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First calculation unifies lepton and photon contributions in hadron production

2026-07-09 03:11 UTC pith:X7XVNZSO

load-bearing objection First application of joint QCD+QED factorization to single-inclusive hadron and jet production in lepton-hadron scattering, with default LDFs and EIC predictions. the 2 major comments →

arxiv 2607.07664 v1 pith:X7XVNZSO submitted 2026-07-08 hep-ph hep-exnucl-exnucl-th

Single inclusive hadron and jet production in lepton-hadron scattering

Jian-Wei Qiu , Kazuhiro Watanabe This is my paper
classification hep-ph hep-exnucl-exnucl-th
keywords lepton distribution functionsQCD+QED factorizationsingle inclusive hadron productionDGLAP evolutionphotoproductionleptoproductionfragmentation functionsElectron-Ion Collider
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

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.

Core claim

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.

What carries the argument

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

Load-bearing premise

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

What would settle it

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.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • 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.

Where Pith is reading between the lines

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

  • 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.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 8 minor

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.

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 (2)
  1. 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
  2. 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
minor comments (8)
  1. 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.
  2. 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.
  3. 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.
  4. 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.
  5. 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.
  6. 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.
  7. 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.
  8. 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.

Simulated Author's Rebuttal

1 responses · 0 unresolved

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.

read point-by-point responses

  1. Referee: 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].

    Authors: 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: no

Circularity Check

0 steps flagged

No significant circularity found; derivation chain is self-contained with transparent model inputs

full rationale

The paper's derivation chain proceeds as follows: (1) The factorization formula (Eq. 2) is argued by extending the published QCD factorization proof of Ref. [41] (Nayak, Qiu, Sterman, 2005) to the QCD+QED case, using the fact that photons commute with gluons. While Ref. [42] (Qiu, Sterman, Yu) is listed as 'in preparation,' this represents an incomplete proof, not a circular one — the argument structure is 'QCD factorization was proven in [41]; QED photons commute with gluons; therefore the same proof extends,' which is a logical extension argument, not a self-referential reduction. (2) The DGLAP evolution equations (Eq. 7) are derived from RG invariance of the physical cross section (Eq. 5), which is standard and not circular. (3) The default LDFs at input scale μ₀ = m_c are constructed from a model ansatz (Eq. 27) with parameters fitted to perturbative QED Mellin moments (Eqs. 29-31), not to the observables being predicted. The paper is transparent that these are model distributions representing lower limits, not predictions. (4) The numerical results (Figs. 6-10) use these model LDFs as inputs, and the paper explicitly frames them as illustrations of the framework's sensitivity rather than first-principles predictions. No step in the chain reduces to its own inputs by construction. The self-citations to [41, 11, 14, 15] are to published, peer-reviewed work that provides independent (if author-overlapping) support. The only concern is the unpublished Ref. [42], which is a completeness gap rather than a circularity. Score 1 reflects this minor self-citation without circular reduction.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 1 invented entities

The free parameters are all in the model LDF ansatz and are fitted to perturbative QED calculations (not to the observables being predicted). The axioms are standard factorization assumptions with one ad hoc simplification (single lepton family). The LDFs are the main new entity, justified by the factorization framework rather than postulated independently.

free parameters (6)
  • alpha_V (valence electron LDF shape) = 60
    Fitted by minimizing cost function J in Eq. (31) against perturbative NLO QED Mellin moments. Chosen over alternative value 50.
  • beta_V (valence electron LDF shape) = 0.1
    Fitted jointly with alpha_V via Eq. (31). Chosen over alternative value 0.125.
  • alpha_0 (initial electron LDF at me, for generating photon/sea) = 70
    Determined by requiring evolved valence electron at mc to reproduce the fitted valence distribution.
  • beta_0 (initial electron LDF at me, for generating photon/sea) = 0.1
    Determined jointly with alpha_0 by matching to perturbatively generated LDFs at mc.
  • N_i (normalization for each LDF flavor) = varies (see Table 1)
    Determined by sum rules (electron number conservation, momentum conservation) and fitting to perturbatively generated distributions.
  • alpha_i, beta_i (shape parameters for photon, positron LDFs) = varies (see Table 1)
    Fitted to perturbatively generated photon and positron distributions at mc.
axioms (5)
  • domain assumption Collinear factorization holds for single inclusive high-PT hadron production in lepton-hadron scattering with joint QCD+QED evolution.
    Invoked in Sec. 2 (Eq. 2). The full proof is stated to follow Refs. [41, 42], with [42] listed as 'in preparation'.
  • domain assumption The input factorization scale µ0 = mc is sufficiently large for perturbative QCD evolution kernels to be reliable.
    Stated in Sec. 3: 'we choose to solve the evolution equations with an input factorization scale µ0 = mc, since the evolution kernels for a photon to split into a light quark-antiquark pair could be non-perturbative in QCD if the scale is less than mc.'
  • domain assumption QED corrections to hadron PDFs and FFs are negligibly small for the energies considered.
    Sec. 3 and Sec. 4: 'the numerical impact of QED corrections to the evolution of hadron PDFs and FFs is much smaller than that of QCD corrections to the evolution of LDFs.' Used available JAM20 PDFs/FFs without QED evolution.
  • domain assumption Power corrections of order 1/PT^2 are negligible for PT values considered.
    Eq. (2) and Sec. 4.6: higher-twist corrections are assumed negligible when PT is sufficiently large, including in lepton-nucleus collisions where they could be enhanced by nuclear size.
  • ad hoc to paper Only a single lepton family (nl=1) contributes to LDF evolution.
    Sec. 3.1: 'we neglected other lepton flavors in this paper.' This is a simplification valid when lepton flavor thresholds are not crossed.
invented entities (1)
  • Lepton Distribution Functions (LDFs) independent evidence
    purpose: Universal, process-independent non-perturbative functions describing the momentum distribution of leptons, photons, and partons inside a colliding lepton beam.
    LDFs are a necessary consequence of joint QCD+QED factorization. They are not independently postulated but derived from the factorization framework. Their universality is testable by comparing different observables. The paper provides falsifiable predictions: once LDFs are extracted from one process, they predict QED radiative corrections for others. However, no direct experimental extraction has yet been performed.

pith-pipeline@v1.1.0-glm · 38859 in / 3920 out tokens · 649106 ms · 2026-07-09T03:11:32.873485+00:00 · methodology

0 comments
read the original abstract

We present the first calculation of single inclusive hadron and jet production at large transverse momentum in lepton-hadron scattering in a joint QCD+QED factorization approach. The scattering cross section is factorized into a convolution of infrared-safe hard coefficient functions with universal lepton distribution functions (LDFs) and parton distribution functions (PDFs) of the colliding lepton and hadron, respectively, together with fragmentation functions (FFs) of the observed hadron (or jet). With joint QCD+QED factorization, the DGLAP-type evolution equations for LDFs, PDFs, and FFs necessarily have evolution kernels calculated in both QCD and QED. We derive a default set of LDFs for our calculations and discuss a strategy to extract universal, non-perturbative LDFs from future data. We present our calculations for single inclusive hadron and/or jet production at the energies of Jefferson Lab and the future Electron-Ion Collider.

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