Hadronic vacuum polarization effect in muonic atoms

Moritz Thierfeld; Natalia S. Oreshkina; Zoia A. Mandrykina

REVIEW 3 major objections 4 minor 23 references

For muonic atoms from Z=20 to Z=100, hadronic vacuum polarization shifts grow with nuclear charge, track measured nuclear-radius fluctuations, and stay within 10% of the 67% non-relativistic ratio to the muonic vacuum polarization.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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Hadronic vacuum polarization shifts in muonic atoms across the periodic table remain near 67% of the muonic vacuum polarization shift, with few-percent nuclear model dependence.

T0 review reviewed 2026-08-05 challenge →

load-bearing objection Useful systematic scan of HVP shifts in muonic atoms, but the unstated choice for the Dirac wavefunctions—point nucleus or finite-size—needs to be pinned down before the high-Z numbers can be trusted. the 3 major comments →

arxiv 2509.01311 v1 pith:GZSUGQF3 submitted 2025-09-01 physics.atom-ph nucl-thquant-ph

Hadronic vacuum polarization effect in muonic atoms

Zoia A. Mandrykina , Moritz Thierfeld , Natalia S. Oreshkina This is my paper

classification physics.atom-ph nucl-thquant-ph

keywords hadronic vacuum polarizationmuonic atomsvacuum polarization rationuclear charge radiiDirac equationfinite nuclear sizeenergy levelsQED corrections

verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

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 computes hadronic vacuum polarization (HVP) energy shifts for the 1s, 2s, 2p1/2, and 2p3/2 states of muonic atoms with nuclear charges Z=20 to 100, using a semi-empirical potential fitted to e+e- annihilation data and two nuclear charge distribution models. It finds that HVP shifts grow with Z but show irregular spikes that disappear when a smooth radius formula replaces measured nuclear charge radii, so the spikes trace the non-monotonic experimental radii. The paper then compares HVP with the muonic vacuum polarization (muVP) shift, finding the ratio stays between 0.60 and 0.70 for all studied states and Z values, within 10% of the non-relativistic 67% benchmark. The two nuclear models agree to a few percent for most nuclei, with differences up to 5-6% for the heaviest 2s states.

Core claim

The central claim is that HVP corrections to energy levels of muonic atoms can be reliably computed across the whole periodic table with a dispersion-relation-based potential, and that two quantitative results hold. First, the Z-dependence of the HVP shift tracks the measured nuclear charge radii, not a smooth Z^(1/3) law: the observed spikes in the high-Z region correlate directly with the non-monotonic behavior of experimental nuclear charge radii, and replacing the radii by a smooth formula restores a monotonic profile. Second, the ratio of HVP to muonic vacuum polarization remains remarkably stable, within 10% of the established non-relativistic 67% value, despite strong relativistic eff

What carries the argument

The machinery is the semi-empirical hadronic Uehling potential, Eq. (2)/(3): a one-dimensional integral over the hadronic polarization function, parameterized piecewise by constants A, B, C fitted to e+e- to hadrons data and restricted here to the low-energy (0-0.7 GeV) domain, folded with the nuclear charge density rho(x). For a homogeneously charged sphere the integral reduces to closed exponential-integral expressions, Eqs. (5)-(6); for the Fermi distribution it is evaluated numerically. First-order energy shifts come from the radial overlap of this potential with Dirac wavefunctions, Eq. (7). The ratio to the muonic vacuum polarization (Uehling) shift provides the benchmark comparison.

Load-bearing premise

The load-bearing premise is that the Dirac radial wavefunctions used in Eq. (7) adequately represent the muon in the actual finite-size nuclear potential; the paper does not state whether they include the finite nuclear charge distribution, and for Z near 100 the 1s wavefunction is strongly modified by nuclear size, so the numerical HVP shifts depend on this silent choice.

