On the finite size corrections of anti-ferromagnetic anomalous dimensions in ${\cal N}=4$ SYM

TL;DR

This work develops Non-Linear Integral Equations (NLIE) from Bethe Ansatz to compute finite-size corrections to anti-ferromagnetic anomalous dimensions in ${\cal N}=4$ SYM, focusing on the SU(2) sector (multi-loop) and the SO(6) scalar sector (one loop). By encoding the state data in a counting function $Z(x)$ and its holes, the authors derive compact NLIEs that yield exact expressions for energy and momentum, with explicit leading and sub-leading finite-size terms. They show the leading finite-size energy scales as $E \sim 2L\big(\tfrac{\pi}{2}+\ln 2\big)$, with $1/L$ and potential logarithmic corrections governed by derivative lemmas and kink NLIEs, and they validate these findings through thorough numerical analysis. The method provides a flexible, general framework for finite-$L$ spectral data in integrable gauge-theory sectors and connects to Hubbard-model insights, with prospects for extensions to other sectors and to QCD contexts.

Abstract

Non-linear integral equations derived from Bethe Ansatz are used to evaluate finite size corrections to the highest (i.e. {\it anti-ferromagnetic}) and immediately lower anomalous dimensions of scalar operators in ${\cal N}=4$ SYM. In specific, multi-loop corrections are computed in the SU(2) operator subspace, whereas in the general SO(6) case only one loop calculations have been finalised. In these cases, the leading finite size corrections are given by means of explicit formulæand compared with the exact numerical evaluation. In addition, the method here proposed is quite general and especially suitable for numerical evaluations.

Figures (4)

• Figure 1: Plot of $\Delta E(L)L-\frac{\pi^2}{6}$ versus $L$ for the state $I=(-1,\,0,\,1,\,2)$ of the Heisenberg spin chain.
• Figure 2: Plot of the counting function $Z(x)$ versus $x$ for the Heisenberg spin chain with $L=12$ sites. The position of the four holes quantised by $I=(-1,\,0,\,1,\,2)$ is indicated by the small crosses.
• Figure 3: Plot of $\Delta E(L,\lambda) \, L$ versus $L$ for the state $I=(-1\,,0\,,1\,,2)$ of the many-loops spin chain with $\lambda=50$.
• Figure 4: Plot of $\Delta E(L) \, L-\frac{\pi^2}{2}$ versus $L$ for the anti-ferromagnetic state of the $SO(6)$ spin chain.