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
We treat the δfunctional in Eq.(7)in its Fourier representation,
δ(χ)→ D φe ?i φ·?µW µ+φ·C µn ×W µ+(φ·n )(?µn ·W µ),(9)
where the second term in the exponent,the triple vertex,can actually be neglected,because it leads only to nonlocal terms (ter)or terms of higher order in derivatives.Inserting Eq.(9)into Eq.(7),we end up with three functional integrals over C ,W and φ,which can successively be performed,leading to three determinants,
e ??Γk [n ]→e ?S cl [n ]?S [n ]?FP [n ] det M C ?1/2
det M W )?1
µνQ φν ?1/2,
(10)
where we have omitted several nonlocal terms that arise from the completion of the square in the exponent during the Gaussian integration.In Appendix B,we argue that these nonlocal terms are unimportant in the present Wilsonian investigation.Again,details about the various operators in Eq.(10)are given in App.A.
The determinants are functionals of n only and have to be evaluated over the space of test functions with momenta between k and Λ.The determinants depend also on the gauge parameter α.Only for the Landau gauge α=0is the gauge-?xing δfunctional implemented exactly;in fact,α=0appears to be a ?xed point of the renormalization group ?ow [11].But this in turn ensures that the choice of α=α(k )≡αk at the cuto?scale k →Λis to some extent arbitrary,since αk ?ows to zero anyway as k is lowered.This allows us to conveniently choose αk =Λ=1at the cuto?scale and evaluate the determinants with this parameter choice.
As mentioned above,we evaluate the determinants in a derivative expansion based on the assumption that the low-order derivatives of n represent the essential degrees of freedom in the low-energy domain.There are various techniques for the calculation at our disposal;it turns out that a direct momentum expansion of the operators is most e?cient.4We shall demonstrate this method by means of the third determinant of Eq.(10),the “C determinant”;the key observation is that derivatives acting on the space of test functions create momenta of the order of p with k <p <Λ,whereas derivatives of the n ?eld are assumed to obey |?n |?k in agreement with the Faddeev-Niemi conjecture.This suggests an expansion of the form
ln det M C 1/2=?1
2Tr ln(??2L )+ln L +
?n ·?n
2Tr ln(??2L )?1
??2+1
??2 2
+O ((?n )6),