The electron-proton bound state in the continuum with the positive binding energy of 1.531 of the electron mass

Abstract

In the bound states in the continuum (BIC) the binding energy is positive, and the mass of a composite particle is greater than the total mass of its constituents. In this work the BIC state is studied for the electron-proton system with using the ladder Bethe-Salpeter equation. We demonstrate that there are two momentum space regions in which the electromagnetic interaction between the particles is strongly enhanced, and the effective coupling constant is equal to $\alpha \sqrt{m_{p}/m_{e}}=0.313$, where $\alpha$ is the fine structure constant, $m_{p}$ and $m_{e}$ are the proton and the electron masses. This interaction resonance causes the confinement of the pair in the BIC state with the positive binding energy of 1.531 of the electron mass. The integral equation for the bispinor wave function is derived. This normalized wave function which must be complex, was found numerically in the momentum and coordinate spaces. It turned out that in the BIC state, the average radius for the electron is equal to 48Fm, and the average radius for the proton is equal to 1.1Fm. This composite particle can exist exclusively in the free state, in which its properties, such as its form-factors, should only be studied. In bound states with other particles, the composite loses its individuality.