VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998

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Confronting Particle Emission Scenarios with Strangeness Data

Frédérique Grassi1,2 and Otavio Socolowski, Jr.3

1Instituto de Fı´sica, Universidade de São Paulo, C.P. 66318, 05315-970São Paulo-SP, Brazil
2Instituto de Fı´sica, Universidade Estadual de Campinas, C.P. 1170, 13083-970 Campinas-SP, Bra

3Instituto de Fı´sica Teórica, UNESP, Rua Pamplona 145, 01405-901 São Paulo-SP, Brazil
(Received 30 April 1997)

We show that a hadron gas model with continuous particle emission instead of freeze-out may s
some of the problems (high values of the freeze-out density and specific net charge) that one enco
in the latter case when studying strange particle ratios such as those from the experiment WA85.
underlines the necessity to understand better particle emission in hydrodynamics to be able to an
data. It also reopens the possibility of a quark-hadron transition occurring with phase equilibr
instead of explosively. [S0031-9007(97)05187-9]

PACS numbers: 25.75.–q, 12.38.Mh, 24.10.Nz, 24.10.Pa
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An enhancement of strangeness production in relativ
tic nuclear collisions (compared to, e.g., proton-proto
collisions at the same energy) is a possible signature
of the much sought-after quark-gluon plasma. It is ther
fore particularly interesting that current data at AGS (A
ternating Gradient Synchroton) and SPS (Super Prot
Synchrotron) energies do show an increase in strangen
production (see, e.g., [2]). At SPS energies, this increa
seems to imply that something new is happening: In m
croscopical models, one has to postulate some previou
unseen reaction mechanism (color rope formation in t
RQMD code [3], multiquark clusters in theVENUS code [4],
etc.) while hydrodynamical models have their own prob
lems (be it those with a rapidly hadronizing plasma [5] o
those with an equilibrated hadronic phase, preceded or
by a plasma phase). In this paper, we examine the sho
comings of the latter class of hydrodynamical models an
suggest that they might be due to a too rough descripti
of particle emission. (The main problem for the forme
class of hydrodynamical models is the difficulty to yield
enough entropy after hadronization.)

To be more precise, let us assume that a hadro
fireball (region filled with a hadron gas, or HG, in loca
thermal and chemical equilibrium) is formed in heavy io
collisions at SPS energies and that particles are emit
from it at freeze-out (i.e., when they stop interacting du
to matter dilution). One then runs into (at least) thre
kinds of problems when discussing strange particle ratio

First, the temperature (Tf. out , 200 MeV) and bary-
onic potential (mbf. out , few 100 MeV) needed at
freeze-out[6–10] to reproduce strangeness data of th
WA85 [11] and NA35 [12] experiments actually corre
spond tohigh particle densities: This is inconsistent with
the very notion of freeze-out.sssWhile WA85 and NA35
data for strange particle ratios are comparable and le
to high T ’s and mb ’s, NA36 data are different and lead
to lower T ’s [Phys. Lett. B327, 433 (1994)] for similar
targets but a somewhat different kinematic window
However, the rapidity distribution forL’s [E. G. Judd
et al., Nucl. Phys.A590, 291c (1995)] as well asL’s and
0031-9007y98y80(6)y1170(4)$15.00
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K0
s ’s [J. Eschkeet al.,Heavy Ion Phys.4, 105 (1996)] for

NA36 are quite below that of NA35; NA44 midrapidity
data forK6 agree with that of NA35.ddd

Second, to reproduce strange particle ratios, it turns
that the strange quark potentialms must be small and the
strangeness saturation factorgs of order 1 (this quantity,
with value usually between 0 and 1, measures how
from chemical equilibrium the strange particles are,
corresponds to full chemical equilibrium of the strang
particles). Both facts are expected in a quark-gluon plas
hadronizing suddenly, not normally in a hadronic fireba
[13,14].

Third, using the values at freeze-out of the temperatu
baryonic potential, and saturation factor extracted
reproduce WA85 strange particle ratios, one can pred
the value of another quantity, the specific net char
(ratio of the net charge multiplicity to the total charg
multiplicity). This quantity has been measured not b
WA85, but in experimental conditions similar to that o
WA85 by EMU05 [15]. It turns out that the predicte
value is too high (while it might be smaller if a quark
gluon plasma fireball had been formed) [5,16].

