Off-Mass-Shell Corrections to Current-Algebra Calculation ofπNS-Wave Scattering Lengths

Abstract

Current-algebra techniques are used to calculate $a^{\mathrm{}(+)\mathrm{}}$, the even-crossing $\pi N$ $S$-wave scattering length. The off-mass-shell extrapolation includes sizable terms ($\sim 0.1{m_{\pi }}^{-1\mathrm{}}$) of order $q^2$ and of all orders in $\nu$. These contributions are found by using a pole model for axial-vector-current-nucleon scattering and on-mass-shell dispersion relations. The experimental result, ${a_{\mathrm{exper}}}^{\mathrm{}(+)\mathrm{}}=(-0.001\mathrm{}\pm 0.004){m_{\pi }}^{-1\mathrm{}}$, is matched if the $\sigma$ term is vanishing or very small: $\sigma =(0.06\mathrm{}\pm 0.14)m_{\pi }$. Alternatively, if we take $\sigma =0\mathrm{}$, then the calculation predicts $a^{\mathrm{}(+)\mathrm{}}=(-0.011\mathrm{}\pm 0.022){m_{\pi }}^{-1\mathrm{}}$, which agrees with experiment. In the case of $a^{\mathrm{}(-)\mathrm{}}$, the off-shell extrapolation is identical to that used in the derivation of the Adler-Weisberger sum rule. The predicted value of $a^{\mathrm{}(-)\mathrm{}}$ also agrees with experiment.