Strangeness in nuclear matter under extreme conditions

Strangeness in Nuclear Matter under Extreme Conditions

PETER KOCH

lnstitut far Theoretische Physik, Universitiit Regensburg, D.8400 Regensburg, F.R.G.

Keywords: Quark Gluon Plasma, Relativistic Nuclear Collisions, Strangelet, H-dibaryon

A b s t r a c t : We survey the role of strangeness in the context of dense hadronic matter including strangeness as a probe of the dynanaics of relativistic heavy ion collisions and its importance in astrophysics.

Contents

1 I n t r o d u c t i o n 255

2 S t r a n g e n e s s in T h e r m o d y n a m i c a l E q u i l i b r i u m 256 2.1 Strange Quarks in a QGP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256 2.2 Strangeness in an Ideal Gas of Hadrons . . . . . . . . . . . . . . . . . . . . . . . . . . 259

3 S t r a n g e Q u a r k P r o d u c t i o n in H a d r o n i e Col l is ions 261 3.1 Phenomenology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 261 3.2 Strange Quark Production from Strings/Flux Tubes . . . . . . . . . . . . . . . . . . . 261

4 H e a v y Q u a r k P r o d u c t i o n in a Q G P - T h e A p p r o a c h to E q u i l i b r i u m 263 4.1 Perturbative Production of 8~ Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263 4.2 Additional Sources (Pre-equilibrium and Non-perturbative Effects) . . . . . . . . . . . 267

5 K ine t i c s of S t r a n g e H a d r o n s 268

6 H a d r o n i z a t i o n o f S t r a n g e H a d r o n s 271 6.1 Gluon Fragmentation plus Quark Recombination . . . . . . . . . . . . . . . . . . . . . 271 6.2 Model Calculations and Predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

6.2.1 The K/zr-Ratio . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272

253

254

7

P. K o c h

Strange Hadron Product ion in Nuclear Collision Experiments 274 7.1 Sub th resho ld S t rangeness P roduc t ion at BEVALAC Energies . . . . . . . . . . . . . . 276

7.1.1 Cross-sect ions and Equi l ibra t ion Ra tes . . . . . . . . . . . . . . . . . . . . . . . 278 7.2 K / r - R a t i o s at Brookhaven AGS Energies . . . . . . . . . . . . . . . . . . . . . . . . . 283 7.3 A, ,~ and K , E n h a n c e m e n t at C E R N SPS Energies . . . . . . . . . . . . . . . . . . . . 285 7.4 C-Enhancement and J / C - S u p p r e s s i o n . . . . . . . . . . . . . . . . . . . . . . . . . . . 286 7.5 S t rangeness E n h a n c e m e n t in ~-Nucleus collisions . . . . . . . . . . . . . . . . . . . . . 289

7.5.1 Eper imen ta l D a t a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 7.5.2 Simple Es t ima te s of A, ,g, and K~ Cross Sections . . . . . . . . . . . . . . . . . 290

S t r a n g e Q u a r k M a t t e r i n A s t r o p h y s i c s a n d H e a v y I o n C o l l i s i o n s 293 8.1 Cold S t range Qua rk M a t t e r . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 8.2 Strmage Qua rk Mat t e r at finite T e m p e r a t u r e . . . . . . . . . . . . . . . . . . . . . . . . 295 8.3 St rangele t P roduc t ion in Nuclear Collisions . . . . . . . . . . . . . . . . . . . . . . . . 297 8.4 Othe r Aspects of S t rangeness Separa t ion . . . . . . . . . . . . . . . . . . . . . . . . . . 301 8.5 K a o n Condensa t ion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

H - D i b a r y o n P r o d u c t i o n in N u c l e a r C o l l i s i o n s 303 9.1 P roduc t ion of H-Dibaryon f rom a Baryon-r ich Hadron Gas . . . . . . . . . . . . . . . . 303 9.2 Detect ion of the H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

9.2.1 Weak Decays of the H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 9.2.2 Dissociat ion of the H . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 9.2.3 Nuclear F r agmen t s conta in ing the H . . . . . . . . . . . . . . . . . . . . . . . . 305

10 E p i l o g u e 308

Strangeness in Nuclear Mat te r 255

1 I n t r o d u c t i o n

Nearly all strongly interacting matter around us is built up of up (u) and down (d) quarks. However as soon as adequate exitation energy becomes available in in hadronic interactions, it becomes apparent that further quark flavors exist. The lightest of the heavy flavors of matter where labeled "strange", and this name is nowadays associated with the .~ quark. An important feature for later reference is that strange quarks (s) are identical in their properties (charge q = - 1 / 3 , baryon number B = +1/3, spin 1/2) to the d quark but different in mass.

Speculations concerning the potential role of strangeness in dense matter began already after the discovery of strangeness itself and were mainly concentrated on the presence of hyperons (A, E0,-~, E0,- and ~ - ) in dense neutron stars.

After it was realized that hadmns are composite objects of quarks and gluons many new questions about the physics of strangeness in dense matter were raised including the presence of quark matter in the early universe and possibly in neutron star cores.

Furthermore, relativistic heavy ion collisions have opened up the possibility to explore the physics of matter at extremely high densities and temperatures in the laboratory, with strangeness as a potentially important probe of the underlying dynamics.

In particular the prospect that at sufficiently high beam energies the matter formed in the initial state of the collision process might exist as a dense plasma of partially deconfled quarks, anti-quarks and gluons and might be observed by an1 enhancement in certain strange particle yields has placed strangeness at the foremost front in theoretical as well experimental research.

In this review I would like to survey recent work on the role of strangeness in the above mentioned various physical situations, and I shall give special emphasis to strangeness production in nuclear collisions.

The review is organized in the following way. To motivate the subject I will introduce the basic idea behind the "strangeness signal" for quark gluon plasma (QGP) formation in nuclear collisions in chapter 2 and introduce some important notions relevant for the thermodynamics of strangeness.

In chapter 3 I will shortly discuss the phenomenology of strangeness production in e + - e - , lepton- nucleon and hadron-hadron collisions where a rather universal strangeness suppression is observed which finds its natural explanation in string fragmentation or flux tube models for so called "soft hadronization".

Since strange particles (quarks and hadrons) produced in nuclear collisions will strongly interact with their environment, it is necessary to study the fate of strangeness during the whole dynaanical process which starts with its production and ends when strange hadrons decouple from the system. This issue will be discussed in sections 4 and 5 where we focus on the non-equilibrium properties of strangeness production in nuclear collisions.

The next section will be concerned with the question how the hadronization process from a QGP back to the hadron world can modify the initial flavor composition of the QGP. Since this part of the evolution process of strangeness in nuclear collisions is the most uncertain one I will constrain myself in discussing the different Ans£tze made and how they lead to quite different final predictions.

In chapter 7 I will discuss some recent interesting measurements of strange hadrons in collision experiments in order to see if they show indications of "new" physics.

The rest of the review will be devoted to the question of the existence of strange quark matter, its relevance in the astrophysical context, and finally the possibility to produce strange quark matter droplets ("strangelets ') in nuclear collisions. The production of the so called H-dibaryon -a genuine six quark state- which might be considered as the smallest strangelet will be discussed in the final section 10.

Since a detailed review [1] on part of the material covered here already exists, I shall keep com- putational details as short as possible and will concentrate on qualitative understanding. This also meets the fact that calculations performed by different groups working in this field are still under controversial debate.

256 P. Koch

2 Strangeness in Thermodynamleal Equilibrium

2.1 Strange Quarks in a QGP

Since most of the phenomena discussed in this review article are closely connected with a new form of mat ter which is usually termed quark gluon plasma I will give a short and perhaps naive description of what is ment by this notion (for references see [2]).

According to our experience a useful set of degrees of freedom to describe cold nuclear matter are protons and neutrons plus mesons as exchange particles to describe their interaction. The fact that each nucleon consists of three valence quarks and a whole sea of gluons and q~ pairs seems to be of minor importance in this energy region, since the effective coupling constant between these constituents is large, leading finMly to confinement of all colored quanta inside hadrons. The space between the nucleons, the vacuum of Quantmn Chromo Dynamics (QCD) has a rather complicated structure and expels color electric flux and is therefore responsible for color confinement. In this sense cold nuclear mat ter is a color insulator forbidding color to propagate over distances larger than typical hadron sizes. The situation is expected to change when we heat up or compress the cold nucleus. The density of hadrons increases and the excitation energy manifests itself in increased thermal motion as well as creation of new hadrons. At a certain point a situation will be reached where the density is so high that the space becomes closely packed with hadrons tha t they begin to overlap and the internal structure of the hadrons becomes relevant. Compressing and heating further the individual hadrons loose their identity. It is now much more economical to switch to the quark and gluon degrees of freedom. Furthermore, since the QCD coupling constant is energy dependent and decreases for larger energies, the coupling between the quarks and gluons becomes weaker and weaker as the thermal and fermi energy increases. Color confinement can no longer be upheld and quarks and gluons can move around nearly freely throughout the whole dense and hot region. Inside this mat ter called quark gluon plasma the vacuum is called the "perturbat ive vacumn", on which quarks and gluons axe constructed as weakly interacting thermal excitations. In this sense a quark gluon plasma is a color conductor allowing hadrons to move freely over larger distances.

This prospect of a possible phase transition from hadron matter to quark mat ter at a critical energy density (e > 1 - 2 G e V / f m 3) has inspired physicists to search for this phenomenon in particular in the field of nuclear physics. Short of recreating the early universe or probing the interior of the core of large neutron stars, the hope rests now on the possibility to reach extreme high energy densities in the laboratory via relativistic nuclear collisions, by converting large amounts of beam energy into random thermal motion within a compact collision zone.

The idea tha t strangeness could be a useful signM for QGP formation in relativistic heavy ion collisions was first proposed almost a decade ago[3]. This idea was motivated by the observation that in baryon-dense quark mat ter in thermodynamical equilibrium the production of s$ pairs is enhanced compared to the production of light quark flavors u (up) and d (down), as soon as the Fermi energy of the already present u and d quarks is higher than the mass of the strange quark.

Disregarding for a moment possible losses of quarks due to hadron radiation from the surface and the slow weak decay process, a quark gluon pla~snaa will contain an equal number of strange and antistrange quarks since they can only be created or annihilated in pairs. For a given temperature T we find for the density of strange or antistrange quarks in aal ideal quark gas:

Ps = P~ ~ g, 27r-"" ~ K2 (2.1)

where gs = 6 for spin and three color degrees of freedom, and ms is the strange quark mass, that is the effective mass parameter of the basic reaction process.

In the high temperature QGP phase with restored chiral symmetry, we expect the strange quark mass to be of the order of the current quark mass m s "" 170 MeV [4,5] which is roughly of the same magnitude as the critical temperature Tc ~ 160 - 200 MeV [6].


! !

I _ m

I0

0 I0

I I

0 I 2 5 laq (v)

F i g u r e 2.1: Abundance of anti-strange quarks relative to the light anti-quark abundance g or -d as a function of #q /T for ~everal choices of rns/T.

Within the ideal gas approximation we can compare to the light antiquark density (~ stands for either fi or d) using Boltzmann statistics:

T 3

Here we have introduced the chemical potential for nonstrange quarks which is a measure of the baryon density because of the relation 3#q = /~B. In order to illustrate the qualitative feature we expand the Bessel function in Eq.(2.1) for large argument and find

p, ~ e×p -~ (m, - m) (2.3)

for the ratio of antistrange to, leCs say, T-quarks. We see from Eq.(2.3) that for /Q > ms - for substantial baryon densities - the production of $ quarks is favored over production of ~-quarks.

The newly paii~produced u or d quark would have to go on top of the Fermi sea which is unfavorable in view of an empty strange Fermi sea. The price to pay is the strange quark mass.

Even in the limit of a q - ~ symmetric system, i.e. net baryon number equal to zero, one finds a similar number of $ quarks, and fi and d quarks as long as T >_ ms. This is illustrated in Fig. 2.1 where we have depicted the g/~ ratio as a function of #~ including the correct Fermi statistics for the quarks. In the case PB ---- 0 the relevant quantity which determines this ratio is the temperature, or the energy density e = a T 4 ( a ~- 377r2/30 for an ideal gas of massless partons). The higher the energy density, the more closly this ratio approaches one.

We remark here that our crude ideal gas estimate has to be taken with caution since lattice calculations of hot SU(2) and SU(3) matter [6] seem to indicate that quark matter might not be an ensemble of ideal partons close to the critical temperature. A recent calculation [7], simulating SU(3)

258 P. Koch

7 r

I

5

L,

3

2

1

T

Tc

• c~ .d /T ~'

• r . . /T ~'

, , , I ~ [ . ~ , J

5.0 5S 6.0 10.0

F i g u r e 2.2: Fermionic energy density of light u-and d-quarks and heavy s-quark in units of T 4 versus fl on a 4 x 123 lattice. Data are taken from [7]. The dashed line gives the lowest order ideal gas behaviour on a finite lattice.

Gauge Theory with a light isodoublet and a heavier strange quark on a 4 x 8 a lattice found a energy density ratio

, ,(re, I T = 1.0) e.(---A-~./T - 6 - ~ ) -~ 0.5 (2.4)

On the basis of the ideal gas relation for massive fermions one would have expected [8]

es(rn~/T = 1.0) e . ( m . / T = 0.0) - 0.8S (2.5)

which in the simulation is approached only very slowly at high temperatures as can be seen in fig.2.2. On the other hand, in view of large finite size effects [8] this result should also be taken with some

caution. Even more important, however, is the problem that it is not obvious how to separate the different contributions to the total energy density. Looking only at the fermionic par t is not entirely correct, as the presence of fermions also influences the gluonic contribution and vice versa.

The expectation, however, that the QGP very close to the phase boundary might not be close to ideal gas behavior is also seen in effective theories for long wavelength QCD like Nambu-Jona- Lasinio type models. In such a model [9] including the UA(1) anomaly it was shown that the rate of approach to the chirally resored phase might be very different between non-strange and strange quark sector. Whereas the chiral symmetry in the non-strange sector gets restored at temperatures of ~ 150-200 MeV, the strange quark condensate keeps most of its zero temperature value even at those temperatures. Consequently the dynamical strange quark mass which depends on the quark condensates might not be close to the current quark mass.

Such a picture seems to be in accord with the lattice finding s and also indicates that one should be careful in using ideal gas relations close to the phase transition point.


2.2 Strangeness in an Ideal Gas of Hadrons

In order to understand what the difference to a gas of hadrons in thermodynamic equilibrium would be, we shall similarly estimate the 5/~ ratio. In this case we have to add up all hadrons containing quarks and divide by those which contain either a '5 or d quark. Restricting to the lowest mass states only and neglecting the antibaryons which give only a small contribution we find approximately:

ttG ~ Zq~)~q)~j (2.6) - T - x

Here we have introduced the one-particle partition functions in the Boltzmann approximation labelled by the quark composition of the corresponding hadron. Usually they ave given by

V T 3 {mi'~2r.

The fugacities Ai = e x p ( ~ ) obey the following relations:

A# = A~-1 (2.8)

and A~ = A~ "I (2.9)

since p# = -#q and #~ = -Ps. As shown in [I0] the strangeness fugacity A 8 can be expressed as a function of Aq by invoking strangeness conservation, i.e. net-strangeness equal to zero. For large baryon densities (large Aq) one finds approximately

311/2 A~ "I'~ 1 + ~ Aq] , (2.10)

where the second term in the bracket accounts for the hyperons E °,± and A. In the limit 6f large baryon densities we find to a good approximation

[ ~ 3-1112 g Z ~ i Zqq~ i+ , (2.11)

Y HG

This shows that a pure gas of hadrons also prefers to produce g quarks as soon as the baryon density, i.e. A~, becomes large. This can be traced back to the dominance of the associated production in a baryon dense system. Although the physical mechanism, that is, flavor suppression of light quark flavors, is essentially the same, the scale when this happens is different and is deeply connected to chiral symmetry breaking. This can be seen by observing that

Zj. ! ~ m K ^

where we have assumed that the mass of the kaon is mainly determined by the constituent quark mass of the stra~ge quark and neglected the pion mass as appropriate for the pion being a massless fiavour SU(2) Goldstone bosom Hence

8 HG ['~l'ts'~312 [ 1 _ ] - ~ ( m s - pq) (2.13) T kT)

which is formally the same as the QGP ratio but contains now the constituent quark mass which is taken usually as rns "J 2 - - 3 ms, where rn s is the formerly used current quark mass. The scale which

260 P. Koch

determines strangeness production in the case of the pure hadron gas is now given by ~ , / T > 2 - 3

and results in a smaller hadron gas ~_ ratio. In the case of chiral symmetry restoration r~, -+ m, we recover the QGP ratio.

We may conclude from this tha t restoration of chiral symmetry is one reason which favors strange quark production in the QGP over the hadron gas. However, it was pointed out [11,12,13] that although the strange quark density in the QGP phase is much higher than in the hadron gas phase, the total content of strange quarks is only slightly enhanced if a QGP is compared with a fully equilibrated hadron gas at any same entropy content. The reason is [11,12,13] tha t the volume associated with a gas of individual hadrons of the same total entropy is much larger due to the smaller number of available degrees of freedom. Consequently one mtmt expect tha t the abundance of strange quarks becomes diluted during the hadronization process and strange hadron yields from a QGP might not be very much above those of a fully chemical equilibrated hadron gas.

