Decomposition of Feynman integrals by multivariate intersection numbers

We present a detailed description of the recent idea for a direct decomposition of Feynman integrals onto a basis of master integrals by projections, as well as a direct derivation of the differential equations satisfied by the master integrals, employing multivariate intersection numbers. We discuss a recursive algorithm for the computation of multivariate intersection numbers, and provide three different approaches for a direct decomposition of Feynman integrals, which we dub the straight decomposition, the bottom-up decomposition, and the top-down decomposition. These algorithms exploit the unitarity structure of Feynman integrals by computing intersection numbers supported on cuts, in various orders, thus showing the synthesis of the intersection-theory concepts with unitarity-based methods and integrand decomposition. We perform explicit computations to exemplify all of these approaches applied to Feynman integrals, paving a way towards potential applications to generic multi-loop integrals.

Automated summary

We present a new approach for decomposing Feynman integrals onto a basis of master integrals by projections and deriving the differential equations satisfied by them using multivariate intersection numbers. Our recursive algorithm for calculating these numbers and three different decomposition methods (straight, bottom-up, top-down) showcase the synthesis of intersection-theory concepts with unitarity-based methods. Computation examples illustrate the potential applications to generic multi-loop integrals.