Effect of a repulsive three-body interaction on the $DD^{(*)}K$ molecule

Atsushi Hosaka; Emiko Hiyama; Jun-Xu Lu; Li-Sheng Geng; Ya-Wen Pan

REVIEW 3 major objections 4 minor 1 cited by

Effect of a repulsive three-body interaction on the $DD^{(*)}K$ molecule

T0 review · 3 major / 4 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read A sufficiently strong repulsive three-body interaction does not unbind the $DD^{(*)}K$ molecule; it splits it into a $D^{(*)}K$ bound pair and a distant $D$ meson.

desk verdict Solid GEM follow-up with a useful spatial-mechanism story, but the infinite-repulsion 'break-up' is extrapolated, not computed. read the letter →

arxiv 2502.00438 v1 pith:6T5J5ACO submitted 2025-02-01 nucl-th hep-lathep-ph

Ya-Wen Pan , Jun-Xu Lu , Emiko Hiyama , Li-Sheng Geng , Atsushi Hosaka This is my paper · ORCID

classification nucl-thhep-lathep-ph

keywords DD*Kmoleculethree-bodyinteractionGaussianexpansionmethodhadronicexoticstatesrepulsiveforcebindingenergyspatialconfiguration

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 asks what an arbitrarily strong repulsive three-body force does to a hadronic molecule built from two charmed mesons and a kaon ($DD^{(*)}K$). Solving the three-body Schrödinger equation with the Gaussian expansion method, it finds that increasing the repulsion first swells the bound state and then reorganizes it: in the infinite-strength limit the three-body state does not unbind but turns into a $D^{(*)}K$ two-body bound state, with the remaining $D$ pushed far away. The binding energy falls from about 77.8 MeV ($DD^*K$) and 71.2 MeV ($DDK$) and asymptotically approaches the corresponding two-body $D^{(*)}K$ binding energies, about 46.1 and 45.0 MeV. This matters because it gives a concrete mechanism by which three-body forces select which two-body cluster survives inside a molecular state, and it ties the asymptotic energies to the $DK$/$D^*K$ molecular binding energies familiar from the $D_{s0}^*(2317)$ and $D_{s1}(2460)$ states.

What carries the argument

The load-bearing object is the local, purely repulsive three-body potential of Eq. (11), $$V_{123}(r,R)=C\,\frac{$e^{{-r^2/a_1^2}}$}{\$pi^{{3/2}}$$a_1^{3}$}\frac{$e^{{-R^2/a_2^2}}$}{\$pi^{{3/2}}$$a_2^{3}$},$$ with fixed ranges $a_1=0.5$ fm and $a_2=0.4$ fm and strength $C$ scanned from $0$ to $\infty$. It is added to the two-body $DD^{(*)}$ one-boson-exchange potential and the Weinberg–Tomozawa $D^{(*)}K$ potential, and the three-body Schrödinger equation is solved by expanding the wavefunction in Gaussian basis functions on the three Jacobi coordinate channels, with symmetrization for the two identical $D$'s in $DDK$. The potential's role is to suppress configurations with all three hadrons overlapping; the paper shows this pushes the system through a saddle-point bifurcation in the effective $K$–$DD^{(*)}$ potential, selecting the $D^{(*)}K$ cluster with the smaller reduced mass as the survivor.

What would settle it

Solve the same $DD^{(*)}K$ problem with a three-body potential of a different functional form, for example a momentum-dependent term or a Gaussian with different ranges, and check whether the infinite-strength limit still produces a $D^{(*)}K$ two-body cluster plus an infinitely distant $D$ meson with binding energy equal to the two-body value. If the asymptotic dissociation disappears or the saturation value shifts, the breakup picture is specific to the chosen contact form rather than a general property of repulsive three-body forces.

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Extended reading notes

Core claim

The central claim is that a repulsive three-body force in the $DD^{(*)}K$ systems does not simply weaken the molecule: it drives a structural transition. Within the Gaussian expansion method, the three-body interaction is a local repulsive Gaussian added to the two-body $DD^{(*)}$ and $D^{(*)}K$ potentials, and the ground state is followed as the strength $C$ grows from zero to infinity. In the first stage ($C \lesssim 2 \times 10^5$ MeV) all inter-hadron distances grow and the binding energy drops from 77.8 MeV ($DD^*K$) or 71.2 MeV ($DDK$). In the second stage the $D^{(*)}K$ distance starts to shrink again while the other $D$ moves away; at $C \to \infty$ the three-body binding energy approaches the two-body $D^{(*)}K$ binding energy (46.1 MeV for $D^*K$, 45.0 MeV for $DK$), the $D^{(*)}K$ pair has the same rms radius as the isolated two-body molecule (1.27 fm for $D^*K$), and the spectator $D$ is infinitely far away. The mechanism is an effective potential between the kaon and the $DD^{(*)}$ cluster that develops a saddle point and splits into separate minima, suppressing transitions between the $[D^{(*)}K]D$ and $[DK]D^{(*)}$ configurations. For $DDK$, the two identical $D$ mesons keep the two configurations equally probable, so the expanding system retains an isosceles-triangle shape.

