Polarization loop

The effects connected with the electron vacuum polarization contributions in muonic atoms were first quantitatively discussed in [4]. In electronic hydrogen polarization loops of other leptons and hadrons considered in Subsect. 3.2.5 played a relatively minor role, because they were additionally suppressed by the typical factors (mg/m). In the case of muonic hydrogen we have to deal with the polarization loops of the light electron, which are not suppressed at all. Moreover, characteristic exchange momenta mZa in muonic atoms are not small in comparison with the electron mass rUg, which determines the momentum scale of the polarization insertions m Za)jme 1.5). We see that even in the simplest case the polarization loops cannot be expanded in the exchange momenta, and the radiative corrections in muonic atoms induced by the electron loops should be calculated exactly in the parameter m Za)/me-... 

In the case of the polarization insertions the calculations may be simplified by simultaneous consideration of the insertions of both the electron and muon polarization loops [18, 19]. In such an approach one explicitly takes into account internal symmetry of the problem at hand with respect to both particles. So, let us preserve the factor 1/(1 - - m/M) in (9.9), even in calculation of the nonrecoil polarization operator contribution. Then we will obtain an extra factor m /m on the right hand side in (9.12). To facilitate further recoil calculations we could simply declare that the polarization operator contribution with this extra factor m /m is the result of the nonrecoil calculation but there exists a better choice. Insertion in the external photon lines of the polarization loop of a heavy particle with mass M generates correction to HFS suppressed by an extra recoil factor m/M in comparison with the electron loop contribution. Corrections induced by such heavy particles polarization loop insertions clearly should be discussed together with other radiative-recoil... 

Recoil corrections induced by the polarization loops containing other heavy particles will be considered below in Sect. 10.2 together with other radiative-recoil corrections. 

Note that in the parenthesis we have parted with our usual practice of considering the muon as a particle with charge Ze, and assumed Z = 1. Technically this is inspired by the cancellation of certain contributions between the electron and muon polarization loops mentioned above, and from the physical point of view it is not necessary to preserve a nontrivial factor Z here, since we need it only as a reference to an interaction with the constituent muon and not with the one emerging in the polarization loops. 

The contribution of the muon polarization operator was already considered above. One might expect that contributions of the diagrams in Fig. 10.8 with the heavy particle polarization loops are of the same order of magnitude as the contribution of the muon loop, so it is natural to consider this contribution here. Respective corrections could easily be calculated by substituting the expressions for the heavy particle polarizations in the unsubtracted skeleton integral in (10.3). The contribution of the heavy lepton t polarization operator was obtained in [37, 38] both numerically and analytically... 

The dominant interaction within the muonium atom is electromagnetic. This can be treated most accurately within the framework of bound state Quantum Electrodynamics (QED). There are also contributions from weak interaction which arise from Z°-boson exchange and from strong interaction due to vacuum polarization loops with hadronic content. Standard theory, which encompasses all these forces, allows to calculate the level energies of muonium to the required level of accuracy for all modern precision experiments1. 

A mft/me) and (m /mT) have contributions from 24 Feynman diagrams containing vacuum-polarization loops or an l-l scattering subdiagram. They have been evaluated very precisely by an asymptotic expansion and by a power series expansion, respectively [23,24,49] ... 

It is not enough to consider the free vacuum polarization. The relativistic corrections to the free vacuum polarization in Eqs. (11-14) are of the same order as the so-called Wichmann-Kroll term due to Coulomb effects inside the electronic vacuum-polarization loop. To estimate this term we fitted its numerical values from Ref. [17], which are more accurate for some higher Z 30, by expression... 

The bottom open side of the P-barrel (opposite the hairpin in the y-subunit) can interact with the c-subunits of Fq. Cysteines were introduced by Zhang and Fillingame " into the polar loop region surrounding Gln-42 in the c-subunits and at Glu-31 in the e-subunit. They were able to form disulfide bond in membranes of three double mutants ... 

These results indicate that Glu-31 on the s-subunit is physically close to the polar loop ofthe c-subunit in Fq. Aggeler, Weinreich and Capaldi also found an interaction between Glu-31 ofthe s-subunit and residues 3943 ofthe c-subunit. In addition. Watts, Tang and Capaldi ° also reported that when Tyr-205 on the y-subunit is replaced by cysteine it can also form a disulfide bond with cysteines introduced into residues 39, 42 or 43 of the c-subunit. 

Subunit c ofEcFo [or subunit III ofchloroplast CFo] consists of 79 amino-acid residues and has a molecular mass of 8.3 kDa. Because of its hydrophobic character it is soluble only in non-polar solvents. Girvin and Fillingame examined the monomeric subunit c of E. coli Fq by NMR spectroscopy and were able to make a partial structure determination ofresidues 9-25 and 52-79 in the N- and C-terminal regions, respectively. Subunit c appears to span the membrane as a hairpin of two hydrophobic helices separated by a polar loop of about 18 amino-acid residues exposed to the cytoplasmic side ofthe membrane, as shown in Fig. 3 8 (C). NMR spectroscopic measurement shows subunit c to consist oftwo gently curved a-helices crossing at a slight angle of -30° [also see Fig. 39 (A)]. 

Y Zhang and RH Fillingame (1995) Subunits couplinghF transport and ATP synthesis in the Escherichia coll ATP synthase. Cys-Cys cross-linking of F subunit epsilon to the polar loop of Fq subunit c. J Biol Chem 26770 24609-24619... 

To find the conductivity it is convenient to calculate the polarization loop. We shall suppose that the impurity scattering is sufficiently weak. Under these conditions in the 3d case ladder diagrams of the type in Fig. la dominate, and the contribution of the diagrams with intersections /Fig. lb and 1c/ may be neglected. As a result, the usual kinetic equation is valid. On the contrary, in the Id case all these diagrams give comparable contributions. 

There are additional corrections from muon and tau vacuum polarization loops [29, 30]... 

Fig. 6. Expansion of the vacuum polarization loop into different powere of Za. The first term of the expansion corresponds to the Uehling contribution, the remaining terms are known as Wichmann-Kroll contribution.

The polarization hysteresis loop measured at room temperature for the P (VDF-TrFE) 68/32 mol% copolymer changes with the irradiation dose (Cheng et al. 2002). With increased dosage, flie near square polarization hysteresis loop, characteristic for a normal ferroelectric material, is transformed to a slim polarization loop (at 75 Mrads). At very high dose (175 Mrads), the polymer becomes a Unear dielectric in which the crystallinity is near zero. 

---