Algebraic Realization of Weinberg's Superconvergence Conditions

We investigate the algebraic realization of a pair of sum rules proposed by Weinberg for forward scattering of massless pions, for the case when only $p$-wave pions are taken into account. It is shown that the algebraic structure of the first superconvergence condition is given by the Lie algebra of the group $\mathrm{SU}\left(2\right)\mathrm{}\otimes \mathrm{SU}\left(4\right)\mathrm{}$. A few $p$-wave pion decay widths are calculated and found to agree satisfactorily with experiment. It is further shown that the mass spectra obtained from the second superconvergence conditon are unsatisfactory, as they predict that the hadron masses decrease with increasing isospin (or spin). Various possible reasons for this defect are discussed.