Baikov-Lee representations of cut Feynman integrals

Abstract

We develop a general framework for the evaluation of d-dimensional cut Feynman integrals based on the Baikov-Lee representation of purely-virtual Feynman integrals. We implement the generalized Cutkosky cutting rule using Cauchy's residue theorem and identify a set of constraints which determine the integration domain. The method applies equally well to Feynman integrals with a unitarity cut in a single kinematic channel and to maximally-cut Feynman integrals. Our cut Baikov-Lee representation reproduces the expected relation between cuts and discontinuities in a given kinematic channel and furthermore makes the dependence on the kinematic variables manifest from the beginning. By combining the Baikov-Lee representation of maximally-cut Feynman integrals and the properties of periods of algebraic curves, we are able to obtain complete solution sets for the homogeneous differential equations satisfied by Feynman integrals which go beyond multiple polylogarithms. We apply our formalism to the direct evaluation of a number of interesting cut Feynman integrals.

Five-loop four-point integrand of N=8 supergravity as a generalized double copy

“…We will find the two coefficients to be both zero, which is consistent In order to explain the machinery that we use at five loops, we briefly review the treatment in Ref. [66] of IBP relations needed to demonstrate the ultraviolet finiteness of half-maximal supergravity at two loops in D = 5, and in addition give a more intuitive treatment by computing cut integrals [46,47,68,69] following the method of Ref. [47].…”

“…We follow the standard strategy of expanding the integrand at large loop momenta or equivalently at small external momenta, as used in earlier supergravity calculations [11,14]. The main difference is that the integral relations needed to simplify the results are much more complex, so we employ modern unitarity compatible integration ideas [7,9,46,47] to streamline the computations.…”

“…Instead, we will exploit recent advances in IBP reduction on unitarity cuts [7,9,46,47] to quickly obtain the coefficients of the two top-level master integralsmaster integrals corresponding to vacuum graphs with only cubic vertices-via the maximal cut of vacuum integrals. We will find the two coefficients to be both zero, which is consistent In order to explain the machinery that we use at five loops, we briefly review the treatment in Ref.…”

“…We also present nontrivial checks based on integrating the expressions in spacetime dimension D = 22/5, where we expect it to be finite, yet individual terms in our expression diverge. To carry out these checks we develop techniques based on modern developments in integration [7,9,46,47].…”

Differential equations for Feynman integrals beyond multiple polylogarithms

“…the second option obtains an elliptic curve from the maximal cut [50][51][52][53][54][55][56] of the sunrise integral…”