What would settle it

Recompute the 1s HVP shift for Z=90-100 with Dirac wavefunctions obtained from a Fermi-distribution nucleus versus a point nucleus; if the two differ by more than about 30% at Z=100, the quoted values are not robust to the wavefunction model. In parallel, measure the 2s-2p Lamb shift of a heavy muonic atom (e.g., Z about 80) to enough precision that the HVP contribution is resolved; if the implied HVP-to-muVP ratio deviates by more than 10% from 0.67, the stability claim fails.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

If this is right

For light and medium muonic atoms, using either the Fermi or homogeneous-sphere nuclear model changes HVP shifts by less than a few percent, so the simpler sphere model is adequate there.
At high Z the HVP shift is isotope-sensitive: measured nuclear radius fluctuations imprint directly on the shift, so precise calculations need the actual radius of the isotope under study rather than a smooth global formula.
The HVP-to-muVP ratio stays within 10% of 0.67 across Z=20-100 for the four lowest states, so the non-relativistic 67% benchmark remains a reasonable planning baseline, but p-states at low Z sit about 5-7% lower and require state-specific values.
For heavy elements the HVP contribution reaches up to 68% of the muVP shift, making HVP a non-negligible part of the Lamb-shift budget in precision muonic x-ray spectroscopy.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The paper leaves unspecified whether the Dirac wavefunctions in Eq. (7) are eigenstates of a Hamiltonian that includes the finite nuclear charge distribution; redoing the calculation with wavefunctions computed in a finite-size Fermi potential versus a point nucleus would likely change the high-Z 1s numbers, so this should be pinned down before using the values in fits.
The stability of the HVP-to-muVP ratio suggests the two corrections share the same short-distance weighting; if future higher-order hadronic corrections preserve this ratio, HVP uncertainties could be absorbed into an effective vacuum-polarization coupling in heavy muonic atoms.
The correlation with nuclear radii could be turned around: precise muonic-atom HVP measurements at high Z, combined with muVP and other QED corrections, might serve as a probe of the nuclear charge radius independent of electron scattering.
The method is restricted to q^2 below 0.7 GeV, so higher-energy hadronic contributions are dropped; if those matter at the percent level, the quoted few-percent model agreement may not cover the full hadronic uncertainty.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

Useful systematic scan of HVP shifts in muonic atoms, but the unstated choice for the Dirac wavefunctions—point nucleus or finite-size—needs to be pinned down before the high-Z numbers can be trusted.

read the letter

This is a clean, practical calculation: HVP shifts for 1s, 2s, 2p1/2, 2p3/2 in muonic atoms from Z=20 to 100, using the Burkhardt–Pietrzyk semiphenomenological hadronic polarization function and two nuclear charge models. The analytic sphere formulas are a nice touch, and the Fermi-model numerical integration is a sensible cross-check. The main new result is systematic: the HVP/muVP ratio stays within 0.60–0.70 across the whole Z range, with s-states near 0.68 and p-states climbing from 0.62 toward the nonrelativistic 67%—that is genuinely new and useful for muonic-atom spectroscopy, even if it doesn't change the theoretical framework.

The soft spot the stress-test flagged is real. Equation (7) evaluates the HVP potential in Dirac wavefunctions G and F, but the paper never says what Hamiltonian produced those wavefunctions. The abstract claims the calculation includes finite nuclear size, and the perturbing potential is built from a finite-size charge distribution, but that's not the same as using Dirac eigenstates of a finite-size Coulomb potential. At Z around 100 the muon 1s wavefunction is strongly pulled into the nucleus; the HVP potential is short-ranged and peaks right there. If those wavefunctions are point-nucleus, the high-Z shifts and the 'irregularities' in Fig. 1 would change materially. The difference matters, and the paper needs to state it explicitly.

Other issues are milder but still worth fixing. There are no error bars anywhere—the B1, C1 fit parameters carry uncertainties, the experimental charge radii carry uncertainties, and those propagate directly into the shifts. The comparison with Sundaresan and Watson (1975) is asserted but not quantified, so 'improved' is hard to evaluate. No code or tabulated data is provided, which limits reuse.

