QCD properties in strong background magnetic fields

The study of the properties of the strongly interacting matter in the presence of a strong magnetic field background is relevant from both a phenomenological and a theoretical point of view. It is crucial to understand the behavior of a wide variety of system, such as the magnetars, the non-central heavy ion scatterings, or the Early Universe soon after the electroweak transition. However, as usual in the field of strong interactions, this kind of systems cannot be studied by means of perturbation theory, because of the asymptotic freedom of the theory. Thus, in the last decades, many studies were performed by means of effective models and lattice computations, and our knowledge on the QCD into a magnetic background has increased. Nevertheless, many questions are left open: in particular, on the confinement properties of the vacuum in the presence of magnetic fields $eB \gtrsim 4~{\rm GeV}^2$ and on the phase diagram, which is expected to exhibit a critical endpoint at $eB \simeq 10~{\rm GeV}^2$ and at a temperature $T \simeq 100~{\rm MeV}$. In this thesis we address these questions by means of Monte Carlo simulations on the lattice. We simulate $2+1$ flavors QCD with physical quark masses using a tree-level improved Symanzik action in the gauge sector, with a stout-improved rooted staggered quark discretization. We simulate two values of the magnetic field, namely $eB=4$ and 9~GeV$^2$, to directly probe the physics in a never-simulated region, where only speculations were made before this work. This choice implies a great numerical effort, to keep under control discretization effects which are made worse by the presence of the external magnetic field. Our results can be divided into two categories: the vanishing and finite temperature effects. In the former case, we found that the vacuum properties of the QCD such as the chiral symmetry breaking and confinement, are not disrupted by the external magnetic fields, not even in the stronger one: we found that the chiral catalysis follows, with impressive precision, the behavior predicted by the Lowest Landau Level approximation; moreover, the confining potential, as well as the color flux tube, is only modified by the background field which introduces an anisotropy in these quantities that grows with the magnetic field intensity, but the state never exhibits the ``anisotropic deconfinement'' which was expected for such strong magnetic fields. On the finite temperature side, we found that the transition temperature drops lower than expected in the former literature; we also found in accordance with former speculations that a first order phase transition happens in the stronger magnetic background. This result represents the first direct detection of a first order phase transition in $2+1$ flavors QCD in a background magnetic field at the physical point. We also studied the confinement and transport properties of the high temperature phase, obtaining evidence on the absence of confinement and the presence of Chiral Magnetic Effect.