Quantum Probes of the Standard Model: Muon g-2 and Higgs Physics

The main uncertainty of the Standard Model prediction of the muon g-2 originates from its hadronic vacuum polarization (HVP) contribution, which cannot be reliably calculated perturbatively in QCD and relies on experimental data as input to dispersion relations. Traditionally, this contribution has been computed via a dispersive, or time-like, integral using hadronic production cross-sections in low energy electron-positron annihilation. A novel approach to determine the HVP contribution to the muon g-2 is desirable. In the first part of this work we present simple analytic expressions to compute the HVP contribution to the muon g-2 in the space-like region up to next-to-next-to-leading order. After that, we derive approximations for the euclidean-time kernel obtained from the HVP contributions in the space-like region up to next-to-leading order. The series expansions presented in this work overcome the problems given by their asymptotic behaviour for large value of the time. These results can be employed in lattice QCD calculations of this contribution as well as in space-like determinations based on scattering data, like that expected from the proposed MUonE experiment at CERN. For the second part of this thesis, we focus on the constraints of the Yukawa couplings of the first-generation quarks, which are notoriously challenging to determine due to their small values within the framework of the Standard Model. We propose the utilization of off-shell Higgs production, where the Higgs boson decays into four leptons, as a method to investigate the up- and down-quark Yukawa couplings. Employing kinematic discriminants akin to those utilized in Higgs width measurements, we have found that the down (up) Yukawa coupling can be constrained to approximately 156 (260) times its Standard Model value, considering only experimental systematic uncertainties, in a high-luminosity LHC scenario. This approach to off-shell Higgs production demonstrates superior sensitivity to the first-generation quark Yukawa couplings compared to other methods, such as Higgs+jet or Higgs pair production.