add the beginning of the cross section calculation

latex/document.tex

@@ -1,6 +1,6 @@
 \documentclass[fontsize=12pt,paper=a4,open=any,parskip=half,
 twoside=false,toc=listof,toc=bibliography,fleqn,leqno,
-captions=nooneline,captions=tableabove,german]{scrbook}
+captions=nooneline,captions=tableabove,english]{scrbook}
 
 \usepackage{hirostyle}
 \addbibresource{thesis.bib}
@@ -11,6 +11,9 @@ captions=nooneline,captions=tableabove,german]{scrbook}
 \begin{document}
 \input{./tex/title.tex}
 
+\input{./tex/qqgammagamma.tex}
+\input{./tex/qqgammagamma/calculation.tex}
+
 \tableofcontents
 \listoffigures
 \listoftables

latex/hirostyle.sty

@@ -2,15 +2,68 @@
 \usepackage[utf8]{inputenc} % load early
 \usepackage[T1]{fontenc}
 % load early
-\usepackage[ngerman]{babel}
+\usepackage[english]{babel}
 \usepackage[autostyle=true]{csquotes}
-\usepackage{palatino}
+\usepackage{newpxtext,newpxmath}
 \usepackage{physics}
 \usepackage{graphicx, booktabs, float, scrhack}
 \usepackage{amsmath,amssymb}
 \usepackage[automark]{scrlayer-scrpage}
-\usepackage[backend=biber,style=verbose,sortcase=false,
+\usepackage[backend=biber,style=verbose,sortcase=false,language=english]{biblatex}
-language=british]{biblatex}
 \PassOptionsToPackage{hyphens}{url}
+\usepackage{siunitx}
+\usepackage[pdfencoding=auto]{hyperref} % load late
+% \usepackage[activate={true,nocompatibility},final,tracking=true,spacing=true,factor=1100,stretch=10,shrink=10]{microtype}
+\usepackage{tikz-feynman}
+\usepackage{caption}
+\usepackage[list=true, font=small,
+labelformat=brace, position=top]{subcaption}
+\usepackage{tikz}
+\usepackage{wrapfig}
+\usepackage{slashed}
 
-\usepackage[hidelinks]{hyperref} % load late
+%% Tikz
+\usetikzlibrary{arrows,shapes,angles,quotes,arrows.meta}
+
+%% Hyperref
+\hypersetup{
+    colorlinks,
+    linkcolor={blue!50!black},
+    citecolor={red!50!black},
+    urlcolor={green!80!black}
+}
+
+%% Captions
+\captionsetup{justification=centering}
+
+%% Labels
+\labelformat{section}{section #1}
+\labelformat{figure}{figure #1}
+\labelformat{table}{table #1}
+
+% Macros
+
+%% qqgg
+\newcommand{\qqgg}[0]{q\bar{q}\rightarrow\gamma\gamma}
+
+%% Impulses and Polarization Vectors convenience
+\DeclareMathOperator{\ps}{\slashed{p}}
+
+\DeclareMathOperator{\pe}{\varepsilon}
+\DeclareMathOperator{\pes}{\slashed{\pe}}
+
+\DeclareMathOperator{\pse}{\varepsilon^{*}}
+\DeclareMathOperator{\pses}{\slashed{\pe}^{*}}
+
+%% Spinor convenience
+\DeclareMathOperator{\us}{u}
+\DeclareMathOperator{\usb}{\bar{u}}
+
+\DeclareMathOperator{\vs}{v}
+\DeclareMathOperator{\vsb}{\bar{v}}
+
+%% Fast Slash
+\let\sl\slashed
+
+%% Notes on Equations
+\newcommand{\shorteqnote}[1]{ &  & \text{\small\llap{#1}}}

latex/makefile

@@ -1,4 +1,4 @@
-LATEXMKFLAGS=-pdfxe -interaction=nonstopmode
+LATEXMKFLAGS=-pdflua -interaction=nonstopmode
 OUTDIR=build
 