Load-bearing premise

The calculation assumes the three-hadron interaction is a local, purely repulsive Gaussian contact force with fixed ranges, and that letting its strength go to infinity is a physically meaningful limit.

Editorial extensions

If this is right

At infinite three-body repulsion, each of the two systems studied remains bound, with the binding energy approaching the corresponding two-body $D^{(*)}K$ binding energy rather than zero.
The $DD^*K$ ground state becomes purely the $[D^*K]D$ configuration ($P_1=100\%$), because the smaller reduced mass of the $D^*K$ pair makes it kinetically favored over $[DK]D^*$.
The $DDK$ system stays an equal 50/50 superposition of $[D(1)K]D(2)$ and $[D(2)K]D(1)$, and its spatial shape is always an isosceles triangle, expanding as the repulsion grows.
The two-stage evolution (swelling followed by breakup) is governed by a saddle point that appears in the effective $K$–$DD^{(*)}$ potential; once the saddle rises above zero, the transition between the two cluster configurations is suppressed.
The asymptotic binding energies match the $D^{(*)}K$ two-body values, so the dissociation is not a loss of binding but a transfer of all binding into the surviving pair.

Reading between the lines

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

This dissociation mechanism should be generic: in any three-body molecule where one pair is more tightly bound than the others, a strong repulsive three-body force will drive the system toward the most kinematically favored two-body pair plus a distant spectator.
The coincidence of the asymptotic $DD^{(*)}K$ binding energies with the $DK$/$D^*K$ molecular binding energies of $D_{s0}^*(2317)$ and $D_{s1}(2460)$ hints that strong repulsive three-body effects in nature could be suppressing the full three-body states in favor of the two-body molecules; a direct lattice determination of the three-body force strength would turn this coincidence into a quantitati
Finite-volume or momentum-correlation measurements could look for the predicted signature: a nearly free $D$ meson with a growing separation from a $D^{(*)}K$ cluster, at binding energy nearly independent of the three-body strength in the second stage.

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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

Two things you should know up front. The paper is a solid GEM calculation that reproduces the lattice EFT binding-energy trend for DD*K and DDK, and it adds a genuinely new layer: the spatial evolution of the system (rms radii, Jacobi-channel probabilities, an illustrative effective potential) as the repulsive three-body strength C grows. The second thing: the central claim that the three-body bound state 'breaks into' a D(*)K two-body bound state plus a distant meson at C=∞ is not actually computed. The C=∞ rows in Tables I and II are extrapolations. The finite Gaussian basis with rmax=50 fm cannot represent the threshold continuum, and at the largest finite C shown (8×10^5 MeV), ⟨rDK⟩=18.93 fm — far from the quoted ∞. The binding energy has converged, but the radii have not. So the qualitative picture is plausible, but the abstract overstates the result.

What is new and useful: the two-stage behavior — the system first expands, then re-clusters around the more attractive D*K pair — is a helpful way to understand why the binding energy saturates at the two-body value. The isosceles-triangle discussion for DDK follows cleanly from the identical D's. The code reproduces prior binding energies (77.8 vs 79.1 MeV), and the authors are transparent that the channel probabilities are approximate (through ⟨V⟩ expectation values) and that the effective potential uses an illustrative k=1.3. That is honest.

The soft spots, in proportion: the three-body potential is a parameterized local Gaussian with unconstrained strength and shape; the authors check range sensitivity but do not pin C to any data. The 'dissociation' language should be softened to a threshold picture — what the calculation shows is the wavefunction approaching a two-body cluster plus a delocalized spectator, which the basis can only mimic. A short discussion of the finite-basis/continuum limitation would fix the main issue. The effective-potential approximation is minor and clearly labeled.

Bottom line: a worthwhile extension of an established program, not a breakthrough. I would send it to peer review, asking for a softened abstract and an explicit statement of the basis limitation. Specialists in hadronic molecules will get value from it; others can skim.

Referee Report

3 major / 4 minor

Summary. This manuscript uses the Gaussian expansion method to examine how a repulsive three-body interaction affects the DD*K and DDK hadronic molecules. The two-body potentials are taken from previous work; the three-body force is a local Gaussian with strength C. Without C, the authors reproduce binding energies of 77.8 MeV for DD*K and 71.2 MeV for DDK, consistent with earlier GEM and lattice results. As C increases, binding energies decrease monotonically toward the D(*)K two-body values. The authors compute rms radii and effective potentials, and interpret the results as a two-stage evolution: first the three-body system expands, then it dissociates into a D(*)K two-body bound state with a distant D meson. For DDK, the symmetry of the two identical D mesons leads to an isosceles-triangle configuration. The paper concludes that an infinitely strong repulsive three-body interaction does not destroy the two-body cluster but rather separates the third meson.

