A variational calculation of particle--antiparticle bound states in the scalar Yukawa model

The perturbative expression for the particle--antiparticle bound state energy for the massless exchange case, equation (34) of our previous paper, is given without the retardation contribution, that is with in the first, one-chion exchange term of equation (29), and hence in equation (30). In fact, the retardation contribution is where with and where is the derivative of the Legendre function of the second kind. Therefore, the complete expression is The terms on the right are the rest energy, the non-relativistic Balmer term, the correction to the kinetic energy, one-chion exchange interaction (including retardation) and the virtual annihilation interaction, respectively. Note that for s -states, the retardation and virtual annihilation contributions cancel identically at .