Isospin breaking corrections to the hadronic vacuum polarization with stochastic coordinate sampling

This paper presents the RBC/UKQCD collaborations' current status on calculating isospin breaking corrections to the hadronic vacuum polarization using stochastic coordinate sampling to efficiently construct all necessary Wick contractions and employing various lattice QED formulations to better estimate finite-volume uncertainties.

The Gist

The Big Picture: The Muon's "Wobble"

Imagine the Muon as a tiny, spinning top. In the world of particle physics, this top doesn't just spin; it wobbles. Scientists call this wobble the "anomalous magnetic moment" (or $g-2$).

For decades, physicists have tried to predict exactly how much this top should wobble based on the Standard Model (the rulebook of the universe). Recently, experiments at Fermilab measured the wobble with incredible precision. The result? The real world wobble is slightly different from the rulebook's prediction. This suggests there might be new, undiscovered particles or forces at play.

However, to be sure, we need to make sure our calculation of the "rulebook" prediction is perfect. The biggest source of error in that calculation comes from something called Hadronic Vacuum Polarization (HVP).

The Problem: The "Noisy" Background

Think of the vacuum (empty space) not as truly empty, but as a bubbling soup of virtual particles popping in and out of existence. When a muon moves through this soup, it interacts with these bubbles.

Calculating this interaction is like trying to hear a whisper in a hurricane. The "hurricane" is the strong nuclear force, which is incredibly complex and doesn't follow simple math rules (it's "non-perturbative"). To solve this, scientists use Lattice QCD, which is like building a giant 3D grid (a lattice) and simulating the particles on a supercomputer.

The Catch: Most of these simulations assume the universe is perfectly symmetrical. They pretend the Up quark and the Down quark (two types of building blocks for protons and neutrons) have the exact same mass and that there are no photons (light particles) messing things up.

In reality, the Up quark is slightly lighter than the Down quark, and photons do exist. These tiny differences are called Isospin Breaking. While they seem small (about 1%), they are huge enough to ruin the precision needed to match the new experimental data.

The Solution: Stochastic Coordinate Sampling (SCS)

The paper by the RBC/UKQCD collaboration is about fixing this 1% error.

The Challenge:
To calculate these tiny corrections, you have to draw millions of complex diagrams (like Feynman diagrams) representing how particles interact. Some of these diagrams are "connected" (easy to trace), but others are "disconnected" (like a ghostly cloud where particles pop in and out of existence without a clear path).

• The Analogy: Imagine trying to count every single grain of sand on a beach to find a few specific, slightly different grains. If you try to count them one by one, it would take forever. If you try to count them all at once, the noise drowns out the signal.

The Innovation: Stochastic Coordinate Sampling (SCS)
Instead of trying to calculate every single interaction point on the grid (which is computationally impossible), the authors use a clever trick called Stochastic Coordinate Sampling.

• The Metaphor: Imagine you want to know the average temperature of a huge city. You could put a thermometer in every single house (impossible). Instead, you randomly pick 1,000 houses, measure the temperature there, and use that data to estimate the whole city.
• How it works here: The team generates a massive dataset of "propagators" (paths particles take) at random points on the grid. They then use these random samples to reconstruct the complex "disconnected" diagrams. It's like using a few well-placed snapshots to reconstruct a 3D movie of the particle soup.

The Results: Taming the Noise

The paper presents "preliminary results" (early drafts of the final answer). They tested their method on different types of simulations:

1. QED$_L$, QED$_r$, QED$_\infty$: These are different ways of handling the "box" the simulation lives in. Since the computer grid is finite, the photons bounce off the walls. The team tested different rules for these walls to ensure their results aren't just an artifact of the box size.
2. The "Tail" Problem: In these calculations, the signal gets very noisy at large distances (the "tail" of the data). The authors used a technique to "reconstruct" this tail based on known physics (like how pions and rho mesons behave), effectively smoothing out the noise.

