Factorized QED and QCD Contribution to Deeply Inelastic Scattering

Jian-Wei Qiu; Jia-Yue Zhang; Justin Cammarota; Kazuhiro Watanabe

REVIEW 3 major objections 4 minor 3 cited by

Factorized QED and QCD Contribution to Deeply Inelastic Scattering

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper computes the next-to-leading-order QED and QCD short-distance hard coefficients for inclusive lepton-hadron deep inelastic scattering in a joint QED-QCD factorization, and claims the NLO QED contribution is completely infrared…

desk verdict First NLO QED hard coefficient in the joint QED-QCD factorization scheme, likely right but with a technical gauge-link error that needs fixing. read the letter →

arxiv 2505.23487 v1 pith:6LMI24YG submitted 2025-05-29 hep-ph hep-exnucl-exnucl-th

Justin Cammarota , Jian-Wei Qiu , Kazuhiro Watanabe , Jia-Yue Zhang This is my paper · ORCID

classification hep-phhep-exnucl-exnucl-th MSC 81V0581V1081T18 PACS 12.38.-t13.60.Hb12.20.-m

keywords DeepinelasticscatteringQEDfactorizationQCDLeptondistributionfunctionfragmentationPhotonPDFInfraredsafetyNLOhardcoefficients

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper tries to establish that collision-induced QED radiation in inclusive lepton-hadron deep inelastic scattering can be handled by the same factorization machinery as QCD radiation, rather than by radiative-correction factors with adjustable parameters. It presents the first next-to-leading-order calculation of the short-distance hard coefficients in a joint QED-QCD factorization for the leading electron-quark subprocess, and the central result is that the NLO QED hard part is infrared safe and depends on no free parameter except the standard factorization scale. If the result is right, analyses of DIS and related lepton-hadron observables can include QED radiation without arbitrary photon-phase-space cuts, and the same universal lepton distribution and fragmentation functions can be used across observables. The paper also gives a procedure for converting existing higher-order QCD structure-function results into this framework, and it stresses that the relevant hard scale is the observed lepton transverse momentum rather than $Q^2$.

What carries the argument

The load-bearing object is the joint QED-QCD factorization formula in Eq. (6), which writes the DIS cross section as convolutions of universal nonperturbative functions, namely lepton distribution functions (LDFs), lepton fragmentation functions (LFFs), and parton distribution functions (PDFs), with short-distance partonic hard coefficients $\widehat H_{ia\to jX}$. The derivation of the NLO QED coefficient uses the subtraction identity in Eq. (36), where the dimensionally regularized partonic cross section $\sigma^{(3,0)}_{eq\to eX}$ is freed of all collinear and pinch divergences by subtracting first-order LDF, LFF, and PDF terms, including a photon-PDF term whose convolution absorbs the $q^2\to 0$ region. The hard coefficient is then organized in powers of quark charge in Eq. (55), and the MS scheme plus DGLAP-type joint evolution equations supply the only scale dependence.

What would settle it

A direct check is to repeat the NLO QED calculation with a different infrared regulator, for example a small photon mass or a photon-energy cutoff instead of dimensional regularization, and verify that the combination in Eq. (36) is finite and independent of the regulator in the limit; any leftover $1/\epsilon$ pole or cutoff dependence in $\widehat H^{(3,0)}_{eq\to eX}$ would show the infrared-safety claim fails. A second, observable check is to measure the inclusive $e+p$ cross section in a kinematic region where the photon-PDF subtraction term is large and compare with the prediction using an independently extracted photon PDF.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the NLO QED contribution to the leading partonic channel of inclusive DIS, the hard coefficient $\widehat H^{(3,0)}_{eq\to eX}$ given in Eq. (55) with coefficients in Appendix A, is completely infrared safe after collinear-sensitive pieces are subtracted through Eq. (36). The subtraction combines the first-order lepton fragmentation function $D^{(1,0)}_{e/e}$, electron and quark distribution functions $f^{(1,0)}_{e/e}$ and $f^{(1,0)}_{q/q}$, and the photon distribution $f^{(1,0)}_{\gamma/q}$ convoluted with the lowest-order $e\gamma\to eX$ hard part; the last term is what removes the perturbative pinch when the exchanged photon goes on shell and collinear to the quark. Grouping the result by quark charge shows the $e_q^2$ (electron-radiation), $e_q^3$ (two-photon-exchange/interference), and $e_q^4$ (quark-radiation) pieces are each separately finite, and the NLO QCD hard part Eq. (34) is likewise infrared safe with only the factorization-scale dependence. The claim is that no extra parameter is needed beyond the standard factorization scale in the MS scheme.

