Portraits of Charmoniumlike States

Chunjiang Shi; Qingyang Liu; Wei Sun; Xiangyu Jiang; Ying Chen

REVIEW 4 major objections 6 minor 43 references

Portraits of Charmoniumlike States

T0 review · 4 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Lattice QCD computes gauge-invariant charm density correlations and reads them as three-dimensional spatial wave functions of charmonium and charmoniumlike states, revealing relativistic Dirac dynamics and an S-wave hybrid structure.

desk verdict Genuine first lattice correlation data for these states, but the wave-function bridge in Eq. (1) is not just uncontrolled; it is wrong as written, so the portraits and hybrid conclusions are conditional on a corrected derivation. read the letter →

arxiv 2505.21193 v2 pith:4UVMRIBE submitted 2025-05-27 hep-ph hep-lat

Qingyang Liu , Xiangyu Jiang , Ying Chen , Chunjiang Shi , Wei Sun This is my paper

classification hep-phhep-lat

keywords charmoniumcharmoniumlikehybridslatticeQCDdensity-densitycorrelationDiracequationrelativisticquarkmodelexoticquantumnumbersBorn-Oppenheimerapproximation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper uses lattice QCD to compute gauge-invariant charm quark density-density correlations $C_H(r)=\langle H|\rho(0)\rho(r)|H\rangle$ in the 1S and 1P charmonia and in the exotic charmoniumlike states $\eta_{c1}(1^{-+})$ and $h_{c0}(0^{+-})$, reading these correlations as three-dimensional spatial wave functions of the $c\bar{c}$ pair under a two-body factorization approximation. The angular and radial shapes of the conventional states match the picture of relativistic two-body Dirac bound states rather than the nonrelativistic quark model: the lower components of the Dirac wave functions produce nonzero density at $r=0$ and $l=0$ or $l=2$ angular admixtures that the nonrelativistic model cannot generate. For $\eta_{c1}$, the spherical relative distribution of the charm pair implies its $c\bar{c}$ core is a color-octet $1^{--}$ object bound to a $1^{+-}$ chromomagnetic gluonic component in $S$-wave, which contradicts the flux-tube Born-Oppenheimer picture. If correct, these are the first gauge-invariant, model-independent three-dimensional portraits of charmonium and charmoniumlike states.

What carries the argument

The load-bearing object is the density-density correlation $C_H(r)=\int d^3x\,|\Phi_{c'}(x)|^2|\Phi_c(x+r)|^2$ of Eq. (1), which under the paper's two-body factorization is read as the squared spatial wave function of the $c\bar{c}$ pair in each state. The interpretive engine is the Dirac Coulomb bound-state solution $\Psi_{n,\kappa,m}(r)=\big(F_{n,\kappa}(r)\chi^{(l)}_{j,m},\, iG_{n,\kappa}(r)\chi^{(l')}_{j,m}\big)$, whose lower component $G_{n,\kappa}$ is of order $\alpha_s$ relative to $F_{n,\kappa}$ for charm quarks; its small-$r$ behavior $r^{s-1}$ produces the central densities and partial-wave admixtures absent from the nonrelativistic quark model. For the exotic states, the same correlation data are decoded by the quantum-number decomposition of a color-octet $c\bar{c}$ pair plus a chromomagnetic gluonic component in a relative $S$-wave.

What would settle it

Compute the same four-point correlation on the same ensemble with the full $C_H^{(4)}$ without invoking the factorized product in Eq. (1), and compare the extracted radial and angular shapes to the Dirac predictions; in particular, if $\chi_{c1}$'s angular distribution shows no $l=2$ component, or if $\eta_{c1}$'s relative distribution acquires a $P$-wave node under a different interpolating operator, the paper's relativistic hybrid portrait is refuted.

