22 February 1996

Physics Letters B 369 (1996) 117- 122

PHYSICS LElTERS B

Gauge-independent thermal /3 function in Yang-Mills theory

Ken Sasaki * Dept. of Physics, New York Universi& 4 Washington Place, New York, NY 10003, USA

and Dept. of Physics, Yokohama National University. Yokohama 240, Japan

Received 18 October 1995 Editor: H. Georgi

Abstract

It is proposed to use the pinch technique (PT) to obtain the gauge-independent thermal p function in a hot Yang-Mills

gas. Calculations of the thermal /I function are performed at one-loop level in four different gauges, (i) the background

field method with an arbitrary gauge, (ii) the Feynman gauge, (iii) the Coulomb gauge, and (iv) the temporal axial gauge, and they yield the same result in all four cases.

It is important for the study of the quark-gluon plasma and/or the evolution of the early Universe to fully understand the behaviour of the effective cou- pling constant cy,(= g*/4=) in QCD at high temper- ature. The running of (Y, with the temperature T and the external momentum K(= (k]) is governed by the thermal p function PT [ 1 I. However, the previous cal- culations of PT have exposed VariOUS problems [2], a serious one of which is that the results are gauge- fixing dependent [ 3 1,

The background field method (BFM) has been ap- plied to the calculation of PT at one-loop [2,4,5]. First introduced by Dewitt [ 61, BFM is a technique for quantizing gauge field theories while retaining ex- plicit gauge invariance for the background fields. Since the Green’s functions constructed by BFM manifestly maintain gauge invariance, they obey the naive QED-

like Ward identities. As a result, the spatial part of the three-gluon vertex, for static and symmetric ex- ternal momenta, is related to the transverse function

’ E-mail address: sasaki@ed.ynu.ac.jp.

srsaki@mafalda.physics.nyu.edu or

IIr(T, ko = 0, K = ]k]) of the polarization tensor IIPV, and thus PT is obtained in BFM from [ 21

MT, io ,& ES TdT = -&T~‘$’ K). (1)

Due to the O(3) invariance, the spatial part of the gluon plarization tensor II, is expressed as follows:

H,,(k) = nT(sij - k’k’) + I-& k* k*

(2)

and IIr can be extracted by applying the projection operator

pi = ; (8, _ k’k,) k*

t0 II,.

The thermal p function has been calculated in BFM at one-loop level for the cases of the gauge parameter [Q = 0 [4], &Q = 1 [ 51 and (Q =an arbitrary num- ber [ 2,7]. The results are expressed in a form,

0370-2693/96/$12.00 @ 1996 Elsevier Science B.V. All rights reserved

SSDIO370-2693(95)01522-l

118 K. Sasaki/Physics Letters B 369 (1996) 117-122

Fig. I. The self-energy diagram for the quark-quark scattering.

(4)

where N is the number of colors. Contrary to the case of the QCD p function at zero temperature, j3FFM is dependent on the gauge-parameter 5~. The reason why we have obtained [Q-dependent PT in BFM is that the contributions to j$ come from the finite part of the gluon polarization tensor IIr,, and that BFM gives [Q-dependent finite part for H,, [ 8,9].

In this paper I propose to use the pinch technique (PT) to obtain the gauge-independent PT in a hot Yang-Mills gas. The PT was proposed some time ago by Cornwall [ lo] for an algorithm to form new gauge- independent proper vertices and new propagators with gauge-independent self-energies. First it was used to obtain the one-loop gauge-independent effective gluon self-energy and vertices in QCD [ 111 and then it has been applied to the standard model [ 121.

