NBSIR 83-2725

Perspectives in High Energy

Nuclear Collisions

Institut fur Theoretische Physik der Universitat Frankfurt

D-6000 Frankfurt, West Germany

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Radiation Research

Washington, DC 20234

June 1983

U.S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS

national bubt.al Of STANDARDS

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PERSPECTIVES IN HIGH ENERGY

NUCLEAR COLLISIONS

Johann Rafelski

Institut fur Theoretische Physik der Universitat Frankfurt

D-6000 Frankfurt, West Germany

Michael Danos

U.S. DEPARTMENT OF COMMERCE

National Bureau of Standards

Center for Radiation Research

Washington, DC 20234

June 1 983

U.S. DEPARTMENT OF COMMERCE, Malcolm Baldrige, Secretary

NATIONAL BUREAU OF STANDARDS, Ernest Ambler, Doctor

PERSPECTIVES IN HIGH ENERGY NUCLEAR COLLISIONS

Johann Rafelski

GSI, Darmstadt

and

Institut ftfr Theoretische Physik der University Frankfurt,

D-6000 Frankfurt+

and

Michael Danos

National Bureau of Standards, Washington, D.C. 20234

This report gives an overview of some aspects of hadronic physics

relevant for the conception of a research facility devoted to the study

of high energy nuclear collisions. Several concepts to be studied in

nuclear collisions are selected, with emphasis placed on the properties

and nature of the quark-gluon plasma, the formation of the plasma state

in the central region and its anticipated lifetime, and the observability,

through strangeness content of this new form of nuclear matter.

^Permanent address. Supported in part by Deutsche Forschugsgemei nschaft.

1. INTRODUCTION

Until now most of the investigations on the fundamental properties of

matter have been performed in the lightest possible system. While this is an essential first step, there exist important phenomena based on many-body

effects and which are not observable in the simple systems. At this time such

a phenomenon is the hypothetical new phase of matter, viz ., the quark-gluon

plasma, which is that state where owing to a suitable combi nation of baryon

density and temperature the individual hadrons have melted together, or, said

differently, have dissolved, freeing the hadron constituents to form a weakly

interacting Fermi and Bose gas. The existence of such a state of matter is an almost inevitable consequence of quantum chromodynamics (QCD) and the study of

its properties is clearly of utmost importance [l]. 1 In this report we intend to present in reasonable detail our view of the present state of these

developments, and also to provide some speculative extrapolations as guides to

the planning of future experiments.

This new and exciting field of high energy physics is based on the

hypothesis that the energy available in the collision of two relativistic

heavy nuclei, at least in part of the system is equipartitioned among the

accessible degrees of freedom. This means that there exists a domain in space

in which, in a suitable Lorentz frame, the energy of the longitudinal motion

has been largely transformed to transverse degrees of freedom. We call this

region the 'fireball'. The physical variables characterizing a fireball are: energy density, baryon number density, and total volume. The basic question

concerns the internal structure of the fireball. It can consist either of

individual hadrons, or instead, of quarks and gluons in a new physical phase,

the plasma, in which they are deconfined and can move freely over the volume

of the fireball. It appears that the phase transition from the hadronic gas

phase to the quark-gluon plasma is controlled mainly by the energy density of

the fireball. Several estimates [2] lead to 0.6-1 GeV/fm 3 for the critical

energy density, to be compared with ca. 0.16 GeV/fm 3 in nuclear matter. Many

theoretical questions about strong ilffer act ions will be settled when the

parameters and the nature of the phase transition are determined. We turn to

these problems further below.

An important aspect of the developments in this field concerns the interaction of the experimentalists with the plasma. It seems that in order to

observe the characteristics of the plasma one must either use electromagnet - ically interacting particles [3] (photons, lepton pairs) which can rather

easily leave the plasma, or study the heavy flavor abundance generated in the

collision [4]. To understand the latter point imagine that strange quarks are very abundant in the plasma (and indeed they are!). Then, since the (sss)-

state is bound and stable in the perturbative QCD-vacuum, it would be the most

abundant baryon to emerge from the plasma. Surely the observation of such an "omegai zation" of nuclear matter could not leave any doubts about the

occurrence of the phase transition. The observation of other exotic hadrons

[5] such as, e.£. s csq, 2 cs, etc. would support this conclusion. But even the

enhancement oT the more accessible abundance of A may already be sufficient

at least for demonstrating the existence of a plasma.

lumbers in brackets indicate literature references at the end of this paper.

2 This so called A

+ Baryon has recently been observed in Z--Be interactions at

135 GeV/C [S. F. Biagi et al .

,

Phys. Lett. 122B (1983) 455].

1
To continue to higher energy densities, one may speculate that the

restoration of the perturbative QCD vacuum may be followed by the restoration

of chiral symmetry, as shown qualitatively in figure 1.1, then of SU ( 2

)

symmetry, and finally of SU (5) symmetry. This way one could trace back the

evolution of the universe [6] in the laboratory.

Another speculation concerns the fundamental aspect of the possible

catalyzation of the baryon decay in the plasma [5]. A possible mechanism

which has been discussed in the recent literature [7] involves the presence of

magnetic monopoles. The quark-gluon plasma might just be the proper environment in which the catalyzer could continue to burn the baryon number at a rate

sufficient to maintain the necessary particle densities and temperatures.

However, in view of our ignorance of how precisely SU (5) tumbles down to

SU(3) x SU(2) x 11(1) we should be prepared for great surprises in this matter.

Certainly, it would be most challenging to unlock the energy which had

orginated in the Big Bang and has since remained frozen in the baryon number.

Coming back to earth we begin by recalling that in a statistical description of matter the un-handy microscopic variables, viz., energy, baryon

number, etc. are replaced by thermodynamical quantities. To wit, the temperature T is a measure of energy per degree of freedom; the baryon chemical

potential u controls the mean baryon density (see fig. 1.1). The statistical

quantities such as entropy (= measure of the number of available states),

pressure, heat capacity, etc. also will be functions of T and u and will have

to be determined. The theoretical techniques required for the description of

the two quite different phases, viz ., the hadronic gas and the quark-gluon

plasma, must allow for the formation of numerous hadronic resonances on the one side [8], which then at sufficiently high energy density dissolve into the

state consisting of their constituents. At this point we must appreciate the

importance and help in reaching the transition to the quark-gluon plasma

2
provided by a finite, , non-zero temperature. To obtain a high particle

density, instead of compressing the matter (which as it turns out is quite

difficult), we may heat it up; many pions are generated easily, allowing the

transition to occur at moderate, even vanishing baryon density [9],

The paper is organized as follows. We begin by considering in section 2

the state of hadronic matter formed by individual baryons and mesons, which we call the hadronic gas phase. The present summary of the theoretical development of this field is based on the work of Hagedorn and Rafelski, to be

published in greater detail elsewhere [9], We content ourselves here with the

presentation of the main results in so far as they influence our thinking

about the phase transition to the quark-gluon plasma phase. We then turn in

section 3 to a summary of the relevant postulates and results that characterize the current understanding of strong interactions in QCD. The most

important postulate is that the true physical vacuum state in QCD is not the

trivial perturbative state which is little changed when the interactions

between quarks and gluons are turned off or on. In QCD the true vacuum state

is believed to have a complicated structure which originates in the glue

(gauge) sector of the theory. It is supposed not to permit the presence of

color fields. The perturbative vacuum is an excited state with an energy

density B above the true vacuum. It is to be found inside hadrons where

perturbative quanta of the theory, in particular quarks, can therefore exist.

The occurrence of the true vacuum state is intimately connected with the glueglue interaction; gluons also carry the color charge that is responsible for

the quark-quark interaction. The confinement of quarks is a natural conse- quence of the hypothetical structure of the true vacuum.

An important feature which arises as a consequence of the energy density

B of the perturbative vacuum is that the true vacuum exercises a pressure on the surface of the region of the perturbative vacuum. Indeed, this is just

the idea of the original MIT bag model [10]. The Fermi pressure of the con- fined almost massless light quarks is in equilibrium with the vacuum pressure

B. When many quarks are combined to form a giant quark bag then their

properties inside the bag can be obtained using standard methods of many-body

theory [2]. In particular, this also allows the inclusion of the effect of

internal excitation through a finite temperature and through a change in the

chemical composition. This will be discussed in section 4.

The attentive reader might question our simultaneous use of the bootstrap

model and the bag model to describe hadronic states. We will indeed find that

the description of hadrons in terms of bound quark states on the one hand and

the statistical bootstrap for hadrons on the other hand have many common properties and are quite complementary. Both, the statistical bootstrap and

the bag model of quarks are based on quite equivalent phenomenological observations. While it would be most interesting to derive the phenomenological

models quantitatively from the accepted fundamental basis - the Lagrangian

quantum field theory of a nonabelian SU ( 3) "glue" gauge field coupled to

colored quarks - we will have to content ourselves in this report with a qualitative understanding only. Already this will allow us to study the

properties of hadronic matter in both aggregate states - with the emphasis put

in this report in particular on the state in which individual hadrons have

dissolved into the plasma consisting of quarks and of the gauge field quanta,

the gluons.

3
Having described the properties of both hadronic phases, we present in

section 5 a discussion of the possible production and lifetime of the baryon

rich plasma in nuclear collisions in the central kinematic region and describe

the phase transition between the hadron gas and the plasma in section 6. As

opposed to the prevailing opinion [11] according to which the baryons populate

the fragmentation region while the plasma has very low baryon density and is

formed at central rapidity, we describe a likely, even if perhaps rare,

reaction mechanism in which both energy and baryon number remain together in

the central rapidity region [12]. The conventional description is based on extrapolations of pp and pA collisions, which in our view cannot lead to the

pileup of matter, i.£. , baryon number, which is needed in our description. In order to estimate The evolution of the plasma state we must consider, also

contrary to popular belief [13] that hydrodynamical expansion dominates the

plasma evolution, we show that the losses arising from particle radiation

through the plasma surface [14] are important and, indeed, determine the time

evolution of the baryon-rich plasma.

In section 7 we discuss the role of strangeness as a characteristic

observable of the plasma state [4] with particular emphasis on strangeness

generation in the plasma by elementary processes, and on expectations about

the normal hadronic gas phase. From the comparison of the expectations for

both phases of hadronic matter we are lead to propose the study of strangeness

abundance as a possible approach to the observation of the properties and

parameters of the quark-gluon plasma created in nuclear collisions. This

situation is emphasized in section 8, in which it is suggested that generally

speaking, for several quantitatively known effects strangeness is an excellent

experimental trigger for the presence of plasma droplets in light-ion-nucleus,

or slow p-nucleus collisions.

2. THERMODYNAMICS OF INTERACTING HADRONS

The main hypothesis which allows us to simplify the description is to

postulate the dominance of the hadron-hadron interaction by the hadron resonances [8]. In this case the hadronic gas phase is essentially a superposition of an infinity of different hadronic gases and all information

about the interaction is hidden in the mass spectrum x(m 2 ,b) which describes

the number of hadrons of baryon number b in a mass interval dm 2 [9].

Let us first assume that the mass spectrum x(m 2 ,b) is already known.

Then, following the developments of Refs. [8], [9], the grand canonical level

density a is given by an invariant phase space integral. The extreme richness

of the spectrum, x(m 2 ,b) ~exp(m/T 0 ), enables us to neglect Fermi and Bose

statistics above T » 50 MeV and to treat all particles as "Boltzmannions" in

the external volume V ex

. We find for given p

y

= (E,fT) and baryon number b

(6« = Kronecker 5-function)

o(p,V

ex

,b) = «4 (p)6

K

(b)

«**Cp - z

Pi ) l V ( b - 1=1 1 (bj)

K

N

l b.) n —1i-'- t(p?, b.)d“p (2.1)

1=1 ’ 1=1 ( 2 n ) 3

1 1 1

N N 2Ap. u

V U 1 TT M l

4
Here, the first term corresponds to the vacuum state. The N th term is the sum

over all possible partitions of the total baryon number and the total momentum

p among N Bol tzmannions, each having an internal number of quantum states

given by x(p2,b

1 ). These Boltzmannions are hadronic resonances of baryon

number b.j(- 00 < b^ < <») . Every resonance can move freely in the remaining

volume A left over after subtracting the proper volumes V

c of all hadrons from

the external volume V ex

:

( 2 . 2 )

V y is a covariant generalization of ; in the rest frame = (V,0).

In the generalization (2.1) of the familiar phase space formula, three

essential features of the hadronic interactions are now explicitly included:

a) The dominance of the particle scattering by the dense set of hadronic

resonances via T(m?,b.j).

b) The proper natural volumes of hadronic resonances via A

y

.

c) The conservation of baryon number and the clustering of hadrons into

lumps of matter with |b| >1.

The thermodynamic properties of a hot hadronic gas follow from the study

of the grand partition function Z(B,V,X), as obtained from the level density

a(p,V,b):

Z(8,V,X) . l

Xb

/e' 6

’ p ofp.v.bld-p (2.3)

b=-o°

Here the covariant generalization of thermodynamics with the inverse temperature four-vector 3 has been used. In the rest frame of the relativistic

baryon the chemical potential u is defined by

X = exp( u/T) ; (2.4)

it is introduced in order to conserve the baryon number in the statistical

ensemble. All quantities of physical interest can be derived as usual by

differentiating £nZ with respect to its variables.

Equations (2.1 )-(2.3) leave us with the task of finding the mass spectrum

r. Experimental knowledge of t is limited to low excitations and/or low

baryon number. Hagedorn [8] has introduced a theoretical model: "the statistical bootstrap," in order to obtain a mass spectrum consistent with direct

and indirect experimental evidence. The qualitative arguments leading to an integral equation for x(m 2 ,b) are the following. When V 0X

in eq (2.1) is just

the proper volume V

c of a hadronic cluster then, up to a normalization factor

5
o in eq (2.1) is essentially the mass spectrum t. Indeed, how could we distinguish between a composite system as described by eq (2.1) compressed to

the natural volume of a hadronic cluster and an "elementary" cluster having

the same quantum numbers? Thus we demand

o(p,V,b)l

v=v iHt(p2,b) (2.5)

c

where the "bootstrap constant" H is to be determined below. It is not

sufficient simply to insert eq (2.5) into eq (2.1) to obtain the bootstrap

equation for t; more involved arguments are necessary [9b] in order to obtain

the following "bootstrap equation" for the mass spectrum:

Hx(p 2 ,b) = Hz

b

6Q

(p

2 - M

2

)

00 -

N=2

1_

N!

/<5 4 (p - I pj I « (b - l b.) n Hi(p 2 b.)d 4 p i=l

1 (b x } * i=l1 i=l11 1 ( 2 . 6 )

The first term is the lowest one-particle contribution to the mass spectrum,

z

b

is its statistical weight (21 + 1) (2J + 1). The index "o" restricts the 6 function to the_positive root. Only terms with b = 0, ±1, corresponding to

lowest energy qq (pion) and qqq (nucleon) states contribute in the first term

of eq (2.6). All excitations are contained in the second term since arbitrary

quark configurations can be achieved by combining [ ( qq)

n

( qqq) m

] . The small

influence of heavy flavors is ignored at this point but easily can be

i ntroduced.

In the course of deriving the bootstrap equation (2.6) it turns out that

the cluster volume V

c grows proportional to the invariant cluster mass [9],

V

c (p2) = /FV(4B) . (2.7)

The proportionality constant has been called 4B in order to establish a close

relationship with the quark bag model [10]. The value of B can be derived

from different considerations involving the true and perturbative QCD states.

While the original MIT-bag fit has been B

1/I+ = 145 MeV, the (unweighted)

average of different fits is today

B 1/4 = 190 MeV

B = 170 MeV/fm 3

( 2 . 8 )

As far as the bootstrap is concerned the constant H and the bag constant

B are free parameters. However, as just pointed out, B is determined from

other considerations, while H turns out to be inversly proportional to B [9b].

6
Hence, if one wishes to believe the statistical bootstrap approach to the last

detail there remains no free parameter in this approach. The implications of

this for the transition gas to plasma will now be discussed.

