Abstract
The large N limit of {mathrm {SU}}(N) gauge theories is well understood in perturbation theory. Also non-perturbative lattice 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 theories in 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-perturbatively and 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 expansion are of mathrm{O}(1), but we found that large (smoothed) Wilson loops have 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.
Highlights
This 1/N scaling is obtained perturbatively, lattice computations provide evidence that it holds at the non-perturbative level, both in D = 4 space-time dimensions [2,3,4,5,6,7,8,9] and in D = 3 [9,10,11,12,13,14]
For quantities which only probe distances below the typical confinement length scale, the coefficients of the 1/N expansion are of O(1), but we found that large Wilson loops have rather large O(1/N 2) corrections
Bμ(t, x) dxμ Choosing 8t to be of a typical quantum chromodynamics (QCD) size, say of the order of the inverse string tension, they benefit from small statistical errors even for large loops [43]
Summary
This 1/N scaling is obtained perturbatively, lattice computations provide evidence that it holds at the non-perturbative level, both in D = 4 space-time dimensions [2,3,4,5,6,7,8,9] and in D = 3 [9,10,11,12,13,14]. [15], this fact can be put in analogy with the classical limit of a quantum theory, where 1/N plays the role of h. Related to this is the concept of the “master field”, i.e., the idea that the path integral is dominated by a single gauge configuration (or rather a gauge orbit) [16,17].
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