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Collinear resummations for the non-linear evolution in QCD at high energy

When computed to next-to-leading order in perturbative QCD, the non-linear Balitsky-Kovchegov (BK) equation for the high-energy evolution of the dipole-hadron scattering appears to be unstable. We show that this instability can be avoided by using the rapidity of the dense hadronic target (instead of that of the dilute dipole projectile) as the evolution time. Using this variable, we construct a collinearly-improved version of the BK equation, where the dominant radiative corrections to the kernel -- those enhanced by double collinear logarithms -- are resummed to all orders.