None of this sinks the paper. The calculation is straightforward, the two nuclear models agree to a few percent, and the ratio stability is a solid empirical observation. It just needs a revision that pins down the wavefunction prescription, adds uncertainties, and gives the earlier comparison concretely.

This deserves a serious referee. It's not a breakthrough, but it's a reference-quality calculation for a community that will use these numbers.

Referee Report

3 major / 4 minor

Summary. The manuscript computes first-order hadronic vacuum polarization (HVP) energy shifts for 1s1/2, 2s1/2, 2p1/2, and 2p3/2 states of muonic atoms with Z = 20–100, using a semi-empirical HVP potential fitted to e+e− → hadrons data. Two nuclear charge distributions are compared: a two-parameter Fermi model (numerical integration) and a homogeneously charged sphere model (partly analytical). The authors report that the HVP shift grows with Z, shows high-Z irregularities correlated with nuclear charge radii, and that the HVP-to-muonic-VP ratio stays near 0.60–0.70, within 10% of the nonrelativistic 67% benchmark. They conclude that their results improve on the earlier Sundaresan-Watson treatment.

Significance. If the central numerical claims are correct, the paper provides a systematic reference dataset for HVP corrections in medium- and high-Z muonic atoms, extending earlier work that focused on low-Z systems. The two-model comparison and the stability analysis of the HVP/muonic-VP ratio are useful for ongoing muonic-atom spectroscopy programs. The approach is transparent and the analytic reduction for the sphere model is a practical asset. The main strength is the breadth of the survey and the explicit correlation of high-Z irregularities with measured nuclear radii. However, the paper does not establish the reliability of the underlying Dirac wavefunctions, gives no uncertainty estimates, and does not quantitatively benchmark against the earlier result it claims to improve.

major comments (3)

[Theory, Eq. (7)] The manuscript does not state how the Dirac radial wavefunctions G and F entering Eq. (7) are generated. This is load-bearing: at Z ≳ 60 the muon 1s wavefunction is strongly modified by finite nuclear size, and V_HVP is short-ranged and peaked near or inside the nuclear surface. If point-nucleus Dirac wavefunctions were used, the high-Z shifts in Fig. 1 would differ materially, and the claimed correlation of the shifts with nuclear radii would be an artifact of the perturbing potential only. Please specify the Dirac Hamiltonian (including the nuclear charge distribution) used to obtain G and F, and ideally demonstrate the sensitivity by comparing point-nucleus and finite-size wavefunctions for a few high-Z cases.
[Results and Discussion, Fig. 1] No uncertainties are reported for any numerical result. The final shifts are quotes to many digits (e.g., −250 eV scale in Fig. 1), yet the input HVP fit parameters B1 and C1 carry fit errors from the e+e− analysis, and the Fermi-model integration and radii data introduce additional uncertainty. A quantitative error budget is needed before these results can serve as a reference for high-precision muonic spectroscopy.
[Introduction / Conclusions] The paper claims 'improved theoretical predictions for low-lying energy levels in muonic systems compared to [18]', but no numerical comparison with Sundaresan-Watson is shown. The reader cannot verify whether the differences are due to updated HVP parameters, different wavefunctions, or different nuclear models. A table comparing representative Z values with Ref. [18] would make the improvement claim concrete and falsifiable.

minor comments (4)

[Theory, Eq. (3)] The derivation from Eq. (2) to Eq. (3) is only sketched. The integral over q is not shown, and the sign convention for the exponential integral E_n may confuse readers. A short derivation in an appendix would improve reproducibility.
[Results and Discussion, Eq. (8)] The caption of Fig. 1 defines δ but the main text references it only implicitly. Please state explicitly that δ is the relative difference between Fermi and sphere models and clarify its sign convention in the main text.
[Conclusions] The conclusion states that HVP reaches up to 68% of muonic VP, but Fig. 1 shows ratios near 0.675–0.68 for s states and lower values for p states. The sentence should distinguish the state-dependence, otherwise it overstates a single representative value.
[References] Equation (1) and the parameter values A1, B1, C1 are attributed to Refs. [17,19], but the reader must infer that A1=0 from Ref. [19]. Please indicate explicitly that these are the low-energy Burkhardt-Pietrzyk parameters and specify their fit uncertainty.

