0.1 Callan-symanzik Equation

10 Renormalization Group The renormalization procedure in the last chapter has eliminated all UV-divergences from the Feynman integrals arising from large momenta in D = 4 − ε dimensions. This was necessary to obtain finite correlation functions in the limit ε → 0. We have seen in Chapter 7 that the dependence on the cutoff or any other mass scale, introduced in the regularization process, changes the Ward identities derived from scale transformations by an additional term—the anomaly of scale invariance. The precise consequences of this term for the renormalized proper vertex functions were first investigated independently by Callan and Symanzik [1]. The original derivation of the scaling properties of interacting theories did not quite proceed along the lines discussed in Section 7.4. Callan and Symanzik wanted to find the behavior of the renormalized proper vertex functions ¯ Γ (n) (k 1 ,. .. , k n) for n ≥ 1 under a change of the renormalized mass m. They did this in the context of cutoff regularization and renormalization conditions at a subtraction point as in Eqs. (9.23)–(9.33). To keep track of the parameters of the theory, we shall from now on enter these explicitly into the list of arguments, and write the proper vertex functions as ¯ Γ the bare proper vertex functions depend on the cutoff, or the deviation ε = 4 − D from the dimension D = 4, and will be denoted by ¯ Γ (n)