Extrapolation of Nucleon Electromagnetic Form Factors

We use conformal transformations to facilitate extrapolation of experimental values of the two isovector and two isoscalar nucleon form factors, to find their spectral functions. Both isovector spectral functions peak around 600 MeV, or somewhat below the energy of the isovector $\rho$ resonance. Both isoscalar spectral functions peak around 700 MeV, somewhat below the $\omega$ resonance. We also analyze the electric form factor for elastic electron-deuteron scattering in terms of the difference between the measured value and the calculated value. The latter uses assumptions concerning the deuteron wave function, and concerning the isoscalar electric nucleon form factor $G_{\mathrm{ES}}$. By extrapolating this difference, we find $G_{\mathrm{ES}}$ in the time-like region $t_A<t<\mathrm{}9\mathrm{}{{\mu }_{\pi }}^2$, where $t_A$ is the anomalous threshold, and the upper limit is the normal isoscalar threshold. These values of $G_{\mathrm{ES}}$ are statistically consistent with the measurements for space-like momentum transfer. Extrapolation of the combined set of values for $G_{\mathrm{ES}}$ (space-like and time-like momentum transfers) gives a spectral function again peaking at 700 MeV, with indications of a dip at 1150 MeV. The dip may be due to the isoscalar $\phi$ resonance.