Hydrogenic rotational levels with spin-0 or spin-1/2 constituent particles

Krzysztof Pachucki; Mateusz Pa\'ntak; Vojt\v{e}ch Patk\'o\v{s}

REVIEW 2 major objections 5 minor 19 references

Hydrogenic rotational levels with spin-0 or spin-1/2 constituent particles

T0 review · 2 major / 5 minor · reviewed 2026-07-11 · grok-4.5

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Pith's one-line read A unified NRQED treatment of rotational levels for spin-0 and spin-1/2 exotic atoms yields state-of-the-art energies and a clear path to two-order theory gains.

desk verdict Solid, usable NRQED tables and open code for L>1 exotic atoms; the polarizability estimate is the only soft spot and they already isolate it. read the letter →

arxiv 2607.04815 v1 pith:HFYD2EEX submitted 2026-07-06 physics.atom-ph

Vojtěch Patkóš , Mateusz Pańtak , Krzysztof Pachucki This is my paper · ORCID

classification physics.atom-ph

keywords NRQEDexoticatomsmuonickaonicantiprotonicvacuumpolarizationnuclearpolarizabilityrotationallevels

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

Exotic atoms that bind a muon, kaon or antiproton to a light nucleus sit outside the reach of the ordinary Dirac or Klein–Gordon equations once the two masses become comparable. This paper collects and implements the complete set of nonrelativistic-QED formulas that work for any mass ratio and for either spin-0 or spin-1/2 constituents, restricted only to states with orbital angular momentum L greater than 1. The resulting code supplies the most accurate theoretical energies currently available for the transitions that experiments are already measuring or planning. The authors further show that two still-missing pieces—three-loop vacuum polarization and the vacuum-polarization correction to the leading recoil term—would shrink the residual theory error by roughly two orders of magnitude. With that improvement, precision spectroscopy of these systems could extract nuclear charge radii and electric-dipole polarizabilities at high accuracy and could place tight limits on long-range hadronic forces beyond the Standard Model.

What carries the argument

The NRQED power series for the energy, E = E² + E⁴ + E⁵ + E⁶ + …, evaluated with non-perturbative inclusion of the Uehling and two-loop vacuum-polarization potentials inside the Schrödinger equation and the Breit Hamiltonian, and with exact mass-ratio dependence retained through order α⁶.

What would settle it

Measure any of the listed transitions (for example the 6h–5g line in antiprotonic silicon or the 5g–4f line in muonic neon) at the few-meV level and compare with the tabulated NRQED prediction after the three-loop vacuum-polarization term has been added; a persistent discrepancy larger than the remaining nuclear-polarizability uncertainty would falsify the claimed accuracy.

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Extended reading notes

Core claim

Nonrelativistic quantum electrodynamics, applied uniformly to two-body systems of spin-0 or spin-1/2 particles with arbitrary masses and magnetic moments, already furnishes state-of-the-art predictions for the rotational (L>1) levels of muonic, kaonic and antiprotonic atoms; the same framework can be tightened by two further orders of magnitude once three-loop vacuum polarization and the vacuum-polarization correction to the O(α⁵) recoil term are included.

Load-bearing premise

Nuclear electric-dipole polarizabilities are taken from a simple phenomenological formula that carries an arbitrary 50 percent uncertainty; that estimate currently dominates the second error bar on every tabulated energy for the heavier systems.

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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

This is a clean, practical extension of the authors’ NRQED program. They take the existing mass-exact operators through α^6 (their earlier papers plus the Breit Hamiltonian with finite-size and g-factors), put them into an updated PbarSpectr code that now handles spin-0 or 1/2 on either particle with arbitrary magnetic moments, and produce concrete numbers for the systems people are actually measuring: μ20Ne, p-bar Si isotopes, K19F, K20Ne.

What is new is the unified implementation and the tables. The Schrödinger solution includes Uehling + two-loop EVP non-perturbatively in both E(2) and E(4); every contribution is broken out with two separate uncertainty sources (three-loop EVP estimated conservatively as α^{2}V(1), and nuclear polarizability). The isotope shift for p-bar 29Si–28Si is a nice, clean test case. Comparisons with earlier Dirac/Klein-Gordon-plus-reduced-mass work show the expected small but real mass-ratio differences once relativistic corrections matter. The code is shipped as supplemental material, so the results are reproducible.

The soft spot is exactly what the reader flagged: nuclear static polarizability is taken from a phenomenological formula with an ad-hoc 50 % error bar. That term dominates the second uncertainty for the heavier systems. They quote it separately and do not hide it, so it does not undermine the “state-of-the-art and improvable by two orders with three-loop EVP” claim. E(7) for spin-0 is borrowed from the spin-1/2 result; that is a minor approximation they acknowledge. Strong-interaction shifts for the lowest kaonic levels are left out, again stated.

