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Studies for the PIENU experiment and on the direct radiative capture of muons in zirconium vom Bruch, Dorothea 2013

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Studies for the PIENU Experiment and on the DirectRadiative Capture of Muons in ZirconiumbyDorothea vom BruchA THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE REQUIREMENTS FOR THE DEGREE OFMaster of ScienceinTHE FACULTY OF GRADUATE AND POSTDOCTORALSTUDIES(Physics)The University Of British Columbia(Vancouver)December 2013c? Dorothea vom Bruch, 2013AbstractThe branching ratio of pions decaying to positrons and muons R = pi+?e+?e+pi+?e+?e?pi+??+??+pi+??+?? ?has been calculated with very high precision in the Standard Model of particlephysics. So far, the theoretical value of R = 1.2352(1) ? 10?4 is 40 times moreprecise than the current experimental value of R = 1.230(4) ? 10?4. To test thisvariable with respect to deviations from the Standard Model, the experimental pre-cision needs to be improved, which is why the PIENU experiment aims at a pre-cision of less than 10?3, i.e. an improvement of an order of magnitude over thecurrent precision. At this level, mass scales ? 1000TeV/c2 can be probed for evi-dence of new pseudo-scalar interactions. The data collected with the experimentalsetup also allows for a search of sterile neutrinos. When determining the branchingratio, various systematic corrections are applied. The largest among these is dueto electro-magnetic shower leakage out of the calorimeters and radiative decays. Itwas calculated to be (2.25?0.06)% in this thesis.In the second part of the thesis, an experiment on the direct radiative captureof muons in zirconium is described. One theoretical extension to the StandardModel involves a new light and weakly interacting particle in the muon sectorwhich does not conserve parity. This can be studied experimentally with polarizedmuons that undergo the direct radiative capture into the 2S state of a medium masstarget nucleus. During this capture, longitudinal muon polarization is conservedand the muons instantly undergo the 2S-1S transition emitting a second photon.Studying the angular distribution of this second photon indicates whether or notthe process is parity violating, which would manifest physics beyond the StandardModel. The direct radiative capture of a muon into an atom in the 1S or 2S statehas not been observed yet. Therefore, data was taken in 2012 to study the radiativeiicapture of muons in zirconium. The analysis method of this data set is describedwith a blind analysis technique.iiiPrefaceIn February 2012, I joined the PIENU collaboration, which has about 25 membersfrom 11 institutions. I actively participated in the run period between May andDecember 2012 by taking shifts in the experiment control room. In terms of dataanalysis, I focused on determining a lower limit on the systematic correction tothe branching ratio that accounts for energy leakage out of the calorimeters andradiative decays. The basic procedure was established by Chloe? Malbrunot and isdescribed in her doctoral thesis [1]. I improved the method by optimizing the cutsthat were applied (indicated in the main text), and by adding a new correction toaccount for radiative decays. Tristan Sullivan determined a second estimate for thecorrection [2] and I implemented a procedure to combine the two estimates.In November 2012, an experiment to study the direct radiative capture of neg-ative muons into zirconium was proposed. I assessed the feasibility of this experi-ment with the PIENU detector and the M13 beamline at TRIUMF. After conclud-ing that it was realistic, I proposed an experimental run which was realized duringfour days in December 2012. I established the procedure for analyzing this data setemploying a blind analysis method. By implementing a new fitting procedure forthe waveform of the calorimeter, I improved its time resolution by a factor of four.I also used this fit for pulse shape discrimination and was able to identify neutronsand photons within the calorimeter. These new tools allowed me to decide on cutsto select the signal process of the direct muon capture and suppress backgrounds.Anthony Fradette and Maxim Pospelov provided the necessary theoretical calcula-tions to characterize the signal events which I used to simulate the signal processin a GEANT4 simulation. Based on this simulation, I estimated the significance ofthe signal with respect to the background events in the data and developed differentivprocedures to extract the signal from the background.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Outline of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Motivation for the PIENU Experiment . . . . . . . . . . . . . . . 11.3 Motivation for the Muon Capture Experiment . . . . . . . . . . . 32 Description of the PIENU Experiment . . . . . . . . . . . . . . . . . 42.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 The Beamline and Detector . . . . . . . . . . . . . . . . . . . . . 62.3 The Trigger and Data Acquisition System . . . . . . . . . . . . . 83 Data Analysis of the PIENU Experiment . . . . . . . . . . . . . . . . 103.1 Raw Branching Ratio and Systematic Effects . . . . . . . . . . . 10vi3.2 Lower Limit Determination . . . . . . . . . . . . . . . . . . . . . 143.2.1 Suppressed Spectrum . . . . . . . . . . . . . . . . . . . . 143.2.2 Determining the Lower Limit . . . . . . . . . . . . . . . 183.2.3 Muon Decay in Flight Correction to the Lower Limit . . . 223.2.4 Radiative Decay Correction . . . . . . . . . . . . . . . . 243.2.5 Bhabha Correction . . . . . . . . . . . . . . . . . . . . . 273.2.6 Error Estimation for the Lower Limit . . . . . . . . . . . 273.2.7 Consistency Checks . . . . . . . . . . . . . . . . . . . . 293.2.8 Statistical Effects . . . . . . . . . . . . . . . . . . . . . . 293.3 Lineshape Measurement . . . . . . . . . . . . . . . . . . . . . . 293.4 Combination of Lineshape Measurement and Lower Limit . . . . 334 Muon Capture Experiment . . . . . . . . . . . . . . . . . . . . . . . 374.1 Introduction and Theoretical Background . . . . . . . . . . . . . 374.2 Experimental Realization . . . . . . . . . . . . . . . . . . . . . . 434.2.1 Beamline . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . 454.2.3 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . . 464.2.4 Data Taking . . . . . . . . . . . . . . . . . . . . . . . . . 474.3 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3.1 Simulation of the Signal . . . . . . . . . . . . . . . . . . 504.3.2 Energy Calibration . . . . . . . . . . . . . . . . . . . . . 524.3.3 Selection Cuts . . . . . . . . . . . . . . . . . . . . . . . 534.3.4 Background from Muon Capture At Rest . . . . . . . . . 554.3.5 Time Resolution . . . . . . . . . . . . . . . . . . . . . . 564.3.6 Blind Analysis . . . . . . . . . . . . . . . . . . . . . . . 594.3.7 Particle Identification . . . . . . . . . . . . . . . . . . . . 604.3.8 Rejection of Neutrons . . . . . . . . . . . . . . . . . . . 644.3.9 Time Cut . . . . . . . . . . . . . . . . . . . . . . . . . . 654.3.10 Analysis of the 74 MeV/c Data Set . . . . . . . . . . . . . 664.3.11 Analysis of the Mylar Data Set . . . . . . . . . . . . . . . 674.3.12 Background Spectra . . . . . . . . . . . . . . . . . . . . 674.4 Signal versus Background Prediction . . . . . . . . . . . . . . . . 69vii5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 785.1 PIENU Experiment . . . . . . . . . . . . . . . . . . . . . . . . . 785.2 Muon Capture Experiment . . . . . . . . . . . . . . . . . . . . . 79Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81viiiList of TablesTable 3.1 Summary of the cuts used to produce the suppressed spectrum. 18Table 4.1 Signal photon energies. . . . . . . . . . . . . . . . . . . . . . 53Table 4.2 Predicted numbers of signal events in the 1S state. . . . . . . . 70ixList of FiguresFigure 2.1 Feynman diagrams for pi ? ?? and pi ? e? decays. . . . . . 5Figure 2.2 Time and energy spectrum for pi ? ?? and pi ? e? decays. . 6Figure 2.3 The M13 beamline at TRIUMF. . . . . . . . . . . . . . . . . 7Figure 2.4 Diagram of the experimental set-up of the PIENU detector. . . 7Figure 3.1 Blinding procedure using the target energy deposit. . . . . . . 11Figure 3.2 Simultaneous time fit of the high and low energy regions. . . . 12Figure 3.3 Energy deposit for MDIF versus MDAR. . . . . . . . . . . . 13Figure 3.4 Energy deposit in the upstream detectors with suppression cut. 16Figure 3.5 Energy deposit in the upstream counters versus ??2 . . . . . . 17Figure 3.6 MDIF before Tg detector. . . . . . . . . . . . . . . . . . . . 17Figure 3.7 Angle from the beam direction for PDIF and PDAR. . . . . . 18Figure 3.8 Energy spectrum with different suppression cuts. . . . . . . . 19Figure 3.9 Evaluation procedure for the lower limit. . . . . . . . . . . . 20Figure 3.10 Lower Limit versus upper integration limit. . . . . . . . . . . 21Figure 3.11 Energy deposit of pi ? e? events vs. MDIF decays. . . . . . . 22Figure 3.12 Simultaneous energy-time fit. . . . . . . . . . . . . . . . . . 23Figure 3.13 Energy spectrum of radiative and non-radiative decays. . . . . 25Figure 3.14 Lower limit with radiative correction versus integration limit. . 26Figure 3.15 Lower limit shape for four statistically independent samples. . 30Figure 3.16 Lineshape spectrum at 0 degrees. . . . . . . . . . . . . . . . . 31Figure 3.17 Lineshape spectrum at 48 degrees. . . . . . . . . . . . . . . . 32Figure 3.18 Tail fraction as a function of angle. . . . . . . . . . . . . . . . 33Figure 3.19 Probability distributions of the tail fraction and the lower limit. 34xFigure 3.20 Allowed region for tail fraction and lower limit. . . . . . . . . 35Figure 4.1 Level diagram of a typical muonic atom. . . . . . . . . . . . . 40Figure 4.2 Capture cross section versus muon momentum. . . . . . . . . 44Figure 4.3 Muon momentum at Zr foil for 72 MeV/c beam. . . . . . . . 46Figure 4.4 Muon momentum at Zr foil for 74 MeV/c beam. . . . . . . . 47Figure 4.5 Detector diagram including zirconium foil. . . . . . . . . . . 48Figure 4.6 T1 counter with and without zirconium foil. . . . . . . . . . . 49Figure 4.7 Polar Angle of the emitted photon during the ARC process. . . 51Figure 4.8 Kinetic energy in slices in Zr foil for 72 MeV/c set. . . . . . . 52Figure 4.9 Kinetic energy in slices in Zr foil for 74 MeV/c set. . . . . . . 52Figure 4.10 Comparison of MC and electron triggered data. . . . . . . . . 54Figure 4.11 Charge in B1 versus TOF . . . . . . . . . . . . . . . . . . . . 55Figure 4.12 Time resolution without fit. . . . . . . . . . . . . . . . . . . . 57Figure 4.13 Waveform in the NaI crystal. . . . . . . . . . . . . . . . . . . 58Figure 4.14 Time resolution with function fit. . . . . . . . . . . . . . . . . 59Figure 4.15 Waveform with definitions of tail and total integral. . . . . . . 62Figure 4.16 Tail versus total integral for PIENU data. . . . . . . . . . . . 63Figure 4.17 Tail versus total integral for muon capture data. . . . . . . . . 63Figure 4.18 Cuts on the waveform shape. . . . . . . . . . . . . . . . . . . 64Figure 4.19 Neutron suppression efficiency. . . . . . . . . . . . . . . . . 65Figure 4.20 Acceptance vs. rejection for waveform cut. . . . . . . . . . . 66Figure 4.21 Tail versus total integral for 74 MeV/c data. . . . . . . . . . . 67Figure 4.22 Tail versus total integral for the Mylar data. . . . . . . . . . . 68Figure 4.23 Energy spectrum for Mylar after selection cuts. . . . . . . . . 69Figure 4.24 Energy-blinded time spectrum for 72 MeV/c data. . . . . . . . 70Figure 4.25 Background energy spectra for 72 MeV/c data. . . . . . . . . 71Figure 4.26 Background energy spectra for 74 MeV/c data. . . . . . . . . 71Figure 4.27 Energy- and prompt-blinded spectra for 74 MeV/c data. . . . 72Figure 4.28 Signal extraction method for 74 MeV/c data. . . . . . . . . . 73Figure 4.29 Signal statistics from integration for the 74 MeV/c data. . . . 74Figure 4.30 Signal statistics from a Gaussian fit for the 74 MeV/c data. . . 74Figure 4.31 Energy- and prompt-blinded spectra for 72 MeV/c data. . . . 75xiFigure 4.32 Signal statistics from integration for the 72 MeV/c data. . . . 76Figure 4.33 Signal statistics from a Gaussian fit for the 72 MeV/c data. . . 77xiiGlossaryADC Analog to Digital Converter.ARC Atomic Radiative Capture.B1 and B2 Two plastic scintillator counters in front of the target of the PIENUexperiment.CsI Cesium Iodide.DAQ Data Acquisition System.FADC Flash Analog to Digital Converter.GEANT4 GEometry ANd Tracking: Toolkit which uses Monte Carlo Methods tosimulate the passage of particles through matter, written in C++..M13 Pion beam line in the meson hall at TRIUMF.MC Monte Carlo.MDAR Muon Decays At Rest.MDIF Muon Decays In Flight.NaI Sodium Iodide.NDF Number of Degrees of Freedom.xiiiNP New Physics.PDAR Pion Decays At Rest.PDIF Pion Decays In Flight.PMT Photo Multiplier Tube.RF Radio Frequency.SM Standard Model.T1 and T2 Two plastic scintillator counters before and after the third wire cham-ber of the PIENU experiment.Tg Target.TOF Time Of Flight.TRIUMF Canada?s National Laboratory for Particle and Nuclear Physics.WC Wire Chamber.xivAcknowledgmentsIn the first place, I want to thank my supervisor Doug Bryman for his support andadvice, and especially for making the muon capture experiment possible, and forguiding me throughout the experimental procedure as well as the data analysis.Furthermore, I am very grateful for discussions with Toshio Numao on thelower limit, for the explanations and help provided by Alexey Sher, Luca Doria,and Dima Vavilov, and for the collaboration with the PhD students Tristan Sullivan,Shintaro Ito and Saul Cuen-Rochin. Special thanks goes to Luca Doria for his quickand efficient proof-reading of this thesis. In particular, I want to acknowledge theinsightful explanations and discussions with Richard Mischke during my very firstshifts and his comments on my analysis as well as the proof-reading of this thesis.A word of thanks is due to Maxim Pospelov and Anthony Fradette from theUniversity of Victoria for their contributions to the theoretical predictions for themuon capture experiment, and their inspiring comments and clear explanations.Finally, I would like to thank Tegan Macdonald for the proof-reading of thethesis from an outside perspective, and for the many lunch breaks with the TITANgroup.xvChapter 1Introduction1.1 Outline of the ThesisThis thesis describes work performed on the data analysis of the PIENU experi-ment as well as a muon capture experiment which was conducted using the samedetector. After motivating both experiments in Chapter 1 the setup of the PIENUexperiment is described in Chapter 2. Chapter 3 deals with the procedure for dataanalysis with special focus on systematic effects. Finally, chapter Chapter 4 ex-plains how the detector was modified to accommodate the muon capture experi-ment, describes the analysis technique and presents the preliminary results. Chap-ter 5 summarizes the results for both the PIENU experiment and the muon captureexperiment, proposes further studies and outlines the impacts of the results.1.2 Motivation for the PIENU ExperimentResearch in particle physics is pursued at three different frontiers nowadays: atthe high energy scale, in the cosmic sector, and with high precision and intensityexperiments. The PIENU (pi ? e?) experiment belongs to the last category; itaims to explore physics beyond the Standard Model (SM) by studying rare piondecays to positrons and muons. The measured branching ratio of pions decayingto positrons and to muons will be compared to the theoretically predicted valueR = 1.2352(1) ?10?4 [3] which has been very precisely calculated within the SM.1So far, the theoretical value is 40 times more precise than the experimental value ofR= 1.230(4) ?10?4 [4], which is why PIENU aspires at a level of relative precisionof less than 10?3, i.e. an improvement of an order of magnitude over the currentprecision.Validating the SM with experimental results has been the task of elementaryparticle physicists for the past five decades. In particular, the discovery of a Higgslike particle last year at the Large Hadron Collider was a big success for the SM [5].However, many questions and puzzles remain unsolved in the domain of particlephysics. Firstly, the discovery of neutrino oscillations [6] requires neutrinos to bemassive particles. Their hierarchy however, and the mechanism of how they obtaintheir mass are still unknown and not predicted within the current standard theory.In addition, the reason for the number of lepton families (electrons, muons, taus)remains a mystery, as well as the large range of mass scales among the elementaryparticles. Furthermore, observations of the rotation speed of galaxies [7], and otherobservations such as galactic clusters [8] and the cosmic microwave background[9], imply that normal matter cannot solely account for the behaviour of stars andgas circling around the center of a galaxy. Hence, a different explanation, such asthe existence of dark matter, is needed to explain these phenomena. Elementaryparticles not included in the SM are possible candidates to explain dark matter.Evidently, there is need for extensions or modifications to the existing SM inorder to consistently explain the processes in our universe. At the level of precisionthat the PIENU experiment aims for, mass scales? 