We present lattice results for the flavor diagonal charges of the proton from the analysis of eight ensembles generated using $2+1+1$-flavors of highly improved staggered quarks by the MILC Collaboration. The calculation includes all the needed connected and disconnected contributions to nucleon three-point function. For extracting matrix elements using fits to the spectral decomposition of these correlation functions, two strategies to remove excited state contributions are employed and compared. To renormalize these charges, the $2+1$-flavor mixing matrix is calculated in the regularization independent symmetric momentum subtraction intermediate scheme on the lattice. The final results are presented in the $\overline{\mathrm{MS}}$ scheme at scale 2 GeV. The axial charges for the proton are $g_A^u=0.781(25)$, $g_A^d=-0.440(39)$, and $g_A^s=-0.055(9)$; the tensor charges are $g_T^u=0.782(28)$, $g_T^d=-0.195(16)$, and $g_T^s=-0.0016(12)$; and the scalar charges are $g_S^u=9.39(88)$, $g_S^d=8.84(93)$, and $g_S^s=0.37(14)$. Results for the neutron are given by the $u\leftrightarrow d$ interchange. Results for the sigma terms are ${\sigma }_{\pi N}{|}_{\text{standard}}=42(6)\mathrm{MeV}$ from a “standard” analysis and ${\sigma }_{\pi N}{|}_{N\pi }=61(6)\mathrm{MeV}$ from an “$N\pi$” analysis that includes the contributions of multihadron $N\pi$ excited states as motivated by chiral perturbation theory. Our preferred value ${\sigma }_{\pi N}{|}_{N\pi }$ is consistent with the phenomenological extraction from $\pi -N$ scattering data. The strangeness content of the proton, for which the standard analysis is appropriate, is ${\sigma }_s{|}_{\text{standard}}=35(13)\mathrm{MeV}$.

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• Lattice field theory
• Quantum chromodynamics