Uniqueness of two-loop master contours

Abstract

Generalized-unitarity calculations of two-loop amplitudes are performed by expanding the amplitude in a basis of master integrals and then determining the coefficients by taking a number of generalized cuts. In this paper, we present a complete classification of the solutions to the maximal cut of integrals with the double-box topology. The ideas presented here are expected to be relevant for all two-loop topologies as well. We find that these maximal-cut solutions are naturally associated with Riemann surfaces whose topology is determined by the number of states at the vertices of the double-box graph. In the case of four massless external momenta we find that, once the geometry of these Riemann surfaces is properly understood, there are uniquely defined master contours producing the coefficients of the double-box integrals in the basis decomposition of the two-loop amplitude. This is in perfect analogy with the situation in one-loop generalized unitarity.In addition, we point out that the chiral integrals recently introduced by Arkani-Hamed et al. can be used as master integrals for the double-box contributions to the two-loop amplitudes in any gauge theory. The infrared finiteness of these integrals allow for their coefficients as well as their integrated expressions to be evaluated in strictly four dimensions, providing significant technical simplification. We evaluate these integrals at four points and obtain remarkably compact results.

Hexagon functions and the three-loop remainder function

“…For example, an elliptic integral, in which the kernel is not rational but contains a square root, enters the two-loop equal-mass sunrise graph [58,59], and it has been shown that a very similar type of integral enters a particular N 3 MHV 10-point scattering amplitude in planar N = 4 super-Yang-Mills theory [60]. However, it has been argued [61], based on a novel form of the planar loop integrand, that MHV and NMHV amplitudes can all be described in terms of multiple polylogarithms alone.…”

On the reduction of generalized polylogarithms to Li n and Li2,2 and on the evaluation thereof

“…At two or more loops many Feynman integrals can be likewise expressed in terms of GPLs [7][8][9][10][11][12][13][14][15][16][17][18][19] (for further references, see [20,21] and the references therein), but there are also integrals which are counter examples, such as notably that of the fully massive sunset graph [22][23][24][25][26]. Certain graphs without massive propagators are also believed to be counter examples [27]. In this paper we will restrict the discussion to GPLs.…”

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…”