Circularity Check

0 steps flagged

No significant circularity: the calculation uses external fitted hadronic parameters and independently computes HVP shifts and ratios; the 67% benchmark is compared, not assumed.

full rationale

The paper's energy shifts are computed from Eq. (7), a first-order expectation value of an HVP potential whose parameters B1, C1 are taken from an external fit to e+e− → hadrons data (refs. [17,19]). The paper does not fit any parameter to its own target quantities, so there is no fitted-input-called-prediction pattern. The HVP-to-muVP ratio is not defined to equal 67%; it is obtained by computing both corrections independently, and the 67% value from Friar and Martorell is used only as a benchmark for comparison. No self-citation is load-bearing: the cited works by Oreshkina (refs. [11,14]) concern nuclear polarization and self-energy, not the HVP potential or the ratio claim. The use of external experimental nuclear charge radii and the semi-empirical hadronic polarization function is a normal dependence on external data, not circularity. The only notable weakness is an unstated modeling choice in Eq. (7), namely whether the Dirac wavefunctions G and F are generated with point-nucleus or finite-size nuclear potentials; this affects high-Z accuracy but is a correctness/documentation issue, not a circularity of the derivation. Therefore the analysis is self-contained in the relevant sense and receives score 0.

Axiom & Free-Parameter Ledger

1 free parameters · 5 axioms · 0 invented entities

The calculation rests on an externally fitted polarization function, a finite-size Dirac wavefunction treatment that is not fully specified, and experimental nuclear radii. No new entities are introduced.

free parameters (1)

Hadronic polarization fit parameters B1 and C1 = B1 = 0.0023092, C1 = 3.9925370 GeV^-2
Taken from Burkhardt and Pietrzyk [19], fitted to e+e- to hadrons cross section data. The HVP potential in Eq. (2) and all derived energy shifts scale with these values, and no fit uncertainty is propagated.

axioms (5)

domain assumption The hadronic vacuum polarization function is represented in the low-energy domain by Re PI_Had(q^2) = B1 ln(1 + C1 q^2), and contributions above 0.7 GeV are neglected.
Invoked in the Theory section, Eqs. (1)-(2). All calculated shifts inherit this truncation.
domain assumption The Dirac wavefunctions used in Eq. (7) include the finite nuclear size potential.
Not explicitly stated, but the matrix element in Eq. (7) requires G and F, and at high Z finite nuclear size strongly modifies them.
standard math First-order perturbation theory is adequate for the HVP potential.
The HVP shift is small compared to the binding energy, making Eq. (7) standard.
domain assumption Experimental nuclear charge radii from Angeli and Marinova [20] are accurate for all elements considered.
The Z dependence of dE_HVP and the spiky structure in Fig. 1 inherit the radii data.
domain assumption The Uehling-type integral representation of the HVP potential, Eq. (2), is valid for an extended nuclear charge distribution.
The potential is obtained by folding the polarization function with the nuclear charge form factor, assuming one-photon-exchange treatment.

reviewed 2026-08-05 · how reviews work

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Cite this review

Pith. "Pith review of Hadronic vacuum polarization effect in muonic atoms." pith.science (2026). https://pith.science/paper/GZSUGQF3

@misc{pith2026250901311,
  author       = {Pith},
  title        = {Pith review of: Hadronic vacuum polarization effect in muonic atoms},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GZSUGQF3}},
  note         = {Machine review of arXiv:2509.01311}
}

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abstract

The hadronic vacuum polarization (HVP) corrections to energy levels in muonic atoms are studied systematically across the periodic table. Two nuclear charge distribution models have been considered, with partly analytical solution for an homogeneously charged sphere model and direct numerical integration for a Fermi nuclear charge distribution. Our comparison reveals a good agreement between both approaches with differences typically below a few percent. The magnitude of HVP corrections increases with nuclear charge, displaying irregularities that correlate directly with the non-monotonic behavior of nuclear radii. Despite strong relativistic effects at high $Z$, the ratio of hadronic to muonic vacuum polarization remains remarkably stable, deviating by less than $10\%$ from the established $67\%$ non-relativistic benchmark value

