The Self-Energy of the Scalar Nucleon

Abstract

The analogue of the variational principle was applied on the path integral to obtain the self-energy of the scalar nucleon interacting with the neutral scalar meson (with vanishing mass) field for variable coupling constant g[superscript 2]. To obtain the lowest of the nucleon from the asymptotic form of its kernel it is essential to replace it (t:time) by another variable [tau], say. Therefore it was first shown that the equation for a scalar nucleon in a given external field has the form which can be obtained from the Klein-Gordon equation by formally changing it by [tau]. It is known that the Klein-Gordon equation may be described, though indirectly, by Lagrangian form by introducing a fifth parameter u and a differential equation for a new function of the space-time variables and u. Then the kernel of this equation is represented by a path integral over trajectories in space time. To apply the variational technique on this kernel and find the best value of the lowest energy corresponding to the Hamiltonian of this equation, here again we need to replace i u by [sigma]. The kernel corresponding to the transition in which there are no mesons present initially and finally was obtained by integrating out the meson field in the kernel for the case of a given potential. The energy contains the logarithmic divergence which was cut off by the analogy to electrodynamics. The kernel of a nucleon for the same transition can be given in terms of the kernel just explained. From the asymptotic form of this kernel the best value of the lowest (or self-) M of the nucleon was obtained. M was given in terms of preliminarily renormalized mass m'. It was found that no solution exists for too large value of g[superscript 2]. For practical purposes the procedure to find the theoretical mass m in terms of M and g[superscript 2] was also explained.