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
by Faddeev and Niemi.This term is relevant in the renormalization group sense and perturbatively exhibits a quadratic dependence on the UV cuto?Λ.
Furthermore,we studied the renormalization group?ow of the marginal couplings of the n?eld self-interactions given by the Yang-Mills coupling and the gauge parameter. These terms are responsible for the stabilization of possible topological excitations of the n?eld,as suggested by the Skyrme-Faddeev model.In total,the di?erence between?Γk andΓk is only of quantitative nature:the inclusion of hard n?eld?uctuations increases the running of the marginal couplings and reduces the new mass scale;qualitative features such as stability of possible solitons remain untouched.
In fact,the question of stability turns out to be delicate:truncating our results for?Γk or Γk in Eqs.(14)or(21)at the level of the original Faddeev-Niemi proposal Eq.(1)(the?rst lines of Eqs.(14)and(21),respectively),we?nd an action that allows for stable knotlike solitons,since the coe?cients of both terms are positive(as long as we stay away from the Landau pole,which we consider as unphysical).Taking additionally the(?n)4term of?Γk orΓk into account,which is also marginal and does not contain second-order derivatives on n,stability is lost,since the coupling coe?cient is negative in Eqs.(14)and(21);for stable solitons,a strictly positive coe?cient would be required for this truncation,as was shown in[12].
Finally dropping the demand for?rst-order derivatives,we can include one further marginal term～?2n·?2n as given in Eq.(21)forΓk.With the aid of the identity
x(?2n×n)2= x[?2n·?2n?(?µn)4],(22)
we?nd that the second line of Eq.(21)represents a strictly positive contribution to the action which again stabilizes possible solitons.5
Of course,this game could be continued by including further destabilizing and stabiliz-ing higher-order terms again and again,but such terms are irrelevant in a renormalization group sense;that means their corresponding couplings are accompanied by inverse powers of the UV cuto?Λand are thereby expected to vanish in the limit of large cuto?.
To summarize,our perturbative renormalization group analysis suggests enlarging the Faddeev-Niemi proposal for the e?ective low-energy action of Yang-Mills theory by taking all marginal operators of a derivative expansion into account.The original proposal of Eq.(1)was inspired by a desired Hamiltonian interpretation of the action that demands the absence of third-or higher-order time derivatives.But,as demonstrated,the covariant renormalization group does not care about a Hamiltonian interpretation of the?nal result. In some sense,the desired“simplicity”of the?nal result is spoiled by the presence of higher-derivative terms;moreover,it remains questionable as to whether the importance of the ?2n·?2n term is still consistent with the derivative expansion of the action.Unfortunately, this cannot be checked within the present approach.
It should be stressed once again that the perturbative investigation performed here hardly su?ces to con?rm results about the infrared domain of Yang-Mills theories.On the