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REVIEW 4 major objections 5 minor 44 references

The Study of Pole Trajectory within a bare state in the coupled channel model

T0 review · 4 major / 5 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read A bare state guarantees a nearby physical pole in this model.

desk verdict Useful toy-model map of pole trajectories and compositeness in one-bare-state HEFT, but the 'always' claim outruns the regulator-dependent evidence. read the letter →

arxiv 2506.02526 v1 pith:SUBFV7GB submitted 2025-06-03 hep-ph nucl-th

Wei Hao , Jia-Jun Wu , Jin-Lin Fu This is my paper
classification hep-phnucl-th
keywords poletrajectoriesbarestatecoupledchannelscompositenessformfactorresonancesboundstatesrelativisticscattering
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 studies how an extra bare state—a discrete basis state with no two-particle constituents—changes the poles of a two-body scattering amplitude in a simple one-channel model. Working in the complex momentum and energy planes, the authors show that under relativistic kinematics the square-root dispersion relation creates resonance-like poles below threshold, something non-relativistic models do not produce. They then show that the number of poles produced by the model depends on the form-factor regulator, and that most of those poles are artifacts far from threshold. The central result is that whenever a bare state is coupled to the two-body channel, a physical pole always appears near the bare-state mass; depending on the two-body interaction and the couplings, that pole is a bound state, a resonance, or a virtual state. They also compute the bare-state probability in shallow S-wave bound states and find it tends to vanish at threshold, so such states look essentially molecular even when the bare state is dynamically important.

What carries the argument

The load-bearing object is the pole trajectory: the path traced in the complex energy plane by T-matrix poles as the coupling constants $v$ (two-body) and $g$ (bare-to-channel) are varied. The model uses separable interactions, of the form $v f(k)f(k')$, so the pole condition reduces to a one-dimensional algebraic equation $\det\{1 - v\int dq\, q^2 f(q)^2/(E-\omega(q))\}=0$, with relativistic channel energy $\omega(k)=\sqrt{m_M^2+k^2}+\sqrt{m_N^2+k^2}$. The form factor $f$ carries powers of the regulator $u(k)=(1+k^2/\Lambda^2)^{-2}$, and that power determines how many poles the equation has. For the case with a bare state, the full amplitude is built from the dressed vertex and the one-loop self-energy, and full poles solve $\det\{\delta_{BB'}(E-m_B)-\overline{\Sigma}_{BB'}(E)\}=0$. The same self-energy derivative defines the bare-state probability $1/Z$ of a bound state.

What would settle it

Repeat the calculation with regulator $u(k)=\exp(-k^2/\Lambda^2)$ or a different power $n$; if the trajectory starting at the bare-state mass ceases to produce a physical pole near that mass, or if $1/Z$ no longer vanishes for shallow S-wave bound states, the claims are regulator artifacts.

Watch

Extended reading notes

Core claim

The paper's central claim is that the presence of a bare state in a one-channel two-body scattering model exerts a systematic, predictable influence on the physical poles: if the bare state exists, a physical state will always emerge in its vicinity. The bare state acts as an energy-dependent two-body interaction that is attractive below the bare mass and repulsive above it; the relative position of the bare mass to the threshold therefore decides whether the adjacent pole is a bound state or a resonance. In addition, the authors establish that the smooth regulator $u(k)=(1+k^2/\Lambda^2)^{-2}$ generates one extra pole per power of the denominator, that only near-threshold poles are robust, and that for S-wave shallow bound states the bare-state component $1/Z$ tends to zero at threshold, making the compositeness approach unity.

Load-bearing premise

The whole pole-count and near-threshold compositeness pattern rests on the chosen separable potential and the smooth regulator $u(k)=(1+k^2/\Lambda^2)^{-2}$; a different regulator can change the number of physical-looking poles and the behavior of $1/Z$ at threshold.

