ALP production in Lepton Flavour Violating meson, tau and gauge boson decays

This paper investigates the production and prompt decay of axion-like particles (ALPs) with lepton-flavour-violating couplings in the mass regime above the muon threshold, proposing new search strategies across meson, gauge boson, quarkonium, and tau decays to leverage the distinctive $a \to e\mu$ signature and assess the discovery potential of future high-energy and fixed-target experiments.

The Gist

Imagine the universe as a giant, bustling city. For decades, physicists have been mapping the "known neighborhoods" of this city, governed by the Standard Model—a rulebook that explains how particles like electrons and muons (heavy cousins of electrons) interact. But they suspect there are hidden alleyways and secret tunnels where new, mysterious residents live.

One of the most intriguing suspects is the Axion-Like Particle (ALP). Think of an ALP as a "ghostly messenger." It's incredibly light, barely interacts with anything, and might be the key to solving some of the universe's biggest mysteries, like why matter exists or what dark matter is made of.

The Problem: The "Muons" are Too Heavy

For a long time, scientists tried to catch these ALPs by looking at muons (a type of heavy electron). If an ALP exists and is very light, a muon might decay into an electron and an ALP. This is like a heavy bouncer (the muon) slipping a secret note (the ALP) to a lighter guest (the electron) before leaving the party.

However, there's a catch. If the ALP is heavier than a specific threshold (heavier than a muon), this "note-passing" trick stops working. The bouncer is too heavy to pass the note to a lighter guest if the note itself is too heavy. This created a "blind spot" in our search: a region of the universe where we thought ALPs might hide, but our old search methods couldn't see them.

The New Strategy: The "Virtual" Detour

This paper proposes a clever new way to find these heavier ALPs. Instead of waiting for a real muon to decay, the authors suggest looking at virtual muons.

The Analogy:
Imagine you are trying to catch a rare bird (the ALP) that only flies when a specific wind gust (the muon) passes by.

• Old Method: You wait for a real wind gust to blow a leaf (the ALP) out of a tree. But if the bird is too heavy, the wind can't lift it.
• New Method: You realize that even when the wind isn't blowing hard enough to lift the bird, the air pressure of the wind (the virtual muon) can still briefly create a "ghost wind" that kicks the bird into the air.

In the world of particle physics, this means looking at particles like Mesons (unstable particles made of quarks) or W and Z bosons (force carriers). When these particles decay, they often create a "virtual" muon for a split second. If an ALP exists, it can steal energy from this virtual muon and pop into existence as a real, detectable particle.

The "Smoking Gun": A Perfect Crime Scene

The beauty of this new strategy is the signature it leaves behind.

1. The Production: A heavy particle (like a Kaon or a Z boson) decays, creating a virtual muon that instantly turns into an electron and an ALP.
2. The Decay: Because the ALP is heavy enough, it doesn't hang around as a ghost. It immediately decays into an electron and a muon.
3. The Result: You end up with a very strange, chaotic scene: Two electrons and one muon (or four charged particles in other scenarios) appearing out of nowhere.

Why is this special?
In the Standard Model (the known rulebook), nature is very strict about "Lepton Flavor." It's like a bouncer at a club who never lets a "Muon" turn into an "Electron" without a very specific, rare reason. Seeing a Muon turn into an Electron (and vice versa) in this specific pattern is a Lepton Flavor Violation (LFV).

It's like walking into a bank and seeing a dollar bill turn into a Euro coin, and then back into a dollar, all in a split second. The Standard Model says this is impossible. If you see it, you know for a fact that New Physics (the ALP) is at work. Furthermore, because this process is so forbidden in the known universe, the "background noise" (false alarms) is practically zero. It's a "background-free" search.

Where are we looking?

The authors suggest checking several "hotspots" where these virtual muons are abundant:

• Particle Colliders (The Big Accelerators): Places like the future FCC-ee or CEPC (which will smash electrons and positrons together) will produce trillions of Z bosons. It's like having a massive factory churning out the exact ingredients needed to bake this rare cake.
• Flavor Factories (Belle II & STCF): These are specialized labs that produce huge numbers of heavy particles (like Tau leptons and J/psi particles) to study their decay.
• Beam Dumps (SHiP & NA62): These are experiments where a high-energy proton beam hits a block of metal (a "dump"). This creates a shower of particles. The idea is to look for ALPs that are produced in the dump, travel through a thick wall of shielding (blocking everything else), and then decay inside a detector on the other side.

