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
and Wµ,the connection Aµis uniquely determined by Eq.(2).In the opposite direction, there is still some arbitrariness:for a given Aµ,n can generally be chosen at will,but then Cµand Wµare?xed(e.g.,Wµ=n×Dµ(A)n,where Dµdenotes the covariant derivative). While the LHS of Eq.(2)describes3color×4Lorentz=12o?-shell and gauge-un?xed degrees of freedom,the RHS up to now allows for(Cµ:)4Lorentz+(n:)2color+(Wµ:)3color×4Lorentz?4n·Wµ=0=14degrees of freedom.Two degrees of freedom on the RHS remain to
be?xed.For example,by?xing n to point along a certain direction and imposing gauge conditions on Wµ,we arrive at the class of abelian gauges which are known to induce monopole degrees of freedom in Cµ.In order to avoid these topological defects,we let n vary in spacetime and impose a general condition on Cµ,n and Wµ,
χ(n,Cµ,Wµ)=0,withχ·n=0,(3) which?xes the redundant two degrees of freedom on the RHS of Eq.(2).Moreover, Eq.(3)also determines how n,Cµand Wµtransform under gauge transformations of Aµ: by demanding thatδχ(n,Cµ(A),Wµ(A))=0(andδ(χ·n)=0),the transformationδn of n is uniquely determined,from whichδCµandδWµare also obtainable.
The thus established one-to-one correspondence between Aµand its decomposition(2) allows us to rewrite the generating functional of YM theory in terms of a functional integral over the new?elds[9,10]:
Z= D n D C D Wδ(χ)?S?FP e?S YM?S gf.(4) Beyond the usual Faddeev-Popov determinant?FP,the YM action S YM and the gauge ?xing action S gf,we?nd one further determinant introduced by Shabanov,?S;this de-terminant accompanies theδfunctional which enforces the constraintχ=0,in complete analogy to the Faddeev-Popov procedure:
?S:=det δχ