Dirac form factors

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

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... 

The two-loop slope was considered in the early pioneer works [17, 18], and for the first time the correct result was obtained numerically in [19]. This last work triggered a flurry of theoretical activity [20, 21, 22, 23], followed by the first completely analytical calculation in [24]. The same analytical result for the slope of the Dirac form factor was derived in [25] from the total e+e cross section and the unitarity condition. 

Calculation of the corrections of order a Za) is similar to calculation of the contributions of order a Za). Respective corrections depend only on the values of the three-loop form factors or their derivatives at vanishing transferred momentum. The three-loop contribution to the slope of the Dirac form factor (Fig. 3.5) was calculated anal3dically [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 second order perturbation theory term with two one-loop self-energy operators does not generate any logarithm squared contribution for the state with nonzero angular momentum since the respective nonrelativistic wave function vanishes at the origin. Only the two-loop vertex in Fig. 3.24 produces a logarithm squared term in this case. The respective perturbation potential determined by the second term in the low-momentum expansion of the two-loop Dirac form factor [111] has the form... 

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... 

The graph in Fig. 7.9 is gauge invariant and generates a correction to the slope of the Dirac form factor, which was calculated in [40]... 

Respective corrections are written via the slope of the Dirac form factor and the anomalous magnetic moment exactly as in Subsect. 7.3.4. The only difference is that the contributions to the form factors are produced by the hadronic vacuum polarization. 

Three-Loop Slope of the Dirac Form Factor and the IS Lamb Shift in Hydrogen... 

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. 

On the theoretical side, the last unknown piece at 0(ma7), the contribution of the three loop slope of the Dirac form factor, has been computed recently [8]. In what follows that calculation is described. 

Three-loop slope of the Dirac form factor 345... 

We are interested in the value of the Dirac form factor when the momentum transfer q is small as compared to the electron mass. Then ... 

To match experimental precision, the energy shift has to be computed up to AE ma7 and even higher order corrections should sometime be included. For this reason the three loop contribution to Ff°pe is required. We have previously reported on our calculation of the three loop slope of the Dirac form factor in [8], where the emphasize was on the phenomenological aspects. Here we would like to discuss some of the methods employed in that calculation in a more detailed way. Nevertheless, all important consequences of our result are presented as well. 

Amazingly, this simple logic generalizes on to much more difficult cases and is now the main tool in perturbative calculations in high energy physics. We use it for the required calculation of the three loop slope of the Dirac form factor. 

All the above steps have been automated (in simple terms this means, that getting either the three loop g — 2 or the slope of the Dirac form factor is just a matter of changing a single line in the computer code). The program is written in Form, a symbolic manipulation language created by J.A.M. Vermaseren. Organization of the program is as follows ... 

The contribution due to the three-loop slope of the Dirac form factor was the last unknown contribution to the hydrogen energy levels at order a3(Za)4. The two other contributions come from the three-loop electron anomalous magnetic moment and the three-loop vacuum polarization correction to the Coulomb propagator. These contributions can be extracted from the literature [10,13]. 

We have displayed the contributions due to the three-loop slope of the Dirac form factor, the three-loop anomalous magnetic moment of the electron and the three-loop photon vacuum polarization separately. Thanks to the cancellation between these contributions, the correction turns out to be quite small numerically. 

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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