Photon interactions with quark-gluon plasma at finite temperature

Nuclear Physics B (Proc. Suppl.) 15 (1990) 215-224 North-Holland

PHOTON

INTERACTIONS

WITH

215

QUARK-GLUON

PLASMA

AT FINITE

TEMPERATURE

J. C L E Y M A NS Department

of Physics,

University

of Capetown,

7700 Rondebosch,

Cape,

South Africa

I. DADIC Ruder Bo~kovid

Institute,

YU-41OO1

Zagreb,

Yugoslavia

Photon interactions w i t h q u a r k - g l u o n plasma at finite temperatures and densities are studied to first order in th~ strong coupling constant a . The real gluon s . p r o c e s s e s are c a l c u l a t e d and c l a s s i f i e d according to the crosslng symmetry. The s i n g u l a r i t y structure is classified and it is shown how the singularities cancel w h e n under the c o n d i t i o n of deta i l e d balance. For this, it is necessary to include all the processes that occur in p l a s m a in first order in a . s processes

i. I N T R O D U C T I O N The e x p e r i m e n t a l relativistic servation

programme

ion c o l l i s i o n s

of phase

ical simulations

in numer-

of lattice QCD have mo-

tivated

renewed

field theory

renewed

and the ob-

transitions

quantum

temperature

on ultra-

interest

in relativ i s t i c

(QFT)

and chemical

interest

at finite

potential.

luckily

This

and densities, form simple

which

Here we shall deep

inelastic

quark-gluon duction duction bath

enables

(but tedious)

of t i m e - d e p e n d e n t

one to per-

"hard gluon

scattering

pro-

plasma 6. The pro-

at rest in a heat-

Pire and Schiff 7 and by Altherr A u r a n c h e 8. Using a similar

loops with soft external mo-

In this paper we follow the notation of Ref.

5. To regularize

by Baier,

is organized

2 the crossing

real gluon processes sect.

0920-5632/90/$03.50 © (North-Ilolland)

This work was

malization

procedure

expressions tribution

are given.

is discussed

In sect.

principle

1990 - Elsevier Science Publishers B.V.

by KFA J~lich.

In

renor-

for the v i r t u a l - g l u o n

singularity-free

supported

symmetry of

is established.

cancel all singularities

formalism,

as follows.

3 the finite-temperature

tailed-balance

and

have been treated by sev-

by I. Dadid.

infrared di-

vergences, we introduce a small gluon 2 mass m w h i c h is allowed to go to zero

eral groups: primordial nucleosynthe9-11 12 sis , decay of scalar particles , *Presented

that the

(which could eventually contrib19 ute to the same order in a ) are abs sorbed in masses to this order in as.]

In sect.

of leptons off

frame has been c a l c u l a t e d

is

menta"

The paper

to

plasma 4'5 and dilepton

the p r o c e s s e s

justified by the evidence

in plasma.

limit our interest

of dileptons

treatment of the present calculation

at the end of calculation.

calculations

processes

in q u a r k - g l u o n

tron a n n i h i l a t i o n 14. [The perturbative

coincides w i t h

the d e v e l o p m e n t of the real time formu1-3 lation of QFT at finite temperatures

in ¢3 QFT 13, electron-posi-

and con-

4 the de-

is used to and calculate

final results.

216

J'. Cle3"maas,L Dadi6/ Plioto. iateractio.s

2. THE CROSSING SYMMETRY

(REAL GLUON

PROCESSES) The processes contributing to either deep inelastic scattering

(q2
k

i

lepton production

(q2>O) are shown in

k'

(a)

Fig. 1, where the time direction for dilepton production is opposite to that for deep inelastic scattering. 7

g

~( ~ a )

k

q

g

q - 7= \

--q

q

k

k'

k

q k

k'

(b)

g - ~

_/

7"

(b)

k'

.:

FIGURE 2 Diagrams contributing to deep inelastic scattering and dilepton production.

