Preprint CHIBAEP99REV
hepth/9709109(revised)
September 1997
AbelianProjected Effective Gauge Theory of QCD
with Asymptotic Freedom and Quark Confinement
KeiIchi Kondo

Department of Physics, Faculty of Science, Chiba University, Chiba 263, Japan

Email:
Abstract
Starting from SU(2) YangMills theory in 3+1 dimensions,
we prove that the abelianprojected effective gauge
theories are written in terms of the maximal abelian gauge
field and the dual abelian gauge field interacting with
monopole current. This is performed by integrating out
all the remaining nonAbelian gauge field belonging to
SU(2)/U(1). We show that the resulting abelian gauge
theory recovers exactly the same oneloop beta function as
the original YangMills theory. Moreover, the dual abelian
gauge field becomes massive if the monopole condensation
occurs. This result supports the dual superconductor
scenario for quark confinement in QCD. We give a
criterion of dual superconductivity and point out that
the monopole condensation can be estimated from the
classical instanton configuration. Therefore there can
exist the effective abelian gauge theory which shows both
asymptotic freedom and quark confinement based on the dual
Meissner mechanism. Inclusion of arbitrary number of
fermion flavors is straightforward in this approach. Some
implications to lower dimensional case will also be
discussed.
1 Introduction
It is one of the most important problems in particle physics to clarify the physical mechanism which realizes the quark and gluon confinement. An important question is what is the most relevant degrees of freedom to describe the confinement. In the mid1970’s, an idea of the dual Meissner vacuum of quantum chromodynamics (QCD) was proposed by Nambu [1], ’t Hooft [2] and Mandelstam [3]. In this scenario, the monopole degrees of freedom plays the most important role in the confinement. This aspect can be seen explicitly through a procedure called the abelian projection by ’t Hooft [2]. Under the abelian projection, the nonAbelian gauge theory can be regarded as an abelian gauge theory with magnetic monopole [4]. For the confinement mechanism, there are other proposals [5] which we do not discuss in this paper.
The abelian projection [2] is to fix the gauge in such a way that the maximal torus group of the gauge group remains unbroken. It goes on as follows for the gauge group SU(N),

One chooses a gaugedependent local quantity which transforms adjointly under the gauge transformation, i.e.,
(1.1) 
One performs the gauge rotation so that becomes diagonal,
(1.2) where are eigenvalues.

At the spacetime point where the eigenvalues are degenerate the monopolelike (hedgehog) singularity appears. The singularity does appear in the abelian gauge field extracted from the nonAbelian gauge field . The monopole singularity is characterized as a topological quantity.

At generic point where the eigenvalues do not coincide, the gauge is not determined completely, since any diagonal gauge rotation (an element of the largest abelian subgroup, , the maximal torus group)
(1.3) leaves invariant. Therefore, within this gauge, the theory reduces to an (N1) fold abelian gaugeinvariant theory.
The Monte Carlo studies of the abelian projection was initiated by the work [6] and the maximal abelian gauge (MAG) was adopted in the simulation on the lattice [7]. Recent extensive studies of abelian projection (see [8] for a review) have confirmed the abelian dominance proposed in Ref.[9]. This states that the nonAbelian gauge field in , behaving as a charged field under the residual gauge rotation, is not important in the low energy physics and the maximal abelian part plays the dominant role in the quark and gluon confinement. In analytical studies, the abelian dominance was assumed from the beginning to construct the effective low energy theory of QCD [9, 10]. Assuming the abelian dominance, one can show that, if the monopole condensation occurs, charged quarks and gluons are confined due to dual Meissner effect. The monopole condensation is expected to bring the mass for the dual gauge field. An effective theory of monopole currents was investigated also on the lattice [11]. In fact, recent Monte Carlo simulations [12] support the abelian dominance and furthermore monopole dominance. However, there seems to be no analytical proof of abelian dominance.
An deficit of the abelian projection is the gaugedependence of the procedure of abelian projection. The quantity is a gaugedependent quantity and the field variable in which the monopole appears is not a gauge invariant quantity. Therefore the result seems to depend crucially on the gauge selected in the abelian projection. However, this would not be a real problem, since it is possible to put the abelian projection in a gaugeinvariant form, if we desire to do so, see [13, 14].
