roundup comparison

latex/document.tex

@@ -14,12 +14,13 @@ captions=nooneline,captions=tableabove,english]{scrbook}
 
 \input{./tex/qqgammagamma.tex}
 \input{./tex/qqgammagamma/calculation.tex}
+\input{./tex/qqgammagamma/comparison.tex}
 
 \listoffigures
 \listoftables
 
-% \appendix
-% \input{./tex/myappendix.tex}
+ \appendix
+\input{./tex/appendix.tex}
 
 
 % Bibliography:

latex/hirostyle.sty

@@ -23,9 +23,16 @@ labelformat=brace, position=top]{subcaption}
 \usepackage{wrapfig}
 \usepackage{slashed}
 \usepackage[toc]{multitoc}
+\usepackage{minted}
+
+%% multicolumn toc
 \renewcommand*{\multicolumntoc}{2}
 \setlength{\columnseprule}{0.5pt}
 
+%% minted
+\usemintedstyle{colorful}
+\newmintedfile{yaml}{linenos,mathescape=true}
+
 %% fix p slash
 \declareslashed{}{/}{0}{.15}{p}
 
@@ -69,6 +76,12 @@ labelformat=brace, position=top]{subcaption}
 \DeclareMathOperator{\vs}{v}
 \DeclareMathOperator*{\vsb}{\overline{v}}
 
+%% Center of Mass energy
+\DeclareMathOperator{\ecm}{E_{\text{CM}}}
+
+%% area hyperbolicus
+\DeclareMathOperator{\artanh}{artanh}
+
 %% Fast Slash
 \let\sl\slashed
 

latex/makefile

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

latex/tex/appendix.tex

@@ -0,0 +1,9 @@
+\chapter{Appendix}%
+\label{chap:appendix}
+
+\section{Sherpa Runcards}%
+\label{sec:runcards}
+
+\subsection{Quark Antiquark Anihilation}%
+\label{sec:qqggruncard}
+\yamlfile{../prog/runcards/qqgg/Sherpa.yaml}

latex/tex/qqgammagamma/calculation.tex

@@ -77,16 +77,21 @@ The total matrix element (the minus sign has been dropped) is given in~\eqref{eq
   \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 \(\bar{\Gamma}_i=\gamma^0\Gamma^\dagger_i\gamma^0\).
+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
+\(\bar{\Gamma}_i=\gamma^0\Gamma^\dagger_i\gamma^0\). An additional
+factor of \(\frac{1}{3}\) arises from the color averaging. There are
+total of \emph{nine} color combinations, but only \emph{three}
+contribute (different color states are orthogonal, no preferred colors
+in the beams).
 
 \begin{equation}
   \label{eq:averagedm}
   \langle\abs{\mathcal{M}}^2\rangle = \frac{1}{4}\sum_{s_1 s_2}\sum_{\lambda_1
-    \lambda_2} \abs{\mathcal{M}}^2=\overbrace{\frac{1}{4}\frac{(gQ)^4}{(2p)^4}}^\mathfrak{F}\sum_{\lambda_1
+    \lambda_2} \abs{\mathcal{M}}^2=\overbrace{\frac{1}{3}\frac{1}{4}\frac{(gQ)^4}{(2p)^4}}^\mathfrak{F}\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}
@@ -145,41 +150,34 @@ can be utilized for the case \(i\neq j\) and lead to
   \label{eq:gij}
   \begin{split}
 \Gamma_{ii} &=
-\tr[\gamma_\mu(\ps_1-\ps(a_i))\gamma_\nu\ps_2\gamma^\mu(\ps_1-\ps(a_j))\gamma^\nu\ps_1)]\\
-&= 2\tr[\ps_2(\ps_1-\ps(a_i))\gamma_v(\ps_1-\ps(a_j))\gamma^\nu\ps_1]\\
-&\hphantom{=} - 2\tr[\gamma_\mu(\ps_1-\ps(a_i))\ps_2(\ps_1-\ps(a_j))\gamma^\mu\ps_1]\\
-&\hphantom{=} +
-\tr[\gamma_\mu(\ps_1-\ps(a_i))\gamma^\mu\gamma_\nu\ps_2(\ps_1-\ps(a_j))\gamma^\nu\ps_1]\\
-&= 32 \qty[2p^2(p(a_j)\cdot p_1) - (p(a_i)\cdot p_2)(p(a_j)\cdot p_1)]
+\tr[\gamma_\mu(\ps_1-\ps(a_i))\gamma^\nu\ps_2\gamma^\mu(\ps_1-\ps(a_j))\gamma_\nu\ps_1)]\\
+&= -2\tr[\ps_2\gamma^\nu(\ps_1-\ps(a_i))(\ps_1-\ps(a_j))\gamma_\nu\ps_1)]
    \\
-&\hphantom{=} + 32\qty[(p(a_i)\cdot p(a_j))(p_1\cdot p_2) - (p(a_i)\cdot p_1)(p(a_j)\cdot p_2)]
+&=-16\qty[(p_1p_2)(2(p_ap(a_j)) - p(a_i)p(a_j)) +
+(p_1p(a_i))(p_2p(a_j)) - (p_1p(a_j))(p_2p(a_i))]
 \end{split}
 \end{equation}
 
