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For spin-0 quarkonia, the quark–antiquark entanglement entropy reduces to the Shannon entropy of the unpolarized TMD f1 plus log(2Nc); for spin-1 quarkonia it depends on polarization and is expressed through polarized and tensor-polarized T

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2026-07-31 23:04 UTC pith:FSWWFI5S

load-bearing objection Clean analytic TMD-entropy relations for quarkonium, with a new spin-1 polarization dependence, but the numerical polarization splitting needs a convergence test before it is trusted. the 3 major comments →

arxiv 2607.24068 v1 pith:FSWWFI5S submitted 2026-07-27 hep-ph quant-ph

Quantum entanglement within quarkonium

Wenyu Zhang , Yiyu Zhou , Yang Li , Qun Wang This is my paper
classification hep-ph quant-ph
keywords entanglement entropyquarkoniumtransverse momentum dependent distributionslight-front wave functionsvon Neumann entropyspin polarizationcharmoniumbottomonium
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper tries to show that the quantum entanglement between a quark and its antiquark inside a heavy quarkonium meson is not an abstract quantum-information curiosity but a computable, measurable partonic quantity. Tracing out the antiquark from the meson's light-front wave function leaves a mixed state for the quark; the paper derives that the von Neumann entropy of this state is, for spin-0 mesons, exactly the Shannon entropy of the unpolarized transverse-momentum-dependent parton distribution (TMD) plus constant log(2Nc) contributions from color and spin. For spin-1 mesons, the entropy acquires a polarization dependence and is expressed through the unpolarized, polarized, and tensor-polarized TMDs, with explicit formulas for m_J=0 and m_J=±1 eigenstates. Using numerically obtained light-front wave functions for charmonium and bottomonium, the paper predicts a pronounced difference in entanglement entropy between m_J=0 and m_J=1 vector mesons. If correct, this turns entanglement entropy into a probe of nonperturbative hadron structure that could be connected to future TMD measurements.

Core claim

The central claim is that, within the valence quark-antiquark Fock sector and with the gauge link set to unity, the reduced density matrix of the quark in a quarkonium state is fully encoded by leading-twist TMDs. For spin-0 quarkonia this makes the entanglement entropy S = log(2Nc) + H(f1), where H(f1) is the Shannon entropy of the unpolarized TMD f1. For spin-1 quarkonia the entropy is polarization dependent: for m_J=0 it is log2Nc + H(f1 + (2/3)f1LL), and for m_J=±1 it involves f1, f1LL, g1L, and h⊥1L through the two spin-density-matrix eigenvalues of Eq. (29). The numerical evaluation shows a pronounced m_J=0 versus m_J=1 entropy difference for vector mesons, with the ordering reversed f

What carries the argument

The load-bearing object is the reduced density matrix of the quark subsystem, built from the valence light-front wave function after tracing over the antiquark. The narrow-wavepacket limit makes this density matrix diagonal in momentum, and the quark-quark correlator functions that appear are parameterized by TMDs (with the gauge link approximated as unity), so the entropy becomes a functional of TMDs. For spin-1 states the spin part of the density matrix is diagonalized by the two eigenvalues in Eq. (29), which combine unpolarized, polarized, and tensor-polarized TMDs. Numerically, the light-front wave functions come from diagonalizing a truncated light-front Hamiltonian in a harmonic-oscil

Load-bearing premise

The derivation assumes the meson is a pure valence quark-antiquark pair with no gluons or sea quarks and with the TMD gauge link set to unity, and the authors themselves state that adding higher Fock sectors or dynamical gluons breaks the direct correspondence.

What would settle it

Compute the same entanglement entropy in the same light-front framework but including the first gluon Fock sector; if the result differs from log(2Nc)+H(f1) by more than the natural size of the omitted gluon contribution, the central identity fails. Equivalently, from future TMD measurements of J/ψ, evaluate H(f1)+log(2Nc) and compare with the predicted m_J=0 entropy; a mismatch beyond estimated uncertainties would refute the claim.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Entanglement entropy of valence-dominated quarkonia can be predicted directly from TMDs, so future measurements of transverse momentum distributions in lepton-hadron scattering would determine a quantum information measure of the bound state.
  • For vector quarkonia, the entropy difference between m_J=0 and m_J=1 states is a concrete prediction that can be tested against polarization-dependent production or decay observables.
  • The matching of the infrared parameter fixes the previously ambiguous logarithmic volume term, making the numerical entropy finite and comparable across charmonium and bottomonium.
  • The transverse entropy density peaks track the excitation pattern of the state, so entanglement distributions offer a momentum-resolved view of radial and orbital quantum numbers.
  • The direct TMD correspondence is limited to the valence sector; the paper states that adding sea quarks or gluons requires returning to the full density matrix, so the formulas apply only to heavy quarkonia below open-flavor threshold.

