Quark-gluon transport theory

603c

NuclearPhysics A418 (1984) 603c-612~ North-Holland.Amsterdam

QUARK-GLUON

TRANSPORT

THEORY+)*)

Ulrich HEINZ') Institut fur Theoretische a. M. 11, WEST GERMANY

1.

Physik, Postfach

11 19 32, D-6000 Frankfurt

INTRODUCTION The discussion

about the possible

in high energy heavy ion collisions cated by the fact that suggested and the literature

signatures

quoted therein)

the system expands from hydrodynamical collisions

plasma phase

detection

is compli-

of this new phase (see refs. 1, 2

are affected

by its space-time

and cools down after the collision.

history. A very

or not the system

Nearly equally

An understanding

level requires

knowledge

The natural

port on an attempt

to formulate

non-Abelian

to correctly

In this

such a kinetic theory.

task since the plasma

gauge field theory

account very carefully.

to approach

functions.

2,556 .

part of the his-

tooter&

local equili-

phase even on a qualitative

of the transport

framework

theory for the plasma distribution

out not to be a trivial

by the approach

is the question

equilibrium

imply that a considerable

of that pre-equilibrium

an approximate

plasma.

important

is in local thermal and chemical

tory of the plasma will be characterized

quark-gluon

It has been estimated

calculations 3y4 $0 be of the order of 4-10 fm/c for U + U

at ~50 GeV/u c.m. energy.

The short time scales just mentioned

brium.

of a quark-gluon

role is played by the lifetime of the plasma which depends on how fast

important

whether

occurrence

and its experimental

properties

is a kinetic

contribution

I will re-

Conceptually,

interactions

(QCD), and gauge invariance

But also from the practical

of the

this problem

this turns

are governed

by a

has to be taken into

side it appears necessary

treat all the color degrees of freedom since in the plasma phase

color can move around freely, allowing

for the possibility

of macroscopic

color

fluctuations. On the conference (1982) I presented meantime, problems.

on ultrarelativistic

a classical

some progress

heavy ion collisions

in Bielefeld

kinetic theory for a colored plasma'.

could be made on a quantum

I will review these two approaches

(QCD) treatment

In the

of this

and add a few remarks on

+Work supported by Deutsche Forschungsgemeinschaft. *Talk given at the Third International Conference on Ultra-Relativistic NucleusNucleus Collisions (Quark Matter '83), Brookhaven Nat'l. Lab., Sept. 26-29, 1983. BPresent address: Department of Physics & Astronomy, Vanderbilt University, Nashville, TN 37235, USA.

037S-9474/84/$03.00 0 Elsevier Science Publishers B.V.

(North-Holland Physics Publishing Division)

604C

U, Heinz / Quark-Gluon Transport Theory

their further development

2.

CLASSICAL

KINETIC THEORY FOR A COLORED

The classical cal colored

and applications.

formulation

particles

that, due to exchange

moving

in an external

of color between

Q,(a=l,..., 8) of the particle being constants

of motion.

shown7'8

for classi-

These tell you

and field, the color charge

rotates in color space, with QaQa and dabc QaQbQC

into the definition

of the plasmal'.

and ?(x,p,Q)

of motion'

color field AZ(x),

particles

Hence Q, is a dynamical

and has to be included lI:cription

PLASMA

starts from the equations

The l-particle

for the classical

variable

distribution

functions

"quarks" and "antiquarks',

to obey the Vlasov-Boltzmann

like the ~mentum

of phase space in a statistical f(x,p,Q)

respectively,

were

equations

[P~~~~QaF~~(x)p~a~fabcpvA~(x)Qc~~~f(x,p,Q)

=

Ctf,~lfx,p,Q);

[p'aU-QaF;v(x)pV$+fabcP"A;(x)Qc$:(x,p,Q)

=

~[f,i+l(x,p,QI .

