Structure of the Gluon Propagator at Finite Temperature

Annals of Physics 271, 141156 (1999) Article ID aphy.1998.5873, available online at http:www.idealibrary.com on

Structure of the Gluon Propagator at Finite Temperature H. Arthur Weldon Department of Physics, West Virginia University, Morgantown, West Virginia 26506-6315 Received July 21, 1998

The thermal self-energy of gluons generally depends on four Lorentz-invariant functions when the gauge-fixing condition is rotationally invariant. Only two of these occur in the hard thermal loop approximation of Braaten and Pisarski because of the abelian Ward identity +& K + 6 +& htl =0. For the exact self-energy it is known that K + 6 {0 and that the SlavnovTaylor identity requires a non-linear relation among three of the Lorentz-invariant self-energy functions: (6 C ) 2 =(K 2 &6 L ) 6 D . This relation simplifies the exact gluon propagator to a form containing only two types of poles: one that determines the behavior of transverse electric and magnetic gluons and one that controls the longitudinally polarized electric gluons. It agrees with known cases in the literature.  1999 Academic Press

I. INTRODUCTION A. Background The constraint imposed by local gauge invariance on the various Green functions of gauge theories is contained in the SlavnovTaylor identities [1, 2]. These identities were originally formulated in covariant gauge, which has a gauge-fixing term Lgf =&( + A + ) 22!. In covariant gauge the identity satisfied by the photon propagator of QED is +& K + D$qed (K )=&!

K& , K2

(1.1)

whereas the identity satisfied by the gluon propagator of QCD is slightly weaker: K + K & D$ +&(K )=&!.

(1.2)

At zero temperature, the abelian identity (1.1) has the same consequence as the nonabelian version (1.2) for the respective self-energy tensors. In both cases 6 +& must be a a linear combination of the available tensors g +& and K +K &. When the propagator is expressed in terms of the self-energy, application of (1.1) yields +& K + 6 +& qed =0 for the abelian case and application of (1.2) yields K + 6 =0 for QCD. At non-zero temperature the SlavnovTaylor identities are unchanged, as shown in [3] or Appendix C. However, even in covariant gauges the self-energy tensor will 141 0003-491699 30.00 Copyright  1999 by Academic Press All rights of reproduction in any form reserved.

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depend on the four-velocity u + of the plasma. Consequently in both QED and QCD the self-energy tensor can be a linear combination of four possible tensors: g +&, K + K &, u + u &, and K + u & +u + K &. It is convenient to choose particular linear combinations of these as the standard basis set [46] denoted A +&, B +&, C +&, D +& and defined in Appendix A. By parametrizing the tensor structure of the self-energy one can then calculate the form of the exact propagator and apply the SlavnovTaylor indentity. For QED the result remains K + 6 +& qed =0, since the electromagnetic current operator satisfies  + j + =0 even at non-zero temperature. Only two of the four basis tensors satisfy this. Therefore qed +& qed +& 6 +& qed =6 T A +6 L B ,

(1.3)

where the subscripts T and L denote transverse and longitudinal with respect to the spatial component k9 of the wave-vector. For QCD Braaten and Pisarski [7, 8] discovered a consistent high-temperature approximation which behaves in the same fashion as quantum electrodynamics. In this hard thermal loop approximation the quarks are massless and only the order T 2 part of one-loop diagrams is retained. The order T 2 part of the one-loop gluon +& self-energy is gauge invariant and satisfies K + 6 +& htl =0. In the standard basis 6 htl = htl +& htl +& 6 T A +6 L B , and the functions 6 T , 6 L are gauge invariant. This was discovered by explicit calculation in covariant gauges [9] and was shown by Braaten and Pisarski to be a consequence of the fact that the hard thermal loops with any number of external gluons all satisfy abelian Ward identities [7, 8, 10, 11]. In hard thermal loop approximation the gluon propagator has the same structure as the photon thermal propagator. If one goes beyond the hard thermal loop approximation these simplifications do not apply. Inclusion of sub-dominant powers of T, quark masses, or higher loop effects all spoil the abelian features. Explicit perturbative calculations [1215] show that K + 6 +& {0.

