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

Building on the recent derivation of a bare factorization theorem for the $b$-quark induced contribution to the $h\to\gamma\gamma$ decay amplitude based on soft-collinear effective theory, we derive the first renormalized factorization theorem for a process described at subleading power in scale ratios, where $\lambda=m_b/M_h\ll 1$ in our case. We prove two refactorization conditions for a matching coefficient and an operator matrix element in the endpoint region, where they exhibit singularities giving rise to divergent convolution integrals. The refactorization conditions ensure that the dependence of the decay amplitude on the rapidity regulator, which regularizes the endpoint singularities, cancels out to all orders of perturbation theory. We establish the renormalized form of the factorization formula, proving that extra contributions arising from the fact that "endpoint regularization" does not commute with renormalization can be absorbed, to all orders, by a redefinition of one of the matching coefficients. We derive the renormalization-group evolution equation satisfied by all quantities in the factorization 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 decay amplitude, where $L=\ln(-M_h^2/m_b^2)$ and $k=6,5,4,3$. We find perfect agreement with existing numerical results for the amplitude and analytical results for the three-loop contributions involving a massless quark loop. On the other hand, we disagree with the results of previous attempts to predict the series of subleading logarithms $\sim\alpha{\hspace{0.3mm}}\alpha_s^n{\hspace{0.3mm}} L^{2n+1}$.