Large $$N$$ N scaling and factorization in $${\mathrm {SU}}(N)$$ SU ( N ) Yang–Mills gauge theory

The large \(N\) limit of \({\mathrm {SU}}(N)\) gauge theoriesis well understood in perturbation theory. Also non-perturbativelattice studies have yielded important positive evidence that ’t Hooft’s predictions are valid. We go far beyond the statistical and systematic precision of previous studies by making use of the Yang–Mills gradient flow and detailed Monte Carlo simulations of \({\mathrm {SU}}(N)\) pure gauge theoriesin 4 dimensions. With results for \(N=3,4,5,6,8\) we study the limit and the approach to it. We pay particular attention to observables which test the expected factorization in the large \(N\) limit. The investigations are carried out both in the continuum limit and at finite lattice spacing. Large \(N\) scaling is verified non-perturbativelyand with high precision; in particular, factorization is confirmed. For quantities which only probe distances below the typical confinement length scale, the coefficients of the \( 1/N\) expansionare of \(\mathrm{O}(1)\), but we found that large (smoothed) Wilson loopshave rather large \(\mathrm{O}(1/N^2)\) corrections. The exact size of such corrections does, of course, also depend on what is kept fixed when the limit is taken.