New Reduction of the Faddeev Equations and Its Application to the Pion as a Three-Particle Bound State

A new separation of the angular momentum in the Faddeev equations is given. This separation makes use of the relative angular momentum of two particles, which is combined with the angular momentum of the third particle in the over-all center-of-mass system. With the assumption that the two-body amplitudes factorize in the initial and final momenta, the Faddeev equations are reduced to a coupled set of integral equations in one variable. This set is furthermore simplified in the case of identical particles to only one integral equation. Thereby the statistics is correctly taken into account. The resulting equation is used to investigate possible bound states of three pions with total angular momentum zero, isospin one, and odd parity. The two-body amplitude which determines the kernel is approximated by the isospin-zero, $s$-wave effective-range formula of Chew and Mandelstam. Use is also made of relativistic kinematics. The pion is found as a bound state of three pions in this model. The outcome is, however, strongly dependent on a physical cutoff parameter in the two-body form factor. As a result a detailed investigation of the form factor is desirable.