Renormalization with dimensional regularization

Two important consequences follow from the fact of the atomic. sccdc of length a being indi.stinguishable in experimentally observed quantities (Oono and Freed, 1981a Oono et al., 1981). First, the macroscopic properties do not depend on the existence of the natural minimum on the molecular scale a, so, they must be well-defined in the limit o — 0. In the excluded volume problem of a molecular coil, the limit a — 0 should be regarded as a rejection of the consideration how a segment interacts with itself (the self-excluded volume). 

Second, there is no natural unit length in macroscopic theory, and such a length L can be chosen arbitrarily regardless of the microscopic natural unit of length o.  

Gell-Mann-Low-Oono-Ohta-Freed s renormalization approximation (Oono et al., 1981) provides a method of constructing, in terms of a microscopic model, such macroscopic quantities which arc well-defined in the limit o —t 0. 

The scaling invariance of microscopic theory to the choice of the macroscopic unit of length L follows, which, together with the renormalization relationships of well-defined iiiaci uscupic quantities, leads to scaling laws. 

and Freed have developed a renormalization method for conformational space of Gell-Mann and Low s (195 1) type for the excluded volume problem. In a number of cases, this approach proves to be more effective than Kadanoff-Wilson s approximation (see subsection 5.1.1) it provides a higher accurau y of the calculation of such quantities as the end-to-end vector distribution function, the scattering function, the conformation of a mricromolecule as a function of polymer concentration, etc. 

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The cssencxr of the renormalization with dimensional regularization is in introduction of some relationships between microscopic and macroscopic (juantitics to reduce (absorb) these singularities and to make macroscopic quantities regular in at = 0. 

The es.sence of the renormalization method with dimensional regularization is the introduction of relationships between microscopic and macroscopic values, which absorb these singularities, so at e = 0, the macroscopic quantities turn out to be regular in e. 

In a very general way, divergences in diagrams can be eliminated with the help of counter-terms these counter-terms can be interpreted as resulting from additional interaction terms and the latter are absorbed by renormalization of the partition functions and of the bare interactions. This is exactly what we did in Chapter 10, Section 4.2.6, for the model with purely repulsive two-body interactions, and, in this case, it is easy to see that the process amounts to dimensional regularization. 

Some details of the renormalization group approach for polymers as done in Sec. 4.2.1 are given here [12]. We consider the problem of two interacting directed polymers and study the second virial coefficient. The second virial coefficient is related to the two-chain partition function with all the ends free. Dimensional regularization is to be used here. 

At d = 1 one has a completely stretched chain with ly = 1. At d = 2 the exact result v = 3/4) [13] is obtained. The upper critical dimension is d = 4, above which the polymer behaves as a random walker. The values of the universal exponents for SAWs on d - dimensional regular lattices have also been calculated by the methods of exact enumerations and Monte Carlo simulations. In particular, at the space dimension d = 3 in the frames of field-theoretical renormalization group approach one has (v = 0.5882 0.0011 [11]) and Monte Carlo simulation gives (i/ = 0.592 0.003 [12]), both values being in a good agreement. 

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