Structure of radiative corrections in a strong constant crossed field in the bubble-chain approximation

We present a calculation of the electron mass operator combining the bubble-chain photon and the leading order electron propagators in a strong constant crossed electromagnetic field. The photon propagator is obtained by a summation of the Dyson series over 1PI polarization loop insertions and is accounted for in general form. By applying the derived mass operator, we find the bubble-chain electron propagator and the elastic scattering amplitude. The results are expressed in the general form of expansion over $\gamma$-matrix structures, each multiplied by a scalar invariant function of the electron virtuality and the quantum dynamical parameter $\chi$. We perform their asymptotic analysis and identify the dominant contribution at $\chi\gg 1$. The presented calculations can be generalized by replacing the leading order electron propagator with the bubble-chain one. This will allow a consistent study the Dyson-Schwinger equations in the fully nonperturbative regime $\alpha\chi^{2/3}\gtrsim1$ within the bubble-chain approximation. In addition, we accompany our paper with the open-access computer-algebraic scripts containing all the presented computations [https://github.com/ArsenyMironov/SFQED-Loops].