A new method for the solution of the two-body Dirac equation: Estimation of the weights of C, P and $CP$ symmetry violating terms in positronium

A new theoretical method is developed for the solution of the two-body bound-state Dirac equation for positronium. Only Coulomb potential was included in the Dirac Hamiltonian. It is shown that the two-body Dirac Hamiltonian can be written in the Hermitian matrix form of the 4× 4 size and contains terms, responsible for the violation of the P, C, and $CP$ symmetries. Numerical results for the energy spectrum of the para- and ortho-positronium ground states performed within the variational method using the harmonic oscillator basis functions are in good agreement with a high-precision finite-element method of T.C. Scott et al. The weights of the P and $CP$ symmetry-violating components in the para-positronium ground state are identical to the weight of the P symmetry-violating component of the ortho-Ps and are estimated to be 6.6E-6. The weights of the C and $CP$ symmetry-violating components of the ortho-Ps are equal to the 2/3 and 1/3 parts of this value, respectively. These numbers are less by two orders of magnitude than the precision limit of current experimental facilities.