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Gluon Pair Production From Space-Time Dependent Chromo?eld

Gouranga C. Nayak and Walter Greiner

Institut f¨r Theoretische Physik, Robert-Mayer Str. 8-10, Johann Wolfgang Goethe-Universit¨t, u a 60054 Frankfurt am Main, Germany

Abstract

arXiv:hep-th/0001009v4 13 Sep 2000

We compute the probabilty for the processes A → q q, gg via vacuum ? polarization in the presence of a classical space-time dependent non-abelian ?eld A by applying the background ?eld method of QCD. The processes we consider are leading order in gA and are simillar to A → e+ e? in QED. Gluons are treated like a matter ?eld and gauge transformations of the quantum gluonic ?eld are di?erent from those of the classical chromo?eld. To obtain the correct physical polarization of gluons we deduct the corresponding ghost contributions. We ?nd that the expression for the probability of the leading order process A → gg is transverse with respect to the momentum of the ?eld. We observe that the contributions from higher order processes to gluon pair production need to be added to the this leading order process. Our result presented here is a part of the expression for the total probability for gluon pair production from a space-time dependent chrom?eld. The result for q q pair production is similar to that of the e+ e? pair production in QED. ? Quark and gluon production from a space-time dependent chromo?eld will play an important role in the study of the production and equilibration of the quark-gluon plasma in ultra relativistic heavy-ion collisions. PACS: 12.38.Aw; 11.55.-q; 11.15.Kc; 12.38.Bx

Typeset using REVTEX 1

I. INTRODUCTION

Over the years there have been several investigations on the production of charged particles from a classical electro-magnetic ?eld, a phenomenon which was discovered nearly ?fty years ago by Schwinger [1]. By now, the production of electron-positron pairs from the abelian ?eld is extensively studied both theoretically and experimentally [2]. The subject of quark/anti-quark and gluon pair production from the non-abelian ?eld is relatively new and is not fully solved. It might be important for the production of the quark-gluon plasma (QGP) in the laboratory by high energy heavy-ion collisions. Lattice QCD predicts the existence of such a state of matter at high temperatures (? 200 MeV) and densities [3]. In high energy heavy-ion collisions at RHIC and LHC [4] the receding nuclei might produce a strong chromo?eld [5,6] which would then polarize the QCD vacuum and produce quark/anti-quark pairs and gluons. These produced quarks and gluons collide with each other to form a thermalized quark-gluon plasma. The space-time evolution of the quarkgluon plasma in the presence of a background chromo?eld is studied by solving relativistic non-abelian transport equation of quarks and gluons with all the dynamical e?ects taken into account. As color is a dynamical variable in the non-abelian theory, the relativistic non-abelian transport equation for quark and gluon is [5,7]

c ? a p? ? ? + gQa F?ν pν ?p + gf abc Qa Ab p? ?Q f (x, p, Q) = C(x, p, Q) + S(x, p, Q). ?

(1)

Here f (x, p, Q) is the single-particle distribution function of the parton in the 14 dimensional extended phase space which includes coordinate, momentum and color in SU(3). The ?rst term on the LHS of Eq. (1) corresponds to the usual convective ?ow, the second term is the non-Abelian version of the Lorentz-force term and the third term corresponds to the precession of the color charge, as described by Wong’s equation [8]. C and S on the RHS of Eq. (1) are the collision and the source terms, respectively. Note that there are separate transport equations for quarks, antiquarks and gluons [5,7]. The source term S contains the detailed information about the production of quarks and gluons from the chromo?eld. It is de?ned as the probability W for the parton production per unit time per unit volume of the phase space. Hence the space-time evolution of the quark-gluon plasma in a high energy heavy-ion collision crucially depends on how quarks and gluons are produced from the chromo?eld [5]. The production of fermion pairs from the ?eld is studied in two di?erent cases: 1) from a constant, uniform electric ?eld and 2) from a space-time dependent ?eld. The fermion pair production from a constant, uniform ?eld is computed by Schwinger. This is an exact one-loop nonperturbative result [1]. This result can also be understood in terms of a semiclassical tunneling across the mass gap [9]. However, a space-time dependent ?eld A can directly excite the negative energy particles to levels above the mass gap, by a perturbative mechanism A → q q , without recourse to any tunneling or barrier penetration ? mechanism [10]. The particle production from this mechanism is important in high energy heavy-ion collisions. As shown by numerical studies [5], the chromo?eld acquires a spacetime dependence as soon as it starts producing partons and so Schwinger’s non-perturbative formula for particle production from a constant ?eld is not applicable. In this case the treatment of parton production from a space-time dependent chromo?eld is necessary. It can be mentioned here that, so far the transport equation is solved with parton productions 2

