Direct renormalization in four dimensions

As has been pointed out in this chapter and in Chapter 10, the space dimension d = 4 is a limiting dimension. This can easily be seen, for instance, by looking at the definition (10.1.7) of the parameter z. For d = 4, b is a pure number we have 

This can be seen by putting e = 0 in the expansions (12.3.106). In the same way, g vanishes for d = 4. Chains, for d = 4, have a quasi-Brownian behaviour. 

There still remain logarithmic factors which appear when the critical domain is reached. It is interesting to study the origin of these logarithmic terms and we shall verify that, strictly speaking, they cannot be considered as mere scaling law corrections since they are dominant when N - oo. 

In order to understand the situation in four dimensions, we place ourselves just below four dimensions and we let e = 4 — d go to zero. Looking at the results of Section 3.3.3, we observe 

That the series expansions in z of the partition functions become infinite when e- 0 and this fact was also conspicuous in Chapter 10 [see, for instance, (10.7.7)]. 

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