An update on the HVP contribution to $g_μ{-}2$ in isoQCD from ETMC

The Extended Twisted Mass Collaboration presents an updated determination of the leading-order hadronic vacuum polarisation contribution to the muon anomalous magnetic moment in isospin-symmetric QCD, utilizing five $N_f=2+1+1$ gauge ensembles with near-physical pion masses and employing two distinct valence-quark regularisations to calculate both isovector and isoscalar components.

The Gist

Here is an explanation of the paper, translated from complex physics jargon into everyday language using analogies.

The Big Picture: The "Magnetic Spin" Mystery

Imagine a muon (a particle similar to an electron, but heavier) as a tiny, spinning top. Because it's charged, this spinning top acts like a tiny magnet. Physicists have a very precise formula to predict how strong this magnet should be. This prediction is called the "Standard Model."

However, when scientists actually measure these spinning tops in a lab (at Fermilab and Brookhaven), they spin slightly faster than the formula predicts. It's like calculating exactly how fast a car should go on a highway, but the car is actually going 5 mph faster.

This difference is called the Muon Anomalous Magnetic Moment (or $g-2$). The question is: Why is the car going faster?

There are two main suspects:

1. New Physics: There might be invisible particles or forces we haven't discovered yet pushing the car.
2. Bad Math: Our calculation of the "road conditions" (the vacuum of space) might be wrong.

This paper is about checking the "road conditions" using a super-computer simulation called Lattice QCD.

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The "Road Conditions": The Hadronic Vacuum Polarization (HVP)

In quantum physics, empty space isn't actually empty. It's a bubbling soup of virtual particles popping in and out of existence. When our spinning muon top moves through this soup, it interacts with these virtual particles, which changes how fast it spins.

The biggest part of this "soup" interaction comes from Hadrons (particles made of quarks, like protons and neutrons). This interaction is called the Hadronic Vacuum Polarization (HVP).

Think of the HVP like a crowded dance floor.

• The muon is the dancer.
• The virtual particles are the crowd.
• The HVP is the difficulty the dancer has moving through the crowd.

To solve the mystery of the muon's speed, we need to calculate exactly how "crowded" this dance floor is.

The Problem: Two Different Maps

For a long time, there were two ways to map this dance floor:

1. The Experimental Map: Scientists look at real-world data from particle colliders to guess how crowded the floor is.
2. The Computer Map: Scientists use supercomputers to simulate the dance floor from scratch using the laws of physics (Quantum Chromodynamics, or QCD).

Recently, the Computer Map and the Experimental Map started disagreeing. The Computer Map suggested the crowd is less dense than the Experimental Map thought. If the Computer Map is right, the "missing speed" of the muon might just be a math error, not new physics.

This paper is the Extended Twisted Mass Collaboration (ETMC) updating their Computer Map.

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How They Built the Map: The "Pixelated" Universe

To simulate the dance floor, the team didn't use a smooth, continuous universe. They broke space and time into a giant grid of tiny pixels, like a video game world. This is called a Lattice.

• The Grid: They used five different grid sizes (some coarse, some very fine) to make sure their results didn't depend on the size of the pixels.
• The Ingredients: They simulated a universe with four types of "quarks" (the building blocks of hadrons): Up, Down, Strange, and Charm.
• The Twist: They used a special mathematical trick called "Twisted Mass" fermions. Imagine trying to walk in a straight line on a slippery floor; this trick ensures they don't slip and their math stays accurate without needing extra corrections.

The Two Main Challenges

The team had to solve two specific problems to get a precise number:

1. The "Foggy Window" Problem (Long Distances)

When calculating the HVP, the most important part happens when the virtual particles are far apart in time (the "long-distance" part).

• The Analogy: Imagine trying to hear a whisper in a noisy room. The further away the whisperer is, the harder it is to hear, and the static (noise) drowns them out.
• The Solution: They used a technique called Low-Mode Averaging (LMA). Think of this as putting on noise-canceling headphones that filter out the static, allowing them to hear the "whisper" of the long-distance interactions clearly. They also used a "Bounding" technique, which is like putting a safety net under the calculation to ensure the answer doesn't drift wildly.

2. The "Ghost" Problem (Disconnected Diagrams)

Some interactions happen where the virtual particles don't connect directly to the muon; they just float around in the background. These are called "disconnected" diagrams.

