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.
Highlights
On-shell method for handling colour information at the multi-loop level
In this paper we have considered the colour structure of tree and one-loop QCD amplitudes involving any number of distinctly flavoured quark-antiquark pairs
We have derived new bases of ordered amplitudes that are independent under the KleissKuijf relations [23] and found decompositions of an n-point colour-dressed amplitude into these bases
Summary
We use an amplitude with two quark pairs and one gluon to show how a new colour decomposition can be constructed from factorisation properties. To constrain the unknown colour factors in eq (1.2), we consider the factorisation limits of the full amplitude on the poles that include gluon 5 and one of the unfixed quarks:. Applying the limits (1.3) both to the five-point decomposition (1.2) as a whole and to the constituent ordered amplitudes therein, we can directly read off the colour coefficients as. The kinematic limits here became identities, since the colour factors are independent of any momenta. In this way, we have used the factorisation properties of the amplitude to derive a new explicit colour decomposition
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