Recoil corrections to $\mu$H hyperfine splitting

Andrzej Maro\'n; Krzysztof Pachucki; Mateusz Pa\'ntak

arxiv: 2604.06930 · v1 · submitted 2026-04-08 · ⚛️ physics.atom-ph

Recoil corrections to μH hyperfine splitting

Andrzej Maro\'n , Mateusz Pa\'ntak , Krzysztof Pachucki This is my paper

Pith reviewed 2026-05-10 16:53 UTC · model grok-4.3

classification ⚛️ physics.atom-ph

keywords muonic hydrogenhyperfine splittingrecoil correctionsQED correctionsproton structuremuonic atoms

0 comments

The pith

The full theory of muonic hydrogen hyperfine splitting, with direct recoil and QED corrections and proton structure taken from ordinary hydrogen, yields 182626(5) μeV for the ground state.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper calculates every contribution above 1 ppm to the hyperfine splitting in the ground state of muonic hydrogen. Quantum electrodynamic and recoil corrections are evaluated directly from first principles, while the proton structure correction is obtained by subtracting those same corrections from the measured splitting in ordinary hydrogen. The result is a predicted energy of 182626(5) μeV. A sympathetic reader would care because muonic hydrogen places the muon much closer to the proton than an electron does, making the splitting a sharp test of both QED and the proton's internal structure once the recoil terms are under control.

Core claim

The authors present a complete calculation of the hyperfine splitting in muonic hydrogen that includes all terms larger than 1 ppm. Recoil corrections are computed directly for the muonic system, QED contributions are evaluated to the required order, and the proton structure correction is extracted from the known experimental value in electronic hydrogen after removing the differences due to reduced mass and wave-function scale. This procedure produces the numerical prediction 182626(5) μeV.

What carries the argument

Direct evaluation of recoil corrections to the hyperfine interaction in muonic hydrogen, combined with transfer of the proton structure term from ordinary hydrogen after subtracting calculated QED and recoil pieces.

If this is right

The 5 μeV uncertainty sets the precision target for any future measurement of the muonic hydrogen hyperfine splitting.
Agreement between this prediction and experiment would confirm that recoil effects are fully accounted for at the ppm level.
A discrepancy larger than 5 μeV would indicate either missing higher-order recoil terms or an unrecognized difference in the proton structure contribution between the two systems.
The calculation removes the dominant theoretical uncertainty that previously limited comparisons with muonic-atom data.

Where Pith is reading between the lines

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

The same subtraction technique could be applied to hyperfine splittings in other light muonic atoms to isolate nuclear-structure effects.
If measurements reach the quoted precision, the result would provide an independent cross-check on the proton magnetic radius extracted from ordinary hydrogen.
The approach underscores that recoil corrections, though small, must be calculated explicitly rather than absorbed into structure parameters when comparing electronic and muonic systems.

Load-bearing premise

The proton structure correction extracted from ordinary hydrogen hyperfine splitting can be applied to muonic hydrogen with only the quoted 5 μeV uncertainty and without additional model-dependent errors from the different reduced-mass and wave-function regimes.

What would settle it

An experimental measurement of the muonic hydrogen ground-state hyperfine splitting that lies outside the interval 182621–182631 μeV would show that the theoretical prediction is incorrect.

read the original abstract

This work attempts to present a complete theory of the $\mu$H hyperfine splitting, including all contributions above 1 ppm. Quantum electrodynamic and recoil corrections are calculated directly, while the proton structure correction is obtained with the help of the H hyperfine splitting. The resulting theoretical prediction for the ground state of $\mu$H is $E_\mathrm{hfs} = 182\,626(5)$ $\mu$eV.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper computes a 182626(5) μeV prediction for μH hyperfine splitting by handling QED and recoil directly while importing the proton-structure piece from ordinary H data.

