A Note on Daughter Poles at Nonvanishing Energy

Abstract

Equations which are satisfied by each daughter pole at any value of s are obtained for spinless particles by using the Bethe-Salpeter equation. The main assumptions are that Regge trajectories are analytic functions of A at the physical value, where A is a parameter which appears in the B-S equation, and that the residue of a parent Lorentz pole is analytic at s=O, in addition to the usual analyticity requirements. In this paper, we study how daughter trajectories are determined at S=1=O for spinless particles in the framework of the 13-S equation. Regge poles are obtained from the zeroes of the F'redholm determinant for the partial wave am­ plitude. At s = 0, the factors which determine daughter poles are separated from the determinant by using the 0 (4) symmetry. We prove that even at s=\=O, these factors are separated from the determinant. In this case, the analyticity of the scattering amplitude at s = ° is essential. The assumptions we require are that Regge trajectories are analytic functions of A at its physical value, where }, is a parameter which appears in the B-S equation, a.nd that the residue of a parent Lorentz pole is analytic at s = 0, in addition to the usual analyticity requirements. In § 2, we investigate a B-S equation with a Fredholm kernel. We expand the amplitude by the four-dimensional spherical functions. 4 ) By decomposing the Fredholm determinant for the partial wave amplitude, we have the zeroes corre­ sponding to the daughter poles at s = 0. In § 3, we continue the partial wave amplitude to unphysical values of l. From the analyticity requirements, we obtain equations which are satisfied by each daughter trajectory at any value of s.