Open Access

Two-photon transition form factor of $η_{c, b} (n S)$ and $χ_{c 0, b 0} (n P)$ via relativized mock meson states

Tian-Cheng Ding, Jian Huang, and Muyang Chen ^*

Department of Physics, Hunan Normal University, and Key Laboratory of Low-Dimensional Quantum Structures and Quantum Control of Ministry of Education, Changsha 410081, China

^*Contact author: muyang@hunnu.edu.cn

Phys. Rev. D 111, 094042 – Published 30 May, 2025

DOI: https://doi.org/10.1103/vwg1-nz37

Abstract

We construct relativized mock meson states for heavy quarkonium, where the Dirac spinors import kinetic relativistic correction and the dynamic wave functions are the same as those solved from the Schrödinger equation. We find that the relativistic correction imported by Dirac spinors is crucial to study the two-photon transition form factors. Using relativized mock meson states, we give credible predictions for the two-photon transition form factors and decay widths of $η_{c, b} (n {{}^{1}S}_{0})$ and $χ_{c, b} (n {{}^{3}P}_{0})$ ( $n = 1$ , 2, 3).

Physics Subject Headings (PhySH)

Article Text

References (25)

K. M. Ecklund et al., Two-photon widths of the $χ_{c J}$ states of charmonium, Phys. Rev. D 78, 091501 (2008).
J. P. Lees et al., Measurement of the $γ γ * \to η_{c}$ transition form factor, Phys. Rev. D 81, 052010 (2010).
M. Ablikim et al., Two-photon widths of the $χ_{c 0, 2}$ states and helicity analysis for $χ_{c 2} \to γ γ$ , Phys. Rev. D 85, 112008 (2012).
M. Masuda et al., Study of $K_{s}^{0}$ pair production in single-tag two-photon collisions, Phys. Rev. D 97, 052003 (2018).
Carlos A. Bertulani, Reinaldo Francener, Victor P. Goncalves, and Juciene T. de Souza, Particle production by $γ − γ$ interactions in future electron-ion colliders, Phys. Rev. C 111, 025201 (2025).
Jozef J. Dudek and Robert G. Edwards, Two photon decays of charmonia from lattice QCD, Phys. Rev. Lett. 97, 172001 (2006).
Brian Colquhoun, Laurence J. Cooper, Christine T. H. Davies, and G. Peter Lepage, Precise determination of decay rates for $η_{c} \to γ γ$ , $J / ψ \to γ η_{c}$ , and $J / ψ \to η_{c} e^{+} e^{-}$ from lattice QCD, Phys. Rev. D 108, 014513 (2023).
Yu Meng, Xu Feng, Chuan Liu, Teng Wang, and Zuoheng Zou, First-principle calculation of the $η_{c}$ $2 γ$ decay width from lattice QCD, Sci. Bull. 68, 1880 (2023).
Jing Chen, Minghui Ding, Lei Chang, and Yu-xin Liu, Two photon transition form factor of $\bar{c} c$ quarkonia, Phys. Rev. D 95, 016010 (2017).
Wolfgang Lucha and Dmitri Melikhov, The $γ^{*} γ^{*} \to η_{c}$ transition form factor, Phys. Rev. D 86, 016001 (2012).
Feng Feng, Yu Jia, and Wen-Long Sang, Can nonrelativistic QCD explain the $γ γ^{*} \to η_{c}$ transition form factor data?, Phys. Rev. Lett. 115, 222001 (2015).
Wen-Long Sang, Feng Feng, Yu Jia, and Shuang-Ran Liang, Next-to-next-to-leading-order QCD corrections to $χ_{c 0, 2} \to γ γ$ , Phys. Rev. D 94, 111501 (2016).
Li Yang, Wen-Long Sang, Hong-Fei Zhang, Yu-Dong Zhang, and Ming-Zhen Zhou, Study on $η_{c 2} (η_{b 2})$ electromagnetic decay into double photons, Phys. Rev. D 103, 034018 (2021).
Fu-guang Cao and Tao Huang, Large corrections to asymptotic $F_{η_{c} γ}$ and $F_{η_{b} γ}$ in the light cone perturbative QCD, Phys. Rev. D 59, 093004 (1999).
Martin Hoferichter and Peter Stoffer, Asymptotic behavior of meson transition form factors, J. High Energy Phys. 05 (2020) 159.
Hui-Young Ryu, Ho-Meoyng Choi, and Chueng-Ryong Ji, Systematic twist expansion of $(η_{c}, η_{b}) \to γ^{*} γ$ transition form factors in light-front quark model, Phys. Rev. D 98, 034018 (2018).
Izabela Babiarz, Victor P. Goncalves, Roman Pasechnik, Wolfgang Schäfer, and Antoni Szczurek, $γ^{*} γ^{*} \to η_{c} (1 s, 2 s)$ transition form factors for spacelike photons, Phys. Rev. D 100, 054018 (2019).
Yang Li, Meijian Li, and James P. Vary, Two-photon transitions of charmonia on the light front, Phys. Rev. D 105, L071901 (2022).
Chao Sun, Ru-Hui Ni, and Muyang Chen, Decay constants of $B_{c} (nS)$ and $B_{c}^{*} (nS)$ , Chin. Phys. C 47, 023101 (2023).
Olga Lakhina and Eric S. Swanson, Dynamic properties of charmonium, Phys. Rev. D 74, 014012 (2006).
S. Godfrey and Nathan Isgur, Mesons in a relativized quark model with chromodynamics, Phys. Rev. D 32, 189 (1985).
Qian Deng, Ru-Hui Ni, Qi Li, and Xian-Hui Zhong, Charmonia in an unquenched quark model, Phys. Rev. D 110, 056034 (2024).
S. Navas et al., Review of particle physics, Phys. Rev. D 110, 030001 (2024).
D. Ebert, R. N. Faustov, and V. O. Galkin, Two photon decay rates of heavy quarkonia in the relativistic quark model, Mod. Phys. Lett. A 18, 601 (2003).
J. P. Lansberg and T. N. Pham, Two-photon width of $η_{b}$ , $η_{b}^{'}$ and $η_{b}^{''}$ from heavy-quark spin symmetry, Phys. Rev. D 75, 017501 (2007).

Phys. Rev. D 111, 094042– Published 30 May, 2025

Vol. 111, Iss. 9 — 1 May 2025

Received 1 April 2025
Accepted 12 May 2025

DOI: https://doi.org/10.1103/vwg1-nz37

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.

Published by the American Physical Society

Physical Review D

Reuse & Permissions

Export citation

Export citation

Two-photon transition form factor of $η_{c, b} (n S)$ and $χ_{c 0, b 0} (n P)$ via relativized mock meson states

Abstract

Physics Subject Headings (PhySH)

Article Text

References (25)

Outline

Information

Filter

Filter

Article Lookup

Enter a citation

Two-photon transition form factor of ηc,b(nS) and χc0,b0(nP) via relativized mock meson states

Abstract

Physics Subject Headings (PhySH)

Article Text

References (25)

Two-photon transition form factor of $η_{c, b} (n S)$ and $χ_{c 0, b 0} (n P)$ via relativized mock meson states