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I've bumped into the study of the $SU(N)$ theory in the large-$N$ limit. I'm wondering in which way the study of this Yang-Mills theory, can give contribution to QCD with gauge group $SU(3)$, i.e. $N=3$ Any suggestion?

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Maybe I've figure out something about that. QCD is an asymptotically free theory, so while for the high energy regime, consider the perturbative method, expanding around $g=0$, is perfectly consistent, for the low energy scales this approach becomes useless. An attempt to manage this problem, is to consider an expansion in terms of the parameter $\frac{1}{N}$. Doing so the leading term is equal for every $SU(N)$ theory, and so for QCD, and this term is the only contribution in the large-$N$ case.

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The large $N$ limit ('t Hooft) of QCD (or in general, theories where the gauge field is in the adjoint representation, matrix models) was introduced in 1974 as a hope to analytically understand QCD outside perturbative regime. The large $N$ QCD is also asymptotically free and captures some of the important features of SU(3). This large $N$ idea also gave rise to the concept of master field (or master orbit), but till date it has not been constructed for d=4 QCD. There does exist large $N$ exact analytical results in two dimensions QCD. It was hoped that by doing expansion in 1/N, one can know about QCD. Also note that there exists many different ways in which one can take this limit (Corrigan-Ramond, Veneziano).

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