Antiprotonic atoms with nonperturbative inclusion of vacuum polarization and finite nuclear mass

K. Pachucki; V. Patk\'o\v{s}

REVIEW 3 major objections 4 minor 26 references

Antiprotonic atoms with nonperturbative inclusion of vacuum polarization and finite nuclear mass

T0 review · 3 major / 4 minor · reviewed 2026-08-04 · deepseek-v4-flash

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Pith's one-line read This paper computes antiprotonic-atom transition energies to meV accuracy by solving the Schrödinger equation with vacuum polarization built in.

desk verdict Solid NRQED calculation with a genuine new dataset, but the unspecified antiproton g-factor is a load-bearing omission that must be fixed before the results can be taken at face value. read the letter →

arxiv 2509.07738 v1 pith:I22LFBVZ submitted 2025-09-09 physics.atom-ph

V. Patkóš , K. Pachucki This is my paper · ORCID

classification physics.atom-ph

keywords antiprotonicatomsvacuumpolarizationNRQEDcircularRydbergstatesfinitenuclearmasschargeradiusBreit-PauliHamiltonianUehlingpotential

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

The paper claims that for highly excited rotational states (l>1) of an antiproton circling a spinless nucleus, nonrelativistic QED still converges even at large nuclear charge because the effective expansion parameter is Z alpha / n rather than Z alpha. The authors place the one- and two-loop electron vacuum-polarization potentials directly into the radial Schrödinger equation, solve it numerically, and use the resulting wave function to evaluate relativistic and higher-order QED corrections with exact finite-nuclear-mass dependence through order (Z alpha)^6. Their tabulated transition energies, such as 180553.71(2) eV for the 12o to 11n transition in antiprotonic 184W, are claimed to be the most accurate theoretical predictions to date for these systems. This matters because precise antiprotonic X-ray spectroscopy could then determine nuclear charge radii more accurately than electronic or muonic atoms, and because the systems test QED in a high-field, two-body setting.

What carries the argument

The central object is the radial Schrödinger equation with the potential V(r) = -Z alpha/r + V_VP(r), where V_VP contains the one-loop Uehling potential, the two-loop Källén-Sabry potential, and the Wichmann-Kroll correction. Solving this equation numerically, with small-r logarithmic singularities handled by a power-log series ansatz and large-r by an asymptotic expansion, gives nonrelativistic energies and wave functions that absorb vacuum polarization nonperturbatively. Those wave functions then evaluate the Breit-Pauli Hamiltonian with vacuum polarization and the analytic (Z alpha)^5 and (Z alpha)^6 corrections. The expansion parameter Z alpha / n keeps the series convergent for circular

What would settle it

Recompute the E(4) and E(6) contributions in Table I with the antiproton g-factor set to its measured value g ≈ 5.58 rather than whatever value was silently adopted, and compare the resulting 12o to 11n transition in 184W: if the shift exceeds the quoted 0.02 eV uncertainty, the paper's accuracy claim fails. A direct measurement of the same transition at meV precision would settle the issue empirically.

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Extended reading notes

Core claim

Using NRQED, the paper demonstrates that circular Rydberg states of antiprotonic atoms can be treated nonperturbatively with respect to vacuum polarization: the Uehling, Källén-Sabry, and Wichmann-Kroll potentials are included in the Schrödinger equation, and the Breit-Pauli Hamiltonian modified by those potentials supplies the leading relativistic correction. Finite nuclear mass is included exactly to order (Z alpha)^6 through analytic formulas valid for arbitrary mass ratio. The result is a set of theoretical transition energies, computed by the accompanying PbarSpectr code, that are claimed to be the most accurate to date for l>1 states of antiprotonic atoms with a spinless nucleus, with

Load-bearing premise

The fine-structure energies depend on the antiproton's magnetic moment through spin-orbit terms, but the paper never states which g-factor value was used; if it is not the real antiproton value of about 5.58, the tabulated 'most accurate' transition energies are wrong.

