Absorption Picture as a Consequence of Multi-Regge Iteration of Low-Energy Peripheral Resonances

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Using the empirical fact that the dominant low-energy resonances are peripheral (i.e., they satisfy the relation ${j}_{s}+\frac{1}{2}=kR\ensuremath{\approx}\frac{1}{2}R\sqrt{s}$), the imaginary part, ${A}_{0}(s,t)$, of the low-energy amplitude is written in the factorized form $F(s){J}_{0}(R\sqrt{\ensuremath{-}t})$. The resonance contribution to $F(s)$ near the resonance position is $\ensuremath{\approx}(2{j}_{s}+1)$, which is $\ensuremath{\approx}{s}^{\frac{1}{2}}$; and thus the well-known value for the vector-tensor trajectory intercept of $\ensuremath{\approx}\frac{1}{2}$ follows simply from peripherality. From finite-energy sum rules the residue of the trajectory is $\ensuremath{\approx}{J}_{0}(R\sqrt{\ensuremath{-}t})$. The quantity ${A}_{0}(s,t)$, which in the $t$ channel corresponds to a fixed pole at ${j}_{t}=\frac{1}{2}$, through the multi-Regge iteration gives rise to a moving pole; and the imaginary part of the total $\ensuremath{\pi}\ensuremath{\pi}$ amplitude $A(s,t)$ with ${I}_{t}=1$ satisfies the dual absorption result, ${s}^{\ensuremath{\alpha}}{J}_{0}(R\sqrt{\ensuremath{-}t})$. A crude estimate gives $\ensuremath{\alpha}(0)\ensuremath{\approx}\frac{1}{2}+(\frac{{\ensuremath{\Gamma}}_{\ensuremath{\rho}}}{{m}_{\ensuremath{\rho}}}) (=0.6)$ and ${\ensuremath{\alpha}}^{\ensuremath{'}}(0)\ensuremath{\approx}\frac{1}{4}(\frac{{\ensuremath{\Gamma}}_{\ensuremath{\rho}}}{{m}_{\ensuremath{\rho}}}){R}^{2}(=1.0 {\mathrm{BeV}}^{\ensuremath{-}2} \mathrm{for} R=1\mathrm{F})$. Thus, for non-diffractive processes, the multi-Regge (or multiperipheral) model is consistent with the absorption picture and as a consequence an approximate bootstrap solution is obtained. The physical interpretation in terms of $j$-plane cuts is discussed.