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

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.