QED Lepton PDFs in QCD+QED Factorization

• QED lepton PDFs are the collinear probability densities that describe the distribution of charged leptons and photons inside hadrons or leptons.
• They evolve through joint QCD+QED DGLAP equations, ensuring a universal treatment of initial-state radiation in high-energy processes.
• Analytic methods using SCET and fixed-order calculations underpin their precise extraction, impacting predictions in lepton and hadron collider experiments.

QED lepton parton distribution functions (PDFs) are the collinear probability densities that arise when electrodynamic radiation is factorized in the same way as QCD radiation. In one standard usage, they are the charged-lepton PDFs $e^\pm,\mu^\pm,\tau^\pm$ inside a hadron, introduced consistently together with the photon PDF once leading-order QED corrections are included in DGLAP evolution. In a second usage, often denoted lepton PDFs or lepton distribution functions (LDFs), they describe the probability to find, inside a parent lepton, a parton $i=\ell,\bar\ell,\gamma,q,\bar q,g$ carrying a light-cone momentum fraction $x$ or $\xi$. For sufficiently inclusive observables, these functions absorb universal collinear initial-state radiation and enter factorized cross sections in the same convolutional form as hadron PDFs (Bertone et al., 2015, Cammarota et al., 2024, Qiu et al., 8 Jul 2026).

1. Definitions and operator formulations

The modern formulation treats QED lepton PDFs exactly as in QCD, except that the parent hadron may be replaced by a physical electron or photon, so that the corresponding distributions are perturbatively calculable (Schnubel et al., 11 Sep 2025). In soft-collinear effective theory (SCET), the bare parton-in-electron PDF for $i=\{e,\bar e,\gamma\}$ is defined by the matrix element of a gauge-invariant operator,

$$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$

with explicit fermion and photon operators built from collinear fields and Wilson lines (Stahlhofen, 23 Aug 2025). In the light-cone-gauge formulation, one may equivalently define

$$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$

whose forward matrix elements yield the bare PDFs in an electron or photon state (Schnubel et al., 11 Sep 2025).

For a parent lepton in joint QCD+QED collinear factorization, one introduces a universal set of LDFs

$f_{i/\ell}(\xi,\mu^2),$

giving the probability to find, inside a parent lepton $\ell$, a “parton” $i=e,\bar e,\gamma,q,\bar q,g$ with light-cone momentum fraction $i=\ell,\bar\ell,\gamma,q,\bar q,g$0 (Qiu et al., 8 Jul 2026). In the analogous Standard Model notation of LePDF, the same logic is extended beyond pure QED to include electroweak gauge bosons and polarization effects; below the electroweak scale, however, only QED and, through $i=\ell,\bar\ell,\gamma,q,\bar q,g$1, QCD interactions contribute (Garosi et al., 2023).

The basic normalization is fixed by momentum conservation. For lepton PDFs in a lepton, the momentum sum rule takes the form

$i=\ell,\bar\ell,\gamma,q,\bar q,g$2

and at tree level one has $i=\ell,\bar\ell,\gamma,q,\bar q,g$3 (Garosi et al., 2023, Schnubel et al., 11 Sep 2025).

2. Mixed QCD+QED evolution and splitting kernels

QED lepton PDFs satisfy DGLAP-type integro-differential equations in direct analogy with ordinary hadronic PDFs. In the joint QCD+QED factorization used for lepton-hadron scattering, the LDF evolution is

$i=\ell,\bar\ell,\gamma,q,\bar q,g$4

with splitting kernels expanded as

$i=\ell,\bar\ell,\gamma,q,\bar q,g$5

At leading non-trivial order one retains only the pure-QED $i=\ell,\bar\ell,\gamma,q,\bar q,g$6 and pure-QCD $i=\ell,\bar\ell,\gamma,q,\bar q,g$7 terms (Qiu et al., 8 Jul 2026).

In the conventions used for the lepton-parent system, the pure-QED leading kernels include

$i=\ell,\bar\ell,\gamma,q,\bar q,g$8

$i=\ell,\bar\ell,\gamma,q,\bar q,g$9

with mixed quark-photon kernels proportional to the quark charge,

$x$0

The pure-QCD kernels $x$1 are the standard leading-order Altarelli-Parisi functions (Qiu et al., 8 Jul 2026).

For hadronic PDFs with QED corrections, the singlet-sector evolution written in APFEL includes quarks, antiquarks, gluon, photon, and charged leptons simultaneously. In that framework, the photon and each charged lepton satisfy

$x$2

$x$3

APFEL 2.4.0 and later solves the resulting twenty coupled integro-differential equations with QCD $x$4 up to NNLO and QED $x$5 at LO, while checking that the total momentum fraction stays unity to better than $x$6 (Bertone et al., 2015).