read the letter

The central result is a numerical value for the muonic-hydrogen ground-state hyperfine splitting that claims to include every contribution larger than 1 ppm. QED and recoil pieces are evaluated from first principles for the muonic system, and the proton-structure correction is taken from the experimental ordinary-hydrogen splitting with a simple kinematic rescaling. That produces the quoted 182626(5) μeV figure. The explicit recoil treatment at this precision is the concrete step forward; earlier μH work left those terms less fully worked out. The authors also keep the calculation transparent by separating the pieces clearly and stating the uncertainty budget up front. The approach avoids introducing new model dependence for the structure term itself, which is a practical advantage. The soft spot is the transfer of the structure correction. Muonic hydrogen has a reduced mass roughly 186 times larger than electronic hydrogen, so the lepton wave function samples the proton at much smaller distances. Two-photon exchange and higher multipole contributions can therefore pick up extra reduced-mass dependence that a direct rescaling does not automatically capture. The paper quotes a 5 μeV uncertainty on the transferred term, but without an explicit bound on those extra pieces the error bar could be optimistic. If the full text shows that those higher-order terms were estimated and shown to lie below a few μeV, the concern shrinks; otherwise it remains the main place where the result could shift. The work is aimed at atomic physicists who need a high-precision anchor for muonic-hydrogen spectroscopy or for extracting proton-structure parameters from experiment. A reader already familiar with QED bound-state methods will find the calculation straightforward to follow and the numerical result useful for comparison. It is narrow enough that it does not claim broad new physics, but the technical execution is solid enough to merit referee attention. I would send it to peer review rather than desk-reject it.

Referee Report

2 major / 1 minor

Summary. The manuscript presents a calculation of the ground-state hyperfine splitting in muonic hydrogen, with QED and recoil corrections evaluated directly while the proton-structure contribution is extracted from the experimental hyperfine splitting in ordinary hydrogen. It asserts that all terms above 1 ppm are included and reports the prediction E_hfs = 182626(5) μeV.

Significance. If the transfer of the proton-structure term is shown to introduce no additional model-dependent errors beyond the quoted uncertainty, the result would supply a precise benchmark for QED tests in muonic atoms and for comparisons with forthcoming μH experiments. The hybrid approach of direct recoil/QED plus data-driven structure is efficient when the scaling assumptions hold.

major comments (2)

[Abstract] Abstract: the claim that the proton-structure correction is obtained from the H hyperfine splitting and inserted into the μH calculation with only kinematic rescaling does not address reduced-mass-dependent higher-order terms (two-photon exchange, higher multipoles) whose relative weight changes because the μH Bohr radius is ~186 times smaller; this directly affects the validity of the 5 μeV uncertainty.
[Abstract] The central prediction E_hfs = 182626(5) μeV rests on the unproven assertion that structure effects isolated from ordinary hydrogen can be applied to μH without extra model dependence; an explicit bound or re-derivation of these terms using the same proton form-factor input is required to support the “all contributions above 1 ppm” statement.

minor comments (1)

Notation for the reduced-mass scaling factors should be defined explicitly in the first section where they appear.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the insightful comments on our manuscript concerning the treatment of proton-structure corrections in the muonic hydrogen hyperfine splitting. We provide detailed responses to each major comment below and indicate where revisions will be made.

read point-by-point responses

Referee: [Abstract] Abstract: the claim that the proton-structure correction is obtained from the H hyperfine splitting and inserted into the μH calculation with only kinematic rescaling does not address reduced-mass-dependent higher-order terms (two-photon exchange, higher multipoles) whose relative weight changes because the μH Bohr radius is ~186 times smaller; this directly affects the validity of the 5 μeV uncertainty.

Authors: The referee correctly identifies a potential subtlety in our hybrid approach. The extraction from ordinary hydrogen isolates the structure contribution after subtracting QED and recoil terms calculated for the hydrogen reduced mass. When transferring to μH, the leading structure term (primarily the Zemach radius contribution) is rescaled by the ratio of reduced masses and wave function densities at the origin. Higher-order terms involving two-photon exchange and higher multipoles do have different relative weights due to the smaller Bohr radius (~186 times smaller). However, these terms are of higher order in the fine-structure constant and the proton size parameter, and our analysis indicates they remain below the 1 ppm level or are absorbed into the conservative 5 μeV uncertainty assigned to the structure term. We will revise the manuscript to include an explicit discussion of these scaling considerations and a rough estimate of the size of the neglected contributions. revision: partial
Referee: [Abstract] The central prediction E_hfs = 182626(5) μeV rests on the unproven assertion that structure effects isolated from ordinary hydrogen can be applied to μH without extra model dependence; an explicit bound or re-derivation of these terms using the same proton form-factor input is required to support the “all contributions above 1 ppm” statement.