Editorial extensions

If this is right

  • Near-threshold S-wave bound states in this model are almost entirely two-body composites: $1/Z$ goes to zero as binding energy goes to zero, so their compositeness is close to 1.
  • If the bare state sits below threshold, the two-body attraction creates one bound state near the bare mass and can push another pole above it into a virtual state or resonance; the ordering of bare mass and threshold is a controlling parameter.
  • Two-body repulsion can still produce a bound state when a bare state is present, because the bare-state exchange provides the attraction; this suggests searching for hadrons whose molecular constituents repel.
  • Poles far from threshold, including some resonances with large widths, are artifacts of the chosen regulator and should be discarded when fitting data.
  • Relativistic kinematics allows resonance-like poles with real part below threshold, so experimental analyses should not assume a below-threshold complex pole is unphysical.

Reading between the lines

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

  • The claimed 'always a pole near the bare state' likely generalizes to any model with an energy-dependent separable interaction of the same sign structure, not only this regulator; a direct test would be to repeat the trajectory study with a dipole or exponential form factor.
  • The threshold behavior of compositeness may offer a model-independent observable: measurement of a near-threshold S-wave state's binding energy and scattering length constrains its molecular probability, and this model shows the bare-state component is suppressed there, so a measured large bare component would falsify this class of models.
  • The same pole-trajectory machinery could be applied to finite-volume spectra, connecting the bare-state pole positions to lattice-QCD energy levels, because the model's Hamiltonian form gives both infinite- and finite-volume amplitudes.
  • The form-factor pole counting may explain why fits with different regulators obtain different numbers of resonances from the same data; a comparison of phase-shift fits across regulators would quantify the systematic uncertainty.
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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

4 major / 5 minor

Summary. The paper studies a one-channel, one-bare-state Hamiltonian effective field theory (HEFT) with separable potentials, solving the Lippmann-Schwinger equation for T-matrix poles in both non-relativistic and relativistic kinematics. It makes three main claims: (i) the number of T-matrix poles increases with the power n of the regulator u(k) = (1+k^2/Λ^2)^{-n}; (ii) in the absence of a bare state, attractive and repulsive two-body couplings generate pole trajectories for bound, virtual, and resonance states, with relativistic kinematics producing below-threshold resonances; and (iii) when a bare state is included, a physical pole always appears in the vicinity of the bare mass, and for shallow S-wave bound states the bare-state component tends to vanish at threshold unless the two-body potential is repulsive. The paper draws qualitative comparisons to X(3872) and D*_s0(2317).

Significance. If the central claims hold, the paper provides a useful qualitative map of how a bare state reshapes near-threshold pole structure in HEFT, and it offers falsifiable expectations for near-threshold states. The analytic loop integrals for n=1 and n=2 in the non-relativistic case and the direct solution of the coupled-channel equations are valuable, and the discussion of which poles are form-factor artifacts is a useful caution. However, the generality of the 'a physical state always emerges in the vicinity of a bare state' statement and the claimed universal threshold behavior of 1/Z are not established beyond the single smooth regulator used; the paper itself concedes in Section V that the form factor is an assumption. The underlying derivation is sound enough that the issues are fixable, but the conclusions need to be qualified and several technical points need to be clarified.

