$B_{sJ}(6114)$ as a Higher-Partial-Wave $\bar{K}^{*}B^{(*)}$ Molecular Candidate

Dan Jiang; Shiquan Zhang; Yin Huang

REVIEW

$B_{sJ}(6114)$ as a Higher-Partial-Wave $\bar{K}^{}B^{()}$ Molecular Candidate

Shiquan Zhang, Dan Jiang, Yin Huang

High Energy Physics - Phenomenology · Submitted 21 Oct 2026 · 55 references

Pith's verdict: The width-based spin-parity ranking is a sensible new step, but the missing cutoff values behind the E=-57 MeV wave functions make the headline preference for J^P=1^- unreproducible as written.

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T1 Audit Automatic checks that the references and files the paper relies on exist and match.
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arXiv ID2610.29354 (version 1)

Pith IDZGDHAFU7, Pith's permanent record of this paper and its review

CitationDan Jiang, Shiquan Zhang, Yin Huang. "$B_{sJ}(6114)$ as a Higher-Partial-Wave $\bar{K}^{*}B^{(*)}$ Molecular Candidate." arXiv:2610.29354 (2026). https://pith.science/paper/ZGDHAFU7

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Claim paper

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Editor's note

read the letter

Quick take: this is standard one-boson-exchange machinery applied systematically to BsJ(6114), and the genuinely new piece is using computed strong decay widths to rank the spin-parity candidates. If the width calculation holds up, J^P=1^- becomes a concrete experimental target. But the key numbers needed to check that ranking are not in the paper, so I read the result as conditional, not established.

Credit where it is due: the partial-wave matrix elements and the coupled-channel setup are laid out in detail, the Gaussian expansion solution of the Schrödinger equation is standard, and the authors are upfront that the mass spectrum alone cannot fix the spin-parity (b31) and that alpha is a phenomenological parameter (b55). The literature survey of quark-model assignments is fair, and the width-based discrimination does go beyond the earlier S-wave molecular mass interpretation in Ref. [37].

The soft spots, in order of importance. First, the per-J^P alpha values used for the width calculation are never reported. Table V tabulates shallow binding energies of a few MeV, but Fig. 4 is built at E=-57 MeV, and Fig. 5 reads widths off at that point. No passage states which alpha produces those wave functions for each J^P. Since alpha changes the wave-function normalization entering Eq. (26), the width ranking is a comparison of separately tuned models as printed. This is fixable by adding a table like Table V for the alpha values at E=-57, but without it the central claim cannot be reproduced.

Second, the widths carry no uncertainty, while the experimental band is 66±18±21 MeV. “Best agreement” is a point comparison with no theory error attached. This is minor-to-moderate: it matters mostly because the paper is claiming discrimination among several channels.

Third, as printed, the decay amplitudes in Eqs. (29)–(35) contain no monopole form factors, while the potentials in Eqs. (11)–(17) are dressed by F^2(q^2). If this is literal, the decay vertices are inconsistent with the bound-state vertices and the absolute widths shift channel-dependently. It may be a notation choice, but it needs a sentence either way.

I do not see a load-bearing mathematical error, and the molecular interpretation plus the J^P=1^- preference remain plausible. The paper deserves a serious referee, not a desk rejection, and I would bring it to a hadron-spectroscopy reading group as a useful case of a plausible claim whose decisive input is underreported. My recommendation: send it to review and ask for the alpha values behind Fig. 4, an uncertainty estimate on the widths, and a consistency check on the decay form factors before the ranking is advertised.

Referee report

Significance

If correct, this would be the first systematic molecular interpretation of BsJ(6114), complementing (and, for the favored assignments, competing with) the conventional 1D bbar-s quark-model interpretations in the literature. The technical apparatus — the explicit partial-wave matrix elements of Tables II-IV, the coupled-channel treatment that includes higher partial waves beyond the earlier hidden-gauge S-wave analysis (Ref. [37]), and the hadronic-loop decay formalism — is a useful resource, and the central claim is in principle falsifiable: the width ranking is a definite observable-level prediction once the input parameters are fixed. The paper states plainly that the mass spectrum alone cannot determine J^P (sentence b31) and that alpha is a free parameter fixed phenomenologically (sentence b55). The decisive weakness is that the width calculation at the physical mass rests on wave functions whose generating cutoff values are never reported, so the J^P ranking is not reproducible as submitted; this is a reporting gap that can be fixed in revision rather than an evident internal inconsistency.

