Statistical Bootstrap Model of Hadrons

Abstract

The hadron is considered to be a compound with two or more constituents circulating freely in a box of radius $\ensuremath{\approx}{10}^{\ensuremath{-}13}$ cm. The density of hadron levels, $\ensuremath{\rho}(m)$, is estimated from the number of states in the box (statistical condition) and is also required to be consistent with the spectrum of constituents, which are assumed to be the hadrons themselves (bootstrap condition). This type of model was first considered by Hagedorn, who obtained a solution of form $\ensuremath{\rho}m\ensuremath{\sim}c{m}^{a}{e}^{\mathrm{bm}}$ with $a=\ensuremath{-}\frac{5}{2}$ which satisfied the bootstrap condition asymptotically to within a power of $m$. We obtain a solution with $a<\ensuremath{-}\frac{5}{2}$ which satisfies the bootstrap condition exactly in the high-mass limit. The constituents in the box are distributed with probability $P(n)=\frac{{(\mathrm{ln}2)}^{n\ensuremath{-}1}}{(n\ensuremath{-}1)}$!; i.e., an average high-mass resonance decays (in the first generation of its decay chain) to two hadrons (69% probability) or three (24% probability). We also review briefly the thermodynamic applications of this model to high-energy scattering and astrophysics.