temperatures cannot be distinguished by eye from $T_{\mathrm{cm}}$. \emph{Bottom panel:} Relative difference between the temperatures and $T_{\mathrm{cm}}$, showing the higher reheating of $\nu_e$ compared to $\nu_{\mu,\tau}$.}
\end{figure}
We plot on Figure~\ref{fig:gnuNoMix} the final non-thermal distortions $\delta g_{\nu_\alpha}$. The same comments can be made as for the final effective temperatures. Note that the typical size of the corrections to the ID limit is $\sim 1 \, \%$.
\begin{figure}[!ht]
\centering
\includegraphics{figs/dgnu_NoMixing.pdf}
\caption[Final non-thermal distortions of the neutrino spectra]{\label{fig:gnuNoMix} Final non-thermal distortions of the neutrino spectra. Since they have exactly the same interactions, $\delta g_{\nu_\mu} = \delta g_{\nu_\tau}$. The larger amplitude of distortions for electronic neutrinos is due to the charged current processes.}
\end{figure}
For completeness, we give the value of $N_{\mathrm{eff}}$ deduced from the results shown on Figure~\ref{fig:TnuNoMix}: $N_{\mathrm{eff}} \simeq 3.043$. It is close to the values previously obtained in the literature, notably the previous reference $N_{\mathrm{eff}} = 3.046$~\cite{Relic2016_revisited}. However, these values strongly depend on the inclusion of QED corrections, of a proper treatment of flavour mixing (which we haven't introduced yet), etc. We postpone the dedicated computation of $N_{\mathrm{eff}}$ to chapter~\ref{chap:Decoupling}.
\subsubsection{Experimental constraints}
We gather in Table~\ref{Table:constraints_Neff} some of the latest bounds on $N_{\mathrm{eff}}$ obtained from different CMB experiments~\cite{WMAP2012,Planck18,ACT:2020,SPT-3G:2021}.\footnote{A detailed discussion of the effects of neutrinos on CMB can be found in, e.g.,~\cite{Neutrino_Cosmology}.} These results notably confirm the existence of only 3 neutrino species being thermally populated close to decoupling, which constrains the properties of possible sterile states. For instance, light sterile neutrinos with a $\sim \mathrm{eV}$ mass, favoured to fit some experimental anomalies, would unavoidably bring $N_{\mathrm{eff}} \simeq 4$~\cite{Gariazzo_2019} thanks to oscillation mechanisms introduced in section~\ref{sec:Intro_massive_nu}.
\begin{table}[!htb]
\centering
\begin{tabular}{|M{4cm} |M{2cm} l |}
\hline
Experiment & $N_{\mathrm{eff}}$ & \\
\hline \hline
WMAP~\cite{WMAP2012} & $3.84 \pm 0.40$ & [WMAP + ACT + SPT + BAO + $H_0$] \\
Planck~\cite{Planck18} & $2.99 \pm 0.17$ & [Planck + BAO] \\
ACT~\cite{ACT:2020} & $2.74 \pm 0.17$ & [ACT + Planck] \\
SPT-3G~\cite{SPT-3G:2021} & $2.95 \pm 0.17$ & [SPT + Planck] \\ \hline
\end{tabular}
\caption[Recent constraints on $N_{\mathrm{eff}}$]{Recent constraints on the value of $N_{\mathrm{eff}}$ obtained from different experiments and combined datasets. The uncertainties are 68 \% confidence intervals.
\label{Table:constraints_Neff}}
\end{table}
One of the main results of this PhD is the reevaluation of the cosmological observable $N_{\mathrm{eff}}$, including all relevant effects to reach a $10^{-4}$ precision, also including the effect of neutrino masses (to be introduced in the following section~\ref{sec:Intro_massive_nu}). To this aim, we first derive the neutrino evolution equations in chapter~\ref{chap:QKE}, extending the work of~\cite{Volpe_2013} for astrophysical environments, and implement two-body collisions in an isotropic and homogeneous environment, including neutrino self-interactions with their full matrix structure. Then, we numerically solve these equations, but also present an approximate solution where an adiabatic evolution is considered, exploiting the different timescales involved in the problem. This procedure allows to maintain the required precision while decreasing substantially the computation time, gaining some physical insight on the role of the phenomenon of flavour oscillations in neutrino decoupling. The numerical results we present correspond to the case of zero chemical potential. Finally we investigate the impact of neutrino masses and mixings on BBN predictions in chapter~\ref{chap:BBN}, going beyond works available in the literature~\cite{Mangano2005,Gava:2010kz,Gava_corr}.
\subsection{Big Bang Nucleosynthesis}
\label{subsec:intro_BBN}
Big Bang Nucleosynthesis (BBN) is one of the historical pillars of the Big Bang model, together with the expansion of the Universe and the Cosmic Microwave Background. It corresponds to the period when the temperature was small enough to enable the formation of light elements by combining neutrons and protons. The idea first appeared in the seminal paper by Alpher, Bethe and Gamow~\cite{AlpherBetheGamow}, and later studies showed that BBN was responsible for the primordial production of deuterium ($\mathrm{D}={}^2\mathrm{H}$), helium-3, helium-4 and lithium-7.
We do not present in-depth the different steps of BBN, referring for more details to books like~\cite{PeterUzan} or reviews like~\cite{Pitrou_2018PhysRept}, and to chapter~\ref{chap:BBN}. There are nevertheless three big steps to keep in mind:
\begin{enumerate}
\item At high temperatures, neutrons and protons are kept in chemical equilibrium by weak interactions:
\begin{equation}
\label{eq:weakrates_BBN_intro}
\begin{aligned}
n + \nu_e &\leftrightarrow p + e^- \\
n &\leftrightarrow p + e^- + \bar{\nu}_e \\
n + e^+ &\leftrightarrow p + \bar{\nu}_e
\end{aligned}
\end{equation}
Similarly to neutrino decoupling, these reactions freeze-out at a temperature $T_{\mathrm{FO}} \sim 0.7 \, \mathrm{MeV}$.
\item Below $T_{\mathrm{FO}}$, the only reaction left\footnote{This is actually an oversimplification, as evidenced for instance in~\cite{Grohs:2016vef}. We discuss this point later in chapter~\ref{chap:BBN}.} is neutron decay $n \to p + e^- + \bar{\nu}_e$, until the beginning of nucleosynthesis at $T_{\mathrm{Nuc}}$, the first nuclear reaction being $n + p \to \mathrm{D} + \gamma$.
\item A whole set of out-of-equilibrium nuclear reactions take place, producing heavier nuclei until BBN eventually stops when the temperature is too low to maintain high enough nuclear rates. This leads to the primordial abundances represented on Figure~\ref{fig:IntroBBN} (the results were obtained with a numerical code, \texttt{PRIMAT}, whose principle is presented in chapter~\ref{chap:BBN}).
\end{enumerate}
The standard notations are the following: $n_i$ is the number density of isotope $i$ and the \emph{number} fraction of isotope $i$ is $X_i \equiv n_i/n_b$, with $n_b$ the baryon density. The \emph{mass} fraction is $Y_i \equiv A_i X_i$, where $A_i$ is the nucleon number. It is customary to define:
\begin{equation}
Y_{\mathrm{p}} \equiv Y_{\He4} = 4 \frac{n_{\He4}}{n_b} \qquad \text{and} \qquad i/\mathrm{H} \equiv \frac{X_i}{X_\mathrm{H}} = \frac{n_i}{n_\mathrm{H}} \, .
\end{equation}
Note that, in addition to the species previously mentioned, we also plot the evolution of the abundances of $\mathrm{T} = {}^3\mathrm{H}$ and ${}^7{\mathrm{Be}}$. These nuclei are actually unstable: tritium decays into helium-3 with a half-life of 12.32 years, and the half-life of the decay ${}^7{\mathrm{Be}} \to {}^7{\mathrm{Li}}$ is 53.22 days. Since these periods are much higher than the time-scale on Figure~\ref{fig:IntroBBN}, and that the next major event in the history of the Universe is the formation of the CMB 380 000 years later, we add the abundances $(\mathrm{T} + \He{3})$ and $({}^7{\mathrm{Be}} + {}^7{\mathrm{Li}})$ in the final abundances reported (cf.~Table~\ref{Table:General_BBN}).
\begin{figure}[!ht]
\centering
\includegraphics{figs/evolution_abundances_bbn.pdf}
\caption[Evolution of light element abundances during BBN]{\label{fig:IntroBBN} Evolution of light element abundances, computed with the code \texttt{PRIMAT}. At the end of BBN, the baryonic content of the Universe is mainly made of hydrogen and helium-4. Deuterium is also an important cosmological probe (see text).}
\end{figure}
Neutrinos affect BBN at various levels: via $N_{\mathrm{eff}}$, the Hubble expansion rate is modified, affecting the "clock" of BBN. Then, the neutron-to-proton ratio, set by~\eqref{eq:weakrates_BBN_intro}, is modified when the electronic neutrino distribution functions get distorted, which is parameterized by $z_{\nu_e}$ and $\delta g_{\nu_e}$.
As of today, the experimental measurements of primordial abundances are precise enough to be compared with numerical predictions are $Y_{\mathrm{p}}$ and $\mathrm{D}/\mathrm{H}$, and the excellent agreement is seen as a decisive proof of the hot Big Bang model (see chapter~\ref{chap:BBN}). We can however make two remarks:
\begin{itemize}
\item the predicted abundance of lithium-7, given the baryon density otherwise measured by Planck and in agreement with helium-4 and deuterium measurements, is three times larger than the observed one: this is called the \emph{cosmological lithium problem}. Many solutions have been proposed, but it is for instance very hard to change the abundance of ${}^7{\mathrm{Li}}$ without dramatically affecting the very well constrained $\mathrm{D}/\mathrm{H}$. For a review, see for instance~\cite{Fields2011} ;
\item the uncertainty on the prediction of the deuterium abundance has recently been reduced thanks to the updated rates of the reaction $\mathrm{D} + p \leftrightarrow \He{3} + \gamma$ from the LUNA experiment~\cite{Mossa2020}. On the one hand, a series of works~\cite{Pisanti2020,Yeh2020} confirm the agreement between predictions and measurements, while the analysis of~\cite{Pitrou2020} hints for a possible tension. As discussed in~\cite{Pitrou2021}, the difference between these results is due to the values selected for the nuclear rates of the reactions $\mathrm{D} + \mathrm{D} \leftrightarrow n + \He{3}$ and $\mathrm{D} + \mathrm{D} \leftrightarrow p + \mathrm{T}$. When the same rates are used, the different numerical codes agree. However, there is no definitive argument in favour of one selection or the other, which calls for higher precision measurements in the future.
\end{itemize}
These issues show how crucial it is to understand the physics at play during BBN. In chapter~\ref{chap:BBN}, we study in detail how incomplete neutrino decoupling affects the primordial abundances, combining a numerical and a theoretical analyses.
\section{From massless to massive neutrinos}
\label{sec:Intro_massive_nu}
Up to this point in the discussion, we have considered that neutrinos were the particles predicted by the Standard Model of particle physics --- cf.~appendix~\ref{App:StandMod}. However, there is now a substantial amount of experimental evidence for the fact that neutrinos are not massless and can undergo \emph{neutrino oscillations}, that is the possible change of flavour as neutrinos propagate. The concept of neutrino oscillations was first proposed by Pontecorvo~\cite{Pontecorvo1957,Pontecorvo1958} but for $\nu \leftrightarrow \bar{\nu}$ mixing, as he had been misled by some experimental results. In 1962, Maki, Nakagawa and Sakata~\cite{MNS} proposed that the "weak neutrinos"\footnote{At that time, the $\tau$ lepton had not yet been discovered.} $\nu_e$ and $\nu_\mu$ were quantum superpositions of the "true neutrinos" $\nu_1$ and $\nu_2$:
\begin{equation}
\label{eq:mixing_2D}
\begin{aligned}
\nu_e &= \cos{\theta} \nu_1 + \sin{\theta} \nu_2 \\
\nu_\mu &= - \sin{\theta} \nu_1 + \cos{\theta} \nu_2
\end{aligned} \, .
\end{equation}
However, it was about a decade later that the current theory of neutrino oscillations was truly developed, cf.~for instance~\cite{Eliezer1975}.
We shall not review the history neutrino oscillations, but we will present it through the prism of a particular topic: the so-called \emph{solar neutrino problem}. This will allow us to introduce all the elements we will later need to take into account neutrino mixing in our calculations.
Moreover, the theoretical description of neutrino masses and mixings is presented in appendix~\ref{App:StandMod}. We do not discuss models of neutrino masses, such as the see-saw mechanisms, and refer to, e.g.,~\cite{GiuntiKim,MohapatraPal,King2015}.
\subsection{Massive neutrinos: the example of the solar neutrino problem}
The Sun is a powerful source of electronic neutrinos with energy of the order of $1 \, \mathrm{MeV}$, produced in the thermonuclear reactions which generate solar energy. One of the two main chains of reactions, the so-called $pp$ chain, involves the following reactions:
\begin{equation}
\label{eq:ppchain}
\begin{aligned}
p + p &\to \mathrm{D} + e^+ + \nu_e &(pp) \\
p + e^- + p &\to \mathrm{D} + \nu_e &(pep) \\
\He{3} + p &\to \He{4} + e^+ + \nu_e &(hep) \\
{}^7{\mathrm{Be}} + e^- &\to {}^7{\mathrm{Li}} + \nu_e &({}^7{\mathrm{Be}}) \\
{}^8\mathrm{B} &\to {}^8\mathrm{Be}^* + e^+ + \nu_e &({}^8\mathrm{B})
\end{aligned}
\end{equation}
Each of these reactions produces neutrinos with different energy distributions, and in particular different average energies (see for instance the spectrum of solar neutrino fluxes in~\cite{PDG} or~\cite{Vitagliano_Review}).
The flux of solar neutrinos on the Earth is about $6 \times 10^{10} \, \mathrm{cm^{-2}s^{-1}}$. Such neutrinos were detected for the first time in 1970 in the Homestake experiment. However, the observed number of neutrinos was about one third of what the Standard Solar Model\footnote{A SSM is a "solar model that is constructed with the best available physics and input data" and is "required to fit the observed luminosity and radius of the Sun at the present epoch, as well as the observed heavy-element-to-hydrogen ratio at the surface of the Sun"~\cite{Bahcall}.} (SSM) predicted. This discrepancy was called the \emph{solar neutrino problem}, in particular after the disagreement between the SSM and experimental counts had been confirmed by several experiments (Kamiokande, GALLEX/GNO, SAGE, Super-Kamiokande), which probed different parts of the solar neutrino energy spectrum.
All these experiments measured a smaller number of electronic neutrinos compared to the expected flux. A solution to this problem can be obtained with the phenomenon of neutrino oscillations. Assuming that the \emph{mass states} $\nu_1$ and $\nu_2$ introduced in~\eqref{eq:mixing_2D} have a squared mass-difference $\Delta m^2 = m_2^2 - m_1^2$, one can show that the probability that a neutrino of energy $E$, initially in a state $\ket{\nu_e}$, be measured in a state of the same flavour $\ket{\nu_e}$ (the \emph{survival probability}) after a distance $L$ is~\cite{GiuntiKim}\footnote{This formula is easily derived in standard quantum mechanics with the so-called "equal momentum" assumption, which states that neutrinos propagate with identical momenta but different energies due to their different masses. This assumption is highly questionable, but does not affect the result, see~\cite{GiuntiKim} for a thorough discussion.}
\begin{equation}
\mathcal{P}_{e \to e} = 1 - \sin^2(2 \theta) \sin^2\left(\frac{\Delta m^2 L}{4 E}\right) \, .
\end{equation}
For neutrinos coming from the sun, the "source-detector" distance is huge and it happens that $\Delta m^2$ is not too small, such that the measurable quantity over the energy resolution of the detector is the average probability
\begin{equation}
\label{eq:survival_vacuum}
\mathcal{P}_{e \to e}^\mathrm{vacuum} = 1 - \frac12 \sin^2(2 \theta) \, .
\end{equation}
The superscript $^\mathrm{vacuum}$ highlights the fact that we do not consider any \emph{matter effect}, which comes from the different behaviour of neutrinos propagating in matter and in vacuum, as evidenced in the following.
The first experiments on solar neutrinos were only able to measure a deficit in the number of detected electronic neutrinos. The observations of the Sudbury Neutrino Observatory (SNO)~\cite{SNO_2002} were crucial in proving the validity of the mechanism of neutrino oscillations. In this heavy-water Cherenkov detector located in the Creighton mine near Sudbury (Ontario, Canada), solar neutrinos are detected through both charged and neutral currents on deuterium (and also via elastic scattering):
\begin{equation*}
\begin{aligned}
\nu_e + \mathrm{D} &\to p + p + e^- &\mathrm{(CC)} \\
\nu_\alpha + \mathrm{D} &\to p + n + \nu_\alpha &\mathrm{(NC)}
\end{aligned}
\end{equation*}
The neutral-current reaction on deuterium is equally sensitive to all neutrinos, while the charged-current one is only sensitive to electronic neutrinos: this provides a direct way to check the flavour transformation and the survival probability by comparing the flux from CC to the total neutrino flux.
We show on Figure~\ref{fig:solar_neutrinos} the measured electron neutrino survival probability for different neutrino energies. These different energies correspond to different production channels of solar neutrinos~\eqref{eq:ppchain}, and the results plotted on this Figure come from two experiments: SNO, already mentioned, and Borexino, a liquid scintillator experiment based at Gran Sasso (Italy).
\begin{figure}[!ht]
\centering
\includegraphics{figs/solar_neutrinos.pdf}
\caption[$\nu_e$ survival probability as a function of neutrino energy]{\label{fig:solar_neutrinos} Electron neutrino survival probability as a function of neutrino energy. The points represent, from left to right, the Borexino $pp$, ${}^7\mathrm{Be}$, $pep$, and ${}^8\mathrm{B}$ data (red points)~\cite{Borexino} and the SNO ${}^8\mathrm{B}$ data (green point)~\cite{SNO_2016}. The error bars represent the $\pm 1 \sigma$ experimental + theoretical uncertainties. The blue curve corresponds to the prediction of the MSW-LMA solution using the formulae given in~\cite{Vissani2017a,Vissani2017b,Borexino} and the parameters from~\cite{PDG}. The dashed grey line is the vacuum prediction (equation~\eqref{eq:survival_vacuum}, corrected to account for three-neutrino mixing).}
\end{figure}
First and foremost, $\mathcal{P}_{e \to e} \neq 1$: there is a large conversion of electronic neutrinos into the other flavours. At low energies, the prediction\footnote{This formula is actually corrected in the three-neutrino case and reads (the mixing angles are introduced in appendix~\ref{App:StandMod})
\[ \mathcal{P}_{e \to e}^\mathrm{vacuum} = \cos^4{\theta_{13}} \left(1 - \frac12 \sin^2(2 \theta_{12})\right) + \sin^4{\theta_{13}} \, . \] } of the vacuum solution~\eqref{eq:survival_vacuum} is in agreement with experimental data. However, this solution does not explain the results at higher energies, notably the SNO one. The answer to this discrepancy requires to take into account \emph{matter effects}.
\paragraph{Matter effects and MSW resonance} Flavour oscillation is a consequence of the fact that flavour and mass eigenstates are not identical (flavour mixing) and that mass eigenstates propagate with different velocities (mass differences). The propagation in a medium different from vacuum is thus expected to change the oscillation mechanism, similarly to the refraction index of a dielectric medium for photons (the matter effects we discuss here are sometimes called \emph{refractive effects}). Wolfenstein~\cite{MSW_W} discovered in 1978 that neutrinos propagating in matter were subject to a mean-field potential which modifies their mixing.
The effective potential due to charged-current interactions, $V_\text{CC}$, is obtained by averaging the weak Hamiltonian~\eqref{eq:hcc_app} over the background particle distribution. For a homogeneous and isotropic gas of unpolarized electrons, we obtain an average effective Hamiltonian (we assume we are at sufficiently low energy to use the effective four-Fermi theory)
\begin{equation}
\label{eq:VCC_intro}
\langle H_\text{eff}^{\mathrm{(CC)}} \rangle = V_\text{CC} \overline{\nu_{eL}}(x) \gamma^0 \nu_{eL}(x) \quad \text{with} \quad \boxed{V_\text{CC} = \sqrt{2} G_F n_{e^-}} \, .
\end{equation}
A background of positrons leads to $V_\text{CC} = - \sqrt{2} G_F n_{e^+}$, because of the anticommutation of annihilation/creation operators in the Hamiltonian when dealing with antiparticles. The same procedure can be applied to neutral-current interactions, and the effective potential experienced by a neutrino of any flavour in a background of unpolarized fermions $f$ reads
\begin{equation}
V_\text{NC}^f = \sqrt{2} G_F n_f g_V^f \, ,
\end{equation}
with $g_V^f$ the weak vector coupling of fermion $f$. Given the values of the different $g_V^f$, in an environment of electrons, positrons, protons and neutrons such as the early Universe, and because of charge neutrality, all contributions cancel except that of neutrons:
\begin{equation}
V_\text{NC} = - \frac12 \sqrt{2} G_F n_n \, .
\end{equation}
In summary, the effective potential felt by a neutrino of flavour $\alpha$ is~\cite{Neutrino_Cosmology,GiuntiKim}
\begin{equation}
V_\alpha = \sqrt{2} G_F \left[ (n_{e^-} - n_{e^+})\delta_{\alpha e} - \frac12 n_n \right] \, .
\end{equation}
Note that in astrophysical environments, there are usually no positrons and the effective potential is the one boxed in~\eqref{eq:VCC_intro}. Since $V_\text{NC}$ is flavour-independent, it does not affect mixing.\footnote{This is true as long as we only consider active-active oscillations. The presence of sterile neutrino species would break this simplification.}
Due to this extra term in the Hamiltonian, the effective mass states are modified --- in other words, the \emph{mass basis} becomes the \emph{matter basis}. In the two-neutrino simplified case, the eigenstates of the total Hamiltonian, $\nu_1^m$ and $\nu_2^m$, are related to the flavour eigenstates via the same relation as~\eqref{eq:mixing_2D}, but the effective mixing angle in matter
\begin{equation}
\label{eq:thetam}
\tan{2 \theta_m} = \frac{\tan{2 \theta}}{ 1 - \frac{2 E V_\text{CC}}{\Delta m^2 \cos{2 \theta}}} \, ,
\end{equation}
and the effective mass-squared difference is
\begin{equation}
\Delta m_m^2 = \sqrt{(\Delta m^2 \cos{2 \theta} - 2 E V_\text{CC})^2 + (\Delta m^2 \sin{2 \theta})^2} \, .
\end{equation}
It was first pointed out in 1985 by Mikheev and Smirnov~\cite{MSW_MS} that in a medium with varying density, there was a region in which the effective mixing angle passes through the maximal mixing value of $\pi/4$. From~\eqref{eq:thetam}, this \emph{resonance} occurs for
\begin{equation}
\Delta m^2 \cos{2 \theta} = 2 p V_\text{CC} \quad \iff \quad n_{e^-}^\mathrm{res} = \frac{\Delta m^2 \cos{2 \theta}}{2 \sqrt{2} G_F E} \, ,
\end{equation}
if we assume there are no positrons (standard case in astrophysical environments where this effect was first discussed). This mechanism is now commonly referred to as the \emph{MSW effect}.
The possibility of large flavour conversion will depend on the hierarchy between two timescales: the oscillation frequency $\Delta m^2/2E$ and the rate of variation of the mixing angle $\mathrm{d}{\theta_m}/\mathrm{d} \mathrm{x}$, where we parameterize the neutrino trajectory by a position in space $\mathrm{x}$. In particular, if $\Delta m^2/2E \gg \abs{\mathrm{d}{\theta_m}/\mathrm{d} \mathrm{x}}$, the evolution is \emph{adiabatic} and transitions between $\nu_1^m$ and $\nu_2^m$ are suppressed. The criterion of adiabaticity, which suppresses the transitions between matter eigenstates, will be at the core of the approximation developed to study neutrino decoupling including flavour oscillations in chapters~\ref{chap:Decoupling} and~\ref{chap:Asymmetry}.
For solar neutrinos, assuming $\Delta m^2 \sim 7 \times 10^{-5} \, \mathrm{eV^2}$ and $\tan^2{\theta} \simeq 0.4$, we obtain the survival probability represented by the blue curve on Figure~\ref{fig:solar_neutrinos}. This set of parameters, that is well-measured today in the 3-neutrino framework, was historically called the "Large Mixing Angle" (LMA) solution, and was for instance in competition with the "Small Mixing Angle" (SMA) solution, for which $\Delta m^2 \sim 5 \times 10^{-6} \, \mathrm{eV}^2$, $\tan^2{\theta} \sim 5 \times 10^{-4}$. The agreement with the experimental points is a proof of the validity of the flavour oscillation mechanism, with the additional complexity of matter effects.
\subsection{Three-neutrino mixing}
In our discussion of the solar neutrino problem, we have reduced the problem to the mixing of two neutrino states. After many other experiments (atmospheric neutrinos, reactors and accelerators), the "standard" model of neutrino mixing involves the three active flavour species and the associated three mass eigenstates. The parameterization of the mixing is presented in appendix~\ref{subsec:Values_Mixing}, and the values of all parameters are now well measured, except for two:
\begin{itemize}
\item the \emph{mass ordering} (we also talk about the mass hierarchy) is unknown. Defining $\Delta m_{ji}^2 = m_{\nu_j}^2 - m_{\nu_i}^2$, there are two possibilities concerning the sign of $\Delta m_{31}^2$, represented on Figure~\ref{fig:hierarchies};
\begin{figure}[!ht]
\centering
\includegraphics{figs/hierarchies.pdf}
\caption[Normal and inverted neutrino mass orderings]{\label{fig:hierarchies} The two possible mass orderings, normal (\emph{left}) and inverted (\emph{right}). The small mass gap $\Delta m_{21}^2$ is the one involved in solar neutrino oscillations.}
\end{figure}
\item the value of the Dirac CP phase: a non-vanishing phase would indicate a difference between the mixing of neutrinos and antineutrinos. An important result of this thesis is the strong independence of neutrino physics in the MeV era, as long as beyond-the-Standard-Model mechanisms (outside of neutrino masses and mixings) are not invoked.
\end{itemize}
This standard "three-neutrino mixing" model will be used throughout the manuscript.
\subsection{Massive neutrinos in cosmology}
A robust and precise prediction of the consequences of incomplete neutrino decoupling is crucial since neutrinos impact many cosmological stages.
\begin{enumerate}
\item During Big Bang Nucleosynthesis (BBN), neutrinos control neutron/proton conversions as they participate to weak interactions, and the frozen neutron abundance subsequently affects nuclear reactions and light element relics --- see~\ref{subsec:intro_BBN}.
\item During the Cosmic Microwave Background (CMB) formation, the free streaming of neutrinos is crucial to predict the CMB angular spectrum. Also, the value of $N_{\mathrm{eff}}$ affects the cosmological expansion, and thus also the radiative transfer of CMB. From these effects, CMB alone can be used to place constraints on $N_{\mathrm{eff}}$ or in combination with BBN constraints on primordial light elements.
\item In the late universe, neutrino free streaming also affects structure formation, via its effect on the growth of perturbations. This is used to place the constraint $\sum_i m_{\nu_i} < 0.12 \, \mathrm{eV}$ (see e.g.~\cite{Planck18,eBOSS}) on the sum of neutrino masses.
\end{enumerate}
It is striking that neutrino masses play a key role in both the earliest stage 1 and the latest stage 3 for very different reasons.
In stage 1, neutrino oscillations, which are due to small neutrino mass-squared differences and mixing angles, affect the non-thermal part of the spectra, as they lead to less distortion in electron-type neutrinos and more distortion in other types than if there were no oscillations at all. Also, oscillations lead to a mild modification of $N_{\mathrm{eff}}$ --- see chapter~\ref{chap:Decoupling}. In stage 3, and due to cosmological redshifting, all massive neutrinos undergo at some point a transition from being very relativistic (they behave gravitationally like decoupled photons) to being non-relativistic. This transition depends only on neutrino masses and not on mixing angles, since frozen neutrino spectra inherited from stage 1 are generated incoherently in the mass basis.
Finally, stage 2 would also be affected beyond the standard cosmological model, if we were to consider exotic physics with increased neutrino self-interactions, so that they would still behave effectively as a perfect fluid around CMB formation~\cite{Kreisch:2019yzn,Grohs:2020xxd}.
This interplay between the various cosmological eras implies that it is crucial to understand neutrino decoupling as precisely as possible, in order to use these predictions as initial conditions for the subsequent eras. For instance, current constraints from CMB on cosmological parameters~\cite{Planck18} were placed using $N_{\mathrm{eff}}=3.046$ when solving numerically for the linear evolution of cosmological perturbations.
\end{document}
\chapter*{Introduction}
\addcontentsline{toc}{chapter}{\protect\numberline{}Introduction}
\begin{quote}
\emph{I have done a terrible thing today, something which no theoretical physicist should ever do. I have suggested something that can never be verified experimentally.}
\sourceatright{Wolfgang Pauli, 1930, quoted in~\cite{Hoyle1967}}
\end{quote}
W. Pauli is reported to have confessed this "terrible thing" to his friend Walter Baade, after having proposed the existence of a particle in order to save the principle of energy conservation: the \emph{neutrino}. Indeed, the measurement of the energy of electrons emitted in beta decays was in complete disagreement with predictions. For instance, the predicted decay $\ce{^{14}_6\mathrm{C} -> ^{14}_7\mathrm{N} + e^-}$ should lead to a very peaked electron energy $E_e = (m_{^{14}\mathrm{C}} - m_{^{14}\mathrm{N}})c^2$. On the contrary, scientists measured a continuous spectrum of energies... Pauli thus proposed that the final state actually contained a third particle, neutral and very light, that would take away part of the disintegration energy, such that the decay actually reads $\ce{_6^{14}\mathrm{C} -> _7^{14}\mathrm{N} + e^- + \bar{\nu}_e}$. However, with such properties, a neutrino (actually here, an antineutrino) should be very difficult to detect, hence the quotation at the start of this introduction. Yet, a few decades later, in 1956, F. Reines and C. L. Cowan sent a telegram informing Pauli of the discovery of the electronic antineutrino. Reines was awarded half the Nobel Prize in Physics "for the detection of the neutrino" in 1995, Cowan having passed away.
Today, we associate one neutrino to each charged lepton, bringing to $3$ the number of known neutrinos: $\nu_e$, $\nu_\mu$ and $\nu_\tau$. In the Standard Model of particle physics, they are \emph{massless} particles which only interact via the \emph{weak} interaction. However, there is now a large body of experimental evidence that neutrinos have properties that are not predicted by the Standard Model. In particular, they undergo \emph{flavour oscillations}, a property that cannot be understood with massless species, whose discovery led to another Nobel Prize in 2015, awarded to T. Kajita (from the Super-Kamiokande experiment) and A. B. McDonald (from the Sudbury Neutrino Observatory).
These exciting developments in particle physics have a particular resonance at an incredibly wider scaler --- in cosmology. Indeed, neutrinos play a key role at various stages of the evolution of the Universe, and the imprints they leave on cosmological observables will be uncovered more and more precisely as new detectors are being developed. In this period of "precision cosmology", there is therefore an important need for accurate theoretical predictions on the values of these cosmological observables, in order to be able to pinpoint potential hints for beyond-the-Standard-Model physics.
During this PhD, we have focused on the epoch when the temperature of the Universe was about $10^{10} \, \mathrm{K} \sim 1 \, \mathrm{MeV}$, the so-called "\emph{MeV age}". As we will detail in the forthcoming chapters, this period is extremely rich in physical events involving neutrinos: they decouple from the electromagnetic plasma, electrons and positrons annihilate, and primordial nucleosynthesis starts. It is crucial to accurately predict the features of neutrino distributions at this epoch, as the MeV era is the neutrino "Grand Finale" before their behaviour is simply described by that of a free streaming particle bath, the Cosmic Neutrino Background. Our goal was twofold: develop new theoretical tools, regarding the derivation of the resolution of the evolution equations, and apply a numerical code to investigate neutrino evolution in various frameworks. In particular, we have studied in-depth the cases of "standard" neutrino decoupling, which involves solely Standard Model physics with the known results about flavour oscillations, and the situation of potentially large primordial neutrino/antineutrino asymmetry, which could largely affect the cosmological expansion.
\noindent This PhD is based on the following publications:
\begin{itemize}
\item \cite{Froustey2019} focusing on the various ways in which incomplete neutrino decoupling (without taking into account flavour mixing) affects the abundances of light elements produced during Big Bang Nucleosynthesis, in order to understand semi-analytically the physics at play;
\item \cite{Froustey2020} in which we performed the first calculation of neutrino decoupling including all known physical effects required to reach a few $10^{-4}$ accuracy on the parameter $N_{\mathrm{eff}}$ prediction;
\item \cite{FrousteyTAUP2021} Proceedings of the TAUP2021 conference where the results of the above paper were presented, and a discussion on the neutrino energy density parameter $\Omega_\nu$ was added;
\item \cite{Froustey2021} extending the study of neutrino evolution to account for the possibility of non-zero neutrino/antineutrino asymmetry.
\end{itemize}
\noindent The manuscript is organised as follows. In chapter~\ref{chap:IntroCosmo}, we provide a wide introduction to cosmology and, in particular, neutrino physics in connection with cosmology. This gives the basis for the in-depth analyses of the next chapters. In order to perform the precision calculation of $N_{\mathrm{eff}}$, we first present a new derivation of the evolution equation of neutrinos and antineutrinos (the "Quantum Kinetic Equations") in chapter~\ref{chap:QKE}. Chapter~\ref{chap:Decoupling} is dedicated to the standard calculation of $N_{\mathrm{eff}}$. We then use these results to study their consequences on Big Bang Nucleosynthesis in chapter~\ref{chap:BBN}. Finally, we extend the previous analytical and numerical tools to the case of non-zero asymmetries in chapter~\ref{chap:Asymmetry}.
\end{document}
\chapter[Standard neutrino decoupling including flavour oscillations][Standard neutrino decoupling]{Standard neutrino decoupling including flavour oscillations}
\label{chap:Decoupling}
\setlength{\epigraphwidth}{0.44\textwidth}
\epigraph{Sometimes I'll start a sentence and I don't even know where it's going. I just hope I find it along the way.}{Michael Scott, \emph{The Office} [S05E12]}
{
\hypersetup{linkcolor=black}
\minitoc
}
\boxabstract{The material of this chapter has been partly published in~\cite{Froustey2020}.}
Several effects take place during neutrino decoupling, with some of them leaving signatures on cosmological observables. From the numerical resolution of the QKE that was derived in the previous chapter, we can obtain the evolution of the density matrix $\varrho$ and fully characterize the neutrino spectra during BBN and later cosmological stages. In particular, we present in this chapter the calculation of "standard" neutrino decoupling and notably the resulting value of $N_{\mathrm{eff}}$. Its previous reference value was $N_{\mathrm{eff}} = 3.045$~\cite{Relic2016_revisited}, but this calculation did not include some important finite-temperature QED corrections (see below) and approximated the off-diagonal components of the self-interaction collision term as damping factors. We present the first calculation relaxing completely the damping approximation and including all (known) QED corrections.
\noindent This standard calculation is thus based on the following assumptions.
\begin{itemize}
\item The early Universe is considered homogeneous and isotropic. Hence, the dynamical evolution of spacetime is entirely described by the evolution of the scale factor through Friedmann equation. This hypothesis was made in the derivation of the QKEs in chapter~\ref{chap:QKE}, and is discussed in section~\ref{subsec:limitations_NuDec}.
\item There is no asymmetry between neutrinos and antineutrinos. Therefore, we will only follow the evolution of the neutrino density matrix $\varrho$, which is justified in section~\ref{sec:set_of_equations}.\footnote{We have checked numerically that solving additionally the QKE on $\bar{\varrho}$ gives consistent results.}
\item We do not include a CP phase in the PMNS matrix (mostly because of the uncertainty on its value). However, as detailed in section~\ref{subsec:Decoupling_CP}, its value does not affect $\varrho^e_e$ nor $N_{\mathrm{eff}}$, that is the key physical parameters for BBN. More generally, the cosmological observables are not affected by the value of the Dirac CP-violating phase --- a wider discussion of this property when including asymmetries is presented in chapter~\ref{chap:Asymmetry}.
\item The QED plasma differs from an ideal gas because of finite-temperature corrections~\cite{Dicus1982,Heckler_PhRvD1994,Fornengo1997,LopezTurner1998,BrownSawyer,Mangano2002,Bennett2020}. The associated corrections to the equation of state are included in the energy conservation equation (see below). The modifications to the scattering rates are yet to be included, since they add a considerable layer of complexity to the computation of the collision integrals~\cite{Bennett2021}.
\end{itemize}
Under these assumptions, we can specify the set of equations we solve and the features of the numerical code, \texttt{NEVO}, developed for that purpose. Our main focus being the role of flavour oscillations in neutrino evolution, we additionally introduce a particular approximation of the evolution equations, based on the large separation of time scales in the problem. Its excellent accuracy provides a powerful framework to understand how the final neutrino distributions depend on physical parameters.
\section{Set of equations}
\label{sec:set_of_equations}
We present here the set of differential equations one needs to solve to determine the evolution of all relevant quantities (notably, the neutrino spectra) across the decoupling era.
\subsection{Neutrino sector}
On the neutrino side, we are interested in determining the evolution of the density matrix $\varrho$, which is given by the QKE~\eqref{eq:QKE_fullfinal}. Given the above hypotheses, this equation can be simplified, reducing the number of terms in the oscillation Hamiltonian.
\paragraph{Neutrino asymmetry mean-field} Except in the subsection~\ref{subsec:Decoupling_CP} specifically dedicated to it, we neglect the Dirac CP phase in the PMNS matrix~\eqref{eq:PMNS}, therefore the Hamiltonian is a real symmetric matrix (instead of a hermitian matrix in the general case). We show in the appendix~\ref{subsec:QKE_consistency} that the structure of the QKEs preserves the equality $\varrho = \bar{\varrho}^*$ if it is true initially (note that $\bar{\varrho}^* = \bar{\varrho}^T$ where $^T$ stands for the transposed of the matrix). A key result of this chapter (cf.~section~\ref{sec:ATAO}) is that $\varrho$ is diagonal in the "matter" basis, that is the basis in which the Hamiltonian is diagonal. Since the Hamiltonian is a real symmetric matrix, the effective mixing matrix between the matter and flavour bases is orthogonal. Therefore, $\varrho$ and $\bar{\varrho}$ cannot have imaginary components and we can conclude that $\varrho = \bar{\varrho}$ at all times.\footnote{There may be a caveat in this reasoning in the inverted hierarchy case for which an instability might lead to the growing of the imaginary off-diagonal components of $\varrho - \bar{\varrho}$~\cite{Hansen_Isotropy} --- but this is only the case if $\varrho$ is not exactly diagonal in the matter basis. We discuss this case in section~\ref{subsec:Inverted_Hierarchy}. }
We will thus neglect the term $\overline{\mathbb{N}}_\nu - \overline{\mathbb{N}}_{\bar{\nu}}$ in the mean-field Hamiltonian, and solve only the QKE for $\varrho$ instead of $\varrho$ and $\bar{\varrho}$.
\paragraph{Neutrino energy density mean-field} We will also discard the mean-field term proportional to $\bar{\mathbb{E}}_\nu + \bar{\mathbb{E}}_{\bar{\nu}}$, as the deviations of $\varrho$ from the equilibrium distribution $\propto \mathbb{1}$ are very small (cf. numerical results below). Thus, such a mean-field term will give a negligible contribution within the commutator compared to $\bar{\mathbb{E}}_\mathrm{lep}$.
\paragraph{Summary} Therefore, the QKE for standard neutrino decoupling in the early Universe reduces to
\begin{equation}
\frac{\partial \varrho(x,y_1)}{\partial x} = - \frac{{\mathrm i}}{xH} \left(\frac{x}{m_e}\right) \left[ U \frac{\mathbb{M}^2}{2y_1}U^\dagger, \varrho \right] + {\mathrm i} \frac{2 \sqrt{2} G_F}{xH} y_1 \left(\frac{m_e}{x}\right)^5 \left[ \frac{\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}}{m_W^2} ,\varrho \right ] + \frac{1}{xH} \mathcal{I} \, . \label{eq:QKE_final}
\end{equation}
For convenience in the forthcoming discussion, we write the effective Hamiltonian $\mathcal{V} \equiv \mathcal{H}_0 + \mathcal{H}_\mathrm{lep}$, with
\begin{itemize}
\item the vacuum contribution
\begin{equation}
\label{eq:Hvac}
\mathcal{H}_0 \equiv \frac{1}{xH} \left(\frac{x}{m_e}\right) U \frac{\mathbb{M}^2}{2 y_1} U^\dagger \, ,
\end{equation}
which is inversely proportional to the momentum $y_1$ ;
\item the lepton mean-field part
\begin{equation}
\label{eq:Hlep}
\mathcal{H}_\mathrm{lep} \equiv - \frac{1}{xH} \left(\frac{m_e}{x}\right)^5 2 \sqrt{2} G_F y_1 \frac{\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}}{m_W^2} \, ,
\end{equation}
which depends linearly on $y_1$.
\end{itemize}
Introducing the dimensionless collision term $\mathcal{K} \equiv \mathcal{I}/xH$, the QKE can be rewritten as
\begin{equation}
\label{eq:QKE_compact}
\boxed{\frac{\partial \varrho}{\partial x} = - {\mathrm i} [\mathcal{V},\varrho] + \mathcal{K}} \, .
\end{equation}
The mass matrix $\mathbb{M}^2$ differs depending on the mass ordering considered. Except in the subsection~\ref{subsec:Inverted_Hierarchy}, we will systematically consider a normal mass ordering, favoured by current neutrino oscillation data~\cite{deSalas_Mixing,Esteban2020}.
\subsection{Electromagnetic plasma sector}
\label{subsec:QED}
In principle, one could also follow the evolution of the distribution functions of photons, electrons and positrons throughout the decoupling era via a Boltzmann equation. The collision term would then contain QED interactions, which are extremely efficient compared to weak ones. In other words, photons, electrons and positrons will always be kept at equilibrium by these interactions. This explains why we have taken the $e^\pm$ distribution functions to be equilibrium Fermi-Dirac ones in the derivation of the QKEs.
Therefore, all the information on the statistical distribution of these particles is contained in the plasma temperature $T_\gamma$, or equivalently the comoving temperature $z = T_\gamma / T_{\mathrm{cm}}$. Its evolution is most simply obtained from the continuity equation $\dot{\rho} = - 3 H (\rho + P)$ with $\rho$ and $P$ the \emph{total} energy density and pressure, that is $\rho = \rho_\gamma + \rho_\nu + \rho_{\bar{\nu}} + \rho_{e^\pm}$. Note that we do not include the (anti)muon energy density in this equation as it is negligible compared to $\rho_{e^\pm}$ in the decoupling era. This choice is consistent with the absence of interactions with muons and antimuons in the collision integral: the only role $\mu^\pm$ play in this problem is at the Hamiltonian level, ensuring the coincidence between the flavour and matter bases at high temperature. The reheating due to $\mu^- \mu^+$ annihilations affects equally neutrinos and the QED plasma (as it happens when all species are still coupled), so we do not take it into account --- it amounts to a different normalization of $T_{\mathrm{cm}}$.
\subsubsection{Energy conservation and QED equation of state}
We rewrite the continuity equation as an equation on the dimensionless photon temperature $z(x)$ \cite{Mangano2002,Bennett2020}:
\begin{equation}
\frac{\mathrm{d} z}{\mathrm{d} x} = \frac{\displaystyle \frac{x}{z}J(x/z) - \frac{1}{2 \pi^2 z^3} \frac{1}{xH} \int_{0}^{\infty}{\mathrm{d} y \, y^3 \, \mathrm{Tr} \left[\mathcal{I}\right]} + G_1(x/z)}{ \displaystyle \frac{x^2}{z^2}J(x/z) + Y(x/z) + \frac{2 \pi^2}{15} + G_2(x/z)} \, , \label{eq:zQED}
\end{equation}
with
\begin{align}
J(\tau) &\equiv \frac{1}{\pi^2} \int_{0}^{\infty}{\mathrm{d} \omega \, \omega^2 \frac{\exp{(\sqrt{\omega^2 + \tau^2})}}{(\exp{(\sqrt{\omega^2 + \tau^2})}+1)^2}} \, , \\
Y(\tau) &\equiv \frac{1}{\pi^2} \int_{0}^{\infty}{\mathrm{d} \omega \, \omega^4 \frac{\exp{(\sqrt{\omega^2 + \tau^2})}}{(\exp{(\sqrt{\omega^2 + \tau^2})}+1)^2}} \, .
\end{align}
In~\eqref{eq:zQED}, the integral involving the neutrino collision integral comes from $\mathrm{d} \bar{\rho}_\nu / \mathrm{d} x$, given that $\bar{\rho}_\nu = \frac{1}{2 \pi^2} \int{\mathrm{d}{y} \, y^3 \, \mathrm{Tr}[\varrho]}$.
The $G_1$ and $G_2$ functions account for the modifications of the plasma equation of state (departure from an ideal gas) due to finite-temperature QED corrections \cite{Heckler_PhRvD1994,Mangano2002,Bennett2020}. Indeed, this is part of the features encountered in interacting quantum fields at finite temperature; often interpreted as a modification of the dispersion relation of electrons/positrons and photons which get extra "thermal masses".\footnote{One must however be very careful with this interpretation which can lead to a missing $1/2$ factor in the pressure correction as in~\cite{Grohs2015}. This factor is emphasized in~\cite{Mangano2002} and notably discussed in~\cite{Bennett2020}.} They can be calculated order by order in an expansion in powers of $\alpha = e^2/4\pi$: starting from an expansion of the partition function $Z$ and therefore the free energy $F$~\cite{KapustaGale}, one gets the thermodynamical quantities $\rho$ and $P$ at the desired order, and the $G-$functions after implementing these modified energy density/pressure in the continuity equation. The expressions read
\begin{align}
G_1^{(2)}(\tau) &= 2 \pi \alpha \left[\frac{K'(\tau)}{3} + \frac{J'(\tau)}{6} + J'(\tau) K(\tau) + J(\tau) K'(\tau) \right] \, , \label{eq:G1e2} \\
G_2^{(2)}(\tau) &= - 8 \pi \alpha \left[\frac{K(\tau)}{6} + \frac{J(\tau)}{6} - \frac12 K(\tau)^2 + K(\tau)J(\tau)\right] \nonumber \\
&\phantom{=} + 2 \pi \alpha \tau \left[\frac{K'(\tau)}{6} - K(\tau)K'(\tau) + \frac{J'(\tau)}{6} + J'(\tau) K(\tau) + J(\tau) K'(\tau) \right] \, , \label{eq:G2e2} \\
G_1^{(3)}(\tau) &= - \sqrt{2 \pi} \alpha^{3/2} \sqrt{J(\tau)} \times \tau \left[2 j(\tau) - \tau j'(\tau) + \frac{\tau^2 j(\tau)^2}{2 J(\tau)} \right] \, , \label{eq:G1e3} \\
G_2^{(3)}(\tau) &= \sqrt{2 \pi} \alpha^{3/2} \sqrt{J(\tau)} \left[\frac{\left(2 J(\tau) + \tau^2 j(\tau)\right)^2}{2 J(\tau)} + 6 J(\tau) + \tau^2 \left( 3 j(\tau) - \tau j'(\tau)\right) \right] \, , \label{eq:G2e3}
\end{align}
where $(\cdots)' = \mathrm{d}(\cdots)/\mathrm{d} \tau$, and with the additional functions
\begin{align}
j(\tau) &\equiv \frac{1}{\pi^2} \int_{0}^{\infty}{\mathrm{d} \omega \, \frac{\exp{(\sqrt{\omega^2 + \tau^2})}}{(\exp{(\sqrt{\omega^2 + \tau^2})}+1)^2}} \, , \\
K(\tau) &\equiv \frac{1}{\pi^2} \int_{0}^{\infty}{\mathrm{d} \omega \, \frac{\omega^2}{\sqrt{\omega^2 + \tau^2}} \frac{1}{\exp{(\sqrt{\omega^2 + \tau^2})}+1}} \, , \\
k(\tau) &\equiv \frac{1}{\pi^2} \int_{0}^{\infty}{\mathrm{d} \omega \, \frac{1}{\sqrt{\omega^2 + \tau^2}} \frac{1}{\exp{(\sqrt{\omega^2 + \tau^2})}+1}} \, .
\end{align}
We discarded a logarithmic contribution to $G_{1,2}^{(2)}$ that is insignificant compared to the dominant contribution to $G_{1,2}^{(2)}$ and even compared to $G_{1,2}^{(3)}$ \cite{Bennett2020}. Note that our expressions look formally different from those of previous literature. For instance \eqref{eq:G1e2} is formally different from the equivalent equation in \cite{Mangano2002,Bennett2020}, while \eqref{eq:G2e2} matches formally with \cite{Mangano2002}, but not with \cite{Bennett2020}. Finally, \eqref{eq:G1e3} and \eqref{eq:G2e3} slightly differ from expressions reported in \cite{Bennett2020}. All expressions are in fact identical, since one can prove (after integrations by parts and rearrangements) the following identities:
\begin{equation}
J'(\tau) = - \tau j(\tau) \; , \
K'(\tau) = - \tau k(\tau) \; , \
Y'(\tau) = - 3 \tau J(\tau) \; , \
2 K(\tau) + \tau^2 k(\tau) = J(\tau) \, .
\end{equation}
\section{Adiabatic transfer of averaged oscillations}
\label{sec:ATAO}
Solving the full QKE \eqref{eq:QKE_final} is \emph{a priori} a considerable numerical challenge because of the need to resolve numerically both the effect of the mean-field terms and of computationally expensive collision integrals. However, the previous numerical results including flavour mixing~\cite{Mangano2005,Relic2016_revisited,Akita2020} seem to indicate that the expected oscillations are somehow "averaged" while there is a comparatively slow evolution due to collisions. Indeed, these studies solved the full QKE (except some approximations in the collision term), hence the vacuum and matter effects must be fully included in their results.
We expect a clear separation of time-scales to hold between the fast oscillations and the secular evolution due to the change of Hamiltonian and the collision term, which would allow for an effective description correctly capturing the salient features of the dynamical evolution. Let us start from the QKE written in its compact form~\eqref{eq:QKE_compact}. We treat the $y$ dependence of $\mathcal{V}$ implicitly, as the following procedure must be applied for each $y$. Since the Hamiltonian $\mathcal{V}$ is Hermitian, it can be diagonalized by the unitary transformation
\begin{equation}
\mathcal{V} = U_\mathcal{V} D_\mathcal{V} U_\mathcal{V}^\dagger \qquad \text{with} \qquad (D_\mathcal{V})^j_k = (D_\mathcal{V})^j_j \, \delta^{j}_{k} \, .
\end{equation}
The density matrix in the matter basis reads $\varrho_\mathcal{V} = U_\mathcal{V}^\dagger \, \varrho \, U_\mathcal{V}$, and evolves according to
\begin{equation}
\label{eq:QKEmatt}
\frac{\partial \varrho_\mathcal{V}}{\partial x} = -{\mathrm i} \left[ D_\mathcal{V},\varrho_\mathcal{V} \right] - \left[U_\mathcal{V}^\dagger \frac{\partial U_\mathcal{V}}{\partial x},\varrho_\mathcal{V} \right] + U_\mathcal{V}^\dagger \mathcal{K} U_\mathcal{V} \, .
