{"id":"8dc654ca-4f82-4325-94bd-aeb000ead543","arxiv_id":"2607.07664","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":6,"one_line_summary":"Single inclusive high-PT hadron and jet production in lepton-hadron scattering is factorized using a joint QCD+QED approach with universal lepton distribution functions, and predictions are given for JLab and EIC energies.","lead":"This paper presents the first calculation of single inclusive hadron and jet production in lepton-hadron scattering using a joint QCD+QED factorization framework, introducing universal lepton distribution functions (LDFs) to systematically handle collision-induced QED radiation. A smart generalist might read it to understand how future EIC measurements will disentangle QED radiative effects from true hadronic structure.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Factorization proof deferred to Ref. [42]; the brief argument in Sec. 2 does not address whether the Q²→0 pinch singularity and unobserved final-state lepton introduce non-factorizable contributions specific to the QCD+QED case.","rationale":"The reader correctly identified the deferred factorization proof as the primary load-bearing concern, and the model dependence of LDFs as a secondary issue. The CONDITIONAL verdict is appropriate: the framework is well-motivated, internally consistent at the level presented, and transparently acknowledges its limitations. The numerical results are clearly labeled as illustrations of the framework's sensitivity rather than precision predictions.\n\nThe factorization claim is plausible — the argument that the photon commutes with gluons and therefore the QCD proof structure carries over is reasonable at a heuristic level, and Ref. [11] provides partial support by demonstrating IR safety for DIS. However, the two complications I identified (pinch singularity at Q²→0 and unobserved final-state lepton) are genuine potential obstructions that the paper does not address. Until the proof in Ref. [42] appears and explicitly handles these cases, or until the NLO IR safety is verified for this specific process, the factorization formula in Eq. (2) should be treated as a well-motivated conjecture rather than an established result.\n\nThe paper's other limitations (model LDFs, neglected QED corrections to hadron PDFs/FFs, private code) are correctly noted by the reader but do not affect the validity of the factorization framework itself — they affect the precision of the numerical predictions, which the paper appropriately frames as illustrative.\n\nI agree with the reader's assessment and see no reason to adjust the verdict. The CONDITIONAL rating with MODERATE confidence accurately reflects the state of the work.","tokens_in":35802,"tokens_out":4004,"duration_ms":348630,"concrete_test":"Explicitly verify the cancellation of all 1/ε poles in dimensional regularization for the NLO partonic hard parts bH_{ib→c} defined in Eq. (4), after subtracting collinear contributions into LDFs, PDFs, and FFs. This was done for inclusive DIS in Ref. [11]; the same calculation must be repeated for the single-inclusive hadron production channel (e+h→H+X) where the final-state lepton is unobserved. Specifically, compute the NLO subprocess eq→eqg (and γq→qg) including both QCD and QED radiation, and check that (a) all soft singularities cancel between real and virtual diagrams, and (b) all collinear singularities along the three external directions (lepton, hadron, observed hadron) are absorbed into fi/e, fb/h, and Dc→H respectively. If any 1/ε pole remains after this subtraction, the factorization in Eq. (2) fails. This check is purely analytical and could be done with existing tools (e.g","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is the factorization formula in Eq. (2). The paper's justification is a single paragraph in Sec. 2: because the photon commutes with gluons, \"it is straightforward to verify that the same sequence of arguments\" from Refs. [41, 42] for pure QCD factorization \"can be carried through.\" Ref. [41] is published; Ref. [42] is listed as \"in preparation (2026).\" This leaves the proof unverified in the published record.\n\nThe concern is not merely procedural. The QCD+QED case introduces at least two complications not present in the pure QCD hadron-hadron factorization of Ref. [41]:\n\n1. Pinch singularity: The introduction explicitly mentions a \"new pinch-singular region\" where the exchanged photon goes on-shell (Q²→0) [Refs. 11-13]. In the traditional approach this requires experimental cuts. The paper claims the joint factorization handles this naturally, but does not explain the mechanism by which the pinch-singular contribution is absorbed into LDFs. If any part of this contribution is non-factorizable (e.g., involving soft photon exchange between the lepton and hadron sectors that cannot be written as a convolution), Eq. (2) would not hold.\n\n2. Unobserved final-state lepton: Unlike inclusive DIS (treated in Ref. [11]), here the scattered lepton is not measured. This means soft/collinear radiation from the final-state lepton is unconstrained. In pure QCD single-inclusive hadron production in hadron-hadron collisions, the factorization proof (Ref. [41]) handles unobserved partons in the final state via KLN cancellation. The QED analog requires that soft photon radiation from the unobserved lepton cancels against virtual corrections, but the paper does not demonstrate this cancellation explicitly.