Intersection matrices associated to geometric-ordered bases of Feynman integrals

Abstract

In integration-by-parts reduction of Feynman integrals, the order relation in the Laporta algorithm determines a set of master integrals. In this paper we investigate the intersection matrices of the integrands of the master integrals that are obtained from a geometric order relation. With an appropriate definition of integrands and their duals, we find that the intersection matrices are simpler than expected: For a filtration-compatible basis, the entries of the intersection matrix are Laurent polynomials in the dimensional regularisation parameter $\varepsilon$. For an $\varepsilon$-factorised basis, the entries are instead integers, up to an overall power of $\varepsilon$, if the boundary values for the auxiliary functions of the rotation are chosen appropriately. This has practical consequences: We can systematically eliminate certain auxiliary transcendental functions, introduced in going from a filtration-compatible basis to an $\varepsilon$-factorised basis. We provide an algorithm that performs this elimination while minimising the number of required calculations.