Unveiling the Higgs sector with tau-leptons: differential cross-section measurements and searches for lepton-flavor-violating decays with the ATLAS detector

The Standard Model (SM) of particle physics provides the theoretical framework for understanding fundamental particles and the forces that interact among them, with the exception of gravity. This relativistic quantum field theory describes the behavior and interactions of fermions, which constitute matter, and gauge bosons, which mediate the electromagnetic, weak, and strong interactions. Within the SM, the Higgs boson, a scalar particle predicted by the theory, plays a central role in confirming the existence of the Higgs field. This field is responsible for giving elementary particles their masses through the Higgs mechanism, which involves the spontaneous breaking of the electroweak symmetry. The discovery of the Higgs boson in 2012 by the ATLAS and CMS collaborations at the Large Hadron Collider (LHC) at CERN marked the culmination of the SM. The Higgs boson has a unique role within the SM, coupling to fermions proportionally to their masses and to gauge bosons with a strength proportional to the square of their masses. The experimental measurement of these couplings allows for comparisons with theoretical predictions. The discovery of a Higgs boson with a mass close to 125 GeV not only validated the SM but also enabled precise measurements of its production and decay channels, allowing to test theoretical predictions and search for possible deviations. However, the SM has limitations that motivate the search for new physics beyond the Standard Model (BSM), with the Higgs boson being a key element due to its potential connection to BSM phenomena. This thesis explores two main research approaches related to the Higgs sector. The first involves the measurement of differential cross-sections in the Higgs boson decay channel to two tau leptons. This channel provides direct access to the Higgs boson's Yukawa coupling to leptons. Additionally, it is particularly sensitive to Higgs boson production via vector boson fusion, a process in which two vector bosons (W or Z) emitted by quarks in the colliding protons at the LHC fuse to produce a Higgs boson. The differential cross-section analysis in this channel employs the Simplified Template Cross-Sections (STXS) framework, which allows for the measurement of Higgs boson production in specific phase-space regions defined by the kinematics of the final-state particles. This approach, combined with the luminosity of the Run 2 LHC dataset, consisting of 140 $\mathrm{fb}^{-1}$ of proton-proton collisions at a center-of-mass energy of $\sqrt{s}=13$ TeV, enhances the precision and extends the sensitivity of Higgs boson measurements beyond previously explored inclusive methods. Furthermore, this thesis presents, for the first time in the ATLAS experiment, a measurement of differential cross-sections in the tau-lepton decay channel within a fiducial region. The results of this measurement are interpreted within the framework of the SM Effective Field Theory (SMEFT). The second research approach in this thesis is the search for Higgs boson decays that violate lepton flavor conservation (LFV) in final states with an electron and a tau lepton or a muon and a tau lepton, also using the full dataset collected during LHC Run 2. While the SM conserves lepton flavor in the charged lepton sector, observations of neutrino oscillations demonstrate that lepton flavor is not conserved in nature, raising the possibility that this violation could also manifest in Higgs boson decays. The detection of such decays would constitute direct evidence of BSM physics and provide insight into possible mechanisms of lepton flavor violation in the charged lepton sector. Although current experimental limits have constrained the possible branching fractions of these processes, the small decay width of the Higgs boson does not rule out small contributions beyond SM predictions. This thesis also addresses the development of simulation software for the readout electronics of the ATLAS Tile Calorimeter, the experiment's central hadronic calorimeter, in preparation for the high-luminosity phase of the LHC (HL-LHC). This software enables the validation of signal reconstruction algorithms to ensure the accuracy of calorimeter measurements under the HL-LHC conditions.