{"id":"9509b069-da5b-46af-b6f1-596e5f869548","arxiv_id":"2412.01507","paper_version":1,"verdict":"REJECT","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"high","formal_verification":"none","parameter_count":3,"one_line_summary":"It reports variational energies and binding energies for muonic He-p, He-d, Li-p, and Li-d three-body molecules, but it does not tabulate resonance widths.","lead":"The paper computes nonrelativistic energies and leading corrections for four muonic molecules containing helium or lithium plus a proton or deuteron. It uses two variational bases and the Complex Coordinate Rotation method to identify quasibound states.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper claims resonance energies but reports only real bound-state energies and never quotes the complex pole or width; the two bases disagree by 0.026 a.u. for 3Hepmu, so the binding energies in Eqs. (17)–(20) are not established as converged resonance parameters.","rationale":"The reader's rejection is justified by the same core defect: the paper does not report resonance parameters. My concern is more specific and, I think, more load-bearing: even if one accepts CCR as a method, the numbers in Tables 1 and 2 are real and the two bases do not converge to a common value at the only system where a comparison can be made from the text. In particular, the 3Hepmu discrepancy is 0.0263 a.u., while the sum of all leading corrections is −0.0044 a.u.; attributing such a spread to numerical error would undercut the claimed precision of the binding energies. A single recomputation that quotes the complex stationary point for both bases, with N and theta convergence, would settle this. If the complex poles agree, the paper's central numbers may be recoverable; if not, the central claim fails. Since neither the widths nor a convergence analysis are present, the correct preprint verdict is REJECT rather than ACCEPT or CONDITIONAL.","tokens_in":6263,"tokens_out":3566,"duration_ms":32723,"concrete_test":"Recompute 3Hepmu with both basis sets at N=1500, 2000, and 2500, scanning theta=(0.98,1.00,1.02) and phi=0.10–0.30; extract the full complex eigenvalue for each variational basis, locate the stationary point in the complex plane, and quote Re E, −Im E/2, and the scatter across theta and basis. Then compare those complex pole values with the real Table 1 entries. If the two bases do not agree at the complex pole to within about 0.001 a.u. and the stationary point is not stable with N, Eqs. (17)–(20) do not describe a converged resonance.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim is that CCR determines resonance energies of four muonic three-particle systems. The decisive gap is that the numerical results in Tables 1 and 2 are real, are captioned as “bound state energies,” and are used in Eqs. (17)–(20) as if they were the full resonance energy. CCR, as set up in Eq. (2), produces a complex eigenvalue for a resonance; the imaginary part is the half-width and must be quoted to identify the pole. Figure 1 plots rotation paths in the complex plane but gives no numerical value for the stationary point, so the reader cannot check the resonance position or width. Worse, for 3Hepmu the exponential basis gives −95.6365412927 a.u. and the Gaussian basis −95.6628556322 a.u., a 0.0263 a.u. disagreement that is about six times the total leading correction of −0.0044 a.u. This is not numerical noise at the stated accuracy, and without a convergence study in N or a demonstration that both bases approach the same complex pole, the real parts cannot be interpreted as converged resonance energies. Thus the load-bearing assumption, that the stationary point in Fig. 1 lies at the energies in Tables 1/2, is unsupported by the reported evidence.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports variational calculations for the four three-particle muonic systems (He-p-μ), (He-d-μ), (Li-p-μ), and (Li-d-μ), using both exponential and Gaussian trial bases. A complex coordinate rotation (CCR) is introduced in Eq. (2), and the nonrelativistic energies together with relativistic, recoil, nuclear-size, and contact corrections are listed in Tables 1 and 2. Binding energies are then quoted in Eqs. (17)–(20). The abstract claims that these are energies of resonant states of the listed molecules.","tokens_in":6415,"tokens_out":10936,"duration_ms":94465,"significance":"If established, these numbers would provide useful independent benchmarks for quasibound muonic molecules relevant to muon-catalyzed fusion, and the two-basis cross-check is a sound idea. The inclusion of leading relativistic and finite-nuclear-size corrections is a strength, and the analytical matrix elements in Eqs. (7)–(12) make the variational formalism transparent. However, the central claim is not currently evidenced: no complex resonance energy or width is reported, the two basis sets disagree at the 0.02–0.05 a.u. level for several systems, and the binding-energy formula appears inconsistent with the quoted results.","major_comments":[{"comment":"The paper never reports the imaginary part of the resonance energy. Under the CCR transformation of Eq. (2), a resonance is identified by a complex eigenvalue whose imaginary part is the half-width; Figure 1 even labels the vertical axis as the resonance half-width. Yet Tables 1 and 2 are captioned “bound state energies” and contain only real numbers, and no numerical value of the stationary point in Fig. 1 is given. The abstract's claim to calculate energies