Faddeev equations for the $J/\psi\,NN$ and $\phi\,NN$ three-body systems in momentum space

Xu Zhang

REVIEW 3 major objections 7 minor 127 references

Faddeev equations for the $J/\psi\,NN$ and $\phi\,NN$ three-body systems in momentum space

T0 review · 3 major / 7 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read Solving three-body Faddeev equations with lattice QCD two-body inputs, the paper finds phi-N-N bound states in three isospin-zero channels and no J/psi-N-N bound state.

desk verdict Solid momentum-space Faddeev calculation, but the phi NN bound states it advertises are inherited from earlier work and rest on the least reliable input in the model. read the letter →

arxiv 2608.03162 v1 pith:LBQBRBMX submitted 2026-08-04 hep-ph nucl-th

Xu Zhang This is my paper · ORCID

classification hep-phnucl-th

keywords Faddeevequationsthree-bodyboundstatesphiNNsystemJ/psiphi-nucleoninteractionLippmann-SchwingerequationlatticeQCDpotentialsS-waveinteractions

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 whether a heavy quark-antiquark meson (a phi or a J/psi) can bind to a pair of nucleons to form a three-body state. It sets up exact Faddeev equations in momentum space, feeding them two-body interactions: a standard two-Yukawa nucleon-nucleon potential, lattice QCD potentials for J/psi-nucleon and spin-3/2 phi-nucleon, and two constructed versions of the poorly known spin-1/2 phi-nucleon force. Solving the equations, it finds no J/psi-N-N bound state but bound phi-N-N states in isospin-zero channels with $(I)J^P=(0)0^-$, $(0)1^-$, and $(0)2^-$, with binding energies from about 2.4 to 50 MeV depending on the spin-1/2 input. If true, a phi meson could bind to a deuteron near threshold, giving an observable three-body hadronic molecule.

What carries the argument

The engine is the Faddeev equations in momentum space, the exact coupled integral equations for three-body bound states. The two-body subsystem t-matrix comes from the Lippmann-Schwinger equation, solved by matrix inversion; the three-body equations are then discretized with cubic-spline interpolation and Gauss-Legendre quadrature, and the bound-state condition is recast as an auxiliary eigenvalue problem $K(z)|\varphi(z)\rangle = \lambda(z)|\varphi(z)\rangle$, with the physical binding energy found where $\lambda(E)=1$. Only S-wave two-body interactions are kept. The two controls are the constructed spin-1/2 phi-nucleon potentials: scheme I scales the spin-3/2 lattice potential by a factor fitted to measured phi-proton correlation functions, while scheme II assumes the spin-spin part of the phi-N and J/psi-N interactions is inversely proportional to the hadron masses.

What would settle it

Measure the near-threshold $\gamma d \to \phi d$ cross section; a bound phi-N-N state would appear as a sharp near-threshold enhancement whose energy sets the binding energy. Alternatively, a lattice QCD calculation of the spin-1/2 phi-N potential including the Lambda-K and Sigma-K coupled channels would determine whether the $0^-$ and $1^-$ bound states survive.

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Extended reading notes

Core claim

The central numerical discovery is a sharp asymmetry between the charmonium and strange-meson three-body systems: with the two-body inputs adopted, the J/psi-N-N system has no three-body bound state, while the phi-N-N system supports bound states in the $(I)J^P=(0)0^-$, $(0)1^-$, and $(0)2^-$ channels in both schemes for the spin-1/2 phi-N force. The deepest state (about 50 MeV in scheme I) comes from the strongly attractive spin-1/2 phi-N interaction, which in that scheme even binds the two-body phi-N system; the shallowest (about 2.4 MeV) is the $2^-$ state, driven by the deuteron plus the spin-3/2 phi-N force and essentially identical in both schemes. In the isospin-one $1^-$ channel, where the nucleon pair is in the unbound $^1S_0$ channel, no bound state is found.

Load-bearing premise

The prediction rests on the strength of the spin-1/2 phi-nucleon interaction, a poorly known quantity for which lattice QCD provides no reliable potential because that channel is strongly coupled to open strange channels such as Lambda-K and Sigma-K, and both schemes are constructed guesses rather than measured inputs.