Load-bearing premise

The load-bearing premise is that the joint QED-QCD factorization formula in Eq. (6) is genuinely valid for inclusive lepton-hadron DIS; the paper assumes this theorem rather than proving it, citing [16] and an analogy to the QCD single-hadron proof [18], so if the combined gauge link or the on-shell-photon pinch region were mishandled, the subtracted hard coefficient would not be the true short-distance quantity.

Editorial extensions

If this is right

NLO QED radiative effects in inclusive DIS can be included without introducing a photon-energy or photon-angle cut, so the predictive uncertainty of radiative corrections is reduced to the standard factorization scale.
The pinch singularity that appears when the exchanged photon is nearly on shell and collinear to the quark is absorbed into the hadron's photon PDF, so the same framework produces the photon-initiated contribution as part of the same calculation.
Existing and future higher-order QCD results for DIS structure functions can be converted into joint-factorization hard coefficients through Eqs. (15) and (18), so the QCD part of the framework is not limited to NLO.
The same LDFs and LFFs, with their nonperturbative endpoint behavior, must be used consistently with the hard coefficients: using perturbative delta-function endpoint distributions together with beyond-LO hard parts produces a spurious divergence, as Eq. (68) shows.
Because the true hard scale is $\ell'^2_T=(1-y)Q^2$, predictions at large $y$ require a correspondingly larger $Q^2$, otherwise inverse-power corrections to the factorization formula are not small.

Reading between the lines

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

Inference: if the joint factorization theorem holds, the same subtraction pattern should extend to semi-inclusive DIS and TMD observables, where QED radiation changes the orientation of the leptonic plane relative to the hadronic plane; this could remove one model dependence from TMD extractions.
Inference: the claim that LDFs and LFFs are nonperturbative and must vanish at momentum fraction 1 is testable: global fits combining inclusive DIS, SIDIS, and electron-positron annihilation data could extract these functions and check that they agree across observables.
Inference: the photon-PDF subtraction makes a definite prediction at high energies: the photon-initiated $e\gamma\to eX$ channel contributes at a level fixed by the evolving photon content of the proton, so tagging low-$Q^2$ or forward-photon events would directly probe the mechanism that removes the $q^2\to 0$ pinch.
Inference: the emphasis on $\ell'_T$ rather than $Q^2$ as the hard scale implies that existing global PDF analyses that cut on $Q^2$ alone may include kinematic regions with sizable power corrections; repeating fits with an $\ell'_T$-based cut could shift extracted distributions at large $y$.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

I read the Cammarota et al. paper on factorized QED/QCD DIS. Bottom line: the calculation is likely right and worth taking seriously, but there's a technical error in the operator definition that needs to be fixed before I'd trust the formal claim.

What's new: Eq. (55) with Appendix A is the first explicit NLO QED hard part for e+q->e+X in the joint factorization scheme. The coefficients are all written out, so the result is reproducible. The QCD part is repackaged known structure-function results, but the conversion procedure (Eqs. 15, 18) is clean and useful. The numerical section is honest: the LDF/LFF inputs are explicitly models, not fits, and they show the resulting uncertainty. I also like that they caught the divergence from using perturbative LDF/LFFs together with NLO hard parts—that's a real subtlety that would trip people up.

