HU TFT
THE ROLE OF STRANGENESS IN
a
ULTRARELATIVISTIC NUCLEAR COLLISIONS
Josef Sollfrank
Research Institute for Theoretical Physics P O Box
FIN University of Helsinki Finland
and
Ulrich Heinz
Institut f ur Theoretische Physik Universit at Regensburg
D Regensburg Germany
ABSTRACT
We review the progress in understanding the strange particle yields in nuclear colli
sions and their role in signalling quark gluon plasma formation We rep ort on new
insights into the formation mechanisms of strange particles during ultrarelativistic
heavy ion collisions and discuss interesting new details of the strangeness phase di
agram In the main part of the review we show how the measured multi strange
particle abundances can b e used as a testing ground for chemical equilibration in
nuclear collisions and how the results of such an analysis lead to imp ortant con
straints on the collision dynamics and space time evolution of high energy heavy ion
reactions
a
To b e published in Quark Gluon Plasma R C Hwa Eds World Scienti c
Contents
Introduction
Strangeness Pro duction Mechanisms
Quark Gluon Pro duction Mechanisms
Hadronic Pro duction Mechanisms
Thermal Mo dels
Thermal Parameters
Relative and Absolute Chemical Equilibrium
The Partition Function
The Phase Diagram of Strange Matter
The Strange Matter Iglo o
Isentropic Expansion Trajectories
The T ! Limit of the Phase Diagram
Hadronization of Strange Quark Matter at T 6
Ingredients and Extensions of the Thermal Mo del
The Strangeness Neutrality Condition
Transverse Flow and Transverse Momentum Sp ectra
Resonance Decays
Particle Ratios
The Sp eci c Entropy
Chemical Parameters from Strange Particle Ratios
Thermal and Chemical Parameters from the WA Exp eriment
Thermal and Chemical Parameters from the NA Exp eriment
Thermal and Chemical Parameters from the NA Exp eriment
Thermal and Chemical Parameters from the AGS Exp eriments
Sp eci c Entropy from Heavy Ion Exp eriments
Discussion of the Results
Consequences for Freeze Out Mo dels
Sequential Freeze out
Sudden Hadronization Scenarios
Conclusions and Outlo ok
A App endix Literature Guide to Strange Particle Measurements
References
THE ROLE OF STRANGENESS IN
ULTRARELATIVISTIC NUCLEAR COLLISIONS
Josef Sollfrank
Research Institute for Theoretical Physics P O Box
FIN University of Helsinki Finland
and
Ulrich Heinz
Institut f ur Theoretische Physik Universit at Regensburg
D Regensburg Germany
ABSTRACT
We review the progress in understanding the strange particle yields in nuclear colli
sions and their role in signalling quark gluon plasma formation We rep ort on new
insights into the formation mechanisms of strange particles during ultrarelativistic
heavy ion collisions and discuss interesting new details of the strangeness phase di
agram In the main part of the review we show how the measured multi strange
particle abundances can b e used as a testing ground for chemical equilibration in
nuclear collisions and how the results of such an analysis lead to imp ortant con
straints on the collision dynamics and space time evolution of high energy heavy ion
reactions
Introduction
The high energy heavy ion program at the Bro okhaven Alternating Gradient Syn
chrotron AGS and at the CERN Sup er Proton Synchrotron SPS is devoted to the
investigation of nuclear matter under extreme conditions high temp eratures and or
densities The most exciting prosp ect is the p ossible observation of a phase tran
1
sition from normal nuclear matter to a so called Quark Gluon Plasma QGP A
large amount of exp erimental and theoretical e ort has gone into working out clear
signatures for the QGP Although the ultimate hop e is to eventually see the QGP
2
shine via direct electromagnetic radiation of photons and lepton pairs for prac
tical reasons larger signals and b etter understanding of backgrounds exp erimental
e orts have up to now fo cussed on the pro duction of hadrons with newly pro duced
quark avors strangeness charm etc At present the largest b o dy of data with the
so far most promising indications for unconventional physics has b een accumulated
3
in the sector of strange particles following the early suggestions of Rafelski well over
years ago
There are two ma jor asp ects of the so called strangeness signal The more sp ec
ulative one is connected with the search for exotic strange matter which requires the
intermediate pro duction of a QGP state The list of these so far undetected phenom
4 6
ena contains strangelets small droplets of metastable strange quark matter the
7
ultimate pro of of the existence the quark gluon plasma but also the H dibaryon
8
and more recently the so called MEMOs Metastable Exotic Multistrange Ob jects
For this asp ect of the strangeness signal we refer the reader to Refs and to the
contribution by C Greiner and J Scha ner in this volume
The other more conventional asp ect of strangeness as a QGP signature was in
tro duced by Rafelski who predicted an enhanced pro duction of strange hadrons in
heavy ion collisions when compared to nucleon nucleon or nucleon nucleus collisions
3 11
at the same energy The idea is simple and is based on the di erent pro duc
tion mechanisms of strangeness in a QGP individual strange quark pair pro duction
from a dense system of gluons and light quarks and in a hadron gas pro duction
of hadrons with opp osite strangeness in inelastic hadron hadron interactions which
have strongly di erent intrinsic time scales But the quantitative interpretation of the
observed enhanced nuclear strangeness pro duction as a QGP signal is a subtle issue
Clear and unambiguous statements require a quantitative theoretical understanding
of strangeness pro duction in a QGP and in a hadronic environment including the
initial pre equilibrium stages of the evolution of strangeness during the reball life
time in particular during the di cult hadronization stage and of the variations of
the non strange multiplicities as one go es from nucleon nucleon to nuclear collisions
Some of these issues are still not satisfactorily resolved but signi cant progress has
b een made in the last few years which we will review in this article
An imp ortant not always appreciated advantage of strangeness is the large variety
of strange particles sp ecies Due to the long life time of strange quarks against weak
decay one has by now b een able to identify and measure the yields and sp ectra of
di erent sp ecies of strange hadrons in heavy ion exp eriments K K K
s
An up to date list of references is given in Tables and in
the App endix an account of the latest results can b e found in Refs These
data triggered a lot of studies within various mo dels for nuclear collisions Esp ecially
14 26
thermal mo dels which are characterized by a rather small number of adjustable
parameters and thus predict various restrictive relations b etween the various particle
sp ecies can thus b e tested very precisely We will devote a large fraction of this article
to these investigations not only b ecause large progress has b een made in particular
in this sector but also b ecause these studies provide a simple and intuitive picture
which gives valuable qualitative and quantitative insight into the physics of hadron
pro duction in nuclear collisions The thermal picture allows for a straightforward
investigation of the question of chemical equilibration in the strange and non strange
sectors and provides a simple characterization of the nal stages of the the heavy ion
collision in terms of freeze out parameters entropy pro duction and collective ow
dynamics It provides an intuitive bridge b etween the data on the one hand and
complete kinetic simulations of the phase space evolution of the collision region on
the other hand
In this review we concentrate on the developments after the publication of the
1 9 27
rst volume of this b o ok for the earlier work we refer to the reviews by Ko ch
In addition to the discussion of strange hadron yields and momentum sp ectra we
also rep ort on some recent progress on the the microscopic mechanisms of strange
particle pro duction in a hot and dense environment and on the general structure of
the strangeness phase diagram
The article is divided into the following parts In Section we cover some new
developments on strangeness pro duction mechanisms Section provides the basic
equations for the thermo dynamics of hadronic matter These are used in Section
to study the phase diagram of strange matter In Section we discuss additional
ingredients like collective ow and deviations from chemical equilibrium which are
necessary when testing the thermal mo del against exp erimental data This is then
done in Section for various heavy exp eriments on strange particle pro duction In
Section we shortly discuss the asp ect of entropy pro duction in these heavy ion
exp eriments A comprehensive discussion of the outcome of the thermal analysis
follows in Section In Section we draw some sp eculative conclusions from this
discussion on the nature of the hadronization and freeze out pro cesses and discuss
the internal consistency of our interpretation of the strange particle abundances We
conclude with an outlo ok in Section
Strangeness Pro duction Mechanisms
The usefulness of strange particles as prob es for the physics of a nuclear collision
arises from certain characteristic features in their pro duction and kinetic evolution
In contrast to pions for example which are the most e cient carriers of entropy
and whose nal abundance is thus more or less determined already in the very rst
28
hard collision stage of the reaction where most of the entropy is pro duced the
transparency of nuclear matter is the imp ortant ingredient the abundance of strange
quarks continues to evolve until the very end of the collision reball Thus all stages
contribute to the nal strangeness yield and must b e considered
The rst pre equilibrium stage is dominated by the physics of hard pro cesses
The amount of strange particles created in these hard collisions is rather uncertain
p
For the present highest energy range s A GeV this phase is di cult to
handle quantitatively b ecause it is still dominated by non p erturbative QCD At
p
s A GeV for higher energies the parton mo del can b e more reliably used at
example the amount of strangeness pro duced in the very early collisions has b een
28
calculated in the Parton Cascade Mo del to b e ab out of the equilibrium value in
the HIJING mo del which uses a di erent soft momentum cuto pro cedure the initial
29
strangeness fraction is somewhat lower For the lower AGS and SPS energies one
must instead make an educated guess and one generally assumes an initial fraction
of strange particles as observed in the nal state of p p collisions at the same energy
30
after correcting for resonance decays which corresp onds to roughly of the
20
equilibrium value
If the collision region is initially in the decon ned phase thermal equilibration
28 31
o ccurs at a very rapid rate Chemical equilibration of gluons and light quarks
31
although this question is still debated among is exp ected to follow shortly after
28 29
the various high energy event generators Ideally one would thus after ab out
fm c create a thermally and chemically equilibrated QGP consisting of mostly light
quarks and gluons Due to the strange quark mass threshold strangeness pro duction
only follows later Given the ab ove scenario the pro duction rate and strangeness
equilibration time scale can b e calculated with the metho ds of thermal equilibrium
eld theory and if the temp erature is su ciently high even p erturbative QCD can
32 33
b e applied While the rst calculations used bare quark and gluon propagators
recent investigations have b egun to incorp orate the new knowledge of how to include
certain collective and non p erturbative e ects at high temp erature via resummation
of the p erturbative series These new developments will b e describ ed in Section
An alternative scenario is a nuclear collision without a decon ning phase transi
tion Then the relevant degrees of freedom are the hadrons High temp eratures and
densities will a ect the hadron dynamics their masses and their resonance widths
and thus the strangeness pro duction Of particular relevance is a p ossible decrease of
the hadron masses In Section we will rep ort some calculations which take into
account this e ect for strangeness pro duction
Quark Gluon Production Mechanisms
The pro duction rates of strange quarks in a QGP are calculated by p erturbative QCD
At nite temp erature however any massless theory is harmed by severe infrared
singularities which are di cult to deal with esp ecially for gauge theories Braaten
34
and Pisarski developed a resummation technique for the endangered low momentum
mo des at nite temp erature and density calculations The main idea is to use for
light particles with momenta of order q g T g a resummed propagator which
contains self energy corrections due to the medium In lowest order of the g the low
momentum particles in a thermal environment thereby acquire a thermal mass m
th
which is given by
m N N g T
g c f
for gluons and by
g T m m
q
q
for quarks These massive quark and gluon mo des of nite temp erature QCD are
analogues of the well known plasmon in the theory of electron gases and electromag
netic plasmas At next to leading order in g these mo des also acquire a thermal
width In addition to the low momentum propagators also some vertices in which
34
all momenta are soft of order g T must b e resummed
The new technique was used to recalculate the pro duction rates for strange quarks
35 36
in a QGP The Feynman diagrams for ss pro duction in lowest order QCD are
shown in Figure There are three di erent groups quark anti quark collisions a
gluon fusion b and thermal gluon decay c The third pro cess which is not allowed
for a bare gluon propagator m is now p ossible b ecause of the nite mass and
g
width of the thermal gluons
36
The expressions for the rates of the pro cesses depicted in Figure are given by
Z
d p d p d p d p
q q s s
R P P P P
q q ss q q s s
E E E E
q q s s
X
f E f E f E f E jM q q ss j
F q F q F s F s
Figure Feynman diagrams for
ss pro duction in lowest order QCD
a q q ss
b g g ss
c g ss
Z
d p d p d p d p
s s
P P P P R
s s g g ss
E E E E
s s
X
f E f E f E f E jM g g ss j
B B F s F s
and
Z
d q d p d p
s s
R Q P P
g ss s s
E E E
g s s
X
f E f E f E jM g ss j
B g F s F s
Capital letters corresp ond to four momenta and f f denote the Bose Einstein
B F
Fermi Dirac distributions resp ectively
An extensive discussion of the calculation of the rst two pro cesses of Figures a
and b with bare propagators and vertices can b e found in the early literature see
Refs A complete new calculation of the relevant matrix elements with
resummed propagators and vertices is not easy and therefore has so far not b een
37
presented work in this direction is in progress The new calculations published
35 38
so far simply replace all bare propagators in Figure by thermal propagators
with a thermal mass and width In this approximation the thermal gluon decay
Figure c is then the most interesting new phenomenon and has therefore received
most of the attention In view of the incompleteness of these calculations it is not
surprising that the results are controversial The thermal gluon decay into strange
quark pairs was rst evaluated in Ref The motivation for a nite gluon mass was
39
at that time taken from lattice QCD results rather than from resummed thermal
p erturbation theory Gluons with masses around MeV t the data on energy
39
density and pressure of a SU lattice calculation ab ove T very well The rate
c c
Figure Partial and total rates
A d N dx for strange quark
s
pro duction according to the calcu
lation in Ref q q ss dashed
line g g ss dotted line g ss
dashed dotted line and M denotes
the used constant thermal gluon
mass
was calculated with a constant temp erature indep endent gluon mass and the
quarks have b een assumed to have their constituent masses
32 40
The result is shown in Figure The upp er plot shows the old calculation with
m bare gluon propagator the lower plot a calculation with m MeV In
g g
the rst case the gluon fusion pro cess is by far the dominant source of strange quark
pro duction dominating the pro duction from light quarks by an order of magnitude
A nite gluon mass decreases the gluon fusion rate b ecause of the strongly reduced
Bose factors see Eq for the incoming gluons On the other hand it raises the
strangeness pro duction rate from light q q pairs slightly b ecause the intermediate gluon
is now less o shell due to its thermal mass The dominant pro cess however is now
the thermal gluon decay b ecause in contrast to the gluon fusion only one thermal
distribution function app ears in the incoming channel see Eq Note also that
this pro cess is of order g while the other pro cesses are of order g Surprisingly in
spite of these drastic changes in the individual pro duction channels the total rate
A R R R R
tot q q ss g g ss g ss
38
stays nearly unchanged The authors conclude that the statements ab out the equi
libration time scales in Ref are nearly unchanged and the gluons remain the main
strangeness pro ducer
35 36
Two recent calculations tried to improve this analysis by using the thermal
propagators of Ref We will shortly describ e the calculation of Ref The
thermal gluon propagator may b e written as
h i
T L ab ab
P q q P q q iD q q i
T L
T L
where P and P are transverse and longitudinal pro jectors resp ectively and the
corresp onding propagators are
q q
T L
Q q q
T L
34
q q denotes the self energy correction due to a thermal lo op The p oles in
T L
the propagator are determined by the disp ersion relation
q q q q
T L
The p oles are lo cated at
q i
T L T L
where the imaginary shift of the p ole is related to the imaginary part of the self
T L
energy through
Res Im
T L T L T L
with the residue of the propagator at the p ole given by
T L
Res
T L
q
T L
The expression for the gluon propagator can then b e rewritten as
Q Re i Res
Q
Q Re Res Q Re Res
where we suppressed the subscripts T L This expression is used to replace the mass
shell function for thermal gluons
Res
T L T L
Q m
g
Res Q Re
T L T L
T L
36
The evaluation of Eq is then done neglecting the Pauli blo cking factors The
35 36
result is
p s
Z Z
q m
s
g m
s
T
R dq f q dq q Q m
B
g ss s
Q
m
s
Res
T T
Q Re Res
T T
T
plus a similar expression for the decay rate of the longitudinal gluons
The biggest uncertainty in such a calculation is the determination of the imaginary
part of the p ole in the gluon propagator is the damping rate of transverse
T L T
plasma oscillations in a QGP at high temp erature For plasmons with nonzero spatial
momentum relative to the heat bath is not p erturbatively calculable but can b e
T
41
related to the so called magnetic mass m inverse magnetic screening length in
mag
the limit m by the following self consistency condition
mag T
m
g N T
c
g
ln
T
m m
mag T
mag
m itself is given at high temp eratures by
mag
m c g T
mag N
c