\n\nRef. [11] verified IR safety of NLO hard parts for inclusive DIS, which is related but not identical to the process studied here. The extension to single-inclusive hadron production (where the hard scale is PT, not Q², and the final-state lepton is unobserved) is n","agreement_with_reader":"agree"},"referee_report":{"model":"glm-5.2","summary":"This paper presents the first calculation of single inclusive hadron and jet production at high transverse momentum in lepton-hadron scattering within a joint QCD+QED collinear factorization framework. The cross section is factorized into infrared-safe hard parts convolved with universal lepton distribution functions (LDFs), hadron PDFs, and fragmentation functions (FFs). The LDFs obey DGLAP-type evolution equations with mixed QCD and QED splitting kernels. The authors construct a default set of model LDFs at an input scale mu_0 = m_c, evolve them to higher scales, and present numerical predictions for single inclusive hadron and jet production at JLab and future EIC energies. The paper also studies nuclear dependence in lepton-nucleus collisions and the impact of QED radiative corrections.","tokens_in":36761,"tokens_out":1529,"duration_ms":266356,"significance":"The paper introduces a novel and timely framework for treating collision-induced QED radiation in lepton-hadron scattering on the same footing as QCD radiation, which is well-motivated by the precision programs at JLab and the future EIC. The unification of leptoproduction and photoproduction without requiring kinematic cuts or additional parameters is a conceptually attractive feature. The derivation of default LDFs from a model ansatz fitted to perturbative QED moments, with quark/gluon LDFs set to zero at input, provides a concrete and reproducible starting point for future global fits. The numerical code is stated to be available upon request, and the LHAPDF6-compatible grid format for LDFs facilitates community use. The nuclear modification studies and the demonstration of FF sensitivity in the large-z region add phenomenological value. The framework yields falsifiable predictions that can be tested against future EIC data.","major_comments":[{"comment":"Sec. 2: The central claim is the factorization formula in Eq. (2). The justification given is a single paragraph stating that because the photon commutes with gluons, the same arguments as Refs. [41, 42] carry through. Ref. [42] is listed as 'in preparation (2026),' so the full proof is not available in the published record. More importantly, the QCD+QED case introduces complications not present in the pure QCD hadron-hadron factorization of Ref. [41]: (1) the pinch-singular region where the exchanged photon goes on-shell (Q^2 -> 0), explicitly mentioned in the Introduction, is claimed to be handled naturally but the mechanism by which the pinch-singular contribution is absorbed into LDFs is not explained; (2) the scattered lepton is unobserved, meaning soft/collinear radiation from the final-state lepton is unconstrained, which differs from the inclusive DIS case treated in Ref. [11]. A","section":null},{"comment":"Sec. 3.2, Eq. (27) and Table 1: The default LDFs are constructed from a model ansatz with parameters (alpha_V, beta_V) = (60, 0.1) fitted to reproduce perturbative QED Mellin moments up to n ~ 4, with quark/gluon LDFs set to zero at input. The paper acknowledges these are model distributions representing lower limits. Since the central numerical predictions (Figs. 6-10) depend on these model LDFs, the quantitative results are illustrations of the framework's sensitivity rather than first-principles predictions. This is acceptable for a first exploration, but the manuscript should more clearly state the range of uncertainty associated with the LDF model choice, e.g., by showing how the alternative parameter set (alpha_V, beta_V) = (50, 0.125) affects the cross sections in Fig. 6 or Fig. 10. Without this, it is difficult for the reader to assess how much the phenomenological conclusions (e","section":null}],"minor_comments":[{"comment":"Sec. 2, Eq. (2): The notation bH_{ib->c} for the hard parts is introduced without explicit definition of the superscript H. A clarifying remark would help.","section":null},{"comment":"Sec. 3.1, Eq. (7): The matrix of splitting functions is large and the notation P^{(m,n)} with powers of alpha_em and alpha_s is compact but dense. A brief sentence explaining which entries are retained at LO and which are neglected would improve readability.","section":null},{"comment":"Sec. 3.2: The choice mu_0 = m_c is motivated by the perturbative reliability of QCD splitting functions, but the sensitivity of the evolved LDFs to this choice is not quantified. A brief comment on how changing mu_0 to, e.g., 1 GeV or 2m_c would affect the results would strengthen the discussion.","section":null},{"comment":"Fig. 4: The comparison between the WW photon distributions and the photon LDF is informative, but the linear-scale panels make it difficult to see the differences at intermediate xi. Consider adding a ratio panel.","section":null},{"comment":"Sec. 4.4, Fig. 10: The ratio dσ(RC)/dσ(NR) is defined with LDFs set to delta(1-xi) for the 'NR' case. It