Froissart Bound from Gluon Saturation

We demonstrate that the dipole-hadron cross-section computed from the non-linear evolution equation for the Colour Glass Condensate saturates the Froissart bound in the case of a fixed coupling and for a small dipole (Q^2 >> Lambda_{QCD}^2). That is, the cross-section increases as the logarithm squared of the energy, with a proportionality coefficient involving the pion mass and the BFKL intercept (alpha_s N_c/pi)4 ln 2. The pion mass enters via the non-perturbative initial conditions at low energy. The BFKL equation emerges as a limit of the non-linear evolution equation valid in the tail of the hadron wavefunction. We provide a physical picture for the transverse expansion of the hadron with increasing energy, and emphasize the importance of the colour correlations among the saturated gluons in suppressing non-unitary contributions due to long-range Coulomb tails. We present the first calculation of the saturation scale including the impact parameter dependence. We show that the cross-section at high energy exhibits geometric scaling with a different scaling variable as compared to the intermediate energy regime.