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arXiv:1106.5231v1 [nucl-th] 26 Jun 2011

QUANTUM SPECTRUM OF CHERENKOV GLUE

M. N. Alfimov∗, A. V. Leonidov†

P.N. Lebedev Physical Institute

119991 Leninsky pr. 53, Moscow, Russia

Abstract

Full quantum calculation of Cherenkov gluon radiation by quark and gluon currents and a

Cherenkov decay of a gluon into a pair of Cherenkov gluons in transparent media is performed.

Energy losses due to Cherenkov gluon radiation in high energy nuclear collisions are calculated. The

angular distribution of the energy flow due to the radiation of Cherenkov gluons is analyzed.

∗Also at the Moscow Institute of Physics and Technology.

†Also at the Institute of Theoretical and Experimental Physics.

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1Introduction

Experimental observation of the two-humped structure of dihadron angular correlations in ultrarel-

ativistic heavy ion collisions at RHIC [1, 2, 3, 4, 5, 6] bearing a remarkable likelihood to the angular

distribution of Cherenkov photons [7] has brought into the focus of attention a possible existence of the

phenomenon of Cherenkov radiation of gluons, an idea formulated in [8, 9] and applied to the analysis

of ring-like structures in cosmic ray events in [10].

Interpretation of the experimental data in terms of the Cherenkov radiation of gluons is not unique.

Theoretical descriptions aiming at describing the double-humped angular pattern of two-particle az-

imuthal correlations include that in terms of the Mach cone generated by jets propagating in dense

medium, see e.g. the recent analysis in [11], as well as in terms of originating from dynamical fluctu-

ations of the expanding hot and dense fireball [12]. From the experimental point of view it has been

demonstrated [13] that in the case of large rapidity interval between the two particles and for one specific

choice of transverse momenta bins for trigger and associated particles the resulting angular pattern can

be completely described by the azimuthal asymmetries of the collective flow. In the case of narrow

rapidity interval the situation looks different, see a detailed argumentation in [11], and the problem of

finding an appropriate description for the experimental data at the level of detalization of [3] is, in our

opinion, still open.

The Cherenkov radiation of gluons is a manifestation of nontrivial properties of nonabelian medium

created in ultrarelativistic heavy ion collisions [14].

Cherenkov gluon radiation and a summary of earlier work was presented in [15]. The analysis of [15]

was based on a straightforward generalization of the classical Tamm-Frank theory [16]. A simple field-

theoretical model of two interacting scalar fields leading to Cherenkov excitations was considered in [17].

A model taking into account the opacity of the medium and rescattering of Cherenkov gluons consid-

ered in [18] was shown to successfully reproduce the experimental data on double-humped correlations

[1, 2, 3, 4, 5, 6]. A new line of studies was started in [19, 20] where a theory of Cherenkov radiation of

mesons was constructed in the framework of holographic approach to strong interactions.

To develop a more reliable theoretical picture for Cherenkov radiation of gluons one has to generalize

the classical approach of [8, 9, 15] and the simple scalar field model of [17] to a quantum field theory

description based on in-medium QCD. The main goal of the present paper is to develop such an approach

to Cherenkov gluon radiation of quark and gluon currents1. Our consideration is essentially based on

the quantum theory of electromagnetic Cherenkov radiation developed in [22], see also [23]. Recently

the approach of [23] was generalized to the case of a moving medium [24].

The calculation of Cherenkov gluon radiation by quark currents presented below is a straightforward

generalization of the abelian case considered in [22]. The calculation of Cherenkov radiation of gluon

currents and of the gluon decay into a pair of Cherenkov gluons are new. The corresponding expressions

and the resulting qualitative picture of the pattern of energy loss related to the Cherenov radiation

present the main results of the present paper.

The plan of the paper is as follows.

In the paragraph 2.1 we give some general remarks on the physics of Cherenkov radiation.

In the paragraph 2.2 we compute the rate of the single Cherenkov decay of the quark current.

In the paragraph 2.3 we compute the rate of the single Cherenkov decay of the gluon current.

In the paragraph 2.4 we compute the rate of the double Cherenkov decay of the gluon current.

In the Appendix A we describe a simple field-theoretical model justifying the Feynman rules for

in-medium QCD used in the paper.

