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

SU(2) gauge theory in the nonlinear gauge of the Curci-Ferrari type is studied in low-energy region. We give a classical solution that connects color electric charges. Its dual solution, which has a configuration of monopole, is also presented. Due to the gluon condensation subsequent to the ghost condensation, these classical fields become massive. The massive Lagrangian with the classical solution and that with the dual solution are derived. We show that these Lagrangians produce a linear potential between a quark and an antiquark. This is the mechanism of quark confinement that is different from the magnetic monopole condensation.


Introduction
In the dual superconductor picture of quark confinement, magnetic monopoles are necessary (see, e.g., [1]). Just like a Cooper pair in superconductivity, they must condense in the vacuum. This condensation gives a mass for non-Abelian gauge fields, and a linear potential between a quark and an antiquark is expected.
In Ref. [2], we studied the SU(2) gauge theory in the nonlinear gauge of the Curci-Ferrari type, and proposed another mechanism that gives a mass for gauge fields. In the low-energy region below the QCD scale parameter Λ QCD , the ghost condensation happens, and the SU(2) gauge theory breaks down to the U(1) theory [3]. If we choose the unbroken U (1) in the A = 3 direction in SU (2), an additional condensate A + µ A −µ appears. Because of this condensate, although the quantum U(1) gauge field a 3 µ is massless, the classical part b 3 µ acquires the mass m.
In the previous paper [4], we considered the magnetic potentialC µ as the classical part b 3 µ . It was shown that the color magnetic charges Q m and −Q m are confined by the linear potential. We also introduced the dual magnetic potential C µ consistently, and derived the same linear potential.
In this paper, based on Ref. [4], we study the confinement of the color electric charges Q e and −Q e . In the next section, we briefly review Ref. [4]. In Sect. 3, we introduce the classical gauge fieldB µ which couples with the color electric current j µ . We call it the electric potential. Its dual potential B µ is also defined. Referring toC µ and C µ , we present the relation betweenB µ and B µ . From this relation, the Lagrangian forB µ and that for B µ are given. Using these Lagrangians, the linear potential between Q e and −Q e is derived in Sect. 4. The origin of the linear potential is discussed in Sect. 5, and the configuration which yields the quark confinement is discussed in Sect. 6. In Sect. 7, the present theory is compared with the dual Ginzburg-Landau model of dual superconductor. Section 8 is devoted to a summary and comments. In Appendix A, notations and formulas are summarized. For a static magnetic charge, the solution of the equation of motion and its dual solution are presented in Appendix B. The solution and its dual solution for a static electric charge are also given. In Appendix C, we calculate the integral which gives the linear potential. To make the manuscript self-contained, the existence of the ghost condensation in Minkowski space is explained in Appendix D.
2 Magnetic potential and its dual potential We review Ref. [4] briefly. Let us consider the SU(2) gauge theory with structure constants f ABC . Using the notations the Lagrangian is This Lagrangian requires gauge fixing, and an appropriate gauge-fixing term and a ghost term are necessary. The Lagrangian for these terms is written as L ϕ (A).

