We consider a nonlinear evolution equation recently proposed to describe the small-x hadronic physics in the regime of very high gluon density. This is a functional Fokker-Planck equation in terms of a classical random color source, which represents the color charge density of the partons with large x. In the saturation regime of interest, the coefficients of this equation must be known to all orders in the source strength. In this first paper of a series of two, we carefully derive the evolution equation, via a matching between classical and quantum correlations, and set up the framework for the exact background source calculation of its coefficients. We address and clarify many of the subtleties and ambiguities which have plagued past attempts at an explicit construction of this equation. We also introduce the physical interpretation of the saturation regime at small x as a Color Glass Condensate. In the second paper we shall evaluate the expressions derived here, and compare them to known results in various limits. 5 See Sect. 2.1 for the definition of light-cone coordinates and momenta.• The background fields are independent of the light-cone time x + , but inhomogeneous, and even singular, in x − . To cope with that, we find that it is more convenient to integrate the quantum fluctuations in layers of p − (the light-cone energy), rather than of p + [cf. Sect. 3.4]. Accordingly, we have no ambiguity associated with possible singularities at p − = 0.• The gauge-invariant action describing the coupling of the quantum gluons to the classical color source is non-local in time [10]. Thus, the proper formulation of the quantum theory is along a Schwinger-Keldysh contour in the complex time plane [cf. Sect. 4]. However, in the approximations of interest, and given the specific nature of the background, the contour structure turns out not to be essential, so one can restrict oneself to the previous formulations in real time [10,11].• We carefully fix the gauge in the quantum calculations. In the light-cone gauge, the gluon propagator has singularities at p + = 0 associated with the residual gauge freedom under x − -independent gauge transformations. We resolve these ambiguities by using a retarded iǫ prescription [cf. eq. (3.84) and Sect. 6.3], which is chosen for consistency with the boundary conditions imposed on the classical background field. We could have equally well used an advanced prescription. However, other gauge conditions, such as Leibbrandt-Mandelstam or principal value prescriptions, are for a variety of technical reasons shown to be unacceptable [32].• The choice of a gauge prescription has the interesting consequence to affect the spatial distribution of the source in x − , and thus to influence the way we visualize the generation of the source via quantum evolution. With our retarded prescription, the source has support only at positive x − [see Sect. 5.1].• We verify explicitly that the obtention of the BFKL equation as the weak-field limit of the general renormalization group equation is independe...

We discuss the Wilson renormalization group approach to the effective action for low x physics. It is shown that in the linearized, weak field regime the RG equation reduces to the BFKL equation for the evolution of the unintegrated gluon density. We discuss the relation of this approach with that of Lipatov.

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We present an explicit and simple form of the renormalization group equation which governs the quantum evolution of the effective theory for the Color Glass Condensate (CGC). This is a functional Fokker-Planck equation for the probability density of the color field which describes the CGC in the covariant gauge. It is equivalent to the Euclidean time evolution equation for a second quantized current-current Hamiltonian in two spatial dimensions. The quantum corrections are included in the leading log approximation, but the equation is fully non-linear with respect to the generally strong beckground field. In the weak field limit, it reduces to the BFKL equation, while in the general non-linear case it generates the evolution equations for eikonal-line operators previously derived by Balitsky and Kovchegov within perturbative QCD.