from a constant ?eld taken into account [2] because gluon pair production from the spacetime dependent chromo?eld is not computed. Aim of this paper is to calculate the probability for the process A → gg which is similar to the process A → q q . Quark and gluon production ? from a space-time dependent chromo?eld is needed to study the production and equilibration of a quark-gluon plasma in ultra relativistic heavy-ion collisions at RHIC and LHC. The production of q q pairs from a space-time dependent non-abelian ?eld is almost the ? same to that of the production of e+ e? pairs from the abelian ?eld. This is because, apart from the color factors, the interaction of the quantized Dirac ?eld with the classical potential is the same in both cases. All the methods used to obtain the probability for the production of the e+ e? pair from a space-time dependent Maxwell ?eld can be applied to obtain the probability for the production of the q q pair from a space-time dependent Yang-Mills ?eld. ? On the other hand, the production of gluon pairs from a space-time dependent Yang-Mills ?eld is not straight forward and there is no counter part to this in the abelian theory. In contrast to abelian theory the quantized Yang-Mills ?eld interacts with the classical nonabelian potential. Because of this interaction there is gluon production from the QCD vacuum in the presence of an external chromo?eld. This phenomenon is absent in the Maxwell theory. We again mention here that the fermion pair production from the space-time dependent ?eld is studied in the literature [1,11,10] but gluon production from space-time dependent chromo?eld is not studied so far. This is because a consistent theory involving the interaction of the gluons with the classical chromo?eld is not available in the conventional theory of QCD. In the case of fermions one knows exactly the theory for the interaction of fermions with the classical ?eld. This is given by the Dirac equation (see section II). However, a consistent theory of gluons in the presence of external classical chromo?eld is not available in the conventional QCD. This problem is addressed properly within the background ?eld method of QCD which was introduced by DeWitt and ’t Hooft. We will brie?y mention the main di?erences between the background ?eld method of QCD and the conventional method of QCD in section III. Unlike in conventional QCD, the Feynman diagrams obtained in the background ?eld method contain classical chromo?elds and gluons. We will use the Feynman rules derived in the background ?eld method of QCD to obtain the probability for the processes A → q q, gg via vacuum polarization. The study of the production and ? evolution of the quark-gluon plasma at RHIC and LHC by solving the relativistic non-abelian transport equation [5] with parton production from a space-time dependent chromo?eld will be undertaken in the future. This paper is organized as follows. We compute the probability for the process A → q q ? in section II. The probability for the process A → gg is computed in section III. A brief description about future research and conclusions can be found in section IV.

? II. PROBABILITY FOR THE PROCESS ACL → QQ

The production of q q pairs from a non-abelian ?eld via vacuum polarization is simillar to ? that of the production of e+ e? pairs from the abelian ?eld. This is because the interaction lagrangian of the quantized Dirac ?eld with the classical gauge potential is similar in both the cases. The amplitude for the lowest order process A → e+ e? which contributes to the

3

production of e+ e? pair from a space-time dependent classical abelian ?eld A? is given by [10] M =< k1 , k2|S (1) |0 >= ?ie?(k1 )γ? A? (K)v(k2 ). u (2)

In the above expression k1 , k2 are the four momenta of the produced electron and positron respectively and A(K) is the Fourier transform of the space-time dependent ?eld A? (x) with K = k1 + k2 . The probability for pair production is: W (1) = where T = Σspin |M|2 . Performing the spin sum one obtains [11] We+ e? =

(1)

d3 k1 0 (2π)3 2k1

d3 k2 0 (2π)3 2k2

d4 K (2π)4 δ (4) (K ? k1 ? k2 ) T

(3)

(4)

α 3

K 2 >4m2 e

d4 K (1 ?

