⚛️ phenomenology

Leggett-Garg Inequality Violation in Muon $g-2$ Experiments

This study presents the first observation of Leggett-Garg inequality violation in polarized muon spin precession by analyzing Fermilab Muon $g-2$ data, achieving a $5.5\sigma$ significance that demonstrates the potential for high-precision measurements of temporal quantum correlations with improved detector modeling.

Original authors: Brian Batell, Morgan Cassidy, Kun Cheng

Published 2026-06-23

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Brian Batell, Morgan Cassidy, Kun Cheng

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

The Big Idea: Testing if the Future "Remembers" the Past

Imagine you have a magical spinning top. In our everyday world (the "classical" world), if you look at the top, it has a definite position. If you look at it again a second later, its position is determined by how it was spinning before. You could, in theory, peek at it without stopping it or changing its spin. This is how our brains usually work: things have a fixed reality, and looking at them doesn't change them.

However, in the quantum world (the world of tiny particles), things are different. Particles can exist in a "superposition," meaning they are in multiple states at once until measured. Furthermore, measuring a particle changes it.

Physicists use a test called the Leggett-Garg Inequality (LGI) to see if a system is behaving like a normal, predictable top or a weird, quantum spinning top.

If the LGI holds true: The system is "classical." It has a definite state at all times, and looking at it doesn't mess it up.
If the LGI is broken (violated): The system is "quantum." It relies on the weird rules of superposition, and the act of measuring it is part of the story.

The Experiment: The "Muon" Spinning Top

The authors of this paper decided to test this using muons.

What is a muon? Think of it as a heavy, unstable cousin of an electron. It's like a tiny, spinning top that is born with its spin pointing in a specific direction.
The Setup: In the famous Muon g-2 experiment at Fermilab, these muons are shot into a giant, perfectly controlled magnetic ring (a storage ring).
The Dance: Once inside the ring, the muons don't just spin; they wobble. Their spin direction rotates (precesses) like a wobbling gyroscope. This wobble happens at a very specific, predictable rhythm.

How They "Saw" the Spin

You can't see a muon's spin directly. It's too small. But the muon has a trick: when it dies (decays), it shoots out a particle called a positron (a positive electron).

The Analogy: Imagine the muon is a lighthouse. When the light (the spin) points one way, it shoots a bright beam of light (positrons) in that direction. When the light points the other way, the beam goes elsewhere.
The Measurement: By counting how many high-energy positrons hit the detectors at different times, the scientists could figure out exactly where the muon's spin was pointing at that moment.

The "Time Travel" Test

To test the Leggett-Garg Inequality, the scientists needed to look at the muon's spin at three different moments in time:

Time A: They check the spin.
Time B: They check the spin again.
Time C: They check it a third time.

They then asked a specific question: If the spin was pointing "Down" at Time A, what is the probability it will be "Up" at Time B, and how does that relate to where it was at Time C?

In a normal, classical world, the math says the relationship between these three times must follow a strict rule (the inequality). If the math breaks that rule, it proves the system is behaving quantumly.

What They Found

The team took data from the Fermilab experiment, which contains about 10 billion muon deaths. That is a massive amount of data—like watching a billion spinning tops wobble.

Reconstruction: They used a computer model to translate the positron counts into the muon's spin direction over time.
The Result: When they crunched the numbers, the Leggett-Garg Inequality was violated.
The Significance: The violation was incredibly strong, with a statistical certainty of 5.5 sigma. In the world of physics, this is like flipping a coin 100 times and getting heads every single time, but with even more certainty. It is a definitive "yes," the system is quantum.

The "Blurry Camera" Problem

The paper admits one major limitation. The scientists used a simplified model to account for how their detectors work.

The Analogy: Imagine trying to take a photo of a fast-spinning top with a slightly blurry camera. You know the top is spinning, but the photo isn't perfectly sharp. The authors had to guess how much the "blur" (detector imperfections) affected their results.
The Impact: Because they used a simplified guess for the "blur," they had to be very conservative. Even with this conservative guess, the quantum violation was still huge (5.5 sigma).
The Future: The authors suggest that if the actual Fermilab team re-analyzed the data using their real camera specs (full detector information), the "blur" would be removed. This would make the measurement even more precise, potentially becoming one of the most accurate tests of quantum time-correlations ever done.

Summary

This paper is the first time scientists have used the massive data from the Muon g-2 experiment to prove that spinning muons obey the weird rules of quantum mechanics over time. They showed that these particles don't have a fixed "reality" at every moment in time but exist in a quantum dance that breaks the rules of classical physics. While their current method had some "blur" due to simplified modeling, the result was so strong that it confirms the quantum nature of these particles beyond any reasonable doubt.

