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

Abstract

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 framingzero 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.

BPS Wilson loops and Bremsstrahlung function in ABJ(M): a two loop analysis

“…This generalizes the well-known relation linking the 1/2−BPS and the 1/6−BPS vev's when computed perturbatively, at framing zero [26,29,30,34,37].…”

“…Therefore, its vev localizes to the same matrix model and a prediction for its exact value can be easily obtained from the 1/6−BPS vev. 1 Perturbative results [29,30,37] not only agree with this prediction but also confirm the correct identification of the framing factor [38] arising from the matrix model calculation [31]. It is interesting to note that in the 1/2−BPS case the appearance at perturbative level of non-trivial contributions from the fermionic sector is instrumental to recover the correct framing factors.…”

“…Perturbative results for 1/6−BPS Wilson loops [24,25,29,30] on the maximal circle have been proved to match the exact prediction obtained by using localization techniques [31]. At variance with N = 4 SYM [32], the corresponding matrix model is no longer gaussian due to non-trivial contributions from the vector and the matter multiplets.…”

“…The simplest contribution originates from expanding the exponential of the superconnection at second order in the fermionic fields and contracting them with the one-loop corrected fermion propagator (A.11) as depicted in figure 4. In the 1/2−BPS case the exact cancellation between the contributions of the upper-left and the lower-right blocks of L(τ 1 )L(τ 2 ) leads to a vanishing result [30,37]. For our deformed Wilson loop instead, such a mechanism no longer occurs because of the nontrivial dependence on the parameter ν, which weights the combination of the two blocks.…”

“…Details of the calculation are given in appendix D. Diagram (d) emerges from the one-loop correction to the fermion propagator. While for the fermionic 1/2−BPS Wilson loop such a contribution vanishes identically for symmetry reasons [29,30,37], its latitude deformation spoils those arguments and yields a finite result…”