Master Equations for Master Amplitudes

TL;DR

The paper develops a general framework of master equations for master amplitudes associated with Feynman graphs, derived from integration-by-parts identities and differential relations, and demonstrates their use by applying to the two-loop 4-propagator self-mass graph. It shows how to expand these equations in the dimensional regulator around $n=4$ and to obtain large-$p^2$ asymptotics, including exact relations among coefficients like $F^{( extinfty,1)}=S^{( extinfty)}(n)/((n-2)(n-4))$. The results provide a systematic, numerically stable approach to evaluating master amplitudes, with potential for analytic quadrature in simple mass configurations and extensions to graphs with more propagators.

Abstract

The general lines of the derivation and the main properties of the master equations for the master amplitudes associated to a given Feynman graph are recalled. Some results for the 2-loop self-mass graph with 4 propagators are then presented.

Figures (1)

• Figure 1: $G(n,m_1^2,m_2^2,m_3^2,m_4^2,p^2)$, the 2-loop 4-propagator self-mass graph.