content_image/309386.jpg content_image/309385.jpg content_image/309384.jpg
the result is 

<img src='content_image/309386.jpg'>

where γ$_{E}$ is the Euler-Mascheroni constant (see Appendix B), Q 2 = − q 2 is the Euclidean momentum and f is a function of the dimensionless ratio Q $^{2}$/m$^{2}$. The divergence has been regulated and appears as a simple pole at ϵ = 0. 

Theories in which divergences can be removed systematically order by order in pertur- bation theory are called renormalizable. The proof that QCD is renormalizable was given in Refs. [115, 116]. Renormalization is the process of canceling these divergences. Formally, this can be achieved by rescaling the parameters of the QCD Lagrangian (1.40) as follows: 

<img src='content_image/309385.jpg'>

where we have used the notations of Ref. [97]. The constants Z$_{i}$ are called renormalization factors, and the subscripts B and R denote bare and renormalized quantities, respectively. The bare couplings [ g$_{YM}$ ]$_{B}$, [˜ g ]$_{B}$, [ g$_{F}$ ]$_{B}$, and [ g$_{5}$ ]$_{B}$ are associated with the three-gluon, ghost- gluon, quark-gluon and four-gluon interaction terms in the bare QCD Lagrangian (1.40). However, all of these couplings must be identical in order for the QCD Lagrangian to be BRST invariant. This implies the that renormalization factors satisfy 

<img src='content_image/309384.jpg'>

These identities are closely related to the Slavnov-Taylor identities that are needed in order to prove that QCD is renormalizable. 

The Lagrangian given above in Eq. (1.40) is implicitly in terms of bare parameters, hence it is called the bare Lagrangian. The renormalized Lagrangian has the same form as the bare Lagrangian, and can be obtained using the relations given in Eq. (1.45). Because QCD is a 

