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Enhanced Signal and Excited-State Control in Nonlocal Lattice Correlators

The paper proposes a new fitting strategy that utilizes the ratio of nonlocal to local correlators to significantly improve signal quality and excited-state control in lattice calculations of parton physics, thereby enabling more precise and stable extractions of hadron structure functions like PDFs, GPDs, and TMDPDFs.

Original authors: Shi-Yi Zhong, Jian-Hui Zhang, Dian-Jun Zhao

Published 2026-10-30

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CC BY 4.0

Original authors: Shi-Yi Zhong, Jian-Hui Zhang, Dian-Jun Zhao

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

Inside every proton and neutron, the building blocks of ordinary matter, quarks and gluons are not static particles but a seething, complex fluid. To understand how these particles organize themselves, physicists rely on maps called distribution functions. These maps tell us how much momentum a quark carries, how it spins, and where it is likely to be found within the particle. While scientists have long measured these distributions by smashing particles together in massive accelerators, a different approach has emerged to see the structure from the inside out. This method uses supercomputers to simulate the laws of physics on a grid of space and time, a technique known as lattice quantum chromodynamics. By calculating how quarks and gluons interact on this grid, researchers hope to predict the properties of matter directly from the fundamental equations of nature, without needing to rely solely on experimental data.

However, this computational approach faces a stubborn obstacle. As the simulations try to probe deeper into the structure of the particle, the signal they are looking for becomes buried under a rising tide of noise. Imagine trying to hear a single whisper in a room that is slowly filling with the roar of a crowd; the further you try to listen, the harder it becomes to distinguish the voice you want from the background chaos. In these computer simulations, this "noise" comes from two main sources. First, the computer struggles to separate the true, stable state of the particle from fleeting, excited states that look similar but are not the real thing. Second, as the distance between the points being measured increases, the statistical uncertainty grows so large that the data becomes unreliable. This limitation has kept physicists from seeing the full picture of how quarks and gluons behave, especially when the particles are moving at high speeds.

A team of researchers has now proposed a new way to cut through this noise, offering a cleaner path to these fundamental maps. Instead of trying to measure the signal in isolation, they realized that by comparing two specific measurements taken at the same time, they could cancel out the overwhelming background interference. In their new method, the researchers take the complex measurement of a particle's internal structure and divide it by a simpler, related measurement taken at the same moment. Because both measurements are affected by the same sources of noise and the same unwanted excited states, these errors cancel each other out almost perfectly when the two are divided. The result is a signal that remains clear and stable even when the researchers look at much larger distances or higher speeds than was previously possible.

The team tested this strategy on four different types of particle data, including the spin of quarks inside a proton, the distribution of gluons, and the internal structure of a pion. In every case, the new method produced results that matched the established, reliable data but with significantly sharper precision. Where the old methods began to fail, with error bars growing so large that the data became meaningless, the new approach held steady. For instance, when looking at the proton moving at a speed of 3.23 GeV, the traditional method lost its signal after a distance of about five units of the simulation grid, while the new method remained clear and usable up to twelve units. This doubling of the effective range means physicists can now explore parts of the particle's structure that were previously invisible.

This improvement is not just about getting slightly better numbers; it fundamentally changes what questions can be asked. The new technique simplifies the analysis for complex scenarios where the particle changes its momentum, a situation that previously required a complicated and error-prone series of calculations. By removing the need for extra, separate measurements to correct for noise, the method makes the entire process more efficient and robust. The researchers found that this approach works for various types of particle interactions and can be applied immediately to existing data sets without needing new computer runs. While the work is still in the stage of demonstrating that the method works, the results suggest a practical way to extend the reach of these simulations, potentially leading to more accurate predictions of how the universe's most fundamental particles hold themselves together.