-The crucial step here was to move the \(\gamma^\mu\) past
-\(\gamma_\nu\ps_2\) as shown in~\eqref{eq:movepast}.
+The crucial step here was to sum over \(\mu\) and utilizing
+\(\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho }\gamma ^{\sigma }\gamma
+_{\mu }=-2\gamma ^{\sigma }\gamma ^{\rho }\gamma ^{\nu }\).
 
-\begin{equation}
-  \label{eq:movepast}
-  \gamma_\nu\ps\gamma^\mu = 2\gamma_\nu p^\mu - 2\delta_\nu^\mu\ps +
-  \gamma^\mu\gamma_\nu\ps
-\end{equation}
 
 After multiplying out the terms in~\eqref{eq:averagedm} and applying
 the \eqref{eq:gii} and~\eqref{eq:gij} there results (after
 rather technical simplifications) the
-averaged matrix element of~\eqref{eq:averagedm_final}.
+averaged matrix element of~\eqref{eq:averagedm_final}. It is
+noteworthy that the mixing terms cancel out, in other terms:
+\(\Gamma_{12} + \Gamma_{21} = 0\). The result is also expressed in
+terms of the pseudo-rapidity \(\eta \equiv -\ln[\tan(\frac{\theta}{2})]\).
 
 \begin{equation}
   \label{eq:averagedm_final}
   \begin{split}
   \langle\abs{\mathcal{M}}^2\rangle &= p^4\cdot\mathfrak{F}\cdot
-  32\cdot\qty[\frac{(1-c)(1+c)}{s'^4}] + \qty[\frac{2(1+c) -
-    (1+c)(1+c)+4-(1-c)(1-c)}{s'^2c'^2}] \\
-  &\hphantom{=} +\qty[\frac{2(1+c) -
-    (1+c)(1+c)+4-(1-c)(1-c)}{s'^2c'^2}] +
-  \qty[\frac{(1-c)(1+c)}{c'^4}] \\
-  &= 4(gQ)^4 \cdot\frac{2+\sin^2(\theta)}{\sin^2(\theta)}
+  32\cdot\qty[\frac{(1-c)(1+c)}{s'^4}] + \qty[\frac{(1-c)(1+c)}{c'^4}] \\
+  &= \frac{4}{3}(gQ)^4 \cdot\frac{1+\cos^2(\theta)}{\sin^2(\theta)} =
+  \frac{4}{3}(gQ)^4\cdot(2\cosh(\eta) - 1)
   \end{split}
 \end{equation}
 

