The generalised scaling function: a note

TL;DR

This paper introduces a method to determine generalized scaling functions in the high spin limit of anomalous dimensions in planar N=4 SYM, revealing connections to the O(6) Non-Linear Sigma Model and providing a framework for further theoretical checks.

Contribution

It proposes a novel approach to compute all-order perturbative expansions of generalized scaling functions in N=4 SYM, linking them to the NLSM mass-gap and extending the understanding of high spin behavior.

Findings

Derivation of the third and fourth scaling functions at all orders.

Emergence of the O(6) NLSM mass-gap from SYM functions.

Identification of the fourth function's unique contributions from non-BES densities.

Abstract

A method for determining the generalised scaling function(s) arising in the high spin behaviour of long operator anomalous dimensions in the planar $sl(2)$ sector of ${\cal N}=4$ SYM is proposed. The all-order perturbative expansion around the strong coupling is detailed for the prototypical third and fourth scaling functions, showing the emergence of the O(6) Non-Linear Sigma Model mass-gap from different SYM 'mass' functions. Remarkably, only the fourth one gains contribution from the non-BES reducible densities and also shows up, as first, NLSM interaction and specific model dependence. Finally, the computation of the $n$-th generalised function is sketched and might be easily finalised for checks versus the computations in the sigma model or the complete string theory.