would be useful to clarify whether this 'NR' baseline also removes the photon LDF or only the electron LDF, since the photon contribution is part of the factorized cross section.","section":null},{"comment":"Sec. 5, Eq. (45): The schematic expression for the jet cross section omits O(alpha_s^2) terms for perturbative consistency, but the g -> J channel with J_g^{(1)} is mentioned as omitted. A brief comment on the expected size of this omission at EIC energies would help assess the consistency of the jet results.","section":null},{"comment":"Appendix A, Eq. (A4): The photon splitting function P^{(1,0)}_{gamma gamma} is proportional to -2/3 n_l delta(1-xi), which vanishes for n_l = 1. This is correct but could be confusing; a note that the photon does not self-split at this order would help.","section":null},{"comment":"The manuscript lists Ref. [42] as 'in preparation (2026).' If this paper is accepted before Ref. [42] appears, the authors should update the citation or provide a more self-contained summary of the proof strategy.","section":null}],"recommendation":"major_revision","confidential_remarks":"The stress-test concern about the deferred factorization proof is legitimate and load-bearing. The paper's claim that the QCD+QED factorization 'straightforwardly' follows from Ref. [41] is not fully convincing without addressing the pinch singularity and unobserved final-state lepton. However, the authors have a strong track record on factorization (Qiu-Sterman and collaborators), and the framework has been partially validated in Ref. [11] for inclusive DIS. I would recommend major revision rather than rejection: the authors should be able to address the proof sketch within the manuscript, and the numerical results are valuable for experimental planning even if the LDFs are model-dependent. The reader's concern about circularity (LDFs fitted to observables being predicted) does not apply here since the default LDFs are fitted to perturbative QED moments, not to the hadron production cross sections."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful reading and constructive comments. We address both major comments below and commit to revisions in the revised manuscript.","responses":[{"response":"We agree that the justification in Sec. 2 is too compressed and that the two specific issues raised by the referee deserve explicit discussion. We will expand Sec. 2 in the revised manuscript to address both points, as follows. (1) Regarding the pinch-singular region: In the joint QCD+QED factorization approach, the collinear modes along the beam lepton direction include the contribution where the exchanged photon becomes quasi-real (Q^2 → 0). This contribution is precisely the collinear photon radiation from the beam lepton that is absorbed into the photon LDF f_{γ/e}(ξ, μ^2), on the same footing as collinear gluon radiation being absorbed into PDFs in QCD factorization. The pinch singularity is regulated by the factorization scale μ_e, which separates the perturbatively calculable hard parts from the non-perturbative collinear contributions encoded in the LDFs. This is analogous to how collinear divergences in hadron-hadron scattering are absorbed into PDFs. The key point is that the LDFs are defined to include all collinear-sensitive contributions along the beam lepton direction, including the quasi-real photon region, making the hard parts infrared safe. We will add an explicit paragraph explaining this mechanism. (2) Regarding the unobserved scattered lepton: In single inclusive hadron production at high P_T, the hard scale is P_T, not Q^2. The observed hadron's transverse momentum P_T defines the hard scattering, and the scattered lepton carries a transverse momentum of order P_T to balance it. Because the final-state lepton has a large transverse momentum, there is no collinear enhancement along the final-state lepton direction — the collinear singularities are confined to the three observed external directions: the beam lepton (absorbed into LDFs), the beam hadr","revision_made":"no","referee_comment":"Sec. 2: The central claim is the factorization formula in Eq. (2). The justification given is a single paragraph stating that because the photon commutes with gluons, the same arguments as Refs. [41, 42] carry through. Ref. [42] is listed as 'in preparation (2026),' so the full proof is not available in the published record. More importantly, the QCD+QED case introduces complications not present in the pure QCD hadron-hadron factorization of Ref. [41]: (1) the pinch-singular region where the exchanged photon goes on-shell (Q^2 -> 0), explicitly mentioned in the Introduction, is claimed to be handled naturally but the mechanism by which the pinch-singular contribution is absorbed into LDFs is not explained; (2) the scattered lepton is unobserved, meaning soft/collinear radiation from the final-state lepton is unconstrained, which differs from the inclusive DIS case treated in Ref. [11]."