Soft spots. The biggest one: the gauge link in Eq. (20) is missing the quark fractional charge e_q. As written, it cancels a charge-1 phase, so the quark PDF defined in Eq. (19) is not QED gauge invariant. Same issue appears in the quark LFF of Eq. (44). This matters because the factorization theorem they rely on requires gauge-invariant operator definitions. The perturbative calculations themselves use e_q correctly, so the hard coefficients are probably fine, but the formal setup is wrong as stated. It's likely a typo—the link should be P exp[-i∫(g_s A_g t^a - e e_q A_γ)]—but it has to be corrected. The paper also does not prove the factorization theorem; it cites Ref. [16] and says the proof is the same as for single-hadron production. That's acceptable as a division of labor, but it means the IR-safety claim is conditional on that theorem.

Two smaller issues. There's no independent cross-check of the Appendix coefficients—a comparison with the traditional RC result for the pure-electron-radiation term in an appropriate limit would be reassuring. And the numerics use CT18 PDFs that are not the joint QED-QCD PDFs defined in Eq. (19); the authors acknowledge the difference is small, but it's an approximation, not the exact framework.

Overall: the main result is new and the coefficients are explicit enough to be tested. The gauge-link issue is a genuine defect that a referee should catch and the authors should fix. I'd send it to peer review, but I'd want the operator definition corrected and ideally a cross-check added before publication.

Referee Report

3 major / 4 minor

Summary. The paper presents the first next-to-leading-order (NLO) factorized QED hard coefficient for the leading partonic subprocess e+q -> e+X of inclusive lepton-hadron DIS in the joint QED and QCD factorization framework. Section III derives the NLO QCD hard part bH^(2,1) by converting standard structure-function results into the joint-factorization language. Section IV computes the NLO QED hard part bH^(3,0), given in Eq. (55) with coefficients in Appendix A, by subtracting, order by order in alpha_em, the collinear contributions absorbed into lepton distribution functions (LDFs), lepton fragmentation functions (LFFs), and quark/photon PDFs. The paper argues that all collinear and pinch singularities cancel, so that bH^(3,0) is infrared safe and depends only on the factorization scale. Section V provides numerical estimates of the impact of collision-induced QED radiation at JLab and EIC kinematics, using CT18 PDFs and model LDF/LFF input shapes.

Significance. If the calculation is correct, this is a useful first step toward treating collision-induced QED radiation in inclusive DIS with the same factorization logic as QCD, avoiding ad hoc radiative-correction cuts. The organization of Eq. (55) by powers of the quark charge e_q is physically illuminating, and the explicit demonstration that the photon-PDF subtraction removes the q^2 -> 0 pinch is valuable. The paper also makes a legitimate point that the hard scale for the factorized formula is the observed-lepton transverse momentum l'_T, not Q^2, and it is candid about the nonperturbative nature of LDFs and LFFs. The manuscript is not yet fully supported, however: the operator definitions of the PDFs and LFFs have a gauge-invariance problem, the central NLO QED coefficient is presented without derivation or cross-check, and the factorization theorem on which the IR-safety claim rests is only cited, not established.

major comments (3)

