Massive dual gauge field and confinement in Minkowski space : Magnetic charge

Gauge field configuration for a magnetic monopole and its dual configuration are studied in SU(2) gauge theory. We present a relation between the monopole field and its dual field. Since these fields can become massive, their massive Lagrangians are derived. In the dual case, an additional term appears. We show this term is necessary to produce a linear potential between a monopole charge and an antimonopole charge.


Introduction
Magnetic monopoles are considered to play an important role in quark confinement.
Models based on the dual superconductor require monopoles and their condensation (see, e.g., [1]). In the Zwanziger's formulation [2], two gauge fields, namely the usual gauge field and the dual gauge field, are used. To describe monopole condensation, a monopole field is introduced. Its vacuum expectation value (VEV) makes the dual gauge field massive. This massive field leads to a linear potential between a quark and an anti-quark.
In the extended QCD [3], a unit color vectorn A (x) in the internal space [4,5] appears.
Non-Abelian magnetic potential defined by C A µ = −(n × ∂ µn ) A /g describes the non-Abelian monopole [6]. In Ref. [7], we studied the SU(2) gauge theory in a nonlinear gauge, and derived the extended QCD with the massive magnetic potential C A µ . In this paper, we study the relation between the Abelian magnetic potential and its dual potential. Based on this relation, we consider the confinement of magnetic charges.
In Sect.2, we introduce the dual magnetic potential for magnetic monopoles. Next, by using the dual magnetic potential, we rewrite the Abelian part of the SU(2) Lagrangian with the monopole field. Massless magnetic potential is considered in Sect. 3, and massive magnetic potential is studied in Sect. 4. In Sect. 5, SU (2) gauge theory in the low energy region is studied. Because of the ghost condensation [8,9] and the condensate A + µ A −µ , the massive magnetic potential appears in this region [7]. Applying the result in Sect. 4, we obtain the low energy SU(2) Lagrangian with the dual magnetic potential. In Sect. 6, using the Lagrangian in Sect. 5, it is shown that the static potential between a magnetic monopole and an anti-monopole is a linear confining potential. Section 7 is devoted to summary. In Appendix A, notations are summarized. For both the massless case and the massive case, the monopole solutions and their dual solutions for a static magnetic charge are given in Appendix B. We also show that the relation between the Abelian magnetic potential and its dual potential, which is presented in the Sect. 2, is satisfied for the Dirac monopole. To obtain the Lagrangian in Sect. 5, the ghost condensation is necessary. In Ref. [10], we have shown that it happens in Euclidean space. In Appendix C, we show it in Minkowski space.

Dual magnetic potential
Let us consider a space-like gauge fieldC µ which satisfies the equation of motion where (∂ ∧C) µν = ∂ µCν − ∂ νCµ , the space-like vector n α satisfies n α n α = −1, and the magnetic current k β satisfies ∂ β k β = 0. We callC µ magnetic potential. The dual field strength is defined by Now we introduce a dual magnetic potential C µ . Using the formula Therefore, if we define C µ as H µν = (∂ ∧ C) µν , the kinetic term for C µ has the wrong sign [4]. We change this relation to If we impose the conditions Eq.(2.6) holds if we choose Thus we obtain the relation This expression will be used in the following sections.
We note H µν is invariant under the transformations If we choose the gauges n µC µ = 0, n µ C µ = 0, (2.10) Eq.(2.9) is solved asC In Appendix B, as an example,C µ and C µ for a static magnetic charge k β ∝ δ β 0 δ(x)δ(y)δ(z) are presented. The magnetic potentialC µ is the Dirac monopole and its dual potential C µ is the Coulomb potential. Eqs.(2.5) and (2.8) are fulfilled by these potentials, and the term Λ µν represents the Dirac string.

