Analytic two-Loop four-point form factor of the stress-tensor supermultiplet in ${\cal N}=4$ SYM

TL;DR

This paper computes the two-loop four-point MHV form factor of the stress-tensor supermultiplet in planar ${ m N}=4$ SYM, providing explicit integrand expressions, finite remainders, and confirming bootstrap predictions, with implications for QCD amplitudes.

Contribution

It presents the first complete two-loop integrand and finite remainder for the four-point form factor in ${ m N}=4$ SYM, including non-planar contributions and symmetry analyses.

Findings

Explicit two-loop integrand obtained via unitarity cuts.

Finite remainder expressed in terms of pentagon functions.

Confirmation of bootstrap symbol results and dual conformal symmetry.

Abstract

We compute the two-loop four-point MHV form factor of the stress-tensor supermultiplet in planar ${\cal N}=4$ super Yang-Mills (SYM). This form factor is analogous to the Higgs plus four-gluon amplitudes in the heavy-top limit of QCD when translated to the ${\cal N}=4$ SYM context. We obtain the full $D$-dimensional integrands up to two loops via unitarity-cut methods. Subsequently, we utilize IBP reduction to express the result in terms of a set of uniformly transcendental basis integrals, incorporating the two-loop non-planar five-point one-mass integrals recently given by Abreu et al. [PRL 132 (2024) 14]. We obtain the two-loop finite remainder in the functional form in terms of the pentagon functions. The symbol of our remainder confirms the bootstrap results reported by Dixon et al. [PRL 130 (2023) 11]. We perform various non-trivial checks of our results, including the triple-collinear limit, which recovers the two-loop six-gluon remainder. We also show that the form factor has a directional dual conformal symmetry at the integrand level. Our results are expected to shed further light on the study of antipodal dualities and the computation of Higgs plus four-parton amplitudes in QCD.