The imaginary-$\theta$ dependence of the SU($N$) spectrum

TL;DR

This study investigates how the spectrum of four-dimensional SU(N) Yang--Mills theories depends on the imaginary $ heta$ parameter, focusing on string tension and mass gap at N=3 and N=6 using lattice simulations.

Contribution

It provides the first lattice determination of the $ heta$-dependence of the spectrum for SU(3) and SU(6) Yang--Mills theories at imaginary $ heta$, including the $ heta^2$ coefficient in the continuum limit.

Findings

Determined the $ heta$-dependence of the spectrum at N=3 and N=6.

Extracted the $ heta^2$ coefficient in the continuum limit for N=3.

Used Parallel Tempering on Boundary Conditions to avoid topological freezing.

Abstract

In this talk we will report on a study of the $\theta$-dependence of the string tension and of the mass gap of four-dimensional SU($N$) Yang--Mills theories. The spectrum at $N=3$ and $N=6$ was obtained on the lattice at various imaginary values of the $\theta$-parameter, using Parallel Tempering on Boundary Conditions to avoid topological freezing at fine lattice spacings. The coefficient of the $\mathcal{O}(\theta^2)$ term in the Taylor expansion of the spectrum around $\theta=0$ could be obtained in the continuum limit for $N=3$, and on two fairly fine lattices for $N=6$.