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 \( L=\ln \left(-{M}_h^2/{m}_b^2\right) \) 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 \( \sim {\alpha \alpha}_s^n{L}^{2n+1} \).
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
C.W. Bauer, D. Pirjol and I.W. Stewart, Soft collinear factorization in effective field theory, Phys. Rev. D 65 (2002) 054022 [hep-ph/0109045] [INSPIRE].
C.W. Bauer, S. Fleming, D. Pirjol, I.Z. Rothstein and I.W. Stewart, Hard scattering factorization from effective field theory, Phys. Rev. D 66 (2002) 014017 [hep-ph/0202088] [INSPIRE].
M. Beneke, A.P. Chapovsky, M. Diehl and T. Feldmann, Soft collinear effective theory and heavy to light currents beyond leading power, Nucl. Phys. B 643 (2002) 431 [hep-ph/0206152] [INSPIRE].
I. Moult, I.W. Stewart, G. Vita and H.X. Zhu, First subleading power resummation for event shapes, JHEP 08 (2018) 013 [arXiv:1804.04665] [INSPIRE].
M.A. Ebert, I. Moult, I.W. Stewart, F.J. Tackmann, G. Vita and H.X. Zhu, Subleading power rapidity divergences and power corrections for qT, JHEP 04 (2019) 123 [arXiv:1812.08189] [INSPIRE].
I. Moult, G. Vita and K. Yan, Subleading power resummation of rapidity logarithms: the energy-energy correlator in N = 4 SYM, JHEP 07 (2020) 005 [arXiv:1912.02188] [INSPIRE].
M. Beneke et al., Leading-logarithmic threshold resummation of the Drell-Yan process at next-to-leading power, JHEP 03 (2019) 043 [arXiv:1809.10631] [INSPIRE].
M. Beneke, A. Broggio, S. Jaskiewicz and L. Vernazza, Threshold factorization of the Drell-Yan process at next-to-leading power, JHEP 07 (2020) 078 [arXiv:1912.01585] [INSPIRE].
Z.L. Liu and M. Neubert, Factorization at subleading power and endpoint-divergent convolutions in h → γγ decay, JHEP 04 (2020) 033 [arXiv:1912.08818] [INSPIRE].
J. Wang, Resummation of double logarithms in loop-induced processes with effective field theory, arXiv:1912.09920 [INSPIRE].
I. Moult, I.W. Stewart and G. Vita, Subleading power factorization with radiative functions, JHEP 11 (2019) 153 [arXiv:1905.07411] [INSPIRE].
M. Beneke, M. Garny, R. Szafron and J. Wang, Violation of the Kluberg-Stern-Zuber theorem in SCET, JHEP 09 (2019) 101 [arXiv:1907.05463] [INSPIRE].
I. Moult, I.W. Stewart, G. Vita and H.X. Zhu, The soft quark Sudakov, JHEP 05 (2020) 089 [arXiv:1910.14038] [INSPIRE].
M. Beneke, M. Garny, S. Jaskiewicz, R. Szafron, L. Vernazza and J. Wang, Large-x resummation of off-diagonal deep-inelastic parton scattering from d-dimensional refactorization, JHEP 10 (2020) 196 [arXiv:2008.04943] [INSPIRE].
M.I. Kotsky and O.I. Yakovlev, On the resummation of double logarithms in the process Higgs → γγ, Phys. Lett. B 418 (1998) 335 [hep-ph/9708485] [INSPIRE].
R. Akhoury, H. Wang and O.I. Yakovlev, On the resummation of large QCD logarithms in Higgs → γγ decay, Phys. Rev. D 64 (2001) 113008 [hep-ph/0102105] [INSPIRE].
T. Liu and A.A. Penin, High-energy limit of QCD beyond the Sudakov approximation, Phys. Rev. Lett. 119 (2017) 262001 [arXiv:1709.01092] [INSPIRE].
T. Liu and A. Penin, High-energy limit of mass-suppressed amplitudes in gauge theories, JHEP 11 (2018) 158 [arXiv:1809.04950] [INSPIRE].
R.V. Harlander, M. Prausa and J. Usovitsch, The light-fermion contribution to the exact Higgs-gluon form factor in QCD, JHEP 10 (2019) 148 [Erratum ibid. 08 (2020) 101] [arXiv:1907.06957] [INSPIRE].
M.L. Czakon and M. Niggetiedt, Exact quark-mass dependence of the Higgs-gluon form factor at three loops in QCD, JHEP 05 (2020) 149 [arXiv:2001.03008] [INSPIRE].
C. Anastasiou and A. Penin, Light quark mediated Higgs boson threshold production in the next-to-leading logarithmic approximation, JHEP 07 (2020) 195 [arXiv:2004.03602] [INSPIRE].
Z.L. Liu, B. Mecaj, M. Neubert and X. Wang, Factorization at subleading power and endpoint divergences in soft-collinear effective theory, arXiv:2009.04456 [INSPIRE].
R.J. Hill and M. Neubert, Spectator interactions in soft collinear effective theory, Nucl. Phys. B 657 (2003) 229 [hep-ph/0211018] [INSPIRE].
M. Beneke and T. Feldmann, Multipole expanded soft collinear effective theory with non-Abelian gauge symmetry, Phys. Lett. B 553 (2003) 267 [hep-ph/0211358] [INSPIRE].
