Unambiguous spin–parity and structure assignment for the strange hidden-charm pentaquark candidates $P_{cs}(4338)$ and $P_{cs}(4459)$

Two narrow structures in the $J/\psi\Lambda$ invariant-mass spectrum — the $P_{\psi s}^{\Lambda}(4338)^0$, observed by LHCb in $B^-\to J/\psi\Lambda\bar{p}$ decays, and the $P_{cs}(4459)^0$, first seen by LHCb in $\Xi_b^-\to J/\psi\Lambda K^-$ decays and independently supported at $3.3\sigma$ by Belle/Belle II in $\Upsilon(1S,2S)$ inclusive decays — are candidates for the first pentaquarks carrying both hidden charm and strangeness (quark content $udsc\bar{c}$). The spin of the 4338 state was measured to be $1/2$ with negative parity preferred, but the spin–parity of the 4459 state is undetermined, and the internal structure of both states is unsettled: compact diquark–diquark–antiquark pentaquarks, meson–baryon molecules such as $\bar{D}\Xi_c$ and $\bar{D}^{*}\Xi_c$, and coupled-channel threshold effects (bound, virtual, or triangle-singularity origins) all yield nearly degenerate mass predictions. The open problem, stated as such by the source work (Wang and Xin, 2026), is to make an unambiguous assignment of the spin–parity and internal structure of both candidates, requiring new amplitude analyses and theoretical frameworks with controlled systematics that discriminate among the competing interpretations; isospin-partner tests would shed further light on the assumed $I=0$ assignments.

Background

In quantum chromodynamics (QCD), ordinary hadrons are either mesons (a quark–antiquark pair) or baryons (three quarks); a pentaquark is a hypothetical five-quark hadron that would extend this classification. In 2015 the LHCb collaboration observed narrow resonances in the $J/\psi p$ system ($J/\psi$ is the ground-state charmonium $c\bar{c}$) in $\Lambda_b^0\to J/\psi K^-p$ decays, establishing the hidden-charm pentaquark candidates $P_c$ (Aaij et al., Phys. Rev. Lett. 115, 072001 (2015)). Replacing one light quark by a strange quark gives strangeness $S=-1$ candidates $P_{cs}$ with quark content $udsc\bar{c}$, which would appear as structures in the $J/\psi\Lambda$ invariant-mass spectrum, where $\Lambda$ is the lightest strange baryon.

Two such structures are known. In 2021 LHCb reported evidence, at $3.1\sigma$ including systematic uncertainties and the look-elsewhere effect, for a structure called $P_{cs}(4459)^0$ in an amplitude analysis of $\Xi_b^-\to J/\psi\Lambda K^-$ decays, with mass $4458.8\pm2.9^{+4.7}_{-1.1}$ MeV and width $17.3\pm6.5^{+8.0}_{-5.7}$ MeV; its spin–parity $J^P$ was not determined, and the data were also consistent with two overlapping resonances (Aaij et al., Sci. Bull. 66, 1278 (2021)). In 2023 LHCb observed a narrower state, $P_{\psi s}^{\Lambda}(4338)^0$, in $B^-\to J/\psi\Lambda\bar{p}$ decays with mass $4338.2\pm0.7\pm0.4$ MeV and width $7.0\pm1.2\pm1.3$ MeV; its spin was determined to be $1/2$ and negative parity was preferred, but not established beyond preference (Aaij et al., Phys. Rev. Lett. 131, 031901 (2023)). In 2025, Belle and Belle II found $3.3\sigma$ evidence for the $P_{cs}(4459)^0$ in $\Upsilon(1S,2S)$ inclusive decays, with mass $4471.7\pm4.8\pm0.6$ MeV, again without quantum numbers (Adachi et al., Phys. Rev. Lett. 135, 041901 (2025)). Because both states decay to $J/\psi\Lambda$, strong decays (which conserve isospin to good approximation) imply the isospin assignment $I=0$; this assignment is untested, since no charged or isovector partner states decaying to $J/\psi\Sigma$ channels have been observed.

