Fixed-Angle Dispersion Approach to the Induction of Strong-Interaction Symmetries

Abstract

Dispersion relations for forward and backward $\mathrm{MM}$ and $\mathrm{MB}$ scattering amplitudes are considered, where the $M$ and $B$ are hypothetical, separately degenerate sets of mesons and baryons. Self-consistency conditions for the quantum numbers and three-particle interaction constants for the particles are obtained from the hypothesis that the dispersion integrals are saturated with poles associated with the $M$ and $B$ states in the $s$, $t$, and $u$ channels. The spin components are treated on the same footing as the internal quantum numbers. If a subset of the mesons is connected with interactions antisymmetric in the exchange of two mesons, this subset must correspond to the regular representation of a compact, simple Lie group, as in Cutkosky's model of vector mesons. On the other hand, mesons and baryons of both parities must be present in this model. If all the helicity states of the odd-parity mesons have nonzero interactions, the mesons must be vector and pseudoscalar mesons, and the Lie group must contain $\mathrm{SU}\left(2\right)\mathrm{}$ as a noninvariant subgroup applied to the spins according to the $W$-spin prescription. An $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$-symmetric solution to the model is discussed briefly. This solution is similar to that obtained previously from partial-wave dispersion relations, and is in agreement with the experimental hadron spectrum.