Reconstructing the full kinematic dependence of GPDs from pseudo-distributions

This paper presents the first direct extraction of double distributions from lattice QCD data to reconstruct the full $(x, \xi, t)$ dependence of unpolarized isovector proton GPDs using a Gaussian process regression framework, thereby enforcing Lorentz symmetry and assessing model dependence uncertainties.

The Gist

Imagine the proton, the tiny particle at the heart of every atom, not as a solid marble, but as a bustling, three-dimensional city. Inside this city, there are millions of tiny citizens (quarks and gluons) zooming around. For decades, physicists have tried to map this city. They know how many citizens live there (total charge) and how fast they move on average (momentum), but they've struggled to create a full, 3D "Google Earth" view that shows exactly where everyone is and how they are moving at the same time.

This paper is a major step forward in creating that map. Here is the story of how they did it, explained simply.

1. The Problem: The "Shadow" vs. The "Object"

Think of the proton's internal structure like a complex sculpture.

• The Goal: We want to see the whole sculpture from every angle (this is called a Generalized Parton Distribution, or GPD).
• The Obstacle: We can't look at the sculpture directly. We can only shine a light on it and look at the shadows it casts. In physics, these "shadows" are mathematical data points called matrix elements.
• The Difficulty: Reconstructing the 3D sculpture from a few 2D shadows is a classic "inverse problem." It's like trying to guess the shape of a hidden object just by looking at its shadow on the wall. If you only have a few shadows, there are infinite shapes that could fit. It's a guessing game with too many wrong answers.

2. The New Tool: The "Double Distribution" Map

In the past, physicists tried to guess the shape of the sculpture by assuming it looked like a specific type of clay (a specific mathematical model). But what if the clay is different?

In this paper, the team decided to stop guessing the shape and instead map the ingredients of the sculpture. They used a concept called Double Distributions (DDs).

• The Analogy: Imagine the sculpture is a cake. Instead of trying to guess the final shape of the cake, they mapped the distribution of flour, sugar, and eggs inside the batter.
• Why it's better: The "ingredients" (the Double Distributions) follow strict rules of symmetry (Lorentz symmetry) that the final cake shape must obey. By mapping the ingredients first, they guarantee that the final cake (the GPD) will be physically possible and mathematically correct, without forcing it into a pre-made mold.

3. The Method: The "Smart Rubber Sheet" (Gaussian Processes)

To turn their "shadow" data into a 3D map, they used a technique called Gaussian Process Regression (GPR).

• The Analogy: Imagine you have a few pins stuck in a large, stretchy rubber sheet. You know exactly where the pins are (the data points), but you don't know the shape of the sheet between them.
• The Challenge: If you just stretch the sheet to touch the pins, it might wiggle wildly and create impossible bumps (noise).
• The Solution: The team used a "smart" rubber sheet. They programmed it with physical rules: "The sheet must be smooth," "It must flatten out at the edges," and "It can't have infinite spikes."
• The Result: The sheet stretches to touch the pins but stays smooth and realistic in between. This allowed them to fill in the gaps in their data without making up fake physics.

4. The Data: A Bigger Net

To catch enough "shadows" to make a good map, they needed more data.

• Previous attempts: They used a net with small holes, catching only the slow-moving citizens (low momentum).
• This paper: They used a net with much larger holes, catching citizens zooming at incredibly high speeds (up to 2.7 GeV).
• Why it matters: Seeing the fast-moving citizens allowed them to see details of the city that were previously blurry. They could finally see how the structure changes when the proton is moving fast.

5. The Result: A Clearer Picture

After processing all this data, they produced the first-ever map of the proton's internal structure that:

1. Is 3D: It shows how the particles are distributed in space (x), how the proton is moving (ξ), and how the momentum is transferred (t).
2. Is Model-Free: They didn't force the data to fit a specific theory; they let the data speak for itself, guided only by the laws of physics.
3. Passes the "Positivity" Test: They checked their map against a fundamental rule of nature (positivity bounds) and confirmed that their map makes physical sense.

The Big Picture

Think of this paper as the first time someone successfully built a 3D hologram of a proton using only a few blurry photos and a super-smart computer algorithm.

Before, we had a sketch. Now, we have a high-definition model. This doesn't just satisfy our curiosity; it helps us understand how the proton gets its mass and spin, which are fundamental questions about how our universe is built. The authors have even made their "blueprints" (the data) available online so other scientists can use them to build even better models in the future.

Technical Summary

1. Problem Statement

Generalized Parton Distributions (GPDs) provide a three-dimensional tomographic picture of hadron structure, encoding information on the longitudinal momentum ($x$), transverse spatial distribution (via momentum transfer $t$), and the skewness ($\xi$) of partons. However, extracting the full kinematic dependence $(x, \xi, t)$ of GPDs is notoriously difficult:

• Experimental Limitations: Experimental extraction requires deconvolution of exclusive scattering data, which is ill-posed and suffers from small cross-sections.
• Lattice QCD Limitations: Traditional lattice QCD approaches rely on local operators to compute Mellin moments, which only provide discrete points and cannot reconstruct the continuous function. Recent non-local approaches (pseudo-distributions) allow access to the full $x$-dependence but face a severe inverse problem: reconstructing a continuous function from a limited number of noisy Fourier harmonics (Ioffe-time data).
• Theoretical Constraints: GPDs must satisfy fundamental properties like polynomiality (Mellin moments are polynomials in $\xi$) and positivity, which are difficult to enforce in standard reconstruction methods.

