tion factors. As an example, consider the quark self-energy represented by Fig. 1.5. The renormalized and bare quark self-energy are related by where Σ$_{B}$ denotes the bare quark self-energy (1.44) which is in terms of bare parameters. Using the relationships between the bare and renormalized parameters given in Eq. (1.45), the bare parameters can be expressed in terms of renormalized parameters along with the corresponding renormalization factors. The renormalization factors are defined such that the limit in Eq. (1.48) can be taken order by order in α . Ref. [97] provides expressions for the renormalization factors in Eq. (1.45) to two-loop order. So far, we have only considered correlation functions that involve multiple quark or gluon fields at distinct spacetime locations. However, in QSR calculations correlation functions in- volving composite local operators are needed. For instance, the following correlation function can be used to study a heavy-light pseudoscalar meson: The current J ( x ) = i ¯ q ( x ) γ$_{5}$Q ( x ), where q and Q respectively denote light and heavy quark fields, is a composite local operator that couples to the heavy-light pseudoscalar mesons [66]. In order to extend QSR calculations to higher orders, we must consider the renormalization of correlation functions that involve composite operators. The renormalization of composite operators is complicated by the fact that multiple com- posite operators may share the same quantum numbers. Because the fields in the QCD Lagrangian (1.40) have distinct quantum numbers, they must renormalize separately. How- ever, this is not the case with composite operators: those with the same quantum numbers can mix under renormalization. The renormalization of composite operators is discussed in detail in Ref. [39]. In general, in order to study the renormalization of an operator O$_{a}$ with dimension a , one must also consider operators O$_{b}$ with the same quantum numbers and dimension b < a (the dimensions of quark and gluon fields are given in Appendix A). The 27 Π $^{(}$Q 2 ) = i$^{∫}$ d $^{4}$x e iq · $^{x}$〈 Ω | T $^{[}$J ( x ) J $^{†}$(0) $^{]}$| Ω 〉 , Q 2 = − q $^{2}$. (1.49) Σ$_{R}$ ( q ; m$_{R}$ , a$_{R}$ , α$_{R}$ ) = lim ϵ → 0 $^{[}$Z − 1 $_{2F}$Σ$_{B}$ ( q ; m$_{B}$ , a$_{B}$ , α$_{B}$ ) ] , (1.48)