We report on a lattice QCD calculation of the nucleon axial charge, $g_A$, using Mobius Domain-Wall fermions solved on the dynamical $N_f=2+1+1$ HISQ ensembles after they are smeared using the gradient-flow algorithm. The calculation is performed with three pion masses, $m_\pi\sim{310,220,130}$ MeV. Three lattice spacings ($a\sim{0.15,0.12,0.09}$ fm) are used with the heaviest pion mass, while the coarsest two are used on the middle pion mass and only the coarsest one is used with the near physical pion mass. On the $m_\pi\sim220$ MeV, $a\sim0.12$ fm point, a dedicated volume study is performed with $m_\pi L \sim {3.22,4.29,5.36}$. Using a new strategy motivated by the Feynman-Hellmann Theorem, we achieve a precise determination of $g_A$ with relatively low statistics, and demonstrable control over the excited state, continuum, infinite volume and chiral extrapolation systematic uncertainties, the latter of which remains the dominant uncertainty. Our final determination at 2.6% total uncertainty is $g_A = 1.278(21)(26)$, with the first uncertainty including statistical and systematic uncertainties from fitting and the second including systematic uncertainties related to the chiral and continuum extrapolation. The largest reduction of the second uncertainty will come from more pion mass points and more precise physical point mass points.

Hadron Structure