{"id":"35e0a761-4a4e-4a3c-9be2-b74a5da54de5","arxiv_id":"2607.08040","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":5,"one_line_summary":"The CMR method combined with Green's functions provides a unified description of DD* exotic hadron states, identifying G(3900) as a P-wave resonance and extracting its scattering phase shifts and cross sections.","lead":"This paper applies the complex momentum representation (CMR) method combined with Green's functions to study exotic hadron states in the DD* system, identifying X(3872), T_cc, and Z_c(3900) as bound states and G(3900) as a P-wave resonance. It offers a unified framework for calculating bound states, resonances, and scattering observables directly in momentum space.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"G(3900) resonance parameters are not anchored to experimental data; the cutoff Λ is chosen without a stated physical criterion, and the resulting pole (mass ~3879 MeV, width ~38 MeV) does not match the experimental G(3900) (mass ~3900 MeV, width ~100 MeV).","rationale":"The reader identified the right general area — parameter dependence — but focused on the bound states, where fitting Λ to data is standard and defensible. The more load-bearing concern is with G(3900), where the cutoff is not fitted to any experimental quantity and the resulting resonance parameters do not match the experimental G(3900). The paper's methodological contribution (CMR + Green's function for scattering observables) has independent value and is not undermined. However, the physical claim about G(3900) is weaker than the bound-state claims because it lacks an experimental anchor. The verdict remains CONDITIONAL, but the specific condition that needs checking is the G(3900) parameter matching, not the bound-state fitting. If the Λ scan cannot reproduce experimental G(3900) properties, the G(3900) interpretation should be softened from 'can be interpreted as' to 'produces a P-wave resonance near threshold that may be related to.'","tokens_in":24682,"tokens_out":3326,"duration_ms":120920,"concrete_test":"Scan the cutoff Λ over the range 0.8–1.3 GeV (with δ) and extract the resonance pole position (E_r, Γ) at each value. Plot these against the experimental G(3900) mass and width. If no value of Λ in a physically reasonable range reproduces both the experimental mass (~3900 MeV, i.e., ~20–28 MeV above threshold) and width (~100 MeV) simultaneously, the identification of the calculated P-wave pole with G(3900) is not supported.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The reader correctly identifies parameter-fitting as a concern, but the more load-bearing issue is specific to G(3900). For the bound states X(3872), T_cc, and Z_c(3900), the cutoff Λ is fitted to reproduce known binding energies (Table II) — standard phenomenological practice. For G(3900), however, the cutoff Λ = 0.9205 GeV (with δ) or 1.0082 GeV (without δ) is selected without any stated experimental anchor. The resulting resonance sits at 4.25 MeV above the D̄D* threshold (≈3876–3883 MeV depending on charge channel) with a width of ~37.7 MeV. The experimental G(3900), observed by BaBar in e+e- → DD̄, has a mass near 3900 MeV and a width of order 100 MeV or more. The paper never compares its calculated resonance parameters to experimental G(3900) properties, nor justifies why this particular Λ is chosen. This means the claim that G(3900) 'can be interpreted as a P-wave resonant state' rests on an essentially free parameter choice that does not reproduce the experimental state it claims to describe. The bound-state claims are conditional but honest; the G(3900) claim is the weakest link.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This manuscript applies the complex momentum representation (CMR) combined with the Green's function method to the $DD^*$ system within a one-boson-exchange (OBE) framework. The authors employ a projection operator method to construct momentum-space partial-wave potentials, solve the Schrödinger equation directly in momentum space, and extract continuum level densities, scattering phase shifts, and cross sections. They interpret $X(3872)$, $T_{cc}^+$, and $Z_c(3900)$ as $S$–$D$ coupled bound states and $G(3900)$ as a $P$-wave resonance. The methodological framework is clearly presented, and the bound-state results are validated against the complex scaling method (Ref. [60]).","tokens_in":25007,"tokens_out":1113,"duration_ms":126832,"significance":"The integration of CMR with the Green's function method to extract decomposed scattering observables (phase shifts, cross sections) in hadronic physics is a useful technical contribution. The projection operator approach for handling complex spin structures in momentum space is a practical advance over partial-wave expansions requiring Fourier transformation. The systematic comparison of results with and without the short-range contact term $delta$ provides useful phenomenological insight into the sensitivity of $S$-wave bound states versus $P$-wave resonances to short-range physics.","major_comments":[{"comment":"§IV, discussion of $G(3900)$: The cutoff parameter for the $G(3900)$ resonance is set to $Lambda = 0.9205$ GeV (with $delta$) or $1.0082$ GeV (without $delta$), chosen to reproduce a resonance energy of $4.25$ MeV above threshold. However, no comparison to the experimentally observed $G(3900)$ properties is provided. The experimental state reported by BaBar and BESIII has a mass near $3900$ MeV (roughly $20$–$30$ MeV above the $Dbar{D}^*$ threshold) and a width of order $100$ MeV or more, whereas the calculated pole sits at $approx 4.25$ MeV above threshold with a width of $sim 37.7$ MeV. The claim that $G(3900)$ 'can be interpreted as a $P$-wave resonant state' is weakened by the absence of any quantitative comparison to experimental data or justification for the chosen $Lambda$. The authors should either (a) state clearly what experimental observables they are targeting and demonstrate","section":null},{"comment":"§IV, Fig. 