Rising Regge Trajectories and Finite Energy Sum Rules

Abstract

It is shown in the narrow-resonance approximation that in order to satisfy the finite energy sum rules (FESR), all Regge trajectories except for the Pomeranchukon must be ac­ companied by an infinite number of daughter trajectories, which are asymptotically parallel to the leading ones. In the presence of an infinite number of daughter trajectories, the narrow-resonance saturation of the FESR cannot give effective constraints on the asymptotic energy-dependence of the trajectory functions, without aid of additional assumptions on the Regge parameters. give dynamical equations for the bootstrap scheme based on rising Regge trajec­ tories. The FESR strongly suggests the existence of interlocking between the direct-channel resonances and the crossed-channel Regge trajectories. In fact, 4 ) have shown that even in the region where well­ established resonances appear in the direct channel, the Regge amplitude gives the local average of the physical amplitude. This property (the famous D-H-S duality) permits us to perform low cutoff calculations of the FESR integrals. Under the standard assumptions, an infinite number of resonances lying on an indefinitely rising tower can be resumed a la Van Hove as a Regge trajec­ tory carrying the same quantum numbers as those of the tower. 6 ) We can there­ fore legitimately assume the saturation of the FESR with resonances lying on indefinitely rising Regge trajectories. Hence, the FESR reduces to consistency conditions on the Regge parameters in the direct and crossed channels. It has been shown in the narrow-resonance approximation that these consistency condi­ tions cannot be satisfied by a single ;resonance-tower m the direct channeL