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massive overhaul, based on christians comments
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%% Creator: Inkscape 1.0 (4035a4fb49, 2020-05-01), www.inkscape.org
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\endgroup%
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@ -12,7 +12,7 @@
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\usepackage[backend=biber, language=english, style=phys]{biblatex}
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\usepackage{siunitx}
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\usepackage[pdfencoding=auto]{hyperref} % load late
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\usepackage{cleveref}
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\usepackage[capitalize]{cleveref}
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% \usepackage[activate={true,nocompatibility},final,tracking=true,spacing=true,factor=1100,stretch=10,shrink=10]{microtype}
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\usepackage{caption}
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\usepackage[list=true, font=small,
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@ -28,6 +28,7 @@ labelformat=brace, position=top]{subcaption}
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\usepackage{fancyvrb}
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\usepackage[autostyle=true]{csquotes}
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\usepackage{setspace}
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\usepackage{newunicodechar}
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%% use the current pgfplots
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\pgfplotsset{compat=1.16}
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@ -81,6 +82,9 @@ labelformat=brace, position=top]{subcaption}
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%% Font for headings
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\addtokomafont{disposition}{\rmfamily}
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%% Minus Sign for Matplotlib
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\newunicodechar{−}{-}
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% Macros
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%% qqgg
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@ -110,6 +114,7 @@ labelformat=brace, position=top]{subcaption}
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%% area hyperbolicus
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\DeclareMathOperator{\artanh}{artanh}
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\DeclareMathOperator{\arcosh}{arcosh}
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%% Fast Slash
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\let\sl\slashed
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@ -1 +0,0 @@
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\(\sigma = \SI{0.0537937\pm 2.55202e-06}{\pico\barn}\)
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@ -3,31 +3,32 @@
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MC (MC) methods have been and still are one of the most important
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tools for numerical calculations in particle physics. Be it for
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validating the well established standard model or for making
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validating the well established Standard Model or for making
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predictions about new theories, MC simulations are the
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crucial interface of theory and experimental data, making them
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directly comparable. Furthermore horizontal scaling is almost trivial
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to implement in MC algorithms, making them well adapted to
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modern parallel computing. In this thesis, the use of MC
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methods will be traced through from simple integration to the
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simulation of proton-proton scattering.
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directly comparable.% Furthermore horizontal scaling is almost trivial
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% to implement in MC algorithms, making them well adapted to
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% modern parallel computing.
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In this thesis, the use of MC methods will be traced through from
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simple integration to the simulation of proton-proton scattering.
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The ``Drosophila'' of this thesis is the quark annihilation into two
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The ``Guinea Pig'' of this thesis is the quark annihilation into two
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photons \(\qqgg\), henceforth called the diphoton process. It forms an
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important background to the higgs decay channel
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\(H\rightarrow \gamma\gamma\) and to a dihiggs decay
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important background to the Higgs decay channel
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\(H\rightarrow \gamma\gamma\) (which was instrumental in its
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discovery) and to a dihiggs decay
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\(HH\rightarrow b\bar{b}\gamma\gamma\)~\cite{aaboud2018:sf}, while
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still being a pure QED process and thus calculable by hand within the
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scope of this thesis. The differential and total cross section of this
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process is being calculated in leading order in \cref{chap:qqgg} and
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the obtained result is compared to the total cross section obtained
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with the \sherpa~\cite{Gleisberg:2008ta} event generator, used as
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matrix element integrator. In \cref{chap:mc} some simple MC
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methods are discussed, implemented and their results
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compared. Beginning with a study of MC integration the
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\vegas\ algorithm~\cite{Lepage:19781an} is implemented and
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evaluated. Subsequently MC sampling methods are explored and
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the output of \vegas\ is used to improve the sampling
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still being a pure QED process at leading order and thus calculable by
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hand within the scope of this thesis. The differential and total cross
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section of this process is being calculated in leading order in
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\cref{chap:qqgg} and the obtained result is compared to the total
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cross section obtained with the \sherpa~\cite{Gleisberg:2008ta} event
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generator, used as matrix element integrator. In \cref{chap:mc} some
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simple MC methods are discussed, implemented and their results
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compared. Beginning with a study of MC integration the \vegas\
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algorithm~\cite{Lepage:19781an} is implemented and
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evaluated. Subsequently MC sampling methods are explored and the
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output of \vegas\ is used to improve the sampling
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efficiency. Histograms of observables are generated and compared to
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histograms from \sherpa\ using the \rivet~\cite{Bierlich:2019rhm}
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analysis framework. \Cref{chap:pdf} deals with proton-proton
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@ -46,15 +47,13 @@ effects. The impact of those effects on observables is studied in
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\label{sec:convent}
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Throughout natural units with \(c=1, \hbar = 1, k_B=1, \varepsilon_0
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= 1\) are used unless stated otherwise. Histograms without label on
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the y-axis are normalized to unity and to be interpreted as
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probability densities.
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= 1\) are used unless stated otherwise.
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\section{Source Code}%
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\label{sec:source}
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The (literate) python code, used to generate most of the results and
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figures can be found on under
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figures can be found under
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\url{https://github.com/vale981/bachelor_thesis/} and more
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specifically in the subdirectory \texttt{prog/python/qqgg}.
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@ -64,8 +63,8 @@ algorithm related functionality as a module. The file
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that generates all the results of \cref{chap:mc}. The file
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\texttt{parton\_density\_function\_stuff.org} contains all the
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computations for \cref{chap:pdf}. The python code makes heavy use of
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\href{https://www.scipy.org/}{scipy} (and of course
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\href{https://numpy.org/}{numpy}).
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\href{https://www.scipy.org/}{scipy}~\cite{2020Virtanen:Sc} (and of
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course \href{https://numpy.org/}{numpy}).
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%%% Local Variables: ***
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%%% mode: latex ***
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@ -3,12 +3,15 @@
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%%% TeX-master: "../document.tex" ***
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%%% End: ***
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\chapter{Survey of elementary Monte-Carlo Methods}%
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\chapter{Survey of Elementary Monte Carlo Methods}%
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\label{chap:mc}
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Monte-carlo methods for multidimensional integration and sampling of
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Monte Carlo methods for multidimensional integration and sampling of
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probability distributions are central tools of modern particle
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physics. Therefore some simple methods and algorithms are being
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studied and implemented here and will be applied to the results
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from \cref{chap:qqgg}. The \verb|python| code for the implementation
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can be found in \cref{sec:mcpy}.
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studied and implemented here and will be applied to the results from
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\cref{chap:qqgg,sec:compsher}. The \verb|python| code for the
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implementation can be found as described in \cref{sec:source}. The
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sampling and integration intervals, as well as other parameters have
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been chosen as in \cref{sec:compsher} been chosen, so that
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\result{xs/python/eta} and \result{xs/python/ecm}\!.
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@ -1,4 +1,4 @@
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\section{Monte-Carlo Integration}%
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\section{Monte Carlo Integration}%
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\label{sec:mcint}
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Consider a function
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@ -6,8 +6,8 @@ Consider a function
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probability density on
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\(\rho\colon \vb{x}\in\Omega\mapsto\mathbb{R}_{\geq 0}\) with
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\(\int_{\Omega}\rho(\vb{x})\dd{\vb{x}} = 1\). By multiplying \(f\)
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with a one in the fashion of \cref{eq:baseintegral}, the integral
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of \(f\) over \(\Omega\) can be interpreted as the expected value
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with a one in the fashion of \cref{eq:baseintegral}, the integral of
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\(f\) over \(\Omega\) can be interpreted as the expected value
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\(\EX{F/\Rho}\) of the random variable \(F/\Rho\) under the
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distribution \(\rho\). This is the key to most MC methods.
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@ -58,16 +58,20 @@ value of \cref{eq:approxexp} varies around \(I\) with the variance
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\frac{f(\vb{x_i})}{\rho(\vb{x_i})}]^2 \label{eq:varI-approx}
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\end{align}
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The name of the game now is to reduce \(\VAR{F/\Rho}\) to speed up the
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convergence of \cref{eq:approxexp} and achieve higher accuracy with
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fewer function evaluations. Some ways variance reductions can be
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The goal now is to reduce \(\VAR{F/\Rho}\) to speed up the convergence
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of \cref{eq:approxexp} and achieve higher accuracy with fewer function
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evaluations. There are at least three angles of attack
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in~\ref{eq:baseintegral}, namely the distribution \(\rho\), the
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variable \(\vb{x}\), and the integration volume
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\(\Omega\). Accordingly some ways variance reductions can be
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accomplished are choosing a suitable \(\rho\) (importance sampling),
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by transforming the integral onto another variable, a combination of
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both approaches or by subdividing integration volume into several
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sub-volumes of different size while keeping the sample size constant
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in all sub-volumes (stratified sampling).\footnote{There are of course
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still other methods like the multi-channel method.} Combining ideas
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from importance sampling and stratified sampling leads to the \vegas\
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by transforming the integral onto another variable or by subdividing
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integration volume into several sub-volumes of different size while
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keeping the sample size constant in all sub-volumes (stratified
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sampling).\footnote{There are of course still other methods like the
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multi-channel method.}\footnote{Of course, combinations of these
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methods can be applied as well.} Combining ideas from importance
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sampling and stratified sampling leads to the \vegas\
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algorithm~\cite{Lepage:19781an} that approximates the optimal
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distribution of importance sampling by adaptive subdivision of the
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integration volume into a grid.
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@ -75,17 +79,18 @@ integration volume into a grid.
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The convergence of \cref{eq:approxexp} is not dependent on the
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dimensionality of the integration volume as opposed to many other
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numerical integration algorithms (trapezoid rule, Simpsons rule) that
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usually converge like \(N^{-\frac{k}{n}}\) with \(k\in\mathbb{N}\) as
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opposed to \(N^{-\frac{k}{n}}\) with MC. Because phase space
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integrals in particle physics usually have a high dimensionality,
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MC integration is suitable approach there. When implementing
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MC methods, the random samples can be obtained through
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hardware or software random number generators (RNGs). Most
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implementations utilize software RNGs because supply pseudo-random
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numbers in a reproducible way, which facilitates deniability and
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comparability.~\cite{buckley:2011ge}
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usually converge like \(N^{-\frac{k}{n}}\) with
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\(k\in\mathbb{N}_{>0}\) and \(n\) being the dimensionality as opposed
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to \(N^{-\frac{k}{n}}\) with MC. Because phase space integrals in
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particle physics usually have a high dimensionality, MC integration is
|
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a suitable approach there. When implementing MC methods, the random
|
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samples can be obtained through hardware or software random number
|
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generators (RNGs). Most implementations utilize software RNGs because
|
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they supply pseudo-random numbers in a reproducible way, which
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facilitates deniability and comparability~\cite{buckley:2011ge}.
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% TODO: maybe remove, ask Frank
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\subsection{Naive Monte-Carlo Integration and Change of Variables}
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\subsection{Naive Monte Carlo Integration and Change of Variables}
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\label{sec:naivechange}
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The simplest choice for \(\rho\) is given
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@ -115,16 +120,15 @@ squared when approximating the integral by the sum.
