The Color Glass Condensate is a theory of the dynamical properties of partons in the Regge limit of QCD: xBj → 0, Q 2 >> Λ 2 QCD = fixed and the center of mass energy squared s → ∞. We provide a brief introduction to the theoretical ideas underlying the Color Glass Condensate and discuss the application of these ideas to high energy scattering in QCD. PACS. PACS-key describing text of that key-PACS-key describing text of that key

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 show that the HERA data for the inclusive structure function F 2 (x, Q 2 ) for x ≤ 10 −2 and 0.045 ≤ Q 2 ≤ 45 GeV 2 can be well described within the color dipole picture, with a simple analytic expression for the dipole-proton scattering amplitude, which is an approximate solution to the non-linear evolution equations in QCD. For dipole sizes less than the inverse saturation momentum 1/Q s (x), the scattering amplitude is the solution to the BFKL equation in the vicinity of the saturation line. It exhibits geometric scaling and scaling violations by the diffusion term. For dipole sizes larger than 1/Q s (x), the scattering amplitude saturates to one. The fit involves three parameters: the proton radius R, the value x 0 of x at which the saturation scale Q s equals 1GeV, and the logarithmic derivative of the saturation momentum λ. The value of λ extracted from the fit turns out to be consistent with a recent calculation using the next-to-leading order BFKL formalism.

No abstract

We present a unified description of the high temperature phase of QCD, the so-called quark-gluon plasma, in a regime where the effective gauge coupling g is sufficiently small to allow for weak coupling calculations. The main focuss is the construction of the effective theory for the collective excitations which develop at a typical scale gT , which is well separated from the typical energy of single particle excitations which is the temperature T . We show that the short wavelength thermal fluctuations, i.e., the plasma particles, provide a source for long wavelength oscillations of average fields which carry the quantum numbers of the plasma constituents, the quarks and the gluons. To leading order in g, the plasma particles obey simple gauge-covariant kinetic equations, whose derivation from the general Dyson-Schwinger equations is outlined. By solving these equations, we effectively integrate out the hard degrees of freedom, and are left with an effective theory for the soft collective excitations. As a by-product, the "hard thermal loops" emerge naturally in a physically transparent framework. We show that the collective excitations can be described in terms of classical fields, and develop for these a Hamiltonian formalism. This can be used to estimate the effect of the soft thermal fluctuations on the correlation functions. The effect of collisions among the hard particles is also studied. In particular we discuss how the collisions affect the lifetimes of quasiparticle excitations in a regime where the mean free path is comparable with the range of the relevant interactions. Collisions play also a decisive role in the construction a Member of CNRS. E-mail: blaizot@spht.saclay.cea.fr b Member of CNRS. E-mail: iancu@spht.saclay.cea.fr c Laboratoire de la Direction des Sciences de la Matière du Commissariatà l'Energie Atomique of the effective theory for ultrasoft excitations, with momenta ∼ g 2 T , a topic which is briefly addressed at the end of this paper.

We show that the evolution equations in QCD predict geometric scaling for quark and gluon distribution functions in a large kinematical window, which extends above the saturation scale up to momenta Q 2 of order 100 GeV 2 . For Q 2 < Q 2 s , with Q s the saturation momentum, this is the scaling predicted by the Colour Glass Condensate and by phenomenological saturation models., we show that the solution to the BFKL equation shows approximate scaling, with the scale set by Q s . At larger Q 2 , this solution does not scale any longer. We argue that for the intermediate values of Q 2 where we find scaling, the BFKL rather than the double logarithmic approximation to the DGLAP equation properly describes the dynamics. We consider both fixed and running couplings, with the scale for running set by the saturation momentum. The anomalous dimension which characterizes the approach of the gluon distribution function towards saturation is found to be close to, but lower than, one half.

No abstract

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.