\end{equation}
The first approximation that we consider is the \emph{adiabatic approximation} \cite{HannestadTamborra,GiuntiKim} which consists in neglecting the time evolution of the matter PMNS matrix compared to the inverse effective oscillation frequency:
\begin{align}
&\text{\textsc{ Adiabatic approximation}}& \norm{U_\mathcal{V}^\dagger \frac{\partial U_\mathcal{V}}{\partial x}} &\ll \norm{D_\mathcal{V}} \, . \label{eq:adiab_cond}
\\
\intertext{This condition means that the effective mixing matrix elements vary very slowly compared to the effective oscillation frequencies, so that the matter basis evolves adiabatically. More specifically, we need to check that $\abs{\left(U_\mathcal{V}^\dagger \frac{\partial U_\mathcal{V}}{\partial x}\right)^j_k} \ll \abs{(D_\mathcal{V})^j_j - (D_\mathcal{V})^k_k}$. Such adiabaticity condition is particularly important in presence of Mikheev-Smirnov-Wolfenstein (MSW) resonances \cite{MSW_MS,MSW_W}. Note that the sign of the mean-field contribution to $\mathcal{V}$ \eqref{eq:Hlep} is opposite to the one encountered due to charged-current neutrino-electron scattering at lowest order, important for astrophysical environments (Sun, supernovae, binary neutron star mergers).
We numerically check (Figure~\ref{fig:ODG_adiab}) that the condition \eqref{eq:adiab_cond} is indeed satisfied throughout the range of temperatures of interest. \endgraf
If we now assume that many oscillations take place before the collision term varies substantially and write the collision term in matter basis $\mathcal{K}_\mathcal{V} \equiv U_\mathcal{V}^\dagger \mathcal{K} U_\mathcal{V}$, its variation frequency $\sim \mathcal{K}_\mathcal{V}^{-1} (\partial \mathcal{K}_\mathcal{V}/\partial x)$ must be small compared to the effective oscillation frequency $D_\mathcal{V}$. We also assume that the collision rate itself is small compared to the oscillation frequencies, namely}
&\text{\textsc{ Averaged oscillations}}& \norm{\mathcal{K}_\mathcal{V}}, \norm{\mathcal{K}_\mathcal{V}^{-1} \frac{\partial \mathcal{K}_\mathcal{V}}{\partial x}} &\ll \norm{D_\mathcal{V}} \label{eq:collis_cond}
\, .
\end{align}
We check on Figure~\ref{fig:ODG_coll} that this separation of time-scales holds. Therefore, it is possible to \emph{average} the evolution of $\varrho_\mathcal{V}$ over many oscillations (the collision term produces at constant rate neutrinos with random initial phases). The non-diagonal parts will then be washed out if the collision rate is not too strong. More precisely, we can write
\begin{equation}
\label{eq:solveapprox}
(\varrho_\mathcal{V})^j_k(x,y) \equiv e^{- {\mathrm i} (D_\mathcal{V})^j_j x} R^{j}_{k}(x,y) e^{{\mathrm i} (D_\mathcal{V})^k_k x} \ \implies \ \frac{\partial R^{j}_{k}}{\partial x} = e^{{\mathrm i} (D_\mathcal{V})^j_j x}(\mathcal{K}_\mathcal{V})^j_k e^{- {\mathrm i} (D_\mathcal{V})^k_k x} \, ,
\end{equation}
where we also assumed a slow variation of $D_\mathcal{V}$, as a consequence of the adiabatic approximation. If \eqref{eq:collis_cond} holds, $\partial R^{j}_{k}/\partial x$ is integrated over many oscillations and the off-diagonal parts vanish. This leaves us with the effective equation in matter basis:
\begin{equation}
\label{eq:ATAO}
\text{\textsc{\bfseries Adiabatic Transfer of Averaged Oscillations}} \qquad \left\{ \begin{aligned}
\frac{\partial \tilde{\varrho}_\mathcal{V}}{\partial x} &= \reallywidetilde{U_\mathcal{V}^\dagger \mathcal{K} U_\mathcal{V}} \\
\varrho_\mathcal{V} &= \tilde{\varrho}_\mathcal{V} \end{aligned} \right. \, ,
\end{equation}
where the tilde means that we only keep the diagonal terms of $\varrho_\mathcal{V}$, then convert it to the flavour basis to compute the collision term $\mathcal{K}$ and only keep the diagonal part of the collision term $U_\mathcal{V}^\dagger \mathcal{K} U_\mathcal{V}$ when transforming back to the matter basis. In the flavour basis, the density matrix $\varrho = U_\mathcal{V} \tilde{\varrho}_\mathcal{V} U_\mathcal{V}^\dagger$ has off-diagonal components, while $\tilde{\varrho}_\mathcal{V}$ is diagonal. Therefore the collision term destroys the coherence between these components (since it aims at a diagonal $\varrho$ in flavour space, with equilibrium distributions), which modifies in turn the diagonal values of $\varrho_\mathcal{V}$ (whose off-diagonal terms average out).
\begin{figure}[!ht]
\centering
\includegraphics{figs/Adiabaticity_condition_NH.pdf}
\caption[Checking the Hamiltonian adiabaticity condition]{\label{fig:ODG_adiab} Evolution of the different quantities appearing in \eqref{eq:QKEmatt} in the normal hierarchy of masses, for a comoving momentum $y=5$. The condition~\eqref{eq:adiab_cond} is satisfied throughout the evolution.}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics{figs/Collision_condition_NH.pdf}
\caption[Checking the slowness of the rates associated to the collision term]{\label{fig:ODG_coll} Comparison of the evolution of the collision term, its relative variation and the effective oscillation frequencies in the normal hierarchy of masses, for a comoving momentum $y=5$. We check that the condition \eqref{eq:collis_cond} is satisfied with several orders of magnitude.}
\end{figure}
For clarity, we refer to this approximate numerical scheme to determine the neutrino evolution "Adiabatic Transfer of Averaged Oscillations'' (ATAO) and we then solve \eqref{eq:ATAO} instead of \eqref{eq:QKE_compact}. Note that we explained the procedure with $\mathcal{V}$, but it could be carried out with any Hamiltonian $\mathcal{H}$, which will be relevant when dealing with asymmetries (cf.~chapter~\ref{chap:Asymmetry}). The full designation of this scheme is thus rather {ATAO}-$\mathcal{V}\,$ but we will simply call it "ATAO" in this chapter since there cannot be any confusion.
In the following section, we will numerically solve the QKEs in both the full "exact" case and the ATAO approximation and discuss the validity of the approximate numerical solution.
\newpage
\section{Numerical implementation}
\label{sec:numeric_Dec}
We integrate numerically the QKE for neutrinos~\eqref{eq:QKE_final}, or \eqref{eq:ATAO} in the ATAO approximation, along with the energy conservation equation~\eqref{eq:zQED}. We developed the code \texttt{NEVO} (Neutrino EVOlver), written in Python with the \texttt{scipy} and \texttt{numpy} libraries.\footnote{Time consuming functions are compiled with the just-in-time compiler \texttt{numba}.}
\paragraph{Solver and initial conditions}
The collision term consists most of the time in nearly compensating gain and loss terms, and for temperatures larger than $0.1\, \text{MeV}$, the system is very stiff. Hence, one must rely on an implicit method. We chose the \texttt{LSODA} method which consists in a Backward Differentiation Formula (BDF) method (with adaptative order and adaptative step) when the system is stiff, which switches to an explicit method when not stiff (the Adams method). It was first distributed within the \texttt{ODEPACK} Fortran library~\cite{ODEPACK}, but we used the Python wrapper \texttt{solve\_ivp} distributed with the Python \texttt{scipy} module. We noticed that when setting the absolute and relative error tolerances to $10^{-n}$, the spectra are typically obtained with precision better than $10^{-n+2}$, in agreement with section B.5 of \cite{Gariazzo_2019}. Hence we fixed these error tolerances to $10^{-7}$ so as to obtain results with numerical errors below $10^{-5}$.
The initial common temperature of all species, that is all types of neutrinos and the electromagnetic plasma, is inferred from the conservation of total entropy. Choosing the initial comoving temperature $T_{\mathrm{cm},\mathrm{in}} = 20 \, \text{MeV}$, the initial common temperature of all species is slightly larger because of early $e^\pm$ annihilations, and given by $T_\mathrm{in} = z_\mathrm{in} T_{\mathrm{cm},\mathrm{in}}$ with $z_\mathrm{in}-1 = 7.42\times 10^{-6}$. Had we chosen to start at $T_{\mathrm{cm},\mathrm{in}} = 10 \, \text{MeV}$, the initial comoving temperature would have been $z_\mathrm{in}-1 = 2.98\times 10^{-5}$, in agreement with~\cite{Mangano2005,Dolgov_NuPhB1999}.
As initial condition for the density matrix we take
\begin{equation}
\label{eq:initial_condition}
\varrho(x_{\mathrm{in}},y) = \begin{pmatrix}
f_\nu^{({\mathrm{in}})}(y) & 0 & 0 \\
0 & f_\nu^{(\mathrm{in})}(y) & 0 \\
0 & 0 & f_\nu^{(\mathrm{in})}(y)
\end{pmatrix} \quad , \quad \text{with} \quad f_\nu^{(\mathrm{in})}(y) \equiv \frac{1}{e^{y/z_\mathrm{in}} + 1} \, .
\end{equation}
\paragraph{Momentum grid}
The neutrino spectra are sampled with $N$ points on a grid in the reduced momentum $y$. When choosing a linear grid, we use the range $0.01 \leq y\leq 16 + [N/20]$, and integrals are evaluated with the Simpson method. However, for functions which decay exponentially for large $y$, it is motivated to use the Gauss-Laguerre quadrature which was already proposed in~\cite{Gariazzo_2019}. We confirm that this method typically requires half of the grid points to reach the same precision as the one obtained with a linear spacing. In practice, when choosing the nodes and weights of the quadrature, we restrict to $y\leq 20+[N/5]$. When using $N=80$, we have thus restricted nodes to $y \leq 36$, and we used Laguerre polynomials of order $439$ to compute the weights with Eq.~(B.14) of \cite{Gariazzo_2019}. Since the tools provided in \texttt{numpy} are restricted to much lower polynomial orders, we used \emph{Mathematica} to precompute once and for all in a few hours the nodes and weights. The results reported in the following were performed with $N=80$ and the Gauss-Laguerre quadrature, checking that with $N=100$ the differences are smaller than the desired precision.
For each momentum $y_i$ of the grid, and with $N_{\nu}$ flavours, each density matrix has $N_{\nu}^2$ independent degrees of freedom ($N_\nu(N_{\nu} +1)$ real parts and $N_\nu(N_{\nu} -1)$ imaginary parts). In practice we reorganize these independent matrix entries into a vector $A^j(y_i)$ with $j=1,\dots,N_{\nu}^2$ and we concatenate them with the $y_i$ spanning the momentum grid. We thus solve for serialized variables, that is a giant vector of length $N N_\nu^2$. When using the ATAO approximation, one needs only to keep the diagonal part in the matter basis, and the giant vector is of size $N N_\nu$.\footnote{Results are then only converted at the very end in the flavour basis if desired.} Note that we do not store the binned density matrix components $\varrho^\alpha_\beta(y_i)$, which would be sub-optimal. Indeed, if neutrinos decoupled instantaneously, their distribution function would then be
\begin{equation}
\label{eq:fnueq}
f_{\nu}^{\mathrm{(eq)}}(x,y) \equiv \frac{1}{e^{y}+1} \, .
\end{equation}
Therefore, we can parametrize the density matrix $\varrho^\alpha_\beta(x,y) = \left[\delta^{\alpha}_{\beta}+a^\alpha_\beta(x,y)\right]\times f_\nu^{\mathrm{(eq)}}(x,y)$, and we store the values of $a^\alpha_\beta$, which encapsulate the deviation from instantaneous decoupling.
\paragraph{Mixing parameters} As specified at the end of section~\ref{sec:set_of_equations}, we will solve the QKEs in the normal ordering case and without a Dirac CP phase, therefore using the parameters given in~\ref{subsec:Values_Mixing}. We specifically include the CP phase in section~\ref{subsec:Decoupling_CP}, and discuss the case of the inverted hierarchy of masses in section~\ref{subsec:Inverted_Hierarchy}.
\paragraph{Numerical optimization via Jacobian computation.} The implicit method requires to solve algebraic equations and thus to obtain the Jacobian of the differential system. For the sake of this discussion, and to alleviate the notation, we ignore the different flavours and consider that we have only one neutrino flavour with spectrum $f(y)$. Noting the grid points $y_i$ and the values of the spectra $f_i = f(y_i)$ on the grid, the differential system is of the type $\partial_x f_i = \mathcal{K}_i(x, f_j)$. The implicit method requires the Jacobian $J_{ij} \equiv \partial \mathcal{K}_i/\partial f_j$. If no expression is provided, it is evaluated by finite differences in the $\{ f_i\}$ at a given $x$. Since the collision term involves a two-dimensional integral for each point of the grid, its computation on the whole grid is of order $\mathcal{O}(N^3)$. Hence, the computation of the Jacobian with finite differences is of order $\mathcal{O}(N^4)$. Since algebraic manipulations (mostly the LU decomposition) are at most of order $\mathcal{O}(N^3)$, reducing the cost of the Jacobian numerical evaluation is crucial to improve the speed of the implicit method. Fortunately, it is possible to compute the Jacobian with an $\mathcal{O}(N^3)$ complexity. To use a simple example, let us only consider the contribution from the loss part of the neutrino self-interactions, without including Pauli-blocking factors. This component of the collision term, once computed numerically with a quadrature, is of the form
\begin{equation}
\label{ContriC}
\mathcal{K}_i(x, f_j) = -\sum_{j,k} w_j w_k g(y_i,y_j,y_k) f_i f_j\,.
\end{equation}
In this expression $\sum_j w_j$ (resp. $\sum_k w_k$) accounts for the integration on $y_2$ (resp.~$y_3$) in~\eqref{eq:I_full} using a quadrature, and the function $g$ takes into account the specific form of the factor multiplying the statistical function (which is for the contribution considered $f_i f_j$).
Noting then that
\begin{equation}
\partial f_i /\partial f_j = \delta_{ij} \, ,\label{dfdf}
\end{equation}
the Jacobian associated with the contribution~\eqref{ContriC} is
\begin{equation}\label{Jimtwocontrib}
J_{i m} = \partial \mathcal{K}_i/\partial f_m = -\delta_{im} \sum_{j,k} w_j w_k g(y_i,y_j,y_k) f_j - \sum_{k} w_m w_k g(y_i,y_m,y_k) f_i \, .
\end{equation}
The complexity of the second sum is of order $\mathcal{O}(N)$, and since the Jacobian has $N^2$ entries, it leads to a complexity of order $\mathcal{O}(N^3)$. The first term is not worse even though the double sum is of order $\mathcal{O}(N^2)$, because it concerns only the diagonal entries of the Jacobian due to the prefactor $\delta_{im}$. More generally for all contributions to the collision term, the complexity when computing the associated Jacobian is always of order $\mathcal{O}(N^3)$, even when taking into account Pauli-blocking factors which bring terms which are cubic or quartic in the density matrix. For instance, terms similar to~\eqref{ContriC}, but with factors $f_i f_j f_k$, are handled with the same method and would lead to three contributions instead of two in~\eqref{Jimtwocontrib}. As for terms with factor $f_i f_j f_l$, they would be handled using total energy conservation $y_i + y_j = y_k + y_l$, which allows for instance to replace the variables of summations (e.g. $\sum_{j,k} \to \sum_{j,l}$) when varying with respect to $f_l$. Following these arguments, one notices that the exponent of the complexity for both the collision term and its associated Jacobian is given by the number of independent momenta magnitudes, given that integrations on momenta directions have all been removed with the integration reduction method using the isotropy of momentum distribution. In the case at hand, we have only two-body collisions, for which total energy conservation implies that only three momenta magnitudes are independent, hence the complexity in $\mathcal{O}(N^3)$.
When restoring the fact that we do not have a single flavour but density matrices, the discussion is similar when using the serialized variables described above, and again the complexity is of order $\mathcal{O}(N^3)$. In practice, we found that it takes roughly five times more time to compute a Jacobian than a collision term. Hence, when compared with the finite difference method, providing a numerical method for the Jacobian leads to a factor $N/5$ speed-up. Note that we must also integrate $z$ with eq.~\eqref{eq:zQED} jointly with the density matrices, so that we must pad the Jacobian obtained with the previous description with one extra line and one extra column. Again, the corresponding entries can be deduced using~\eqref{dfdf} and their computation is also of order $\mathcal{O}(N^3)$.
It is worth mentioning that providing a method for the Jacobian is not specific to the ATAO approximation. Indeed, when solving the full QKE one can also compute the Jacobian of the collision term, and one only needs to add the contribution from the vacuum and mean-field commutators whose complexity is simply of order $\mathcal{O}(N^2)$. The precise description of the Jacobian calculation, also adequate for the case of non-zero asymmetries (cf.~chapter~\ref{chap:Asymmetry}) is done in appendix~\ref{App:Numerics}.
When compared with the full QKE method, the ATAO numerical resolution allows to gain at least a factor 5 in time. Hence when using both a method for the Jacobian and the ATAO approximation, we gain typically a factor $N$ and computations that would otherwise last days on CPU clusters, are reduced to just few hours on a single CPU. Moreover, nothing prevents the computation of collision terms and Jacobians to be parallelized on the momentum grid, as we checked on the 4 or 8 CPUs of desktop machines, reducing even further the computation time.
\section{Neutrino temperature and spectra after decoupling}
\label{sec:results_nudec}
We use our code \texttt{NEVO} with the parameters given in the previous section to follow neutrino evolution across the MeV era. The quantities we are most interested in are the frozen-out spectra of neutrinos after decoupling, which allow to compute in particular their energy density (hence $N_{\mathrm{eff}}$ as we also know the final photon temperature).
\subsection[Numerical results: benchmark value for $N_{\mathrm{eff}}$]{Numerical results: benchmark value for $\bm{N_{\mathrm{eff}}}$}
\label{subsec:results_Neff}
Since the diagonal entries of the neutrino density matrix correspond to the generalization of the distribution functions, we can use the same parameterization as~\eqref{eq:param_fnu} to separate effective temperatures and residual distortions, namely,
\begin{equation}
\label{eq:param_rho}
\varrho^\alpha_\alpha(x,y) \equiv \frac{1}{e^{y/z_{\nu_\alpha}} + 1} \left[1 + \delta g_{\nu_\alpha}(x,y)\right] \, ,
\end{equation}
where we recall that the reduced effective temperature $z_{\nu_\alpha} \equiv T_{\nu_\alpha}/T_{\mathrm{cm}}$ is the reduced temperature of the Fermi-Dirac spectrum with zero chemical potential which has the same energy density as the real distribution:
\begin{equation}
\frac{1}{2 \pi^2} \int{\mathrm{d}{y} y^3 \varrho^\alpha_\alpha(y)} = \bar{\rho}_{\nu_\alpha} \equiv \frac78 \frac{\pi^2}{30} z_{\nu_\alpha}^4 \,.
\end{equation}
The appeal of this parameterization lies in the clear separation between energy changes compared to instantaneous decoupling (hence gravitational effects through the Hubble parameter), and non-thermal distortions which can only affect the reaction rates. This will be very useful to study the consequences of incomplete neutrino decoupling on BBN in chapter~\ref{chap:BBN}.
We plot in Figure~\ref{fig:Tnu} the evolution of the neutrino effective temperatures, with and without flavour oscillations (including all QED corrections to the plasma thermodynamics), and in Figure~\ref{fig:deltagnu} the non-thermal residual distortions. In the absence of flavour mixing (dashed lines on Figures~\ref{fig:Tnu} and \ref{fig:deltagnu}), we cannot distinguish between $\nu_\mu$ and $\nu_\tau$ since they have exactly the same interactions with $e^\pm$ (neutral-currents only) and with (anti)neutrinos, thus nothing distinguishes these two flavours in the no-mixing case. On the contrary, the higher effective temperatures or non-thermal distortions for the electronic flavour are due to the charged-current processes, which increase the transfer of entropy from electrons and positrons.
\begin{figure}[!ht]
\centering
\includegraphics{figs/Evolution_Tnu.pdf}
\caption[Evolution of the effective neutrino temperatures]{\label{fig:Tnu} Evolution of the effective neutrino temperatures, with and without oscillations. Long before decoupling, they remain equal to the photon temperature $z$, before freezing-out at different values depending on the interaction with the electromagnetic plasma. Without mixing, the distribution functions (and thus, the effective temperatures) are identical for $\nu_\mu$ and $\nu_\tau$.}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics{figs/deltagnu_final.pdf}
\caption[Frozen-out effective spectral distortions]{\label{fig:deltagnu} Frozen-out effective spectral distortions, with and without oscillations, for $x_f \simeq 51$ (corresponding to $T_{\mathrm{cm},f} = 0.01 \, \mathrm{MeV}$). The full QKE results are indistinguishable from the ATAO approximate ones.}
\end{figure}
The final values of the comoving temperatures and $N_{\mathrm{eff}}$ are given in Table~\ref{Table:Res_NuDec}. We find that including the different corrections shifts $N_{\mathrm{eff}}$ in agreement with the results quoted in Table 5 of~\cite{Bennett2021}:
\begin{itemize}
\item the $\mathcal{O}(e^2)$ finite-temperature QED correction to the plasma equation of state increases $N_{\mathrm{eff}}$ by $\sim 0.01$ (fourth and fifth rows in Table~\ref{Table:Res_NuDec}) ;
\item the next order in the QED corrections, $\mathcal{O}(e^3)$ reduces $N_{\mathrm{eff}}$ by $\sim 0.001$ (fifth and sixth rows in Table~\ref{Table:Res_NuDec}), as predicted in~\cite{Bennett2020}, and also observed in~\cite{Akita2020} ;
\item flavour oscillations have a subdominant contribution compared to QED corrections, namely $+ \, 6 \times 10^{-4}$ (third and sixth rows in Table~\ref{Table:Res_NuDec}).
\end{itemize}
\renewcommand{\arraystretch}{1.3}
\begin{table}[!htb]
\centering
\begin{tabular}{|l|ccccc|}
\hline
Final values & $z$ & $z_{\nu_e}$ & $z_{\nu_\mu}$ & $z_{\nu_\tau}$ &$N_{\mathrm{eff}}$ \\
\hline \hline
Instantaneous decoupling, no QED & $1.40102$ & $1.00000$ & $1.00000$ &$1.00000$ & $3.00000$ \\ \hline
No oscillations (NO), QED $\mathcal{O}(e^3)$ &$1.39800$ & $1.00234$ & $1.00098$ & $1.00098$ &$3.04338$ \\
ATAO, no QED & $1.39907$ & $1.00177$ & $1.00134$ & $1.00132$ & $3.03463$ \\
ATAO, QED $\mathcal{O}(e^2)$ & $1.39786$ & $1.00175$ & $1.00132$ & $1.00130$ & $3.04491$ \\ \hline
ATAO, QED $\mathcal{O}(e^3)$ & $1.39797$ & $1.00175$ & $1.00132$ & $1.00130$ & $3.04396$ \\
Full QKE, QED $\mathcal{O}(e^3)$ & $1.39797$ & $1.00175$ & $1.00132$ & $1.00130$ & $3.04396$ \\ \hline
\end{tabular}
\caption[Final (effective) temperatures after decoupling]{Frozen-out values of the dimensionless photon and neutrino temperatures, and the effective number of neutrino species. We detail the results of different implementations in order to assess the contribution of each correction w.r.t.~the instantaneous decoupling approximation. $N_{\mathrm{eff}}$ differs between the ATAO and full QKE calculations by a few $10^{-6}$, which we attribute mainly to numerical errors. The values quoted here differ slightly from~\cite{Froustey2020} due to the updated value of $\mathcal{G}$~\cite{PDG}, which affects the Hubble expansion rate.
\label{Table:Res_NuDec}}
\end{table}
Flavour oscillations reduce the discrepancy between the different flavours, thus $z_{\nu_e}$ is reduced while $z_{\nu_\mu}$ and $z_{\nu_\tau}$ are increased, with a very slightly higher value for $z_{\nu_\mu}$. This enhanced entropy transfer towards $\nu_\mu$ compared to $\nu_\tau$ is due to the more important $\nu_e-\nu_\mu$ mixing (cf.~Figure~\ref{fig:ATAO_Transfer} and the corresponding discussion).
\paragraph{Comparison with the literature}
The deviation of the dimensionless temperatures with respect to $1$ can be expressed as a relative change in the energy density, $\delta \bar{\rho}_{\nu_\alpha} \simeq 4 (z_{\nu_\alpha} - 1)$. Our values for the increase in the neutrino energy density are $\delta \bar{\rho}_{\nu_e} \simeq 0.70 \, \%$, $\delta \bar{\rho}_{\nu_\mu} \simeq 0.53 \, \%$ and $\delta \bar{\rho}_{\nu_e} \simeq 0.52 \, \%$. This is in agreement with the results of~\cite{Relic2016_revisited} (Table 1) or~\cite{Akita2020} (Table 2), except for the relative variation of muon and tau flavours: these works obtain a higher reheating of $\nu_\tau$ compared to $\nu_\mu$, while we find the opposite. This is due to a difference in the values of the mixing angles.\footnote{For instance, the older values used in \cite{Mangano2005} lead to higher distortions for $\nu_\mu$ than for $\nu_\tau$.} Nevertheless, if we use the mixing angles from \cite{Relic2016_revisited}, we obtain $\delta \bar{\rho}_{\nu_e} \simeq 0.694 \, \%$, $\delta \bar{\rho}_{\nu_\mu} \simeq 0.525 \, \%$ and $\delta \bar{\rho}_{\nu_\tau} \simeq 0.530 \, \%$. Furthermore, if $\mathcal{O}(e^3)$ QED corrections are not included and only the diagonal components of the self-interaction collision term are kept, the spectra reach less flavour equilibration and the results of \cite{Relic2016_revisited} are recovered (at the level of a few $10^{-5}$): $\delta \bar{\rho}_{\nu_e} \simeq 0.706 \, \%$, $\delta \bar{\rho}_{\nu_\mu} \simeq 0.515 \, \%$ and $\delta \bar{\rho}_{\nu_\tau} \simeq 0.522 \, \%$.
We emphasize that our work is the first to include the full form of the self-interaction collision term and QED corrections to the plasma thermodynamics up to $\mathcal{O}(e^3)$ order. Compared to~\cite{Relic2016_revisited}, the closeness of our results is due to a compensation between the update in the value of physical parameters (namely $\mathcal{G}$, $G_F$) and the new ingredients of our calculation. Our results have been independently confirmed in~\cite{Bennett2021}, providing the new reference value for $N_{\mathrm{eff}}$~\cite{PDG}.
\paragraph{Validity of the ATAO approximation} Finally, the results in Table~\ref{Table:Res_NuDec} show the striking accuracy of the ATAO approximation, as expected since the conditions \eqref{eq:adiab_cond} and \eqref{eq:collis_cond} are satisfied by several orders of magnitude (Figures~\ref{fig:ODG_adiab} and \ref{fig:ODG_coll}). The frozen-out values of the comoving temperatures and of $N_{\mathrm{eff}}$ differ by $10^{-6}$, which is beyond our desired accuracy, and beyond the expected effect of neglected contributions.
In chapter~\ref{chap:Asymmetry}, we extend the ATAO approximation to the case of a non-zero neutrino/antineutrino asymmetry, showing once again its accuracy and how it provides a very efficient numerical way to tackle the problem of neutrino evolution.
\paragraph{Increase of $\bm{N_{\mathrm{eff}}}$ due to mixing} The numerical solution of the QKE shows a larger $N_{\mathrm{eff}}$ value (Table~\ref{Table:Res_NuDec}) compared to the no-oscillation case. To understand this slight increase of the total energy density of neutrinos, one should keep in mind that electron-positron annihilations, which is the dominant energy-transferring process during decoupling~\cite{Grohs2015}, are more efficient in producing electronic type neutrinos (because of the existence of charged-current processes). Now the mixing and mean-field terms tend to depopulate $\nu_e$ and populate the other flavours, which frees some phase space for the reactions which create $\nu_e$, while increasing the effect of Pauli-blocking factors for reactions creating $\nu_{\mu,\tau}$. Since the former are the dominant reactions, the net effect is a larger entropy transfer from $e^\pm$, hence the larger value of $N_{\mathrm{eff}}$. We further clarify the effect of mixing and mean-field terms in the light of the ATAO approximation in the section~\ref{sec:dependence_parameters}.
\vspace{0.5 cm}
Before discussing the various limitations to this calculation in the next section, we can quote the main result, namely the value of $N_{\mathrm{eff}}$ in the Standard Model of particle physics and in the $\Lambda$CDM model of cosmology,
\begin{equation}
\label{eq:result_Neff}
\boxed{N_{\mathrm{eff}} = 3.0440} \ ,
\end{equation}
read on the two last rows of Table~\ref{Table:Res_NuDec}. The uncertainty on this result will be discussed in section~\ref{sec:dependence_parameters}.
\subsection{Limitations of the calculation}
\label{subsec:limitations_NuDec}
Let us discuss the some physical features that are not taken into account in this calculation of neutrino decoupling.
\paragraph{Neglected corrections to the plasma thermodynamics}
As explained in section~\ref{subsec:QED}, the energy density and pressure of $e^\pm$ and photons are modified at finite temperature, which is taken into account via the functions $G_1$ and $G_2$ in~\eqref{eq:zQED}. These functions are computed from the partition function of the QED plasma, in an expansion in powers of $e$. In our calculation, we included the contributions at order $\mathcal{O}(e^2)$ and $\mathcal{O}(e^3)$.
However, we discarded at order $\mathcal{O}(e^2)$ a "log-dependent" term, that is a contribution to the energy density and pressure involving a double integral with a logarithmic dependence on momenta. It was indeed estimated that it should lead to a negligible variation $\Delta N_{\mathrm{eff}} \sim - 5 \times 10^{-5}$ (see Eq.~(4.21) in~\cite{Bennett2020}), a result confirmed in~\cite{Bennett2021} (Table 3).
We have shown that the $\mathcal{O}(e^3)$ contribution leads to a reduction of $N_{\mathrm{eff}}$ by $10^{-3}$, which naturally asks the question of the higher order contributions. We can safely ignore them according to the estimate of~\cite{Bennett2020}, which found for the $\mathcal{O}(e^4)$ correction a contribution $\Delta N_{\mathrm{eff}} \sim 3.5 \times 10^{-6}$ in the ultra-relativistic limit. This departure from $N_{\mathrm{eff}} = 3.044$ is way beyond our uncertainty goal.
Therefore, the finite-temperature QED corrections to the plasma thermodynamics can be considered as fully taken into account at the level of $10^{-4}$ for the value of $N_{\mathrm{eff}}$.
\paragraph{Finite-temperature corrections to the scattering rates} In the problem we consider, finite-temperature corrections do not appear only in bulk thermodynamic quantities, but also in weak scattering rates. Therefore, they should be accounted for in the collision integral $\mathcal{I}$, in reactions involving electrons and positrons. There are four types of such corrections to the weak rates~\cite{Tomalak2019,Bennett2021}: modification to the dispersion relation (which also leads to the modifications of the energy density and pressure), vertex corrections, real emission or absorption of photons, and closed fermion loops. Including them consistently in the collision term is a very tedious task that has not been done yet.
The only estimate of these effects in the literature has been done using the value of the difference in the energy loss rate in $e^- + e^+ \to \nu + \bar{\nu}$ due to the rate corrections in~\cite{Esposito_QED}, which would lead to $\Delta N_{\mathrm{eff}} \sim - 0.001$ according to~\cite{Escudero_2020}. However, the estimate of~\cite{Esposito_QED} uses "temperature-dependent wave-function renormalization" techniques, which have led to disagreements in the literature (notably with detailed balance requirements being missed~\cite{BrownSawyer}). We thus conclude that a specific study of finite-temperature corrections to the scattering rates and their effect on neutrino decoupling must be undertaken.
\paragraph{Hypothesis of isotropy} There is, finally, a global underlying assumption that simplifies considerably the problem: isotropy. Indeed, without this hypothesis the density matrices would depend on both the direction and the magnitude of $\vec{p}$. This leads, for instance, to non-vanishing angular integrals in the self-interaction mean-field (that is important in the asymmetric case, see chapter~\ref{chap:Asymmetry}). Moreover, isotropy is also used in the reduction of the collision integral down to two dimensions. For these reasons, the calculation would be considerably more involved if one released this assumption.
We can quote the work~\cite{Hansen_Isotropy}, which tried to tackle this problem with a simplified framework: two bins of neutrinos (left-moving and right-moving, as a very crude anisotropic model), and a simplified collision term. Their results seem to indicate that the mean-field asymmetric neutrino contribution that we discarded might sometimes play a role. However, their treatment of anisotropies remains incomplete: even though a simple anisotropic model might be necessary at this stage, one should also consistently consider the anisotropic degrees of freedom in the metric, whose dynamics is governed by Einstein equations.
\subsection{Neutrinos today}
The neutrino spectra obtained after decoupling remain frozen from that point onwards, so our numerical results allow us to calculate the thermodynamic quantities expected for neutrinos today.
\subsubsection{Neutrino energy density}
\label{subsec:Omeganu}
Given the value of the CMB temperature today $T_\mathrm{CMB}=2.7255 \pm 0.0006 \, \mathrm{K} = (2.3487 \pm 0.0005) \times 10^{-4} \, \mathrm{eV}$~\cite{Fixsen:2009,PDG}, we can estimate the temperature of the C$\nu$B in the instantaneous decoupling limit:\footnote{We do not transfer the uncertainties from $T_\mathrm{CMB}$ since they are below the $\sim 0.1 \, \%$ variation of $T_\nu$ due to incomplete neutrino decoupling, and thus meaningless.}
\begin{equation}
T_\mathrm{C\nu B} = \left(\frac{4}{11}\right)^{1/3} T_\mathrm{CMB} = 1.945 \, \mathrm{K} = 1.676 \times 10^{-4} \, \mathrm{eV} \, .
\end{equation}
Given the values of the mass-squared differences~\eqref{ValuesStandard} $\Delta m_{21}^2 \simeq 7.5 \times 10^{-5} \, \mathrm{eV}^2$ and $\lvert \Delta m_{31}^2 \rvert \simeq 2.5 \times 10^{-3} \, \mathrm{eV}^2$ , the two heaviest mass eigenstates have necessarily masses $m_{\nu_i} \gg T_\mathrm{C\nu B}$ today. We could be more precise by using the results from section~\ref{subsec:results_Neff} and compare $m_{\nu_i}$ and the effective $T_{\nu_i}$ today, but the conclusion would be identical since $T_{\nu_i}$ and $T_\mathrm{C\nu B}$ differ by $\simeq 0.1 \, \%$ (cf.~Table~\ref{Table:Res_NuDec}).
We can then write the total neutrino energy density parameter as
\begin{equation}
\label{eq:calcul_Omeganu}
\Omega_\nu = \frac{\rho_\nu + \rho_{\bar{\nu}}}{\rho_\mathrm{crit}} \simeq \frac{2 \sum_{i}{(m_{\nu_i} n_{\nu_i})}}{\rho_\mathrm{crit}} \, ,
\end{equation}
where we assume a zero asymmetry and that all mass species are non-relativistic today. Should this be wrong, the error made would be completely negligible since $\rho_\nu$ is strongly dominated, in the matter-dominated era, by the contribution from the non-relativistic neutrinos.
The critical density today is \cite{PDG}
\begin{equation*}
\rho_\text{crit} = \frac{3 H_0^2}{8 \pi \mathcal{G}} = 8.0959 \times 10^{-11} \times h^2 \, \mathrm{eV^4} \equiv \rho_\text{crit}^{100} \times h^2
\end{equation*}
This value corresponds to the updated value for Newton's constant of gravitation $\mathcal{G} = 6.67430 \times 10^{-11} \, \mathrm{m^3 \cdot kg^{-1} \cdot s^{-2}}$. $h$ is the present value of the Hubble parameter in units of $100 \, \mathrm{km \cdot s^{-1} \cdot Mpc^{-1}}$.
The conversion between the comoving quantities and the physical ones is done in the following way:
\begin{equation*}
n_{\nu_i} = \bar{n}_{\nu_i} \times T_{\mathrm{cm}}^3 = \bar{n}_{\nu_i} \times \left(\frac{T_\gamma}{z_\gamma}\right)^3
\end{equation*}
Today $T_\gamma = T_\mathrm{CMB}$, so we rewrite
\begin{equation}
\label{eq:Omeganu}
\Omega_\nu = \frac{2 \sum_{i}{m_{\nu_i} \bar{n}_{\nu_i}}}{\rho_\text{crit}^{100} \times h^2} \times \frac{T_\mathrm{CMB}^3}{z_\gamma^3} \, .
\end{equation}
\paragraph{Instantaneous decoupling}
In the instantaneous decoupling limit,
\begin{equation*}
\boxed{\text{Inst. Dec.}} \quad \left\{
\begin{aligned}
z_\gamma &= \left(\frac{11}{4}\right)^{1/3} \\
\bar{n}_\mathrm{ID} &= \frac{3 \zeta(3)}{4 \pi^2}
\end{aligned} \right. \, .
\end{equation*}
From that, we can compute
\[ \frac{\rho_\text{crit}^{100} z_\gamma^3}{2 \bar{n}_\mathrm{ID} T_\mathrm{CMB}^3} = \frac{\rho_\text{crit}^{100} \times \frac{11}{4}}{2 \times \frac{3 \zeta(3)}{4 \pi^2} \times T_\mathrm{CMB}^3} \simeq 94.06 \, \mathrm{eV} \, . \]
Therefore \eqref{eq:Omeganu} becomes:
\begin{equation}
\label{eq:Omeganu_ID}
\boxed{\Omega_\nu^\mathrm{ID} = \frac{\sum_{i}{m_{\nu_i}}}{94.06 \, \mathrm{eV} \times h^2}} \, .
\end{equation}
\paragraph{Incomplete neutrino decoupling} Thanks to our results, rewritten in the mass basis in Table~\ref{Table:Resmass_NuDec}, we can actually make a precise prediction for $\Omega_\nu$.
\begin{table}[!htb]
\centering
\begin{tabular}{|l|ccccc|}
\hline
Final values & $z$ & $z_{\nu_1}$ & $z_{\nu_2}$ & $z_{\nu_3}$ &$N_{\mathrm{eff}}$ \\
\hline \hline
ATAO, QED $\mathcal{O}(e^3)$ & $1.39797$ & $1.00191$ & $1.00143$ & $1.00102$ & $3.04396$ \\ \hline
\end{tabular}
\caption[Frozen-out values of the comoving temperatures for the mass eigenstates]{Frozen-out values of the dimensionless photon and neutrino effective temperatures, defined for the \emph{mass} eigenstates. We only report the ATAO values, which differ from the full QKE results at the level of $10^{-6}$.
\label{Table:Resmass_NuDec}}
\end{table}
As the differences between the different $z_{\nu_i}$ are very small (see Table~\ref{Table:Resmass_NuDec}), the approximation $n_{\nu_1} \simeq n_{\nu_2} \simeq n_{\nu_3}$ is often made, allowing to factorize out the sum of neutrino masses $\sum{m_{\nu_i}}$ in~\eqref{eq:calcul_Omeganu}. Alternatively, one can assume the neutrino masses to be quasidegenerate, i.e.~$m_0 \equiv m_{\nu_1} \simeq m_{\nu_2} \simeq m_{\nu_3} \gg T_\nu^0$. In this case, the differences of masses can be neglected and we can once again factorize them. We then have
\[ \frac{\rho_\text{crit}^{100} z_\gamma^3}{2(\bar{n}_{\nu_1} + \bar{n}_{\nu_2} + \bar{n}_{\nu_3}) T_\mathrm{CMB}^3} \simeq 31.04 \, \mathrm{eV} \, . \]
Thus we can write the result as
\begin{equation}
\label{eq:Omeganu_degenerate}
\boxed{\Omega_\nu = \frac{3 m_0}{93.12 \, \mathrm{eV} \times h^2} = \frac{\sum_{i}{m_{\nu_i}}}{93.12 \, \mathrm{eV} \times h^2}} \, ,
\end{equation}
as long as we do not make a difference between the masses today. We can compare this to the previous result~\cite{Mangano2005}, whose denominator value was $93.14 \, \mathrm{eV}$. Although this is extremely close to what our much more precise study of neutrino decoupling gives, one must remember that some differences are hidden in the new values of the physical constants. For instance, the instantaneous decoupling value derived in Eq.~\eqref{eq:Omeganu_ID} above ($94.06 \, \mathrm{eV}$) was quoted to be $94.12 \, \mathrm{eV}$ at the time of the study~\cite{Mangano2005}.
In order to fully exploit the results of our study, we plot in Figure~\ref{fig:Omeganu} the generalization of this coefficient to any value of the masses. We vary the minimal neutrino mass and deduce the other two given the values~\eqref{ValuesStandard}, depending on the choice of mass ordering. The endpoints of each line correspond to the minimal sum of masses, respectively reached for $m_{\nu_1} = 0$ (normal ordering) and $m_{\nu_3} = 0$ (inverted ordering). We also show the exclusion zone obtained in~\cite{Planck18}, showing the current range of variation of this coefficient, and thus, of $\Omega_\nu$.
\begin{figure}[!ht]
\centering
\includegraphics{figs/Energy_density_masses.pdf}
\caption[Dependence of the neutrino energy density parameter today with the sum of neutrino masses]{\label{fig:Omeganu} Dependence of the neutrino energy density parameter today with the sum of neutrino masses. The grey area is the 95 \% excluded zone by~\cite{Planck18}, $\sum{m_{\nu_i}} < 0.12 \, \mathrm{eV}$. The brown dash-dotted line corresponds to the value quoted in~\cite{Mangano2005,PDG}, for which $m_{\nu_1} \simeq m_{\nu_2} \simeq m_{\nu_3} \gg T_{\nu}^0$, cf.~equation~\eqref{eq:Omeganu_degenerate}.}
\end{figure}
\subsubsection{Neutrino number density}
To obtain the neutrino energy density today, we have actually computed their number density for each mass eigenstate. Summing over all states, the total number density reads
\begin{equation}
\sum_{i}{(n_{\nu_i} + n_{\bar{\nu}_i})} \simeq 339.5 \, \mathrm{cm^{-3}} \, .
\end{equation}
This is, among all astrophysical and cosmological sources, the largest neutrino density at Earth (cf.~the Grand unified neutrino spectrum in~\cite{Vitagliano_Review}). However, due to the very small energy of these neutrinos today, their direct detection is a considerable task~\cite{Neutrino_Cosmology,PTOLEMY2018,PTOLEMY2019}.
\section{Dependence on the physical parameters}
\label{sec:dependence_parameters}
In this section, we discuss how the results obtained in section~\ref{sec:results_nudec} depend on the various physical parameters that enter in the calculation. This allows us to get a nominal uncertainty on the main result~\eqref{eq:result_Neff}. Moreover, we also explore the physical role played by flavour oscillations, aided by the ATAO approximation.
\subsection{Effect of the CP phase}
\label{subsec:Decoupling_CP}
The standard calculation we presented does not include a non-zero Dirac CP phase, while this is not excluded by oscillation data. Indeed, the analysis of the appearance channels $\nu_\mu \to \nu_e$ and $\bar{\nu}_\mu \to \bar{\nu}_e$ (whose oscillation probabilities get opposite shifts due to the CP phase) in long-baseline accelerator experiments (T2K, NO$\nu$A) and in neutrino atmospheric data (Super-Kamiokande) show a preference\footnote{Note that there is a relative tension between the determinations of $\delta$ obtained from T2K and NO$\nu$A data in the normal mass ordering case, while there is an excellent agreement in the inverted ordering scenario~\cite{Esteban2020,Kelly2020}.} for values $\delta \neq 0^{\circ}, \, 180^{\circ}$. We show in this section why this is nevertheless a safe choice, as such a phase does not affect our results.
The generalized parameterization of the PMNS matrix~\eqref{eq:PMNS} when including a CP violating phase\footnote{We do not include possible Majorana phases that have no effect on neutrino oscillations.} reads
\begin{equation}
\label{eq:PMNS_CP}
U = R_{23} S R_{13} S^\dagger R_{12} = \begin{pmatrix}
c_{12} c_{13} & s_{12} c_{13} & s_{13} e^{-{\mathrm i} \delta} \\
- s_{12}c_{23} - c_{12}s_{23}s_{13} e^{{\mathrm i} \delta} & c_{12} c_{23} - s_{12}s_{23}s_{13} e^{{\mathrm i} \delta} & s_{23} c_{13} \\
s_{12}s_{23} - c_{12}c_{23}s_{13} e^{{\mathrm i} \delta} & -c_{12}s_{23} - s_{12}c_{23}s_{13} e^{{\mathrm i} \delta} & c_{23} c_{13}
\end{pmatrix} \, ,
\end{equation}
where $S = \mathrm{diag}(1,1,e^{{\mathrm i} \delta})$. The best-fit value quoted in~\cite{PDG} is $\delta = 1.36 \, \pi \, \mathrm{rad}$, while~\cite{deSalas_Mixing} quotes $\delta_\mathrm{NO} = 1.08 \, \pi \, \mathrm{rad}$ and $\delta_\mathrm{IO} = 1.58 \, \pi \, \mathrm{rad}$. We will use, as always in this manuscript, the value from~\cite{PDG}, but the conclusions remain the same regardless of the value chosen.
Although this new phase affects the vacuum oscillation term in the QKEs, it is actually possible to factorize this dependence and reduce the problem to the case $\delta = 0$, in some limits that we expose below. We revisit the derivation of \cite{Balantekin:2007es,Gava:2008rp,Gava:2010kz,Gava_corr}, where conditions under which the CP phase has an impact on the evolution of $\varrho$ in matter were first uncovered.
In this section, we will note with a superscript $^0$ the quantities in the $\delta =0$ case. We introduce a convenient unitary transformation
\[ \check{S} \equiv R_{23} S R_{23}^\dagger \, , \]
and define $\check{\varrho} \equiv \check{S}^\dagger \varrho \check{S}$ (likewise for $\bar{\varrho}$). Let us now prove that $\check{\varrho} = \varrho^0$. First, we need to show that $\check{\varrho}$ has the same evolution equation as $\varrho^0$. Let us rewrite the QKE~\eqref{eq:QKE_final} in a very compact way:
\begin{equation}
\label{eq:QKE_compact_CP}
{\mathrm i} \frac{\partial \varrho}{\partial x} = \lambda [U \mathbb{M}^2 U^\dagger,\varrho] + \mu [\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}, \varrho] + {\mathrm i} \mathcal{K}(\varrho,\bar{\varrho}) \, ,
\end{equation}
with coefficients $\lambda, \mu$ which can be read from \eqref{eq:QKE_final}. Applying $\check{S}^\dagger (\cdots) \check{S}$ on both sides of the QKE gives the evolution equation for $\check{\varrho}$.
First, using that $\check{S}^\dagger U = U^0 S^\dagger$ (we recall that $U^0$ is the PMNS matrix without CP phase) and that $\mathbb{M}^2$ and $S$ commute since they are diagonal, the vacuum term reads $\check{S}^\dagger [U \mathbb{M}^2 U^\dagger,\varrho] \check{S} = [U^0 \mathbb{M}^2 {U^0}^\dagger, \check{\varrho}] $.
Then, the mean-field term satisfies
\begin{equation}
\label{eq:meanfield_Sdagger}
\check{S}^\dagger [\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}, \varrho] \check{S} \simeq [\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}, \check{\varrho}] \, .
\end{equation}
This equality would be exact if we completely neglected the mean-field contribution of the background muons, as the energy density would read $\bar{\mathbb{E}}_\mathrm{lep} \simeq \mathrm{diag}(\rho_{e^-} + \rho_{e^+}, 0, 0)$ (likewise for the pressure). This is justified since the energy density of muons is negligible compared to the electron one across the decoupling era, and it results in muon and tau neutrinos having the very same interactions, a condition evidenced in~\cite{Gava:2010kz}. Moreover, in the region of high temperatures where the muon energy density get closer to the electron one (even though, at $20 \, \mathrm{MeV}$, muons are still largely non-relativistic), the commutator of $\bar{\mathbb{E}}_\mathrm{lep}$ with $\varrho$ vanishes as $\varrho \propto \mathbb{1}$. The density matrix differs from $\mathbb{1}$ only when $e^\pm$ annihilations are effective, which is "too late" for the muon mean-field to have an effect. Therefore, we can safely take~\eqref{eq:meanfield_Sdagger} as exact.
Finally, the collision term contains products of density matrices and $G_{L,R}$ coupling matrices for the scattering/annihilation terms with electrons and positrons. Since $[G_{L,R}, \check{S}^{(\dagger)}]=0$ (as we only consider standard interactions), we can write $\check{S}^\dagger \mathcal{K}(\varrho, \bar{\varrho}) \check{S} = \mathcal{K}(\check{\varrho},\check{\bar{\varrho}})$. Once again, the fact that $\nu_\mu$ and $\nu_\tau$ have identical interactions is key to this factorization, as pointed out in~\cite{Gava:2010kz} and previously in~\cite{Balantekin:2007es,Gava:2008rp} in the astrophysical context. In~\cite{Gava:2010kz}, the collision term is approximated by a damping factor ; the factorization then holds since the damping coefficients are identical whether they involve $\nu_\mu$ or $\nu_\tau$.
Therefore, the QKE for $\check{\varrho}$ reads:
\begin{equation}
\label{eq:QKE_compact_CP_S}
{\mathrm i} \frac{\partial \check{\varrho}}{\partial x} = \lambda [U^0 \mathbb{M}^2 {U^0}^\dagger,\check{\varrho}] + \mu [\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}, \check{\varrho}] + {\mathrm i} \mathcal{K}(\check{\varrho},\check{\bar{\varrho}}) \, ,
\end{equation}
which is exactly the QKE for $\varrho^0$, i.e.the QKE without CP phase. Moreover, the initial condition~\eqref{eq:initial_condition} is unaffected by the $\check{S}$ transformation: $\check{\varrho}(x_\mathrm{in},y) = \varrho^0(x_\mathrm{in},y)$. Since the initial conditions and the evolution equations are identical for $\check{\varrho}$ and $\varrho^0$, then at all times $\varrho^0(x,y) = \check{\varrho}(x,y)$. We can therefore write the relation between the density matrices with and without CP phase,
\begin{equation}
\label{eq:CP_flavour}
\varrho(x,y) = \check{S} \varrho^0(x,y) \check{S}^\dagger \, .
\end{equation}
This relation has two major consequences:
\begin{enumerate}
\item The trace of $\varrho$ is unaffected by $\delta$, therefore $N_{\mathrm{eff}} = N_{\mathrm{eff}}(\delta = 0)$ ;
\item The first diagonal component is unchanged $\varrho^e_e = (\varrho^0)^e_e$. Equivalently with the parameterization~\eqref{eq:param_rho}, $z_{\nu_e} = z_{\nu_e}^0$ and $\delta g_{\nu_e} = \delta g_{\nu_e}^0$.