What they found:

• They successfully calculated the contributions of all the different diagrams (the "connected" ones and the tricky "disconnected" ones).
• They confirmed that the "disconnected" diagrams (the ghostly clouds) cancel out a lot of the "connected" ones. This cancellation is delicate; if you get the math slightly wrong, the whole result blows up.
• Their method works! They can now calculate these corrections with enough precision to help solve the Muon mystery.

Why This Matters

This paper is a crucial step in the "Muon $g-2$" detective story.

• Before: We had a great measurement of the wobble, but our theoretical calculation was too fuzzy to say for sure if new physics was involved.
• Now: This team has built a better microscope (SCS) to look at the tiny 1% differences caused by quark mass and photons.

By refining these corrections, they are helping to sharpen the theoretical prediction. If the gap between the prediction and the experiment remains after these corrections are applied, it will be strong evidence that we have discovered something new about the universe. If the gap closes, it means the Standard Model is still holding up, and the "new physics" might be hiding elsewhere.

In short: They found a smarter, faster way to count the invisible particles in the vacuum, ensuring we don't miss the tiny clues that could rewrite the laws of physics.

Technical Summary

1. Problem Statement

The anomalous magnetic moment of the muon ($a_\mu$) is a critical test of the Standard Model (SM). Recent experimental measurements at Fermilab have reached a precision of 127 parts-per-billion (ppb), creating a tension with theoretical predictions. The dominant source of uncertainty in the SM prediction arises from the Hadronic Vacuum Polarization (HVP) contribution.

While the dispersive approach (using experimental $e^+e^-$ data) and Lattice QCD are the two methods to calculate HVP, recent lattice results have been adopted as the primary reference. However, standard lattice simulations are typically performed in isospin-symmetric QCD (isoQCD), where the up ($u$) and down ($d$) quarks have equal masses and photons are absent.

• The Challenge: To match experimental precision, isospin-breaking (IB) effects—arising from the $u-d$ quark mass difference and electromagnetic (QED) interactions—must be included. These effects are of order 1% but introduce significant computational complexity.
• Specific Difficulties:
  • Calculating all necessary Wick contractions for QED and strong IB effects is computationally expensive.
  • Signal-to-noise degradation: Quark-disconnected diagrams (where quark loops are not connected to the external currents) suffer from severe noise, particularly at large time separations.
  • Finite Volume Effects: The inclusion of photons introduces infrared divergences, requiring careful treatment of finite-volume regularization schemes (e.g., $QED_L$, $QED_r$, $QED_\infty$).

2. Methodology

The authors employ the RM123 approach, which treats isospin breaking as a perturbation around the isoQCD baseline. The HVP correlator $C(t)$ is expanded as:
$$C(t) = C_{\text{isoQCD}}(t) + C_{\text{QED}}(t) + C_{\text{SIB}}(t) + O(\alpha^2, (\Delta m_f)^2)$$

Key Technical Components:

