Extracting Mellin moments of double parton distributions from lattice data

This paper investigates the impact of kinematic skewness on reconstructing Mellin moments of double parton distributions from Euclidean lattice data by analyzing the skewness dependence of hadronic correlation functions through various models.

The Gist

Imagine the proton not as a solid marble, but as a bustling, chaotic city filled with tiny, invisible residents called partons (quarks and gluons). Physicists have spent decades mapping out the "population density" of this city—knowing how many residents live in different neighborhoods and how fast they are moving. This map is called a Parton Distribution Function (PDF).

However, the city is so complex that sometimes, two residents interact with the outside world at the exact same time. This is called Double Parton Scattering. To understand this, we need a new, much more complex map called a Double Parton Distribution (DPD). This map doesn't just tell us where one resident is; it tells us the probability of finding two specific residents in specific spots, moving at specific speeds, all at once.

The problem? This new map is incredibly hard to draw. We can't just look at the proton with a microscope; the rules of quantum mechanics make it impossible to see everything at once.

The Lattice "Time Machine"

To solve this, physicists use a supercomputer method called Lattice QCD. Think of this as taking a series of frozen snapshots of the proton city. Because of the way these snapshots work (they exist in "Euclidean" time, which is a bit like a mathematical mirror of our real world), the computer can only see the residents at specific, limited distances from each other.

To get the full picture of the DPD, the physicists need to combine all these snapshots into a single, continuous movie. Mathematically, this requires adding up (integrating) information over a variable they call Ioffe time (let's call it "Time-Shift").

Here's the catch: The computer can only take snapshots for a short period of "Time-Shift." It's like trying to reconstruct a 2-hour movie when you only have 10 minutes of footage. You have to guess what happens in the missing parts.

The "Skewness" Twist

In their previous work, the authors tried to guess the missing parts by assuming a simple, smooth curve (a polynomial). They introduced a variable called Skewness (let's call it "Tilt").

• Tilt = 0: This is the normal state we care about for Double Parton Scattering.
• Tilt = 1: This is a weird, extreme state where the two residents are carrying almost the entire momentum of the proton, leaving nothing for the rest of the city.

The authors realized their previous "smooth curve" guess had two major flaws:

1. The Edge Problem: Their smooth curve didn't drop to zero fast enough when the "Tilt" got extreme (near 1). Physics suggests that in this extreme state, the probability of finding such a configuration should vanish almost instantly, like a cliff edge, not a gentle slope.
2. The Smoothness Problem: They assumed the curve was perfectly smooth everywhere. But physics suggests that at the very center (Tilt = 0) and the very edge (Tilt = 1), the curve might have a "kink" or a sharp point, much like how a shadow changes abruptly when a light source moves.

The New Models

In this paper, the team tried out four new ways to guess the missing footage, designed to respect these "cliff edges" and "kinks":

1. The Power-Law: A curve that drops off sharply at the edges.
2. The Integral Model: A shape based on how particles split apart in high-energy collisions.
3. The Cosine Model: A wave-like shape that can be tweaked to have sharp or smooth edges.
4. The Polynomial (Old Way): The smooth curve they used before, kept for comparison.

The Results: A Puzzle with Missing Pieces

The team fed their computer data into these new models to see which one fit best.

• The Good News: All the new models fit the available computer data very well. They all agreed on the "middle" of the story (the behavior at moderate Tilt).
• The Bad News: When they tried to use these models to reconstruct the most important part of the map—Tilt = 0 (the actual Double Parton Scattering event)—the results were wildly uncertain.
  • Because the computer data gets very "noisy" (blurry) at the extreme ends of the "Time-Shift" (where the missing footage is), the different models gave very different answers for the center.
  • Some models predicted a value of 2 (which is what a fundamental rule called the "Number Sum Rule" says it should be).
  • Others predicted values that were way off, or had huge error bars (uncertainty ranges) that were hundreds of times larger than the value itself.

The Conclusion

The authors conclude that we cannot yet perfectly reconstruct the Double Parton Distribution at the most critical point (Tilt = 0) using only the current computer data.

It's like having a puzzle where you have all the corner pieces and the middle pieces, but the pieces connecting them are missing. You can guess the shape of the puzzle, but you can't be sure exactly how the pieces fit together in the center without more information.

To fix this, they say we need better computer data that is less "noisy" at the extreme ends of the "Time-Shift." Until then, they have to rely on extra theoretical rules (like the Number Sum Rule) to force the answer to be correct, rather than letting the data speak for itself.

In short: They built better tools to guess the shape of a complex quantum map, but they found that their current "photos" of the proton aren't sharp enough to see the most important detail clearly. They need sharper photos to finish the job.

