Is $N=2$ Large?

TL;DR

This paper investigates the dependence of the vacuum energy on the theta parameter in 4d SU(2) Yang-Mills theory using lattice simulations, revealing insights into topological features, large N scaling, and CP symmetry breaking at theta equals pi.

Contribution

It provides the first detailed lattice determination of the theta dependence of SU(2) Yang-Mills vacuum energy and explores the phase structure as a function of N and theta.

Findings

SU(2) results are consistent with large N scaling.

The vacuum at theta = pi is gapped with spontaneous CP breaking.

Quantitative estimates of topological susceptibility and coefficients in the theta expansion.

Abstract

We study $\theta$ dependence of the vacuum energy for the 4d SU(2) pure Yang-Mills theory by lattice numerical simulations. The response of topological excitations to the smearing procedure is investigated in detail, in order to extract topological information from smeared gauge configurations. We determine the first two coefficients in the $\theta$ expansion of the vacuum energy, the topological susceptibility $\chi$ and the first dimensionless coefficient $b_2$, in the continuum limit. We find consistency of the SU(2) results with the large $N$ scaling. By analytic continuing the number of colors, $N$, to non-integer values, we infer the phase diagram of the vacuum structure of SU(N) gauge theory as a function of $N$ and $\theta$. Based on the numerical results, we provide quantitative evidence that 4d SU(2) Yang-Mills theory at $\theta = \pi$ is gapped with spontaneous breaking of the CP symmetry.