Lepton Flavor Violation: From Muon Decays to Muon Colliders

Technical Summary: Lepton Flavor Violation: From Muon Decays to Muon Colliders

Problem Statement
The Standard Model (SM) of particle physics, while successful, leaves several fundamental puzzles unresolved, including the flavor puzzle—the observed patterns of masses and mixings between fermion generations. New physics (BSM) in the ultraviolet (UV) generically induces flavor-changing processes. While low-energy, high-intensity experiments have placed stringent constraints on charged lepton-flavor violation (LFV), particularly in $\mu-e$ transitions, they typically probe a narrow range of flavor observables and BSM models. Furthermore, if a signal is discovered in these precision experiments, a high-energy collider is required to comprehensively characterize the underlying physics. The paper investigates the unique potential of a future high-energy muon collider (specifically a benchmark 10 TeV machine with 10 ab$^{-1}$ integrated luminosity) to probe LFV signals, complementing and extending the reach of low-energy precision searches.

Methodology
The authors adopt a model-independent approach using the Standard Model Effective Field Theory (SMEFT). They focus on the nine independent dimension-six operators involving only leptons, electroweak gauge bosons, and the Higgs field that violate lepton flavor (specifically $\tau-\mu$, $\tau-e$, and $\mu-e$ transitions) and do not involve quark fields.

1. Low-Energy Probes: The authors systematically match SMEFT operators onto the Low-Energy Effective Field Theory (LEFT) at the weak scale, incorporating renormalization group evolution (RGE) and one- and two-loop matching effects. They calculate constraints from existing and projected low-energy observables, including:

   • Radiative decays ($\ell \to \ell' \gamma$).
   • Three-body decays ($\ell \to 3\ell'$).
   • Muon-to-electron conversion in nuclei ($\mu \to e$).
   • Muonium-antimuonium conversion.
   • Flavor-violating decays of the $Z$ and Higgs bosons.

2. High-Energy Probes (Muon Collider): The authors simulate high-energy scattering processes unique to a muon collider environment. They utilize MadGraph5 for signal and background generation, Pythia8 for hadronization, and Delphes with a parameterized muon collider detector card for fast simulation. Key processes analyzed include:

   • LFV Higgs Decays: $h \to \ell_i \ell_j$ via vector boson fusion (VBF).
   • Scattering Processes: $\mu V \to \tau h$, $\mu W \to \tau W$, $\mu \mu \to \mu \tau$, $\mu V \to \tau \tau \tau$, and $\mu^\mp V \to \mu^\pm \tau^\mp \tau^\mp$ (where $V = \gamma, Z$).
   • Backgrounds: Irreducible backgrounds (e.g., $W/Z$ production, $\tau$ pair production) and reducible backgrounds (e.g., jet mistagging) are simulated and suppressed using kinematic cuts (e.g., transverse momentum $p_T$, invariant mass, stransverse mass $M_{T2}$, and missing transverse energy $E_T^{miss}$).

3. Flavor Ansätze: To compare the sensitivity of different probes across generations, the authors test various assumptions regarding the flavor structure of Wilson coefficients, including "flavor anarchy" ($C_{ij} \sim 1$), mass-proportional scaling ($C_{ij} \propto m_i m_j$ or $\sqrt{m_i m_j}$), and Minimal Lepton Flavor Violation (MLFV).

Key Contributions and Results

• SMEFT Mapping and Low-Energy Constraints: The paper provides a comprehensive recasting of low-energy limits (e.g., $\mu \to e \gamma$, $\mu \to e$ conversion, $Z \to \ell_i \ell_j$) in terms of the specific SMEFT operators. It highlights that while $\mu-e$ transitions are extremely well-constrained by low-energy experiments, constraints on $\tau-e$ and $\tau-\mu$ transitions are significantly weaker, leaving substantial parameter space open.
• LFV Higgs Decays at a Muon Collider: The authors project that a 10 TeV muon collider can improve sensitivity to LFV Higgs decays ($h \to \tau e$ and $h \to \tau \mu$) by approximately one order of magnitude compared to High-Luminosity LHC (HL-LHC) projections. For $h \to \mu e$, the sensitivity remains comparable to HL-LHC due to the lack of hadronic backgrounds in proton collisions, but the muon collider offers superior reach for $\tau$-involving channels.
• Scattering Process Sensitivity:
  • $\mu V \to \tau h$ and $\mu W \to \tau W$: These processes probe dipole operators ($O_{eB}, O_{eW}$) and Higgs-current operators ($O_{He}, O_{Hl}^{(1,3)}$). The muon collider can probe new physics scales up to $\mathcal{O}(10\text{--}100)$ TeV.
  • $\mu \mu \to \mu \tau$: This process, sensitive to four-fermion operators ($O_{ee}, O_{ll}, O_{le}$), benefits from the $\sqrt{s}$ enhancement of contact interactions. It achieves the highest reach, probing scales up to $\Lambda \sim 160\text{--}225$ TeV, vastly exceeding the reach of future low-energy $\tau \to 3\mu$ searches (limited to $\sim 40\text{--}50$ TeV).
  • $\mu V \to \tau \tau \tau$ and $\mu^\mp V \to \mu^\pm \tau^\mp \tau^\mp$: These processes probe specific four-fermion combinations and $\Delta L = 2$ operators, reaching scales of $\sim 20\text{--}35$ TeV.
• Complementarity and Flavor Structures:
  • Confirmation vs. Extension: For many operators, if a signal is detected in future low-energy experiments, the muon collider can confirm and characterize it. For certain flavor combinations (e.g., specific four-fermion operators or $\tau-\mu$ transitions with mass-suppressed coefficients), the muon collider extends the reach to scales inaccessible to low-energy experiments.
  • Flavor Ansätze: Under "flavor anarchy," low-energy $\mu-e$ constraints are generally dominant. However, under hierarchical assumptions (e.g., $C_{ij} \propto m_i m_j$ or MLFV), $\tau-\mu$ transitions become more constraining. In these scenarios, the muon collider often outperforms projected low-energy limits, particularly for four-fermion operators.
  • Multi-Operator Constraints: The paper demonstrates that combining multiple collider channels (e.g., $\mu \mu \to \mu \tau$ and $\mu W \to \tau W$) with low-energy data allows for the disentanglement of degenerate operator combinations, providing a nearly complete characterization of the underlying BSM physics.

Significance
The paper argues that a high-energy muon collider is uniquely positioned to probe lepton flavor violation, particularly in the $\tau-\mu$ and $\tau-e$ sectors where low-energy constraints are weaker. Unlike proton colliders, the muon collider offers a clean environment with high center-of-mass energy and initial states dominated by second-generation leptons, enabling the study of high-momentum scattering processes that are inaccessible at lower energies.

The authors emphasize that while low-energy experiments may provide the first hints of new physics, a muon collider is essential for:

1. Confirming signals found in precision experiments.
2. Extending the reach to energy scales well beyond the kinematic limits of low-energy decays.
3. Characterizing the underlying operator structure through complementary high-energy scattering channels.

The study concludes that targeted analyses of LFV at a muon collider are critical for guiding the design of future detectors and shaping the long-term physics case for the machine, ensuring that the discovery potential of next-generation colliders is fully realized.