[Sec. II, Eqs. (19) and (20); Sec. IV, Eq. (44)] The quark PDF in Eq. (19) and the quark-to-electron LFF in Eq. (44) are not QED gauge invariant as written. Under a U(1) transformation the quark field transforms with a phase involving the fractional quark charge e_q, whereas the QED part of the gauge link in Eq. (20), P exp[-i integral (g_s A_g^+ t^a - e A_gamma^+)], is constructed to cancel only a unit-charge phase. The same defect carries over to Eq. (44), whose gauge link is also taken from Eq. (20). Because the subtraction terms in Eq. (36) are supposed to represent the collinear divergences of gauge-invariant, universal distributions, the factorization underpinning the IR-safety claim of Eq. (55) is not established as stated. Please insert the quark fractional charge e_q in the QED part of the link, or state an explicit charge convention and demonstrate that the bilocal operators are invariant under both QCD and QED gauge transformations.
[Sec. IV, Eq. (55) and Appendix A] The central result of the paper, Eq. (55), is presented with the coefficients a_i, b_i, and c_i relegated to Appendix A and with no derivation shown. The text does not describe how the d-dimensional phase-space integrals for the diagrams in Figs. 11 and 12 are evaluated, how the 1/epsilon poles of sigma^(3,0) are distributed among the subtraction terms in Eq. (36), or how the cancellation between the two-photon-exchange diagrams and the real-interference diagrams is verified. No independent cross-check is provided against known QED radiative-correction results, even in a limiting case. As a calculation paper, this leaves the main claim unsupported. Please provide a derivation summary and at least one nontrivial consistency check, such as a demonstration of the 1/epsilon pole cancellation order by order or a comparison with a standard radiative-correction expression in a well-defined limit.
[Sec. II, Eq. (6); Sec. IV, Eqs. (35)-(36)] The IR-safety claim for bH^(3,0) is conditional on the joint QED-QCD factorization formula in Eq. (6), but this theorem is neither proved nor precisely stated. The text says that the proof is 'effectively the same' as that in Ref. [18] and is 'straightforward to verify', and the treatment of the q^2 -> 0 pinch region is only heuristic. Since Eq. (36) is exactly the order-by-order implementation of this theorem, the identification of Eq. (55) as the correct short-distance coefficient requires either a proof sketch for the inclusive e h -> e X case, including the q^2 -> 0 region and the combined QED-QCD gauge links, or an explicit statement of the theorem from Ref. [16] and the conditions under which it applies.

minor comments (4)

[Sec. IV and Sec. V, general] There are several typographical errors: 'contributiosn' in the first paragraph of Sec. IV, 'infra-safe' in the paragraph after Eq. (55), and 'purbatively' in Sec. V. A careful proofreading pass is needed.
[Eq. (61)] The logarithm in the perturbative LDF f_{e/e}(xi) is written with (1 - zeta) in the denominator; it should presumably be (1 - xi). Please confirm and correct.
[Eq. (56)] The notation e_l^2 is confusing, because it denotes a sum over all fermion flavors in the photon vacuum polarization, not the electron charge squared. Consider renaming this quantity, for example C_gamma or Sigma_f n_c e_f^2, to avoid confusion with the lepton charge.
[Figs. 15 and 16] The captions and text should state explicitly that the shaded bands in Figs. 15 and 16 are model estimates based on the ad hoc input LDFs and LFFs of Eqs. (63) and (64), not extracted universal functions, and that the horizontal axis region near the dotted vertical line corresponds to small l'_T where the factorized formula is not expected to be reliable.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the NLO QED hard part is obtained by an explicit subtraction calculation with no fitted parameters; the factorization theorem is an external input, not the paper's output.

full rationale

The central coefficient bH^(3,0) in Eq. (55) is computed from the NLO partonic cross section sigma^(3,0) (real and virtual diagrams in Figs. 11 and 12) minus convolutions of the LO hard parts with O(alpha_em) LDF, LFF, quark-PDF, and photon-PDF splitting functions, as written in Eq. (36). The subtraction terms are obtained from operator definitions (Eqs. (19)-(20), (38), (41), (44)-(45)) and are not fitted to sigma^(3,0); the finite coefficients a_i, b_i, c_i in Appendix A are the genuinely new perturbative content. No fitted parameter enters the derivation of bH^(3,0); the Beta-function parameters for LDFs and LFFs in Sec. V are explicitly a model for numerical illustration and do not feed back into Eq. (55). The joint factorization theorem in Eq. (6) is cited to Refs. [15,16], with the proof stated to be 'effectively the same as' Ref. [18]; this is a self-citation chain because J.-W. Qiu is a common author. The theorem is load-bearing in the conditional sense: if it failed, the subtraction in Eq. (36) would not define the true short-distance coefficient. However, the theorem is a general external result and does not assert the specific NLO hard part; the paper's new contribution is the explicit evaluation and pole cancellation leading to Eq. (55). This is reliance on a prior theorem, not a reduction by construction, and the citation is independent support in the sense that the theorem is not defined in terms of the target coefficient. A technical correctness concern is flagged separately: the gauge link in Eq. (20) is written with -e A_gamma^+, which appears to lack the quark fractional charge e_q, so the operator definitions in Eqs. (19) and (44) may not be QED-gauge invariant for quark fields as written. This is a validity concern for the cited factorization theorem's operator definitions, not a circular step. The paper also omits a detailed proof of Eq. (6), delegating to prior work; that is a completeness gap, not circularity. Overall, the derivation is self-contained in the sense that the claimed hard part is computed, not assumed, and no element of the derivation is equivalent to its own input by definition.

