further write-up of the pdf stuff

latex/document.tex

@@ -23,6 +23,7 @@ captions=nooneline,captions=tableabove,english]{scrbook}
 \input{./tex/pdf.tex}
 \input{./tex/pdf/pdf_basics.tex}
 \input{./tex/pdf/lab_xs.tex}
+\input{./tex/pdf/results.tex}
 
 \appendix
 \input{./tex/appendix.tex}

latex/figs/pdf

@@ -0,0 +1 @@
+../../prog/python/qqgg/figs/pdf/

latex/hirostyle.sty

@@ -108,6 +108,7 @@ labelformat=brace, position=top]{subcaption}
 \newcommand{\sherpa}{\texttt{Sherpa}}
 \newcommand{\rivet}{\texttt{Rivet}}
 \newcommand{\vegas}{\texttt{VEGAS}}
+\newcommand{\lhapdf}{\texttt{LHAPDF6}}
 
 %% Expected Value and Variance
 \newcommand{\EX}[1]{\operatorname{E}\qty[#1]}

latex/tex/monte-carlo/sampling.tex

@@ -247,6 +247,12 @@ reference histograms created by generating events with \sherpa\ and
 analyzing them with the \rivet toolkit~\cite{Bierlich:2019rhm}. The
 utilized analysis can be found in~\ref{sec:simpdiphotriv}.
 
+\begin{figure}[hb]
+  \centering \plot{xs_sampling/diff_xs_p_t}
+  \caption{\label{fig:diff-xs-pt} The differential cross section
+    transformed to \(\pt\).}
+\end{figure}
+
 \begin{figure}[p]
   \centering
 
@@ -265,12 +271,6 @@ utilized analysis can be found in~\ref{sec:simpdiphotriv}.
     include histograms generated by \sherpa\ and \rivet.}
 \end{figure}
 
-\begin{figure}[p]
-  \centering \plot{xs_sampling/diff_xs_p_t}
-  \caption{\label{fig:diff-xs-pt} The differential cross section
-    transformed to \(\pt\).}
-\end{figure}
-
 Where~\ref{fig:histeta} shows clear resemblance
 of~\ref{fig:xs-int-eta}, the sharp peak in~\ref{fig:histpt} around
 \(\pt=\SI{100}{\giga\electronvolt}\) seems surprising. When

latex/tex/pdf/lab_xs.tex

@@ -43,7 +43,7 @@ the pseudo-rapidity one photon.
 
 \begin{equation}
   \label{eq:xs-eta-lab}
-  \dv{\sigma}{\eta} = 2\pi\cdot\frac{\alpha^2Q^4}{24 E_p^2
+  \dv{\sigma}{\eta} = 2\pi\cdot\frac{\alpha^2Z^4}{24 E_p^2
     x_1x_2}\cdot\qty(\tanh(\eta - w)^2 + 1)
 \end{equation}
 

latex/tex/pdf/pdf_basics.tex

@@ -6,7 +6,7 @@
 \section{Parton Density Functions}%
 \label{sec:pdf_basics}
 
-Parton Density Functions give, restricting the considerations to
+Parton Density Functions encode, restricting the considerations to
 leading order, the probability to ``encounter'' a constituent parton
 (quark or gluon) of a hadron with a certain momentum fraction \(x\) at
 a certain factorization scale \(Q^2\). PDFs are normalized according
@@ -48,3 +48,8 @@ scale\footnote{More appropriately: The factorization scale depends on
 
 Summing~\eqref{eq:pdf-xs} over all partons in the hadron gives
 the total scattering cross section for the hadron.
+
+PDFs can not be derived from first principles and have to be
+determined experimentally for low \(Q^2\) and can be evolved to higher
+\(Q^2\) through the \emph{DGLAP} equations~\cite{altarelli:1977af} at
+different orders of perturbation theory.

