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62 lines
2.9 KiB
62 lines
2.9 KiB
\hypertarget{plotting-and-transition}{%
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\midheading{Plotting routine and transition function}\label{plotting-and-transition}}
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The following transition function is implemented for the example
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input files. For more details we refer to our publication. The fully
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matched cross-section is described in general by
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\includegraphics[width=0.9\textwidth]{./sections/Matching.png}
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%\begin{equation}\label{eq:matchingmod}
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% \;
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% \left.\frac{\mathrm{d}\sigma^{\text{N$^3$LL}}}{\mathrm{d}q_T}\right|_{\text{matched to \NNLO{}}}
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% = t(x) \left( \frac{\mathrm{d}\sigma^{\text{N$^3$LL}}}{\mathrm{d}q_T} +
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% \left.\Delta\sigma\right|_{q_T>q_0} \right)
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% + (1-t(x)) \frac{\mathrm{d}\sigma^\NNLO{}}{\mathrm{d}q_T}\,
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%\end{equation}
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using a transition function $t(x)$. We have implemented a transition function $t$
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with $x=q_T^2/Q^2$ that smoothly switches between 1 and 0 like a sigmoid function.
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Following a choice in CuTe, we first define
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\includegraphics[width=0.9\textwidth]{./sections/sfunction.png}
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%\begin{equation}
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%s(x;l,r,u) = \left (1 + \exp\left(\log\left(\frac{1-u}{u}\right) \frac{x-m}{w}\right) \right )^{-1}\,,\quad
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%m = (r+l)/2\,,\quad w = (r-l)/2\,.
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%\end{equation}
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The function $s(x)$, parametrized by $l,r,u$, is defined to be $s(l)=1-u$ and $s(r)=u$.
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In terms of this sigmoid, our transition function $t(x; x^{\text{min}},x^{\text{max}},u)$, where
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$x=q_T^2/Q^2$, is then defined by
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\begin{equation}\label{eq:transition}
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t(x; x^{\text{min}},x^{\text{max}},u) = \left\{\begin{array}{lr}
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1 , & \text{for } x < x^{\text{min}}\\
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\frac{s(x; x^{\text{min}}, x^{\text{max}},u)}
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{s(x^{\text{min}}; x^{\text{min}}, x^{\text{max}},u)}, &
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\text{otherwise}
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\end{array}\right\}\,.
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\end{equation}
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This ensures that below $x^{\text{min}}=(q_T^{\text{min}}/Q)^2$ only the naively matched result is
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used, and at
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$x^{\text{max}}$
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for small $u\ll1$ the transition function is approximately $u$. In practice it makes sense to set
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the transition
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function to zero below a small threshold like $10^{-3}$ without a noticeable discontinuity.
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This has the advantage that the deteriorating resummation and matching corrections do not impact
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the region of
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large $q_T$ at all.
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Our example plotting routines use $x^{\text{min}}=0.001$, and $u=0.001$, and the parameter
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$x^{\text{max}}$ corresponds to the value of \texttt{transitionswitch} set in the input file. The
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transition function can be changed or completely replaced by just modifying the plotting routines.
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The following figure illustrates this transition function.
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\begin{figure}[t!]
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\centering
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\includegraphics[width=0.8\textwidth]{transition.pdf}
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\caption{The transition function defined in eq.~\eqref{eq:transition} for different values of
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the parameter $x^{\text{max}}$ which determines the position of the
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transition. The $x$-axis is displayed on a square-root scale
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to guide the eye on
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the quadratic $q_T$-dependence.}
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\label{fig:transition}
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\end{figure}
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