Chiba Univ. Preprint CHIBAEP123
hepth/0009152
September 2000
Dual superconductivity,
monopole condensation and confining string
in lowenergy YangMills theory. Part I.
KeiIchi Kondo

Department of Physics, Faculty of Science, Chiba University, Chiba 2638522, Japan
Abstract
We show that the QCD vacuum (without dynamical quarks) is a dual superconductor at least in the lowenergy region in the sense that monopole condensation does really occur. In fact, we derive the dual GinzburgLandau theory (i.e., dual Abelian Higgs model) directly from the SU(2) YangMills theory by adopting the maximal Abelian gauge. The dual superconductor can be on the border between type II and type I, excluding the London limit. The masses of the dual Abelian gauge field is expressed by the YangMills gauge coupling constant and the mass of the offdiagonal gluon of the original YangMills theory. Moreover, we can rewrite the YangMills theory into an theory written in terms of the Abelian magnetic monopole alone at least in the lowenergy region. Magnetic monopole condensation originates in the nonzero mass of offdiagonal gluons. Finally, we derive the confining string theory describing the lowenergy Gluodynamics. Then the area law of the large Wilson loop is an immediate consequence of these constructions. Three lowenergy effective theories give the same string tension.
Key words: quark confinement, magnetic monopole, QCD, confining string, monopole condensation, dual superconductivity,
PACS: 12.38.Aw, 12.38.Lg Email:
Contents
 1 Introduction

2 Dual superconductivity in lowenergy Gluodynamics
 2.1 Conventions
 2.2 Step 1: NonAbelian Stokes theorem for the Wilson loop
 2.3 Step 2: Cumulant expansion
 2.4 Step 3: Maximal Abelian gauge fixing
 2.5 Step 4: Dynamical mass generation for offdiagonal components
 2.6 Step 5: Lowenergy effective theory for diagonal fields
 2.7 Step 6: Dynamical generation of the kinetic term of
 2.8 Step 7: Dual transformations
 2.9 Step 8: Recovery of hypergauge symmetry and gauge fixing
 2.10 Step 9: Change of variables (pathintegral duality transformation)
 2.11 Step 10: Dual GinzburgLandau theory in the London limit
 3 Final step: Dual GinzburgLandau theory of the general type
 4 Estimation of neglected higherorder terms
 5 Magnetic monopole condensation and area law
 6 Confining string theory and string tension
 7 Parameter fitting for numerical estimation
 8 Conclusion and discussion
 A Useful formulae
 B Derivation of a version of nonAbelian Stokes theorem
 C Calculation of the vacuum polarization for tensor fields
 D Manifest covariant quantization of the second rank antisymmetric tensor gauge field
 E Renormalization of dual Abelian Higgs model
 F Calculation of the Wilson loop
 G Derivative expansion of the string
1 Introduction
The main aim of this paper is to discuss how the dual superconductor picture for explaining quark confinement is derived directly from Quantum Chromodynamics (QCD). It is believed that quark and gluon are confined into the inside of hadrons by the strong interaction described by QCD and that they can not be observed in isolation. Quark confinement can be understood based on an idea of the electromagnetic duality of ordinary superconductivity, i.e., the dual superconductor picture[1]. In this picture, the color electric flux can be excluded from QCD vacuum as a dual superconductor (the dual Meissner effect), just as the magnetic field can not penetrate into the superconductor (the socalled Meissner effect). In this context, the dual is used as implying electromagnetic duality. However, QCD is a nonAbelian gauge theory and its gluonic part is described by the YangMills theory, whereas the usual superconductivity is described by the Abelian gauge theory. Therefore we must make clear the precise meaning of dual superconductivity in QCD. It is possible to assume that the diagonal component of color electric field can be identified with the dual of magnetic field in the ordinary superconductivity. If so, we must answer what is the role played by the offdiagonal component of gluons? Finally, we must show that QCD vacuum has a tendency to exclude the (diagonal component of) color electric field. If a pair of heavy quark and antiquark is immersed in the QCD vacuum, the color electric flux connecting a pair is squeezed into the tubelike region, leading to the formation of QCD (gluon) string between a quark and an antiquark. As a result, the interquark potential is proportional to the interquark distance . To separate a quark from an antiquark, we need infinite energy. In this sense, quark confinement is achieved.
The dual superconductor picture of QCD vacuum is based on the following assumptions:

Existence of magnetic monopole: QCD has magnetic monopole.

