Recently, Li and Pagels indicated that there might be nonanalytic corrections as bad as $m_{\pi }^{}{}_{}{}^2\ln m_{\pi }^{}{}_{}{}^2$ in the matrix elements. We study the nonanalytic corrections in a renormalized linear $\sigma$ model in the one-loop approximation. Although we observe these corrections [of the form $\ln m_{\pi }^{}{}_{}{}^2$, $m_{\pi }^{}{}_{}{}^2\ln m_{\pi }^{}{}_{}{}^2$, $(\stackrel{/}{p}-m)\ln m_{\pi }^{}{}_{}{}^2$, $({\stackrel{/}{p}}_1-m)\times ({\stackrel{/}{p}}_2-m)\ln m_{\pi }^{}{}_{}{}^2$ in individual diagrams, they sum up to zero in the physically interesting amplitudes (pion and nucleon propagators, $\pi \pi$ and $\pi N$ scattering amplitudes). We conclude that the nonanalytic correction is of higher order and thus does not invalidate the low-energy results derived from a phenomenological Lagrangian for these amplitudes.

• Received 10 November 1972

DOI:https://doi.org/10.1103/PhysRevD.7.2467