Improvements on dipole shower colour

The dipole formalism provides a powerful framework from which parton showers can be constructed. In a recent paper (Forshaw et al. 2020), we proposed a dipole shower with improved colour accuracy and in this paper we show how it can be further improved. After an explicit check at $${\mathcal {O}}(\alpha _{\mathrm {s}}^{2})$$ we confirm that our original shower performs as it was designed to, i.e. inheriting its handling of angular-ordered radiation from a coherent branching algorithm. We also show how other dipole shower algorithms fail to achieve this. Nevertheless, there is an $${\mathcal {O}}(\alpha _{\mathrm {s}}^{2})$$ topology where it differs at sub-leading $$N_{\mathrm {c}}$$ from a coherent branching algorithm. This erroneous topology can contribute a leading logarithm to some observables and corresponds to emissions that are ordered in $$k_t$$ but not angle. We propose a simple, computationally efficient way to correct this and assign colour factors in accordance with the coherence properties of QCD to all orders in $$\alpha _{\mathrm {s}}$$