Beyond leading order in QED, the electron splitting kernels can be systematically extended. In SCET, the QED kernels are expanded as

$x$7

with one-loop kernels $x$8, $x$9, and $\xi$0, and two-loop kernels $\xi$1 obtained by “Abelianizing” the QCD two-loop results (Stahlhofen, 23 Aug 2025). The complete QED NNLO kernels $\xi$2 are incorporated in the fully analytic two-loop calculation of electron and photon structure functions (Schnubel et al., 11 Sep 2025).

3. Factorization formulas and observable content

The central role of QED lepton PDFs is to factorize collinear radiation from hard scattering. For sufficiently inclusive processes with incoming $\xi$3, one has at leading power

$\xi$4

where the partonic cross section is computed with massless external partons and all mass singularities are absorbed into $\xi$5 and $\xi$6 (Stahlhofen, 23 Aug 2025).

In inclusive lepton-hadron deep inelastic scattering, the joint QCD$\xi$7QED factorization formula introduces both lepton PDFs and lepton fragmentation functions,

$\xi$8

up to power-suppressed terms (Cammarota et al., 2024). The explicit NLO subtraction formula removes the universal collinear contributions from the bare partonic cross section by convoluting the hard part with the order-$\xi$9 lepton PDF, lepton fragmentation function, quark PDF, and photon-in-quark PDF (Cammarota et al., 2024).

For single-inclusive hadron production in lepton-hadron scattering at large transverse momentum $i=\{e,\bar e,\gamma\}$0,

$i=\{e,\bar e,\gamma\}$1

The short-distance hard part is infrared-safe in both QCD and QED after collinear subtraction into the distribution and fragmentation functions. In this formulation, leptoproduction channels with $i=\{e,\bar e,\gamma\}$2 start at $i=\{e,\bar e,\gamma\}$3, while photoproduction with $i=\{e,\bar e,\gamma\}$4 enters at $i=\{e,\bar e,\gamma\}$5. For single-inclusive jets, the hadron fragmentation function is replaced by parton-to-jet functions $i=\{e,\bar e,\gamma\}$6 with the same convolution structure (Qiu et al., 8 Jul 2026).

A recurrent conceptual point is that this formalism treats collinear QED radiation from the incoming lepton as a genuine factorized ingredient rather than as a mere “add-on” radiative correction. The joint-factorization papers state this explicitly: all perturbative collinear sensitivities of partonic scattering in both QCD and QED are factorized into corresponding universal hadron and lepton distribution functions without the need of any parameters other than the standard factorization scale (Cammarota et al., 2024).

4. Boundary conditions, nonperturbative input, and public implementations

Initial conditions depend on the parent state and on whether one is treating a purely perturbative lepton system or a mixed QCD+QED system. For hadronic PDFs with QED corrections, one may start at $i=\{e,\bar e,\gamma\}$7 GeV from either a “zero photon” boundary or an existing LO-QED PDF set, and for the light leptons $i=\{e,\bar e,\gamma\}$8 one may assume that they are entirely generated by collinear $i=\{e,\bar e,\gamma\}$9 splitting above threshold,

$$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$0

Equivalently, one may set $$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$1 and let the evolution build them up. The $$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$2 PDFs are turned on dynamically in the variable-flavour scheme at $$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$3 with zero boundary condition (Bertone et al., 2015). The same two starting prescriptions were implemented and tested in APFEL for proton PDFs with QED corrections (Carrazza, 2015).

For a parent muon, the leading-logarithmic setup of Frixione and Stagnitto uses

$$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$4

at $$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$5, and then evolves the full QED$$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$6QCD system for the lepton, photon, quark singlets, and gluon (Frixione et al., 2023). In the purely perturbative QED discussion of inclusive DIS, one similarly takes $$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$7 with $$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$8 and vanishing $$f_{i/e}^{\rm bare}(x) = \frac{1}{2}\sum_{s=1,2} \big\langle e_n^-(p^-,s)\big| \mathcal Q_i^{\rm bare}(x p^-) \big|e_n^-(p^-,s)\big\rangle, \qquad p^-=\bar n\!\cdot\! p,$$9, $$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$0, together with the possibility of a nonperturbative parameterization if an extraction from data is envisaged (Cammarota et al., 2024).