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\begin{equation}
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\label{eq:approxvar-I}
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\VAR{I} = \abs{\Omega}\int_\Omega f(\vb{x})^2 -
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I^2 \dd{\vb{x}} \equiv \abs{\Omega}\cdot\sigma_f^2 \approx
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\underbrace{\qty(\frac{I}{\abs{\Omega}})^2}_{=\bar{f}} \dd{\vb{x}} \equiv \abs{\Omega}\cdot\sigma_f^2 \approx
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\frac{\abs{\Omega}^2}{N-1}\sum_{i}\qty[f(\vb{x}_i) - \bar{f}]^2
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\end{equation}
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Applying this method to integrate
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\(2\pi\sin(\theta)\cdot\dv{\sigma}{\Omega}\) (see \cref{eq:crossec})
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over a \(\theta\) interval, equivalent to \(\eta\in [-2.5, 2.5]\) with
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a target accuracy of \(\varepsilon=10^{-3}\) results in
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\result{xs/python/xs_mc} with a sample size of
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\result{xs/python/xs_mc_N}.
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Applying this method to integrate the \(\qqgg\) cross section from
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\cref{eq:crossec} over a \(\theta\) interval, equivalent to
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\(\eta\in [-2.5, 2.5]\) with a target accuracy of
|
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\(\varepsilon=10^{-3}\) results in \result{xs/python/xs_mc} with a
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sample size of \result{xs/python/xs_mc_N}.
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Changing variables and integrating \cref{eq:xs-eta} over \(\eta\) with
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the same target accuracy yields~\result{xs/python/xs_mc_eta} with a
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@ -133,7 +137,7 @@ reduction in variance and sample size can be understood qualitatively
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by studying \cref{fig:xs-int-comp}. The differential cross section in
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terms of \(\eta\)~(\cref{fig:xs-int-eta}) is less steep than the
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differential cross section in terms of
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\(\theta\)~(\cref{fig:xs-int-theta}) and takes on significant values
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\(\theta\)~(\cref{fig:xs-int-theta}) and takes on large values
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over most of the integration interval. In general, the Jacobian
|
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arising in variable transformation has the same effect as the
|
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probability density in importance sampling. It can be shown that
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@ -168,7 +172,7 @@ sub-volume is the same~\cite{Lepage:19781an}. In importance sampling,
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the optimal probability distribution is given
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by \cref{eq:optimalrho}, where \(f(\Omega) \geq 0\) is presumed
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without loss of generality. When applying \vegas\ to multi dimensional
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integrals, \cref{eq:optimalrho} is usually modified to factorize into
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integrals,~\cref{eq:optimalrho} is usually modified to factorize into
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distributions for each variable to simplify calculations.
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\begin{equation}
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@ -177,17 +181,24 @@ distributions for each variable to simplify calculations.
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\end{equation}
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The idea behind \vegas\ is to subdivide \(\Omega\) into hypercubes
|
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(create a grid), define \(\rho\) as step-function on those hypercubes
|
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and iteratively approximating \cref{eq:optimalrho}, instead of trying
|
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to minimize the variance directly~\cite{Lepage:19781an}. In the end,
|
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the samples are concentrated where \(f\) takes on the highest values
|
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and changes most rapidly. This is done by subdividing the hypercubes
|
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into smaller chunks, based on their contribution to the integral and
|
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then varying the hypercube borders until all hypercubes contain the
|
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same number of chunks. Note that no knowledge about the integrand is
|
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required. The probability density used by \vegas\ is given in
|
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\cref{eq:vegasrho} with \(K\) being the number of hypercubes and
|
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\(\Omega_i\) being the hypercubes themselves.
|
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(create a grid), define \(\rho\) as step-function with constant value
|
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on those hypercubes and iteratively approximating
|
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\cref{eq:optimalrho}, instead of trying to minimize the variance
|
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directly. In the end, the samples are concentrated where \(f\) takes
|
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on the highest values and changes most rapidly. This is done by
|
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subdividing the hypercubes into smaller chunks, based on their
|
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contribution to the integral, which is calculated through MC sampling,
|
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and then varying the hypercube borders until all hypercubes contain
|
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the same number of chunks. So if the contribution of a hypercube is
|
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large, it will be divided into more chunks than others. When the
|
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hypercube borders are then shifted, this hypercube will shrink, while
|
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others will grow by consuming the remaining chunks. Repeating this
|
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step in so called \vegas\ iterations will converge the contribution of
|
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each hypercube to the same value. More details about the algorithm can
|
||||
be found in~\cite{Lepage:19781an}. Note that no knowledge about the
|
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integrand is required. The probability density used by \vegas\ is
|
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given in \cref{eq:vegasrho} with \(K\) being the number of hypercubes
|
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and \(\Omega_i\) being the hypercubes themselves.
|
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|
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\begin{equation}
|
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\label{eq:vegasrho}
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|
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@ -206,18 +217,20 @@ required. The probability density used by \vegas\ is given in
|
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weighting applied (\(f/\rho\)).}
|
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\end{figure}
|
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|
||||
In one dimension the hypercubes become simple interval
|
||||
increments. Applying \vegas\ to \cref{eq:crossec} with
|
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This algorithm has been implemented in python and applied to
|
||||
\cref{eq:crossec}. In one dimension the hypercubes become simple
|
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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
|
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\result{xs/python/xs_mc_θ_vegas_N} samples. This result is comparable
|
||||
with tho 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 \cref{fig:xs-int-vegas}, along
|
||||
with the original integrand and it is intuitively clear, that the
|
||||
variance is being reduced. Smaller increments correspond to higher
|
||||
sample density and lower weights, flattening out the integrand.
|
||||
\result{xs/python/xs_mc_θ_vegas_N} function evaluations (including
|
||||
\vegas\ iterations). This result is comparable with tho 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
|
||||
\cref{fig:xs-int-vegas}, along with the original integrand and it is
|
||||
intuitively clear, that the variance is being reduced. Smaller
|
||||
increments correspond to higher sample density and lower weights,
|
||||
flattening out the integrand.
|
||||
|
||||
Generally the result gets better with more increments, but at the cost
|
||||
of more \vegas\ iterations. The intermediate values of those
|
||||
|
|
|
|||
|
|
@ -1,26 +1,32 @@
|
|||
\section{Monte-Carlo Sampling}%
|
||||
\section{Monte Carlo Sampling}%
|
||||
\label{sec:mcsamp}
|
||||
|
||||
Drawing representative samples from a probability distribution (for
|
||||
example a differential cross section) results in a set of
|
||||
\emph{events}, the same kind of data, that is gathered in experiments
|
||||
and from which one can the calculate samples from the distribution of
|
||||
other observables without explicit transformation of the
|
||||
distribution. Here the one-dimensional case is discussed.
|
||||
\emph{events}. This procedure is called sampling and produces the same
|
||||
kind of data, that is gathered in experiments and from which one can
|
||||
then calculate samples from the distribution of other observables
|
||||
event-by-event without explicit transformation of the
|
||||
distribution. Furthermore, these samples can be used as the basis for
|
||||
more involved simulations (see \cref{chap:pheno}). Sampling shares
|
||||
many characteristics with integration and thus the methods discussed
|
||||
here use similar terminology and often inherit their fundamental ideas
|
||||
from the integration methods.
|
||||
|
||||
Sampling a multi-dimensional distribution is equivalent to sampling
|
||||
one dimensional distributions by reducing the distribution itself to
|
||||
one variable through integration over the remaining variables and
|
||||
then, keeping the first variable fixed, sampling the other variables
|
||||
in a likewise manner.
|
||||
Here the one-dimensional case is discussed. Sampling a
|
||||
multi-dimensional distribution is equivalent to sampling one
|
||||
dimensional distributions by reducing the distribution itself to one
|
||||
variable through integration over the remaining variables and then,
|
||||
keeping the first variable fixed, sampling the other variables in a
|
||||
likewise manner.
|
||||
|
||||
Consider a function \(f\colon x\in\Omega\mapsto\mathbb{R}_{\geq 0}\)
|
||||
where \(\Omega = [0, 1]\) without loss of generality. Such a function
|
||||
is proportional to a probability density \(\tilde{f}\). When \(X\) is
|
||||
a uniformly distributed random variable on~\([0, 1]\) (which can be
|
||||
easily generated) then a sample \({x_i}\) of this variable can be
|
||||
easily generated), then a sample \({x_i}\) of this variable can be
|
||||
transformed into a sample of \(Y\sim\tilde{f}\). Let \(x\) be a single
|
||||
sample of \(X\), then a sample \(y\) of \(Y\) can be obtained by
|
||||
sample of \(X\). A sample \(y\) of \(Y\) can be obtained by
|
||||
solving \cref{eq:takesample} for \(y\).
|
||||
|
||||
\begin{equation}
|
||||
|
|
@ -35,10 +41,11 @@ to \cref{eq:takesample}, the probability that
|
|||
\int_{0}^{y'+\dd{y}'}f(x')\dd{x'}]\) which is
|
||||
\(A^{-1}\qty(\int_{0}^{y'+\dd{y}'}f(x')\dd{x'} -
|
||||
\int_{0}^{y'}f(x')\dd{x'}) = A^{-1} f(y')\dd{y}'\). So \(y\) is really
|
||||
distributed according to \(f/A\).
|
||||
distributed according to \(f/A=\tilde{f}\).
|
||||
|
||||
If the antiderivative \(F\) of is known, then the solution
|
||||
of \cref{eq:takesample} is given by \cref{eq:solutionsamp}.
|
||||
If the antiderivative \(F\) (and its inverse) of \(f\) is known, then
|
||||
the solution of \cref{eq:takesample} is given by
|
||||
\cref{eq:solutionsamp}.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:solutionsamp}
|
||||
|
|
@ -51,9 +58,9 @@ obtained numerically or one can change variables to simplify.
|
|||
|
||||
\subsection{Importance Sampling and Hit or Miss}%
|
||||
\label{sec:hitmiss}
|
||||
If integrating \(f\) and/or inverting \(F\) is too expensive or a
|
||||
fully \(f\)-agnostic method is desired, the problem can be
|
||||
reformulated by introducing a positive function
|
||||
If integrating \(f\) or inverting \(F\) is too expensive or a fully
|
||||
\(f\)-agnostic method is desired, the problem can be reformulated by
|
||||
introducing a positive function
|
||||
\(g\colon x\in\Omega\mapsto\mathbb{R}_{\geq 0}\) with
|
||||
\(\forall x\in\Omega\colon g(x)\geq f(x)\).
|
||||
|
||||
|
|
@ -71,8 +78,9 @@ probability~\(f/g\), so that \(g\) cancels out. This is known as the
|
|||
\end{equation}
|
||||
|
||||
The thus obtained samples are then distributed according to \(f/B\)
|
||||
and the total probability of accepting a sample (efficiency
|
||||
\(\mathfrak{e}\)) is given by hat \cref{eq:impsampeff} holds.
|
||||
and the total probability of accepting a sample, also called the
|
||||
efficiency \(\mathfrak{e}\), is given by hat \cref{eq:impsampeff}
|
||||
holds.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:impsampeff}
|
||||
|
|
@ -80,13 +88,7 @@ and the total probability of accepting a sample (efficiency
|
|||
\end{equation}
|
||||
|
||||
The closer the volumes enclosed by \(g\) and \(f\) are to each other,
|
||||
higher is \(\mathfrak{e}\). This method is called importance sampling
|
||||
|
||||
\begin{wrapfigure}[15]{l}{.5\textwidth}
|
||||
\plot{xs_sampling/upper_bound}
|
||||
\caption{\label{fig:distcos} The distribution \cref{eq:distcos} and an upper bound of
|
||||
the form \(a + b\cdot x^2\).}
|
||||
\end{wrapfigure}
|
||||
higher is \(\mathfrak{e}\).