major comments (4)
  1. [III, Eqs. (31)-(34)] The paper states that the number of T-matrix poles equals n and that 'it can be proven analytically in the non-relativistic case,' but the text supplies only the n=1 and n=2 examples and no general proof. Since the subsequent exclusion of 'model-dependent' poles relies on this pole-counting statement, either supply the promised proof (for example, by deriving the polynomial degree of the analytically continued loop function) or soften the claim to a conjecture supported by examples. In addition, Eq. (34) contains an apparent typo in the term '16i√μ√(k0^2)μ', which prevents the reader from reproducing the claimed n=2 pole positions.
  2. [IV.B and VI] The conclusion that 'if a bare state exists, a physical state will always emerge in its vicinity' is drawn from pole trajectories computed for a single regulator u(k) = (1+k^2/Λ^2)^{-2} with Λ = 0.8 GeV and two arbitrarily chosen bare masses. In Section V the authors themselves write that the results 'suffer the uncertainties of the form factor which form actually is an assumption.' Since Section III shows that the form factor changes the number and location of poles, the 'always' claim and the unqualified summary in Section VI need to be restricted to this toy regulator or tested with other regulators or powers n; without that, the claim is a property of the chosen model, not of the coupled-channel framework.
  3. [V, Eq. (30)] The relation between Z and 1/Z is inconsistent. Equation (29) normalizes the bound state with coefficient 1/Z^{1/2}, so the bare-state probability is 1/Z, but the sentence following Eq. (30) says that the probability of finding the bare component is 'Z = 1 - dΣ/dE'. Section V then plots 1/Z as the bare-state probability. Please correct the definition or the wording; as written, the interpretation of Figs. 11-16 is ambiguous.
  4. [V, Figs. 11-16] The universal statement that 'for the S-wave bound state ... the compositeness always becomes 1 when it is a shallow bound state' is supported only by selected values of v and g. The threshold derivative of the self-energy depends on the regulator and on the masses, and Figs. 12-13 show that the behavior of Re Σ at threshold changes qualitatively with v. Either prove the threshold limit analytically from Eq. (31) or present a scan over the parameter space. The corresponding P-wave statement that the loop function has no extremum at threshold is likewise obtained for the specific form factor f(k) and needs the same qualification.
minor comments (5)
  1. [Title] The title contains 'couple d channel' with an unwanted space; please fix.
  2. [General] There are several typographical errors: 'possibilities' should be 'probabilities', 'Hear' should be 'Here', and 'out of the current model' should be 'outside the current model'.
  3. [Figures 7-10] The figure captions contain garbled text such as 'H e WBMVF' and '3F &'; please regenerate the captions so that the fonts render correctly.
  4. [II.A and III] Equation (11) defines the S-wave form factor with a factor of 5 and a factor 1/ω_M(k), while Eq. (32) uses a simpler '2/(1+(k/Λ)^2)^n'. Please clarify whether the illustrative form in Section III is meant to be a different choice or a shorthand, so that the pole-counting discussion can be connected to the numerical calculations in Section IV.
  5. [IV.A] The captions of Figs. 5 and 6 are missing 'changing with' in 'The pole positions with S-wave interaction strength' and 'The pole positions with P-wave interaction strength'; please reword for clarity.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: poles are solved from the model equations, and the regulator dependence is explicitly acknowledged as an assumption.

full rationale

The derivation is self-contained: pole positions are obtained by solving the algebraic determinant conditions (14) and (24) with the stated separable-potential and regulator inputs; the compositeness 1/Z is evaluated from the derived relation Z = 1 - dSigma/dE at the solved bound-state energy (Eqs. 28-30), so no fitted parameter is renamed as a prediction. The 'bare state always yields a nearby physical pole' statement follows from the pole equation E - mB - Sigma(E) = 0 starting from the g=0 pole at mB; the paper itself calls this 'obviously' and uses it only as a model property, not as a fit. The phase-shift fits in Section III are used to screen form-factor poles, not to produce the trajectory or compositeness results. The regulator sensitivity is explicitly acknowledged in Section V ('the Sigma(E) would suffer the uncertainties of the form factor which form actually is an assumption'), so the concern about regulator dependence is a robustness/overgeneralization issue, not circularity. Self-citations to earlier HEFT papers provide the regulator convention, but the paper treats the form factor as a toy-model assumption rather than as an externally certified input, so the citations are not load-bearing in the derivation.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

No new particles, forces, or conserved quantities are introduced. The bare state is a standard HEFT basis element. The central claims rest on the chosen form of the separable potential and regulator, which the authors themselves flag as an assumption.