For example, let us consider the S-matrix element T for the elastic quark-quark scattering at one-loop or- der. Besides the self-energy diagram in Fig. 1, the ver- tex diagrams of the first kind and the second kind, and the box diagrams, which are shown in Fig. 2a, Fig. 3a, and Fig. 4a, respectively, contribute to T. Such con- tributions are, in general, gauge-dependent while the sum is gauge-independent. Then we can extract the “pinch parts” of the vertex and box diagrams, which are dipicted in Fig. 2b, Fig. 3b, and Fig. 4b. They emerge when a rp matrix on the quark line is con-

tracted with a four-momentum kp offered by a gluon propagator or an elementary three-gluon vertex. Such a term triggers an elementary Ward identity of the form

g=(#+It-m) - (#-m). (5)

The first term removes (pinches out) the internal fermion propagator, whereas the second term vanishes on shell, or vice versa. This leads to contributions to T with one less fermion propagator and, hence, these contributions are called “pinch parts”. The contribu- tion of the self-energy diagram, when added pinch parts from the vertex and box diagrams, become gauge-independent. In this way we can construct the gauge-independent effective gluon polarization tensor (self-energy).

As in the case of BFM, the effective gluon polar- ization tensor and vertices constructed by PT obey the naive QED-like Ward identities. Thus we can use the same Eq. ( 1) to obtain PT in the framework of PT. More importantly, PT gives the gauge-independent re- sults up to the finite terms, since they are constructed from s-matrix. It was shown recently [ 131 that BFM with the gauge parameter 5~ = 1 reproduces the PT results at one-loop order. However, for 5~ $1, this co- incidence does not hold any more. In fact, BFM gives at one-loop order the gluon polarization tensor whose finite part is SQ-dependent. Interestingly enough, Pa- pavassiliou [ 81 showed that when PT is applied to BFM for 5~ # 1 to construct the effective gluon po-

larization tensor, the gauge dependence of the finite part disappears and the previous 6~ = 1 result (or the universal PT result) is obtained.

To my knowledge, there exists, so far, only one approach, i.e. FT, which gives the gauge-independent gluon polarization tensor up tofinite terms. And these finite terms give contributions to PT. This notion in- spires the use of PT for the calculations of PT. In the

Y I

(4 (b)

Fig. 2. (a) The vertex diagrams of the first kind for the quark-quark scattering. (b) Their pinch contribution.

K. Sasaki/Physics Letters B 369 (19%) 117-122 119

(4 (b)

Fig. 3. (a) The venex diagram of the second kind for the quark-quark scattering. (b) Its pinch contribution.

following 1 will show that we obtain the same & in the framework of PT even when we calculate in four different gauges, (i) the background field method with an arbitrary gauge, (ii) the Feynman gauge, (iii) the Coulomb gauge, and (iv) the temporal axial gauge.

(i) The backgroundfield method. In the background field method with an arbitrary gauge, the gluon propa- gator, iDr&,,, = -i&hD$&,,,, and the three-gluon

vertex with one background gluon field, ?$,, are given, respectively, as follows [ 141:

DC”” (BFM) d”-<I -,,,y,

and

ti;v(p, k, q)

=gfboc [ (1 - +‘:&pAq) +r:Jp,k,q) , 1 (7)

where

I$p,, (P, kv q) = pAg,v - qvgl,.

~!JP, k-q) = 2kAg,v - 2k,gA, - (2~ + k),gA,.

(8)

In the vertex, k, is taken to be the momentum of the background field and each momentum flows inward and, thus, p + k + q = 0.

A one-loop calculation of the polarization tensor was performed in Ref. [ 73, from which the transverse function II?“’ (ko = 0, K = lkl ) in the static limit can be extracted for K << T as follows:

lTIcBFM) (T, K) T

+ o(K2c2>. (9)

Using this expression for nT in Eq. ( 1 ), Elmfors and Kobes obtained Eq. (4) for @sFM which is indeed gauge-parameter 6~ dependent [ 21.

Now we evaluate the pinch contributions to &. We consider the quark-quark scattering at one-loop order in the Minkowski space. We use the gluon propagator and the three-gluon vertex given in Eqs. (6) -( 8). The pinch contributions come from the vertex diagrams of the first kind [Fig. 2b and its mirror graph], the vertex diagrams of the second kind [Fig. 3b and its mirror graph] and the box-diagrams [Fig. 4b]. We can extract from them the pinch contribution to the polarization tensor, which is expressed as [ 81

in;;,FM, = N2(1 -&)k2 J

d4p -2kp -_---_d (277)4 P492

N2 + zg (I- iW2k4 J

d4p -p’p”

m=’ (10)

where it is understood that the loop variables are re- lated by k + p + q = 0.