Where Pith is reading between the lines

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

  • If this valence-sector relation survives in full QCD at large x, entanglement entropy could be retroactively extracted from existing TMD fits, giving the quantum information content of mesons without new experiments.
  • The strong m_J dependence suggests a two-particle entanglement analog of the J/ψ polarization puzzle: the ratio of m_J=0 to m_J=1 entropy might be correlated with the measured polarization parameter λ_θ, a comparison not made in the paper.
  • A direct numerical falsification test would be to include a one-gluon Fock sector in the same light-front Hamiltonian; if the entropy shifts by more than O(α_s) from the valence-only value, the TMD-entropy identity fails beyond a toy model.
  • The harmonic-oscillator matching procedure could be validated by checking whether the extracted infrared cutoff scales as κ/√N_max when N_max is varied; if it does not, the entropy is scheme-dependent in a way the paper does not control.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 5 minor

Summary. The paper derives the quark-antiquark entanglement entropy in heavy quarkonium from the valence light-front wave function. For spin-0 quarkonia, it claims the entanglement entropy reduces to the Shannon entropy of the unpolarized TMD f1 plus constant color/spin contributions, Eq. (23). For spin-1 quarkonia, it derives a polarization-dependent entropy expressed through polarized and tensor-polarized TMDs, Eqs. (28)-(34). The authors evaluate these entropies with BLFQ wave functions for charmonium and bottomonium, report a pronounced dependence on mJ for vector mesons (Fig. 4) and on a transverse rotation angle (Fig. 6), and propose an IR-parameter matching between the momentum-space result and a harmonic-oscillator representation (Sec. III.B).

Significance. If the central claims hold, the paper provides a concrete, nonperturbative bridge between quantum-information measures and TMD observables in quarkonium, and it identifies polarization dependence of partonic entanglement as a potentially measurable effect. The analytic derivation from the two-particle LFWF density matrix is clean and standard: the NWL diagonalization and the TMD parametrization are presented carefully, and the spin-0 formula is a simple and appealing result. The paper also deserves credit for explicitly discussing, in Sec. IV, where the valence/Fock-space and W=1 approximations limit the correspondence. The main unresolved risk is numerical: the headline polarization dependence rests on single-truncation BLFQ results with no convergence study, and the IR matching procedure has ad hoc elements. These issues are fixable and do not invalidate the analytic core, but they currently prevent the numerical claims from being accepted as robust.

major comments (3)
  1. [Sec. III.C, Figs. 4 and 6] The central numerical claim—pronounced entanglement entropy differences between |mJ=0> and |mJ=1> vector quarkonia, and the rotation-angle dependence—is obtained from single basis truncations (Nmax=8 for charmonium, Nmax=32 for bottomonium) with no convergence test. The light-front Hamiltonian and BLFQ basis do not preserve full rotational invariance, and the mJ=0 and mJ=1 states are not demonstrated to be degenerate as required for members of a J=1 multiplet. If rotational symmetry is not restored at these truncations, the reported mJ dependence could be an artifact of the basis cutoff rather than a property of the continuum quarkonium state. Please provide a convergence study in Nmax and Lmax (or, at minimum, a check of the J/psi and Upsilon mass degeneracy between mJ=0 and mJ=1), and state the sensitivity of the mJ-entropy difference to the truncation. If such a study is not feasible,
  2. [Sec. III.B, Eqs. (40)-(44)] The determination of the IR parameter is not as unique as claimed. Eq. (43) fixes P0+ L = 2 Nmax × 2π without a derivation, and the extrapolation to sigma=0 uses a purely phenomenological fit f(x)=a e^{-bx}+cx+d over a sigma range that explicitly excludes the strict NWL. The 'matching' in Eq. (44) then determines L_perp from the difference between the extrapolated harmonic-oscillator entropy and the momentum-space entropy with the IR term removed. The resulting IR cutoff is therefore sensitive to the choice of longitudinal box rule, the fit ansatz, the fitting range, and the basis truncation. Please show the sensitivity of Λ_IR and of the absolute entropies to these choices, or explain why Eq. (43) is not an arbitrary normalization.
  3. [Secs. II.A and IV] The derivation of the TMD-entropy identity uses the valence Fock sector only and sets the Wilson line W=1 in Eqs. (16)-(17). As the authors acknowledge in Sec. IV, higher Fock sectors and dynamical gluons break the direct correspondence between entanglement entropy and quark TMDs. Since the abstract and title state the result without this qualifier, the paper should make the domain of validity explicit in the abstract and conclusion: the central TMD-entropy relation is established for a valence-dominated, two-particle effective theory. This does not undermine the derivation within that model, but it is load-bearing for the claim that the entropy 'is' the TMD Shannon entropy in QCD.
minor comments (5)
  1. [Sec. III.C] Typo: 'charmonoium' and 'For charmonoium' should read 'charmonium'.
  2. [Eq. (23) and Table II] The Shannon entropy H(f1) is introduced without an explicit convention for the measure in the logarithm (dimensionful k⊥ vs. dimensionless x,k⊥). Since the logarithmic IR term is subtracted in Table II, please state the normalization convention used so the reader can reproduce the numerical values.
  3. [Eq. (41)] The notation in the Talmi-Moshinsky transformed expression is hard to parse: the subscripts of the transformation coefficient and the placement of the Kronecker delta should be clarified, and the normalization of c_{NM} in Eq. (42) should be specified.
  4. [Figs. 3 and 5] The figure labels such as 'SE c', 'SE c0' and the axis 'k × S_T^D(k)' are not fully typeset. Also, in Fig. 3 the legend '3σ confidence band' should be '3σ' or '99.73%' consistently.
  5. [Refs. [4] and [29]] The DOIs '10.1103/3yg7-r5s9' and '10.1103/wlm7-x1wn' appear nonstandard; please verify they resolve correctly.