|
||||
|
||||
Choosing \(g\) like \cref{eq:primitiveg} and looking back at
|
||||
\cref{eq:solutionsamp} yields \(y = x\cdot A\), so that the sampling
|
||||
|
|
@ -95,14 +97,20 @@ 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
|
||||
\(f\). Minimizing variance will therefore improve sampling
|
||||
performance. The method can also in higher dimensions be used without
|
||||
modification.
|
||||
performance. The method can also be used in higher dimensions without
|
||||
modification and has again been implemented and evaluated.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:primitiveg}
|
||||
g=\max_{x\in\Omega}f(x)=f_{\text{max}}
|
||||
\end{equation}
|
||||
|
||||
\begin{wrapfigure}[15]{l}{.5\textwidth}
|
||||
\plot{xs_sampling/upper_bound}
|
||||
\caption{\label{fig:distcos} The distribution \cref{eq:distcos} and an upper bound of
|
||||
the form \(a + b\cdot x^2\).}
|
||||
\end{wrapfigure}
|
||||
|
||||
Using the upper bound defined in \cref{eq:primitiveg} with the
|
||||
distribution for \(\cos\theta\) derived from the differential cross
|
||||
section \cref{eq:crossec} given in \cref{eq:distcos}
|
||||
|
|
@ -115,23 +123,21 @@ of~\result{xs/python/naive_th_samp}.
|
|||
\end{equation}
|
||||
|
||||
This very low efficiency stems from the fact, that \(f_{\cos\theta}\)
|
||||
is a lot smaller than its upper bound for most of the sampling
|
||||
interval.
|
||||
is a lot smaller than its maximum in most of the sampling interval.
|
||||
|
||||
Utilizing an upper bound of the form \(a + b\cdot x^2\) with \(a, b\)
|
||||
constant improves the efficiency
|
||||
to~\result{xs/python/tuned_th_samp}. The distribution, as well as the
|
||||
upper bound are depicted in \cref{fig:distcos}.
|
||||
|
||||
|
||||
\subsection{Importance Sampling through Change of Variables}%
|
||||
\label{sec:importsamp}
|
||||
|
||||
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 the context of sampling a distribution is called
|
||||
\emph{importance sampling}.
|
||||
way in is an alternative method of performing \emph{importance
|
||||
sampling}.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:transff}
|
||||
|
|
@ -140,31 +146,30 @@ way in the context of sampling a distribution is called
|
|||
|
||||
The optimal transformation would be the solution of \(y = F(x)\)
|
||||
(\(F\) being the antiderivative), so that
|
||||
\(f(x(y)) \cdot \dv{x(y)}{y} = 1\), which is the same as sampling in
|
||||
the way described in \cref{eq:solutionsamp}. This is no coincident as
|
||||
it can be shown, that this method is equivalent with the method
|
||||
described in \cref{sec:hitmiss} \cite{Lepage:19781an}. The technical
|
||||
trade-off of this method is that one has to chain it with some other
|
||||
sampling technique (\(\tilde{f}\) still has to be sampled). On the
|
||||
other hand the step of generating samples distributed according to
|
||||
\(g\) falls away.
|
||||
\(f(x(y)) \cdot \dv{x(y)}{y} = 1\). But transforming \(f\) in this way
|
||||
is the same as solving \cref{eq:takesample} which is a problem that
|
||||
has been addressed in \cref{sec:hitmiss}. The difference here is, that
|
||||
we restate the sampling problem in \(y\) space, which separates the
|
||||
step of converting our \(y\) samples to \(x\) samples away from the
|
||||
sampling process (see \cref{sec:obs}). Solving \cref{eq:takesample}
|
||||
may now be easier, or applying the hit or miss method may be more
|
||||
efficient.
|
||||
|
||||
When transforming the differential cross-section to the pseudo
|
||||
rapidity \(\eta\) the efficiency of the hit or miss method rises
|
||||
to~\result{xs/python/eta_eff} so applying this method in conjunction
|
||||
with others is worthwhile.
|
||||
with others is worthwhile (see \cref{fig:xs-int-comp}).
|
||||
|
||||
\subsection{Hit or Miss and \vegas}%
|
||||
\label{sec:stratsamp}
|
||||
|
||||
Finding a suitable upper bound or variable transform requires effort
|
||||
and detail knowledge about the distribution and is hard to
|
||||
automate\footnote{Sherpa does in fact do this by looking at the
|
||||
propagators in the matrix elements.}. Revisiting the idea
|
||||
behind \cref{eq:takesample2d} but looking at probability density
|
||||
\(\rho\) on \(\Omega\) leads to a slight reformulation of the method
|
||||
discussed in \cref{sec:hitmiss}. Note that without loss of generality
|
||||
one can again choose \(\Omega = [0, 1]\).
|
||||
Finding a suitable upper bound or variable transformation requires
|
||||
effort and detail knowledge about the distribution and is hard to
|
||||
automate. Revisiting the idea behind \cref{eq:takesample2d} but
|
||||
looking at a probability density \(\rho\) on \(\Omega\) leads to a
|
||||
slight reformulation of the method discussed in
|
||||
\cref{sec:hitmiss}. Note that without loss of generality one can again
|
||||
choose \(\Omega = [0, 1]\).
|
||||
|
||||
Define \(h=\max_{x\in\Omega}f(x)/\rho(x)\), take a sample
|
||||
\(\{\tilde{x}_i\}\sim\rho\) distributed according to \(\rho\) and
|
||||
|
|
@ -172,11 +177,7 @@ accept each sample point with the probability
|
|||
\(f(x_i)/(\rho(x_i)\cdot h)\). This is very similar to the procedure
|
||||
described in \cref{sec:hitmiss} with \(g=\rho\cdot h\), but here the
|
||||
step of generating samples distributed according to \(\rho\) is left
|
||||
out.
|
||||
|
||||
The important benefit of this method is, that step of generating
|
||||
samples according to some other function \(g\) is no longer
|
||||
necessary. This is useful when samples of \(\rho\) can be obtained
|
||||
out as these samples are assumed to be available or can be obtained
|
||||
with little effort (see below). The efficiency of this method is given
|
||||
by \cref{eq:strateff}.
|
||||
|
||||
|
|
@ -200,20 +201,21 @@ constraint \(\int_0^1\rho(x)\dd{x}=1\) to be considered.
|
|||
|
||||
The closer \(h\) approaches \(A\) the better the efficiency gets. In
|
||||
the optimal case \(\rho=f/A\) and thus \(h=A\) or
|
||||
\(\mathfrak{e} = 1\). Now this distribution can be approximated in the
|
||||
way discussed in \cref{sec:mcintvegas} by using the hypercubes found
|
||||
by~\vegas and simply generating the same number of uniformly
|
||||
distributed samples in each hypercube (stratified sampling). The
|
||||
distribution \(\rho\) takes on the form \cref{eq:vegasrho}. The
|
||||
effect of this approach is visualized in \cref{fig:vegasdist} and the
|
||||
resulting sampling efficiency \result{xs/python/strat_th_samp} (using
|
||||
\(\mathfrak{e} = 1\). This distribution can be approximated in the way
|
||||
discussed in \cref{sec:mcintvegas} by using the hypercubes found
|
||||
by~\vegas\ and simply generating the same number of uniformly
|
||||
distributed samples in each hypercube. The distribution \(\rho\) takes
|
||||
on the form \cref{eq:vegasrho}. The effect of this approach is
|
||||
visualized in \cref{fig:vegasdist} and the resulting sampling
|
||||
efficiency \result{xs/python/strat_th_samp} (using
|
||||
\result{xs/python/vegas_samp_num_increments} increments) is a great
|
||||
improvement over the hit or miss method in \cref{sec:hitmiss}. By using
|
||||
more increments better efficiencies can be achieved, although the
|
||||
run-time of \vegas\ increases. The advantage of \vegas\ in this
|
||||
situation is, that the computation of the increments has to be done
|
||||
only once and can be reused. Furthermore, no special knowledge about
|
||||
the distribution \(f\) is required.
|
||||
improvement over the hit or miss method in \cref{sec:hitmiss} and even
|
||||
surpasses the variable transformation to \(\eta\). By using more
|
||||
increments better efficiencies can be achieved, although the run-time
|
||||
of \vegas\ increases. The advantage of \vegas\ in this situation is,
|
||||
that the computation of the increments has to be done only once and
|
||||
can be reused. Furthermore, no special knowledge about the
|
||||
distribution \(f\) is required.
|
||||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
|
|
@ -223,8 +225,7 @@ the distribution \(f\) is required.
|
|||
differential cross-section and the \vegas-weighted
|
||||
distribution]{\label{fig:vegasdist} The distribution for
|
||||
\(\cos\theta\) (see \cref{eq:distcos}) and the \vegas-weighted
|
||||
distribution. The inc It is intuitively clear, how variance is
|
||||
being reduced.}
|
||||
distribution.}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}{.49\textwidth}
|
||||
\plot{xs_sampling/vegas_rho}
|
||||
|
|
@ -237,6 +238,45 @@ the distribution \(f\) is required.
|
|||
and weighting distribution.}
|
||||
\end{figure}
|
||||
|
||||
|
||||
|
||||
\subsection{Stratified Sampling}
|
||||
\label{sec:stratsamp-real}
|
||||
|
||||
Yet another approach is to subdivide the sampling volume \(\Omega\)
|
||||
into \(K\) sub-volumes \(\Omega_i\subset\Omega\) and then take a
|
||||
number of samples from each volume proportional to the integral of the
|
||||
function \(f\) in that volume. This is known as \emph{stratified
|
||||
sampling}. The advantage of this method is, that it is now possible
|
||||
to optimize the sampling in each sub-volume independently.
|
||||
|
||||
Let \(N\) be the total sample count (\(N\gg 1\)),
|
||||
\(A_i = \int_{\Omega_i}f(x)\dd{x}\) and \(A=\sum_iA_i\).
|
||||
Then we can calculate the total efficiency of taking \(N_i=A_i/A \cdot N\)
|
||||
samples in each is then given by \cref{eq:rstrateff}.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:rstrateff}
|
||||
\mathfrak{e} = \frac{\sum_i N_i}{\sum_i N_i/\mathfrak{e}_i} =
|
||||
\frac{\sum_i A_i/A\cdot N}{\sum_i A_i/A\cdot N/ \mathfrak{e}_i} = \frac{\sum_i A_i}{\sum_i A_i/\mathfrak{e}_i}
|
||||
\end{equation}
|
||||
|
||||
In the case when all \(\mathfrak{e}_i\) are the same, the total
|
||||
efficiency is the same as the individual efficiencies. In the case of
|
||||
one efficiency being much smaller then the others, this efficiency
|
||||
dominates the overall efficiency (assuming somewhat similar
|
||||
\(A_i\)). So in general one should optimize so that the individual
|
||||
efficiencies are roughly the same. Using the \(\Omega_i\) generated by
|
||||
\vegas\ has the advantage, that this requirement can be approximated
|
||||
and the \(A_i\) have already been obtained by \vegas. The technical
|
||||
difficulty in the implementation is the way in which sample points get
|
||||
distributed among the sub-volumes. The pitfall here is that the
|
||||
\(A_i\) (and the upper bounds for the hit-or-miss method) have to be
|
||||
determined increasingly accurate with growing sample size.
