Euclidean SU(2) gauge theory is studied in a nonlinear gauge. In this theory, ghost condensation happens and gauge fields acquire tachyonic masses. It is shown that these tachyonic masses are removed by a gauge field condensate . Because of the ghost condensation, monopole solutions are included naturally. We find that the condensate makes the magnetic potential massive.
B00B03B061. Introduction
In non-Abelian gauge theories, a magnetic monopole may play an important role. In the dual superconductor model of color confinement (see, e.g., Ref. [1]), monopole condensation is expected. As a result, a dual gauge field [2] acquires a mass, and color confinement may happen.
How can monopoles appear without Higgs fields? One way is the Abelian projection by ’t Hooft [3]. Let us consider a field , which belongs to the adjoint representation, and perform the Abelian gauge fixing. When we diagonalize , there may be points at which the eigenvalues of degenerate. Monopoles appear at these points. Another way is to introduce a unit color vector in the internal space [4,5]. A non-Abelian magnetic potential is defined by
and the gauge field is decomposed by using . This model is called extended quantum chromodynamics (QCD) [6].
What is the origin of ? Is it possible to make the magnetic potential massive? In this paper, we propose a scenario that produces and massive . For simplicity, we consider the SU(2) gauge theory in Euclidean space-time, and sometimes call the gauge field a gluon. In the next section, we briefly review ghost condensation and tachyonic gluon masses. In Sect. 3, it is shown that these tachyonic masses are removed by the vacuum expectation value (VEV) . Under the ghost condensation, we can include a magnetic monopole as a classical solution. In Sect. 4, we introduce it in the Abelian gauge. In the presence of the VEV , a magnetic potential is expected to become massive. By using the background covariant gauge, we confirm this expectation in Sect. 5. In Sect. 6, to remove the Dirac string, we perform a singular gauge transformation, and derive the extended QCD with massive . In Sect. 7, the case of 3D space-time is discussed briefly. Section 8 is devoted to a summary and comments. In Appendix A, it is shown that vanishes. The tachyonic gluon mass is derived in the background covariant gauge in Appendix B. The singular gauge transformation used in Sect. 6 is summarized in Appendix C. The Becchi–Rouet–Stora (BRS) symmetry and the breakdown of the global gauge symmetry are explained in Appendix D.
2. Ghost condensation and tachyonic gluon mass
We consider the SU(2) gauge theory with structure constants . Using the notations
the Lagrangian in the nonlinear gauge is given by [7]
where , and are gauge parameters, and is a constant. Introducing the auxiliary field , which represents , is rewritten as
In Ref. [8], we have shown that acquires the VEV under an energy scale . Dividing the quantum fluctuation from , we substitute into Eq. (2.2), and choose the constant . This choice is necessary to maintain the BRS symmetry [9].
Then Eq. (2.2) becomes
Because of the VEV , the global SU(2) symmetry breaks down to U(1). The relation between this breaking and BRS-invariant Green functions is explained in Ref. [10].1
Under the ghost condensation, the ghost-loop contribution to the gluon propagator was calculated. When the gluon momentum becomes small, it is found that the gluon acquires tachyonic mass [7,11]. The term in the ghost propagator gives rise to the tachyonic mass. Now we choose the VEV in the third direction as , and use the notation . Then, the scale , the value of the VEV , and the tachyonic gluon mass term in 4D Euclidean space () are summarized as
where is a momentum cut-off, and is the momentum scale satisfying . The coupling constant and the gauge parameter in Eq. (2.4) are the quantities at the scale . In the same way, the corresponding quantities in 3D Euclidean space () are
where and are the coupling constant and the VEV in , respectively.
From now on, we concentrate on the case, and try to remove the tachyonic gluon mass. We write Eq. (2.4) as
where .