The question of "how much" and "if any" enhancement of certain strange hadrons above the hadron gas equilibrium level can be expected depends very strongly on the hadronization scenario which is presently not under control due to its essential nonperturbative nature. Staying conservative one can say that a QGP leads to hadron abundancies close to their equilibrium hadron gas yields.

This indicates that questions concerning the space time evolution of hot dense matter , the approach to flavor chemical equilibrium, and in particular the hadronization process are very important in understanding strangeness production in nuclear collisions. Only if the time scale for strangeness equilibration in a hadron gas is much larger when compared to the lifetime of the hadronic system can we expect that enhanced strange ha<irons are indicative for QGP formation. This conclusion rests on the observation, tha t in usual hadronic interactions strange particle yields are far below their hadron gas equilibrium yield.

Strangeness in Nuclear Matter 261

e: o i

~D

.8 Z o u3 .6 o'3 UJ CO.

13.. 4

U')

(.,3 .2 u') UJ Z uJ 0 ~D Z <~ (2: F-

Figure 3.1: /15]:

o PP • ~v*p

n K'p

• K+p • ~ DECAY

e * e -

x (Ly)N

t, ot

5 I

t+ ........ t, t .... ......................... tl ......

t ' , _ _ ~ _ _ _ 4 . . _ + _ ~ ~ _ F . I . . . . . . ~ - - - 4 - - 4 - -

2 5 EO 20 50

V~s,, (GeV)

Strangeness ~uppression factor )%~ a.s a function of the effective c.m. energy

3 Strange Quark Production in Hadronlc Coll is ions

3.1 P h e n o m e n o l o g y

The observation that SU(3) flavor symmetry is badly broken in hadron-hadron, lepton-nucleon, and electron-positron collisions is quite old, as it is indicated by a reduction in the yields of strange hadrons when compared to nonstrange hadrons. This fact is usually accounted for by introducing a phenomenological parameter As~, called strangeness suppression factor, which describes the production probability of sg-pairs as compared to light quark pairs[14,15,16]:

< "hr~ > (3.1) As~ = 1 < Aru~ +Ndd >

Extracting this parameter from measured hadron yields is not uneanbiguous sinee a large fraction of stable hadrons observed in the detector result from resonance decays which mainly produce pions and usually don't change the number of strange hadrons. This is of particular relevance for the ratios of kaons to pions. Although in different papers assumptions are made concerning the resonance composition, the extracted value of ~s~ seems to be quite unique and stable within a range of 0.25 -0 .3 [14,15,16]. A typical example of such an aalalysis is shown in Fig. 3.1.

3.2 Strange Quark Production from Strings/Flux Tubes

The remarkable internal consistency in the value of As, deduced from hadron-hadron, lepton-hadron, and lepton-lepton interactions suggests that there is a universal SU(3) violating strangeness suppression in all of these processes. This A,s value of the order of 0.2 - 0.3 has a quite natural interpretation in the flux-tube model for soft hadrolfization[17,18,19,20]. Here a quasi one-dimensional confinement

262 P. Koch

force field is formed between a q and a q moving in the opposite directions. If in such a color force field a quark-antiquark pair appears at one point along the string, energy conservation is violated by 2 mq, and the system is in a short-lived, virtual state. Further, if the quark pair has the right color to screen the initial color field, the quarks may end up on-shell leading to the breaking of the string [21]. This process is basically a tunneling phenomenon, and the quark production rate depends strongly on the barrier heights, which is given essentially by the masses of the newly created qq pairs. In close analogy to Schwinger's[22] expression for the creation of e+e - pairs in a constant uniform electric field E, one assumes that the rate of qq production is proportional to

d,x --- ~-~ n=l ~ ~ exp \ 2~ } (3.2)

with mq the quark mass and s the so-called string tension. One finds a value for ~ ~ I GeV/fm, a value which one can also derive from the slope of Regge trajectories and other observations[17].

Whether one should evaluate (3.2) with current or constituent quark masses leads to some uncertainty in the predictions for As~. Using current quark masses, for example, a value As~ "~ 0.5 results [2O].

However, since the produced quarks are confined within a color flux tubc of the transverse dimen- sions similar to hadron sizes, the general expectation is that constituent ("dressed") quark masses of about 350 MeV for light quarks and 550 MeV for strange quarks should be used. In this case one obtains As~ "~ 0.3 as observed. However, finite size effects may be important and as shown by [23] even current quark masses can reproduce the experimentally observed strange quark suppression factor.

We point out that within such an approach the rate of strange quark production as compared to light quark pairs depends on the energy density in the color field, i.e. the "string constant." In the case of the quark gluon plasma however it seems possible to achieve higher energy densities, which consequently can result in a higher g/q ratio, e-~ventually up to 0.8-0.9, in the case of a q - q symmetric plasma.

The possibility that stronger color fields are formed in nuclear collisions was discussed by Biro, Nielsen and Knoll [24]. This approach basically results in a change ~ ~ t¢~ H with ~l.f > s- Con- sequently one expects [25] an enhanced massive quark pair production. On the other hand, these non-elementary flux tubes would decay much faster and would allow for a very rapid thermalization of the produced partons. In this respect such a scenario is very strongly correlated to quark gluon plasma formation.


o) b) C)

Figure 4.1: Lowest order QCD diagrams for s~ production (a-e) gg --* s$; (d) q~ --* s$.

4 Heavy Quark Product ion in a QGP - The Approach to Equil ibrium

In this section we study processes for strange quark pair production and annihilation to investigate the approach to chemical equilibrium of the strange quark abundance in the plasma phase.

4.1 Perturbative Production of s~ Pairs

The general procedure to study strange quark evolution would involve a detailed calculation of different processes that can contribute to the production as well as annihilation of strange quarks. First of all it should include pre-equilibrium production of quarks during the initial stage of the collision process which seems best described by an independent fragmentation of strings. If those produced quarks find themselves in an environment where the energy density (or particle density) is large enough to allow for partial deconfinement, the further strangeness evolution might be governed by processes which usually [26,27,28,29,30] are assumed to be described reasonably well by lowest order QCD processes where either two gluons or a light quark-antiquark pair annihilate to create a sg pair. It was found [28] that the gluonic process leads to a much faster production rate and dominates strangeness production in the quark gluon plasma.

Neglecting quantum statistics and Pauli blocking effects, the number of sg-pairs per unit volume and time is

foo f d3kl [ dSk2 ~, __dN = Ja ds j ~ J ~ - ~ o t s - (k, + k2) 2) Z f i (k , , x) f j (k2, x)-fflj(s)vij (4.1) d4x m2 i,j

Here 8ra~ (1 + ~C-)w(~) (4.2a) a q ~ s ~ - 27s

and 27ra~ 4N 2 M 4 7 31 M S

Yag~s, = - - ~ s [(1 + s + -~ - ) tanh-1 W(s) - (g + -~- --~-)W(s)] (4.2b)

axe the spin and color averaged cross sections calculated in first order perturbative QCD [31,32], corresponding to the graphs shown in Fig. 4.1. The factor W(s) is given by

264 P. Koch

and vii is the relative velocity of the initial particles, which are taken to be massless. The sum over the initial states includes the discrete quantum nmnbers i , j (color, spin, etc.) over which the cross sections were averaged. Usually the phase-space densities f i(x, ki) for light quarks and gluons are approximated by the statistical Fermi, or Bose momentum distribution functions, respectively:

fg(x, k) = (exp(fl • k) - 1 ) - ' (4.3a)

and fq,~(x, k) = (exp(fl • k q: ,%IT) + 1) -1 (4.3b)

where fl- k = fl~k ~ and (fl~, . fl~)-1/2 = T, the temperature in the local rest frame. The statistical momentum distributions fg(k), fq(k), f¢(k) of gluons, quarks, and antiquarks are

assumed to be spatially homogenous throughout the plasma but can still have a parametric time dependence, i.e. through T(x) or /Q(z) .

-Once strange quark-antiquark pairs have been created, the sg-annihilation reaction will deplete the strange quark population. Under the condition of statistical independence of the creation and annihilation process, this loss term is proportional to the square of the density p, of strange and antistrange quarks. With p ~ being the saturation density at large times, the following differential equation then approximately describes the evolution of Ps as a function of time:

dps 1 d V d--'t = -P" V -d~ + A [1 - (ps/pT) 2] (4.4)

where A and p ~ are time dependent quantities through the time dependence of the temperature. However, Ag ~ A and p ~ do not depend on the baryon chemical potential of the light quarks. The first term on the r ighthand side of Eq. 4.4 takes into account tha t the volume occupied by the quark gluon plasma changes in time by hydrodynamical expaaasion, diluting the density of strange quarks. The total number of strange quarks (Ns = V • ps) will increase until phase space saturation, i.e. chemical equilibrium is reached.

The basic quantity which determines the rate of equilibration is the relaxation time ~" = ~ A '

shown in Fig. 4.2 as a function of temperature. We note a strong dependence of the relaxation time on the temperature T; the higher the temper-

ature of the plasma, the faster the approach to chemical equilibrium. Typical times for temperatures above T > 200 MeV are r ~ 2 - 3 fm/c, which are comparable to the lifetime of the quark gluon plasma created in nuclear collisions.

The dependence of ~- on the strange quark mass has been studied by Munehisa and Munehisa [33] and their results are shown in Fig. 4.3.

At a fixed temperature T = 200 MeV and as = 0.6 the strangeness relaxation time increases rapidly with ms. A chaalge of approximately 100 MeV in the effective mass of the strange quark would result in almost an order of magnitude change of the equilibration time v. The small window, ms "" T, makes the strangeness formation time similar to the expected lifespan of the quark gluon plasma.

In a realistic collision process between two heavy nuclei the temperature will not stay constant during the lifetime of the quark gluon plasma, and we have to integrate Eq. 4.4 over the space-time history of the plasma, i.e. from formation to hadrolfization. This is usually done in the framework of simplified hydrodynamical models. Depending on the chosen initial conditions, ra ther different evolution scenarios can be obtained. Assuming that at very high projectile energies the nuclei are basically transparent to each other, and imposing longitudinal boost invariance on the hydrodynamical equations [34] one can find in the central rapidity the following behavior for the temperature

T(t) ~_ To( ~ ) - ' /~ , (4.5)

reflecting the fact that the expansion is basically only longitudinal. No stopping of baryon number in the central rapidity takes place, and the quark gluon plasma hadronizes at a critical temperature


5o~- E ~ Notsui etoL | . . . . Rofelski • HOUer ~ : ~ , , \ ............ Bottzmonn approx.

' r

E \ "',. 1! \ ~ \ \ "L'"'"'.. 100 200 300 400

T(MeV)

Figure 4.2: Equilibration time constant 7" as a function of temperature. The solid line includes Pauli Blocking [80]; whereas the dashed line not [~8]. The dotted line makes use of the Boltzmann approzimation [1].

10 2

I0 ~

I.( 1.0

i I ~ I _1 b)

T- 200 Pier - ]0 "2 I l I

I00 200 300 400 500 msi~4eV}

Figure 4.3: Dependence of the strange quark evolution on the strange quark mass [33] for fized temperature T = 200MeV and as = 0.6. (a) Relazation time for the process gg -~ s~; (b) density of strange quarks after 7" = 20fra/c.

266 P. Koch

P~(tc) psequ(Z(tc)) Tc=160 MeV, pc = O MeV

1.0 . . . . . . . . . . . .

/ / / / " 3 ~ / / , ~ / ~ . / /

/ / ~ 2frr~

/ / / ~ / / / /

~ V~t _ _ _ _ V . t 3

0--I i i , I I ~ i I I

200 250 'o""'J~'"e"l 300 v

F i g u r e 4 . 4 : Degree of strangeness equilibration as a function of initial temperature To for one- dimensional (V ,,. t) and three-dimensional (V ..~ t 3) expansion. The radius of the plasma is a parameter Ro = 2, 3, 4fro.

Tc which is usually estimated by constructing a first-order phase transit ion by matching the pressure of an ideal gas of quarks and gluons with tha t of an ideal gas of massless pions at zero net baryon density. The lifetime te of the pure quark gluon plasma is then obtained as

/T0~3 t~ -- ~o~ -~ ) , (4 .6)

and strongly depends on the formation time to of the plasma and the ratio of the initial temperature To to the critical temperature To.

Recent experiments at the Brookhaven AGS and the CERN SPS show, however, tha t a baryon free central region can not be reached with beam energies up to 200 AGeV. In this case a scenario which takes into account "stopping" of baryons in the central rapidity region seems to be more appropriate.

As a consequence of a finite net baryon density the hadronization phase transition takes place at a lower critical temperature To, and the formation time constant ~'0 can get larger due to initial state interactions. In Ref. [1] the simple parametrization To = Ro/vs was used with R0 the radius parameter of the target nucleus and v, the shock velocity. This finally results in longer lifetimes of the pure quark gluon plasma phase, typically of the order of ~ 3 - 5 fm. A realistic scenario will lie between both extremes.

The influence of the two parameters To and R0 on the degree of strangeness equilibration has been systematically investigated in [1] and is shown in Fig. 4.4.

One observes tha t to within a factor of two, strangeness can always achieve phase space saturation, of course more easily for the long-lived large plasma or hot initial environment. Also shown is a scenario where the temperature drops much faster, T ~- To(t/to) -1 , which corresponds to an expansion of the plasma volume V ~ t 3 [1].


4.2 Additional Sources (Pre-equil ibrium and Non-perturbative Effects)

Before we close this section we have to add a few remarks which might be important in understanding flavor equilibration in a QGP:

• The question of Panli-blocking (presence of already produced strange quarks) h ~ been studied by Matsui et al. [30] and found to be small.

• The question of pre-equilibrium production of sg pairs in the presence of a strong color electro- magnetic field coupled to transport equations for QGP formation were investigated by Dyrek et al. [35]. Here it is assumed that during the very initial stage of the collision process a strong chromo-electric field is formed which allows quark-antiquark and gluon pairs to tunnel from the vacuum [36]. The so-produced partons are described subsequently by the Boltzmann-Vlasov equation. Already after few fractions of a fermi an q-ratio was found very close to the plasma equilibrium ratio, depending somewhat on the strength of the chromo-electric field.

• Due to the color octet nature of gluons and overall entropy conservation, any realistic model for the hadronization process of a QGP must allow for direct hadronization of gluons (see chapter 6.1). The conversion of gluons into quark-antiquark pairs was explored by Koch et al. [1] and later elaborated upon by Barz et al. [37] who found that such a mechanism will lead to a strange quark abundance which is close to the chemical equilibrium value, for nearly all dynamical scenarios studied.

• Very recently Biro et al. [38] considered perturbative strange quark production in a QGP containing gluons that are effectively "massive" due to medium effects. In such a case also the process g "--' sg is possible if the gluon mass is larger than twice the strange quark mass. It was found that for a gluon mass around 500 MeV the decay channel is dominating. Furthermore, even before this threshold is reached, the qq process gains importance relative to the gluonic process, since now the intermediate gluon in diagram 4.1.d) is less off-shell. This causes an overall rise of the s,~ rate, quite in contrast to first naive expectations. Although one might criticize this approach because it assumes equilibrium for the initial heavy gluonic modes but not for the lighter strange quarks it shows that non-perturbative modes for strange quark production in the QGP might be quite important. Such processes could counteract the conjecture that a QGP close to the phase boundary might not contain low momentmn massless gluonic modes which would eventually reduce the the two gluon perturbative production rate. Much further work has to be done here to clarify the situation.

In summarizing, we conclude that a quark gluon plasma formed in nuclear collisions most likely results in strange quark equilibration, one of the backbones of the proposal that strange hadrons are useful probes for quark matter formation.

268 P. Koch

5 Kinetics of Strange Hadrons

Since strange quarks interact strongly and do not decouple early from the highly excited matter, any quantitative prediction of experimental observables requires a thorough understanding of the hadronic flavor chemistry in a pure hadron matter phase. Such a state can be realized whether or not a quark gluon plasma is formed. One of the important results discussed here is the fact that during expected nuclear collision reaction times of approximately ~ 10 -22 sec in particular strange antibaryons will not reach their chemical equilibrium abundances unless a quark gluon plasma is formed [39,1].

The dynamics of strangeness production and redistribution of strange hadrons has been studied recently by several authors [1,37] in the framework of chemical reaction (kinetic) equations, including all stable hadrons (~r, K; A, E, E, fl and their antiparticles) and all possible reactions among them. Since only a few of them are known experimentally:

• strangeness production 7rN --~ K Y , (Y = A, ~)

• strangeness exchange K N -* ~rY

• baryon annihilation N N ~ nlr

all reactions of the same type were assumed to proceed by the same invariant matrix element and differ only in the available phase space (for details consult [39,1]). The evolution equations take the following generic form.

d N , = k _ , ( 5 . 1 ) d4x

j , k t

where the sum has to be taken over processes j + k --~ i + l, which produce species i and over processes i + ~ --~ j + k, which destroys species i. The p's are the particle densities whereas Ni represents the total number. The (av)'s are averaged cross sections which are usually given as:

fd3pj d3Pk f j (P j ) f k ( P k ) i .

