Extended partially conserved axial-vector current hypothesis and chiral-symmetry breaking

An extended partially conserved axial-vector current (PCAC) hypothesis that incorporates a family of heavy bosons in a model-independent way is proposed. This is motivated by the impossibility of accounting for the corrections to Goldberger-Treiman relations, both in SU(2) × SU(2) and SU(3) × SU(3), by means of ordinary dynamical mechanisms (many-particle intermediate states). This new hypothesis coupled with an assumption on the strong-coupling constants of the heavy bosons leads to the following results: (i) A universality among the corrections to Goldberger-Treiman relations for $\Delta S=0\mathrm{}$ decays, ${\Delta }_{\pi }$, on the one hand and for $\Delta S\ne 0$ decays, ${\Delta }_K$, on the other. (ii) From this universality there follow two sets of sum rules involving masses and strong and weak coupling constants. These sum rules become identities in the chiral as well as in the SU(3) limit and although a definite check has to await for the advent of accurate hyperon data, there are indications that they might be saturated. (iii) By studying the Dashen-Weinstein sum rules, new sets of sum rules involving only strong coupling constants are predicted as well as an expression for $\frac{{\Delta }_{\pi }}{{\Delta }_K}$ in good agreement with present data. (iv) It is found that ${\Delta }_{\pi }$ and ${\Delta }_K$, which are a measure of chiral-symmetry breaking, determine completely the on-mass-shell corrections to soft-meson theorems. Since both ${\Delta }_{\pi }$ and ${\Delta }_K$ are known experimentally, a calculation is made of the on-mass-shell amplitudes for ${\pi }^0\to \gamma \gamma$, $\eta \to \gamma \gamma$, $\eta \to \pi \pi \gamma$, $\gamma \to \pi \pi \pi$, $\gamma \gamma \to \pi \pi \pi$ starting from the zero-mass limits implied by anomalous Ward identities. In particular, it is found that the results for the radiative $\eta$ decays are in agreement with present experimental data without the need for invoking, $\eta -{\eta }^{\prime }$ mixing. Finally, the corrections to the soft-pion and soft-kaon theorems on $K_{l3\mathrm{}}$ decay are also obtained and it turns out that in the last case the ratio $\frac{f_K}{f_{\pi }}$ comes into agreement with present data after the derived chiral-symmetry-breaking correction is performed. In summary the proposed extended PCAC hypothesis links many chiral-symmetry-breaking problems together in a unified fashion.