An Algebro-Geometric Approach to Ansätze for Amplitudes

by Ben Page (CERN)



Over the coming decade, the experimental program at the LHC will reach
unprecedented levels of precision. To match this on the theory side, extremely
complicated amplitude calculations must be performed. Recently, we have
witnessed a boom in analytic calculations of scattering amplitudes, pushed
forward by the application of finite fields and Ansatz methodology. Unlike in
supersymmetric theories, the relevant space of rational functions in QCD is
poorly understood and so frontier calculations have made use of large, generic
Ansätze. As we look toward more complex processes, Ansatz dimension increases
and this threatens further progress.

In this talk, we introduce an approach to address this problem by making use of
observed simplification in both physical and spurious singular limits. We
discuss how tools from algebraic geometry and the use of p-adic numbers allow us
to investigate and understand the behaviour of scattering amplitudes in such
configurations. We then use these tools to systematically construct Ansätze that
exhibit the correct singular behaviour term by term. These refined Ansätze can
then be fit with relatively few finite-field samples when compared to functional
reconstruction methods. As a proof-of-concept application of our algorithm, we
reconstruct the two-loop 0🠆q𐨸qɣɣɣ pentagon-function coefficients with less than 1000 numerical samples.