Quantum radiation reaction: Analytical approximations and obtaining the spectrum from moments

We derive analytical $\chi\ll1$ approximations for spin-dependent quantum radiation reaction for locally constant and locally monochromatic fields. We show how to factor out fast spin oscillations and obtain the degree of polarization in the plane orthogonal to the magnetic field from the Frobenius norm of the Mueller matrix. We show that spin effects lead to a transseries in $\chi$, with powers $\chi^k$, logarithms $(\ln\chi)^k$ and oscillating terms, $\cos(\dots/\chi)$ and $\sin(\dots/\chi)$. In our approach we can obtain each moment, $\langle(kP)^m\rangle$, of the lightfront longitudinal momentum independently of the other moments and without considering the spectrum. We show how to obtain a low-energy expansion of the spectrum from the moments by treating $m$ as a continuous, complex parameter and performing an inverse Mellin transform. We also show how to obtain the spectrum, without making a low-energy approximation, from a handful of moments using the principle of maximum entropy.