latex/tex/pdf/results.tex

@@ -0,0 +1,79 @@
+%%% Local Variables: ***
+%%% mode: latex ***
+%%% TeX-master: "../../document.tex"  ***
+%%% End: ***
+
+\section{Implementation and Results}%
+\label{sec:pdf_results}
+
+The considerations of~\ref{sec:pdf_basics} and~\ref{sec:lab_xs} can
+now be applied to obtain a cross section and histograms of observables
+for the scattering of two protons into two photons. Because the PDF is
+not available in closed form, event generation is the only viable way
+to verify theory against experiment, even in this simple leading-order
+process.
+
+The integrand in~\eqref{eq:pdf-xs} can be concretised
+into~\eqref{eq:weighteddist}, where \(q\) runs over all quarks (except
+the top quark). The averaged sum accounts for the fact, that the two
+protons are indistinguishable. The choice of \(Q^2\) was explained
+in~\ref{sec:pdf_basics} and is being given in~\eqref{eq:q2-explicit}.
+
+\begin{gather}
+  \label{eq:weighteddist}
+  \frac{\dd[3]{\sigma}}{\dd{\eta}\dd{x_1}\dd{x_2}} =
+  \sum_q \frac{1}{2}\qty[f_q\qty(x_1;Q^2) f_{\bar{q}}\qty(x_2;Q^2) + f_q\qty(x_2;Q^2) f_{\bar{q}}\qty(x_1;Q^2)] \dv{\sigma(x_1,
+    x_2, Z_q)}{\eta} \\
+  \label{eq:q2-explicit}
+  Q^2 = 2x_1x_2E_p^2
+\end{gather}
+
+This distribution can now be integrated to obtain a total
+cross-section as described in~\ref{sec:mcint}. Sampling a
+multi-dimensional distribution can be reduced to the sampling of one
+dimensional distributions by reducing the distribution itself to one
+variable through integration over the remaining ones and then, keeping
+the first variable fixed, sampling the other variables in a likewise
+manner. The hit-or-miss method described in~\ref{sec:hitmiss} has to
+be modified only in so far as to choose the sampling points in an
+\(n\)-dimensional volume. As before, the range of the \(\eta\)
+parameter has to be constrained to obtain physical results. Because
+the absolute values of the pseudo rapidities of the two final state
+photons are not equal in the lab frame, the shape of the
+integration/sampling volume differs from a simple
+hypercube \(\Omega\). Furthermore, for the massless limit to be applicable the
+center of mass energy of the partonic system must be much greater than
+the quark masses. This can be implemented by demanding the
+transverse momentum \(p_T\) of a final state photon to be greater than
+approximately~\SI{20}{\giga\electronvolt}. A restriction (cut) on
+\(p_T\) is suitable because detectors are usually only sensitive above
+a certain \(p_T\) threshold. %TODO CITE%
+Such cuts can be implemented simply by multiplying the distribution
+in~\eqref{eq:weighteddist} by the characteristic function
+\(\chi_\Omega\) of \(\Omega\) which is equal to one if \(x\in\Omega\)
+and zero otherwise.
+
+The PDF being used in the following has been determined at leading
+order and is the central member of the PDF set
+\verb|NNPDF31_lo_as_0118| provided by \emph{NNPDF} collaboration and
+accessed through the \lhapdf\
+library\cite{NNPDF:2017pd}\cite{Buckley:2015lh}.
+% TODO clean separation of pdf, pdf set %
+
+The resulting distribution (without cuts) is depicted for fixed
+\(x_2\) in~\ref{fig:dist-pdf}.
+
+\begin{figure}[hb]
+  \centering
+  \plot{pdf/dist3d_x2_const}
+  \caption{\label{fig:dist-pdf}Differential cross section convolved with PDFs for fixed
+    \result{xs/python/pdf/second_x} in picobarn.}
+\end{figure}
+
+For \(x_1 = x_2\) the distribution retains some likeness with the
+partonic distribution (see~\ref{fig:xs-int-eta}) but gets suppressed
+for greater values of \(x_1\). The overall shape of the distribution
+is clearly highly sub-optimal for hit-or-miss sampling, only having
+significant values when \(x_1\) or \(x_2\) are small and being very
+steep. To remedy that, one has to use a more efficient sampling
+algorithm (\vegas) or impose very restrictive cuts.

latex/tex/qqgammagamma/calculation.tex

@@ -8,7 +8,7 @@
 
 After labeling the incoming quarks and outcoming photons, as well as
 the momenta according to~\ref{fig:qqggfeyn}, the feynman rules yield
-the matrix elements in~\eqref{eq:matel}, where \(Q\) is the electric
+the matrix elements in~\eqref{eq:matel}, where \(Z\) is the electric
 charge of the quark and \(g\) is the QED coupling constant. The
 respective spinors and polarisation vectors are \(\us,\vs\) and
 \(\pe\).  The matrix element for~\ref{fig:qqggfeyn2} is obtained by
@@ -17,9 +17,9 @@ whenever indices would clutter the notation.
 
 \begin{align}
   \label{eq:matel}
-  \mathcal{M}_1 &= \frac{(gQ)^2}{\qty(p_1 - p_4)^2}\vsb(1)\pses(4)(\ps_1 -
+  \mathcal{M}_1 &= \frac{(gZ)^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)
+  \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}
 
 \begin{wrapfigure}{R}{0.4\textwidth}
@@ -79,7 +79,7 @@ and \(\Gamma_1\) as in~\eqref{eq:gammadef}.
 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)
+  \mathcal{M} = \mathcal{M}_1 + \mathcal{M}_2 = \frac{(gZ)^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
@@ -96,7 +96,7 @@ 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}{3}\frac{1}{4}\frac{\qty(gQ)^4}{\qty(2p)^4}}^\mathfrak{F}\sum_{\lambda_1
+    \lambda_2} \abs{\mathcal{M}}^2=\overbrace{\frac{1}{3}\frac{1}{4}\frac{\qty(gZ)^4}{\qty(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}
@@ -181,8 +181,8 @@ terms of the pseudo-rapidity \(\eta \equiv -\ln[\tan(\frac{\theta}{2})]\).
   \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{(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)
+  &= \frac{4}{3}(gZ)^4 \cdot\frac{1+\cos^2(\theta)}{\sin^2(\theta)} =
+  \frac{4}{3}(gZ)^4\cdot(2\cosh(\eta) - 1)
   \end{split}
 \end{equation}
 