1000TeV/c2 can be probed forevidence of new pseudo-scalar interactions. In addition, the data collected with theexperimental setup allows for a search of sterile neutrinos. These sterile neutrinosare a possible candidate for dark matter [10].Thorough understanding of the experimental setup, background processes, andsystematic effects is required in order to obtain the desired precision. The study ofone of these systematic effects, due to the electro-magnetic shower leakage out ofthe calorimeters and radiative decays, is the main focus in this thesis concerningthe PIENU experiment .21.3 Motivation for the Muon Capture ExperimentIn the search for physics beyond the SM the experiments at the Large HadronCollider (LHC) at CERN are very crucial in probing for new hypothetical particles.However, they can miss new physics effects and particles such as light and weaklycoupling particles not included in the SM. This is where precision experiments areneeded for complementary studies in the search for new physics.Inspired by the discrepancies between theory and experiment for the muonanomalous magnetic moment [11] as well as between the results for the protoncharge radius determined from spectroscopy of atomic hydrogen and from muonichydrogen [12], David McKeen and Maxim Pospelov suggested to study muonphysics more closely. In particular, they suggested to investigate muons interactingwith thin targets in order to probe for a new light and weakly interacting particle[13]. More explicitly, they recommended an experiment where low energy nega-tive muons are captured in a thin target with Z ? 30. The muons enter directly intothe atomic 2S state which preserves the longitudinal muon polarization, and theyinstantly undergo the 2S1/2?1S1/2 transition. During this process two photons areemitted: one when the muon enters the atom, the other one during the deexcitation.The direction in which the second photon is emitted with respect to the directionof the muon spin indicates whether or not the process is parity-violating.So far, the direct capture of a negative muon into the 1S or 2S state has notbeen observed. That is why the PIENU detector was modified slightly in December2012 and data was collected with a negative muon beam to detect the direct captureprocess. The procedure of this experiment, the data analysis and a comparison tothe theoretical prediction are presented in the second part of this thesis.3Chapter 2Description of the PIENUExperiment2.1 OverviewThe PIENU experiment was carried out from 2008 until 2012 in the meson hallof Canada?s National Laboratory for Particle and Nuclear Physics (TRIUMF). Itscyclotron delivered a 500 MeV proton beam from which pions were produced andguided through a beamline to the PIENU detector. During an experimental run,positively charged pions entered the detector and stopped in an active target wherethe pion decayed to a neutrino and a positron or to a neutrino and a muon. Thelatter stopped within the target as well and decayed further to a positron and twoneutrinos. Feynman diagrams for the decays are shown in Figure 2.1. In bothcases, positrons were emitted in the final state, but their different features in timeand energy were utilized to distinguish between them. Since the lifetime of a pion(?pi = 26ns) is two orders of magnitude smaller than that of a muon (?? = 2.2?s),measuring the time distribution of the outgoing positrons with respect to the in-coming pions indicated which type of decays occurred within the target. The leftpanel of Figure 2.2 shows the simulated time spectra for both decays. Furthermore,the energy spectra of the two decay modes differed substantially since one of themwas a two body decay (e+ and ?e) at rest, whereas the other one involved three de-cay products (e+, ?e and ?? ). The positron and neutrino from the direct pion decay4Figure 2.1: Feynman diagrams for the decay of a pion to a lepton and a neu-trino (left panel) and for a muon decaying to a positron and two neutri-nos (right panel).shared the momentum of the pion?s rest mass, so that in theory the positron shouldhave had an energy of 69.8 MeV. However, because of energy deposited in the de-tector components, leakage out of the calorimeter, and radiative decays the energyspectrum peaked at ? 65 MeV and had a low energy tail, as shown on the righthand side of Figure 2.2. On the other hand, the energy spectrum of the positronfrom the muon decay, the so called Michel spectrum, had the typical broad shapeof a three body decay and a sharp edge at ? 52 MeV.In the analysis, the energy spectrum was used to divide the data sample into alow (E < 50 MeV) and a high (E > 50 MeV) energy part, and a simultaneous timefit was applied to these two data portions including backgrounds to determine theraw branching ratio. Corrections had to be applied to account for systematic effectssuch as the energy-dependent acceptance of the detector, the fact that muon decaysin flight (MDIF) could reach energies above 50 MeV and were then misidentifiedas pi ? e?e decays, and finally the leakage out of the calorimeter, the so called tailcorrection.In this thesis, only the detector elements which are crucial for the study of thetail correction and for the muon capture experiment are described. For a detaileddescription of the theoretical background and the experimental setup of PIENUrefer to Chloe? Malbrunot?s doctoral thesis [1].5Time [ns]0 100 200 300 400 500 600Normalized Amplitudes00.20.40.60.81?+ e?+pi?+? ?+piEnergy [MeV]0 20 40 60 80Normalized Amplitudes00.20.40.60.81?+ e?+pi?+? ?+piFigure 2.2: Time spectra (left panel) and energy spectra (right panel) for pi?? ? e (red solid line) and pi ? e (blue dashed line) decay.2.2 The Beamline and DetectorThe cyclotron at TRIUMF produced 500 MeV protons with a Radio Frequency(RF) of ? 23.1 MHz which hit a 12 mm beryllium production target emitting var-ious particles, among which were pions, electrons, and muons. These particleswere directed towards the experimental area of PIENU through the momentumand particle selecting beamline M13 shown in Figure 2.3.Bending magnets, quadrupoles, and a combination of slits and absorbers se-lected the desired momentum of 75 MeV/c. Furthermore, a degrader placed nearthe focus F1 caused a momentum spread between positrons, muons, and pions sothat the collimator placed at focus F3 after the second bending magnet could betuned to select pions only. Therefore, backgrounds due to positrons and muonswere highly suppressed [14]. After passing through the beamline, the particlesentered the PIENU detector; a diagram of its experimental set-up is shown in Fig-ure 2.4.The pions traversed two multi-wire proportional chambers (WC1 and WC2)for tracking purposes, then they reached a set of plastic scintillator counters; B1and B2 were used for particle identification through energy deposit and the TimeOf Flight (TOF) with respect to the cyclotron RF timing. In the active scintillator6vacuum valvebeam blockerhorizontal slit (F1SL)absorbervertical jaw (F1JA)horizontal slit (F2SL)vertical jaw (F2JA)horizontal slit (F0SL) & vertical jaws (F0JA)collimatorzxFigure 2.3: The M13 beamline at TRIUMF including the extension built in2008. Picture taken from reference [1].WC1WC2B1B2S1S2TgT1S3WC3T2NaI(Tl)CsI ringCsI ring?eepi+Figure 2.4: Diagram of the experimental set-up of the PIENU detector (notto scale).7target (Tg), the pions came to a stop and decayed. Two silicon strip detectors wereplaced before the target (S1 & S2), one behind it (S3) to reconstruct the tracks ofthe in- and outgoing particles. Each of these detectors consisted of two layers ofstrips, placed perpendicular to each other for two dimensional position information.A third wire chamber (WC3) was placed behind the target to obtain tracking incombination with S3 and to define the angular acceptance. This wire chamber wassandwiched between two more plastic scintillator counters (T1 & T2), which wereused to define the trigger. After T2, the particles entered a cylindric solid (Thalliumdoped) Sodium Iodide (NaI(Tl)) crystal, 48 cm in diameter and 48 cm long, whichserved as calorimeter. Two layers of Cesium Iodide (CsI) crystals surrounded theNaI(Tl) crystal in order to reduce the amount of undetected shower leakage. Eachof the plastic scintillators was read out by four Photo Multiplier Tubes (PMTs),whereas 19 PMTs were attached to the NaI(Tl) crystal for light collection and eachof the 97 CsI crystals was attached to one PMT [15].2.3 The Trigger and Data Acquisition SystemAn event was only recorded if an incoming pion came in coincidence with an out-going positron. The entrance of a pion was defined by a coincidence of the beamcounters B1, B2 and Tg, whereas a positron leaving the Tg in the direction of theNaI(Tl) crystal was represented by a coincidence of T1 and T2. If these two sig-nals occurred within a time window of [-300 ns, 500 ns] the event was selected.The lower boundary of this window was chosen to characterize the background ofmuons present in the detector before the pion entry, and the region between 0 nsand 500 ns allowed to study the two decay modes with their different decay con-stants. Since the probability for pi? ??? decay is four orders of magnitude largerthan that for pi? e?e decay, specialized triggers were implemented to enhance thepi ? e?e decay mode. This was later taken into account when analyzing the data.Two special triggers selected pi? e?e events, the so called early and TIGC triggers.If the signal of the outgoing positron occurred between 7 ns and 40 ns the event wasselected by the early trigger. In case the online sum of the energy deposited in thecalorimeters was above a threshold of 46 MeV the event was recorded by the TIGCtrigger. These two triggers exploited the characteristics of the positron originating8from pi ? e?e decay since the pion has a short lifetime and the emitted positronleft a high energy deposit in the crystals. Furthermore, the ?normal? trigger defin-ing the incoming pion and emitted positron, which recorded both pi ? e?e andpi ? ? ? e events, was prescaled by a factor of 1/16 to reduce pi ? ? ? e events[15].In addition to the triggers mentioned above, three more triggers were used forcalibration purposes: one trigger selected cosmic-ray events for CsI calibration;a Xe lamp was flashed twice per second illuminating the CsI PMTs to monitorchanges in their gains; and data collected with a beam positron trigger providedinformation for the calibration of the NaI(Tl) and the plastic scintillators T1 andT2 [1].Once an event was triggered, the data was processed using a Data AcquisitionSystem (DAQ). A 60 MHz Analog to Digital Converter (ADC) read out the signalsfrom the CsI crystals, from each single NaI(Tl) PMT as well as from the onlinesum of the PMTs. Furthermore, it digitized the signals from the silicon detectorsin a time window of 1 ?s around the timing of the decay positron. The signalsfrom the plastic scintillators were read out by a 500 MHz Flash Analog to DigitalConverter (FADC) covering a time window of 8 ?s around the pion timing. ThisFADC also read out the online sum of the NaI(Tl) tubes providing more samples ina longer time window than the 60 MHz ADC. The MIDAS data acquisition system[16] provided a web interface to control the data taking process. It also facilitatedthe monitoring process by allowing programs to check the quality of data online.9Chapter 3Data Analysis of the PIENUExperimentAfter the data collection, the stability within the recorded variables was studied forall runs and only those with smooth data taking conditions were chosen for furtheranalysis. The procedure of extracting the branching ratio between pi ? e?e andpi ? ? ? e decays is described in this chapter.3.1 Raw Branching Ratio and Systematic EffectsThe raw signals obtained from the PMTs of each individual detector had to becalibrated in order to measure the energy deposit in MeV. The information fromthe cosmic ray triggered events was used to calibrate the CsI crystals by comparingthe data to a detailed simulation of the system. The NaI(Tl) crystal was calibratedby using the beam positron trigger, whereas the tracking detectors? energy depositwas compared to that calculated for minimum ionizing particles and to a MonteCarlo (MC) simulation. The signal time for the beam counters B1, Target (Tg) andT1 as well as that for the NaI(Tl) crystal were either obtained from a hit-findingalgorithm or from a fit of the waveform in the ADC. During the fit procedure, thewaveform was compared to a template for an average PMT signal and thus thetime of the maximum energy deposit was determined. In the special case of the Tgcounter the fit procedure was applied for one, two, and three hits. This provided10Figure 3.1: Blinding procedure: pi? e? decays were selected via the energydeposit in the target by the cuts indicated by the vertical red lines, and aninefficiency function between 0.99 and 1.00 was applied to the selectedevents.a handle to distinguish between the two hit pi ? e? and the three hit pi ? ? ? edecays [17].In order to minimize human bias, the data was modified to accommodate ablind analysis. As shown in figure 3.1, the pi ? ? ? e decay deposited an extra4 MeV of energy in the Tg compared to the pi ? e? decay mode due to the muon,which lost all of its energy and came to a stop within the Tg. By applying a cut onthe energy deposit in the target, pi ? e? events were selected, and an inefficiencyfunction between 0.99 and 1.0 was applied to this selection so that the ratio betweenthe decay modes was altered artificially at a significant level at the precision thePIENU experiment is aiming at in the end. The value of this function was randomlychosen and is not known to the scientists involved in the analysis. Therefore, theexact result for the branching ratio cannot be compared to any known values untilall of the systematic effects and backgrounds are well understood. In the end, thedata is unblinded by removing the inefficiency function [18].So as to determine the raw branching ratio, a simultaneous time fit of boththe low energy region (ENaI +ECsI < 50MeV) and the high energy region (ENaI +ECsI > 50MeV) was performed, as shown in figure 3.2. The fitting functions in-cluded the exponential decays due to the lifetimes of the muon and the pion respec-tively, as well as a number of background shapes. These backgrounds had variousorigins: in the low energy region they were mainly caused by muons sitting in thedetector material and decaying at a time that was not correlated to the incoming11Figure 3.2: Simultaneous time fit of the high (left panel) and low (right panel)energy regions including background shapes.pion and by pions decaying in flight prior to the arrival at the target. In the highenergy region on the other hand, one of the origins was the pile up of two muondecays: one muon arriving from the beam, the other one sitting in the detector al-ready. Other possibilities for pi ? ? ? e decays to have an energy above 50 MeVwere the simultaneous detection of the photon in a radiative muon or pion decaythat increased the energy deposit in the calorimeter, and the finite energy resolutionof the latter due to which some events actually located below the 50 MeV thresholdwere detected as higher energy events. While taking all of these backgrounds intoconsideration, the ratio R = pi?e?(?)pi???