{"id":"4d0245de-00f2-4eeb-8496-8b416b288e83","arxiv_id":"2411.09727","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The hadronic light-by-light and related meson-exchange corrections to the muonium 1S-2S interval are computed to be about 0.1 to 11 Hz, far below the 10 kHz experimental target.","lead":"This paper calculates the hadronic light-by-light scattering correction to the 1S-2S transition in muonium, an atom made of a muon and an electron. It finds the correction is about 0.1 Hz for the main diagrams and around 11 Hz when extra hadronic diagrams are included, both far below the planned 10 kHz experimental precision.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified: the computed hadronic effects (−11.3 Hz total from Eqs. 45–46 plus Table I) are three orders of magnitude below the 10 kHz target, so even the flagged parameter uncertainties cannot flip the central conclusion.","rationale":"The reader's conditional verdict is driven by missing uncertainty estimates and estimated scalar couplings. Those are legitimate quality concerns, but they are not load-bearing for the paper's central scientific claim. The target accuracy is 10 kHz, while the complete hadronic estimate is about −11 Hz from the single-photon amplitudes plus +0.086 Hz from Figs. 1–2. Order-of-magnitude uncertainties in the estimated couplings or form-factor parameters cannot bridge a gap of roughly three orders of magnitude. The largest plausible parametric variation, even a factor of ten on the σ contribution, yields about −100 Hz, still two orders of magnitude below the target. The paper would be improved by adding error bars, by presenting the derivation of Eqs. (45)–(46), and by clarifying that the full hadronic estimate is about −11 Hz rather than the 0.086 Hz total shown in Table I, but these are presentation and robustness issues rather than a defect in the central argument. I therefore see no change to the reader's conditional verdict, and no basis for rejection or for a stronger accept recommendation without the requested numerical documentation.","tokens_in":20543,"tokens_out":17833,"duration_ms":183028,"concrete_test":"Independently re-derive Eqs. (45)–(46) from the polarization operator (43)–(44), applying the 1/n^3 scaling explicitly to the 1S−2S interval, and recompute the quoted −1.79 Hz and −9.53 Hz values. Separately, reproduce Table I with an independent implementation of the traces and multidimensional integrations in Eqs. (3), (16), and (19), using a doubled cutoff for convergence checking. If the σ channel shifts by more than a factor of two, the numerical headline requires revision, but the negligible-vs-10 kHz conclusion would still stand.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim—that hadronic light-by-light and the related single-photon hadronic amplitudes shift the muonium 1S−2S interval by at most tens of Hz, far below the 10 kHz Mu-MASS goal—does not rest on a fragile assumption. The largest computed item is the σ-meson single-photon contribution of −9.53 Hz (Eq. 46), which uses an estimated A_S coupling (Section II.1, Eq. 21). Even a factor-of-three change in that coupling, or a different off-shell suppression factor for axial mesons (Eq. 14), leaves the total below roughly 50 Hz. The reader's weakest-assumption concern about scalar couplings and VDM monopole parameters is therefore a numerical-accuracy issue, not a load-bearing one. Two presentation caveats remain: Table I's 'Total 0.086 Hz' excludes the −11.3 Hz from Fig. 3, and Eqs. (45)–(46) are presented without derivation or uncertainty estimates. Also, Eqs. (45)–(46) contain 1/n^3 scaling; the quoted interval values should be checked for the 7/8 factor from subtracting 2S from 1S. None of these issues threatens the qualitative conclusion that hadronic contributions are negligible at the planned 10 kHz accuracy.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":20834,"tokens_out":15375,"duration_ms":133439,"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":[{"comment":"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":"Table I and Section III"},{"comment":"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":"Section III, Eqs. (45)-(46)"},{"comment":"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.","section":"Section III, Eqs. (43)-(46) and Table I"},{"comment":"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.","section":"Eqs. (29) and (36)"}],"minor_comments":[{"comment":"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.","section":"Table I"},{"comment":"References [10] and [31] lack complete publication information (volume, year, and pages); please update them.","section":"References"},{"comment":"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":"Integration measure notation"},{"comment":"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.","section":"Section III"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and the qualitative conclusion is very likely correct. The main issues are internal numerical consistency and missing derivations/uncertainties for the largest terms; these are fixable but should be addressed before publication. I do not see a need to request the full code, but providing the numerics in a reproducible form would substantially strengthen the manuscript."