\end{enumerate}
Therefore, under the assumptions made above (in particular, the initial distribution has no chemical potentials), the CP phase will have no effect on BBN, since light element abundances are only sensitive to $N_{\mathrm{eff}}$, $z_{\nu_e}$ and $\delta g_{\nu_e}$ (see chapter~\ref{chap:BBN}). Note that in presence of initial degeneracies, the initial conditions do not necessarily coincide $\check{\varrho}(x_\mathrm{in},y) \neq \varrho^0(x_\mathrm{in},y)$ and signatures of a CP phase could in principle be found in the primordial abundances~\cite{Gava:2010kz,Gava_corr}. We discuss this topic in section~\ref{SecDiracPhase}.
A useful rewriting of~\eqref{eq:CP_flavour} can be made with the final distributions ($x = x_f$), when mean-field effects are negligible. The correspondence between the $\delta =0$ and $\delta \neq 0$ cases reads in the matter basis (which is then the mass basis):
\begin{equation}
\varrho_{\mathcal{H}_0}(x_f,y) = S \varrho_{\mathcal{H}_0}^0(x_f,y) S^\dagger \ . \label{eq:CP_mass}
\end{equation}
Note that the transformation involves now $S$ instead of $\check{S}$ (this is linked to the fact that the transformation between $\varrho$ and $\varrho_{\mathcal{H}_0}$ is made through $U$, while the transformation between $\varrho^0$ and $\varrho_{\mathcal{H}_0}^0$ involves $U^0$). We can go further using the ATAO approximation, which constrains the form of $\varrho$ and allows to analytically estimate the effect of the CP phase. \emph{In the ATAO approximation}, $\varrho_{\mathcal{H}_0}^0$ is diagonal, such that we get the result:
\begin{equation}
\text{\textsc{\bfseries ATAO}} \qquad \qquad \varrho_{\mathcal{H}_0}(x_f,y) = \varrho_{\mathcal{H}_0}^0(x_f,y) \, . \qquad \qquad
\end{equation}
Defining effective temperatures $z_{\nu_i}$ for the mass states ($i=1,2,3$), we have then $z_{\nu_i} = z_{\nu_i}^0$. Using the PMNS matrix to express the results in the flavour basis, the effective temperatures read:\footnote{These expressions are rigorously exact for the energy densities, and they can be rewritten for the effective temperatures since $z_\nu - 1 \ll 1$.}
\begin{equation}
\label{eq:resultsCP}
\begin{aligned}
z_{\nu_e} &= z_{\nu_e}^0 \ , \\
z_{\nu_\mu} &= z_{\nu_\mu}^0 - \frac12 (z_{\nu_1} - z_{\nu_2}) \sin{(2 \theta_{12})} \sin{(\theta_{13})}\sin{(2 \theta_{23})} \left[ 1 - \cos{(\delta)}\right] \ , \\
z_{\nu_\tau} &= z_{\nu_\tau}^0 + \frac12 (z_{\nu_1} - z_{\nu_2}) \sin{(2 \theta_{12})} \sin{(\theta_{13})}\sin{(2 \theta_{23})} \left[ 1 - \cos{(\delta)}\right] \ .
\end{aligned}
\end{equation}
These relations show that the CP phase only affects the muon and tau neutrino distribution functions, with a $[\cos{(\delta)} - 1]$ dependence. For the preferred values of $\delta = 1.36 \, \pi \, \mathrm{rad}$ and the mixing angles~\cite{PDG}, and with the results for $\delta=0$ from section~\ref{sec:results_nudec}, we expect $\lvert z_{\nu_\mu} - z_{\nu_\mu}^0 \rvert = \lvert z_{\nu_\tau} - z_{\nu_\tau}^0 \rvert \simeq 4.7 \times 10^{-5}$. This is in excellent agreement with the numerical results obtained solving the QKE with a CP phase (see Table~\ref{Table:Res_NuDec_CP}).
\begin{table}[!htb]
\centering
\begin{tabular}{|l|ccccc|}
\hline
Final values & $z$ & $z_{\nu_e}$ & $z_{\nu_\mu}$ & $z_{\nu_\tau}$ &$N_{\mathrm{eff}}$ \\
\hline \hline
$\delta = 0$ & $1.39797$ & $1.00175$ & $1.00132$ & $1.00130$ & $3.04396$ \\
$\delta = 1.36 \, \pi \, \mathrm{rad}$ & $1.39797$ & $1.00175$ & $1.00127$ & $1.00135$ & $3.04396$ \\ \hline
\end{tabular}
\caption[Final effective temperatures with and without CP phase]{Frozen-out values of the dimensionless photon and neutrino temperatures, and the effective number of neutrino species. We compare the results without CP phase (see also Table~\ref{Table:Res_NuDec}) and with the average value for $\delta$ from~\cite{PDG}.
\label{Table:Res_NuDec_CP}}
\end{table}
Finally, the antineutrino density matrices satisfy the same relation as for neutrinos~\eqref{eq:CP_flavour} $\bar{\varrho}(x,y) = \check{S} \bar{\varrho}^0(x,y) \check{S}^\dagger$. The QKEs in the absence of CP phase preserve the property $\varrho^0 = {(\bar{\varrho}^0)}^{*}$ if it is true initially. The asymmetry with $\delta \neq 0$ would then read $\varrho - \bar{\varrho} = \check{S}(\varrho^0 - {\varrho^0}^*)\check{S}^\dagger$. Therefore, CP violation effects in the $\nu_\mu$ and $\nu_\tau$ distributions (which would be contributions $\propto \sin{\delta}$) can arise from the complex components of $\varrho^0$, thus requiring the ATAO approximation to break down. Since in the cosmological context without initial degeneracies the approximation is very well satisfied, there can be no additional CP violation and the formulae~\eqref{eq:resultsCP} are equally valid for antineutrinos.
Let us mention that we also performed a calculation solving the full QKEs for both neutrinos and antineutrinos, with a non-zero CP phase, and the results were once again the same as the ATAO approximate ones and consistent with~\eqref{eq:CP_mass}.
\subsection{Role of flavour oscillations}
As we have shown in section~\ref{subsec:results_Neff}, flavour oscillations do not significantly affect the value of $N_{\mathrm{eff}}$ compared to, for instance, the inclusion of $\mathcal{O}(e^3)$ finite-temperature QED corrections to the plasma thermodynamics, although they affect much more importantly the neutrino spectra (see Figures~\ref{fig:Tnu} and~\ref{fig:deltagnu}). Therefore, if one is interested in understanding how flavour oscillations modify the physics at play during decoupling --- for instance because $N_{\mathrm{eff}}$ is not the only relevant quantity (for BBN, for instance) --- a deeper study is called for. Notably, we will study precisely the changes brought by the vacuum + mean-field terms in the QKE, and how the mixing parameters play a role (through their values or their sign, for instance considering the inverted hierarchy of masses).
\paragraph{ATAO transfer functions}
The ATAO approximation allows to get some insight on the impact of the mixings and mean-field terms, as its extreme accuracy shows that flavour oscillations "only act as" changes of matter basis across the evolution.
Let us define the \emph{"ATAO transfer function"}
\begin{equation}
\label{eq:defATAO}
\mathcal{T}(\alpha \to \beta, x \to x', y) = \left[U_\mathcal{V}(x',y) \,\reallywidetilde{\left(U_\mathcal{V}^\dagger(x,y) D(\alpha) U_\mathcal{V}(x,y)\right)} \,U_\mathcal{V}^\dagger(x',y)\right]^{\beta}_{\beta} \, ,
\end{equation}
where $D(\alpha)$ is a diagonal matrix with a non-vanishing (unit) component, that is $\left[D(\alpha)\right]^\beta_{\gamma} \equiv \delta_\alpha^\beta \delta^\alpha_\gamma$ (no summation). Equation \eqref{eq:defATAO} corresponds to the probability for a state of flavour $\alpha$ and momentum $y$ generated at a pseudo scale factor $x$, ``averaged'' according to the ATAO approximation, to re-emerge as a flavour $\beta$ at later $x'$, if it is not affected by collisions in the meantime.
When evaluated at $x' \to \infty$, the asymptotic $\mathcal{T}(\alpha \to \beta, x, y) \equiv \mathcal{T}(\alpha \to \beta, x \to \infty, y)$ provide information on neutrino flavour conversion from their last scattering with other species, until all neutrino spectra are frozen since mean-field and collisions are then negligible. These asymptotic functions are shown on Figure~\ref{fig:ATAO_Transfer}.
If mean-field effects can be ignored, the asymptotic ATAO transfer function converges to the following expression
\begin{equation}
\label{VacuumAverage}
\mathcal{T}^\mathrm{vac}(\alpha \to \beta) \equiv \left[U \,\reallywidetilde{\left(U^\dagger D(\alpha) U \right)}\,U^\dagger\right]^{\beta}_{\beta}\,,
\end{equation}
which is independent of $y$ and where the PMNS matter matrix is replaced by the vacuum one. Note that $\mathcal{T}^\mathrm{vac}(\alpha \to \beta) = \mathcal{T}^\mathrm{vac}(\beta \to \alpha)$, as can be seen on Figure~\ref{fig:ATAO_Transfer} at small temperatures.
\begin{figure}[ht]
\centering
\includegraphics{figs/Transfer_functions_NH.pdf}
\caption[Asymptotic ATAO transfer functions (normal ordering)]{\label{fig:ATAO_Transfer} Asymptotic ATAO transfer function $\mathcal{T}(\alpha \to \beta, x, y)$ for $y = 5$. The asymptotic values for large $x$ correspond to the vacuum oscillation averages~\eqref{VacuumAverage}.}
\end{figure}
\subsubsection{Proof that (only) mixing in vacuum matters}
Due to the particular features of this standard calculation (no asymmetries, very small corrections compared to the instantaneous decoupling limit), we will show that the role of flavour mixing on neutrino spectra is essentially captured by considering solely vacuum mixing. To gain this insight on the impact of the mixing and mean-field terms, we have performed two schematic calculations, including either the neutrino probabilities at the end of the evolution, i.e. $T_{\mathrm{cm}, f} = 0.01 \, \mathrm{MeV}$ (``No osc., post-aver.''), or keeping only the mixing and collision terms during the evolution (``Without mean-field''). The corresponding results are shown in Table~\ref{Table:Res_NuDec_Transfer}.
\begin{table}[!htb]
\centering
\begin{tabular}{|l|ccccc|}
\hline
Final values & $z$ & $z_{\nu_e}$ & $z_{\nu_\mu}$ & $z_{\nu_\tau}$ &$N_{\mathrm{eff}}$ \\
\hline \hline
No oscillations, QED $\mathcal{O}(e^3)$ &$1.39800$ & $1.00234$ & $1.00098$ & $1.00098$ &$3.04338$ \\ \hline
No osc., post-averaging, QED $\mathcal{O}(e^3)$ & $1.39800$ & $1.00173$ & $1.00130$ & $1.00127$ & $3.04340$ \\
W/o mean-field, QED $\mathcal{O}(e^3)$ & $1.39796$ & $1.00175$ & $1.00132$ & $1.00131$ & $3.04405$ \\ \hline
ATAO, QED $\mathcal{O}(e^3)$ & $1.39797$ & $1.00175$ & $1.00132$ & $1.00130$ & $3.04396$ \\ \hline
\end{tabular}
\caption[Results of schematic calculations with an approximate inclusion of flavour oscillation physics]{\label{Table:Res_NuDec_Transfer} Frozen-out values of the dimensionless photon and neutrino temperatures, and the effective number of neutrino species. The no oscillations result is presented here to facilitate the discussion. The post-averaging result corresponds to Eq.~\eqref{eq:DefPost}.}
\end{table}
\paragraph{A crude treatment of flavour mixing: post-averaging method} The goal of the first schematic calculation is to start from the no-mixing results (Boltzmann equation, second row in Table~\ref{Table:Res_NuDec}), and "post-average" them (in the ATAO sense), namely,
\begin{equation}
\label{eq:DefPost}
(\varrho^\mathrm{post})^\beta_{\beta} \equiv \sum_\alpha (\varrho^\mathrm{NO})^\alpha_\alpha \, \mathcal{T}^\mathrm{vac}(\alpha \to \beta) \, .
\end{equation}
From Table~\ref{Table:Res_NuDec_Transfer} one can see that the electronic spectra are suppressed and other neutrino types spectra are enhanced by this vacuum averaging procedure. Since one can nearly recover the exact oscillation case results by averaging the final results found without oscillations, it proves that the different values of the effective neutrino temperatures between the no-oscillation case and the full oscillation case are likely to be essentially due to the effect of the mixings. However, the post-averaging of the no-oscillation case preserves, by construction, the trace of $\varrho$ hence the energy density $\rho_\nu$. Therefore, it cannot capture the enhancement of $N_{\mathrm{eff}}$ discussed at the end of section~\ref{subsec:results_Neff}.
\paragraph{Role of the mean-field term}
In the second schematic calculation we have solved the QKEs \eqref{eq:QKE_final} without the mean-field term, i.e., keeping only the vacuum and collision terms.\footnote{We thus have, at all times, $U_\mathcal{V} = U$ and the matter basis coincides with the mass basis.} This is somehow an improvement of the ``post averaging'' procedure, since it neglects the variation of the transfer functions (which always have their asymptotic vacuum values), but accounts correctly for the effect of collisions. The accuracy of the results compared to the full treatment shows once more that the effect of the mean-field is very mild in this case. Indeed, the mean-field contribution becomes effective when $\varrho$ deviates from a matrix proportional to the identity, which only happens when $x \sim 3\times10^{-1}$: however at this point the mean-field contribution is becoming negligible compared to the vacuum one (cf.~Figure~\ref{fig:ATAO_Transfer}). Note that this would not hold if we introduced chemical potentials \cite{Bell98,Dolgov_NuPhB2002,Gava:2010kz,Mirizzi2012,Saviano2013,HannestadTamborra}. The higher value obtained for $N_{\mathrm{eff}}$ in this case can be qualitatively understood. Since $\mathcal{T}^\mathrm{vac}(e \to e) < \mathcal{T}(x \ll1, e \to e)$, $\nu_e$ produced by collisions will be more converted into other flavours (in particular $\nu_\tau$) at early times compared to the full calculation. This frees some phase space for the reheating of $\nu_e$, which is the dominant process. More entropy is transferred from $e^\pm$ annihilations, which increases slightly $N_{\mathrm{eff}}$.
\subsection{Dependence on the mixing angles}
\label{subsec:dependence_mixing}
The transfer functions introduced in the previous section also shed some light on the importance of the precise value of the mixing angles, which explain some discrepancy with previous results (see section~\ref{subsec:results_Neff}). Indeed, varying $\theta_{ij}$ within their uncertainty ranges slightly modify the $\mathcal{T}(\alpha \to \beta)$ curves, which can cross each other. For instance, with the set of parameters used in \cite{Relic2016_revisited}, the asymptotic value $\mathcal{T}^\mathrm{vac}(e \to \tau)$ is higher than $\mathcal{T}^\mathrm{vac}(e \to \mu)$, contrary to Figure~\ref{fig:ATAO_Transfer}. This higher conversion of electron neutrinos into tau neutrinos explains why their final temperatures are $z_{\nu_\tau} \gtrsim z_{\nu_\mu}$ (the values remaining very close).
The experimental uncertainties on the values of the mixing angles~\cite{PDG} lead to small variations of the neutrino distribution functions and $N_{\mathrm{eff}}$. The numerical sensitivity of $N_{\mathrm{eff}}$ to the variation of the mixing angles around their preferred values is found to be:
\begin{equation}
\frac{\partial N_{\mathrm{eff}}}{\partial \theta_{12}} \simeq 8 \times 10^{-4} \ \mathrm{rad^{-1}} \quad ; \quad
\frac{\partial N_{\mathrm{eff}}}{\partial \theta_{13}} \simeq 9 \times 10^{-4} \ \mathrm{rad^{-1}} \quad ; \quad
\abs{\frac{\partial N_{\mathrm{eff}}}{\partial \theta_{23}}} \ll \abs{\frac{\partial N_{\mathrm{eff}}}{\partial \theta_{12}}}, \abs{\frac{\partial N_{\mathrm{eff}}}{\partial \theta_{13}}} \, .
\end{equation}
The sensitivity with respect to $\theta_{23}$ is much smaller than for the other mixing angles, and cannot be separated from numerical noise. Given the uncertainties at $\pm 1 \sigma$ on the mixing angles~\cite{PDG}, we estimate the associated variation of $N_{\mathrm{eff}}$ to be $\Delta N_{\mathrm{eff}} \simeq 10^{-5}$, beyond our accuracy goal.
\subsection{Inverted mass ordering case}
\label{subsec:Inverted_Hierarchy}
In the inverted mass ordering, for which $\Delta m_{31}^2 < 0$, we obtain an increase of $N_{\mathrm{eff}}$ by $5\times10^{-6}$ when solving the QKE~\eqref{eq:QKE_final}. To understand this, we plot on Figure~\ref{fig:ATAO_Transfer_IH} the transfer functions $\mathcal{T}(\alpha \to \beta, x, y)$ in the inverted hierarchy case. Comparing this plot with Figure~\ref{fig:ATAO_Transfer}, we see that electronic neutrinos can be generated above an MSW resonance (e.g.~at about $4\, \mathrm{MeV}$ for $y=5$), and are converted nearly entirely into $\nu_\mu$ and $\nu_\tau$ (solid lines on Figure~\ref{fig:ATAO_Transfer_IH}). Again, this impacts subsequent collisions because it frees some phase space for $\nu_e$, which is beneficial for the total production of neutrinos. However, since neutrino decoupling occurs mainly at temperatures which are below the MSW resonance,\footnote{This is not the case for very large $y$ but they are subdominant in the total energy density budget.} the differences between normal and inverted hierarchies are extremely small.
\begin{figure}[!h]
\centering
\includegraphics{figs/Transfer_functions_IH.pdf}
\caption[Asymptotic ATAO transfer functions (inverted ordering)]{\label{fig:ATAO_Transfer_IH} Asymptotic ATAO transfer function $\mathcal{T}(\alpha \to \beta, x, y)$ for $y = 5$ in the inverted hierarchy of neutrino masses.}
\end{figure}
Given the discussion of the previous section, it appears that neutrino decoupling is mostly sensitive to the neutrino mixings, whereas it has little sensitivity to the mass-squared differences and therefore to the neutrino mass hierarchy.
However, a recent work by \emph{Hansen et al.}~\cite{Hansen_Isotropy} uncovered potential instabilities that could occur during neutrino evolution. In particular, in the inverted ordering case with symmetric initial conditions, the number densities of neutrinos and antineutrinos are the same, but there can be an effect of different off-diagonal terms that build up and trigger collective effects. Indeed, if there are non-zero off-diagonal imaginary terms in $\varrho$, then $\varrho-\bar{\varrho}$ will be non-zero even though the number densities would be equal. Therefore, including the asymmetric neutrino term in the QKE (which is not the case in our calculation and in other standard studies of neutrino decoupling~\cite{Relic2016_revisited,Bennett2021}), it could be possible to see such imaginary parts grow exponentially. However, in the "standard" calculation of $N_{\mathrm{eff}}$ we performed there should be no such imaginary parts: they are zero initially, and the validity of our ATAO approximation (which amounts to say that $\varrho$ is diagonal in the matter basis, hence notably real in the flavour basis) shows that they should keep vanishing.
Imaginary parts in the flavour basis can physically only appear from the oscillatory phases of $\varrho$ (that is, from departures from the ATAO approximation). Given the values of the oscillation frequencies and the fact that they are $y$-dependent, the oscillations of the different $y$-modes will be dephased so one would need extremely many grid points to follow the evolution and capture the cancellation --- or not --- of the imaginary parts in $\mathbb{N}_\nu - \mathbb{N}_{\bar{\nu}}$.
The results of~\cite{Hansen_Isotropy} nevertheless indicate that, even when the instability occurs, $N_{\mathrm{eff}}$ is changed at most by $5 \times 10^{-4}$. This is due to the very small difference between the distributions of $\nu_e$, $\nu_\mu$ and $\nu_\tau$: even if there are large flavour conversions due to this instability, it will remain a higher order effect compared to the global reheating of the three neutrino flavours.
\end{document}
\chapter{The Quantum Kinetic Equations}
\label{chap:QKE}
\epigraph{You think this is hard? Try being waterboarded, \emph{that's} hard!}{Sue Sylvester, \emph{Glee} [S01E01]}
{
\hypersetup{linkcolor=black}
\minitoc
}
\boxabstract{The material of this chapter was published in~\cite{Froustey2020}.}
A precision calculation of neutrino evolution requires to take into account the phenomenon of flavour oscillations. This means that the Boltzmann kinetic equation must be generalized to account for \emph{flavour coherence}, that is the possibility to have a non-vanishing statistical average of mixed-flavour states. A convenient formalism consists in promoting the set of distribution functions to a one-body density matrix, a strategy notably introduced in a seminal paper by Sigl and Raffelt~\cite{SiglRaffelt}. They obtained the so-called "Quantum Kinetic Equation" (QKE) through a perturbative expansion of this "matrix of densities" on the interaction parameter (i.e.~the Fermi constant $G_F$). In the following, we present a formalism that follows quite closely this historical approach, but in the more general framework of a hierarchy of equations. Alternatively to this operator approach, we can quote the functional approach\footnote{We do not compare here these approaches, but simply note that the functional formalism contains information about the spectrum of the theory (i.e., which states are available)~\cite{Berges:2004,Vlasenko_PhRevD2014,Berges:2015,Drewes:2017}. However, for neutrinos in the early Universe the spectral function is, at the order we are interested in, the same as the massless, free-field one, such that the two formalisms can be used interchangeably.} of~\cite{BlaschkeCirigliano} which uses the Closed-Time Path formalism.
After deriving the QKEs in the general case of a system subject only to two-body interactions, we apply the formalism to the specific case of neutrinos in the early Universe, which leads to many simplifications.
\section{Hierarchy of evolution equations}
\label{sec:BBGKY}
In this section, we derive from first principles the neutrino quantum kinetic equations, which generalize the Boltzmann kinetic equation for distribution functions~\eqref{eq:boltzmann_nu}, to account for neutrino masses and mixings. We present the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy~\cite{Bogoliubov,BornGreen,Kirkwood,Yvon} that was historically derived for a non-relativistic $N-$body system and heavily used in nuclear physics~\cite{WangCassing1985,Cassing1990,Reinhard1994,Lac04,Simenel,Lac14}, but which can also be applied to a relativistic system such as the bath of neutrinos and antineutrinos in the early Universe. We extend the formalism of~\cite{Volpe_2013}, where the BBGKY formalism was applied to derive extended mean-field equations for astrophysical applications, and include the collision term.
\subsection{BBGKY formalism}
The exact evolution of a $N-$body system under the Hamiltonian $\hat{H}$ is given by the Liouville-von Neumann equation for the many-body density matrix
\begin{equation}
\label{eq:vonneumann}
{\mathrm i} \frac{\mathrm{d} \hat{D}}{\mathrm{d} t} = [\hat{H}, \hat{D}] \, ,
\end{equation}
where $\hat{D} = \ket{\Psi}\bra{\Psi}$, with $\ket{\Psi}$ the quantum state, from which we define the $s$-body reduced density matrices,
\begin{equation}
\label{eq:defrhos}
\hat{\varrho}^{(1 \cdots s)} \equiv \frac{N !}{(N-s)!} \mathrm{Tr}_{s+1\dots N} \hat{D} \, .
\end{equation}
Its components (we drop the superscript $^{(1\cdots s)}$, redundant with the number of indices) read, as detailed in the appendix~\ref{app:vrho_components}:
\begin{equation}
\label{eq:defrhos2}
\varrho^{i_1 \cdots i_s}_{j_1 \cdots j_s} \equiv \langle \hat{a}_{j_s}^\dagger \cdots \hat{a}_{j_1}^\dagger \hat{a}_{i_1} \cdots \hat{a}_{i_s} \rangle \, ,
\end{equation}
where the indices $i,j$ label a set of quantum numbers (species $\phi_i$, momentum $\vec{p}_i$, helicity $h_i$) which describe a one-particle quantum state, and $\langle \cdots \rangle$ is shorthand for $\bra{\Psi} \cdots \ket{\Psi}$. Let us give one example, where the summing rules are also made explicit:\footnote{Reminder: the factors of $2E$ in the denominator of $\ddp{}$ arise from relativistic phase-space constraints. More precisely, the phase space integrals should be four-dimensional, with the on-shell condition:
\[\int{\mathrm{d}^3 \vec{p}}\int_{0}^{\infty}{\mathrm{d}E \, \delta(E^2 - p^2 - m^2)} = \int{\mathrm{d}^3 \vec{p}}\int_{0}^{\infty}{\mathrm{d} E \, \frac{\delta(E - \sqrt{p^2 + m^2})}{2 E}} \, ,\]
and the additional $(2\pi)^3$ are actually the fundamental phase space volumes $(2 \pi \hbar)^3$ with $\hbar=1$.}
\begin{equation}
\label{eq:set_quantum_numbers}
\sum_i{\hat{a}^\dagger_i}= \sum_{\phi_i} \sum_{h_i} \int{\ddp{i} \, \hat{a}^\dagger_{\phi_i}(\vec{p}_i,h_i)} \qquad \text{with} \qquad \ddp{i} \equiv \frac{\mathrm{d}^3 \vec{p}_i}{(2 \pi)^3 2 E_i} \, .
\end{equation}
The creation and annihilation operators satisfy the fermionic anticommutation rules $\{ \hat{a}^\dagger_i, \hat{a}_j \} = \delta_{ij}$, with $\delta$ the Kronecker delta (generalized to the set of quantum numbers precised above)
\begin{equation}
\delta_{ij} \equiv (2 \pi)^3 \, 2 E_i \, \delta^{(3)}(\vec{p}_i - \vec{p}_j \,) \delta_{h_i h_j} \delta_{\phi_i \phi_j} \, .
\end{equation}
The central object is the one-body reduced density matrix \cite{SiglRaffelt},
\begin{equation}
\label{eq:defrho}
\varrho^i_j \equiv \langle \hat{a}_j^\dagger \hat{a}_i \rangle \, ,
\end{equation}
whose diagonal entries correspond to the standard occupation numbers.
The Hamiltonian for this system is given by the sum of the kinetic and the two-body interaction terms (such an interaction Hamiltonian being adequate for neutrinos whose interactions are described by Fermi theory),
\begin{equation}
\hat{H} = \hat{H}_0 + \hat{H}_{\mathrm{int}} = \sum_{i,j}{t^{i}_{j} \, \hat{a}^\dagger_i \hat{a}_j} + \frac14 \sum_{i,j,k,l}{\tilde{v}^{ik}_{jl} \, \hat{a}^\dagger_i \hat{a}^\dagger_k \hat{a}_l \hat{a}_j} \label{eq:defHint} \, .
\end{equation}
The interaction matrix elements are fully anti-symmetrized by construction:
\begin{equation}
\label{eq:defvint}
\bra{ik} \hat{H}_{\mathrm{int}} \ket{jl} \equiv \tilde{v}^{ik}_{jl}= - \tilde{v}^{ki}_{jl} = \tilde{v}^{ki}_{lj} \, .
\end{equation}
The traditional presentation of the BBGKY formalism in nuclear physics is sometimes based on tensor products of one-particle states rather than fully antisymmetrized states (defined as $\ket{i_1 \cdots i_s} = \hat{a}^\dagger_{i_1} \cdots \hat{a}^\dagger_{i_s} \ket{0}$ with $\ket{0}$ the quantum vacuum state). The equivalence between both approaches is discussed for completeness in the appendix~\ref{app:BBGKY_Antisym}.
This set of definitions ensures proper transformation laws under a unitary transformation of the one-particle quantum state $\psi^i = \mathcal{U}^{i}_{a} \psi^a$: all lower indices are covariant while upper indices are contravariant, namely,
\begin{equation}
\label{eq:transfo}
\varrho^a_b = {\mathcal{U}^\dagger}^a_i \, \varrho^i_j \, \mathcal{U}^j_b \quad , \quad t^a_b = {\mathcal{U}^\dagger}^a_i \, t^i_j \, \mathcal{U}^j_b \quad , \quad \tilde{v}^{ac}_{bd} = {\mathcal{U}^\dagger}^a_i {\mathcal{U}^\dagger}^c_k \, \tilde{v}^{ik}_{jl} \, \mathcal{U}^j_b \mathcal{U}^l_d \, .
\end{equation}
One must keep in mind that the unitary transformation "matrix" $\mathcal{U}$ is extremely complicated a priori, all the complexity being hidden in the use of the generalized indices.
\paragraph{BBGKY hierarchy} The evolution equation for $\varrho$ can be obtained directly via the Ehrenfest theorem. One can also apply partial traces to \eqref{eq:vonneumann}, which leads to the well-known \emph{BBGKY hierarchy}~\cite{Bogoliubov,BornGreen,Kirkwood,Yvon}, whose first two equations read\footnote{We explicitly wrote the components of the tensors compared to the expressions found in~\cite{Volpe_2013} or~\cite{Lac04,Simenel}.} (Einstein summation convention implied):
\begin{subequations}
\label{eq:hierarchy}
\begin{empheq}[left=\empheqlbrace]{align}
{\mathrm i} \frac{\mathrm{d} \varrho^{i}_{j}}{\mathrm{d} t} &= \left( t^{i}_{k} \varrho^{k}_{j} - \varrho^{i}_{k} t^{k}_{j} \right) + \frac12 \left(\tilde{v}^{ik}_{ml} \varrho^{ml}_{jk} - \varrho^{ik}_{ml} \tilde{v}^{ml}_{jk} \right)\,, \label{eq:hierarchy_1} \\
{\mathrm i} \frac{\mathrm{d} \varrho^{ik}_{jl}}{\mathrm{d} t} &= \left(t^{i}_{r} \varrho^{rk}_{jl} + t^{k}_{p} \varrho^{ip}_{jl} + \frac12 \tilde{v}^{ik}_{rp} \varrho^{rp}_{jl} - \varrho^{ik}_{rl} t^{r}_{j} - \varrho^{ik}_{jp} t^{p}_{l} - \frac12 \varrho^{ik}_{rp} \tilde{v}^{rp}_{jl} \right) \label{eq:hierarchy_2} \\
&\qquad \qquad + \frac12 \left(\tilde{v}^{im}_{rn} \varrho^{rkn}_{jlm} + \tilde{v}^{km}_{pn} \varrho^{ipn}_{jlm} - \varrho^{ikm}_{rln} \tilde{v}^{rn}_{jm} - \varrho^{ikm}_{jpn} \tilde{v}^{pn}_{lm} \right) \, . \nonumber
\end{empheq}
\end{subequations}
More than simply recasting in a less compact form the very complicated problem \eqref{eq:vonneumann}, this hierarchy furnishes a set of evolution equations which depend on higher-order reduced density matrices. In order to solve these equations, one necessarily needs to \emph{truncate} this hierarchy. Different truncation schemes exist, and we will only discuss the useful ones for neutrino evolution in the early Universe.
\subsection{Hartree-Fock approximation and mean-field terms}
It proves convenient to split the two-body density matrix into its uncorrelated (i.e., products of one-body density matrices) and correlated contributions \cite{Lac04,Simenel,Lac14}:
\begin{equation}
\label{eq:splitrho}
\varrho^{ik}_{jl} \equiv 2 \varrho^{i}_{[j} \varrho^{k}_{l]} + C^{ik}_{jl} \equiv \varrho^{i}_{j} \varrho^{k}_{l} - \varrho^{i}_{l} \varrho^{k}_{j} + C^{ik}_{jl} \, .
\end{equation}
Inserting this decomposition into \eqref{eq:hierarchy_1}, we get
\begin{equation}
\label{eq:eqvrho}
{\mathrm i} \frac{\mathrm{d} \varrho^{i}_{j}}{\mathrm{d} t} = \left[ \left(t^{i}_{k} + \Gamma^{i}_{k}\right) \varrho^{k}_{j} - \varrho^{i}_{k} \left(t^{k}_{j} + \Gamma^{k}_{j}\right) \right] + \frac12 \left(\tilde{v}^{ik}_{ml} C^{ml}_{jk} - C^{ik}_{ml} \tilde{v}^{ml}_{jk} \right)
= \left [ \hat{t} + \hat{\Gamma} , \hat{\varrho} \right]^{i}_{j} + {\mathrm i} \, \mathcal{C}^{i}_{j} \, ,
\end{equation}
which defines the collision term $\hat{C}$ (discussed later) and the \emph{mean-field potential} $\hat{\Gamma}$ (for once, we explicit the summation)
\begin{equation}
\label{eq:Gamma}
\Gamma^{i}_{j} = \sum_{k,l}{\tilde{v}^{ik}_{jl} \varrho^{l}_{k}} \, .
\end{equation}
It accounts for the effective potential "felt" by the particles when propagating in a non-vacuum background.
\subsubsection{Mean-field approximation}
The simplest non-trivial closure of the BBGKY hierarchy is the so-called Hartree-Fock or \emph{mean-field} approximation. It consists in neglecting $C^{ik}_{jl} \simeq 0$ and keeping only the commutator part in \eqref{eq:eqvrho}.
However, in the context of neutrino decoupling in the early Universe, one seeks a generalization of the Boltzmann equation for neutrino distribution functions \cite{Dolgov_NuPhB1997,Esposito_NuPhB2000,Mangano2002,Grohs2015,Froustey2019}, which describes the evolution of densities under two-body collisions. In other words, we need to truncate the hierarchy~\eqref{eq:hierarchy} assuming the \emph{molecular chaos} ansatz: correlations between the one-body density matrices arise from two-body interactions between uncorrelated matrices~\cite{Lac04}. This prescribes the form of $C^{ik}_{jl}(t)$, leading to a formal expression for $\hat{\mathcal{C}}$, which we establish in the following section.
\subsection{Derivation of the structure of the collision term}
\label{subsec:derivation_collision}
Compared to the Boltzmann treatment of neutrino evolution, which neglects flavour mixing, the QKE contains mean-field terms, and the collision term has a richer matrix structure with non-zero off-diagonal components. To derive this collision term, i.e., the contribution to the evolution of the one-body density matrix from two-body correlations, one needs an expression for the correlated part $\hat{C}$ in \eqref{eq:eqvrho}. It is obtained from the evolution equation for $\hat{\varrho}^{(12)}$, where we separate correlated and uncorrelated parts \cite{Lac04}.
To do so, we need a splitting similar to \eqref{eq:splitrho} for the three-body density matrix,
\begin{equation}
\label{eq:splitrho123}
\varrho^{ikm}_{jln} = 6 \varrho^{i}_{[j}\varrho^{k}_{l}\varrho^{m}_{n]} + 9 \varrho^{[i}_{[j} C^{km]}_{ln]} + C^{ikm}_{jln} \, .
\end{equation}
This allows to rewrite~\eqref{eq:hierarchy_2} as an equation for the two-body correlation function \cite{Volpe_2013}. In the \emph{molecular chaos} ansatz, correlations are built through collisions between uncorrelated particles. These correlations then evolve ``freely'', i.e., we do not take into account a mean-field background for $\hat{C}$. The evolution equation is thus greatly simplified and reads
\begin{equation}
\begin{aligned}
{\mathrm i} \frac{\mathrm{d} C^{ik}_{jl}}{\mathrm{d} t} &\simeq \left[t^{i}_{r} C^{rk}_{jl} + t^{k}_{p} C^{ip}_{jl} - C^{ik}_{rl} t^{r}_{j} - C^{ik}_{jp} t^{p}_{l} \right] \\
&\quad + \underbrace{(\hat{\mathbb{1}} - \varrho)^i_r (\hat{\mathbb{1}}-\varrho)^k_p \, \tilde{v}^{rp}_{sq} \, \varrho^{s}_{j} \varrho^{q}_{l} - \varrho^i_r \varrho^k_p \, \tilde{v}^{rp}_{sq} \, (\hat{\mathbb{1}} -\varrho)^{s}_{j} (\hat{\mathbb{1}} - \varrho)^{q}_{l}}_{\displaystyle \equiv B^{ik}_{jl}} \, ,
\end{aligned}
\end{equation}
The commutator in the first row is a "vacuum term" which accounts for the evolution of correlations in the vacuum, hence depending only on the kinetic part of the Hamiltonian~\eqref{eq:defHint}. The second row is the "Born term" which only involves the uncorrelated part of~\eqref{eq:splitrho}. The other neglected terms can be found in e.g.~\cite{WangCassing1985,Lac04,Volpe_2013}.
We can solve this equation, starting from $C(t=0)=0$,
\begin{equation}
\label{eq:solveC}
C^{ik}_{jl}(t) = - {\mathrm i} \int_{0}^{t}{\mathrm{d} s \, T^{ik}_{mp}(t,s) B^{mp}_{nq}(s) {T^\dagger}^{nq}_{jl}(t,s)} \, ,
\end{equation}
with the evolution operator
\begin{equation}
T^{ik}_{jl}(s,s') = \exp{\left(-{\mathrm i} \int_{s'}^{s}{\mathrm{d} \tau \, \hat{t}(\tau)}\right)}^{i}_{j} \exp{\left(-{\mathrm i} \int_{s'}^{s}{\mathrm{d} \tau \, \hat{t}(\tau)}\right)}^{k}_{l} \, .
\end{equation}
Now we consider that there is a clear separation of scales \cite{SiglRaffelt}, i.e.the duration of one collision is very small compared to the variation timescale of the density matrices (i.e., compared to the duration between two collisions, and the typical inverse oscillation frequency). Therefore, the argument inside the integral of \eqref{eq:solveC} is only non-zero for $s \simeq 0$: we can extend the integration domain to $+ \infty$, while the operators keep their $t=0$ value. Finally we symmetrize the integration domain\footnote{See section 6.1~\cite{FidlerPitrou} for a detailed discussion of this procedure, which amounts to separate the macroscopic evolution from the microphysics processes.} with respect to 0 (with an extra factor of $1/2$), which leads to the equation with collision term:\footnote{The product $\hat{T} \hat{B} \hat{T}^\dagger$ must be done from left to right, as can be seen with the components in~\eqref{eq:solveC}.}
\begin{subequations}
\begin{align}
{\mathrm i} \frac{\mathrm{d} \varrho^{i}_{j}}{\mathrm{d} t} &= \left[ \hat{t} + \hat{\Gamma}, \hat{\varrho}\right]^{i}_{j} - \frac{{\mathrm i}}{4} \int_{-\infty}^{+\infty}{\mathrm{d} t\, \left[ \tilde{v},\hat{T}(t,0)\hat{B}(0) \hat{T}^\dagger(t,0) \right]}^{ik}_{jk} \\
&= [(t^i_k + \Gamma^i_k)\varrho^k_j - \varrho^i_k (t^k_j + \Gamma^k_j)] \nonumber \\
&\qquad - \frac{{\mathrm i}}{4} \underbrace{\int_{-\infty}^{+\infty}{\mathrm{d} t \, e^{-{\mathrm i} (E_m+E_l-E_j-E_k)t}}}_{(2\pi) \, \delta(E_m + E_l - E_j - E_k)}\left[\tilde{v}^{ik}_{rl} B^{rl}_{jk} - B^{ik}_{rl} \tilde{v}^{rl}_{jk} \right] \, , \\
&\equiv \left[ \hat{t} + \hat{\Gamma}, \hat{\varrho}\right]^{i}_{j} + {\mathrm i} \, \mathcal{C}^{i}_{j}
\end{align}
\end{subequations}
The exponential of energies comes from the $\hat{T}$ terms, using that the density matrix for a given momentum $\hat{\varrho}(p)$ satisfies\footnote{We anticipate the simplification of the density matrices due to homogeneity and isotropy~\eqref{eq:homogeneity} in order to get a practical result. Note that we do not take into account the contribution of masses to neutrino energies in the collision term, since they are completely negligible compared to the ultrarelativistic part.} $\hat{t}\hat{\varrho}(p) = p \,\hat{\varrho}(p)$. The general form of the collision term is then
\begin{multline}
\label{eq:C11}
\mathcal{C}^{i_1}_{i_1'} = \frac14 \sum_{i_2, i_3, i_4} \sum_{j_1, j_2, j_3, j_4} \left(\tilde{v}^{i_1 i_2}_{i_3 i_4} \varrho^{i_3}_{j_3} \varrho^{i_4}_{j_4} \tilde{v}^{j_3 j_4}_{j_1 j_2} (\hat{\mathbb{1}} -\varrho)^{j_1}_{i_1'} (\hat{\mathbb{1}} - \varrho)^{j_2}_{i_2} - \tilde{v}^{i_1 i_2}_{i_3 i_4} (\hat{\mathbb{1}} -\varrho)^{i_3}_{j_3} (\hat{\mathbb{1}} - \varrho)^{i_4}_{j_4} \tilde{v}^{j_3 j_4}_{j_1 j_2} \varrho^{j_1}_{i_1'} \varrho^{j_2}_{i_2} \right. \\
\left. + (\hat{\mathbb{1}} -\varrho)^{i_1}_{j_1} (\hat{\mathbb{1}} - \varrho)^{i_2}_{j_2} \tilde{v}^{j_1 j_2}_{j_3 j_4} \varrho^{j_3}_{i_3} \varrho^{j_4}_{i_4} \tilde{v}^{i_3 i_4}_{i_1' i_2} - \varrho^{i_1}_{j_1} \varrho^{i_2}_{j_2} \tilde{v}^{j_1 j_2}_{j_3 j_4} (\hat{\mathbb{1}} - \varrho)^{j_3}_{i_3} (\hat{\mathbb{1}}-\varrho)^{j_4}_{i_4} \tilde{v}^{i_3 i_4}_{i_1' i_2} \right) \\
\times (2 \pi) \, \delta(E_{i_1} + E_{i_2} - E_{i_3} - E_{i_4}) \, .
\end{multline}
It has the standard structure ''gain $-$ loss $+$ h.c.'', which will be made more explicit when we give the full expressions for a system of neutrinos and antineutrinos interacting with standard model weak interactions, cf.~section~\ref{subsec:collision_integral}.
In the following sections, we will focus on the case of the early Universe and consider three active species of neutrinos in a background of electrons and positrons, muons and antimuons (in traces), and photons. The influence of baryons can be discarded given their negligible density compared to relativistic species (the baryon-to-photon ratio is $\eta \equiv n_b/n_\gamma \simeq 6.1\times 10^{-10}$ from the most recent measurement of the baryon density~\cite{Fields:2019pfx}). Therefore, we focus on the lepton sector evolution, and baryons will only be dealt with when we discuss BBN in chapter~\ref{chap:BBN}.
\section{Application to neutrinos in the early Universe}
Assuming the Universe to be homogeneous and isotropic in the period of interest, the density matrices read,\footnote{The annihilation and creation operators satisfy the equal time anticommutation rules
\[\{\hat{a}_{\nu_\alpha}(\vec{p}, h),\hat{a}^\dagger_{\nu_\beta}(\vec{p}', h') \} = (2\pi)^3 \, 2E_p \, \delta^{(3)}(\vec{p} - \vec{p}') \, \delta_{h h'} \, \delta_{\alpha \beta} \ ; \ \{\hat{a}^\dagger_{\nu_\alpha}(\vec{p}, h),\hat{a}^\dagger_{\nu_\beta}(\vec{p}', h') \} = \{\hat{a}_{\nu_\alpha}(\vec{p}, h),\hat{a}_{\nu_\beta}(\vec{p}', h') \} = 0 \]
Similar relations hold for the antiparticle operators.}
\begin{subequations}
\label{eq:homogeneity}
\begin{align}
\langle \hat{a}^\dagger_{\nu_\beta}(\vec{p}', h') \hat{a}_{\nu_\alpha}(\vec{p}, h) \rangle &= (2 \pi)^3 \, 2 E_p \, \delta^{(3)}(\vec{p} - \vec{p}') \delta_{h h'} \, \varrho^{\nu_\alpha}_{\nu_\beta}(p,t) \, \delta_{h-} \, , \\
\langle \hat{b}^\dagger_{\nu_\alpha}(\vec{p}, h) \hat{b}_{\nu_\beta}(\vec{p}', h') \rangle &= (2 \pi)^3 \, 2 E_p \, \delta^{(3)}(\vec{p} - \vec{p}') \delta_{h h'} \, \bar{\varrho}^{\nu_\alpha}_{\nu_\beta}(p,t) \, \delta_{h+} \, .
\end{align}
\end{subequations}
The Kronecker delta ensures that only left-handed neutrinos and right-handed antineutrinos are included. However, in anisotropic environments, "wrong-helicity" densities or \emph{pairing} densities (like the non-lepton-number-violating correlator, in the Dirac neutrino case, $\langle \hat{b}_{\nu_\alpha} \hat{a}_{\nu_\beta} \rangle$) can be sourced~\cite{Volpe_2013,SerreauVolpe,Volpe_2015,KartavtsevRaffelt}. We do not consider such terms here. The energy function is $E_p = p$ for neutrinos,\footnote{We always neglect the small neutrino masses compared to their typical momentum, except for the vacuum term since the diagonal momentum contribution disappears from the evolution equation (section~\ref{subsec:vacuum}).} and $E_p = \sqrt{p^2 + m_{e,\mu}^2}$ for $e^\pm, \, \mu^\pm$.
In the following, we will apply the BBGKY formalism to a system of neutrinos, leaving the details of the inclusion of antineutrinos\footnote{Note that the antineutrino density matrix $\bar{\varrho}^i_j \equiv \langle \hat{b}_i^\dagger \hat{b}_j \rangle$ is defined with a transposed convention, compared to the neutrino density matrix, to have similar evolution equations and transformation properties.} to appendix~\ref{app:antiparticles}. Note that, for a relativistic system, the hierarchy is given by an infinite set of equations~\cite{CalzettaHu} (basically, $N \to \infty$ in \eqref{eq:defrhos}, but this does not affect the reduced equations for the one-body density matrix).
We emphasize that we use the notation $\varrho$ for a slightly different object from the one defined in equation~\eqref{eq:defrho}. $\varrho^{\nu_\alpha}_{\nu_\beta}(p,t)$ is a reduced part of the full $\varrho$, namely the diagonal values in helicity and momentum space. It nevertheless possesses a matrix structure in flavour space. However, the previous formalism must be applied with the \emph{full} density matrix,\footnote{In practice, this is hardly a problem: one must just remember to sum over all possible species.} thus one must also look at the charged lepton indices. Since only neutrinos mix between themselves (i.e., the flavour structure is strictly diagonal except in the neutrino-neutrino subspace of $\varrho$), we can distinguish two "blocks" in $\varrho$: the lepton part, purely diagonal
\[\varrho_\text{lep}(p,t) = \mathrm{diag}(f_{e^-}(p,t), f_{\mu^-}(p,t),0) \qquad \text{and} \qquad \bar{\varrho}_\text{lep} = \mathrm{diag}(f_{e^+}(p,t), f_{\mu^+}(p,t),0) \, , \]
and the neutrino part for which we will use the simplified notation $\varrho^\alpha_\beta \equiv \varrho^{\nu_\alpha}_{\nu_\beta}$. In the following, we will only use the distribution functions when it comes to leptons. Furthermore, and as expected from homogeneity and isotropy assumptions, all quantities are diagonal in momentum space, such that in the end, only the diagonal values $A(p)$ of operators $A^{\vec{p}}_{\vec{p}'} = A(p) \bm{\delta}_{\vec{p} \vec{p}'}$ will be dealt with, where the ''Kronecker symbol'' in momentum space is $\bm{\delta}_{\vec{p} \vec{p}'} = (2 \pi)^3 \, 2 E_p \, \delta^{(3)}(\vec{p} - \vec{p}')$. This will be made explicit in the upcoming calculations.
To determine the equation of evolution of the statistical ensemble of neutrinos, we now have to calculate the relevant expressions of the vacuum, the mean-field~\eqref{eq:Gamma} and collision~\eqref{eq:C11} terms.
\paragraph{Remark: Majorana and Dirac neutrinos} In the case of Dirac neutrinos, the discussion above can be directly applied: right-handed (RH) neutrinos and left-handed (LH) antineutrinos are not present in the early Universe~\cite{Shapiro1980,LesgourguesPastor,Dolgov_2002PhysRep}, such that $\varrho$ and $\bar{\varrho}$ correspond respectively to the LH part of the full neutrino density matrix in helicity space, and the RH part of full antineutrino density matrix in helicity space.
In the Majorana case, what we call antineutrinos are actually the right-handed neutrinos. More precisely, since the energy scale is much higher than the mass of neutrinos, helicity-flip processes are suppressed and one can match these definitions safely. The interactions of the neutrino field have the same form regardless of the mass mechanism, see for instance Eq.~(14.23) in~\cite{GiuntiKim}. Therefore, all the following calculations, including the mean-field and collision terms, can be done without considering the nature of neutrinos. Note that this would not hold in an anisotropic setup: taking into account \emph{spin coherence} effects shows differences between the Majorana and Dirac cases~\cite{Vlasenko_PhRevD2014,SerreauVolpe,Cirigliano2014,BlaschkeCirigliano}.
\subsection{Vacuum term}
\label{subsec:vacuum}
The neutrino kinetic term is, by definition, diagonal in the mass basis (the basis elements being the eigenstates of the vacuum Hamiltonian $\hat{H}_0$):
\begin{equation}
\left. t^{a}_{b}(p)\right|_{\text{mass basis}} = \sqrt{p^2 + m_a^2} \, \delta^a_b \simeq p \delta^a_b + \frac{m_a^2}{2p} \delta^a_b \, .
\end{equation}
Since terms proportional to the identity do not contribute to flavour evolution (their commutator with $\varrho$ vanishes), the first term will later disappear from the evolution equation. In the flavour basis, the vacuum term is obtained following the transformation laws \eqref{eq:transfo}:
\begin{equation}
t^i_j = p \delta^i_j + \left( U \frac{\mathbb{M}^2}{2p} U^\dagger \right)^i_j \, ,
\end{equation}
with $\mathbb{M}^2$ the matrix of mass-squared differences and $U$ the Pontercorvo-Maki-Nakagawa-Sakata (PMNS) mixing matrix \cite{GiuntiKim,PDG}.
\subsection{Weak interaction matrix elements}
\label{subsec:weak_matrix_el_QKE}
Neutrinos and antineutrinos in the early Universe interact with each others and with the charged leptons forming the homogeneous and isotropic plasma. The interaction Hamiltonian is thus given by the charged- and neutral-current terms from the standard model of weak interactions, expanded at low energies compared to the gauge boson masses. All expressions and subsequent interaction matrix elements~\eqref{eq:defvint} are gathered in the appendix~\ref{app:matrix_el_MF}, while we give here the details of the calculation for the charged-current processes $\nu_e - e^\pm$ so as to illustrate how the formalism works.
