Magnon dispersion to four loops in the ABJM and ABJ models

TL;DR

The paper computes the four-loop magnon-dispersion function $h^2(\bar{\lambda},\sigma)$ for ABJM/ABJ, showing $h^2(\bar{\lambda},\sigma)=\bar{\lambda}^2+\bar{\lambda}^4\big(-4\zeta(2)-\sigma^2\zeta(2)\big)$ and confirming maximal transcendentality. By performing an explicit four-loop Feynman-diagram calculation in the $SU(2)\times SU(2)$ sector, the authors determine the four-loop dilatation operator, fix a sign via two-loop renormalization, and derive the four-loop wrapping corrections for a length-four $SU(4)$ operator in the 20 representation. The wrapping result agrees with the ABJM Y-system prediction of Gromov, Kazakov, and Vieira, providing a nontrivial check on the integrability-based framework. The work also identifies a perturbatively integrable ABJ limit with a short-range Hamiltonian in a certain parameter regime, and discusses implications for parity, branch cuts in $h^2$, and potential extensions to higher loops and other sectors.

Abstract

The ABJM model is a superconformal Chern-Simons theory with N=6 supersymmetry which is believed to be integrable in the planar limit. However, there is a coupling dependent function that appears in the magnon dispersion relation and the asymptotic Bethe ansatz that is only known to leading order at strong and weak coupling. We compute this function to four loops in perturbation theory by an explicit Feynman diagram calculation for both the ABJM model and the ABJ extension. We find that all coefficients have maximal transcendentality. We then compute the four-loop wrapping correction for a scalar operator in the 20 of SU(4) and find that it agrees with a recent prediction from the ABJM Y-system of Gromov, Kazakov and Vieira. We also propose a limit of the ABJ model that might be perturbatively integrable at all loop orders but has a short range Hamiltonian.