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Precise measurement of rare pion decay Cuen-Rochin, Saul 2019

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Precise Measurement of Rare PionDecaybySaul Cuen-RochinB.Sc.E.E., Centro de Ensen˜anza Te´cnica y Superior, 2005M.Phys., Universidad Autonoma de Sinaloa, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Physics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)April 2019© Saul Cuen-Rochin 2019The following individuals certify that they have read, and recommend to theFaculty of Graduate and Postdoctoral Studies for acceptance, a thesis/dis-sertation entitled:Precise Measurement of Rare Pion Decaysubmitted by Saul Cuen-Rochin in partial fulfillment of the requirements forthe degree of Doctor of Philosophyin PhysicsExamining Committee:Omer Angel, MathematicsChairDouglas Bryman, PhysicsResearch SupervisorChristopher Hearty, PhysicsSupervisory Committee MemberMichael Hasinoff, PhysicsUniversity ExaminerTakamasa Momose, ChemistryUniversity ExaminerKlaus Kirch, Physics - ETH Zu¨richExternal ExaminerAdditional Supervisory Committee Members:Douglas Scott, PhysicsSupervisory Committee MemberOliver Stelzer-Chilton, PhysicsSupervisory Committee MemberHirohisa Tanaka, PhysicsSupervisory Committee MemberiiAbstractThe PIENU1 experiment at TRIUMF2 aims to measure the pion decaybranching ratio, defined as the relative rate of decay of pions into elec-trons over muons including associated neutrinos and radiative components(denoted Rpi) to a precision level of O(0.1%). This Standard Model (SM)observable provides a sensitive test of lepton universality, where weak cou-pling strengths are assumed to be equal for all leptons (g = ge = gµ = gτ ).Comparing the measured experimental (Rexppi ) and calculated SM (RSMpi ) ra-tios, the ratio of the coupling constants can be extracted and compared withthe SM expectation ge/gµ = 1 as follows ge/gµ = (Rexppi /RSMpi )1/2.The current theoretical calculation of the SM predictionRSMpi = (1.2352±0.0002)× 10−4 with a precision of 0.016% is more precise than the measure-ments of previous generation experiments by a factor of 30; thus, there isscope for significant improvement. If the measurement is consistent with theSM, new constraints could be set on new physics scenarios for SM exten-sions, such as R-parity-violating super-symmetry, leptoquarks, and heavyneutrinos lighter than the pion. Most remarkably, a deviation from the SMcould result from a new pseudo-scalar interaction with an energy scale of upto O(1000 TeV) which would enhance the branching ratio by O(0.1%). Insome instances, these constraints can far exceed the reach of direct searchesat colliders.Between 2009 and 2012 around 6.5 million pi+ → e+νe events were gath-ered. The analysis of a subset of the 2010 data with 0.4 million events waspublished in 2015, giving Rexppi = (1.2344 ± 0.0023(stat.) ± 0.0019(syst.)) ×10−4, with a precision of 0.24%. This is in agreement with the SM, represent-ing a 0.12% measurement of lepton universality at ge/gµ = 0.9996± 0.0012.The analysis presented in this thesis is blinded but includes the highestquality data portion available, around 3 million pi+ → e+νe events. For thiswork, major experimental systematic problems have been solved allowingfor increased precision up to 0.12% for Rexppi and up to 0.06% for leptonuniversality.1Acronym for pi+ → e+νe decay mode.2Canada’s particle accelerator centre.iiiLay SummaryThe Standard Model (SM) of particle physics is the best available the-oretical framework to predict the subatomic interactions between the fun-damental elements of known matter. The PIENU experiment at TRIUMFmakes a precise measurement for one of the SM’s best calculated predictionsinvolving the force that governs radioactive decays. If a measurement is con-sistent with the SM, better constraints can be set on theories which extendthe SM. In some instances, these constraints can far exceed the reach of di-rect searches at high energy colliding beam facilities like the Large HadronCollider at CERN. Most remarkably, a deviation from the SM expectationcould imply the presence of new physics effects not included in the SM. Thisthesis describes the analysis of a dataset including 3 million pi+ → e+νeevents. Major experimental systematic problems have been solved, allowingfor increased precision by a factor of two over PIENU’s previous measure-ment from 2015.ivPrefaceThe PIENU collaboration consists of a team of around 20 people fromseveral countries. TRIUMF approved the experiment’s proposal in 2005,and the PIENU detector was designed and prototyped the following yearsin TRIUMF’s Meson Hall. The prototype had initial beam tests in 2007,and finally during 2008 the final version of the PIENU detector was installedat the end of M13-beam-line with most of the components assembled. Af-ter the assembly was completed, data-taking Run I and Run II and themain calorimeter’s energy response test measurements were performed in2009, culminating in the beam-line paper [1]. In 2010, Run III and Run IVwere performed, and the calorimeter paper was published [2]. In 2011, thecollaboration collected special data for beam-line studies and improved thecalorimeters’ energy response test measurement. Subsequently, Run V wascompleted and a heavy neutrino search analysis paper using runs just from2009 was published [3].The author joined the collaboration as a Ph.D. student in the summerof 2012, and took many shifts during Run VI, the most significant datataking period. In addition, the author participated in data monitoring andcollection, and ensured quality data taking with onsite-online-offline beam-line-trigger-detector-computer maintenance. Later in the same year he par-ticipated in data taking of additional special runs for beam-line studies.Finally in 2013, the collaboration dismantled the experiment. From 2012 to2015, the collaboration worked intensively to unravel all aspects of the de-tector performance and available data. In 2015, the collaboration publishedthe detector design and performance [4] as well as an analysis of the Run IVdata [5], setting a record for precision measurement of the branching ratioat a level of 0.24% and in the e-µ universality test at a level of 0.12%. Since2012, the author has been responsible for the beam-line low-momenta par-ticle contamination study (Section 6.1.2 and Appendix D), performing dataanalysis and simulations [6] to avoid uncertainty in the main correctionsand sources of systematic uncertainties. The author also conducted studieson possible energy bias and or acceptance in the calorimeter (pi+ → e+νevPrefaceover pi+ → µ+νµ → e+νeν¯µ or vice-versa) from multiple pulse elimination inscintillator T1 [7], the counter responsible for the positron timing after thepion-stopping target. The author was in charge of and compiled compre-hensive documentation [8], including all aspects of the analysis and resultsused in the first PIENU paper [5].In addition, the author inherited all the legacy code and frameworks forthe analysis, and served as a system administrator for the local cluster andthe PIENU web-page [9]. Furthermore, the author re-coded and reprocessedall raw data to the current version and pushed the entire dataset migrationfrom the Westgrid-Bugaboo to the ComputeCanada-Cedar cluster. The au-thor also compiled an extensive initial report [10] on the Run VI dataset,which was approximately five times larger than the Run IV dataset. Theauthor participated in the collaboration’s three-year effort to eliminate theprimary source of systematic uncertainty, the calorimeter’s radial acceptancein the branching ratio, in order to achieve the current level of precision. Fi-nally, through this thesis, the author has been in charge of the comprehensivedocumentation for the current preliminary final result, including all aspectsof the analysis for the upcoming final PIENU paper.From 2016 to 2018, the collaboration was engaged in full analysis for alldatasets. The group published a new improved heavy neutrino search pa-per [11] in 2018 using the full PIENU dataset, i.e., all runs from 2009 to2012. The experiment is now in the last stage prior to unblinding the finalbranching ratio Rexppi result, currently having an estimated precision level ofup to 0.12% (0.06% for the e-µ universality test), using the highest qualitydata portion available, around 3 million pi+ → e+νe events. Representinga factor of approximately 30 improvement from previous generation experi-ments [12] [13] and a factor of 2 from a subset of PIENU data (0.4 millionevents) published [5] in 2015. For the 2012 dataset (2 million events), thisthesis presents the total reduced χ2/d.o.f. (d.o.f. = 1557) of 1.19, and 1.13for the pulse-height (PH) and charge-integration (Q) based time spectrumanalysis, from which the raw branching ratio is extracted. For the 2011dataset (0.5 million events) the χ2/d.o.f. is 1.08, and 1.06 for the PH andQ based analysis. For the 2010-November dataset (0.4 million events) theχ2/d.o.f. is 1.00, and 1.07 for the PH and Q based analysis. Since the SMbranching ratio prediction RSMpi is at a precision level of 0.016%, there isstill scope for improvement in the next generation of experiments. In recentyears, most PIENU collaborators have left TRIUMF, making the author theviPrefacelast Ph.D. student working full time on the experiment. Through this the-sis the author shares the latest breakthroughs and improvements in majorsystematic problems recently solved with the collective contributions fromthe collaboration; specifically (in order) D. Bryman, L. Doria, S. Ito, R.Mischke, T. Numao, A. Sher, and T. Sullivan.The author gave a talk [14] in 2016 to promote the initial results fromthe PIENU experiments and holds co-authorship with the collaboration forthree peer-reviewed articles ([4], [5], [11]), and six proceedings. In addition,the author produced four technical-notes ([6], [7], [8], [10]) and more thanone-hundred documents for PIENU’s internal archives, most of them pre-sented across six years of weekly meetings. The PIENU collaboration plansto publish up to six more peer-reviewed articles in the near future: regard-ing massive neutrino searches in pion-stopping scintillator target pi+ → µ+νenergy spectra; Majorana neutrino searches in the calorimeter’s pi+ → e+νeenergy spectrum; direct muon capture in Zirconium from a special set ofmuon runs; the calorimeter’s energy response; and the final branching ra-tio analysis (both short and extended versions) for PIENU. The goals andmilestones reached so far for the PIENU experiment are shown in Table3.3. The author is indebted to the previous PIENU theses, which were thefoundation of this dissertation: specially (and chronologically) those of, K.Yamada, Ph.D. 2010 [15], C. Malbrunot, Ph.D. 2012 [16], D. vom Bruch,M.Sc. 2013 [17], S. Ito, Ph.D. 2016 [18], T. Sullivan, Ph.D. 2017 [19], R.Nuttall, B.Sc. 2018 [20], and L. Doria’s Habilitationsschrift (in preparation).Nevertheless, the writing of this thesis is from the author alone.viiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xivAlgorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviiGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxii1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Previous Measurements . . . . . . . . . . . . . . . . . . . . . 31.2 Experimental Technique . . . . . . . . . . . . . . . . . . . . . 51.2.1 Lessons from the E248 experiment . . . . . . . . . . . 61.2.2 PIENU technique . . . . . . . . . . . . . . . . . . . . 81.3 Blind Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.1 The Standard Model of Particle Physics . . . . . . . . . . . . 142.1.1 Electroweak interactions . . . . . . . . . . . . . . . . 162.1.2 Strong interactions . . . . . . . . . . . . . . . . . . . 162.2 Pion Decay Theory . . . . . . . . . . . . . . . . . . . . . . . 17viiiTable of Contents2.2.1 Vector-Axial-Vector (V-A) Weak Interaction . . . . . 182.2.2 Helicity Suppression . . . . . . . . . . . . . . . . . . . 212.2.3 Radiative Corrections . . . . . . . . . . . . . . . . . . 222.3 Motivation Beyond the Standard Model . . . . . . . . . . . . 262.3.1 Lepton Universality . . . . . . . . . . . . . . . . . . . 262.3.2 New-Pseudo-scalar Interactions . . . . . . . . . . . . 302.3.3 Partial Compositeness . . . . . . . . . . . . . . . . . . 352.3.4 Heavy Neutrino . . . . . . . . . . . . . . . . . . . . . 353 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.1 Cyclotron and Beam-line . . . . . . . . . . . . . . . . . . . . 393.1.1 Beam-line Extension . . . . . . . . . . . . . . . . . . 423.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.2.1 Scintillators . . . . . . . . . . . . . . . . . . . . . . . 463.2.2 Wire Chambers . . . . . . . . . . . . . . . . . . . . . 473.2.3 Silicon Detectors . . . . . . . . . . . . . . . . . . . . . 483.2.4 Bina . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.2.5 CsI . . . . . . . . . . . . . . . . . . . . . . . . . . . . 503.2.6 Tracking . . . . . . . . . . . . . . . . . . . . . . . . . 513.3 Final Detector Assembly . . . . . . . . . . . . . . . . . . . . 553.4 Data Acquisition System . . . . . . . . . . . . . . . . . . . . 563.4.1 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . 563.4.2 Boards . . . . . . . . . . . . . . . . . . . . . . . . . . 603.4.3 Software . . . . . . . . . . . . . . . . . . . . . . . . . 643.5 Data-taking History and Milestones . . . . . . . . . . . . . . 653.5.1 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . 653.5.2 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5.3 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5.4 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5.5 Full analysis . . . . . . . . . . . . . . . . . . . . . . . 674 Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 694.1 Variable Extraction and Calibration . . . . . . . . . . . . . . 694.1.1 Run Selection . . . . . . . . . . . . . . . . . . . . . . 694.1.2 Scintillators . . . . . . . . . . . . . . . . . . . . . . . 694.1.3 Silicon Detectors and Calorimeter . . . . . . . . . . . 714.1.4 Wire-chambers . . . . . . . . . . . . . . . . . . . . . . 734.2 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . 764.2.1 Pion Identification . . . . . . . . . . . . . . . . . . . . 764.2.2 Pileup T1, and T2 . . . . . . . . . . . . . . . . . . . . 78ixTable of Contents4.2.3 Early Time . . . . . . . . . . . . . . . . . . . . . . . . 794.2.4 Calorimeter Acceptance Radius AR . . . . . . . . . . 804.2.5 Summary of Event Selection . . . . . . . . . . . . . . 804.3 Energy Spectra . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3.1 Monte-Carlo Calibration . . . . . . . . . . . . . . . . 815 Time Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 945.2 Signal and Background . . . . . . . . . . . . . . . . . . . . . 985.2.1 Signals . . . . . . . . . . . . . . . . . . . . . . . . . . 985.2.2 Pion Decay-In-Flight and Muon from Previous Event(Old-muon) in Target B3 . . . . . . . . . . . . . . . . 995.2.3 T1 Double Pulse Resolution. . . . . . . . . . . . . . . 995.2.4 Muon from Previous Event (Old-muon) No-T1-Hit . 1055.2.5 Radiative Pion Decay . . . . . . . . . . . . . . . . . . 1055.3 The Fitting Function . . . . . . . . . . . . . . . . . . . . . . 1065.3.1 Time-Independent Addition of Energy . . . . . . . . 1075.3.2 Low-Energy Components . . . . . . . . . . . . . . . . 1085.3.3 High-Energy Components . . . . . . . . . . . . . . . . 1085.3.4 Fit Parameters . . . . . . . . . . . . . . . . . . . . . . 1095.3.5 Signal Overlay and Residuals . . . . . . . . . . . . . . 1116 Corrections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.1 Calorimeter’s Low Energy Tail . . . . . . . . . . . . . . . . . 1246.1.1 Response Function Measurement . . . . . . . . . . . . 1276.1.2 Beam-line’s Intrinsic Tail . . . . . . . . . . . . . . . . 1316.2 Acceptance . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.3 Muon Decay in Flight . . . . . . . . . . . . . . . . . . . . . . 1376.4 t0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1387 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1407.1 Stability and Systematic Errors . . . . . . . . . . . . . . . . 1407.1.1 Fit Tests . . . . . . . . . . . . . . . . . . . . . . . . . 1417.1.2 LET tests . . . . . . . . . . . . . . . . . . . . . . . . 1497.1.3 Charge- vs. Pulse-height-based Rpi . . . . . . . . . . . 1507.2 Error Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . 1537.3 Combination of Datasets . . . . . . . . . . . . . . . . . . . . 1547.4 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . 1577.4.1 Current PIENU experiment . . . . . . . . . . . . . . 1577.4.2 Next generation PIENU . . . . . . . . . . . . . . . . 157xTable of Contents8 Limits on New Physics . . . . . . . . . . . . . . . . . . . . . . 1598.1 The pi+ → e+νe branching ratio . . . . . . . . . . . . . . . . 1598.2 Lepton Universality . . . . . . . . . . . . . . . . . . . . . . . 1608.3 New Pseudo-scalar Interactions . . . . . . . . . . . . . . . . . 1608.3.1 R-Parity violating SUSY . . . . . . . . . . . . . . . . 1618.3.2 Charged Higgs Boson . . . . . . . . . . . . . . . . . . 1618.4 Search for Massive Neutrinos in the pi+ → e+νe Decay . . . . 1628.5 Summary and Forward-looking for SM deviation scenarios . 163Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165AppendicesA Time spectrum for pi → µ→ e . . . . . . . . . . . . . . . . . . 176B Cuts for Pion Data . . . . . . . . . . . . . . . . . . . . . . . . . 177C Cuts for Positron Data . . . . . . . . . . . . . . . . . . . . . . 179D Beam-line Simulation . . . . . . . . . . . . . . . . . . . . . . . 184E Trigger Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 191F Technical Drawings . . . . . . . . . . . . . . . . . . . . . . . . . 192xiList of Tables2.1 Bosons (integer spin). . . . . . . . . . . . . . . . . . . . . . . 152.2 Fermions (spin 1/2 integers). . . . . . . . . . . . . . . . . . . 162.3 Measured pion decay modes [21]. The radiative energy (Eγ1)restrictions are concerning the cited experiment, not the PIENUexperiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.4 Measured muon decay modes [21]. The radiative energy (Eγ1)restrictions are concerning the cited experiment, not the PIENUexperiment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.5 Summary of the electroweak corrections for Rpi0 . . . . . . . . 252.6 Experimental results on lepton universality (LU) tests fromstudies of pi, K, τ , µ and W decay. In some cases, µ and τ ’slifetime (τµ, and ττ ) measurements were used in combinationfor LU tests. Here, B represents the branching fraction of aparticular decay mode. . . . . . . . . . . . . . . . . . . . . . . 283.1 Parameters for the PIENU detector [4]. . . . . . . . . . . . . 453.2 Rates for all triggers [4]. . . . . . . . . . . . . . . . . . . . . . 603.3 Run history and milestones of the PIENU experiment. . . . . 684.1 Cut flow for event selection. The number of events beforecuts is 2.027× 109 for the 2012 dataset. . . . . . . . . . . . . 82xiiList of Tables5.1 Results from the timing spectra for the three data-taking pe-riods, presented for both integrated-charge (Q) and pulse-height (PH) calorimeter variables. The exact fit values aretruncated for a more compact presentation. The errors arestatistical only as obtained by the MINUIT [22] fit, and theparameters marked as fixed were kept fixed during the fit.The errors in the Rrawpi reflect the magnitude of the data sam-ples collected in the three periods. The acceptance radiusused was RA = 40 mm, and the nominal range for our fittingfunction (FF) for both high- and low-energy time spectra isfrom −290 to 520 ns, excluding prompt events from −20 to10 ns. Using 1 ns bins for the time spectrum, the total degreesof freedom (d.o.f.) are 1557. . . . . . . . . . . . . . . . . . . 1106.1 Low energy tail fraction (T ) percentage for nominal pionbeam configuration as a function of the maximum acceptanceradius AR with Ecut = 52 MeV, and as a function of Ecut withAR = 60 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.2 Upper limit to beam-line’s contribution to tail fraction (Tbeam)percentage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1346.3 Acceptance correction CAcc for different AR values. . . . . . . 1376.4 Muon decay in flight correction CµDIF for different Ecut andAR values. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397.1 Stability tests and systematic errors from the fit, followingstandard methodology [23]. Non-negligible deviations are inred. See Section 7.1.1 for discussion. Units in the branchingratio change are ∆R [10−8], with uncorrelated errors unlessspecified otherwise. . . . . . . . . . . . . . . . . . . . . . . . 1467.2 Error budget in [10−8] branching ratio units. . . . . . . . . . 1537.3 Combination of 2010, 2011, and 2012 datasets for AR = 40mm. The branching ratios for all datasets are still blinded.See Section 7.3 and Table 7.2 for nomenclature. The PHversion was chosen over the Q based branching ratio sincethe global systematic error is (marginally) better. . . . . . . . 156B.1 List of cuts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177B.2 Year dependent cut values. . . . . . . . . . . . . . . . . . . . 178D.1 Beam-line’s settings for positron runs . . . . . . . . . . . . . . 185xiiiList of Figures1.1 History of the Rexppi branching ratio measurements. Red line:SM calculation [24]. Black dashed line: PDG experimentalaverage [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 Schematic illustration of experimental technique: two pionicdecays in a scintillator target; and decay positrons collectedby the calorimeter. . . . . . . . . . . . . . . . . . . . . . . . . 51.3 (Top) Experimental setup of the E248 experiment at TRI-UMF [25]. (Bottom) Positron energy spectrum obtained bysuppressing pi+ → µ+νµ → e+νeν¯µ events, the x-axis is en-ergy in ADC counts (channel 3400 corresponds to 56.4 MeV). 71.4 Energy deposited in the target B3 for pi+ → e+νe (blue line)and pi+ → µ+νµ → e+νeν¯µ (red line) events from GEANT4. . 91.5 (Left) Time spectra and (right) energy spectra in the calorime-ters of pi+ → e+νe and pi+ → µ+νµ → e+νeν¯µ decays ob-tained from simulations. The spectra are normalized to thesame amplitude. Using an energy cut-off (Ecut) above thepi+ → µ+νµ → e+νeν¯µ energy spectrum edge (dashed line),we divide the energy spectrum into low-energy (LE) and high-energy (HE) parts The low energy tail (LET) from the pi+ →e+νe energy spectrum is not visible due to scale and the piondecay in flight (piDIF) contribution was deactivated. Thepi+ → e+νe time distribution peaks near t = 0 because of therelatively short pion lifetime. . . . . . . . . . . . . . . . . . . 91.6 Evolution in time (years) of the neutron’s lifetime experimen-tal result [21]. . . . . . . . . . . . . . . . . . . . . . . . . . . 121.7 PIENU’s blinding technique. A smooth inefficiency func-tion (unknown to the experimenters) removes events basedon their energy deposited in the target, lowering (case a) orraising (case b) the branching ratio. . . . . . . . . . . . . . . 122.1 Feynman diagram for the pi+→l+νl decay, where l = e, µ. . . 19xivList of Figures2.2 Fermi-point-like interpretation for pi+→l+νl decay. . . . . . . 232.3 Feynman diagrams for the radiative corrections to pion decay,from real (a) and virtual (b) photons. . . . . . . . . . . . . . 242.4 The limits on ∆µτ and ∆eτ from (a) W -decay, (b) τ -decay, (c)pi and K-decay, and (d) all decays combined. The 1σ bandsare shown for each coupling constant ratio, ignoring correla-tions. The shaded areas represent the 68% (dark grey) and90% (light grey) confidence contours, including correlations(Figure from ref. [26]). . . . . . . . . . . . . . . . . . . . . . . 292.5 Comparison of measurements with SM predictions: The branch-ing fraction B is B− → τ−ντ (left), the ratio R(D) is B →Dτ−ντ overB → De−νe (center), andR(D∗) isB → D∗τ−ντover B → D∗e−νe (right) by BABAR, Belle, and LHCb.The data points indicate statistical and total uncertainties.ST and HT refer to the measurements with semileptonic andhadronic tags, respectively. The average values of the mea-surements and their combined uncertainties, obtained by theHeavy Flavor Averaging Group, are shown in red as verticallines and bands, and the expectations from the SM calcula-tions are shown in blue. Image and data from ref. [27]. . . . . 302.6 Feynman diagrams for pseudo-scalar interactions induced atone loop including three classes of diagrams: scalar-dressedZ exchange box diagrams (top), scalar-dressed W exchangebox diagrams (middle) and radiative corrections to the quarkvertex (bottom). . . . . . . . . . . . . . . . . . . . . . . . . . 322.7 Tree level RPV contributions to Rpi [28]. . . . . . . . . . . . . 332.8 Constraints on RPV parameters from Rpi . . . . . . . . . . . 342.9 The 90% C.L. upper limit on the heavy-neutrino mixing pa-rameter, as a function of its mass. The dashed line showsthe result from the previous PIENU experiment [29], and thecircles and triangles are the limits from a subset of PIENUdata, published in 2011 [3]. The circles indicate a restrictedangular region was used when constructing the pi+ → e+νeenergy spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . 383.1 Schematic illustration of TRIUMF’s cyclotron, primary beam-lines, and Meson Hall’s secondary beam-lines [30]. . . . . . . 403.2 M13 channel with the extension [1] . . . . . . . . . . . . . . . 41xvList of Figures3.3 Left: Position distribution of pi+, µ+, and e+ at F3. The solidlines are Gaussian fits. Right: pi+ and e+ rates at F4 as afunction of the selected momentum [1]. The PIENU detectorwas placed at final focus point F4. . . . . . . . . . . . . . . . 413.4 The end of the M13 beam-line, before (left) and after (right)the extension. Part of the detector was in place to measurethe particle content of the beam. . . . . . . . . . . . . . . . . 423.5 Left: Fraction of beam positrons as a function of the se-lected momentum. Right: Fit of the delayed component ofthe positrons time-of-flight showing consistency with the piondecay time [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . 443.6 Schematic illustration of the PIENU detector [4]. The targetregion is magnified in the inset. . . . . . . . . . . . . . . . . . 463.7 (Left) B1, (also B2, B3, and T1) plastic scintillator is readout with 4 PMTs (grey cylinders); Light was collected by fouracrylic light guides (light green). (Right) Readout schemewith wavelength-shifting fibers of the T2 plastic scintillator. . 473.8 (Left) WC1/2 wire chamber plane and its preamplifier board.Each chamber consisted of three planes. (Right) WC1/2 afterinstallation on the beam pipe [9]. . . . . . . . . . . . . . . . . 483.9 (Left) Image of the wire chamber WC3 placed in front of theNaI(Tl) calorimeter. (Right) S1 and S2 assembly on theirsupport structure [9] [31]. . . . . . . . . . . . . . . . . . . . . 493.10 (Left) Back side of the NaI(Tl) crystal on the test bench.(Right) The NaI(Tl) crystal and the 97 CsI crystals while thecalorimeter was under construction [9]. . . . . . . . . . . . . . 503.11 Schematic of the tracking devices, the pi+ → e+νe signal, andthe different decay-in-flight backgrounds (the sizes are not toscale). piDAR → µDIF: In pi+ → µ+νµ → e+νeν¯µ decay, themuon decays in flight in the target. piDIF upstream of target(“up.”) → µDAR: The pion decays in flight before enteringthe target. Part of these decays can be detected by trackingthrough the kink variable (Kθ). piDAR → µDAR: Both thepion and the muon in the pi+ → µ+νµ → e+νeν¯µ channeldecay at rest in the target. piDIF inside target (“it.”) →µDAR: Pion decay-in-flight in the target and muon decay atrest. The “u” orientation of a WC plane corresponds to arotation of +60◦ while “v”=−60◦. . . . . . . . . . . . . . . . 523.12 Simulation of the kink angle Kθ for different pion decay modes. 53xviList of Figures3.13 Bottom: Beam goes from right → left. The PIENU detectorand beam-line after the last bending magnet, showing thesteel wall used for radiation shielding. Top-Left: PIENU-1assembly of scintillators, wire-chambers, and silicon detectors.Top-Right: PIENU-2 detector calorimeter assembly, imagefrom [9]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 573.14 Beam-line-Detector CAD drawing [9]. . . . . . . . . . . . . . 583.15 Schematic of the trigger diagram for the three physics trig-gers. The rates of the triggers are listed in Table 3.2. Imagefrom [18]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.16 Picture of main COPPER board mounted with four FINESSEmodules. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 613.17 A waveform digitized by COPPER. The red circles and bluecrosses show the digitization of each 250 MHz ADCs, whichproduce a 500 MHz waveform. . . . . . . . . . . . . . . . . . 623.18 Web interface of the MIDAS data acquisition system. All theVME modules were integrated and easily controlled via thisinterface. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.1 COPPER’s signals timing. Image from [18]. . . . . . . . . . . 714.2 Comparison prior to calibration between data (black) takenwith the cosmic ray trigger and an MC simulation (red) basedon the CRY [32] simulation package. The spectra are relativeto the 21 crystals in the inner-upstream CsI ring. Horizontalaxes are ADC counts. The peak positions vary up to 20%in energy with the position of the crystal in the detector,but they are well emulated in MC. The energy deposited byminimum ionizing particles in a single CsI crystal is about50 MeV. Image from [18]. . . . . . . . . . . . . . . . . . . . . 754.3 Pion Cut: B1 (top) and B2 (bottom) energy distributionwithout cuts in black, energy distribution with all cuts in red(excluding cut being discussed), and cut values are shown inblue. No normalization. Peaks from left to right in B1 (andB2): positrons at 1.1 (0.5), muons at 3.2 (1.5), pions at 4.5(2.5), and two pions arriving at the same time at 9 (4.7) MeV. 844.4 Acceptance for WC1 (top) and WC2 (bottom); beam halo isremoved. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85xviiList of Figures4.5 Charge (top) and time (bottom) distributions without cuts inblack, distributions with all cuts in red (excluding cut beingdiscussed), cut values are shown in blue. No normalization.Top: B1 short gate/wide gate integrated charge. Bottom:Trigger Consistency Cut. . . . . . . . . . . . . . . . . . . . . 864.6 Energy in the NaI (Bina) versus minimum energy loss in thedownstream counters. Protons are above the red line indicat-ing the cut position. The red blob represents pi+ → µ+νµ →e+νeν¯µ events and the small yellow blob pi+ → e+νe events. . 874.7 The ratio of integrated charge in the T1 PMTs to the fittedpulse height as a function of the fitted pulse height. Thered line indicates the cut used to separate real pileup (above)from pileup due to fake hits (below). . . . . . . . . . . . . . . 874.8 The false trigger cut rejects events when positrons from piDIFmake false trigger. The positrons are found at (B3t−B1t) > 4ns and B3charge < 200 ADC counts (∼3 MeV). The threebands on the left represent pileup related to 4, 3, and 2 PMTsat 450, 300, and 150 ADC counts (10, 4, and 2 MeV), respec-tively. The main red blob in the center represents good beampion events stopping in the center of target B3 and the blob’sdownward tail represents pions barely entering B3, while theupward tail is pions stopping at the end of B3. . . . . . . . . 884.9 Calorimeter’s acceptance radius (AR) cut distribution with-out cuts in black, radius distribution with all cuts in red (ex-cluding the cut being discussed), and cut value is shown inblue. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 894.10 Combined energy spectrum of the NaI plus CsI detectors forthe 2012 dataset. The vertical red line indicates Ecut = 52MeV. The composition of the high energy tail beyond 70 MeVis due to pileup events. . . . . . . . . . . . . . . . . . . . . . 89xviiiList of Figures4.11 Alignment of scintillators and calorimeter’s energy scales toMonte-Carlo. a) Etot’s “suppressed” energy in black, Etot’s“late” energy in blue, and MC’s Etot energy for pi+ → e+νeevents only in red. b) Calorimeter’s “suppressed” energy inblack, “late” energy in blue, and MC’s for pi+ → e+νe eventsonly in red. c) Calorimeter’s “suppressed” and “late” energywith Etot’s cut to select pi+ → e+νe events. For all subplotsthe “suppressed” (black) energy distribution was normalizedto the MC’s (red) pi+ → e+νe peak, for proper comparisonsuch normalization was used for the “late” (blue) spectrumas well. The alignment coefficient between data and MCwas done to match the pi+ → e+νe peak in the calorimeter(Bina+CsI). The resultant alignment uncertainty of the scin-tillators’ total energy and the calorimeter’s peak to MC is be-low the calibration’s uncertainty 0.1 MeV. The normalizationfor each subplot (left, center, and right) was done indepen-dently, thus the vertical axes between subplots don’t match,i.e., Etot’s plots were normalized to 1, and the calorimeter’senergies to the total number of events. . . . . . . . . . . . . 904.12 Early triggers for all years including charge-integration (Q)and pulse-height (PH) calorimeter (NaI+CaI) energy basedvariables defined by Eq. 4.5 and Eq. 4.6. The three datasets(2010, 2011, and 2012) make two groups in the tail above 70MeV, in such region all three PH versions have less pileupthan the Q version group. . . . . . . . . . . . . . . . . . . . 914.13 Prescale triggers for all years including charge-integration (Q)and pulse-height (PH) calorimeter (NaI+CaI) energy basedvariables defined by Eq. 4.5 and Eq. 4.6. The three datasets(2010, 2011, and 2012) make two groups in the tail above 70MeV, in such region all three PH versions have less pileupthan the Q version group. . . . . . . . . . . . . . . . . . . . 924.14 TIGC triggers for all years including charge-integration (Q)and pulse-height (PH) calorimeter (NaI+CaI) energy basedvariables defined by Eq. 4.5 and Eq. 4.6. The three datasets(2010, 2011, and 2012) make two groups in the tail above 70MeV, in such region all three PH versions have less pileupthan the Q version group. . . . . . . . . . . . . . . . . . . . 93xixList of Figures5.1 2012 Dataset - Low-Energy (LE) Time-Spectrum (TS), Tposfor ENaI+CsI < Ecut: Using the pulse-height “PH” EPHNaI+CsIand charge-integrated “Q” EQNaI+CsI calorimeter based vari-able to construct the LE TS “tsL”. Overlaying TS with dif-ferent levels of cuts: The “raw” (orange and black) spectrum(no cuts), “L1” (violet and red) with “Pion Identification”cuts, “L2” (light-blue and yellow) with “Pileup T1, and T2”cuts, and the final “clean” (dark-green and navy-blue) TSwith “Early Time and Acceptance” (All) cuts. See Section4.2 for discussion on cuts. The PH and Q versions overlap. . 965.2 2012 Dataset - High-Energy (HE) Time-Spectrum (TS), Tposfor ENaI+CsI >= Ecut: Using the pulse-height “PH” EPHNaI+CsIand charge-integrated “Q” EQNaI+CsI calorimeter based vari-able to construct the HE TS “tsH”. Overlaying TS with dif-ferent levels of cuts: The “raw” (orange and black) spectrum(no cuts), “L1” (violet and red) with “Pion Identification”cuts, “L2” (light-blue and yellow) with “Pileup T1, and T2”cuts, and the final “clean” (dark-green and navy-blue) TSwith “Early Time and Acceptance” (All) cuts. See Section4.2 for discussion on cuts. The PH version is shown to be lesssensitive to pileup compared to the Q based branching ratio. 975.3 The time difference between subsequent hits in each T1 PMT.The leading times are fitted with an error function. The peakaround 30 ns is due to an after-pulse hit at a characteristictime after the real hit. . . . . . . . . . . . . . . . . . . . . . . 1015.4 a) T1 resolution function F2A(t) evaluated with ∆T = 15.7ns. b) T1 resolution pileup events with artificial ∆T = 100 ns. 1035.5 a, b, c, d, and e) Amplitudes of F2(t) from 2012 dataset’spileup events vs. artificial T1 double pulse resolutions (∆T)for different pre-pileup windows; points fitted with a quadraticcurve. If the double pulse resolution time ∆T was zero, theamount of pileup would not be negative below 15.7 ns. f)Each intercept at ∆T = 15.7 ns from subplots a) to e) is cor-related to the number of old-muon events from the LE timeregion. See Section 5.2.3 for discussion. . . . . . . . . . . . . 1045.6 The shape used in the time spectrum fit from positrons en-tering the calorimeter, missing the T1-hit requirement. Inte-grated charge (Q) based in blue and pulse-height (PH) basedin black. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106xxList of Figures5.7 The shape used in the time spectrum fit for pi+ → µ+νµγevents. contribution from NaI in red, CsI crystals in black,and the sum in blue. . . . . . . . . . . . . . . . . . . . . . . 1075.8 Time Spectra for 2012 dataset pulse-height (PH) Rpibased time fit. Left: LE time spectrum on a logarithmicscale (black line). Right: HE time spectrum on a logarithmicscale (black line). . . . . . . . . . . . . . . . . . . . . . . . . 1125.9 Residuals (data - fit) for 2012 dataset pulse-height(PH) Rpi based time fit. Top-Left: HE, negative times.Top-Right: HE, positive times. Bottom-Left: LE, negativetimes. Bottom-Right: LE, positive times. . . . . . . . . . . . 1135.10 Time Spectra for 2012 dataset integrated-charge (Q)Rpi based time fit. Left: LE time spectrum on a logarithmicscale (black line). Right: HE time spectrum on a logarithmicscale (black line). . . . . . . . . . . . . . . . . . . . . . . . . 1145.11 Residuals (data - fit) for 2012 dataset integrated-charge (Q) Rpi based time fit. Top-Left: HE, negativetimes. Top-Right: HE, positive times. Bottom-Left: LE,negative times. Bottom-Right: LE, positive times. . . . . . . 