Expansion parameter of perturbation theory in intense-field quantum electrodynamics

The asymptotic behavior of electron and photon elastic scattering amplitudes of order ${\alpha }^3$ in an intense constant electromagnetic field are studied. It is shown that the expansion parameter of perturbation theory in intense-field quantum electrodynamics is $\alpha {\chi }^{\frac{2}{3}}$ when $\chi \gg 1$, where $\chi =\frac{{\left[{\left(e{F}_{\mu \nu }{p}_{\nu }\right)}^2\right]}^{\frac{1}{2}}}{m^3}$ with $p_{\mu }$ the momentum of the particle and $F_{\mu \nu }$ the electromagnetic field tensor.