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.Comment: 9 pages, 2 figure

Adcox

Adler

Afanasev

Anderlik

Appelshäuser

Bertsch

Bjorken

Bugaev

Csernai

Csörgő

Gorenstein

Gyulassy

Heinz

I. Lovas

Jenkovszky

Keranen

L.L. Jenkovszky

L.P. Csernai

M.I. Gorenstein

Magas

Pratt

Rischke

Shuryak

Sinyukov

Stock

Stöcker

Taub

Teaney

V.K. Magas

Weiner

Physics Letters B

English

arXiv

Crossref

AbstractPossible 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 (Rout/Rside≈1). HBT data also show that the expansion time before freeze out is very short (∼6–10 fm/c). In this Letter we discuss the question of supercooled QGP and the timescale of the reaction

Csernai, L.P.

Gorenstein, M.I.

Jenkovszky, L.L.

Lovas, I.

Magas, V.K.

Elsevier - Publisher Connector 

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 (Rout/Rside≈1). HBT data also show that the expansion time before freeze out is very short (∼6–10 fm/c). In this Letter we discuss the question of supercooled QGP and the timescale of the reaction

Csernai, László Pal

Gorenstein, Mark I.

Jenkovszky, László

Lovas, István

Magas, Volodymyr K.

Hochschulschriftenserver - Universität Frankfurt am Main

Can supercooling explain the HBT puzzle?