 thesis: document.tex

latex/tex/qqgammagamma.tex

@@ -0,0 +1,55 @@
+\chapter{Quark-Antiquark Annihilation into
+  two Photons}%
+\label{sec:qqgg}
+
+Consider the scattering reaction \(\qqgg\). The first order expansion
+of this process is described by the Feynman diagrams
+in~\ref{fig:qqggfeyn}.  Because there is only QED involved, the color
+degrees of freedom average out and will not be considered henceforth.
+Furthermore a high energy regime will be supposed and therefor masses
+will be neglected.
+
+\begin{figure}[h]
+  \centering
+  \begin{subfigure}[c]{.4\textwidth}
+  \centering
+  \begin{tikzpicture}
+    \begin{feynman}
+    \diagram [small,horizontal=i2 to a] {
+      i2 [particle=\(q\)] -- [fermion, momentum=\(p_2\)] a --
+      [fermion, reversed momentum=\(q\)] b,
+      i1 [particle=\(\bar{q}\)] -- [anti fermion, momentum'=\(p_1\)] b,
+      i2 -- [opacity=0] i1,
+      a -- [photon, momentum=\(p_3\)] f1 [particle=\(\gamma\)],
+      b -- [photon, momentum'=\(p_4\)] f2 [particle=\(\gamma\)],
+      f1 -- [opacity=0] f2,
+    };
+    \end{feynman}
+  \end{tikzpicture}
+  \subcaption{u channel}
+  \end{subfigure}
+  \begin{subfigure}[c]{.4\textwidth}
+  \centering
+  \begin{tikzpicture}
+    \begin{feynman}
+      \diagram [small,horizontal=i2 to a] {
+      i2 [particle=\(q\)] -- [fermion, momentum=\(p_2\)] a --
+      [fermion, reversed momentum'=\(q\)] b,
+      i1 [particle=\(\bar{q}\)] -- [anti fermion, momentum'=\(p_1\)] b,
+      i2 -- [opacity=0] i1,
+      a -- [draw=none] f2 [particle=\(\gamma\)],
+      b -- [draw=none] f1 [particle=\(\gamma\)],
+      f1 -- [opacity=0] f2,
+    };
+    \diagram* {
+      (a) -- [photon] (f1),
+      (b) -- [photon] (f2),
+    };
+    \end{feynman}
+  \end{tikzpicture}
+    \subcaption{\label{fig:qqggfeyn2}t channel}
+  \end{subfigure}
+
+  \caption{First order diagrams for \(\qqgg\).}%
+  \label{fig:qqggfeyn}
+\end{figure}

latex/tex/qqgammagamma/calculation.rip

@@ -0,0 +1,9 @@
+% -*- mode: reftex-index-phrases; TeX-master: "calculation.tex" -*-
+%                              Key      Macro Format            Repeat
+%---------------------------------------------------------------------
+>>>INDEX_MACRO_DEFINITION:      i       \index{%s}              t
+>>>INDEX_MACRO_DEFINITION:      g       \glossary{%s}           t
+%---------------------------------------------------------------------
+
+
+