Figures

Figures reproduced from arXiv: 2509.01311 by Moritz Thierfeld, Natalia S. Oreshkina, Zoia A. Mandrykina.

discussion (0)

Reference graph

Works this paper leans on

23 extracted references · 23 canonical work pages · 1 internal anchor

[1]

For all four states, we observe a general mono- tonic increase in the magnitude of the energy shift with increasing Z; however in the high- Z region ( Z ≥ 60) we observe a spiky, slightly irregular behavior, especially f or the 1s state. Since these anomalies correlate with the non-monotonic behavior of nuclear charge radii [ 20], we repeated our calculat...

work page
[2]

has been used. Another model is the less realistic but more simple homogeneous charged sphere model where the corresponding radius is related to the root-mean-square (rms) nuclear charge ra- dius through R = √ 5/3⟨r2⟩. In the case of the spherical charge distribution, integral (3) can be further simpliﬁed and represented as a sum for two diﬀerent regions,...

work page
[3]

Knecht, A

A. Knecht, A. Skawran, and S. Vogiatzi, Study of nu- clear properties with muonic atoms, Eur. Phys. J. Plus 135, https://doi.org/10.1140/epjp/s13360-020-00777-y (2020)

work page doi:10.1140/epjp/s13360-020-00777-y 2020
[4]

T. Y. Saito, M. Niikura, T. Matsuzaki, and et al., Muonic-atom spectroscopy and impact on nuclear structure and precision QED theory https://doi.org/10.48550/arXiv.2210.16929

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2210.16929
[5]

T. Y. Saito, M. Niikura, T. Matsuzaki, and et al., Muonic x-ray measurement for the nuclear charge distribution: The case of stable palladium isotopes, Phys. Rev. C 111, 034313 (2025)

work page 2025
[6]

Morgner, B

J. Morgner, B. Tu, C. König, and et al., Stringent test of QED with hydrogen-like tin, Nature 622, 53 (2023)

work page 2023
[7]

Okumura, T

T. Okumura, T. Azuma, D. A. Bennett, and et al., Proof-of-principle experiment for testing strong- ﬁeld quantum electrodynamics with exotic atoms: High precision X-ray spectroscopy of muonic neon, Phys. Rev. Lett. 130, 173001 (2023)

work page 2023
[8]

Antognini, N

A. Antognini, N. Berger, T. E. Cocolios, R. Dressler, R. Eichler, A. Eggenberger, P. Indelicato, K. Jung- mann, C. H. Keitel, K. Kirch, A. Knecht, N. Michel, J. Nuber, N. S. Oreshkina, A. Ouf, A. Papa, R. Pohl, M. Pospelov, E. Rapisarda, N. Ritjoho, S. Roccia, N. Sev- erijns, A. Skawran, S. M. Vogiatzi, F. Wauters, and L. Willmann, Measurement of the quad...

work page 2020
[9]

Karshenboim, V

S. Karshenboim, V. Ivanov, and E. Korzinin, Study of nuclear properties with muonic atoms, Eur. Phys. J. D 39, 351 (2006)

work page 2006
[10]

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)

work page 2024
[11]

Borie, Lamb shift in light muonic atoms – Revisited, Annals of Physics 327, 733 (2012)

E. Borie, Lamb shift in light muonic atoms – Revisited, Annals of Physics 327, 733 (2012)

work page 2012
[12]

Chen, Nuclear polarization in muonic atoms of de- formed nuclei, Phys

M. Chen, Nuclear polarization in muonic atoms of de- formed nuclei, Phys. Rev. C 1, 1176 (1970)

work page 1970
[13]

I. A. Valuev and N. S. Oreshkina, Full leading- order nuclear polarization in highly charged ions, Phys. Rev. A 109, 042811 (2024)

work page 2024
[14]