1155.12 Time Spectra for 2011 dataset pulse-height (PH) Rpibased time fit. Left: LE time spectrum on a logarithmicscale (black line). Right: HE time spectrum on a logarithmicscale (black line). . . . . . . . . . . . . . . . . . . . . . . . . 1165.13 Residuals (data - fit) for 2011 dataset pulse-height(PH) Rpi based time fit. Top-Left: HE, negative times.Top-Right: HE, positive times. Bottom-Left: LE, negativetimes. Bottom-Right: LE, positive times. . . . . . . . . . . . 1175.14 Time Spectra for 2011 dataset integrated-charge (Q)Rpi based time fit. Left: LE time spectrum on a logarithmicscale (black line). Right: HE time spectrum on a logarithmicscale (black line). . . . . . . . . . . . . . . . . . . . . . . . . 1185.15 Residuals (data - fit) for 2011 dataset integrated-charge (Q) Rpi based time fit. Top-Left: HE, negativetimes. Top-Right: HE, positive times. Bottom-Left: LE,negative times. Bottom-Right: LE, positive times. . . . . . . 1195.16 Time Spectra for November 2010 dataset pulse-height(PH) Rpi based time fit. Left: LE time spectrum on alogarithmic scale (black line). Right: HE time spectrum on alogarithmic scale (black line). . . . . . . . . . . . . . . . . . 120xxiList of Figures5.17 Residuals (data - fit) for November 2010 dataset pulse-height (PH) Rpi based time fit. Top-Left: HE, negativetimes. Top-Right: HE, positive times. Bottom-Left: LE,negative times. Bottom-Right: LE, positive times. . . . . . . 1215.18 Time Spectra for November 2010 dataset integrated-charge (Q) Rpi based time fit. Left: LE time spectrum ona logarithmic scale (black line). Right: HE time spectrum ona logarithmic scale (black line). . . . . . . . . . . . . . . . . 1225.19 Residuals (data - fit) for November 2010 dataset integrated-charge (Q) Rpi based time fit. Top-Left: HE, negativetimes. Top-Right: HE, positive times. Bottom-Left: LE,negative times. Bottom-Right: LE, positive times. . . . . . . 1236.1 Schematic drawing of the detector setup for special positronruns, showing rotating angle θ between the beam and calorime-ter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2 The energy spectrum from a 70 MeV positron beam parallelto the crystal axis. Data is shown in black and simulationis shown in red. The histograms are normalized to have thesame total number of events. . . . . . . . . . . . . . . . . . . 1266.3 Tail fraction below 53.7 MeV vs angle for the positron data(blue) and MC (red), equivalent to the 52 MeV cutoff in thepi+ → e+νe data. The 1σ error band for the tail fractions indata and MC overlap at all angles. . . . . . . . . . . . . . . 1286.4 Simulated Bina+CsI spectrum from pi+ → e+νe decay includ-ing radiative components and events that underwent Bhabhascattering in the target. . . . . . . . . . . . . . . . . . . . . . 1296.5 The BINA spectrum for events with a late hit (450 to 670ns) in CsI. Data in black, MC in red. The two photo-nuclearpeaks are enhanced. . . . . . . . . . . . . . . . . . . . . . . . 1316.6 Positron momentum distribution at F4 (target B3), for the75 MeV/c positron beam (run #54880). . . . . . . . . . . . . 1326.7 Positron momentum vs. angle distribution at F4 (target B3),for the 75 MeV/c positron beam (run #54880). . . . . . . . . 1336.8 Left: Sum of the energies in B1, B2, S1, S2, and B3. Right: Z-vertex for events with positron energy Ecut < 52 MeV (shadedhistogram) and Ecut > 52 MeV (blue full line). The twodistributions are normalized to the same number of events,and cuts applied are indicated by the red vertical dashed lines.Image from [11]. . . . . . . . . . . . . . . . . . . . . . . . . . 135xxiiList of Figures6.9 The pion stopping position Zv distribution from data. . . . . 1366.10 Acceptance correction CAcc as a function of the AR radius forthe 2012 dataset. Error bars are only statistical. . . . . . . . 1376.11 Time and energy spectra for µDIF. . . . . . . . . . . . . . . . 1397.1 Change in the branching ratio ∆R vs time resolution: Thex-axis is the time resolution from the scintillators. The y-axis is in ∆R units, with zero change representing 2012(PH)’snominal analysis (without time resolution effects). The un-correlated statistical error is zero for all points since thereis no change in statistics for this test. The blue solid linerepresents the 2012 dataset pulse-height (PH) based branch-ing ratio. The blue dashed line represents the actual timeresolution from the scintillators (B1 and T1) from which thetiming signal is extracted. The change in the branching ra-tio ∆R is < 1 [10−8] for time resolutions < 2 ns. Since thetime difference between B1 and T1 has the time resolution ofσ = (0.3± 0.1) ns the time resolution effects are negligible forthe branching ratio to our level of precision. . . . . . . . . . . 1427.2 ∆R ± ∆e (Eq. 7.1) vs. PrePU: The x-axis is the PrePUwindow in ns units (Figure 4.1). The y-axis is in ∆R units,with zero change representing 2012(PH)’s nominal analysis(PrePU cut enabled), the error bars (∆e) on each point rep-resent the uncorrelated statistical error between the point inquestion and the nominal point with the error bars going up,when there is a statistical increase, and down otherwise. Thehorizontal dashed black lines, both at the same distance fromnominal, represent the raw statistical error from the 2012dataset. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1477.3 ∆R±∆e (Eq. 7.1) vs Bin size: The x-axis is the bin size in nsunits. The y-axis is in ∆R units, with zero change represent-ing 2012(PH)’s nominal analysis (binning 1 ns), the error bars(∆e) on each point represent the uncorrelated statistical errorbetween the point in question and the nominal point with theerror bars going up, when there is a statistical increase, anddown otherwise. The horizontal dashed black lines, both atthe same distance from nominal, represent the raw statisticalerror from the 2012 dataset. . . . . . . . . . . . . . . . . . . 148xxiiiList of Figures7.4 ∆R ± ∆e (Eq. 7.1) vs. AR, Charge Integration and Pulse-height: The x-axis is the AR value in mm units. The y-axis is in ∆R (corrected) units, with zero change represent-ing 2012(PH)’s analysis using anchor point with cuts AR =60 mm and Ecut = 52 MeV, the error bars (∆e) on each pointrepresent the uncorrelated statistical error between the pointin question and the anchor point with the error bars going upwhen there is an statistical increase and down otherwise. Thehorizontal dashed black lines both at the same distance fromanchor represent the calorimeter’s LET systematic error. Thebottom part shows the total χ2 from the fitting function foreach point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1517.5 ∆R ±∆e (Eq. 7.1) vs. Ecut, Charge Integration and Pulse-height: The x-axis is the Ecut value in MeV units. The y-axisis in ∆R units, with zero change representing 2012(PH)’sanalysis using anchor point with cuts AR = 60 mm andEcut = 52 MeV, the error bars (∆e) on each point repre-sent the uncorrelated statistical error between the point inquestion and the anchor point with the error bars going upwhen there is an statistical increase and down otherwise. Thehorizontal dashed black lines both at the same distance fromanchor represent the calorimeter’s LET systematic error. Thebottom part shows the total χ2 from the fitting function foreach point. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1528.1 The 95% C.L. upper limit on the heavy neutrino mixing pa-rameter, as a function of its mass. The blue line shows theresult from the derived branching ratio upper limit from asubset of data (Run IV) published in 2015 [5]. . . . . . . . . . 1628.2 Background-suppressed pi+ → e+νe positron energy spectrum(black histogram). Fitted components include muon decaysin flight (thick blue line, from MC), pi+ → e+νe (green, dot-dashed line, fit to MC), and pi+ → µ+νµ → e+νeν¯µ (reddashed line, from late-time data events). The insert showsthe (rebinned) residuals (Data-Fit) with statistical error barsand the signal shape (massive neutrino search) in the case ofEe+ = 40 MeV and |Uei|2 = 10−8 [11]. . . . . . . . . . . . . . 163xxivList of Figures8.3 90% C.L. upper limits on the square of the mixing matrixelements |Uei|2 of heavy neutrinos coupled to electrons (thickred line) regarding the full PIENU dataset, , i.e., all runsfrom 2009 to 2012 [11]. The black dashed line shows theresults from the previous generation PIENU experiment [29]. 164C.1 The 0 degree positron energy spectrum cleanse trough WC12spatial and timing cuts. . . . . . . . . . . . . . . . . . . . . . 180C.2 The energy in Bina + CsI vs. the energy in T2. Blobs cor-responding to positrons (∼70 MeV), muons (∼18 MeV), andpions (∼14 MeV) can be clearly seen. There is also a struc-ture around 30 MeV in Bina + CsI, with energy loss in T2between positrons and beam muons. A similar structure ap-pears in simulated pion events, from decays in flight. . . . . . 181C.3 The time of flight vs. the energy in BINA + CsI. Blobs cor-responding to positrons and muons can be clearly seen. Theregion with essentially no events is due to the trigger condi-tion excluding part of the RF window. . . . . . . . . . . . . . 181C.4 The energy spectrum of positrons in BINA + CsI, selectedby time of flight. . . . . . . . . . . . . . . . . . . . . . . . . . 182C.5 The energy spectrum of muons in BINA + CsI, selected bytime of flight. . . . . . . . . . . . . . . . . . . . . . . . . . . . 182D.1 T1 production target apparatus for M13 beam extension. . . 184D.2 Beam input simulation . . . . . . . . . . . . . . . . . . . . . . 185D.3 Beam low momenta cleaning sequentially through differentbeam components and the final beam spot at F4. . . . . . . . 187D.4 Position profiles from MC and from positron run #54880 atF4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188D.5 Position profiles from MC and from positron run #81633 atF4. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188D.6 Focus point F1 slit simulation . . . . . . . . . . . . . . . . . . 189xxvList of FiguresD.7 Aerial view of beam-line simulation including all main com-ponents. Please refer to Figure 3.2 for blueprint. a) Rightto Left: Starting from the T1 production target 75 MeVwidth ±12 MeV positrons (red) are isotropically simulatedand go trough the first two focusing quadrupoles Q2. Only asmall solid angle is displayed. Positron passes horizontal slit(F0SL) and vertical jaws (F0JA) combo, then the first bend-ing dipole steers the beam CW, then low momenta cleanseis done trough F1SL/JA. Beam gets re-focused with threequads Q3, Q4, and Q5, enters another F2SL/JA to thenget bended CC and further focused by quads Q6 and Q7.Positrons enter the beam-line extension and positrons bendCW trough dipole B3 and final focusing is done with Q8, Q9and Q10. b) Same configuration but in this case only pions(green) are produced initially. Muons (blue) and positrons areproduced along each pion tree event but limited to one ver-tex. Additionally, a Lucite absorber is inserted after F1SLto separate the beam composition to enable magnetic selec-tion of pions further downstream and finally a collimator atthe beginning of the beam-line extension to filter the pions. . 190E.1 Complete trigger diagram of the PIENU Experiment [19]. . . 191F.1 Side view of the PIENU Detector. The pion beam comes fromthe right side. . . . . . . . . . . . . . . . . . . . . . . . . . . 192F.2 Cross section of the PIENU Detector. . . . . . . . . . . . . . 193F.3 The PIENU detector mounted to TRIUMF’s M13 beam-line. 194xxviAlgorithms5.1 Time Spectra Algorithm. . . . . . . . . . . . . . . . . . . . . 95xxviiGlossaryEach entry is followed by “(G)” if it is a term in general use or “(E)” if it isspecific to the experiment.Rpi (E)Γ(pi+→e+νe+pi+→e+νeγ)Γ(pi+→µ+νµ+pi+→µ+νµγ) branching ratio.SM (G)Standard Model of particle physics.piDIF (E)Pion Decay in Flight.piDAR (E)Pion Decay at Rest.LET (E)Low Energy Tail.LE (E)Low Energy.HE (E)High Energy.B1 (E)Scintillator target for the beam.NaI(Tl), NaI or Bina (E)The main calorimeter, a monolith crystal made of thallium-dopedsodium iodide.CsI (E)The 97 pure cesium iodide crystals surrounding Bina.Bina+CsI (E)The full PIENU calorimeter.xxviiiGlossaryGEANT4 (E)(for GEometry ANd Tracking) a platform for “the simulation of thepassage of particles through matter,” using Monte Carlo methods.Prescale (E)One of the three physics triggers. Pre-scaled by 16 to save redundantdata.Early (E)One of the three physics triggers. Selects events from an early timewindow.TIGC or BinaHigh (E)One of the three physics triggers. Selects events above an energythreshold.QFT (G)Quantum Field Theory.EM (G)Electro-Magnetic.QCD (G)Quantum Chromo Dynamics.U(N) (G)Unitary matrix of order N.SU(N) (G)Special Unitary matrix of order N.QED (G)Quantum Electro-Dynamics.QFD (G)Quantum Flavor-Dynamics.EWT (G)Electro-Weak Theory.CKM (G)CabibboKobayashiMaskawa matrix.CMB (G)Cosmic Microwave Background.xxixGlossaryΛCDM (G)Lambda Cold Dark Matter model.WIMPs (G)Weakly-interacting massive particles.ChPT (G)Chiral Perturbation Theory.BSM (G)Beyond the Standard Model.MSSM (G)Minimal Super-symmetric Standard Model.RPV (G)R-Parity Violation.xxxAcknowledgementsTo the entire PIENU collaboration and the Rare Decay Group at TRI-UMF for their contributions, followed by guidance, and comradeship alongthe way from those who I met personally: alphabetically, Alexis A. Aguilar-Arevalo, Dorothea vom Bruch, Luca Doria?, Shintaro Ito, Matek Lewczuk,Richard Mischke?,†, Toshio Numao?,†, Rohan Nuttall, Yevgeniy Petrov, Alek-sey Sher?, Tristan Sullivan?, Dima Vavilov, and Bob Velghe.To my advisor Douglas Bryman?,† for his trust, mentorship and financialsupport.To † for writing reference letters to secure my next job with the T2K-Canada group.To my committee, alphabetically, Omer Angel, Michael Hasinoff?, Christo-pher Hearty?, Klaus Kirch?, Takamasa Momose, Douglas Scott?, OliverStelzer-Chilton?,, Hirohisa Tanaka, also to faculty members, Gary Hin-shaw, William McCutcheon, and Scott Oser, and fellow graduate studentsTerry Buck, Ricardo Chavez-Gonzalez, Joochun (Jason) Park, Itamar Reis,and Rodrigo A. Vargas-Hernandez for their assistance in achieving candi-dacy and graduation. To  and Donald Witt for entrusting me with theirlectures on several occasions, which provided me with invaluable teachingexperience.To ? for giving critical comments on the draft stages for this thesis.To UBC, TRIUMF, CONACYT, SEP-DGRI, Universidad Autonoma deSinaloa, and Industrias Rochin SA de CV for their financial support. Tomembers of the Brock Hall Ballers, and the Strathcona Stevedores for thegood times throughout the years. To my family and friends for their un-conditional support, especially Hernelda Felix-Avendan˜o, Maria J. Rochin-Uriarte, and Ernelda Ramos-Felix.xxxiTo Nelly, Sara, and Lucia.xxxiiChapter 1IntroductionThe Prophecy. The pion is the lightest meson (quark-anti-quark boundstate) with a mass of 139 MeV/c2, and it was first predicted by Yukawa,when he published his theory of mesons in 1935 [33], as the carrier of a strongand short-range force that can bind nucleons in nuclei. In 1947, Powell andhis collaborators discovered the pion [34] by exposing photographic platesto cosmic rays at a high altitude, i.e., at the tops of mountains. Yukawa andPowell received the Nobel Prize in Physics in 1949 and 1950, respectively[35]. Another light particle, the muon has a mass of 105.7 MeV/c2; it wasdiscovered in 1936, 10 years before the pion, and as it was in the same massrange, it was initially thought to be Yukawa’s particle. The community hadshared confusion for years before realizing that the muon was some heavyelectron that was unable to interact with strong forces. Before the muon, thescientific community had only come across photons, protons, electrons, andneutrons. The particle physics revolution was still underway. Years later,Yukawa reflected on his seminal paper on particle interaction published in1934:“I felt like a traveler who rests himself at a small tea shop at the top of amountain slope. At that time I was not thinking about whether there wereany more mountains ahead.” Tabibito [36].Puzzles. The first puzzle was the observation of the pi+ → µ+νµ →e+νeν¯µ decay chain, but never the direct pi+ → e+νe decay3. From purephase space considerations, if the electron at 0.511 MeV/c2 is two ordersof magnitude smaller in mass than the muon, why do pions not decay di-rectly into positrons or electrons? In 1955 and 1957, two experiments, oneat Columbia University [37] and the other at the E. Fermi Institute [38],reported no direct electronic decay from pions, setting an upper limit on thebranching ratio defined as the relative rate of decay of pions into electrons3 pi+ → µ+νµ → e+νeν¯µ refers to a pion decaying to a muon and then to a positronwith their respective neutrinos; pi+ → e+νe refers to a pion decaying to a positron directly.1Chapter 1. Introductionover muons (including associated neutrinos and radiative components),Rpi =Γ(pi+ → e+νe + pi+ → e+νeγ)Γ(pi+ → µ+νµ + pi+ → µ+νµγ) . (1.1)The upper limit was set to Rexp1957 ∼ 10−6. Another puzzle at the time was theevidence for parity violation in weak interactions; C. Wu et al. confirmed itwith their beta-decay experiment in 1956 [39].At the time, parity violation could only be explained by the contemporaryvector-axial-vector (V-A) theory of weak interactions proposed by E.C.G.Sudarshan and R.E. Marshak [40]. In 1958, parity violation and the conceptof a universal form of weak interaction were combined into one theory byR.P. Feynman and M. Gell-Mann [41]. The approach predicted a branchingratio of pions decaying directly to positrons over muons of the order ofR(V-A)1958 ∼ 10−4 in contradiction with the experimental upper limit at thattime. The V-A theory explains how the mass dependent helicity suppression(Section 2.2.2) favors the muonic decay over the positron by four orders ofmagnitude.“These theoretical arguments seem to the authors to be strong enoughto suggest that the disagreement with the He6 recoil (a double focusingmagnetic spectrometer used by Anderson et. al.) experiment and with someother less accurate experiments indicates that these experiments are wrong.The pi → e+ ν problem may have a more subtle solution.” - Feynman andGell-Mann [41].Redemption. Later in 1958, the pi+ → e+νe decay mode was fi-nally discovered at CERN [42] and Columbia University [43]. Later, in1960, H.L. Anderson et al. obtained the first precise measurement [44] withRexp1960 = (1.21±0.07)×10−4, cementing and establishing the new V-A theoryas the correct description of the weak interaction, which was subsequentlyadopted into the Standard Model (SM) of particle physics. Since pions wereused to establish the SM, we can now use them to challenge it, measur-ing its properties with high precision and trying to detect deviations frompredictions. Bryman et al. reported the latest theoretical ratio update in2011 [24] at RSM2011 = (1.2352 ± 0.0002) × 10−4 which represents one of themost precisely calculated SM observable involving quarks.4 By contrast, the4Discussed in Chapter 2.21.1. Previous Measurementscurrent experimental value reported in 2015 by the PIENU experiment isRexp2015 = (1.2344±0.0023(stat.)±0.0019(syst.))×10−4 [5], representing onlyabout a tenth of our data, which is less precise than the theory by an orderof magnitude. Therefore, further precision is required. The PIENU experi-ment at TRIUMF was planned with the aim of improving the precision levelto 0.1%.Motivation and Status. Deviations from the SM prediction may im-ply a violation of lepton universality, the SM hypothesis that electrons andmuons have the same weak interactions; heavy neutrinos lighter than thepion [45]; and the presence of new physics beyond the SM, such as newpseudo-scalar interactions, i.e., R-parity violating super-symmetry pseudo-scalars [28], leptoquarks [46], and charged Higgs bosons [24]. In some in-stances, these indirect constraints can far exceed the reach of direct searchesat colliders. Most remarkably, a deviation from SM could imply the existenceof a new pseudo-scalar interaction with an energy scale up to O(1000 TeV),which would enhance the branching ratio by O(0.1%) [47].This dissertation represents the latest experimental measurement effortby the PIENU collaboration. The PIENU datasets contain four years ofdata, taken between 2009 and 2012, with 6.5 million (M) pi+ → e+νe events.The current analysis presented in this thesis is blinded, but includes thehighest quality data portion available, 3 M pi+ → e+νe events. Moreover,major experimental systematic problems have been solved recently, allowingfor increased precision up to 0.12% in Rexppi .1.1 Previous MeasurementsThe first precise measurement of the branching ratio was performed in1960 by Anderson et al, using a magnetic spectrometer [44]. The experi-mental ratio Rexp1960 = (1.21± 0.07)× 10−4 represents a precision level of 5%,and it was in complete agreement with the SM and the V-A structure ofthe weak interaction. The next milestone came in 1964, when Di Capua etal. [48] used a NaI (Tl) detector (length, 23 cm; diameter, 23 cm) sensi-tive to positrons as well as photons from radiative decays; the experimentcollected around 11k pi+ → e+νe events at Rexp1964 = (1.247 ± 0.028) × 10−4representing a precision level of 2%. Di Capua’s ratio was later revised toRexp1975 = (1.274 ± 0.024) × 10−4 [49], owing to a more accurate determina-tion of the pion lifetime and remained within 2σ from the SM’s theoretical31.1. Previous Measurementscalculation ratio RSMpi .Another generation of experiments was initiated in 1983 at TRIUMFby Bryman et al. [50] using a larger NaI(Tl) crystal, measuring Rexp1983 =(1.218 ± 0.014) × 10−4 from a sample of 0.032 M events. Such a ratio at aprecision level of 1% was within 1σ of RSMpi . In the 1990s, two subsequentexperiments were carried out in TRIUMF [13] (experiment E248, see Section1.2.1) and PSI [12], both collecting around 0.190 M events. The 1992 TRI-UMF experiment used a NaI(Tl) crystal as the main calorimeter, while the1993 PSI experiment used a 4pi steradian BGO5 calorimeter; both achievedcomparable levels of statistical and systematic uncertainties yielding as acombined result of RexpPDG1994 = (1.231± 0.005)× 10−4 [21]. This ratio, at aprecision level of 0.5%, was again within 1σ from RSMpi .Figure 1.1: History of the Rexppi branching ratio measurements. Red line:SM calculation [24]. Black dashed line: PDG experimental average [21].The current generation of Rexppi measurements is being performed at TRI-UMF and PSI [51] with similar precision goals. Recently, in 2015, thePIENU experiment at TRIUMF reported a subset of its data resultingRexp2015 = (1.2344 ± 0.0023(stat.) ± 0.0019(syst.)) × 10−4 [5], with only 0.4M events out of the 5 M available; this represents a 0.24% precision level,5Bismuth germanium oxide.41.2. Experimental Techniqueand is in agreement with the SM. The current average reported by the Par-ticle Data Group (PDG) is RexpPDG2018 = (1.2327 ± 0.0023) × 10−4 [21] withprecision of 0.19%. This weighted average includes all measurements from1986 to 2015. The PIENU experiment once finalized, will have an expectedprecision level of 0.1%. Figure 1.1 shows the experimental ratio time evolu-tion.Figure 1.2: Schematic illustration of experimental technique: two pionic de-cays in a scintillator target; and decay positrons collected by the calorimeter.1.2 Experimental TechniqueSince 1964, following Di Capua’s experiment [48], the same fundamentaltechnique has been used for every branching ratio measurement. One stops acharged pion beam in a scintillator target, thick enough to allow the pion todecay within it to either the pi+ → µ+νµ → e+νeν¯µ decay chain or directlyto pi+ → e+νe ; then, a calorimeter measures positrons from both piondecays (Figure 1.2). Muons deposit Tµ = 4.12 MeV of kinetic energy in thescintillator and decay within the target. pi+ → µ+νµ → e+νeν¯µ positrons,produce a broad energy distribution (referred to as muon spectrum) betweenits rest mass 0.511 MeV and a sharp endpoint at half the muon’s mass of 52.8MeV. The pi+ → e+νe positrons give rise to a mono-energetic peak at 69.8MeV. Most positrons (Ee+ > 5 MeV) from both decay channels, traverse51.2. Experimental Techniquehalf of target, and the rest of the remaining detector components traversingabout 8 cm upstream of the front of the calorimeter’s face. Positrons on thebeam’s axis traverse plastic scintillator, silicon, and aluminum depositingabout 3.7 MeV on average before entering the calorimeter. Time-wise, thepion lifetime at τpi = 26.0 ns is two orders of magnitude shorter than thatof the muon at τµ = 2.197µs. Different energies and timings allow fordistinction between the two decay modes.1.2.1 Lessons from the E248 experimentThe setup of the previous TRIUMF experiment E248 for the pion branch-ing ratio measurement is shown in Figure 1.3. Pions were stopped in ascintillator target, and the decay positrons were detected in a cylindri-cal NaI(Tl) crystal named “Tina”, whose axis of rotation was orientatedat 90◦ with respect to the beam to avoid beam-related backgrounds. Al-though the solid angle was only 2% of 4pi steradians, 0.190 M pi+ → e+νeevents were collected during six months of data-taking, resulting in Rexp1992 =(1.2265 ± 0.0034(stat.)) ± 0.0044(syst.)) × 10−4 [25]. The main systematicuncertainty came from the estimation of the pi+ → e+νe low energy tail(LET) “buried” under the broad pi+ → µ+νµ → e+νeν¯µ energy spectrum.The LET comes mainly due to energy loss in the calorimeter (Section 6.1).The LET needs to be estimated precisely in order to correct the branchingratio for those misidentified low energy positrons. In order to suppress thepi+ → µ+νµ → e+νeν¯µ events and estimate the size of this tail, tight cutswere used on the target energy to reject the muonic decay. The calorimeter’s“suppressed spectrum” is shown in Figure 1.3 (right). Clearly, a pi+ →µ+νµ → e+νeν¯µ component is still visible; these were mostly events wherethe pion decayed in flight (piDIF) before the target. The piDIF componentwas identified with the slightly higher energy deposit in target by the muondue to the Lorentz boost effect, thus through this mechanism the event ismisidentified as a pi+ → e+νe event and is carried over to the “suppressedspectrum” shown in Figure 1.3 (right).The calorimeter energy threshold was set at 56.4 MeV (3400 channel inFigure 1.3). The fraction of events below the energy threshold comparedto the total number of pi+ → e+νe events was around 20% and was domi-nated by these piDIF events. The large remaining tail and the low statisticswere limiting factors for precise estimation of the low energy tail (LET),61.2. Experimental TechniqueFigure 1.3: (Top) Experimental setup of the E248 experiment at TRIUMF[25]. (Bottom) Positron energy spectrum obtained by suppressing pi+ →µ+νµ → e+νeν¯µ events, the x-axis is energy in ADC counts (channel 3400corresponds to 56.4 MeV).71.2. Experimental Techniquewhich is the main correction for the branching ratio and therefore a lead-ing contributor to the final error. The LET arises because of energy lossesdue to electromagnetic shower6 leakage in the calorimeter measuring thepositron energy and from radiative decays ([25], and [13]). To increase thestatistics in the new PIENU experiment, the calorimeter was placed directlydownstream following the target scintillator, thereby increasing the angularacceptance of the isotropic positron tracks. Particle tracking hardware wasadded before the target to identify piDIF in order to reduce the uncertaintyin the LET. Also, the new PIENU detector was designed with the ability torotate the calorimeter setup relative to the beam angle to help characterizethe calorimeters response to a direct positron beam to further reduce theexperiment’s main correction and source of systematics for the branchingratio.1.2.2 PIENU techniqueIn the PIENU experiment 520 MeV protons from TRIUMF’s cyclotronprimary beam-line BL1 strike a Beryllium production target to generatepions that are subsequently collected by a secondary beam-line. Pions areselected with a momentum of 75 ± 1 MeV/c [1]; then, the beam is aimedat PIENU’s 8-mm-thick plastic scintillator target named “B3”. The beammomentum is tuned so that the pions will stop in the middle of target B3.Muons from piDAR have a penetration range of 1 mm within B3. Therefore,there is sufficient material to contain the decay vertex from both decayspi+ → µ+νµ → e+νeν¯µ and pi+ → e+νe . Figure 1.4 shows a GEANT47energy distribution in the target for both pion decay modes. Positrons fromboth decays enter the 48 cm × 48 cm (19 radiation-length long) single crystalPIENU calorimeter named “Bina”, made of Thallium-doped sodium iodideNaI(Tl) loaned from Brookhaven National Laboratory. To further containthe radiative shower energy leakage and reduce uncertainty in the LET, Binais surrounded by 97 pure CsI crystals. The PIENU calorimeter is named“Bina+CsI”. Figure 1.5(left) shows GEANT4 energy distributions for both6 An electromagnetic shower begins when a high-energy electron, positron or photonenters a material. At high energies (above a few MeV), photons interact with matterprimarily via pair production (electron-positron) by interacting with an atomic nucleus orelectron in order to conserve momentum. High-energy electrons and positrons primarilyemit photons, a process called bremsstrahlung. When photons fall bellow the pair produc-tion threshold, then energy losses of electrons (and positrons) from photoelectric effectsand Compton scattering start to dominate.7For GEometry ANd Tracking is a platform for “the simulation of the passage ofparticles through matter,” using Monte Carlo method [52].81.2. Experimental Techniquesignals.Figure 1.4: Energy deposited in the target B3 for pi+ → e+νe (blue line)and pi+ → µ+νµ → e+νeν¯µ (red line) events from GEANT4.Figure 1.5: (Left) Time spectra and (right) energy spectra in the calorime-ters of pi+ → e+νe and pi+ → µ+νµ → e+νeν¯µ decays obtained from sim-ulations. The spectra are normalized to the same amplitude. Using anenergy cut-off (Ecut) above the pi+ → µ+νµ → e+νeν¯µ energy spectrumedge (dashed line), we divide the energy spectrum into low-energy (LE) andhigh-energy (HE) parts The low energy tail (LET) from the pi+ → e+νe en-ergy spectrum is not visible due to scale and the pion decay in flight (piDIF)contribution was deactivated. The pi+ → e+νe time distribution peaks neart = 0 because of the relatively short pion lifetime.91.2. Experimental TechniqueIf the two decay modes and the various backgrounds are known precisely,then counting the number of events above and below the pi+ → µ+νµ →e+νeν¯µ energy spectrum edge can provide an estimate of the branchingratio. Such an estimate would ignore the background due to pile-up ef-fects (Chapter 4) and the dominant correction of the LET (Section 6.1).The main sources of background in the PIENU experiment are the beam-related background, pileup of muons from previous pion decays, pileup ofneutral particles, and photons emitted during the decay chains that couldshift energies in the detector leading to misidentification of events. Chap-ter 4 will address all backgrounds. As the exact energy distribution fromthe two main decay modes and backgrounds cannot be known with suffi-cient accuracy, we extract the number of events from the time spectrumdistributions, most of which are analytically well known. Figure 1.5(right)shows GEANT4 simulations of time distributions for both decays. Thepi+ → e+νe decay time distribution is an exponential with the pion life-time ∝ e−t/τpi . The pi+ → µ+νµ → e+νeν¯µ decay chain time distribution(derived in Appendix A) rises up to around 100 ns, and then falls with themuon lifetime ∝ (e−t/τpi − e−t/τµ). Using an energy cut-off (Ecut) abovethe pi+ → µ+νµ → e+νeν¯µ energy spectrum edge (dashed line in Fig-ure 1.5(Left)), we divide the energy spectrum into low-energy (LE) andhigh-energy (HE) parts, and we then build the two separate time spectra.The LE time spectrum contains mainly pi+ → µ+νµ → e+νeν¯µ and the HEtime spectrum mainly pi+ → e+νe events.The raw branching ratio can be extracted by performing a simultaneousfit of signal and background shapes from both the LE and HE time spec-tra. The raw ratio is corrected by the amount of LET calculated (the LETshape is not included in the time spectrum analysis) plus other correctionsrelated to the detector’s acceptance and the pion stopping position withinthe target. Chapter 6 describes the calculation of the corrections and theseparate experiment for LET calculation involving the rotation of Bina+CsIagainst a direct positron beam to obtain the energy response. Measuring theratio of the decay rates or the energy distributions does not require knowl-edge of the total number of incoming pions, as positrons from both decaychains are measured regardless of the mode. Most efficiencies of the cutsand triggers cancel in the measured ratio of decay, thus reducing the sys-tematic uncertainty. The geometrical acceptance and its correction are due101.3. Blind Analysisto energy-dependent multiple scattering, Bhabha scattering8, and pair pro-duction9. The time spectrum analysis (Chapter 4) is chosen over the energyanalysis, as it eliminates or reduces most sources of systematic error.1.3 Blind AnalysisA blind analysis is recognized as an important tool to reduce the impactof human conscious or unconscious bias, especially in a high precision mea-surement that will then be compared with a precise theoretical prediction.A well-known example of possible bias would be the experimental results ofthe neutron lifetime through the years as shown in Figure 1.6 [21]; the goodagreement of the central value for sets of consecutive experiments may be in-terpreted as bias. Several scenarios for blind analysis in particle physics havebeen executed, as discussed in many papers (e.g., [53] [54]). However, theblinding technique is fully dependent on the experiment and can sometimesbe difficult to implement. The blinding procedure should not artificiallyhide or create new systematic effects that would sabotage the analysis.In the PIENU experiment, the energy information in the target was usedto blind the branching ratio (Rpi). The Rpi value was changed withoutdistorting the time spectrum in which the fitting was performed. Figure 1.7shows the schematic of the blinding method in the PIENU experiment. Asmooth rectangular function (red line in Figure 1.7) with hidden efficiencywas used to remove pi+ → e+νe events. As pi+ → e+νe events were randomlyrejected, Rpi was changed without distortion of the time spectrum or thecalorimeter’s energy spectrum. This inefficiency factor was produced bya uniform random number between 0 and 0.5%. The same procedure wasapplied uniformly to all datasets so that they can be compared in systematictests such as Rpi vs. acceptance, energy cut-off (Ecut) for the HE/LE regime,and pileup. The position of the edge of the rectangular function was alignedto the position of valley between pi+ → e+νe and pi+ → µ+νµ peaks sothat the edge would be hidden under the statistical fluctuation of the lowstatistics region. The blinded events will be included in the analysis once allthe event selection cuts, shapes used in the time spectrum fit, and branching8 In quantum electrodynamics, Bhabha scattering is the electron-positron scatteringprocess mediated by the photon. The Bhabha scattering rate is used as a luminositymonitor in electron-positron colliders.9The creation of a subatomic particle and its antiparticle from a neutral boson. Ex-amples include creating an electron and a positron, a muon and an antimuon, or a protonand an antiproton.111.3. Blind AnalysisFigure 1.6: Evolution in time (years) of the neutron’s lifetime experimentalresult [21].ratio corrections are finalized. Moreover, the blinded branching ratio mustbe stable as we vary the parameters in the analysis, and all systematicerrors must be assigned before unblinding the result. The current analysispresented in this thesis is blinded.Figure 1.7: PIENU’s blinding technique. A smooth inefficiency function(unknown to the experimenters) removes events based on their energy de-posited in the target, lowering (case a) or raising (case b) the branchingratio.121.4. Thesis Outline1.4 Thesis OutlineIn Section 1.1, previous pion decay measurements are listed, and in Sec-tion 1.2, an overview of PIENU’s experimental technique is provided. Chap-ter 2 briefly explains the Standard Model (SM), i.e., the main theoreticalbackground for our experiment, provides a historical review of the currentbranching ratio calculation, beyond SM candidates for deviations and pa-rameter space limits linked to the experimental value. Chapter 3 providesa comprehensive description of the experimental setup with regard to thebeam-line, instrumentation, detectors, and software architecture involved.Chapter 4 briefly describes the variable extraction procedures, event selec-tion, and the energy spectra for the calorimeter. Chapter 5 outlines the timespectrum analysis procedure for obtaining the first-order or raw branchingratio. Chapter 6 explains the main corrections for the raw ratio. Chapter7 presents the results for the branching ratio, including systematic stabil-ity tests, total error budget, the dataset combination procedure, and futureprospects for further precision measurement improvement. Finally, Chap-ter 8 presents the final blinded branching ratio, and the new physics limitsreached assuming the central value from the branching ratio published bythe PIENU experiment in 2015 [5].13Chapter 2TheoryThe Standard Model (SM) is the theoretical framework for describing thepi+ → e+νe decay. A brief description is given in Section 