(?) was calculated. This was, however, the rawbranching ratio that still needed to be corrected for a number of systematic effects.One of these effects was a correction for MDIF. A positron emerging from thedecay of a pion at rest and a muon in flight can have energies above 50 MeV, andits timing is shorter than the muon lifetime. Therefore, the ratio between pi ? e?eand pi ? ? ? e decays was distorted. The difference between the energy spectrafor muon decays at rest (MDAR) and decays in flight is shown in figure 3.3. Thecorrection to the branching ratio due to pi ? ? ? e events above 50 MeV wascalculated from the probability of muon decay in flight and the proportion of theseevents above 50 MeV, which was obtained from MC. Combining these two leads toa multiplicative correction to the branching ratio of CorrMDIF = 0.9976?0.0002.A second systematic effect was due to the acceptance of the detector. There12NaI + CsI Energy [MeV]0 10 20 30 40 50Counts05001000150020002500 MDAR (Data)MDIF (MC)Figure 3.3: Energy deposit of the positron from muon decays at rest (solidblack line) versus muon decays in flight (dashed red line). The dottedblack line indicates the energy cut off at 50 MeVwere two energy dependent processes that deflected the positron after the Tg counter:Multiple scattering caused by the Coulomb force of the nucleus could deflect thepath of a positron and Bhabha scattering resulted in an electron and a positronbeing emitted in a direction different from the one the initial positron had. In ad-dition, annihilation in flight could occur before or in the trigger counters T1 andT2 and some low energy positrons did not reach T1 or T2. Consequently, theseevents did not cause a trigger. All of the processes mentioned above have energydependent cross sections which means that the effects differ for low and high en-ergy positrons, so a correction had to be applied to the raw branching ratio. It wascalculated for an angular acceptance defined by a 60 mm radius in the third wirechamber.13Finally, a tail of pi ? e? events existed in the low energy part of the spectrumbecause of low energy photons leaking out of the calorimeters, radiative photonsescaping the detector, and Bhabha scattered electrons or positrons that were notrecorded in the detector system. The correction for the tail was relatively large(at the order of ? 2%), so it was not only determined from MC but also by dataanalysis. There were two approaches in estimating this correction: firstly, an esti-mate of the low energy tail was derived from a special measurement with 70 MeVpositrons; secondly, we obtained a lower limit on the tail correction by analyzingthe PIENU data set and suppressing pi? ?? e decays. The procedure of estimat-ing this lower limit is the main work for this thesis and is described in the followingsections. Subsequently, the lineshape measurement and data analysis are explainedand a method of combining the two is presented.3.2 Lower Limit DeterminationTo estimate the fraction of pi ? e?e events below 50 MeV, the so called ?pi ? e?etail ?, a number of suppression cuts was applied to the energy spectrum whichselected mostly pi ? e? events and suppressed the large pi ? ? ? e backgroundthat was present in the low energy region. What remained after these cuts was thepi ? e? tail buried under a background of the pi ? ? ? e events that were notsuppressed. The following sections describe the procedure of determining a lowerlimit on the pi ? e? tail as well as the corrections that had to be applied.3.2.1 Suppressed SpectrumThe pi ? ? ? e events in the low energy region (< 50MeV) had to be suppressedto obtain an estimate for the tail of pi? e?e events. To do so, different observableswere extracted from the data and the different properties of pi? e? and pi? ?? edecays were exploited to place cuts on these observables and select mostly pi? e?events. To improve on the work in [1], the following cuts were studied with respectto their efficiency (ratio of high energy events before and after the cut) and the lowenergy fraction (number of low energy events divided by all events). During theoptimization process, the efficiency for each cut was maximized while minimizingthe low energy fraction and keeping enough events to have sufficient statistics.14As mentioned in section 2.1, the time spectra of pi ? e? and pi ? ? ? e de-cays differ substantially. This fact was used to set a cut on the decay time betweenthe incoming pion and the outgoing positron from the Tg scintillator. The require-ment of a short decay time of the pion mostly selected pi ? e? events, so the firstrequirement for the suppressed spectrum was: 4ns < t < 35ns, this selection re-sulted in a decrease in the low energy fraction to 98.83 %, which is 0.2 % lowerthan in reference [1], while remaining at the same efficiency.The second major difference between the two decay modes was their energydeposit in the upstream detectors B1, B2 and Tg. We call the cumulative energydeposit in these three detectors the ?total beam energy?. As shown in figure 3.4,the extra 4 MeV energy deposit from the muon clearly separated the two decays.So a cut on the total energy between 15.9 MeV and 16.8 MeV suppressed morepi ? ? ? e decays. These boundaries ensured a more symmetric cut on the totalenergy than in reference [1] and the low energy fraction was decreased by 4 % to26.55 %.The difference in the total energy spectrum was also exploited in a differentway: As mentioned in section 3.1, a fit to the waveform in the target was performedto check for a two hit (pion and positron) or three hit (pion, muon, and positron)event [1]. The ?2 for each case provided information on the goodness of the fit andserved as a handle to determine which decay occurred. The difference betweenthe values of ?2 for the two and three hit fits is shown in figure 3.5 versus thetotal energy. There is a clear separation between pi ? e? and pi ? ? ? e eventsdepending on the sign of the difference, defined as ??2 = ?22hits? ?23hits. A cuton this variable at ??2 < 0 extracted pi ? e? events, this selection differs slightlyfrom the one in [1].The cuts mentioned above removed most of the background due to decays atrest. However, some pi ? ? ? e events involved pion decays in flight (PDIF)before the target and had a similar energy deposit in the target as a pi ? e? event,but a different signature in the upstream tracking detectors. As shown in figure 3.6,events in flight entered the target at an angle. This angle was calculated from thecoordinates in the tracking detectors and is shown in figure 3.7 for PDIF and piondecays at rest (PDAR). Placing a cut for ? < 12? suppressed the decays-in-flightof pi ? ? ? e events.15Figure 3.4: Energy deposit in the upstream tracking detectors (?total energy?,solid black line) and suppression cut selecting pi ? e? events (bold redlines).However, the angle cut did not suppress the decays in flight occurring afterthe upstream tracking detectors. Since muons originating from a pion decay inflight had a higher energy than those from decays at rest, some of them leakedout of the target and reached the silicon detector S3. They had a greater energydeposit in the silicon than a positron from a pion or muon decaying in the target.Therefore, requiring a low energy deposit in the x- and y-planes of S3 accordingto E2X + E2Y < (1.2MeV)2 suppressed the decays in flight taking place after theupstream tracking detectors.Table 3.1 summarizes the cuts described above, which produced the suppressedspectrum shown in figure 3.8. For each cut, the fraction of low energy events (LEfraction) is included, defined as the number of events below 50 MeV divided by allevents.165.1. Tail correction2? ?-10 0 10 20 30 40 50 60 70 80Total Energy [MeV]1415161718192021? + e?+pi+ e? +? ?+piFigure 5.5: Contour plot of the Total Energy as a function of ??2 =?22-pulse ? ?23-pulse for pi+ ? e+?e (ENaI > 55 MeV) and pi+ ? ?+ ? e+(amplitude of the muon in the 3-pulse fit is larger than 0) events. The redhorizontal lines show the Total Energy cut. The vertical red line gives thevalue of the pulse shape cut. This figure shows events before any suppressioncuts.Table 5.1: Summary of the suppressed spectrum cuts. The low energy frac-tion represents the integral of events below 50 MeV divided by the integralof the full energy spectrum. The signal efficiencies are non-cumulative whichmeans that they are representative of the efficiency of each cut while the lowenergy fraction is cumulative. The * indicates potentially energy-dependentcuts.Cuts Low energy fraction [%] Signal efficiency [%]Time 99.0 82.85Total Energy * 30.7 76.80Kink 18.4 93.14S3 * 17.7 99.99Pulse Shape * 16.7 99.46119Figure 3.5: Energy deposit in the upstream counters (B1, B2, Tg) versus??2 = ?22hits ? ?23hits. The red lines indicate the cut on ??2 and theenergy deposit. (Figure taken from ref. [1].)WC1WC2B1S1S2Tgpi+ ?B2?Figure 3.6: Schematic drawing of a PDIF in front of the target with its decayangle compared to the straight track of a pion stopping in the target.175.1. Tail correction [deg]?0 10 20 30 40 50Normalized Counts00.0050.010.0150.020.025 events? e ?? e events????Figure 5.2: Kink angle for ?+ ? e+?e events (ENaI > 55 MeV) and ?+ ??+ ? e+ (ENaI < 30 MeV) events. The vertical red line indicates theposition of the cut.115Figure 3.7: Angle between the particle track and the beam direction in solidblack for pi? e? (ENaI > 55MeV) events, in dashed red for pi? ?? e(ENaI < 30MeV) events. The vertical red line indicates the cut placedat ? = 12?. Figure taken from ref. [1]Variable Cut LE fraction [%]Time 4ns < t < 35ns 98.83Tg Energy 15.9MeV < E < 16.8MeV 26.55Angle ? ? < 12 ? 14.06Energy in x- and y-plane of Si3 E2X +E2Y < (1.2MeV)2 12.91Pulse Shape in Tg ??2 < 0 12.87Table 3.1: Summary of the cuts used to produce the suppressed spectrum.The low energy fraction (LE fraction) is defined as the number of lowenergy events divided by the total number of events after each cut.3.2.2 Determining the Lower LimitThe cuts described above suppressed most of the pi ? ? ? e events in the lowenergy region. However, a fraction of them still remained. Therefore, the tail ofpi ? e?e events in this energy region had to be extracted. To calculate this tail, wefirst estimated the amount of pi ? ? ? e events within the suppressed spectrum18Energy [MeV]0 20 40 60 80 100Counts110210310410510TimeTotal Energy?Angle Si3 Energy^2? ?Figure 3.8: Effect of the different suppression cuts on the energy spectrum.by assuming that there was no tail present below an energy i which was lower thanthe Michel edge (i < 50MeV), and by comparing its shape to the Michel spectrumbelow this energy i. Subsequently, we subtracted this amount from the total numberof events in the suppressed spectrum. Due to the assumption of no tail below theenergy i, we obtained a lower limit on the amount of pi? e?e events below 50 MeV;the integral up to i was called a[i]. The total number of pi ? ? ? e events in thelow energy region was evaluated from the region below i by integrating the Michelspectrum up to the same energy i, this integral was called b[i]. b[i] divided by thetotal number of events present in the Michel spectrum below 50 MeV, called B, isequal to the fraction of the total number of events that a[i] should represent. a[i]divided by this ratio provided us with the total number of events below 50 MeV inthe suppressed spectrum originating from pi ? ? ? e background. By subtractingthis number from the number of events in the low energy part of the suppressedspectrum (A), we obtained a lower limit on the tail fraction, called Ll[i] [1]:Ll[i] = A - a[i] Bb[i](3.1)19low energytail50unsuppressedbackground50Energy [MeV] Energy [MeV]i ib[i]a[i]BAFigure 3.9: Evaluation procedure for the lower limit. Left: suppressed spec-trum with the low energy tail in green and the integration region a[i] inblue. Right: Michel spectrum with the integration region b[i] in blue.Dashed lines indicate integration limits. A and B are the integrals upto 50 MeV of the suppressed spectrum and the Michel spectrum respec-tively.Figure 3.9 illustrates the different integrals and the evaluation procedure. TheMichel spectrum was obtained from the data by applying the same suppressioncuts as for the suppressed spectrum, the only differences were the timing cut andthe pulse shape cut as the latter would introduce time distortions. For a sample ofpi ? ? ? e decays, t > 100ns was required instead of the early cut for pi ? e?events.After calculating the lower limit for the upper integration limit i varying be-tween 1 MeV and 50 MeV, we chose the highest of these values to avoid an un-derestimation. Figure 3.10 shows the dependence of the lower limit on the upperintegration limit i. The assumption of zero tail is a good approximation at low en-ergies, although the statistics in this region are poor. At higher energies, the actualtail is non-zero but there are more events in the spectra. By construction, the lower20Figure 3.10: Lower Limit versus upper integration limit i. At 50 MeV thelower limit is zero, as indicated by the red line. The MDIF correctionhas been applied for this figure.limit is zero at 50 MeV since a[i] = A and b[i] = B; the red line in figure 3.10 in-dicates this. Therefore, a value at an intermediate energy is the best approximationfor the lower limit. The MDIF correction described in the next section has beenapplied to the lower limit in figure 3.10.As shown in figure 3.10 the lower limit takes negative values below an upperintegration limit of ? 20MeV. This occurs because of an oversubtraction in thelower limit formula 3.1 when the pi ? ? ? e background estimate is too large.However, the effect can be explained by statistical fluctuations as outlined in sec-tion 3.2.8.The tail fraction was calculated by dividing the lower limit Ll[i] by the totalnumber of events in the suppressed spectrum and was used as a multiplicativecorrection to the raw branching ratio.215.1. Tail correctionTotal Energy [MeV]12 14 16 18 20 22 24Normalized counts020406080100120310?MDIF (MC) (MC) ?+ e?+?Figure 5.17: Total beam energy deposited by MDIF events compared to?+ ? e+?e events.selecting late muon decays while the shapes of MDIF and ?+ ? e+?e areobtained from MC. The fit range is 3 to 57 MeV in the energy spectrumand 10 to 35 ns64 in the time spectrum. The combined ?2 is indicated onthe figure. The fit gives a contamination of MDIF in the low energy tail of10.77 ?2.81% which is in good agreement with the result of the calculationmade above. MDIF is thus found to make up 1.8% ((10.77?16.7)%) of thesuppressed spectrum which is in very good agreement with an independentanalysis on data taken in 2009 dealing with a search for massive neutrinosin the suppressed spectrum. This analysis, detailed in chapter 8, found acontamination of the suppressed spectrum by MDIF of 1.7%. From ?4.9,the error of the MDIF fraction above 50 MeV (fMDIF ) is estimated to be?0.002. Therefore, the error on the fit (inflated by??2/NDF ) summedin quadrature with the error on fMDIF translates to an error of 9.0?10?4on the ?+ ? e+?e tail correction from MDIF. Finally, the upper and lowerlimit tail fraction with MDIF correction gives a tail fraction (fUL+MDIF )of:fUL+MDIF = 0.0097+0.0032??0.00262 + 0.00092 = (1.29?0.28)% (5.7)64The region closer to the prompt cannot be fitted due to the distortions introduced inthe Early region by the Pulse Shape cut.142Figure 3.11: Energy deposit in the tectors upstream of th target f r pi ?e? decay in solid black and for MDIF events in dashed red. Thedashed vertical lines indicate the cut on the energy deposit. Figuretaken from ref. [1]3.2.3 Mu n Decay in Flight Correction to the Lower LimitAs described in the previous section, we used a selection of late events to obtain thebackground shape of pi ? ? ? e decays. However, this left out the MDIF whichcould not be selected separately within the d ta b t were present in the suppressedspect um. Figure 3.11 shows the energ deposit of MDIF compared to that ofpi ? e?e events and the effect of the total energy cut.We included those events in the background estimation by applying a correc-tion to the background Michel spectrum based on an energy spectrum for MDIF ob-tained from MC. 1 In order to add the two spectra with the correct proportions, wecalculated the fraction of MDIF events compared to PDIF. A fit to the energy andtime distributions of the supp essed spectrum provided information on the comp -sition of the suppressed spectrum in terms of MDIF, PDIF and the pi? e? tail. By1Note that there are two different corrections for MDIF: The one mentioned in section 3.1 isapplied to the branching ratio to account for the fact that some MDIF events have energies above50 MeV and are therefore contributing to the ?wrong? part of the ratio. The correction described inthis section corrects the lower limit for the fact that MDIF events are not included in the selection oflate events for the background spectrum, even though they are present in the suppressed spectrum.22minimizing the ?2 for both the energy and the time distribution simultaneously, thefitting procedure determined the most likely composition of the suppressed spec-trum. The shapes used in the fit were the MDIF distribution obtained from MC,the pi ? e? tail above 50 MeV from MC and the PDIF spectrum originating fromthe late muon selection in the data used as background spectrum in the lower limitmethod.Energy in NaI+CsI [MeV]0 10 20 30 40 50 60Counts0200400600800100012001400/NDF=2? 