3.1. The effective potential for <inline-formula><tex-math notation="LaTeX" id="ImEquation54"><![CDATA[$\langle A_{\mu}^+A_{\mu}^-\rangle$]]></tex-math></inline-formula>
When the tachyonic mass term (3.1) exists, the gluon propagator blows up as . To avoid this, we return to the Lagrangian , and introduce the source term for the local composite operator (LCO) .2 The partition function is
where the usual source terms are
Dividing the action into free and interaction parts as , and performing the Gaussian integration, we obtain
where is , and we set after constructing . From , we find
and we define the effective action as
If we write the -related part in as , the -dependent term in becomes
Thus, at , Eq. (3.2) gives
From , the diagrams depicted in Fig. 1 contribute to . Since the kinetic term contains the terms
The diagrams that contribute to the effective potential at . The solid line represents , the dotted line is , and the dashed line represents .
the diagram in Fig. 1(a) comes from the vertex in Eq. (3.5). Applying Eq. (3.4), it gives . The ghost loop in Fig. 1(b) yields the term , which makes tachyonic. Using Eq. (3.4) again, Fig. 1(b) gives . The diagrams in Fig. 1(c) contain the loops with . In the dimensional regularization, as , these diagrams vanish. If the source is constant, it plays the role of mass-squared for the . So the diagrams in Fig. 1(d) can produce the term , where is a divergent constant. This divergence should be removed by the renormalization of . By summing up these results, becomes , where
Thus, up to , we obtain
This satisfies
where Eq. (3.4) has been used. Since satisfies , from Eq. (3.8), we obtain .
In Fig. 2(a), the diagrams that contribute to are depicted. The closed loop with a solid line represents . As we show in Fig. 2(b), by removing one closed loop, the diagrams that correspond to are obtained. If we cut one closed loop as in Fig. 2(c), the diagrams that contribute to the mass term for are obtained. Therefore, when the condition holds, the two diagrams in Fig. 2(c) cancel out. In the same way, in , the diagrams depicted in Fig. 1(d) cancel out. Namely, the diagrams in Figs. 2(c) and (d) do not contribute to the masses for and , respectively.
The diagrams that contribute to , , and the mass terms for and .
We summarize the results. Up to , the effective potential is Eq. (3.7). As it satisfies , we obtain Eq. (3.4) and . Using Eq. (3.6), gives
and Eqs. (3.4) and (3.9) give
where we set . Thus, when , the condensate exists, and the mass for is determined by Eq. (3.10).
As contains a gauge parameter, Eq. (3.10) is gauge-dependent and requires regularization. So, although Eq. (3.10) implies that , the determination of this must be done carefully. We do not try to determine in this paper, and make it a future task.
3.2. Expedient procedure
The above results are obtained simply by the following procedure. Add the tachyonic mass term (3.1) to the Lagrangian. Then replace as
and add the source term . Then we obtain the following terms that contain , or :
where Eq. (3.5) has been used. From , Eq. (3.9) holds and Eq. (3.12) becomes . Namely, the tachyonic masses are removed and, whereas is massless, acquire the mass . In the following sections, we apply this procedure for simplicity.
We note, although the component has the tachyonic mass in Eq. (3.1) as well, we did not take the source term for and the VEV into account. In Appendix A, it is shown that the VEV vanishes.
4. Inclusion of monopoles
The Lagrangian is invariant under the BRS transformation , if the field transforms as . That is, behaves like a Higgs field in the adjoint representation. As acquires the VEV , we can introduce Abelian monopoles in the direction [14]. Let us consider the classical field , which satisfies the equation of motion
Except for a string singularity, it is known that the configuration
satisfies this equation, where is an integer, and are spherical coordinates. If (), the Dirac string appears on the negative (positive) -axis. In Sect. 6, we choose as a concrete example, i.e.,
This monopole satisfies the equation of motion
where
We call a magnetic potential.
In Sect. 3, we have shown that the VEV removes the tachyonic mass for the gluon. What happens for the classical solution ? If we substitute into Eq. (3.5), Eq. (3.12) changes to
As is given by Eq. (3.5), the mass term for the magnetic potential appears as
In the next section, we study the mass for by using the background covariant gauge.
5. Massive magnetic potential
First we derive the Lagrangian with the field and the magnetic potential in the background covariant gauge [15]. We divide into the classical part and the quantum fluctuation as3
Then the gauge transformation
is satisfied by
where the notation has been used. Choosing the gauge-fixing function
and applying the transformation (5.2), we obtain the Lagrangian
where is a constant. We require that belongs to the adjoint representation, and it transforms as . The last term is added because it is gauge-invariant and necessary to yield the quartic ghost interaction term in Eq. (2.1).
Since the ghost condensation happens by the quartic ghost interaction [16], the condensate appears irrespective of . Thus, after the ghost condensation, becomes
where and . We note that Eq. (5.4) is obtained from Eq. (2.3) by replacing to . As we explained in Sect. 4, the VEV selects the unbroken U(1) direction, and the Abelian monopole is included in the Lagrangian consistently.
