Mass-Shell Evaluation of the Amplitude, Partial Conservation of Axial-Vector Current, and Equal-Time Commutators in Low-EnergyK±pScattering

Abstract

For the on-mass-shell calculation of the scattering amplitude, we have used the following method. On the basis of the hypothesis of partially conserved axial-vector current (PCAC), we introduce the axial-vector current $J=A+c\ensuremath{\partial}\ensuremath{\varphi}$, where $A$ is the current appearing in the PCAC relation, and $\ensuremath{\varphi}$ is the pion field. Using this current in the Lehmann-Symanzik-Zimmermann formalism, the amplitude is decomposed into two terms ${W}^{00}$ and ${W}^{0}$ due to the equal-time commutators (ETC) of the divergence of the current $J$, respectively; a term ${W}^{1}$ due to an ETC leading to a vector current; and a term $W$ giving contributions of the discrete intermediate states. The ETC's of the current octet $J$ are assumed to be similar in form, though not equivalent, to those normally used for the current $A$. For kaon-proton (${K}^{\ifmmode\pm\else\textpm\fi{}}p$) scattering, we have used this formula in conjunction with dispersion relations and the $\mathrm{SU}(3)\ensuremath{\bigotimes}\mathrm{SU}(3)$ scheme, and derived sum rules for ${W}^{0}$ and ${W}^{1}$. Using the scattering length, the sum ${W}^{00}+{W}^{0}$ is evaluated. It is found that in order to obtain the correct signs and magnitudes of the ${K}^{\ifmmode\pm\else\textpm\fi{}}p$ scattering lengths, with a single coefficient $c$ in the PCAC relation, we must take into account the sum ${W}^{00}+{W}^{0}$, which seemed to be negligible in $\ensuremath{\pi}N$ scattering. The difference between the on- and off-mass-shell amplitudes is derived, and seen to depend on the type of particles involved. Assuming that only the sum ${W}^{00}+{W}^{0}$ is a smooth function of the squares of the kaon four-momenta, this difference is found to be negligible in the ${K}^{\ensuremath{-}}p$ case, while it is 40% of the ${K}^{+}p$ scattering amplitude at threshold.