Significance. The paper is a useful model study in the hadronic-molecule program: it connects the DD*K three-body binding energy to the D*K two-body binding energy and offers a transparent effective-potential explanation. The reproduction of the lattice EFT binding energy (77.8 vs 79.1 MeV) and the systematic GEM study of the C-dependence are strengths, as is the explicit discussion of the spatial configuration. However, the central dissociation claim is currently presented more strongly than the calculation supports, because the infinite-C limit is an extrapolation from a finite basis rather than a computed threshold. With a careful revision that either establishes the limit or softens the claim, the paper would be a solid contribution.

major comments (3)

[Sec. III, Tables I and II; Sec. IV] The rows labeled C=∞ are not obtained by direct GEM diagonalization. A finite Gaussian basis with rmax=50 fm cannot represent a scattering threshold consisting of a two-body bound state times a zero-energy spectator plane wave. At the largest finite strength shown (C=8×10^5 MeV), Table I gives ⟨rD∗K⟩=1.93 fm and ⟨rDK⟩=18.93 fm, still far from the claimed ∞ values of 1.27 fm and ∞, even though the binding energy has essentially converged (46.1 MeV). The same issue affects Table II, where ⟨R1⟩1 at C=8×10^5 MeV is 3.37 fm while the ∞ row states ∞. The assertion in the abstract and Sec. IV that the system 'breaks into a D(*)K two-body bound state with a distant D meson' is therefore an extrapolation beyond the computed regime. The authors should either provide a convergence study at large C (e.g., increasing rmax and Nmax, or explicitly treating the asymptotic channel) or rephrase the conclusion as a plausible inference from the trends.
[Sec. III, Table I and surrounding text] The channel probabilities P1 and P2 are not true probabilities: they are computed from the expectation values of the two-body potentials (⟨VD∗K⟩ and ⟨VDK⟩) in a non-orthogonal Gaussian basis, as the authors note. The C=∞ row reports P1=100%, and this value is used to support the conclusion that the final state is a pure D∗K cluster. Because the basis is non-orthogonal, these expectation values are not a proper decomposition of the wave function, and the 100% figure is an approximation at best. The authors should either provide an orthogonal decomposition or explicitly label these quantities as potential-weighted measures and avoid drawing exact conclusions from them.
[Sec. II, Eq. (11)] The three-body potential is assumed to be a local Gaussian with fixed ranges a1=0.5 fm and a2=0.4 fm, and the infinite-C limit is taken within this form. The authors state that varying the effective ranges does not alter the qualitative conclusions, but no sensitivity study is shown. Since the C→∞ limit of a repulsive contact potential is regulator-sensitive, the claim that the system dissociates into a two-body bound state plus a far-separated spectator could depend on the chosen Gaussian shape. Please provide a quantitative test of this assumption, for example by varying a1 and a2 (or using a different regulator) for a representative large-C case, or by explaining why the limit is independent of the regulator.

minor comments (4)

[Sec. III] The text contains several typos: 'Sine the attraction' should be 'Since the attraction', 'in the second state' should be 'in the second stage', and one sentence refers to the '[D∗K]K configuration' instead of '[D∗K]D'.
[Tables I and II] The table headers use P1 and P2 but the captions do not define them; a definition in the caption or in the text would improve readability.
[Figs. 4 and 6] The C values for panels (d)–(f) are hard to read because the superscript notation is partially cut off; consider reporting the values in the caption or using a clearer format.
[Sec. III] The symbol ⟨r⟩ is introduced as the rms radius of the Jacobi coordinate, but for the DDK system the authors later clarify that ⟨rc⟩c does not equal the physical interparticle distance; this caveat should appear earlier, before the physical interpretation is given.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: externally calibrated two-body inputs, scanned three-body strength, and a model consequence stated as such.

full rationale

The paper's central observable is the dependence of the DD(∗)K binding energy and geometry on the strength C of a repulsive three-body contact potential. The two-body D(∗)K and DD(∗) potentials are taken from earlier work but were calibrated to the external states Ds0(2317) and X(3872), so they provide independent inputs rather than encoding the target result. The three-body strength C is scanned, not fitted to the reported binding energies or radii; the asymptotic approach to the two-body binding energy is a consequence of the model Hamiltonian in which the repulsive potential vanishes when one particle is far away, and the paper presents it as an implication rather than a fitted value. There is no equation in which the output is defined in terms of the conclusion, no fitted parameter renamed as a prediction, and no load-bearing uniqueness claim resting on self-citations. The possible numerical concern that the finite Gaussian basis cannot represent a true continuum at C=∞ is a question of extrapolation and convergence, not of circularity.