Three classes of interpretation compete. (a) Compact pentaquarks: bound states of two diquarks (correlated quark pairs) and an antiquark, here $[ud][sc]\bar{c}$ or $[us][dc]\bar{c}$-type configurations. The source work (Wang and Xin, Eur. Phys. J. C 86, 472 (2026)) studies these with QCD sum rules — a method that relates a hadron's properties to vacuum condensates by matching an interpolating current's correlator, expanded via the operator product expansion (OPE), to a hadronic spectral representation, optimized through a Borel transform and a continuum-threshold parameter $s_0$. Separating parity channels and isospin, and including condensates up to dimension 13, they obtain an $I=0$, $J^P=1/2^-,3/2^-,5/2^-$ mass spectrum and assign $P_{cs}(4338)$ to a $J^P=1/2^-$ compact state and $P_{cs}(4459)$ to either a $1/2^-$ or a $3/2^-$ compact state. (b) Hadronic molecules: loosely bound meson–baryon systems, here an anti-charmed meson ($\bar{D}$, its vector partner $\bar{D}^{*}$, or the strange $\bar{D}_s$) bound to a charmed baryon ($\Xi_c$, which carries strangeness, or the non-strange $\Lambda_c$); the 4338 peak sits essentially on the $\bar{D}\Xi_c$ threshold and the 4459 peak near the $\bar{D}^{*}\Xi_c$ threshold, so both can be read as near-threshold molecular bound states (e.g. Wang and Liu, Phys. Lett. B 835, 137583 (2022); Feijoo et al., Phys. Lett. B 839, 137760 (2023)). (c) Coupled-channel and threshold effects: in unitarized coupled-channel amplitudes, a pole on the second Riemann sheet below threshold (a bound state) or just below on the real axis (a virtual state), or a kinematic triangle singularity, can each generate a peak resembling a resonance; a coupled-channel fit to the LHCb $B^-\to J/\psi\Lambda\bar{p}$ data places a pole at $4338.2-1.9i$ MeV, essentially a $\bar{D}\Xi_c$ bound state (Nakamura and Wu, Phys. Rev. D 108, L011501 (2023)), while a machine-learning model-selection study of the same line shape favors a virtual-state pole (Co and Sombillo, PoS(HADRON2025) 255 (2025)).

The obstacle is degeneracy: several distinct compact-current constructions and several molecular channels predict masses within a few tens of MeV of both peaks — a recent diquark-triquark calculation, for example, assigns $J^P=3/2^-$ to the 4459 state (Cao et al., arXiv:2604.20618), and a molecular-current QCD-sum-rule study finds no $J^P=1/2^-$ bound state in the molecular sector at all (Ben and Zhang, Phys. Rev. D 114, 034030 (2026)) — so mass spectroscopy alone cannot discriminate them. The known systematic limitations of the sum-rule program (the empirical modified energy-scale formula $\mu=\sqrt{M_P^2-(2\mathbb{M}_c)^2}-\mathbb{M}_s$ used to set the OPE scale, the continuum thresholds $s_0$, Borel windows, and OPE truncation) and the absence of a fully coupled-channel treatment further limit the discriminating power of current theory.

Problem Statement

Make an unambiguous assignment of the spin–parity quantum numbers and the internal structure of the two strange hidden-charm pentaquark candidates observed in the $J/\psi\Lambda$ invariant-mass spectrum, the $P_{\psi s}^{\Lambda}(4338)^0$ and the $P_{cs}(4459)^0$ — the assignment that Wang and Xin (arXiv:2508.17373) conclude still requires more experimental data and further theoretical study. The single objective is to establish, for each candidate, (a) its $J^P$ and (b) which interpretation — compact diquark–diquark–antiquark pentaquark, meson–baryon molecule ($\bar{D}\Xi_c$, $\bar{D}^{*}\Xi_c$, $\bar{D}_s\Lambda_c$), or coupled-channel threshold effect (a virtual-state pole or a triangle singularity) — it corresponds to. The three interpretations are a partition: a bound pole in a $\bar{D}\Xi_c$, $\bar{D}^{*}\Xi_c$, or $\bar{D}_s\Lambda_c$ channel — including a bound pole obtained from a coupled-channel amplitude — is counted as the molecular interpretation, and the coupled-channel category is reserved for the non-molecular origins, a virtual-state pole or a triangle singularity. Concretely, an unambiguous assignment requires jointly reconciling: (i) a determination of the $J^P$ of $P_{cs}(4459)$ (undetermined since its 2021 observation) and of the favored $J^P=1/2^-$ of $P_{\psi s}^{\Lambda}(4338)$, whether from amplitude analyses of $J/\psi\Lambda$ spectra in additional or higher-statistics production channels ($\Xi_b$ decays, $B$ decays, $\Upsilon$ decays, photoproduction or kaon-induced production) with full angular observables or from a theoretical framework with quantified uncertainties; (ii) discriminating evidence among the compact, molecular, and threshold/coupled-channel interpretations — via pole positions and line shapes across production channels, decay patterns, radiative transitions, or a theoretical framework with quantified uncertainties (a fully coupled-channel QCD sum rule treatment, a unitarized coupled-channel amplitude, or lattice QCD) whose predictions select a unique assignment for the observed spectrum. Searches for isovector partner states (for example in $J/\psi\Sigma$ channels) testing the assumed $I=0$ assignments, which would help resolve the near-degeneracy among competing theoretical current constructions and model predictions, are advisory corroboration and are not required for completion. The problem is complete when both candidates have unambiguous $J^P$ assignments and quantified evidence that discriminates the competing structure hypotheses for each state; a $J^P$ determination for the 4459 state alone, or a structure argument for one state alone, constitutes partial progress rather than completion.

References

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