Asymptotic behavior of form factors in a class of field theories

It is shown that for all renormalizable field theories, excluding gluon and gauge theories, the asymptotic behaviors of the on-mass-shell form factors are related to the short-distance behaviors of the theories. In particular, if the Gell-Mann-Low eigenvalue conditions are satisfied, the asymptotic behavior of the on-mass-shell electromagnetic form factor is ${\left(\frac{m^2}{-q^2}\right)}^{\gamma }$, where $\gamma$ is the anomalous dimension of the proton field. We also show that the form factor of the axial-vector current has the same asymptotic behavior as that of the vector current.