Vacuum polarization of a charged massless scalar field on cosmic string spacetime in the presence of a magnetic field

In this paper, we consider a charged massless scalar quantum field operator in the spacetime of an idealized cosmic string, i.e., an infinitely long, straight and static cosmic string, which presents a magnetic field confined in a cylindrical tube of finite radius. Three distinct situations are taken into account in this analysis: (i) a homogeneous field inside the tube, (ii) a magnetic field proportional to 1/r and (iii) a cylindrical shell with δ-function. In these three cases, the axis of the infinitely long tube of radius R coincides with the cosmic string. In order to study the vacuum polarization effects outside the tube, we explicitly calculate the Euclidean Green function associated with this system for the three above situations, considering points in the region outside the tube. Having these Green functions we calculate the renormalized vacuum expectation values, ⟨Φ̂*(x)Φ̂(x)⟩Ren and ⟨T̂νμ(x)⟩Ren, associated with the charged field. In the evaluation of these vacuum polarization effects, two contributions appear for the three models. The first are the standard ones due to the conical geometry of the spacetime and the magnetic flux. The second contributions appear as extra terms. They are corrections due to the finite thickness of the radius of the tube. These extra terms provide relevant contributions, even for points very far away from the system, like a long-range effect.