Global Analyses of Generalized Parton Distributions with Diverse PDF Inputs

This paper presents six global analyses of generalized parton distributions (GPDs) at zero skewness using various modern PDF sets and factorization scales, demonstrating that NNPDF40 at NLO provides the best fit to elastic electron scattering data while revealing moderate sensitivity to PDF inputs and perturbative orders.

The Gist

Imagine the proton (the tiny particle inside an atom's nucleus) not as a solid marble, but as a bustling, chaotic city filled with millions of tiny citizens called partons (quarks and gluons). These citizens are constantly zooming around, carrying different amounts of energy and momentum.

To understand how this city works, physicists use two main maps:

1. PDFs (Parton Distribution Functions): A map showing how these citizens are distributed based on how much energy they carry.
2. GPDs (Generalized Parton Distributions): A much more detailed 3D map. It doesn't just show how much energy a citizen has, but also where they are located inside the city and how they move when the city is jostled.

The Big Problem: The "Map Scale" Issue

The paper tackles a tricky problem with these maps. The PDF maps are usually drawn for a specific "zoom level" (called the energy scale, $\mu$). Think of it like a street map:

• High Zoom (High Energy): You see the highways and major roads clearly.
• Low Zoom (Low Energy): You see the tiny alleyways and backstreets.

Usually, scientists have high-quality maps for the "high zoom" levels. But sometimes, to study specific low-energy experiments, they try to use a high-zoom map and just "zoom out" (mathematically evolve it backward) to see the low-energy details.

The authors discovered a trap: If you try to use a map designed for a high zoom level and force it to work at a lower zoom level than it was built for, the map starts to glitch. The "citizen count" stops making sense (violating conservation laws), and the predictions become unphysical.

What They Did: The "Six-Map" Experiment

To fix this and find the best way to draw the 3D GPD map, the team (The MMGPDs Collaboration) ran six different simulations. They used three different, modern "PDF map sets" (NNPDF40, CT18, and MSHT20) and tested them at three different "zoom levels" (2 GeV, 1.3 GeV, and 1 GeV).

They compared all these simulations against a massive collection of real-world data from electron scattering experiments (basically, shooting electrons at protons to see how they bounce off).

The Key Findings (In Plain English)

1. The "Best" Map
They found that the NNPDF40 map set, used at a specific zoom level (2 GeV), gave the most accurate picture of reality. It matched the experimental data better than the others. Interestingly, making the math more complex (using "Next-to-Next-to-Leading Order" calculations) didn't necessarily make the map better; sometimes it even made it slightly worse. This suggests the current methods are already quite stable.

2. The "Backward Evolution" Warning
The study confirmed that you cannot just take a map built for high energy and blindly use it for low energy. If you try to use a map below its "minimum zoom" (like using the NNPDF40 map at 1 GeV when it was built for 1.65 GeV), the math breaks down. The "citizen count" (quark numbers) becomes wrong. It's like trying to use a satellite photo to navigate a tiny alleyway; the resolution is wrong, and you'll get lost.

3. The "Up" vs. "Down" Quark Mystery
The proton is made mostly of "Up" and "Down" quarks.

• Up Quarks: Their distribution was consistent across the different maps.
• Down Quarks: These were much harder to pin down. The maps disagreed more on where the Down quarks were, especially when the proton was jostled hard (high momentum transfer).
• The Twist: As the "jostling" got harder, the Down quarks seemed to disappear (get suppressed) much faster than the Up quarks. This suggests our understanding of the Down quark's role inside the proton might need a rewrite.

Why This Matters

This paper is like a quality control check for the tools physicists use to understand the universe's building blocks.

• For Scientists: It provides six different, reliable sets of GPD maps to use for future research, depending on what specific question they are asking.
• For the Field: It warns everyone: "Don't use these maps outside their intended range, or your results will be garbage."

In a nutshell: The authors built better 3D models of the proton's interior by carefully choosing the right "base maps" (PDFs) and the right "zoom level." They found that while the Up quarks are well-understood, the Down quarks are still a bit of a mystery, and they warned everyone not to use high-energy maps for low-energy problems without doing the math correctly first.

Technical Summary

1. Problem Statement

The paper addresses two critical issues in the phenomenological extraction of Generalized Parton Distributions (GPDs) at zero skewness ($\xi = 0$):

1. Dependence on PDF Inputs: GPDs at the forward limit ($\xi \to 0, t \to 0$) reduce to standard Parton Distribution Functions (PDFs). However, the sensitivity of extracted GPDs to the choice of modern PDF sets (NNPDF40, CT18, MSHT20) and their associated factorization scales ($\mu$) has not been systematically investigated.
2. Validity of Backward Evolution: A common practice involves using PDFs at energy scales ($\mu$) lower than their initial parametrization scale ($\mu_0$). The authors investigate whether this "backward evolution" leads to unphysical results, specifically testing the preservation of quark number sum rules when PDFs are evaluated below their nominal input scales.

2. Methodology

The authors performed six global QCD analyses of GPDs at zero skewness using a consistent phenomenological framework but varying the input PDF sets and perturbative orders.

