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 ${\mathcal{W}}_{sphere}[J_{\mathcal{O}},\hat{I}»]\sim {\mathcal{W}}_{asymsphere}[J_{\mathcal{O}},\hat{I}»]$ and ${\mathcal{W}}_{torus}[J_{\mathcal{O}},\hat{I}»]\sim {\mathcal{W}}_{asymtorus}[J_{\mathcal{O}},\hat{I}»]$ in the coordinate representation as all the coordinates of the correlators are uniformly rescaled by a factor $\hat{I}»\to 0$. Remarkably, we verify the above prediction that ${\mathcal{W}}_{asymtorus}[J_{\mathcal{O}},\hat{I}»]$ -- being asymptotic in the UV to ${\mathcal{W}}_{torus}[J_{\mathcal{O}},\hat{I}»]$ -- 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.

latex, 36 pages; introduction and section 7 expanded, section 4 simplified, minor changes and a few typos corrected