Axion-like Particles and Lepton Flavor Violation in Muonic Atoms

This paper investigates the potential for the Mu2e experiment to detect lepton-flavor-violating $\mu^- e^- \to e^- e^-$ transitions in muonic atoms mediated by axion-like particles, concluding that while light mediators can parametrically enhance the rate, stringent constraints from electron anomalous magnetic moments and other flavor-violating processes severely limit the observable branching ratio to at most $\mathcal{O}(10^{-20})$, making the upcoming Mu3e experiment the most promising probe for the viable parameter space.

The Gist

The Big Picture: A Cosmic Game of "Musical Chairs"

Imagine the universe as a giant dance floor where particles are the dancers. In the Standard Model (our current best rulebook for physics), there is a strict rule: Leptons (a family of particles including electrons and muons) must stay in their own lanes. An electron can't just turn into a muon, and a muon can't just turn into an electron. They are like different species of dancers who never swap partners.

However, we know neutrinos (a shy cousin of these particles) do break this rule. This suggests there is a "secret rulebook" (New Physics) that allows these swaps, but we haven't found the evidence yet.

This paper asks a very specific question: Can we catch a muon breaking the rules inside an atom, and could a mysterious ghost particle called an "Axion-like Particle" (ALP) be the culprit?

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The Setup: The Muonic Atom (The VIP Lounge)

Usually, atoms have electrons orbiting a nucleus. But in this experiment, scientists replace one of those electrons with a muon.

• The Muon: Think of the muon as a "heavy electron." It's about 200 times heavier, so it doesn't orbit far out; it dives deep into the "VIP lounge" right next to the nucleus.
• The Interaction: Because the muon is so close to the nucleus and the other electrons, it's like a VIP crashing a party. The paper looks at a scenario where this VIP muon bumps into a regular electron, and instead of just bouncing off, they both turn into two electrons.
• The Crime: This is a "flavor violation." A muon (flavor A) and an electron (flavor B) went in, and two electrons (flavor B) came out. The muon vanished.

The Suspect: The Axion-like Particle (The Invisible Matchmaker)

If this swap happens, something must have facilitated it. The paper investigates a suspect called an Axion-like Particle (ALP).

• The Analogy: Imagine the muon and electron are two people who want to swap dance partners, but they can't touch each other directly. An ALP is like a ghostly matchmaker that flies between them, whispers a secret, and makes the swap happen.
• The Twist: If this matchmaker is very light (low mass), it makes the swap much easier and faster. If it's heavy, the swap is incredibly slow.

The Investigation: Why We Haven't Caught Them Yet

The authors ran a massive simulation (a "numerical scan") to see if this scenario is possible. They asked: "If we assume this ghost matchmaker exists, how often would we see this swap happen in a real experiment?"

Here is what they found, using a few metaphors:

1. The "Heavy Target" Advantage

The paper notes that this process happens much more often in heavy atoms (like Gold) than light ones (like Aluminum).

• Analogy: Imagine trying to catch a thief in a crowded stadium versus an empty room. In a heavy atom (Gold), the nucleus has a huge positive charge (like a massive magnet). This pulls the muon and electron so close together that the "ghost matchmaker" has a much easier time connecting them. The rate of this event scales with the cube of the nuclear charge—so a slightly heavier atom makes the event happen much more often.

2. The "Speed Bump" Problem (Constraints)

This is the most important part of the paper. The authors checked if this scenario fits with everything else we know about physics. They found that existing laws of physics are acting like a giant speed bump.

• The Electron's Magnetic Personality ($\Delta a_e$):
  The electron has a tiny magnetic personality (its magnetic moment). The "ghost matchmaker" (ALP) interacts with electrons. If the matchmaker is too strong, it would change the electron's magnetic personality in a way that we have already measured and said, "No, that's not right."

  • Result: This single measurement (the electron's magnetic moment) is the biggest filter. It ruled out 62% of all the possible scenarios the scientists tested. It's like a bouncer at a club who checks IDs so strictly that most people can't get in.

• The "Decay" Trap:
  If the ALP is light enough to be produced, it might decay into other particles (like two electrons or photons). We have very sensitive detectors looking for these specific decays (like the Mu3e experiment). If the ALP exists, it would likely show up in these other experiments first.