A distinct situation arises in the joint QCD+QED treatment of high-$$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$1 lepton-hadron scattering. Because QCD splitting into light quarks becomes nonperturbative below $$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$2, the input scale is chosen as $$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$3. At that scale the nonperturbative LDFs are parameterized by a simple Beta-function ansatz,

$$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$4

while quark, antiquark, and gluon LDFs are set to zero at $$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$5 and generated purely by evolution above $$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$6. The parameters are fixed by valence-electron number, momentum sum, and matching the valence LDF to the known NLO perturbative result at $$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$7 for Mellin moments up to $$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$8. The default set is

• $$O_e(x)=\tfrac12\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,\bar\psi(\xi^-)\gamma^+\psi(0),\qquad O_\gamma(x)=\frac1{P^+}\int\!\frac{d\xi^-}{2\pi}\,e^{-i xP^+\xi^-}\,F^{+}{}_{\alpha}(\xi^-)\,F^{+\alpha}(0),$$9: $f_{i/\ell}(\xi,\mu^2),$0, $f_{i/\ell}(\xi,\mu^2),$1, $f_{i/\ell}(\xi,\mu^2),$2,
• $f_{i/\ell}(\xi,\mu^2),$3: $f_{i/\ell}(\xi,\mu^2),$4, $f_{i/\ell}(\xi,\mu^2),$5, $f_{i/\ell}(\xi,\mu^2),$6,
• $f_{i/\ell}(\xi,\mu^2),$7: $f_{i/\ell}(\xi,\mu^2),$8, $f_{i/\ell}(\xi,\mu^2),$9, $\ell$0. The evolved grids are distributed in LHAPDF format for $\ell$1 and $\ell$2 (Qiu et al., 8 Jul 2026).

On the implementation side, APFEL stores PDFs on an $\ell$3-grid, performs convolutions with Gauss-Legendre quadrature and high-order interpolation in $\ell$4, solves the coupled QCD+QED equations simultaneously, runs $\ell$5 at LO, and implements smooth heavy-flavor and heavy-lepton matching across charm, bottom, and $\ell$6 thresholds (Bertone et al., 2015). In the broader Standard Model extension, LePDF provides public LHAPDF6 files for both muons and electrons, including polarization effects (Garosi et al., 2023).

5. Fixed-order structure functions through two loops

A substantial part of the literature concerns explicit perturbative calculations of lepton-parent PDFs. For the electron, the fixed-order expansion is written as

$\ell$7

with

$\ell$8

At NLO, using $\ell$9, one finds

$i=e,\bar e,\gamma,q,\bar q,g$0

At NNLO, the result contains a one-flavor contribution $i=e,\bar e,\gamma,q,\bar q,g$1 and a genuinely new extra-flavor contribution $i=e,\bar e,\gamma,q,\bar q,g$2 containing logarithms and dilogarithms of the mass ratio $i=e,\bar e,\gamma,q,\bar q,g$3 (Stahlhofen, 23 Aug 2025).

The fully analytic two-loop calculation extends this program to all five QED lepton/photon channels: $i=e,\bar e,\gamma,q,\bar q,g$4 The computation is performed in $i=e,\bar e,\gamma,q,\bar q,g$5-space with reduction to master integrals and the differential-equation method, using $i=e,\bar e,\gamma,q,\bar q,g$6 renormalization for wave functions and charge and on-shell renormalization for the mass (Schnubel et al., 11 Sep 2025). The results are organized as

$i=e,\bar e,\gamma,q,\bar q,g$7

with the coefficients of the large logarithms $i=e,\bar e,\gamma,q,\bar q,g$8 controlled by the renormalization-group convolution relations. The one-loop terms reproduce the results of Frixione and Llauret, while the two-loop $i=e,\bar e,\gamma,q,\bar q,g$9-space expressions for $i=\ell,\bar\ell,\gamma,q,\bar q,g$00, $i=\ell,\bar\ell,\gamma,q,\bar q,g$01, and $i=\ell,\bar\ell,\gamma,q,\bar q,g$02 agree pointwise with the recent SCET calculation (Schnubel et al., 11 Sep 2025).

The calculational frameworks are complementary. The SCET analysis emphasizes the operator definition and renormalization-group origin of the $i=\ell,\bar\ell,\gamma,q,\bar q,g$03 towers, making DGLAP-type resummation of $i=\ell,\bar\ell,\gamma,q,\bar q,g$04, $i=\ell,\bar\ell,\gamma,q,\bar q,g$05, and higher towers straightforward (Stahlhofen, 23 Aug 2025). The direct two-loop analytic calculation emphasizes IBP reduction, differential equations in $i=\ell,\bar\ell,\gamma,q,\bar q,g$06, boundary conditions from inclusive integrals, and the cancellation of spurious rapidity divergences in the sum of graphs (Schnubel et al., 11 Sep 2025).

The available numerical statements are limited but definite. The SCET paper states that for one-flavor QED at a typical scale $i=\ell,\bar\ell,\gamma,q,\bar q,g$07, with $i=\ell,\bar\ell,\gamma,q,\bar q,g$08 and $i=\ell,\bar\ell,\gamma,q,\bar q,g$09, the NLO corrections $i=\ell,\bar\ell,\gamma,q,\bar q,g$10 amount to a few percent of the Born $i=\ell,\bar\ell,\gamma,q,\bar q,g$11 term, the genuine NNLO corrections are at the per-mille level, and additional fermion-flavor effects can shift the NNLO result by up to $i=\ell,\bar\ell,\gamma,q,\bar q,g$12 of the NNLO term itself (Stahlhofen, 23 Aug 2025).

