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$θ$ dependence of 4D $SU(N)$ gauge theories in the large-$N$ limit

We study the large-$N$ scaling behavior of the $θ$ dependence of the ground-state energy density $E(θ)$ of four-dimensional (4D) $SU(N)$ gauge theories and two-dimensional (2D) $CP^{N-1}$ models, where $θ$ is the parameter associated with the Lagrangian topological term. We consider its $θ$ expansion around $θ=0$, $E(θ)-E(0) = {1\over 2}χ\,θ^2 ( 1 + b_2 θ^2 + b_4θ^4 +\cdots)$ where $χ$ is the topological susceptibility and $b_{2n}$ are dimensionless coefficients. We focus on the first few coefficients $b_{2n}$, which parametrize the deviation from a simple Gaussian distribution of the topological charge at $θ=0$. We present a numerical analysis of Monte Carlo simulations of 4D $SU(N)$ lattice gauge theories for $N=3,\,4,\,6$ in the presence of an imaginary $θ$ term. The results provide a robust evidence of the large-$N$ behavior predicted by standard large-$N$ scaling arguments, i.e. $b_{2n}= O(N^{-2n})$. In particular, we obtain $b_2=\bar{b}_2/N^2 + O(1/N^4)$ with $\bar{b}_2=-0.23(3)$. We also show that the large-$N$ scaling scenario applies to 2D $CP^{N-1}$ models as well, by an analytic computation of the leading large-$N$ dependence.