|
||||
|
||||
This method will be applied to multi-dimensional sampling in
|
||||
\cref{sec:pdf_results}.
|
||||
|
||||
\subsection{Observables}%
|
||||
\label{sec:obs}
|
||||
|
||||
|
|
@ -245,11 +285,10 @@ observables can be calculated from those samples without having to
|
|||
transform the distribution into new variables. This is due to the
|
||||
discrete nature of the samples. Suppose there is an observable
|
||||
\(\gamma\colon\Omega\mapsto\mathbb{R}\). Now to take a sample
|
||||
\(\{x_i\}\) of \(\gamma\) we sample \(f\)\footnote{As defined
|
||||
in \cref{sec:mcsamp}.} and convert the sample values by simply
|
||||
applying \(\gamma\). This is equivalent to
|
||||
substituting \(y=\gamma^{-1}(z)\) in \cref{eq:takesample} and solving
|
||||
for \(z\).
|
||||
\(\{x_i\}\) of \(\gamma\) we sample \(f\) and convert the sample
|
||||
values by simply applying \(\gamma\), where \(f\) is as defined in
|
||||
\cref{sec:mcsamp}. This is equivalent to substituting
|
||||
\(y=\gamma^{-1}(z)\) in \cref{eq:takesample} and solving for \(z\).
|
||||
|
||||
The probability that \(z\in[z', z'+\dd{z'}]\) now is the same as the
|
||||
probability that
|
||||
|
|
@ -273,8 +312,8 @@ as described in \cref{eq:observables}.
|
|||
|
||||
\begin{align}
|
||||
\label{eq:observables}
|
||||
\pt &= \sqrt{(p_1)^2+(p_2)^2} & \eta &=
|
||||
\frac{\abs{\vb{p}}}{\pt}\cdot\sign(p^3)
|
||||
\pt &= \sqrt{(p_x)^2+(p_y)^2} & \eta &=
|
||||
\arcosh\qty(\frac{\abs{\vb{p}}}{\pt})\cdot\sign(p_z)
|
||||
\end{align}
|
||||
|
||||
The histograms in \cref{fig:histos} have been created by generating
|
||||
|
|
@ -284,7 +323,7 @@ contains reference histograms created by generating events with
|
|||
toolkit~\cite{Bierlich:2019rhm}. The utilized analysis can be found
|
||||
in \cref{sec:simpdiphotriv}.
|
||||
|
||||
\begin{figure}[hb]
|
||||
\begin{figure}[ht]
|
||||
\centering \plot{xs_sampling/diff_xs_p_t}
|
||||
\caption{\label{fig:diff-xs-pt} The differential cross section
|
||||
transformed to \(\pt\).}
|
||||
|
|
@ -309,53 +348,17 @@ in \cref{sec:simpdiphotriv}.
|
|||
\end{figure}
|
||||
|
||||
Where \cref{fig:histeta} shows clear resemblance of
|
||||
\cref{fig:xs-int-eta}, the sharp peak in \cref{fig:histpt} around
|
||||
\(\pt=\SI{100}{\giga\electronvolt}\) seems surprising. When
|
||||
transforming the differential cross section to \(\pt\) it can be seen
|
||||
in \cref{fig:diff-xs-pt} that there really is a singularity at
|
||||
\(\pt =\abs{\vb{p}}\). This singularity will vanish once considering a
|
||||
more realistic process (see \cref{chap:pdf}). Furthermore the
|
||||
histograms \cref{fig:histeta,fig:histpt} are consistent
|
||||
with their \rivet-generated counterparts and are therefore considered
|
||||
valid.
|
||||
|
||||
|
||||
\subsection{Stratified Sampling}
|
||||
\label{sec:stratsamp-real}
|
||||
|
||||
A different approach is to subdivide the sampling volume \(\Omega\)
|
||||
into \(K\) sub-volumes \(\Omega_i\subset\Omega\) and then take a
|
||||
number of samples from each volume proportional to the integral of the
|
||||
function \(f\)\footnote{As defined in \cref{sec:mcsamp}.} in that
|
||||
volume. The advantage of this method is, that it is now possible to
|
||||
optimize the sampling in each sub-volume independently.
|
||||
|
||||
Let \(N\) be the total sample count (\(N\gg 1\)),
|
||||
\(A_i = \int_{\Omega_i}f(x)\dd{x}\) and \(A=\sum_iA_i\).
|
||||
Then we can calculate the total efficiency of taking \(N_i=A_i/A \cdot N\)
|
||||
samples in each is then given by \cref{eq:rstrateff}.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:rstrateff}
|
||||
\mathfrak{e} = \frac{\sum_i N_i}{\sum_i N_i/\mathfrak{e}_i} =
|
||||
\frac{\sum_i A_i/A\cdot N}{\sum_i A_i/A\cdot N/ \mathfrak{e}_i} = \frac{\sum_i A_i}{\sum_i A_i/\mathfrak{e}_i}
|
||||
\end{equation}
|
||||
|
||||
In the case when all \(\mathfrak{e}_i\) are the same, the total
|
||||
efficiency is the same as the individual efficiencies. In the case of
|
||||
one efficiency being much smaller then the others, this efficiency
|
||||
dominates the overall efficiency (assuming somewhat similar
|
||||
\(A_i\)). So in general one should optimize so that the individual
|
||||
efficiencies are roughly the same. Using the \(\Omega_i\) generated by
|
||||
\vegas\ has the advantage, that exactly this requirement is fulfilled
|
||||
and the \(A_i\) have already been obtained by \vegas. The technical
|
||||
difficulty in the implementation is the way in which sample points get
|
||||
distributed among the sub-volumes. The pitfall here is that the
|
||||
\(A_i\) (and the upper bounds for the hit-or-miss method) have to be
|
||||
determined increasingly accurate with growing sample size.
|
||||
|
||||
This method will be applied to multi-dimensional sampling in
|
||||
\cref{sec:pdf_results}.
|
||||
\cref{fig:xs-int-eta}, the peak and the rise before this peak in
|
||||
\cref{fig:histpt} around \(\pt=\SI{100}{\giga\electronvolt}\) seems
|
||||
surprising and opens the possibility for the production of hard
|
||||
transverse photons. When transforming the differential cross section
|
||||
to \(\pt\) it can be seen in \cref{fig:diff-xs-pt} that there really
|
||||
is a singularity at \(\pt = \ecm\), due to a term
|
||||
\(1/\sqrt{1-(2\cdot \pt/\ecm)^2}\) stemming from the Jacobian
|
||||
determinant. This singularity will vanish once considering a more
|
||||
realistic process (see \cref{chap:pdf}). Furthermore the histograms
|
||||
\cref{fig:histeta,fig:histpt} are consistent with their
|
||||
\rivet-generated counterparts and are therefore considered valid.
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
|
|
|
|||
|
|
@ -5,13 +5,13 @@
|
|||
Because free quarks do not occur in nature, one has to study the
|
||||
scattering of hadrons to obtain experimentally verifiable
|
||||
results. Hadrons are usually modeled as consisting of multiple
|
||||
\emph{partons} using Parton Density Functions (PDFs). By applying a
|
||||
simple PDF, the cross section for the process \(\ppgg\) on the
|
||||
matrix-element~\cite[14]{buckley:2011ge} level\footnote{Neglecting the
|
||||
remnants and other processes like parton showers, primordial
|
||||
transverse momentum and multiple interactions.} and event samples of
|
||||
that process have been obtained. These results are once again be
|
||||
compared with results from \sherpa.
|
||||
\emph{partons} (i.e. quarks and gluons) using Parton Density Functions
|
||||
(PDFs). By using a leading order PDF, the cross section for the
|
||||
process \(\ppgg\) on the matrix-element~\cite[14]{buckley:2011ge}
|
||||
level\footnote{Neglecting the remnants and other processes like parton
|
||||
showers, primordial transverse momentum and multiple interactions.}
|
||||
and event samples of that process are obtained. These results are
|
||||
being compared with results from \sherpa.
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
|
|
|
|||
|
|
@ -2,7 +2,7 @@
|
|||
\label{sec:lab_xs}
|
||||
|
||||
To utilize \cref{eq:pdf-xs} for modeling the~\(\ppgg\) process and to
|
||||
generate event samples the results of \cref{chap:qqgg} have to be
|
||||
generate event samples, the results of \cref{chap:qqgg} have to be
|
||||
transformed into the center of momentum frame of the colliding
|
||||
protons. Quantities in the center of momentum frame of the partons
|
||||
will be starred (like \(x^\ast\)).
|
||||
|
|
|
|||
|
|
@ -2,10 +2,11 @@
|
|||
\label{sec:pdf_basics}
|
||||
|
||||
Parton Density Functions encode, restricting considerations to leading
|
||||
order, the probability to \emph{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 to
|
||||
\cref{eq:pdf-norm}, where the sum runs over all partons.
|
||||
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 \(Q^2\) in a scattering process. PDFs are
|
||||
normalized according to \cref{eq:pdf-norm}, where the sum runs over
|
||||
all partons.
|
||||
|
||||
\begin{equation}
|
||||
\label{eq:pdf-norm}
|
||||
|
|
@ -14,23 +15,24 @@ certain factorization scale \(Q^2\). PDFs are normalized according to
|
|||
|
||||
More precisely \({f_i}\) denotes a PDF set, which is referred to
|
||||
simply as PDF in the following. PDFs can not be derived from first
|
||||
principles (at the moment) and have to be determined experimentally
|
||||
for low \(Q^2\) and are evolved to higher \(Q^2\) through the
|
||||
\emph{DGLAP} equations~\cite{altarelli:1977af} at different orders of
|
||||
perturbation theory. In deep inelastic scattering \(Q^2\) is just the
|
||||
negative over the momentum transfer \(-q^2\). For more complicated
|
||||
processes \(Q^2\) has to be chosen in a way that reflects the
|
||||
\emph{momentum resolution} of the process. If the perturbation series
|
||||
behind the process would be expanded to the exact solution, the
|
||||
principles and have to be determined experimentally for low \(Q^2\)
|
||||
and are evolved to higher \(Q^2\) through the \emph{DGLAP}
|
||||
equations~\cite{altarelli:1977af} at different orders of perturbation
|
||||
theory. In deep inelastic scattering \(Q^2\) is just the negative
|
||||
over the momentum transfer \(-q^2\). For more complicated processes
|
||||
\(Q^2\) has to be chosen in a way that reflects the
|
||||
\emph{energy-momentum scale} of the process. If the perturbation
|
||||
series behind the process would be expanded to the exact solution, the
|
||||
dependence on the factorization scale vanishes. In lower orders, one
|
||||
has to choose the scale in a \emph{physically
|
||||
meaningful}\footnote{That means: not in an arbitrary way.} way,
|
||||
which reflects characteristics of the process~\cite{altarelli:1977af}.
