Non-perturbative signatures of non-linear Compton scattering

The probabilities of various elementary laser - photon - electron/positron interactions display in selected phase space and parameter regions typical non-perturbative dependencies such as $\propto {\cal P} \exp\{- a E_{crit} /E\}$, where ${\cal P}$ is a pre-exponential factor, $E_{crit}$ denotes the critical Sauter-Schwinger field strength, and $E$ characterizes the (laser) field strength. While the Schwinger process with $a = a_S \equiv \pi$ and the non-linear Breit-Wheeler process in the tunneling regime with $a = a_{n \ell BW} \equiv 4 m / 3 \omega'$ (with $\omega'$ the probe photon energy and $m$ the electron/positron mass) are famous results, we point out here that also the non-linear Compton scattering exhibits a similar behavior when focusing on high harmonics. Using a suitable cut-off $c > 0$, the factor $a$ becomes $a = a_{n \ell C} \equiv \frac23 c m /(p_0 + \sqrt{p_0^2 -m^2)}$. This opens the avenue towards a new signature of the boiling point of the vacuum even for field strengths $E$ below $E_{crit}$ by employing a high electron beam-energy $p_0$ to counter balance the large ratio $E_{crit} / E$ by a small factor $a$ to achieve $E / a \to E_{crit}$. In the weak-field regime, the cut-off facilitates a threshold leading to multi-photon signatures showing up in the total cross section at sub-threshold energies.