Instead of solving eq (2.6), which leads to the exponential mass spectrum

[ 8 ],

x(m 2 ,b) ~ em ^o (2.9)

we wish to concentrate here on the double integral, j_.e., the Laplace

transform of eq (2.6) which will be all we need to establish the physical

properties of the hadronic gas phase. Introducing the transforms of the one- particle term, eq (2.6)

<Kb,X) = l

X

b

Hz

b

60

(p

2 - M

b

2 )e' 6

' d d-p (2.10)

b=-°°

with pions and nucleons only (K^ is the modified Bessel function)

m . m

M *(M) = 2itHT + 4(x + ^)m

N

K 1 (y

1

)] , (2.11)

and the mass spectrum:

*(B,X) = l

X

b /HT(p 2 ,b)e' B

'd

d

4

p , (2.12)

b=-oo

we find for the entire eq (2.6) the simple relation

4>(M) = *(S,A) + exp[ <}>( 3, A)] - <)>(b. A) - 1 . (2.13)

To study the behavior of 4>( B, A) we make use of the apparent implicit dependence:

<t>( 6, A) = G(*(e,A)) (2.14a)

with the function G being defined by eq (2.13)

v> = 2G + 1 -exp[G] (2.14b)

7
This function G(v>) is shown in figure 2.1. As is apparent there is a maximal

value v>0

= *n(4/e) = 0.3863 (2.14c)

Fig. 2.1 Bootstrap function G(v>) . The dashed line represents the unphysical

branch.

beyond which the function G has no real solutions. Recalling the physical meaning of G, eq (2.12, 2.14a) we conclude that eq (2.14c) establishes a boundary for the values of A, i._e. , y, and T beyond which the hadronic gas phase cannot exist. This boundary is implicitly given by the relation (2.11):

£n(4/e) m 2-HI

cr

[3m^K 1 (T

^

cr

) + 8cosh

cr

)] (2.15)

shown in figure 2.2. our current approach,

that

The region denoted

With H correlated

"Hadronic Gas Phase

to B as given by eq

is described by

(2.8) we find

Terser = 0) = To ~ 160-170 MeV (2.16)

8
>

<D

Fig. 2.2 Boundary of the "hadronic gas phase" in the bootstrap model. In the

shaded region quantum statistics cannot be neglected.

However, in view of the uncertainties involved it is more prudent to argue

that the value T

cr ~ 160-170 MeV which is required in the description of

hadronic reactions determines the value of the parameter H. Note that y

= 0

implies zero baryon number of the plasma state. For yQ

= ucr

(Tcr

= 0) the

solution of eq (2.15) is simply y.- r ~ mN since no quantum statistics effects

have been included. Thus the dashed region in figure 2.2 "nuclear matter" is

excluded from our considerations. As we shall see shortly, the boundary to

the hadronic gas phase is also characterized by a constant energy density

e = 4B.

Given the function G(^) =

<J>( 3 , A) we can in principle study the form of

the hadronic mass spectrum. As it turns out we can obtain the partition

function directly from <}>. Namely, the formal similarity between eq (2.3) and

eq ( 2 . 12 ) can be exploited to derive a relation between their integral

transforms [9] (from here on: 3 = ^ 3 ^

3 ^)

2A(V ) a *nZ(3,V A) 4,( 3, A) (2.17)

ex H(2 tt) 3 30

9
which can also be written in a form which makes more explicit the different

physical inputs:

A(V

p

J 8G(*)

*nZ(Mex

,A) * —vf Z a ( 6, A,V) (2.18)

In the absence of a finite hadronic volume and of the interactions described

by the first two terms respectively, we would simply have an ideal Soltzman

gas described by the one-particle partition function

Z,= Z

q

- + 2cosh(„/T)Z

qqq

(2.19)

where

VT 3 m

z qq/qqq

(21 + 1)(2S + n f_2[/N

11 2

^

m

) Krf) ( 2 . 20 )

The remainder of the discussion of the hadronic gas is an application of

the rules of statistical thermodynamics. However, when working out the

relevant physical consequences we must always remember that the fireball is an isolated physical system for which the statistical approach has been taken in

view of the internal disorder (high number of available states) rather than

because of a coupling to a heat bath. Let us first discuss the role of the

available volume. As we have explicitly assumed, all hadrons have an internal

energy density 4B (actually at finite pressure there is a small correction,

see Ref. [4a] for details). Hence the total energy of the fireball Ep can be

written as

E c

= eV = 4B(V - a)

F ex

v ex

J ( 2 . 21 )

where V - A is the volume occupied by the hadrons. We thus find

ex

4 = V ex

- E p /4B = Vex

(l - e/(4B)) . (2.22)

By investigating the meaning of the thermodynamic averages it turns out

that the apparent (8, A) dependence of the available volume A in eq (2.22) must

be disregarded when differentiating £nZ with respect to 8 and A. As eq (2.1)

shows explicitly, the density of states of extended particles in V gx

is the same as that of point particles in A. Therefore also

10
*nZ(e,V (2.23)

ex

,X) = AnZ

pt (6,A,X) .

We thus first calculate the point particle energy, and baryon number

densities, pressure, and entropy density

ept

= ‘ A 36

*nZ

pt " H(2 tt) 3 Jb 2 ^ 6 » X) (2.24)

1 _ 3

0 7 vpt ' A

X 3X

*nZ

pt

* 2T20 <f>( *) H(2tt) 3 3X36 (2.25)

P . = — iinZ pt ‘ A* ,z-pt "

“ H(2up 36

2T

7 4? <*>( B, X) (2.26)

e . - yv

S pt

= A 3T ^ TJlnZ

pt^

= + ^“7 21

pt (2.27)

From this, we easily find the energy density as

e = <E>

ex

V ex36 *nZ(B,Vex’

x) = VexEpt (2.28)

Inserting eq (2.22) into eq (2.28) and solving for e we find:

£

Dt (M)

e(6,X) =

1 + e

t (6,X)/4B •

Hence we can write eq (2.22) also in another form:

v ex

= A U + ept

(B,X)/4B) .

(2.29)

(2.30)

11
Using eq (2.30) we find for the baryon density, pressure, and entropy density:

v

Pt

1 + e . /4B

pt

(2.31)

P _&L

1 + V

/4B (2.32)

s t

1 + ept

/4B * (2.33)

We now have a complete set of equations of state for the observable

quantities as functions of the chemical potential u, the temperature T, and

the external volume V ex

. While these equations are semi-analytic, one has to

evaluate the different quantities numerically owing to the implicit definition

of <f>( 8, X) that determines Uni. However, when 8, \ approach the critical curve,

figure 2.2, we easily find from the singularity of <|> that e

t diverges and

therefore

e > 4B

p > 0 (2.34)

A > 0

These limits indicate that at the critical line matter has lumped into one large cluster with the energy density 4B. No free volume is left and as only one cluster is present the pressure has vanished. However, the baryon

density varies along the critical curve; it falls with increasing temperature.

This is easily understood: as the temperature is increased more mesons are produced that take up some of the available space. Therefore hadronic matter

can saturate at lower baryon density. We further note here that in order to

properly understand the apoproach to the phase boundary one has to incorporate

and understand the properties of the hadronic world beyond the critical curve. Therefore we now turn to the study of the world of quarks and gluons and

ultimately of the phase of matter consisting of these quanta.

3. THE WORLD OF QUARKS AND GLUONS

From the study of hadronic spectra as well as from hadron-hadron and

hadron-lepton interactions there has emerged convincing evidence for the

description of hadronic structure in terms of quarks [15]. For many purposes

it is entirely satisfactory to consider baryons as bound states of three

fractionally charged particles, while mesons are quark-antiquark bound states.

One of the central aims of this and the next section is to show how this

picture of hadrons can be reconciled with the description of hot hadronic

matter consisting of individual particles described in the preceeding section.

12
We now recall some fundamental assumptions about the strong interactions,

as needed here. The elementary quantum fields which appear in quantum chromodynamics are:

Spin 1: gauge bosons - gluons i = 1 . . .8

Spin 1/2: baryomc matter - quarks

a a - R,G,B = color

Y = d,u,s,c,b,(t) = flavor

The octet of gauge bosons G

y mediates the quark-quark and -antiquark interactions between the color triplets {Red, Green, _Blue} and antitriplets. The

gauge vector fields are written as

8

1 X

1 G i E G

1 (3.1)

u

i = l u2

where X

1 are the generators of the SU ( 3) algebra [16]

[x\xj

] - 21 f

iJk

X

k (3.2)

Only quarks and antiquarks carry baryon number (± 1/3). The flavor of quarks

represents all internal quantum numbers conserved in strong interactions - the

_up and down quarks carry ± 1/2 units of I z (isospin) and combine to form the

lowest baryonic isospin doublet

(!)•(£) <>•>>

and the mesonic isospin triplet

13
These were the input particles of the statistical bootstrap model. The

heavier flavors of quarks include the strange, £harm, bottom and perhaps the

as yet undiscovered top quark. The electric charge of”u, c, t is + 2/3 and

that of d,s,b is - 173.

It is the color-charge of the flavored quarks that introduces the quark

quark interactions. The important fact is that all known hadrons are color

neutral (j_.e., color singlets). Including color into the wave functions

eq (3.3) ancf ignoring the space and spin degrees of freedom we have, _e.£.

,

1 , R G .B G R .B . G B .R B G,R L B R ,G R B .G x p

= — (u ud - u u d + u u d - u u d + u u d - u u d ) /6

(3.4)

7T+ = ft

/ R"R

(u d +

Gt G ud + u d ) (3.5)

where the p, and baryons in general, are color-antisymmetric and it is color

symmetric. The antisymmetry of baryonic wave functions in a hidden degree of

freedom has been one of the original reasons for the introduction of color.

Otherwise, e.cj_., (a++ )*

= 3/2 = (uuu)*= 3/2 could not have an antisymmetric

quark wave Junction as required for Fermions. Further experimental evidence

[17] of color includes the it 0 2y decay rate and the size of the e +e“

hadrons annihilation cross section. However, the evidence for color as a dynamical degree of freedom, in particular, as being responsible for quarkquark interactions, is derived from deep-inelastic lepton-nucleon scattering,

from a detailed study of e +

e* annihilation into hadrons, and in particular,

from the flavor independence of the charmonium and upsilonium potential with

quantitative agreement between the experimental and the theoretical excitation

spectra.

The Lagrangian of quarks and gluons is very similar to that of electrons

and photons

L QED

* *(Y‘(p-eA) - m) * - J

F

u /

uv

(3.6)

except for the required summations over flavor and color:

f 1 avors

L QCD

= ^ q“(Y'P-m

r )q“ +

3 8

9 I q

r°Vl

a, 3=1 i =1

AslSI

1 2

I

4

8

I a=l

F

1 F yv

VlV 1 + herm. conj. + gauge fixing (3.7)

14
The flavor dependent masses m

f of the quarks are small. For u,d flavors one estimates m u a ~ 5-20 MeV when the strange quark mass is chosen in the range

150-280 MeV.’ In particular [18],

m

d

- % i

m

d + mu

3

(3.8a)

m

u jj- = 0.38 ± .13 ; (3.8b)

d

0.045 ± 0.011 (3.8c)

The heavy quark mass differences can be obtained reliably from the detailed

study of quarkonium spectra [19], [20]

m,be- m = 3400 MeV; m - m = 1280 MeV

c s (3.9)

The color field strengths are now

F

1 = 3 G1 - 3 G1 + gf

1jk

G

J G

k

yv y v v y

y y v (3.10)

We record the nonlinearity of F, which is required to secure the invariance

under local non-abelian gauge transformations. The presence of this glue-glue

interaction leads to major differences between the properties of QED and QCD.

As an example let us consider briefly the asymptotic freedom of gauge theories

[ 21 ].

To introduce the subject we note that it is often convenient to define a q-dependent coupling constant, £.£. , through

D(q 2

) = - a(q 2

) ^7 (3.11)

15
where D in this case is the QED longitudinal photon propagator, ignoring for

the moment the transveral photon degrees of freedom. In terms of the polarization function n(q 2

) we have

a(q 2

)

q

2 > m 2e

(3.12)

or equivalently with the more complete form of the polarization function

a" 1

(

q

2

) = or

l {0) + n(q 2 )/a(0)

O 00 = “-'( 0 ) - £- / OTT 0 (3.13)

The electron-loop polarization function n(q 2

) follows from the iteration

of the standard lowest order diagram:

(3.14)

As easily can be seen, a-1

(q

2

) decreases with increasing q

2 > 4m

e

2

. This means that for short distances the effective strength of the QED interaction

increases. Only because of the magnitude of or^O) = 137 is this effect

usually unimportant. However, it has been quite precisely measured.

16
In QCD additional contributions originate in the glue-glue interaction;

Since gluons are massless we cannot select the point q

2 = 0 as a reference

point. We have [21],

a-‘(q 2

) = 0,-'(-g2) ^ [ll - -^] Kn ^ (3.16)

with a certain space-like q

2 = - u

2 * 0 serving now as reference. n q

is the

number of light quark flavors (m 2r < |q

2 |). For large q

2

, absorbing the first

term on the right hand side in eq (3.16) in the definition of m

2 we have the

so called asympototic freedom formula:

a

s (q

2

) 12 TT 1 33 - 2

n

q

£n(- q

2/

A

z

f (3.17)

which, unlike the case of QED

leads to falling a

s with rising

!

q

2

1 for the likely case

n < 16. Hence, at asymptotically short distances the

interaction diminishes and the

theory becomes free. We

emphasize that therefore the

chain of approximations leading

to eq (3.17) here, , in

QCD, becomes more and more

consistent. In figure 3.1 the

running coupling constant is

shown for space-1 ike, q

2 < 0

and time-like q

2 > 0, momenta.

In the latter case we show

Re a

$

:

Fig. 3.1 a (q

2

) for space-1 ike and

Re (a (q

2

) for time-like momenta.

17
(3.18)

We notice that at the presently accessible momenta, i.e., up to 100A (A ~ 200-

400 MeV), a

$

is considerably smaller for time-like q^Than for space-like q

2

. For large a

$ (q

2

) other than order g

2 diagrams must be included into the

determination of ^(q 2

). This may change the value of A which at this stage

is a phenomenological parameter fitted to the experiment and which reflects in

its value the order of the expansion. At present the actual value of A is

rather uncertain since as can be seen in eq (3.17) only small logarithmic

corrections are involved. Quarkonium fits (space-like q

2

) favor A ~ 400 MeV

[19] while deep-inelastic experiments (time-like q

2

) lead to A = 100 ± 100 MeV

As we have seen above, the strength of the qlue-glue interaction influ- ences significantly the glue propagation in the (perturbative) vacuum. Little

is known about the behavior of the glue propagator, i._e. , a(q 2 ), at small q

2

, that is at large distances. Attractive channels in The glue-glue interaction

are expected to induce a gluonic structure onto the vacuum state [22], [23].

To appreciate this remark let us imagine a box of size R filled with a gas of

N gluons. Including a 1/R kinetic energy and an attractive long range interaction we have for the energy density E/V:

and hence for some N = N cr

it would cost no energy to fill the box with

gluons. Hence the empty box (perturbative vacuum) and the box with Ncr particles would be degenerate. We conclude that an improved gound state,

the true vacuum, has to be constructed. Such a state would have a lower

energy density than the value of the perturbative state.

The energy density of the perturbative state is defined with respect to

the true vacuum state and hence is by definition a positive quantity, denoted

by B. This notion has been introduced originally in the MIT bag model [10],

but initially in a different context. The value of B is derived phenomenologically from a fit to the hadronic spectrum [10], [24] or from sum rules

considerations [25] which give

The central assumption of the quark-bag approach is that inside a hadron

where quarks are found the true vacuum structure is displaced or destroyed.

One can turn this point around: quarks can only propagate in domains of space

in which the true vacuum structure is absent. This statement is a resolution

of the quark confinement problem. The remaining difficult problem is to show

the incompatibility of quarks with the true vacuum structure. Examples of

[17].

eBox (N) ~ n/r4 " n2 9

2/R 4 (3.19)

B = [(140 - 210) MeV] 4

= (50 - 250) MeV/fm 3

. (3.20)

18
such behavior in ordinary physics are easily found: , a light wave is

reflected from a mirror surface; magnetic field lines are expelled from superconductors; etc.

In this spirit we may argue that all color-charged particles are reflected at the true vacuum surface (stationary waves) or alternatively, may

under certain circumstances deform the surface. Whatever is the case, the

presence of color electric fields in a volume element is incompatible with the

presence of the vacuum structure. It is interesting to note that the Lorentz

covariance of the theory requires that a negative pressure p

= -B as seen from

the perturbative vacuum acts on the surface between the true and the perturbative vacuum. Hence, in the absence of other forces the excited space domain

containing the perturbative vacuum would quickly vanish.

In this picture of hadronic structure and quark confinement all colorless

assemblies of quarks, antiquarks, and gluons can form stationary states,

called a quark bag. In particular all higher combinations of the three-quark

baryons (qqq) and quark-antiquark mesons (qq) form a permitted state, i.e., a hadronic resonance, much in the spirit of the statistical bootstrap mocTeT of

section 2.

The energy of a hadronic bag of radius R including the particle and the

volume bag terms is:

E(R) = (I xp/R + |

hR 3 B (3.21)

where X^/R are the appropriate eigenvalues, , single particle energies of

"confined" particles and the sum is over all quanta in the bag. Effects of

interactions can be considered to be included in X^, in which case the X^

become functions of the interaction strength and the number of particles

present. For massive particles an additional dependence on mR is recorded.