Major objections (2)

[§III, Figs. 4-5 and Table V] The paper's central claim — that the decay-width analysis selects J^P = 1- among the compatible assignments — is evaluated at the physical mass 6114 MeV, corresponding to binding energy E = -57 MeV relative to the B Kbar* threshold, as stated in the Fig. 4 caption. However, the manuscript never reports the values of the cutoff parameter alpha (Eq. (10)) that produce the E = -57 MeV wave functions for any J^P. Table V tabulates coupled-channel bound states only down to |E| ≈ 7 MeV (J^P = 1-: E = -0.18, -0.85, -2.03 MeV at alpha = 0.8, 0.9, 1.0; J^P = 2-: -6.96 MeV at alpha = 1.3; J^P = 3+: -6.84 MeV at alpha = 2.2), with no E = -57 MeV row, and no passage states which value of alpha generates the Fig. 4 wave functions for each J^P or how the curves of Fig. 5 are produced. Since alpha controls the binding energy, the wave-function normalization entering Eq. (26), and the channel composition (the B Kbar* fraction of the coupled 1- state drops from 49.9% at alpha = 0.8 to 40.3% at alpha = 1.0 in Table V), the classification of assignments as 'disfavored' versus 'compatible' (abstract; §IV) is a comparison of differently tuned models, and the preference for J^P = 1- is not reproducible as reported. The authors acknowledge that alpha is a free parameter that cannot be determined from first principles (sentence b55), but the specific values used in the decisive calculation are absent; a table reporting per-J^P the alpha, binding energy, channel composition, and width at 6114 MeV is required. Extrapolating the near-threshold slopes of Table V suggests that alpha values of order 2-3 or larger are needed to reach E = -57 MeV, i.e., cutoffs well above the tabulated regime, where the OBE model's validity should also be discussed.
[§III, Fig. 5 and Eq. (25)] The width predictions are compared with the experimental value as single unrounded points, and no theoretical uncertainty is attached to any predicted width. The measured constraint is broad — Gamma = 66 ± 18 ± 21 MeV, i.e., roughly 27-105 MeV at 1σ — and the model depends on the OBE couplings of Eqs. (3)-(6), the monopole form-factor shape, and the tuned values of alpha, each of which carries substantial uncertainty. Terms such as 'disfavored', 'slightly exceed the upper limit', and 'best agreement' (Sec. III) are therefore not yet supported as robust statements. The authors should provide a sensitivity estimate — for example, widths evaluated at two or three neighboring alpha values for each J^P, or under alternative coupling choices — and use it to demarcate the compatible/disfavored split. Without such an estimate, the discrimination power of the width analysis cannot be assessed, and the abstract's 'compatible versus disfavored' claim is a single-point comparison against a wide experimental band.

Minor objections (6)

[§III] The text refers to the bound-state table as 'Table VI' in two places ('In Table VI, we present the properties...' and '...accounting for nearly 100% of its composition (see Table VI)'), but the table is labelled 'TABLE V'; the numbering should be made consistent.
[§II] The decay diagrams are said to be 'illustrated in Fig. 4' in the paragraph following Eq. (28), but the two-body decay diagrams appear in Fig. 2; the cross-reference should be corrected.
[§III and Fig. 5 caption] The sentence 'As the molecular-state mass approaches that of BsJ(6114), the B Kbar component exceeds 89% in all cases' is ambiguous: the molecular wave function is expanded in B Kbar* and B* Kbar* channels (Eq. (26)), so 'B Kbar component' presumably refers to the share of the total decay width carried by the B Kbar final state, and the caption phrase 'molecular-state masses are computed relative to the threshold of the dominant B Kbar channel' is also unclear since the binding energies in Table V are quoted relative to the B(*)Kbar* thresholds; please clarify both.
[Fig. 5] The horizontal-axis tick labels of Fig. 5 appear misprinted (they read '1086, 1126, 1166, 1200' rather than values near 6100-6200 MeV); please check the axis labeling, and state in the caption whether the cyan band represents the 1σ combined experimental uncertainty.
[§III] Minor typos: 'the interaction potentials are the sam' should read 'the same' (end of the 2+/3+ B Kbar* discussion), and the Fig. 5 caption 'obtained form Ref. [1]' should read 'from'.
[References and §II] Reference [36] carries the DOI '10.1103/m2xt-pkxm', which appears malformed and should be verified, and the sentence fragment 'MP (V )i1i2→f1f2 corresponds to the amplitude of scattering of the process by exchanging the hadron P (V )' following Eq. (35) should be rewritten as a complete sentence.

