Two-loop amplitudes with nested sums: Fermionic contributions toe+e−→qq¯g

We present the calculation of the $n_f$ contributions to the two-loop amplitude for $e^{+}{e}^{-}\to q\overline{q}g$ and give results for the full one-loop amplitude to order ${\varepsilon }^2$ in the dimensional regularization parameter. Our results agree with those recently obtained by Garland et al. The calculation makes extensive use of an efficient method based on nested sums to calculate two-loop integrals with arbitrary powers of the propagators. The use of nested sums leads in a natural way to multiple polylogarithms with simple arguments, which allow a straightforward analytic continuation.