Testing strong-field QED close to the fully non-perturbative regime using aligned crystals

Processes occurring in the strong-field regime of QED are characterized by background electromagnetic fields of the order of the critical field $F_{cr}=m^2c^3/\hbar|e|$ in the rest frame of participating charges. It has been conjectured that if in their rest frame electrons/positrons experience field strengths of the order of $F_{cr}/\alpha^{3/2}\approx 1600\,F_{cr}$, with $\alpha\approx 1/137$ being the fine-structure constant, their effective coupling with radiation becomes of the order of unity. Here we show that channeling radiation by ultrarelativistic electrons with energies of the order of a few TeV on thin tungsten crystals allows to test the predictions of QED close to this fully non-perturbative regime by measuring the angularly resolved single photon intensity spectrum. The proposed setup features the unique characteristics that essentially all electrons 1) undergo at most a single photon emission and 2) experience at the moment of emission and in the angular region of interest the maximum allowed value of the field strength, which at $2\;\text{TeV}$ exceeds $F_{cr}$ by more than two orders of magnitudes in their rest frame.