implement falkos tipps

latex/tex/abstract.tex

@@ -15,7 +15,7 @@ with the \sherpa\ event generator.
 
 \section*{Zusammenfassung}
 
-Der analytische Wirkungsquerschnitt fuer den \(\qqgg\) diphoton
+Der analytische Wirkungsquerschnitt f\"ur den \(\qqgg\) diphoton
 Prozess wird in f\"uhrender Ordnung berechnet. In Hinblick auf die
 Anwendung auf diesen Prozess werden Monte Carlo Methoden f\"ur
 Integration und zur Generierung von Stichproben aus

latex/tex/erklaerung.tex

@@ -1,7 +1,7 @@
 \thispagestyle{empty}
 \section*{Erklärung}
 
-Hiermit erkläre ich, dass ich diese Arbeit im Rahmender Betreuung am
+Hiermit erkläre ich, dass ich diese Arbeit im Rahmen der Betreuung am
 Institut für Kern- und Teilchenphysik ohne unzulässige Hilfe Dritter
 verfasst und alle Quellen als solche gekennzeichnet habe.
 

latex/tex/introduction.tex

@@ -66,8 +66,8 @@ algorithm related functionality as a module. The file
 that generates all the results of \cref{chap:mc}. The file
 \texttt{parton\_density\_function\_stuff.org} contains all the
 computations for \cref{chap:pdf}. The python code makes heavy use of
-\href{https://www.scipy.org/}{scipy}~\cite{2020Virtanen:Sc} (and of
-course \href{https://numpy.org/}{numpy}).
+\href{https://www.scipy.org/}{scipy}~\cite{2020Virtanen:Sc} and
+its component \href{https://numpy.org/}{numpy}).
 
 \section{Software Versions}%
 \label{sec:versions}

latex/tex/monte-carlo/integration.tex

@@ -229,7 +229,7 @@ interval increments and applying \vegas\ to \cref{eq:crossec} with
 \result{xs/python/xs_mc_θ_vegas_K} increments yields
 \result{xs/python/xs_mc_θ_vegas} with
 \result{xs/python/xs_mc_θ_vegas_N} function evaluations (including
-\vegas\ iterations). This result is comparable with tho one obtained
+\vegas\ iterations). This result is comparable with the one obtained
 by parameter transformation in \cref{sec:naivechange}.  The sample
 count \(N\) is the total number of evaluations of \(f\). The resulting
 increments and the weighted integrand \(f/\rho\) are depicted in

latex/tex/monte-carlo/sampling.tex

@@ -77,8 +77,7 @@ probability~\(f/g\), so that \(g\) cancels out. This is known as the
 %
 The thus obtained samples are then distributed according to \(f/B\)
 and the total probability of accepting a sample, also called the
-efficiency \(\mathfrak{e}\), is given by hat \cref{eq:impsampeff}
-holds.
+efficiency \(\mathfrak{e}\), is given by \cref{eq:impsampeff}.
 %
 \begin{equation}
   \label{eq:impsampeff}
@@ -93,7 +92,7 @@ Choosing \(g\) like \cref{eq:primitiveg} and looking back at
 procedure simplifies to choosing random numbers \(x\in [0,1]\) and
 accepting them with the probability \(f(x)/g(x)\). The efficiency of
 this approach is related to how much \(f\) differs from
-\(f_{\text{max}}\) which in turn related to the variance of
+\(f_{\text{max}}\) which in turn is related to the variance of
 \(f\). Minimizing variance will therefore improve sampling
 performance. The method can also be used in higher dimensions without
 modification and has again been implemented and evaluated.
@@ -135,8 +134,7 @@ upper bound are depicted in \cref{fig:distcos}.
 When transforming \(f\) to a new variable \(y=y(x)\) one arrives at
 \cref{eq:transff} and may reduce variance, analogous to
 \cref{sec:naivechange}. Transforming the distribution in a beneficial
-way in is an alternative method of performing \emph{importance
-  sampling}.
+way is an alternative method of performing \emph{importance sampling}.
 %
 \begin{equation}
   \label{eq:transff}

