Graded transcendental functions: an application to four-point amplitudes with one off-shell leg

TL;DR

This paper develops a systematic approach to construct minimal bases of transcendental functions for scattering amplitudes, simplifying calculations and avoiding complex cancellations, with applications to three-loop amplitudes and form factors in quantum field theory.

Contribution

It introduces a method to organize and rotate basis functions for scattering amplitudes, reducing spurious features and facilitating boundary condition fixing without full functional forms.

Findings

Successfully evaluated three-loop amplitude topologies in specific limits.

Confirmed the maximal transcendentality conjecture for certain amplitudes.

Provided the first analytic computation of a three-point form factor in planar N=4 sYM.

Abstract

Several recent works have demonstrated the powerful algebraic simplifications that can be achieved for scattering amplitudes through a systematic grading of transcendental quantities. We develop these concepts to construct a minimal basis of functions tailored to a scattering amplitude in a general way. Starting with formal solutions for all master integral topologies, we organise the appearing functions by properties such as their symbol alphabet or letter adjacency. We rotate the basis such that functions with spurious features appear in the least possible number of basis elements. Since their coefficients must vanish for physical quantities, this approach avoids complex cancellations. As a first application, we evaluate all integral topologies relevant to the three-loop $Hggg$ and $Hgq\bar{q}$ amplitudes in the leading-colour approximation and heavy-top limit. We describe the derivation of canonical differential equation systems and present a method for fixing boundary conditions without the need for a full functional representation. Using multiple numerical reductions, we test the maximal transcendentality conjecture for $Hggg$ and identify a new letter which appears in functions of weight 4 and 5. In addition, we provide the first direct analytic computation of a three-point form factor of the operator $\mathrm{Tr}(\phi^2)$ in planar $\mathcal{N}=4$ sYM and find agreement with numerical and bootstrapped results.