where c is a p erturbatively uncalculable number which so far can only b e determined
N
c
from lattice QCD calculations
The further evaluation of is done di erently in Refs In b oth pap ers
the approximation is assumed but in Ref Eq was approximated
L T
by the m indep endent expression
mag
N T ln
s c s
while in Ref Eq was expanded in p owers of m
mag
m
g N T
c
g
ln
T
m
mag
where corresp onds to leading order and N c to next to leading
c N
c
36
order In the calculation c c was chosen Further Altherr et al
N
c
used for Re q m the asymptotic limit for large q and q while Bili c et
T
g
al evaluated the disp ersion relation numerically in the high temp erature hard
thermal lo op approximation
The results of b oth calculations are shown in Figure with g and N
c
Figure a shows a clear dominance of the thermal gluon decay in the mass region
relevant for strangeness pro duction m T But the other two channels are prob
s
ably underestimated They were calculated in the limit M T from and
36
resp ectively An exact numerical treatment of Eq and shows a higher rate
for these pro cesses as seen in Figure b The solid line corresp onds to the sum of
light quark antiquark scattering and gluon fusion the gluon decay is shown as dashed
lines for various approximations for the damping rate We see that the q q and gluon
fusion pro cesses still dominate the strangeness pro duction rate According to Ref
the gluon fusion pro cess is the more imp ortant one of the two channels exactly as in
32
the original tree level calculation Note the large in uence of the so far unknown
gluon damping rate on the gluon decay into ss pairs a large damping rate i e a
large gluon width enhances the strangeness pro duction rate drastically In view of 4 10-1 R/T 10-1 g decay -2 g fusion 10 -3 q fusion 10 4 10-3 R/T 10-5 10-4
10-5 10-7 0 1 2 3 4 1.0 2.0 3.0 4.0 m/T
m s (0)/T
Figure a Strange quark pro duction rates Figure b Strange quark pro duction rates
for di erent channels according to the calcu for thermal gluon decay with di erent damp
lations of Ref from where this gure was ing rates The upp er short dashed line
taken For details see text corresp onds to the damping rate of Ref
Eq the next lower dashed line to
and the lowest dashed one to N c
c
N
c
in Eq The solid line is the sum of the
gluon fusion and quark anti quark collision
rates Figure from Ref
this sensitivity the approximation should b e reinvestigated and if necessary
T L
improved
The physically most relevant quantity is the total strangeness equilibration time
The additional channel of thermal gluon decay can only shorten but quantita
s s
35 38
tively its in uence is not really very drastic For temp eratures around MeV
is of the order of fm c it strongly dep ends on T the strange quark mass and
s
the coupling constant The controversy ab out the imp ortance of the thermal gluon
s
decay do es not a ect the conclusion that it won t b e able to reduce the strangeness
equilibration time to values well b elow fm c Thus even in a heavy ion induced
QGP full strangeness equilibration remains questionable Calculating the thermal
gluon decay is a rst attempt to include non p erturbative features in the strangeness
pro duction At temp eratures around MeV such non p erturbative asp ects
play an imp ortant role and therefore many more detailed investigations of this ques
tion in particular the consistent implementation of the real and imaginary parts of
37
the thermal propagators also in the diagrams a b are necessary to obtain a reliable
picture
Hadronic Production Mechanisms
It is generally assumed that with the b eam energies available at the AGS and SPS we
may reach energy densities close to but certainly not very far ab ove the decon nement
phase transition It is therefore a p ossibility that in the present heavy ion exp eriments
the collision zone sp ends all or a large fraction of its lifetime in the con ned hadronic
phase With the critical temp erature and energy density from lattice QCD now
42
apparently having settled rmly at T MeV and GeV fm we must
c c
however exp ect some deviations from the often used picture of a non interacting
43
mixture of hadronic resonances Being close to the p oint of decon nement and
chiral symmetry restoration the masses and widths of the hadrons as well as their
interaction cross sections may already b e strongly mo di ed by the medium compared
to their vacuum values
Most authors working on this problem agree that the masses of most of the hadrons
44 47
decrease with density and temp erature The Goldstone b osons of chiral sym
metry breaking i e the pions and kaons are an exception their character protects
48 49 50
their masses from large changes with temp erature and density The reason for
that is an approximate comp ensation of two medium e ects One can view these par
ticles as b ound states of constituent quarks with a very large negative binding energy
This results in a very low total energy i e mass of these particles The constituent
quark mass is generated by chiral symmetry breaking By heating the system the
constituent quark mass decreases b ecause chiral symmetry gets restored but so do es
the absolute value of the binding energy Altogether the Goldstone masses are nearly
48 50
stable and if anything their masses tend to slightly increase as the chiral phase
transition is approached
45 51
Another controversial particle is the meson whose mass some authors exp ect
52
to drop in the medium while others exp ect it to rise Decreasing masses of heavier
particles and resonances op en the phase space for their pro duction This leads to the
exp ectation that the mostly heavy strange particles are pro duced with larger rates in
53
hot and dense hadronic matter compared to calculations using free cross sections
and mass thresholds
The e ect of the medium on the hadronic pro duction of strange particles has b een
54
studied mainly by Ko and his group their work includes the pro duction of kaons
55 56
mesons and s and s
Kaon production
Hadronic kaon pro duction in heavy ion collisions pro ceeds mainly through the follow
ing three pro duction channels M non strange meson B non strange baryon
a BB KYB
b MB KY
c MM K K
The channel a is the dominant pro duction mechanism in collisions of protons with
hydrogen or light nuclear e g Be targets There the K ratio at AGS energies
57
A GeV incident momentum is ab out For heavier targets e g Au
129
the ratio rises at the same incident momentum to K indicating that
probably also channel b b egins to contribute Channel c will b ecome imp or
tant once a substantial pion density has emerged This is supp osed to b e the case
128
in heavy ion collisions such as Si Au where indeed K at the AGS
b 58 59
Semi hydrodynamical calculations and also some Monte Carlo calculations us
ing free strangeness pro duction cross sections from exp eriment fail to repro duce the
K ratio of the AGS Si Au exp eriment The event generators for nuclear colli
sions either repro duce the pions correctly underestimating somewhat the K yield
60
as e g ARC or correctly repro duce the kaon yields at the exp ense of slightly over
61
predicting the pions as RQMD So far these calculations have ignored medium
e ects on the involved interactions it is conceivable that somewhat larger elementary
cross sections might give b etter agreement with the data
Ko et al have parametrized the dep endence of the hadronic masses on T and
B
54
baryon density by
n
m T
B
m T
c
where the detailed b ehavior is mo del dep endent T is the decon nement temp erature
c
taken as T MeV in the studies describ ed b elow is the nuclear ground state
c
density fm and is an empirical parameter determined as
62
from K p and e scattering on nuclei The p ower n dep ends on the mo del in
which the temp erature dep endence is studied n is obtained from QCD sum
46 47
rule calculations while n is favored by chiral mo dels
63 53
The density dep endence of the kaon mass di ers from the mass scaling
53
Within linear chiral p erturbation theory the kaon mass b ehaves like
m
B
K
m
K c
KN
is the critical density for kaon condensation f MeV is m where f
K c
K K
KN
the kaon decay constant MeV is the kaon nucleon sigma term Eq
re ects an exceptional b ehavior of the kaons although the kaon b elongs to the Gold
stone o ctet in a SU representation for the avor group its mass decreases with
f
density in contrast to the density scaling of Goldstone b osons discussed b efore Ac
53
cording to the authors the reason is the large explicit chiral symmetry breaking
by the strange quark mass This issue is however still under debate Within the
Nambu Jona Lasinio mo del the tendency for kaon condensation is not supp orted
Lutz et al get an increase in mass for the K while the K stays nearly unchanged
63
as a function of density More recent calculations in the framework of chiral p er
64
turbation theory lead to a rising K and a decreasing K mass including higher
orders of the approximation within a coupled channels approach reduces somewhat
Figure Temperature and density de
p endence of h v i for K K dotted
curve K K short dashed curve
and K K long dashed curve The
solid curve is for MM K K averag
ing over initial meson distributions Note
the di erent vertical scales in the upp er
and lower row The gure was taken from
Ref
65
the decrease of the K mass and additionally coupling in the resonance
66
essentially destroys any indications for K condensation at high baryon densities
Thermally averaged cross sections h v i of the strangeness pro ducing reactions are
54
needed in the calculation describ ed b elow For a pro cess n they are
de ned by
Z Z
d k f k f k k k v h v i d k
n n
where f denote the normalized thermal distribution functions for the incoming
i
particles and v is their relative velocity The elementary medium cross section
k k is calculated using established form factors and coupling constants
n
but masses which satisfy the scaling relations and An example is shown
in Figure from Ref where the temp erature and density dep endence of mesonic
pro duction cross sections are plotted One sees a clear rise of the cross sections with
temp erature and baryon density
The medium cross sections may now b e used to calculate the nal strange particle
abundances in a heavy ion collision This requires a mo del for the time evolution of
67
the nuclear collision In Ref the hydrodynamical mo del of Biro et al was used
It employs an ideal hadron gas equation of state without phase transition In addition
to the hydrodynamical equations which determine the overall time evolution of the
initial reball carried mainly by pions and nucleons rate equations for the evolution
b
The expression semi hydrodynamical will b e de ned b elow
Figure The rapidity distributions of
di erent particles at AGS The gure is
from Ref where the calculation vari
ous curves was made while the data are
from Ref
of the strange particle densities were solved The rate equation for a particle sp ecies
s has the general form
N
nal
Q
B C
f s
i j
X
B C
d V
f
s
B C
h v i
i j s X i j
B C
N
nal
Q
V dt
A
i j s X
i j
s f
f
The summation is meant to extend over all considered pro cesses for pro ducing elim
inating particles of kind s The sup erscript indicates the equilibrium densities
Eq contains gain and loss terms and takes care of the expansion of the re
ball volume V Since hydrodynamical mo dels for nuclear collisions usually assume
complete chemical equilibrium even for strange particles throughout the dynamical
evolution we use the expression semi hydrodynamical for these calculations which
decouple some particle sp ecies from the bulk evolution and treat them separately
As initial conditions for Si Au collisions at A GeV a baryon density of
and an energy density of GeV fm were chosen corresp onding to an initial
temp erature of ab out MeV Figure shows the rapidity distribution of pions
kaons and protons after freeze out Freeze out is assumed to o ccur sp ontaneously at
a common temp erature of T MeV
f
The agreement with the data is reasonable the kaon abundances are well repro
duced The authors conclude in contrast to the earlier work in Ref that no
QGP phase transition is needed in order to understand the kaon yields at the AGS
It should b e stressed that the main mechanism for kaon pro duction in the describ ed
54
analysis is the pro cess K K The assumed reduction of the mass gives the
necessary high initial density however as p ointed out ab ove this b ehavior of the
must still b e considered as controversial
production
The strangeness enhancement from p A to A A collisions is also clearly seen in the
meson yield The NA collab oration rep orted a rise of the ratio from p U
68
to S U by a factor of in the dimyon sp ectrum most central collisions As origi
70
nally suggested by Shor it is p ossible to explain this enhancement as a consequence
71
of QGP formation However hadronic rescattering mechanisms with di erent ab
71 72
sorption cross sections for the and mesons by the nuclear medium can
also account for the observed systematics of NA although the required rather large
71
reball densities and lifetimes present some problems on the conceptional level as
well as in practice since the exp ected signi cant rise of the dilepton yield with
73
the reball lifetime is in contradiction with the observed indep endence of the
74
dilepton p eak with collision centrality
There are also theoretical studies predicting a drastic mass reduction of the
69
meson in the medium In combination with reduced kaon masses it is sp eculated
55
that this could drastically enhance pro duction The main hadronic pro duction
mechanisms are here
a K N
b K K
these are the only channels taken into account in Ref in the rate equation
for the The channel MM K K violates the OZI Okub o Zweig Izuka rule and
is therefore suppressed
The calculation of Ref takes similar initial conditions as in Sec but assumes
a higher freeze out temp erature of T MeV at CERN SPS energies The result is
f
55
a ratio of which was claimed to b e in agreement with the measured one
68
of NA in A GeV S U collisions But a direct comparison with data obtained
from the leptonic decay channel is problematic First the and cannot b e resolved
exp erimentally therefore the should b e included in the theoretical calculation to
predict the ratio Secondly dilepton sp ectra are integrated over the whole
history of the collision which is imp ortant in particular for the dilepton yield under
73
the p eak while the calculation in Ref considers only ratios at freeze out Thus
hadronic mechanisms for the observed enhancement remain a subtle issue
It is exp ected that in the medium also the widths of the particles change The
calculation of Ref within the vector dominance mo del shows a large broadening of
the width with increasing density and in addition mesons and kaons have reduced
e ective masses Contrary the authors of Ref claim that within the Nambu Jona
Lasinio mo del the width of the decreases with increasing temp erature b ecause
the mass go es down while the kaon mass stays nearly unchanged such that the
decay phase space for the dominant channel K K closes On the other hand an
76
increase of the particle widths in the medium by collision broadening is a generic
feature which should b e taken into account in any realistic calculation
In the dilepton sp ectra no change in mass or width of the has b een seen so far
within the rather p o or exp erimental resolution In the invariant mass sp ectra of
K K pairs from Si Au collisions at the AGS an indication for a small mass shift
153
of the p eak has recently b een rep orted the observed shift amounts to a mass
decrease by MeV for low p s from very central nuclear collisions A surprising
T
feature of this rep orted mass shift is that in the selected kinematic region al l s
app ear to b e shifted i e the width of the shifted p eak is smaller than the mass
shift and shows no traces of an unshifted comp onent Clearly this situation requires
further clari cation
production
In the context of strangeness enhancement as a QGP signal the sp ecial role of
11
strange antibaryons was p ointed out very early For pro ducing antibaryons an
energy threshold of at least GeV must b e overcome taking free hadronic masses
The observed high yield in A GeV collisions is thus hard to explain by ordinary
77
hadronic mechanisms Hydro dynamical calculations based on a hadron gas equa
c
tion of state with a freeze out temp erature around MeV fail to repro duce the
s The same is true for event generators based on string mo dels unless string fusion
78 79
the formation of color rop es or of quark gluon clusters is taken into account
Thus an explanation of the yield on a microscopic level would give much insight to
the nuclear reaction dynamics
56 157 165
Ko et al have investigated the multiplicities from the NA exp eriment
The analysis was done in the same manner as in Sections and An enhancement
of s is exp ected mainly due to a mass decrease with rising density and temp erature
80
The mass scaling of the was deduced from the SU Walecka mo del as
f
m g g
N N
S
m m
p
and a meson mass of m MeV with the couplings g g
N N
is the scalar nuclear density
S
The main contribution to the pro duction rate comes from meson collisions which
are not suppressed by the OZI rule
a K N
b K N
56
As initial conditions a temp erature of T MeV and a density of twice the
nuclear ground state density were chosen The mo del results in a multiplicity of s
p er S S collision NA Ref Ref but underestimates
the pion kaon and multiplicities by and resp ectively
c
Higher temp eratures would solve the problem but they are hard to justify
We see that with certain initial condition and a mass reduction for the
a reasonable value for the multiplicity results But the calculation do es not give
a consistent overall picture Medium e ects for the s are certainly an imp ortant
ingredient but whether the mass reduction alone can account for the observed nal
yields remains to b e proven
Further remarks on hadronic strangeness production
The introduced mo del raises the question how the particles get on the mass shell
at freeze out At the chosen freeze out temp erature the particles still exp erience a
considerable reduction of their free masses Strictly sp eaking this is inconsistent with
the naive freeze out concept which assumes a more or less sudden transition from
interacting to free streaming particles The re ned picture b ehind these calculations
assumes that after the freeze out p oint all two bo dy collisions have ceased but the
particles still feel an attractive mean eld which arises from virtual pro cesses It was
53 81
argued that on their way out the particles recover their vacuum masses by giving
up kinetic energy as they move up the mean eld p otential well In this pro cess their
momentum changes according to the equation
m k m k
where the quantities with a star corresp ond to the in medium values at freeze out
This leads to an accumulation of particles with low asymptotic momenta Shuryak
81
