• tau
• lifetime
• Belle II
• lepton flavor universality

The Belle II experiment, operating at the SuperKEKB electron-positron collider, is going to collect $\SI{50}{\atto\barn}^{-1}$ of integrated luminosity by 2030, a factor 50 more than its predecessors. The large cross section for $e^+e^- \rightarrow \tau^+\tau^-$ at the $\Upsilon(4S)$ resonance energy, will yield to about 46 billions of produced $\tau$ pairs, making Belle II, besides a $B$-factory, a $\tau$-factory as well.

The $\tau$-lepton is a powerful laboratory for both Standard Model (SM) and beyond SM measurements. A precise $\tau$ lifetime measurement, in particular, can contribute to the test of lepton flavor universality within the SM. Also, lifetime measurements are very effective tools to monitor detector performance and alignment. Furthermore, they are challenging in case of a particle such as $\tau$-lepton, whose lifetime uncertainty is below 0.2\% at present. To achieve a competitive precision a very good understanding of detector performance is needed. At Belle II, $\tau$ pairs have an expected mean decay length of about $\SI{250}{\micro\meter}$.

In this thesis, a new method for $\tau$ lifetime measurement is studied. The method aims to fully reconstruct $e^+e^- \rightarrow \tau^+\tau^-$ events, where both taus decay with one neutrino only, i.e. hadronically. For this class of events, which potentially represents 42\% of the total $\tau$ pair decays, $\tau$ four-momenta can be reconstructed up to a two-fold ambiguity, just measuring the momenta of all hadronic final states and requiring four-momentum conservation.
Thanks to the nano-beam collision scheme adopted by SuperKEKB machine, the beam spot size at the interaction point is much smaller than in any of the previous collider experiments. The primary vertex position on the transverse plane can hence be treated as a fixed constraint.
Using the information on the impact parameter of reconstructed tracks, the vertex relations can be included in a unique system of equations together with the kinematic relationships. The system can be numerically solved through a $\chi^2$-like minimization where the $\tau$ decay lengths and momenta, as well as the primary vertex position along the beam direction, are considered as free parameters.

The method proposed is tested for the 1-prong$\times$3-prong decay topology, where the vertex position of the 3-prong $\tau$ can be independently fitted. A selection of these events in Belle II had been developed on Monte Carlo and applied on $\SI{34.6}{\femto\barn}^{-1}$ data, collected during 2019 and early 2020 runs. The full event reconstruction method gives a proper decay time resolution $\sigma(t) = \SI{65}{\femto\second}$, obtained for the 3-prong side. In addition, the interaction point along the beam direction ($z$ direction) is also reconstructed with a resolution $\sigma(\textsc{ip}_z) = \SI{20}{\micro\meter}$\,, thus making possible a precise event per event monitoring of the beam spot position.