Next we study mass terms. Without , a ghost loop brings about the tachyonic gluon mass term (3.1). We show that Eq. (3.1) is obtained even if exists. To show this, we introduce the background gauge transformation
which satisfies the transformation (5.1). When , the Lagrangian and in Eq. (5.4) are invariant under the transformation (5.5), if , , and transform in the adjoint representation. When , the ghost determinant
breaks the symmetry (5.5). However, if we restrict Eq. (5.5) to the U(1) transformation
the determinant (5.6) is invariant. As tachyonic mass terms come from Eq. (5.6), they must respect this U(1) symmetry. So the terms and are forbidden, and the terms and are allowed. We calculate the determinant (5.6) in Appendix B, and obtain the tachyonic mass term
as expected.
Finally, to remove the tachyonic mass, the procedure in Sect. 3.2 is applied. We replace with , where , and add the source term . As contains the term
Eqs. (5.7) and (5.8) lead to
As takes the minimum value at , Eq. (5.9) becomes
Namely, the magnetic potential has the mass term (4.3).
Except for string singularity, the magnetic potential must be modified to satisfy the massive equation of motion4
Then the term linear with respect to
vanishes. Thus the final Lagrangian with massive magnetic potential becomes
where interaction terms with and are neglected.
6. Massive non-Abelian magnetic potential
The magnetic potential has string singularity. We remove it by a singular gauge transformation [14]. In this section, we use the matrix notation with . We perform a singular gauge transformation for and as
The matrix , which corresponds to the monopole in Eq. (4.1), is . As we explain in Appendix C, using this matrix and the monopole solution , Eq. (6.1) becomes
and
where , and are defined in Eq. (C.2) with and . We note that the right-hand side of Eq. (6.2) is the gauge field decomposition by Cho [6], and Eq. (6.3) shows that the color vector comes from the VEV .
Now we regard the transformation (6.1) as the background gauge transformation. The classical field has the inhomogeneous part as
Then, as we show in Appendix C, Eqs. (6.2) and (6.4) indicate that the background gauge transformation is realized simply by the replacement
We call a non-Abelian magnetic potential. Let us give an example. If we choose , and become
Although this configuration found by Wu and Yang [17] is singular at the origin, it has no string.
Applying Eq. (6.5) to the Lagrangian in Eq. (5.3), we obtain
where the following notation in Ref. [6] has been used:
Equation (6.6) is the Lagrangian of the extended QCD [6]. Next we apply the transformation (6.5) to in Eq. (5.4) as well. However, as , the tachyonic mass term (5.7) is obtained again. Replacing with , and adding the source term , the part that depends on , or becomes
Equation (6.8) is derived directly by applying the transformation (6.5) to Eq. (5.9). However, there is a difference between Eqs. (5.9) and (6.8). Since , the cross-term vanishes without using the equation of motion for . Now we set . Then, from Eqs. (6.6) and (6.8), we obtain the Lagrangian
where has been used. Equation (6.9) is the extended QCD with the massive non-Abelian magnetic potential and the massive off-diagonal gluons.
We make a comment. Neglecting the quantum fields, the Lagrangian (6.9) becomes
This is the Skyrme–Faddeev Lagrangian [18]. The equation of motion for comes from
As satisfies , this equation means that the component of that is perpendicular to vanishes. So, from Eq. (6.10), we obtain
and the perpendicular component gives [6]
Now we give an example. The color vector in Eq. (C.7) satisfies . When , it must be modified to satisfy Eq. (6.12). However, we find that it satisfies
So, if we choose , the equations
hold, and Eq. (6.12) is fulfilled. As , Eq. (6.9) becomes
7. Comment on the <inline-formula><tex-math notation="LaTeX" id="ImEquation247"><![CDATA[$\boldsymbol{D=3}$]]></tex-math></inline-formula> case
In , ghost condensation happens as well. From Eq. (2.5), the tachyonic gluon mass term is
As in Sect. 5, we divide the gluon as , add the source term , and introduce as
The terms that contain , or are
where
The minimum of determines the VEV and the mass as
Substituting Eq. (7.2) into Eq. (7.1), we find the mass terms
Namely, the components have the mass , and the classical part acquires the mass . However, unlike the case, the component acquires the mass . The value is crucial.