In what follows, we study how problems 1 and 3 a
related to the mechanism for particle emission norma
used, freeze-out, and suggest that the use of continu
emission instead of freeze-out might shed some light
these questions. (We also rediscuss problem 2.) T
underlines the necessity to understand better part
emission in hydrodynamics and reopens perspectives
conclusion) for scenarios of the quark-hadron transition

Fluid behavior and particle spectra.—First let us see
in more detail what the two particle emission mechanis
just mentioned are. In the usual freeze-out scena
hadrons are kept in thermal equilibrium until som
decoupling criterion has become satisfied (then they fr
stream toward the detectors). For example, in the pap
mentioned above where experimental strange part
ratios are reproduced, the freeze-out criterion is tha
certain temperature and baryonic potential have be
reached. The formula for the emitted particle spec
© 1998 The American Physical Society



VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998

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used normally is the Cooper-Frye formula [17]. In
the particular case of a gas expanding longitudinal
only in a boost invariant way, freezing out at som
fixed temperature and chemical potential, the Cooper-Fr
formula reads

dN
dyp'dp'

­
gR2

2p
tf. outm'

X̀
n­1

s7dn11 expsnmf. outyTf. outd

3 K1snm'yTf. outd . (1)
(The plus sign corresponds to bosons, and minus
fermions.) It depends only on the conditions at freez
out: Tf. out and mf. out ­ mbf. outB 1 mSf. outS, with B
and S the baryon number and strangeness of the hadr
species considered, andmSf. outsmbf. out, Tf. outd obtained
by imposing strangeness neutrality. So the experimen
spectra of particles teach us in that case only what t
conditions were at freeze-out.

In the continuous emission scenario developed
[18,19], the basic idea is the following: Because of th
finite dimensions and lifetime of the fluid, a particle a
space-time pointx has some chanceP to have already
made its last collision. In that case, it will fly freely
towards the detector, carrying with it memory of what th
conditions were in the fluid atx. Therefore the spectrum
of emitted particles contains an integral over all spac
and time, counting particles where they last interacte
In other words, the experimental spectra will give u
in principle information about the whole fluid history,
not just the freeze-out conditions. For the case of
fluid expanding longitudinally only in a boost invarian
way with continuous particle emission, the formula tha
parallels (1) is

dN
dyp'dp'

,
2g

s2pd2

Z
P ­0.5

df dh

3
m' coshhtFrdr 1 p' cosfrFt dt

expfsm' coshh 2 mdyT g 6 1
,

(2)
wheretFsr, f, h; y'd [rFst, f, h; y'd] is the time [ra-
dius] where the probability to escape without collisio
P ­ 0.5 is reached. P is given by a Glauber formula,
expf2

R
syrelnst0d dt0g, and depends in particular on lo-

cation and direction of motion. We are using both (1
and (2) in the following. Clearly, in (2), variousT and
m ­ mbB 1 mSS appear [againmSsmb , Td is obtained
from strangeness neutrality], reflecting the whole fluid hi
tory, not justTf. out andmbf. out.

So to predict particle spectra, in the case of continuo
emission, we need to know the fluid history. To ge
it, we fix some initial conditionsT st0, rd ­ T0 and
mbst0, rd ­ mb 0 and solve the equations of conservatio
of momentum energy and baryon number for a mixture
free and interacting particles, using the equation of sta
of a resonance gas (including the 207 known lowest ma
particles) and imposing strangeness neutrality. As a res
we getTst, rd, mbst, rd and we can use these as input i
the formula for the particle spectra (2). The procedu
ly
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ye

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re

is similar to that of a massless pion gas [18,19] but
numerically more involved.

An important result [18,19] for the following is tha
for heavy particles, the spectrum (2) is dominated by t
initial conditions, precisely a formula similar to (1) with
freeze-out quantities replaced by initial conditions cou
be used as an approximation (particularly at highp');
for light particles the whole fluid history matters. T
understand this fact, one can consider Eq. (2) and co
pare particles emitted atT st, rd ­ 200 and 100 MeV.
For particles with mass of 1 GeV, the exponential ter
gives a thermal suppression above 100 between these
temperatures. The multiplicative factors in front of th
exponential are in principle larger at the lower tempe
ture but do not compensate for such a big decrease. T
is why heavy particles are abundantly emitted at hi
temperatures. On the other side for pions, the therm
suppression is only a factor of 2. This is why ligh
particles are emitted significantly in a larger interval
temperatures.

