functions must be calculated. For instance, the β function is calculated to O ( α $^{3}$) in Ref. [97]. We will now outline the calculation of the leading order term in the expansion. First, the Slavnov-Taylor identities (1.46) allow us to write The renormalization factor Z$_{α}$ can be determined using the methods described previously. To one-loop order, where n$_{f}$ denotes the number of quark flavours. In Ref. [97] it is shown that to lowest order in α , where Z (1) α denotes the divergent term in Eq. (1.64). In a similar fashion it can be shown that to lowest order The differential equations defining the renormalization group functions in Eq (1.62) can be solved to determine how the QCD Lagrangian parameters depend on the renormalization scale. For instance, using the one-loop expression for the β function (1.65), we find The value of β$_{1}$ depends on the number of active quark flavours n$_{f}$ , as can be seen from Eq. (1.65). In QSR calculations n$_{f}$ is chosen to encompass the heaviest quark in the hadron being studied. For instance, if the heaviest quark is the charm quark n$_{f}$ = 4, whereas if it is the bottom quark n$_{f}$ = 5. This is justified by the decoupling theorem, which states that 31 µ dα dµ = α $_{2}$β$_{1}$ π → α ( µ ) = α ( M ) 1 − β$_{1}$α ( M ) 2 π log [ µ 2 M 2 ] , (1.67) γ ( α ) = α π γ$_{1}$ , γ$_{1}$ = 2 . (1.66) β ( α ) = − 2 α ∂Z (1) α ∂α , β ( α ) = α π β$_{1}$ , β$_{1}$ = − 11 2 + n$_{f}$ 3 , (1.65) Z$_{α}$ = 1 + α π $^{[}$11 4 − n$_{f}$ 6 ] 1 ϵ , (1.64) α = ( gµ ) 2 ϵ 4 π , α$_{R}$ = Z − 1 α α$_{B}$ , Z$_{α}$ = ˜ Z 2 1 ˜ Z − 2 3 Z − 1 3YM . (1.63)