Pith. sign in

REVIEW 3 major objections 4 minor 38 references

Recoil corrections with finite nuclear size in hydrogenic systems

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read This paper derives exact finite-radius formulas for recoil finite-size corrections at orders (Zα)^5 and (Zα)^6, and shows the linear nuclear-radius term at (Zα)^5 in the Breit approximation is spurious.

desk verdict A solid, technically demanding QED derivation that earns referee time; the main caveat is that the headline-level 'complete formulas' claim really means 'complete for the exponential charge model,' and a few overstatements need softening. read the letter →

arxiv 2502.09455 v2 pith:DZPZZ4R3 submitted 2025-02-13 physics.atom-ph

Krzysztof Pachucki , Vojtěch Patkóš , Vladimir A. Yerokhin This is my paper · ORCID
classification physics.atom-ph
keywords nuclearrecoilfinitesizeQEDcorrectionsmuonicatomshydrogenicchargeradiusBreitapproximationisotopeshifts
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 derives formulas for the combined nuclear-recoil and finite-nuclear-size energy shifts in hydrogen-like atoms at orders $(Z\alpha)^5$ and $(Z\alpha)^6$, keeping the nuclear charge radius $r_C$ arbitrary instead of expanding in the small parameter $m r_C$. That is what makes the formulas applicable to both electronic and muonic atoms, since for muonic atoms $m r_C$ is not small. The paper also shows that the widely used Breit approximation—adding nuclear kinetic energy and the magnetic Breit interaction to the Dirac equation—produces a finite-size recoil term linear in $r_C$ at order $(Z\alpha)^5$, and that this term is an artifact: in full QED it cancels, leaving a much smaller term quadratic in $r_C$ up to logarithms. If correct, the results remove a known inconsistency between approximate and rigorous QED treatments and sharpen the extraction of nuclear charge radii from muonic-atom spectroscopy.

What carries the argument

The central object is the recoil integral of the heavy-particle QED formulation, Eq. (33), in which the transverse photon propagator carries the nuclear charge form factor $\rho(-k^2)$ with an $\omega$-dependent argument. The derivation works by splitting the $\omega$-integration contour into a pole part and a high-energy part, evaluating the momentum integrals analytically in $d=3-2\epsilon$ dimensions, and using the analytic continuation of $\rho(-k^2)$ to rotate the contour. The pole part is exactly the piece that, when treated in the Breit approximation, produces the spurious linear-$r_C$ term; the high-energy part cancels it in the full QED result.

What would settle it

Numerically evaluate the order-$(Z\alpha)^5$ two-photon recoil integral, Eq. (48), for a dipole form factor along the real $\omega$ axis without contour rotation at a low $Z$, and compare with the closed form in Eq. (49). If a term linear in $r_C$ survives, the claimed full-QED cancellation fails; if the two agree to all orders in $m r_C$, the analytic-continuation assumption is supported.

Watch

Extended reading notes

Core claim

The central claim is that the recoil contribution to the finite-nuclear-size shift can be computed exactly in $r_C$ at the next two orders of the $Z\alpha$ expansion, and that doing so eliminates a long-standing spurious effect. In the full QED treatment, the apparent $(Z\alpha)^5$ term linear in $r_C$ that appears in the Breit approximation vanishes; the true correction is proportional to $r_C^2$ (with logarithmic factors). The final formulas, Eqs. (49), (73), and (75), show that the total finite-size shift contains odd powers of $r_C$, logarithms of $m r_C Z\alpha$, and model-dependent constants, and that for electronic atoms the $(Z\alpha)^6$ recoil term is the dominant correction to the leading finite-size effect.

Load-bearing premise

The nuclear charge form factor $\rho(-k^2)$ must be analytically continuable over the whole complex plane apart from the negative real axis and must vanish at complex infinity; the dipole parametrization satisfies this, while Gaussian and Fermi models do not.

Editorial extensions

If this is right

  • For electronic atoms, the $(Z\alpha)^6$ recoil finite-size correction is numerically the dominant correction to the leading finite-size effect (about 585 Hz in the hydrogen 2S-1S interval) and must be included in high-precision isotope-shift and charge-radius determinations.
  • The linear-in-$r_C$ term at order $(Z\alpha)^5$ that comes from the Breit approximation is unphysical, so many-body calculations using a Breit-type recoil operator for an extended nucleus need a compensating correction or a QED-based recoil operator.
  • Because no expansion in $m r_C$ is made, the formulas apply directly to muonic atoms and can be used to extract root-mean-square charge radii from muonic-atom spectroscopy for light nuclei.
  • The appearance of odd powers of $r_C$ and logarithms means that Seltzer-moment parametrizations, which assume only even multipoles of $r_C$, are not adequate for these finite-size corrections.

