Hadronic light-by-light scattering contribution to 1S-2S transition in muonium

A. P. Martynenko; A. V. Eskin; F. A. Martynenko; V. I. Korobov

REVIEW 4 major objections 4 minor 56 references

Hadronic light-by-light scattering contribution to 1S-2S transition in muonium

T0 review · 4 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Hadronic light-by-light shifts muonium 1S-2S by roughly -11 Hz in total.

desk verdict A narrow but honest hadronic-correction calculation whose bottom line—tens of Hz at most, far below the 10 kHz Mu-MASS goal—is robust; referee it, but ask for error bars, reproducibility, and a clearer total. read the letter →

arxiv 2411.09727 v1 pith:E6PIX6SO submitted 2024-11-14 hep-ph physics.atom-ph

V. I. Korobov , F. A. Martynenko , A. P. Martynenko , A. V. Eskin This is my paper · ORCID

classification hep-phphysics.atom-ph

keywords hadroniclight-by-lightscatteringmuonium1S-2Stransitionformfactorvectormesondominancescalarmesonspseudoscalaraxial-vector

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

The paper aims to pin down the hadronic light-by-light contribution to the 1S-2S energy interval in muonium, a term of seventh order in the fine-structure constant that must be known if upcoming 10 kHz-level measurements are to be interpreted cleanly. It constructs the relevant electron-muon interaction amplitudes for meson production by two photons, expresses the S-state shifts as multidimensional integrals over Euclidean momenta, and evaluates them numerically. The central numbers are tiny: the amplitudes of Figs. 1 and 2 add to 0.086 Hz, while the additional single-photon hadronic amplitudes of Fig. 3 contribute about -11.3 Hz. If the calculation is right, the whole hadronic correction is negligible at the planned experimental sensitivity, so it will not interfere with extracting the electron-muon mass ratio from the 1S-2S frequency.

What carries the argument

The load-bearing object is the transition form factor for two virtual photons fusing into a meson, modeled through vector-meson dominance as simple monopole forms such as $F(k_1^2,k_2^2)=\Lambda_V^4/[(k_1^2-\Lambda_V^2)(k_2^2-\Lambda_V^2)]$, with analogous cutoff forms for scalars and axial vectors. Working in Euclidean space with projection operators that select $S=0$ and $S=1$ lepton-antilepton states, the authors reduce each class of diagrams to convergent multiple integrals over loop momenta and angular variables, with the meson mass acting as a regulator. Vertical exchanges and the three-photon amplitudes are handled the same way, some analytically because the vertical scalar loop factorizes, and all are evaluated numerically to about one percent accuracy.

What would settle it

Measure the two-photon decay width of the sigma meson: if it exceeds the adopted estimate by about three orders of magnitude, the hadronic correction would rise to the 10 kHz level and the paper's main conclusion would fail, whereas any smaller discrepancy leaves the conclusion intact.

Watch

Extended reading notes

Core claim

The authors claim that the hadronic light-by-light correction to the muonium 1S-2S splitting is a few hertz at most, not the tens of kilohertz scale one might fear from the naive estimate $m\alpha^7$. Summing pseudoscalar, scalar, and axial-vector meson exchanges in horizontal, vertical, and three-photon configurations gives a total of 0.086 Hz from the two-photon fusion amplitudes; adding the polarization-operator insertions of a pion or a $\sigma$ meson in a single-photon exchange changes the interval by about $-1.79$ Hz and $-9.53$ Hz, respectively. The conclusion is that at the 10 kHz target of the Mu-MASS experiment the hadronic contribution can be neglected, and the 1S-2S measurement can be used to extract the electron-muon mass ratio without a hadronic correction.

Load-bearing premise

The numerical results rest on the vector-dominance monopole form factors and on estimates of scalar-meson two-photon couplings; if the true couplings or off-shell behavior differ substantially, the individual contributions, especially the -9.5 Hz sigma term, could change, though the overall smallness next to 10 kHz would likely survive.

