Non-linear pair production in scattering of photons on ultra-short laser pulses at high energy

We consider scattering of a photon on a short intense laser pulse at high energy. We argue that for ultra-short laser pulses the interaction is coherent over the entire length of the pulse. At low pulse intensity I the total cross section for electron-positron pair production is proportional to I. However, at pulse intensities higher than the characteristic value Is, the total cross section saturates - it becomes proportional to the logarithm of intensity. This non-linear effect is due to multi-photon interactions. We derive the total cross section for pair production at high energies by resuming the multi-photon amplitudes to all orders in intensity. We calculate the saturation intensity Is and show that it is significantly lower than the Schwinger's critical value. We discuss possible experimental tests.