The Conclusion: Opening a New Door

The paper concludes that by using these new strategies, we can finally explore the "heavy" region of ALP masses that was previously invisible.

• If the ALP is light: It escapes detection (like a ghost).
• If the ALP is heavy (the focus of this paper): It decays instantly into a visible, weird pattern of electrons and muons that the Standard Model cannot explain.

By combining data from future colliders, specialized flavor factories, and beam-dump experiments, scientists hope to either find this "ghostly messenger" or rule out a huge chunk of the possible places it could be hiding. It's a comprehensive net, cast wide to catch a particle that has been slipping through our fingers for too long.

Technical Summary

1. Problem Statement

The paper addresses a specific gap in the search for Axion-Like Particles (ALPs) coupled to charged leptons.

• The Regime: The study focuses on the mass regime $m_a > m_\mu$ (above the muon mass threshold).
• The Challenge: In this regime, the standard, highly sensitive constraints from exotic muon decays (e.g., $\mu \to e a$) are kinematically forbidden. Consequently, current bounds on Lepton Flavour Violating (LFV) ALP couplings ($e\mu$) are significantly weaker, typically limited to $f_a \approx 10$ TeV.
• The Opportunity: For $m_a > m_\mu + m_e$, the decay channel $a \to e\mu$ becomes kinematically accessible. Because the coupling is proportional to the muon mass, the ALP typically decays promptly (short lifetime) rather than escaping detection. This necessitates new search strategies that do not rely on missing energy but instead look for striking, background-free LFV signatures involving prompt ALP decays.

2. Methodology

The authors employ an Effective Field Theory (EFT) approach to model ALP interactions and calculate production rates across various high-energy processes.

A. Theoretical Framework

• Lagrangian: The ALP ($a$) interactions with charged leptons are governed by derivative couplings:
  $$\mathcal{L}_{a\ell\ell} \supset \frac{\partial_\mu a}{2f_a} \bar{\ell}_i \gamma^\mu (C^V_{ij} + C^A_{ij}\gamma_5) \ell_j$$
  where $f_a$ is the decay constant and $C^{V,A}_{ij}$ are dimensionless coefficients.
• SU(2)$_L$ Breaking: A critical aspect of their analysis is the potential departure from the $SU(2)_L$-invariant limit. If the coupling to left-handed charged leptons differs from that to neutrinos ($C^V_{ij} - C^A_{ij} - C^\nu_{ij} \neq 0$), an effective ALP-W vertex is generated. This removes the usual chiral (helicity) suppression in meson and $W$ decays, significantly enhancing production rates.
• Decay Widths: The ALP decay width into charged leptons ($a \to \ell_i \ell_j$) is calculated, noting that for $m_a > m_\mu$, the $a \to e\mu$ mode dominates (in the absence of large $\mu\mu$ couplings), leading to prompt decays.

B. Production Channels Analyzed

The paper computes production rates for ALPs emitted from virtual muons in the following processes:

1. Charged Meson Decays: $M^- \to e^- \bar{\nu}_\mu a$ (where $M = K, D_s, B$). The ALP is emitted from the virtual muon line ($M \to \mu^* \bar{\nu} \to e a \bar{\nu}$).
2. Gauge Boson Decays:
   • $W^- \to e^- \bar{\nu}_\mu a$
   • $Z \to e^\pm \mu^\mp a$
3. Quarkonium Decays: $V \to e^\pm \mu^\mp a$ (where $V = J/\psi, \Upsilon$).
4. Tau Decays: $\tau \to \ell a$ (where $\ell = e, \mu$), followed by $a \to e\mu$. This provides a complementary probe if the ALP couples to the third generation.

C. Experimental Scenarios & Simulations

The authors assess sensitivity for:

• Future $e^+e^-$ Colliders: CEPC and FCC-ee (Tera-Z factories), producing $\mathcal{O}(10^{12})$ $Z$ bosons and large samples of $W$, $\tau$, and $D_s$ mesons.
• Flavor Factories: Belle II and the proposed STCF (Super Tau Charm Factory), focusing on $J/\psi$ and $\tau$ decays.
• Fixed-Target/Beam-Dump: NA62 (rare kaon decays) and SHiP (Search for Hidden Particles).
  • Simulation Detail: For SHiP, the authors performed a detailed Monte Carlo simulation (using GEANT4 and MadGraph5) to model the production of $D_s$ and $K$ mesons in a proton target, their decay into ALPs, and the geometric acceptance of the ALP decaying within the fiducial volume (34m downstream, 50m long).

