The Lamb Shift Experiment in Muonic Hydrogen Dissertation submitted to the Physics Faculty of the Ludwig{Maximilians{University Munich by Aldo Sady Antognini from Bellinzona, Switzerland Munich, November 2005 1st Referee : Prof. Dr. Theodor W. H¨ansch 2nd Referee : Prof. Dr. Dietrich Habs Date of the Oral Examination : December 21, 2005 Even if I don't think, I am. Itsuo Tsuda Je suis ou` je ne pense pas, je pense ou` je ne suis pas. Jacques Lacan A mia mamma e mio papa` con tanto amore Abstract The subject of this thesis is the muonic hydrogen (µ−p) Lamb shift experiment being performed at the Paul Scherrer Institute, Switzerland. Its goal is to measure the 2S 2P − energy difference in µp atoms by laser spectroscopy and to deduce the proton root{mean{ −3 square (rms) charge radius rp with 10 precision, an order of magnitude better than presently known. This would make it possible to test bound{state quantum electrody- namics (QED) in hydrogen at the relative accuracy level of 10−7, and will lead to an improvement in the determination of the Rydberg constant by more than a factor of seven. Moreover it will represent a benchmark for QCD theories. The experiment is based on the measurement of the energy difference between the F=1 F=2 2S1=2 and 2P3=2 levels in µp atoms to a precision of 30 ppm, using a pulsed laser tunable at wavelengths around 6 µm. Negative muons from a unique low{energy muon beam are −1 stopped at a rate of 70 s in 0.6 hPa of H2 gas. Highly excited µp atoms are formed, and most of them promptly deexcite to the ground state within 100 ns. However, there is a ∼ roughly 1% probability that long{lived µp2S atoms with a lifetime of 1:3 µs are formed. An incoming muon triggers a pulsed, multi{stage laser system which delivers 0:2 mJ per pulse at λ 6 µm with 55 s−1 repetition rate. It consists of two XeCl excimer lasers ' followed by dye lasers which pump an oscillator–amplifier frequency{controlled Ti:Sa laser. Its 6 ns long pulse at 708 nm is then frequency shifted to 6 µm via third Stokes production in a Raman cell filled with hydrogen. The laser pulse has a delay of about 1:5 µs with respect to the prompt muon cascade. If the laser is on resonance, it induces 2S 2P transitions. The subsequent deexcitation − to the 1S state emits a 1.9 keV Kα x ray which is detected by large area avalanche photodiodes. The resonance frequency, and hence the Lamb shift and rp, are determined by measuring the intensity of these x rays as a function of the laser wavelength. A search for the 2S 2P resonance line was performed in November 2003 when a broad − range of laser frequencies was scanned (49:7409 49:8757 THz), corresponding to proton − radii between 0.844 and 0.905 fm. The result of the data analysis is that no significant 2S 2P resonance was observed. The negative result is with high probability due to the − low statistics and not to an incorrect search region. The first part of this thesis reports on the present status of the Lamb shift theory in µp. Following, there is a detailed description of the apparatus and analysis of the data. An estimate of the present and future laser{induced event rates are given, together with a study of the present and future background. In the Appendices are discussed: the energy levels in H, the proton radius definition, the relevance of this experiment, the 2S state population and lifetime, and the spectroscopic properties of the 2S 2P transition. − i ii Contents Abstract i Contents ii List of Tables vi List of Figures vii 1 Overview of the muonic hydrogen Lamb shift experiment 1 2 Present status of the 2S − 2P Lamb shift in muonic hydrogen 7 2.1 Vacuum polarization . 8 2.2 Finite nuclear size . 11 2.3 Relativistic recoil corrections . 12 2.4 Fine and hyperfine structure . 