SU(2) gauge theory in the low-energy region
In Refs. [3,5], we employed the nonlinear gauge of the Curci-Ferrari type [6]. Using the Nakanishi-Lautrup field B A , the ghost c A and the antighostc A , and the gauge parameter α 2 , we introduced the field ϕ A = α 2 (−B + igc × c) A . At the one-loop level, it was shown that ϕ A acquires the vacuum expectation value (VEV) ϕ 0 = | ϕ A | = 0 below the scale Λ QCD [3]. This phenomenon is called the ghost condensation [5,7,8]. 1 Choosing the VEV in the A = 3 direction, we write ϕ A = ϕ 0 δ A3 . Next we divided the gauge field A A µ into the classical part b A µ and the quantum part a A µ as In the presence of the VEV ϕ 0 δ A3 , ghost loops yield the tachyonic gluon masses for a A µ [5,9]. In Ref. [2], we have shown that the VEV A + µ A −µ appears and the tachyonic gluon masses are removed. Thus we obtained the Lagrangian Here, we used the notations . We find, although the quantum part a 3 µ is massless, the classical part b 3 µ acquires the mass m = g 3 ϕ 0 /(32π). At the one-loop level, the quantum parts a a µ (a = 1, 2) also acquire the mass M defined by − m 2 2g 2 = i x|tr ∆ + M 2 −1 |x . The gauge-fixing and ghost part becomes is the quantum fluctuation, and α 1 is another gauge parameter. We note, if ϕ A is integrated out, L ϕ (a, b) gives When the classical field b A µ = 0, L NL represents the nonlinear gauge of the Curci-Ferrari type [6], and the last term is required to keep the BRS symmetry in the presence of ϕ 0 [2].

Lagrangian with the magnetic potentialC µ
First, we consider the magnetic potentialC µ , and set b 3 µ =C µ . This field satisfies the equation of motion where k β is the magnetic current, and the space-like vector n α satisfies n α n α = −1. When the mass term forC µ exists, the equation of motion changes from Eq. (2.2) to As an example, n α and k ν for the Dirac monopole are presented in Appendix B. The solutions C µ for Eqs. (2.2) and (2.3) are also given in this appendix.
To incorporate the current k ν in the Lagrangian, we replace (∂ ∧C) µν with 2 Then, performing this replacement and neglecting the components a a µ (a = 1, 2), Eq. (2.1) leads to the Abelian Lagrangian where, because of the equation of motion (2.3), the linear term of a 3 µ vanishes. Now we neglect a 3 µ . In Ref. [4], it is shown that Eq. (2.5) gives the magnetic currentcurrent correlation We choose the current where the magnetic charge is Q m , and the position of the static magnetic monopole (antimonopole) is a (b). We write r = a − b, r = |r| and n µ = (0, n), and follow the procedure in Refs. [11][12][13][14]. Then, when r n, the correlation (2.6) gives the magnetic monopole-antimonopole potential where m χ is the ultraviolet cutoff for the momentum components q T that is perpendicular to n. 3 Thus the magnetic monopoles are confined by the linear potential V mL (r).
The derivation of the linear potential will be discussed in Sect. 4.

Lagrangian with the dual magnetic potential C µ
If we consider the magnetic monopole,C µ is the space-like potential. So, Cho introduced a time-like potential, which is called the dual magnetic potential C µ [15]. We define a dual field strength by and give the dual magnetic potential C µ by the relation [4] As we show in Appendix B, in the case of the Dirac monopole, Λ µν m represents the Dirac string. Since Eq. (2.10) is invariant under the transformationsC µ →C µ + ∂ µ ε and C µ → C µ + ∂ µ ϑ, we choose the gauges n µC µ = 0, n µ C µ = 0.
3 Electric potential and its dual potential Let us consider the color electric current j A µ = j µ δ A3 , which usually couples with the gauge field as −A A µ j Aµ = −(a 3 µ + b 3 µ )j µ . If the magnetic monopole solution in Eq. (B1) is chosen as the classical part b 3 µ , this field cannot couple with the static current j µ = (j 0 , 0). Therefore, to study the confinement of color electric charges, we introduce the electric potentialB µ and its dual potential B µ . It is natural to assume the dual relation Then Eq. (2.10) gives the relation From this equation, we obtain As in Sect. 2,B µ and B µ have U(1) symmetries. If we choose the gauges Namely, the dual electric potential B µ has the string singularity. The term Λ µν e represents the string, which we call the electric string.
Based on the dual relation in Eq. (3.1), we can repeat the procedure in Ref. [4]. So, by applying Eq. (3.1), the Lagrangians forB µ and B µ are obtained from those for C µ and C µ . However, in this section, we derive them directly. To incorporate the electric current j µ , we add ǫ µναβ (n ρ ∂ ρ ) −1 n α j β to (∂ ∧ B) µν . 4 In addition, taking the London current in From the left-hand side (LHS) of Eq.(3.5), using Eqs. (3.4) and (A2), we obtain In the same way, using the formula We note, because of Eq. (A1), the term − 1 gives the kinetic term with the wrong sign [15], i.e., 1 4 (∂ ∧B) 2 . The cross term changes the sign of this term, and the correct kinetic term is derived [4]. If we move the last two terms in the LHS (Eq. (3.6)) to the RHS (Eq. (3.7)), we obtain where The LHS of Eq. (3.8), i.e., is the Lagrangian for B µ . It gives the equation of motion From the RHS of Eq. (3.8), we obtain the equation of motion forB µ as We note, if we multiply Eq. (3.13) by −ǫ λκσµ (n ρ ∂ ρ ) −1 n κ ∂ σ , Eq. (3.11) is obtained.

