Superconvergent bootstrap, exchange degeneracy and fixed singularities in the complex J-plane

Abstract

We consider the dynamical scheme based on the generalized superconvergence relations (GSCR) or the finite energy sum rules (FESR). We call this scheme the superconvergent bootstrap, where crossing is imposed by the GSCR (or FESR). The fundamental approximation is that the scattering amplitude can be approximated by a finite number of Regge poles. The bootstrap scheme investigated here is based on the narrow-resonance approximation which allows one to make the simple and systematic parametrization of the Regge parameters based on analiticity and unitarity. The bootstrap program is then to find a set of mutually consistent Regge trajectories and determine their parameters self-consistently. We carry out the superconvergent bootstrap for the reaction P + P → P + V, where P and V stand for the pseudoscalar and vector-meson octet respectively. We assume exact SU(3) symmetry. Solving the bootstrap equations algebraically, we show that the coupled vector- and tensor-octet trajectories can bootstrap themselves quite satisfactorily. We also obtained the interesting mass formula 3 m V 2 = m T 2 + 3 m P 2, where m M denotes the mass of a M-meson octet and T stands for the tensor-meson octet. The formula is well satisfied experimentally. As was shown by Mandelstam and Wang, the singular behavior of the Regge residues when the trajectories pass through wrong-signature nonsense points is critically dependent on whether or not we have fixed singularities in the Froissart-Gribov partial-wave amplitude at nonsential values of J with the wrong signature. or equivalently, whether we neglect or take into account the contribution of the third double-spectral function of the scattering amplitude. From the bootstrap equations we find the close connection between exchange degeneracy and the above mentioned fixed singularities: in the limit of exchange degeneracy any contribution from the wrong-signature singularities of the Regge residues at nonsense points of α vanishes, while conversely, exchange degeneracy comes out when we require the vanishing of the above contribution. This statement is proved quite generally based on the crossing properties of the scattering amplitudes. We also applied the dynamical scheme to the reaction P + P → P + ω, where ω stands for the vector-meson singlet. This reaction is dominated by only the vector-octet trajectory in each channel. Solving the bootstrap equation the single vector-octet trajectory is also shown to bootstrap itself quite satisfactorily.