Hydrodynamics of a quark-gluon plasma

Nuclear Physics A428 (1984) 34%356~ North-Holland, Amsterdam

HYDRODYNAMICS Jean-Paul

34%

OF A QUARK-GLUON

PLASMA

BLAIZOT CEN-SACLAY,

Service de Physique Theorique,

91191 Gif-sur-Yvette

Cedex,France

The main arguments leading to the conclusion that a quark-gluon plasma could be formed in ultrarelativistic heavy-ion collisions are reviewed. The evolution of a plasma formed in a central collision is described using a simple hydrodynamical model.

1. INTRODUCTION There is a growing belief that it might be possible to achieve the deconfinement transition

leading to the formation

tivistic heavy-ion

collisions

nucleon. The existence

of this phase transition,

simole phenomenological namic calculations

of a quark-gluon

plasma in ultrarela-

with center of mass energy &L

models

lo-50 GeV per

which could be inferred from

is most strongly supported by quantum chromody-

on a lattice. Those calculations

indicate that when the

energy density exceeds a critical value of the order of several times the energy density of ordinary elementary

nuclear matter,

constituents,

ting particles

the usual hadrons dissolve

quarks and gluons, which form a gas of weakly interac-

: the quark-gluon plasma. The conditions for the formation of

the plasma (energy density E - 2 GeV/fm3, to those which prevailed

temperature

in the very early universe,

the big bang. Our experimental

information

T - 200 MeV) are similar a few microseconds

concerning

is still very limited. Heavy ion collisions

sive information.

after

this new phase of matter

have been performed until now at

too low energy or with too light ions to be really relevant. ray events have been observed,

Interesting

cosmic

but they are too few in number to provide deci-

In order to get conclusive evidence for the existence of the

plasma and to be able to study its properties, density over an extended These conditions

into their

one needs to reach a high energy

domain, many times larger than the size of a hadron.

could be achieved

in collisions

of large nuclei at sufficiently

high energy. In this talk, I shall briefly outline simple arguments which lead to estimates of the critical

parameters

finement

transition.

I shall also review

(temperatureandbaryon

expected

in ultrarelativistic

estimates

heavy ion collisions.

evolution

density) forthe

decon-

for the energy densities Then I shall describe the

of a quark-gluon plasma formed in a central collision using a simple \ hydrodynamical model 5'6 developed in collaboration with G. Baym, W. Czyz, B.L.

03759474/84/$03.00 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

346~

J-P. Blaizot / Hydrodynamics

Friman and M. Soyeur. Obviously tion of this fast developing

of a quark-&on

plasma

this talk provides a very incomplete

field. Excellent

presenta-

reviews can be found in referen-

ces 1 and 2. 2. THE TRANSITION

TO THE QUARK-GLUON

Several phase transitions temperature

PLASMA

are expected to occur in hadronic matter at high

T and/or high baryon density7

ment transition, cal temperature For nb=O,

nb. Concentrating

we shall give simple phenomenological and density

in the two limiting cases nb=O

a simple model for hadronic matter consists

interacting massless

pions, whose pressure

on the deconfine-

estimates

is given by

for the criti-

and T=O. in a gas of non

:

(2.1) This pressure

is to be compared to that of the quark-gluon PQG = 37 ;

where B, the "bag constant", ma and 37 is the effective

given by

For T
(2.2)

is the pressure exerted by the vacuum on the plas-

(2.1) and (2.2) show that there is a cross-

: II4 B1/4

the phase of maximum

nit phase, while for T>Tc

T4 - El

:

number of degrees of freedom in the plasma,assuming

two quark flavors. The expressions over temperature

plasma given by

N 0.72 B1'4

pressure

(= minimum

the quark-gluon

.

(2.3)

free energy)

is the hadro-

plasma is thermodynamically

favored.

