{"id":"d362782f-344c-4219-8f87-902300ca44a7","arxiv_id":"2507.22097","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"The authors compute one-loop, quadratic, and reducible two-loop electroweak corrections to the parity-violating asymmetry in elastic lepton-proton scattering and find percent-level NNLO effects at the kinematics of several planned experiments.","lead":"This paper calculates electroweak radiative corrections, including quadratic and reducible two-loop terms, to the parity-violating asymmetry in electron and muon scattering off protons. It provides numerical predictions for several planned precision experiments such as Qweak, P2, MOLLER, MUSE, and EIC.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Missing γ-Z box and hard-bremsstrahlung contributions are an unquantified same-order effect, so the claim that the computed NNLO corrections must be included is not yet established.","rationale":"The reader's weakest assumption correctly identifies the omission of box diagrams and hard-photon bremsstrahlung as the load-bearing issue. The central scientific claim is that the newly computed quadratic and reducible two-loop corrections are significant enough to require inclusion in future precision programs. This is only true if the remaining NNLO-order contributions are small by comparison. The paper explicitly discloses the omission of boxes and hard real emission, but it never gives an estimate of their size or an argument for their smallness. Because the computed NNLO increment over NLO is itself only a few ppb, the possibility that the missing terms are comparable is concrete, not speculative. A second, related weakness is the arbitrary soft-photon cut ΔE = 0.05√s in Sec. VI; without hard bremsstrahlung the physical result should depend on this cut, and no cut-independence test is shown. I do not see an internal algebraic inconsistency that would make the computed quadratic and reducible two-loop terms wrong as far as they go; the issue is completeness of the NNLO set. The reader's CONDITIONAL verdict already reflects this, so no verdict change is needed. My concrete test would quantify the missing contributions at the most relevant kinematics and would settle whether the 'significant corrections' claim survives.","tokens_in":22826,"tokens_out":4948,"duration_ms":70368,"concrete_test":"Recompute the total A_PV at P2 kinematics (E_beam = 155 MeV, θ_lab = 35°) with the γ-Z box amplitude evaluated in the dispersive formalism of Gorchtein et al. and with hard-photon bremsstrahlung included above the ΔE = 0.05√s cut, and then check whether the result is unchanged when ΔE is varied from 0.01√s to 0.1√s. If the shift exceeds ~0.6 ppb or the total depends on ΔE, the omitted contributions are not negligible and the quoted NNLO value of -65.09 ppb is not a reliable full-NNLO prediction.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central claim is that the quadratic plus reducible two-loop electroweak corrections to A_PV are 'quite significant' and must be included in precision BSM searches. For that claim to hold, all other contributions at comparable order must be subdominant. The manuscript itself states this is not known: Sec. VI says 'we need to account boxes and hard photon bremsstrahlung cross-section,' and Sec. III/Conclusions defer the gauge-invariant box set to a separate paper. No numerical magnitude for these missing terms is given anywhere. The omitted γ-Z box diagrams are known from the dispersive literature (refs. [27,28] and related work) to shift PV electron-proton scattering at low Q^2, and hard bremsstrahlung is required to remove the dependence of the soft-photon result on the arbitrary cut ΔE = 0.05√s that is fixed in Sec. VI without a stability test. The reported NNLO effect beyond NLO is only a few ppb at the key kinematics (Qweak: -217.11 → -221.19 ppb; P2: -65.14 → -65.09 ppb), so even an omitted contribution at the level of P2's 0.56 ppb projected precision could change the conclusion. Thus the headline assertion rests on an unquantified assumption about the size of the missing diagrams.