4m2 e ) K2

1/2

(1 +

2m2 e ) [K · A(K)K · A(?K) K2 ?K 2 A(K) · A(?K)].

(5)

In the above expression α =

e2 4π

is the coupling constant. Using (6)

F ?ν (K) = ?i(K ? Aν ? K ν A? ) one obtains

(1) We+ e?

α = 3

4m2 d K (1 ? 2e ) K K 2 >4m2 e

4

1/2

(1 +

2m2 e ) [|E(K)|2 ? |B(K)|2 ]. K2

(7)

This is the probability for the production of the e+ e? pair from a space-time dependent abelian ?eld. This result was obtained for the ?rst time by Schwinger (see Eq. (6.33) in [1]). Now we proceed to compute the probability for the production of a q q pair from a space? time dependent non-abelian ?eld. The amplitude for the lowest order process A → q q which ? contributes to q q pair production from a space-time dependent classical non-abelian ?eld ? a? A is given by

a M = ig ui (k1 )γ? Tij Aa? (K)v j (k2 ). ?

(8)

In the above expression k1 and k2 are the four momenta of the quark and anti-quark respectively and Aa? (K) is the Fourier transform of the space-time dependent non-abelian ?eld a Aa? (x) with K = k1 + k2 . Tij are the generators in the fundamental representation, i, j are the color indices for the quarks and a is the color index of the non-abelian ?eld. Following the above procedure and choosing SU(3) gauge group one obtains

(1) Wqq ?

αs = tr[T T ] 3

a b

4m2 d K (1 ? 2q ) K K 2 >4m2 q

4

1/2

(1 +

2m2 q ) [K · Aa (K)K · Ab (?K) 2 K ?K 2 Aa (K) · Ab (?K)]. (9)

4

In the above expression αs =

(1) Wqq ?

g2 . 4π

1 Using tr[T a T b ] = 2 δ ab one obtains 1/2

αs = 6

4m2 d K (1 ? 2q ) 2 >4m2 K K q

4

2m2 (1 + 2q ) [K · Aa (K)K · Aa (?K) K ?K 2 Aa (K) · Aa (?K)].

(10)

This is the probability for the production of a q q pair from a space-time dependent chro? mo?eld to the lowest order in the coupling constant αs . It can be noted that Eq. (10) is valid only for a single ?avor of quarks. This equation is simillar to Eq. (5) in the abelian theory.

III. PROBABILITY FOR THE PROCESS A → GG

Now we proceed to compute the probability for the process Acl → gg via vacuum polarization. The amplitude for this process (see Fig. 1(a)) is given by

abc M = ?bν (k1 )V?νλ (K, k1 , k2 )Aa? (K)?cλ (k2 ).

(11)

In the above expression k1 , k2 are the four momenta of the produced gluons, Aa? (K) is the abc Fourier transform of the non-abelian ?eld Aa? (x) and V?νλ (K, k1 , k2) is the vertex involving a single classical ?eld and two gluons with K = k1 + k2 . In the conventional method of QCD such vertices which involve classical ?eld and gluons are not calculated. Hence, the gluon production from a space-time dependent chromo?eld is not available. However, such a calculation is possible in the background ?eld method of QCD which was introduced by DeWitt and ’t Hooft [12,13]. This is because the Feynman diagrams involving a classical chromo?eld and gluons are obtained in the background ?eld method of QCD. We use the Feynman rules obtained by the background ?eld method of QCD to compute the probability for the process Acl → gg via vacuum polarization. First of all we will brie?y describe the main di?erences between conventional QCD and the background ?eld method of QCD before studying the above process. In the conventional theory of QCD the generating function is Z[J] = [dA] detMG exp(i[S[A] ? 1 G · G + J · A]), 2α (12)

