Connection between Regge Poles and Elementary-Particle Poles

Abstract

It is shown that if in a field theory the proper vertex function $\ensuremath{\Gamma}(s)$ vanishes under a condition that the proper vertex-function poles are not the poles of scattering amplitudes, then the elementary particle lies smoothly on the Regge trajectory. [The condition $\ensuremath{\Gamma}(s)\ensuremath{\equiv}0$ does not always mean the vanishing of the coupling constant, when the vertex function poles exist.] The bootstrap equations are immediate consequences of the above condition. We formulate our problem using multichannel theory. Other related results are: (a) It is found that in the multichannel case as well as in the single-channel case, the proper vertex poles are not the poles of scattering amplitudes if the poles come up from the second Riemann sheet. (b) The finite self-mass condition of the composite particle, due to Gerstein and Deshpande, is not applicable when the proper vertex pole appears below the elementary-particle mass, i.e., the modified propagator needs one substraction. (c) Vanishing renormalization constants, ${Z}_{1}={Z}_{3}=0$, are not sufficient to Reggeize the elementary particle. (d) All our conclusions remain true even when additional bound-state poles are included. In the latter case, however, we can also use a different condition, due to Kaus and Zachariasen, that the form factor (improper vertex function) and ${Z}_{3}$ both vanish; then our main results again follow, that the elementary particle lies on the Regge trajectory and that the bootstrap equations are obtained. (The vanishing form factor does not always mean the vanishing of the coupling constant, when the bound-state poles exist.) But this last condition implies that the unsubtracted dispersion relations in weak interactions are not valid.