Odorico's bootstrap of pseudoscalar mesons

Odorico has derived bootstrap conditions for $P$ (pseudoscalar) meson interactions from the hypothesis that the zeros in $PP$ scattering amplitudes are linear in the Mandelstam plane. We explain the relation between this hypothesis and an earlier bootstrap hypothesis based on duality. If the solution proposed by Odorico for the $PP$ scattering bootstrap is correct, other bootsrap conditions involving virtual $P$ mesons suggest that a tenth member should be associated with the $P$ nonet. If the 1422-MeV $E$ meson is pseudoscalar, it is a possible candidate for the tenth member. It is shown that in some, but not all, hadron-hadron scattering amplitudes, a simple condition based on duality and isotopic-spin invariance is almost sufficient to predict linear zeros.