Stability of an electron embedded in Higgs condensate

Abstract

We study stability of an electron distributed on the surface of a spherical cavity in Higgs condensate. The surface tension of the cavity prevents the electron from flying apart due to Coulomb repulsion. A similar model was introduced by Dirac in 1962, though without reference to Higgs condensate. In his model, the equilibrium radius of the electron equals the classical electron radius, $R^{c}_{e} \simeq 2.8 \times 10^{-13}$ cm, that is about $10^{5}$ times the radius consistent with experimental data. To address this problem, we replace the Coulomb term in the total energy of the electron by fermion self-energy involving screening by electrons occupying the negative energies of the vacuum. The tension of the cavity is obtained using the approximation $\xi_{0} \ll R_{0}$ where $\xi_{0}$ is the coherence length. For $\xi_{0} = 10^{-3} R_{0}$, the equilibrium radius in this model is $R_{0} \simeq 9.2 \times 10^{-32}$ cm. For such a small radius, we find the gravitational energy of the electron to be large enough to cancel the energy $\hbar c/R$, coming from the vibrational zero point energy and the kinetic energy of the embedded electron.