Ansätze for scattering amplitudes from p-adic numbers and algebraic geometry

Abstract

Rational coefficients of special functions in scattering amplitudes are known to simplify on singular surfaces, often diverging less strongly than the naïve expectation. To systematically study these surfaces and rational functions on them, we employ tools from algebraic geometry. We show how the divergences of a rational function constrain its numerator to belong to symbolic powers of ideals associated to the singular surfaces. To study the divergences of the coefficients, we make use of p-adic numbers, closely related to finite fields. These allow us to perform numerical evaluations close to the singular surfaces in a stable manner and thereby characterize the divergences of the coefficients. We then use this information to construct low-dimensional Ansätze for the rational coefficients. As a proof-of-concept application of our algorithm, we reconstruct the two-loop 0 → q$$ \overline{q} $$ q ¯ γγγ pentagon-function coefficients with fewer than 1000 numerical evaluations.

FIRE6: Feynman Integral REduction with modular arithmetic

“…We mention in passing that an entirely different approach is reconstructing rational functions from finite-field numerical evaluations [40,41,42,43,44,45,46,37], bypassing the need to simplify rational functions in intermediate steps. A related technique is the further simplification of rational functions by multivariate partial fractioning [47,48], after analytic results for IBP reduction tables or complete loop amplitudes have already been obtained.…”

Two-loop QCD corrections for three-photon production at hadron colliders

“…refs. [36,75], we will consider the r i to be rational functions of the spinor-helicity variables λ and λ already introduced in section 2. 3 The coefficients admit a least common denominator representation, which reads…”

“…Analytic expressions can then be reconstructed using multivariate functional reconstruction techniques [32,33] (see also refs. [34][35][36][37][38] for related developments). In order to perform our calculation within this framework, we make a number of theoretical developments.…”

“…[47]) indicate that further developments will be important to tackle amplitudes with an increased number of scales. For this reason, we explore new techniques to construct ansätze whose analytic structure better exhibit the physical properties of the scattering amplitudes, performing the analytic reconstruction using spinor-helicity ansatz techniques [35,36]. In contrast to a more traditional reconstruction using a set of independent kinematic invariants (as e.g.…”

Les Houches 2021—physics at TeV colliders: report on the standard model precision wishlist

“…This technique has been used to compute analytic one-loop amplitudes for H + 4j [60], the pp → W( → lν) + γ [61] process and ¢ ¢ qqll l l g ¯¯¯ [ 62]. In [63], a related approach to reconstructing analytic expressions from evaluations using p-adic numbers was presented.…”