Technical Summary: Enhanced Signal and Excited-State Control in Nonlocal Lattice Correlators

Problem Statement
Lattice QCD calculations of parton physics, including Parton Distribution Functions (PDFs), Generalized Parton Distributions (GPDs), and Transverse-Momentum-Dependent distributions (TMDs), rely on nonlocal correlation functions. The extraction of these quantities is currently hindered by two primary systematic issues:

Excited-state contamination: The convergence to the ground-state plateau is controlled by energy gaps ( $E_n - E_0$ ) which are small (hundreds of MeV), leading to significant contamination from excited states, particularly at large source-sink separations and high hadron momenta.
Signal-to-noise degradation: As Euclidean separation ( $z$ ) and hadron momentum increase, the signal deteriorates rapidly.
The conventional approach involves fitting the ratio of a three-point correlator ( $C_3$ ) to a two-point correlator ( $C_2$ ). This method suffers because the normalization by $C_2$ amplifies statistical noise (as $C_2$ decays exponentially) and retains overlap factors dependent on the interpolating field, which are difficult to constrain. Furthermore, for non-forward kinematics (required for GPDs), the standard ratio method requires complex constructions involving multiple correlators and introduces sign ambiguities and square-root fluctuations.

Methodology: The Self-Ratio Strategy
The authors propose a new fitting strategy termed the self-ratio. Instead of forming the ratio $C_3(t, \tau, z) / C_2(t)$ , the method constructs the ratio of two three-point correlators measured at the same source-sink separation ( $t$ ) and insertion time ( $\tau$ ), but with different operator separations:
$R_{\text{self}}(t, \tau, z) \equiv \frac{C_3(t, \tau, z)}{C_3(t, \tau, 0)}$
Here, the numerator contains the nonlocal operator with separation $z$ , and the denominator contains the local operator ( $z=0$ ).

Theoretical Basis: The authors derive the spectral decomposition of this ratio. Because the numerator and denominator share identical interpolating fields, source-sink structures, and energy spectra, the overlap factors ( $\zeta_n$ ) and energy denominators cancel term-by-term.
Residual Contamination: In the conventional ratio, residual contamination is governed by the ratio of overlaps ( $\zeta_n/\zeta_0$ ) and the full operator matrix elements. In the self-ratio, the residual is governed solely by the difference in the operator's matrix elements between $z$ and $0$. The contamination vanishes at $z=0$ and is significantly suppressed elsewhere.
Applicability: The method is derived for forward kinematics and extended to non-forward kinematics (GPDs) and TMDs (using staple-shaped Wilson lines). For non-forward cases, it avoids the complex six-correlator construction and sign ambiguities associated with standard non-forward ratios.

Key Contributions and Results
The paper demonstrates the efficacy of the self-ratio method through four specific case studies using existing lattice data from CLS and CLQCD ensembles:

Nucleon Isovector Quark Transversity PDF: Reanalysis of data at $P_z = 3.23$ GeV shows that while the conventional ratio loses signal beyond $z \approx 5a$ due to noise amplification, the self-ratio remains controlled and precise up to $z \approx 12a$ .
Nucleon Unpolarized Gluon PDF: Comparisons of the Feynman-Hellmann method, conventional ratio, and self-ratio show agreement in central values. However, the self-ratio achieves comparable or smaller statistical errors and reaches the same plateau with greater stability.
Pion Quark GPD (Non-forward): The conventional ratio exhibits a "dip" and sign change at intermediate separations ( $z \approx 3-6a$ ) due to contamination. The self-ratio eliminates this artifact, yielding a smooth extraction of the matrix element without requiring the complex normalization factors of standard non-forward methods.
Nucleon Boer-Mulders Function (TMD): For transverse separations $b/a = 1$ and $2$, the self-ratio results agree with the conventional ratio but exhibit visibly smaller statistical uncertainties across the entire range of $z$ .

Significance and Claims
The authors claim that the self-ratio method offers a practical and immediately deployable improvement for lattice parton physics programs. Its primary advantages are:

Enhanced Precision and Stability: By canceling common exponential decays and overlap factors, the method significantly reduces statistical noise and excited-state contamination.
Extended Kinematic Reach: The method allows for reliable extraction of matrix elements at larger hadron momenta and larger spatial separations ( $z$ ), which are critical for accessing the full $x$ -range of parton distributions.
Simplified Analysis: For non-forward matrix elements (GPDs), the method simplifies the analysis procedure by removing the need for complex multi-correlator constructions and avoiding sign ambiguities.
Cost-Effectiveness: The method utilizes correlator data already generated in standard lattice calculations (specifically the $z=0$ local correlator), requiring no additional measurements or expensive new bases (unlike Variational methods or Lanczos approaches).

The paper concludes that while the method improves statistical precision, future work should focus on systematically quantifying the remaining $z$ -dependent residual excited-state effects to further refine ground-state extraction.

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