Editorial extensions

If this is right

Table I gives meV-level predictions for antiprotonic transitions in 20Ne, 40Ar, 132Xe, and 184W that upcoming X-ray experiments can test directly.
The finite-size contribution E_fns = c r_C^2 to each transition means circular-state energies directly probe mean-square nuclear charge radii.
If experimental precision reaches the meV level, extracted nuclear radii could compete with values from muonic atoms and electron scattering.
Including the three-loop vacuum polarization potential should improve the theoretical accuracy by about two orders of magnitude.
The same method extends naturally to rotational states of muonic atoms, and later to l=0,1 states once additional QED contributions are added.

Reading between the lines

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

The PbarSpectr code could be adapted to expose the antiproton g-factor as a tunable input; scanning it would reveal how strongly the quoted fine-structure energies depend on this parameter.
Because the dominant uncertainty is estimated rather than computed, a direct numerical inclusion of the recently derived three-loop vacuum-polarization density would settle whether the stated meV accuracy holds.
The analytic (Z alpha)^6 formulas used here are valid for arbitrary constituent masses, so the same machinery may apply to other exotic two-body atoms with heavy orbiting particles, not only antiprotonic ones.
The l-dependence of transition energies gives a cross-check on the assumed suppression of strong-interaction effects: if radius extraction from different l values disagrees, hadronic corrections would need to be reintroduced.

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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

Dear colleague,

This paper makes a genuine contribution: it extends NRQED predictions for antiprotonic atoms to l>1 rotational states, includes one- and two-loop vacuum polarization nonperturbatively in the Schrödinger equation, and treats finite nuclear mass exactly through (Zα)^6. The transition energies in Table I are new, the code is included, and the numbers agree with the less detailed results of Ref. [8] at the expected lower accuracy. The expansion parameter Zα/n for circular states is a sensible way to keep the perturbation series under control for high Z. That part deserves credit.

The soft spots are real. The most serious is that the antiproton g-factor is never stated. The Breit Hamiltonian in Eq. (19) and the (Zα)^6 coefficients in Eqs. (28)-(32) depend on g1, and for l>1 the spin-orbit terms are dominated by (g1-1)/(2m^2). The antiproton's g-factor is about 5.58, not 2. If the authors used g1=2 anywhere in E(4) or E(6), the fine-structure splittings in Table I would be off by large multiples of the quoted 10^-6 accuracy. The text explicitly uses g=2 only for the approximate E(7) estimate, which is fine. But for the main calculation no value is given. This makes the calculation irreproducible and the central 'most accurate to date' claim uncheckable. This is not a minor omission; it affects every term in the table.

A second, lesser issue is that nuclear electric dipole polarizabilities are neglected without an uncertainty estimate. The paper itself notes these enter at (Zα)^6 and can be accounted for, so a simple bound or a numerical estimate should be there. A third issue is the unresolved dispute with Adkins and Jentschura over the completeness of the Breit-Pauli Hamiltonian from Ref. [6]. The authors defend their formula, and the historical verification is decent, but a short note addressing the specific 'gauge' criticism would help an independent referee.

The numerics and the structure of the expansion look sound. The code exists and the agreement with Ref. [8] is external. If the g-factor is clarified—and the table recomputed if needed—the results will be a solid reference for the PAX program. As written, the paper deserves peer review, but with a request for a major revision on the g-factor before it can be trusted.

I'd bring it to a reading group focused on precision calculations, but I'd flag the g-factor at the start.

Recommendation: send to peer review, but the referee should demand the g-factor value and a demonstration that the fine structure is insensitive or corrected.

Referee Report

3 major / 4 minor

Summary. The manuscript presents an NRQED treatment of antiprotonic atoms with spinless nuclei, aimed at circular (l>1) states where the effective expansion parameter is Zalpha/n. The authors solve the radial Schrödinger equation numerically with the Coulomb potential plus the Uehling, two-loop vacuum-polarization, and Wichmann-Kroll potentials included nonperturbatively. Using the resulting wave functions they evaluate the Breit-Pauli relativistic correction, combine this with analytic E5 and E6 results with full mass dependence, and estimate E7 and E8. Table I gives transition energies for 20Ne, 40Ar, 132Xe, and 184W and claims these are the most accurate predictions to date for l>1 antiprotonic atoms. A Mathematica code, PbarSpectr, is provided in the supplemental material.

