Virtues and disadvantages of the e-expansion

Consider the e-expansion (12.27) of the renormalized end-to-end distribution. It contains the constant b and thus depends on our renormalization scheme. This dependence can be eliminated by replacing the chain length, which is a microscopic parameter, by the end-to-end distance 

However, the result still depends on / = u/V5, which is not an observable quantity. We can eliminate this parameter in favor of the interpenetration ratio 

This quantity will be analyzed in detail in Sect. 15.4. The e-expansion result reads 

This calculation illustrates a general feature we may write down scaling laws for normalized quantities in terms of scaled momenta qRg (or qf e, equivalently), scaled concentrations cpRand ip replacing the coupling. Such relations involve only physically observable macroscopic quantities. They must have a uniquely defined -expansion, where ip = 0(e) acts as an expansion parameter. The result is necessarily independent of any conventions of the renormalization scheme. Not even the form of the RG flow equations matters. Furthermore, in establishing such results, we never have to invoke a condition like hr = 1. These are the great virtues of consistent e-expansion. 

There are more problems. The result (12.34), for instance, yields no information on chain-length or temperature dependence, which is hidden in R2 or ip To extract it, we have to write down RG flow equations for these variables. This results in the so-called direct renormalization7 scheme [CI08I], which however has not been pushed to the same high order as the flow equations derived for u, Nr in minimal subtraction. 

We may ask why we do not work exclusively within that scheme, which gives seemingly unambiguous results. The answer is that the above statement - correct as it is - shows the matter in too rosy a light. Consider again Eqs. (12.33), (12.31). We have chosen to keep some factors of d, the reason being that these factors come in naturally in the definition of the variables This clearly is some prejudice. Expanding out these factors 

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