{"id":"ee47c587-9612-410f-8107-5d4b63ee4fad","arxiv_id":"2608.03162","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A momentum-space Faddeev calculation with Malfliet-Tjon and HAL QCD potentials predicts phi-nucleon-nucleon bound states and no J/psi-nucleon-nucleon bound state.","lead":"This paper calculates whether a J/psi or a phi particle can bind together with two nucleons into a three-body state. It finds no J/psi-nucleon-nucleon bound state, but predicts bound phi-nucleon-nucleon states, confirming earlier calculations with a different method.","discovery_kind":"replication","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Predicted phiNN bound states rest on the unvalidated 2S1/2 phiN input; a strength scan would show whether the (0)0- and (0)1- states survive.","rationale":"The paper is a clean momentum-space Faddeev implementation with standard numerical methods, and it reproduces known deuteron properties and previous phiNN results, so the machinery is not the issue. The load-bearing weakness is the physical input: the phiN 2S1/2 channel is explicitly flagged by HAL QCD as unreliable, and the two ad hoc schemes are not cross-checked against each other except in the final binding energies. Since the abstract claims bound states in phiNN in both schemes, the claim inherits the uncertainty of the 2S1/2 input. The reader's weakest assumption identifies the same concern; my proposed strength scan is a minimal, concrete way to determine whether the claim is stable. I therefore endorse the CONDITIONAL verdict without further adjustment.","tokens_in":20121,"tokens_out":6459,"duration_ms":62935,"concrete_test":"Multiply the 2S1/2 phiN potential in Eq. (38) and in Eq. (41) by a global factor f ranging from 0.5 to 1.5 and recompute B3 for the (0)0- and (0)1- channels in both schemes, keeping all other inputs fixed. Determine the critical f at which each eigenvalue crosses the three-body threshold, and compare that f with the 1-sigma band of beta from the ALICE fit. If the critical f lies within that band, the bound-state prediction is not robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central existence claim for the (0)0- and (0)1- phiNN bound states is carried by the 2S1/2 phiN potential, and that potential is the least secure input in the calculation. HAL QCD (Ref. [90]) states explicitly that in the 2S1/2 channel the phiN potential is strongly coupled to Lambda K and Sigma K and that lattice information there is unreliable; the present calculation nevertheless uses a single-channel S-wave t-matrix. Scheme I (Eq. 38) manufactures the 2S1/2 potential by scaling the 4S3/2 short-range terms with beta=6.9 from an ALICE phi-p correlation-function fit, with no propagated uncertainty. Scheme II (Eqs. 39-41) assumes the spin-spin parts of J/psiN and phiN scale as m_J/psi/m_phi, an untested hypothesis. In both schemes the (0)0- and (0)1- three-body states could disappear if this channel is even moderately weaker, and because no strength variation or coupled-channel estimate is reported, the calculation does not establish that those bound states exist beyond the chosen inputs. The (0)2- state, which uses only the well-determined 4S3/2 channel, is on firmer ground.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20447,"tokens_out":8301,"duration_ms":77194,"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":[{"comment":"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":"Section III C, Eq. (38), Tables V and VI"},{"comment":"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":"Section IV C, Tables V and VI"},{"comment":"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.","section":"Section III C, Eq. (23)"}],"minor_comments":[{"comment":"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.","section":"Table III"},{"comment":"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.","section":"Eq. (38)"},{"comment":"The notation '(I=1, JP=1- - (-))' is unclear; replace it with 'no bound state' or a consistent dash.","section":"Tables V and VI"},{"comment":"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.","section":"Section IV B"},{"comment":"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":"Conclusions"},{"comment":"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).","section":"Section IV C"},{"comment":"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.","section":"Eq. (38)"}],"recommendation":"major_revision","confidential_remarks":"I see no evidence of circularity: the three-body binding energies are computed and not fitted to the three-body output. The main reservation is that the headline states, especially (0)0- and (0)1-, depend on an unconstrained two-body input. This can be addressed within the manuscript's scope by adding a strength scan and numerical uncertainty estimates. If the author is unwilling to add those, the paper should be reframed as a technical momentum-space Faddeev application with the existence claims for the 2S1/2-driven states marked as strongly model-dependent."