The generalised scaling function: a systematic study

TL;DR

This paper systematically studies the generalized scaling functions in N=4 SYM, providing a recursive method to compute them at all couplings and analyzing their strong coupling behavior and relation to the O(6) sigma model.

Contribution

It introduces a recursive integral equation approach to determine the generalized scaling functions at all couplings, including strong coupling analysis.

Findings

Explicit recursive solutions for $f_n(g)$ functions.

Demonstration of convergence to the O(6) sigma model mass gap.

Analysis of next-to-leading order corrections at strong coupling.

Abstract

We describe a procedure for determining the generalised scaling functions $f_n(g)$ at all the values of the coupling constant. These functions describe the high spin contribution to the anomalous dimension of large twist operators (in the $sl(2)$ sector) of ${\cal N}=4$ SYM. At fixed $n$, $f_n(g)$ can be obtained by solving a linear integral equation (or, equivalently, a linear system with an infinite number of equations), whose inhomogeneous term only depends on the solutions at smaller $n$. In other words, the solution can be written in a recursive form and then explicitly worked out in the strong coupling regime. In this regime, we also emphasise the peculiar convergence of different quantities ('masses', related to the $f_n(g)$) to the unique mass gap of the $O(6)$ nonlinear sigma model and analyse the first next-to-leading order corrections.