Monopole condensation: Magnetic monopole is condensed in QCD.

Infrared Abelian dominance: Charged gluon (i.e., offdiagonal gluon) can be neglected at least in the lowenergy region of QCD.

Absence of quantum effect: Classical configuration, e.g., the magnetic monopole, is dominant. Hence quantum effect can be neglected.
Each of these statements should be explained based on QCD to really confirm the dual superconductor picture of QCD vacuum. As a first step, a prescription of extracting the field configuration corresponding to magnetic monopole was proposed by ’t Hooft [2]. This idea is called the Abelian projection. Immediately after the proposal of this idea, infrared Abelian dominance was suggested by Ezawa and Iwazaki [3]. However, recent simulations revealed that the Abelian dominance is not necessarily realized in all the Abelian gauges, see the review [4]. To author’s knowledge, the best covariant gauge fixing condition realizing Abelian dominance is given by the maximal Abelian (MA) gauge proposed by Kronfeld et al. [5]. In fact, the infrared Abelian dominance is first confirmed in MA gauge based on Monte Carlo simulations by Suzuki and Yotsuyanagi [6]. Subsequent simulations have also confirmed the monopole dominance in lowenergy QCD[7]. An analytical derivation of the dual GinzburgLandau (DGL) theory was tried by Suzuki [8] by way of Zwanziger formulation [9] by neglecting the offdiagonal components of gluon fields by virtue of the Abelian dominance and by assuming condensation of magnetic monopoles. However, the DGL theory is not yet derived directly from the underlying theory, i.e. QCD, since the Abelian dominance and monopole condensation themselves must be derived in the same framework of the theory. According to the recent Monte Carlo simulations [10], the presumed dual GinzburgLandau theory is on the border of type II and type I.
In this paper we discuss how to derive the DGL theory as a lowenergy effective theory (LEET) of QCD. The LEET is not unique and we can derive various LEET’s. Of course, they should be equivalent to each other, if they are to be derived directly from QCD. Even if a specific LEET of QCD is assumed, the parameters included in the LEET can be adjusted so as to meet the data of experiments or Monte Carlo simulation on a lattice. In the first paper [11] of a series of papers [12, 13, 14, 15, 16] on the quark confinement in YangMills theory in the Abelian gauge, the author has given a scenario of deriving the dual superconductivity in lowenergy region of QCD and demonstrated that the DGL theory, i.e., the dual Abelian Higgs (DAH) theory in the London limit can be derived from QCD, provided that the condensation of magnetic monopole takes place. In this scenario, the mass of the dual gluon (dual Abelian gauge field) is given directly by the nonzero condensate of the magnetic monopole current , i.e., . Since the DGL theory is a LEET written in term of only the Abelian (diagonal) component extracted from the original nonAbelian field, the treatment of the offdiagonal component is crucial for deriving the LEET and also for giving reasonable interpretation of the result. In the previous paper [11], all the offdiagonal components are integrated out to write down the LEET in terms of the diagonal components alone. The resulting LEET is an Abelian gauge theory preserving a characteristic feature of nonAbelian gauge theory in the sense that the the function for the coupling constant has the same form as the original YangMills theory, exhibiting the asymptotic freedom [17, 11]. For this procedure to be meaningful, the mass of the offdiagonal components must be heavier than the mass of the diagonal components. This procedure can be justified based on the Wilsonian renormalization group or the decoupling theorem [18]. As a result, the LEET is valid in the energy region below .
The aim of this paper is to show that the dual superconductivity can be derived at least in the lowenergy region of Gluodynamics (i.e., the gluonic sector of QCD). The most important ingredient in deriving the dual superconductivity is the existence of nonzero mass of the offdiagonal gluons. Recently, it has been shown [19, 20] that the offdiagonal gluons and offdiagonal ghosts (antighosts) acquire nonzero masses in YangMills theory in the modified MA gauge [20]. This result strongly supports the Abelian dominance in lowenergy Gluodynamics. The modified MA gauge was already proposed by the author from a different viewpoint in the paper [12]. A remarkable difference of MA gauge from the usual Lorentz type gauge lies in the fact that MA gauge is a nonlinear gauge. In order to preserve the renormalizability of YangMills theory in the MA gauge, it