Citation pattern is normal for this group; the self-cites are to the operator papers that are independently checkable. No free parameters are being sold as predictions.

This is for people doing precision exotic-atom spectroscopy or extracting nuclear radii/polarizabilities. It deserves a serious referee. I would accept it, use the tables, and watch for the three-loop EVP update.

Referee Report

2 major / 5 minor

Summary. The manuscript presents a unified NRQED treatment of two-body hydrogenic systems with spin-0 or spin-1/2 constituents, arbitrary masses and magnetic moments, restricted to rotational states with L>1. It collects and implements the complete set of operators through O(α^6) (with nonperturbative vacuum polarization in E^(2) and E^(4)), extends the PbarSpectr code accordingly, and supplies state-of-the-art numerical predictions for selected levels and transitions in μ^20Ne, p-bar^28,29Si, K^19F and K^20Ne. Explicit formulas for the Breit Hamiltonian, the O(α^5) recoil and Bethe-logarithm terms, the spin-dependent O(α^6) recoil corrections, and the electric-dipole polarizability contribution are given; residual uncertainties are isolated into three-loop vacuum polarization and nuclear polarizability. The authors argue that inclusion of three-loop EVP and the EVP correction to E^(5) would improve theory by roughly two orders of magnitude, enabling high-accuracy extractions of nuclear charge radii, polarizabilities and possible long-range hadronic forces.

Significance. The work supplies mass-exact, spin-generalized NRQED predictions for exotic atoms that are currently or soon to be measured (J-PARC muonic neon, PAX antiprotonic silicon, SIDDHARTA-type kaonic systems). The explicit operators (Eqs. 7–18), the tabulated breakdowns with two distinct uncertainty sources, and the publicly released PbarSpectr code constitute a concrete, reproducible advance over reduced-mass Dirac/Klein-Gordon calculations. If the residual theory error can indeed be reduced by the two orders claimed, precision spectroscopy of these systems becomes a competitive route to nuclear radii, polarizabilities and tests of long-range hadronic interactions. The paper therefore has clear and timely impact for both atomic and nuclear physics.

major comments (2)

Section II, paragraph after Eq. (18) and the uncertainty discussion: the claim that three-loop EVP plus the EVP correction to E^(5) would improve accuracy by “approximately two additional orders of magnitude” is asserted but not quantified for any of the concrete systems. A short estimate (or a reference to an existing three-loop calculation for a related system) for at least one transition (e.g., 5g–4f in μ^20Ne or 6h–5g in p-bar Si) would make the central “improvable-by-two-orders” statement falsifiable rather than qualitative.
Eq. (19) and the accompanying 50 % uncertainty: the phenomenological polarizability formula is used for every tabulated energy and dominates the second uncertainty for the heavier systems. While the manuscript correctly isolates this contribution, the text should state more clearly that the formula is taken from Ref. [7] without re-derivation and that the 50 % figure is an ad-hoc assignment; otherwise readers may misinterpret the second error bar as a controlled theoretical uncertainty.

minor comments (5)

Table I caption: the kaon mass is quoted from the 2008 PDG; a more recent value (or an explicit statement that the 2008 value is retained for consistency with earlier work) would avoid confusion.
Eqs. (11)–(12): the lengthy spin-dependent O(α^6) expressions are given only for the NS component; a brief pointer that the remaining spin-orbit, spin-spin and tensor pieces are taken unchanged from Ref. [6] would help readers who do not have that paper open.
Section III.C (K^19F): the strong-interaction shift of −2 eV for the 4f–3d transition is mentioned only in the text and not in Table V; adding a footnote or a separate column would make the comparison with experiment transparent.
Typographical: “EXP ANSION” in the section heading II; “RESUL TS” in heading III; and the inconsistent use of “Wichman-Kroll” versus the more common “Wichmann-Kroll”.
The Supplemental Material is cited as [4] but the arXiv version does not yet contain a permanent link or DOI; a stable repository identifier would improve long-term reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: NRQED expansion and tabulated predictions rest on explicit published operators and a shipped code; self-citations supply independently checkable formulas rather than closing a definitional loop.

full rationale

The paper’s central results are numerical evaluations of the standard NRQED power series (Eqs. 3–18) for L>1 two-body systems of arbitrary mass and spin 0 or 1/2. E^(2) is obtained by direct numerical solution of the Schrödinger equation with the Uehling + two-loop + Wichmann–Kroll potential; E^(4) is the expectation value of the Breit Hamiltonian (Eq. 7) that already incorporates the exact g-factors and finite-size terms; E^(5) and E^(6) are taken from the closed-form expressions of Refs. [5,6] (which reduce to the known Klein–Gordon/Dirac limits); higher-order pieces are the infinite-mass QED terms of CODATA. All of these operators are written out explicitly, the code that evaluates them is supplied as Supplemental Material, and the only free phenomenological input (nuclear polarizability, Eq. 19) is used solely for a 50 % uncertainty estimate that is quoted separately and never enters the central “state-of-the-art” energy values. Self-citations to the authors’ earlier NRQED papers and to the previous version of PbarSpectr are therefore ordinary literature references to published, machine-reproducible formulas; they do not define the target quantities in terms of themselves, do not re-label a fit as a prediction, and do not invoke an unverified uniqueness theorem. Consequently the derivation chain is self-contained against external benchmarks and exhibits no circular reduction.

