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Constructing heavy-quark sum rule for $b \to c$ meson and baryon decays

Motoi Endo ^1,2,3, Syuhei Iguro ^4,3, Satoshi Mishima ⁵, and Ryoutaro Watanabe ⁶

¹KEK Theory Center, IPNS, KEK, Tsukuba 305–0801, Japan
²Graduate Institute for Advanced Studies, SOKENDAI, Tsukuba, Ibaraki 305–0801, Japan
³Kobayashi-Maskawa Institute (KMI) for the Origin of Particles and the Universe, Nagoya University, Nagoya 464–8602, Japan
⁴Institute for Advanced Research (IAR), Nagoya University, Nagoya 464–8601, Japan
⁵Department of Liberal Arts, Saitama Medical University, Moroyama, Saitama 350-0495, Japan
⁶Institute of Particle Physics and Key Laboratory of Quark and Lepton Physics (MOE), Central China Normal University, Wuhan, Hubei 430079, China

Phys. Rev. D 113, 014021 – Published 20 January, 2026

DOI: https://doi.org/10.1103/xs7x-jcrd

Abstract

We study heavy-hadron semileptonic decays proceeding via $b \to c$ transition, such as $B \to D^{(*)} τ {\bar{ν}}_{τ}$ and $Λ_{b} \to Λ_{c} τ {\bar{ν}}_{τ}$ . In the heavy-quark limit, where the heavy-quark symmetry holds, we provide a fundamental framework for heavy-quark sum rules among these decays based on the spin decomposition picture. The relation holds directly for the squared amplitudes without requiring phase-space integration. We then apply this relation to reproduce the sum rule among $B \to D^{(*)} τ {\bar{ν}}_{τ}$ and $Λ_{b} \to Λ_{c} τ {\bar{ν}}_{τ}$ . Furthermore, we derive new sum rules for $Ω_{b} \to Ω_{c}^{(*)}$ transitions and those involving excited states, such as $B \to {D_{0}^{*}, D_{1}^{*}}$ and $B \to {D_{1}, D_{2}^{*}}$ .

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References (41)

