Parton branching at amplitude level

We present an algorithm that evolves hard processes at the amplitude level by dressing them iteratively with (massless) quarks and gluons. The algorithm interleaves collinear emissions with soft emissions and includes Coulomb/Glauber exchanges. It includes all orders in $N_{\mathrm{c}}$, is spin dependent and is able to accommodate kinematic recoils. Although it is specified at leading logarithmic accuracy, the framework should be sufficient to go beyond. Coulomb exchanges make the factorisation of collinear and soft emissions highly non-trivial. In the absence of Coulomb exchanges, we show how factorisation works out and how a partial factorisation is manifest in the presence of Coulomb exchanges. Finally, we illustrate the use of the algorithm by deriving DGLAP evolution and computing the resummed thrust, hemisphere jet mass and gaps-between-jets distributions in $e^+ e^-$.