Muonium Hyperfine Splitting Uncertainty Revisited

This paper re-examines the uncertainty of the theoretical prediction for the ground-state hyperfine splitting in muonium and compares it with the assessments found in the two most recent CODATA adjustments of fundamental physical constants.

The Gist

Here is an explanation of the paper using simple language and creative analogies.

The Big Picture: A Cosmic Ruler and a Broken Tape Measure

Imagine physicists are trying to measure the universe with extreme precision. They use a "ruler" made of fundamental laws of physics (Quantum Electrodynamics, or QED) to predict how atoms should behave. One of the most important atoms they study is Muonium.

Muonium is a tiny, exotic atom made of a positive muon (a heavy cousin of an electron) and a negative electron. It's like a "test drive" for the laws of physics because it's simple enough to calculate perfectly, yet complex enough to reveal if our laws are wrong.

For decades, scientists have been trying to measure the Hyperfine Splitting (HFS) of Muonium. Think of HFS as the "hum" or the specific vibration frequency of this atom.

• The Goal: Compare the measured hum (from experiments) with the predicted hum (from math).
• The Stakes: If the two don't match, it means there is "New Physics" hiding in the shadows—something we don't understand yet.

The Problem: A Confusing "Official" Number

The author of this paper, Michael Eides, is pointing out a serious mix-up in the "official rulebook" of physics, known as CODATA.

CODATA is like the "Bureau of Standards" that sets the official values for constants like the speed of light or the mass of an electron. In their most recent updates (2018 and 2022), they listed a value for the Muonium HFS.

Here is the confusion:

1. The Old Way: For 20 years, everyone agreed: To get the "Theoretical Prediction," you take the math formula and plug in the best-known numbers (like the mass of the muon). This gives you a prediction with a certain "margin of error" (uncertainty).
2. The New Way (CODATA 2018/2022): CODATA took the actual experimental measurement from 1999 (which was very precise) and labeled it as the "Recommended Theoretical Value."

The Analogy:
Imagine you are baking a cake.

• The Recipe (Theory): You calculate exactly how much sugar you need based on the weight of the flour.
• The Taste Test (Experiment): You taste the cake and say, "It needs exactly 100 grams of sugar."
• The Mistake: The "Official Baking Bureau" (CODATA) looks at your taste test, writes it down, and says, "This is the Theoretical Calculation of how much sugar is needed."

This is wrong. A taste test is an observation, not a calculation.

Why Does This Matter? (The "False Alarm" Danger)

The author argues that this mix-up is dangerous, especially because a new, super-precise experiment called MuSEUM is currently running at J-PARC in Japan. They hope to measure the Muonium "hum" 10 times more accurately than before.

Here is the danger scenario:

1. The new experiment measures the hum.
2. They compare it to the "Official CODATA Value."
3. Because CODATA treated the old 1999 experiment as a "perfect theory," they listed the uncertainty (the margin of error) as incredibly small (very tight).
4. The Trap: If the new experiment's result is even slightly different from the old 1999 result, it will fall outside the tiny error bars of the CODATA value.
5. The False Conclusion: Physicists might panic and say, "Aha! The math doesn't match the experiment! There is New Physics!"

The Reality:
The author says, "Wait a minute!" The reason the numbers don't match isn't because of New Physics. It's because the "Official Value" had a fake, too-tight error bar.

The real theoretical prediction has a much larger margin of error (about 10 times bigger) because we don't know the mass of the muon perfectly yet. If you use the real error bar, the new experiment will likely fit perfectly inside the range, and there is no crisis.

The Core Conflict: Two Different Uncertainties

The paper highlights a disagreement that has lasted 20 years:

• Group A (The Author): Says the uncertainty is large (around 515 Hz) because it depends on how well we know the mass of the muon.
• Group B (CODATA): Says the uncertainty is tiny (51 Hz) because they are treating the old experimental result as a "fact" rather than a "prediction."

The author argues that CODATA is confusing Experimental Data (what we measured) with Theoretical Prediction (what the math says).

The Takeaway

The author is writing this short note to act as a "correction flag." He wants the physics community to realize:

"Don't treat the old experimental measurement as a perfect theoretical law. If you do, you might think you've discovered a new force of nature when you've actually just made a math error in how you calculated the uncertainty."

He hopes that by clarifying this, the upcoming MuSEUM experiment results will be interpreted correctly, preventing false alarms about "New Physics" and ensuring that when we do find something new, it's real.

Technical Summary

Based on the paper "Muonium Hyperfine Splitting Uncertainty Revisited" by Michael I. Eides, here is a detailed technical summary:

1. Problem Statement

The paper addresses a critical discrepancy in the treatment of the theoretical uncertainty for the ground-state hyperfine splitting (HFS) of muonium ($\Delta\nu_{HFS}$) within the two most recent CODATA adjustments of fundamental physical constants (2018 and 2022).