Electric charge confinement
Since L ecl is equivalent to L ′ ecl , we consider L ecl first. Using the equation of motion SubstitutingB µ = D µν m j ν into Eq. (4.1), we obtain the electric current-current correlation [11][12][13][14] To derive the static potential between the electric charges Q e and −Q e , we insert the static electric current Then the first term in Eq. (4.2) leads to where r = a − b. If we write r = |r|, by removing constants, it gives the Yukawa potential e −mr r .
The factor 1 2 j ν N νµ ∂ σ (∂ ∧B) σ µ comes from the electric string Λ µν e in Eq. (3.2). This factor becomes − m 2 2 j ν N νµB µ only when m = 0. Therefore there are two causes of the linear potential. One is the electric string and the other is the mass for the electric potential. 6 Classical configuration for confinement In Eq. (2.1), we can choose any classical solution as b 3 µ . First we chooseB µ as b 3 µ . The coupling with the electric current is supposed to be −j µB µ . Then the classical part of Eq. (2.1) gives and the equation of motion (D −1 m ) µνB ν = j µ is satisfied.
Next we consider another solution B µ (B, n), which containsB µ and n µ . To couple with B µ , the current j µ may be modified. This modified current, which depends on j µ and n µ , is denoted by J µ (j, n). Then, by setting b 3 µ = B µ , Eq. (2.1) gives Now we assume that Eq. (6.2) is rewritten as µB µ − j µB µ + ∆L(B, n, j).
and the term Ω j yields the confining potential.
Thus we can conclude that the classical configuration which yields the quark confinement is the monopole solution of the dual gauge field B µ .
In the same way, we can derive the equivalent Lagrangian To study the quark confinement, L(B) is often used. In the dual Ginzburg-Landau model of dual superconductor, introducing the monopole field χ, replace the term −k µ B µ with the covariant derivative of χ [17]. This part contains the term |B µ χ| 2 . Adding an appropriate potential V (χ), the VEV χ appears, and B µ becomes massive. Then the interaction (∂ ∧ 2 produces the linear potential [12]. In this case, we can identify A µ and B µ withC µ = (C 0 ,C) and C µ = (C 0 , C), respectively.
Let us consider the current j µ = (j 0 , 0). Since j 0 couples with not C 0 but C, the component C is indispensable to produce the linear potential V eL . The coupling between the magnetic current k µ = (k 0 , k) and C µ is k µ C µ , the space component k is also necessary. Furthermore, to make C massive, some additional mechanism like the introduction of χ and V (χ) is inevitable.
In the present approach, we can identify A µ and B µ withB µ = (B 0 , 0) and B µ = (0, B). Since the ghost condensation and the VEV A + µ A −µ produce the mass for any classical solution, the magnetic current k µ and additional fields like χ are unnecessary to yield V eL . The Lagrangian L(B) holds by adding the mass term m 2 B 2 /2. However, as we showed in Sect. 3, in addition to the mass term m 2 A 2 /2, the term Ω j in Eq. (3.9) should be added to the Lagrangian L(A).