\subsubsection{Example: matrix elements for charged-currents with $e^\pm$}
The part of the interaction Hamiltonian corresponding to charged-current processes with electrons and positrons is~\eqref{eq:hcc_app}, which we recall here (cf.~also equation (4.10) in~\cite{SiglRaffelt}):
\begin{multline}
\label{eq:hcc}
\hat{H}_{CC} = 2 \sqrt{2} G_F m_W^2 \int{\ddp{1} \ddp{2} \ddp{3} \ddp{4}} \ (2\pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\ \times [\overline{\psi}_{\nu_e}(\vec{p}_1)\gamma_\mu P_L\psi_e(\vec{p}_4)] W^{\mu \nu}(\Delta) [\overline{\psi}_e(\vec{p}_2) \gamma_\nu P_L \psi_{\nu_e}(\vec{p}_3)] \, ,
\end{multline}
with $\psi(\vec{p}) = \sum_{h} \left[ \hat{a}(\vec{p},h) u^h(\vec{p})+ \hat{b}^\dagger(-\vec{p},h) v^h(-\vec{p}) \right]$ the Fourier transform of the quantum fields, $P_L = (1-\gamma^5)/2$ the left-handed projection operator, and the gauge boson propagator
\begin{equation}
\label{eq:propagator}
W^{\mu \nu}(\Delta) = \frac{\eta^{\mu \nu} - \frac{\Delta^\mu \Delta^\nu}{m_W^2}}{m_W^2 - \Delta^2} \simeq \frac{\eta^{\mu \nu}}{m_W^2} + \frac{1}{m_W^2}\left(\frac{\Delta^2 \eta^{\mu \nu}}{m_W^2} - \frac{\Delta^\mu \Delta^\nu}{m_W^2}\right) \, .
\end{equation}
The lowest order in the previous expansion is the Fermi four-fermion effective theory. The momentum transfer is $\Delta = p_1-p_4$ for a $t$-channel ($\nu_e-e^-$ scattering), and $\Delta= p_1 + p_2$ for the $s$-channel ($\nu_e-e^+$), see figure~\ref{fig:Feynman-diagram-CC}. More precisely, going from $t$-channel to $s$-channel corresponds to the change of variables $- p_2 \leftrightarrow p_4$ in the integral~\eqref{eq:hcc}, which will be important to keep track of the correct signs when computing the interaction matrix elements. Finally, note that we only discuss the interaction between $\nu_e - e^{\pm}$, but the exact same calculation can be carried out with $\nu_\mu - \mu^{\pm}$.
\begin{figure}[!ht]
\vspace{0.3cm}
\begin{fmffile}{CC-Feynman}
\begin{equation*}
\begin{gathered}
\begin{fmfgraph*}(140,70)
\fmfleft{i1,i2}
\fmfright{o1,o2}
\fmflabel{$e^-$}{i1}
\fmflabel{$\nu_{e}$}{o1}
\fmflabel{$\nu_{e}$}{i2}
\fmflabel{$e^-$}{o2}
\fmf{fermion,label=$p_2$,label.side=left}{i1,v1}
\fmf{fermion,label=$p_3$,label.side=left}{v1,o1}
\fmf{photon, label=$W$}{v1,v2}
\fmf{fermion,label=$p_1$,label.side=left}{i2,v2}
\fmf{fermion,label=$p_4$,label.side=left}{v2,o2}
\fmfdot{v1,v2}
\fmfv{label=$g$}{v1,v2}
\end{fmfgraph*}
\end{gathered}
\qquad \qquad \qquad
\begin{gathered}
\begin{fmfgraph*}(140,70)
\fmfleft{i1,i2}
\fmfright{o1,o2}
\fmflabel{$e^+$}{i1}
\fmflabel{$e^+$}{o1}
\fmflabel{$\nu_e$}{i2}
\fmflabel{$\nu_e$}{o2}
\fmf{fermion,label=$p_2$,label.side=left}{v1,i1}
\fmf{fermion,label=$p_4$,label.side=left}{o1,v2}
\fmf{photon, label=$W$}{v1,v2}
\fmf{fermion,label=$p_1$,label.side=left}{i2,v1}
\fmf{fermion,label=$p_3$,label.side=left}{v2,o2}
\fmfdot{v1,v2}
\fmfv{label=$g$}{v1,v2}
\end{fmfgraph*}
\end{gathered}
\end{equation*}
\end{fmffile}
\vspace{0.3cm}
\caption[Charged-current processes]{\label{fig:Feynman-diagram-CC} Charged-current processes.}
\end{figure}
Our goal is to identify the $\tilde{v}$ coefficients by matching the expression~\eqref{eq:hcc} with the general definition~\eqref{eq:defHint}. Note that it is important to write the sums and integrals in the form~\eqref{eq:set_quantum_numbers} to ensure the proper identification.
To compute the coefficient $\tilde{v}^{\nu_e e}_{\nu_e e}$, we need to extract from~\eqref{eq:hcc} the terms involving $\hat{a}_e^{(\dagger)}$ and $\hat{a}_{\nu_e}^{(\dagger)}$. It reads:
\begin{multline}
\label{eq:hcc_matrixel}
\hat{H}_{CC} \supset 2 \sqrt{2} G_F m_W^2 \sum_{h_1, \cdots} \int{\ddp{1} \cdots} \ (2\pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \times [\bar{u}_{\nu_e}^{h_1}(\vec{p}_1) \gamma_\mu P_L u_e^{h_4}(\vec{p}_4)] \\ \times W^{\mu \nu}(p_4-p_1) [\bar{u}_e^{h_2}(\vec{p}_2) \gamma_\nu P_L u_{\nu_e}^{h_3}(\vec{p}_3)] \times \underbrace{\hat{a}^\dagger_{\nu_e}(\vec{p}_1,h_1)\hat{a}_e(\vec{p}_4,h_4)\hat{a}^\dagger_e(\vec{p}_2,h_2)\hat{a}_{\nu_e}(\vec{p}_3,h_3)}_{\displaystyle = - \hat{a}^\dagger_{\nu_e}(1)\hat{a}^\dagger_e(2) \hat{a}_e(4) \hat{a}_{\nu_e}(3)} \, .
\end{multline}
In anticommuting $\hat{a}^\dagger_e$ and $\hat{a}_e$ we were slightly careless regarding the delta-function that appears if $\vec{p}_2 = \vec{p}_4$ and $h_2 = h_4$, but it corresponds to disconnected parts in the Hamiltonian which only affect the ground-state energy. In other words, it disappears when prescribing the normal ordering of the Hamiltonian, which we have not mentioned here for brevity. Note that we explicitly replaced the gauge boson momentum $\Delta = p_4 - p_1$ since we look at the interaction between $\nu_e$ and $e^-$.
Finally, in order to fit with the general definition~\eqref{eq:set_quantum_numbers} so that we can directly read off the coefficients $\tilde{v}$ from its expression, the Hamiltonian~\eqref{eq:hcc_matrixel} must explicitly show a sum on the different species. This is a crucial step to get the numerical prefactor right: indeed, for now the coefficients appearing in the expression~\eqref{eq:hcc_matrixel} are not $\tilde{v}$, as it does not contain all the necessary (normal) orderings of the annihilation/creation operators, like for instance $\hat{a}^\dagger_{\nu_e}(1)\hat{a}^\dagger_e(2) \hat{a}_{\nu_e}(3) \hat{a}_{e}(4)$. In other words, as long as we do not properly antisymmetrize the ordering of $\hat{a}, \hat{a}^\dagger$ operators in the Hamiltonian, we would get coefficients like $\tilde{v}^{\nu_e(1) e(2)}_{\nu_e(3) e(4)} \neq 0$ but $\tilde{v}^{\nu_e(1) e(2)}_{e(4) \nu_e(3)} = 0$, which does not respect the antisymmetrization property~\eqref{eq:defvint}. We thus write (using $i$ instead of $(\vec{p}_i,h_i)$ for brevity):
\begin{align*}
\hat{a}^\dagger_{\nu_e}(1)\hat{a}^\dagger_e(2) \hat{a}_e(4) \hat{a}_{\nu_e}(3) = \frac14 \Big( &\hat{a}^\dagger_{\nu_e}(1)\hat{a}^\dagger_e(2) \hat{a}_e(4) \hat{a}_{\nu_e}(3) - \hat{a}^\dagger_{\nu_e}(1)\hat{a}^\dagger_e(2) \hat{a}_{\nu_e}(3)\hat{a}_e(4) \\
+ \, &\hat{a}^\dagger_e(2) \hat{a}^\dagger_{\nu_e}(1) \hat{a}_{\nu_e}(3)\hat{a}_e(4) - \hat{a}^\dagger_e(2) \hat{a}^\dagger_{\nu_e}(1)\hat{a}_e(4) \hat{a}_{\nu_e}(3)\Big) \, ,
\end{align*}
Given the factor $1/4$ in the definition~\eqref{eq:defHint} we obtain
\begin{multline}
\tilde{v}^{\nu_e(1) e(2)}_{\nu_e(3) e(4)} = - 2 \sqrt{2} G_F m_W^2 \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_e}^{h_1}(\vec{p}_1) \gamma_\mu P_L u_e^{h_4}(\vec{p}_4)] \ W^{\mu \nu}(p_4-p_1) \ [\bar{u}_e^{h_2}(\vec{p}_2) \gamma_\nu P_L u_{\nu_e}^{h_3}(\vec{p}_3)] \, .
\end{multline}
With the expression of the propagator~\eqref{eq:propagator}, we have three contributions to $\tilde{v}$.
\paragraph{Fermi order} Keeping only the lowest order in~\eqref{eq:propagator}, the exchange of a $W$ boson is reduced to a contact interaction. We can then perform a Fierz transformation~\cite{GiuntiKim,PeskinSchroeder} (it amounts to rewriting this charged-current process as a neutral-current one):
\begin{equation}
\label{eq:Fierz_CC}
[\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{e}^{h_4} (\vec{p}_4)] \ [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu P_L ] = - [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] \ [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] \, .
\end{equation}
This leads us to the final expression of the weak interaction matrix element for the charged-current processes, at Fermi order:
\begin{multline}
\label{eq:vtildeCC_Fermiorder}
\tilde{v}^{\nu_e(1) e(2)}_{\nu_e(3) e(4)} \underset{\text{CC, Fermi}}{=} 2 \sqrt{2} G_F \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] \ [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu P_L u_{e}^{h_4} (\vec{p}_4)] \, .
\end{multline}
\paragraph{First post-Fermi order} Note that we can perform a Fierz transformation with the part of the propagator $\propto \eta^{\mu \nu}$, but not with the last term. Therefore, we have:
\begin{multline}
\label{eq:vtildeCC_Delta2}
\tilde{v}^{\nu_e(1) e(2)}_{\nu_e(3) e(4)} \underset{\text{CC, $\Delta^2/m_W^2$}}{=} 2 \sqrt{2} G_F \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] \ \frac{(p_4 - p_1)^\nu (p_4-p_1)_\nu}{m_W^2} [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu P_L u_{e}^{h_4} (\vec{p}_4)] \, ,
\end{multline}
and
\begin{multline}
\label{eq:vtildeCC_DmuDnu}
\tilde{v}^{\nu_e(1) e(2)}_{\nu_e(3) e(4)} \underset{\text{CC, $-\Delta^\mu \Delta^\nu/m_W^2$}}{=} 2 \sqrt{2} G_F \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma_\mu P_L u_{e}^{h_4} (\vec{p}_4)] \ \frac{(p_4 - p_1)^\mu (p_4 - p_1)^\nu}{m_W^2} [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\nu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] \, .
\end{multline}
We chose to use Fierz identities to avoid remaining minus signs and to make the calculation of the mean-field potentials more straightforward (see next subsection).
\paragraph{Positron background} If we are interested in the interaction between electronic neutrinos and positrons, we extract the contributions in $\hat{H}_{CC}$ involving $\hat{b}_e, \hat{b}^\dagger_e$. It reads:\footnote{Recall the change of variables compared to~\eqref{eq:hcc_matrixel} corresponding to the crossing symmetry $p_2 \leftrightarrow - p_4$.}
\begin{multline}
\label{eq:hcc_matrixel_bar}
\hat{H}_{CC} \supset 2 \sqrt{2} G_F m_W^2 \sum_{h_1, \cdots} \int{\ddp{1} \cdots} \ (2\pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \times [\bar{u}_{\nu_e}^{h_1}(\vec{p}_1) \gamma_\mu P_L v_e^{h_2}(\vec{p}_2)] \\ \times W^{\mu \nu}(p_1+p_2) [\bar{v}_e^{h_4}(\vec{p}_4) \gamma_\nu P_L u_{\nu_e}^{h_3}(\vec{p}_3)] \times \hat{a}^\dagger_{\nu_e}(\vec{p}_1,h_1)\hat{b}^\dagger_e(\vec{p}_2,h_2)\hat{b}_e(\vec{p}_4,h_4)\hat{a}_{\nu_e}(\vec{p}_3,h_3) \, ,
\end{multline}
which has the opposite sign compared to~\eqref{eq:hcc_matrixel}, since there is no need to anticommute the antiparticle creation/annihilation operators which already appear "in the reference order". Of course, the full antisymmetrization is still necessary to read the $\tilde{v}$ coefficients and get the prefactor right. For instance, at Fermi order, we get following the same steps as before (including a Fierz transformation):
\begin{multline}
\tilde{v}^{\nu_e(1) \bar{e}(2)}_{\nu_e(3) \bar{e}(4)} \underset{\text{CC, Fermi}}{=} - 2 \sqrt{2} G_F \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] \ [\bar{v}_{e}^{h_4} (\vec{p}_4)\gamma_\mu P_L v_{e}^{h_2} (\vec{p}_2)] \, .
\end{multline}
\subsubsection{Set of matrix elements at Fermi order}
We show in table~\ref{Table:MatrixElements} the set of interaction matrix elements derived from the Hamiltonians~\ref{subsec:Hamiltonians}, at Fermi order. They are of special importance, since they give the first non-zero contribution to the collision term: in other words, we will only use these matrix elements to compute $\mathcal{C}$. Conversely, to compute the mean-field potentials at order $\Delta^2/m_{W,Z}^2$, one needs the matrix elements from the expansion of the propagator \eqref{eq:propagator}, which are obtained similarly (cf. the example detailed above) and not reproduced here for the sake of brevity.
\renewcommand{\arraystretch}{1.5}
\begin{table}[!h]
\centering
\begin{tabular}{|l|r|}
\hline
Interaction process & $\tilde{v}^{12}_{34}/\left[\sqrt{2} G_F (2 \pi)^3 \delta^{(3)}(\vec{p}_1+\vec{p}_2-\vec{p}_3-\vec{p}_4)\right]$ \\
\hline \hline
$CC$ & \\ \hline
$\nu_e(1) e(2) \nu_e(3) e(4)$ & $2 \times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu P_L u_{e}^{h_4} (\vec{p}_4)]$ \\
$\nu_e(1) \bar{e}(2) \nu_e(3) \bar{e}(4)$ & $- 2 \times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] [\bar{v}_{e}^{h_4} (\vec{p}_4)\gamma_\mu P_L v_{e}^{h_2} (\vec{p}_2)]$ \\
$\nu_e(1) \bar{\nu}_e(2) e(3) \bar{e}(4)$ & $2 \times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L v_{\nu_e}^{h_2} (\vec{p}_2)] [\bar{v}_{e}^{h_4} (\vec{p}_4)\gamma_\mu P_L u_{e}^{h_3} (\vec{p}_3)]$ \\ \hline \hline
$NC, \text{matter}$ & \\ \hline
$\nu_e(1) e(2) \nu_e(3) e(4)$ & $2 \times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu (g_L P_L + g_R P_R) u_{e}^{h_4} (\vec{p}_4)]$ \\
$\nu_e(1) \bar{e}(2) \nu_e(3) \bar{e}(4)$ & $- 2 \times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] [\bar{v}_{e}^{h_4} (\vec{p}_4)\gamma_\mu (g_L P_L + g_R P_R) v_{e}^{h_2} (\vec{p}_2)]$ \\
$\nu_e(1) \bar{\nu}_e(2) e(3) \bar{e}(4)$ & $2 \times [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L v_{\nu_e}^{h_2} (\vec{p}_2)] [\bar{v}_{e}^{h_4} (\vec{p}_4)\gamma_\mu (g_L P_L + g_R P_R) u_{e}^{h_3} (\vec{p}_3)]$ \\ \hline \hline
$NC, \text{self-interactions}$ & \\ \hline
$\nu_\alpha(1) \nu_\beta(2) \nu_\alpha(3) \nu_\beta(4)$ & $(1+\delta_{\alpha \beta})
\times [\bar{u}_{\nu_\alpha}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_\alpha}^{h_3} (\vec{p}_3)] [\bar{u}_{\nu_\beta}^{h_2} (\vec{p}_2)\gamma_\mu P_L u_{\nu_\beta}^{h_4} (\vec{p}_4)]$ \\
$\nu_\alpha(1) \bar{\nu}_\beta(2) \nu_\alpha(3) \bar{\nu}_\beta(4)$ & $- (1 + \delta_{\alpha \beta}) \times [\bar{u}_{\nu_\alpha}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_\alpha}^{h_3} (\vec{p}_3)] [\bar{v}_{\nu_\beta}^{h_4} (\vec{p}_4) \gamma_\mu P_L v_{\nu_\beta}^{h_2} (\vec{p}_2)]$ \\
$\nu_\alpha(1) \bar{\nu}_\alpha(2) \nu_\beta(3) \bar{\nu}_\beta(4)$ & $(1+ \delta_{\alpha \beta}) \times [\bar{u}_{\nu_\alpha}^{h_1} (\vec{p}_1) \gamma^\mu P_L v_{\nu_\alpha}^{h_2} (\vec{p}_2)] [\bar{v}_{\nu_\beta}^{h_4} (\vec{p}_4) \gamma_\mu P_L u_{\nu_\beta}^{h_3} (\vec{p}_3)]$ \\ \hline
\end{tabular}
\caption[Interaction matrix elements at Fermi order]{Interaction matrix elements at lowest order in the expansion of the gauge boson propagators (Fermi effective theory of weak interactions). We have not written the matrix elements with (anti)muons which are exactly similar to the ones with electrons/positrons with $\nu_e \leftrightarrow \nu_\mu$. The neutral-current couplings are $g_L = -1/2 + \sin^2{\theta_W}$ and $g_R = \sin^2{\theta_W}$, where $\sin^2{\theta_W} \simeq 0.231$ is the weak-mixing angle. These expressions are derived in the appendix~\ref{app:matrix_el_MF}.
\label{Table:MatrixElements}}
\end{table}
At leading order, the charged-current processes have been written as neutral-current ones thanks to Fierz rearrangement identities (cf. example above). Therefore one can write the global expression\footnote{To be precise, we should call these coupling matrices $G_{L,(e)}$ and $G_{R,(e)}$, with similarly for interactions with muons $G_{L,(\mu)} = \mathrm{diag}(g_L,g_L+1,g_L)$ and $G_{R,(\mu)} = \mathrm{diag}(g_R,g_R,g_R)$. We omit this heavy notation since only interactions with $e^\pm$ are considered in the collision term.} for all interactions between $\nu_e$ and $e^-$:
\begin{multline}
\label{eq:vtilde_nue_full}
\tilde{v}^{\nu_\alpha(1) e(2)}_{\nu_\beta(3) e(4)} = 2 \sqrt{2} G_F \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \\
\times [\bar{u}_{\nu_\alpha}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_\beta}^{h_3} (\vec{p}_3)] \ [\bar{u}_{e}^{h_2} (\vec{p}_2)\gamma_\mu (G_L^{\alpha \beta} P_L + G_R^{\alpha \beta} P_R ) u_{e}^{h_4} (\vec{p}_4)] \, ,
\end{multline}
with, in the Standard Model,
\begin{equation}
\label{eq:couplingmatrices}
G_L = \mathrm{diag}(g_L+1,g_L,g_L) \quad , \quad G_R = \mathrm{diag}(g_R,g_R,g_R) \, .
\end{equation}
One can also introduce non-standard interactions which promote those couplings to non-diagonal matrices \cite{Relic2016_revisited}.
\subsection{Mean-field potential}
With the set of all relevant $\tilde{v}^{ik}_{jl}$, one can compute the mean-field potential from \eqref{eq:Gamma}. This procedure is outlined in \cite{Volpe_2013}, and we continue the example of the charged-current processes with electrons and positrons, leaving the other cases to the appendix~\ref{app:matrix_el_MF}.
\subsubsection{Example: mean-field due to charged-current interactions with $e^{\pm}$}
Following the definition~\eqref{eq:Gamma}, we have:
\begin{align*}
\Gamma^{\nu_e(\vec{p}_1,h_1)}_{\nu_e(\vec{p}_3,h_3)} &= \sum_{h_2,h_4} \int{\ddp{2} \ddp{4}} \tilde{v}^{\nu_e(1) e(2)}_{\nu_e(3) e(4)} \times \langle \hat{a}^\dagger_e(\vec{p}_2, h_2) \hat{a}_e(\vec{p}_4, h_4)\rangle \\
&= \sum_{h_2,h_4} \int{\ddp{2} \ddp{4}} \tilde{v}^{\nu_e(1) e(2)}_{\nu_e(3) e(4)} \times (2\pi)^3 \, 2E_{p_2} \, \delta^{(3)}(\vec{p}_2 - \vec{p}_4) \, \delta_{h_2 h_4} \, f_{e^-}(p_2) \, .
\end{align*}
Let us now calculate the potentials arising from the three contributions to $\tilde{v}$.
\paragraph{Fermi order} We use the matrix element~\eqref{eq:vtildeCC_Fermiorder}, and write $\vec{p} = \vec{p}_2 = \vec{p}_4$ (which is enforced by the delta-function), such that
\begin{multline*}
\Gamma^{\nu_e(\vec{p}_1,h_1)}_{\nu_e(\vec{p}_3,h_3)} \underset{\text{CC, Fermi}}{=} 2 \sqrt{2} G_F \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \sum_{h} \int{\ddp{}} [\bar{u}_{\nu_e}^{h_1} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{h_3} (\vec{p}_3)] \\
\times [\bar{u}_{e}^{h} (\vec{p})\gamma_\mu P_L u_{e}^{h} (\vec{p})] \, f_{e^-}(p)
\end{multline*}
The spinor products can be simplified thanks to trace technology:
\begin{align*}
\sum_{h} [\bar{u}_{e}^{h} (\vec{p})\gamma_\mu P_L u_{e}^{h} (\vec{p})] &= \sum_{h} [\bar{u}_{e,i}^{h} (\vec{p})\left(\gamma_\mu P_L\right)_{ij} u_{e,j}^{h} (\vec{p})] \\
&= \left(\sum_{h}{u_{e}^{h}(\vec{p}) \bar{u}_{e}^{h}(\vec{p})}\right)_{ji} \left(\gamma_\mu P_L \right)_{ij}
\end{align*}
Next, we use the spin sum~\cite{PeskinSchroeder} $\sum_{h}{u_e^h(\vec{p})\bar{u}_e^h(\vec{p})} = \slashed{p} + m_e$. Moreover, we are dealing with ultra-relativistic neutrinos which have only one possible helicity state ($-$), the useful formula being then the projection of the spin sum on spinors with definite helicity in the ultra-relativistic limit\footnote{See for instance equation~(38.31) in~\cite{Srednicki}, being careful that it uses the metric $(-,+,+,+)$ and the convention $\{\gamma^\mu,\gamma^\nu\} = - 2 \eta^{\mu \nu}$, hence the different sign for $\slashed{k}$.} $u_{\nu_e}^{(-)}(\vec{k}) \bar{u}_{\nu_e}^{(-)}(\vec{k}) = P_L \slashed{k}$. With this, we can rewrite
\begin{align*}
\sum_{h} [\bar{u}_{e}^{h} (\vec{p})\gamma_\mu P_L u_{e}^{h} (\vec{p})] = \mathrm{tr} \left[(\gamma_\nu p^\nu + m_e) \gamma_\mu P_L \right] &= 2 p_\mu \, , \\
\intertext{and}
[\bar{u}_{\nu_e}^{(-)} (\vec{p}_1) \gamma^\mu P_L u_{\nu_e}^{(-)} (\vec{p}_1)] = \mathrm{tr} [P_L \gamma^\nu \gamma^\mu P_L] p_{1, \nu} &= 2 p_1^\nu \, ,
\end{align*}
where we have used the identities recalled in equation~\eqref{eq:trace_identities}. All in all, we have
\begin{equation}
\label{eq:Gamma_CC_Fermi_temp}
\Gamma^{\nu_e(\vec{p}_1,-)}_{\nu_e(\vec{p}_3,-)} \underset{\text{CC, Fermi}}{=} 2 \sqrt{2} G_F \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times 4 \times \int{\frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3 \, 2 E_p} } \underbrace{(p_1 \cdot p)}_{p_1 E_p - \vec{p}_1 \cdot \vec{p}} f_{e^-}(p) \, .
\end{equation}
Because of isotropy, the integral $\int{\mathrm{d}^3 \vec{p} \cdots (\vec{p}_1 \cdot \vec{p})} = 0$,
\begin{equation}
\label{eq:meanfield_CC_Fermi}
\Gamma^{\nu_e(\vec{p}_1,-)}_{\nu_e(\vec{p}_3,-)} \underset{\text{CC, Fermi}}{=} \sqrt{2} G_F \, (2 \pi)^3 \, 2 p_1 \, \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times 2 \int{\frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} \, f_{e^-}(p)} = \sqrt{2} G_F n_{e^-} \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, ,
\end{equation}
where we recognized the electron number density from~\eqref{eq:thermo_intro}. We find back the well-known mean-field potential~\eqref{eq:VCC_intro} responsible, for instance, for the MSW effect~\cite{MSW_W,MSW_MS} in stars.
\paragraph{Post-Fermi order, $\bm{\Delta^2/m_W^2}$ term} With the matrix element~\eqref{eq:vtildeCC_Delta2}, we can follow the same steps as before to get
\begin{equation*}
\Gamma^{\nu_e(\vec{p}_1,-)}_{\nu_e(\vec{p}_3,-)} \underset{\text{CC, $\Delta^2/m_W^2$}}{=} 8 \sqrt{2} \frac{G_F}{m_W^2} \, (2 \pi)^3 \, \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \times \int{\frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3 \, 2 E_p} \, (p_1 \cdot p) \times (p_1-p)^2 \times f_{e^-}(p)} \, .
\end{equation*}
We have $(p_1 - p)^2 = m_e^2 - 2 (p_1 \cdot p)$. The first term is identical to the Fermi order calculation with an overall multiplication by $(m_e/m_W)^2$. To compute the second contribution we use spherical coordinates aligned with $\vec{p_1}$, such that $p_1 \cdot p = p_1 E_p - p_1 p \cos{\theta}$. Therefore we are left with the integral
\begin{equation*}
- 2 p_1^2 \int{\frac{2 \pi p^2 \mathrm{d} p}{(2 \pi)^3 \, 2 E_p}} \int_{0}^{\pi}{\sin{\theta} \mathrm{d} \theta} \, (E_p - p \cos{\theta})^2 f_{e^-}(p) \, .
\end{equation*}
We need the three angular integrals:
\begin{equation*}
\int_{0}^{\pi}{\sin{\theta} \mathrm{d}{\theta}} = 2 \ , \quad \int_{0}^{\pi}{\sin{\theta} \cos{\theta} \mathrm{d}{\theta}} = 0 \ , \quad \int_{0}^{\pi}{\sin{\theta} \cos^2{\theta} \mathrm{d}{\theta}} = \frac23 \, ,
\end{equation*}
with which we obtain, using the thermodynamic formulas~\eqref{eq:thermo_intro},
\begin{align*}
- 2 p_1^2 \int{\frac{2 \pi p^2 \mathrm{d} p}{(2 \pi)^3 \, 2 E_p}} \int_{0}^{\pi}{\sin{\theta} \mathrm{d} \theta} \, (E_p - p \cos{\theta})^2 f_{e^-}(p) &= - p_1^2 \int{\frac{4 \pi p^2 \mathrm{d} p}{(2 \pi)^3}} \left(E_p + \frac{p^2}{3 E_p}\right) f_{e^-}(p) \\
&= - \frac12 p_1^2 \left(\rho_{e^-} + P_{e^-}\right) \, .
\end{align*}
Hence the second contribution to the mean-field potential
\begin{equation}
\label{eq:meanfield_CC_Delta2}
\Gamma^{\nu_e(\vec{p}_1,-)}_{\nu_e(\vec{p}_3,-)} \underset{\text{CC, $\Delta^2/m_W^2$}}{=} \sqrt{2} G_F n_{e^-} \left(\frac{m_e}{m_W}\right)^2 \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} - \frac{2 \sqrt{2} G_F p_1}{m_W^2}(\rho_{e^-} + P_{e^-}) \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, .
\end{equation}
The mean-field potentials up to first order in $\Delta^2/m_W^2$ do not usually take into account the non-relativistic nature of electrons and positrons \cite{SiglRaffelt,Mangano2005,Relic2016_revisited,Gariazzo_2019,Akita2020}. Instead, our expression involves both the energy density and the pressure of charged leptons, as mentioned for instance in \cite{NotzoldRaffelt_NuPhB1988}. As expected, we recover the more common expression in the ultra-relativistic limit $\rho_{e^-} + P_{e^-} \to (4/3) \rho_{e^-}$.
\paragraph{Post-Fermi order, $\bm{- \Delta^\mu \Delta^\nu/m_W^2}$ term} We finally include the matrix element~\eqref{eq:vtildeCC_DmuDnu} in the general expression of $\Gamma$. It reads
\begin{multline*}
\Gamma^{\nu_e(\vec{p}_1,-)}_{\nu_e(\vec{p}_3,-)} \underset{\text{CC, $-\Delta^\mu \Delta^\nu/m_W^2$}}{=} 2 \sqrt{2} \frac{G_F}{m_W^2} \, (2 \pi)^3 \delta^{(3)}(\vec{p}_1 - \vec{p}_3) \sum_{h} \int{\ddp{}} \, (p - p_1)_\mu (p - p_1)_\nu \\ \times [\bar{u}_{\nu_e}^{-} (\vec{p}_1) \gamma^\mu P_L u_{e}^{h} (\vec{p})] [\bar{u}_{e}^{h} (\vec{p})\gamma^\nu P_L u_{\nu_e}^{-} (\vec{p}_3)] \times f_{e^-}(p) \, .
\end{multline*}
As before, we rewrite the spinor product (enforcing once again $\vec{p}_1 = \vec{p}_3$):
\begin{align*}
\sum_{h} [\bar{u}_{\nu_e}^{-} (\vec{p}_1) \gamma^\mu P_L u_{e}^{h} (\vec{p})] [\bar{u}_{e}^{h} (\vec{p})\gamma^\nu P_L u_{\nu_e}^{-} (\vec{p}_1)] &= \mathrm{tr} [ P_L \gamma^\sigma p_{1,\sigma} \gamma^\mu P_L (\gamma^\lambda p_\lambda + m_e) \gamma^\nu P_L ] \\
&= m_e p_{1, \sigma} \underbrace{\mathrm{tr} [\gamma^\sigma \gamma^\mu \gamma^\nu P_L]}_{=0} + \, p_{1, \sigma} p_\lambda \mathrm{tr} [\gamma^\sigma \gamma^\mu \gamma^\lambda \gamma^\nu P_L ]
\end{align*}
With the trace identities~\eqref{eq:trace_identities}, we split the result into its imaginary and its real part.
\begin{itemize}
\item The imaginary part of the mean-field contains the product:
\[ \underbrace{(p-p_1)_{\mu} (p-p_1)_\nu}_{\text{sym. $\mu \leftrightarrow \nu$}} \ \ \times \underbrace{\epsilon^{\sigma \mu \lambda \nu}}_{\text{asym. $\mu \leftrightarrow \nu$}} = 0 \, . \]
\item The real part reads after calculation:
\begin{equation*}
(p-p_1)_\mu (p-p_1)_\nu p_{1,\sigma} p_\lambda \times 2 \left(\eta^{\sigma \mu} \eta^{\lambda \nu} - \eta^{\sigma \lambda} \eta^{\mu \nu} + \eta^{\sigma \nu} \eta^{\mu \lambda} \right) = 2 m_e^2 (p_1 \cdot p) \, .
\end{equation*}
\end{itemize}
Comparing with the expression~\eqref{eq:Gamma_CC_Fermi_temp} in the Fermi order calculation, we get
\begin{equation}
\label{eq:meanfield_CC_DmuDnu}
\Gamma^{\nu_e(\vec{p}_1,-)}_{\nu_e(\vec{p}_3,-)} \underset{\text{CC, $-\Delta^\mu \Delta^\nu/m_W^2$}}{=} \frac{G_F}{\sqrt{2}} n_{e^-} \left(\frac{m_e}{m_W}\right)^2 \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, .
\end{equation}
\paragraph{Positron background} Up until now, we have only considered the mean-field due to the interaction with the bath of electrons. Without any additional calculation, we can obtain the potential due to the positron background, simply looking at the additional signs that appear along the derivation:
\begin{itemize}
\item due to the minus sign coming from the anti-commutation of $\hat{b}_e, \, \hat{b}^\dagger_e$ (cf.~for instance the expression of the matrix element at Fermi order in table~\ref{Table:MatrixElements}), the Fermi order result for the $e^+$ background is exactly the opposite of the $e^-$ one, with $n_{e^-}$ replaced by $n_{e^+}$:
\begin{equation*}
\Gamma^{\nu_e(\vec{p}_1,-)}_{\nu_e(\vec{p}_3,-)} \overset{[e^-]}{ \underset{\text{CC, Fermi}}{=}} - \sqrt{2} G_F n_{e^+} \, \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, ;
\end{equation*}
\item beyond Fermi order, the interaction being now an $s-$channel instead of a $t-$channel, $\Delta = (p_1 - p)$ is replaced by $(p_1 + p)$. This changes the sign inside $\Delta^2 = m_e^2 - 2 (p_1 \cdot p)$. Therefore the energy density and pressure contributions to the mean-field get an additional minus sign (and end up with the same sign whether it is an $e^-$ or $e^+$ background), but not the density terms.
\end{itemize}
\paragraph{Full CC mean-field} We can now gather all the previous contributions \eqref{eq:meanfield_CC_Fermi}, \eqref{eq:meanfield_CC_Delta2}, \eqref{eq:meanfield_CC_DmuDnu} (and the corresponding potentials due to the interactions with positrons), which leads to
\begin{equation}
\Gamma^{\nu_e(\vec{p}_1, -)}_{\nu_e(\vec{p}_3, -)} = \left\{ \sqrt{2} G_F (n_{e^-} - n_{e^+}) \left[1 + \mathcal{O}\left(\frac{m_e^2}{m_W^2}\right) \right] - \frac{2 \sqrt{2} G_F p_1}{m_W^2}(\rho_{e^-} + P_{e^-} + \rho_{e^+} + P_{e^+}) \right \} \bm{\delta}_{\vec{p}_1 \vec{p}_3} \, .
\end{equation}
The post-Fermi order contribution proportional to the charged lepton densities is negligible compared to the Fermi order one. Moreover, the asymmetry $(n_{e^-}-n_{e-+})/n_\gamma \simeq \eta \sim 6 \times 10^{-10}$ is of the order of the baryon-to-photon ratio, hence completely negligible compared to the (yet "higher order") energy density/pressure term.
Since, as expected given the assumptions of homogeneity and isotropy, $\Gamma$ is diagonal in momentum space, we only deal from now on with its diagonal part $\Gamma(p)$ (such that $\Gamma^{\vec{p},-}_{\vec{p}',-} = \Gamma(p) \bm{\delta}_{\vec{p} \vec{p}'}$). Moreover, we restrict ourselves to the $(-,-)$ helicity subspace (otherwise we would just need to add some helicity Kronecker symbols). Finally, as mentioned before, the exact same calculation can be made with $\nu_\mu - \mu^{\pm}$, which allows to display the full charged-current contribution, showing the flavour matrix structure:
\begin{multline}
\Gamma^{\nu_\alpha}_{\nu_\beta}(p) \underset{\text{CC}}{=} \sqrt{2} G_F (n_{e^-} - n_{e^+}) \delta^{\alpha}_{e} \delta^e_{\beta} - \frac{2 \sqrt{2} G_F p}{m_W^2} \left(\rho_{e^-} + P_{e^-} + \rho_{e^+} + P_{e^+}\right) \delta^{\alpha}_{e} \delta^e_{\beta} \\
+ \sqrt{2} G_F (n_{\mu^-} - n_{\mu^+}) \delta^{\alpha}_{\mu} \delta^\mu_{\beta} - \frac{2 \sqrt{2} G_F p}{m_W^2} \left(\rho_{\mu^-} + P_{\mu^-} + \rho_{\mu^+} + P_{\mu^+}\right) \delta^{\alpha}_{\mu} \delta^\mu_{\beta} \, .
\end{multline}
\subsubsection{Complete mean-field expression}
In addition to the charged-current processes, one needs to take into account the neutral-current processes with the background leptons and with (anti)neutrinos (so-called self-interactions). Some elements of the calculation are outlined in the appendix~\ref{app:matrix_el_MF}, and lead to the following final expression:\footnote{The absence of extra complex conjugation on $\mathbb{N}_{\bar{\nu}}$, i.e.~on $\bar{\varrho}$, compared to \cite{Volpe_2013} is due to the transposed definition of the antineutrino density matrix~\eqref{eq:homogeneity}.}
\begin{multline}
\label{eq:Gamma_potential}
\Gamma^{\alpha}_{\beta} = \sqrt{2} G_F (\mathbb{N}_\mathrm{lep^-} - \mathbb{N}_\mathrm{lep^+})^\alpha_\beta + \sqrt{2} G_F \left(\mathbb{N}_\nu - \mathbb{N}_{\bar{\nu}}\right)^\alpha_\beta \\ - \frac{2 \sqrt{2} G_F p}{m_W^2}(\mathbb{E}_\mathrm{lep^-} + \mathbb{P}_\mathrm{lep^-} + \mathbb{E}_\mathrm{lep^+} + \mathbb{P}_\mathrm{lep^+})^\alpha_\beta- \frac{8 \sqrt{2} G_F p}{m_Z^2}\left(\mathbb{E}_\nu + \mathbb{E}_{\bar{\nu}} \right)^{\alpha}_{\beta} \, .
\end{multline}
The first two terms are the particle/antiparticle asymmetric mean-field potentials arising from the V$-$A Hamiltonian. There is no contribution from the neutral-current processes with the matter background as they are flavour-independent (see appendix~\ref{app:matrix_el_MF}). As shown in the example above, expanding the gauge boson propagators to next-to-leading order in the exchange momentum leads to the symmetric terms proportional to the neutrino momentum $p$. This expression is derived in the flavour basis in which $\delta^{\alpha}_{e}$ is the Kronecker symbol. However it can be directly read in any basis, through the contravariant (covariant) transformation of upper (lower) indices \eqref{eq:transfo}.
The various thermodynamic quantities involved are, where we use the standard definitions~\eqref{eq:thermo_intro} for the charged leptons,
\begin{equation}
\begin{aligned}
\mathbb{N}_\mathrm{lep^-} &= \mathrm{diag}(n_{e^-} , n_{\mu^-}, 0) \\
\left. \mathbb{N}_\nu \right|^\alpha_\beta &= \int{\frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} \varrho^\alpha_\beta(p)}
\end{aligned} \qquad
\begin{aligned} \mathbb{E}_\mathrm{lep^-} &= \mathrm{diag}(\rho_{e^-} , \rho_{\mu^-}, 0) \, , \\
\left. \mathbb{E}_\nu \right|^\alpha_\beta &= \int{\frac{\mathrm{d}^3 \vec{p}}{(2 \pi)^3} \, p \, \varrho^\alpha_\beta(p)} \, ,
\end{aligned}
\end{equation}
and the corresponding quantities for antiparticles are obtained by replacing $f_{e^-} \to f_{e^+}$ and $\varrho^\alpha_\beta \to \bar{\varrho}^\alpha_\beta$. We define the total charged lepton energy density matrix $\mathbb{E}_\mathrm{lep} \equiv \mathbb{E}_\mathrm{lep^-} + \mathbb{E}_\mathrm{lep^+}$, and likewise for the pressure. In the following, we will systematically neglect the very small asymmetry of electrons/positrons, which is constrained to be of the order of the baryon-to-photon ratio $\eta \sim 10^{-9}$. Since electrons and positrons undergo very efficient electromagnetic interactions with the photon background, ensuring that their distribution function remains a Fermi-Dirac one at the photon temperature $T_\gamma$ \cite{Grohs2019}., we will use as electron distribution function
\begin{equation}
\label{eq:distrib_electrons}
f_{e^-}(p) = f_{e^+}(p) = \dfrac{1}{e^{\sqrt{p^2 + m_e^2}/{T_\gamma}}+1} \equiv f_e(p) \, .
\end{equation}
Note that the total neutrino energy density, that is the sum of the diagonal contribution of $\mathbb{E}_\nu$, reads:
\begin{equation}
\label{eq:rhonu_matrix}
\rho_\nu = \mathrm{Tr}(\mathbb{E}_\nu) \, .
\end{equation}
\subsection{Collision integral}
\label{subsec:collision_integral}
The remaining part of the QKE is the collision term, which is derived by inserting all possible matrix elements in the general expression \eqref{eq:C11}.
This leads to collision integrals previously derived in \cite{SiglRaffelt,BlaschkeCirigliano}, and progressively included in numerical computations, except for the self-interactions, whose off-diagonal components were approximated by damping terms or discarded \cite{Mangano2005,Gava:2010kz,Gava_corr,Relic2016_revisited,Gariazzo_2019}. In the following, we illustrate how our formalism applies by carrying out an explicit derivation for neutrino-neutrino scattering, displaying the full matrix structure of the statistical factor. The other contributions to the collision term are discussed in the appendix~\ref{app:collision_term}.
\subsubsection{Neutrino self-interactions collision term}
As an illustration of the use of the BBGKY formalism to derive the collision integrals, we detail the steps to obtain the neutrino-neutrino scattering contribution to the expression to come~\eqref{eq:C_nn}.
Neutrino-neutrino scattering processes correspond to the terms in \eqref{eq:C11} for which the inner matrix elements are scattering ones $\tilde{v}^{\nu_\delta \nu_\sigma}_{\nu_\delta \nu_\sigma}$. For simplicity, we focus here on the first term in the expression of $\mathcal{C}^{i_1}_{i_1'}$ \eqref{eq:C11}. Here, the index $i_1$ will refer to $\nu_\alpha(\vec{p}_1)$ and $i_1'$ to $\nu_\beta(\vec{p}_{\underline{1}})$. We do not specify the helicity, which is necessarily $h=(-)$ for ultra-relativistic neutrinos. Finally, we impose $\vec{p}_k = \vec{p}'_k$ for all $k$, which is enforced by the assumption of homogeneity~\eqref{eq:homogeneity}. There are two non-zero contributions to this part of the collision matrix.
\begin{itemize}
\item when 1 and 3 have the same flavour, that is for the following term:
\[ \frac14 \tilde{v}^{\nu_\alpha(1) \nu_\gamma(2)}_{\nu_\alpha(3) \nu_\gamma(4)}\times \varrho^{\alpha(3)}_{\delta(3)} \varrho^{\gamma(4)}_{\sigma(4)} \times \tilde{v}^{\nu_\delta(3') \nu_\sigma(4')}_{\nu_\delta(1') \nu_\sigma(2')} \times (\mathbb{1}- \varrho)^{\delta(1)}_{\beta(1)} (\mathbb{1}- \varrho)^{\sigma(2)}_{\gamma(2)} \, . \]
The scattering amplitude is then
\begin{align}
&\tilde{v}^{\nu_\alpha(1) \nu_\gamma(2)}_{\nu_\alpha(3) \nu_\gamma(4)}\times \tilde{v}^{\nu_\delta(3') \nu_\sigma(4')}_{\nu_\delta(1') \nu_\sigma(2')} \nonumber \\
&\quad = 2 G_F^2 \times (2 \pi)^6 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \delta^{(3)}(\vec{p}_1 - \vec{p}_{\underline{1}}) \nonumber \\
&\quad \quad \times [\bar{u}_{\nu_\alpha}(1) \gamma^\mu P_L u_{\nu_\alpha}(3)][\bar{u}_{\nu_\delta}(3) \gamma^\nu P_L u_{\nu_\delta}(1)] \times [\bar{u}_{\nu_\gamma}(2) \gamma_\mu P_L u_{\nu_\gamma}(4)][\bar{u}_{\nu_\sigma}(4) \gamma_\nu P_L u_{\nu_\sigma}(2)] \nonumber\\
&\quad = 2 G_F^2 \times (2 \pi)^6 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \delta^{(3)}(\vec{p}_1 - \vec{p}_{\underline{1}}) \nonumber\\
&\quad \quad \times p_{3 \eta} p_{1 \rho} \mathrm{tr}[\gamma^\rho \gamma^\mu P_L \gamma^\eta \gamma^\nu P_L] \times p_4^\lambda p_2^\tau \mathrm{tr}[\gamma_\tau \gamma_\mu P_L \gamma_\lambda \gamma_\nu P_L] \nonumber \\
&\quad = 2^5 G_F^2 \times (2 \pi)^6 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \delta^{(3)}(\vec{p}_1 - \vec{p}_{\underline{1}}) \times (p_1 \cdot p_2) (p_3 \cdot p_4) \, , \label{eq:scatt_ampl_13}
\end{align}
while the matrix product associated to this scattering amplitude is
\[\varrho^{\alpha(3)}_{\delta(3)} \varrho^{\gamma(4)}_{\sigma(4)} (\mathbb{1}- \varrho)^{\delta(1)}_{\beta(1)} (\mathbb{1}- \varrho)^{\sigma(2)}_{\gamma(2)} = \Big[ \mathrm{Tr}[ \varrho_4 \cdot (\mathbb{1}-\varrho_2) ] \cdot \varrho_3 \cdot (\mathbb{1}-\varrho_1) \Big]^\alpha_\beta \, . \]
\item when 1 and 4 have the same flavour, the scattering amplitude is
\begin{align}
&\tilde{v}^{\nu_\alpha(1) \nu_\gamma(2)}_{\nu_\gamma(3) \nu_\alpha(4)} \times \tilde{v}^{\nu_\delta(3') \nu_\sigma(4')}_{\nu_\delta(1') \nu_\sigma(2')} \nonumber \\ &\quad = - 2 G_F^2 \times (2 \pi)^6 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \delta^{(3)}(\vec{p}_1 - \vec{p}_{\underline{1}}) \nonumber \\
&\quad \quad \times [\bar{u}_{\nu_\alpha}(1) \gamma^\mu P_L u_{\nu_\alpha}(4)][\bar{u}_{\nu_\sigma}(4) \gamma^\nu P_L u_{\nu_\sigma}(2)] [\bar{u}_{\nu_\gamma}(2) \gamma_\mu P_L u_{\nu_\gamma}(3)][\bar{u}_{\nu_\delta}(3) \gamma_\nu P_L u_{\nu_\delta}(1)] \nonumber \\
&\quad = - 2 G_F^2 \times (2 \pi)^6 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \delta^{(3)}(\vec{p}_1 - \vec{p}_{\underline{1}}) \nonumber \\
&\quad \quad \times p_{3 \lambda} p_{1 \rho} p_{4 \eta} p_{2 \tau} \mathrm{tr}[\gamma^\mu P_L \gamma^\eta \gamma^\nu P_L \gamma^\tau \gamma_\mu P_L \gamma^\lambda \gamma_\nu P_L \gamma^\rho] \nonumber \\
&\quad = 2^5 G_F^2 \times (2 \pi)^6 \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) \delta^{(3)}(\vec{p}_1 - \vec{p}_{\underline{1}}) \times (p_1 \cdot p_2) (p_3 \cdot p_4) \, , \label{eq:scatt_ampl_14}
\end{align}
and the matrix product reads
\[\varrho^{\gamma(3)}_{\delta(3)} \varrho^{\alpha(4)}_{\sigma(4)} (\mathbb{1}- \varrho)^{\delta(1)}_{\beta(1)} (\mathbb{1}- \varrho)^{\sigma(2)}_{\gamma(2)} = \Big[ \varrho_4 \cdot (\mathbb{1}-\varrho_2) \cdot \varrho_3 \cdot (\mathbb{1}-\varrho_1) \Big]^\alpha_\beta \, . \]
\end{itemize}
We chose the compact notation $\varrho_k \equiv \varrho(p_k)$ for brevity, and used $\varrho_1 = \varrho_{\underline 1}$ thanks to the momentum-conserving function $\delta^{(3)}(\vec{p}_1 - \vec{p}_{\underline{1}})$.
The scattering amplitudes of the four terms in~\eqref{eq:C11} are identical, and their matrix products arrange such that the final result has the expected "gain $-$ loss $+$ h.c." structure. Note that we considered here a particular ordering of the indices, while the full expression is symmetric through the exchange of the indices $(1',2')$. In other words, one must take twice the previous results~\eqref{eq:scatt_ampl_13} and \eqref{eq:scatt_ampl_14} to account for all non-zero combinations.\footnote{This symmetry vanishes if $\delta$ and $\sigma$ represent the same flavour. However, this is exactly compensated by the extra factor of $2$ in the matrix elements for identical flavour, cf.~table~\ref{Table:MatrixElements}. This extra factor of $2$ is already accounted for regarding the couple $(\alpha, \gamma)$ as it allows to treat the case $\alpha=\gamma$ like the others (i.e.~separating a "trace" and a "non-trace" contributions).} Therefore,
\begin{equation}
\label{eq:C_nnscatt}
\begin{aligned}
\mathcal{C}^{[\nu\nu \leftrightarrow \nu \nu]} = &(2 \pi)^3 \delta^{(3)}(\vec{p}_1-\vec{p}_{\underline{1}}) \frac{2^5 G_F^2}{2}\int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_3] [\mathrm{d}^3 \vec{p}_4] (2 \pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)} \\
&\times (p_1 \cdot p_2)(p_3 \cdot p_4) \times F_\mathrm{sc}(\nu^{(1)},\nu^{(2)},\nu^{(3)},\nu^{(4)})
\end{aligned}
\end{equation}
with the statistical factor:
\begin{multline}
F_\mathrm{sc}(\nu^{(1)},\nu^{(2)},\nu^{(3)},\nu^{(4)}) = \left[ \varrho_4 (\mathbb{1} - \varrho_2) + \mathrm{Tr}(\cdots) \right] \varrho_3 (\mathbb{1} -\varrho_1) + (\mathbb{1} - \varrho_1) \varrho_3 \left[ (\mathbb{1} - \varrho_2) \varrho_4 + \mathrm{Tr}(\cdots)\right] \\
- \left[ (\mathbb{1} - \varrho_4) \varrho_2 + \mathrm{Tr}(\cdots)\right] (\mathbb{1} -\varrho_3) \varrho_1 - \varrho_1 (\mathbb{1} -\varrho_3) \left[\varrho_2(\mathbb{1} -\varrho_4) + \mathrm{Tr}(\cdots)\right] \, ,
\end{multline}
where $\mathrm{Tr}(\cdots)$ means the trace of the term in front of it.
A useful check consists in neglecting flavour mixing, i.e., assuming that the neutrino density matrices are diagonal, the diagonal entries being the distribution functions. In this case, we have $\varrho_4 (\mathbb{1} - \varrho_2) + \mathrm{Tr}(\cdots) \to 4 f_4 (1-f_2)$, hence a total amplitude for neutrino-neutrino scatterings $2^5 G_F^2 \times 4 \times (p_1 \cdot p_2)(p_3 \cdot p_4) = 2^7 G_F^2 (p_1 \cdot p_2)(p_3 \cdot p_4)$. This is in agreement with the results quoted in~\cite{Grohs2015} (Table I), \cite{Dolgov_NuPhB1997} (Tables 1 and 2) or \cite{FidlerPitrou}.