Technical Summary

Technical Summary: Extracting Mellin Moments of Double Parton Distributions from Lattice Data

Problem Statement
Double parton distributions (DPDs) describe quantum correlations between two partons within a proton and are essential for understanding multiparton interactions, particularly double parton scattering (DPS) at colliders like the LHC. While single-parton distributions (PDFs) are well-constrained, DPDs remain poorly understood due to their complexity and experimental extraction difficulties. Lattice QCD offers a non-perturbative approach to constrain DPDs. Specifically, the Mellin moments of DPDs can be related to hadronic matrix elements of two local currents at different spatial positions. However, reconstructing these moments from Euclidean lattice data requires integrating over a variable $\omega$ (Ioffe time) that ranges from $-\infty$ to $+\infty$. Lattice simulations are restricted to a finite range of $\omega$. To address this, the authors previously introduced a "skewness" parameter $\zeta$, Fourier conjugate to $\omega$, where the physical DPDs correspond to $\zeta = 0$. The challenge lies in determining the functional dependence of the Mellin moments on $\zeta$ to perform the necessary inverse Fourier transform without introducing uncontrolled biases.

Methodology
The authors revisit the extraction of the lowest Mellin moment of DPDs, $I_{q_1q_2}(\zeta, y^2)$, using existing lattice data from the CLS H102 ensemble ($N_f=2+1$). The methodology proceeds in three main stages:

1. Theoretical Constraints: The paper establishes two critical physical properties for the $\zeta$-dependence of Mellin moments that were not fully satisfied by previous polynomial ansätze:

   • The moments must vanish rapidly as $|\zeta| \to 1$.
   • The moments may exhibit non-analytic behavior at $\zeta = 0$ and $\zeta = 1$, analogous to the behavior of PDFs at momentum fractions $x=0$ and $x=1$.
     These properties are derived from the support regions of skewed DPDs and perturbative splitting mechanisms in the short-distance limit.

2. Lattice Data Analysis: The study utilizes four-point correlation functions of two vector currents in a proton state. The invariant function $A_{q_1q_2}(\omega, y^2)$ is extracted from the lattice data. The analysis focuses on the flavor combination $ud$ due to superior statistical precision compared to equal-flavor combinations. The data is restricted to a distance range $4a \le y \le 14a$ to minimize discretization and finite-size effects.

3. Modeling and Fitting: The authors assume a factorization of the $\zeta$ and $y$ dependence, $I(\zeta, y^2) = J(\zeta)K(y^2)$. They fit the normalized $\omega$-dependence of the lattice data, $\hat{A}(\omega)$, to the Fourier transforms of several new ansätze for $J(\zeta)$:

   • Polynomial Model: The previous approach (polynomial in $\zeta^2$).
   • Power-law Model: Proportional to $(1-|\zeta|)^a$.
   • Integral Model: Based on an integral representation inspired by perturbative splitting.
   • Cosine Model: Proportional to $\cos^b(\frac{\pi}{2}|\zeta|^a)$.

   Fits are performed both globally and in narrow bins of $y$ to test the factorization hypothesis. Additionally, constrained fits are performed where parameters are fixed to satisfy the number sum rule ($S_{ud} = 2$).

Key Results

• Model Comparison: The polynomial model, while providing a good fit to the lattice data in $\omega$-space, fails to satisfy the physical boundary conditions (non-vanishing at $\zeta=1$ and smoothness at $\zeta=0$). The new models (power-law, integral, cosine) satisfy these conditions and provide equally good fits to the data.
• Uncertainty in Reconstruction: When fitting with two free parameters, the new flexible models yield extremely large uncertainties for the Mellin moment at $\zeta=0$ (the physical DPD limit). For instance, the integral model yields a value of $4.4 \pm 938.4$ for the sum rule quantity, despite the theoretical expectation of 2. This indicates that the lattice data, particularly at large $|\omega|$, is insufficient to constrain the behavior of the function near $\zeta=0$ without additional theoretical input.
• Constrained Fits: Fixing one parameter in the new models to enforce the number sum rule ($S_{ud}=2$) results in stable, smooth reconstructions of the Mellin moments that are consistent with the data.
• Average Skewness: All fits consistently yield a small average squared skewness, $\langle \zeta^2 \rangle \approx 0.07$, suggesting that parton configurations with small skewness are dominant.
• Factorization: The study confirms that the factorization of $\zeta$ and $y$ dependence holds within the statistical precision of the current data.

Significance and Claims
The paper claims that while lattice QCD can provide high-precision data for the invariant functions related to DPDs, the reconstruction of the physical Mellin moments at zero skewness ($\zeta=0$) is currently limited by the finite range of accessible Ioffe times ($\omega$). The authors demonstrate that assuming a specific functional form (like a polynomial) can lead to physically inconsistent results, whereas physically motivated models (power-law, integral, cosine) are necessary but introduce large statistical uncertainties in the $\zeta=0$ limit when left unconstrained.

The primary conclusion is modest: With the lattice data currently available, Mellin moments at $\zeta=0$ cannot be reconstructed without additional theory input (such as enforcing the number sum rule). The authors emphasize that improving this situation requires lattice simulations with significantly better precision at large proton momenta and large distances to extend the accessible range of $\omega$. However, the results for non-zero $\zeta$ remain valuable, providing information on the joint distribution of partons and confirming that small skewness configurations are more probable. The study also suggests that the lessons learned regarding the $\zeta$-dependence and the necessity of theory input are relevant for future lattice studies of DPDs using quasi-distributions or pseudo-distributions.