Watch

Extended reading notes

Core claim

The central claim is that the charm density-density correlation $C_H(r)$, computed ab initio on a $16^3\times 128$ lattice with two degenerate light quark flavors and a tuned valence charm mass, resolves the internal spatial structure of charmonium and charmoniumlike states well enough to distinguish competing dynamical pictures. For 1S and 1P charmonia the small-$r$ behavior deviates from the nonrelativistic quark model and follows the Dirac asymptotic form $(F_{n,\kappa},G_{n,\kappa})\sim r^{s-1}$ with $s=\sqrt{\kappa^2-\alpha_s^2}$, so the lower component is not negligible because $\alpha_s\sim 0.3$; this explains the nonzero or weakly divergent density at the origin for $\eta_c$, $J/\psi$, $h_c$, $\chi_{c0}$, $\chi_{c1}$ and the small $l=2$ admixture seen in $\chi_{c1}$ and $\chi_{c2}$. For the hybrid $\eta_{c1}(1^{-+})$, the $c\bar{c}$ distribution is spherical with an $S$-wave radial profile, indicating that the charm pair is in $1^{--}$ and the $1^{+-}$ gluonic component sits in a relative $S$-wave, a configuration incompatible with the Born-Oppenheimer flux-tube picture.

Load-bearing premise

The whole reading rests on Eq. (1), which treats each charmonium state as a two-body $c\bar c$ bound state with factorized single-particle densities, neglects $c\bar c$ annihilation, and integrates out all other degrees of freedom; if this two-body factorization is inaccurate, the computed $C_H(r)$ remains a gauge-invariant lattice correlation but cannot be interpreted as the spatial wave function, or portrait, that the paper claims.

Editorial extensions

If this is right

For 1S and 1P charmonia, the spatial wave function at $r=0$ is nonzero and weakly divergent (except for $\chi_{c2}$), so any nonrelativistic quark model prediction that forces $|\phi(r)|^2\sim r^2$ for P-waves must be revised.
The $\eta_{c1}(1^{-+})$ hybrid should be described as a color-octet $1^{--}$ $c\bar{c}$ core in $S$-wave with a $1^{+-}$ chromomagnetic gluonic component, not as a flux tube with the $c\bar{c}$ pair in an effective $P$-wave.
The octet Cornell potential with Casimir scaling gives a $1S$-$2S$ mass splitting of about $1.27$ GeV for this hybrid, matching the lattice spacing of $1.1$-$1.3$ GeV rather than the Born-Oppenheimer estimate near $350$ MeV.
The measured RMS charm separations, $0.42$-$0.67$ fm across the states studied, provide a model-independent size scale against which future quark-model and effective-field-theory predictions can be tested.

Reading between the lines

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

If the two-body factorization is robust, the same density-density observable could classify tetraquark and $D\bar D^*$ molecular candidates by showing whether their heavy-quark pair is compact or spatially extended.
An implication the paper leaves implicit is that the chromomagnetic gluonic component behaves like a constituent with a mass near $0.8$ GeV; a testable extension is to predict radiative transitions of $\eta_{c1}$ from the compact $1^{--}$ core and compare with experiment.
The conflict with the flux-tube picture suggests that other hybrid multiplets may also be organized by Casimir-scaled octet potentials rather than by string excitations; computing $C_H(r)$ for the $0^{+-}$ state and higher radial excitations would test this systematically.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

The four-point function here is a real lattice computation, gauge-invariant and new for these channels. The angular trends (spherical 1S, the quadrupole in chi_c2, the S-wave-looking c-cbar distribution for the 1-+ hybrid) are suggestive and worth taking seriously. The Dirac interpretation of the 1P small-r behavior is also a genuinely new way to organize the data. Credit where it is due: this is the first time these correlations have been computed for conventional and exotic charmonium in this form, and the raw observable is legitimate.

The problem is the bridge from C_H(r) to a spatial wave function. Eq. (1) inserts a single-particle completeness relation between two one-body currents; for a two-body state the intermediate state must be two-particle. And as written, for a zero-momentum state the one-body densities are translation-invariant, so the convolution shown would be flat in r. That is not what the figures show. So the lattice data are fine, but the interpretation of C_H(r) as |psi(r)|^2 has no valid derivation here. The old papers they cite (Wilcox-Liu and others) do give a proper relation, and the authors should fix the derivation instead of writing a single-particle insertion that cannot produce the observed r-dependence.

Because every physical claim passes through this step, the letters are conditional. The r^{-alpha_s^2} divergence is inferred from fits that exclude r=0, with alpha_s~0.3 put in by hand; the 1-- assignment for the c-cbar in eta_c1 and the disfavoring of the flux tube both rest on reading the angular shapes as relative wave functions. Those readings might survive a correct derivation, but they have not been established here.