2.1. Section 2.2deals with the electroweak theory for pion decay, and Section 2.3 presents themotivation beyond the Standard Model for the pi+ → e+νe measurement.2.1 The Standard Model of Particle PhysicsThe SM is a relativistic Quantum Field Theory (QFT) for particle physics.The SM was developed in the 20th century with ideas for unification, sym-metries, and gauge theories to describe the basic structure of matter andvacuum. The SM considers the fundamental particles (or fields) as indi-visible entities and their interactions are governed by known forces in theuniverse, i.e., Electromagnetic (EM), Weak, Strong, but not including Grav-ity. Table 2.1 shows the force mediators, called bosons. Topologically, theSM is a gauge theory based on the groupSU(3)︸ ︷︷ ︸strong×SU(2)× U(1)︸ ︷︷ ︸electroweak. (2.1)The model unifies the weak and electromagnetic force (“electroweak”force) within the SU(2) × U(1) groups, respectively. Mass is provided bya spontaneous symmetry-breaking mechanism driven by the presence of ascalar Higgs field. The Higgs mechanism has four degrees of freedom andafter spontaneous symmetry breaking three of them are “eaten up” by 3 ofthe 4 SU(2)× U(1) generators, leading to 3 new massive particles W± andZ0, while leaving the fourth massless particle identified with the photon (γ)of the electromagnetic interaction and their respective coupling constants gand α [55] [56] [57] [58].All known fundamental particles and some of their properties are summa-rized in Table 2.1 for full-integer spin bosons, and Table 2.2 for half-integer142.1. The Standard Model of Particle Physicsspin fermions. Fermions are the building blocks of all known nuclei, thus re-sponsible for all known elements. The SM bosons and fermions are massive,with the exception of the massless photons (γ) and neutrinos (ν). Never-theless, current experimental observations indicate ν do have mass. Eachcolumn in the Table 2.2 represents a generation10. Each particle has it isown antiparticle; in theory, they should have the same mass as one anotherbut opposite electric charges and differences in other quantum numbers. Forexample, an electron (e−) has a positive partner, the positron (e+). Otherneutral charged entities, such as the electron neutrino (νe), has a counterpart electron anti-neutrino (νe), and they differ by having opposite signsof lepton number11 and chirality12. A detailed historical and theoreticaldescription and the limits of the SM can be found in [55], [56], [57], and[58].Table 2.1: Bosons (integer spin).Mediator Coupling at ∼100MeV Range Behaviourgluon (massless) αs = 1.7 (energy dependent) 10−15m ∼ r (confinement)γ < 1× 10−18 eV α = 1/137 ∞ 1/r2W± = 80.38(1) GeV/c2 GFermi ∼ g2/m2WZ0 = 91.1876(21) GeV/c2 GFermi ≈ 10−5GeV−2H = 125 GeV/c210 Between generations, particles differ by their flavour quantum number and mass,but their interactions are identical.11Lepton number is a conserved quantum number representing the difference betweenthe number of leptons and the number of anti-leptons in an elementary particle reaction12 Chirality is a fundamental property of a particle; particles which differ in terms ofchirality can be viewed as an entirely different type of particle. It refers to how a particle’squantum mechanical wave function behaves when a particle is rotated (or looked at froma different angle). For example, a spin 1/2 (fermion) particle’s wavefunction will gain aminus sign under a 360 degree rotation, as the rotation changes the complex phase ofthe wavefunction. The particle’s chirality determines in a sense which way around thecomplex plain this phase travels to reach the -1, traveling in either a left handed way from1 to -1, or a right handed way from 1 to -1. A massive particle can have either left- orright-handed helicity dependent on the reference frame, but can only have one chiralityeither left- or right-handed. A massless particle helicity and chirality is the same for allframes. Only left-handed fermions and right-handed antifermions interact with the weakforce.152.1. The Standard Model of Particle PhysicsTable 2.2: Fermions (spin 1/2 integers).Generations ChargeI II III (Q/|e|)Leptonse = 0.511 MeV/c2 µ = 105.658 MeV/c2 τ = 1.77686(12) GeV/c2 −1νe < 2 eV/c2 νµ < 0.17 MeV/c2 ντ < 18.2 MeV/c2 0Quarksu = 2.2(5) MeV/c2 c = 1.275(35) GeV/c2 t =1 73.0(4) GeV/c2 +2/3d = 4.7(5) MeV/c2 s = 95(9) MeV/c2 b = 4.18(4) GeV/c2 −1/32.1.1 Electroweak interactionsWeak interactions are less familiar in everyday life than EM and actthough a mediator W± for a charged current decay channel or Z0 bo-son for a neutral-current decay channel on all known particles. Quantumflavour-dynamics (QFD) is the fundamental framework for the weak forcewith the SU(2) group; however, the electroweak theory (EWT) is used overQFD as it provides the best understanding of the weak processes under theSU(2) × U(1) groups. The weak interaction is responsible for radioactivedecay, such as beta decay, n → p + e− + νe. Characteristics of the weakinteraction are flavour -changing 13 of quarks, parity-symmetry P violation,and charge-parity CP violation [55] [56] [57] [58].The electromagnetic (EM) force acts using the photon (γ) boson as themediator between electric charge particles such as protons, electrons, muons(µ), and charged pions. The EM force is best described with quantum elec-trodynamics (QED) theory using the U(1) group. The EM force explainsthe structure of atoms, crystals, molecules, and chemistry in general, as wellas EM radiation or light. QED gives exceptionally accurate predictions forquantities such as the magnetic moment of the electron and the Lamb shiftof the hydrogen energy level [59].2.1.2 Strong interactionsStrong interactions act using a gauge boson called a gluon as the mediatoron some subset of the particles, i.e., hadrons, or anything made of quarks(q), such as protons (p), neutrons (n), and pions (pi0 or pi±). The stronginteraction is described with Quantum Chromo-Dynamics (QCD) using the13 Flavour refers to the species of an elementary particle. The Standard Model countssix flavours of quarks and three flavours of leptons. They are conventionally parameterizedwith flavour quantum numbers that are assigned to all subatomic particles.162.2. Pion Decay TheorySU(3) Lie groups. The role of the strong interaction in nature results innuclear binding or fusion, i.e., allowing the variety of nuclei to be formedin the stars, indispensable for their life cycle, and representing their mainsource of energy. The strong interaction is also responsible for nuclear fission,which is the ultimate source of energy of nuclear reactors, and weapons.An important feature of QCD is “asymptotic freedom”, meaning that thestrong coupling becomes smaller with increasing momentum transfer in par-ticle interactions. One of the consequences of the QCD running coupling isthat at low energies quarks are confined in uncoloured bound states (baryonsand mesons). In the low energy regime the QCD coupling constant cannotbe considered small and this implies that a perturbative treatment is notpossible. Non-perturbative methods such as Chiral Perturbation Theory(ChPT) and Lattice QCD have been used for strong interaction calculationsin the low energy regime, allowing for an expansion of the decay rates inpowers of the pion mass (Section 2.2.3) and the electromagnetic coupling,through which the uncertainty on the ratio can be tightly constrained.2.2 Pion Decay TheoryMesons are quark-anti-quark pairs bound together by strong forces. Pionsare mesons of the first generation; pi+ is made of an up (u) and anti-down(d) quark; pi− is made of a down (d) and anti-up (u) quark; and pi0 is acombination of a u with u or d with an d quark. As pions are the lightestparticles made of quarks, they can only decay via weak interactions. Thepi± has mass 139.57 MeV/c2 and a mean lifetime of 26.033ns. It onlydecays into lighter leptons, i.e., either a muon or an electron and a neutrino.Muons (µ) have mass of 105.658 MeV/c2 and a mean lifetime of 2.2µs. Theydecay through weak interactions principally to electrons and neutrino-anti-neutrino pairs.According to [21], the measured pi+ and µ+ decay modes are shownin Table 2.3 and 2.4, respectively. The PIENU experiment is sensible topi+→µ+νµ (Γpi1 ), pi+→µ+νµγ (Γpi2 ), pi+→e+νe (Γpi3 ), and pi+→e+νeγ (Γpi4 ).All other pion decay rate channels are below 10−7, negligible for our 0.1%level of precision and thus can be ignored here. Section 2.2.1 explains thedecay mode theory for Γpi1 and Γpi3 . Section 2.2.2 discusses helicity suppres-sion, which shows why the pi+ → µ+νµ → e+νeν¯µ decay is preferred over172.2. Pion Decay TheoryTable 2.3: Measured pion decay modes [21]. The radiative energy (Eγ1)restrictions are concerning the cited experiment, not the PIENU experiment.Decay mode Fraction (Γpii /Γpi)Γpi1 pi+→µ+νµ 0.9998770±0.00004Γpi2 pi+→µ+νµγ (2.00±0.25)×10−4 (Eγ > 1 MeV)Γpi3 pi+→e+νe (1.230±0.004)×10−4Γpi4 pi+→e+νeγ (7.39±0.05)×10−7 (Eγ > 10 MeV)Γpi5 pi+→pi0e+νe (1.036±0.006)×10−8Γpi6 pi+→e+νee+e− (3.2±0.5)×10−9Table 2.4: Measured muon decay modes [21]. The radiative energy (Eγ1)restrictions are concerning the cited experiment, not the PIENU experiment.Decay mode Fraction (Γµi /Γµ)Γµ1 µ+→e+νeνµ ≈100%Γµ2 µ+→e+νeνµγ (1.4±0.4)% (Eγ > 10 MeV)Γµ3 µ+→e+νeνµe+e− (3.4±0.4)×10−5the pi+ → e+νe one, and Section 2.2.3 comments on the radiative correctioncalculations concerning the Γpi2 and Γpi4 decay modes.2.2.1 Vector-Axial-Vector (V-A) Weak InteractionThe pi+ decay is described with the Feynman diagram [55] shown in Fig-ure 2.1. It illustrates the scattering process of pi+, u + d → l+ + νl de-cay, where l = e, µ. The internal wavy line represents the W+ boson asthe intermediate particle and the flavour changing processes, q, pl, and pνare the four-momenta for W+, anti-lepton l+, and ν, as indicated in Fig-ure 2.1. Each vertex has 4-momentum conservation using delta functions,−ig(2pi)4δ4(pu−pd− q) on the left and −ig(2pi)4δ4(q+pl−pν) on the right.4-momenta entering the vertex are positive, while those leaving are negative.The factors at each vertex and internal line are multiplied by the amplitudeintegral. The direction of time goes from negative left to positive right. Thearrows going right are particles and arrows going left are anti-particles. Theweak interaction coupling constant (g), is expected to be the same for allleptons in the SM.182.2. Pion Decay Theorydu l+νlqW+plpvpi+Figure 2.1: Feynman diagram for the pi+→l+νl decay, where l = e, µ. TheTikZ-Feynman package [60] was used.Expressing Fermi’s golden rule [55], the differential decay rate for pi+ →l+νl (where l = e or µ) can be written asdΓ =12mpi|M|2 1ElEνd3pl(2pi)3d3pν(2pi)3(2pi)4δ4(q − pl − pν), (2.2)where mpi is the mass of a pion. The matrix element M is the productof the propagator and the leptonic and hadronic currents, where the pionand lepton vertex currents Jµpi and Jlν are products of the particles’ wave-functions (Ψ) with 4-vector operators (O):M = igµνM2W − q2JµpiJlν ∼ (Ψ¯bOΨb)(Ψ¯lOΨl). (2.3)However, in our case, the momentum transfer is small compared to themass of the W boson so that the momentum transfer q in the propagator’sdenominator can be ignored. This is equivalent to assuming a Fermi point-like interaction, and the matrix element is M = 〈l+νl|L|pi+〉 and L is thecharged current Lagrangian [55]:LW+ =ig2√2W+µ (νmγµ(1− γ5)em + Vmnu′mγµ(1− γ5)d′m). (2.4)Dirac gamma matrices are γµ, where the summation index µ goes fromzero to three and γ5 = iγ0γ1γ2γ3. The index m goes from one to three,representing the particle generation with e1 = e, e2 = µ, and e3 = τ , asin the case of neutrinos νm. The term Vmnu′mγµ(1 − γ5)d′m = umγµ(1 −γ5)dm represents the flavor quark change due to the weak interaction and192.2. Pion Decay TheoryVmn comes from the Cabbibo-Kobayashi-Maskawa (CKM) matrix in thefollowing manner: d′s′b′ =Vud Vus VubVcd Vcs VcbVtd Vts Vtbdsb . (2.5)The primes indicate particles in the interaction basis, while the unprimedvector represents particles in the mass basis. The operator (1−γ5) is respon-sible for the parity-violating nature of the weak interaction. Parity violationis most easily seen by writing γ5 and the spinor representing a fermion inthe Weyl or chiral basis [61]:γ5 =1 0 0 00 1 0 00 0 −1 00 0 0 −1 , ψ = [ψLψR]. (2.6)Here, ψL and ψR are two-component objects, where the components rep-resent the two possible spin states of a spin 1/2 particle. The operator1−γ52 when multiplied against a spinor selects left-handed chiral particlesand right-handed chiral anti-particles as components.Expanding the matrix element into a hadronic and leptonic part withEq. 2.3 and Eq. 2.4, respectively. Together with the Fermi coupling constantrelation GF√2=g2l8M2Wgives [55],M = iGFVud√2〈0|d(γµ − γµγ5)u|pi+〉 l(pl)γµ(1− γ5)ν(pν), (2.7)where pl and pν are the momenta carried by the outgoing anti-lepton andneutrino, respectively. The first part of the brackets in Eq. 2.7 connectsthe pseudo-scalar pion to the scalar vacuum. The vector part of the weakinteraction becomes an expression with odd parity and therefore vanisheswhen the integral is completed. The remaining hadronic part in the bracketis the axial-vector component of the weak interaction. This term is not easyto calculate as it involves the strong interaction. In any case, we know thisterm should be a Lorentz 4-vector and the only one available is momentumtransfer qµ to the virtual W+ boson shown in Figure 2.1. The bracket isthen202.2. Pion Decay Theory〈0|dγµγ5u|pi+〉 = iFpiqµ (2.8)The term Fpi is the constant that parameterizes the strong interaction,i.e., the so-called pion decay constant. Then, the matrix element squaredafter summation over the spin states, and integrating Eq. 2.2 over outgoingparticle energies gives an expression for the decay rate:Γ0pi→l =G2FV2udmpiF2pim2l4pi(1− m2lm2pi)2. (2.9)To first order, the pion branching ratio R0pi is the ratio of decay ratesfrom pions to positrons and muons, which can be calculated using Eq. 2.9as follows:R0pi =Γpi→eΓpi→µ=g2eg2µm2em2µ(m2pi −m2em2pi −m2µ)2. (2.10)The strong interaction coefficients are canceled in the branching ratio,leaving the final equation as a ratio of lepton masses and the weak interactioncouplings g2e/g2µ. As the SM assumes “lepton universality”, i.e., ge = ge,then,R0pi = (1.28336±0.00002)×10−4 (2.11)The error in the first-order branching ratio R0pi not including the radiative(QED) corrections, comes from the uncertainty in the muon and positronmasses.2.2.2 Helicity SuppressionThis section explains how the charged pion decay plays the role of helicityin the weak interaction. As the muon is two orders of magnitude larger inmass, naively, we could say that the pi+→l+νl decay illustrated in Figure 2.1should have electronic mode dominance from pure phase-space considera-tions. However, the opposite happens because helicity suppresses electrondecay. Experiments have verified the establishment of the V-A form of theweak interaction. The operator 1−γ52 (2.4) selects only left-handed chiralparticles and right-handed chiral antiparticles, ultimately explaining parityviolation at a fundamental level.212.2. Pion Decay TheoryThe helicity of a particle is right-handed if the direction of its spin isthe same as the direction of its motion and left-handed if the directions areopposite; e.g., if a standard clock is tossed with its face facing forwards withits hands rotating as the spin vector, it has left-handed helicity. Formally,helicity is the sign of the projection of the spin vector onto the momentumvector: left is negative, right is positive. For massless spin 1/2 particles orantiparticles, helicity is equivalent to chirality. For massive particles, distinctchirality states have both right-handed and left-handed helicity componentsproportional to the mass of the particle.Considering the kinematics of positively charged pion decay at rest, asshown in Figure 2.2, the following can be deduced.ˆ As the pi+ spin is zero, the spins from the anti-lepton l+ and associatedneutrino νl must be opposite. Their momenta are anti-parallel or back-to-back.ˆ The neutrino mass is very small mν ≈ 0 and our energy frameworkgives the condition E  mν . We can approximate neutrinos as mass-less. Therefore, the associated neutrino νl must have left-handed he-licity and chirality.ˆ Angular momentum (helicity) must be conserved. Therefore, the anti-lepton l+ must have left-handed helicity.ˆ Weak interaction restrictions force anti-lepton l+ to have right-handedchirality.ˆ The matrix element given by Eq. 2.7 is proportional to the right-handed chiral component and left-handed helicity for the anti-leptonl+ spinor, M∝ mlmpi+ml .ˆ Hence, as the positron mass is much smaller than the muon mass, thepi+ → e+νe decay suffers heavily from “helicity suppression”, leadingto the 10−4 factor coming from the positron mass squared ∼ m2e, asshown in Eq. 2.10 and Eq. 2.11.2.2.3 Radiative CorrectionsThe first-order branching ratio R0pi calculated in Section 2.2.1 does notinclude radiative corrections. In leading order, radiative decay Feynman222.2. Pion Decay TheoryFigure 2.2: Fermi-point-like interpretation for pi+→l+νl decay: pion (mid-dle); anti-lepton (right); and neutrino (left). The helicity suppression mech-anism in the pion (spin zero) decay is illustrated: pl and pνl are the particles’momenta; the black arrows over the decay particles describe their spin state,which according to angular momentum conservation are opposite; and thehelicity states (in the case of massless neutrinos) both forced to be left-handed by the chiral V-A structure of the weak interactions (see text). Thismechanism leads to the suppression of the positron mode relative to themuon.diagrams based on to the emission of real photons are named Inner Bremm-strahlung (IBγ), as shown in Figure 2.3(a). Decays from the emission andre-absorption of virtual photons (ERγ) are shown in Figure 2.3(b). Thefirst attempt in the late 1950s to calculate the IBγ and ERγ radiative cor-rections for the branching ratio assuming a point-like pion were made byKinoshita [62] and Berman [63]. Although the calculation of these diagramsrequires both infrared and ultraviolet cutoffs14 to be imposed, their effecton Rpi can still be rigorously computed. The term involving the infraredcutoff cancels exactly for IBγ and ERγ processes, and the ultraviolet cutoffcancels in the branching ratio (equal contributions from both pi+ → e+νeand pi+ → µ+νµ decays were assumed), although it affects the individualdecay rates. Ultimately, a correction of -3.929% to Rpi was obtained.In the late 1970s, the pion was well known to have a structure, leadingto attempts at the calculation using proper gauge theories. First, Goldmanand Wilson [64] and later, Marciano and Sirlin [65], expanded the piondecay in a power series and found that structure-dependent contributions14 An infrared cutoff is the minimal value of energy or equivalently, the maximal wave-length that will be taken into account in a calculation, typically an integral.At the opposite end of the energy scale, an ultraviolet cutoff is the maximal allowedenergy or the shortest wavelength.232.2. Pion Decay TheoryFigure 2.3: Feynman diagrams for the radiative corrections to pion decay,from real (a) and virtual (b) photons.from ERγ and common interference IBγ components canceled each other;this allowed for a high precision calculation. It was also found that theleading lepton mass term is independent of strong interactions. Such aterm is in agreement with the Kinoshita and Berman calculations. In 1993,Marciano and Sirlin repeated the prediction with a proper assessment forthe uncertainty; the radiative correction constrained the branching ratio toRpi = (1.2352± 0.0005)× 10−4 [66].In 2007, Cirigliano and Rosell [47] recalculated the corrections using ChiralPerturbation Theory (ChPT). ChPT uses a low-energy effective field theoryfor QCD, allowing for strong interaction calculations. ChPT enabled a powerseries solution for the radiative corrections,Rpi = R0pi[1 + ∆e2p2 + ∆e2p4 + ∆e2p6 + ...][1 + ∆LL] . (2.12)The terms ∆e2pn represent decay-rate expansions in powers of p pro-portional to the pion mass and electromagnetic coupling constant e. The242.2. Pion Decay Theorypion point-like calculation is equivalent to the leading electromagnetic term∆e2p2 ; the next term ∆e2p4 represents the structure-dependent correctionwith prominent uncertainty in the prediction. The ∆e2p6 term arises fromthe emission of a photon by the decaying pion, which evades the helicity sup-pression and must thus be taken into account despite being of higher order.Photons emitted at any other part of the pion decay diagram, such as realbremsstrahlung from the decay lepton or a loop starting on the W line, donot affect the helicity suppression. Finally, ∆LL represents the lepton masscorrections of order αn lnn(mµ/me). Table 2.5 lists the values for Eq. 2.12and the branching ratio sums up to Rpi = (1.2352± 0.0001)× 10−4 [47].Table 2.5: Summary of the electroweak corrections for Rpi0Power counting Corrections (%) from [47]∆e2p2 −3.929∆e2p4 0.053±0.011∆e2p6 0.073∆LL 0.054aa The original correction in [47] is 0.055%, but because of a shift in thepion’s mass, it has become 0.054% [24].In 2011, Bryman et al. [24] reported an additional 0.01% uncertaintyfrom two-loop diagrams contributions from O(α2) terms. After includingthe point-like (∆e2p2) and structure-dependent (∆e2p4) radiative correctionsterms, together with the higher order final state photon (∆e2p4), and leptonmass corrections (∆LL) to the vector-axial first-order branching ratio andadding their respective uncertainties in quadrature, the final branching ratiotakes the value [24]Rpi = (1.2352± 0.0002)× 10−4, (2.13)which is in agreement with all previous calculations. The theoretical uncer-tainty prediction of Rpi is 0.016%. Such a level of precision in a hadronicdecay is possible because the strong interaction dynamics cancels out thebranching ratio and the structure dependence appears only through elec-troweak radiative corrections. The next section will explain the new physics252.3. Motivation Beyond the Standard Modelthat can be found if a measurement deviates from theRpi calculation (Eq. 2.13);if it is in agreement, then new constraints could be set on SM extensions.2.3 Motivation Beyond the Standard ModelThe Standard Model (SM) has been extremely successful at describinginteractions among the known particles. However, there are still unsolvedmysteries, e.g., the three generations and different mixing in the lepton andquark sector, the nature of neutrinos and their masses, the large range ofparticle masses from <eV to GeV, and the relative small mass of the Higgsboson. The SM does not provide sufficient CP violation to explain the mat-ter/antimatter asymmetry in the observed universe. There is no explanationfor the presence of dark matter and dark energy, which ultimately affects thestructure and fate of the universe over SM matter-energy. Thus, the SM isbelieved to be an effective low energy approximation of a more fundamentaltheory. Beyond Standard Model (BSM) theory could be discovered by pro-ducing new particles at high-energy colliders. Moreover, BSM theory canalso be manifest through SM predictions by the presence of virtual effectsof new particles [55] [56] [57] [58].BSM or new physics (NP) effects at the weak TeV scale could be foundat the precision level of 0.1%. If the Rpi measurement is consistent withthe SM, new constraints could be set for new physics scenarios on SM ex-tensions. Examples include, lepton universality violation (section 2.3.1),and new pseudo-scalar interactions (section 2.3.2), including R-parity vio-lating super-symmetry, lepto-quarks, and charged Higgs (non-SM coupling).Other BSM possibilities are partial compositeness (section 2.3.3), and mas-sive neutrinos lighter than the pion (section 2.3.4). In some instances, theseconstraints can far exceed the reach of direct searches at colliders; under theassumption that a deviation from the SM is found, a new pseudo-scalar inter-action with an energy scale up to O(1000 TeV) could enhance the branchingratio by O(0.1%) [47]. More recently, an analysis of renormalization-groupevolution has denied that the current precision measurement and calculationof meson decays (i.e, pi+ → e+νe ) sets a scale for BSM at O(500 TeV) [67].2.3.1 Lepton UniversalityThe assumption that the W boson couples with equal strength with everylepton generation, i.e., the coupling is flavour independent, was used to262.3. Motivation Beyond the Standard Modelderive R0pi. Such an assumption is known as lepton universality. The SMleptons differ only by their mass, and their electroweak coupling constant isthe same. Going back to Eq. 2.10, we could introduce the hypothesis thatthe coupling constants are different for each generation (g = ge = gµ = gτ )and then the branching ratio expression becomesRSMpi =(gµge)2Rexppi . (2.14)Hence, using the measured Rexppi and calculated RSMpi branching ratio, thecoupling constants ratio for the electron-muon universality test becomes ac-cessible to the PIENU experiment. Constraints on the ratios of the couplingconstants come from many different types of precision measurement exper-iments using W bosons, τ -lepton, or pi and K meson decays; examples areW → lν, pi → lν, W → lν and τ → lντνl decays. Lepton universalitytests with pi and τ decays give comparable precision, but complementary as-pects: pi+ → e+νe currently provides the most precise test of electron-muonuniversality, although the branching ratio of τ -lepton decays to muons andelectrons is close. These tests are not exactly equivalent; since the pion isspin zero while the tau is spin 1/2, the mediating W boson in the pi case mustbe in the spin zero state, whereas in the τ case all spin states contribute.Table 2.6 summarizes the most recent results.Loinaz et al. [26] parameterized the couplings gl to quantify the currentbounds asgl→g(1− εl2). (2.15)The linear combinations of εl constrained by W , τ , pi, and K decay mea-surements are given bygµge= 1 +εe − εµ2,gτgµ= 1 +εµ − ετ2, andgτge= 1 +εe − ετ2. (2.16)Setting ∆eµ≡εe−εµ, ∆µτ≡εµ−ετ , and ∆eτ≡εe−ετ , the experimental boundscan be evaluated in the parameter space of lepton universality constraints, asshown in Figure 2.4. The PIENU experiment aims to restrict BSM theorieswith a measurement of the pi+ → e+νe branching ratio within 1.0 to 0.01%precision.272.3. Motivation Beyond the Standard ModelTable 2.6: Experimental results on lepton universality (LU) tests from stud-ies of pi, K, τ , µ and W decay. In some cases, µ and τ ’s lifetime (τµ, and ττ )measurements were used in combination for LU tests. Here, B representsthe branching fraction of a particular decay mode.Decay mode, and lifetimes gµ/geΓpi→µ/Γpi→e 1.0004± 0.0012[5]Bτ→µ/Bτ→e 1.0018± 0.0014 [68]BK→µ/BK→e 0.996± 0.005 [69]BK→piµ/BK→pie 1.002± 0.002 [70]BW→µ/BW→e 0.997± 0.010 [70]gτ/gµBτ→e, τµ, ττ 1.0011± 0.0015 [68]Bτ→pi/Bpi→µ 0.9963± 0.0027 [68]Bτ→K/BK→µ 0.9858± 0.0071 [68]BW→τ/BW→µ 1.039± 0.013 [70]gτ/geBτ→µ, τµ, ττ 1.0029± 0.0015 [68]BW→τ/BW→e 1.036± 0.014 [70]Recently, charged current (CC) second-order weak interactions have beenmeasured, pointing towards lepton universality violation. LHCb reportedflavour-changing neutral-current processes B+ → K+l+l− [71], where l =e, µ, and the charged-current processes B0 → D∗+l−νl [72], where l = µ, τ .The first process yielded an excess of 2.8σ in the electron mode, and thesecond gave a surplus of 2.1σ in the τ mode. The BaBar collaborationalso reported a 2.7σ excess in this mode and a 2.0σ excess in the similarB0 → D+τ−ντ [73]. On the other hand, the latest test of flavour universalitythrough measurement of the B0 → D∗−τ+ντ [74] branching ratio to themuon channel is in agreement with the SM prediction and with previousmeasurements. Alternatively, the latest test of lepton universality with theB0 → K∗0l+l− [75] branching ratio (where l = µ, e), is compatible withthe SM expectations. A comparison of second-order measurements with SMpredictions [27] is shown in Figure 2.5. A comprehensive review of leptonuniversality tests in B decays can be found in ref. [76].282.3. Motivation Beyond the Standard ModelFigure 2.4: The limits on ∆µτ and ∆eτ from (a) W -decay, (b) τ -decay, (c) piand K-decay, and (d) all decays combined. The 1σ bands are shown for eachcoupling constant ratio, ignoring correlations. The shaded areas representthe 68% (dark grey) and 90% (light grey) confidence contours, includingcorrelations (Figure from ref. [26]).The LHCb and BaBar second-order weak interaction deviations from uni-versality, required to explain these measurements, are extensive compared tothe uncertainties stated in Table 2.6. To interpret these results concerningnew physics, while remaining consistent with other measurements, generallyrequires the new physics to couple preferentially to the third generation ofparticles [77]. Beyond SM theories have proposed solutions such as newvector bosons W ′, similar to the electroweak ones but more massive, which292.3. Motivation Beyond the Standard ModelFigure 2.5: Comparison of measurements with SM predictions: The branch-ing fraction B is B− → τ−ντ (left), the ratio R(D) is B → Dτ−ντ overB → De−νe (center), and R(D∗) is B → D∗τ−ντ over B → D∗e−νe (right)by BABAR, Belle, and LHCb. The data points indicate statistical and to-tal uncertainties. ST and HT refer to the measurements with semileptonicand hadronic tags, respectively. The average values of the measurementsand their combined uncertainties, obtained by the Heavy Flavor AveragingGroup, are shown in red as vertical lines and bands, and the expectationsfrom the SM calculations are shown in blue. Image and data from ref. [27].couple differently among generations for quarks and leptons. Another possi-bility is a new charged spin-0 Higgs boson. Lastly, SUSY theories genericallypredict the presence of charged Higgs particles [78] [79].2.3.2 New-Pseudo-scalar InteractionsMeasurements of pseudo-scalar meson (pion) decay can provide high pre-cision in searches for new pseudo-scalar interactions for beyond SM theories,as such decays are highly helicity suppressed. Electroweak renormalizationeffects or loop corrections can generate new-pseudo-scalars such as lepto-quarks, super-symmetric (SUSY) particles at loop level, and charged Higgsbosons [78] [79]. Taking the pion decay matrix element for leptonic andhadronic currents from Eq. 2.7,M = iG√2〈0|(V −A)u|pi+〉 l(pl)γµ(1− γ5)ν(pν). (2.17)Here, the bracket 〈0|(V −A)|pi+〉 connects the pseudo-scalar particle to vac-uum. As explained in Section 2.2.1, the vector part vanishes, leaving onlythe vector-axial-vector contribution.302.3. Motivation Beyond the Standard ModelInstead, a general bracket 〈0|O|pi+〉 is proposed, allowing beyond SMphysics, where O can be a scalar (S), pseudo-scalar (P), vector (V), oraxial-vector (A) operator. The pion is P, and since only P and A termsgive non-vanishing contributions, the transition amplitude is [79]〈0|uγ5d|pi〉 = i√2 fpim2pimu +md= i√2f˜pi. (2.18)The effective Fermi pseudo-scalar contact Lagrangian assuming only left-handed neutrinos, isLP = −i ρ2Λ2[l(1− γ5)νl][uγ5d], (2.19)where ρ is the coupling constant for the new pseudo-scalar and Λ is its massscale. The Lagrangian LP leads to a new pseudo-scalar matrix elementMP .The final matrix elementMBSM will be a coherent sum ofMP andM, theSM (V-A) matrix element from Eq. 2.17. After squaring the BSM totalmatrix element and summing over final states, assuming that lepton univer-sality holds for the new interaction, the branching ratio becomes accessible[79],1− RexppiRSMpi∼ ±√2piG1Λ2f˜pime∼(1TeVΛ)2× 103. (2.20)Considering real coupling of approximately the same strength as theweak interaction, the most significant contribution from the BSM matrixcomes from the interference term proportional to 1Λ2. The PIENU exper-iment aims to reach a 0.1% precision measurement; thus, we are sensitiveto a new pseudo-scalar interaction at the 1000 TeV mass scale, well beyondthe reach of any present direct searches at colliders. The pseudoscalar in-teraction can potentially be induced at one loop through three classes ofdiagrams: scalar-dressed Z exchange box diagrams, scalar-dressed W ex-change box diagrams and radiative corrections to the quark vertex (Figure2.6). The weak interactions do not respect parity and the scalar interactionschange chirality, thus diagrams of this form can potentially induce a pseu-doscalar interaction. Pseudo-scalar BSM candidates include leptoquarks,SUSY particles, and charged Higgs bosons. In the following section thesecandidates will be described briefly.R-Parity Violation SUSYThe Minimal Super-symmetric Standard Model (MSSM) is an extensionof the SM. It has been shown that the MSSM can induce non-universal312.3. Motivation Beyond the Standard ModelFigure 2.6: Feynman diagrams for pseudo-scalar interactions induced atone loop including three classes of diagrams: scalar-dressed Z exchangebox diagrams (top), scalar-dressed W exchange box diagrams (middle) andradiative corrections to the quark vertex (bottom).contributions and modify the branching ratio calculation RSMpi by a quan-tity δRSUSYpi , which can arise either at the tree or loop levels [28]. If R-parity is conserved, then the value of δRSUSYpi is negligible for current ex-perimental reach [80] or requires very large mass splitting between the left-handed sfermion [28]. A sfermion is a hypothetical spin-0 super-partnerparticle (sparticle) of its associated fermion. The R-parity definition isPR = (−1)3B+L+2S , where S is spin, B is baryon number, and L is lep-ton number. All SM particles have R-parity of +1, while super-symmetricparticles have R-parity of −1.322.3. Motivation Beyond the Standard ModelFigure 2.7: Tree level RPV contributions to Rpi [28].If we consider R-parity violation (RPV) together with lepton numberconservation violation, then the effects on δRSUSY)pi are measurable at thetree level for the current PIENU experiment’s precision [28]. Alternatively,if no deviation is found, new constraints could be set on MSSM. In thepresence of R-Parity Violation (RPV) interactions, tree level exchanges ofsfermions shown in Figure 2.7 lead to violations of lepton universality withviolation of lepton number (∆L = 1) and no helicity suppression in the Rpi.The magnitude of these tree level contributions is determined by both thesfermion mass and the parameters λ′11k and λ′21k, which are the coefficientsin RPV interactions [28].The RPV interactions are related to Rpi as follows∆RRPVpiRSMpi= 2(∆′11k −∆′21k), (2.21)∆′i1k(f˜) =λ′i1k4√2Gm2f˜i = 1, 2, (2.22)where λ′11k and λ′12k are the parameters related to the RPV interaction forthe decay into a positron or a muon respectively, mf is the mass of theexchange sfermion, and G is the Fermi constant. The allowed regions forλ′11k and λ′12k from precision measurements of electroweak parameters areshown in Figure 2.8, at the 95% confidence level [28]. The dark blue lineencloses current constraints on these parameters using an old 1.0% precisionPDG value of the branching ratio, Rexppi = 1.230(4)× 10−4. The dashed redline shows the future expected experimental 0.1% precision from the PIENUexperiment and the light green line shows the prospective impact of a futuremeasurement of the proton weak-charge at Jefferson Lab [81].332.3. Motivation Beyond the Standard ModelFigure 2.8: Present 95% C.L. constraints on RPV parameters ∆′11k and ∆′21kthat enter Rpi obtained from a fit to precision electroweak observables [28].The dark blue contour shows the current constraints on these parameters(the interior is the allowed region). The dashed red line shows the contourwhen adding the future expected experimental precision (0.1%) from thePIENU experiment, assuming the same central value. The light green curveindicates the prospective impact of a future measurement of the proton weakcharge at Jefferson Lab [81].Charged Higgs BosonSome SM extensions [82] [83], predict the existence of a charged Higgsdoublet boson H±. Assuming an H± coupling constant of g/2√2λud to thepseudo-scalar current〈0|d(γ5)u|pi+〉 and g/2√2λlν to the leptonic currentl¯(1 − γ5)νl, where g is the SU(2)L gauge coupling and λ is the chirality-breaking factor, we can access a deviation from the Rexppi experimental mea-surement [24]:1− RexpRSM=2m2pime(mu +md)m2Wm2H±λud(λeν − memµλµν). (2.23)If we assume lepton universality for the electroweak coupling constants,i.e., λeν/λµν = me/mµ, then RExppi = RSMpi in Eq. 2.23 and no experimen-tal constraints can be reached. On the other hand, if the charged Higgsdoublet couplings are λeν ∼ λµν ∼ λud ∼ α/pi (where α is the electromag-netic coupling constant), then measuring the Rexppi branching ratio at the0.1% level will allow access to a relatively high mass to the charged Higgs,342.3. Motivation Beyond the Standard ModelmH± ∼ 400 GeV [24].LeptoquarkBeyond-SM frameworks postulate leptoquarks as particles carrying bothlepton and baryon quantum numbers; therefore, they can act as mediatorsbetween quarks and leptons. Leptoquarks can be chiral or non-chiral, al-lowing them to couple to both left- and right-handed leptons and quarks.The pi+ → e+νe decay set strong constraints on non-chiral leptoquarks withbounds on the mass MLQ and couplings gL, gR of M2LQ/gLgR ≥ (100 TeV)2[84]. If chiral components that couple left-handed particles are required,the pion decay can still set constraints on pseudo-scalar leptoquarks in asingle representation. Assuming similar coupling as the strong interaction,the bound from the Rpi branching ratio is MLQ/g ≥ 12 TeV [46].2.3.3 Partial CompositenessThe existence of substructure for a particle previously considered ele-mentary is referred to as “compositeness”. The Higgs boson representsan important case of not being an elementary particle, which drives theelectroweak symmetry breaking obtaining a non-zero vacuum expectationvalue. In compositeness scenarios the Higgs boson is instead a pseudo-Nambu-Goldstone 15 particle resulting from the formation of a condensatein a new strong interaction (a new “force”) [85]. Alternatively, partial com-positeness [86] is a model to explain the fermion masses, where the standardmodel (SM) fermions mix with new composite fermions and become mas-sive. Precise measurements of pi/K → eν branching ratios give importantconstraints on the parameter space, since partial compositeness unavoidablyleads to lepton flavour violation.2.3.4 Heavy NeutrinoNeutrino mass is zero according to the SM; however, flavour-oscillationdata indicate that at least two have non-zero values [87]. This is a clearsign of new physics. Their masses and their nature (e.g., are neutrinos theirown anti-particles?) is an area of current research. In order to explain therelatively small observed neutrino masses and to resolve some experimental15In particle and condensed matter physics, Goldstone bosons or NambuGoldstonebosons (NGBs) are bosons that appear necessarily in models exhibiting spontaneous break-down of continuous symmetries.352.3. Motivation Beyond the Standard Modelanomalies [88] observed at LSND and MiniBOONE, for example, the exis-tence of additional neutrino states is hypothesized. The Neutrino MinimalStandard Model (νMSN) [89] is an extension of the SM. This framework, inaddition to the left-handed neutrinos να(α = e, µ, τ), postulates three ad-ditional right-handed “sterile” neutrinos. The right-handed neutrinos havezero electric, weak, and strong charges; therefore, they are called sterile.The (νMSN) Majorana masses of the right-handed neutrinos are chosen tobe below the electroweak scale, and via small Yukawa couplings, the νMSNachieves the smallness of the first left-handed neutrino masses, consistentwith the gauge symmetries and see-saw mechanism of the SM. According toνMSN, the two more massive states of the sterile neutrinos are responsiblefor baryogenesis, and the lightest one can be a candidate for dark matter inthe keV/c2 range, since using a smaller Yukawa coupling would cause the thelight-sterile neutrino lifetime to exceed the age of the universe. In the earlyuniverse sterile neutrinos are produced by their coupling with left-handedneutrinos.Simple re-normalizable dark matter models addressing problems withsmall-scale structure formation of the universe [90] postulate a dark mat-ter candidate that can couple to a sterile heavy neutrino via a new darksector mediator. The model requires heavy neutrinos in the 100 MeV massrange and roughly 10 MeV dark matter particles. The same model hasbeen systematically explored [91] for addressing dark matter annihilationand thermalization via interactions with heavy neutrinos. More generally,for k sterile neutrinos, the weak eigenstates νχk are related to the masseigenstates νi by a unitary transformation matrix Uli, whereνl =3+k∑i=1= Uliνi, (2.24)with l = e, µ, τ, χ1, χ2...