  • Result: The "light" ALPs that would make the muonic atom swap happen fast are already banned by other experiments.

The Verdict: A Very Narrow Window

After applying all the "bouncers" (constraints from other experiments), the authors calculated the maximum possible chance of seeing this event.

• The Number: The probability is incredibly small—about 1 in $10^{20}$.
• The Metaphor: Imagine you are looking for a specific grain of sand on all the beaches on Earth. Even with the "ghost matchmaker" helping, the rules of the universe make it so rare that you would need to watch trillions of muons for a very long time to see it happen once.

The Silver Lining: What's Next?

Even though the paper says "it's probably too rare to see," it's not a dead end.

1. The Mu3e Experiment: The authors point out that the upcoming Mu3e experiment (which looks for muons turning into three electrons) is the key. If Mu3e finds a signal near its current limit, it would tell us that the "ghost matchmaker" exists, and then we might have a chance to see the muonic atom swap.
2. Complementary Proof: Even if we don't see the swap directly, finding it would be like finding a second fingerprint at a crime scene. It would confirm that the "ghost matchmaker" is real and help us understand the rules of the universe better.

Summary in One Sentence

This paper investigates whether a mysterious, light "ghost particle" could help a muon turn into an electron inside an atom, but concludes that while the idea is theoretically possible, strict rules from other experiments make the event so incredibly rare that we likely won't see it unless a future experiment (Mu3e) finds a major clue first.

Technical Summary

1. Problem Statement

The paper investigates the potential for the Mu2e experiment to detect a specific Lepton Flavor Violating (LFV) process: the scattering of a bound muon and an atomic electron into two electrons ($\mu^- e^- \to e^- e^-$) within a muonic atom.

• Context: While the Standard Model (SM) conserves individual lepton flavors, neutrino oscillations imply LFV exists. However, SM predictions for charged LFV (e.g., $\mu \to e\gamma$) are suppressed to unobservable levels ($<10^{-54}$) by the GIM mechanism.
• The Gap: Existing searches focus on free muon decays ($\mu \to 3e$, $\mu \to e\gamma$) or muon-to-electron conversion in nuclei ($\mu N \to eN$). The process $\mu^- e^- \to e^- e^-$ in a muonic atom is an exotic, nuclei-assisted channel that has been studied for heavy New Physics (NP) but less so for light mediators.
• Hypothesis: If the LFV interaction is mediated by a light particle, such as an Axion-like Particle (ALP), the suppression due to a heavy mass scale is lifted, potentially enhancing the rate. The authors aim to determine if this process is observable given current constraints on light ALPs.

2. Methodology

Theoretical Framework

The authors employ a simplified ALP model featuring:

• Flavor-Violating Couplings: Scalar ($g^S_{e\mu}$) and pseudoscalar ($g^P_{e\mu}$) couplings between electrons and muons.
• Flavor-Diagonal Coupling: A pseudoscalar coupling to electrons ($g^P_{ee}$), which is crucial for loop-induced processes and ALP decays.
• Dark Sector: The model allows for invisible ALP decays ($a \to \chi\bar{\chi}$) into dark fermions, which can relax constraints by increasing the ALP total width.
• Lagrangian: The effective Lagrangian includes derivative couplings and is reduced to scalar/pseudoscalar forms using the Dirac equation.

Calculation of the Transition Rate

• Process: The calculation considers the tree-level exchange of an ALP in $t$- and $u$-channels between a bound muon and a $1s$ atomic electron.
• Approximations:
  • Non-relativistic wavefunctions for bound leptons.
  • The muon density is approximated as a Dirac delta function at the nucleus (due to the muon's heavy mass).
  • Scattered electrons are treated as plane waves.
• Key Result: The derived branching ratio $B(\mu^- e^- \to e^- e^-)$ scales as $(Z-1)^3$, where $Z$ is the nuclear charge. This indicates a significant enhancement for heavier nuclei (e.g., Gold vs. Aluminum).
• Resonance: The rate is parametrically enhanced for light ALP masses ($m_a \ll m_\mu$) due to the propagator structure, specifically in the region $2m_e < m_a < m_\mu - m_e$ where resonant production occurs.