When we turn to the imaginary time finite tem- perature formulation, we replace the integral in the Minkowski space with the following one:

(4 (b) Fig. 4. ( f I The box diagrams for the quark-quark scattering. (b) Their pinch contribution.

K. Sasaki/Physics Letters B 369 (1996) 117-122

(11) II$FG’(T,~) = N~‘KT$ + O(K~). (18)

where the summation goes over the n in pc = 2?rinT. Applying the projection operator fij to the spatial part of IIr;aro)(ka = 0, K = )k)), we obtain in the limit K << T,

The pinch contribution to the polarization tensor in FG is very simple. Since the gluon propagator in FG does not have a kpk” term, the only contribu- tion is coming from the vertex diagram of the second kind with the three-gluon vertex TP (and its mirror graph) [ II], and it is given by

$(BFM)(r K) = Ng2( 1 - &)K2

- ;g2( 1 - c&)~K~

&&j/&2 s d4p_!- P’.

(2,rr>4 p%p (19)

= Ng2~T $(I -5~) - {

$1 -5a)2 >

+o(K2). (13)

Adding two contributions we tind that the sum

&(T,K) =~~m’(~K) +n,PtBFM’(T,K)

= Ng2~‘l’i +O(K’) (14)

is gauge-parameter 5~ independent, and this gives a gauge-independent thermmal fi function:

Following the same procedure as we did in (i), we obtain for the pinch contribution to IIFG as

pr = g3N&;. (15)

Note that the result coincides with pFFM in Eq. (4) with te = 1 [5].

(ii) The Feynman gauge (the cnvuriant gauge with 5 = 1) . In the Feynman gauge (FG) the gluon prop- agator, iDfl,“,,, = -i&,,,D$&), is in a very simple form:

D;&, = $g?

The three-gluon vertex is expressed as

I’$& k, 9)

(16)

= gfh"" [ ~j;,,(p,kq) +r:JpAq) 9 1 (17)

where I’IPY (p. k, 9) and r&,(p, k, 9) are given in Eq. (8). From the one-loop polarization tensor in FG, we extract the contribution to the transverse function llkFG) and find in the limit K < T,

n,PtFG’(T,K) = h’g’KT$ + o(K2). (20)

Again when we add two contributions, IIkFG) and IIFcFG), we obtain the same IIT in Eq. ( 14) and, thus, the same fir in Eq. (15).

(iii) The Coulomb gauge. In the frame of a unit vector np = ( 1, 0, 0, 0), the gluon propagator in the Coulomb gauge (CG), iDz&,, = -iSobD~~Gj, is de- fined by

DPV (CG) I ,

where

P*“(k) = gp" - #nv

kfik” - (nNk’+kc”n”)k~+n@n’k~ I

, (22)

kfik”k; -- k2k2 ’

(23)

(Y’(k) = - fiko

L(n’k” + Un”) + wk@k”, JZlkl

(24)

kfik” D”“(k) = -.

k2 (25)

The three-gluon vertex is the same as in FG, that is, I’$(p,k,q) in Eq. (17).

K. Sasaki/Physics Letters B 369 (1996) 117-122 121

The transverse function IIr is related to the polar- ization tensor as

ITIii - -l_kiIIijkj

k= I G (26)

Since kilIi,ikj = 0 in the static limit ke = 0, we have ~I~(T,K) = illii(ko = 0, K). The IIii for general kc and K was evaluated in CG and the temporal axial gauge in Ref. [ 1.51. Using the expression of Eq. (4.38) in Ref. [ 151 for IIE”( kc, K), we can calculate the static limit of IIF and obtain

II$cG’ (T, K) = @o’(o, K)

= N2KT& + U( K2). (27)