Editorial extensions

If this is right

A phi-N-N bound state would appear near threshold in photon-induced phi production on the deuteron, such as $\gamma d \to \phi d$, with the final deuteron energy reflecting the binding energy.
The $(0)2^-$ state is predicted in both schemes with essentially the same binding energy, making it the cleanest prediction to test.
J/psi-N-N is predicted not to bind, so the charmonium-nucleon force remains too weak to form this kind of hadronic molecule.
The spread between scheme I and scheme II in the $0^-$ and $1^-$ channels quantifies how much the unknown spin-spin part of the phi-N force controls the three-body spectrum.

Reading between the lines

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

Because the $(0)2^-$ state is insensitive to the spin-1/2 ambiguity, an experimental search for a phi-N-N bound state should look first in that channel; not finding it would call the spin-3/2 phi-N input into question rather than the spin-1/2 schemes.
The S-wave-only truncation probably underestimates the three-body attraction; adding higher partial waves and the open Lambda-K and Sigma-K channels could shift the binding energies and might even turn the isospin-one channel into a bound state.
Using the Faddeev wave functions to compute the $\gamma d \to \phi d$ amplitude would convert the existence prediction into a quantitative cross-section prediction, giving experiment a sharper target.

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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

Xu Zhang's paper is a competent, clearly written momentum-space Faddeev calculation for J/psi NN and phi NN. The main physics result — phi NN bound states, no J/psi NN bound state — is not new; the author says as much himself, citing Refs. [96-98] and [99]. What is new is the momentum-space formulation and numerical implementation, which is a legitimate technical extension. The derivation of the Faddeev equations and the geometrical coefficients is careful, and the two-body checks (deuteron binding, scattering lengths, effective range) reproduce the input potentials correctly. So as a numerical exercise, this is solid.

The soft spot is the load-bearing 2S1/2 phi N interaction. HAL QCD explicitly says this channel is strongly coupled to Lambda K and Sigma K and that the lattice potential there is unreliable. The paper acknowledges this and tries two schemes: scheme I scales the spin-3/2 potential with beta=6.9 taken from an ALICE correlation-function fit, and scheme II assumes spin-spin forces scale inversely with meson mass. Both schemes produce (0)0- and (0)1- phi NN bound states, but that agreement does not validate the input — it just shows the states are not sensitive to the difference between the two ad hoc prescriptions. There are no uncertainty bands, no convergence tests, and no coupled-channel treatment. The stress-test concern is right: a moderate reduction in the 2S1/2 strength could eliminate those two states. The paper should have scanned the strength of this potential and shown the binding energies versus beta, or at least given an error bar from the beta fit.

The (0)2- state, which uses only the well-determined 4S3/2 channel, is on much firmer ground, and the no-bound-state conclusion for J/psi NN is robust because the J/psi N interaction is genuinely weak. So the calculation is not without value; it just overstates what the two fragile channels establish.

I'd send it to a serious referee, but my own verdict would be conditional: publish if the author adds a strength scan and a convergence study, or when possible a coupled-channel estimate for the 2S1/2 channel. The paper is not circular — the three-body energies are computed, not fitted — and the presentation is honest about the HAL QCD caveat.

For a reading group, it's a maybe: useful if you work on few-body hadronic molecules or momentum-space Faddeev methods, but not a must-read for the general hep-ph audience.

Referee Report

3 major / 7 minor

Summary. This paper formulates momentum-space Faddeev equations for the J/psi NN and phi NN three-body systems with two identical nucleons. The two-body input consists of the Malfliet-Tjon NN potential, the HAL QCD J/psi N potentials in the 4S3/2 and 2S1/2 channels, the HAL QCD phi N potential in the 4S3/2 channel, and two constructed models for the phi N 2S1/2 channel: scheme I scales the HAL QCD 4S3/2 short-range part by beta=6.9 obtained from the ALICE phi p correlation function, and scheme II assumes that spin-spin interactions scale inversely with the hadron mass. The Lippmann-Schwinger equation is solved for the two-body t-matrix and the Faddeev equations are solved by discretization and diagonalization. The author reports no J/psi NN bound state and phi NN bound states in the (0)0-, (0)1-, and (0)2- channels in both schemes, with binding energies given in Tables V and VI.

