Composite operators in non-Abelian gauge theories

The renormalization of the composite gauge field product operators A_μᵃ(x)A_νᵇ(x) is carried out in detail in asymptotically free non-Abelian SU(n) gauge theories. Upon renormalization, these operators mix with similar operators obtained by Lorentz and SU(n) group rotations and with other composite operators formed from ghost fields or derivatives of A. It is shown, using renormalization-group and SU(n)-projection techniques, that this renormalization problem is completely soluble. The renormalization-group equations satisfied by the composite renormalization-constant matrix Z are deduced and solved using the computed second-order expression for Z. For SU(2), Z is put in triangular form so that the effective anomalous dimension eigenvalues can be read off. For the general SU(n) group, it is more convenient to use group projection operators and crossing matrices to explicitly diagonalize the renormalization-group equations. The main results can be most simply stated as an explicit short-distance operator expansion which expresses the product A_μᵃ(x)A_νᵇ(0) for x→0 in terms of the finite composite operators A_αᶜ(0)A_βᵈ(0):. The leading singularity is seen to be associated with the singlet operator δᵃᵇg_μν:A·A:. The results are used to study the invariance of the models under the Abelian gauge transformations A_μᵃ→A_μᵃ+∂_μΛᵃ.