Significance. The approach is significant: it extends practical NRQED calculations to high-Z two-body systems by exploiting the small parameter Zalpha/n for rotational states, and it incorporates vacuum polarization nonperturbatively while retaining exact finite-mass corrections through order (Zalpha)^6. The reported agreement with Ref. [8] at lower accuracy and the release of the code are concrete strengths. However, the central numerical claim cannot be verified from the manuscript as written because the antiproton g-factor entering the spin-dependent parts of the Hamiltonian is never specified; the same is true for the antiproton charge radius in the finite-size correction. These are not cosmetic omissions, because the tabulated transition energies are between states of definite j and therefore depend on the spin-orbit and fine-structure terms.

major comments (3)

[II, Eq. (19) and IV, Table I] The numerical value of the antiproton g-factor g1 is never stated. For l>1 states with definite j, Eq. (19) contains spin-orbit terms proportional to [(g1-1)/(2m^2)+g1/(2mM)] L·s V'/r, and the E6 coefficients in Eqs. (28)-(32) also depend on g1. Table I lists transitions between different j states, so E(4) is fine-structure sensitive. The physical antiproton has |g|≈5.585, not 2; the text only says 'with g=2' in the context of the approximate E(7) estimate in Eq. (34). If g1=2 was used in E(4) and E(6), the Table I entries, e.g. E(4)=37.26 eV for 184W, would be shifted by an amount that swamps the quoted 0.02 eV uncertainty. Please state g1 and its sign convention explicitly, and confirm that Table I was computed with the physical value.
[IV, Table I] E_fns in Table I includes the finite charge radii of both the nucleus and the antiproton, and the rows E_fns(fs N) are used to demonstrate nuclear charge radius determination. However, no numerical value for the antiproton charge radius r_C1 is given in the paper. This makes the 'Total (point N + fs pbar)' entries and the separation between nuclear and antiproton finite-size effects non-reproducible. The value of r_C1 and its uncertainty should be stated explicitly, preferably in a table of input constants.
[II, Eq. (17)] The uncertainty estimate for the omitted three-loop vacuum polarization, δE3loop ≈ (α/π)^2 [E + (Zα)^2/(2n^2)], is not defined in a way that reproduces the quoted digits in Table I. If E is the tabulated E(2) in eV, (α/π)^2 E is about 0.16 eV for 20Ne and about 5 eV for 184W, whereas the table lists uncertainties of 0.001 eV and 0.02 eV. If a different convention is intended (e.g. Hartree units without the reduced mass), it must be stated. This matters because the three-loop VP uncertainty is claimed to dominate and sets the final precision.

minor comments (4)

[II, Eq. (17)] Please define all symbols in the uncertainty formula; in particular, specify the units of E and the origin of the (Zα)^2/(2n^2) term.
[II, Eq. (37)] The sums over i=−1 use notation that is easy to misread; please clarify the lower limits and the meaning of the A_r and B_r terms.
[IV, Table I] The number of decimal places is inconsistent across rows (e.g. E(5) is given to 4 decimals for Xe/W but E(7) to 3 decimals). A consistent convention would improve readability.
[IV, Table I] The paper states that electric dipole polarizabilities are neglected, but the E6 formula in Eq. (28) includes them. A sentence quantifying the expected size of this neglected contribution would be useful.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the calculation solves the Schrödinger equation with vacuum-polarization potentials and uses prior parameter-free NRQED formulas; no fitted inputs or self-defined predictions.