\subsubsection{Final form of the QKE}
The full expression of the collision integral is derived in the appendix~\ref{app:collision_term}. As for the mean-field term, it is diagonal in momentum space: $\mathcal{C}^{\vec{p}_1}_{\vec{p}_{\underline{1}}}$ is proportional to $\bm{\delta}_{\vec{p}_1 \vec{p}_{\underline{1}}}$. Therefore, all the terms in the QKE are momentum-diagonal, and the actual equation is obtained by removing the momentum-conserving delta-functions. Notably, we will denote as the collision integral the quantity $\mathcal{I}$, related to $\mathcal{C}$ via $\mathcal{C}^{\vec{p}_1}_{\vec{p}_{\underline{1}}} = (2 \pi)^3 \, 2 E_1\, \delta^{(3)}(\vec{p}_1 - \vec{p}_{\underline{1}}) \mathcal{I}[\varrho]$, and the QKE reads:
\begin{equation}
\boxed{{\mathrm i} \, \frac{\mathrm{d} \varrho(p)}{\mathrm{d} t} = \left[t + \Gamma, \varrho \right] + {\mathrm i} \, \mathcal{I}} \, .
\end{equation}
\section{QKEs for neutrinos in the early Universe}
In this last section, we gather the previous elements of the QKE to introduce the precise equations that will be numerically solved in various cases.
\subsection{Set of Quantum Kinetic Equations}
We present here the QKE for $\varrho(p,t)$, obtained from \eqref{eq:eqvrho} after dividing each term by the momentum-conserving function $\bm{\delta}_{\vec{p} \vec{p}'}$ from \eqref{eq:homogeneity}. Moreover, the time derivative $\mathrm{d}/\mathrm{d} t$ becomes $\partial/\partial t - H p \, \partial/\partial p$ to account for the expansion of the Universe, $H\equiv \dot a/a$ being the Hubble rate, given by Friedmann's equation $H^2 = (8 \pi \mathcal{G}/3) \rho$. The QKEs read:
\begin{multline}
{\mathrm i} \left[ \frac{\partial}{\partial t} - H p \frac{\partial}{\partial p}\right] \varrho = \Big[ U \frac{\mathbb{M}^2}{2p}U^\dagger, \varrho \Big] + \sqrt{2} G_F \Big[ \mathbb{N}_\nu - \mathbb{N}_{\bar{\nu}}, \varrho \Big] \\
- 2 \sqrt{2} G_F p \Big[ \frac{\mathbb{E}_\text{lep} + \mathbb{P}_\text{lep}}{m_W^2} + \frac43 \frac{\mathbb{E}_{\nu} + \mathbb{E}_{\bar{\nu}}}{m_Z^2},\varrho \Big ] + {\mathrm i} \mathcal{I} \label{eq:QKE_rho}
\end{multline}
where we recall the definitions in flavour space $\mathbb{E}_\text{lep} \equiv \mathrm{diag}(\rho_{e^-} + \rho_{e^+},\rho_{\mu^-} + \rho_{\mu^+},0)$ and likewise for $\mathbb{P}_\text{lep}$. Similarly, the QKEs for the antineutrino density matrix read (cf.~appendix~\ref{app:antiparticles}):
\begin{multline}
{\mathrm i} \left[ \frac{\partial}{\partial t} - H p \frac{\partial}{\partial p}\right] \bar{\varrho} = - \Big[ U \frac{\mathbb{M}^2}{2p}U^\dagger, \bar{\varrho} \Big] + \sqrt{2} G_F \Big[ \mathbb{N}_\nu - \mathbb{N}_{\bar{\nu}}, \bar{\varrho} \Big] \\
+ 2 \sqrt{2} G_F p \Big[ \frac{\mathbb{E}_\text{lep} + \mathbb{P}_\text{lep}}{m_W^2} + \frac43 \frac{\mathbb{E}_{\nu} + \mathbb{E}_{\bar{\nu}}}{m_Z^2},\bar{\varrho} \Big ] + {\mathrm i} \bar{\mathcal{I}} \label{eq:QKE_rhobar}
\end{multline}
The collision term is the sum of the contributions from different physical processes: scattering with charged leptons ($\nu e^\pm\leftrightarrow \nu e^\pm$), annihilation ($\nu \bar{\nu} \leftrightarrow e^+ e^-$) and self-interactions (involving only $\nu$ and $\bar{\nu}$). Note that in the collision integral, we do not take into account the interactions with muons whose number density is negligible in the range of temperatures of interest. The expressions for the processes involving charged leptons are exactly the same as the ones quoted in \cite{Relic2016_revisited} [eqs.~(2.4)--(2.10)], and we do not report them here for brevity. This reference, however, does not contain the full expressions for neutrino self-interactions, of which we derived explicitly a part in the section~\ref{subsec:collision_integral}. Our complete expression for the self-interactions contribution to the collision integral reads:\footnote{It is equivalent to eq.~(96) of~\cite{BlaschkeCirigliano} (one only needs to swap the variables $\vec{p}_3 \leftrightarrow \vec{p}_4$ in the second and fourth terms of \eqref{eq:F_sc_nn}). Our expression highlights the ''gain $-$ loss $+$ h.c.'' structure of this collision term.}
\begin{equation}
\label{eq:C_nn}
\begin{aligned}
\mathcal{I}^{[\nu \nu]} = &\frac12 \frac{2^5 G_F^2}{2 p_1} \int{[\mathrm{d}^3 \vec{p}_2] [\mathrm{d}^3 \vec{p}_3] [\mathrm{d}^3 \vec{p}_4] (2 \pi)^4 \delta^{(4)}(p_1 + p_2 - p_3 - p_4)} \\
&\Big[ (p_1 \cdot p_2)(p_3 \cdot p_4) F_\mathrm{sc}(\nu^{(1)},\nu^{(2)},\nu^{(3)},\nu^{(4)}) \\
&+ (p_1 \cdot p_4)(p_2 \cdot p_3) \left( F_\mathrm{sc}(\nu^{(1)},\bar{\nu}^{(2)},\nu^{(3)},\bar{\nu}^{(4)}) + F_\mathrm{ann}(\nu^{(1)},\bar{\nu}^{(2)},\nu^{(3)},\bar{\nu}^{(4)}) \right) \Big] \, ,
\end{aligned}
\end{equation}
with the statistical factors for scattering and annihilation processes:
\begin{multline}
\label{eq:F_sc_nn}
F_\mathrm{sc}(\nu^{(1)},\nu^{(2)},\nu^{(3)},\nu^{(4)}) = \left[ \varrho_4 (\mathbb{1} - \varrho_2) + \mathrm{Tr}(\cdots) \right] \varrho_3 (\mathbb{1} -\varrho_1) + (\mathbb{1} - \varrho_1) \varrho_3 \left[ (\mathbb{1} - \varrho_2) \varrho_4 + \mathrm{Tr}(\cdots)\right] \\
- \left[ (\mathbb{1} - \varrho_4) \varrho_2 + \mathrm{Tr}(\cdots)\right] (\mathbb{1} -\varrho_3) \varrho_1 - \varrho_1 (\mathbb{1} -\varrho_3) \left[\varrho_2(\mathbb{1} -\varrho_4) + \mathrm{Tr}(\cdots)\right] \, ,
\end{multline}
\begin{multline}
\label{eq:F_sc_nbn}
F_\mathrm{sc}(\nu^{(1)},\bar{\nu}^{(2)},\nu^{(3)},\bar{\nu}^{(4)}) = \left[ (\mathbb{1} - \bar{\varrho}_2) \bar{\varrho}_4 + \mathrm{Tr}(\cdots) \right] \varrho_3 (\mathbb{1} -\varrho_1) + (\mathbb{1} - \varrho_1) \varrho_3 \left[ \bar{\varrho}_4 (\mathbb{1} - \bar{\varrho}_2) + \mathrm{Tr}(\cdots)\right] \\
- \left[ \bar{\varrho}_2 (\mathbb{1} -\bar{\varrho}_4) + \mathrm{Tr}(\cdots) \right] (\mathbb{1} -\varrho_3) \varrho_1 - \varrho_1 (\mathbb{1} -\varrho_3) \left[ (\mathbb{1} -\bar{\varrho}_4) \bar{\varrho}_2 + \mathrm{Tr}(\cdots)\right] \, ,
\end{multline}
\begin{multline}
\label{eq:F_ann_nn}
F_\mathrm{ann}(\nu^{(1)},\bar{\nu}^{(2)},\nu^{(3)},\bar{\nu}^{(4)}) = \left[ \varrho_3 \bar{\varrho}_4 + \mathrm{Tr}(\cdots) \right] (\mathbb{1} -\bar{\varrho}_2) (\mathbb{1} -\varrho_1) + (\mathbb{1} - \varrho_1) (\mathbb{1} -\bar{\varrho}_2) \left[ \bar{\varrho}_4 \varrho_3 + \mathrm{Tr}(\cdots)\right] \\
- \left[ (\mathbb{1} -\varrho_3) (\mathbb{1} -\bar{\varrho}_4) + \mathrm{Tr}(\cdots) \right] \bar{\varrho}_2 \varrho_1 - \varrho_1 \bar{\varrho}_2 \left[ (\mathbb{1} -\bar{\varrho}_4) (\mathbb{1} -\varrho_3) + \mathrm{Tr}(\cdots)\right] \, ,
\end{multline}
where, as before, we chose the more compact notation $\varrho_k = \varrho(p_k)$, and $\mathrm{Tr}(\cdots)$ means the trace of the term in front of it.
\subsection{Reduced equations}
\label{subsec:reduced_equations_QKE}
In this final section, we transform the QKE~\eqref{eq:QKE_rho} and in particular the collision integral into a form suitable for numerical resolution, which will be the topic of the next chapter.
\paragraph{Reduction of the collision integral} The most time consuming part of the QKE is the computation of the collision term. Thanks to the homogeneity and isotropy of the early Universe, and the particular form of the scattering amplitudes, the nine-dimensional collision integrals can be reduced to two-dimensional ones \cite{Hannestad_PhRvD1995,Semikoz_Tkachev,Dolgov_NuPhB1997,Grohs2015}. We follow here the reduction method of~\cite{Dolgov_NuPhB1997}. The first idea is to use the integral representation of the delta function:
\[ \delta^{(3)}(\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4) = \int{\frac{\mathrm{d}^3 \vec{\lambda}}{(2 \pi)^3} e^{{\mathrm i} \vec{\lambda} \cdot (\vec{p}_1 + \vec{p}_2 - \vec{p}_3 - \vec{p}_4)}} \, , \]
and use spherical coordinates defined as follows: the "$\vec{e}_\mathrm{z}$ unit vector" for $\vec{\lambda}$ is aligned with $\vec{p}_1$, while $\vec{\lambda}$ is the "$\vec{e}_\mathrm{z}$ unit vector" for $\vec{p}_{i \geq 2}$, that is,
\[ \cos{\theta_\lambda} \equiv \frac{\vec{p}_1 \cdot \vec{\lambda}}{p_1 \lambda} \qquad ; \qquad \cos{\theta_i} \equiv \frac{\vec{p}_i \cdot \vec{\lambda}}{p_i \lambda} \ \ \text{for } i = 2,3,4 \, , \]
the associated azimuthal angles $\varphi_\lambda, \varphi_{i \geq 2}$ being defined as usual. Then, thanks to the very simple form of the scattering amplitudes in the four-fermion approximation --- cf.~for instance~\eqref{eq:scatt_ampl_13} and~\eqref{eq:scatt_ampl_14} ---, we can perform all the $\varphi$ and $\theta$ integrations. The next step consists in performing the integration on $\lambda$, whose result can be analytically calculated~\cite{Dolgov_NuPhB1997,BlaschkeCirigliano} (these are the so-called "$D-$functions" of Dolgov, Hansen and Semikoz~\cite{Dolgov_NuPhB1997}, which are piecewise quadrivariate polynomials). Finally, we are left with a three-dimensional integral on the momentum moduli $p_2, p_3, p_4$, and one of them is done integrating out the energy delta function $\delta(E_1 + E_2 - E_3 - E_4)$, namely $\int{p_4 \mathrm{d} p_4 \, \delta(E_1+E_2-E_3-E_4)} = E_1+E_2-E_3$, since $p_4 \mathrm{d} p_4 = E_4 \mathrm{d} E_4$. This procedure is outlined in the Appendix~\ref{app:reduc_collision_integral}.
\paragraph{Equation in comoving variables} In view of a numerical implementation, let us use the comoving variables introduced in~\eqref{eq:comoving_variables}.
Therefore, the QKEs are rewritten:
\begin{multline}
\frac{\partial \varrho(x,y_1)}{\partial x} = - \frac{{\mathrm i}}{xH} \left(\frac{x}{m_e}\right) \left[ U \frac{\mathbb{M}^2}{2y_1}U^\dagger, \varrho \right] + {\mathrm i} \frac{2 \sqrt{2} G_F}{xH} y_1 \left(\frac{m_e}{x}\right)^5 \left[ \frac{\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}}{m_W^2} ,\varrho \right ] \\
- {\mathrm i} \frac{\sqrt{2} G_F}{x H} \left(\frac{m_e}{x}\right)^3 \left[ \overline{\mathbb{N}}_\nu - \overline{\mathbb{N}}_{\bar{\nu}}, \varrho \right] + {\mathrm i} \frac{8 \sqrt{2} G_F}{3 x H} y_1 \left( \frac{m_e}{x}\right)^5 \left[\frac{\bar{\mathbb{E}}_\nu + \bar{\mathbb{E}}_{\bar{\nu}}}{m_Z^2}, \varrho\right] + \frac{1}{xH} \mathcal{I} \, , \label{eq:QKE_fullfinal}
\end{multline}
with the two-dimensional collision integral (recall that we assume $f_e = f_{\bar{e}}$, which regroups some terms):
\begin{equation}
\label{eq:I_full}
\begin{aligned}
\mathcal{I} = &\frac{G_F^2}{2 \pi^3 y_1} \left(\frac{m_e}{x}\right)^5 \int{y_2 \mathrm{d} y_2 \, y_3 \mathrm{d} y_3 \, \bar{E}_4 \times \frac12} \\
\times &\Big[ 4 \left[ 2 d_1 + 2 d_3 + d_2(1,2) + d_2(3,4) - d_2(1,4) - d_2(2,3) \right] \\
&\qquad \qquad \times \left(F_\mathrm{sc}^{LL}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) + F_\mathrm{sc}^{RR}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)})\right) \\
&- 4 x^2 \left[d_1 - d_2(1,3) \right]/\bar{E}_2 \bar{E}_4 \times \left(F_\mathrm{sc}^{LR}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) + F_\mathrm{sc}^{RL}(\nu^{(1)},e^{(2)},\nu^{(3)},e^{(4)}) \right) \\
+ \, &4 \left[ d_1 + d_3 - d_2(1,4) - d_2(2,3) \right] \times \left( F_\mathrm{ann}^{LL}(\nu^{(1)},\bar{\nu}^{(2)},e^{(3)},e^{(4)}) + F_\mathrm{ann}^{RR}(\nu^{(1)},\bar{\nu}^{(2)},e^{(3)},e^{(4)}) \right) \\
&+ 2 x^2\left[d_1 + d_2(1,2) \right]/\bar{E}_3 \bar{E}_4 \times \left(F_\mathrm{ann}^{LR}(\nu^{(1)},\bar{\nu}^{(2)},e^{(3)},e^{(4)}) + F_\mathrm{ann}^{RL}(\nu^{(1)},\bar{\nu}^{(2)},e^{(3)},e^{(4)}) \right) \\
+ \, & \left[ d_1 + d_3 + d_2(1,2) + d_2(3,4) \right] \times F_\mathrm{sc}(\nu^{(1)},\nu^{(2)},\nu^{(3)},\nu^{(4)}) \\
&+ \left[ d_1 + d_3 - d_2(1,4) - d_2(2,3) \right] \times \left( F_\mathrm{sc}(\nu^{(1)},\bar{\nu}^{(2)},\nu^{(3)},\bar{\nu}^{(4)}) + F_\mathrm{ann}(\nu^{(1)},\bar{\nu}^{(2)},\nu^{(3)},\bar{\nu}^{(4)}) \right) \Big]
\end{aligned}
\end{equation}
The $d-$functions are $d_i = (x/m_e) d_i^{\mathrm{DHS}}$, with $d_i^{\mathrm{DHS}}$ defined in\cite{Dolgov_NuPhB1997} as functions of the momenta $p$, hence the prefactor $x/m_e$. It should be noted that \cite{Relic2016_revisited,Bennett2021} use a different convention (4 times greater $D-$functions and opposite sign for $D_2$). $\bar{E} \equiv E/T_{\mathrm{cm}}$ is the comoving energy, and $E_4$ stands for $E_1 + E_2 - E_3$ by energy conservation. The full expressions can be found in appendix A of~\cite{Dolgov_NuPhB1997}, appendix D of~\cite{BlaschkeCirigliano} or in the appendix~\ref{app:reduc_collision_integral} of this manuscript.
\paragraph{QKE for $\bm{\bar{\varrho}}$} Similarly to the neutrino density matrix QKE, we rewrite~\eqref{eq:QKE_rhobar} using the comoving variables, which leads to:
\begin{multline}
\frac{\partial \bar{\varrho}(x,y_1)}{\partial x} = + \frac{{\mathrm i}}{xH} \left(\frac{x}{m_e}\right) \left[ U \frac{\mathbb{M}^2}{2y_1}U^\dagger, \bar{\varrho} \right] - {\mathrm i} \frac{2 \sqrt{2} G_F}{xH} y_1 \left(\frac{m_e}{x}\right)^5 \left[ \frac{\bar{\mathbb{E}}_\mathrm{lep} + \bar{\mathbb{P}}_\mathrm{lep}}{m_W^2} ,\bar{\varrho} \right ] \\
- {\mathrm i} \frac{\sqrt{2} G_F}{x H} \left(\frac{m_e}{x}\right)^3 \left[ \overline{\mathbb{N}}_\nu - \overline{\mathbb{N}}_{\bar{\nu}}, \bar{\varrho} \right] - {\mathrm i} \frac{8 \sqrt{2} G_F}{3 x H} y_1 \left( \frac{m_e}{x}\right)^5 \left[\frac{\bar{\mathbb{E}}_\nu + \bar{\mathbb{E}}_{\bar{\nu}}}{m_Z^2}, \bar{\varrho}\right] + \frac{1}{xH} \bar{\mathcal{I}} \, . \label{eq:QKEbar_fullfinal}
\end{multline}
These equations are at the core of any study of (anti)neutrino evolution in the early Universe, and allow to take into account mixing (via the vacuum term), refractive matter effects (mean-field potentials, including the self-interaction one), and collisions which notably drive the transfer of entropy from electron/positron annihilations towards (anti)neutrinos throughout the decoupling era. In the following chapters, we will adapt these equations to particular setups of interest: the "standard" calculation of neutrino decoupling in chapter~\ref{chap:Decoupling}, and the evolution of primordial neutrino asymmetries in chapter~\ref{chap:Asymmetry}. We will systematically be focusing on the final neutrino spectra, which allow to compute the cosmological observables we are interested in (namely, $N_{\mathrm{eff}}$), before exploring further the consequences on BBN in chapter~\ref{chap:BBN}.
\end{document}
\chapter[\protect\numberline{}Remerciements][Remerciements]{Remerciements}
\lettrine[lines=3]{U}{ne} thèse est, avant qu'elle commence, une expérience tout à la fois excitante et effrayante. Excitante d'abord, pour la liberté qu'elle annonce, la possibilité de travailler sur un sujet passionnant, de rencontrer des chercheurs de tous horizons... mais effrayante aussi car il faut faire face à un problème a priori « sans réponse ». Ce manuscrit est le fruit de cette expérience, et il n'aurait pu voir le jour sans l'apport de tant de personnes que je vais essayer bien modestement de mentionner.
Qui dit soutenance de thèse dit jury, dont je remercie les membres pour avoir accepté un rôle si particulier : celui de me faire entrer dans la « communauté de la recherche ». Merci notamment à George Fuller et Pasquale Serpico d'avoir accepté d'être les rapporteurs de cette thèse. Plus généralement, je souhaite remercier ici tous les chercheurs et toutes les chercheuses que j'ai eu le plaisir et la chance de côtoyer ces dernières années, que ce soit à l'IAP ou ailleurs. J'ai notamment pu collaborer avec Cristina Volpe que je remercie pour tout ce qu'elle a pu m'apprendre sur les neutrinos et leur monde si mystérieux. Une mention particulière aux membres du GReCO et à mes parrains de thèse, Silvia Galli et Patrick Boissé, pour leur accompagnement précieux. L'IAP a été un excellent environnement de travail, et ce grâce à ses équipes administratives et techniques, ainsi qu'au travail de l'équipe de direction que je salue ici. À côté de la recherche, l'enseignement a constitué une part de ce doctorat très importante pour moi, et je suis reconnaissant à Arnaud Raoux, Agnès Maître, Christophe Balland, Quentin Grimal, Laurent Coolen et Arnaud Cassan de m'avoir permis d'enseigner dans leurs différentes UE.
Parmi tous ces chercheurs, il y en a évidemment un que je dois remercier tout particulièrement : mon directeur de thèse, Cyril. Plus qu'un simple encadrant, j'ai eu avec toi une relation de quasi-collègue, me donnant de formidables clés pour entrer dans ce monde de la recherche. Nous avons franchi tant bien que mal les différents confinements en gardant toujours un contact étroit, et je ne peux que souligner tout spécialement tes capacités exceptionnelles de relecture de manuscrit dans des lieux inhabituels. Je ne sais pas si je retrouverai ailleurs une telle disponibilité bienveillante, et je resterai longtemps marqué par ta vision géométrique (fort éclairante le plus souvent~!) des choses, ainsi que cette volonté systématique de chercher à comprendre la physique derrière des équations ou des résultats numériques — une attitude qui devrait être, j'en suis bien convaincu, systématique. Pour tout cela, merci.
Les amis, bien sûr. Celles et ceux sans qui trois années de thèse (d'autant plus en rajoutant une surcouche de confinement) seraient bien plus tristes. Merci à mes amis de lycée que je retrouve annuellement en rouge \& blanc, à mes amis de prépa devenus partenaires de brunch. Je pense aux moments passés avec mes amis de l'ENS, qu'ils soient musicaux\footnote{« Chevaleresques », dixit un certain groupe nommé BrassENS.}, pseudo-sportifs (une pensée pour cette brochette d'athlètes olympiques du BDS)... je ne peux pas citer tout le monde, mais tout le monde a joué un rôle dans cette thèse, et j'espère avoir soudé des amitiés qui dureront encore longtemps. Un groupe tout particulier m'aura beaucoup accompagné, car notre amitié a été forgée dans \sout{le sang} des préparations de plans pour l'agrégation (et bien plus que ça depuis~!). Je me permets de mentionner en particulier Hugo et Jules, partenaires d'un projet d'édition un peu fou, mais qui traduit notre amour de la physique.
La vie quotidienne à l'IAP serait bien moins agréable sans la présence d'un formidable groupe de doctorants, organisant (les regrettées) movie nights, nombreuses pauses et autres vendredis à des bars non loin du laboratoire. Mille mercis à ceux que j'ai vus devenir docteurs, ceux qui m'ont accompagné pendant trois ans et ceux qui resteront encore un peu après mon départ\footnote{Félicitations à Pierre qui entre dans ces trois catégories.} : Aline, Amaury, Axel, Clément, Daniel, Denis, Eduardo, Emilie, Emma, Etienne, François, Louis, Lucas, Marko, Pierre, Quentin, Shweta, Virginia, Warren et tous les autres.
Parmi tous mes amis, une mention particulière à celui avec qui je ne pensais pas passer autant de temps\footnote{Vous avez dit confinement ?}, mais qui se sera avéré être un merveilleux colocataire, compagnon de jeux, de boissons \& fromages, de visionnage de séries, de coinche, soutien indéfectible quand cela s'est avéré nécessaire, bref merci infiniment à toi Emilien.
J'en termine avec ceux sans qui, évidemment, je ne serai pas arrivé là tant ils m'ont accompagné depuis (c'est le cas de le dire) le début. Merci à toute ma famille, merci Alexandre, merci Papa, merci Maman.
\end{document}
\chapter[\protect\numberline{}Résumé en français][Résumé]{Résumé}
\lettrine[lines=3]{C}{ette} thèse s'inscrit dans le cadre de la cosmologie moderne, qui est définitivement entrée dans une ère de précision. Par exemple, les dernières mesures du fond diffus cosmologique (CMB) par \emph{Planck} ont fourni une nouvelle validation du modèle standard de la cosmologie $\Lambda$CDM et de ses extensions directes. Un résultat notable de \emph{Planck} est la mesure d'une quantité clé, le \emph{nombre effectif d'espèces de neutrinos} $N_{\mathrm{eff}}$. Ce paramètre quantifie l'excès de densité d'énergie dans le fond cosmique de neutrinos entre l'évolution réelle de l'Univers et l'approximation dite de découplage instantané : puisque le découplage des neutrinos du plasma électromagnétique de photons, d'électrons et de positrons n'est pas entièrement terminé lorsque les annihilations des électrons/positrons prennent place, la prise en compte de ce "chevauchement" conduit à une plus grande densité d'énergie. Plus généralement, une prédiction robuste et précise des conséquences de ce "découplage incomplet des neutrinos" est cruciale car les neutrinos ont un impact sur de nombreuses étapes cosmologiques, de la nucléosynthèse primordiale (BBN) à la formation des structures.
\subsection*{Étude du découplage "standard" des neutrinos}
\paragraph{Aspects formels} Les premières déterminations de $N_{\mathrm{eff}}$ négligeaient les oscillations de saveur des neutrinos, ce qui réduisait le problème physique à la résolution d'une équation de Boltzmann pour les fonctions de distribution des neutrinos. Cependant, puisque les particules n'évoluent pas dans le vide mais dans un bain thermique, des corrections d'électrodynamique quantique (QED) à la thermodynamique du plasma doivent aussi être prises en compte. Enfin, la prise en compte des oscillations de saveur nécessite de remplacer l'ensemble des fonctions de distribution par un objet plus général, une \emph{matrice densité}, et d'introduire de même une généralisation adaptée de l'équation de Boltzmann. Jusqu'à présent, les méthodes permettant d'obtenir la dénommée "équation cinétique quantique" (\emph{Quantum Kinetic Equation}, QKE) étaient une approche opérationnelle reposant sur un développement perturbatif en l'interaction faible~\cite{SiglRaffelt}, ou l'approche fonctionnelle basée sur les fonctions de Green et le formalisme CTP (\emph{Closed-Time-Path})~\cite{BlaschkeCirigliano}.
Dans le chapitre~\ref{chap:QKE}, nous présentons une méthode alternative --- à savoir une \emph{hiérarchie BBGKY généralisée} ---, où le développement perturbatif de~\cite{SiglRaffelt} est remplacé par une séparation contrôlée des contributions (non-)corrélées à la matrice densité à $1-$, $2-$, ... $n-$ corps. Cette méthode avait été utilisée dans~\cite{Volpe_2013} pour obtenir les termes de champ moyen de la QKE dans l'approximation de Hartree-Fock. Nous avons été au-delà de cette approximation et inclus des corrélations d'ordre plus élevé (utilisant l'ansatz du chaos moléculaire) afin d'obtenir le terme de collision, c'est-à-dire les contributions de diffusion et d'annihilation entre $\nu$, $\bar{\nu}$ et avec $e^-$, $e^+$ --- et donc l'équation cinétique complète.
\paragraph{Aspects numériques} Dans le chapitre~\ref{chap:Decoupling}, nous présentons un calcul du découplage des neutrinos avec, pour la première fois, le terme de collision complet (et les corrections QED susmentionnées). À titre d'exemple, l'évolution de la température\footnote{Il s'agit en réalité d'une température \emph{effective} car le processus de découplage est légèrement hors-équilibre, cela est précisé dans la section~\ref{subsec:results_Neff}.} des différentes saveurs de neutrinos est représentée Figure~\ref{fig:evolution_tnu_fr}. Avec ces résultats, nous avons obtenu la nouvelle valeur
\[ N_{\mathrm{eff}} = 3, \! 0440 \, , \]
avec une précision de quelques $10^{-4}$. Cette incertitude est due aux valeurs expérimentales actuelles des paramètres physiques (et notamment l'angle de mélange $\theta_{12}$), ainsi que la variabilité reliée aux paramètres numériques du code \texttt{NEVO} que nous avons développé.
Des calculs précédents, qui tenaient compte des oscillations de saveur et qui ont abouti à la valeur $N_{\mathrm{eff}} \simeq 3, \! 045$~\cite{Relic2016_revisited}, ne tenaient pas compte de l'intégralité du terme de collision : ses composantes hors diagonale étaient approchées par un terme d'amortissement. L'inclusion de ce terme sans aucune approximation est un véritable défi numérique, en particulier à cause de la raideur qu'il apporte à l'équation différentielle et parce qu'il se calcule en un temps $\mathcal{O}(N^3)$ où $N$ est la taille de la grille d'impulsions. Nous assurons un temps de calcul raisonnable grâce à une amélioration majeure, à savoir le calcul direct du jacobien du système différentiel. Notre résultat sur $N_{\mathrm{eff}}$ a été confirmé ultérieurement par~\cite{Bennett2020}.
\begin{figure}[!ht]
\centering
\includegraphics{figs/Evolution_Tnu_FR.pdf}
\caption[Évolution des températures effectives des neutrinos]{\label{fig:evolution_tnu_fr} Évolution de la température (effective) des neutrinos au cours du découplage, en prenant ou non en compte les oscillations de saveur. $T_{\mathrm{cm}}$ est la température comobile, proportionnelle à l'inverse du facteur d'échelle, qui correspond à la température des neutrinos dans l'approximation de découplage instantané.}
\end{figure}
\paragraph{Description approchée de l'évolution} Nous avons également introduit une description effective des oscillations de saveur, qui donne des résultats indiscernables de ceux obtenus en résolvant l'équation exacte. Elle permet également de réduire considérablement le temps de calcul, ce qui constitue une autre amélioration importante de notre code. Cette approximation repose sur l'existence d'une grande séparation d'échelles entre les fréquences d'oscillation et le taux de collision, ce qui permet de faire la moyenne de ces oscillations. En d'autres termes, la matrice densité reste toujours diagonale dans la base de la matière (la base des états propres du Hamiltonien prenant en compte les effets du vide et de champ moyen). Nous avons appelé cette description simplifiée l'approximation de \emph{Transfert Adiabatique d'Oscillations Moyennées} (ATAO pour \emph{Adiabatic Transfer of Averaged Oscillations}). De plus, nous avons utilisé cette approximation afin de mieux comprendre certains résultats comme l'absence d'effets de la phase CP dans le découplage standard des neutrinos.
\subsection*{Nucléosynthèse primordiale et découplage incomplet des neutrinos}
En résolvant la QKE, nous obtenons les distributions gelées des (anti)neutrinos, qui à leur tour donnent accès aux paramètres cosmologiques tels que $N_{\mathrm{eff}}$ (cf.~ci-dessus) ou la densité d'énergie des neutrinos \emph{aujourd'hui} $\Omega_\nu$. Ainsi, nous sommes notamment en possession des deux paramètres qui fixent les différents effets du découplage incomplet des neutrinos sur la plus ancienne sonde de l'histoire de l'Univers dont nous disposons --- la BBN --- : la distribution de $\nu_e$, $\bar{\nu}_e$ et la densité d'énergie paramétrée par $N_{\mathrm{eff}}$.
Le chapitre~\ref{chap:BBN} est dédié à l'évaluation des changements des abondances primordiales d'hélium, de deutérium et de lithium dus au découplage incomplet des neutrinos. Tout d'abord, les abondances des éléments légers dépendent du taux d'expansion de l'Univers (donc de $N_{\mathrm{eff}}$, via l'effet dit d'horloge --- \emph{clock effect} ---). Ensuite, l'abondance des neutrons au début de la BBN est entre autres fixée par le rapport neutrons-protons qui varie si l'on change les distributions de $\nu_e$, $\bar{\nu}_e$. Nous avons étudié en détail comment ces effets interagissaient, en comparant leurs contributions relatives et en fournissant des estimations analytiques lorsque cela était possible. Ce travail théorique est mené conjointement à une étude numérique, en combinant notre code d'évolution des neutrinos et le code de BBN \texttt{PRIMAT}~\cite{Pitrou_2018PhysRept}. En particulier, nous avons pu résoudre un désaccord existant dans la littérature entre~\cite{Mangano2005} et~\cite{Grohs2015} concernant la variation de l'abondance du deutérium due à un découplage incomplet des neutrinos.
\subsection*{Évolution des asymétries primordiales}
Le cas "standard" du découplage des neutrinos suppose que l'asymétrie des leptons est nulle, une approximation justifiée pour les électrons et les positrons (dont la dégénérescence doit être de l'ordre du rapport baryon/photon $\eta \simeq 6 \times 10^{-10}$~\cite{Fields:2019pfx} par neutralité de charge), mais il n'existe pas de telle contrainte pour les neutrinos. Le CMB et, plus important encore, le BBN sont en fait les meilleures sources de limites sur les asymétries des neutrinos, puisque de telles asymétries affecteraient les abondances primordiales par les mêmes mécanismes que ceux présentés précédemment.
La présence d'asymétries de neutrinos non nulles ajoute une couche de complexité considérable à la physique de l'évolution des neutrinos. En effet, il faut désormais prendre en compte un terme supplémentaire de champ moyen d'auto-interaction dans la QKE, qui domine pendant une grande partie de l'ère du découplage des neutrinos pour des asymétries $\mu/T \in [10^{-3},10^{-1}]$. Dans la lignée de nos travaux sur la résolution des QKEs dans le cas standard avec le terme de collision complet, nous étendons notre code au cas asymétrique dans le chapitre~\ref{chap:Asymmetry}.
\begin{figure}[!h]
\centering
\includegraphics{figs/th23_mu1_0p001_NoColl_FR.pdf}
\caption[Exemple d'évolution des neutrinos dans un cas asymétrique à deux saveurs]{\label{fig:asym_resume_fr} Évolution de l'asymétrie dans un cas à deux saveurs $\nu_\mu$ (vert) - $\nu_\tau$ (rouge), avec une différence de masse $\Delta m^2 = 2,45 \times 10^{-3} \, \mathrm{eV}^2$ et un angle de mélange $\theta = 0,831$. Les résultats de la résolution directe de la QKE (trait plein) et via l'approximation ATAO (pointillés) sont indiscernables. \emph{Paramètres initiaux :} $\xi_1 = \mu_1/T = 0,001 \equiv \xi$, $\xi_2 = 0$.}
\end{figure}
De plus, nous avons généralisé l'approximation ATAO pour tenir compte des auto-interactions, qui rendent le Hamiltonien non linéaire. Cette description approchée nous a permis de retrouver analytiquement les résultats connus sur les oscillations collectives dites \emph{synchrones}, mais aussi de découvrir que ce régime est généralement suivi d'oscillations \emph{quasi-synchrones} de fréquences croissant plus rapidement. Nous avons fourni de nombreuses vérifications analytiques et numériques de ce nouveau résultat, dans le cas simplifié à deux saveurs mais aussi dans le cadre général à trois saveurs. À titre d'illustration, on représente Figure~\ref{fig:asym_resume_fr} un exemple d'évolution de l'asymétrie dans un cas simplifié à deux saveurs. Dans cet exemple, le régime d'oscillations synchrones prend place jusqu'à environ $2,8 \, \mathrm{MeV}$, et est suivi par le régime d'oscillations quasi-synchrones.
Nous avons exploré plus avant la dépendance de la configuration finale des neutrinos par rapport aux paramètres de mélange, et nous avons notamment montré que la phase CP de Dirac ne peut pas affecter substantiellement la valeur finale de $N_{\mathrm{eff}}$ ni le spectre électronique final des (anti)neutrinos, et ne devrait donc pas affecter les observables cosmologiques.
\end{document}
\chapter*{Table of Symbols}
\vspace{-1cm}
\renewcommand{\arraystretch}{1.3}
\begin{center}
\begin{longtable}{|lll|}
\hline
Notation & Description & Definition, relation \\\hline\hline
\endhead
$g_{\mu \nu}$ & 4-dimensional spacetime metric & Signature $(+,-,-,-)$ \\
$\gamma_{ij}$ & Spatial metric & $g_{ij} = - a^2 \gamma_{ij}$ \\
$a(t)$ & Scale factor & \\
$H$ & Hubble rate & $H = \dot{a}/a$ \\
$T_{\mathrm{cm}}$ & Comoving temperature & $T_{\mathrm{cm}} \propto a^{-1}$ \\
$x$ & Reduced scale factor & $x = m_e / T_{\mathrm{cm}}$ \\
$y$ & Comoving momentum & $y = p / T_{\mathrm{cm}}$ \\
$z$ or $z_\gamma$ & Dimensionless plasma temperature & $z = T_\gamma / T_{\mathrm{cm}}$ \\
$f_{\nu_\alpha}(p,t)$ & Neutrino distribution function & \\
$z_{\nu_\alpha}$ & Effective temperature of $\nu_\alpha$ & Equations~\eqref{eq:param_fnu} and~\eqref{eq:param_rho} \\
$\delta g_{\nu_\alpha}$ & Non-thermal spectral distortions of $\nu_\alpha$ & \\
$\rho_{\nu_\alpha}$ & Energy density of $\alpha-$flavour neutrinos & $\rho_{\nu_\alpha} = 7/8 \times \pi^2/30 \times (z_{\nu_\alpha} T_{\mathrm{cm}})^4$ \\
$\rho_\nu$ & Total neutrino energy density & $\rho_{\nu} = \rho_{\nu_e} + \rho_{\nu_\mu} + \rho_{\nu_\tau}$ \\
$N_{\mathrm{eff}}$ & Effective number of neutrino species & See Eq.~\eqref{eq:intro_def_Neff} and section~\ref{subsec:results_Neff} \\
$\mathcal{N}$ & Heating rate & Equation~\eqref{eq:Nheating} \\ \hline
\textbf{BBN} & & Section~\ref{subsec:intro_BBN} and chapter~\ref{chap:BBN} \\
$n_b$ & Baryon density & \\
$X_i$ & Number fraction of isotope $i$ & $X_i = n_i / n_b$ \\
$Y_i$ & Mass fraction of isotope $i$ & $Y_i = A_i X_i$ \\
$Y_{\mathrm{p}}$ & Helium-4 primordial abundance & $Y_{\mathrm{p}} = Y_{\He4}$ \\
$i/\mathrm{H}$ & Density ratio & $i/\mathrm{H} = {n_i}/{n_\mathrm{H}}$ \\ \hline
$\Delta m^2_{ij}$ & Mass-squared difference & $\Delta m^2_{ij} = m_{\nu_i}^2 - m_{\nu_j}^2$ \\
$\mathbb{M}^2$ & Matrix of mass-squared differences & $\mathbb{M}^2 = \mathrm{diag}(0,\Delta m^2_{21}, \Delta m^2_{31})$ \\
$\theta_{ij}$ & Mixing angle & \\
$U$ & PMNS mixing matrix & See section~\ref{subsec:Values_Mixing} \\ \hline
\textbf{BBGKY} & & Chapter~\ref{chap:QKE} \\
$\varrho^{i_1 \cdots i_s}_{j_1 \cdots j_s}$ & $s-$body reduced density matrix & $\varrho^{i_1 \cdots i_s}_{j_1 \cdots j_s} = \langle \hat{a}^\dagger_{j_s} \cdots \hat{a}^\dagger_{j_1} \hat{a}_{i_1} \cdots \hat{a}_{i_s} \rangle$ \\
$C^{ik}_{jl}$ & Correlated part of $\varrho^{ik}_{jl}$ & $\varrho^{ik}_{jl} = \varrho^i_j \varrho^k_l - \varrho^i_l \varrho^k_j + C^{ik}_{jl}$ \\
$\tilde{v}^{ik}_{jl}$ & Interaction matrix element & $\tilde{v}^{ik}_{jl} = \bra{ik} \hat{H}_\mathrm{int} \ket{jl}$ \\
$\Gamma^i_j$ & Mean-field potential & $\Gamma^i_j = \sum_{k,l}{\tilde{v}^{ik}_{jl} \varrho^l_k}$ \\
$\mathcal{I}$ & Collision integral (see Eq.~\eqref{eq:C11} for $\mathcal{C}$) & $\mathcal{C}^{\vec{p}}_{\vec{p}'} = (2 \pi)^3 \, 2 E \, \delta^{(3)}(\vec{p} - \vec{p}') \mathcal{I}(p)$ \\ \hline
$\mathcal{H}_0$ & Vacuum Hamiltonian & Equation~\eqref{eq:Hvac} \\
$\mathcal{H}_\mathrm{lep}$ & Lepton mean-field Hamiltonian & Equation~\eqref{eq:Hlep} \\
$\mathcal{V}$ & Effective Hamiltonian (no asymmetry) & $\mathcal{V} = \mathcal{H}_0 + \mathcal{H}_\mathrm{lep}$ \\
$\mathcal{A}$ & Asymmetry matrix & $\mathcal{A} = \int{(\varrho - \bar{\varrho}) y^2 \mathrm{d}{y}/(2 \pi^2)}$ \\
$\mathcal{J}$ & Self-interaction mean-field & Equation~\eqref{DefJ} \\
$\mathcal{K}$ & Dimensionless collision term & $\mathcal{K} = \mathcal{I}/xH$ \\
$\gamma$ & Adiabaticity parameter & Equation~\eqref{Defgammatr1} \\ \hline
\end{longtable}
\end{center}
\renewcommand{\arraystretch}{1}
\pagestyle{simple}
\import{Remerciements/}{remerciements}
\import{Resume/}{resume_fr}
\mainmatter
\import{Introduction/}{Introduction}
\adjustmtc
\clearpage
\import{IntroCosmo/}{IntroCosmo}
\import{QKE/}{QKE}
\import{NeutrinoDecoupling/}{StandardDecoupling}
\import{BBN/}{BBN}
\import{Asymmetry/}{Asymmetry}
\import{Conclusion/}{Conclusion}
\adjustmtc
\clearpage
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
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St. John the Clairvoyant of Egypt was born at the beginning of the 4th Century. He lived in the town of Lycopolis in Central Egypt, and worked as a carpenter. At tl1e age of 25, he accepted monastic tonsure. Over the course of 15 years, St. John struggled in various monasteries. Then, wanting to live in total seclusion, he left for the Hill of Wolves in the Thebaid. St. John spent 50 years in seclusion, never leaving the site of his spiritual struggle. He spoke to visitors through a little window, through which he also received the simple meals brought to him. After St. John had already spent 20 years in seclusion, God made him worthy to receive the gift of clairvoyance. Thus, he prophesied to Emperor Theodosius the Great (379 -395) that he would gain victory over his enemies Maximus and Eugenius, and that a war against the Gauls would bring him victory. To many visitors he told of events to come in their lives, and he provided them with instruction. The holy struggler would give holy oil to ill visitors who, upon anointing themselves, would be healed of their various illnesses.
Venerable St. John prophesied to his hagiographer, the Monk Palladios, that he would become a bishop. The prophecy came to pass, and Palladios was made bishop of Bythinia, in Asia Minor.
Children, love silence, keep your thoughts always on God, and always implore God that He might grant you a pure mind, free of sinful thoughts. Of course, also worthy of praise is the spiritual struggler who, living in the world, practices the virtues, offers hospitality to strangers, or gives alms, or helps those in need, or never rises to anger. Such a person is worthy of praise because he remains virtuous and fulfills the Lord's commandments without abandoning his secular activities. But better and more praiseworthy will be one who, ever mindful of God, ascends from the material to the immaterial, leaving material cares and others' concerns, himself aspires to the Heavenly. Having renounced everything earthly, and loosed from ties to the world through earthly cares, he ever stands before God. Such a person is close to God, Whom he ceaselessly praises in prayer and psalmody.
By such salvific teachings and instructive stories, and by the example of his angelic life, the Saint brought great spiritual help to others. St. John of Egypt departed to the Lord in the year 395 at the age of 90.
In thee, oh, Father, the one created in the image of God was saved, for taking up the Cross, thou didst follow Christ and, by thy deeds didst teach us to overlook the flesh, for it is perishable, but to be attentive to the soul since it is immortal. Therefore, oh, pious Father Daniil, your spirit rejoices with the angels.
The icon is of another John, called the Hermit, who also was from Egypt, but lived, it is presumed, in the 8th-9th century and traveled with 98 other ascetics to Crete where he spent the rest of his life in a cave. The icon depicts how he traverses the sea between the island of Gavdos to the island of Crete on his ryassa, because he had been left behind there by error by his companions. It also shows the way of his death - while collecting herbs for his food he was shot by a hunter who mistook him for an animal. His cave became a famous place of pilgrimage. His holy relics are venerated in the nearby Monastery of Gouverneto (12 km from Hania).
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"redpajama_set_name": "RedPajamaC4"
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{"url":"https:\/\/homework.cpm.org\/category\/CC\/textbook\/cca2\/chapter\/8\/lesson\/8.3.1\/problem\/8-126","text":"Home > CCA2 > Chapter 8 > Lesson 8.3.1 > Problem8-126\n\n8-126.\n\nThe city of Waynesboro is trying to decide whether to initiate a composting project where each residence would be provided with a dumpster for garden and yard waste. The city manager needs some measure of assurance that the citizens will participate before launching the project, so he chooses a random sample of $25$ homes and provides them with the new dumpster for yard and garden waste. After one week the contents of each dumpster is weighed (in pounds) before processing. The sorted data is shown below: 8-126 HW eTool (CPM). Homework Help \u270e\n\n $0$ $0$ $0$ $0$ $1.7$ $2.6$ $2.9$ $4.2$ $4.4$ $5.1$ $5.6$ $6.4$ $8.0$ $8.9$ $9.7$ $10.1$ $11.2$ $13.5$ $15.1$ $16.3$ $17.7$ $21.4$ $22.0$ $22.2$ $36.5$ Checksum $245.5$\n1. Create a combination boxplot and histogram. Use an interval of $0$ to $42$ pounds on the x-axis and a bin width of $6$ pounds.\n\nThe distribution has a right skew and an outlier at $36.5$ pounds so the center is best described by the median of $8.0$ pounds and the spread by the IQR of $12.95$ pounds.\n\n2. Describe the center, shape, spread and outliers.\n\n3. What is a better measure of center for this distribution the mean or median and why?\n\nThe median is better in this case because it is not affected by skewing and outliers.\n\n4. What is a better measure of spread the standard deviation or IQR and why?\n\nThe IQR is better in this case because it is less affected by skewing and outliers than the standard deviation.\n\n5. The city can sell the compost, and engineers estimate the program will be profitable if each home averages at least $9$ pounds of material. The city manager sees the mean is nearly $10$ pounds and is ready to order dumpsters for every residence. What advice would you give him?\n\nRemoving the outlier from the data drops the mean to $8.7$ pounds which is below the profitable minimum. Suggest running the test a few more weeks. Perhaps as people get used to the composting program, they will participate more.\n\nUse the eTool below for help with this problem.\nClick on the link at right for the full eTool version: CCA2 8-126 HW eTool","date":"2020-02-29 07:24:17","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 36, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.2687211334705353, \"perplexity\": 1190.0865124009997}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875148671.99\/warc\/CC-MAIN-20200229053151-20200229083151-00429.warc.gz\"}"}
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{"url":"http:\/\/arnienumbers.blogspot.com\/2010\/10\/equality-of-sums-some-random-matrices.html","text":"## 28.10.10\n\n### Equality of Sums & Some \"Random\" Matrices\n\n(I would like to personally thank Terry Tao for his very helpful blogs on random matrices -- see here. A closer look at his ideas there helped me with my research on OPNs.)\n\nConsider the equation\n\nA + B = C + D (where, for now, we assume all of A, B and C are positive, D is unrestricted)\n\nAssuming a nonzero sum C + D:\n\n(A + B)\/(C + D) = 1 (from which it also follows that the sum A + B is nonzero)\n\nAssuming C is not equal to D:\n\n[(A + B)(C - D)]\/[(C + D)(C - D)] = 1\n\n[ (AC - BD) + (BC - AD) ] \/ [ C^2 - D^2 ] = 1\n\n[ (AC - BD)\/(C^2 - D^2) ] + [ (BC -AD)\/(C^2 - D^2) ] = 1\n\nPassing on to determinant notation:\n\ndet X + det Y = det Z\n\nwhere the 2x2 matrices X, Y and Z are defined as follows:\n\nX\nA B\nD C\n\nY\nB A\nD C\n\nZ\nC D\nD C\n\n(The numbers A, B, C and D were laid out in their respective positions by \"basket-weaving\".)\n\nSome linear algebra:\n\nThe traces of these matrices are:\n\ntr(X) = A + C\ntr(Y) = B + C\ntr(Z) = 2C\n\nSolving for the eigenvalues of each of these 3 matrices:\n\ndet(X - xI) = 0 = (A - x)(C - x) - BD = x^2 - (A + C)x + (AC - BD)\ndet(Y - yI) = 0 = (B - y)(C - y) - AD = y^2 - (B + C)y + (BC - AD)\ndet(Z - zI) = 0 = (C - z)(C - z) - D^2 = z^2 - (2C)z + (C^2 - D^2)\n\nIt is evident that the characteristic polynomial P(m) for a 2x2 matrix M then takes the form:\n\nP(m) = m^2 - [tr(M)]m + [det(M)]\n\nand its discriminant d(P(m)) is then:\n\nd(P(m)) = [tr(M)]^2 - 4[det(M)]\n\nIn particular, if s and t are the eigenvalues of a particular 2x2 matrix M, then:\n\ndet(M) = st\ntr(M) = s + t\n\nConsequently (assuming s <= t):\n\ns = (1\/2)*([tr(M)] - SQRT[d(P(m)])\nt = (1\/2)*([tr(M)] + SQRT[d(P(m)])\n\nFor the matrices X, Y and Z above, the respective discriminants are:\n\nd(P(x)) = [tr(X)]^2 - 4[det(X)] = (A + C)^2 - 4(AC - BD) = (A^2 + 2AC + C^2) - 4AC + 4BD\n= (A - C)^2 + 4BD\n\nd(P(y)) = [tr(Y)]^2 - 4[det(Y)] = (B + C)^2 - 4(BC - AD) = (B^2 + 2BC + C^2) - 4BC + 4AD\n= (B - C)^2 + 4AD\n\nd(P(z)) = [tr(Z)]^2 - 4[det(Z)] = (2C)^2 - 4(C^2 - D^2) = 4C^2 - 4C^2 + 4D^2 = (2D)^2\n\n(Note that, if we want all three discriminants to be perfect squares, we can take D = 0. The original equation under consideration then becomes A + B = C. Indeed, I had the ABC conjecture in mind while trying to do these things the other day -- but it turned out these were closely related to another open problem.)\n\nIf the discriminant d(P(m)) is a (rational) square, it follows that the eigenvalues s and t are also rational, and conversely.\n\nWe now compute the eigenvalues (under the assumption s <= t):\n\ns_X = (1\/2)*(A + C - SQRT((A - C)^2 + 4BD))\nt_X = (1\/2)*(A + C + SQRT((A - C)^2 + 4BD))\n\ns_Y = (1\/2)*(B + C - SQRT((B - C)^2 + 4AD))\nt_Y = (1\/2)*(B + C + SQRT((B - C)^2 + 4AD))\n\ns_Z = (1\/2)*(2C - SQRT((2D)^2)) = (1\/2)*(2C - 2D) = C - D\nt_Z = (1\/2)*(2C + SQRT((2D)^2)) = (1\/2)*(2C + 2D) = C + D\n\nUnder the additional assumption that D = 0, these expressions simplify to:\n\ns_X = A\nt_X = C\n\ns_Y = B\nt_Y = C\n\ns_Z = C\nt_Z = C\n\nWith this assumption about the vanishing of D, the original matrices X, Y and Z become:\n\nX'\nA B\n0 C\n\nY'\nB A\n0 C\n\nZ'\nC 0\n0 C\n\nThus, we have two upper-triangular matrices X' and Y', and Z' which is a scalar multiple of the 2x2 identity matrix. But that is not really the crucial point.\n\nNote that we have shown that:\n\nA = s_X <= t_X = C\nB = s_Y <= t_Y = C\nC = s_Z <= t_Z = C\n\nwithout knowing the actual values of A, B and C. How did this happen?\n\nHint: SQRT(x^2) = ABS(x), where ABS is the absolute value function.\n\nIn summary: Note how the value of D (which was initially unrestricted, and which we later take to be zero for purposes of \"simplifying\" the computations for the eigenvalues) imposed A LOT of STRUCTURAL RESTRICTIONS on the three matrices.\n\nLater during the day, I will present the details of my (dis-)proof for the inequality \"m < p^k\", which was the last stumbling block for my current approach in an attempt to resolve the OPN conjecture.","date":"2018-03-19 10:23:50","metadata":"{\"extraction_info\": {\"found_math\": false, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8135023713111877, \"perplexity\": 1859.9890907431154}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257646875.28\/warc\/CC-MAIN-20180319101207-20180319121207-00022.warc.gz\"}"}
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\section{Introduction}\label{Introduction}
The most fundamental quantity to analyze in the study of the elastic motion of non-relativistic quantum particles in a potential $V({\bf r})$ that undergoes sufficiently fast decays at infinity is the scattering matrix $S(E)$, which describes the distortion of the field-free wave function by the force exerted on the corpuscle by $V({\bf r})$ \cite{Newton1,Landau1,Baz1,Kukulin1}. $S(E)$ is generally considered to be a dimensionless complex function of the energy $E$ of the particle and, as such, can have poles in the ${\rm Re}(E)-{\rm Im}(E)$ plane, which determine the resonances characterized by (usually) {\em complex} energy. If the potential allows the existence of bound states, the scattering matrix also has poles at their negative energies. Another important property of the matrix $S$ is its unitarity, $|S|=1$, at real energies that do not coincide with the energies of the bound states; this physically expresses the conservation of the number of particles in the elastic collisions. It is natural to expect that for the {\em complex} unitary function $S(E)$ its {\em real} values $\pm1$ can have a special meaning. Additionally, due to the unitarity, the scattering matrix can be expressed in the form
\begin{equation}\label{Unitary1}
S(E)=e^{i\varphi_S},
\end{equation}
with the real phase $\varphi_S$ being a function of the energy, $\varphi_S\equiv\varphi_S(E)$. It is known \cite{Newton1,Kukulin1} that the region where the fastest change of $\varphi_S$ occurs is most significant, and the Wigner delay time $\tau_W$ \cite{Wigner1} is arrived at in this manner. $\tau_W$ is defined as a derivative of the phase $\varphi_S$ with respect to energy
\begin{equation}\label{Wigner1}
\tau_W(E)=\hbar\frac{d\varphi_S}{dE}
\end{equation}
and is an essential characteristic of the scattering process \cite{deCarvalho1,Maquet1}, with the extreme energies at which it achieves its maxima being the most important. Thus, three sets of energy have been identified, all of which are significant for the scattering matrix $S$
\begin{itemize}
\item (in general) {\em complex} energies, at which $S(E)=\infty$,
\item {\em real} energies, at which $S(E)=\pm1$,
\item {\em real} energies, at which $d\tau_W/dE=0$ and $d^2\tau_W/dE^2<0$.
