Nucleon mass: From lattice QCD to the chiral limit

Previous extrapolations of lattice QCD results for the nucleon mass to the physically relevant region of small quark masses, using chiral effective field theory, are extended and expanded in several directions. A detailed error analysis is performed. An approach with explicit $\Delta (1232)$ degrees of freedom is compared to a calculation with only pion and nucleon degrees of freedom. The role of the $\Delta (1232)$ for the low-energy constants of the latter theory is elucidated. The consistency with the chiral perturbation theory analysis of pion-nucleon scattering data is examined. It is demonstrated that this consistency can indeed be achieved if the $\Delta (1232)$ dominance of the $P$-wave pion-nucleon low-energy constant $c_3$ is accounted for. Introduction of the $\Delta (1232)$ as an explicit propagating degree of freedom is not crucial in order to describe the quark-mass dependence of the nucleon mass, in contrast to the situation with spin observables of the nucleon. The dependence on finite lattice volume is shown to yield valuable additional constraints. What emerges is a consistent and stable extrapolation scheme for pion masses below 0.6 GeV.