• % k v J k (5.2) (av)~k = fdZPj d3pk f j (P j ) fk(Pk)

where the elementary cross section times the relative velocity vik is averaged over thermal momentum distributions f (p) .

The thermally averaged cross sections then have characteristic forms for each reaction type [1]:

• Strangeness production cross sections rise steeply with temperature T since the final state hadrons carrying the produced strange quarks are usually heavier than the initial chmmel leading to a dominant behavior (av) ,,, e x p ( - A Q / T ) where AQ is the threshold energy AQ = ~"~final m -- ~initial m. These processes are therefore very sensitive to the temperature, and a sizable amount of additional strange hadrons is only produced if the excited hadronic matter system can stay for rather long times at high temperature.

• The exchange cross sections do not change very much with temperature, since in most cases AQ ~ 0, i.e. ~l i , ,a t m - ~i,,itial m. As a result, strangeness is redistributed between different strange quarks carrying hadrons quite fast.

We point out here that the important quantity relevant for equilibration is the mean thermal cross section times the density of the scattering particles; both are relatively small when compared to the quark gluon plasma, leading to rather long equilibration times for the hadron gas. Some typical examples are shown in Fig. 5.1 for fixed temperature T, chemical potential/-tB, and a static hadron gas is assumed.

Initially, all strange quark-carrying hadrons are set equal to zero, whereas pions, nucleons, and antinucleons are assumed to have reached chemical equilibrium. The important point to note here


10 "1 ( a ) p - % p ~

lo ~ . ~7op7 / /

/ / . . . . i I ~ I I i

' ," pV'~ I I I

/ / / / / /

~ _ ii1~ ii i I I I I

/ i i I / i

~o ~ / /

/ / l/ !

~ fl ii it

10 z~ 10 ~ 10 n

I0 ~

10 f

10; T - 160 MeV

I J . , / U

l i rne I s ] ~ i i i 10 71 iO zo 10 ~4

pi(t)ll~-3 ] ' - - ~ - p,~o

16 ( b ) , - J ~ - - /

/

"~- Z /

10' ,./ ~ /

/

i i i / / / / / / /

/ / / /

//P~ / / / / /

/ / / /

/ / / /

/ / I o 160 MeV /

I / /

I / I l ~ I I(] -~3 10 n

/ /

10"1 /

168 /~: time[s] / i

L I l ,

I0 71 !0 ~° I0 ~4 I0 n

16 / / / ¢~.

I I I I 1~ I/F~.I,, f i i

I I i I I/ I~-"~--

"~ / 1 ~ ii I I / / /

, /

/ I / I /I I

10' / I / t l • t50 MeV I p.b= 45{] MeY

I

/P,, / l i m e [ s ]

L I t I - _ . l .

iO-n lO-21 lO-~

F i g u r e 5.1: Approach to equilibrium of various strange particle densities in hot at fixed temperature T = 160 MeV and (a) ktb = 0 MeV and (b-c) #b = 450 MeV. density $-hadrons are shown in (b) whilst s-hadrons are shown in (c).

hadronic matter At finite baryon

is the extremely slow equilibration rate for strange antibaryons, which is due to the combination of rather small cross sections for strangeness and baryon-antibaryon pair production.

These long equilibration times have the important consequences that approach to chemical equilibrium is slow from both below and above the chenfical equilibrium abundances. It was found [1,37] that the chemical composition as generated by the hadrolfization process of a quark gluon plasma barely changes anymore. The reason is that hadrolfization of a quark gluon plasma results in hadron abundances very close to chemical equilibrium.

However, any nuclear collision process at high energy proceeds through a pre-equilibrium phase, during which substantial strangeness production might occur (e.g. initial independent fragmentation of strings or scattering between "hard" secondaries). This aspect has not been exhaustively investigated, 'so far, and requires the application of a detailed dynamical model of the complete collision process which couples the pre-equilibrium production to, possibly, final state ha@on gas interactions.

Here we would like to stress one important point: the initial conditions for a ha@on gas phase have to be chosen carefully. Conditions requiring temperatures of order T > 200 MeV and assuming an ideal gas of pions seem to be rather unrealistic in view of the fact that at temperatures of T ~ 180 MeV a large amount of the available energy goes into excitation of resonances as already pointed out by Hagedorn [40]. What one has to consider instead is the effects from interactions of produced resonances on strange hadron production. However, the most important point in estimating final state interactions seems to be that the lifetime of the ha(Iron gas is rather short. Even in the most optimistic case of purely boost invariant longitudinal expansion, the decoupling time is approximately

To 3 ~d~c ~ t 0 ( r z ; ) ' (5.3)

resulting in tdec <_ 10 fro/c, which has to be compared to typical equilibration time scales requ 100 fm/c [1] for strangeness equilibration. From this we would expect that final state interactions

270 P. Koch

contribute on the level of 10-20 %. This would mean that practically all strangeness observed after the collision process can be traced back to the initial stage of the collision process.

We close this section with a remark about the 7rr ~ K K process which very often had been considered as the most relevant one in strangeness production in the baryon-free [41,30,1] as well as baryon-rich [42,1,37] environment. The only information available concerning this process comes from partial wave analyses of the ~r+p ---* K - K N process under the assumption that the latter process can be described by one-pion exchange only. In the case of a low energy pion, the propagator of the exchanged pion is assumed to be basically on-shell, which in turn could allow to extract the 7r~r ---* K K vertex. Assuming that this is a valid procedure, one finds this process to be dominated by two bumps in the threshold region for K K production corresponding to the f0(975) (formerly S*(975) and f0(1400) (formerly E(1300)) resonances. Both resonances have isospin I = 0 and are therefore only accessible for the initial 7r-~r- and r%r ° state in the ratio 2 : 1. Taking into account that strangeness production in hadronic collisions is dominated by intermediate resonance production, as can be seen most clearly in data on r N --* KA(E) reactions, one must expect that these two channels are the only pion channels which contribute to K K production. This gives a factor of 3 for the a,~o~ilable pion degrees

of freedom in the rate equation as compared to (3~3) =~9 when all pion degrees of freedom are taken into account. Furthermore, the unitarity limit for the f0(975) maximum of the cross section is given by a(s) ~ 1 mb, which is smaller than the cross section often used [41,30,42].

And, finally, the assumption of an energy-independent cross section leads to a gross overestimate of the mean thermal cross section, since the available final phasespace is much too large. Based on a reasonable estimate about the asymptotic behavior of the rTr ~ K K cross section, combined with low energy dominance by the above-mentioned resonances, results in much larger equilibration times for this particular process [43]. Even at temperatures as high as T ~ 200 MeV,r,q~ is of the order of 100 fm/c, which is much too long to contribute essentially to strangeness production.

Resonances which could be excited by rp-collisions usually have very small branching ratios into the KK-channel, so that one has to expect that the corresponding cross sections are also small.

As it turns out, the problem of final state interactions in dense hadron matter has to be handled with care, and much further work has to be done.

St rangeness in Nuc lea r M a t t e r 271

6 Hadronization of Strange Hadrons

Very little is known to this date about the process of hadron formation from a cooling QGP basically because of its non-perturbative nature. It is not even clear whether hadronization in the context of nuclear collisions proceeds in bulk, by formation of bubbles of hadronic mat ter coexisting with the QGP phase, or by evaporation of hadrons from the surface. A detailed knowledge, however, of this process is necessary in order to see how the observed hadrons reflect the flavor content of the initial QGP. Several papers have been devoted to the study of this problem.

Among the recent ones, the works of Kapusta and Mekjian [41], Matsui et al. [30], and Kajantie et al. [44] considered hadronization in bulk in t h e baryon poor regime, assuming either that the nucleation rate for passing from the quark-gluon to the hadron phase is very large or very small compared to the expansion rate of the system. In the first case, the transit ion process is modeled by an equilibrium phase transit ion determined by the Maxwell condition for two coexisting phases. In the second case, the plasma emerges supercooled before the transit ion begins. Although several important features are taken into account in these approaches, the phase transit ion process is not treated as a kinetic process.

To fill this gap, two recent approaches have been undertaken to study hadronization as a genuine non-equilibrium process [1,37]. We will not discuss these models in detail, however, we will point out how different assumptions about the hadron formation process lead finally to different predictions. It is basically the hadronization process which introduces uncertainties and ambiguities into predictions about the "right" strangeness signal.

6.1 Gluon Fragmentation plus Quark Recombination

To establish a connection between the flavor composition of the QGP and the hadron phase, one needs a hadronization model. Unfortunately, the simplest version, where quarks and anti-quarks combine to form mesons, three quarks to form baryons, and so on, produces severe problems with respect to the second law of thermodynamics. Either entropy conservation is violated when considering a phase transition in complete chemical equilibrium or the final hadron gas volmne has to be enormuosly large when allowing for hadronization out of chemical equilibrimn. In the latter case hadrons are well below their chemical equilibrium abundaacies.

A possible solution to this difficulty was proposed by [1] and requires tha t partons, i.e. quarks and gluons, fragment during the hadronization process, thereby balancing the loss of particles due to the recombination. In our model [1], this was aclfieved by converting all gluons into quark-antiquark pairs, whose flavor ratio was determined in analogy to the Santer formula

( - ~ ) f i = fo exp , i = u, d, s (6.1)

(with n ~ 1 GeV/fm being the QCD string constant) in order to account for the SU(3) flavor breaking. With mu = m d = 0 GeV, m, = 170 GeV, this yields a straaageness ratio f s / ( f , , + fa) = 1/6. The number of quarks effectively available for recombination are

N; f$ = Ni + l i N g . (6.2)

where Ng denotes the number of gluons. Two constants c~,/~ introduced to determine the probabilities for meson mad baryon formation, respectively are uniquely fixed by strangeness and baryon number conservation, so tha t no adjustable parameter besides the f i ' s are contained in the model.

A slightly different model was studied by Barz et al. [37] who described hadronization in terms of string breaking. The same relation as above was applied to the s t range/non-strange ratio of produced quark pairs. The main difference, however, between the models of Koch et al. [1] and Barz et al. [37] is tha t hadronization is treated as a reversible process by the latter. The balance equations are very similar, but conversion between QGP and hadron gas can proceed in bo th directions, and is

272 P. Koch

governed by an overall conversion rate r. The entropy increase due to the nonequilibrium nature of the phase transit ion was found to be small, of order 10 per cent. But contrary to global chemical equilibrium models [13,45] no reheating takes place, i.e. the temperature stays nearly constant during the coexistence phase.

6.2 Model Calculations and Predictions

Reviewing the at tempts to describe the flavor chemistry in relativistic nuclear collisions with or without initial QGP formation, we begin the discussion with those hadrons which are usually produced most abundantly and are measured most easily.

6.2.1 The K/~-Rat io .

The K/~r-ratio is the quantity which has received the most scrutiny. The earliest calculations [41,30,44] were done for a QGP formed at net baryon number equal to zero. In the hadronic phase, only pions and kaons were considered, but no baryons. Hadronization was described by overall conservation of the number of strange quarks and entropy. No detailed hadrolfization mechanism was studied by these authors, although some underlying mechanism was assumed to keep the entropy balance. The K/zr- ratio comes out smaller than that of a HG in full chemical equilibrium, but about a factor three larger than in pp collisions. Since no gluon conversion into quark-antiquark pairs was taken into account during the hadronization and the equilibration of strangeness during the ra ther short lifetime of the plasma was not completed, the major contribution of additional kaons where produced in the hadron part of the mixed phase by the process

~r + 7r ~ K/-(

which does not allow the kaons to reach chemical equilibriunl. A typical number obtained is K+/Tr + = K - / ~ r - ~ 0.2 - 0.3.

In the model of Barz et al. [37] somewhat higher K / r - r a t io s are obtained depending on the lifetime of the hadronic mat ter system and the unknown reaction coefficient which determines the speed of plasma conversion. The basic reason for the higher ratios is the fact that gluon conversion into quark- antiquark pairs was taken into account which finally provides as least as many strange quarks as in chemical equilibrium. Their results are shown in Fig.6.2.

For the present experiments done at the Brookhaven AGS and the CERN SPS it has been observed that the central rapidi ty region in these collisions is not baryon free. Model calculations which also take into account a net finite baryon density in the central rapidity region were undertaken by Koch, Mfiller and Rafelski [1], by Barz, Friman, Knoll and Schulz [37], m~d also by Ko and Xia [42]. In the first of these calculations [1], the full hadro- and quarko-chemistry was solved in bo th phases, and the conversion between QGP and HG was described in the fragmentation-recombination model without detailed balance. Volume and longitudinal fireball expansion were studied schematically, assuming the intermediate existence of a mixed QGP/HG phase. Since no reinteraction of produced hadrons with the quark gluon plasma was assumed no reheating takes place in this model. Hadronization was treated basically as plasma volume conversion into hadron matter volume. The evolution was stopped when the hadronization process was completed and the hadrons where assumed to freeze out at the temperature and chemical potential of the coexistence phase.

As shown in Fig. 6.1 the K/zc ratio was found to increase with baryon density, i.e. baryo-chemical potential whereas the / ( / r - r a t i o decreases. This is due to the suppression of light flavours by the baryochemical potential as discussed in section 2. Depending on the baryon density, a K+/Tr+-ratio of the order of 0.5-0.7 (note: K+/~r + = ~K/~r) can be reached [1] in agreement with [37]. In both models the higher kaon rate can be traced back to the extra strange quarks produced by gluon conversion.

On the other hand, a calculation by Ko and Xia [42] predicts a K+/Tr+-ratio never larger thtm 0.2 for reasonable baryon densities.


0.6 ̧

0.4 o)

. ~ ,.

ot o3

0.2

0.1

0 o

QSP

b)

2 0 400 ~B[MeV ] 600

F i g u r e 6.1: The K/~( top) and K/Tr(bottom) ragios along the phase boundary between QGP and hadron gas. The temperature drops with increasing baryondensity from T = 160 MeV at #B = OMeV to T ~ 120 MeV at PB = 600 MeV.

Strange A n t i b a r y o n s . In the model of [1] a very strong enhancement was predicted for strange antibaryons, in particular in the baryon rich QGP ratios like A/A, or ~/E. emerged as excellent probes of QGP formation [10]. This result can be traced back to the strong enhancement of antiquarks from ghion decay in an environment where antiquarks are usually suppressed. It is the entropy content of the ghions which shows up in the enhanced antibaryon production. The quarks on the other hand are relatively insensitive to extra quarks from gluon decay in a baryon-rich plasma.

This effect can be seen clearly in Fig. 6.3-4 which shows the various particle densities at the point when the hadronization process is completed. Unfortunately, this result depends strongly on the fact that , (a) the entropy content of the QGP is much larger than that of the also shown hadron gas when compared at same temperature and baryochemicai potential #B and, (b) detailed balance during the hadronization process was neglected. If one compares the QGP and hadron gas yields at the same entropy no such a strong enhancement is seen rmymore. Furthermore, if the QGP - hadron gas system has enough time to interact so strongly that the hadronization rate of hadrons is balanced by the reabsorption, the antibaryon abundances can change very drastically, they are reabsorbed. This was investigated by Barz et al. [37], and antibaryon to baryon ratios were predicted which always were below those predicted by Koch et al. [1]. This discrepancy again points towards the need of a bet ter understanding of the hadronization of a QGP, and much further work is needed.

It seems fair to say that there is presently no way to decide which scenario is the most likely one. Staying conservative one might say that an initially formed QGP results in at least hadron gas chemical equilibrium abundancies. On the other hand we have shown that this limit most likely will not be reached in the case of a nuclear collision without initial QGP formation. This should be true for strange antibaryons at least.

274

4-

P. Koch

1 . 5 ~

1.0

I

k,

0.0

1.0

0.5

I l I

0 . 0 J I I

0 5 10 15

TIME [fruit] Figure 6.2: The Kilt-ratios as a function of time for different reaction coefficients r, both for baryon rich plasma (full lines (r = 20fro-2), dashed lines (7" --- l fro-2)) and a baryon poor plasma (dashed-dotted lines (r = 20fro-2), dotted lines (r = l fm-2)) [37].

7 S t range H a d r o n P r o d u c t i o n in Nuc lea r Coll is ion E x p e r i m e n t s

In this section I will focus on the most interesting and promising aspects of recent experimental results on strangeness production in nucleus- nucleus collisions (rather than attempting to give a complete overview).

The first experimental data on A, K :e and Ks production in nucleus-nucleus collisions were obtained in Berkeley at the BEVALAC and shortly afterwards at slightly higher energy, on A and K~, in Dubna at the Synehrophasotron (for recent reviews see [46,47]). In these experiments and subsequent analyses the main emphasis was on separation of effects caused by quasi-free independent nucleon- nucleon collisions from those caused by collective nuclear processes. At BEVALAC energies, up to 2.1 GeV/A, we are either below or at best barely above the threshold for strange particle production. Consequently one has to ask if the observed considerably strong strange particle signal is due to final state reseattering processes whose understanding would provide a testing ground for final state interaction models also relevant at higher beam energies.

