Impact of space-time curvature coupling on the vacuum energy induced by a magnetic topological defect in flat space-time of arbitrary dimension

TL;DR

The paper investigates whether the curvature coupling $ξ$ of a scalar field leaves a measurable imprint on the total vacuum energy induced by a finite-thickness magnetic flux tube in flat $d+1$ dimensional space-time. It derives a general decomposition of the renormalized energy into a canonical part and a $ξ$-dependent part, and evaluates the $ξ$-dependent contribution in $(2+1)$- and higher dimensions for Robin boundary conditions parameterized by $θ$ and fractional flux $F$. For Robin BCs, the $ξ$-dependent term $E^{(d+1)}_{ξ}$ is nonzero and can be comparable to or larger than the canonical energy, especially for thin tubes, with distinct asymptotics in the thin- and thick-tube limits. These results suggest vacuum-polarization measurements near magnetic topological defects may provide a novel probe of the curvature coupling $ξ$, with implications for both high-energy and condensed-matter contexts.

Abstract

We have investigated vacuum polarization of a quantized charged massive scalar field in the presence of a magnetic topological defect, modeled as an impenetrable tube of finite thickness carrying magnetic flux. At the tube's surface, we imposed a general Robin boundary condition. Our analysis demonstrates that, in flat space-time, the total induced vacuum energy is independent of the coupling $ξ$ of the scalar field's interaction with the space-time curvature only in the special cases of Dirichlet and Neumann boundary conditions. For general Robin boundary conditions, however, the total induced vacuum energy depends on the coupling $ξ$ in a flat space-time and exhibits a nontrivial dependence on the parameter of the Robin boundary condition. We investigated the dependence of this effect not only on Robin's boundary condition parameter, but also on the tube thickness and the space-time dimensionality. We conclude that careful measurements of vacuum polarization effects in flat space-time may, in principle, provide an independent way to probe the $ξ$ coupling.

Figures (3)

• Figure 1: The total induced dimensionless vacuum energy $E_\xi^{2+1}/m$\ref{['m5']} in $(2+1)$-dimensional space-time as the function of parameter $\theta$ of Robin boundary conditions on the edge of the magnetic impenetrable tube is presented for different thicknesses of the tube. For convenience, the above functions are multiplied by the coefficient $c$. For example, the value of the function presented in the figure for $mr_0=1/10$ is multiplied by a factor of $c=200$.
• Figure 2: The induced dimensionless vacuum energy $E_\xi^{d+1}/m^{d-1}$\ref{['4s5']} within the plane transverse to the tube in $(d+1)$-dimensional space-time as the function of parameter $\theta$ of Robin boundary conditions on the edge of the magnetic impenetrable tube is presented for different thicknesses of the tube: a) $d=3$, b) $d=4$. For convenience, the above functions are multiplied by the coefficient $c$. For example, the value of the function presented in the figure a) for $mr_0=10^{-1}$ is multiplied by a factor of $c=8000$.
• Figure 3: The induced dimensionless vacuum energy $E_{can}^{d+1}/m^{d-1}$ within the plane transverse to the tube in $(d+1)$-dimensional space-time as the function of the thicknesses of the magnetic impenetrable tube $mr_0$ for the cases of a) Dirichlet and b) Neumann boundary conditions.