1.52% of the tail: tail  = ? e ?pi 8.42 +/- 0.13MDIF = 8.11 +/- 2.71PDIF = 78.85 +/- 2.85?+ e?+piMDIFPDIFCombined fitTime after prompt [ns]10 15 20 25 30 35Counts0100200300400500600700800900  + MDIF?+ e?+piPDIFCombined fitEnergy [MeV]10 20 30 40 50-3-2-1012Energy Spectrum ResidualsTime [ns]12 14 16 18 20 22 24 26 28-1.5-1-0.500.511.522.5Time Spectrum ResidualsFigure 3.12: Upper panel: Simultaneous fit to the energy (left) and time(right) distributions of the suppressed spectrum. The different con-tributions of PDIF (green), MDIF (red) and pi? e? tail (black) as wellas the combined fit (blue) are shown. The percentages indicate theamount of each process within the suppressed spectrum. Lower panel:Residuals from the energy (left) and time (right) spectra.Figure 3.12 shows the result of the fit with the contributions of the differentprocesses to the suppressed spectrum. The value of the combined ?2 divided by23the Number of Degrees of Freedom (NDF) ?2/NDF = 1.52 indicates that there isstill room for optimization within this fitting procedure. Studies have shown thatthe result of the fit depends largely on the fitting range as well as the pi ? e? tailspectrum. The upper limit of the energy range used for the fit shown in figure 3.12is 57 MeV, when decreasing the upper limit to 55 MeV, the MDIF fraction takes avalue of 7.8 %, and for an upper limit of 60 MeV the fraction is 5.0 %. This in-dicates that the fit is not stable with respect to the energy range used for fitting.Additionally, the MC simulation does not reproduce the photonuclear interactionscorrectly at present time. These two issues, as well as the fact that the PDIF spec-trum has low statistics since it is obtained from the data, can cause the value of?2/NDF to be larger than one.From this fit, we extracted the ratio of MDIF to PDIF ? = 0.10? 0.03. Theuncertainty was calculated from the errors determined within the fitting procedure.Subsequently, the MDIF spectrum was added to the background spectrum withthis proportion and the resulting spectrum was used to determine the integrals Band b[i] in the lower limit formula 3.1.We checked the amount of MDIF predicted by the fit by calculating the fractionof MDIF to pi ? e? events. According to the MC simulation, 15.4 % of MDIFsurvived the Tg energy cut. The probability for MDIF is 8 ?10?6, as calculated in[1]. Assuming the theoretical branching ratio between pi ? e? and pi ? ? ? eevents of 1.24 ? 10?4, we estimated the amount of MDIF versus pi ? e? events,called Q, knowing that the Tg cut had an efficiency of 89.25 % as follows:Q = 8 ?10?6?0.1540.8925?1.24 ?10?4 = 1.11?10?2. (3.2)Since the fraction of low energy events compared to the total number of events inthe suppressed spectrum is 12.87 %, 1.11/12.87 = 8.65% of these events are dueto MDIF. This agrees well with the fraction determined from the fitting procedure.3.2.4 Radiative Decay CorrectionThe lower limit method is based on the assumption that there is no tail at lowenergies and that it emerges at intermediate energies. As described above, the24Energy (NaI+CsI) [MeV]05101520253035404550Counts 200400600800100012001400Radiative DecaysEnergy (NaI+CsI) [MeV]05101520253035404550Counts050010001500200025003000Non-radiative DecaysFigure 3.13: Low energy part of the Monte Carlo pi ? e? spectrum. Leftpanel: Radiative decays only. Right panel: Non-radiative decays only.highest value for the lower limit is chosen, which is 37 MeV. Consequently no tailis expected below this energy. However, MC studies showed that the spectrum ofpi ? e?? decays (so called radiative decays) differs from that of pi ? e? decays.In the radiative case there is an excess of events around 1 MeV and a constant tailexists below 40 MeV due to events where the photon was missed in the NaI(Tl);this tail is shown in figure 3.13. On the other hand, the tail consisting of non-radiative decays only, shown in the right panel, becomes negligible at very lowenergies. To account for this difference, an additional term was introduced to thelower limit formula (see equation 3.1), which corrected for the radiative decays.This correction was not applied in reference [1]. The non-zero component at lowenergies was described by a correction to the lower limit formula by adding a termr[i]:.Ll[i] = A - (a[i]-r[i]) Bb[i], (3.3)where r[i] is the amount of the radiative decay spectrum integrated up to the energyi. The shape of radiative decays from MC was used to determine the integral r[i];25the spectrum was scaled to the pi ? e? peak of the suppressed spectrum. Thescaling factor was then multiplied by 6.2 % since this is the fraction of radiativedecays compared to the total of pi ? e? events [2].The additional term accounted for the fact that the portion of radiative decaysbelow 37 MeV would otherwise be counted as background resulting in an under-estimation of the lower limit. After applying this correction, the shape of the tailfraction depending on the integration energy i did not approach zero at 50 MeVany more. In the limit of i= 50MeV, a[i] = A, b[i] = B, and r[i] = R, therefore thevalue of the tail fraction at 50 MeV was equal to the amount of radiative decaysbelow 50 MeV. This is indicated by the red line in figure 3.14, which implies thatthere is a contribution of 0.32 % radiative decays below 50 MeV.Figure 3.14: Lower limit shape with radiative decay correction (blue graph).Red Line: Value of tail fraction at 50 MeV.263.2.5 Bhabha CorrectionThe cut applied on the total energy deposit of the upstream detectors removedevents with an energy deposit below 15.9 MeV and above 16.8 MeV, therefore se-lecting most of the pi ? e? events. However, a portion of the pi ? e? eventsunderwent Bhabha scattering in the target. This caused additional energy to be de-posited in the detectors by the scattered electron which is why the energy deposit ofBhabha events was greater than that of non-scattered events. These Bhabha eventswere cut out by the selection cut despite the fact that they contributed to the lowenergy tail. Consequently, a correction was applied to the lower limit, which addedback the portion of Bhabha events that were removed by the total energy cut.According to an MC simulation, the correction amounts to (1.186?0.013(stat)?0.119(sys))%. Conservatively, the systematic uncertainty was estimated to be10 % of the correction itself. For details on the Bhabha correction and its cor-responding error as well as a comparison with Bhabha events selected within thedata, refer to the PIENU technical report [19].3.2.6 Error Estimation for the Lower LimitFor an estimate of the uncertainty of the lower limit, both the statistical and thesystematic error were calculated. This procedure differs from reference [1].Statistical ErrorWe estimated the statistical error on the tail fraction by assuming that the numberof high energy events (HE) is Poisson-distributed, whereas the ratios of a[i] and A,b[i] and B, and r[i] and A follow a Binomial distribution since a[i], b[i] and r[i] aresmall samples of A and B. The expression of the lower limit in equation 3.3 can berewritten in terms of efficiencies in the following way:Ll[i] = A? (a[i]? r[i]) Bb[i] = A(1? ?a?b+ ?r?b)(3.4)?a =a[i]A, ?b =b[i]B, ?r =r[i]A(3.5)27Assuming a Binomial distribution for all of the efficiencies, we obtained thefollowing uncertainty on the tail fraction TF:?TF =?(HE(Ll[i]+HE)2 ??Ll[i])2+(Ll[i](Ll[i]+HE)2 ??HE)2(3.6)Systematic ErrorThe uncertainties of the MDIF and radiative decay corrections as well as the (in)stabilityof the tail fraction with respect to rebinning introduced systematic uncertainties.Those were determined by calculating the TF for the different scenarios and takingthe largest difference within the tail fraction ?TF which was assumed to be thesystematic error.? vary MDIF coefficient ? = (0.10?0.03) within errors: ?TF = 0.07%? vary radiative decay contribution of (6.2? 0.2)% within errors: ?TF =0.02%? rebin all histograms by factors of 2,4,6: ?TF = 0.01%When summed up in quadrature, those effects resulted in a systematic error of0.07 % (absolute error on the tail fraction). This provided us with a lower limit onthe tail fraction with statistical and systematic uncertainty:TF < (1.05?0.07(stat)?0.07(sys))%Adding the Bhabha correction mentioned in section 3.2.5, resulted in:TF < (2.24?0.07(stat)?0.14(sys))%283.2.7 Consistency ChecksThe branching ratio, including all its corrections, is being studied with respect tosystematic effects before unblinding the data set. These systematic effects includethe energy cut off between the low and high energy region and the definition ofacceptance. Therefore, these studies were also performed on the lower limit on thetail fraction itself in order to apply the varied correction to the branching ratio. Theresults for the systematics checks as well as a more detailed description of studieson the lower limit are contained in a PIENU technical report [20].3.2.8 Statistical EffectsIn order to understand the impact of statistical fluctuations on the shape of the lowerlimit curve, MC simulated spectra were used to calculate the lower limit multipletimes and to compare the behaviour of the curves. The suppressed spectrum wasconstructed by adding 87 % of the pi ? e? tail to 13 % of the Michel spectrum asbackground. These numbers correspond to the percentages in the suppressed spec-trum obtained from data (compare to table 3.1). In order to introduce statisticalfluctuations, a histogram was filled randomly using the MC spectrum as a proba-bility density function. The number of entries in the data spectrum determined thenumber of times the filling procedure was repeated. This ensured approximatelythe same level of statistical fluctuation in the simulated spectrum as in the dataspectrum. This procedure was repeated 1000 times, calculating the lower limitevery time, producing a distribution of lower limit values. Figure 3.15 shows thelower limit curve for four statistically independent samples, each of which showsa different behaviour at low energies. This indicates that the decrease of the lowerlimit shape from data towards low energies can be explained by statistical fluctua-tions.3.3 Lineshape MeasurementComplimentary to the estimation of the pi ? e? tail via the lower limit method,Tristan Sullivan determined an estimate of the tail using results from a special mea-surement from September 2011 [2]. With a 70 MeV/c positron beam, the crystalresponse function was measured by directing the particles at the NaI(Tl) at various29Energy [MeV]01020304050Lower Limit -0.04-0.0200.020.04Energy [MeV]01020304050Lower Limit -0.1-0.08-0.06-0.04-0.0200.02Energy [MeV]01020304050Lower Limit -0.06-0.05-0.04-0.03-0.02-0.0100.01Energy [MeV]01020304050Lower Limit -0.03-0.02-0.0100.010.020.030.040.05Figure 3.15: Lower limit shape for four statistically independent samplesproduced from MC.angles. Only the three wire chambers and the counter T2 were used to characterizethe beam in order to reduce any scattering effects. This special setup imitated thepositrons emitted from the target at different angles and was therefore suitable toestimate the low energy tail of pi ? e? decays. Figure 3.16 shows the measuredenergy spectrum at 0 ? with selection cuts applied to choose positrons only.The bumps on the left side of the positron beam peak at 68 MeV are due tophotonuclear interactions in the NaI that resulted in one, two, or three neutronsescaping the crystal and carrying away ? 9MeV each [21]. Because of these in-teractions, the energy deposit of some positrons with an energy above 50 MeV wasmeasured to be lower since the energy deposit detected by the calorimeter wasmissing the energy of the escaped neutrons. It was therefore important to take this30Energy (NaI+CsI) [MeV]10 20 30 40 50 60 70Counts110210310410510Figure 3.16: Energy spectrum of a 70 MeV positron beam aimed at the centerof the NaI(Tl) crystal at 0 ?. Cuts are applied to select positrons only.The ?bumps? originate from one, two, and three neutrons escaping thecrystal after photonuclear interactions.effect into account when estimating the low energy tail. Some differences betweenthe actual PIENU experiment and the special run had to be considered when calcu-lating the tail fraction: since some of the detectors were removed for the lineshapemeasurement, less material was present in the path of the beam, resulting in lessdegradation in energy and less scattering effects. Also, the positron beam did notimitate the decay positrons exactly as differences arose from spatial and momentumdivergence in the beamline as well as imprecision in the setup and uncertainty inthe information from the detectors that were used for the lineshape analysis. Onlya simulation could overcome these differences. However, the escaping neutrons,which were first observed in the PIENU experiment, were not sufficiently well re-produced by the GEANT4 simulation used for this study. Therefore, the simulationwas corrected for the photonuclear interactions by studying the difference betweena simulation of the lineshape measurement and the lineshape.Comparing the lineshape spectrum at 0 ? (figure 3.16) to the spectrum at 48 ?31Energy (NaI+CsI) [MeV]0 10 20 30 40 50 60 70Counts110210310410510Figure 3.17: Energy spectrum of a 70 MeV positron beam aimed at the centerof the NaI(Tl) crystal at 48 ?.