Higgs mass and QCD axion properties at high precision

This thesis is dedicated to study the physical properties at high precision of two scalars of well-motivated beyond the standard model theories: the Higgs boson in the Minimal Supersymmetric Standard Model (MSSM) and the axion of the Peccei-Quinn mechanism. The discussion is divided in two parts. We make use of the advantages of the effective field theory framework. In the first part, we consider the state-of-the-art of the effective field theory computation of the MSSM Higgs mass, improving the existing ones by including extra threshold corrections. We perform a detailed estimate of the theoretical uncertainty. We study the large tangent beta region and we put emphasis on the allowed parameter space reproducing the experimental value of the Higgs mass. We present SusyHD, a fast computer code that computes the Higgs mass and its uncertainty for any supersymmetry (SUSY) scale, from the TeV to the Planck scale, even in Split SUSY, both in the DRbar and in the on-shell (OS) schemes. Finally, we apply our results to derive bounds on some well motivated SUSY models, in particular we show how the value of the Higgs mass allows to determine the complete spectrum in minimal gauge mediation. In the second part, we discuss how to extract several properties of the axion of Quantum Chromodynamics (QCD) with great accuracy using only first principle QCD computations. We obtained the axion potential, the mass and the coupling to photons by combining next to leading order (NLO) calculations in chiral perturbation theory (ChPT) with recent Lattice QCD results. Axion-nucleon interactions are also derived reliably. The method we have followed allows to further improve the precision as uncertainties on the light quark masses and the effective field theory couplings are reduced. We have also studied the finite temperature dependence of the axion potential and its mass, in connection with its role in determining the axion relic abundance.