Abstract
Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces. We present a general algorithm for the construction of multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter. We apply it to the decomposition of a few Feynman integrals at one and two loops, as first steps toward potential applications to generic multiloop integrals. The proposed method can be more generally employed for the derivation of contiguity relations for special functions admitting multifold integral representations.
Highlights
Introduction.—Scattering amplitudes encode crucial information about collision phenomena in our Universe, from the smallest to the largest scales
Feynman integrals obey linear relations governed by intersection numbers, which act as scalar products between vector spaces
We present a general algorithm for the construction of multivariate intersection numbers relevant to Feynman integrals, and show for the first time how they can be used to solve the problem of integral reduction to a basis of master integrals by projections, and to directly derive functional equations fulfilled by the latter
Summary
An exception is made for those cases where a limited number of kinematic invariants yields the use of direct integration techniques; the evaluation of multiloop Feynman integrals requires the exploitation of linear relations among integrals, in order to simplify the otherwise impossible calculations Those relations can be used both for decomposing scattering amplitudes in terms of a basis of functions, referred to as master integrals (MIs), and for the evaluation of the latter. In the initial studies, [9,11], this novel decomposition method was applied to the realm of special mathematical functions falling in the class of Lauricella FD functions, as well as to Feynman integrals on maximal cuts, i.e., with on-shell internal lines, mostly admitting a onefold integral representation Those results concerned a partial construction of Feynman integral relations, mainly limited to the determination of the coefficients of the MIs with the same number of denominators as the decomposed integral, which was achieved by means of intersection numbers for univariate forms. Their complete decomposition requires the evaluation of intersection numbers for multivariate rational
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