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Production of $D_{s} {\bar{D}}_{s}$ and $D \bar{D}$ bound states in the $B$ decays within the Bethe-Salpeter framework

Zhen-Yang Wang ^1,*, Jing-Juan Qi^2,1,†, Zhen-Hua Zhang^3,‡, and Xin-Heng Guo ^4,§

¹Physics Department, Ningbo University, Zhejiang 315211, China
²College of Information and Intelligence Engineering, Zhejiang Wanli University, Zhejiang 315101, China
³School of Nuclear Science and Technology, University of South China, Hengyang, 421001, Hunan, China
⁴School of Physics Science and Technology, Kunming University, Kunming 650214, China

^*Contact author: wangzhenyang@nbu.edu.cn
^†Contact author: qijj@mail.bnu.edu.cn
^‡Contact author: zhangzh@usc.edu.cn
^§Contact author: xhguo@bnu.edu.cn

Phys. Rev. D 114, 034002 – Published 3 August, 2026

DOI: https://doi.org/10.1103/qnmq-cqvc

Abstract

Within a Bethe-Salpeter model, we investigate the production of possible $D_{s} {\bar{D}}_{s}$ $(X_{s \bar{s}})$ and $D \bar{D}$ $(X_{q \bar{q}})$ bound states in $B$ decays. The bound state properties of the two heavy meson systems are studied in the one-boson-exchange kernel, and the resulting normalized Bethe-Salpeter wave functions are used to calculate the branching fractions of $B^{+} \to X_{s \bar{s}} K^{+}$ and $B^{+} \to X_{q \bar{q}} K^{+}$ . We find that bound-state solutions for the $D \bar{D}$ system exist for all three coupling sets considered, whereas the $D_{s} {\bar{D}}_{s}$ system supports a bound state solution only in a restricted parameter region. The predicted branching fractions are in the ranges of $1.09 \times 10^{- 5} - 20.06 \times 10^{- 4}$ for the $D_{s} {\bar{D}}_{s}$ bound state and $1.56 \times 10^{- 6} - 4.14 \times 10^{- 4}$ for the $D \bar{D}$ bound state. In particular, if $X (3915)$ is interpreted as a predominantly $D_{s} {\bar{D}}_{s}$ bound state, its production branching fraction in $B$ decay is found to be $5.79 \times 10^{- 4}$ .

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Phys. Rev. D 114, 034002– Published 3 August, 2026

Vol. 114, Iss. 3 — 1 August 2026

Received 5 May 2026
Accepted 14 July 2026

DOI: https://doi.org/10.1103/qnmq-cqvc

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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Production of $D_{s} {\bar{D}}_{s}$ and $D \bar{D}$ bound states in the $B$ decays within the Bethe-Salpeter framework

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Production of DsD¯s and DD¯ bound states in the B decays within the Bethe-Salpeter framework

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Production of $D_{s} {\bar{D}}_{s}$ and $D \bar{D}$ bound states in the $B$ decays within the Bethe-Salpeter framework