Physics Letters B 551 (2003) 121–126
www.elsevier.com/locate/npe
Can supercooling explain the HBT puzzle?
L.P. Csernai a,b, M.I. Gorenstein c,d, L.L. Jenkovszky c, I. Lovas e, V.K. Magas c,f
a Section for Theoretical and Computational Physics, Department of Physics, University of Bergen,
Allegaten 55, N-5007 Bergen, Norway
b KFKI Research Institute for Particle and Nuclear Physics, P.O. Box 49, 1525 Budapest, Hungary
c Bogoliubov Inst. for Theor. Phys., Ac. of Sciences of Ukraine, Metrologichna str. 14-b, 03143 Kiev, Ukraine
d University of Frankfurt, Robert-Mayer St. 8-10, D-60054 Frankfurt am Main, Germany
e University of Debrecen, Egytem-ter 1-3, H-4010 Debrecen, Hungary
f Center for Physics of Fundamental Interactions (CFIF), Instituto Superior Tecnico, Av. Rovisco Pais, 1049-001 Lisbon, Portugal
Received 23 October 2002; accepted 11 November 2002
Editor: P.V. Landshoff
Abstract
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 (Rout/Rside ≈ 1).
HBT data also show that the expansion time before freeze out is very short (∼ 6–10 fm/c). In this Letter we discuss the question
of supercooled QGP and the timescale of the reaction.
 2002 Elsevier Science B.V.
PACS: 24.10.Pa; 24.10.Nz; 25.75.Dw; 25.75.Ld
Keywords: Ultra-relativistic heavy ion collisions; RHIC; HBT; Supercooled QGP; Hadronization
1. 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 Pb +
Pb 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
E-mail addresses: csernai@fi.uib.no (L.P. Csernai),
goren@th.physik.umi-frankfurt.de (M.I. Gorenstein),
jenk@gluk.org (L.L. Jenkovszky), lovas@dtp.atomki.hu (I. Lovas),
vladimir@cfif.ist.utl.pt (V.K. Magas).
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
0370-2693/02  2002 Elsevier Science B.V.
PII: S0370-2693(02)0 30 03 -4
Open access under CC BY license.
Open access under CC BY license.
122 L.P. Csernai et al. / Physics Letters B 551 (2003) 121–126
expansion would lead to long lifetimes of the particle
source [2,6,7],1 which would manifest itself as a large
Rout/Rside ratio. Now this type of hadronization is
excluded by experimental data.
An alternative possibility, discussed in Refs. [9–13],
is the hadronization from the supercooled QGP
(sQGP). This is expected to be a very fast shock-
like process. If the hadronization from sQGP 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 (Rout/Rside ≈ 1). In this
work we are asking the following question—can the
hadronization from sQGP explain also the another part
of the HBT puzzle, i.e., a very short (∼ 6 [14]–10
[5,15] fm/c) expansion time before freeze out?
2. Shock hadronization of the sQGP
Relativistic shock phenomena were widely dis-
cussed with respect to their connection to high-energy
heavy ion collisions (see, for example, [16]). In ther-
mal 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 rarefac-
tion shocks. The system evolution in relativistic hydro-
dynamics is governed by the energy–momentum ten-
sor T µν = (+p)uµuν −pgµν and conserved charge
currents (in our applications to heavy ion collisions we
consider only the baryonic current nuµ). They consist
of local thermodynamical fluid quantities (the energy
density , pressure p, baryonic density n) and the col-
lective four-velocity uµ =√1− v2 (1,v). Continuous
flows are the solutions of the hydrodynamical equa-
tions:
(1)∂µT µν = 0, ∂µnuµ = 0,
with specified initial and boundary conditions. These
equations are nothing more than the differential form
of the energy–momentum and baryonic number con-
servation laws. Along with these continuous flows, the
conservation laws can also be realized in the form of
1 If we use some microscopic model for hadronization, for ex-
ample, nucleation of relativistic first-order phase transition [8], the
lifetime is even longer—it was estimated to be about 50–100 fm/c
in Ref. [8].
discontinuous hydrodynamical flows which are called
shock waves and satisfy the following equations:
(2)T µν0 dσν = T µνdσν, n0uµ0 dσµ = nuµdσµ,
where dσµ is the unit 4-vector normal to the discon-
tinuity hypersurface. In Eq. (2) the zero index corre-
sponds 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 time-like normal vectors
dσµ) was given in Ref. [17].
The important constraint on the transitions (2)
(thermodynamical stability condition) is the require-
ment of non-decreasing entropy (s is the entropy den-
sity):
(3)suµdσµ  s0uµ0 dσµ.
To simplify our consideration and make our ar-
guments 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 al-
ways choose the Lorentz frame where the shock front
is at rest. Then dσµ = (0,1) at the surface of shock
discontinuity, and Eq. (2) in this (standard) case be-
comes:
(4)T 010 = T 01, T 110 = T 11, n0u10 = nu1.
Solving Eq. (4) one obtains
v20 =
(p− p0)( + p0)
( − 0)(0 + p) ,
(5)v2 = (p− p0)(0 + p)
( − 0)( + p0) ,
and the well-known Taub adiabat (TA) [18]
(6)n2X2 − n20X20 − (p− p0)(X+X0)= 0,
where X ≡ ( + p)/n2.2
For discontinuities on a hypersurface with a time-
like (t.l.) normal vector dσµ (we call them t.l. shocks)
one can always choose another convenient Lorentz
frame (“simultaneous system”) where dσµ = (1,0).
2 It has been shown in a series of works [19], that freeze out
through the space-like hypersurface leads to non-equilibrium post
FO distribution.
L.P. Csernai et al. / Physics Letters B 551 (2003) 121–126 123