of resonant states therefore cannot be checked. Please quote E_r and Γ for each of the eight systems and specify the relevant dissociation threshold.","section":"Section 3, Tables 1 and 2, Fig. 1"},{"comment":"The two variational bases are not converged to a common result. For 3Hepμ, Table 1 gives −95.6365412927 a.u. (exponential) and −95.6628556322 a.u. (Gaussian), a difference of 0.0263 a.u. ≈ 0.72 eV. This is roughly six times the total leading correction of about −0.0044 a.u. and is far larger than the 10-decimal precision printed. Large differences also occur for 4Hepμ, 3Hedμ, and 4Hedμ. Because no convergence study in the basis size N (N=1500 is stated but not varied) is provided, the real parts used in Eqs. (17)–(20) are not established as converged resonance parameters. The authors should either reconcile the two bases or state which value enters Eqs. (17)–(20) and assign a realistic uncertainty.","section":"Tables 1 and 2, Eqs. (17)–(20)"},{"comment":"The binding-energy formula as written is inconsistent with the quoted numbers. With E and E_cl both negative, as implied by E_cl = -μ/(2n^2), the quantity −(E + E_cl) is a large positive number; for 3Hepμ one obtains roughly +188 a.u. ≈ 5100 eV, not −73.762 eV. The listed values near −20 to −81 eV correspond instead to E − E_cl (or to E_cl taken with the opposite sign). The sign convention in Eq. (16) must be corrected, and the principal quantum number n used for the pμ or dμ cluster must be stated explicitly.","section":"Eq. (16) and Eqs. (17)–(20)"},{"comment":"Figure 1 does not quantitatively support the resonance identification. The caption asserts that the node in the center corresponds to the stationary point defining the resonance, but no coordinates of this point are printed, and the plotted real-energy ranges are not reconciled with Tables 1 and 2 (for example, the first panel's real-energy axis is near −73.91 a.u., while Table 1 lists −95.64 a.u. for 3Hepμ). If the plotted quantity is a shifted or relative energy, that must be stated. Please give the stationary-point energies and widths numerically for all four panels.","section":"Fig. 1 and surrounding text"}],"minor_comments":[{"comment":"The quantities F_1^13, F_2^13, F_1^23, and F_2^23 in Eq. (12) are never defined; please define them or provide a reference containing the full expressions.","section":"Eq. (12)"},{"comment":"The text says the corrections are obtained using exponential basis functions, but Tables 1 and 2 do not indicate whether the same is true for all rows; please state this explicitly and, if possible, give a Gaussian-basis estimate for at least the dominant correction.","section":"Section 2, after Eq. (15)"},{"comment":"The axis label “a.e.” should be “a.u.”, and the figure should specify whether the plotted real energy is the total nonrelativistic energy or some shifted quantity.","section":"Fig. 1"},{"comment":"The phrase “accuracy of up to three significant digits after the decimal point” should be rephrased as “three decimal places,” and an uncertainty estimate should accompany the binding energies in Eqs. (17)–(20).","section":"Section 3, last paragraph"},{"comment":"Reference [15] has a garbled author list (“H. A. Bethe E. E. and Salpeter”), and reference [17] is incomplete; please correct these entries.","section":"References"},{"comment":"The tables are captioned “bound state energies” while the abstract and introduction describe resonant states embedded in the continuum; please harmonize the terminology and clarify what is being reported.","section":"Tables 1 and 2"}],"recommendation":"major_revision","confidential_remarks":"The reader's core concern is valid and lands: the resonance claim is not supported by the data as presented. I recommend major revision rather than rejection because the missing items (complex energies, widths, a convergence study, and a corrected binding-energy definition) appear obtainable within the manuscript's scope. The two-basis approach and the inclusion of leading corrections are genuine strengths, but the paper should not be accepted until the numerical disagreement between the bases is resolved or explicitly quantified and the pole positions are reported."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThe paper is a legitimate extension of an established variational program to six muonic three-particle systems, and the numbers in Tables 1 and 2 are new. The authors carefully include relativistic, recoil, finite-nuclear-size, and contact corrections, and they cross-check with two different bases. That part is real work and will be useful to people tracking few-body muonic molecule calculations.\n\nBut the paper as written does not support its headline claim. The abstract says 'energies of resonant states,' and the method is CCR, which yields a complex pole whose imaginary part is the half-width. Nowhere in the paper are the complex energies or widths reported. Tables 1 and 2 are captioned 'bound state energies,' real numbers, and they are fed into Eq. (16) as if they were the full resonance energy. Figure 1 shows rotation paths with a stationary point, but no numerical value for that point is given. That is not a cosmetic omission; it is the observable that distinguishes a resonance from a bound state.