Infrared divergence

Concluding Remarks.—We have come to the end of our exposition of some aspects of quantum electrodynamics. We have not delved in some of the more technical and difficult facets of the subject matter. Mention should, however, be made of what some of the difficulties are. Foremost at the technical level is perhaps the role played by the infrared divergences. The fact that the photon has zero mass not only gives rise to divergences in various matrix elements,20 but also implies... 

In order to obtain this estimate of the electron radius we have taken into account that the electron is slightly off mass shell in the bound state. Hence, the would be infrared divergence in the electron charge radius is cut off by its virtuality p = m which is of order of the nonrelativistic binding... 

Direct substitution of the radiatively corrected electron factor C k) in the skeleton integral in (3.33) would lead to an infrared divergence. This divergence reflects existence in this case of the correction of the previous order in Za generated by the two-loop insertions in the electron line. The magnitude of this previous order correction is determined by the nonvanishing value of the electron factor C k) at zero... 

Calculation of the same contribution with the help of the Braun formula was made in [4]. In the Braun formula approach one also makes the substitution in (3.35) in the propagators of the exchange photons, factorizes external wave functions as was explained above (see Subsect. 5.1.1), subtracts the infrared divergent part of the integral corresponding to the correction of previous order in Za, and then calculates the integral. The result of this calculation [4] nicely coincides with the one in (5.5). ... 

The contribution to HFS induced by the skeleton diagram with two external photons in Fig. 9.2 is given by the infrared divergent integral... 

Some of the diagrams in Fig. 9.8 also generate corrections of the previous order in Za, which would naively induce infrared divergent contributions after substitution in the skeleton integral in (9.9). 

Technically the lower order contributions to HFS are produced by the constant terms in the low-frequency asymptotic expansion of the electron factor. These lower order contributions are connected with integration over external photon momenta of the characteristic atomic scale mZa and the approximation based on the skeleton integrals in (9.9) is inadequate for their calculation. In the skeleton integral approach these previous order contributions arise as the infrared divergences induced by the low-frequency terms in the electron factors. We subtract leading low-frequency terms in the low-frequency as Tnp-totic expansions of the electron factors, when necessary, and thus get rid of the previous order contributions. 

This correction is induced by the gauge invariant set of diagrams in Fig. 9.8(d) with the polarization operator insertions in the radiative photon. The two-loop anomalous magnetic moment generates correction of order a Ep to HFS and the respective leading pole term in the infrared asymptotics of the electron factor should be subtracted to avoid infrared divergence and double counting. 

As usual we start consideration of the contributions of order Za)Ep with the infrared divergent integral (9.9) corresponding to the two-photon skeleton diagram in Fig. 9.2. Insertion of factors GE —k ) — 1 or Gm(— )/(l + k) — 1 in one of the external proton legs corresponds to the presence of a nontrivial proton form factor. ... 

Therefore quantum fluctuations in B<3) are accompanied by fluctuations in the transverse electric field. The ultraviolet divergence is probably unimportant [17] because of the co 2 dependence of the fluctuation. The infrared divergence is also damped statistically. The divergences in U(l) electrodynamics [6] can exist as a subset of 0(3) electrodynamics and can be absorbed into integrals that involve photon loop processes associated with quantum fluctuations in B(3 ... 

The first term is infrared divergent for l = 0 and p = 0, if the energy of the intermediate state p coincides with En. A similar term with opposite sign occurs from the vertex term which will be presented next. Therefore the sum of both terms is infrared finite. In performing their numerical evaluation, we explicitly exclude the terms with l = 0, p = 0 from the calculation. The vertex part is given by... 

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