Progress on computing the hadronic vacuum polarization contribution to the muon anomalous magnetic moment with staggered fermions

This paper presents an update on the calculation of the light-quark, connected hadronic vacuum polarization contribution to the muon anomalous magnetic moment using $2+1+1$ flavor HISQ ensembles with physical pion mass, highlighting preliminary results and algorithmic improvements for computing vector-vector correlation functions.

The Gist

The Big Picture: Why Are We Doing This?

Imagine you are trying to weigh a feather on a scale that is incredibly sensitive. You know exactly how heavy the feather should be based on the laws of physics (Theory). You also have a super-precise scale that measures the actual weight (Experiment).

Recently, scientists measured the "weight" of a tiny particle called a muon (a heavy cousin of the electron). The measurement was so precise that it didn't match the theoretical prediction. This mismatch is like finding a ghost on the scale—it suggests there might be a new, invisible particle or force interacting with the muon that we haven't discovered yet.

However, before we can claim we found a "ghost," we have to make sure our theoretical calculation of the "expected weight" is perfect. The biggest source of uncertainty in that calculation comes from a messy, chaotic cloud of particles called Hadronic Vacuum Polarization (HVP).

This paper is a progress report from a team of physicists trying to clean up that calculation. They are using a supercomputer to simulate the universe on a tiny grid (a "lattice") to get a more accurate number for the muon's weight.

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The Challenge: The Noisy Room

To calculate this, the team has to listen to a very specific sound (a signal) coming from a very noisy room.

• The Signal: The interaction of quarks (the building blocks of protons and neutrons) with the muon.
• The Noise: Random statistical fluctuations that make the data look messy.

In the past, their method was like trying to hear a whisper in a crowded stadium by standing in one spot and guessing what everyone else is saying. It worked okay, but the "noise" was still too loud, especially for the parts of the signal that happen far away in time (the "long-distance" parts).

The New Tools: Better Microphones and Smarter Listeners

The team has introduced two major upgrades to their "listening" equipment to reduce the noise and get a clearer picture.

1. The "Low-Mode" Strategy (Hearing the Bass)

Think of the quark interactions like a song. The "Low Modes" are the deep bass notes that carry the most energy and travel the furthest. The "High Modes" are the high-pitched, chaotic static.

• Old Way: They treated the bass and the static together. Because the static was so loud, it drowned out the subtle details of the bass.
• New Way: They separate the bass from the static.
  • They calculate the Bass (Low Modes) perfectly by using a special technique called Low-Mode Averaging (LMA). This is like using a high-quality microphone specifically for the deep notes.
  • They calculate the Static (High Modes) using a different, faster method.
  • The Trick: They realized that if they just averaged the static over a tiny area, it was still noisy. So, they used a "random hit" method. Imagine trying to guess the average temperature of a city. Instead of measuring one house, they take a few random samples from different neighborhoods and average them. By doing this "random sampling" (called "hits") on the low-frequency parts, they can get a much clearer picture of the whole city without measuring every single house.

2. The "Sparse" Grid (Skipping Steps)

The computer grid they use is huge (imagine a 3D checkerboard with millions of squares). Calculating the "bass notes" (Low Modes) on every single square takes forever and uses up all the computer's memory.

• The Analogy: Imagine you are painting a massive mural. You don't need to paint every single pixel perfectly to see the big picture.
• The Solution: They use a technique called Sparsening. Instead of painting every square, they paint every 2nd or 4th square in a regular pattern.
  • Because the "pixels" next to each other are very similar (correlated), skipping some doesn't change the final image much.
  • This makes the calculation 8 to 64 times faster and saves a massive amount of computer memory, allowing them to use even more "bass notes" to get a better result.

The Results: A Clearer Picture

The team tested these new methods on two different "simulated universes" (lattice sizes):

1. The Old One (64³): They compared their old method with the new one. The new method reduced the error (the "fuzziness" of the data) significantly, especially in the long-distance parts where the noise used to be worst. With 10 "random hits," they improved the accuracy by nearly 24%.
2. The New One (144³): This is a much finer, more detailed grid (like a 4K screen vs. an old TV). They are just starting to run calculations here. The preliminary results look promising, showing that their new methods work even on these massive, complex grids.

The Bottom Line

This paper isn't the final answer yet, but it's a massive step forward.

• Before: They were trying to solve a puzzle with half the pieces missing and a lot of fog.
• Now: They have cleared the fog, found a way to get the missing pieces faster, and are building a clearer, more accurate picture of the muon's magnetic moment.

If their final numbers match the experimental "ghost" (the discrepancy), it could mean we have discovered New Physics—something beyond our current understanding of the universe. If they don't match, it means the "ghost" might just be a calculation error, and we need to keep looking. Either way, this work is essential to solving the mystery.

Technical Summary

1. Problem Statement

The paper addresses the theoretical calculation of the Hadronic Vacuum Polarization (HVP) contribution to the muon anomalous magnetic moment ($a_\mu$). This quantity is critical for testing the Standard Model of particle physics, as recent high-precision experiments (Fermilab E989) have shown a tension with theoretical predictions.