full rationale

Walking the derivation chain: E(2) is obtained by numerical solution of Eq. (3) with the Coulomb plus vacuum-polarization potential (Eq. 16); E(4) is an expectation value of the Breit Hamiltonian (Eq. 19), cited to Ref. [6]; E(5) uses Eq. (27) with Bethe logarithms from Ref. [17] plus the authors' own high-n values; E(6) uses the general two-body formula (Eq. 28) from Ref. [7]; E(7) and E(8) are explicit estimates from nonrecoil hydrogenic results and the Dirac equation. None of these inputs is fitted to antiprotonic transition data, and none is defined in terms of the tabulated transition energies. The cited prior works [6,7,15] are parameter-free analytic derivations with stated assumptions; even though they share authors with the present paper, they do not contain the target transition energies, so the citations are genuine evidence rather than a self-citation chain. The disputed correctness of Eq. (19) noted in Ref. [16] and the unspecified antiproton g-factor in Eqs. (19) and (28)-(32) are reproducibility/correctness risks: if the wrong g-factor were used, Table I could be wrong. But no step reduces algebraically to its own input or renames a fitted parameter as a prediction. Therefore no circular step is identified.

Assumptions & free parameters 5 free parameters · 5 assumptions · 0 invented entities

The paper rests on standard NRQED machinery, on previously derived analytic formulas, and on several domain assumptions about the suppression of strong interaction and nuclear polarizability. The most important unaccounted inputs are the antiproton g-factor and antiproton charge radius, which are used in the calculations but never stated. No new physical entities are introduced.

free parameters (5)

antiproton g-factor g1 = not stated
Breit Hamiltonian Eq (19) and E6 coefficients Eqs (29)-(32) depend on g1; the paper never reports the value used. The antiproton has a large magnetic moment anomaly (g approximately 5.58), so tabulated fine-structure transitions depend on this unreported input.
antiproton charge radius r_C1 = not stated
Table I includes finite-size corrections for both nucleus and antiproton, but only nuclear radii from Ref [24] are quoted. The antiproton radius input is not given.
nuclear electric dipole polarizabilities alpha_E1, alpha_E2 = 0 (neglected)
Eq (28) includes an alpha_E term at (Z alpha)^6, but Table I sets these to zero without estimating the resulting shift or uncertainty.
numerical grid parameters r0, h, N = 3e-5, 3e-4, 75000
Chosen in Appendix A to achieve 20-digit convergence; not fitted to experimental data, but the values that deliver the stated accuracy are hand-chosen.
three-loop evp uncertainty coefficient = ~1
Eq (17) assigns deltaE_3loop approximately (alpha/pi)^2 [E + (Z alpha)^2/(2n^2)] as dominant uncertainty; the coefficient is an order-of-magnitude estimate, not a derived bound.

assumptions (5)

standard math NRQED power-series expansion in alpha with the nonrelativistic Schrödinger wave function as zeroth order is valid for the considered states.
The paper uses the NRQED framework throughout, with energies expanded as E(2)+E(4)+... and wave functions from Eq. (3).
domain assumption For l > 1 circular states, strong interactions and nuclear structure effects beyond charge radius and polarizability are negligible.
Introduction states that the paper focuses on excited circular states where strong interaction and nuclear structure effects are 'highly suppressed'; this justifies the pointlike QED treatment.
domain assumption The Breit-Pauli Hamiltonian of Ref [6] (Eq. 19) is complete and correct for a spin-1/2 antiproton, including the unspecified g-factor treatment.
The central relativistic correction E(4) is computed with this Hamiltonian. Ref [16] claims terms were omitted due to gauge; the authors rebut this but do not present a new derivation here.
domain assumption The analytic E(5) and E(6) formulas of Ref [7] (Eqs. 27-32) are valid for arbitrary mass ratio and for the antiproton g-factor used.
The higher-order corrections are taken from the authors' previous derivation; the paper does not rederive them.
ad hoc to paper The omitted three-loop vacuum polarization is bounded by the order-of-magnitude estimate in Eq. (17).
This estimate is the dominant quoted uncertainty but is introduced as an approximation rather than a rigorous bound.