Editorial extensions

If this is right

The hadronic light-by-light contribution to the 1S-2S interval from two-photon fusion amplitudes is about 0.1 Hz, so it need not be included in the theoretical prediction at 10 kHz precision.
The dominant hadronic correction comes from single-photon polarization insertions, about $-1.8$ Hz from the pion and $-9.5$ Hz from the sigma meson, still far below the planned experimental accuracy.
Extraction of the electron-muon mass ratio from the 1S-2S measurement can proceed without a hadronic systematic at the precision the Mu-MASS experiment targets.
Axial-vector mesons give essentially no contribution at this order, and vertical exchanges of pseudoscalar and axial mesons vanish in leading order, so only scalar and pseudoscalar three-photon amplitudes matter numerically.

Reading between the lines

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

If the Mu-MASS experiment reaches 10 kHz, a combined fit of the 1S-2S frequency with other muonium observables could test the hadronic model used here only if the measurement improves by another order of magnitude; at the stated precision the hadronic term is a negligible offset.
The vector-dominance monopole form factors could be replaced by dispersive or lattice-QCD transition form factors to turn the estimate into a precision prediction, since the paper's integral formulas are flexible enough to accommodate such replacements.
The same amplitude construction, with different masses and couplings, can be carried over to hadronic light-by-light shifts of other S-level transitions in muonic atoms, where the hadronic suppression factor and the mass-ratio enhancement differ.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Bottom line: this is a serious, narrow calculation, and the main physical conclusion survives contact with the largest uncertainties. The hadronic correction to the muonium 1S–2S interval is at most tens of Hz, three orders of magnitude below the planned 10 kHz Mu-MASS accuracy. I would send it to a referee, with the expectation that the next version adds error bars, cleans up the presentation, and makes the numerics reproducible.

What is new: the same group had done hadronic light-by-light for muonium hyperfine splitting in [24]; here they extend the machinery to the 1S–2S interval. The genuinely new elements are the Fig. 2 amplitudes with three photons on one lepton (which drop out of the hyperfine structure) and the Fig. 3 single-photon hadronic amplitudes taken from [39]. All are evaluated numerically using VDM form factors. The integral representations are explicit, the trace algebra is documented via FORM and the appendices, and the paper does not oversell: it says plainly that scalar meson couplings are estimates and that the effect is far below the planned accuracy. The citation pattern is honest—the method is carried over from their own work and from standard references—and there is no fitting to the target result.

Soft spots, in decreasing order of seriousness. First, there are no error bars on the final numbers. The largest single item, the sigma-meson contribution of −9.5 Hz, is built from an A_S coupling labeled an estimate. The central conclusion is robust—even a factor-of-three change leaves the total well below 50 Hz—but a precision-theory paper that claims to reduce theoretical uncertainty should quote an uncertainty. Second, no code or data are provided. The traces are long, and the claimed 1% integration accuracy is not independently checkable. Third, Table I's “Total 0.086 Hz” is easy to mistake for the full hadronic correction; the −1.79 Hz and −9.53 Hz from Fig. 3 are given later and never combined in a table. Fourth, the text has typos (the scalar amplitudes in Eqs. 34–36 use a pseudoscalar form-factor label; Eq. 29 has a repeated gamma index). Finally, Eqs. 45–46 carry a 1/n^3 dependence; the paper should show explicitly how the 1S−2S subtraction produces the quoted values, since the 7/8 factor is easy to miss.

This paper is for the muonium precision spectroscopy and few-body QED community. It is not a landmark, but it is a legitimate, useful check that clears a small item in the theory budget. The right outcome is peer review, not desk rejection: a referee can ask for the missing uncertainties and clarifications without overturning the qualitative result.

Referee Report

4 major / 4 minor

Summary. The paper studies hadronic light-by-light scattering and related hadronic single-photon amplitudes in muonium, aiming at the 1S-2S interval that the Mu-MASS experiment plans to measure at 10 kHz accuracy. Using vector-dominance form factors for pseudoscalar, axial-vector, and scalar mesons, the authors derive integral expressions for horizontal-exchange, vertical-exchange, and three-photon-interaction amplitudes, and quote numerical contributions in Table I. A final paragraph adds hadronic single-photon polarization-operator contributions for the pion and sigma meson, giving -1.79 Hz and -9.53 Hz, respectively. The paper concludes that the total hadronic correction is far below the planned experimental accuracy.

