Composite operators in non-Abelian gauge theories

Abstract

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