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BPS vortex from nonpolynomial scalar QED in a $\mathds{C}\mathrm{P}^1$-Maxwell theory

F. C. E. Lima

Abstract

We investigate a generalized gauged $\mathds{C}\mathrm{P}^1$-Maxwell theory in which the electromagnetic sector acquires a field-dependent magnetic permeability generated dynamically through fermionic vacuum polarization. Starting from the gauged $\mathds{C}\mathrm{P}^1$-sigma model, whose dynamics occurs on a curved target space endowed with the Fubini-Study metric, we show that integrating out a Dirac fermion with effective mass induces, at one loop, a non-polynomial magnetic permeability, which after dimensional reduction to $(2+1)$-dimensions yields an effective Maxwell sector takes the form of a logarithmic magnetic permeability. Within this framework, one builds a generalized $\mathds{C}\mathrm{P}^1$-Maxwell model by admitting Bogomol'nyi-Prasad-Sommerfield (BPS) configurations. Taking this into account, we solved the self-dual equations that describe vortex-like solutions with quantized magnetic flux. Furthermore, one highlights the interactions between the target-space geometry and the induced permeability.

BPS vortex from nonpolynomial scalar QED in a $\mathds{C}\mathrm{P}^1$-Maxwell theory

Abstract

We investigate a generalized gauged -Maxwell theory in which the electromagnetic sector acquires a field-dependent magnetic permeability generated dynamically through fermionic vacuum polarization. Starting from the gauged -sigma model, whose dynamics occurs on a curved target space endowed with the Fubini-Study metric, we show that integrating out a Dirac fermion with effective mass induces, at one loop, a non-polynomial magnetic permeability, which after dimensional reduction to -dimensions yields an effective Maxwell sector takes the form of a logarithmic magnetic permeability. Within this framework, one builds a generalized -Maxwell model by admitting Bogomol'nyi-Prasad-Sommerfield (BPS) configurations. Taking this into account, we solved the self-dual equations that describe vortex-like solutions with quantized magnetic flux. Furthermore, one highlights the interactions between the target-space geometry and the induced permeability.
Paper Structure (16 sections, 58 equations, 10 figures)

This paper contains 16 sections, 58 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Formulation of the gauged $\mathbb{C}\mathrm{P}^1$ model
  3. Dynamical Generation of magnetic permeability in the $\mathbb{C}\mathrm{P}^1$ theory
  4. Dimensional reduction of the effective Maxwell sector
  5. Construction in the generalized Maxwell theory
  6. Radially symmetric BPS configurations
  7. On the magnetized configurations
  8. General asymptotic behavior
  9. The case near the origin
  10. The behavior at spatial infinity
  11. On the linear stability
  12. Solutions of the self-dual equations
  13. Example 1: The case without permeability, $m(f)=\left\vert\mu\sqrt{\mathrm{e}^{\frac{32\pi^2}{e^2}}}\right\vert$
  14. Example 2: The logarithmic case $m(f)=\left\vert\mu\sqrt{P(f)^\frac{32\pi^2}{e^2}}\right\vert$
  15. Example 3: The case polynomial: $m(f)=\left\vert\mu\mathrm{e}^{\frac{16\pi^2P(f)^{-1}}{e^2}}\right\vert$
  16. ...and 1 more sections

Figures (10)

  • Figure 1: One-loop fermionic vacuum polarization contributing to the gauge-field two-point function.
  • Figure 2: Numerical solutions of the variable fields with unit winding number.
  • Figure 3: Magnetic field $B(r)$ vs. $r$ with unit winding number.
  • Figure 4: BPS energy density $\mathcal{E}_{\mathrm{BPS}}(r)$ vs. $r$ with unit winding number.
  • Figure 5: Numerical solutions with coupling $e=1$ and unit winding number.
  • ...and 5 more figures