latex/tex/qqgammagamma/comparison.tex

@@ -0,0 +1,60 @@
+\section{Comparison with Sherpa}%
+\label{sec:compsher}
+
+The result obtained in~\ref{sec:qqggcalc} shall now be verified by
+monte-carlo in \verb|Sherpa|. To facilitate this, an expression for
+the total cross section for a range of \(\theta\) or \(\eta\) has to
+be obtained. Using the golden rule for \(2\rightarrow 2\) processes
+and observing that the initial and final impulses are equal
+(\(p_i=p_f\)) and \(g=\sqrt{4\pi\alpha}\), the
+result~\eqref{eq:crossec} arises.
+
+An additional
+factor of \(\frac{1}{2}\) arises from there being two identical
+photons in the final state.
+\begin{equation}
+  \label{eq:crossec}
+  \dv{\sigma}{\Omega} =
+  \frac{1}{2}\frac{1}{(8\pi)^2}\cdot\frac{\abs{\mathcal{M}}^2}{\ecm^2}\cdot\frac{\abs{p_f}}{\abs{p_i}}
+  = \overbrace{\frac{\alpha^2Q^4}{6\ecm^2}}^{\mathfrak{C}}\frac{1+\cos^2(\theta)}{\sin^2(\theta)}
+\end{equation}
+
+The differential cross section~\eqref{eq:crossec} is divergent for
+angles near zero or \(\pi\). Allowing finite mass in the calculation
+may regularize this divergence. Because \(m=0\) is the limit for
+\(\ecm\rightarrow\infty\), the cross section would still have strong
+peaks for angles near \(0,\pi\) at high energies so that the result is
+not altogether nonphysical. It is clearly symmetric around
+\(\theta=\frac{pi}{2}\) as was to be expected, because the photons are
+indistinguishable. To compare the cross section to experiment and to
+simulation an interval around \(\theta=\frac{\pi}{2}\) has to be
+chosen, where the first order, mass-less approximation may yield
+sensible results.
+
+The total cross section in such an interval, given by
+integrating~\eqref{eq:crossec} for \(\theta\in [\theta_1, \theta_2]\)
+or \(\eta\in [\eta_1, \eta_2]\) is given in~\eqref{eq:total-crossec}.
+
+\begin{equation}
+  \label{eq:total-crossec}
+  \begin{split}
+  \sigma &=
+  2\pi\mathfrak{C}\cdot\qty{\cos(\theta_2)-\cos(\theta_1)+2\qty[\artanh(\cos(\theta_1))
+    - \artanh(\cos(\theta_2))]} \\
+  &=2\pi\mathfrak{C}\cdot\qty[\tanh(\eta_1) - \tanh(\eta_2) + 2(\eta_2
+  - \eta_1))] \\
+  &={\frac{\pi\alpha^2Q^4}{3\ecm^2}}\cdot\qty[\tanh(\eta_1) - \tanh(\eta_2) + 2(\eta_2
+  - \eta_1))]
+  \end{split}
+\end{equation}
+
+Choosing \(\eta\in [-2.5,2.5]\) and
+\(\ecm=\SI{100}{\giga\electronvolt}\) the process was monte carlo
+integrated in sherpa using the runcard in~\ref{sec:qqggruncard}. This
+runcard describes the exact same (first order) process as the
+calculated cross section.
+
+Sherpa yields \(\sigma = \SI{0.0538009\pm
+  .00005}{\pico\barn}\). Plugging the same parameters
+into~\eqref{eq:total-crossec} gives \input{../results/xs/xs} which is
+acceptable.

prog/python/qqgg/analytical_xs.org

@@ -1,18 +1,18 @@
-#+PROPERTY: header-args :exports both
+#+PROPERTY: header-args :exports both :output-dir ../../../results/xs
 
 * Init
 ** Required Modules
 #+NAME: e988e3f2-ad1f-49a3-ad60-bedba3863283
-#+BEGIN_SRC ein-python :session :exports both
+#+begin_src ipython :session :exports both
   import numpy as np
   import matplotlib.pyplot as plt
-#+END_SRC
+#+end_src
 