C and t are collision

terms, which

but were here assumed

to be of Boltzmann

in general

involve Z-particle

(la,b)

correlations,

type and hence to depend only on f and

f themselves. All the transport ductivity)

mary interest. purely classical

However,

In the plasma,

appears

-- here a thorough necessary

heat and color con-

therefore

are of prithese by

study in the framework

along the way to be outlined

the mean gluon field AZ(x) entering

of

below.

into (1) is determined

condition

f [a,,&,, - fabcA;(x)l

where ji is the quark-antiquark j:(x)

(viscosity,

terms, which

there is little hope to be able to estimate

the self-consistency

(D,F'v),(x)

of the plasma

by the collision

considerations

quantum chromodynamics

through

properties

are determined

Fr(x)

= j;(x)

color current

t

(2)

in the plasma: (3)

=~'Qa[f(x,p,Q) - f(x,p,Q)ldPdQ .

The measure

for the integration

over the color and momentum

sectors of phase

space in (3)‘is given by dP E 2e(p0)6(p2-mz)d'+p; dQ f a(QaQe-Q2)s(dabcQaQbQc-~3)daQ with d8Q being the invariant

group measure

,

(4)

for the octet representation

of

W(3). The kinetic equations an infinite hierarchy

(1) for the distribution

of equations

functions

are equivalent

for their color and mo~ntum

moments.

Of

to

6052

U. Heinz / Quark-Gluon Transport Theory

practical

importance

b'(x) =Jp'(f ThiT(x)

are the following

and the color current tions which

(energy-momentum

- f)dPdQ (3).

These macroscopical

take the form of conservation

aPb'(x) = 0

of the distribution

(baryon number current)

- f)dPdQ

=lp'lp'(f

moments

function:

;

(5)

of matter);

densities

46)

obey hierarchy

equa-

laws

;

(7)

(D&‘),(x) = 0 ; = -ap

apThiT

T&(x)

and can be obtained terms reflected @(x,p,Q)

I -j:(x) F:"(x)

directly

from (1) using general symmetries

in the vanishing - $x,p,Q)ldPdQ

= 0

;

-

t(x,p,Q)ldPdQ = 0 ;

~P,[C(X,P,Q)

+

i(w,Q)ldPdQ

one truncates

thus arriving equations

Although

lar in view of equation

tion of the chrome-hydrodynamical

Before,

therefore,

derive the infinite

dx,p)

which

function,

higher moments

treatment

hierarchy

by

in particu-

still the definitions functions,

(3,

and in an evalua-

is usually done by expanding the collision

a la (10).

terms creep back

Thus the need to deter-

urgent also for the less de-

of the plasma.

going on to a QCD description

These will prove helpful kinetic theory.

equations,

terms from QCD remains equally

tailed chrome-hydrodynamical

is defined

that by doing this one has completely

distribution

distribution

in the form of non-vanishing

at this point,

of these equations,

terms, this is not true:

around a local equilibrium

equations

of chrome-hydrodynamicswhich

a naive inspection

the full, non-equilibrium

mine the collision

(lOa,b,c)

(l), would suggest

gotten rid of the collision 5,6) contain

=0 .

the infinite set of hierarchy

at the formalism

(2,7-g).

of the collision

of the following moments:

~Q,[C(X,P,Q)

In practice

(9)

,

of the problem,

of color moment equations

in understanding

equivalent

the color structure

let us

to (1).

of the quantum

To this end let us define the color moments

+fb,p,Q)

+ fb,-p,Q)ldQ

gabs) =jQ,[f(wQ) gab(x’p) =jQ,Qb[f(x,P,Q)

+ f(w,Q)ldQ

;

;

+ 7(x,-p,Q)ldQ;

etc.

(11)

U. Heinz / Quark-Ghm

606c

The corresponding

moment

equations

read

p'lapg = ~'Fzv a;ga +j[C(x,p,Q)

= PPF;~ P%,,sac- f,,,A;I P'[au6acbbd

- ~acfbmdA~

Trunsport Tkeory

+

h-p,Q)ldQ ;

a;g,b fJ

Qa[Cb,~.Q) + i(x,-p,Q)ldQ;

- 'amc~bdA~lgcd +

= p"FFV $g,bc

J

=

Q,Q,[C(X,P,Q) +

%w,Q)ldQ ;

etc.

3.

QUANTUM

KINETIC THEORY

PLASMA

analog of the one-particle

The quantum-~chanical generated

FOR A COLORED

by the following

distribution

function

(13)

1

<$X,P

is

operator

with j(y,z)

(14)

Here 6a = -ha/2 is the color charge operator,

and riisolves the QCD Dirac equa-

tion i7'[all + iGaA:(x)]i(x)

= m;(x).