• Data Set: The analysis utilized a wide range of elastic electron scattering data, including:
  • Polarization measurements ($R_p = \mu_p G_E^p/G_M^p$).
  • Neutron form factors ($G_E^n, G_M^n$).
  • Reduced cross-sections ($\sigma_R$) from JLab (GMp12).
  • Differential cross-sections from the PRad experiment.
  • Note: Mainz data were excluded due to inconsistencies with the global dataset.
• Theoretical Framework:
  • Ansatz: Valence quark GPDs ($H_v^q$) were parameterized as the product of unpolarized valence PDFs ($q_v$) and a profile function ($f_v^q$): $H_v^q(x, \mu^2, t) = q_v(x, \mu^2) \exp[t f_v^q(x)]$.
  • Forward Limit: The forward limit distributions were constrained by specific functional forms (Eq. 7) and normalization conditions related to quark anomalous magnetic moments.
  • Optimization: The MINUIT package was used for $\chi^2$ minimization. The Hessian method estimated uncertainties.
• Experimental Variables:
  • PDF Sets: NNPDF40, CT18, and MSHT20.
  • Scales ($\mu$): $\mu = 2$ GeV (for NNPDF40), $\mu = 1.3$ GeV (for CT18), and $\mu = 1$ GeV (for MSHT20).
  • Perturbative Orders: Both Next-to-Leading Order (NLO) and Next-to-Next-to-Leading Order (NNLO).
• Sum Rule Check: The authors explicitly tested the quark number sum rules ($\int [u - \bar{u}] dx = 2$ and $\int [d - \bar{d}] dx = 1$) for each PDF set at scales below their respective $\mu_0$.

3. Key Contributions

• Identification of Unphysical Regimes: The study demonstrates that using PDFs at scales $\mu < \mu_0$ (backward evolution) leads to a violation of quark number sum rules for NNPDF40 and CT18 sets. This violation is attributed to numerical instabilities and interpolation limitations in the PDF grids below their parametrization thresholds, rather than a failure of QCD evolution itself.
• Comprehensive GPD Extraction: The paper provides six distinct, high-quality sets of GPDs extracted at different scales and perturbative orders, offering a robust resource for the community.
• Systematic Sensitivity Analysis: It quantifies how the choice of PDF input and perturbative order affects the extracted GPDs, particularly in the low momentum transfer ($t$) region.

4. Key Results

• Sum Rule Violation:
  • For NNPDF40 ($\mu_0 = 1.65$ GeV) and CT18 ($\mu_0 = 1.3$ GeV), evaluating PDFs below their $\mu_0$ resulted in significant deviations from the quark number sum rules.
  • For MSHT20 ($\mu_0 = 1$ GeV), no violation was observed down to $\mu = 1$ GeV, confirming the validity of its lower scale.
  • This led to a sharp increase in $\chi^2/d.o.f.$ when using NNPDF40 or CT18 at scales below their $\mu_0$, indicating unphysical results.
• Best Fit Performance:
  • The NNPDF40 set at NLO ($\mu = 2$ GeV) provided the best overall description of experimental data, yielding the lowest total $\chi^2/d.o.f.$ (305.6/261).
  • Moving to NNLO for NNPDF40 slightly degraded the fit quality (increasing $\chi^2$ by ~23 units), primarily due to a worse description of AMT07 data.
  • CT18 and MSHT20 showed comparable performance at NLO and NNLO, though MSHT20 yielded higher $\chi^2$ values, largely driven by poor description of low-$t$ PRad data.
• Scale and $t$-Dependence:
  • Stability: The extraction procedure showed mild dependence on the perturbative order, indicating stability.
  • Consistency at High $|t|$: While GPD sets differed at low $|t|$, they converged and became more consistent at larger momentum transfer values ($|t|$).
  • Flavor Suppression: Down-quark GPDs ($H_v^d, E_v^d$) experienced significantly stronger suppression with increasing $|t|$ compared to up-quark GPDs.
  • Comparison with DK13: The authors' results generally agreed with the previous DK13 analysis at $t=0$. However, at higher $|t|$, the DK13 distribution showed greater suppression and a shift toward larger $x$ compared to the new extractions, suggesting the new data imposes weaker constraints on the down-quark GPDs.

5. Significance

• Cautionary Note on PDF Usage: The paper establishes a critical guideline for the community: PDFs should not be used at scales below their initial parametrization scale ($\mu_0$) in precision phenomenology, as this can lead to unphysical violations of fundamental sum rules and misleading GPD extractions.
• Resource for Future Studies: The six extracted GPD sets are made available for future theoretical and phenomenological studies, including proton tomography and the investigation of proton mechanical properties (e.g., pressure and shear forces).
• Framework Robustness: The study confirms that the MMGPDs framework is robust against variations in PDF inputs and perturbative orders, provided the inputs are used within their valid kinematic ranges.
• Insight into Nucleon Structure: The findings highlight specific challenges in constraining the down-quark sector of the nucleon, suggesting a need for further experimental data or theoretical refinement to better understand the down-quark distribution at high momentum transfer.