Decomposition of Feynman integrals on the maximal cut by intersection numbers

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 2 F 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 decompositionby-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.

From positive geometries to a coaction on hypergeometric functions

“…Instead, we must use the framework of 'twisted (co)homology' [20] which is well known in the mathematics literature. Recently, these tools have been applied in several areas of theoretical physics such as in string theory [21,22] or in the study of the integration-by-parts relations satisfied by Feynman integrals [23][24][25].…”

From infinity to four dimensions: higher residue pairings and Feynman integrals

“…It would be fascinating to understand a similar geometric condition that leads to an ε-form differential equations, or decide whether such a basis could even exist. Although we used a representation in terms of Symanzik polynomials, as in (3.12), there is no substantial difficulty in repeating our analysis in other ways, e.g., using the original loop-momentum variables [72] or Baikov representation [30,31,68], where the answer to this question might prove easier.…”

“…As remarked before, many quantities of physical interest can be written as integrals on M of the general form: [5] for an exposition), other than the fact that it leads to the above pairing [58]. For physical applications see, e.g., [5,30,31,[59][60][61][62][63][64][65][66][67][68][69][70][71][72] and references therein.…”

“…where the middle-dimensional integration cycle is Γ := R P + and the hat denotes the fact that we expect (3.14) to agree with (3.1) only after taking the limit δ a → 0. Closely related ways of rewriting Feynman integrals as twisted periods were introduced in [30], see also [31,68,70,72,[74][75][76][77].…”

“…The latter expansion can be consistently carried out using higher residue pairings. As demonstrated in [30,31,68] the intersection numbers in Ω can be also computed exactly in ε, though the techniques used there rely on the knowledge of either stratification of M or its fibration properties. Using higher residue pairings allows us to forget about these technicalities.…”