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper calculates parity-violating asymmetries in elastic lepton-proton scattering (ep and mu p) using a covariant leptonic-tensor approach, including tree-level, one-loop (NLO), quadratic, and reducible two-loop (NNLO) electroweak corrections. Numerical results are presented for the kinematics of Qweak, P2, MOLLER, EIC, and MUSE experiments. The authors report that their total NNLO asymmetry agrees with the Qweak measurement and the P2 proposal, and they conclude that the NNLO corrections are significant enough to be included in future BSM searches. The manuscript explicitly states that electroweak box diagrams and hard-photon bremsstrahlung are not included, with box diagrams deferred to a separate paper.","tokens_in":22942,"tokens_out":6403,"duration_ms":78184,"significance":"If the calculation were complete and validated, it would provide a useful extension of one-loop electroweak corrections to parity-violating lepton-proton scattering, with direct relevance to the Qweak, P2, MOLLER, MUSE, and EIC programs. The work uses standard automated tools (FeynArts, FormCalc, FeynCalc, LoopTools), keeps the lepton mass, and does not fit any parameter to the asymmetry data used for comparison; the Qweak and P2 comparisons are after-the-fact consistency checks. The main significance hinges on the size of the new NNLO terms relative to experimental precision, which is exactly where the paper's evidence is incomplete.","major_comments":[{"comment":"The central quantitative claim is not yet supported because the calculation omits the gamma-Z box diagrams and hard-photon bremsstrahlung. The text states 'we need to account boxes and hard photon bremsstrahlung cross-section' and defers the gauge-invariant box set to a separate paper, but no numerical estimate of the size of these omitted contributions is given. The gamma-Z box is a one-loop correction to A_PV and is therefore needed already at NLO, not only at NNLO; hard bremsstrahlung is needed to remove the dependence of the soft-photon results on the arbitrary cut Delta-E = 0.05 sqrt(s). Since the computed NNLO shift beyond NLO is only 0.05 ppb at P2 kinematics (Table III), even a small omitted contribution can change the conclusion.","section":"Sec. VI and Conclusions"},{"comment":"There is no benchmark of the NLO results against the existing complete one-loop calculations cited in refs. [10]-[15], and the 19 NLO and 21 quadratic structure functions r_i and n_i are not given in the paper or in an ancillary file. The plots in Figs. 5 and 8 show only a subset of these functions. Without a benchmark or explicit expressions, the numerical results cannot be independently checked; this matters because the final ppb-level NNLO statements rely on cancellations among large corrections of about 30%.","section":"Sec. III, Eq. (15)-(17); Sec. VI"},{"comment":"The quantity called the 'NNLO correction percentage' delta_2_APV is defined relative to the tree-level asymmetry and therefore includes the full NLO correction; it does not measure the size of the new NNLO terms. The incremental NNLO effect is +0.05 ppb at P2 kinematics (one-loop -65.14 ppb, total -65.09 ppb in Table III), far below the projected 0.56 ppb precision, while at Qweak it is -4.08 ppb (one-loop -217.11 ppb, total -221.19 ppb in Table I). The statement that NNLO corrections 'have to be included' is therefore overstated for P2 and should be rephrased in terms of the incremental NNLO shift with an uncertainty estimate.","section":"Tables III-IV and Eq. (29)"},{"comment":"The infrared cancellation at NNLO is asserted analytically, but the only numerical demonstration of cancellation of the photon-mass regulator lambda is shown for the NLO MUSE case in Fig. 15. No numerical scan over lambda or over the soft-photon cutoff Delta-E = 0.05 sqrt(s) is presented for the quadratic and reducible two-loop results. Since Eq. (27) involves both delta_SP and delta_SP^2, a stability test over Delta-E is needed to show that the reported NNLO asymmetries are IR finite and cut independent.","section":"Sec. V, Eqs. (26)-(27), Fig. 15"}],"minor_comments":[{"comment":"The quoted Qweak NNLO asymmetry, -221.46 ppb, does not match Table I, which gives -221.19 ppb at theta_lab = 7.9 degrees; please correct and unify the value.","section":"Sec. VII"},{"comment":"The P2 target asymmetry is given as -39.94 ppb in the introduction and -67.34 ppb in Sec. VI B; clarify which angle and Q^2 each number corresponds to.","section":"Sec. I and Sec. VI B"},{"comment":"The column label 'Qud-APV' should be spelled out as 'Quadratic-APV' for clarity, and the tables should state explicitly that all entries are in ppb unless otherwise noted.","section":"Tables I-XIV"},{"comment":"The form-factor notation C_i is introduced but the index i is not defined; additionally, 'Sach' should be 'Sachs' in Sec. II.","section":"Appendix B"},{"comment":"The relation between h_i and H_i appears to have a dimension problem because of the factor 1/q^2, and the connection to the truncated self-energy is not derived; please clarify this equation.","section":"Eq. (20)"},{"comment":"The solid line