Abelian part of the SU(2) gauge theory
We consider the SU(2) gauge theory with structure constants f ABC . Using the notations contains the Abelian part To incorporate the magnetic potential, we divide the gauge field A 3 µ into a classical part b A µ and a quantum part a A µ as For simplicity, we use the notation Next we introduce the magnetic current k β . To reproduce the equation of motion (2.1), we consider the Lagrangian 2 The field strength H + h 3 is the Zwanziger's field strength F = (∂ ∧ A) − (n · ∂) −1 (n ∧ j g ) d in Ref. [2].
Then, using Eq.(2.9), we rewrite Eq.(3.5) as where dxF µν h µν 1 = 0 has been used. As we stated in Eq.(2.3), the part gives the kinetic term with the wrong sign. However, using the current conservation ∂ µ k µ = 0, we obtain Thus we find the cross term 2h 1µν h µν 2 changes the sign of the kinetic term for C µ , 3 and − 1 4 h 1µν (h µν 1 + 2h µν 2 ) yields the correct kinetic term. Thus we obtain Now we neglect the quantum part F µν . Then the classical solution C ν must satisfy the equation of motion However Eq.(3.10) is satisfied by because Eq.(3.11) leads to h µν 2 + h µν 3 = 0. If we insert h µν 2 + h µν 3 = 0 into the Lagrangian (3.9), the term h µν 2 , which is related to the Dirac string Λ αβ , disappears.
4 Massive Abelian part of the SU(2) gauge theory In the previous paper [7], we have shown that there appears the mass terms where the mass squared m 2 is defined in Eq.(5.5), and the derivation of L m is explained briefly in Sect. 5. Using Eq.(2.11) and integration by parts, the first term becomes and the second term becomes where Eq.(A3) and the gauge condition n ν C ν = 0 have been used.
If we apply Eq.(A2) and n ν C ν = 0, Eq.(4.3) is rewritten as Now, combining the kinetic term (3.5) with the mass term (4.1), we consider the This is the massive Abelian part expressed by the dual magnetic potential C µ .
Since the relation Then the Lagrangian (4.5) leads to the following equation of motion for C ν : Eq.(4.6) is satisfied by the equation of motion because Eq.(4.7) leads to h µν 2 + h µν 3 + h µν 4 = 0. For a static magnetic charge, a solution of Eq.(4.7) is presented in Appendix B.
We note, using Eq.(A2), Ω is rewritten as 5 SU(2) gauge theory in the low energy region

Derivation of the massive magnetic potential
In this subsection, we review the derivation of the Lagrangian with the massive magnetic potential [7]. Let us consider the Lagrangian In the background covariant gauge, a gauge-fixing part is chosen as where α 1 and α 2 are gauge parameters, andB = −B + igc × c. Namely Eq.(5.1) gives the Lagrangian in the nonlinear gauge [12] with the constant w A . Although ϕ A is the auxiliary field which represents α 2B A , because of the quartic ghost interaction α 2 2B ·B, it acquires the where v = gϕ 0 , and to preserve the BRS symmetry, w A is chosen as w A = ϕ 0 δ A 3 [7,14,15]. [16]. When v = 0, it is known that ghost loops bring about tachyonic gluon masses [10,17]. In Ref. [7], we have shown that the ghost determinant det where a + µ a −µ = a a µ a aµ /2 (a = 1, 2). We note, contrary to the quantum part a A µ , b A µ does not have tachyonic mass.
To avoid the tachyonic masses, we introduced the source term M 2 a + µ a −µ into the Lagrangian, and constructed the effective potential for Φ = a + µ a −µ [7]. However, at the lowest order, we can obtain the effective potential by the following simple procedure. First add the tachyonic mass terms (5.5) and the source term M 2 a + µ a −µ to the Lagrangian. Next replace a + µ a −µ to Φ + a + µ a −µ . Thus the terms which contain Φ, m 2 , or M 2 are The interaction term The factors g 2 Φ 2 and g 2 Φ in Eq.(5.6) come from the first and the second terms in Eq.(5.7).