Significance. The paper is a technically clean application of standard Faddeev machinery, and the two-body sector is checked by reproducing the input scattering lengths and the deuteron binding energy with the MT potential, which is a genuine internal consistency test. The strongest and least model-dependent result is the (0)2- phi NN bound state, which is driven by the well-determined HAL QCD 4S3/2 channel and is stable across the two schemes; this is a concrete prediction that could be probed in reactions such as gamma d -> phi d. The absence of a J/psi NN bound state is also consistent with the weak J/psi N interactions used. However, the (0)0- and (0)1- states rest on the phi N 2S1/2 interaction, which the author explicitly identifies as the least reliable input; the paper is transparent about this and offers two independent constructions, but neither construction is validated by data or lattice QCD. The falsifiable character of the prediction and the transparency about the unreliable channel are strengths of the manuscript.

major comments (3)

[Section III C, Eq. (38), Tables V and VI] The existence of the (0)0- and (0)1- phi NN bound states rests entirely on the phi N 2S1/2 potential, and that potential is the least secure input in the calculation. Ref. [90] states explicitly that in the 2S1/2 channel the phi N potential is strongly coupled to the S-wave Lambda K and Sigma K channels and that the lattice information in this channel is not reliable. Scheme I replaces this channel with Eq. (38), scaling the 4S3/2 short-range Gaussian terms by beta=6.9 without any uncertainty or sensitivity analysis, while scheme II replaces it with Eq. (41), an untested inverse-mass scaling hypothesis. Because no strength variation or coupled-channel estimate is reported, the paper does not establish that the (0)0- and (0)1- bound states survive a moderate reduction of this interaction. I request a scan in the strength of the 2S1/2 potential, for example varying beta around 6.9 or scaling the whole potential, that identifies the critical strength at which each of these states disappears, or an estimate of the coupled-channel uncertainty.
[Section IV C, Tables V and VI] The central numerical results are quoted as single numbers to two decimals with no propagated uncertainties and no numerical convergence tests. The input potentials carry uncertainties (Tables II and III), the factor beta has an unknown fit error, and the discretization parameters such as the number of Gauss-Legendre points, momentum cutoffs, and spline knots are not stated. For the (0)2- state with B3=2.39 MeV and for the scheme-II states with B3 around 3 MeV, this level of numerical detail is not sufficient to establish that the bound states are significant. Please add a convergence study and propagate at least the potential parameter uncertainties through the Faddeev equations.
[Section III C, Eq. (23)] In both schemes the phi N 2S1/2 t-matrix is computed as a single-channel S-wave t-matrix, yet the dominant issue identified by HAL QCD for this channel is not a parameter uncertainty but missing coupled-channel dynamics due to the S-wave Lambda K and Sigma K channels. A single-channel potential fitted to the elastic phi p correlation function can reproduce an effective scattering length, but it cannot reproduce the coupled-channel physics that, according to Ref. [90], makes the lattice potential unreliable in this channel. The manuscript should state this limitation explicitly and, if possible, estimate the effect using a coupled-channel two-body model or a coupled-channel Faddeev calculation, since the (0)0- and (0)1- phi NN states are directly carried by this channel.

minor comments (7)