Technical Summary: Leggett-Garg Inequality Violation in Muon g −2 Experiments

Problem and Context
The Leggett-Garg Inequality (LGI) serves as a temporal analog to Bell inequalities, testing the compatibility of "macrorealism" (the assumption that a system possesses definite properties at all times and can be measured non-invasively) with quantum mechanics. While LGI violations have been studied in systems like neutrinos and mesons, this work addresses the application of LGI tests to high-energy polarized muon spin precession. The authors identify the Fermilab Muon $g-2$ experiment as an ideal candidate for such a study due to the well-characterized initial polarization of the muon ensemble, the highly controlled storage ring environment, and the availability of approximately $10^{10}$ muon decay events. The central challenge is reconstructing the temporal correlators of longitudinal muon polarization from measured time-dependent decay spectra, which are distorted by detector effects, to test the inequality.

Methodology
The authors develop a procedure to reconstruct the third-order Leggett-Garg variable, $K_3$ , using publicly available data from the Fermilab Muon $g-2$ experiment. The methodology proceeds as follows:

Theoretical Framework: The study utilizes the "stationarity" formulation of the LGI, where the inequality is expressed as $K_3 = 2C(\Delta t) - C(2\Delta t) \leq 1$ . Here, $C(\Delta t)$ represents the autocorrelation function of a dichotomic observable (muon spin projection), defined via conditional probabilities of finding the muon in a spin-up or spin-down state at time $t_0 + \Delta t$ given its state at $t_0$ .
Data Extraction: The longitudinal muon polarization, $b_z(t)$ , is inferred from the energy spectrum of decay positrons ( $\mu^+ \to e^+ \nu_e \bar{\nu}_\mu$ ). The positron energy distribution is spin-dependent; higher energy positrons are preferentially emitted along the muon spin direction.
Detector Modeling: To account for experimental realities, the authors employ a simplified detector response model. They assume the detector efficiency is a linear function of polarization, $\epsilon(E, b_z) = \epsilon_0(E) + b_z \epsilon_1(E)$ . The parameters are derived by comparing the theoretical spin-averaged spectrum and asymmetry with the experimentally reported values ( $N_{exp}$ and $A_{exp}$ ).
Reconstruction: The authors digitize the positron energy spectra from the Fermilab data (spanning two oscillation periods starting at 44.5 $\mu$ s). They fit the extracted spectra against their theoretical template (incorporating the simplified detector model) to determine $b_z(t)$ at each time bin.
Uncertainty Treatment: The residual differences between the simplified model's prediction and the experimental data are treated as systematic uncertainties. The authors use an "offset method," varying the fit by the magnitude of these residuals (with a sign parameter $s = \pm 1$ ) to estimate the total systematic error on $b_z(t)$ and the resulting $K_3$ . Statistical uncertainties are derived from the fit to the binned data.

Key Contributions and Results

First Observation in Muons: This work presents the first determination of LGI violation using oscillating muons.
Significance of Violation: Using the simplified detector model and existing data, the authors find that the LGI is violated with a single-bin significance of 5.5 $\sigma$ .
Statistical Precision: The statistical uncertainty on the reconstructed variable $K_3$ is extremely small, on the order of $O(10^{-3})$ .
Systematic Limitations: The current analysis is limited by systematic uncertainties arising from the simplified detector modeling. The observed 5.5 $\sigma$ violation persists even when these conservative systematic uncertainties are included.
Scaling Potential: The authors note that combining neighboring time bins where the quantum mechanical prediction exceeds the classical bound ( $K_3 > 1$ ) further increases the significance of the violation.

Significance and Claims
The paper claims to provide the first demonstration of LGI violation in the context of muon $g-2$ experiments. The authors emphasize that while their current analysis is constrained by the simplified modeling of detector effects, the underlying data contains sufficient statistical power ( $10^{10}$ events) to enable a much more precise measurement.

The authors assert that a dedicated reanalysis by the Fermilab Muon $g-2$ collaboration, utilizing full experimental information and realistic detector simulations, could reduce the total uncertainty to the statistical level ( $O(10^{-3})$ ). Such an effort would potentially enable one of the most precise measurements of temporal quantum correlations to date, leveraging the high-precision environment of the $g-2$ experiment to probe the "quantumness" of macroscopic spin evolution. The work positions high-energy physics experiments as rich laboratories for quantum information science, specifically for studying temporal correlations in evolving systems.