Lagrangian with the magnetic potentialC µ
In the previous paper [7], to remove the string, we performed the singular gauge transformation [16]. Then the Lagrangian with the massive non-Abelian magnetic potential was obtained. However, since we want to use the dual magnetic potential in this paper, we introduce the magnetic current k β as in Eq.(3.5). Namely replacing H µν with H µν + h µν 3 , Eq.(5.10) gives The classical solutionC µ satisfies the equation of motion If we use it, the linear terms on a 3 µ disappear, andL ′ inv becomes This is the low energy effective Lagrangian with the magnetic potentialC µ . For a static magnetic charge, a solutionC µ which satisfies Eq.(5.11) is presented in Appendix B.
6 Magnetic charge confinement 6.1 The use of the Lagrangian (5.12) The classical part of the Lagrangian (5.12) is 2) and the last term comes from −h 2 3 /4. The equation (6.1) can be written as .
If we apply Eq.(5.11), the first term in Eq.(6.3) vanishes. Using Eq.(A3), ∂ ν K ν = 0 and the current conservation ∂ µ k µ = 0, we find the second term in Eq.(6.3) becomes n µ n ν n σ n σ k ν . (6.5) As = + m 2 − m 2 , Eq.(6.5) is rewritten as The last term cancels out the third term in Eq.(6.3). Thus we obtain the magnetic currentcurrent correlation If we replace the magnetic current k µ with the color electric current j µ , Eq.(6.7) becomes the electric current-current correlation, which was derived in the framework of the dual Ginzburg-Landau model [18][19][20]. So, by replacing electric charges with magnetic charges, we can apply the results in these references . For a static magnetic monopole-antimonopole pair, the current is chosen as where the magnetic charge is Q m , and the position of the monopole (antimonopole) is a (b). We write r = a − b, r = |r| and n µ = (0, n). The vector n is chosen as n r. 5 Then the correlation (6.7) gives the monopole-antimonopole potential [18][19][20] where m χ is the ultraviolet cutoff for the p T , which is the momentum component perpendicular to r. Thus the magnetic monopoles are confined by the linear potential V L (r).
We comment on the scale m χ . In the usual dual superconductor model, m χ is the scale that the QCD-monopole condensation vanishes [20]. In our model, since the ghost condensation happens at the scale µ 0 = Λe −4π 2 /(α 2 g 2 ) and it yields the mass forC µ , m χ is the scale µ 0 . As we showed in Ref. [13], µ 0 coincides with the QCD scale parameter Λ QCD .

The use of the Lagrangian (5.13)
Next we study the classical part of Eq.(5.13), i.e., Using Ω in Eq.(4.8), Eq.(6.10) is rewritten as From the equation of motion (4.7), C µ = (D m ) µβ k β is derived. By substituting it into Eq.(6.11), we find Eq.(6.11) coincides with Eq.(6.7). We note the second term in Eq.(6.7), which yields the linear potential V L , comes from the h 3µν h µν 4 /4 term in Ω.

