pith. sign in

arxiv: 2606.27600 · v1 · pith:6P6V44S3new · submitted 2026-06-25 · ✦ hep-ph

Radiative Corrections in Bound States: Recent Results

Andrzej Czarnecki , Artem O. Davydov This is my paper

Pith reviewed 2026-06-29 01:08 UTC · model grok-4.3

classification ✦ hep-ph
keywords radiative correctionsbound statesmuon decayparapositroniumZ bosondecay ratesQED
0
0 comments X

The pith

Precise calculations resolve long-standing discrepancy in bound muon decay rates for light nuclei and show Z boson suppresses parapositronium three-photon decay by many orders of magnitude.

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

The paper presents two calculations of radiative corrections to bound-state decays. It computes the modification to the muon decay rate when bound to nuclei with atomic numbers from 4 to 9, achieving high precision and settling an analytical-numerical mismatch for oxygen. It also recomputes the three-photon decay of parapositronium with Z-boson contributions, finding a rate suppressed by many orders of magnitude relative to earlier estimates. These refinements matter because precise decay rates test electromagnetic and weak interactions in bound systems and support experimental comparisons. A sympathetic reader cares as they improve reliability of predictions used in precision physics measurements.

Core claim

The change of the decay rate of a muon bound to a light nucleus has been calculated for several light nuclei 4≤Z≤9 with high precision, resolving a long-standing discrepancy between analytical and numerical results for oxygen (Z=8). The decay of parapositronium into three photons has been calculated including effects of the Z boson. The resulting rate is many orders of magnitude smaller than previously estimated.

What carries the argument

Radiative corrections to bound-state decay rates, including electromagnetic and weak Z-boson contributions.

If this is right

  • Updated decay rates for bound muons improve accuracy of lifetime predictions in exotic atoms.
  • Resolution of the oxygen discrepancy validates consistency between analytical and numerical methods.
  • The reduced parapositronium rate shows this channel contributes negligibly compared to prior estimates.
  • Results supply refined benchmarks for testing quantum electrodynamics in bound systems.

Where Pith is reading between the lines

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

  • Similar calculations could be extended to other nuclei to map how corrections scale with Z.
  • Negligible three-photon rate may allow simpler models in positronium lifetime studies by dropping this channel.
  • If confirmed, these corrections could adjust interpretations of precision measurements in atomic and particle physics.

Load-bearing premise

The standard perturbative framework for radiative corrections is complete and no unaccounted higher-order or non-perturbative effects change the precision or the order-of-magnitude suppression.

What would settle it

A high-precision experimental measurement of the muon decay rate for oxygen that deviates from the calculated value beyond the stated uncertainty, or a parapositronium three-photon decay rate not suppressed by many orders of magnitude.

Figures

Figures reproduced from arXiv: 2606.27600 by Andrzej Czarnecki, Artem O. Davydov.

Figure 1
Figure 1. Figure 1: The quantity 1 − Γ/Γ0 as a function of 𝑍. Red circles: present work. Blue solid line: analytical approximation 1 − 𝑓 (𝑥𝑒) (1 − (𝛼𝑍) 2 /2), where 𝑓 (𝑥𝑒) accounts for the finite electron mass. Green cross: result of Ref. [8] for 𝑍 = 8. The systematic excess of the red circles above the solid line reflects higher-order corrections in 𝛼𝑍. decay rates were averaged over different isotopes of each element. The a… view at source ↗
Figure 2
Figure 2. Figure 2: Representative 𝑊-loop diagrams contributing to 𝑍 ★(𝑞) → 𝛾(𝑘1)𝛾(𝑘2)𝛾(𝑘3). Diagrams (𝑎) and (𝑏) contain one 𝐴𝑊𝑊 and one 𝐴𝐴𝑊𝑊 photon vertex, (𝑐) contains one 𝑍 𝐴𝑊𝑊 and two 𝐴𝑊𝑊 vertices, (𝑑) contains one 𝑍 𝐴𝑊𝑊 and one 𝐴𝐴𝑊𝑊 vertex, and (𝑒) contains three 𝐴𝑊𝑊 vertices. The photon permutations are generated by S3. where we denote by 𝐷 𝑎 𝜇𝜈 propagators of vector bosons, 𝑎 = 𝑊, 𝑍. In the unitary gauge we use 𝐷 𝑎 𝛼𝛽… view at source ↗
Figure 3
Figure 3. Figure 3: Unitary-gauge propagator and vertices used in the calculation. All momenta entering the vertex expressions are taken as incoming. The elementary Ward identities following from Eqs. (15) and (16) are 𝑞 𝜌𝑉 𝑍 𝛼𝛽𝜌 (𝑝−, 𝑝+, 𝑞) = 𝑔𝑍 h 𝐷 −1 𝛼𝛽 (𝑝+) − 𝐷 −1 𝛼𝛽 (𝑝−) i , (17) 𝑞 𝜌𝑉 𝑍 𝐴 𝛼𝛽𝜌𝜇 (𝑝−, 𝑝+, 𝑞, 𝑘) = 𝑔𝑍 h 𝑉 𝐴 𝛼𝛽𝜇 (𝑝−, 𝑝+ + 𝑞, 𝑘) − 𝑉 𝐴 𝛼𝛽𝜇 (𝑝− + 𝑞, 𝑝+, 𝑘) i . (18) Placing Eq. (17) between the two adjacent propag… view at source ↗
read the original abstract