• The Core Issue: The 2018 and 2022 CODATA reports cite a value for the "recommended muonium hyperfine splitting" ($4,463,302,776 \pm 51$ Hz) that numerically coincides with the experimental value from 1999 (LAMPF experiments). However, the text of these reports explicitly treats this value as a theoretical QED prediction when discussing its uncertainty.
• The Consequence: By labeling the experimental value as the "theoretical prediction" with an uncertainty of only $51$Hz ($\delta \approx 1.1 \times 10^{-8}$), CODATA significantly underestimates the true theoretical uncertainty. The author argues this misinterpretation could lead to false claims of "New Physics" when comparing future high-precision experimental results against an artificially precise theoretical benchmark.

2. Methodology

Eides employs a comparative analysis of theoretical formulas, uncertainty propagation, and historical data:

• Theoretical Framework: The paper utilizes the standard QED formula for muonium HFS:
  $$\Delta\nu_{HFS} = \nu_F \left[ 1 + F\left(\alpha, Z\alpha, \frac{m_\mu}{m_e}\right) \right] + \Delta\nu_{weak} + \Delta\nu_{hadr} + \Delta\nu_{th}$$
  Where $\nu_F$ is the Fermi frequency, dependent heavily on the muon-electron mass ratio ($m_\mu/m_e$).
• Uncertainty Decomposition: The author breaks down the total uncertainty of the theoretical prediction into three components:
  1. Uncertainty in the experimental mass ratio ($m_\mu/m_e$).
  2. Uncertainty in uncalculated higher-order theoretical terms ($\Delta\nu_{th}$).
  3. Uncertainty in the fine-structure constant ($\alpha$).
• Comparison with CODATA: The paper contrasts the author's calculated theoretical uncertainty (based on the dominance of the mass ratio error) against the value presented in CODATA 2018/2022.
• Contextual Analysis: The author examines the logic used in the CODATA least-squares adjustment, noting that including the 1999 experimental value as a "fundamental constant" to derive a "theoretical" value is circular and statistically dubious given the lack of newer experimental data.

3. Key Contributions

• Identification of a Critical Error: The paper identifies that CODATA 2018 and 2022 have conflated the experimental value with the theoretical prediction. While the central values match (because the theory is constrained by the experiment), the uncertainty attributed to the theory is incorrect.
• Re-evaluation of Theoretical Uncertainty: The author demonstrates that the dominant source of error in the theoretical prediction is not the uncalculated QED terms, but the experimental uncertainty of the muon-electron mass ratio ($m_\mu/m_e$).
  • CODATA 2018/2022 cites a theoretical uncertainty of 51 Hz.
  • Eides recalculates the uncertainty, showing it should be 515 Hz (based on Ref [17]) or 271 Hz (based on Ref [24]).
• Clarification of the "Recommended Value": The paper argues that the "recommended value" in CODATA is effectively the experimental result treated as a constant, not a pure QED prediction. Treating it as a QED prediction with a 51 Hz error bar is physically unjustified.

4. Results

• Corrected Uncertainty: The true theoretical uncertainty for the muonium HFS is approximately $10\times$ larger than the value cited in the latest CODATA reports.
  • CODATA Claim: $\delta \approx 51$ Hz ($1.1 \times 10^{-8}$).
  • Eides Calculation: $\delta \approx 515$ Hz ($1.2 \times 10^{-7}$).
• Source of Dominance: The uncertainty is dominated by the experimental error in the mass ratio ($m_\mu/m_e$), which enters the Fermi frequency formula. The uncertainty from uncalculated theoretical terms ($\Delta\nu_{th}$) is actually smaller (estimated at $\sim 70-85$ Hz).
• Inconsistency in CODATA: The paper highlights that the uncertainty cited in CODATA (51 Hz) is smaller than the estimated uncertainty of the uncalculated theoretical terms themselves (85 Hz), which is a logical contradiction.

5. Significance

• Impact on New Physics Searches: The MuSEUM experiment at J-PARC aims to measure muonium HFS with an order of magnitude higher precision than previous experiments. The goal is to detect weak interaction contributions and potential "New Physics."
  • If the MuSEUM experimental result falls outside the narrow error bars of the CODATA "theoretical" value (51 Hz), it might be incorrectly interpreted as a discovery of New Physics.
  • However, if the correct, larger theoretical uncertainty (515 Hz) is used, the same experimental result might fall well within the Standard Model prediction, preventing false positives.
• Methodological Correction: The paper serves as a necessary correction to the fundamental constants community, urging that theoretical predictions must be calculated using the best available constants and their actual propagated uncertainties, rather than adopting experimental values as theoretical benchmarks with artificially reduced errors.
• Future Outlook: The author calls for an immediate remedy in the presentation of these values to ensure the upcoming MuSEUM results are interpreted correctly within the framework of Quantum Electrodynamics (QED).