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
where we suppressed Lorentz (L)indices.Here,we neglected higher-derivative terms of n ,e.g.,?2n ,which is in the spirit of the Faddeev-Niemi conjecture;of course,this has to be checked later on.Employing the integral formulas given in App.C,we ?nally obtain for the C determinant
ln det M C 1/2??
132π2ln Λ
32π2ln Λ
64π2
x (?µn )2+1k x ?µn ×?νn 2?1k x (?µn )4,ln(det
64π2
x (?µn )2?5k x ?µn ×?νn 2+35k x (?µn )4,ln(det ? Q φ128π2 x (?µn )2+49k x ?µn ×?νn 2?5k x (?µn )4.(13)The determinant ?S does not contribute,because it is independent of n in one-loop ap-proximation.Inserting these results into Eq.
(10)leads us to the desired Wilsonian e?ective action to one-loop order for the n ?eld in a derivative expansion:?Γk [n ]=1316π2 1?e 2t x (?µn )2+1g 2+716π2t x (?µn ×?νn )2
?1αg 2+516π2t x
(?µn )4,(14)where t =ln k/Λ∈]?∞,0]denotes the “renormalization group time”.We would like to stress once more that ?Γ
k [n ]does not contain the result of ?uctuations of the n ?eld itself;in other words,it represents (an approximation to)the “tree-level action”for the complete quantum theory of the n ?eld.
Indeed,the generation of a “kinetic”term ～(?µn )2growing under the ?ow of increasing k as conjectured by Faddeev and Niemi is observed.Moreover,it has the correct sign (+),implying that an “e?ective ?eld theory”interpretation seems possible.The second term which is proportional to the classical action reveals information about the renormalization of the Yang-Mills coupling:
1
g 2+7
16π2t ?
?βg 2:=?t ?g 2R =?716π2?g 4R .(15)7