Direct determinations of the nucleon and pion σ terms at nearly physical quark masses

We present a high statistics study of the pion and nucleon light and strange quark sigma terms using N_f=2 dynamical nonperturbatively improved clover fermions with a range of pion masses down to m_π∼150 MeV and several volumes, L m_π=3.4 up to 6.7, and lattice spacings, a =0.06 - 0.08 fm , enabling a study of finite volume and discretization effects for m_π≳260 MeV . Systematics are found to be reasonably under control. For the nucleon we obtain σ_{π N}=35 (6 ) MeV and σ_s=35 (12 ) MeV , or equivalently in terms of the quark fractions, f_Tu=0.021 (4 ) , f_Td=0.016 (4 ) and f_Ts=0.037 (13 ) , where the errors include estimates of both the systematic and statistical uncertainties. These values, together with perturbative matching in the heavy quark limit, lead to f_Tc=0.075(4 ), f_{T b}=0.072 (2 ) and f_{T t}=0.070 (1 ). In addition, through the use of the (inverse) Feynman-Hellmann theorem our results for σ_{π N} are shown to be consistent with the nucleon masses determined in the analysis. For the pion we implement a method which greatly reduces excited state contamination to the scalar matrix elements from states traveling across the temporal boundary. This enables us to demonstrate the Gell-Mann-Oakes-Renner expectation σ_π=m_π/2 over our range of pion masses.