Berechnung der Pion-Nukleon-Streuphasen aus Dispersionsrelationen. I
AbstractThe small pion-nucleon scattering phase shifts have been calculated byChew, Goldbeeger, Low andNameu, using relativistic dispersion relations and the data of the first resonance. The authors introduced several approximations without going into the details of their validity. It is the aim of this paper to give a more accurate treatment, because it turned out that the approximations used byChew et al. result in pretty large errors, at least for the s-waves.First we retain the neglect of all contributions to the dispersion integral other than the 33-part and consider the s-wave amplitude Re fs(−)=(sin 2α1−sin 2α3)/6q. For 200 MeV (lab.) pions, the correct evaluation of the recoil effects leads to a value 2.8 times lower than the 1/M-approximation and the projection, carried through without an approximation, deviates by 20% from the first terms of the expansion used by CGLN. At zero kinetic energy a comparison with the dispersion relation for forward scattering shows that the above mentioned neglected contributions to the dispersion integral amount to 35±15%.The combination of the s-phases was recalculated, replacing the first two approximations of CGLN by an exact treatment. In order to take care of the main part of the neglected contributions to the dispersion integral, we added the value found at zero kinetic energy. The energy dependence, not accounted for by this procedure, should result in a one-sided deviation from the experimental data.Comparison with these data, however, shows that the absolute values as well as the energy dependence of the calculated curve agree reasonably with the measurements up to 333 MeV.
Cini et al. andHamilton et al. have used an interpolation formula which represents the measured s-wave data by adjusting parameters, whereas in this paper the combination of s-phases is calculated from α33 and σtot. Our result for the s-wave scattering lengths a1−a3=0.255 is compatible with P=1.60 for thePanofsky ratio and with the measured photomeson cross section which near threshold shows no deviations from the perturbation theoretical values for charged pions (f2=0.080). It is doubted that in this energy region the small additional contributions, which follow from the dispersion theory of photoproduction in its present state, are really an improvement of the perturbation theoretical results.The scattering lengths of the p-waves have been calculated, taking into account only the 33-part of the dispersion integral, but without the recoil approximation of CGLN (f2=0.080): a33=0.189, a13=−0.045, a13−a31=0.0007, a11=−0.147. The formulae for these scattering lengths and the corresponding q2-coefficient of the s-wave amplitude fulfilGeffens relation identically, if the total cross sections occuring in the integral are replaced by their 33-parts. This changes the value of the integral by 5 to 10%.The approximations ofChew et al. have been used in the discussion of the influence of the ππ-interaction on the πN-scattering phase shifts. Our result makes it worthwhile to reconsider this question.