Differential equations for loop integrals without squared propagators
Jorrit Bosma, Kasper J. Larsen, Yang Zhang
TL;DR
The paper tackles the problem of constructing differential equations for multi-loop integrals without introducing squared propagators in intermediate steps. It leverages the Baikov representation and a sufficient enhanced ideal membership condition to ensure that derivative terms can be rewritten entirely in terms of $D$-dimensional integrals, enabling unitarity-compatible IBP reductions to be used in the differential equation setup. The authors demonstrate the approach on a nontrivial case—the fully massless planar double-box—obtaining a canonical $oldsymbol{ abla}$-form in $oldsymbol{ abla}$ with explicit integer matrices, validating the method's practicality. They also discuss limitations via a counterexample and outline open problems, notably classifying diagrams for which the enhanced ideal membership holds and deriving closed formulas for cofactors. This work potentially broadens the applicability of the differential equations method to complex multi-loop, multi-scale problems while reducing intermediate computational complexity.
Abstract
We provide a sufficient condition for avoiding squared propagators in the intermediate stages of setting up differential equations for loop integrals. This condition is satisfied in a large class of two- and three-loop diagrams. For these diagrams, the differential equations can thus be computed using "unitarity-compatible" integration-by-parts reductions, which simplify the reduction problem by avoiding integrals with higher-power propagators.