On the integrand-reduction method for two-loop scattering amplitudes

By Pierpaolo Mastrolia, G. Ossola. 2011

138 citations recorded by the source corpus.

We propose a first implementation of the integrand-reduction method for two-loop scattering amplitudes. We show that the residues of the amplitudes on multi-particle cuts are polynomials in the irreducible scalar products involving the loop momenta, and that the reduction of the amplitudes in terms of master integrals can be realized through polynomial fitting of the integrand, without any apriori knowledge of the integral basis. We discuss how the polynomial shapes of the residues determine the basis of master integrals appearing in the final result. We present a four-dimensional constructive algorithm that we apply to planar and non-planar contributions to the 4- and 5-point MHV amplitudes in $ \mathcal{N} = 4 $ SYM. The technique hereby discussed extends the well-established analogous method holding for one-loop amplitudes, and can be considered a preliminary study towards the systematic reduction at the integrand-level of two-loop amplitudes in any gauge theory, suitable for their automated semianalytic evaluation.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/8cb0b3521c44f647ed9c04c61d3f780910f91024

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In Nuclear and High Energy Physics

Citations in the corpus

Uniqueness of two-loop master contours (2012, unknown): “[40, 41], in which the heptacut integrand is reconstructed by polynomial matching in similarity with the OPP approach [21].”
Cuts of Feynman Integrals in Baikov representation (2017, background): “In the recent years, progress has been made also towards the extension of these reduction methods for two-loop amplitudes at the integral [17, 18, 19, 20, 21, 22, 23, 24, 25, 26] as well as the integrand [27, 28, 29, 30, 31, 32] level.”
Integrand reduction of one-loop scattering amplitudes through Laurent series expansion (2012, methodology): “The integrand reduction method has been recently shown to be applicable to scattering amplitudes beyond one-loop as well [15, 16].”
A two-loop five-gluon helicity amplitude in QCD (2013, methodology): “Integrand reduction methods have also been developed to the two-loop level [49, 50] via the polynomial fitting and Gram matrix constraints.”
Scattering Amplitudes from Multivariate Polynomial Division (2012, methodology): “The first extension of the integrand reduction method beyond one-loop was proposed in [14], and it was used to reproduce the results of two-loop 5point planar and non-planar amplitudes in N = 4 SYM [15, 16].”
GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond (2014, methodology): “The principle of an integrand-reduction method, which is valid at any order in perturbation theory [19–23], is the underlying multi-particle pole expansion for the integrand of any scattering amplitude, or, equivalently, a representation where the numerator of each Feynman integral is expressed as a”