Coupled Integral Equations for the Nucleon and Pion Electromagnetic Form Factors

The dispersion relations for the nucleon isotopic vector form factors and the pion form factor which take into account contributions from both the $2\pi$ and $\mathrm{NN}$ intermediate states become a set of coupled integral equations for the form factors if the four amplitudes ($\pi \pi |NN$) ($\pi \pi |\pi \pi$) ($NN|\pi \pi$) ($NN|NN$) are assumed known. If these four amplitudes are replaced by their Born approximation values and spin and certain kinematic factors are neglected, the resulting set of coupled singular integral equations can be solved exactly. Comparison of these exact solutions with the form factors obtained from the usual approximation of retaining only the lowest mass state (i.e., the $2\pi$ state) confirms the hope that high-mass states do not contribute much to dispersion integrals. It is also of interest that these solutions are obtained from dispersion relations without subtractions and satisfy the necessary conditions that they vanish at infinite momentum transfer and take on the value $e$ at the origin for all values of the coupling parameters appearing in the equations.