Search for the lepton-flavour violating decay $\tau\to\mu\phi$ at the LHCb experiment

The subject of the thesis is the search for the lepton-flavour violating decay $\tau\rightarrow\mu\phi$ at the LHCb experiment. In the Standard Model (SM), the charged leptons ($e^-$, $\mu^-$, $\tau^-$) and their associated neutrinos ($\nu_e$, $\nu_\mu$, $\nu_\tau$) carry the three lepton flavours (electron, muon, tau) that are strictly conserved in all reactions. If the SM is modified to include the mixing between the neutrino flavour eigenstates, charged lepton flavour violating (LFV) processes are permitted at higher order, but they are extremely suppressed because of the small neutrino masses. Therefore, the observation of LFV processes such as $\tau^-\rightarrow\mu^-\phi$ would be a clear sign of New Physics. Many Beyond Standard Model theories allow charged lepton flavour violation. Among these, a specific theoretical framework formulated by Isidori and collaborators predicts for the $\tau\rightarrow\mu\phi$ decay branching fraction values up to $10^{-8}$ through the exchange of a vector leptoquark. The predicted branching fraction values are close to the present experimental limit set by the Belle collaboration, $\mathcal{B}(\tau\rightarrow\mu\phi)<8.4 \times 10^{-8} \text{ @ 90\% CL}$. The LHCb collaboration has not measured this decay yet, and this work represents a first attempt to search for the $\tau\rightarrow\mu\phi$ decay at a hadron collider. The search for LFV in $\tau$ decays at LHCb can benefit from a larger $\tau$ production cross-section than at $e^+e^-$ $B$-factories. In the detector acceptance, $\tau$ leptons are copiously produced, almost exclusively from the decays of $D_s$, $D$ mesons and $b$-hadrons. The signal candidates are reconstructed by selecting the $\phi$ decay in the $K^{+}K^{-}$ final state. Therefore, a pair of oppositely charged tracks, identified as kaons and with invariant mass consistent with the $\phi$ mass, are combined with a track identified as muon. Several selection requirements are applied on trigger, kinematic and geometrical quantities. The signal search strategy developed in this work differs substantially from the one adopted by the Belle experiment because of the entirely different nature of backgrounds in the two environments. In particular, at LHCb, the background is dominated by the decay $D_{s}^{+}\rightarrow\phi \mu^{+}\nu_{\mu}$, which has the same final state as the signal (the neutrino is not reconstructed) except for the mass, and it is very abundant. The discrimination between this background and the signal is optimised using a multivariate classifier that exploits the residual differences in their kinematic and geometric properties. In particular, KNN and BDT algorithms are trained and tested on signal and background MC simulations. Finally, the optimal cut on the output of the chosen BDT classifier is set with an approach which allows to optimise the chance of discovery and the tightness of the resulting limits at the same time, here applied for the first time to a non-counting measurement. As typical at hadronic colliders, the branching fraction is measured relative to the known branching fraction of a normalisation channel, and the $D_{s}^{+}\rightarrow\phi \mu^{+}\nu_{\mu}$ background itself has been considered for the current result. The study is performed on the Run 2 dataset, which corresponds to an integrated luminosity of about $6\,\rm{fb^{-1}}$ of proton-proton collisions at a centre-of-mass energy of $\sqrt{s}=13\,\rm{TeV}$. The analysis is conducted without looking at the candidates in the signal $\tau$ mass region (blind analysis in a $\pm 20\, \rm{MeV/c^2}$ mass window around the $\tau$ PDG mass, approximately four times the expected mass resolution). Toy datasets, which simulate the data in the blinded region, are generated in order to study the single event sensitivity. As a result of the present work, the analysis process has been defined and optimised, allowing to evaluate of an expected upper limit from the Run 2 dataset $$\mathcal{B}(\tau\rightarrow\mu\phi)<1.1 \times 10^{-6} \text{ @ 90 \% CL}$$ Although this is less stringent than the current existing limit, it shows that this measurement, while challenging, is not too far out of reach at hadron colliders, and it may become interesting with future larger LHCb datasets and improved trigger selections.