Theory Seminar

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.