Probing Lepton-Flavor-Violating Four-Lepton Operators at a Muon Collider

We investigate charged lepton-flavour violation (LFV) induced by dimension-six four-lepton operators within the Standard Model Effective Field Theory at a proposed high-energy muon collider. We study the processes $\mu^{+}\mu^{-}\to e^{\pm}\tau^{\mp}$, $\mu^{+}\mu^{-}\to e^{\pm}\mu^{\mp}$, and $\mu^{+}\mu^{-}\to \mu^{\pm}\tau^{\mp}$ at $\sqrt{s}=3$, $10$, and $14$~TeV, incorporating beam polarisation and hadronic $\tau$ reconstruction. Using an optimal-observable analysis of the angular distributions, we perform a global fit to the relevant set of four-lepton operators. Projected sensitivities reach $C/\Lambda^{2}\sim(0.6$-$1.6)\times10^{-11}\,\mathrm{GeV}^{-2}$, depending on the flavour and chiral structure of the operator, exceeding current limits by up to an order of magnitude. A combined analysis of multiple centre-of-mass energies and beam polarisations significantly improves the resolution of correlations among the Wilson coefficients. These results highlight the strong sensitivity of a future multi-TeV muon collider to charged lepton flavour violating four-lepton interactions, establishing it as a powerful probe of the SMEFT parameter space.