Weak Interaction Contribution to the Muonium Hyperfine Structure in the Standard Model

This paper investigates the weak interaction's contribution to the ground-state hyperfine splitting of muonium by calculating amplitudes for one- and two-quantum exchanges mediated by Z and W bosons, as well as deriving one-loop corrections to the photon and Z boson propagators.

The Gist

Here is an explanation of the paper "Weak Interaction Contribution to the Muonium Hyperfine Structure in the Standard Model," translated into simple, everyday language with creative analogies.

The Big Picture: A Cosmic Dance Floor

Imagine the universe as a giant dance floor. Usually, particles like electrons and muons (heavy cousins of electrons) dance together, held close by the electromagnetic force—think of this as a strong, invisible magnetic hand holding their hands. This is how atoms are built.

But there are other forces at play too, like the Weak Interaction. In the Standard Model of physics, this force is like a shy, distant relative who only shows up to the dance floor to whisper a secret or nudge someone slightly. Because this force is carried by very heavy particles (the $W$ and $Z$ bosons), it usually doesn't do much in small atoms. It's like trying to hear a whisper from someone standing a mile away.

The Problem: For decades, scientists have been measuring the "dance steps" of a special atom called Muonium (an electron orbiting a muon) with incredible precision. They are so good at measuring that they can now hear the "whisper" of the Weak Interaction. The old math ignored this whisper because it was too quiet to matter. But now, the measurements are so precise that ignoring the whisper would be like trying to tune a radio while ignoring a nearby siren.

What is Muonium?

Think of Muonium as a "ghost atom."

• Normal Hydrogen: A proton (heavy nucleus) holding an electron.
• Muonium: A muon (a heavy electron) holding an electron.
• Why it matters: Because the muon is a fundamental particle with no internal parts (unlike a proton, which is made of quarks), Muonium is a "pure" system. It's the perfect laboratory to test the laws of physics without any "noise" from messy internal structures.

The "Hyperfine Splitting": The Atom's Hum

Every atom has a tiny internal vibration, like a guitar string. In Muonium, the electron and the muon have a property called "spin" (imagine them as tiny spinning tops).

• When they spin in the same direction, the atom has slightly more energy.
• When they spin in opposite directions, it has slightly less energy.

The difference between these two states is called Hyperfine Splitting. Scientists measure this difference as a frequency (a hum). The paper is about calculating how much the "Weak Interaction" changes the pitch of this hum.

The Scientists' Mission: Finding the Ghosts

The authors (Martynenko, Martynenko, and Seredina) wanted to calculate exactly how much the Weak Interaction shifts this frequency. They looked at several ways the "ghosts" (Weak force particles) could interact with the atom:

1. The One-Time Nudge (Single Boson Exchange)

Imagine the electron and muon are talking. Usually, they pass a ball back and forth (a photon). But sometimes, a heavy, invisible messenger ($Z$ boson) drops by and gives them a tiny nudge.

• The Math: They calculated the effect of this single nudge.
• The Result: It shifts the frequency by about -68 Hz. This is the biggest effect, but it's still tiny compared to the total frequency (which is in the billions of Hz).

2. The Loop-the-Loops (Vacuum Polarization)

In quantum physics, the "vacuum" isn't empty. It's like a busy highway where particle-antiparticle pairs pop in and out of existence for a split second.

• The Analogy: Imagine the electron and muon are trying to talk, but the air between them is filled with temporary "traffic jams" made of heavy particles ($W$ bosons, ghosts, etc.). These traffic jams distort the signal.
• The Calculation: The authors calculated how these "traffic jams" (loops of particles) change the interaction.
• The Result: These effects add up to a few more Hertz of shift.

3. The Box Diagrams (The Complex Dance)

Sometimes, the particles don't just exchange one messenger; they exchange two at the same time, forming a square or "box" shape in the math.

• The Analogy: Instead of just passing a ball, the dancers do a complex four-step routine involving two invisible partners simultaneously.
• The Result: These complex routines contribute a tiny bit more (about -0.1 Hz).

