Minimal subtraction

We thus take the following attitude. For technical reasons we calculate the renormalized theory starting from the continuous chain model. By equivalence of the bare theories for fo 1 we know that we can derive the same theory from the discrete chain model. Since the renormalized theory in the way we construct it should show no singularity at u, we can use it for u u. This region however can be interpreted only in terms of the discrete chain model. 

Starting from the continuous chain model as the dimensionally regularized theory we write the renormalization factors as 

We introduced a parameter bu (s), which is assumed to be a smooth function obeying 

In the dimensionally regularized theory the coefficients A are functions only of , which have to be chosen to cancel the singularities of the bare theory occurring for e — 0. This goal can be reached with the ansatz 

We now discuss the resulting properties of the RG flow equations. Writing Eq. (12.3) as 

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In this part,we first explain in general terms the construction of renormalized perturbation theory. We show how the RG results from the arbitrariness of r and establish the general scaling form (Chap. 11). We then turn to the specific technique of minimal subtraction and show how to calculate the scaling functions (Chap. 12). The RG mapping, used in the sequel, is presented and discussed in Chap. 13. We finally (Chap. 14) illustrate the theory with an evaluation of the tree approximation. 

The results of the previous chapter are completely general. They are valid for any field theoretic renormalization scheme, i.e. independent of the specific choice of the renormalization factors, For quantitative calculations we of course have to specify the Z-factors, and as pointed out in Sect. 11,1, we have some freedom there. We will use the scheme of dimensional regularization and minimal subtraction . This scheme is most efficient for actual calcular tions, but its underlying basis is a little bit delicate, It needs some careful explanation. 

These results show that the minimal subtraction scheme eliminates ambiguities inherent in an extrapolation of e-expansion results to physical dimension d = 3. For the flow equations no extrapolation is necessary. Furthermore they are strictly independent of the parameter bu(z). 

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. 

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. 

In field theoretic context the method of dimensional regularization and minimal subtraction has been proposed, in [tHV72]. To the level considered here it is discussed in standard textbooks [ZJ89, Ami84], There the explicit calculation of the -factors can also be found. The method has been applied directly to the polymer system in the continuous chain limit in [Dup86a], where different versions of the approach are compared,... 

For low order calculations of the scaling functions a variety of implementations of the RG have been used. The present formulation has grown out of the work [Sch84]. The basic philosophy is the same, but in this earlier work the renormalization scheme was based on field theoretic renormalization conditions1. This amounts to using a non-minimally subtracted theory, where the Z-factors are determined by imposing specific values to certain renormalized field theoretic vertex functions. The renormalized coupling, for instance, is defined as the value of (qi, qa, qg. qi) at some special momenta of order... 

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 subtraction , where the flow equations depend on d only trivially... 

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