Assumptions & free parameters 3 free parameters · 4 assumptions · 3 invented entities

The central hard-part calculation itself has no fitted parameters, but the numerical impact studies depend on ad hoc LDF/LFF shapes and a flavor-scheme choice. The factorization theorem and the new PDF definition are load-bearing assumptions imported from prior work. The new nonperturbative objects (LDFs, LFFs) have no independent evidence yet.

free parameters (3)

LDF input shape parameters (a,b) = (50,1/8) or (5,1/2)
Chosen by hand at mu0 = mc to model the unknown nonperturbative LDF; used only for numerical impact studies, not in the hard part calculation.
LFF input shape parameters (alpha,beta) = (50,1/8) or (5,1/2)
Same choice as the LDF parameters; an illustrative model for the lepton fragmentation function, used in the numerical section.
Number of active fermion flavors in photon vacuum polarization = two generations, charm mass neglected
Affects the e_l^2 term in Eq. (55) through Eq. (56); a scheme choice for the numerical evaluation, not a fundamental parameter of the hard part.

assumptions (4)

domain assumption Joint QED-QCD factorization theorem for inclusive DIS (Eq. 6)
Assumed from Ref. [16]; not proven in this paper. The paper states the proof is effectively the same as the QCD factorization proof for single hadron production in Ref. [18].
domain assumption Operator definition of PDFs with combined QED and QCD gauge links (Eq. 19)
Defines the PDFs used in the factorization; a new definition slightly extended from standard QCD PDFs, required for QED gauge invariance.
standard math Standard perturbative QED/QCD with MS renormalization and dimensional regularization
Used throughout the NLO calculations; the standard framework for perturbative field theory.
domain assumption LDFs and LFFs are universal and nonperturbative
Assumed by the factorization; their operator definitions are given, but their input shapes are modeled ad hoc in Sec. V.

invented entities (3)

Lepton distribution functions (LDFs)
purpose: Absorb collinear QED and QCD radiation from the incoming lepton
New universal nonperturbative functions introduced in Ref. [16]; not yet extracted from data. Only ad hoc models are used here.
Lepton fragmentation functions (LFFs)
purpose: Absorb collinear radiation from the observed lepton
Same status as LDFs; no independent measurement yet. The paper parametrizes them with Beta functions for numerical illustration.
PDFs with combined QED-QCD gauge links
purpose: Ensure gauge invariance of quark distributions under both QED and QCD
A redefinition of the standard PDF; no independent extraction yet. The numerical study instead uses standard CT18 PDFs, which do not include the QED gauge link.

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Cite this review

Pith. "Pith review of Factorized QED and QCD Contribution to Deeply Inelastic Scattering." pith.science (2026). https://pith.science/paper/6LMI24YG

@misc{pith2026250523487,
  author       = {Pith},
  title        = {Pith review of: Factorized QED and QCD Contribution to Deeply Inelastic Scattering},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6LMI24YG}},
  note         = {Machine review of arXiv:2505.23487}
}

read the original abstract

We present the first calculation of next-to-leading order (NLO) factorized QED and QCD contributions to the short-distance hard coefficients of inclusive lepton-hadron deep inelastic scattering (DIS) in a joint QED and QCD factorization approach. Unlike the traditional radiative correction approach to handle the collision-induced QED contributions to DIS, QED radiation from all charged leptons and quarks are treated equally, and their collinear sensitivities are systematically factorized into corresponding universal lepton and parton distribution functions. We demonstrate that the NLO factorized QED contribution is completely infrared safe and calculable without the need of any parameters other than the standard factorization scale in the same way as the factorized QCD contribution. We discuss the potential impact of this joint factorization approach on the extraction of partonic information from lepton-hadron DIS.

