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

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

References in corpus (10)

• The Uses of Conformal Symmetry in QCD
• Broken conformal invariance and spectrum of anomalous dimensions in QCD
• Dilatation operator in (super-)Yang-Mills theories on the light-cone
• One-Loop QCD Spin Chain and its Spectrum
• Three-loop evolution equation for flavor-nonsinglet operators in off-forward kinematics
• Two-loop evolution equation for the B-meson distribution amplitude
• Three-point correlators of twist-2 operators in N=4 SYM at Born approximation
• On the geometry of operator mixing in massless QCD-like theories
• $n$-point correlators of twist-$2$ operators in $SU(N)$ Yang-Mills theory to the lowest perturbative order
• Operator mixing in massless QCD-like theories and Poincare'-Dulac theorem

Cited by in corpus (3)

• Nonresonant renormalization scheme for twist-$2$ operators in SU($N$) Yang-Mills theory
• On the structure of the large-$N$ expansion in SU($N$) Yang-Mills theory
• Superfield twist-$2$ operators in $\mathcal{N}=1$ SCFTs and their renormalization-group improved generating functional in $\mathcal{N}=1$ SYM theory