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Subtraction method for disconnected diagrams in full QCD

This paper introduces a domain decomposition-based subtraction method that significantly reduces the statistical noise of disconnected diagrams in full QCD by achieving a three-order-of-magnitude variance reduction at a modest computational cost, offering a practical alternative to neglecting these contributions.

Original authors: Lorenzo Barca, Joshua Crawford, Jacob Finkenrath, Stefan Schaefer, Juan Andrés Urrea-Niño

Published 2026-09-30

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CC BY 4.0

Original authors: Lorenzo Barca, Joshua Crawford, Jacob Finkenrath, Stefan Schaefer, Juan Andrés Urrea-Niño

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

In the subatomic world, matter is built from particles called quarks, which are held together by a force so strong it behaves differently than gravity or magnetism. To understand how these particles form the protons and neutrons inside our bodies, or the heavier particles that existed just after the Big Bang, physicists use a technique called lattice quantum chromodynamics. Imagine the universe as a giant, three-dimensional grid of tiny points, like a digital map, where the laws of physics are calculated step by step. This method allows scientists to simulate the behavior of quarks and the force fields that bind them, turning abstract equations into concrete numbers that can be compared with real-world experiments.

However, this digital simulation faces a massive hurdle when trying to calculate certain types of interactions. Some particle behaviors involve "disconnected" loops, where a quark briefly pops into existence, travels in a circle, and vanishes without ever touching the main particle being studied. These loops are like ghostly whispers in the simulation; they are essential for a complete picture, but they are incredibly difficult to hear because they are buried under a storm of random noise. In the language of the simulation, this noise drowns out the signal so quickly that the further one looks into time, the more the useful information disappears. For years, this has forced researchers to ignore these ghostly loops entirely, leaving gaps in our understanding of the heaviest particles and the precise masses of certain exotic forms of matter.

A team of researchers has now developed a clever way to silence that noise without losing the signal. By rethinking how they look at the digital grid, they found a method to subtract the overwhelming background static, revealing the faint whispers that were previously hidden. Their work focuses on a specific type of heavy particle known as charmonium, which is made of a charm quark and its antimatter partner. In the standard approach, calculating the properties of this particle requires trillions of computer calculations to get a single clear answer, and even then, the result is often too fuzzy to be useful. The new method, however, achieves a level of clarity that was previously thought to require millions of times more computing power, all while using a fraction of the time.

The core of the discovery lies in how the researchers divided the digital grid. Instead of treating the entire simulation as one giant, chaotic block, they split the time dimension of their grid into separate, non-overlapping regions. They realized that the noise in the calculation comes largely from parts of the grid that are far apart from each other. By isolating these distant regions, the team could calculate a specific "noise term" that represents the random fluctuations between them. Because this noise term is mathematically linked to the total result, they could subtract it out. It is similar to trying to hear a quiet conversation in a loud room: if you can precisely measure the hum of the air conditioner and the traffic outside, you can subtract those sounds from your recording to hear the voices clearly.

When the team applied this subtraction technique to their simulations of charmonium, the results were striking. The statistical noise, which usually grows so large that it makes the data useless, was reduced by a factor of nearly fifty. This means that the uncertainty in their measurements dropped dramatically, allowing them to see the particle's behavior over a much longer period of time than before. In practical terms, this reduction in noise is equivalent to what would have been achieved by running their simulation on a supercomputer for roughly two thousand five hundred times longer. Yet, the new method only required about twice the computational effort of their previous standard approach. This efficiency gain transforms a task that was once prohibitively expensive into one that is manageable, opening the door to studying these elusive particle interactions with a precision that was previously out of reach.

The researchers tested their method on different types of particle signals and found that it works best when the particles involved are heavy and the interaction is relatively simple. In some cases, the subtraction removes a tiny bit of the actual physical signal along with the noise, creating a small bias. However, the team showed that this bias is negligible for the specific heavy particles they studied, and it can be corrected if necessary using advanced sampling techniques. The method proved particularly effective for one type of particle channel, where the noise reduction was most dramatic. In another channel, the signal itself was so intertwined with the noise that simply subtracting the noise removed too much of the physics, highlighting that the technique is not a universal fix but a powerful tool that must be applied with care depending on the specific particle being studied.

This breakthrough does not just solve a problem for one type of particle; it offers a new strategy for the entire field of lattice physics. By showing that the noise in these complex calculations can be tamed through clever mathematical separation rather than brute force, the researchers have provided a practical alternative to ignoring disconnected diagrams. This means that future studies can include these previously neglected contributions, leading to more accurate predictions for the masses of particles like the eta-prime meson and a deeper understanding of how the strong force shapes the universe. The work demonstrates that sometimes, the key to seeing the faintest details is not to look harder, but to look smarter, filtering out the chaos to reveal the underlying order.

Technical Summary: Subtraction Method for Disconnected Diagrams in Full QCD

Problem Statement
Disconnected diagrams remain a primary bottleneck for precision lattice QCD calculations, particularly for flavor-singlet hadrons. The numerical evaluation of these diagrams is hindered by extreme statistical fluctuations arising from quark loops. In disconnected correlators, the variance is typically dominated by the vacuum expectation values (VEVs) of quark-loop operators, causing the signal-to-noise ratio to deteriorate exponentially with Euclidean time separation. This issue is exacerbated in systems involving heavy quarks, such as charmonium, where the signal decays faster than in light-quark systems. Consequently, disconnected contributions are often neglected in precision studies (e.g., the $\eta_c$ mass or nucleon three-point functions), introducing significant uncertainties. While various noise-reduction techniques exist (stochastic estimators, low-mode averaging, multi-level algorithms), achieving a sufficient signal-to-noise ratio often remains computationally prohibitive.

