Wilsonian effective action for SU(2) Yang-Mills theory with(13)

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

The last determinant in Eq.(10)arises

from

the

φ

integration and

receives contributions from the relevant parts of the exponent of Eq.(9),which we denote by

Q φµ:=i ??µc +?µn ?n ,

(A.5)so that δ(χ)→ D φexp(? W µ·Q φµφ)to one-loop order.Employing a notation similar

to Eq.(A.4),the di?erential operator accompanying the term ～φφin the exponent ?nally

reads Q φµ(

??2+?n ·?n µν

(n ·?κn ×?κνn ).(B.8)

Within the calculation of the determinants,we expanded the inverse operator assuming that ?n ·?n ???2.This was justi?ed,since the derivative operator acts on the test func-tion space with momenta p between k and Λ,which are large compared to the conjectured slow variation of the n ?eld.

In the present case,the situation is di?erent,because the derivative term ??2acts only on the n ?eld and its derivatives to the right (there is no test function to act on).In other words,the nonlocal terms are only numbers,not operators.The derivatives can thus be approximated by the (inverse)scale of variation of the n ?eld or its derivatives which is much smaller than k or Λ.This implies that the nonlocal terms do not depend on k or Λ,so that they cannot contribute to the ?ow of the couplings.

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