Scattering amplitudes over finite fields and multivariate functional reconstruction

Abstract

Several problems in computer algebra can be efficiently solved by reducing them to calculations over finite fields. In this paper, we describe an algorithm for the reconstruction of multivariate polynomials and rational functions from their evaluation over finite fields. Calculations over finite fields can in turn be efficiently performed using machine-size integers in statically-typed languages. We then discuss the application of the algorithm to several techniques related to the computation of scattering amplitudes, such as the four- and six-dimensional spinor-helicity formalism, tree-level recursion relations, and multi-loop integrand reduction via generalized unitarity. The method has good efficiency and scales well with the number of variables and the complexity of the problem. As an example combining these techniques, we present the calculation of full analytic expressions for the two-loop five-point on-shell integrands of the maximal cuts of the planar penta-box and the non-planar double-pentagon topologies in Yang-Mills theory, for a complete set of independent helicity configurations.

44 pages, 2 tables, 5 figures, published version

References in corpus (7)

• Reducing full one-loop amplitudes to scalar integrals at the integrand level
• An Automated Implementation of On-Shell Methods for One-Loop Amplitudes
• Full one-loop amplitudes from tree amplitudes
• GoSam-2.0: a tool for automated one-loop calculations within the Standard Model and beyond
• Recola2: REcursive Computation of One-Loop Amplitudes 2
• Amplitudes and Spinor-Helicity in Six Dimensions
• Cross-Order Integral Relations from Maximal Cuts

Cited by in corpus (9)

• FiniteFlow: multivariate functional reconstruction using finite fields and dataflow graphs
• Analytic Form of the Planar Two-Loop Five-Parton Scattering Amplitudes in QCD
• Decomposition of Feynman Integrals on the Maximal Cut by Intersection Numbers
• Matter dependence of the four-loop cusp anomalous dimension
• The two-loop five-point amplitude in $\mathcal{N}=8$ supergravity
• Quark and gluon form factors to four loop order in QCD: the $N_f^3$ contributions
• The two-loop five-particle amplitude in $\mathcal{N}=8$ supergravity
• Planar master integrals for four-loop form factors
• Multi-loop calculations: numerical methods and applications