The Stratification of Rigidity

TL;DR

This paper demonstrates that all planar, two-loop amplitudes in massless 4D theories can be decomposed into integrands with definite rigidity, either purely polylogarithmic or elliptic-polylogarithmic, structured by a master basis.

Contribution

It introduces a master integrand basis for all such amplitudes, fully stratified by rigidity, unifying polylogarithmic and elliptic-polylogarithmic integrands.

Findings

Existence of a master basis for all planar, two-loop amplitudes.

Classification of integrands by rigidity into polylogarithmic and elliptic types.

Each elliptic integrand involves a single elliptic curve.

Abstract

We show that a master integrand basis exists for all planar, two-loop amplitudes in massless four-dimensional theories which is fully stratified by rigidity -- with each integrand being either pure and strictly polylogarithmic or (pure and) strictly elliptic-polylogarithmic, with each of the later involving a single elliptic curve. Such integrands can be said to have definite rigidity.