Note that since heavy particle and highp' particle
spectra are sensitive mostly to the initial values ofT
and mb , the exact fluid expansion does not matter ve
much for them; in particular, the assumption of boo
invariance should play no part in the forthcoming analy
of strange highp' particle ratios. (Note also that the dat
considered below are in a small rapidity window, ne
midrapidty. Were it not for this fact, boost invarianc
should not be assumed, because the rapidity distributi
do not have this symmetry.) It would be, howeve
interesting to include continuous emission in, e.g.,
hydrodynamical code, to obtain the fluid evolution an
study pion data and lowp' data.

Particle ratios.—Once the spectra have been obtaine
they can be integrated to get particle numbers, taking i
account eventual experimental cutoffs and correcting
resonance decays. Since we had to specify the ini
conditions to solve the conservation equations and
this solution as input into (2), the particle numbers depe
on T0, mb 0. In contrast, for the freeze-out case, partic
numbers depend on the conditions at freeze-out,Tf. out
andmbf. out.

We look for regions in theT0, mb 0 space which permit
one to reproduce the latest WA85 experimental data
strange baryons [11] for2.3 , y , 2.8 and1.0 , p' ,

3.0 GeV: L̄yL ­ 0.20 6 0.01, J2yJ2 ­ 0.41 6 0.05,
and J2yL ­ 0.09 6 0.01 (J2yL̄ ­ 0.20 6 0.03 fol-
lows). In fact, there is no set of initial conditions whic
permits one to reproduce all the above ratios. A simi
situation occurs with freeze-out, as noted in [20].

In the comparison of our model with WA85 data w
have assumed, however, complete chemical equilibri
for strangeness production. As already mentioned in
introduction, this is not expected for a HG. In orde
to account for incomplete strangeness equilibration,
introduce the additional strangeness saturation param
gs by making the substitution expsmSSd ! gjSj

s expsmSSd
1171



VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998

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-

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,
-
.
-

e

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-

-

is
-

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-
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i-

-

d

g.,
in the (Boltzmann) distribution functions [21]. In our
case,a priori, gs depends on the space-time locationx;
however, since as already mentioned, the initial conditio
dominate in the shape and normalization of the spectra
heavy particles (particularly at highm'), we take

dN
dyp'dp'

, gjSj
s st0d

dNeq

dyp'dp'

, (3)

with dNeqydyp'dp' given by (2). In Fig. 1(a), we see
that for gsst0d ­ 0.58, there exists an overlap region
in the T0, mb 0 plane where all the above ratios are
reproduced. For the freeze-out case, a similar situati
occurs as noted in [20], namely, there exists an overl
region forgs ­ 0.7.

In the freeze-out case, the values of the parameters
the overlap region correspond to high particle densitie
and so it is hard to understand how particles hav
ceased to interact: this is the problem 1 mentioned in t
introduction. In the continuous emission case,T0 andmb 0

in the overlap region lead to high initial densities, but th
is, of course, quite reasonable since these are values w
the HG started its hydrodynamical expansion.

FIG. 1. Overlap region in theT0-mb 0 plane for WA85 data,
with ksyrell ­ 1 fm2 (a) without (b) with hadronic volume
corrections.
1172
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hen

The aim of Fig. 1(a) is to allow an easy comparison
with freeze-out results such as [20]; however, it is no
physically complete: so far we have neglected hadron
volume corrections. For freeze-out, this correction can
cels between numerator and denominator in baryon r
tios so it can be ignored [10] but for continuous emission
since we are considering the whole fluid history to get par
ticle numbers (and then their ratio), it must be included
There are various ways to do this (e.g., [10,22–24]). Us
ing the more consistent method of [25,26], we get th
overlap region shown in Fig. 1(b), which is shifted to-
wards smallerT ’s and mb ’s but not very different from
that of Fig. 1(a). Given that simulations of QCD on a
lattice indicate a quark-hadron transition for temperature
around 200 MeV, it seems more reasonable to consid
initial conditions T0 , 190 MeV and mb 0 , 180 MeV,
i.e., the bottom part of the overlap region. Thepreciselo-
cation of the overlap region (and exact value ofgs) is sen-
sitive to changes in the equation of state—as we have ju
seen—as well as in the cross section or cutoffP ­ 0.5
in Eq. (2). Therefore, problem 1 (whether the overlap re
gion is physically reasonable) is taken care of.