\n\nThe second problem is the basis discrepancy. For 3Hepmu the exponential basis gives −95.63654 a.u. and the Gaussian basis −95.66286 a.u., a 0.0263 a.u. gap. That is about six times the total leading correction of −0.0044 a.u. and far beyond the quoted accuracy. No convergence study in basis size N is provided, so the reader cannot tell whether the two bases are approaching the same pole or one of them is missing the state. The text says the results are 'consistent with [1,2]' and then notes the difference is about 10%, which is self-contradictory.\n\nI don't think this is dishonest or sloppy in a malicious sense; the variational machinery is standard and the error estimates for the corrections look plausible. But the central claim — that these are accurate resonance energies — is not established by the reported evidence. The paper needs a major revision: quote the resonance positions and widths from the stationary points, show the imaginary parts, and either reconcile the two bases or explain the disagreement. If that is done, the new numbers for 6Li and 7Li systems would be a modest but solid contribution to muonic molecule calculations.\n\nFor now, I would not cite the binding energies. But I would send it to peer review because the underlying calculation is serious and the missing pieces are identifiable and fixable — a good referee could turn this into a useful paper.","headline":"Useful variational numbers for muonic molecules, but the resonance claim is not backed up: no widths reported and the two bases disagree by 0.026 a.u. for 3Hepmu.","tokens_in":7054,"tokens_out":2893,"would_cite":false,"duration_ms":24737,"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":"The paper reports nonrelativistic energies and leading corrections for resonant states of four muonic three-particle molecules, obtained by complex coordinate rotation with two independent variational bases.","keywords":["muonic molecules","resonant states","complex coordinate rotation","variational method","three-body Coulomb problem","muon-catalyzed fusion","exotic atoms","binding energies"],"falsifier":"A decisive check is to recompute the (3He-p-mu) resonance with a substantially larger or differently parametrized basis and see whether the exponential and Gaussian energies converge to a common value rather than staying about 0.026 atomic units apart; if the gap persists across independent bases, the reported binding energy in Eq. (17) is not reliable. A second, independent check is to vary the rotation angle and dilation parameter far beyond the ranges $a=0.98$ to $1.02$ and $\\phi=0.1$ to $0.3$ and confirm that the stationary point of the rotational paths in Figure 1 does not move.","tokens_in":5968,"feed_emoji":"⚛️","tokens_out":5471,"duration_ms":45907,"temperature":0.7,"pith_summary":"This paper calculates the nonrelativistic energies and the leading corrections for resonant, quasibound states of four muonic three-particle molecules: (He-p-mu), (He-d-mu), (Li-p-mu), and (Li-d-mu). The authors use the complex coordinate rotation method with two independent variational basis sets, exponential and Gaussian, to locate each resonance as a stationary point in the complex energy plane. If the results hold, they provide a consistent set of binding energies, for example -73.762 eV for (3He-p-mu) and -18.432 eV for (6Li-p-mu), for systems relevant to muon-catalyzed fusion and the spectroscopy of exotic molecules. The paper also finds that its binding energies differ by about 10 percent from earlier calculations, so the values are a renewed benchmark rather than a confirmation.","feed_headline":"Complex rotation fixes energies of four muonic molecules","feed_subtitle":"Quasibound states of (He-p-mu), (He-d-mu), (Li-p-mu), (Li-d-mu) now have reported binding energies.","key_machinery":"The central object is the complex-rotated Hamiltonian $H(\\theta)=T\\exp(-2i\\theta)+V\\exp(-i\\theta)$, obtained from the coordinate transformation $r \\to r e^{i\\theta}$. Rotating the coordinates exposes resonant poles of the continuum on the physical sheet, so a resonance shows up as a stationary point in the complex energy plane. The paper finds that stationary point by building variational wavefunctions from exponential and Gaussian basis functions and plotting the rotational paths for dilation parameters $a=0.98$, $1.00$, $1.02$ and rotation angles $\\phi=0.1$ to $0.3$. The same rotated wavefunctions are then used to evaluate relativistic, recoil, nuclear-finite-size, and contact corrections from the Breit-Pauli Hamiltonian.","core_discovery":"The paper's central claim is that the states listed in Tables 1 and 2 are genuine resonant states of the Coulomb three-body problem, and that the energies shown there, together with the binding energies in Eqs. (17) to (20), are accurate to the precision stated. For the helium-containing systems the binding energies range between roughly -70 and -82 eV, while for lithium-containing systems they fall between -18 and -21 eV. The two basis sets agree closely for all systems except (3He-p-mu), where the exponential and Gaussian energies differ by about 0.026 atomic units; the paper treats this as numerical scatter and reports a single high-accuracy value. The authors state that the results are consistent with earlier calculations but that the differences, about 10 percent, are significant for this type of calculation.","pith_inferences":["A direct extension the authors do not pursue is to compute the decay widths, the imaginary parts of the resonance energies, and compare them with the predissociation widths reported for the lithium-deuterium system.","One could test the 10 percent discrepancy against earlier adiabatic or hyperspherical calculations by computing the same binding energies in a