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The paper

The paper claims

The B_sJ(6114) meson, the heaviest bottom-strange state known, is best explained as a higher-partial-wave $B^{(*)}\bar{K}^*$ molecular state with $J^P = 1^-$, the assignment that best matches its measured decay width.

The reading

The paper asks what B_sJ(6114) — the heaviest bottom-strange meson known, first seen in 2021 through its $B^\pm K^\mp$ decay — is made of, and argues that it can be read as a loosely bound $B^{(*)}\bar{K}^*$ molecule. Solving a coupled-channel Schrödinger equation with one-boson-exchange potentials for all eight spin-parity channels ($0^\pm$, $1^\pm$, $2^\pm$, $3^\pm$), the authors find that the mass alone does not discriminate: every assignment can be made to bind near 6114 MeV with a suitable form-factor cutoff. The strong decay width into the observed $B K$ channel and the unobserved $B^* K$ channel is what breaks the degeneracy. Five assignments ($0^+$, $0^-$, $1^+$, $2^+$, $3^-$) predict widths outside the measured $66 \pm 18 \pm 21$ MeV, while $J^P = 1^-$, $2^-$ and $3^+$ survive, and $1^-$ — a P-wave-dominated configuration — comes closest to the central value. If the picture is right, B_sJ(6114) would be the first bottom-strange hadron built from a meson molecule in a higher partial wave rather than from a quark–antiquark pair.

What carries the argument

The working machinery is the one-boson-exchange potential built from $\pi$, $\eta$, $\rho$ and $\omega$ exchange among $B^{(*)}$ and $\bar{K}^{(*)}$ mesons, with vertices from heavy-quark- and chiral-symmetric effective Lagrangians and a monopole form factor whose cutoff is parameterized as $\Lambda_i = m_i + \alpha\Lambda_{\rm QCD}$ with $\Lambda_{\rm QCD} = 220$ MeV and $\alpha$ scanned over 0.5–9.7. That potential, expressed through spin-orbit wave functions that mix partial waves and couple the $B\bar{K}^*$ and $B^*\bar{K}^*$ channels, is inserted into the nonrelativistic Schrödinger equation and solved with the Gaussian expansion method to produce binding energies and wave functions for each $J^P$. The second piece is the molecular decay framework: the amplitude for $B_{sJ} \to B^{(*)} K$ is the overlap of these wave functions with one-boson-exchange transition amplitudes, so the same solution that produces the mass also produces the width. The cutoff relation does the decisive work, because it lets every candidate be bound at the experimental mass, which is why the width, not the spectrum, ends up carrying the physical information.

Extended reading notes

Core claim

The paper's central claim is that B_sJ(6114) is best understood as a $B^{(*)}\bar{K}^*$ hadronic molecule whose spin-parity is fixed not by its mass but by its decay width. Within the one-boson-exchange model, the authors show that the interaction among $B$, $B^*$, $\bar{K}$ and $\bar{K}^*$ supports bound states for every $J^P$ from $0^\pm$ to $3^\pm$, once the monopole form-factor cutoff $\Lambda = m + \alpha\Lambda_{\rm QCD}$ is chosen large enough, so the spectrum alone cannot identify the state. Computing the strong decay widths into $B\bar{K}$ and $B^*\bar{K}$ from the bound-state wave functions, they find that $J^P = 0^\pm$ and $1^+$ greatly overproduce the measured width, $2^+$ and $3^-$ exceed the experimental band, while $1^-$, $2^-$ and $3^+$ remain compatible — with $1^-$ giving the best agreement. The conclusion is that B_sJ(6114) is plausibly a higher-partial-wave $B^{(*)}\bar{K}^*$ molecular state, dominated in the preferred $1^-$ case by its P-wave component.

Load-bearing premise

The ranking of the eight spin-parity candidates depends on choosing the cutoff strength $\alpha$ that makes each candidate bind at the experimental mass, and the paper does not report which $\alpha$ values produce the wave functions used for the widths; if different assignments needed different hand-picked cutoffs, the width comparison compares differently tuned models rather than making a parameter-free prediction.

