Nucleon isovector couplings in Nf = 2 + 1 lattice QCD at the physical point

We present results for the scalar and tensor isovector-couplings (\(g_S\) and \(g_T\)) of the nucleon measured at the physical point (\(M_{\pi}=135\) MeV) with a single lattice spacing of \(0.085\ \mathrm{fm}\) in 2+1 flavor QCD. Our calculations are carried out with two ensembles of gauge configurations generated by the PACS Collaboration with nonperturbatively \({\cal O}(a)\) improved Wilson quark action and Iwasaki gauge action on \((10.9\ {\rm fm})^4\) and \((5.5\ {\rm fm})^4\) lattices, where the finite-size effect on the nucleon mass was not shown at the level of the statistical precision less than 0.5%. We compute the nucleon three-point correlation functions in the vector, axial, scalar, and tensor channels. We confirm that our previous result of the nucleon axial coupling on the large spatial volume of \((10.9\ {\rm fm})^4\) has no finite-size effect at the level of the statistical precision of 1.9%. For the renormalization, we first renormalize \(g_S\) and \(g_T\) nonperturbatively using the RI/SMOM\(_{(\gamma_\mu)}\) scheme, a variant of Rome-Southampton RI/MOM scheme with reduced systematic errors, as the intermediate scheme. We evaluate our final results at the renormalization scale of 2 GeV in the \(\overline{\rm MS}\) scheme through matching procedure between the RI/SMOM\(_{(\gamma_\mu)}\) and \(\overline{\rm MS}\) schemes with the help of perturbation theory, and then obtain \(g_S=0.927(71)_{\rm stat}(22)_{\rm syst}\) and \(g_T=1.036(6)_{\rm stat}(20)_{\rm syst}\).