Present status of the pion-nucleonσterm

The long-standing discrepancy between the predicted value of the $\pi N$ $\sigma$ term in the framework of quantum chromodynamics (QCD) and the value extracted from extrapolated on-mass-shell $\pi N$ scattering data is critically reexamined. Assuming the validity of the Okubo-Zweig-Iizuka (OZI) rule at $t=0\mathrm{}$ and using quark mass ratios extracted from the pseudoscalar-meson mass spectrum one obtains the canonical result ${\sigma }_{\pi N}\simeq 23\pm 5$ MeV. It is argued that the possibility of readjusting the quark mass ratios to give a $\sigma$ term of 60-70 MeV so as to agree with some values extracted from $\pi N$ data is most likely ruled out. In particular this would imply a huge violation of the nonrenormalization theorem in $K_{13}$ decay. Since the OZI rule is unreliable at $t=0\mathrm{}$, we have recalculated the matrix element $\frac{〈p\left|\overline{s}s\right|p〉}{〈p\left|\right(\frac{1}{2}\left)\right(\overline{u}u+\overline{d}d\left)\right|p〉}$ using the Goldstone-boson-pair mechanism and have found a $\sigma$ term of 36±8 MeV. Other evidence supporting a breakdown of the OZI rule at $t=0\mathrm{}$ is also examined. Another theoretical uncertainty is the effect of higher-order terms in SU(3) breaking. Two recent calculations suggest that such terms could increase the $\sigma$ term by another 10 MeV or more. Finally, the whole procedure of extracting a value of the $\sigma$ term from $\pi N$ scattering data is reconsidered. The usual evaluations do not include the uncertainties or their correlations in the $\pi N$ amplitude or in the fixed-$t$ dispersion relations. We perform several fits to the $\pi N$ amplitude at $\nu =0\mathrm{}$ that suggest that values of the $\sigma$ term in the range 30-70 MeV are not ruled out by the existing data. Also, nonlinearities in the $\pi N$ amplitude lead to another ±10 MeV uncertainty in the comparison between theory and experiment. We therefore conclude that both the theoretical and experimental uncertainties in the determination of the $\sigma$ term are sufficiently large for the apparent discrepancy to provide neither evidence against QCD nor against the canonical quark mass ratios that one obtains in the conventional (3,$\overline{3}$) model.