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REVIEW 3 major objections 4 minor 14 references

Reply to Comment on "Is a Trineutron Resonance Lower in Energy than a Tetraneutron Resonance?"

T0 review · 3 major / 4 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This reply maintains that extrapolating trapped few-neutron energies to zero trap depth can still indicate possible resonances, while conceding it cannot distinguish a resonance from a virtual state.

desk verdict Honest reply with useful sanity checks, but the resonance-vs-virtual-state gap survives; fine as a reply, not a resolution. read the letter →

arxiv 1908.06610 v1 pith:HIIVG6GC submitted 2019-08-19 nucl-th nucl-ex

S. Gandolfi , H.-W. Hammer , P. Klos , J. E. Lynn , A. Schwenk This is my paper · ORCID
classification nucl-thnucl-ex
keywords trineutrontetraneutronfew-neutronresonancestrapextrapolationauxiliary-fielddiffusionMonteCarloWoods-Saxonpotentialvirtualstateimaginary-timeevolution
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 reply to a Comment defends the claim that linearly extrapolating the energy of three- or four-neutron states held in an external trap to zero trap depth can indicate possible few-neutron resonances. The authors argue that this trap-based procedure differs from analytic continuation in the coupling constant, and that the states they compute are bound in the conventional sense inside the trap. They add two-body test calculations in which the same extrapolation locates two known resonances within the fit uncertainties, and they report that the four-neutron energy stays flat over long imaginary time in a regime where the dineutron is bound, which they read as evidence against a simple pair-of-dineutrons structure. They explicitly concede that the extrapolation cannot distinguish a resonance from a virtual state and says nothing about widths. The question matters because the existence of few-neutron resonances remains experimentally unsettled.

What carries the argument

The load-bearing object is the linear extrapolation of the trapped ground-state energy to zero trap depth, with the depth $V_0$ of an external Woods-Saxon well as the control parameter. The reply's support comes from two-body checks: two S-wave two-Gaussian interactions tuned to resonances at 0.78 MeV and 4.40 MeV, whose extrapolated intercepts match the known positions within the fit uncertainty. The imaginary-time evolution of the energy serves as a decay diagnostic, with a quickly decaying unbound $^8$Be-like $4^+$ state as the counterexample and the flat four-neutron energy as the claimed stable case.

What would settle it

Take a two-body model whose untrapped spectrum is known to contain a virtual state and no resonance. If the zero-depth extrapolation returns a positive intercept near that virtual-state energy, the positive intercepts reported for three and four neutrons cannot be taken as resonance evidence. Alternatively, evolve the trapped four-neutron state in the $E_{4n}<E_{2n}$ regime for substantially longer imaginary time; a downward bend in the energy would contradict the claimed stability and the conclusion that the state is more complex than separated dineutrons.

Watch

Extended reading notes

Core claim

The central claim of this reply is that the trap-extrapolation analysis of the original Letter remains a valid indicator of possible few-neutron resonances. When the energy of a few-body state in a Woods-Saxon trap is extrapolated linearly to zero well depth, the intercept reproduces the known energies of two S-wave two-body resonances, and the same procedure applied to three- and four-neutron states yields a common positive energy scale. For the four-neutron system, the energy shows no decay over very long imaginary-time evolution in a regime where $E_{4n}<E_{2n}$, which the authors take to mean the state is more complex than two dineutrons or a dineutron plus two neutrons; restricting the extrapolation to points with $E_{4n}<E_{2n}$ still places a possible tetraneutron resonance near 2.5 MeV. The authors do not claim that these resonances definitely exist, do not claim to know their widths, and explicitly state that the extrapolation cannot distinguish a resonance from a virtual state.

Load-bearing premise

The load-bearing premise is that the energy of a bound state in a finite external trap, extrapolated linearly to zero trap depth, lands on a resonance or virtual state of the untrapped Hamiltonian; the authors test this on two-body examples but do not prove it, and they concede the extrapolation cannot tell a resonance from a virtual state.

