An asymptotic solution of large-$N$ $QCD$, and of large-$N$ $\mathcal{N}=1$ $SUSY$ $YM$

Abstract

We find an asymptotic solution for two- and three-point correlators of local gauge-invariant operators, in a lower-spin sector of massless large-$N$ $QCD$, in terms of glueball and meson propagators, by means of a new purely field-theoretical technique that we call the asymptotically-free bootstrap. The asymptotically-free bootstrap exploits the lowest-order conformal invariance of connected correlators of gauge invariant composite operators in perturbation theory, the renormalization-group improvement, and a recently-proved asymptotic structure theorem for glueball and meson propagators, that involves the unknown particle spectrum and the anomalous dimension of operators for fixed spin. In principle the asymptotically-free bootstrap extends to all the higher-spin two- and three-point correlators whose lowest-order conformal limit is non-vanishing in perturbation theory, and by means of the operator product expansion to the corresponding asymptotic multi-point correlators as well. Besides, the asymptotically-free bootstrap provides asymptotic $S$-matrix amplitudes in massless large-$N$ $QCD$ in terms of glueball and meson propagators as opposed to perturbation theory. Remarkably, the asymptotic $S$-matrix depends only on the unknown particle spectrum, but not on the anomalous dimensions. Moreover, the asymptotically-free bootstrap applies to large-$N$ $\mathcal{N}=1$ $SUSY$ $YM$ as well. Practically, as just a few examples among many more, it follows the structure of the light by light scattering amplitude, of the pion form factor, and the associated vector dominance. Theoretically, the asymptotic solution sets the strongest constraints on any actual solution of large-$N$ $QCD$ (and of large-$N$ $\mathcal{N}=1$ $SUSY$ $YM$), and in particular on any string solution.