The interpretation of RHIC data in terms of

2Single and double Cherenkov decays

In this section we compute the spectra of Cherenkov gluons radiated by quark and gluon currents

and the spectrum of Cherenkov gluons created in the decay of a free gluon into two Cherenkov gluons.

1Some preliminary results were discussed in [21].

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2.1General remarks

The phenomenon of Cherenkov radiation has its origin in the nontrivial changes of the dispersion

relation for the excitations (quasiparticles) in the medium (as seen from the poles of the propagators):

1

ω2− k2=⇒

1

ε(ω,k)ω2− k2,(2.1)

where ε(ω,k) is a (chromo)permittivity of the medium under consideration. In what follows we shall

concentrate on the simplified treatment in which the standard in-vacuum quark and gluon currents

interact with the transverse in-medium excitations, the Cherenkov gluons2. In this setting the Cherenkov

radiation is a decay of a free vacuum particle q(g) into a quasiparticle ˜ g and a free particle q(g) possible

for certain special values of the permittivity ε(ω,k) > 1 that allow an existence of transverse massless

excitations, the Cherenkov gluons, so that, e.g., for the Cherenkov radiation of quark current we have

q(ω1,k1) → q(ω2,k2) ⊕ ˜ g(ω3,k3).(2.2)

In the simplest QED case the Cherenkov radiation is a decay of a free electron into a free in-medium

photon and a free electron [22]. Another interesting process to study is a decay of free in-vacuum gluons

g(ω1,k1) → ˜ g(ω2,k2) ⊕ ˜ g(ω3,k3).(2.3)

The Cherenkov gluon emission is of course possible only for special values of energy and momenta

of the three participating gluons so that the energy-momentum conservation for the considered decay is

fulfilled. To give a quantitative description for this possibility one has to consider an explicit model for

the chromopermittivity tensor ε(ω,k). Generically chromopermittivity is a nontrivial matrix in the color

space εab(ω,k). The nontrivial color structure of εab(ω,k) leads, in particular, to the appearance of the

color Cherenkov rainbow [14]. In what follows we shall confine ourselves to the simplest quasiabelian

case, where εab(ω,k) → δabε(ω) and use in our qualitative estimates a model for ε(ω,k):

ε(ω) = ε > 1, ω < ω0

ε(ω) = 1, ω > ω0.

(2.4)

(2.5)

The Cherenkov radiation is then possible for excitations with energies in the interval ω < ω0.

In what follows we shall use in our numerical estimates the values ε = 5 and ω0= 3 GeV obtained

by fitting the experimental data in [18].

2.2Cherenkov decay of quark current

Let us illustrate the approach we use in this paper by presenting a detailed calculation of the spectrum

of Cherenkov gluons radiated by the massless quark current. The process in question is then a decay of

a free quark into a free quark and a Cherenkov gluon,

q(p) → q(q − p) + ˜ g(q),(2.6)

where an incident quark q(p) propagates along the z axis and has the four-momentum pµ= (E,0,0,E)

and ˜ g(q) is a Cherenkov gluon having the four-momentum qµ= (ω,|q|sinθ,0,|q|cosθ) and characterized

by the in-medium dispersion law ω =√ε|q| emitted at the Cherenkov angle θ with respect to the direction

of the incident particle. The corresponding cut diagram is shown in Fig. 13. The final quark has the

four-momentum p′µ= ((E − ω),−(E − ω)sinβ,0,(E − ω)cosβ). The conservation of four-momentum

in the decay leads to the following equalities fixing the Cherenkov and recoil angles θ and β:

?

sinβ =ω

E1 − ω/E

The familiar classical expression for the Cherenkov angle cosθ = 1/√ε follows from (2.7) in the limit

ω/E → 0. Let us note that in the energy range characterizing the trigger and associate particles,

correspondingly E and ω, in correlation measurements in heavy ion collisions, in particular in reference

cosθ =

1

√ε

1 +ε − 1

2

?ε − 1

ω

E

?

,(2.7)

1

ε

?

1 −ω

E−ε − 1

4

?ω

E

?2?1/2

(2.8)

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qq

p − qp − q

pp

Figure 1: Cherenkov decay of the quark current g(p) → g(q − p) + ˜ g(q).

to RHIC data on two-humped azimuthal angular correlations, the energy-dependent term in (2.7) can

be numerically important.

