Vacuum degeneracy of scalar charged field with self-interaction $H'=g\int(\varphi^*\varphi)^2{dx}$ in the case of one spatial degree of freedom

The vacuum degeneration is studied in the quantum theory of a charged scalar field with the self-interaction $H'=g(\varphi^*\varphi)^2$ in the case of one spatial degree of freedom. Besides the Goldstone particle with the rest mass equal to zero, there is another particle with the non-zero rest mass, which decays into the even number of particles with zero rest mass. Impossibility of the decay into the odd number of particles is connected with the existence of a motion integral of the parity type; the particles with the zero and non-zero rest mass belong to the eigen-values of this integral equal to $+1$ and $-1$ respectively.