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.

To $${d}$$ d , or not to $${d}$$ d : recent developments and comparisons of regularization schemes

“…Ref. [68]) which allow one to reconstruct full analytic results from numerical calculations over finite fields.…”

“…[64,66,67], as well as for the first application of multivariate reconstruction techniques to generalized unitarity presented in Ref. [68]. The latter includes the calculation of the on-shell integrands of the maximal cuts of the two-loop planar pentabox and the nonplanar double pentagon topology, for a complete set of independent helicity configurations.…”

First two-loop amplitudes with the numerical unitarity method

“…The problem of how to reconstruct a rational functions has recently received attention in the literature [17,18]. The main observation is that the values of the coefficients can be inferred from the value that the function takes at a finite number of points.…”

“…Therefore, by evaluating the coefficient at a set of (in principle arbitrary) distinct points D i one reconstructs the full function. In practice we use the formula of Thiele [17,19], expressing c(D) in the form of a continued fraction,…”

“…As such, these functions are amenable to the univariate functional reconstruction techniques described in section 2.3. Similar approaches have been applied to the reconstruction from numerical samples of the full integrand [17] as well as IBP relations [18]. We applied this technique to reconstruct the full set of master integral coefficients, extracting analytic results from our numerical approach.…”