Scattering-amplitude ansätze from algebraic geometry and p-adic numbers

by Giuseppe DeLaurentis (Freiburg University)

Thursday 9 Jun 2022, 14:00 → 15:00 Europe/Zurich

Aula Polvani (Dipartimento di Fisica, Università degli Studi di Milano)

Scattering amplitudes in perturbative quantum field theories
have their process-dependent information encoded in rational functions
of the external kinematics. The infra-red behavior of the amplitude is
imprinted on these rational functions as a rich structure of zeros and
poles, which are best understood as varieties in complexified momentum
space. By leveraging this highly non-trivial structure, one can
generate refined ansätze to aid the computation of scattering
amplitudes and enhance their evaluation speed and stability for
subsequent phenomenological studies.  Algebraic geometry provides a
natural language to describe geometric varieties in terms of algebraic
ideals. Since poles and zeros are well defined only on irreducible
sub-varieties, which we call branches, I will discuss how to
systematically identify these branches via primary decompositions of
the respective ideals in spinor space. I will then introduce a tool
from number theory, namely p-adic numbers, to evaluate the rational
functions in proximity to these branches in a stable and efficient
manner, and thus obtain the degree at which they diverge or vanish. In
some sense, p-adic numbers bridge the gap between finite fields and
floating-point numbers by combining the stability of finite fields
with a non-trivial absolute value. Numerical evaluations of the
rational functions lead, by the Zariski-Nagata theorem, to constraints
on the numerators in terms of membership to a particular type of
ideals, the symbolic power. I will show applications to the
pentagon-function coefficients of the two-loop $0\rightarrow
q\bar{q}\gamma\gamma\gamma$ helicity amplitudes and -- time permitting
-- to the master-integral coefficients of the one-loop $0\rightarrow
q\bar{q}V(\rightarrow \ell \bar{\ell})V(\rightarrow \ell'
\bar{\ell}')g$ helicity amplitude.

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