This is in particular the ease for the recently measttred enhancement in the K+/Tr + ratio [48] at the Brookhaven AGS with beam momenta of 14.5 GeV/e per nucleon and the enhwcement in practically all observed strange hadron channels in the S + S collision at 200 GeV/A at the CERN SPS [49].

Finally I will also comment on the recent observed enhancement in the A and Ks yields [50] in fi-Tantalum collisions. Since the annihilation of/5 in nuclei releases 2 GeV in energy, one might expect a local heating of nuclear matter in these collisions leading to a "hot spot". It was suggested [51] that a possible strangeness enhancement in fi-nucleus collision might be traced back to QGP formation inside such "hot spots"


(a) 10 l!pi~ tlm]] / ~

L / / . / t+

l O ~ ~

Z \

i0.~ /

>t "~"L I0 ~ v-,~ \

To. ZSO Mev \ Ro.3 ~ "N c, cO~ tr,~. i'll) MeV

160 158 TtMeV] 148 130 0 200 PBiMeV ] 400 600

10 ;~ P+ 1 Ira'3] (b)

[ K,l+q)

10:

e , - 06 \ \ \ ~. rlot~ \ \ \

m m +=M~Vl !48, \ \I~ I 4 i : . =_

0 ~0 ~lMeV ] 400 600

Figure 6.3: Strange particle abundences assumin 9 fast volume ezpansion (V ~ t 3) and recombination with 91uon fragmentation: (a) s-quark and (b) g-quark carr~.lin9 hadrons as a function of the temperature T and baryochemieal potential Pb along the phase boundary. Dashed lines give the equilibrium abundances in a pure hadronic gas at same values of T and #b.

3 (a) ~! p tin- / lot- ~ /

/ / ' /

l f f Z ~

/

160 i 0

Ro. 3tin \ o c 0.6 rn+• tTO P ~

TIMeV] 158 148 130 i

i0_,I~ 9i lira+3] (b)

L

1 0 3 ~

".Y,\ ,[ mLpov, ev " . \ \ I -,s . . . . . . . T J Me,%/I ~ , ~, \

160 F"+8 " "~" %48\ ",, 130 0 200 ~BIMeVI 4O0 60o

Figure 6.4: Strange particle abundences assuming slow volume ezpansion (V ~ t) and recombine- tion with gluon fragmentation, else see Fi 9. 6.3.

276 P. Koch

7.1 Subthreshold Strangeness Production at BEVALAC Energies

The first exploratory investigations of violent collisions between massive nuclei have been carried out during the last few years at the Berkeley BEVALAC, and numerous intriguing results have emerged. In particular, the data [46] suggest that a considerable degree of equilibration and collective interaction between the colliding nucleons takes place. This led to the success of models which are based on a compression/heating/expansion cycle of the participating hadronic matter [52,53], opening up the possibility to study small pieces of highly compressed and excited nuclear matter in the laboratory.

As a possible test of the validity of this collision scenario, it was suggested that the observation of particles whose production threshold lies close to or even significantly higher than the energy available in individual nucleon-nucleon collisions would clearly signal collective effects. Much of the hope, originally associated with strange particle observables, was inspired by the small total K+-nucleon cross section which led to the belief that these particles may provide information of the early stage of the nuclear collision process. However, these expections turned out to be too optimistic. Although the mean free path of the K + at normal nuclear matter density is approximately 7 fro, only very few collisions are sufficient to wash out the primordial information [54,55]. Moreover, conventional cascade model calculations which assume that kaons are produced in sequences of independent baryon-baryon collisions are able to reproduce the total production cross section of K+ ' s [56,57,58] although they have problems in explaining the apparently thermal Boltzmann momentum distribution. On the other hand, models assuming that during the nuclear collision process a hadronic matter fireball is formed in complete thermal and chemical equilibrium with respect to all hadronic constituents overpredict the strange hadron yields considerably but are able to reproduce the observed thermal momentum distributions.

Consequently more attention was paid to hadrons whose production cross section is small in single nucleon-nucleon collisions and are considerably below threshold at BEVALAC energies to learn more about the nuclear collision dynamics.

A corresponding search [59,60] for antikaon ( K - ) and mltiproton (15) production has been con- ducted in 12C +12 C, 2sSi +~s Si, and 4°Ca +40 Ca collisions with ELAB <_ 2.1 GeV/A, revealing that the observed yields cannot be explained in "first collision" models based on nucleon Fermi motion only.

Most strikingly in the experiment by Shot et M.[59] /5 were measured with_a rate roughly three orders of magnitude larger than expected from the first collision model not including annihilation. Similar results also hold for the production of K - as measured in the same Si + Si system at different beam energies [59,61].

Although there are several sophisticated and elaborate models available for describing the dynamics of relativistic heavy ion collisions [52,62], we shall focus here on simple analytic estimates, in order to exhibit with greater transparency the underlying physical picture. We follow here closely the reasoning of [63] who recently discussed subthreshold K + and/~ production based on a simple thermal fireball model and assume that central collisions between relativistic nuclei lead to the formation of hot compressed nuclear (hadronic) matter, consisting mainly of nucleons, A(1232) resonances, and pions. A collision scenario is assumed in which, during the main part of the collision process, the mesonic degrees of freedom are frozen in nucleonic resonances, e.g., A resonances whose importance has been often pointed out [46,52,53].

A recent study [64] of pion production in the Vlasov - Uehling - Uhlenbeck (VUU) model [52] has shown that A production starts very soon and increases rapidly during the initial compression stage, while the reabsorption of A's starts later. During the high density stage only a few free pions exist; most of the pions are trapped in the A resonances. In particular, it was shown that A production takes place not only during this high density stage, but rather in a broad interval around the maximum density. There is also the possibility that higher mass resonances could be excited by A N collisions(a


possibility which is not included in the VUU code). For instance, processes like

A N ~ AA, AN*, N N *

a r e possible. In some of these processes also nucleon resonances are exited which can decay into a KA(~) final state (e.g. N'(171O)),

In such a picture, the pions appear only in the final stage of the collision process as decay products of the baryon resonances. This scenario was recently shown [65] to be compatible with the observation of two distinct slopes in the momentum spectra of pions, corresponding to two different production mechanisms. One can qualitatively understand this observation by a dominant ~r production from resonance decays and a small contribution of "directly" produced pions; a qualitative explanation is, however, still missing.

Accepting this explanation, one finds that the transverse energy distributions of A's, p's, K+'s and direct r ' s are compatible with an emission temperature of the order of 120 - 130MeV.

We will show here tha t A resonances might also he effective in subthreshold particle production, if one admits the following assumptions about the collision process:

* The participating nucleons and A's achieve thermal and chemical equilibrium during the nuclear collision process. This assumption has been under extensive discussion and appears to be justified. No chemical equilibration for hadrons other than nucleons and A resonances is assumed. The assumption of thermalization is essential, since for K - and /5 production processes the necessary energy for the colliding N N pair can only be supplied by the tail of the momentum distribution, due to the high threshold for production of a K + K - or N N pair. This has the consequence that the predicted K - , / 5 yields strongly depend on the aclfievable temperature T and serves as a sensitive probe of this parameter.

• During the main period of dynamical evolution, the hot nuclear mat ter fireball builds up a high baryon density PB. This is necessary since K - , /5 production increases as p 2 whereas absorption increases only linearly with Pc. Therefore, K - ' s and/5's could also be useful probes in determining the baryon density pB.

• Finally it is assumed that a large part of the available energy goes into the excitation of A resonances. As discussed in detail by [63], the omission of A's reduces the calculated/5 mad K - yields by a factor of roughly ~ 25. The first reason is, tha t the threshold for heavy particle production is easier to overcome because A's have a larger mass than nucleons, and secondly, by including A's one gains due to the larger spin, isospin degrees of freedom. Hence, these hadrons might also be a sensitive probe of the nucleonic resonances which can be excited. The authors of [63] neglected more massive nucleon resonances at BEVALAC energies, since they expected A excitation to be the dominant mechanism. However, higher mass resonances will certainly be important at Alternating Gradient Synchrotron (AGS) energies (15 GeV/A) at Brookhaven.

Appropriate to the assumption of thermal equilib1~um in [63], simplified hadrochemical rate equations for the t ime evolution of the hadron number densities were analysed, describing the following production processes:

N N - - - * N N + X - X ; X = K , N (7.1a)

N A ~ N N + X X (7.1b)

A A ~ N N + X X (7.1c)

and subsequent annihilation processes N X ~ M

A X ~ M ,

where M is a final state to be specified for both particle species separately.

(7.2a)

(7.2b)

278 P. Koch

In the case of K - production it was often assumed [66] that the strangeness exchange reactions

r Y ~ K - N , Y = {A, E} (7.3)

dominate, and that the contributions of the pair production processes (7.1a-c) are negligible. For the first to be true it is required that during the lifetime of the nuclear fireball an appreciable number of pions already exist. This is presently not supported by detailed studies within the VUU codes, where pions are mostly produced in the final expansion stage, as decay products of A and possibly higher-lying resonances.

The key point is tha t the rates for reactions (7.1) grow quadratically with baryon-density, and will be significant in an initial high density phase. The main process which depletes the abundance of K ' s is the reverse of Eq.(7.3), which basically proceeds by intermediate resonance excitation of Y*'s. (Note that this process can proceed in bo th directions, basically without phase space inhibition.)

7.1.1 Cross-sections and Equilibration Rates

We now discuss in some detail the model [63] that was used to calculate reaction rates. The basic quantity within this approach is the mean thermal cross section (av) as already introduced in section 5. Assuming that the momentum distributions of the colliding particles can be represented by thermal Boltzmann distributions, this quantity is

<o'v>ijk = C~I(T) - ~ aijk(s) Aij(s) / ( l ( ) (7.4) 0

for the reaction ij --* k, where we have defined

C~j(T) ~ 2 = 8Tm~ ,~j K~(m,/T) g~(m/T) (7.5)

and Aij(s) = Is -- (mi + mj)2][s -- (mi -- m y ] (7.6)

Here, mi and mj are rest masses, s is the square of the total c.m. energy, and K1,2 are the usual Bessel functions of the second kind. For the cross sections aljk(s) the measured free space cross sections were used.

Within the hadrochemical approach, one can derive the following evolution equation for the abundance in the rest frame of the fireball:

Nk(~) -- ~k ~ . ~ r ~ ( T ( ~ ) ) Nk(t)] (7.7) -- A~k~))tAkA(T(t) )

where k stands for K - or 15. The dot indicates the derivative with respect to time. The production and annihilation rates A~ and AkA are defined by

1 ~6ij)(av)ijkN.~q(T(t)) (7.8) A k = ~- ~ . ( 1 - N;q(T(t)) t 3

Ak A = 1 ~(av)k i j N:q(T(t)) , (7.9) t~3

where V is the t ime dependent volume of the expanding "fireball" and N~ q is the equilibrium abundance for species i = {N, A} for a given temperature T(t) of the fireball. We have explicitly indicated that the temperature T(t) depends on time, reflecting the cooling of the fireball due to the expansion of the system. In the Boltzmann approximation we have

V T 3 N:q(T) d i ( ~ ) (_~)2 mi PB = K~(~) exp(T), (7.10)


where dl corresponds to the degeneracy of spin and isospin states, T is the temperature, and #B is the chemical potential introduced to control the baryon number of the system. We adopt the same expression to describe the A resonances in accord with the idea tha t the complicated interactions in a hot, dense gas of hadrons can be well described as an ideal gas where resonances appear as "elementary" objects [67,40].

If we neglect for a moment the t ime dependence of the temperature and the volume~ which amounts to a stat ionary fireball, we can easily integrate Eq.(7.7) to arrive at

N, ( t ) = A~, _exp(_A~t)l (7.11)

k k We note tha t the actual abundance is determined by the ratio Ap/AA, which in M1 cases we consider is smaller than N~q(T(t)), i.e., the chemical equilibrium abundances of species k. We point out that this is a consequence of the fact tha t we have omitted the backward-going reactions in Eqs. (7.1) and (7.2). Under the conditions of a shortlived fireball, formed in nuclear collisions, this should be justified. But for t ~ e~, the baekward reactions have to be taken into account since we should find Nk(t) ~ N~ q for large t. This is equivalent to reaching chemical equilibrium. However, in the case of a shortlived fireball, only a stationary equilibrium is reached. The maximum abundmlce is given by the ratio of the production and annihilation rate constants, corresponding to the relevant reactions.

The time constant which determines the approach to this stationary equilibrium is given by the process with the larger rate eonstant. In our case, this is the annihilation process. If we neglect the baryon resonances for a moment we see tha t

<(TV)NNk (7.12) gk( t ) ~ N s . ~-]~j(aV)kNj •

Note that the number of K - alld/5 increases linearly with the baryon nmnber NB. At first sight, this appears counterintuitive, since we expect that the larger the baryon number the larger the annihilation rate. However, from Eqs. (7.1) and (7.2), we see that the production of k = { K - , 16} is proportional to N~, whereas the annihilation is only proportional to NB. This indicates tha t most 16's and K - ' s are produced during the high baryon density stage of the collision process. The annihilation rate coefficient which controls the speed of the reaction process decreases very strongly with decreasing baryon density. Thus, during the final expansion stage, the produced 16's and K - ' s will not be strongly depleted.

These considerations imply that i6's and K - ' s are produced during a short interval between the time when the high baryon density stage is reached (which also amounts to the point where the nucleons and A's are practically equilibrated) and the time when the fireball evolution is dominated by the free expansion. Although this time span is rather short, it is nevertheless larger than the time required to a t ta in s tat ionary equilibrium, which is of the order of 2-3 fm/c. Therefore, one might also assume that the change of temperature is very small during this time, allowing to use Eq.(7.11) to describe approximately the time dependence of K - and t6 production. One might also neglect the factor c -xA "~ since the lifetime of the system under consideration is larger than the time needed to establish the stat ionary equilibrium. The yield of K - ' s and 16's can than be estimated from the simplified expression

Ap(td) Nk ~ AA(td-----~ ' (7.13)

where ta is the decoupling time. Using parametrizations of the measured free space cross sections for the processes (7.1) and (7.2)

and estimating the corresponding resonance induced processes [63] arrived at the result which is shown in Fig. 7.1. Here the calculated K - / p ratio is displayed as a function of temperature.

Due to the uncertainty in the unknown cross sections for the N A and A A induced processes, the parameters aog,~g and a K-g 0, z,z~ were varied. The broken curve corresponds to the choice K~ °'o, NN =

280 P. Koch

K~ _ (TK'K = 6 #b. The full line corresponds to the set %, NN = Z°'o, NA -- O,AA (T0,AN-- O,AA K-K - gT( __4(TKT~ = 6 / ~ b w h i c h was chosen to simulate possible effects due to isospin. If the N A and AA channels are neglected, the K - / p ratio decreases by a factor ~ 24, and one can no longer account for the data.

In order to make contact with experiment, Koch and Dover have taken the recently measured excitation function [60,61] of the Si + Si ---) K - + X reaction in the energy range of I(ELAB(2.1 GeV/A, and have assumed tha t the decoupling temperature is given by the value extracted from the proton slopes at the corresponding beam energies, as measured in other experiments [68]. This procedure is consistent with the conjecture tha t K - ' s are produced at high baryon density during the hot stage of the collision process and also compatible with the recent observation by Broekmann et a/.[65] that the apparent differences in the slopes of pion and proton spectra in Ar + K collisions at 1.8 GeV/A can be explained by the decay kinematics of delta resonances in thermal equilibrium. Thus the proton slope more closely reflects the decoupling (freezeout) temperature. In the above reaction [65] a value T ~ 120MeV was obtained. Interestingly enough, one also gets reasonable agreement with the measured K - transverse momentum spectrum in Si + Si collisions at 2.1 GeV/A when one takes into account the K - ' s resulting from Y* decays [66]. Here also a temperature of the order of T "" 120MeV is found. Although the solid curve in Fig. 7.1 provides an excellent description of the data, one should not at tach any special significance to this fit. The important point is that the observed temperature (energy) dependence of K - production, as well as the absolute value of the K - / p ratio, is reproduced for a range of choices of the A cross sections. It is not clear that the energy dependence can be reproduced by a model which does not rely on a thermal mechanism. The picture would break down if nucleons and A's could no longer be considered in thermal equilibrium.

Some comments are necessary regarding the assumption of chemical equilibrittm. Evidence for the breakdown of this assumption for small A can be seen from the K - production data at 2.1 GeV/A for 12 C +12 C, s s s i +2s Si and 4°Ca +40 Ca systems [60]. When the data are fit with a power law (A '~) dependence, the comparison of the Si and Ca cases yields a ~ 1.1, consistent with the here presented picture when strong final state absorption is included. On the other hand, if one compares C and Si, a larger value a ~. 3.5 is found [60]. This apparently very rapid A dependence could be a signature of incomplete chemical equilibration for the rather light 12C +12 C system. Due to large surface effects for 12C, the number of nucleons and A's contributing to K - production is expected to be less than A.

