Regularization dimensional

Here d and n denote the dimension of space and the scale of the dimensional regularization used as in Ref. (Lyubovitskji and Rusetsky, 2000), respectively. 

Any deviation from the ideal three-dimensional regularity of the polymer crystal structure. Note Examples of structural disorder in crystalline polymers are given in Table 2. 

Crystals in which molecules rotate still have three-dimensional regularity they must not be confused with liquid crystals , in which there... 

Crystalline material, as stated above (whether single-crystal, polycrystalline, or even nanocrystalline-amorphous), is made such that its component (constituent base) atoms and molecules are arranged on a three-dimensional regular, repetitive pattern called a lattice. The commonest form among (plated) metals is the... 

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. 

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. 

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... 

Step (iii) together with the substitution j = n Si yields the dimensionally regularized result... 

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,... 

To perform a dimensional regularization of this integral we replace the integral with... 

This calculation demonstrates that the loop fluctuation of a photons, which correlated to a virtual quanta of B field, can be calculated to be finite with out divergence. So the virtual fluctuation of a field does not lead to an ultraviolet divergence, and thus 0(3)h QED is renormalizable by dimensional regularization. 

Let us now explain the main idea behind this technique. An important point is that the applicability of the integration-by-parts requires that Feynman integrals are regularized dimensionally, so that we work in continuous space-time dimension D = 4 — 2e, where e is the regularization parameter. Both, ultraviolet and infra-red divergences show up as poles in e. If dimensional regularization is adopted, one observes that the following relation ... 

In the present work we closely follow the scheme of calculation, applied in [5] to the similar problem in positronium. Namely, we employ the dimensionally regularized nonrelativistic QED (NRQED). In this approach, separation of contributions to the energy coming from the hard ( m) and soft ( rna) scales... 

At the last step, we use equation (CAC) = 0, valid in the dimensional regularization. In order to express the average value... 

An important result is that the new operators Hte] are generated by the hard momentum contributions only. Hence, as follows from the above formula, computation of new operators requires only a Taylor expansion of full QED amplitudes in external momenta no matching is required. This approach provides the required operators in dimensional regularization and hence, for consistency, the rest of the problem (quantum mechanics) must be solved using the same (dimensional) regularization. 

Let us now present an example of simple but striking identity which is valid in dimensionally regularized quantum mechanics but would be wrong in other regularization schemes. Consider the following integral ... 

Such experimental accuracy warrants a calculation of 0(a2) corrections to the decay rate on the theoretical side. To compute them, we again use dimensionally regularized NRQED. In this case, however, finite parts of hard integrals are done numerically because of their complexity nevertheless, divergent parts of hard integrals are obtained analytically. 

Eq. (2) presents the basis for the covariant renormalization approach. The explicit expressions are known for E Ten(E), X u 6 in momentum space. For obtaining these expressions the standard Feynman approach [11,12] or dimensional regularization [13] can be used. They are free from ultraviolet divergencies but acquire infrared divergencies after the renormalization. However, these infrared divergencies, contained in X 1) and cancel due to the Ward identity X -1) = —A1 1 and the use of the Dirac equation for the atomic electron in the reference state a) ... 

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