Editorial extensions

If this is right

  • If the extrapolation is accepted, the common positive energy scale found for trapped three- and four-neutron states remains an indication that few-neutron resonances may exist, but with no width information attached.
  • The two-body benchmark implies that, within this interaction class, the zero-depth intercept carries information about where a resonance sits, not merely about the trap itself.
  • The long flat four-neutron energy curve implies that a trapped four-neutron state in the $E_{4n}<E_{2n}$ regime is not trivially a dineutron pair or a dineutron plus two free neutrons, so a realistic explanation must involve a more collective structure.
  • Because the method cannot separate resonances from virtual states, any claimed 3n or 4n resonance derived from trap extrapolation should be phrased as a localized near-threshold feature until scattering data decide.

Reading between the lines

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

  • A testable extension the authors do not pursue: run the same zero-depth extrapolation on a two-body model known to produce only a virtual state; a matching positive intercept would mean the method locates near-threshold features generally, not resonances specifically.
  • The imaginary-time stability criterion could be turned into a quantitative diagnostic by comparing trapped three- and four-neutron decay curves against a ladder of known unbound cases in the same Monte Carlo setup, which would give a time threshold beyond which non-decay is meaningful.
  • If a future experiment sees a narrow structure near the extrapolated energy, it would retrospectively validate the method; if it sees only a smooth threshold rise, the virtual-state interpretation becomes the more natural reading of the same calculation.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. This manuscript is a reply to the Comment by Deltuva and Lazauskas on the authors' earlier Letter (PRL 118, 232501 (2017)) concerning possible trineutron and tetraneutron resonances. The reply defends the use of an external trapping potential with a linear extrapolation to zero trap depth as a way to indicate possible few-neutron resonances. It presents new imaginary-time evolution results for four neutrons in a Woods-Saxon well, argues that the absence of decay over long imaginary time distinguishes the 4n state from a simple pair of dineutrons, and reports additional two-body benchmark calculations for two S-wave resonances. The authors also argue that the analytic continuation in the coupling constant (ACCC) approach used by the Commenters is not equivalent to applying an external trap. In the final paragraph they acknowledge that their current extrapolation cannot distinguish between a resonance and a virtual state, and they do not quantify resonance widths.

Significance. If the benchmarking and stability checks in this reply are accepted, they provide useful evidence that the trapped-state extrapolation is a reproducible diagnostic for identifying resonance-like structures in specific model cases. The two-body benchmarks in Fig. 2 are a concrete and honest test, and the authors explicitly narrow their original claim by admitting that resonance versus virtual-state discrimination is not possible with the current method. The reply therefore strengthens the methodological side of the original Letter but does not resolve the central ambiguity raised by the Comment, namely whether the extrapolated few-neutron intercepts correspond to S-matrix poles, virtual states, or finite-volume artifacts.

major comments (3)
  1. [Fig. 2 and final paragraph] The benchmark in Fig. 2 tests only two cases that are known to be resonances, and it contains no control case with a virtual state or a very broad resonance. Since the authors state in the final paragraph that 'our current extrapolation cannot distinguish between a resonance and a virtual state,' the benchmark does not establish that the few-neutron extrapolated intercepts are resonance poles rather than virtual states. A virtual-state control case would be needed before the extrapolation can be claimed to indicate possible few-neutron resonances.
  2. [Fig. 1] The imaginary-time stability of the 4n energy in Fig. 1 is computed inside a finite Woods-Saxon well with V0 = -1.25 MeV and RWS = 6.0 fm. For a Hamiltonian with a confining external well, the lowest eigenstate is necessarily long-lived in imaginary time, regardless of whether the untrapped system has a resonance, a virtual state, or no singularity at all. The comparison with the 8Be inset is not made on equal footing because the 8Be evolution is not performed with the same external confinement. Therefore the absence of decay in Fig. 1 does not support the statement that 'this 4n state is more complex than a pair of dineutrons.'
  3. [First paragraph, definition of bound states] The reply asserts that 'Bound states are states whose wave functions have compact support' and uses this to conclude that all calculated 3n and 4n states in the trap are bound. This is not a standard physics definition: bound states are normalizable eigenstates with energy below the continuum threshold, and wave functions in a finite-range external well generally do not have compact support. More importantly, this definition makes 'bound' equivalent to 'confined by the external trap,' which is exactly the point raised by the Commenters. The argument therefore does not rebut the distinction between trapped states and intrinsic bound states.
minor comments (4)
  1. [Fig. 2 and caption] The symbol '□/2' appears to be a font substitution for Γ/2; please use a proper Gamma in the figure and caption.
  2. [Paragraph 2 (RWS = 7.5 fm extrapolation)] The sentence 'including in the extrapolation only the points where E4n < E2n for RWS = 7.5 fm still identifies the potential 4n resonance at approximately 2.5 MeV' would benefit from stating the V0 range used and the uncertainty of the fitted intercept.
  3. [References] Reference [10] is listed as 'to be published,' and reference [12] is cited as already containing the arguments of the Comment; in a formal reply, a published or preprint identifier should be provided where available.
  4. [Fig. 2 benchmark] The parameters of the two-Gaussian S-wave potential used for the Fig. 2 benchmarks are not given in the reply; a reader should not need to consult the original Letter to reproduce the benchmark.

