Parton Distribution Functions from Large Momentum Expansion of Current-Current Correlators

This paper presents a next-to-leading order expansion formula for computing parton distribution functions from current-current correlators, which offer simple renormalization properties and avoid linear power divergences, alongside preliminary numerical results utilizing four-point functions on the lattice.

The Gist

The Big Picture: Trying to See the Invisible Inside a Proton

Imagine a proton (a tiny particle inside an atom) as a busy, chaotic city. Inside this city, there are tiny residents called "partons" (quarks and gluons) zooming around at incredible speeds.

Physicists want to know exactly who lives where and how fast they are moving. This map is called a "Parton Distribution Function" (PDF). Knowing this map is crucial for predicting what happens when we smash particles together in giant machines like the Large Hadron Collider (LHC).

The problem? We can't just take a photo of this city. The laws of physics (specifically, the fact that we live in a "Euclidean" world in our calculations) make it impossible to take a direct snapshot of these fast-moving residents.

The Old Way: The "Wilson Line" Problem

For the last decade, scientists have tried to solve this using a method called LaMET (Large Momentum Effective Theory).

• The Analogy: Imagine trying to measure the speed of cars in that city by stretching a long, invisible rubber band (a "Wilson line") from one car to another.
• The Problem: This rubber band is messy. In the math, it gets "dirty" with infinite errors (divergences) that are very hard to clean up. It's like trying to measure the wind speed while holding a tangled, static-charged balloon; the balloon itself messes up the reading.

The New Idea: The "Current-Current" Shortcut

This paper proposes a new, cleaner way to measure the city's traffic without using that messy rubber band.

Instead of stretching a line, the scientists look at two specific "traffic lights" (currents) placed at different spots in the city and watch how they interact with each other.

• The Analogy: Instead of a rubber band, imagine placing two cameras at different intersections. You don't need a physical connection between them; you just analyze the relationship between the traffic patterns they see.
• The Benefit: Because these "traffic lights" are based on fundamental, conserved laws of physics, they don't get "dirty." They are naturally clean, meaning the math doesn't have those annoying infinite errors that plague the old method.

The Challenge: The "Four-Point" Puzzle

There is a catch.

• The Old Way: Required calculating a "two-point" connection (Start point $\to$ End point). This is like calculating a trip from Home to Work.
• The New Way: Requires calculating a "four-point" connection (Start $\to$ Camera 1 $\to$ Camera 2 $\to$ End).
• The Analogy: This is like trying to calculate a trip that involves Home, two specific intersections, and Work, all at once. It is much more computationally expensive and complex. It's like solving a harder puzzle.

What This Paper Actually Did

The authors didn't just talk about the theory; they actually tried to solve the puzzle using a supercomputer simulation (Lattice QCD).

1. The Setup: They used existing data from a previous project (like re-using a dataset from a past experiment) to simulate a proton moving at high speed.
2. The Calculation: They calculated the interaction between the two "traffic lights" (currents) inside this moving proton.
3. The Translation: They used a mathematical "translator" (called a matching kernel) to convert their simulation results into the actual map of the proton's interior (the PDF).

The Results: A "Proof of Concept"

The paper presents a first draft of this new map for a specific type of proton traffic (the difference between up and down quarks, denoted as $f_{u-d}$).

• The Outcome: The map they drew looks somewhat like the maps made by other methods, but it's a bit "fuzzy" and has some wiggles.
• Why it's fuzzy:
  • Low Resolution: The "camera" (the simulation) wasn't running at the highest possible speed or with enough data points.
  • Heavy Proton: The simulated proton was heavier than a real proton (like simulating a heavy truck instead of a sports car), which distorts the traffic patterns.
• The Conclusion: The authors admit this isn't a perfect, final map yet. However, it proves that the method works. They successfully showed that you can get a map of the proton's interior using these clean "traffic light" interactions without the messy rubber bands.

Summary in One Sentence

This paper demonstrates a new, cleaner way to map the inside of a proton by analyzing the interaction between two points (instead of a messy line), proving that while the math is complex and the current results are rough, the method is viable for future, more precise experiments.

Technical Summary

Technical Summary: Parton Distribution Functions from Large Momentum Expansion of Current-Current Correlators

Problem Statement
Parton Distribution Functions (PDFs) are fundamental non-perturbative quantities in Quantum Chromodynamics (QCD) describing hadronic structure on the light cone. While essential for predicting high-energy collider processes (e.g., at the LHC and EIC), calculating them directly from first principles using Euclidean lattice QCD is challenging due to the light-like nature of the observables. The Large Momentum Effective Theory (LaMET) framework addresses this by expanding Euclidean correlation functions at large momentum ($P_z \gg \Lambda_{QCD}$) to match them to light-cone PDFs.