\end{itemize}
Each of these sets defines its own specific type of behavior of $S$ at and around the corresponding energy, and even within each set the physical processes that are mathematically described by the scattering matrix may be quantitatively different. The overwhelming majority of mathematicians prefer to analyze only the first case where, on the basis of the time-independent Schr\"{o}dinger equation, the complex energies are located and calculated for each particular potential without requiring details of the associated waveforms. These functions, as the first stage of the physical consideration reveals, exhibit unrestricted growth at infinity, known as the 'exponential catastrophe' \cite{Bohm1}. However, even more careful interpretation allows it to be eliminated by proper reasoning involving the temporal evolution of the wave function \cite{Baz1,Bohm1,Holstein1}. The real part of the complex energy is customarily associated with the location of the resonance on the $E$ axis, whereas its imaginary component describes its half width or lifetime. For each specific potential, physicists also analyze the last two energy sets, but quite often this is performed separately for each case and no parallels are drawn between them and the first (complex energy) counterpart.
In the present study, a comparative analysis was performed of the outcomes of the three above-mentioned approaches, as applied to the behavior of electrons in the uniform electric field $\mathscr{E}$ superimposed on (a) a one-dimensional (1D) attractive or repulsive $\delta$-potential, which is the extreme limit of the finite width and depth quantum well (QW) or finite height quantum barrier, and (b) the Robin wall, i.e., a surface ${\cal S}$ that in the general 3D geometry supports the boundary condition (BC) for the wave function $\Psi(\bf r)$ of the form\cite{Gustafson1}
\begin{equation}\label{Robin1}
\left.{\bf n}{\bm\nabla}\Psi\right|_{\cal S}=\left.\frac{1}{\Lambda}\Psi\right|_{\cal S},
\end{equation}
where $\bf n$ is a unit inward vector and the extrapolation length $\Lambda$, which is considered to be real, is called the Robin or de Gennes \cite{deGennes1} distance. The model that represents an extremely localized finite strength interaction in the form of the $\delta$-potential is a very appealing one due to its relative simplicity and ease with which the necessary calculations can be carried out. It also reflects the essential properties of more complicated structures \cite{Belloni1}. The model is characterized by only one parameter, whose continuous variation from positive to negative values describes its repulsive or attractive strength; in particular, it possesses one localized level in the latter configuration. On the other hand, the quantum system, which at zero voltage is able to support bound orbitals, no longer has any discrete stationary levels when placed into the time and space unvarying electric field but instead only a continuum of states with their energies covering the whole axis, $-\infty<E<\infty$. As a result, the corresponding scattering matrix has poles only at the complex energies. A great deal of attention has been devoted to the analysis of the finite-width QW \cite{Moyer1,Bastard1,Austin1,Austin2,Austin3,Singh1,Borondo1,Ahn1,Austin4,Ghatak1,Ghatak2,Barrio1,Bloss1,Nakamura1,Juang1,Glasser2,Yuen1,delaCruz1,Kim1,Kuo1,Panda1,Zambrano1,Emmanouilidou1} and its $\delta$ counterpart \cite{Lukes1,Moyer2,Geltman1,Popov1,Scheffler1,Arrighini1,Fernandez1,Dargys1,Kundrotas1,Ludviksson1,Elberfeld1,Glasser1,Gottlieb1,Cocke1,Moyer3,Carpena1,Nickel1,Emmanouilidou2,Emmanouilidou3,Korsch1,Cavalcanti1,Alvarez1,Deych1,Galitski1,Moyer4,Brown1} subjected to a dc electric field with the outcome that one approach is sometimes completely at odds to the predictions of another scheme; in particular, the peculiarities of the transformation of the resonances into the true bound states at $\mathscr{E}\rightarrow0$ were scrutinized and the formation of new field-induced complex energy quasibound states and resonances was predicted.
Building on previous studies, below a detailed analysis was conducted of the resonances and quasibound states of the $\delta$-potential calculated from the three above-specified requirements. It can be seen that the complex energies derived as solutions of the stationary Schr\"{o}dinger equation with a non-zero field inevitably imply the exponential growth of the associated wave functions that, however, can be correctly construed with the help of the time-dependent picture. The development, based on the assumption that the energies have to remain real, leads to the conclusion that the infinitely small applied voltage at maximal scattering, $S=+1$, generates an infinite number of quasibound states in the positive energy continuum, the first set of which bears the features of electrons while the second exhibits the properties of a positively-charged particle that in solid state physics corresponds to hole excitations; for example, corpuscles residing in these two types of levels move in opposite directions as the field grows, and their evolution with increasing voltage ultimately forces them to coalesce with each other in what can be considered as the electric breakdown of the structure. The highest breakdown field is predicted to occur for mergers involving the level that developed from the zero-field bound state. Amalgamation of the levels is accompanied by the divergence of the associated dipole moments. This phenomenon, previously predicted only for the ground state \cite{Moyer3,Moyer4}, is calculated and analyzed in subsection \ref{QuasiboundStates1} for all quasibound levels. It should be noted that the corresponding eigenvalue equation has, at intensities $\mathscr{E}$ greater than the breakdown voltage, a pair of complex conjugate solutions that can be a mathematical indication of the formation in the electric field of a composite electron-hole-like structure. Corresponding maxima of the Wigner delay time were also computed as a function of the applied voltage. It can be seen that at the vanishing electric intensities, the predictions of the three methods coincide for the field-free bound state; however, even in this regime each approach has its own peculiarities. The difference between the results that coincided at $\mathscr{E}\ll1$ grows with the field. Contrary to the model of the $\delta$-potential that is symmetric at $\mathscr{E}=0$, the motion of the particle in the presence of the Robin wall takes place only on the half line. This lack of spatial symmetry has a drastic effect on the emergence and evolution of the quasibound states when the voltage is applied; in particular, for this system the field induces only hole-like quasibound levels and the lowest of them merges with the state that evolved from the field-free orbital that exists for the negative de Gennes distance, while the higher lying states survive any electric intensity.
\section{$\delta$-potential}\label{Sec_Delta}
The starting point of our analysis is the 1D stationary Schr\"{o}dinger equation
\begin{equation}\label{Schrodinger1}
\hat{H}\Psi(x)=E\Psi(x),
\end{equation}
where the Hamiltonian $\hat{H}$ is given by
\begin{equation}\label{Hamiltonian1}
\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}+V(x)-e\mathscr{E}x
\end{equation}
for the wave function $\Psi(x)$ of the particle with mass $m$ and charge $-e$ (with $e$ being an absolute value of the electronic charge) moving along an infinite straight line $-\infty<x\le+\infty$ in a uniform electric field $\mathscr{E}$ with a potential $V(x)$ being of the $\delta$-like form
\begin{equation}\label{DeltaPotential1}
V(x)=\frac{\hbar^2}{m}\frac{1}{\Lambda}\,\delta(x).
\end{equation}
Here, $\Lambda$ is a real coefficient , which has a dimension of length, being either positive or negative. Due to its presence, the matching conditions at $x=0$ are:
\begin{subequations}\label{MatchingCondtions1}
\begin{eqnarray}\label{MatchingCondtions1_1}
\Psi(0-)&=&\Psi(0+)\\
\label{MatchingCondtions1_2}
\Psi'(0+)-\Psi'(0-)&=&\frac{2}{\Lambda}\Psi(0)
\end{eqnarray}
\end{subequations}
with the prime denoting a derivative of the function with respect to its argument. Eq.~\eqref{MatchingCondtions1_2} demonstrates that there is a jump in the derivative of the wave function at the origin that is inversely proportional to the distance $\Lambda$. In the absence of an electric field, $\mathscr{E}=0$, the attractive potential, $\Lambda<0$, in addition to the continuous spectrum at $E>0$ (which is also characteristic for $\Lambda>0$) binds the particle at negative energy
\begin{equation}\label{EnergyDeltaZeroFields1}
E=-\frac{\hbar^2}{2m\Lambda^2},\quad\Lambda<0,
\end{equation}
while its normalized to unity,
\begin{equation}\label{Normalization1}
\int_{-\infty}^\infty\Psi^2(x)dx=1,
\end{equation}
wave function $\Psi(x)$ exponentially decreases away from the origin:
\begin{equation}\label{FunctionDeltaZeroFields1}
\Psi(x)=\frac{1}{|\Lambda|^{1/2}}\exp\!\left(-\left|\frac{x}{\Lambda}\right|\right).
\end{equation}
An applied electric field changes the charge distribution in the system. The quantitative measure of this influence is provided by the polarization, or dipole moment, $P(\mathscr{E})$, defined as \cite{Nguyen1,Olendski1,Olendski2}
\begin{equation}\label{Polarization1}
P(\mathscr{E})=\left\langle ex\right\rangle_\mathscr{E}-\left\langle ex\right\rangle_{\mathscr{E}=0},
\end{equation}
where the angular brackets denote a quantum mechanical expectation value:
\begin{equation}\label{AngleBrackets1}
\left\langle x\right\rangle=\int x\Psi^2(x)dx
\end{equation}
with the integration carried out over all available space, which in the case of the $\delta$-potential, reduces the polarization to
\begin{equation}\label{Polarization2}
P^\delta(\mathscr{E})=e\int_{-\infty}^\infty x\Psi^2(x)dx.
\end{equation}
As will be shown below, it is possible to calculate this quantity even in the case of open structures, such as the ones considered in the present study.
It is convenient to switch to dimensionless scaling from the outset so that all distances are measured in units of $|\Lambda|$, energies -- in units of $\hbar^2/\left(2m|\Lambda|^2\right)$, time -- in units of $2m|\Lambda|^2/\hbar$, polarization -- in units of $e|\Lambda|$, velocity -- in units of $\hbar/(2m|\Lambda|)$, electric fields -- in units of $\hbar^2/\left(2em|\Lambda|^3\right)$, and current density -- in units of $-e\hbar/\left(m|\Lambda|^2\right)$. Then, Eq.~\eqref{Schrodinger1} using the potential from Eq.~\eqref{DeltaPotential1} takes a universal form:
\begin{equation}\label{Schrodinger2}
-\Psi''(x)\pm\delta(x)\Psi(x)-\mathscr{E}x\Psi(x)=E\Psi(x),\\
\end{equation}
while the matching condition from Eq.~\eqref{MatchingCondtions1_2} is transformed to
\begin{equation}\label{MatchingCondtions2}
\Psi'(0+)-\Psi'(0-)=\pm2\Psi(0),
\end{equation}
where the upper (lower) sign refers to the repulsive (attractive) potential. The same convention will be used, as necessary, throughout the whole section.
Due to its generic definition, the scattering matrix describes the results of wave reflection from the structure when the total function $\Psi_t$ includes both the incoming [first term on the right-hand side of Eq.~\eqref{ScatteringFunction1}] and reflected (second item) components:
\begin{eqnarray}
\Psi_t(\mathscr{E};x)&=&{\rm Ci}^-\!\!\left(\!-\mathscr{E}^{1/3}x-\frac{E}{\mathscr{E}^{2/3}}\!\right)\nonumber\\
\label{ScatteringFunction1}
&+&S{\rm Ci}^+\!\!\left(\!-\mathscr{E}^{1/3}x-\frac{E}{\mathscr{E}^{2/3}}\!\right),\quad x\geq0.
\end{eqnarray}
Here, $${\rm Ci}^\pm(\eta)={\rm Bi}(\eta)\pm i{\rm Ai}(\eta)$$ (obviously, the superscript at ${\rm Ci}$ refers to the sign of its imaginary part and not to the $\delta$-potential), and ${\rm Ai}(\eta)$ and ${\rm Bi}(\eta)$ are Airy functions \cite{Abramowitz1,Vallee1}. To the left of the well, the fading at the negative infinity solution is:
\begin{equation}\label{WaveFunction3}
\Psi_n(x)=A_n{\rm Ai}\!\left(-\mathscr{E}^{1/3}x-\frac{E_n}{\mathscr{E}^{2/3}}\right),\quad x\leq0,
\end{equation}
where the explicitly-included subscript $n=0,1,2,\ldots$, counts the corresponding resonances (see below) and $A_n$ is a normalization constant. Matching according to Eqs.~\eqref{MatchingCondtions1_1} and \eqref{MatchingCondtions2} leads to the scattering matrix:
\begin{eqnarray}
&&S^{\delta\pm}(\mathscr{E};E)=\nonumber\\
\label{DeltaPotentialScatteringMatrix1}
&&-\frac{2{\rm Ai}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right){\rm Bi}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)\pm\frac{\mathscr{E}^{1/3}}{\pi}-i\,2{\rm Ai}^2\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)}{2{\rm Ai}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right){\rm Bi}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)\pm\frac{\mathscr{E}^{1/3}}{\pi}+i\,2{\rm Ai}^2\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)}.
\end{eqnarray}
Note that for real energies, this complex function, which also depends on the parameter $\mathscr{E}$, is unitary.
\subsection{Poles of the Scattering Matrix: Gamow-Siegert States}\label{GamowSiegertStates1}
Zeroing the denominator of the right-hand side of Eq.~\eqref{DeltaPotentialScatteringMatrix1} produces a universal equation for calculating the complex energies $E_{res_n}$:
\begin{equation}\label{DeltaPotentialEigenEquation1}
2\pi{\rm Ai}\!\left(\!-\frac{E_{res_n}}{\mathscr{E}^{2/3}}\!\right)\left[{\rm Bi}\!\left(\!-\frac{E_{res_n}}{\mathscr{E}^{2/3}}\!\right)+i{\rm Ai}\!\left(\!-\frac{E_{res_n}}{\mathscr{E}^{2/3}}\!\right)\right]\pm\mathscr{E}^{1/3}=0.
\end{equation}
Note that this equation can be derived in an alternative way; because the applied field is created by the potential that unrestrictedly decreases with $x\rightarrow+\infty$, it follows that at the non-zero voltage even for the attractive well there are, strictly speaking, no true bound states since the electron localized near the origin at $\mathscr{E}=0$ lowers its potential energy by tunneling away from the attractive center when the electric intensity is not zero. It is then contended \cite{Ahn1,delaCruz1,Kim1,Emmanouilidou1,Deych1,Galitski1,Yuen1} that the solution of the Schr\"{o}dinger equation should represent the outgoing waves at infinity and, since this requirement infers the non-zero imaginary component of the wave function $\Psi$, the energy also becomes complex. This results in a transformation of the true bound level into the resonance state with a finite lifetime
\begin{equation}\label{lifetime1}
\tau=\frac{1}{\Gamma},
\end{equation}
where the positive $\Gamma$ is a half width of the corresponding resonance:
\begin{equation}\label{ComplexEnergy1}
E_{res_n}=E_{r_n}-i\frac{\Gamma_n}{2}
\end{equation}
with $E_{r_n}$ being real. Accordingly, the function $\Psi_n(x)$ for positive $x$ is written as
\begin{equation}\label{WaveFunction2}
\Psi_n(x\geq0)=C_n{\rm Ci}^+\!\!\left(-\mathscr{E}^{1/3}x-\frac{E_{res_n}}{\mathscr{E}^{2/3}}\right),
\end{equation}
where $C_n$ is a normalization constant, and its match with the waveform from Eq.~\eqref{WaveFunction3} leads to Eq.~\eqref{DeltaPotentialEigenEquation1}.
From the properties of the Airy functions \cite{Abramowitz1,Vallee1} it is easy to derive the evolution of the field-free bound level at small electric intensities:
\begin{equation}\label{DeltaPotentialAsymptotics1}
{E_{res}^{\delta-}}_0(\mathscr{E})=-1-\frac{5}{16}\mathscr{E}^2-i\exp\!\left(\!-\frac{4}{3}\frac{1}{\mathscr{E}}\right),\quad\mathscr{E}\ll1.
\end{equation}
It can be seen that the real part of the energy depends quadratically on the voltage, which is a result of the symmetry of the field-free structure with respect to the inversion $x\rightarrow-x$. An exponentially small increase of the half width
\begin{equation}\label{DeltaHalfWidthAsymptotics1}
{\Gamma_{res}^{\delta-}}_0=2\exp\!\left(\!-\frac{4}{3}\frac{1}{\mathscr{E}}\right),\quad\mathscr{E}\ll1,
\end{equation}
which is typical for a wide range of potentials that decay at infinity \cite{Galitski1}, physically means quite a small probability of tunneling away from the well and a particularly long lifetime of the state, as follows from Eq.~\eqref{lifetime1}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{DeltaPotentialComplexEnergyFunction0.eps}
\caption{\label{DeltaPotentialComplexEnergyFunction0}
(a) Real ${E_r}_0$ (solid line, left axis) and negative double imaginary $\Gamma_0$ (dashed curve, right axis) components of the complex energy as a function of the field calculated from Eq.~\eqref{DeltaPotentialEigenEquation1}. (b) Wave function $\Psi_0$ (normalized to its value at $x=0$) in terms of the distance $x$ and electric field $\mathscr{E}$.
}
\end{figure}
Panel (a) of Fig.~\ref{DeltaPotentialComplexEnergyFunction0} shows the zeroth level energy dependence on the field. A quadratic decrease of the real part and an exponentially small increase of the half width at the small intensities derived above are clearly seen in the plot. The real part of the energy reaches a minimum of ${E_r}_{min}=-1.232$ at $\mathscr{E}_{min}=3.125$, after which it demonstrates a permanent growth; in particular, it crosses its zero-field value of $-1$ at $\mathscr{E}=9.864$ and enters the positive part of the spectrum at $\mathscr{E}=26.303$. The half width $\Gamma_0$ after an almost flat profile at small $\mathscr{E}$ exhibits a nearly linear dependence on the applied field, which at higher voltages approaches $\mathscr{E}^{2/3}$. This growth implies a drastic decrease of the lifetime by field-enhanced tunneling.
The evolution of the associated wave function $\Psi_0(\mathscr{E};x)$ is depicted in panel (b) of Fig.~\ref{DeltaPotentialComplexEnergyFunction0}. Its most striking feature is an exponential growth with positive distance $x$ at the non-zero fields. Indeed, the claim that the function from Eq.~\eqref{WaveFunction2} describes the outgoing wave \cite{Ahn1,delaCruz1} is correct only for the {\em real} energies, but for the {\em complex} $E$ it inevitably leads to an exponential increase of the function at large $x$, which can be easily shown by employing the asymptotic behavior of the Airy functions \cite{Abramowitz1,Vallee1}. Complex eigenvalues were first introduced by Thomson \cite{Thomson1} in his analysis of the modes supported by an electric sphere and were later implemented in Gamow's explanation of alpha decay \cite{Gamow1}. The most serious criticism of these Gamow, or Siegert \cite{Siegert1}, states denies their physical legitimacy due to the unrestricted growth of the wave function, with the vivid manifestation of this 'exponential catastrophe' presented in Fig.~\ref{DeltaPotentialComplexEnergyFunction0}. However, this divergence is elegantly eliminated by considering the temporal evolution of the structure; as a matter of fact, if the time decay of the state is considered and even the corresponding lifetime from Eq.~\eqref{lifetime1} is introduced, it is logical to investigate the associated time-dependent function ${\bm \Psi}(\mathscr{E};x,t)=\Psi(\mathscr{E};x)e^{-iE_{res}t}$. For the large negative argument in the waveform from Eq.~\eqref{WaveFunction2}, the total function reads:
\begin{eqnarray}
{\bm \Psi}_0(\mathscr{E};x,t)&=&C_0\frac{\mathscr{E}^{1/6}}{\pi^{1/2}\left(E_{r_0}+\mathscr{E}x-i\frac{\Gamma_0}{2}\right)^{1/4}}\nonumber\\
&\times&\exp\!\left(i\left[\frac{2}{3}\frac{(E_{r_0}+\mathscr{E}x)^{3/2}}{\mathscr{E}}-E_{r_0}t+\frac{\pi}{4}\right]\right)\nonumber\\
\label{WaveFunction4}
&\times&\exp\!\left(-\frac{\Gamma_0}{2}\left[t-T(\mathscr{E};x)\right]\right),\quad\mathscr{E}x\gg1,
\end{eqnarray}
where also a smallness of the half width $\Gamma_0$ has been assumed. The first exponent in Eq.~\eqref{WaveFunction4} is a plane wave that describes free motion in the linear potential and $T(\mathscr{E};x)$ is just the classical time needed for the electron to travel the distance between the quasi classical turning point $x_{qc}=-E_{r_0}/\mathscr{E}$ and the coordinate $x>x_{qc}$:
\begin{subequations}\label{Classical1}
\begin{align}\label{Classical1_Time1}
T(\mathscr{E};x)&=\frac{(E_{r_0}+\mathscr{E}x)^{1/2}}{\mathscr{E}}.
\intertext{This is particularly clear when this expression is rewritten in the alternative equivalent form:}
\label{Classical1_Time2}
T(x)&=2\,\frac{x-x_{qc}}{v(x)},
\end{align}
\end{subequations}
where $v(x)\equiv v(\mathscr{E};x)$ is a field-dependent classical speed at $x$:
\begin{equation}\label{ClassicalVelocity1}
v(\mathscr{E};x)=2\left(E_{r_0}+\mathscr{E}x\right)^{1/2}.
\end{equation}
The above equations in this paragraph, similar to the analysis of alpha decay \cite{Bohm1,Baz1,Holstein1}, can be construed as follows. In the derivation of Eq.~\eqref{WaveFunction4} it was tacitly assumed that it is valid for all times $-\infty<t<\infty$. However, in reality the decay does not start in the infinitely remote past since the corresponding state has to be created first by, say, the adiabatic varying of the field or any other means. Accordingly, it is natural to choose as the origin the moment when the emitted particle emerges at the turning point $x_{qc}$ after tunneling through the triangular barrier, which the electron with the negative energy $E_{r_0}<0$ located inside the $\delta$-well "sees" to its right. This also means that at this point, the prehistory of the formation of the scattering level at $t<0$ is of no concern. Then, at any positive time the particle travels with an average speed $\overline{v}=v(x)/2$ to reach the observation point $x$ at moment $T(x)$ from Eq.~\eqref{Classical1_Time2}. Consequently, it does make sense to talk about measuring the probability density $\rho(\mathscr{E};x,t)\equiv\left|{\bm \Psi}(\mathscr{E};x,t)\right|^2$ at detector position $x$ at times $t\ge T(x)$ only:
\begin{eqnarray}
&&\rho_0(\mathscr{E};x,t)=\nonumber\\
\label{Density1}
&&\frac{|C_0|^2\mathscr{E}^{1/3}}{\pi\!\left[\left(E_{r_0}\!+\!\mathscr{E}x\right)^2\!\!\!+\!\left(\frac{\Gamma_0}{2}\right)^2\right]^{1/4}}\,e^{-\Gamma_0\left[t-T(\mathscr{E};x)\right]},\, t\ge T(\mathscr{E};x).
\end{eqnarray}
The corresponding current density in the $x$ direction
\begin{equation}\label{CurrentDensity2}
j_x=\!\!\!\left[v(x)-\frac{1}{8}\frac{\mathscr{E}\Gamma_0}{\left(E_{r_0}+\mathscr{E}x\right)^2+\left(\frac{\Gamma_0}{2}\right)^2}\right]\!\!\rho_0(\mathscr{E};x,t),
\end{equation}
which is calculated from the general expression \cite{Landau1}
\begin{equation}\label{CurrentDensity1}
{\bf j}={\rm Im}({\bm\Psi}^\ast{\bm\nabla}{\bm\Psi}),
\end{equation}
apart from the familiar velocity-dependent term [first item in Eq.~\eqref{CurrentDensity2}], contains an additional contribution that is proportional to the half width $\Gamma$. Such a viewpoint eliminates the 'exponential catastrophe', yielding instead the anticipated exponential decay law \cite{Bohm1,Baz1,Holstein1} with its lifetime taken from Eq.~\eqref{lifetime1}. However, neglecting all times smaller than $T(x)$ and, in particular, $t<0$, completely ignores the processes of under-barrier tunneling at $0<x<x_{qc}$ and this semi classical reasoning is therefore not a strictly quantum one. To correctly account for the build-up of the levels at earlier times $t<0$, a solution of Eq.~\eqref{DeltaPotentialEigenEquation1} with the positive imaginary component, which is a complex conjugate of its counterpart from Eq.~\eqref{ComplexEnergy1}, is required. Negative half widths $\Gamma$ specify the system in the growing state, which is called antiresonance (see Sec. \ref{QuasiboundStates1}). They are also appropriate to describe a particle traveling back in time towards the past. A separation of the complex energy eigenfunctions into those corresponding to the physical states at the earlier times (with $\Gamma<0$) and those associated with the configuration for the later times, $\Gamma>0$, is a peculiar property of the Gamow vectors with complex energies \cite{Bohm2}. From the point of view of mathematical formalism, the Gamow-Siegert states do not represent vectors from the Hilbert space of the quantum structure under consideration, being instead eigenvectors of the rigged Hilbert space (see Refs. \cite{Bohm2,Civitarese1,deLaMadrid1} and references therein).
In addition to the resonance that, at vanishing electric intensities, transforms into the field-free bound state, Eq.~\eqref{DeltaPotentialEigenEquation1} has other solutions for either attractive \cite{Ludviksson1,Alvarez1} or repulsive potentials. The easiest way to show their existence is to substitute into the equation an Airy functions relation \cite{Abramowitz1,Vallee1}
$$
{\rm Ai}(z)\mp i{\rm Bi}(z)=2e^{\mp i\pi/3}{\rm Ai}\!\left(ze^{\pm i2\pi/3}\right)
$$
and to consider the resulting transcendental formula
\begin{equation}\label{DeltaPotentialEigenEquation3}
4\pi ie^{-i\pi/3}{\rm Ai}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right){\rm Ai}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\,e^{i2\pi/3}\right)\pm\mathscr{E}^{1/3}=0
\end{equation}
in the limit of the low voltages. After some algebra involving the properties of the Airy functions, two sets of solutions are achieved, which at $\mathscr{E}\ll1$ are:
\begin{subequations}\label{DeltaComplexSolutionsTwoSets1}
\begin{eqnarray}\label{DeltaComplexSolutionsTwoSets1_Set1}
E_{res_n}^{(1)\pm}&=&-a_n\mathscr{E}^{2/3}\mp\frac{1}{2}\,\mathscr{E}\!+\!\frac{1}{4}\frac{{\rm Bi}'(a_n)}{{\rm Bi}(a_n)}\mathscr{E}^{4/3}\!-\!i\frac{\mathscr{E}^{4/3}}{4\pi{\rm Bi}^2(a_n)}\\
\label{DeltaComplexSolutionsTwoSets1_Set2}
{E_{res_n}^{(2)\pm}}&=&\frac{1}{2}\,a_n\mathscr{E}^{2/3}\pm\frac{1}{2}\,\mathscr{E}+i\,\frac{3^{1/2}}{2}\,a_n\mathscr{E}^{2/3}.
\end{eqnarray}
\end{subequations}
Here, (all negative) coefficients $a_n$, $n=1,2,\ldots$, are solutions of equation ${\rm Ai}(a_n)=0$ \cite{Abramowitz1,Vallee1}. Note the opposite signs of the real parts of the energies $E_{res_n}^{(1)\pm}$ and $E_{res_n}^{(2)\pm}$ and different powers of the field dependence of their imaginary components. We will address more of the properties of these states while comparing them to the results obtained via other methods.
\subsection{Real-energy Quasibound States: $S=+1$}\label{QuasiboundStates1}
Having seen the properties of the complex Gamow-Siegert states, let us now return to the scattering matrix from Eq.~\eqref{DeltaPotentialScatteringMatrix1}. To remain rigorously within the time-independent quantum treatment without divergences, only the {\em real} energies $E$ will be used in this and the following subsections, unless otherwise stipulated. For our geometry, the matrix $S$ characterizes the influence of the $\delta$-potential on the incident particle; namely, it is well known \cite{Landau1} that without it, $\Lambda=0$, the corresponding waveform is proportional to the Airy function ${\rm Ai}(\eta)$, which corresponds to $S=-1$ in Eq.~\eqref{ScatteringFunction1}:
\begin{equation}\label{ScatteringFunction0}
\Psi_{\Lambda=0}(\mathscr{E};x)=-2i{\rm Ai}\!\left(\!-\mathscr{E}^{1/3}x-\frac{E}{\mathscr{E}^{2/3}}\!\right).
\end{equation}
Then, in the superposition of both forces, the scattered wave $\Psi_{sc}$ is the difference between the total function from Eq.~\eqref{ScatteringFunction1} and its unperturbed counterpart, Eq.~\eqref{ScatteringFunction0}:
\begin{eqnarray}
\Psi_{sc}(\mathscr{E};x)&=&\Psi_t(\mathscr{E};x)-\Psi_{\Lambda=0}(\mathscr{E};x)\nonumber\\
\label{ScatteringFunction2}
&=&(1+S){\rm Ci}^+\!\!\left(\!-\mathscr{E}^{1/3}x-\frac{E}{\mathscr{E}^{2/3}}\!\right).
\end{eqnarray}
This equation shows that $\Psi_{sc}$ is a purely outgoing wave. The squared modulus of its amplitude with its maximum normalized to unity is called the scattering probability:
\begin{equation}\label{ScatProbab1}
p(\mathscr{E};E)=\frac{1}{4}\,\left|1+S(\mathscr{E};E)\right|^2.
\end{equation}
Note that the extremely localized potential, as expected, does not scatter the wave of the arbitrary strength electric field when its position coincides with one of the nodes of the unperturbed orbital $\Psi_{\Lambda=0}$:
\begin{equation}\label{ZeroScattering1}
p^\delta\!\left(\mathscr{E};-a_n\mathscr{E}^{2/3}\right)=0.
\end{equation}
In this way, the significance of the special value $-1$ of the matrix $S$ mentioned in the Introduction is established. Another important property is the identical vanishing of the current density of the total function $\Psi_t$ at any energy, as it easily follows from its substitution into Eq.~\eqref{CurrentDensity1}. Thus, mathematical models of the Gamow-Siegert states and that based on the real energy analysis describe different physical situations: while the former one calculates the temporal leakage from the well of the level that was prepared at the earlier times, the subject of the latter method is the stationary configuration that emerged as a result of the interference between the incident and reflected waves, with the resulting net current being an exact zero. As a consequence of this, {\em real} values of the energy guarantee that the waveform $\Psi_t(x)$ is finite everywhere.
Distortion of the electron motion by the $\delta$-potential is maximal for energies $E_n$, $n=0,1,2,\ldots$, at which the scattering matrix $S^\delta$ changes to positive unity:
\begin{equation}\label{DeltaPotentialEigenEquation2}
2\pi{\rm Ai}\!\left(\!-\frac{E}{\mathscr{E}^{2/3}}\!\right){\rm Bi}\!\left(\!-\frac{E}{\mathscr{E}^{2/3}}\!\right)\pm\mathscr{E}^{1/3}=0,
\end{equation}
and the associated total function $\Psi_t(x)$ to the right degenerates to the Airy function ${\rm Bi}(\eta)$:
\begin{equation}\label{DeltaFunction0}
\Psi_{t_n}(x)=B_n{\rm Bi}\!\left(-\mathscr{E}^{1/3}x-\frac{E_n}{\mathscr{E}^{2/3}}\right),\quad x\geq0.
\end{equation}
For the attractive potential, Eq.~\eqref{DeltaPotentialEigenEquation2} was derived previously with the help of the Green functions \cite{Glasser1,Carpena1} without any detailed analysis. The same configuration was discussed by C. A. Moyer \cite{Moyer3,Moyer4}, who concentrated on finding the energy and associated polarization of its lowest level only, which at the vanishing electric intensities tends to the zero-field bound state:
\begin{subequations}\label{DeltaEnergiesSmallFields1}
\begin{align}\label{DeltaEnergiesSmallFields1B0}
E_0&=-1-\frac{5}{16}\,\mathscr{E}^2,\quad \mathscr{E}\ll1.\\
\intertext{In addition, for either the $\delta$-well or barrier, Eq.~\eqref{DeltaPotentialEigenEquation2} has two infinite sets of positive solutions that will be denoted below by the superscripts $A$ or $B$ corresponding to their behavior at low voltages; namely, for the weak fields, $\mathscr{E}\ll1$, one finds:}
\label{DeltaEnergiesSmallFields1An}
E_n^{A\pm}&=-a_n\mathscr{E}^{2/3}\mp\frac{1}{2}\,\mathscr{E}+\frac{1}{4}\frac{{\rm Bi}'(a_n)}{{\rm Bi}(a_n)}\mathscr{E}^{4/3},\\
\label{DeltaEnergiesSmallFields1Bn}
E_n^{B\pm}&=-b_n\mathscr{E}^{2/3}\pm\frac{1}{2}\,\mathscr{E}+\frac{1}{4}\frac{{\rm Ai}'(b_n)}{{\rm Ai}(b_n)}\mathscr{E}^{4/3},
\end{align}
\end{subequations}
$n=1,2,\ldots$. Here, negative $b_n$ is the $n$th zero of the Airy function ${\rm Bi}(x)$: ${\rm Bi}(b_n)=0$ \cite{Abramowitz1,Vallee1}. It is important to stress that Eq.~\eqref{DeltaPotentialEigenEquation2}, in addition to the real solutions, has complex roots too. This follows from the fact that at low electric intensities the $B$ set is essentially determined by ${\rm Bi}(\eta_0)=0$ with $\eta_0=-E/\mathscr{E}^{2/3}$. This equation is also satisfied by the complex numbers $\beta_n$, in addition to the real coefficients $b_n$ \cite{Abramowitz1,Vallee1}. However, these states are disregarded due to the convention of avoiding the 'exponential catastrophe'. Therefore, under the assumption of keeping the energies {\em real} we have found that quantization results in a countably infinite number of solutions. To place them correctly within the nomenclature of the other solutions of the Schr\"{o}dinger equation, it should be noted that historically, the terms "quasibound (or quasi-stationary) state" and "resonance" were used interchangeably to describe the {\em complex}-energy Gamow-Siegert level. The standard procedure for categorizing the poles of the $S$-matrix implements their location in the complex $k$-plane, where $k\equiv k_r+ik_i=\sqrt{E}$ with real $k_r$ and $k_i$ \cite{Kukulin1,Bang1}. Bound states, $E<0$, lie on the imaginary semi axis in the upper $k$-halfplane, $k_{r_B}=0$, $k_{i_B}=+\sqrt{|E|}>0$, which leads to fading function at large distances, $\Psi_B(x)\xrightarrow[|x|\rightarrow\infty]{}\exp\left(-k_{i_B}|x|\right)$, as expected. A complete mathematical set of solutions should also include those dependencies with the purely imaginary {\em negative} wave vector, $k_{r_{AB}}=0$, $k_{i_{AB}}=-\sqrt{|E|}<0$. Accordingly, the waveforms $\Psi_{AB}$ of such {\em anti}-bound states \cite{Kukulin1,Bang1} diverge at infinity: $\Psi_{AB}(x)\xrightarrow[x\rightarrow\infty]{}\exp\left(\left|k_{i_{AB}}\right|x\right)$. Despite this unlimited growth, these levels can have a physical meaning too \cite{Ohanian1,Heiss1}. All other poles of the scattering matrix lie in the lower $k$-halfplane, $k_i<0$, with the positive real part of the wave vector corresponding to the above-discussed Gamow quasi-stationary states (resonances) while those with $k_r<0$ are called antiresonances and describe an ingoing wave \cite{Kukulin1}. Obviously, solutions from Eqs.~\eqref{DeltaEnergiesSmallFields1} do not fall into one of these categories, since in the first instance they are not poles of the $S$-matrix and, second, all their energies (except the lowest one of the attractive potential) are positive. Moreover, the corresponding functions at large positive $x$ neither exponentially diverge nor fade, presenting instead oscillatory damped modes, as it follows from Eq.~\eqref{DeltaFunction0} and asymptote of the Airy function. Nevertheless, in the recent analysis of the lowest level evolution in the field \cite{Moyer4} this orbital was called the quasibound state and was defined broadly as the level having a connectedness to the true bound state through the variation of some physical parameter. In our extension to all the solutions (including those with $E\geq0$), we found it relevant to call them real energy quasibound (REQB) states. These are bound states, since for each of them the wave function $\Psi_n(x)$ has a finite absolute value everywhere including the point $x=+\infty$ while the prefix 'quasi' in their definition means that due to the slow decrease at large positive $x$ of the function ${\rm Bi}$ from Eq.~\eqref{DeltaFunction0}, they cannot be normalized according to Eq.~\eqref{Normalization1} \cite{Moyer4}. Contrary to these states, the divergent-at-infinity Gamow-Siegert solutions will be called resonances \cite{Kukulin1,Moyer4}. It is important to underline that the REQB states with the discrete energies $E_0$, $E_n^A$, and $E_n^B$ are embedded into the continuum of the delocalized levels, with its energies ranging from the negative to positive infinity; as such, they mathematically represent only a measure zero part of all continuum energy eigenstates of the given Hamiltonian, while physically they describe the largest possible, $p=S=1$, disturbance by the $\delta$-potential of the motion in the uniform electric field. To end this part of the discussion, it should be mentioned that, in addition to the levels discussed above, open systems can also support under very special conditions true bound states in the continuum, i.e., waves that remain localized (with square integrable functions) even though they coexist with a continuous spectrum of radiating oscillations that can carry energy away \cite{Hsu1}.
The first term on the right-hand side of Eq.~\eqref{DeltaEnergiesSmallFields1An} states that the $A$ set of solutions reflects the creation by the applied voltage and the $\delta$-potential of the triangular QW \cite{Olendski2,Olendski1,Katriel1}, with the wave function taken from Eq.~\eqref{WaveFunction3} and the subsequent terms representing an admixture due to its coupling to the right half space. Since the leading term in this formula is independent of the sign of the $\delta$ term, the formation of the triangular well takes place at either a positive or negative electrostatic potential $V(x)$ while the interaction between the left and right semi-infinite areas carries its sign. In the same way, the first expression on the right-hand side of Eq.~\eqref{DeltaEnergiesSmallFields1Bn} describes the formation in the region $x\ge0$ terminated by the impenetrable wall of the standing wave from Eq.~\eqref{DeltaFunction0} with the linear in the field and higher order factors there describing the correction due to coupling to the left-hand territory. The first part of the mathematical inequality chain
\begin{equation}\label{InequalityChain1}
|a_{n+1}|>|b_{n+1}|>|a_n|
\end{equation}
physically means that the $A$ solutions are located to the left of the $\delta$-potential, with the energies $E_n^A$ being larger than those of the corresponding $B$ states with their distribution spreading at $x>0$. The sequence from Eq.~\eqref{InequalityChain1} also states that the $B$ and $A$ levels alternate on the energy axis.
The product of the two Airy functions on the left-hand side of Eq.~\eqref{DeltaPotentialEigenEquation2} is a bounded function of the energy: it decreases to zero as $\mathscr{E}^{1/3}/|E|^{1/2}$ for large negative $E$, reaches its global maximum at $E=0$, and for positive energies it presents sinusoidal oscillations with its amplitude modulated again by the same factor $\mathscr{E}^{1/3}/E^{1/2}$. Hence, for quite large electric intensities, this equation does not have any solutions. The disappearance of the levels at increasing voltage occurs as a coalescence of the two adjacent states at the electric fields $\mathscr{E}_n^\times$ that are different for the well and the barrier:
\begin{subequations}\label{DeltaCoalesceField1}
\begin{eqnarray}\label{DeltaCoalesceFieldMinus1}
\mathscr{E}_n^{\times-}&=&8\pi^3f^3(s_{2n})\\
\label{DeltaCoalesceFieldPlus1}
\mathscr{E}_n^{\times+}&=&-8\pi^3f^3(s_{2n+1})
\end{eqnarray}
\end{subequations}
and the energies $E_n^{\times\pm}$ at the merger are:
\begin{subequations}\label{DeltaCoalesceEnergy1}
\begin{eqnarray}\label{DeltaCoalesceEnergyMinus1}
E_n^{\times-}&=&-s_{2n}\left(\mathscr{E}_n^{\times-}\right)^{2/3}=-4\pi^2s_{2n}f^2(s_{2n})\\
\label{DeltaCoalesceEnergyPlus1}
E_n^{\times+}&=&-s_{2n+1}\left(\mathscr{E}_n^{\times+}\right)^{2/3}=-4\pi^2s_{2n+1}f^2(s_{2n+1}),
\end{eqnarray}
\end{subequations}
where
\begin{equation}\label{function_f1}
f(s)={\rm Ai}(s){\rm Bi}(s)
\end{equation}
and non-positive $s_n$ is the $n$th solution, $n=0,1,2,\ldots$, of equation $f'(s)=0$, or in the expanded form
\begin{equation}\label{DeltaCrossingEquation1}
{\rm Ai}(s){\rm Bi}'(s)+{\rm Ai}'(s){\rm Bi}(s)=0.
\end{equation}
Since $s_0=0$ \cite{Abramowitz1,Vallee1}, the breakdown field of the two lowest levels is
\begin{equation}\label{DeltaCoalesceFieldMinus0}
\mathscr{E}_0^{\times-}=\mathscr{E}_f
\end{equation}
with
\begin{equation}\label{FundamentalField1}
\mathscr{E}_f=\frac{1}{3}\frac{\Gamma^3(1/3)}{\Gamma^3(2/3)}=2.58106\ldots,
\end{equation}
where $\Gamma(x)$ is the $\Gamma$-function \cite{Abramowitz1}. For quite large $n$ it is elementary to derive an asymptote
\begin{equation}\label{AsymptotSn1}
s_n=-\left(\frac{3}{4}\pi n\right)^{\!2/3},\quad n\gg1,
\end{equation}
that leads to the approximate formula for $\mathscr{E}_n^\times$:
\begin{subequations}\label{DeltaCrossingField1}
\begin{eqnarray}\label{DeltaCrossingFieldMinus1}
\mathscr{E}_n^{\times-}&=&\frac{2}{3\pi}\frac{1}{n},\quad n\gg1\\
\label{DeltaCrossingFieldPlus1}
\mathscr{E}_n^{\times+}&=&\frac{4}{3\pi}\frac{1}{2n+1},\quad n\gg1.
\end{eqnarray}
\end{subequations}
Table~\ref{Table1} lists the exact values of $s_n$ and their approximations by Eq.~\eqref{AsymptotSn1} together with the exact and approximate coalescence fields and energies. It shows that the estimates from Eqs.~\eqref{AsymptotSn1} and ~\eqref{DeltaCrossingField1} provide reasonably good accuracy, even for small $n$.
\newpage
\begin{sidewaystable}
\caption{Exact and approximate solutions $s_n$ of Eq.~\eqref{DeltaCrossingEquation1} together with the dissociation fields $\mathscr{E}_n^\times$ and energies $E_n^\times$ for the $\delta$-potential}
\centering
\begin{tabular}{c|c c|c c|c||c c|c c|c}
\hline
\hline
\multirow{2}[3]{*}{$n$}&\multicolumn{2}{c|}{$s_{2n}$}&\multicolumn{2}{c|}{$\mathscr{E}_n^{\times-}$}&\multirow{2}[3]{*}{$E_n^{\times-}$}&\multicolumn{2}{c|}{$s_{2n+1}$}&\multicolumn{2}{c|}{$\mathscr{E}_n^{\times+}$}&\multirow{2}[3]{*}{$E_n^{\times+}$}\\[0.5ex]
\cline{2-5}\cline{7-10}
&Exact&Eq.~\eqref{AsymptotSn1}&Exact&Eq.~\eqref{DeltaCrossingFieldMinus1}& &Exact&Eq.~\eqref{AsymptotSn1}&Exact&Eq.~\eqref{DeltaCrossingFieldPlus1}\\
\hlin
0&0&0&2.58106&$\infty$&0&-1.76475&-1.77068&0.39591&0.42441&0.95150\\
1&-2.80824&-2.81078&0.20773&0.21221&0.98501&-3.68166&-3.68317&0.14006&0.14147&0.99293\\
2&-4.46080&-4.46184&0.10549&0.10610&0.99593&-5.1767&-5.1775&0.084566&0.084883&0.99736\\
3&-5.84606&-5.84667&0.070551&0.070736&0.99815&-6.47897&-6.47947&0.060514&0.060630&0.99864\\
4&-7.08232&-7.08273&0.052973&0.053052&0.99895&-7.66095&-7.66130&0.047102&0.047157&0.99917\\
5&-8.21847&-8.21878&0.042401&0.042441&0.99933&-8.75768&-8.75795&0.038553&0.038583&0.99944\\
6&-9.28076&-9.28100&0.035344&0.035368&0.99953&-9.78949&-9.78971&0.032629&0.032647&0.99960\\
7&-10.28532&-10.28552&0.030300&0.030315&0.99966&-10.76947&-10.76965&0.028282&0.028294&0.99970\\
8&-11.24297&-11.24313&0.026516&0.026526&0.99974&-11.70670&-11.70685&0.024957&0.024965&0.99977\\
9&-12.16141&-12.16155&0.023572&0.023579&0.99979&-12.60778&-12.60791&0.022332&0.022338&0.99981\\
10&-13.04638&-13.04650&0.021216&0.021221&0.99983&-13.47773&-13.47784&0.020206&0.020210&0.99985\\
20&-20.70998&-20.71003&0.010610&0.010610&0.99996&-21.05373&-21.05377&0.010351&0.010352&0.99996\\
50&-38.14819&-38.14820&0.0042441&0.0042441&0.99999&-38.40209&-38.40210&0.0042021&0.0042021&0.99999\\
100&-60.55649&-60.55650&0.0021221&0.0021221&1&-60.75818&-60.75819&0.0021115&0.0021115&1\\
[1ex
\hlin
\end{tabular}
\label{Table1}
\end{sidewaystable}
\begin{figure*}
\centering
\includegraphics[width=\columnwidth]{DeltaEnergies.eps}
\caption{\label{DeltaEnergies}
Energies $E$ of the quasibound states of the attractive $\delta$-potential (solid lines) as a function of the electric field $\mathscr{E}$. The upper edge of each petal corresponds to the $B$ level. Dotted curves depict real components of the complex solutions of Eq.~\eqref{DeltaPotentialEigenEquation1}. The thin vertical dash-dot-dotted line shows the location of the fundamental dissociation field $\mathscr{E}_f$. Solid and dashed lines at $\mathscr{E}>\mathscr{E}_f$ denote real and positive imaginary parts, respectively, of the first complex solution of Eq.~\eqref{DeltaPotentialEigenEquation2}. The inset depicts an enlarged view of several positive energy states, where levels of the repulsive barrier from Eq.~\eqref{DeltaPotentialEigenEquation2} (dash-dotted curves) and ~\eqref{DeltaPotentialEigenEquation1} (dash-dot-dotted lines) are also shown. Contrary to the attractive well, in the latter geometry the lower rim of each petal is formed by the $B$ level.}
\end{figure*}
Fig.~\ref{DeltaEnergies} shows the energies of REQB states calculated from Eq.~\eqref{DeltaPotentialEigenEquation2} together with the real parts of the complex energies being solutions of Eq.~\eqref{DeltaPotentialEigenEquation1}. At the weak fields, the energy $E_0$ decreases quadratically with electric intensity, as derived in Eq.~\eqref{DeltaEnergiesSmallFields1B0}. This can be interpreted as an increase in the binding of the electron by the small voltages. The ground state energy reaches a minimum of ${E_0}_{min}=-1.0806$ at ${\mathscr{E}_0}_{min}=0.739$. A comparison with the corresponding Gamow-Siegert data provided above shows a conspicuous difference at these fields, while for small $\mathscr{E}$ the energies calculated by either method are the same. The deviation of the energies at $\mathscr{E}\gtrsim0.5$ is clearly seen in the figure. A subsequent increase of the voltage leads to the growth of the energy until it approaches zero at $\mathscr{E}_f$, where it amalgamates with the lowest $B$ level. All positive energies grow from their zero value as $\mathscr{E}^{2/3}$ at the weak fields, with the steepness being higher for larger $n$. This energy increase for the electric potential that seemingly has to force them downward is explained for the $A$ levels by the formation of the triangular QW, as discussed above. The lowest $B$ level that does not have its complex counterpart passes at ${\mathscr{E}_1}_{max}=1.372$ through the broad maximum of ${E_1}_{max}=0.6475$, after which it decreases towards zero. To determine the energy behavior close to this merger, it is convenient to represent Eq.~\eqref{DeltaPotentialEigenEquation2} in the parametric form \cite{Moyer3}
\begin{subequations}\label{Parametric1}
\begin{eqnarray}\label{ParametricEnergy1}
E&=&-4\pi^2zf^2(z)\\
\label{ParametricField1}
\mathscr{E}&=&8\pi^3f^3(z),
\end{eqnarray}
\end{subequations}
where the coefficient $z$, which is equal to $z=-E/\mathscr{E}^{2/3}$, varies from zero to positive infinity (for the lower level) or to $b_1$ (for the upper state). Close to the coalescence, this parameter is small, $|z|\ll1$, and the Taylor expansion simplifies Eqs.~\eqref{Parametric1} to
\begin{subequations}\label{Parametric2}
\begin{eqnarray}\label{ParametricEnergy2}
E&=&-4\pi^2f^2(0)z\\
\mathscr{E}&=&8\pi^3\left[f^3(0)+\frac{3}{2}f^2(0)f''(0)z^2\right]\nonumber\\
\label{ParametricField2}
&=&\left[1+\frac{3}{2}\frac{f''(0)}{f(0)}z^2\right]\mathscr{E}_f.
