Color Glass Condensate Primer
We asked Larry McLerran, the world’s leading expert on the Color Glass Condensate, to write this CGC primer for us:
The basic idea of the Color Glass Condensate arose from trying to understand why hadrons could have a roughly fixed size and at the same time a rapidly rising gluon density. There were early discussions that this growth might be tempered by annihilations of gluons when the density began to get large:
- Semihard Processes in QCD
- Gluon Recombination and Shadowing at Small Values of x
- NonAbelian Weizsacker-Williams field and a two-dimensional effective color charge density for a very large nucleus
An explicit mechanism by means of which the rise in the gluon density was understood was introduced in:
- Computing quark and gluon distribution functions for very large nuclei
- Gluon distribution functions for very large nuclei at small transverse momentum
Because the gluon density is large, the coupling constant is weak, so that one can use a weak coupling analysis. Nevertheless, the gluon density is large and coherent, so that the fields, although intrinsically weakly coupled are strong. Mathematically, the problem is similar to quantum electrodynamics in strong fields. The fine structure constant is weak so that there is a weak coupling expansion, but one must include effects to all orders in the strong field.
The idea is that the high density of gluons generates a highly coherent state with properties analogous to condensates. In particular, the phase space density of gluons is so high hat the quantum occupation number of low energy states is very large. The gluons are described with a coherent classical field. This classical field fluctuates from event to event, but because of Lorentz time dilation, there is no quantum interference between these different field configurations. This means there is an incoherent sum over classical fields, as in a spin glass. This was explained most clearly in the following papers:
- Nonlinear gluon evolution in the color glass condensate. 1.
- Nonlinear gluon evolution in the color glass condensate. 2.
The sum over different fields introduces the concept of a density matrix over which one sums individual classical fields.
A consequence of the analysis using classical fields is that saturation does occur for gluons of a fixed size. When the phase space density is full at one size scale, gluons of smaller size scale which have less occupied phase space density begin to become occupied. The total number of gluons grows forever. The saturation of phase space proceeds from larger to smaller size scales, or from smaller momentum to larger momentum scales. The weak coupling analysis is therefore better and better as one goes to higher and higher energy. The momentum scale that separates saturated modes from less occupied modes is called the saturation momentum.
These gluons form a part of the hadron wavefunction that dominates typical high energy processes. The typical gluon separation is small compared to the size of the system.
This matter was first named the Color Glass Condensate (CGC) in the above references.
The renormalization group plays an important role in the theory of this matter. The parameters in the effective action of the CGC evolve as one goes to higher energy, and can be evaluated through renormalization group techniques. This is explained in the above papers by Iancu et. al. and the origins of this discussion may be found in:
- The Intrinsic glue distribution at very small x
- The Wilson renormalization group for low x physics: Towards the high density regime
The renormalization group equation in mean field for the two point function is the Balitsky-Kovchegov equation. This equation explicitly shows saturation:
- Small x F(2) structure function of a nucleus including multiple pomeron exchanges
- Operator expansion for high-energy scattering
An alternative way to think about high energy processes comes from Mueller’s dipole model. Most practitioners actually keep both the CGC and the dipole picture in mind when thinking bout high energy small x processes:
- Parton saturation at small x and in large nuclei
- Single and double BFKL pomeron exchange and a dipole picture of high-energy hard processes
One can show that the dipole picture and the Color Glass Condensate are mathematically equivalent in the limit of a large number of colors.
There is now a wide spread phenomenology of the CGC which works quite well. An excellent review of theoretical and phenomenological developments is given in:
The Color glass condensate and high-energy scattering in QCD
Since the publication of this review there have of course been many more recent developments, but the review is a good beginning to the literature. It is important to remember that the goals of such phenomenology are very ambitious. One tries to describe all of high energy strong interactions. This means that the same basic ingredients and parameters must simultaneously describe processes as diverse as deep inelastic scattering, diffraction and high energy heavy ion collisions. One of the problems is that at current energies, we are probably at the edge where these concepts begin to work well. Nevertheless, it is surprising how well the phenomenology of the CGC describes data. To my mind, the real test of these ideas will be at the LHC, where there will be a wide variety of data at various values of x and Q^2, and the x values are small enough that the ideas associated with the CGC should work well.
The Color Glass Condensate describes the matter in the nuclear wavefunction responsible for dominant high energy processes. During the collision process itself, there is a sudden change at the instant of collision. The two Lorentz contracted nuclei pass through one another rapidly, and when this happens, they get dusted with color electric and color magnetic charge. These charges make a random distribution of longitudinal electric and magnetic fields, which as they decay away make gluons that eventually thermalize. This new matter between the Color Glass Condensate and the Quark Gluon Plasma is called the Glasma. It is matter with both strong gluonic fields, and with gluons thought of as particles.
The formation of such fields was first described in:
- Gluon production from nonAbelian Weizsacker-Williams fields in nucleus-nucleus collisions
- Gluon production at high transverse momentum in the McLerran-Venugopalan model of nuclear structure functions
(the papers wre published in reverse order due to refereeing issues)
A good description of the Glasma and its properties is in:
The equations that describe the evolution of the Glasma may be solved numerically:
- Nonperturbative computation of gluon minijet production in nuclear collisions at very high-energies
- The Initial energy density of gluons produced in very high-energy nuclear collisions
- Production of gluons in the classical field model for heavy ion collisions
There has been much discussion that the boost invariant solutions assumed in the evolution of the Glasma may become turbulent and develop boost non-invariant ripples. This is analogous to original arguments by:
- Collective modes of an anisotropic quark gluon plasma
- Hard-loop dynamics of non-Abelian plasma instabilities.
One has shown that there are such instabilities in the Glasma evolution:
One can also understand something of the origin of such instabilities from quantum noise in the initial wavefunction:
Initial Singularity of the Little Bang
The magnitude of effects associated with such instabilities are not yet fully understood. A search of the recent papers of by authors such as Peter Arnold, Michael Strikland, Adrian Dumitru, Berndt Mueller, M. Asakawa, S.A. Bass, T. Lappi, F. Gelis and R. Venugopalan will give an excellent view on the status of understanding. The hope is that these instabilities may lead to a turbulent thermalization of the system, and perhaps explain the rapid thermalization claimed in the RHIC heavy ion collisions.