To get a qualitative idea of the size of this effect, we return to our expressions (7.12) and (7.13) substi tute

eq eq e--At] NN, ~ "-> NN, A [1 - (7.14)

which simulates a possible deviation from equilibrium. Neglecting the A's for the moment, we can write approximately

NK- (ta) ((TV)NNK- ~q ((T,b- N NN [1 - exp(-~t~)][1 - exp(-~-t~)] (7.15)

which for a small lifetime or size of the "fireball", as appropriate for a C + C collision, gives a suppression which imitates a higher power A-dependence. If this explanation is correct, then "subthreshold" production could provide a measure of the degree of chemical equilibration. The mass dependence A measured for the Si and Ca collision systems presumably indicates that equilibration is nearly achieved. One might suspect that this effect would be even more pronounced in the case of subthreshold 16 production.

Utilizing the same model [63] also achieved reasonable agreement for the i~/p-ratio as shown in Fig.7.2.

Note that here the temperature dependence is nmch more pronounced than in the case of K - production, due to the higher threshold. This renders the/3 a rather sensitive probe of the temperature. A change of 20MeV in T corresponds to an order of magnitude change in the yield. Since the ffs do not arise as decay products of resonances, the transverse momentum distribution should provide


15.~ - i 7 I I

10-5 L L N K" .-rl

10"6 i

10 ̀7 L

10. 8 I 60

i i

KAON / PROTON RATIO /

I " ,..J

! -

I/' L ' i so ~ao izo

T ( MeV )

Figure 7.1 The K - / p ratio as a funct ion of temperature from [63]. The dashed line corresponds to - - K-~ K-~ _ 4aK-E the choice ffO, NNKK = ao ,a = aO, LXLx,K-~ while the solid line corresponds to ao, NN = 2aO, Na -- o,aa"

The dashed-dotted line i~ the result if ao, K-K = aKKo,AZx = O, i.e., no A excitation is included. In each

K ~ = 61~b were chosen. The data points are obtained from the measurement~ of Shot et al. case frO, NN [59].

1°71 A.T,.RO O ,.ROTON Np

10 "1c I I T ' ' I ' 100 120 140 160

T(MeV)

Figure 7.2 The ratio N~/Np as a function of temperature [63]. The data point is obtained from Shot [501.

282 P. Koch

a signature of the freezeout temperature. Therefore, it would be desirable to have data on the P± distribution of antiprotons. It would be also interesting to see how far i6 production can be followed into the lower energy region by the higher luminosity beams which will be available in the near future at GSI (Darmstadt) .

Finally, we demonstrate tha t baryon-baryon-induced processes can also account for the measured K + / K - ratio [46] in order of magnitude. Since K+ ' s are mostly produced by the associated production reactions

NN---* K Y N

N A ~ K Y N

A A --* K Y N

and are not absorbed in nuclear matter, we should have approximately

At NK+ = ~ E (av)~ Y N 'N i ; i, j = N, A (7.16)

Z,J

where At is roughly the lifetime of the hot, high density stage of the collision process where K+ ' s axe produced. We neglect the small number of K + ' s coming from K + K pair creation. We have also neglected the K + ' s coming from ~rN --* K Y, in agreement with our assumptions. Introducing

AK Y i = V E (av)KY N ' N i ' (7.17) Z,}

we find for the K + / K - ratio

where A~ ~ and AA K- are given by

(NK+) AKY ~K- (NK-) ~ A .At, (7.18)

= VE <av)~ICYNiNj (7.19) t ,3

= ( 7 . 2 0 )

Data [69] on K and K production in pp collisions indicate that the "free" cross section for the associated a n d p a i r production is of roughly the same order of magnitude above threshold. Hence the ratio A ~ Y / A ~ K depends mainly on the threshold for the corresponding processes.

A rough estimate is then

(N~+) = expr-('u~ - m K - " ) u K - . ~ A A t . (7.21) (N~-) T

With A K- being the inverse mean free path for K - annihilation and At the lifetime of the high density stage, a number like AA K- • At ~ 3 -- 5 is reasonable. Assuming T ~ 120MeV as before, we estimate K + / K - ~ 30 - 50. This is in the ballpark of the measm'ed ratio. The basic mechanism that leads to more K + production is the lower threshold for associated production.

Summarizing the above considerations one might conclude that despite some uncertainties in the detailed dynamics of the nuclear collision process the creation of strm~ge hadrolls and i6's in medimn energy heavy ion collisions might be understood by a superposltlon of baryon-baryon collision processes. However a full quantitative understanding including, in particular, particle inclusive momentum spec- t ra is still lacking but will provide useful information on the nuclear collision process.


120

BO

°

D / 0.8 1 0,~ 01

A\\,,

F i g u r e 7.3: Lines of constant K+ / r +, K - l i t - and ~r+ /p ratios in the phase diagram for a hadron gas in ~hermodynamic equilibrium

7.2 K/~-Rat ios at hrookhaven AGS Energies

The K/~ ratios measured in Si+Au collisions [48] are among the most striking results which so far have come out of the high energy heavy ion collision progrmn. For a beam energy of 14.6 GeV per nucleon the E 802 collaboration finds in the midrapidity region K+/Tr + ~ 19.2 :t= 3 % and K - / v - ~- 3.6 + 0.8 %. These numbers should be compared with data from proton-proton collisions at similar energies in the nucleon-nucleon e.m. frame (i.e. 4-5 GeV) (and not with data from CERN ISR at much higher energies which show both ratios approximately equal to 10 %). At the lower relevant energies one finds K / r ratios of the order of 4-5 % which implies an enhancement in nuclear collisions of the K+/Tr + ratio of nearly a factor of four, whereas the K - / ~ - ratio seems to agree with the pr0ton-proton data. But such a comparison can be very misleading since the proton-proton data correspond to a baryon poor system whereas the AGS data indicate a significant amount of baryon stopping which would result in a baryon rich fireball. In such an environment, however, the hadron gas predicts an increase in the K + yield a n d a d e c r e a s e of the K - yield when compared to a baryonless hadron gas. As already stressed very often [10], the increase of kaons is a result of the dominance of the associated production (K + together with A or ~.), whereas the decrease in the anti-kaons is due to the strangeness exchange c h a n n e l / ~ N ---+ TrY (Y = A,Z) which favours the hyperons at finite baryon density. In spite of this reasoning one should consider both ratios as remarkable, possibly indicating the approach to strangeness equilibration.

Several studies [70,71,72] along these lines have been undertaken, and it was found that the observed particle ratios measured at the AGS seem to be perfectly consistent with a thermally and chemically equilibrated baryon rich hadronic fireball freezing out at a temperature of around 100 MeV and a baryon density of ~- 0.3p0 (pB -- 600 MeV). This is clearly seen in Fig. 7.3 where the surprising confirmation of this interpretation comes from the fact tha t the line corresponding to the

284 P. Koch

. .Q

tl:l &..~.i

>- -O

i , --

13_ "O

Z

10

1

7 0.1 Q_

0.01

o . exp

_ , , \ _,0.0. \ -

- - b K

' . FRITIOF .. ~1%,,,~

0.0 0.5 1.00.0 0.5 1.0

PT [GeV] PT [GeV]

Figu re 7.4: Transverse momentum spectra of pions and kaons in the rapidity range 1 < y < 1.5. The full lines represent the RQMD results [73].

measured lr+/p ratio also crosses the K/Tr lines in nearly the same point. In light of the possibility discussed above that this might be an indii~ect signature for an inter-

mediate QGP phase in these collisions it is certainly necessary to work towards a more detailed understanding of the dynamics of the equilibration process itself.

Such an attempt was undertaken very recently by Mattiello et al. [73] who investigated the Si + An-collision in the Relativistic Quantum Molecular Dynamics (RQMD) approach and found that non-equilibrium (non-thermal momentum distributions of scattering particles) effects caused by cascading and rescattering of "primary" and "secondary" hadrons might play an important role which finally could lead to a strong enhancement of the total kaon production cross section. Their calculated kaon and pion transverse momentum distributions are in good agreement with the experimental data (see Fig.7.4), without requiring the formation of a QGP.

However, the RQMD model does not agree with all observations of the E-802 collaboration. In particular the model predicts roughly a factor of two more charged mesons as measured experimentally and only half of the observed K + / K - ratio. This indicates that a more careful study is necessa~-y.


x

A a:.

U ^

v

A

+

o p-S op-p ---Frili01

v

^

v

~,, S-S

Ks

c : ~

^

v

5-

v

0 50 180 <h->

I

i I i

5b ' ' so <h-> <h->

Figure 7.5: Ratio of the mean multiplicities < A >ace, < [k >ace, < K o >~¢e observed in the S + S kinematical acceptance regions to the total negative hadron multiplicity < h> for peripheral, intermediate and central collisions. Open circles give p + p data, the dashed line represents the Fritiof Monte Carlo Model [74] prediction. Also shown (squares) are the results for minimum bias p + S collisions.

7.3 A, h and Ks Enhancement at CERN SPS Energies

Nucleus-nucleus collisions at 60 and 200 GeV per nucleon have been studied at the CERN SPS with oxygen and sulphur beams and a variety of targets. Three major experiments, NA35, WA85 and NA36 have investigated strange particle production. The most intriguing result came from an analysis of neutral strange particle production in 200 AGeV S -F S interactions [49] which show a significant enhancement of the strange particle yield in central interactions. The A, Ks° and .~ multiplicities divided by the multiplicity of non-strange negative particles ((n"_~)) as a function of <n"_~/is shown in Fig. 7.5

The important feature of the data from Fig. 7.5 is the steep increase of the strange particle yields with the negative particle multiplicity, which can be taken as a measure of the event centrality or even of the initially attained energy density [34].

Incoherent collision models (Monte Carlo event generators based on independent fragmentation of strings, e.g. the LUND Fritiof version [74]) without cascading or final state rescattering between produced hadrons predict approximate proportionality of the strange particles with negative particle multiplicity with as indicated by the horizontal broken line in Fig. 7.5; this is in obvious contradiction to the experimental results. Even the inclusion of final state interactions in such models (e.g. VENUS [75], RQMD [76]) leads only to minor changes of the order of 20%. Thus the observed strong enhancement of strange particles in the Sq- S experiment is difficult to describe using standard hadronic processes.

On the other hand, "naive" patton gas model calculations [49,77] give predictions quite close to the data. A simple model has been formulated [77], where the only difference between nucleon-nucleon and central nucleus-nucleus collisions is a difference in the volume occupied by the quark-gluon plasma system, the creation of which is assumed in both collisions. The system is expected to be in equilibrium with respect to light quarks and gluons, while the heavier strange quarks are far from equilibrium in nucleon-nucleon collisions but the level of equilibration substantially increases in central heavy-ion

286 P. Koch

collisions because of the larger volume. Within this model [77], the average number of produced strange quarks can increase faster than linearly with the charged multiplicity due to secondary parton collisions in a quark gluon plasma phase. When the strange quark density is much smaller than its equilibrium value (according to the model this should be the case in nucleon-nucleon collisions) the strangeness yield is proportional to the number of secondary parton collisions which have taken place during the whole life-time of the plasma system. The model describes well the data on strangeness enhancement and those concerning the J / ¢ suppression [78]. The model is also compatible with the pion characteristics of nucleus-nucleus collisions.

The NA35 collaboration [49] has also pointed out that their estimated high K/Tr ratio of about 0.15 at midrapidity is similar to that recently observed in p +/5 collisions at V~ = 1.8 TeV at the Fermilab TEVATRON which also increases with the centrality of the collision, i.e. with the charged multiplicity at midrapidity. Estimating an initial energy density from the charged particle rapidity density of the TEVATRON experiment, they find it similar to that encountered in central S + S collisions at 200 AGeV if one assumes that the reaction volume is proportional to A 2/3. Hence they suggest that strangeness enhancement might be a general consequence of high energy density.

7.4 C-Enhancement and J/~-Suppression Asher Shor [79] first suggested that a strong enhmlcement of the ¢/w ratio above the value of ~ observed in pp collisions from v ~ = 7 to ~ = 53 GeV [80] could be a rather clean signature of QGP formation. Shor [79] estimated it by calculating the density ratio between sg pairs (determining the coalescence rate into 5) and light q~ pairs (determining the coalescense rate into p0 and w), using thermal and chemical equlibrium values in the QGP:

( -~)~

However, recent studies modelling the chemistry of the hadronization process make this estimate appear somewhat too high, by a factor 2-3. The reason is [1,37] that during hadroulzation more quark-antiquark pairs are created from gluons and that perhaps about 85% of these extra q~ pairs are light ones. Such a mechanism would lead to a dilution of the ratio (1).

A more pessimistic estimate of the final result for this ra~io at the end of hadronization is a value corresponding to a thermalized hadron resonance gas dose to chemical equilibrium [1,37,81], such that from a QGP we expect

J £ P2dP , __(¢) ~,~ e x p [ ~ / T ~ ] - I (w) J ~ 1,~dp ' (7.31)

where Tc is the temperature of the hadroniza,ion phase trmlsition. Values for this ratio vary ][rom 1/3 to 1/2 for Tc in the range 150-200 MeV. These ratios may be slightly enhanced by final state absorption effects in the hadron gas before chemical fi'eeze-out: while the absorption of ¢ is hindered by a small absorption cross section (for example about 8-10 nab on nucleons [82] and even less on mesons) and it is thus reasonable to assume that the number of e-mesons freezes in at the point of hadronization, the interaction of the p and w with the surrom~ding resonance gas is much stronger. They may remain in chemical equilibrium until the hadron gas has cooled down to a lower chemical freeze-out temperature Tf, resulting in a reduction of their abundance. However, chemical kinetic simulations of the hadronization process [1,37] indicate that for most particle species chemical freeze- out occurs soon after completion of the hadronization, such that these final state effects are expected to be quite small.

We thus conclude that an enhancement of (¢)/((w +p0)) by a factor 6-10 above the pp-value of is the most optimistic scenario we can expect from a QGP. In the present experiments it is unlikely that this highest possible enhancement value could be obtained since most probably only a mixed phase is reached (leading to incomplete chemical equilibration of the strange quarks in the plasma subphase), and even that only in a fraction of all events.

In light of these arguments the recent observation by the NA38 collaboration [83] of an increase of


the ¢ / ( w + p) ratio in central 200 A GeV O+U m~d S+U collisions by about a factor 3 over the value measured in the same experiment in 200 A GeV p+U collision appears interesting.

This is particular true since the same collaboration has also reported the suppression of the J / ¢ meson [78] by a factor of two under the same kinematic conditions which also was predicted as a sign for QGP formation [84,85]. This is due to the deconfining nature of the plasma which is supposed to screen the static potential between heavy quarks. The screening lenght, which can be roughly estimated from lattice Monte Carlo calculations, is so small that no bound state appears in the c~ system inside the quark gluon plasma. Only a fast c~ pair which has left the plasma before its relative separation has reached the bound state radius of a J / ¢ can actually form a J/~b.

However this signal is not uniqe since also alternative explainations for the observed suppression have been put forward [85]. The basic idea is that the J / ~ on its way out of the collision zone might interact with hadronic mat ter constituents in the confined state which can destroy the J / ¢ too (for a detailed account of the ra ther involved reasonings I recommend the two recent reviews cited under [85] and references therein).

The crucial question is therefore: is the explaination by QGP formation for bo th observations, - the ¢ enhancement and the J / ¢ suppression -, unique, or can the observed features also be understood in a hadron scenario?

In [86] the effects of secondary vector meson production and absorption caused by rescattering of hadrons in the central rapidity region, which where produced in the primary nucleon-nucleon collisions, were analized.

They set up rate equations for the secondary production processes of the type i + j --* V + X and absorption processes of the type l + V ---* X

d N v = ~ ( a v ) ~ , ' ~ p i ( x ) p j ( x ) - ~ ( a v ) ~ v p l ( x ) p v ( x ) . (7.32) d4x

i , j I

Here the (av) 's are cross sections suitably averaged over the momentum distributions of the colliding particles which enter via their space-time dependent densities pi(x).