The radial pressure (force/area) on the surface is:

P r -9E/3R

4ttR 2 -B

(sx

i

) + (3.22)

which, combined with eq (3.21) leads to the interesting relation

E(R) = (3P

r + 4B)V . (3.23)

For a radially stable object P r must vanish, or, in other words, E(R) must

have a minimum. From eq (3.22) we have

EX - i/4

R—•- B- 1 "

mi n (i/) (3.24)

19
and from eq (3.23)

E(Rmin ) 4BV = B

l/J

*( SX

1

) 3/<

* (4») 1/4 (3.25)

From eqs (3.24), (3.25) we learn that the radius of the bag grows with

(ZX^) 1/4

, while it decreases as B‘ 1/4

, as could be expected from dimensional

arguments. Similarly, the energy (mass) of the bag grows with (EX^) 3/4

, but

also with B 1/4

, as expected on dimensional grounds. The remarkable relation

E = 4BV is often called the virial relation as it follows from the

dimensionality of space alone. We further notice that the dimensionless

structure constant

Rmin ^ R min^

(zx

i

) (3.26)

can not be directly compared with the values known for example for protons:

R chargedM

proton proton

3.82 (3.27)

since R^p is not the charge radius but the hadronic radius of the bag. To

illustrate the difference between both quantities, assume that the wave function of the quarks inside of the bag is approximately given by

*q,

“ N c °s(Xi (3.28)

with X

0 = tt/ 2 for the lowest node-less state such that <p (r = R) = 0. The

charge radius is

q

^charge

j

2 (3.29)

where n = 3 is the space dimensionality. For consistency with the wave function (3.28) we choose n = 1 and find

20
R charge

/R _ (1 . I,)

172

0.36 . (3.30)

In three dimensions the difference between the charge radius and hadronic

radius is not so pronounced, but the latter is expected to be 40-50% larger

than the charge radius. Returning to eq (3.26) we note that using X

Q = tt/2 one finds for three quarks

^min E (^min) " ^

(3.31)

which is about twice the anticipated charge radius size, eq (3.27), and in

particular leads to a much too large hadronic radius (~ 1.3 fm)

The true lowest eigenvalue Xq omitting for the present the interactions

is found solving the three-dimensional Dirac equation with the bag boundary

condition [10], which leads to

R E = X

0 = 2.04 . (3.32)

When inserted into eq (3.26), we would find for three quarks for the

hadronic radius of a nucleon

Rm1n ~ 4 x 2.04 x 197 MeV fm/940 MeV s 1.7 fm (3.33)

which clearly is an unacceptable result. Obviously, something is missing in

eq (3.22), and it must be added in order for it to give the proper phenomenology of hadronic states. In the original MIT bag approach an additional

zero-point energy

Eo (3.34)

was introduced. This can be taken care of by replacing (EX-;) in above formula

by (£X

i - Z

0 ).

With this we find for the proton

R

min=l< ZXi

< 3 ' 35 >

which requires Z

0 ~ 2 in order to make Rmj D sufficiently small, < 1 fm,

as long as the noninteracting value X

0 = 2.04 is employed. The constraint

arising from the fact that the sum of the bag energy and E

0 must not become

negative has been so far little appreciated. Namely, a negative value is

21
unacceptable, as it leads to stable empty bags which would contradict the

characteristics of the true vacuum. Using the virial relation eq (3.23), this

requirement becomes:

(3.36)

Recalling now eq (3.35) we find from eq (3.36) that we must have

(3.37)

This consideration is equally valid for mesons, but is less conclusive since

other effects intervene, such as restoration of translational invariance to

the quark bound states. For nucleons, eq (3.37) implies

which is usually just barely satisfied once one includes the interactions.

For free quarks the introduction of Z 0 coupled with the constraint (3.36)

reduces the numerical value eq (3.33) at most by a factor 3/4, which is not

enough to yield the empirical value.

Clearly, this discussion shows the necessity to include the quark-quark

interaction and eventually project on transl ational ly invariant states in

order to resolve this apparent contradiction. For our present discussion it

is important to realize that the quark-bag picture can be made internally

consistent only when the quarks are allowed to interact. Unfortunately, for

"small" bags, j_.

_. , for normal hadronic states, this opens the Pandora's box

of all complex self-energy, exchange and other contributions leading to the

large current confusion in the field of how such large corrections can mutually cancel; we do not discuss such problems as the theoretically infinite

values for Z 0 , or the influence of pionic degrees of freedom when the bag

radius is too small. However, we note that most of these problems disappear

in large bags, i.e., those containing many single-particle excitations.

4. FROM QUARK BAG TO QUARK-GLUON PLASMA

A large quark-gluon bag, Ue_. , one which contains many particles, is

characterized by the available modes and their occupation numbers n • . An

important simplification of its description arises if it is possible to use a statistical treatment.

As the u and d quarks are almost massless inside a bag they can be produced in pairs, and at moderate internal excitations, i.e., temperatures, many

qq pairs will be present. Similarly, s5 pairs also wiTl”be produced. We will

(3.38)

22
return to this point at length below. Furthermore, real gluons can be excited

and will be included here in our considerations. We now first convince

ourselves that already a moderate number of quarks justifies the statistical

approach. For the degenerate Fermi gas of quarks the number of light quarks

(u and d) determines the quark Fermi energy y . Omitting for the present the

qq interactions we have

3 b = = 2. x2f.x3.xV

s f c

4tt

(*») 3 3 (4.1)

where the indices s,f,c refer to spin, flavor, and color degeneracies respectively. Equation (4.1) establishes a relation between a given baryon number b

(quarks carry 1/3 unit of baryon number) and the variables V (volume) and y . The energy of the quark bag is easily obtained noting that

E q,gas

= 2

s x2

f

x3

c xV > ^ ^ ^ ‘ m

q> ) (4 ‘ 2 >

Hence in the limit of small quark mass, j^e. , yq

» m

q

, we find, omitting

here again for the sake of simplicity the qq-interaction term.

E(V,p(N_,V) ) = BV + V — + 0(m Jv) . (4.3)

q 2* 2 q q

In order to determine the explicit dependence on a given quark number (baryon

number) we use eq (4.1) to eliminate yq

:

E(V,N

q

) = BV +

q 473

3

1/3 4

l 2> (4.4)

This expression has as before a minimum as function of the volume V, which

corresponds to the equilibrium state:

P

_3E

av

= 0 = -B + s,b

I ("iy

4 v J (4.5)

Combining eqs (4.4) and (4.5), of course we find again

'min

= 4BV min (4.6)

23
and hence we see that the energy density is 4B also in the statistical bag

(virial theorem). Combining eq (4.4) with (4.5), we find furthermore for the

energy per quark the usual result:

(E/Nq)$fp “

( Mq)min B 1/4 (2it 2

) 1/4 = 2.11 B 1/4 (4.7)

Here (u

q

) min is the chemical potential that is found inserting eq (4.5) into

eq (4.1), upon which the N

q -dependence drops out. This result, eq (4.7) can

be compared with a similar result for the smallest closed-shell bag which

contains 12 quarks owing to 2

s x2fx3 c = 12. With X q

= 2.04 we find from

eq (3.25)

(E/12) bag

= '~T2 4

1/4 Bl/4 = 2,3 Bl/4 * ( 4 * 8 )

Thus we conclude that the statistical result, eq (4.6) is in a remarkably good

agreement with a closed shell bag even when its baryon number is only 4. As

the energy per quark in the statistical bag approach is slightly underest i

mated we conclude that the quark ( i . e . , baryon) density

N q

/V = (E/V) (N

q /E) = 4B/(E/Nq

) (4.9)

is somewhat overestimated.

As a final remark we note that eqs (4.3) and (4.6) imply that the energy

per baryon in the bag is just y, i.e., the baryon chemical potential

E/b = E q/(N

q

/3) = 3u

q

= u (4.10)

The factor 3 is necessary to account for the baryon number 1/3 of the quarks:

three quarks form one baryon. We note that from eq (4.10) stems the conventional wisdom that y = m_/3 at T = 0. Omitting the bag term in eq (4.2) one finds the well known rel at i v i Stic ideal-gas result

E q/N = 3/4 y . (4.11)

Thus we see that the bag term is a necessary ingredient for recovering the

hadronic gas limit [9]

E/b

|

t_q = y

where T is the temperature.

24
Quarks will not always form a degenerate Fermi gas, especially inside a large bag. Depending on the creation history of the bag it is very likely

that in an initial stage some of the quarks will be in excited states. In the

statistical approach this situation easily can be described by introducing a quark temperature T = 1/e which describes the internal excitations of each bag

(= hadronic cluster) [26], This does not imply an internal thermodynamic

equilibrium of the quark gas in the bag. However, an assembly of excited bags

in mutual thermal contact that is sharing a certain internal excitations, may

be already similar in nature to the Gibb's ensemble, _Ke . , an infinity of

interacting identical subsystems.

Hence, though each individual bag may be far from thermodynamic equilibrium, an assembly of bags able to scatter several times may be much closer to

it. When making these remarks here we have particularly in mind highly

excited nuclear matter as created in relativistic nuclear collisions, and,

perhaps in antiproton annihilations on nuclei. Other circumstances prevail in

e+e

- * hadrons or even pp reactions. But also in our case the word "kinetic

equilibrium" has to be used with great care: the further the mean kinetic

energy of quarks is from ~ T, the less reliable becomes a priori the equilibrium assumption. We record here, however, that particle spectra from p-p

collisions [8] behave as if a thermal equilibrium were always reached. In

thermodynamical models of hadron reactions therefore the concept of "preformed" equilibrium has been introduced.

With these remarks in mind we now turn to the description of excited

quark bags with the help of quantum statistical methods. We will initially

ignore the effect of quark-quark interactions and return to this problem

further below. In principle, we may avoid the formal development and simply

proceed by including temperature through a Fermi distribution function in

eq (4.2). However, as was outlined in section 2 a complete description of the

thermodynamical behavior of a many-particle system can be derived from the

grand canonical partition function Z. Hence it is more useful for further

developments to obtain right-away the grand partition functions for ideal

Fermi and Bose gases. We follow here initially the standard textbooks [27] in

calculating the grand canonical partition function defined as

where H is the Hamiltonian of the system and Q is the baryon charge operator.

The chemical potential y determines the average baryon number of the system.

The trace is to be carried out over all allowed states of the many-body

system. To appreciate the importance of the quantity (4.12) we note that

Z(B, V...) = Tr(e- B<

fi ‘ $>

(4.12)

<Q> = ITr-2§-Oe6(

^

5)

Tr e -S(M) iii; 9 • • « (4.13)

(4.14)

25
The partition function may depend implicitly on other quantities such as the

volume or even the shape of the considered quantum system.

In the particle number representation the trace, eq (4.12), can be easily

carried out for free quarks. With

H = l e.n- + l e-n. (4.15a)

i , z M

Q = l

b

t (n* - nf ) (4.15b)

1 , i

where nj1 is the number operator of the i th single-particle state of a quark

(n^ for antiquarks) with (discrete) quantum numbers such as flavors, b^

is the baryon charge + 1/3 for quarks and of opposite sign for antiquarks as introduced already explicitly in eq (4.15b). A quantum state is characterized

by occupation numbers n*, n^ of quarks and antiquarks. Hence the trace which

sums only the diagonal matrix elements is

Z_ =

l e

t n *,n*}

-[3 I (ef-1/3 u)n* + J (e?+l/3 u) n*]

i ,Jt i,£

(4.16)

Here the sum runs over all sets of numbers nf, nf. We factorize the partition

function in terms of the discrete quantum numbers Z: V5 Z * (4.17a)

Z. * l ne

r z -ill i

-8(e*-l/3 p)n? B(s*+l/3 p)n*]

(4.17b)

The infinite product over all states can be interchanged with the infinite sum over all occupation numbers leading to:

bfZ = I [(bn l e

B( ef-l/3 p)n*

)+(bn l e

-B(e*+l/3 win*

1

)] . (4.18)

1 z

n

l

- z

n

l

26
We can have only n^ = 0,1 for fermions. Hence we find the well known result

(f = flavor, s = spin, c = color)

-6(e.-l/3 u) - 0( e .+1/3 u)

*nZ * l 2 3 [I tn(l + e ) + I *n(l + e )] (4.19)

H flavor i i

where the spin and color factors count the respective degeneracies. In the

continuum limit

Xjd 3x d

3 p = v |

d

3 p (4.20a)

i ( 2tt) 3

( 2 ir) 3

f

e. -

l —> /p z + mj? (4.20b)

we find

inZ =

j 3 -0(/m|+p ?-l/3 u) - 3( /rrTp+p^+l/S m)

I 2 3 V / [*n(l+e

f

) + *n(l+e

f

)].

f

s c (2 tt)3

(4.21)

For the light u and d quarks, for which usually m^ « y is fulfilled, we can evaluate the momentum integrals analytically [ 2 b], [28]:

(T £nZ ),.

^ q^light flavors

2 x2,x3 xV

s f c

24ir 2 (Uq + 2 y

2

( nT) 2 + 7_

15 (*T)“) (4.22)

where we have used the quark chemical potential

(4.23)

which controls the quark number N

q

- Nq.

In order to recover the limit T 0, eq (4.3), we must introduce a phenomenological bag, i.e., vacuum term

*"Z bag = -BV3 . (4.24)

27
With eq (4.24) and

JlnZ = JlnZq + *nZ vac (4.25)

we indeed find

E <H> - h *nZ + u t;: T lnZ

BV + -2- V [ujj + 2m2(„t)2 + 7- (,!)'.] (4.26)

and at the same time for the baryon number

b/3 -- 4 Q> * Nq - N- = T tnZ

« v C

|,a + M*1 ) 2

) (4 ' 27)

TT 2 M H

Equations (4.26) and (4.27) generalize the T s 0 results, eq (4.1) and (4.3)

to finite temperatures. We note also that for finite T it is possible to

eliminate y analytically from eq (4.26) with the help of Cardan's formulae

[28] for eq^(4.27), and to obtain E(V,b,T).

We have not yet considered one quite important aspect of the excited bag,

viz . , the possible presence of real transverse gluons. At present the evidence

for the existence of perturbative gluons inside bags is not quite conclusive.

Some theoretical calculations [29] indicate that glue could be admixed to the

quark wave functions in bags. However, gluonium (glue-only bags) have not been

conclusively established experimentally [30]. We record here that the color

confining bag boundary condition should be different for perturbative gluons

from that postulated for the perturbative quarks, as a direct consequence of

the fact that the vacuum structure originates in the gauge (glue) sector of

QCD. While this problem remains to be resolved it is very likely that in a large quark bag at finite temperature these transverse gluons will be present

with an abundance corresponding to that expected from the blackbody radiation

law, j_.je., as given by the Stefan-Boltzmann equation. To include these bosonic

degrees of freedom we evaluate eq (4.16) taking into account the fact that the

occupation number of glue modes can be n^ = 1 , 2 ,...», and that they do not

carry baryon charge. We thus find

28
„ & 1 °° -3ne- 1 Z = n n l e = n n r . (4.28)

y A i n=o i i -Be

1 - e

1

Here l counts the N

2 - 1 = 8 color degrees of freedom as well as the two

transverse polarizations. All gluon single-particle energies are degenerate

with respect to A. Taking the continuum limit we find:

tnZ

g

* - 8

c

2

S V ! T2*P *" (1 ' e

’ BlPl

) • (4 ' 29)

This expression is very well known and corresponds up to the color factor to

the standard photon result. We find explicitly

T *"Z g

= 8 ih* (,rT)4 • (4 - 30)

As emphasized at the end of the last section a further effect that must

be taken into account is the quark-quark interaction. We shall use here the

first order contributions in the QCD running coupling constant a

$ (q

2

) = g

2/4ir.

As ct s (q

2

) increases when the average momentum exchanged between quarks

decreases it would seem that this approach will have only limited validity at

relatively low densities and/or temperatures. However, the collective

screening and phonon modes in the plasma are of comparable order of magnitude

and cancel each other [31]. The influence of perturbative contributions which

are governed by the expansion factor 6 = 4/3 o^/ir ~ 0.15 - 0.3 is limited. In

other words, since S 2 < 0.02 - 0.09 the use of first order perturbation theory

may be quite adequate [32]. For the case of the quark-gluon plasma in the

perturbative vacuum, one finds for the partition function an analytic expression through first order in a neglecting the quark masses. We obtain for the

quark Fermi gas [2b], [2g], [28]:

gV 2a . tt 2 50 a 7ir 4 * nZ

q (S,u) = s3 [(l - (us) 4 + T (ue) 2

) + (1 - 21 /) go-] (4.31)

where g= (2s + 1 )

( 2 I + 1)N = 12 counts the number of the components,

the degeneracy of the quark gas. The glue partition function is similarTy

reduced by the interactions:

8 it 2 15a

«nZ

g (B,X) = V w S'

3 (1 - -jf) _ (4.32)

29
We notice above the second relevant difference from the photon gas: aside from

the occurrence of the factor 8 associated with the number of gluons there is

the term associated with the glue-glue interaction since gluons carry colorcharge.