3. Key Contributions

1. New Search Strategy: Proposes utilizing on-shell ALP production in processes involving virtual muons, followed by the prompt decay $a \to e\mu$. This bypasses the kinematic suppression of direct $\mu \to e a$ searches.
2. Background-Free Signatures: Identifies unique final states with negligible Standard Model (SM) backgrounds:
   • Meson/W decays: $M/W \to e^- e^- \mu^+ \bar{\nu}$ (two same-sign electrons, one opposite-sign muon, missing energy).
   • Z/Quarkonium decays: $Z/V \to e^\pm e^\pm \mu^\mp \mu^\mp$ (four charged leptons with a resonant $e\mu$ pair).
   • These signatures violate lepton family number conservation in a way that SM processes cannot mimic without multiple misidentifications or rare multi-neutrino processes.
3. Role of Weak Breaking: Quantifies how $SU(2)_L$-breaking effects can enhance production rates in meson and $W$ decays by orders of magnitude compared to the chirally suppressed $SU(2)_L$-invariant case.
4. Indirect Constraints: Systematically analyzes indirect bounds from $\mu \to e\gamma$, $\mu \to eee$, and muonium-antimuonium oscillation, distinguishing between scenarios where flavor-diagonal muon couplings are negligible (Scenario A) or comparable to LFV couplings (Scenario B).

4. Results

The paper presents projected sensitivities in the $(m_a, f_a)$ plane (or equivalently $C_{e\mu}/f_a$) for two benchmark scenarios:

• Scenario A (Negligible $\mu\mu$ coupling):

  • Dominated by muonium-antimuonium oscillation constraints.
  • Direct searches (Z, $W$, $D_s$, $J/\psi$) can probe $f_a$ up to $\sim 5$ TeV at colliders and $\sim 10^6$ GeV at SHiP.
  • The $a \to e\mu$ branching ratio is $\approx 50\%$.

• Scenario B (Comparable $\mu\mu$ and $e\mu$ couplings):

  • Strongly constrained by loop-induced $\mu \to e\gamma$ and future $\mu \to eee$ searches, which exclude a large fraction of the parameter space.
  • Direct searches are less sensitive due to the reduced branching ratio ($BR(a \to e\mu) \approx 25\%$ due to the competing $a \to \mu\mu$ channel).
  • Exception: In the Left-Handed (LH) coupling scenario with significant $SU(2)_L$ breaking, $W$ boson decays at Tera-Z factories can still probe $f_a$ beyond the reach of indirect $\mu \to e\gamma$ limits for $m_a \gtrsim 10$ GeV.

• Displaced Vertices: If the ALP has LFV couplings to $\tau$ leptons, searches for displaced vertices in $\tau \to \ell a (\to e\mu)$ at Belle II and Tera-Z factories can probe $f_a$ up to $\sim 10^8$ GeV in the mass window $m_\mu \lesssim m_a \lesssim m_\tau$.

5. Significance

• Unexplored Territory: This work opens a largely unexplored avenue for testing ALP interactions above the muon mass threshold, a region where traditional missing-energy searches fail.
• Complementarity: The proposed strategies are complementary to existing low-energy constraints. While indirect bounds (like $\mu \to e\gamma$) are powerful, they rely on specific model assumptions (presence of $\mu\mu$ couplings). The direct production methods proposed here can test the parameter space even if flavor-diagonal couplings are suppressed.
• Feasibility: The paper demonstrates that with the high luminosity of future facilities (CEPC, FCC-ee, STCF, SHiP), it is possible to test ALP scales up to $f_a \approx 10^8$ GeV (via displaced vertices) and $f_a \approx 10^6$ GeV (via beam dumps), covering significant portions of the viable parameter space that are currently unconstrained.
• Model Building Insight: The analysis highlights the phenomenological importance of $SU(2)_L$-breaking effects, showing that they can drastically alter experimental reach, and discusses the UV-completion challenges associated with generating such effects without violating other experimental bounds.

In conclusion, the authors argue that LFV ALP production in meson, gauge-boson, quarkonium, and $\tau$ decays constitutes a promising and necessary strategy for the next generation of flavor and intensity-frontier experiments.