Adler-Weisberger Theorem Reexamined

We consider the nature of the current-algebra constraints on the on-mass-shell amplitudes of pion-nucleon scattering at $s=m_N^{}{}_{}{}^2$, $t=2\mathrm{}m_{\pi }^{}{}_{}{}^2$. We prove that, at this point, an even amplitude is determined up to corrections of order $m_{\pi }^{}{}_{}{}^4$ and that the corrections to an odd amplitude are of order $m_{\pi }^{}{}_{}{}^2$. Thus, the $\sigma$ term in the even amplitude, which is of order $m_{\pi }^{}{}_{}{}^2$, can be determined, confirming the recent work of Cheng and Dashen. However, such a determination cannot be made with a mixture of even and odd amplitudes as von Hippel and Kim have attempted to do. We estimate the actual magnitude of the corrections from what should be the dominant physical mechanisms: the large nucleon size and the proximity of the strong 3-3 resonance state. These corrections are insignificant for the even amplitude. For the odd amplitude, they are separately large, about 10%, but they cancel for the most part and give a much smaller contribution. Corrections due to other resonances are insignificant compared to the 3-3 contribution. The Adler-Weisberger theorem, with our on-shell method, relates the pion-nucleon coupling constant $f$ to the pion decay constant and an integral over the absorptive part of the physical $\pi -N$ scattering amplitude; it does not involve the axial-vector coupling constant $g_A$. We find that this relation gives $f^2=0.077\mathrm{}$, which is to be compared to direct evaluations which range from $f^2=0.082\mathrm{}$ to $f^2=0.076\mathrm{}$. We conclude that these current-algebra theorems may well be satisfied to a considerable degree of accuracy. More precise low-energy pion-nucleon scattering data are needed, however, for a definitive test.