Feynman integrals and intersection theory

Abstract

We introduce the tools of intersection theory to the study of Feynman integrals, which allows for a new way of projecting integrals onto a basis. In order to illustrate this technique, we consider the Baikov representation of maximal cuts in arbitrary space-time dimension. We introduce a minimal basis of differential forms with logarithmic singularities on the boundaries of the corresponding integration cycles. We give an algorithm for computing a basis decomposition of an arbitrary maximal cut using so-called intersection numbers and describe two alternative ways of computing them. Furthermore, we show how to obtain Pfaffian systems of differential equations for the basis integrals using the same technique. All the steps are illustrated on the example of a two-loop non-planar triangle diagram with a massive loop. * pierpaolo.mastrolia@pd.infn.it † smizera@pitp.ca arXiv:1810.03818v2 [hep-th]

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

“…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].…”

“…For our purposes, however, it is more convenient to think of a basis of Feynman integrals as a section of a vector bundle V over the kinematic space. To make this concrete, let us split the differential operator on the total space as where the matrix-valued one-form Ω is given by intersection numbers [30], as a special case of (2.60): Since the integration domain Γ is always kept constant, each fiber of V is isomorphic to a twisted cohomology H P dW , where W is determined by a point K ∈ K on the base space. Connections obtained in the above way are always integrable (flat), meaning that D − Ω∧ squares to zero, i.e., DΩ − Ω ∧ Ω = 0.…”

“…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.…”

“…Thus the problem amounts to finding the matrix Ω. It was recently realised that fibers of the vector bundle can be described by the cohomology of (Ω • M , d+τ dW ∧) and hence the entries of Ω can be computed by the same intersection numbers (1.3) described above [30,31].…”