Magnetic field in vacuum of quantum spinor matter induced by a cosmic string in three-dimensional space

TL;DR

This work analyzes vacuum polarization of a massive spinor field in the background of a cosmic string modeled as an impenetrable finite-radius tube with internal magnetic flux in 3+1 dimensions. By enforcing self-adjointness of the Dirac operator and exploiting discrete-symmetry constraints, the authors show that only the MIT bag boundary condition ($\theta=0$) yields a finite induced vacuum magnetic flux, while other boundary choices lead to divergences. The induced flux is computed via a mode-sum approach, relating the 3+1D result to the known 2+1D flux through $\Phi_I(mr_0) = \int_0^\infty \frac{dk_3}{\pi} \Phi_I^{(2D)}(\sqrt{m^2+k_3^2}\, r_0)$, and exhibits a $\ln(mr_0)$ divergence as the tube thickness vanishes, with a stronger effect in 3+1D than in 2+1D for small $mr_0$. The flux vanishes at $F=1/2$, and the dependence on the interior flux $F$ is sine-like and symmetric under $F \to 1-F$. These results clarify the role of boundary conditions in quantum vacuum phenomena around cosmic strings and confirm the physical viability of the MIT bag boundary condition in 3+1D.

Abstract

A linear magnetic topological defect (cosmic string) is modeled as a magnetic flux-carrying tube that is impenetrable to external spinor matter. The matter field is quantized in the background of this tube, with the most general set of boundary conditions ensuring both the tube's impenetrability and the self-adjointness of the Dirac Hamiltonian operator. We compute the induced vacuum magnetic flux along the tube in (3+1)-dimensional space-time. It was shown that the requirement for the total induced vacuum magnetic flux to be finite restricts the admissible boundary conditions to only one choice: the MIT quark bag boundary condition. The dependence of the effect on the transverse size of the tube and the flux inside the tube was also analyzed.

Figures (2)

• Figure 1: Induced vacuum magnetic flux in dimensionless units as the function of the "magnetic" flux inside the tube $F$ at boundary parameter $\theta=0$ for different values of the tube's thickness $mr_0=10^{-3}$, $10^{-2}$, $10^{-1}$, $1$: a) $e m^{-1}\Phi^{(d=2)}_{\rm I}$ in $(2+1)$-dimensional space-time, b) $e \pi \Phi^{(d=3)}_{\rm I}$ in $(3+1)$-dimensional space-time. For convenience of presentation, the above functions are multiplied by the coefficient $c$.
• Figure 2: Induced vacuum magnetic flux in dimensionless units in 2+1 space-time ($e m^{-1}\Phi^{(d=2)}_{\rm I}$), and in 3+1 space-time $e \pi \Phi^{(d=3)}_{\rm I}$ as function of the tube thickness $mr_0$ for the cases of $mr_0<0.16$ (a) and $mr_0>0.16$ (b).