where we have not included the quark part for simplicity. In the above expression the gauge ?eld action is S[A] = ? with F a?ν = ? ? Aaν ? ? ν Aa? + g f abc Ab? Acν for the group with structure constants f abc . The other two terms are J ·A=

a d4 x J? Aa?

1 4

d4 x (F a?ν )2

(13)

(14)

(15)

5

and G·G= d4 xGa Ga (16)

with Ga being the gauge ?xing term. A typical choice for the gauge ?xing term in QCD is Ga = ?? Aa? . The matrix element of (MG (x, y)) is given by (MG (x, y))ab =

a

(17)

δ(Ga (x)) δθb (y)

(18)

(x)) where δ(Gb (y) is the derivative of the gauge ?xing term under the in?nitesimal gauge transδθ formation

δAa? = ?f abc θb Ac? +

1 ? a ? θ . g

(19)

The detMG in the generating functional is written as functional integral over the FaddeevPopov ghost ?eld χa . We mention here that any physical quantity calculated in this method is gauge invariant and independent of the particular gauge chosen. However, there is still a problem of gauge invariance with some of the quantities like o?-shell Green’s functions or divergent counter terms. This problem arises because in order to quantize the theory one must ?x a gauge. This means that the total lagrangian we actually use in conventional QCD, consisting of the classical lagrangian (which is explicitly gauge invariant) plus a gauge-?xing and ghost terms, is not gauge invariant. The background ?eld method is a technique which allows one to ?x a gauge in quantizing the theory without losing explicit gauge invariance which is present at the classical level of the gauge ?eld theory. In the background ?eld approach, one arranges things so that explicit gauge invariance is still present once gauge ?xing and ghost terms are added. In this way Green’s functions obey the naive Ward identities of gauge invariance and the unphysical quantities like divergent counterterms becomes gauge invariant. However, for our purpose of gluon productions from the space-time dependent chromo?eld to the leading order we do not need to address all these issues here. The details about the explicit gauge invariance of QCD in the presence of background chromo?eld can be found in [12–15]. In our study we require Feynman rules involving the classical chromo?eld and gluons. For this reason we brie?y outline the procedure of background ?eld method in QCD and present the generating functional which generates the Feynman rules involving gluons, ghosts and the classical chromo?elds. In the background ?eld method of QCD the gauge ?xing term is given by [13] Ga = ? ? Aa? + g f abc Ab? Ac? , q cl q (20)

which depends on the classical background ?eld Aa? . The variable of the integration in the cl functional integral is the quantum gauge ?eld Aa? and, following ’t Hooft [13] the background q ?eld is not coupled to the external source J. The generating functional depends on J and Acl and is given by 6

Z[J, Acl ] =

[dAq ] detMG exp(i[S[Aq + Acl ] ?

1 G · G + J · Aq ]). 2α

(21)

The matrix element of MG is given by (MG (x, y))ab =

a

δ(Ga (x)) δθb (y)

(22)

(x)) where δ(Gb (y) is the derivative of the gauge ?xing term under the in?nitesimal gauge transδθ formation

δAa? = ?f abc θb (Ac? + Ac? ) + q q cl

1 ? a ? θ . g

(23)

Writing detMG as functional integral over the ghost ?eld one ?nds Z[J, Acl , ξ, ξ ?] = [dAq ] [dχ] [dχ? ] exp(i[S[Aq + Acl ] + Sghost ? 1 G · G + J · Aq 2α +χ? ξ + ξ ? χ]).

