Scattering amplitudes over finite fields and multivariate functional reconstruction

TL;DR

The paper develops a dense, finite-field–based framework for reconstructing multivariate polynomials and rational functions from modular evaluations, enabling analytic calculations that bypass large intermediate expressions. It combines Newton and Thiele reconstructions for univariate cases with a recursive multivariate extension, and couples this with a multivariate rational-reconstruction strategy using a shift/homogenization trick and the Chinese Remainder Theorem. The methodology is applied to spinor-helicity, Berends-Giele recursion, and generalized-unity–based multi-loop integrand reduction, culminating in full analytic expressions for the two-loop, five-point on-shell integrands of planar penta-box and non-planar double-pentagon topologies in Yang–Mills theory across multiple helicity configurations. The authors implement the approach in a self-contained C++ library using 64-bit finite-field arithmetic and provide publicly available analytic results, illustrating substantial efficiency and scalability for high-multiplicity, multi-loop calculations. The work broadens the toolkit for high-energy phenomenology by enabling exact, analytic reconstructions from fast modular evaluations, with potential extensions to IBP and beyond.

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.

Figures (5)

• Figure 1: Schematic depiction of Berends-Giele recursion relations. The grey blobs $V_k$ represent contractions defined by the $k$-point Feynman rules of the theory.
• Figure 2: Sum of diagrams with gluon (solid lines) and scalar (dashed lines) loops, for the penta-box topology. Each scalar loop should be multiplied by the number of scalar flavours, which in our case is equal to $d_s-\mathcal{D}$.
• Figure 3: The two-loop planar penta-box topology.
• Figure 4: The two-loop non-planar double-pentagon topology.
• Figure 5: Schematic depiction of a unitarity cut. Grey blobs represent tree-level amplitudes and they are joined by the lines corresponding to the on-shell momenta of the cut loop propagators $\ell_i$. The loop momenta are defined as $k_1=\ell_1$, and $k_2=-\ell_{j_1+j_2}$. Double lines represent an arbitrary number of external legs.