The Heavy Flavor Averaging Group, Averages of $b$ -hadron, $c$ -hadron, and $τ$ -lepton properties as of 2023, arXiv:2411.18639, See the web update for Spring 2025 version.
M. Blanke, A. Crivellin, S. de Boer, M. Moscati, U. Nierste, I. Nišandžić, and T. Kitahara, Impact of polarization observables and $B_{c} \to τ ν$ on new physics explanations of the $b \to c τ ν$ anomaly, Phys. Rev. D 99, 075006 (2019).
M. Blanke, A. Crivellin, T. Kitahara, M. Moscati, U. Nierste, and I. Nišandžić, Addendum to “Impact of polarization observables and $B_{c} \to τ ν$ on new physics explanations of the $b \to c τ ν$ anomaly”, Phys. Rev. D 100, 035035 (2019).
M. Endo, S. Iguro, S. Mishima, and R. Watanabe, $b \to c$ semileptonic sum rule: Current status and prospects, arXiv:2508.06322.
M. Fedele, M. Blanke, A. Crivellin, S. Iguro, T. Kitahara, U. Nierste, and R. Watanabe, Impact of $Λ_{b} \to Λ_{c} τ ν$ measurement on new physics in $b \to c l ν$ transitions, Phys. Rev. D 107, 055005 (2023).
W.-F. Duan, S. Iguro, X.-Q. Li, R. Watanabe, and Y.-D. Yang, On sum rules for semi-leptonic $b \to c$ and $b \to u$ decays, J. High Energy Phys. 07 (2025) 166.
M. Endo, S. Iguro, S. Mishima, and R. Watanabe, Heavy quark symmetry behind $b \to c$ semileptonic sum rule, J. High Energy Phys. 05 (2025) 112.
N. Isgur and M. B. Wise, Weak decays of heavy mesons in the static quark approximation, Phys. Lett. B 232, 113 (1989).
N. Isgur and M. B. Wise, Weak transition form factors between heavy mesons, Phys. Lett. B 237, 527 (1990).
N. Isgur and M. B. Wise, Heavy baryon weak form-factors, Nucl. Phys. B348, 276 (1991).
H. Georgi, Comment on heavy baryon weak form-factors, Nucl. Phys. B 348, 293 (1991).
M. Endo, S. Iguro, T. Kretz, S. Mishima, and R. Watanabe, $b \to c$ semileptonic sum rule: Extension to angular observables, Eur. Phys. J. C 85, 961 (2025).
K. Zalewski, Isgur-Wise symmetry and heavy hadron wave functions, Phys. Lett. B 264, 432 (1991).
M. Sadzikowski and K. Zalewski, Isgur-Wise functions from the MIT bag model, Z. Phys. C 59, 677 (1993).
S. Nussinov and W. Wetzel, Comparison of exclusive decay rates for $b \to u$ and $b \to c$ transitions, Phys. Rev. D 36, 130 (1987).
M. A. Shifman and M. B. Voloshin, On production of $D$ and $D^{*}$ mesons in $B$ meson decays, Sov. J. Nucl. Phys. 47, 511 (1988).
E. Eichten and F. L. Feinberg, Spin dependent forces in heavy quark systems, Phys. Rev. Lett. 43, 1205 (1979).
E. Eichten and F. Feinberg, Spin dependent forces in QCD, Phys. Rev. D 23, 2724 (1981).
G. P. Lepage and B. A. Thacker, Effective Lagrangians for simulating heavy quark systems, Nucl. Phys. B, Proc. Suppl. 4, 199 (1988).
J. G. Korner and G. A. Schuler, Exclusive semileptonic decays of bottom mesons in the spectator quark model, Z. Phys. C 38, 511 (1988); 41, 690(E) (1989).
A. F. Falk, H. Georgi, B. Grinstein, and M. B. Wise, Heavy meson form-factors from QCD, Nucl. Phys. B343, 1 (1990).
A. F. Falk, Hadrons of arbitrary spin in the heavy quark effective theory, Nucl. Phys. B378, 79 (1992).
A. V. Manohar and M. B. Wise, Heavy Quark Physics (Cambridge University Press, Cambridge, England, 2000).
H. D. Politzer, Counting form-factors using the Isgur-Wise symmetry, Phys. Lett. B 250, 128 (1990).
T. Amano, D. Jido, and S. Leupold, Sum rule for the partial decay rates of bottom hadrons based on the dynamical supersymmetry of the $\bar{s}$ quark and the $u d$ diquark, Phys. Rev. D 105, L051504 (2022).
Q. P. Xu, A Bjorken sum rule for semileptonic $Ω_{b}$ decays to ground and excited charmed baryon states, Phys. Rev. D 48, 5429 (1993).
J. G. Korner, M. Kramer, and D. Pirjol, Heavy baryons, Prog. Part. Nucl. Phys. 33, 787 (1994).
C.-K. Chow, New universality of the baryon Isgur-Wise form-factor in the large $N_{c}$ limit, Phys. Rev. D 51, 1224 (1995).
D. Ebert, R. N. Faustov, and V. O. Galkin, Semileptonic decays of heavy baryons in the relativistic quark model, Phys. Rev. D 73, 094002 (2006).
N. Isgur and M. B. Wise, Excited charm mesons in semileptonic $\bar{B}$ decay and their contributions to a Bjorken sum rule, Phys. Rev. D 43, 819 (1991).
S. Veseli and M. G. Olsson, Modeling form-factors in HQET, Phys. Lett. B 367, 302 (1996).
M. Bordone, N. Gubernari, D. van Dyk, and M. Jung, Heavy-quark expansion for ${\bar{B}}_{s} \to D_{s}^{(*)}$ form factors and unitarity bounds beyond the $S U (3)_{F}$ limit, Eur. Phys. J. C 80, 347 (2020).
E. Kou and P. Urquijo, eds., The Belle II physics book, Prog. Theor. Exp. Phys. 2019, 123C01 (2019); 2020, 029201(E) (2020).
Belle-II Collaboration, Snowmass white paper: Belle II physics reach and plans for the next decade and beyond, arXiv:2207.06307.
ATLAS, Belle-II, CMS, and LHCb Collaborations, Projections for key measurements in heavy flavour physics, arXiv:2503.24346.
F. U. Bernlochner, M. F. Sevilla, D. J. Robinson, and G. Wormser, Semitauonic $b$ -hadron decays: A lepton flavor universality laboratory, Rev. Mod. Phys. 94, 015003 (2022).
T. S. M. Ho, X.-H. Jiang, T. H. Kwok, L. Li, and T. Liu, Testing lepton flavor universality at future $Z$ factories, Phys. Rev. D 109, 093004 (2024).
X. Ai et al., Flavor physics at the CEPC: A general perspective, Chin. Phys. 49, 103003 (2025).
J. D. Bjorken, New symmetries in heavy flavor physics, in Proceedings of the 4th Rencontres de Physique de la Vallee d’Aoste, La Thuile, Italy (Editions Frontieres, Gif-sur-Yvette, France, 1990).
N. Isgur, M. B. Wise, and M. Youssefmir, Excited charm baryons in semileptonic $Λ_{b}$ decay and their contribution to a Bjorken sum rule, Phys. Lett. B 254, 215 (1991).
F. Hussain, G. Thompson, and J. G. Korner, Hadrons of arbitrary spin and heavy quark symmetry, arXiv:hep-ph/9311309.

Phys. Rev. D 113, 014021– Published 20 January, 2026

Vol. 113, Iss. 1 — 1 January 2026

Received 7 October 2025
Accepted 15 December 2025

DOI: https://doi.org/10.1103/xs7x-jcrd

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

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Constructing heavy-quark sum rule for $b \to c$ meson and baryon decays

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Constructing heavy-quark sum rule for b→c meson and baryon decays

Abstract

Physics Subject Headings (PhySH)

Article Text

References (41)

Constructing heavy-quark sum rule for $b \to c$ meson and baryon decays