is claimed to be the IR-finite sum, but no values of the photon-mass parameter lambda are shown; please state the range of lambda over which the cancellation was tested.","section":"Fig. 15"}],"recommendation":"major_revision","confidential_remarks":"I do not see concerns about the authors' citation pattern or about fitting to data; the issue is completeness of the calculation relative to the claims. I would encourage the editor to require a quantitative estimate of the omitted box and hard-bremsstrahlung contributions, or a softening of the abstract's conclusion, before publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a real calculation of something not done before — quadratic and reducible two-loop electroweak corrections to the PV asymmetry in elastic ep and µp scattering — but the paper overclaims what the numbers support. The NNLO terms are only a few ppb at the key kinematics, and the omitted γ-Z box and hard bremsstrahlung are unquantified contributions of potentially the same size.\n\nWhat is good: the covariant tensor-splitting technique is extended from their earlier Møller work to lepton-proton scattering with the lepton mass kept. They provide tables for Qweak, P2, MOLLER, MUSE, and EIC kinematics, and they are upfront that boxes and hard bremsstrahlung are left for later. The agreement with the Qweak central value is within the experimental error, which is a nontrivial consistency check.\n\nSoft spots, in order of importance. First, there is no benchmark of the NLO result against the existing one-loop calculations (refs [10]–[15]). The one-loop correction is ~30% of the asymmetry, so if the one-loop piece had an error of a few percent, the NNLO discussion is moot. Second, the analytic expressions for the NLO/NNLO structure functions are not given and no code or data is released; the reader cannot check the calculation. Third, the central claim is overstated relative to its own numbers. At P2 kinematics, the NLO-to-NNLO shift is 0.05 ppb against a projected 0.56 ppb precision; at Qweak it is about 4 ppb against a roughly 9 ppb total error. The 'quite significant' and 'have to be included' language is therefore too broad: whether these terms matter depends on the experiment, and for P2 they are irrelevant at the quoted precision. The concern is compounded by the omitted γ-Z box, which the dispersive literature puts at the few-ppb level at low Q^2, i.e., potentially the same size as the computed NNLO shift. Finally, the soft-photon cut ΔE=0.05√s is fixed and no stability test is shown; without hard bremsstrahlung, cut dependence is a real concern.\n\nThe infrastructure here is plausible and this is the kind of calculation a specialist in PV radiative corrections would want to look at. But in current form it is a progress report, not a finished NNLO prediction. A serious referee should ask for a benchmark against an existing one-loop code, an estimate or inclusion of the box and hard-bremsstrahlung contributions, and either public code or writeable expressions.\n\nMy recommendation: send it out, but expect major revision.","headline":"A genuine first calculation of quadratic/reducible two-loop corrections to PV ep/µp scattering, but with unquantified same-order omissions and an overbroad significance claim.","tokens_in":23632,"tokens_out":5214,"would_cite":false,"duration_ms":56720,"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":"The paper claims that quadratic and reducible two-loop electroweak corrections shift the parity-violating asymmetry in elastic lepton-proton scattering by up to several parts per billion, bringing theory into agreement with Qweak and P2…","keywords":["electroweak radiative corrections","parity-violating asymmetry","lepton-proton scattering","NNLO two-loop","covariant approach","soft-photon bremsstrahlung","Qweak","P2 experiment"],"falsifier":"Complete the electroweak box-diagram contribution using the methods of the cited references and compute the hard-photon bremsstrahlung cross section, then check whether the total NNLO $A_{PV}$ changes by more than the size of the quadratic and two-loop corrections; also verify numerically that the result is unchanged when the soft-photon cutoff $\\Delta E = 0.05\\sqrt{s}$ is varied.","tokens_in":22464,"feed_emoji":"⚛️","tokens_out":5024,"duration_ms":52445,"temperature":0.7,"pith_summary":"This paper computes higher-order electroweak radiative corrections to the parity-violating asymmetry in elastic lepton-proton scattering, going beyond the one-loop level to include quadratic and reducible two-loop contributions (NNLO) in a covariant approach. The authors find these