How to choose a good rational basis for elliptic Feynman integrals?

Abstract

Representing multi-loop scattering amplitudes as linear combinations of multivalued transcendental functions with process-dependent rational coefficients has long been understood to be advantageous. In general, these transcendental functions satisfy a system of differential equations with coupled homogeneous blocks. When these coupled blocks can be removed through algebraic basis transformations, the relation between the rational and algebraic bases is universal and minimal. Here, we ask whether an analogous universal and minimal relation exists when decoupling requires transformations involving complete elliptic integrals. We elaborate on the method proposed in ref. arXiv:2504.20897, in which we suggested that a basis constructed using an elliptic generalization of leading singularities may provide an answer to this question. We extend this analysis to the off-diagonal blocks of the rational differential equations satisfied by the bases obtained through this construction. We find that the $\epsilon$ dependence of these blocks can be organized into a universal structure that is preserved under the decoupling transformation used to express the solutions in terms of iterated integrals.