The real problem is another in our view. In the abelianprojected theory, the magnetic monopole degrees of freedom appear as the singularity in the abelian gauge field. The magnetic current is obtained as the divergence of the dual abelian field strength ,
(1.4) 
in the similar way that the equation of motion relates the field strength to the electric current ,
(1.5) 
If the U(1) potential is nonsingular, the abelian field strength leads to vanishing magnetic current, , which is nothing but the Bianchi identity for the U(1) field, . So, if one needs the nonzero magnetic current, the abelian field must include a singularity. However, we do not think that it is sound as a quantum field theory to treat the singularity of the field variable as the essential ingredient from the very outset. In the lattice gauge theory, such a singularity does not appear due to the lattice regularization [15] and the monopole contribution is extracted from the gaugeinvariant magnetic flux, although the monopole dominance is supported in the Monte Carlo simulation on the lattice. Moreover, it should be noted that the magnetic monopole does not exists in the original nonAbelian gauge theory. Magnetic monopole appears only after the abelian projection (see Appendix C).
The purpose of this paper is to derive the abelianprojected effective gauge theory (APEGT) of QCD as a quantum field theory, from which we should start the analysis. For simplicity, we restrict the following argument to the case. case is more involved and will be presented in a subsequent paper. In this paper, without using various assumptions (actually with no assumptions), we derive the APEGT of YangMills (YM) theory and QCD. This is done by integrating out offdiagonal fields belonging to the SU(2)/U(1) based on the functional integral formalism. We use the word ”effective” in the sense of the Wilson renormalization group [16], since the abelianprojected theory is obtained after integrating out the degrees of freedom corresponding to nonabelian gauge fields which behave as massive charged matter fields and don’t play the important role in the low energy physics of confinement. Such a strategy can be exactly performed in the supersymmetric YM and QCD [17].
We show that the offdiagonal field gives rise to the nontrivial magnetic monopole current for the abelian part,
(1.6) 
In other words, the charged offdiagonal gluon field plays the role of the source of the monopole. Although the definition (1.6) of monopole current seems to be different from the usual definition based on the singularity of the abelian field, we show that both are equivalent to each other (apart from the Dirac string singularity). In the APEGT, the singularity does not appear apparently, although we can always include the singularity if necessary.
The effective dual GinzburgLandau (GL) theory derived assuming the abelian dominance does not have sufficient predictive power, since it contains undetermined free parameters. On the contrary, all the quantities in APEGT are calculable and all the effects of the nonAbelian gauge field are included in the APEGT. In fact, we show that the APEGT recovers exactly the same oneloop beta function as that of the original nonAbelian gauge theory. The dual abelian gauge field follows naturally in the course of the derivation of the theory and has a coupling with the monopole current. This interaction leads to the dual Meissner effect due to monopole condensation. The resulting nonzero mass of the dual gauge field gives the nonzero string tension, i.e. linear potential for static quarks. Thus the string tension is determined by the monopole loop condensate, (see section 4 for precise definition). The monopole condensate plays the role of the order parameter for confinement.
Moreover, we discuss a possibility that the nonzero monopole condensation is derived from the instanton configuration. Hence instanton may lead to the confinement against the conventional wisdom [18].
In our approach, the inclusion of fermions is straightforward. Hence APEGT is also a starting point to study the relationship between the confinement and the chiral symmetry breaking (or restoration) [19, 20].
This paper is organized as follows. In section 2, we derive the APEGT for the maximal abelian part by integrating out the remaining nonAbelian gauge field. In this step, we introduce the auxiliary tensor field which is converted to the dual gauge field. The dual gauge field is essential to discuss the dual Meissner effect in section 4. APEGT is first obtained in the form including the logarithmic determinant. The logarithmic determinant is explicitly calculated. It generates the gauge invariant form due to U(1) gauge invariance. An effect of this term is the renormalization of the abelian gauge field. In section 3, we calculate the oneloop beta function without using the Feynman diagram. It is shown to agree with the original nonAbelian gauge theory. In this sense, the effective theory recovers the asymptotic freedom. In section 4, we discuss the dual Meissner effect. If the monopole loop condensation occurs, the dual vector field becomes massive. In section 5, we include the fermion into the APEGT. In section 6, we discuss the lower dimensional case. In the final section we give conclusion and discussion.