In a cylindrically symmetric environment with dominant boost invariant longitudinal motion the densities can be expressed in terms of the measured rapidity densities ~ and, with a few not very dy restrictive simplifying assumptions, a simple analytical expression for the asymptotic rapidity density for vector mesons can be found [86]:

~gv(b, ~)/d~ [~-^v (~,~) ~.Av (~)] dNyNN / d y = Npa~t(b) + R v ( 1 - (7.33)

Here d N N N / d y is the vector mesons density produced in primary NN collisions, T is the total proper t ime available for rescattering (i.e. until freeze-out), and Npa,.t(b) is the impact parameter dependent number of participant nucleons in the collision. The absorption factor

dNch , , r , x--, , , x Av(r) = d y ~ mt 7o) ~ ~o~,,v~, (7.34)

where S e l f is the effective transverse overlap area in the collision, and az is the fraction of the total charged multiplicity contributed by particle species l depends on the impact parameter through the multiplicity density of charged particles which in turn is found to be linearly dependent on the transverse energy produced in a nuclear collision event. This quantity is hard to calculate reliably (due to many unknown absorption cross sections under the sum) and was estimated using the experimental results on J / ¢ suppression. R v is the ratio of secondary production and absorption rates:

~ i , i < av >i v x Plpj (7.35) .RV = ~_,, < av > x p lpv

where all densities are those produced in the primary NN collisions. Koch et a1.[86] estimated that R e ~- 0 (due to the high mass threshold for secondary J / ¢ production), R e ~_ R~ ~- 1 (light non- strange vector mesons are produced and absorbed in equilibrium), and R~ ~ 3 - 4. They assumed that additional production of ¢ mesons proceeds at most with the same probability relative to w and

288 P. Koch

p0 mesons as in the initial nucleon-nucleon scatterings, i.e.

dN~N/dy >w.+p

Using the equilibrium condition for w and p:

Z < av >~+P PiPJ "" Z < av >~w+pPiP.,+p

(7.36)

(7.37)

it was found that R+ is essentially the ratio between hadronic absorption cross sections for non-strange vector mesons and for the ¢ whose absorption in the dominant channels is OZI suppressed.

With R,/, "~ 0 the absorption J /¢ can be expressed for a given value of produced transverse energy E r a s

dNo(ET ,r)/dy = e_A,(Er,r) (7.38) dN~(ET, ro)/dy

with ln( v ) E l < av >tx¢ 4, E~ A¢( ET, r) (7.39)

r0 7rr02 a2/a ~ L p

Here the the observation of the HELIOS collaboration [87] was used that dNch/d~ "" (1/2)E~h/GeV ' ' E ° / G e V in the pseudo-rapidity range 2 < r /< 4 and assumed that for small projectiles Sell " rc.R2p ""

2 ~ 2 / 3 71"to ~L p .

The absorption factors for other vector mesons are given as a power of the one for the J/e , the exponent being just the ratio of absorption cross sections, i.e.

A+(ET, r) = A¢(ET, v)7 + (7.40)

with

E , < ~ >,~ 4, (7.41)

Using experimental data, the ¢ absorption cross section is found to be about 3-4 times bigger than the J/¢ absorption cross section [82], which fixes the remaining parameters in Eq. 7.42. Hence they find [86]:

dY,~+poldy) \ dN~+~oldV ) = n+ + (S[ErIF*(1 - n+) (7.42)

with R+ and 7¢ both being in the range 3-4, and S[ET] being the experimental J l¢ suppression as a function of ET. Inserting the fit

S[ET] = e -°'lE° /A}/s (7.43)

into Eq. 7.43 they get the curve as shown in Fig.7.6b for the ¢ / (w+p °) ratio. A rather good qualitative agreement is achieved with R+ = 3 - 4 and 7¢ = 3.5.

Since the ¢ meson is produced at a low level in in the primary collisions (much below chemical equilibrium), secondary production (which depends quadratically on total multiplicity [86]) wins over absorption (which rises ?nly linearly with multiplicity), and a sizable enhancement of ¢ mesons by secondary interactions appears possible.

However, the [86] analysis depends crucially on the postulated link between the J /¢ absorption and ¢ enhancement: If the former is due to final state rescattering, they are also able to find a way to explain the latter effect. Wether the first part is really true, is still under debate. If it is, then the new data on ¢ production appear not to require QGP formation either, but indicate a well-developed tendency towards chemical equilibration of ¢ mesons via hadronic processes.


17.5

15

1 2 . 5 -

10

75

5

2.5

11~b SUPPRESSION

{o)

o Oxygen- U

A Suiph~-U

200 A GeV

2.5 5. 75 10. 12,5 E°IA 2/3 (5eV)

o -

¢l(ua+¢) ENHANCEMENT

~ p+U

D O*U

A S .U Rv =~..0 H

'

1

f , i , , i , , , , I , , , | I , , t , I , , , ,

0 2 ¢ 6 B 10 ET IAWlOeV)

Figure 7.6 (a) Fit to the J / ¢ suppression as a function of transverse energy measured by NA38. (b) ¢/(w + pO) enhancement as a function of E ° from O+ U and S+ U collisions, normalized to p+ g events [SS].

7.5 Strangeness Enhancement in ~-Nucleus collisions

The annihilation of antinucleons in nuclei leads to a large energy deposition into a small system, opening up numerous possibilities for the study of the dynamical evolution of hot hadronic matter. The study of such systems in an unusual realm of baryon density and temperature is not only potentially very useful in probing the equation of state of heated nuclear matter and studying the mechanism of energy dissipation and the approach to thermodynamical equilibrium but it also might be an arena to look for manifestations of a transition to the QGP [51]. Although a/3-nucleus annihilation event produces initially a smaller localized "hot spot" thm~ possible in central heavy ion collisions, one might still ask wether the subsequent evolution of the system, which leads to the emission of tens of nucleons and other fragments, passes through a QGP phase.

Several models have been applied to the a~lalysis of p-nucleus annihilation processes ranging from the intranuclear cascade model (INC) [88] to a fluid dynamics approach [89].

The multiplicity and momentum spectra [90] of 7r's and nucleons, as well as the distribution of residual nuclei [91], are well described within the INC approach. The search for new phenomena has thus been focused in the domain of strange particle production. However, it was often pointed out, for instance by Cugnon and Vandermeulen [92] for the 15 case, that strangeness enhmlcement is not necessarily a signature for the creation of a "quark gluon soup", but might also occur in a more "conventional" hadronic picture once one includes mtdtinucleon ( A N N and higher order) absorption reactions. But in such a case it seems hard to distinguish between both approaches from a fundamental point of view.

Leaving such questions aside for the moment we will try to understand strange particle production in p-nucleus annihilation in terms of a picture where strangeness is produced and redistributed by a combination of direct and sequential (final state interaction) processes.

Such scenarios have been studied by several attthors [93,94] and the conclusion is that the main features of the data, namely the energy and A dependence of R, as well as the absolute cross section

290 P. Koch

for A production in/5-nucleus collisions for PL ~-- 4GeV/c, can be understood without invoking the notion of an intermediate QGP phase.

7.5.1 Eperimental Data

The data on strange particle production in/5-nucleus collisions is ra ther limited. Condo et a1.[95] have measured the yield Y of A's per incident/5 at low momwntum to be Y ~ 0.02, essentially independent of A (but with sizable errors of 20-30 %). There are also limits [96] of Y < 4 - 5 x 10 -4 for the production of doubly strange AA, A K - or K + K + systems. Miyano et al. [50] have published cross sections for A, ,4. and Ks in/5 + Ta collisions at 4 GeV/c and reported preliminary values [97] at 3 GeV/c.

Ta--.* K s X ) = (~ 43 .4± 10.2mb at 3GeV/c O-(/5 + ( 82 .0±6 .0mb at 4GeV/c

o(/5 -b Ta ~ AX) = / 145.0 4- 20.0rob at 3GeV/c 193.0 :k 12.0rob at 4GeV/c

o-(/5 + Ta ---*/~X) = ( 4.9 i 2.Tmb at 3GeV/c 3.8 -4- 2.0rob at 4GeV/c

At 600 MeV/c, Balestra et al. [98] give

o-(/5+ N e ~ K s X ) = 5.4 ± 1.1rob

a(/5 + Nc --* AX) = 12.3 + 2.8rob

The yield Y = (1.95 ± 0.43) × 10 -2 of A's is consistent with that measured by Condo et al. [95]. The A / K s ratio is

2.3 4- 0.7 ; / 5 + N e at 600MeV/c R = 2 .4±0 .3 ; / 5 + T a at 4 G e V / c

3.3"4-0.9 ; / 3 + T a at 3 GeV/c

These values are an order of magnitude larger than the ratio R ~ 0.25 for/sp at 4 GeV/c [99,100] or R ~ 0.3 for/sd at 2.9 GeV/c [101]. T h e question is whether this order of magnitude enhancement in R is a signal of of new physics, in particular the formation of QGP [51,102,1], or wether it simply reflects the increased probability for/7IN ~ TrY strangeness exchange or even ~rN ~ K Y associated production in the environment of an complex nucleus when compared to the deuteron.

7.5.2 Simple Estimates of A, ft, a n d K, Cros s Sections

In this section we will use simple considerations based on a work by Dover and Koch [93] to discuss initial and final state interactions. Let us consider the various cross sections separately.

The /~ Cross Section From the experiment at 4 GeV/c [50] we have the ratio

a(/5 + Ta ~ A X ) ,~ 7.9 =k 4.2 Rz~= o-(/5 + p --, £ X )

In a simple Glauber model, worked out for inelastic processes by KSlbig and Margolis [103], R£ for a target with A -- N + Z is given by

R£ = Zo-(/5 + p ---* ~,X) + N o ( p + n ~ - o ( / 5+p ~ A X ) A X ) p ' a " ( ) (7.45)


where P(a) is the "survival probability" defined by

with

1 oo

P(a) = -~ fo e-~T(OT(b)db2 (7.46)

t + o ~ T(b) = / - o o dzp(r) (7.47)

where a is the total cross section averaged over the initial state i6p and i6n and the final state ~.p and /~n interactions, and p(r) is the nuclear density (normalized to the mass number A). For a Gaussian density with < r 2 >1/2= r0A1/3 one obtains [93] R h -~ 8.5 which compares well with the experimentally determined quantity.

T h e Ks Cross S e c t i o n There are two components to the K~ production cross section: a "direct" part o'di r reflecting the process pN --* K s X and a part asp generated by "second order" associated production processes ~rN ---* K s X . At low lab momentum the "direct" process dominates and one finds

a(pA ---* K~X) "~ A~lfa(p p ~ K , X ) "~ 5.4rob (7.48)

for the neon target, which is in perfect agreement with the data (see Eq.(7.2)). For the p + T a reaction at 3 GeV/c one obtains

2~rr~ A2/a ,~ 17 (7.49) A e f f : 3a

resulting in a prediction adir -~ 34 mb which is only slightly smaller than the experimental value of 43.4:£10.2 mb indicating that the final state interactions are not strong at these energies. At 4 GeV/c we expect AeI$ "" 18 and adir ~- 36 mb which leaves space for a a,p "~ 46 mb.

This associated production cross section will be discussed in connection with the A production in the next paragraph.

T h e Cross S e c t i o n for A P r o d u c t i o n In a complex nucleus, A's cannot only arise from direct strong A or ~0 production but also via the conversion processes ~=~N --* ~°N, AN. Introducing a conversion probability

2~rr02 Pc = 1 3All3~rc (7.50)

with ac = < a(E+N ---* A N + E°N) > and an effective nmnber of h ' s by A = A + E ° + PoE + we can split the total cross section into several parts,

a(pA ~ £ X ) = adir(fk) + a,p(£) + as~(fk) (7.51)

in terms of direct (d/r), associated production (ap) and strangeness exchange (se) contributions. The different contributions are given by

ad~r(A) = A~II [a(~p ~ AX) + a(pp ~ ~°X) + a(~p ---* ~+X)Pc] (7.52a)

aap(£) = Aely~A(fip ) ~ YiP( iN "* K £ ) (7.52b)

~,~(~) = Ao~A(~p) ~ Y,P(/N --, ~ , ) (7.52c)

where an(~p) is the total inelastic 15p cross section, Yi is the yield of meson i per 16p annihilation, and P( iN ---* jA) is the probability for the final state process iN ---* j A which contains averaging over the momentum distributions of the participating hadrons and the collision geometry.

292 P. Koch

With a few reasonable assumptions Dover and Koch [93] estimated tz(/~ + Ta ~ ~.X;4 GeV/c ) - 215 mb which compares well to the measured 193 4- 12 mb.

They [93] conclude that the absolute cross section for A production in/~+ Ta collisions at 4 GeV/c can be understood in terms of conventional hadronic multistep (find state) processes. The bulk of the A cross section arises from strangeness exchm~ge ( [ ( N ~ ~rY ~ 40%) and associated production (rrN ~ K Y , p N ~ K Y , . . . . 50%) induced by the annihilation mesons which traverse the nuclear medium. However one should note that the quantitative agreement with the experiment rests very strongly on the assumption of strong absorption limit ( Pc --* 1, d l ~+ convert to A). The cross sections for ~ + A --* K - X , E±X which have not been measured are important as a test of these assumptions.


8 S trange Q u a r k M a t t e r in A s t r o p h y s i c s and H e a v y Ion Col l i s ions

The proposal tha t strange quark mat ter droplets ("strangelets") at zero temperature and/~-equilibrium might be absolutely stable [105,106,107,104] has stimulated substantial activity since it could have fundamental importance for cosmological models and fo the physics of neutron stars.

If strange quark mat ter would be absolutely stable, i.e. the ground state of matter , then one would expect the existence of a class of neutron stars made of strange quark matter . At the very high central densities in a neutron star, up to an order of magnitude above normal nuclear mat ter density a seed in form of a strange quark mat ter droplet might form either spontaneously or through a large density fluctuation and subsequently will begin to convert the mat te r around it into strange matter. The "burning" front would first convert the liquid core of the neutron star to exotic matter; the heat ahead of the front would melt the crust of the neutron star, as well as melt the nuclei in the crust into normal fluid nuclear matter , and within an hour or so, the entire star would be converted into a strange neutron star [108,109]. If strange quark stars can be made, than one would expect many, if not all, neutron stars to be strange with gross features which axe not very different from normal neutron stars.

One initial motivation for studying strange stars was an a t tempt to understaald the reported deep underground muons with source in the direction of a compact source composed of a neutron star in close orbit with an ordinary star having the 4.79 hour period of Cygnus X-3 [108]. Such muons were interpreted as the decay products of a new long lived neutral hadron originating in a strange rather than a normal neutron star in Cygnus X-3 [108]. Strange stars have also been proposed as an exotic model for the transient ")'-ray source observed on 5 March 1979 [110].

One very interesting consequence of strange quark mat ter being the lowest energy state of matter would be the possibility of strange quark mat ter droplet formation in the early universe [106]. Such "nuggets" would be ideal candidates for the the present dark mat ter of the universe [111]. Recent work[ll2], however, has shown that strange mat ter could not have survived longer than one second in the early universe, unless the baryon number of the strange mat ter clumps exceeds sire 1046 [113]. Therefore it seems hard to believe that strange quark mat ter will survive as a plausible dark mat ter candidate.

Recently, it was pointed out [114,115,116,118] that strange quark mat ter droplets might as well play an important role in relativistic heavy ion collisions. It was shown [116,118] tha t strangeness separation in the coexistence phase of quark mat ter and hadron mat ter accompanied by pion and kaon radiation from the system can lead to metastable droplets of nearly (uds)-quark symmetric matter . They will then be emitted as long-lived (v > 10 -4 sec) [114], nearly neutral clusters. Such "strangelets" will serve as a qualitatively new structure which, if discovered, would prove the transient existence of a QGP in unique way.

8.1 Cold S t r a n g e Q u a r k M a t t e r

Before we discuss the possibility of strangelet production in nuclear collisions in more detail, we will first discuss the properties of cold strangelets: As in the case of finite temperature, the presence of strange quarks ca~ lead to a lowering of the Fermi energy. The system is either metastable [105,114] or even absolutely stable [106,107]. A necessary condition for the stability of strange quark mat ter against strong decay is tha t the energy per baryon must be smaller than that of the corresponding hadron mat ter ground state. In the case of absolute stability the lat ter would be nuclear matter , but in the case of metastabili ty one has to compare with mat ter built up by hyperons.

Introducing the quantity Is = ns - n~ (8.1)

~[(nq - nq) + (ns - n,)]

294 P. Koch

(E/A)mi~ [MeV] ~ i BI'4.21OMeV, ms-15OMeV . /

1200 - . , . . , . / ' ~ "

. _ . . . . . . J . x . z

.

1000 . /

900

I t , ~ , l , , , , l , , , , I , , ,

0 0.5 1.0 1.5

/<. /- B ~ , % [~V] IBO, 27g

160 , 27g

180 , 150

145, 279

160 , 150

145 , 150

fs

Figure 8.1: Energy per baryon for strange quark matter at zero temperature and zero total pressure as a function of strangeness fraction fs for different bag constants and strange quark masses. The dashed line defines the corresponding mass of the hyperonic matter ground state.

we can compare the ground state energy of quark mat ter at a fixed fs to the corresponding hadronie ground-state energy determined as

f s m A + ( 1 - - f s ) m N - - e B ; O<_fs <_ 1 mhyperoni c = (fs - 1)mE + (2 - f , )mA -- eB ; 1 < fs < 2 (8.2)

( f s - 2 ) m f l + ( 3 - f , ) m z - e s ; 2 < f s <3

where A, E, and f/ are the strange hyperons, and eB is the binding energy per nucleon, which is, for simplicity, taken to be the infinite nuclear matter parameter of 15 MeV. Fig. 8.1 shows a calculation[116] based on a model equation of state of noninteracting particles confined by a bag at temperature T = 0 and total pressure P = 0 for different values of the bag constant B aaxd strange quark mass ms.