(1.4)

The QCD self-energy therefore depends on all four basis tensors, 6 +& =6 T A +& +6 L B +& +6 C C +& +6 D D +&,

(1.5)

where K + C +& {0 and K + D +& {0. In the exact self-energy the functions 6 T , 6 L , 6 C , 6 D are all gauge-dependent. B. Summary of Results The purpose of this paper is to investigate the structure of the full gluon propagator at finite temperature that results from the self-energy (1.5). Various special cases have appeared in the literature [5, 12].

STRUCTURE OF THE GLUON PROPAGATOR

143

General linear gauge-fixing. In a general linear gauge the nonabelian Slavnov Taylor identity (1.2) generalizes to F+ F & D$ +& =&!,

(1.6)

where F + is the gauge-fixing vector. (See [3] or Appendix C.) This includes choices such as axial gauge and light-cone gauge, which are not rotationally invariant. In these gauges the self-energy tensor is more complicated than (1.5). The standard way to implement (1.6) is to compute the full propagator from the Schwinger Dyson equation and then apply the SlavnovTaylor constraint (1.6). This can be rather tedious. There is a simpler approach that will be followed here. The quadratic part of the gluon action is the sum of the kinetic energy, the self-energy, and the gauge-fixing term. The reduced action without the gauge-fixing term will be denoted by a check: 18 +& =&K 2g +& +K +K & +6 +&.

(1.7)

Section III will show that the SlavnovTaylor identity (1.6) can be expressed as the pair of constraints 18 +& G & =0,

F & G & =&!,

(1.8)

which determine the vector G &. The consequences of this are simple. Viewed as a matrix, 18 + & must have four eigenvalues. Three eigenvalues are determined dynamically by the equations of motion. The fourth eigenvalue must be zero because of local gauge invariance as expressed in (1.8). A zero eigenvalue is only possible if the determinant of 18 + & vanishes. This immediately gives a non-linear relation on the gluon self-energy. Rotationally invariant gauge-fixing. The condition Det(18 + & )=0 applies with any linear gauge-fixing condition. If we specialize to gauges that preserve the rotational invariance of the plasma rest frame, then the gauge-fixing vector F + should be a linear combination of K + and the plasma velocity u +. In such cases the selfenergy has the tensor structure (1.5). The requirement Det(18 + & )=0 leads to (6 C ) 2 =(K 2 &6 L ) 6 D ,

(1.9)

which must hold for all K +. This is quite remarkable since the functions 6 L , 6 C , 6 D are all gauge-dependent. This relation is not new. Results equivalent to this were obtained by Kunstatter [3], Gross et al. [4], Kajantie and Kapusta [5], Kobes et al. [14], and Flechsig and Schulz [15]. In a perturbative expansion, 6 C is known to begin at order g 2 and consequently 6 D must begin at order g 4 [14, 15]. The emphasis of the present paper is to use (1.9) to compute the full gluon propagator from the SchwingerDyson equation. The result is that the full propagator is constructed of four tensors but it does not have four poles. Instead there are only two: one pole associated with the two transverse polarizations

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contained in A +& that determines the behavior of transverse electric and magnetic gluons; and one pole associated with a particular combination of B +&, C +&, D +& that controls the longitudinal electric gluons. The entire development is done in the real-time formulation of thermal QCD. $ +&, where a, b=1, 2. The The full gluon propagator has a 2_2 matrix structure D ab formulas displayed throughout this paper are for the diagonal entries of the 2_2 matrix structure. The diagonal entries satisfy SchwingerDyson equations that involve products of functions rather than products of matrices. For example, if k 0 is given a small positive imaginary part, i.e., k 0 +i=, then all formulas in the paper apply to retarded propagators and self-energies. If instead, k 0 is given an imaginary part with alternating sign, i.e., k 0 +i=k 0 , then all formulas apply to propagators and self-energies that satisfy Feynman boundary conditions. Appendix C gives further details. In Section II the exact form of the thermal gluon propagator is displayed for three popular gauges: covariant, Coulomb, and temporal. The proof of these results is delayed until Section III, which derives the general result (3.25) for the thermal gluon propagator for any linear gauge-fixing condition that preserves the rotational invariance of the plasma rest frame. Section IV discusses the gauge invariance of the location of the transverse and longitudinal poles and shows that the two types of poles coincide at k9 =0. Appendix A gives explicit details about the four basis tensors. Appendix B derives the form of the gluon propagator in the hard thermal loop approximation for a general linear gauge. Appendix C derives the Slavnov Taylor identity for general linear gauges at finite temperature.

