• Lattice QCD

It is widely believed that strong interactions are described by the theory of quantum chromodynamics (QCD), which is a non-abelian gauge field theory based on the symmetry group $SU(3)$. As in quantum electrodynamics (QED), we have a theory describing matter fields and gauge fields, although now we have to deal with 8 self-interacting gauge fields, the gluons, and with fermionic fields carrying a color degree of freedom, the quarks. In particular, we have a different behavior for the coupling constants of the two theories: contrary to QED, the QCD coupling constant is large at the low energy scale while it vanishes for asymptotically high energies. This behavior, which is called asymptotic freedom, permits the use of perturbation theory, giving good agreement with deep inelastic scattering experiments.

On the other side, the fact that the coupling constant is large at low energies gives rise to a series of new non-perturbative phenomena, which perturbation field theory is unable to tackle. An example is quark confinement, which is related to the experimental fact that quarks and gluons are not directly observable. In absence of perturbation theory a different approach is required. An effort in this direction was made by Wilson, who proposed a formulation of gauge fields theory on the lattice, introducing a natural regularization on the theory and giving the possibility to carry out numerical calculations in the non-perturbative regime. Using these techniques we are now able to investigate all these new phenomenological problems, thanks also to the increasing technological development of calculators.

The non-perturbative nature of QCD partially arises from the presence of classical solutions with a non-trivial topology, the instantons, which are connected to hadron phenomenology through the equation of the axial anomaly. Theoretically, one can introduce in QCD a new interaction that violates explicitly the CP symmetry, the so called $\theta$-term, which can be written as a parameter $\theta$ coupled to the topological charge $Q$ (which gives the winding number associated to the gauge configuration of the system). None of the experiments done in the last decades has shown any signal of a possible CP violation in the strong sector, from which one can put an upper bound $|\theta|\lesssim 10^{-10}$; the reason of this suppression is still unknown, although many possible mechanisms have been considered. Nevertheless, the dependence of QCD on $\theta$ is of great theoretical and phenomenological interest. $\theta$ derivatives of the vacuum free energy, computed at $\theta\equiv 0$, enter various aspects of hadron phenomenology; an example is the topological susceptibility $\chi \equiv \langle Q^2\rangle/V$ which enters the solution of the so-called $U(1)_A$ problem.

A significant interest has been raised recently by the possibility that local effective variations of $\theta$, corresponding to topological charge fluctuations, may induce detectable phenomena in presence of magnetic fields as strong as those produced in the early phases of non-central heavy ion collisions, which can reach up to $10^{15}$ Tesla at LHC. According to the so-called chiral magnetic effect (CME), the net unbalance of chirality induced by the topological background would lead, in presence of a magnetic field strong enough to align the magnetic moments of quarks, to a net separation of electric charge along the field direction.

So in the CME we have a $CP$-odd gluonic background giving rise to a charge separation, hence to an electric field, parallel to the background magnetic field, which means a $CP$-violating term $\vec{E}\cdot\vec{B}$. In this thesis we study a phenomenon which is in some sense complementary to the CME: we want to understand if and how a $CP$-violating electromagnetic (e.m.) background field gives rise to $CP$-violation in the strong sector, i.e. to an effective $\theta$ term: $\theta_{eff}\simeq\chi_{CP}e^2\vec{E}\cdot\vec{B}$. The purpose of this study is to furnish a first determination of $\chi_{CP}$ based on lattice QCD simulations. To this aim we have done simulations of QCD in presence of uniform e.m. background fields: as we show in the thesis, in order to maintain a positive path integral measure and feasibility of the Monte Carlo simulations, the electric field $\vec{E}$ can be introduced only as imaginary field. As a consequence, the induced $\theta$-term will be imaginary and we will compute it by measuring the shift of the topological charge distribution. Finally, we will rely on analytic continuation in order to estimate $\chi_{CP}$. In the simulation we have considered 2 flavoured degenerate quarks, with the charges corresponding to the quarks \emph{up} and \emph{down}, using various values of lattice spacing to study the continuum limit. Numerical simulations have been performed on graphics cards (GPU), making use of an existing CUDA code for the simulation of QCD, developed in Pisa, which we have modified to permit the introduction of e.m. background fields. We have made use of graphics card clusters provided by INFN and located in Pisa, Genova and Rome.

Because of the limited computing power of the devices, we have study until now only an unphysical regime for the quarks masses, corresponding to $m_\pi$ down to 280 MeV. For these values of the quarks masses, we have found a susceptibility $\chi_{CP}\sim7-8\ (\mbox{GeV})^{-4}$, whose order of magnitude is in agreement with previous phenomenological estimations. As a natural development of this work, in addition to a study at physical values of the quarks masses, it would be important to measure this effect also for finite values of the temperature, to understand if $\chi_{CP}$ can change dramatically as one crosses the deconfinement transition, where quarks and gluons are expected to be liberated. That could have phenomenological consequences for heavy ions collisions and for the early stages of the universe.