2. Methodology

The authors propose a novel framework to reconstruct the full $(x, \xi, t)$ dependence of unpolarized isovector proton GPDs ($H^{u-d}$ and $E^{u-d}$) directly from lattice QCD data using the pseudo-distribution formalism.

A. Theoretical Framework: Double Distributions (DDs)

Instead of reconstructing GPDs directly, the authors extract Double Distributions (DDs), denoted $h(\beta, \alpha, t)$ and $e(\beta, \alpha, t)$.

• Rationale: DDs are fundamental objects that encode GPDs via a Radon transformation (Eq. 18). Crucially, the DD representation automatically enforces the polynomiality property of GPDs, a consequence of Lorentz symmetry.
• Input: The lattice calculates non-local matrix elements with space-like separation $z$. These are matched to the $\overline{\text{MS}}$ scheme and related to the DDs via Fourier transforms in Ioffe time ($\nu, \bar{\nu}$).

B. Gaussian Process Regression (GPR)

To solve the ill-posed inverse problem of reconstructing DDs from limited lattice data, the authors employ Gaussian Process Regression (GPR):

• Non-parametric Approach: Unlike traditional fits with fixed functional forms (e.g., Radyushkin's Ansatz), GPR is flexible and data-driven.
• Regularization: A Bayesian prior is constructed to enforce physical constraints:
  • Small $\beta$ behavior: Modeled to allow for power-law divergence (consistent with PDF behavior).
  • Large $\beta$ behavior: Enforced to vanish as $(1-\beta)^n$.
  • Edge behavior: Constraints on the domain boundaries ($\beta + \alpha = 1$).
  • Smoothness: Controlled by correlation lengths in $\beta, \alpha,$ and $t$.
• Discretization: The DD is represented on a 3D mesh ($30 \times 15 \times 15$ points in $\beta, \alpha, t$). The reconstruction involves solving for the posterior distribution of the DD values on this mesh.

C. Lattice Data and Setup

• Ensemble: $2+1$ flavor Wilson clover fermions with pion mass $m_\pi = 358$ MeV and lattice spacing $a = 0.094$ fm.
• Statistics: Approximately 1490 configurations (4x previous statistics) to handle the signal-to-noise degradation at high momenta.
• Kinematics:
  • Hadron momenta up to 2.7 GeV (doubling the previous range).
  • Maximal Ioffe time $\nu \approx 7$.
  • Denser coverage in skewness $\xi$.
• Analysis: Ground-state matrix elements are extracted using a one-excited-state fit with a cut $t_s = 3$ (source-sink separation), validated against the summation method.

3. Key Contributions

1. First Direct DD Extraction: This is the first time Double Distributions have been extracted directly from lattice QCD data, rather than inferring them from GPD moments or assuming a specific model.
2. Enforcement of Symmetry: By working in the DD basis, the reconstruction inherently satisfies the polynomiality property without ad-hoc constraints.
3. Advanced Regularization: The application of multidimensional GPR with physically motivated priors provides a robust solution to the inverse problem, offering a systematic way to assess model dependence.
4. Extended Kinematics: The study utilizes significantly higher momenta (up to 2.7 GeV) and denser $\xi$ coverage compared to previous works, extending the accessible Ioffe time range.

4. Results

• Reconstruction Quality: The GPR successfully reconstructs the real and imaginary parts of the GPDs ($H^{u-d}$ and $E^{u-d}$) across the full kinematic range. The fits show good agreement with the lattice data (correlated $\chi^2 \approx 1.7$).
• Model Dependence: The authors performed a partial assessment of model dependence by varying hyperparameters. They found that while the baseline uncertainty is optimistic, the variation remains within reasonable bounds, particularly at larger $z$ (larger Ioffe time) where the inverse problem is better constrained.
• Perturbative Matching: One-loop matching and renormalization group evolution were performed to a scale of $\mu = 2$ GeV. The scale variation uncertainty was found to be negligible compared to statistical and model uncertainties.
• Comparison with Phenomenology:
  • The results show broad qualitative agreement with the Goloskokov-Kroll (GK) phenomenological model at $t=0$.
  • Systematic Shifts: The lattice results are systematically larger than the GK model, particularly for $H^{u-d}$. This is attributed to the unphysical pion mass ($358$ MeV vs. physical $\approx 140$ MeV), which increases the average quark momentum fraction $\langle x \rangle$.
• Positivity: The extracted GPDs satisfy positivity bounds (inequalities relating GPDs to PDFs), a long-standing goal for lattice extractions.
• Generalized Form Factors (GFFs): The authors re-extracted GFFs from the new GPDs. They observed discrepancies with their previous work (based on lower momenta) regarding Elastic Form Factors (EFFs), likely due to discretization errors or higher-twist contamination at large momenta, highlighting the need for continuum limit studies.

5. Significance

• Theoretical Milestone: This work demonstrates that lattice QCD can provide a first-principles, model-independent reconstruction of the full 3D structure of the nucleon, satisfying all fundamental theoretical constraints (polynomiality, positivity).
• Phenomenological Impact: The results provide a crucial benchmark for global fits of GPDs and offer the first lattice-based inputs for calculating amplitudes of deep exclusive processes (e.g., Deeply Virtual Compton Scattering) without relying on phenomenological models.
• Future Directions: The authors identify that the next critical steps are performing calculations at the physical pion mass and taking the continuum limit to fully quantify discretization errors and higher-twist effects, which currently limit the precision of the elastic form factors.

Data Availability: The full dataset of reconstructed GPDs is made publicly available via the Zenodo database, allowing the community to use these results for phenomenological applications.