Borie and G

E. Borie and G. A. Rinker, Improved calculation of the muonic-helium lamb shift, Phys. Rev. A 18, 324 (1978)

work page 1978
[15]

V. A. Yerokhin and N. S. Oreshkina, QED calcula- tions of the nuclear recoil eﬀect in muonic atoms, Phys. Rev. A 108, 052824 (2023)

work page 2023
[16]

N. S. Oreshkina, Self-energy correction to the energy levels of heavy muonic atoms, Phys. Rev. Res. 4, L042040 (2022)

work page 2022
[17]

Borie and G

E. Borie and G. A. Rinker, The energy levels of muonic atoms, Rev. Mod. Phys. 54, 67 (1982)

work page 1982
[18]

Breidenbach, E

S. Breidenbach, E. Dizer, H. Cakir, and Z. Harman, Hadronic vacuum polarization correction to atomic en- ergy levels, Phys. Rev. A 106, 042805 (2022)

work page 2022
[19]

Burkhardt, F

H. Burkhardt, F. Jegerlehner, G. Penso, and et al., Uncertainties in the hadronic con- tribution to the qed vacuum polarization, Z. Phys. C - Particles and Fields 43, 497 (1989)

work page 1989
[20]

M. K. Sundaresan and P. J. S. Watson, Hadronic vacuum-polarization corrections in muonic atoms, Phys. Rev. D 11, 230 (1975)

work page 1975
[21]

Burkhardt and B

H. Burkhardt and B. Pietrzyk, Update of the hadronic contribution to the qed vacuum polarization, Physics Letters B 513, 46 (2001)

work page 2001
[22]

Angeli and K

I. Angeli and K. Marinova, Table of experimen- tal nuclear ground state charge radii: An update, Atomic Data and Nuclear Data Tables 99, 69 (2013)

work page 2013
[23]

J. L. Friar and J. Martorell, Hadronic vacuum polariza- tion and the lamb shift, Phys. Rev. A 59, 4061 (1999)

work page 1999

This paper was first reviewed by deepseek-v4-flash on August 5, 2026.

[1] [1]

For all four states, we observe a general mono- tonic increase in the magnitude of the energy shift with increasing Z; however in the high- Z region ( Z ≥ 60) we observe a spiky, slightly irregular behavior, especially f or the 1s state. Since these anomalies correlate with the non-monotonic behavior of nuclear charge radii [ 20], we repeated our calculat...

work page

[2] [2]

has been used. Another model is the less realistic but more simple homogeneous charged sphere model where the corresponding radius is related to the root-mean-square (rms) nuclear charge ra- dius through R = √ 5/3⟨r2⟩. In the case of the spherical charge distribution, integral (3) can be further simpliﬁed and represented as a sum for two diﬀerent regions,...

work page

[3] [3]

Knecht, A

A. Knecht, A. Skawran, and S. Vogiatzi, Study of nu- clear properties with muonic atoms, Eur. Phys. J. Plus 135, https://doi.org/10.1140/epjp/s13360-020-00777-y (2020)

work page doi:10.1140/epjp/s13360-020-00777-y 2020

[4] [4]

T. Y. Saito, M. Niikura, T. Matsuzaki, and et al., Muonic-atom spectroscopy and impact on nuclear structure and precision QED theory https://doi.org/10.48550/arXiv.2210.16929

work page internal anchor Pith review Pith/arXiv arXiv doi:10.48550/arxiv.2210.16929

[5] [5]

T. Y. Saito, M. Niikura, T. Matsuzaki, and et al., Muonic x-ray measurement for the nuclear charge distribution: The case of stable palladium isotopes, Phys. Rev. C 111, 034313 (2025)

work page 2025

[6] [6]

Morgner, B

J. Morgner, B. Tu, C. König, and et al., Stringent test of QED with hydrogen-like tin, Nature 622, 53 (2023)

work page 2023

[7] [7]

