QCD at High Parton Density

The high parton density regime in deep inelastic electronproton (nucleus) scattering is characterized by small values of the Bjorken variable x = Q/s, where Q is the momentum transfer and √ s is the center of mass energy, and represents the challenge of studying the interface between the perturbative and nonperturbative QCD, with the peculiar feature that this transition is taken in a kinematical region where the strong coupling constant αs is small. By the domain of perturbative QCD we mean the region where the parton picture has been developed and the separation between the short and long distance contributions (the collinear factorization) is made possible by the use of the operator product expansion (OPE). The Dokshitzer-GribovLipatov-Altarelli-Parisi (DGLAP) equations [1] are the evolution equations in this kinematical region. In the limit of small values of x (< 10−2) one expects to see new features inside the nucleon: the density of gluons and quarks become very high and an associated new dynamical effect is expected to stop the further growth of the structure functions. In particular, for a fixed hard scale Q À ΛQCD, the OPE eventually breaks down at sufficiently small x [2]. Ultimately, the physics in the region of high parton densities will be described by nonperturbative methods, which is still waiting for a satisfactory solution in QCD. However, the transition from the moderate x region towards the small x limit may possibly be accessible in perturbation theory, and, hence, allows us to test the ideas about the onset of nonperturbative dynamics. The expectation of the transition for the high density regime can be understood considering the physical picture of the deep inelastic scattering. In the infinite momentum frame (IMF) the virtual photon with virtuality Q measures the number density of charged partons having longitudinal momentum fraction x and transverse spatial size ∆x⊥ ≤ 1/Q. When Q is large, αs(Q) is small, so that the struck quark can be treated perturbatively. Also, when Q is large the struck quark is small, so that one can picture the struck quark as being isolated, far away from similar quarks, in the proton. Thus, so long as the parton distributions are not large, the partons in a proton are dilute. However, if the parton distributions get large enough, which happens when x is very small, partons in the proton must begin to overlap. If there is a sufficient amount of parton overlap then a given parton will not act as a free quantum over its lifetime but will interact strongly with the other partons in the proton, even though αs may still be in the perturbative regime. In other words, while for large momentum transfer k⊥, the linear evolution equations (DGLAP/BFKL) predicts that the mechanism g → gg populates the transverse space with a large number of small size gluons per unit of rapidity (the transverse size of a gluon with momentum k⊥ is proportional to 1/k⊥), for small k⊥ the produced gluons overlap and fusion processes, gg → g, are equally important. Considering the latter process, the rise of the gluon distribution below a typical scale is reduced, restoring the unitarity. That typical scale is energy dependent and is called saturation scale Qs. The saturation momentum sets the critical transverse size for the unitarization of the cross sections. Therefore, at sufficient small values of x, one enters in the regime of high density QCD, where partons from neighboring ladders overlap spatially and new dynamical effects associated with the unitarity corrections are expected to stop further growth of the parton densities.