(d)

7"

q convention

is such that all the energies

are positive when the diagram in Fig. 2a corresponds

to the process

in Fig. la,

and the energy of a particle is consid-

(el

g

(f)

ered to be negative when it changes from the left to the right side in Fig. la or vice versa.

Defining the z axis

as the axis of the photon momentum, can define

(g)

(hi

q

q = z -

q~ll

we

for real gluon processes:

--q

(j)

FIGURE I Processing contributing to first order in u s . The diagrams to be calculated are shown in Fig. 2. It should be noted that only crossed vertices in self-energy contributions could be of type I and 2, the others are always of type I. Our

=

(v,O,O,q z), cos

e

= ~/(1~[

(2.1)

q,)~ (koPo),

(2.2)

z, = cos e, = ~,"q/(1~'1 q,)~ (kop o) , (2.3) s -- (q+k) 2= _ 2 ~ l q z(zC (k'oPo )-d) ,

(2 •4)

t - (q-k')2= 21~'hz(Z'¢(koPo)-d'),

(2.5)

d =

2k v + q o

2 ,

(2.6)

.

(2.7)

21~1%

d' = 2k° - q

2

21~'1% The singularity

structure is deter-

J. Cleymans, L Dadi~/ Photon interactions mined by the properties of the phase-

where z U ,L are

space integral

d3~¢ d~=

d3~: '

......

.,_d3p

(1~'1 +_~(koPo)I~'l ) 2- IE 12-q 2 Z e (WOpO) • 2l'klqz

2U,L =

,

(2,)321ko I (2,,)321ko I (2,,)321Po i × (2~) 4 6(4) (q+k-k'-p).

(2.8)

The functions to be integrated over depend on s, t and the energies appearing in the statistical weights.

Thus,

217

(2.15a1

~l~l~¢(koPo)l~I)2-I~'I 2 2 Z!

-qz¢ (koPo) .

--

21~:'1 qz

u,L-

12.15b}

The quantities ZU,L and, similarly, Z ~ L have a very simple geometrical meaning,

except for two energies and two angles

as the limiting values of angles under

the other variables could be integrated

the condition of conservation of momen-

over with the resultS:

tum i~+~ I - ~

dp="

i 4 dkodPo d £, 8(2~)

d R = 2dzdz' [-y(z,z',cos ~)]

( 2.9 ) -1/2

,

(2.101

or ~i.

o~ i ~' +~1

-I1~1_+1~11 • Now one can t r y

t o see w h a t h a p p e n s

at given energies (ko,Po). First of all, the integration region will be non-zero only if the following inequalities are

where

fulfilled: 7(x,y,z)

(2.11)

= x2+y2+z2-2xyz-i

zU > -i,

and COS

=

~'

~(ko k'o )=[ ko2+k2_ I~12+ qz2

ZL_< i,

(2.16)

z& _> -1,

+2I ~ Iqzz ~(k;po1-21E'I q.z'~ (koPo)]~(k ok',o /(21~11~'1).

~£< 1. Z'

(~.12)

The boundary of the angular integration is determined by the condition

!

J z~

7(z,z',cos a)

< O,

(2.13)

of course, I zl<1, Iz'i<_z. ~he integration region is the region in the

where,

->.Z

ZH

Z~

pm

centre of Fig. 3. The straight lines

I

I I

z~

limiting the boundary are given by ZM = min(l,Zu),

(2.14a)

Zm = max(_l,ZL ),

(2.14b)

z~4 = min(l,z~),

(2.14c) FIGURE

' =

zm

max(-l,z~),

(2.14d)

3

The boundary of the region where Y < O.

218

J. Cleymans, L Dadi4 /Photon interactions

Next set of inequalities determines the boundary between different real gluon

\ T4\ ' \

processes contributing to the total rate

R8

is determined by

-~ "^

/ , ./LS ~ 8

ko ~ O, k'o ~ O,

(2.17)

~

-

"

'\/C2 \\ . . . . .~. . .

