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
gauge we shall choose.We face this problem by?xing the gauge in such a way that Lorentz invariance and global color transformations remain as residual symmetries;these are the symmetries of the Skyrme-Faddeev model and must mandatorily be respected.
The Wilsonian e?ective action is characterized by the fact that it governs the dynamics of the low-energy modes below a certain cuto?k;it incorporates the interactions that are induced by high-energy?uctuations with momenta between k and the ultraviolet(UV) cuto?Λwhich have been integrated out.Following the Faddeev-Niemi conjecture,we only retain the n?eld as low-energy degree of freedom.Actually,we integrate over the high-energy modes in two di?erent ways:?rst,we integrate out the k<p<Λ?uctuations of all?elds except for the n?eld,which is left untouched(Sec.3).Secondly,we integrate all ?elds including the n?eld over the same momentum shell(Sec.4).In this way,we can study the e?ect of the n?eld?uctuations on the?ow of the mass scale and the couplings in detail.
The results for both calculations are similar:the mass scale m appearing in Eq.(1)is indeed generated by the renormalization group?ow,and the gauge coupling is asymptoti-cally free.As far as the simplicity of the conjectured e?ective action Eq.(1)is concerned, our results are a bit disappointing:as discussed in Sec.5,further marginal terms(not displayed in Eq.(1))are of the same order as the displayed one and therefore have to be included in Eq.(1).Keeping only those terms that involve single derivatives acting on n results in an action without stable solitons;nevertheless,stability is in fact ensured owing to the presence of higher-derivative terms.The disadvantage is that these terms spoil the desired simplicity of the low-energy e?ective theory.
Of course,our perturbative results represent only a?rst glance at the true infrared behavior of the system and are far from providing qualitatively con?rmed results,not to mention quantitative predictions.To be precise,the one-loop calculation investigates only the form of the renormalization group trajectories of the couplings in the vicinity of the perturbative Gaussian?xed point.Nevertheless,various extrapolations of the perturba-tive trajectories can elucidate the question as to whether the Faddeev-Niemi conjecture is realizable or not.
2Quantum Yang-Mills theory in Cho-Faddeev-Niemi-Shabanov variables
In decomposing the Yang-Mills gauge connection,we follow[2,9,10].Let Aµbe an SU(2) connection where the color degrees of freedom are represented in vector notation.We parametrize Aµas
Aµ=n Cµ+(?µn)×n+Wµ,(2) where the cross product is de?ned via the SU(2)structure constants.Cµis an“abelian”connection,whereas n denotes a unit vector in color space,n·n=1.Wµshall be orthogonal to n in color space,obeying Wµ·n=0,so that Cµ=n·Aµ.For a given n,Cµ
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