|
||||
|
||||
In the case of \(\qqgg\) the mean of the Mandelstam variables \(\hat{t}\)
|
||||
and \(\hat{u}\), which is equal to \(\hat{s}/2\), can be used. This
|
||||
choice is lorentz-invariant and reflects the s/u-channel nature of the
|
||||
process.
|
||||
In the case of \(\qqgg\) the mean of the Mandelstam variables
|
||||
\(\hat{t}\) and \(\hat{u}\), which is equal to \(\hat{s}/2\), can be
|
||||
used. This choice is lorentz-invariant and reflects the s/u-channel
|
||||
nature of the process, although the \(\pt\) of photon would also have
|
||||
been a good choice~\cite[18]{buckley:2011ge}.
|
||||
|
||||
The (differential) hadronic cross section for scattering of two
|
||||
partons in equal hadrons is given in \cref{eq:pdf-xs}. Here \(i,j\)
|
||||
|
|
|
|||
|
|
@ -25,10 +25,10 @@ is justified in \cref{sec:pdf_basics} and formulated in
|
|||
\end{gather}
|
||||
|
||||
The PDF set being used in the following has been fitted (and
|
||||
developed) 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}.
|
||||
\emph{DGLAP} developed) 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,Buckley:2015lh}.
|
||||
|
||||
\subsection{Cross Section}%
|
||||
\label{sec:ppxs}
|
||||
|
|
@ -36,9 +36,9 @@ library~\cite{NNPDF:2017pd}\cite{Buckley:2015lh}.
|
|||
The distribution \cref{eq:weighteddist} can now be integrated to
|
||||
obtain a total cross-section as described in \cref{sec:mcint}. For
|
||||
the numeric analysis a proton beam energy of
|
||||
\result{xs/python/pdf/e_proton} has been chosen, in accordance to
|
||||
\lhc{} beam energies. As for the cuts, \result{xs/python/pdf/eta} and
|
||||
\result{xs/python/pdf/min_pT} have been set. Integrating
|
||||
\result{xs/python/pdf/e_proton} has been chosen, in resemblance to
|
||||
\lhc{} beam energies. The cuts \result{xs/python/pdf/eta} and
|
||||
\result{xs/python/pdf/min_pT} have been imposed. Integrating
|
||||
\cref{eq:weighteddist} with respect to those cuts using \vegas\ yields
|
||||
\result{xs/python/pdf/my_sigma} which is compatible with
|
||||
\result{xs/python/pdf/sherpa_sigma}, the value \sherpa\ gives.
|
||||
|
|
@ -47,19 +47,19 @@ the numeric analysis a proton beam energy of
|
|||
\label{sec:ppevents}
|
||||
|
||||
Generating events of \(\ppgg\) is very similar in principle to
|
||||
sampling partonic cross section. 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
|
||||
sampling the partonic cross section. 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 and the final state particles have to be
|
||||
isolated from the beams.
|
||||
momentum \(p_T\) of a final state photon to be greater
|
||||
than~\SI{20}{\giga\electronvolt}. A restriction (cut) on \(p_T\) is
|
||||
suitable because detectors are usually only sensitive above a certain
|
||||
\(p_T\) threshold and the final state particles have to be isolated
|
||||
from the beams.
|
||||
|
||||
The resulting distribution (without cuts) is depicted in
|
||||
\cref{fig:dist-pdf} for fixed \(x_2\) and in
|
||||
|
|
@ -67,7 +67,7 @@ The resulting distribution (without cuts) is depicted in
|
|||
the distribution retains some likeness with the partonic distribution
|
||||
(see \cref{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
|
||||
sub-optimal for hit-or-miss sampling, only having significant magnitude
|
||||
when \(x_1\) or \(x_2\) are small (\cref{fig:dist-pdf-fixed-eta}) and
|
||||
being very steep.
|
||||
|
||||
|
|
@ -94,17 +94,43 @@ To remedy that, one has to use a more efficient sampling algorithm
|
|||
(\vegas) or impose very restrictive cuts. The self-coded
|
||||
implementation used here can be found in \cref{sec:pycode} and employs
|
||||
stratified sampling (as discussed in \cref{sec:stratsamp-real}) and
|
||||
the hit-or-miss method. Because the stratified sampling requires very
|
||||
accurate upper bounds, they have been overestimated by
|
||||
\result{xs/python/pdf/overesimate}, which lowers the efficiency
|
||||
slightly but reduces bias. The MC integrator was used to
|
||||
estimate the location of the maximum in each hypercube and then this
|
||||
estimate was improved by gradient ascend\footnote{Which becomes
|
||||
problematic, when performed close to a cut.}.
|
||||
the hit-or-miss method. The matrix element (ME) and cuts are
|
||||
implemented using \texttt{cython}~\cite{behnel2011:cy} to obtain
|
||||
better performance as these are evaluated very often. The ME and the
|
||||
cuts are then convolved with the PDF (as in \cref{eq:weighteddist})
|
||||
and wrapped into a simple function with a generic interface and
|
||||
plugged into the \vegas\ implementation which then computes the
|
||||
integral, grid, individual contributions to the grid and rough
|
||||
estimates of the maxima in each hypercube. In principle the code could
|
||||
be generalized to other processes by simply redefining the matrix
|
||||
elements, as no other part of the code is process specific. The cuts
|
||||
work as simple \(\theta\)-functions, which has the advantage, that the
|
||||
maximum for hit or miss can be chosen with respect to those cuts. On
|
||||
the other hand, this method introduces discontinuity into the
|
||||
integrand, which is problematic for numeric maximizers. The estimates
|
||||
of the maxima, provided by the \vegas\ implementation used as the
|
||||
starting point for a gradient ascend maximizer. In this way, the
|
||||
discontinuities introduced by the cuts got circumvented. Because the
|
||||
stratified sampling requires very accurate upper bounds, they have
|
||||
been overestimated by \result{xs/python/pdf/overesimate}\!, which
|
||||
lowers the efficiency slightly but reduces bias. The sampling
|
||||
algorithm chooses hypercubes randomly in accordance to their
|
||||
contribution to the integral by generating a uniformly distributed
|
||||
random number \(r\in [0,1]\) and summing the weights of the hypercubes
|
||||
until the sum exceeds this number. The last hypercube in this sum is
|
||||
then chosen and one sample is obtained. Taking more than one sample
|
||||
can improve performance, but introduces bias, as hypercubes with low
|
||||
weight may be oversampled. At various points, the
|
||||
\texttt{numba}~\cite{lam2015:po} package has been used to just-in-time
|
||||
compile code to increase performance. The python
|
||||
\texttt{multiprocessing} module is used to parallelize the sampling
|
||||
and exploit all CPU cores. Although the \vegas\ step is very time
|
||||
intensive, but the actual sampling performance is in the same order of
|
||||
magnitude as \sherpa, but some parameters have to be manually tuned.
|
||||
|
||||
A sample of \result{xs/python/pdf/sample_size} events has been
|
||||
generated both in \sherpa\ and through own code. The resulting
|
||||
histograms of some observables are depicted in
|
||||
generated both in \sherpa\ (with the same cuts) and through own
|
||||
code. The resulting histograms of some observables are depicted in
|
||||
\cref{fig:pdf-histos}. The sampling efficiency achieved was
|
||||
\result{xs/python/pdf/samp_eff} using a total of
|
||||
\result{xs/python/pdf/num_increments} hypercubes. As can be seen, the
|
||||
|
|
@ -119,42 +145,41 @@ 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
|
||||
\(m_{\gamma\gamma} = (p_{\gamma,1} + p_{\gamma,1})^2\) is the center
|
||||
\(m_{\gamma\gamma} = (p_{\gamma,1} + p_{\gamma,2})^2\) is the center
|
||||
of mass energy of the partonic system that produces the photons (see
|
||||
\cref{eq:ecm_partons}) and proportional to the product of the momentum
|
||||
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. Due
|
||||
to the \(\pt\) cuts the first bin is slightly lower then the second.
|
||||
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.
|
||||
|
||||
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 photons and the z axis in the c.m. frame if
|
||||
this frame can be reached by a boost along the z axis\footnote{Or me
|
||||
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 therefore the system is symmetric around the beam axis
|
||||
and therefore this boost is possible. When allowing transverse parton
|
||||
momenta, as will be done in % TODO: REFERENCE
|
||||
this symmetry goes away. Defining the z-axis as one beam axis in a
|
||||
frame would be a quite arbitrary choice that disrespects the symmetry
|
||||
of the two beams considered here (same energy, identical protons).
|
||||
Also the random direction of the transverse momentum can add noise
|
||||
that does not contain much information. The CS frame is defined as the
|
||||
rest frame of the two outgoing photons in which the z-axis bisects the
|
||||
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
|
||||
beams considered here (same energy, identical protons). Also the
|
||||
random direction of the transverse momentum can add noise that does
|
||||
not contain much information. The CS frame is defined as the rest
|
||||
frame of the two outgoing photons in which the z-axis bisects the
|
||||
angle between the first beam momentum and the inverse momentum of the
|
||||
second beam. The azimuth angle is measure with respect to a vector
|
||||
perpendicular to the plane of the beams (which is parallel to the
|
||||
transverse momentum in the lab frame). In this frame, which was
|
||||
originally chosen to simplify the extension of the Drell-Yan parton
|
||||
model to transverse parton momenta~\cite{collins:1977an}, some
|
||||
symmetry is restored and the study of effects of transverse parton
|
||||
momenta is facilitated. Because of the above-mentioned symmetry, the
|
||||
histograms in \cref{fig:pdf-o-angle,fig:pdf-o-angle-cs} are the
|
||||
same. One would naively expect some likeness to \cref{fig:distcos} but
|
||||
the cuts imposed alter the distribution quite considerably, cutting of
|
||||
the \(\cos\theta^\ast\rightarrow 1\)
|
||||
region. % TODO: come back to that in next chapter
|
||||
second beam. In this frame, which was originally conceived to simplify
|
||||
the extension of the Drell-Yan parton model to transverse parton
|
||||
momenta~\cite{collins:1977an}, some symmetry is restored and the study
|
||||
of effects of transverse parton momenta is facilitated. Because of the
|
||||
above-mentioned symmetry, the histograms in
|
||||
\cref{fig:pdf-o-angle,fig:pdf-o-angle-cs} are the same. One would
|
||||
naively expect some likeness to \cref{fig:distcos} but the cuts
|
||||
imposed alter the distribution quite considerably, cutting of the
|
||||
\(\cos\theta^\ast\rightarrow 1\) region.
|
||||
% TODO: come back to that in next chapter TODO: graphic (tikz)
|
||||
|
||||
\begin{align}
|
||||
\cos\theta^\ast &= \tanh\frac{\eta_1 - \eta_2}{2} \label{eq:sangle}\\
|
||||
|
|
|
|||
|
|
@ -12,18 +12,17 @@ or less hard processes (Multiple Interactions, MI) and affect the hard
|
|||
process through color correlation. All of the processes not directly
|
||||
connected to the hard process are called the underlying event and have
|
||||
to be taken into account to generate events that can be compared with
|
||||
experimental data, as they form a measurable background. Finally the
|
||||
partons from the showers recombine into hadrons (Hadronization) due to
|
||||
confinement. This last effect doesn't produce diphoton-relevant
|
||||
background directly, but affects photon
|
||||
experimental data. Finally the partons from the showers recombine into
|
||||
hadrons (Hadronization) due to confinement. This last effect doesn't
|
||||
produce diphoton-relevant background directly, but affects photon
|
||||
isolation.~\cite[11]{buckley:2011ge} % TODO: describe isolation
|
||||
|
||||
These effects can be calculated or modeled on an per-event base by
|
||||
modern monte-carlo event generators like \sherpa. This has been done
|
||||
modern monte-carlo event generators like \sherpa\footnote{But these
|
||||
calculations and models are always approximations.}. This is done
|
||||
for the diphoton process in a gradual way described in
|
||||
\cref{sec:setupan}. Histograms of observables have been generated and
|
||||
are being discussed in \cref{sec:disco}.
|
||||
|
||||
\cref{sec:setupan}. Histograms of observables are generated and are
|
||||
being discussed in \cref{sec:disco}.
