Theory of the $J=\frac{3}{2}$, $I=\frac{3}{2}$ $\pi N$ Resonance

We calculate the position $W_R$ and width ${\gamma }_{33}$ of the $J=\frac{3}{2}$, $I=\frac{3}{2}$ $P$-wave $\pi N$ resonance, using partial-wave dispersion relations. In the present calculation we treat as given the nucleon and $\rho$-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 $\pi N$ state given by fixed energy dispersion relations, in a region where they are valid without subractions, in a way used by Balázs for the $\pi \pi$ 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 $W_R\approx m+2.35\mathrm{}$ and ${\gamma }_{33}\approx 0.14$. The experimental values are $W_R=m+2.17\mathrm{}$ and ${\gamma }_{33}\approx 0.12$, (where $m$ is the nucleon mass and we use units in which $\hslash =c=m_{\pi }=1\mathrm{}$). 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.