Abstract:We study the θ dependence of four-dimensional SU(N) gauge theories, for N ≥ 3 and in the large-N limit. We use numerical simulations of the Wilson lattice formulation of gauge theories to compute the first few terms of the expansion of the ground-state energyfor sufficiently small values of θ, θ < π. Indeed we verify that the topological susceptibility has a nonzero large-N limit χ ∞ = 2A with corrections of O(1/N 2 ), in substantial agreement with the Witten-Veneziano formula which relates χ ∞ to the η ′ mass. Furthermore, higher order terms in θ are suppressed; in particular, the O(θ 4 ) term b 2 (related to the η ′ − η ′ elastic scattering amplitude) turns out to be quite small: b 2 = −0.023(7) for N = 3, and its absolute value decreases with increasing N, consistently with the expectation b 2 = O(1/N 2 ).

We review results concerning the θ dependence of 4D SU (N ) gauge theories and QCD, where θ is the coefficient of the CP-violating topological term in the Lagrangian. In particular, we discuss θ dependence in the large-N limit.Most results have been obtained within the lattice formulation of the theory via numerical simulations. We review results at zero and finite temperature. We show that the results support the scenario obtained by general large-N scaling arguments, and in particular the Witten-Veneziano mechanism to explain the U (1) A problem. We also compare with results obtained by other approaches, especially in the large-N limit, where the issue has been also addressed using, for example, the AdS/CFT correspondence.We discuss issues related to theta dependence in full QCD: the neutron electric dipole moment, the dependence of the topological susceptibility on the quark masses, the U (1) A symmetry breaking at finite temperature.We also review results in the 2D CP N −1 model, which is an interesting theoretical laboratory to study issues related to topology.Finally, we discuss the main features of the two-point correlation function of the topological charge density.

In this paper we present results for the renormalization of gauge invariant nonlocal fermion operators which contain a Wilson line, to one-loop level in lattice perturbation theory. Our calculations have been performed for Wilson/clover fermions and a wide class of Symanzik improved gluon actions.The extended nature of such 'long-link' operators results in a nontrivial renormalization, including contributions which diverge linearly as well as logarithmically with the lattice spacing, along with additional finite factors.On the lattice there is also mixing among certain subsets of these nonlocal operators; we calculate the corresponding finite mixing coefficients, which are necessary in order to disentangle individual matrix elements for each operator from lattice simulation data. Finally, extending our perturbative setup, we present non-perturbative prescriptions to extract the linearly divergent contributions.

We determine by numerical simulations on a lattice the gauge-invariant two-point correlation function of the gauge field strengths in the QCD vacuum, down to a distance of 0.1 fm.

In this work we present, for the first time, the non-perturbative renormalization for the unpolarized, helicity and transversity quasi-PDFs, in an RI' scheme. The proposed prescription addresses simultaneously all aspects of renormalization: logarithmic divergences, finite renormalization as well as the linear divergence which is present in the matrix elements of fermion operators with Wilson lines. Furthermore, for the case of the unpolarized quasi-PDFs, we describe how to eliminate the unwanted mixing with the twist-3 scalar operator. We utilize perturbation theory for the one-loop conversion factor that brings the renormalization functions to the MS-scheme at a scale of 2 GeV. We also explain how to improve the estimates on the renormalization functions by eliminating lattice artifacts. The latter can be computed in one-loop perturbation theory and to all orders in the lattice spacing. We apply the methodology for the renormalization to an ensemble of twisted mass fermions with Nf=2+1+1 dynamical light quarks, and a pion mass of around 375 MeV.Comment: 24 pages, 10 figures, 2 Tables, Section 3 largely expanded compared to v1. Version accepted for publication in Nucl. Phys. B (invited Frontiers Article