latex/tex/qqgammagamma/comparison.tex

@@ -21,9 +21,9 @@ two identical photons in the final state.
   \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}}
                         =
-                        \underbrace{\frac{\alpha^2Q^4}{6\ecm^2}}_{\mathfrak{C}}\frac{1+\cos^2(\theta)}{\sin^2(\theta)}\label{eq:crossec}
+                        \underbrace{\frac{\alpha^2Z^4}{6\ecm^2}}_{\mathfrak{C}}\frac{1+\cos^2(\theta)}{\sin^2(\theta)}\label{eq:crossec}
   \\
-  \dv{\sigma}{\eta} &= 2\pi\cdot\frac{\alpha^2Q^4}{6\ecm^2}\cdot\qty(\tanh(\eta)^2 + 1)\label{eq:xs-eta}
+  \dv{\sigma}{\eta} &= 2\pi\cdot\frac{\alpha^2Z^4}{6\ecm^2}\cdot\qty(\tanh(\eta)^2 + 1)\label{eq:xs-eta}
 \end{align}
 
 \begin{figure}[ht]
@@ -74,7 +74,7 @@ in~\eqref{eq:total-crossec}.
     - \artanh(\cos(\theta_2))]} \\
   &=2\pi\mathfrak{C}\cdot\qty[\tanh(\eta_2) - \tanh(\eta_1) + 2(\eta_1
   - \eta_2))] \\
-  &={\frac{\pi\alpha^2Q^4}{3\ecm^2}}\cdot\qty[\tanh(\eta_2) - \tanh(\eta_1) + 2(\eta_1
+  &={\frac{\pi\alpha^2Z^4}{3\ecm^2}}\cdot\qty[\tanh(\eta_2) - \tanh(\eta_1) + 2(\eta_1
   - \eta_2))]
   \end{split}
 \end{equation}

latex/thesis.bib

@@ -78,3 +78,59 @@
   volume =       8,
   year =         2020
 }
+
+@article{Buckley:2015lh,
+  title =        {LHAPDF6: parton density access in the LHC precision
+                  era},
+  volume =       {75},
+  ISSN =         {1434-6052},
+  url =          {http://dx.doi.org/10.1140/epjc/s10052-015-3318-8},
+  DOI =          {10.1140/epjc/s10052-015-3318-8},
+  number =       {3},
+  journal =      {The European Physical Journal C},
+  publisher =    {Springer Science and Business Media LLC},
+  author =       {Buckley, Andy and Ferrando, James and Lloyd, Stephen
+                  and Nordström, Karl and Page, Ben and Rüfenacht,
+                  Martin and Schönherr, Marek and Watt, Graeme},
+  year =         {2015},
+  month =        {Mar}
+}
+
+@misc{NNPDF:2017pd,
+  title =        {Parton distributions from high-precision collider
+                  data},
+  author =       {The NNPDF Collaboration and Richard D. Ball and
+                  Valerio Bertone and Stefano Carrazza and Luigi Del
+                  Debbio and Stefano Forte and Patrick Groth-Merrild
+                  and Alberto Guffanti and Nathan P. Hartland and
+                  Zahari Kassabov and José I. Latorre and Emanuele
+                  R. Nocera and Juan Rojo and Luca Rottoli and Emma
+                  Slade and Maria Ubiali},
+  year =         {2017},
+  eprint =       {1706.00428},
+  archivePrefix ={arXiv},
+  primaryClass = {hep-ph}
+}
+
+@article{altarelli:1977af,
+  title =        "Asymptotic freedom in parton language",
+  journal =      "Nuclear Physics B",
+  volume =       "126",
+  number =       "2",
+  pages =        "298 - 318",
+  year =         "1977",
+  issn =         "0550-3213",
+  doi =          "https://doi.org/10.1016/0550-3213(77)90384-4",
+  url =
+                  "http://www.sciencedirect.com/science/article/pii/0550321377903844",
+  author =       "G. Altarelli and G. Parisi",
+  abstract =     "A novel derivation of the Q2 dependence of quark and
+                  gluon densities (of given helicity) as predicted by
+                  quantum chromodynamics is presented. The main body
+                  of predictions of the theory for deep-inleastic
+                  scattering on either unpolarized or polarized
+                  targets is re-obtained by a method which only makes
+                  use of the simplest tree diagrams and is entirely
+                  phrased in parton language with no reference to the
+                  conventional operator formalism."
+}

notes.org

@@ -134,6 +134,8 @@ Viele Gruesse, Frank
 ** Sind quark verhaeltnisse in PDF enthalten (2:1 fuer proton)
 ** beide finalstate photonen behalten?
 ** TODO PDF members
+** TODO Sensitivity detectors cite!
+** TODO was fuer eine pdf ist das NNPDF31lo
 * Work Log
 ** 18.03
  - habe mich in manche konzeptionelle Dinge ziemlich verrannt!

prog/python/qqgg/results/pdf/second_x.tex

@@ -0,0 +1 @@
+\(x_2 = 0.01\)