As the gauge field in (or ) is expected to have masses of this order (see, e.g., Refs. [19,20]). It is known that the gauge theory at high temperature becomes the effective gauge theory with the coupling constant [21]. The mass of , which is called the magnetic mass, is expected [22]. The mass terms and may imply the existence of the magnetic mass at high .
8. Summary and comments
We have studied the SU(2) gauge theory in the nonlinear gauge. In 4D Euclidean space, the ghost condensation happens under the scale , and the ghost loop yields the tachyonic gluon mass term . We considered the effective potential for the LCO up to , and showed that gives the minimum of the potential. The VEV makes the diagonal gluon massless. In contrast, the off-diagonal gluons acquire the mass determined by Eq. (3.10).
The field belongs to the adjoint representation of SU(2). Since the VEV selects the direction, the Abelian monopole solution can be introduced in this direction. Combining the VEV and , we find that the magnetic potential becomes massive.
To remove the string singularity of the monopole solution , we performed the singular gauge transformation in the background covariant gauge. As transforms into , the field that specifies the color direction appears, and the extended QCD [6] is derived. In addition, the non-Abelian magnetic potential transformed from becomes massive. If quantum fields are neglected, the Skyrme–Faddeev Lagrangian is obtained.
In 3D Euclidean space, phenomena similar to the case happen. However, unlike the case, the diagonal component becomes massive.
We make some comments.
(1) In the maximally Abelian gauge, the lattice simulation shows that the off-diagonal gluons are massive and the diagonal gluons are nearly massless [23,24]. In the present model, the VEV breaks the global SU(2) symmetry to U(1), so the result that and have different masses may be reasonable. In addition, the masslessness of the component may be related to the remaining U(1) symmetry.
(2) Since the magnetic potential couples with the off-diagonal components , can become massive by the VEV . Namely we can obtain a massive magnetic potential without monopole condensation.
(3) To show the quark confinement, Abelian dominance is often assumed [25,26]. In this assumption, the off-diagonal components are neglected. Our result insists that these components play an important role in realizing the Abelian dominance.5
(4) In Ref. [27], based on the LCO formalism in Refs. [28,29], the LCO is studied in the Landau gauge, together with the LCO . This formalism introduces sources and for and , respectively, in a BRS-exact form. From the renormalizability, the terms and are necessary. To delete these terms, the Hubbard–Stratonovich transformation is applied, and the fields and , which correspond to and , respectively, are introduced. The final Lagrangian they used is , where
The one-loop effective potential was calculated by using the free propagators derived from and the interactions in . Then the conditions and give the VEVs and , respectively.
When the VEV exists, the ghost loop yields the tachyonic gluon masses. As in Sect. 2, the magnitude for is twice that for . In the presence of the VEV , gluons acquire a mass. The latter mass compensates the former tachyonic masses. Thus gluons have the usual masses, and the mass for is different from that for .
In their approach, the interactions that yield these VEVs are introduced irrelevant to the original Lagrangian . As a result, the VEV appears irrespective of the VEV .
(5) In our procedure, we need the action in the nonlinear gauge. The quartic ghost interaction yields the VEV , and the ghost loop brings about the tachyonic gluon masses. The VEV appears to eliminate the tachyonic mass for .
In the Landau gauge, there is no quartic ghost interaction. However, in the low energy region, gauge field configurations on the Gribov horizon contribute to the partition function . These configurations give rise to zero-modes of the ghost operator , and make vanish.
However, these zero-modes can produce the quartic ghost interaction [10]. The Landau gauge changes automatically to the nonlinear gauge. Thus, even if we start from the Landau gauge, effective quartic ghost interaction is produced, and we can apply the scenario proposed in this article.
We note that the scale in Eq. (2.4) depends on the parameter at the cut-off scale . One-loop calculation shows that has the ultraviolet fixed point , where is the coefficient of the beta function . By substituting into , we find that coincides with the QCD scale parameter [8].
Funding
Open Access funding: SCOAP.
Appendix A. <inline-formula><tex-math notation="LaTeX" id="ImEquation356"><![CDATA[$\boldsymbol{\langle (A_{\mu}^3)^2\rangle=0}$]]></tex-math></inline-formula>
To eliminate the tachyonic mass terms
the VEV is necessary. In this appendix, we show, in contrast with , the VEV vanishes.
