Factorization at Subleading Power and Endpoint-Divergent Convolutions in $h\to\gamma\gamma$ Decay

It is by now well known that, at subleading power in scale ratios, factorization theorems for high-energy cross sections and decay amplitudes contain endpoint-divergent convolution integrals. The presence of these divergences hints at a violation of simple scale separation, as a result of the so-called collinear anomaly. At the technical level, endpoint divergences indicate an unexpected failure of dimensional regularization and the $\bar{\rm MS}$ subtraction scheme. In this paper we start a comprehensive discussion of factorization at subleading power within the framework of soft-collinear effective theory. As a concrete example, we factorize the decay amplitude for the radiative Higgs-boson decay $h\to \gamma\gamma$ mediated by a $b$-quark loop, for which endpoint-divergent convolution integrals require both dimensional and rapidity regulators. We derive a factorization theorem for the decay amplitude in terms of bare Wilson coefficients and operator matrix elements. We show that endpoint divergences caused by rapidity divergences cancel to all orders in perturbation theory, while endpoint divergences that are regularized dimensionally can be removed by rearranging the terms in the factorization theorem. We use our result to resum the leading double-logarithmic corrections of order $\alpha_s^n\ln^{2n+2}(-M_h^2/m_b^2)$ to all orders of perturbation theory.