The large-N Yang-Mills S matrix is ultraviolet finite, but the large-N QCD S matrix is only renormalizable

摘要

Yang-Mills (YM) theory and QCD are known to be renormalizable, but not ultraviolet (UV) finite, order by order, in perturbation theory. It is a fundamental question whether YM theory or QCD is UV finite, or only renormalizable, order by order, in the large-N ’t Hooft or Veneziano expansions. We demonstrate that the renormalization group (RG) and asymptotic freedom imply that in ’t Hooft large-N expansion the S matrix in YM theory is UV finite, while in both ’t Hooft and Veneziano large-N expansions, the S matrix in confining massless QCD is renormalizable but not UV finite. By the same argument, the large-N N = 1 supersymmetry (SUSY) YM S matrix is UV finite as well. Besides, we demonstrate that, in both ’t Hooft and Veneziano large-N expansions, the correlators of local gauge-invariant operators, as opposed to the S matrix, are renormalizable but, in general, not UV finite, either in YM theory and N = 1 SUSY YM theory or a fortiori in massless QCD. Moreover, we compute explicitly the counterterms that arise from renormalizing the ’t Hooft and Veneziano expansions by deriving in confining massless QCD-like theories a low-energy theorem of the Novikov-Shifman-Vainshtein-Zakharov type that relates the log derivative with respect to the gauge coupling of a k-point correlator, or the log derivative with respect to the RGinvariant scale, to a (k + 1)-point correlator with the insertion of TrF~2 at zero momentum. Finally, we argue that similar results hold in the large-N limit of a vast class of confining massive QCD-like theories, provided a renormalization scheme exists-as, for example, MS-in which the beta function is not dependent on the masses. Specifically, in both ’t Hooft and Veneziano large-N expansions, the S matrix in confining massive QCD and massive N = 1 SUSY QCD is renormalizable but not UV finite.