Multi-Quark Colour Decompositions from Unitarity
Alexander Ochirov, Ben Page
TL;DR
<3-5 sentence high-level summary>The paper addresses the challenge of constructing full-colour QCD amplitudes with an arbitrary number of quarks and gluons within a unitarity-based on-shell framework. It introduces co-unitary, KK-independent tree-level bases and derives exact colour decompositions whose colour factors factorise across factorisation channels, enabling systematic reconstruction of loop integrands. The authors provide explicit constructions for like-flavour, distinct-flavour, and gluon-stretched bases, along with closed-form JO/DDM-building blocks and proofs of co-unitarity and linear independence. They further extend the framework to one-loop level, showing how full-colour decompositions can be assembled from tree-level colour-ordered amplitudes, with implications for multi-loop QCD and potential connections to gravity via colour-kinematics duality.
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.