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nThe 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.\n\nThe (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.\n\nI'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.\n\nFor 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.","headline":"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.","tokens_in":20971,"tokens_out":2838,"would_cite":false,"duration_ms":25797,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["Faddeev equations","three-body bound states","phi NN system","J/psi NN system","phi-nucleon interaction","Lippmann-Schwinger equation","lattice QCD potentials","S-wave interactions"],"falsifier":"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.","tokens_in":19842,"feed_emoji":"⚛️","tokens_out":10244,"duration_ms":83664,"temperature":0.7,"pith_summary":"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.","feed_headline":"Predicting phi-NN bound states, none for J/psi-NN","feed_subtitle":"Momentum-space Faddeev solutions yield 0-, 1-, and 2- bound phi-N-N states at a few to 50 MeV.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the J/psi-nucleon potentials in both S-wave channels from lattice QCD, whose scattering lengths are reproduced.","marker":"[76]"},{"why":"Supplies the spin-3/2 phi-nucleon potential and records that the spin-1/2 channel is strongly coupled to open strange channels and unreliable on the lattice.","marker":"[90]"},{"why":"Fixes the scaling factor in scheme I's spin-1/2 phi-nucleon potential through measured phi-proton correlation functions.","marker":"[100]"},{"why":"Provides the mass-scaling assumption used in scheme II and earlier Gaussian-expansion three-body results used for comparison.","marker":"[96]"},{"why":"Provides independent configuration-space Faddeev results for the phi-N-N system that the present binding energies are compared with.","marker":"[97]"},{"why":"Provides a further independent phi-N-N Faddeev calculation in configuration space used for comparison.","marker":"[98]"},{"why":"Supplies the momentum-space Faddeev formalism and partial-wave machinery the equations are built on.","marker":"[79]"},{"why":"Earlier J/psi-N-N three-body calculation whose no-bound-state conclusion this work reproduces.","marker":"[99]"},{"why":"Supplies the two-Yukawa nucleon-nucleon potential used for the NN subsystem.","marker":"[116]"}],"fun_headline_variants":["Phi-NN bound states predicted, J/psi-NN none","Faddeev predicts phi-NN binds, J/psi-NN not","Asymmetric binding: phi-NN bound, J/psi-NN not","Phi-NN binds (up to 50 MeV), J/psi-NN free","No J/psi-NN bound states, phi-NN does"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Phi-NN bound states predicted, J/psi-NN none","Faddeev predicts phi-NN binds, J/psi-NN not","Asymmetric binding: phi-NN bound, J/psi-NN not","Phi-NN binds (up to 50 MeV), J/psi-NN free","No J/psi-NN bound states, phi-NN does"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000905,"raw_usage":{"total_tokens":3929,"prompt_tokens":1020,"completion_tokens":2909,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":636,"completion_tokens_details":{"reasoning_tokens":2809}},"tokens_in":636,"tokens_out":2909,"duration_ms":16913,"temperature":1.0,"reasoning_tokens":2809,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T14:51:21.173890+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Nonexistence of hidden-charm pentaquarks in $J/\\psi$ photoproduction","cited_arxiv_id":"2606.11808","evidence_quote":"Supplies the J/psi-nucleon potentials in both S-wave channels from lattice QCD, whose scattering lengths are reproduced."},{"cited_title":"Examination of the $ \\phi-NN $ bound-state problem with lattice QCD $ N-\\phi $ potentials","cited_arxiv_id":"2402.06914","evidence_quote":"Provides independent configuration-space Faddeev results for the phi-N-N system that the present binding energies are compared with."}],"review_version":2}