Other soft spots are fixable: one Nf=2 ensemble, one lattice spacing, no continuum limit, and an internal inconsistency in the geometry statement (isotropic lattice vs. as/at~5). The text itself flags some of these. None of that by itself kills the paper.

Bottom line: the raw correlations are a useful lattice observable and the phenomenology is interesting, but the central interpretive step needs to be rewritten. I would send it to peer review, because the calculation is genuine and the issues are correctable; a good referee can push for a proper wave-function derivation and more ensembles rather than a desk rejection.

Referee Report

4 major / 6 minor

Summary. The paper reports a lattice QCD calculation of the charm quark density-density correlation C_H(r) in the conventional charmonia eta_c, J/psi, hc, chi_c0, chi_c1, chi_c2 and the charmoniumlike exotic states eta_c1(1^-+) and hc0(0^+-). Using N_f=2 gauge configurations with m_pi about 420 MeV, the authors extract C_H(r) from a four-point function and interpret it, via Eq. (1), as the squared spatial wavefunction of a two-body c-cbar bound state. On this basis they claim that the 1S and 1P charmonia show substantial relativistic effects described by the Dirac equation, that the c-cbar component of eta_c1 has quantum numbers 1-- and is in an S-wave relative to a 1^+- gluonic (chromomagnetic) component, and that this configuration disfavors the Born-Oppenheimer flux tube picture.

Significance. If the wavefunction identification were justified, this would be a novel, gauge-invariant visualization of quark distributions in charmonia and the first lattice-based portrait of the internal spatial structure of a charmoniumlike hybrid. The underlying four-point functions are legitimate lattice observables, and the idea of using density-density correlations to access hadron spatial structure is well motivated. However, the significance is conditional: the central physical conclusions all depend on the validity of Eq. (1), which is not properly derived. The raw correlation data could still be useful if reinterpreted, but the paper's headline claims as they stand are not supported.

major comments (4)

[Introduction, Eq. (1)] Equation (1) is not a valid bridge from the lattice four-point function to a two-body spatial wavefunction. Inserting a single-particle position basis between the two density operators and identifying the resulting overlaps as one-body charge densities leads to a convolution of one-body densities. For the zero-momentum states used in Eq. (3), translation invariance makes each one-body density uniform, so the convolution is independent of r; this cannot reproduce the non-trivial r dependence in Figs. 2 and 3. The correct two-body relative distribution would be obtained from a two-body density matrix or an equal-time Bethe-Salpeter amplitude, and the connection to C_H(r) must be derived explicitly. Because the r->0 Dirac behavior, the 1-- assignment for the c-cbar component of eta_c1, the S-wave assignment for the gluonic component, and the flux-tube disfavoring all pass through this identification, this is a load-bearing issue that must be addressed before the results can be interpreted as spatial wavefunctions.
[Portraits of 1S and 1P charmonia, Fig. 2 and Table I] The claimed small-r divergence is not directly measured. The fits to |phi_H(r)|^2 use the ad hoc functional form A exp[-(r/r0)^alpha] + B r exp[-(r/r0)^beta] and explicitly exclude r=0, and the paper does not report fit qualities, parameter uncertainties, or an analysis of the first few lattice separations. The comparison between the raw lattice value P(0) and the extrapolated |phi(0)|^2 in Table I is sensitive to the fit ansatz and to discretization effects at the shortest distance, and no continuum extrapolation is attempted. The statement that the Dirac prediction r^{-alpha_s^2} is "observed" therefore goes beyond what the data demonstrate; this matters because the radial behavior is central to the relativistic-effect claim.
[Supplemental Material S1, Eq. (S17) and main text] The explicit total wavefunction for chi_c0 in Eq. (S17) contains only the large component F2,1 combined with Y1M S1M terms, whereas the main text (bullet list after Eq. (6)) states that the small-r behavior of chi_c0 is dominated by the lower component G2,1 with G2,1 ~ r^{s-1}. The lower component of the quark spinor, iG2,1 Y00 in the appropriate spin coupling, can combine with the antiquark spin to form a J=0 state and should appear in Psi_00(0++). This should either be included in Eq. (S17) or the omission explicitly explained; as written the two statements are inconsistent.
[1-+ and 0+- charmoniumlike states, hybrid mass-splitting estimate] The quantitative estimate of the 1S-2S hybrid splitting is illustrative rather than a robust result. It relies on assumed effective masses of 3.0 GeV and 0.8 GeV, alpha_s about 0.3, sigma about 0.25 GeV^2, Casimir scaling, and a non-relativistic Schrodinger equation. The agreement with the lattice range 1.1-1.3 GeV should be presented as a consistency check of the model, not as a prediction of the lattice calculation reported here. This does not affect the raw correlator results but should be contextualized.