has shown that such an assumption can explain the low p rise of pions and for
147
the recently observed cold low momentum kaons at the AGS the same mech
82
anism was suggested But there are also alternative explanations for the low p
83
enhancement discussed in the literature and in particular for the cold kaons the
13
exp erimental situation is not yet completely settled Thus for the time b eing the
direct exp erimental evidence for medium e ects on the particles widths and masses
remains shaky
The calculations of Ko et al show that such e ects are able to account for a large
part of measured strange particle yields without assuming QGP formation In these
calculations the chemical equilibration is driven by the strongly decreased masses
which was not taken into account in the earlier calculations of Ref However
up to now these investigations su er from the fact that each calculation fo cuses on
one particular asp ect of the observations without reaching a coherent and internally
consistent description of all observables and their characteristic mutual relationships
simultaneously For example an additional test for the mo del would b e the multi
strange anti baryons measured by the NA and WA collab orations at CERN
for references see Table In spite of the extremely high mass thresholds their mul
tiplicities are high and present a particular challenge to any hadronic kinetic mo del
Other p oints need improvement to o the study of Ko et al su ers from a ther
54
mo dynamical inconsistency The reduction of the masses has only b een taken into
account in the particle distribution functions and cross sections for the kinetic rate
equations but not in the equation of state for the hydrodynamic evolution One ex
p ects that the mass reduction should also lower the energy density and the pressure
of the system
Last but not least the sub ject of medium e ects on hadron widths and masses
is altogether still very controversial Di erent approaches lead to qualitatively dif
ferent and mutually contradictory results A more fundamental and comprehensive
approach to the problem of in medium masses based for example on QCD inspired
low energy e ective hadron Lagrangians should eventually replace the rather phe
nomenological scaling law Eq Since these e ective masses enter the chemical
equilibration rates in a crucial way and the whole issue of strangeness equilibration as
a signature for interesting new physics is mostly a question of time scales this should
b e a high priority pro ject
Thermal Mo dels
In this chapter we explain in more detail the physics of thermal mo dels and their
application to relativistic nuclear collisions and discuss the conclusions that can b e
drawn by lo oking at strange particle pro duction yields
Thermal Parameters
The thermal mo del starts with the assumption of a lo cally equilibrated not neces
sarily stationary reball The reball could consist of a QGP a hadron gas or an
equilibrated mixture of b oth The following set of thermal parameters can b e used
for all these phases
The reball is describ ed by a volume V a temp erature T and chemical p otentials
for the conserved quantum numbers In general the intensive thermal parameters
are lo cal quantities i e dep end on space and time This description corresp onds
to a grand canonical ansatz i e the quantum number densities parametrized by the
chemical p otentials are not conserved exactly but only in the average The corrections
compared to a canonical description with exact quantum number conservation are of
p
order N with N b eing the number of particles with corresp onding quantum
numbers Thus the corrections are exp ected to b e negligible for heavy nuclei
On the time scale of a nuclear reaction the imp ortant forces are the strong and
electro magnetic interactions b oth conserving the quark avors Flavor changing
weak interactions can b e completely neglected in nuclear collisions This is a rst
example of the entrance of time scales in the cosmological context where the dynam
ics happ ens on the time scale of microseconds weak interactions can b e considered as
chemically equilibrated and the corresp onding parametrization in terms of chemical
p otentials and conserved quantum numbers is corresp ondingly di erent Therefore
we introduce chemical p otentials and for the three avors considered here
u d s
Note that this ansatz automatically takes care of baryon number and electric charge
conservation In view of the approximate isospin symmetry of the colliding nuclei
which carries over to the reball it is common to combine the two light avors
q u d
The small breaking of the isospin symmetry can b e parametrized by
d u
The nuclei considered here will b e either exactly isospin symmetric e g S or
25
the breaking is small leading to Therefore isospin e ects can usually b e
neglected and we use always the averaged light quark chemical p otential Flavor
conservation is expressed on the level of hadrons as baryon number and strangeness as
well as isospin conservation It is also common to introduce the chemical p otentials
on this level i e a baryon chemical p otential and a chemical p otential for the
B
quantum number strangeness These two equivalent sets of chemical p otentials
S
are related by
q B
s B S
S q s
where the minus signs are due to the conventional assignment of strangeness to
the strange quark
If B S are the baryon number and strangeness of hadron h its chemical p otential
h h
can thus b e written either as
B S
h h B h S
in terms of and or as
B S
q
s
h q s
h h
q
s
count the number of light and strange valence quarks inside the hadron where
h h
resp ectively with anti quarks counted with a minus sign
The particle numbers are more directly given in terms of the fugacities related to
the chemical p otentials by
T
i
e
i
The fugacity of each hadronic sp ecies is simply the pro duct of the valence quark
etc fugacities viz
u s p d
K
u
Relative and Absolute Chemical Equilibrium
On the nuclear time scale the state of absolute chemical equilibrium is de ned as the
state in which all inelastic pro cesses which transform di erent hadron sp ecies into
each other sub ject only to the conservation laws for baryon number and strangeness
have come to equilibrium On much larger time scales this is actually only a state
of relative chemical equilibrium b ecause weak avor changing pro cesses have not
b een equilibrated Since the strong and weak interaction time scales are so widely
separated the concept chemical equilibrium with resp ect to strong interaction pro
cesses only is very useful In such a state the chemical p otentials for particles and
anti particles are opp osite to each other implying for the resp ective fugacities
i
i
However the lifetime of a nuclear collision is actually so short that not all allowed
chemical pro cesses among the di erent sp ecies of hadrons may have time to fully
equilibrate The assumption of absolute chemical equilibrium is thus even at freeze
out the most questionable one in a thermal reball approach Thermal equilibration
is much easier to justify since every type of collision elastic and inelastic ones in
volves exchange of momentum and thus leads to thermalization of the momentum
sp ectrum No thresholds or small cross sections inhibit thermal equilibration Chem
ical pro cesses on the other hand involve only certain inelastic channels with some
times small branching ratios cross sections and may b e suppressed by large mass
thresholds in particular in the case of baryon antibaryon or strange antistrange pair
pro duction
33
Lo oking at the systematics of hadronic chemical reactions it is however p ossible
to sub divide them into two classes with generically well separated time constants The
rst class of pro cesses involves the creation or destruction of only light quarks or
hadrons containing only light quarks these pro cesses have low mass thresholds and
happ en relatively fast a few fm c Such pro cesses include the exchange of existing
strange quarks or antiquarks among di erent hadron sp ecies e g in the pro cess
KN Y which is characterized by a strong absorption cross section of antikaons
by nucleons Much slower are pro cesses involving the creation or destruction of one
or more strange quark antiquark pairs several fm c A quite analogous situation
is found in a QGP environment q q pairs are created from gluons with a much faster
rate than ss pairs from gluons or light quark pairs
Given these di erent time scales it is thus sensible to assume that the state of
absolute chemical equilibrium with resp ect to strong interactions is reached in stages
rst a state of relative chemical equilibrium is reached in which gluons and light
quarks and antiquarks are thermally and chemically equilibrated but strange quarks
and antiquarks are only thermally equilibrated but haven t completely reached their
equilibrium density On the hadronic level this state corresp onds to chemical equi
libration of all non strange hadrons as well as a partial chemical equilibrium with
resp ect to all strangeness exchanging pro cesses while at the same time the absolute
amount of strange valence quarks and antiquarks remains b elow its saturation limit
In other words all fast chemical pro cesses are equilibrated while the slow strange
pair pro duction pro cesses are out of equilibrium Only in a second much later stage
also these latter pro cesses equilibrate and a state of full chemical equilibration o ccurs
The crucial question is how these various chemical time scales compare to the
lifetime of a nuclear collision reball As far as we know from the discussions in
Section this hierarchy b etween light and strange quark pro ducing pro cesses is
generic and true in b oth hadronic and QGP environments the absolute time scales
for b oth types of pro cesses are however much faster in the QGP than in the hadron
gas b oth b ecause of the larger densities of scatterers and lower mass thresholds This
is the deep er reason why strangeness is such a useful observable with an initial QGP
phase relative chemical equilibrium in the ab ove sense should b e a valid concept
in any case and the nuclear collision lifetimes may even b e at the threshold for
complete chemical equilibration Without the QGP as a catalyst absolute chemical
equilibrium seems out of reach in nuclear collisions and even the concept of relative
chemical equilibrium may not b e very well realized
The idea of relative chemical equilibrium was rst introduced heuristically by
84
Rafelski and later exploited in more detail by several groups see Refs
and Only recently it was put on a solid theoretical foundation and gener
alized to other situations e g the p ossible breaking of chemical equilibrium b etween
mesons and baryons after sudden hadronization of a QGP followed by rapid freeze
25 85
out by C Slotta He used the principle of maximum entropy to derive the distri
bution function for relative chemical equilibrium In the case of relative strangeness
equilibration it turns out to b e su cient to introduce one common strangeness satu
ration factor with where means full chemical equilibrium resp
s s s
complete saturation of the strangeness phase space The strange quark and antiquark
densities are then regulated by the e ective fugacities
e
s s
s
e
s
s s
In the hadron language these enter the hadron fugacities according to Eqs
The individual hadron fugacities are thus given by a pro duct of e ective valence
h
quark fugacities
Y
e
h
j
j
and the approach to equilibrium is controlled by the factor
n h s
s h
where the p ower n h s counts the total number of valence strange quarks plus anti
quarks in the hadron sp ecies h From the treatment in Ref it follows that must
s
b e treated as an additional fugacity i e it can b e written in the form
T
jsj
e
s
where is an additional chemical p otential for strangeness carrying quarks and
jsj
equal for strange and anti strange quarks Its deviation from zero parametrizes the
deviation of the thermo dynamic state from absolute chemical equilibrium The hidden
temp erature dep endence on in Eq is imp ortant for the calculation of the
s
85
entropy of the system
Since parametrizes the relative equilibrium b etween the two subgroups of non
s
strange and strange particles it can b e used as a quantitative measure for the
20
strangeness enhancement from p p over p A to A A collisions Due to its ther
mal de nition however this requires an application of the thermal mo del even to
low multiplicity p p collisions The viability of such an approach has b een discussed
for more than years see Ref for a historical overview and Ref for a recent
careful investigation of the pp annihilation system at rest and can up to now only
b e justi ed by its phenomenological success
The Partition Function
Partition function for the QGP
The calculation of all thermo dynamical variables starts from the grand canonical
partition function Z It is related to the thermo dynamical p otential through
T ln Z
This can then further b e expressed by the pressure P V For the QGP with
two massless avors u and d and a massive m strange quark we have
s
T
Q Q
P T ln Z V T
q s s q s s
V
T T B
q q
Z
dE E m
s
E E
s s
e e
s s
m
s
Here B is the MIT bag constant which is needed to simulate the background con ne
ment pressure when we investigate the strangeness phase space diagram For other
thermal quantities it has no relevance From the partition function one derives the
net particle densities for light quarks strange quarks and the baryon density
q s
as
B
Q Q Q
T P ln Z ln Z
q
q
V V
q q q
Q Q Q
P ln Z ln Z T
s
s
V V
s s s
q s B
Partition function for the hadron gas
The partition function for the hadron gas is given as a pro duct of the one particle
partition functions of the di erent hadrons h
Y
H
Z T V exp Z T V
q s s h h h
h
The latter is given by
Z
E m g V
h
pt
h
dE ln Z T V V P
h h h
h
E
h
e
h
m
h
g is the degeneracy factor m is the mass is the chemical p otential according to
h h h
Eq and is the strangeness suppression factor of hadron sp ecies h given by
h
Eq The upp er index pt stands for point like b ecause so far we treated the
H
hadrons as p oint like particles The real pressure P involves an additional prop er
87
volume correction and is given by
X
pt
H
P P
h
pt
B
h
pt
where is the energy density calculated for p oint like particles
In the analysis of measured strange particle yields the Boltzmann approximation is
su cient Quantum statistical corrections are imp ortant only for the very light pions
or for fermions at very large baryon densities which will not o ccur in the actual
applications discussed here The emphasis of a chemical analysis of the data lies on
the fugacities and the strangeness suppression factor We sp ell these factors out
s
explicitly and obtain in Boltzmann approximation
h
VT
H
F F F ln Z
q s K N s M
s q q q
i
F F F
q s Y s
s s s s q s s q s q
This expression takes into account all non strange mesons M for mesons with hid
den strangeness see b elow kaons K non strange baryons i e nucleons N and
deltas hyperons Y cascades and the omegas together with their anti
particles and all their resonance excitations The factors
X
m
h
F g K m T
f h h
T
h
where K is the mo di ed Bessel function of second order result from a summation
over all hadrons ground state and resonances h in the hadron family f where f
M N K Y or
In order to describ e consistently a strongly interacting hadron gas the full diver
43
gent hadron sp ectrum should b e taken into account as describ ed by Hagedorn For
a comparison with data we need however the nal decays into stable measurable
particles These branching ratios are only known for the lowest states Therefore one
must cut the hadronic mass sp ectrum somewhere The bias from this arti cial cut
will b e investigated when we analyze the NA data from S S collisions
The e ect of the strangeness suppression factor on mesons with hidden strange
s
ness via Eq requires a careful discussion According to the SU quark mo del
f
88
classi cation we take f f f and as pure
ss states with a suppression factor The physical and states are mixed states
s
88
consisting of uu dd and ss We take a mixing angle b etween and of
This results in an ss content of in the and in the It follows that the
multiplicities of these two mesons incorp orate the following strangeness saturation
factors
s s
Therefore the term F contains some factors not explicitly written down in Eq
M s
The Phase Diagram of Strange Matter
In this Section we describ e the phase diagram for strange hadronic matter For sys
tems with vanishing net strangeness it was rst studied in Refs Due to the
conservation of strangeness by the strong interaction this is the most relevant case for
heavy ion collisions Due to surface radiation of strange particles from the expanding
collision reball it is however p ossible that the net strangeness of the system uc
91
tuates around zero or even acquires an appreciable non zero value Furthermore
hadronic systems with nite net strangeness are relevant for early cosmology and the
structure of neutron stars with or without a quark core where weak interactions
are equilibrated Therefore the authors of Ref generalized the phase diagram to
systems with non zero net strangeness
Such systems can b e characterized by their strangeness fraction
T
s q s
f
s
T
B q s
the net number of strange quarks or minus the strangeness p er baryon
Strange hadronic matter like matter with zero net strangeness undergo es a phase
transition to a quark gluon plasma if its density is su ciently high High particle
density can b e achieved either via quark pair creation by heating the system or via
compression high net baryon density In the latter case it do esn t matter whether
the baryon density is generated by strange or non strange baryons nor is the sign of
the net baryon density imp ortant only its mo dulus must b e large In the following
subsection we discuss this phase transition
The Strange Matter Igloo
Go o d mo del equations of state for hadronic matter near this phase transition are
hard to obtain The usual phenomenological pro cedure is to estimate the lo cation
of the phase transition by matching two di erent mo del equations of state one for
the hadronic phase which is valid at low temp eratures and normal baryon densities
where we have information from conventional nuclear physics and one for the quark
gluon plasma ab ove the phase transition which is derived from QCD at very high
temp eratures where it can b e calculated p erturbatively Such a matching pro cedure
pro duces almost by necessity a rst order phase transition
Two commonly used equations of state are in this context the ones de ned in
Eqs and resp ectively The phase transition line is then obtained from
the Gibbs stability conditions for phase co existence
P P T T and
H Q H Q q H q Q s H s Q
By setting the temp eratures and chemical p otentials in the two phases equal one
obtains the pressure balance relation
P T P T
H q s Q q s
Its solution de nes a dimensional phase co existence surface in the dimensional
