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

Abstract

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.

Figures (22)

• Figure 1: Lattice results for the spectrum compared with the experimental values, from Monte Carlo simulations of QCD with $N_f=2+1$ dynamical $O(a)$ improved Wilson quarks, taken from Ref. PACS-CS-07. In each pair of data, the left (right) point has been expressed in GeV using as input the mass of $\Omega\ (\phi)$.
• Figure 2: Histograms of the lattice topological charge as measured by cooling, using the standard twisted double plaquette operator in Eq. (\ref{['qL']}), at $g_0^2=1$ ($\beta=6$) and on a $16^4$ lattice for the $SU(3)$ gauge theory, after $n=4$ (dashed line) and $n=12$ (full line) cooling steps. The data clearly tend to cluster around integer values as the number of cooling steps is increased.
• Figure 3: The lattice topological susceptibility $\chi_L\equiv \langle Q_L^2\rangle/V$ versus cooling step at $g_0^2=1$ ($\beta=6$) for the $SU(3)$ gauge theory, on a lattice of size $16^3\times 36$. The data indicated by full circles are obtained using the operator in Eq. (\ref{['qL']}). The data indicated by crosses have been obtained using the procedure of Ref. DPV-02, which reads integer values of $Q$ from the cooled configurations. The latter set of data show better convergence; the dotted line indicates the plateau where the value of of $\chi_L$ is taken.
• Figure 4: This figure shows the ratio $Q_{L,n}/Q_{L,0}$ versus $n$, where $Q_{L,n}$ is the lattice topological charge as measured by the operator (\ref{['qL']}) after $n$ steps of a local heat-bath algorithm, when heating an instanton-like configuration of charge $Q_{L,0}\approx 1$ and averaged over many independent trajectories, using the Wilson action at $g_0^2=1$ ($\beta=6$) for the $SU(3)$ gauge theory. The observed plateau provides an estimate of the multiplicative renormalization $Z$. From Ref. ACDDPV-94.
• Figure 5: This figure shows $\chi_{L,n}$ versus $n$, where $\chi_{L,n}$ is the lattice topological susceptibility as measured by the operator (\ref{['qL']}) after $n$ steps of a local heat-bath algorithm at $g_0^2=1$ ($\beta=6$) for the $SU(3)$ gauge theory, starting from a flat configuration and averaged over many independent trajectories. The dot-dashed line indicates the equilibrium value of $\chi_{L}$ (the dotted lines indicate the error). The dashed line shows the estimate of the background $B$ obtained by averaging data on the plateau. From Ref. v-95.