# Loop-level double-copy for massive fermions in the fundamental

## TL;DR

This paper develops a method to construct color-dual scattering amplitudes for massive fermions in QCD, enabling straightforward generation of gravitational amplitudes through the double-copy approach, with explicit calculations up to six points and one-loop corrections.

## Contribution

It introduces a novel color-dual representation for massive fermions in QCD, extending double-copy techniques to include fundamental matter and clarifying fermionic kinematic ansatze.

## Key findings

- Constructed color-dual amplitudes for six-point tree-level processes.
- Derived the four-point one-loop correction for massive fermions.
- Enabled gravitational amplitude generation via double-copy from QCD amplitudes.

## Abstract

We find that unitarity cuts and the duality between color and kinematics are sufficient constraints to bootstrap $D$-dimensional QCD scattering amplitudes starting from three-particle tree-level. Specifically, we calculate tree level amplitudes through six-points, as well as the four-point one-loop correction for massive fermions in the fundamental representation of the gauge group -- constructing a color-dual representation of the latter for the first time. To do so we clarify a prescription for functional kinematic ansatze involving fermionic matter. The advantages of color-dual calculation, familiar from particles in the adjoint, also apply here: only a small number of basis topologies must be constrained via physical information of the theory, and algebraic relations propagate this to a full solution. As all the QCD amplitudes we construct here are color-dual, they trivially generate $D$-dimensional amplitudes in gravitational theories via double-copy construction.