Effective Lepton Flavor Violating couplings at Muon Collider

Technical Summary: Effective Lepton Flavor Violating couplings at Muon Collider

Problem Statement
While the Standard Model (SM) is highly successful, it fails to explain phenomena such as baryon asymmetry, neutrino oscillations, and the absence of dark matter candidates. Although Lepton Flavor Violation (LFV) is absent in the SM, small contributions may arise from Beyond Standard Model (BSM) scenarios or low-energy effective interactions. This paper investigates the potential of a proposed $\mu^+\mu^-$ collider to detect LFV signatures induced by dimension-six effective four-fermionic operators, specifically the process $\mu^+\mu^- \to e^\pm \tau^\mp$. The study aims to estimate the sensitivity of Wilson coefficients for these operators and compare the projected limits with existing constraints from low-energy experiments (such as $\tau \to \mu\mu e$ decays) and high-energy colliders like the LHC.

Methodology
The authors employ an Effective Field Theory (EFT) approach, defining an interaction Lagrangian ($\mathcal{L}_{LFV}$) containing six dimension-six operators involving scalar and vector currents with various chiral structures ($LL, LR, RL, RR$). Due to the initial state of the muon collider, the first two flavor indices are restricted to the same generation, allowing the authors to tag couplings by the final state indices ($e, \tau$). Using Fierz identities, the six effective couplings are reduced to three linearly independent Wilson coefficients: $C_{LL}/\Lambda^2$, $C_{LR}/\Lambda^2$, and $C_{RR}/\Lambda^2$.

The analysis proceeds through the following steps:

1. Simulation: The effective Lagrangian is implemented in FeynRules to generate Feynman rules, which are fed into the event generator MadGraph. Signal ($\mu^+\mu^- \to e^\pm \tau^\mp$) and background processes ($\mu^+\mu^- \to \tau^+\tau^-, W^+W^-, \nu\bar{\nu}Z$) are simulated.
2. Detector Simulation: Generated events undergo parton showering via Pythia8 and detector simulation using Delphes3.
3. Kinematic Selection: To suppress backgrounds (where taus decay hadronically), specific cuts are applied: a veto on final-state muons ($N_\mu=0$), exactly one electron ($N_e=1$), and one hadronically decaying tau ($N_{\tau_h}=1$). A transverse momentum cut of $p_T > 1$ TeV is imposed on the electron to maximize signal retention while minimizing background.
4. Statistical Analysis:
   • Significance: Signal significance ($S = N_S/\sqrt{N_B}$) is calculated for 5$\sigma$ discovery contours.
   • Optimal Observable Method: To constrain the Wilson coefficients ($c_i$) at the 1$\sigma$ level, the authors utilize the optimal observable method. This technique exploits the full shape profile of differential distributions (specifically $d^2N/d(\cos\theta)d(p_T)$) to construct a covariance matrix and calculate $\Delta\chi^2$ contours.

Key Results
The study evaluates the collider performance at center-of-mass energies ($\sqrt{s}$) of 3 TeV and 10 TeV, with integrated luminosities ($\mathcal{L}_{int}$) of 1 ab$^{-1}$ and 10 ab$^{-1}$, respectively. Both unpolarized and polarized ($\pm 80\%$) muon beams are considered.

• Cross-Sections and Polarization: At $\sqrt{s} = 3$ TeV, the cross-sections for LFV processes vary significantly with beam polarization. For instance, with $C_{LL}/\Lambda^2 = 1 \times 10^{-9}$ GeV$^{-2}$, the cross-section increases from 0.74 fb (unpolarized) to 1.3 fb with $P_{\mu^-} = -80\%$, while the background cross-section drops from 2554 fb to 4588 fb (note: background behavior is complex, but polarization generally aids in signal-to-background optimization).
• Discovery Potential: Contour plots for 5$\sigma$ significance demonstrate that the muon collider can probe Wilson coefficients well beyond current limits derived from the Belle experiment ($B(\tau \to \mu\mu e) \le 2.7 \times 10^{-8}$).
• Measurement Precision: Using the optimal observable method, the authors map the 1$\sigma$ confidence regions ($\Delta\chi^2 = 2.3$) for the Wilson coefficients. The analysis indicates that polarization plays a crucial role in tightening these constraints.
• Energy and Luminosity Scaling: A comparison between the 3 TeV (1 ab$^{-1}$) and 10 TeV (10 ab$^{-1}$) scenarios reveals that the higher energy and luminosity configuration can provide upper limits on Wilson coefficients that are one order of magnitude or more stringent than those achievable at 3 TeV.

Significance and Claims
The paper claims that the proposed muon collider offers a high-precision probe for effective LFV vertices. The authors assert that at $\sqrt{s} = 3$ TeV with $\mathcal{L}_{int} \sim 1$ ab$^{-1}$, the collider can probe these couplings with an accuracy superior to existing limits from the LHC, electroweak physics, and B-meson decays. Furthermore, the study highlights that beam polarization is a significant factor in enhancing sensitivity to these effective couplings. The results suggest that a 10 TeV muon collider with 10 ab$^{-1}$ luminosity could establish constraints on Wilson coefficients that are substantially tighter (by an order of magnitude or more) than current capabilities, providing a powerful test for BSM physics scenarios involving lepton flavor violation.