Can supercooling explain the HBT puzzle?

Possible hadronization of supercooled QGP, created in heavy ion collisions at RHIC and SPS, is discussed within a Bjorken hydrodynamic model. Such a hadronization is expected to be a very fast shock-like process, what, if hadronization coincides or shortly followed by freeze out, could explain a part of the HBT puzzle, i.e. the flash-like particle emission ($R_{out}/R_{side}\approx 1$). HBT data also show that the expansion time before freeze out is very short ($\sim 6-10 fm/c$). In this work we discuss question of supercooled QGP and the timescale of the reaction.


Introduction
Two-particle interferometry has become a powerful tool for studying the size and duration of particle production from elementary collisions (e + e − , pp and pp) to heavy ions like Au + Au at RHIC or P b + P b at SPS [1,2]. For the case of nuclear collisions, the interest mainly focuses on the possible transient formation of a deconfined state of matter. This could affect the size of the region from where the hadrons (mostly pions) are emitted as well as the time for particle production.
Comparing recent data [3] from RHIC with SPS data one finds a "puzzle" [4]: all the HBT radii are pretty similar although the center of mass energy is changed by an order of magnitude. Discussions at "Quark Matter 2002" [5] lead to the conclusion that the duration of particle emission, as well as the lifetime of the system before freeze out, appear to be shorter than the predictions of most of the model at the physics market.
It was demonstrated that a strong first-order QCD phase transition within continuous hydrodynamical expansion would lead to long lifetimes of the particle source [2,6,7] 1 , which would manifest itself as a large R out /R side ratio. Now this type of hadronization is excluded by experimental data.
An alternative possibility, discussed in Refs. [9,10,11,12,13], is the hadronization from the supercooled QGP. This is expected to be a very fast shock-like process. If the hadronization from supercooled QGP coincides with freeze out, like it was assumed in Ref. [12], then this could explain a part of the HBT puzzle, i.e. the flash-like particle emission (R out /R side ≈ 1). In this work we are asking the following question -can the hadronization from supercooled QGP explain also the another part of the HBT puzzle, i.e. a very short (∼ 6 [14] − 10 [5,15] f m/c) expansion time before freeze out?

Shock hadronization of a sQGP
Relativistic shock phenomena were widely discussed with respect to their connection to high-energy heavy ion collisions (see, for example, [16]). In thermal equilibrium by admitting the existence of the sQGP and the superheated hadronic matter (HM) we have essentially richer picture of discontinuity-like transitions than in standard compression and rarefaction shocks. The system 1 If we use some microscopic model for hadronization, for example nucleation of relativistic first-order phase transition [8], the lifetime is even longer -it was estimated to be about 50 − 100 f m/c in Ref. [8]. evolution in relativistic hydrodynamics is governed by the energy-momentum tensor T µν = (ǫ+ p)u µ u ν −pg µν and conserved charge currents (in our applications to heavy ion collisions we consider only the baryonic current nu µ ). They consist of local thermodynamical fluid quantities (the energy density ǫ, pressure p, baryonic density n) and the collective four-velocity u µ = √ 1 − v 2 (1, v). Continuous flows are the solutions of the hydrodynamical equations: with specified initial and boundary conditions. These equations are nothing more than the differential form of the energy-momentum and baryonic number conservation laws. Along with these continuous flows, the conservation laws can also be realized in the form of discontinuous hydrodynamical flows which are called shock waves and satisfy the following equations: where dσ µ is the unit 4-vector normal to the discontinuity hypersurface. In eq. (2) the zero index corresponds to the initial state ahead of the shock front and quantities without an index are the final state values behind it. A general derivation of the shock equations (valid for both space-like and timelike normal vectors dσ µ ) was given in Ref. [17].
The important constraint on the transitions (2) (thermodynamical stability condition) is the requirement of non-decreasing entropy (s is the entropy density): To simplify our consideration and make our arguments more transparent we consider only one-dimensional hydrodynamical motion. To study the shock transitions at the surface with space-like (s.l.) normal vector (we call them s.l. shocks) one can always choose the Lorentz frame where the shock front is at rest. Then dσ µ = (0, 1) at the surface of shock discontinuity, and eq. (2) in this (standard) case becomes: Solving eq. (4) one obtains where For discontinuities on a hypersurface with a time-like (t.l.) normal vector dσ µ (we call them t.l. shocks) one can always choose another convenient Lorentz frame ("simultaneous system") where dσ µ = (1, 0). Equation (2) is then Solving eq. (7) we find where we use the " ∼ " sign to distinguish the t.l. shock case (8) from the standard s.l. shocks of (5). Another relation contains only the thermodynamical variables. It appears to be identical to the TA of eq. (6). Eqs. (8) and (5) are connected to each other by simple relations [11]: These relations show that only one kind of transition can be realized for a given initial state and final state. The physical regions [0, 1) for v 2 o , v 2 (5) and forṽ 2 o ,ṽ 2 (8) can be easily found in (ǫ-p)-plane [11]. For a given initial state (ǫ o , p o ) they are shown in Fig. 1. For supercooled initial QGP states the TA no longer passes through the point (ǫ o , p o ) and new possibilities of t.l. shock hadronization transitions to regions III and VI in Fig. 1 appear.