latex/tex/qqgammagamma/calculation.tex

@@ -0,0 +1,142 @@
+\section{Calulation of the Cross Section to first Order}%
+\label{sec:qqggcalc}
+
+After labeling the incoming quarks and outcoming photons, as well as
+the impulses according to~\ref{fig:qqggfeyn}, the feynman rules yield
+the matrix elements in~\eqref{eq:matel}. The matrix element
+for~\ref{fig:qqggfeyn2} is obtained by simply renaming
+\(3\leftrightarrow 4\).
+
+\begin{align}
+  \label{eq:matel}
+  \mathcal{M}_1 &= \frac{(gQ)^2}{\qty(p_1 - p_4)^2}\vsb(1)\pses(4)(\ps_1 -
+                \ps_4)\pses(3)\us(2)\\
+  \mathcal{M}_2 &= \frac{(gQ)^2}{\qty(p_1 - p_4)^2}\vsb(1)\pses(3)(\ps_1 - \ps_3)\pses(4)\us(2)
+\end{align}
+
+\begin{wrapfigure}{R}{0.4\textwidth}
+\centering
+\begin{tikzpicture}
+  \coordinate (origin) at (0,0);
+
+  \draw[-Latex] (origin) -- (-2,0) node[left] {\(p_3\)};
+  \draw[-Latex] (origin) -- (2,0) coordinate (p4) node[right] {\(p_4\)};
+  \draw[Latex-,rotate=40] (origin) -- (2,0) coordinate (p2) node[right] {\(p_2\)};
+  \draw[Latex-,rotate=40] (origin) -- (-2,0) node[left] {\(p_1\)};
+  \draw[fill=black] (origin) circle (.03);
+
+  \draw pic["$\Theta$", draw=black, <->, angle eccentricity=1.2, angle radius=1cm] {angle=p4--origin--p2};
+\end{tikzpicture}
+\caption{\label{fig:qqimpulses} Momentum diagram for the proces
+  \(\qqgg\) in the massles limit.}
+\end{wrapfigure}
+
+
+To simplify notation, the some shorthands are intruduced
+in~\eqref{eq:scshort}.
+\begin{align}
+  \label{eq:scshort}
+  s(x) &= \sin(x) & c(x) &= \cos(x) \\ s'(x) &= \sin(\frac{x}{2}) & c'(x) &= \cos(\frac{x}{2})
+\end{align}
+
+All calculations are made in the center of momentum frame unless
+stated otherwise.  The impulses used in the center of momentum frame
+are concretised to in~\eqref{eq:pchoice} as well
+as~\ref{fig:qqimpulses}.  Note that the photons are aligned to the
+z-axis as this led to a simple choice for the polarization vectors,
+when calculating the matrix element directly. Here casimir's trick is
+being used but the labeling was kept.
+
+
+\begin{align}
+  \label{eq:pchoice}
+  p_1&=p\cdot\mqty(1 \\ s \\ 0 \\ c)
+     & p_2&=p\cdot\mqty(1 \\ -s \\ 0 \\ -c)
+     & p_3&=p\cdot\mqty(1 \\ 0 \\ 0 \\ -1)
+     & p_4&=p\cdot\mqty(1 \\ 0 \\ 0 \\ 1)
+\end{align}
+
+Now observe that \((p_1-p_4)^2=-4p^2s'^2\) and
+\((p_1-p_3)^2=-4p^2c'^2\) (Minkowski metric) and define \(\Gamma_1\)
+and \(\Gamma_1\) as in~\eqref{eq:gammadef}.
+
+\begin{align}
+  \label{eq:gammadef}
+  \Gamma_1 &= \pses(4)(\ps_1 - \ps_4)\pses(3) &
+  \Gamma_2 &= \pses(3)(\ps_1 - \ps_3)\pses(4)
+\end{align}
+
+The total matrix element (the minus sign has been dropped) is given in~\eqref{eq:totalm}.
+\begin{equation}
+  \label{eq:totalm}
+  \mathcal{M} = \mathcal{M}_1 + \mathcal{M}_2 = \frac{(gQ)^2}{\qty(2p)^2}\vsb(1)\qty(\frac{\Gamma_1}{s'^2}+\frac{\Gamma_2}{c'^2})\us(2)
+\end{equation}
+
+To obtain an experimentally verifiable cross section the absolute square of the
+matrix element will averaged over incoming helicities and summed over
+all photon polarisations.  Using casimir's trick, the averaging can be
+simplified to the calculation of a trace as in where \(s_i\) are
+helicities, \(\lambda_i\) are the polarisations and \(\bar{\Gamma_i}=\gamma^0\Gamma^\dagger_i\gamma^0\).
+
+\begin{equation}
+  \label{eq:averagedm}
+  \langle\mathcal{M}\rangle = \frac{1}{4}\sum_{s_1 s_2}\sum_{\lambda_1
+    \lambda_2} \abs{\mathcal{M}}^2=\frac{1}{4}\sum_{\lambda_1
+    \lambda_2}\tr[\qty(\frac{\Gamma_1}{s'^2}+\frac{\Gamma_2}{c'^2})
+  \ps_2\qty(\frac{\bar{\Gamma}_1}{s'^2}+\frac{\bar{\Gamma}_2}{c'^2})\ps_1]
+\end{equation}
+
+With the definition \(a_1=4,b_1=3,a_2=3,b_2=4\) the \(\Gamma\)
+matrices and their bared variants can be written as in~\eqref{eq:shortgamma}.
+
+\begin{align}
+  \label{eq:shortgamma}
+  \Gamma_i &= \pses(a_i)(\ps_1 - \ps(a_i))\pses(b_i) & \bar{\Gamma}_i &= \pes(b_i)(\ps_1 - \ps(a_i))\pes(a_i)
+\end{align}
+
+To work out~\eqref{eq:averagedm} one must evaluate terms of the
+form~\eqref{eq:gbricks}.
+
+\begin{align}
+  \label{eq:gbricks}
+  \Gamma_{ij} = \sum_{\lambda_1\lambda_2} \tr(\Gamma_i\ps_2\Gamma_j\ps_1) =
+  \sum_{\lambda_1\lambda_2} \tr[\pses(a_i)(\ps_1-\ps(a_i))\pses(b_i)\ps_2\pes(b_i)(\ps_1 - \ps(a_i))\pes(a_i)\ps_1]
+\end{align}
+
+The sum over plarisation can be simplified by utilizing the
+completeness relation for polarisation vectors for \emph{external}
+photons~\eqref{eq:polcomp}.
+
+\begin{equation}
+  \label{eq:polcomp}
+  \sum_{\lambda=1}^{2}\pe_{(\lambda)}^\mu\pe_{(\lambda)}^{*\nu} = -g^{\mu\nu}
+\end{equation}
+
+For \(i=j\) and by utilizing \(\gamma_\mu\sl{a}\gamma^\mu=-2\sl{a}\),
+ \(\gamma_\mu\sl{a}\sl{b}\sl{c}\gamma^\mu=-2\sl{c}\sl{b}\sl{a}\) as
+well as \(\ps_i\ps_i=p_i\cdot p_i = 0\) and the well known trace
+theorems for the gamma matrices~\eqref{eq:gii} follows.
+\begin{equation}
+  \label{eq:gii}
+  \begin{split}
+\Gamma_{ii} &=
+\tr[\gamma_\mu(\ps_1-\ps(a_i))\gamma_\nu\ps_2\gamma^\nu(\ps_1-\ps(a_j))\gamma^\mu\ps_1)]
+\\
+&= 4\tr[(\ps_1-\ps(a_i))\ps_2(\ps_1-\ps(a_i))\ps_1]\\
+&= 32\qty[(p(a_i)\cdot p_2)(p(a_i)\cdot p_1)]
+\end{split}
+\end{equation}
+
+The same tricks as well as the commutation relation for gamma matrices
+can be utilized for the case \(i\neq j\) and lead to
+
+\begin{equation}
+  \label{eq:gii}
+  \begin{split}
+\Gamma_{ii} &=
+\tr[\gamma_\mu(\ps_1-\ps(a_i))\gamma_\nu\ps_2\gamma^\nu(\ps_1-\ps(a_j))\gamma^\mu\ps_1)]
+\\
+&= 4\tr[(\ps_1-\ps(a_i))\ps_2(\ps_1-\ps(a_i))\ps_1]\\
+&= 32\qty[(p(a_i)\cdot p_2)(p(a_i)\cdot p_1)]
+\end{split}
+\end{equation}