(15)

The color indices A, B run from 1 to 3, the spin indices a, B from 1 to 4. path ordered

exponentials

have been included

under gauge transfor~tions the cancellation variants

like an operator

of gauge transformations

operators

tion of the Wigner

sitting

at point z.

in expectation

This leads to

values of gauge in-

6, cTr i 6,. and hence (13) is a gauge invariant Its expectation

operator I2 for quark fields.

is as close as one can get to a positive

definite

distribution

generaliza-

value

function

in quan-

tum mechanics. The Wigner operator

can be used to express

the macroscopical

bM, je and TV": b'(x) z <:~(x)~$(x):>

=J4p
= d'+p p' Jj:(x) 1 <:$(X)&~;(X):> =

J

TV F(x,p)>

+ spin terms

=Jd4p

d4p pp(Tr 3a F(x,p)>

The

in order to make j(y,z) transform

;

+ spin terms:

observables

U. Heinz / Quark-Glum

Transport Theory

60%

T;iT(x) 5 f <:;(x)~%~;(x) - $(x)b"'&(x):>

J

= d4p p' = I spin terms .

+

d4p p'lpv

fl6c)

Here, spin effects have been isolated through a Gordon decompositionas terms involving CT

and are henceforth neglected. They can be treated analogously to

the color d&ees

of freedom (see below), but there is no classical analogue to

them in the classical formulation of Section 2.

However, the latter can be

correspondinglygeneralized by using the classical equations of motion for colored, spinning particlesI to generate a Vlasov-Boltzmannequation for the classical distribution function ffx,p,s,Q) (s denoting spin). In fact, spincolor couplings may be important for the description of chromomagneticeffects in a colored plasma. Comparing equations (16) with (3,5,6) we recognize a similarity in structure, with the only differences being that in (16) the mo~ntum integrals are not on mass shell, and the color integrals of the classical theory have been replaced by sums over color indices. We will now discuss these differences in some more detail.

i can be split up in the following way:

F(x,p) = i(+)(x,p) +

F(-+X.P)

+ F(Zf(x,P)

(174

with

fw(x,p) i

e(~po)e(P2)i$x,Pl;

&+X,P)

s e(-p2G?x,p)

.

(17b)

The last term, containing spacelike momenta, is attributed to the Zitterbewegu~ and vanishes classically. ,(+) and ,c-) are usually peaked near the mass shell'* and classically correspond to 6(po)~(p2-m2)f(x,p,Q)and e(-p,)s(p2-m2)f(x,-p,Q),respectively. Thus we see the particle-antiparticle structure emerge from the quantum theory. This correspondenceshould become explicit in an* + 0 approximation. Next we discuss the color structure of P, neglecting the spin effects and taking the trace over spin indices. The color 3 x 3 matrix & as

with

dxd

'L

5

TrC ; +a(x9P) 9

may be expanded

608~

U. Heinz f Quark-G&on Transport Theory

Inserting

this expansion

dence between

the singlet

their classical g(x,p) -

za(x.p) -

function

higher ~ments

and octet quantum

eb,?lf(x,p,QMQ

1

Q, fbw,QNQ

@(PO)

J

(T)(x,p,Q).


distribution

and octet quantum

equation

(@AZ>

distribution

the following

equation

for

(204 terms.

(20b)

= I
of the path-ordered

in obtaining

mean non-Abelian

g ,(ab)

the correct

field.

6Ai and higher momentum into a form treatable

out and used for a computation

exponentials

terms in (20) involve

deviations

of i(x,p).

by Feynman diagrams

of transport

equations

the

They still

in order to be

coefficients.

a moment close our eyes for the SU(3) algebra transport

in (13, 14)

force terms due to the se'if-

The collision

have to be brought

p"

function8.

value and a fluctuating

functions:

gluon fluctuations

(redundant)

of the

terms:

(20) closes.

tet'sfor

dis-

the

with g(ab) in (20b) mean symmetrization:

Note that the introduction

and continue

for the higher color moments.

deriving

We find

- fame 6bd A: - 'ac fbmd AF % g(cd) I = pFI Fzv a~~((abjcJ

etc.

expectation

'ab - fame AC g = p" 'iv ai z(ab) + collision I %c

here proves crucial

worked

(19b)

their knowledge

for the QCD Wigner

= 0), we obtain

= p' Fiv .3; g + collision %a

consistent

.

of the classical

for-the h?gher ~ments

priori

The color algebra zan be used to express system

the Wigner-function,

the gluon field into its ensemble

The brackets

f

The reason is that quantum mechanically

exists a

the singlet

P'I

(1%)

function.