Constraints and Parameter Scan

The authors perform a global parameter scan over:

• Couplings: $g^S_{e\mu}, g^P_{e\mu}, g^P_{ee} \in [10^{-18}, 10^2]$.
• Mass: $m_a \in [10^{-4} \text{ GeV}, 100 \text{ GeV}]$.
• Constraints Applied:
  1. LFV Decays: $\mu \to e\gamma$, $\mu \to 3e$, $\mu \to e\gamma\gamma$, and $\mu \to e + \text{inv}$.
  2. Oscillations: Muonium-antimuonium conversion ($M_\mu \to \bar{M}_\mu$).
  3. Magnetic Moments: Anomalous magnetic moments of the electron ($\Delta a_e$) and muon ($\Delta a_\mu$).
  4. Astrophysical/Beam-Dump: Stellar cooling (Red Giants, White Dwarfs), Supernova 1987A, and beam-dump limits on $g^P_{ee}$.

3. Key Contributions

1. Analytical Derivation: Provided a closed-form expression for the ALP-mediated $\mu^- e^- \to e^- e^-$ branching ratio, explicitly showing the $(Z-1)^3$ scaling and dependence on ALP mass and couplings.
2. Global Constraint Analysis: Conducted a comprehensive scan of the ALP parameter space, integrating constraints from diverse sources (LFV decays, magnetic moments, astrophysics) rather than just a single benchmark.
3. Identification of the Dominant Constraint: Discovered that the electron anomalous magnetic moment ($\Delta a_e$) is the most stringent constraint on this specific model, excluding the largest fraction of the parameter space (62% of sampled points). This is because $\Delta a_e$ directly constrains the flavor-diagonal coupling $g^P_{ee}$, which is essential for the $\mu^- e^- \to e^- e^-$ process.
4. Correlation with Mu3e: Established a strong correlation between the branching ratio of the muonic atom process and the standard muon decay $\mu \to 3e$.

4. Results

• Branching Ratio Limits: After applying all experimental constraints, the maximum achievable branching ratio for $\mu^- e^- \to e^- e^-$ in an Aluminum target is suppressed to $O(10^{-20})$.
  • In the resonant region ($2m_e < m_a < m_\mu - m_e$), the rate is even more heavily suppressed due to tight bounds from $\mu \to 3e$ and $\mu \to e\gamma\gamma$.
  • For $m_a > m_\mu$, limits are slightly looser but still far below current experimental sensitivities for Mu2e.
• Role of $\Delta a_e$: The scan revealed that $\Delta a_e$ is the "killer" constraint. It limits $g^P_{ee}$ so severely that even if LFV couplings ($g^S_{e\mu}, g^P_{e\mu}$) are large, the overall process rate is suppressed.
• Dark Sector Impact: Including a dark sector decay channel ($a \to \chi\bar{\chi}$) does not significantly improve the viable branching ratio, as the constraints from $\mu \to e + \text{inv}$ and $\Delta a_e$ remain dominant.
• Target Dependence: While Gold ($Z=79$) offers a $(Z-1)^3$ enhancement over Aluminum ($Z=13$), the absolute limits on couplings prevent the rate from reaching observable levels in current or near-future experiments.

5. Significance and Conclusion

• Complementary Probe: The paper concludes that while $\mu^- e^- \to e^- e^-$ is theoretically interesting and benefits from nuclear enhancement, it is not a primary discovery channel for light ALPs in the minimal model considered. The process is too heavily suppressed by existing bounds on $\Delta a_e$ and $\mu \to 3e$.
• Future Outlook: The viable parameter space for this process is tightly linked to the current limits on $\mu \to 3e$. Therefore, the upcoming Mu3e experiment (aiming for $10^{-16}$ sensitivity) will explore the most promising region relevant for this muonic-atom signal. If Mu3e observes a signal, $\mu^- e^- \to e^- e^-$ could serve as a valuable complementary probe to confirm the LFV structure.
• Methodological Value: The study highlights the critical importance of including flavor-diagonal constraints (like $\Delta a_e$) when analyzing LFV processes mediated by light particles, as they can qualitatively alter the phenomenological picture compared to analyses relying solely on LFV decay bounds.

In summary, the paper demonstrates that while light ALPs can parametrically enhance $\mu^- e^- \to e^- e^-$, the stringent constraints from the electron's magnetic moment and other LFV processes render the observable rate unattainable for current experiments like Mu2e, shifting the focus of discovery potential to the $\mu \to 3e$ channel.