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Discussion: Second Order Phase Transition

The specific experimental signature of second order phase transition (known since the discovery of critical opalescence [18]) is the emergence of critical fluctuations of the order parameter with an enormous increase of the correlation lengths. Scalapino and Sugar formulated a statistical theory of multiple particle production in hadron-hadron collisions[14], based on the phenomenological free energy functional of the type used by Ginzburg and Landau [96] to describe the second order phase transition in superconductors. This theory predicts particle emission in clusters, coherently over large rapidity range.

In an SU(2) theory with massless $u$ and $d$ and infinitely massive $s$ quark, the phase transition is of second order [100,101]. SU(3) with three massless quarks results in a fluctuation driven[101] first order [100,101] phase transition. There are some indications[106], based on lattice QCD work, that for a finite $s$-quark mass, the phase transition is of second order. Moreover, for a certain class of lattice gauge theories, a tricritical point is expected to exist for a particular coupling strength, separating the first order confinement phase transition from a second order transition with the same critical exponents as the 3D Ising model[100]. In the massless SU(2) case, critical local fluctuations in the order parameter (which in this case can be parametrized as a four-component $(\sigma,\vec{\pi})$ field) should result in detectable observables: local fluctuations of isospin and enhanced correlation lengths [104]. Possibly, the Centauro event[2] represents just such an occurrence. However, for physical quark masses Rajagopal and Wilczek [104,105] argued that due to closeness of the pion mass to the critical temperature, it would be unlikely for the correlation volumes to include large numbers of pions, if the cooling of the plasma and hadronization proceeds in an equilibrated manner. If, on the contrary, the high temperature configuration suddenly finds itself at a low temperature, a self-organized criticality regime settles in, and the critical local fluctuations develop fully[104,105].

How can the NA44 experimental data just presented clarify this complex and uncertain picture ? The data signifies absence of dynamical fluctuations on the scales probed (which are the relevant scales), within the limit of sensitivity discussed in Section 7.3. Convincing evidence of thermal equilibration can be provided best by event-by-event observables, due to the very nature of the problem. Our data is consistent with local thermal equilibrium, understood as an absence of physically distinguished scales between the scale of a hadron and the scale of the system, or scale invariance of fluctuations [107] (``white noise''). However to probe equilibration directly with this method, texture sensitivity at least down to the typical fireball (cluster) sizes observed in $pN$ collisions in cosmic rays and accelerator experiments [5,7] is necessary, but lacking. In the absence of such direct evidence, the non-observation of critical fluctuations can imply either absence of the second order phase transition or thermal equilibration - the latter voids the criticality signature, according to Rajagopal and Wilczek [104].