Decomposition of Feynman integrals by multivariate intersection numbers

Abstract

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.

Leading singularities in Baikov representation and Feynman integrals with uniform transcendental weight

“…One crucial step in applying the canonical differential equation approach is to find an integral basis with uniform transcendental weight. Significant effort has been put in designing methods and algorithms for finding a UT basis, for example, by the four-dimensional leading singularity analysis [14,15], by the Magnus series [16], by the dlog integrand construction [17] with the four-dimensional integrand or the Baikov representation [18,19], by the initial algorithm [20], by the Poincare index computations (Lee's algorithm) [21][22][23], by the intersection theory [24,25] and etc. In recent years, there is a great progress of the UT basis determination, and there are several publicly available packages for determining a UT basis, like Canonica [26,27], Fuchsia [28], epsilon [29], initial [20] and libra [23].…”

“…This computation is straightforward and fast with the help of modern computational algebraic geometry softwares. See [24,25] for the different approach for converting these integrand via the intersection theory.…”

Coaction and double-copy properties of configuration-space integrals at genus zero

“…For hypergeometric functions associated to dimensionally-regularized Feynman integrals, however, the analogous expansion is with respect to the dimensionalregularization parameter . The formal analogy between α and has already been noticed by comparing differential equations of Feynman integrals and configuration-space integrals of string amplitudes at genus zero [22,23] and at genus one [24][25][26], as well as in the context of twisted cohomology [27][28][29][30][31][32][33]. The discussion of this work only applies to the genus-zero case while leaving important extensions to non-polylogarithmic integrals to the future.…”

Mixed QCD-EW corrections for Higgs leptonic decay via HW+W− vertex

“…These candidates can be converted to Feynman integrals of the form (3.5) using intersection theory [41][42][43][44][45][46][47] or via IBP relations. The results are given by…”