Electron-electron bound states in Maxwell-Chern-Simons-Proca QED

We start from a parity-breaking MCS QED3 model with spontaneous breaking of the gauge symmetry as a framework for evaluation of the electron-electron interaction potential and for attainment of numerical values for the $e^{-}e^{-}-$ bound state. Three expressions ($V_{{{\rm eff}}_{\uparrow \uparrow }}$, $V_{{{\rm eff}}_{\uparrow \downarrow }}$, $V_{{{\rm eff}}_{\downarrow \downarrow}})$ are obtained according to the polarization state of the scattered electrons. In an energy scale compatible with condensed matter electronic excitations, these three potentials become degenerated. The resulting potential is implemented in the Schrödinger equation and the variational method is applied to carry out the electronic binding energy. The resulting binding energies in the scale of $10{-}100$ meV and a correlation length in the scale of $10{-}30$ Å are possible indications that the MCS-QED3 model adopted may be suitable to address an eventual case of $e^{-}e^{-}$ pairing in the presence of parity-symmetry breakdown. The data analyzed here suggest an energy scale of $10{-}100$ meV to fix the breaking of the U(1)-symmetry