Logarithmic corrections

It should also be noted that ternary and higher order polymer-polymer interactions persist in the theta condition. In fact, the three-parameter theoretical treatment of flexible chains in the theta state shows that in real polymers with finite units, the theta point corresponds to the cancellation of effective binary interactions which include both two body and fundamentally repulsive three body terms [26]. This causes a shift of the theta point and an increase of the chain mean size, with respect to Eq. (2). However, the power-law dependence, Eq. (3), is still valid. The RG calculations in the theta (tricritical) state [26] show that size effect deviations from this law are only manifested in linear chains through logarithmic corrections, in agreement with the previous arguments sketched by de Gennes [16]. The presence of these corrections in the macroscopic properties of experimental samples of linear chains is very difficult to detect. 

The Darwin potential generates the logarithmic correction to the nonrela-tivistic Schrodinger-Coulomb wave function in (3.65), and the result in (3.97) could be obtained by taking into account this correction to the wave function in calculation of the contribution to the Lamb shift of order a Za.ym. This logarithmic correction is numerically equal 14.43 kHz for the IS -level in hydrogen, and 1.80 kHz for the 2S level. 

Corrections of Order a Za) Ep 9.4.2.1 Leading Double Logarithm Corrections... 

Calculation of the leading logarithmic corrections of order a Za) Ep to HFS parallels the calculation of the leading logarithmic corrections of order a(Za) to the Lamb shift, described above in Subsect. 3.5.1. Again all leading logarithmic contributions may be calculated with the help of second order perturbation theory (see (3.71)). 

The leading logarithmic correction induced by the radiative insertions in the external photon in Fig. 9.16 is calculated in exactly the same way as was done above in the case of radiative insertions in the electron line. The only difference is that instead of the potential Vi in (3.95) we have to use the respective potential in (3.99), and instead of the potential V p in (9.38) we have to use the potential... 

The single-logarithmic correction of order Z a Za) m/M)Ep originating from radiative insertion in the muon line was calculated in [23, 24]... 

The calculation is based on a result in Ref. [24] for the singe logarithmic correction due to the one-loop self energy and one-loop vacuum polarization. 

We also found two higher-order logarithmic corrections... 

An important point is that the difference is sensitive to 4th order corrections and so is competitive with the muonium hfs as a test of the QED. The difference between the QED part of the theory and the experiment is an indication of higher-order corrections due to the QED and the nuclear structure, which have to be studied in detail. In particular, we have to mention that while we expect that we have a complete result on logarithmic corrections and on the vacuum-... 

At the moment, it is not possible to compute ma7 corrections completely. Nevertheless, the leading logarithmic correction, 0(ma7In2 a), to positronium energy levels can be computed. This provides an estimate of higher order effects and hence of the uncertainty in the current theoretical prediction. In what follows we briefly describe our calculation of these corrections [3],... 

Examples of diagrams that lead to double logarithmic corrections are shown in Fig.2. The details of our calculation can be found in [3]. The final result, that agrees with the independent calculation in Refs. [17,18], reads ... 

Hausdorff-Besicovitch dimension of A is 2, although its area in phase space converges to zero as the number of kicks approaches infinity. In order to resolve this paradox, the definition (8.2.2) has to be extended by including logarithmic corrections (Hausdorff (1919), Umberger et al. (1986)). The idea is to retain the general structure of (8.2.1), but to admit a larger class of functions than to counterbalance the proliferation of the number of boxes B e) for e 0. We define... 

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