3 and surrounding text: The pole trajectory of $G(3900)$ is shown as a function of $Lambda$, but the physical criterion for selecting the specific $Lambda$ value used in the scattering calculations is not stated beyond reproducing a chosen resonance energy. Since the pole position varies significantly with $Lambda$ (from $(-7.73, -21.14)$ to $(1.80, -0.49)$ MeV), the physical content of the result depends entirely on this choice. The authors should clarify whether there exists a first-principles or phenomenological criterion for fixing $Lambda$ for resonances (as opposed to fitting to known binding energies for bound states), or acknowledge that the resonance prediction is exploratory rather than definitive.","section":null}],"minor_comments":[{"comment":"Table II caption: the cutoff parameters are listed in units of GeV, but the text immediately below states '0.8272, 0.749 and 0.9998 MeV' — these should be GeV, not MeV.","section":null},{"comment":"Fig. 1: the axis labels and legend are rendered as garbled character sequences (e.g., '/s48/s46/s48...'). This appears to be a font or encoding issue that should be fixed for readability.","section":null},{"comment":"§II, Eq. (7): the notation $V_D$ and $V_C$ for direct and cross diagrams is introduced but the superscript $C$ is also used for charge conjugation in §II. Clarifying the notation would avoid confusion.","section":null},{"comment":"§IV: the statement 'the cutoff parameters without the contact term $delta$ are all around 1.0 GeV with smaller variations, which appears better suited to describe hadronic molecular states' is a qualitative judgment. The authors should either provide a quantitative criterion for what constitutes 'better suited' or soften the language.","section":null},{"comment":"References [32]–[35] and [40]–[41] discuss $Z_c(3900)$ and $G(3900)$ interpretations but are cited without detailed comparison to the present results. A brief discussion of how the present findings relate to these prior works would strengthen the paper.","section":null}],"recommendation":"major_revision","confidential_remarks":"The reader's concern about parameter-fitting for bound states is standard phenomenological practice and not, in my view, a load-bearing objection. The more serious issue is the $G(3900)$ claim: the calculated resonance does not match experimental properties, and the cutoff is chosen without a stated physical anchor. This is the central weakness that should be addressed before publication. The methodological framework itself is sound and potentially publishable if the $G(3900)$ claims are appropriately qualified or the comparison to data is made explicit."},"author_rebuttal":null,"desk_editor":{"model":"glm-5.2","letter":"The methodological contribution here is real. The authors apply the complex momentum representation (CMR) combined with the Green's function method to the DD* system, and the projection operator technique for handling the spin structure in momentum space is a clean piece of work. The unified treatment of bound states, resonances, and continuum — plus the extraction of scattering phase shifts and cross sections with a resonance/background decomposition — is a useful framework. Validation against the complex scaling method (Ref. [60]) for bound states checks out, and the treatment of the contact term δ is handled self-consistently in momentum space, which is a genuine advantage of the approach. Credit where it's due: the machinery is well-built and the paper is clearly written. The bound-state results for X(3872), T_cc, and Z_c(3900) are honest phenomenology — the cutoff Λ is fitted to reproduce known binding energies, which is standard practice in OBE models. The observation that P-wave resonances are less sensitive to the contact term than S-wave bound states is a useful, if modest, finding. The soft spot is the G(3900) claim, and it's load-bearing. For the bound states, the cutoff is anchored to experimental data. For G(3900), the cutoff Λ = 0.9205 GeV (with δ) or 1.0082 GeV (without δ) is selected without any stated physical criterion. The resulting pole sits at about 4.25 MeV above the DD* threshold with a width of roughly 37–38 MeV. The experimental G(3900) has a mass near 3900 MeV and a width of order 100 MeV or more. The paper never compares its calculated resonance parameters to the experimental G(3900) properties, nor does it justify why this particular Λ was chosen. So the claim that G(3900) 'can be interpreted as a P-wave resonant state' rests on an essentially free parameter choice that doesn't reproduce the experimental state it claims to describe. The bound-state claims are conditional but transparent. The G(3900) claim is the weakest link and needs either a physical justification for the cutoff or an honest acknowledgment that the calculated pole doesn't match the experimental G(3900). This paper deserves a serious referee. The methodology is sound and independently useful, but the G(3900) interpretation needs to be either substantiated or walked back before publication.","headline":"The G(3900) claim is the weak link — the cutoff is chosen without an experimental anchor and the resulting pole doesn't match the data.","tokens_in":25538,"tokens_out":585,"would_cite":false,"duration_ms":88898,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Four exotic hadrons unified under one scattering framework","keywords":[],"falsifier":"If the scattering phase shifts or cross sections predicted by the CMR Green's function method for the DD* system disagree with future experimental measurements of D D* scattering observables, or if the G(3900) resonance parameters extracted from the pole position fail to match independent amplitude analyses, the unified molecular interpretation would be undermined.","tokens_in":24979,"feed_emoji":"🔬","tokens_out":1072,"duration_ms":138422,"temperature":0.7,"pith_summary":"The paper introduces a computational framework that combines the complex momentum representation (CMR) with the