Assumptions & free parameters 1 free parameters · 3 assumptions · 0 invented entities

The calculation rests on the standard NRQED expansion, the Schrödinger equation with Uehling + two-loop + Wichmann–Kroll potentials, and a set of previously derived Breit and higher-order operators. The only free numerical input that is not a measured constant is the phenomenological nuclear-polarizability formula used for uncertainty estimates.

free parameters (1)

phenomenological nuclear polarizability α_E2 = 8 (A/132)^2 [(A/132)^{1/3}–0.31] fm^{3}
Eq. (19) supplies a rough estimate with an assigned 50 % uncertainty; it is not fitted to the spectroscopic data of the paper but is used to quote the second error bar on every energy.

assumptions (3)

domain assumption NRQED power counting in α with exact mass-ratio dependence at each order
Invoked throughout Sec. II; the expansion E = E^(2)+E^(4)+… is taken as the organizing principle.
ad hoc to paper Three-loop vacuum-polarization contribution can be bounded by α^{2} V^(1)
Stated in Sec. II as a more conservative estimate than the previous α^{2}/π^{2} V^(1); used for the first uncertainty in all tables.
ad hoc to paper E^(7) for a spin-0 particle is numerically close to the known spin-1/2 result
Explicitly adopted in Sec. II for the higher-order QED terms of kaonic atoms.

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Cite this review

Pith. "Pith review of Hydrogenic rotational levels with spin-0 or spin-1/2 constituent particles." pith.science (2026). https://pith.science/paper/HFYD2EEX

@misc{pith2026260704815,
  author       = {Pith},
  title        = {Pith review of: Hydrogenic rotational levels with spin-0 or spin-1/2 constituent particles},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HFYD2EEX}},
  note         = {Machine review of arXiv:2607.04815}
}

abstract

We employ nonrelativistic quantum electrodynamics with a unified description of two-body systems with spin-0 or spin-1/2 constituents, arbitrary masses, and arbitrary magnetic moments in rotational states with $L>1$, to present state-of-the-art theoretical predictions for muonic, kaonic, and antiprotonic atoms that have recently been measured or are targeted by upcoming experiments. We show that the theoretical accuracy can further be improved, opening the possibility of using precision spectroscopy of muonic and hadronic atoms for high-accuracy determinations of nuclear charge radii and nuclear electric dipole polarizabilities, and for testing the existence of hypothetical long-range hadronic interactions.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed July 11, 2026 · model on record in the stance chip above.

[1] [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)

2022

[2] [1]

The other compo- nents can be found in Ref

µ3 2m3 2 + µ4 m4 2 + 3λ0 4n 4 −g 2 2 µ2 2m2 2 +g 2 µ3 m3 2 − µ4 2m4 2 + 1 n3 g2 2 λ1 − 9 2 µ2 4m2 2 + 3(2 + 5g2 −g 2 2)−2g 2λ1 µ3 4m3 2 + (λ1 −9) µ4 4m4 2 ,(11) and for the cases 1 =s 2 = 1/2 withL >1 is E(6) NS,s1=s2=1/2 −E (6) s1=s2=0 = µ(Zα) 6 ∆ 1 n5 l(l+ 1) (g 1 −2) 4 g1 + 2 2 µ2 m2 1 + (g1 −1) µ3 m3 1 + µ4 m2 1m2 2 l(l+ 1) 2 + (1↔2) + 1 n4 − 3λ0 16 −...

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[3] [2]

Patk´ oˇ s and K

V. Patk´ oˇ s and K. Pachucki, Antiprotonic atoms with nonperturbative inclusion of vacuum polarization and fi- nite nuclear mass, Phys. Rev. A112, 052808 (2025). 7

2025

[4] [3]

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] [4]

W. R. Johnson,Atomic Structure Theory, Berlin, Heidel- berg: Springer-Verlag (2007)

2007

[6] [5]

Supplemental Material: PbarSpectr II code

[7] [6]

Veitia and K

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

2004

[8] [8]

H. Liu, B. Ohayon, O. Shtaif, Y. Soreq, Probing New Hadronic Forces with Heavy Exotic Atoms, Phys. Rev. Lett.135, 131803 (2025)

2025