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99 lines
4.2 KiB
TeX
99 lines
4.2 KiB
TeX
\section{Discussion and Comparison with \sherpa}%
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\label{sec:compsher}
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The result obtained in \cref{sec:qqggcalc} shall now be verified by the
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monte-carlo event generator \sherpa{}~\cite{Gleisberg:2008ta}. To
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facilitate this, an expression for the total cross section for a range
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of \(\theta\) or \(\eta\) has to be obtained. Using the golden rule
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for \(2\rightarrow 2\) processes and observing that the initial and
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final momenta are equal (\(p_i=p_f\)) and \(g=\sqrt{4\pi\alpha}\), the
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result \cref{eq:crossec} arises. The differential cross section has
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also been calculated in terms of the pseudo-rapidity
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in \cref{eq:xs-eta}.
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An additional factor of \(\frac{1}{2}\) comes in due to there being
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two identical photons in the final state.
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\begin{equation}
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\label{eq:crossec}
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\dv{\sigma}{\Omega} =
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\frac{1}{2}\frac{1}{(8\pi)^2}\cdot\frac{\abs{\mathcal{M}}^2}{\ecm^2}\cdot\frac{\abs{p_f}}{\abs{p_i}}
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= \underbrace{\frac{\alpha^2Z^4}{6\ecm^2}}_{\mathfrak{C}}\frac{1+\cos^2(\theta)}{\sin^2(\theta)}
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\end{equation}
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\begin{equation}
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\label{eq:xs-eta}
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\dv{\sigma}{\eta} = 2\pi\cdot\frac{\alpha^2Z^4}{6\ecm^2}\cdot\qty(\tanh(\eta)^2 + 1)
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\end{equation}
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\begin{figure}[ht]
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\centering
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\begin{subfigure}[c]{.45\textwidth}
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\centering \plot{xs/diff_xs}
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\caption[Plot of the differential cross section of the \(\qqgg\)
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process.]{\label{fig:diffxs} The differential cross section as a
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function of the polar angle \(\theta\). }
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\end{subfigure}
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\begin{subfigure}[c]{.45\textwidth}
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\centering
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\plot{xs/total_xs}
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\caption[Plot of the total cross section of the \(\qqgg\)
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process.]{\label{fig:totxs} The total cross section
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(\cref{eq:total-crossec}) of the process for a pseudo-rapidity
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range of \([-\eta, \eta]\).}
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\end{subfigure}
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\end{figure}
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The differential cross section \cref{eq:crossec} (see
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also \cref{fig:diffxs}) is divergent for angles near zero or
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\(\pi\). Allowing finite mass in the calculation may regularize this
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divergence. Because \(m=0\) is the limit for
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\(\ecm\rightarrow\infty\), the cross section would still have strong
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peaks for angles near \(0,\pi\) at high energies so that the result is
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not altogether nonphysical. The divergence of the cross section itself
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is also not the problem here, because it can be transformed into a
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form where the divergence does not occur (see \cref{eq:xs-eta}).
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The differential cross section is clearly symmetric around
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\(\theta=\frac{\pi}{2}\) as was to be expected\footnote{Such
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properties are very handy and intuitive checks for calculations
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along the way.}, because the photons are indistinguishable. To
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compare the cross section to experiment and to simulation an interval
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around \(\theta=\frac{\pi}{2}\) has to be chosen, where the first
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order, mass-less approximation may yield a physical result.
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The total cross section in such an interval, given by
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integrating \cref{eq:crossec} for \(\theta\in [\theta_1, \theta_2]\)
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or \(\eta\in [\eta_1, \eta_2]\) is given
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in \cref{eq:total-crossec}.
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\begin{equation}
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\label{eq:total-crossec}
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\begin{split}
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\sigma &=
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2\pi\mathfrak{C}\cdot\qty{\cos(\theta_2)-\cos(\theta_1)+2\qty[\artanh(\cos(\theta_1))
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- \artanh(\cos(\theta_2))]} \\
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&=2\pi\mathfrak{C}\cdot\qty[\tanh(\eta_2) - \tanh(\eta_1) + 2(\eta_1
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- \eta_2))] \\
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&={\frac{\pi\alpha^2Z^4}{3\ecm^2}}\cdot\qty[\tanh(\eta_2) - \tanh(\eta_1) + 2(\eta_1
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- \eta_2))]
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\end{split}
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\end{equation}
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As can be seen in \cref{fig:totxs}, the cross section, integrated over
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an interval of \([-\eta, \eta]\), is dominated by the linear
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contributions in \cref{eq:total-crossec} and would result in an
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infinity if no cut on \(\eta\) would be made. Choosing
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\(\eta\in [-2.5,2.5]\) and \(\ecm=\SI{100}{\giga\electronvolt}\) the
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process was monte carlo integrated in \sherpa\ using the runcard
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in \cref{sec:qqggruncard}. This runcard describes the exact same (first
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order) process as the calculated cross section.
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\sherpa\ arrives at the cross section \result{xs/sherpa_xs}. Plugging
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the same parameters into \cref{eq:total-crossec} gives
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\result{xs/python/xs} which is within the uncertainty range of the
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\sherpa\ value. This verifies the result for the total cross section.
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%%% Local Variables:
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%%% mode: latex
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%%% TeX-master: "../../document"
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%%% End:
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