UNA MIRADA A LA REALIDAD DEL CONTEXTO DE LA EDUCACIÓN RURAL EN EL DEPARTAMENTO DE CUNDINAMARCA

Rydberg Constant R∞

Following the argumentation in Section 1.7, one can extract a new value for the Rydberg constant R∞. Assuming the correctness of the Lamb shift calculations in

µp (Section 1.5) and in ordinary hydrogen, we use the precise measurement of the 1S −2S transition in hydrogen [8, 10], the H(1S) and H(2S) Lamb shifts newly cal- culated with our value for rp, and the most recent value of the fine structure constant

(1/α= 137.035 999 084 [129]) and obtain

R∞= 10 973 731.568 161 (16) m−1. (4.60)

This value is −110 kHz/c or 4.9σ away from the CODATA value (Equation (1.23)), but with a relative accuracy of 1.5×10−12, it is 4.6 times more precise.

4.6

Other

µp and

µd Resonances

In addition to the focus of this thesis, the µp(2SF=1

1/2 −2P3/2F=2) resonance, three other

resonances in µp and µd were measured and one more line in µd were observed in the 2009 beam time. The careful analysis of these four lines has yet to be performed. Therefore, all reported data here is preliminary and subject to change. Nevertheless, they are presented here.

Table 4.8 lists all the resonances measured together with thestatistical accuracy of the position determination of the preliminary online analysis. Systematic studies are yet to be performed. Figure 4.16 presents the resonance lines apart from the first transition in µp. All lines are plotted versus the laser frequency and are scaled along the frequency axis such that the resonance widths match and the scanning regions can be compared. Note that all data in this section is preliminary and subject to be changed by a refined analysis.

Starting from both µp transitions, several scenarios can be considered. First of all, two values for the proton charge radius rp can be calculated using the theoretically

known energy differences of the two transitions. Moreover, a value for the 2S1/2−2P1/2

Lamb shift independent of the 2S HFS and the Zemach radius rZ can be extracted if

one adds bothµptransitions in a correctly weighted manner and considers the 2P fine and hyperfine structure to be well known [130] (cf. Equation (A.5)). The difference of the twoµptransition frequencies sheds light on the 2S HFS: with the HFS of the 2P3/2

state being calculated accurately [130], the frequency difference is simply equivalent to the 2SF=0

1/2 −2S1/2F=1 transition frequency. Since the Zemach radiusrZ [131] describes the

first order nuclear structure correction to the HFS, a new value forrZ can be deduced

from this with about 3 % relative accuracy.

In µd, a major puzzle needs to be solved: neither the absolute line positions nor the relative positions of the lines among each other seem to be correct. It has to be

laser frequency [THz]

54.55 54.6 54.65 54.7

]

-4

delayed / prompt events [10

0 1 2 3 4 5 6 PRELIMINARY µp(2S₁F_/=0₂ −2P₃F_/=1₂ ) laser frequency [THz] 50.75 50.8 50.85 50.9 ] -4

delayed / prompt events [10

0 2 4 6 8 10 PRELIMINARY µd(2S₁F_/=3₂ /2−2P₃F_/=5₂ /2) laser frequency [THz] 52 52.05 52.1 52.15 ] -4

delayed / prompt events [10

0 1 2 3 4 5 6 7 PRELIMINARY µd(2S₁F_/=1₂ /2−2P₃F_/=3₂ /2) µd(2S₁F_/=1₂ /2−2P₃F_/=1₂ /2)

Figure 4.16: The Other Resonances in µp and µd. The other four transitions measured in the 2009 campaign are plotted versus the laser frequency. Since all resonances possess the same linewidth, they are scaled so that their widths agree. The empty circles

80 4.6. OTHERµP ANDµDRESONANCES

Muonic Transition Position [GHz] Status Rel. Uncertainty

µp(2SF=1 1/2 −2P F=2 3/2 ) 49881.70 (71) final 15 ppm µp(2SF=0 1/2 −2P3/2F=1) 54612.6(1.2) statistical only 22 ppm µd(2S_1/2F=3/2−2P_3/2F=5/2) 50815.4(1.0) statistical only 20 ppm µd(2S_1/2F=1/2−2P_3/2F=3/2) 52061.7(2.2) statistical only 42 ppm µd(2S_1/2F=1/2−2P_3/2F=1/2) 52155.7(3.5) statistical only 67 ppm

Table 4.8: The µp and µd Transitions. All 2S −2P transitions in muonic hydrogen

and deuterium examined during the 2009 campaign. Apart from the first transition in µp

all values presented here are preliminary and the stated uncertainties are purely statistical.

The last two transitions in µd appear together in the bottom plot in Fig. 4.16.

mentioned that the theory of the Lamb shift in µd lacks redundancy. So far, only the paper of Borie [132] is available.

As in µp, the deuteron charge radius rd can be extracted from the Lamb Shift

measurements in µd. Apart from the vacuum polarization effects, the largest contri- butions to the µd Lamb shift are the finite nuclear size corrections and the nuclear polarizability corrections [132]. Based on correct QED calculations, on reliable val- ues for the fine and hyperfine structure, and on a theoretical value for the nuclear polarizability [48, 133, 134], one can determinerdfrom every µdtransition individually.

On the other hand, one can also derive a prediction for the deuteron charge radius using hydrogen 1S−2S spectroscopy. In order to do so, the value forrp deduced from

our µpmeasurements needs to be inserted into the difference of the charge radiir2_d−r2_p

obtained from the isotope shift in electronic hydrogen and deuterium [135, 136]. This leads to a prediction for rd which can then be compared to the values for rd obtained

from our µdmeasurements.

The argumentation could also be turned around: one could verify the nuclear po- larizability contribution to theµdLamb shift. By supplying the predictive value forrd

to the µdLamb shift, and assuming that the other QED terms are correct, the nuclear polarizability term is left as the only relevant free parameter. In such a way the nuclear polarizability calculations [48, 133, 134] could be tested.

However, there seem to be inconsistencies between the results of the online data and theoretical calculations. This has yet to be examined carefully. Hence, in order to resolve the puzzles raised by muonic deuterium, the data has to be analysed rigorously and the theory behind the Lamb shift has to be revised carefully.