\end{eqnarray}
\end{subequations}
Eliminating $z$ from these equations, one gets after some simple algebra
\begin{equation}\label{AsymptotCriticalField1}
E_{\left\{^0_1\right\}}=\mp\left(\frac{\mathscr{E}_f}{3}\right)^{\!\!1/2}\left(\mathscr{E}_f-\mathscr{E}\right)^{1/2},\quad\mathscr{E}\rightarrow\mathscr{E}_f.
\end{equation}
For the higher lying amalgamations, this expression is generalized as
\begin{eqnarray}
E=E_n^\times\left[1\mp\frac{1}{s_n}\!\left(\!-\frac{2}{3}\frac{f(s_n)}{f''(s_n)}\frac{1}{\mathscr{E}_n^\times}\!\right)^{\!\!1/2}\!\!\left(\mathscr{E}_n^\times-\mathscr{E}\right)^{1/2}\right],\nonumber\\
\label{AsymptotCriticalField1_1}
\mathscr{E}\rightarrow\mathscr{E}_n^\times,\quad n\geq1,
\end{eqnarray}
which is also valid for the repulsive potential. The coalescence of the two states physically results in ionization of the structure by the growing field when the $\delta$-potential can no longer bind the charged particle. Higher lying states dissociate at weaker electric fields, as they are less bound by the potential. As Fig.~\ref{DeltaEnergies} demonstrates, for larger quantum numbers $n$ the energies of the $A$ levels deviate less from their complex-Airy-functions counterparts, while the petals formed by the $A$ and $B$ energies become narrower. For comparison, the curves of the repulsive potential are also shown in the inset. In this case, the energies can take positive values only with all basic features described for the QW being observed too. One major difference lies in the fact that for the barrier, the lower edge of each petal is formed by the $B$ level. Additionally, it should be noted that to the right of the field $\mathscr{E}_n^\times$ at which the levels with real energies merge, Eq.~\eqref{DeltaPotentialEigenEquation2} has two complex conjugate solutions. Real and positive imaginary components of the lowest levels are also shown in the figure. The real part grows with the field from its value $E_n^\times$ at the breakdown while the magnitude of the imaginary part increases more rapidly from its zero value. The physical interpretation of these mathematically correct solutions of Eq.~\eqref{DeltaPotentialEigenEquation2} will be discussed below.
A clear manifestation of the electric breakdown of the QW is revealed by the analysis of the polarization $P$ that, in the coordinate representation, can be written as
\begin{eqnarray}
P_n^\delta(\mathscr{E})=\frac{{\rm Bi}^2\!\left(\!-\frac{E_n}{\mathscr{E}^{2/3}}\right)\!\!\int_{-\infty}^0\!x{\rm Ai}^2\!\left(-\mathscr{E}^{1/3}x\!-\!\!\frac{E_n}{\mathscr{E}^{2/3}}\right)\!dx}{{\rm Bi}^2\!\left(\!-\frac{E_n}{\mathscr{E}^{2/3}}\right)\!\!\int_{-\infty}^0\!{\rm Ai}^2\!\left(-\mathscr{E}^{1/3}x\!-\!\!\frac{E_n}{\mathscr{E}^{2/3}}\right)\!dx}\nonumber\\
\label{DeltaPolarization1}
\frac{+\!{\rm Ai}^2\!\left(\!-\frac{E_n}{\mathscr{E}^{2/3}}\right)\!\!\int_0^\infty\!x{\rm Bi}^2\!\left(-\mathscr{E}^{1/3}x\!-\!\!\frac{E_n}{\mathscr{E}^{2/3}}\right)\!dx}{+\!{\rm Ai}^2\!\left(\!-\frac{E_n}{\mathscr{E}^{2/3}}\right)\!\!\int_0^\infty\!{\rm Bi}^2\!\left(-\mathscr{E}^{1/3}x\!-\!\!\frac{E_n}{\mathscr{E}^{2/3}}\right)\!dx}.
\end{eqnarray}
Note that the primitives in Eq.~\eqref{DeltaPolarization1} can be readily calculated analytically \cite{Vallee1} but applying the limits of integration to the second terms in the numerator and denominator leads to their divergence. Scattering theory has developed special regularization procedures for treating such integrals \cite{Kukulin1} that go back to early efforts in the 1960s \cite{Zeldovich1,Berggren1}. However, to avoid handling of the divergent integrals and their subsequent division in our particular case, it is much easier to use another method for finding the dipole moment that employs the Hellmann-Feynman theorem, which on application to the Hamiltonian from Eq.~\eqref{Hamiltonian1} is
\begin{equation}\label{HellmannFeynman1}
\frac{dE_n}{d\mathscr{E}}=\left\langle\frac{\partial\hat{H}}{\partial\mathscr{E}}\right\rangle=-\left\langle x\right\rangle.
\end{equation}
This immediately leads to the following reworking of Eq.~\eqref{Polarization1} \cite{Olendski1,Moyer3}:
\begin{equation}\label{Polarization3}
P_n(\mathscr{E})=-\frac{dE_n}{d\mathscr{E}}-\left\langle x\right\rangle_{\mathscr{E}=0}.
\end{equation}
The field-free $\delta$-potential is symmetric with respect to the change $x\rightarrow-x$, which results in $\left\langle x\right\rangle_{\mathscr{E}=0}^\delta\equiv0$. Applying the rule of differentiation of the implicit functions to Eq.~\eqref{DeltaPotentialEigenEquation2} leads to
\begin{equation}\label{DeltaPolarization2}
P_n^\delta(\mathscr{E})=-\frac{1}{6}\left[4\frac{E_n}{\mathscr{E}}-\frac{1}{\pi}\frac{1}{f'\left(-E_n/\mathscr{E}^{2/3}\right)}\right].
\end{equation}
The implementation of Eq.~\eqref{Polarization3} to the ground state weak-field limit from Eq.~\eqref{DeltaEnergiesSmallFields1B0} shows that in this regime, the dipole moment increases linearly with applied voltage
\begin{equation}\label{DeltaPolarization3}
P_0^{\delta-}(\mathscr{E})=\frac{5}{8}\,\mathscr{E},\quad\mathscr{E}\ll1,
\end{equation}
meaning that the electron moves in the direction of the external force acting upon it. However, close to the fundamental critical field $\mathscr{E}_f$, the negative divergence of the ground state polarization is gained from Eqs.~\eqref{Polarization3} and \eqref{AsymptotCriticalField1}
\begin{equation}\label{DeltaPolarization4}
P_0^{\delta-}(\mathscr{E})=-\frac{3^{1/2}}{6}\frac{\mathscr{E}_f^{1/2}}{\left(\mathscr{E}_f-\mathscr{E}\right)^{1/2}},\quad\mathscr{E}\rightarrow\mathscr{E}_f.
\end{equation}
To understand this reversal of polarization, the structure of the associated wave functions needs to be considered. Starting from the $A$ levels, the behavior in the very weak fields to the left of the potential is determined by the Airy functions entering the first integrands in Eq.~\eqref{DeltaPolarization1}, which together with the asymptotes from Eq.~\eqref{DeltaEnergiesSmallFields1An} yields:
\begin{equation}\label{DeltaFunctionA1}
\Psi_{A_n}^\delta(x)={\rm Bi}(a_n){\rm Ai}\!\left(-\mathscr{E}^{1/3}x+a_n-\frac{\mathscr{E}^{1/3}}{2}\right),\,x\leq0,\,\mathscr{E}\ll1.
\end{equation}
Applying the properties of the Airy functions \cite{Abramowitz1,Vallee1}, it is found that this waveform has $n$ extrema that are located at
\begin{equation}\label{DeltaExtremaA1}
x_{A_{nm}}^{ext}=\frac{a_n-a_m'}{\mathscr{E}^{1/3}}-\frac{1}{2},\quad\mathscr{E}\ll1,\quad m=1,2,\ldots,n
\end{equation}
[the negative $a_n'$ is the $n$th zero of the derivative of the Ai Airy function, ${\rm Ai}'(a_n')=0$], and the value of the functions at these points is:
\begin{equation}\label{DeltaFunctionA2}
\Psi_{A_n}^\delta\!\left(x_{A_{nm}}^{ext}\right)={\rm Bi}(a_n){\rm Ai}(a_m').
\end{equation}
The wave function to the right possesses an infinite number of fading oscillations, whose amplitudes at low voltages are $\mathscr{E}^{-1/3}$ times smaller than their counterpart(s) at $x<0$:
\begin{eqnarray}
\Psi_{A_n}^\delta(x)=-\frac{1}{2}\,\mathscr{E}^{1/3}{\rm Ai}'(a_n){\rm Bi}\!\left(-\mathscr{E}^{1/3}x+a_n-\frac{\mathscr{E}^{1/3}}{2}\right),\nonumber\\
\label{DeltaFunctionA3}x\geq0,\quad\mathscr{E}\ll1.
\end{eqnarray}
Hence, at extremely weak fields, the $A$ function is located far to the left, which is in accordance with the corresponding negatively diverging polarization derived from Eqs.~\eqref{Polarization3} and \eqref{DeltaEnergiesSmallFields1An}:
\begin{equation}\label{DeltaPolarizationAlimit1}
P_{A_n}^\delta(\mathscr{E})=\frac{2}{3}\frac{a_n}{\mathscr{E}^{1/3}},\quad\mathscr{E}\ll1.
\end{equation}
Note that with the increase of the small voltage the particle, as follows from Eq.~\eqref{DeltaExtremaA1}, moves to the right: this results in the growth of the dipole moment from Eq.~\eqref{DeltaPolarizationAlimit1} and agrees with the electron behavior in the electric field. This shift in the positive $x$ direction is clearly seen in panel (a) of Fig.~\ref{DeltaFunction1and2}. On the contrary, the $B$ level under the same assumption of the small electric intensities resides mainly to the right of the $\delta$-potential with its wave function
\begin{eqnarray}
\Psi_{B_n}^\delta(x)={\rm Ai}(b_n){\rm Bi}\!\left(-\mathscr{E}^{1/3}x+b_n+\frac{\mathscr{E}^{1/3}}{2}\right),\nonumber\\
\label{DeltaFunctionB1}x\geq0,\quad\mathscr{E}\ll1
\end{eqnarray}
exhibiting an infinite number of peaks and dips with the following values
\begin{equation}\label{DeltaFunctionB2}
\Psi_{B_n}^\delta\!\left(x_{B_{nm}}^{ext}\right)={\rm Ai}\left(b_n\right){\rm Bi}\left(b_m'\right)
\end{equation}
[negative coefficients $b_n'$ form an infinite set of roots of the derivative of the Bi Airy function, ${\rm Bi}'(b_n')=0$] located at
\begin{equation}\label{DeltaExtremaB1}
x_{B_{nm}}^{ext}=\frac{b_n-b_m'}{\mathscr{E}^{1/3}}+\frac{1}{2},\quad\mathscr{E}\ll1,\quad m=n,n+1,\ldots.
\end{equation}
Observe that the period of swinging and the distance between extrema increase for decreasing electric intensity. Oscillations to the left, if they exist, are characterized by the amplitudes that are $\mathscr{E}^{-1/3}$ times smaller, after which the waveform exponentially decays with $x$ tending to negative infinity:
\begin{eqnarray}
\Psi_{B_n}^\delta(x)=\frac{1}{2}\,\mathscr{E}^{1/3}{\rm Bi}'(b_n){\rm Ai}\!\left(-\mathscr{E}^{1/3}x+b_n+\frac{\mathscr{E}^{1/3}}{2}\right),\nonumber\\
\label{DeltaFunctionB3}x\leq0,\quad\mathscr{E}\ll1.
\end{eqnarray}
The most important properties to deduce from Eq.~\eqref{DeltaExtremaB1} are: i) at extremely weak fields the particle is located far to the right and ii) with the increase of the (still small) voltage it moves to the {\em left}, which is the opposite direction compared to the $A$ state, and which has just been associated with the electron. This can only be possible if the charge of particle dwelling at the $B$ level is opposite that of its $A$ counterpart. In this way, the natural conclusion is that the $B$ levels correspond to the hole states carrying positive charge. This explains the growth of their energies at small intensities, Eq.~\eqref{DeltaEnergiesSmallFields1Bn}; namely, as the particle moves into the area of the higher potential, its energy increases correspondingly. The general definition of the polarization, Eq.~\eqref{Polarization3}, has to be amended to take into account the different charges of the electrons and holes:
\begin{subequations}\label{PolarizationAmended1}
\begin{eqnarray}\label{PolarizationAmended1A1}
P_{A_n}^\delta(\mathscr{E})&=&-\frac{dE_{A_n}^\delta}{d\mathscr{E}}\\
\label{PolarizationAmended1B1}
P_{B_n}^\delta(\mathscr{E})&=&\frac{dE_{B_n}^\delta}{d\mathscr{E}}.
\end{eqnarray}
\end{subequations}
The last formula together with Eq.~\eqref{DeltaEnergiesSmallFields1Bn} results in positively diverging dipole moments at the weak fields:
\begin{equation}\label{DeltaPolarizationBlimit1}
P_{B_n}^\delta(\mathscr{E})=-\frac{2}{3}\frac{b_n}{\mathscr{E}^{1/3}},\quad\mathscr{E}\ll1,
\end{equation}
conforming to our earlier conclusion regarding the hole locations in this electric regime. Fig.~\ref{DeltaFunction1and2}(b) exemplifies the wave function shift in the left-hand direction for the lowest $B$ state at small $\mathscr{E}$. Note that in this regime, the $A$ and $B$ waveforms do not exhibit a mutual mirror symmetry with respect to $x=0$ since, as discussed above, the former (latter) is characterized at $x<0$ ($x>0$) by a finite (infinite) number of extrema and with the distance from the origin growing fades as $e^{-\mathscr{E}^{1/2}|x|^{3/2}}$ $\left(\mathscr{E}^{-1/12}x^{-1/4}\right)$.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{DeltaFunction1and2.eps}
\caption{\label{DeltaFunction1and2}
The waveforms $\Psi(x)$ of the lowest $A$ [panel (a)] and $B$ [panel (b)] levels at several electric fields $\mathscr{E}$, normalized to the maximum of their absolute values. In the upper plot, the dash-dot-dotted line represents $\mathscr{E}=10^{-4}$, the dotted curve represents $\mathscr{E}=10^{-2}$, the dashed curve represents $\mathscr{E}=0.1$, the dash-dotted curve represents $\mathscr{E}=0.162$, which corresponds to the maximum value of the associated polarization (see Fig.~\ref{DeltaPolarizationFig1}), and the solid line describes the wave function $\Psi_{A_1}(x)$ at the coalescence field $\mathscr{E}_1^{\times-}=0.20773$. For panel (b) the thin solid line represents $\mathscr{E}=10^{-4}$, dotted and dashed curves denote the same voltages as in the upper part, the dash-dotted line denotes $\mathscr{E}=1$, the dash-dot-dotted line denotes $\mathscr{E}=2$, and the thick solid line corresponds to the fundamental field $\mathscr{E}_f$. Note different $x$ and $\Psi$ ranges for each of the panels.
}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{DeltaFunction0_3D.eps}
\caption{\label{DeltaFunction0_3D}
Evolution of the wave function $\Psi_0(x)$ of the lowest quasibound state (normalized to its value at $x=0$) with the electric field $\mathscr{E}$ (in units of $\mathscr{E}_f$).}
\end{figure}
It has thus been shown that, while the particle localization at $E>0$ is completely undetermined for the flat geometry with a short-range potential, the vanishingly weak electric intensity creates electron- and hole-like REQB states in the positive energy continuum that are split in opposite directions. It is important to stress that we use the word "hole" just to underline that the corresponding level behaves like the particle with the positive charge. However, this interesting analogy has its limitations since in the considered system there is no positive charge whatever while in semiconductors the hole carries it because its host atom has a missing electron. From this point of view, it is better to use for our configuration the terms "hole-like state" or quasi-hole, what is tacitly assumed below.
Spatial electron-hole separation $\Delta x_n^{e-h}$ can be defined as the distance between their nearest largest extrema, which for the low voltages reduces to
\begin{equation}\label{DeltaXseparation1}
\Delta x_n^{e-h}=\frac{b_n+a_n'-b_n'-a_n}{\mathscr{E}^{1/3}}+1,\quad\mathscr{E}\ll1.
\end{equation}
This length decreases for larger quantum numbers:
\begin{equation}\label{DeltaXseparation2}
\Delta x_n^{e-h}=\left(\frac{2\pi^2}{3n\mathscr{E}}\right)^{1/3}+1,\quad\mathscr{E}\ll1,\quad n\gg1.
\end{equation}
These equations, together with Fig.~\ref{DeltaFunction1and2}, manifest that at weak electric intensities the electron and hole states are well separated and, accordingly, barely affect each other. The growing field decreases the partition and, as a result, their mutual distortion increases with the corresponding changing shape of the wave functions. The same remains true for the interaction of the ground state with its closest $B$ counterpart. Evolution with the field of the ground level wave function is shown in Fig.~\ref{DeltaFunction0_3D}. At very low voltages, it exhibits fading trigonometric oscillations with the largest amplitude of $\sim\mathscr{E}^{1/6}$ after $x\gtrsim1/\mathscr{E}$ only; accordingly, the associated polarization $P_0$, which, together with its counterparts for several higher lying quasibound states, is shown in Fig.~\ref{DeltaPolarizationFig1}, is determined by the redistribution of the charges near $x=0$, which results in the linear dependence from Eq.~\eqref{DeltaPolarization3}. At the same time, the nearest $B_1$ state is rapidly moving to the left, causing the steep decrease in the dipole moment $P_{B_1}$ seen in Fig.~\ref{DeltaPolarizationFig1}. By pushing the two states closer to each other, the increasing electric intensity forces them to interact more strongly with the concomitant larger deformations of the wave functions that exhibit higher frequencies of oscillations. This increase in the voltage leading to mutual interference of the electron- and hole-like states, which can be construed as an attraction of opposite charges, also affects their energies, compelling them to change direction and move towards each other, which, according to Eq.~\eqref{Polarization3}, influences the ground dipole moment in such a way that at $\mathscr{E}=0.299$ it passes through the maximum of $P_{0_{max}}^{\delta-}=0.1943$, after which it decreases and moves closer and closer to its $B_1$ partner. As the field approaches the coalescence value, the squeezing of the waveforms gets stronger, which leads to a drastic decrease in the polarizations. Simultaneously, there is an increasing degree of similarity between the two levels and, at the merger, one has two identical solutions with the same energies and wave functions. This collision of levels at the coalescence field can be considered as an electron-hole recombination and, as mentioned above, the voltage $\mathscr{E}_f$ is the breakdown field beyond which the $\delta$-potential cannot bind the electron \cite{Moyer3,Moyer4}. However, the existence at $\mathscr{E}>\mathscr{E}_f$ of the complex conjugate solutions of Eq.~\eqref{DeltaPotentialEigenEquation2} suggests another interpretation; namely, a creation at these electric intensities of a composite particle when the electron and hole stick together, forming an exciton; their individual motions cannot be considered independently since they correlate with each other. Note that we have considered the one-particle Schr\"{o}dinger equation [see Eqs.~\eqref{Schrodinger1} and \eqref{Hamiltonian1}] but even in this simplest form it hints at the emergence of the many-particle phenomena in the electric field. More advanced theories should be employed to address this issue, which lies beyond the scope of the present research. Of course, what has been said above applies also to the higher lying states for which the coalescence fields are lower due to the smaller electron-hole separation, see Eq.~\eqref{DeltaXseparation2}. At the end of this paragraph, we will point out that a comparison of the corresponding wave functions from Figs.~\ref{DeltaPotentialComplexEnergyFunction0}(b) and \ref{DeltaFunction0_3D} vividly underlines the difference between the two approaches.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{DeltaPolarizationFig1.eps}
\caption{\label{DeltaPolarizationFig1}
Polarizations $P$ of the quasibound states of the attractive $\delta$-potential, Eq.~\eqref{DeltaPolarization2}, as a function of the electric field $\mathscr{E}$ where the thick solid line describes the evolution of the zero-field bound state, dotted, dash-dotted, and thin solid lines represent the $B$ levels with $n=1$, $n=2$, and $n=3$, respectively, while the dashed and dash-dot-dotted curves represent the $A$ states with the quantum numbers $n=1$ and $2$, respectively. The upper inset enlarges the view of the ground state polarization at the weak electric intensities and the lower panel depicts the dipole moments in the fields up to $\mathscr{E}_1^{\times-}$.}
\end{figure}
Under some conditions specified below, {\em real} energies $E_n$, which are solutions of Eq.~\eqref{DeltaPotentialEigenEquation2}, can be supplemented by the associated half widths $\Gamma_n^{BW}$, and the set of these two quantities can be construed again as the {\em complex} energy; namely, applying a Taylor expansion to the real part of the numerator and denominator on the right-hand side of Eq.~\eqref{DeltaPotentialScatteringMatrix1} around its zeros, the expression for the scattering matrix can be recast in the well-known Breit-Wigner form \cite{Newton1,Landau1,Baz1,Kukulin1,Breit1}
\begin{subequations}\label{Auxiliary1}
\begin{align}\label{BreitWigner1}
S(E)&=e^{i2\varphi_P}\frac{E-E_n-i\left.\Gamma_n^{BW}\!\right/2}{E-E_n+i\left.\Gamma_n^{BW}\!\right/2},
\intertext{
where a slowly varying potential phase $\varphi_P$ takes, for the current geometry, a constant value of $\pi/2$ resulting in the following expression for the scattering probability:}
\label{ScatProbab2}
p(E)&=\frac{\left(\left.\Gamma_n^{BW}\!\!\right/2\right)^2}{\left(E-E_n\right)^2+\left(\left.\Gamma_n^{BW}\!\!\right/2\right)^2}.
\intertext{These equations are valid at}
\label{Condition1}
\left|\Gamma_n^{BW}\right|&\ll1,\,\left|E-E_n\right|\!\ll\!\left|E_n-E_{n+1}\right|,\,\left|E-E_n\right|\!\ll\!\left|E_n-E_{n-1}\right|.
\end{align}
\end{subequations}
The Breit-Wigner half width $\Gamma_n^{BW}$, which, according to Eq.~\eqref{ScatProbab2}, defines a full width at half maximum (FWHM) of the scattering probability $p(\mathscr{E};E)$, is given by
\begin{equation}\label{DeltaPotentialHalfWidth1}
\Gamma_n^{BW}=-2\mathscr{E}^{2/3}\frac{{\rm Ai}^2\!\left(\!\left.-E_n\!\right/\!\mathscr{E}^{2/3}\right)}{f'\!\left(\!\left.-E_n\!\right/\!\mathscr{E}^{2/3}\right)}.
\end{equation}
For the weak fields, $\mathscr{E}\ll1$,
\begin{subequations}\label{DeltaHalfWidthSmallFields1}
\begin{eqnarray}\label{DeltaHalfWidthSmallFields1B0}
\Gamma_0^{BW}&=&2\exp\!\left(\!-\frac{4}{3}\frac{1}{\mathscr{E}}\right)\\
\label{DeltaHalfWidthSmallFields1An}
\Gamma_{A_n^\pm}^{BW}&=&\frac{1}{2\pi{\rm Bi}^2(a_n)}\,\mathscr{E}^{4/3}\\
\label{DeltaHalfWidthSmallFields1Bn}
\Gamma_{B_n^\pm}^{BW}&=&-2\frac{{\rm Ai}(b_n)}{{\rm Bi}'(b_n)}\,\mathscr{E}^{2/3}.
\end{eqnarray}
\end{subequations}
First, again note, similar to Eqs.~\eqref{DeltaComplexSolutionsTwoSets1}, the different powers of the field for the $A$ and $B$ resonances. This is explained by the fact that the corresponding waveforms are not exact replicas of each other under the transformation $x\rightarrow-x$. Next, Eqs.~\eqref{DeltaEnergiesSmallFields1B0} and \eqref{DeltaHalfWidthSmallFields1B0} are, obviously, just the expression for the complex energy from Eq.~\eqref{DeltaPotentialAsymptotics1} derived by the Gamow-Siegert method, while Eqs.~\eqref{DeltaEnergiesSmallFields1An} and \eqref{DeltaHalfWidthSmallFields1An} compose complex solutions from Eq.~\eqref{DeltaComplexSolutionsTwoSets1_Set1} that, accordingly, describe the states located at the far left (not taking into account, of course, the divergence at large positive $x$). However, real and imaginary parts of the second set, Eq.~\eqref{DeltaComplexSolutionsTwoSets1_Set2}, of the complex solutions of Eq.~\eqref{DeltaPotentialEigenEquation1} bear no resemblance to the corresponding numbers from Eqs.~\eqref{DeltaEnergiesSmallFields1Bn} and \eqref{DeltaHalfWidthSmallFields1Bn}. Moreover, the results that coincided at small electric intensities diverge considerably from each other at stronger $\mathscr{E}$. Mathematically, this is due to the fact that {\em complex} solutions of Eq.~\eqref{DeltaPotentialEigenEquation1} are, in general, different from the complex numbers whose {\em real} parts are determined from Eq.~\eqref{DeltaPotentialEigenEquation2} and whose {\em imaginary} components obey Eq.~\eqref{DeltaPotentialHalfWidth1}.
Eqs.~\eqref{DeltaHalfWidthSmallFields1An} and \eqref{DeltaHalfWidthSmallFields1Bn} can be further simplified for large quantum numbers \cite{Alvarez1} when the approximate analytic expressions for the roots $a_n$ and $b_n$ do exist \cite{Abramowitz1,Vallee1}:
\begin{subequations}\label{AiryZeros1}
\begin{eqnarray}\label{AiryAiZeros1}
a_n=-\left[\frac{3\pi}{8}(4n-1)\right]^{2/3},\quad n\gg1\\
\label{AiryBiZeros1}
b_n=-\left[\frac{3\pi}{8}(4n-3)\right]^{2/3},\quad n\gg1.
\end{eqnarray}
\end{subequations}
Even though these asymptotic formulae were derived for large $n$, they also provide reasonably good accuracy for small $n$. For example, in the extreme opposite limit of $n=1$ the exact values are \cite{Abramowitz1} $a_1=-2.3381$ and $b_1=-1.1737$ while their approximations from Eqs.~\eqref{AiryZeros1} yield $-2.3203$ and $-1.1155$, respectively. These expressions also lead to a drastic reduction of the products and ratio of the Airy functions in the above equations:
\begin{subequations}\label{DeltaEnergiesSmallFieldsLargeN1}
\begin{eqnarray}
\label{DeltaEnergiesSmallFieldsLargeN1An}
E_n^{A\pm}&=&\left[\frac{3\pi}{8}(4n-1)\mathscr{E}\right]^{2/3}\mp\frac{1}{2}\,\mathscr{E},\quad\mathscr{E}\ll1,\,n\gg1\\
\label{DeltaEnergiesSmallFieldsLargeN1Bn}
E_n^{B\pm}&=&\left[\frac{3\pi}{8}(4n-3)\mathscr{E}\right]^{2/3}\pm\frac{1}{2}\,\mathscr{E},\quad\mathscr{E}\ll1,\,n\gg1\\
\label{DeltaHalfWidthSmallFieldsLargeN1An}
\Gamma_{A_n^\pm}^{BW}&=&\frac{1}{2}\left[\frac{3\pi}{8}(4n-1)\right]^{1/3}\mathscr{E}^{4/3},\quad\mathscr{E}\ll1,\,n\gg1\\
\label{DeltaHalfWidthSmallFieldsLargeN1Bn}
\Gamma_{B_n^\pm}^{BW}&=&-2\left[\frac{3\pi}{8}(4n-3)\right]^{-1/3}\mathscr{E}^{2/3},\quad\mathscr{E}\ll1,\,n\gg1.
\end{eqnarray}
\end{subequations}
The first thing to notice is that the half widths of the $B$ resonances are negative; this superficially contradicts a commonly accepted view regarding the positiveness of $\Gamma$. However, in the wake of the previous discussion in this subsection, this result is not surprising as it means that the corresponding states leak in opposite directions: at low voltage the $A$ level is localized mainly to the left of the $\delta$-potential and as the field grows it tries to expand to its right, which is considered to be a positive route with the same sign of the half width. On the other hand, the $B$ orbital, which at $\mathscr{E}\ll1$ is found mainly at $x\gg1$, tries to make its way to the left of the potential. This opposing direction of the leakage is mathematically reflected in the negativeness of the factor $\Gamma$. Note that the scattering probability $p$ from Eq.~\eqref{ScatProbab2} depends on the square of the half width what means that it can be experimentally measured for $\Gamma_n^{BW}<0$. It is instructive to draw parallels with the Gamow-Siegert states, where the opposite signs of the half widths describe the behavior of the {\em same} level at earlier and later times, as discussed in Sec.~\ref{GamowSiegertStates1}; however, in the time-independent Breit-Wigner picture they are associated with different localizations of the {\em two} states. Observe also that the Gamow half widths are the imaginary parts of the complex conjugate solutions of Eq.~\eqref{DeltaPotentialEigenEquation1}; therefore, their absolute values are equal to each other, whereas for the Breit-Wigner configuration the magnitudes are different, as Eqs.~\eqref{DeltaHalfWidthSmallFieldsLargeN1An} and \eqref{DeltaHalfWidthSmallFieldsLargeN1Bn} exemplify.
With the growth of the field, the amplitudes of the half widths increase too, narrowing in this way the conditions of applicability of the Breit-Wigner approximation from Eq.~\eqref{Condition1}; in particular, and in a similar way to the derivation of Eq.~\eqref{AsymptotCriticalField1}, it can be shown that in the near vicinity of the fundamental critical field $\mathscr{E}_f$ the half width dependencies on the applied voltage are:
\begin{equation}\label{AsymptotCriticalField2}
\Gamma_{0,B_1}^{BW}=\pm\frac{\mathscr{E}_f^{3/2}}{\left(\mathscr{E}_f-\mathscr{E}\right)^{1/2}},\quad\mathscr{E}\rightarrow\mathscr{E}_f.
\end{equation}
For the smaller critical fields the divergence formula can be derived in a similar way to Eq.~\eqref{AsymptotCriticalField1_1}. Of course, for such wide resonances the Breit-Wigner approximation can no longer be used and instead the original exact equation~\eqref{DeltaPotentialScatteringMatrix1} should be applied.
As a final part of this subsection, let us point out that coalescences characterized by the fields $\mathscr{E}_n^{\times\pm}$ and energies $E_n^{\times\pm}$ present a special case of so-called exceptional points (EPs) \cite{Heiss2} where a merging of the two (or more) levels with the variation of some physical parameter is governed by the square root singularities from Eqs.~\eqref{AsymptotCriticalField1} or \eqref{AsymptotCriticalField1_1}. Contrary to usual degeneracy, EP exhibits not only equal energies of the different states but also linear dependent eigenfunctions, as discussed above. These spectral singularities, which are ubiquitous in nature, produce dramatic effects in, e.g., multichannel scattering, anomalous time behavior \cite{Heiss3}, etc.; for instance, for our geometry, EPs are characterized by the diverging polarizations, as exemplified by Eq.~\eqref{DeltaPolarization4} and Fig.~\ref{DeltaPolarizationFig1}. Note that, in general, the physical parameter variation of which leads to the coalescence is {\em complex} with the corresponding {\em complex} eigenvalues \cite{Heiss2} while for the $\delta$-potential it is the electric field with the {\em real} magnitude that causes a merging of the two REQB levels with {\em real} energies and their subsequent motion into the {\em complex} plane when the voltage is increased.
\subsection{Time Delay}
The expression for the time delay of the $\delta$-potential in the electric field is:
\begin{eqnarray}
\tau_W^{\delta\pm}&=&-\frac{8\pi}{\mathscr{E}^{2/3}}\nonumber\\
\label{DeltaTimeDelay1}
&\times&\frac{{\rm Ai}_0\left({\rm Ai}_0\mp\mathscr{E}^{1/3}{\rm Ai}_0'\right)}{4\pi^2{\rm Ai}_0^2\left({\rm Ai}_0^2+{\rm Bi}_0^2\right)\pm4\pi\mathscr{E}^{1/3}{\rm Ai}_0{\rm Bi}_0+\mathscr{E}^{2/3}}.
\end{eqnarray}
This formula, where, for brevity, the subscript '$0$' at each of the functions means that they are evaluated at the value of $\eta_0$ defined above [${\rm Ai}_0\equiv{\rm Ai}\left(-E/\mathscr{E}^{2/3}\right)$, etc.], was derived from the corresponding counterpart for the phase $\varphi_S$
\begin{equation}\label{DeltaPhase1}
\varphi_S^{\delta\pm}=\pi+\arctan\!\!\left(4\frac{{\rm Ai}_0^2\left(2{\rm Ai}_0{\rm Bi}_0\pm\mathscr{E}^{1/3}/\pi\right)}{4{\rm Ai}_0^4-\left(2{\rm Ai}_0{\rm Bi}_0\pm\mathscr{E}^{1/3}/\pi\right)^2}\right)
\end{equation}
with the use of the properties of the Airy functions. For large negative energies, the delay time very abruptly approaches zero:
\begin{subequations}\label{DeltaTimeDelayAsymptotics1}
\begin{align}\label{DeltaTimeDelayAsymptotics1_NegativeEnergy}
\tau_W^{\delta\pm}=\mp\frac{2}{\mathscr{E}}\exp\!\left(-\frac{4}{3}\frac{|E|^{3/2}}{\mathscr{E}}\right),\quad E\ll-1,
\intertext{which physically means that the incident particle at such energies does not 'see' the $\delta$-potential and is reflected from the tilted potential almost immediately \cite{Emmanouilidou2}. In the opposite limit the Wigner time degenerates to}
\label{DeltaTimeDelayAsymptotics1_PositiveEnergy}
\tau_W^{\delta\pm}=\mp\frac{4}{\mathscr{E}}\cos\!\left(\frac{4}{3}\frac{E^{3/2}}{\mathscr{E}}\right),\quad E\gg1,
\end{align}
\end{subequations}
meaning that it reaches its maxima
\begin{equation}\label{AsymptotDeltaStrongField1}
\tau_{max_n}^{\delta\pm}=\frac{4}{\mathscr{E}}
\end{equation}
at
\begin{subequations}\label{DeltaTimeDelayAsymptotics3}
\begin{eqnarray}\label{DeltaTimeDelayAsymptotics3Plus}
E^{\delta+}_{max_n}&=&\left[\frac{3\pi}{4}\left(2n+1\right)\mathscr{E}\right]^{2/3}\\
\label{DeltaTimeDelayAsymptotics3Minus}
E^{\delta-}_{max_n}&=&\left(\frac{3\pi}{2}n\mathscr{E}\right)^{2/3},
\end{eqnarray}
\end{subequations}
where the non-negative integer $n$ and the field $\mathscr{E}$ are such that the condition $E\gg1$ is satisfied basically reducing it to the requirements $\mathscr{E}\gg1$ or/and $n\gg1$. This oscillating behavior is explained by the interference of the incident and reflected waves in the region to the left of the well where, as stated above, the interplay of the electric field and $\delta$-potential creates a triangular QW with a transparency of its right wall depending on the field and energy. This interaction of the incoming and outgoing fluxes also explains the negative delay times that directly follow from Eq.~\eqref{DeltaTimeDelayAsymptotics1_PositiveEnergy}; namely, at some particular energies the interference makes the right wall of the triangular QW an impenetrable surface, meaning that the electron is reflected from the point $x=0$ and not from $x=-E/\mathscr{E}$, which is a quasi-classical turning point without the $\delta$-potential. Note that the distance between the maxima of the Wigner times increases with voltage while their peak values are inversely proportional to it.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{DeltaWellWignerTime.eps}
\caption{\label{DeltaWellWignerTime}
Wigner delay time $\tau_W^{\delta-}$ of the attractive $\delta$-potential as a function of energy $E$ at several electric fields where the solid line is for $\mathscr{E}=0.5$, dotted -- for $\mathscr{E}=2/3$, dashed -- for $\mathscr{E}=1$, dash-dotted -- for $\mathscr{E}=2$, and dash-dot-dotted curve -- for $\mathscr{E}=4$. Thin horizontal line denotes zero time.}
\end{figure}
Fig.~\ref{DeltaWellWignerTime} shows the delay time as a function of energy at several different electric intensities. At each fixed field an infinite number of maxima can be observed with their sharpness and peak value being field-dependent. Accordingly, each resonance is characterized by three parameters: the location of the delay time peak, its maximum at this energy, and the corresponding FWHM. For weak fields, it can be shown that the energies of the resonances in the positive part of the spectrum are:
\begin{eqnarray}
E^{\delta\pm}_{max_n}=-a_n\mathscr{E}^{2/3}\mp\frac{1}{2}\,\mathscr{E}-\frac{1}{16}\frac{{\rm Ai}'(a_n)^2{\rm Bi}'(a_n)}{{\rm Bi}^3(a_n)}\,\mathscr{E}^2,\nonumber\\
\label{DeltaResonancesPositiveEnergiesSmallField1}\mathscr{E}\ll1,\quad n\geq1,
\end{eqnarray}
i.e., they are close to the energies of the $A$-type quasibound states from Eq.~\eqref{DeltaEnergiesSmallFields1An}. Physically, this proximity to only the $A$ (and not to the $B$) levels is again explained by the formation to the left of the $\delta$-potential of the triangular QW that captures the electron for some time, while from the mathematical point of view both terms in the denominator of $\arctan$ in Eq.~\eqref{DeltaPhase1} are small and the tiny variation in the energy range close to that from Eq.~\eqref{DeltaResonancesPositiveEnergiesSmallField1} leads to subtle interplay between them, resulting in pronounced resonance with the maximum
\begin{equation}\label{DeltaResonancesPositiveEnergiesSmallField2}
\tau^{\delta\pm}_{max_n}=-8\frac{{\rm Bi}(a_n)}{{\rm Ai}'(a_n)}\frac{1}{\mathscr{E}^{4/3}},\quad\mathscr{E}\ll1,\quad n\geq1,
\end{equation}
which, according to the previous reasoning, for the rather large $n$ can be approximated as:
\begin{equation}\label{DeltaResonancesPositiveEnergiesSmallField3}
\tau^{\delta\pm}_{max_n}=8\left[\frac{3\pi}{8}(4n-1)\right]^{-1/3}\frac{1}{\mathscr{E}^{4/3}},\quad\mathscr{E}\ll1,\,n\gg1.
\end{equation}
Note that the peak decreases with growing $n$ and tends to infinity for the vanishing fields as $\mathscr{E}^{-4/3}$. For $E<0$ and small electric intensities, the resonance location $E^{\delta-}_{max_0}$ is exponentially close to the lowest quasibound state energy from Eq.~\eqref{DeltaEnergiesSmallFields1B0} [or, alternatively, to the negative real part of the Gamow-Siegert energy, Eq.~\eqref{DeltaPotentialAsymptotics1}]; thus, assuming their equality, the phase shift is determined as
\begin{equation}\label{DeltaPhase2}
\varphi_S=\pi-\arctan\frac{\Gamma_0(E-E_0)}{(E-E_0)^2-(\Gamma_0/2)^2},\quad\mathscr{E}\ll1,
\end{equation}
resulting in the Lorentzian shape of the time delay:
\begin{equation}\label{DeltaResonanceNegativeEnergySmallField1}
\tau_0^{\delta-}=\frac{\Gamma_0}{(E-E_0)^2+(\Gamma_0/2)^2},
\end{equation}
whose maximum exponentially approaches infinity with the disappearing fields:
\begin{equation}\label{DeltaResonanceNegativeEnergySmallField2}
\tau_{max_0}^{\delta-}=\frac{4}{\Gamma_0}=2\exp\!\left(\frac{4}{3}\frac{1}{\mathscr{E}}\right),\quad\mathscr{E}\ll1.
\end{equation}
Its FWHM in this regime is equal to $\Gamma_0$ from Eqs.~\eqref{DeltaHalfWidthAsymptotics1} or \eqref{DeltaHalfWidthSmallFields1B0}. In the opposite limit of $\mathscr{E}\gg1$, Eq.~\eqref{DeltaTimeDelayAsymptotics1_PositiveEnergy} gives
\begin{subequations}\label{DeltaResonanceHalfWidth1}
\begin{eqnarray}\label{DeltaResonanceHalfWidth1Minus}
\Gamma_n^{\delta-}=\left(\frac{3\pi}{4}\,\mathscr{E}\right)^{2/3}\left[\left(2n+\frac{1}{2}\right)^{2/3}-\left(2n-\frac{1}{2}\right)^{2/3}\right],\,\mathscr{E}\gg1\\
\label{DeltaResonanceHalfWidth1Plus}
\Gamma_n^{\delta+}=\left(\frac{3\pi}{4}\,\mathscr{E}\right)^{2/3}\left[\left(2n+\frac{3}{2}\right)^{2/3}-\left(2n+\frac{1}{2}\right)^{2/3}\right],\,\mathscr{E}\gg1,
\end{eqnarray}
\end{subequations}
which for large $n$ degenerates to
\begin{equation}\label{DeltaResonanceHalfWidth2}
\Gamma_n^{\delta\pm}=\frac{1}{3}\left(\frac{3\pi}{2}\,\mathscr{E}\right)^{2/3}\frac{1}{n^{1/3}},\quad\mathscr{E}\gg1,\quad n\gg1.
\end{equation}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{DeltaAllEnergies.eps}
\caption{\label{DeltaAllEnergies}
Comparison of the different methods for calculating the energies of the attractive $\delta$-potential in the electric field $\mathscr{E}$ where the solid lines show solutions of Eq.~\eqref{DeltaPotentialEigenEquation2}, dashed curves denote locations of the maxima of the Wigner delay time, and the dotted and dash-dotted lines are the real parts of the first and second sets, respectively, of solutions of Eq.~\eqref{DeltaPotentialEigenEquation1}. The inset shows the lowest levels at a different scale.}
\end{figure}
Fig.~\ref{DeltaAllEnergies} compares the energies calculated by the three methods. It can be seen that the positive real parts of the complex Gamow-Siegert energies are indistinguishable from the locations of the time delay maxima with $n\geq1$: due to their closeness, they are not resolved in the figure. However, the lowest maximum position and the corresponding real part of the complex energy, which are the same at weak fields, deviate from each other with increasing voltage; for example, $E^{\delta-}_{max_0}$ reaches a minimum of $-1.2685$ at $\mathscr{E}=4.57$ which, if compared to the analogous data of the other methods detailed above, means that the Wigner time calculations provide the lowest estimate. The divergence between the energies increases for stronger fields, as the inset demonstrates. As emphasized above, this increase with the field of the difference between the outcomes of the two approaches is mathematically explained by the different equations, while the physical reason lies in the fact that Gamow-Siegert solutions describe the outgoing oscillation while the real energy scattering approach operates with the standing waves that do not carry current. For completeness, the plot also shows the evolution of the real components of the second set of solutions of Eq.~\eqref{DeltaPotentialEigenEquation1}, which for the weak fields are described by Eq.~\eqref{DeltaComplexSolutionsTwoSets1_Set2}. Their characteristic feature is the fact that with the increase of the electric intensity they cross the level that evolved from the zero-field bound state \cite{Emmanouilidou2}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{DeltaHalfWidthResonances1.eps}
\caption{\label{DeltaHalfWidthResonances1}
Half widths $\Gamma$ calculated for the Gamow-Siegert (GS) states, and as FWHMs of the Wigner resonance peaks denoted by the corresponding acronyms. For comparison, the $n=0$ curve for the Breit-Wigner (BW) half width, Eq.~\eqref{DeltaPotentialHalfWidth1}, is also shown. Solid lines denote $n=0$, dotted lines are for $n=1$, dashed curves -- $n=2$, and dash-dotted lines denote $n=3$.}
\end{figure}
The variation of the half widths with the field calculated with the help of the complex Airy functions and as FWHMs of the scattering configuration is depicted in Fig.~\ref{DeltaHalfWidthResonances1} where a positive $\Gamma_0^{BW}$ is also shown. Qualitatively, the transformation from the weak field regime described by Eqs.~\eqref{DeltaHalfWidthSmallFields1B0} and \eqref{DeltaHalfWidthSmallFieldsLargeN1An}, when the half widths of the states with the larger $n$ are greater than their counterparts with the smaller quantum numbers, to the high voltage configuration with its dependence from Eq.~\eqref{DeltaResonanceHalfWidth2} is typical for either approach: crossings of the lines for different $n$ are clearly seen in the plot. Quantitatively, the Breit-Wigner approximation produces the larger estimate of $\Gamma$ followed by the Gamow model with increasing difference with the field: the half widths with $n\geq1$ are approximately equal at $\mathscr{E}\lesssim0.02$ only while, for $n=0$, they remain approximately the same up to $\mathscr{E}\sim0.6$. Thus, in general, the complex energy method leads to smaller lifetimes compared to those obtained by the scattering approach with real $E$.
\section{Robin Wall}\label{Sec_Robin}
Consider the motion of the particle on the half line $0\leq x<\infty$ subject at the left edge to the BC \cite{Olendski2,Seba1,Pazma1,Fulop1,Belchev1,Georgiou1}
\begin{equation}\label{Robin2}
\Lambda\Psi'(0)=\Psi(0),
\end{equation}
which is just the 1D analogue of Eq.~\eqref{Robin1}. Well-known particular cases of this general interface demand are the Dirichlet requirement, $\Lambda_D=0$, with the vanishing function at the left end,
\begin{subequations}\label{BCRobin1}
\begin{align}\label{BCRobin1_D}
\Psi_D(0)=0,\\
\intertext{and the Neumann one, $\Lambda_N=\infty$, when its spatial derivative vanishes at the confining surface,}
\label{BCRobin1_N}
\Psi_N'(0)=0.
\end{align}
\end{subequations}
In the absence of the fields, $\mathscr{E}=0$, the BC from Eq.~\eqref{Robin2} is formally identical to Eq.~\eqref{MatchingCondtions1_2}, which allows us to immediately conclude that the Robin wall with a negative extrapolation length acts as an attractive interface, creating a bound state with the energy from Eq.~\eqref{EnergyDeltaZeroFields1} and the wave function
\begin{equation}\label{FunctionRobinZeroFields1}
\Psi(x)=\left(\frac{2}{|\Lambda|}\right)^{1/2}\exp\!\left(-\frac{x}{|\Lambda|}\right),
\end{equation}
which satisfies the normalization
\begin{equation}\label{Normalization2}
\int_0^\infty\Psi^2(x)dx=1.
\end{equation}
Experimentally, the surfaces with negative de Gennes distances were fabricated with the help of superconductors \cite{Fink1,Kozhevnikov1}. This model also approximates, as the limiting case, the finite continual potentials \cite{Pazma1,Fulop1}, which are created by using thin layers of different types of semiconductors. More relevant references relating to the Robin structures can be found in Refs.~\cite{Olendski3,Olendski4,Olendski5,Grebenkov1}.
The wall is highly asymmetric with respect to the sign of the applied field: for negative electric intensities, $\mathscr{E}<0$, the spectrum is completely discrete, while for positive voltages it stays continuous. The former configuration is discussed elsewhere \cite{Olendski2}. In this section, we will address the case of the electric force that attempts to push the electron away from the wall, $\mathscr{E}>0$. For the finite non-vanishing $\Lambda$ the same dimensionless units as those in Sect.~\ref{Sec_Delta} will be used, while for the Dirichlet or Neumann BC the most appropriate unit of distance is the reduced Compton wavelength $\lambda\kern-1ex\raise0.55ex\hbox{--}=\hbar/(mc)$, which naturally leads to the units of energy $mc^2$, and electric field -- $m^2c^3/(e\hbar)$, with $c$ being the speed of light. As a result, the equation of motion takes the form
\begin{equation}\label{Schrodinger3}
-\Psi''(x)-\mathscr{E}x\Psi(x)=E\Psi(x),
\end{equation}
and the BC changes to either
\begin{equation}\label{BCRobin2}
\pm\Psi'(0)=\Psi(0)
\end{equation}
with the sign corresponding to that of the de Gennes distance, or Eqs.~\eqref{BCRobin1}. Below, we will use the superscript D (N) to denote the Dirichlet \cite{Dean1} (Neumann) type of BC at the interface, while the character R followed, if necessary, by the plus (minus) sign will refer to the Robin surface with a positive (negative) extrapolation length.