The angle β characterizes the straggling of the incident particle in the transverse plane. From (2.8)

we see that in the classical limit ω/E → 0 the leading contribution to the transverse momentum of the

final quark reads

?ε − 1

ε

?ε − 1

ε

so that, at given ω, the relative importance of transverse struggling is directly dependent on the value

of ε.

Let us also note that from (2.7) there follows the restriction on the energy of Cherenkov gluon

|p′

T| = ω

?

1 −ω

E−ε − 1

4

?ω

E

?2?1/2

,(2.9)

|p′

T|ω/E→0∼ ω

,(2.10)

ω

E≤

2

√ε + 1.(2.11)

The matrix element for the Cherenkov decay (2.6) q → q˜ g reads

iM

s→s′j

q→q˜ g i→ka= us′(p − q)(−igγl)(ta)kius(p)˜ e(j)

l(q).(2.12)

The polarization vectors of the Cherenkov gluon ? e(j)should satisfy the in-medium transversality condi-

? e(1)(q) =

Summation and averaging over the spin and color indices of the matrix element squared gives

tion q? e(i)(q) = 0 (in the present paper we use the Coulomb gauge) and can be chosen in the form4

√ε(0,1,0),

1

? e(2)(q) =

1

√ε(cosθ,0,−sinθ).(2.13)

1

2Nc

?

s,s′,j,i,k,a

|M

s→s′j

q→q˜ g i→ka|2=g2(N2

c− 1)

Ncε

?2|p|2sin2θ + 2|p||p′|(1 − cosβ)?,(2.14)

from which, taking into account the dispersion law for the Cherenkov gluon, it is straightforward to

compute the differential decay rate into an interval [ω,ω + dω]:

?

γq→q˜ g(ω|E) = αs(N2

c− 1)

2Nc

1 −1

ε

??

1 −ω

E+ε + 1

4

ω2

E2

?

.(2.15)

As expected, it differs from the QED answer [22] only by the Casimir invariant for the fundamental

representation of SU(Nc), CF = (N2

is simply given by

Pq→q˜ g(ω|E) = ω γq→q˜ g(ω|E).

2Let us note that a more complete treatment of the problem at hand would involve a trilinear interaction of quasiparticles.

A sketch of the corresponding field-theoretical formalism is given in the Appendix A.

3The actual calculations in the paper are performed by straightforward computation of |M|2.

4See Appendix A for details.

c−1)/2Nc. The corresponding differential energy loss per unit time

(2.16)

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The differential energy loss spectrum (2.16) can be used for computing two observables of physical

interest.

First, using the fact that there exists a one-to-one correspondence between the Cherenkov angle θ

and the energy of the Cherenkov gluon ω, it is straightforward to reinterpret (2.16) as describing the

energy flow into an angular interval [θ,θ + dθ]

P(θ) = P(ω(θ)|E)∂ω(θ)

∂θ

. (2.17)

The resulting distribution is shown in Fig. 2. We see that the energy flow is confined to an angular

interval [θ0,θc] (shaded region in Fig. 2) where the lower limit θ0is obtained from

?

and the upper limit θc corresponds to the classical Cherenkov angle cosθ0 = 1/√ε corresponding to

taking the limit ω/E → 0 in (2.16).

Second, by integrating the differential spectrum (2.16) over ω, one gets an expression for the energy

loss per unit time:

dEq→q˜ g

dt

0

Note that the energy loss per unit time can be easily converted into the energy loss per unit length. The

loss per unit time and per unit length are connected via the relation

cosθ0=

1

√ε

1 +ε − 1

2

ω0

E

?

,(2.18)

(E|ω0,ε) =

?min{ω0,

2E

√ε+1}

dωPq→q˜ g(ω|E).(2.19)

dE

dl

=1

v

dE

dt,

(2.20)

where v is the speed of the incident particle. In the chosen system of units the speed of quark and gluon

is v = 1, so we have the result for the energy loss per unit length

dEq→q˜ g

dl

(E|ω0,ε) =

?min{ω0,

2E

√ε+1}

0

dωPq→q˜ g(ω|E).(2.21)

The resulting energy loss is plotted in Fig. 3. We see that the Cherenkov energy loss rate for the

quark current is quite substantial.

