The theorem of renormalizability

rthermore this result generoXizts to arhittary order of renormalized perturbation theory. The renormalized expansion therefore yields microstructnre-independent scaling functions reproducing the physical observables up to negligible corrections. 

This result is of central importance, and we formulate it in precise terms in a separate section. 

The general result of renormalization is best expressed in terms of the cumu-lants introduced in Sect. 4,3. (Recall that given by the 

The precise form of Eq. (11,10) results from the field theoretic analysis sketched in Appendix A 11,1. 

The renormalization factors can be chosen such that the limit i —f 0 for exists for all dimensions d 4, the corrections being of canonical 

This result is valid in perturbation theory to all orders in a simultaneous expansion in u and e. 

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The theorem of renormalizability can be read in two ways. With the renormalized theory taken to be fixed, it implies the existence of a one-parameter class, parameterized by , of bare theories, all equivalent to the given renormalized theory and thus equivalent to each other. This aspect is related to universality a whole class of microscopic models yields the same scaling functions. In the next chapter we will use this aspect to get rid of the technical complications of the discrete chain model. We can however also interpret the theorem as establishing the existence of a one-parameter class of renormalized theories, all equivalent to a given bare theory. This class is parameterized by the length scale r or the scaling parameter... 

By virtue of Eq. (11.10) this is identical to a ratio of renormalized cumulants, and invoking the theorem of renormalizability we find... 

This derivation is easily extended to all correlation functions of interest. Any grand-canonical cumulant can be expressed by summing appropriate M-chain cumulants. We then use the theorem of renormalizability, and we... 

For d < 4 in the theorem of renormalizability we can take the naive continuous chain limit, addressed here as NCL for shortness. 

We know that the additively renormalized bare theory (the left hand side in Eq. (11.10)) exists in that limit. Also the -factors attain finite limits Z(ti), Zu(u), Zn(u). Indeed they are constructed as power series in u, the coefficients for t > 0,6 > 0 taking the form of polynomials in / r) . Thus no problem results from setting t = 0. Since the renormalized theory is finite for d = 4, whereas the bare continuous chain model diverges for d —> 4, showing poles in 5, also the Z-factors must diverge for d —> 4. In the NCL we can therefore formulate the theorem of renormalizability as follows ... 

In Chap. 7 we have shown that the bare discrete chain or continuous chain models are naively equivalent only close to the 0-point. We thus might wonder whether the equivalence of the two models, shown above to one loop order, can hold generally. We thus have to show that starting from these different bare theories we nevertheless can construct identical renormalized theories. We consider the renormalized continuous chain limit (RCL), used in the theorem of renormalizability. 

In the construction of the RGf dimension d = 4 plays a special role as upper critical dimension of the thebry. This for instance shows up in the estimate of the nonuniversal corrections to the theorem of renormalizability, or in the feature that the nontrivial fixed point u merges with the Gaussian fixed point for d — 4. It naturally leads to the e-expansion. However, the RG mapping constructed in minimal subtraction only trivially depends on e. Also results of renormalized perturbation theory do not necessarily ask for further expansion in e. Equation (12.25) gives an example. We should thus consider the practical implications of the -expansion in some more detail. 

Let us very clearly express the important and nontrivial content of the Theorem of Renormalizability going to higher orders we in p-th order the unrenormalized expansion find microstructuro-dependentxorrections of type for all Q < < p. All such terms are eliminated by renormaliza-... 

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