SU(6)W-Symmetric Meson Bootstrap Model

A bootstrap model of the pseudoscalar-meson octet and singlet and the vector-meson octet and singlet is constructed. The mesons ($M$) are assumed to be $\mathrm{MM}$ bound states, produced by one-$M$-exchange forces. The input $\mathrm{MMM}$ interaction is assumed $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$-symmetric, and the forces in all $P$-wave $\mathrm{MM}$ states are examined. The forces are most attractive in 36 states of the quantum numbers of the mesons; identification of these states with the $M$ leads to output $\mathrm{MMM}$ interactions consistent with those assumed originally. However, the $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$ symmetry requires that the total coupling of the singlet vector meson should be only $\frac{2}{3}$ that of the other mesons. The method of calculation involves continuing the interaction off the mass shell in an $\mathrm{SU}{\mathrm{}\left(6\right)\mathrm{}}_W$-symmetric fashion, and comparing the $P$-wave amplitudes at threshold in Born approximation. The technique can also be used to predict heavier resonances that do not correspond to $\mathrm{SU}\left(6\right)\mathrm{}$ representations. This fact is encouraging, since it appears that only the lighter observed hadrons form complete representations of $\mathrm{SU}\left(6\right)\mathrm{}$.