(figure 3.17) illustrates the difference in tail fraction for different incident angles.Consequently, the fraction was determined as a function of angle, which is shownin figure 3.18 for the case when only considering the energy deposit in the NaI(Tl)and when including the CsI energy as well. In the latter case, a considerable amountof leakage out of the NaI(Tl) at higher angles is detected by the CsI. The weightedaverage of the NaI(Tl) plus CsI tail fraction within the acceptance region was usedas correction to the simulation.This correction also accounted for the crystal resolution of the NaI(Tl) detectoras measured in the lineshape setup. The simulation provided us with a tail fractionof 1.93 % consisting of leakage out of the crystal, Bhabha events and radiative de-cays. The difference in tail fractions between the lineshape data and the lineshapesimulation produced a correction of + 0.32 %, resulting in a tail fraction of [2]TF = (2.25?0.06)%.32Angle (deg)0 10 20 30 40 50Tail Fraction <50 MeV (%)024681012NaI(Tl)+CsINaI(Tl)Figure 3.18: Lineshape tail fraction estimate as a function of angle for theenergy deposit in the NaI(Tl) only, and for the combination of NaI(Tl)and CsI.Taking into account the scattering in the beamline and momentum dispersion, thisestimate could be considered as an ?upper limit? of the tail fraction. However, asimulation of the M13 beamline using the program g4beamline was used to studythe scattering effects. Before entering the PIENU detector the fraction of positronsbelow 52 MeV was found to be 0.03 % for a radial cut of 22 mm, which is the sameas the selection cut used when analyzing the lineshape data [22]. Since this value issmaller than the uncertainty on the tail fraction estimated from the lineshape mea-surement, the scattering effects in the beamline are considered negligible. How-ever, this should still be confirmed by analysis of special runs that were taken tocharacterize the beamline effects.3.4 Combination of Lineshape Measurement and LowerLimitThe lineshape measurement and the lower limit were combined into one estimateof the low energy tail fraction. Taking into account their respective probabilityfunctions, we obtained a combined estimate for the probability function of the tail33fraction [23]. In the previous chapters we calculated the values of the tail fractionand the lower limit (TF and LL) with their corresponding uncertainties:TF = (2.25?0.06)% (3.7)LL = (2.24?0.07(stat)?0.14(sys))% = (2.24?0.14)% (3.8)We assumed that the estimated uncertainties for both values are distributedaccording to a Gaussian function. Consequently, the value and its uncertainty cor-respond to the mean x? and the variance ? of a Gaussian distribution:f (x) = e? 12( x?x?? )2(3.9)Tail Fraction [%]1.61.822.22.42.62.83Normalized Amplitude00.20.40.60.81Figure 3.19: Probability distributions of the tail fraction estimate from thelineshape measurement (dotted blue line) and the lower limit (solidred line) assuming they have a Gaussian shape.The distributions for both measured limits are shown in figure 3.19. The prob-ability of the true value to be x corresponds to f (x) ? x. The physical meaning of34the lower limit is that the tail fraction should take a value higher than the lowerlimit itself. Since the probability for the true value to take the value x accumulatesfrom all of the lower values, the integral up to x corresponds to the acceptance atthat point. Therefore, we calculated the probability region for the lower limit viathe error functionf (x) = er f (x) = 2?pi? x0d?t2dt. (3.10)Substituting?2t = x?x?? , this corresponds to the integral of the Gaussian dis-tribution, up to a normalization constant. The allowed regions for the lineshapeestimate and the lower limit are shown in figure 3.20 in blue and red respectively.Tail Fraction [%]1.61.822.22.42.62.83Normalized Amplitude00.20.40.60.81Lower Limit allowed regionTF estimateCombined regionFigure 3.20: The allowed region for the tail fraction obtained from lineshapedata, estimated by a Gaussian distribution is shown in blue. The regionfor the lower limit is obtained from the error function and indicated inred. The black shaded region represents the combined allowed regionof the tail fraction.We took the product of these two allowed regions to find the combined proba-bility function of the tail fraction. The latter is shown in figure 3.20 in black. The35statistical analysis of the moments of this probability function served as a handleto find the mean and variance of the distribution. The first and second moment (?1and ?2) of a probability distribution within the interval [a,b] are defined as?1 =? bax ? f (x)dx (3.11)?2 =? bax2 ? f (x)dx (3.12)With these definitions, the mean x? corresponds to the first moment ?1 and thevariance is defined as ?2 = ?2? ?2 . Using the combined probability function ofthe tail fraction as f(x), we obtained the following result: 2TF = (2.25137?0.05996)% (3.13)2More than the standard significant digits are printed to indicate the difference of the combinedvalue from the tail fraction obtained from the lineshape measurement.36Chapter 4Muon Capture Experiment4.1 Introduction and Theoretical BackgroundThe LHC offers a unique opportunity to search for new hypothetical particles. Atthe energy scale of heavy resonances, it is the most promising accelerator in thesearch of parity-violating processes. Nevertheless, weakly coupled particles withlow mass might be missed at the high energies and luminosities present at theLHC. Therefore, complimentary experiments at low energies with high intensityare crucial in order to cover this part of the energy scale [13].One of the puzzles brought to light by low energy experiments is the dis-crepancy in the charge radius of the proton rp. It was recently measured to berp = 0.84184(67) fm on a muon-proton system by determining the value fromthe muonic hydrogen Lamb shift [12]. The currently most accurate value fromelectron-proton systems, published by the CODATA compilation of physical con-stants, rp = 0.8768(69) fm [24], reveals a striking difference of five standard devi-ations compared to the measurement in the muon-proton system. This discrepancycould be due to missing elements within the SM or experimental mistakes, howeverit could also bring to light New Physics (NP).Another unresolved issue concerns the anomalous magnetic moment of themuon ?? . The theoretical calculation of this observable deviates from the valueobtained by experiments; in the E821 experiment at Brookhaven National Labora-tory the precession of ?+ and ?? in a constant external magnetic field was studied37while the muons were circulating in a storage ring. Combining their values into anaverage and assuming CPT invariance resulted in ?exp? = 11659208.9(5.4)(3.3)?10?10, where the first error originates from statistics, the second one from sys-tematic effects [11] 1. The theoretical value on the other hand, is calculated tobe ?SM? = 116591802(2)(42)(26)? 10?11 combining electroweak, hadronic, andQED contributions. e+e? data provides a value for the hadronic vacuum polariza-tion and the errors in the quoted SM value are due to electroweak, lowest-orderhadronic, and higher-order hadronic contributions [25]. The difference betweentheory and measurement of the muon anomalous magnetic moment is 3.6? .As the two examples above illustrate, muon physics presents a field of interestwhen studying discrepancies between predictions by the SM and tests with exper-iments. Consequently, muons are well suited for studies probing new physics in-teractions and hypothetical particles. As discussed in reference [26], such physicsbeyond the SM might originate from a force carrier in the MeV energy range orlower, preferentially coupling to muons.A new light vector mediator seems promising when comparing the hierarchyof rp values obtained from atomic hydrogen and deuterium, e-p scattering andmuonic hydrogen, and when considering the data on neutrino scattering in the lowMeV energy range as well as neutrino oscillations [26]. If we divide the newinteraction into couplings with left and right handed SM fermions, the left-handedfermion interaction can be excluded since it involves a neutrino field and we knowfrom experimental data that no new interactions between neutrinos and electrons ornucleons occur at a level higher than the Fermi constant. This leaves only the right-handed fermion current. However, parity non-conservation tests in the electronsector exclude large neutral right-handed currents for electrons, therefore the mostpromising candidate is a vector particle coupling to right-handed muons [26].Focusing on a newU(1)R gauge symmetry, this leads to parity-violating muon-proton neutral current interactions. McKeen and Pospelov consider a low-energyeffective neutral current Lagrangian in reference [13] as follows:1The latest value for the absolute muon-to-proton magnetic ratio was used to determine ?exp? asfor citation in [25].38L =LSM +LNP (4.1)LSM =?GF2?2?????5?(gnn???n+gp p??? p) (4.2)LNP = ?????5?4pi?gNP?m2V +(gNPn n???n+gNPp p??? p)(4.3)Both SM and NP contributions are included in this Lagrangian, and the vectorcouplings to nucleons take the values gn =?12 , gp = 12?2sin2 ?W , where ?W is theWeinberg angle; = ? ??? is the d?Alembert operator. When studying this model,the most freedom in the parameter space corresponds to a mass of the mediatorgauge boson of mV ' 30MeV. Assuming this value for mV and fitting to the protoncharge radius, the strength for the muon proton interactions is at the order of [13]4pi?gNP? gNPpm2V' 2?10?5(30MeV)2  GF (4.4)A possible way to test the manifestation of equation 4.1 is to study the AtomicRadiative Capture (ARC) of ?? into medium Z atoms. In particular, the directradiative capture of a muon into the 2S state is of interest: ??+Z? (??Z)2S+ ? .The level diagram for a typical muonic atom is shown in figure 4.1. If we measurea non-zero expectation value of a pseudoscalar observable, we have an indicationof a neutral current. Experimentally, this can be realized by measuring the angulardistribution of photons emitted during the atomic transitions. The customary ter-minology to describe the state of the photons is as follows: the photon has a totalangular momentum j, composed of the spin s and the orbital angular momentuml, which can take the values 1,2,3,.... For each value of j, two states exist distin-guished by their parity of either (?1) j or (?1) j+1. In the first case, the photon iscalled an electric 2 j?pole photon (E j), in the second case it is called a magnetic2 j?pole photon (M j) [27]. The observable we can detect in an experiment is pro-portional to the ratio between the transition amplitudes of two states. Therefore,choosing a transition where the parity-allowed process is suppressed will increase39the chance to detect a parity-odd process. Due to the selection rule for the orbitalangular momentum l, ?l = ?1, the parity-allowed transition from the 2S state tothe 1S state via a magnetic dipole transition (M1) is suppressed because of the or-bital quantum number. However, the transition of the 2P state into the 1S stateoccurs via a normal electric dipole transition (E1) with a change of ?l =?1. Con-sequently, the 2S state is preferred for an experiment with focus on observing aparity-odd process [28].E1E1nP2S1S2P E1,M1Figure 4.1: A level diagram of a typical muonic atom. Transitions betweenthe 2S and 2P state to the 1S state are shown as well as those from higherlevels (nP).So far, the single photon transition from the 2S state to the 1S state has not beenmeasured in any muonic atom. This is mostly due to the small branching ratio ofthe one photon decay of the 2S state in light elements and the fact that the 2P-1Stransition is by far more dominant. In atoms with higher charge number (Z ? 30) itis considerably easier to distinguish between the 2S-1S and 2P-1S transitions as theenergy difference between the 2S and 2P states is substantially larger. However,there is a high background due to transitions from higher levels into the 1S state,the so called cascade process, which occurs for muons at rest. Therefore, the 2S-1Stransition has not been observed until now in heavier atoms either [13].Consequently, McKeen and Pospelov suggested an experimental setup to ob-serve the 2S-1S transition which was slightly different from the ones used thus far.In contrast to previous attempts where the incoming muons were stopped com-pletely, their approach was to use thin targets with Z ? 30 in which the muons are40only slowed down, decreasing the probability for muonic cascade processes con-siderably. Some of the muons will be directly captured into the 2S state, emittingphotons in two steps [13]:???+Z? (???Z)2S1/2 + ?1 (4.5)2S1/2? 1S1/2 + ?2 (4.6)The notation ??? indicates a longitudinally polarized muon, ?1 and ?2 are thetwo emitted photons. The energy of ?1 is the sum of the kinetic energy of themuon and the binding energy of the atom in the 2S state. The energy of ?2 isapproximately 2 MeV, and its angular distribution represents the expectation valueof the pseudoscalar observable exhibiting parity violation [13].The cross section of the ARC process into the 2S state is calculated analo-gously to electron-nucleus photorecombination, derived from the hydrogen-likephotoelectric ionization cross section ? (0)PE , using a dipole approximation and as-suming a point-like nucleus. With the dipole approximation, the probability of thetransition from one state to a lower state with emission of a photon is estimated.The electrons are assumed to be non-relativistic and the interaction between theradiation field and the electron is estimated as a small perturbation. Furthermore,it is assumed that the electrons are at a distance from the nucleus at the order ofthe Bohr radius (? 10?8 cm), so that the wave number of the emitted light (forvisible light ? 105 cm?1) is considerably smaller than 10?8 cm. This leads to anexpression of the matrix element similar to dipole radiation, hence the name ofthe approximation. The angular distribution of dipole radiation is proportional tosin2(?) where ? is the angle between the dipole moment and the direction of ob-servation [29]. When accounting for the finite nuclear charge radius and departingfrom the dipole approximation, the cross section takes the form [13]:41?ARC =2?2p2?PE (4.7)?PE = ?(p,Rc,Z,n, l)?? (0)PE (nl) (4.8)? (0)PE (2S) =214pi2?a2E423?4[1+ 3E2?] exp(? 4pa cot?1 12pa)1? exp(?2pi/pa) , (4.9)where p represents the incoming muon momentum, E2 is the binding energy of amuon in the 2S state, ? is the Bohr radius, ? = p2/2m? +E2 is the energy of thephoton emitted during the ARC process, a= (Z?m?)?1 and n and l are the princi-pal and orbital quantum number respectively. The factor in front of the photoelec-tric ionization cross section ? is obtained by numerically solving the Schroedingerequation under the assumption that the muon is moving in the field of the nucleuswhich has a uniform charge distribution of radius Rc. The capture cross sectiondecreases for increasing momentum, consequently relatively slow muons are re-quired for an experiment with a high signal rate. However, at very low energies,the background process of a capture at rest dominates, so the trade off betweensignal and background has to be considered when choosing the muon momentumfor an experiment.The probability for the ARC process into the 2S state PARC,2S depends on theinitial and final momentum of the muons passing through the target, pmin and pmax,as well as the number density of the target material n as follows [13]:PARC,2S =? pmaxpmindpn ??ARC,2S|dp/dx| . (4.10)The emission rate dN2S?1Sdt of the next step involving the 2S-1S transition and emit-tance of the second photon ?2 is estimated from the probability for the captureprocess (4.10), the muon flux ??? and the branching ratio for single photons in the2S state Br1? [13]:42dN2S?1Sdt= PARC?Br1? ???? , (4.11)for Z ? 30, Br1? is given by (4.12)Br1? ??2S?1S+1??2S?2P? 2?10?3. (4.13)Given the specifics of an experimental setup concerning the target and muon flux,the emission rate of the two photons can be calculated.4.2 Experimental RealizationThe experimental realization of observing the ARC process described above re-quires a beam of polarized muons and an adequate detector system which canmeasure the energy of the emitted photons and their timing as well as their spa-tial distribution. The PIENU experiment at TRIUMF provided a high resolutioncalorimeter for the energy measurement and tracking detectors to characterize thebeam and set up an appropriate trigger. In addition, the M13 beamline delivered notonly pions, but also muons. However, these muons had a low level of polarizationsince they were mostly produced as cloud muons [30] near the production target.Therefore it was impossible to use polarized muons at TRIUMF with the currentsetup. In spite of that, the first part of the ARC process involving the emittanceof the photon ?1 could be realized with the existing equipment. This partial studyis of high interest since the direct capture of a muon into the 1S or 2S state of anatom has not been observed yet. Therefore, we collected data in a special run inDecember 2012 in order to study this direct capture process.For this run, we added a medium atomic number target in the detector systemclose to the NaI(Tl) crystal, which measured the energy of the emitted photon. Aplastic scintillator in between the new target and the crystal served as veto counterto reject any events where charged particles were emitted from the target in order toselect only neutral particles. The components of the PIENU detector are shown infigure 2.4. The cosmic ray trigger from the PIENU experiment vetoed cosmic ray43muons. Since a gaseous target involves a considerable amount of equipment anda complicated handling procedure, a metal foil was more suitable for the PIENUdetector setup than the krypton target suggested in reference [13]. The chargenumber of zirconium ZZr = 40 is close to that of krypton ZKr = 36, which is why azirconium foil was chosen as target.The cross section of the radiative capture into the 1S state is approximately fivetimes larger than that into the 2S state, so the main focus of the experiment wasto detect photons emitted during the 1S capture. Figure 4.2 shows the ARC crosssection into the 1S state of zirconium versus the muon momentum, as calculatedby M. Pospelov and A. Fradette [31].Momentum [MeV/c]30 32 34 36 38 40 42 44Cross section [cm^2]0.30.40.50.60.70.80.911.1-2710?Figure 4.2: Cross section of ARC into the 1S state of zirconium versus muonmomentum, calculated by M. Pospelov and A. Fradette.By tuning the M13 beamline to negative muons and adjusting the momentum,all of the requirements for the direct capture process were met. The experimentalprocedure and data analysis are described in the following sections.444.2.1 BeamlineA full GEANT4 simulation of the PIENU detector existed already, including aparameterized pion beam entering the first detector element and then simulatingthe tracks of particles through all of the different elements until they reached thecalorimeters. This simulation was used in studies for the muon capture experiment.Changing the incoming particle beam from pi+ to ?? allowed the study of a muon?senergy deposit in the detector elements. This study showed that a Zr target wasbest placed in front of the third wire chamber (WC3) (see figure 2.4). With anincoming muon momentum of (72.0 ? 