Resummation of the $\alpha$ expansion for nonlinear pair production

We show how to resum the Furry-picture $\alpha$ expansion in order to take quantum radiation reaction and spin transition into account in the nonlinear trident process in (pulsed) plane-wave background fields. The results are therefore nontrivial functions of both the background field strength, $eE$, and the coupling to the quantized photon field, $\alpha =e^2\mathrm{/}4\pi$. The effective expansion parameter, $T$, is $\alpha$ times $eE\mathrm{/}m\omega \gg 1$, which makes higher orders important. We show that they can change the sign of the spin dependent part already at $T<1$, which will be experimentally accessible.