Strong-Interaction Sum Rules for Pion-Hadron Scattering

Using Regge high-energy behavior, the chiral algebra of charges, and pion pole dominance of the divergence of the axial-vector current, all the strong-interaction sum rules which hold for elastic pion-hadron scattering amplitudes at $t=0\mathrm{}$ are derived. These include charge-algebra sum rules as well as superconvergence relations. We distinguish between "pure" $t=0\mathrm{}$ sum rules (class I) and "extrapolated" sum rules (class II) and relate them to the evenness or oddness of the helicity flip in the $t$ channel. Using the explicit crossing relations for the relevant helicity amplitudes, the connection of class-I superconvergence relations to the charge-algebra sum rules is established, and the algebraic structure of class-I sum rules is then discussed in terms of representations of the $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ chiral algebra of charges, for particle states moving with infinite momentum. The properties of the mass operator in $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ are analyzed, and it is shown that even in the presence of $\mathrm{SU}\left(2\right)\mathrm{}\times \mathrm{SU}\left(2\right)\mathrm{}$ symmetry breaking the ${\mathrm{}\left(\mathrm{mass}\right)\mathrm{}}^2$ values of all the (mixed) eigenstates of an irreducible representation of the algebra are predicted to be equal. Since these eigenstates can be determined from the matrix elements of the vector and axial-vector charges, a large number of nontrivial mass relations are obtained. Class-II superconvergence relations, sum rules for $t\ne 0$, and sum rules for the derivative with respect to $t$ of the scattering amplitude at $t=0\mathrm{}$ are briefly discussed. Many applications of the strong-interaction sum rules are presented, including a model consisting of $I=0\mathrm{}$ and $I=1\mathrm{}$ scalar, pseudoscalar, vector, and axial-vector mesons. The predictions of the model, as well as those of the other sum rules, are derived and are found to be in satisfactory agreement with experiment.