90
half space de ned by T and the so called strange matter iglo o
q s
Along the iglo o surface hadronic and quark matter can co exist with variable volume
H Q
fractions but in general with di erent strangeness fractions f and f in the two
s s
subphases If the hadronic subphase o ccupies the volume fraction V V V
H H Q
the strangeness fraction of the total system dep ends on and is given by
Q H
T T f f T f
q s q s s q s
s s
Systems with a xed strangeness fraction f will thus move along the iglo o surface
s
as they hadronize as increases from QGP to hadron gas the condition of
constant strangeness fraction cuts out a strip from the iglo o surface as shown
in Figure If the system is further constrained to p ossess a certain xed baryon
number B and total entropy S and is allowed to hadronize slowly and adiabatically
i e at constant sp eci c entropy S B it will follow a sp eci c line the so called
isentropic expansion tra jectory through that strip
Isentropic Expansion Trajectories
A discussion of these tra jectories is very interesting Supp ose that our system reaches
a state of thermo dynamic equilibrium somewhere in the quark phase characterized
by a p oint P in Figure outside the iglo o and b egins to expand hydrodynamically
In the absence of sho ck discontinuities the hydrodynamic equations conserve entropy
Since baryon number and strangeness are also conserved by the strong interactions
the system will expand at constant strangeness fraction f and sp eci c entropy S B
s
These two conditions together de ne a line through the three dimensional phase dia
gram This is the isentropic expansion tra jectory just mentioned It b egins at P in
the quark phase reaches the iglo o surface at a p oint P at the edge of the strip on
that surface which corresp onds to the chosen f and then b egins to cross that strip
s
Along the tra jectory b oth the temp erature and baryon density are continuously de
creasing functions If allowed to stay in lo cal thermal equilibrium forever in reality
the system will decouple once the density b ecomes to o low the hydrodynamic ex
pansion will only stop at zero temp erature and zero baryon density after the system
has expanded over an in nite volume There is no other p ossibility to reach a state
of zero temp erature at nite total entropy
The interesting question is which p oint on the T plane the tra jectory will
nally approach There are two p ossibilities
Figure The phase diagram of strongly interacting matter in T space for B
q s
MeV The iglo o type surface describ es the phase co existence region mixed phase b etween quark
matter outside and hadronic matter inside Figures a through f show various sections
through this surface corresp onding to systems with xed strangeness fraction f Also shown are
s
the pro jections of the mixed phase region onto the three co ordinate planes From Ref
The tra jectory reaches the other side of the phase co existence strip b efore hit
ting the T plane In this case it will leave the iglo o surface at a p oint P
on the other side of the strip and pro ceed into the interior of the iglo o until it
nally reaches a p oint P on the T plane
The tra jectory reaches the T plane b efore the crossing is completed at a
p oint P somewhere on the line in which the strip and the zero temp erature
plane intersect
These two cases corresp ond to two completely di erent physical situations In
order to discuss them we must understand the T limit of the phase diagram in
more detail
The T Limit of the Phase Diagram
Non strange hadronic matter has at zero temp erature a quite simple structure mesons
are absent b ecause they don t carry any conserved charges and thus their chemical
p otentials vanish and only those baryons exist whose mass is smaller than the baryon
chemical p otential i e the Fermi energy If is smaller than the mass of the
B B
lightest baryon the nucleon then no particles exist at all The hadronic state corre
sp onding to T and m is thus the vacuum it has zero pressure In other
B N
words as T any state with nite net baryon number has to approach a p oint
with m
B N
If strangeness is added things get more complicated The T phase diagram
is now a dimensional plane spanned by and Its structure phase
q B s
co existence line lines of constant strangeness fraction etc is very rich and highly
nontrivial A detailed discussion is given in Ref Here we can only summarize the
most imp ortant features
Generally sp eaking mesons are still absent at T But there are exceptions
since kaons are b osons which carry a conserved charge strangeness they can form
a Bose condensate if their chemical p otential is equal to their rest mass
m m or
s q K K q s K
K
These conditions de ne two straight lines in the plane where the rst corre
q s
sp onds to kaon the second to antikaon condensation Figure These equations
delineate the physical limits of the hadron phase at T which can only exist
b etween these two lines
Baryons exist at T only if their chemical p otentials exceed their rest masses
The baryon chemical p otentials have the form s s where s is
i i s i q i
the number of strange quarks in the baryon For s nucleons hyperons
i
cascades and baryons the condition m de nes four straight lines Their
i i
envelope separates the region with zero baryon density and zero pressure from the
region with nite baryon density see Figure
One can now solve the pressure balance equation in this plane For the three
values B MeV for the QCD bag constant one obtains the three
solid curves shown in Figure One sees that for the two smaller bag constants
part or all of the co existence line lies inside the region of zero baryon density in the
hadronic phase these parts of the line thus corresp ond to a mixed phase where the
hadronic subphase is just the vacuum Another way of describing such a state is
that it consists of quark matter bubbles separated by vacuum This state is stable
b ecause it solves the pressure balance equation the pressure of the hadronic phase
vacuum is zero and the quark bubbles have also zero pressure b ecause the Pauli
Figure Phase structure at T The
three solid curves are the phase co existence
lines for B MeV The
thick solid and dashed lines separate the re
gions of zero baryon density left from re
gions of nite baryon density right The
left region which at T is empty is sub
divided into smaller regions indicating
which particle sp ecies dominate at small
but non zero temp erature The small num
b ers along the lines refer to equation num
b ers in Ref from where this gure was
taken
pressure of the quarks is balanced by the bag pressure on the surface If the quark
matter in this state has f these bubbles are called strangelets
s
92
One can show that for B MeV some part of the phase co existence line
lies in the region of zero baryon density in the hadronic phase Actually practically
all p oints corresp onding to a non zero strangeness fraction lie on this section of the
phase co existence line For these values of the bag constant cold strange quark matter
is thus mechanically stable and stops expanding it cannot hadronize
Hadronization of Strange Quark Matter at T
Another way of plotting these results is shown in Figure Here we show the phase
diagram in the T plane eliminating and The previous phase co existence
B q s
line is here resolved into a whole shaded domain the mixed phase M Di erent
p oints in this mixed phase at the same temp erature T corresp ond to di erent volume
fractions of the hadron and quark subphases
One sees that for the particular value of the bag constant chosen in this Figure
at low temp eratures the mixed phase extends all the way to zero baryon density even
for systems with very small amounts of net strangeness This is just the situation
discussed at the end of the previous subsection If we start at nite temp erature
and let the system expand adiabatically we see that some of the isentropic expan
sion tra jectories never exit the mixed phase This corresp onds to the situation
mentioned at the end of Section Strangelets form On the other hand if the
sp eci c entropy S B was su ciently large to b egin with this do es not happ en and
the system hadronizes completely this corresp onds to situation mentioned at the
end of Section
Figure Isentropic expansion tra jectories in the T plane for B MeV and several
B
f values For f at T the mixed phase extends to zero baryon density Systems with
s s
su ciently large f and or small sp eci c entropy S B reach T without leaving the mixed
s
phase forming strangelets The full and op en circles indicate pion freeze out for spherical reballs
with and fm radius resp ectively For B MeV the diagrams lo ok like Figure a for
al l values of f and strangelet formation is imp ossible From Ref
s
It is of course interesting to ask which are the largest allowed values for S B
which still lead to strangelet formation in a system with given f We already
s
know that we need bag constants b elow B MeV B MeV fm
for strangelets to form at any value of S B and f For B MeV B
s
MeV fm on the other hand even non strange quark matter would b e stable which
contradicts nuclear phenomenology Within this narrow window of the bag constant
92
the values of f and S B play a crucial role E g for B MeV strange
s
matter with f hyperon matter completely hadronizes as so on as S B Typ
s
ical values for S B from heavy ion collisions at Bro okhaven or CERN range from
to see Section
We would like to close this section with one imp ortant remark if strange
quark matter undergo es adiabatic hadronization at xed sp eci c entropy S B and
strangeness fraction f which may b e zero the chemical p otentials usually change
s
during the phase transition Thus a strangeness neutral QGP will start to hadronize
with but at the end of the hadronization pro cess will take on a di er
s s
ent value which dep ends on T and and corresp onds to the equilibrium value in
B
a strangeness neutral hadron gas To draw conclusions from measured values of the
chemical p otentials in the nal hadronic state ab out their values in the preceding
QGP state is thus highly nontrivial In particular a vanishing strange quark chem
ical p otential in the nal state can have survived from a primordial QGP
s
phase only if hadronization did not happ en adiabatically but suddenly
Ingredients and Extensions of the Thermal Mo del
The Strangeness Neutrality Condition
An imp ortant constraint on the thermal mo del parameters arises from the fact that
in strong interactions strange quarks are only created in pairs and that therefore the
nuclear collision reball up to small uctuations due to p ossibly asymmetric surface
6
radiation of strange and antistrange hadrons or due to strangeness distillation
89
during hadronization of a QGP always remains approximately strangeness neutral
The condition of vanishing total strangeness according to Eq takes the form
hsi hs i ln Z
s s
s
For the hadron gas Eq is an implicit equation relating to in a way which
s q
89 93
strongly dep ends on the temp erature
In Figure we show the relation b etween and for di erent temp eratures
q s
which ensures vanishing net strangeness in a hadron gas One recognizes a slight
dep endence on the number of hadronic resonances taken into account The curves
have a characteristic b ehavior and for not to high temp eratures intersect the
s
line twice
At high temp erature the heavy resonance states b ecome imp ortant Unfortu
nately the high mass resonance states are p o orly known esp ecially in the mesonic
sector where the states ab ove ab out GeV b ecome very broad and are di cult to
identify exp erimentally This triggers an arti cial in uence of the known resonance
mass sp ectrum on the strangeness neutrality relation b etween mesons and baryons at 0.06 0.06 0.04 0.04 0.02 0.02
[GeV] 0 [GeV] 0 s s µ −0.02 µ −0.02 −0.04 −0.04 −0.06 −0.06 −0.08 −0.08 −0.1 −0.1 −0.12 −0.12 −0.14 −0.14 −0.16 −0.16 −0.18 −0.18 −0.2 −0.2 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 µ µ
q [GeV] q [GeV]
Figure Strangeness neutrality curves for a hadron gas for di erent temp eratures The dashed
line corresp onds to T MeV the solid line to T MeV and the dotted line to T
MeV In the left gure the hadron mass sp ectrum was cut at m GeV in right gure at
cut
m GeV
cut
high temp erature We b elieve that the strangeness neutrality curves are unreliable
ab ove temp eratures of ab out MeV
Generally Eq relates the three parameters and T resp and T
s q s q
If two of them are given the third can b e determined from this equation which thus
reduces the number of indep endent parameters The strangeness neutrality condition
dep ends also somewhat on but this in uence is weak b ecause it really enters only
s
n
n which survives in through the multistrange hadrons They carry a factor
s
Eq after the leading linear dep endence has b een factored out
s
For a QGP strangeness neutrality is a matter of the strange quarks alone in
dep endent of the amount of light quarks b ecause the strange quarks are no longer
b ound together with other non strange quarks into hadrons Strangeness neutrality
in the QGP is thus given by indep endent of This can b e easily derived by
s q
inserting Eq into Eq
As we will see later the exp eriments at A GeV suggest a value for very
s
close to zero While for a strangeness neutral hadron gas this solution is only p ossible
for T MeV and a particular value of a strangeness neutral QGP leads to this
q
solution naturally and indep endent of and T This is the characteristic di erence
q
in the thermo dynamics of a hadron gas and a QGP
A more detailed discussion of this issue will follow later In order to b e able to
discuss violations of the strangeness neutrality condition we de ne
hs i hsi
hsi
as a measure for the net strangeness
Transverse Flow and Transverse Momentum Spectra
If the system is in lo cal thermal equilibrium then the observed momentum sp ectra
should re ect the average temp erature However since a thermalized source which
is surrounded only by vacuum must necessarily b egin to expand the thermal motion
is sup erimp osed in the sp ectra by a dynamical comp onent arising from the collec
tive expansion This is most clearly seen in the measured longitudinal momentum
or rapidity sp ectra of hadrons from heavy ion collisions which for b oth Bro okhaven
and CERN energies are much broader than a thermal distribution from a stationary
26 94 96
reball the strong collective ow comp onent here is partially due to incomplete
stopping of the colliding nuclei and partially due to additional hydrodynamical expan
sion generated in the later stages of the collision In the transverse momentum sp ectra
which lo ok approximately exp onential like a thermal distribution the identi cation of
a collective ow comp onent is more involved the observed convex shap e of the sp ectra
at low momenta which do es not agree with the concave b ehavior of a thermal distri
95
bution near p can b e asso ciated b oth with transverse collective ow and with
83
non thermal contributions from resonance decays after freeze out This ambiguity
was systematically studied and resolved in Ref theoretical arguments based on
95 97
the consistency of the observations with the freeze out criterium and direct ex
26 147
p erimental reconstruction of the contribution of decays to the pion sp ectrum
con rm the existence of a strong collective ow comp onent h i also in
f
the transverse momentum sp ectra
For a purely thermal distribution the m sp ectra and the double di erential dis
are given in the Boltzmann approximation by tribution dN dy dm
dN g V
h h
m cosh y
h
m cosh y e
dy dm
q
T m
m g V g V dN
h h h
m T
h h
p
e m K m T e e
dm T
where K x is a mo di ed Bessel function If one ts the measured m sp ectra
with the expression an apparent temp erature can directly b e read o from
the exp onential fall o This is a very common pro cedure which is easily applied
by b oth theorists and exp erimentalists But due to the problem of resonance decay
and ow contamination just mentioned care should b e taken not to denote these
slop e parameters indiscriminately by temp eratures At AGS and SPS energies
the resulting apparent temp eratures are always of the order of MeV or higher
43
which is well ab ove Hagedorn s limiting temp erature for a hadron gas and ab ove
42
the decon nement temp erature from lattice QCD This is true even though the
contributions from resonance decays tend to steep en the sp ectra and thus reduce
the apparent temp eratures as will b e discussed in Section It app ears to b e
theoretically inconsistent to asso ciate such a large temp erature with the observed
nal hadronic state from heavy ion collisions
Transverse collective ow provides a simple resolution for this dilemma It is
97
a necessary dynamical consequence of early thermalization of the collision zone
The amount of hydrodynamically generated transverse ow dep ends among other
things on the initial energy entropy density and the equation of state These are
p o orly known so far and the resulting theoretical freedom must b e removed by the
exp eriment
Transverse momentum sp ectra from a collectively expanding thermalized source
96
can b e parametrized by
R
f
Z
ow
L p sinh r m cosh r dN
h
g I r dr m K
h h h
dm T T
where I x is also a mo di ed Bessel function R the freeze out radius and L is some
f
normalization length r stands for the transverse ow rapidity
r tanh r
and the transverse ow velocity pro le r is usually assumed to have the form
r
r
f
R
f
99
If the system expands longitudinally in a b o ost invariant way the double di erential
sp ectrum is simply given by
ow ow
dN dN
dy dm dm
Eq represents an approximation which assumes a particular shap e of the freeze
96 97
out hypersurface A detailed study within a global hydrodynamical evolution
96 97
mo del shows that represents well the transverse momentum sp ectra even if
they are calculated with a more sophisticated kinematical freeze out criterium For
not to o heavy particles and not to o low values of p the t is also rather insensitive to
95 96 97
the p ower in Eq Thus we take here for simplicity a constant transverse
ow velocity
For large m the sp ectrum falls o exp onentially but with a slop e which do es
not re ect the true temp erature T of the lo cally equilibrated system at freeze out but
rather an apparent blue shifted temp erature T which in the ab ove approximation
app
95 96
is simply given by
s
f
T T
app
f
In the case Eq reduces the purely thermal sp ectrum Eq
f
But since the exp erimental sp ectra x only the asymptotic inverse slop e accu
rately while at lower values of p the details of the sp ectra in particular the observed
strongly concave shap e of the pion sp ectrum are governed by an intricate interplay
of transverse ow and resonance decays the true temp erature and transverse ow (MeV) 250
200
150 T π- without resonances 100 p
π-
Figure All t pairs T compati
0 f
K s
m sp ectra Every
50 ble with the measured Λ ?
_ p oint on these curves results in a go o d t of
Λ
the computed m sp ectrum including res ?