[Table III] The caption states that 'alpha3 m_pi^4 and beta3 are in units of fm', which is dimensionally confusing because alpha3 m_pi^4 multiplies a 1/r^2 term; please clarify the units of the reported combination.
[Eq. (38)] The factor beta is introduced without stating its normalization; please state that it is a dimensionless scaling factor and give its uncertainty from the fit to the ALICE phi p correlation function.
[Tables V and VI] The notation '(I=1, JP=1- - (-))' is unclear; replace it with 'no bound state' or a consistent dash.
[Section IV B] The absence of a J/psi NN bound state is reported without showing the largest eigenvalue of the Faddeev kernel in the relevant channels; a short table or a statement about the convergence of lambda(E) would strengthen this negative result.
[Conclusions] The statement that the predicted bound states agree with Refs. [96-98] is not quantified; please give the corresponding binding energies from those references and explain what the present momentum-space treatment adds beyond them.
[Section IV C] The binding energies are defined relative to the three-body threshold, but the threshold is not defined in the text; please state it explicitly as m_phi + 2 m_N (or as appropriate for the HAL QCD masses).
[Eq. (38)] The two-pion exchange tail is not scaled by beta in scheme I; since the text justifies the tail as spin-independent, it would be helpful to state explicitly why only the short-range Gaussian terms are rescaled.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the phiNN binding energies are computed outputs of a fixed-input Faddeev calculation, with the only fitted parameter (beta in scheme I) entering the two-body input potential and never tuned to the three-body result.

full rationale

The paper's central claim is produced by solving the homogeneous Faddeev equation K(E)|psi> = |psi> (Eqs. 28-31), where the bound-state energy is fixed by the eigenvalue condition lambda(E)=1. All two-body inputs -- the Malfliet-Tjon NN potential, the HAL QCD J/psi N potentials, the HAL QCD phi N (4S3/2) potential, and the two constructed phi N (2S1/2) potentials -- are fixed before the three-body diagonalization. No parameter is adjusted to reproduce the reported B3 values. In particular, scheme I's beta=6.9 is determined by fitting the ALICE phi-p correlation function (Section III C, Eq. 38) and is an ingredient of the two-body phi N potential; the three-body output is not used to tune anything. This is a model-input uncertainty, not a fitted quantity renamed as a prediction. Scheme II's inverse-mass scaling assumption (Eqs. 39-41) is also an input hypothesis, not a recirculated output. The reproduction of HAL QCD scattering lengths in Table IV is a sanity check, not circularity. The agreement with Refs. [96-98] is an external cross-check by other groups using different methods; the author's only self-citation, Ref. [72], is peripheral to the three-body claim. The genuine weakness -- that the (0)0- and (0)1- phiNN states depend on the poorly constrained 2S1/2 phi N force, which HAL QCD itself flags as unreliable due to Lambda K / Sigma K coupling (Section III C) -- is a robustness concern, and the paper explicitly states the HAL QCD limitation. A strength scan would test stability, but the absence of one does not make the derivation circular. Therefore the derivation chain is self-contained: outputs are computed from stated inputs, with no equation reducing a prediction to its inputs by construction.

Assumptions & free parameters 4 free parameters · 6 assumptions · 0 invented entities

The central claim rests entirely on externally fitted two-body potentials: the MT NN potential, the HAL QCD J/psi N and phi N spin-3/2 potentials, and the beta=6.9 scale from the ALICE phi p correlation function. No new dynamical degrees of freedom are introduced. The only paper-specific modeling choices are the two constructions of the uncertain phi N spin-1/2 potential (schemes I and II), both of which are ad hoc and load-bearing.

free parameters (4)

beta (scheme I phi N spin-1/2 scale) = 6.9
Multiplicative factor in Eq. (38) applied to the HAL QCD spin-3/2 Gaussians to build the spin-1/2 phi N potential; determined by fitting the ALICE phi p correlation function (Ref. [100]). Directly controls the strength of the spin-1/2 phi N interaction and the deep (0)0- and (0)1- three-body binding energies.
MT potential strength parameters (1S0, 3S1) = 1S0: C1=-514 MeV fm, C2=1439 MeV fm; 3S1: C1=-627 MeV fm, C2=1439 MeV fm; mu1=1.55 fn^-1, mu2=3.11 fm^-1
Malfliet-Tjon parameters from Table I, originally fitted to NN scattering; used as input for the NN subsystem. They set the deuteron binding energy and scattering lengths.
HAL QCD J/psi N potential parameters (Table II) = See Table II, from Ref. [76]
Three-range Gaussian fits to HAL QCD lattice potentials for the spin-3/2 and spin-1/2 channels; used directly as input.
HAL QCD phi N spin-3/2 potential parameters (Table III) = alpha1=-371(27) MeV, beta1=0.13(1) fm, alpha2=-119(39) MeV, beta2=0.30(5) fm, alpha3 m_pi^4=-97(14), beta3=0.63(4) fm
Fit to the HAL QCD phi N potential (Ref. [90]); used as input and also as the basis for scheme I's spin-1/2 potential.

