Electron-muon conversion in nuclei and rare decays induced by LFV dark photon

Technical Summary: Electron-Muon Conversion and LFV Decays Induced by Dark Photons

Problem Statement
The paper addresses the search for Lepton Flavor Violation (LFV), a phenomenon not permitted in the Standard Model (SM) but predicted by various Beyond the Standard Model (BSM) theories. While significant experimental progress has been made in constraining LFV via muon decays (e.g., $\mu \to e\gamma$, $\mu \to 3e$) and coherent $\mu-e$ conversion in nuclei, the authors investigate the specific role of a sub-GeV mass vector mediator, the dark photon ($A'$), in these processes. The study focuses on two primary avenues:

1. $e-\mu$ conversion in nuclei: Specifically, the conversion of an electron beam into a muon via interaction with a nucleus ($e^- + N \to \mu^- + N$) in fixed-target experiments.
2. Rare radiative decays: The LFV decays of light pseudoscalar mesons, specifically $\eta(\eta') \to \gamma \mu e$.

The authors aim to derive bounds on the diagonal ($g_{ll}$) and non-diagonal ($g_{lf}$) couplings of the dark photon to SM fermions and determine the sensitivity of current and future experiments to these parameters.

Methodology
The authors employ a quantum field formalism incorporating a dark photon model developed in previous literature (Ref. [19]). The phenomenological Lagrangian includes a singlet scalar field, the dark photon, and Dirac dark fermions, with the dark photon mass generated via the Stueckelberg mechanism.

Key methodological components include:

• Coupling Scenarios: The analysis considers three distinct interaction scenarios for the dark photon:
  1. General Portal: Based on kinetic mixing ($\epsilon$) between the SM photon and the dark photon, allowing interaction with all fermions.
  2. Leptophilic Portal: The dark photon couples directly only to leptons; quark interactions are induced radiatively via loops.
  3. Hadrophilic Portal: The dark photon couples directly only to quarks; lepton interactions are induced radiatively.
• Process Calculation:
  • $\mu \to e\gamma$ and $\mu \to 3e$: The authors calculate decay widths using gauge-invariant sets of Feynman diagrams involving the dark photon mediator. Form factors are derived to handle divergences, ensuring renormalized results.
  • $e-\mu$ Conversion in Nuclei: The differential cross-section for $e^- N \to \mu^- N$ is calculated. The nuclear interaction is modeled using a spin-0 form factor (Tsai/Bjorken parametrization) to account for elastic and inelastic effects. The process is described by two diagrams: one induced by QED-dark photon mixing and another involving a vertex function relevant to the $\mu \to e\gamma$ transition.
  • $\eta(\eta') \to \gamma \mu e$ Decays: The decay width is computed using the Wess-Zumino-Witten (WZW) chiral anomaly framework. The authors relate the $P\gamma A'$ coupling to the standard $P\gamma\gamma$ coupling via the mixing parameter or loop-induced mechanisms, depending on the scenario. The calculation includes contributions from intermediate vector mesons ($\rho^0, \omega, \phi$) and utilizes the narrow-width approximation for the dark photon resonance region ($m_{A'} < m_\eta$).

Key Contributions

1. Derivation of Coupling Bounds: The paper derives constraints on the product of diagonal and non-diagonal dark photon couplings ($g_{ll}g_{lf}$) by analyzing rare LFV processes.
2. Fixed-Target Feasibility Analysis: The authors estimate the number of muons produced in $e-\mu$ conversion for specific fixed-target experiments: NA64e (CERN), LDMX (SLAC), BDX (FermiLab), and DarkSHINE (China). They utilize parameters such as beam energy, target density, and projected electrons on target (EoT).
3. Rare Decay Estimates: The study provides estimates for the branching ratios of $\eta \to \gamma \mu e$ and $\eta' \to \gamma \mu e$ under the three interaction scenarios, incorporating existing experimental limits from BABAR, NA64, COHERENT, and MiniBooNE.

Results

• $e-\mu$ Conversion Sensitivity: The analysis concludes that current and planned fixed-target experiments using electron beams (NA64e, LDMX, etc.) are unlikely to reach the sensitivity required to compete with existing bounds derived from LFV muon decays. The bounds on LFV couplings derived from $e-\mu$ conversion are less stringent than those from $\mu \to e\gamma$ and $\mu \to 3e$ decays. The authors state that achieving the necessary sensitivity would require increasing statistics by orders of magnitude.
• Rare Decay Branching Ratios:
  • Resonance Region ($m_{A'} < m_\eta$): In the sub-GeV mass region where resonance production is possible, the upper bounds for the branching ratios are estimated as:
    • $Br(\eta \to \gamma \mu e) \lesssim 2.7 \times 10^{-7}$ (Hadrophilic scenario).
    • $Br(\eta' \to \gamma \mu e) \lesssim 2 \times 10^{-8}$.
    • For the general scenario, bounds are $\lesssim 2 \times 10^{-7}$ and $\lesssim 10^{-9}$ for $\eta$ and $\eta'$ respectively.
  • Non-Resonance/Heavy Mass ($m_{A'} > m_\eta$): The branching ratios are significantly suppressed (e.g., $\lesssim 10^{-18}$) due to existing constraints from muon decays.
  • Diagonal Coupling Dominance: If diagonal couplings dominate, the bounds for $\eta$ and $\eta'$ decays are tighter, reaching levels of $\lesssim 2 \times 10^{-12}$ and $\lesssim 10^{-13}$, respectively, but this requires high-statistics meson yields.
• Leptophilic Suppression: In the leptophilic scenario, the branching ratios are suppressed by a factor of $\alpha/64\pi^3$ compared to the general dark photon scenario.

Significance and Claims
The paper claims to provide a comprehensive phenomenological assessment of LFV effects mediated by a dark photon in both fixed-target conversion and rare meson decays.

• On Fixed-Target Experiments: The authors modestly conclude that studying $e-\mu$ conversion in nuclei with current electron beam intensities is "impossible" to achieve the sensitivity levels of muon decay studies. The signal is expected to be too weak given the current statistical limits.
• On Rare Decays: The authors suggest that rare decays of $\eta$ and $\eta'$ mesons could be more sensitive to LFV effects than lepton conversion at GeV electron beam energies, particularly in the resonance region. They highlight that future facilities like the REDTOP and eta-HIAF factories, which plan to produce $10^{12} \div 10^{13}$ mesons, offer a viable testbed for these rare decays.
• Specific Signatures: The paper notes that the decay $\eta \to \gamma \mu e$ possesses a specific signature (a final state with one muon) that could be detected and analyzed in "super-$\eta$" factories, provided the vector mediator is feebly interacting.

In summary, the work shifts the focus from electron-beam induced conversion (deemed less promising with current stats) to the potential of high-statistics meson factories to probe LFV dark photon models through rare radiative decays.