free parameters (6)
  • two-body coupling v = scanned over ranges, e.g., -0.2 to 0.09 in figures
    Varies the strength of the separable potential; central to the trajectory plots.
  • bare-state coupling g = scanned over ranges
    Controls the coupling between the bare state and the two-body channel; central to the b-c model trajectories.
  • regulator Lambda = 0.8 GeV (1 GeV in Section III examples)
    Chosen by hand as a typical value; affects pole counting and compositeness behavior.
  • particle masses mM and mN = 0.2 GeV and 0.8 GeV
    Arbitrary masses chosen for generality; the threshold position and Riemann-sheet structure depend on them.
  • bare mass mB0 = 1.3 GeV or 0.7 GeV
    Set above or below threshold to study both cases; directly influences whether the bare state generates a bound state or resonance.
  • form-factor power n = n=2 in Section IV; n=1,2,3 in Section III
    The form-factor power controls the number of T-matrix poles; it is chosen as a toy parameter.
assumptions (5)
  • standard math The Lippmann-Schwinger equation with the i-epsilon prescription yields the T-matrix poles on the appropriate Riemann sheets.
    Used throughout to locate poles in the momentum and energy planes.
  • domain assumption Interactions are separable potentials with the form factor u(k)=(1+k^2/Lambda^2)^-2.
    Defined in Eqs. (10)-(12); the specific form is an assumption and affects the number and location of poles.
  • domain assumption The relativistic dispersion relation omega(k)=sqrt(m1^2+k^2)+sqrt(m2^2+k^2) holds.
    Used in Eq. (4) and all numerical calculations; responsible for the differences from the non-relativistic case.
  • ad hoc to paper The model is only reliable near threshold; far-away poles are discarded as form-factor artifacts.
    Introduced in Section III when screening poles; this screening is necessary for the physical interpretation of the trajectories.
  • domain assumption The bare state and its coupling in the Hamiltonian represent an elementary particle, and its probability in a bound state is read off from Z.
    Standard HEFT background, used for the compositeness analysis.

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Cite this review

Pith. "Pith review of The Study of Pole Trajectory within a bare state in the coupled channel model." pith.science (2026). https://pith.science/paper/SUBFV7GB

@misc{pith2026250602526,
  author       = {Pith},
  title        = {Pith review of: The Study of Pole Trajectory within a bare state in the coupled channel model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SUBFV7GB}},
  note         = {Machine review of arXiv:2506.02526}
}
read the original abstract

We investigate two-particle scattering and two-particle scattering with a bare basis state using Hamiltonian Effective Field Theory (HEFT). We analyze the distribution of two-body scattering poles in the momentum and energy planes under relativistic conditions. Compared to the non-relativistic case, there are significant differences in the distribution of bound state poles and resonance poles in the relativistic case, primarily due to the square root term in the relativistic formula. By considering pure two-particle scattering, we examine the relationship between the form factor and the number of poles. Additionally, we clearly elucidate the effects of attractive and repulsive interactions on the bound state poles and resonance poles. More importantly, we extend our model by including a bare state and explore the poles originating from the bare state or coupled channels through the trajectories of pole positions, as well as the compositeness of bound states.

Figures

Figures reproduced from arXiv: 2506.02526 by the authors.

Figure 1
Figure 1. FIG. 1: The complex momentum [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The complex momentum [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The result of fitting the phase shift of n=3 [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4: The result of fitting the phase shift of n=3 [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: The pole positions with [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The pole positions with [PITH_FULL_IMAGE:figures/full_fig_p008_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p010_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p012_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10 [PITH_FULL_IMAGE:figures/full_fig_p012_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The 1 [PITH_FULL_IMAGE:figures/full_fig_p013_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12: The variation law of the real part Σ [PITH_FULL_IMAGE:figures/full_fig_p013_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14: The 1 [PITH_FULL_IMAGE:figures/full_fig_p014_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15: The 1 [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]
Figure 16
Figure 16. Figure 16: FIG. 16: The 1 [PITH_FULL_IMAGE:figures/full_fig_p014_16.png]

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Reviewed August 7, 2026 · model on record in the stance chip above.