Reading between the lines

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

  • The closed formulas require $\rho(-k^2)$ to be analytically continuable and vanishing at complex infinity, so Gaussian or Fermi nuclear charge distributions need either a dipole-like approximation or a direct numerical evaluation of the $\omega$ integral; the paper states the restriction but does not quantify its impact on radius extractions.
  • The exact-in-$r_C$ formulas provide a benchmark for many-body isotope-shift codes: a code should reproduce Eqs. (49) and (71) in the hydrogenic limit, and any residual linear-$r_C$ term at order $(Z\alpha)^5$ would signal the Breit-approximation artifact.
  • The same contour-splitting machinery is likely transferable to radiative recoil finite-size corrections and to the hyperfine splitting of muonic hydrogen, applications the paper names as future work; such calculations would offer low-energy Standard Model tests by comparing electronic and muonic atoms.
Share X Bluesky LinkedIn Reddit HN

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. The paper derives analytic formulas for the combined nuclear-recoil and finite-nuclear-size (fns) corrections of order (Zα)^5 and (Zα)^6 in hydrogenic atoms, working within the heavy-particle QED formalism and the exponential (dipole) nuclear charge form factor. The results are not expanded in the nuclear radius r_C, which is intended to make them applicable to muonic atoms. The paper also demonstrates that the linear-in-r_C recoil fns term at order (Zα)^5 obtained in the Breit approximation is spurious and cancels in the full QED treatment, leaving a term ∝ r_C^2 ln r_C.

Significance. This is a strong analytic calculation. It shows explicit cancellation of dimensional-regularization singularities, reduces the recoil fns corrections to compact formulas plus a tabulated function δf, and checks the Zα expansion against all-order numerical results in Fig. 2. The identification of the spurious linear-in-r_C term in the Breit approximation and its removal by the full QED treatment is a valuable clarification with direct consequences for isotope-shift and King-plot analyses. The central weakness is the restriction to the exponential charge model; because the paper acknowledges this restriction but does not quantify its impact on muonic charge-radius extraction, the practical reach of the results is not fully established.

major comments (3)
  1. [§VII, Eq. (73) and Eq. (75)] The recoil parts of Eqs. (73) and (75) are derived exclusively for the exponential (dipole) charge form factor, as the paper itself states after Eq. (73). Since the analytic-continuation condition used in the derivation fails for Gaussian and Fermi distributions, the paper does not provide values for the recoil fns correction for realistic nuclear models, and Table III is therefore model-specific. In the regime m_μ r_C ~ 1, where the recoil fns correction is most relevant for charge-radius extraction, no estimate of the model dependence of E(5,1) and E(6,1) is given. This leaves the abstract's claim that the results are 'particularly relevant for high-precision determinations' of muonic charge radii partially unsupported. Please either evaluate Eq. (48) (and the analogous E(6) term) for a second realistic charge model such as the Helm or Fermi distribution, or provide a quantitative estimate of the model dependence and state it as an uncertainty in Table III.
  2. [§VI, Eq. (55) and Table I] The function δf(m r_C) is introduced in Eq. (55) and computed numerically with Wolfram Mathematica, but no integral representation, integration method, precision estimate, or code is provided. For muonic atoms the (m r_C)^3 δf term contributes significantly to f and therefore to E(6,1)_fns in Table III, so the result is not reproducible as presented. Please document the numerical evaluation (or provide an ancillary file with the code and tabulated values), and state the numerical accuracy of the δf values in Table I.
  3. [§VIII, Fig. 2] The all-order comparison in Fig. 2 is a central check of the Zα-expansion formulas, but the figure caption and text do not specify which nuclear charge distribution is used in the all-order calculation of Ref. [27]. If the all-order calculation uses the exponential model, the check is internal to that model; if it uses a different distribution, the comparison may conflate model dependence with the Zα-expansion error. Please state the charge-model assumption for both the all-order and the Zα-expansion curves.
minor comments (4)
  1. [Abstract] The phrase 'without any expansion in the nuclear charge radius r_C' should be qualified by 'within the exponential (dipole) charge model' to avoid overstating the model independence of the final formulas.
  2. [§III, Eq. (30)] Please state explicitly before Eq. (30) that κ1 and κ2 are computed for the dipole model and that Table II contains the corresponding values for the Gaussian model, since these parameters are model-dependent.
  3. [§VII, text after Eq. (77)] The sentence 'It is truly astonishing that the expansion in terms of Seltzer moments ... continues to appear in the literature' is editorializing; a factual statement about the mathematical inconsistency of that expansion would be more appropriate.
  4. [§VIII, Eq. (78)] The quoted uncertainty of E(6,0)_fns (H, 2S−1S) = −585(5) Hz is described as twice the difference between the exponential and Gaussian models; please specify how the sign of the difference is handled when converting it into a symmetric uncertainty.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the recoil finite-size derivation is self-contained, with independent prior QED foundations and no fitted parameters.

