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
contrary,it is only a valid approximation in the vicinity of the Gaussian UV?xed point of
the theory.Nevertheless,our study might lend some intuition to possible nonperturbative scenarios:for example,let us assume that the Landau gaugeα=0indeed is an infrared
?xed point in covariant gauges.Then the stabilizing term～(?2n×n)2is enhanced in the
infrared,provided that the increase of the running coupling g obeysαg2→0for k→0; this would be realized,e.g.,if g approached an infrared?xed point.Such a scenario thus
supports the idea of topological knotlike solitons as important infrared degrees of freedom
of Yang-Mills theories.
Perhaps the main drawback of our study lies in the fact that the new mass scale is not
renormalization-group invariant;for example,we can read o?from Eq.(21)that
m2k=1
Λ≤0.(23)
The new mass scale m k is necessarily proportional toΛ,because there simply is no other
scale in our system.But contrary to the gauge coupling or the gauge parameter,which can be made independent ofΛby adjusting the bare parameters,theΛdependence of m k
persists,since there is no bare mass parameter to adjust.One may speculate that this problem is solved by“renormalization group improvement”of the kind
Λ2→Λ2e?3·16π2
q m,where q denotes the value of the coe?cient in front of the(?µn×?νn)2term[12,14].For couplings of order1,
we end up with soliton masses of the order of M～O(1)GeV;this is in accordance with
lattice results for glue ball masses:e.g.,M GB?1.5GeV for the lowest lying state in SU(2) [15].Of course,this rough and speculative estimate should not be viewed as a“serious
prediction”of our work.
With all these reservations in mind,the Faddeev-Niemi conjecture about possible low-energy degrees of freedom of Yang-Mills theories provides an interesting working hypothesis which deserves further exploration.
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