YM and QCD are known to be renormalizable, but not ultraviolet finite, orderby order in perturbation theory. It is a fundamental question as to whether YMor QCD are ultraviolet finite, or only renormalizable, order by order in thelarge-N 't Hooft or Veneziano expansions. We demonstrate that RenormalizationGroup and Asymptotic Freedom imply that in 't Hooft large-N expansion theS-matrix in YM is ultraviolet finite, while in both 't Hooft and Venezianolarge-N expansions the S-matrix in confining QCD with massless quarks isrenormalizable but not ultraviolet finite. By the same argument it follows thatthe large-N $\mathcal{N}=1$ SUSY YM S-matrix is ultraviolet finite as well.Besides, we demonstrate that the correlators of local gauge-invariantoperators, as opposed to the S-matrix, are renormalizable but in general notultraviolet finite in the large-N 't Hooft and Veneziano expansions, neither inpure YM and $\mathcal{N}=1$ SUSY YM nor a fortiori in massless QCD. Moreover,we compute explicitly the counterterms that arise renormalizing the large-N 'tHooft and Veneziano expansions, by deriving in confining massless QCD-liketheories a low-energy theorem of NSVZ type, that relates the log derivativewith respect to the gauge coupling of a $k$-point correlator, or the logderivative with respect to the RG-invariant scale, to a $k+1$-point correlatorwith the insertion of $Tr F^2$ at zero momentum. Finally, we argue that similarresults hold in the large-N limit of a vast class of confining QCD-liketheories with massive matter fields, provided a renormalization scheme exists,as for example $\overline{MS}$, in which the beta function is independent onthe masses. In particular, in both 't Hooft and Veneziano large-N expansionsthe S-matrix in confining massive QCD and massive $\mathcal{N}=1$ SUSY QCD isrenormalizable but not ultraviolet finite.
展开▼