full rationale

The paper's central derivation starts from the heavy-particle QED recoil formula, Eq. (33), taken from the authors' prior work [17,18]. This is genuine independent support: it is a parameter-free all-orders-in-Zα expression for nuclear recoil with finite nuclear size, derived previously and not constructed to produce the present (Zα)^5 and (Zα)^6 expansions. The subsequent steps are analytic evaluations of specified integrals, with no parameter fitted to data and no target quantity built into the starting definitions. The claim that the linear-in-r_C term at order (Zα)^5 is an artifact of the Breit approximation is demonstrated by computing the pole contribution in Eq. (51) and comparing it with the full QED result of Eq. (49), so the conclusion reverses the approximate result rather than assuming it. The restriction to the dipole nuclear form factor is an openly stated limitation forced by analytic continuation, not a circular move. Self-citations to Refs. [13,17,18,24] provide the framework and cross-checks, but the final formulas are independently derived and verified against all-order numerical calculations; no prediction reduces by construction to an input.

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

The central claim rests on the heavy-particle QED recoil formula from prior work, the dipole parametrization of the nuclear form factor, and the analytic-continuation requirement. No free parameters or invented entities are introduced; r_C and related radii are physical inputs.

assumptions (3)
  • domain assumption The nuclear charge form factor ρ(-k^2) can be analytically continued to the whole complex plane apart from the negative real axis, and vanishes at complex infinity.
    Invoked in Sec. IV to justify the ω-contour manipulations in Eq. (33); only the dipole parametrization satisfies this.
  • domain assumption The heavy-particle QED expression for the recoil correction E_rec (Eq. 33) is exact to first order in m/M.
    Taken from prior work [18]; the paper relies on it without re-deriving it.
  • standard math The nonrecoil fns corrections of order (Zα)^6 are obtained correctly using dimensional regularization and the low-energy expansion of the Dirac-Coulomb Hamiltonian.
    Standard QED technique; the paper presents the derivation in Sec. III.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Recoil corrections with finite nuclear size in hydrogenic systems." pith.science (2026). https://pith.science/paper/DZPZZ4R3

@misc{pith2026250209455,
  author       = {Pith},
  title        = {Pith review of: Recoil corrections with finite nuclear size in hydrogenic systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DZPZZ4R3}},
  note         = {Machine review of arXiv:2502.09455}
}
abstract

Formulas for the combined nuclear-recoil and finite-nuclear-size effects of order $(Z\,\alpha)^5$ and $(Z\,\alpha)^6$ are derived without any expansion in the nuclear charge radius $r_C$, making them applicable to both electronic and muonic atoms. The obtained results are particularly relevant for high-precision determinations of root-mean-square charge radii from muonic atom spectroscopy. We demonstrate that calculations of the atomic isotope shift based on the widely used Breit approximation give rise to an unphysical nuclear-size contribution that is linear in the nuclear charge radius $r_C$ at order $(Z\,\alpha)^5$. This spurious term vanishes in a full QED treatment, leaving the correct contribution quadratic in $r_C$. For electronic atoms, this quadratic term is significantly smaller than the spurious linear contribution.

Figures

Figures reproduced from arXiv: 2502.09455 by the authors.