0 ow to the
0 0.25 0.5 0.75 1 onance decays and transverse
resp ective measured particle sp ectra β
s From Ref
velocity cannot b e easily separated by the t pro cedure and must b e determined by
additional theoretical input Figure shows this ambiguity of the tting pro cedure
for the transverse momentum sp ectra of ve di erent hadron sp ecies measured in
98 157
A GeV S S collisions by the NA collab oration The exp erimental m sp ectra
can b e repro duced within a large range of temp eratures MeV by adjusting
97
accordingly Hydro dynamical studies show that dynamical consistency b etween
f
the longitudinal and transverse momentum sp ectra and with the freeze out kinetics
restrict the allowed freeze out parameters to a rather narrow domain around T
MeV and h i c In detail there remains however certainly some mo del de
f
p endence to these statements mostly related to the equation of state used for the
hydrodynamical expansion In spite of our theoretical prejudice towards freeze out
97 42
temp eratures in the MeV region we will therefore treat for the chemical
analysis T as a free parameter
Resonance Decays
At temp eratures ab ove MeV hadronic resonance pro duction b ecomes very imp or
tant After freeze out the resonances decay into stable particles which are indistin
guishable from their directly emitted brothers and sisters The measured distribution
of a particle sp ecies i is thus given by
thermal ow
R
X
dN dN T dN T
i
i
i
b
R i
dy dm dy dm dy dm
R
with the direct thermal contribution given by Eq or The contribution from
decays of resonances R i n with branching ratio b into the observed
R i
83
channel i is calculated according to
M
Y s
Z Z Z
R
dN
i
d Y ds g s dM
n
dy dm
s
Y
M
dN M
R
q
d Y dM
P p ME M m cosh Y y
p
Capital letters indicate variables asso ciated with the resonance R s is the invariant
mass of the unobserved decay pro ducts n and the kinematic limits are given
by
n
X
s m s M m
k i
k
p
Y y sinh
m
q
p sinh Y y m E m cosh Y y p
M M
sinh Y y m m
i
q
m E M m s p E
i
i
M
In Eq g s is the decay phase space for the n b o dy decay R i n for
n
the dominant b o dy decay assuming isotropic decay in the resonance rest frame it
is given by
g s s m
p
For n see Ref
The resonance sp ectrum dN dy dM entering the r h s of Eq is in general
R
itself a sum of a thermal or ow sp ectrum like Eq or Eq and of decay
sp ectra from still higher lying resonances for example most decays of high lying
nucleon resonances into protons or anti protons pro ceed through the or its
anti particle e g N p This has b een taken into
account in the calculations Extracting the degeneracy factors and fugacities of the
decaying resonances we write shortly
Z Z
R
X
dN
i R R
g b dm N N dy
R R i R R R R
i i
cut
dy dm
m y window
R
with the sum now including only resonances with identical quantum numbers if these
quantum numbers agree with those of sp ecies i the sum is meant to also include the
thermal contribution We will use this notation b elow
Between particles and anti particles we have the relation
R R R
N N N
R
R i R i
i
Particle Ratios
A quantitative comparison of the exp erimental strange particle yields or ratios with
the thermal mo del prediction must take into account the kinematic acceptance win
dows of the exp eriments The measured ratio of two di erent particles a and b is in
general given by
RR
dN
a
dy dm
dy dm
N
a R
a
RR
dN
b
N
dy dm
b
dy dm
R
b
Here R denotes the exp erimental kinematic acceptance window for particle sp ecies i
i
The advantage of considering particle ratios is the cancellation of the p o orly known
reball volume V at freeze out assuming that b oth particle sp ecies freeze out simul
taneously Of course in general one must consider the di erent resonance decay
contributions to the numerator and denominator in according to Eq A
rst naive estimate can however b e obtained by neglecting the resonance contribu
the purely thermal distribution Then tions and further taking for dN dy dm
i
the integration kernels of are indep endent of the particle rest masses m If we
a b
then choose R R i e cut the data to the same rapidity and m window the
a b
84
integrals over the Boltzmann factors cancel in the ratio which then simply reduces
to a quotient of fugacities and factors
s
N
a a a
N
b b b
est
With prop er ow distributions the integral kernels in are no longer inde
p endent of the rest masses of the particles and the simple estimate must b e
corrected by a prop er numerical calculation Similar comments generally apply to
the contributions from resonance decays
The Speci c Entropy
A useful quantity for studying the evolution of a nuclear reaction is the sp eci c entropy
S B as already discussed in Section If we knew S B and the temp erature we
would b e able to decide whether we are in the QGP phase or in the hadron phase
b ecause of the characteristic high sp eci c entropy of the plasma compared to the
hadron gas at the same temp erature Thus it would b e go o d to have a measure for
S B A rough estimate is provided by the following considerations
In the ultrarelativistic limit T m the entropy p er particle is indep endent of
temp erature and approximately S N a little less for b osons a little more for
fermions If the three charge states in which the stable nal state hadrons
tot
arise are equally distributed we have N N N The net charged
multiplicity N N is given by the number of incoming protons Therefore the
total baryon number is approximately B N N Putting all this together
one can write
tot
N N
S N S N N
tot
B N B N N N N
Figure The pro duct D S B as a
Q
function of for di erent values of
q s
Solid line dashed line
s s
dash dotted line The ad
s
ditional parameters are and T is
s
xed by the strangeness neutrality relation
The upp er curves are for D with
Q
out resonance decays the lower curves in
clude the decay of resonances after freeze
out Figure from Ref
De ning the sp eci c net charge D as
Q
N N
D
Q
N N
we can rewrite this as
S B D K
Q
D is an easily measurable quantity and most precisely accessible in emulsion exp er
Q
iments
Of course in a realistic hadron resonance gas at T MeV the ultra
relativistic limit S N is not realized We can test the accuracy of the estimate
for the sp eci c net charge in the Eq numerically We introduce the notation D
Q
hadron gas b efore p ost freeze out resonance decays In the upp er curves of Figure
for a hadron resonance gas at di erent values of near we plot S B D
s s
Q
as a function of The temp erature varies with according to the strangeness
q q
neutrality relation and actually covers a wide range over the interval shown
q
Obviously the pro duct S B D is indeed roughly constant as suggested by the es
Q
timate but the constant is approximately rather than The reason is the
larger sp eci c entropy of the heavier particles compared to the assumed value of
However after adding the resonance decay contributions to the total charged par
ticle multiplicity in the denominator of the lower set of curves in Figure is
obtained It is surprising to see how the decays bring the approximate constant K
again down to a value around as in the simple estimate Eq We thus have
with D a reasonably accurate measure for S B from a thermalized and chemically
Q
equilibrated hadron gas For hadronic systems far away from chemical equilibrium
however the estimate should not b e applied without further scrutiny
Chemical Parameters from Strange Particle Ratios
In this part we want to confront the thermal description with exp erimental data
This will b e done in several steps rst the input parameters for the thermal mo del
T V will b e determined from a su ciently large subset of the measured
s q s
quantities Due to the non linearity of the equations to b e solved see b elow it is
not a priori obvious how large such a minimal set of observables must b e Second the
validity of the thermal ansatz will b e sub jected to a test by making predictions for
additional observables for which data exist or can b e measured in future exp eriments
Third systematic deviations of the mo del predictions with the exp erimental data will
b e analyzed and interpreted
Before pro ceeding further we should comment on the determination of the freeze
96 97
out temp erature T It can b e extracted either from the hadronic m sp ectra or
f
17 20 19
from the particle ratios However it should b e stressed that these two pro ce
dures lead conceptually to di erent freeze out temp eratures The momentum sp ectra
determine mo dulo p ossible ambiguities from collective transverse ow the thermal
freeze out temp erature at which al l collisions stop and the momentum distributions
are frozen in The temp erature extraction from particle ratios relies on the assumption
of chemical equilibrium and thus determines the p oint of chemical freeze out where
the number changing inelastic collisions cease and the particle abundances are frozen
in Generically chemical freeze out o ccurs prior to thermal freeze out but in excep
tional situations like the sudden hadronization of a QGP into a dilute free streaming
hadronic system b oth freeze out pro cesses can o ccur simultaneously
We would like to stress that in the context of the following analysis temp erature
values far ab ove T MeV generally p ose a threat to its theoretical consistency
c
taking such temp eratures seriously one is necessarily led to doubt the lattice QCD
42
evidence for a decon ning phase transition around T MeV and one must face
c
serious questions ab out the applicability of the non interacting hadron resonance gas
picture underlying the analysis at T MeV a hadron resonance gas is already
b eyond its dense packing limit
Thermal and Chemical Parameters from the WA Experiment
The WA exp eriment provides data on multistrange baryon and antibaryon pro duc
tion on sulphur tungsten collisions at A GeV see Table The WA collab ora
tion has measured and very recently see WA contributions
to Refs and Ref they also extracted the sp ectra and yields of neutral and
charged kaons The data on strange baryons have b een analyzed indep endently by
two di erent groups Refs and Refs with mutually consistent results
We will follow here the ma jor steps of the work of Ref and comment on the other
work later
We concentrate here on the dataset published in Refs There we have
the four ratios
R for y and m GeV
for y and m GeV R
R for y and m GeV
R for y and m GeV
Note that these ratios are all measured in the same kinematic region and that only
three of the four ratios are indep endent b ecause of the relation
R R R R
First estimates for WA
84
The common kinematical region of the ratios in Eqs allows a rst estimate
via Eq Sp eci cally we obtain
q
s q s q s q s
s s
R R R R
s s q s s
s s q q q
This system of equations is easily solved for and
q s s
R R
q
R R
s
R R
s
from which one obtains
q s s
As we will see in the following subsection the rst two values are roughly correct
but the estimate for is wrong by ab out a factor due to uncorrected resonance
s
decays
Resonance gas analysis for WA
Incorp orating resonance decays and allowing for collective ow needs a numerical
treatment Because for the particle ratios we need the chemical freeze out temperature
which is not necessarily the same as the thermal freeze out temperature re ected in
the momentum sp ectra we take T as a free parameter and investigate three cases
f
A a thermal mo del without ow where the freeze out temp erature T is
f f
assumed to corresp ond directly to the value T MeV deduced from the
app
185 186
slop e of the transverse mass sp ectra of high m strange anti baryons
B a mo del with a freeze out temp erature of T MeV i e a value consistent
f
with the kinetic freeze out criterium developed in Ref This temp erature
also agrees with the QCD phase transition temp erature from lattice Monte
Carlo simulations It entails a ow velocity at freeze out of
f
C in order to maintain zero net strangeness in a hadron gas reball at relative
chemical equilibrium b etween strange meson and baryon abundances we con
sider also freeze out at T MeV with
f f
The system of equations to b e solved may b e written as
N N
s q s s s
R
N N
cut
q
s s s s
m m
N Y
N N N
s
q q s s q s
R
N Y
N N N
cut
s s q
q q s s
m m
N N
q
s s s s
R
s
Y N
N N N
cut
q s s
s s q q
m m
R
where the N are only functions of the rest masses and the given temp erature T as
i
can b e seen from The result of the numerical solution of is given in
Table
Since the probability to excite the various resonances contributing to the observed
nal particle abundances dep ends on T it is not surprising that slightly di erent
f
values of the chemical parameters result from the same data in the three di erent
scenarios Comparing the numerical results of Table with the simple analytical
approximation of Eq which contained no contributions from higher resonances
we can draw the following conclusions
The extraction of and according to Eq is only very weakly a ected by
q s
resonance decays and by the origin of the slop e of the m sp ectrum thermal or
ow The absolute value of the asso ciated chemical p otentials do es of course
dep end on the choice of the freeze out temp erature The reason is that in
Eq and resp ectively the rst resp ectively second term in the sums
o ccurring in the numerator and denominator completely dominates Neglecting
then the other terms leads to the same equation as in Therefore the
determination of and via Eqs and leads already to a precise
q s
result
Since the ratio is the crucial ingredient for the determination of the
strangeness saturation factor the e ects from resonance decays and ow can
s
b e clearly seen in this number From Eq we had extracted
s
This could b e drastically improved by the simple observation that the which
Table Thermal reball parameters extracted from the WA data on strange baryon and
antibaryon pro duction for three di erent interpretations of the measured m slop e Resonance
?
decays were included For details see text
A B C
thermal thermal ow strangeness balance
T MeV T MeV T MeV
f f f
s
T
s
MeV
s
q
T
q
MeV
B
s
S B
D
Q
is approximately as abundant as the decays into and this fast electro
magnetic decay cannot b e resolved exp erimentally Therefore ab out of the
observed s result from decays which improves the naive estimate by a
factor of to a value of Taking into account the full resonance sp ectrum
raises the value of even further to values around surprisingly close
s
to full equilibrium i e
s
We see from Eq and Table that is small and compatible with zero
s
This con rms the results of Ref based on the simple estimates
and implies that the WA data on strange baryon and antibaryon pro duction
from GeV A S W collisions establish a vanishing strange quark chemical
p otential This result is stable against large variations in the temp erature and
transverse ow velocities and thus insensitive to the pro duction and decay of
high mass resonances The reason is that these parameters are extracted from
particle antiparticle ratios in which the resonance decays largely cancel A
vanishing strange quark chemical p otential is natural in a strangeness neutral
QGP Note also that also the light quark fugacity is practically una ected by
the complex pattern of resonance formation and decay as well as by the amount
of transverse ow in the sp ectra
Assuming that at freeze out the hadronic system is in a state of thermal and
relative chemical equilibrium and thus fully characterized by the parameters
given in Table all other hadron abundances can now b e predicted and the
strangeness neutrality can b e checked The second to last row in Table shows
that under these assumptions strangeness neutrality is only established in sce
nario C Although it is not necessary that strangeness neutrality is satis ed
locally the conservation laws ensure strangeness neutrality only for the total
reball it is hard to see how lo cal variations from strangeness neutrality ab ove
j j could b e generated dynamically or by uctuations Thus we get into
serious problems with scenarios A and B
One should keep in mind that the strange mesons play a crucial role in the
strangeness neutrality relation in order to resurrect for example scenario
B b ecause it features a reasonably low freeze out temp erature it would b e nec
essary to break chemical equilibrium at least b etween the mesons and baryons
This would require the introduction of at least one further non equilibrium pa
rameter in addition to the global strangeness undersaturation factor As
s
we will see b elow and in Section entropy considerations p oint in a similar
direction Under these conditions the applicability of the concept of relative
chemical equilibrium in a thermalized hadron gas state to the observed nal
state in heavy ion collisions at CERN energies app ears highly questionable We
will however continue a little further down this line by asking whether the
concept works at least within the subspace of baryons and antibaryons The
following subsection will show that strikingly the predictions arising from this
picture are in qualitative agreement with exp eriment
A similar extraction of thermal and chemical parameters from the WA exp er
iment was done in Refs Because Refs deal with older data of
the WA exp eriment we only want to comment on Ref which uses the same
input as the calculation describ ed ab ove The strategy in Ref is somewhat
di erent They extrap olate the ratios of Ref to p and compare these
values to fully integrated thermal particle ratios They also force the system to
b e strangeness neutral using the strangeness balance equation to uniquely
determine the chemical freeze out temp erature Their results are T
MeV MeV and in agreement with the scenario C
B s
presented in Table ab ove
Table Input and predicted high m particle ratios near central rapidity y
?
for A GeV S W collisions The parameters for the three thermal thermal ow and
balanced strangeness scenarios are as sp eci ed in Table
A B C
N N
i j
cut
m GeV thermal thermal ow strangeness
balance
p
p
p p
K K
p GeV
K
s
K
s
Predictions resulting from the WA analysis
It is evident that the extraction of three thermo dynamic parameters and
q s s
from three indep endent particle ratios cannot b e considered a test of the generalized
thermal mo del considered here in particular since the extracted values dep end on
how the temp erature is xed in the analysis However with the thermal parameters
now xed according to scenarios A B or C further particle ratios can b e predicted
and are given in Table
Some of these ratios can b e compared with new data from A GeV S W col
185 188 189 190
lisions WA has published the following partially preliminary values
R for y and m GeV
for y and m GeV R
K
s
R for y and p GeV
K
K
s
R for y and p GeV
K
K
R for y and p GeV
K
K
The ratio agrees within error bars with the prediction but the large exp erimental
uncertainty limits its usefulness In the mo del is given by and thus should
s
b e if the strange quark chemical p otential vanished exactly R disagrees with
the prediction only for scenario A with the highest freeze out temp erature the two
values overlap within their error bars For larger the situation would improve
s
However again the exp erimental error bar is very large and a de nite conclusion
requires b etter data statistics The K K ratio is much more accurately known
its central value is slightly higher than the prediction but considering the error bars
and the fact that the exp erimental ratio includes all kaons down to m GeV
where the resonance contribution in particular to the K b egins to b ecome imp ortant
the agreement is go o d
The kaon to anti lambda ratios which are the only two ratios combining mesons
with baryons are quite badly o the predicted values only for the lowest assumed
freeze out temp erature the prediction is anywhere close to the data This raises the
question how strangeness neutrality is realized in the exp erimental data Clearly more
work is required to clarify these issues
There is also a recent exp erimental value for p for the similar S Au collision
166
system from the NA collab oration at CERN Unfortunately it was measured in
a di erent kinematic domain namely forward of central rapidity y and
integrated over all p GeV Therefore the comparison with WA is somewhat
problematic The exp erimental value from NA is p The ratios
predicted from the thermal parameters in Table for the same p range are
A B and C resp ectively for the three scenarios Again
there is agreement but due to the rather large error bars and the di erent y window
of the data the value of this statement is limited
From this comparison taking into account the partially still very large exp eri
mental uncertainties we conclude that the concept of relative chemical equilibrium
works reasonably well within the baryonic and mesonic sectors separately but fails
badly when mesonic and baryonic yields are compared with each other We will en
counter a similar problem with pions relative to the baryons when we analyse entropy
pro duction in Section This indicates a breaking of chemical equilibrium b etween
mesons and baryons
Thermal and Chemical Parameters from the NA Experiment
For the strange particle pro duction data of the NA collab oration a similar analysis
was done in Ref but now for the lighter symmetric system S S Although this
exp eriment has not measured any multistrange baryons or antibaryons it has accu
mulated a complete set of m and y sp ectra for all the hadrons which they could
identify Due to the symmetry of the system data taken at rapidities y can
b e simply re ected around the nucleon nucleon center of mass at y to obtain
cm
a complete rapidity sp ectrum Therefore this exp eriment can b e used to p erform a
global chemical analysis using the phase space integrated particle yields data or
to study the rapidity dep endence of the chemical parameters
The advantage of using full phase space data lies in the insensitivity to the ow
dynamics at freeze out In the WA analysis longitudinal and transverse ow com
p onents had to b e taken into account by cutting the theoretical ow sp ectra to the
exp erimental kinematic acceptance range For the data this is not necessary How
ever to achieve this simpli cation one must assume that the system is characterized
by a single set of thermal freeze out parameters While the observed rapidity in
dep endence of the slop e of the m sp ectra suggests a roughly rapidity indep endent
freeze out temp erature and transverse ow velocity the assumption of rapidity in
dep endent chemical p otentials cannot b e justi ed A analysis can thus only give
information on the rapidity averaged chemical p otentials
In Ref a chemical analysis of the data was p erformed and compared with an
analysis of the particle ratios from the midrapidity bin only Except for the extracted
values for the baryon chemical p otential whose average came out larger than the
central rapidity value in agreement with exp ectations based on the central rapidity
dip of the proton distribution seen in this collision system the results and conclusion
85
were the same More recently Slotta et al p erformed a t to the complete rapidity
dep endence of all particle yields and ratios We will comment on that work at the
end of this subsection and show that it mostly con rms the conclusions of Ref
Within the global ansatz for the integrated data one can also easily compare
the total hadron pro duction from S S and from elementary nucleon nucleon N N
101
collisions at the same collision energy p er nucleon Ga zdzicki et al have extracted
total multiplicities for an isospin symmetric nucleon nucleon collision at A GeV by
isospin averaging available p p and p n p d results using a reasonable extrap ola
tion to the n n case A comparison with the isospin symmetric S S collision system
illustrates very clearly the genuine nuclear medium e ects on particle pro duction
We will p erform such a comparison within the thermal mo del and parametrize these
