This can easily be derived by recalling that the Lagrangian is $O(N^2)$ and the large-$N$ limit is performed keeping the coupling $g^{2}N$ fixed [2].

The estimates of $b_2$ and $b_4$ for $N=6$ reported in Table I are obtained by a high-statistics MC simulation at $\beta /(2N^2)=0.348$ on a ${12}^4$ lattice, averaging over $\approx 3\times {10}^4$ independent measurements of $Q$ [corresponding to $O({10}^8)$ MC sweeps], which also gives ${\chi }_l=0.97(2)\times {10}^{-4}$. They should be taken with some caution since the continuum limit has not been checked.

The formula for $b_4$ appeared with the wrong sign in Ref. [5].

We use a mixture of heat-bath and overrelaxation algorithms for the link-variable updating.

Our choice of the $L_t$ values was motivated by the fact that the cooling method is not sufficiently stable if the temporal extent of the lattice is too small (see, e.g., [11]). The values of $L_t$ considered here appear safe, providing stable results. Larger values would prevent us from obtaining the high statistics required in this study.

In our analysis, and in particular to determine the reduced temperature, we use results for the transition points at $N=3$ taken from Ref. [23]. For the SU(6) gauge theory we use ${\beta }_c/(2N^2)=0.34508(5)$ for $L_t=6$ (from [23]), $\sigma ({\beta }_c)=0.276(2)$ and $t(L_t,\beta )=\sqrt{\sigma \mathbf{(}{\beta }_c(L_t)\mathbf{)}/\sigma (\beta )}-1$.

The MC results for $N=3$ are obtained by averaging $O({10}^5)$ independent measurements of $Q$, corresponding to approximately ${10}^7–{10}^8$ sweeps due to the large autocorrelation time ${\tau }_Q$ of $Q$ [20] (e.g., ${\tau }_Q\approx 2\times {10}^3$ for the largest $\beta$ value, $\beta =6.437$). In the case of $N=6$, they are obtained from $O({10}^4)$ independent measurements, which corresponds to $O({10}^7)$ sweeps (e.g., ${\tau }_Q\approx 400$ for the runs at $\beta /(2N^2)=0.348$, 0.350).