/ ET\ /

,' /I

', H8

Po ~ O,

I6

,z- _ _ -v /]

v ~qz,

in agreement with our convention. Finally, the analytic form of the contribution will be classified according to the particular choice of solutions of (2.14a-d). For convenienc~ of the cancellation of collinear singularities, instead of the energy variables (ko,Po) we use more appropriate variables (K,Po), where K = k o + (v - po)/2.

(2.18)

Fcr real photon processes, K is simply the average of two quark energies

/N5 / q-,+--÷-+ q 9 7* /.--.4~'

M8 \

•

04 \ \

FIGURE 5 Crossing symmetry in dilepton production

(K = (ko + k o' ) / 2 ) The classification of contributions, with all the inequalities taken into ac-

In this classification,

count, is represented graphically in

the meaning given in Table i.

the Latin letters

indicate the region, and the numbers label the class of the contribution with

Fig. 4 for deep inelastic scattering and in Fig. 5 for dilepton production.

u7 ,, ,

h7 F2

Z

/

Label

(ZM,Zm,Z~,Z~)

Fig.4

Fig.5

1

(Zu,ZL,z~,z~) A,I,T,W

A,V

2

(Zu,ZL,I,-I)

D,F,P,Q

C,U

3

(Zu,-l,z~,-l)

B,G,O,J,S,Z

G

4

(Zu,-I,I,z ~)

C,R

B,O,T

5

(I,ZL,Z~,-I)

K,V

F,L,N

6

(I,ZL,I,z ~)

M,E

D,I,S

7 8

(l,-l,z~,z L') (i,-i,i,-I)

H,L,N,U

E,K HI~,R

TABLE 1 Classes of contributions related by crossing symmetry In Fig. 5, region G3 is absent because of qz
For qz>V/2, Fig. 5 is slightregion H3 disappears and a

region classified as G3 appears

219

J. Cleymans, !o Dadi~ / Photon interactions in the form of a triangle

turned upsideMZ = _812(q2+m2)2 l___+stt--s+ _t _ 2(q2+m 2)

down.

s

It is evident that the crossing metry is valid only w i t h i n belonging elastic

tion together!}. antisymmetry

2 (q2+m2)

the regions

to the same class

scattering

sym-

(deep in-

and dilepton produc-

In addition,

2q2[ ~11 - m I (k'-q)

t

s

12

12 ].

+ I ~11 (k+q)

(2.2!)

there is

related to the identity of

This expression

is identical to the cor-

quarks which joins pairs of classes 2

responding one at zero temperature ex-

and 7, 4 and 5.

cept for the ~11 terms containing the

To discuss the singularities, observe

let us

that

%

m

62 terms multiplied by the

statistical

factors at finite temperature

and chemical potential

2 Po+ko s (z U)

ill-defined

Po

(2.19)

'

For definiteness, the angular

t(z~) %. m 2 P°-k° Po From

•

(2.201

(2.201

and Table 1

we can directly read the collinear-sinstructure:

taining either

singularity

integrals over the most sin-

gular term 1/(st). tions : A = 2K + Po + v,

(2.22a)

a = 2K + Po - u ,

(2.22b)

the integrals con-

s

or

t

in the denomi-

nator will be singular whenever respectively,

we shall present

First, we introduce a few abbrevia-

(2.19) and

gularity

(Ref. 5).

z~ is chosen.

appears

z U or,

Thus,

in classes

4, while the t singularity

the s

Po + ~'

(2.22c)

b = 2K - Po - u'

(2.22d)

B

=

2K

-

l, 2, 3,

appears

classes i, 3, 5, 7. Furthermore, the ill-defined

in

62 ex-

|

C = 2K + PO + qze(kokoPo }'

(2.22e)

c = 2K + P o -

(2.22f)

qze(kokoPo }'

D = 2K - Po + qze(kokoPo)'

(2.22g)

' )" d = 2K - Po - qz e( k okoPo

(2.22h)

pressions will appear whenever either S(ZM)S(Zm)
i.e.