0 0

0.01 0.01

0.02 0.02

0.03 0.03

0.04 0.04

0.05 0.05

0.06 0.06

0.07 0.07

00

π/4

θθ

π/2

π/2

P(θ)/E2

π/4

P(θ)/E2

Figure 2: The angular differential energy flow

of quark Cherenkov radiation, ε = 5, ω0 =

3 GeV.

0

1

2

3

4

5

6

0 3 6 9 12 15

dE/dl, GeV/fm

E, GeV

Figure 3: The quark Cherenkov energy loss,

ε = 5, ω0= 3 GeV.

2.3Cherenkov decay of gluon current

Let us now turn to the consideration of the Cherenkov gluon radiation by the gluon current. Analo-

gously to (2.6) this process is a decay of in-vacuum gluon into in-vacuum and Cherenkov gluons:

g(p) → g(q − p) + ˜ g(q),(2.22)

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qq

p − qp − q

pp

Figure 4: Cherenkov decay of the gluon current q(p) → q(q − p) + ˜ g(q).

0 0

0.05 0.05

0.1 0.1

0.15 0.15

0.2 0.2

0.25 0.25

0.3 0.3

00

π/4

θe

π/2

π/2

P(θe)/E2

π/4

θe

P(θe)/E2

Figure 5: The angular differential energy flow

of gluon Cherenkov radiation, ε = 5, ω0 =

3 GeV.

0

2

4

6

8

10

12

14

16

3 4 5 6 7 8 9 10 11 12 13 14 15

E, GeV

dE/dl, GeV/fm

Figure 6: The gluon Cherenkov energy loss, ε =

5, ω0= 3 GeV.

The corresponding cut diagram for this process is shown in Fig. 4. The kinematics of the gluon

Cherenkov decay (2.22) is completely equivalent to that of (2.6) and is described by the equations

(2.7,2.8).

The matrix element Mg→g˜ gof the decay (2.22) reads

g→g˜ g= −igfabc?

+ 2(qe(i)(p))(e(j)(p′)? e(k)(q))−

Mi→jk

2(p? e(k)(q))(e(i)(p)e(j)(p′))+

−2(qe(j)(p′))(e(i)(p)? e(k)(q))

?

,(2.23)

where the polarization vectors ? e(1,2)are described in Eq. (2.13). A straightforward computation leads

γg→g˜ g(ω|E)

ε

?

21 −ω

to the following expression for the differential decay rate

=αsNc

?

1 −1

?

??

1 −ω

E−ε − 1

4

ω2

E2

?

?

ω2

E2+

× 1 +1

ε +ε + 1

E

+

ε

?1 −ω

E

?2

(ε + 1)2

8?1 −ω

E

?2

ω4

E4

?

,(2.24)

from which one can compute in complete analogy with the calculations described in the previous para-

graph. The only difference in the formula for the energy loss

dEg→g˜ g

dl

(E|ω0,ε) =

?min{ω0,E−ω0,

2E

√ε+1}

0

dωPg→g˜ g(ω|E),(2.25)

where Pg→g˜ g(ω|E) = ωγg→g˜ g(ω|E), is the existence of the additional restriction on the energy of the

emitted Cherenkov gluon ω < E −ω0, which is determined by the fact that the energy of the ”ordinary”

gluon after the emission is greater than ω0. The angular distribution of the energy flow and the rate of

the energy loss shown in Fig. 5 and Fig. 6 correspondingly.

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qq

p − qp − q

pp

Figure 7: Double Cherenkov decay of the gluon current g(p) → ˜ g(q − p) + ˜ g(q).

0

0.05

0.1

0.15

0.2

0.25

0.3

0

π/2

θ

π

P(θ)/E2

Figure 8: The angular differential energy flow

of the gluonic double Cherenkov decay, ε = 5,

ω0= 3 GeV.