= AL(x) + e??(x)

%

g, ;a and 'L

etc. are related to g and g, via the SU(3) algebra,

part, am

p%,9

correspon-

;

i

+ e(-p,)ji,f(x.-p,9)dQ

specify

We wilf now derive a transport Splitting

functions

only the two lowest color moments

no such relationship

classical

distribution

f e(-po$b,-P.Q)dQ

g and g, completely

allows us to de?ermin$

whereas

i

6(p2-m2)

Thus, although

(16) reveals the following

counterparts:

6(p2-m2f

%

tribution

into expression

t collision

Up to the only partial symmetrization

is formally

identical

to the classical

of color indices,

analog

(2Ocl

terms,

(12).

this

hierarchy

In the quantum case the

hierarchy

is generated

by the color algebra, whereas

tion it is due to the non-Abelian

terms in (1).

This demonstrates

tance of this term and shows that the classical in a natural way to the correct QCD transport In view of this, two ways of proceeding to deduce the collision

Vlasov equation

Vlasov-Boltzmann

between both approaches

treated like a usual partial dependence

results

transport equation

fi)

equations (I) which

(2Qa,b).

equation,

whereas

twu types of distribution

Try

(20a,b) then re-

The major

is that in (i) the color dependence

differential

in considering

the impor-

(1) is related

further offer themselves:

mains to be solved; or (ii) work directly with equations difference

formula-

equation.

terms C, ?Z from the quantum

and insert them into the classical

in the classical

can be

in (ii) the color functions

(singlet

and octet),

which are coupled with each other, but contain no further color

variables.

I am not yet sure which approach will finally prove simpler.

the following

4.

section we will see an approximate

THE TWO-STREAM

the Vlasov equation

plasmas

can develop

modes are usually

found through an analysis

linearized

some given homogeneous

around

this procedure

i la method

fi),

INSTABILITY

From the theory of electrodynamic situations

treatment

In

it is known that for specific

unstable

solutions.

Such unstable

in which the Vlasov equation

background

configuration,

is

Applying

to our case, we set

(-) f (x,p,Q) = ($(,,Q) AZ(x) = AZ e-iksx;

+ (;;(p,Q)e*ik'x

(kU e (w, 1))

;

(21a

;

(21b)

where At and f (% ) are to be treated as perturbations. (21b) assumes that the 1 I background configuration should not contain a macroscopic color field, leading to the requirement (j:),(x)

"jQapp(fo-?,)dPdQ

For the purpose

= 0

(22)

.

of this chapter we will focus on the color eZe&ric

only by requiring

Ftj(x) = 0, which

now (21) into the coflisionless

in &X&X&

Vlasov equation

second order in the perturbations,

one finds

gauge yields i;" = 0, (If and neglecting

ssctor Inserting terms of

U. Heinz / Quark-Gluon Transport Theory

610~

Qa(k*p Ai aifo - po AZ k*anfo)

7, = -

&-

- i fabc P, Ai

Qa(k'p Ai ai?, - P, At k*apFo)

These expressions

can be used to eliminate

tion for A:, which

PC 3; f, ; I

f i fabc p, Ai Qc aa 7

(23)

Q oI *

fI and ?I from the Yang-Mills

equa-

in Coulomb gauge reads

.I- Q,Q,a~$l,(~.Q) I.

- i f,,de dQ

There the po-integretion

was already

Go (go) are the sum (difference) distribution Go6

functions

(24)

performed yielding

of the particle

p, = m,

and

and antiparticle

background

on mass shell:

;

(25a)

Qf 5 If,fp,Q) - ~o(~,Q)lpo=~~ .

