The 1/2 BPS Wilson loop in ABJ(M) at two loops: The details

TL;DR

This paper calculates the two-loop expectation value of the 1/2 BPS Wilson loop in ABJ(M) theory, confirming localization predictions and clarifying the framing factor and divergence cancellations.

Contribution

It provides a detailed two-loop perturbative calculation of the 1/2 BPS Wilson loop in ABJ(M), including non-planar contributions and the framing factor, with explicit integral solutions.

Findings

Agreement with localization predictions at weak coupling

Confirmation of the framing factor identification

Detailed method for solving fermion-gauge integrals in DRED

Abstract

We compute the expectation value of the 1/2 BPS circular Wilson loop operator in ABJ(M) theory at two loops in perturbation theory. Our result turns out to be in exact agreement with the weak coupling limit of the prediction coming from localization, including finite N contributions associated to non-planar diagrams. It also confirms the identification of the correct framing factor that connects framing-zero and framing-one expressions, previously proposed. The evaluation of the 1/2 BPS operator is made technically difficult in comparison with other observables of ABJ(M) theory by the appearance of integrals involving the coupling between fermions and gauge fields, which are absent for instance in the 1/6 BPS case. We describe in detail how to analytically solve these integrals in dimensional regularization with dimensional reduction (DRED). By suitably performing the physical limit to three dimensions we clarify the role played by short distance divergences on the final result and the mechanism of their cancellation.