assumptions (6)

standard math Faddeev decomposition and Lippmann-Schwinger equations are valid for these three-body systems
Section II A-B; standard quantum mechanical few-body formalism taken from Refs. [77-79].
domain assumption Only S-wave two-body interactions contribute; higher partial waves are neglected
Section II B: 'In this work, only S-wave interactions are considered.' This truncation is standard for near-threshold bound states but omits possible D-wave effects.
ad hoc to paper The phi N spin-1/2 interaction can be represented by a single-channel potential
Section III C: HAL QCD finds strong coupling to Lambda K and Sigma K in the spin-1/2 channel, yet the Faddeev calculation uses an effective single-channel potential. This is a modeling assumption that could affect the existence of bound states.
ad hoc to paper Spin-spin parts of phi N and J/psi N interactions are inversely proportional to hadron masses (scheme II)
Eq. (40): V^Spin_phiN = (m_J/psi / m_phi) V^Spin_J/psiN. This assumption, taken from Ref. [96], determines the entire scheme II spin-1/2 potential and is not derived from data.
ad hoc to paper The spin-1/2 phi N potential in scheme I is obtained by scaling the spin-3/2 HAL QCD potential by beta=6.9 and adding the same two-pion tail
Eq. (38): beta multiplies the two Gaussian terms; the two-pion exchange tail is assumed spin-independent. This construction is not uniquely determined by the phi p correlation function.
domain assumption Non-relativistic kinematics with static potentials is adequate near threshold
The Hamiltonian in Eq. (1) uses non-relativistic kinetic energies; relativistic effects are neglected. Standard for these near-threshold systems.

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Cite this review

Pith. "Pith review of Faddeev equations for the $J/\psi\,NN$ and $\phi\,NN$ three-body systems in momentum space." pith.science (2026). https://pith.science/paper/LBQBRBMX

@misc{pith2026260803162,
  author       = {Pith},
  title        = {Pith review of: Faddeev equations for the $J/\psi\,NN$ and $\phi\,NN$ three-body systems in momentum space},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/LBQBRBMX}},
  note         = {Machine review of arXiv:2608.03162}
}

abstract

We construct Faddeev equations for the $J/\psi\,NN$ and $\phi\,NN$ three-body systems in momentum space. The two-body subsystem $t$-matrix is obtained by solving the Lippmann-Schwinger equations. The $NN$ interactions are constructed using the Malfliet-Tjon potential. The $J/\psi\,N$ interaction potentials in the ${}^4S_{3/2}$ and ${}^2S_{1/2}$ channels are taken from HAL QCD. The $\phi\,N$ interaction potential in the ${}^4S_{3/2}$ channel is also taken from HAL QCD. The ${}^2S_{1/2}$ $\phi N$ interaction is obtained from the combination of the $\phi p$ correlation function analysis and the HAL QCD results, as well as from the assumption that the spin-spin parts of the $\phi N$ and $J/\psi N$ interactions are inversely proportional to the respective hadron masses. The Faddeev equations are solved to find bound states in the $J/\psi\,NN$ and $\phi\,NN$ three-body systems. The numerical results suggest that there exist bound states in the $\phi\,NN$ three-body system, while there is no bound state in the $J/\psi\,NN$ system.

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SchemeI In scheme I, there exists a deep bound state in the (I)JP = (0)0− channel. In this channel, the two-body subsystemNNis in the 3S1 channel, andϕNis in the 2S1/2 channel. TheNNinteraction in the 3S1 channel is strong and forms the deuteron. TheϕNinteraction in the 2S1/2 channel is even stronger, and can form a bound state with binding energy around ...

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