Okumura, T

T. Okumura, T. Azuma, D. A. Bennett, and et al., Proof-of-principle experiment for testing strong- ﬁeld quantum electrodynamics with exotic atoms: High precision X-ray spectroscopy of muonic neon, Phys. Rev. Lett. 130, 173001 (2023)

work page 2023

[8] [8]

Antognini, N

A. Antognini, N. Berger, T. E. Cocolios, R. Dressler, R. Eichler, A. Eggenberger, P. Indelicato, K. Jung- mann, C. H. Keitel, K. Kirch, A. Knecht, N. Michel, J. Nuber, N. S. Oreshkina, A. Ouf, A. Papa, R. Pohl, M. Pospelov, E. Rapisarda, N. Ritjoho, S. Roccia, N. Sev- erijns, A. Skawran, S. M. Vogiatzi, F. Wauters, and L. Willmann, Measurement of the quad...

work page 2020

[9] [9]

Karshenboim, V

S. Karshenboim, V. Ivanov, and E. Korzinin, Study of nuclear properties with muonic atoms, Eur. Phys. J. D 39, 351 (2006)

work page 2006

[10] [10]

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)

work page 2024

[11] [11]

Borie, Lamb shift in light muonic atoms – Revisited, Annals of Physics 327, 733 (2012)

E. Borie, Lamb shift in light muonic atoms – Revisited, Annals of Physics 327, 733 (2012)

work page 2012

[12] [12]

Chen, Nuclear polarization in muonic atoms of de- formed nuclei, Phys

M. Chen, Nuclear polarization in muonic atoms of de- formed nuclei, Phys. Rev. C 1, 1176 (1970)

work page 1970

[13] [13]

I. A. Valuev and N. S. Oreshkina, Full leading- order nuclear polarization in highly charged ions, Phys. Rev. A 109, 042811 (2024)

work page 2024

[14] [14]

Borie and G

E. Borie and G. A. Rinker, Improved calculation of the muonic-helium lamb shift, Phys. Rev. A 18, 324 (1978)

work page 1978

[15] [15]

V. A. Yerokhin and N. S. Oreshkina, QED calcula- tions of the nuclear recoil eﬀect in muonic atoms, Phys. Rev. A 108, 052824 (2023)

work page 2023

[16] [16]

N. S. Oreshkina, Self-energy correction to the energy levels of heavy muonic atoms, Phys. Rev. Res. 4, L042040 (2022)

work page 2022

[17] [17]

Borie and G

E. Borie and G. A. Rinker, The energy levels of muonic atoms, Rev. Mod. Phys. 54, 67 (1982)

work page 1982

[18] [18]

Breidenbach, E

S. Breidenbach, E. Dizer, H. Cakir, and Z. Harman, Hadronic vacuum polarization correction to atomic en- ergy levels, Phys. Rev. A 106, 042805 (2022)

work page 2022

[19] [19]

Burkhardt, F

H. Burkhardt, F. Jegerlehner, G. Penso, and et al., Uncertainties in the hadronic con- tribution to the qed vacuum polarization, Z. Phys. C - Particles and Fields 43, 497 (1989)

work page 1989

[20] [20]

M. K. Sundaresan and P. J. S. Watson, Hadronic vacuum-polarization corrections in muonic atoms, Phys. Rev. D 11, 230 (1975)

work page 1975

[21] [21]

Burkhardt and B

H. Burkhardt and B. Pietrzyk, Update of the hadronic contribution to the qed vacuum polarization, Physics Letters B 513, 46 (2001)

work page 2001

[22] [22]

Angeli and K

I. Angeli and K. Marinova, Table of experimen- tal nuclear ground state charge radii: An update, Atomic Data and Nuclear Data Tables 99, 69 (2013)

work page 2013

[23] [23]

J. L. Friar and J. Martorell, Hadronic vacuum polariza- tion and the lamb shift, Phys. Rev. A 59, 4061 (1999)

work page 1999

Hadronic vacuum polarization effect in muonic atoms

The reading

Core claim §

What carries the argument

Load-bearing premise §

What would settle it §

If this is right

Where Pith is reading between the lines

discussion (0)

Core claim

Load-bearing premise

What would settle it