Finally, let us recall the true-vacuum term An Z

va(-, eq (4.24). This

leads to the required positive energy density B within the volume occupied by

the colored quarks and gluons and to a negative pressure on the surface of

this region. At this stage, this term is entirely phenomenological as discussed above. The equations of state for the quark-gluon plasma are easily

obtained by differentiating

JtnZ = JlnZq + JtnZg + JlnZ vac (4.33)

with respect to e, y, and V. The energy density, baryon number density, pressure and entropy density of u,d quarks and gluons are respectively, written in

terms of the baryonic chemical potential y and the temperature T

2a

£ = Id - -r)(i (§) 1 (•$ (*T) 2

) + (i

^

(*)*] 2 ^3

50

21 '60

1 5 TT

‘ (*T)“ (1 - if-?) + B (4.34)

V = 2 a 3 S . JJ 3 TT id +?(*)*) (4.35)

P = 3

(e - 4B) (4.36)

50

21

•)(7tT)

32

45 it (1 - T“)(wT) 3

. (4.37)

In eqs (4.34) and (4.37) the second T

4 and T3 terms originate in the gluonic

degrees of freedom. In eq (4.36) we have right away used the relativistic

relation between the quark and gluon energy density and pressure

D 1 1 q 3 q* K

g 3 g (4.38)

in order to derive this simple form of the equation-of-state. This simple

equation-of-state of the quark-gluon plasma is slightly modified when the

finite quark masses are considered, or when the QCD coupling constant a

$

is

dependent on the dimensional parameter A.

30
As we have already seen in the discussion of the hadronic bag structure

in section 3, an assembly of quarks will assume a geometric configuration such

as to make the total energy E(V,b,S) as small as possible at fixed given

baryon number and fixed total entropy. As is apparent from the first law of

thermodynamics

dE = - PdV + TdS + ydb

we have

PK

. &LL&SL '

3 V

Hence, the geometrically stable configuration 3E/3V = 0 corresponds also to

the configuration with vanishing pressure P. Rather than to work in the

microcanonical ensemble with fixed b and S, we exploit the advantages of the

grand cononical ensemble and consider P as a function of y and T:

P -

-§V fT *nZ(u.T.V) ) (4.41)

with the result as given by eq (4.36). From eq (4.36) it follows that when

the pressure vanishes in a static configuration the energy density is 4B,

independent of the values of y and T which fix the line P = 0. We recall that

this has been precisely the kind of behavior found for the hadronic gas. For

P > 0 we have s > 4B. We recall that in the hadronic gas we always had

e < 4B. Thus, in this domain P > 0 of the y - T plane we have the quark-gluon

plasma exposed to an external force.

In order to obtain an idea of the form and location of the P = 0 configuration in the y - T plane for the quark-gluon plasma, we rewrite eq (4.36)

for P = 0:

(4.39)

(4.40)

2 a

(1 -) TSt 2 5 a 15a

B = L_ [ y

2 + ( 3 ttT ) 2

] 2 [(1 -) 12 - (1 - —-) 8 ]

. ( 4 . 42 ) 162 7T 2 45 3 tt 4tt

Here, the last term is the contribution of the glue to the pressure. We find

that the greatest lower bound on the temperature T at y - 0 is about

K = 1/2)

q

T

q * .83B 1/4 ~ 160 MeV » T 0 (4.43)

31
This result shows the expected order of magnitude. The most remarkable point

is that it leads to a numerically similar value as that found in the study of

the hadronic gas of section 2. Another point worth mentioning is the

influence of the strange quarks: they increase the quark pressure just by the

amount needed to counter the effect of the interaction in eq (4.42). Hence we indeed have T ~ B 1/4

, including the strange quarks (see the discussion below

eq (4.46)).

Let us here further note that for T « y the baryon chemical potential

tends to

y = 3y

q ==> 3B 1/4

[-

2tt‘

2 ct.

•] 1/4

= 1320 MeV [ae

(i --r)

1/2, B 1/4 = 190 MeV] .

(4.44)

In concluding this discussion of the P = 0 line of the quark-gluon plasma, let

us note that the choice a

s ~ 1/2 is motivated by the fits to the charmonium

and upsilomum spectra as well as to deep inelastic scattering. In both these cases space-like domains of momentum transfer are explored. The much smaller

value of a

s ~ 0.2 is found in time-like regions of momentum transfer in e+e-+

hadron experiments. We recall that this was the behavior derived from

eq (3.17) (see fig. 3.1). In the quark-gluon plasma, described up to first

order perturbation theory, positive and negative momentum transfers occur:

the perturbative corrections to the radiative T

4 contribution is dominated by

time-like momentum transfers, while the correction to the y

4 term originates

from space-like quark-quark scattering. Hence it is preferable that two

different values of a be used in the above given expressions.

Consider now the energy density at y = 0. We find the simple result,

restating again some factors

tt 2 15 a

7 50 a e(u = 0) = B +

30

T " [2

S 8 C

(l- r -^ + 2

I 2s 3C * (1-^)] (4.45)

We note that in both quarks and gluons the interaction conspires to reduce the

effective available number of degrees of freedom. At a

$ = 0 we find the handy

rel ation

eq + eg

= (l6oW • (4 - 46)

At a = 1/2 we are seemingly left with only ~ 50% of the degrees, of freedom,

and the temperature "unit" in the above formula drops to 135 MeV. However, as mentioned above, we should rather use a

$ ~ .2 in eq (4.45) in which case the

contribution of strange quarks, which is about 30% of the last term in (4.45)

just compensates these interaction effects. Hence (4.46) is the proper rough

estimate to be kept in mind.

32
So far we have mostly neglected to include heavy flavors into the

description. For a charm quark with a mass of about 1500 MeV the thermodynamic abundance is sufficiently low to ignore its influence on the

properties of the plasma. While its production is exceedingly slow, even the

influence of its equilibrium abundance on the thermodynamic properties of the

plasma is quite negligible. Evaluating the phase-space integrals we find that

the ratio of charm to light antiflavor (either u or d) is

-(m -u/3)/T

c/q = c/q = e 3/2 i2

(4.47)

Taking as a numerical example m = 1500 MeV, T = 200 MeV, y = 0 one finds

c/q = 7 . 10- 3. Thus the energy fraction carried by the intrinsic plasma charm

here would be ~ 0.2% and unimportant for the thermodynamic properties of the

plasma, but quite significant in direct charm detection experiments. However,

the approach to chemical equilibrium (see below) is too slow in nuclear collisions to saturate the phase space even within the most optimistic scenarios,

except in circumstances in which T ~

c were reached.

Clearly, we must turn our attention to strangness. With a current quark mass of about 150-180 MeV we are actually at the threshold T = m and indeed

one finds that there is a quite appreciable s-abundance. An explicit

calculation [4b] has shown that chemical equilibrium will be reached during

the short time of a heavy ion reaction. The motion of the particles being

already semirel ativistic, a significant increase of the number of available

degrees of freedom of quarks in eq (4.45) is recorded due to ss production.

In the case that T > m

$ we have to increase the number of flavors to 3 while

at T ~ m

$ the effective flavor number is 2.8. The appearance of strangeness

is a very important qualitative feature and we will return to its discussion

in section 7.

As a final aspect of the perturbative quark-gluon plasma we consider now

the role of color charge in the statistical description. We note that for

finite-size bags it is essential to ensure the color neutrality of the con- sidered states: much of hadronic structure is a consequence of the requirement

of color neutrality and of symmetries of the quark wave functions in the bags.

This effect of color interactions we have, however, not yet included into our considerations. The requirement that all physical states accessible to the

quark-gluon plasma are color neutral is a very strict requirement. As long as only very few particles are present, color neutral states can be constructed

explicitly. But how can we treat an excited, relativistic many-body system?

The answer of principle is quite simple: in eq (4.12) the trace has to include

only color neutral states. That is, we should consider

Z c=o

Tr c=o

-B(H-p Q) e

q (4.48)

However, in order to arrive at a manageable result we had to allow all states

in the trace. In order to solve this problem [33] we borrow the main

technical idea from the work of Redlich and Turko [34], as first applied in

reference [35]. Each state of the Hamiltonian H can be classified within the

33
irreducible subspaces according to its transformation properties under the

representations of the color SU ( 3) group. In order to compute the singlet

contribution eq (4.48) we first introduce the generating functional

l T z

c

c c (4.49)

where the sum is carried out over all irreducible representations characterized by the index c. The variables of the coefficient functions, _i_.e.,

group characters Xc will permit the inversion of eq (4.49) through a reTation

of the type found when solving for a set of complete orthogonal functions,

Hence d

c in eq (4.49) is a suitable normalization constant (dimension of the

representation), while M(x-j) is a function defining the Haar measure. With

these relations we have

The problem is to obtain a suitable set of functions 4>c

(x-j) such that

1) Z eq (4.49) can be explicitly computed, _e. £. , in the particle number

representation, and

2) eq (4.50) is satisfied.

A hint how it is possible to proceed is contained in eq (4.12), since the

baryon number operator commutes with the Hamiltonian we could use Q in the

exponent in order to divide the Hilbert space into sectors of a given baryon

number. We proceed now in this fashion with the non-abelian group SU ( 3 )

.

There are two mutually commuting charges, in the standard representation of

SU ( 3 ) these are the 3 and 8 directions of the color space. We therefore

consider the following Ansatz for the generating function

/

d

n

<{> M( ) x

c

i(<!>

1 ) Xc

(4>

i

) = 5CC (4.50a)

I x

c ( ^) x

c

( j) = «"( <J>' * <t>) c (4.50b)

Z

c

= d

c /d% M( ^ ) ^(4,) • • • y . (4.51)

-B(H-uQ)^

l = Tr (e q u(4 3 ,4 0) ) (4.52a)

here

\ A (4.52b)

34
where we have introduced the new factor u. Since the norm of u is bounded by

unity we have no trouble to establish the existance of the generation function

(4.52): its absolute value must always be smaller than Z, obtained replacing

u by unity.

It is our first aim now to show that eq (4.52) obtains the desired form

(4.49). The Hilbert space spanned by H is a direct product of orthogonal

subspaces characterized by the color c of states belonging to each sector

(_!_.£., transformation properties under the SU ( 3

)

c group). Hence we can write:

— $( H- u Q ) i a n -i j. n

Z = l Tr[e q (4.53)

c c

where the sum c is a symbolic sum over all irreducible representations of

SU(3), usually characteri zed by two positive integers (p,q) e (0,°°). We can

now constrain our discussion to the "good color" subsectors of the entire

Hilbert space of the color space. Within each sector, a complete orthonormal

set of states is generated in the particle number representation of H:

1 c

l l |v ,5 > <K ,V

|

1 V

c c c c c c

(4.54)

where d

c is the degeneracy, _i_. e_. , dimensionality of each color multiplet,

while Cc counts all quantum numbers within a given irreducible representation

which are related to the internal symmetry. v

c is a short-hand notation for

the set of states determined by the occupantion numbers n^ or i th momentum

state. Inserting eq (4.54) into (4.53) we find

Z

= -B(H-u Q)

HI I I I <C

c> v | e

«

c 5

c vc

5c v

c

' c* c

X <5^1 |v

c ,C

c >l (4.55)

The first factor in eq (4.55) cannot depend on Kc as we had explicitly assumed

exact color symmetry; hence it is diogonal in £. Similarly, the second factor

is diagonal in v. Thus we have both and <$ vv

, as factors and two of the sums collapse. Dropping the irrelevant indices we find

„ . rr -S(H- u Q)

Z HI <H e

q |v >

c v.

I <5,1

V1 (4.56)

35
Recalling that by definition

Zc

-B(H-m.Q)

Tr e

q

c

-B(H-ti.Q)

d

c l ^c 1 e

q IV v

c

(4.57)

we find the desired decomposition (4.49) with

Xc( <J> 3 » 4

> 8 ) <

5|_| e

_1 *303-1 * 8Q 8

I £ > (4.58)

being now recognized as the character of the irreducible representation. The

relations (4.50) are automatically satisfied for all compact semi-simple Lie

groups. Even without the use of methods of group theory one can easily verify

in particular cases that eq (4.58) indeed defines a suitable set of functions.

Regressing for a short moment to the SU ( 2 ) Lie group we easily find

SU(2),., H -^z s4n(2J + 1 ) 4.3 Xi U3) = L e m =- J sin 1/2 4 * z 3

(4.59)

Hence, with the Haar measure [36]

M

SU( 2 ) (* 3

) * 7

sin 2

4, 3/2 , 4

, 36 ( 0 , 2 *) (4.60)

equations (4.50) are easily .verified. At this point the surprise is that

alone the eigenvalues of e

- 1 4*3*3 fix the irreducible representation of SU(2).

However, we note that the rank of the representation enters explicitly in

eq (4.59) and, of course, into the expression (4.58) for SU ( 3 )

.

The crucial point of this approach is the fact that the generating

function Z, eq (4.53) can be explicitly determined! Actually the steps are identical to those used above, c.f_. , eqs (4. 16) -( 4.21) . The reader is invited

to repeat the derivation now wiTh complex quantities i <j> instead of the

chemical potential. One simply finds for quarks the analogue of eq (4.21):

AnZ = I

2 V I An [1 + exp (- b(

/

p"

tf>) - i« )] (4.61)

M flavor v ' c,b

36
for colored quarks. Instead of a color factor 3,. we now find the sum over the

eigenvalues ou of the charges Q 3

,

f) 8 ; in the triplet (c e (1,0) ) and antitriplet (c e (0,1)) representation. In the triplet representation for which

b * + 1/3 we have

= tj/2 + 4> 2/3

a

G

= "4> i/ 2 +

a

g

= -2/3 <|) 2

(4.62a)

(4.62b)

(4.62c)

where R,B,G, refers to the usual red, blue, green colors. In the antitriplet

representation in which b = - 1/3 the sign of all the three angles reverses.

We note that except when y = 0, the generating function Z, eq (4.61) will not

be real. However, the integration over the group with the proper Ha ar measure

leads to a real result for the partition function Z. This measure is in the case given here

d

2

<j>M( 4> ! , <j> 2 ) = 3-2 d(~) d(|^) [sin \ Q- + <f> 2 ) sin ^ sin j

(- -p + <J> 2 )]

2

; <h

<f>i, <p 2 e (4.63)

We note that for massless quarks, j_.e., neglecting the strange quark fraction

eq (4.61) can be evaluated analytically following the results of eq (4.22) and

replacing y - y ± i a. In order to judge the influence of the color conservation it is sufficient to consider computationally the particular case of a baryon-less, y = 0, plasma droplet. In this case we find, substituting

y/T ia in eq (4.22)

(4.64a)

(4.64b)

VT i [

i =R , G,

hr)

1 (4.64c)

37
Here fnzi 0 ^ is the partition function without the color constraint while

fcnZjj 1

' vanishes at aj = 0.

It is important to appreciate that in the quark-gluon plasma the

projection, eq (4.51) on a good color sector has to be carried out for both

quarks and gluons simultaneously . Hence one has to carry out the integral

z

c * /

d

2

* H(* lt *2 ) Xc () Zq

( B,V, u; <(>) Z

g

(S,V;*) (4.65)

where for the singlet, c = 0, the character Xo is just unity. In order to

obtain 7

g we have to evaluate the analogue of eq (4.29)

8 a 3 n

- 6 1 p | - i a

= 2

s ^

V f " e ) (4.66)

where the color sum i runs over the octet of gluons with the eigenvalues of

the Q 3 ,Q 8 charges in the octet representation (1,1).

As the octet results from the product of the triplet and anti-triplet

representations of SU(3), omitting the singlet, we have simply

a

i = a

j ' “k * J,k e ( R * B » G ) (4.67a)

taking the six non-zero and two zero angles oq. Explicitly

“i

= (a

R

- °G' “G - V “B * V

” ( 01^ - Otg ) , - ( - Olg ) , - ( Olg - ap)» 0 » 0 )

•

We therefore find

ini = inl^ °) + inl^ ^

9 9

inZ* o) = VT 3

ir 2 |r

g 45

inl^ ^ = VT 3 1

g 6

(4.67b)

(4.68a)

(4.68b)

(4.68c)

38
We note that the color sum in eq (4.68) was simplified and includes now only

the first three terms of eq (4.67b). Again, z(°) is the partition function

without the color constraint eq (4.30), while JnZ^ 1 ' vanishes at aj = 0.

