Existence of the equivalent continuous chain model

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. 

We thus move along a set of bare theories, equivalent in the sense that they yield the same renormalized theory. To relate this way of taking the limit — 0 to the NCL from Eqs. (11.27), (11.33) we note that the set of all equivalent bare theories is defined by fixed values of the parameter sp and of the combination sn n. Now assuming a starting value /o 1 we note from Eq. (11.28) that S = co-nst implies the asymptotic behavior 

Thus asymptotically the NCL and the RCL, if applied to the bare functions for d 4, are identical. In other words, for fo 1 the set of equivalent theories for i — 0 reaches the 0-region, where the continuous chain model and the discrete chain model coincide. The same renormalized theory can therefore be constructed from both models. 

Concerning the first question we note that the result of any renormalization scheme based on the continuous chain model via a finite renormalization can be mapped on the renormalized theory derived from the discrete chain model, and vice versa. After renormalization the models are completely equivalent. 

The problem, however, does not ruin our construction of the renormalized theory, and it does not keep us from using the results in some region u = fu u. Concerning the RG flow we note that we will use the special scheme of minimal subtraction1, where the flow equations depend on d only trivially 

We now turn to the restriction /o 1, which in view of Eq. (11.26) implies / 1, i.e. w. A priori this is a serious point of concern. Equation (11.28) shows that starting from /o 1 (which implies / 1) and decreasing i for 

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