is indispensable to introduce quartic ghost interaction term as a piece of gauge fixing term [21]. The modified MA gauge fixes the strength of the quartic ghost interaction by imposing the symmetry, i.e., orthosymplectic group . The implications of the symmetry have been discussed in the previous papers [12, 13, 14]. The attractive quartic ghost interaction causes the ghostantighost condensation. Consequently, the offdiagonal gluons become massive,^{1}^{1}1The offdiagonal gluon mass in the MA gauge has been calculated on a lattice by Amemiya and Suganuma [23], GeV for . whereas the diagonal gluons remain massless in this gauge. Since the classical YangMills theory is a scale invariant theory, the mass scale must be generated due to quantum effect. In this paper, we show that the monopole condensation does really occur due to existence of nonzero mass of offdiagonal gluons. The offdiagonal gluon mass also provides the mass of dual gauge field in the DGL theory.
It should be remarked that in the conventional approaches the offdiagonal components are completely neglected from the beginning in deriving the effective Abelian gauge theory by virtue of the Abelian dominance. As a result, the conventional approach can not predict the physical quantities without some fitting of the parameters introduced by hand. The purpose of this paper is to bridge between the perturbative QCD in the highenergy region and the lowenergy effective Abelian gauge theory. Consequently, the undetermined parameters in the lowenergy effective theory can in principle be expressed by the parameters of the original YangMills theory, i.e., the gauge coupling constant and the renormalization group (RG) invariant scale .
In this paper we pay attention to the vacuum expectation value (VEV) of the Wilson loop operator , which is written in the framework of the functional integration as
(1.1) 
where is the total YangMills Lagrangian, i.e., YangMills Lagrangian plus the gauge fixing (GF) term including the FaddeevPopov (FP) ghost term in the modified MA gauge. Our derivation is based on the BecchiRouetStoraTyutin [22](BRST) formulation of the gauge theory. Hence, is the integration measure which is BRST invariant, where is the NakanishiLautrap (NL) Lagrangian multiplier field. We regard the Wilson loop as a special choice of the source term in general YangMills theory. The Wilson loop is used as a probe to see the QCD vacuum. The shape of the Wilson loop is arbitrary at this stage.
We seek other theories (with the corresponding source term) which is equivalent to the original YangMills theory at least in the lowenergy region in the sense that it gives the same VEV of the Wilson loop operator as the original YangMills theory for large loop . From this viewpoint, we obtain three LEET’s, i.e., DGL theory (DAH theory), magnetic monopole theory and confining string theory. For this derivation to be successful, the existence of quantum corrections coming from offdiagonal components of gluons and ghost (and antighost) is indispensable. Without quantum corrections, we can not derive magnetic monopole condensation. This is plausible, since it is the quantum correction that can introduce the scale into the gauge theory which is scale invariant at the classical level.
This paper is organized as follows. In section 2, we give a strategy (steps) of deriving the LEET of Gluodynamics. In fact, we demonstrate that at least the London limit of the DGL theory (i.e., DAH model) of type II can be obtained as a very special limit of the resulting LEET of Gluodynamics. However, our method is able to treat more general situation, not restricted to the London limit. Rather, our result suggests that the DGL theory can not be in the London limit.
In section 3, a more general case of the DGL theory is discussed. It is shown that the LEET of the supposed DGL theory agrees with the LEET derived from the YangMills theory according to the above steps, in the energy region less than the mass of the dual Higgs mass. This way of showing equivalence is a little bit indirect. The reason is as follows. We know that the DAH model or the DGL theory is a renormalizable theory (within perturbation theory in the magnetic coupling constant ) and hence is a meaningful theory at arbitrary energy scale. On the other hand, we know that the high energy region of Gluodynamics is correctly described by the nonAbelian YangMills theory with asymptotic freedom. Therefore, the DGL theory is at best meaningful only in the lowenergy region in this context, although it is a renormalizable theory. In this sense, we must be careful in saying that the DGL theory is regarded as a LEET of Gluodynamics.
