An Integrand Reconstruction Method for Three-Loop Amplitudes
Simon Badger, Hjalte Frellesvig, Yang Zhang
TL;DR
The paper addresses the challenge of computing three-loop massless 2→2 amplitudes by developing a three-step integrand-reduction framework that combines generalized unitarity with computational algebraic geometry. It solves the on-shell constraints of the planar triple box using primary decomposition to obtain 14 branches parameterized by $ au_1$ and $ au_2$, fits the integrand basis in a seven-ISP space by separating NS and S contributions, and reconstructs the full integrand via branch-by-branch Gröbner-basis methods. After mapping to an IBP-reducible basis, the full expression reduces to three master integrals $I_{10}[1]$, $I_{10}[(k_1+p_4)^2]$, and $I_{10}[(k_3-p_4)^2]$, with explicit MI coefficients computed for gluon-gluon scattering in Yang–Mills theories with adjoint matter across $ ext{N}=4,2,1,0$. The results show consistency with known $ ext{N}=4$ limits and reveal simplifications at lower supersymmetry; the method demonstrates the viability of applying algebraic-geometry techniques to three-loop amplitudes and paves the way for $D$-dimensional extensions and reductions of lower-point topologies.
Abstract
We consider the maximal cut of a three-loop four point function with massless kinematics. By applying Groebner bases and primary decomposition we develop a method which extracts all ten propagator master integral coefficients for an arbitrary triple-box configuration via generalized unitarity cuts. As an example we present analytic results for the three loop triple-box contribution to gluon-gluon scattering in Yang-Mills with adjoint fermions and scalars in terms of three master integrals.