Anomalous dimensions at four loops in N=6 superconformal Chern-Simons theories

TL;DR

This work determines the four-loop correction to the magnon dispersion function h^2(\bar{\lambda},\sigma) in N=6 Chern-Simons theories and computes the wrapping corrections for a length-4 operator in the SU(2)×SU(2) sector. By extracting h^2 from the four-loop dilatation operator and employing dimensional reduction with explicit Feynman-diagrammatics, the authors obtain h^2(\bar{\lambda},\sigma)=\bar{\lambda}^2+\bar{\lambda}^4(h_4(\sigma)) with h_4(\sigma)=-(4+\sigma^2)\zeta(2), i.e., h_4=-4\zeta(2) and h_{4,\sigma}=-\zeta(2). They confirm that parity is preserved at four loops and show that the four-loop wrapping correction to the SU(4) 20 representation matches the Y-system prediction, with the total anomalous dimension gamma_{\mathbf{20}}=4+8\bar{\lambda}^2-8(6+\sigma^2)\zeta(2)\bar{\lambda}^4. The results highlight maximal transcendentality in the four-loop sector and solidify integrability-based predictions for ABJM/ABJ models.

Abstract

In arXiv:0908.2463 we computed the four-loop correction to a function depending on the 't Hooft coupling(s) that appears in the magnon dispersion relation of the spin chains derived from single trace operators in N=6 superconformal Chern-Simons theories. In this paper we give detailed descriptions of this calculation and the computation of the four-loop wrapping corrections for a length four operator in the 20 of SU(4), the R-symmetry group for these theories. Here, we give all relevant Feynman diagrams and loop integrals explicitly, and also demonstrate the cancellation of double poles in the logarithm of the renormalization constant.