Maximal Abelian gauge and a generalized BRST transformation

We apply a generalized Becchi-Rouet-Stora-Tyutin (BRST) formulation to establish a connection between the gauge-fixed $SU(2)$ Yang-Mills (YM) theories formulated in the Lorenz gauge and in the Maximal Abelian (MA) gauge. It is shown that the generating functional corresponding to the Faddeev-Popov (FP) effective action in the MA gauge can be obtained from that in the Lorenz gauge by carrying out an appropriate finite and field-dependent BRST (FFBRST) transformation. In this procedure, the FP effective action in the MA gauge is found from that in the Lorenz gauge by incorporating the contribution of non-trivial Jacobian due to the FFBRST transformation of the path integral measure. The present FFBRST formulation might be useful to see how Abelian dominance in the MA gauge is realized in the Lorenz gauge.

In SU(N) YM theory, the MA gauge has been exploited to investigate its nonperturbative features, such as quark confinement [16]. The MA gauge is a nonlinear gauge for a partial gauge fixing imposed to maintain only the maximal Abelian gauge symmetry specified by U(1) N −1 . This gauge enables us to extract Abelian degrees of freedom latent in SU(N) YM theory. In fact, in the MA gauge, Abelian dominance [10,[17][18][19][20] and the emergence of magnetic monopoles [3][4][5]11] are realized as remarkable phenomena in the non-perturbative infrared region. Abelian dominance is known as a low energy phenomenon in which only the diagonal YM fields associated with U(1) N −1 dominate, behaving as Abelian gauge fields, while effects of the off-diagonal YM fields associated with SU(N)/U(1) N −1 are strongly suppressed because of their large effective mass of about 1GeV [18][19][20]. (If we consider massive off-diagonal YM fields at the classical Lagrangian level, the MA gauge condition can be derived as the Euler-Lagrange equation for an additional scalar field [21].) Magnetic monopoles emerge as topological objects characterized by the nontrivial homotopy group π 2 SU(N)/U(1) N −1 = Z N −1 [4]. The resulting effective Abelian gauge theory leads to the dual-superconductor picture for the YM vacuum upon assuming condensation of the monopoles [22][23][24]. In this picture, the electric flux defined from the Abelian gauge fields is squeezed into a string-like tube owing to the dual Meissner effect; as a result, (anti-)quarks are confined by a linear potential due to the electric flux tube [25,26]. In this way, quark confinement is well explicated in SU(N) YM theory formulated in the MA gauge.
However, since quark confinement is a physical phenomenon, it should be explicated independent of choices of gauge. We therefore need to explore how quark confinement is analytically demonstrated in terms of another gauge, for instance, the Lorenz gauge [27].
For this purpose, it will be useful to clarify the connection between different gauge-fixed SU(N) YM theories formulated in the MA gauge and another gauge. If such a connection is established, it may become possible to see how Abelian dominance and the emergence of magnetic monopoles are realized in another gauge. A universal formulation for connecting two different effective gauge theories has been developed by Joglekar and Mandal by means of the finite field dependent Becchi-Rouet-Stora-Tyutin (FFBRST) transformation [28].
In this formulation, the usual (infinitesimal) BRST transformation [29,30] is generalized by allowing the parameter finite and field-dependent [28]. The FFBRST transformation enjoys the properties of the usual BRST transformation except it does not leave the path integral measure invariant due to its finiteness. Under a certain condition, the non-trivial Jacobian caused by the FFBRST transformation of the path integral measure is expressed as a local functional of fields, which eventually modifies the effective action of the theory [28]. Due to this remarkable feature, the FFBRST transformation is capable of relating the generating functionals in different gauge-fixed YM theories. The FFBRST formulation has found various applications in gauge field theories over last two decades [28,[31][32][33][34][35][36][37].
In this paper, we apply the FFBRST formulation to establish a connection between the generating functional corresponding to the Faddeev-Popov (FP) effective action in the Lorenz gauge and that in the MA gauge. 1 For this purpose, we start with the FP effective action in the Lorenz gauge [38][39][40] and construct the FFBRST transformation with an appropriate finite field dependent parameter. Then we show that the generating functional corresponding to the FP effective action in the MA gauge [12,14,15] can be derived from that in the Lorenz gauge by carrying out the FFBRST transformation. In this process, we see that the FP effective action in the MA gauge is obtained by incorporating a non-trivial contribution of the Jacobian arising from the FFBRST transformation of the path integral measure. For convenience, we treat the case of N = 2 only. However, our approach can 1 In this paper, the FP effective action means the sum of the pure YM action and the gauge-fixing and FP ghost term that can be written in the BRST and anti-BRST exact form. be generalized for arbitrary N.
This paper is organized as follows. In the next section, we briefly discuss the BRST and Here, g is a coupling constant. The action S YM remains invariant under the infinitesimal gauge transformation where λ a (a = 1, 2, 3) are infinitesimal real functions and ǫ abc is the Levi-Civita symbol in We can decompose the gauge transformation (2.3) into the SU(2)/U(1) part specified by λ i (i = 1, 2) and the U(1) part specified by λ 3 in such a way that and We see that ∇ µ is the covariant derivative for the U(1) gauge transformation (2.6). The fields A i µ are identified as the off-diagonal YM fields and A 3 µ is identified as the diagonal YM field.
Next, introducing the FP ghost fields c a (x), the FP anti-ghost fieldsc a (x), and the Nakanishi-Lautrup (NL) fields B a (x), we define the BRST transformation [29,30] and the anti-BRST transformation The (anti-)BRST transformations with a constant Grassmann parameter δΛ are defined by δ B := δΛs andδ B := δΛs. Then Eq. (2.7) is expressed as