Hadronization of the sQGP within Bjorken hydrodynamics
In Bjorken model the natural choice of the freeze out hypersurface is τ = const hypersurface, where normal vector is parallel to the Bjorken flow velocity, v = z/t. Thus, dσ ν = (1, 0) in the rest frames of each fluid element. This leads to the simple solution of the t.l. shock equations (7): The entropy condition (3) is reduced to  H , eq. (14). Subplot C shows several possibilities according to scenario 2 with shock-like hadronization into superheated HM. Time τ (2) H (16) can be smaller or larger than τ (1) H , depending on details of the EoS, but always larger than τ H . Now let us try to answer the main question of this work -can QGP expansion with t.l. shock hadronization of supercooled state be faster than the hadronization through the mixed phase? The initial state is given at the proper time τ init ≡ τ Q , when the local thermal equilibrium is achieved in the QGP state Q ≡ (ǫ Q , p Q , s Q ). The final equilibrium hadron state is also fixed, by experiment or otherwise, as H ≡ (ǫ H , p H , s H ). For the continuous expansion given by eq. (10) the proper time for the Q → H transition is (see Fig. 2

-subplot A)
: If our system enters the sQGP phase and the particle emission is flash-like, i.e. the system hadronizes and freezes out at the same time, then eq. (10) is also valid all the time with final t.l. shock transition to the same H state. We call this as a scenario number one (see Fig. 2 -subplot A). Our system should go into supercooled phase to the point where ǫ (1) o = ǫ H , n (1) o = n H , as it is required by eq. (11). At this point our sQGP has entropy density s (1) o . It's value depend on the EoS, but t.l. shock transition is only possible if s (1) o ≤ s H according to eq. (12). Thus, for the proper time of Q → H transition according to first scenario we have: We can also study a scenario number two when our system supercools to the state (ǫ (2) o , p (2) o , s (2) o ), then hadronizes to a superheated HM state (ǫ (2) , p (2) , s (2) ), and then this HM state expands to the same freeze out state H ≡ (ǫ H , p H , s H ). (see Fig. 2 -subplot C). At the point of the shock transition one has: Then we have a t.l. shock transition satisfying eq. (11), and following the HM branch of the hydrodynamical expansion we find: since s (2) ≥ s (2) o due to non-decreasing entropy condition (12). In this second scenario the value of the of entropy density s (2) o of sQGP can be both smaller and larger than HM final value s H . Depending on details of the EoS the proper time τ

Conclusions
The conclusion of our analysis seems to be a rather general one: the system's evolution through a supercooled phase and time-like shock hadronization can not be shorter than a continuous expansion within the perfect fluid hydrodynamics independently of the details of EoS and the parameter values of the initial, Q, and final, H, states. Although in such a way we may achieve flash-like particle emission, supported by the HBT data, the expansion time becomes longer, making it harder to reproduce the experimental HBT radii.
So, how can we achieve shorter freeze out time than the minimal one coming from (very fast) Bjorken expansion via thermal and phase equilibrium? Any delay in the phase equilibration (see assignment 9 in Ref. [21]) or/and any dissipative process in our system lead to the entropy production, what increases the time needed to reduce entropy density to s o ≤ s H (for the flash-like particle emission).
The system may, nevertheless, freeze out and hadronize into a non-equilibrated hadron gas well before τ H . This is possible e.g. through a dominantly s.l. hypersurface with non-decreasing entropy condition, eq. (12) [19], at earlier times from a slightly supercooled QGP. On the other hand a dominantly s.l. hypersurface gives a finite duration of the particle emission, making it harder to reproduce experimental R out /R side ratio.
The construction of a full reaction model, which simultaneously describes data on two particle interferometry, hadron spectra and hadron abundances is a formidable task which is still ahead of us.