Editorial extensions

If this is right

If the $1^-$ assignment survives direct measurement, B_sJ(6114) becomes the first established bottom-strange hadronic molecule, with its structure dominated by a P-wave component rather than a quark–antiquark core.
The $2^-$ and $3^+$ assignments remain viable under the current width constraint, so they must be treated as live alternatives until spin-parity is measured directly.
The unobserved $B_{sJ}(6114) \to B^* K$ decay has a calculable width in the molecular picture for each surviving assignment, giving experiments a second observable to discriminate among $1^-$, $2^-$ and $3^+$.
Because the same interaction binds all eight channels near the experimental mass, the analysis implies that mass spectroscopy alone cannot certify a molecular state; decay observables are the discriminating information.
S-wave-dominated molecular configurations ($0^+$, $1^+$) are disfavored by the width analysis, which is the specific sense in which the favored picture is a higher-partial-wave molecule.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The width ranking is only as trustworthy as the cutoff tuning: the authors state that $\alpha$ cannot be fixed from first principles, and the paper never reports the specific $\alpha$ values behind the wave functions used for the decay widths, so a reader cannot tell whether each $J^P$ candidate needed a different hand-picked cutoff; if it did, the comparison is between differently tuned models.
A lattice QCD calculation of the coupled $B\bar{K}^*$/$B^*\bar{K}^*$ scattering amplitude near 6114 MeV would test whether the one-boson-exchange bound state is a genuine near-threshold pole or an artifact of the regulated potential; the paper does not attempt such a test.
If the $1^-$ molecule is confirmed, heavy-quark spin symmetry suggests that other $B^{(*)}\bar{K}^*$ spin combinations should form near-threshold partners, and searching the bottom-strange spectrum for those partners would be a natural extension the paper leaves implicit.
The result would sharpen the contrast between the charm-strange sector, where molecular candidates like $D_{s0}^*(2317)$ sit in S waves, and the bottom-strange sector, where the leading molecular candidate would live in a P wave — a pattern that could guide searches for other hidden heavy-flavor molecules.

Original abstract

In 2021, the LHCb Collaboration observed the $B_{sJ}(6114)$ state in the $B^\pm K^\mp$ invariant mass spectrum. It is currently the highest-mass bottom-strange state listed by the Particle Data Group (PDG). However, its spin-parity quantum numbers have not yet been experimentally determined, and its internal structure remains under debate. Although several quark-model interpretations have been proposed, including the $1^3D_1$ state and a member of the $1D(2^-,3^-)$ doublet, the possibility of a hadronic molecular structure for this state has not yet been systematically investigated. We investigate the possibility of interpreting $B_{sJ}(6114)$ as a $B^{(*)}\bar{K}^{*}$ molecular state within the one-boson-exchange model. By solving the Schr\"odinger equation using the Gaussian expansion method, we study the bound-state properties with different spin-parity assignments, taking into account partial-wave mixing and coupled-channel effects. We further calculate the strong decay widths into the allowed $B^{(*)}\bar{K}$ channels and compare them with the experimental data. Our results show that the mass spectrum alone cannot uniquely determine the spin-parity of $B_{sJ}(6114)$. However, the decay-width analysis disfavors the $J^P=0^+$, $0^-$, $1^+$, $2^+$, and $3^-$ assignments, while the $J^P=1^-$, $2^-$, and $3^+$ configurations remain compatible with the experimental constraints. Among these possibilities, the $J^P=1^-$ assignment provides the best agreement with the measured decay width. These findings suggest that $B_{sJ}(6114)$ may be a higher-partial-wave $B^{(*)}\bar{K}^{*}$ molecular state.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-22: 1 request, 8,543 input tokens.

What the abstract itself states, as probabilities: claims first 0.41 · states limitation 0.85 · human participants 0.01 · controlled comparison 0.05 · randomized assignment 0.01 · states quantitative result 0.38 · compares with prior results 0.66 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