Figure 1
Figure 1. FIG. 1. Integration contours [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Finite-size recoil correction, calculated to all orders in [PITH_FULL_IMAGE:figures/full_fig_p011_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

38 extracted references · 26 canonical work pages

  1. [27]

    Pachucki and H

    K. Pachucki and H. Grotch, Pure recoil corrections to hydrogen energy levels, Phys. Rev. A51, 1854 (1995)

    1995

  2. [1]

    C. G. Parthey, A. Matveev, J. Alnis, B. Bernhardt, A. Beyer, R. Holzwarth, A. Maistrou, R. Pohl, K. Predehl, T. Udem, T. Wilken, N. Kolachevsky, M. Abgrall, D. Rovera, C. Sa- 14 lomon, P. Laurent, and T. W. H ¨ansch, Improved measurement of the hydrogen 1S-2S transition frequency, Phys. Rev. Lett. 107, 203001 (2011)

    work page 2011

  3. [2]

    We perform first R ddk1 ddk2 integration analytically using formulas from Appendix A, and next integration over ω along the CH contour with the ε → 0 limit

    ρ(⃗k 2 3 ) − 1 k2 1 k2 2 ⃗k 2 3 × δik − ki 2 kk 2 ω2 δjk − kj 1 kk 1 ω2 Tr γi 1 ̸ k2+ ̸ t − m γ0 1 ̸ k1+ ̸ t − m γj + γ0 1 ̸ k1− ̸k2+ ̸ t − m γi 1 ̸ k1+ ̸ t − m γj + γi 1 ̸ k2+ ̸ t − m γj 1 ̸ k2− ̸k1+ ̸ t − m γ0 γ0 + I 4 , (53) where ⃗k3 = ⃗k1 − ⃗k2 and k2 i = ω2 − ⃗k2 i . We perform first R ddk1 ddk2 integration analytically using formulas from Appendix ...

    work page

  4. [3]

    (60) where ⃗ q3 = ⃗ q1 − ⃗ q2

    Gij C (0, ⃗ q3) q2 3 ρ(q2 3) + Tr γ0 1 (̸ p1 − m) γ0 1 (̸ p2 − m) γi (γ0 + I) 4 qj 2 Gij C (0, ⃗ q2) q2 2 ρ(q2 2) . (60) where ⃗ q3 = ⃗ q1 − ⃗ q2. After performing traces we obtain EP 2 = − (4 π Z α)3 ϕ2(0) Z ddq1 (2 π)d Z ddq2 (2 π)d ρ(q2 1) q4 1 ρ(q2 2) q4 2 ρ(q2 3) q4 3 1 M × 4 q2 1 q2 2 − 4 (⃗ q1 · ⃗ q2)2 + ρ′(q2 3) ρ(q2

    work page

  5. [4]

    q2 3 (q2 1 − q2 2)2 + q2 2 q2 3 + q2 1 q2 3 . (61) The complete fns recoil pole high-energy part E(1) P = EP 1 + EP 2 is E(1) P = − (5 − 8 ϵ) m M EH1 − (3 − 8 ϵ) m M EH2 + δEP , (62) where δEP = − (4 π Z α)3 ϕ2(0) m M Z ddq1 (2 π)d Z ddq2 (2 π)d ∂ ∂q 2 3 ρ(q2 1) q4 1 ρ(q2 2) q4 2 ρ(q2 3) (q2 1 − q2 2)2 q2 3 + q2 2 + q2 1 = (Z α)6 n3 m2 M 2471 2592 m2 r2 C...

    work page

  6. [5]

    (A5), but later we have to separate out 1/mi 1 terms and the remainder coincides with the general formula in Eq

    (A6) In fact, when m1 = 0 , we can still use differentiation as in Eq. (A5), but later we have to separate out 1/mi 1 terms and the remainder coincides with the general formula in Eq. (A6). Two non obvious examples are f (0, m2, m3, −1, 1, 1) = − 4 π m2 m3 (m2 2 + m2

    work page

  7. [6]

    , (A7) f (0, m2, m3, −2, 1, 1) = 4 π 3 m2 m3 (3 m2 2 + m2

    work page

  8. [7]

    (A8) Appendix B: Matrix elements for S states The following matrix elements contain 1/ϵ singularity 1 r4 ϵ ≡ ⃗∇ 1 r ϵ 2 = 1 r4 + ⟨π δd(r)⟩ − 2 ϵ + 8 , (B1) 1 r3 ϵ ≡ 1 r 3 ϵ = 1 r3 + ⟨π δd(r)⟩ 1 ϵ + 2 . (B2) Here 1 r3 = lim a→0 Z ∞ a dr r f (r) + f (0) (γ + ln a) = 4 n3 1 2 − 1 2 n − γ − Ψ(n) + ln n 2 , (B3) 1 r4 = lim a→0 Z ∞ a dr r2 f (r) − f (0) a + f ′...

    work page

Show all 38 references
  1. [8]

    Matveev, C

    A. Matveev, C. G. Parthey, K. Predehl, J. Alnis, A. Beyer, R. Holzwarth, T. Udem, T. Wilken, N. Kolachevsky, M. Ab- grall, D. Rovera, C. Salomon, P. Laurent, G. Grosche, O. Terra, T. Legero, H. Schnatz, S. Weyers, B. Altschul, and T. W. H¨ansch, Precision measurement of the hy...

