# Multi-Quark Colour Decompositions from Unitarity

## TL;DR

This paper develops new color decompositions for multi-quark amplitudes in QCD using unitarity principles, enabling more efficient calculations of loop amplitudes with arbitrary quark and gluon configurations.

## Contribution

It introduces novel multi-quark bases and closed-form decompositions for loop QCD amplitudes, advancing unitarity-based amplitude construction methods.

## Key findings

- Derived new color decompositions for multi-quark amplitudes.
- Formulated closed-form expressions for these decompositions.
- Enhanced the framework for loop amplitude calculations in QCD.

## Abstract

Any loop QCD amplitude at full colour is constructed from kinematic and gauge-group building blocks. In a unitarity-based on-shell framework, both objects can be reconstructed from their respective counterparts in tree-level amplitudes. This procedure is at its most powerful when aligned with flexible colour decompositions of tree-level QCD amplitudes. In this note we derive such decompositions for amplitudes with an arbitrary number of quarks and gluons from the same principle that is used to bootstrap kinematics - unitarity factorisation. In the process we formulate new multi-quark bases and provide closed-form expressions for the new decompositions. We then elaborate upon their application in colour decompositions of loop multi-quark amplitudes.