The precise value of the bag constant is still uncertain (see below). Taking a l/4 "typical" value B N 200 MeV, one gets Tc N 150 MeV. Note that in this simple picture, the transition ence namely

is first order, with a latent heat equal to the differ-

between the energy densities

of the quark-gluon

phase and the pion phase,

: 2 AE = B + & zF

(37-3)

= 48

(2.4)

that is AE - 1 GeV fm -3 for B1'4 - 200 MeV. At T =0 we may assume hadronic matter to be composed, densities,

of ordinary

nucleons. When the baryon density

nucleons start to overlap, which eventually

at "normal" baryon is increased,

leads to deconfinement.

the

The density

nc at which this occurs is related to the density no of ordinary nuclear matter 3 where rN is an effective nucleon radius and r. the average by nc'no = (ro /r N ) , distance between nucleons in ordinary nuclear matter (r. - 1.2 fm). For rN-lfm

one finds nc N 2n, while for rN-0.5 These simple estimates density and temperature properties

at which one might expect qualitative

of hadronic matter.

In order to get a more precise

must resort to a more fundamental indications associated

approach.

ties concerning

T=Tdec

and that associaof pions at

for having Tch 2 Tdec and some calculations'

indi-

could be very close if not equal. For pure gauge

transition

appears as a strong first order transition

latent heat. However there remains considerable

parameters

obtained

are not settled yet, due to uncertainties

approximation

and the approximate

deconfinement

temperature

corresponding

valuesfor

uncertain-

in the presence of quarks *'. Also,

the nature of the transition

the precise values of the critical calculations

one

provide

in hot hadronic matter. That

occuring at temperature

cate that the two temperatures fields, the deconfin~ent with a substantial

information,

of chiral symmetry and the disappearance

There are arguments8

of the

changes in the

Lattice gauge calculations

for at least two phase transitions with deconfinement

ted with the restoration T=Tch.

fm, nc-15no.

give us a feeling for the order of magnitude

in these lattice QCD arising from the lattice

treatment of the quarks.

Predictions

for the

range from 200 MeV (ref.lO) to 370 MeV (ref.ll), the

the latent heat being 1.5 GeV/fm3 and 10 GeV/fm3 res-

pectively. 3. HOW TO REACH HIGH ENERGY DENSITIES High energy densities were reached in the hot early universe. They may also possibly be reached in strong gravitational

collapses.

The first indications

that high energy density may be reached in nuclear collisions the spectacular

events observed

collaborations.

From the huge (- 1000) multiplicities

in these energetic

collisions

the order of 4 GeV.fmq3 tremely important densities

Collisions

could have been obtained.

are ex-

of energy

in nuclear collision. lead to the fo~ation

region

(- the size of a nucleus) with large energy density. One 19 two regimes in relativistic heavy ion collisions. At moderate

lElab - 10 GeV per nucleon) the colliding

nuclei are expected

against each other thus producing highly compressed density as high as 10 no. At higher energy become "transparent"

(typically

obtained

to stop

nuclear matter, with baryon

(Flab N I TeV/A) nuclear matter

to nucleons and the colliding

other. The nuclear compression energy3

an energy density of

These observations

of large nuclei at high energy may possibly

may distinguish energy

of the particle produced

(E/A - TeV) one infersthat

since they provide support for current estimates

reachable

of an extended

are provided by

in cosmic rays by the JACEE12 and ECHOS13

nuclei pass through each

in this regime is smaller than at lower

a factor 3 to 5) while the energy density achieved

is

348~

J-P. Blaizot / Hydrodymanics

estimated334

plasma

to be of order 2 GeV/fm3. These estimates

range for the formation

4. SPACE-TIME

PICTURE OF CENTRAL COLLISIONS

sions. Such collisions considerations

are expected

to be produced

are rare, but not extremely

indicate for example that-l%

with impact parameter

region

which contain

regions

left in between the two

baryon number but a substantial

region

sity may be so high that a quark-gluon

energy den-

and the central region, the energy denplasma could be formed. We shall follow

of the central region which is the simplest to describe. The phy-

sics of the fragmentation

regions is complicated 14,15 in the nuclei .

by the presence of baryons and

of fragments

The basic space-time the coordinate

nuclei pass through each

two highly excited nuclear fragmentation

receding nuclei contains negligible

viewed in the

heavy ion collision.

the net baryon number of the system. The centra2

sity. In both the fragmentation

are head-or?

in fig.1. Owing to the nuclear transparency

expected at high energy the two Lorentz contracted

the production

: simple geometrical

of U-U collisions

FIGURE 1 view of a central ultrarelativistic

other producing

rare

in head-on colli-

b 2 1 fm. The geometry of such collisions

center of mass frame is illustrated

the evolution

are in the appropriate

of a quark gluon plasma.

