RG Solutions for α s at large N c in d = 3 + 1 QCD

QCD is known to simplify at large Nc: i) the perturbation theory dictates that only planar diagrams survive in the leading order [1]; ii) assuming that confinement exists also in the limit Nc → ∞, the spectrum of mesons and glueballs consists of bound-state poles, and the decay width vanishes at large Nc[1, 2, 3]. In the 1+1 QCD this property was proved both analytically and numerically [3] (for a review see [4]) since confinement in this case is induced by the perturbative gluon exchange. On the lattice numerical calculations have confirmed that the large Nc limit is achieved with few percent accuracy already at Nc = 3 for several tested observables [5]. It is thus likely that the large Nc QCD is a good first approximation for the realistic Nc = 3 case, which has an accuracy of 10% or better. From theoretical point of view the large Nc limit of QCD is very useful since it drastically simplifies the analytic structure of amplitudes, e.g. one has sums of simple poles in the two-point function, and for the four-point function one expects formulas of the Veneziano type. Thus one expects the large Nc amplitudes as functions of external momenta to be meromorphic. On the other hand the perturbation series yields the typical logarithmic dependencies and the RG equation prescribes for αs(Q) the structure with unphysical poles and cuts which are incompatible with unitarity and analyticity. E.g. the one-loop expression for αs(Q) has the form