II. GLUON PROPAGATOR IN THREE GAUGES The final result for the full gluon propagator in a general linear gauge is given in (3.25). Before deriving the general form it may be helpful to display the final result in familiar gauges, viz. covariant, Coulomb, and temporal. To give these results, some notation is necessary. Instead of the four-vectors K + and u + it is simpler to use K + and K + where K + =(K +K } u&u +K 2 )k,

(2.1)

and k 2 #(K } u) 2 &K 2 0. This vector satisfies K } K =0,

K 2 =&K 2.

(2.2)

It is often convenient to use the plasma rest frame u + =(1, 0, 0, 0) in which case K + =(k, kk 0 ). In this frame the transverse tensor A +& has the value A +& | rest =

0

\0

0 . &$ mn +k m k n

+

(2.3)

STRUCTURE OF THE GLUON PROPAGATOR

145

A second notation requires solving (1.9) for 6 C , 6 C =_ - (K 2 &6 L ) 6 D ,

(2.4)

where _=\1. In terms of this, it will be useful to employ the four&vector H + =K + - K 2 &6 L +_K + - 6 D ,

(2.5)

which has components H + =(h 0 , kh). A. Covariant Gauges In covariant gauges the free propagator is

_

+& D +& free = &g +(1&!)

K+K& 1 , K2 K2

&

(2.6)

or equivalently A +& K + K & K +K & D +& + &! . free =& K 2 (K 2 ) 2 (K 2 ) 2

(2.7)

In this gauge the exact propagator (3.25) reduces to D$ +& =

&A +& K + K & !H + H & + & . K 2 &6 T K 2(K 2 &6 L ) (K 2 ) 2 (K 2 &6 L )

(2.8)

This agrees with Kajantie and Kapusta [5]. B. Coulomb gauges A second example is provided by generalized Coulomb gauges with gauge-fixing term Lgf =&(% } A) 22!. The free gluon propagator is D +& free =&

A +& 1 1 0 K+K& + 2 &! . 2 K k 0 0 k4

\ +

(2.9)

In this case the D 00 propagator is instantaneous and only the transverse potentials have time-dependence. The true Coulomb gauge { } A=0 corresponds to !=0. The full propagator that results from (3.25) is D$ +& =

&A +& K2 1 0 H+H& + 2 &! 2 2 , 2 K &6 T h 0 0 h k

\ +

(2.10)

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H. ARTHUR WELDON

where h=k - K 2 &6 L +_k 0 - 6 D is the spatial component of H +. A more useful expression is h 2 =K 2(k 2 +6 00 ).

(2.11)

Heinz et al. [12] obtained (2.10) in the strict Coulomb gauge !=0. C. Temporal gauges A third type of example is the generalized temporal gauge propagator. The gauge-fixing term is Lgf =&(A 0 ) 22!, where ! has dimensions of length squared. The free propagator is A +& 1 0 =& + D +& free K 2 k 20 0

\

0 K+K& &! . k m k n k 20

+

(2.12)

The exact propagator (3.25) reduces in this gauge to D$ +& =&

0 A +& K2 0 H+H& + 2 &! , m n K &6 T h 0 0 k k h 20

\

2

+

(2.13)

where h 0 =k 0 - K 2 &6 L +_k - 6 D is the time component of H +. A more useful relation is (h 0 ) 2 =K 2(k 20 +k m k n 6 mn ).

(2.14)

Heinz et al. [12] obtained (2.13) in strict temporal gauge !=0.

III. PROOF FOR GENERAL LINEAR GAUGES A. Gauge fixing condition The investigation pertains to any linear gauge-fixing condition that does not break rotational invariance in the plasma rest frame. With a gauge-fixing term Lgf =&(F+ A + ) 22! the quadratic part of the momentum-space action is 1 2

d 4K

| (2?)

4

A +(K ) 1 +& free A &(&K ),

(3.1)

where 2 +& + & 1 +& free =&K g +K K &

F +F & . !

(3.2)

STRUCTURE OF THE GLUON PROPAGATOR

147

The most general F + that does not spoil the rotational invariance is a linear combination of two vectors, F + = f 1 K + +f 2 K +,

(3.3)

where f 1 and f 2 are functions of k 0 and k. In the plasma rest frame K + =(k 0 , k9 ) and K + =(k, k 0 k ). It will be helpful to also introduce the vector F + = f 1 K + + f 2 K +,

(3.4)

F } F =0.

