# Large $N_{c}$ QCD: motivation

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Suppose that we want to study QCD - theory with massive quarks which interact with itself through $SU_{c}(3)$ gauge bosons. By using perturbation theory we find that at some scale this theory enters the regime of strong coupling, and then perturbative treatment fails. We know, however, that below this scale approximate symmetry $SU_{L}(3)\times SU_{R}(3)$ breaks down spontaneously to $SU_{V}(3)$, and octet of pseudogoldstone mesons $U$ arise. By using this fact we want to evaluate corresponding effective field theory action, starting from QCD generating functional.

As I know, this direct method fails for QCD, and instead of it we use general method of constructing of effective field theory based on construction of invariant forms of coset $SU_{L}(3)\times SU_{R}(3)/SU_{V}(3)$ elements, whose parametrization defines goldstone degrees of freedom. Such approach leaves parametrical degrees of freedom (coupling constants etc.). However, I know, that if we assume QCD-like theory based on $SU_{c}(N)$ local gauge group, where $N \to \infty$, then, in assumption that such theory enters strong coupling regime and confinement, we may derive, using some simplifications, the chiral effective field theory with Wess-Zumino term.

According to this, I have the few questions.

• How can be $SU_{c}(N)$ results comparable with $SU_{c}(3)$ ones? I.e., can we make some general predictions for $SU_{c}(3)$ theory by using results for $SU_{c}(N)$ one?
• If yes, what is the reason for this? Is the requirement of strong coupling and confinement sufficient to be the reason?
One expects the $N=3$ case to be qualitatively similar to the large $N$ case; since the latter is much more tractable, it gives valuable (though not fully certain) insigt into the case realized in Nature.
The large $N$ treatment essentially gives the leading few terms of a power series in $1/N$. It depends on the series whether this is a good or a poor approximation when $N=3$ is substituted; sometimes it can be quite good. Try the low order expansions of $\sin x$, $e^x$ and $\log(1+x)$ for $x=1/N=1/3$ and compare.
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