Multiple Poles in the Off-Shell Scattering Amplitude
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概要
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We consider, in the ladder approximation, the scattering problem of two spinless particles interacting through a massive scalar field. The off-shell scattering amplitudes of these particles 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 rigorously 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 amplitudes in the massless case P_μ= (0,0, P_0, P_0) can be obtained in a series expansion in powers of P_0, by starting from the scattering amplitudes in the zero energy-momentum case. Then, it is shown that only simple and mu ltiple poles exist in the massless amplitudes at those values of the coupling constants, for which the zero energy-momentum amplitudes are singular. The order of the above multiple poles is equal to or lower than L-|m|+1. Here, m is the helicity of the massless bound state and L is the maximum value of |m|. Residues of the N-th order multiple poles are expressed in a power series in P_0, by starting from the power P_0^<2(N-1)>. 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.
- 理論物理学刊行会の論文
- 1969-02-25
著者
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NAITO Seichi
Department of Physics Osaka City University
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NAITO Seichi
Department of Physics, University of Tokyo