Summary and comment
We studied the low energy effective SU(2) gauge theory in Minkowski space. In the low energy region, the ghost condensation gϕ 0 = 0 happens, and the SU(2) symmetry breaks down to the U(1) symmetry. We introduced the Abelian magnetic potentialC µ as a classical solution, and presented the relation betweenC µ and its dual potential C µ in Minkowski space. It was shown that the term h µν 2 = − 1 2 ǫ µναβ Λ αβ , which is the Dirac string essentially, plays an important role to derive the correct Lagrangian for C µ .
When gϕ 0 = 0, the quantum parts of the gauge field acquire the tachyonic masses. These tachyonic masses are removed by the condensate A + µ A −µ = 1 2 (A 1 µ A 1µ + A 2 µ A 2µ ) . At the same time, this condensate makes classical parts of the gauge field massive. Thus the magnetic potentialC µ and its dual potential C µ become massive. The effective low energy Lagrangian withC µ is presented in Eq. (5.12), and that with C µ is Eq.(5.13).
If there are static magnetic charges Q m and −Q m , the classical fieldC µ connects them. The static potential between them is V Y (r) + V L (r), where V Y is the Yukawa potential and V L is the linear potential. Namely the linear confining potential appears. If we use the dual potential C µ , the Lagrangian is not Eq. (5.12) but Eq.(5.13). However, because of the term Ω in Eq.(4.8), the same static potential is obtained.
Usually the Dirac string is considered to be unphysical. We cannot detect it. In fact, in the massless case, the equation of motion (3.11) for C µ leads to h µν 2 + h µν 3 = 0, and h µν 2 disappears from the Lagrangian (3.9). However, h µν 2 is necessary to produce the correct kinetic term for the dual gauge field. Namely theoretical consistency requires the string term.
When the field C µ becomes massive, this situation changes a little. The equation of motion (4.7) for C µ leads to h µν 2 + h µν 3 + h µν 4 = 0, and h µν 2 can be removed from the Lagrangian (4.5). However there is the remnant h µν 3 h 4µν /4 in Ω. This term is the origin of the linear potential.
A The ǫ symbol and notation In this paper, we employ the metric g µν = diag(1, −1, −1, −1). The antisymmetric ǫ symbol defined by ǫ 0123 = 1 satisfies the formulae From Eq.(A1), the following relations are obtained: For simplicity, we use the notations and where H µν is an antisymmetric tensor.
Next, to calculate H µν , we use the dual potential C µ . From Eqs.(2.8) and (B5), H µν becomes We follow the Zwanziger's definition [2] 1 and, to put the Dirac string on the nagative z-axis, we set a = 0. This choice gives and we find Thus Λ µν term represents the Dirac string part, and Eq.(B7) coincides with Eq. (B6).

B.2 The massive case
In this case,C µ fulfills the equation By modifying Eq.(B2), we find Eq.(B9) is satisfied by the solutioñ In the same way, the dual potential In Ref. [10], we calculated V gh directly, and showed Eq.(C1) becomes Therefore the condition V ′ M (v) = 0 gives v = 0. However, when v = 0, the integrand ln[(−p 2 ) 2 + v 2 ] diverges at p 2 = 0, and the calculation in Ref. [10] is inapplicable. So we should replace p 2 with p 2 + iǫ as usual, and set ǫ → 0 after the p-integration. Thus we consider When ǫ < v, if we take the limit ǫ → 0, the result (C2) is obtained. In the p 0 plane, since the pole {|p| 2 − iǫ + iv} 1/2 (−{|p| 2 − iǫ + iv} 1/2 ) is in the first quadrant (the third quadrant), the usual Wick rotation is inapplicable. On the other hand, when ǫ > v, the poles {|p| 2 − iǫ ± iv} 1/2 are in the fourth quadrant, and −{|p| 2 − iǫ ± iv} 1/2 are in the second quadrant. Then + + + iv −iv Fig. C1 The one-loop ghost diagrams. The dashed line is the ghost propagator c ∓c± , and the blobs represent ±iv.
we can apply the usual Wick rotation, which is performed by the replacement p 0 → −ip 4 and dp 0 → i dp 4 . After that, we can take the limit ǫ → 0, and we find where (p E ) µ = (p, p 4 ) is the Euclidean four-momentum. This is the usual Euclidean potential, and its minimum is at v = 0 [13]. Namely the ghost condensation happens in Minkowski space as well.
We make a comment. The one-loop diagrams in Fig.C1 lead to the series Under the condition v 2 (p 2 + iǫ) 2 < 1, this series converges as This expression gives the potential V gh . For an arbitrary value of p 2 , the condition (C6) is satisfied if ǫ > v. Namely, the condition ǫ > v is required for convergence.