On the maximal cut of Feynman integrals and the solution of their differential equations

Abstract

The standard procedure for computing scalar multi-loop Feynman integrals consists in reducing them to a basis of so-called master integrals, derive differential equations in the external invariants satisfied by the latter and, finally, try to solve them as a Laurent series in ϵ=(4−d)/2 , where d are the space–time dimensions. The differential equations are, in general, coupled and can be solved using Euler's variation of constants, provided that a set of homogeneous solutions is known. Given an arbitrary differential equation of order higher than one, there exists no general method for finding its homogeneous solutions. In this paper we show that the maximal cut of the integrals under consideration provides one set of homogeneous solutions, simplifying substantially the solution of the differential equations

Cuts from residues: the one-loop case

“…Since uncut and cut integrals satisfy the same IBP identities, we immediately conclude that cut integrals satisfy the same differential equations as uncut Feynman integrals. We stress that this argument is independent of the loop order, and it agrees with the reverse-unitarity approach to the computation of inclusive cross sections [57][58][59][60][61] and recent approaches to solve homogeneous differential equations by maximal cuts [15,16]. As a consequence, the bases of one-loop cut integrals in eq.…”

“…(7.12)). The resulting discontinuity function is still multi-valued, and our goal is to compute the discontinuities of the discontinuity function 16 Disc C I D n . We only discuss singularities of the first type, because all discontinuities around Landau varieties associated to singularities of the second type can be expressed in terms of those of JHEP06 (2017)114 the first type.…”

“…Modern unitarity methods build on this observation and, in a nutshell, use cuts to construct projectors onto a basis of master integrals [5][6][7][8][9][10][11]. More recently, there has been a renewal in the interest in cut integrals in the study of integration-by-parts identities [12][13][14] or differential equations [15][16][17][18] satisfied by Feynman integrals, and in applications of the solutions to Landau conditions [19,20].…”

Differential equations for Feynman integrals beyond multiple polylogarithms

“…The periods ψ 1 , ψ 2 of the elliptic curve are solutions of the homogeneous differential equation [38]. In general, the maximal cut of a Feynman integral is a solution of the homogeneous differential equation for this Feynman integral [57]. We define the new variables τ and q by…”

Evaluating ‘elliptic’ master integrals at special kinematic values: using differential equations and their solutions via expansions near singular points

“…[20][21][22][23]. On the other hand, it is quite natural to try to introduce new functions which would enable us to present results, in elliptic cases, in an analytical form.…”