Factorization at Next-to-Leading Power and Endpoint Divergences in $gg\to h$ Production

We derive a factorization theorem for the Higgs-boson production amplitude in gluon-gluon fusion induced by a light-quark loop, working at next-to-leading power in soft-collinear effective theory. The factorization is structurally similar to that obtained for the $h\to \gamma \gamma$ decay amplitude induced by a light-quark loop, but additional complications arise because of external color charges. We show how the refactorization-based subtraction scheme developed in previous work leads to a factorization theorem free of endpoint divergences. We use renormalization-group techniques to predict the logarithmically enhanced terms in the three-loop $gg\to h$ form factor of order ${\alpha }_s^3{\ln }^k(-M_h^2\mathrm{/}{m}_b^2)$ with $k=6,5,4,3$. We also resum the first three towers of leading logarithms, ${\alpha }_s^n{\ln }^{2n-k}(-M_h^2\mathrm{/}{m}_b^2)$ with $k=0,1,2$, to all orders of perturbation theory.