0

0.5

1

1.5

2

2.5

3 4 5 6

τ, fm

E, GeV

Figure 9: The lifetime of the gluon decaying

through double Cherenkov decay, ε = 5, ω0=

3 GeV.

The most important feature of the gluonic Cherenkov decay is the large value of the energy loss, see

Fig. 6. This is to be expected if the cutoff energy ω is not too small and ε is not too close to 1 which is

definitely not the case for the values taken from the fit made in [18].

2.4 Double Cherenkov decay of gluon current

Let us now turn to the analysis of the another purely non-abelian mechanism coupling ordinary

gluons to the Cherenkov transverse gluonic excitations, the double Cherenkov decay

g(p) → ˜ g(q − p) + ˜ g(q), (2.26)

The corresponding cut diagram for this process is shown in Fig. 7.

Kinematics of the double decay (2.26) differs from that of (2.6) and (2.22).

First, the double decay (2.26) is possible only for ω0< E < 2ω0, i.e. in the restricted interval of the

energy of the decaying gluon.

Second, the energy-momentum conservation laws impose restrictions on the decay angles

cosθ =√ε −ε − 1

cosβ =√ε −ε − 1

2√ε

E

ω

(2.27)

2√ε

E

E − ω,

(2.28)

from which we obtain the following restrictions on the energy of the emitted in-medium gluons

1

2−

1

2√ε<ω

E<1

2+

1

2√ε.

(2.29)

Let us note that, as follows from (2.27), for the typical value E/ω = 2 corresponding to the center of

the allowed interval (2.29) one has cosθ = 1/√ε, i.e. the angle equals the ”classical” Cherenkov one.

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Calculation of the double Cherenkov decay is completely analogous to that for the Cherenkov decay

considered in the previous paragraph. The matrix element now reads

Mi→jk

g→˜ g˜ g= −igfabc?

2(p? e(k)(q))(e(i)(p)? e(j)(p′))+

+ 2(qe(i)(p))(? e(j)(p′)? e(k)(q))−

−2(q? e(j)(p′))(e(i)(p)? e(k)(q))

?

2ε

?

(2.30)

and the corresponding expression for the differential decay rate takes the form

γg→˜ g˜ g(ω|E) =αsNc

2

?

1 −

?√ε −ε − 1

2√ε

E

ω

?2?

1 + εω2

E2+

ω2

E2

(1 −ω

E)2+ ε1 −ε − 1

1

1 −ω

E

+

ω2

E2

1 −ω

E

?2

(2.31)

.

The angular distribution of the energy flow for the double Cherenkov decay is shown in Fig. 8. The

corresponding energy loss is so large5(approximately three times larger than for the single Cherenkov

decay considered in the previous paragraph) that it makes more sense to plot, instead of it, the lifetime

of a decaying gluon as a function of its energy shown in Fig. 9. We see that unless the energy is not too

close to the decay boundary of 2ω0, the decay turns out to be extremely fast.

2.5Cherenkov energy losses

The results of studying the single Cherenkov decays of quark and gluon currents and the double

Cherenkov decay of gluon currents lead to the following generic picture for the Cherenkov-related energy

loss.

• For quark currents the only available decay channel is the single Cherenkov decay. The correspond-

ing energy loss is non-negligible but subleading with respect to that of the gluon current.

• For incident gluons with energy in the interval ω0< E < 2ω0the leading contribution to the energy

loss comes from the double Cherenkov decay. The corresponding pattern of angular correlations

corresponds to two peaks around the direction of propagation of the decaying gluon. There also

exists a small contribution due to single Cherenkov decay.

• At the threshold energy E = 2ω0 there takes place a regime switch between the predominant

double Cherenkov decay at E < 2ω0to the single Cherenkov decay of quark and gluon currents

at E > 2ω0 where one expects the possible appearance of the third hump corresponding to the

incident particle. Besides that, as follows from (2.7) and (2.27), at the threshold E = 2ω0there

takes place the following change in the Cherenkov angle:

cosθ =√ε −ε − 1

2√ε

2ω0

ω

→ cosθ =

1

√ε

?

1 +ε − 1

2

ω

2ω0

?

.(2.32)

It is easy to verify that the Cherenkov angle of the double Cherenkov decay is greater than the

Cherenkov angle for the single Cherenkov decay for all ω. This means that in addition to the regime

switch between the dominant energy loss processes we also have a sharp change of the Cherenkov

angles of emitted gluons at E = 2ω0.