2 Abelianprojected effective gauge theory
2.1 Separation of the abelian part and introduction of the dual field
First, we decompose the field into the diagonal (maximal abelian U(1)) and the offdiagonal part SU(2)/U(1),
(2.1) 
We adopt the following convention. The generators of the Lie algebra for the gauge group are taken to be hermitian satisfying and normalized as The generators in the adjoint representation is given by We define the quadratic Casimir by For SU(2), with Pauli matrices and the structure constant . The indices denote the offdiagonal parts.
Then the field strength
(2.2) 
is decomposed as
(2.3) 
where the derivative is defined by
(2.4) 
Hence the diagonal part of the field strength is given by
(2.5) 
Next, we rewrite the YangMills (YM) action
(2.6) 
By using
(2.7) 
the YM action is rewritten as
(2.8) 
Here we introduce an antisymmetric auxiliary tensor field in order to linearize the term. This procedure enables us to perform the Gaussian integration over the offdiagonal gluon fields . ^{1}^{1}1 This procedure is similar to the field strength approach for nonAbelian gauge theory [21]. It turns out that the tensor field plays the role of the ”dual” field to the abelian gluon field . We find that there are two ways to introduce the ”dual” tensor field.
One way is to introduce the tensor field such that the tensor is the dual of the diagonal field strength ,
(2.9) 
This is achieved in the tree level by the following action
(2.10) 
This theory is equivalent to the BFYM theory,
(2.11) 
Actually, by identifying , the action (2.10) is obtained from (2.11) by separating the diagonal part from the offdiagonal part and integrating out the offdiagonal auxiliary tensor field . Quite recently, equivalence of the BFYM theory with the YM theory has been proved in the quantum level, see [22]. This theory is interesting from the topological point of view.
Another way is to introduce the tensor field as a dual to at the tree level,
(2.12) 
Thus we are lead to the action,
(2.13)  
In this case, is generated through the radiative correction as shown in section 2.4. In either case, Gaussian integration over recovers the action (2.8) and hence the original YM action. This model (2.13) is simpler than the model (2.10) in the actual treatment, since the topological theory need some delicate treatment [22]. (Equivalence of two formulations is shown in Appendix A.) In what follows, we focus on the action (2.13) which is essentially equivalent to that derived by Quandt and Reinhardt [23].
2.2 Gaugefixing
We discuss the gaugefixing term. This is independent from the choice of the action. The gaugefixing term is constructed based on the BRST formalism. We consider the gauge given by
(2.14)  
(2.15) 
where we have used the basis, ^{2}^{2}2 In this basis,
(2.16) 
The gauge fixing with is the Lorentz gauge, . In particular, corresponds to the differential form of the maximal abelian gauge (MAG) which is expressed as the minimization of the functional
(2.17) 
The differential MAG condition (2.14) corresponds to a local minimum of the gauge fixing functional , while the MAG condition (2.17) requires the global (absolute) minimum. The differential MAG condition (2.14) fixes gauge degrees of freedom in SU(2)/U(1) and is invariant under the residual U(1) gauge transformation. An additional condition (2.15) fixes the residual U(1) invariance. Both conditions (2.14) and (2.15) then completely fix the gauge except possibly for the Gribov problem. It is known that the differential MAG (2.14) does not spoil renormalizability of YM theory [24]. An implication of this fact is shown in Appendix B.
From physical point of view, we expect that MAG introduces the nonzero mass for the offdiagonal gluons, . This is suggested from the form (2.17) which is equal to the mass term for , although we need an independent proof of this statement. This motivates us to integrate out the offdiagonal gluons in the sense of Wilsonian renormalization group and allows us to regard the resulting theory as the lowenergy effective gauge theory written in terms of massless fields alone which describes the physics in the length scale . The abelian dominance will be realized in the physical phenomena occurring in the scale . In this sense the choice of MAG is not unique in realizing abelian dominance. We can equally take the gauge so that the offdiagonal gluon fields acquire nonzero masses. Then the abelianprojected effective gauge theory obtained by integrating out the massive offdiagonal gluons will be valid in the low energy region below the energy scale given by the offdiagonal gluon mass.