Authors: We disagree that the assertion is unproven, as the method relies on the dominance of the leading structure correction, which is model-independently extracted from hydrogen data. A full re-derivation using proton form factors for μH would indeed provide a more rigorous check but is computationally intensive and outside the primary focus of this paper on recoil corrections. The 5 μeV uncertainty is intended to reflect the precision of the input from hydrogen and any unaccounted scaling effects. To address the referee's request, we will add a section or paragraph providing a bound on the additional model dependence using scaling arguments from the literature on proton form factors. revision: partial

Circularity Check

0 steps flagged

No significant circularity; derivation uses independent external H data

full rationale

The paper calculates QED and recoil corrections directly from first principles for μH. The proton-structure correction is extracted from the experimental hyperfine splitting of ordinary hydrogen, an independent external measurement. This extraction and kinematic transfer does not reduce any μH result to the paper's own inputs or predictions by construction. No self-definitional steps, fitted-input predictions on the same dataset, or load-bearing self-citations appear in the derivation chain. The result remains falsifiable against future μH measurements and is self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The calculation rests on standard QED perturbation theory and the assumption that proton-structure effects extracted from ordinary hydrogen transfer directly to muonic hydrogen.

axioms (2)

domain assumption QED perturbation theory remains valid at the 1 ppm level for muonic hydrogen hyperfine structure
Invoked when stating that all QED and recoil corrections are calculated directly.
ad hoc to paper Proton structure contributions can be isolated from ordinary hydrogen data and applied to muonic hydrogen without extra model dependence beyond the quoted uncertainty
Explicitly used to obtain the structure correction from the H hyperfine splitting.

pith-pipeline@v0.9.0 · 5361 in / 1356 out tokens · 61434 ms · 2026-05-10T16:53:20.255436+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The proton structure correction is obtained with the help of the H hyperfine splitting... Ehfs = 182626(5) μeV
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

δ(1)fns = −2Zα m rZ ... using dipole parametrization

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

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Bodwin, D.R

G.T. Bodwin, D.R. Yennie, Some recoil corrections to the hy- drogen hyperfine splitting, Phys. Rev. D37, 498 (1988)

work page 1988
[2]

V olotka, V .M

A.V . V olotka, V .M. Shabaev, G. Plunien, and G. Soff, Zemach and magnetic radius of the proton from the hyperfine splitting in hydrogen, Eur. Phys. J. D33, 23 (2005)

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Hevler, A

J. Hevler, A. Maro ´n, and K. Pachucki, Relativistic nuclear recoil effects in hyperfine splitting of hydrogenic systems, arXiv:2602.15616 [physics.atom-ph] (2026)

work page arXiv 2026
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L. Essen, R.W. Donaldson, E.G. Hope, M.J. Bangham, Fre- quency of the Hydrogen Maser, Nature229, 110 (1971)

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L. Essen, R.W. Donaldson, E.G. Hope, M.J. Bangham, Hydro- gen Maser Work at the National Physical Laboratory, Metrolo- gia9, 128 (1973)

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Amaro, A

P. Amaro, A. Adamczak, M. Abdou Ahmed, L. Affolter, F. D. Amaro, P. Carvalho, T. -L. Chen, L. M. P. Fernandes, M. Ferro, D. Goeldiet al., Laser excitation of the 1s-hyperfine transition in muonic hydrogen, SciPost Phys.13, 020 (2022)

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A. Vacchi, E. Mocchiutti, A. Adamczak, D. Bakalov, G. Bal- dazzi, M. Baruzzo, R. Benocci, R. Bertoni, M. Bonesini, H. Cabreraet al., Investigating the Proton Structure: The FAMU Experiment, Nuclear Physics News,339 (2023)

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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)

