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Charged lepton mixing via heavy sterile neutrinos

Pseudoscalar meson decay leads to an entangled state of charged leptons ($μ,e$) and massive neutrinos. Tracing out the neutrino degrees of freedom leads to a reduced density matrix for the charged leptons whose off-diagonal elements reveal \emph{charged lepton oscillations}. Although these decohere on unobservably small time scales $ \lesssim 10^{-23} s $ they indicate charged lepton \emph{mixing} as a result of common intermediate states. The charged lepton self energy up to one loop features flavor off-diagonal terms responsible for charged lepton mixing: a dominant ``short distance'' contribution with $W$ bosons and massive neutrinos in the intermediate state, and a subdominant ``large distance'' contribution with pseudoscalar mesons and massive neutrinos in the intermediate state. Mixing angle(s) are GIM suppressed, and are \emph{momentum and chirality dependent}. The difference of negative chirality mixing angles near the muon and electron mass shells is $θ_L(M^2_μ) -θ_L(M^2_e)\propto G_F \sum U_{μj} m^2_j U^*_{j e}$ with $m_j$ the mass of the neutrino in the intermediate state. Recent results from TRIUMF, suggest an upper bound $θ_L(p^2\simeq M^2_μ)-θ_L(p^2 \simeq M^2_e) < 10^{-14}\,\Big(M_S/\mathrm{100}\,MeV\Big)^2$ for one generation of a heavy sterile neutrino with mass $M_S$. We obtain the wavefunctions for the propagating modes, and discuss the relation between the lepton flavor violating process $μ\rightarrow eγ$ and charged lepton mixing, highlighting that a measurement of such process implies a mixed propagator $μ, e$. Furthermore writing flavor diagonal vertices in terms of mass eigenstates yields novel interactions suggesting further contributions to lepton flavor violating process as a consequence of momentum and chirality dependent mixing angles.