Hydrogen Lamb shift

Having in mind that the data from the muonic hydrogen Lamb shift experiment will be used for measurement of the rms proton charge radius [2] it is useful to write this correction in the form... 

Beams of slow negative muons have been developed and are for example employed in the muonic hydrogen Lamb shift experiment at PSI. Other developments are under way at the Rutherford Appleton Laboratory (RAL) in Chilton,... 

As can be seen from Table 1, the CH result for the transition, 2787.997 eV, is consistent with all other potentials within 0.01 eV. Adding in small three-photon and recoil corrections of -0.033 eV gives a final theoretical prediction of 2787.964 eV for the CH potential. As this disagrees with experiment, new physics must be present. The source of this physics is well known from the hydrogen Lamb shift, where two-loop effects are known to be quite important, and in fact are at present the dominant source of uncertainty, with the only other major unknown in the calculation being the precise size of the proton. We infer, then,... 

An experiment has been proposed at LAMPF to observe the 2S state and to measure the Lamb shift. To get to 10 - Ippm errors requires line narrowing techniques like the ones used in the hydrogen Lamb shift measurement. 

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. 

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 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 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. 

This value is two times larger than the result cited above in Table 3.6, and taken at face value shifts the value of the IB Lamb shift in hydrogen by 7 kHz. However, the authors of [123] show that extrapolation of their numerical data to Za) —> 0 leads to the value of Beo twice as large as the anal3dic result for Bgo in [109, 110]. Clearly any phenomenological conclusions in this situation are premature, and the problem of corrections of order a Za) "m requires... 

Concluding our discussion of the purely radiative corrections to the Lamb shift let us mention once more that the main sources of the theoretical uncertainty in these contributions is connected with the nonlogarithmic corrections of order a (Za)" and uncalculated contributions of orders a Za), which may be as large as a few kHz for IS-state and a few tenths of kHz for the 2S-state in hydrogen. All other unknown purely radiative contributions to the Lamb shift are much smaller. 

Prom the practical point of view, the difference between the results in (5.6) and (5.8) is about 0.18 kHz for lA level in hydrogen and at the current level of experimental precision the distinctions between the expressions in (5.6) and (5.8) may be ignored in the discussion of the Lamb shift measurements. These distinctions should, however, be taken into account in the discussion of the hydrogen-deuterium isotope shift (see below Subsect. 12.1.7). 

Then we easily obtain the weak interaction contribution to the Lamb shift in hydrogen [56]... 

Numerically, contribution to the 2P — 2S Lamb shift in muonic hydrogen is equal to... 

Despite the difference between the two cases, discussion of the proton size and structure corrections to HFS in hydrogen below is in many respects parallel to the discussion of the respective corrections to the Lamb shift in Chap. 6. 

We used the same value of A in Subsect. 5.1.3 for calculation of the correction to the Lamb shift in hydrogen generated by the radiative insertions in the proton hne. Due to the logarithmic dependence of this correction on A small changes of its value do not affect the result for the proton line contribution to the Lamb shift. 

For example, measurement of the IS Lamb shift in [34] is disentangled from the measurement of the Rydberg constant by using the experimental data on two different intervals of the hydrogen gross structure [34]... 

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