NNLO corrections are sizable at the low-energy kinematics of existing and proposed experiments, shifting the asymmetry by up to several parts per billion. They show that including these terms brings their computed asymmetry into good agreement with the measured Qweak value and the proposed P2 projection. The paper argues that these corrections therefore have to be included in precision searches for physics beyond the Standard Model.","feed_headline":"Two-loop electroweak shift hits 4 ppb at Qweak, must be included","feed_subtitle":"Quantum corrections bring computed parity-violating asymmetries in line with Qweak and P2 targets.","key_machinery":"The central mechanism is the covariant approach of Bardin and Shumeiko, which gives a factorized amplitude-squared built from a contracted leptonic tensor $L_{\\mu\\nu}$ and hadronic tensor $W_{\\mu\\nu}$, allowing infrared divergences from vertex corrections to be cancelled analytically by adding soft-photon bremsstrahlung contributions. The NNLO leptonic tensor is constructed from quadratic one-loop graphs and reducible two-loop graphs, with all Passarino-Veltman integrals evaluated numerically using LoopTools; the same tensor machinery also provides one-loop hadronic self-energy corrections attached to the hadronic side.","core_discovery":"The central claim is that the parity-violating asymmetry $A_{PV}$ in elastic lepton-proton scattering receives significant NNLO electroweak corrections from quadratic one-loop terms and reducible two-loop diagrams, and that these corrections must be included to match the precision of current and future experiments such as Qweak, P2, MOLLER, MUSE, and the EIC. Using a covariant approach that separates the leptonic and hadronic tensors, the authors compute the corrected asymmetry for electron-proton and muon-proton scattering. Their final NNLO results are $-221.19$ ppb at Qweak kinematics compared with the measured $-226.5 \\pm 7.3 \\pm 5.8$ ppb, and $-65.09$ ppb at P2 kinematics compared with the proposed $-67.34$ ppb.","pith_inferences":["If the omitted electroweak box diagrams and hard-photon bremsstrahlung shift $A_{PV}$ by an amount comparable to the computed quadratic and two-loop terms, the quoted agreement with Qweak and P2 could move by a few ppb; completing the box calculation, which the authors identify as their next step, would settle this.","The same leptonic-tensor machinery should extend naturally to a polarized proton target and inelastic kinematics, and the authors indicate those directions; the structure-function decomposition is target-agnostic apart from the hadronic form factors.","The near-equal percentage corrections for electron and muon scattering in MUSE kinematics suggest lepton-universality tests will be sensitive mainly to the remaining box-diagram terms rather than to the computed NNLO contributions.","If future P2 data fall closest to one of the intermediate values in Table III, the soft-photon cutoff choice $\\Delta E = 0.05\\sqrt{s}$ could be pinned down empirically, effectively turning this cutoff from a residual uncertainty into a fixed parameter of the calculation."],"forward_implications":["The total NNLO corrected asymmetry at Qweak kinematics is roughly $-221.19$ ppb, within about 5 ppb of the measured central value, with the quadratic and reducible two-loop terms together shifting the one-loop result by about 4 ppb at that angle.","At P2 kinematics the computed NNLO value of $-65.09$ ppb is close to the proposed target of $-67.34$ ppb, giving a concrete benchmark for the upcoming measurement.","The paper provides tables of tree, NLO, and NNLO $A_{PV}$ values for MOLLER, EIC, and MUSE kinematics, usable as signal predictions or background estimates for those programs.","Because the NNLO contributions are as large as several ppb while future experiments aim at sub-ppb precision, the paper concludes these corrections must be included in searches for physics beyond the Standard Model."],"supporting_citations":[{"why":"Introduces the covariant approach used to extract the infrared-divergent bremsstrahlung contribution and structure the calculation.","marker":"[22]"},{"why":"Provides the Qweak experimental result for the weak charge of the proton used as the main benchmark.","marker":"[3]"},{"why":"Provides the final Qweak measurement of $A_{PV}$ equal to $-226.5 \\pm 7.3 \\pm 5.8$ ppb, the central experimental comparison.","marker":"[5]"},{"why":"Defines the P2 experiment's proposed kinematics and target asymmetry value of $-67.34$ ppb used for comparison.","marker":"[6]"},{"why":"Previous quadratic electroweak corrections for polarized Møller