14 2.5 The ∆E(2P F=2 2SF=1) energy splitting . 14 3=2 − 1=2 3 Muon beam, target, and electronics 17 3.1 Low{energy muon beam . 17 3.1.1 Cyclotron trap . 18 3.1.2 Muon extraction channel . 22 3.1.3 The 5 T solenoid with the muon detector . 23 3.2 Gas target . 26 3.2.1 Detectors for the 1.9 keV energy x rays: the LAAPDs . 27 3.2.2 Electron detectors . 30 3.2.3 Anti{coincidence detector . 30 3.2.4 Intermezzo about foils . 31 3.3 Electronics of the data acquisition system . 33 4 The laser system 35 4.1 Excimer lasers . 37 4.2 Dye lasers . 40 4.3 Continuous wave Ti:Sa laser . 42 4.3.1 Wavelength control . 43 4.3.2 Frequency stabilization . 44 4.4 Pulsed Ti:Sa oscillator and amplifier . 44 iii 4.4.1 Oscillator . 44 4.4.2 Injection seeding . 45 4.4.3 Chirp in the Ti:Sa oscillator . 46 4.4.4 Ti:Sa amplifier . 50 4.5 Raman cell . 51 4.6 Q01(1) Stokes{shift in H2 . 55 4.7 Water absorption . 56 4.8 Frequency calibration of the 6 µm light . 57 4.9 FP calibration and stability . 59 4.10 Summary of the frequency control of our laser system . 62 4.11 The 6 µm multipass cavity . 64 4.12 Summary . 68 5 Measurements 71 5.1 Analysis of the waveform digitizer signals . 71 5.2 Event classification . 73 5.2.1 Signal versus particle identification . 73 5.2.2 Event construction . 73 5.3 X-ray and electron energy spectra: Kα energy cut and delayed electron cut 74 5.4 Second{muon cut . 77 5.5 Electron time spectrum . 78 5.6 Time calibration and resolution . 81 5.7 LAAPDs x-ray efficiency . 83 5.7.1 Detector efficiency "x . 83 5.7.2 Total efficiency ηx . 85 6 Search for the 2S − 2P resonance 87 6.1 \Laser ON" and \Laser OFF" data . 87 6.2 2 keV x-ray energy and time spectra . 87 6.3 Resonance line . 91 6.4 Problems during 2003{run . 92 6.5 Event and background rate in 2003 beam time . 95 7 Future improvements of the apparatus 97 7.1 Thin{disk laser . 97 7.2 Future event rate . 101 7.3 Background rate . 102 7.4 Possible background reduction . 103 7.5 Measuring time and search of the resonance . 104 7.6 Future extension of the experiment . 106 iv A Theory of hydrogen energy levels 107 A.1 Bohr energy levels . 107 A.2 Dirac energy levels . 108 A.3 The Lamb shift . 109 A.4 Radiative corrections . 112 A.5 Perturbative and all{order approaches to the self{energy . 113 A.6 One{loop self{energy . 115 A.7 Two{loop self{energy . 115 A.8 Finite nuclear size and nuclear structure corrections . 118 A.9 Summary . 121 B Bound{state QED test and extraction of Rydberg constant 123 B.1 Bound{state QED test . 124 B.2 Rydberg constant . 125 B.3 Lamb shift and R1 uncertainty related to α and m=M . 126 B.3.1 Lamb shift uncertainty caused by α . 126 B.3.2 Uncertainty of R1 caused by the uncertainty of m=M . 126 C Electron{proton scattering experiments 129 C.1 Elastic scattering cross sections . 129 C.2 Measurements and extraction of the nuclear structure . 131 C.3 Structure functions . 132 D Proton radius 135 D.1 Proton radius in scattering experiments . 135 D.2 Proton radius in atomic hydrogen . 136 D.3 Problems related to the definition of the proton radius . 136 E 2S − 2P transition probability 137 E.1 2S 2P transition probabilities and matrix elements . 137 − E.2 2S 2P linewidth . 139 − E.3 Laser intensity . 141 E.4 Two{level system . 141 E.5 Summary . 143 F 2S − 2P transition systematics 145 F.1 Doppler broadening . 145 F.2 Zeeman effect . 145 F.2.1 Linear versus quadratic Zeeman effect . 146 F.2.2 Orbital and spin magnetic moments . 146 F.2.3 Zeeman effect of the hyperfine levels . 147 F.2.4 Anomalous Zeeman effect versus Breit{Rabi solution . 148 F.3 Collisional shift and broadening . 149 F.3.1 Model . 149 v F.3.2 2S 2P energy shift . 150 − F.3.3 Electric field and interatomic potential . 151 F.3.4 Numerical results . 152 G Population and lifetime of.