Direct determinations of the nucleon and pion $\sigma$ 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 non-perturbatively improved clover fermions with a range of pion masses down to $m_\pi\sim 150$ MeV and several volumes, $Lm_\pi=3.4$ up to $6.7$, and lattice spacings, $a=0.06-0.08$ fm, enabling a study of finite volume and discretisation effects for $m_\pi\gtrsim 260$ MeV. Systematics are found to be reasonably under control. For the nucleon we obtain $\sigma_{\pi N}=35(6)$ MeV and $\sigma_s=35(12)$ MeV, or equivalently in terms of the quark fractions, $f_{T_u}=0.021(4)$, $f_{T_d}=0.016(4)$ and $f_{T_s}=0.037(13)$, 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 $f_{T_c}=0.075(4)$, $f_{T_b}=0.072(2)$ and $f_{T_t}=0.070(1)$. In addition, through the use of the (inverse) Feynman-Hellmann theorem our results for $\sigma_{\pi N}$ 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 travelling across the temporal boundary. This enables us to demonstrate the Gell-Mann-Oakes-Renner expectation $\sigma_\pi=m_\pi/2$ over our range of pion masses.