The highest energy densities

Schematic

of a quark-&on

picture of the collision

along the collision axis, z=t=O

very short times following

the collision,

is illustrated

point). At

the degrees of freedom excited

posedly quarks and gluons) are, because of asymptotic ting and "free streaming".

in fig.2 (z is

is the collision

(sup-

freedom, weakly interac-

Those reaching a point z,t have velocity z/t and

proper time T = (t*-z*)I'*. As time goes, collisions

become more and more

349c

FIGURE 2 Space time diagram of a central ultrarelativistic important and lead eventually

heavy ion

to thermal equilibrium

to be ho - 1 fm. At that proper time beg hydrodynamic the observation

hydrodynamic

at a proper time estimated expansion

of a plateau in the rapidity distribution

in p-p or p-A collisions,

sets in.. From

of particles

Bjorken argues' that the initial conditions

expansion are invariant

axis. In the simplified

COfiiSiOn.

under Lore&z

produced for the

boosts along the collision

picture where one ignores transverse motion,

this condi-

tion implies that the energy density s(z,t) should be a function only of the proper time T. In particular, is constant on the hyperbola

at the initial proper time to, the energy density 2 2 I/2 ho = (t -z ) . During the expansion quarks and

gluons will recombine into hadrons tion may be accompanied verse expansion. cease to interact

(mostly pions). This ha&onizatZon transi-

by the fo~atiun

The hydrodynamic

of a rarefaction

flow continues

: this ~freeae out” transition

shock6

in

the trans-

until the produced hadrons occurs

when the temperature

of

the plasma falls below a certain value Tfo. (Typically Tfo N m,). Our ability to determine

the properties

of the matter produced

in the early

35oc

J-P. Blaizor / Hydrody~~ics of a q~~k~~l~o~plasm

stages of the collision depends on our quantitative

understanding

of the evolu-

tion of the plasma which has been sketched in the previous paragraph. tant thermodynamic

diagnostic

of the state produced

One impor-

in the initial collision

volume is the total entropy of the matter. To the extent that entropy is conserved in the subsequent

expansion,thefinaldistributionof

in phase space is related directly

in the initial collision. Two mechanisms, tion during the evolution a deconfined

quark-gluon

detected particles

to the number of degrees of freedom excited

of the plasma

however, may lead to entropy genera-

: the first order phase transition from

plasma to confined hadronic matter,

out" of this hadronic matter. We shall give below arguments entropy generated

in the phase transition

and the "freeze showing that the

is expected to be small.

We end this section by giving an estimate of the energy density at the beginning

of the hydrodynamic

expansion.

served charged pionmultiplicities

This is obtained by extrapolating

inthecentral

ob-

rapidity regime in p-p and prj

collisions4S5.

In the energy range N 30-270 GeV per nucleon, the energy per unit rapidity carried by the pions is dE/dy - 1.2-1.8 GeV. In a central AA collision, this energy is increased by a factor - A so that the energy density is initially

:

where R is the nuclear radius

(R - I.2 A

Y"- i and $$ = : = +

:

so that

l/3 fm). Near the collision plane

Al/3 EO - (0.3-0.4) 7 For ~~ - 1 fm and A - 238, ~~ - 2 GeV/fm3. consists of a gas of thermalized

GeV/fm'

.

If we assume that the matter at ~~

quarks and gluons with equal numbers of u,;,

d and ?! present, then an energy density of 2 GeV/fm3 corresponds to a tempera-3 . This large d;nsitS; ture T - 200 MeV and a total number of quanta no - 4 fm of quanta implies that they have relatively fm where ufm is a mean scattering

&opt

mean j%ee paths h = ~-4o

cross section in fm2 (ofm-1

leads us to expect the validity of a hydrodynamic

description.

questions

equilibriumis

after

of whether,

proper

tigations.

time

and how,local thermodynamic

ftn

Of course the really achieved

~~ remain important questions which deserve further inves-

(For a recent calculation

linearized Boltzmann

fm2). This

equation,

of the approach to equilibria

see reference

using a

16).

