Conspiracy and Evasion: Property of Regge Poles

Abstract

It is shown that if standard methods are used to apply the Regge-pole theory to relativistic problems in which the external particles have nonzero spin, then there exist constraint equations which enforce relationships between the residue and trajectory functions of the participating poles in the region $t\sim 0$. The constraint equations follow directly from general quantum-mechanical principles and it is therefore essential to satisfy them. Moreover, the number of constraint equations increases roughly as the fourth power of the spin of the external particles. The structure of the constraint equations also differs radically according to whether the $t$-channel process has equal-mass particles in both the initial and the final state or unequal-mass particles in both the initial and the final state. A separate treatment of the various situations is given. Several examples are worked out in detail: $\pi N\to \pi N$, $\pi N^{*}\to \pi N^{*}$, $\pi \rho \to \pi \rho$, $\mathrm{NN}\to \mathrm{NN}$, and $\rho \rho \to \rho \rho$. The discussion of the general case in which the external particles have arbitrary spins requires a slight extension of previously given methods for the Reggeization of processes with spin. A very simple, and completely general scheme for the Reggeization, and for the classification of the Regge poles involved, is given. Finally, a discussion is given of the fundamental underlying group-theoretical origin of the constraint equations, and it is suggested that the necessity to satisfy them artificially represents a weakness in our present methods of applying Regge-pole theory to relativistic processes.