Low-energy behavior of the few-body scattering amplitudes

We know that the two-body scattering amplitude at low energy goes to a constant. Only recently, it has been shown by Amado and Rubin that the three-body scattering amplitude at low energy diverges like $a{E}^{-1}+b{E}^{-\frac{1}{2}}+c\ln E$, where E is the three-body center-of-mass energy. We here calculate the coefficients a, b, c in terms of the two-body s-wave scattering length and some kinematical factors. We extend our analysis for the four-body case and find that at low energy the four-body scattering amplitude also diverges, and we classify the divergent terms and indicate how to calculate their coefficients. Beyond the four-body problem the scattering amplitude also diverges at low energy, and we indicate what is needed to calculate the coefficients of the divergent terms in general.