Renormalization constants

Each resonance-enhanced Raman mode has a counterpart in the IR spectrum, due to the pinning of charge oscillations. The amplitude mode model can be used to derive the doping-induced modes if a pinning constant a for the renormalization of the bare phonons is employed instead of the renormalization constant A for rc-electron interaction (Ehrenfreund et al., 1987). This procedure is very similar to the evaluation of Raman modes by Eq. 4.8-10, as indicated in Fig. 4.8-10. The high frequency conductivity cr(iu) is obtained from... 

To proceed, we have to reckon with the renormalization of the interspecies coupling constant, [Lifshitz 1980 Belyaev 1958], The renormalized constant gi2 is expressed in terms of the bare coupling constant gn as... 

In (7.57) p(k) are polariton energies, and F are renormalized constants determining the polariton-phonon interaction. Making use of (7.57) we find by... 

The critical exponents y, v, and at were calculated to third-order in e independently by Brezin, Le Guillou, and Zinn-Justin, and by Nickel. The outcome was published jointly by these physicists in 1973.29 To obtain the result, Nickel used Wilson s original method conversely, Brezin, Le Guillou, and Zinn-Justin used another approach 30 they eliminated the cut-off by renormalization, and in order to calculate simpler integrals, they worked directly at the critical point the critical theory and the renormalization constants were then defined by introducing adequate renormalization conditions.2... 

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. 

For the charge renormalization it is convenient to introduce the renormalization constants Zi and Z3. Then ... 

The renormalization constants Zj, Z2, and 8m have to be understood as functions of the finite physical charge e and mass m of the electrons, which will have to be constructed order by order in the perturbation series. In other words The original fields and parameters in mren are no longer interpreted as the correct physical fields and parameters, but rather as bare, unrenormalized quantities. 

As perturbation theory on the basis of (222) yields the irreducible functions in terms of the bare parameters the expressions on the right-hand sides of (228), (229), (237) and (238) are obtained by use of (220) and (221) and thus depend themselves on the renormalization constants. The renormalization thus has to proceed in a recursive fashion, i.e. order by order in perturbation theory. 

The geometry of the experiment is taken into account by rotating pq(j) onto the probe laser frame. The rotated pq j) moments are proportional to the Aq(j) moments. In order to obtain the proper form for Ak(j), the bQ(kl, k2) moments must be renormalized to Pq(kv, k2) moments according to Dixon [18] the renormalization constants and the ranges of jSq(/c1,/c2) are shown in Tables la and lb. 

An alternative derivation of Eq. (5.112) can be established using the classical method of renormalization constants. To illustrate this, we note first that the quantity for the corresponding ideal system satisfies... 

The quantities Zi,Z, Yl, and are called renormalization constants. According to the above argument, they should be universal functions of the respective variables. 

We must know Yl u) and Y u) to convert QeC-t Oi o) into Q L,u,X). In the primary chain, excluded-volume interactions can act between infinitesimally close points on the chain, since the chain is allowed to take on conformations which contain infinitesimally small loops. Owing to this property, unless vo is zero, Qb diverges to infinity as e —> 0, as will be illustrated below. However, no such singularity appears in a coarse-grained chain with cut-off length A because intrachain interactions in it do not act between contour points separated by less than A. Therefore, Q(L, u, A) should remain finite in the limit e 0 if A 0. The renormalization constants Yl and can be determined from this physical requirement, as exemplified below. Because of this operation, renormalization is sometimes misunderstood as a mere mathematical maneuver of removing singular behavior of the primary chain in the limit of e —> 0. 

The renormalized constants eliminate all the singularities in Ob completely by means of defining the coefficients a,-, 6, ... in subsequent orders of perturbation theory. 

When choosing normal lattice modes at the F-point as a basis in the space of the displacement vectors of sublattices, we obtain renormalized constants of coupling with macroscopic deformation (eq. 105) in the following form ... 

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