Neutrino Zero Modes and Stability of Electroweak Strings

We discuss massless and massive neutrino zero modes in the background of an electroweak string. We argue that the eventual absence of the neutrino zero mode implies the existence of topologically stable strings where the required non-trivial topology has been induced by the fermionic sector. Recent experimental evidence strongly suggests that neutrinos are very light, but not massless. A simple way to extend the standard model such that it incorporates light neutrino masses is to add a singlet right-handed neutrino. The origin of MR is a vacuum expectation value of some scalar field, acquired at some energy scale much greater than that of the electroweak scale. MR can therefore be taken to be spatially homogeneous. The Lagrangian is: LSM = − 4 W a μνW aμν − 1 4 FμνF μν + (DμΦ) † (DμΦ)− λ ( Φ†Φ− η )2 − iΨ̄γDμΨ− ieRγDμeR + h′ ( eRΦ †Ψ + Ψ̄ΦeR ) (1) + h [ Ψ̄iτ2Φ ∗νR + νRΦ (iτ2)†Ψ ] + 1 2 ν RM ∗ RνR + 1 2 νRMRν c R with Ψ ≡ (νL, eL)T , Φ ≡ (φ, φ) , ν L = (α, β,−α,−β), ν R = (γ, δ, γ, δ), ν ≡ Cν̄ and with the usual definition of the gauge and Higgs fields and covariant derivatives [1]. In the Z-string ansatz, all the gauge and Higgs fields are zero except: Z ∼ ( 0,− r ~eθ ) , φ = ηf(r)e (2) where v(r) and f(r) are the Z-string profile functions. The neutrino equations of motion are: