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Nonlinear 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 $I_s$, the total cross section saturates -- it becomes proportional to the logarithm of intensity. This nonlinear 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 $I_s$ and show that it is significantly lower than the Schwinger's critical value. We discuss possible experimental tests.