latex/tex/pdf/lab_xs.tex

@@ -61,7 +61,7 @@ the pseudo-rapidity one photon.
 The 4-momenta of the final state photons can be obtained by explicit
 Lorentz transformation and are listed in \cref{eq:lab-momenta-fs}
 where \(\theta\) is the polar angle of the ``first'' final state
-photon.  These relation are easily transcribed to \(\eta\) dependence
+photon. These relations are easily transcribed to \(\eta\) dependence
 by using the identity \(\cos(\theta) = \tanh(\eta)\).
 
 %

latex/tex/pdf/results.tex

@@ -141,8 +141,8 @@ the analysis used to produce the histograms can be found in
 \cref{fig:pdf-eta,fig:histeta} it becomes apparent, that the PDF has
 substantial influence on the resulting distribution. Also the center
 of momentum energy is not constant anymore and has a steep peak at low
-energies due to the steepness of the PDF. The convolution with the pdf
-has also smoothed out the jacobian peak seen in \cref{fig:histpt}.
+energies due to the steepness of the PDF. The convolution with the PDF
+has also smoothed out the Jacobian peak seen in \cref{fig:histpt}.
 
 Furthermore new observables have been introduced.  The invariant mass
 of the photon pair
@@ -152,17 +152,17 @@ of mass energy of the partonic system that produces the photons (see
 fractions of the partons. \Cref{fig:pdf-inv-m} shows, that the vast
 majority of the reactions take place at a rather low c.m. energy,
 owing to the high weights of the PDF at small \(x\) values. Due to the
-\(\pt\) cuts the first bin is slightly lower then the second.
+\(\pt\) cuts the first bin is slightly lower than the second.
 
 The cosines of the scattering angles in the labe frame and the
 Collins-Soper (CS) frame are defined in
 \cref{eq:sangle,eq:sangle-cs}. The scattering angle is just the angle
 between one photon and the z-axis (beam axis) in the c.m. frame if
-this frame can be reached by a boost along the z-\footnote{Or me
-  generally, in a z-boosted frame where the angles of the two photons
-  are the same.}. Here, the partons are assumed to have no transverse
-momentum and the system is symmetric around the beam axis and
-therefore this boost is possible. When allowing transverse parton
+this frame can be reached by a boost along the z-axis\footnote{Or more
+  generally, in a z-boosted frame where the polar angles of the two
+  photons are the same.}. Here, the partons are assumed to have no
+transverse momentum and the system is symmetric around the beam axis
+and therefore this boost is possible. When allowing transverse parton
 momenta, as will be done in \cref{chap:pheno} this symmetry goes
 away. Defining the z-axis as one beam axis in the c.m. frame would be
 quite an arbitrary choice that disrespects the symmetry of the two

latex/tex/qqgammagamma.tex

@@ -13,48 +13,3 @@ process is being described by the Feynman diagrams in
 degrees of freedom average out and will not be considered henceforth.
 Furthermore a high energy regime will be supposed and therefore 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{Leading order diagrams for \(\qqgg\).}%
-  \label{fig:qqggfeyn}
-\end{figure}

latex/tex/qqgammagamma/calculation.tex

@@ -1,6 +1,23 @@
 \section{Calculation of the Cross Section to Leading Order}%
 \label{sec:qqggcalc}
 
