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REVIEW 3 major objections 4 minor 143 references

A femtoscopic tale of two $C$-parities: the $Z_c(3900)$ and the isovector partner of the $X(3872)$

T0 review · 3 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Femtoscopy of charged D-D* pairs can expose the predicted Wc1 exotic state.

desk verdict The C-parity mixing argument is real and robust, but the '>2.5σ' enhancement claim is conditional on the Wc1 pole prediction whose uncertainty is excluded. read the letter →

arxiv 2608.01237 v1 pith:D5C2NKQB submitted 2026-08-02 hep-ph hep-exnucl-exnucl-th

Pan-Pan Shi , Miguel Albaladejo , Feng-Kun Guo , Juan Nieves This is my paper · ORCID
classification hep-phhep-exnucl-exnucl-th
keywords femtoscopyexotichadronsZc(3900)Zcs(3985)Wc1correlationfunctionsC-paritycoupledchannels
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

The paper claims that femtoscopic correlation functions of charged charm-meson–antimeson pairs such as $D^0D^{*-}$ and $D^{*0}D_s^-$ receive contributions from two different charge-conjugation (C) sectors at once: the C-odd interaction that generates the $Z_c(3900)$ and $Z_{cs}(3985)$ states, and the C-even interaction that hosts the predicted isovector partner $W_{c1}$ of the $X(3872)$. Because these pairs are not eigenstates of G-parity, the measured correlation function is approximately the average of the C-odd and C-even eigenchannel correlation functions, so the C-even sector enters with the same weight as the C-odd sector. Including that C-even admixture enhances the low-momentum correlation functions by more than $2.5\sigma$ near threshold. This opens the first direct femtoscopic window onto the unobserved $W_{c1}$ state and lets a single LHC measurement address both the $Z_c$ and $Z_{cs}$ pole scenarios and the existence of $W_{c1}$ at once.

What carries the argument

The load-bearing object is the coupled-channel Koonin–Pratt correlation function, with the relative wave function obtained from the scattering $T$-matrix via a once-subtracted Lippmann–Schwinger equation. The interaction is a $3\times3$ potential matrix built from heavy-quark spin symmetry and light-flavor SU(3), with the C-odd sector governed by the energy-dependent constant $C'_{1Z}$ taken from the $Z_c/Z_{cs}$ analysis and the C-even sector by $C_{1X}$, set to $-0.294(18)$ fm$^2$ from the predicted $W_{c1}$ pole. The identity carrying the argument is Eq. (8): for degenerate thresholds and equal production weights, the physical correlation function is the arithmetic mean of the C-odd and C-even eigenchannel correlation functions, so the C-even sector is not a small correction but a half-weight partner in the observable.

What would settle it

Measure the $D^0D^{*-}$ correlation function in high-multiplicity $pp$ collisions at the LHC in the relative-momentum range $k\lesssim100$ MeV and compare the threshold value and low-momentum slope with the three-channel prediction (including $C_{1X}=-0.294$ fm$^2$) versus the two-channel reference that keeps only the C-odd interaction; agreement with the two-channel curve would indicate the C-even admixture is absent or much weaker than assumed. Independently, a lattice QCD determination of the $W_{c1}$ pole position that places it outside $8^{+8}_{-5}$ MeV below threshold would require re-fitting $C_{1X}$ and would invalidate the quantitative significance claim if no pole exists there.