χk. In particular, sterile neutrinos with MeV/c2 toGeV/c2 masses can have measurable effects on meson decays that can beexplored by precisely measuring their decay branching ratios or by searchingfor extra peaks in the energy spectrum of their leptonic two-body decays(e.g., pi,K,B → lν) [45]. The presence of any neutrino heavier than a fewMeV will weaken the helicity-suppression mechanism and thus modify theRpi branching ratio. Thus, the PIENU experiment is sensitive to neutrinosbelow the pion mass range, i.e., 0 to 130 MeV, and particularly to thoseabove 55 MeV.362.3. Motivation Beyond the Standard ModelBelow 55 MeV. The rate of the decay pi+ → e+νi, where νi is a heavyneutrino, relative to the rate of the pi+ → e+νe decay, is given by [92]Γ(pi+ → e+νi)Γ(pi+ → e+νe) = |Uei|2ρe, (2.25)where Uei is the mixing parameter between νe and νi, and the kinematicfactor isρe =√1 + δ2e + δ2i − 2(δi + δe + δiδe)δe(1− δe)2 × δi + δe − (δi − δe)2. (2.26)with δe = m2e/m2pi, δi = m2νi/m2pi, and the massive neutrino mass mνi isrestricted by the two-body decay mechanism to bemνi =√m2pi − 2mpiEe+ +m2e. (2.27)The presence of a massive neutrino will modify the branching ratioRexp =N(pi → eν) +N(pi → eνM )N(pi → µ→ e)= RSM +N(pi → eνM )N(pi → eν)N(pi → eν)N(pi → µ→ e)= RSM + |Uei|2ρeRSM,(2.28)leading to|Uei|2 = r − 1ρe − 1 , (2.29)where r = Rexp/RSM. Thus, the limits on the mixing matrix |Uei|2 can becalculated as a function of neutrino mass mνi .Above 55 MeV. In the case of the pi+ → e+νe decay, heavy neutrinostates with masses below the pion mpi can be searched for with a peaksearch on the decay lepton energy spectrum. In particular, leptonic two-body decays like pi+ → e+νe have a fixed kinematics which results in aprecise final state energy for the lepton, given the pion four-vector. In thecase of the PIENU experiment, the pion is at rest and therefore the leptonenergy was fixed by energy-momentum conservationEe+ =m2pi +m2e −m2ν2mpi. (2.30)372.3. Motivation Beyond the Standard ModelNear 55 MeV and above, the positron energy is low enough that an extrapeak (sterile massive neutrino) would appear in the pi+ → e+νe energyspectrum. Figure 2.9 shows the upper limit on |Uei|2 obtained through asearch for extra peaks in PIENU data taken in 2009, compared with thelimits from the previous PIENU experiment [3] [29].Figure 2.9: The 90% C.L. upper limit on the heavy-neutrino mixing param-eter, as a function of its mass. The dashed line shows the result from theprevious PIENU experiment [29], and the circles and triangles are the limitsfrom a subset of PIENU data, published in 2011 [3]. The circles indicate arestricted angular region was used when constructing the pi+ → e+νe energyspectrum.38Chapter 3Experiment3.1 Cyclotron and Beam-lineThe PIENU experiment used a 520 MeV proton beam from TRIUMF’s cy-clotron with an average intensity up to 400 µA, divided among four primarybeam-lines. The cyclotron with a diameter of 18 m and main magnet weigh-ing 4000 tons has an accelerating gradient provided by a 23.05 MHz 93 kVradio-frequency (RF) field and delivers 4-ns wide bunches every 43.4 ns withan intensity of 100µA through the primary beam-line (BL1A). The protonbeam was aimed at a 1cm-thick Beryllium production target T116 locatedin the Meson Hall, as shown in Figure 3.1. The proton beam hitting T1produces several types of particles, such as photons, neutrons, protons, andpions, each with wide energy distribution. The secondary beam-line M13delivers particles from the T1 production target in vacuum to the PIENUdetector. The M13 low energy beam-line (0-130 MeV/c) was tuned to selectpositively or negatively charged particles of momentum 75 MeV/c with a1% spread. The final beam composition used for the PIENU detector wasapproximately 85% pion, 14% muon, and 1% positron [1].The original M13 beam-line [93] begins from the BL1A at an angle of135◦ from T1 with a maximum angular acceptance of 29 mili-steradian.M13 is a low-momentum achromatic channel with −60◦ (B1 magnet) and+60◦ (B2 magnet) bends. M13 has a quadrupole 17 doublet (Q1-Q2) be-tween the production target and B1 for collecting pions, a quadrupole triplet(Q3-Q4-Q5) between the two bends, and a quadrupole doublet (Q6-Q7)downstream of B2 for the F3 focal point. Figure 3.2 shows the location16Note that the beam-line components in bold are not to be confused with the T1,B1, B2, or B3 scintillators from the PIENU detector.17 A quadrupole consist of groups of four magnets laid out so that in the planar multi-pole expansion of the field, the dipole terms cancel and where the lowest significant termsin the field equations are quadrupole. Quadrupole magnets are useful as they create amagnetic field whose magnitude grows rapidly with the radial distance from its longitu-dinal axis. This is used in particle beam focusing.393.1. Cyclotron and Beam-lineFigure 3.1: Schematic illustration of TRIUMF’s cyclotron, primary beam-lines, and Meson Hall’s secondary beam-lines [30].of the M13 components. Before the beam-line extension, M13 had threefoci: F1 between B1 and Q3; F2 between Q5 and B2; and F3 after Q7.Beam acceptance-defining slits SL0 are located just upstream of the firstbending magnet B1, and there are momentum-defining slits SL1 and SL2at F1 and F2, respectively. Around 10 cm downstream of SL1, two wheelshold different absorbers/slits.By placing one of the absorber materials in the beam in combinationwith a collimator just before bending magnet B3, M13 works as an energy-loss based particle separator. In our case pions, muons, and positrons passthrough a 1.45-mm-thick Lucite18 absorber; thus, owing to their differentmasses and energy deposited in the material (dEdx ), there is a sufficient mo-mentum change to obtain a clean separation of the pions and positronsmagnetically. Figure 3.3 (left) shows the pion and positron separation atF3. The data were taken in 2008 to verify the beam dynamics calculationsperformed with the REVMOC package [94] used for designing the beam-line [95]. The test setup consisted of a calorimeter named “Tina”19 and twoplastic scintillators for triggering and particle identification via energy and18Poly(methil methacrylate) or PMMA. Lucite is one of the commercial names for thismaterial, commonly referred to as acrylic.19The calorimeter used in a previous experiment; see Section 1.2.1.403.1. Cyclotron and Beam-lineFigure 3.2: M13 channel with the extension [1]time-of-flight (TOF) with respect to the cyclotron’s radio frequency (RF)phase. In front of the calorimeter, there was a 3-layer wire chamber.Figure 3.3: Left: Position distribution of pi+, µ+, and e+ at F3. The solidlines are Gaussian fits. Right: pi+ and e+ rates at F4 as a function of theselected momentum [1]. The PIENU detector was placed at final focus pointF4.For reducing the statistical error on the branching ratio Rexppi , a largesample of pion decays must be collected. By placing the calorimeter in thebeam a larger acceptance is achieved. However, positrons in the beam canplace severe limitations on data collection. The original M13 beam-line de-413.1. Cyclotron and Beam-linelivered a pion beam with 25% contamination of positrons, which severelyincreased detector and trigger rates. Furthermore, in the 2008 test, with atwo plastic scintillator setup, it was shown that after offline analysis cuts,2% of positrons with respect to pions events remained in the data. Anotherbackground was identified in the form of neutrons and gamma rays fromthe beam-line (the source being the T1 production target) that raised theenergy of the pi+ → µ+νµ → e+νeν¯µ decay chain to pi+ → e+νe energiesadding another background [96]. All the previous considerations and ex-perimental predictions pointed toward a modification of the M13 beam-linefor dealing with these unacceptably high levels of beam background. Onceit was verified that a clean separation between pions and positrons couldbe achieved, an extension of the beam-line was installed for further beampurification, as shown in Figure 3.2 and 3.4.Figure 3.4: The end of the M13 beam-line, before (left) and after (right) theextension. Part of the detector was in place to measure the particle contentof the beam.3.1.1 Beam-line ExtensionThe extension starts at the F3 focus and consists of an additional −70◦dipole (B3) and a 30 cm aperture quadrupole triplet (Q8, Q9, Q10) after423.1. Cyclotron and Beam-lineB3. A 5-cm-thick lead collimator with a 3-cm square hole was placed at F3,blocking the spatially separated positrons. The beam-line extension defineda new focus, F4, 1.5 m after Q10, where the PIENU detector was placed.The B3 magnet bent the beam for cleaning the electromagnetic radiationarising from the collimator. Figure 3.3 (right) shows the obtained particlerates as a function of the selected beam-line momentum. The rates wereconsistent with beam dynamics calculations, and it was demonstrated thatthe positron rate could be suppressed by a factor 60 with respect to the pionrate [1].The momentum calibration of the beam-line is challenging to achieve withhigh accuracy, owing to the presence of fringe fields of the dipoles. Never-theless, to obtain a proper calibration, we rely upon physics processes suchas the endpoint of the muon decay spectrum µ+ → e+νe and the peak of thepi+ → e+νe decay. The positrons from the two chains come from the decaysoccurring in the primary target. Above 55 MeV/c, the primary source ofpositrons is γ pair production from pi0 → γγ decays inside the primary tar-get. The momentum distribution of these prompt positrons (with respectto the proton beam bunch) is nearly flat [97].Instead, by selecting positrons delayed with respect to the primary beamRF structure, it is possible to eliminate the prompt background and observepion and muon decays. Figure 3.5 (left) shows the result of the momentumscan where the endpoint of the muon decay and the peak of the pi+ →e+νe decay are clearly visible, this peak is used for calibration. In Figure3.5 (right), it is shown that the delayed positrons have a time spectrumconsistent with the pion decay time [1].The beam-line was also tested with negative polarity, yielding a ratio ofdelayed to prompt positrons of (3.4±0.4)×10−3, consistent with an estimatebased on the yield ratio Npi−/Npi+ = 1/5 in this energy region and the 1%fraction of pion decays in flight in which muons stop in the target [98]. Thebeam-line extension ended by going through a 20-cm-thick steel wall forshielding the experiment from the remaining γ and neutron backgrounds.The PIENU detector was attached to the end of the beam after the wall.433.2. DetectorFigure 3.5: Left: Fraction of beam positrons as a function of the selectedmomentum. Right: Fit of the delayed component of the positrons time-of-flight showing consistency with the pion decay time [1].3.2 DetectorThe PIENU detector design has been reported in detail in Ref. [4]. Fig-ure 3.6 shows a schematic of the detector that was conveniently designed astwo main assemblies PIENU-1 and PIENU-2. PIENU-1 contains the detec-tion, identification, and tracking capabilities for the incoming pion beam.In downstream order, the beam first passes through a pair of 3-layer wirechambers (WC1, WC2) that provide the beam profile. Then, the beamis degraded by two plastic scintillator counters (B1, B2), followed by twosilicon-strip detectors (S1, S2), each of them with X-Y planes. Finally, thebeam enters and stops in the target plastic scintillator (B3) where most ofthe pions will decay, and an isotropic positron “aura” will emerge. Decaypositrons after B3 go into another double-sided silicon microstrip detector(S3) and a scintillator (T1) for positron timing.PIENU-2, which follows PIENU-1, is contained inside a steel cylinderaligned with the beam-line to allow internal sub-detector rotation (alongbeam axis) capabilities for special positron runs to get the calorimeter’s en-ergy response. PIENU-2 is a positron telescope with a 3-layer wire chamber(WC3) and a scintillator (T2) covering the front face of a NaI(Tl) calorime-ter (Bina). To contain the electromagnetic shower produced, four ringsmade of 97 pure CsI crystals surrounded Bina. Veto scintillators were in-stalled to cover the flanges for T2 (V2), and the calorimeter (V3). Table 3.1summarizes the details of all the main detector components.443.2. DetectorTable 3.1: Parameters for the PIENU detector [4].Plastic scintillator countersTrigger counters B1 B2 B3 T1 T2Size in X (inner radius) 100 mm 45 mm 70 mm 80 mm (0) mmSize in Y (outer radius) 100 mm 45 mm 70 mm 80 mm (171.45) mmSize in Z 6.604 mm 3.07 mm 8.05 mm 3.04 mm 6.6 mmZ position −39.03 mm −30.02 mm 0 mm 19.92 mm 72.18 mmPhotomultiplier model/ H3178-51 83112-511 XP2262B 83112-511 H3165-10manufacturer Hamamatsu Burle Photonis Burle HamamatsuPhoto-cathode diameter 34 mm 22 mm 44 mm 22 mm 10 mmVeto counters V1 V2 V3Inner radius 40 mm 107.95 mm 177.8 mmOuter radius 52 mm 150.65 mm 241.3 mmSize in Z 3.175 mm 6.35 mm 6.35 mmPhotomultiplier model/ H3164-10 H3165-10Photomultiplier manufacturer Hamamatsu HamamatsuPhotomultiplier photo-cathode diameter 8 mm 10 mmTracking detectorsMulti-wire proportional chambers WC1 WC2 WC3Wire spacing 0.8 mm 2.4 mmNumber of planes/wires/readout channels 3/120/40 3/96/48Active area diameter 96.0 mm 230.4 mmCathode plane to anode wire spacing 1.6 mm 2.0 mmAnode wire diameter 15 µmWire orientation 0◦, +120◦, −120◦Silicon strip detector pair (X and Y oriented strips) S1/S2/S3Active area 61 × 61 mm2Silicon strip pitch 80 µmEffective pitch after binding 4 strips 320 µmNumber of planes/readout channels per plane 2/48Thickness (size in Z) 0.285 mmSeparation between X and Y strip detectors 12 mmElectromagnetic calorimeterCrystal NaI(T`) CsINumber used 1 97Energy resolution (FWHM) at 70 MeV 2.2% 10%Thickness (size in Z) 480 mm 250 mmOuter radius 240 mm . . .Approximate width × height for pentagon shaped CsI crystals . . . 90× 80 mm2Number of PMTs per crystal 19 1Hamamatsu PMT model (central PMT for NaI(T`) was R1911-07) R1911 R5543Photomultiplier photo-cathode diameter 76.2 mm453.2. DetectorFigure 3.6: Schematic illustration of the PIENU detector [4]. The targetregion is magnified in the inset.3.2.1 ScintillatorsB1 and B2 are two square beam counters placed downstream of WC1 andWC2. Only B1 covers WC1 and WC2’s full aperture. B1 and B2 were placedupstream close to B3 to select pions of their energy deposited (B2 is smallerthan the target). B1 and B2 served to measure the time and energy loss forparticle identification, and most importantly to improve the signal to noiseratio (Section 4.2.5). B3 is followed by the positron telescope counters T1and T2. B3 and T1 were rotated with respect to B1 and B2 by an angle of45◦ around the beam axis. T1 defines the timing of the decayed positronswith respect to the incoming pion time measured by B1. After B3, it isessential to have a compact assembly to maximize solid angle acceptance;therefore, T2 was placed directly in front of the NaI(Tl) calorimeter.463.2. DetectorFigure 3.7: (Left) B1, (also B2, B3, and T1) plastic scintillator is read outwith 4 PMTs (grey cylinders); Light was collected by four acrylic light guides(light green). (Right) Readout scheme with wavelength-shifting fibers of theT2 plastic scintillator.The plastic scintillator counters were made of Bicron BC-408 (polyvinyltoluene) scintillator.20 Each, except T2 and vetos, was read out by fourPMTs through acrylic light guides. Owing to T2 and the veto’s circularshape and limited space, they were read out by wavelength-shifting (WLS)fibers having a diameter of 1 mm (Kuraray Y-11). The schematic configu-ration of the scintillator readouts is showed in Figure 3.7.3.2.2 Wire ChambersBeam particles were tracked using WC1 and WC2. On the other hand,WC3 is part of the tracking devices for decay positrons and defines Bina’sacceptance at the entrance of the calorimeter enclosure. The three wirechambers used for the PIENU detector were constructed similarly to thesuccessful design from the E949 (TWIST) experiment [100]. Each wirechamber consisted of three wire planes rotated by an angle of 120◦ to eachother to form an X-U-V assembly. The chambers used a gas mixture of80% tetrafluoromethane (CF4) and 20% isobutane (C4H10) at atmosphericpressure.Figure 3.8 shows the assembly for WC1 and WC2. They had 120 wiresin each of the three planes. For WC1-2, the effective pitch is 2.4 mm, andthe total active diameter is 96 mm. Figure 3.9 (left) shows the assembly20Light output: 10240 photons/MeV, attenuation 380 cm, decay time 2.1 ns, anddensity 1.032 g/cm3 [99]473.2. DetectorFigure 3.8: (Left) WC1/2 wire chamber plane and its preamplifier board.Each chamber consisted of three planes. (Right) WC1/2 after installationon the beam pipe [9].for WC3, which has the same design; it only differs by being larger, with 96wires for each of the three planes. For WC3, the effective wire pitch is 4.8mm, and the total active diameter is 23.04 cm. Signals from the wires werefed in to a multi-hit TDC channel after preamplifiers and discriminators.The efficiency of every plane was measured to be larger than 99% for beampositrons.3.2.3 Silicon DetectorsEach silicon detector (S1, S2, S3) had two planes (48 channels per plane)of strips to measure the X and Y coordinates, and each detector was a single-sided AC-coupled micro-strip device of the same type as the ones used in theATLAS central tracker [31]. S1 and S2 were placed immediately upstreamof B3, while S3 was placed immediately downstream of it to provide posi-tion and angle information of the incoming pion and the outgoing positron,respectively. Figure 3.9 (bottom-right) shows one visible plane from the S1and S2 assembly; each plane of the silicon detector has an active volume of61 mm × 61 mm × 285µm.The strip pitch was 80µm, and as the PIENU experiment required a reso-lution of 300µm, the design was modified by binding four silicon strips to oneread-out line. The read-out lines were interconnected with capacitors, andonly every fourth line was read out by an amplifier. Figure 3.9 (top-right)shows a schematic of the silicon strip read-out. By adequately weighting the483.2. DetectorFigure 3.9: (Left) Image of the wire chamber WC3 placed in front of theNaI(Tl) calorimeter. (Right) S1 and S2 assembly on their support structure[9] [31].channels that fire during an event, the capacitors form a charge division lineto reconstruct amplitude and position. The signals were read out by VF4860 MHz ADCs, where predefined thresholds for pulse-signal waveforms wereadjusted to reduce the data size and to suppress channels with no hits. S1and S2 were tuned for pions, and S3 thresholds were set lower to ensure thatthe efficiency for decay positrons for at least one plane was higher than 99%.3.2.4 BinaFigure 3.10 (left) shows the back side of PIENU’s main calorimeter duringPMT installation. It is a single crystal of Thallium-doped Sodium Iodide(NaI(Tl)) and it is the largest ever grown of this kind. The NaI(Tl) was ob-tained from the Brookhaven National Laboratory (Upton, NY, USA), whereit was used by the LEGS collaboration [101] [102]. A reflective material wasused to cover the surface of the crystal and it was enclosed in a 3-mm-thickaluminum enclosure having 19 circular quartz windows at the rear end. Tominimize the amount of material crossed by the incoming particles, a 0.5-mm-thick aluminum front face was installed instead of the 3-mm enclosure.493.2. DetectorFigure 3.10: (Left) Back side of the NaI(Tl) crystal on the test bench.(Right) The NaI(Tl) crystal and the 97 CsI crystals while the calorimeterwas under construction [9].On each circular window was mounted a Hamamatsu R1911 PMT havinga diameter of 3 inches, except the centre PMT which is of type R1911-07. AllPMTs used for Bina and the CsI crystals were wrapped with a µ-meter thinmetal shield to reduce the cyclotron’s 2 G fringe fields. Further, an opticalsimulation was performed with the software Detect2000 [103]. Results fromsimulations showed that light was uniformly reflected [104], and this wasconfirmed within 2% by bench tests with a 22Na radioactive source [105].3.2.5 CsITo further contain radiative shower energy leakage and reduce uncertaintyin the LET (discussed in Section 1.2), Bina was surrounded by a total of 97pure CsI crystals, 25 cm in length (13.5 radiation length), with a pentagonalcross-section and around 9 radiation length radially (two layers). Figure 3.6and 3.10(right) show how the CsI crystals were arranged in four concentriclayers around the NaI(Tl) calorimeter. Layers are divided in an upstreamand downstream part, each further divided into inner or outer part, forminga total of 4 rings [106]. CsI crystals and photo-tubes, obtained from BNL,had been used in the E949 experiment [107]; Hamamatsu R5543 PMTs [108]having a diameter of 3 inches PMT, are designed to operate in high parallelmagnetic fields and used for the CsI crystals. As the pure CsI crystalsare slightly hygroscopic, they were flushed continuously by nitrogen gas tomaintain low humidity levels.503.2. DetectorFor light output and PMT gain performance traceability, each crystalhad a YalO3:Ce245 light pulser attached to it [109], to emit about 8 MeVequivalent 50 Hz light pulses with similar wavelength and pulse width asthe CsI scintillation. Furthermore, to independently monitor the PMT gainswithout exciting the crystals, they were connected to a reference Xenon lampvia a merging quartz fiber to trigger the crystals at 2 Hz during data taking[110]. The Xenon lamp also sent pulses to seven reference PMTs (of thesame type) enclosed in an incubator maintained at a constant temperatureof 24.0 °C. Such external PMTs gave reference measurements for correctingthe light-output changes of the Xenon-lamp that was located in an identicalincubator. To obtain information on the light collection efficiency of thecrystals, we compared data from both the YalO3:Ce245 and the Xenon lamp.The instability of the light yield from the Xenon lamp was measured to beless than 1%.3.2.6 TrackingThe PIENU tracking system consists of three subsystems that can provideparticle positions and angle information of a track in three dimensions. Ourfirst “tracker” (Trk1) uses both of the beam wire chambers WC1 and 2,the second tracker (Trk2) uses the first two silicon detectors S1 and 2, andthe third tracker (Trk3) joins S3 and WC3. Trk1 provides six positionmeasurements (6 wires), while Trk2 provides four (2 x-y planes) and Trk3five (1 x-y and 3 wires). Figure 3.11 shows a schematic of the trackingdevices and the different track topologies, i.e., pion decay-at-rest (piDAR),muon decay-at-rest (µDAR), pion decay-in-flight (piDIF), and muon decay-in-flight (µDIF).Trk3 is used for tracking decay positrons entering the calorimeter, andtherefore, it defines the acceptance radius (AR). To suppress pion decays inflight, Trk1 and Trk2 can be used for detecting pion decays before the target.Further suppression and removal of background events can be achieved bymatching the tracks from Trk1 and Trk2 with the positron tracks from Trk3and checking if the decay vertex lies within the target (Zv). Some tracktopologies are as follows.ˆ pi+ → e+νe : The pion stops in the target and decays directly to apositron.ˆ piDAR → µDIF: In pi+ → µ+νµ → e+νeν¯µ decay, the muon decaysin flight in the target. These events are a problem because the muon513.2. DetectorFigure 3.11: Schematic of the tracking devices, the pi+ → e+νe signal, andthe different decay-in-flight backgrounds (the sizes are not to scale). piDAR→ µDIF: In pi+ → µ+νµ → e+νeν¯µ decay, the muon decays in flight in thetarget. piDIF upstream of target (“up.”) → µDAR: The pion decays in flightbefore entering the target. Part of these decays can be detected by trackingthrough the kink variable (Kθ). piDAR → µDAR: Both the pion and themuon in the pi+ → µ+νµ → e+νeν¯µ channel decay at rest in the target.piDIF inside target (“it.”) → µDAR: Pion decay-in-flight in the target andmuon decay at rest. The “u” orientation of a WC plane corresponds to arotation of +60◦ while “v”=−60◦.energy can boost the LE positrons into HE events. Such a topologyhas the same timing distribution as the direct pi+ → e+νe decay. Theseevents cannot be detected and separated from the pi+ → e+νe events;therefore, a correction is needed. Such correction will be discussed inSection 6.3.ˆ piDIF upstream of target (“up.”) → µDAR: The pion decays in flight523.2. Detectorbefore entering the target. Part of these decays can be detected bytracking through the kink variable (Kθ), shown in Figure 3.11.ˆ piDAR → µDAR: Both the pion and the muon in the pi+ → µ+νµ →e+νeν¯µ channel decay at rest in the target.ˆ piDIF inside target (“it.”) → µDAR: Pion decay-in-flight in the targetand muon decay at rest.From Geant4 calculations, the probabilities of decays in flight (DIF) withrespect to the decays at rest (DAR) for the pi+ → µ+νµ → e+νeν¯µ eventsare: about 0.6% for piDIF up. → µDAR; and 0.6% for piDIF it. → µDAR.Figure 3.12 shows the kink angle Kθ distribution of piDAR events and piDIFevents obtained with simulations. The probability of both piDIF → µDIF isnegligible.Figure 3.12: Simulation of the kink angle Kθ for different pion decay modes.Track definitionThe goal is to find an algorithm for calculating the track parameters ofcharged particles traversing three or more planes. The tracking algorithm533.2. Detectoris described in [111], where without magnetic fields, tracks are straight linesparameterized asx = x0 + vxt,y = y0 + vyt,z = z0 + vzt.(3.1)The reference frame is the center of target B3, defined as point {x0, y0, z0}where z-axis points downstream, x-axis goes horizontally and y-axis verti-cally. The point {x, y, z} is for the position of the particle at a given timet, with velocity vector ~v = {vx, vy, vz}. The number of parameters is six;however, only four are independent. We can choose z0 = 0 and set a nor-malization for the vector ~v. Choosing vz = 1 gives ~v = {tx, ty, 1}, wheretx = vx/vz and ty = vy/vz. With these choices, and z = t, we get the newparameterizationx = x0 + txz,y = y0 + tyz.(3.2)The parameterization choice is convenient, as the particles are mainly goingin one direction, which we choose to be the beam direction z, and the pa-rameters x0 and y0 identify the point where the track intersects the planeat z = 0 (the center of the target). A drawback of the chosen parameteriza-tion is that it is not able to describe lines parallel to the xy plane, but thissituation is not relevant in this case.Track FittingThe measurement of one hit in a tracking detector plane correspondsideally to one wire chamber wire or to a silicon detector channel, for whichthe position is known. In reality, more wires or strips can be active and thetracking software used only “hits”, which were consistent with the correcttrigger timings. Consider now a coordinate system uv in a tracker’s plane(z is fixed), where the axis u is orthogonal to the wires/strips. In this way,the coordinate u is proportional to the wire/strip number. With rotation, itis possible to transform the uv system to the xy system of the experimentalhall, as we know the fixed angles for each plane. In addition, every trackhas a χ2 function, which is the squared deviation of the tracks from themeasurements, weighted with the errors in each measurement.543.3. Final Detector AssemblyTracking QuantitiesFor defining the acceptance and suppress the pi+ → µ+νµ → e+νeν¯µbackground in the suppressed spectrum,21 the following variables were con-structed:ˆ Acceptance Radius:AR =√(txzWC3 + x0)2 + (tyzWC3 + y0)2, (3.3)where zWC3 was the location of the centre of WC3 along the beamdirection z.ˆ Kink angle:Kθ = arccostxAtxB + tyAtyB + 1√(t2xA + t2yA + 1)(t2xB + t2yB + 1), (3.4)where the track A was reconstructed by Trk1, while the track B wasreconstructed by Trk2.ˆ Z-vertex (pi stopping position in target B3):Zv =(x0A − x0B)(txA − txB) + (y0A − y0B)(tyA − tyB)(txA − txB)2 + (tyA − tyB)2 , (3.5)where track A was reconstructed by Trk2 and track B by Trk3.3.3 Final Detector AssemblyThe final detector assembly is shown in Figures 3.13, and 3.14. Addi-tional technical drawings are shown in Appendix F. PIENU-1 was mountedto the beam pipe and PIENU-2 enclosed by a steel cylinder was mountedon a supporting structure on wheels which were guided by rails to ensurecorrect alignment to PIENU-1. This flexible system allowed removing of thePIENU-1 assembly to enable rotation for PIENU-2 with respect to the beamaxis for investigation of the calorimeter response to a positron beam at dif-ferent entrance angles. This information is crucial for the determination ofthe pi+ → e+νe LET. Following the 2009 data-taking, it was observed that21In the calorimeter, the low energy tail from the pi+ → e+νe energy distribution isburied under the pi+ → µ+νµ → e+νeν¯µ energy. The pi+ → µ+νµ → e+νeν¯µ events canbe suppressed with specialized cuts to access just the pi+ → e+νe distribution.553.4. Data Acquisition Systemtemperature variations in the experimental hall resulted in gain variationsin the PMTs. Therefore, a temperature-controlled enclosure housing for thedetector was constructed to maintain temperatures at 20◦C within ±0.5◦Cto keep the gain variations whitin acceptable limits.3.4 Data Acquisition System3.4.1 TriggerThe PIENU trigger (full diagram in Appendix E) system was assembledusing NIM22 modules for the most part. The trigger logic was designedfor two main functions; physics and detector calibration data. Figure 3.15shows a schematic of the PIENU trigger diagram. Particle identification forthe incoming beam composed mainly of pions is made by requiring a triggercoincidence between beam counters B1, B2, and target scintillator B3, andproper energy cuts in B1-B2 to ensure a pion particle. Such coincidence iscalled pion signal. If needed, beam muons or positrons could be selectedfor sub-detector calibration. Positrons from either pi+ → e+νe or pi+ →µ+νµ → e+νeν¯µ decays downstream from target (B3) are detected witha T1-T2 counter coincidence, which defines the decay-positron-signal. Apion-decay-positron-signal coincidence within the time window of −300 nsto 540 ns with respect to the pion stop in target B3 (t0) is the basis of thetrigger logic. We call such events “PIE” events.We used three main trigger configurations for normal physics data tak-ing, named: Prescale, Early and TIGC aka BinaHigh trigger; we call themphysics triggers,ˆ Prescale: As pi+ → µ+νµ → e+νeν¯µ events dominate pi+ → e+νeevents by four orders of magnitude, a Prescale unbiased trigger selectsonly 1/16 of PIE events. The PIE events include pi+ → e+νe eventsas well, thus an event can have more than one trigger tag.ˆ Early: As the pion has a very short decay time relative to the muon,26 ns vs. 2.2µs, respectively, around 70% of the pi+ → e+νe decaypositron events can be selected in an early time window, 6 ns to 46 ns22 The Nuclear Instrumentation Module (NIM) standard defines mechanical and electri-cal specifications for electronics modules used in experimental particle and nuclear physics.The concept of modules in electronic systems offers enormous advantages in flexibility, in-terchange of instruments, reduced design effort, ease in updating and maintaining theinstruments.563.4. Data Acquisition SystemFigure 3.13: Bottom: Beam goes from right → left. The PIENU detectorand beam-line after the last bending magnet, showing the steel wall usedfor radiation shielding. Top-Left: PIENU-1 assembly of scintillators, wire-chambers, and silicon detectors. Top-Right: PIENU-2 detector calorimeterassembly, image from [9].573.4. Data Acquisition SystemFigure 3.14: Beam-line-Detector CAD drawing [9].after pion stop time t0. We used such Early trigger configuration toenhance those events.ˆ TIGC or BinaHigh: Another pi+ → e+νe event enhancer is the TIGCor BinaHigh trigger. It selects events that have a high energy depositin the calorimeters (Bina and CsI rings). The energy (TIGC) thresholdis set at the upper edge of the pi+ → µ+νµ → e+νeν¯µ spectrum. Thistrigger selects nearly all the pi+ → e+νe events (with the exclusion ofthe tail events which extend below the TIGC threshold).The other three triggers Cosmic, Xe-lamp, and Beam-Positron were usedfor calibration and data quality checks.ˆ Cosmic: The Cosmic trigger selected cosmic-ray events. Mostly highenergy cosmic muons were selected by the requirement of a high-energydeposit in the CsI outer layer or the coincidence of inner and outerlayers. A prescaling factor of 16 is applied to reduce the rate of thistrigger. These events are used for the calibration of the CsI calorimeteras it is the only detector not directly exposed to the beam. This trigger583.4. Data Acquisition SystemFigure 3.15: Schematic of the trigger diagram for the three physics triggers.The rates of the triggers are listed in Table 3.2. Image from [18].provides an energy calibration for the crystals as well as the monitoringof the crystal and PMT gains.ˆ Xe-lamp: The Xe lamp provided flashes to all the CsI crystals (Xetrigger) for monitoring PMT variations. This lamp was triggered bya pulse generator twice in a second.