Two recent studies of radiative corrections to bound state properties are discussed. The change of the decay rate of a muon bound to a light nucleus has been calculated for several light nuclei $4\leq Z \leq 9$ with high precision, resolving a long-standing discrepancy between analytical and numerical results for oxygen ($Z=8$). The decay of parapositronium into three photons has been calculated including effects of the $Z$ boson. The resulting rate is many orders of magnitude smaller than previously estimated.

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.

Referee Report

0 major / 1 minor

Summary. The paper discusses two recent studies of radiative corrections to bound state properties. The change of the decay rate of a muon bound to a light nucleus has been calculated for several light nuclei 4≤Z≤9 with high precision, resolving a long-standing discrepancy between analytical and numerical results for oxygen (Z=8). The decay of parapositronium into three photons has been calculated including effects of the Z boson. The resulting rate is many orders of magnitude smaller than previously estimated.

Significance. If the underlying external calculations hold, the results are significant for precision QED in bound systems: they resolve an analytical-numerical discrepancy in bound-muon decay rates for light nuclei and demonstrate that Z-boson contributions to parapositronium three-photon decay are suppressed by many orders of magnitude, rendering them negligible at current experimental precisions. The manuscript itself functions as a concise summary of external work rather than presenting new derivations, error budgets, or numerical methods.

minor comments (1)
  1. [Abstract] The abstract could include explicit citations to the two external studies to improve immediate accessibility for readers.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment and the recommendation to accept the manuscript. The report accurately summarizes the content and significance of the work.

Circularity Check

0 steps flagged

No significant circularity; paper reports external results only

full rationale

The manuscript is a short discussion summarizing results from two external recent studies on radiative corrections (bound-muon decay rates for 4≤Z≤9 and parapositronium→3γ including Z-boson effects). No derivation chain, equations, fitted parameters, or self-citations are presented within the paper itself that could reduce to inputs by construction. The central claims are reports of calculations performed elsewhere, with no internal steps that qualify as self-definitional, fitted-input predictions, or load-bearing self-citations. This is a normal non-finding for a results-summary paper.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available; no free parameters, axioms, or invented entities are described.

pith-pipeline@v0.9.1-grok · 5598 in / 1199 out tokens · 30777 ms · 2026-06-29T01:08:04.065342+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

26 extracted references · 4 linked inside Pith

  1. [1]

    Uesaka, M

    Y. Uesaka, M. Yamanaka, and Y. Kuno,𝜇− →𝑒 −𝛾in a muonic atom as a probe for effective lepton flavor-violating operators involving photon fields, Phys. Rev. D111, 035017 (2025), arXiv:2411.10304

    arXiv 2025

  2. [2]

    J. E. J. Matias, A. S. Lemos, and F. Dahia,Probing Short-Distance Modifications of Gravity viaSpin-IndependentandSpin-DependentEffectsinMuonicAtoms(2025),arxiv:2511.00719

    arXiv 2025

  3. [3]

    Miscetti,Status of the Mu2e experiment, Nucl

    S. Miscetti,Status of the Mu2e experiment, Nucl. Instrum. Meth. A1073, 170257 (2025). 8 Radiative Corrections in Bound StatesAndrzej Czarnecki

    2025

  4. [4]

    Aoki et al.,Charged Lepton Flavour Violations searches with muons: present and future (2025), arxiv:2503.22461

    M. Aoki et al.,Charged Lepton Flavour Violations searches with muons: present and future (2025), arxiv:2503.22461

    arXiv 2025

  5. [5]

    H.Nishiguchi,Asearchformuon-to-electronconversionatJ-PARC:TheCOMETexperiment, PoSICHEP2024, 469 (2025)

    2025

  6. [6]

    Yamamoto,DeeMe – Muon-Electron Conversion Search Experiment, Phys

    K. Yamamoto,DeeMe – Muon-Electron Conversion Search Experiment, Phys. Sci. Forum8, 39 (2023)

    2023

  7. [7]