(24)

where ξ and ξ ? are source functions for the ghosts and Sghost = ? ← ? d4 x χ? [22 δ ab ? g ? ? f abc (Ac? + Ac? ) + gf abc Ac? ?? a cl q + g 2 f ace f edb Ac (Ad? + Ad? )]χb . q cl? cl (25)

The Feynman rules [14] for QCD in the presence of a classical background chromo?eld are constructed from the generating functional which is given by Eq. (24) with the gauge ?xing term given by Eq. (20). We use the Feynman rules obtained in the background ?eld method to compute the probability for the process Acl → gg. Within this method the vertex abc V?νλ (K, k1, k2 ), involving a single classical chromo?eld and two gluons which appear in the amplitude (See Fig. 1(a) and Eq. (11)) in Feynman gauge is given by

abc V?νλ (K, k1, k2 ) = gf abc [2g?ν Kλ ? 2g?λ Kν ? gνλ k? ],

(26)

where K = k1 + k2 and k = k1 ? k2 . The above vertex is di?erent from the three gluon vertex usually used in conventional QCD. Using Eq. (11) in (4) we ?nd 1 ′ ′ ′ ′ ′ ′ abc ?a′ ′ ′ Tgl = Σspin Aa? (K)Aa ? (?K)V?νλ (K, k)V?′ ν ′b ′c (K, k)?bν (k1 )??b ν (k1 )?cλ (k2 )??c λ (k2 ), (27) λ 4 where we have used A? (K) = A(?K). The factor 1 is the weight factor given in the Feynman 4 amplitude in order to obtain a correct gauge ?eld and gauge-?xing action. To obtain correct and physical results we have to use the appropriate projection operators for the transverse polarization states of the gluons. For this purpose we proceed as follows. First of all we use the polarization sum Σspin ?ν ??ν = Σspin ?ν ??ν = ?g νν 2 2 1 1

′ ′ ′

(28)

and then substract the corresponding ghost contributions (see Fig. 1(b)). 7

Using Eq. (26), (28) and performing the calculation we ?nd 1 ′ ′ ′ ′ abc ?a′ Tgl = Aa? (K)Aa ? (?K)V?νλ (K, k)V?′ ν ′bc′ (K, k)g νν g λλ = Ng 2 [4k1 · k2 Aa (K) · Aa (?K) λ 4 ?3k1 · Aa (K)k2 · Aa (?K) ? 3k2 · Aa (K)k1 · Aa (?K) ? k1 · Aa (K)k1 · Aa (?K) ?k2 · Aa (K)k2 · Aa (?K)], (29) where we have used f abc f a bc = Nδ aa in SU(N) gauge group. Now we proceed to compute the probability for the process Acl → gg. To obtain the probability for the above process we require that K 2 = (k1 + k2 )2 > 0 with K 0 > 0. We 2 2 recall that for real gluons k1 = k2 = 0. With the above requirements we proceed to perform the integration in Eq. (3) using Tgl from Eq. (29). As gluons are similar particles we 1 multiply a factor 2 in the phase space and ?nd Wgl =

(1)

′ ′

1 Ng 2 4 32π 2

K 2 >0

d4 K θ(K 0 )

d3 k1 d3 k2 (4) δ (K ? k1 ? k2 )Aa (K)Aa (?K) ? ν 0 0 k1 k2 [8g ?ν K 2 ? 8K ? K ν + 4k ? k ν ]

(30)

where k = k1 ? k2 . Now, we evaluate the corresponding ghost diagrams which have to be substracted from the above result. The amplitude for the ghost part (see Fig. 1b ) is given by

bc Mgh = gf abc k ? Aa . ?

