Abstract
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.
Highlights
In the high energy region, Yang-Mills (YM) theory enjoys the asymptotic freedom and can be used perturbatively to describe physical systems [1, 2]
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 maximal Abelian (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 π2 SU (N )/U (1)N−1 = ZN−1 [4]
We have applied the finite and field-dependent BRST (FFBRST) formulation developed in Ref. [28] to clarify the connection between the gauge-fixed SU(2) YM theories formulated in the Lorenz and MA gauges
Summary
In the high energy region, Yang-Mills (YM) theory enjoys the asymptotic freedom and can be used perturbatively to describe physical systems [1, 2]. We 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. 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. 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..
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