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Single inclusive hadron and jet production in lepton-hadron scattering

This paper presents the first calculation of single inclusive hadron and jet production at large transverse momentum in lepton-hadron scattering using a joint QCD+QED factorization framework, which introduces universal lepton distribution functions with combined evolution kernels and provides predictions for experiments at Jefferson Lab and the future Electron-Ion Collider.

Original authors: Jian-Wei Qiu, Kazuhiro Watanabe

Published 2026-07-09

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Jian-Wei Qiu, Kazuhiro Watanabe

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand the internal structure of a proton (a tiny building block of matter) by smashing a high-speed electron into it. This is like firing a bullet at a complex machine to see how the gears inside are arranged. In the past, scientists have been very good at this, but they usually ignored a specific problem: when the electron hits the proton, it doesn't just bounce off cleanly. It often emits a flash of light (a photon) or interacts with other particles in a way that creates a "cloud" of radiation.

For a long time, scientists treated the "bullet" (the electron) as a perfect, solid object and only worried about the "machine" (the proton) being messy. This paper changes the rules. It argues that we must treat the electron just as carefully as the proton, acknowledging that the electron itself can break apart into a cloud of particles (like a photon turning into a quark-antiquark pair) before it even hits the target.

Here is a breakdown of what the authors did, using simple analogies:

1. The New "Joint" Rulebook

Think of the laws of physics governing these collisions as two different rulebooks: one for the strong force (QCD, which holds the proton together) and one for electromagnetism (QED, which governs the electron and light).

Old Way: Scientists used the QCD rulebook for the proton and the QED rulebook for the electron, but they kept them separate. They tried to fix the electron's messiness later with "patches" (radiative corrections).
New Way (This Paper): The authors created a joint rulebook (QCD+QED). They realized that the electron and the proton are part of the same messy system. They developed a new mathematical framework where the electron's "cloud" is treated as a universal ingredient, just like the proton's internal parts.

2. The "Lepton Distribution Function" (LDF)

In the old days, we had a map called a "Parton Distribution Function" (PDF) that told us where the quarks were inside a proton.

The New Map: The authors created a new map called the Lepton Distribution Function (LDF). This map tells us the probability of finding an electron, a photon, or even a quark inside the incoming electron beam.
The Analogy: Imagine you are looking at a flashlight beam. You used to think the beam was just pure light. This paper says, "Actually, if you look closely, the beam is a mix of light, some heat, and even tiny sparks." The LDF is the map that tells you exactly how much of each is in the beam at any given moment.

3. The "Recipe" for Predictions

To predict what happens when these particles collide, you need a recipe.

Ingredients: You need the map of the proton (PDF), the map of the electron (LDF), and a map of how particles turn into the final products (Fragmentation Functions).
The Cooking Process: The authors showed how to "cook" these maps together. They proved that these maps change (evolve) as the energy of the collision changes, and they calculated exactly how the electron's map changes when you mix the strong force and electromagnetic force rules.

4. What They Actually Calculated

The authors didn't just write theory; they ran the numbers for two specific places:

Jefferson Lab (JLab): A current facility in Virginia.
The Electron-Ion Collider (EIC): A future, massive collider being built.

They calculated the production of two things:

Single Hadrons: Like a single pion or kaon flying out of the collision.
Jets: A spray of particles that looks like a cone of debris, which acts like a "jet" of energy.

5. Key Findings (The "Flavor" of the Results)

The Electron Matters More Than You Thought: When they included the new "electron map" (LDF) in their calculations, the results changed significantly. For high-energy collisions, the electron's internal cloud reduces the chance of finding a high-energy electron, which changes the final outcome of the crash by up to 50% in some cases.
The "Photon" Surprise: They found that the electron often acts like a source of real photons (light particles) that then hit the proton. This "photoproduction" is a natural part of the process and doesn't need special experimental cuts to separate it.
Uncertainty in the "Recipe": They discovered that the biggest uncertainty in their predictions doesn't come from the electron or the math, but from the Fragmentation Functions.
- Analogy: Imagine you know exactly how to mix the ingredients (the electron and proton), but you aren't sure exactly how the final cake will look when it comes out of the oven. The "cake" is the final particle (like a pion). Different recipes for the cake give different results, and this is currently the biggest source of error.
Nuclear Effects: They also looked at what happens if the target is a heavy nucleus (like Lead) instead of a single proton. They found that the "cloud" of particles inside the nucleus changes the results, acting as a probe to study how nuclear matter behaves under pressure.