(25b)

Q) 5 [fo(PsQ) + ?,(P,Q)I, =J" 0

so&

It is now easy to show that the last term in (24) does not contribute linearized

approximation:

from its color structure

(symmetrical

in the

in a and b) it

has to be of the form

= C+ibade+ C2dadnfnea J- QaQda&, s dQ

which

vanishes

dQ Q, 9, +

after contraction

first color integral J where

(26)

9

with fbde due to the constraint

(22).

The

is of the form

(27)

dQ QaQbGo(~,Q) = Gtifb,b + C3dabcpQ9, Go&Q) + ofA,) the second

configuration J-

o(Ao)

last term does not necessarily

is color neutral with respect

vanish.

Only if the background

to both particles

and antiparticles,

(281

dQ 9, f, =J-dQ Q, f. = 0 ,

this term vanishes,

too.

In order to be able to further analyse eq. (24) analy-

tically, we will here assume

(28).

indices and allows us to eliminate

This renders A:.

(24) diagonal

in the color

One finds the dispersion

relation

61IC

d3 c2 = +-it*?

I

P

!iT_. PO

p'

where G(c) is proportional persion

relation

(29)

G(+)

to f, + a

on mass shell

[see (25, 27)l.

is very similar

his famous discovery

to the non-relativistic 14 . of the so-called Landau damping

This dis-

one found by Landau in In general,

for real

l?, w need not be real:

w=w However,

r

(30)

+ iw.1 .

An analysis very similar to the only modes with wi 2 0 are stable. 14 reveals that unstable modes may develop plasmas

one known from electrodynamic when Gfff) is anisotropic Fig. 1 (independent

in mo~ntum

space with a double peak structure

like in

of px, py):

FIGURE 1 This is the well~known

W-stream

if in the plasma two streams instabiZity:

flow through each other, this configuration particles electric

and antiparticles, field.

all colors

through

growth

separation

increasing

of

(color-}

is only stopped when the nonlineari-

equations

its approximations

symmetrically,

against

which create an exponentially

(The exponential

ties of the Vlasov or Yang-Mills our treatment

is unstable

become important.)

(in particular

hence what we obtained

Unfortunately,

eq. (28)) has treated

is a complete Abelian

analogue

to the electrodynamic

case.

genuine color separation,

This unstable mode will therefore

but only to particle-antiparticle

this might happen in heavy ion collisions, other.

If I may speculate

nism might have something central

rapidity

not lead to

separation.

a little bit, it is even conceivable to do with the creation

region, but certainly

Still,

when the two nuclei fly through each that this mecha-

of the (baryon number poor)

a closer study is necessary

before this

can be finally decided.

ACKNOWLEDGEMENT I would Vanderbilt

like to thank the Department University,

of Physics and Astronomy

where this manuscript

was finished,

at

for their kind hos-

pitality,

REFERENCES I)

M. Gyulassy,

Signatures

2)

J. Rafelski, ference,

Strangeness

3)

G. Domokos and J. Goldman, Phys. Rev, D23 (198If 203.

4)

J. D. Bjorken,

5)

K. Kajantie, Hydrodynamics plasma, this conference.

6)

G. Baym, this conference.

7)

U. Heinz,

in:

of new phenomena, production

this conference.

in the quark-gluon

plasma,

this con-

Phys. Rev. D27 (1983) 140. and approach

Quark matter

Jacob and K. Satz

(tlorfd

Phys. Rev, Lett.

for~~i~~

Scientific,

to equilibrium

and heavy ion

Sinaapore.

in the quark-gluon

collisions,

eds. M,

1982). D. 439.

8)

8. Heinz,

9)

S. K. Wong, Nuovo Cimento 65A (1970) 689.

IO)

This was first pointed out by V. Moncrief

11)

3. M. Stewart, Non-equilibrium Notes in Physics 10 (Springer,

12)

S, R, de Groat, W, A. van Leeuwen, and Ch. G. van Wee&, Kinetic Theory (North-Ho~land~ Amsterdam, 1980).

IS)

H. Arod;, Phys.

14)

L. D. Landau, J. Phys. U.S.S.R. IO (1946) 25; R. Balescu$ Statistical Mechanics of Charged Particles (Interscience, London, 1963).

Lett.

51 (1983) 351.

(private communication).

relativistic kinetic theory, New York, 1971).

in: Lecture

Relativistic

1168 (1982) 251.