Maximal unitarity for the four-mass double box

Abstract

We extend the maximal-unitarity formalism at two loops to double-box integrals with four massive external legs. These are relevant for higher-point processes, as well as for heavy vector rescattering, VV → VV. In this formalism, the two-loop amplitude is expanded over a basis of integrals. We obtain formulas for the coefficients of the double-box integrals, expressing them as products of tree-level amplitudes integrated over specific complex multidimensional contours. The contours are subject to the consistency condition that integrals over them annihilate any integrand whose integral over real Minkowski space vanishes. These include integrals over parity-odd integrands and total derivatives arising from integration-by-parts (IBP) identities. We find that, unlike the zero-through three-mass cases, the IBP identities impose no constraints on the contours in the four-mass case. We also discuss the algebraic varieties connected with various double-box integrals and show how discrete symmetries of these varieties largely determine the constraints.

Local integrand representations of all two-loop amplitudes in planar SYM

“…And as with the contributions (3.9), including the terms, 13) will automatically ensure that many other physical cuts match via residue theorems. There is one final class of composite leading singularities of amplitudes associated with soft-collinear divergences (see section 2.3):…”

“…One of its earliest triumphs was to show that any one-loop amplitude could be represented in terms of a basis of pre-chosen integrals, with coefficients computed in terms of treeamplitudes (glued together into 'on-shell functions'), [1][2][3][4][5][6]. Despite the enormous success of generalized unitarity at one-loop order, its extension to two or more loops -while straight-forward in principle -proved surprisingly difficult in practice until quite recently, when renewed interest from collider experiments was met with more powerful theoretical techniques (and more powerful computers), [7][8][9][10][11][12][13][14][15][16].…”

Cuts from residues: the one-loop case

“…(5.8) does not involve the hyperplane P ∞ , the structure of the homology group H D (Σ − 10 The homology groups are specific to the compactification, and it is not clear if different compactifications may lead to different homology groups [32]. 11 We recall that we work in a complexified space, where for each point of a hypersurface we can define a complex plane transverse to the hypersurface Pj. We can then consider a loop around this point in the transverse space that encircle the hyperplane Pj without touching it.…”

“…Singularities and branch cuts of Feynman integrals are classified by the solutions to the Landau conditions [4], a set of necessary conditions on the external data of an integral for a pinch singularity to occur. 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].…”

Integral reduction by unitarity method for two-loop amplitudes: a case study

“…The unitarity method has been proven to be very successful in determining coefficients for one-loop amplitudes (see reviews [73,74]). For some subsets of bases (such as box topology for one-loop and double-box topology for planar two-loop), more efficient method, the so called "generalized unitarity method" (or "maximum unitarity cut" or "leading singularity"), has been developed [48,[75][76][77][78][79][80][81][82][83][84][85][86][87]. The applicability of reduction method is based on the valid expression of expansion (1.1).…”