(3.5)

with the property

For the covariant, Coulomb, and temporal gauges the vectors (3.3) and (3.4) are F+ Covariant (k 0 , k9 )

F +

f1

(k, k 0 k )

1 2

f2 0

Coulomb

(0, k9 )

(k, 0)

&k K

Temporal

(1, 0)

(0, k )

k 0 K 2

2

k 0 kK 2 &kK 2.

(3.6)

The free propagator is the inverse of (3.2): +& =& D free

A +& F +F & K+K& + &! . 2 2 2 K ( f1 K ) ( f1 K 2)2

(3.7)

B. Exact Gluon Propagator For the full gluon propagator it is necessary to add the self-energy to (3.2) to obtain the quadratic part of the full effective action: 1 +& =&K 2g +& +K + K & +6 +& &

F +F & . !

(3.8)

The full gluon propagator, denoted D$&: , is the inverse 1 +& D$&: =$ + : .

(3.9)

The full propagator must satisfy the nonabelian SlavnovTaylor identity F &F : D$&: =&!,

(3.10)

which is the generalization of (1.2) to general linear gauges. Appendix C and [3] show that this identity must hold even at finite temperature. To determine how

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H. ARTHUR WELDON

(3.10) directly constrains the proper self-energy, it is useful to define a four-vector G & by G & #D$&: F :.

(3.11)

1 +& G & =F +,

(3.12)

F & G & =&!.

(3.13)

Then (3.9) implies that

and (3.10) implies

Consequently (3.12) can be written

\

1 +& +

F +F & G & =0. !

+

(3.14)

This condition applies even if F + is not a combination of K + and K + as, for example, in axial or light-cone gauge. The tensor in parentheses will be denoted by a check: 18 +& #1 +& +F + F &! =&K 2 g +& +K + K & +6 +&.

(3.15)

In terms of the tensor decomposition (1.5) for rotationally invariant gauge-fixing, the explicit form is 18 + & =(6 T &K 2 ) A + & +(6 L &K 2 ) B + & +6 C C + & +6 D D + & .

(3.16)

Although 18 has the gauge-fixing term removed it is still gauge-dependent because of the self-energy functions. It must satisfy 18 +& G & =0,

(3.17)

where G & is an unknown four-vector satisfying (3.13). The first step in implementing this is to require that 18 have determinant zero. Since 18 is a 4_4 matrix, it must have four eigenvectors. Two of the eigenvectors are the two transverse polarization vectors = & which satisfy K } ==0 and u } ==0. The eigenvalue for each of these is the same: 18 +& = & =(6 T &K 2 ) = +. The remaining two eigenvectors are each linear combinations of K & and K &. The properties of the basis tensors A, B, C, D give 18 +& K & =6 C K + +6 D K + 18 +& K & =(6 L &K 2 ) K + &6 C K +,

(3.18)

STRUCTURE OF THE GLUON PROPAGATOR

149

which yields a quadratic equation for the remaining two eigenvalues. The product of all four eigenvalues is Det(18 +& )=(6 T &K 2 ) 2 [(6 L &K 2 ) 6 D +(6 C ) 2 ].

(3.19)

The SlavnovTaylor identity forces the quantity in square brackets to vanish for all K + : (6 L &K 2 ) 6 D +(6 C ) 2 =0.

(3.20)

It seems most natural to solve this for 6 C , 6 C =_ - (K 2 &6 L ) 6 D ,

(3.21)

where _=\1. When this condition is subsituted into (3.16), the particular linear combination of the tensors B, C, D can be written in terms of a single four-vector H + as 18 +& =(6 T &K 2 ) A +& + &

&

2

H + H & K2

(3.22)

&

H #K - K &6 L +_K - 6 D . The null vector of 18 + & is the vector H & =K & - K 2 &6 L +_K & - 6 D .

(3.23)

It satisfies 18 +& H & =0 because H } H=0. (The null vector G & normalized by (3.13) is G & =&!H &F } H.) Now that the SlavnovTaylor constraint has been satisfied and it is straightforward to obtain the full propagator. First add back the gauge-fixing term to 18 : 1 +& =(6 T &K 2 ) A +& +

H + H & F + F & & . K2 !