Methodology
The authors propose a subtraction method designed to reduce statistical fluctuations by removing a term that is strongly correlated with the full disconnected correlator but contributes negligibly to the physical signal in specific channels. The core of the method relies on the domain decomposition of the Dirac operator.

Domain Decomposition: The temporal lattice extent $\Lambda$ is partitioned into two non-overlapping regions, $\Lambda_0$ and $\Lambda_1$ . The Dirac operator is written in block form, allowing the exact quark propagator $D^{-1}(x,x)$ to be decomposed into a term local to the region ( $D^{-1}_{\Lambda_i}$ ) and correction terms involving propagation across boundaries.
Factorization: The quark loop $L_\Gamma(x_0)$ is separated into a local contribution $L_{\Gamma, \Lambda_0}(x_0)$ and a correction $\delta L_{\Gamma, \Lambda_0}(x_0)$ . As the size of the region increases, the local term becomes an increasingly accurate approximation of the full loop, while the correction is suppressed exponentially with the distance from the boundaries.
The Subtraction: The disconnected correlator $C^{(d)}_\Gamma$ $C_{Γ}^{(d)}$ is decomposed into a "completely factorized" term $C^{(d)}_{\Gamma, \Lambda_0, \Lambda_1}$ $C_{Γ, Λ_{0}, Λ_{1}}^{(d)}$ and a remainder $\delta C^{(d)}_{\Gamma, \Lambda_0, \Lambda_1}$ $δ C_{Γ, Λ_{0}, Λ_{1}}^{(d)}$ .
- The factorized term involves quark loops restricted to non-overlapping regions. Physically, these loops communicate only through sea gauge-field fluctuations, making their correlation exponentially suppressed with the separation between regions.
- The authors define a noise-reduced estimator by subtracting this factorized term: $\delta C^{(d)}_{\Gamma, \Lambda_0, \Lambda_1} = C^{(d)}_\Gamma - C^{(d)}_{\Gamma, \Lambda_0, \Lambda_1}$ .
Bias vs. Variance Trade-off: The resulting estimator is biased because the factorized term is neglected. However, the variance of the subtracted estimator is free of the dominant constant term in the variance equation (the product of the VEVs of the local loops). The size of the non-overlapping regions ( $\Delta$ ) controls the trade-off: larger regions better approximate the full loop (reducing variance) but decrease the separation between regions (increasing the bias of the neglected term).

Key Contributions and Results
The method was tested on flavor-singlet charmonium two-point functions using $N_f = 3+1$ ensembles with Wilson-Clover action and open boundary conditions.

Standard Distillation Analysis:
- Using standard distillation with 200 Laplacian eigenvectors, the authors applied the subtraction method with varying block sizes ( $\Delta/a = 7, 9, 11, 13$ ).
- Variance Reduction: For the largest block size ( $\Delta/a = 13$ ), the statistical error of the disconnected contribution was reduced by a factor of nearly 50 compared to the standard estimator. This corresponds to a variance reduction of approximately three orders of magnitude ( $O(2500)$ ).
- Computational Cost: This reduction was achieved with only a factor of two increase in computational cost. The authors note that achieving the same error reduction via standard statistics would require roughly $O(10^6)$ gauge configurations, which is computationally prohibitive.
- Bias: The bias introduced by neglecting the factorized term was found to be small for the $\eta_c$ channel ( $\Gamma = \gamma_5$ ) at large source-sink separations. However, for the $\Gamma = I$ channel, the bias was more significant, indicating a larger contribution from the factorized term.
Gaussian-Profiled Distillation:
- The authors also tested the method using Gaussian-profiled distillation sources to modulate eigenmode contributions.
- For the $\Gamma = \gamma_5$ channel, the narrowest Gaussian profile ( $\omega_0 = 0.05$ ) combined with the subtraction method provided an additional factor of two reduction in statistical uncertainty compared to standard distillation.
- For the $\Gamma = I$ channel, the narrowest profile resulted in the signal being dominated by the completely factorized block-block correlator. In this specific case, neglecting the factorized term removed most of the physical signal, rendering the subtraction method ineffective without further correction.
Multilevel Sampling Potential:
- The analysis revealed that for the $\Gamma = I$ channel with narrow profiles, both the signal and the dominant statistical noise are concentrated in the completely factorized term. The authors suggest that this specific term is ideally suited for multilevel sampling algorithms. By computing the dominant factorized term with a multilevel algorithm and the remainder with standard sampling, one could construct an unbiased estimator with reduced variance, avoiding the need for more elaborate propagator approximations.

Significance and Claims
The paper claims to provide a practical alternative to completely neglecting disconnected contributions in lattice QCD. By exploiting the locality of quark propagation, the method offers a substantial reduction in statistical noise (up to three orders of magnitude in variance) at a modest computational cost (factor of two).

The authors emphasize that the method is channel-dependent:

It is highly effective for channels where the physical signal is small compared to the variance (e.g., $\eta_c$ ), allowing for the resolution of correlators at larger Euclidean time separations.
It introduces a bias that must be managed; however, this bias can be estimated efficiently using two-level sampling techniques.
The approach offers a pathway to efficient, unbiased estimators by combining the subtraction method with multilevel algorithms for the dominant factorized terms, particularly in channels where the subtraction approximation alone is insufficient.

The work does not claim to solve the signal-to-noise problem for all observables universally but demonstrates a viable strategy for specific heavy-quark flavor-singlet observables where the factorized contribution is physically negligible or can be treated separately.

Technical Summary: Subtraction Method for Disconnected Diagrams in Full QCD

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