A study of $B^0 \to D^{*-} \tau^+ \nu_{\tau}$ with three-prong $\tau$ decays at LHCb

The $B^0 \to D^{*-} \tau^+ \nu_{\tau}$ decay is sensitive to physics processes beyond the Standard Model. It has recently gained attention after the measurement, performed at BaBar, of $R_{D^\ast} = \mathcal B (B^0 \to D^{*-} \tau^+ \nu_{\tau}) / \mathcal B (B^0 \to D^{*-} \ell^+ \nu)$, which exceeds the SM prediction at the $3\sigma$ level. Therefore other measurements of this process are crucial to confirm or disprove this discrepancy. The purpose of this thesis is to study the feasibility at LHCb of the precise measurement of $\mathcal B (B^0 \to D^{*-} \tau^+ \nu_{\tau})$, using the tau hadronic decay with three charged particles in the final state, $\it{i.e.}$ $\tau^+ \to \pi^+ \pi^- \pi^+ (\pi^0) \overline {\nu} _\tau$. The study of this tau decay mode is complementary in many ways to similar studies carried with the more abundant tau leptonic decays. In particular, three-prong tau decays allow the reconstruction of the tau decay vertex, which gives a powerful criterium to discriminate between signal and the most abundant background source due to hadronic $B$ decays. The study has been performed on the data taken by LHCb collaboration in 2011, corresponding to $1 \text{fb}^{-1}$ of integrated luminosity. The data sample has been subjected to a first cut-based selection, in order to suppress the most dominant background due to $B^0 \to D^{*-} \pi^+ \pi^- \pi^+ (\pi^0)$ decays. One more stage necessary to isolate the signal component consists in applying a partial reconstruction technique, in order to get new dynamical variables which are used as input for a MultiVariate Analysis (MVA). This MVA is useful to discriminate signal from the dominant background surviving after the first cut-based selection, which is given by $B^0 \to D^{*-} D^+_{(s)} X$, where the $D^+_{(s)}$ decays in three charged pions and one or more undetected particles, and to get an estimate of the expected number of signal events. It has been shown that a sample of about 360 $B^0 \to D^{*-} \tau^+ (\to \pi^+ \pi^- \pi^+ (\pi^0)) \nu_{\tau}$ decays with a 60% purity can be isolated from the data sample. Extending this study to the other $2 \text{fb}^{-1}$ data taken at LHCb in 2012 will increase the sample by a factor 3, giving at the end more than 1000 signal events, resulting in a relative statistical uncertainty of the order of 3%.