03 and P2 in Fig.

in regions G3, F2, 4

5, or t(z~)t(z~)
and U2, T4 in Fig. i.e.

H7, 03 and N7 in Fig. in Fig.

in regions G3,

4 and K7 and L5

5.

for eight regions corresponding different classes.

Thus,

to eight

in the limit of

small m 2 we obtain

Infrared divergences

appear when

Po÷O. Again, we have the s and t divergence.

It is enough to write down the integrals

e (koko)I

(i/(st) = e (kok o) I i/(st)d~

In comparison with the zero-tem-

perature

case,

the divergence

is strong-

er by one power of p~i owing to the Bose distribution After

summation over virtual photon

polarizations, lated is

of gluons.

the trace to be calcu-

:

-~

in IA 41,

lqzpoq21

(2.23)

where A 4 takes the value~ given in Table 2 (see Refs.

4-6 for details}:

.I. Cleymans, I. Dadi~/ Photon interactions

220

Value of A 4

TABLE 2

Class

16 p4q4/(m4AaBb)

tions, while

it contributes

1

4 po2 q 2 B/(m2ADd)

2

16 po4q4/(m4AbcD)

3

4 po2q 2 c/(m2Aad)

of the vertices

could also be of type 2.

The terms with one delta function are as B, F, F', depending on !

5

this delta function refers to

the boson

6

4 po2q2a/(m2bCc)

line, the fermion line or the

fermion primed line. The self-energy will generally be a

7

2x2 matrix, cDI

(Cd)

only to the ill-defined In the latter case, one

whether aB/(Cd)

corrections

62 expressions.

indicated

4

4 po2q2D/(m2BbC)

for self-energy

8

diagonal

the real part of it being

to the order calculated,

and

we may follow the procedure of Donoghue, Holstein

and Robinett (see Refs.

The procedure The ill-defined

62 expressions

obey this crossing-symmetry

do not

rule.

To obtain the final result, we have to multiply the expression

(2.23) by

a statistical

factor and integrate over

two energies.

The statistical

factor

12, 15).

is based on the T=O re-

normalization as an intermediate step. 11 For this, E is broken into the zerotemperature

and finite-temperature

parts: 11 + E 11 = E o

11

E8

(3.1)

•

for y + q ÷ q'+g is The zero-temperature nF(Xko ) (l-nF(Xko})

(2.24)

(i+nB([P°l) '

wave-function where Xko = I k o ~ -

part is decom-

posed into the contribution

renormalization

and the mass renormalization ~¢(k O)

12.25)

- mo

_

to the constant as

Re Zo11 _~ z-l(M_mo_6mo} o '

(3 "2)

and similarly for the other processes where the

with a real gluon.

zol- terms cancel out the

terms in 6.3, while the 6 m o terms can3. VIRTUAL-GLUON

cel with the mass counter terms. The

CORRECTIONS

To this order, virtual-gluon butions correspond

contri-

to the interference

term between the lowest-order

diagram

(Mo) and the sum of vertex and selfenergy corrections

finite-temperature

part is decomposed

as

Esll(k) = B(k)+C(k) (){-mph(l~l))+l~(k) (3.3)

(M1}, as shown in

Fig. 2b. The terms with zero and two

where B, C and D are some known func-

delta functions appearing on internal

tions and mph(k)

lines are pure imaginary and therefore

finite temperature.

do not contribute

tails, we merely state the results:

to this order.

term witn three delta functions nematically

forbidden

The is ki-

in vertex correc-

m ph 2 (I~1)

=

+

is the fermion mass at Omitting

the de-

M ÷2MR

2k.O( o, I 1),

(3.4)

3. Cleymaas, I. Dsdi6 / Photon interactions

where M R is the renormalized mass at zero temperature, and T o is the energy satisfying the mass-shell condition -2 ko -

l£12 2k-D

221

: 2 Re M o M o [ Z-B 1 / 2 ( k ) + z ~-I / 2 ( k ' ) - 2 ] , - (3.9) .