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Extended reading notes

Core claim

The central claim is that the physically observed correlation function for $D^0D^{*-}$ (and its strange counterpart $D^{*0}D_s^-$) is not the C-odd correlation function alone but approximately $\big(\tilde{C}_{1Z}(k)+\tilde{C}_{1X}(k)\big)/2$, where $\tilde{C}_{1Z}$ is the unphysical C-odd eigenchannel correlation function and $\tilde{C}_{1X}$ the C-even one. Working in a three-channel system ($J/\psi\pi$, $D^0D^{*-}$, $D^{*0}D^-$, and the analogous $J/\psi K$ system) with a heavy-quark-spin and light-flavor SU(3) symmetric potential, and fixing the C-even coupling $C_{1X}$ to reproduce the predicted $W_{c1}$ pole at $8^{+8}_{-5}$ MeV below the $D^0D^{*-}$ threshold, the authors find that the full three-channel correlation functions exceed their two-channel counterparts (which treat only the C-odd interaction) by more than $2.5\sigma$ near threshold. They conclude that the earlier results of Ref. [94] should be read as the unphysical C-odd eigenchannel correlation function rather than as the physical charge-state correlation function, and that measuring these correlation functions can simultaneously probe $Z_c(3900)$, $Z_{cs}(3985)$, and $W_{c1}$.

Load-bearing premise

The quantitative size of the C-even enhancement rests on the assumed contact coupling $C_{1X}=-0.294(18)$ fm$^2$, fixed by reproducing the predicted $W_{c1}$ pole at $8^{+8}_{-5}$ MeV below the $D^0D^{*-}$ threshold from Refs. [29, 30]; if $W_{c1}$ does not exist near that position, or if the pole lies outside the quoted range, the claimed more-than-$2.5\sigma$ enhancement can weaken or vanish, and the paper does not propagate the $C_{1X}$ uncertainty into that significance claim.

Editorial extensions

If this is right

  • A measurement of the $D^0D^{*-}$ or $D^{*0}D_s^-$ correlation function in high-multiplicity $pp$ collisions at the LHC would simultaneously test the pole structure of $Z_c(3900)$, $Z_{cs}(3985)$, and the existence of $W_{c1}$; the paper argues these channels are free of Coulomb distortions and accessible with projected ALICE3 statistics.
  • The virtual-state and resonance interpretations of $Z_c(3900)$ and $Z_{cs}(3985)$ leave distinguishable imprints in up to four independent correlation functions, so the two scenarios can be separated rather than inferred from one line shape.
  • If the interpretation of Ref. [94] is corrected, the discriminating power among bound, virtual, and resonant scenarios reported there is reduced by roughly a factor of two, because in the C-even-free limit the physical correlation function is $[1+\tilde{C}_{1Z}(k)]/2$.
  • The coupled-channel cusps at the $D^{*0}D^-$ and $D^0D_s^{*-}$ thresholds, located at $k\simeq50$–$60$ MeV in the lowest-threshold correlation functions, provide signatures that single-channel descriptions cannot produce, acting as independent checks of the mechanism.
  • The same mechanism should apply to other G-parity-mixed pairs; the paper specifically notes that $D^0D^{*0}$ femtoscopy requires the full coupled-channel dynamics of $D^0D^{*0}$, $D^{*0}D^0$, $D^+D^{*-}$, and $D^{*+}D^-$.
  • Editorial inference: the strength of the claimed more-than-$2.5\sigma$ enhancement depends almost entirely on the assumed $W_{c1}$ pole position; if future lattice or experimental work moves that pole away from $8^{+8}_{-5}$ MeV below threshold, the significance estimate would need to be recomputed rather than reinterpreted.
  • Editorial inference: the same two-C-parity averaging applies to any femtoscopic measurement of charged meson–antimeson pairs that are not G-parity eigenstates, so earlier or future correlation-function analyses in other channels may need a similar decomposition before their scattering-length extractions can be trusted.
  • Editorial inference: because the relation $a_2=(a_X+a_Z)/2$ connects the physical scattering length to the C-even and C-odd eigenchannel scattering lengths, a precise measurement of the threshold correlation function could be inverted to determine $C_{1X}$ and hence the $W_{c1}$ pole position, rather than assuming it.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper studies femtoscopic correlation functions (CFs) of the D0D*-, D*0D-, D0D*_s-, and D*0D_s- channels in a coupled-channel framework with J/psi pi (K), D(star)0 D(*)-, and charge-conjugate channels, using heavy-quark spin and SU(3) flavor symmetry. The authors observe that the charged D(star)0 D(*)- pairs measured in femtoscopy are not G-parity eigenstates, so their CFs receive equal-weight contributions from both the C-odd and C-even sectors. With the C-even low-energy constant C1X fixed to reproduce the predicted Wc1 pole about 8 MeV below the D0D*- threshold, they find that the full three-channel CFs exceed two-channel reference CFs by more than 2.5 sigma near threshold. They further argue that the results of Ref. [94] should be interpreted as the unphysical C-odd eigenchannel CF, and that the physical CF is approximately (1 + tilde-C_Z)/2, reducing the discriminating power of that earlier analysis. Additional predictions include coupled-channel cusps and a distinction between virtual-state and resonance scenarios for the Zc(3900) and Zcs(3985).

