In this work, we present new extraction of the proton $\langle r_p\rangle$ and neutron $\langle r_n^2\rangle$ root-mean-square charge radius using only two-photon-exchange corrected proton's $G_{(E,M)}^p\left(Q^2\right)$ form-factor data from I. A. Qattan and J. Arrington [Phys. Rev. C 86, 065210 (2012)] and I. A. Qattan, J. Arrington, and A. Alsaad [Phys. Rev. C 91, 065203 (2015)] covering the range of $0.0155<Q^2<5.250{\mathrm{GeV}}^2$. Assuming isospin and charge symmetry and neglecting the strange quarks contributions, $G_{(E,M)}^p$ are first expressed in terms of the flavor-separated $G_{(E,M)}^{(u,d)}$ form factors and then fitted separately up to $Q^2=5.25{\mathrm{GeV}}^2$ using different rational functional forms of $G_{(E,M)}^{(u,d)}$. The best achieved $G_{(E,M)}^{(u,d)}$ parametrizations are used to extract $G_{(E,M)}^n$ extending the range of extraction of $G_E^n$ and the Dirac flavor-separated $F_1^{(u,d)}$ up to $Q^2=5.25{\mathrm{GeV}}^2$. The charge radius are calculated using the two-dimensional transverse quark charge distributions based on the slopes of $F_1^{(u,d)}\left(Q^2\right)$ at $Q^2=0$ through model-independent relations. We also have investigated the impact of inclusion of high-$Q^2$ data points, functional forms fitted, and fitting range on the radius extraction. For the proton, we find $\langle r_p\rangle =0.8354\pm 0.{0020}_{\text{(stat.)}}\pm 0.{0060}_{\text{(sys.)}}$, which agrees with the PRad I collaboration measurements and ultrahigh precise muonic hydrogen $\mu \mathrm{H}$ Lamb shift results but disagrees with the Mainz collaboration extractions and the Particle Data Book (CODATA 2010-2014) results. For the neutron, on the other hand, our obtained value of $\langle r_n^2\rangle =-0.0998\pm 0.{0030}_{\text{(stat.)}}\pm 0.{0040}_{\text{(sys.)}}$ is $\approx 10.22\%$ below in magnitude the value reported by the recent precise measurements of the low-$Q^2 G_E^n$ in connection with the $N\to \mathrm{\Delta }$ quadrupole transitions and $\approx 15.43\%$ below the new adjusted $\langle r_n^2\rangle$ world data value.

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• Elastic scattering reactions
• Form factors
• Lepton induced nuclear reactions
• Nuclear radii
• Particle interactions
• Polarization phenomena
• QCD in nuclear reactions
• Baryons
• Down quark
• Up quark
• Polarization