We introduce the Lagrange multiplier field and for the gaugefixing function and , respectively. It is well known that the gauge fixing term in the BRST quantization is given by [25]
(2.18) 
where carries the ghost number and is a hermitian function of Lagrange multiplier field , ghost field , antighost field , and the remaining field variables of the original lagrangian. In this paper we consider a simple gauge given by
(2.19) 
For the most general gauge fixing, see [26].
The BRST transformation in the usual basis is
(2.20) 
Then the BRST transformation in the basis is given by
(2.21) 
Under the local U(1) gauge transformation,
(2.22) 
Hence transforms as a U(1) gauge field, while and behave as charged matter fields under the U(1) gauge transformation. It turns out that and
(2.23) 
are U(1) gauge invariant as expected.
In the usual basis, we can write
(2.24) 
where
(2.25) 
For the gauge fixing function (2.19) with the BRST transformation (2.21), or (2.24) with (2.20), straightforward calculation leads to the gauge fixing lagrangian (2.18),
(2.26)  
This reduces to the usual form in the Lorentz gauge, .
Finally we introduce the source term,
(2.27) 
which will be necessary to calculate the correlation functions.
2.3 Integration over SU(2)/U(1)
Our strategy is to integrate out the offdiagonal fields, (and for BFYM case) belonging to the Lie algebra of SU(2)/U(1) and to obtain the effective abelian gauge theory written in terms of the diagonal fields (and ghost fields if we need a completely gaugefixed theory also for the residual U(1) gauge invariance).
First of all, when , ^{3}^{3}3 The case of should be treated separately. Since is linear in , the integration can be performed finally after integrating out the field. However, it generates the additional complicated logarithmic determinant, Such a case was treated in [23]. The choice of the gaugefixing parameter should not change the physics, since it appears due to a gauge choice. Therefore we don’t treat this case in this paper. the Lagrange multiplier field can be easily integrated out. The result is
(2.28) 
Next, as a preliminary procedure to integrate out , we rewrite the last term in the action (2.13) into
(2.29) 
where we have used
(2.30) 
Discarding the surface term, ^{4}^{4}4 This will be justified, since the offdiagonal gluons become massive due to MAG. we arrive at
(2.31)  
(2.32)  
(2.33)  
(2.34)  
(2.35)  
(2.36) 
where we have rescaled the parameter to absorb the dependence.
All the terms appearing in the resulting YM action are at most quadratic in . Therefore the field in can be eliminated using the Gaussian integration and we obtain
(2.37)  
Thus we obtain the effective abelian gauge theory
(2.38)  
As will be shown in the next subsection, gives the renormalization of the fields and . The residual U(1) invariant theory is obtained by putting and (hence ). Therefore, the resulting APEGT is greatly simplified.
On the other hand, the effective abelian BFYM theory is obtained if and in are replaced by
(2.39)  
where the is the same as (2.36). This case is discussed in Appendix A.
2.4 Calculation of logarithmic determinant
In MAG (), the last two terms in cancel by taking (they disappear also for [23]),
(2.40)  
In order to calculate the , we use the function regularization or heat kernel method (see e.g. [27]),
(2.41) 
where is understood in the functional sense. In this subsection the calculations are performed in Euclidean formulation.
First, we calculate the trace of . To estimate this quantity, we use the plane wave basis,
(2.42) 
By making use of the relation,
(2.43) 
we find
(2.44) 
Furthermore the rescaling of , , leads to
(2.45)  
where we have omitted the unit operator, . It is obvious that all terms odd w.r.t. in the expansion go to zero in the integration. Thus we obtain ^{5}^{5}5 The zeroorder term of the expansion with respect to is equal to the free term
(2.47)  
where we have used the cyclicity of trace and the replacement
(2.48) 
which is applied in the integrand of the integration formula
(2.49) 
Separating the first term from the other terms in ,
(2.50) 
we see
(2.51)  