A. Lorenz gauge
The Lorenz gauge condition ∂ µ A a µ = 0 [27] can be used to completely break the SU(2) gauge invariance of the YM action (2.1). This gauge condition can be incorporated into the following gauge-fixing and FP ghost term in a BRST and anti-BRST invariant manner [38][39][40]: (2.13b) (2.14) (Another generalized Lorenz gauge condition ∂ µ A a µ − αB a = 0 is often adopted in literature.) When α = 0, the gauge condition (2.14) reduces to the (original) Lorenz gauge condition. The FP effective action in the Lorenz gauge is given by which is, of course, invariant under the BRST and anti-BRST transformations.

B. MA gauge
The MA gauge condition is a nonlinear gauge condition and is defined by This condition partially breakes the SU(2) gauge invariance of the YM action (2.1) so as to be maintaining its gauge invariance under the U(1) gauge transformation (2.6). In fact, under the gauge transformation (2.6), ∇ µ A i µ transforms covariantly as It is easy to show that δ B S MA =δ B S MA = 0. Variation of S MA with respect to B i yields a generalized MA gauge condition (2.20) When β = 0, this condition reduces to the (original) MA gauge condition (2.16). The FP effective action in the MA gauge is given by which is obviously both BRST and anti-BRST invariant.
In the light of Eq. (2.6a), we can consistently impose the U(1) gauge transformation rules In this section, we recapitulate the FFBRST formulation for YM theory developed in Ref. [28]. For this purpose, we first write the usual BRST transformation (2.10) as where δΛ is an infinitesimal and field-independent Grassmann parameter, 2 and φ I is the generic notation of the fields (A a µ , c a ,c a , B a ) involved in the theory. The index I distinguishes the fields as well as their components. The basic properties of BRST transformation do not depend on whether the parameter δΛ is (i) finite or infinitesimal and/or (ii) field-dependent or not, as long as it is anti-commuting and spacetime independent. This renders us a freedom to construct the BRST transformation with the parameter finite and field-dependent without affecting its basic features. First we make the infinitesimal parameter field-dependent by interpolating a continuous parameter, κ (0 ≤ κ ≤ 1), in the theory. The generic field, φ I (x, κ), depends on κ such that φ I (x, κ = 0) = φ I (x) is the initial field and φ I (x, κ = 1) = φ ′ I (x) is the transformed field.
It has been shown [28] that the Jacobian J(κ) can be replaced within the functional integral as iff the following condition is satisfied [28]: Here, S 1 [φ(κ), κ] is a local functional of the fields, and S denotes either the FP effective action S L or S MA . The infinitesimal change in the Jacobian J(κ) can be calculated with the following formula [28] 1 where |I| is defined as |I| = 0 for bosonic fields φ I and as |I| = 1 for fermionic fields φ I .
Once we know J −1 (dJ/dκ), we can find S 1 from the condition in Eq. (3.8).

