Resonant states of muonic three-particle systems with lithium, helium and hydrogen nuclei

A. P. Martynenko; A. V. Eskin; F. A. Martynenko; V. I. Korobov

REVIEW 4 major objections 6 minor 21 references

Resonant states of muonic three-particle systems with lithium, helium and hydrogen nuclei

T0 review · 4 major / 6 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read 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.

desk verdict 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. read the letter →

arxiv 2412.01507 v1 pith:K5K4X77U submitted 2024-12-02 hep-ph physics.atom-ph

A. V. Eskin , V. I. Korobov , A. P. Martynenko , F. A. Martynenko This is my paper · ORCID

classification hep-phphysics.atom-ph

keywords muonicmoleculesresonantstatescomplexcoordinaterotationvariationalmethodthree-bodyCoulombproblemmuon-catalyzedfusionexoticatomsbindingenergies

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 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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

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

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.

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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

Colleague,

The 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.

But 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.

The 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.

I 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.

For 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.

Referee Report

4 major / 6 minor

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.

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 (4)

[Section 3, Tables 1 and 2, Fig. 1] 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.
[Tables 1 and 2, Eqs. (17)–(20)] 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.
[Eq. (16) and Eqs. (17)–(20)] 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.
[Fig. 1 and surrounding text] 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.

minor comments (6)

[Eq. (12)] 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 2, after Eq. (15)] 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.
[Fig. 1] 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 3, last paragraph] 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).
[References] Reference [15] has a garbled author list (“H. A. Bethe E. E. and Salpeter”), and reference [17] is incomplete; please correct these entries.
[Tables 1 and 2] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the variational CCR calculation is self-contained and does not reduce to its inputs.

full rationale

The paper's derivation chain is a direct variational solution of the three-body Coulomb problem: it builds the nonrelativistic Hamiltonian in Eq. (1), applies the standard complex coordinate rotation in Eq. (2), constructs exponential and Gaussian trial bases, solves the generalized eigenvalue problem in Eq. (5), and evaluates Breit-Pauli and finite-nuclear-size corrections as expectation values in Eqs. (13)-(15). No parameter is fitted to the target binding energies; the nonlinear basis parameters are optimization variables in the variational procedure, and the quoted binding energies in Eqs. (17)-(20) are arithmetic consequences of Eq. (16). Using the stationary point of the rotated energy curves in Fig. 1 to identify the resonance energy is the standard CCR analytic-continuation procedure, not a circular reduction. The self-citations [5], [12]-[14] are references for basis forms and numerical methods rather than load-bearing uniqueness claims, and the final results are compared with the independent earlier calculations [1,2] rather than being derived from them. The 0.026 a.u. disagreement between the exponential and Gaussian energies for 3Hepmu and the omission of explicit imaginary parts are convergence and reporting concerns, but they are not circularity: the calculation would stand or fall on numerical accuracy, not on an equation reducing to itself.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The calculation rests on standard nonrelativistic quantum mechanics and the complex coordinate rotation method. No new particles, forces, or physical entities are introduced. The main burdens are the variational convergence of the basis sets and the identification of the resonance pole from the CCR stabilization curves, both of which are treated as assumptions rather than demonstrated.

free parameters (3)

Nonlinear variational parameters in Gaussian and exponential bases = not reported
These are optimized by the stochastic variational method to minimize the energy. The final energies depend on them, and the 0.026 a.u. spread between basis sets shows that convergence is not established.
Basis size N = 1500
The basis size is chosen by hand. No convergence study versus N is reported, so the reader cannot check whether 1500 terms are sufficient.
CCR parameter grid = a=0.98, 1.00, 1.02; phi=0.1,...,0.3
The resonance position is read from rotational paths built on this grid. The paper assumes the stationary point is stable, but does not demonstrate grid independence.