third independent basis or with a finite-element method; a converged three-way agreement would settle which of the older numbers was off.","The near-10 percent change in binding energy is large enough to shift the kinetics of muon-catalyzed fusion in helium-lithium mixtures, so the paper implicitly calls for updated reaction-rate estimates."],"forward_implications":["If these energies are right, they give muon-catalyzed fusion calculations specific benchmarks for the quasibound states that form when muonic hydrogen collides with helium or lithium nuclei.","The roughly 10 percent shift from earlier binding energies would change predicted formation and decay rates for these molecules, not just their level positions.","The 0.026 atomic unit two-basis spread for (3He-p-mu) marks the current numerical uncertainty and sets a target: a converged calculation should land between the two basis values or explain the gap.","The agreement of the two basis sets for the other seven systems supports the use of complex coordinate rotation for charge-asymmetric muonic resonances."],"supporting_citations":[{"why":"Supplies the complex coordinate rotation method used to expose resonant poles in the energy plane.","marker":"[6]"},{"why":"Establishes the theory of complex coordinates for resonance energies and widths in atomic and molecular structure.","marker":"[7]"},{"why":"Supplies the stochastic variational method and the computational framework adapted for constructing the Hamiltonian matrix.","marker":"[8]"},{"why":"Provides earlier binding and decay results for hydrogen-helium mesic molecules that the paper compares against.","marker":"[1]"},{"why":"Provides earlier binding energies and nonradiative decay rates for (Hed-mu) that set the comparison baseline.","marker":"[2]"},{"why":"Reports previous quasibound-state energies for the lithium-deuterium muonic system that the present calculation extends.","marker":"[5]"},{"why":"Reviews variational methods for the Coulomb three-body problem that underlie both basis sets used here.","marker":"[10]"}],"fun_headline_variants":["Muonic He, Li, H systems get precise resonance energies","Four muonic molecules: resonance energies now certain","Quasibound states of muonic p, d, He, Li resolved","Resonant states of muonic three-body systems pinned","New precise energies for muonic molecule resonances"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation's load-bearing premise is that the exponential and Gaussian bases have converged to the same resonance pole, so the 0.026 atomic unit gap between their energies for (3He-p-mu) is numerical noise rather than evidence that one or both calculations miss the state.","fun_headline_variants_meta":{"raw":{"variants":["Muonic He, Li, H systems get precise resonance energies","Four muonic molecules: resonance energies now certain","Quasibound states of muonic p, d, He, Li resolved","Resonant states of muonic three-body systems pinned","New precise energies for muonic molecule resonances"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000508,"raw_usage":{"total_tokens":2384,"prompt_tokens":760,"completion_tokens":1624,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":376,"completion_tokens_details":{"reasoning_tokens":1541}},"tokens_in":376,"tokens_out":1624,"duration_ms":14986,"temperature":1.0,"reasoning_tokens":1541,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T04:17:50.264086+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"A decisive check is to recompute the (3He-p-mu) resonance with a substantially larger or differently parametrized basis and see whether the exponential and Gaussian energies converge to a common value rather than staying about 0.026 atomic units apart; if the gap persists across independent bases, the reported binding energy in Eq. (17) is not reliable. A second, independent check is to vary the rotation angle and dilation parameter far beyond the ranges $a=0.98$ to $1.02$ and $\\phi=0.1$ to $0.3$ and confirm that the stationary point of the rotational paths in Figure 1 does not move.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the complex coordinate rotation method used to expose resonant poles in the energy plane."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the theory of complex coordinates for resonance energies and widths in atomic and molecular structure."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the stochastic variational method and the computational framework adapted for constructing the Hamiltonian matrix."},{"cited_title":"The study of the reactions of formation of such molec ules and their characteristics is important for calculating the probabilities of muon cataly sis reactions","cited_arxiv_id":null,"evidence_quote":"Provides earlier binding and decay results for hydrogen-helium mesic molecules that the paper compares against."},{"cited_title":"The particles are numbered as follows: 1 - for the helium or lithium nucleus, 2 - for the muon and 3 - for the proton or deuteron, correspondingly","cited_arxiv_id":null,"evidence_quote":"Provides earlier binding energies and nonradiative decay rates for (Hed-mu) that set the comparison baseline."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reports previous quasibound-state energies for the lithium-deuterium muonic system that the present calculation extends."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Reviews variational methods for the Coulomb three-body problem that underlie both basis sets used here."}],"review_version":1}