Hubbard's Adventures in ${\cal N}=4$ SYM-land? Some non-perturbative considerations on finite length operators

TL;DR

This paper derives an exact non-perturbative expression for the Hubbard energy in the SU(2) sector of planar ${ m extbf{N}}=4$ SYM, using coupled non-linear integral equations to analyze finite length operators at strong coupling.

Contribution

It introduces a novel non-perturbative approach using NLIEs to study finite length operators in ${ m extbf{N}}=4$ SYM, improving upon previous Bethe Ansatz methods.

Findings

Exact non-perturbative Hubbard energy expression derived.

Large $L$ expansion matches BDS Bethe Ansatz asymptotics.

Numerical integration of NLIEs shows high precision.

Abstract

As the Hubbard energy at half filling is believed to reproduce at strong coupling (part of) the all loop expansion of the dimensions in the SU(2) sector of the planar $ {\cal N}=4$ SYM, we compute an exact non-perturbative expression for it. For this aim, we use the effective and well-known idea in 2D statistical field theory to convert the Bethe Ansatz equations into two coupled non-linear integral equations (NLIEs). We focus our attention on the highest anomalous dimension for fixed bare dimension or length, $L$, analysing the many advantages of this method for extracting exact behaviours varying the length and the 't Hooft coupling, $\lambda$. For instance, we will show that the large $L$ (asymptotic) expansion is exactly reproduced by its analogue in the BDS Bethe Ansatz, though the exact expression clearly differs from the BDS one (by non-analytic terms). Performing the limits on $L$ and $\lambda$ in different orders is also under strict control. Eventually, the precision of numerical integration of the NLIEs is as much impressive as in other easier-looking theories.