• Stochastic Coordinate Sampling (SCS):
  • To handle the "all-to-all" propagator requirements for 4-point functions (necessary for QED corrections) without prohibitive cost, the authors use SCS.
  • They utilize a pre-computed dataset of propagators with random source and sink positions (tuples of $N$ sources and $M$ sinks) generated on three gauge ensembles (48I, C, and 4) from the RBC/UKQCD collaborations.
  • This allows the construction of estimators for complex diagrams (like Fig. 1 in the paper) by summing over sampled coordinates, with correction factors ($\zeta$) applied to account for sampling density.
• Diagram Decomposition:
  • QED Corrections ($O(\alpha)$): Involves 14 distinct diagrams (Fig. 1), including connected (V, S) and disconnected (F, D1, D2, etc.) topologies. The disconnected diagrams require subtracting vacuum expectation values (VEV) to avoid double-counting.
  • Strong Isospin Breaking (SIB): Involves mass derivatives ($\Delta m_u, \Delta m_d$) inserted as scalar operators, resulting in diagrams shown in Fig. 2 (M, R, O, Rd).
• Photon Regularization:
  • The study compares different lattice QED formulations to estimate finite-volume uncertainties:
    • $QED_L$: Standard finite volume with zero-mode removal.
    • $QED_r$: Redistributes zero modes to the lowest non-zero modes.
    • $QED_\infty$: Simulates infinite volume by increasing the QED box size ($L \to \infty$).
• Noise Reduction Techniques:
  • VEV Subtraction: For disconnected diagrams (e.g., diagram F), the VEV of subdiagrams is subtracted on a per-source basis to significantly reduce statistical noise.
  • Low-Mode Reconstruction: For tadpole diagrams (T, D2, D3), the authors use multi-grid Lanczos to compute exact low-mode fields and estimate high-mode contributions using sparse $Z_2$ grids.
  • Tail Reconstruction: To mitigate the large noise-to-signal ratio at large time separations ($t > 1.5$ fm), the authors reconstruct the correlator tail using asymptotic intermediate states ($\pi\gamma$ and $\rho$).

3. Key Contributions

• Implementation of SCS for IB Corrections: The paper presents the first application of stochastic coordinate sampling to the full set of isospin-breaking corrections for HVP, covering both QED and strong IB effects.
• Systematic Validation: The authors validated their implementation by computing all diagrams on a small $4^3 \times 8$ lattice using full all-to-all propagators, confirming that the stochastic estimators converge to the exact results.
• Finite Volume Analysis: By employing $QED_L$, $QED_r$, and $QED_\infty$, the study provides a robust framework for quantifying finite-volume systematic errors, which is crucial for precision physics.
• Kaon Mass Splitting: The methodology is extended to calculate the neutral-charged kaon mass difference ($\Delta(m_{K^0} - m_{K^\pm})$), a necessary input for fixing the renormalization scheme of the IB corrections.

4. Preliminary Results

The authors present blinded preliminary results for the integrand of the HVP contribution:

• Connected Contributions (V, S):
  • A strong cancellation is observed between the vector (V) and self-energy (S) diagrams, consistent with previous RBC/UKQCD and Mainz results.
  • The statistical uncertainty is dominated by diagram (S) at large $t$. The tail reconstruction method successfully reduces this uncertainty.
• Disconnected Contributions (F):
  • Diagram (F) contributes significantly with an opposite sign to connected terms, leading to large cancellations in the total sum.
  • VEV Subtraction: Performing VEV subtraction on individual SCS summands drastically reduces statistical errors.
  • Volume Dependence: In $QED_L$, the contribution is largely contained within $0.7$ fm, whereas $QED_\infty$ shows a longer range, highlighting the importance of volume corrections.
• Tadpole and SIB Diagrams:
  • Tadpole diagrams (T, D2) and Strong IB diagrams (M, R, O) are resolved with sufficient accuracy.
  • SIB diagrams exhibit similar noise problems at large $t$ as QED diagrams.
• Kaon Mass Splitting: Ratios of contributing diagrams for the kaon mass splitting are presented, showing that tadpole contributions are subleading compared to connected and self-energy terms.

5. Significance

This work represents a major step toward a sub-1% precision determination of the HVP contribution to the muon $g-2$ using Lattice QCD.

• Feasibility: It demonstrates that stochastic coordinate sampling is a viable and efficient method for computing the computationally prohibitive all-to-all propagator contractions required for IB corrections.
• Precision: By combining SCS with noise-reduction techniques (VEV subtraction, tail reconstruction, low-mode reconstruction), the collaboration aims to overcome the signal-to-noise barriers that have historically limited the precision of disconnected diagrams.
• Future Outlook: The authors plan to expand the dataset to include finer lattice spacings (ensembles 64I and 96I) and perform dynamical QED simulations to further refine the treatment of noisy disconnected contributions. This work is essential for resolving the current tension between experimental $a_\mu$ measurements and Standard Model predictions.