(3.24)

It is easy to invert this using F } F =H } H =0 and F } H =&F } H to obtain the full propagator: D$ +& =&

F + F & A +& H+H& 2 +K &! . K 2 &6 T (F } H ) 2 (F } H ) 2

(3.25)

Naturally it satisfies the SlavnovTaylor identity (3.10). Section II has displayed examples of this result in covariant, Coulomb, and temporal gauges. The denominator in the nontransverse sector contains F } H=K 2[ f 1 - K 2 &6 L &_f 2 - 6 D ].

(3.26)

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H. ARTHUR WELDON

However, a simpler expression for the full denominator is obtained from using (3.8) and (3.24): (F } H ) 2 =F + F & 1 +& =(F } K ) 2 +F + F & 6 +&. K2

(3.27)

IV. DISCUSSION A. Gauge Invariance of Poles The tensor structure of the gluon propagator (3.25) is explicitly dependent upon ! and upon the gauge-fixing parameters f 1 and f 2 contained in F + and F +. Moreover the self-energy 6 +& is dependent upon these same gauge parameters since they occur in the free propagator (3.7). Thus all four functions 6 T , 6 L , 6 C , and 6 D are gauge-dependent. Kobes et al. proved a remarkable theorem [16]. If the exact propagator is expressed as a linear combination of the four basis tensors A +&, B +&, C +&, D +& then the location of the poles in the A +& and B +& part of the propagator are gauge-fixing independent. Therefore the solutions to K 2 =6 T and (F } H ) 2 =0 are gauge invariant. At momenta for which (F } H ) 2 Ä 0, H + Ä F + - 6 D  f 1 so that the tensor structure of the propagator at the pole is simple:

_

(F } H ) 2 D$ +& | pole Ä K 2 &!

6D F + F &. f 21

&

(4.1)

The nontransverse function 6 D modifies the residue but not the polarization structure. B. Zero Three-Momentum The pole in the transverse sector occurs at K 2 =6 T . The pole in the longitudinal sector occurs when (3.27) vanishes. It is easy to show that at k9 =0 these poles coincide and that they have the same residue. This comes about because the proper self-energy 6 +&(k 0 , k9 ) has the property that when |k9 | =0, it does not depend upon k. The proof of this is quite simple. The self-energy is computed by integrations over fermion and boson propagators. Let P : denote a generic loop momenta to be integrated and K : denote the external four-momentum of the gluon. The fermion and boson propagators to be integrated over have arguments of the form P : +c j K :, where c j depends upon where the propagator occurs in the diagram. At |k9 | =0, the arguments of the propagators are ( p 0 +c j k 0 , p ) and all information about the direction k disappears. To apply this result, begin with the general self-energy 6 +&(k 0 , k9 )=6 T A +& +6 L B +& +6 C C +& +6 D D +&.

(4.2)

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STRUCTURE OF THE GLUON PROPAGATOR

According to Appendix A the |k9 | =0 limit of the four basis tensors gives for the spatial components of the self-energy 6 mn(k 0 , 0)=(&$ mn +k m k n ) 6 T (k 0 , 0)&k m k n 6 L(k 0 , 0).

(4.3)

Since this cannot depend upon k 6 T (k 0 , 0)=6 L(k 0 , 0).

(4.4)

Similarly 6 0n(k 0 , 0)=k n 6 C (k 0 , 0) and since this cannot depend on k it is necessary that 6 C (k 0 , 0)=0. Because of relation (3.21) this automatically means that 6 D (k 0 , 0)=0. Thus the self-energy tensor is 6 +&(k 0 , 0)=

\

0 0 . mn 0 &$ 6 T (k 0 , 0)

+

(4.5)

The full gluon propagator is quite simple in this limit. For example, in covariant gauge the propagator is D +&(k 0 , 0)=

\

&!k 20 0 . mn 2 0 $ [k 0 &6 T (k 0 , 0)]

+

(4.6)

It is also simple in the temporal gauge. In the Coulomb gauge the limit |k9 | Ä 0 does not exist for either the free or the full propagator.

APPENDIX A: TENSOR BASIS There is a standard set of four tensors [46] constructed out of g +&, K +, and K =(K } uK + &K 2u + )k that will be used throughout: +

A +& = g +& &B +& &D +& B +& =&

K + K & K2

(A1)

K + K & +K + K & C = K2 +&

D +& =

K+K& K2

The squares of these tensors are A 2 =A,

B 2 =B,

C 2 =&B&D,

D 2 =D.

