CALCULATION OF THE PION-NUCLEON SCATTERING PHASES FROM DISPERSION RELATIONS (in German)
The small pion-nucleon scattering phase shifts were calculated by Chew, Goldberger, Low, and Nannbu, 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 by Chew et al. result in pretty large errors, at least for the swaves. First retain the neglect of all contributions to the dispersion integral other than the 33- part and consider the s-wave amplitude Re f/sub s//sup (-)/ = (sin 2 alpha /sub 1/ -- sin 2 alpha /sub 3/)/6q. For 200Mev (lab.) pions, the correct evaluation of the recoil effects leads to a value 2.8 times lower than the l/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 neglected contributions to the dispersion integral amount to 35 plus or minus 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, the value found at zero kinetic energy was added. 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. and Hamilton 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 alpha /sub 33/ and sigma /sub tot/. The result for the s-wave scattering lengths a/sub 1/ - a/sub 3/ = 0.255 is compatible with P = 1.60 for the Panofsky ratio and with the measured photomeson cross section, which near threshold shows no deviations from the perturbation theoretical values for charged pions (f/sup 2/ = 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 pwaves were calculated, taking into account only the 33-part of the dispersion integral but without the recoil approximation of CGLN (f/sup 2/ = 0.080): a/sub 33/ = 0.l89, a/sub 13/ = --0.045, a/sub 1/ i/sub 3/ - a/sub 31/ = 0.0007, a/sub 11/ = -0.147. The formulas for these scattering lengths and the corresponding q/sup 2/-coefficient of the s-wave amplitude fulfill Geffens relation identically if the total cross sections occurring in the integral are replaced by their 33parts. This changes the value of the integral by 5 to 10%. The approximations of Chew et al. were used in the discussion of the influence of the pi - pi interaction on the pi -N scattering phase shifts. The result makes it worthwhile to reconsider this question. (auth)
- Research Organization:
- Technische Hochschule, Karlsruhe
- NSA Number:
- NSA-15-007976
- OSTI ID:
- 4113535
- Journal Information:
- Z. Physik, Journal Name: Z. Physik Journal Issue: 4 Vol. Vol: 160; ISSN 1434-6001
- Country of Publication:
- Country unknown/Code not available
- Language:
- German
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