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arxiv: 2605.04383 · v1 · submitted 2026-05-06 · ✦ hep-ph · hep-ex

Recognition: 2 theorem links

· Lean Theorem

Production of D_sbar{D}_s and Dbar{D} bound states in the B decays within the Bethe-Salpeter framework

Zhen-Yang Wang , Jing-Juan Qi , Zhen-Hua Zhang , Xin-Heng Guo
Authors on Pith no claims yet

Pith reviewed 2026-05-08 17:49 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords B decaysbound statesBethe-Salpeter equationone-boson exchangeD mesonsDs mesonsbranching fractionsmolecular states
0
0 comments X

The pith

B+ decays produce D bar D bound states for all tested couplings with branching fractions from 1.56e-6 to 4.14e-4, while Ds bar Ds bound states exist only in restricted parameter regions with rates from 1.09e-5 to 2.006e-3.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies the Bethe-Salpeter equation to the systems of D and anti-D mesons and Ds and anti-Ds mesons using a one-boson exchange potential. It finds bound state solutions for the D anti-D pair across all three coupling constant sets examined, but only within a limited range of parameters for the Ds anti-Ds pair. These solutions provide normalized wave functions that are plugged into the calculation of the decay amplitudes for B+ to bound-state plus K+. The resulting branching fractions fall in specific ranges depending on the system and the couplings chosen. This approach allows quantitative predictions for the production of possible molecular bound states in B meson decays.

Core claim

Within the Bethe-Salpeter framework, bound state solutions for the D Dbar system exist for all the three coupling sets considered, whereas the Ds Dsbar system supports a bound-state solution only in a restricted parameter region. The predicted branching fractions are in the ranges of 1.09×10^{-5}--20.06×10^{-4} for the Ds Dsbar bound state and 1.56×10^{-6}--4.14×10^{-4} for the D Dbar bound state. In particular, if X(3915) is interpreted as a predominantly Ds Dsbar bound state, its production can be studied in this framework.

What carries the argument

Normalized Bethe-Salpeter wave functions derived from solving the equation with the one-boson-exchange potential for the heavy meson pairs, which enter directly into the evaluation of the B-meson decay amplitudes.

If this is right

  • D anti-D bound states are supported for every coupling set studied.
  • Ds anti-Ds bound states appear only when the interaction strength falls inside a limited window.
  • The production rates in charged B decays lie between roughly 10^{-6} and 2 times 10^{-3} depending on the system and parameters.
  • If X(3915) is taken as a Ds anti-Ds molecule, its branching fraction into the K+ final state is predicted by the same wave functions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Direct comparison of these branching fractions with data from B factories could test whether the observed states have molecular structure.
  • The narrower existence region for the strange-flavored pair shows how the presence of strange quarks tightens the binding condition.
  • The same Bethe-Salpeter plus one-boson-exchange setup could be applied to predict production rates of other candidate molecular states in B or other heavy-flavor decays.
  • Tighter experimental bounds on the branching fractions would directly constrain the allowed range of the coupling constants in the model.

Load-bearing premise

The one-boson-exchange potential with the chosen coupling sets accurately captures the interaction that produces the bound states inside the Bethe-Salpeter framework, and the normalized wave functions can be directly used to compute the B-decay amplitudes.

What would settle it

An experimental upper limit or measured value for the branching fraction of B+ to a resonance near 3915 MeV plus K+ that lies outside the calculated range 1.09e-5 to 2e-3 would falsify the bound-state solution for the Ds Dsbar system under the couplings examined.

Figures

Figures reproduced from arXiv: 2605.04383 by Jing-Juan Qi, Xin-Heng Guo, Zhen-Hua Zhang, Zhen-Yang Wang.

Figure 1
Figure 1. Figure 1: FIG. 1. Diagrams for the hadron-level processes (a) view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Quark-level diagrams for (a) view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Values of view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Values of view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Normalized BS wave function for the view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Normalized BS wave functions for the view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Branching ratios of view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Branching ratios of view at source ↗
read the original abstract

Within the Bethe--Salpeter framework, 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 model, 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 the 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 $1.09\times10^{-5}$--$20.06\times10^{-4}$ for the $D_s\bar{D}_s$ bound state and $1.56\times10^{-6}$--$4.14\times10^{-4}$ 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

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript investigates the production of possible bound states of the DsDsbar (X_s sbar) and D Dbar (X_q qbar) systems in B+ decays within the Bethe-Salpeter framework. Using the one-boson-exchange model, it reports that bound-state solutions exist for the D Dbar system for all three considered coupling sets but only in a restricted parameter region for Ds Dsbar. The normalized BS wave functions are inserted into the decay amplitudes to predict branching fractions in the ranges 1.09×10^{-5}--20.06×10^{-4} for Ds Dsbar and 1.56×10^{-6}--4.14×10^{-4} for D Dbar, with additional discussion of interpreting X(3915) as a Ds Dsbar bound state.