scattering, whose approach the present work extends to lepton-proton scattering.","marker":"[19]"},{"why":"Earlier two-loop parity-violating Møller scattering calculation that provides the basis for the reducible two-loop treatment.","marker":"[20]"},{"why":"Earlier two-loop effects in low-energy electroweak measurements that motivate the need for NNLO corrections.","marker":"[21]"},{"why":"Supplies the analytic soft-photon integral used to make the bremsstrahlung and vertex corrections infrared finite.","marker":"[33]"},{"why":"Provides the standard one-loop interpretation framework for parity-violating electron scattering that this work extends.","marker":"[10]"}],"fun_headline_variants":["NNLO electroweak corrections essential for Qweak parity-asymmetry","Covariant two-loop terms bring lepton-proton asymmetry theory in line","Reducible two-loop electroweak diagrams shift parity-violating asymmetry","Higher-order electroweak corrections must match upcoming precision experiments","Two-loop NNLO effects significant for lepton-proton scattering at Qweak"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The calculation assumes the omitted electroweak box diagrams and hard-photon bremsstrahlung do not shift the asymmetry by as much as the computed quadratic and two-loop terms, and that the soft-photon cutoff choice does not bias the infrared-finite result.","fun_headline_variants_meta":{"raw":{"variants":["NNLO electroweak corrections essential for Qweak parity-asymmetry","Covariant two-loop terms bring lepton-proton asymmetry theory in line","Reducible two-loop electroweak diagrams shift parity-violating asymmetry","Higher-order electroweak corrections must match upcoming precision experiments","Two-loop NNLO effects significant for lepton-proton scattering at Qweak"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000755,"raw_usage":{"total_tokens":3294,"prompt_tokens":816,"completion_tokens":2478,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":432,"completion_tokens_details":{"reasoning_tokens":2386}},"tokens_in":432,"tokens_out":2478,"duration_ms":19624,"temperature":1.0,"reasoning_tokens":2386,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T12:10:31.817228+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Complete the electroweak box-diagram contribution using the methods of the cited references and compute the hard-photon bremsstrahlung cross section, then check whether the total NNLO $A_{PV}$ changes by more than the size of the quadratic and two-loop corrections; also verify numerically that the result is unchanged when the soft-photon cutoff $\\Delta E = 0.05\\sqrt{s}$ is varied.","supporting_citations":[{"cited_title":"Aleksejevs, S","cited_arxiv_id":null,"evidence_quote":"Introduces the covariant approach used to extract the infrared-divergent bremsstrahlung contribution and structure the calculation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Qweak experimental result for the weak charge of the proton used as the main benchmark."},{"cited_title":"First Determination of the Weak Charge of the Proton","cited_arxiv_id":"1307.5275","evidence_quote":"Provides the final Qweak measurement of $A_{PV}$ equal to $-226.5 \\pm 7.3 \\pm 5.8$ ppb, the central experimental comparison."},{"cited_title":"The Q_weak Experimental Apparatus","cited_arxiv_id":"1409.7100","evidence_quote":"Defines the P2 experiment's proposed kinematics and target asymmetry value of $-67.34$ ppb used for comparison."},{"cited_title":"One-loop electroweak corrections for polarized Moller scattering at different renormalization schemes and conditions","cited_arxiv_id":"1010.4185","evidence_quote":"Previous quadratic electroweak corrections for polarized Møller scattering, whose approach the present work extends to lepton-proton scattering."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Earlier two-loop parity-violating Møller scattering calculation that provides the basis for the reducible two-loop treatment."},{"cited_title":"Aleksejevs, S","cited_arxiv_id":null,"evidence_quote":"Earlier two-loop effects in low-energy electroweak measurements that motivate the need for NNLO corrections."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the analytic soft-photon integral used to make the bremsstrahlung and vertex corrections infrared finite."},{"cited_title":"Accardi et al., Electron-ion collider: The next qcd frontier - understanding the glue that binds us all, Eur","cited_arxiv_id":null,"evidence_quote":"Provides the standard one-loop interpretation framework for parity-violating electron scattering that this work extends."}],"review_version":1}