Colour Evolution and Infrared Physics

We give a complete account of how soft gluon, massless quark, evolution equations in colour space originate, from a factorization into a hard cross section density operator and a soft function encoding measurements and the projection on definite colours. We detail this formalism up to the two loop level and we demonstrate how the evolution kernels relate to infrared subtractions, and how the resolution of infrared singular regions conspires with the structure of observables the algorithm should be able to predict. The latter allows us to address evolution in different kinematic variables, including energy ordering and angular cutoffs in non-global observables. The soft factor and its evolution resembles a hadronization model including effects such as colour reconnection, and could give insight into the structure of power corrections in observables which require soft gluon evolution.