Differential Equations for Two-Loop Four-Point Functions

TL;DR

The paper tackles the difficulty of computing exclusive observables at two loops by extending inclusive-tools: integration-by-parts and Lorentz-invariance identities reduce the vast space of two-loop four-point integrals to a small set of master integrals. It then derives differential equations in the external invariants for these master integrals and solves them (often in terms of generalized hypergeometric functions) with boundary conditions from simpler subtopologies. The authors demonstrate the approach with a detailed one-loop four-point example and provide substantial results for several two-loop topologies up to $t=5$, with partial results for $t=6$ and $t=7$, highlighting both the method’s power and the remaining computational challenges. The framework offers a practical path toward computing two-loop virtual corrections to exclusive quantities, such as jet observables, and sets the stage for completing the remaining master integrals in higher-topology diagrams.

Abstract

At variance with fully inclusive quantities, which have been computed already at the two- or three-loop level, most exclusive observables are still known only at one-loop, as further progress was hampered so far by the greater computational problems encountered in the study of multi-leg amplitudes beyond one loop. We show in this paper how the use of tools already employed in inclusive calculations can be suitably extended to the computation of loop integrals appearing in the virtual corrections to exclusive observables, namely two-loop four-point functions with massless propagators and up to one off-shell leg. We find that multi-leg integrals, in addition to integration-by-parts identities, obey also identities resulting from Lorentz-invariance. The combined set of these identities can be used to reduce the large number of integrals appearing in an actual calculation to a small number of master integrals. We then write down explicitly the differential equations in the external invariants fulfilled by these master integrals, and point out that the equations can be used as an efficient method of evaluating the master integrals themselves. We outline strategies for the solution of the differential equations, and demonstrate the application of the method on several examples.