Significance. The conceptual observation is sound and potentially important: femtoscopic measurements of these neutral-charged meson pairs are not projections onto a single C-parity sector, and Eq. (8) makes the equal-weight mixing explicit. The reinterpretation of Ref. [94] is supported by the supplemental comparison (Fig. S1), where [1 + C_Liu]/2 reproduces the C1X=0 coupled-channel results. If the Wc1 prediction is correct, this paper provides concrete, falsifiable predictions: threshold enhancements in the D0D*- and D*0D_s- CFs beyond the C-odd-only expectation, plus cusp structures at coupled-channel thresholds, accessible in high-multiplicity pp collisions at the LHC. The paper is careful about source radii, production weights, and cutoff variation. However, the central quantitative '>2.5 sigma' claim depends on an unpropagated uncertainty and on the assumed existence and pole position of Wc1; the strength of the claim needs to be re-expressed as a conditional prediction.

major comments (3)
  1. [Fixing the theory parameters; Appendix B, Fig. 4] The headline claim that the three-channel CFs exceed their two-channel counterparts by more than 2.5 sigma is computed while deliberately excluding the uncertainty of C1X, the parameter that controls the C-even effect. The text states: 'Rather than propagating its uncertainty into the CF predictions, we explicitly investigate the dependence...' and Fig. 4 shows that the threshold CF C_D0D*-(0) grows rapidly and nonlinearly as C1X becomes more negative and the Wc1 pole approaches threshold. Since the three-vs-two-channel difference is approximately (tilde-C_X - tilde-C_Z)/2 in the degenerate-threshold limit, an O(0.02 fm^2) shift in C1X, well within its quoted 1-sigma range, changes the numerator directly and can move the difference by an amount comparable to the displayed uncertainty bands. The 'more than 2.5 sigma' statement is therefore not a robust quantitative result; please propagate the C1X uncertainty, or rephrase the claim as explicitly conditional on the central Wc1 pole position and remove the sigma language.
  2. [Full-model prediction (definition of the two-channel reference)] The two-channel reference calculation is obtained by setting C1X = C'_1Z, which makes the C-even and C-odd interactions degenerate, not by switching off the C-even sector. The physically motivated C-even-free reference [1 + tilde-C_Z]/2 appears only in the Supplemental Material (Fig. S1). Consequently, the reported '>2.5 sigma enhancement' isolates the asymmetry between the C-parity sectors under the Wc1-tuned C-even interaction, rather than the mere presence of a C-even contribution. The main text should state this clearly and should quantify how the two different references change the size and significance of the claimed effect.
  3. [Fixing the theory parameters (C1X from Wc1)] The central input C1X = -0.294(18) fm^2 is fixed by reproducing the Wc1 pole predicted in Refs. [29,30], two papers sharing authors with the present work. This makes the quantitative enhancement conditional on the existence, quantum numbers, and shallow-virtual nature of Wc1; if Wc1 does not exist or its pole lies outside the quoted 8(+8,-5) MeV range, Fig. 4 indicates that the threshold CF can change dramatically. The paper should present the enhancement as a prediction under the Wc1 hypothesis, and should separate the model-independent C-parity mixing argument (Eq. (8)) from the model-dependent quantitative claim. The external lattice support [31] is mentioned only in passing; a short summary of the current evidence for Wc1 would help the reader assess the assumption.
minor comments (4)
  1. [Acknowledgments] There is a typo: 'M.A.acknwoledges' should read 'M.A. acknowledges'.
  2. [Eq. (8) and notation] The tilde notation in Eq. (8) (printed as 'eC' in the text) is not explicitly defined; please define tilde-C_1Z and tilde-C_1X as the CFs in the unphysical C-parity eigenchannels.
  3. [Eqs. (6)-(7) and parameter notation] The notation C1Z versus C'_1Z is easy to confuse: C'_1Z is introduced as the energy-dependent C-odd coupling in Eq. (7), while Eq. (6b) uses both C1Z and C'_1Z. A short table or explicit statement of which parameters are fitted and which are derived would improve readability.
  4. [Full-model prediction] The text says the C-even effect is 'predicted here for the first time'; since the Wc1 state itself was predicted in Refs. [29,30], it would be more precise to say that this paper presents the first femtoscopic prediction of the C-even admixture.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the C-parity averaging is an algebraic identity, and the C-even LEC is a transparent, externally anchored input rather than a fitted prediction.