Summary and comments
In the previous papers [3][4][5], we studied the SU(2) gauge theory in the nonlinear gauge of the Curci-Ferrari type. It was shown that, because of the ghost condensation ϕ 0 = 0, the SU(2) gauge theory breaks down to the U(1) theory in the low-energy region [3]. In Ref. [2], we found that, although the quantum U(1) gauge field a 3 µ is massless, the classical part b 3 µ acquires the mass m = g 3 ϕ 0 /(32π) through the VEV A + µ A −µ . Then, in Ref. [4], we considered the magnetic potentialC µ as the classical part b 3 µ . It was shown that the magnetic charges Q m and −Q m are confined by the linear potential. We also showed that the linear potential is derived by using the dual magnetic potential C µ consistently.
In this paper, we considered the electric potentialB µ and the dual electric potential B µ as the classical part b 3 µ . The dual relation betweenB µ and B µ requires the expression (3.2), which contains the string term Λ µν e . In fact, for a point color electric charge,B µ is the usual Coulomb potential, B µ is the monopole-type potential, and Λ µν e is the electric Dirac string. Using Eq. (3.2), we derived the Lagrangians L ecl withB µ and L ′ ecl with B µ , and showed the relation L ecl = L ′ ecl . We note, by applying the duality (3.1), the Lagrangians L ′ mAbel and L mAbel in Sect. 2 lead to the Lagrangians L ecl and L ′ ecl , respectively. From the Lagrangian L ecl , the linear potential V eL between the color electric charges Q e and −Q e is obtained. The operator 1/(n ρ ∂ ρ ), which yields the unphysical string, and the mass m are necessary to give V eL . The Lagrangian L ecl contains the term Ω j in Eq. (3.9).
This term, which is the origin of the linear potential, comes from the electric string Λ µν e and the mass term forB µ . We can also use the Lagrangian L ′ ecl to derive V eL . For a point color electric charge, B µ is the monopole-type solution. So we can say that the classical configuration which yields the quark confinement is the monopole solution of the dual gauge field B µ .
In the dual Ginzburg-Landau model of dual superconductor, the operator 1/(n ρ ∂ ρ ) exists as well. However, there are two different points. One is the fields that contribute, and the other is the mechanism to produce the mass m. In the dual superconductor model, the field (dual field) isC µ (C µ ), and the mechanism is the monopole condensation. In the present approach, the field (dual field) isB µ (B µ ), and the mechanism is the condensation A + µ A −µ subsequent to the ghost condensation ϕ 0 = 0.
We make some comments.
(1). Below the scale Λ QCD , the ghost c A and the antighostc A make a bound state igf ABCcB c C [18], and the ghost condensate ϕ 0 appears. This is the origin of the mass m. This condensation happens in the non-Abelian gauge theory. Without m, the term Ω j vanishes, and the Lagrangian L ecl in Eq. (3.14) reduces to the usual U(1) Lagrangian. Thus the confinement by the mechanism presented here does not happen in QED.
(2). The operator 1/(n ρ ∂ ρ ) yields the string singularity. This singularity should not be detected. However, as we stated in Sect. 3 and stressed in Ref. [4], the string term Λ µν e is important to yield the correct kinetic term −(∂ ∧B) 2 /4. In addition, the effect of the string exists energetically. The energy of the string is proportional to its length [19]. This is the infrared divergence ∝ 1/ε in Appendix C. To get a finite energy, the color electric charges Q e and −Q e must be on the line determined by n, where n µ = (0, n). Then the energy becomes finite, and is proportional to the distance r = |r| between them.
(3). Physical quantities should not depend on n µ . For example, the equation of motion forB µ presented in Eq. (3.12) contains n µ . However, it reduces to the usual equation of motion (D −1 m ) µνB ν = j µ . The next example is the linear potential. The positions of the charges Q e and −Q e must satisfy r n energetically. However, as n can be chosen in an arbitrary direction, we can put Q e and −Q e in arbitrary positions, and the potential V eL (r) is independent of n µ .

