⚛️ phenomenology

Higher order perturbative and nonperturbative QCD corrections on the proton structure functions and parity violating electron asymmetry

This paper investigates the impact of higher-order perturbative (up to NNLO) and nonperturbative (target mass and higher-twist) QCD corrections on proton structure functions and parity-violating electron asymmetry, providing numerical results for JLab energies to aid future analyses at the Electron Ion Collider and EicC.

Original authors: F. Zaidi, M. Sajjad Athar, S. K. Singh

Published 2026-03-17

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: F. Zaidi, M. Sajjad Athar, S. K. Singh

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 the proton not as a solid marble, but as a bustling, chaotic city inside a tiny bubble. This city is filled with tiny citizens called quarks and gluons (the glue holding them together). Scientists want to understand the "traffic patterns" of this city: How many up-quarks are there? How many down-quarks? How do they share the proton's momentum?

To map this city, scientists shoot high-speed electrons at protons. Sometimes, these electrons bounce off purely via electricity (like magnets repelling). Other times, they interact via the "weak force" (a different, more subtle kind of interaction). By comparing how the electrons bounce in these two different ways, scientists can measure a tiny difference called Parity Violating Asymmetry. Think of this asymmetry as a tiny "tilt" in the data that reveals the hidden secrets of the quark city.

However, there's a problem. The math used to predict these bounces is like a recipe. For a long time, scientists used a "Basic Recipe" (Leading Order) that assumed the city was perfectly still and the citizens didn't interact much. But in reality, the city is noisy, the citizens are running around, and the bubble itself has mass.

This paper is essentially a team of physicists (F. Zaidi, M. Sajjad Athar, and S. K. Singh) saying: "Let's upgrade our recipe." They are adding "extra ingredients" to their calculations to make the prediction match reality much better.

Here is a breakdown of what they did, using everyday analogies:

1. The "Extra Ingredients" (The Corrections)

The authors added three main types of "seasoning" to their mathematical soup to make it taste more like the real world:

The "Traffic Jams" (Higher Order Perturbative Corrections):
Imagine the quarks are cars on a highway. The basic recipe assumes they drive in straight lines without talking to each other. But in reality, they honk, swerve, and interact (quarks talk to gluons). The authors calculated these interactions up to the "Next-Next-to-Leading Order" (NNLO).
- Analogy: It's the difference between predicting traffic flow by assuming everyone drives at a constant speed versus accounting for red lights, accidents, and lane changes. The more complex the calculation, the more accurate the prediction.
The "Heavy Backpack" (Target Mass Corrections - TMC):
In the simplest math, scientists often pretend the proton is weightless or that the electron is hitting a stationary, massless target. But the proton has a real, heavy mass. When the electron hits it, the proton wobbles.
- Analogy: Imagine throwing a tennis ball at a ping-pong ball (weightless) versus a bowling ball (heavy). The way the ball bounces back is totally different. The authors corrected for the fact that the proton is a "bowling ball," not a ping-pong ball. This is crucial when the electron doesn't have infinite energy.
The "Crowded Room" (Higher Twist Effects - HT):
The basic recipe assumes quarks act like individuals. But sometimes, they act as a group. A "Higher Twist" effect is like a group of friends in a crowded room pushing against each other simultaneously.
- Analogy: If you try to push one person in a crowd, they move easily. If you try to push a group holding hands, it's much harder. These "group effects" become important when the electron hits the proton at specific angles and energies.

2. The "Broken Rule" (Callan-Gross Relation)

For decades, physicists relied on a rule called the Callan-Gross relation. It was like a law of physics that said: "If you know how the proton moves sideways, you automatically know how it moves forward."

The authors found that this rule breaks down when you get into the "messy" parts of the proton (specifically at high energies and specific angles).

Analogy: It's like a rule that says "If a car is moving fast, its wheels must be spinning at a specific speed." This works on a smooth highway, but if the car is driving off-road in mud (the non-perturbative region), the wheels might spin differently than the rule predicts. The authors showed exactly how and when this rule breaks, which is vital for accurate measurements.

3. The Goal: The "d/u Ratio"

Why do all these calculations matter? The ultimate goal is to find the d/u ratio.

u = Up quarks
d = Down quarks
Ratio = How many down quarks are there compared to up quarks?

In the "Basic Recipe," this ratio is easy to guess. But with the new "seasoning" (TMC and HT), the ratio changes, especially when looking at the "edge" of the proton (high x values).

Analogy: Imagine trying to count the number of red and blue marbles in a jar. If you just shake the jar gently (basic recipe), you get a rough count. But if you shake it violently (high energy) and the marbles clump together (higher twist), your count changes. The authors are refining the count so that future experiments get the exact number right.

