Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers

TL;DR

This work develops a decomposition-by-intersections framework that directly expresses a given Feynman integral in terms of a chosen basis of master integrals on maximal cuts, using intersection numbers between differential forms within Baikov representations. By extending to both 1-form and, prospectively, 2-form representations, the authors derive a master-decomposition formula and demonstrate its consistency with IBP-based reductions across a wide array of multi-loop examples, from Beta and Gauss hypergeometric functions to complex two-loop and multi-loop topologies. They show that choosing monomial or $d\log$ bases yields canonical differential equations and facilitates dimensional-recurrence relations, often with fewer masters than traditional IBP counts. The methodology is validated by extensive cross-checks against public IBP tools, and the results point to potential computational advantages for high-multiplicity amplitudes in perturbation theory. The paper thus provides a unifying, geometrically motivated route to master integral reductions that complements and, in some cases, surpasses standard IBP techniques.

Abstract

We elaborate on the recent idea of a direct decomposition of Feynman integrals onto a basis of master integrals on maximal cuts using intersection numbers. We begin by showing an application of the method to the derivation of contiguity relations for special functions, such as the Euler beta function, the Gauss ${}_2F_1$ hypergeometric function, and the Appell $F_1$ function. Then, we apply the new method to decompose Feynman integrals whose maximal cuts admit 1-form integral representations, including examples that have from two to an arbitrary number of loops, and/or from zero to an arbitrary number of legs. Direct constructions of differential equations and dimensional recurrence relations for Feynman integrals are also discussed. We present two novel approaches to decomposition-by-intersections in cases where the maximal cuts admit a 2-form integral representation, with a view towards the extension of the formalism to $n$-form representations. The decomposition formulae computed through the use of intersection numbers are directly verified to agree with the ones obtained using integration-by-parts identities.

Figures (29)

• Figure 1: Four-loop vacuum diagram.
• Figure 2: Triple-cross two-point function.
• Figure 3: Massless sunrise.
• Figure 4: Massive sunrise.
• Figure 5: Non-planar triangle.