  We compute the renormalization and evolution of the soft-quark function that
appears in the factorization theorem for Higgs-boson decays to two photons
through a $b$-quark loop. Our computation confirms a conjecture by Liu, Mecaj,
Neubert, Wang, and Fleming for the form of the renormalization and evolution of
the soft-quark function in order $\alpha_s$. We also work out the analyticity
structure of the soft-quark function by making use of light-cone perturbation
theory.
