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JIMWLK evolution for multi-particle production in Langevin form

Within the effective theory for the Color Glass Condensate, we study multi-particle production with rapidity correlations in proton-nucleus collisions at high energy. The high-energy evolution responsible for such correlations is governed by a generalization of the JIMWLK equation which describes the simultaneous evolution of the (strong) nuclear color fields in the direct amplitude and the complex conjugate amplitude. This functional equation can be used to derive ordinary evolution equations for the cross-sections for particle production (a generalization of the Balitsky hierarchy). However, the ensuing equations appear to be too complicated to be useful in practice, including in the limit where the number of colors is large. To circumvent this problem, we propose an alternative formulation of the high-energy evolution as a Langevin process, which is better suited for numerical implementations. This process is directly oriented towards the calculation of the cross-sections, so its detailed structure depends upon the nature of the final state. We present the stochastic equations appropriate for two gluon production, and also for three gluon production, with generic rapidity differences.