In section 4, we try to estimate the neglected terms in the derivation of the LEET based on the large argument and the decoupling theorem. It is shown that in the large expansion the higherorder terms are suppressed by compared to the leading order term. Thus the LEET obtained above is considered as the leadingorder result of the large expansion.
The LEET of Gluodynamics is not unique. In fact, we can obtain various LEET’s which are equivalent to each other. Other LEET’s can be more convenient for the purpose of calculating some kinds of physical quantities.
In section 5, we derive the magnetic monopole action as another LEET. The magnetic monopole action is written entirely in terms of the magnetic monopole current .
(1.2) 
by introducing the fourdimensional solid angle . We show that the VEV of the Wilson loop exhibits area law decay for large loop ,
(1.3) 
where is the area of the minimal surface bounded by . The string tension is explicitly obtained and it agrees with that predicted by the DGL theory.. Then we show that the magnetic monopole condensation really occurs in the lowenergy region in the sense that the currentcurrent correlation at the same spacetime point, , has nonvanishing expectation value,
(1.4) 
The result is consistent with that of the monopole action on a lattice[24].
In section 6, we derive a string theory which is equivalent to the LEET’s derived above,
(1.5) 
The action of the confining string is equal to the NambuGoto action for the world sheet with the Wilson loop as the boundary. The string tension in the NambuGoto action is the same as evaluated by the monopole action derived in section 5. The Wilson loop exhibits area law with the string tension . Therefore, the obtained string theory is regarded as a lowenergy limit of the socalled confining string theory proposed by Polyakov [25].
In section 7, we discuss what values of the parameters in the LEET’s should be chosen to reproduce the numerical results.
The final section 8 is devoted to summarizing the result obtained in this paper and discussing the future directions of our investigations. The details of calculations are collected in Appendices, together with useful formulae.
2 Dual superconductivity in lowenergy Gluodynamics
2.1 Conventions
The gauge potential is written as
(2.1) 
where the generators of the Lie algebra of the gauge group are taken to be Hermitian satisfying and normalized as For a closed loop , we define the Wilson loop operator by
(2.2) 
where denotes the pathordered product and is the YangMills coupling constant.
We begin with the vacuum expectation value (VEV) of the Wilson loop operator in the YangMills theory with a gauge group defined by the functional integral,
(2.3) 
where is a normalization factor (or a partition function) to guarantee and is the total action obtained by adding the gauge fixing (GF) and FaddeevPopov (FP) ghost term to the YangMills action ,
(2.4) 
The YangMills action is of the usual form,
(2.5) 
where is the field strength defined by
(2.6) 
The GF+FP action is specified later (see section 2.4). Finally, is the integration measure,
(2.7) 
which is invariant under the BecchiRouetStoraTyutin (BRST) transformation,
(2.8) 
where is the NakanishiLautrap (NL) field and () is the ghost (antighost) field.
2.2 Step 1: NonAbelian Stokes theorem for the Wilson loop
We make use of a version of the nonAbelian Stokes theorem (NAST) [26, 14, 27] to rewrite the Wilson loop operator in terms of the diagonal components. This version of NAST was first obtained by Diakonov and Petrov for [26]. It is possible to generalize their result to ().
Theorem:[27] For a closed loop , we define the nonAbelian Wilson loop operator by
(2.9) 
where is the pathordered product. Then it is rewritten as
(2.10) 
where
(2.11) 
and^{2}^{2}2 Note that is not equal to the diagonal component of .
(2.12) 
Here is the highestweight state of the representation defining the Wilson loop and the measure is the product measure along the loop , , where is the invariant Haar measure on with the maximal stability group . The maximal stability group is the subgroup leaving the highestweight state invariant (up to a phase factor), i.e.,
(2.13) 
for and . It depends on the group and the representation in question.
For , the is given by the maximal torus subgroup irrespective of the representation. Hence . For , however, does not necessarily agree with the maximal torus group depending on the representation. For , all the representations can be classified by the Dynkin index . If or , and . On the other hand, when and , and . Here is the complex projective space and the flag space. This NAST is obtained by making use of the generalized coherent state. For details, see the reference[27].