latex/tex/title.tex

@@ -1,2 +1 @@
 \maketitle
-test

notes.org

@@ -2,6 +2,8 @@
 ** Latex
 *** Latex/KOMA Ref-Sheet
  - [[file:literature/prog/LaTeX_RefSheet.pdf][Refsheet]]
+*** KOMA Docs
+ - [[file:literature/prog/scrguide.pdf][KOMA Docs]]
 *** AUCTeX
  - [[file:literature/prog/tex-ref.pdf][Auctex]]
 *** Modular Documents
@@ -26,9 +28,18 @@
     :END:
  - [[file:literature/feynman/Thomson.pdf][Modern Particle Physics]]
  - [[file:literature/feynman/Thomson.pdf::100][Spinors]]
+ - [[file:literature/feynman/Thomson.pdf::107][Spinors, Helicity Eigenstates]]
+ - [[file:literature/feynman/Thomson.pdf::533][Completeness Pol. Vectors]]
 * Aufgaben
-** Erste Aufgaben
+** Erste Aufgabenp
-**** Mail von Siegert
+   :LOGBOOK:
+   CLOCK: [2020-03-20 Fri 09:30]
+   :END:
+*** Mail von Siegert
+     :LOGBOOK:
+     CLOCK: [2020-03-19 Thu 15:21]--[2020-03-19 Thu 17:25] =>  2:04
+     CLOCK: [2020-03-19 Thu 10:05]--[2020-03-19 Thu 11:56] =>  1:51
+     :END:
 Hi Valentin,
 
 alles klar. Das Formular machen wir dann einfach im Nachhinein und
@@ -56,7 +67,13 @@ Dann kannst Du mal qq->yy rechnen.
 Klingt das OK fuer den Start?
 
 Viele Gruesse, Frank
-
+** Berechnung qq -> γγ
+ - 4 Anlaeufe :). Idiotischerweise 4-Vektor negiert
+ - letzter Anlauf mit Casimir Trick erfolgreich
+ - gute tricks:
+   - γ auf z Achse
+   - Symmetrien Beachten -> spart die Haelfte beim umdrehen der Spins
+ - Vollstaendigkeitsrelation von pol. Vektoren in Form: [[file:literature/feynman/Thomson.pdf::533][Completeness Pol. Vectors]]
 * Clock Table
 #+BEGIN: clocktable :scope file :maxlevel 2
 #+CAPTION: Clock summary at [2020-03-18 Wed 21:01]
@@ -74,6 +91,10 @@ Viele Gruesse, Frank
  - Ich stand ganz schoen auf dem Schlauch: Lorentz Invar = selbe Form
    in allen BS (muss nicht unb. konst bei LT sein), lorentzskalarfeld
 
+** Impulserhaltung aus dem Gefuehl... (ohne deltas) ok?
+** Normierung Photonenfeld?
+** Globaler Spin bei pol. Vektoren?
+** Spin nicht erhalten?
 * Work Log
 ** 18.03
  - habe mich in manche konzeptionelle Dinge ziemlich verrannt!