minor comments (6)

[Throughout] There are several typographical errors: "componet" for "component", "samll r" for "small r", "nad" for "and", "view" for "viewed", and "ηc1(1−+ is" missing the closing parenthesis. These should be corrected.
[Numerical results] The text says "N_s^3 x N_t = 16^3 x 128 isotropic lattice with the aspect ratio as/at about 5". If as/at is about 5, the lattice is not isotropic; please clarify whether the intended statement is about the temporal extent or the lattice spacing ratio.
[Fig. 2 caption] The caption states that all panels share a common logarithmic colour scale, but the lower six panels are one-dimensional radial plots, not color-scale renderings. Please specify that the color scale applies only to the three-dimensional renderings.
[Numerical results, Eq. (3)] The normalization in Eq. (3) should be derived more explicitly: the factor 2m_H and the cancellation of the source-sink overlap are not shown. A reader cannot verify that the ratio is the ground-state matrix element without additional intermediate steps.
[Numerical results, spin projection] The statement that the spin state is |J J_z> = |10> for J/psi, hc, chi_c1 and proportional to (|22> - |2-2>) for chi_c2 refers to continuum quantum numbers, but the lattice operators used are in irreducible representations of the cubic group. Please clarify how the continuum quantum numbers are assigned from the cubic irreps used.
[Abstract and Summary] The phrase "first gauge invariant and model-independent three-dimensional portraits" overstates the results: Eq. (1) is an uncontrolled approximation, and the Dirac interpretation introduces model assumptions. The wording should be softened to match what is actually established.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the lattice density-density correlations are independent observables, the Dirac interpretation uses external inputs, and the same-group citation [34] is corroborative rather than load-bearing.

full rationale

The central derivation is self-contained. The lattice four-point function C_H^(4) in Eq. (2) and the ratio in Eq. (3) are first-principles QCD observables; Eq. (1) is an interpretive identification of the density-density correlation with a two-body relative distribution, not a reduction of the result to a fitted input. The Dirac explanation uses the textbook Coulomb-Dirac solution [29] with an external value alpha_s ~ 0.3, not parameters fitted to C_H(r), so the r -> 0 behavior and angular multipole expectations are genuine model comparisons. The 1-- / S-wave assignment for eta_c1 follows from the measured spherical angular distribution and standard C-parity/angular-momentum rules; the same-group citation [34] is used only as corroboration, while the current lattice data independently exhibit the same shape, so the self-citation is not load-bearing. The Supplemental S2 mass-splitting is an explicit postdiction with stated external masses and couplings, and it is not used to derive the main portrait claims. The two-body factorization in Eq. (1) is an uncontrolled approximation and may threaten the wavefunction interpretation, but that is a validity concern, not circularity, because no equation or fitted parameter is recycled as a prediction.

Assumptions & free parameters 5 free parameters · 5 assumptions · 2 invented entities

The central lattice observable is independent, but the interpretation layers add assumptions: the two-body factorization of C_H(r), the Dirac Coulomb-potential model, the radial-angular factorization, and the hybrid quantum-number assignment. The hybrid mass-splitting estimate introduces several hand-picked parameters (alpha_s, sigma, effective masses) that are not fitted to the lattice data in this paper.

