Efficiently unquenching QCD+QED at O($\alpha$)

This paper outlines a strategy to efficiently compute electromagnetic sea-quark effects in QCD+QED by analyzing stochastic estimator variances to improve the precision of challenging disconnected diagrams, supported by preliminary results from $N_\mathrm{f}=2+1$ domain-wall fermion ensembles.

The Gist

Imagine you are trying to bake the perfect cake to understand how the universe works. In the world of particle physics, this "cake" is a model of matter called QCD (Quantum Chromodynamics). For a long time, scientists have been baking this cake using a recipe that assumes all the ingredients are perfectly identical twins. They assumed the "up" and "down" quarks (the basic ingredients) were exactly the same, just like two identical eggs.

However, in reality, these ingredients aren't twins. One is slightly heavier, and one has a tiny bit of electric charge while the other doesn't. This difference is called iso-spin breaking. To get a truly perfect cake (a precise prediction for things like the magnetic strength of a muon), you have to account for these tiny differences.

This paper is about a new, efficient way to mix those tiny differences into the batter without ruining the whole batch.

The Problem: The "Ghost" Ingredients

When scientists try to add the electric charge of the "sea quarks" (the virtual particles that pop in and out of existence inside the cake) to their calculations, they run into a massive computational headache.

Think of it like this: To calculate the effect of these sea quarks, you have to trace every possible path a particle could take through the cake. Some of these paths are "connected" (like a direct line from the start to the finish). But others are "disconnected"—imagine a ghostly loop that floats in the middle of the cake, touching nothing else.

These disconnected loops are notoriously noisy. If you try to measure them, the signal is so weak and the background noise so loud that it's like trying to hear a whisper in a hurricane. In the past, scientists often ignored these "ghost" loops (a method called "electroquenching"), but that leaves a hidden error in their results.

The Solution: Smarter Math Tricks

The authors of this paper, Tim Harris and his team, propose a strategy to hear that whisper clearly without needing a supercomputer the size of a planet. They use a method called RM123, which is like a mathematical expansion that breaks the problem down into small, manageable pieces.

They focus on two specific types of "ghost" loops (diagrams labeled $W_1$ and $W_2$) and apply two clever tricks:

1. The "Cancellation" Trick (For Diagram $W_1$)

In the first type of loop, the noise from the "up" quarks and the "strange" quarks naturally cancels each other out, much like how two people pushing a car in opposite directions might keep it still.

• The Analogy: Imagine you are trying to measure the wind speed by holding a flag. If the wind blows the flag left, it's hard to measure. But if you have two flags, one that blows left and one that blows right with the exact same force, they cancel out, and the remaining movement is very small and easy to measure.
• The Result: The authors found that by combining the flavors of quarks in a specific way, the "noise" drops by a factor of 10,000. They also used a special mathematical shortcut (called a "split-even estimator") that acts like a noise-canceling headphone, making the calculation incredibly efficient.

2. The "Zoom-In" Trick (For Diagram $W_2$)

The second type of loop doesn't have that natural cancellation. The noise is loud and comes mostly from the very center of the loop (the short-distance part).

• The Analogy: Imagine trying to measure the temperature of a room. The temperature near the heater (the center) is wild and fluctuating, but the temperature in the corners (the long-distance part) is calm and steady.
• The Strategy: Instead of trying to measure the whole room with one expensive, high-tech thermometer, they split the job.
  • The Heater Zone: They use a powerful, fast computer method to measure the chaotic center very precisely.
  • The Corners: They use a simple, cheap method (just taking a few random samples) to measure the calm corners.
• The Result: This "frequency splitting" allows them to get a precise answer without wasting energy measuring the calm parts too many times.

The Ingredients Used

To test this, they didn't just use theory; they ran actual simulations on a supercomputer using a specific type of "cake batter" (called domain-wall fermions) generated by the RBC/UKQCD collaboration.

The Bottom Line

The paper shows that by using these specific mathematical tricks—cancelling out noise for some parts and splitting the work for others—it is possible to include the electric charges of sea quarks in our models of matter.

This means we can finally stop ignoring the "ghost" loops and get a much clearer, more accurate picture of how the universe works, all without needing to wait for a thousand years of computer time. It's a way to make the "whisper" of the sea quarks loud enough to be heard, ensuring our predictions for the Standard Model are as precise as possible.

Technical Summary

1. Problem Statement

Precision tests of the Standard Model (e.g., the muon $g-2$, decay constants $f_K/f_\pi$, and the Wilson flow scale) require lattice QCD predictions with uncertainties at or below the 1% level. Achieving this precision necessitates including electromagnetic (QED) interactions and isospin-breaking effects (arising from the mass difference between up and down quarks and their different electric charges).

Current computational strategies often rely on the electroquenched approximation, where the electric charges of sea quarks (dynamical quarks) are neglected. This introduces uncontrolled systematic errors. The RM123 approach allows for the inclusion of sea-quark electric charges by expanding physical observables in the electromagnetic coupling $\alpha$ and the quark mass difference. However, this expansion introduces quark-line disconnected diagrams (specifically Wick contractions $W_1$ and $W_2$) that involve traces of quark propagators. These diagrams are notoriously difficult to compute precisely due to:

• Large statistical noise (variance).
• The need for stochastic estimators, which introduce auxiliary fields and additional fluctuations.
• The computational cost scaling with the number of stochastic sources.

