Vacuum polarization of a charged massless fermionic field by a magnetic flux in the cosmic string spacetime

We calculate the vacuum averages of the energy-momentum tensor associated with a massless left-handed spinor fields due to magnetic fluxes on idealized cosmic string space-time. In this analysis three distinct configurations of magnetic fields are considered: (i) a homogeneous field inside the tube, (ii) a magnetic field proportional to 1/r, and (iii) a cylindrical shell with delta-function. In these three cases the axis of the infinitely long tubes of radius R coincides with the cosmic string. In order to proceed with these calculations we explicitly obtain the Euclidean Feynman propagators associated with these physical systems. As we shall see, these propagators possess two distinct parts. The first are the standard ones, i.e. corresponding to the spinor Green's functions associated with the massless fermionic fields on the idealized cosmic string spacetime with a magnetic flux running through the line singularity. The second parts are new, they axe due to the finite thickness of the radius of the tubes. As we shall see these extra parts provide relevant contributions to the vacuum averages of the energy-momentum tensor.