On the logarithmic powers of $sl(2)$ SYM$_4$

TL;DR

This paper investigates the high spin limit of twist operators in planar ${ m f N}=4$ SYM, deriving recursive integral equations for sub-logarithmic corrections to anomalous dimensions and connecting them to scaling functions at various couplings.

Contribution

It introduces a recursive integral equation framework to compute sub-logarithmic corrections in the high spin limit of ${ m f N}=4$ SYM and relates these to generalized scaling functions.

Findings

Explicit recursive formulas for $oldsymbol{\gamma^{(n)}(g,L)}$

Zero sub-logarithmic corrections for $L=2,3$

First weak and strong coupling orders computed

Abstract

In the high spin limit the minimal anomalous dimension of (fixed) twist operators in the $sl(2)$ sector of planar ${\cal N}=4$ Super Yang-Mills theory expands as $\gamma(g,s,L)=f(g) \ln s + f_{sl}(g,L) + \sum \limits_{n=1}^\infty \gamma^{(n)}(g,L) (\ln s)^{-n} + ... $. We find that the sub-logarithmic contribution $\gamma^{(n)}(g,L) $ is governed by a linear integral equation, depending on the solution of the linear integral equations appearing at the steps $n'\leq n-3$. We work out this recursive procedure and determine explicitly $\gamma^{(n)}(g,L) $ (in particular $\gamma^{(1)}(g,L)=0$ and $\gamma^{(n)}(g,2)=\gamma^{(n)}(g,3)=0$). Furthermore, we connect the $\gamma^{(n)}(g,L) $ (for finite $L$) to the generalised scaling functions, $f^{(r)}_n(g)$, appearing in the limit of large twist $L\sim\ln s$. Finally, we provide the first orders of weak and strong coupling for the first $\gamma^{(n)}(g,L)$ (and hence $f^{(r)}_n(g)$).