Singularity-free model of electrically charged fermionic particles and gauged $Q$-balls

We propose a model of an electrically charged fermion as a regular localized solution of electromagnetic and spinor fields interacting with a physical vacuum, which is phenomenologically described as a logarithmic superfluid. We analytically study the asymptotic behavior of the solution, while obtaining its form by numerical methods. The solution has physically plausible properties, such as finite size, self-energy, total charge, and mass. In the case of spherical symmetry, its electric field obeys the Coulomb asymptotics at large distances from its core. It is shown that the observable rest mass of the fermion arises as a result of interaction of the fields with the physical vacuum. The spinor and scalar field components of the solution decay exponentially outside the core; therefore, they can be regarded as internal degrees of freedom which can only be probed at sufficiently large scales of energy and momentum. Apart from conventional Fermi particles, our model can find applications in a theory of exotic localized objects, such as $U(1)$ gauged $Q$-balls with half-integer spin.

Figure 1

Electrostatic field ${\tilde{\varphi }}^{\prime }$ (a), scalar field $\tilde{\psi }$ (b), spinor field components $\tilde{f}$ (c), and $\tilde{h}$ (d) vs the dimensionless distance from the origin, evaluated at $\tilde{g}=0.1$, ${\tilde{\varphi }}_0=1$, $\tilde{e}=1$, ${\tilde{\psi }}_0=1$, ${\tilde{f}}_0=1$, and the following values of $\tilde{m}$: 0.4 (solid curve), 1.5 (dotted curve), 3.0 (dash-dotted curve), 6.0 (dashed curve), and 10.0 (dash-double dotted curve).

Figure 2

Dimensionless energy density $\tilde{H}$ (upper panel), charge density $\tilde{\rho }$ (middle panel), and TAM’s $z$-component density ${\tilde{\mathcal{M}}}_z$ (lower panel) vs the dimensionless distance from the origin, evaluated at $\tilde{g}=0.1$, ${\tilde{\varphi }}_0=1$, $\tilde{e}=1$, ${\tilde{\psi }}_0=1$, ${\tilde{f}}_0=1$, and the following values of $\tilde{m}$: 0.4 (solid curve), 1.5 (dotted curve), 3.0 (dash-dotted curve), 6.0 (dashed curve), and 10.0 (dash-double dotted curve).

Figure 3

Cumulative plot of an eigenvalue $\tilde{E}$ (solid curve, vertical axis’ scale $1:1.86$), eigenvalue $\tilde{\epsilon }$ (dotted curve, scale $1:9.81$), total rest mass-energy $\tilde{\mathrm{M}}$ (dot-dashed curve, scale $1:121.5$), total charge $\tilde{Q}$ (dashed curve, scale $1:42.1$), and TAM’s $z$-component ${\tilde{M}}_z$ (double-dot-dashed curve, scale $1:20.55$) vs $\tilde{m}$. Plotted values correspond to solutions of Eqs. (28)–(31), evaluated at $\tilde{g}=0.1$, ${\tilde{\varphi }}_0=1$, $\tilde{e}=1$, ${\tilde{\psi }}_0=1$, ${\tilde{f}}_0=1$.

Figure 4

Electrostatic field ${\tilde{\varphi }}^{\prime }$ (a), scalar field $\tilde{\psi }$ (b), spinor field components $\tilde{f}$ (c) and $\tilde{h}$ (d) vs the dimensionless distance from the origin, evaluated at $\tilde{g}=1$, ${\tilde{\varphi }}_0=1$, $\tilde{e}=1$, ${\tilde{\psi }}_0=1$, ${\tilde{f}}_0=1$, and the following values of $\tilde{m}$: 0.7 (solid curve), 1.5 (dotted curve), 3.0 (dash-dotted curve), 6.0 (dashed curve) and 10.0 (dash-double dotted curve).

Figure 5

Dimensionless energy density $\tilde{H}$ (upper panel), charge density $\tilde{\rho }$ (middle panel), and TAM’s $z$-component density ${\tilde{\mathcal{M}}}_z$ (lower panel) vs the dimensionless distance from the origin, evaluated at $\tilde{g}=1$, ${\tilde{\varphi }}_0=1$, $\tilde{e}=1$, ${\tilde{\psi }}_0=1$, ${\tilde{f}}_0=1$, and the following values of $\tilde{m}$: 0.7 (solid curve), 1.5 (dotted curve), 3.0 (dash-dotted curve), 6.0 (dashed curve), and 10.0 (dash-double dotted curve).

Figure 6

Cumulative plot of the eigenvalue $\tilde{E}$ (solid curve, vertical axis’ scale $1:2.48$), eigenvalue $\tilde{\epsilon }$ (dotted curve, scale $1:9.43$), total rest mass-energy $\tilde{\mathrm{M}}$ (dot-dashed curve, scale $1:461.1$), total charge $\tilde{Q}$ (dashed curve, scale $1:42.1$), and TAM’s $z$-component ${\tilde{M}}_z$ (double-dot-dashed curve, scale $1:45.6$) vs $\tilde{m}$. Plotted values correspond to solutions of Eqs. (28)–(31), evaluated at $\tilde{g}=1$, ${\tilde{\varphi }}_0=1$, $\tilde{e}=1$, ${\tilde{\psi }}_0=1$, ${\tilde{f}}_0=1$.