Circularity Check

1 steps flagged · score 2.0 of 10

No central circularity: two-body benchmarks are independent; only minor redefinition of 'true bound states' is definitional, not load-bearing.

  1. self definitional [Second paragraph of the Reply (discussion of 'true bound states')]
    ""It is not clear what the authors mean by 'true bound states.' Bound states are states whose wave functions have compact support. This is the case for all of our calculated 3n and 4n states in the trap.""

    The reply defines 'bound states' by compact support, a property automatically enforced by the external Woods-Saxon trap, and then asserts that all its trapped 3n/4n states are bound states. The conclusion that these are 'true bound states' therefore follows from the definition rather than from a calculation of the untrapped Hamiltonian. This definitional move is used to dismiss the Comment's distinction, but it is not load-bearing for the extrapolated resonance energies, which are supported by the independent two-body benchmarks.

full rationale

The central claim of the Reply—that a linear extrapolation of trapped few-neutron energies to zero trap depth can indicate possible few-neutron resonances—is not circular. The two-body benchmarks in Fig. 2 are independent checks: trapped two-neutron energies are computed for an S-wave two-Gaussian potential and extrapolated to V0=0, with the intercepts compared with known resonance positions. The few-neutron extrapolations are not fitted to the 3n/4n data, and no fitted parameter is renamed as a prediction. Self-citations to the original Letter and to Pieper establish the method's provenance, but the validation presented in this Reply is new and external to the few-neutron results. The final concession that the extrapolation 'cannot distinguish between a resonance and a virtual state' is a limitation that weakens the resonance interpretation, but it does not reduce the result to its inputs. The only definitional strain is the compact-support redefinition of 'true bound states,' which is a rhetorical move and does not enter the extrapolated-energy derivation. Overall, the derivation chain retains independent content; hence the circularity score is low.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the unproven assumption that trapped bound-state energies, linearly extrapolated to zero trap depth, locate resonances of the untrapped system. The reply gives two-body examples and an imaginary-time stability check, but no general proof, and the authors explicitly concede they cannot distinguish a resonance from a virtual state.

free parameters (1)
  • Two-Gaussian S-wave potential strength parameters = Tuned to yield two-body resonances at ER = 0.78 MeV and ER = 4.40 MeV
    Used in the benchmark tests to verify the extrapolation method; these are free parameters of the test potential, not of the physics claim.
assumptions (4)
  • domain assumption Bound states are states whose wave functions have compact support.
    Invoked to argue that all trapped 3n and 4n states are bound states. This does not address whether the zero-trap-depth limit is a resonance.
  • ad hoc to paper Linear extrapolation of trapped-state energy to zero trap depth identifies resonances of the untrapped Hamiltonian.
    The central methodological assumption of the original Letter and this reply. It is supported by two-body examples, not a derivation, and the authors concede it cannot distinguish resonance from virtual state.
  • ad hoc to paper The difference between ACCC and an external trap invalidates the Comment's bound-dineutron argument.
    The reply asserts the two methods are different, but does not prove why the trap-based extrapolation is the correct analytic continuation to the continuum.
  • domain assumption AFDMC converges to the lowest energy eigenstate with the relevant quantum numbers of the trapped Hamiltonian.
    Standard quantum Monte Carlo assumption, invoked to interpret the imaginary-time energy plateaus as physical eigenstates.

