The Long Way to the Statistical Bootstrap Model

Abstract

The statistical bootstrap model (SBM) is a statistical model of strong interactions based on the observation that hadrons not only form bound and resonance states but also decay statistically into such states if they are heavy enough. This leads to the concept of a possibly unlimited sequence of heavier and heavier bound and resonance states, each being a possible constituent of a still heavier resonance, while at the same time being itself composed of lighter ones. We call these states clusters (in the older literature heavier clusters are called fireballs; the pion is the lightest “one-particle-cluster”) and label them by their masses. Let ρ(m)dm be the number of such states in the mass interval {m, dm}; we call ρ(m) the “SBM mass spectrum”. Bound and resonance states are due to strong interactions; if introduced as new, independent particles in a statistical model, they also simulate the strong interactions to which they owe their existence. To simulate all attractive strong interactions we need all of them (including the not yet discovered ones), that is: we need the complete mass spectrum ρ(m). To simulate repulsive forces we may use proper cluster volumes à la Van der Waals. In order to obtain the full mass spectrum, we require that the above picture, namely that a cluster is composed of clusters, be self-consistent. This leads to the “bootstrap condition and/or bootstrap equation” for the mass spectrum ρ(m). The bootstrap equation (BE) is an integral equation embracing all hadrons of all masses. It can be solved analytically with the result that the mass spectrum ρ(m) has to grow exponentially. Consequently any thermodynamics employing this mass spectrum has a singular temperature T ₀ generated by the asymptotic mass spectrum: ρ(m) ∼ exp(m/T ₀). Today this singular temperature is interpreted as the temperature where (for baryon chemical potential zero) the phase transition (hadron gas) ⇔ (quark-gluon plasma) occurs.