Scattering amplitudes over finite fields and multivariate functional reconstruction

By Tiziano Peraro. 2016

271 citations recorded by the source corpus.

A bstractSeveral 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.

Subfields: Artificial Intelligence, Computer Networks and Communications, Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/c43fd107664d5f45b9262667ce1a3287812cca7d

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Citations in the corpus

Decomposition of Feynman integrals on the maximal cut by intersection numbers (2019, methodology): “It would be interesting to study connections to the recent applications of closelyrelated mathematical topics to Feynman integrals, such as D-module theory [62], Hopf algebras [91, 121, 122], computational algebraic geometry [21, 63], finite fields arithmetic [20], and the theory of special function”
Feynman integrals and intersection theory (2018, methodology): “[18] T.”
Integral reduction with Kira 2.0 and finite field methods (2020, unknown): “In the last few years, new techniques have been explored and applied to integral reduction problems, employing syzygy equations [7–12], algebraic geometry [13–15], intersection numbers [16–20], finite field and interpolation techniques [21–26], or special integral representations [27–30].”
Collider physics at the precision frontier (2020, methodology): “Meanwhile public tools are available for functional reconstruction using finite fields, such as FiniteFlow [468] or FireFly [469, 470], implementations in Fire [462] and Kira [466] are also available.”