The generalised scaling function: a note

Understand

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.