Significance. The central qualitative conclusion is significant and likely robust: hadronic corrections to the muonium 1S-2S interval appear to be at the few-hertz level, three orders of magnitude below the 10 kHz Mu-MASS target. Even with the admitted uncertainties in scalar-meson couplings and form-factor cutoffs, the conclusion that these effects can be neglected for the planned electron-muon mass-ratio measurement is credible. The paper makes a useful extension of the authors' previous muonium hyperfine work and gives explicit integral representations that could be checked independently. The main weaknesses are numerical and presentational: the table's 'total' excludes the largest contributions, the sign and n-dependence of the final single-photon formulas are not clearly reconciled with the 1S-2S interval convention, and no uncertainty estimates accompany the quoted numbers.

major comments (4)

[Table I and Section III] Table I is titled 'Hadronic light-by-light contribution' and ends with 'Total 0.086 Hz', but the single-photon hadronic amplitudes introduced in the Conclusion (Eqs. (45) and (46)) contribute -1.79 Hz and -9.53 Hz and are not included in this total. The paper therefore never states its own final hadronic correction to the 1S-2S interval; with the quoted numbers it is about -11.3 Hz, not 0.086 Hz. Please specify the scope of Table I and give the aggregate total, or the numerical summary is internally inconsistent.
[Section III, Eqs. (45)-(46)] The sign and the n-dependence of the quoted interval values need clarification. If Delta E(nS) = -C/n^3 as written, with C > 0, then the correction to the transition frequency E(2S)-E(1S) is +7C/8, whereas the text quotes negative values (-1.79 Hz and -9.53 Hz) for the 1S-2S interval. Either the sign convention for the interval differs from the positive experimental frequency in Eq. (1), or the quoted values are level shifts for n=1 rather than the interval. Please state the convention and apply the 1 - 1/8 factor consistently.
[Section III, Eqs. (43)-(46) and Table I] The two largest hadronic contributions (the pion and sigma single-photon terms) are introduced in the final paragraph without a derivation of the energy shift from the polarization operator, and no uncertainty estimates are given. Since the sigma value depends on the estimated coupling A_S (Section II.1) and on the cutoff Lambda_S, please provide at least a derivation sketch and an estimate of the numerical accuracy, including the sensitivity of the -9.53 Hz value to the uncertainty in A_S.
[Eqs. (29) and (36)] In the displayed three-photon amplitudes, the trace contains two gamma_sigma matrices and no gamma_lambda matrix: Eq. (29) reads ... gamma_sigma (k2-p2+m2) ... gamma_sigma (-q2-k1+m2) ... gamma_nu, and Eq. (36) has the same structure. The expected sequence following the vertex tensors is gamma_mu, gamma_sigma, gamma_lambda, gamma_nu. Please correct the repeated index, since the printed formulas are otherwise not a valid representation of the amplitudes.

minor comments (4)

[Table I] Table I entries are quoted to six decimal places without uncertainties; the text only states approximately one-percent accuracy for the integrals in Eq. (3), not for the other contributions. Please add error bars or at least specify the numerical precision of each entry.
[References] References [10] and [31] lack complete publication information (volume, year, and pages); please update them.
[Integration measure notation] The integration measure notation in Eqs. (3), (16), and (19) (e.g., 'integral dk1 dOmega_1 / pi^2') is ambiguous; please define the Euclidean angular integration measures explicitly in one place.
[Section III] The sentence 'The series of values in Table I is 0' should be reworded; it presumably means that zero entries denote contributions of higher order in alpha.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the hadronic 1S-2S shifts are computed from external form-factor inputs and stated parameterizations, with no fitting to the target interval; self-citations supply method only.