+\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 process
+  \(\qqgg\) in the massles limit.}
+\end{wrapfigure}
+%
 After labeling the incoming quarks and outcoming photons, as well as
 the momenta according to \cref{fig:qqggfeyn}, the Feynman rules for
 QED yield the matrix elements in \cref{eq:matel}, where \(Z\) is the
@@ -18,7 +35,52 @@ would clutter the notation. The matrix element for
   \mathcal{M}_2 &= \frac{(gZ)^2}{\qty(p_1 - p_4)^2}\vsb(1)\pses(3)(\ps_1 - \ps_3)\pses(4)\us(2)
 \end{align}
 %
-To simplify notation, some shorthands are intruduced
+\begin{figure}[hb]
+  \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{Leading order diagrams for \(\qqgg\).}%
+  \label{fig:qqggfeyn}
+\end{figure}
+
+To simplify notation, some shorthands are introduced
 in \cref{eq:scshort}.
 \begin{equation}
   \label{eq:scshort}
@@ -26,23 +88,6 @@ in \cref{eq:scshort}.
     s(x) &= \sin(x) & c(x) &= \cos(x) \\ s'(x) &= \sin(\frac{x}{2}) & c'(x) &= \cos(\frac{x}{2})
   \end{split}
 \end{equation}
-
-\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}
 %
 All calculations are made with respect to the center of momentum
 (c.m.) frame unless stated otherwise. The momenta in the c.m. frame
@@ -118,7 +163,7 @@ form \cref{eq:gbricks}.
   \end{split}
 \end{equation}
 %
-The sum over plarisation can be simplified by utilizing the
+The sum over polarization can be simplified by utilizing the
 completeness relation for polarization vectors for \emph{real} photons
 \cref{eq:polcomp}.
 %
@@ -145,8 +190,7 @@ theorems for the gamma matrices.
 %
 The same tricks as well as the commutation relation for gamma matrices
 can be utilized for the case \(i\neq j\) and lead to \cref{eq:gij},
-albeit with more technical effort.\footnote{If I learned one thing, it
-  is the importance of doing calculations with the utmost verbosity.}
+albeit with more technical effort.
 %
 \begin{equation}
   \label{eq:gij}
@@ -155,8 +199,8 @@ albeit with more technical effort.\footnote{If I learned one thing, it
 \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)]
    \\
-&=-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))]
+&=-16\qty[(p_1\cdot p_2)(2(p_a\cdot p(a_j)) - p(a_i)\cdot p(a_j)) +
+(p_1\cdot p(a_i))(p_2\cdot p(a_j)) - (p_1\cdot p(a_j))(p_2\cdot p(a_i))]
 \end{split}
 \end{equation}
 %
@@ -164,9 +208,8 @@ The crucial step here was to sum over \(\mu\) and utilizing
 \(\gamma ^{\mu }\gamma ^{\nu }\gamma ^{\rho }\gamma ^{\sigma }\gamma
 _{\mu }=-2\gamma ^{\sigma }\gamma ^{\rho }\gamma ^{\nu }\).
 
-
 After multiplying out the terms in \cref{eq:averagedm} and plugging in
-\cref{eq:gii,eq:gij} there results (with some rather technical
+\cref{eq:gii,eq:gij} their results (with some rather technical
 simplifications) the averaged matrix element of
 \cref{eq:averagedm_final}. It is noteworthy that the mixing terms
 cancel out, in other terms: \(\Gamma_{12} + \Gamma_{21} = 0\). The

latex/tex/qqgammagamma/comparison.tex

@@ -49,7 +49,7 @@ two identical photons in the final state.
     \cref{eq:total-crossec} of the process for a pseudo-rapidity
     integrated over \([-\eta, \eta]\).}
 \end{subfigure}
-\caption{\label{fig:xsfirst} Plots of the differntial and total cross section
+\caption{\label{fig:xsfirst} Plots of the differential and total cross section
   for \(\qqgg\).}
 \end{figure}
 %
@@ -65,11 +65,9 @@ into a form where the divergence does not occur (see
 \cref{eq:xs-eta}).
 
 The differential cross section clearly is symmetric around
-\(\theta=\frac{\pi}{2}\) as was to be expected\footnote{Such
-  properties have been very handy and intuitive checks for
-  calculations along the way.}, 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
+\(\theta=\frac{\pi}{2}\) 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 leading order, mass-less approximation may yield a
 physical result.