(31)

so that 1 bc bc ? Tgh = Mgh Mgh 4 Using Eq. (32) in (3) we ?nd Wgh =

(1)

(32)

1 Ng 2 4 16π 2

K 2 >0

d4 K θ(K 0 )

d3 k1 d3 k2 (4) δ (K ? k1 ? k2 )Aa (K)Aa (?K)[k ? k ν ] ? ν 0 0 k1 k2

(33)

Substracting Eq. (33) from (30) we ?nd WA→gg =

(1)

1 Ng 2 4 32π 2

K 2 >0

d4 K θ(K 0 )

d3 k1 d3 k2 (4) δ (K ? k1 ? k2 )Aa (K)Aa (?K) ? ν 0 0 k1 k2 [8g ?ν ? 8K ? K ν + 2k ? k ν ]

(34)

To perform the integration in the above equation we proceed as follows. First of all using d3 k2 = 0 2k2 we check that d3 k1 d3 k2 (4) δ (K ? k1 ? k2 )g ?ν = 2πg ?ν . 0 0 k1 k2 8 (36)

2 0 d4 k2 δ(k2 ) θ(k2 )

(35)

Using Eq. (36) we ?nd d3 k1 d3 k2 (4) ? ν δ (K ? k1 ? k2 )k1 k2 = 0 0 k1 k2 d3 k1 d3 k2 (4) ? ν δ (K ? k1 ? k2 )k2 k1 0 0 k1 k2 π = [K 2 g ?ν + 2K ? K ν ], 6

(37)

and d3 k1 d3 k2 (4) ? ν δ (K ? k1 ? k2 )k1 k1 = 0 0 k1 k2 d3 k1 d3 k2 (4) ? ν δ (K ? k1 ? k2 )k2 k2 0 0 k1 k2 π = [?K 2 g ?ν + 4K ? K ν ]. 6

(38)

With the help of the above relations the integration in Eq. (34) can be easily performed. Using (36), (37) and (38) we obtain from Eq. (34) WAgg =

(1)

11Nαs 24

K 2 >0

d4 K θ(K 0 ) [K 2 Aa (K) · Aa (?K) ? K · Aa (K)K · Aa (?K)]

(39)

In the above expression the repeated indices are summed from a = 1, ...(N 2 ? 1). For SU(3) gauge group we obtain WAgg =

(1)

11αs 8

K 2 >0

d4 K θ(K 0 ) [K 2 Aa (K) · Aa (?K) ? K · Aa (K)K · Aa (?K)]

(40)

This is the probability computed from the process A → gg via vacuum polarization. It can be seen that the above expression is transverse with respect to the momentum of the ?eld (1) K, i.e, when A? is replaced by K ? we get WAgg = 0. This result is a part of the expression for the total probability (including higher order terms in gA) for the production of gluon pairs.

IV. CONCLUSIONS

Using the background ?eld method of QCD, we have computed the probability for the processes A → q q , gg via vacuum polarization, in the presence of a space-time dependent ? chromo?eld A. These processes are similar to A → e+ e? in QED. For any arbitary classical non-abelian chromo?eld one has to check the gauge invariance of the result with respect to the general non-abelian local gauge transformation which is known as type I gauge transformation [16]. Under this gauge transformation the classical chromo?eld transform like: i A′? → UA? U ?1 ? (?? U)U ?1 . cl cl g The gluonic and ghost ?elds transform like: A′? → UA? U ?1 , q q and 9 (42) (41)

χ′ → UχU ?1

(43)

respectively. In these situations one has to add the results of the higher order processes (for the gluon pair production) to the leading order results obtained in this paper. The gluon pair production amplitude which is obtained from the ?rst order S matrix contains higher order terms in gA. This is due to the non-abelian nature of the gluonic and classical chromo?eld. The result of the gluon pair production probability which contains both leading order and higher order terms in gA (but ?rst order in S matrix) will be reported elsewhere [17]. In this paper we have presented only the result from the leading order processes A → gg. This result is a part of the expression for the total probability of the gluon production from a spacetime dependent chromo?eld [17]. Quark and gluon production from a space-time dependent chromo?eld will play a crucial role in the production and equilibration of the quark-gluon plasma in ultra relativistic heavy-ion collsions. Our main goal is to study the production and equilibration of the quark-gluon plasma by solving relativistic non-abelian transport equations for partons (see Eq. (1)) with parton production from a space-time dependent chromo?eld taken into account. Such work requires extensive numerical computation [18] and needs a separate publication.

