Library

Notes: Abstract The pion-nucleon forward scattering amplitude has been calculated from new data of the total cross sections, using several assumptions on the energy dependence above 20 GeV. The results are presented as complex diagrams of the forward amplitude, which are of interest for the discussion of the nucleonic resonances and the non-resonant background scattering. In addition the predictions for the elastic forward cross sections are given as well as the contributions of the real parts to this quantity. A comparison with the new Saclay data of the charge exchange forward cross section leads to the estimate αρ≈0.6, if a Regge behaviour is assumed above 20 GeV. There are indications in favour of a new $$T = \tfrac{1}{2}$$ resonance at the total c.m. energyW≈2.8 GeV.

Notes: Abstract The difference between theπ + p andπ − p diffraction peaks is used for an estimate of the imaginary part of the charge exchange scattering amplitude. The imaginary part has a narrow peak in the forward direction and passes over to negative values at a momentum transfert of about −0.15(GeV/c)2. If the charge exchange amplitude is dominated by the contribution of theρ Regge pole, the peak is mainly due to thet-dependence of the residue function and a narrow forward peak is expected in the charge exchange angular distribution.

Notes: Abstract The total cross sections up to 65 GeV/c and the experimental real parts of the isospin evenπN forward amplitude are compatible with the dispersion relation. However the arguments which led to the introduction of theP′ Regge pole (α p′≈0.5) are not valid. The data are compatible with the existence of the integral in Lehmann's sum rule, which implies a finite real partD +(∞) at infinity. — The isospin odd combinationσ − is still consistent with the reggeizedρ-exchange model and with Pomeranchuk's theorem.

Notes: Abstract The effective range parameterb − for πN s-wave scattering is calculated from fixed-t dispersion relations, using experimental information for the imaginary parts. The result is compared with that of other calculations, based on Ward identities and phenomenological Lagrangians. It is pointed out that a direct determination ofb − from experimental data is not possible. Since the Δ-propagator is not unique, the Δ-exchange amplitudes are given for a one-parameter class of propagators (Appendix).

Notes: Abstract The 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 f s (−) =(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.

Notes: Abstract The real part of the invariant isospin evenπ N amplitudeC +≡A′ + at ¦t¦≲0.4 (GeV/c)2 is determined from the condition thatC + fulfills the fixed-t dispersion relation, is compatible with this amplitude as reconstructed from phase shifts up to 1.5 GeV/c and leads to differential cross sections above 3 GeV/c in agreement with the experimental data.-The result for ReC +/ImC + is remarkably large (−0.5 att=−0.3 (GeV/c)2 and 6 GeV/c) and up to 20 GeV/c its uncertainty is comparable with that in the upper part of the phase shift region.-Even at 40 GeV/c ReC + is far from being asymptotic because of large energy independent low energy contributions. The deviations from our simple high energy ansatz (k〉4GeV/c): ImC +(k, t)∼(k + m)exp(9t/2) are almost comparable with the experimental errors.

Notes: Abstract The real parts of the photoamplitudesE 1S 1/2,M 1P 1/2,M 1P 3/2 have been calculated from the angular distribution of the reactionγ+p→p+π0 recently measured byGoldansky et al. at 160 to 240 MeV. One of the solutions fits pretty well to the theoretical prediction for theM 1P 3/2-amplitude according to the dispersion method ofChew, Goldberger, Low andNambu. There is a discrepancy forM 1P 1/2 ifα 11 is taken from the effective range formula, but the positive values ofα 11, necessary to give agreement, are not excluded by the results of the phase shift analysis, especially sincePontecorvo et al. have recently found positive values at higher energies. The prediction for the real part of theE 1 S1/2-amplitude agrees with the experimental data, if pretty large recoil corrections are added which had been neglected byChew et al.

Notes: Abstract The full one-particle approximation to the Ward identities for πN scattering is investigated, taking into account elastic unitarity in thet-channel for πN- and ππ-amplitudes. The results for the one-particle contributions are compared with the corresponding expressions in the dispersion relation approach, wheret-channel exchanges are introduced into fixed-t dispersion relations by subtracting the lowest ππN¯N-partial wave amplitudes. Our method leads to a prescription for an isobaric model, in which double counting does not occur althoughs-,t- andu-channel exchanges are superimposed.

Notes: Abstract Impact parameter transforms of isospin even no-flip πN-amplitudes and of the overlap function are calculated from the new CERN phase shifts. The results are compared with different models.—The approximate equality of the partial wave and the impact transform forbq=(l+ 1/2) is valid for smalll and not for largel as frequently assumed. The “optical potential” has a repulsive core in the real part and rather flat tails beyond 0.5 fermi in the real and imaginary parts.

Notes: Abstract The πN charge-exchange forward scattering amplitude has been recalculated, using the most recent total cross section data, charge independence and the forward dispersion relation. In general the prediction agrees rather well with the data, but there are several discrepancies which are considerably larger than the errors. Some of these discrepancies follow from the fact that a structure at small ¦t¦ has been ignored in the extrapolation. — The new fit gives an accurate value for the subtraction constant of the integral. It agrees with the evaluation of the unsubtracted integral at threshold, if, at high energies, the Regge behaviour for σ(π− p)—σ(π+ p) is inserted. This result favours the validity of Pomeranchuk's theorem. — An improved value forf 2 cannot be obtained at present because of the uncertainty following from charge-dependent effects at low energies.