87 lines
3.8 KiB
TeX
87 lines
3.8 KiB
TeX
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% Appendices related to the theory
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\section[Theoretical introduction to the \texorpdfstring{${\BuToKstmmBF}$}{BuToKstmumu} decay]
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{Theoretical introduction to the \texorpdfstring{\BuToKstmmBF}{BuToKstmumu} decay}
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\label{app:ANA-Theo}
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\subsection{Decay rate}
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The full form of \refEq{decay_rate} with explicitly stated $f_i$ is a rather lengthy \refEq{decay_rate_full}:
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%%
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\begin{align}\label{eq:decay_rate_full}\begin{split}
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\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)} =\\
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= \frac{9}{32\pi}& \left\{\right. J_{1s} \sin^2\thetak \\
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&+J_{1c} \cos^2\thetak \\
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&+J_{2s} \sin^2\thetak \cos 2\thetal \\
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&+J_{2c} \cos^2\thetak \cos 2\thetal \\
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&+J_{3} \sin^2\thetak \sin^2\thetal \cos 2\phi \\
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&+J_{4} \sin 2\thetak \sin 2\thetal \cos \phi \\
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&+J_{5} \sin 2\thetak \sin \thetal \cos \phi \\
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&+J_{6s} \sin^2\thetak \cos \thetal \\
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&+J_{7} \sin 2\thetak \sin \thetal \sin \phi \\
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&+J_{8} \sin 2\thetak \sin 2\thetal \sin \phi \\
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&+J_{9} \sin^2\thetak \sin^2\thetal \sin 2\phi \left.\right\}\,. \\
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\end{split}\end{align}
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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:
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\begin{align}\begin{split}
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J_1^c &= 1-\frac{4}{3}J_1^s\,,\\
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J_2^s &= \frac{1}{3}J_1^s\,,\\
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J_2^c &= \frac{4}{3}J_1^s-1\,.\\
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\end{split}\end{align}
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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})
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\begin{equation}\label{eq:Ai_definition}
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A_i = \frac{J_i-\bar{J_i}}{\Gamma+\bar{\Gamma}}\,.
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\end{equation}
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\clearpage
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Rewriting the \refEq{decay_rate_full} using the \emph{CP-symmetric} $S_i$ variables results in the following formula:
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\begin{align}\label{eq:decay_rate2}\begin{split}
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&\left.
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\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\thetal\,{\rm dcos}\thetak\,{\rm d}\phi}
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\right
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|_{\rm P} = \tfrac{9}{32\pi}\Bigl[\tfrac{3}{4} (1-\textcolor{red}{F_{\rm L}})\sin^2\thetak\\
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&+ \textcolor{red}{F_{\rm L}}\cos^2\thetak + \tfrac{1}{4}(1-\textcolor{red}{F_{\rm L}})\sin^2\thetak\cos 2\thetal\nonumber\\
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&- \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\\
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&+ \textcolor{red}{S_4} \sin 2\thetak \sin 2\thetal \cos\phi + \textcolor{red}{S_5}\sin 2\thetak \sin \thetal \cos \phi\nonumber\\
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&+ \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\\
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&+ \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
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\Bigr]
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\end{split}\end{align}
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\subsection{\swave decay rate}
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The decay rate of the \swave is
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\begin{equation}\label{eq:decay_rate_S_app}
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\left.
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\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
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\right |_{\rm S}
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=\frac{3}{16\pi}{F_S}\sin^2\theta_L\,.
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\end{equation}
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%
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The \pwave and \swave interference term can be parameterized as follows:
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\begin{equation}
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\label{eq:decay_rate_PS_full}
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\begin{aligned}
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\left.
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\frac{{\rm d}(\Gamma+\bar{\Gamma})}{{\rm dcos}\theta_L\,{\rm dcos}\theta_K\,{\rm d}\phi}
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\right |_{\rm PS}
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=\frac{3}{16\pi}\left[\right.
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&{S_{S1}}\sin^2\theta_L\cos\theta_K \\
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+& {S_{S2}}\sin2\theta_L \sin \theta_K \cos\phi\\
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+& {S_{S3}}\sin\theta_L \sin \theta_K \cos\phi \\
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+& {S_{S4}}\sin\theta_L \sin \theta_K \sin\phi \\
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+& {S_{S5}}\sin2\theta_L \sin \theta_K \sin\phi
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\left.\right]\,.
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\end{aligned}
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\end{equation}
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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}.
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\clearpage |