• The Analogy: Imagine trying to count how many people are in a room by looking at the shadows they cast on the wall, but the shadows are faint and overlapping. It's incredibly hard to count them.
• The Solution: They used a clever math trick called Frequency Splitting. Imagine listening to a song and separating the bass, the drums, and the vocals into different tracks. By breaking the calculation into different "frequency" layers, they could reduce the noise and get a much clearer count of these "ghost" particles.

The Results: A New Number

After running these massive simulations on supercomputers (like the Leonardo and LUMI supercomputers), the team calculated the HVP contribution.

• The Outcome: Their result is very precise. It suggests that the "crowded dance floor" is indeed less dense than the old experimental maps suggested.
• The Implication: If their number is correct, it means the "gap" between the Standard Model prediction and the experimental measurement of the muon's speed might be smaller than we thought. This puts pressure on the idea that "New Physics" is the answer. It suggests we might just need to refine our old maps (the experimental data) rather than inventing new laws of physics.

Summary in a Nutshell

The ETMC team built a high-definition, pixel-perfect simulation of the quantum vacuum to see how it affects a spinning muon. By using clever math tricks to cut through the "noise" of the simulation, they produced an updated, highly accurate number. This number challenges the idea that we have discovered new particles, suggesting instead that our understanding of the "old" particles might need a slight tune-up.

It's like a team of cartographers redrawing a map of a mountain range. They found that the peak isn't as high as the old maps said. If they are right, the "treasure" (New Physics) we thought was at the top might not be there after all.

Technical Summary

Here is a detailed technical summary of the paper "An update on the HVP contribution to $g_\mu-2$ in isoQCD from ETMC" by the Extended Twisted Mass Collaboration (ETMC).

1. Problem Statement

The muon anomalous magnetic moment ($a_\mu$) is a precision test of the Standard Model. A significant discrepancy exists between experimental measurements (Fermilab/Brookhaven) and theoretical predictions based on dispersive evaluations of the Hadronic Vacuum Polarization (HVP) contribution. The HVP term is the dominant source of theoretical uncertainty.

• Goal: To provide a first-principles determination of the leading-order HVP contribution ($a_\mu^{\text{HVP, LO}}$) in isospin-symmetric QCD (isoQCD) using Lattice QCD, aiming to resolve the tension with experimental data and validate or challenge dispersive methods.
• Specific Challenge: Achieving sub-percent precision requires controlling statistical noise at large Euclidean times (where the integration kernel is large) and systematic errors arising from discretization (lattice spacing), finite volume, and quark-disconnected diagrams.

2. Methodology

The ETMC employs a rigorous lattice QCD framework with the following key components:

• Gauge Ensembles: Calculations are performed on five $N_f=2+1+1$ gauge ensembles generated with Wilson–clover twisted-mass quarks at maximal twist. This setup ensures automatic $\mathcal{O}(a)$ improvement.
  • Parameters: Four lattice spacings ($a \approx 0.079$ to $0.050$fm) and two physical volumes ($L \approx 5.1$to$7.6$ fm).
  • Tuning: Ensembles are tuned close to the FLAG/Edinburgh consensus point for isoQCD ($m_\pi=135$ MeV, $f_\pi=130.5$ MeV, etc.).
• Mixed-Action Strategy: The valence action uses twisted-mass fermions, allowing for two distinct vector current regularizations:
  1. Twisted Mass (tm): $\hat{V}^{tm}_\mu \propto \bar{q}_\pm \gamma_\mu q_\mp$
  2. Osterwalder-Seiler (OS): $\hat{V}^{OS}_\mu \propto \bar{q}_\pm \gamma_\mu q_\pm$
  • Significance: These two regularizations differ only by lattice artifacts, providing a powerful cross-check for discretization effects in the connected quark contributions.
• Isospin Decomposition: The total HVP is split into isovector ($I=1$) and isoscalar ($I=0$) components:
  • $I=1$: Dominated by light quarks ($u, d$).
  • $I=0$: Includes light, strange, charm, and crucially, quark-disconnected diagrams.
• Blinding: To prevent unconscious bias, an additive blinding procedure is applied to the vector-vector correlators ($C(t) \to C(t) + C_{\text{blind}}(t)$) before analysis. The blinding function is constructed to preserve finite-volume and cutoff effects.
• Signal Enhancement Techniques:
  • Low-Mode Averaging (LMA): Used to improve the signal-to-noise ratio for the connected correlator at large Euclidean times by separating low-lying eigenmodes of the Dirac operator.
  • Frequency-Splitting: Applied to disconnected diagrams to reduce variance by decomposing the difference between light and strange quark contributions into sums of differences probing distinct frequency regions.
  • Bounding Procedure: Instead of integrating the noisy tail of the correlator directly, the authors replace the tail ($t \ge t_c$) with bounds derived from the lowest-energy state ($E_0$) and the effective mass ($E_{\text{eff}}$). The final result is averaged over a stable time window $[t_c^{\text{opt}}, t_c^{\text{opt}} + 0.25 \text{ fm}]$.

