Complete analysis of the eta N S-wave scattering-length values and its natural limitations in any single-resonance model

We show that the S-wave eta N scattering length can be extracted in a model-independent way only if the restricting assumption is adopted that one resonance describes the behaviour of the S-wave entirely. To obtain the eta N S-wave scattering length in a model-independent way one needs only two well-known quantities: the pi N elastic S-wave T-matrix at the eta-production threshold, and the near threshold pi(-)p --> eta n total cross section slope. The results are independent of the particular parametrization of the elastic pi N S-wave T-matrix and of the number of channels used. These assumptions are more general than the assumptions of the existing single-resonance models employed up to now for extracting the eta N S-wave scattering length. We show that the eta N S-wave scattering-length value of other single-resonance models agrees with the model-independent estimate if the input data agree with the commonly accepted values. The existence of the upper limit of the real part and the almost fixed value of the imaginary part of the eta N S-wave scattering length are demonstrated, and the nature of limitations originating directly from first principles is explained as the reduction of the full model to the single resonance in the S-wave. The expected differences from the single-resonance estimate for models that are more general is shown. A simple criterion for recognizing the importance of parts of the model that are added in addition to the single resonance in the S-wave is given as well.