CGC initial conditions for nuclear collisions

Posted by Adrian Dumitru (July 7, 2008)

Models for the thermalization stage or hydrodynamic expansion of QCD matter produced in heavy-ion collisions require initial conditions. This refers, for example, to the distribution of partons in the transverse plane, in rapidity, and in transverse momentum. It is determined by the wave functions of the colliding nuclei.

It is common to assume, according to the Glauber model, that the density of produced particles in impact parameter space is proportional to the density of wounded nucleons. However, this may change near the unitarity limit (high energies and large nuclei). For example, Hirano et al (nucl-th/0511046) discovered that the kt-factorization formula with a particular model for the unintegrated gluon distribution (uGD) functions proposed by Kharzeev, Levin and Nardi leads to a larger eccentricity $\epsilon ={\frac {\langle y^{2}-x^{2}\rangle }{\langle y^{2}+x^{2}\rangle }}$ than the Glauber model; hydrodynamics translates this into a larger azimuthal asymmetry of the particle momentum distributions in the final state, as measured by the second Fourier coefficient v2.

Shortly after, it was shown in nucl-th/0605012 that this result is recovered with other models for the uGD, and a general argument was given that this should be a generic consequence of the unitarity limit. Also, a non-trivial evolution of the eccentricity (and of other moments) with rapidity was predicted using the kt-factorization approach and the KLN uGDs.

Nara and Drescher realized that it is important that the uGD of a nucleus approaches that of a single nucleon at the surface, which leads to more reliable and accurate predictions for the eccentricity, in particular for more peripheral A+A collisions and/or smaller nuclei. This implies, in particular, that Qs(nucleus) does not drop to zero at the surface but is bound from below by Qs(nucleon), resulting in more stable and realistic predictions for the eccentricity. They described their so-called fKLN model in nucl-th/0611017.

In arXiv:0707.0249, finally, they also implemented fluctuations of the "hard sources", that is, of the transverse positions of nucleons in the colliding nuclei. The uGDs of the colliding nuclei and the distribution of produced partons is constructed on an "event-by-event" basis. As a consequence, the eccentricity of the "overlap" zone does not vanish at impact parameter b=0 (which affects the limiting value of v2 in central collisions of large nuclei, see, for example, arXiv:0704.3553). This is the so-called MC-KLN model.

The following FORTRAN code implements both the mean-field fKLN model as well as the MC-KLN model with fluctuations. The code has been written by Hajo Drescher.

To use the code, unpack the .tar.gz archive and follow the instructions in README. More details may be provided here later; at this point, if you need to dig deeper please read the papers mentioned above and look at the source code.

---

Note that what you obtain by convoluting two uGDs is supposed to be the particle distribution at the very earliest time, say of order 1/Qs. This is the distribution of partons that have been "released" from the wave functions of the colliding nuclei. Local thermal equilibrium need not be established yet.

Indeed, if you take the energy density and the number density at midrapidity obtained from fKLN/MC-KLN and convert them to temperatures, you get two different results. In general, therefore, it is necessary to follow the evolution of those initial gluons towards thermalization with some other model.

On the other hand, the number of gluons per unit rapidity turns out to be very close to the measured multiplicity of hadrons in the final state. Therefore, one could perhaps assume that the gluon number is essentially conserved, implying that the local density at some arbitrary time $\tau$ is given approximately by $n(\tau )=n(1/Q_{s})/(\tau Q_{s})$ Also, if local thermal and chemical equilibrium is established by then, then the temperature T could be deduced from the number density via standard thermodynamic relations.

Back to Pre-equilibrium dynamics