To be complete, we also examined the more re
cent ratios obtained by WA85 [27] (at midrapid-
ity): V2yV

2
m'$2.3 GeV ­ 0.57 6 0.41, sV2 1 V2dy

sJ2 1 J2dm'$2.3 GeV ­ 1.7 6 0.9, K0
s yLp'.1.0 GeV ­

1.43 6 0.10, K0
s yL̄p'.1.0 GeV ­ 6.45 6 0.61, and

K1yK2
p'.0.9 GeV ­ 1.67 6 0.15. We looked for a region

in the T0, mb 0 plane whereV2yV
2
m'$2.3 GeV is repro-

duced: Because of the large experimental error bars, th
does not bring new restrictions to Fig. 1(b). We also cal
culated our value forsV2 1 V2dysJ2 1 J2dm'$2.3 GeV
in the overlapping region and found,0.7, in marginal
agreement with the above experimental values. Th
three ratios involving kaons depend on more than jus
initial conditions (kaons are intermediate in mass be
tween pions and lambdas, so part of the fluid therma
history must be reflected in their spectra), in particula
gssxd , gsst0d , cst may not be a good approximation,
and we are still working on this. The above experimenta
ratios concern SW collisions, data with SS are not s
extensive yet but not very different [28] so a similar
overlapping region can be found.

The apparent temperature extracted from the exper
mentalp' spectra forL, L, J2, andJ2 is ,230 MeV
[11]. Given that we extracted from ratios of these par
ticles, temperaturesT0 $ 190 MeV, we conclude that
heavy particles exhibit little transverse flow, which is
compatible with the fact that they are emitted early during
the hydrodynamical expansion.

Specific net charge.—We now turn to

Dq ­ sN1 2 N2dysN1 1 N2d (4)

using the continuous emission scenario. As mentione
in the introduction, for HG models with freeze-out the
predicted Dq is too high, when using values of the
freeze-out parameters that fit strangeness data, e.



VOLUME 80, NUMBER 6 P H Y S I C A L R E V I E W L E T T E R S 9 FEBRUARY 1998

n
al
.

s

,

d
f

Tf. out , 200 MeV, mbf. out , 200 MeV, and gs , 0.7.
For continuous emission, due to thermal suppressio
particlesheavier than the pionare approximately emitted
at T0 , 200 MeV, mb 0 , 200 MeV, and gs , 0.49
[Fig. 1(b)], soDq so far is similar to that of freeze-out.
However, there is an additional source of particles th
enters the denominator of (4), namely pions are emitted
T0 and then on(since they are not thermally suppressed
So we expect to get a lower value forDq in the continuous
emission case than in the freeze-out case. (We recall t
pions are the dominant contribution inN1 1 N2.) This
would go into the direction of solving problem 3; it is stil
under investigation.

Conclusion.—Our present description is simplified
For example, we do not include the transverse expans
of the fluid, use similar interaction cross sections fo
all types of particles, etc. In addition, we need to loo
systematically at strangeness data from other collabo
tions as well as other types of data such as Bose-Einst
correlations. Nevertheless, we have seen that the c
tinuous emission scenario with a HG may shed light o
problems 1 and 3 (discussed in the introduction) that
freeze-out model with a HG encounters. Namely, in th
overlap region of the parameters needed to reprodu
WA85 data, the density of particles is high, and th
is consistent with the emission mechanism, since it
the initial density of the thermalized fluid. We also
expect Dq to be smaller for continuous emission tha
freeze-out. But (problem 2) the value of the strangene
saturation parameter may be high for a HG, partic
larly at the beginning of its hydrodynamical expansion
However, what we really need to get Fig. 1(b), is th
J2yL ­ gJJ2ygLLjeq. and J2yL ­ gJJ2ygLLjeq
with gJygL ­ 0.49. We expect indeed that multistrang
J2 and J2 are more far off chemical equilibrium than
singlestrangeL and L so thatgJygL , 1. The result
gs ­ 0.49 arises if one makes theadditional hypothe-
sis that quarks are independent degrees of freedo
inside the hadrons so that one has factorizations of
type gL expsmLyT d ­ gs exp2smqyT d expsmsyTd, and
gJ expsmJyTd ­ g2

s expsmqyT d exps2msyTd. Therefore
problem 2 may not be so serious.