4. Why This Matters for the Future

The paper concludes that these "extra ingredients" are small for some experiments but huge for others, specifically for upcoming giant machines like:

JLab (Jefferson Lab): A lab in the US.
EIC (Electron Ion Collider): A massive new collider being built in the US.
EicC: A similar collider being built in China.

These machines will shoot electrons at protons with incredible precision. If scientists use the "Basic Recipe" to analyze the data, they will get the wrong answer. The authors are essentially providing the updated instruction manual for these future experiments.

Summary

Think of this paper as a team of expert mechanics upgrading the engine of a race car.

The Car: The proton.
The Race: The collision of electrons and protons.
The Old Engine: Simple math that ignores friction, weight, and group dynamics.
The New Engine: Complex math that accounts for the proton's weight, the "traffic" of quarks, and their "group hugs."

By fixing the engine, they ensure that when the new race cars (EIC and EicC) hit the track, the data they collect will tell us the true story of what the proton is made of, rather than just a rough approximation. This is a crucial step toward understanding the fundamental building blocks of our universe.

1. Problem Statement

The paper addresses the theoretical precision required for interpreting future high-energy experiments (JLab 12 GeV, EIC, EicC) involving the Parity Violating (PV) electron asymmetry ( $A_{PV}^{(e)}$ ) in deep inelastic scattering (DIS) of longitudinally polarized electrons off unpolarized protons.

Core Challenge: Extracting fundamental parameters like the weak mixing angle ( $\theta_W$ ) and the quark flavor ratio ( $d/u$ ) from $A_{PV}^{(e)}$ requires precise knowledge of proton structure functions ( $F_{1,2,3}^{\gamma}, F_{1,2,3}^{\gamma Z}$ ).
Limitations of Current Models: Standard Leading Order (LO) Quark-Parton Model (QPM) calculations assume Bjorken scaling ( $Q^2 \to \infty$ $Q^{2} \to \infty$ ). However, current and future experiments operate at finite $Q^2$ $Q^{2}$ (typically $1 \le Q^2 \le 20$ $1 \leq Q^{2} \leq 20$ GeV $^2$ $^{2}$ ) and high $x$ $x$ ( $0.1 \le x \le 0.8$ $0.1 \leq x \leq 0.8$ ). In this regime, the Callan-Gross relation ( $2xF_1 = F_2$ $2 x F_{1} = F_{2}$ ) is violated due to:
1. Higher-order perturbative QCD corrections (NLO, NNLO).
2. Nonperturbative effects: Target Mass Corrections (TMC) and Higher Twist (HT, specifically twist-4) effects arising from multi-parton correlations.
Goal: To quantify the impact of these corrections on structure functions and the resulting PV asymmetry to assess their feasibility for extracting $d/u$ ratios and $\theta_W$ .

2. Methodology

The authors employ a comprehensive theoretical framework combining perturbative QCD evolution with nonperturbative corrections.

Formalism:
- The differential cross-section is derived for polarized electron scattering, separating contributions from pure photon exchange ( $\gamma$ ), pure $Z$ -boson exchange, and $\gamma-Z$ interference.
- The PV asymmetry $A_{PV}^{(e)}$ is expressed as a ratio of helicity-dependent cross-sections, dominated by the $\gamma-Z$ interference term.
- The Callan-Gross relation is generalized to include a longitudinal structure function $F_L$ , defined via the ratio $R = F_L / 2xF_1$ .
Parton Distribution Functions (PDFs):
- Uses the MMHT14 parameterization in the $\overline{\text{MS}}$ scheme.
- PDFs are evolved up to Next-to-Next-to-Leading Order (NNLO) using splitting functions from Vermaseren et al. and Moch et al.
Corrections Incorporated:
- Perturbative: Calculations performed at LO, NLO, and NNLO.
- Target Mass Corrections (TMC): Implemented following the prescription of Schienbein et al., accounting for finite nucleon mass effects ( $M^2/Q^2$ terms).
- Higher Twist (HT): Twist-4 effects included following Dasgupta et al. and Stein et al., representing multi-parton correlations ( $1/Q^2$ terms).
Kinematic Range:
- $0.1 \le x \le 0.8$
- $1 \le Q^2 \le 20$ GeV $^2$
- Beam energies: $E = 6, 12, 22$ GeV (relevant for JLab, EIC, and EicC).
- Beam polarization: $|P_e| = 0.8$ .