0.4) MeV/c, the particles were degraded to(37 ? 3) MeV/c when entering the new target, as shown in figure 4.3. At this lowmomentum the cross section for the direct capture process was larger than at highermomenta and most muons still had sufficient kinetic energy to traverse the targetand potentially undergo the muon capture process followed by the cascade. With ahigher beam momentum of (74.0 ? 0.4) MeV/c the muons? momentum decreasedto (44 ? 2) MeV/c at the entrance to the Zr foil, as shown in figure 4.4. At thismomentum, the probability for both a capture at rest and a direct radiative capturewas smaller, resulting in less background but also a smaller signal cross section.We took data with both momentum settings, as discussed in section 4.2.4.For the collection of data in December 2012, the fields of the bending magnetswere reversed to guide negatively charged particles through the beamline insteadof the positively charged particles used in the PIENU experiment. As describedin section 2.2, a degrader was placed in the beam producing a spread between thedifferent particle kinds after the second bending magnet. Therefore, the collimatorafter this magnet was placed in the correct position to select muons instead of pions.Furthermore, the strength of the magnetic fields in the quadrupoles and bendingmagnets was adjusted for a momentum of 72 MeV/c or 74 MeV/c respectively. Bytuning the slits and absorbers, we focused the beam in the center of WC1 where itentered the detector.4.2.2 DetectorThe only modification to the PIENU detector system was the addition of a newtarget where the radiative capture took place. During the preparation of the ex-45Momentum [MeV/c]0 10 20 30 40 50 60Counts0100200300400500600700800900Figure 4.3: Momentum distribution of muons entering the Zr foil for a beammomentum of 72 MeV/c, obtained from the MC simulation.periment and the studies with the GEANT4 detector simulation, several differentmaterials for a possible target were studied. Comparing molybdenum (Z = 42),copper (Z = 29) and zirconium (Z = 40) showed that zirconium would best meetthe requirements of momentum degradation with an acceptable spread in the mo-mentum distribution. To determine the thickness, the momentum before and afterthe zirconium was obtained from the simulation. A thickness of 0.25 mm degradedthe incoming 37 MeV/c muons by ? 3 MeV/c in momentum. A zirconium foilwith this thickness was placed in front of WC3, as shown in figure 4.5. Figure 4.6shows a photograph of the detector elements upstream of the third wire chamberwith and without the zirconium foil.4.2.3 TriggerThe type of event we were interested in was an incoming muon absorbed in thezirconium, where the radiative capture occurred and only a photon was emitted.46Momentum [MeV/c]0 10 20 30 40 50Counts0100200300400500600Figure 4.4: Momentum distribution of muons entering the Zr foil for a beammomentum of 74 MeV/c, obtained from the MC simulation.Consequently, we required a charged particle to pass through all of the detectorcomponents in front of the zirconium, namely B1, B2, Tg and T1. In addition,we rejected any charged particles after the zirconium. The only counters presentafter the foil were T2 and WC3, so they both would be used as a veto counter. Thespecial physics triggers for the PIENU experiment described in section 2.3 wereturned off, only the electron and cosmics triggers were used in order to calibrate thescintillators and calorimeters for the data analysis and reject cosmic muon events.4.2.4 Data TakingWhile tuning the beam and monitoring the trigger rates in the various counters, wenoticed that the majority of ?? selected with our trigger condition stopped in frontof the T2 counter when entering the detector system at 72 MeV/c. Most likely, thiswas due to the momentum spread and low energy tail of the muons (see figure 4.3).Since the stopping muons increased the chance of a muon capture at rest followed47WC1WC2B1B2S1S2TgT1S3WC3T2CsI ringCsI ringNaI(Tl)???ZrFigure 4.5: Diagram of the detector including the zirconium foil in front ofthe third wire chamber (WC3). (Schematic drawing, not to scale)by the cascade process and therefore enhanced the amount of background, we tookdata in this condition for only 22 h, with a muon rate of 8.4? 103 Hz. A secondset of data was taken at 74 MeV/c for 16 h with a muon rate of 6.8? 103 Hz andsignificantly fewer muons stopped in the zirconium target. Nevertheless, this wasa trade-off between a high background rate and a high cross section of the radiativecapture process since the latter increases with lower momentum, as shown in figure4.2.The simulation studies had shown that 0.71 mm of Mylar degrades the momen-tum by the same amount as 0.25 mm of zirconium. Mylar was therefore used totake background data. For 22 h a Mylar foil was placed in front of WC3 instead ofthe zirconium and we took data with a 74 MeV/c beam.4.3 Data AnalysisThe signal we were aiming to detect was a gamma ray originating from the directradiative capture. Its energy spectrum in the NaI(Tl) crystal was simulated withGEANT4 to gain knowledge about what to expect within the data. In order to48Figure 4.6: Detector parts downstream of the scintillator counter T1 which isthe outermost component seen in the picture. The left panel shows thedetector without the zirconium, the right panel shows the zirconium foilattached to T1.extract the capture photons, the first requirement was to calibrate the energy outputfrom the NaI(Tl) crystal. After that, selection cuts were applied to suppress thebackground mostly due to the muon capture at rest. Since the largest differencebetween the radiative capture and the capture at rest is their timing, sufficient timeresolution from the NaI(Tl) was necessary. Furthermore, neutrons originating fromthe background process had to be identified and suppressed.To avoid bias, a blind analysis was performed and the background was pre-dicted from the blinded energy spectrum. Comparing the background to the simu-lation of the signal process assessed whether or not the signal would be detectable.49Finally, a procedure to extract the signal from the background was developed andsteps to take after the unblinding were planned. The following sections describethe procedure of data analysis.4.3.1 Simulation of the SignalWe obtain the energy of the ARC photon by summing up the muonic binding en-ergy of the excited state and the kinetic energy of the muon. For zirconium, themuonic binding energies of the 1S and 2S states are the following [32]:E1S = 3643keV (4.14)E2S = 1021keV (4.15)Due to the larger cross section of the 1S state, we focused on the signal producedby the capture into this state only. The kinetic energy of the muons was determinedfrom the GEANT4 simulation. Figures 4.3 and 4.4 show the momentum distribu-tions for muons entering the zirconium foil at the two different settings. Due to thewidth of the distributions and the energy loss within the foil, the cross section ofthe ARC capture varied within the zirconium. To account for this, the zirconiumfoil was segmented into five slices in the beam direction. In a simulation withmuons entering the detector at the two different momentum settings, the kineticenergy distributions for each of the slices were determined. Based on these and the1S binding energy, the energy distributions for emitted photons were calculated.In a second simulation, photons departing from within each of the slices accordingto the determined energy distribution were simulated. The azimuthal angle hada flat distribution between zero and 2pi , whereas the distribution of the photons?polar angle was obtained from the cross section calculated by M. Pospelov and A.Fradette and is shown in figure 4.7 for both momentum sets. The distribution canbe explained by the fact that not only the electric dipole transition E1 contributesto the cross section, but also higher order transitions are taken into account. Whenonly considering E1, the distribution of the polar angle is proportional to sin3(?),which is similar to the electric dipole radiation mentioned in section 4.1. However,as the photon is traveling with momentum p, also transverse components of other50wave functions contribute and the angular distribution departs from the sin3(?)dependence and is a function of the momentum. Consequently, the polar angle dis-tribution shown in figure 4.7 takes a slightly different shape for the two momentumsettings. Polar Angle [rad]00.511.522.53Normalized Probability00.10.20.30.40.50.60.70.80.9Figure 4.7: Probability of the ARC photon being emitted at a certain polarangle. The red line is for the 74 MeV/c momentum set, the blue line forthe 72 MeV/c set.From the photon simulation, the energy deposit in the NaI(Tl) crystal was ex-tracted for events with no hit in the T2 counter. It is shown for the five differentslices in the left panel of figure 4.8 for the 72 MeV/c momentum set. These fiveenergy distributions were then added together, weighted by their respective crosssections and the resulting energy distribution is shown in the right panel of figure4.8. Figure 4.9 shows the energy slices as well as the total signal energy in theNaI(Tl) for the 74 MeV/c momentum set. Based on the results from the simula-tion, the signal energies from the 1S state expected for the two momentum sets aresummarized in table 4.2.51Energy in NaI [MeV]510152025Counts051015202530Slice 1Slice 2Slice 3Slice 4Slice 5Energy in NaI [MeV]0510152025Counts020406080100120Figure 4.8: Left Panel: The kinetic energy of the muons in the five slices ofthe Zr foil for the 72 MeV/c momentum set. Right panel: All slicesadded together, weighted by their cross section.Energy in NaI [MeV]510152025Counts051015202530Slice 1Slice 2Slice 3Slice 4Slice 5Energy in NaI [MeV]510152025Counts020406080100120140Figure 4.9: Left Panel: The kinetic energy of the muons in the five slices ofthe Zr foil for the 74 MeV/c momentum set. Right panel: All slicesadded together, weighted by their cross section.4.3.2 Energy CalibrationThe data from the electron trigger was used for calibration purposes. This triggerselected beam electrons which entered the NaI(Tl) crystal with an energy corre-sponding to the beam energy minus the deposit in the detector components beforethe crystal. The spectrum from the data was compared to a simulation of e? pass-ing through the PIENU detector including the zirconium foil. For comparison of5272 MeV/c beam 74 MeV/c beam1S state 9.7 ? 1.1 MeV 12.3 ? 1 MeVTable 4.1: Signal photon energies for the two beam momenta for the 1S state.the two spectra, they were both fitted to the Crystal Ball function, which combinesa Gaussian with an exponential function as follows:f (x;?,n, x?,?) = N ????exp(? (x?x?)22?2), for x?x?? >??A ?(B? x?x??)?n , for x?x?? ???(4.16)A =(n|?|)n? exp(?|?|22)(4.17)B = n|?| ? |?| (4.18)N is a normalization constant, x? corresponds to the mean of the Gaussian part and? represents its width. The variables ? and n describe the ratio of the exponentialpart to the Gaussian portion and the decay constant of the exponential.By comparing the mean values obtained from the fit, the optimum calibrationfactor was determined. Figure 4.10 shows the energy spectra from MC and thedata with the fit after the calibration procedure for the 72 MeV/c data set. Themean value determined from the fit is ? = 65.8MeV/c for both spectra, so thedata is correctly calibrated. The same procedure was repeated for the 74 MeV/cdata set to identify its calibration factor.4.3.3 Selection CutsAfter calibrating the data, we applied a number of selection cuts ensuring that onlyevents with one incoming muon were considered that did not reach the counter T2.First, events triggered by one of the calibration triggers were rejected to reduce thecontamination of electrons. Then, a pile up cut was applied requiring one hit onlyin the upstream counters B1, B2, Tg and T1 as well as no hits in these countersbefore the incoming particle caused the trigger. In addition, no hits were allowed53Energy in NaI [MeV]0 10 20 30 40 50 60 70 80 90 100Counts02004006008001000 = 65.082139 +/- 0.020920 MeV?Energy in NaI [MeV]0 10 20 30 40 50 60 70 80 90 100Counts020406080100120140160180200220 = 65.078036 +/- 0.037606 MeV?Figure 4.10: The comparison of a 72 MeV/c e? beam in the detector simula-tion (left panel) with the calibrated electron triggered data (right panel)is shown in black. The red dashed line is a fit to a Crystal Ball functionwhich results in a mean of ? = 65.8MeV/c in both cases.in WC3 or the T2 counter, and only one hit was allowed in the NaI(Tl) crystal sincethe signal only contains one photon. These cuts ensured that one particle traversedthe detector up to the counter T1 and no charged particle was detected after it,but one hit in the NaI(Tl) crystal was measured. The condition of only one hit inthe NaI(Tl) might have killed potenial events where the muon underwent the ARCprocess into the 1S state and was subsequently captured by the zirconium. Thiscase remains for further studies.Furthermore, the incoming particle could be identified by its energy deposit inthe upstream counters and the TOF with respect to the cyclotron RF timing. Figure4.11 shows the charge collected by one PMT of the B1 counter versus the TOF forthe 72 MeV/c data set. The three most populated areas in the plot correspond topi?, e? and ?? particles. The energy loss per length is the largest for pions, so theycorrespond to the left area with the highest charge deposit, electrons deposit theleast energy and muons range in between the two. Therefore, the mostly populatedarea to the far right corresponds to the muons within the beam. It was selected bya cut on the charge 130 < Q < 330 (ADC channels) and the TOF 25 < tTOF < 32(ns) as indicated by the red box in figure 4.11. Similarly, the charge deposit of54Figure 4.11: Charge collected by one PMT of the B1 counter versus the TOFwith respect to the cyclotron RF timing. The red box indicates the cutsselecting muons.?? for the other PMTs and the remaining counters was determined and cuts wereplaced accordingly. Analogously, the selection cuts for the 74 MeV/c data set wereplaced.4.3.4 Background from Muon Capture At RestAfter selecting events with the signature of the ARC process, events resemblingthis process but originating from backgrounds had to be suppressed. The domi-nating background process was the muon capture at rest into a higher excited statewhich is followed by the cascade into the 1S state and the capture of the ?? intothe nucleus:??