content_image/313651.jpg content_image/313650.jpg content_image/313649.jpg content_image/313648.jpg
renormalization factors of these operators are formally defined as 

<img src='content_image/313651.jpg'>

where the vector O$_{i}$ = $^{(}$O$_{a}$ , Λ a − $^{b}$O$_{b}$ , . . . ) and the parameters Λ ensure that all elements of the vector have the same dimension. The matrix Z$_{ij}$ is an upper diagonal matrix containing the renormalization factors. The renormalization factors Z$_{ij}$ can be determined by calculating correlation functions composed of the operators O$_{i}$ . 

In Chapter 6 mixing between scalar $^{(}$J P C = 0 ++ ) glueballs and quark mesons is studied using the currents 

<img src='content_image/313650.jpg'>

The scalar glueball operator J$_{g}$ mixes under renormalization with the scalar quark meson operator J$_{q}$ , which has the same dimension and quantum numbers. In Refs. [97, 89] the renormalization of the scalar glueball operator is studied using background field techniques. The resulting renormalized scalar glueball operator is given by 

<img src='content_image/313649.jpg'>

where n$_{f}$ is the number of active quark flavours. The mixing between the operators J$_{g}$ and J$_{q}$ under renormalization is signaled by the second term in Eq. (1.52). This second term leads to a crucial renormalization-induced contribution in the mixing analysis in Chapter 6. 

A somewhat simpler example of composite operator renormalization is considered in Chapter 5. There the renormalization of the scalar diquark current is considered, which is given by 

<img src='content_image/313648.jpg'>

where α , β , γ are colour indices, C is the charge conjugation operator and γ$_{5}$ is a Dirac ma- trix (both are defined in Appendix A). There are no composite operators of lower dimension 