The general line of investigation is the same as that used for the $\delta$-potential, but to make the exposition shorter the expression for the scattering matrix $S$ is written from the outset as:
\begin{subequations}\label{RobinScatteringMatrix1}
\begin{eqnarray}
S^{R\pm}(\mathscr{E};E)=\nonumber\\
\label{RobinScatteringMatrix1_R}
\!\!-\!\frac{{\rm Bi}\!\left(\!-\!\frac{E}{\mathscr{E}^{2/3}}\!\!\right)\!\!\pm\!\!\mathscr{E}^{1/3}\!{\rm Bi}'\!\left(\!-\!\frac{E}{\mathscr{E}^{2/3}}\!\!\right)\!\!-\!\!i\!\!\left[{\rm Ai}\!\left(\!-\!\frac{E}{\mathscr{E}^{2/3}}\!\!\right)\!\!\pm\!\!\mathscr{E}^{1/3}\!{\rm Ai}'\!\left(\!-\!\frac{E}{\mathscr{E}^{2/3}}\!\!\right)\!\!\right]}{{\rm Bi}\!\left(\!-\!\frac{E}{\mathscr{E}^{2/3}}\!\!\right)\!\!\pm\!\!\mathscr{E}^{1/3}\!{\rm Bi}'\!\left(\!-\!\frac{E}{\mathscr{E}^{2/3}}\!\!\right)\!\!+\!\!i\!\!\left[{\rm Ai}\!\left(\!-\!\frac{E}{\mathscr{E}^{2/3}}\!\!\right)\!\!\pm\!\!\mathscr{E}^{1/3}\!{\rm Ai}'\!\left(\!-\!\frac{E}{\mathscr{E}^{2/3}}\!\!\right)\!\!\right]}\\
\label{RobinScatteringMatrix1_D}
S^D(\mathscr{E};E)=-\frac{{\rm Bi}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)-i{\rm Ai}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)}{{\rm Bi}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)+i{\rm Ai}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)}\\
\label{RobinScatteringMatrix1_N}
S^N(\mathscr{E};E)=-\frac{{\rm Bi}'\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)-i{\rm Ai}'\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)}{{\rm Bi}'\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)+i{\rm Ai}'\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)}.
\end{eqnarray}
\end{subequations}
A condition of zeroing its denominator produces the equations for determining the complex resonance energies $E_{res}$, which are written in the form:
\begin{subequations}\label{RobinEigenEquation1}
\begin{eqnarray}\label{RobinEigenEquation1_R}
{\rm Ai}\!\left(-\frac{E^{R\pm}}{\mathscr{E}^{2/3}}\,e^{i2\pi/3}\!\right)&\mp&\mathscr{E}^{1/3}e^{i2\pi/3}{\rm Ai}'\!\!\left(-\frac{E^{R\pm}}{\mathscr{E}^{2/3}}\,e^{i2\pi/3}\!\right)=0\\
\label{RobinEigenEquation1_D}
{\rm Ai}\!\left(-\frac{E^D}{\mathscr{E}^{2/3}}\,e^{i2\pi/3}\!\right)&=&0\\
\label{RobinEigenEquation1_N}
{\rm Ai}'\!\left(-\frac{E^N}{\mathscr{E}^{2/3}}\,e^{i2\pi/3}\!\right)&=&0.
\end{eqnarray}
\end{subequations}
From the last two equations it follows immediately that
\begin{eqnarray}
E_{res_n}^{\left\{_N^D\right\}}&=&-\left\{\begin{array}{c}a_n\\a_n'\end{array}\right\}\mathscr{E}^{2/3}e^{-i2\pi/3}\nonumber\\
\label{RobinEigenEnergiesDN1}
&=&\frac{1}{2}\left\{\begin{array}{c}a_n\\a_n'\end{array}\right\}\mathscr{E}^{2/3}\left(1+i3^{1/2}\right).
\end{eqnarray}
Eq.~\eqref{RobinEigenEnergiesDN1} means that the real parts of the resonance energies for the Dirichlet and Neumann BCs are always negative, while the corresponding half widths, as always in the method of zeroing the resolvent $\left(\hat{H}-E\right)^{-1}$, remain positive and both of them depend on the field as $\mathscr{E}^{2/3}$. For the Robin surface, the following asymptotes are taken from Eq.~\eqref{RobinEigenEquation1_R}:
\begin{subequations}\label{RobinEigenEnergiesAsymptot1}
\begin{eqnarray}\label{RobinEigenEnergiesAsymptot1SmallFields}
{E_{res}^{R\pm}}_n&=&\frac{1}{2}\,a_n\mathscr{E}^{2/3}\pm\mathscr{E}+i\frac{3^{1/2}}{2}\,a_n\mathscr{E}^{2/3},\quad\mathscr{E}\ll1\\
\label{RobinEigenEnergiesAsymptot1LargeFields}
{E_{res}^{R\pm}}_n&=&\frac{1}{2}\,a_n'\mathscr{E}^{2/3}\left(1\!\!+\!\!i3^{1/2}\right)\!\left(1\mp\!\frac{1}{a_n'^{\,\,2}}\frac{1}{\mathscr{E}^{1/3}}\right),\,\mathscr{E}\gg1.
\end{eqnarray}
\end{subequations}
A comparison of Eqs.~\eqref{RobinEigenEnergiesDN1} and \eqref{RobinEigenEnergiesAsymptot1} manifests that at the weak fields the Robin BC reduces basically to the Dirichlet one with a small admixture due to the interplay between the intensity $\mathscr{E}$ and non-zero extrapolation length $\Lambda$, while the high voltages essentially turn it into the Neumann surface where the higher order items, contrary to the previous limit, are the same for both the real and imaginary parts. Asymptotes from Eqs.~\eqref{RobinEigenEnergiesAsymptot1} present a general property of the interaction between the Robin BCs and the electric fields, which will be encountered below for the quasibound states as well as the true bound levels for the opposite direction of the field \cite{Olendski2}. It is also instructive to draw parallels with the similar states of the extremely localized potential whose energies at the weak fields are determined by Eq.~\eqref{DeltaComplexSolutionsTwoSets1_Set2}. It can be seen that in this limit they are almost identical to the difference, which is due to particle penetration to the left of the $\delta$ perturbation, being in the higher-order corrections. It is worth noting here that the Robin wall does not have the Gamow-Siegert resonances with the positive real parts of their energies since they are developed, as shown in the previous section, at $x<0$ where the motion for the present geometry is forbidden. In addition, the energy of the zero-field bound state at the small electric intensities is calculated as
\begin{equation}\label{RobinEigenEnergiesAsymptot2}
{E_{res}^{R-}}_0=-1-\frac{1}{2}\,\mathscr{E}-\frac{1}{8}\,\mathscr{E}^2-2i\exp\!\left(\!-\frac{4}{3}\frac{1}{\mathscr{E}}\right),\quad\mathscr{E}\ll1.
\end{equation}
The above formula shows that in this regime, due to the spatial asymmetry of the structure at $\mathscr{E}=0$, the real part of the energy is a linear function of the applied voltage while for the $\delta$-potential, which in the absence of the field is symmetric with respect to the transformation $x\rightarrow-x$, it depends quadratically on the electric intensity [see Eq.~\eqref{DeltaPotentialAsymptotics1}]. Note also a different pre-exponential factor in the expression for the half width in comparison to the attractive $\delta$-potential, Eq.~\eqref{DeltaHalfWidthAsymptotics1}:
\begin{equation}\label{RobinHalfWidthAsymptotics1}
{\Gamma_{res}^{R-}}_0=4\exp\!\left(\!-\frac{4}{3}\frac{1}{\mathscr{E}}\right),\quad\mathscr{E}\ll1.
\end{equation}
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{RobinComplexEnergyFunction0.eps}
\caption{\label{RobinComplexEnergyFunction0}
As Fig.~\ref{DeltaPotentialComplexEnergyFunction0} but for the Robin wall with negative extrapolation length.
}
\end{figure}
Fig.~\ref{RobinComplexEnergyFunction0} shows the evolution with the field of the resonance energy ${E_{res}^{R-}}_0$ and the associated waveform $\Psi_0(x)$. It can be seen that, contrary to the $\delta$-potential, the real part of the energy of the negative Robin wall in the whole range of the electric intensity is a decreasing function of $\mathscr{E}$: for low voltages, this decline is linear while for strong fields it asymptotically transforms to $a_1'\mathscr{E}^{2/3}/2$. The exponentially small increase of the half width at small fields is converted into the $3^{1/2}a_1'\mathscr{E}^{2/3}/2$ dependence at high $\mathscr{E}$. Of course, the 'exponential catastrophe' is an essential feature of the complex wave function of the Robin wall as well, as panel (b) of Fig.~\ref{RobinComplexEnergyFunction0} exemplifies. Its physical interpretation is the same as for the $\delta$-potential.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{RobinEnergiesFig1.eps}
\caption{\label{RobinEnergiesFig1}
Energy spectrum $E$ of the negative Robin wall as a function of the applied field $\mathscr{E}$ where the solid lines show REQB states that are solutions of Eq.~\eqref{RobinResonanceB1}, dotted curves depict energies satisfying Eq.~\eqref{RobinResonanceA1}, dashed lines are the Gamow-Siegert solutions from Eq.~\eqref{RobinEigenEquation1_R}, and the dash-dotted curve describes the evolution of the location of the maximum of the Wigner delay time $\tau_W^{R-}$, Eq.~\eqref{WignerRobin1_R}. The dash-dot-dotted line represents a negative imaginary component of the complex solution of Eq.~\eqref{RobinResonanceB1} at $\mathscr{E}\geq\mathscr{E}^{\times R-}$. The inset compares the complex-energy method and the location of the Wigner time maximum at the strong fields.
}
\end{figure}
To not deal with the divergences, the REQB states and delay time resonances have to be considered. Analysis of Eq.~\eqref{RobinScatteringMatrix1_R} reveals that for the Robin surface there exist two sets of quasibound levels that at low voltages are approximated by the Breit-Wigner formula. The first of these corresponds to the maximal distortion, $p^{R\pm}\left(\mathscr{E};E_n^{B\pm}\right)=1$, by the de Gennes interface of the wall-free function from Eq.~\eqref{ScatteringFunction0} when, as follows from Eq.~\eqref{RobinScatteringMatrix1_R}, the scattering matrix is a positive unity, $S^{R\pm}\left(\mathscr{E};E_n^{B\pm}\right)=1$, $n^\pm=\left\{\begin{array}{c}
1,2,\ldots\\
0,1,\ldots
\end{array}\right.$, and, accordingly, the total solution from Eq.~\eqref{ScatteringFunction1} degenerates to the Airy ${\rm Bi}$ function:
\begin{equation}\label{RobinFunctionB1}
\Psi_n^B(x)\sim {\rm Bi}\left(-\mathscr{E}^{1/3}x-\frac{E_n^B}{\mathscr{E}^{2/3}}\right).
\end{equation}
The corresponding energies $E_n^{B\pm}$ are real solutions of equation
\begin{equation}\label{RobinResonanceB1}
{\rm Bi}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)\pm\mathscr{E}^{1/3}{\rm Bi}'\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)=0.
\end{equation}
The energies $E_n^{A\pm}$ of the second set of REQB states, which are found from equation
\begin{equation}\label{RobinResonanceA1}
{\rm Ai}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)\pm\mathscr{E}^{1/3}{\rm Ai}'\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)=0,
\end{equation}
guarantee that the Robin wall does not disturb the free particle motion in the uniform electric field to its right: $p^{R\pm}\left(\mathscr{E};E_n^{A\pm}\right)=0$, $n=1,2,\ldots$. Note that Eq.~\eqref{RobinResonanceA1} describes the evolution of the true bound states for the opposite direction of the field \cite{Olendski2}. Corresponding Breit-Wigner half widths are calculated as:
\begin{subequations}\label{RobinHalfWidth1}
\begin{eqnarray}
\Gamma_n^{B\pm}(\mathscr{E})&=&\!-\!2\mathscr{E}\frac{{\rm Ai}\!\left(-E_n^B\!\left/\!\mathscr{E}^{2/3}\right.\!\!\right)\pm\mathscr{E}^{1/3}{\rm Ai}'\!\left(-E_n^B\!\left/\!\mathscr{E}^{2/3}\right.\!\!\right)}{\mathscr{E}^{1/3}{\rm Bi}'\!\left(-E_n^B\!\left/\!\mathscr{E}^{2/3}\right.\!\!\right)\mp E_n^B\,{\rm Bi}\!\left(-E_n^B\!\left/\!\mathscr{E}^{2/3}\right.\!\!\right)},\nonumber\\
\label{RobinHalfWidthB1}
&&n=\left\{\begin{array}{c}
1,2,\ldots\\
0,1,\ldots
\end{array}\right.\\
\Gamma_n^{A\pm}(\mathscr{E})&=&2\mathscr{E}\frac{{\rm Bi}\!\left(-E_n^A\!\left/\!\mathscr{E}^{2/3}\right.\!\!\right)\pm\mathscr{E}^{1/3}{\rm Bi}'\!\left(-E_n^A\!\left/\!\mathscr{E}^{2/3}\right.\!\!\right)}{\mathscr{E}^{1/3}{\rm Ai}'\!\left(-E_n^A\!\left/\!\mathscr{E}^{2/3}\right.\!\!\right)\mp E_n^A\,{\rm Ai}\!\left(-E_n^A\!\left/\!\mathscr{E}^{2/3}\right.\!\!\right)},\nonumber\\
\label{RobinHalfWidthA1}
&&n=1,2,\ldots.
\end{eqnarray}
\end{subequations}
In the limiting cases one has:
for the weak intensities, $\mathscr{E}\ll1$:
\begin{subequations}\label{AsymptotSmallFieldsScattering1}
\begin{align}\label{AsymptotSmallFieldsScatteringBE0}
E_0^{B-}&=-1-\frac{1}{2}\,\mathscr{E}-\frac{1}{8}\,\mathscr{E}^2
\intertext{(this equation under the transformation $\mathscr{E}\rightarrow-\mathscr{E}$ describes the lowest true bound state \cite{Olendski2})}
\label{AsymptotSmallFieldsScatteringBEn}
E_n^{B\pm}=&-b_n\mathscr{E}^{2/3}\pm\mathscr{E},\quad n=1,2,\ldots\\
\label{AsymptotSmallFieldsScatteringBG0}
\Gamma_0^{B-}&=4\exp\!\left(\!-\frac{4}{3}\frac{1}{\mathscr{E}}\right)\\
\label{AsymptotSmallFieldsScatteringBGn}
\Gamma_n^{B\pm}&=-2\frac{{\rm Ai}(b_n)}{{\rm Bi}'(b_n)}\mathscr{E}^{2/3}\\
\label{AsymptotSmallFieldsScatteringAEn}
E_n^{A\pm}&=-a_n\mathscr{E}^{2/3}\pm\mathscr{E}
\intertext{(comparing again with the reversed field configuration \cite{Olendski2}, it is important to stress that the only negative solution of Eq.~\eqref{RobinResonanceA1} is dropped for the present geometry, since it does not correspond to the free particle motion in the tilted potential)}
\label{AsymptotSmallFieldsScatteringAGn}
\Gamma_n^{A\pm}&=2\frac{{\rm Bi}(a_n)}{{\rm Ai}'(a_n)}\mathscr{E}^{2/3};
\end{align}
\end{subequations}
for the high voltages, $\mathscr{E}\gg1$:
\begin{subequations}\label{AsymptotLargeFieldsScattering1}
\begin{eqnarray}
\label{AsymptotLargeFieldsScatteringBEn}
E_n^{B\pm}&=&-b_n'\left(1\mp\frac{1}{b_n'^{\,\,2}\mathscr{E}^{1/3}}\right)\mathscr{E}^{2/3}\\
\label{AsymptotLargeFieldsScatteringBGn}
\Gamma_n^{B\pm}&=&-\frac{2}{b_n'}\frac{{\rm Ai}'(b_n')}{{\rm Bi}(b_n')}\left(1\pm\frac{1}{b_n'^{\,\,2}\mathscr{E}^{1/3}}\right)\mathscr{E}^{2/3}\\
\label{AsymptotLargeFieldsScatteringAEn}
E_n^{A\pm}&=&-a_n'\left(1\mp\frac{1}{a_n'^{\,\,2}\mathscr{E}^{1/3}}\right)\mathscr{E}^{2/3}\\
\label{AsymptotLargeFieldsScatteringAGn}
\Gamma_n^A&=&\frac{2}{a_n'}\frac{{\rm Bi}'(a_n')}{{\rm Ai}(a_n')}\left(1\pm\frac{1}{a_n'^{\,\,2}\mathscr{E}^{1/3}}\right)\mathscr{E}^{2/3}.
\end{eqnarray}
\end{subequations}
In the first limit, the outcomes of the two methods for the zero-field ground state are the same, as a comparison of Eqs.~\eqref{RobinEigenEnergiesAsymptot2} and \eqref{RobinHalfWidthAsymptotics1} with Eqs.~\eqref{AsymptotSmallFieldsScatteringBE0} and \eqref{AsymptotSmallFieldsScatteringBG0}, respectively, shows. The Gamow-Siegert states with the negative real components of their energies, which are described by Eqs.~\eqref{RobinEigenEnergiesAsymptot1}, do not have their quasibound counterparts, as was also the case for the $\delta$-potential. Next, we see once again that the asymptotes of the weak, Eqs.~\eqref{AsymptotSmallFieldsScatteringBEn} and \eqref{AsymptotSmallFieldsScatteringAEn} [strong, Eqs.~\eqref{AsymptotLargeFieldsScatteringBEn} and \eqref{AsymptotLargeFieldsScatteringAEn}], fields simplify the BC to the Dirichlet (Neumann) requirement with the small admixtures due to the tiny interaction between the electric intensity and the non-zero (finite) de Gennes distance. Moreover, all $A$ and non-zero $B$ states are characterized by negative half widths. This is seen from their expressions for large $n$:
\begin{subequations}\label{RobinAsymptotLargeN1}
\begin{eqnarray}\label{RobinAsymptotLargeN1_SmallField}
\Gamma_n^{\left\{_B^A\right\}}&=&-2\left[\frac{3\pi}{8}\left(4n\!-\!\left\{
\begin{array}{c}
1\\3
\end{array}
\right\}\right)\right]^{-1/3}\!\!\!\!\mathscr{E}^{2/3},\,\mathscr{E}\ll1,\,n\gg1\\
\label{RobinAsymptotLargeN1_LargeField}
\Gamma_n^{\left\{_B^A\right\}}&=&-2\left[\frac{3\pi}{8}\left(4n\!-\!\left\{
\begin{array}{c}
3\\1
\end{array}
\right\}\right)\right]^{-1/3}\!\!\!\!\mathscr{E}^{2/3},\,\mathscr{E}\gg1,\,n\gg1.
\end{eqnarray}
\end{subequations}
Negative half widths mean that for the Robin wall in the repulsive tilted potential, all field-induced REQB states are hole-like. It is easy to understand the absence of the electron-like excitations for the present geometry; namely, for the $\delta$-potential they are formed and modified in the area that lies on the opposite (when compared to the holes) side of the $x$ axis, but for the structure under consideration, the motion there is forbidden by the impenetrable Robin wall; accordingly, electron-like quasibound states have no room to be created and developed. This also means that with the hole being varied by the field, quasibound states have no partners with whom they can interact and therefore, in the framework of this model, they survive any electric intensity. The only exception is the lowest field-induced $B$ level, which interacts and ultimately collides with the state developed from the zero-field level. All these features are seen in Fig.~\ref{RobinEnergiesFig1}, which shows energies of the negative Robin wall as functions of the applied voltage. The zero-field level decreases its energy to the wide minimum of $E_{{B_0}_{min}}^{R-}=-1.678$ that is achieved at $\mathscr{E}=2.275$, after which it moves upwards towards the $B_1$ level, which has a broad maximum of ${E_{{B_1}_{max}}^{R-}}=0.314$ at $\mathscr{E}=0.825$ and crosses zero at $\mathscr{E}_{E=0}^{R-}=\mathscr{E}_f$; this is easily derived from Eq.~\eqref{RobinResonanceB1}. At the merger, and in addition to this equation, the derivative with respect to the energy of the left-hand side should change to zero, which means that the energy at the amalgamation is $E^{R_-^\times}=-1$ while the corresponding voltage $\mathscr{E}^{R_-^\times}=3.94827\ldots$ is found numerically from the following equation
\begin{equation}\label{RobinMergerField1}
{\rm Bi}\!\left(\mathscr{E}^{-2/3}\right)-\mathscr{E}^{1/3}{\rm Bi}'\!\left(\mathscr{E}^{-2/3}\right)=0.
\end{equation}
Thus, even though the lowest field-induced quasibound state of the de Gennes wall turns to zero at the same electric intensity as its $\delta$-well counterpart, the coalescence of the levels takes place at higher voltages, which can be interpreted as the stronger binding of the particles by the negative Robin surface. Close to amalgamation, the energies of the merging levels are:
\begin{equation}\label{AsymptotRobinCriticalField1}
E_{\left\{_{B_0}^{B_1}\right\}}=-1\pm\left[\frac{2}{3}\left(\mathscr{E}^{R_-^\times}-\mathscr{E}\right)\right]^{1/2},\quad\mathscr{E}\rightarrow\mathscr{E}^{R_-^\times}.
\end{equation}
Beyond the coalescence, the system again exhibits two complex conjugate solutions whose evolution with the voltage is also shown in Fig.~\ref{RobinEnergiesFig1}. Their existence is another implication of the electron-hole coupling and interaction at the strong fields. The energies of all other higher-lying quasibound states monotonically increase in the whole range of the voltage change from the Dirichlet-like dependence at the small $\mathscr{E}$, Eqs.~\eqref{AsymptotSmallFieldsScatteringBEn} and \eqref{AsymptotSmallFieldsScatteringAEn}, to the almost Neumann BC, Eqs.~\eqref{AsymptotLargeFieldsScatteringBEn} and \eqref{AsymptotLargeFieldsScatteringAEn}, at large electric intensities. As mentioned above, these positive energy levels, contrary to the $\delta$-well, do not have their Gamow counterparts. Several negative real parts of the field-induced complex energies are also plotted in the figure. Qualitatively, their behavior is similar to the $\delta$-geometry. In particular, all of them cross the level evolved from the zero-field bound state whose energy is a monotonically decreasing function of the field; namely, at low voltages it is a negative linear function of the electric intensity according to Eq.~\eqref{RobinEigenEnergiesAsymptot2} while at strong $\mathscr{E}$ it obeys the dependence $a_1'\mathscr{E}^{2/3}/2$ from Eq.~\eqref{RobinEigenEnergiesAsymptot1LargeFields} with $n=1$.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{RobinWignerTime.eps}
\caption{\label{RobinWignerTime}
Wigner delay time $\tau_W^{R-}$ of the negative Robin wall as a function of the energy $E$ for several electric fields where solid line is for $\mathscr{E}=0.5$, dotted line is for $\mathscr{E}=2/3$, dashed curve -- for $\mathscr{E}=1$, dash-dotted line -- for $\mathscr{E}=2$, and dash-dot-dotted one -- for $\mathscr{E}=4$. All curves intersect at $(-1,0)$, as follows from Eq.~\eqref{WignerRobin1_R}.
}
\end{figure}
For the delay time, explicit evaluation yields:
\begin{subequations}\label{WignerRobin1}
\begin{eqnarray}
&&\tau_W^{R\pm}(\mathscr{E};E)=-\frac{2}{\pi\mathscr{E}^{2/3}}\times\nonumber\\
\label{WignerRobin1_R}
&&\frac{E+1}{{\rm Ai}_0^2\!+\!\!{\rm Bi}_0^2\pm2\mathscr{E}^{1/3}\!\left({\rm Ai}_0{\rm Ai}_0'\!+\!\!{\rm Bi}_0{\rm Bi}_0'\right)\!+\!\!\mathscr{E}^{2/3}\!\left({{\rm Ai}_0'}^2\!\!+{{\rm Bi}_0'}^2\!\right)}\\
\label{WignerRobin1_D}
&&\tau_W^D(\mathscr{E};E)=-\frac{2}{\pi\mathscr{E}^{2/3}}\frac{1}{{\rm Ai}_0^2+{\rm Bi}_0^2}\\
\label{WignerRobin1_N}
&&\tau_W^N(\mathscr{E};E)=-\frac{2}{\pi\mathscr{E}^{4/3}}\frac{E}{{{\rm Ai}_0'}^2+{{\rm Bi}_0'}^2},
\end{eqnarray}
\end{subequations}
where the same convention regarding the subscript '$0$' in Eq.~\eqref{DeltaTimeDelay1} is used. It can be seen that the Dirichlet delay time is always negative, which means that the corresponding wall cannot capture the particle for a prolonged period but rather is in a hurry at any $E$ to reflect it back, whereas its Neumann counterpart bears the opposite sign to that of the energy. The delay time decays exponentially for large negative energies
\begin{subequations}\label{WignerRobinAsymptot1}
\begin{align}
\tau_W^{R\pm}(\mathscr{E};E)&=\tau_W^N(\mathscr{E};E)=-\tau_W^D(\mathscr{E};E)\nonumber\\
\label{WignerRobinAsymptot1_Negative}
&=2\frac{|E|^{1/2}}{\mathscr{E}}\exp\!\left(-\frac{4}{3}\frac{|E|^{3/2}}{\mathscr{E}}\right),\quad E\ll-1,\\
\intertext{while in the opposite case it is transformed into the BC-independent smoothly varying negative quantity:}
\tau_W^{R\pm}(\mathscr{E};E)&=\tau_W^N(\mathscr{E};E)=\tau_W^D(\mathscr{E};E)\nonumber\\
\label{WignerRobinAsymptot1_Positive}
&=-2\frac{E^{1/2}}{\mathscr{E}},\quad E\gg1.
\end{align}
\end{subequations}
Observe that the leading term of the first limit coincides with the one from the corresponding expression for the $\delta$-potential, Eq.~\eqref{DeltaTimeDelayAsymptotics1_NegativeEnergy}. However, for large positive energies the dependencies of the two geometries are very different, since the Robin wall forbids motion in the area $x<0$ where the $\delta$-well resonances described by Eqs.~\eqref{AsymptotDeltaStrongField1}, \eqref{DeltaTimeDelayAsymptotics3} and \eqref{DeltaResonanceHalfWidth1} are formed. Note that the absolute value of the right-hand side of Eq.~\eqref{WignerRobinAsymptot1_Positive} is just the time needed for the classical particle to travel from the Robin surface to the wall-free quasi-classical turning point $x_{qc}=-E/\mathscr{E}$ and return at $x=0$.
Fig.~\ref{RobinWignerTime} depicts the Wigner delay time as a function of energy for several voltages. Field-dependent asymptotes from Eqs.~\eqref{WignerRobinAsymptot1} are clearly seen. Another feature that follows from Eq.~\eqref{WignerRobin1_R} and is shown in the plot is the fact that $\tau_W^{R-}$ at arbitrary electric intensities vanishes if the energy is a negative unity, $E=-1$. As the figure depicts, the asymmetric structure of the attractive Robin surface is characterized by one resonance only, which stems from the zero-field bound state: at very small $\mathscr{E}$ it is located at the energies from Eq.~\eqref{AsymptotSmallFieldsScatteringBE0} with the narrow FWHM coinciding with the half width from Eq.~\eqref{AsymptotSmallFieldsScatteringBG0} while its peak value in this regime is
\begin{equation}\label{WignerLimitMax1}
{\tau_W^{R-}}_{\!\!max}(\mathscr{E})=\exp\left(\frac{4}{3}\frac{1}{\mathscr{E}}\right),\quad\mathscr{E}\rightarrow0.
\end{equation}
The growing field decreases the maximum and widens the FWHM of the resonance. A change in its location with the applied voltage is shown by the dash-dotted line in Fig.~\ref{RobinEnergiesFig1}. At small electric intensities, $\mathscr{E}\ll1$, the three energies describing the transformation of the zero-field bound state by the three different methods are practically equal to each other. At $\mathscr{E}\gtrsim0.7$ the quasibound state starts to deviate upwards from the other two energies, which, contrary to the symmetric $\delta$-structure, monotonically decrease across the whole range of the increasing voltage. Another difference between the two geometries is the fact that for the Robin interface the Gamow energy lies above its Wigner counterpart.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{RobinPolarizationFig1.eps}
\caption{\label{RobinPolarizationFig1}
Dipole moments $P^{R-}$ of the quasibound states for the negative Robin wall as a function of the electric field $\mathscr{E}$.
The same line convention as in Fig.~\ref{DeltaPolarizationFig1} is used. Inset shows the enlarged view at low voltages.}
\end{figure}
As the considered structure is asymmetric, the zero-field mean coordinate $x$ needs to be evaluated to correctly calculate the polarization. For the ground state, an elementary computation where the integrand contains the square of the function $\Psi$ from Eq.~\eqref{FunctionRobinZeroFields1} produces $\langle x\rangle_{\mathscr{E}=0}=1/2$. Then, at low voltages, this term is exactly compensated by the linear contribution to the energy, as an application of the Hellmann-Feynman theorem, Eq.~\eqref{Polarization3}, to Eq.~\eqref{AsymptotSmallFieldsScatteringBE0} shows:
\begin{equation}\label{RobinPolarizationSmallFields1}
P_0^{R-}(\mathscr{E})=\frac{1}{4}\mathscr{E},\quad\mathscr{E}\ll1.
\end{equation}
Note the different slope of this polarization compared to the $\delta$-well mean coordinate, Eq.~\eqref{DeltaPolarization3}. Eq.~\eqref{RobinPolarizationSmallFields1} remains the same for the opposite direction of the field \cite{Olendski2}. In turn, for the excited REQB states, a calculation of the zero-field $\langle x\rangle$ has to be performed at $E=0$ since the energies of all these levels approach zero in the limit of the vanishing electric intensities; see Eqs.~\eqref{AsymptotSmallFieldsScatteringBEn} and \eqref{AsymptotSmallFieldsScatteringAEn} and Fig.~\ref{RobinEnergiesFig1}. Then, the solution of the corresponding Schr\"{o}dinger equation, Eq.~\eqref{Schrodinger3}, with $\mathscr{E}=E=0$ that satisfies the BC from Eq.~\eqref{BCRobin2}, reads:
\begin{equation}\label{FakeSolution1}
\Psi_{n\geq1}^{R\pm}(x)\sim x\pm1\quad {\rm at}\quad\mathscr{E}=E=0.
\end{equation}
This waveform has to be discarded since it diverges at infinity. The only remaining trivial solution $\Psi=0$ means that $\langle x\rangle_{\mathscr{E}=0}=0$ for $n\geq1$. Accordingly, the expression for the dipole moment reads for, e.g., the $B$ levels:
\begin{equation}\label{RobinPolarization1}
P_{B_n}^{R-}=\frac{1}{3}\frac{2\frac{E^2}{\mathscr{E}}{\rm Bi}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)\!+\!\mathscr{E}^{1/3}\left(2\frac{E}{\mathscr{E}}-\!\!1\right){\rm Bi}'\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)}{E\,{\rm Bi}\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)+\mathscr{E}^{1/3}{\rm Bi}'\!\left(-\frac{E}{\mathscr{E}^{2/3}}\right)},\,n\geq1.
\end{equation}
The limiting cases, in addition to Eq.~\eqref{RobinPolarizationSmallFields1}, are:
\begin{subequations}\label{RobinPolarizationAsymptote1}
\begin{eqnarray}\label{RobinPolarizationAsymptote1_ABsmall1}
P_{\left\{_{B_n}^{A_n}\right\}}^{R-}&=&-\frac{2}{3}\left\{\!\!\begin{array}{c}
a_n\\b_n
\end{array}\!\!\right\}\frac{1}{\mathscr{E}^{1/3}},\quad n\geq1,\,\mathscr{E}\ll1\\
\label{RobinPolarizationAsymptote1_ABlarge1}
P_{\left\{_{B_n}^{A_n}\right\}}^{R-}&=&-\frac{2}{3}\left\{\!\!\begin{array}{c}
a_n'\\b_n'
\end{array}\!\!\right\}\frac{1}{\mathscr{E}^{1/3}},\quad\left\{\!\begin{array}{c}
n\geq1\\n\geq2
\end{array}\!\right\},\,\mathscr{E}\gg1\\
\label{RobinPolarizationAsymptote1_Bcoalesc1}
P_{B_n}^{R-}&=&\!\!-\!\!\left[6\left(\mathscr{E}^{R_-^\times}\!\!-\!\!\mathscr{E}\right)\right]^{-1/2}\!\!\!-\frac{1}{2}\delta_{n0},\,n=0,1,\,\mathscr{E}\!\!\rightarrow\!\!\mathscr{E}^{R_-^\times},
\end{eqnarray}
\end{subequations}
where $\delta_{nm}$ is the Kronnecker $\delta$.
Fig.~\ref{RobinPolarizationFig1} shows several polarizations for the attractive surface in the electric field. The qualitative behavior of the dipole moments of the two lowest states is similar to their counterparts considered in Sec.~\ref{Sec_Delta}, with the quantitative differences, one of which was discussed in the previous paragraph, being due to the inability of the particle to penetrate into the area to the left of the wall; namely, the initial linear growth of the ground-state polarization with the field is followed by its maximum of ${P_0^{R-}}_{\!\!\!\!\!max}=0.1087$, which is achieved at $\mathscr{E}=0.281$. As a result of a subsequent decrease, the dipole moment returns to zero at $\mathscr{E}_{P_0=0}^{R-}=0.567$ compared to $\mathscr{E}_{P_0=0}^{\delta-}=0.739$ for the $\delta$-well. The turnaround behavior of the ground state polarization is caused by its interaction with the neighboring hole-like level, with its dipole moment decreasing from the infinitely large values at intensities $\mathscr{E}$ close to zero, which correspond to its location far to the right, to the large negative magnitudes when it approaches the coalescence field $\mathscr{E}^{R_-^\times}$. At the amalgamation, the two waveforms coincide. Their evolution with the field is shown in the two lowest panels of Fig.~\ref{RobinFunctionsFig1}. Higher-lying hole-like states, contrary to the $B_1$ level, do not have counterparts with whom they collide at increasing voltage; accordingly, their polarizations monotonically decrease to zero as the electric intensity grows and this corresponds to the squeezing of the wave functions closer to the wall. This shift to the left is depicted in the upper plot of Fig.~\ref{RobinFunctionsFig1}.
\begin{figure}
\centering
\includegraphics[width=0.7\columnwidth]{RobinFunctionsFig1.eps}
\caption{\label{RobinFunctionsFig1}
Wave functions $\Psi(\mathscr{E};x)$ of the first three lowest states of the attractive Robin wall, normalized to the maximum of their absolute values in terms of the coordinate $x$ and normalized electric field $\nu\equiv\mathscr{E}/\mathscr{E}^{R_-^\times}$. Note the different ranges of the $\nu$ axis of the upper panel compared to those for $B_0$ and $B_1$. The lower limit of the vertical axis for the $B_0$ state also differs from its counterparts for $B_1$ and $A_1$ levels.}
\end{figure}
\section{Concluding Remarks}\label{Conclusions}
To correctly describe a physical phenomenon, scientists need to use a proper mathematical model. Here, to overcome a definite discord in the existing literature, an attempt was made to identify and analyze three different processes taking place in 1D quantum nanostructures in the uniform electric field. Mathematically, they are described by the distinct properties of the corresponding scattering matrix; namely, its poles, real values, and zeros of the second derivative of its phase. Using the examples of the 1D $\delta$-potential and Robin wall, it was demonstrated that the zero-resolvent method produces mathematically correct solutions in the form of the {\em complex} eigen energies $E$ and eigen functions $\Psi(x)$ that, at $\mathscr{E}\neq0$, show a divergence at large distances [see panels (b) of Figs.~\ref{DeltaPotentialComplexEnergyFunction0} and \ref{RobinComplexEnergyFunction0}]. This is referred to as the 'exponential catastrophe' \cite{Bohm1}, which can, however, be cured by the time-dependent interpretation. The non-zero current density for this model denotes a leakage of the electron away from the QW. For the real energies, the total net current is zero, and conditions of the formation of the quasibound states are formulated as a requirement of the largest distortion by the potential of the free particle motion in the electric field. Analysis of this model reveals that the weak field splits the positive-energy continuum into electron- and hole-like quasibound levels that, with increasing voltage, strongly interact between themselves and ultimately collide with each other at the breakdown voltage. At even stronger electric intensities, the corresponding equation possesses two complex-conjugate solutions that may correspond to the formation of the composite exciton-like structure, where the motion of one part is correlated with a second constituent. Amalgamation of the levels is accompanied by the divergence of the associated dipole moments. The total number of each kind of levels is specified by the zero-field geometry; for example, the $\delta$-potential, which is symmetric with respect to the inversion $x\rightarrow-x$, has a countably infinite number of both electron and hole excitations while the asymmetric Robin wall, which forbids motion to its left, is characterized by the only electron state developed from the zero-field bound level, in addition to the infinite set of hole orbitals.
Another set of resonances is associated with the maxima of the Wigner delay time $\tau_W$, which is a derivative of the phase $\varphi_S$ of the scattering matrix with respect to energy. The number of these extrema is again determined by the zero-field symmetry of the structure and is infinite for the $\delta$-potential and just one for the asymmetric attractive Robin wall. It was shown that at low voltages, the results of the calculation of the resonances and quasibound states developed from those of the field-free geometry are identical, but they diverge from each other for stronger electric intensities. Other similarities and differences between the three models were also discussed.
Having theoretically seen the peculiarities of the electric field effect, it is natural to wonder whether they could be experimentally verified. First, let us consider the atomic system; namely, the hydrogen ion ${\rm H}^-$ near the-one electron threshold \cite{Stewart1}. Of course, modeling a 3D structure with a 1D attractive potential is a rather crude approximation \cite{Emmanouilidou2}, but we are only interested in estimating the orders of magnitude. The reported binding energy $E=-0.7542$ eV \cite{Stewart1} corresponds to the very deep $\delta$ well with length $\Lambda=-2.2\times10^{-15}$ m [see Eq.~\eqref{EnergyDeltaZeroFields1}]. Accordingly, converting the dimensionless value of the fundamental dissociation field from Eq.~\eqref{FundamentalField1} into regular, normalized units, static electric intensities $\sim10^{16}$ V/m ($\sim10^5$ a.u.) that far exceed the experimentally available voltages of $\sim10^8$ V/m ($\sim10^{-3}$ a.u.) are gained \cite{Stewart1}. Thus, the behavior near the breakdown point $\mathscr{E}_f$ is hardly verifiable for atomic systems. The situation, however, changes dramatically for man-made semiconductor structures. Impressive advances in nanotechnology over the last two decades have allowed the development of low-dimensional artificial patterns of almost any desirable shape. From this point of view, it should be noted that early conjectures \cite{Pazma1,Fulop1} regarding the possibility of mimicking the Robin wall with $\Lambda<0$ by the limit of finite regularized potentials were corroborated by consideration of the following term $V(x)$ in Eq.~\eqref{Hamiltonian1} at $\mathscr{E}=0$:
\begin{equation}\label{Potential2}
V(x)=\left\{\begin{array}{cc}
\infty,&x<0\\
-V_0,&0<x<d\\
0,&d<x<\infty
\end{array}\right.
\end{equation}
with positive $V_0$ and $d$ \cite{Olendski6}. It was shown that at
\begin{equation}\label{Potential4}
\frac{1}{8}\frac{\pi^2\hbar^2}{md^2}<V_0<\frac{9}{8}\frac{\pi^2\hbar^2}{md^2}
\end{equation}
only one negative-energy state does exist (which is a necessary characteristic of the negative de Gennes surface) with its depth being controlled by the interrelation between $V_0$ and $d$. Accordingly, by applying to it appropriately directed voltage, it is possible to reach the dissociation field at any desired intensity.
Idealized models of the $\delta$-potential and Robin wall allow us to gain quite simple analytical results that facilitate their physical interpretation. A more realistic approach should take into account the finite width $d$ and depth $V_0$ of the structures resulting, in particular, in several zero-field bound states, a number of which depends on the ratio $V_0d^2$. Contradictory \cite{Austin4} complex-Airy-function \cite{Ahn1,Emmanouilidou1} and real energy \cite{Austin2} calculations have been carried out for the symmetric finite QW. Another possibility to get at least two bound states for the voltage-free configuration is to use more than one $\delta$-potential. Analysis of the location of the poles in the $E$ plane revealed the dependencies of the {\em complex} energy resonances on the field and the separation between the two extremely localized wells \cite{Korsch1,Alvarez2}. For the same system, even a superficial analysis based on the {\em real} Airy functions revealed a very complicated structure for the positive energy part of the spectrum \cite{Glasser1,Carpena1}. Relevant to our research, we mention here in passing that at $\mathscr{E}=0$, the Robin BCs are equivalent to a pair of 1D Dirac $\delta(x)$-$\delta'(x)$ interactions for a couple of critical values of the $\delta'$ coupling \cite{Kurasov1,Gadella1}. In light of the results presented above, it does make sense to readdress the problem of the two extremely localized QWs or barriers in the electric field in the same way as was carried out above for the single $\delta$-potential.
\section{Acknowledgement}\label{sec_6}
Constructive comments of anonymous referees are gratefully acknowledged.
|
{
"redpajama_set_name": "RedPajamaArXiv"
}
| 9,226
|
#include "choosebuildingtiledialog.h"
#include "ui_choosebuildingtiledialog.h"
#include "buildingeditorwindow.h"
#include "buildingpreferences.h"
#include "buildingtiles.h"
#include "buildingtilesdialog.h"
#include "zoomable.h"
#include <QSettings>
using namespace BuildingEditor;
using namespace Tiled::Internal;
ChooseBuildingTileDialog::ChooseBuildingTileDialog(const QString &prompt,
BuildingTileCategory *category,
BuildingTileEntry *initialTile,
QWidget *parent) :
QDialog(parent),
ui(new Ui::ChooseBuildingTileDialog),
mCategory(category),
mZoomable(new Zoomable(this))
{
ui->setupUi(this);
ui->prompt->setText(prompt);
ui->tableView->setZoomable(mZoomable);
mZoomable->connectToComboBox(ui->scaleCombo);
connect(ui->tilesButton, &QAbstractButton::clicked, this, &ChooseBuildingTileDialog::tilesDialog);
setTilesList(mCategory, initialTile);
connect(ui->tableView, &QAbstractItemView::activated, this, &ChooseBuildingTileDialog::accept);
QSettings &settings = BuildingPreferences::instance()->settings();
settings.beginGroup(QLatin1String("ChooseBuildingTileDialog"));
QByteArray geom = settings.value(QLatin1String("geometry")).toByteArray();
if (!geom.isEmpty())
restoreGeometry(geom);
qreal scale = settings.value(QLatin1String("scale"),
BuildingPreferences::instance()->tileScale()).toReal();
mZoomable->setScale(scale);
settings.endGroup();
}
ChooseBuildingTileDialog::~ChooseBuildingTileDialog()
{
delete ui;
}
BuildingTileEntry *ChooseBuildingTileDialog::selectedTile() const
{
QModelIndexList selection = ui->tableView->selectionModel()->selectedIndexes();
if (selection.count()) {
QModelIndex index = selection.first();
#if 1
return ui->tableView->entry(index);
#elif 0
return static_cast<BuildingTileEntry*>(ui->tableView->model()->userDataAt(index));
#else
Tiled::Tile *tile = ui->tableView->model()->tileAt(index);
return mBuildingTiles.at(mTiles.indexOf(tile));
#endif
}
return 0;
}
void ChooseBuildingTileDialog::setTilesList(BuildingTileCategory *category,
BuildingTileEntry *initialTile)
{
#if 1
QList<BuildingTileEntry*> entries;
if (category->canAssignNone())
entries += BuildingTilesMgr::instance()->noneTileEntry();
QMap<QString,BuildingTileEntry*> entryMap;
int i = 0;
foreach (BuildingTileEntry *entry, category->entries()) {
QString key = entry->displayTile()->name() + QString::number(i++);
entryMap[key] = entry;
}
entries += entryMap.values();
ui->tableView->setEntries(entries);
if (entries.contains(initialTile))
ui->tableView->setCurrentIndex(ui->tableView->index(initialTile));
#else
Tiled::Tile *tile = 0;
mTiles.clear();
mBuildingTiles.clear();
QList<void*> userData;
if (category->canAssignNone()) {
mTiles += BuildingTilesMgr::instance()->noneTiledTile();
mBuildingTiles += BuildingTilesMgr::instance()->noneTileEntry();
userData += BuildingTilesMgr::instance()->noneTileEntry();
if (initialTile == mBuildingTiles[0])
tile = mTiles[0];
}
MixedTilesetView *v = ui->tableView;
QMap<QString,BuildingTileEntry*> entryMap;
int i = 0;
foreach (BuildingTileEntry *entry, category->entries()) {
QString key = entry->displayTile()->name() + QString::number(i++);
entryMap[key] = entry;
}
foreach (BuildingTileEntry *entry, entryMap.values()) {
mTiles += BuildingTilesMgr::instance()->tileFor(entry->displayTile());
userData += entry;
mBuildingTiles += entry;
if (entry == initialTile)
tile = mTiles.last();
}
v->setTiles(mTiles, userData);
if (tile != 0)
v->setCurrentIndex(v->model()->index(tile));
else
v->setCurrentIndex(v->model()->index(0, 0));
#endif
}
void ChooseBuildingTileDialog::saveSettings()
{
QSettings &settings = BuildingPreferences::instance()->settings();
settings.beginGroup(QLatin1String("ChooseBuildingTileDialog"));
settings.setValue(QLatin1String("geometry"), saveGeometry());
settings.setValue(QLatin1String("scale"), mZoomable->scale());
settings.endGroup();
}
void ChooseBuildingTileDialog::tilesDialog()
{
BuildingTilesDialog *dialog = BuildingTilesDialog::instance();
dialog->selectCategory(mCategory);
QWidget *saveParent = dialog->parentWidget();
dialog->reparent(this);
dialog->exec();
dialog->reparent(saveParent);
if (dialog->changes()) {
setTilesList(mCategory);
}
}
void ChooseBuildingTileDialog::accept()
{
saveSettings();
QDialog::accept();
}
void ChooseBuildingTileDialog::reject()
{
saveSettings();
QDialog::reject();
}
|
{
"redpajama_set_name": "RedPajamaGithub"
}
| 5,790
|
Events • Regions • Wanaka
Cutting Edge Adventure
Written by Rebecca Williamson
For three decades, world-renowned Wanaka mountaineer Guy Cotter has been scaling the globe's most challenging peaks, from the backyard playgrounds of New Zealand's Mount Aspiring and Mount Cook to the ultimate conquest of Mount Everest.
His innate passion spearheaded the Wanaka mountain guiding company Adventure Consultants and for more than 20 years the company has specialised in providing extraordinary experiences for keen skiers, snowboarders and explorers offering mountain expeditions, wilderness adventures and treks in the most remote corners of the world. Guy, his wife Suze Kelly and their team of mountaineers and ski guides, offer 72 different experiences in far flung places including the Arctic, Antarctica, the Himalayas, South America and the Seven Summits. Multi-day backcountry skiing tours – in which advanced skiers or boarders glide through remote areas – are particularly popular. Guy says the tours are a remarkable way to explore New Zealand's backcountry, as well as Adventure Consultants' international destinations.
"Getting into our backcountry to ride is a real privilege because you get to enjoy absolutely superb terrain and most of the time there is no-one else around. Unlike the ski resort experience where you'd be lucky to get in two powder runs before it's all trashed, you can just session the best parts of a run or do complete top-to-bottom runs at your leisure when you are out ski touring."
The Southern Lakes is Adventure Consultants' main stamping ground. The team knows where the snow is best on any given day and a favourite is the picturesque McKerrow Range, with hundreds of kilometres of free terrain. Other New Zealand tours include a helicopter flight to the head of the Tasman Glacier at Mount Cook or to the Fox and Franz Josef Glaciers. Guy says guides are highly skilled and qualified and they're experts not only in how to enjoy the best snow and terrain but how to do it safely. This mantra is also adopted by the guides who lead their backcountry skiing tours in other parts of the world such as Antarctica, India and the European Alps. "We also operate trips on the famous Haute Route in the French and Swiss Alps as well as week-long, hotel-based ski touring packages. Our most exotic trip is a yacht-based ski-touring expedition to the Antarctic Peninsula, where it's not unusual to ski past penguins and sea lions before heading back to the yacht to watch humpback whales. We also run ski touring packages from the remote village of Gulmarg in India. The scope of where we go ski touring is expanding all the time."
STUNNING SKYLINE STARGAZING
Wild Food Foodie
Architecture • Design • Home • Lifestyle • Wanaka
Building a legacy
Community • Home • Lifestyle • Property • Queenstown • Wanaka
Fitness • Health • Wanaka • Wellbeing
Inspiring Business Performance
Rebecca Williamson
Embracing the Change
|
{
"redpajama_set_name": "RedPajamaCommonCrawl"
}
| 8,828
|
Q: how to make simple PHP base quiz I want to make 15 questions, and any time user visits the page it shows random 5 questions and each question has 4 answers and 1 is the correct. The marks are 20, 15, 10 and 0.
How can i make it?
A: I always find it best to start learning by Googling for tutorials.
Here are a few:
*
*simple php quiz
*select random list items in php
I'm sorry to tell you go Googling, but I think the tutorials and examples you'll find there, will be far more helpful to you than any answer here on StackOverflow.
A: Ok well nobody is going to write the full code for you I think, but as a general schema design, you'd want something like this...
questions table (q_id, q_question)
questionoptions (qo_optionid, qo_questionid, qa_option)
useranswers (ua_userid, ua_questionid, ua_optionid)
To get the choices for a given question (question 1 lets say)
select
*
from
questions
inner join questionoptions on (qo_questionid = q_id)
order by
qo_optionid
To get a report of the options each user chose...
select
*
from
questions
inner join questionoptions on (qo_questionid = q_id)
inner join useranswers on (ua_questionid = q_id and ua_optionid = qo_optionid)
order by
ua_userid, ua_questionid
Note that im not advocating the use of SELECT *, but its there for simplicity of example.
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 4,983
|
Hopefully you already have some sort of tagging or category system set up for your blog posts. I won't get into the subtle differences between tags and categories here, but you should be using one or the other or both to keep your posts organized and be able to easily link to a specific topic.
Does Anyone Care About Proofreading Anymore?
|
{
"redpajama_set_name": "RedPajamaC4"
}
| 7,803
|
Q: Labels for grouped bar plot with uneven length of data? I am working with a plot that contains an uneven length of data. I created another group of females (green bars), and I would like to label these two female groups F1 and F2.
Here is my code:
import matplotlib
import matplotlib.pyplot as plt
import numpy as np
labels = ['G1', 'G2', 'G3', 'G4']
labels2 = ['F1', 'F2']
male = [1, 3, 10, 20]
female = [2, 7]
female_2 = [3, 11]
x_male = np.arange(len(male))
x_female = np.arange(len(female))
offset_male = np.zeros(len(male))
offset_female = np.zeros(len(female))
shorter = min(len(x_male), len(x_female))
width = 0.25 # the width of the bars
offset_male[:shorter] = width/2
offset_female[:shorter] = width/2
fig, ax = plt.subplots()
rects1 = ax.bar(x_male - offset_male, male, width, label='male')
rects2 = ax.bar(x_female + offset_female, female, width, label='female')
rects3 = ax.bar(x_female + 3 * offset_female, female_2, width, label='female')
# Add some text for labels, title and custom x-axis tick labels, etc.
ax.set_xticks(x_male)
ax.set_xticklabels(labels)
ax.legend()
fig.tight_layout()
plt.show()
Do you have any idea how I can do it?
A: blend all ticks together
ax.set_xticks(list(x_male)+list(x_female + 3 * offset_female))
ax.set_xticklabels(labels+labels2)
|
{
"redpajama_set_name": "RedPajamaStackExchange"
}
| 510
|
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