-

-

Eo, l

2MRB(Eo, l)

= o.

(3.S)

As shown by Weldon 16, mph will be different from zero even in the case when MR=O. The renormalization constant is Z-l(k) = Z: IZ6 l(k) and Z -1 6 (k) = [ 1-C (k)

DO ( k )

+

+

~oo

)

¢'~'[MRB (k) +MRB (k') +k' * D (k) +k- D (k,) t

- 2¢ "k' ¢'Re D(k)-2¢ *k ¢*Re D(k')}. (3.10) Adding all these corrections, one obtains for the sum over polarizations:

k o ~) ~k---~[MRB (k) +k-D (k) ]

Do(k) + ko ]mass " shell

(3.6)

The counter terms consist of scalar parts and parts including 7 matrices:

2 Re MoM 1 vertex+self-enerw1+corr. 2 = q $ d4p 1 , (21)3 k--~o{-2q4+2q2[lp-k) + (p-k') 2-2p2]+2[p2 (p-k) 2+p2 (p-k') 2 -(p-k) 2(p'-k)2-p2(k'-p)

£11-c "t = Re ( ZB (k)-I) ( M - M R- Z11 ( k ) ) + i Im

DO ( k ' )

MFT S : R e { MoMo( ko

I)

Z11,

(3.7)

where the real parts refer to scalar quantities and coefficients of ¥ matrices. The imaginary parts 17'18 give the

(-

q2 _ koV 2q: 2 ) qz

ko ~..+q 2 [ ko -1)(k'-p)2 _p2(k_p) 2(_ q 2 + --~U 2q: qz PO -1) (k-p) 2+p2 ] - ~ (ko+k O ) + (k-'~ qz

contribution containing the product of 6 functions (MR=O case only):

x (ko+ko-2Po)p2}A

Re(i~ 11 Im ~11 + i~12 Im Z21)

where by Awe have denoted the expression = AB + aF + AF' '

= ~6 (k 2) [(1-2nF(x k) ) Im Z11 6(Xk0+,)/2

- 2n F(x k)¢(k O)e

Im

,

£21],

(3.8)

the other 6 being in the phase-space integration. There are further corrections to the lowest-order diagram coming from renormalization of the fermion wave function (WFR) and use of the finitetemperature spinors (FTS}:

1+2n B ([PoJ)

AB =

aF

AF'

2 2 6(p2-m2)' (k-p) (k'-p}

(3.11)

(3.12)

(3.13a)

=

l-2nF (Xko-Po) 2 6 ((k-P)), (k' -p) 2 (p2_m2)

(3-13b)

=

l-2nF (Xk~-P0) 2 6 ((k'-p)), (k-p)2 (p2_m2)

(3-13c)

In the integrals related to the B con-

J. CJeyma,s, L DadJ6 / Photon interactions

222

tribution we shall refer to s B, t B,

and the factors

where

tions o f

sB

included

in the defini-

AB, AF, A F , , c o r r e s p o n d i n g l y ,

2 =

(k'

-

p)

,

t B = (k - p)2.

13.14)

4. C A N C E L L A T I O N

OF SINGULARITIES

Now we have to discuss Similarly,

singularities

we shall refer to s F, for

the F' contribution contribution,

separately.

UV infinities

and t F for the F

are identical

those at zero temperature

where

od is the same. p2

m2

[SF,

-

for F'

=

.

tF

(3.15)

for F

methods sect.

is performed

analogous

using

2, but the results

in

separate

zero and non-zero

= lqzpoq21~ in A B,

~3.16)

(3.17)

counter

IF(

AF

W

2) tFlk'-p) =

2

=

2

(3.19)

4q Pob/(m2aDd)

W 1 I F, ( 2 ) = lqzpoq21 c (po B) in AF, , SF, (k-p) (3.20)

A F, = 4q2p2B/(m2ADd).

