Wilsonian effective action for SU(2) Yang-Mills theory with(5)

The Cho-Faddeev-Niemi-Shabanov decomposition of the SU(2) Yang-Mills field is employed for the calculation of the corresponding Wilsonian effective action to one-loop order with covariant gauge fixing. The generation of a mass scale is observed, and the fl

Incidentally,the gauge transformation properties of Cµand Wµalso become very simple with the choice(6):Wµalso transforms homogeneously,andδCµ=n·?µ?.

Finally,the choice of the gauge-?xing condition must also be viewed as being part of the de?nition of the decomposition.Not only does the functional form of?FP and S gf depend on this choice,but the discrimination of high-and low-momentum modes is also determined by the gauge?xing.In fact,this gauge dependence of the mode momenta usually is the main obstacle against setting up a Wilsonian renormalization group study.But in the present context,it belongs to the conjecture that the particular gauge that we shall choose singles out those low-momentum modes which?nally provide for a simple description of low-energy QCD;in a di?erent gauge,we would encounter di?erent low-momentum modes, but we also would not expect to?nd the same simple description.

In this work,we choose the covariant gauge condition?µAµ=0.This automatically ensures covariance of the resulting e?ective action and,moreover,allows for the residual symmetry of global gauge transformations,?=const.Together with the choice(6),this residual symmetry coincides with the desired global color symmetry of the Skyrme-Faddeev model(1).This means that the demand for color and Lorentz symmetry of the action(1) is satis?ed exactly by a covariant gauge and condition(6).

3One-loop e?ective action without n?uctuations Our aim is the construction of the one-loop Wilsonian e?ective action for the n?eld by integrating out the C and W?eld over a momentum shell between the UV cuto?Λand an infrared cuto?k<Λ.In general,this will induce nonlinear and nonlocal self-interactions of the n?eld;since we are looking for an action of the type(1),we represent these interactions in a derivative expansion and neglect higher derivative terms of order O(?2n?2n)(later, we shall question this approach).

Furthermore,we do not integrate out n?eld?uctuations in this section(see Sect.4)and disregard any induced C or W interactions below the infrared cuto?k.From a technical viewpoint,the one-loop approximation of the desired e?ective actionΓk[n]is obtained by a Gaussian integration of the quadratic C and W terms in Eq.(4),neglecting higher-order terms of the action:

e??Γk[n]=e?S cl[n] k D C D W?S[n]?FP[n]δ(χ)(7)

×e?12M CµνCν+Wµ1

4g2(?µn×?νn)2+

1

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