Reciprocity and self-tuning relations without wrapping

TL;DR

This paper investigates the high-spin behavior of scalar Wilson operators in ${ m extbf{N=4}}$ SYM, revealing reciprocity and self-tuning relations in integral equations and suggesting how wrapping corrections restore these relations at higher orders.

Contribution

It demonstrates that reciprocity and self-tuning relations hold for high-spin operators in ${ m extbf{N=4}}$ SYM up to all orders in $1/s$, and proposes how wrapping corrections may restore these relations at next order.

Findings

Reciprocity and self-tuning relations hold up to all orders in $1/s$ for high-spin operators.

Wrapping corrections are expected to enter at the next order to restore these relations.

Relations for anomalous dimensions are consistent with the integral equations at the same order.

Abstract

We consider scalar Wilson operators of ${\cal N}=4$ SYM at high spin, $s$, and generic twist in the multi-color limit. We show that the corresponding (non)linear integral equations (originating from the asymptotic Bethe Ansatz equations) respect certain 'reciprocity' and functional 'self-tuning' relations up to all terms $\frac{1}{s(\ln s)^n}$ (inclusive) at any fixed 't Hooft coupling $\lambda$. Of course, this relation entails straightforwardly the well-known (homonymous) relations for the anomalous dimension at the same order in $s$. On this basis we give some evidence that wrapping corrections should enter the non-linear integral equation and anomalous dimension expansions at the next order $\frac{(\ln s)^{2}}{s^2}$, at fixed 't Hooft coupling, in such a way to re-establish the aforementioned relation (which fails otherwise).