Theta dependence of SU(N) gauge theories in the presence of a topological term (0803.1593v5)

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.

Essay on the $\theta$ Dependence of $SU(N)$ Gauge Theories with a Topological Term

The paper under review examines the $\theta$ dependence of $SU(N)$ gauge theories in four dimensions, exploring both theoretical frameworks and numerical results. The parameter $\theta$ refers to the coefficient of a CP-violating topological term in the Lagrangian, an essential concept in understanding phenomena related to the QCD vacuum and topological contributions such as instantons.

Theoretical Background

The paper of $\theta$ dependence is anchored in the complex nature of the Euclidean Lagrangian, which includes terms that can break parity (P) and time reversal (T) symmetries. In the continuum, these $\theta$ terms do not contribute perturbatively, suggesting a nontrivial $\theta$ dependence rooted in nonperturbative effects. Classical analysis of solutions like instantons has demonstrated the quantization of topological charge, leading to multi-vacuum structures parametrized by $\theta$.

A cornerstone of the analysis is examining these theories in the large $N$ limit, where $N$ represents the number of colors. Large $N$ scaling arguments provide insight into the relevant scaling variables, suggesting that $\bar{\theta} = \theta/N$ governs the theory's behavior. This insight helps justify an approach where $\theta$-dependence can potentially affect physical observables like the string tension $\sigma$ and glueball masses.

Lattice Calculations and Numerical Simulations

Most numerical results regarding $\theta$ dependence are obtained from lattice simulations, which discretize spacetime into a grid upon which quantum field theories like QCD can be simulated in a nonperturbative manner. The lattice approximation allows for the computation of $\theta$ dependent quantities, especially near $\theta=0$ using expansions. Monte Carlo methods are typically employed to explore these theories' dynamics and extract information about the ground state energy, the spectrum's $\theta$ dependence, and the topological susceptibility.

Numerical studies on 4D $SU(3)$ gauge theories reveal that significant $\theta$ dependence can be seen in observables such as the topological susceptibility. Recent advancements have incorporated overlap fermions, circumventing issues of chiral symmetry on the lattice and offering improved control over theoretical ambiguities. The renormalization of topological quantities can thus be robustly handled, enhancing the reliability of numerical results.

Large-$N$ Limit and Theoretical Implications

Results across several studies support a scenario where $\theta$ dependence in gauge theories displays intriguing behavior in the large-$N$ limit. Observables like topological susceptibility demonstrate stable large-$N$ behavior that aligns with theoretical predictions based on scaling arguments. For instance, the Witten-Veneziano mechanism, a critical aspect of the $U(1)_A$ problem in QCD, is further corroborated through these findings, linking the suppression of $\eta'$ mass to topological charge dynamics.

In the large-$N$ approach, the analysis of 2D $CP^{N-1}$ models offers additional theoretical insight, demonstrating striking parallels to the 4D cases. They provide a theoretical laboratory that captures nuances related to $\theta$ dependence through a controlled $1/N$ expansion, elucidating aspects such as topological susceptibility and vacuum stability.

Conclusion and Future Directions

This paper significantly enhances our understanding of the $\theta$ dependence of $SU(N)$ gauge theories, integrating theoretical analysis with computational results. The implications reach both the practical field, impacting how simulations are conducted, and the theoretical domain, informing discussions on topological features of the QCD vacuum. Future studies are suggested to improve lattice techniques and investigate further implications in the presence of fermions and at finite temperature, broadening the scope and understanding of $\theta$ dependence in quantum field theories. These efforts could lead to refined insights into physical phenomena like the strong CP problem and vacuum dynamics in quantum chromodynamics.