ˆ Beam-Positron: The beam positrons are accepted by this trigger withpre-scaling by a factor of 32. Beam positron trigger was used for theBina and plastic scintillators calibration.During a typical data taking run, all six triggers were used, and severalof them could be triggered at the same time. To distinguish the associatedtrigger types to a particular event, the trigger logic pulses were also recorded593.4. Data Acquisition Systemwith a multi-hit Time-to-Digital Converter (TDC) named VT4823. Therates of the triggers are listed in Table 3.2. The total trigger rate wasabout 600 Hz. The trigger signal made by any of the six triggers enabledmeasurement of the pion (tpi+) and the positron (te+) timing. These latchedsignals triggered the data acquisition. te+ was used for the trigger of theVME24 modules (VF48 and VT48)23, while tpi+ triggered the COPPER23board data acquisition. Details of those modules will be described in thenext section.Table 3.2: Rates for all triggers [4].Trigger Rate (in Hz)Pion stop in Target 5×104Physics TriggersEarly trigger 160TIGC trigger 170Prescale trigger 240Other TriggersCosmic trigger 15Beam Positron trigger 5Xe lamp trigger 2Total Trigger ∼6003.4.2 BoardsCOPPERPIENU featured a 500 MHz Flash-ADC system, named The COmmonPipelined Platform for Electronics Readout (COPPER) ([4], [15] and [112]).It was used for all plastic scintillators; B1, B2, B3, T1, and T2. COPPERwas initially designed for the Belle experiment at KEK. The main COPPERboard was a 9U-size VME board. The significant advantage of the systemwas its onboard data processing capability featuring a CPU able to host23Details in Section 3.4.2.24 VMEbus (Versa Module Europa bus) is a computer bus standard widely used todayin particle physics. It is physically based on Eurocard sizes, and connectors (DIN 41612),but uses its own signaling system.603.4. Data Acquisition SystemLINUX on board to allow data suppression with embedded software. OneCOPPER board had four frontend digitization modules called “FINESSE”(Figure 3.16). Each frontend can receive two analogue inputs, and the back-end data process was handled on the COPPER main board; therefore, eachCOPPER board can receive a total of eight signals to digitize.Figure 3.16: Picture of main COPPER board mounted with four FINESSEmodules.Each FINESSE card had four 250-MHz Fast-Analogue-to-Digital-Converters(FADC) devices that were driven in alternating phases to realize 500-MHzsampling. The gain of these two synchronized FADCs were monitored andadjusted on a run-by-run basis using beam particle signals to be able tosample the signal at 500 MHz correctly. Figure 3.17 shows the digitizedwaveform from a PMT obtained with COPPER. The PIENU experimentused 4 COPPER boards to digitize the signals coming from the 23 PMTsof all plastic scintillators and a few other additional signals.The FINESSE cards were driven and synchronized by a 250 MHz ClockDistribution module by providing gate, reset, and busy signals to theboards. A General Purpose Input Output (GPIO) module developed by613.4. Data Acquisition SystemFigure 3.17: A waveform digitized by COPPER. The red circles and bluecrosses show the digitization of each 250 MHz ADCs, which produce a500 MHz waveform.KEK provided gate and reset signals into Clock Distributor module. Ad-ditionally, GPIO module received the busy signal from COPPER boardsand provided it to the trigger logic, and received the trigger signal from thetrigger logic to distribute it to the COPPER system. In short, GPIO is theinterface between the COPPER system and the trigger. The time windowof the signals recorded by COPPER covers approximately 8µs (1.35 µs afterand 7.75 µs before the trigger timing) to be able to detect pre and post-pileup particles. Data below a certain threshold was suppressed to reduce theamount of data except for a given region around detected peaks to be ableto record pedestals. For the PIENU experiment, the dynamic range of theFADC was set from −950 mV to 50 mV.VF48The VF48 is a 60 MHz flash-ADC with 10 bits and a dynamic range of±250 mV. VF48 is a 6U-size VME module designed at the University ofMontreal in 2004 [113]. All the Bina and CsI PMT signals, as well as all thesilicon detectors channels, were read out by VF48 modules. A total of 404channels (NaI: 19, CsI: 97, Silicon: 288) were read out by 10 VF48 modules.623.4. Data Acquisition SystemAll VF48 modules received a 20-MHz clock signal provided by the TIGCmodule. This clock is multiplied internally to reach 60 MHz. Owing tothe large number of channels we needed further data suppression. The fullwaveform is recorded with zero-suppression only with the following logic: iftwo subsequent samples have a pulse height difference higher than a giventhreshold. Except for Bina signals which were always recorded, but to areduced rate of 30 MHz since the waveforms were 1.3µs long. In order tosuppress electronic noise the data suppression threshold for the CsI channelswas set at 2 MeV, while it was 0.2 MeV for S1 and S2, and 0.1 MeV for S3.The number of samples recorded by the VF48 is different for each detector:40 (666 ns), 40 (1333 ns) and 70 (1162 ns) samples are recorded for the CsI,Bina, and Silicon channels, respectively.TIGCThe Tigress Collector (TIGC) is a VME module built and developed bythe University of Montreal and TRIUMF for the TIGRESS experiment atTRIUMF [114]. This module allows on-the-fly summing of VF48 signalsbefore the read-out. Every 250 ns, the highest sample of each waveform of allCsI and Bina channels went to TIGC, which then sums them and comparesit to a predefined threshold. Before the sum, a multiplicative factor wasapplied to take into account the different gains of the two detectors. For2010 and 2011 the TIGC threshold was set to be about 2 MeV below thepi+ → µ+νµ → e+νeν¯µ energy upper edge. For 2012 the TIGC thresholdwas lowered. A TIGC trigger is issued if a threshold is passed in coincidencewith a valid pion-positron-decay signal, enabling the readout. TIGC alsoprovided the synchronized clock to all the VF48 modules.VT48VT48 multi-hit TDC modules were used to read out the Wire Chamberwires, logic signals from PMTs after discrimination, and some trigger logicsignals. VT48 is a single width VME 6U-size module [115] designed atTRIUMF in 2006 for the KOPIO experiment [116]. The VT48 module usesthe AMT3 chip [117] which was initially developed for reading out ATLASmuon detectors channels. An onboard 25-MHz clock is multiplied to achieve0.625 ns resolution. All VT48s are fed with an external 25-MHz clock tosynchronize each of the modules. One board can read out 48 channels for upto 20µs. In 2012, two channels were read out with the full-time window todetect long lifetime backgrounds, while the other channels were read out with633.4. Data Acquisition SystemFigure 3.18: Web interface of the MIDAS data acquisition system. All theVME modules were integrated and easily controlled via this interface.an 8.0 µs window before the trigger signal to reduce dead-time. However,because of the delay induced by the TIGC decision time, the trigger signalarrives in those latter channels at the middle of the VT48 recording window;this means that signals up to 4.0 µs before and after the trigger time wereread out. The PIENU experiment employed eleven VT48 modules.3.4.3 SoftwareThe PIENU data acquisition system consisted of three VME crates. TwoVME crates were used for the VF48 and VT48 modules while the third wasused for Slow Control modules and COPPER boards with a processor oneach board. The slow control modules recorded many quantities such as thehigh voltage of PMTs, pressure of WC gas, magnet NMR, and other similarhardware to monitor the data-taking conditions. Collection of the data was643.5. Data-taking History and Milestonesdone via the MIDAS data acquisition system [118] which incorporates anintegrated slow control system with a fast on-line database (ODB) and ahistory. To ensure scalability, MIDAS was designed to integrate multipledata sources from multiple computers through a TCP/IP network. ThePIENU DAQ system made use of this advantage of MIDAS to integrate allthe VME modules. The MIDAS server computer could be controlled via aweb interface; see Figure 3.18. All the information and errors from the DAQmodules were displayed on the web page. MIDAS also controlled programsto make on-line histograms for the data quality check during data-taking.3.5 Data-taking History and MilestonesThe PIENU datasets contain four years of data, taken between 2009 and2012, with around 5M pi+ → e+νe events. A summary of the data takinghistory and milestones is presented in Table 3.3. The DAQ system was set torecord runs containing approximately 300k events at an incident pion rate of50–60 kHz in around 10-min-long MIDAS files of about 1.8-GB in size. TheMIDAS files or “raw” data had to be processed with the PIENU analysisframework to produce ROOT [119] “tree” files of about the same size. Afterall cuts from the analysis, each run had approximately 150 pi+ → e+νe“clean” events depending on the beam rate and hardware configuration. ThePIENU proposal was approved by TRIUMF in 2005, the PIENU detectorwas designed in 2006, concept tested with the M9 beam-line in 2007, andthe full detector constructed and fully tested with M13 beam-line in 2008.3.5.1 2009During 2009, the first stable runs with physics data were recorded. Thedataset was divided as Run I and Run II, with about 1 M and 0.5 M cleanpi+ → e+νe events respectively. As the digital module “TIGC” was notyet installed, a discriminator for BinaHigh triggers (high-energy events) wasused to determine the pulse height of the sum of the NaI(Tl) and CsI PMTs.The analogue sum of the PMTs was recorded without gain correction, lead-ing to unstable trigger conditions and potential loss of pi+ → e+νe events. Inthis period, the trigger for recording cosmic rays in coincidence with the CsIrings to calibrate them properly was not yet available. The CsI ring calibra-tion was attempted using the external Xenon lamp/trigger and the internalYalO3:Ce245 crystals, but such an attempt was inadequate. Such constraintslimited the usability of the data to initial measurements of the detector re-sponse with special positron beam setup and exotic neutrino search in the653.5. Data-taking History and Milestonespi+ → e+νe energy spectrum. Preliminary results for massive neutrino anal-ysis of the 2009 dataset were disseminated in the Ph.D. thesis of K. Yamada[15] and published in ref. [3], and the beam-line design and performancewas published in ref. [1].3.5.2 2010The final trigger configuration was available starting 2010. The datasetwas divided as Run III and Run IV with about 2 M and 0.4 M cleanpi+ → e+νe events, respectively. However, the CsI PMTs were out of tim-ing in Run III; therefore, no CsI information is available. Thus, the largestsource of systematic error in the experiment, the estimated uncertainty inthe low-energy tail of the measured pi+ → e+νe energy spectrum, was largerby approximately a factor of 2 for Run III. Run IV was the first high-qualitydata with all triggers and detector capabilities available. An initial analysisof Run IV was presented in the Ph.D. thesis of C. Malbrunot [16] whichafter further investigation resulted in the first publication of an improvedvalue of the branching ratio [5]. Furthermore, the calorimeter design andperformance was published in ref. [2]. The branching ratio uncertaintyreached 0.24% precision with similar contributions from statistics and sys-tematics; an improvement by a factor of 2 over the previous measurementswas achieved.3.5.3 2011During 2011, an improved measurement of the response function of thedetector was taken. This special set of runs replaces the 2009 special runs.Physics data were taken and named Run V with about 0.5 M clean pi+ →e+νe events. Preliminary results of the analysis of the 2011 dataset werepublished in the Ph.D. thesis of S. Ito [18].3.5.4 2012The 2012 dataset represents the largest and most easily usable high qualitydata recorded. Physics data were taken and named Run VI with around 2 Mclean pi+ → e+νe events. At the start of this run, the energy threshold of theTIGC trigger was lowered, to ensure that no pi+ → e+νe decays were beingmissed. This resulted in additional pi+ → µ+νµ → e+νeν¯µ events causingTIGC triggers. As these events are not used in the analysis, the number ofevents per run is around 1.5 times lower in 2012 compared with 2010 and663.5. Data-taking History and Milestones2011. Preliminary results of the analysis of the 2012 dataset were publishedin the Ph.D. thesis of T. Sullivan [19].3.5.5 Full analysisThe full analysis, including all datasets for the PIENU branching ratio, iscurrently in progress. The current analysis presented in this thesis is blinded,but includes the highest quality data portion available: Run IV, V and VIwith a total around 3M pi+ → e+νe events. From this point forward Run IV,V and VI will be addressed, respectively as the 2010, 2011 and 2012 datasets.The massive neutrino search in the pi+ → e+νe energy spectrum includingall 5M events from the PIENU datasets was recently published [11], andmore collateral studies are being considered for publication, including thedetector’s energy response, an exotic neutrino decay pi → µν search in thescintillator target (B3), a search for 3-body decays pi → eνM where M is aMajorana [120] in the positron energy spectrum, and finally, an analysis ondirect muon capture in zirconium for a special set of muon runs.673.5. Data-taking History and MilestonesTable 3.3: Run history and milestones of the PIENU experiment.Year Month Events Run Range2005 Dec. Proposal approved by TRIUMF2006/07 Detector designed and prototyped in Meson Hall2008 May Beam test in M13Oct. M13 beam channel extension completedOct.-Nov. Test in M13 with most of the detectors2009 May PIENU detector completed 5365May-Sep. Run I (1 M pi+ → e+νe) 5365–19123Oct.-Dec. Run II (0.5 M) 19126–25751Nov 26 Lineshape tests 26021–26244Nov Lineshape measurements 26245–26955Beam-line NIM paper published [1]2010 March Temperature controlled enclosure completedApr.-Sep. Run III (2 M) 29412–45780Oct.-Dec. Run IV (0.4 M) 49669–52003Calorimeter NIM paper published [2]2011 Aug. Systematic studies with beamSept-Oct. Lineshape measurements 54879–56496Nov. Run V (0.5 M) 57420–61179Neutrino Analysis for 2009’s data-sets published [3]2012 Apr.-Dec. Run VI (2 M) 62492–81560Dec. Special Runs for systematic Studies 81566–824892015 Detector NIM paper published [4]2010’s Run IV Rexppi analysis [5]2018 Massive neutrino in pi+ → e+νe spectrum, all datasets [11]Ongoing Massive Neutrino search pi → µν in target (B3)pi → eνM search [120]Direct muon capture in Zirconium [17]Detector’s energy responseFinal Rexppi publication (short version)Final Rexppi publication (long version)68Chapter 4Analysis4.1 Variable Extraction and Calibration4.1.1 Run SelectionThe 2012 data-set range goes from run #62000 to 82000, a total of ∼20000runs. The Midas (Section 3.4.3) log was checked for any DAQ system relatederrors, and high-voltage wire-chamber planes trips. Run durations outsidethe normal range were excluded; the type of excluded runs were periodswith no beam, pure cosmic rays data taking periods, or special beam testconditions. Additionally, the electronic run logs were manually inspectedto exclude bad runs due to DAQ errors, rack temperature outside workingconditions, or any other special condition not appropriate for the analysis.After the run selection, there are 13211 good runs available for analysis. Forthe 2011 and 2010 data-set, the suggested lists were taken from [18] and [5]respectively.4.1.2 ScintillatorsEach scintillator has four PMTs read out (except T2 read out with wave-length-shifting fibers) by the 500-MHz COPPER system (See Section 3.4.2).Before the extraction of charge and pulse-height variables, the pedestal issubtracted from the waveforms. The gain correction factors for each ADCpair are monitored and adjusted on a run-by-run basis based on pulse-heightfrom the physics trigger signals. The pedestal procedure is calculated as themean of the distribution of the first three samples of the waveforms over anentire run, thus is insensitive to random signal pulses that may change thelevel of baseline for each ADC (COPPER and VF48).To automatically correct the gain against fluctuations, beam pion (for B1and B2) and beam muon (for B3 and T1) energy distributions are used on arun-by-run basis. The strong position dependence of signals in T2 because ofthe Wavelength Shifting Fibers (WLSF) geometry calls for gain calibration694.1. Variable Extraction and Calibrationwith decay positrons from the pi+ → µ+νµ → e+νeν¯µ decay chain selectingtheir entrance with WC3. The energy calibration is based on the amount ofenergy deposited by a minimum ionizing particle along the known amountof material of each scintillator (Polyvinyltolulene) using the PDG value andverified with an MC (Geant4) including the corresponding Birks’ correction[121]. All scintillators had the Birks’ correction applied. The light yield perpath length is generally proportional to the the energy loss per path length:dY/dx ∝ dE/dx. Birks law takes into account saturation and quenchingeffects with the correctiondY/dx = SdE/dx1 + kB(dE/dx), (4.1)where S is the scintillation efficiency and kB the Birks constant (which istypically of the order 10−1–10−2 mm/MeV).Figure 4.1 shows charge variables with different integration times. The“prompt” signal and timing is defined as a “simultaneous” coincidence be-tween B1 and T1. The main trigger25 enables the pion-decay-positron signal“PIE” when there is a coincidence between the pion signal timing tpi+ andpositron timing te+ within an 840 ns window, specifically 300 ns before and540 ns after prompt. The COPPER system records −6.4µs prior and 1.35µsafter the prompt, for a total integration window of 7.75 µs. The main triggertiming t = −1.35µs corresponds to the pion timing tpi+ = 0µs or prompttime. The signal-region is defined within −3.5 < t < 0µs, and the pre-region(pileup detection) from −7.75 < t < −3.5µs.The number of hits in each region (NSig, NPre) were identified by a hit-finding algorithm based on the highest point before a drop. In the signal-region, the charge Q[i] of each hit is obtained by integrating the pulse be-tween −20 and +20 ns around the pulse peak; nominally in target B3 api+ → µ+νµ → e+νeν¯µ decay Q[i=0] is a pion, Q[i=1] a muon and Q[i=2] apositron. Similarly, wider integration windows are available as Qw[i] with−20 ns to +80 ns and Qww[i] with −20 ns to +600 ns around the pulsepeak. Additionally, Qfull[i] is integrated in the whole 7.75 ns window. Thetime of the peak point (t), the pulse height of the peak point (PH), and thecharge deposits (Q, Qw, and Qww) were recorded as array variables. Forexample, if three hits (NSig = 3) were found in the signal region, the time,pulse height, and charge deposit were respectively stored as t[3] = {t0th,25Section 3.4.1704.1. Variable Extraction and Calibrationt1st, t2nd}, PH[3] = {PH0th, PH1st, PH2nd}, and Q[3] = {Q0th, Q1st, Q2nd}.In the pre-region, the charge Pre.Q[i] variable stores the pulse between −20and +20 ns around the peak and Pre.Qw[i] similarly stores a pulse between−20 and +80 ns. The pre-region charge variables (Pre.Q, Pre.Qw), theirpeak point times and pulse heights were also stored in the array.Figure 4.1: COPPER’s signals timing. Image from [18].4.1.3 Silicon Detectors and CalorimeterThe VF48 was used for the silicon detectors, Bina, and CsI crystals, inte-grating typically -300 ns to 540 ns (∼ 1 µs window) with respect to prompt.The COPPER pedestal methodology was used for VF48, but different in-tegration ranges were used: Q : thit − 5 < t < thit + 5 samples, Qw :thit − 10 < t < thit + 10 samples, and Qww : thit − 10 < t < thit + 25samples. Hits are identified by a hit finding algorithm; the charge, pulseheight, and time are recorded for each hit. The number of samples recordedby the VF48 is different for each detector: 40 (666 ns), 40 (1333 ns), and70 (1162 ns) samples for the CsI, Bina, and silicon channels, respectively.Information on the charge deposited before (Qpre : thit − 15 < t < thit − 5samples) and after (Qpos : thit + 5 < t < thit + 15 samples) the pulse is alsostored in the tree.714.1. Variable Extraction and CalibrationSilicon DetectorsThe extraction of the charge deposited and the position of the hit in thesilicon is more complex owing to the charge division circuit. Hits on adjacentstrips are clustered. For each cluster, the two strips with the highest charge(Q: thit−128 < t < thit+128 ns) are tagged (they will be called “high-strips”in the rest of this thesis). Amplitudes of the two high strips are comparedto estimate the position of the hit with a resolution of ∼95 µm.26 The timeof the hit is the average of the time recorded in the high strips weighted bytheir respective charge. The number and size (how many strips were hit) ofclusters are also recorded.A calibration pulser was connected to the amplifiers of all silicon detec-tor channels. Run by run, a correction factor is calculated from the pulsertaking a specific run as a reference. In total, 288 correction factors corre-sponding to all the silicon detector channels are calculated every run. Thiscalibration procedure corrects only changes in the amplification electronicsand is therefore not sensitive to changes in the silicon detector itself. Theenergy scale calibration is expected to change because of temperature fluc-tuations, voltage fluctuations, or degradation in the silicon due to radiationdamage. Voltages and temperatures in the area were recorded for every runin order to make corrections off-line if needed. Such residual fluctuationscould be identified and corrected during the offline analysis. The energycalibration is based on the amount of energy deposited in the “high-strips”by a minimum ionizing particle traversing a silicon wafer. As for other de-tectors, this energy calibration factor has been calculated from PDG dataand checked against MC predictions.NaI(Tl) Crystal “Bina”Bina uses the same pedestal procedure as COPPER’s scintillators. Forevery event, hits found within the 1 µs window around prompt are fittedfor all 19 NaI PMTs. The amplitude, time, χ2, and the value of the fittedpedestal are recorded in the tree. The energy in Bina for all years was cal-ibrated by using long charge-integrated variable Qww and the pulse-height(PH) from VF48 in order to reduce the pileup effect. The energy calibrationin Bina is based on the total energy deposited in the detectors downstream26 This resolution is reached if at least two strips are fired. It corresponds to theresolution on the readout strip (1.28mm/√12) divided by four. If only one strip is hit,the resolutions is 1.28mm/√12=0.37 mm.724.1. Variable Extraction and Calibrationof the target by the pi+ → e+νe events. The total energy should be equalto 70.3 MeV: the positron kinetic energy is 69.3 MeV, plus the 0.511 MeVmass of the positron and 0.511 MeV mass of the electron with which thepositron is annihilated. The energies recorded by S3, T1, and T2 amountto ∼2.5 MeV, while the mean energy deposited in the target (∼1 MeV) andin the front aluminum face (0.22 MeV) of Bina are obtained from MC (de-pendent on average pion stopping position). The sum of all these energies isused to fix the energy calibration for the NaI(Tl) calorimeter. As for otherdetectors, this energy calibration factor has been calculated from PDG dataand checked against MC predictions.CsI CrystalsAs the CsI crystals are not directly exposed to the beam, they are cali-brated using cosmic rays. A cosmic ray trigger was operated in parallel tothe other triggers, enabling a new calibration every 20 runs (needed to col-lect sufficient statistics). The peak due to the passage of minimum ionizingcosmic muons in each crystal was compared with the energy deposit pre-dicted by a simulation made using the CRY package [32]. CRY generatedcosmic-rays at the altitude (sea level) and at the geographic coordinatesof the PIENU experiment, and the resulting particles are injected in theGeant4 simulation of the detectors. The charge deposit in each CsI crystalwas converted to the energy deposit by using a multiplicative factor ff =∆ECosmicMC∆QCosmicDataQData[Xeref ]QData[Xe], (4.2)where ∆ECosmicMC is the peak position of the energy deposit obtained by MC,∆QCosmicData is the charge deposit from the Cosmic trigger, QData[Xeref ] is thereference charge of Xe lamp (Section 3.4.1) trigger event, and QData[Xe]is the charge of Xe lamp trigger event for each run. The precision of theenergy calibration in the calorimeters (NaI plus CsI) is at least 0.1 MeV. InFigure 4.2, the comparison between simulation and the data is shown. Thepeak positions vary up to 20% in energy with the position of the crystal inthe detector, but they are well emulated in MC. The energy deposited byminimum ionizing particles in a single CsI crystal is ∼50 MeV.4.1.4 Wire-chambersThe VT48 was used for the wire chambers. The VT48 records hits in awindow of -3.6 µs and +4.4 µs with respect to pion timing tpi+ . For the734.1. Variable Extraction and Calibrationwire chambers, the wire hit indicates the spatial position of the hit. If twoadjacent WC wires are fired the track is assumed to have passed in betweenthe two wires giving a twice better position resolution. Based on the WireChamber (or Silicon detector) channels that fired, a track is reconstructed.In case of multiple hits for WCs (or clusters for Silicon), tracks are con-structed with all possible combinations of hits. For each reconstructed track,the χ2, number of degrees of freedom, residuals, and position informationare stored.744.1.VariableExtractionandCalibrationUpstream IN 1500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.014Upstream IN 2500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.014Upstream IN 3500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.014Upstream IN 4500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.016Upstream IN 5500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.016Upstream IN 6500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.0160.018Upstream IN 7500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.0160.018Upstream IN 8500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.016Upstream IN 9500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.016Upstream IN 10500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.016Upstream IN 11500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.0160.0180.020.022Upstream IN 12500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.0160.0180.020.022Upstream IN 13500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.0160.0180.020.022Upstream IN 14500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.016Upstream IN 15500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.016Upstream IN 16500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.014Upstream IN 17500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.016Upstream IN 18500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.0160.018Upstream IN 19500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.0140.016Upstream IN 20500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.014Upstream IN 21500 1000 1500 2000 2500 3000 3500 400000.0020.0040.0060.0080.010.0120.014Figure 4.2: Comparison prior to calibration between data (black) taken with the cosmic ray trigger and an MCsimulation (red) based on the CRY [32] simulation package. The spectra are relative to the 21 crystals in theinner-upstream CsI ring. Horizontal axes are ADC counts. The peak positions vary up to 20% in energy withthe position of the crystal in the detector, but they are well emulated in MC. The energy deposited by minimumionizing particles in a single CsI crystal is about 50 MeV. Image from [18].754.2. Event Selection4.2 Event SelectionIn the following section the cuts are briefly described. In general, a pionneeds to be selected from the incoming beam using scintillators for energyidentification and wire chambers to restrict acceptance. Positron tracksare calculated from pions decaying in the center of target which then en-ter the calorimeter. Such events are checked for trigger timing consistencyand pileup effects among the detector’s scintillators, wire-chambers, silicon-detectors, and calorimeters. A summary of the selection cuts is presentedat the end of this section (Section 4.2.5) accompanied by a cut flow shownin Table 4.1.4.2.1 Pion IdentificationEnergy and Beam Acceptance. The energy information in B1 andB2 from the incoming beam identifies the type of particle. Figure 4.3(a)and 4.3(b) show the four main distributions from left to right, positrons,muons, pions, and two pions arriving at the same time. An energy cutwindow requirement is made for both B1 and B2 to select pions, 3.8 to5.2 MeV, and 2.0 to 3.1 MeV, as shown in Figure 4.3(a) and 4.3(b). The5.2 MeV energy cut in B1 trims out pileup and represents a non-negligiblesystematic uncertainty according to Ref. [122]. In the results Chapter 7,such uncertainty is tabulated in the final error budget for the final branchingratio. Furthermore, cuts on the beam profile in WC1 and WC2 (Figure 4.4)were applied to remove particles (mostly positrons and muons) from thebeam halo. Beam profile restrictions for the 2012 dataset are -23 to 19 mmon the x-axis and -17 to 19 mm on the y-axis for both WC1 and WC2. Thelatter is a run dependent cut (see Table B.2).Pileup in Scintillators. The waveforms digitized by COPPER in theplastic scintillators were used for reconstructing timing, studying energydeposits, and detecting the presence of multiple pulses generated by differentparticles (pileup). In principle, to reject pileup events, only single pulse(or “hit”) events should be accepted, but we also need to account for thepresence of additional pulses in the waveforms owing to optical reflectionsand electronic noise. The condition to reject only true pileup in B1, B2,and T1 was that each scintillator was required to have one hit in the signal764.2. Event Selectionregion is at least one PMT as follows,{(NB1 1Sig = 1)∪(NB1 2Sig = 1)∪(NB1 3Sig = 1)∪(NB1 4Sig = 1)}∩{(NB2 1Sig = 1)∪(NB2 2Sig = 1)∪(NB2 3Sig = 1)∪(NB2 4Sig = 1)}∩{(NT1 1Sig = 1)∪(NT1 2Sig = 1)∪(NT1 3Sig = 1)∪(NT1 4Sig = 1)}. (4.3)Where NB1 1Sig stands for the number of hits for the first of four PMTs inscintillator B1 in the signal region (Figure 4.1), similarly for B2 and T1. Onthe other hand, requiring one hit on all PMTs gave an unacceptable numberof signal events rejected with such tight requirements. Target B3 presentsmultiple hits from the decay of the pions; therefore, no hit requirements weremade. The number of hits in T2 scintillator were not inspected since thereis a position and energy dependence associated to the optic-fiber detectortopology.There are pulses in B1 and B2 that are too close in time to each other;thus, a hit-based pile-up rejection scheme would not work properly. Pile-upcan be rejected using the ratio between the charge pulse areaQmeasured in ashort time ([−20, 20] ns) window over another oneQw with longer integrationtime ([−20, 80] ns). To optimize the beam pile-up rejection, the requiredcondition was 0.75 ≤ Q/Qw ≤ 1.05 for each of B1 PMTs (Figure 4.5(a))and 0.75 ≤ Q/Qw ≤ 0.97 for B2. The decay time variable is obtained withthe time difference between B1 and T1. The time in B1 or T1 is calculatedas the average of the time of the four PMTs. The time is extracted from awaveform fit. As all the PMTs are used, we require all B1 channels to have agood χ2 to eliminate cases in which two pulses are very close to one anotherand therefore not detected as multiple hits by the hit finding algorithm.B1 Timing and Trigger Consistency. The time consistency for thepion-signal (B1-B2-B3 coincidence) is verified by looping over all the hits ofthe B1 PMT. Figure 4.5(b) shows the trigger consistency cut window for aspecific run period27 with −4399 ≤ (B1.Time − pion.trigger) ≤ −4380 ns,time distribution without cuts in black, time distribution with all cuts inred (excluding cut being discussed), and cut values are shown in blue, withno normalization.27 Cut values for all run periods are available in Table B.2.774.2. Event Selection4.2.2 Pileup T1, and T2Pion Trigger and T1-T2 Sync. For every PMT of T1, all the pulses areinspected. Only the pulse (for each PMT) that is closest to the pion-signaltime is fitted. This requirement ensures that the fitted pulse is the first seenby the PMT after a pion stops in target B3. Events with additional pulsesbefore the fitted one are discarded. Additionally, a coincidence of ±20 ns isrequired between the T1 and T2 counters.Proton Cut. The pion beam nuclear interaction with target B3 producedprotons with energies up to ∼100 MeV; this is due to pion absorption by nu-clei. Since their large energy was deposited along the downstream counters,they were easily identified from the minimum-ionizing positrons. Figure 4.6show the correlation between the minimum dE/dx in the downstream coun-ters (S3, T1, T2) and energy deposit in the NaI. Because decay positronscould undergo Bhabha scattering in the counters to produce rather higherenergy deposit, the minimum dE/dx in three counters was used for protonrejection.T1 Prompt Time. Hits in T1 were rejected if an event was found to bein time coincidence (±2 ns) with the pion time. This cut mostly kills beampositrons, muons, and protons. This cut also helps in rejecting events wherean old-muon28 decays from the target B3 and hits T1. Beam particles mayopen T1’s gate waiting for the decay positron to effectively blind T1, thusthe positron from the signal can be missed. This cut eliminates both promptpi+ → µ+νµ → e+νeν¯µ and pi+ → e+νe events. Such events are not requiredin our time spectrum analysis (Chapter 5) used to extract the branchingratio.T1 Fake Pileup. For T1 pile-up, the one hit requirement in at least oneof the PMTs does not address the presence of reflections and fluctuations(fake hits), raising the concern of preferentially rejecting earlier decay eventsand thus biasing the branching ratio. The identification of fake hits wasachieved via the ratio of the full integrated charge over the pulse height ofthe triggering hit as a function of the pulse height. Figure 4.7 shows thefake hits and real pileup separated clearly into two bands; by only rejectingevents where the ratio of integrated charge to pulse height is higher than thered line, only real pileup will be removed, and events with fake pileup will28Muon from a previous event.784.2. Event Selectionbe preserved. This protects the event being rejected by the pileup cut fromdepending on the positron decay time. The possible energy dependence was< 1× 10−8 branching ratio units [7], thus negligible for our current level ofprecision.4.2.3 Early TimePre-Pileup. The Pre-Pileup (Pre-PU) cut normally rejects events in a-6.4 µs to -2.2 µs window before the arrival of the pion (-7.7 µs to -3.5 µswindow with respect to trigger time, see Figure 4.1). The implementationrequires no hits in the Pre-PU window for all PMTs: B1, B2, and B3 toexclude pion pre-pileup, and T1 and T2 for positron pre-pileup.Beam Muons and Multiple Pions. There is a muon selection logicfor B1 connected to a special VT48 channel readout. With this signal, thepresence of an extra incoming beam muon can be inspected up to 16µsbefore the pion trigger signal. Events with hits in this channel up to 8.5µsbefore and 1.25µs after prompt were rejected, keeping events in a 0.1µswindow centered at prompt. These beam muons stop in T2, decay and mayenter the calorimeter’s time window. In this case, the energy they depositin the calorimeters is added to that of the decay positron creating a pile-upevent in the calorimeter, which was not detected by the scintillator counters.This cut reduces by a factor of two the level of beam muons. Similarly forthe 2012 data taking period, in order to detect out-of-time pion structuresthere was an additional pion selection logic for B1 connected to a specialVT48 channel readout. The same cut range was applied.False Triggers. Certain processes (e.g., nuclear interactions, range strag-gling, low momentum pions, decays in flight) can enable false triggers. Falsetriggers [123] occur, e.g., if the pion traverses only B1, and B2, and thenstops before B3. Then, the target (B3) is fired by a positron from a pionwhich decayed upstream of the target. Such triggers can be seen by observ-ing the event pulse time difference between the target and B1 (B3t − B1t)versus the total charge in the target (B3charge). The false trigger cut spacecan be seen in Figure 4.8. The main distribution of positrons making falsetrigger can be separated at (B3t − B1t) > 4 ns and B3charge < 200 ADCcounts (∼3 MeV). The three bands on the left represent pileup related to 4,3, and 2 PMTs at 450, 300, and 150 ADC counts (10, 4, and 2 MeV), respec-tively. The main red blob in the center represents good beam pion eventsstopping in the center of target B3 and the blob’s downward tail represents794.2. Event Selectionpions barely entering B3, while the upward tail is pions stopping at the endof B3. This tail, and some of the few false positron triggers that go below4 ns overlap representing a non-negligible systematic uncertainty accordingto Ref. [124]. In the results Chapter 7, such uncertainty is tabulated in thefinal error budget for final branching ratio.4.2.4 Calorimeter Acceptance Radius ARTo ensure that the decay positrons hit all the downstream counters, usingtrack reconstruction (Section 3.2.6) the calorimeter acceptance radius (AR)was defined using the radial distribution from the middle of WC3 (see Eq.3.3 for definition). Thus, AR is referred to as the track radius from WC3.Such a cut is shown in Figure 4.9 together with the acceptance cut. It isimportant to note that in order to keep Bhabha scattering events, there areno requirements on the number of tracks in the downstream tracker. Forevents with multiple tracks, the track with the minimum distance from thecenter is taken. The measured energy spectrum is highly dependent on theangle at which decay positrons enter the calorimeter assembly (NaI(Tl) andCsI). The choice of the cutoff value for AR results from a trade-off betweenthe increasing systematic error as the low energy tail of the pi+ → e+νe decayincreases and the decreasing statistical error as more events are included athigher values ofAR. Considering these arguments, the radial cut was found29to be AR = 40 mm for the full analysis using all combined data sets. Ideallywe would like to make the acceptance as large as possible, but setting toAR = 90 mm increases the systematic uncertainty in the branching ratio bya factor of 5 and almost a factor of 2 in the total (syst. and stat.) combinederror.4.2.5 Summary of Event SelectionAbout 90% of the events were removed by the event selection cuts dis-cussed above. The major cuts were the pion energy cuts, the pileup cuts,and the pre pileup cuts. The combination of those three cuts rejected about70% of all the events. The acceptance radius cut after all other cuts re-moved about 20% more of all the events. After all cuts have been appliedwe have about 11% at AR = 60 mm or 7% at AR = 40 mm left of all theevents, which are used for the time spectrum analysis described in Chapter5. Table 4.1 is a summary of the event selection cuts with the ratio of event29To be discussed in Chapter 7804.3. Energy Spectrareduction. A comprehensive list of cuts used for the analysis is shown inTable B.1, and run dependent cut values listed on Table B.2.4.3 Energy SpectraOnly events that passed all cuts are selected to produce the histogramsused for the time spectrum fit (Chapter 5) to obtain the blinded raw branch-ing ratio Rraw. In nominal data-taking settings, a pion beam stops near thecenter of the target B3. The pi+ → µ+νµ → e+νeν¯µ energy spectrum isbelow 52 MeV. Events are separated in “high-energy” (HE), for which thesum of the energy deposited in the NaI and CsI (ENaI+CsI) is larger than52 MeV, and “low-energy” (LE) region otherwise (ENaI+CsI ≤ 52 MeV).The LE and HE threshold is called Ecut. The energy spectrum is shownin Figure 4.10 as the sum of NaI and CsI energies for all physics triggerscombined after all event selection cuts. The branching ratio vs.Ecut wasshown to be stable within ±3 MeV (Chapter 7). The pi+ → e+νe events areemitted isotropically from the center of the target with a kinetic energy of69.3 MeV about 8 cm upstream of the front of the calorimeter’s face. Mostpositrons (about Ee+ > 5 MeV) traverse half of B3, and all of S3, T1, T2and Bina’s aluminum face. Positrons on the beam’s axis traverse plasticscintillator, silicon, and aluminum depositing ∼3.7 MeV on average. Thusthe pi+ → e+νe peak is at ∼65.6 MeV as shown in Figure 4.10.4.3.1 Monte-Carlo CalibrationA crucial step for the analysis was to match the scintillators and calorime-ter’s energy scale to Monte-Carlo (MC). The sum of calibrated energies fromscintillators B1, B2, B3, and the silicon detectors S1, and S2 is the Pion’sTotal Energy,Etot = EB1 + EB2 + EB3 + ES1 + ES2, (4.4)shown in Figure 4.11(a). As explained earlier in Section 1.2, the pi+ → e+νeand pi+ → µ+νµ → e+νeν¯µ energy distribution peaks are about ∼4 MeVapart. The black line represents the Etot’s “suppressed” energy, since a setof specialized cuts are used to enhance pi+ → e+νe events, and suppresspi+ → µ+νµ → e+νeν¯µ i.e., Early triggers, 7 < positron time < 42 ns, andkink angle (Section 3.2.6) < 12 degrees. The blue line is the opposite, Etot’s“late” energy uses Prescale triggers, and positron times greater than 100 ns.The red line represents MC’s Etot’s energy for pi+ → e+νe events only. Fig-ure 4.11(b) represents the “suppressed, “late”, and MC’s pi+ → e+νe energy814.3. Energy SpectraTable 4.1: Cut flow for event selection. The number of eventsbefore cuts is 2.027× 109 for the 2012 dataset.Cut Events after each cut (%)Blinding §1.3 and Integrity §3.4.2 99.33Physics Triggers §3.4.1 99.25Pion Energy §4.2.1 75.69WC1,2’s Halo 72.30B1,2 PU 55.55B1 Waveform 55.42B1 prompt 55.40Proton 54.07TrCons 53.87T1prompt 52.15T1 fake PU 50.28T1 Waveform 50.06PionTrig §4.2.2 49.62T1-T2 sync 49.59Pre-PU §4.2.3 35.54Beam Muons 34.30Two Pionsa 33.22FalseTrig 33.20Acceptance radius (AR)§4.2.4AR = 60mm 11.35AR = 40mm 6.60a Two pion detection only available for 2012 dataset.824.3. Energy Spectrain the calorimeter (Bina+CsI). Figure 4.11(c) both the “suppressed”, and“late” spectrum are shown with an extra cut in the pion total energy toselect pi+ → e+νe events i.e., ±1 MeV around Etot’s pi+ → e+νe peak. Forall subplots in Figure 4.11, the “suppressed” (black) energy distribution wasnormalized to the MC’s (red) pi+ → e+νe peak, for proper comparison suchnormalization was used for the “late” (blue) spectrum as well. The align-ment coefficient between data and MC was done to match the pi+ → e+νepeak in the calorimeter (Bina+CsI). The resultant alignment uncertainty ofthe scintillators’ total energy and the calorimeter’s peak to MC is below thecalibration’s uncertainty 0.1 MeV.Furthermore, also crucial is to check the alignment between the calorime-ter’s energy for all physics triggers for all years (databases). All physicstriggers were added together in the calorimeter energy distribution shownin Figure 4.10. In contrast, Figure 4.12, Figure 4.13, and Figure 4.14 showthe calorimeter’s energy for Early, Prescale, and TIGC (physics) triggers(Section 3.4.1) respectively. For both, the integrated-charge (Q) and pulse-height (PH) calorimeter (NaI+CsI) based variables defined as,EQNaI+CsI = EQNaICQNaI +97∑i=1EQCsI i (4.5)andEPHNaI+CsI = EPHNaICPHNaI +97∑i=1EQCsI i, (4.6)respectively. Where EQNaI is the calibrated NaI calorimeter Q based variable,CQNaI is the corresponding MC alignment coefficient, and EQCsI i is the cali-brated energy of the i-th CsI crystal. Similarly goes for the PH superscripts.Note the CsI Q based energies are used in both cases. The coefficients canbe found in the Appendices, in Table B.2. In each plot from Figure 4.12,4.13, and 4.14 for the 2010, 2011, and 2012 energies were normalized topi+ → e+νe peak. There is an improvement in pileup reduction on the PHversions over the Q, specifically fewer events in the low signal-to-backgroundratio region around Ecut (50 to 55 MeV), and after the pi+ → e+νe peak.Figure ??