3. Key Contributions and Analysis

A. Isovector Contribution ($I=1$)

• Analysis: The $I=1$ channel is dominated by the $\rho$ resonance (two-pion states).
• Finite Volume Effects (FVE): FVEs are non-negligible. The authors validate a modified Gounaris–Sakurai (GS) model that incorporates twisted-mass isospin-breaking effects to correct for finite volume. The modified model reproduces correlator differences between volumes ($L \approx 5.1$ fm vs $7.6$ fm) at the 10–15% level.
• Continuum Extrapolation: Results from both tm and OS regularizations are extrapolated to the continuum limit ($a \to 0$). The data shows a flat behavior consistent with a constant, though a linear fit in $a^2$ is also presented.

B. Isoscalar Contribution ($I=0$)

• Disconnected Diagrams: The dominant uncertainty in the $I=0$ channel comes from quark-disconnected diagrams (light-strange sector).
• Ward Identity (WI) Trick: The authors utilize the space-time integrated light-strange axial Ward identity to compute disconnected Wick contractions efficiently. This generalizes the "one-end trick" and significantly reduces variance.
• Intermediate States: The leading intermediate states are three-pion configurations (dominated by the $\omega$ resonance).
  • Note: The three-pion state with specific momentum configurations is unique to parity-breaking regularizations like twisted mass; in the continuum, momentum conservation would forbid this specific configuration.
• Volume Effects: Unlike the $I=1$ channel, FVEs in the $I=0$ channel are found to be negligible within current precision, consistent with the suppression expected from three-pion intermediate states.

4. Results

The paper presents preliminary, blinded results obtained by one analysis group:

• $I=1$ Contribution:
  • The continuum extrapolation yields a value around **$608.9(5.4) \times 10^{-10}$** (based on the constant fit$P_0$ in Fig. 3).
  • The absolute error is approximately $(3.5\text{--}4) \times 10^{-10}$.
  • The tm and OS regularizations show consistent behavior, validating the control of discretization effects.
• $I=0$ Contribution:
  • The preliminary continuum result (Bayesian AIC combination of linear and constant fits) is $127.5(2.0) \times 10^{-10}$.
  • The data exhibits weak dependence on lattice spacing, indicating small cutoff effects.
  • Finite volume effects are confirmed to be negligible.

Note: The final unblinded total $a_\mu^{\text{HVP}}$ is not explicitly stated in the text as the analysis is still in the "preliminary/blinded" stage, but the components are being determined with high precision.

5. Significance

• Precision Benchmark: This work represents a significant step toward sub-percent precision in lattice QCD calculations of the muon $g-2$, a critical benchmark for the Standard Model.
• Methodological Robustness: The use of two distinct valence regularizations (tm and OS) provides a unique and rigorous check against discretization artifacts, a common source of systematic error in lattice calculations.
• Handling Disconnected Diagrams: The successful application of the Ward identity and frequency-splitting techniques to the light-strange disconnected sector demonstrates a viable path to reducing the dominant uncertainty in the isoscalar channel.
• Resolving Tensions: By providing a first-principles calculation that is consistent with experimental values (as hinted by the introduction's reference to other collaborations), this work supports the hypothesis that the discrepancy between experiment and dispersive evaluations may stem from issues in the data-driven dispersive analyses rather than New Physics.
• Community Standard: The adoption of blinding and advanced variance reduction techniques (LMA, frequency splitting) sets a high standard for future lattice determinations of $a_\mu$.

In summary, the ETMC has successfully established a robust framework for calculating both connected and disconnected contributions to the HVP in isoQCD, achieving control over major systematic uncertainties and providing preliminary results that are competitive with other leading lattice collaborations.