Green's function method to solve the two-body Schrödinger equation directly in momentum space for the DD* system of charmed mesons. Using a projection operator technique to decompose the one-boson-exchange potential into partial waves, the approach simultaneously captures bound states, resonances, and the scattering continuum in a single calculation. The authors fit the cutoff parameter of a monopole form factor to reproduce the known binding energies of X(3872), T_cc^+, and Z_c(3900) as S-wave dominated molecular bound states of D and D* mesons, then use the same machinery to identify G(3900) as a P-wave resonant state. By extracting the continuum level density from the complex-momentum Green's function, they derive scattering phase shifts and cross sections, decomposing them into resonant and background contributions. The key finding is that all four exotic hadrons near the DD* threshold can be consistently described within a single molecular picture: three as bound states and one as a P-wave resonance, with the P-wave resonance showing less sensitivity to short-range contact interactions than the S-wave bound states.","feed_headline":"Four exotic hadrons unified under one scattering framework","feed_subtitle":"A complex-momentum method binds X(3872), T_cc, Z_c as molecules and pins G(3900) as a P-wave resonance, all with decomposed scattering phase","key_machinery":"The central mechanism is the complex momentum representation (CMR), which deforms the momentum integration contour into the complex plane so that bound-state poles, resonance poles, and continuum states all appear as discrete eigenvalues of a single Hamiltonian matrix. A projection operator method decomposes the momentum-space one-boson-exchange potential into partial waves for systems with coupled spin channels. The Green's function constructed from these complex eigenvalues yields the continuum level density, whose integral gives the scattering phase shift and whose decomposition separates the pure resonance contribution from the continuum background. A monopole form factor with cutoff Λ ~","core_discovery":"The CMR combined with the Green's function method provides a single, parameter-stable framework that unifies the description of bound states, resonances, and continuum scattering for the DD* system, allowing X(3872), T_cc^+, and Z_c(3900) to be identified as S-D coupled bound states and G(3900) as a P-wave resonant state, with scattering observables decomposed into resonant and background components.","pith_inferences":["The fact that different cutoff values are needed for different bound states (0.83 GeV for X(3872) vs 1.0 GeV for Z_c(3900) with contact terms) may indicate that a single universal short-range interaction cannot describe all DD* molecular states simultaneously, which could challenge the 'consistent explanation' claim if one demands a single cutoff.","The near-threshold narrowing of the G(3900) resonance as the cutoff increases suggests that experimental width measurements could constrain the form-factor cutoff and thereby the spatial extent of the DD* molecular system.","Extending the CMR framework to coupled-channel problems (e.g., including D*D*, J/psi pi, or eta_c pi channels) would test whether the single-channel molecular interpretation survives when additional decay channels compete."],"forward_implications":["If the CMR framework is valid, scattering phase shifts and cross sections for other exotic hadron candidates near thresholds can be extracted without separate bound-state and scattering calculations.","The decomposition of phase shifts into resonant and background components could be applied to distinguish genuine resonant poles from threshold cusps in controversial exotic states.","The finding that P-wave resonances are less sensitive to short-range contact terms than S-wave bound states suggests that P-wave exotic hadron predictions may be more robust against model uncertainties in the short-range potential.","The method can be extended to other two-hadron systems with coupled spin channels, such as hidden-charm pentaquark candidates or doubly-heavy baryon-meson systems."],"fun_headline_variants":["DD* system yields four exotic hadrons in one framework","X(3872), T_cc, Z_c bound and G(3900) resonant under unified CMR method","Complex-momentum approach classifies four exotic DD* states","Green's function and CMR unite bound and resonant DD* hadrons","Four exotic hadrons classified via decomposed DD* scattering phases"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The binding energies of the three bound states are reproduced by fitting the cutoff parameter of a monopole form factor individually for each state, so the claim of a 'consistent' molecular description rests on the physical validity of a phenomenological one-boson-exchange potential with state-dependent cutoffs rather than a derivation from first principles.","fun_headline_variants_meta":{"raw":{"variants":["DD* system yields four exotic hadrons in one framework","X(3872), T_cc, Z_c bound and G(3900) resonant under unified CMR method","Complex-momentum approach classifies four exotic DD* states","Green's function and CMR unite bound and resonant DD* hadrons","Four exotic hadrons classified via decomposed DD* scattering phases"]},"model":"glm-5.2","effort":"low","cost_usd":0.0,"raw_usage":{"total_tokens":596,"prompt_tokens":512,"completion_tokens":84,"prompt_tokens_details":null},"tokens_in":512,"tokens_out":84,"duration_ms":20860,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-10T01:09:49.821924+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If the scattering phase shifts or cross sections predicted by the CMR Green's function method for the DD* system disagree with future experimental measurements of D D* scattering observables, or if the G(3900) resonance parameters extracted from the pole position fail to match independent amplitude analyses, the unified molecular interpretation would be undermined.","supporting_citations":[],"review_version":1}