Fermions on the electroweak string

We construct a simple class of exact solutions of the electroweak theory including the naked Z-string and fermion fields. It consists in the Z-string configuration (φ, Zθ), the time and z components of the neutral gauge bosons Z0,3, A0,3) and a fermion condensate (lepton or quark) zero mode. The Z-string is not altered (no feedback from the rest of fields on the Z-string) while fermion condensates are zero modes of the Dirac equation in the presence of the Z-string background (no feedback from the time and z components of the neutral gauge bosons on the fermion fields). For the case of the n-vortexZ-string the number of zero modes found for charged leptons and quarks is (according to previous results by Jackiw and Rossi) equal to |n|, while for (massless) neutrinos it is |n| − 1. The presence of fernion fields in its core makes the obtained configuration a superconducting string, but their presence (as well as that of Z0,3, A0,3) does not enhance the stability of the Z-string.