Freeze out in Hydrodynamical models

My Master of Science Thesis - MSc_Magas.ps.gz.

This very brief presentation bases on Refs. [4-7,9 (see CV)].

Main points

1. Conservation laws should be taken into account exactly across freeze out hypersurfaces with both time-like and space-like normals.

2. The post freeze out distribution can not be a thermal distribution in the case of surface with space-like normal! We still can use Cooper-Frye formula for space-like normal, but we have to assume cut post freeze out distribution - it is non zero only for particle moving outside the freeze out front, pn>0. p is four-momentum of particle, n is a normal four-vector to the freeze out surface.

The idea of cut disrtibution was initially introduced by K.A. Bugaev, Nucl. Phys. A 606 (1996) 559.

Three models were discussed in our articles

1. Idealized kinetic model with drain term. We can describe the freeze out kinetics assuming that we have two components of our momentum distribution, free and interacting components. If we dramatically oversimplify the freeze out process and take into account only gradual transfer from interacting to free component - such a model gives a cut Juttner distribution as a post freeze out distribution. Main problem is that not all particles can freeze out, so physical freeze out does not happen.

Figure 1

2. Freeze out distribution with rescattering, immediately re-thermalization limit. This improved model gives cut distribution, which does not any more look like cut Juttner distribution. The final momentum distribution of emitted particles, for freeze out surface with space-like normal, shows a non-exponential transverse momentum spectrum. The slope parameter of the transverse momentum distribution increases with increasing transverse momentum, in agreement with recently measured SPS pion and h^- spectra. All particles can freeze out but after infinitely long time, this also my be a problem. Model gives results which qualitatively agree with experimental data.

Figure 2

Figure 3

Figure 4

3.Volume emission model is most understandable. This model is based on the escape probability calculated according to a simple gas model. Such a consideration seems to be more realistic, than the two discussed above. Its particular adventage is that freeze out in a layer of finite width is discribed by the model. Unfortunately, it leads to a nontrivial integro-differential equation. Using an approximation approach we can obtain solutions, and we could see that these satisfied conservation laws. This model leads to incomplete freeze out, but this problem can be cured by introducing rescattering in the model as we demonstrated in the previous case.

Figure 5