Chiral Symmetry Breaking, Partial Conservation of Axial-Vector Currents, and theσCommutator in the Kaon-Nucleon System

Chiral-symmetry-breaking effects are studied in the kaon-nucleon system, and discussed and compared with various symmetry-breaking schemes. We first derive a relation between low-energy parameters of the kaon-nucleon scattering amplitude and $s$- and $p$-wave scattering lengths, where contributions from $\overline{K}N$ unphysical regions and nearby singularities are calculated from field theory. This sum rule is nearly saturated. Together with consistency conditions of the isospin-even amplitude, the nucleon expectation value of the $\sigma$ commutator is expressed in terms of $s$- and $p$-wave scattering lengths and a rather well-known integral over $K^{\pm }N$ cross sections. Although definitely not compatible with the recent estimate of Cheng and Dashen, our result for the magnitude of the $\sigma$ term is in agreement with most of the calculations for the $\pi N$ system and favors the (conventional) (3,¯3) + (¯3,3) breaking scheme of SU(3) × SU(3) of Gell-Mann, Oakes, and Renner. The same conclusion has been reached by using a dispersive approach. Since these $\sigma$ terms turn out to be slightly larger (by about a factor of 2) than conventional (3,¯3) estimates, mechanisms and models are discussed in order to explain this (possible) enhancement. Nonsmooth higher-order effects due to the ${\Lambda }^{\prime } \mathrm{}\left(1520\right)\mathrm{}$ are found to be small, suggesting that even in the kaon-nucleon system second-order effects in chiral symmetry breaking, i.e., $O\left(m_K^{}{}_{}{}^4\right)$ can be neglected. Finally, PCAC (partial conservation of axial-vector current) for kaons has been directly compared with experiment using most recent phase-shift analyses, and its compatibility with the data has been confirmed, as suggested by generalized Goldberger-Treiman relations, contrary to previous calculations.