Neutral Pseudoscalar Mesons Research Articles

A covariant formulation of the non-adiabatic method for the relativistic two-body problem (II)

It has been shown in Part I of this work how the Tamm-Dancoff non adiabatic treatment of the relativistic problem of two particles interacting through a neutral pseudoscalar meson field can be put in a covariant form. The integral equations obtained involve infinite kernels corresponding to radiative processes. In the present Part II the techniques of Dyson and Matthews for the renormalization of theS-matrix have been applied in order to separate from the kernel of the integral equations a convergent part, showing that the divergent contributions can be absorbed in unobservable mass and coupling constant renormalization. If we assume the normalized masses and charge to be defined by the same formal series with divergent constant terms as in theS-matrix, as required if a physical meaning is to be attached to the renormalization procedure, finite contributions arise also from self energy kernels of non interacting particles. The terms arising from disconnected loops in the vacuum can be eliminated if the same approximation is used as in the (S-matrix theory, namely that the time dependence of the state vector can be neglected in these integrals.

The non-adiabatic method with a neutral pseudo-scalar meson field

Dancoff's non-adiabatic procedure for the two-nucleon system interacting with a meson field is applied to the neutral pseudoscalar meson field, taking into account both types of couplings, the pseudoscalar and the pseudo-vector. The procedure is limited to the 2nd order of the coupling constants. Using the Hamiltonian containing the directinteraction term, the Schrodinger equation of motion contains interaction potentials, one of which introduces into the equation a high singularity. The reason is that the term arising from the direct-interaction Hamiltonian does not cancel completely with the so-called contact-term, which arises from the nucleon-meson interaction Hamiltonian, as it is the case in the adiabatic procedure. If the direct-interaction Hamiltonian is neglected from the beginning, the Schrodinger equation contains a singular potential, which corresponds to the contact-term of the adiabatic approximation. This potential behaves, however, as the corresponding potential in the previous case, and not as a δ-function, as is the case in the diabatic procedure. The remaining interaction potentials, which are the same in the two cases, are less singular than the potentials in the adiabatic procedure. In the non-relativistic limit, the interaction functions lead to the same functions given by the conventional perturbation procedure.