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Building on the recent derivation of a bare factorization theorem for the$b$-quark induced contribution to the $h\to\gamma\gamma$ decay amplitude basedon soft-collinear effective theory, we derive the first renormalizedfactorization theorem for a process described at subleading power in scaleratios, where $\lambda=m_b/M_h\ll 1$ in our case. We prove two refactorizationconditions for a matching coefficient and an operator matrix element in theendpoint region, where they exhibit singularities giving rise to divergentconvolution integrals. The refactorization conditions ensure that thedependence of the decay amplitude on the rapidity regulator, which regularizesthe endpoint singularities, cancels out to all orders of perturbation theory.We establish the renormalized form of the factorization formula, proving thatextra contributions arising from the fact that "endpoint regularization" doesnot commute with renormalization can be absorbed, to all orders, by aredefinition of one of the matching coefficients. We derive therenormalization-group evolution equation satisfied by all quantities in thefactorization formula and use them to predict the large logarithms of order$\alpha{\hspace{0.3mm}}\alpha_s^2{\hspace{0.3mm}} L^k$ in the three-loop decayamplitude, where $L=\ln(-M_h^2/m_b^2)$ and $k=6,5,4,3$. We find perfectagreement with existing numerical results for the amplitude and analyticalresults for the three-loop contributions involving a massless quark loop. Onthe other hand, we disagree with the results of previous attempts to predictthe series of subleading logarithms$\sim\alpha{\hspace{0.3mm}}\alpha_s^n{\hspace{0.3mm}} L^{2n+1}$.

Factorization at Subleading Power and Endpoint Divergences in $h\to\gamma\gamma$ Decay: II. Renormalization and Scale Evolution