renormalizable theory, all correlation functions calculated with the renormalized Lagrangian must be free of divergences. The renormalization factors are of the form where A is a constant and dimensional regularization is used with d = 4 + 2 ϵ . In practice, a correlation function can be calculated using the bare Lagrangian (1.40), and the result in terms of the bare parameters will contain divergences. The bare parameters can be rewritten in terms of the renormalization factors and renormalized parameters using (1.45). When this is done, the divergences in the bare correlation function are canceled by compensating divergences in the renormalization factors. The resulting renormalized correlation function is purely in terms of the renormalized parameters and is free of divergences. This approach is called bare perturbation theory, because it involves calculating correlation functions in terms of bare parameters which are then renormalized. A complementary approach is renormalized perturbation theory, where correlation functions are calculated in terms of renormalized parameters. This procedure involves inverting the relations (1.45), leading to the introduction of counterterms. Renormalized perturbation theory is described in Ref. [100]. The renormalization process is somewhat arbitrary because there are many ways that it can be implemented. The minimal subtraction (MS) scheme is often used in conjunction with dimensional regularization. In the MS scheme the renormalization factors Z$_{i}$ are defined such that only the poles at ϵ = 0 are canceled, while the γ$_{E}$ and log (4 π ) terms appearing in Eq. (1.44) remain. However, these terms are merely artifacts of dimensional regularization and are unphysical. In the modified minimal subtraction (MS) scheme the renormalization constants are defined so that these terms are also canceled. A convenient method of partially implementing the MS renormalization scheme is discussed in Chapter 2. Renormalization factors can be calculated as an expansion in the coupling α . This can be done to one-loop order as follows. First, all possible one-loop connected correlation functions are calculated in terms of bare parameters. Second, the bare parameters are eliminated in favor of the renormalized parameters using Eq. (1.45). Finally, the requirement that the renormalized correlation functions must be finite can be used to determine the renormaliza- 26 Z = 1 + α π A ϵ + O $^{(}$α 2 ) , (1.47)