full rationale

Walking the derivation chain: Eqs. (1)-(5) are standard femtoscopy and Lippmann-Schwinger machinery, and Eq. (8), C = (C~_1Z + C~_1X)/2, is an algebraic projection identity following from expressing the physical charged D0D*- state as an equal superposition of C-odd and C-even eigenstates. It is not a fit and its assumptions (degenerate thresholds, equal production weights) are explicitly stated. The quantitative enhancement is governed by the C-even LEC C1X = -0.294(18) fm^2, which is fixed by reproducing the Wc1 pole predicted in Refs [29,30]. Those papers overlap in authorship, but the Wc1 pole is itself anchored to X(3872) observables and is independently supported by the lattice calculation [31]; the present paper does not fit C1X to any femtoscopic CF. The claimed enhancement is therefore a forward consequence of a stated theoretical input, not a refit of the target quantity. Appendix B transparently displays the strong, nonlinear dependence of the threshold CF on C1X, and the supplemental C1X=0 comparison supports the reinterpretation of Ref [94] as the C-odd eigenchannel CF, so that reinterpretation is not a bare renaming. The main caveat is that the '>2.5 sigma' significance statement deliberately excludes the uncertainty of C1X, the very parameter controlling the effect; this weakens the robustness of the quantitative claim, and Fig. 4 shows the threshold CF varies strongly with C1X. That is a statistical reporting limitation, not a circularity: no equation in the paper reduces to its own output by construction, and the central G-parity mixing result has independent content.

Assumptions & free parameters 7 free parameters · 5 assumptions · 1 invented entities

The central calculation rests on an effective coupled-channel model whose LECs are taken from prior fits (Ref [54]) and on a C-even LEC fixed by a predicted Wc1 pole (Refs [29,30]) from the same community. The only genuinely new input is the C-parity mixing argument and its application to correlation functions.