Existing LaMET approaches, such as the quasi-PDF, rely on operators containing spatial Wilson lines. These suffer from linear power divergences and complex renormalization issues related to Wilson-line self-energies. Alternative approaches, like Coulomb gauge distributions, avoid Wilson lines but lack all-order factorization proofs. Current-current correlators offer a promising alternative: they possess simple renormalization properties (no linear power divergences) and are theoretically well-understood. However, their application has been limited because they require the computationally expensive calculation of four-point correlation functions on the lattice.

Methodology
This work develops an $x$-space formulation for the large momentum expansion of matrix elements involving two currents (vector $V$ and axial-vector $A$) to extract PDFs. The core theoretical framework involves:

1. Operator Definition: The study considers forward matrix elements of two local currents, $J^\mu_{X,ab}(z)$ and $J^\nu_{Y,ba}(0)$, separated by a spatial distance $z$ parallel to the proton momentum $P$. The currents are defined as $J^\mu_V = \bar{q}_a \gamma^\mu q_b$ and $J^\mu_A = \bar{q}_a \gamma^\mu \gamma^5 q_b$.
2. Universality and Matching: The authors demonstrate that these current-current correlators belong to the same universality class as quasi-PDFs. The light-cone PDF $f(x, \mu^2)$ is factorized into a perturbative matching kernel $C$ and a lattice-computable momentum-space distribution $\tilde{h}(y, P_z)$ derived from the Fourier transform of the Euclidean correlator.
3. Renormalization Advantages: Because the correlators are constructed from conserved currents and do not involve Wilson lines, they are free from power UV divergences and renormalon ambiguities at the one-loop level. The matching kernels are calculated up to one-loop order in the $\overline{\text{MS}}$ scheme.
4. Lattice Implementation:
   • Data Source: The analysis re-uses existing lattice data from the CLS collaboration ($n_f=2+1$ Wilson-Clover ensembles, ensemble H102, $m_\pi = 355$ MeV, $a=0.085$ fm).
   • Four-Point Functions: The matrix elements are extracted from four-point correlation functions. To isolate the non-singlet contribution ($f_{u-d}$), the authors select an auxiliary quark flavor $b$ different from the valence flavors, effectively canceling disconnected diagrams and sea-quark contributions ($S_2$).
   • Extraction: The relevant Lorentz structure ($B$) is extracted by decomposing the matrix element $M^{\mu\nu}$ and solving an over-determined system of equations using data where the momentum and separation vectors are aligned or nearly aligned.
   • Extrapolation: Due to the contact singularity at $z=0$, the value is fixed perturbatively. For large $|z|$, where lattice data is dominated by noise, an extrapolation ansatz based on asymptotic behavior ($A|z|^{5/2}e^{-m|z|}$) is employed before performing the Fourier transform to momentum space.

Key Results

• First Numerical Demonstration: The paper presents the first numerical calculation of the isovector PDF $f_{u-d}$ using the current-current correlator approach within the LaMET framework.
• Momentum Distribution: The Fourier transform yields the momentum-space distribution $\tilde{h}(y, P_z)$, which shows significant oscillations outside the physical region, attributed to limited statistics and residual extrapolation uncertainties.
• PDF Extraction: After applying the one-loop matching kernel and Renormalization Group Resummation (RGR), the resulting PDF at $\mu = 2$ GeV is compared to the CT18NNLO global fit.
• Discrepancies: The lattice result shows a noticeable deviation from the global fit, particularly at larger $x$. The authors attribute this to the limited maximum proton momentum used ($|P| \approx 1.52$ GeV) and the unphysical pion mass ($355$ MeV). As $P_z$ increases, parton momentum is expected to shift to smaller $x$, improving agreement.

Significance and Claims
The paper positions this work as a "proof of concept" rather than a precision determination. Its primary significance lies in demonstrating the feasibility of extracting $x$-dependent PDFs from lattice QCD using current-current correlators.

• Conceptual Advantages: The approach successfully avoids the renormalization complications (linear power divergences) inherent in Wilson-line-based quasi-PDF calculations.
• Feasibility: Despite the computational cost of four-point functions, the study confirms that with existing techniques (momentum smearing, stochastic sources) and re-used data, such calculations are routine and viable.
• Future Outlook: The authors emphasize that the observed discrepancies are expected to be resolved by future calculations utilizing higher hadron momenta ($|P| > 2$ GeV) and physical pion masses, potentially aided by kinematically enhanced interpolators. The current analysis serves to validate the theoretical framework and methodology for this specific class of distributions.