For the fundamental representation, the expression (2.11) is greatly simplified as
(2.14) 
Therefore,
(2.15)  
(2.16) 
This implies that the nonAbelian Wilson loop can be expressed by the diagonal (Abelian) components. This is suggestive of the Abelian dominance in the expectation value of the Wilson loop.
The monopole dominance in the Wilson loop is also expected to hold as shown follows. We can rewrite in the NAST as
(2.17) 
where
(2.18) 
The is invariant under the full gauge transformation as well as the residual gauge transformation. (Indeed, we can write a manifestly gauge invariant form, see ref.[28].) The is a generalization of the ’t HooftPolyakov tensor of the nonAbelian magnetic monopole, if we identify with the unit vector of the elementary Higgs scalar field in the gaugeHiggs theory:
(2.19) 
This implies that is identified with the composite scalar field and plays the same role as the scalar field in the gaugeHiggs model, even though QCD has no elementary scalar field. This fact could explain why the QCD vacuum can be dual superconductor due to magnetic monopole condensation.
By introducing the magnetic monopole current by (see Appendix A), we have another expression,
(2.20) 
where is the Laplacian and is a twoform determined by the surface element of the surface spanned by the Wilson loop . The derivation is given in Appendix B. Hence, the Wilson loop can also be expressed by the magnetic monopole current . The above results hold irrespective of which gauge theory we consider.
In the case of , the Wilson loop in an arbitrary representation is written in the form [26, 14],
(2.21) 
where is the Abelian gauge field (or the diagonal component of ) defined by
(2.22) 
for an element . Here is a character which distinguishes the different representation defining the Wilson loop, . Moreover, the use of the usual Stokes theorem leads to
(2.23) 
where is the Abelian field strength defined by and is an arbitrary twodimensional surface with the loop as the boundary. Here it should be remarked that is invariant under the full gauge transformation as well as the residual gauge transformation, since it has the same form as the usual ’tHooft–Polyakov tensor describing the magnetic monopole, see [14].
2.3 Step 2: Cumulant expansion
Now we specify the gauge theory in terms of which the VEV of the Wilson loop operator is evaluated. We consider the YangMills theory with gauge fixing term. The gauge fixing is discussed in the next step.
By applying the cumulant expansion to the VEV of the Wilson loop,
(2.24) 
the VEV of the exponential is replaced by the exponential of the connected expectation as
(2.25) 
where we have used and denotes the higherorder cumulants.
The cumulant expansion is a wellknown technique in statistical mechanics and quantum field theory. In what follows, we will neglect the higherorder cumulants in (2.25) as in the analysis of the stochastic vacuum model.^{3}^{3}3In the nonperturbative study of QCD, the cumulant expansion is extensively utilized by the stochastic vacuum model (SVM) [29] where the different version of the nonAbelian Stokes theorem is adopted. In the SVM, the approximation of neglecting higher order cumulants is called the bilocal approximation. The validity of bilocal approximation in SVM was confirmed by Monte Carlo simulation on a lattice [30, 31]. The author would like to thank Dmitri Antonov for this information. The validity of this approximation, i.e, the truncation of the cumulant expansion, can be examined by Monte Carlo simulations on a lattice, as performed for the stochastic vacuum model, see [30]. In the framework of our approach, this approximation can be justified in the sense that the approximation is selfconsistent within the APEGT derived below. See section 4.1 for more details.
2.4 Step 3: Maximal Abelian gauge fixing
First of all, we define the decomposition of the gauge potential into the diagonal and offdiagonal components,
(2.26) 
where and with being the Cartan subalgebra. As a gauge fixing condition for the offdiagonal component, we adopt the modified version of the maximal Abelian (MA) gauge proposed by the author [12],
(2.27) 
where corresponds to the gauge fixing parameter for the offdiagonal components, since the explicit calculation of the antiBRST transformation yields
(2.28) 
In order to see the effect of ghost selfinteraction, we take
(2.29) 
where we must put to recover Eq.(2.27). The most general form of the MA gauge was obtained by Hata and Niigata [32].
By performing the BRST transformation explicitly, we obtain
(2.30)  
In particular, the case is greatly simplified as
(2.31)  
Integrating out the NL field leads to
(2.32)  
The advantages of the modified MA gauge (2.27) are as follows.

