Superconducting cosmic string: Equation of state for spacelike and timelike current in the neutral limit

The equation of state relating the tension T and the energy per unit length U of a cosmic string is investigated in the simplest nontrivial case, namely, that of a field theory with U(1${)}^{\mathrm{local}}$×U(1${)}^{\mathrm{global}}$ invariance, in four dimensions, which is interpretable as the zero-charge-coupling-constant limit of the more general superconducting string models that have been previously investigated. This limit has the advantage of giving vacuum vortex defects that are strictly local so that the quantities such as U and T that are relevant for the macroscopic description can be computed without ambiguity. In the case of ‘‘electric’’ states (with timelike current) for which no comparable previous calculations exist, it is shown there is a critical frequency ${\mathit{w}}_{\mathit{c}}$ beyond which the vortex becomes unstable due to ‘‘charge’’ carrier emission. In the case of ‘‘magnetic’’ states (with spacelike current), the present analysis provides more precise results than those of previous investigations, whose predictions are broadly confirmed for typical moderate models in which the tension T remains comparable to the energy density U though not for extreme models, in which serious discrepancies are revealed.