%==========================================
%
%   Appendices related to the theory 
%
%==========================================

\section[Theoretical introduction to the \texorpdfstring{${\BuToKstmmBF}$}{BuToKstmumu} decay]
{Theoretical introduction to the \texorpdfstring{\BuToKstmmBF}{BuToKstmumu} decay}
\label{app:ANA-Theo}

\subsection{Decay rate}

The full form of \refEq{decay_rate} with explicitly stated $f_i$ is a rather lengthy \refEq{decay_rate_full}:
%%
\begin{align}\label{eq:decay_rate_full}\begin{split}
\frac{\deriv^4\Gamma}{\deriv\cos\thetak\deriv\cos\thetak\deriv\phi\deriv q^2} =  \frac{9}{32\pi}&\sum_i{ J_i\left(q^2\right)f_i\left(\cos\thetal,\cos\thetak,\phi\right)} =\\
= \frac{9}{32\pi}& \left\{\right. J_{1s}  \sin^2\thetak \\
&+J_{1c} \cos^2\thetak \\
&+J_{2s} \sin^2\thetak \cos 2\thetal \\
&+J_{2c} \cos^2\thetak \cos 2\thetal \\
&+J_{3}  \sin^2\thetak \sin^2\thetal \cos 2\phi \\
&+J_{4}  \sin 2\thetak \sin 2\thetal \cos \phi  \\
&+J_{5}  \sin 2\thetak \sin \thetal  \cos \phi  \\
&+J_{6s} \sin^2\thetak \cos \thetal  \\
&+J_{7}  \sin 2\thetak \sin \thetal  \sin \phi  \\
&+J_{8}  \sin 2\thetak \sin 2\thetal \sin \phi  \\
&+J_{9}  \sin^2\thetak \sin^2\thetal \sin 2\phi \left.\right\}\,. \\
\end{split}\end{align} 
%
This lengthy formula can be simplified by neglecting the muon mass. This is a good assumptions for $\qsq\gtrsim1\gevgev$. Under this assumption, the following relations can be obtained:
\begin{align}\begin{split}
J_1^c &= 1-\frac{4}{3}J_1^s\,,\\
J_2^s &= \frac{1}{3}J_1^s\,,\\
J_2^c &= \frac{4}{3}J_1^s-1\,.\\
\end{split}\end{align} 

In some cases it is also convenient to define \emph{CP-asymmetric} angular observables $A_i$ besides the usual $S_i$ variables (see \refEq{Si_definition})
\begin{equation}\label{eq:Ai_definition}
A_i = \frac{J_i-\bar{J_i}}{\Gamma+\bar{\Gamma}}\,.
\end{equation}

\clearpage
Rewriting the \refEq{decay_rate_full} using the \emph{CP-symmetric} $S_i$ variables results in the following formula:
%
\begin{align}\label{eq:decay_rate2}\begin{split}
&\left.
\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\thetal\,{\rm dcos}\thetak\,{\rm d}\phi}
\right
|_{\rm P} = \tfrac{9}{32\pi}\Bigl[\tfrac{3}{4} (1-\textcolor{red}{F_{\rm L}})\sin^2\thetak\\
&+ \textcolor{red}{F_{\rm L}}\cos^2\thetak + \tfrac{1}{4}(1-\textcolor{red}{F_{\rm L}})\sin^2\thetak\cos 2\thetal\nonumber\\
&- \textcolor{red}{F_{\rm L}} \cos^2\thetak\cos 2\thetal + \textcolor{red}{S_3}\sin^2\thetak \sin^2\thetal \cos 2\phi\nonumber\\
&+ \textcolor{red}{S_4} \sin 2\thetak \sin 2\thetal \cos\phi + \textcolor{red}{S_5}\sin 2\thetak \sin \thetal \cos \phi\nonumber\\
&+ \tfrac{4}{3} \textcolor{red}{A_{\rm FB}} \sin^2\thetak \cos\thetal + \textcolor{red}{S_7} \sin 2\thetak \sin\thetal \sin\phi\nonumber\\
&+ \textcolor{red}{S_8} \sin 2\thetak \sin 2\thetal \sin\phi + \textcolor{red}{S_9}\sin^2\thetak \sin^2\thetal \sin 2\phi \nonumber
\Bigr]
\end{split}\end{align} 

\subsection{\swave decay rate}

The decay rate of the \swave is
\begin{equation}\label{eq:decay_rate_S_app}
\left.
\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
\right |_{\rm S} 
=\frac{3}{16\pi}{F_S}\sin^2\theta_L\,.
\end{equation}
%
The \pwave and \swave interference term can be parameterized as follows:
%
\begin{equation}
\label{eq:decay_rate_PS_full}
\begin{aligned}
\left.
\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
\right |_{\rm PS} 
=\frac{3}{16\pi}\left[\right.
&{S_{S1}}\sin^2\theta_L\cos\theta_K \\
+& {S_{S2}}\sin2\theta_L \sin \theta_K \cos\phi\\ 
+& {S_{S3}}\sin\theta_L \sin \theta_K \cos\phi \\
+& {S_{S4}}\sin\theta_L \sin \theta_K \sin\phi \\
+& {S_{S5}}\sin2\theta_L \sin \theta_K \sin\phi
\left.\right]\,.
\end{aligned}
\end{equation}
%
These two terms need to be added to the full measured decay rate. The full angular description of the decay is then in \refEq{decay_rate_final}.
\clearpage