S.W. Bosch, R.J. Hill, B.O. Lange and M. Neubert, Factorization and Sudakov resummation in leptonic radiative B decay, Phys. Rev. D 67 (2003) 094014 [hep-ph/0301123] [INSPIRE].
T. Becher and M. Neubert, Drell-Yan production at small qT, transverse parton distributions and the collinear anomaly, Eur. Phys. J. C 71 (2011) 1665 [arXiv:1007.4005] [INSPIRE].
J.-Y. Chiu, A. Jain, D. Neill and I.Z. Rothstein, A formalism for the systematic treatment of rapidity logarithms in quantum field theory, JHEP 05 (2012) 084 [arXiv:1202.0814] [INSPIRE].
Y. Li, D. Neill and H.X. Zhu, An exponential regulator for rapidity divergences, Nucl. Phys. B 960 (2020) 115193 [arXiv:1604.00392] [INSPIRE].
S. Alte, M. König and M. Neubert, Effective field theory after a new-physics discovery, JHEP 08 (2018) 095 [arXiv:1806.01278] [INSPIRE].
R. Tarrach, The pole mass in perturbative QCD, Nucl. Phys. B 183 (1981) 384 [INSPIRE].
Z.L. Liu and M. Neubert, Two-loop radiative jet function for exclusive B-meson and Higgs decays, JHEP 06 (2020) 060 [arXiv:2003.03393] [INSPIRE].
Z.L. Liu, B. Mecaj, M. Neubert, X. Wang and S. Fleming, Renormalization and scale evolution of the soft-quark soft function, JHEP 07 (2020) 104 [arXiv:2005.03013] [INSPIRE].
G. Lepage and S.J. Brodsky, Exclusive processes in quantum chromodynamics: evolution equations for hadronic wave functions and the form-factors of mesons, Phys. Lett. B 87 (1979) 359 [INSPIRE].
V.L. Chernyak and A.R. Zhitnitsky, Asymptotic behavior of exclusive processes in QCD, Phys. Rept. 112 (1984) 173 [INSPIRE].
F.M. Dittes and A.V. Radyushkin, Two loop contribution to the evolution of the pion wave function, Phys. Lett. B 134 (1984) 359 [INSPIRE].
S.V. Mikhailov and A.V. Radyushkin, Evolution kernels in QCD: two loop calculation in Feynman gauge, Nucl. Phys. B 254 (1985) 89 [INSPIRE].
S.J. Brodsky, P. Damgaard, Y. Frishman and G. Lepage, Conformal symmetry: exclusive processes beyond leading order, Phys. Rev. D 33 (1986) 1881 [INSPIRE].
D. Mueller, Conformal constraints and the evolution of the nonsinglet meson distribution amplitude, Phys. Rev. D 49 (1994) 2525 [INSPIRE].
D. Mueller, The evolution of the pion distribution amplitude in next-to-leading-order, Phys. Rev. D 51 (1995) 3855 [hep-ph/9411338] [INSPIRE].
M. Beneke, M. Garny, R. Szafron and J. Wang, Anomalous dimension of subleading-power N-jet operators. Part II, JHEP 11 (2018) 112 [arXiv:1808.04742] [INSPIRE].
T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [INSPIRE].
B.O. Lange and M. Neubert, Renormalization group evolution of the B meson light cone distribution amplitude, Phys. Rev. Lett. 91 (2003) 102001 [hep-ph/0303082] [INSPIRE].
T. Luthe, A. Maier, P. Marquard and Y. Schröder, Five-loop quark mass and field anomalous dimensions for a general gauge group, JHEP 01 (2017) 081 [arXiv:1612.05512] [INSPIRE].
I.A. Korchemskaya and G.P. Korchemsky, On lightlike Wilson loops, Phys. Lett. B 287 (1992) 169 [INSPIRE].
J.M. Henn, G.P. Korchemsky and B. Mistlberger, The full four-loop cusp anomalous dimension in N = 4 super Yang-Mills and QCD, JHEP 04 (2020) 018 [arXiv:1911.10174] [INSPIRE].
S. Moch, J.A.M. Vermaseren and A. Vogt, The quark form-factor at higher orders, JHEP 08 (2005) 049 [hep-ph/0507039] [INSPIRE].
T. Becher, M. Neubert and B.D. Pecjak, Factorization and momentum-space resummation in deep-inelastic scattering, JHEP 01 (2007) 076 [hep-ph/0607228] [INSPIRE].
V.M. Braun, Y. Ji and A.N. Manashov, Two-loop evolution equation for the B-meson distribution amplitude, Phys. Rev. D 100 (2019) 014023 [arXiv:1905.04498] [INSPIRE].
B.A. Ovrut and H.J. Schnitzer, Gauge theories with minimal subtraction and the decoupling theorem, Nucl. Phys. B 179 (1981) 381 [INSPIRE].
A.V. Smilga, Next-to-leading logarithms in the high-energy asymptotics of the quark form-factor and the jet cross-section, Nucl. Phys. B 161 (1979) 449 [INSPIRE].
M. Niggetiedt, Exact quark-mass dependence of the Higgs-photon form factor at three loops in QCD, arXiv:2009.10556 [INSPIRE].
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2009.06779
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Liu, Z.L., Mecaj, B., Neubert, M. et al. Factorization at subleading power and endpoint divergences in h → γγ decay. Part II. Renormalization and scale evolution. J. High Energ. Phys. 2021, 77 (2021). https://doi.org/10.1007/JHEP01(2021)077
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP01(2021)077