}],"tokens_in":35476,"tokens_out":2010,"duration_ms":164810,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Short version: this is a solid, well-motivated paper that extends the joint QCD+QED factorization framework to single-inclusive hadron and jet production in lepton-hadron scattering. The central factorization formula is plausible and the numerical results are useful for EIC planning. The main weakness is that the factorization proof is deferred to a forthcoming reference, and the default LDFs are model inputs, not predictions. Neither gap is hidden — the paper is transparent about both. I think it deserves a serious referee, with the proof gap as the central question to press on. The stress-test concern about the pinch singularity and unobserved final-state lepton is legitimate but somewhat overblown — the paper's argument that the QED case inherits the QCD factorization structure is reasonable given that Ref. [11] already verified IR safety of NLO hard parts for inclusive DIS in this framework, and the extension to single-inclusive hadron production follows the same collinear factorization logic. The real issue is that the argument is asserted in one paragraph rather than demonstrated, and Ref. [42] is not available. A referee should ask the authors to at least sketch how the pinch-singular contribution is absorbed into LDFs and why the unobserved final-state lepton does not introduce non-factorizable pieces. The default LDF construction is honest about being a model — the ansatz is fitted to perturbative QED moments, not to the observables being predicted, so there is no circularity. The numerical results (Figs. 6-10) are illustrations of framework sensitivity, not precision predictions, and the paper says so. The QED radiative correction effects are large (up to 50% in the PT spectrum), which is the main phenomenological takeaway. The FF uncertainty comparison (MAP1.0 vs. JAM20) and the nuclear modification factor calculations are useful and well-executed. The jet production section with NLO jet functions is straightforward. One minor concern: QED corrections to hadron PDFs and FFs are neglected without a quantitative error estimate, though the paper argues this is a small effect at lepton-hadron energies. This paper is for theorists and experimentalists working on EIC physics and precision QCD. It deserves a serious referee who should focus on the factorization proof gap and push for more discussion of the pinch singularity mechanism.","headline":"First application of joint QCD+QED factorization to single-inclusive hadron and jet production in lepton-hadron scattering, with default LDFs and EIC predictions.","tokens_in":37029,"tokens_out":570,"would_cite":true,"duration_ms":186808,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"First calculation unifies lepton and photon contributions in hadron production","keywords":["lepton distribution functions","QCD+QED factorization","single inclusive hadron production","DGLAP evolution","photoproduction","leptoproduction","fragmentation functions","Electron-Ion Collider"],"falsifier":"If future EIC data on single inclusive hadron production, once stripped of traditional radiative corrections, cannot be fit by a universal set of LDFs that simultaneously describes both hadron and jet production channels, the joint factorization framework would fail its universality test.","tokens_in":35777,"feed_emoji":"⚛️","tokens_out":1579,"duration_ms":165239,"temperature":0.7,"pith_summary":"This paper presents the first calculation of single inclusive hadron and jet production at high transverse momentum in lepton-hadron scattering within a framework that treats collision-induced QED radiation from the lepton beam on the same footing as QCD radiation from the hadron. The cross section is factorized into perturbatively calculable hard parts convolved with three types of universal, non-perturbative functions: lepton distribution functions (LDFs) describing how the beam lepton's momentum is redistributed by radiation, standard parton distribution functions (PDFs) for the hadron, and fragmentation functions (FFs) for the observed hadron. The LDFs obey DGLAP-type evolution equations with splitting kernels that mix QCD and QED channels — a lepton can radiate a photon that splits into quark-antiquark pairs, and quarks can radiate photons that produce lepton pairs. The authors construct a default model set of LDFs at an input scale equal to the charm quark mass, evolve them to higher scales, and compute numerical predictions for pion and kaon production at Jefferson Lab and future Electron-Ion Collider energies. A central result is that this framework naturally unifies what were previously treated as separate processes — leptoproduction (via virtual photon exchange) and photoproduction (via quasi-real photons) — without requiring experimental cuts on the scattered lepton or additional parameters to separate direct and resolved photon contributions. The authors show that QED radiative corrections encoded in the LDFs can change the hadron production cross section by up to 50% depending on kinematics, and that the dominant theoretical uncertainty currently comes from the choice of fragmentation functions rather than scale dependence.","feed_headline":"Lepton radiation folded into universal distributions in new factorization","feed_subtitle":"First QCD+QED calculation of single hadron and jet production in lepton-hadron scattering unifies leptoproduction and photoproduction, with ","key_machinery":"The central object is the lepton distribution function (LDF), a non-perturbative function analogous to a parton distribution function but defined for the beam lepton. It describes the probability of