IV. CONNECTION BETWEEN GENERATING FUNCTIONALS IN THE LORENZ AND MA GAUGES
In this section, we construct the FFBRST transformation with an appropriate finite parameter to obtain the generating functional corresponding to S MA from that corresponding to S L . We calculate the Jacobian corresponding to such a FFBRST transformation following the method outlined in Sec III and show that it is a local functional of fields and accounts for the differences of the two FP effective actions.
The generating functional corresponding to the FP effective action S L is written as Here, γ p (p = 1, 2, 3, 4, 5) are arbitrary constant parameters and all the fields depend on the parameter κ. The infinitesimal change in the Jacobian corresponding to this FFBRST transformation is calculated using Eq. (3.9) to obtain To express the Jacobian contribution in terms of a local functional of fields, we make an ansatz for S 1 by considering all possible terms that could arise from such a transformation as + ξ 7 gǫ abc B a c bcc + ξ 8 g 2 ǫ abc ǫ adecbcc c d c e + ξ 9 gǫ ij B icj c 3 + ξ 10 g 2 ǫ ij ǫ klcicj c k c l , where all the fields are considered to be κ dependent and we have introduced arbitrary κ dependent parameters ξ n = ξ n (κ) (n = 1, 2 . . . , 10). It is straight to calculate with ξ ′ n := dξ n /dκ by using Eqs. (3.2) and (2.7) and the nilpotency s 2 = 0. We substitute Eqs. (4.3) and (4.5) into Eq. (3.8) with S = S L to find the condition to replace the Jacobian contribution in terms of a local functional of the fields as This can be written as The terms proportional to Θ ′ , which are regarded in Eq. (4.7) as nonlocal terms due to Θ ′ , independently vanish if ξ 1 + ξ 6 = 0 , (4.8a) ξ 2 + ξ 5 = 0 , (4.8b) ξ 7 + 4ξ 8 + ξ 9 + 4ξ 10 = 0 , (4.8c) To make the remaining local terms in Eq. (4.7) vanish, we need the following conditions: from which we also have The differential equations for ξ n (κ) can indeed be solved with the initial conditions ξ n (0) = 0 to obtain the solutions It should be noted that the solutions in Eq (4.11) also satisfy Eqs. (4.8a)-(4.8d). The conditions in Eqs. (4.8) and (4.9) are thus compatible with each other.
Since γ p (p = 1, 2, 3, 4, 5) are arbitrary constant parameters, we can chose them as follows: Substituting the solutions found in Eq. (4.11) into Eq. (4.4) and considering the specific values of the parameters in Eq. (4.12), we obtain (4.13) Thus the FFBRST transformation with the finite parameter Θ that is defined by Eq.
(3.4) with Eq. (4.2) changes the generating functional Z L as where m denotes an effective mass of the off-diagonal YM fields A i µ . 3 The mass term S m remains invariant under the U(1) gauge transformation (2.6a), so that it does not break the U(1) gauge invariance of S MA . Being introduced S m , Eq. (5.1) is modified as where S ′ m is defined as the inverse FFBRST transformation of S m . As expected S ′ m is highly nonlocal and will not be easy to deal with. However, S ′ m must describe a phenomenon corresponding to Abelian dominance, and we would be able to see with S ′ m how Abelian dominance is realized in the Lorenz gauge. We therefore hope to investigate the details of S ′ m in the near future.