• Specific Challenge: The authors focus on the light-quark, connected contribution using Highly Improved Staggered Quark (HISQ) action for $2+1+1$ flavors.
• Computational Bottleneck: Calculating the Euclidean time correlation function $C(t)$ with sufficient precision is computationally expensive, particularly at long Euclidean times where statistical noise dominates. Previous methods using Low-Mode Averaging (LMA) and All-Mode Averaging (AMA) were insufficient for the target lattice size ($144^3 \times 288$) and physical pion mass required to match experimental precision.

2. Methodology

The authors employ Lattice QCD combined with continuum perturbative QED. The calculation relies on the time-momentum representation of the HVP integral, where $a_\mu$ is derived from the weighted sum of the vector-vector correlation function $C(t)$.

To improve efficiency and reduce statistical errors, the authors implemented several algorithmic advancements:

A. Spectral Decomposition and Separation of Terms

The quark propagator $S(x,y)$ is decomposed into low modes ($S_L$) and high modes ($S_H$) based on the eigenvalues of the Dirac operator. Consequently, the correlation function $C(t)$ is split into four components:

1. LL (Low-Low): Both propagators are low modes.
2. HL (High-Low) & LH (Low-High): One propagator is low, the other is high.
3. HH (High-High): Both propagators are high modes.

B. Algorithmic Improvements

1. Separation of HL and HH Contributions:

   • Previous Approach: HL and HH were computed together as a "rest" term in an AMA framework, averaging over only a fraction of source points.
   • New Approach: The HL contribution is computed separately. Each low-mode eigenvector is used as a source for the high-mode part of the propagator. This allows for a full-volume average of the HL term, significantly reducing noise compared to the previous partial averaging.
   • Cost Reduction: To mitigate the high cost of inverting the Dirac operator for every eigenvector, the authors use a randomized "hit" method. They form linear combinations of low-mode sources using unique random numbers and contract them at the sink with the same random numbers. This cancels cross-terms on average while adding manageable random noise.

2. Sparsening for LL Contributions:

   • The LL term requires constructing "meson fields" from eigenvectors, which scales quadratically with the number of eigenvectors and linearly with lattice volume. This is prohibitive for large lattices ($144^3 \times 288$).
   • Solution: The authors implement sparsening, where they omit a regular pattern of lattice sites (e.g., every $s$-th point) when computing meson fields. Since nearby points are highly correlated, this reduces the memory footprint and computational cost (by factors of 8 or 64) without significantly degrading statistical precision.

3. Conserved Currents:

   • To ensure gauge invariance and current conservation on the lattice, a point-split conserved current is used instead of the local current.

3. Key Contributions

• Novel HL Treatment: The decoupling of the High-Low (HL) term from the High-High (HH) term to enable full-volume averaging, which was previously a major source of statistical noise.
• Sparsening Technique: A new method to accelerate the calculation of Low-Low (LL) meson fields on large lattices by selectively omitting lattice sites while preserving the spin-taste structure of staggered fermions.
• Randomized Source Averaging: The implementation of a "hit" strategy to reduce the computational cost of the HL full-volume average while maintaining statistical control.
• New Lattice Ensemble: Presentation of preliminary results on a physical pion mass ensemble with a very fine lattice spacing ($a = 0.042$ fm) and large volume ($144^3 \times 288$), generated by the MILC collaboration.

4. Results

The authors compared their new methodology against their previous work on a coarser $64^3 \times 96$ ensemble and applied it to the new $144^3 \times 288$ ensemble.

• Error Reduction (643 Ensemble):
  • The new method reduced the total statistical error by 6.78% using a "1 hit" approach.
  • Using a "10 hits" approach, the error in the long-distance window ($2.3 - 3.3$ fm) was reduced by 23.7%.
  • The HL contribution, previously a dominant noise source, was significantly suppressed relative to the LL part.
• Preliminary Results (1443 Ensemble):
  • Preliminary values for the intermediate and long-distance windows of $a_\mu^{HVP}$ were computed.
  • The results show consistency with previous calculations at coarser lattice spacings, though the statistical errors are currently larger due to the limited number of configurations measured so far.
  • The summand $w(t)C(t)$ shows the expected behavior, with the new method effectively handling the long-distance noise.

5. Significance

• Matching Experimental Precision: As experimental precision on the muon $g-2$ improves (aiming for 0.14 ppm), theoretical calculations must match this accuracy. The proposed algorithmic improvements are essential to make calculations on fine, large-volume lattices computationally feasible.
• Controlling Systematic Errors: By moving to physical pion masses and finer lattice spacings ($0.042$ fm), the collaboration aims to control discretization errors and finite-volume effects, which are critical for a definitive Standard Model prediction.
• Scalability: The combination of HL separation and LL sparsening provides a scalable framework for future calculations on even larger and finer lattices, potentially resolving the current tension between theory and experiment in the muon $g-2$ sector.

In conclusion, this paper presents a significant step forward in Lattice QCD methodology, offering specific, tested techniques to overcome the statistical and computational barriers in calculating the HVP contribution to the muon anomalous magnetic moment.