One-loop QCD corrections to ūd → $$ t\overline{t}W $$ at $$ \mathcal{O}\left({\varepsilon}^2\right) $$

A bstract We present a computation of the one-loop QCD corrections to top-quark pair production in association with a W boson, including terms up to order ε 2 in dimensional regularization. Providing a first glimpse into the complexity of the corresponding two-loop amplitude, this result is a first step towards a description of this process at next-to-next-to-leading order (NNLO) in QCD. We perform a tensor decomposition and express the corresponding form factors in terms of a basis of independent special functions with compact rational coefficients, providing a structured framework for future developments. In addition, we derive an explicit analytic representation of the form factors, valid up to order ε 0 , expressed in terms of logarithms and dilogarithms. For the complete set of special functions required, we obtain a semi-numerical solution based on generalized power series expansion.