e ects via the thermal and chemical parameters In spite of all the caveats ab out
applying a thermal mo del to single N N collisions the result provides a very nice
intuitive understanding of the basic di erence b etween nucleon nucleon and nucleus
nucleus collisions
101 102 157 165
The relevant data are summarized in Table From the quoted
multiplicities we build three ratios avoiding meson to baryon ratios b ecause of the
indications for a meson baryon non equilibrium from the WA analysis ab ove
Table List of measured particle multiplicities and corresp onding ratios The S S data are from
Refs the N N data from Ref The values listed under S S midrapidity are
the hadron multiplicities in the rapidity interval y
Multiplicity S S S S N N
Ratio midrap
K
K
p p
N
R
K
R
K
K
R
p
p p
D
Q
First estimates for NA
We b egin again by using the approximation to obtain a rst estimate of the
thermal parameters From the two relations
R R
K
s q s q
one easily gets
R R R R
q s
K K
With the available data cannot b e estimated by simple algebra For this we would
s
p in order to combine it with R such that the rest masses need the ratio R
p
p
166
and the temp erature cancel This ratio was recently published but only in a
restricted kinematic region Therefore we must determine numerically from a
s
complete resonance gas analysis
Resonance gas analysis for NA
The treatment of particle pro duction in an equilibrated hadron gas is always sensitive
to its mass sp ectrum fi g m g It should contain all indep endent hadronic degrees
i i
of freedom at freeze out However resonance data b ecome sparse ab ove GeV
and thus we don t have a complete set of hadronic states In contrast to the WA
88
analysis where we always used all well known resonance states up to a mass of
GeV we will here also explore the sensitivity of the chemical analysis to this limit
of our knowledge by restricting the exp erimental mass sp ectrum in alternative ways
lab eled by A B and C m
cut
A Only the measured particles K p n plus the are taken into account
the is added b ecause a large fraction of the measured stems from
B We include the pseudoscalar and pseudovector meson nonet and the baryon
spin o ctet and spin decuplet
C We include the full particle sp ectrum of all known hadronic resonances with
m m
i cut
We also treat the temp erature di erently than b efore Whereas in the WA
analysis it was essentially taken as a free parameter we x it here from the condition
of strangeness neutrality since certainly within the full phase space strangeness must
b e conserved
To obtain the parameters T and we now must solve a similar set
q s s
of equations as in plus the strangeness neutrality condition The
contributions from strong and electromagnetic decays after freeze out are taken into
account Decay pro ducts from weak decays can b e neglected b ecause they are strongly
discriminated against in the exp eriment by the requirement that the reconstructed
tracks p oint back to the target The results are given in Table Although for
the measured particle sp ecies the resonance decay contributions are appreciable the
simple rst estimates are again not strongly a ected b ecause of a large amount
of cancellation of these contributions from particle antiparticle ratios is mostly
s
sensitive to the exp erimental p p ratio for which no such cancellation o ccurs
Accordingly we see a strong sensitivity of to the chosen resonance sp ectrum This
s
is also graphically presented in Figure The errors on the freeze out parameters
were calculated by error propagation from the measured multiplicities
We now shortly discuss these results with resp ect to their physical and systematic
reliability The physical validity of our pro cedure can b e tested by comparing these
full phase space results with a similar analysis of midrapidity ratios Of course the
midrapidity results are correlated to the full phase space results since the same data
set is used an estimate of the systematic uncertainties of our pro cedure can however
still b e obtained from this comparison Since the midrapidity ratios have much larger
statistical errors error propagation b ecomes a problem and we thus refrain from
giving error estimates for the midrapidity results in Table
Table Generalized thermal mo del parameters resulting from our analysis of the particle ratios
given in Table for the various tested scenarios as describ ed in the text
T
q s s
MeV
A
B
S S C
C
C
C
A
B
S S C
midrap C
C
C
A
B
N N C
C
C
C
The observed di erences to the full phase space results are consistent with simple
102
exp ectations from the shap e of the proton rapidity sp ectra it is known that the
rather small sulphur nuclei cannot fully stop each other resulting in a baryon number
de ciency at midrapidity On the other hand excess strangeness pro duction was
157 165
found to b e maximal near midrapidity Corresp ondingly we nd a reduced
value of and an increased value of at midrapidity Values of ab ove unity
q s s
should not b e taken to o seriously the use of the strangeness neutrality condition
in a restricted rapidity window is not absolutely safe and the large exp erimental
error on the exp erimental p p ratio also directly propagates into the
determination of
s
Altogether the di erences b etween the midrapidity and the values are small
and for and T the di erences are even b elow This was recently con rmed in
s
85
an analysis by Slotta of the full rapidity sp ectra from this exp eriment Assuming a
b o ost invariant longitudinal expansion velocity pro le with a maximum ow rapidity
of relative to the center of mass this value yields an excellent t to the shap e
of the measured pion rapidity sp ectrum a rapidity indep endent freeze out temp er
ature T and strange quark chemical p otential he was able to repro duce the
s
Figure The dep endence of the extracted thermal parameters T and on the mass
q s s
sp ectrum of the hadron system at freeze out for central S S collisions at A GeV data For
details see text Also one set of values for N N collisions is plotted
p and rapidity sp ectra of the measured proton and all strange hadron sp ectra from
T
the NA S S exp eriment simultaneously in shap e as well as normalization The
calculation contains three free parameters the value of the baryon chemical p otential
at midrapidity one parameter A describing its dep endence on the longitudinal
B
co ordinate resp the space time rapidity and a strangeness suppression factor
and A were xed from the relative normalization and shap e of the and
s
B
A where is measured rapidity sp ectrum using the parametrization
B
B
from midrapidity was determined by adjusting the relative normalization of the
s
proton and sp ectra Assuming a freeze out temp erature of T MeV Slotta
found MeV A MeV and see Figure
s
B
Averaged over rapidity the baryon chemical p otential comes out as MeV in
go o d agreement with the analysis which gives MeV see Table Over
all strangeness neutrality is satis ed at the level of a few p ercent p erfect neutrality
could b e reached by a slight adjustment of the temp erature Problems arise with the
normalization of the pion sp ectrum which is underpredicted by ab out a factor
this problem is generic and will b e discussed extensively later in this review
In Figure an attempt is shown to t the same rapidity sp ectra with a lower
freeze out temp erature of T MeV comp ensating the for the slop e of the exp er
imental m sp ectra which for the S S data corresp onds to an apparent temp erature
2.5 1
dNe (a) dNe (b) dy Λ dy Λ 2 0.75
1.5
0.5
1
0.25 0.5
0.3 GeV < pT < 2.0 GeV 0.3 GeV < pT < 2.0 GeV 0 0 0 1 2 3 4 5 y 6 0 1 2 3 4 5 y 6
2
dNe (c) + dNe (d) − dy 3 K dy K
1.5
2 1
1 0.5
0.5 GeV < mT < 1.2 GeV 0.5 GeV < mT < 1.2 GeV 0 0 0 1 2 3 4 5 y 6 0 1 2 3 4 5 y 6
dN (e) dN (f) e 4 0 e 5 dy K s dy p−p
4 3
3
2 2
1 1 γ s = 0.75
0.1 GeV < pT < 2.0 GeV 0.2 GeV < pT < 2.0 GeV 0 0
0 1 2 3 4 5 y 6 0 1 2 3 4 5 y 6
Figure Rapidity distributions of K K K and p p from the NA S S
s
exp eriment The theoretical curves assume T MeV and ev
s
erywhere The other parameters are describ ed in the text The theoretical sp ectra are cut to the
exp erimental acceptance windows indicated in the gures by the index e on the vertical axis
of ab out MeV by a transverse ow velocity This is in the spirit of sce
f
nario B of the WA analysis The rapidity dep endent baryon chemical p otential is
now given by MeV and A MeV In this case the proton sp ectra are au
B
tomatically correctly normalized with but the strangeness neutrality and kaon
s
normalization are badly o very similar to the observations made in the WA S W
case It is striking to see that one additional parameter namely a meson baryon sup
85
pression factor repairs b oth de ciencies very economically Figs d e
m
85
The problem of underpredicting the pions by a factor remains however
Let us now return to the sensitivity of the chemical analysis to the hadron mass
sp ectrum This is illustrated in Figure for the analysis of the data of NA
From left to right the assumed sp ectrum of hadron resonances gets more and more
rich While T and app ear to b e quite insensitive to this shows appreciable
q s s
variations with the hadron mass sp ectrum However as more and more known reso
nances are included in the sp ectrum all values seem to converge to asymptotic limits
given by and T MeV
s q s
We conclude that chemical freeze out in central S S collisions at A GeV is
characterized by
a vanishing strange quark chemical p otential or
s s
a largely saturated strange quark phase space
s
The chemical freeze out temp erature is around T MeV if relative chemical
equilibrium and strangeness neutrality of the hadron gas are assumed However we
argued b efore that this temp erature is uncomfortably high Furthermore the hadron
gas picture cannot correctly repro duce the pion multiplicity Reducing the freeze out
temp erature to more reasonable values by p ostulating transverse ow leads to a
violation of the strangeness neutrality This can b e repaired by assuming a global
meson baryon suppression i e by giving up relative chemical equilibrium b etween
mesons and baryons but this do es not resolve the pion de ciency problem All of
these features are in qualitative and quantitative agreement b etween the S W and
S S collision systems
An interesting additional result is the di erence b etween the S S and the N N
data shown in the right sector of Figure for the C scenario Although one
must b e careful when applying the thermal description to N N collisions we think
that the extracted results still show clearly the imp ortant qualitative di erence b e
tween nucleon nucleon and nucleus nucleus collisions While takes similar values
q
intermediate b etween the S S and midrapidity results the strange quark fugac
ity tends to deviate from unity The temp erature is somewhat lower than in S S
case it is close to the canonical Hagedorn value of MeV this is also seen in
the slop e of the m sp ectra A clear qualitative di erence is observed however for
the strangeness saturation factor it is times larger in central S S collisions than
in N N interactions The strangeness enhancement is therefore clearly seen in the
thermal description by an increased factor
s 2.5 1 dNe (a) dNe (b) dy Λ dy Λ 2 0.75
1.5
0.5
1
0.25 0.5
β β T = 0.3c T = 0.3c 0 0 0 1 2 3 4 5 y 6 0 1 2 3 4 5 y 6
dNe 5 (c) dNe (d) + dy p−p dy 3 K
4
2 3
2 1
1 β T = 0.3c β γ T = 0.3c m = 0.37 0 0 0 1 2 3 4 5 y 6 0 1 2 3 4 5 y 6
2
Figure Similar to Figure but
dNe (e) −
assuming T MeV together
dy K now
with a transverse ow velocity
1.5 f
In this case the protons are cor
rectly repro duced with but
s the correct kaon normalization and over
1
all strangeness neutrality require a me
son baryon suppression factor
m
0.5
β T = 0.3c γ m = 0.37 0
0 1 2 3 4 5 y 6
Table Predicted total particle multiplicities in A GeV S S collisions for scenario C
with the freeze out volume xed by the total K multiplicity In the lower part of the table we
give the abundances of hadronic resonances b efore their strong decays but including the feed down
contributions from higher resonances
Total S S S S N N
multiplicities midrap
K
N
K
p p
p
Predictions resulting from NA
Similar to the WA case we can use the extracted thermal parameters to predict
20
further particle ratios Table lists the predicted total multiplicities from which
pt
was any ratio can b e constructed The p oint particle volume parameter V
xed by the total K yield and then used to normalize all other multiplicities
Comparing the predictions of Table with the measured values of Table the
following p oints emerge
The K yield is exact b ecause the K yield and the K K ratio were tted
For the baryons and p p data and prediction agree within error bars
this indicates that the K ratio is correctly repro duced all other ratios were
already used in the thermal analysis
Table gives a p ratio of and midrapidity The recently
166
published central rapidity value from the NA collab oration of p
is somewhat on the high side but also a ected by large statistical errors
Table Particle ratios and acceptance windows covered by the NA S Pb exp eriment at CERN
The ratios in the lower part of the table were obtained by extrap olating the data to a common
m window using a thermal ansatz with the measured m slop es but without correction for
? ?
the di erent y acceptances All values are taken from Ref
Ratio value y range p range
R y p GeV
R y p GeV
R y p GeV
R y p GeV
Ratio value y range m range
R y m GeV
R y m GeV
R y m GeV
R y m GeV
The negative particle multiplicity N is strongly underestimated This will b e
discussed in Section in the context of entropy pro duction
The resonance yields listed in Table show the imp ortance of resonance con
tributions For example the four isobars contribute ab out to the nal
proton yield and similarly for the antiprotons
Thermal and Chemical Parameters from the NA Experiment
172 173
The NA collab oration has published strange baryon ratios from S Pb col
lisions at A GeV They study the same set of ratios as the WA collab oration
Eqs and since the target nuclei are not to o di erent the results should in
principle b e comparable Unfortunately the NA ratios listed in Table are not
all given in the same kinematic window which makes the analysis more di cult The
rapidity sp ectra from S S collisions in Figure show that in general the particle
ratios have a strong rapidity dep endence and since for the asymmetric S W and
S Pb collisions the shap e of the rapidity sp ectra is not known a straightforward
comparison b etween the two exp eriments is not p ossible
Although the ratios may b e safe we should still mention that the absolute nor
malization of the NA strange particle yields has b een controversial see NA and
NA contributions in Ref for so far unknown reasons the rapidity distri
butions from the NA and NA exp eriments disagree by ab out a factor in nor
malization and also show a di erent shap e In view of these unsettled questions all
theoretical interpretations of these data sets should b e taken with a grain of salt
First estimates for NA
Similar estimates as in Section can b e extracted from these data only if the rapidity
dep endence of the ratios is neglected As already p ointed out this is very dangerous
and indeed a comparison of the results given in Refs and shows that at least
R is very sensitive to the selected rapidity interval
y R Ref
y R Ref
The p range for b oth ratios is the same p GeV
Since the particle antiparticle ratios in Table were obtained in the same rapidity
173
interval one can at least apply Eqs and to extract the fugacities
q s
The estimate for from Eq is less reliable due to the di erent rapidity intervals
s
R is violated of the required ratios Indeed the identity R R R
by the ratios given in Table by Neglecting this problem one obtains
s
173
and after multiplication by a factor for the contribution one gets
This is b elow the naive estimate from the WA data but we
s
already know that it will b e corrected upward once resonance decays are included
Resonance gas analysis for NA
173 17
The NA collab oration have analyzed their data following the work of Cleymans
et al This incorp orates the resonance contributions In this description strangeness
neutrality is taken as a constraint and full chemical equilibration is assumed
s
The result is shown in Figure All ratios are compatible with a fully equilibrated
hadron resonance gas at T MeV and MeV This result can
B
b e summarized by the parameters
q s
T MeV
s
After the rst naive estimate for in Section came out b elow the corresp onding
s
one for the WA results it is surprising to see that this resonance gas analysis
yields compatibility of the NA data with while a similar analysis of the
s
19
WA data nds strong incompatibility with and rather con rms the value
s
extracted in Section On the other hand the resulting values for and
s q
T are also di erent These di erences are obviously a consequence of the di erent
exp erimental particle ratios in particular of R which di ers from the WA value by
more than a factor As long as quantitative exp erimental information on the crucial
rapidity dep endence of these ratios is not available more detailed statements ab out
the consistency of the two exp eriments and their resp ective theoretical interpretations
cannot b e made It is however interesting to observe that the strange quark chemical
p otential seems to vanish in all these analyses of heavy ion data at CERN energies
indep endent of the selected rapidity window and is always quite large
s
Figure The T plane for a
B
strangeness neutral resonance gas with
The values compatible with the
s
measured particle ratios within one stan
dard deviation after resonance decays are
shown as hatched areas The common
overlap region de nes the b oundary of al
lowed values for T and The Figure was
B
taken from Ref
Thermal and Chemical Parameters from the AGS Experiments
The heavy ion exp eriments at the AGS in Bro okhaven are p erformed at a lower b eam
momentum of ab out A GeV Comparing these exp eriments with the ones at
CERN energies of A GeV allows to study the b eam energy dep endence
of particle pro duction We are interested in the di erences in strange hadron pro
duction and in the resulting changes in the thermal reball parameters An analysis
along these lines of the available data at the AGS from Si Au Pb collisions was
recently done in Ref Let us shortly discuss some small di erences relative to the
investigation of the CERN data presented ab ove
The hadron gas used in Ref incorp orates baryons up to masses of GeV
but mesons only up to GeV In view of the lower temp erature at the AGS
see b elow this seems justi ed
103
In Ref a nite size correction to the density of states in the partition
function was applied This correction is of the order R where
th th
q
mT is the thermal wavelength of a particle with mass m and R is the linear
dimension of the system This correction is imp ortant for low temp eratures
and for light particles and mostly a ects the ratios b etween a heavy and a light
particle e g p n
For extensive observables the volume correction was taken into account via the
87
excluded volume description This drops out from particle ratios
The calculation was done assuming absolute chemical equilibrium from
s
the b eginning i e no strangeness suppression was allowed for
Table Particle ratios calculated in a thermal mo del for two di erent temp eratures a baryon
chemical p otential of GeV and a strange quark chemical p otential xed by the condition
B s
of strangeness neutrality in comparison with exp erimental data statistical errors in parentheses
from central Si Au Pb collisions at A GeV The table was taken from Ref
Particles Thermal Mo del Data
T MeV T MeV exp ratio rapidity Ref
p n
d p n
p p
K
K
K
s
K K
p n
K K
d p
Together with the requirement of strangeness neutrality this last assumption re
duces the number of input parameters to two T and The temp erature
f B q
was xed by an analysis of the measured contribution to the proton yield
104
An investigation showed that the p ratio is compatible with a temp erature in
the range of T MeV This very useful information is unfortunately not
available from the CERN exp eriments If measured well it allows an accurate extrac
tion of the chemical freeze out temp erature indep endent of the baryon and strange
chemical p otentials
The baryon chemical p otential is xed by a compromise b etween the measured
26
p n ratio and considerations regarding the particle density at freeze out The
latter can b e xed from the size of the freeze out volume as measured by HBT
105
interferometry and the measured baryon multiplicity The resulting value was
26
given as MeV
B
The results of this analysis are shown in Table In general the agreement b e
tween the thermal mo del ratios and the exp erimental ones is very go o d The an
tibaryon baryon ratios are very sensitive to the temp erature for xed in the given
B
temp erature interval they vary by a factor of and cover a range which includes the
exp erimental value
In view of the di erent rapidity windows for the various measurements and the
generic rapidity dep endence of particle ratios such an agreement is not trivial How
d
ever most of the data cover a rather wide rapidity range around the central rapidity
of and stopping is more e cient and longitudinal ow less imp ortant at AGS
106 107
energies To a large extent the analysis of Ref thus resembles in character
the global analysis of the NA data at CERN in Section
Taking a closer lo ok at the strange particle ratios one observes an interesting sys
tematic b ehavior While within the chosen temp erature interval the predictions for
the two particle antiparticle ratios K K and overlap nicely with the exp er
imental values inside the error bars this is not the case for some of the other ratios
containing strange quarks Let us therefore lo ok at those and divide the exp erimental
mean values by the two mo del values Neglecting the ratio see b elow we obtain
ratio T MeV T MeV
K
exp value
K
mo del value
K
s
p n
K K
The mo del ratios are systematically b elow the exp erimental values by a factor
Now all these ratios involve particles with one more strange quark or antiquark in the
numerator than in the denominator This indicates incomplete strangeness satura
tion and thus a violation of the mo del assumption of absolute chemical equilibrium
We can soften this assumption by assuming only relative chemical equilibrium and
introducing as b efore the parameter through Eq From the ab ove table we
s
extract a value of thus the level of strangeness saturation reached at
s
the AGS is similar to the CERN value
We left out the ratio for a particular reason neither do es it follow the ab ove
systematics nor do es it agree with the predictions of any of the available event
146
generators So far there is no mo del which can explain such a large ratio con
sistently with the other observations This strange result cries out for indep endent
exp erimental con rmation
26
Braun Munzinger et al have also tried to con rm the thermal parameters by
comparing the measured transverse momentum sp ectra with the mo del predictions
from a thermalized source with the same temp erature undergoing longitudinal and
96 97
transverse hydrodynamical ow Since the mo del was developed for symmetric
137 140
systems and the new momentum sp ectra for Au Au collisions were not yet
e
publicly available this comparison was done in Ref for the lighter Si Al system
The m sp ectra were describ ed by Eq with After adding resonance decay
26
contributions excellent agreement is obtained for al l particle sp ectra from the pions
to the deuterons The presence of transverse ow is very clearly seen in the slop e of
d 139
We de ne as central rapidity the value where the pion pro duction p eaks
e
A similar comparison of the m sp ectra with E data from Si Au collisions is shown in
?