It is straightforward to calculate the pinch contribu- tions to the polarization tensor in CG. Again we con- sider the quark-quark scattering at one-loop order. We discard the terms which are proportional to $ and It, where k, is the momentum transfered in the scatter- ing, and we make use of the Dirac equations satisfied by the external quark fields, such as, (i - m)u(p) = 0 and ii(p) (4 - m) = 0. The pinch contribution to the polarization tensor in CG is expressed as

i%‘;,ci,

= Ng2k2 1 4k.p @y

p’42p2-p2p2-p2q2p2 g 1 + Gg2k2

J A+

2k2

p2 p2d * 1 (28) Applying the projection operator tij to the spatial

part II&co,, we obtain in the static limit and for K <

T

@.(CG) (T, K)

= -Ng2~2

-TV N = =Jpgl-y$]

{ k2 4

X ---

p2q2q2 p2q2

= Ng’tcT; +o(K2).

(29)

(30)

Adding the two contributions, IIkCG) and I’$?‘, we find that the sum is equal to IIr in Eq. (14) and, thus, we obtain the same pr in Eq. ( 15).

(iv) The temporal axial gauge. The gluon propa- gator in the temporal axial gauge (TAG), iD$&AGj =

-iSo~D&Gj, is defined by

D$” P””

(TAG) = k2

+ $ Ikl Cp” _ !!$,,v Q"'+ ~zk 1 k; ’ (31)

0

where Pfi”, Qkv, CCL” and Dfiv are given in Eqs. (22)- (25). The three-gluon vertex is given by Y$ (p, k, q)

in Eq. (17). The static limit of IIjiTAG’ was calucu- lated in Ref. [ 151. Using the result of Eq. (4.44) in

Ref. [15],wefindfor~<T

nT ‘TAG’(T,K) = ~n~~G’(O,K)

= N?KT$ +C>(K~). (32)

Following the same procedure as we extracted the pinch parts from the one-loop quark-quark scattering diagrams in CG, we obtain for the pinch contribution to the polarization tensor in TAG,

2 -I-

4k.p --- q2p; P2q2P02 1 d

4+-. 2k2

pi p;q; 1 (33)

Then in the static limit, @(TAG) is expressed as

IIF’TAG’(T, K) = -N~=K~

x J$TT{~;~:;;~+&-&} _ ;g=K=/$T~[p=_ y]

k2 X --

P2q2P;q; (34)

122 K. Sasaki/Physics Leiters B 369 (19%) 117-122

Due to the 1 /p,’ and 1 /qi terms coming from the TAG propagator, the above integrand contains a k2/p2 sin- gularity at the lower limit of the integration. This sin- gularity is circumvented by the principal value pre- scription [ 15,161. The result is for K < T

I?;(TAC)(Q) =N~*KT;+B(K*). (351

Again the sum of HAAG’ and @TAG) coincides with Hr in Eq. (14) and yields the same Pr in Eq. (15).

I have demonstrated the calculation of the thermal p function j3r in four different gauges, that is, in BFM with an arbitrary gauge, in FG, in CG, and in TAG. When the pinch contributions are taken care of, the same result PT = g3 N$ f was obtained at one-loop order in all four cases. More details will be reported elsewhere [ 171. However, this is not the end of the story. Elmfors and Kobes pointed out [2] that the leading contribution to j3r. which gives a term T/K,

does not come from the hard part of the loop integral, responsible for a T*/K* term, but from soft loop inte- gral. Hence they emphasized that it is not consistent to stop the calculation at one-loop order for soft internal momenta and that the resummed propagator and ver- tices [ 181 must be used to get the complete leading contribution. Since the corrections to the bare propa- gator and vertices, which come from the hard thermal loops, are gauge-independent and satisfy simple Ward identities [ 181, it is well-expected that we will obtain the gauge-independent thermal /3 function even when we use the resummed propagator and vertices in the framework of PT. Study along this direction is under way.

I would like to thank Rob Pisarski for the hospitality extended to me at Brookhaven National Laboratory in the summer of 1995. I was inspired to start this work by the “hot” atmosphere created at the laboratory. I would also like to thank A. Sirlin for the hospitality extended to me at New York University where this work was completed. Finally I am very grateful to J. Papavassiliou for helpful discussions. This work is partially supported by Yokohama National University Foundation.

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