Determination of Asymptotic Parameters in the Statistical Bootstrap Model

Abstract

The statistical bootstrap model predicts that the density of hadron states $\rho \mathrm{}\left(m\right)\mathrm{}$ approaches $c{m}^a{e}^{\mathrm{bm}}$ asymptotically. We consider the consequences of extending the bootstrap condition in the model from asymptotic down to finite masses. This allows us to determine the parameters $a$, $b$, and $c$ for various assignments of the hadron volume and low-mass input spectrum, and for the extreme cases of excluding all exotic particles or including all of them. In all cases, $a=-3\mathrm{}$ for ${\rho }_{\mathrm{tot}}$ (summed over all internal quantum numbers). The parameter $b$ varies somewhat from case to case but is always of order $m_{\pi }^{}{}_{}{}^{-1\mathrm{}}$; thus we predict the maximum temperature $T_0=b^{-1\mathrm{}}\approx m_{\pi }$ in rough agreement with Hagedorn's empirical determination. The inclusion of exotic states has little effect on ${\rho }_{\mathrm{tot}}$ but does redistribute the partial level densities according to a simple rule. The predicted level densities (excluding exotic states) are compared with present data.