Off-Mass-Shell Approach to Nucleon Resonances of Spin32and Isospin ½

Abstract

The off-mass-shell self-consistency approach studied previously in the case of the strong decays among the nucleon resonances of both spin and isospin \textonehalf{} is extended to include the nucleon resonances of spin $\frac{3}{2}$ and isospin \textonehalf{}. The off-mass-shell decay amplitudes are defined with respect to the spin-$\frac{3}{2}$ resonance in terms of the field which is a sum of products of the isovector currents and the isospinor nucleon field. The above field satisfies the same equal-time commutation relation with the time component of the axial-vector current as does the nucleon field. Four sets of self-consistency conditions are obtained by saturating the unsubtracted and also the once-subtracted dispersion relations satisfied by various off-mass-shell decay amplitudes, in which the subtraction refers to the point where the external pion four-momentum vanishes. It is shown that, in order for a solution to exist that is compatible with all the self-consistency conditions, including the ones obtained previously, at least three spin-$\frac{3}{2}$ resonances have to be assumed, together with at least four spin-\textonehalf{} resonances including the nucleon itself. However, the solution of the self-consistency conditions in the case of these seven resonances gives some decay amplitudes which are too large in magnitude to be consistent with experiments. It is shown that the addition of the fifth nucleon, which is actually seen in the most recent phase-shift analysis by Lovelace et al., removes this difficulty. Thus, the self-consistency conditions obtained in the present and previous papers need all the nucleon resonances of spin \textonehalf{} and $\frac{3}{2}$ and isospin \textonehalf{} that are seen by Lovelace et al.