(A2)

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H. ARTHUR WELDON

Note that requiring A, B, and D to be idempotent fixes their sign and magnitude, but there is no such universal choice for C. The twelve mixed products are all simple: AB=AC=AD=BD=0 BA=CA=DA=DB=0

(A3)

(BC ) +& =(CD) +& =K + K &K 2 (CB) +& =(DC) +& =K + K &K 2.

It is often convenient to work in the rest frame of the plasma, where u + = (1, 0, 0, 0) and the vector K + takes the simple form (k, kk 0 ). In this frame the four basis tensors have the values A +& | rest =

C | rest D +& | rest

0 &$ +k m k n mn

&k 2 1 K 2 &k m k 0

+

&k 0 k n &k 20 k m k n

\ + 1 (k +k ) k 2k k = \  K k (k +k ) 2k kk k + k k k 1 = . K \k k k k +

B +& | rest = +&

0

\0

2 0

0

2

m

2 0

m

2

n

(A4)

n

0

n

2 0

2

2

0 m

m

n

0

APPENDIX B: HARD THERMAL LOOP APPROXIMATION Although the structure of the gluon propagator in the hard thermal loop approximation is well known in standard gauges, it is useful to obtain it in a general linear gauge using the method employed in Section III. Braaten and Pisarski showed that the gluon propagator in the hard thermal loop approximation obeys abelian Ward identities [7, 8, 10, 11]. In a general linear gauge the abelian Ward identity is D$&:htl F : =&

! K& , F}K

(B1)

as shown in Appendix C. This constraint can be rewritten by multiplying (B1) by 1 +& htl to obtain F + =&

! 1 +& K & . F } K htl

(B2)

STRUCTURE OF THE GLUON PROPAGATOR

153

This is equivalent to

\

1 +& htl +

F +F & K & =0. !

+

(B3)

+& + & As before, denote the tensor in parentheses by a check: 18 +& htl #1 htl +F F !. In terms of the tensor decomposition (1.5) 2 + htl 2 + htl + htl + 18 + htl & =(6 htl T &K ) A & +(6 L &K ) B & +6 C C & +6 D D & .

(B4)

Condition (B3) requires 18 + & htlK & =0. From the tensor definitions in Appendix A, +  + +6 htl A + & K & =0 and B + & K & =0 so that 18 + & htl K & =6 htl C K D K . Because the vectors + + htl K and K are linearly independent, the only solution is 6 htl C =0, 6 D =0. To obtain the propagator in the hard thermal loop approximation, we must add the gauge-fixing term to (B.4): htl 2 +& htl 2 +& 1 +& htl =(6 T &K ) A +(6 L &K ) B &

F+F& . !

(B5)

The inverse of this is the hard thermal loop propagator. Using F } F =K } K =0 and F } K=&F } K = f 1 K 2 gives the result +& D$htl =

&A +& F + F & !K + K & + . & 2 htl 2 2 2 htl K &6 T ( f 1 ) K (K &6 L ) ( f 1 K 2 ) 2

(B6)

APPENDIX C: SLAVNOVTAYLOR IDENTITY IN GENERAL LINEAR GAUGES This appendix will derive the SlavnovTaylor identity for the gluon propagator at finite temperature. The generating functional for the real-time Green functions of finite-temperature QCD is [6]

|

Z(J )= D0 exp(iS J ),

(C1)

where SJ =

|

\

d 4 x L& C

1 (F A & ) 2 +J & A & . 2! &

+

(C2)

The label C denotes a contour in the complex time plane that runs from an initial time t 0 and ends at t 0 &i; along a path that will be specified later. The gauge-fixing vector F& determines the FadeevPopov matrix M(x, y)=&Fx& D &x $ 4(x& y)

(C3)

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H. ARTHUR WELDON

and the invariant integration measure D0=D D DA det(M).

(C4)

The generating functional is invariant under a change of the integration variables A +. If the change of variables is a gauge transformation A + Ä A + +D +4 then the measure D0 and the quark-gluon lagrangian L will not change. The only change comes from the gauge-fixing and source terms. Thus

|

0= D0 $S J exp(iS J ),

(C5)

where $S J =

|

\

d 4x &

C

(F+ A + ) F& D &4 +J & D &4 . !