B Monopole solutions and dual solutions
For a magnetic charge and an electric charge, we present monopole solutions and dual solutions in the massless case and the massive case.

B.1 Magnetic potential and its dual potential
In the massless case, we choose the magnetic potential, which describes a magnetic monopole, asC where N is an integer, and ρ = x 2 + y 2 . This field satisfies the equation The corresponding dual magnetic potential C µ and its equation of motion are From Eq. (B2), the dual field strength H µν in Eq. (2.10) becomes We follow Zwanziger's definition [10] 1 and, to put the Dirac string on the negative z-axis, set a = 0. This choice gives and we find Namely, Λ µν m represents the Dirac string part.
In the massive case, Eq. (B1) changes tõ C µ = N g z − r rρ 2 e −mr (0, −y, x, 0), and it fulfills the equation The dual magnetic potential and its equation of motion change from Eq. (B2) to B.2 Electric potential and its dual potential Next we consider the color electric current and apply the dual relation (3.1). In the massless case, the electric potentialB µ and its dual potential B µ given byB UsingB µ in Eq. (B4), we find the RHS of Eq. (3.2), i.e., (∂ ∧B) 0j + Λ 0j e becomes (∂ ∧B) 0j = − g 4π Namely, Λ µν e represents the string part. In the massive case, the potentialB µ and the dual potential B µ in Eq. (B4) change tõ Instead of Eqs. (B5) and (B6), they fulfill the equations respectively. First, we calculate the integral C e izrn (z 2 + ω 2 )z 2 dz along the path C in Fig. C1. In the limit R → ∞ and ε → +0, this integral gives where we set r n ≥ 0, and P means the Cauchy principal value. Using I 1 with ω = q 2 T + m 2 , Eq. (4.4) becomes where q T · n = 0 and r T · n = 0. We note, when r T = |r T | = 0, the infrared divergences 2/(εω 2 ) in I 1 (0, ω) and I 1 (r n , ω) cancel out. Using the Bessel function J 0 (ax), we define K 0 (ya, m χ ) as where K 0 (ya) is the modified Bessel function. Then Eq. (C1) becomes V 1 (r n , r T ) = Q 2 e m 2 4π 2 ε ln m 2 χ + m 2 m 2 − 2K 0 (mr T , m χ ) + Q 2 e m 2 4π K 0 (mr T , m χ )r n + I 2 (r n ) + C 1 , Fig. C2 The relation between the length r T and the potential V 1 (r n , r T ) in Eq. (C2). The cases with r T = 0 and r T = 0 are depicted in (a) and (b), respectively.
We note the damping behaviors K 0 (mr T ) and e −mrn come from the propagator 1/( + m 2 ) in j µ N µν /( + m 2 )j ν , and the behavior ∝ r n is from the operator N µν .
To see the meaning of the above infrared divergence, we choose n = (0, 0, 1), a = (a T , a 3 ), and b = (b T , b 3 ). Using Eq. (B3), the current (4.3) gives When a T = b T , the strings from Q e and −Q e with infinite length exist. This is the origin of the infrared divergence. However, when a T = b T , the strings with infinite length disappear, and there remains the string with the length |a − b|. This situation is depicted in Fig. C2. Fig. D1 The one-loop ghost diagrams. The dashed line is the ghost propagator c ∓c± , and the blobs represent ±iv.

D On the ghost condensation in Minkowski space
In Ref. [5], without ǫ, we calculated Eq. (D1) directly. This implies that Eq. (D4) with ǫ = 0 was obtained. However this result is not equivalent to Eq. (D5). The contribution of the residues at p 0 = ± p 2 + iǫ is missing.