Composite operators in non-Abelian gauge theories

Abstract

The renormalization of the composite gauge field product operators $A_{\mu }^a\mathrm{}\left(x\right)\mathrm{}{A}_{\nu }^b\mathrm{}\left(x\right)\mathrm{}$ is carried out in detail in asymptotically free non-Abelian $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ gauge theories. Upon renormalization, these operators mix with similar operators obtained by Lorentz and $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ group rotations and with other composite operators formed from ghost fields or derivatives of $A$. It is shown, using renormalization-group and $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$-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 $\mathrm{SU}\mathrm{}\left(n\right)\mathrm{}$ 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 $A_{\mu }^a\mathrm{}\left(x\right)\mathrm{}{A}_{\nu }^b\mathrm{}\left(0\right)\mathrm{}$ for $x\to 0$ in terms of the finite composite operators $A_{\alpha }^c\mathrm{}\left(0\right)\mathrm{}{A}_{\beta }^d\mathrm{}\left(0\right)\mathrm{}$:. The leading singularity is seen to be associated with the singlet operator ${\delta }^{\mathrm{ab}}{g}_{\mu \nu }:A·A:$. The results are used to study the invariance of the models under the Abelian gauge transformations $A_{\mu }^a\to A_{\mu }^a+{\partial }_{\mu }{\Lambda }^a$.