Chebyshev Approximations of Feynman Integrals for Collider Physics

Abstract

We present a novel approach for solving canonical differential equations for Feynman integrals based on an approximation of the integrals with Chebyshev polynomials. By exploiting the analyticity properties of Feynman integrals, the method constructs rapidly converging polynomial approximations along a path, enabling highly efficient numerical evaluation. Moreover, we introduce an adaptive approximation method that dynamically samples to optimise convergence. We implement this framework in double-precision arithmetic and demonstrate its stability across physical phase space using a series of two-loop, five-point examples. Our proof-of-principle implementation proves competitive with state-of-the-art one-fold integral methods, while requiring little to no case-by-case intervention to handle spurious singularities.