From the preceeding derivations the enormous influence of color

neutrality is apparent. Practically all states have to be rejected when

evaluating the trace (4.48) constrained to color singlets. A global quarkgluon color correlation is introduced in view of eq (4.65). Both quark and

gluon fractions of the plasma separately can have color as long as overall

neutrality is assured. In the limit V » one can advance arguments that

invalidate this conclusion concerning the importance of color neutrality. To

wit, color fluctuations can not be large; hence, the influence of the

requirement of exact color conservation should not be felt. While this

argument can be proven easily analytically it turns out [33] that the limit

V

00 actually means

VT 3 > 10 (4.69)

To illustrate this point we show in figure 4.1 the

energy density as derived

from eq (4.65) and divided

by aT 4

Fig. 4.1 Relative degeneracy D

e ff of the quarkgluon plasma, as a function of TV 173

.

o = (2 x 8 + 2 x 2 x 2 x 3 x |) tt 2/30 (4.70)

for quark-gluon plasma, c.f. eq (4.45). We observe that at TV 1/3 ~ 1.5 only

half of the expected number of excitations is available. That means that

since we anticipate T ~ 150-170 MeV, one has to deal with the finite volume

problems and color constraints whenever clumps of hadronic matter are produced

with baryon number b < 10. In particular this concerns the study of the

transition from the plasma state to the hadronic gas; color configurations and

correlations here will be of great practical relevance. Work along these

lines is in progress. Therefore the discussion of different hardonic phases

given below by necessity can be only qualitative in nature. But before we turn to this issue we will consider the kinematic conditions for the formation

of the plasma in nuclear collisions.

39
5. QUARK-GLUON PLASMA FORMATION IN NUCLEUS-NUCLEUS COLLISIONS

From the limited information coming from cosmic ray data [37], and from

our knowledge of the nucleon-nucleus interaction [38] we take as the starting

point the observation that the normal hadronic cross sections are not large

enough, j^.e., the mean free path in hadronic matter is too large, to confine a high-denTiTy region by multiple scattering. We record, however, here the

recently observed rather narrow rapidity distributions at /s = 540 GeV from

the CERN pp collider [39] which indicate non-transparency at a level not anticipated before. In order to create a larger size high-density region a quark

gluon plasma seed [12] must have been formed by a statistical fluctuation.

Thereafter the plasma can begin to grow by capture of the trailing nucleons of

the colliding nuclei. In such a scenario, the densest plasma will result when

the seed is formed early in the collision, _i_.e., in the central rapidity

region for symmetric collisions (A = A

t

). However, plasma production will occur according to this mechanism with a non-negl igible distribution towards

projectile and target rapidity limits. In events with early plasma seed the

baryon number content of the plasma would be appreciable for large nuclei,

peaking in the central rapidity region . A different scenario would arise in

events in which the plasma state Ts not ignited; here the baryons would be

found mostly in projectile-target regions of the rapidity, owing to the

expected substantial transparancy of normal nuclei to high energy particles

[38]. Still, the high radiation energy density reached in such collisions

could also lead to a baryonless plasma, as discussed by others [11].

In order to fulfill its role the above introduced seed must indeed be a high particle density region similar to the quark-gluon plasma, albeit small

in size, with sufficiently thermal ized momentum distributions and with some color deconfinement; however, chemical equilibrium between different particle

species, j_.e_. , quark flavors, is not required. In such a case the quark mean

free path, X, can become comparable to the seed size, R, and we can have

R/X ^ 1. Occasional formation of such seeds is assured by inspection of

actual numerical results obtained with relativistic cascade calculations [40].

We have good reason to believe not only in occasional, but perhaps even in

relatively frequent, creation of such a seed, through an accidental local

large fluctuation of particle density over the hadronic volume.

The energy influx to the plasma seed is controlled by the nuclear matter

inflow. We consider here a) the kinematic conditions for the occurrence of

the instability, seed + plasma, and b) the maximum achievable temperature in

the most favorable case. For this purpose we do not need to consider the

influence of the likely increase of the energy and particle density of the

projectile or target in their rest frames arising from the entrance channel

interactions. In order to err on the conservative side we compute as if all

of the interacting region would instantly turn into the plasma state without

compressions of nuclear degrees of freedom. Namely, if the formation of the

seed is delayed, the increase of the densities would make the environment even more suitable for the occurrence of the plasma seed. However, the crucial

condition to be respected follows from the observation that once the seed is

there it can lead to a large-scale plasma state only if the energy loss of the

seed is exceeded by its energy gain. Even below this "sharp" boundary defined

as the instability without nuclear compression in target or projectile.

40
occasional formation of plasma drops in the dense regions of compressed

nuclear matter will occur. These precursor phenomena will smear out the

kinematic limit, otherwise already spread out by fluctuations of the seed

location, range of the impact parameters, etc. We believe that a detailed

discussion of these effects is premature. Therefore we now determine the

conditions which must be fulfilled for the ignition of a large-scale central

plasma state.

While the plasma receives energy and baryon number by the nucleons

impacting on it, it also inevitably loses energy by thermal radiation. Thus,

in order to grow there must hold for the total plasma energy E,

AC .r-A ,,-R dE _ dE _ dE

dt dt ~ dt (5.1)

where dEA /dt is the heating by the incoming nucleons absorbed in the seed, and

dE^/dt is the energy loss by thermal radiation. If dE/dt is negative the

plasma will fizzle rather than grow. We now discuss the two terms, beginning

with the gain term.

The energy influx into the plasma is controlled by the nuclear fourvelocity u v

= y(l,v), the plasma surface normal vector as seen from the CMframe, n

M = (0,ft) the nuclear energy-momentum tensor T

pv

and the probability

for the absorption of an incoming nucleon by the plasma, a. Thus we have,

with d

2A the surface element.

= J

d

2

A( _ u

M

n

v

a) _ (5.2)

As is well known

T =e 0 uu+p(uu-g) (5.3)

yv u y v y v

J yv

where e

Q and p are the energy density and the pressure (p included here for

completness only) in the rest frame of the projectile or target nucleus,

respectively. Hence we have

v

n p 0 m y n • v (5.4)

where p 0 is the equilibrium nuclear density, j_.e., Po = 1/6 fm~ 3

. Furthermore, seen from the CM frame and expressed in Terms of the projectile

laboratory energy per nucleon, E , we have

41
V

=

1/2

(5.5)

Y

(2E

d m + 2m 2

) 1/2

2m (5.6)

The absorption coefficient a is assumed, as usual, to be

a(z) = 1 (5.7)

where z is the thickness of the plasma region and X is the absorption length

of a hadron in the plasma. When weighted with ft • vf over the plasma surface

this leads to

m + 2e’

2R/X

[|r (^)*] - 2(^)7} . (5.8)

The overall factor 1/2 above reflects the ratio between the surface of a circle with radius R and a half sphere, for X/R + 0. The absorption

coefficient a(R) is indeed the average absorption probability . Through X it

depends on the particle density in the pi asma, i.eV, temperature and baryon

density. Still, the gain term in eq (5.1) depends mainly on the projectile

energy. The final expression is, in detail,

d

3 EA

I

d

2Adt

' 2

P °( 2E

p m + 2m 2

) 1/2

x \

[1 + 2e" 2R/X

+ (|^)7) - 2(^)7] . (5.9)

We now turn to the description of the energy loss term of eq (5.1). In

general, two mechanisms for energy loss from a plasma are possible, viz.,

adiabatic expansion and thermal radiation. At least in the beginning, j_.e.,

at the time of decision between ignite and fizzle, the expansion should pTay

no role as the impacting nucleons provide an inertial confinement for the

plasma. However, pion evaporation from the plasma is still possible, and the

cooling associated with this process provides the energy loss of eq (5.1). Of

course, some of the emitted pions will be returned to the plasma by the

42
incoming nucleons. However, this return will be too late to have an impact on the question fizzle or grow: once the process has fizzled, j_.e., the plasma

seed has hadronized, the collision is back to the hadron cascade regime. On

the other hand, if plasma growth has taken place the returning pions will of

course return their evaporation energy to the plasma and contribute to the

ultimate energy density of the plasma. Also, the influence of expansion has

to be reconsidered then.

We now describe the microscopic mechanism of the pion evaporation process

and estimate the magnitude of the energy loss [14]. To wit: when a fast quark

or antiquark hits the boundary it can form a jet-like structure filled with

color field flux, , a fluxtube. For sufficiently high quark momentum this

tube instead of retracting tears apart by qq pair creation. The leading

particle associates with the antiparticle of the pair to form a meson, while

the remaining pair particle may retract into the plasma. By this process in

effect a fast quark of the plasma is replaced by a slow quark, the total

number of quarks in the plasma having remained constant. If the so emitted

particle is a pion, rather than a "heavy" meson, the probability for the

flux tube to break is greatly influenced by the gain in binding of the final

qq state. In this case the necessary momentum is determined by kinematic

considerations, Ue_. , the produced pion must be able to take away not only the

energy but also The appropriate momentum.

We now develop a quantitative model suitable for surface temperatures of

150 - 220 MeV and moderate baryon densities, such that the particle density is

less than ~ 10 particles/fm 3

. Under these circumstances surface collisions

involving more than one particle per fm 2 are rare. Hence we can limit

ourselves to consider sequential one-particle events. In order for the

surface collision to lead to pion emission the particle momentum normal to the

surface must exceed a certain threshold. In particular, this momemtum has to

be larger than the normal momentum of the emitted pion. We take this

threshold momentum to be of the order of 1/4 GeV/c for quarks leading to

pions; our results are quite insensitive to the precise choice, as well as to

the actual shape of the threshold function 0 describing the probability of

pion emission. Hence we will use:

We note that the average energy (momentum) of practically massless quarks is

about 3T ~ 500-650 MeV and that the particle densities peak at ~ 2T. Hence

almost half of all quarks and antiquarks can participate in the radiation

cool i ng.

The energy per unit surface and unit time that leaves the quark-gluon

plasma is now simply given by

(5.10)

(5.11)

43
where g is the degeneracy. As only light quarks lead to the dominant pion

channel we have g

= 3

c x2

s x2f = 12. p(p) is the phase space quark and antiquark particle density

p(p) = [exp ((p - uq )/T) + 1]“ 1 + [exp ((p + uq

)/T) + l]- 1 (5.12)

The differential is simply the normal velocity of particles impinging on the

plasma surface

Since the energy leaving the plasma region is not the total energy contained

in the leading particle we have in (5.11) included the efficiency factor f. In the present case only one pair is created to form the emitted pion. A

naive degree-of-freedom counting leads to f - 2/3. f probably approaches

unity for very high energy leading particles. We disregard the energy dependence of f; choosing the value f = 2/3 we obtain a lower limit on the energy

transfer.

In view of the uncertainties it is sufficient to expand in eq (5.12) the

Fermi distributions and to retain only the Boltzmann term for the q,q distribution:

We find that this overestimates the final result by about 5-10%. Combining

eqs (5.10) and (5.13) with eq (5.14) we obtain the generalized StefanBoltzmann law:

(5.13)

p(p) « 2cosh( u/T) e /p

| 2+p

i

1/1

(5.14)

d

2Adt

d

3 E

44
In figure 5.1 we show the cooling

rate calculated from eq (5.15) as a function of the surface temperature

T, choosing y /T =1. For y « 0

the values arelower by about H 30%: in

a baryonless plasma at a given

temperature the reduced q-density is

not entirely compensated by the

enhanced 3 density. We see that

indeed the precise value of P M

, or,

said differently, the precise form

of the threshold function 9,

eq (5.10), does not matter.

Concerning the choice of y

Qwe note

that even though equilibrium baryon

density does not very likely

prevail, the best choice for y in

the initial stages of the plasma

formation would be y < M^/3 » 2T,

consistent with the non-degeneracy

assumption for T ~-£MN. As local

thermal ization occurs, yq

diminishes

and approaches T. temperature.

Before returning to the ignition condition given by the inequality (5.1) we discuss our result in terms of a numerical example chosen to represent a typical case of a quark-gluon plasma. Our example is a spherical plasma

droplet of R = 4 fm, a surface temperature of T = 180 MeV, and y/T = 1. The

energy density then is 2.1 GeV/fm 3 according to eq (4.34) and recalling that

strange quarks compensate for a large part of the interaction which is of

order 0(a

$ ). The baryon density is according to eq (4.34), ~ .5/fm 3

, Uja.

,

about 3 p 0

. The baryon number exceeds 150 if T is larger in the interior.

Since 0.7 GeV/fm 3 is needed for creation of the final baryons implied by the

assumed value of y, the available energy density is about 1.4 GeV/fm 3 and the

total available energy is ca. 400 GeV. For this example we find for the rate

of energy loss through the surface

f

= A?0- 29 Bk = A 0 - 58 f" 10+23 sec

- 3

•

We note that this confirms the assumption of a sequential i ndividual-particl

e

process: when one particle of 0.29 GeV impinges on a surface area of 1 fm 2 the

next particle following it with light velocity would be behind by a distance

of one fm ( i . e . , several mean free paths). On the other hand, this indeed is

a very large enerqy loss rate. In our example, the enerqy loss in the first

10" 23 sec is (A = 200 fm 2

)

At = 120 GeV ,

Fig. 5.1 Pion radiation surface

brightness as function of

45

U

Ffm

TP82-

278
which represents a substantial fraction of the total available energy of about

400 GeV. Clearly the smaller the plasma droplet, the more relevant becomes

the radiation loss for the lifetime of the plasma. As the available

excitation energy scales with R

3 and the radiation loss with R, a small plasma

droplet of b ~ 18, R ~ 2 fm and available energy 40 GeV radiates 30 GeV in the

first 10-23 sec. Hence we are led to urge that experiments involving very

heavy nuclei be performed to allow for the creation of sufficiently longlived

(j_._e. , large) plasma regions.

We now return to the discussion of

the ignition conditions: we set ^

= 0

in eq (5.1). In figure 5.2 we show

the minimum size a plasma seed must

have in order for it to grow, , the minimum seed size for plasma

ignition, as a function of projectile

energy for a selection of plasma

ignition temperatures, Tj, computed

taking u/T = 2. In the initial stages

of the nuclear collision this is the more likely nature of the parameters,

q dominate q and we err on the

conservative side by enhancing the

radiation losses by that choice.

While at density of 2 GeV/fm 3 in the

plasma the particle density is about

. ii

4/fnr leading to X fm, we anticipate that in the initial stages

of the collision we have a particle

density of about 1/fm 3 and hence

X ~ 1 - 1.5 fm. For R/ X ~ 1 we notice that at Tj ~ 150 - 160 MeV beam

energies of 20 - 50 GeV/nucleon should

suffice to lead to plasma ignition

with R ~ X. We note that the seed size considered is of the order of the

nucleon size. We note that the obtained lower limits for the heavy ion

kinetic energy is above the kinematic limit obtained neglecting the loss term

in eq (5.1) and requiring an ignition termperature of 160 MeV. On the other

hand, it seems rather unlikely that ignition can be achieved at much lower

beam energies if the phase transition is of first order. Thus below our limit

the collision will fizzle and we just achieve a superheated nucleon gas.

Once the plasma has ignited the temperature of the plasma will grow until

the nuclear collision terminates or until the temperature has risen to a level

at which the pion radiation overwhelms the energy influx. At this point one must re-examine the question of the cooling mechanism, i.e_. , first, whether

the evaporation of pions will lead to a cooling off of The surface and hence

to a shut-off of the evaporation process, or whether the plasma heat conductivity is sufficiently large to maintain a surface temperature high enough for

pion radiation to continue; second, whether other processes, principally

expansion, contribute substantially to the cooling process.

Fig. 5.2 Minimum size of a plasma

seed as function of beam

laboratory energy for

different radiation

temperatures.

46
We begin by considering the heat conductivity. Since the plasma consists

of rather free particles the naive expectation is that a sufficently higtj.

conductivity obtains. Indeed, the basic relation between the heat flow Q and

the energy density e is

$ = * 7 e(T;u

q

/T) (5.16)

where l is the mean free path. Assuming that only a radial gradient of T

exists, with y / T ~ const over the volume, the radiation equilibrium

requires

d

3 E

d

2Adt

3T _3e _ I JTT 3r 3T

" 1

T 3r (5.17)

In our numerical example the required temperature gradient at the surface is,

with Z in the range 1/2 - 1/3 fm:

il = I 0.215 GeV/fm 3

, MeV

3r "

l 4 x 2.1 GeV/fm^ " fm

It appears that this temperature gradient is just within sensible bounds,

leading for a plasma radius of 4 fm to a temperature differential between the

origin and the surface of ~ 20-30 MeV. We further note that unlike in non- relativistic gases, the mean free path l here is inversely proportional to

3e/3T since it is inversely proportional to the particle density. For

y /T < 2 the energy per particle in the plasma is just 3T and hence the

particle density p = e/3T. Therefore the necessary temperature gradient,

eq (5.17), turns out to be

3J. _ d

3 E l_ - 8r

d

2Adt

12 (5.18)

where a is the average particle-particle cross section. The range of values

3T - 1 2 given above for corresponds to a ~ j

to jfm 2

.

We now turn to the discussion of the adiabatic expansion of the plasma.

To begin with one must recognize that in contrast to the above discussed pion

radiation process the expansion requires a collective flow, _i_.e., a flow in

which a hydrodynamic velocity is superimposed over the random Thermal motion

of all the quarks and gluons. Therefore the relevant time constant is given

by the speed of sound and thus is about three times larger than the radiation

time constant. Furthermore, the expansion is driven by the excess of the

internal pressure over that exerted on the surface by the physical vacuum.

