The photon-energy spectrum in B → X$_s$γ to N$^3$LO: light-fermion and large-N$_c$ corrections

We consider the photon-energy spectrum of the inclusive radiative decay B -> X$_s$γ, induced by the electromagnetic dipole operator O$_7$, to next-to-next-to-next-to-leading order and calculate the complete corrections for light fermions, for the contributions with two closed massive fermion loops, and for the limit of large QCD colour factors N$_c$ in the remaining part. We discuss the integrated decay rate both without and with a cut on the photon energy. In addition to the on-shell renormalization of the bottom-quark mass, we also consider the kinetic mass and the MSR mass schemes. The latter two lead to an improved perturbative behaviour of the decay rate.