Illustration of renormalized perturbation theory

In Chap. 8 we have constructed the renormalization group, starting from bare perturbation theory for the discrete chain model. This expansion involves nonuniversal microstructure corrections which we will now absorb into renormalization factors, introduced via a redefinition of the interaction constant and the chain length, According to Eq. (11.1) we write 

With the ansatz (11.2) we reorder bare perturbation theory in powers of u. For simplicity taking A = 1, from Eq. (4.16 ii) we find 

This is just a reordered expansion for the given microscopic model. In Eq. (11-5) we now take the limit l — 0 with u, tir, fixed, which amounts to dropping the terms of order Since tir 2r is of order 

Ri rtf these terms truly axe negligible. They arise from the terms of 

We now repeat these steps, starting from Eq. (4.32) for A%. Choosing 

The constant a(e) is given in Eq, (A 4.16). Again the renormalized expression is finite in four dimensions and does not refer to a specific microstructure. 

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