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V . Patkos and K. Pachucki, Antiprotonic atoms with nonpertur- bative inclusion of vacuum polarization and finite nuclear mass, Phys. Rev. A112, 052808 (2025)

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V . Shabaev, Virial relations for the Dirac equation and their ap- plications to calculations of hydrogen-like atoms, in Precision Physics of Simple Atomic Systems, edited by S. G. Karshen- boim and V . B. Smirnov, Lecture Notes in Physics, pages 97 – 113, Berlin, 2003, Springer

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Pachucki, Nuclear recoil correction to the hyperfine splitting in atomic systems, Phys

K. Pachucki, Nuclear recoil correction to the hyperfine splitting in atomic systems, Phys. Rev. A106, 022802 (2022)

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M. I. Eides, H. Grotch, V . A. Shelyuto, Theory of light hydro- genic ions, Phys. Rep.342, 63 (2001)

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A. C. Zemach, Proton Structure and the Hyperfine Shift in Hy- drogen, Phys. Rev.104, 1771 (1956)

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Meißner, Precision cal- culation of the recoil–finite-size correction for the hyperfine splitting in muonic and electronic hydrogen, Phys

Aldo Antognini, Yong-Hui Lin, Ulf-G. Meißner, Precision cal- culation of the recoil–finite-size correction for the hyperfine splitting in muonic and electronic hydrogen, Phys. Lett. B835, 137575 (2022)

work page 2022
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Karshenboim, Nuclear structure-dependent radiative cor- rections to the hydrogen hyperfine splitting, Phys

S.G. Karshenboim, Nuclear structure-dependent radiative cor- rections to the hydrogen hyperfine splitting, Phys. Lett. A225, 97 (1997)

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P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, CO- DATA recommended values of the fundamental physical con- stants: 2022, Rev. Mod. Phys.97, 025002 (2025)

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Patkos, V

V . Patkos, V . A. Yerokhin, and K. Pachucki, Nuclear polariz- ability effects in 3He+ hyperfine splitting, Phys. Rev. A107, 052802 (2023)

work page 2023
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Meißner, New Insights into the Nucleon’s Electromagnetic Structure, Phys

Yong-Hui Lin, Hans-Werner Hammer, and Ulf-G. Meißner, New Insights into the Nucleon’s Electromagnetic Structure, Phys. Rev. Lett.128, 052002 (2022)

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David Ruth, Karl Slifer, Jian-Ping Chen, Carl E. Carlson, Franziska Hagelstein, Vladimir Pascalutsa, Alexandre Deur, Sebastian Kuhn, Marco Ripani, Xiaochao Zheng, Ryan Zielin- ski, Chao Gu, New spin structure constraints on hyperfine split- ting and proton Zemach radius, Phys. Lett. B859, 139116 (2024)

work page 2024
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Kalinowski, K

M. Kalinowski, K. Pachucki, and V .A. Yerokhin, Nuclear- structure corrections to the hyperfine splitting in muonic deu- terium, Phys. Rev. A98, 062513 (2018)

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Hagelstein, V

F. Hagelstein, V . Lensky, B. Malaescu, and V . Pascalutsa, in preparation (2026)

work page 2026
[25]

Ivanov, Evgeny Yu

Vladimir G. Ivanov, Evgeny Yu. Korzinin, Savely G. Karshen- boim, Second-order corrections to the wave function at origin in muonic hydrogen and pionium, Phys. Rev. D80, 027702 (2009). 11

work page 2009
[26]

Eides, Weak-interaction contributions to hyperfine splitting and Lamb shift, Phys

Michael I. Eides, Weak-interaction contributions to hyperfine splitting and Lamb shift, Phys. Rev. A53, 2953 (1996)

work page 1996
[27]

R. N. Faustov, A. P. Martynenko, Muonic hydrogen ground state hyperfine splitting, J. Exp. Theor. Phys.98, 39 (2004)

work page 2004
[28]

F. V . Tkachov, A theorem on analytical calculability of 4-loop renormalization group functions, Phys. Lett.100B, 65 (1981)

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[29]

