Nucleon form factors in $N_f=2+1$ lattice QCD at the physical point: Finite lattice spacing effect on the root-mean-square radii

We present results for the nucleon form factors: electric ($G_E$), magnetic ($G_M$), axial ($F_A$), induced pseudoscalar ($F_P$), and pseudoscalar ($G_P$) form factors, using the second PACS10 ensemble that is one of three sets of $2+1$ flavor lattice QCD configurations at physical quark masses in large spatial volumes [exceeding $(10\mathrm{fm}{)}^3$]. The second PACS10 gauge configurations are generated by the PACS Collaboration with the six stout-smeared $O\left(a\right)$ improved Wilson quark action and Iwasaki gauge action at the second gauge coupling $\beta =2.00$ corresponding to the lattice spacing of $a=0.063\mathrm{fm}$. We determine the isovector electric, magnetic and axial radii, and magnetic moment from the corresponding form factors, as well as the axial-vector coupling $g_A$. Combining our previous results for the coarser lattice spacing [E. Shintani et al., Phys. Rev. D 99, 014510 (2019); Phys. Rev. D 102, 019902(E) (2020)], the finite lattice spacing effects on the isovector radii, magnetic moment, and axial-vector coupling are investigated using the difference between the two results. It was found that the effect on $g_A$ is kept smaller than the statistical error of 2% while the effect on the isovector radii was observed as a possible discretization error of about 10%, regardless of the channel. We also report the partially conserved axial-vector current relation using a set of nucleon three-point correlation functions in order to verify the effect by $O\left(a\right)$ improvement of the axial-vector current.