6. Phenomenology, extraction strategies, and recurring misconceptions

The phenomenology of QED lepton PDFs depends strongly on the parent state. For charged leptons inside the proton, the numerical impact is generally very small. In the APFEL-based study, representative evolved proton PDFs at $i=\ell,\bar\ell,\gamma,q,\bar q,g$13 GeV yield lepton distributions suppressed by a factor $i=\ell,\bar\ell,\gamma,q,\bar q,g$14 relative to the photon PDF, with momentum fractions

$i=\ell,\bar\ell,\gamma,q,\bar q,g$15

so that the lepton momentum is two orders of magnitude below the photon momentum (Bertone et al., 2015). Accordingly, in hadronic phenomenology the $i=\ell,\bar\ell,\gamma,q,\bar q,g$16-initiated channel is stated to be almost always negligible once realistic cuts are applied, whereas photon-initiated processes can be $i=\ell,\bar\ell,\gamma,q,\bar q,g$17 or more of standard quark-antiquark interactions at multi-TeV scales (Bertone et al., 2015). This addresses a common misconception: the formal necessity of including lepton PDFs in a complete LO-QED PDF basis does not imply that they are numerically competitive with photon PDFs in ordinary hadron-collider observables.

For a parent muon, the partonic content can nevertheless be sizable enough to matter in lepton-collider applications. Frixione and Stagnitto find at $i=\ell,\bar\ell,\gamma,q,\bar q,g$18 GeV the momentum fractions

$i=\ell,\bar\ell,\gamma,q,\bar q,g$19

and use these PDFs for dijet cross sections at a $i=\ell,\bar\ell,\gamma,q,\bar q,g$20 TeV $i=\ell,\bar\ell,\gamma,q,\bar q,g$21 collider (Frixione et al., 2023). A plausible implication is that once QCD is radiatively induced from the photon component, quark and gluon densities become relevant for specific collider final states even though the parent state is elementary.

In lepton-hadron scattering, the phenomenological status is different again. The NLO factorized-QED DIS analysis compares three scenarios—LO-NR, LO-Pert, and LO-Model+NLO—and finds that including QED LDFs and LFFs typically reduces the cross section by up to $i=\ell,\bar\ell,\gamma,q,\bar q,g$22–$i=\ell,\bar\ell,\gamma,q,\bar q,g$23 relative to LO-NR, depending on kinematics, while the NLO hard-part correction induces further changes at the few-percent to ten-percent level (Cammarota et al., 2024). The study concludes that a precision extraction of hadron PDFs from DIS at the few-percent level will require simultaneous extraction of the nonperturbative LDFs $i=\ell,\bar\ell,\gamma,q,\bar q,g$24, $i=\ell,\bar\ell,\gamma,q,\bar q,g$25, and related functions (Cammarota et al., 2024).

The most explicit extraction strategy has been formulated for high-$i=\ell,\bar\ell,\gamma,q,\bar q,g$26 single-inclusive hadron and jet production at Jefferson Lab and the future Electron-Ion Collider. In that proposal, the universal LDFs $i=\ell,\bar\ell,\gamma,q,\bar q,g$27 are to be constrained from high-$i=\ell,\bar\ell,\gamma,q,\bar q,g$28 single-inclusive hadron and jet measurements without imposing “radiative” cuts on the final-state lepton. Comparing measured spectra $i=\ell,\bar\ell,\gamma,q,\bar q,g$29 with theory for varying LDF parameters would permit a global fit in close analogy to hadron PDF and fragmentation-function analyses, thereby extracting nonperturbative LDFs at $i=\ell,\bar\ell,\gamma,q,\bar q,g$30 and evolving them perturbatively to any scale (Qiu et al., 8 Jul 2026). The same work states that, once determined, these LDFs will provide model-independent, leading-power QED “radiative corrections” for all lepton-initiated processes in the Standard Model and beyond (Qiu et al., 8 Jul 2026).

A second recurring misconception concerns the universality of lepton-side radiation. The joint-factorization program argues that sufficiently inclusive observables admit the same kind of universal collinear factorization on the lepton side as on the hadron side, with infrared-safe hard parts after subtraction (Cammarota et al., 2024, Qiu et al., 8 Jul 2026). This does not eliminate the need for input conditions: in purely perturbative settings the boundary conditions are fixed near the lepton mass, while in mixed QCD+QED settings involving quark and gluon content of a parent lepton, a nonperturbative input at $i=\ell,\bar\ell,\gamma,q,\bar q,g$31 is explicitly introduced and must ultimately be constrained by data (Qiu et al., 8 Jul 2026).