    2013

  2. [9]

    P. Mohr, D. Newell, B. Taylor, and E. Tiesinga, CODATA rec- ommended values of the fundamental physical constants: 2022 (2024), arXiv:2409.03787 [hep-ph]

    2024 arXiv

  3. [10]

    R. Pohl, A. Antognini, F. Nez, F. D. Amaro, F. Biraben, J. a. M. R. Cardoso, D. S. Covita, A. Dax, S. Dhawan, L. M. P. Fernandes, A. Giesen, T. Graf, T. W. H ¨ansch, P. Indelicato, L. Julien, C.-Y . Kao, P. Knowles, E.-O. Le Bigot, Y .-W. Liu, J. A. M. Lopes, L. Ludhova, C. M....

    2010

  4. [11]

    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. [12]

    Schuhmann, L

    K. Schuhmann, L. M. Fernandes, F. Nez, M. A. Ahmed, F. D. Amaro, P. Amaro, F. Biraben, T.-L. Chen, D. S. Covita, A. J. Dax, et al. , The helion charge radius from laser spectroscopy of muonic helium-3 ions, arXiv preprint arXiv:2305.11679 (2023)

    2023 arXiv

  6. [13]

    van der Werf, K

    Y . van der Werf, K. Steinebach, R. Jannin, H. Bethlem, and K. Eikema, The alpha and helion particle charge radius differ- ence from spectroscopy of quantum-degenerate helium, arXiv preprint arXiv:2306.02333 (2023)

    2023 arXiv

  7. [14]

    Qi, P.-P

    X.-Q. Qi, P.-P. Zhang, Z.-C. Yan, L.-Y . Tang, A.-X. Chen, T.-Y . Shi, and Z.-X. Zhong, Revised3He nuclear charge radius due to electronic hyperfine mixing, arXiv preprint arXiv:2409.09279 (2024)

    2024 arXiv

  8. [15]

    Pachucki, Patk ´oˇs, and V

    K. Pachucki, Patk ´oˇs, and V . A. Yerokhin, Second-order hyper- fine correction to H, D, and 3He energy levels, Phys. Rev. A 110, 062806 (2024)

    2024

  9. [16]

    Ohayon, A

    B. Ohayon, A. Abeln, S. Bara, T. E. Cocolios, O. Eizen- berg, A. Fleischmann, L. Gastaldo, C. Godinho, M. Heines, D. Hengstler, G. Hupin, P. Indelicato, K. Kirch, A. Knecht, D. Kreuzberger, J. Machado, P. Navratil, N. Paul, R. Pohl, D. Unger, S. M. V ogiatzi, K. v. Schoeler, a...

    2024

  10. [17]

    J. L. Friar, Nuclear finite size effects in light muonic atoms, Ann. Phys. (NY) 122, 151 (1979)

    1979

  11. [18]

    V . M. Shabaev, Finite nuclear size corrections to the energy lev- els of the multicharged ions, J. Phys. B 26, 1103 (1993)

    1993

  12. [19]

    Pachucki, V

    K. Pachucki, V . Patk ´oˇs, and V . A. Yerokhin, Three-photon- exchange nuclear structure correction in hydrogenic systems, Phys. Rev. A 97, 062511 (2018)

    2018

  13. [20]

    Borie and G

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

    1982

  14. [21]

    Borie, Lamb shift in light muonic atoms — Revisited, Ann

    E. Borie, Lamb shift in light muonic atoms — Revisited, Ann. Phys. 327, 733 (2012)

    2012

  15. [22]

    V . M. Shabaev, A. N. Artemyev, T. Beier, G. Plunien, V . A. Yerokhin, and G. Soff, Recoil correction to the ground-state en- ergy of hydrogenlike ions, Phys. Rev. A57, 4235 (1998)

    1998

  16. [23]