Circularity Check

2 steps flagged

The entropy–TMD identities are tautological in the valence W=1 truncation; the BLFQ polarization numbers are an independent but unconverged model output.

specific steps
  1. self definitional [Sec. II.B, Eqs. (14), (21)–(23)]
    "f1(x, k⊥) = Φ[γ+](x, ⃗k⊥) = Σ_{s,¯s} |ψ_{s¯s}(x, ⃗k⊥)|^2 / (2x(1−x)(2π)^3) ... ρq = ½ Σ_s ∫ d^3p/((2π)^3 2p+) f1(x,k⊥) ... SvN(ρq) = log(P0+ V Nc/(4π^3)) − ∫ dx ∫ d^2k⊥ f1 log f1 = log(2Nc)+H(f1)."

    The reduced density matrix eigenvalue density is, by Eq. (14), the same LFWF overlap that Eq. (21) defines as the TMD f1. Thus Eq. (23) is obtained by substituting the definition of f1 into the definition of ρq; no independent f1 input enters, so the advertised entropy–TMD relation is an identity within the valence W=1 approximation, not a derived prediction that could fail.

  2. self definitional [Sec. II.C, Eqs. (24)–(34)]
    "λ^{mJ=0}_1 = λ^{mJ=0}_2 = ½ f1(x, ⃗k⊥) + ⅓ f1LL(x, ⃗k⊥) ... S_q(mJ=0) = log 2Nc + H(f1 + ⅔ f1LL)."

    The λ_a are, through Eqs. (14)–(17), eigenvalues of the LFWF-built density matrix, while the TMDs f1, f1LL, g1L, h⊥1L are defined from the same LFWFs in the W=1 truncation. The polarization-dependent entropy formulas are therefore a re-expression of the density-matrix eigenvalues in TMD notation, not an independent connection between separately computed or measured quantities.

full rationale

The central formal results of the paper—the spin-0 identity Eq. (23) and the spin-1 formulas Eqs. (28)–(34)—are constructed from the same light-front wave functions: Eq. (14) makes the reduced density matrix eigenvalue density equal to the LFWF overlap, and Eq. (17)/(21) define the TMDs as the identical overlap under the W=1 valence-sector approximation. Substituting the TMD definitions into the density matrix yields the entropy formulas without any independent TMD input, so these parts of the paper are definitional identities rather than falsifiable predictions. The numerical BLFQ evaluation is not circular in the same way: the LFWFs come from diagonalizing the effective Hamiltonian Heff of Ref. [80], which the paper cites as benchmarked against charmonium spectra and radiative widths, and the mJ=0 versus mJ=1 entropy differences are genuine outputs of that model. The IR-parameter fixing (Sec. III.B) is a self-calibration: the momentum-space entropy is matched to the harmonic-oscillator representation using the ad hoc choice P0+L=32π and a nonlinear σ→0 extrapolation. This fixes a state-independent additive offset in the entropy and does not drive the polarization differences highlighted in Figs. 4 and 6; the absence of a convergence test is a robustness/correctness concern, not circularity. Self-citations [7] and [80] are not load-bearing circularity because the narrow-wavepacket diagonalization is re-derived in the text and [80] is externally benchmarked. Overall, the paper has partial circularity in its formal dictionary, but the numerical model output retains independent content.