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
|
|
|
|||
|
|
@ -7,15 +7,15 @@ are incremental in the sense that each subsequent configuration
|
|||
extents the previous one.
|
||||
|
||||
\begin{description}
|
||||
\item[Basic] The hard process on parton level as used in \cref{sec:pdf_results}.
|
||||
\item[PS] The shower generator of \sherpa, \emph{CSS} (dipole-shower),
|
||||
\item[LO] The hard process on parton level as used in \cref{sec:pdf_results}.
|
||||
\item[LO+PS] The shower generator of \sherpa, \emph{CSS} (dipole-shower),
|
||||
is activated and simulates initial state radiation, as there are no
|
||||
partons in the final state yet.
|
||||
\item[PS+pT] The beam remnants and intrinsic parton
|
||||
\item[LO+PS+pT] The beam remnants and intrinsic parton
|
||||
\(\pt\) are simulated, giving rise to final state radiation.
|
||||
\item[PS+pT+Hadronization] A cluster hadronization model
|
||||
\item[LO+PS+pT+Hadronization] A cluster hadronization model
|
||||
implemented in \emph{Ahadic} is activated.
|
||||
\item[PS+pT+Hadronization+MI] Multiple interactions based on the
|
||||
\item[LO+PS+pT+Hadronization+MI] Multiple interactions based on the
|
||||
Sj\"ostrand-van-Zijl are simulated.
|
||||
\end{description}
|
||||
|
||||
|
|
@ -31,14 +31,19 @@ as \SI{6500}{\giga\electronvolt} to resemble \lhc\ conditions.
|
|||
The analysis is similar to the one used in \cref{sec:ppevents} with
|
||||
the addition of the observable of total transverse momentum of the
|
||||
photon pair, which now can be non-zero. Because the final state now
|
||||
contains multiple photons, the two photons with the highest \(\pt\)
|
||||
(leading photons) are selected. Furthermore a cone of
|
||||
potentially contains additional photons from hadron decays, the
|
||||
analysis only selects prompt photons with the highest \(\pt\) (leading
|
||||
photons). Furthermore a cone of
|
||||
\(R = \sqrt{\qty(\Delta\varphi)^2 + \qty(\Delta\eta)^2} = 0.4\) around
|
||||
each photon must not contain more than \SI{4.5}{\percent}
|
||||
(\(+ \SI{6}{\giga\electronvolt}\)), so that photons stemming from
|
||||
showers are filtered out. For similar reasons the leading photons are
|
||||
required to have \(\Delta R > 0.45\). The code of the analysis is
|
||||
listed in \cref{sec:ppanalysisfull}.
|
||||
each photon must not contain more than \SI{4.5}{\percent} of the
|
||||
photon transverse momentum (\(+ \SI{6}{\giga\electronvolt}\)),
|
||||
attempting to exclude photons stemming from hadron decay are filtered
|
||||
out. The leading photons are required to have \(\Delta R > 0.45\), to
|
||||
filter out colinear photons, as they likely stem from hadron
|
||||
decays. In truth, the analysis already excludes such photons, but to
|
||||
be compatible with experimental data, which must rely on such
|
||||
criteria, they have been included. The code of the analysis is listed
|
||||
in \cref{sec:ppanalysisfull}.
|
||||
|
||||
% TODO: refer back to this in discussion
|
||||
% TODO: cite analysis template
|
||||
|
|
|
|||
|
|
@ -1,10 +1,10 @@
|
|||
\section{Calculation of the Cross Section to first Order}%
|
||||
\section{Calculation of the Cross Section to Leading Order}%
|
||||
\label{sec:qqggcalc}
|
||||
|
||||
After labeling the incoming quarks and outcoming photons, as well as
|
||||
the momenta according to \cref{fig:qqggfeyn}, the feynman rules yield
|
||||
the matrix elements in \cref{eq:matel}, where \(Z\) is the electric
|
||||
charge of the quark and \(g\) is the QED coupling constant. The
|
||||
charge number of the quark and \(g\) is the QED coupling constant. The
|
||||
respective spinors and polarisation vectors are \(\us,\vs\) and
|
||||
\(\pe\). Parenthesis are being used, whenever indices would clutter
|
||||
the notation. The matrix element for \cref{fig:qqggfeyn2} is obtained
|
||||
|
|
|
|||
|
|
@ -27,24 +27,35 @@ two identical photons in the final state.
|
|||
|
||||
\begin{figure}[ht]
|
||||
\centering
|
||||
\begin{subfigure}[c]{.45\textwidth}
|
||||
\begin{subfigure}[t]{.49\textwidth}
|
||||
\centering \plot{xs/diff_xs_zoom}
|
||||
|
||||
\caption[Plot of the differential cross section of the \(\qqgg\)
|
||||
process.]{\label{fig:diffxs_zoom} The differential cross section as a
|
||||
function of the polar angle \(\theta\) in the crucial region.}
|
||||
\end{subfigure}
|
||||
\begin{subfigure}[t]{.49\textwidth}
|
||||
\centering \plot{xs/diff_xs}
|
||||
\caption[Plot of the differential cross section of the \(\qqgg\)
|
||||
process.]{\label{fig:diffxs} The differential cross section as a
|
||||
function of the polar angle \(\theta\). }
|
||||
function of the polar angle \(\theta\) over the full integration
|
||||
interval. }
|
||||
\end{subfigure}
|
||||
\begin{subfigure}[c]{.45\textwidth}
|
||||
\begin{subfigure}[t]{.49\textwidth}
|
||||
\centering
|
||||
\plot{xs/total_xs}
|
||||
\caption[Plot of the total cross section of the \(\qqgg\)
|
||||
process.]{\label{fig:totxs} The cross section
|
||||
(\cref{eq:total-crossec}) of the process for a pseudo-rapidity
|
||||
\cref{eq:total-crossec} of the process for a pseudo-rapidity
|
||||
integrated over \([-\eta, \eta]\).}
|
||||
\end{subfigure}
|
||||
\end{subfigure}
|
||||
\caption{\label{fig:xsfirst} Plots of the differntial and total cross section
|
||||
for \(\qqgg\).}
|
||||
\end{figure}
|
||||
|
||||
The differential cross section \cref{eq:crossec} (see also
|
||||
\cref{fig:diffxs}) is divergent for angles near zero or \(\pi\). At
|
||||
\cref{fig:diffxs}) is divergent for angles near zero or \(\pi\) but
|
||||
remains finite in the physical region (see \cref{fig:diffxs_zoom}). At
|
||||
small scattering angles the absolute square of the momentum carried by
|
||||
the virtual quark in \cref{fig:qqggfeyn} goes to zero, which in the
|
||||
mass-less limit means that the virtual quark comes \emph{on-shell} and
|
||||
|
|
@ -84,15 +95,16 @@ As can be seen in \cref{fig:totxs}, the cross section, integrated over
|
|||
an interval of \([-\eta, \eta]\), is dominated by the linear
|
||||
contributions in \cref{eq:total-crossec} and would result in an
|
||||
infinity if no cut on \(\eta\) would be imposed. Choosing
|
||||
\(\eta\in [-2.5,2.5]\) and \(\ecm=\SI{100}{\giga\electronvolt}\) the
|
||||
\result{xs/python/eta} and \result{xs/python/ecm} the
|
||||
process was MC integrated in \sherpa\ using the runcard in
|
||||
\cref{sec:qqggruncard}. This runcard describes the exact same (leading
|
||||
order) process as the calculated cross section.
|
||||
|
||||
\sherpa\ arrives at the cross section \result{xs/sherpa_xs}. Plugging
|
||||
the same parameters into \cref{eq:total-crossec} gives
|
||||
\result{xs/python/xs} which is within the uncertainty range of the
|
||||
\sherpa\ value. This verifies the result for the total cross section.
|
||||
\sherpa\ arrives at the cross section
|
||||
\result{xs/python/xs_sherpa}. Plugging the same parameters into
|
||||
\cref{eq:total-crossec} gives \result{xs/python/xs} which is within
|
||||
the uncertainty range of the \sherpa\ value. This verifies the result
|
||||
for the total cross section.
|
||||
|
||||
%%% Local Variables:
|
||||
%%% mode: latex
|
||||
|
|
|
|||
|
|
@ -202,3 +202,65 @@
|
|||
collision data collected by the ATLAS experiment},
|
||||
journal = {Journal of High Energy Physics}
|
||||
}
|
||||
|
||||
@article{2020Virtanen:Sc,
|
||||
author = {{Virtanen}, Pauli and {Gommers}, Ralf and
|
||||
{Oliphant}, Travis E. and {Haberland}, Matt and
|
||||
{Reddy}, Tyler and {Cournapeau}, David and
|
||||
{Burovski}, Evgeni and {Peterson}, Pearu and
|
||||
{Weckesser}, Warren and {Bright}, Jonathan and {van
|
||||
der Walt}, St{\'e}fan J. and {Brett}, Matthew and
|
||||
{Wilson}, Joshua and {Jarrod Millman}, K. and
|
||||
{Mayorov}, Nikolay and {Nelson}, Andrew R.~J. and
|
||||
{Jones}, Eric and {Kern}, Robert and {Larson}, Eric
|
||||
and {Carey}, CJ and {Polat}, {\.I}lhan and {Feng},
|
||||
Yu and {Moore}, Eric W. and {Vand erPlas}, Jake and
|
||||
{Laxalde}, Denis and {Perktold}, Josef and
|
||||
{Cimrman}, Robert and {Henriksen}, Ian and
|
||||
{Quintero}, E.~A. and {Harris}, Charles R and
|
||||
{Archibald}, Anne M. and {Ribeiro}, Ant{\^o}nio
|
||||
H. and {Pedregosa}, Fabian and {van Mulbregt}, Paul
|
||||
and {Contributors}, SciPy 1. 0},
|
||||
title = "{SciPy 1.0: Fundamental Algorithms for Scientific
|
||||
Computing in Python}",
|
||||
journal = {Nature Methods},
|
||||
year = "2020",
|
||||
volume = {17},
|
||||
pages = {261--272},
|
||||
adsurl = {https://rdcu.be/b08Wh},
|
||||
doi = {https://doi.org/10.1038/s41592-019-0686-2},
|
||||
}
|
||||
|
||||
@article{behnel2011:cy,
|
||||
title = {Cython: The best of both worlds},
|
||||
author = {Behnel, Stefan and Bradshaw, Robert and Citro, Craig
|
||||
and Dalcin, Lisandro and Seljebotn, Dag Sverre and
|
||||
Smith, Kurt},
|
||||
journal = {Computing in Science \& Engineering},
|
||||
volume = {13},
|
||||
number = {2},
|
||||
pages = {31--39},
|
||||
year = {2011},
|
||||
publisher = {IEEE}
|
||||
}
|
||||
|
||||
@inproceedings{lam2015:po,
|
||||
author = {Lam, Siu Kwan and Pitrou, Antoine and Seibert,
|
||||
Stanley},
|
||||
title = {Numba: A LLVM-Based Python JIT Compiler},
|
||||
year = {2015},
|
||||
isbn = {9781450340052},
|
||||
publisher = {Association for Computing Machinery},
|
||||
address = {New York, NY, USA},
|
||||
url = {https://doi.org/10.1145/2833157.2833162},
|
||||
doi = {10.1145/2833157.2833162},
|
||||
booktitle = {Proceedings of the Second Workshop on the LLVM
|
||||
Compiler Infrastructure in HPC},
|
||||
articleno = {7},
|
||||
numpages = {6},
|
||||
keywords = {LLVM, Python, compiler},
|
||||
location = {Austin, Texas},
|
||||
series = {LLVM ’15}
|
||||
}
|
||||
|
||||
|
||||
|
|
|
|||
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|
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|
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|
|
@ -156,13 +156,24 @@ esp = 200 # GeV
|
|||
|
||||
#+RESULTS:
|
||||
|
||||
Let's save that stuff.