is BRST invariant, i.e., , due to nilpotency of the BRST transformation .

is antiBRST invariant, i.e., , due to nilpotency of the antiBRST transformation .

is supersymmetric, i.e., invariant under the rotation among the component fields in the supermultiplet defined on the superspace . The hidden supersymmetry causes the dimensional reduction in the sense of ParisiSourlas. Then the 4dimensional GF+FP sector reduces to the 2dimensional coset (G/H) nonlinear sigma model. See ref.[12] for more details.

is invariant under the FP ghost conjugation,
(2.33) Therefore, we can treat and on equal footing. In other words, the theory is totally symmetric under the exchange of and .

The YangMills theory in the modified MA gauge (with the total action ) is renormalizable. The naive MA gauge
(2.35) spoils the renormalizability, since radiative corrections induce (even for ) the fourghost interaction,
(2.36) owing to the existence of the vertex , see eq.(2.52) and Appendix B of the paper [11]. This is because the MA gauge is a nonlinear gauge. For the renormalizability of the YangMills theory in the MA gauge, therefore, we need the fourghost interaction from the beginning. In fact, the renormalizability of the YangMills theory supplemented with the fourghost interaction was proved to all orders in perturbation theory [21].
In order to completely fix the gauge degrees of freedom, we add the GF+FP term for the diagonal component to (2.28) or (2.29):
(2.37) 
where we have adopted the gauge fixing condition of the Lorentz type, . The choice of the modified MA gauge is essential in deriving the offdiagonal gluon mass.
2.5 Step 4: Dynamical mass generation for offdiagonal components
It is shown that the fourghost selfinteraction is indispensable for the renormalizability of YangMills theory in the MA gauge. Moreover, it has been shown [19, 20] that the attractive fourghost interaction in the modified MA gauge causes the ghost–antighost condensation and that this condensation provides masses for the offdiagonal gluons and offdiagonal ghosts and antighosts. The massive offdiagonal fields do not propagate in the long distance. Therefore, we can neglect offdiagonal components in the lowenergy or longdistance region, except for the renormalization of the remaining diagonal fields. This result strongly supports the infrared Abelian dominance conjectured by Ezawa and Iwazaki [3].
The dynamical mass generation of the offdiagonal components is understood based on the argument of ColemanWeinberg type. See the paper[20]. The offdiagonal gluon propagator is given by
(2.38)  
(2.39) 
The mass of the offdiagonal gluon comes from the ghostantighost condensation caused by the fourghost interaction. Especially, in the SU(2) case, we obtain^{4}^{4}4Here we have used the minimal subtraction scheme (MS) in the dimensional regularization.
(2.40) 
The SU(3) case is more complicated, see [20] for details.
2.6 Step 5: Lowenergy effective theory for diagonal fields
We are going to calculate the VEV of the Abelian components in the given nonAbelian gauge theory. If the nonAbelian gauge theory can be rewritten into the effective Abelian gauge theory which is expressed exclusively in terms of the Abelian components only, we can calculate the above VEV in the resulting effective Abelian theory.
In the previous work [11], the author has derived an effective Abelian gauge theory by integrating out the offdiagonal components. The resulting theory was called the Abelianprojected effective gauge theory (APEGT). The APEGT is expected to be able to describe the lowenergy region of gluodynamics or QCD. In order to obtain APEGT, we have introduced an antisymmetric tensor field which enables us to derive the dual (magnetic) theory which is expected to be more efficient for describing the lowenergy region. The magnetic theory can be obtained by the electromagnetic duality transformation from the electric theory and vice versa. We have imposed the following duality in the tree level,
(2.41) 
where denotes the Hodge star operation (or duality transformation) [33, 34] defined by
(2.42) 
First, the YangMills Lagrangian is decomposed as
(2.43) 
The first piece is expanded as
(2.44)  
