Renormalized scaling functions

L. Schafer, T. A. Witten. Renormalization field theory of polymer solutions. I. Scaling laws. J Chem Phys 66 2121-2130, 1977 A. Knoll, L. Schafer, T. A. Witten. The thermodynamic scaling function of polymer solution. J Physique 42 161-m, 1981. 

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. 

Furthermore this result generalizes to arbitrary order of renormalized perturbation theory. The renormalized expansion therefore yields microstructure-independent scaling functions reproducing the physical observables up to negligible corrections. 

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... 

To summarize the renormalization group proves two parameter scaling. The two parameters J q3 z however show a more complicated temperature dependence than assumed in the naive two-parameter scheme. The latter is correct only close to the 0-point. Furthermore the scaling functions take two different forms, representing the weak or the strong coupling branch. 

On a deeper level we observe that the e-expansion does not properly respect the structure of the theory. As discussed before (see Sects. 8.3 or 10.2, in particular), we should first use the RG to map the system to an uncritical manifold, which for dilute systems is determined by Nr = 0(1), for instance. In a second step the scaling functions on the uncritical manifold can be calculated by renormalized perturbation theory. These two steps are well separated conceptually. The exponents, for instance, are properties of the RG, independent of specific scaling function. Strict e-expansion mixes these two steps since it simultaneously expands exponents and scaling functions. The expected scaling structure then has to be put in by hand at the end of the calculation. 

The evaluation of the scaling functions directly in d = 3 has been advocated in [SD89, SD90], where the method has been shown to work for small momenta. Also the treatment of the additive mass renormalization (corresponding to th introduction of p. J has been carefully considered there. 

The resulting value of u (Eq, (13,18)) is u — 8.107, It must be stressed that the parameter values (13.22) are adjusted to a one-loop evaluation of the scaling functions within the present formulation of renormalized perturbation theory in d — 3 and would have to be readjusted if for instance we would work to two-loop order. 

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... 

Again we can easily calculate the full crossover. As an example Fig. 14.3 shows the scaling function V/s as function of s in the excluded volume limit. In unrenormalized tree approximation this ratio would be a constant proportional to the second virial coefficient. In renormalized theory we see a pronounced variation which rapidly approaches the asymptotic power law. 

Fig. 9.3. The scaling function R (N,Cp)/R (A, Cj, — 0) (excluded volume limit, monodisperse) hh function of s = CpRg N,Cp = 0). The full line is the renormalization group result. (See Chap. 18.) The broken lijies give approximations as motivated bv the blob model...

To calculate the scaling functions we now follow the flow up to some fixed surface chosen far from the critical manifold, such that i/ C- The resulting macroscopic Hamiltonian allows for a perturbative treatment, all critical effects being absorbed into the renormalization of the couplings. On the purely formal level the precise choice of tliis manifold where the scaling... 

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