full rationale

The paper's central numerical result, the total hadronic light-by-light contribution to the muonium 1S-2S interval, is obtained by writing out meson-exchange and three-photon lepton-interaction amplitudes in integral form (Eqs. 3, 16, 19, 27-29, 34-40), inserting transition form factors, and then performing numerical integration. The form-factor parameters are taken from experiment or from stated models: the VDM monopole expression in Eq. 10 uses the rho-meson mass, the axial-vector form factor in Eq. 13 is quoted from prior work [22,29], and the scalar couplings AS are estimated from two-photon decay widths through Eq. 21. None of these inputs is adjusted to reproduce the 1S-2S transition frequency or any hadronic shift of that interval. The paper does not fit a parameter to the target result, and the final values are predictions rather than rearrangements of inputs. The self-citations to [24] and [38] supply the calculational approach and a monopole parameterization, but the derivation in this paper is written out explicitly and the cited results are not invoked as a uniqueness theorem or as the justification for the final numbers. The authors themselves flag the uncertainty in the scalar couplings: 'Although there are no precise experimental data on the two-photon decay widths of scalar mesons yet, it is nevertheless possible to make some estimates' (Section II.1, after Eq. 21). This is a numerical-accuracy limitation, not circularity. A presentation caveat is that the Table I total of 0.086 Hz does not include the additional -11.3 Hz from Eqs. 45-46, so the final hadronic shift is about -11.2 Hz if those are added; this affects the reporting of the total, not the logical independence of the calculation. Overall, the central claim that hadronic contributions are far below the 10 kHz Mu-MASS target does not reduce to any fitted or self-referential quantity.

Assumptions & free parameters 6 free parameters · 4 assumptions · 0 invented entities

The calculation introduces no new particles or forces. It relies on known meson couplings and form factors as external inputs, plus modeling assumptions for their momentum dependence. The main free parameters are the form-factor cutoffs and meson-photon couplings, which are taken from experiment or from prior estimates, not fitted to the 1S-2S transition being studied.

free parameters (6)

Lambda_V (VDM cutoff for pseudoscalar form factor) = M_rho = 0.7693 GeV
Eq. (10): the transition form factor F_PSgamma-gamma uses Lambda_V = M_rho. This is a model choice, not a fit to the muonium transition.
Lambda_A for axial mesons = 1040 MeV (f1, a1); 926 MeV (f1(1420))
Table I and Eq. (13): cutoffs in the axial-vector form factor are taken from prior fits.
A(M_A^2,0,0) axial-vector couplings = 0.266, 0.160, 0.193 GeV^-2
Table I: extracted from experimental two-photon transition form factors for f1(1285), a1(1260), f1(1420).
Lambda_S (scalar cutoff) = 2000 MeV
Table I and Eq. (20): the same monopole cutoff is assumed for all scalar mesons, chosen by hand.
A_S scalar couplings = -0.596, -0.085, -0.086, -0.036 GeV^-1
Table I: estimated from two-photon decay widths via Eq. (21); the paper notes there are no precise experimental data.
F_eta and F_eta' decay constants = derived from Eq. (11) using measured two-photon widths
These enter the pseudoscalar meson contributions; they are external experimental inputs.

assumptions (4)

domain assumption Hadronic light-by-light scattering is dominated by exchanges of pseudoscalar, scalar, and axial-vector mesons with transition form factors
Section II: the amplitudes are built from two-photon-to-meson vertices; non-resonant quark/gluon contributions are not considered.
ad hoc to paper Vector dominance parameterization of transition form factors (Eqs. 10, 13, 20) with cutoffs from Table I
This is a modeling assumption chosen for simplicity; the paper states other parameterizations change results by about one percent but does not quantify this for the final total.
domain assumption Leading-order truncation in alpha and neglect of particle momenta in the numerators
Section II.1, statement that only leading contributions in alpha are kept, neglecting transferred momenta.
standard math The single-photon polarization operator results from Ref. [39] (Eqs. 43-46) are correct and applicable to muonium
The concluding Fig. 3 amplitudes use J(0) from Czarnecki and Karshenboim without rederivation.