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Cite this review

Pith. "Pith review of Antiprotonic atoms with nonperturbative inclusion of vacuum polarization and finite nuclear mass." pith.science (2026). https://pith.science/paper/I22LFBVZ

@misc{pith2026250907738,
  author       = {Pith},
  title        = {Pith review of: Antiprotonic atoms with nonperturbative inclusion of vacuum polarization and finite nuclear mass},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/I22LFBVZ}},
  note         = {Machine review of arXiv:2509.07738}
}

abstract

We demonstrate that energy levels of excited states in a hydrogenic system consisting of an arbitrary nucleus and an antiproton can be calculated within the framework of nonrelativistic quantum electrodynamics, even for a large nuclear charge $Z$. It is because for rotational states the expansion parameter is $Z\,\alpha/n$. The main advantage of this approach is the possibility of exact inclusion of the finite nuclear mass, which we achieve up to the $(Z\,\alpha)^6$ order. In addition, we include unperturbatively the one-loop and two-loop electron vacuum polarization (evp) potentials in the nonrelativistic Hamiltonian, as well as in the leading relativistic correction. The obtained results for $l>1$ states of antiprotonic atoms with spinless nucleus are the most accurate to date. We make available a user-friendly {\sl Mathematica} code for antiprotonic atoms {\sl PbarSpectr}, which can be further improved by combining evp potentials with $(Z\,\alpha)^5$ QED effects, by adding three-loop evp, and by extending to an arbitrary nuclear spin. Finally, we note that rotational states of antiprotonic atoms can be used to determine the mean square nuclear charge radius much more accurately than from electronic or muonic atoms.

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

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Pith tools

Reviewed August 4, 2026 · model on record in the stance chip above.

[1] [8]

Baptista, S

G. Baptista, S. Rathi, M. Roosa, Q. Senetaire, J. Som- merfeldt, T. Azuma, et al., Towards Precision Spec- troscopy of Antiprotonic Atoms for Probing Strong-field QED, arXiv:2501.08893 [physics.atom-ph] (2025)

arXiv 2025

[2] [6]

Veitia and K

A. Veitia and K. Pachucki, Nuclear recoil effects in an- tiprotonic and muonic atoms, Phys. Rev. A69, 042501 (2004)

work page 2004

[3] [1]

Bezginov, T

N. Bezginov, T. Valdez, M. Horbatsch, A. Marsman, A.C. Vutha, and E.A. Hessels, A measurement of the atomic hydrogen Lamb shift and the proton charge ra- dius, Science3651007 (2019)

work page 2019

[4] [2]

Pachucki, V

K. Pachucki, V. Lensky, F. Hagelstein, S. S. Li Muli, S. Bacca, and R. Pohl, Comprehensive theory of the Lamb shift in light muonic atoms, Rev. Mod. Phys.96, 015001 (2024)

2024

[5] [3]

Ohayon, G

B. Ohayon, G. Janka, I. Cortinovis, Z. Burkley, L. de Sousa Borges, E. Depero, A. Golovizin, X. Ni, Z. Salman, A. Suter, C. Vigo, T. Prokscha, and P. Crivelli, Precision Measurement of the Lamb shift in Muonium, Phys. Rev. Lett.128, 011802 (2022)

work page 2022

[6] [4]

P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, CODATA recommended values of the fundamental phys- ical constants: 2022, Rev. Mod. Phys.97, 025002 (2025)

work page 2022

[7] [5]

Borie, Vacuum polarization corrections and spin-orbit splitting in antiprotonic atoms, Phys

E. Borie, Vacuum polarization corrections and spin-orbit splitting in antiprotonic atoms, Phys. Rev. A28, 555 (1983)

work page 1983

[8] [7]

Zatorski, V

J. Zatorski, V. Patk´ oˇ s, and K. Pachucki, Quantum elec- trodynamics of two-body systems with arbitrary masses up toα 6 order, Phys. Rev. A106, 042804 (2022)

work page 2022