$\theta$ 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 $\theta =0$, $E(\theta )-E(0)=\frac{1}{2}\chi {\theta }^2(1+b_2{\theta }^2+b_4{\theta }^4+\dots )$, 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 $b_2={\overline{b}}_2/N^2+O(1/N^4)$ with ${\overline{b}}_2=-0.23(3)$. We also show that the large-$N$ scaling scenario applies to 2D $C{P}^{N-1}$ models as well, by an analytical computation of the leading large-$N$ $\theta$ dependence around $\theta =0$.

Figure 1

An example of the global fit procedure, with a truncation including $O({\theta }_L^4)$ terms: data refer to the $14^4$ lattice at coupling $\beta =11.008$ for the $SU(4)$ gauge theory. Continuous lines are the result of a combined fit of the first four cumulants.

Figure 2

Comparison of the results obtained for $b_2$ in $SU(6)$ using different truncations of Eq. (3).

Figure 3

Dependence of $\chi$ (in lattice units) on the lattice size. From top to bottom results are displayed for $SU(3)$ at coupling $\beta =6.2$ (from [43]), $SU(4)$ at $\beta =11.104$ and $SU(6)$ at $\beta =24.500$. Horizontal dashed lines are the results of fits to constant and are plotted in order to better appreciate the absence of statistically significant deviations.

Figure 4

Dependence of $b_2$ on the lattice size. From top to bottom results are displayed for $SU(3)$ at coupling $\beta =6.2$ (from [43]), $SU(4)$ at $\beta =11.104$ and $SU(6)$ at $\beta =24.500$. Horizontal dashed lines are the results of fits to constant and are plotted in order to better appreciate the absence of statistically significant deviations.

Figure 5

Continuum limit of the dimensionless ratio $\chi /{\sigma }^2$ for $SU(6)$ gauge theory. The results obtained in this work are compared with the determination of [37] (data have been slightly shifted horizontally to improve the readability).

Figure 6

Scaling of the dimensionless ratio $\chi /{\sigma }^2$ with the number of colors. The dashed line is the result of a best fit with a linear functional dependence.

Figure 7

Dependence of the $b_2$ values on the lattice spacing for the case of three, four and six colors. See the main text for the details of the fitting procedure.

Figure 8

Scaling of $b_2$ with the number of colors. Lines are result of a best fit performed using the linear dependence expected from large-$N$ arguments (dashed line fitting all data, full line fitting only those for $N=4$, 6) and adding also a quadratic contribution (dotted-dashed line).

Figure 9

Estimates of $b_4$ for $N=3$, 4, 6.

Figure 10

Dependence of the renormalization constant $Z$ values on the lattice spacing for the case of three, four and six colors.

Figure 11

Behavior of $b_2$ across the deconfinement transition for $SU(3)$ and $SU(6)$ [$t$ is the reduced temperature defined by $t=(T-T_c)/T_c$]. The horizontal bands denote the zero temperature values. Updated version of the figure originally presented in [76].

Figure 12

Comparison of cooling and gradient flow evolutions for two $SU(6)$ configurations.

Figure 13

Autocorrelation times (in units of measure) of the square of the topological charge for the standard run and for the two tests with parallel tempering. In run 1 an exchange was proposed every four measures, while in run 2 it was proposed every 40 measures.