Bootstrap of Meson Trajectories from Superconvergence

Abstract

In this paper we study the reactions $\pi \pi \to \pi \omega \left(1^{-}\right)$, $\pi \pi \to \pi A_2\left(2^{+}\right)$, and $\pi \pi \to \pi {\omega }_3\left(3^{-}\right)$ as a bootstrap system for natural-parity trajectories. We start from the solution of our previous work that gave, among other results, expressions for the trajectory and residue functions as well as mass formulas, in agreement with experiment. Here we study in detail the sum rules as a function of momentum transfer $t$. We find a set of residue functions $\beta \mathrm{}\left(t\right)\mathrm{}$ that are self-consistent and such that the Regge and resonance sides of the equations are almost equal in a large region of $t$. We study also a step-by-step approximation that, at each stage, enlarges the region where the equations are valid. We find, however, that the leading Regge trajectories, even if infinitely rising, cannot bootstrap themselves. We outline two possible (not incompatible) ways of implementing the bootstrap. The first way demands an optimized choice of the cutoff parameter and considers the whole family of reactions $\pi \pi \to \pi X_J$ ($X_J$ being a normal-parity state of spin $J$). Our results for $J\le 3$ show that this is a definite possibility. The second way is to consider a whole family (parent and daughters) as participating in the bootstrap. We find this possibility also attractive, and as a consequence we find that daughters must be parallel to the parent, for linear trajectories. The properties of our parametrization are also discussed—in particular, the Khuri paradox and the coupling of high-spin resonances to the system. We also compare our results with experiment whenever possible. Our $A_2$ trajectory, for instance, follows the Gell-Mann mechanism, and the exponential $t$ dependence of our residue functions is perfectly consistent with the one found in recent phenomenological fits to inelastic reactions.