UV asymptotics of $n$-point correlators of twist-$2$ operators in SU($N$) Yang-Mills theory

TL;DR

This paper derives the ultraviolet asymptotics of n-point correlators of twist-2 operators in SU(N) Yang-Mills theory, revealing their structure and providing constraints for nonperturbative solutions at large N.

Contribution

It explicitly computes the UV asymptotics of generating functionals for correlators, confirming their structure as functional determinants and offering insights into nonperturbative large-N Yang-Mills theory.

Findings

UV asymptotics of sphere and torus correlators derived

Asymptotic correlators exhibit functional determinant structure

Provides constraints for nonperturbative large-N Yang-Mills solutions

Abstract

The generating functional $\mathcal{W}[J_{\mathcal O}]$ of Euclidean correlators of twist-$2$ operators in SU($N$) Yang-Mills theory admits the 't Hooft large-$N$ expansion: $\mathcal{W}[J_{\mathcal O}]=\mathcal{W}_{sphere}\,\,\,\,[J_{\mathcal O}]+\mathcal{W}_{torus} \,\,\,[J_{\mathcal O}]+ \cdots$. Nonperturbatively, $\mathcal{W}_{sphere} \,\,\,\,[J_{\mathcal O}]$ is a sum of tree diagrams involving glueball propagators and vertices, while $\mathcal{W}_{torus} \,\,\,[J_{\mathcal O}]$ is a sum of glueball one-loop diagrams. Moreover, it has been predicted that $\mathcal{W}_{torus } \,\,\,[J_{\mathcal O}]$ should admit the structure of the logarithm of a functional determinant summing glueball one-loop diagrams. We work out in a closed form the ultraviolet (UV) asymptotics of $\mathcal{W}_{sphere} \,\,\,\,[J_{\mathcal O},\lambda] \sim \mathcal{W}_{asym \, sphere} \,\,\,\,\,\,\,[J_{\mathcal O},\lambda]$ and $\mathcal{W}_{torus} \,\,\,[J_{\mathcal O},\lambda] \sim \mathcal{W}_{asym \, torus} \,\,\,\,\,\,[J_{\mathcal O},\lambda]$ in the coordinate representation as all the coordinates of the correlators are uniformly rescaled by a factor $\lambda \rightarrow 0$. Remarkably, we verify the above prediction that $\mathcal{W}_{asym \, torus} \,\,\,\,\,\,[J_{\mathcal O},\lambda]$ -- being asymptotic in the UV to $\mathcal{W}_{torus} \,\,\,[J_{\mathcal O}, \lambda]$ -- admits the structure of the logarithm of a functional determinant as well. Hence, the computation above sets strong UV asymptotic constraints on the nonperturbative solution of large-$N$ YM theory and it may be a pivotal guide for the search of such a solution.