The Grand Total

After adding up all the whispers, nudges, traffic jams, and complex dances, the authors found that the Weak Interaction shifts the Muonium frequency by a total of -70.12 Hz.

Why Does This Matter?

You might ask, "Who cares about 70 Hertz?"

1. Precision is King: The new experiments (like the MuSEUM collaboration at J-PARC in Japan) are measuring the Muonium frequency with an accuracy of 1 part per billion. They are so precise that they can now detect this 70 Hz shift. If the theory doesn't include it, the theory will look "wrong" compared to the experiment.
2. Testing the Standard Model: By comparing the new, ultra-precise measurements with this new, ultra-precise calculation, scientists can check if the Standard Model is perfect. If the numbers don't match, it might mean there is new physics out there—something we haven't discovered yet!
3. Cleaning the Lens: Before this paper, the "Weak Interaction" was a blurry smudge on the lens of our theoretical calculations. This paper sharpens the lens, allowing us to see the true picture of how the universe works at the smallest scales.

Summary in a Nutshell

• The Atom: Muonium (a pure, simple atom).
• The Measurement: How fast the atom "hums" when its parts spin differently.
• The New Discovery: Scientists finally calculated the tiny "whisper" of the Weak Interaction that was previously too small to notice.
• The Result: The whisper shifts the hum by about 70 Hz.
• The Goal: To ensure our theoretical maps of the universe match the incredibly precise GPS of modern experiments. If they match, the Standard Model wins. If they don't, we might find a new law of physics!

Technical Summary

Here is a detailed technical summary of the paper "Weak Interaction Contribution to the Muonium Hyperfine Structure in the Standard Model" by F. A. Martynenko, A. P. Martynenko, and K. A. Seredina.

1. Problem Statement

Muonium (a bound state of a positive muon and an electron) is a purely leptonic atom, making it an ideal system for testing Quantum Electrodynamics (QED) and the Standard Model (SM). While the theoretical prediction for the ground-state hyperfine splitting (HFS) of muonium is highly precise ($\Delta E_{th} \approx 4,463,302,872$ Hz), the uncertainty is currently dominated by the experimental measurement of the lepton mass ratio ($m_\mu/m_e$).

Recent and upcoming experiments (e.g., MuSEUM at J-PARC) aim to measure the HFS with an accuracy of 1 part per billion (ppb), corresponding to an uncertainty of roughly 4.5 Hz. At this level of precision, contributions previously considered negligible—specifically electroweak corrections (weak interaction effects mediated by $W^\pm$ and $Z$ bosons)—become significant. The paper addresses the need to calculate these weak interaction contributions to the muonium HFS to match the new experimental precision.

2. Methodology

The authors employ a combination of Quantum Field Theory (QFT) techniques, specifically focusing on bound-state perturbation theory. The calculation involves determining the scattering amplitudes for electron-muon interactions mediated by weak bosons and converting these into effective potentials.

Key methodological approaches include:

• Dispersion Relations: Used extensively to calculate one-loop corrections to photon and $Z$-boson propagators. This involves calculating the imaginary parts of polarization operators (via Cutkosky rules) and reconstructing the real parts via dispersion integrals.
• Gauge Choices:
  • Coulomb Gauge: Used for the $Z$-boson propagator to facilitate bound-state calculations.
  • 't Hooft-Feynman Gauge ($\xi=1$): Used for calculating loop diagrams involving $W$ bosons, ghosts, and pseudo-Goldstone bosons to simplify propagator forms.
  • Unitary Gauge: Referenced for physical interpretation, though calculations are performed in renormalizable gauges.
• Direct Integration: For "box-type" diagrams (two-boson exchange), the authors perform direct four-dimensional integration in Euclidean space rather than using dispersion methods, as these integrals are finite.
• Effective Potentials: The scattering amplitudes are transformed into coordinate-space potentials (involving $\delta$-functions and Yukawa-type terms) and averaged over the muonium ground-state wave function ($1S$) to obtain energy shifts.