The fact that we may cure some of the problems of t
HG scenario does not mean that no quark-gluon plas
has been created before the HG, in fact it may open n
possibilities for scenarios of the quark-hadron transitio
(e.g., an equilibrated quark-gluon plasma evolving into
equilibrated HG with continuous emission); in particula
it may not be necessary to assume an explosive transit
[5] or a deflagration-detonation scenario [29–31].

But our main conclusion is that the emission mechanis
may modify profoundly our interpretation of data. (Fo
example, does the slope in transverse mass spectrum
something about freeze-out or initial conditions?) In tur
this modifies our discussion of what potential problem
(such as 1 and 3) are emerging. Therefore we believe i
necessary to devote more work to get a realistic descript
n,

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of particle emission in hydrodynamics, [18,19] being a
first step in that direction. We remind the reader that the
idea that particles are emitted continuously and not o
a sharp freeze-out surface is supported by microscopic
simulations at AGS energies [32] and SPS energies [33]

The authors wish to thank U. Ornik for providing
some of the computer programs to start working on thi
problem. This work was partially supported by FAPESP
(Proc. No. 95/4635-0), CNPq (Proc. No. 300054/92-0)
and CAPES.

Note added.—After completing this paper, we learned
that G. D. Yen, M. I. Gorenstein, W. Greiner and S. N.
Yang suggested [34] another solution to problems 1 an
3 above, in terms of the excluded volume approach o
[25], for the preliminary Au1 Au (AGS) and Pb1 Pb
(SPS) data.

[1] P. Koch, B. Müller, and J. Rafelski, Phys. Rep.142, 167
(1986).

[2] Proceedings of Quark Matter ’96[Nucl. Phys. A610,
(1996)], as well as proceedings from previous years.

[3] H. Sorge, M. Berenguer, H. Stöcker, and W. Greiner,
Phys. Lett. B289, 6 (1992).

[4] J. Aichelin and K. Werner, Phys. Lett. B300, 158 (1993).
[5] J. Letessieret al., Phys. Rev. D51, 3408 (1995).
[6] N. J. Davidsonet al., Phys. Lett. B255, 105 (1991).
[7] N. J. Davidsonet al., Phys. Lett. B256, 554 (1991).
[8] D. W. von Oertzenet al., Phys. Lett. B274, 128 (1992).
[9] N. J. Davidsonet al., Z. Phys. C56, 319 (1992).

[10] J. Cleymans and H. Satz, Z. Phys. C57, 135 (1993).
[11] S. Abatziset al., Nucl. Phys.A566, 225c (1994).
[12] J. Bartkeet al., Z. Phys. C48, 191 (1990).
[13] J. Letessier, A. Tounsi, and J. Rafelski, Phys. Lett. B292,

417 (1992).
[14] J. Sollfranket al., Z. Phys. C61, 659 (1994).
[15] Y. Takahashiet al. (to be published), quoted in [5] and

[16].
[16] J. Letessieret al., Phys. Rev. Lett.70, 3530 (1993).
[17] F. Cooper and G. Frye, Phys. Rev. D10, 186 (1974).
[18] F. Grassi, Y. Hama, and T. Kodama, Phys. Lett. B355, 9

(1995).
[19] F. Grassi, Y. Hama, and T. Kodama, Z. Phys. C73, 153

(1996).
[20] K. Redlichet al., Nucl. Phys.A566, 391c (1994).
[21] J. Rafelski, Phys. Lett. B262, 333 (1991).
[22] J. Cleymanset al., Z. Phys. C33, 151 (1986).
[23] J. Cleymanset al., Phys. Lett. B242, 111 (1990).
[24] J. Cleymanset al., Z. Phys. C55, 317 (1992).
[25] D. H. Rischkeet al., Z. Phys. C51, 485 (1991).
[26] J. Cleymanset al., Phys. Scr.48, 277 (1993).
[27] S. Abatziset al., Phys. Lett. B316, 615 (1993);347, 158

(1995);376, 251 (1996);355, 401 (1995).
[28] S. Abatziset al., Phys. Lett. B354, 178 (1995).
[29] N. Bilić et al., Phys. Lett. B311, 266 (1993).
[30] N. Bilić et al., Z. Phys. C63, 525 (1994).
[31] T. Csörgő and L. P. Csernai, Phys. Lett. B333, 494

(1994).
[32] L. V. Bravinhaet al., Phys. Lett. B354, 196 (1995) .
[33] H. Sorge, Phys. Lett. B373, 16 (1996).
[34] G. D. Yenet al., Phys. Rev. C56, 2210 (1997).
1173