Physical Meaning of Redundant Poles ofS-Matrix

Abstract

As was pointed out by Ma,l) when we want to determine the energy levels of the bound states from the poles in the positive imaginary axis of the complex k-plane (k= I k I, k is momentem), there may appear the redundant poles having no connection with the bound states. He discussed the motion of a particle in the potential V(r) = -Vo exp(-Iff).(1)The S-matrix for the S-wave is S(k)= J;.(a)r(ip-}--1) (a/z)-2ip, (Z) J-;.(a)r( -zp+ 1)where p=2k/p., a=2../2mVo/p., m is the mass of the considered particle. Therefore, S(k) has poles in the positive imaginary axis, when and J-;.(a)=O r(ip+1)=oo.(3)The condition (3) gives the correct results for the bound states, but (4) the redundant poles. Namely the redundant poles appear at

Effective Field Theory for Two-Body Systems with Shallow S-Wave Resonances

“…The interpretation of a redundant pole is unclear: it has a non-normalizable wavefunction [40,42], but carries information about the asymptotic behavior of continuum states [43]. Since at least in a nonrelativistic setting its position is determined by the range of the potential [35,44,45], it is comforting that renormalization of our EFT prevents a shallow redundant pole.…”