+N(A,Z)? ?? +N(A,Z?1) (4.19)55When the muon is captured in the nucleus, the muon and a proton combine toform a neutron and a muon neutrino, reducing the proton number Z of the nucleusby one. Usually, photons, neutrons, and charged particles (protons) are also pro-duced during this capture process [33]. The photons are emitted during the lowerlevels of the cascade process or originate from radiative muon decay. They couldeither reach the NaI(Tl) or undergo the photo-electric effect, pair-production orCompton scattering producing electrons which were rejected by the trigger condi-tion. The low energy protons most likely did not exit the zirconium foil due to theirhigh energy loss rate and would also have been vetoed by the trigger. The neutronsare caused by direct or evaporative processes, their energy ranges from very lowenergies to some 50 MeV. However, their emission after a muon capture is not verywell understood and described by theoretical models yet and data is only availablefor a limited number of materials [33]. One theoretical model predicts resonancestates analogously to photonuclear giant-resonance states, and broad energy peaksat low energies are also observed within data from muon capture in 12C and 16O[34]. The multiplicity of neutrons emitted in one capture process ranges from zeroto three neutrons [33]. In the ARC experiment, any charged particles were rejectedby the trigger and selection cuts; neutrons and photons however remained. There-fore, this background was reduced by distinguishing photons from neutrons usingthe pulse shape in the NaI(Tl) and by selecting photons only (see section 4.3.7).The lifetime of the capture at rest in zirconium has been measured to be ? =(110?1)ns [35], so it could be further suppressed by applying a time cut since theARC process occurs promptly.4.3.5 Time ResolutionSince the timing was important to reduce the background detailed in the sectionabove (4.3.4), it was crucial to have a good time resolution in order to place a cutdirectly around the prompt time. Data from the PIENU experiment was used tostudy the time resolution. First, the online sum of all the NaI(Tl) PMTs read outby the 500 MHz digitizer mentioned in section 2.3 was analyzed. An algorithmidentifying hits in the pulseshape of the PMTs attributed a time to each hit withinthe crystal, and the time spectrum of the first hit was studied in this first study.56Events within the Michel spectrum were selected by requiring the prescaled trigger(see section 2.3) and a decay time t > 100ns. Since these decays range between0 MeV and 50 MeV, they offered a good data sample for a wide range of energiesincluding the signal region of interest. For slices of 2 MeV each, the time of thedecay positron was determined and a Gaussian was fitted to the time distributionfor this energy selection. The mean of the Gaussian corresponds to the positrontime, its width represents the resolution at that particular energy. The resolutionversus energy is plotted in figure 4.12 in the left panel, whereas the right panelshows the positron time. The time resolution around 10 MeV was ? 18ns.Energy [MeV]0 5 10 15 20 25 30 35 40Resolution [ns]0510152025Energy [MeV]0 5 10 15 20 25 30 35 40Mean of Gaussian fit [ns]200210220230240250260270280Figure 4.12: Time resolution (left panel) and positron time (right panel) de-termined from the hit finding algorithm for the 500 MHz digitizer. Theuncertainty on the resolution is obtained from the error of the Gaus-sian fit and the error bars are very small. The width of the Gaussiandistribution was used as estimate for the uncertainty of its mean.In a second study, a different time variable was studied with respect to its res-olution. As mentioned in section 3.1, a fit to the waveform was applied to thePIENU data based on a template. The input for this fit was the waveform from the60 MHz digitizer for the online sum of the PMTs. Studying its resolution with thesame method described above resulted in a time resolution of? 16ns at the energy57region of interest. This was an improvement of 2 ns compared to the resolution ofthe time determined from the hit finding algorithm of the 500 MHz digitizer. Sincethe muon capture at rest process follows an exponential decay with a lifetime of110 ns, ? 24% of the decays have occurred after 16 ns. Consequently, for morebackground suppression, a better time resolution was needed to further restrict theregion of the prompt time.That is why a different fitting procedure was developed for the waveform inthe NaI(Tl) crystal. It is not based on a template but on a fit to the Crystal Ballfunction, defined in equation 4.16. This fitting function was applied to the sum ofthe PMTs read out by the 500 MHz digitizer, which is the same waveform used inthe first study. An example of such a waveform is shown in figure 4.13, also shownis the fit to the Crystal Ball function. The mean of the Gaussian part of the CrystalBall function, x?, was used as definition for the pulse timing.Time [ns]6600 6800 7000 7200 7400 7600 7800Charge in arbitrary units0510152025303540Figure 4.13: Waveform produced by a particle in the NaI(Tl) crystal (solidblack line) shown with a fit to the Crystal Ball function (dashed redline).58This fitting procedure was used on the pi ? ? ? e data set mentioned abovefor the resolution study which was performed similarly for the new fit. Figure 4.14shows the time resolution versus energy when using the Crystal Ball function fit.For this fitting routine, the time resolution in the signal energy region was ? 4ns,which is an improvement of a factor of four compared to the previous fit and thelower digitization frequency. Therefore, this new procedure was used to analyzethe muon capture data and to place a tight time cut around the prompt as describedlater in section 4.3.9. The prompt time varied within the resolution of? 2ns around20 MeV and was quite stable at the region of interest around 10 MeV.Energy [MeV]0 5 10 15 20 25 30 35 40Resolution [ns]-20246Energy [MeV]0 5 10 15 20 25 30 35 40Mean of Gaussian fit [ns]160165170175180185190195200Figure 4.14: Time resolution versus energy for Crystal Ball function fit (leftpanel) and positron time determined from the mean of the Gaussian(right panel). As in figure 4.12, the uncertainty on the resolution isobtained from the error of the Gaussian fit and the error bars are verysmall. The width of the Gaussian distribution was used as estimate forthe uncertainty of its mean.4.3.6 Blind AnalysisFor the remaining analysis, we applied a blinding technique to minimize bias. Thesignal region was concealed at the expected energy, taking into account a spread of59one ? and adding an additional 2MeV region to both the lower and upper border.After this blinding, we could study the whole time spectrum and investigate theenergy regions above and below the expected signal. In addition, we analyzed thedata by blinding the prompt region in a window of 12 ns, corresponding to three? for a time resolution of 4 ns at 10 MeV. With this blinding, we could study thewhole energy region without the prompt photons from the ARC process. In thefollowing sections, these two differently blinded data sets will be referred to as?energy-blinded? and ?prompt-blinded? respectively.First, we focused on the 72 MeV/c momentum set since we expected morebackground events at this lower kinetic energy. Consequently, the energy and timespectrum due to the background muon capture process could be studied. The lowmomentum data set was blinded between 6.6 MeV and 12.8 MeV.4.3.7 Particle IdentificationAs mentioned in section 4.3.4, some of the particles emitted by the backgroundmuon capture process are neutrons. Since they are neutral they have the same sig-nature as photons in the trigger. However, their waveform in the scintillator hasa different shape. When interacting with the crystal atoms, the neutrons mostlyknock out protons, which means that the waveform is characterized by the energyloss of a proton in NaI(Tl). Photons on the other hand, mostly interact via thephotoelectric effect, Compton scattering and pair production, therefore producingelectrons [36]. The scintillation process for these two particles differs slightly inNaI(Tl). In inorganic crystals, scintillation light is mostly produced by the acti-vator impurities added to the crystal by doping. The energy deposit of a particleeither produces electron hole pairs resulting in holes traveling in the valence bandand electrons moving freely in the conduction band, or creates an exciton. Thisis a bound state of an electron and a hole, traveling through the so-called excitonband which is located slightly lower than the conduction band in the band struc-ture of the crystal. After moving for some time, an electron and a hole are trappedin a doping center, or at a lattice defect or impurity, located between the valenceand the conduction band, therefore exciting the center and emitting scintillationlight during the de-excitation. If excitons were produced, they recombine after60some time when reaching a trap and also emit scintillation light. An explanationof the different behaviour for neutrons, protons and alpha particles versus that ofelectrons and photons can be found in reference [37]. The assumption is that ei-ther the time scale for electron-hole recombination or the diffusion to the activatorcenters is quite long when caused by electrons since their energy loss per distancedE/dx is small. Heavier particles on the other hand deposit considerably more en-ergy, therefore the ionization density is higher and free particles are trapped morequickly. The scintillation pulse is characterized by a quick rise time of ? 60ns anda slow decrease with a decay constant of? 230ns for both particle types. However,a pulse caused by an electron remains flat after initiation for about 150ns due to thelonger time-scale of recombination or diffusion [37]. Therefore, the pulse shapedue to an electron or photon has a larger amplitude at late times compared to thatcaused by an alpha particle, a neutron, or a proton. Consequently, the waveformrepresents a handle to distinguish between different particle types. Pulse shape dis-crimination was successfully applied to distinguish between photons and neutronsin a NaI(Tl) crystal by P. Doll et.al [38] and for discrimination between alphas,deuterons, protons, and alphas and muons by G.H. Share [39].We used a similar method to study the shape of the waveforms in the NaI(Tl)crystal of the PIENU detector. The parameter ? , extracted by the fitting procedureoutlined in the section above (4.3.5), provides an estimate of the width of the mainpeak. We used it to define a tail region, as shown in figure 4.15. Since some waveforms from high energy pulses extended beyond the time window of the detectorand their tail was truncated, we also defined the main pulse region in terms of itswidth to obtain a scalable definition of the main pulse. The integration limits aredefined with respect to the mean x? as follows:Lower integration limit for total integral: x??1 ?? (4.20)Lower integration limit for tail integral: x?+2 ?? (4.21)Upper integration limit for both integrals: x?+5 ?? (4.22)First, the selection of Michel events from the PIENU experiment used for thetime resolution study was analyzed with respect to the waveform shape. The in-61Figure 4.15: Waveform produced by a particle in the NaI(Tl) crystal (solidblack line), with a fit to the Crystal Ball function (dashed red line).The vertical green line indicates the mean of the Gaussian, the verticalblack line at one ? from the mean is the lower integration limit for thetotal waveform. The vertical blue line two ? to the right of the meanis the lower integration limit of the tail. Five ? away from the mean,indicated by the black line, is the upper integration limit.tegral of the tail region versus the total integral is plotted in figure 4.16. We onlyexpected photons and electrons after ?? ? e???e decay. Since photons deposittheir energy by interactions emitting electrons, the light in the scintillator was onlyproduced by one particle kind which is confirmed by the fact that we only see oneband in figure 4.16.In a second step, we analyzed the 72 MeV/c prompt-blinded muon capture dataset. Its tail versus total integral is shown in figure 4.17. The appearance of a secondband compared to figure 4.16 suggests that there was a second type of particlesproducing scintillation light. Most likely, these originated from the neutrons from62Figure 4.16: Integral of the tail region versus total integral of the waveformfor PIENU data.the muon capture background.Figure 4.17: Integral of the tail region versus total integral of the waveformfor muon capture data (72 MeV/c data set).634.3.8 Rejection of NeutronsFigure 4.18: Integral of the tail region versus total integral of the waveformfor PIENU data (left panel) and muon capture data (right panel) withvarious cuts indicated by the colored lines.To suppress background events with neutrons, we made use of their differentpulse shape in the NaI(Tl), compared to that of photons. Due to the differencebetween figures 4.16 and 4.17 we placed a straight line cut in between the photonand the neutron band. The PIENU data set provided us with a sample of electronsand photons, so this could be used as reference to decide which wave forms toaccept. Various different selections are shown in figure 4.18 for both the PIENUdata and the muon capture data. In order to determine which one rejected themost neutrons while preserving a sufficient amount of photons, the effect on theacceptance within the PIENU data set and on the rejection in both data sets werestudied for the signal region in a one ? interval: ?ESignal = 8.6? 10.8MeV. Theacceptance A was defined as the number of events after applying a selection cut,divided by the number of events in the same region before any selection. Similarly,the rejection R was the number of events without applying a selection cut minus thenumber of events after applying the cut, normalized to the number of events withouta selection. In figure 4.19 the rejection in the muon capture data minus the rejectionin the PIENU data is shown for various cuts. Since any additional rejection in themuon capture data removes more events in the neutron band, the highest value for64the difference between the two indicates the highest neutron rejection efficiency.In figure 4.19 the cut with the highest efficiency is marked by a red square, and islabelled by number ?8?. Figure 4.20 shows the acceptance versus rejection for thesame cuts. Again, cut number ?8? is indicated by a red square. The acceptancewithin the PIENU data set for this cut is 84 % and it is chosen as optimum cut forthe remaining steps of the analysis.Figure 4.19: Neutron suppression efficiency, defined as the rejection in themuon capture data minus the rejection in the PIENU data, plotted forvarious cuts. The red square indicates the cut with the highest neutronsuppression efficiency.4.3.9 Time CutFinally, the last cut applied to the muon capture data was a time cut selecting theprompt region. Before the prompt time at 186 ns, a region of three ? of the Gaus-sian prompt peak was accepted in order to include a large portion of prompt events,but reject any random hits before the prompt time. After the prompt time, a tightcut at one ? was chosen in order to considerably reduce events originating from65Figure 4.20: Acceptance (within PIENU data) versus rejection (withinprompt-blinded muon capture data) for the same cuts used in figure4.19. The red square indicates the same cut as the one in figure 4.19.the muon capture cascade process with its long lifetime of 110 ns. However, thisonly cut out 15 % of the events in the prompt Gaussian, therefore accepting 85 %of them.4.3.10 Analysis of the 74 MeV/c Data SetThe analysis of the 74 MeV/c data followed the same procedure as that for the72 MeV/c data set. The energy-blinding was applied between 9.3 MeV and 15.3 MeVwhereas the prompt-blinding was the same as for the 72 MeV/c data. The tail ver-sus total integral of the pulse shape for the prompt-blinded data is shown in fig-ure 4.21. At the higher momentum, fewer events were registered in total, so wehad fewer statistics for this momentum set, but the two bands due to neutrons andphotons are recognizable similarly as in the 72 MeV/c data. After studying the ac-ceptance and rejection for several cuts on the waveform, the one with the highestneutron rejection efficiency was chosen again and then the same prompt cut as forthe 72 MeV/c data set was applied.66Total integral in arbitrary units0 500 1000 1500 2000 2500 3000 3500 4000Tail integral in arbitrary units200400600800100012001400Figure 4.21: Tail versus total integral for the 74 MeV/c data set. Note that adifferent plotting style was used for this plot compared to the 72 MeV/cpulse shape plots due to the low statistics.4.3.11 Analysis of the Mylar Data SetThe data taken with a Mylar foil instead of the zirconium foil was intended forbackground studies. As it was taken at the higher momentum of 74 MeV/c, thenumber of events stopping in the foil was not very high, so the statistics for thisrun are low. Figure 4.22 shows the integral of the tail versus the total integral ofthe waveform in the NaI(Tl) crystal before the prompt cut. Since the figure showsthe same one-band structure as in the plot from pi ? ? ? e decays, the particlesdetected in the NaI(Tl) are likely photons.The same selection cuts as for the 74 MeV/c data set including the promptcut were applied to the Mylar data. The energy spectrum in the NaI(Tl) crystalis shown in figure 4.23. Only 35 events were left after the selection cuts, conse-quently, the Mylar data was not used for the direct analysis of the zirconium data.4.3.12 Background SpectraAfter applying all of the selection cuts described above to both of the zirconiumdata sets, the remaining background was studied. Figure 4.24 shows the energy-blinded time spectrum for the 72 MeV/c momentum set. It is composed of a prompt67Total integral in arbitrary units0 500 1000 1500 2000 2500 3000 3500 4000Tail integral in arbitrary units200400600800100012001400Figure 4.22: Tail versus total integral of the pulse shape in NaI(Tl) for theMylar data before the prompt cut. Note that a different plotting stylewas used for this plot compared to the 72 MeV/c pulse shape plots dueto the low statistics.component, fitted by a Gaussian function in blue in the plot, and by an exponentialcomponent, fitted to an exponential function in red, with a lifetime of ? = 113ns.This lifetime agrees well with the one measured for muon capture in zirconium(? = (110? 