Dimensional regularization 4-theory

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

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... 

The step of first taking the limit f —> 0 for d < 4 to arrive at a finite unrenormalized theory without cut-off is known as dimensionai regularization in contrast to the cut-off regularization of the discrete chain model. The breakdown of the dimensionally regularized theory as d — 4 shows up in pole terms v. These terms have to be absorbed into the renormal-... 

In Refs. [82,83] the elegant method of renormalization at zero mass and non-zero external momenta was developed, which avoids the additional renormalization conditions. Here, the vertex functions are analytically continued in the dimensional parameter d leading to a so called dimensionally regularized theory, where the cutoff Aq in 69 can be removed. 

The renormalization factors can be chosen to absorb all the pole terms of the dimensionally regularized bare theory to yield a renormalized theory finite for d < 4. 

We first restrict ourselves to the two-dimensional case and take advantage of the fact that Levi-Civita s regularizing transformation (Levi-Civita 1920) has the agreeable property of transforming perturbed Kepler problems into perturbed harmonic oscillators, i.e. into perturbed linear problems. For a recent account of regularization theory see the article (Celletti 2002) and other contributions in the same volume. 

Thus, for 4 > d > 2, the value of 1p(k) given by (10.4.2) is just the analytic continuation with respect to d, of the integral giving 2p k) for d < 2. As was seen above, this analytic continuation can be obtained with the help of subtractions, and the process is quite general (see Appendices G and H) in field theory, it is called dimensional regularization . 

The importance of dimensional regularization comes from the fact that it is equivalent to a renormalization. Actually, the fact has not been rigorously proved in polymer theory, but it seems very likely to be true, and in Appendix... 

We shall assume that this expression remains valid when it is extrapolated to the poor solvent domain. Thus, b 0 for T = TF (34.0 °C) and b < 0 for T < TF. This does not really mean that the true two-body interactions vanish for T = TF and that at T = TF there are only three-body interactions. Actually, the interaction b is a bare interaction but is nevertheless an interaction which is additively renormalized (see Chapter 14, Section 6) and its value depends partly on the true three-body interactions. Incidentally, this remark also applies to good solutions for this reason, if b is measured in good solvent, it is legitimate to extrapolate the result thus obtained to determine the value of b in poor solvent. Thus, in perturbation theory, when the diagram contributions are calculated by dimensional regularization, we may say that b = 0 for T = TF ... 

In held theory, the dimensional regularization can also be used but, in general, one prefers other renormalization methods which always give finite results for all values of d. These methods are equivalent to a direct introduction of 91 s then, the renormalization constants depend on the masses, but this is not a drawback, since in field theory, the masses are considered as constant. 

As the divergences a —> 0 are found when the principal parts in the c expansion of appear, tin- very principal parts must be absorbc d in the relationships between the microscopic and macroscopic quantities. Therein lies the method of dimensional regularization proposed by t llooft and Veltmrui (1972) in field theory. 

Since then, the theory of the tricritical phenomena has been developed by the direct renormalization methods (des Cloizcaux, 1981) (see section 5.4) in (l)uplantier, 1986bd), by the dimensional regularization in momentum. sp lce (see section 5.2) in (Kholodenko and Freed, 1984 Cherayil ct al., 1985 Douglas and Freed, 1985). 

Duplanticr (1987) provides expressions for Z f S) in the versions of theory with a cut-off and dimensional regularization. 

The theory of tricritical phenomena was developed by different methods the direct renormalization, the dimensional regularization in momentum p2tce, etc. There are some contradictions between different authors which can be explained by certain specific features of different approsiches. However, Duplantier performed comparative calculations and an analysis of the tricritical state of polymer systems by different methods (in particular, by the method of dimensional regularization and using a cut-off) and achieved equivalent results. 

In books on field theory one starts with the Lagrangian in which all the parameters, coupling and masses are called bare parameters and are labelled eo, 50, "lo etc. These bare parameters are not what one measures to make the theory finite they have to be allowed to depend on a cut-off A temporarily introduced into the theory and most of them become infinite when at the end one lets A -> 00. What A is depends upon the method of regularization . Thus A may literally be a cut-off, i.e. the upper limit of some loop integration, or it may be 1/e where one uses dimensional regularization to work in 4 — e dimensions and at the end lets c 0. 

Regular solution theory, the solubility parameter, and the three-dimensional solubility parameters are commonly used in the paints and coatings industry to predict the miscibility of pigments and solvents in polymers. In some applications quantitative predictions have been obtained. Generally, however, the results are only qualitative since entropic effects are not considered, and it is clear that entropic effects are extremely important in polymer solutions. Because of their limited usefulness, a method using solubility parameters is not given in this Handbook. Nevertheless, this approach is still of some use since solubility parameters are reported for a number of groups that are not treated by the more sophisticated models. 

An important feature of the band system is that electrons are delocalised or spread over the lattice. Some delocalisation is naturally expected when an atomic orbital of any atom overlaps appreciably with those of more than one of its neighbours, but delocalisation reaches an extreme form in the case of a regular, 3-dimensional lattice. We can understand this best if we choose to think of the wave nature of electrons, and from that point of view we can formulate band theory as follows. 

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