assumptions (6)

domain assumption The nonrelativistic three-body Coulomb Hamiltonian in Eq. (1) with point charges is the starting point.
Relativistic and QED effects are added later as perturbations, so the base model assumes nonrelativistic point-charge Coulomb dynamics.
standard math Complex Coordinate Rotation via H(theta)=T exp(-2i theta)+V exp(-i theta) exposes resonance poles for sufficiently large theta.
This is the standard CCR result cited to [6,7]. The paper assumes the stationary point in the theta path gives the resonance.
standard math The smallest eigenvalue of the generalized eigenvalue problem HC=E lambda B C is an upper bound for the state energy.
This is the standard variational principle, used without further proof.
domain assumption The Breit-Pauli spin-independent terms in Eq. (13) give the leading relativistic and recoil corrections.
The paper adopts this Hamiltonian from [15,16] without derivation or justification for muonic systems.
domain assumption The nuclear finite-size correction in Eq. (14) uses rms radii of He and Li nuclei.
Nuclear radii are taken from prior data, and no uncertainty from those radii is propagated or discussed.
domain assumption The stationary point in the complex energy plane for three dilation parameters identifies the true resonance.
This is the standard CCR stabilization criterion, but the paper provides no proof that the curves in Figure 1 are converged to the pole.

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Cite this review

Pith. "Pith review of Resonant states of muonic three-particle systems with lithium, helium and hydrogen nuclei." pith.science (2026). https://pith.science/paper/K5K4X77U

@misc{pith2026241201507,
  author       = {Pith},
  title        = {Pith review of: Resonant states of muonic three-particle systems with lithium, helium and hydrogen nuclei},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/K5K4X77U}},
  note         = {Machine review of arXiv:2412.01507}
}

read the original abstract

We study the energy spectrum of three-particle systems (He-p-\mu), (He-d-\mu), (Li-p-\mu) and (Li-d-\mu) on the basis of variational approach with exponential and Gaussian basis. Using the Complex Coordinate Rotation (CCR) method we calculate energies of resonant states of listed molecules.

Figures

Figures reproduced from arXiv: 2412.01507 by the authors.

**Figure 1.** Rotational paths for resonance in the systems (He − µ − p), (He − µ − d), (Li − µ − p), (Li − µ − d). The node in the center of the graph corresponds to the stationary point defining the position of the resonance E in the complex energy plane. The results we obtained within the framework of the complex coordinate rotation method are presented both in Tables 1,2 and in [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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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

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

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

INTRODUCTION In this paper we consider three-particle muonic systems of t he type (µpHe ), (µdHe ), (µpLi), (µdLi) which include nuclei of hydrogen isotopes, as well as helium and lithium nuclei. 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....

work page

[2] [2]

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

GENERAL FORMALISM To calculate the energy levels of three-particle molecules , we introduce the Hamiltonian of the system, which in the center-of-mass system in the non- relativistic approximation is constructed from paired Coulomb interactions of particles : H =T +V = p2 1 2m1 + p2 2 2m2 + p2 3 2m3 + z1z2 R + z1z3 r1 + z2z3 r2 , (1) whereT andV denote th...

work page

[3] [3]

In such systems, the two particles have a large positive charge, which leads to a repulsive potential

RESULTS AND DISCUSSION In this paper, we have investigated the energy levels of thre e-particle muonic molecules with helium, lithium, and hydrogen nuclei. In such systems, the two particles have a large positive charge, which leads to a repulsive potential. Howe ver, as studies in previous papers have shown, there are resonant states in such systems. Sin...

work page

[4] [4]

A. V. Kravtsov, A. I. Mikhailov, V. I. Savichev, Calculati on of the decay rates of hydrogen- helium mesic molecules, Hyp. Inter. 82, 205 (1993)

work page 1993

[5] [5]

V. B. Belyaev, O. I. Kartavtsev, V. I. Kochkin, E. A. Kolganov a, Binding energies and nonra- diative decay rates of (Hedµ ) molecular ions, Phys. Rev. A 52, 1765 (1995)

work page 1995

[6] [6]

A. V. Kravtsov, N. P. Popov and G. E. Solyakin, Yu. A. Aristo v, M. P. Faifman, N. F. Truskova, Decay of µ -mesic molecules formed by isotopes of hydrogen and helium n uclei, Phys. Lett. A 83, 379 (1981). 8

work page 1981

[7] [7]

V. M. Bystritskii, A. V. Kravtsov, N. P. Popov, Kinetics of e xcited muonic hydrogen in mixtures of hydrogen and helium isotopes, Zh. Eksp. Teor. Fiz. 97, 73 (1989)

work page 1989

[8] [8]

V. I. Korobov, A. V. Eskin, F. A. Martynenko, O. S. Sukhoruk ova, Nonrelativistic energies and predissociation widths of quasibound states in the (Li3+ − d − µ ) molecular ions, J. Phys. B 53, 6, 065001 (2020)

work page 2020