5. HYDRODYNAMICS Once local thermodynamic

equilibrium

is established

in the central collision

J-P, Blaizot / Hydrodymanics

volume, the evolution momentum

where

summarized

of a quark&on

is governed by the conservation

in the equation

plasma

351c

laws for energy and

: a T'" = 0 lJ

(5-I)

T'" = (E+P) u" u" + P g'"

(5.2)

:

E being the energy density

that the equation

(in the local rest frame), P the pressure, gVV the

~(1,;) thefourthvelocityofthe

metric tensor and u" =

fluid.

(5.1) contains the entropy conservation

&

law

Let us emphasize

:

(sy) + G.(syG) = 0

(5.3)

where s is the entropy per unit proper volume. For a simple one-dimensional

s cash y) +

A( & where y = tanh-Iv, to Eqs.(5.4)

motion, the equation

&(

(T sinh y) + 2

(5.1) reduces to

s sinh y) = 0 (T cash y ) = 0

(5.4)

is the rapidity of the fluid element. Two types of solutions

are interesting

- ScaZing SoZution,

in the present discussion.

obtained for Lorentz invariant boundary

posedatproper

timeTo. This leads in particular

of the entropy

:

conditions

which may be understood

im-

to the followingtimedependence

S(T) = So.To/T

of entropy

:

(5.5)

from the fact that the volume over which a given amount

is spread grows as the proper time T. In this regime the flow ve-

locity is given simply by vz = z/t. - ~iemann sohtion

corresponding

infinite slab of matter This solution

to the one-dimensional

initially at temperature

is given by5

expansion

To between z=-m

of a semi-

and z=R(

>O).

: z-R+Cst

‘S/*

T=T >

v(z,t) = v

(5.5)

where Cs is the (constant) speed of sound ('=$ Cs These two typical solutions refer to a numerical z=O

to z=R,

solution5

are illustrated of eqs.(5.4)

on figs. 3 and 4. These figures

for a finite slab extending

with an ideal equation of state (Cs = l/a).

from

The initial condition

352~

J-P. Blaizot / Hydrodymunics of a quark-&on

plasma

'0l-T---r---1

0

2

r/R

0

6

4

FIGURE 3 Temperature distribution fortheonedimensionalexpansionofafinite slab.

2

is v(z=R)

= 0. Each curve is marked by the corresponding

recognizes

the two types of solutions

that t/R
The plasma produced geometry.

expansion

on this motion

Lorentz invariance

greatly simplifies

motions,

by the Riemann solution, while for

is the transverse

of a rarefaction

and transverse

can be described

wave

in cylindrical

The underlying

the problem of coupling the longitudinal the transverse motion

to other frames. The evolution

direction

for a cylindrical

in

of the tempera-

expansion

played in fig.5. Here, r (fm) is the distance from the collision tial radius of the cylinder

(5.5).

expansion which begins with the

(Riemann solution).

by allowing one to determine

ture profile in the transverse

value of t/R. One

above. For short times t such

obeys a simple scaling solution

inward propagation

only one frame and then boosting

6

at small r.

in a head-on collision

The longitudinal

Superimposed

described

J5, the system is described

larger t, a scaling solution develops

r/R

FIGURE 4 Same as fig. 3, for velicity.

is dis-

axis. The ini-

is R = 7 fm. Notice that the longitudinal

expansion

causes a uniform cooling of the matter at small r, interior to the rarefaction front. The figure 6 shows a snapshots t=9

of the isotherms

in the z,r plane at

fm. Each curve is marked by the ratio T/To. The rarefaction

front is indi-

cated by a dashed line. The essential expansion

quantity

one would like to determine

is the final distribution

from the hydrodynamic

of particle multiplicities

Here we shall just report the main result of a calculation

and momenta.

described

in detail

in reference 5. We assume that the freeze out occurs at a given temperature For R-

7 fm and Tfo - 0.7 To (where To is the initial temperature)

tial fraction of the matter

remains unaffected

by the transverse

to freeze out, and as a result, the emitted particles verse momentum

due to hydrodynamic

motion.(In

Tfo.

a substan-

expansion

prior

acquire very limited trans-

the case considered,

90 % of the

of a q~~rk-gI~on

J-P. Blairof / ~yd~ody~nics

plasma

7ffml

r(fm) FIGURE 5 Temperature profile for cylindrical expansioncoupledtolongitudinalexpansion. particles

have transverse

6. DYNAMICAL

QUARK-HADRON

As mentioned

ary

rapidities

FIGURE 6 Isotherms in the z,r plane at time t=9 fm.

smaller than 0.25).