Now, the effect of the internal pressure on the surface is reduced by the pion

47
radiation. The point Is that those particles which penetrate the surface do

not exert their full force on the surface. We now demonstrate that they are responsible for a substantial fraction of the Internal surface pressure.

Balancing the momenta at the surface we find that Instead of 2p ± the momentum

transferred to the surface Is

Ap

2P, : P

x

< PM

2PJU - f) : P x

> P

f

(5.19)

where f Is the fraction of the normal momentum carried away by the emitted

pion. We now recompute the effective pressure on the plasma surface:

Pq

dp

i

( 2 ")

2pi

v i

p

> dp.

( 2 -rr) 2 P(P)

+

00 dp

(1 - f) /

± 2p v

PM ( 2 *)

00 p „dp

p(p) ] (5.20)

where we have used eq (5.13). We notice that the effective pressure P is

equal to the expected quark pressure Pq

= 1/3 e_, reduced by the contribution

of high normal momentum particles, weighted by the factor f:

P q

°o dp oo p dp

fg / ± 2p i

v i / -L-f p(p)

PM ( 2 *) (2w)

(5.21)

The important point to realize is that the contributions of particles with

p, > p M to the particle pressure P are dominant. To see this we evaluate, in

obvious notation,

00 00 P ..dp 00 -P./T

P (d > d 1 JD

dPiPl2 J

o IFr^F* p(P) !

n dPiPl2 6

VP

1

> PM> = /p l^p l

P .dP

/. dp p

2

/., ~~~~~~ p(p) / dp p

2

e Pi

0 0 0 1 1

- p M

/T

r l / P M .

2

+ , P

M> . 1 (5.22)

48
This is a monotonical ly falling function of pM

/T; for pM/T ~ 1 - 1.5 we find

that the ratio eq (5.21) varies between .92 and .81. Hence, inserting

eq (5.22) into eq (5.21) we find for f ~ 2/3

P (p > p ) pq

= P

q (1 - f -) = 0.4 Pq

. (6.23)

Since we used the lower limit on f it is likely that P q

is even further

reduced. Since the quark pressure at u/T ~ 1 is about

M twice as large as the

glue pressure, 40% of the total pressure is relieved by radiation and only

about 60% of the internal pressure acts on the surface. Thus, in effect, the

time constant relevant for the cooling process through expansions is extended

by a factor of almost two. Thus we are led to the conclusion that the

expansion contributes only about 20% to the cooling of the plasma. Even for

baryonless plasma, i.e., u = 0, though this effect is somewhat reduced it

still relieves 1/3 of the total pressure.

The physical distinction between the cooling by pion radiation and by

expansion resides in the reduction of the temperature without a significant

increase of the plasma volume. This, of course, has important consequences in

the dynamics of the plasma development, and, in particular, eventually on the

observable quantities. In particular, cooling by radiation seems to convert

more efficiently the internal energy into pions. In an expansion this energy

through collective motion becomes additional kinetic energy of the final

particles. Hence in the radiation cooling the available entropy is used to

create more new particles, j_.e., pions, while in the adiabatic expansion it is

essentially contained in the "kinetic motion. In both instances cooling is

approximately adiabatic.

We next discuss the maximally obtainable plasma temperature, neglecting

the effect of the cooling by expansion. As already remarked, once the plasma

has ignited a fraction of the radiated pions will be swept along by the

incoming nucleons and re-enter the plasma. This process introduces a dependence of the loss term on the beam characteristics. Even though this

turn-around of the pions does not change the ignition conditions it influences

the maximal achievable plasma energy density. Since the thermal radiation is

isotropic this returned fraction, n, will be of the order n < 1/2. To obtain

an estimate of this maximum plasma energy density one has to multiply the

energy radiation term, eq (5.15), with (1 - n) and balance it with the

unmodified gain term, eq (5.9). We recall that in the derivation of eq (5.15)

a non-degenerate quark-gas has been assumed, and u/T is expected to be less

than 2. As the collision process continues the temperature of the plasma will

grow until the nuclear collision terminates or until the temperature has risen

to a level at which the pion radiation overwhelms the energy influx. This

49
maximum achievable temperature is shown in figure 5.3 for a few choices of the

pion turn-around coefficient n, as a function of projectile beam energy. In view of high plasma density here we have used R/X = 5, u/T = 1. As one can

see the maximal temperature achievable in the collision does not depend too

sensitively on the choice of the parameters and reaches for 50 GeV a value

around 230 MeV. Hence, once a plasma has ignited one can expect that a fullfledged quark-gluon plasma event will take place, with energy density

reaching 4-5 GeV/fm 3

. However, we note that underlying this scenario is the

requirement that the collisions take place between two quite heavy nuclei.

Fig. 5.3 Maximum achievable plasma temperature as function of beam laboratory

energy for two values of the pion turn-around coefficient.

Considering experiments at beam

energies 20 - 50 GeV/A one should

recognize that even in symmetric

collision, _e.g_., Pb on Pb, the plasma

will not necessarily always occur at

central rapidity. Namely, if the seed occurs relatively late in the collision

then even for head-on collisions plasma

would be created in a non-symmetric

configuration, as illustrated in

figure 5.4 which is drawn in the

CM frame. In this example the plasma

will appear shifted towards the target

rapidity. However, the largest and

hottest plasma events of necessity will

arise at about central rapidity since

they must be seeded early in the

collision. This circumstance could be

utilized in the design of the

experiment. We note that this type of

dicussion requires event by event

analysis.

Fig. 5.4 Non-symmetric plasma

arising from late seed.

50
6. PHASE TRANSITION OR PERHAPS TRANSFORMATION: HADRONIC GAS TO THE QUARK

GLUON PLASMA

We have described two inherently different descriptions leading to the

prediction of a qualitatively similar thermodynamic region for the transition

between both phases of hadronic matter. As we shall see in a moment the

physics which went into these theoretical approaches requires that this is a first order phase transition. However, of course, we cannot actually deduce

the order of the transition in the presented considerations. We record here

that recent Monte-Carlo simulations on a lattice show phase coexistence in

SU(3) gauge theories which is characteristic of first order phase transitions

[6a], [41]. This is contrary to results found in SU ( 2) simulations [42].

Fig. 6.1 P-V diagram for the gas-plasma first order transition.

Consider the P-V diagram shown in figure 6.1. Here we distinguish three

domains. The hadronic gas region is simply a Boltzmann gas where the pressure

raises with reduction of the volume. When the internal excitation rises, the

individual hadrons begin to cluster. This reduces the increase in the

Boltzmann pressure since a smaller number of particles exercises a smaller

pressure. In a complete description of the different phases we have to allow

for a coexistence of hadrons with the plasma state in the sense that the

internal degrees of freedom of each cluster, j_.e., quarks and gluons con- tribute to the total pressure even before the dissolution of individual

hadrons. This indeed becomes necessary when the clustering overtakes the

compressive effects and the hadronic gas pressure falls to zero as V reaches

the proper volume of hadronic matter. At this point the pressure rises again

very quickly, since in absence of individual hadrons we now compress only the

hadronic constituents. By performing the Maxwell construction between volumes

V

x and V

2 as indicated in figure 6.1 we can in part account for the complex

process of hadronic compressibility alluded to above. We find this way the

most likely path taken by the compressed hadronic gas in a nuclear collision.

This discussion shows that in our approach we are straightforwardly led to a first order phase transition [43]. We should remember that on the way out,

during the final step, presumably an expansion of the plasma state, the

51
entropy generated and still present in

the plasma may require that the

(isolated) plasma state must expand to

close to P = 0, , under-cool

before it can disintegrate into

individual hadrons. This disintegration may be a quite slow process of

successive fissions, when color

neutrality, j_.e., the door to freedom,

can no longer Fe achieved by particle

production.

It is interesting to follow the

path taken by an isolated quark-gluon

plasma fireball in the u-T plane, or equivalently in the v-T plane.

Several cases are depicted in figure

6.2. In the Big Bang expansion the

cooling shown by the dashed line occurs in a universe in which most of

the energy is in the radiation.

Hence, we have for the chemical potential y « T. Similarly, the baryon

density v is quite small. In normal

stellar collapse leading to cold

neutron stars we follow the dashdotted line parallel to the y-resp.

v-axis. The compression is

accompanied by little heating.

In contrast, in nuclear collisions almost the entire y-T and v-T can be

explored by varying the parameters of the colliding nuclei. We show an example by the full line, and we show only the path corresponding to the cooling of the plasma, i.e., the part of the time evolution after the termination

of the nuclear collTsTon. The figure reflects the circumstance that in the

beginning of the cooling phase, j_.e_. , for 1 - 1.5 x 10.

23

sec, the cooling

happens almost exclusively by the mechanism of pion radiation. In terms of our numerical example of section 5 this means that during that time about half

of the available energy has been radiated away. More precisely, at t = 1.5 x 10- 23

sec, 200 GeV has been radiated, and the initial surface temperature of

180 MeV has decreased to T = 150 MeV. This is close to the temperature of the

transition to the hadronic phase. Hence a possible, perhaps even likely,

scenario is that in which the freezing-out and the expansion happen simultaneously. If that in fact takes place then the expanding hadronic gas will be

quite dilute and a hadron will not undergo many scattering events before

reaching an asymptotic distance, U

e

_. , before becoming accessible to detection

by an experiment. Of course, this would be ideal in that the detected

particles then would reflect the chemical composition of the plasma at the

freeze-out point. These highly speculative remarks are obviously made in the

absence of experimental guidance. A careful study of the hadronization

process most certainly remains to be performed.

Fig. 6.2 Paths taken in the (a) y-T

plane and (b) v-T plane by

different physical events.

52
Let us now consider the question: is the transition hadronic-gas «+ quark-gluon plasma in principle a phase transition or is it only a change in

the nature of hadronic matter, not associated with any kind of singularity of

the partition function in the limit of infinite volume? The spirit of the

theoretical approaches taken here requires a first order transition. However,

this conclusion is only preliminary. Contrary arguments can be found [44] by

arguing that only a finite number of incompressible hadrons can be fitted into

a given volume. Here it turns out that one must very carefully study the

meaning of thermodynamical limits (see references [9d] and [45]). Even worse:

for compressible individual hadrons we might find a second order phase

transition [43]. We see that the theory is highly sensitive at this point.

It is only the experiment which will teach us this important aspect of strong

interactions. Numerical digital experiments concluded recently [41] allow one

to believe that we have been on the right track with our description of the

hadronic world. Unfortunately we probably will have to wait for some years

for the needed experiments in view of the newest emphasis on less relevant

research fields that lack the "disadvantage" of being interdisciplinary. As

the reader further notes, this is the shortest section of this work - and we hope that as soon as the just mentioned required experimental advances occur,

we will review and re-analyze its content, we hope before the end of this

century. To encourage these experiments we therefore describe in the next

section our current favorite observable.

7. STRANGENESS IN QUARK-GLUON PLASMA

We now show in some detail why the strange particle abundances are so helpful [4] in observing the formation and the properties of the quark-gluon

plasma. First we note that at a given temperature the quark-gluon plasma will

contain an equal number of strange (s) quarks and antistrange (5) quarks,

naturally assuming that the hadronic collision time is much too short to allow

for conversion to strangeness of the light flavors by the weak interaction.

Thus, assuming chemical and thermal equilibrium in the quark plasma we find

the density of the strange quarks to be (two spins and three colors):

neglecting, for the time being, the QCD perturbative corrections. We recall

that the mass of the strange quarks, m , in the perturbative vacuum is

believed to be of the order of 140-200 MeV [18]. In eq (7.1) we were able to use the Boltzmann limit since the phase space density of strangeness is not

too high. Similarly, there is a certain light antiquark density (q stands for

either u or d)

:

d

3 p 1 - 3 —S- K (m

s /T) (7.1)

Tm 2 s/V = s/V = 6 /

(7.2)

53
The chemical potential of quarks surpresses the q density. This phenomenon

reflects on the chemical equilibrium between q-q and the presence of a liqhtquark density associated with the net baryon number. Alternative but

physically equivalent ways to understand this factor are the following two

statements

1) q is Fermi -blocked, since in its production the partner q-quark has

to go on top of the Fermi sphere (T 0 limit);

2) q are easily destroyed by the abundant q's when the q-density is

1 arge.

We now intend to show that often more s quarks are present than antiquarks of either light flavor. Indeed:

i m o m /ot

s/q = j (y

1

) K 2 (y

5

-) e p/3T

. (7.3)

This ratio is shown in figure 7.1. Thus, we almost always have more § than q quarks and,_in many cases of interest, S/q ~ 5. For y + 0 there are about as many u and d quarks as there are s quarks at T ~ms

.

Fig. 7.1 Abundance of strange (or antistrange) quarks relative to the light

quark abundance as function of y for several choices of the

temperature T and of the strange quark mass m

s

.

54
When the quark matter dissociates into hadrons some of the numerous s, instead of being bound in a q5 kaon {K+ ,K

0 ^, may enter into a (qj|5) or (q§5)

antibaryon and, in particular, a A, z, or s. The probability for this

process seems to be of similar magnitude to the production of A, E, or s by

the quarks present in the plasma. It is particularly noteworthy about the 5 carrying antibaryons that conventionally, i.e., in pp collisions, they can be

produced only in direct pair production reactions. This process is suppressed

by energy-momentum conservation up to high energies and phase space considerations since the final state has to contain four particles. This leads to the

argument that a study of the A, E, 5 in high-energy nuclear collisions could

shed light on the early stages of the reaction in which a quark-gluon plasma

may be formed.

As is apparent from the above remark, the crucial aspects of the proposal

to use strangeness as a signature for the quark-gluon plasma involve:

a) assumption of thermal and chemical equilibrium

b) comparison between results anticipated in both hadronic phases at

given T and y, where the chemical potential must be determined by

other considerations.

We now turn to the discussion of both these points and begin by calculating

the abundance of strangeness as function of the lifetime and excitation of the

plasma state [4b].

In lowest order in perturbative QCD, ss-quark pairs can be created by

annihilation of light quark-antiquark pairs (fig. 7.2a) and in collisions of

two gluons (fig. 7.2b). The averaged total cross sections for these processes

were calculated by Combridge [46]. For fixed invariant mass-squared

s = (k x + k 2 ) 2

, where k^ are the four-momenta of the incoming particles

(„(s) = (1

Fig. 7.2 Lowest order QCD diagrams for ss production: a) qq •* ss; b) gg - ss.

55
0 (7.4)

qq+ss

8tt<x 2 2M 2 5- (1 + ) w ( s

)

27s s

ogg+$s

2Tra

s

2

3s Id + i^ + LKs s tanh-1 w(s) - (| +

31 M£

8 s

) w(s) ] (7.5)

For the mass of the strange quark we will assume, a) the value [10] fitted

within the MIT bag model: m = 280 MeV and, b) the typical value [18] found

in the study of quark currents: m

$ = 150 MeV. When discussing light quark

production below we will use m

q

= 15 MeV. The effective QCD coupling constant

a

s = g

2/4 tt is an average over space-like and time-like domains of momentum

transfers in the reactions shown in figure 7.2 as discussed in section 3. We

use: (a) a = 2.2, the value consistent with m = 280 MeV in the MIT bag

model, and (b) the value = 0.6, expected at the involved momentum

transfers. We consider the choice (b) of the parameters to be realistic and

consistent with the spirit of this work. The choice (a) is used as a reference; even when m

$ = 280 MeV it shows that the chemical equilibrium will

be reached.

Given the averaged cross sections it is easy to calculate the rate of

events per unit time, summed over all final and initial states:

f /« 3* /

d

3 k

j

(2») 3 |k

1

| il P-j (k i»x)

d

3 k

2

00 / r I Pi (k 2.x) / ds 5(s - (kj + k

2 ) 2

) k^ 2 o(s) . (7.6)

( 2 it) 3

1 k

2

1 i

1 4M 2 y

The sum over initial states involves the discrete quantum numbers i (color,

k

i *k2 spin, etc.) over which eq (7.5) was averaged. The factor

]^j"j"kr~j' the

relative velocity for massless particles, and we have introduced a dummy

integration over s in order to facilitate the calculations. We now replace

the phase space densities p^(k,x) by momentum distributions f

q (k), f ( k )

,

f _( k ) of gluons, quarks, ana antiquarks that can still have a

9 paramet?ric

x-dependence through a space-dependence of the temperature T = T(x) and the

chemical potential u = u(x). The invariant rate per unit time and volume for

the elementary processes shown in figure 7.2 is then:

56
X {(2 X 8)

2 f (kj) f (k 2 ) a Js)

+ 2x(2 x 3)

2 f

q

(kj) f

_ (k 2 ) a_ Js)} , q qq^ss

(7.7)

where the numerical factors count the spin, color and isospin degrees of

freedom.

