Lifetime measurements of the $\mathrm{B_{s}^{0} \rightarrow K^{+}K^{-}}$ system and RICH studies at LHCb

Lifetime analyses in heavy flavour systems can provide excellent opportunities to make precision measurements of Standard Model (SM) parameters that probe for new physics. This thesis presents lifetime measurements from the $\mathrm{B_{d}^{0}}$ and $\mathrm{B_{s}^{0}}$ mesons to the charmless two-body hadron family of decays, known as $\mathrm{B \rightarrow h^{+}h^{'-}}$, where $h$ can be either a kaon or a pion. A particular interest is given to the $\mathrm{B_{s}^{0} \rightarrow K^{+}K^{-}}$ decay channel. The $\mathrm{B_{s}^{0} \rightarrow K^{+}K^{-}}$ channel is interesting as it decays into a $CP$ even final state, with a small amount of $CP$-violation predicted by the Standard Model. This can be quantified using the decay rate asymmetry parameter ${\cal A}_{\Delta\Gamma}$. The decay is also loop dominated, and thus sensitive to New Physics (NP) that could enter in these processes. These effects can be studied by constraining the values of the decay rate difference, $\Delta\Gamma_{s}$, and ${\cal A}_{\Delta\Gamma}$ from a lifetime measurement of the $\mathrm{B_{s}^{0} \rightarrow K^{+}K^{-}}$. A method used to measure the effective lifetime of the $\mathrm{B \rightarrow h^{+}h^{'-}}$ channels, which removes the acceptance bias induced by the event selection, is described. This includes fits to the invariant mass and reconstructed lifetime spectrums, together with the methods employed to determine the per-event acceptance functions from data and treatment of non-parametric backgrounds. The application of this method to two datasets is then presented. Both datasets are collected using a centre of mass energy $\mathrm{\sqrt{s} = 7~TeV}$ with the first dataset from the 2010 run comprising a total integrated luminosity of $\mathrm{37~pb^{-1}}$, and the second from the 2011 run comprising a total integrated luminosity of $\mathrm{1.0~fb^{-1}}$. The fit methods are verified using various techniques, primarily using a simplified MC simulation and full LHCb Monte Carlo (MC). The fit to the mass spectrum of the $\mathrm{B_{d}^{0} \rightarrow K^{+}\pi^{-}}$ from the 2011 dataset is used both for the lifetime measurement, and for the normalisation in a search for the $\mathrm{B_{d/s}^{0} \rightarrow p\bar{p}}$ decay. As the details of the lifetime analysis methods are subtly different between datasets, the sources of uncertainty differ between the analyses. The lifetime measurement resulting from the 2010 dataset is found to be \begin{equation} \label{eq:bsresult2010_abs} \tau_{\mathrm{ B_{s}^{0} \rightarrow K^{+}K^{-} }} = {\mathrm{1.440 \pm 0.096 ~ ps ~ (stat) ~ \pm ~ 0.008 ~ ps ~ (syst).}} \end{equation} The resulting $\mathrm{B \rightarrow h^{+}h^{'-}}$ lifetimes, measured from the 2011 dataset are found to be \begin{equation} \label{eq:bsresult2011_abs} \tau_{\mathrm{B_{s}^{0} \rightarrow K^{+}K^{-}}} = {\mathrm{1.407 \pm 0.016 ~ ps ~ (stat) ~ \pm ~ 0.007 ~ ps ~ (syst),}} \end{equation} \begin{equation} \label{eq:bd2kpi_result2011_abs} \tau_{\mathrm{B_{d}^{0} \rightarrow K^{+}\pi^{-}}} = {\mathrm{1.524 \pm 0.011 ~ ps ~ (stat) ~ \pm ~ 0.004 ~ ps ~ (syst),}} \end{equation} \begin{equation} \label{eq:bs2piK_result2011_abs} \tau_{\mathrm{B_{s}^{0} \rightarrow \pi^{+}K^{-}}} = {\mathrm{1.597 \pm 0.056 ~ ps ~ (stat) ~ \pm ~ 0.012 ~ ps ~ (syst).}} \end{equation} These measurements are consistent with previous measurements, with the value of $\tau_{\mathrm{B_{s}^{0} \rightarrow K^{+}K^{-}}}$ measured from the 2011 dataset having a greater precision than the current world average. From this, we are able to directly measure the quantity ${\cal A}_{\Delta\Gamma}$ for the first time, with the value evaluated found to be \begin{equation} \label{eq:ADG_val_conclusions} {\cal A}_{\Delta\Gamma} = {\mathrm{-0.87 \pm 0.17 ~ (stat) ~ \pm ~ 0.13 ~ (syst).}} \end{equation} Particle IDentification (PID) of the family of $\mathrm{B \rightarrow h^{+}h^{'-}}$ decays is crucial for distinguishing the many different final states that are kinematically similar. These final states tend to overlap in mass when reconstructed, providing considerable difficulties when making measurements with them. The Ring Imaging CHerenkov (RICH) detectors provide the experiment with PID capability, which allow high precision measurements from these channels to be made. During data taking periods, it is important to monitor the performance of the many subsystems that allow the RICH operations to run smoothly. Two studies are presented in this thesis that pertain specifically to RICH 2. The first studies the stability of the spherical and flat mirrors, using the Laser Alignment Monitoring System (LAMS). A direct correlation between observed movement and temperature is found, which is in the order $10-70~\rm{\mu rad/K}$ and agrees with estimates based on the expansion coefficients of materials and the mirror supports. This is an order of magnitude lower than the angular resolution of RICH 2, so does not affect the resolution or performance of the detector. The second study suggests an alternative method to constrain the refractive index of the RICH gas. This method allows for continuous monitoring of the refractive index, with a greater frequency in time than is currently available. The method is purely data driven and is found to be in agreement with the currently employed RICH method to within $1\%$.