N-Point Functions in ChiralSU(3)×SU(3)Current Algebra

Hard-meson techniques are presented for calculating processes involving octets (nonets) of mesons under the assumptions of chiral $\mathrm{SU}\left(3\right)\mathrm{}\times \mathrm{SU}\left(3\right)\mathrm{} \mathrm{}\left[U\right(3\mathrm{})\mathrm{}\times U(3\left)\right]$ current-algebra commutation relations, partial conservation of axial-vector currents, conservation and partial conservation of vector currents, and single-meson saturation of intermediate sums. Using these conditions and the usual smoothness hypothesis, the general procedure for constructing the arbitrary $N$-point function is given. If, in addition, one assumes that the "$\sigma$ commutators" (i.e., the commutators of the time components of the currents with the scalar fields) are single-particle dominated, it is inconsistent to assume only octets of particles. A consistent formalism involving nonets of particles can be constructed, however. The spin-zero mesons must then belong to the (3,$3^{*}$) + ($3^{*}$,3) representation. Hence the (3,$3^{*}$) + ($3^{*}$,3) symmetry-breaking condition is deduced from the current-algebra conditions when combined with pole dominance.