Nuclear $μ-e$ conversion via Lorentz and CPT violation

This paper investigates Lorentz- and CPT-violating contributions to nuclear muon-to-electron conversion within the Standard-Model Extension, deriving the first experimental bounds on relevant quark-lepton operators from SINDRUM II data and exploring potential improvements from future experiments.

The Gist

Imagine the universe as a giant, perfectly tuned orchestra. For decades, physicists believed the rules of this orchestra were absolute: the music (physics) sounds the same no matter which direction you face (Lorentz symmetry) or whether you play the song forward or backward in time (CPT symmetry).

However, recent discoveries about neutrinos (tiny, ghostly particles) suggest that the "sheet music" of the universe might have a few hidden notes we missed. This paper explores what happens if we assume those hidden notes exist—specifically, if the universe has a slight "tilt" or a "glitch" in its rules.

Here is the story of the paper, broken down into simple concepts:

1. The "Impossible" Magic Trick

In the world of subatomic particles, there is a rule called Lepton Flavor Conservation. Think of "Lepton Flavor" like a specific color of paint. A muon is painted "blue," and an electron is painted "red." The rules of the Standard Model (our current best physics textbook) say a blue muon can never magically turn into a red electron.

If a blue muon did turn into a red electron, it would be like seeing a red ball spontaneously turn into a blue ball. This is called Charged Lepton Flavor Violation (CLFV).

While we know neutrinos can change colors (oscillate), seeing a muon turn into an electron is incredibly rare. In fact, if it happens only because of the known neutrino tricks, it's so rare it would take longer than the age of the universe to see it once. So, if we do see it, it's not a glitch in the neutrino; it's a sign of brand new physics.

2. The Experiment: The Atomic Trap

The paper focuses on a specific experiment called $\mu-e$ conversion.

• The Setup: Scientists shoot a beam of muons (blue particles) at a heavy metal target, like gold or aluminum.
• The Capture: The muons get stuck in orbit around the nucleus of the atom, forming a "muonic atom."
• The Goal: Usually, the muon just disappears by turning into a neutrino. But the scientists are watching for the "magic trick": the muon turning into an electron without making a neutrino.
• The Signal: If this happens, the electron flies out with a very specific, unique energy—like a coin dropping from a specific height. It's a "monoenergetic" signal that is impossible to miss if it's there.

3. The "Tilted Universe" Theory (Lorentz & CPT Violation)

The authors ask: What if the universe isn't perfectly symmetrical?
Imagine the universe is a giant, flat floor. If you roll a ball, it goes straight. But what if the floor is slightly tilted? The ball would curve, even if you pushed it straight.

In physics, this "tilt" is called Lorentz violation. It means the laws of physics might change slightly depending on:

• Which way you are facing (direction).
• What time of day it is (as the Earth rotates).
• Whether time is moving forward or backward.

The paper uses a theoretical framework called the Standard-Model Extension (SME). Think of the SME as a "master control panel" for the universe. It has knobs and dials that represent these possible tilts. The authors are trying to figure out how tight those dials need to be to explain why we haven't seen the magic trick yet.

4. The Detective Work: Using Old Data to Find New Clues

The authors didn't build a new machine. Instead, they acted like detectives re-examining old evidence. They looked at data from an experiment called SINDRUM II, which ran in the 1990s using a gold target.

They asked: "If the universe had these specific 'tilts' (Lorentz/CPT violations), would SINDRUM II have seen them?"

By running the numbers, they found that SINDRUM II didn't see the magic trick. This means the "tilts" in the universe must be very, very small. They used this lack of evidence to set new limits (boundaries) on how big these violations can be.

The Analogy: Imagine you are looking for a specific type of bird in a forest. You didn't see it. That doesn't mean the bird doesn't exist, but it does mean the bird must be very rare, or it must be hiding in a very specific spot. This paper calculates exactly how rare the bird must be.

5. The Big Discovery: New Types of Clues

The most exciting part of this paper is that they found a way to look for a specific type of "tilt" that other experiments can't see.

• Other experiments (like looking for a muon turning into an electron and a photon) are like looking for a bird in the trees.
• This experiment (muon turning into an electron inside a nucleus) is like looking for a bird in the roots.

The authors found that this specific "root" channel is the only place where we can test certain interactions between quarks (the building blocks of the nucleus) and leptons (muons/electrons). They set the first-ever limits on these specific interactions.

6. The Future: Sharper Eyes

The paper concludes by looking ahead. Two massive new experiments are coming online: COMET (in Japan) and Mu2e (in the US).

• These new machines will be like upgrading from a pair of binoculars to a high-powered telescope.
• They will be able to see signals that are 100 times fainter than what SINDRUM II could see.
• If the "tilted universe" theory is true, these new experiments might finally catch the magic trick.

Summary

This paper is a theoretical detective story. It takes old data from a gold experiment and uses it to say, "Okay, if the universe is tilted, it can't be tilted this much." It opens a new window to look for cracks in the laws of physics, specifically interactions between the nucleus and electrons that no one has been able to measure before. If we ever find these cracks, it will rewrite our understanding of how the universe works.

Technical Summary

Here is a detailed technical summary of the paper "Nuclear $\mu-e$ conversion via Lorentz and CPT violation" by William P. McNulty, presented at the Tenth Meeting on CPT and Lorentz Symmetry (CPT'25).