|
||||
#+begin_src jupyter-python :exports both :results raw drawer
|
||||
tex_value(η, prefix=r"\abs{\eta}\leq ", prec=1, save=("results", "eta.tex"))
|
||||
tex_value(
|
||||
esp, prefix=r"\ecm = ", unit=r"\giga\electronvolt", save=("results", "ecm.tex")
|
||||
)
|
||||
#+end_src
|
||||
|
||||
|
||||
#+RESULTS:
|
||||
: \(\ecm = \SI{200}{\giga\electronvolt}\)
|
||||
|
||||
Set up the integration and plot intervals.
|
||||
#+begin_src jupyter-python :exports both :results raw drawer
|
||||
interval_η = [-η, η]
|
||||
interval = η_to_θ([-η, η])
|
||||
interval_cosθ = np.cos(interval)
|
||||
interval_pt = np.sort(η_to_pt([0, η], esp/2))
|
||||
plot_interval = [0.1, np.pi-.1]
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
|
|
@ -173,6 +184,22 @@ but that doen't reduce variance and would complicate things now.
|
|||
#+end_note
|
||||
|
||||
** Analytical Integration
|
||||
Let's plot a more detailed view of the xs.
|
||||
#+begin_src jupyter-python :exports both :results raw drawer
|
||||
plot_points = np.linspace(np.pi/2 - 0.5, np.pi/2 + 0.5, 1000)
|
||||
plot_points = plot_points[plot_points > 0]
|
||||
|
||||
fig, ax = set_up_plot()
|
||||
ax.plot(plot_points, gev_to_pb(diff_xs(plot_points, charge=charge, esp=esp)))
|
||||
ax.set_xlabel(r"$\theta$")
|
||||
ax.set_ylabel(r"$d\sigma/d\Omega$ [pb]")
|
||||
ax.set_xlim([plot_points.min(), plot_points.max()])
|
||||
save_fig(fig, "diff_xs_zoom", "xs", size=[2.5, 2.5])
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:./.ob-jupyter/3986a139c4a6c3a27b1ef12a26b2e8f3ce473547.png]]
|
||||
|
||||
And now calculate the cross section in picobarn.
|
||||
#+BEGIN_SRC jupyter-python :exports both :results raw file :file xs.tex
|
||||
xs_gev = total_xs_eta(η, charge, esp)
|
||||
|
|
@ -203,12 +230,15 @@ but that doen't reduce variance and would complicate things now.
|
|||
Compared to sherpa, it's pretty close.
|
||||
#+NAME: 81b5ed93-0312-45dc-beec-e2ba92e22626
|
||||
#+BEGIN_SRC jupyter-python :exports both :results raw drawer
|
||||
sherpa = 0.05380
|
||||
xs_pb - sherpa
|
||||
sherpa = np.loadtxt("../../runcards/qqgg/sherpa_xs", delimiter=",")
|
||||
tex_value(
|
||||
,*sherpa, unit=r"\pico\barn", prefix=r"\sigma = ", prec=6, save=("results", "xs_sherpa.tex")
|
||||
)
|
||||
xs_pb - sherpa[0]
|
||||
#+END_SRC
|
||||
|
||||
#+RESULTS: 81b5ed93-0312-45dc-beec-e2ba92e22626
|
||||
: -6.7112594623469635e-06
|
||||
: -5.112594623490896e-07
|
||||
|
||||
I had to set the runcard option ~EW_SCHEME: alpha0~ to use the pure
|
||||
QED coupling constant.
|
||||
|
|
@ -216,21 +246,19 @@ but that doen't reduce variance and would complicate things now.
|
|||
** Numerical Integration
|
||||
Plot our nice distribution:
|
||||
#+begin_src jupyter-python :exports both :results raw drawer
|
||||
plot_points = np.linspace(*plot_interval, 1000)
|
||||
plot_points = np.linspace(*np.arccos(interval_cosθ), 1000)
|
||||
plot_points = plot_points[plot_points > 0]
|
||||
|
||||
fig, ax = set_up_plot()
|
||||
ax.plot(plot_points, gev_to_pb(diff_xs(plot_points, charge=charge, esp=esp)))
|
||||
ax.set_xlabel(r'$\theta$')
|
||||
ax.set_ylabel(r'$d\sigma/d\Omega$ [pb]')
|
||||
ax.set_xlim([plot_points.min(), plot_points.max()])
|
||||
ax.axvline(interval[0], color='gray', linestyle='--')
|
||||
ax.axvline(interval[1], color='gray', linestyle='--', label=rf'$|\eta|={η}$')
|
||||
ax.legend()
|
||||
save_fig(fig, 'diff_xs', 'xs', size=[2.5, 2.5])
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:./.ob-jupyter/3dd905e7608b91a9d89503cb41660152f3b4b55c.png]]
|
||||
[[file:./.ob-jupyter/ea9069041c3e2ccd18c7642001c20d374696498d.png]]
|
||||
|
||||
Define the integrand.
|
||||
#+begin_src jupyter-python :exports both :results raw drawer
|
||||
|
|
@ -248,7 +276,7 @@ Plot the integrand. # TODO: remove duplication
|
|||
fig, ax = set_up_plot()
|
||||
ax.plot(plot_points, xs_pb_int(plot_points))
|
||||
ax.set_xlabel(r'$\theta$')
|
||||
ax.set_ylabel(r'$2\pi\cdot d\sigma/d\theta [pb]')
|
||||
ax.set_ylabel(r'$2\pi\cdot d\sigma/d\theta$ [pb]')
|
||||
ax.set_xlim([plot_points.min(), plot_points.max()])
|
||||
ax.axvline(interval[0], color='gray', linestyle='--')
|
||||
ax.axvline(interval[1], color='gray', linestyle='--', label=rf'$|\eta|={η}$')
|
||||
|
|
@ -256,7 +284,7 @@ Plot the integrand. # TODO: remove duplication
|
|||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:./.ob-jupyter/ccb6653162c81c3f3e843225cb8d759178f497e0.png]]
|
||||
[[file:./.ob-jupyter/0faa37f24e5e531a55c6679794b5ad84f98ed47b.png]]
|
||||
*** Integral over θ
|
||||
Intergrate σ with the mc method.
|
||||
#+begin_src jupyter-python :exports both :results raw drawer
|
||||
|
|
@ -326,9 +354,9 @@ Now we use =VEGAS= on the θ parametrisation and see what happens.
|
|||
xs_pb_int,
|
||||
interval,
|
||||
num_increments=num_increments,
|
||||
alpha=1.5,
|
||||
increment_epsilon=0.01,
|
||||
vegas_point_density=15,
|
||||
alpha=2,
|
||||
increment_epsilon=0.02,
|
||||
vegas_point_density=20,
|
||||
epsilon=.001,
|
||||
acumulate=False,
|
||||
)
|
||||
|
|
@ -336,11 +364,11 @@ Now we use =VEGAS= on the θ parametrisation and see what happens.
|
|||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: VegasIntegrationResult(result=0.052773271883732584, sigma=0.0006119771217887033, N=920, increment_borders=array([0.16380276, 0.20109846, 0.24530873, 0.30103045, 0.37106657,
|
||||
: 0.45713119, 0.56758289, 0.71432935, 0.91252993, 1.17768957,
|
||||
: 1.55420491, 1.91978804, 2.20362647, 2.40547652, 2.55765843,
|
||||
: 2.67560835, 2.76436094, 2.83801142, 2.89499662, 2.94066386,
|
||||
: 2.9777899 ]), vegas_iterations=23)
|
||||
: VegasIntegrationResult(result=0.053415671595191345, sigma=0.0006275962280404523, N=280, increment_borders=array([0.16380276, 0.20536326, 0.25384714, 0.31480502, 0.39193629,
|
||||
: 0.48757604, 0.60550147, 0.75723929, 0.96215207, 1.23107803,
|
||||
: 1.56182395, 1.89731315, 2.17107801, 2.37597223, 2.52898466,
|
||||
: 2.64874341, 2.74453741, 2.82025926, 2.88209227, 2.93279579,
|
||||
: 2.9777899 ]), vegas_iterations=7)
|
||||
|
||||
This is pretty good, although the variance reduction may be achieved
|
||||
partially by accumulating the results from all runns. Here this gives
|
||||
|
|
@ -365,8 +393,8 @@ This depends, of course, on the iteration count.
|
|||
xs_pb_int,
|
||||
interval,
|
||||
num_increments=num_increments,
|
||||
alpha=1.5,
|
||||
increment_epsilon=0.01,
|
||||
alpha=2,
|
||||
increment_epsilon=0.02,
|
||||
vegas_point_density=20,
|
||||
epsilon=.001,
|
||||
acumulate=True,
|
||||
|
|
@ -374,11 +402,11 @@ This depends, of course, on the iteration count.
|
|||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
: VegasIntegrationResult(result=0.05379385150190898, sigma=0.0003068472335040267, N=760, increment_borders=array([0.16380276, 0.20176444, 0.24723471, 0.30169898, 0.3725654 ,
|
||||
: 0.46249133, 0.57260516, 0.71753884, 0.9047832 , 1.17676749,
|
||||
: 1.53514921, 1.91009792, 2.19262501, 2.40344474, 2.55656491,
|
||||
: 2.67302165, 2.76296776, 2.83550718, 2.89246912, 2.93998605,
|
||||
: 2.9777899 ]), vegas_iterations=19)
|
||||
: VegasIntegrationResult(result=0.053638937795766034, sigma=0.00041459931173526546, N=280, increment_borders=array([0.16380276, 0.20595251, 0.25473743, 0.31600983, 0.39036863,
|
||||
: 0.48642444, 0.61192596, 0.76541368, 0.96544251, 1.24106758,
|
||||
: 1.57551962, 1.90430764, 2.16425611, 2.36635898, 2.52157247,
|
||||
: 2.64333433, 2.7431946 , 2.82557121, 2.88798018, 2.93770475,
|
||||
: 2.9777899 ]), vegas_iterations=7)
|
||||
|
||||
Let's define some little helpers.
