By G. Höhler (auth.), H. Schopper (eds.)
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Extra info for Pion Nucleon Scattering. Part 2: Methods and Results of Phenomenological Analyses
2N roots of the polynomials and IF(l Since the polynomials are real functions, the roots occur in pairs zi. e. only 2N of the complex roots are independent and we have again 4N+ 1 red poromaters. The only direrence is that the amplitudes have an additional real parameter, argF(l), which cannot be determined from the Z-data. 2)shows that the zeros of the amplitude can bc derived from the zeros of the data but, unfortunately,thcre is a 22Nfold ambiguity because,for each pair of zeros zi, z: of C, one has the choice whether zi or z: belongs to the amplitude F.
ZIDELL et al. assumedtheoretical constraints for the threshold behaviour and by using the prediction for the forward amplitude as part of the input. It is necessaryto employ in addition the theoretical information on partial wave dispersion relations (Sect. 1) in order to check, whether the energy dependenceof the nonresonant partial waves can be described correctly by simple effective range type formulas. 1shows that a more complicated energy dependencecan be expectedbecause,in the s-plane,the point T, = 350MeV is twice as far from threshold as two important left hand cut singularities : the short nucleon cut and the front of the circle.
2. fJ,,= T,,,/q is another notation for fd*. The 2N zeros wi follow from the z’ , i = 1.. N. Again, we have 2N + 1 complex parameters: wi and F(l), which can be expressedby 2N+ 1 partial wave amplitudes ft*. 4) For the evaluation of the integral it is useful to have the formula aw”lsin0ld0=” 4/(1- n’), if n is odd, if n is even. 5) If one writes the expression for Z(w), one should notice that zi-zT belongs to wi+l/wT, because the first operation is carried out within the same sheet of the z-plane (see Fig.
Pion Nucleon Scattering. Part 2: Methods and Results of Phenomenological Analyses by G. Höhler (auth.), H. Schopper (eds.)