3. Key Contributions

Systematic Quantification: The first comprehensive study to simultaneously include NLO/NNLO perturbative corrections, TMC, and HT effects on both electromagnetic ( $F^\gamma$ ) and weak-interference ( $F^{\gamma Z}$ ) structure functions.
Callan-Gross Violation Analysis: Detailed analysis of the ratio $r = F_2 / 2xF_1$ and $R = F_L / 2xF_1$ across the kinematic range, demonstrating how nonperturbative effects drive the violation of the Callan-Gross relation at finite $Q^2$ .
Impact on $d/u$ Ratio: Explicitly links the corrected structure functions to the extraction of the $d(x)/u(x)$ quark distribution ratio, a critical parameter for understanding nucleon structure.
Benchmarking: Provides numerical benchmarks for future experiments (JLab SoLID, EIC, EicC) by comparing LO, NLO, and NNLO results with and without nonperturbative corrections.

4. Key Results

A. Structure Functions ( $F_{1,2,3}$ )

Perturbative Effects: NLO corrections are small at low $x$ but become significant at high $x$ and low $Q^2$ . NNLO corrections are generally smaller than NLO but non-negligible.
Target Mass Corrections (TMC):
- TMC significantly enhances $F_1$ and $F_2$ at high $x$ ( $x > 0.5$ ) and low $Q^2$ .
- At $Q^2 = 2$ GeV $^2$ and $x=0.7$ , TMC enhances $F_1$ by ~78% and $F_2$ by ~26% relative to NLO.
- For the parity-violating structure function $xF_3$ , TMC causes a reduction at low $x$ and enhancement at high $x$ .
Higher Twist (HT):
- HT effects are most pronounced at high $x$ ( $x > 0.6$ ) and low $Q^2$ .
- In $F_3$ , HT effects lead to significant reductions (~16% at $x=0.3$ ) or enhancements depending on the specific $x$ and $Q^2$ .
- Equivalence: The authors find that NLO + TMC + HT results are often numerically similar to NNLO + TMC results in the low-to-intermediate $x$ region, suggesting that including nonperturbative effects at lower perturbative orders can mimic higher-order perturbative terms.

B. Callan-Gross Relation Violation

The ratio $r = F_2 / 2xF_1$ deviates significantly from unity (the CG limit) at low $x$ and low $Q^2$ .
TMC and HT play a crucial role: TMC reduces $r$ at intermediate-to-high $x$ , while HT enhances it.
The violation is quantitatively different for electromagnetic ( $\gamma$ ) and weak-interference ( $\gamma Z$ ) sectors, though the qualitative behavior is similar.

C. Parity Violating Asymmetry ( $A_{PV}^{(e)}$ )

Magnitude: $A_{PV}^{(e)}$ is negative, and its absolute value increases with beam energy.
Correction Impact:
- NLO/NNLO: Corrections are small (< 6%) across most of the $x$ range.
- TMC: Significant only at high $x$ ( $x \gtrsim 0.6$ ) and low beam energies (e.g., 6 GeV). At $x=0.6$ , TMC changes the asymmetry by ~6% at 6 GeV, dropping to ~3% at 22 GeV.
- HT: Becomes significant at very high $x$ ( $x \approx 0.8$ ). At $x=0.8$ , HT adds a ~5-7% correction relative to TMC-only results.
Conclusion: For most kinematic regions, perturbative corrections dominate, but nonperturbative effects are essential for precision at high $x$ .

D. The $d/u$ Ratio

The ratio $a_{1p}(x, Q^2) \propto F_1^{\gamma Z} / F_1^{\gamma}$ is directly sensitive to the $d/u$ ratio.
Sensitivity to Corrections:
- Including NLO terms reduces the extracted $d/u$ ratio by up to 48% at $x=0.8$ compared to LO.
- Including TMC enhances the $d/u$ ratio significantly (e.g., +73% at $x=0.8$ for $Q^2=2$ GeV $^2$ ) relative to NLO without TMC.
- HT effects on $d/u$ are found to be negligible.
Implication: Ignoring TMC leads to a severe underestimation of the $d/u$ ratio at high $x$ .

5. Significance and Conclusion

Experimental Relevance: The study provides essential theoretical inputs for upcoming experiments at JLab, EIC, and EicC. It demonstrates that Target Mass Corrections are non-negligible for precision measurements of the $d/u$ ratio and weak mixing angle, particularly in the high- $x$ region.
Theoretical Insight: The work highlights that the violation of the Callan-Gross relation is driven by both perturbative scaling violations and nonperturbative mass/twist effects.
Future Outlook: The results suggest that to accurately extract fundamental parameters from future high-precision data, theoretical analyses must go beyond LO and include a consistent treatment of TMC and HT, or equivalently, utilize NNLO perturbative calculations with TMC. The study confirms that parity-violating DIS on proton targets is a viable, "clean" method (free from nuclear effects) to determine the $d/u$ ratio and test isospin symmetry violation.