|
||||
#+begin_src jupyter-python :exports both :tangle tangled/plot_utils.py
|
||||
|
|
@ -459,7 +487,7 @@ And now we plot the integrand with the incremens.
|
|||
ax.set_xlabel(r"$\theta$")
|
||||
ax.set_ylabel(r"$2\pi\cdot d\sigma/d\theta$ [pb]")
|
||||
ax.set_ylim([0, 0.09])
|
||||
|
||||
plot_points = np.linspace(*interval, 1000)
|
||||
ax.plot(plot_points, xs_pb_int(plot_points), label="Distribution")
|
||||
|
||||
plot_increments(
|
||||
|
|
@ -483,7 +511,7 @@ And now we plot the integrand with the incremens.
|
|||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:./.ob-jupyter/d3b48be87b26058e6083e6f3a138e935436e3a87.png]]
|
||||
[[file:./.ob-jupyter/1c53bfcbf349269eff9f54eb58b9639ed9e6ce21.png]]
|
||||
*** Testing the Statistics
|
||||
Let's battle test the statistics.
|
||||
#+begin_src jupyter-python :exports both :results raw drawer
|
||||
|
|
@ -981,15 +1009,15 @@ That looks somewhat fishy, but it isn't.
|
|||
fig, ax = set_up_plot()
|
||||
points = np.linspace(interval_pt[0], interval_pt[1] - .01, 1000)
|
||||
ax.plot(points, gev_to_pb(diff_xs_p_t(points, charge, esp)))
|
||||
ax.set_xlabel(r'$p_T$')
|
||||
ax.set_xlabel(r'$p_\mathrm{T}$')
|
||||
ax.set_xlim(interval_pt[0], interval_pt[1] + 1)
|
||||
ax.set_ylim([0, gev_to_pb(diff_xs_p_t(interval_pt[1] -.01, charge, esp))])
|
||||
ax.set_ylabel(r'$\frac{d\sigma}{dp_t}$ [pb]')
|
||||
ax.set_ylabel(r'$\frac{\mathrm{d}\sigma}{\mathrm{d}p_\mathrm{T}}$ [pb]')
|
||||
save_fig(fig, 'diff_xs_p_t', 'xs_sampling', size=[4, 2])
|
||||
#+end_src
|
||||
|
||||
#+RESULTS:
|
||||
[[file:./.ob-jupyter/29724b8c1f2b0005a05f64f999cf95d248ee0082.png]]
|
||||
[[file:./.ob-jupyter/5c39a14515ced9b3f1d5d0cdd0c4fe75921ee3a7.png]]
|
||||
this is strongly peaked at p_t=100GeV. (The jacobian goes like 1/x there!)
|
||||
|
||||
*** Sampling the η cross section
|
||||
|
|
|
|||
Binary file not shown.
File diff suppressed because it is too large
Load diff
BIN
prog/python/qqgg/figs/xs/diff_xs_zoom.pdf
Normal file
BIN
prog/python/qqgg/figs/xs/diff_xs_zoom.pdf
Normal file
Binary file not shown.
4122
prog/python/qqgg/figs/xs/diff_xs_zoom.pgf
Normal file
4122
prog/python/qqgg/figs/xs/diff_xs_zoom.pgf
Normal file
File diff suppressed because it is too large
Load diff
Binary file not shown.
Binary file not shown.
File diff suppressed because it is too large
Load diff
Binary file not shown.
|
|
@ -3868,83 +3868,82 @@
|
|||
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|
||||
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|
||||
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||||
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||||
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|
||||
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|
||||
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|
||||
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|
||||
\pgfpathlineto{\pgfqpoint{1.525675in}{1.066265in}}%
|
||||
\pgfpathlineto{\pgfqpoint{1.476621in}{1.101086in}}%
|
||||
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||||
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|
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||||
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||||
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|
||||
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|
||||
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||||
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|
||||
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|
||||
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|
||||
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|
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|
||||
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|
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|
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|
||||
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||||
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|
||||
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|
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|
||||
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|
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|
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|
||||
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|
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||||
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||||
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|
||||
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|
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|
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|
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|
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|
||||
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|
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|
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\pgfpathlineto{\pgfqpoint{4.695805in}{1.938929in}}%
|
||||
\pgfpathlineto{\pgfqpoint{4.719268in}{2.044562in}}%
|
||||
\pgfpathlineto{\pgfqpoint{4.738821in}{2.145441in}}%
|
||||
\pgfpathlineto{\pgfqpoint{4.758373in}{2.260702in}}%
|
||||
\pgfpathlineto{\pgfqpoint{4.777926in}{2.393706in}}%
|
||||
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|
||||
\pgfpathlineto{\pgfqpoint{4.801389in}{2.583173in}}%
|
||||
\pgfpathlineto{\pgfqpoint{4.801389in}{2.583173in}}%
|
||||
\pgfusepath{stroke}%
|
||||
\end{pgfscope}%
|
||||
\begin{pgfscope}%
|
||||
|
|
@ -3956,8 +3955,8 @@
|
|||
\definecolor{currentstroke}{rgb}{0.501961,0.501961,0.501961}%
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\pgfpathmoveto{\pgfqpoint{4.743691in}{0.594444in}}%
|
||||
\pgfpathlineto{\pgfqpoint{4.743691in}{2.801389in}}%
|
||||
\pgfusepath{stroke}%
|
||||
\end{pgfscope}%
|
||||
\begin{pgfscope}%
|
||||
|
|
@ -3969,8 +3968,8 @@
|
|||
\definecolor{currentstroke}{rgb}{0.501961,0.501961,0.501961}%
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\pgfpathlineto{\pgfqpoint{4.743691in}{2.801389in}}%
|
||||
\pgfusepath{stroke}%
|
||||
\end{pgfscope}%
|
||||
\begin{pgfscope}%
|
||||
|
|
@ -3982,8 +3981,8 @@
|
|||
\definecolor{currentstroke}{rgb}{0.501961,0.501961,0.501961}%
|
||||
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|
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|
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|
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|
||||
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|
||||
\pgfpathlineto{\pgfqpoint{4.676382in}{2.801389in}}%
|
||||
\pgfusepath{stroke}%
|
||||
\end{pgfscope}%
|
||||
\begin{pgfscope}%
|
||||
|
|
@ -3995,8 +3994,8 @@
|
|||
\definecolor{currentstroke}{rgb}{0.501961,0.501961,0.501961}%
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|
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|
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|
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|
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|
||||
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|
||||
|
|
@ -4008,8 +4007,8 @@
|
|||
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|
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|
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|
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|
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|
||||
|
|
@ -4021,8 +4020,8 @@
|
|||
\definecolor{currentstroke}{rgb}{0.501961,0.501961,0.501961}%
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|
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|
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|
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|
|
@ -4034,8 +4033,8 @@
|
|||
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|
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|
|
@ -4047,8 +4046,8 @@
|
|||
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|
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|
|
@ -4060,8 +4059,8 @@
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|
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|
|
@ -4073,8 +4072,8 @@
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|
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|
|
@ -4086,8 +4085,8 @@
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|
|
@ -4099,8 +4098,8 @@
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|
|
@ -4112,8 +4111,8 @@
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|
|
@ -4125,8 +4124,8 @@
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|
|
@ -4138,8 +4137,8 @@
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|
|
@ -4151,8 +4150,8 @@
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|
|
@ -4164,8 +4163,8 @@
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|
|
@ -4177,8 +4176,8 @@
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|
@ -4190,8 +4189,8 @@
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@ -4203,8 +4202,8 @@
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\pgfpathlineto{\pgfqpoint{4.355591in}{1.856747in}}%
|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
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|
||||
\pgfpathlineto{\pgfqpoint{4.785747in}{2.216393in}}%
|
||||
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|
||||
\pgfpathlineto{\pgfqpoint{4.801389in}{2.330148in}}%
|
||||
\pgfusepath{stroke}%
|
||||
\end{pgfscope}%
|
||||
\begin{pgfscope}%
|
||||
|
|
|
|||
Binary file not shown.
|
|
@ -1857,7 +1857,7 @@
|
|||
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
|
||||
\pgfsetstrokecolor{textcolor}%
|
||||
\pgfsetfillcolor{textcolor}%
|
||||
\pgftext[x=2.406374in,y=0.318333in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle p_T\)}%
|
||||
\pgftext[x=2.406374in,y=0.318333in,,top]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle p_\mathrm{T}\)}%
|
||||
\end{pgfscope}%
|
||||
\begin{pgfscope}%
|
||||
\pgfpathrectangle{\pgfqpoint{1.058958in}{0.594444in}}{\pgfqpoint{2.694832in}{1.206944in}}%
|
||||
|
|
@ -3863,7 +3863,7 @@
|
|||
\definecolor{textcolor}{rgb}{0.000000,0.000000,0.000000}%
|
||||
\pgfsetstrokecolor{textcolor}%
|
||||
\pgfsetfillcolor{textcolor}%
|
||||
\pgftext[x=0.520347in,y=1.197917in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle \frac{d\sigma}{dp_t}\) [pb]}%
|
||||
\pgftext[x=0.520347in,y=1.197917in,,bottom,rotate=90.000000]{\color{textcolor}\rmfamily\fontsize{10.000000}{12.000000}\selectfont \(\displaystyle \frac{\mathrm{d}\sigma}{\mathrm{d}p_\mathrm{T}}\) [pb]}%
|
||||
\end{pgfscope}%
|
||||
\begin{pgfscope}%
|
||||
\pgfpathrectangle{\pgfqpoint{1.058958in}{0.594444in}}{\pgfqpoint{2.694832in}{1.206944in}}%
|
||||
|
|
|
|||
1
prog/python/qqgg/results/ecm.tex
Normal file
1
prog/python/qqgg/results/ecm.tex
Normal file
|
|
@ -0,0 +1 @@
|
|||
\(\ecm = \SI{200}{\giga\electronvolt}\)
|
||||
1
prog/python/qqgg/results/eta.tex
Normal file
1
prog/python/qqgg/results/eta.tex
Normal file
|
|
@ -0,0 +1 @@
|
|||
\(\abs{\eta}\leq 2.5\)
|
||||
|
|
@ -1 +1 @@
|
|||
\(\sigma = \SI{0.0528\pm 0.0006}{\pico\barn}\)
|
||||
\(\sigma = \SI{0.0534\pm 0.0006}{\pico\barn}\)
|
||||
|
|
@ -1 +1 @@
|
|||
\(N = 920\)
|
||||
\(N = 280\)
|
||||
|
|
@ -1 +1 @@
|
|||
\(\times23\)
|
||||
\(\times7\)
|
||||
1
prog/python/qqgg/results/xs_sherpa.tex
Normal file
1
prog/python/qqgg/results/xs_sherpa.tex
Normal file
|
|
@ -0,0 +1 @@
|
|||
\(\sigma = \SI{0.0537938\pm 0.0000025}{\pico\barn}\)
|
||||
Loading…
Add table
Reference in a new issue