Study of Possible Practical Relativistic Generalizations of the Faddeev Equations in an Actual Model

Abstract

The relativistic Faddeev equations derived from the Bethe-Salpeter equation are considered with the alternative rules of Blankenbecler and Sugar which put the intermediate particles on the mass shell. The above rules give rise to three alternative sets of equations. Another set of equations is obtained by introducing relativistic kinematics and phase-space factors in the (nonrelativistic) Faddeev equations. Each alternative set of equations is applied to the problem of the possible existence of the pion as a bound state of three pions, considered previously. Although qualitative results are similar, quantitative results from the various alternatives differ considerably. Because of the phase-space factors, no cutoff is needed in these calculations. It is also found that the three pions can form a bound state at the pion mass only if the scattering length in the two-body amplitude is negative.