One should note tha t absolute stability would require ( ~ ) , , i , < 940 MeV, which can be achieved only for a very restricted set of B 1/4 < 150 MeV and ms < 170 MeV. This is quite different in the case of metastability as indicated by the dashed dotted curve which shows that strange quark matter can be metastable up to B ]/4 < 190 MeV and ms < 180 MeV. Consequently, metastable strangelets will decay by weak interaction, but heavy ion collisions with much shorter reaction times would allow the detection of these exotic objects.

The conclusion that strange quark mat ter can be the absolute ground state depends critically on how interactions are taken into account, in particular at lower densities where neither the ideal gas equation of state for quark mat ter nor pertubation theory might be valid anymore. This issue was discussed by Bethe et al. [122] which argue that strange quark mat ter cannot be the lowest energy state. The interaction s trength in their calculation, which increases the energy of quark matter , is substantially larger than that by Farhi and Jaffe [107]. On the other hand, by neglecting the strange

Strangeness in Nuclear Mat ter 295

quark mass in their calculations, they leave out the attractive of interactions which favors the presence of massive quarks. At the present state of calculation one cannot rule out the possibility that strange quark matter is the absolute ground state of matter.

8.2 Strange Quark Matter at finite Temperature

Let us now turn to the basic mechanism of strangeness separation, which can result in the formation of strange quark matter droplets in heavy ion collisions. Assuming B 1/4 = 160 MeV and ms = 150 MeV, we would argue that strange quark matter with fs = 0.4 (see Fig. 8.1) is the metastable ground state. However, the total energy per baryon can be lowered by additional ~ 50 MeV by assembling the nonstrange quarks into pure nucleonic degrees of freedom, leaving the strange quarks in a strange matter droplet, its strangeness fraction enriched to fs ~ 1. The minimal total energy is obtained with the tangent construction used in Fig. 8.1. Now the system consists of two phases, 60 % of the baryon number contained in nucleons and 40 % forming a strange quark matter cluster with f~ ~ 1. Hence, one finds that for 0 < fs < 1 the "true" ground state is a mixture of two phases, pure nucleonic matter and strange quark matter.

This separation of strange from nonstrange quarks and the occurrence of two different phases is preserved also at finite temperature. In this case the deconfinement phase tremsition between a large number of mesons and baryon resonances and nearly massless quarks and gluons was assumed [118,117] to be of first order, implying that the relaxation times for chemical transmutations as well as the hadronization time are small when compared to the lifetime the hadrorfic matter stays in the coexistence phase.

Entropy conservation was imposed, i.e.

a V = aQG P VQG P Jr" a HG VHG (8.3)

where a and V are the entropy densities and volmuina of the different phases. The total volume is given by V = VQGp + VHG. The Gibbs equilibrium between the two phases is given by

T q G p = THG (8.4a)

PQGP -~ PHG (8.4b)

Pq,QGP -~ ~g,gG (8.4C)"

and P'a,QGP = ~s,HG (8.4d)

The requirement of total strangeness and baryon number conservation yields two additional con- stralnts.

The flavor composition of strange hadrons and quarks yields

PY,V = -t-2#q 4- #s

Ps,~ = q-I~s

(8.5a)

(8.~b)

(8.56 Let us point out again that in the present approach the net strangeness in each separate phase need not vanish, although Stota~ = O.

During the coexistence region, baryon number and entropy conservation forces the system to expand along the critical curve (T, #) by converting plasma volume into hadron gas volume. For a given entropy and baryon number, the temperature and the baryochemical potential specify the volumina occupied by the two phases. The strange chemical potential #~ is determined to ensure Stotat = O. If the hadron phase is eliminated (VHa = 0), the well known result Ps = 0 is recovered:

296 P. Koch

10( £

B% " %~8 MeV rn S = 279 MeV / /

100 200 300 (gq),,,, (bteY)

3

B ~& • 210 bier rn s • 150 MeV

/ J o~ - _ 0.05

100 ~ 3OO 4OO 5OO (~ql=,lHeV)

(a) (b)

F i g u r e 8.2: Ratio of strange to antistrange quarks in the QGP as a function of the chemical potential along the phase boundary. The parameters are (a) B 1/4 = 148 MeV, ms = 279 MeV and (b) B 1/4 = 210 MeV, m, = 150 MeV [118].

in the QGP s and ~ quarks are produced in pairs only. On the other hand, for vanishing QGP (VQap = 0), zero net strangeness leads to a nonzero value of the strange chemical potential, P8 ~ 0: different strange-particle production modes show up in the dominance of the associated production over the direct pair production at finite baryon density [10].

During the coexistence of the QGP and HG, an additional channel opens up for the strangeness: Besides the associated production and pair production in the hadron gas it is possible to have, for example, associated production of a K meson in the sector of the hadron phase and the s quark staying in the QGP. Consequently, the strange quarks in the plasma phase do acquire a chemical potential #, different from zero. This leads to a ratio of s to g quarks larger than unity in the QGP and to a diminished hyperon abundance in the hadronic sector. This phenomenon only occurs for finite net-baryon densities, PB, #q 7 ~ O. For #q = 0, i.e., #s = 0 and s/g = 1. A typical example is shown in Fig. 8.2 for temperatures and baryochemical potentials along the phase boundary, assuming different values of the bag constant and of strange quark mass.

Observe that in Fig. 8.2(a), where the original MIT bag-model constant is used, s /g exceeds 10. Fig. 8.2(b) shows the analogous result obtained with a larger bag constant and smaller strange quark mass. Note tha t the ratio does not exceed 3 for this parameter set. Also shown is the path tha t the system must take in an isentropic expansion with ~r/A = 10. Of special importance is the fact that the accumulation of s quarks in the plasma phase grows with decreasing plasma volume (the fraction X, introduced in Fig. 8.2 denotes the fraction the hadron gas volume occupies compared to the total volume). This opens up the possibility that s quarks may be bound not only in hyperons and strange mesons: They could form strange quark clusters which might be metastable objects.

For small temperatures and large/~q, the ratio p~/p~ becomes very large even for small X, although the net strangeness fraction fs becomes very small. The ratio fs gives the net straalge-quark concen- trat ion in the separating quark droplets. For X = 1 values of 0.85 for fs can be reached, i.e., about one strange quark per baryon number. Rapid expansion can therefore result in the formation of droplets of strange-quark matter.

The possibility of separating strange quarks from antistrange quarks in the QGP-HG transition can lead - towards the late stage of the phase transition - to a tremendous enrichment of strange


quarks in the QGP.

8.3 Strangelet Production in Nuclear Collisions

Up to now we have examined the fate of strange particles during the phase transit ion in a system with zero net strangeness. However, one can also develop a scenario of the phase transition, which takes into account the pre-freeze-out meson radiation from the fireball during its whole evolution. Pious and K +, K ° are easily radiated off the hot surface of the quark phase. The f i - , d - and g-quarks in the initial quark phase can easily find u- or d-quarks in the baryon-rich environment to form pious and K + , K °. In contrast, the equally abundant s-quarks need to find the rare light antiquarks, which

are suppressed by the positive quark chemical potemial #q. Hence K - , K -° are formed much less numerous than the K +, K ° mesons.

Moreover, during the phase coexistence the q u a r k p h a s e is charged up with strangeness. K +- and K°-mesons, which absorb the g-quarks in the hadronic sector, are enriched while the hyperons are suppressed. Pious and K +, K ° are lighter than the baryons, their thermal velocity is higher and therefore they are emitted more rapidly from the hadron gas during the phase coexistence. Thus the system cools by thermal evaporation in addition to the cooling due to expansion.

The meson evaporation in both stages of the expansion just described carries away entropy, energy and antistrangeness. Therefore the residual expanding fireball, which is in the mixed phase, loses entropy and is charged up with net strangeness. The entropy in the quark phase drops quite strongly, while the hadron phase dilutes immensely and hence absorbs the remaining entropy. This is illustrated in Fig.8.3.

To prime this picture, let us give a rough estimate of the entropy loss and the kaon emission during the expansion process of the fireball: The pion and kaon densities in a thermalized system are

3 p~ = ~-~2 2.4T 3 (8.6a)

, T 3 [¥ ,mI( ,~/2 _~,¢ p,~. = z~-~ 2 ~ ] ~ - t ~ - - ) - , - e T e r (8.6b)

Here rn/~- ---- 500 MeV is the mass of the kaon and #K = t~q --#s is its chemical potential. The antikaons are suppressed by a Boltzmann-factor exp(- -pK/T) as compared to the k~ons and are thus neglected here.

In the quark phase at least p,~/2 pions and pK/2 kaons, respectively, leave a surface layer per unit time and area. However, during the phase coexistence nearly all pions and kaons spatter off the outer surface of the hadron phase because the outer layer of mesons moves with nearly speed of light, rapidly losing the thermal contact to the rest of the remaining fireball. Each pion carries ,,~ 4 units of entropy and each kaon carries one unit of antistrangeness, hence the loss of entropy per time is given by the size of the surface times four times the outgoing pion density. The gain of strangeness is obtained analogously:

d(S/A) ~ _ 1__47r(3 NB ).z/34p ~ (S.7a) dt Nb rTr pB

dfs 1 47r'-:---3 NB) dt NB ( 47r pB

(S.Vb)

Let us now investigate the consequences of the above mentioned non-equilibrium meson and kaon evaporation. Rather than analysing the complete time-dependent microscopic evolution, we vary the typical increase of net strangeness fraction Af~ and entropy loss A(S /A) estimated above to follow the evolution of the system in a simple, quasi-isentropic expansion calculation.

We constrain ourselves to B1/4 = 145 MeV and m~ = 150 MeV to discuss some typical scenarios. Fig. 8.4(a) shows the fraction A~GP ~rQGP/nrtot = " 'B / " 'B of baryons in the quark phase as a function of time for an entropy per baryon of 10 for different net strangeness fractions fs of the total system.

298 P. Koch

fl

./}

L

b.

/"

K ° K °

Koo~odioilon K'.~.+ Irom pure quork phose

corries owoy onii-s ond entropy

! K , l~°

~ K * ~ Koonrodiolion _~~~A_~ J"~'~-J/ from lhe coexislence region

¥ T'5--'-- ~ . E . . ~

K °

/ B )190 MeV

I B ~I90Mey

C. -----a,,-

/ /

f

Figure 8.3: Sketch of kaon(K +, K °) radiation from a baryon rich hot fireball (a) from a pure QGP, (b) from the surrounding hadron phase after entering the phasetranaition region, (c) the two possibilities at the end of ezpansion, i.e. a "strangelet" might be formed or multiple strange hadrons are created in the last stage of the transition.


1.0

=

0.5

t :5

B~J~I45 MeV, SIA.10

fs" 1.5

oc~u0t poth 0 6 e ° ° e ' ~ " O 0 e e e Q 0 • Q e o e o 0 e e (I, O Q Q m

, \ , , , , , ,

100 200 300 400 f[fm/c]

F i g u r e 8.4: Time evolution of baryon number in the quark phase assuming isentropic, one- dimensional expansion with S /A = 10 for different initial strangeness fraction fs and B 1/4 = 145 Me V, ms = 150 Me V.

We remark that for zero total strangeness (fs(to) = 0) the quark phase finally totally converts into hadrons. However, the pre-freeze-out kaon radiation discussed above forces the system from an initial net strangeness of zero to finite f~(t) values. We see tha t even for small fs-values (_~ 0.25) the conversion from quark mat te r into hadrons comes to a halt in a prolonged phase transit ion at a point where the pressure of the system approaches zero. The remaining quark droplet has the properties of a "strangelet".

In Fig. 8.5 we fix the total net strangeness at fs = 0.25 and show the evolution of the baryon number in the quark droplet, As qGP as a function of time for various entropy per baryon ratios ranging from 3 to 25.

Note tha t only for values S/A < 20, the phase transition actually comes to a halt. Then strangelets are formed which have masses in the range from 60 to 5 for S/A = 3 to S/A = 15. However, the in i t i a l entropy must have been considerably larger than these S/A-values, because of the loss of entropy due to the meson emission discussed above. On the other hand taldng into account radiation of nucleons [120] could decrease the effective baryon number A much faster than the entropy S which would mean that the quark mat ter blob might not cool further than a certain temperature which finally would allow the quark mat te r blob to disintegrate by radiation. In this calculation [120], however, no initial net strangeness in the pure QGP phase was taken into account which was shown in [118] to be essential for bringing the hadronization process to a halt during the mixed phase. Even more important is tha t [120] use a ra ther large Bag constant B1/4 = 235MeV in their calculation which already at temperature equal to zero would not allow for a stable straagelet. Hence no final conclusion is possible yet.

The time evolution for the entropy in the two individual phases while passing through the coexistence region is shown in Fig. 8.6.

The entropy per baryon of the quark mat ter phase decreases dramatically because of its finite density which lies around two times the density of normal nuclear matter . Nearly all entropy is absorbed by the surrounding dilute hadron gas, mainly nucleons and pions. At some critical point thermal contact cannot be further maintained, - freeze-out happens - leaving a ra ther cold strangelet, with typical values of fs ~- 0.8 and a baryon density of about 1.5 - 2.0 normal nuclear density. Thus we conclude that there are two processes which would suggest the formation of "strangelets" in relativistic heavy ion collisions in the baryon rich regime.

300 P. K o c h

1.0

0.5

; ' : B~J~=IL5 HeY. is :0 25

SIA- 3

• O O * O I $ , O O O i O ~ Q Q e

0 Io:5 100 200 300 400 f [fm/c ]

Figure 8.5: Same as Fig.8.4, but for fized initial strangeness fraction f~ = 0.25 and different entropy per baryon

QGP/HNt (S/A)

25

20

i5

10

5

\ \\

\ \ \ \

\ \

B~I~:145 MeV, fs :025

H M SIA :I 5

"" ~ lIGP SIA:25

- ..._QGP SIA =]5 HM S/A: 5

O i , I I I 010:5 100 200 300 400

I [fmlc]

Figure 8.6: Time evolution of the entropy per baryon in the two individual phase8 while passing through the coexistence region.


SIA 20

I0

0.1

BY4= 180 MeY q / / / / C: 1800MeV ~ / / / / / / :

phase coexistence / . i / /

#° , i [ t I I I i i

] lO kin I OU ELa b[G eVlA]

F i g u r e 8.7: The entropy per baryon produced in a hydrodynamical model as a function of the bombarding energy for hadron matter (h) and QGP (q).

The first is due to s - J-separation ~nd second, the evaporation of pions and kuons from the system, in particular during the phase transition which cools the matter and further charges it up with net strangeness. When the thermal freeze-out stage is reached, a cold strange quark matter droplet is left - a unique signal for the formation of a quark-gluon plasma.

In order the answer the question concerning the experimental conditions which can lead to strangelet formation, we show the entropy per baryon, as calculated in a one dimensional hydrodynamical model [119,121] as a function of the bombarding energy in Fig.8.7.

One observes that according to these calculations a quark-gluon plasma eouM be created at rather moderate bombarding energies. For bombarding energies ~ki,~ = 10 - 50 GeV/N the resulting low ~ L a b entropy per baryon values would favour strangelet formation. In collisions with bombarding energies of 200 GeV/N the entropy in the target fragmentation regions would also be moderate enough for quark matter formation and thus for strangelet creation. It could be detected by its unusually small charge to mass ratio. Also, if a strangelet would undergo secondary interactions with nuclei, the multiple production of A's in such a secondary reaction would serve as a signal for the strange quark matter droplet [115,124].

The possible decay modes of strangelets have been investigated by Chin and Kerman [114] with the most probable weak leptonic decay Q --~ Q' + e- + u and lifetimes of the order of 104sec.

Presently several experiments are underway, both at Brookhaven as well as at CERN, searching for nearly neutral, unusually massive objects. First low statistics result at AGS energies have already been published [123]. An upper limit on the production rate in 14.5 GeV/A Si + Cu has been determined to be 1 in 103 interactions for Z = 1 and 1 in 104 interactions for Z > 2 strmlgelets. In the near future sensitivities beyond 1 in 106 interactions will be achieved.

8.4 Other A s p e c t s of S t r a n g e n e s s Separa t ion

In the case that QGP formation does not lead to stable strangelets, it was proposed to observe the separation effect by:

(i) Strange Particle Correlations Measuring the pair correlations of neutral strange particles, in particular A - A and K~ - Ks [124].

Due to the separation mechanism the hadrons with negative strmageness are expected to be produced

302 P. Koch

mainly at the last stage of the phase transition when the quark phase has become quite small. The produced hyperons will not freeze out immediately, but will interact strongly with the surrounding hadrons while the surface expands roughly with the speed of light. As the thermal velocity of hyperons is considerably smaller, they will still be localized near the center of the system when the fireball has expanded to the freeze-out density. Strong localization in space-time implies that a pair of produced A's should be strongly correlated. H deconfinement does not take place the hyperons are created and thermalized over a larger region in space-time, namely throughout the whole hadron fireball. The two-particle correlation function then has a different shape.