1. Problem Statement

The paper addresses the search for Charged Lepton Flavor Violation (CLFV), specifically the coherent conversion of a muon to an electron in the presence of a nucleus ($\mu^- + A \to e^- + A$).

• Standard Model Context: In the Standard Model (SM), lepton flavor is conserved. While neutrino oscillations imply CLFV, the branching ratios for processes like $\mu-e$ conversion are suppressed to $\lesssim 10^{-54}$ by the GIM mechanism and small neutrino masses, rendering them unobservable with current technology.
• New Physics Signal: Any observation of CLFV would constitute a clear signal of physics beyond the SM.
• Specific Gap: While electromagnetic operators (probed in $\mu \to e \gamma$) are well-studied, four-point quark-lepton operators are uniquely accessible in the nuclear $\mu-e$ conversion channel at tree level. Furthermore, the potential for Lorentz and CPT violation in these specific operators had not been systematically constrained using existing experimental data.

2. Methodology

The author utilizes the Standard-Model Extension (SME), an effective field theory framework that incorporates all possible Lorentz and CPT-violating terms into the SM.

• Theoretical Framework:

  • The study focuses on leading contributions from electromagnetic operators (mass dimension 5) and four-point quark-lepton operators (mass dimension 6).
  • The conversion rate is calculated by integrating the interaction over the nuclear volume, utilizing the muon's bound-state wavefunction ($\psi^{(\mu)}$) and the electron's continuum wavefunction ($\psi^{(e)}$).
  • The wavefunctions are derived by numerically solving the Dirac equation with spherically symmetric nuclear charge distributions.

• Reference Frames and Transformations:

  • Calculations are initially performed in the laboratory frame but must be transformed to the Sun-Centered Frame (SCF) to account for Earth's rotation and the resulting sidereal variations in Lorentz violation signals.
  • The SCF is defined with the $X$-axis pointing to the Sun at the vernal equinox and the $Z$-axis parallel to Earth's rotation axis.
  • The analysis assumes $2\pi$ azimuthal coverage and specific polar angle acceptance symmetric about the beamline.

• Experimental Data Used:

  • The study utilizes the current best experimental limit set by the SINDRUM II experiment at the Paul Scherrer Institute (PSI): $R_{\mu e} < 7 \times 10^{-13}$ (90% CL) on a Gold ($^{197}\text{Au}$) target.
  • The ratio $R_{\mu e}$ is defined as the rate of CLFV conversion divided by the standard muon capture rate.

3. Key Contributions

This paper makes several novel theoretical and phenomenological contributions:

1. First Bounds on SME Quark-Lepton Operators: It provides the first reported constraints on the SME coefficients for four-point quark-lepton operators using nuclear $\mu-e$ conversion data. These operators are inaccessible in other "golden channels" (like $\mu \to e \gamma$) at tree level.
2. Lorentz/CPT Violation Analysis: It explicitly derives the leading Lorentz- and CPT-violating contributions to the conversion rate within the SME framework.
3. Complementarity: It establishes the nuclear conversion channel as a unique and complementary probe to other CLFV searches, specifically for constraining quark-level interactions.
4. Future Projections: It offers projections for upcoming experiments (COMET and Mu2e), estimating the potential improvement in sensitivity.

4. Results

Using the SINDRUM II data, the author derived the following constraints (at 90% confidence level):

Operator Class	Constraint (GeV units)
Electromagnetic	$< (6–8) \times 10^{-12} \text{ GeV}^{-1}$
Scalar Quark-Lepton (u, d)	$< (6–7) \times 10^{-13} \text{ GeV}^{-2}$
Scalar Quark-Lepton (s)	$< (10–15) \times 10^{-13} \text{ GeV}^{-2}$
$\gamma^0$ Quark-Lepton (u, d)	$< (20–30) \times 10^{-13} \text{ GeV}^{-2}$

• Comparison with MEG: The constraints on electromagnetic operators derived here are approximately one order of magnitude weaker than those previously obtained from the MEG experiment ($\mu^+ \to e^+ \gamma$). However, the MEG II experiment is expected to improve these bounds further.
• Future Sensitivity:
  • Initial Phases: The upcoming COMET (Phase I) and Mu2e (Run I) experiments are projected to improve constraints on these operators by at least one order of magnitude.
  • Final Goals: The final sensitivity goals of COMET ($R_{\mu e} \sim 2.6 \times 10^{-17}$) and Mu2e ($R_{\mu e} \sim 8 \times 10^{-17}$) suggest a potential improvement of two orders of magnitude over current limits.

5. Significance

• New Physics Reach: By constraining quark-lepton operators, this work opens a new window for testing the Standard Model that is distinct from purely leptonic processes.
• Lorentz Symmetry Testing: It extends the search for Lorentz and CPT violation into the hadronic sector of CLFV, providing a more comprehensive test of fundamental symmetries.
• Experimental Guidance: The results provide a theoretical baseline for the next generation of experiments (COMET and Mu2e), demonstrating that these facilities will not only search for CLFV but also significantly tighten the bounds on fundamental symmetry violations in the quark-lepton sector.
• Complementarity: The paper reinforces the necessity of a multi-channel approach to CLFV, as different channels probe different combinations of SME operators.