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Cite this review

Pith. "Pith review of Reply to Comment on "Is a Trineutron Resonance Lower in Energy than a Tetraneutron Resonance?"." pith.science (2026). https://pith.science/paper/HIIVG6GC

@misc{pith2026190806610,
  author       = {Pith},
  title        = {Pith review of: Reply to Comment on "Is a Trineutron Resonance Lower in Energy than a Tetraneutron Resonance?"},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HIIVG6GC}},
  note         = {Machine review of arXiv:1908.06610}
}
read the original abstract

We reply to a Comment on our Letter [Phys. Rev. Lett. 118, 232501 (2017), arXiv:1612.01502] by A. Deltuva and R. Lazauskas [Phys. Rev. Lett 123, 069201 (2019), arXiv:1904.00925].

Figures

Figures reproduced from arXiv: 1908.06610 by the authors.

Figure 1
Figure 1. FIG. 1. The 4 [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The energy of two neutrons trapped in various [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

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Reference graph

Works this paper leans on

14 extracted references · 13 canonical work pages

  1. [1]

    Gandolfi, H

    S. Gandolfi, H. W. Hammer, P. Klos, J. E. Lynn, and A. Schwenk, Phys. Rev. Lett. 118, 232501 (2017)

    work page 2017

  2. [2]

    S. C. Pieper, Phys. Rev. Lett. 90, 252501 (2003)

    work page 2003

  3. [3]

    Kisamori et al

    K. Kisamori et al. , Phys. Rev. Lett. 116, 052501 (2016)

    work page 2016

  4. [4]

    C. A. Bertulani and V. Zelevinsky, J. Phys. G 29, 2431 (2003)

    work page 2003

  5. [5]

    N. K. Timofeyuk, J. Phys. G 29, L9 (2003)

    work page 2003

  6. [6]

    Lazauskas and J

    R. Lazauskas and J. Carbonell, Phys. Rev. C 72, 034003 (2005)

    work page 2005

  7. [7]

    Hiyama, R

    E. Hiyama, R. Lazauskas, J. Carbonell, and M. Kamimura, Phys. Rev. C 93, 044004 (2016)

    2016

  8. [8]

    A. M. Shirokov, G. Papadimitriou, A. I. Mazur, I. A. Mazur, R. Roth, and J. P. Vary, Phys. Rev. Lett. 117, 182502 (2016)

    work page 2016

Show all 14 references
  1. [9]

    Fossez, J

    K. Fossez, J. Rotureau, N. Michel, and M. P loszajczak, Phys. Rev. Lett. 119, 032501 (2017)

    2017

  2. [10]

    Adiabatic hyperspherical picture of 3 n and 4n states,

    C. Greene, “Adiabatic hyperspherical picture of 3 n and 4n states,” in Proceedings of the XXII International Con- ference on Few-Body Problems in Physics, Caen, France (to be published.)

  3. [11]

    Lazauskas and J

    R. Lazauskas and J. Carbonell, Phys. Rev. C 71, 044004 (2005)

    2005

  4. [12]

    Deltuva, Phys

    A. Deltuva, Phys. Lett. B 782, 238 (2018)

    2018

  5. [13]

    Deltuva and R

    A. Deltuva and R. Lazauskas, Phys. Rev. Lett. 123, 069201 (2019)

    2019

  6. [14]

    Pastore, R

    S. Pastore, R. B. Wiringa, S. C. Pieper, and R. Schiav- illa, Phys. Rev. C 90, 024321 (2014)

    2014

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Reviewed August 14, 2026 · model on record in the stance chip above.