The simplest form satisfying the duality requirement (2.41) is given by
(2.45)  
On the other hand, by defining the covariant derivative with respect to the Abelian gauge filed,
(2.46) 
the second piece is rewritten as
(2.47) 
The tensor field is introduced in such a way that integration recovers the original YangMills theory. is an auxiliary field, since it doesn’t have the corresponding kinetic term. However, the duality requirement leads to ambiguities for the identification as to what is the dual of . In fact, existence of two parameters in (2.41) reflects this ambiguity.
In particular, when , is nothing but the diagonal component of the nonAbelian field strength, , and hence, . In view of this, the choice (2.41) is a generalization of that of the previous paper [11] in which two special cases, and for have been discussed as eq. (2.9) and eq. (2.12) respectively [11]. The latter case has been first discussed by Quandt and Reinhardt[17]. In the case where , the choice completely eliminates the quartic selfinteraction among offdiagonal gluons.
The derivation of the APEGT was improved recently [35] so as to obtain the APEGT in the systematic way to the desired order where we have required the renormalizability of the resulting effective Abelian gauge theory as a guiding principle. We will discuss how to choose and in subsection 3.4.
The strategy of deriving the APEGT is not unique. A way for obtaining the APEGT is to separate each field into the highenergy mode and the lowenergy mode (i.e., ) and then to integrate out the highenergy modes,
of all the fields according to the idea of the Wilsonian renormalization group (RG). The resulting theory will be written in terms of the lowenergy modes . However, we can neglect the lowenergy modes of and due to Abelian dominance and the final theory can be written in terms of the lowenergy modes, . In other words, the offdiagonal components and have only the highenergy modes. To oneloop level, the highenergy modes of the diagonal components don’t contribute to the results. Therefore, we can identify and with the highenergy modes to be integrated out for obtaining the LEET. This strategy was adopted in the paper [35]. We do not adopt this method in this paper.
Another way is to integrate out all the massive fields
(2.48) 
Then the resultant theory will be written in terms of the massless or light fields . The effect of the massive fields will appear only through the renormalization of the resultant theory. This is an example of the decoupling theorem [18]. The only role of the heavy fields in the low momentum behavior of graphs without external heavy fields is their contribution to coupling constant and fieldstrength renormalization. The heavy fields effectively decouple and the lowmomentum behavior of the theory is described by a renormalizable Lagrangian consisting of the massless fields only. The decoupling theorem applies not only to theories with massless fields but in fact to any renormalizable theory with different mass scales. At momentum smaller compared to the larger masses, the dynamics is determined by the light sector of the theory. In this paper we adopt this strategy. See also section 4.2.
Consequently, the APEGT which was heuristically obtained in the paper [11] and improved systematically in the paper [35] is further modified by taking into account the mass of offdiagonal field components. The simplest derivation of the modified APEGT is to replace the massless offdiagonal propagator given in the previous paper [35] with the massive offdiagonal propagator in the QCD vacuum with ghost condensation.^{5}^{5}5However, the following steps can be performed irrespective of the origin of the offdiagonal gluon mass.
The Feynman rules are given as follows, see Fig. 1. We enumerate only a part of the rules which are necessary for the renormalization at oneloop level. The twoloop result will be reported in a subsequent paper[36].
2.6.1 Feynman rules
Propagators:

Offdiagonal gluon propagators:
(2.49)
Threepoint vertices:

one diagonal and two offdiagonal gluons:
(2.51) where we have introduced the abbreviated notation,
(2.52) 
diagonal gluon, offdiagonal ghost and antighost:
(2.53) 
diagonal tensor and two offdiagonal gluons: