Theory of theJ=32,I=32πNResonance

Abstract

We calculate the position ${W}_{R}$ and width ${\ensuremath{\gamma}}_{33}$ of the $J=\frac{3}{2}$, $I=\frac{3}{2}$ $P$-wave $\ensuremath{\pi}N$ resonance, using partial-wave dispersion relations. In the present calculation we treat as given the nucleon and $\ensuremath{\rho}$-meson masses and coupling constants, which determine the long-range part of the forces. The parameters, which characterize the distant part of the left-hand cut, are fixed by using the expressions for the ($\frac{3}{2}, \frac{3}{2}$) $P$-wave $\ensuremath{\pi}N$ state given by fixed energy dispersion relations, in a region where they are valid without subractions, in a way used by Bal\'azs for the $\ensuremath{\pi}\ensuremath{\pi}$ problem. We then impose the self-consistency demand that the position and width of the ($\frac{3}{2}, \frac{3}{2}$) resonance used as input values in the crossed channel in the fixed-energy dispersion relation be the same as the calculated values of the position and width. The preliminary results of the calculation are ${W}_{R}\ensuremath{\approx}m+2.35$ and ${\ensuremath{\gamma}}_{33}\ensuremath{\approx}0.14$. The experimental values are ${W}_{R}=m+2.17$ and ${\ensuremath{\gamma}}_{33}\ensuremath{\approx}0.12$, (where $m$ is the nucleon mass and we use units in which $\ensuremath{\hbar}=c={m}_{\ensuremath{\pi}}=1$). These results constitute the first part of the intended selfconsistent calculation of the nucleon mass and ($\frac{3}{2}, \frac{3}{2}$) resonance position, exploiting the "reciprocal bootstrap" mechanism discussed by Chew.