(bottom) shows how the different thresholds in the TIGC triggeracross the datasets defines the “rising energy” between 30 and 45 MeV.834.3. Energy Spectra(a) Cut values for B1 scintillator(b) Cut values for B2 scintillatorFigure 4.3: Pion Cut: B1 (top) and B2 (bottom) energy distribution withoutcuts in black, energy distribution with all cuts in red (excluding cut beingdiscussed), and cut values are shown in blue. No normalization. Peaks fromleft to right in B1 (and B2): positrons at 1.1 (0.5), muons at 3.2 (1.5), pionsat 4.5 (2.5), and two pions arriving at the same time at 9 (4.7) MeV.844.3. Energy SpectraFigure 4.4: Acceptance for WC1 (top) and WC2 (bottom); beam halo isremoved.854.3. Energy SpectraE.B1_x_WF_Q[0]/E.B1_x_WF_Qw[0]0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2Counts110210310410510610710(a) B1 short gate / wide gate integrated chargeE.B1_i_VT_t[j]-E.Upstr_VT_t[0], (ns)-4440 -4420 -4400 -4380 -4360 -4340 -4320 -4300Counts310410510610710810910(b) B1 time minus pion trigger timingFigure 4.5: Charge (top) and time (bottom) distributions without cuts inblack, distributions with all cuts in red (excluding cut being discussed), cutvalues are shown in blue. No normalization. Top: B1 short gate/wide gateintegrated charge. Bottom: Trigger Consistency Cut.864.3. Energy SpectraFigure 4.6: Energy in the NaI (Bina) versus minimum energy loss in thedownstream counters. Protons are above the red line indicating the cutposition. The red blob represents pi+ → µ+νµ → e+νeν¯µ events and thesmall yellow blob pi+ → e+νe events.Figure 4.7: The ratio of integrated charge in the T1 PMTs to the fittedpulse height as a function of the fitted pulse height. The red line indicatesthe cut used to separate real pileup (above) from pileup due to fake hits(below).874.3. Energy SpectraFigure 4.8: The false trigger cut rejects events when positrons from piDIFmake false trigger. The positrons are found at (B3t − B1t) > 4 ns andB3charge < 200 ADC counts (∼3 MeV). The three bands on the left representpileup related to 4, 3, and 2 PMTs at 450, 300, and 150 ADC counts (10,4, and 2 MeV), respectively. The main red blob in the center representsgood beam pion events stopping in the center of target B3 and the blob’sdownward tail represents pions barely entering B3, while the upward tail ispions stopping at the end of B3.884.3. Energy SpectraFigure 4.9: Calorimeter’s acceptance radius (AR) cut distribution withoutcuts in black, radius distribution with all cuts in red (excluding the cut beingdiscussed), and cut value is shown in blue.Figure 4.10: Combined energy spectrum of the NaI plus CsI detectors forthe 2012 dataset. The vertical red line indicates Ecut = 52 MeV. Thecomposition of the high energy tail beyond 70 MeV is due to pileup events.894.3.EnergySpectraFigure 4.11: Alignment of scintillators and calorimeter’s energy scales to Monte-Carlo. a) Etot’s “suppressed”energy in black, Etot’s “late” energy in blue, and MC’s Etot energy for pi+ → e+νe events only in red. b)Calorimeter’s “suppressed” energy in black, “late” energy in blue, and MC’s for pi+ → e+νe events only in red.c) Calorimeter’s “suppressed” and “late” energy with Etot’s cut to select pi+ → e+νe events. For all subplots the“suppressed” (black) energy distribution was normalized to the MC’s (red) pi+ → e+νe peak, for proper comparisonsuch normalization was used for the “late” (blue) spectrum as well. The alignment coefficient between data andMC was done to match the pi+ → e+νe peak in the calorimeter (Bina+CsI). The resultant alignment uncertaintyof the scintillators’ total energy and the calorimeter’s peak to MC is below the calibration’s uncertainty 0.1 MeV.The normalization for each subplot (left, center, and right) was done independently, thus the vertical axes betweensubplots don’t match, i.e., Etot’s plots were normalized to 1, and the calorimeter’s energies to the total number ofevents.904.3.EnergySpectraFigure 4.12: Early triggers for all years including charge-integration (Q) and pulse-height (PH) calorimeter(NaI+CaI) energy based variables defined by Eq. 4.5 and Eq. 4.6. The three datasets (2010, 2011, and 2012)make two groups in the tail above 70 MeV, in such region all three PH versions have less pileup than the Q versiongroup.914.3.EnergySpectraFigure 4.13: Prescale triggers for all years including charge-integration (Q) and pulse-height (PH) calorimeter(NaI+CaI) energy based variables defined by Eq. 4.5 and Eq. 4.6. The three datasets (2010, 2011, and 2012) maketwo groups in the tail above 70 MeV, in such region all three PH versions have less pileup than the Q versiongroup.924.3.EnergySpectraFigure 4.14: TIGC triggers for all years including charge-integration (Q) and pulse-height (PH) calorimeter(NaI+CaI) energy based variables defined by Eq. 4.5 and Eq. 4.6. The three datasets (2010, 2011, and 2012)make two groups in the tail above 70 MeV, in such region all three PH versions have less pileup than the Q versiongroup.93Chapter 5Time Spectrum5.1 ConstructionThe time spectrum (TS) is constructed from the decay pion time andpositron time. The time was obtained on the basis of the time differenceaverage of fitted pulses between B1 and T1 scintillators for all four PMTsas follows,Tpos =144∑i=1(tT1 iFit − tB1 iFit ). (5.1)Where tB1 iFit refers to the time from B1’s first fitted pulse from the i-th PMT,similarly for T1 with tT1 iFit . Signals, background functions, and processes toconstruct the TS will be explained later in this chapter. Only “physics”trigger (TIGC, Early or Prescale) events (section 3.4.1) are included and all“calibration” triggers (cosmic, beam-positrons, Xe-lamp) are omitted. Onlyevents which passed all cuts are selected to produce the histograms usedfor the TS fit to obtain the blinded raw branching ratio. Events are sepa-rated into “high-energy” (HE), for which the sum of the energy deposited inthe NaI and CsI (ENaI+CsI) is larger than 52 MeV, and “low-energy” (LE)(ENaI+CsI ≤ 52 MeV) regions. The LE and HE threshold is called Ecutnominally at 52 MeV.The LE and HE region are associated with pi+ → µ+νµ → e+νeν¯µ andpi+ → e+νe decays respectively. Events in the HE region were requiredto have fired the TIGC trigger. Events in the LE region were triggeredby the Early trigger in the early time window while the rest of the eventswere Prescale. Outside the boundaries of the early time region, the Prescaleevents were added 16 times30 to the spectrum and the errors on each timebin were scaled accordingly. The Early trigger efficiency was ∼100%. Therewere no distortions observed at the boundary between Prescaled and Early30 As pi+ → µ+νµ → e+νeν¯µ events dominate pi+ → e+νe events by four orders ofmagnitude, a Prescale unbiased trigger selects only 1/16 of PIE events. See Section 3.4.1for more detail.945.1. Constructionevents. The addition of Early to the Prescaled events reduced the statisticaluncertainty in the raw branching ratio by 10%. The implementation togenerate the TS for LE and HE events is the Algorithm 5.1.1 bool Bhigh = E. Cal eBina * C[ year ] + E. Cal CsISum >= 52 ;2 bool Blow = E. Cal eBina * C[ year ] + E. Cal CsISum < 52 ;34 // tsH and tsL : time spectrum histogram f o r HE and LE5 i f ( ALL CUTS ) { // app ly a l l s tandard c u t s67 // F i l l h is togram with E. Tpos ( T1 time − B1 time )8 // TIGC t r i g g e r ONLY in HE,9 i f ( Bhigh && E. BinaHTrig VT N>0) H. F i l l ( tsH ,E. Tpos , 1 . 0 ) ;10 // PreScale (P) and Early (E) t r i g g e r ONLY in LE11 i f ( Blow ) {12 i f (E. PreScaleTrig VT N>0 | | E. EarlyTrig VT N>0){13 i f (E. EarlyTrig VT N>0) H. F i l l ( tsL ,E. Tpos , 1 . 0 ) ;14 else H. F i l l ( tsL ,E. Tpos , 1 6 . 0 ) ; // i f P, we igh t *1615 }}}Algorithm 5.1: Time Spectra Algorithm.The charge-integrated (Q) based TS is populated with Tpos, using EQNaI+CsI(Eq. 4.5) to distinguish between LE and HE events. On the other hand, thepulse-height (PH) based TS uses EPHNaI+CsI (Eq. 4.6). The LE and HE TSare labeled as “tsL” and “tsH” for the 2012 dataset shown in Figure 5.1 and5.2 respectively. The raw TS with the selected good runs list and only dataintegrity checked out from the DAQ system is labeled “raw”, orange andblack for PH and Q respectively. Level 1 TS with “Pion Identification” plusprevious cuts from Section 4.2.1 is labeled “L1”, violet and red for PH andQ. Level 2 TS with “Pileup T1, and T2” plus previous cuts from Section4.2.2 is labeled “L2”, light-blue and yellow for PH and Q. The final TS withthe “Early Time and Acceptance” plus previous cuts from Section 4.2.3 and4.2.4 is labeled “clean”, dark-green and navy-blue for PH and Q. There is noclear differences between the PH and Q version on the LE TS on the otherhand the PH version of the HE TS is clearly less receptive to pile-up. The∼44 ns structures in the raw TS are due to beam pileup from the cyclotron’sRF which are eliminated from the analysis with event selection discussed inSection 4.2; therefore they do not affect the branching ratio.955.1.ConstructionFigure 5.1: 2012 Dataset - Low-Energy (LE) Time-Spectrum (TS), Tpos for ENaI+CsI < Ecut: Using the pulse-height “PH” EPHNaI+CsI and charge-integrated “Q” EQNaI+CsI calorimeter based variable to construct the LE TS“tsL”. Overlaying TS with different levels of cuts: The “raw” (orange and black) spectrum (no cuts), “L1” (violetand red) with “Pion Identification” cuts, “L2” (light-blue and yellow) with “Pileup T1, and T2” cuts, and thefinal “clean” (dark-green and navy-blue) TS with “Early Time and Acceptance” (All) cuts. See Section 4.2 fordiscussion on cuts. The PH and Q versions overlap.965.1.ConstructionFigure 5.2: 2012 Dataset - High-Energy (HE) Time-Spectrum (TS), Tpos for ENaI+CsI >= Ecut: Using the pulse-height “PH” EPHNaI+CsI and charge-integrated “Q” EQNaI+CsI calorimeter based variable to construct the HE TS“tsH”. Overlaying TS with different levels of cuts: The “raw” (orange and black) spectrum (no cuts), “L1” (violetand red) with “Pion Identification” cuts, “L2” (light-blue and yellow) with “Pileup T1, and T2” cuts, and thefinal “clean” (dark-green and navy-blue) TS with “Early Time and Acceptance” (All) cuts. See Section 4.2 fordiscussion on cuts. The PH version is shown to be less sensitive to pileup compared to the Q based branchingratio.975.2. Signal and Background5.2 Signal and BackgroundPrompt events from the HE (Figure 5.2) and LE (Figure 5.1) TS at t =0 ns refers to beam pions (after selection cuts from Section 4.2) stopping inthe center of scintillator target B3. The negative time region reflects the levelof background in the positive region. Before detailing the fit functions used,we will briefly describe the different signals backgrounds present in eachspectrum. Each signal and background will be presented as a probabilitydistribution function and it’s amplitude. The amplitudes can be either afixed or a free parameter.5.2.1 SignalsThe low and high energy regions are mostly associated with pi+ → µ+νµ →e+νeν¯µ and pi+ → e+νe events, respectively. The two-way leakage of eventsfrom one region to another will be described later in this chapter. The timeprobability distribution functions (PDFs) for the signals in the LE and HEtime spectra are given by Eq. 5.2 and Eq. 5.3, respectively. Both signals areonly valid for positive times,Epi→µ→e(t) =exp(− tτµ )− exp(− tτpi )τµ − τpi , (5.2)Epi→e(t) =exp(− tτpi )τpi. (5.3)To introduce time resolution effects each signal was convoluted with a Gaus-sian kernel G′(t) = 1√2piσ2exp(−t22σ2); this technique was used in the previousexperiment [25]. The signals with time resolution effects are,E ′pi→µ→e(t, σ) =∫ ∞0Epi→µ→e(x)G′(t− x)dx = R(t, σ, τµ)−R(t, σ, τpi)2(τµ − τpi) (5.4)E ′pi→e(t)(t, σ) =∫ ∞0Epi→e(x)G′(t− x)dx = R(t, σ, τpi)2τpi, (5.5)where E ′pi→µ→e(t, σ) and E ′pi→e(t, σ) are the signals with time resolution σeffects included, term R(t, σ, τ) = exp(σ2−2τt2τ2)erfc(σ2−τt√2στ), and erfc(t) is thecomplementary error function.31 The time difference between scintillators31 The complementary error function is defined as erfc(t) = 2√pi∫∞te−x2dx.985.2. Signal and BackgroundB1 and T1 used to determine t was measured to have resolution σ = (0.3±0.1) ns. The effects of including the time resolution are further discussed inSection 7.1.5.2.2 Pion Decay-In-Flight and Muon from Previous Event(Old-muon) in Target B3Only two backgrounds remain at a non-negligible level in the LE timespectra after all selection cuts: pion decay-in-flight (piDIF) for positive timesand old-muons32 in target B3 for positive and negative times. Both havethe same probability distribution function as shown in Eq. 5.6.Eµ→e(t) =exp(− tτµ )τµ. (5.6)For negative times, the spectrum is mainly old-muons (muon from previousevent) coming from beam muons or decayed from beam pions which stoppedin the target or surroundings materials. Such beam contribution is replen-ished every 43 ns thus should be a flat component to the time spectrumTS. However, an early time rejection cut (Section 4.2.3) of 8.5µs before theprompt (t < 0) means no beam particles can add to the background, thusthe remaining background follows an exponential decay with the muon life-time. The old-muon component is also contaminating the positive region.For positive times, the spectrum is mainly signal pi+ → µ+νµ → e+νeν¯µevents, more specifically piDAR → µDAR (Section 3.2.6). A non-negligiblefraction of piDIF → µDAR (about 2% of piDAR → µDAR) is present in thespectrum. This piDIF background starts at t = 0 and decays with the muonlifetime. piDIF events are only included in the fit if the decay muon stopsbefore T1, if it stops in T1 or beyond, the event is prompt and thus outsidethe fitting range.5.2.3 T1 Double Pulse Resolution.Following the discussion of the T1 pileup cut from Section 4.2.2, the casewhen two hits occur within T1’s double pulse time resolution is not takeninto account; therefore, this should be modeled into the time spectrum fitas a background. Let ∆T be the time for two positrons to pass throughT1 sufficiently close together in time, where the waveforms will overlap, andonly a single hit is recorded. Case A, defined as F2A(t) (Eq. 5.7) is when T132A muon from a previous event.995.2. Signal and Backgroundis triggered by a positron from an old-muon decay in positive and negativetimes. This means the real pion gives signal in B1 but an old-muon hitsT1 before the true positron. The positron from old-muon is followed onlyin positive time by a positron from a primary pion event. Such a case ismodeled as the product of the old-muon probability distribution function(Eq. 5.6) and the probability that the pi+ → µ+νµ → e+νeν¯µ positron(Eq. 5.2) will emerge within ∆T :F2A(t) =0 t < −∆TEµ→e(t)∫ t+∆T0 Epi→µ→e(y)dy −∆T < t < 0Eµ→e(t)∫ t+∆Tt Epi→µ→e(y)dy t > 0.(5.7)Case B, defined as F2B(t) (Eq. 5.8) is the opposite and is therefore mod-eled as the product of the pi+ → µ+νµ → e+νeν¯µ shape and the probabilityof old-muon decaying within ∆T as follows:F2B(t) ={0 t < 0Epi→µ→e(t)∫ t+∆Tt Eµ→e(y)dy t > 0.(5.8)∆T is a fixed value and the extraction procedure is to plot the time differencebetween the first and second hits for each PMT. Then, each distribution isfitted on the edge to a step function with Gaussian resolution (Eq. 5.9) asshown in Figure 5.3. The average of the four tubes is ∆T = 15.7 ns and theshape with this value is shown in Figure 5.4(a):12{1 + erf(t−∆T√2σ)}, (5.9)where erf(t) is the error function, also called the Gauss error function.3333 The error or Gauss error function is defined as erf(t) = 2√pi∫ t0e−x2dx.1005.2.SignalandBackgroundFigure 5.3: The time difference between subsequent hits in each T1 PMT. The leading times are fitted with anerror function. The peak around 30 ns is due to an after-pulse hit at a characteristic time after the real hit.1015.2. Signal and BackgroundThe effect of pileup coming within the double pulse resolution time inT1 was estimated by artificially increasing the double pulse resolution time∆T in the data up to 100 ns. This is done by rejecting events in T1 afterinitial pulse. Figure 5.4(b) shows the time spectrum of the pileup eventsfor the case of 100 ns double pulse resolution. The time distribution ofthe pileup events was fitted to the Eqs. 5.7 (F2A(t)), 5.8 (F2B(t)), and5.6 (Eµ→e(t)). The amplitudes of F2A(t) and F2B(t) should be the samemagnitude, therefore a common amplitude for the fitting parameter wasused for the fit, F2(t) = F2A(t) + F2B(t).The amplitude (number of events) of F2(t) is obtained from the timespectrum pileup events vs. different artificial double pulse time resolutions(∆T) are shown in Figure 5.5(a,b,c,d, and e) for increasing pre-pileup win-dows -7500, -6500, -5500, -4500, and -3500 ns from trigger time (Figure 4.1)as upper edge respectively and lower edge is fixed at -7750 ns. The plotsare fitted to a quadratic curve and the fitted functions evaluated at ∆T =15.7 ns to extract the amplitude (absolute values of the intercepts) for eachpre-pileup window. As shown in Figure 5.5(f), the extracted amplitudes arecorrelated to the number of old-muon events in the LE region for each pre-pileup window. The points are fitted to a line; the slope (f ′) and y-intercept(yf ) are obtained. Finally the amplitude (f) for T1 resolution (F2(t)) isnormalized as f = f ′c+yf , where c is the number (amplitude) of old-muonsevents in the LE region (Eq. 5.6) independent from the pre-pileup window.The c amplitude is a free parameter to be found with the time spectrumtechnique to be discussed in Section 5.3. For the 2012 dataset, the values off ′ and yf were extracted and found to be independent of acceptance radiusAR at f′ = (4.08±0.22)×10−4 and yf = 803. Such values were used globallyfor all datasets since the 2012 dataset is the most significant, statistically.1025.2.SignalandBackground(a) The shape used in the fit for pileup events that pass the T1pileup cut due to the double-pulse resolution of the T1 counter.(b) Pileup events with fitting functions from the 2011 dataset[18]. The artificial ∆T was 100 ns and the lower edge of Pre-region was at -5500 ns.Figure 5.4: a) T1 resolution function F2A(t) evaluated with ∆T = 15.7 ns. b) T1 resolution pileup events withartificial ∆T = 100 ns.1035.2.SignalandBackgroundFigure 5.5: a, b, c, d, and e) Amplitudes of F2(t) from 2012 dataset’s pileup events vs. artificial T1 doublepulse resolutions (∆T) for different pre-pileup windows; points fitted with a quadratic curve. If the double pulseresolution time ∆T was zero, the amount of pileup would not be negative below 15.7 ns. f) Each intercept at ∆T= 15.7 ns from subplots a) to e) is correlated to the number of old-muon events from the LE time region. SeeSection 5.2.3 for discussion.1045.2. Signal and Background5.2.4 Muon from Previous Event (Old-muon) No-T1-HitAnother mechanism for an old-muon34 pileup to appear in the HE regionis when the positrons from a nominal event or from old-muons may enterthe calorimeters without traversing the T1 counter. Geometrically, this ispossible for muons in the target as there is some solid angle allowing such atrajectory to enter Bina or CsI without hitting T1. Since the positrons enterthe calorimeter with no-T1-hit, the pileup cuts for T1 and T2 do not takecare of a such case. Hence, when combined with a signal pi+ → µ+νµ →e+νeν¯µ event, the energy may get bumped up to the HE region.As this background is HE (the LE component is negligible), the event isfired by a TIGC trigger that requires a sum of the Bina + CsI pulse heightabove a certain threshold in a 250 ns window. However, as the integrationtime for Bina energy calibration was 1µs, if the pileup and signal eventsare separated in time by more than the trigger time window, the calibratedenergy could be above Ecut and a TIGC trigger would still not be present,meaning no trigger or event registered. This particular type of event wasrestricted when generating the MC shape for the time spectrum fit. Theshape of the time spectrum for this background was obtained by Geant4simulation using the waveform templates of the NaI and CsI detectors withthe same pileup cut and trigger requirement as the data. Figure 5.6 showsthe simulated time spectrum and it is represented in the HE region as F1(t).5.2.5 Radiative Pion DecayIf the decay positron was produced in association with a γ via µ+ →e+νeνµγ the energy spectrum of the positron was altered, but the timedependence was not, and a separate shape is not required. In the othercase, if the pion decayed radiatively to a muon as pi+ → µ+νµγ, followedby µ+ → e+νeν¯µdecay, the measured energy could get boosted to the HEregion above Ecut. The γ has the time of a pion decay and the positron hasthe muon decay time. The probability of this happening is dependent onthe time difference between the γ and the positron entering the calorimeter;further, the TIGC trigger integration times comes into play just as in theOld-muon No-T1-Hit component described previously. If recorded in thecalorimeter, the radiative γ will look like a pre-pileup event since it carriesthe time of the pion decay instead of the muon decay. Due to the long NaIpulse, the effect of such a pre-pileup in the HE region can persist long after34A muon from a previous event.1055.3. The Fitting FunctionFigure 5.6: The shape used in the time spectrum fit from positrons enter-ing the calorimeter, missing the T1-hit requirement. Integrated charge (Q)based in blue and pulse-height (PH) based in black.the pion decay time. The CsI crystals have better time resolution than theNaI monolith, thus they can reject events with smaller time difference. Theshape generated with Geant4 is shown in Figure 5.7, the contribution fromNaI in red (G1(t)), CsI crystals in black (G2(t)), and the sum in blue (G(t)).The fixed amplitudes from Geant4 are d1 = 3.62×10−7, and d2 = 1.26×10−7for NaI and CsI crystals respectively. Thus, the amplitude for the totalradiative pion decay component G(t) is d = d1 + d2 = 4.88× 10−7.5.3 The Fitting FunctionThe package MINUIT [22] is used over the time fitting function to fit alldescribed signals and backgrounds and to extract the raw branching ratioRrawpi . The current fitting limits are -290 to -20 and 10 to 520 ns in bothHE and LE spectra. The gap from -20 to 10 ns is called prompt time. TheT1 prompt cut from Section 4.2.2 kills beam positrons, muons, and protons,but the prompt region is excluded because of distortions to the time analysisowing to nuclear reactions which generate gamma rays, and pair production.In the time fit, t = t′ − t0, where t′ is the measured time and t0 is the offset1065.3. The Fitting FunctionFigure 5.7: The shape used in the time spectrum fit for pi+ → µ+νµγ events.contribution from NaI in red, CsI crystals in black, and the sum in blue.in the time spectrum (pion stop time in B3), which is included and fixedin the time spectrum fitting function. t0 is determined by fitting the risingtime from the time spectrum from a special set of muon runs (through-goingparticles). The pion stopping time t0 was found to be 1.68 [16], 2.24 [18], and2.15 [19] ns for 2010, 2011, and 2012 respectively. Although, the 2012 valuewas used for all datasets since its the most representative (statistically), thedifferences did not change the branching ratio result more than 1 [10−8] Runits.5.3.1 Time-Independent Addition of EnergyThe low-energy time spectrum backgrounds are present in some portionin the HE time region as well. Time-independent mechanisms bump theenergy of low-energy components and push events into high energy. Some ofthese LE events are promoted to the high-energy time spectrum due to poorenergy resolutions of Bina and CsI crystals, cosmic rays, radiative muondecays µ+ → e+νeνµγ in which the γ-ray increases the apparent positronenergy, and pileup events in the calorimeter with a flat time distribution (e.g.due to neutrons coming from the production target). The free parameterr is the proportion of the low energy time spectrum that is present in the1075.3. The Fitting Functionhigh energy time spectrum. The fitting implements such a degree of freedomwith a free parameter, namely r, present in both LE ∼ (1− r) and HE ∼ r.5.3.2 Low-Energy ComponentsThe major low-energy components are muon decays: pi+ → µ+νµ →e+νeν¯µ , piDIF, and old-muons. There is also a negligible portion pi+ → e+νetail and µDIF which both decay with the pion lifetime can be ignored in thelow-energy time-spectrum fit [16]. The fitting function used in the low energytime spectrum (ΦLE(t)) is shown in Eq. 5.10, where H is the Heavisidefunction, τµ and τpi are the muon and pion lifetimes, (a) is the total numberof pi+ → µ+νµ → e+νeν¯µ events, (1 − r) is a correction for the loss of LEevents that are boosted to HE, (b) is the LE amplitude of the piDIF shape,and (c) is the LE amplitude of the old-muon background. The overall LEfitting function is,ΦLE(t) = H(t)[a(1− r)Epi→µ→e(t)︸ ︷︷ ︸LE signal+ bEµ→e(t)︸ ︷︷ ︸LE piDIF]+ cEµ→e(t).︸ ︷︷ ︸LE old-muon from Tg(5.10)5.3.3 High-Energy ComponentsThe fitting function used in the high energy time spectrum (ΦHE(t))shown in Eq. 5.11 consists of all of the shapes previously discussed. Therest of the parameters are as follows: Rrawpi for the raw branching ratio be-fore corrections, (a × r) represents the boosted LE pi+ → µ+νµ → e+νeν¯µevents to HE, (b′) for HE amplitude of the pion decay-in-flight shape, (c′)for HE amplitude of the old muon background, (d) for HE amplitude ofradiative pion, (e) for HE amplitude of the old-muon with No-T1-Hit, (f)for HE amplitude of T1 resolution. The correlation between piDIF decaysand old-muon decays was significant; therefore, parameter b′ was scaled tothe amplitude of piDIF in the low-energy region, i.e. b′ = rb. The parameterCµDIF was the corrected amplitude for µDIF events in the target, which willbe discussed in Section 6.3. The HE fitting function is,ΦHE(t) = H(t)[a{(Rrawpi + CµDIF)Epi→e(t)︸ ︷︷ ︸HE signal+ dG(t)︸ ︷︷ ︸Radiative Pion+ rEpi→µ→e(t)︸ ︷︷ ︸LE signal}+ b′Eµ→e(t)︸ ︷︷ ︸HE piDIF]+ c′Eµ→e(t)︸ ︷︷ ︸HE Old-muon from B3+ eF1(t)︸ ︷︷ ︸Old-muon No-T1-Hit+ fF2(t).︸ ︷︷ ︸T1 Resolution(5.11)1085.3. The Fitting Function5.3.4 Fit ParametersResults from the time spectrum fitting for the three data taking periodsare shown in Table 5.1. The first column shows the fit parameters, and thefirst row shows the datasets. The errors are statistical only as obtained bythe MINUIT [22] fit, and the parameters marked as fixed were kept fixedduring the fit. The total number of low-energy events is labeled NLE , andthe total number of high-energy events is NHE .The parameter amplitudes for the pi+ → µ+νµ → e+νeν¯µ events “a”, thepiDIF “b”, and the LE old-muon component “c” are consistent within errorfor integrated-charge (Q) and pulse-height (PH) based Rpi for each year,but increases from 2010 to 2011, and from 2011 to 2012. The increase in“a”, “b”, and “c” throughout the years is expected when comparing the sizeof each dataset. The only inconsistency is “b” being slightly lower in 2011compared to 2010. This could be explained by fact the of parameter “b” andt0 being degenerate. As mentioned earlier in section 5.3, t0 is slightly lowerin 2010 than in 2011. The 2012’s t0 value was used for all datasets since itis the most representative (statistically); the differences did not changed thebranching ratio result more than 1 [10−8] Rrawpi units.The amplitude for the amount of LE events being boosted to HE “r”is larger for the Q compared to PH versions. This is expected since PHshould be less sensitive to pile-up events responsible to push LE events toHE regime. The errors in the Rrawpi reflect the magnitude of the data samplescollected in the three periods. The amplitudes for old-muon (c’) parametersin HE spectra are consistent with zero for the Q version datasets: within1σ for all three years. The amplitudes (c’) for the PH versions are alllarger than any Q version. The largest (events) dataset 2012-PH has thelargest amplitude, and the smaller dataset 2010-PH has the smallest. Theamplitude for the amount of old-muon no-T1-hit (e) events is larger forthe Q calorimeter based branching ratio compared to the PH version. Thelargest (events) dataset 2012 (Q or PH) has the largest amplitude, and thesmaller dataset 2010 (Q or PH) has the smallest.The total χ2 over the total degrees of freedom (χ2/d.o.f.) is larger forthe PH versions. This is mainly due to the ∼44 ns structures in the HEt < 0 region in the PH version due to beam pileup from the cyclotron’sRF. These events in such region causes distortions in the time spectrum andinflates the total χ2/d.o.f. Such distortions and pile-up at negative timesare negligible for positive times, and do not affect the branching ratio. Analternative could be introducing a ∼44 ns cyclic term in the fitting functionto improve the fit.1095.3.TheFittingFunctionTable 5.1: Results from the timing spectra for the three data-taking periods, presented for both integrated-charge (Q) and pulse-height (PH) calorimeter variables. The exact fit values are truncated for a more compactpresentation. The errors are statistical only as obtained by the MINUIT [22] fit, and the parameters marked asfixed were kept fixed during the fit. The errors in the Rrawpi reflect the magnitude of the data samples collected inthe three periods. The acceptance radius used was RA = 40 mm, and the nominal range for our fitting function(FF) for both high- and low-energy time spectra is from −290 to 520 ns, excluding prompt events from −20 to10 ns. Using 1 ns bins for the time spectrum, the total degrees of freedom (d.o.f.) are 1557.Parameter ↓, Dataset → 2012 (PH) 2012 (Q) 2011 (PH) 2011 (Q) 2010 (PH) 2010 (Q)NLE [108] “Total Low-Energy events” §5.3.2 1.323 1.323 0.424 0.424 0.301 0.301NHE [106] “Total High-Energy events” §5.3.3 1.556 1.775 0.521 0.609 0.363 0.415a [109] “pi+ → µ+νµ → e+νeν¯µ ” §5.2.1 9.4502± 0.0020 9.4498± 0.0020 3.0441± 0.0011 3.0439± 0.0011 2.1252± 0.0009 2.1251± 0.0009b [108] “piDIF” §5.2.2 1.666± 0.014 1.665± 0.014 0.570± 0.008 0.569± 0.008 0.671± 0.007 0.671± 0.007c [107] “LE’s old-muon” §5.2.2 1.811± 0.005 1.805± 0.005 1.555± 0.004 1.549± 0.004 1.073± 0.004 1.069± 0.004r [10−4] “boosted to HE” §5.3 2.126± 0.018 3.087± 0.018 1.914± 0.050 2.827± 0.053 2.028± 0.056 2.922± 0.050Rrawpi [10−4] 1.2∗∗∗ ± 0.0014 1.2∗∗∗ ± 0.0014 1.2∗∗∗ ± 0.0025 1.2∗∗∗ ± 0.0026 1.2∗∗∗ ± 0.0030 1.2∗∗∗ ± 0.0031c′ [103] “HE’s old-muon” §5.2.2 6.49± 0.27 1.28± 3.14 4.99± 0.24 1.72± 3.01 3.77± 0.21 1.44± 1.97d [10−8] “pi+ → µ+νµγ” §5.2.5 48.8 (fixed)e [104] “oldmuon-no-T1-hit” §5.2.4 5.85± 0.42 8.96± 0.54 4.64± 0.37 8.17± 0.51 2.41± 0.29 3.80± 0.34f [10−4] “T1Res” §5.2.3 4.08 (fixed)t0 [ns] §5.3 2.15 (fixed)CµDIF [10−7] §6.3 2.406 (fixed)τµ [ns] 2197.03 (fixed)τpi [ns] 26.033 (fixed)χ2/d.o.f §5.3.4 1.19 1.13 1.08 1.06 1.00 1.07HE t < 0 1.62 1.37 1.21 1.19 0.85 1.15HE t > 0 1.14 1.09 1.04 0.99 1.05 1.10LE t < 0 1.07 1.07 1.15 1.15 1.10 1.11LE t > 0 1.13 1.13 1.07 1.07 1.01 1.011105.3. The Fitting Function5.3.5 Signal Overlay and ResidualsThe fitted amplitudes superimposed in the LE (ΦLE(t)) and HE (ΦHE(t))time spectrum for all datasets after all event selection cuts for the pulse-height (PH) and integrated-charge (Q) calorimeter based variables are shownin Figure 5.8, 5.10, 5.12, 5.14, 5.16, and 5.18 for 2012-PH, 2012-Q, 2011-PH,2011-Q, 2010-PH, and 2010-Q respectively. The residuals (data - fit) for HEt < 0, HE t > 0, LE t < 0, and LE t > 0 are shown in Figure 5.9, 5.11, 5.13,5.15, 5.17, and 5.19 for 2012-PH, 2012-Q, 2011-PH, 2011-Q, 2010-PH, and2010-Q respectively.The solid red line, dashed dark blue line, and dashed pink lines indicatepi+ → µ+νµ → e+νeν¯µ , piDIF, and old-muon decays, respectively. Thesum of the two LE backgrounds is shown as a solid green line. The high-energy (HE) spectrum is more complex than the low-energy (LE) spectrum:pi+ → e+νe signal in red, pi+ → µ+νµ → e+νeν¯µ events in blue (boostedfrom LE), pileup from T1’s resolution in dashed red, positrons from pi+ →µ+νµγ radiative decay in dashed black, positrons from piDIF in dashed blue,positrons from old-muons coming from B3 in dashed pink, and decayedpositron from old-muon-no-T1-hit component in dotted blue.1115.3.TheFittingFunctionFigure 5.8: Time Spectra for 2012 dataset pulse-height (PH) Rpi based time fit. Left: LE time spectrumon a logarithmic scale (black line). Right: HE time spectrum on a logarithmic scale (black line).1125.3.TheFittingFunctionFigure 5.9: Residuals (data - fit) for 2012 dataset pulse-height (PH) Rpi based time fit. Top-Left: HE,negative times. Top-Right: HE, positive times. Bottom-Left: LE, negative times. Bottom-Right: LE, positivetimes.1135.3.TheFittingFunctionFigure 5.10: Time Spectra for 2012 dataset integrated-charge (Q) Rpi based time fit. Left: LE timespectrum on a logarithmic scale (black line). Right: HE time spectrum on a logarithmic scale (black line).1145.3.TheFittingFunctionFigure 5.11: Residuals (data - fit) for 2012 dataset integrated-charge (Q) Rpi based time fit. Top-Left:HE, negative times. Top-Right: HE, positive times. Bottom-Left: LE, negative times. Bottom-Right: LE, positivetimes.1155.3.TheFittingFunctionFigure 5.12: Time Spectra for 2011 dataset pulse-height (PH) Rpi based time fit. Left: LE time spectrumon a logarithmic scale (black line). Right: HE time spectrum on a logarithmic scale (black line).1165.3.TheFittingFunctionFigure 5.13: Residuals (data - fit) for 2011 dataset pulse-height (PH) Rpi based time fit. Top-Left: HE,negative times. Top-Right: HE, positive times. Bottom-Left: LE, negative times. Bottom-Right: LE, positivetimes.1175.3.TheFittingFunctionFigure 5.14: Time Spectra for 2011 dataset integrated-charge (Q) Rpi based time fit. Left: LE timespectrum on a logarithmic scale (black line). Right: HE time spectrum on a logarithmic scale (black line).1185.3.TheFittingFunctionFigure 5.15: Residuals (data - fit) for 2011 dataset integrated-charge (Q) Rpi based time fit. Top-Left:HE, negative times. Top-Right: HE, positive times. Bottom-Left: LE, negative times. Bottom-Right: LE, positivetimes.1195.3.TheFittingFunctionFigure 5.16: Time Spectra for November 2010 dataset pulse-height (PH) Rpi based time fit. Left: LEtime spectrum on a logarithmic scale (black line). Right: HE time spectrum on a logarithmic scale (black line).1205.3.TheFittingFunctionFigure 5.17: Residuals (data - fit) for November 2010 dataset pulse-height (PH) Rpi based time fit.Top-Left: HE, negative times. Top-Right: HE, positive times. Bottom-Left: LE, negative times. Bottom-Right:LE, positive times.1215.3.TheFittingFunctionFigure 5.18: Time Spectra for November 2010 dataset integrated-charge (Q) Rpi based time fit. Left:LE time spectrum on a logarithmic scale (black line). Right: HE time spectrum on a logarithmic scale (blackline).1225.3.TheFittingFunctionFigure 5.19: Residuals (data - fit) for November 2010 dataset integrated-charge (Q) Rpi based time fit.Top-Left: HE, negative times. Top-Right: HE, positive times. Bottom-Left: LE, negative times. Bottom-Right:LE, positive times.123Chapter 6CorrectionsThe raw branching ratio Rrawpi needs to be corrected for the calorimeter’slow energy tail from the pi+ → e+νe events buried under the pi+ → µ+νµ →e+νeν¯µ energy distribution (CT ) §6.1, the calorimeter’s energy-dependentacceptance (CAcc) §6.2, the effect of muons decaying in flight (CµDIF ) §6.3,and the energy-dependent effects in the determination of the timings be-tween the two decay modes (Ct0) §6.4. All of them are multiplicative cor-rections to Rrawpi , except the additive CµDIF which is embedded in the highenergy time spectrum fitting function (Equation 5.11). In this chapter allcorrections are described.6.1 Calorimeter’s Low Energy TailThe Low Energy Tail (LET) from the pi+ → e+νe energy spectrum arisesmainly due to electromagnetic shower leakage and energy loss upstreamof the calorimeter. Another small contribution arises from photo-nuclearinteractions within Bina. N(E) is defined as the pi+ → e+νe energy spectrumand the tail fraction T is defined as the proportion of this spectrum belowthe cutoff energy Ecut = 52 MeV over all events:T =∫ Ecut0 N(E)dE∫∞0 N(E)dE. (6.1)The raw branching ratio obtained from the fit (Section 5.3) is thus relatedto the actual branching ratio by Rpi = Rrawpi /(1−T ), then we can define themultiplicative LET correction as,CT =11− T . (6.2)There are two different methods used to obtain the tail fraction. Thefirst one is called the Response Function Measurement : a 70 MeV positronbeam imitating pi+ → e+νe decay positrons in the calorimeter (Bina+CsI)1246.1. Calorimeter’s Low Energy TailFigure 6.1: Schematic drawing of the detector setup for special positronruns, showing rotating angle θ between the beam and calorimeter.is injected at several angles (Figure 6.1) to measure the proportion of thespectrum below Ecut directly. This method was initially referred to as anupper-limit because of the potential for the positrons to scatter in the beam-line, giving an intrinsic low momentum tail coming from the beam-line. TheResponse Function Measurement method was originally described in detail[19] with updates [126] and [127] including the proper photo-nuclear crosssection scaling, better cuts, and a 3.2-mm layer of powdered aluminum oxide(Al2O3) on the front face of Bina missing from the previous analysis [5].Currently, we can reproduce the simulated tail fraction T at several angles(Figure 6.1) from the 70 MeV positron beam data to a level of precisionthat is sufficient to keep us within our precision goals. Figure 6.2 showsthe agreement between MC35 and data corresponding to angle 0.0◦, i.e., no35All MC is Geant4 based unless stated otherwise.1256.1. Calorimeter’s Low Energy Tailrotation. The disagreement between data and MC above 68 MeV is due topileup being deactivated in the simulation, resulting in the difference to theright of the main peak. The nature of the negligible deactivated pileup inthe MC is mainly due to out of time beam muons and beam pions, whichdoes not affect the peak location in MC (Appendix C). The small to nulldisagreement below 35 MeV is also a pileup effect negligible for our levelof precision. The normalization is to the total number of events. The setsof cuts used to clean the positron beam data are described in Appendix C.The peaks at 58 and 50.5 MeV are due to photo-nuclear interactions. UsingMC, it was determined that they were caused by either one or two neutronsbeing emitted from iodine and escaping Bina [2]. Such agreement validatesour positron beam MC and gives us confidence to use the Response FunctionMeasurement to calculate the true tail fraction directly from our nominalpi+ → e+νe MC. Section 6.1.1 discusses the agreement to the tail for allavailable angles, the uncertainties for the positron data, nominal pi+ → e+νedata, and MC.Figure 6.2: The energy spectrum from a 70 MeV positron beam parallel tothe crystal axis. Data is shown in black and simulation is shown in red. Thehistograms are normalized to have the same total number of events.1266.1. Calorimeter’s Low Energy TailThe second method to calculate T was done similarly to what the previousgeneration experiment did to estimate the tail fraction (Section 1.2.1) bysuppressing pi+ → µ+νµ → e+νeν¯µ events with specialized cuts from theenergy spectrum itself to uncover the positrons from the pi+ → e+νe events,also called the suppressed spectrum. This approach assumes that the pi+ →e+νe tail is negligible at very low energies, which leads to a slight over-subtraction of pi+ → µ+νµ → e+νeν¯µ events from the measured energyspectrum. Thus, the suppressed spectrum results in an underestimation ofT ; therefore, we refer to it as the lower limit. The lower limit was describedin detail in [18]. At a previous point of the analysis, the upper and lowerlimits were combined to give the best estimate of T , but as the ResponseFunction Measurement gives the best estimate of T , such combination isno longer needed. In this section, we present a brief description of the tailfraction estimate and the possible intrinsic tail coming from the beam-line.6.1.1 Response Function MeasurementSince the calorimeter is finite, additional energy will be lost owing toelectromagnetic (EM) shower leakage (mainly via Bremsstrahlung and pairproduction until the initial positron runs out of energy) and a small contri-bution from photo-nuclear interactions within Bina [2]. If sufficient energyescaped from the calorimeter, the measured energy of a pi+ → e+νe eventcould fall below Ecut, putting the event in the tail or low energy (LE) region.The positron’s shower leakage is dependent on the entrance angle becausethe amount of material in the path of the beam changes resulting in a vary-ing tail fraction T . The beam-line settings were adjusted to produce a 70MeV positron beam (collimator and absorber were removed) to measure thecalorimeter’s response function by rotating it from the beam reference atseveral angles as shown in Figure 6.1. In order to geometrically allow rota-tion of the calorimeter against the beam axis, material was removed for thepositron beam configuration: B1, B2, S1, S2, B3, S3, and T1 were removed,leaving only the wire chambers, T2, and the calorimeter. This reduces themomentum and position divergence of the positron beam and allows formore accurate measurement of the crystal response.We confidently rely on Geant4 MC for the contribution to the tail fractionT due to energy loss upstream of the calorimeter, as energy loss in thematerial is well understood and reproduced with Geant4 [52]. As the averageenergy loss for pi+ → e+νe positrons in B3, T1, and S3 is 1.7 MeV, thismeant that the effective value of Ecut for the response function measurement1276.1. Calorimeter’s Low Energy TailFigure 6.3: Tail fraction below 53.7 MeV vs angle for the positron data(blue) and MC (red), equivalent to the 52 MeV cutoff in the pi+ → e+νedata. The 1σ error band for the tail fractions in data and MC overlap at allangles.was 53.7 MeV compared to its value for the pi+ → e+νe case at 52.0 MeVin the nominal pion beam configuration, thereby shifting the peak of theenergy spectrum. Positron beam data was first obtained in 2009 and lateragain in 2011 with higher quality. Only the 2011 positron beam data wasused for the tail fraction calculation, with subsets of data for each angle:0.0, 6.0, 11.8, 16.5, 20.9, 24.4, 30.8, 36.2, 41.6, and 47.7 degrees with anaccuracy of 0.1 degrees. The angles of 41.6 and 47.7 degrees correspondto the calorimeter’s acceptance radius AR (Section 4.2.4) equal to 50 and60 mm, respectively. Figure 6.3 shows the tail fraction below 53.7 MeV(equivalent to the 52 MeV cutoff in the pi+ → e+νe data) vs. angle for thepositron data (blue) and MC (red).Uncertainty. The 1σ error bands for the positron beam tail fractionsin data and MC from Figure 6.3 overlap at all angles. The seven sourcesof systematic uncertainties are: the energy calibration (±0.1 MeV), thephoto-nuclear cross section scaling constant (1.1±0.1), the muon correction(Appendix C), the angle between the crystal axis and the incident positron1286.1. Calorimeter’s Low Energy TailFigure 6.4: Simulated Bina+CsI spectrum from pi+ → e+νe decay includingradiative components and events that underwent Bhabha scattering in thetarget.Table 6.1: Low energy tail fraction (T ) percentage for nominal pion beamconfiguration as a function of the maximum acceptance radius AR with Ecut= 52 MeV, and as a function of Ecut with AR = 60 mm.Max AR [mm] Tail fraction T [%] Stat. Error [%] Syst. Error [%]30 2.140 0.023 0.02840 2.540 0.020 0.04550 3.030 0.019 0.06860 3.580 0.018 0.09570 4.220 0.018 0.11580 4.960 0.018 0.15490 5.850 0.019 0.198Ecut [MeV]50 2.780 0.016 0.06351 3.130 0.017 0.07652 3.580 0.018 0.09553 4.140 0.020 0.11654 4.850 0.021 0.14255 5.740 0.023 0.1731296.1. Calorimeter’s Low Energy Tailbeam (±0.1◦), the centre of rotation of the crystal array (±0.25 mm), thebeam momentum (0.5%), and the beam divergence.36 As our MC effectivelyreproduces the energy spectrum and tail fraction for all angles from thepositron beam configuration, we are now confident to calculate the nominalpion beam tail fraction from our nominal pi+ → e+νe validated MC.The systematic error in the pi+ → e+νe tail is different from the systematicerror in the Response Function Measurement (RFM) tail. The error fromthe pi+ → e+νe tail shares two items from the RFM: energy calibration(±0.1 MeV), and photo-nuclear scaling (1.1±0.1). The photo-nuclear crosssection scaling was done within experimental uncertainty [128]. There isan additional geometrical error coming from the uncertainty in the WC3position (±1 mm). Figure 6.4 shows the MC generated pi+ → e+νe energyspectrum in the calorimeter (Bina + CsI), including radiative componentsand events that underwent Bhabha scattering in the target B3. The cutsapplied are energy deposit > 0.1 MeV in T1 and T2, pion decay at restwithin the target, B1 energy between 3.8 and 5.2 MeV, and B2 energybetween 2.0 and 3.1 MeV. The tail fraction is simply the number of countsbelow 52 MeV divided by the total number of counts; the value obtainedwith acceptance radius AR < 60 mm is 3.58% with ± 0.05%, ± 0.05%, and± 0.07% uncertainty from the calorimeter energy calibration, photo-nuclearscaling, and WC3’s position, respectively. In the same order, the value forAR < 40 mm is 2.54% ± 0.03% ± 0.02% ± 0.02%. The tail correction forAR < 40 mm has better uncertainty than AR < 60 mm. Ten MC trees ofone million events each were used for this result. Table 6.1 shows the tailfraction as a function of the maximum calorimeter’s acceptance radius ARand the Ecut energy.Photo-nuclear interactions. The agreement between data and MC forall the tail fraction measurements from the positron data does not suggest byitself that low-energy beam positron contamination is negligible. In principlea low-energy tail in the beam could be masking some disagreement betweenMC and data. Thus, we need some way of validating that the agreement be-tween data and MC for all the tail fraction measurements from the positrondata is insensitive to the low-energy beam positron contribution. As shownin Ref. [127] this was done by selecting events from the positron beam data36 In electromagnetism, especially in optics, beam divergence is an angular measureof the increase in beam diameter or radius with distance from the optical aperture orantenna aperture from which the beam emerges.1306.1. Calorimeter’s Low Energy TailFigure 6.5: The BINA spectrum for events with a late hit (450 to 670 ns) inCsI. Data in black, MC in red. The two photo-nuclear peaks are enhanced.with a late hit (450 to 670 ns) in CsI and looking for MC agreement; areasonable fraction of the delayed neutrons produced in photo-nuclear inter-actions that escape Bina will deposit their energy in CsI [2]. The majorityof CsI hits are at ∼3 ns. Figure 6.5 shows the BINA spectrum in data andMC, for events with a delayed hit in CsI, with the photo-nuclear cross sec-tion scaled by 1.1. 0◦ data is used, where the shower leakage is smallest andthus the photo-nuclear peaks are best defined. Such agreement says thatphoto-nuclear effects are simulated properly, which suggest the low-energybeam positron contamination is not present. Additionally, an independentG4beamline simulation was implemented for the M13 beam-line channel togenerate from production target (T1) through beam-line components theplausible positron contamination at focus point F4 (target B3). Such con-tamination was shown to be negligible for different beam-line configurationsas shown in [6], and [20]. Section 6.1.2 will describe this attempt.6.1.2 Beam-line’s Intrinsic TailThis section briefly explores the intrinsic positron low-momentum taildistribution coming from TRIUMFs M13 beam-line used for the PIENUexperiment. This is relevant because of the potential for the positrons toscatter in the beamline, giving an intrinsic low momentum tail coming fromthe beam-line, as we assume such contribution is negligible it could mask a1316.1. Calorimeter’s Low Energy Tailportion of the low-energy tail due to the detector’s energy leakage. This in-trinsic beam positron low-momentum tail events cannot be excluded by cutsfrom the nominal pion beam analysis since the positron beam detector con-figuration only uses the wire-chambers, T2 scintillator, and the calorimeters.The approach is to simulate the production of the beam, transport throughall beam-line components and spatial-energy distributions at the final fo-cus point (F4) using the specific MC tool G4beamline [129]. Previous workon how positron distributions affect the experiment can be found in [130]and [131]. There was a set of special experimental runs to obtain the PIENUdetector’s response where the beam-line components were adjusted to ob-tain 75 and 70 MeV/c positron beams corresponding to runs #54880 and#81633 (Table D.1). Both configurations were tested with MC. Appendix Ddescribes how the beam production target T1 and beam-line componentswere implemented in the simulation, including the results of the simulation,comparison with run data, and systematic tests.Figure 6.6: Positron momentum distribution at F4 (target B3), for the 75MeV/c positron beam (run #54880).The M13-beam-line output at F4 generated by G4beamline is shown inFigure 6.6 representing the positron beam-line configuration intrinsic mo-1326.1. Calorimeter’s Low Energy Tailmentum distribution. The momentum distributions at F4 include a squarecut of ±20 mm to exclude the beam halo similarly to beam cuts discussedfor nominal pion data and positron data input into the calorimeter. Themomentum vs. angle (between X and Z coordinates) distributions at F4 isshown in Figure 6.7 with a square cut of ±20 mm. As F4’s square cut isrelaxed, the intrinsic beam contribution rises but such events can be safelyignored since their angle distribution does not point directly to the calorime-ter; the maximum angle to enter the calorimeter is roughly 57 degrees cor-responding to an acceptance radius AR = 90 mm. The possibility for theseexcluded halo beam events or any others to reach the calorimeter by scat-tering with the beam-line’s exit foil or air in between was not included inthe simulation. After taking into consideration all settings and systematicsmentioned in this section and appendix D, we set an upper limit to theBeam-line’s contribution to the tail fraction Tbeam at (2.8±0.5)×10−4. Thetwo different positron beam-line settings, 75 and 70 MeV/c beams shownegligible differences in Tbeam as shown in Table 6.2.Figure 6.7: Positron momentum vs. angle distribution at F4 (target B3),for the 75 MeV/c positron beam (run #54880).1336.2. AcceptanceTable 6.2: Upper limit to beam-line’s contribution to tail fraction (Tbeam)percentage.Tbeam [%] Error [%] Condition0.028 < 0.005 75 MeV/c positron beam, 1010 simulated events, run #548800.027 0.005 70 MeV/c positron beam, 108 simulated events, run #816336.2 AcceptanceTo first order, the detector’s acceptance for the pi+ → e+νe and pi+ →µ+νµ → e+νeν¯µ decays is the same, as they are both measured with thesame detector and time interval. In second order, two effects may changethe acceptance ratio: the extra spread in the starting position distributionof the decay positron caused by the O(1 mm) distance traveled within targetby the 4.1 MeV muon, and energy dependent interactions upstream of Bina.Figure 6.8(left) shows a cut on the full combined datasets of the total energyseen by the sum of B1, B2, B3, S1, and S2. This sum was defined as thepion total energy (Etot) in Eq. 4.4. B3 used a longer (100 ns) integrationwindow with respect to the branching ratio analysis. This choice allowed tointegrate also the muon energy deposit. Because of the presence of the muon,the pi+ → µ+νµ → e+νeν¯µ decay deposits more energy in B3 with respect tothe pi+ → e+νe decay. The Etot distribution is used to identify LE and HEevents in the pion stopping position distribution (Zv) previously defined inSection 3.2.6. Figure 6.8(right) shows the pion stopping distribution for thecombined datasets for both decay types.Although, Zv and Etot parameters were not used to select events in thenominal analysis, it served as a diagnostic measure for beam momentumchanges. It also turned out be a good complementary cut to separate thepi+ → e+νe and pi+ → µ+νµ → e+νeν¯µ energy distribution in the calorime-ters. In Figure 6.9(a) the average peak position after all event selectioncuts for 2010 in black, 2011 in red, and 2012 in blue are 0.2, -0.3, and0.2 mm respectively. The different Zv are due to beam momentum changesthroughout the years as shown in Figure 6.9(b) and 6.9(c) for 2010 and 2012respectively. On the other hand the 2011 dataset had an overall shift notbecause beam momentum change but due to an extra piece of material leftnear target B3 for a special set of runs for direct muon capture [17].1346.2. AcceptanceFigure 6.8: Left: Sum of the energies in B1, B2, S1, S2, and B3. Right:Z-vertex for events with positron energy Ecut < 52 MeV (shaded histogram)and Ecut > 52 MeV (blue full line). The two distributions are normalizedto the same number of events, and cuts applied are indicated by the redvertical dashed lines. Image from [11].Processes such as multiple Coulomb scattering, Bhabha scattering, andpair production occurring in the materials traversed by the decay positronsare energy dependent. As the two decay modes have different energy dis-tributions, a small change in the acceptance is expected. Furthermore, thethree main data-taking periods from 2010, 2011 and 2012 had slightly dif-ferent input beam momenta and detector geometry. The beam differencesmay alter the pion stopping position; therefore, the correction may change.As all related processes are well-understood electromagnetic physics, theacceptance correction was estimated with MC independently for all threeperiods. A total of 109 events for each decay mode and data-taking periodwere simulated and the correction factorCAcc(AR) =N(pi+ → µ+ → e+, AR)N(pi+ → e+νe, AR) (6.3)was calculated. N is the number of events for the specified decay channelin the detector’s energy spectrum (Bina+CsI) for a maximum value of theradius AR. The branching ratio is corrected as,Rpi = Rrawpi × CAcc. (6.4)The results are shown in Figure 6.10 for the 2012 dataset, the most signif-icant data-taking period. It was found that the small differences in beammomentum and geometry in our datasets are negligible concerning accep-tance correction CAcc at our level of precision. The systematic error on CAcc1356.2. Acceptance(a) The pion stopping position within tar-get B3 or Z vertex (Zv). The averages peakafter all event selection cuts for cuts for2010 in black, 2011 in red, and 2012 in blueare 0.2, -0.3, and 0.2 mm respectively.(b) Zv vs. run number for the 2010 dataset[16].(c) Zv vs. run number for the 2012 dataset [19].Figure 6.9: The pion stopping position Zv distribution from data.was obtained by varying several parameters in the simulation: the positionand width of the pion stopping distribution (Zv), the positions and thick-nesses of various detectors, and the trigger thresholds in T1 and T2. All theuncertainties, both statistical and systematic are at the 10−8 level and aretherefore negligible for the branching ratio error budget. The acceptancecorrection for various AR values is shown in Table 6.3. The change in the1366.3. Muon Decay in Flightacceptance correction with different Ecut values is negligible at our level ofprecision.Figure 6.10: Acceptance correction CAcc as a function of the AR radius forthe 2012 dataset. Error bars are only statistical.Table 6.3: Acceptance correction CAcc for different AR values.Max AR (mm) Correction CAcc Stat. Error30 0.99703 0.0002340 0.99782 0.0001850 0.99846 0.0001560 0.99907 0.0001370 0.99980 0.0001280 1.00050 0.0001190 1.00004 0.000106.3 Muon Decay in FlightThe 4.1 MeV muons coming from the pion beam stopping in the centreof target B3 can decay in flight (µDIF) inside the target. These eventsare a problem because the µDIF kinetic energy can boost the LE positrons1376.4. t0above Ecut. Such a topology has the same timing distribution as the directpi+ → e+νe decay. These events cannot be detected and separated from thepi+ → e+νe events; therefore, a correction is needed. The MC simulationshown in Figure 6.11(a) indicates that the decay time distribution of suchDIF muons that were not at rest in the target is approximately flat between0 and 19 ps. The probability of a muon decay in flight can be estimated as1− e−τµDIF /γτµ = 8.3× 10−6. (6.5)Where γ = 1/√1− v2/c2 = 1.039 for the muon kinetic energy Tµ = 4.1MeV, τµDIF is the time that the muon travels before it stops, about 19 ps.Typically decays in flight will have lost some energy, but the muon’s 19 psflight path is too short to lose a significant amount of kinetic energy Tµ.The proportion of these events above Ecut = 52 MeV and AR < 40 mmwas estimated to be 2.90% (Figure 6.11(b)), giving a total correction factorCµDIF = 0.0290×8.3×10−6 = 2.406×10−7 for this case. Taking into accountthe level of agreement in the measured energy spectra between Monte-Carloand data for both pi+ → µ+νµ → e+νeν¯µ events and the positron beam, therelative error on the proportion of the spectrum above 52 MeV is on theorder of a few percent, resulting in an uncertainty on the correction of lessthan 10−8. The error on CµDIF is negligible for our level of precision mea-surement. The values for different AR and Ecut values are in Table 6.4. Theadditive correction CµDIF was embedded in the high energy time spectrumfitting function (Equation 5.11).6.4 t0The starting point of the time spectrum analysis is t0. The timing of thepositron signal from the main decays pi+ → e+νe and pi+ → µ+νµ → e+νeν¯µis calculated by fitting the waveforms from the T1 scintillator. If the shapeof the waveform depends on the positron energy, the extracted time canbe energy-dependent, thereby affecting the branching ratio. To investigatethis effect, special muon runs at 62 MeV/c making the muons stop at thecentre of target B3, then the time spectra for different energy regions wereconstructed and t0 was obtained by fitting the edge with a step functionwith Gaussian resolution. The correction used for this global analysis wasobtained using data runs from 2011. The multiplicative correction was Ct0= 1.0006±0.0003 [18]. As the error is only due to statistics, further precisioncould be achieved using more runs from the 2012 dataset. For the moment,the 2011 value is used for all datasets.1386.4. t0(a) Decay time of muons in the target withnon-zero kinetic energy at the time of thedecay.(b) Simulated energy distributions ofpositrons coming from the pi+ → µ+νµ →e+νeν¯µ decay chain. The black histogramcorresponds to positrons from stoppedmuons, while the red histogram corre-sponds to events where the muon decayedin flight in the target.Figure 6.11: Time and energy spectra for µDIF.Table 6.4: Muon decay in flight correction CµDIF for different Ecut and ARvalues.Max AR (mm) Correction CµDIF30 2.533E-740 2.406E-750 2.219E-760 2.071E-770 1.937E-780 1.826E-790 1.727E-7Ecut (MeV) for AR < 60mm50 3.374E-751 2.655E-752 2.071E-753 1.560E-754 1.175E-755 0.852E-7139Chapter 7ResultsResults from the branching ratio analysis are presented in this chapter.The previous two chapters described the fitting technique of the time spec-tra for the extraction of the raw branching ratio and its corrections. Inthis chapter, the results of the fits are reported together with the (blinded)corrected results for the branching ratio. The simultaneous fit to the high-energy and low-energy timing spectra allowed for the extraction of the rawbranching ratio (Rrawpi ), which had to be corrected with the corrections de-scribed in the previous chapter for obtaining the final branching ratio Rexppi .At the time of writing, the analysis is in its final stages and the datablinding has not been removed yet. The first analysis of the 2010 datasethas already been published [5], with an outdated event selection analysis anda different procedure for estimating the LET. For this thesis, we present theresults of an improved re-analysis of the 2010 data combined with the resultsfor the 2011 and 2012 datasets. Before results (and unblinding the data),some crucial tests on the Rpi stability are presented in this chapter, i.e.,branching ratio versus bin size, pileup time windows, acceptance (AR), andhigh energy threshold (Ecut).7.1 Stability and Systematic ErrorsIn general, all sensitivities of the branching ratio against other parameterswere calculated following standard methodology [23], which were used to as-sign systematic errors by varying the parameters within reasonable ranges.Each particular case will be discussed in the rest of this section. The branch-ing ratio difference (∆R) and the uncorrelated statistical error (∆e) betweentwo different branching ratio calculations is defined as,∆R±∆e = (R−R′)±√|e2stat. − e′2stat.| (7.1)where estat is the raw statistical error from the test point and e′stat from theanchor or nominal point to be tested against, with ∆R being the branching1407.1. Stability and Systematic Errorsratio difference from the test pointR and anchorR′. The uncorrelated statis-tical error (∆e) between two different branching ratio calculations is definedas√|e2stat. − e′2stat.| to show how much statistical difference there is betweenthe two. Normally the errors are summed in quadrature, i.e.,√e2stat. + e′2stat.but we are interested in difference between two different branching ratio cal-culations not in adding errors from the same calculation. For the rest of thechapter all quoted branching ratio changes are in 10−8 units for simplicity,unless specified otherwise.7.1.1 Fit TestsThe shapes derived from MC were modified to assess the dependence ofthe Rrawpi on them. Additional background shapes were included in the fitfor checking the sensitivity of the Rrawpi to small unaccounted backgrounds.The shapes tested were a flat background, and a faster (τµ/2) decay timecomponent. The Rrawpi must be stable against changes in the fit conditions.The fit results were tested by changing the fitting range, bin width, andtime resolution effects.. The stability of the Rrawpi was checked when theparameters of the fit were changed. This includes changing the pion andmuon lifetimes, the contribution of the radiative decay, and the variation oft0. Results are shown in Table 7.1.Time Resolution Effects: The fitting function nominally uses the proba-bility distribution functions (PDFs) Epi→µ→e(t) (Eq. 5.2) and Epi→e(t) (Eq. 5.3)for the pi+ → µ+νµ → e+νeν¯µ and pi+ → e+νe signals, respectively. In orderto test the effects of including the time resolution (σ) from the scintilla-tors (B1 and T1) from which the timing signal is extracted, the PDFs arereplaced by E ′pi→µ→e(t, σ) (Eq. 5.4) and E ′pi→e(t, σ) (Eq. 5.5). The timedifference between B1 and T1 has the time resolution of σ = (0.3± 0.1) ns.Figure 7.1 shows ∆R vs time resolution (σ) where the x-axis is the time res-olution from the scintillators in ns and the y-axis is in ∆R units, with zerochange representing 2012(PH)’s nominal analysis result for σ = 0 ns. Theuncorrelated statistical error is zero for all points since there is no changein statistics for this test. The blue solid line represents the 2012 datasetpulse-height (PH) based branching ratio. The blue dashed line representsthe actual time resolution from the scintillators. Since time resolution effectsare negligible for the branching ratio to our level of precision, i.e., ∆R < 1change. Eq. 5.2 and 5.3 have been used for the PDFs.1417.1. Stability and Systematic ErrorsFigure 7.1: Change in the branching ratio ∆R vs time resolution: Thex-axis is the time resolution from the scintillators. The y-axis is in ∆Runits, with zero change representing 2012(PH)’s nominal analysis (withouttime resolution effects). The uncorrelated statistical error is zero for allpoints since there is no change in statistics for this test. The blue solid linerepresents the 2012 dataset pulse-height (PH) based branching ratio. Theblue dashed line represents the actual time resolution from the scintillators(B1 and T1) from which the timing signal is extracted. The change in thebranching ratio ∆R is < 1 [10−8] for time resolutions < 2 ns. Since the timedifference between B1 and T1 has the time resolution of σ = (0.3 ± 0.1) nsthe time resolution effects are negligible for the branching ratio to our levelof precision.Triggers: The variation of the Rrawpi between the time spectrum with onlythe Prescale trigger and the combined triggers was ∆R < 1 [18], i.e., belowour precision goal. The combined triggers improved the statistics by 10%.Fitting Range: The nominal range for our fitting function (FF) for bothhigh and low energy time spectra is from −290 to 520 ns, excluding promptevents from −20 to 10 ns. In order to test the stability of the FF, these valueswere shifted. Ideally, the FF should report no change in the raw branching1427.1. Stability and Systematic Errorsratio (Rrawpi ) within uncorrelated statistical error, since such shifts effectivelydo change the level of statistics in the analysis. All Rrawpi are changes eitherof 1σ deviation or are at the level of ∆R < 1, thus not significant to ourlevel of precision. This is shown in Table 7.1, where the bottom limit goesfrom −290 to −250 ns, the bottom prompt limit from −20 to −30 ns, thetop prompt limit from 10 to 8 ns, and the top limit from 520 to 490 ns.Lifetimes: Nominally, both the muon and pion lifetimes (LT) in the fittingfunction (FF) are fixed to PDG values. Ideally, the FF should find bothLTs at PDG values and report no change in the raw branching ratio (Rrawpi )within the uncorrelated statistical uncertainty. As shown in Table 7.1, whenlifetimes are set free for the 2012 charge-integration (Q) based Rrawpi there is achange of 1.4 ± 3.7 ns (0.4σ variation) in the muon LT, no change in the pionLT, and a change in ∆R±∆e of 5.0 ± 18.3 (0.3σ variation). Such variationsfor the muon LT, pion LT, and ∆R ±∆e when the LT parameters are setfree on the fitting function are acceptable since the changes are consistentwith zero, thus there is no need to add a systematic error to the final errorbudget. The other datasets behave similarly.Fixed Parameters: Nominally, the amplitude of pi+ → µ+νµγ and T1resolution background energy distributions are fixed in the fitting function(FF). To test their sensitivity to the FF, pi+ → µ+νµγ was varied by ±20%and the T1 resolution by ±50%. The results are shown in Table 7.1. Forpi+ → µ+νµγ there is a change in the raw branching ratio of about ∆R = ±3for all three datasets in both the integration-charge (Q) and pulse-height(PH) branching ratio calorimeter based variables. The uncertainty of pi+ →µ+νµγ is also ±20% [18], thus a global systematic uncertainty of ±3 isassigned to the experiment. The uncertainty of the T1 resolution is only±10%, therefore after proper weighting all errors fall to either below 1σdeviation from error or at a level ∆R < 1 change in Rrawpi . The fixed pionstopping time (t0) in target B3 was extracted to be 2.15, 2.24, and 1.68 nsfor 2012, 2011 and 2010 respectively ([19], [18], [16]); thus t0 was shifted ±1ns to cover such uncertainty; such changes were found to be ∆R < 1 changein Rrawpi . It was observed that the changes in t0 are compensated by thepiDIF parameter from the FF. This is expected since t0 is degenerate withpiDIF parameter. The error in the muon decay in-flight correction (CµDIF)used in the FF is negligible [18], thus there is no systematic error to reporthere.1437.1. Stability and Systematic ErrorsOld-muon No-T1-Hit MC shape: The old-muon time distribution de-scribed in Section 5.2.4 and shown in Figure 5.6 was generated with Geant4Monte Carlo. In order to test the sensitivity to the raw branching ratio(Rrawpi ) the shape was binned to 1, 2, 3, and 4 ns and shifted ±1 ns indepen-dently from the time spectra, Table 7.1 shows the non-negligible deviations∆R of 3.5, 5.4, 2.6, and 3.5 units for the 2011-PH, 2011-Q, 2010-PH, and2010-Q datasets.Other backgrounds: Table 7.1 shows tests for additional backgroundshapes included in the fit for checking the sensitivity of Rrawpi to smallunaccounted for backgrounds. Activating a flat component (due to time-independent backgrounds) shows a negligible deviations for the 2012-PHand 2012-Q datasets, but there are non-negligible deviations ∆R of -3.9,-3.5, -4.7, and -4.5 units for the 2011-PH, 2011-Q, 2010-PH, and 2010-Qdatasets. On the other hand when a double µ lifetime (falling twice as fast)as Eµ→e(2t) (Eq. 5.6 evaluated with 2t) time spectra is enabled, there is nosignificant change in Rrawpi .Pre-pileup: Nominally as discussed in Section 4.2.3, no hits are allowedin the scintillators before the arrival of the pion (pre-region). An importantdiagnostic test was the stability of the Rrawpi as more pileup is allowed in thetrigger window. This test shows how robust and precise is the PIENU timespectrum analysis to identify and model pileup correctly. Toward this end,the PrePileup (PrePU) window identified by the PrePileup Cut was varied.This cut normally rejects events in a −6.4µs to −2.2µs window before thearrival of the pion (−7.7µs to −3.5µs window with respect to trigger time,as shown in Figure 4.1). The rejection window was varied in B1, B2, B3,T1, and T2 scintillators to be sensitive to both sections, before and after thepion to lepton vertex. The dependence of the Rrawpi from the PrePU windowwidth was studied.Figure 7.2 shows the stability of the charge-integration (Q) and pulse-height (PH) based ∆R vs. PrePU window for the 2010 (yellow), 2011 (or-ange) and 2012 (blue) datasets. The x-axis is the PrePU window in ns units.The y-axis is in ∆R change units, with zero change representing 2012(PH)’snominal analysis (PrePU cut enabled). The error bars on each point rep-resent the uncorrelated statistical error between the point in question andthe nominal point with the error bars going up when there is a statisticalincrease and down otherwise. For comparison, the horizontal dashed black1447.1. Stability and Systematic Errorslines both at the same distance from nominal represent the raw statisticalerror from the 2012 dataset. PrePU is allowed in from left to right, the pointat −7,500 ns trigger time (−7.5µs) is the closest to the nominal prePU cut,therefore as the inclusion window is relaxed further right, there is an increasein statistics. When the window is completely relaxed at -3,500 ns triggertime (-3.5 µs) it is equivalent to not applying the PrePU cut.The PH versions are better than Q-based branching ratios, and it is alsoclear that both versions of 2012 are better than the rest. The χ2 values for2010 and 2011 shown in the bottom of Figure 7.2 do increase as the pre-pile-up window is relaxed to the right. The 2012 data advantage is the extradetection mechanism for out of time pions and related pileup as describedin Section 4.2.3. The branching ratio varies considerably as pileup is addedto the spectrum, indicating the presence of an incorrect shape or missingcomponent in the time spectrum fit. However, the impact on the branchingratio monotonically becomes negligible especially for the 2012 dataset (bothQ and PH versions) as the cut approaches its nominal value. Since the 2012dataset is the most significant statistically, no systematic error was includedin the final branching ratio.Binning: The fitting function uses 1-ns bins for the nominal analysis.Ideally the raw branching ratio (Rrawpi ) should be independent of the binsize within a reasonable range. Figure 7.3 shows ∆R change vs. bin sizefrom 1 to 8 ns bins for all three datasets in both the integration-charge (Q)and pulse-height (PH) based Rrawpi . The ∆R change is stable within 1 to8 ns binning. The χ2 values are best when the fitting function uses 1-nsbins and grows monotonically for greater binning, as shown in Figure 7.3.The negligible ∆R variations with different bin sizes may be due to differentsampling rates from different DAQ hardware components. No systematicerror was applied for this effect.1457.1.StabilityandSystematicErrorsTable 7.1: Stability tests and systematic errors from the fit, following standard methodology [23]. Non-negligibledeviations are in red. See Section 7.1.1 for discussion. Units in the branching ratio change are ∆R [10−8], withuncorrelated errors unless specified otherwise.