free parameters (7)
  • C_1X = -0.294(18) fm^2
    C-even contact interaction; fixed by reproducing the predicted Wc1 virtual-state pole 8(+8,-5) MeV below the D0D*- threshold (Refs [29,30]). Controls the magnitude of the claimed C-even enhancement.
  • C_12 = 0.006(1) or 0.005(1) fm^2
    Mixing LEC from the combined fit to Zc(3900)/Zcs(3985) data in Ref [54]; used as input here.
  • C_1Z = -0.217(10) or -0.203(7) fm^2
    C-odd LEC from Ref [54] fits to Zc and Zcs invariant-mass spectra.
  • b = 0 or -0.473(45) fm^3
    Energy-dependent slope of the C-odd interaction in Eq. (7), from Ref [54].
  • subtraction constants a_i = -2.77 (J/psi pi), -2.5 (heavy meson channels)
    Loop-function subtraction constants at scale mu=1 GeV, taken from Ref [54].
  • source radii R_eff = 0.84(7) fm (pp), 2 fm (pA), 5 fm (AA)
    pp radii derived from average transverse mass scaling with Pythia 8; pA and AA use representative values.
  • UV cutoff Lambda = 0.6-1.4 GeV
    Regulator in the half-off-shell T-matrix factorization, Eq. (4); varied to estimate systematic uncertainty.
assumptions (5)
  • domain assumption Heavy-quark spin and light-flavor SU(3) symmetries determine the coupled-channel potential matrix of Eq. (6).
    The entire amplitude construction relies on HQS and SU(3) relations to connect the C-odd Zc/Zcs interactions to the C-even Wc1 interaction.
  • domain assumption The on-shell T-matrix factorization with step-function cutoff (Eq. 4) represents the off-shell behavior.
    The CF wave function uses a separable approximation for the half-off-shell T-matrix; the authors note the off-shell part is unobservable and estimate the cutoff uncertainty.
  • domain assumption The Koonin-Pratt formula with a Gaussian source (Eqs. 1-2) is valid for these channels.
    Standard femtoscopic formalism; assumes a Gaussian source with radius from transverse-mass scaling.
  • domain assumption The production weight w1=0 for J/psi pi and J/psi K channels.
    J/psi production is strongly suppressed in pp collisions; the authors check that setting w1=1 changes the CFs by less than 3e-5.
  • ad hoc to paper Wc1 exists as a shallow virtual state with pole 8(+8,-5) MeV below threshold.
    The C-even LEC C1X is fixed by this assumed pole from Refs [29,30]; the paper's quantitative enhancement follows from this input, so it is a load-bearing assumption rather than a derived result.
invented entities (1)
  • W_c1 (isovector C-even partner of X(3872))
    purpose: Target of the proposed femtoscopic probe; its assumed pole position fixes C1X and generates the C-even enhancement in the correlation functions.
    Predicted in Refs [29,30] and supported by a lattice calculation [31], but not yet observed experimentally; no directly falsifiable mass or width measurement outside the model is available. It is not invented in this paper but is used as an input.

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Cite this review

Pith. "Pith review of A femtoscopic tale of two $C$-parities: the $Z_c(3900)$ and the isovector partner of the $X(3872)$." pith.science (2026). https://pith.science/paper/D5C2NKQB

@misc{pith2026260801237,
  author       = {Pith},
  title        = {Pith review of: A femtoscopic tale of two $C$-parities: the $Z_c(3900)$ and the isovector partner of the $X(3872)$},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D5C2NKQB}},
  note         = {Machine review of arXiv:2608.01237}
}
abstract

Understanding the nature of the exotic $Z_c(3900)$ and $Z_{cs}(3985)$ states, and searching for the predicted isovector partner $W_{c1}$ of the $X(3872)$, remain central challenges in exotic-hadron spectroscopy. We investigate the femtoscopic correlation functions (CFs) of the $D^{(\ast)0}D_{(s)}^{(\ast)-}$ systems. Since these charm-meson--antimeson pairs are not $G$-parity eigenstates, their CFs contain contributions from both the $C$-odd and $C$-even sectors, providing direct access to the dynamics underlying the $Z_c$, $Z_{cs}$, and the predicted isovector exotic $W_{c1}$. Within a heavy-quark-spin-symmetric coupled-channel framework, we show that the $C$-even admixture enhances the low-momentum CFs by more than $2.5\sigma$ in the vicinity of their thresholds. Free from Coulomb distortions and accessible in high-multiplicity $pp$ collisions at the LHC, these channels offer the first direct femtoscopic probe of the isovector $C$-even sector and of the elusive $W_{c1}$ state.

Figures

Figures reproduced from arXiv: 2608.01237 by the authors.

Figure 1
Figure 1. FIG. 1. CFs for the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Same as Fig [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Transverse momentum distributions of [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Dependence of the non-strange scattering lengths (upper [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]

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