Proton charge radius

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. 

An alternative treatment of the correction of order Z a) Za) m/M)m was given in [4]. The idea of this work was to modify the standard definition of the proton charge radius, and include the first order quantum electrodynamic radiative correction into the proton radius determined by the strong interactions. Prom the practical point of view for the nS levels in hydrogen the recipe of [4] reduces to elimination of the constant 11/72 in (5.6) and omission of the Pauli correction in (5.7). Numerically such a modification reduces the contribution to the lA energy level in hydrogen by 0.14 kHz in comparison with the naive result in (5.6), and increases it by 0.03 kHz in comparison with the result in (5.8). Hence, for all practical needs at the current level of experimental precision there are no contradictions between our result above in (5.8), and the result in [4]. 

We see that the correction to the energy level induced by the finiteness of the proton charge radius shifts the energy level upwards, and is nonvanishing only for the S states Physically the finite radius of the proton means that the proton charge is smeared over a finite volume, and the electron spends some time inside the proton charge cloud and experiences a smaller attraction than in the case of the pointlike nucleus (Compare similar arguments in relation with the finite radius of the electron below (2.4)). 

However, this is not yet the end of the story, since the proton charge radius is usually defined via the Sachs electric form factor Gp, rather than the Dirac form factor Fi... 

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... 

The Zemach correction is essentially a nontrivial weighted integral of the product of electric and magnetic densities, normalized to unity. It cannot be measured directly, like the rms proton charge radius which determines the main proton size correction to the Lamb shift (compare the case of the proton size correction to the Lamb shift of order Za) in (6.13) which depends on the third Zemach moment). This means that the correction in (11.4) may only conditionally be called the proton size contribution. 

The experimental results in the first five lines in Table 12.3 seem to be compatible with the theoretical value in (12.11). This compatibility crucially depends on the value of the proton charge radius [27]. We presented in Table 12.3 also theoretical values of the IS Lamb shift calculated with the once popular earlier lower values of the radius, and they are clearly incompatible with the experimental data on the IS Lamb shift. However, it is necessary to remember that the experimental results in the first five lines in the Table are biased , namely they depend on the experimental value of the 2Si/2 — Pil2 Lamb shift [15, 24, 25]. In view of a rather large scattering of the results for the classic Lamb shift such dependence is unwelcome. 

Since the main contribution to the uncertainty of the theoretical value of the IS Lamb shift comes from the uncertainty of the proton charge radius we can invert the problem and calculate the proton radius using the average of the self-consistent Lamb shifts in Table 12.3 L IS) = 8 172 847 (14) as input. Then we obtain the optical value of the proton charge radius... 

Individual uncertainties of the proton and deuteron charge radii introduce by far the largest contributions in the uncertainty of the theoretical value of the isotope shift. Uncertainty of the charge radii are much larger than the experimental error of the isotope shift measurement or the uncertainties of other theoretical contributions. It is sufficient to recall that uncertainty of the 15 Lamb shift due to the experimental error of the proton charge radius is as large as 50 kHz (see (12.11)), even if we ignore all problems connected with the proton radius contribution (see discussion in Subsects. 12.1.5, 12.1.6). 

The current surge of interest in muonic hydrogen is mainly inspired by the desire to obtain a new more precise value of the proton charge radius as a result of measurement of the 2P — 25 Lamb shift [64]. As we have seen... 

We can write the 2P — 25 Lamb shift in muonic hydrogen as a difference of a theoretical number and a term proportional to the proton charge radius squared... 

We see from this equation that when the experiment achieves the planned accuracy of about 0.008 meV [64] this would allow determination of the proton charge radius with relative accuracy about 0.1% which is about an order of magnitude better than the accuracy of the available experimental results. 

The most likely sources for this difference are a deviation of the value of the proton charge radius and/or the deuteron charge radius predicted by the spectroscopic data from the values deduced from scattering experiments, an uncalculated contribution to the energy levels from the two-photon QED correction that exceeds the estimated uncertainty for this term, or a combination of these. 

Historically, measurements of the IS Lamb shift in hydrogen have constituted the most accurate tests of bound-state QED. However, these recently calculated terms are obscured in hydrogen by the experimental error in the proton charge radius. This is because a non-QED correction to the Dirac levels due to the finite size of the nucleus is included in the Lamb shift, and the uncertainty in this term for the proton is comparable to the two-loop correction in the IS state. In He+, the error introduced by the experimental uncertainty in the alpha particle radius is relatively much smaller [14,15], making Lamb shift measure-... 

Abstract. The calculation of the last unknown contribution to hydrogen energy levels at order ma7, due to the three loop slope of the Dirac form factor, is described. The resulting shift of the nS energy level is found to be 3.16/n3 kHz. Adding this result to many known contributions to the 1S Lamb shift and comparing with experimental value, we derive the value of the proton charge radius rp = 0.883 0.014 fm. 

We now present our result for the IS Lamb shift in hydrogen. A detailed description of all the corrections that we include to obtain the final result as well as references to the original papers are given in [8], The final value for the lS -shift strongly depends on the value of the proton charge radius ... 

In conclusion, we have computed the three-loop slope of the Dirac form factor. Thanks to this calculation the theoretical uncertainty in the predictions for the IS1 Lamb shift is reduced. Comparison of the theoretical and experimental results for the IS1 level shift permits an accurate determination of the proton charge radius. Further improvements in theoretical predictions for the IS1 level shift would be possible if subleading a2(Za)6 log2 a corrections are calculated. Only then can the theoretical uncertainty be brought down to several kHz and can the potential of the recent measurement of the 15 — 2S transition frequency [6] be fully exploited. 

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