To apply the procedure in Sect. 3.2, we substitute
into Eqs. (3.5) and (A.1). Then the part that contains or is given by
If we set , we obtain
The potential has the minimum value at . As , and is linear with respect to , the coefficient must satisfy . Thus we find that
Since the local minimum condition leads to
from Eqs. (A.4) and (A.5), we obtain
Now we show that . When , as , and have their minimum values at , and Eq. (A.5) leads to . When , there are two possibilities to minimize . As , one way is to choose . However, since V_1(\lambda=1)$]]>, does not have a minimum value. Another way is to set . Then Eq. (A.5) gives , and is minimized.
We make a comment. Historically, the VEV was discussed in the framework of the operator product expansion (OPE) [30,31], and the relation with monopoles was considered [32]. When , the OPE gives [30]
However, the VEV is different from , because appears as , and the component does not condense. To make massive, the VEVs and were considered in Ref. [33]. In our approach, although becomes massive, the significant role of is to remove the tachyonic masses.
Appendix B. Derivation of Eq. (5.7)
We calculate the ghost determinant (5.6) given by
where the classical part is , and . Up to the second order of the fields, we find
The term in Eq. (B.1) gives
Using
we obtain [7]
In the same way, the term in Eq. (B.1) gives
and the term in Eq. (B.1) gives
Thus, substituting and , we obtain
Appendix C. Singular gauge transformation
Following Ref. [4], we consider the gauge transformation with the matrix
where . This matrix satisfies
and , where
These color vectors satisfy the orthonormality
Under this transformation, the gauge field transforms as
If we write
Eq. (C.3) becomes
If we regard the transformation (C.5) as the background gauge transformation
Eq. (C.5) implies that and transform as
Now, using the spherical coordinates and integer , we set the angles as . Then and become
where is the Abelian monopole in Eq. (4.1). The corresponding non-Abelian monopole is
The field strength is defined in Eq. (6.7). If we use Eq. (C.8), it becomes
and it satisfies . In the same way, in Eq. (C.7) satisfies
Appendix D. BRS symmetry and global gauge symmetry
In this appendix, for the sake of explanation, we use the operator formalism [34]. The BRS transformation is and the BRS charge is . A state in the physical subspace satisfies the condition .
D.1. BRS symmetry
First we show that the constant in Eq. (2.2) must be chosen as to preserve the BRS symmetry. The Lagrangian (2.2) is invariant under the BRS transformation
Using the equation of motion for , we obtain
where and have been used. As , we must set to obtain .
Next we consider the VEV . Since the operator satisfies
it is not BRS-invariant. If there is an operator that satisfies , the BRS symmetry is broken spontaneously by the VEV . However, as , Eq. (D.2) implies that such an operator does not exist.6 So the VEV does not contradict the BRS invariance of the vacuum.
We make two comments. First, since there is no ghost-number-violating interaction nor such a condensate, the ghost number should be conserved. Therefore, although , its VEV satisfies
Second, the anti-BRS symmetry is broken in this model [10]. However, the unitarity of the model is guaranteed by the BRS symmetry.
D.2. Global SU(2) symmetry
Using the constant small parameter , the global color transformation is defined by , where represents all the fields in . Since , this symmetry breaks down spontaneously to U(1), and the fields are Goldstone bosons. In addition, as , the nonzero constant breaks this symmetry at the Lagrangian level. However, as , this breaking term is BRS-exact. Since the physical states satisfy , we find that
Thus the breaking term does not contribute to amplitudes between physical states [10].
Footnotes
1 In Appendix D, the necessity of is explained. The BRS symmetry and the broken global gauge symmetry are also discussed.
2 In this subsection, we write explicitly. For scalar fields , the effective potential for the LCO is studied in Refs. [12,13].
3 In this paper, as we are interested in the massive magnetic potential, we choose . However, instead of , we can introduce other classical solutions in the form .
4 Equation (4.1) no longer satisfies Eq. (5.10). If we use a dual potential, a modified solution is obtained. A concrete solution will be given in the next paper.
5 The components and in Ref. [18] correspond to in this paper. Condensations related to and are discussed in Ref. [18].
6 In the case of , the equation of motion for gives Eq. (D.1). However, in the case of , any colored fields do not give the equation of motion with .
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