Abstract
Building on the recent derivation of a bare factorization theorem for the b-quark induced contribution to the h → γγ 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 λ = mb/Mh « 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 alpha}_s^2{L}^k in the three-loop decay amplitude, where L=ln left(-{M}_h^2/{m}_b^2right) 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 alpha}_s^n{L}^{2n+1} .
Highlights
Soft-collinear effective theory (SCET) [1,2,3] provides a convenient framework for addressing the problems of scale separation and factorization in high-energy physics using the powerful tools of effective field theory
We have focused on the contribution to the radiative Higgs-boson decay amplitude induced by the Higgs coupling to light bottom quarks; the methods we have developed are more general and can be applied to other subleading-power factorization theorems
Endpoint-divergent convolution integrals arising when the factorized decay amplitude is expressed in terms of bare matching coefficients and bare operator matrix elements have been tamed by introducing rapidity regulators on the convolution integrals
Summary
Soft-collinear effective theory (SCET) [1,2,3] provides a convenient framework for addressing the problems of scale separation and factorization in high-energy physics using the powerful tools of effective field theory. Specific applications discussed in the literature include the study of power corrections to event shapes [4] and transverse-momentum distributions [5, 6], the threshold factorization for the Drell-Yan process [7, 8], and the factorization of power-suppressed contributions to Higgs-boson decays [9, 10] One finds that such factorization theorems contain a sum over convolutions of Wilson coefficients with operator matrix elements, where the relevant SCET operators mix under renormalization. A complete resummation of large logarithms in RG-improved perturbation theory is left for future work
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