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

The central calculation rests on inherited two-body potentials fitted to X(3872) and Ds0(2317), an ad hoc Gaussian three-body potential with unconstrained strength, and a finite Gaussian basis. No new particles or conserved quantities are invented. The qualitative conclusions are presented as robust under changes of the three-body ranges, but the overall form of the three-body force is assumed.

free parameters (6)

C (three-body strength) = 0 to infinity (scanned)
Strength of the repulsive three-body contact potential in Eq. (11); it is scanned over (0, infinity) and not fitted to the target binding energies. The central claim is a statement about this scan.
a1 (three-body Gaussian range for r) = 0.5 fm
Ad hoc choice for the Jacobi coordinate r; the paper states varying it does not change qualitative conclusions, but no sensitivity data are shown.
a2 (three-body Gaussian range for R) = 0.4 fm
Ad hoc choice for the Jacobi coordinate R, same caveat as a1.
Lambda (DD* form-factor cutoff) = 1.01 GeV
Inherited from Ref. [43], where it was fit to reproduce X(3872) as a D D* bound state with 4 MeV binding.
C_L (D(*)K contact strength) = -320.1 MeV
Inherited from Ref. [43], fit to reproduce Ds0(2317) as a DK molecule.
b (D(*)K Gaussian range) = 1 fm
Set in Ref. [43] together with C_L; used as the coordinate-space regulator for the Weinberg-Tomozawa potential.

assumptions (5)

domain assumption The GEM variational basis with nmax=40, Nmax=30 and ranges 0.01-50 fm converges to the exact bound-state solution of the three-body Schrödinger equation.
Invoked in Sec. II; the paper states that further optimization does not change the results, but no systematic convergence study is shown.
domain assumption The two-body interactions for DD*, DK, and D*K obtained in Refs. [43-45] are valid inputs for the DD(∗)K systems.
Used throughout; these potentials were calibrated to X(3872) and Ds0(2317), and they reproduce prior DD*K and DDK binding energies.
ad hoc to paper The three-body interaction has the local Gaussian contact form of Eq. (11) and is purely repulsive, and the infinite-strength limit C to infinity is a legitimate model limit.
Introduced in Sec. II following the LO Lagrangian of Ref. [46]; the regulator shape and ranges are chosen for convenience and are not derived from data.
standard math Identical D mesons in DDK are fully symmetrized under exchange, and the total wavefunction can be approximated by the sum over Jacobi channels c=1-3 with only S-wave couplings.
Sec. II, Eqs. (2)-(5); spin and higher partial waves are neglected.
domain assumption The channel c=3 (DD* cluster) configuration contributes negligibly to the DD*K bound state.
Sec. III states 'the contribution from channel c = 3 can be neglected'; this reduces the configuration analysis to two channels.

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Cite this review

Pith. "Pith review of Effect of a repulsive three-body interaction on the $DD^{(*)}K$ molecule." pith.science (2026). https://pith.science/paper/6T5J5ACO

@misc{pith2026250200438,
  author       = {Pith},
  title        = {Pith review of: Effect of a repulsive three-body interaction on the $DD^(*)K$ molecule},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6T5J5ACO}},
  note         = {Machine review of arXiv:2502.00438}
}

abstract

The hadronic molecular picture of the observed exotic states has inspired numerous investigations into few-body systems. Recently, the lattice effective field theory studied the effect of a three-body interaction on the binding energy of the $DD^{*}K$ system, revealing an intriguing phenomenon in the binding energy. This work uses the Gaussian expansion method to explore the underlying physics. Our results show that as the repulsive three-body interaction strengthens, the spatial size of the $DD^{(*)}K$ bound state gradually increases. Further enhancement of the three-body interaction causes the $DD^{(*)}K$ three-body bound state to break into a $D^{(*)}K$ two-body bound state, accompanied by a distant $D$ meson. The identical nature of the two $D$ mesons leads to the fact that the $DDK$ system consistently resembles an isosceles triangle-shaped spatial configuration.

Figures

Figures reproduced from arXiv: 2502.00438 by the authors.

**Figure 3.** There are two stages in the evolution. In the first stage, [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 2.** FIG. 2: Binding energies of the [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: FIG. 4: The effective potential between the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]

**Figure 6.** Figure 6: FIG. 6: The effective potential between the [PITH_FULL_IMAGE:figures/full_fig_p007_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Low-energy $DD$ scattering in lattice QCD
hep-lat 2025-02 conditional novelty 6.0 of 10

The first lattice QCD calculation of single-channel DD scattering finds a weakly repulsive S-wave isovector interaction and a slightly attractive P-wave isoscalar interaction.

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Reviewed August 9, 2026 · model on record in the stance chip above.

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