3. Key Contributions and Calculated Effects

The paper calculates various electroweak corrections, categorized by the type of boson exchange and loop order:

A. Single Boson Exchange ($Z$-boson)

• Tree Level: The leading contribution comes from the exchange of a single $Z$ boson. The authors separate the vector and axial-vector current contributions.
  • Vector contribution: $\approx -3.15$ Hz.
  • Axial-vector contribution: $\approx -64.88$ Hz.
  • Total Tree Level: $\approx -68.03$ Hz.

B. One-Loop Propagator Corrections (Self-Energy)

• Photon Self-Energy ($\gamma\gamma$): Corrections involving $W$-boson loops in the photon propagator.
  • Contribution: $\approx -1.22$ Hz.
• Mixed Propagator ($Z\gamma$): Corrections where a photon and $Z$ boson mix via fermion loops (leptons/quarks) or boson loops ($W$, ghosts).
  • Fermion loops (e, $\mu$, $\tau$, c, b, t): Range from negligible ($e$-loop) to $\approx -0.68$ Hz ($t$-quark loop).
  • Boson loops ($W$, $\omega$, ghosts): $\approx +0.36$ Hz.
  • Total $Z\gamma$ contribution: $\approx -0.54$ Hz (dominated by heavy quarks).

C. Vertex Corrections

• $ZWW$ Vertex: One-loop corrections where the $Z$ boson couples to a pair of $W$ bosons.
  • Vector part: $\approx +0.14$ Hz.
  • Axial-vector part: $\approx +0.08$ Hz.

D. Box-Type Diagrams (Two-Boson Exchange)

These involve the simultaneous exchange of two bosons between the electron and muon.

• $\gamma Z$ Box: Exchange of one photon and one $Z$ boson.
  • Contribution: $\approx -0.038$ Hz.
• $ZZ$ Box: Exchange of two $Z$ bosons.
  • Contribution: $\approx -0.040$ Hz.
• $WW$ Box: Exchange of two $W$ bosons (involving intermediate neutrinos).
  • Contribution: $\approx -0.103$ Hz.

4. Results

The authors sum all calculated electroweak contributions to determine the total weak interaction shift to the muonium ground-state hyperfine splitting.

• Total Electroweak Shift:
  $$\Delta E_{hfs}^{tot} (1S) = -70.1217 \text{ Hz}$$
• Breakdown:
  • The dominant term is the tree-level single $Z$-boson exchange ($\approx -68$ Hz).
  • Propagator corrections (vacuum polarization) contribute several Hz (specifically the $Z\gamma$ and $\gamma\gamma$ loops).
  • Box diagrams and vertex corrections contribute less than 1 Hz in total but are included for completeness.

The paper notes that while the result is presented to four decimal places, unaccounted higher-order terms may contribute at the level of $10^{-4}$to$10^{-3}$ Hz, which is well below the current experimental uncertainty but relevant for future precision goals.

5. Significance

• Theoretical Completeness: This work provides the first comprehensive calculation of all leading-order electroweak corrections to muonium HFS, moving beyond previous studies that often treated these effects as negligible or provided only summary estimates.
• Experimental Relevance: With the MuSEUM collaboration aiming for 1 ppb precision ($\sim 4.5$ Hz), the calculated weak interaction shift of $\approx -70$ Hz is a critical correction. Without accounting for this, the comparison between theory and experiment would be flawed, potentially leading to incorrect conclusions about physics beyond the Standard Model.
• Methodological Validation: The paper demonstrates the effectiveness of combining dispersion relations with direct integration for bound-state problems involving heavy bosons, providing a robust framework for future precision calculations in light atomic systems.
• Comparison with Previous Work: The authors compare their box-diagram results with a previous study (Asaka et al., arXiv:1810.05429), finding agreement on box contributions but noting significant differences in vacuum polarization terms, highlighting the importance of their detailed calculation.

In conclusion, the paper establishes that electroweak corrections are no longer negligible in muonium spectroscopy and must be included in the theoretical standard model prediction to interpret the next generation of high-precision experimental data.