The $\theta$-dependence of the Yang-Mills spectrum from analytic continuation

We study the $\theta$-dependence of the string tension and of the lightest glueball mass in four-dimensional $\mathrm{SU}(N)$ Yang-Mills theories. More precisely, we focus on the coefficients parametrizing the $\mathcal{O}(\theta^2)$ dependence of these quantities, which we investigate by means of numerical simulations of the lattice-discretized theory, carried out using imaginary values of the $\theta$ parameter. Topological freezing at large $N$ is avoided using the Parallel Tempering on Boundary Conditions algorithm. We provide controlled continuum extrapolations of such coefficients in the $N=3$ case, and we report the results obtained on two fairly fine lattice spacings for $N=6$.