Nucleon electromagnetic form-factors in a minimal Gari-Krümpelmann model including the explicit two-pion continuum

The Gari--Kr\"umpelmann (GK) model provided one of the first semiphenomenological descriptions of the electromagnetic nucleon form-factors over a wide range of momentum transfer. It combined vector-meson dominance at low and intermediate momentum transfer with perturbative-QCD constraints at large momentum transfer. The present work extends this framework by including the dispersive two-pion continuum explicitly in the isovector channel. The two-pion contribution is taken from modern dispersive analyses and embedded into the intrinsic high-$Q^2$ structure of the GK ansatz. The physical $\rho$ dynamics is then interpreted as part of the correlated two-pion spectral function. The direct coupling terms remain essential for the transition to the asymptotic regime. This gives a cleaner separation between long-range continuum dynamics, coherent vector-meson contributions, and direct quark-current dynamics. With just 5 parameters, we can describe the large body on form-factor data and differential cross sections from electron-proton scattering in the space-like region.