free parameters (5)

strong coupling alpha_s = about 0.3 (chosen, not fitted to lattice data here)
Used for the Dirac asymptotics r^{-alpha_s^2} in the 1S and 1P analysis and for the octet Cornell potential in Eq. (8).
string tension sigma = about 0.25 GeV^2 (chosen)
Input to the hybrid 1S-2S mass-splitting estimate in Eq. (S25).
effective mass of color-octet c cbar block = 3.0 GeV (chosen)
Input to the Schroedinger estimate of the hybrid mass splitting in Supplemental Material S2.
effective mass of chromomagnetic excitation = 0.8 GeV (chosen)
Input to the same Schroedinger estimate of the hybrid mass splitting.
per-state radial fit parameters (A, B, r0, alpha, beta) = fitted to lattice |phi_H(r)|^2 for each of eight states
Used to quantify the r to 0 behavior; the fits exclude r=0, so the fitted origin value is an extrapolation rather than a direct measurement.

assumptions (5)

domain assumption C_H(r) as defined in Eq. (1) equals the convolution of the charm and anticharm single-particle densities of the state, which requires treating the charmonium as a two-body c c' bound state with factorized densities and neglecting annihilation.
Stated in the Introduction as an approximation, but it is the bridge from a gauge-invariant four-point function to a spatial wave function.
domain assumption Near r=0 the interaction is dominated by the one-gluon-exchange Coulomb potential -alpha_s/r, so the confining term can be neglected in the Dirac equation.
Used in the 1S and 1P section to derive the r^{-alpha_s^2} behavior; no lattice input for V(r) near the origin is given.
ad hoc to paper The radial and angular parts factorize as C_H(r) = |phi_H(r)|^2 |xi_H(theta, phi)|^2.
Assumed to define radial and angular distributions; not explicitly tested for each state.
domain assumption The 1-+ and 0+- states are c cbar g hybrids with a single gluon such that C(g) = - and the c cbar component has negative C-parity.
Used to infer the 1-- character of the c cbar component from the spherical angular distribution; the paper cites Refs. [32-34] for this classification.
domain assumption Casimir scaling of the static potential between octet charges with factor 9/4 and a Cornell form applies to the c cbar plus gluelump system.
Eq. (8) and Supplemental Material S2; the resulting 1S-2S splitting depends on this scaling.

invented entities (2)

1+- chromomagnetic gluonic component (gluelump) independent evidence
purpose: Constituent of the charmoniumlike hybrid in the proposed S-wave bound state and in the Schroedinger mass-splitting estimate.
The 1+- gluelump is a known QCD excitation studied on the lattice (Refs. [32, 34]); the paper does not invent it, though its effective mass of 0.8 GeV is an input, not measured here.
color-octet c cbar block
purpose: Effective two-body constituent of the hybrid whose relative S-wave with the gluelump is inferred from the spherical density-density correlation.
No independent handle outside the hybrid context; its quantum numbers 1-- are inferred from the C-parity argument and the spherical distribution, not measured directly.

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Cite this review

Pith. "Pith review of Portraits of Charmoniumlike States." pith.science (2026). https://pith.science/paper/4UVMRIBE

@misc{pith2026250521193,
  author       = {Pith},
  title        = {Pith review of: Portraits of Charmoniumlike States},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4UVMRIBE}},
  note         = {Machine review of arXiv:2505.21193}
}

abstract

The charm quark density-density correlation is calculated for $1S$ and $1P$ conventional charmonia and $J^{PC}=1^{-+},0^{+-}$ charmoniumlike states from lattice QCD and are interpreted as spatial wave functions of these states with some approximations. The angular distributions of $c\bar{c}$ in conventional charmonia are found to be in accordance with the expectation of two-body systems, while that of the $1^{-+}$ state exhibits an $S$-wave feature. However, the $c\bar{c}$ radial distributions turn out to be strikingly different from the non-relativistic quark model and can be understood by the Dirac theory of two-body bound states. These results provide the first gauge invariant and model-independent three-dimensional portraits of charmonium(like) states.

Figures

Figures reproduced from arXiv: 2505.21193 by the authors.

**Figure 2.** , where the z-axis is vertical and passes through the center of the distribution. The angular distributions of 1S states (ηc and J/ψ) are spherical as expected from NRQMs which describes them to be the L = 0 states. The charmonium states hc(1+−), χc0(0++), χc1(1++) and χc2(2++) are usually assigned by NRQMs to be 1P states, namely, n 2S+1LJ = 11P1 and 13P0,1,2 states, respectively. The spherical shape of CH(⃗r) for … view at source ↗

**Figure 3.** FIG. 3: Three-dimensional probability-density ren [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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