Summary

This paper is the first to provide a complete, unified mathematical recipe for smashing electrons into protons where the electron's own internal messiness is treated with the same seriousness as the proton's. They provided the first set of "maps" (LDFs) for the electron and used them to predict what we will see at current and future particle accelerators. They showed that ignoring the electron's internal structure leads to big errors, and that understanding the final "cake" (the fragmentation of particles) is the next big challenge for physicists.

Technical Summary: Single Inclusive Hadron and Jet Production in Lepton-Hadron Scattering

Problem Statement
The paper addresses the theoretical challenge of describing single inclusive hadron and jet production at large transverse momentum ( $P_T$ ) in lepton-hadron scattering ( $e(\ell) + h(p) \to H(P) + X$ ). While Deep Inelastic Scattering (DIS) has historically been the primary tool for probing hadron structure, large momentum transfer in these collisions inevitably generates both Quantum Chromodynamics (QCD) and Quantum Electrodynamics (QED) collision-induced radiation. Traditional approaches often treat QED radiative corrections (RC) as separate, process-dependent perturbations or rely on experimental cuts to isolate "photoproduction" events. However, recent developments have identified pinch-singular regions where the exchanged virtual photon can go on-shell ( $Q^2 \to 0$ ), complicating the definition of the hard probe. Without a unified framework, these collision-induced effects can obscure the true partonic structure of the hadron and introduce uncertainties in determining the hard probe's momentum. The specific problem is to formulate a consistent, infrared-safe calculation for high- $P_T$ hadron and jet production that treats QED and QCD radiation on equal footing, thereby enabling the extraction of universal Lepton Distribution Functions (LDFs) and testing the joint factorization approach.

Methodology
The authors employ a joint QCD+QED collinear factorization approach. The scattering cross section is factorized into a convolution of infrared-safe hard coefficient functions with universal distribution functions:

Lepton Distribution Functions (LDFs): $f_{i/e}(\xi, \mu^2)$ , describing the probability of finding a parton $i$ (lepton, photon, quark, gluon) with momentum fraction $\xi$ inside the colliding lepton.
Parton Distribution Functions (PDFs): $f_{b/h}(x, \mu^2)$ , for the target hadron.
Fragmentation Functions (FFs): $D_{c \to H}(z, \mu^2)$ , for the observed hadron (or Jet Functions for jets).

Key methodological steps include:

Factorization Formalism: The cross section is derived to be independent of the factorization scale $\mu$ , leading to DGLAP-type evolution equations for LDFs, PDFs, and FFs. Crucially, the evolution kernels ( $P_{ij}$ ) are calculated in both QCD and QED, including mixing terms (e.g., a photon splitting into a quark-antiquark pair, or a lepton radiating a photon that splits into quarks).
LDF Evolution Strategy: Since existing experimental data have already been processed with traditional RCs, they cannot be directly used to extract LDFs without re-analysis. Consequently, the authors derive a default set of model LDFs at an input scale $\mu_0 = m_c$ (charm quark mass). This scale is chosen to ensure the perturbative reliability of QCD splitting functions (avoiding non-perturbative photon-to-light-quark splitting at lower scales).
Input Parameterization: At $\mu_0 = m_c$ , the valence electron distribution is approximated using Next-to-Leading Order (NLO) perturbative QED results, while sea quarks, gluons, and the photon distribution are assumed to be perturbatively generated from the evolution equations starting from $\mu = m_e$ . The functional forms are parameterized to satisfy electron number and momentum conservation sum rules.
Numerical Implementation: The authors calculate cross sections for single inclusive $\pi^+$ and $K^+$ production, as well as inclusive jet production, at Jefferson Lab (JLab) and future Electron-Ion Collider (EIC) energies. They utilize available sets of hadron PDFs (JAM20, CT18) and FFs (JAM20, MAP1.0) to assess uncertainties. For jets, they employ Jet Functions (JFs) up to NLO accuracy with the anti- $k_T$ clustering algorithm.

