Leading Contribution to the Lamb Shift

1 Radiative Insertions in the Electron Line and the Dirac Form Factor Contribution 

The main contribution to the Lamb shift was first estimated in the nonrela-tivistic approximation by Bethe [8], and calculated by Kroll and Lamb [9], and by French and Weisskopf [10]. We have already discussed above qualitatively the nature of this contribution. In the effective Dirac equation framework the 

The mass dependence of the correction of order a Za) beyond the reduced mass factor is properly described by the expression in (3.7) as was proved in [11, 12]. In the same way as for the case of the leading relativistic correction in (3.4), the result in (3.7) is exact in the small mass ratio m/M, since in the framework of the effective Dirac equation all corrections of order Za) are generated by the kernels with one-photon exchange. In some earlier papers the reduced mass factors in (3.7) were expanded up to first order in the small mass ratio m/M. Nowadays it is important to preserve an exact mass dependence in (3.7) because current experiments may be able to detect quadratic mass corrections (about 2 kHz for the IS level in hydrogen) to the leading nonrecoil Lamb shift contribution. 

The Bethe logarithm is formally defined as a certain normalized infinite sum of matrix elements of the coordinate operator over the Schrodinger-Coulomb wave functions. It is a pure number which can in principle be calculated with arbitrary accuracy, and high accuracy results for the Bethe logarithm can be found in the literature (see, e.g. [13, 14] and references therein). For convenience we have collected some values for the Bethe logarithms [14] in Table 3.1. 

The Pauli form factor also generates a small contribution to the Lamb shift. This form factor does not produce any contribution if one neglects the lower components of the unperturbed wave functions, since the respective matrix element is identically zero between the upper components in the standard representation for the Dirac matrices which we use everywhere. Taking into account lower components in the nonrelativistic approximation we easily obtain an explicit expression for the respective perturbation 

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Note the emergence of the last term in (3.4) which lifts the characteristic degeneracy in the Dirac spectrum between levels with the same j and / = j 1/2. This means that the expression for the energy levels in (3.4) already predicts a nonvanishing contribution to the classical Lamb shift E 2Si) — E 2Pi). Due to the smallness of the electron-proton mass ratio this extra term is extremely small in hydrogen. The leading contribution to the Lamb shift, induced by the QED radiative correction, is much larger. 

Let us start systematic discussion of such corrections with the recoil corrections to the leading contribution to the Lamb shift. The most important observation here is that the mass dependence of all corrections of order a." Za.Y obtained above is exact, as was proved in [1, 2], and there is no additional mass dependence beyond the one already present in (3.7)-(3.24). This conclusion resembles the similar conclusion about the exact mass dependence of the contributions to the energy levels of order (Za) m discussed above, and it is valid essentially for the same reason. The high frequency part of these corrections is generated only by the one photon exchanges, for which we know the exact mass dependence, and the only mass scale in the low frequency part, which depends also on multiphoton exchanges, is the reduced mass. 

Another obvious contribution to the Lamb shift of the same leading order is connected with the polarization insertion in the photon propagator (see Fig. 2.2). This correction also induces a correction to the Coulomb potential... 

The leading polarization operator contribution to the Lamb shift in Fig. 2.2 was already calculated above in (2.6). Restoring the reduced mass factors which were omitted in that qualitative discussion, we easily obtain... 

Only the value of the leading coefficient in the low energy expansion of the hadronic vacuum polarization is needed for calculation of the hadronic contribution to the Lamb shift (see the LHS of (3.32)). A model independent value of this coefficient may be obtained for the analysis of the experimental data on the low energy e+e annihilation. Respective contribution to the 15 Lamb shift [39] is —3.40(7) kHz. This value is compatible but more accurate than the result in (3.32). ... 

We have seen above that calculation of the corrections of order a"(Za) m (n > 1) reduces to calculation of higher order corrections to the properties of a free electron and to the photon propagator, namely to calculation of the slope of the electron Dirac form factor and anomalous magnetic moment, and to calculation of the leading term in the low-frequency expansion of the polarization operator. Hence, these contributions to the Lamb shift are independent of any features of the bound state. A nontrivial interplay between radiative corrections and binding effects arises first in calculation of contributions of order a Za) m, and in calculations of higher order terms in the combined expansion over a and Za. 

The main part of the nuclear size (Za) contribution which is proportional to the nuclear charge radius squared may also be easily obtained in a simpler way, which clearly demonstrates the source of the logarithmic enhancement of this contribution. We will first discuss in some detail this simple-minded approach, which essentially coincides with the arguments used above to obtain the main contribution to the Lamb shift in (2.4), and the leading proton radius contribution in (6.3). 

The leading nuclear size correction of order Za) m r )EF may easily be calculated in the framework of nonrelativistic perturbation theory if one takes as one of the perturbation potentials the potential corresponding to the main proton size contribution to the Lamb shift in (6.3). The other perturbation potential is the potential in (9.28) responsible for the main Fermi contribution... 

We cannot calculate the nuclear structure but we can describe the leading correction to the Lamb shift with the help of a simple delta-like potential, which depends on a single parameter, the nuclear charge radius R, calculated as fW) The parameter must be found experimentally. Usually this contribution is small enough and if necessary some corrections can be calculated. 

Fig. 1. Feynman diagrams representing various contributions to the Lamb shift. A solid line represents an electron, a wavy line a virtual photon and a cross denotes exchange of a Coulomb photon (a) Leading self-energy term (b) One-loop vacuum polarisation term. The loop represents a virtual electron-positron pair (c) Some diagrams contributing to the two-loop binding correction...

Table 1 Comparison of Za-expansion values to numerical calculcations including all orders in Za for a -order QED contributions to the Lamb shift of the lsi/2-state in lead and uranium. The Kalldn-Sabry contribution VPVP b) c) and the S(VP)E contribution are considered in the Uehling approximation only.

The exchange of a muon for the electron in hydrogen leads to an atomic system having considerably enlarged QED effects. However, in the case of pp and pHe the major contribution to the Lamb shift comes now from the vacuum polarisation rather than the self-energy... 

Calculation of the contribution of order a Za) induced by the radiative photon insertions in the electron line is even simpler than the respective calculation of the leading order contribution. The point is that the second and higher order contributions to the slope of the Dirac form factor are infrared finite, and hence, the total contribution of order Za) to the Lamb shift is given by the slope of the Dirac form factor. Hence, there is no need to sum an infinite number of diagrams. One readily obtains for the respective contribution... 

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