How to extract the P_{33}(1232) resonance contributions from the amplitudes M_{1+}^{3/2},E_{1+}^{3/2},S_{1+}^{3/2} of pion electroproduction on nucleons

Abstract

Within the dispersion relation approach, solutions of integral equations for the multipoles M_{1+}^{3/2},E_{1+}^{3/2},S_{1+}^{3/2} are found at 0 < Q^2 < 3 GeV^2. These solutions should be used as input for the resonance and nonresonance contributions in the analyses of pion electroproduction data in the P_{33}(1232) resonance region. It is shown that the traditional identification of the amplitude M_{1+}^{3/2} (as well of the amplitudes E_{1+}^{3/2},S_{1+}^{3/2}) with the P_{33}(1232) resonance contribution is not right; there is a contribution in these amplitudes which has a nonresonance nature and is produced by rescattering effects in the diagrams corresponding to the nucleon and pion poles. This contribution is reproduced by the dispersion relations. Taking into account nonresonance contributions in the amplitudes M_{1+}^{3/2},E_{1+}^{3/2}, the helicity amplitudes A_p^{1/2},~A_p^{3/2} and the ratio E2/M1 for the gamma N -> P_{33}(1232) transition are extracted from experiment at Q^2=0. They are in good agreement with quark model predictions.