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

We study the large-$N$ scaling behavior of the $\theta$ dependence of the ground-state energy density $E(\theta )$ of four-dimensional (4D) $SU(N)$ gauge theories and two-dimensional (2D) $C{P}^{N-1}$ models, where $\theta$ is the parameter associated with the Lagrangian topological term. We consider its $\theta$ expansion around $\theta =0$, $E(\theta)-E(0) = {1\over 2}\chi ,\theta^2 ( 1 + b_2 \theta^2 + b_4\theta^4 +\cdots)$where$\chi$ 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 $\theta =0$. 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 $\theta$ 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 $b_2={\bar{b}}_2\mathrm{/}{N}^2+O(1\mathrm{/}{N}^4)$ with ${\bar{b}}_2=-0.23(3)$. We also show that the large-$N$ scaling scenario applies to 2D $C{P}^{N-1}$ models as well, by an analytic computation of the leading large-$N$ dependence.