+

(C6)

It is convenient to choose 4 as 4(x)=

|

d 4y[F+ D + ] &1 (x, y) *( y)

(C7)

C

for an arbitrary function *( y) so that 1 $S J =& F& A &( y)+ $*( y) !

|

C

d 4 x J &(x) D &x[F } D] &1 (x, y).

(C8)

The invariance (C5) becomes F&( y) i$Z = Z(0) $J &( y)

|

d 4 x J &(x) G &(x, y)

(C9)

C

G &(x, y)#&!

| D0 D [F } D] & x

&1

(x, y) exp(iS J )Z(0).

(C10)

This has the property that F+(x) G +(x, y)| J=0 =&! $ 4C(x& y).

(C11)

Now apply $$J +(x) to (C9) i$ 2Z F&( y) =G +(x, y)+ Z(0) $J +(x) $J &( y)

|

d zyJ :(z) C

$G :(z, y) $J +(x)

(C12)

and set J=0: F&( y) D$ +&(x, y)=G +(x, y)| J=0 .

(C13)

STRUCTURE OF THE GLUON PROPAGATOR

155

In QCD the covariant derivative in (C10) depends on the vector potential. The SlavnovTaylor identity follows from applying Fx+ to (C13): F+(x) F&( y) D +&(x, y)=&! $ 4C(x& y).

(C14)

To convert this to momentum space it is necessary to know where the time components of x and y lie on the contour C in the complex time plane. For field theory in Minkowski space the contour C is taken as the union of four parts [6] that depends on a parameter _ with 0_;: C 1 runs from &t 0 to t 0 , C 3 runs from t 0 to t 0 &i_, C 2 runs from t 0 &i_ to &t 0 &i_, and C 4 runs from &t 0 &i_ to &t 0 &i;. As t 0 Ä  the contributions of C 3 and C 4 can be neglected. The physical field A +1 (x)# A+(x) and the auxiliary field A +2 (x)#A +(t&i_, x ) lie on C 1 and C 2 , respectively. Since the time components of x and y may be taken along C 1 or C 2 independently, the propagator becomes a 2_2 matrix in the internal space. The contour-ordered propagator contains four time-ordered parts. The explicit form of (C14) is F+(x) F&( y) D ij$ +&(x, y)=&! $ 4(x& y)({ 3 ) ij ,

(C15)

where now x and y have real time components. In momentum space this becomes F + F & D $ij+&(K )=&!({ 3 ) ij .

(C16)

For covariant gauges F + =K +, for Coulomb gauges F + =(0, k9 ), and for the temporal gauge F + =(1, 0, 0, 0). Feynman Basis. The 2_2 matrix structure of the time-ordered thermal propagator may be diagonalized in terms of functions which have Feynman-like analytic structure (analytic in the upper half-plane of complex (k 0 ) 2 ) as follows [6, 17], D ij$ +&(K )=U il

\

D $F+&(K ) 0

0 +& F

&D $ *(K )

+

V mj ,

(C17)

lm

where U and V are nonsingular matrices that satisfy U{ 3 V={ 3 . The matrix equation (C16) therefore reduces to the single condition F + F & D F$ +&(K )=&!.

(C18)

Condition (C16) imposes no additional constraint on the off-diagonal parts of the $ +&. Therefore F + F & D $12+& B F + F & Im D $11+& B propagator because D $12+& B Im D 11 Im !=0. Retarded Basis: The time-ordered propagator may also be diagonalized in terms of retarded propagators (analytic in the upper half-plane of complex k 0 ) and advanced propagators (analytic in the lower half-plane of complex k 0 ) [18], D ij$ +&(K )=U il

\

DR $ +&(K ) 0

0 &D $A+&(K )

+

V mj , lm

(C19)

156

H. ARTHUR WELDON

where U and V are different from those in (C17) but still satisfy U{ 3 V={ 3 . As before the matrix equation (C16) reduces to a single condition on the retarded function F+ F& D R $ +&(K )=&!.

(C20)

Abelian Ward Identity. For an abelian theory the covariant derivative in (C10) is independent of the vector potential so that (C13) becomes F&( y) D$ +&(x, y)=&!

 [F } ] &1 (x, y). x &

(C21)

In momentum space this is F& D $ij+&(K )=&!

K+ ({ 3 ) ij , F}K

(C22)

which may be diagonalized in either the Feynman or the retarded basis. ACKNOWLEDGMENT This work was supported in part by the National Science Foundation under Grant PHY-9630149.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18.

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