Multiple Poles in the Off-Shell Scattering Amplitude

Abstract

We consider, in the ladder appro'ximation, the scattering problem of two spinless part~cles interacting through a massive scalar field. The off-shell scattering amplitudes of these par­ ticles are analytically continued from the real axis to the imaginary axis in both the initial and final relative energy planes. In particular, in this' paper, we deal with amplitudes on the imaginary axes, assuming that they satisfy the Bethe-Salpeter equation in a Euclidean space. Then, the scattering amplitudes in the zero total energy-momentum case are obtained rigor­ ously as functions of coupling constants. These functions have no singularities except for simple poles in the ,variable of coupling constants. On the other hand, the scattering ampli­ tudes in the massless case Pp, = (0, 0, Po, Po) can be obtained in a series expansion in powers of Po, by starting from the scattering amplitudes in the zero energy-momentum case. Then, it is shown that only simple and multiple poles exi~t in the massless amplitudes at those values of the coupling constants, for which the zero energy-momentum amplitudes are singu­ lar. The order of the ~bove multiple poles is equal to or lower than L-Iml + 1. Here, m is the helicity of the massless bound state and L is the maximum value of Iml. Residues of the N-th order multiple poles are expressed in a power series in Po, by starting from the power P02(N-l). Finally, it is shown that multiple poles always exist in the equal-mass case, while they exist in the unequal-mass case only when massive bound states are not degenerate with respect to angular momentum.