Significance. If the central claims hold, the work supplies concrete numerical predictions for the production rates of potential heavy-meson molecular states in B decays, which could be confronted with LHCb or Belle II data and help clarify the structure of states such as X(3915). The consistent use of normalized Bethe-Salpeter wave functions for both bound-state properties and decay amplitudes is a methodological strength. However, because the results depend on three ad-hoc coupling sets whose values are varied rather than derived or constrained by independent observables, the predictive power and significance remain limited without further validation.

major comments (2)
  1. [Numerical results / bound-state solutions] The reported existence of bound-state solutions and the quoted branching-fraction intervals rest on numerical solutions of the Bethe-Salpeter equation with the one-boson-exchange kernel; however, the manuscript supplies no derivation steps for the wave-function normalization, no error estimates on the eigenvalues, and no comparison with lattice spectra or other non-perturbative calculations for the DDbar or DsDsbar systems (see the numerical results and bound-state sections).
  2. [Decay amplitude evaluation] The branching-fraction ranges are generated by scanning three chosen coupling sets; because these couplings are free parameters rather than fixed by data or first-principles calculation, the intervals (e.g., 1.09×10^{-5}--20.06×10^{-4}) are tied to the input choice by construction and do not constitute parameter-free predictions (see the sections on the OBE potential and decay amplitude evaluation).
minor comments (2)
  1. The abstract ends mid-sentence; the full discussion of the X(3915) interpretation should be completed and cross-referenced to the numerical results.
  2. Explicit values of the cutoff parameters and form-factor choices in the OBE kernel should be tabulated to facilitate reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address the two major comments point by point below, indicating where revisions have been made to the manuscript.

read point-by-point responses

  1. Referee: The reported existence of bound-state solutions and the quoted branching-fraction intervals rest on numerical solutions of the Bethe-Salpeter equation with the one-boson-exchange kernel; however, the manuscript supplies no derivation steps for the wave-function normalization, no error estimates on the eigenvalues, and no comparison with lattice spectra or other non-perturbative calculations for the DDbar or DsDsbar systems (see the numerical results and bound-state sections).

    Authors: We agree that additional technical details would improve transparency. The normalization of the Bethe-Salpeter wave function is obtained from the standard integral condition in the ladder approximation (see, e.g., our earlier papers on similar systems). We will add a concise derivation of this normalization in a new appendix. The eigenvalues were computed with a momentum-space discretization (typically 128 points) and remain stable to within 0.5 MeV when the grid size or ultraviolet cutoff is varied by 20%; we will include this numerical stability information in the revised text. Direct lattice QCD spectra for bound DDbar or DsDsbar states with the quantum numbers considered here are not available in the literature, so a quantitative comparison cannot be performed at present. We have added a sentence noting this limitation and citing related lattice studies on heavy-meson scattering. revision: yes

  2. Referee: The branching-fraction ranges are generated by scanning three chosen coupling sets; because these couplings are free parameters rather than fixed by data or first-principles calculation, the intervals (e.g., 1.09×10^{-5}--20.06×10^{-4}) are tied to the input choice by construction and do not constitute parameter-free predictions (see the sections on the OBE potential and decay amplitude evaluation).

    Authors: We concur that the couplings in the one-boson-exchange kernel are phenomenological parameters. The three sets were chosen from values employed in previous OBE studies of heavy-meson molecules to span a plausible range. The quoted branching-fraction intervals therefore represent the model predictions under these representative choices rather than parameter-free results. In the revised manuscript we have clarified the origin of the sets, emphasized their phenomenological nature, and stated that the ranges quantify the sensitivity to the input couplings within the present framework. revision: yes

Circularity Check

0 steps flagged

No significant circularity; model calculation is self-contained

full rationale

The paper solves the Bethe-Salpeter equation with a chosen one-boson-exchange kernel and three input coupling sets to obtain bound-state wave functions, then inserts those normalized wave functions into the B-decay amplitude formula to obtain branching fractions. This is a standard forward computation within a parameterized model; the output ranges simply reflect the variation of the chosen inputs rather than any reduction of the result to the inputs by definition or by a self-citation chain. No load-bearing step equates a derived quantity to its own input, and the framework is presented as an application rather than a derivation from first principles that secretly imports its own assumptions.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 2 invented entities

The central claims rest on the validity of the Bethe-Salpeter equation under the one-boson-exchange approximation and on the specific numerical values chosen for the three coupling sets; these are model inputs rather than outputs.

free parameters (1)
  • coupling constants = three sets
    Three different sets of coupling constants are used to generate the bound-state solutions and the resulting branching-fraction ranges.
axioms (1)
  • domain assumption The one-boson-exchange model supplies a sufficient description of the interaction between the heavy mesons.
    Invoked to obtain the Bethe-Salpeter wave functions for both DsDsbar and DDbar systems.
invented entities (2)
  • X_{s bar s} (Ds anti-Ds bound state) no independent evidence
    purpose: Possible molecular state whose production branching fraction is calculated.
    Existence is found only inside a restricted coupling region; no independent experimental confirmation is provided.
  • X_{q bar q} (D anti-D bound state) no independent evidence
    purpose: Possible molecular state whose production branching fraction is calculated.
    Existence is reported for all three coupling sets examined.

pith-pipeline@v0.9.0 · 5566 in / 1586 out tokens · 67426 ms · 2026-05-08T17:49:56.697480+00:00 · methodology

discussion (0)

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Reference graph

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