Taming the N3LO corrections to semileptonic $b \to u$ decay

We review the calculation of the QCD corrections of order $\alpha_s^3$ to the decay rate of $b \to u l \bar \nu_l$, originating from diagrams with closed fermion loops and neglecting the mass of the up quark. Our calculation takes advantage of integration-by-parts reduction of Feynman integrals with one propagator raised to a symbolic power in Kira. The five-loop master integrals are then evaluated numerically with the auxiliary mass flow method as implemented in AMFlow, in combination with an ad hoc interface to Kira. We obtain results for the fermionic contributions to the total semileptonic rate with an accuracy of more than thirty digits.