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Cite this review

Pith. "Pith review of Hadronic light-by-light scattering contribution to 1S-2S transition in muonium." pith.science (2026). https://pith.science/paper/E6PIX6SO

@misc{pith2026241109727,
  author       = {Pith},
  title        = {Pith review of: Hadronic light-by-light scattering contribution to 1S-2S transition in muonium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/E6PIX6SO}},
  note         = {Machine review of arXiv:2411.09727}
}

read the original abstract

We study hadronic light-by-light scattering contribution to the energy interval (1S-2S) in muonium. Various amplitudes of interaction of a muon and an electron are constructed, in which the effect of hadronic scattering of light-by-light is determined using the transition form factor of two photons into a meson. Their contributions to the particle interaction operator in the case of S-states are obtained in integral form, and to the energy spectrum in numerical form. The contributions of pseudoscalar, scalar, axial vector mesons are taken into account.

Figures

Figures reproduced from arXiv: 2411.09727 by the authors.

**Figure 2.** Amplitudes of hadronic scattering of light-by-light with thr [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗

**Figure 3.** Hadronic light-by-light scattering amplitudes in one photon [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

56 extracted references · 54 canonical work pages

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(A1) +k3 1k3 2( 2 3 sin2ψ 1 + 2 3 cos2ψ 2 sin2ψ 1 + 2 cosψ 2 1 + 4 3 cosψ 2 1 sin2ψ 2 + 2 cosψ 2 1 cos2ψ 2− 2 3 cos Ω cosψ 1 cosψ 2 − 2 3 cos Ω cosψ 1 cosψ 2 sin2ψ 1 − 2 3 cos Ω cos 3ψ 1 cosψ 2+ 2 3 cos2 Ω − 4 3 cos2 Ω cos ψ 2

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2024 doi arXiv

Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

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R. Iwai, M. Abe, S. Fukumura, and al. (MuSEUM Collaborati on), J. Phys. Conf. Ser. 2462, 012019 (2023)

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[3] [1]

(A1) +k3 1k3 2( 2 3 sin2ψ 1 + 2 3 cos2ψ 2 sin2ψ 1 + 2 cosψ 2 1 + 4 3 cosψ 2 1 sin2ψ 2 + 2 cosψ 2 1 cos2ψ 2− 2 3 cos Ω cosψ 1 cosψ 2 − 2 3 cos Ω cosψ 1 cosψ 2 sin2ψ 1 − 2 3 cos Ω cos 3ψ 1 cosψ 2+ 2 3 cos2 Ω − 4 3 cos2 Ω cos ψ 2

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[4] [2]

7693 GeV is the mass of ρ meson for the case of π 0 meson

= Λ 4 V (k2 1 − Λ 2 V )(k2 2 − Λ 2 V ), (10) where Λ V = MV = Mρ = 0. 7693 GeV is the mass of ρ meson for the case of π 0 meson. Transition form factor data from the BaBar, Belle, CLEO experimen ts are obtained mainly for the singly virtual case. Other pseudoscalar mesons η,η′ make contributions determined by the same formula ( 3). In this case, the decay...

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[5] [3]

The form of the transition form factor A(t2,k 2 1,k 2

is a scalar function of the four-momentum transfer of the virtua l photons k1,k2. The form of the transition form factor A(t2,k 2 1,k 2

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[6] [4]

In this calculation we use the same 6 Table I: Hadronic light-by-light contribution to muonium energy inter val (1S-2S)

of 1 ++ meson to two photons was studied experimentally [ 30–32] in diﬀerent reactions. In this calculation we use the same 6 Table I: Hadronic light-by-light contribution to muonium energy inter val (1S-2S). The last three columns in the Table I correspond to a horizontal exchange (H), to vertical exchange (V) and to a lepton three-photon interaction. Me...

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[7] [5]

= A(M 2 A, 0, 0)F 2 AV (k2 1,k 2 2), F AV (k2 1,k 2

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[8] [6]

A(t2,k 2 1,k 2

= Λ 4 A (Λ 2 A +k2 1)(Λ 2 A +k2 2), (13) where t = k1 +k2 is the four-momentum of the meson. A(t2,k 2 1,k 2

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