On the Solutions of the Bethe-Salpeter Equation in the Unequal-Mass Wick-Cutkosky Model in the Moving System

Abstract

The stereographic projection method is applied to obtain the bound state amplitude in the moving system of the Bethe-Salpeter equation in the unequal-mass Wick-Cutkoskymod­ el. The timelike, spacelike, lightlike and spurionlike solutions are derived from the one In the moving system. The normalization properties of these solutions are also discussed. It has been found in an early paperl) that the stereographic projection method, which was first introduced by Cutkosky,2) is extremely useful to solve the Bethe-Salpeter (B-S) equation in the unequal-mass Wick-Cutkosky modePl,3l for the bound state amplitude in the rest system. The method leads us to the 0(4) symmetry in the converted momentum space, and hence it becomes possible to solve the equation by use of the four-dimensional solid harmonic. In this paper, the same method is applied to solve the bound state equation 111 the moving system by the extended ~se of the prescriptions given in the early paper. It will be shown that the method is also useful ~or this case and the solution in the moving system obtained in this way contains the timelike, space­ like, lightlike and spurionlike solutions which are the analytic continuities of one another with respect to the total momentum of the bound state. The solutions have also been presented by Nakanishi without using the stereographic. projection method;4 l however, the method adopted in this paper to derive these solutions is much simpler than that of Nakanishi's and the connection between these solu­ tions can easily be seen. In the next section we apply the stereographic projection method to the B-Sequation in the moving system and its solutions are explicitly given for the time1ike, spacelike, lightlike and spurionlike bound states in § 3. The B-S equa­ tion in the unequal-mass Wick-Cutkosky model presents a useful mathematica'l tool to reveal the analytical behaviors of the bound state such as the negative norm and the multiple poles of the corresponding Green's function. These ana­ lytical behaviors arc discussed in the fmal section.