(3.21)

the result is

that appear

singularities

stead,

(i.e., those

as m2÷O at finite po } are to cancel,

summation

the straight

of real- and virtual-

gluon contributions (3.18)

will not work.

contributions

(3.16)-(3.21)

define

the

in the regions which,

ance identities

(Refs.

4-8}:

nB(x-y)[ nF(x)-nF(Y)] +nF(X)[l-nF(Y)] = O, x >y>O,

(4. i)

n B (x-y)[ n F (x) +n_ (-y)-l] +n F (x} n_ (-y)= O, F F

The c a n c e l l a t i o n terms:

the

gluon processes,

gluon processes,

gluon boson

gion of classes 4 and 5. Outside tribution obtained (3.11).

with the re-

1, 2, 3 and 4 in Figs. this region the con-

is zero.

Similar

results

for the other contributions The contribution

by the statistical

in

is m u l t i p l i e d

factor

nF(x k )[i - nF(x k +~)] o o

There

primed

three

to real

and the third

either

fermion or

contribution.

tions that cancel to the scheme ty Po
includes

are four groups of contribuseparately

in Fig.

to s -1 singularity (3.22)

(4.2)

the second to virtual-

contribution

to the virtual-gluon fermion

are

scheme

first corresponds

when put on the same plot with real coincide

In-

we have to use the detailed-bal-

x>O>y. Expressions

and in-

UV finite.

forward

¢(po b) in A F,

lqzpoq21

term, wave and finite-

over the angles,

more c o m p l i c a t e d i

temperature

spinor contributions

Collinear = 16 p4q4/(m4AaBb),

we do not

but when we add the ver-

renormalization

temperature tegrate

IB(s~~)

procedure:

self-energy,

function

are simpler:

and the meth-

different

tex,

to those applied

with

Here we perform a

slightly

contributions, The integration

each type of

6 and correspond

Po>O,

t -I singularity

singula-"ity Po
according

s -1 singulariPo>O and t -I

Together

with the

J. Cleyman$, L Dadi~/ Photon interactions

223

given in Table 3.

po~ 0

TABLE 3 Value of A4

Class

1

1

aB d--6

2

aB c--6

3

J J

c

po,0

D C

3 PARTICLE

4 PARTICLE

4

F

8

FIGURE 6 Scheme of cancellations of various contributions to collinear singularities

--

5

aB c-5

6

aB c-~

7

cD d--C

8

in m 2 terms, additional kinematical factors could be cancelled.

For example,

one can introduce 422 Po q

classes 1,2,3,4

m2AB =

As

(4.3)

1

otherwise

A B = I,

(4.6a)

_ b2 AF = d-D '

(4.6b)

B2 AF' = d-~ '

(4.6c)

It is evident that all collinear divergences are explicitly cancelled.

In ad-

dition, the limit Po+O is safe. SimilarI 4p~q2 m2ab

classes 1,3,5,7

At =

ly, the cancellation is performed for (4.4)

1

the other contributions to the process. Finally, the ill-defined 62 expres-

otherwise

sions will be cancelled by the prescripto obtain the singularity-free contribu-

tion that E in the propagators should be

tions

kept finite until the end of calculation. After summing carefully the real-

A4 = A4/(AsAt)'

(4.5a)

A4 = AB/(AsAt)'

(4.5b)

AF = AF/At '

(4.5c)

AF' = AF'/As "

(4.5d)

gluon contribution and the self-energy contribution, where in one of the vertices one has to take both field i and

The results of these cancellations are

field 2, one can perform angular inte-i gration; then the c contribution cancels. The final result is free from any

J. Cleymans, I. Dadi6 / Photon interactions

224

ambiguity and could be safely integrated over the plane

(K,Po).

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