PHASE TRANSITION

earlier, when the deconfin~ent

the hyd~dyn~ic

353c

expansion,

a shock-like

transition

discontinuity

takes place during

may developatthe

bound-

between the hadron phase and the quark phase6. Also the latent heat re-

leased in the transition

is converted

matter. We shall illustrate sional expansion.

on a simple,

Figure 7 shows a typical temperature

Example of a one-dimensional tion. of a phase transition. given temperature

into kinetic energy of expansion

these phenomena

FKGURE 7 Riemann wave without

Riemann-like,

of the one-dimen-

profile without effects

effect of the phase transi-

The fluid flows in the positive x direction.

A point with

T moves with the velocity of sound relative to the matter flow:

;c(t) = (v-Cs)/(l-VC,)

(6-I)

where Cs is the speed of sound and v the fluid velocity. We assume that the equation of state has a discontinuity jumps from sh in the hadronic In the two-phase

region, where sh
Thus,as explained spontaneously

Tc, at which the entropy plasma (see fig.8).

and T=Tr_, the sound velocity vanishes.

in detail in reference! 6, the entropy pattern will develop

the structure

py of the incoming matter of the outgoing matter the discontinuity. tinuity

at temperature

phase to sq in the quark-gluon

exhibited

in fig.9. At the discontinuity,

is that of the quark-gluon

the entro-

plasma, while the entropy

is that which leads to flow at sonic velocity just beyond

The decrease

in entropy and temperature

across the discon-

is shown as the dashed line in fig.8.

sl

sh:il _

%-

*I

FIGURE 8 Model equation of state with first order phase transition.

x2

FIGURE 9 Condensation discontinuity induced by a first order phase transition.

The state of the matter emerging from the discontinuity solving the two equations components

which express the continuity

is determined

by

of the stress tensor

To' and TX' accross the "shock", in the rest frame of the "shock".

This allows one to estimate as the acceleration general conclusion phase transition

the entropy produced

of the matter

in the phase transition

in passing through the discontinuity.

reached in reference

6 is that the entropy production

as well The in the

is at most of the order of 6 to 7 %, and is thus not likely

to be a significant

complication

in relating observed particle distributions

to the initial state of the matter across

*

the discontinuity,

in a collision.

The acceleration

although small, is not negligible.

of particles

For an ideal

J-P. Blaizot / Hydrodymanics

equation of state one finds an increase

of a quark-&on

in transverse

plasma

355c

rapidity of the order of

0.6.

7. CONCLUDING

REMARKS

There is now growing evidence

that collisions

ions can produce high energy densities study of the properties in the laboratory. phase of matter,

of matter under extreme

In particular

could be observed

recent past, a lot of enthousiasm In particular

discussion

nuclei to center of mass energies

which arise when one tries to construct

detailed quantitative

a consistent

Let we finish by emphasizing

a nuclear col-

work. The

some of the many questions picture of the collision

work remains to be done before

could be made.

one important aspect of this fast developing

field. The study of ultrarelativistic

heavy ion collisions

is a new frontier

in physics, at the border line between high energy and nuclear physics. to fundamental

theoretical

questions

lenge. It gives a unique opportunity join their ccmpetences

in the

nuclear

of the order

a lot of theoretical

heavy ions. Considerable

predictions

This possibi-

has motivated,

for the study of ultrarelativistic

It has also stimulated

new

is a direct consequence

in such collisions.

in the laboratory

of the previous sections has illustrated

of two ultrarelativistic

never reached before

it has led to proposals for building

lider capable of accelerating of 50 GeV per nucleon.

conditions

plasma, whose existence

lity of producing dense and hot matter

collisionsl.

heavy

there is a good hope that a completely

the quark-gluon

of quantum chromodynamics,

of ultrarelativistic

over extended domains, thus allowing the

and efforts

and offers an exciting experimental

It leads chal-

for nuclear and high energy physicists

to

in a common project.

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Phys.Rev. D27 (1983) 140

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Phys.Lett.

1328 (1983) 291.

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J-P. Bhizot / Hydrodymmics

356~

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G. Baym, in Quark Matter Formation and Heavy Ion Collisions, Proc. Bielefeld Workshop, ed. M.Jacob and H.Satz (World Scientific, Singapore, 1982) 17.

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