    H.Überall,Decayof𝜇 − MesonsBoundintheKShellofLightNuclei,Phys.Rev.119,365–376 (1960)

    1960

  8. [8]

    Watanabe, K

    R. Watanabe, K. Muto, T. Oda, T. Niwa, H. Ohtsubo, R. Morita, and M. Morita,Asymmetry and energy spectrum of electrons in bound-muon decay, Atomic Data and Nucl. Data Tables 54, 165 (1993)

    1993

  9. [9]

    Czarnecki, A

    A. Czarnecki, A. O. Davydov, and M. Y. Kaygorodov,Total decay rate of a muon bound to a light nucleus, Phys. Rev. D113, 036028 (2026),2512.23023

    arXiv 2026

  10. [10]

    Gilinsky and J

    V. Gilinsky and J. Mathews,Decay of bound muons, Phys. Rev.120, 1450 (1960)

    1960

  11. [11]

    M.J.Aslam,A.Czarnecki,G.Zhang,andA.Morozova,Decayofaboundmuonintoabound electron, Phys. Rev. D102, 073001 (2020),2005.07276

    arXiv 2020

  12. [12]

    Czarnecki, X

    A. Czarnecki, X. Garcia i Tormo, and W. J. Marciano,Muon decay in orbit: spectrum of high-energy electrons, Phys. Rev.D84, 013006 (2011),1106.4756

    Pith/arXiv arXiv 2011

  13. [13]

    Watanabe, M

    R. Watanabe, M. Fukui, H. Ohtsubo, and M. Morita,Angular distribution of electrons in bound muon decay, Prog. Theor. Phys.78, 114 (1987)

    1987

  14. [14]

    M.Y.Kaygorodov,Y.S.Kozhedub,A.V.Malyshev,A.O.Davydov,Y.Wu,andS.B.Zhang, Studyofatomiceffectsonelectronspectruminbound-muondecayprocess,Chin.Phys.C50, 063103 (2026),2506.02416

    arXiv 2026

  15. [15]

    M. E. Rose,Relativistic Electron Theory, John Wiley, New York (1961)

    1961

  16. [16]

    Salvat, J

    F. Salvat, J. M. Fernández-Varea, and W. Williamson Jr,Accurate numerical solution of the radial Schrödinger and Dirac wave equations, Computer Physics Communications90, 151–168 (1995)

    1995

  17. [17]

    Salvat and J

    F. Salvat and J. M. Fernández-Varea,RADIAL: A Fortran subroutine package for the solution oftheradialSchrödingerandDiracwaveequations,ComputerPhysicsCommunications240, 165–177 (2019)

    2019

  18. [18]

    Piessens, E

    R. Piessens, E. de Doncker-Kapenga, C. W. Überhuber, and D. K. Kahaner,QUADPACK: a subroutine package for automatic integration, Springer, Berlin (2012)

    2012

  19. [19]

    9 Radiative Corrections in Bound StatesAndrzej Czarnecki

    Y.Uesaka,T.Naito,S.Ebata,andM.Niikura,ComprehensivetableofcalculatedHufffactors, Atomic Data and Nuclear Data Tables page 101809 (2026), arXiv:2602.07501. 9 Radiative Corrections in Bound StatesAndrzej Czarnecki

    arXiv 2026

  20. [20]

    N. U. Sani, M. J. Aslam, and I. Ahmed,Weak decay of the positronium ion(2026), arXiv:2606.25433

    Pith/arXiv arXiv 2026

  21. [21]

    S. D. Bass,QED and Fundamental Symmetries in Positronium Decays, Acta Phys. Polon. B 50, 1319 (2019),1902.01355

    Pith/arXiv arXiv 2019

  22. [22]

    Bernreuther and O

    W. Bernreuther and O. Nachtmann,Weak Interaction Effects in Positronium, Z. Phys.C11, 235 (1981)

    1981

  23. [23]

    A.PokrakaandA.Czarnecki,Parapositroniumcandecayintothreephotons,Phys.Rev.D96, 093002 (2017),1707.09466

    Pith/arXiv arXiv 2017

  24. [24]

    Czarnecki, D

    A. Czarnecki, D. Dagia, T. Gao, and R. Toor,Parapositronium decay into three photons and implications for the neutral pion, Phys. Rev. D113, 073002 (2026),2602.14899

    arXiv 2026

  25. [25]

    E. W. N. Glover and A. G. Morgan,𝑍boson decay into photons, Z. Phys.C60, 175–180 (1993)

    1993

  26. [26]

    Denner and S

    A. Denner and S. Dittmaier,Electroweak Radiative Corrections for Collider Physics, Phys. Rept.864, 1–163 (2020),1912.06823. 10

    arXiv 2020