Assuming that in the rest frame of the plasma the distribution functions

f depend only on the absolute value of the momentum, |k| = k

0 = k, we can evaluate the angular integrals in eq (7.7):

where the step function 0 requires that k

x k2

> > M

2

. We now turn to the

discussion of the momentum distribution and related questions. We note that

the anticipated lifetime of the plasma created in nuclear collisions, as discussed in section 5, is about 6 fm/c = 2 x 10" 23 sec. After this time the

high internal excitation will most likely have dissipated to below the energy

density required for the global restoration of the perturbative QCD vacuum

state. We recall again that the transition between the hadronic and the

quark-gluon phase is expected at an energy density of approximately 0.6 -

1 GeV/fm 3

. Under these conditions we note that each perturbative quantum

(light quark, gluon) in the plasma state will rescatter several times during

the lifetime of the plasma. Hence the momentum distribution functions f(p) can be approximated by the statistical Bose or Fermi distribution functions,

regardless of the shortness of time:

oo A = “x/ sds o .[/“ dk x /“ dk 20(4k 1 k 2

- s) f (kj) f (k 2 )]

AM 2 nn+cc 0 0 3 3 4M 2 gg+ss 0 0

00 9

f

g (p) - (e e

’ p - l)' 1

, (gluons) (7.9a)

57
f (p) * (e

0

* p X* + l)" 1

, (quarks-antiquarks) (7.9b)

q/q

where again 6 *p = 8

0 |?| - for massless particles, (e* 8)" 1/2 T is the

temperature and X 1

is the baryon number (antibaryon number) fugacity. In the

rest frame of the plasma, B*p = |?|/T. The distributions (7.9) can be taken

seriously only for |jT| not very much larger than T; to populate the high

energy tail of the distributions too many collisions are required for which

there may not be enough time during the lifetime of the plasma. Furthermore,

we note that while in each individual nuclear collision the momentum

distribution may vary, the ensemble of many collisions may lead to better

statistical distributions.

Finally, consider the values of the fugacities X* in eq (7.9b). As we

will show the gg > qq reaction time is much shorter than that for qq s5

production since the light quark masses are only of the order of ~ 15 MeV.

Consequently we may assume chemical equilibrium between q and q, i.e.,

and the baryon density is given by eq (4.35) omitting for the present the 0(a)

corrections, i.e.;

v(T,w

q

) = 3I7 Uq 3 + uq

( nT) 2

) .

We note that since gluons dominate the s§ production in the plasma state, the

conditions at the phase transition, such as abundance of q and q, will not

matter for the s5 abundances at times comparable to the lifetime of the

pi asma.

We now return to the evaluation of the rate integrals, eq (7.8). In the

glue part of the rate A, eq (7.8), the k

1 ,k 2 integral can be carried out

exactly by expanding the Bose function in a power series in exp(-k/T):

* 4t /V s3,2 '« _(s) l (nn ')- 1 ' 2 K, ) . (7.11)

9 4M 2 gg-^s n,n'=l

In the quark contribution an analytic treatment of the Fermi function is not

feasible and the integrals must be evaluated numerically. It is found that

the gluon contribution, eq (7.11), dominates the rate A. For T/M > 1 we find:

58
A « A

7 2 MT 3 e "2M/T

+ (7.12)

The abundance of s5-pairs cannot grow forever; at some point the

sS'annhi 1 ation reaction will deplete the strange quark population. It is

important to appreciate that the s5-pair annihilations may not proceed via the

two gluon channel, but instead occasionally through yG final states [47]. The

noteworthy feature of such a reaction is the production of relatively high

energetic y's at an energy of about 700-900 MeV (T = 160 MeV) stimulated by

coherent glue emission. These y's will leave the plasma without further

interactions and provide an independent confirmation of the s-abundance in the

pi asma.

The loss term of the strangeness population is proportional to the square

of the density n of strange and antistrange quarks. With n

s (~) being the

saturation density at large times, the following differential equation

determines n

$ as function of time:

We note that eq (7.13) in principle should also include a term linear in

n (t). Namely, when the plasma density is sufficiently high the produced

strange quarks have difficulty to quickly get away from each other. With a scattering length of the order of 1/3 fm in extreme cases one has to consider

diffusion rather than free motion. Hence in this limiting case we have

always a s5 pair in a given unit volume, leading to [48]

dn

" A U - ("

s (t)/ns (»)) 2

]

• (7.13a)

dn

* A(1 - n

s (t)/ns (») )

. (7.13b)

The solutions of eq (7.13) are, respectively

n

$

(t) = n

s H tanh(t/ t) (7.14a)

n

$

(t) = n

s H (1 - e'

t/r

(7.14b)

with

t = n

s (“)/A (7.14c)

59
Both solutions are monotonical ly rising saturating functions with similar

behavior, controlled by the characteristic time constant t. In a thermally

equilibrated plasma the asymptotic strangeness density, n

$ («), is that of a chemically unconstrained relativistic Fermi gas (X = 1):

n

s H=f$TM2 l -^^K2 (nM/T) , (7.15)

We find for the relaxation time (7.14c) from eq (7.12), (7.15)

T - Tg (f)

1 ' 2 a

s -2 T-3'2 eM/T

(1 -ff (7.16)

which falls rapidly with increasing temperature.

Fig. 7.3 Chemical relaxation times as functions of the temperature T. Full

lines: qq s5 and gg

-»• s5; dashed lines qq + s5; dotted lines

gg -* qq (m = 15 MeV). Curves marked I are for = 2.2 and m s = 280 MeV, those marked II are for a

$ = 0.6 and ms = 150 MeV: a)

rates A; b) time constants t.

We now discuss the numerical results for the rates, time constants, and

the expected strangeness abundance. In figure 7.3a we compare the rates for

strangeness production by the processes depicted in figure 7.2 for the two

different choices of parameters discussed above, after eq (7.5). The rate for

qq

*> s5 alone (shown separately) contributes less than 10 percent to the total

60
rate. In figure 7.3b we show the corresponding characteristic relaxation

times toward chemical equilibrium, t, defined in eq (7.14). While our results

for strangeness production by light quarks agree only in order of magnitude

with those of Bird and Zimdnyi [49] owing to the difference in the chosen

values of the parameters, it is obvious from our results that gluonic

strangeness production, which was not discussed initially by these authors

[50], is the dominant process. If we compare the time constant t with the

estimated lifetime of the plasma state we find that the strangeness abundance

will be chemically saturated for temperatures of 160 MeV and above, , for

an energy density above 1 GeV/fm 3

. We note that t is quite sensitive to the

choice of the strange quark mass parameter and the coupling constant a

$

which

must, however, be chosen consistently. A measure of the uncertainty

associated with the choice of parameters is illustrated by the difference

between our results for the two parameter sets taken here.

Also included in figures 7.3a, and 7.3b are our results for gluon

conversion into light quark-antiquark pairs. The shortness of t for this

process indicates that gluons and light quarks reach chemical equilibrium

during the beginning stage of the plasma state, even if the quark/antiquark,

i.£., baryon/meson ratio was quite different in the prior hadronic compression

phase.

Fig. 7.4 Time evolution of the relative strange quark to baryon number

abundance in the plasma for various temperatures T. m

s = 150 MeV,

a

s = 0.6.

The evolution of the density of strange quarks, eq (7.14), relative to

the baryon number content of the plasma state, is shown in figure 7.4 for

various temperatures. The saturation of the abundance is clearly visible for

T > 160 MeV. To obtain the experimentally accessible abundance of strange

61
quarks, the corresponding values reached after the typical lifetime of the

plasma state, 2 x 10" 23 sec, can be read off in figure 7.4 as a function of

temperature. The strangeness abundance shows a pronounced threshold behavior

at T ~ 120 -160 MeV.

We thus conclude that strangeness abundance saturates in a sufficiently

excited quark-gluon plasma with T > 160 MeV, e > 1 GeV/fm 3

, allowing

strangeness to be an important observable. We hence turn to the study of the

strangeness in normal nuclear matter in order to gain insight into the

relevance of strangeness as a characteristic signature of the quark-gluon

pi asma.

To this end we must first establish [4c, 51] the relevant relative

strange particle rates originating from highly excited matter consisting of

individual hadrons, the hadronic gas phase. The main hypothesis which allows

us to simplify the situation is to postulate the resonance dominance of

hadron-hadron interactions (see section 2). In this case the hadronic gas

phase is practically a superposition of an infinity of different hadronic

gases and all information about the interaction is hidden in the mass spectrum

x(m 2 ,b) which describes the number of hadrons of baryon number b in a mass

interval dm 2

. When considering strangeness carrying particles, all we then

need to include is the influence of the nonstrange hadrons in the baryon

chemical potential established by the nonstrange particles. The total

partition function is approximately additive in these degrees of freedom:

z = in z nonstran9 e + in z stran 9 e (7.17)

For our purposes i.e., in order to determine the particle abundances it is

sufficient to lisTThe strange particles separately and we find

in z strange (j^,^) = C {2W(x

fc ) [x,.X-> + x-lx

q

]

+ 2[W(x

a

) + 3W(x

s

) ] [X

$

X2+ X-'X- 2

]} (7.18a)

m . 2 m • W^) = (y

1

) k 2 Gfl) . (7.18b)

We have C = VT 3/2 tt 2 for a fully equilibrated state. The case of chemical non- equilibrium can be effectively taken care of by using smaller values of C. As

the strangeness-exchange cross sections are very large, strangeness will be

always distributed among all particles in (7.18a) according to the values of

the fugacities A = Ag 1/3 and A

s

. Hence we can speak of relative strangeness

chemical equilibrium, see below. We neglected to write down quantum

statistics corrections as well as the multi -strange particles, S and ft", as our considerations remain valid in this simple approximation [51]. Interactions are effectively included through explicit reference to the baryon

62
number content of the strange particles as just discussed. Nonstrange hadrons

influence the strange fraction by establishing the value of A at the given

temperature and baryon density.

The fugacities A

s and \

q as introduced here control the strangeness and

the baryon number, respectively. While A

s counts the strange quark content

the up- and down-quark content is counted by Aq

= Ag 1/3

.

Using the partition function eq (7.18a) and (7.18b) we calculate for

given T, and V the mean strangeness by evaluating

8 9A,

*n Z stran9e(T,V,X

s>

Xq

) (7.19)

which is the difference between strange and anti-strange components. This

expression must be equal to zero due to the fact that the strangeness is a conserved quantum number with respect to strong interactions. From this

condition we get:

A = A

s q

W(x

k

) + V [W(x

A

) + 3W(x v) ] W < x lJ

+ X

B [W(x.) + 3W(xTT

1/2

V (7.20)

We notice a strong dependence of y on the baryon number. For large ud the

term with An -1 will tend to zero and the term with Ag will dominate the

expression for A

s and y. As a consequence the particles with fugacity A. and

strangeness S = -1 (note that by convention strange quarks s carry S

= -I,

while strange antiquarks 5 carry S = 1) are suppressed by a factor y which is

always smaller than unity. Conversely, the production of particles which

carry the strangeness S = +1 will be favored by y- 1

. This is the consequence

of the presence of nuclear matter; for u = 0 we find y

= 1.

In order to calculate the mean abundance of strange particles we must

introduce for each species its own fugacity which subsequently must be set

equal to unity since all different strange particles are in mutual chemical

equilibrium by assumption. This assumption is made as a consequence "of The

large strangeness exchange cross sections, in reactions such as

N + K <-- Y + tt (7.21a)

where Y stands for a hyperon A, z. These are much larger then the strangeness

production cross sections, such as

N + N+ N + A+ K (7.21b)

or even

63
tt + N + E + K (7.21c)

when considered at moderate temperatures (energy threshold > 500 MeV). Hence

in nuclear collisions the mutual chemical equilibrium, that is, a proper

distribution of strangness among the strange hadrons, most likely will be

achieved. By studying the relative yields we can exploit this fact and

eliminate the absolute normalization, C, cf., eq (7.18) from our considerations. We recall that the value of C is uncertain for several reasons: (i) V is unknown, (ii) T

3 is strongly T(t,r)-dependent, and (iii) most importantly,

the absolute normalization assumes chemical saturation which is not achieved

owing to the shortness of the collision. Indeed we have (cf., eq (7.3))

$ - A

h

(1 - C(t)2/C 2 H) , (7.22)

and the time constant iu = C(°°)/A|, for strangness production in nuclear

matter can be estimated to be 10" sec. [52]. The generation of strangeness

is most likely driven by reaction (7.21c). Thus C does not reach C(°°) in

plasma-less nuclear collisions. If the plasma state is formed, then the

relevant C > C( °»)

.

We now compute the relative strangness abundances. Using eq (7.20) we

find from eq (7.18) the grand cononical partition sum for zero average

strangeness:

£n grange = C {2W(x ) [y X + y

-

1

X ] + 2W(x ) [yX X + y

-1

X -1X KK ] R ABA BA

+ 6W(x ) [yX X + ]} . £ BE BE

(7.23)

The strange particle multiplicities follow from (i = K,K, A, A, E, E)

<"i>

•

*1 « Zo

tra "9e

*i . 1 (7.24)

Explicitly we find

<n +

> = C y

+1

W(xJ

K~

K (7.25a)

<n > = C A/

A

±1

W(x

A

) VT

(7.25b)

64
and hence the ratio <n

K+>/<n)<,-> = y

2

. This is shown in figure 7.5 as function of the baryo-chemical potential u B for several temperatures.

Fig. 7.5 The ratio <n^+Xn|

< -> = y~ 2 as a function of the baryo-chemical

potential for several temperatures.

We note that this particular particle ratio is a good measure of the

baryon chemical potential in the hadronic gas phase, provided that the

temperatures are approximately known. The mechanism for this process is: the

strangeness exchange reaction (eq (7.21a)) tilts to the left (K~) or the right

(abundance y ~ K+

) depending on the value of the baryo-chemical potential.

We turn our further interest to the rarest of all singly strange

particles, and show in figure 7.6 the ratio <n.>/<n >. We notice an expected

suppression of A due to the baryo-chemical potential as well as the strangeness chemistry. This ratio exhibits both a strong temperature and ug dependence. The remarkably small abundance of A, e^.g,, 10~ 4

A, under conditions

likely to be reached in an experiment at the ericT of the dissociation phase

(T ~ 120 - 180 MeV, ~ (4-6)T) is characteristic of the nuclear nature of

the hot hadronic matter phase. Our estimates for the quark-gluon plasma based

on flavor content are two to three orders of magnitude higher. One may

observe that the formation of A in nuclear matter will probably be even much

less than shown here since A will be much further away from the equilibrium

abundance than A's. Hence the ratio of figure 7.6 may be viewed as an upper

limit for the case of hot nuclear matter.

65
Fig. 7.6 The ratio <n->/<n > as a function of ug for several temperatures as

an upper limit for A abundance in the hadronic gas phase.

We have already shown that the strangeness abundance is chemically

equilibrated in the quark-gluon plasma phase and indicated that this is not

the case in the hadronic gas phase. We now further note that even assuming,

probably much too optimistically, absolute chemical equilibrium in the gas

phase, we find 3 to 5 times more strangness in the plasma at comparable

thermodynamic circumstances, i.£. , equal p,T. This is shown in figure 7.7 as function of u at some selected values of T and m

$

, where the conversion from u as a variable to baryon density has been done using perturbative QCD. Thus

the simplest of all observations pointing to the quark-gluon plasma is the

measurement of an anomolously high strange particle abundance as function of

CM energy in the colliding nuclei, i.e., preferably at high p x

. Furthermore

we have argued that_A are strongly suppressed in the nuclear gas. Thus an anomalous yield of A's is an even more characteristic observable of the

pi asma.

66
More speculative is the observation that strangeness may cluster in

the dilute plasma to form strangeness

impurities such as 55, 555, etc., owing

to the attractive QCD-Coulomb interaction. Although the 5s state would be more bound than 55 or ss state, its

statistical weight associated with the

color selection factors is so much

smaller that we believe the latter

state to be the dominant strangeness

cluster. This coagulation of strangeness can proceed even further with 555

(or sss) states (free ft's) being

formed. It is very hard to make a quantitative prediction of this

process, but clearly one must look out

to measure the abundances of such rare baryons as ft, possibly on event by

event basis. However we observe that

hadrochemical calculations, j_,e., those

in hot nuclear matter, along tlTe lines

outlined above eq (7.18) also lead to

anomalous abundances of multistrange

baryons [53] quite similar to the

results of a recent hyperon beam

experiment [54].

8. STRANGENESS AS AN EXPERIMENTAL TRIGGER FOR PLASMA DROPLETS

In the above discussion, the rapid production of strangeness in the

plasma phase and its higher statistical abundance as compared with the

hadronic gas phase, are the central features of strangeness as a characteristic observable for the quark-gluon plasma. While we have discussed in this

work mainly the large plasma domains, we now will turn our attention to

"small" plasma droplets which may be either created in collisions of light

nuclei [55] or perhaps in antiproton annihilations on nuclear targets [2h],

[56]. Before turning to these phenomenological details we first derive

another effect that further enhances the role of strangeness as a signature

for plasma droplets.

