Generalized Multiperipheral Models, Pion Exchange, and the Van der Waals Gas

It is shown that the ABFST (Amati, Bertocchi, Fubini, Stanghellini, and Tonin) multiperipheral model with a resonance kernel possesses an effective cutoff in ${\sigma }_n\mathrm{}\left(s\right)\mathrm{}$ at a maximum value of $n$ related to lns by $n_{max}-1=\left(\frac{1}{b}\right)\ln \left(\frac{s}{{m_0}^2}\right)$, where $\ln 2<b<\mathrm{}\ln 4$, and $n$ is the number of pion pairs (resonances). It is shown that this is analogous to a Van der Walls-like (VdW) gas (in the Feynman gas analogy) with the parameter $b$ occurring in the equation of state $p(v-b)=1\mathrm{}$, the pressure $p$ being taken equal to the Pomeron intercept $\alpha$, and $v^{-1\mathrm{}}=\frac{〈n〉}{\ln s}$. The choice of $p=\alpha$ is shown to correspond to massless-pion exchange. The results are strong-coupling results and cannot be obtained in a weak-coupling (Poisson) limit in which $pv=1\mathrm{}$. If the VdW parameter $a$ is introduced via the prescription $(p+\frac{4a}{v^2}) \mathrm{}(v-b)=1\mathrm{}$, we can establish a relationship with a generalized multiperipheral ABFST model, in which pairwise pion final-state interactions are taken into account in an average, though nontrival, way. We also examine an iterated resonance-plus-diffraction multiperipheral model (the "schizophrenic Pomeron" model), and find similar results for cutoffs, though an explicit analogy with the $a=0\mathrm{}$ VdW model is not possible. The introduction of $a\ne 0$ is shown to mix the resonance and diffractive couplings in this model. In all cases, leading-Regge-pole behavior is obtained if it is assumed that the critical point is not reached.