Ref with very similar parameters
the exp erimental sp ectra for m m GeV i e b eyond the resonance decay
95
region which attens considerably with increasing mass of the hadrons in the data
presented in Ref the deuteron and pion slop es di er by ab out This is well
repro duced by the assumed average ow velocity h i for T
f f
MeV resp ectively
We conclude that a thermal mo del for Si Au Pb Al collisions at AGS energies
works well and leads to the following thermal parameters
T MeV MeV
B
MeV
s S
In terms of the fugacities used b efore this translates as
q s
This result can b e compared with the earlier less elab orate thermal mo del in
terpretations of Refs The authors of Ref consider only the two parti
cle antiparticle ratios K K and from Table as input As discussed b efore
resonance decay contributions can b e neglected for these ratios Using the require
ment of strangeness neutrality the fugacities and also the freeze out temp erature can
24
b e extracted
T MeV
q s f
This agrees very nicely with Eqs the small di erences are probably due
to the fact that in contrast to Ref the ratio was not tted exactly in Ref
Sp eci c Entropy from Heavy Ion Exp eriments
The careful reader will have noticed the conspicuous absence of any discussion of
the pion yields so far Although the pion is the most abundantly pro duced hadron
esp ecially so at CERN energies it carries no conserved quantum numbers and is
thus unsuitable for extracting information ab out the chemical p otentials In chemical
equilibrium the pion multiplicity is a function of the temp erature alone However of
all the thermo dynamic parameters used in Section the chemical freeze out temp er
ature T is the most uncertain one This uncertainty a ects the prediction of total
f
pion multiplicities in a very direct way
On the other hand the pions b eing the most abundant of all pro duced hadrons are
the dominant carriers of entropy in the collision reball This was already discussed
in Section where we related the intensive quantity S B to the sp eci c net charge
D which is normalized to the total charged particle multiplicity and thus dominated
Q
by pion pro duction
In Section we discussed extensively whether at freeze out the collision zone can
b e describ ed as a hadron resonance gas in thermal and some kind of relative chemical
equilibrium We will now discuss and test an additional asp ect of this picture namely
its sp eci c entropy content
A measurement of D as a function of rapidity was p erformed for the S Pb system
Q
109
at A GeV by the EMU collab oration The value at central rapidity is D
Q
This can b e compared with values calculated from the hadron gas mo del
with the thermal parameters extracted from the WA data Both collision systems
feature nearly the same number of target participants No detailed comparison has
b een made for the NA parameter values but since the thermal parameters extracted
from this exp eriment are similar the conclusions would b e the same
The thermal mo del results for D and S B corresp onding to the thermal param
Q
eters extracted from the chemical analysis are given in the last row Table Only
for scenario B which is plagued by a strong violation of strangeness neutrality the
mo del pro duces a value of D which is compatible with the exp erimental value of
Q
In all other cases the calculated D is higher then the measured one
Q
implying that the data contain a higher total charged multiplicity and larger sp eci c
100
entropy S B than provided by the hadron gas mo del The discrepancy is of the
order of a factor
The situation is similar for the NA exp eriments The measured D is given in
Q
Table The hadron gas values for the parameters from the scenario C would
b e D for the data and D at central rapidity In other words the
Q Q
measured negative particle multiplicity N from Table is higher than
the predicted one N from Table again the discrepancy is large of the
15
order of Similar conclusions have b een reached earlier by Davidson et al
An indep endent way to arrive at this statement has recently b een presented by
110
Ga zdzicki He lo oked at the systematics of pion pro duction in symmetric nuclear
reactions at b eam energies from A GeV BEVALAC to A GeV CERN and
compared it with pion pro duction in elementary nucleon nucleon collisions in the
111
same energy range According to his analysis one sees for BEVALAC and AGS
energies a suppression of pion pro duction in nuclear collisions relative to nucleon
110
nucleon interactions
h i h i h i
AA NN
hN i hN i hN i
P P AA P NN
Here h i is the mean pion multiplicity for symmetric nuclear collisions A A or
nucleon nucleon reactions N N resp ectively and hN i is the number of partici
P
pating nucleons Note that in the studied energy range turns out to b e valid
p
s Pro ceeding to the higher CERN indep endent of the center of mass energy
NN
110
energies he obtains from an analysis of the NA data on S S collisions a value
of
h i
hN i
P
Ga zdzicki also derives an absolute value for the entropy in the form S hN i
P
110
where S is the entropy in units of the entropy p er pion Taking for
the latter the thermal value of ab out units p er pion we obtain S B
This is again ab out higher than the hadron gas value resulting from the thermal
mo del parameters for the S S collisions given in Table in agreement with our ab ove
conclusion
This feature is generic for all heavy ion exp eriments at CERN energies The
hadron gas mo del assuming relative chemical equilibrium with only the absolute
strangeness level out of equilibrium is not consistent with all exp erimental observa
tions As long as the mo del is forced to yield zero net strangeness it predicts values
of D well ab ove the measured values irresp ective of the particular choice for the
Q
25
freeze out temp erature In other words the sp eci c entropy S B in the CERN data
is large and cannot b e repro duced by the hadron gas mo del
At the lower AGS energies D has not b een investigated exp erimentally But we
Q
can consider the absolute pion pro duction directly Table shows that the thermal
mo del slightly overestimates the measured pion nucleon ratio but not at a level which
would indicate serious disagreement From the extracted thermal mo del parameters
a value of S B was calculated in Ref this is signi cantly lower than
the CERN values We conclude that at AGS energies the entropy pro duction is low
and can b e describ ed by the thermal mo del with a hadron resonance gas equation of
state while at CERN energies entropy pro duction is dramatically higher and cannot
b e repro duced by the hadron gas mo del
Discussion of the Results
We have shown the results of an interpretation of exp erimental particle abundances
in a thermal hadron gas mo del with sp ecial emphasis on strange particles For the
lower AGS energies the mo del yields an excellent description of all hadron sp ectra
and particle abundances from Si Al and Si Au collisions It will b e interesting to
see whether this can b e con rmed for Au Au collisions where some of the necessary
data have already b een taken this largest presently available collision system at the
AGS is exp ected to show even clearer signs of thermal and chemical equilibration
sup erimp osed by collective expansion
At the higher CERN energies on the other hand some features of the data can
not b e repro duced by the mo del What exactly go es wrong is not yet quite clear the
interpretation of how the mo del fails dep ends to some extent on how it is applied to
the data The crucial parameter is the freeze out temp erature whose determination is
a b ottleneck in the analysis Whereas the baryon and strange fugacities can b e very
precisely given by a few particle antiparticle ratios the temp erature is much
harder to determine One of the reasons is that one must distinguish b etween the
thermal and the chemical freeze out temp erature The thermal freeze out temp erature
can b e extracted from the slop e of the m sp ectra although the separation of thermal
97
and collective motion is not quite mo del indep endent Dynamical arguments suggest
thermal freeze out around T MeV The chemical freeze out temp erature is
f
re ected by the particle ratios ratios b etween heavy baryons and a light hadrons
mesons show the largest sensitivity to the freeze out temp erature
This sensitivity can however b e exploited only if prior to freeze out chemical
equilibrium was established to b egin with Whether this is true or not is an imp ortant
dynamical question whose answer would provide crucial information ab out various
time scales in nuclear collisions We have argued that chemical equilibration happ ens
in stages and that the equilibration of the overall strangeness level and in a hadronic
environment of the baryon antibaryon meson pro cesses should o ccur most slowly
If the latter pro cesses have not equilibrated the ab ovementioned meson baryon ratios
are useless for a determination of the freeze out temp erature
We have therefore based our chemical analysis on particle ratios taken entirely
from within the baryonic or from within the mesonic sector It turns out that the
baryonic ratios by themselves do not x the freeze out temp erature very well if there
is a preference at all it is that the not very accurately known triple double strange
ratios prefer a high freeze out temp erature of the order of the m slop e of
MeV
Another in principle imp ortant constraint is the condition of overall strangeness
neutrality for the decaying collision region Unfortunately it can also b e exploited only
if the system is chemically equilibrated or if chemical equilibrium is broken in par
ticular well controlled ways which we have lab eled by relative chemical equilibrium
Assuming relative chemical equilibrium with resp ect to all pro cesses except strange
pair pro duction the strange and non strange baryon antibaryon and meson meson
ratios except for those involving pions are compatible with strangeness neutrality in
a hadron resonance gas if the freeze out temp erature is around MeV Lower
temp eratures are inconsistent with strangeness neutrality in a hadron gas unless the
meson baryon equilibrium is broken but even by allowing for such a p ossibility the
disagreement with the measured pion yields cannot b e resolved
Summarizing these comments we conclude that the CERN data do not allow for a
consistent and comprehensive interpretation within the hadron gas mo del even after
allowing for some simple patterns for breaking absolute chemical equilibrium Still
the chemical parameters extracted from the analysis presented in the previous chap
ters show a number of strikingly consistent features which are worth b eing exp osed in
a systematic way In spite of all the caveats ab ove these features provide an intuitive
and internally consistent picture of some of the main characteristics of high energy
nuclear collisions
A priori nuclear collisions are characterized by the two controllable parameters
p
s i e the center of mass energy p er participating nucleon pair and B the
part
number of participating pro jectile and target baryons which is related to the size of
target and pro jectile nuclei and the impact parameter Accordingly we will sub divide
p
s and those the observables into two categories those which scale dominantly with
which show mostly a scaling with B We call these di erent scaling laws energy
part
scaling and size scaling resp ectively
Figure summarizes graphically the thermal parameters extracted in Sections
and The initial reball temp erature is related to the initial energy density and
thus exp ected to exhibit mostly an energy scaling a weaker size scaling arises from
the di erent stopping p ower of light and heavy nuclei which transforms b eam energy
into internal excitation energy But what is the exp ected scaling of the freeze out
temp erature T
f 220 Tf [MeV]180 140 100 1.6 λ s 1.4 1.2 1
λ 4 q 3 2 1
γ 1 s 0.8 0.6 0.4 0.2
β 0.6 < f> [c] 0.4 0.2 0 40 S/B 30 20 10 0 Si − Au(Pb) N − N S − S S − W(Pb)
14.5 A GeV 200 A GeV 200 A GeV 200 A GeV
Figure Summary of the thermal parameters extracted in Section In the last row of
diagrams the triangles corresp ond to the value of S B in a hadron gas with the thermal parameters
shown in the rst rows while the crosses corresp ond to measured values of this quantity The
values of S B in the rst three columns are from Ref note that the value plotted by a cross in
the Si Au column is really from Si Al collisions The exp erimental S B value in the last column
is from the D measurement using Eq
Q
If we assume that T is determined by geometry and o ccurs when the mean free
f
path is of the order of the transverse radius R of the reball
F
R T T
f F f
n T h v i T
f f
where n is the reball density and h v i the thermal cross section at freeze out
then we exp ect T to scale with the inverse of the system size The reason is that
f
b oth the particle density and the thermal cross sections rise with temp erature see
e g Figure Eq can dep end on the b eam energy only through the baryon
stopping p ower via the relationship b etween the reball density n and T its energy
f
scaling is thus exp ected to b e very weak
This exp ectation is in stark contrast to the b ehavior seen in Figure where T
f
shows a clear scaling with b oth increasing b eam energy and increasing size of the
system Note that we plotted in Figure the chemical freeze out temp erature as
determined by the particle ratios and the strangeness neutrality condition and not
the measured slop e parameter of the m sp ectra As seen T follows the scaling b e
f
havior exp ected for the initial temp erature T as discussed ab ove Somehow the nal
particle ratios seem to have a memory of the initial conditions of the reball Naively
this would seem to contradict the thermal interpretation b ecause thermalization by
assumption implies memory loss via an increase in entropy
However in our discussion of thermal vs chemical freeze out we noted the rather
strong evidence in the transverse momentum sp ectra for the presence of collective
transverse ow at freeze out In the presence of ow the geometric freeze out cri
terium should b e replaced by a dynamical one b ecause collective expansion leads
95 112
to dynamical decoupling b efore the geometric freeze out criterium is satis ed
Hydro dynamical simulations suggest that in nuclear collisions thermal freeze out is
96 97
entirely dominated by the collective dynamics It is more than likely that the same
is true for chemical freeze out b ecause the only di erence b etween the two freeze out
criteria results from the substitution of the total by the corresp onding inelastic par
tial cross section Since higher initial energy densities generate larger hydrodynamic
ow and larger hydrodynamic ow leads to earlier dynamical freeze out at a higher
96 97
freeze out temp eratures the qualitative b ehavior of T seen in Figure can b e
f
understo o d in this picture In this sense the development of collective ow helps the
system to remember the initial conditions of the reball in spite of thermalization
The two fugacities and fall o going from left to right in Figure At
s q
the AGS the large value of re ects the high baryon density caused by the nearly
q
complete baryon stopping at these energies A part but not all of the strong decrease
of at CERN energies can b e understo o d by the larger freeze out temp eratures there
q
on top of this trivial e ect it re ects however the onset of partial transparency at
CERN energies Compared to the strong jump b etween the lower rst column
and the higher b eam energies second to fourth column varies very little with
q
the system size It should b e noted however that the values in the second and
third column are from particle ratios whereas the last column is from the WA
midrapidity ratios The analysis of Ref showed that dep ends on rapidity and
q
at CERN energies is lower at midrapidity than in the nuclear fragmentation regions
Comparing the values at midrapidity from S S NA S W WA and S Pb
q
collisions NA as they were given in Section one can see a rising tendency with
the size of the nuclear target This re ects again the increasing baryon stopping
p ower of larger nuclei which counteracts the onset of nuclear transparency at CERN
energies
Hence is a measure for the e ciency of baryon number stopping of di erent
q
p
size nuclei at di erent energies At the relativistic heavy ion collider RHIC s
GeV one exp ects at central rapidity indep endent of the size of the colliding
q
nuclei
In rst approximation must follow the b ehavior of b ecause of the strangeness
s q
neutrality relation However it is not all trivial that for all nuclear collisions at SPS
energies the value of is approximately corresp onding to a vanishing strange
s
quark chemical p otential In N N collisions this seems to b e di erent but due to
the large error bar a safe statement cannot b e made Indeed this app ears to b e the
most stable result of the chemical analysis and it is indep endent of the actual value
of the freeze out temp erature and how the strangeness neutrality is actually realized
b etween the baryons and mesons We already commented b efore on the intriguing
agreement of this value with the one prevalent in a strangeness neutral quark gluon
plasma To what extent this may b e more than a diab olical coincidence will b e
discussed in Section
The strangeness suppression factor shown in the fourth row of Figure demon
s
strates clearly that the physics of nuclear collisions is di erent from N N collisions
For nuclear collisions is close to while in the N N case is much smaller
s s
This size scaling of was worked out in more detail in Ref The increase of
s s
with B is a clear sign that strangeness equilibration is caused by secondary in
part
teractions The large degree of strangeness saturation in nuclear collisions which is
re ected by is a strong indication that unusually e cient and rapid mechanisms
s
for strangeness pro duction are at work
In the second but last row of Figure we have plotted the average transverse
collective ow velocity at the p oint of thermal freeze out i e at the p oint of decou
pling of the momentum sp ectra For this we need to know the exp erimental slop e of
the m sp ectra and the thermal freeze out temp erature At the AGS the transverse
ow and the thermal freeze out temp erature can b e determined separately from the
momentum sp ectra b ecause fortunately also momentum sp ectra of very heavy parti
26
cles like the deuteron are available still the determination is not yet very accurate
and lower thermal freeze out temp eratures down to T MeV comp ensated by
corresp ondingly larger transverse ow velocities still give acceptable ts to the m
sp ectra The identity of the thermal and chemical freeze out temp eratures at the
AGS is thus not rigorously proven although it seems to work quite nicely At CERN
energies such a clear separation of thermal and collective motion from the sp ectra is
f
not yet p ossible The transverse ow must b e determined with the help of a theo
retical mo del for the dynamics of the reball and involves a freeze out criterium Such
96
calculations are only available for the S S case at SPS energies These calculations
suggest thermal freeze out at T MeV with h i this value
f
has b een indicated in the S S column For the N N case collective hydrodynamical
expansion is not exp ected so we entered the value h i in the second column
f
The value in the last column was obtained by assuming for lack of b etter knowledge
f
A rst deuteron sp ectrum was presented by the NA collab oration in Ref but these data
have not yet b een sub jected to a full thermal ow analysis
that thermal freeze out o ccurs also in the heavier systems at T MeV see sce
185 186
nario B in Section by averaging over the m slop es quoted by WA and
173
NA
Generically the exp erimental m sp ectra atten for larger collision systems in
dicating earlier thermal decoupling at a higher temp erature and or larger collective
96
transverse ow Hydro dynamical calculations suggest a combination of b oth Com
paring Si Au collisions at A GeV and S W collisions at A GeV b oth similar