    Pachucki and V

    K. Pachucki and V . A. Yerokhin, QED theory of the nuclear recoil with finite size, Phys. Rev. Lett. 130, 053002 (2023)

    2023

  17. [24]

    Pachucki and V

    K. Pachucki and V . A. Yerokhin, Heavy-particle quantum elec- trodynamics, Phys. Rev. A 110, 032804 (2024)

    2024

  18. [25]

    V . M. Shabaev, Mass corrections in a strong nuclear field, Teor. Mat. Fiz. 63, 394 (1985), [Theor. Math. Phys. 63, 588 – 596 (1985)]

    1985

  19. [26]

    V . M. Shabaev, Nuclear recoil effect in the relativistic theory of multiply charged ions, Sov. J. Nucl. Phys. 47, 69 – 72 (1988)

    1988

  20. [28]

    B. K. Sahoo, S. Blundell, A. V . Oleynichenko, R. F. G. Ruiz, L. V . Skripnikov, and B. Ohayon, Recent advancements in atomic many-body methods for high-precision studies of isotope shifts, J. Phys. B: At. Mol. Opt. Phys. 58, 042001 (2025)

    2025

  21. [29]

    I. S. Anisimova, A. V . Malyshev, D. A. Glazov, M. Y . Kay- gorodov, Y . S. Kozhedub, G. Plunien, and V . M. Shabaev, Model-QED-operator approach to relativistic calculations of the nuclear recoil effect in many-electron atoms and ions, Phys. Rev. A 106, 062823 (2022)

    2022

  22. [30]

    Patk ´oˇs, V . A. Yerokhin, and K. Pachucki, Two-bodyp-state en- ergies at order α6, Phys. Rev. A 109, 022819 (2024)

    2024

  23. [31]

    E. C. Seltzer, k x-ray isotope shifts, Phys. Rev. 188, 1916 (1969)

    1969

  24. [32]

    J. Hur, D. P. L. Aude Craik, I. Counts, E. Knyazev, L. Cald- well, C. Leung, S. Pandey, J. C. Berengut, A. Geddes, W. Nazarewicz, P.-G. Reinhard, A. Kawasaki, H. Jeon, W. Jhe, and V . Vuleti´c, Evidence of two-source King plot nonlinearity in spectroscopic search for new boson...

    2022

  25. [33]

    V . A. Yerokhin and N. S. Oreshkina, QED calculations of the nuclear recoil effect in muonic atoms, Phys. Rev. A108, 052824 (2023)

    2023

  26. [34]

    Tomalak, Two-photon exchange correction to the Lamb shift and hyperfine splitting of S levels, Eur

    O. Tomalak, Two-photon exchange correction to the Lamb shift and hyperfine splitting of S levels, Eur. Phys. J. A55, 1 (2019)

    2019

  27. [35]

    Gorchtein, A hitchhiker’s guide to nuclear polarization in muonic atoms (2025), arXiv:2501.15274 [nucl-th]

    M. Gorchtein, A hitchhiker’s guide to nuclear polarization in muonic atoms (2025), arXiv:2501.15274 [nucl-th]

    2025

  28. [36]

    Wilzewski, L

    A. Wilzewski, L. I. Huber, M. Door, J. Richter, A. Mariotti, L. J. Spieß, M. Wehrheim, S. Chen, S. A. King, P. Micke, M. Filzinger, M. R. Steinel, N. Huntemann, E. Benkler, P. O. Schmidt, J. Flannery, R. Matt, M. Stadler, R. Os- wald, F. Schmid, D. Kienzler, J. Home, D. P. L. ...

    2024 arXiv

  29. [37]

    Nuber, A

    J. Nuber, A. Adamczak, M. A. Ahmed, L. Affolter, F. D. Amaro, P. Amaro, P. Carvalho, Y . H. Chang, T. L. Chen, W. L. Chen, L. M. P. Fernandes, M. Ferro, D. Goeldi, T. Graf, M. Guerra, T. W. H¨ ansch, C. A. O. Henriques, M. Hildebrandt, P. Indelicato, O. Kara, Diffusion of muon...

    2023

  30. [38]

    Vacchi, E

    A. Vacchi, E. Mocchiutti, A. Adamczak, D. Bakalov, G. Bal- dazzi, M. Baruzzo, R. Benocci, R. Bertoni, M. Bonesini, H. Cabrera, et al. , Investigating the proton structure: The FAMU experiment, Nucl. Phys. News 33, 9 (2023)

    2023

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.