The nucleon sigma term from lattice QCD.

We present an analysis of the pion--nucleon $\sigma$-term, $\sigma_{\pi N}$, using six ensembles with 2+1+1-flavor highly improved staggered quark action generated by the MILC collaboration. The most serious systematic effect in lattice calculations of nucleon correlation functions is the contribution of excited states. We estimate these using chiral perturbation theory ($\chi$PT), and show that the leading contribution to the isoscalar scalar charge comes from $N \pi $ and $N\pi\pi$ states. Therefore, we carry out two analyses of lattice data to remove excited-state contamination, the standard one and a new one including $N\pi$ and $N\pi\pi$ states. We find that the standard analysis gives $\sigma_{\pi N} = 40.4(4.7)$ MeV, consistent with previous lattice calculations, while the $\chi$PT-motivated analysis gives $\sigma_{\pi N} = 61.6(6.4)$ MeV, which is consistent with phenomenological values obtained using $\pi N$ scattering data. Our data on one physical pion mass ensemble was crucial for exposing this difference, therefore, calculations on additional physical mass ensembles are needed to confirm our result and resolve the tension between lattice QCD and phenomenology.