in size we see a decrease in h i combined with an increase in the thermal freeze out
f
temp erature re ected in the atter m slop e This is connected with the increas
ing nuclear transparency at higher energies the resulting fast longitudinal expansion
leads to earlier freeze out when transverse motion has not yet developed as strongly
Finally we consider the entropy p er baryon S B in the last row of Figure
There are two marks in each plot The crosses corresp ond to the measured entropy
110
obtained from the pion multiplicity or D The triangles corresp ond to the values
Q
in a thermal mo del with the parameters plotted in the rst rows The exp erimental
values for S B increase continuously from left to right in Figure The strong rise of
entropy pro duction from AGS to SPS energies is exp ected The observed size scaling
can b e understo o d in terms of increasing stopping p ower We already discussed in
Section but want to p oint out again that the thermal mo del predictions lie above
the measured values in Si Au AGS collisions and N N collisions but below the
measured values in the sulphur induced reactions at CERN
In summary the thermal quantities can roughly b e divided into two classes T
f
and S B scale dominantly with energy while and h i scale mostly with
q s s f
size We have seen that the qualitative b ehavior of these parameters is fully consistent
with the exp ectations based on our present understanding of the heavy ion collision
dynamics However quantitatively the thermal hadron gas picture is plagued by
internal consistencies In the following Section we will pick up on these problems and
use them to extract some further constraints on the early collision dynamics and the
freeze out pro cess
Consequences for Freeze Out Mo dels
In the preceding Sections we presented an extensive study of strange parti
cle abundances in thermal mo dels Since a thermalized system surrounded only by
vacuum necessarily b egins after a while to expand hydrodynamically and any ther
mal mo del analysis should for consistency b e done on the basis of a hydrodynamical
evolution background This introduces the ow velocity as an additional essential pa
rameter into the analysis which must b e determined consistently with the rest of the
thermal parameters and with the dynamical evolution and freeze out kinetics In this
light the strangeness signal for QGP formation acquires additional facets b eyond the
mere question of enhanced strange hadron pro duction in ultrarelativistic heavy ion
33
As we showed also the temp erature and chemical collisions as originally discussed
p otentials at the chemical freeze out p oint can b e extracted from strange particle
abundances once the e ects from the decay of unstable resonances after freeze out
are prop erly taken into account Combining the exp erimental information from the
various strange particle abundances and from their momentum sp ectra we obtain
valuable insight into the state of the system at freeze out using dynamical mo dels
we can then draw further conclusions also ab out earlier stages of the collision
We saw that contrary to the situation at the AGS at CERN energies the assump
tion of a thermalized hadronic system in relative chemical equilibrium cannot describ e
al l observables There is a crucial inconsistency with the observed pion pro duction
which we interpreted in terms of a lack of entropy in the hadron gas mo del compared
to the data Furthermore the chemical analysis p oints towards chemical freeze out
temp eratures near MeV which are problematic for any weakly interacting hadron
gas picture and at variance with lattice QCD results on thermally equilibrated sys
tems of strongly interacting particles Up to now no fully consistent solution to these
problems has b een provided However there are a number of interesting theoretical
suggestions in this connection which we would like to shortly discuss in this Section
Sequential Freeze out
An implicit assumption in the thermal and chemical analyses describ ed in this review
is that all particles freeze out simultaneously when the critical freeze out temp erature
T T is reached This is a technical requirement related to the problem that
f
it hard to include the mo di cations of partial freeze out of some selected particle
sp ecies on the subsequent hydrodynamical evolution of the remaining system But
it is nevertheless p ossible that di erent particle sp ecies decouple from the system at
di erent times or temp eratures dep ending on their average cross sections with the
114 115
medium which at CERN energies consists mainly of pions Such a sequential
freeze out can o ccur b oth for the chemical comp osition of the reball and for the
momentum distributions
In Ref the idea of sequential freeze out was applied to the measured strange
and non strange particle abundances Freeze out criteria of the type lead to
di erent freeze out temp eratures T for particles with di erent cross sections h v i T
f
18
Cleymans et al suggested that generically strange particles interact more weakly
than non strange hadrons and thus freeze out earlier Assuming a typical factor
b etween these two types of cross sections and adiabatic spherical expansion they
argue that freeze out of strange particles at T MeV is consistent with freeze out
of non strange particles at T MeV The conceptual di erence b etween chemical
and thermal freeze out temp eratures was not considered in this work and thus no
theoretical justi cation for the observed similar m slop es of al l hadrons could b e
given This mo del is able to resolve the excess entropy problem in terms of a larger
sp eci c entropy S B of the hadron gas at the lower freeze out temp erature of the non
strange hadrons at the higher freeze out temp erature of the strange hadrons it agrees
with the values around S B quoted ab ove It is not clear how this additional
nal state entropy arises in the mo del implicitly it is generated b etween the strange
particle and non strange particle decoupling p oints The question of viability of the
hadron gas picture at the high strangeness freeze out temp erature is not addressed
It is natural to ask whether there is any direct evidence for sequential freeze out
114 115
in nuclear collisions It was suggested many years ago that kaons should freeze
out from a baryon rich environment earlier than most other hadrons based on the
very small K N cross section As far as thermal freeze out is concerned this idea
can b e tested by lo oking at the two particle momentum distributions Hanbury
Brown Twiss HBT interferometry naively if the kaons decouple earlier than
say the pions and the reball is expanding the K K correlation function should
re ect a smaller HBT radius than the correlation function Measurements of
152 153 116 176 177
kaon and pion correlations at the AGS and at the SPS at rst sight
indeed seem to show such an e ect However it is not clear whether this is indeed
evidence for a smaller kaon freeze out radius Pion HBT radii are generically larger
than kaon radii due to a larger contribution from delayed resonance decays which
create a halo to the source Furthermore transverse collective expansion generates a
p
117 118
characteristic m dep endence of the HBT radii which causes kaon radii to
b e smaller than pion radii if measured at the same p The situation at the AGS has
not yet b een fully analysed at the SPS no statistically signi cant di erences have
176
b een observed b etween K K and K K correlations for K the earlier freeze
out argument do es not apply b ecause it has a large cross section with nucleons and
177
the smaller kaon radius compared with the pion one is claimed to b e consistent
with the exp ected e ects from resonance decays and transverse ow
Thus there is presently no clear direct exp erimental evidence for sequential thermal
freeze out To test this idea further for chemical freeze out requires extensive kinetic
and hydrodynamic simulations
Sudden Hadronization Scenarios
Another p ossibility to solve the inconsistencies of the thermal analysis is to p ostulate
that that the particles do not decouple from an equilibrated system However even
in that case one still has to explain the approximately thermal nature of the momen
tum sp ectra and the remaining order in the particle ratios which is describ ed by the
vanishing strange quark chemical p otential A completely non equilibrium
s
119 120
approach to hadronization was recently initiated by the Budap est group Un
fortunately this mo del cannot predict the nal hadronic momentum sp ectra and is
not yet in a form where it can b e implemented into dynamical calculations
In Ref it was argued that the apparently large sp eci c entropy and vanishing
could b e due to the existence of a QGP in the early stages of the collision which
s
hadronizes suddenly into a hadronic system which immediately decouples The exis
100
tence of a QGP would naturally explain the excess sp eci c entropy and the van
ishing Sudden hadronization followed by immediate decoupling was required in
s
order to preserve the values of the chemical p otentials across the phase transition and
avoid their re equilibration towards equilibrium hadron gas values b efore freeze out
However no consistent dynamical mo del which would pro duce the desired results was
found in Ref
The idea of sudden QGP hadronization also called QGP break up mo del was
121 123 124
recently picked up by several groups They revived an old idea by van Hove
who suggested that if the hadronization phase transition is of rst order hadroniza
g
tion could o ccur dynamically through a de agration or detonation sho ck front Such
a violent pro cess results in very rapid hadronization in which freely moving particles
are ejected into the surrounding vacuum with large average velocities
The hadronization of a QGP through a sho ck front must satisfy energy momentum
122
and baryon number conservation across the front This de nes the Taub adiabat
n P P
QGP QGP QGP QGP H
n P P
H H H H QGP
where n is the baryon number density the energy density and P the pressure in
the initial QGP and nal hadronic state resp ectively Even if for the nal state an
equilibrium hadron gas equation of state is assumed Eq is not su cient to
determine the temp erature and chemical p otential change through the hadronization
front As a further constraint one often assumes entropy conservation which leads to
the additional equation Poisson adiabat
S S
QGP H
B B
QGP H
For sho ck fronts Eq is generally not a go o d assumption but it can b e used to
give upp er or lower limits on the changes of thermo dynamic parameters during the
hadronization
The idea b ehind the approaches in Refs is to see whether a detonation
or de agration scenario can provide a dynamical setting in which a sup erheated equi
librated hadron gas can arise in the nal state which has T MeV and
s
is strangeness neutral but which successfully ghts its inherent tendency to return
into the QGP state and rather decouples immediately due to its rapid dynamical
expansion However these pap ers do not address the problem that such an equili
brated hadron gas would again yield to o little sp eci c entropy In this sense they are
complementary to the approaches of the previous subsection No attempts have b een
made yet to include suitable deviations from thermo dynamic equilibrium in the nal
state
Eqs and de ne a detonation or de agration front in space time which
123 121 122
can b e either timelike or spacelike The time evolution is roughly as follows
The system starts from a QGP with a high initial temp erature assumed to b e around
MeV or higher Then it sup erco ols to ab out T Rapid longitudinal
c
125
expansion and co oling of the QGP combined with a very large nucleation time of
around fm c for hadron gas bubbles prevent the system from hadronization at
T The strong sup erco oling is basically caused by the requirement only if the
c
QGP is strongly sup erco oled and the hadron gas is strongly sup erheated the larger
sp eci c entropy in the QGP phase can b e absorb ed by the outgoing hadrons
g
A sho ck front is called a detonation if in its rest frame the velocity of the incoming QGP
matter is lower than that of the outgoing one hadron gas otherwise it is called a de agration
The sup erco oled QGP is mechanically unstable Within the bag mo del the QGP
123
pressure turns negative already at T T At T sudden hadronization
c c
via a de agration detonation sho ck sets in The outcome is a sup erheated hadron gas
with a temp erature around MeV After the de agration detonation the particles
122 123
can considered as frozen out
Mo dels of this type explain naturally some of the puzzling exp erimental results
121
and are nearly conserved during the detonating hadronization Thus
B s
the naturally vanishing strange quark chemical p otential of a strangeness neutral
QGP can b e dynamically transferred to the nal hadronic state
42
Lattice QCD calculations favor a T around MeV at However
c B
strangeness neutrality in the thermal hadron gas mo del with requires
s
temp eratures around T MeV The m slop es of all particles measured at
f
CERN energies result in e ective temp eratures ab ove MeV The sup erheat
ing to ab out T in the calculation of Ref combined with the collective
c
ejection of the hadrons from the sho ck front could explain all these features
100 110
A QGP is a natural source of a large sp eci c entropy Instead of
a realistic detonation or de agration scenario would lead to further entropy
124
increase However as already mentioned this additional entropy can only
b e used pro tably in comparison with the data if the nal state is not mo deled
by an equilibrium hadron gas In this resp ect all suggested mo dels fail so far
160
Assuming b o ost invariant longitudinal expansion HBT interferometry results
suggest for sulphur induced collisions at A GeV a total collision lifetime of
ab out fm c This agrees with the lifetime of the QGP phase until
detonation from the sup erco oled state in Ref Assuming near equilibrium
125
hadronization by bubble nucleation would take much longer
The detonation into freely moving particles would naturally explain the high
antiparticle yields seen in the exp eriments see references in Table Dur
ing slow hadronization the pro duced antiparticles would b e absorb ed in the
11
longlived hadron gas subphase during and after the phase transition
These p oints seem make the detonation de agration scenario an attractive can
didate for the dynamics of heavy ion collisions at CERN energies However as far as
we know it only works if the hadronization is a strong rst order phase transition
This is not supp orted by mo dern lattice QCD calculations even when including nite
42
net baryon number according to the b est present knowledge Clearly much work
is needed to supplement the present qualitative considerations by a fully consistent
dynamical hadronization calculation
Conclusions and Outlo ok
We have reviewed new developments concerning the strangeness signal in heavy ion
exp eriments In spite of tremendous exp erimental progress a clear pro of of QGP for
mation still do es not exist However circumstantial evidence in favor of a decon ned
quark gluon state in the initial stages of the collision in particular at CERN ener
gies keeps accumulating A high degree of strangeness equilibration
s
indicates the op ening of new unusually rapid strangeness pro duction channels The
mechanism could b e QGP formation but strong medium e ects in a dense hadronic
environment leading to drastically reduced strange hadron masses and mo di ed in
elastic cross sections have also b een shown to repro duce some of the observations
More detailed and comprehensive studies should b e p erformed for further di erenti
ation
The rather stable result of a vanishing strange quark chemical p otential
s
in all of the CERN exp eriments further p oints in the direction of a state in which the
phase space for strange quarks and antiquarks is symmetric In any type of con ned
state this symmetry is sp oiled by a nite net baryon density b ecause the excess of
light quarks over light antiquarks rubs o on the strange quarks b ecause they are
clustered together with them into hadrons
The observation of unexp ectedly high pion pro duction in heavy ion collisions at
110
CERN p oints to a new e cient mechanism for entropy pro duction as one passes
from the lower BEVALAC and AGS energies to the SPS
We showed that all of the AGS data and most of the SPS data can b e e ciently
describ ed by a thermal hadron gas mo del The mo del contains only a few parameters
with obvious intuitive interpretation As shown in Section they provide an e cient
parametrization of the multitude of data resulting in a comprehensive understanding
of the basic features of the collision dynamics in ultrarelativistic heavy ion collisions
The mo del can b e tested with high accuracy using the large b o dy of available data on
particle yields and momentum sp ectra This test is successfully passed by the data
from Si Al Au collisions at the AGS providing convincing evidence for the formation
of a thermally and chemically equilibrated state with a large degree of strangeness
saturation and showing clear signs of collective b ehavior But even where the mo del
fails in its application to hadron pro duction data at CERN it still provides a useful
parametrization of the failure thus leading to useful insights and stimulating further
mo del building as discussed in Section
The future will provide us with many new sets of data esp ecially from the larger
collision systems Au Au at the AGS and Pb Pb at CERN These will allow for fur
ther tests and con rmation of the picture developed so far We exp ect even stronger
evidence for equilibration and collective b ehavior at the AGS and p erhaps a clari
cation of the remaining puzzles ab out the nature of the reball at CERN
On the theoretical side a ma jor e ort in constructing realistic dynamical mo d
els for the hadronization of a QGP is required in order to reach a comprehensive
qualitative and quantitative understanding of the particle pro duction pro cess and to
28
supplement the microscopic kinetic mo dels presently under development by a suc
cessful semiphenomenological macroscopic approach Also the kinetics of the hadronic
freeze out pro cess and its prop er implementation into macroscopic mo dels should b e
studied in more detail
Finally more e orts are needed to b etter understand the p erturbative and non
p erturbative contributions to strangeness pro duction in a decon ned equilibrium or
non equilibrium quark gluon state
Acknowledgments
We would like to thank Marek Ga zdzicki Jean Letessier Ahmed Tounsi and
esp ecially Johann Rafelski for several years of collab oration on this sub ject Many of
new results rep orted in this overview were obtained by them or in collab oration with
them This work was supp orted by DFG BMBF and GSI
A App endix Literature Guide to Strange Particle Mea
surements
Although this article is meant as a theoretical review on strangeness the phe
nomenology of strangeness requires the input from and the comparison with exp eri
ment For this reason we present here a list of exp erimental publications on strange
collisions organized by particle sp ecies and exp eriment particle pro duction in nuclear
number The two tables summarize the exp eriments at the Bro okhaven AGS and the
CERN SPS resp ectively We concentrate on strange hadron sp ectra and multiplicities
and left out articles on e g kaon interferometry
The list mainly contains articles in refereed journals Conference pro ceedings are
126 127
mostly neglected with the exception of Quark Matter and We recom
13
mend for up dated strangeness data also the pro ceedings of Quark Matter and
12
Strangeness of which a few contributions have already b een included according
the notes taken by one of the authors UH and which will b e published so on
The tables should contain all relevant publications from the last few years known
to the authors it is however quite likely that we missed one or the other imp ortant
contribution for which we would like to ap ologize
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