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

## TL;DR

This paper investigates the ultraviolet properties of the S-matrix in large-N Yang-Mills and QCD theories, demonstrating finiteness in YM but only renormalizability in QCD, with implications for supersymmetric theories and correlator behaviors.

## Contribution

It proves that in the large-N limit, YM S-matrix is ultraviolet finite, while QCD S-matrix is only renormalizable, and derives explicit counterterms and low-energy theorems for these theories.

## Key findings

- YM S-matrix is ultraviolet finite in large-N 't Hooft expansion.
- QCD S-matrix is renormalizable but not ultraviolet finite in large-N expansions.
- Correlators of local gauge-invariant operators are renormalizable but generally not ultraviolet finite.

## Abstract

YM and QCD are known to be renormalizable, but not ultraviolet finite, order by order in perturbation theory. It is a fundamental question as to whether YM or QCD are ultraviolet finite, or only renormalizable, order by order in the large-N 't Hooft or Veneziano expansions. We demonstrate that Renormalization Group and Asymptotic Freedom imply that in 't Hooft large-N expansion the S-matrix in YM is ultraviolet finite, while in both 't Hooft and Veneziano large-N expansions the S-matrix in confining QCD with massless quarks is renormalizable but not ultraviolet finite. By the same argument it follows that the large-N $\mathcal{N}=1$ SUSY YM S-matrix is ultraviolet finite as well. Besides, we demonstrate that the correlators of local gauge-invariant operators, as opposed to the S-matrix, are renormalizable but in general not ultraviolet finite in the large-N 't Hooft and Veneziano expansions, neither in pure 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 't Hooft and Veneziano expansions, by deriving in confining massless QCD-like theories a low-energy theorem of NSVZ 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 RG-invariant scale, to a $k+1$-point correlator with the insertion of $Tr F^2$ at zero momentum. Finally, we argue that similar results hold in the large-N limit of a vast class of confining QCD-like theories with massive matter fields, provided a renormalization scheme exists, as for example $\overline{MS}$, in which the beta function is independent on the masses. In particular, in both 't Hooft and Veneziano large-N expansions the S-matrix in confining massive QCD and massive $\mathcal{N}=1$ SUSY QCD is renormalizable but not ultraviolet finite.