Axiom & Free-Parameter Ledger

3 free parameters · 4 axioms · 1 invented entities

The central claims rely on the valence/truncated effective-theory setup and on a calibration of the IR volume via matching two representations of the same entropy. The derived formulas themselves are analytical consequences of the LFWFs and known TMD decompositions.

free parameters (3)
  • IR parameter (box volume) P0^+ V / (2pi)^3 = 396 GeV^-2 for charmonium Nmax=8; 3186 GeV^-2 for bottomonium Nmax=32
    Determined by matching momentum-space entropy to harmonic-oscillator entropy, with ad hoc P0^+ L = 32pi (Eq. 43). This choice affects all reported entropy values.
  • Wavepacket width sigma extrapolation parameters a,b,c,d = not stated explicitly; Table I fit values
    f(x) = a e^(-bx) + cx + d is fitted to entropy vs sigma and extrapolated to sigma=0.
  • BLFQ basis parameters (kappa, Nmax, Lmax) = kappa not stated in this paper; Nmax=8 (charm), 32 (bottom); Lmax from ref. [80]
    These control the effective Hamiltonian and wave functions; the quarkonium spectrum was tuned in prior work [80].
axioms (4)
  • domain assumption Valence Fock sector approximation, truncating the state to |q qbar> only
    Eq. (7) with ellipsis; justified as an effective theory below open flavor threshold. The paper's own Section IV admits higher Fock sectors break the TMD-entropy correspondence.
  • domain assumption W=1 approximation for the TMD Wilson line
    Used in Eq. (17) to identify quark-quark correlators with LFWF overlaps. The paper notes Wilson lines are needed for gauge invariance and their effect is not quantified.
  • domain assumption Light-front wavefunction normalization and Gaussian wavepacket (Eq. 12) with narrow-wavepacket limit sigma -> 0
    Central to making the reduced density matrix diagonal and to obtaining the IR-dependent log term. The wavepacket is asserted not to change intrinsic structure.
  • domain assumption The BLFQ effective Hamiltonian Heff = T + V_conf + V_OGE (Eq. 35) provides accurate charmonium/bottomonium LFWFs
    Adopted from ref. [80]. The accuracy of the entropy predictions depends on these wave functions.
invented entities (1)
  • None no independent evidence
    purpose: No new particles, fields, or forces are introduced.
    The framework uses only known QCD degrees of freedom and LFWFs.

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read the original abstract

We investigate quark-antiquark entanglement in heavy quarkonium within a nonperturbative light-front Hamiltonian framework. By tracing over the antiquark degrees of freedom in the hadronic state vector, we construct the reduced density matrix of the quark subsystem and compute the associated von Neumann entropy. For spin-0 quarkonia, we show that this entropy reduces to the Shannon entropy of the unpolarized transverse momentum dependent parton distribution (TMD), up to constant color and spin contributions. For spin-1 quarkonia, we derive the explicit polarization dependence of the entropy and connect it to polarized and tensor-polarized TMDs. Using light-front wave functions obtained via basis light-front quantization (BLFQ), we evaluate the entanglement entropy for charmonium and bottomonium states, revealing a pronounced sensitivity to the polarization of vector mesons. Furthermore, we resolve the infrared parameter by matching the momentum-space entropy to a harmonic-oscillator representation. Ultimately, these results establish entanglement entropy as a novel probe of nonperturbative quarkonium structure, forging a direct link between quantum information measures and partonic observables.

Figures

Figures reproduced from arXiv: 2607.24068 by Qun Wang, Wenyu Zhang, Yang Li, Yiyu Zhou.

Figure 1
Figure 1. Figure 1: FIG. 1. 3D images of the unpolarized TMDs [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. 3D images of the eigenvalues of the quark spin density matrix [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Entanglement entropy between quark and anti-quark d.o.f’s in charmonium as a function of Gaussian wave packet [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Distribution of entanglement entropy in transverse [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. The entanglement entropy between quark [PITH_FULL_IMAGE:figures/full_fig_p009_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The entanglement entropy as a function of [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

discussion (0)

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