K. G. Chetyrkin and F. V . Tkachov, Integration by parts: The algorithm to calculateβ-functions in 4 loops, Nucl. Phys. B 192, 159 (1981)

work page 1981
[30]

Pachucki, Radiative corrections to the nuclear size and po- larizability effects in atomic systems, Phys

K. Pachucki, Radiative corrections to the nuclear size and po- larizability effects in atomic systems, Phys. Rev. A112, 012816 (2025)

work page 2025

[1] [1]

Bodwin, D.R

G.T. Bodwin, D.R. Yennie, Some recoil corrections to the hy- drogen hyperfine splitting, Phys. Rev. D37, 498 (1988)

work page 1988

[2] [2]

V olotka, V .M

A.V . V olotka, V .M. Shabaev, G. Plunien, and G. Soff, Zemach and magnetic radius of the proton from the hyperfine splitting in hydrogen, Eur. Phys. J. D33, 23 (2005)

work page 2005

[3] [3]

Karshenboim, Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants, Phys

Savely G. Karshenboim, Precision physics of simple atoms: QED tests, nuclear structure and fundamental constants, Phys. Rep.422, 1 (2005)

work page 2005

[4] [4]

Hevler, A

J. Hevler, A. Maro ´n, and K. Pachucki, Relativistic nuclear recoil effects in hyperfine splitting of hydrogenic systems, arXiv:2602.15616 [physics.atom-ph] (2026)

work page arXiv 2026

[5] [5]

Essen, R.W

L. Essen, R.W. Donaldson, E.G. Hope, M.J. Bangham, Fre- quency of the Hydrogen Maser, Nature229, 110 (1971)

work page 1971

[6] [6]

Essen, R.W

L. Essen, R.W. Donaldson, E.G. Hope, M.J. Bangham, Hydro- gen Maser Work at the National Physical Laboratory, Metrolo- gia9, 128 (1973)

work page 1973

[7] [7]

Amaro, A

P. Amaro, A. Adamczak, M. Abdou Ahmed, L. Affolter, F. D. Amaro, P. Carvalho, T. -L. Chen, L. M. P. Fernandes, M. Ferro, D. Goeldiet al., Laser excitation of the 1s-hyperfine transition in muonic hydrogen, SciPost Phys.13, 020 (2022)

work page 2022

[8] [8]

Vacchi, E

A. Vacchi, E. Mocchiutti, A. Adamczak, D. Bakalov, G. Bal- dazzi, M. Baruzzo, R. Benocci, R. Bertoni, M. Bonesini, H. Cabreraet al., Investigating the Proton Structure: The FAMU Experiment, Nuclear Physics News,339 (2023)

work page 2023

[9] [9]

Itzykson and J

C. Itzykson and J. B. Zuber, Quantum Field Theory, Interna- tional Series In Pure and Applied Physics (McGraw-Hill, New York) (1980)

work page 1980

[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]

Patkos and K

V . Patkos and K. Pachucki, Antiprotonic atoms with nonpertur- bative inclusion of vacuum polarization and finite nuclear mass, Phys. Rev. A112, 052808 (2025)

work page 2025

[12] [12]

V . M. Shabaev, Generalizations of the virial relations for the Dirac equation in a central field and their applications to the Coulomb field, J. Phys. B24, 4479 (1991)

work page 1991

[13] [13]

Shabaev, Virial relations for the Dirac equation and their ap- plications to calculations of hydrogen-like atoms, in Precision Physics of Simple Atomic Systems, edited by S

V . Shabaev, Virial relations for the Dirac equation and their ap- plications to calculations of hydrogen-like atoms, in Precision Physics of Simple Atomic Systems, edited by S. G. Karshen- boim and V . B. Smirnov, Lecture Notes in Physics, pages 97 – 113, Berlin, 2003, Springer

work page 2003

[14] [14]

Pachucki, Nuclear recoil correction to the hyperfine splitting in atomic systems, Phys

K. Pachucki, Nuclear recoil correction to the hyperfine splitting in atomic systems, Phys. Rev. A106, 022802 (2022)

work page 2022

[15] [15]