Key Contributions

First Calculation: This paper presents the first calculation of single inclusive hadron and jet production at large $P_T$ in lepton-hadron scattering within a joint QCD+QED factorization framework.
Unified Treatment of Radiation: The work demonstrates that "leptoproduction" and "photoproduction" (Direct and Resolved photon contributions) are naturally unified in this formalism. No experimental cuts on the scattered lepton are required to separate these contributions, eliminating uncertainties associated with arbitrary kinematic cuts.
Derivation of Default LDFs: The authors provide a self-consistent, default set of LDFs evolved from $\mu_0 = m_c$ using mixed QCD+QED kernels. This set serves as a baseline for future phenomenological studies and data analysis.
Quantification of QED Effects: The paper quantifies the impact of leading-power QED radiative corrections. It shows that these corrections, encoded in the LDFs, significantly alter the $P_T$ spectrum of produced hadrons and jets.
Nuclear Dependence: The formalism is extended to lepton-nucleus collisions to study cold nuclear matter effects (nPDFs), specifically investigating nuclear modification factors ( $R_{A/D}$ ) for various nuclei (He, C, Al, Fe, Pb).

Results

Hadron Production ( $P_T$ Spectra): At JLab energies, the cross section is dominated by leptoproduction at high $P_T$ , while photoproduction dominates at lower $P_T$ . At EIC energies, photoproduction remains a significant contributor even at higher $P_T$ . The inclusion of QED radiative corrections (via evolved LDFs) reduces the cross section at high $P_T$ by up to 50% or more compared to calculations without these corrections (where LDFs are set to $\delta(1-\xi)$ ), while enhancing the cross section at lower $P_T$ .
Scale and FF Uncertainties: Theoretical uncertainties are found to be dominated by the choice of Fragmentation Functions (FFs) rather than the renormalization/factorization scale. Comparisons between JAM20 and MAP1.0 FF sets show significant deviations (ratios far from unity), particularly for $\pi^+$ and $K^+$ production, highlighting the sensitivity of high- $P_T$ hadron production to the large- $z$ region of FFs.
Jet Production: For inclusive jet production, the NLO Jet Function corrections reduce the total cross section by approximately 50% for typical EIC jet cone sizes ( $R \sim 1$ ). QED radiative corrections further deplete the high- $P_T$ jet cross section by 10–20%.
Nuclear Effects: In lepton-nucleus collisions, the nuclear modification factor exhibits suppression at moderate $P_T$ (attributed to the EMC effect) and enhancement at high $P_T$ (attributed to Fermi motion). The magnitude of these effects depends strongly on the choice of nuclear PDF (nPDF) sets (EPPS21, nCTEQ15, nNNPDF, TUJU21), confirming that high- $P_T$ hadron production is a sensitive probe for constraining nPDFs in the large- $x$ region.

Significance and Claims
The paper claims that the joint QCD+QED factorization approach provides a systematic and predictive framework for lepton-hadron scattering. By treating collision-induced QED radiation as a universal, process-independent effect absorbed into LDFs, the approach eliminates the need for ad-hoc experimental cuts to define "photoproduction."

The authors assert that:

The derived default LDFs allow for the first quantitative predictions of single inclusive hadron and jet production that include leading-power QED effects.
High- $P_T$ hadron production in lepton-hadron collisions is a complementary and crucial observable for extracting universal FFs, particularly in the large- $z$ region, which is less constrained by $e^+e^-$ annihilation data.
The process serves as a powerful probe for nuclear PDFs, especially in the EMC region, offering a method to constrain nPDFs that is complementary to inclusive DIS.
Once universal LDFs are extracted from future precise data, the size of QED radiative corrections in lepton-hadron collisions will be uniquely fixed, enhancing the precision of future studies at JLab, the EIC, and potentially the LHC (via lepton distributions of hadrons).

The work lays the groundwork for future global analyses of LDFs, FFs, and nPDFs, and suggests extensions to polarized DIS, parity-violating DIS, and heavy flavor production as critical next steps for the EIC science program.