2. Methodology

The authors propose a strategy to efficiently compute the leading-order isospin-breaking corrections ($O(\alpha)$) by analyzing the variance of stochastic estimators for the relevant Wick contractions. They utilize an ensemble of $N_f = 2+1$ domain-wall fermions generated by the RBC/UKQCD collaboration.

The methodology focuses on two specific disconnected subdiagrams arising from sea-quark charges:

A. The Disconnected Subdiagram ($W_1$)

• Structure: This diagram involves a convolution of two traces of quark propagators, $T_\mu(x)$ and $T_\nu(y)$, connected by a photon propagator.
• Flavor Cancellation: In an $N_f=2+1$ theory with isospin symmetry ($m_u=m_d$), the trace $T_\mu$ is proportional to the difference between light-quark ($u,d$) and strange-quark ($s$) propagators: $S_{ud} - S_s$.
• Identity Utilization: The authors exploit the identity $S_{ud} - S_s = (m_s - m_{ud}) S_{ud} S_s$. This explicitly suppresses the signal by the isospin-breaking parameter $(m_s - m_{ud})$ and, crucially, suppresses the variance by $(m_s - m_{ud})^4$.
• Estimator Strategy: They compare two stochastic estimators for the trace:
  1. Standard Estimator: Uses a single set of noise vectors.
  2. Split-Even Estimator: Splits the trace calculation into two parts using independent noise vectors ($\Theta_\mu$ vs. $T_\mu$).
• Implementation: The convolution with the photon propagator is computed efficiently using Fast Fourier Transforms (FFT).

B. The Connected Subdiagram ($W_2$)

• Structure: This diagram involves a single trace of a product of two propagators at different spacetime points, $S(x, y)S(y, x)$. Unlike $W_1$, there is no flavor cancellation in the variance; the light and strange quark contributions add constructively.
• Variance Behavior: The variance is dominated by short-distance contributions and diverges as $a^{-4}$ (where $a$ is the lattice spacing).
• Hybrid Estimation Strategy: To manage the noise, the authors propose decomposing the diagram based on the separation $r = |x-y|$:
  1. Short-Distance ($|r| \leq R$): Estimated using a stochastic "all-to-all" estimator (using noise vectors) because the signal is strong and the variance scales favorably ($N_s^{-2}$).
  2. Long-Distance ($|r| > R$): Estimated using position-space sampling (randomly selected point sources). This avoids the expensive all-to-all calculation for large separations where the signal is weak.
• Photon Propagator: The photon field is treated exactly (using the QED$_L$ formulation in Feynman gauge) rather than stochastically, eliminating one source of variance.

3. Key Contributions

1. Generalization of Split-Even Estimators: The authors demonstrate that the "split-even" technique, previously used for Wilson fermions, applies exactly to domain-wall fermions (which satisfy the Ginsparg-Wilson relation). This allows for the exploitation of flavor cancellations in the variance.
2. Variance Reduction via Flavor Decomposition: They show that for $W_1$, the specific combination of quark flavors leads to a massive suppression of variance (by a factor of $\sim 10^4$) when using the split-even estimator compared to the standard estimator.
3. Hybrid Decomposition for $W_2$: They introduce a novel decomposition strategy for the connected diagram $W_2$, splitting it into short-distance (stochastic) and long-distance (point-source) components. This balances computational cost and statistical precision.
4. Exact Photon Treatment: By convolving with the exact photon propagator rather than stochastically sampling the U(1) gauge field, they remove an entire layer of stochastic noise.

4. Results

The authors present preliminary numerical results using the C1 ensemble ($N_f=2+1$, $m_\pi \approx 340$ MeV, $L/a=24$):

• For $W_1$ (Disconnected):

  • The variance of the split-even estimator is reduced by a factor of $10^4$ compared to the standard estimator.
  • The variance scales as $1/N_s^2$ (where $N_s$ is the number of stochastic sources) until it plateaus at the gauge-field variance level (around $N_s \approx 100$).
  • The flavor cancellation mechanism works as predicted, suppressing the noise significantly.

• For $W_2$ (Connected):

  • The variance is dominated by the short-distance region ($|r| \approx 0$).
  • The short-distance component scales as $N_{inv}^{-2}$ (where $N_{inv}$ is the number of inversions), while the long-distance component scales as $N_{inv}^{-1}$.
  • By choosing a cutoff $R/a = 4$, the total variance is sufficiently reduced to reach the gauge variance limit with a reasonable computational cost ($N_{inv} \sim 1000$).
  • Translation averaging (via the all-to-all estimator for short distances) proves highly effective, suppressing the gauge variance by a factor of $(L^3 T)/a^4$.

5. Significance

• Systematic Error Control: This work provides a viable path to "unquench" QCD+QED simulations at $O(\alpha)$, allowing for the inclusion of sea-quark electric charges without incurring prohibitive computational costs. This eliminates the uncontrolled systematic error associated with the electroquenched approximation.
• Scalability: The proposed decompositions are universal and can be applied to any isospin-breaking correction, not just specific observables.
• Future Applications: The methods are being integrated into the RBC/UKQCD collaboration's large-scale simulations to refine calculations of meson decay rates and other precision observables.
• Algorithmic Efficiency: The combination of flavor-cancellation identities, split-even estimators, and hybrid stochastic/position-space sampling offers a blueprint for handling high-variance disconnected diagrams in lattice field theory.

In summary, the paper successfully outlines and validates a strategy to efficiently compute the challenging sea-quark electromagnetic contributions in lattice QCD, making high-precision Standard Model tests more robust.