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Discussion: First Order Phase Transition

In 1985, L. Van Hove formulated a model of quark-gluon plasma hadronization [87] which implied very specific experimental signatures observable on EbyE basis. He developed a picture of the hadronization dynamics for a first order phase transition: a QGP cylinder expands longitudinally, until, near transition, its color field assumes a longitudinal topology with strings and color flux tubes. The string network so formed continues stretching, thus slowing down the expansion, while some strings suffer break-up. String breaking creates plasma droplets, as large as a few fm across, which no longer expand, but retain the longitudinal motion. They hadronize by deflagration[89]. This is expected to result in peculiar bumpy $\,dN/\,dy$ distributions.

Mardor and Svetitsky [90] performed a calculation of the free energy of a hadronic gas bubble, embedded in the QGP phase, and of a QGP droplet in a hadronic gas, in the MIT bag model. They found that at temperatures of 150 MeV and below, growth of the hadron gas bubbles (and evaporation of the QGP droplets) becomes irresistible and QGP hadronizes.

In the absence of a direct, event-by-event observable-based test of these predictions, the picture had been further developed [91,92] in order to connect it with the traditional observables such as the $m_T$ slope parameter $T$ and the baryon and strangeness chemical potentials: the hadron ``temperatures'' $T$ in the SPS data are higher than lattice QCD predictions for a phase transition temperature. Using a first order phase transition hydrodynamical model with a sharp front between the phases, Bilic et al. [91,92] concluded that a QGP supercooling and hadron gas superheating is a consequence of the continuity equations and of the requirement that the entropy be increased in the transition. In the case of bubbles in the QGP phase, the plasma deflagrates; otherwise, it detonates. The statement that a Van Hove type of a hadronization scenario explains some observations is, however, by no means a verification of the hypothesis. An alternative explanation of the high hadron ``temperatures'' which does not involve overheating is the collective flow [93]. Our dynamic texture measurement tests the QGP droplet hadronization hypothesis [87] in a more direct way, because, as we have shown quantitatively, the measurement is sensitive to the presense or absence of the droplets in course of the hadronization (with the necessary caveat that some fraction of the hadronization texture can be washed out by rescattering in the post-hadronization phase [88]). Our result can be used to constrain phenomenological quantities which represent basic QCD properties and affect texture formation in this class of hadronization models [87,90,92]. Such quantities are

• the energy flux, or rate at which the QGP transmits its energy to hadrons. The Van Hove hypothesis is based on the estimates of a low flux ( $\ll m_{\pi}/(fm^2 fm/c)$) [94,95]. If the flux is low, the droplet is preserved until it deflagrates. If the flux is high, the droplet emits mesons continuously as it moves, and the signal is washed out.
• typical initial size of the QGP droplet. At a fixed upper energy density of the transition $\epsilon_0^\prime$, this parameter determines the number of hadrons per fireball after hadronization - the ``clustering parameter'', whose experimental manifestation is explored by Fig.7.5. For the phase transitions out of the QGP phase, this parameter will be determined by the percolation of the growing hadron gas bubbles.
• initial upper energy density of the transition $\epsilon_0^\prime$. At a fixed initial size of the QGP droplet, $\epsilon_0^\prime$ controls the ``clustering parameter'' in much the same way as the initial size does.

The event generator results on Fig.7.5 are calculated for pure cases, that is, when all events originate from a parent distribuion with the same clustering parameter. A real situation may well be a combination of a variety of pure cases, for example, the case of a certain mean clustering parameter in $x$ fraction of events, and no phase transition in $1-x$ fraction of events. To conclude this section, we remark that the operation of the power spectrum averaging over events is linear, therefore in this case

$$P^{\lambda}(m)_{true} = xP^{\lambda}_{X}(m)_{true} + (1-x) P^{\lambda}_{0}(m)_{true}.$$ (96)

Ideally, event mixing should destroy dynamic textures fully, in that case

$$P^{\lambda}(m)_{mix} = P^{\lambda}_{X}(m)_{mix} = P^{\lambda}_{0}(m)_{mix}$$ (97)

It however is not the case if the subevent granularity of the mixing is coarse compared with the correlation length of the dynamic texture. (Spikes which are narrower than pixel size can not be removed by mixing the pixels.) Because only very contrived ``dynamic textures'' have precise scale localization (a chessboard is an example), Eq.7.5 is merely an approximation in all practical cases. In this approximate sense one can say, based on Eq.7.4 and Eq.7.5, that the DWT dynamic texture of a composite sample is a weighted average of the respective dynamic textures of the component samples. This means that the non-observation of clustering texture signifies smallness of the clustering parameter or rarity of the phenomenon.