These features appear to be in qualitative agreement with the pattern of angular correlations mea-

sured at RHIC [3]. A detailed comparison with the data will be published separately.

Acknowledgements

We are grateful to I.M. Dremin for useful discussions.

The work of A.L. was supported by RFBR grant 09-02-00741 and CERN-RAS program. The work

of M.A. was supported by the 2010 Dynasty Foundation Grant.

5For the initial gluon with the energy E = 5 GeV it is about 35 GeV/fm.

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Appendix

A Field theory model for in-medium QCD

In this appendix we consider a simple field theory model justifying the Feynman rules for in-medium

QCD used in the present paper. Our consideration will be confined to the case of QCD matter in its

rest system.

The notion of dielectric permittivity in in-medium QED in the case of the homogenous, isotropic

medium with temporal dispersion arises in describing a response to an external electric field:

?t

or, in the Fourier space:

D(t,x) =

−∞

dt′ε(t − t′)E(t′,x) =

?+∞

0

dτε(τ)E(t − τ,x), (A.1)

D(ω,r) = ε(ω)E(ω,r),ε(ω) =

?+∞

0

dτε(τ)eiωτ

(A.2)

Let introduce the following simple action for the in-medium QCD (note that we use the quasiabelian

model εab(t − t′) = δabε(t − t′)):

?

where

W(t,t′) = Pexp−ig

d4xTr

??t

−∞

dt′ε(t − t′) F0i(t′,x) W(t,t′) F0i(t,x)W†(t,t′) −1

2FijFij

?

(A.3)

??

?t

t′A0(τ,x)dτ

??

(A.4)

is the Wilson line introduced to preserve the local gauge invariance. It is convenient to choose the

Coulomb gauge Aa0= 0, ∂iAai= 0 in which the action (A.3) simplifies to

?1

S =

?

d4x

2

?t

−∞

dt′ε(t − t′)∂0Aai(t′,x)∂0Aai(t,x) −1

4FaijFaij

?

(A.5)

Let us now divide the action (A.5) into the free field and interaction contributions:

S0=

?

d4x

?1

2

?

?t

−gfabc(∂iAai)AbiAcj−g2

−∞

dt′ε(t − t′)∂0Aai(t′,x)∂0Aai(t,x) −1

4(∂iAaj− ∂jAai)(∂iAaj− ∂jAai)

?

?

(A.6)

Sint=

?

d4x

4fabcfaedAbiAcjAeiAdj

(A.7)

From the decomposition (A.6) there follows that in the Coulomb gauge the form of the triple-gluon

interaction remains unchanged. Turning now to the free field contribution we obtain after integration

by parts:

?

Varying the action (A.8) with respect to the field Aai(x) we arrive at the following equations of motion

S0=d4x1

2

??t

−∞

dt′ε(t − t′)∂0Aai(t′,x)∂0Aai(t,x) − (∂iAaj)(∂iAaj)

?

(A.8)

?+∞

0

dτε(τ)(∂0)2Aai(t − τ,x) − (∂j)2Aai(t,x) = 0(A.9)

or, in the Fourier space,

?ε(k0)(k0)2− k2?? Aai(k) = 0(A.10)

The above consideration shows that within the chosen model of chromoelectric permittivity we have two

different branches of the dispersion relation (two different types of excitations):

|k| =√εω, ω < ω0

|k| = ω, ω > ω0

(A.11)

(A.12)

9

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and, therefore, the following decomposition for Aai:

Aai(x) = Aai

(1)(x) + Aai

?

?

(2)(x)

d3k

(2π)3

(A.13)

Aai

(1)(x) =

|k|<√εω0

1

√2k0

?

?

λ

?

k(λ)e−ikx+ ca†

ba

k(λ)e−ikx+ ba†

k(λ)eikx?

k(λ)eikx?

? ei

(λ)(k), |k| = k0;

(λ)(k), |k| =√εk0;(A.14)

Aai

(2)(x) =

|k|>ω0

d3k

(2π)3

1

√2k0

?

λ

ca

ei

(A.15)

It is easy to check that in order to be consistent with the commutation relations for the vector potential

the gluon polarization vectors should be normalized as follows: (ei

and ε(? ei

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