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 𝒪(α _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 𝒪(α _s^2) topology where it differs at sub-leading N_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 α _s .