Figures

Figures reproduced from arXiv: 2505.23487 by the authors.

**Figure 3.** FIG. 3. The cut diagram representation of the QED radiative [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

**Figure 2.** FIG. 2. Diagrams contribute to the RC of inclusive DIS cross [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

Figures from the paper (11 more)

**Figure 4.** Figure 4: FIG. 4. The cut-diagram for quark distribution of momentum [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. The lowest order cut diagram contributing to the [PITH_FULL_IMAGE:figures/full_fig_p006_5.png]

**Figure 7.** Figure 7: FIG. 7. The cut-diagrams contribute to the [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 9.** Figure 9: FIG. 9. The cut-diagram for photon distribution of momen [PITH_FULL_IMAGE:figures/full_fig_p008_9.png]

**Figure 10.** Figure 10: FIG. 10. The cut-diagram for lepton fragmentation function [PITH_FULL_IMAGE:figures/full_fig_p009_10.png]

**Figure 11.** Figure 11: FIG. 11. NLO Feynman diagrams for real QED contributions [PITH_FULL_IMAGE:figures/full_fig_p009_11.png]

**Figure 12.** Figure 12: FIG. 12. NLO Feynman diagrams for virtual QED contribu [PITH_FULL_IMAGE:figures/full_fig_p009_12.png]

**Figure 14.** Figure 14: FIG. 14 [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

**Figure 13.** Figure 13: FIG. 13. Two very choices of input LDF and LFF at [PITH_FULL_IMAGE:figures/full_fig_p013_13.png]

**Figure 15.** Figure 15: FIG. 15 [PITH_FULL_IMAGE:figures/full_fig_p014_15.png]

**Figure 16.** Figure 16: , which are supposed to be derived by using perturbatively calculated LDF and LFF in Eqs. (61) and (62), respectively, to evaluate the DIS cross sections. This is due to an intrinsic problem for evaluating a factorized expression with multiple perturbatively derived…

Discussion (0). Continue with ORCID to comment.

Forward citations

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Reference graph

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Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

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= ξa(1−ξ) b B(1 +a,1 +b) ,(63) De/e(ζ, µ2

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(61) and (62)

= ζ α(1−ζ) β B(1 +α,1 +β) .(64) With the Beta function, these parametrized LDFs and LFFs are normalized to have the same integrated prob- ability as those in Eqs. (61) and (62). While these non- perturbative and universal input distributions should be extracted by fitting experimental data, for the simplic- ity of the current discussion, we choosef e/e(ξ, µ2

work page

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13 to cover a wide range of LDFs and LFFs and their impact on the cross sections

with two sets of parameters: (a, b) = (α, β) = (50,1/8) or (5, 1/2), as shown in Fig. 13 to cover a wide range of LDFs and LFFs and their impact on the cross sections. For the other channels, we setf ¯e,γ/e(ξ, µ2

work page

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= 0 andD e/¯e,γ(ζ, µ2

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radiative correction

= 0. 0.0 0.5 1.00 1 2 3 4fe/e(ξ, m2 c)/De/e(ζ, m2 c) (a, b) = (α, β) = (5, 1/2) 0.5 1.00 10 20 30 40(a, b) = (α, β) = (50, 1/8) ξ / ζ FIG. 13. Two very choices of input LDF and LFF atµ0 =m c. To test the impact of evolution, we plot thex B- dependence of the ratio of DIS cross sectiondσ LO−RC eh→eX in Eq. (60), calculated with evolved over non-evolved LDF...

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