Studies of rare $B$-meson decays to muons at the LHCb experiment

In this dissertation, very rare decays of $B$-mesons to purely muonic final states, both dimuon and four-muon, are studied at the LHCb experiment using data recorded during the first and second data-taking periods of the Large Hadron Collider, spanning the years 2011-2012 and 2015-2018, respectively. LHCb is a specialised experiment dedicated to the study of $b$- and $c$-hadrons produced in high energy proton-proton collisions at the Large Hadron Collider. This is in order to test the predictions of the Standard Model of particle physics, search for indirect evidence for new physics beyond the Standard Model and search for new sources of CP violation. The studies documented in this dissertation are performed using proton-proton collision data, corresponding to integrated luminosities of $1.11$, $2.08$ and $5.90\ \text{fb}^{-1}$ taken at centre-of-mass energies $\sqrt{s} = 7, 8 \ \text{and} \ 13 \ \text{TeV}$, respectively. The $B^{0}_{s} \to \mu^+ \mu^-$ effective lifetime is measured to be $\tau^{\mu^{+}\mu^{-}}_{\text{eff}} = 2.07 \pm 0.29 \pm 0.03 \ \text{ps}$, where the first uncertainty is statistical and the second is systematic. Searches for $B$-meson decays to the final state $\mu^{+}\mu^{-}\mu^{+}\mu^{-}$ are also performed, where both direct decays and decays via light scalar and $J/\psi$ resonances are considered. No evidence is found for any of the six decays searched for and upper limits are set on their branching fractions at $95\%$ confidence level, ranging between $2.6 \times 10^{-9}$ and $1.8 \times 10^{-10}$. All results are consistent with the predictions of the Standard Model.