Zeros of Scattering Amplitude and Consistency of Dispersion Relation

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Abstract
The validity of the necessary condition for the existence of solutions of pion-nucleon forward dispersion relation is examined using experimental data. The position of two zeros of scattering amplitude is determined on the imaginary axis. The position of zeros on the real axis depends critically on the values of the S-wave scattering lengths. If we use Hamilton-Woolcock's data, two zeros must exist in the unphysical region and not in the physical region. If we use Orear's data, there is a little possibility of existence of zeros on the real axis. By imposing the further condition which can be derived by requiring consistency of dispersion relation at resonance energy, it is shown that the data of Orear are more promising than that of Hamilton-Woolcock. With Orear's data on the S-wave scattering lengths and with existence of appropriate zeros, the necessary condition can be satisfied by the present experimental data.

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