Abstract
This thesis reports the branching fraction measurement of the rare decay $B^+ \to K^+ \mu^+ \mu^-$ in the di-lepton invariant mass, $q = m(\mu^+ \mu^-)$, region above narrow $c \overline{c}$, $J/\psi$ and $\psi{(2S)}$. The region of interest is defined as $q^2 > 14.3 \, \mathrm{GeV}^2/c^4$. The measurement is performed using 6 fb$^{-1}$ of proton-proton collision data recorded by the LHCb~detector between 2015 and 2018. The measured value of the branching fraction:
\begin{align} \mathcal{B}(B^+ \to K^+ \mu^+ \mu^-) = \mathrm{(10.7 \pm 0.2~(stat) \pm 0.3~(syst)) \times 10^{-8}}.\nonumber \end{align}
\noindent is obtained using $B^+ \to K^+ J/\psi (\to \mu^+ \mu^-)$ as a normalization channel. The statistical uncertainty is marked as "stat" and the systematic uncertainty is marked as "syst". The systematic uncertainty includes the uncertainty on the branching fractions of the normalization mode which is the component with the highest contribution.
The studies of occupancy and forward tracking for future upgrades of the downstream tracking detector of LHCb, known as Mighty Tracker, are discussed in the thesis as well.
From the initial occupancy studies, the downstream tracking detector layout has been studied. It has been shown that the maximum occupancy per fiber per event in the outer part of the Mighty Tracker can be achieved in Upgrade~II conditions, matching the maximum occupancy in the downstream tracker of the current Upgrade~I detector. The considered detector geometry includes a silicon made inner detector with a large area of coverage with a drawback of a large potential cost.
Regarding the forward tracking studies, it has been found that the average efficiency of 91.0$\%$ and a ghost rate of 80.0$\%$ can be achieved if no optimization cuts on found tracks are applied. The optimal point of the cut can be chosen based on the distribution of the efficiency and ghost rate as a function of the $\chi^2_{track}$ cut.
Denysenko, Vadym