(ii) mT-~lopes of Kaon~ K + and K - mesons may probe the difference in the expansion trajectories of QGP and HG

through their different slope parameters [125] of their energy spectra.

8.5 K a o n C o n d e n s a t i o n

Independent of whether or not strange quark matter is the absolute ground state, one can ask if the ground state of hadronic matter can develop a strange component, via kaon condensation, an issue raised by Lee et al. [117] in a non-interacting hadron gas and by Kaplan and Nelson [126,127] introducing interactions in the framework of an effective chiral meson theory. In the first case kaon condensation is found to take place at unrealistic (for hadron matter) large baryondensities and disappears if one introduces a repulsive interaction for kaons.

The latter authors find the kaon condensed state analogously to the pion condensed state [128] of nuclear matter: in pion condensation the nucleons spontaneously undergo a chiral SU(2) ® SU(2) rotation to become linear superpositions of neutron and proton states; in kaon condensation one has rather a chiral SU(3) ® SU(3) rotation in which, e.g., a neutron state becomes a linear superposition of a neutron and a E - , while a proton is rotated into a linear superposition of a proton, ~.0 and A. Such a rotation leads to a non-zero field expectation value in matter, < .~u >, with the quantum numbers of the kaon.

Kaon condensation is a phenomenon in the hadronic state for which chiral SU(3) ® SU(3) is spontaneously broken; the question of its existence is not immediately coupled to the earlier question of the energetics of strange quark matter.

The basic argument suggesting kaon condensation is that the underlying SU(3) ® SU(3) symmetry of strong interactions implies that kaons should have an effective interaction with nucleons of the form

HKRNN ~ - ~ ( N / V ) ( K / - ( ) (8.8)

where N is the nucleon field, K = (K +, K°), ~/(N is the kaon- nucleon sigma term, and f g is the "kaon decay constant". This interaction acts as an effective mass term for kaons of order - m ~ ( p s / p c ) [ ( K , where m g is the kaon mass and pc is of order three times the nuclear matter density. Thus the effective mass of kaons in matter is reduced by a factor N (1 - PB/Pe), which vanishes at PB = Pc; above this density the system develops a kaon condensate.

However, in order to observe a K + condensate in relativistic collisions one needs a huge number of hyperons to allow for strangeness conservation which also means a large baryon density which might be never reached in nuclear collisions.


9 H - D i b a r y o n P r o d u c t i o n in N u c l e a r Co l l i s i ons

The spectroscopy of multi-quark systems Qmc~ with m + n >_ 4 has provoked considerable interest, on bo th the theoretical and experimental side. However, there is yet no conclusive experimenta evidence for long-lived objects of such structure.

The system which has received the most scrutiny from the theoretical side is the H-particle, with the quark structure (uuddss) ( J " = 0 +, I = 0). For this state, which was originally proposed by Jaffe [129] a variety of predictions are available concerning its mass (for further reference consult [130,131]), and its existence is not yet ruled out.

We shall discuss the possibility of searching for the H in nuclear collisions. In particular, it was suggested [131] tha t the H would be produced quite numerously in relativistic heavy ion collisions which are presently performed at the Brookhaven AGS and the CERN SPS.

As the production of an object with baryon number B = 2 requires a sufficiently large baryon density, ideal places to search for the H would be the central region of the AGS experiments and the fragmentation regions of the CERN experiments. An encouraging feature of heavy ion collisions viz k viz hadron-hadron collisions is the ra ther copious fraction of events, especially at the AGS energies, which involve the creation of more thai1 one unit of strangeness [48] within a relatively small rapidity interval. For a 2aS/ -4- 197Au collision at ~ 14.5 GeV/A, preliminary results [48] indicate that , in a typical central collision, at least one AA pair is produced.

The H could then be produced from baryon-baryon fusion, according to the wave function [132,131]

-5 CH = (AA+H°H°+H+H-+H-H++nE°+~°n+":-p+p': -)

We will, in particular, address H formation in heavy ion collisions from a hadronic fireball. In the case of quark gluon plasma formation and s - g separation [116,117] through the complicated dynamical evolution of the hadronization process, we expect the production of multistrange objects [118] such as the H to be enhanced.

Finally, we point out several possibilities to detect the H. These include the search for the weak decay mode H --~ H-p, the dissociation of the H via the process H + p --+ A + A + p, and the more speculative possibility to detect nuclear fragments of anomalously low charge / mass ratio, which could arise as bound states of the H and a light nuclear core.

9.1 P r o d u c t i o n o f H - D i b a r y o n f r o m a B a r y o n - r i c h H a d r o n G a s

We estimate now the formation of the H in a hot, dense baryon-lich hadronic fireball. Such a picture can be motivated by inspection of the large trmlsverse energy production observed recently in nuclear collisions which, at least at the AGS energies Ebeam ~ 10 -- 15 GeV/A, suggests a large degree of stopping of energy as well as baryon number in the central rapidity region. Assuming an initially collision tube geometry which develops into a fireball in thermodynmnic equilibrium, we can estimate the mean number of hyperons for a given baryon density and temperature T. For reasonable temperatures T ,,~ 1OO - l lO MeV, and baryon densities pB/pO = 43- and ~ (P0 = 0.15fm -3) the number of produced A's and H's are very close to what we have estimated from the preliminary AGS data [131].

In a central Au -4- Au collision approximately 400 nucleons could participate, and at a baryon density PB ~- P0, one could expect roughly 10 - 20A's, 15 - 40H's (summed over all isospin states) and even 1 - 5"Z's. The formation of the H we envisage as a fusion of two 3-quark bags into one 6-quark bag. Following [132] the fusion amplitude Aij for a baryon pair BiB i into the H is taken as the momentum space overlap

1 [ d3Pl d3P¢ ~3H(Pl . . . . 1-f6)¢i(l'~l,f2,ff3)¢j(ff4,ffb,ff¢) (9.1) Aij = (27r) is j 2E, "'" 2E~ '

304 P. Koch

5xIO -I

iO-I

N H

i0-2

I0-]0 0

I I I I I I I I

H's PER CENTRAL COLLISION

SI + Au Au + Au TOTAL/ R:4fm R:8.5 fm "k~.

pB/PO :2.? pB/PO:I ~ . . ~ / ~

TOTAL / / ~?-'~"

/ Y I I I ] I r I

120 140 160 180 120 140 160 180 200 T (MeV) T(MeV)

F i g u r e 9.1: The total number of H's and the individual baryon-baryon fus ion contribution is shown as a func t ion of temperature. Part (a) displays a typical Si 4- Au central collision whereas (b) models a Au + A u collision.

For Gaussian wave functions, and assuming RH = Ri = R j for simplicity (RH = v.m.s, radius of the H), we estimate the total number of H's (in the rest system of the fireball) by overlapping the fusion probability IA~j 12 with Boltzmann momentum distribution functions f~ and f j for the baryons:

with

= ~'~ f d3pi d3pj d3pH ¢.{,7.~ NH ~...¢j(2a.lS (2~r)3 ~ , , , ~ . , , f j (~ j )A~j

3 ,3

(9.2)

1 . 2 = 6, + m, - ,, - (9.3)

YB and #s axe chemical potentials, si is the strangeness quantum number of baryon i, di is the spin degeneracy factor, and T the temperature. The final result can be written as [131]:

NH = 2 ( ~ ) 1/2 ~ .RHea ± 1 + R 2 T "~*~- 7}2J]-3/2 (9.4) ~.. N i g J c l j ( - - ~ - ) H 3 J z$

and m i T 3/2 i

Ni = di(-~-~--) V e x p [ - ~ ( m l - #B - # s sl)] (9.5)

Here V = ~ R 3 is the fireball volume, and Cij axe the appropriate color-spin factors. In Fig. 9.1 we show some typical numbers for the H-abundancies as a function of temperature

for two different collision systems and baryon densities. Since we assmne that each central collision leads to the formation of a hadronic fireball, the estimated munbers are expected to represent the H-abundance per central collision. We have also indicated sepaxately the contributions of the different single fusion contributions.


Usually the number of H's lies in the range 10 - s < N H < 10 -2 for reasonable parameters of PB and T. This number results in a rather substantial cross section and indicates that relativistic heavy ion collisions provide a very promising way to produce the H dibaryon.

9.2 D e t e c t i o n o f the H

Because of the high multiplicity of particles (principally nucleons and pions) usually generated in nuclear collisions, the detection of the H poses a severe problem; after all the H must be detected directly. Dover, Koch and May [131] have discussed several possible methods in some depth. Here we summarize their considerations.

9.2.1 W e a k D e c a y s o f the H

The only calculation of H weak decay which takes full account of the microscopic quark structure of the H wave function is due to Donoghue et al. [133]. Their estimate of the H lifetime rH, as a function of mass, is shown in Fig. 9.2. This calculation has the remarkable feature that T H is at least a factor of 10 larger than the A lifetime rA = 2.7 x 10 - l ° sec, even when the H lies close to the AA threshold. If this is true, the possibilities for experimental detection of the H are greatly improved, since the decay products of the H will be found rather far from the production point.

Experimentally, the charged decay mode H ~ ~ - p is most favorahle for detection. Neglecting phase space factors, Donoghue et al. [133] predict P (E-p) -~ r (E °n ) ~ 3r(An), so the E - p decay mode of the H will be prominent in the range 2134 < mF/ < 2231 MeV. The signature for the H would be a "V" originating far from the production point. For an H produced at rest in the c.m. system for ~ s s i +19v A u at 15 GeV/A, the decay length of the H is about 1.5 meters for rH = 0.5 x 10 - s sec. This is a favorable situation for detection of the E - p "V". On the other hand, if the H mass drops below 2134 MeV, H decay produces one or two neutrons, and detection becomes very difficult.

9.2 .2 D i s s o c i a t i o n o f the H

If the H is deeply bound, one could look for the process of dissociation on a hydrogen or complex nuclear target. Shahbazian et al. [134] have in fact reported two events which are interpreted as H dissociation. The prototype reactions are H + p ~ A + h + p or E - + p + p. After weak decays A ~ pTr- or E - ~ ATr- ~ p r 7r , a five-prong final state 3p + 2~r- is produced. If one is able to detect the recoil proton [135] as well as the charged decay products, one could determine the mass of the H. The dissociation of the H into ~ + ~ - or ~0Z0 is not favorable for detection, because of the presence of two or more neutral particles in the finM state.

The presence of a large neutron flux could constitute an important background which might mask the signal due to H dissociation. The cross section for the process n p ---* 5 prongs varies from 0.2 to 0.6 mb in the 3 - 4 GeV region [136], which is much smaller than the estimated dissociation cross section a ~ 3 - 10 mb. However, since N , , / N H is very large, a detector would need to distinguish between the 3p~r- final state (the signal) and events containing combinations like 2pTr + 27r-, which could arise from n p interactions.

9 .2 .3 N u c l e a r F r a g m e n t s c o n t a i n i n g the H

If the H is very long-lived, as would be the case if m H < 2055 MeV, it would not be practical to look for the weak decay H --* n n in the debris of a heavy ion collision. An alternative strategy, although speculative, is to look for nuclear fragments [131] of abnormal charge to mass ratio Q / M , consisting of an H bound to a nuclear core consisting of Z protons and N neutrons (with A = N -I-Z). Writing rn H = (2 + x ) m n , where O < z < 2 ( m h / m , ~ - 1), and neglecting the binding energy, we have (M in units of m , )

O z w, - - ( 9 . 6 )

M A + 2 + x

306 P. Koch

Figure 9.2: (~om pss/).

,o ~

io 6

~661

'S

- r I -

I I l I 1 I

WEAK DECAY LIFETIME r H

nn NNTr An ~N AN~r AA'

1.9 1.95 2.0 2.05 2.1 2.15 2.2

m H (GeVlc 2)

The weak decay lifetime rH of the K dibaryon as a function of the K-mass, mH


For the systems H + d(4HH), H + t(SH H) and H + o~(~HHe), we have

Q ~ o . 1 1 o - o . 1 2 5 ( b H ) - - = 0 . 1 8 0 - 0.200 (~,H) (9.7) M ~ 0.313- 0.330 (~He)

A few neutron-rich metastable nuclear species also have a low Q/M (ex. SHe, with Q/M ~ ¼). If necessary, these could be distinguished from H -nuclei by a measurement of M.

Since the H is much more massive than the nucleon, only a ra ther shallow attractive potential is needed for it to bind to a nucleus. The production of H-nuclei may proceed via a second order coalescence mechanism in which {d, t, a ) clusters are formed, subsequently fusing with the H to form ~H, ~H, ~He. Band5 et al. [137] have performed such calculations for AA hypernuclear formation. There will be a substantial penalty factor for such a higher order fusion process; in [131], a rate of order 10 -4 is estimated for ~ H formation, relative to tha t for the H itself. Thus such objects will be rare, but their signature of an anomalously low Q/M ratio is quite distinctive.

Another ra ther speculative possibility is tha t of an H 2 = (HH)L=0 bound state [J~ = 0 +, I = 0], a S = - 4 analog to the a particle. Only a small amount of at t ract ion is required to bind such an object. Some of this would be provided by second order pion exchange (HH ~ H~ H~ ~r HH, analogous to NN --* AA --* NN). In heavy ion collisions, such an object could be formed by a second order fusion process, with a rate estimated [131] at 10 -6 to 10 -7 per central collision at AGS energies. The weak decays of the HH system could provide spectacular experimental signatures, for instance H 2 --* 4A --* 4 p + 4~r- (an eight-prong event) or H 2 ---* ~-pH ~ ~ - + 3p + 27r- (a six-prong event).

308 P. Koch

10 Epilogue

I shall not try here to comprehensively repeat the discussion set forth ill the preceeding sections, but instead concentrate on the most important results.

In the first part of this review I have tried to lay out the ideas and theoretical approaches behind the proposal tha t strangeness might be a useful probe for discovering QGP formation in relativistic nuclear collisions. I have argued that one of the often debated questions, the chemical equilibration of strange quarks in the QGP, seems to converge to its affirmative answer. Although there are still questions open concerning the initial state process in nuclear collisions and possible non-perturbative plasma effects it is fair to say that taking into account pre-equilibration, s t -product ion in the QGP and during the hadronization proces, strange quarks most likely reach their chemical equilibrium abundance.

In the following sections we discussed the approach to chemical equilibrium in nuclear collisions via hadronic final state interactions and the influence of the hadrol~ization phase transit ion on strange hadron yields. We showed that scenarios based on different hadrolfization scenarios predict different results for strange hadron abundances, in particular for strange antibaryons. In the case of the often discussed K / r - r a t i o s most calculations converge to a value around 0.25-0.3 in baryon poor mat ter which is a factor 3-4 higher than in p - p collisions but below the predictions of a chemical equilibrated gas of hadrons.

The situation looks bet ter for the rare strange antibaryons. Two recent model calculations indicate that initial QGP formation results at least in chemical equilibrium abundances. We argued however, that in nuclear collisions without initial QGP formation the level of chemical equilibration will by far not be reached. In the case tha t hadrons produced from the plasma decouple early from the system even higher abundances might be expected.

We have also discussed some selected data on strange particle production in nuclear collisions ranging from medium BEVALAC enegies up to the presently highest beam energies at the CERN SPS which might indicate "new" physics. We have pointed out that the enhancement of ~, K, , A and ,~ as seen in CERN experiments are in accord with the QGP hypothesis. But we also showed, that in particular for the ¢ enhancement an explanation in terms of hadronic final state interactions can not be ruled out, provided the latter are also responsible for the observed J/¢ suppression. However this explanation is not unambiguous mid a failure of the hadronic scenario for the J/¢ suppression would also put a strong strain on the hadronic "explanation" for the ¢ enhancement. Taking into account the fact tha t presently no hadronic scenario exists for the strange particle enhancement seen in the S + S experiment we might be tempted to shift the balance towards the QGP scenario. But such a conclusion has to walt for more detailed models and confirmation of the data. All in all, there is mounting evidence that in these collisions a new state of vel T high energy density and particle density is created and strangeness seems to be an appropriate probe to study this matter . Wether it also satisfies the criteria for a QGP, further studies have to show.

In the final par t of the review we have discussed some aspects of the proposal tha t strange quark mat ter might be either absolutely stable with respect to weak interactions or metastable, decaying by weak interaction. We have also discussed in some depth the possibility to create (long-lived) droplets of strange quark mat ter in high energy nuclear collisions which by itself would offer a qualitatively new signature for QGP formation. That nuclear collisions might also be a suitable place to look for a possibly stable H-dibaryon (the smallest strangelet) was pointed out, and several avenues to detect the H were presented.

A c k n o w l e d g e m e n t s : I would like to thank my friends and colleagues C. B. Dover, C. Greiner, U. Heinz, B. M~ller, J. Rafelski, H. StJcker and K. Werner without whose help and support this work would never have been possible. In particular I want to thank U. Heinz for critically reading the manuscript. I also acknowledge ~upport by the Bunde~ministerium f~r For~chung und Technologie (BMFT), grant 06 OR 764.


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Strangeness in nuclear matter under extreme conditions

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