As we have described at length in section 4, exact conservation of

quantum numbers, here of the total strangeness, greatly influences the actual

partition function. Rewriting eq (4.61) for strange quarks we have the

generating partition function

Fig. 7.7 Ratio of strangeness along

the transition line between

the plasma and the hadronic

gas phase as a function of

assumed baryon density on the plasma side.

Jin Z = g V / —-

3-P— (£n(l + exp[- e/p ^+m 5 - + i $])

q

s S

(2it)3

J

+ ln( 1 + exp[- B/p^Hn 5 + ^

- i ])

}

( 8 . 1 )
where the statistical degeneracy of strange quarks is g s = 2

$ x 3

C

= 6. In eq (8.1) we have included the baryochemical potential associated with strange

quarks. The angle 4> is associated with the U(l) group and will allow to ensure exact strangeness conservation.

From eq (8.1) we can extract the partition function of given strangeness

s by a simple integration

which in the Boltzmann limit can be carried out analytically. We have in this case

where as usual W(x) = x

2K

2 (x).

We now recall the generating series of the modified Bessel functions

( 8 . 2 )

+ 9

S

V / expf-B/pV + I

11

] e'

1 *

W(m/T) [e* Su/3

e

1 * + e Bu/3

e'

1 *] (8.3)

g

z(t+Vt)/2

(8.4)

We introduce the definitions

t = g

- Bu/3+i <f> (8.5a)

(8.5b)

where ^ is the one-particle partition function. For the generating

partition function in the Boltzman approximation we have hence

68
Z? ( 8 . 6 ) = l e"

0lik/3 e

lk<,)

I. (2Z ( )

. q

s k=-°°

K

The integral (8.2) can now be carried out without further delay

- 6un /3 V e

s " I (2Z

(1)

)

. (8.7)

For finite n

$

the chemical potential regulates the particle abundances in the

conventional fashion. For n

s = 0 we have a very interesting special case [57]

Zo(T.V) - I 0 (2Z

(1) (T,V)) ( 8 . 8 )

Z

0 (T,V) is now u-i ndependent and describes an arbitrary number of ss pairs in

the volume V and at temperature T, but with the number of s-quarks being

exactly equal to that of 5 quarks. However, having used the Boltzman limit,

we have constrained the validity of eq (8.8) to temperatures not much larger

than the strange quark mass; only then is the expansion (8.3) valid, j^e., the

phase space sufficiently thinly populated allowing the neglect of quantum

symmetry effects.

Next we notice that the argument in the I 0 function is the number of

strange and antistrange quarks, computed as if we had neglected the influence

of exact strangeness conservation. In the limit that this number is large we can employ the asymptotic expansion

i 0 ( z ) U + h + • ••) (8.9)

/2ttz

to recover the usual result

in Z 0 (T,V) —> 2Z

(1)

(T,V) = Zp(T.V) + Z

(1)

(T,V)

V~" * s

( 8 . 10 )

In the above equation we have the volume V as the only quantity that controls

the argument of the partition function at fixed temperature.

The actual number of s-pairs present in the plasma is

<n

s > = X-fj in I 0 (7‘722Z ( 1 ))

\=1 I 0 (2Z

(l)

)

Z (l)(T,V) ( 8 . 11 )

69
where we note the appearence of a suppression factor

Il(2Z

(l)

)

This factor is shown in figure 8.1

for T = 160 MeV and strange quark mass m = 160 MeV as a function of

the volume. The volume is measured

in terms of the elementary hadronic

volume V

h

=

-f1 (1 fm) 3

. The

important aspect of this result is

that as the volume goes from 1/2 V

h to 3 V ^ , n more than triples.

Hence, we expect that in the plasma

droplet strangeness would be

enhanced by the relaxation of phase

space constraints arising for small

volumes from the fact that

strangeness is generated in s5

production which is the physical

fact expressed by the quenching

factor (8.12).

This way we have two effects leading to a significant increase in the

strangeness abundance even in small plasma droplets:

(a) The nonlinear volume effect: the abundance of strangeness is not

only proportional to V through z( ) eq (8.10), but in addition there is the

disappearance of the phase space quenching factor n eq (8.12) for volumes

exceeding V^. Even for small droplets with V « 2V

H this effect leads to an enhancement by a factor ~ (V/V h ) 2 = 4.

(b) The word "plasma" implies the equidistribution of the energy into

the available degrees of freedom and a lifetime of the droplet of more than

10" 23 sec. As the QCD cross sections indicate (see preceeding sections) this

time almost allows reaching the chemical equilibrium state. As discussed in

section 7 this means that the strangeness abundance found would be larger by a factor of 2 to 4 than in the equilibrated hadronic gas phase. As the equilibration is also unlikely in the gas phase, we must also expect at least a factor 5 or so more strangeness from the plasma droplet than from the gas

phase.

Taken together, both (a) and (b) imply that the strangeness originating

from plasma droplets is more than ten times as abundant than that expected

from the hadronic gas phase, making the appearance of plasma easily visible as

a plasma production threshold is passed in suitable experiments.

I„(2Z ( ')

Fig. 8.1 Suppression factor for strangeness production as a function

of the reaction volume .

70
While no systematic experimental information is available as of now, we have found one piece of data which seems to confirm these considerations.

Namely, instead of using high-energy nuclear collisions, one can employ antiproton annihilation in nuclei in order to produce a local plasma droplet [56].

We would like to argue that when slow i.£., LEAR antiprotons penetrate into a nucleus the first step in the annihi lafion process will be the formation of a baryon-number zero fireball, filled with colored gluons and quark-antiquark

pairs. As it turns out, this picture allows us to describe satisfactorily the

Tt-multipl icities in annihilations [58]. In Pp reactions such a state would

then break up into several mesons, a process that may last sufficiently long

to allow the fireball to sometimes collide with one or more of the nearby

nucleons in a nucleus. Very likely, this will lead to the absorbtion of some

as yet unspecified number of nucleons into the fireball. What do equations of

state, cf. section 4, tell us about the physical properties of such a state?

We have the energy density of a (relaxed) droplet, eq (4.6):

4B = E/V = (E/b) (b/V) , b = A - 1 (8.13)

where b is the baryon number of the droplet, i.e^. , one less than the total

number of nucleons A that have reacted with tTTe incoming antiproton. We also

know the total energy E of the fireball.

E = (A + l)m

N

(8.14)

and hence can solve eq (8.13) for the baryon density v of the droplet:

v « b/V = 4B £E = ^

m^ §4A+l (8.15)

At low energy the last factor containing

the kinetic energy of p may be dropped.

Figure 8.2 shows the "compression" v/v

Q as function of A, where v

0 as usual is the

normal nuclear density in heavy nuclei,

v

0 = .145/fm 3

.

We see that the baryon density of the

droplet remains similar to that of the normal

nuclear matter, i.e_. 9 we find no appreciable

compression [56b7.

Certainly, the quark droplet will

disintegrate into A-l baryons and several mesons. Our understanding of this process is

still unsatisfactory but some properties of

Fig. 8.2 Nuclear matter

compression as a function of A.

71
the emerging particles can nonetheless be estimated. First we observe that

inside the low density droplet a temperature of the order of about 160 MeV

(A = 1) prevails. This value is actually slightly A-dependent but will be

about 140 MeV even for large A, owing to the low baryon density. Particles

originating from the disintegration of the droplet will in the event-ensemble

show a momentum distribution with characteristic slope (~ 1/T). Parallel to

this effect we should expect a significant enhancement of the strange particle

abundance, as just described.

Apparently, an experiment to

observe a plasma droplet would consist of using a strangeness trigger

and measuring the momentum distribution of high-momentum so called

spectator protons. Such an experiment has actually been carried out

for the lightest nuclei. In figure

8.3 we show the results of the p-d

annihilation taken from reference

[59]. Here the event rate as a function of the recoiling proton

momentum is shown if the_annihi 1 a- tion is accompanied by KK production.

Indeed we see a strong enhancement at proton momenta p > .3 GeV

which nicely follows a T = 160 MeV

slope. It is very interesting to

note that this clear signal seems to

disappear in the background when the

KK trigger is not used (see fig. 2

of reference [60]). Another confirmation of this interpretation is

obtained by considering the reaction

Mom. of Spectotor P (GeV/c)

Fig. 8.3 Momentum distribution of

spectator protons in

coincidence with strangeness

production.

A + X . (8.16)

The strangeness is now attached to the nucleon and the reaction is selfanalyzing in the sense that the recoiling particle has the trigger quantum

number. Indeed, in reference [61] Oh and Smith record that the A p x spectrum

is identical to their p spectrum in the bump above p ± > .3 GeV. Recent

measurements of the reaction (8.16) [62] present also an alternative

interprtation in terms of mass shell K-exchange which, however* seems to fall

short of the data. Another experimental evidence against the K-exchange

mechanism is the anomalous enhancement of "K-d reactions" when the spectator

momentum exceeds 200 MeV/c [63].

We can thus conclude that in the p-d reaction a first signal for the

annihilation on two nucleons leading to a b = 1 plasma droplet may have been

seen. It would be of great interest to see if a similar signal can be

obtained, e.g_., in p

- a annihilations. Here in particular a 4ir geometry

would be oT great help in order to select events in which all three remaining

nucleons share the annihilation energy. A simultaneous enhancement of the s- yield would give a confirmation of the presented arguments.

72
9. SUMMARY AND OUTLOOK

Our aim has been to obtain a description of highly excited hadronic

matter. By considering matter in kinetic and chemical equilibrium we have

been able to develop a thermodynamic description valid for high temperatures

and different chemical compositions. In the present work we have described

two physically different domains: first, the hadronic gas phase, in which

individual hadrons can exist as separate entities, but are sometimes combined

to larger hadronic clusters; and second, the quark-gluon plasma, where

individual hadrons dissolve into one large cluster consisting of the hadron

constitutents.

In order to obtain a theoretical description of both phases we have used some "common" knowledge and plausible interpretations of the currently available experimental observations. In particular, in the case of the hadronic

gas we have completely abandoned a more conventional Lagrangian approach in

favor of a semi -phenomenological statistical bootstrap model of hadronic

matter that incorporates those properties of hadronic interaction that are, in our opinion, the most important.

In particular, the attractive interactions are included through the rich

exponentially growing hadronic mass spectrum x(m 2 ,b) while the introduction of

the finite volume of each hadron is responsible for an effective short-range

repulsion. We neglect quantum statistics in the hadronic gas phase since a quantitative study has revealed that this is allowed above T - 50 MeV. But we allow particle production, which introduces a quantum physical aspect into the

otherwise classical theory of Boltzmann particles.

Our considerations lead us to the equations of state of hadronic matter

which reflect what we have included in our considerations. It is the

quantitative nature of our work that allows a detailed comparison with

experiment. It is important to observe that the predicted temperatures and mean transverse momenta of particles agree with the experimental results

available at E

k ]ab /A = 2 GeV [BEVELAC] and at 100 GeV [ISR] as far as a comparison is permitted [4a].

The internal theoretical consistency of our description of the gas phase

leads us in a straightforward fashion to the postulate of a first order phase

transition to the quark-gluon plasma. This second phase is treated by a quite

different method; in addition to the standard Lagrangian quantum field theory

of "weakly" interacting particles at finite temperature and density, we also

introduce the phenomenological vacuum pressure and energy density B. This

term is required in a consistent theory of hadronic structure. It turns out

that B 1/4 ~ 190 MeV is just, to within 20%, the temperature of the plasma

phase before condensation into hadrons. This is similar to the maximal

hadronic temperature T

Q = 160 MeV.

Perhaps the most interesting aspect of our work is the realization that

the transition to the quark-gluon plasma matter will occur at a very much

lower baryon density for highly excited hadronic matter than for matter in the

ground state (T =0). Using the currently accepted value for B we find that

at v ~ 2 - 3 v

0

, T = 150 MeV, the plasma phase may indeed already be formed.

73
These plasma characteristics were used to evaluate both the radiation

cooling of the plasma by the emission of pions, and the conditions under which

a plasma seed will grow in a high energy nuclear collision. Using the above

value T = 150-160 MeV as the phase transition temperature we find that 20 - 50 GeV/nucleon on a fixed target should suffice to lead to a quark-qluon

plasma. The frequency of such events is controlled by the probability that a plasma seed of adequate size be formed early in the collision. A plasma seed

is needed since the hadron cross sections are too small to lead to energy

confinement for the required length of time. The cross sections in a plasma

can be much larger since the phase space there is much larger than in the

hadronic gas phase owing to the absence of the color constraint; for example,

a 3 quark system in the plasma has 27 color states while only the color

singlet is permitted in the hadronic phase.

We also discuss in a somewhat speculative manner the phase transition

plasma hadronic gas in order to estimate the parameters u,T needed in the

study of possible signatures of the plasma in high energy nuclear collisions.

One of the possible signatures is the strangeness production in the

plasma. We show that it has a sufficiently large reaction rate for the

strangeness abundance in the plasma to reach chemical equilibrium during the

lifetime of the plasma. The subsequent depletion of the strangeness during

the plasma hadronization as well as its preferred hadronization channels have

not yet been studied in detail. However, only if the plasma hadronization is an extremely slow process, lasting on the order of 10“ 22 sec., a significant

depletion of the high s-abundance created at the maximal temperature reached

in the collision can be anticipated. As shown in figure 8.3 the invariant

rates drop quite rapidly with decreasing temperature, leading to a rapid

increase of the equilibrium time constant t. Hence the strangeness abundance

decouples from the equilibrium and remains a witness of the hot collision

period. We have further shown that strangness may be a useful trigger on plasma formation.

While we can not yet discuss in detail the abundance of multi-strange

antihadrons which are influenced also by the possible strange bound states in

the plasma, it is apparent from our results that the measurement of production

cross sections of antistrange baryons already could be quite helpful in the

observation of the phase transition. The high suppression of these degrees of

freedom in the hadronic gas phase is not maintained in the plasma phase where

5 abundance is larger than D,d abundance. A measurement of the relative K+/K - yield, while indicative of the value of the chemical potential in a hot

nuclear gas may carry less specific information about the plasma. The K/tt

ratio may also contain relevant information. However, since the tt originates

from diverse sources its abundance is controlled by the total entropy created

in N-N collisions. Hence, it will be much more difficult to decipher the

message. Perhaps a steep rise of K/tt ratio at high Pj^ could be helpful 1.

On the other hand it appears that the abundances of otherwise quite rare strange hadrons will be enhanced, on the one hand by the relative high phase

space density of strangeness in the plasma, on the other hand in view of the

attractive ss-QCD interaction in the 3

q and ss in l c channels. Hence we should search for the strangeness abundance in the yields of particles like

74
2,S,ft,fi, $, rather than in the K-channels. It may be that such experiments

would uniquely determine the existence, and eventually the characteristics, of

the phase transition to the quark-gluon plasma.

It is important to appreciate that the experiments discussed above would

certainly be quite complementary to the measurements utilizing electromagnetically interacting probes, e.£., dileptons, direct photons. Strangeness

based measurements have the advanFage that they are much more abundant than

the electromagnetic particles since they involve the observation of a strongly

interacting particle (s,S quark) which happened to be a direct constituent of

the hot plasma phase.

10. ACKNOWLEDGMENTS

While much of the work is still left to the future, we would like to

thank all who have through collaboration or stimulating discussions helped in

the development of this research field: H.-Th. Elze, W. Greiner, R. Haqedorn,

P. Koch, B. MCIller, H. Rafelski, and G. Staadt. One of us (J.R.) would like

to thank Prof. R. Bock for his hospitality at the GSI laboratory where much of

this work has been done. Several research periods at CERN-Theory Division by one of us (J.R.) have contributed essentially to the theoretical developments

presented here.

75
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80
NBS-114A (REV. 2-BO)

3. Publication Date

June 1983

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NBSIR 83-2725

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4. TITLE AND SUBTITLE

PERSPECTIVES IN HIGH ENERGY NUCLEAR COLLISIONS

5. AUTHOR(S)

Johann Rafelski and Michael Danos

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NATIONAL BUREAU OF STANDARDS

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| | Document describes a computer program; SF-185, FIPS Software Summary, is attached.

11. ABSTRACT (A 200-word or less factual summary of most significant information. If document includes a significant

bi bl iography or literature survey, mention it here)

This report has been prepared as a working document for the

conception of a research facility devoted to the study of high

energy nuclear collisions. Different aspects of hadronic physics to

be studied in nuclear collisions are selected, with emphasis placed

on the properties and nature of the quark-gluon plasma, the formation of the plasma state in the central region and its anticipated

lifetime, and the observability and strangeness content of this new form of nuclear matter.

12. KEY WORDS (Six to twelve entries; alphabetical order; capitalize only proper names; and separate key words by semicolon s) elementary particles; gluon; hadron; nuclear collisions; nuclear plasma;

quark

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