Factorization at subleading power and endpoint divergences in h → γγ decay. Part II. Renormalization and scale evolution

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 $$ {\alpha \alpha}_s^2{L}^k $$ αα s 2 L k in the three-loop decay amplitude, where $$ L=\ln \left(-{M}_h^2/{m}_b^2\right) $$ 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 $$ \sim {\alpha \alpha}_s^n{L}^{2n+1} $$ ∼ αα s n L 2 n + 1 .

Erratum to: Light quark mediated Higgs boson threshold production in the next-to-leading logarithmic approximation

“…The numerical values of the NNLO and N 3 LO NLL corrections in table 1 should be changed from −2.170 and −2.189 to −2.183 and −2.204, respectively. After the correction the abelian part of our result agrees with the results [1][2][3] for the Higgs boson two-photon decay amplitude. The numerical impact of the correction on the threshold cross section in NNLO and N 3 LO is very small.…”

Exact quark-mass dependence of the Higgs-photon form factor at three loops in QCD

“…The radii of convergence of the three expansions are limited due to singularities located at z = 0, z = 1 and 1/z = 0. The supplementary material HaaNiggetiedt.m attached to this paper shipped with this publication contains the large-mass expansion with exact coefficients truncated at O z 100 , the threshold expansion truncated at O (1 − z) 20 and the high-energy expansion truncated at O 1/z 8 . The latter ones are expansions with numerical coefficients.…”

“…The parts of the numerical coefficients of terms proportional to L k s /z for k ∈ {6, 5, 4, 3}, which stem from C (2,0) , comply with the exact coefficients predicted recently in refs. [19,20].…”

“…The paper at hand was motivated by the authors of refs. [19,20], who kindly requested the availability of the Higgs-photon form factor expanded in the high-energy limit to perform consistency-checks with their own results. Since the diagrams that account for the Higgs-photon form factor form a subset of diagrams contributing to the Higgs-gluon form factor, we closely follow our previous publication ref.…”

“…Figure 2. Comparison of the large-mass expansion (LME) up to O z 100 , threshold expansion (THR) up to O (1 − z)20 and high-energy expansion (HE) up to O 1/z 8 with the sum C (2,0) + C(2,1) evaluated numerically (L µ = 0). The absolute difference between the exact result and the expansions is shown in the bottom panel.…”