Theta dependence of SU(N) gauge theories in the presence of a topological term

We review results concerning the theta dependence of 4D SU(N) gauge theories and QCD, where theta is the coefficient of the CP-violating topological term in the Lagrangian. In particular, we discuss theta dependence in the large-N limit. Most results have been obtained within the lattice formulation of the theory via numerical simulations, which allow to investigate the theta dependence of the ground-state energy and the spectrum around theta=0 by determining the moments of the topological charge distribution, and their correlations with other observables. We discuss the various methods which have been employed to determine the topological susceptibility, and higher-order terms of the theta expansion. 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 consider the 2D CP(N) model, which is an interesting theoretical laboratory to study issues related to topology. We review analytical results in the large-N limit, and numerical results within its lattice formulation. Finally, we discuss the main features of the two-point correlation function of the topological charge density.

O(a^2) corrections to the propagator and bilinears of Wilson / clover fermions

We present the corrections to the fermion propagator, to second order in the lattice spacing, O(a^2), in 1-loop perturbation theory. The fermions are described by the clover action and for the gluons we use a 3-parameter family of Symanzik improved actions. Our calculation has been carried out in a general covariant gauge. The results are provided as a polynomial of the clover parameter, and are tabulated for 10 popular sets of the Symanzik coefficients (Plaquette, Tree-level Symanzik, Iwasaki, TILW and DBW2 action). We also study the O(a^2) corrections to matrix elements of fermion bilinear operators that have the form $barPsiGammaPsi$, where $Gamma$ denotes all possible distinct products of Dirac matrices. These correction terms are essential ingredients for improving, to O(a^2), the matrix elements of the fermion operators. Our results are applicable also to the case of twisted mass fermions. A longer write-up of this work, including non-perturbative results, is in preparation together with V. Gimenez, V. Lubicz and D. Palao.

Higher representations on the lattice: perturbative studies

We present analytical results to guide numerical simulations with Wilson fermions in higher representations of the colour group. The ratio of $Lambda$ parameters, the additive renormalization of the fermion mass, and the renormalization of fermion bilinears are computed in perturbation theory, including cactus resummation. We recall the chiral Lagrangian for the different patterns of symmetry breaking that can take place with fermions in higher representations, and discuss the possibility of an Aoki phase as the fermion mass is reduced at finite lattice spacing.

The three-loop beta-function of SU(N) lattice gauge theories with overlap fermions

We briefly report our calculation of the 2-loop coefficient of the coupling constant renormalization function Z_g in lattice perturbation theory. The quantity under study is defined through g_0 = Z_g g, where g_0 (g) is the bare (renormalized) coupling constant. The 2-loop expression for Z_g can be directly related to the 3-loop bare beta-function beta_L(g_0). Our calculation is performed using overlap fermions and Wilson gluons, and the background field technique has been chosen for convenience. Our results depend explicitly on the number of fermion flavors (N_f) and colors (N). Since the dependence of Z_g on the overlap parameter rho cannot be extracted analytically, we tabulate our results for different values of rho in the allowed range (0<rho<2), focusing on values which are being used most frequently in simulations. Plots of the 1- and 2-loop results for Z_g versus rho exhibit a nontrivial dependence on the overlap parameter. A longer write-up of this work may be found in 0709.4368.

SU(N) gauge theories in the presence of a topological term

We review recent results on the theta dependence of the ground-state energy and spectrum of four-dimensional SU(N) gauge theories, where theta is the coefficient of the CP-violating topological term F-Fdual in the Lagrangian. In particular, we discuss the results obtained by Monte Carlo simulations of the lattice formulation of QCD, which allow the investigation of theta dependence around theta=0 by determining the moments of the topological charge distribution, and their correlations with other observables. The results for N=3 and larger values of N support the scenario obtained by general large-N scaling arguments.