renormalization factors of these operators are formally defined as 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 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 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 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 28 J d $_{α}$( x ) = ϵ$_{αβγ}$Q$_{β}$ ( x ) Cγ$_{5}$q$_{γ}$ ( x ) , (1.53) G 2 R = [ 1 + 1 ϵ α π $^{(}$11 4 − n$_{f}$ 6 )] G 2 B − 4 ϵ α π $^{[}$m$_{u}$ ¯ uu + m$_{d}$ ¯ dd ] B , (1.52) J$_{g}$ = αG $^{2}$, G 2 = G a $_{µν}$G µν a , J$_{q}$ = m$_{q}$ $^{(}$¯ uu + ¯ dd ) . (1.51) [ O$_{i}$ ]$_{R}$ = Z$_{ij}$ [ O$_{j}$ ]$_{B}$ (1.50)