In 2021, the LHCb Collaboration observed the $B_{sJ}(6114)$ state in the $B^\pm K^\mp$ invariant mass spectrum. It is currently the highest-mass bottom-strange state listed by the Particle Data Group (PDG). However, its spin-parity quantum numbers have not yet been experimentally determined, and its internal structure remains under debate. Although several quark-model interpretations have been proposed, including the $1^3D_1$ state and a member of the $1D(2^-,3^-)$ doublet, the possibility of a hadronic molecular structure for this state has not yet been systematically investigated. We investigate the possibility of interpreting $B_{sJ}(6114)$ as a $B^{(*)}\bar{K}^{*}$ molecular state within the one-boson-exchange model. By solving the Schr\"odinger equation using the Gaussian expansion method, we study the bound-state properties with different spin-parity assignments, taking into account partial-wave mixing and coupled-channel effects. We further calculate the strong decay widths into the allowed $B^{(*)}\bar{K}$ channels and compare them with the experimental data. Our results show that the mass spectrum alone cannot uniquely determine the spin-parity of $B_{sJ}(6114)$. However, the decay-width analysis disfavors the $J^P=0^+$, $0^-$, $1^+$, $2^+$, and $3^-$ assignments, while the $J^P=1^-$, $2^-$, and $3^+$ configurations remain compatible with the experimental constraints. Among these possibilities, the $J^P=1^-$ assignment provides the best agreement with the measured decay width. These findings suggest that $B_{sJ}(6114)$ may be a higher-partial-wave $B^{(*)}\bar{K}^{*}$ molecular state.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	In 2021, the LHCb Collaboration observed the $B_{sJ}(6114)$ state in the $B^\pm K^\mp$ invariant mass spectrum.	background 1.00	0.04	0.01	asserted 1.00
s1	It is currently the highest-mass bottom-strange state listed by the Particle Data Group (PDG).	background 1.00	0.04	0.02	asserted 1.00
s2	However, its spin-parity quantum numbers have not yet been experimentally determined, and its internal structure remains under debate.	gap 0.84	0.04	0.02	asserted 0.99
s3	Although several quark-model interpretations have been proposed, including the $1^3D_1$ state and a member of the $1D(2^-,3^-)$ doublet, the possibility of a hadronic molecular structure for this state has not yet been systematically investigated.	gap 0.98	0.04	0.02	asserted 0.69
s4	We investigate the possibility of interpreting $B_{sJ}(6114)$ as a $B^{()}\bar{K}^{}$ molecular state within the one-boson-exchange model.	contribution 0.96	0.85	0.04	hedged possibility 0.79
s5	By solving the Schr\"odinger equation using the Gaussian expansion method, we study the bound-state properties with different spin-parity assignments, taking into account partial-wave mixing and coupled-channel effects.	method 0.99	0.96	0.03	asserted 0.95
s6	We further calculate the strong decay widths into the allowed $B^{(*)}\bar{K}$ channels and compare them with the experimental data.	method 0.79	0.95	0.03	asserted 0.97
s7	Our results show that the mass spectrum alone cannot uniquely determine the spin-parity of $B_{sJ}(6114)$.	result 0.99	0.94	0.13	asserted 1.00
s8	However, the decay-width analysis disfavors the $J^P=0^+$, $0^-$, $1^+$, $2^+$, and $3^-$ assignments, while the $J^P=1^-$, $2^-$, and $3^+$ configurations remain compatible with the experimental constraints.	result 1.00	0.94	0.22	asserted 0.89
s9	Among these possibilities, the $J^P=1^-$ assignment provides the best agreement with the measured decay width.	result 0.97	0.95	0.39	asserted 0.99
s10	These findings suggest that $B_{sJ}(6114)$ may be a higher-partial-wave $B^{()}\bar{K}^{}$ molecular state.	interpretation 1.00	0.84	0.90	hedged possibility 1.00

What kind of paper the classifier reports this is: theoretical result 0.54 · evidence from simulation or computation 0.99.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1: Feynman diagrams for the processes

**Figure 2.** FIG. 2: Two-body strong decay diagrams for the

**Figure 3.** FIG. 3: The interaction potentials for the

**Figure 4.** FIG. 4: The radial wave functions for the

**Figure 5.** , where the molecular-state masses are computed relative to the threshold of the dominant BK¯ channel. As the molecular-state mass approaches that of BsJ (6114), the BK¯ component exceeds 89% in all cases. As shown in the figure, the predicted total decay width for the J P = 1+ assignment is much larger than the experimental value, disfavoring the interpretation of BsJ (6114) as a K¯ ∗B–K¯ ∗B∗ molecular state with J P = 1+. This assignment corresponds to an S-wave-dominated molecular state. Similarly, the J P = 0+ and J P = 0− assignments

Figures reproduced from arXiv:2610.29354 by the authors.

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