M. I. Eides, H. Grotch, V . A. Shelyuto, Theory of light hydro- genic ions, Phys. Rep.342, 63 (2001)

work page 2001

[16] [16]

A. C. Zemach, Proton Structure and the Hyperfine Shift in Hy- drogen, Phys. Rev.104, 1771 (1956)

work page 1956

[17] [17]

Meißner, Precision cal- culation of the recoil–finite-size correction for the hyperfine splitting in muonic and electronic hydrogen, Phys

Aldo Antognini, Yong-Hui Lin, Ulf-G. Meißner, Precision cal- culation of the recoil–finite-size correction for the hyperfine splitting in muonic and electronic hydrogen, Phys. Lett. B835, 137575 (2022)

work page 2022

[18] [18]

Karshenboim, Nuclear structure-dependent radiative cor- rections to the hydrogen hyperfine splitting, Phys

S.G. Karshenboim, Nuclear structure-dependent radiative cor- rections to the hydrogen hyperfine splitting, Phys. Lett. A225, 97 (1997)

work page 1997

[19] [19]

P. J. Mohr, D. B. Newell, B. N. Taylor, and E. Tiesinga, CO- DATA recommended values of the fundamental physical con- stants: 2022, Rev. Mod. Phys.97, 025002 (2025)

work page 2022

[20] [20]

Patkos, V

V . Patkos, V . A. Yerokhin, and K. Pachucki, Nuclear polariz- ability effects in 3He+ hyperfine splitting, Phys. Rev. A107, 052802 (2023)

work page 2023

[21] [21]

Meißner, New Insights into the Nucleon’s Electromagnetic Structure, Phys

Yong-Hui Lin, Hans-Werner Hammer, and Ulf-G. Meißner, New Insights into the Nucleon’s Electromagnetic Structure, Phys. Rev. Lett.128, 052002 (2022)

work page 2022

[22] [22]

David Ruth, Karl Slifer, Jian-Ping Chen, Carl E. Carlson, Franziska Hagelstein, Vladimir Pascalutsa, Alexandre Deur, Sebastian Kuhn, Marco Ripani, Xiaochao Zheng, Ryan Zielin- ski, Chao Gu, New spin structure constraints on hyperfine split- ting and proton Zemach radius, Phys. Lett. B859, 139116 (2024)

work page 2024

[23] [23]

Kalinowski, K

M. Kalinowski, K. Pachucki, and V .A. Yerokhin, Nuclear- structure corrections to the hyperfine splitting in muonic deu- terium, Phys. Rev. A98, 062513 (2018)

work page 2018

[24] [24]

Hagelstein, V

F. Hagelstein, V . Lensky, B. Malaescu, and V . Pascalutsa, in preparation (2026)

work page 2026

[25] [25]

Ivanov, Evgeny Yu

Vladimir G. Ivanov, Evgeny Yu. Korzinin, Savely G. Karshen- boim, Second-order corrections to the wave function at origin in muonic hydrogen and pionium, Phys. Rev. D80, 027702 (2009). 11

work page 2009

[26] [26]

Eides, Weak-interaction contributions to hyperfine splitting and Lamb shift, Phys

Michael I. Eides, Weak-interaction contributions to hyperfine splitting and Lamb shift, Phys. Rev. A53, 2953 (1996)

work page 1996

[27] [27]

R. N. Faustov, A. P. Martynenko, Muonic hydrogen ground state hyperfine splitting, J. Exp. Theor. Phys.98, 39 (2004)

work page 2004

[28] [28]

F. V . Tkachov, A theorem on analytical calculability of 4-loop renormalization group functions, Phys. Lett.100B, 65 (1981)

work page 1981

[29] [29]

K. G. Chetyrkin and F. V . Tkachov, Integration by parts: The algorithm to calculateβ-functions in 4 loops, Nucl. Phys. B 192, 159 (1981)

work page 1981

[30] [30]

Pachucki, Radiative corrections to the nuclear size and po- larizability effects in atomic systems, Phys

K. Pachucki, Radiative corrections to the nuclear size and po- larizability effects in atomic systems, Phys. Rev. A112, 012816 (2025)

work page 2025