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105 lines
4.5 KiB
105 lines
4.5 KiB
%==========================================
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% Appendices related to the selection
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%==========================================
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\section{Event selection}
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\subsection{Crystal Ball function}\label{app:CrystalBall}
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The Crystal Ball function is a probability density function widely used to model processes with losses \cite{APP-CB}.
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It consists of a gaussian core and one power-law low end tail, that describes the loss, typically from the final state radiation. The function got its name from the Crystal Ball collaboration \cite{APP-CBCollab}.
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The experiment was placed at the SPEAR accelerator at SLAC National Laboratory and designed as a spark chamber surrounded by an almost complete sphere (covering 98\% of the solid angle) made of scintillating crystals. Therefore, the detector got its prophetic name. The detector is operating until today.
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It is located in Mainz, placed at the MAMI microtron \cite{APP-CBMainz}.
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The Crystal Ball function is then defined as
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\begin{equation}
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\mathcal{P}(x;\alpha,n,\bar x,\sigma) = N \cdot
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\begin{cases}
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\exp(- \frac{(x - \bar x)^2}{2 \sigma^2}), & \mbox{for }\frac{x - \bar x}{\sigma} > -\alpha \\
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A \cdot (B - \frac{x - \bar x}{\sigma})^{-n} & \mbox{for }\frac{x - \bar x}{\sigma} \leqslant -\alpha
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\end{cases}\,,
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\end{equation}
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where $A$ and $\alpha$ and $n$ describe the tail, $\mu$ and $\sigma$ are the mean and the width of the peak. $N$ is a normalization factor, $A$ and $B$ are constants defined as:
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\begin{align}
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\begin{split}
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A &= \left(\frac{n}{\left| \alpha \right|}\right)^n \cdot \exp\left(- \frac {\left| \alpha \right|^2}{2}\right)\,,\\
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B &= \frac{n}{\left| \alpha \right|} - \left| \alpha \right|\,.\\
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%N &= \frac{1}{\sigma (C + D)}\,,\\
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%C &= \frac{n}{\left| \alpha \right|} \cdot \frac{1}{n-1} \cdot \exp\left(- \frac {\left| \alpha \right|^2}{2}\right)\,,\\
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%D &= \sqrt{\frac{\pi}{2}} \left(1 + \operatorname{erf}\left(\frac{\left| \alpha \right|}{\sqrt 2}\right)\right)\,.
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\end{split}
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\end{align}
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%
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%The $\operatorname{erf}$ is Gauss error function defined as
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%
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%\begin{equation}
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% erf(z) =\frac{2}{\sqrt\pi}\int_0^z e^{-t^2}\,dt\,.
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%\end{equation}
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\subsubsection{Double sided Crystal Ball function}
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The Crystal Ball function can be extended to contain a gaussian core and two power-law low end tails.
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The double sided Crystal Ball function is then defined as
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%
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\begin{align}
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\mathcal{P}_{CB}(x; x_{peak}, \sigma, n_1, n_2, \alpha_1, \alpha_2) = N \cdot
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\begin{cases}
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A_1\cdot(B_1 - \frac{x-x_{peak}}{\sigma})^{-n_1} & $for $ \frac{x - x_{peak}}{\sigma} \leq -\alpha_1
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\vspace*{0.3cm}\\
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\exp(\frac{-(x - x_{peak})^{2}}{2\sigma^{2}}) & $for $ -\alpha_1 \leq \frac{x - x_{peak}}{\sigma} \leq \alpha_2
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\vspace*{0.3cm}\\
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A_2\cdot(B_2 - \frac{x-x_{peak}}{\sigma})^{-n_2} & $for $ \alpha_2 \leq \frac{x - x_{peak}}{\sigma}
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\end{cases}
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\,,
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\end{align}
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%
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$N$ is a normalization factor, $A_{1,2}$ and $B_{1,2}$ are constants defined as:
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\begin{align}
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\begin{split}
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A_{1,2}= & ( \frac{n_{1,2}}{\abs{n_{1,2}}} )^{n_{1,2}} \cdot \exp(\pm\frac{\alpha_{1,2}^{2}}{2}),
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\vspace*{0.3cm}\\
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B_{1,2}= & \frac{n_{1,2}}{\abs{\alpha_{1,2}}} - \abs{\alpha_{1,2}}.
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\end{split}
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\end{align}
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\subsection{ExpGaus function}\label{app:ExpGaus}
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ExpGaus function is a function used to describe partially reconstructed backgrounds in \B~meson decays. The definition is in \refEq{App-ExpGaus}. The $\mu$ denotes the mean of the distribution, $\sigma$ is the variance of the function, D is a constant representing the decay of the B meson.
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%
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\begin{align}\label{eq:App-ExpGaus}
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f_{EG}(x) =
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\begin{cases}
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\exp\left(-\frac{\mu-D}{\sigma^2} \left(x-D\right)\right) \exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right) & \text{if } x \leq D\\
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\exp\left(-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2\right) & \text{otherwise}
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\end{cases}
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\,.
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\end{align}
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%\clearpage
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%\subsection{Correction to the simulation}\label{app:SimulationCorrection}
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%\todo[inline]{\piz pseudorapidity resolution: data does not agree with MC, see talk from 2018\_05\_14}
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%
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%\clearpage
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\subsection{Reweighted distributions of parameters used for the MLP training}\label{app:CompareVariables}
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\input{./figures/fig_CompareVariables}
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%\input{./figures/fig_CompareVariables_sig}
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%\clearpage
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%\subsection{\lone trigger efficiency}\label{app:L0Eff}
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%\input{Chapters/EventSelection/L0Efficiency}
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%
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\clearpage
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\subsection[Signal yield in bins of the dimuon invariant mass squared]{Signal yield in bins of the dimuon invariant mass squared}\label{app:yield_q2}
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\input{Chapters/EventSelection/FitsInQ2}
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\clearpage
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