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","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.\n\nWhat 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.\n\nSoft 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.\n\nThis 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.","headline":"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.","tokens_in":21397,"tokens_out":5456,"would_cite":false,"duration_ms":51855,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Hadronic light-by-light shifts muonium 1S-2S by roughly -11 Hz in total.","keywords":["hadronic light-by-light scattering","muonium","1S-2S transition","transition form factor","vector meson dominance","scalar mesons","pseudoscalar mesons","axial-vector mesons"],"falsifier":"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.","tokens_in":20312,"feed_emoji":"⚛️","tokens_out":5192,"duration_ms":43175,"temperature":0.7,"pith_summary":"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.","feed_headline":"Hadronic light-by-light shifts muonium 1S-2S by roughly -11 Hz","feed_subtitle":"The hadronic correction is far below the 10 kHz precision goal and safe to ignore for the electron-muon mass ratio.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the currently measured 1S-2S frequency that the planned experiments aim to improve by a thousandfold.","marker":"[4]"},{"why":"Defines the Mu-MASS collaboration's 10 kHz target that sets the precision benchmark for the paper's conclusion.","marker":"[6]"},{"why":"Provides the CLEO singly-virtual transition form factor data underlying the vector-dominance parameterization for pseudoscalar mesons.","marker":"[13]"},{"why":"Earlier calculation of hadronic light-by-light in muonium hyperfine structure whose approach is extended here to the 1S-2S interval.","marker":"[16]"},{"why":"Source of the axial-vector transition form factor parameterization and the off-shell suppression factor used in Eq. (14).","marker":"[22]"},{"why":"Previous work by the same group on the hadronic contribution to muonium hyperfine splitting, providing the calculation framework and integral reductions.","marker":"[24]"},{"why":"Used for the scalar meson contribution to the muon anomalous magnetic moment and for estimating scalar-meson couplings.","marker":"[25]"},{"why":"Provides the pion polarization operator $J(k^2)$ that underlies the single-photon contributions in Eqs. (44)-(45).","marker":"[39]"},{"why":"Gives the current electron-muon mass ratio that a 10 kHz measurement of the 1S-2S transition would improve.","marker":"[41]"}],"fun_headline_variants":["Hadronic light-by-light shifts muonium 1S-2S by -11 Hz","Muonium 1S-2S hadronic correction: only -11 Hz","Muonium 1S-2S hadronic shift: -11 Hz, negligible","Hadronic effect on muonium 1S-2S is just -11 Hz"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Hadronic light-by-light shifts muonium 1S-2S by -11 Hz","Muonium 1S-2S hadronic correction: only -11 Hz","Muonium 1S-2S hadronic shift: -11 Hz, negligible","Hadronic effect on muonium 1S-2S is just -11 Hz"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000672,"raw_usage":{"total_tokens":2999,"prompt_tokens":825,"completion_tokens":2174,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":441,"completion_tokens_details":{"reasoning_tokens":2083}},"tokens_in":441,"tokens_out":2174,"duration_ms":15367,"temperature":1.0,"reasoning_tokens":2083,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T20:29:39.818219+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"In this calculation we use the same 6 Table I: Hadronic light-by-light contribution to muonium energy inter val (1S-2S)","cited_arxiv_id":null,"evidence_quote":"Supplies the currently measured 1S-2S frequency that the planned experiments aim to improve by a thousandfold."},{"cited_title":"A(t2,k 2 1,k 2","cited_arxiv_id":null,"evidence_quote":"Defines the Mu-MASS collaboration's 10 kHz target that sets the precision benchmark for the paper's conclusion."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the CLEO singly-virtual transition form factor data underlying the vector-dominance parameterization for pseudoscalar mesons."},{"cited_title":"Ohayon, G","cited_arxiv_id":null,"evidence_quote":"Source of the axial-vector transition form factor parameterization and the off-shell suppression factor used in Eq. (14)."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Previous work by the same group on the hadronic contribution to muonium hyperfine splitting, providing the calculation framework and integral reductions."},{"cited_title":"Babusci, D","cited_arxiv_id":null,"evidence_quote":"Used for the scalar meson contribution to the muon anomalous magnetic moment and for estimating scalar-meson couplings."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the pion polarization operator $J(k^2)$ that underlies the single-photon contributions in Eqs. (44)-(45)."},{"cited_title":"Pauk and M","cited_arxiv_id":null,"evidence_quote":"Gives the current electron-muon mass ratio that a 10 kHz measurement of the 1S-2S transition would improve."}],"review_version":1}