finding an electron, positron, photon, quark, or gluon inside a parent electron at a given momentum scale. LDFs evolve via DGLAP-type equations whose splitting kernel matrix has four blocks: pure QED evolution (lepton/photon sector), pure QCD evolution (quark/gluon sector), and two mixing blocks that transfer probability between the QED and QCD sectors. The factorization formula (Eq. 2) expresses the physical cross section as a triple convolution of LDFs, hadron PDFs, and hadron FFs with infrared","core_discovery":"The paper establishes that collision-induced QED radiation in lepton-hadron scattering can be systematically absorbed into universal lepton distribution functions that evolve with mixed QCD+QED splitting kernels, and that this absorption simultaneously handles leptoproduction and photoproduction in a single factorization formula without kinematic cuts or extra parameters. The photon distribution of the electron, when evolved with both QCD and QED kernels, differs significantly from the traditionally used Weizsäcker-Williams distribution — it is smaller at low momentum fraction and vanishes at high momentum fraction, which can lead to quantitatively different predictions for photoproduction.","pith_inferences":["The fact that quark and gluon LDFs are set to zero at the input scale and only generated perturbatively means the numerical predictions represent lower bounds on hadron production rates from these channels; if non-perturbative quark/gluon content exists in the electron at the charm scale, the true rates could be higher.","The framework could be extended to parity-violating deep inelastic scattering and beyond-Standard-Model searches, where percent-level QED effects from LDFs could mimic or obscure new physics signals if not properly accounted for.","A practical test of the framework would be to compare LDF-extracted photon distributions with direct measurements of quasi-real photon spectra at HERA or future EIC, checking whether the QCD+QED evolved photon LDF gives a better description than Weizsäcker-Williams distributions."],"forward_implications":["If the framework is correct, future EIC measurements of single hadron and jet production at high transverse momentum can serve as a direct channel for extracting universal LDFs, provided existing radiative corrections are removed from the data first.","The unification of leptoproduction and photoproduction into a single formula eliminates the need for the direct/resolved photon decomposition used at HERA, potentially simplifying cross-section predictions and reducing systematic uncertainties from kinematic cuts.","The sensitivity of the cross section to the large-z region of fragmentation functions (Appendix B) means lepton-hadron data would complement electron-positron annihilation data, which probes the small-z region, enabling tighter constraints on FFs across the full momentum-fraction range.","High-transverse-momentum hadron production in lepton-nucleus collisions could constrain nuclear PDFs in the EMC region, with the nuclear modification factor showing distinctive suppression and enhancement patterns that differ across nPDF sets.","The 10-20% depletion of jet production cross sections from QED radiative corrections suggests that precision jet measurements at the EIC will require LDFs to be determined to at least that accuracy to avoid systematic bias in extracting other quantities."],"fun_headline_variants":["Mixed QCD+QED evolution alters electron photon predictions","Unified factorization replaces Weizsäcker-Williams in scattering","Lepton distributions absorb QED radiation in joint factorization","Joint QCD+QED factorization unifies lepton and photoproduction","Electron photon distribution revised under QCD+QED evolution"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The default lepton distribution functions at the input scale are constructed from a simple two-parameter model fitted to reproduce perturbative QED moments, with all quark, antiquark, and gluon content of the electron set to zero. The authors acknowledge these are model distributions representing lower limits, and that true LDFs can only be extracted from future data after removing existing radiative corrections. The full factorization proof is also stated to follow arguments","fun_headline_variants_meta":{"raw":{"variants":["Mixed QCD+QED evolution alters electron photon predictions","Unified factorization replaces Weizsäcker-Williams in scattering","Lepton distributions absorb QED radiation in joint factorization","Joint QCD+QED factorization unifies lepton and photoproduction","Electron photon distribution revised under QCD+QED evolution"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":1085,"prompt_tokens":494,"completion_tokens":591,"prompt_tokens_details":null},"tokens_in":494,"tokens_out":591,"duration_ms":62418,"temperature":1.0,"reasoning_tokens":514,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-09T03:11:32.873485+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If future EIC data on single inclusive hadron production, once stripped of traditional radiative corrections, cannot be fit by a universal set of LDFs that simultaneously describes both hadron and jet production channels, the joint factorization framework would fail its universality test.","supporting_citations":[],"review_version":1}