Fig. 1. Possible final states in the (energy density–pressure)-plane
for shock transitions from the initial state (0,p0). I and IV are the
physical regions for s.l. shocks, III and VI for t.l. shocks. II and
V are unphysical regions for both types of shocks. Note, that only
states with p   are possible for any physical equation of state in
the relativistic theory.
Eq. (2) is then
(7)T 000 = T 00, T 100 = T 10, n0u00 = nu0.
Solving Eq. (7) we find
v˜20 =
( − 0)(0 + p)
(p− p0)( +p0) ,
(8)v˜2 = ( − 0)( + p0)
(p− p0)(0 + p),
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]:
(9)v˜20 =
1
v20
, v˜2 = 1
v2
.
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 v20 , v2 (5) and for v˜20, v˜2
(8) can be easily found in (–p)-plane [11]. For a
given initial state (0,p0) they are shown in Fig. 1. For
supercooled initial QGP states the TA no longer passes
through the point (0,p0) and new possibilities of t.l.
shock hadronization transitions to regions III and VI
in Fig. 1 appear.
3. Hadronization of the sQGP within Bjorken
hydrodynamics
For a study of the expanding QGP we have chosen a
framework of the one-dimensional Bjorken model [20]
(actually our principal results will not change if we use
3D Bjorken model). Within the Bjorken model all the
thermodynamical quantities are constant along con-
stant proper time curves, τ = √t2 − z2 = const. The
important result of Bjorken hydrodynamics (which as-
sumes a perfect fluid) is that the evolution of the en-
tropy density, is independent of the Equation of State
(EoS), namely,
(10)s(τ )= s(τinit)τinit
τ
.
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, dσν = (1,0) in the rest frames of each fluid
element. This leads to the simple solution of the t.l.
shock equations (7):
v˜2 = v˜20 = 0,  = 0,
(11)n= n0, p = p0.
The entropy condition (3) is reduced to
(12)s  s0.
Now let us try to answer the main question of this
Letter—can QGP expansion with t.l. shock hadroniza-
tion of supercooled state be faster than the hadroniza-
tion through the mixed phase? The initial state is
given at the proper time τinit ≡ τQ, when the lo-
cal thermal equilibrium is achieved in the QGP state
Q≡ (Q,pQ, sQ). The final equilibrium hadron state
is also fixed, by experiment or otherwise, as H ≡
(H ,pH , sH ). For the continuous expansion given by
Eq. (10) the proper time for the Q→ H transition is
(see Fig. 2(a)):
(13)τH = sQτQ
sH
.
If our system enters the sQGP phase and the par-
ticle emission is flash-like, i.e., the system hadronizes
124 L.P. Csernai et al. / Physics Letters B 551 (2003) 121–126
(a) (b)
(c)
Fig. 2. Different ways for a system to go from Q state (sQ) to H state (sH ) are presented on {s, τ } plane. (a) Shows continuous expansion,
which takes time τH , Eq. (13). (b) Presents flash-like particle emission, i.e., simultaneous hadronization and freeze out; which takes time τ (1)H ,
Eq. (14). (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 τH .
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(a)). Our system should go into supercooled
phase to the point where (1)0 = H ,n(1)0 = nH , as it is
required by Eq. (11). At this point our sQGP has en-
tropy density s(1)0 . It is value depend on the EoS, but t.l.
shock transition is only possible if s(1)0  sH according
to Eq. (12). Thus, for the proper time of Q→H tran-
sition according to first scenario we have:
(14)τ (1)H =
sQτQ
s
(1)
0
 τH .
We can also study a scenario number two when our
system supercools to the state ((2)0 ,p
(2)
0 , s
(2)
0 ), then
hadronizes to a superheated HM state ((2), p(2), s(2)),
and then this HM state expands to the same freeze out
state H ≡ (H ,pH , sH ). (see Fig. 2(c)). At the point
of the shock transition one has:
(15)τ (2)0 =
sQτQ
s
(2)
0
.
Then we have a t.l. shock transition satisfying Eq. (11),
and following the HM branch of the hydrodynamical
L.P. Csernai et al. / Physics Letters B 551 (2003) 121–126 125
expansion we find:
(16)τ (2)H =
s(2)τ (2)0
sH
= sQτQ
sH
s(2)
s
(2)
0
 τH ,
since s(2)  s(2)0 due to non-decreasing entropy condi-
tion (12). In this second scenario the value of the of
entropy density s(2)0 of sQGP can be both smaller and
larger than HM final value sH . Depending on details of
the EoS the proper time τ (2)H (16) of the Q→H tran-
sition can also be smaller as well as larger than τ (1)H
(14), but always larger than τH .
4. 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
cannot 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 s0  sH (for the
flash-like particle emission).
The system may, nevertheless, freeze out and
hadronize into a non-equilibrated hadron gas well be-
fore τH . This is possible, e.g., through a dominantly
s.l. hypersurface with non-decreasing entropy condi-
tion, 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
Rout/Rside ratio.
The construction of a full reaction model, which si-
multaneously describes data on two particle interfer-
ometry, hadron spectra and hadron abundances is a
formidable task which is still ahead of us.
Acknowledgements
We have benefitted from discussions with I. Mishus-
tin, L. Satarov and Yu.M. Sinyukov. The financial sup-
port from the Humboldt Foundation and INTAS grant
00-00366 are acknowledged.
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Can supercooling explain the HBT puzzle?

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