 #+RESULTS: e988e3f2-ad1f-49a3-ad60-bedba3863283
 
 ** Utilities
 #+NAME: 53548778-a4c1-461a-9b1f-0f401df12b08
-#+BEGIN_SRC ein-python :session :exports both :results raw drawer
+#+BEGIN_SRC ipython :session :exports both :results raw drawer
 %run ../utility.py
 #+END_SRC
 
@@ -22,7 +22,7 @@ import matplotlib.pyplot as plt
 
 * Implementation
 #+NAME: 777a013b-6c20-44bd-b58b-6a7690c21c0e
-#+BEGIN_SRC ein-python :session :exports both :results raw drawer :exports code :tangle xs.py
+#+BEGIN_SRC ipython :session :exports both :results raw drawer :exports code :tangle xs.py
   """
   Implementation of the analytical cross section for q q_bar ->
   gamma gamma
@@ -108,7 +108,7 @@ import matplotlib.pyplot as plt
 ** XS qq -> gamma gamma
 First, set up the input parameters.
 #+NAME: 7e62918a-2935-41ac-94e0-f0e7c3af8e0d
-#+BEGIN_SRC ein-python :session :exports both :results raw drawer
+#+BEGIN_SRC ipython :session :exports both :results raw drawer
 eta = 2.5
 charge = 1/3
 esp = 200  # GeV
@@ -120,20 +120,20 @@ esp = 200  # GeV
 
 And now calculate the cross section in picobarn.
 #+NAME: cf853fb6-d338-482e-bc55-bd9f8e796495
-#+BEGIN_SRC ein-python :session :exports both :results raw drawer
+#+BEGIN_SRC ipython :session :exports both :results raw drawer file :file xs.tex :output-dir ../../../results/xs
 xs_gev = total_xs_eta(eta, charge, esp)
 xs_pb = gev_to_pb(xs_gev)
-xs_pb
+print(tex_value(xs_pb, unit=r'\pico\barn', prefix=r'\eta = '))
 #+END_SRC
 
 #+RESULTS: cf853fb6-d338-482e-bc55-bd9f8e796495
 :RESULTS:
-0.053793289459925515
+[[file:../../../results/xs/xs.tex]]
 :END:
 
 Compared to sherpa, it's pretty close.
 #+NAME: 81b5ed93-0312-45dc-beec-e2ba92e22626
-#+BEGIN_SRC ein-python :session :exports both :results raw drawer
+#+BEGIN_SRC ipython :session :exports both :results raw drawer
   sherpa = 0.0538009
   xs_pb/sherpa
 #+END_SRC
@@ -143,5 +143,7 @@ Compared to sherpa, it's pretty close.
 0.9998585425137037
 :END:
 
+
+
 I had to set the runcard option ~EW_SCHEME: alpha0~ to use the pure
 QED coupling constant.

prog/python/utility.py

@@ -2,6 +2,7 @@ import matplotlib
 import matplotlib.pyplot as plt
 from SecondaryValue import SecondaryValue
 from scipy.constants import hbar, c, electron_volt
+import numpy as np
 
 ###############################################################################
 #                                   Utility                                   #
@@ -11,6 +12,12 @@ def gev_to_pb(xs):
     """Converts a cross section from 1/GeV^2 to pb."""
     return xs/(electron_volt**2)*(hbar*c)**2*1e22
 
+def tex_value(val, unit='', prefix='', prec=10, err=None):
+    """Generates LaTeX output of a value with units and error."""
+
+    val = np.round(val, prec)
+    return fr'\({prefix}\SI{{{val}}}{{{unit}}}\)'
+
 ###############################################################################
 #                                  Plot Porn                                  #
 ###############################################################################

results/xs/sherpa_xs.tex

@@ -0,0 +1 @@
+../../prog/runcards/qqgg/sherpa_xs

results/xs/xs.tex

@@ -0,0 +1 @@
+\(\eta = \SI{0.0537932}{\pico\barn}\)