1)ns [35]), so this component most likely originates from the cap-ture at rest process. The prompt component could be due to bremsstrahlung frommuons stopping before reaching the T2 counter, as they would have been rejectedby the trigger otherwise. (It was not possible to address this issue at present.) Theenergy-blinded and prompt-blinded energy spectra for the lower momentum caseare shown in figure 4.25. There are significantly fewer background events withinthe signal region in the energy-blinded spectrum with the prompt cut applied. Inthe case of the higher momentum set, fewer muons stopped in the zirconium foil,leading to fewer background events due to the muon capture at rest. The two dif-ferently blinded energy spectra are shown in figure 4.26.68Energy in NaI [MeV]0510152025Counts 00.20.40.60.811.21.41.61.82Figure 4.23: Energy spectrum in the NaI(Tl) for the Mylar data after theprompt cut.4.4 Signal versus Background PredictionThe final step was to select the signal from within the remaining background. Todo so, we developed a procedure to extract the signal peak by using the simulatedsignal energy spectrum, and the background spectrum from the blinded data. Com-bining them resulted in an energy spectrum that we expected if the signal eventswere observed in the data as predicted theoretically. From this combined spec-trum, the signal events were then recovered and their statistical significance wasassessed.For a realistic representation of the signal, the energy spectrum from the sim-ulation described in section 4.3.1 was used. In addition, the expected number ofsignal events for our experimental setup was needed. The latter was calculatedfrom the theoretical cross section by integrating from the initial momentum of themuons entering the zirconium pi to their momentum when exiting the foil p f andby integrating over the whole angular range. The angular acceptance of the NaI(Tl)crystal, A, obtained from the simulation was taken into account, as well as the totalnumber of muons entering the zirconium foil, N? , for each momentum set, calcu-lated from the rate of particles passing through the counters B1 through T1 and thecomposition of the beam to select only muons:69Time [ns]100 200 300 400 500 600 700 800 900Counts050001000015000200002500030000350004000045000Figure 4.24: Energy-blinded time spectrum for the 72 MeV/c momentum set.A Gaussian function is fitted to the prompt region (solid blue) and anexponential to the delayed region (solid red) with a lifetime of ? =113ns. The dashed lines show the extensions of the fitted functionsbeyond the fitting range.Nsignal = A?N? ?? pip f?? d?dp (4.23)ARC events into the 1S state72 MeV/c momentum set 195 events74 MeV/c momentum set 110 eventsTable 4.2: Predicted numbers of signal events in the 1S state.The amount of signal events predicted for the two different momentum settings70Energy [MeV]0 5 10 15 20 25Counts02004006008001000120014001600Energy [MeV]0 5 10 15 20 25Counts010002000300040005000600070008000Figure 4.25: Background energy spectra for the 72 MeV/c momentum set.Left: prompt-blinded spectrum. Right: energy-blinded spectrum withan exponential fit to the blinded region.Energy [MeV]0 5 10 15 20 25Counts0100200300400500600700Energy [MeV]0 5 10 15 20 25Counts050100150200250Figure 4.26: Background energy spectra for the 74 MeV/c momentum set.Left: prompt-blinded spectrum. Right: energy-blinded spectrum withan exponential fit to the blinded region.are listed in table 4.2. Based on the amount of signal events expected within thedata, a histogram was filled according to the distribution of the signal energy in theNaI(Tl). This histogram represented the signal energy with the correct distributionand statistics.For the background spectrum, the shape of the prompt-blinded spectrum wasused since it has higher statistics than the energy-blinded distribution, and the two71agree above an energy of 8 MeV, as shown in figure 4.27 for the 74 MeV/c momen-tum set. In the region around 4 MeV, the energy-blinded spectrum has significantlymore events than the prompt-blinded spectrum, suggesting that these events orig-inated from the prompt background component. The energy of interest for thesignal, however, lies at 12.3 MeV, so the prompt-blinded shape could be used tomodel the background in that energy region.Energy in NaI [MeV]0 5 10 15 20 25 30 35 40Counts05101520253035Figure 4.27: Energy-blinded spectrum (red histogram) with fit to blindedregion (smooth black line) and prompt-blinded spectrum (black his-togram) for 74 MeV/c data set, scaled to the energy-blinded spectrumabove 8 MeV.A histogram was filled according to the number of events in the energy-blindedspectrum above 8 MeV and the shape of the prompt-blinded spectrum to representthe background in that energy region; it is shown in the left panel of figure 4.28.By adding the signal energy distribution to this background spectrum we obtaineda combined spectrum representing the data with the expected signal, shown in themiddle panel of figure 4.28. Now, a second background histogram was constructedaccording to the prompt-blinded shape and with the statistics of the prompt-blindedspectrum, scaled down to the number of events in the energy-blinded spectrum.72This was used as background shape to subtract the background from the combinedspectrum. The higher statistics were used for the filling procedure, since this spec-trum will be available from the data when performing the method on the unblindeddata set. For the combined spectrum, the background was filled with lower statis-tics as this corresponds to the expected data spectrum after all the selection cuts.The difference between the combined spectrum and the background shape, called?difference spectrum? from now on, is shown in the right panel of figure 4.28.For the extraction of the signal amount from the difference spectrum, two methodswere studied:1. The integral of the difference spectrum within one ? of the expected signalenergy was calculated.2. The difference spectrum was fitted to a Gaussian function, and its mean andintegral were extracted.Energy [MeV]8101214161820Counts 05101520253035Energy [MeV]8101214161820Counts 051015202530354045Energy [MeV]8101214161820Counts 051015202530Figure 4.28: Left: constructed background spectrum. Middle: combinedsimulated signal and background spectrum. Right: difference betweencombined and background spectrum. (74 MeV/c data set)For both of these cases, the procedure was repeated 1000 times to study thestatistical effects. For the first method, the number of events in the differencespectrum within one ? of the expected signal energy is shown in the left panel offigure 4.29. Its right panel shows the number of events divided by their uncertainty,this serves as a measure for the significance of the extracted signal, i.e. a value of5 means that the signal was extracted with a significance of 5? .For the second method involving the Gaussian fit to the difference spectrum,73Nr. of Signal Events020406080100120140160180200Counts02004006008001000Significance: (Integral) / (Uncertainty of Integral)024681012Counts020406080100120Figure 4.29: Left: integral of the difference spectrum within one ? of theexpected signal. Right: significance, defined as integral / (uncertaintyof integral). (74 MeV/c data set)the integral of the Gaussian within one ? of the expected signal energy is shownin the left panel of figure 4.30. The significance for this method is defined as theamplitude of the Gaussian, divided by its uncertainty determined from the fit. Thisquantity is shown in the right panel of figure 4.30.Nr. of Signal Events020406080100120140160180200Counts0100200300400500600700Significance: (Amplitude) / (Uncertainty of Amplitude)024681012Counts020406080100120Figure 4.30: Left: integral of the Gaussian within one ? of the expected sig-nal. Right: significance, defined as (amplitude of the Gaussian) / (un-certainty of the amplitude). (74 MeV/c data set)74When comparing figures 4.29 and 4.30, we notice that the two methods producevery similar results for the 74 MeV/c data set. With both methods, the amount ofextracted signal events has a mean of 70 events, which agrees with the amount ofsignal events from the input spectrum within one ? around the signal energy. Thesignificance varies between 4 and 7 ? .For the lower momentum set, the same methods were applied to extract thesignal. Figure 4.31 shows the energy-blinded spectrum together with the prompt-blinded spectrum, scaled to the number of events in the energy-blinded spectrumabove 8 MeV. The two spectra differ at a higher energy more than in the 74 MeV/cmomentum set, but at the signal energy of 9.3 MeV they agree sufficiently well, sothe same procedure was used for the signal extraction.Energy in NaI [MeV]4 6 8 10 12 14 16Counts010002000300040005000600070008000Figure 4.31: Energy-blinded spectrum (red histogram) with fit to blindedregion (smooth black line) and prompt-blinded spectrum (black his-togram) for 72 MeV/c data set, scaled to the energy-blinded spectrumabove 8 MeV.The results from the statistical study are shown in figure 4.32 for the integralmethod and in figure 4.33 for the Gaussian fit method. Using the integral method,the amount of signal is recovered with a significance of approximately one ? , vary-75Nr. of Signal Events-300-200-1000100200300400500Counts05001000150020002500Significance: (Integral) / (Uncertainty of Integral)-6-4-20246Counts020406080100120140Figure 4.32: Left: integral of the difference spectrum within one ? of theexpected signal. Right: significance, defined as integral / (uncertaintyof integral). (72 MeV/c data set)ing between no recovery and four ? . When the Gaussian fit is applied, the signif-icance is zero for 42 % of the cases, meaning that the signal was not recovered.Accordingly, the amount of signal calculated for these cases is zero. Due to thehigh background in the 72 MeV/c momentum case, the signal can therefore onlybe recovered in ? 2/3 of the cases. However, when the fitting procedure works, itprovides additional information about the position of the peak, namely the signalenergy, which is not obtained by simply integrating at the expected region as in theintegral method.76Nr. of Signal Events-300-200-1000100200300400500Counts020406080100120140160180Significance: (Amplitude) / (Uncertainty of Amplitude)-6-4-20246Counts050100150200250300Figure 4.33: Left: integral of the Gaussian within one ? of the expected sig-nal. Right: significance, defined as (amplitude of the Gaussian) / (un-certainty of the amplitude). (72 MeV/c data set)77Chapter 5Conclusions5.1 PIENU ExperimentThe analysis of the PIENU data is currently underway and therefore the data is stillblinded. Consequently, a final result for the branching ratio has not been deter-mined yet. For this thesis, a lower limit was estimated for the largest systematiccorrection to the branching ratio due to leakage out of the calorimeters and radia-tive decays. It amounts to TF = (2.24? 0.07(stat)? 0.14(sys))%. This agreeswell with the estimate from the lineshape measurement of TF = (2.25?0.06)%.Both values provide us with an estimate of the correction, which we obtained fromthe data, only using the simulation for corrections that cannot be extracted fromthe data. This is important, since this multiplicative correction is the largest amongother corrections that are applied to the branching ratio.The largest contribution to the uncertainty of the lower limit originates mainlyfrom the systematic uncertainty of the Bhabha correction. Since this is due to themismatch between MC and data, there is most likely room for improvement thatwill decrease the systematic error on the lower limit considerably. The estimateof the MDIF fraction causes the second largest contribution to the systematic un-certainty. By studying the simultaneous fit of the energy and time spectrum of thesuppressed spectrum in more detail, this can likely be reduced as well.To reach the goal of the PIENU experiment of determining the branching ratioR = pi+?e+?e+pi+?e+?e?pi+??+??+pi+??+?? ? with a precision of at least 10?3, the main challenge con-78stitutes in understanding all of the systematic effects and backgrounds very wellto decrease the systematic uncertainty. The work for this thesis contributed to theunderstanding of the largest systematic effect.5.2 Muon Capture ExperimentThe experiment described in the second part of this thesis is the first attempt ofmeasuring the direct radiative capture of muons into the 1S state of zirconium.The background process due to muon capture at rest was sufficiently suppressedin the higher momentum data set by achieving a time resolution of 4 ns in theNaI(Tl) crystal and by applying a pulse shape discrimination method successfully.With a simulation of the energy spectrum combining signal and background, therecovery of the signal was shown to be possible with a significance of five ? forthe 74 MeV/c momentum set if the signal has the characteristics as predicted bythe theory.In the lower momentum set with a higher level of background, the signal couldonly be recovered in 2/3 of the cases. When unblinding the data set, the results forthe 72 MeV/c data set should be compared to the 74 MeV/c data set and studied forconsistency. Consequently, the data was not unblinded yet, and further attemptsof decreasing the background and increasing the signal significance are currentlyunderway. One possibility of identifying stopping muons which contribute to thebackground is to study the energy deposit in the T1 counter, which is located di-rectly in front of the zirconium foil and compare it to the total energy loss in thedetectors upstream of the foil. The same study can be done with the GEANT4simulation, where the fact whether or not a muon stops in the zirconium is known.Since slow muons have a higher dE/dx in the T1 counter than those with higher ki-netic energy, the ones stopping in the zirconium could be identified. Consequently,a further cut decreasing the amount of stopped muons in the zirconium could beapplied to suppress more background events.After unblinding the data, the same procedure for signal recovery as explainedfor the simulated signal in section 4.4 will be applied. The only difference will beto use the unblinded data spectrum instead of the constructed combined spectrum.One possible scenario is to extract a signal with exactly the same properties as79predicted by the theory. However, is also possible that no signal is observed orit could have a different magnitude or be extracted at a different energy. In thosecases, the spectrum can be probed for a signal at different energies or an upper limiton the cross section for the direct radiative capture into the 1S state of zirconiumcan be set.Whether the predicted cross section is confirmed by this experiment, or anupper limit is set on the cross section of the ARC process, this study will provideimportant information for future experiments in the field. It will provide crucialknowledge on the cross section of the direct radiative capture for the case that theexperiment suggested in reference [13] is carried out in total to probe for a parity-violating process. If a signal is extracted, this would be the first measurement of amuon capture directly into the 1S state. In case a limit is set, essential informationfor future attempts of measuring the direct radiative capture is provided by thisexperiment.80Bibliography[1] C. Malbrunot, Ph.D. thesis, University of British Columbia (2012). ? pages[2] T. Sullivan, Tech. 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