ACKNOWLEDGMENTS

We thank Dennis D. Dietrich for useful discussions, reading of the manuscript and drawing the Feynman diagrams. We also thank Dr. Qun Wang and Dr. Chung-Wen Kao for useful discussions. G.C.N. acknowledges the ?nancial support from Alexander von Humboldt Foundation.

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REFERENCES

[1] J. Schwinger, Phys. Rev. 82, 664 (1951). [2] A. Karman, T. Matsui and B. Svetitsky, Phys. Rev. Lett. 56, 219 (1986) and references therein. [3] See, e.g., L. McLerran and B. Svetitsky, Phys. Rev. D24 (1981) 450; L. McLerran, Phys. Rev. D36 (1987) 3291; R.V. Gavai, in Quantum Fields on the Computer, ed. M. Creutz, (World Scienti?c, 1992), p. 51; F. Karsch and E. Laermann, Rep. Prog. Phys. 56 (1993) 1347; M. Oevers, F. Karsch, E. Laermann and P. Schmidt, in Proc. of Lattice ’97: Nucl. Phys. Proc. Suppl. 63 (1998) 394. [4] Proceedings of 14th International Conference on Ultrarelativistic Nucleus-Nucleus Collisions (Quark Matter 99), Torino, Italy, 10-15 May 1999, Nucl. Phys. A 661 (1999). [5] Gouranga C. Nayak and V. Ravishankar, Phys. Rev. D 55, 6877 (1997); Phys. Rev. C 58, 356 (1998). [6] Larry McLerran and Raju Venugopalan, Phys. ReV. D 49, 2233 (1994); Alex Kovner, Larry McLerran and Heribert Weigert, Phys. Rev. D 52, 6231 (1995). [7] H-T Elze and U. Heinz, Phys. Rep. 183, 81 (1989). [8] S.K. Wong, Nuovo Cimento A 65, 689 (1970). [9] A. Casher, H. Neuberger and S. Nussinov, Phys. Rev. D 20, 179 (1979). [10] R. S. Bhalerao and V. Ravishankar, Phys. Lett. B409, 38 (1997). [11] C. Itzykson and J. Zuber, Quantum Field Theory (McGraw-Hill Inc., 1980). [12] B. S. Dewitt, Phys. Rev. 162, 1195 and 1239 (1967); in Dynamic theory of groups and ?elds (Gordon and Breach, 1965). [13] G. ’t Hooft, Nucl. Phys. B62, 444 (1973). [14] L. F. Abbott, Nucl. Phys. B185, 189 (1981). [15] J. Honerkamp, Nucl. Phys. B36, 130 (1971); B48, 269 (1972); R. Kallosh, Nucl. Phys. B78, 293 (1974); S. Sarkar, Nucl. Phys. B82, 447 (1975); S. Sarkar and H. Strubbe, Nucl. Phys. B90, 45 (1975); H. Kluberg-Stern and J. B. Zuber, Phys. Rev. D 12, 3159 (1975); M. Grisaru, P. van Nieuwenhuizen and C. C. Wu, Phys. Rev. D 12, 3203 (1975). [16] B. W. Lee and J. Zinn-Justin, Phys. Rev. D7, 1049 (1973); H. Kluberg-Stern and J. B. Zuber, Phys. Rev. D12, 482 (1975). [17] D. D. Dietrich, G. C. Nayak and W. Greiner, ”Leading order gluon pair production from space-time dependent chromo?eld” To be communicated. [18] R. S. Bhalerao and G. C. Nayak, Phys. Rev. C 61, (2000) 054907.

11

(a) K A a,m

b,n K A a,m

(b)

b

k1 k2

k1 k2

c,l

c

Fig. 1 Lowest order vacuum polarization diagrams. Fig. 1(a) is the gluon diagram and Fig. 1(b) is the corresponding ghost diagram.

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