The Lamb Shift

These fluctuations will affect the motion of charged particles. A major part of the Lamb shift in a hydrogen atom can be understood as the contribution to the energy from the interaction of the electron with these zero point oscillations of the electromagnetic field. The qualitative explanation runs as follows the mean square of the electric and magnetic field intensities in the vacuum state is equal to... 

We have assumed the potential to be spherically symmetric.) It is precisely the perturbation -J(Aq) 0V2F(q) that gives rise to the major part of the Lamb shift in the 2s state of the hydrogenic atom. 

Soon after the Schrodinger equation was introduced in 1926, several works appeared dealing with the fundamental problem of the nuclear motion in molecules. Very soon after, the relativistic equations were introduced for one-and two-electron systems. The experiments on the Lamb shift stimulated... 

L(0) = Z In 7, where I is the mean excitation potential appearing in Bethe s stopping power equation [Eq. (4)]. 7,(2) is proportional to the logarithm of average excitation energy, which is also involved in the Lamb shift [26]. 7,(—1) has been shown to be an optical... 

Calculation of the self-energy part of the Lamb shift... 

VP vacuum polarization SE self-energy part of the Lamb shift LS = VP + SEE Lamb shift RC nucleus recoil correction, polarization Relativistic PT accounts for the main relativistic and correlation effects HOPT higher-order PT contributions. Data are from refs [1-10]. 

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... 

It is clear now that the scale setting factor which should be used for qualitative estimates of the high order corrections to the Lamb shift is equal to Am Za) /n. Note the characteristic dependence on the principal quantum number 1/n which originates from the square of the wave function at the origin V (0) 1/n . All corrections induced at small distances (or at high... 

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. 

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 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... 

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... 

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... 

Then one readily obtains for the Lamb shift contribution... 

For calculation of the Pauli form factor contribution to the Lamb shift the third order contribution to the Pauli form factor (Fig. 3.5), calculated numerically in [33], and analytically in [34] is used... 

The hadronic polarization contribution to the Lamb shift was discussed in a number of papers [37, 38, 39]. The light hadron contribution to the polarization operator may easily be estimated with the help of vector dominance... 

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). ... 

It is not obvious that the hadronic vacuum polarization contribution should be included in the phenomenological analysis of the Lamb shift measurements, since experimentally it is indistinguishable from an additional contribution to the proton charge radius. We will return to this problem below in Sect. 6.1.3. 

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. 

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