Two-loop amplitude for the associated production of a top-anti-top pair and a $W$ boson at hadron colliders

In this contribution I present a calculation of the inclusive cross-section for the associated production of a top-anti-top pair and a W boson at next-to-next-to-leading order in QCD, based on the first exact computation of the two-loop QCD amplitude at leading colour. I discuss a strategy to express the amplitude in terms of a set of special functions with rational coefficients, overcoming the obstacle posed by the presence of elliptic functions. The special functions are evaluated through the method of differential equations, while the rational coefficients are computed using finite field techniques. For the computation of the inclusive cross-section, the double-virtual correction is evaluated by interpolating a five-dimensional grid.