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
Acknowledgment
The author wishes to thank W.Dittrich for helpful conversations and for carefully read-ing the manuscript.Furthermore,the author pro?ted from discussions with T.Tok, ngfeld and A.Sch¨a fke.This work was supported in part by the Deutsche Forschungs-gemeinschaft under DFG GI328/1-1.
Appendix
A Di?erential operators,tensors,currents,etc.
This appendix represents a collection of di?erential operators and other tensorial quantities which are required in the main text.
The Faddeev-Popov determinant?FP in Eq.(7)and(10)for covariant gauges involves the operator(in one-loop approximation)
??µDµ(A) C=0=W=??2c+(?2n?n?n??2n)+(?µn?n?n??µn)?µ,(A.1) so that?FP=det ??µDµ(A) C=0=W .
The objects occurring in the exponent of Eq.(7)are de?ned as follows:
M Cµν:=??2δµν+?µ?ν?1α?µn·?νn
M Wµν:=??2δµνc+?µ?νc?1
α ?µn?ν+?νn?µ+?µ?νn
1
K Cµ:=?ν(n·?νn×?µn)+
?µ(n×?2n).(A.2)
α
The determinants in Eq.(10)employ several composites of these operators.Since we?rst perform the C integration,the resulting determinant involves only M C,whereas the W determinant also receives contributions from the mixing term Q C,