∆R ± ∆e [10−8] 2012(PH) 2012(Q) 2011(PH) 2011(Q) 2010(PH) 2010(Q)Stability testsFitting range, tpositive limit: 520→ 490 0.1± 0.7 0.5± 0.7 −1.5± 1.2 −0.3± 1.3 −1.0± 1.5 0.2± 1.6prompt positive: 10→ 8 −2.9± 4.1 −2.9± 4.3 2.4± 7.4 2.2± 7.6 8.9± 8.9 9.3± 9.1prompt negative: −20→ −30 0.1± 0.1 0.0± 0.1 0.2± 0.2 0.2± 0.3 0.1± 0.2 0.1± 0.2negative limit: −290→ −250 0.0± 0.0 −0.1± 0.1 0.0± 0.0 0.0± 0.3 0.0± 0.1 0.0± 0.1Lifetimes∆R, τµ and τpi free 4.6± 18.1 5.0± 18.3 −38.9± 38.2 −47.1± 38.8 8.8± 46.3 1.3± 47.0τfitµ − τPDGµ [ns] 1.4± 3.7 1.4± 3.7 −6.8± 6.4 −8.2± 6.4 1.3± 7.7 −1.8± 7.7τfitpi − τPDGpi [ns] 0.0± 0.0 0.0± 0.0 0.1± 0.0 0.1± 0.0 0.0± 0.0 0.0± 0.0Systematic errors from sensitivitiesFixed Parameterpi → µγ (±20%) ±3.2± 0.0 ±2.9± 0.0 ±3.1± 0.0 ±2.8± 0.1 ±3.1± 0.0 ±2.8± 0.0T1 resolution (±50%) ±1.0± 2.5 ±0.9± 0.0 ±2.5± 4.9 ±2.3± 0.1 ±2.4± 5.5 ±2.2± 0.0T1’s effective error [∆R] 0.2± 2.5 0.2± 0.0 0.5± 4.9 0.5± 0.1 0.5± 5.5 0.4± 0.0(±10% uncertainty)Old-muon MC shape2 ns bin −0.4± 0.0 −0.3± 0.1 −1.1± 0.1 −0.9± 0.2 −0.7± 0.1 −0.7± 0.13 ns bin −0.5± 0.0 −1.0± 0.1 −1.0± 0.1 −2.8± 0.2 −0.1± 0.2 −1.8± 0.2−1 ns shift −1.1± 0.1 −1.2± 0.1 −2.6± 0.1 −3.3± 0.3 −1.9± 0.2 −2.2± 0.2+1 ns shift 0.9± 0.1 1.1± 0.1 2.3± 0.3 3.2± 0.3 1.8± 2.5 2.1± 0.3effective syst. error [∆R] 0.0 0.0 3.5 5.4 2.6 3.5Other backgroundsFlat component −0.9± 0.5 −0.8± 0.5 −3.9± 1.5 −3.5± 1.6 −4.7± 1.8 −4.5± 1.8Eµ→e(2t) (Eq. 5.6) 0.0± 0.0 0.0± 0.0 0.0± 0.1 0.2± 0.3 0.1± 1.3 0.2± 0.41467.1.StabilityandSystematicErrorsFigure 7.2: ∆R±∆e (Eq. 7.1) vs. PrePU: The x-axis is the PrePU window in ns units (Figure 4.1). The y-axis isin ∆R units, with zero change representing 2012(PH)’s nominal analysis (PrePU cut enabled), the error bars (∆e)on each point represent the uncorrelated statistical error between the point in question and the nominal pointwith the error bars going up, when there is a statistical increase, and down otherwise. The horizontal dashedblack lines, both at the same distance from nominal, represent the raw statistical error from the 2012 dataset.1477.1.StabilityandSystematicErrorsFigure 7.3: ∆R±∆e (Eq. 7.1) vs Bin size: The x-axis is the bin size in ns units. The y-axis is in ∆R units, withzero change representing 2012(PH)’s nominal analysis (binning 1 ns), the error bars (∆e) on each point representthe uncorrelated statistical error between the point in question and the nominal point with the error bars goingup, when there is a statistical increase, and down otherwise. The horizontal dashed black lines, both at the samedistance from nominal, represent the raw statistical error from the 2012 dataset.1487.1. Stability and Systematic Errors7.1.2 LET testsAs the LET is the main correction for Rpi, two tests involving it areparticularly important. The LET changes if a different acceptance radiusAR is chosen, as well as a different energy threshold Ecut for separatingthe two energy regions. If Rpi is calculated with different AR and Ecut,a different LET correction has to be applied. If the LET-corrected Rpi isstable against the change in AR and Ecut, there is confidence that the LETis globally known. The stability of Rpi with respect to variations of AR andEcut has already been demonstrated in the first published results [5]; here itis currently finalized for the other datasets.Figure 7.4 shows the stability of the charge-integration (Q) and pulse-height (PH) based Rpi vs. AR for all datasets. The x-axis is the AR value inmm units. The y-axis is in Rpi (corrected) change units. The horizontal lineat zero represents 2012(PH)’s analysis using anchor point with cuts Ecut =52 MeV and AR = 60 mm.37 The error bars on each point represents theuncorrelated statistical error between the point in question and the anchorpoint with the error bars going up when there is a statistical increase anddown otherwise. The dashed black lines both at the same distance fromanchor represent the calorimeter’s LET systematic error (Table 6.1). Thebottom part of Figure 7.4 shows the total χ2 from the fitting function foreach point.There is a slight but clear downward trend after AR > 60 mm, althoughthose point’s statistical bars are within the LET’s systematic error envelope.The calorimeter’s response function measurement (Section 6.1.1) correctionsas a function of acceptance (AR) were confirmed against GEANT4 untilAR = 60 mm (effective maximum rotation angle for the calorimeter); forAR > 60 mm the corrections are only GEANT4 predictions presented forcompleteness. It is also worth pointing out that χ2 also grows monotonicallyas the acceptance AR is relaxed. There is no AR dependence, since points forboth Q- and PH-based Rpi are within statistical error from the systematicuncertainties envelope (Table 6.1) from the horizontal anchor.37 The nominal acceptance radius cut for this thesis is AR = 40 mm. But for the LETtests the value of Ecut = 52 MeV and AR = 60 mm was kept as the anchor comparisonpoint for historical reasons, i.e., the 2010 dataset branching ratio publication [5] used60 mm.1497.1. Stability and Systematic ErrorsFigure 7.5 shows the stability of the charge-integration and pulse-heightbased Rpi vs. high and low energy threshold (Ecut) for all datasets. Thex-axis is the Ecut value in MeV units. The y-axis is in Rpi change units. Thehorizontal line at zero represents the 2012(PH)’s analysis using an anchorpoint with cuts Ecut = 52 MeV and AR = 60 mm. The error bars oneach point represent the uncorrelated statistical error between the pointin question and the anchor point with the error bars going up when thereis a statistical increase and down otherwise. The horizontal dashed blacklines both at the same distance from the anchor represent the calorimeter’sLET systematic error. The bottom part of Figure 7.5 shows the total χ2of the fitting function for each point. There is no Ecut dependence, sincepoints for both Q- and PH-based Rpi are within the statistical error from thesystematic uncertainties envelope (Table 6.1) from the horizontal anchor.7.1.3 Charge- vs. Pulse-height-based RpiThe charge (Q) and pulse-height (PH) based Rpi for 2012, 2011, and 2010datasets can be compared in different stability and systematic tests shown inFigure 7.2 for Rrawpi change vs. PrePU, Figure 7.3 for Rrawpi change vs. binning,Figure 7.4 for Rpi change vs. AR, and Figure 7.5 for Rpi change vs. Ecut.The Q-based Rpi is consistently higher for all points. To assign a systematicerror to the difference between Q and R, all points from the Rpi vs. AR andvs. Ecut were taken into account. Half of the average from the difference foreach point was found to be 3.0, 4.2, and 5.9 [Rrawpi ] units for 2012, 2011, and2010, respectively. These differences are due to PH being less sensitive topileup. Such non-negligible variations are included in the final error budgetpresented in the following section (Section 7.2).1507.1.StabilityandSystematicErrorsFigure 7.4: ∆R ± ∆e (Eq. 7.1) vs. AR, Charge Integration and Pulse-height: The x-axis is the AR value inmm units. The y-axis is in ∆R (corrected) units, with zero change representing 2012(PH)’s analysis using anchorpoint with cuts AR = 60 mm and Ecut = 52 MeV, the error bars (∆e) on each point represent the uncorrelatedstatistical error between the point in question and the anchor point with the error bars going up when there is anstatistical increase and down otherwise. The horizontal dashed black lines both at the same distance from anchorrepresent the calorimeter’s LET systematic error. The bottom part shows the total χ2 from the fitting functionfor each point.1517.1.StabilityandSystematicErrorsFigure 7.5: ∆R ± ∆e (Eq. 7.1) vs. Ecut, Charge Integration and Pulse-height: The x-axis is the Ecut value inMeV units. The y-axis is in ∆R units, with zero change representing 2012(PH)’s analysis using anchor point withcuts AR = 60 mm and Ecut = 52 MeV, the error bars (∆e) on each point represent the uncorrelated statisticalerror between the point in question and the anchor point with the error bars going up when there is an statisticalincrease and down otherwise. The horizontal dashed black lines both at the same distance from anchor representthe calorimeter’s LET systematic error. The bottom part shows the total χ2 from the fitting function for eachpoint.1527.2. Error Budget7.2 Error BudgetAll year-dependent and common errors are shown in Table 7.2. The firstrow is the header for the year-dependent systematic errors, the second rowcomes from the flat component test described in section 7.1.1, the third rowold-muon shape (Section 5.2.4) test (Section 7.1.1), the fourth row is theassigned error for the differences between Q- vs. PH-based branching ratios(Section 7.1.3), and the fifth row is the quadrature sum of the previous threerows (√Σ2α), which will be used as the total year-dependent systematic errorin the final branching ratio calculation.The sixth row displays the statistical uncertainties for each dataset forboth PH- and Q-based R. The seventh row is the header for the commonsystematic uncertainties grouped in two categories, one for PH- and anotherfor Q-based R. The eighth row corresponds to the pi → µγ MC generatedshape (Section 5.2.5) test (Section 7.1.1). The ninth and tenth rows are thesystematic errors assigned to the pion energy (Section 4.2.1) and false trigger(Section 4.2.3) cuts, respectively. The eleventh row is the quadrature sumof the previous three rows (√Σ2β), which will be used as common systematicerror in the final branching ratio calculation. Finally, the common statisticaland systematic errors coming from the corrections (Chapter 6) are listed,specially the low energy tail, acceptance, and t0.Table 7.2: Error budget in [10−8] branching ratio units.Dependent (syst) 2012(PH) 2012(Q) 2011(PH) 2011(Q) 2010(PH) 2010(Q)Flat §7.1.1 0.0 0.0 3.9 3.5 4.7 4.5Oldmuon-No-T1-Hit §7.1.1 0.0 0.0 2.5 4.2 1.9 2.8PH vs. Q §7.1.3 3.0 4.2 5.9√Σ2α 3.0 3.0 6.2 6.9 7.8 7.9Statistics (RA = 40 mm) §5.3.4 14 14 25 25 30 31Common (syst) (PH) (Q)pi → µγ §7.1.1 3.1 2.8Pion cut §4.2.1 3 5FalseTrig cut §4.2.3 3 3√Σ2β (RA = 40 mm) 5.2 6.4LET (RA = 40 mm) §6.1 2 (stat), 5 (syst)Acceptance §6.2 2 (stat)Ct0 §6.4 3 (stat)1537.3. Combination of Datasets7.3 Combination of DatasetsThe three different datasets were collected in similar conditions, but dif-ferences are present besides the statistics collected. The differences do notallow for a global fit of all the data available. Therefore, the three branchingratios have to be combined after the separate fits to the timing spectra. Inref. [132], a procedure for combining the Rpi is outlined. The raw branchingratios (Rrawpi ) for each year with their respective statistical and systematicerrors are Yi± δY st.i ± δY sy.i , labeling the data taking periods with the indexi = 1, 2, 3 the 2010, 2011, and 2012, respectively. The dataset-dependentcorrections with their uncertainties are Cij± δCst.ij ± δCsy.ij , labeling the mul-tiplicative corrections with index j = 1, ..., J .In the present analysis there are no dataset-dependent corrections, butthey are presented for completeness or for future use if needed. The dataset-independent (common) corrections with their uncertainties are Ck± δCst.k ±δCsy.k , using index k = 1, 2, 3 for the LET (CT ), Acceptance (CAcc), andt0 (Ct0), respectively. The common global systematic uncertainties are (seeSection 7.1, and Table 7.2) ±δS =√ΣS2β with index β = 1, 2, 3 for pi → µγ,pion cut, and false trigger cut, respectively.Defining the branching ratio for each year corrected for dataset-dependentcorrections with uncertainties as Ri± δRst.i ± δRsy.i with index i = 1, 2, 3 forthe 2010, 2011, and 2012, respectively,Ri = YiΠCij , (7.2)δRi =√(Ri)2[(δYi/Yi)2 + Σ(δCij/Cij)2]. (7.3)Since there is no dataset-dependent correction, then Cij = 1 and δCst.ij =δCsy.ij = 0 for all i and j, effectively making Ri = Yi. The branching-ratio-weighted average with uncertainties before common corrections is Rs±δRst.s ± δRsy.s , defined as,Rs = ΣRiwi/Σwi, (7.4)δRs =√Σ(δRiwi/Σwi)2, (7.5)where the combined statistical plus systematic weight for each dataset is,wi = 1/{(δY st.i )2 + (Ri)2[(δY sy.i /Yi)2 + Σ(δCst.ij /Cij)2 + Σ(δCsy.ij /Cij)2]}. (7.6)1547.3. Combination of DatasetsThe final combined and weighted branching ratio, with uncertainties in-cluding global systematics, year-dependent, and year-independent correc-tions, is Rf ± δRst.f ± δRsyt.f , defined as,Rf = RsΠCk ± δRst.f ± δRsy.f , (7.7)δRstf = RsΠCk√(δRst.s /Rs)2 + Σ(δCst.k /Ck)2, (7.8)δRsy.f =√(RsΠCk)2((δRsys /Rs)2 + Σ(δCsy.k /Ck)2) + δS2. (7.9)The combination of datasets is implemented in Table 7.3. The optimalacceptance cut (AR) for the best combined statistical and systematic errorwas found to be 40 mm, giving a 0.12% precision measurement instead of0.14% at 60 mm.1557.3. Combination of DatasetsTable 7.3: Combination of 2010, 2011, and 2012 datasets for AR = 40 mm.The branching ratios for all datasets are still blinded. See Section 7.3 andTable 7.2 for nomenclature. The PH version was chosen over the Q basedbranching ratio since the global systematic error is (marginally) better.Value Stat. error Syst. errorRraw[10−4] §5.3.4 Yi δY st.i δYsy.i =√Σ2α2012 (PH) 1.2∗∗∗ 0.0014 0.0003(Q) 1.2∗∗∗ 0.0014 0.00032011 (PH) 1.2∗∗∗ 0.0025 0.0006(Q) 1.2∗∗∗ 0.0025 0.00072010 (PH) 1.2∗∗∗ 0.0030 0.0008(Q) 1.2∗∗∗ 0.0031 0.0008Common Corrections Ck δCst.k δCsy.kLET §6.1.1 1.0261 0.0002 0.0005Acceptance §6.2 0.9978 0.0002t0 §6.4 1.0006 0.0003Common systematics Sl√Σ2β (PH) 0.0005(Q) 0.0006Rfinal[10−4]2012 (PH) 1.2∗∗∗ 0.0015 0.0008(Q) 1.2∗∗∗ 0.0015 0.00092011 (PH) 1.2∗∗∗ 0.0026 0.0010(Q) 1.2∗∗∗ 0.0026 0.00112010 (PH) 1.2∗∗∗ 0.0030 0.0011(Q) 1.2∗∗∗ 0.0031 0.0012Weighted avg. Rf δRst.f δRsy.f(PH) 1.2∗∗∗ 0.0013 0.0008(Q) 1.2∗∗∗ 0.0013 0.00091567.4. Future prospects7.4 Future prospects7.4.1 Current PIENU experimentThe current dominant source of error is statistical, at 13 [10−8] Rpi units.Another set with about 3.5 M pi+ → e+νe events38 is available from Run I(1 M), II (0.5 M), and III (2 M) collected from 2009 and prior to Novem-ber 2010 which if added to the analysis could potentially bring down thestatistical error below 10 [10−8] Rpi units. Those extra 3.5 M events arelower quality data because the CsI crystal information is not available, thusmaking the systematic error on the LET bigger. Additionally, the triggerwas still a work in progress during the 2009 dataset, which could bring inextra systematic uncertainties. Also, all MC would have to be re-generatedindependently for those early datasets, since the pion stopping position wassignificantly different from the ones analyzed in this thesis. All correctionsand shapes are dependent on the pion stopping position.Another possibility is relaxing the acceptance AR up to 60 mm, reducingthe statistical error to around 10 [10−8] Rpi units using only Run IV, V andVI. However, it was verified that when AR = 60 mm the systematic errorfrom the LET inflates the total systematic error from the analysis from 8[10−8] Rpi units to 14. If the uncertainty on the wire-chamber-3 (WC3)position along the beam axis currently at ±1 mm is proved to be actually±0.5 mm, and the calorimeter energy uncertainty currently at 100 keV isreduced to 50 keV, the systematic error on the LET correction at higherangles will shrink, thus allowing events up to AR = 60 mm without increasingthe global systematic error. The implementation of the earlier datasetsor the refinement of WC3 position and the calorimeter energy calibrationuncertainty would improve the branching ratio measurement uncertaintyfrom 0.12% to 0.10%, and if both are executed properly the analysis couldaccess a measurement uncertainty of 0.09% or better.7.4.2 Next generation PIENUThe next generation PIENU experiment would have to aim for a higherprecision measurement goal close to the current theoretical calculation pre-cision at 0.016%. To allow further experimental statistical precision whilekeeping the current stopping-pion technique, a higher number of pi+ → e+νe38 The number of pi+ → e+νe events quoted for the rest of the chapter are for acceptanceradius AR = 60 mm.1577.4. Future prospectsevents must be collected by running for longer periods of time, or a calorime-ter setup with a bigger acceptance (currently around 20%) such as the 4piacceptance calorimeter used in the PEN experiment at PSI [51]. To al-low further experimental systematical precision while keeping the currentstopping-pion technique, the LET energy correction must be calculated tohigher precision. The LET precision is limited by the detector geometricaccuracy, calorimeter’s photo-nuclear (PN) interactions Monte-Carlo calcu-lation accuracy, and calorimeter’s energy resolution. Therefore better ma-chinery and assembly techniques for the detector’s components, improvedtheoretical PN interaction Monte-Carlo implementations, and better energyresolution are required to reach a new level of systematic precision measure-ment.158Chapter 8Limits on New Physics8.1 The pi+ → e+νe branching ratioThe blinded39 branching ratio Rpi =Γ(pi+→e+νe+pi+→e+νeγ)Γ(pi+→µ+νµ+pi+→µ+νµγ) calculatedfor this thesis regarding the highest quality data available from PIENU’sdatasets (Run IV, V and VI) with about 3 million pi+ → e+νe events40collected between 2010 and 2012 isRblindpi = (1.2∗∗∗ ± 0.0013(stat.)± 0.0008(syst.))× 10−4. (8.1)For the 2012 dataset the total reduced χ2/d.o.f. (where d.o.f. = 1557) is:1.19, and 1.13 for the pulse-height (PH) and charge-integration (Q) basedtime spectrum analysis, from which the raw branching ratio is extracted;1.08, 1.06, 1.00, and 1.07 for 2011; and 2010-November datasets. Although,the PH and Q analyses are consistent with each other, PH was chosen overthe Q-based branching ratio for being less sensitive to pileup events. Rblindpirepresents a 0.12% precision measurement, a factor of 30 improvement fromprevious generation experiments [12] [13] and a factor 2 from a subset ofdata (Run IV) published [5] in 2015 asR2015pi = (1.2344± 0.0023(stat.)± 0.0019(syst.))× 10−4. (8.2)Limits on new physics can be obtained starting from an upper limit to thebranching ratio RUL, which can be calculated for example with the Feldman-Cousins “unified approach” frequentist method [133] (Rexp−RSM)/σ whereRexp is the measured branching ratio, σ the total error, and RSM is the SMprediction. Consulting Table X of ref. [133], an upper limit can be obtained.For the published result R2015pi , with a combined (statistical+systematic)error σ = 0.003 × 10−4, the upper limit at 95% confidence level is 1.67standard deviations above the SM value39 If the blinding (Section 1.3) is to be removed from this analysis, the branching ratiowill move within ±0.5%.40 3 million events when acceptance radius AR = 60 mm.1598.2. Lepton UniversalityRUL = 1.2402× 10−4. (8.3)For comparison, using the improved combined error from this thesis σ =0.002 × 10−4, and the R2015pi value for Rexp shrinks the upper limit to1.2384 × 10−4. New physics would not necessarily increase the branch-ing ratio, it could also decrease it. Thus, a lower limit could be calculatedsimilarly.8.2 Lepton UniversalityLepton universality (LU) is the assumption that the W boson couples withthe same strength to each lepton generation, i.e., ge = gµ = gτ . If there isa difference in the couplings, we can quantify it with the three differentcoupling constants ge, gµ, and gτ . In the case of the pi+ → e+νe branchingratio we have Rexppi = (ge/gµ)2RSMpi (see Section 2.3.1) where Rexppi is themeasured branching ratio and RSMpi is the SM prediction. Since the yieldsdepend on the square of the coupling constants, the measurement of thebranching ratio is a particularly powerful test of LU. Using the publishedresult of R2015pi , a 0.24% precision measurement, the following result wasobtained,ge/gµ = 0.9996± 0.0012, (8.4)translating into a 0.12% precision of the lepton universality test. Using thecurrent estimates for the errors from Rblindpi (0.12% precision) would improvethe errors of the ratio of the coupling constants to ±0.0006, thus reaching a0.06% precision test of LU.This would make pion decay the most sensitive test of lepton universality,and improve the already stringent constraints on models attempting to ex-plain the hints of possible lepton non-universality seen by the LHCb [71] [72]and BaBar [73] experiments. Essentially, the models must include the prop-erty that the mechanism that couples differently to the different generationsbe greatly enhanced for the third generation [77].8.3 New Pseudo-scalar InteractionsThe branching ratio is very sensitive to the presence of new pseudo-scalarinteractions. By substituting the SM prediction and the value from the1608.3. New Pseudo-scalar Interactionsestimated upper limit RUL (Eq. 8.3) into the Eq. 2.20 gives1.24021.2352− 1 ∼(1 TeVΛ)2× 103, (8.5)which gives the estimateΛ ∼ 497 TeV. (8.6)Thus, the mass scale of a new fundamental pseudo-scalar, with the samecoupling strength to quarks and leptons as the weak interaction, must be> 500 TeV at 95% C.L. Using the upper limit derived with the improved es-timated error raises the new pseudo-scalar interaction limit to Λ = 621 TeV.8.3.1 R-Parity violating SUSYThe relationship between Rpi and the R-parity violating parameters ∆′11kand ∆′21k (see Section 2.3.2) is∆RpiRSMpi= 2(∆′11k −∆′21k). (8.7)Rpi itself does not provide any constraint on the size of ∆′11k and ∆′21k in thecase where they are equal in value. According to Figure 2.8, a 0.1% levelprecision measurement of the branching ratio and in the extreme case where∆′11k = 0 then ∆′21k should be restricted to 0.002± 0.001, at 95% C.L.8.3.2 Charged Higgs BosonAs discussed in Section 2.3.2, if the coupling of the charged Higgs bosonto leptons is proportional to the lepton mass, as with the SM Higgs boson,Rpi is unaffected by the presence of a charged Higgs boson. However, ifthe coupling is independent of the lepton mass, this is no longer the case.Assuming couplings of the order λeν ∼ λµν ∼ λud ∼ α/pi we havemH± ∼mpimWαpi√2me(mu +md)(1− memµ)RSMRSM −Rexp . (8.8)The limit at 95% C.L. for the upper limit RUL (Eq. 8.3) isMH± ≥ 182 GeV. (8.9)Using the upper limit derived with the improved estimated error raises themass limit to MH± ≥ 227 GeV.1618.4. Search for Massive Neutrinos in the pi+ → e+νe Decay8.4 Search for Massive Neutrinos in thepi+ → e+νe DecayLimits for massive neutrinos described in Section 2.3.4 below 50 MeV/c2can be set by using the Feldman-Cousins upper limit to the branching ratioRUL at 95% C.L. and Equation 2.29,|Uei|2 = RUL/RSM − 1ρe − 1 . (8.10)Thus, the limits on the mixing matrix |Uei|2 can be calculated as a functionof neutrino mass mνi . Figure 8.1 shows the 95% C.L. upper limit on theheavy-neutrino mixing parameter, as a function of its mass. The blue lineshows the result from the derived branching ratio upper limit from a subsetof PIENU data (Run IV) published [5] in 2015, i.e., a heavy neutrino massmνi of 50 MeV/c2 has a limit of approximately 10−6 in the mixing parameterand the limit increases as mνi goes to zero.Figure 8.1: The 95% C.L. upper limit on the heavy neutrino mixing parame-ter, as a function of its mass. The blue line shows the result from the derivedbranching ratio upper limit from a subset of data (Run IV) published in 2015[5].Above 55 MeV, a search has been performed [11] for the mixing of heavyneutrinos coupled to electrons in the decay pi+ → e+νh using the full PIENU1628.5. Summary and Forward-looking for SM deviation scenariosdataset, i.e., all runs from 2009 to 2012. No extra peaks due to heavyneutrinos were found in the positron energy spectrum as shown in Figure8.2, resulting in upper limits set on the square of the mixing matrix elements|Uei|2 from 10−8 to 10−7 for neutrino masses in the range 60 to 135 MeV/c2.See Figure 8.3. These results assume coupling to e+ but are independent ofassumptions about the nature of the heavy neutrino and are complementaryto limits from neutrino-less double beta decay found in Ref. [134], whichassume that massive neutrinos are Majorana in nature.Figure 8.2: Background-suppressed pi+ → e+νe positron energy spectrum(black histogram). Fitted components include muon decays in flight (thickblue line, from MC), pi+ → e+νe (green, dot-dashed line, fit to MC), andpi+ → µ+νµ → e+νeν¯µ (red dashed line, from late-time data events). Theinsert shows the (rebinned) residuals (Data-Fit) with statistical error barsand the signal shape (massive neutrino search) in the case of Ee+ = 40 MeVand |Uei|2 = 10−8 [11].8.5 Summary and Forward-looking for SMdeviation scenariosIn the scenario where the PIENU experiment gives a mild deviation fromthe SM result, what beyond-SM explanation is right for some future experi-ment? This thesis had already discussed direct access to Lepton Universality1638.5. Summary and Forward-looking for SM deviation scenariosFigure 8.3: 90% C.L. upper limits on the square of the mixing matrix el-ements |Uei|2 of heavy neutrinos coupled to electrons (thick red line) re-garding the full PIENU dataset, , i.e., all runs from 2009 to 2012 [11]. Theblack dashed line shows the results from the previous generation PIENUexperiment [29].test of a first order weak interaction using the measured pi+ → e+νe branch-ing ratio. There is also the search of massive neutrinos lighter than the pi+in the pi+ → e+νe energy spectrum. Thus, the next generation PIENU ex-periment (see Section 7.4) would be a sensible test for beyond-SM deviationsby delivering a higher precision pi+ → e+νe branching ratio measurement,i.e. O(0.01%). On the other hand, a direct detection of a charged Higgs bo-son (H±) is not within the capabilities of the current PIENU experimentaltechnique. Nevertheless, the ATLAS collaboration has reported a search forcharged Higgs bosons H± → tb decay channel in proton-proton (pp) colli-sions at 8 TeV and H± → τ±ντ of pp collision at 13 TeV in ref. [82] and[83], respectively. The H± → tb search explored the H± mass range from200 to 600 GeV but no significant candidates were found. The H± → τ±ντsearch reported no evidence of a charged Higgs boson for the mass range90–2000 GeV at a 95% confidence level. 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Rev. D, 57:3873–3889, 1998.[134] A. de Gouveˆa and A. Kobach. Global constraints on a heavy neutrino.Phys. Rev. D, 93:033005, 2016.175Appendix ATime spectrum for pi → µ→ eThe pi → µ → e process is a decay chain composed by the two decayscharacterized by the decay times τpi = 1/λpi and τµ = 1/λµ. The piondecays with a rate dNpi/dt given bydNpidt= −λpiNpi, (A.1)where Npi is the number of pion at time t. Assuming that all the pions decayinto muons, the formation rate of the muons equals the decay rate of thepionsdNµdt= +λpiNpi. (A.2)At the same time, the muons decay according todNµdt= −λµNµ. (A.3)The overall change in the muon population is therefore given bydNµdt= λpiNpi − λµNµ. (A.4)Assuming a known initial amount of pions N0pi and muons N0µ, the solutionsof Eq. A.4 isNµ = N0piλpiλµ − λpi (e−λpit − e−λµt) +N0µe−λµt. (A.5)From the last result, assuming no initial muons present, the normalizedpi → µ→ e time spectrum has a shape described byf(t) =1τµ − τpi (e− tτµ − e− tτpi ), (A.6)Where N is the total number of events.176Appendix BCuts for Pion DataTable B.1: List of cuts.Cut NotesBlinding §1.3 ActivatedIntegrity §3.4.2 Error signals from COPPER system clearedPhysics Triggers §3.4.1 Only Prescale, Early, and TIGC.Pion Identification §4.2.1Pion Energy B1 from 3.8 to 5.2 MeV,B2 from 2.0 to 3.1 MeVWC1,2’s HaloaB1,2 PU Only one hit in one of four PMTs,0.75 < Q/Qw < 1.05B1 Waveform B1 pulse fitting activated, χ2 ≥ 0B1 prompt −1380 < B1t < −1340 nsTrConsaPileup After Target §4.2.2PionTrigaT1-T2 sync abs(T1− T2) < ±20 nsProton See Figure 4.6T1prompta T1 hits within ±2 ns of pion timing (tpi+)T1 fake PU See Figure 4.7T1 Waveform T1 pulse fitting activated, χ2 ≥ 0Post-PUb Any VT48 hit with: 1335 < T1t[i]−B1t[j] < 1480 ns,COPPER’s fitted avg: T1t,avg[i]−B1t,avg[i] > 420 nsEarly Time Pileup §4.2.3Pre-PU No hits in pre-region for B1, B2, B3, T1, and T2Beam Muons and No hits from 8000 to 16400 andTwo Pionsc 16600 to 17850 ns (prompt at 16500).FalseTrig See Figure 4.8Acceptance §4.2.4 AR < 40 mm, See Figure 4.9a Year dependent, see Table B.2.b Only used for integrated charged (Q) based branching ratio.c Two pion detection only available for 2012 dataset.177AppendixB.CutsforPionDataTable B.2: Year dependent cut values.Run# ↓, Cut → WC1,2 TrCons, T1prompt T1prompt PionTrig Bina’s Alignment>= < x L x H y L y H p L p H av[0] av[1] av[2] av[3] t 1 t 2 t low (Q) (PH)29000 -4399 -4380 11.2 13.1 11.8 13.1 1020 1040 -379731000 42250 -22 16 -12 18 -4395 -4375 7.9 8.7 9.1 9.1 0.98298 0.9969142250 54819 -4409 -4390 7.3 8.1 8.4 8.3 1030 105045819 46816 -4410 -4380 2.9 3.7 3.9 3.846816 47133 -4390 -4370 3.3 5.2 3.9 5.1 1020 104047133 49005 7.2 9 7.8 9.149005 52006 -4399 -4380 11.2 13.1 11.8 13.157418 61179 -20 18 -4399 -4380 11.9 13.6 12.6 13.6 1010 1040 0.98086 0.9938562491 70025 -23 19 -17 19 -4399 -4350 11.2 13.1 11.8 13.1 1000 1040 -3820 0.97953 0.9945870025 81560 -4375 -4350 12.1 13.6 12.8 13.6178Appendix CCuts for Positron DataThis appendix is complementary to the discussion of the Response Func-tion Measurement described in Section 6.1.1. Details for event selectioncuts to cleanse the 70 MeV positron beam for proper calorimeter responsecharacterization and MC comparison are discussed here. Before any cuts,the raw calorimeter’s response for the beam aligned with the crystal axis(0 degrees), is shown in Figure C.1(a) in black. The energy spectrum isstructure rich as only the T2 scintillator was used for triggering, therebyallowing several types of events to be included.The raw calorimeter’s response shown in Figure C.1(a) (black) has richstructure: the peak at 1 MeV are events without hits in the wire chamberscorresponding to the beam spot in the calorimeter. Such a pileup must becoming out backward from the calorimeter and not from the beam itself.The beam pions and muons are at 14 and 18 MeV, respectively. Using MC,it was determined that structure near 30 MeV appear to be pions decaying inflight. Photo-nuclear effects due to photons kicking out one or two neutronsare visible near 50 and 60 MeV. The main peak around 70 MeV is dueto beam positrons. When a positron and pion arrive simultaneously, theyform the peak near 78 MeV. The structure to the right of that peak is for apositron and muon arriving together. Finally, the peak around 130 MeV isdue to two positron events. The last three structures composition is knownfor the correspondence of the peaks to the sum of individual particles inconjunction to the energy deposited in T2.WC halo and timing. The beam’s reconstructed x and y positiondistributions using WC12 tracker (section 3.2.6) are shown in Figure C.1(b)with red lines indicating the cut values. Events not due to beam particlesmust be excluded. The beam’s halo needs to be trimmed out with thetracking information, leaving just the beam spot. The resultant energyspectrum is shown in Figure C.1(a) in red. Further suppressing for non-beam particles; the background was reduced by eliminating events without-of-time hits in all three wire chambers; such a cut was implemented by179Appendix C. Cuts for Positron Data(a) The BINA + CsI energy before (black)and after (red) the WC12 X and Y cuts.(b) Y vs. X position profiles as recon-structed by WC12. The red lines indicatesthe cut values.(c) Time of the first plane in WC1 in VT48counts. The red lines indicate the cut val-ues.(d) The BINA + CsI energy before (black)and after (red) the WC timing cuts.Figure C.1: The 0 degree positron energy spectrum cleanse trough WC12spatial and timing cuts.keeping only the peak at ∼7470 ADC counts (∼4700 ns). The WC1 timing isshown in Figure C.1(c) with red lines indicating the cut values. Similar cutswere made in WC2 and WC3. Results for the WC timing cut are reflectedin the energy spectrum, i.e., Figure C.1(d) in black before and red after thetiming cut.Muon correction. Following these cuts, the spectrum contained eventsdue to beam positrons and beam muons. Assuming no shower leakage fromthe crystals, the total positron energy is the sum of the energy deposited inT2 and the energy deposited in the calorimeter (Bina+CsI). It is possible toremove beam muons completely using a cut on the energy deposited on theT2 scintillator, but this changes the tail significantly, as such a constraintremoves some beam positrons with a direct dependence on the calorimeter180Appendix C. Cuts for Positron DataFigure C.2: The energy in Bina + CsI vs. the energy in T2. Blobs cor-responding to positrons (∼70 MeV), muons (∼18 MeV), and pions (∼14MeV) can be clearly seen. There is also a structure around 30 MeV in Bina+ CsI, with energy loss in T2 between positrons and beam muons. A similarstructure appears in simulated pion events, from decays in flight.Figure C.3: The time of flight vs. the energy in BINA + CsI. Blobs cor-responding to positrons and muons can be clearly seen. The region withessentially no events is due to the trigger condition excluding part of the RFwindow.response, thereby introducing a bias to the response function measurement.Alternatively, the calorimeter vs. T2 energy distribution is shown in FigureC.2, which clearly identifies beam backgrounds below 35 MeV in Bina+CsI181Appendix C. Cuts for Positron Dataenergy, where the positron tail is tiny, beam pions are at 14 MeV, beammuons are at 18 MeV, and piDIF are at 30 MeV. Implementation was carriedout by removing events with more than 400 ADC counts in T2 and less than35 MeV in Bina.Figure C.4: The energy spectrum of positrons in BINA + CsI, selected bytime of flight.Figure C.5: The energy spectrum of muons in BINA + CsI, selected by timeof flight.182Appendix C. Cuts for Positron DataThe remaining muon trail above 35 MeV are events in which muons de-cayed within Bina’s 1 µs integration window. Such muons are identifiedwith the RF time window vs. calorimeter energy distribution, as shown inFigure C.3. Muons are selected within 12 to 15 ns and positrons between4 to 11 ns. The trigger was limited to a portion of the cyclotron’s 43.3 nsRF window to record only where most positrons were present. The timeregion of 20 to 40 ns was not sampled as it contained mostly pions. Thecalorimeter’s energy distribution after calorimeter vs T2 energy (Figure C.2)and calorimeter vs RF (Figure C.3) cuts is shown in Figure C.4 and Fig-ure C.5, for positron and muon selection respectively. There is still a muonpeak in the positron’s spectrum but there is at best a negligible amount ofpositrons in the muon’s distribution, as the near 70 MeV positron peak isgone. These conditions allow the muon spectrum to be subtracted from thepositron spectrum and cleanse the muon contribution completely withoutcompromising the response function. The procedure for muon subtractionis as follows:ˆ The muon spectrum is normalized to the muon peak from the positronspectrum.ˆ The T2 vs. calorimeter cut is applied to the positron spectrum.ˆ Set the muon spectrum to zero up to 35 MeV.ˆ Subtract the muon spectrum from the positron spectrum.The result is shown in Figure 6.2 in black and the corresponding simu-lated spectrum is shown in red with the same cuts applied. There still aresome muons left and piDIF in the positron spectrum, but it represents anegligible contribution to the tail fraction < 0.01%.183Appendix DBeam-line SimulationTRIUMFs primary 520 MeV proton beam-line (BL1A) with 120 µA im-pulses on the Be production target apparatus shown in Figure D.1(a) withproton bunches of 4 ns width every 43 ns. Figure D.1(b) show the Beproduction target in detail. The cassette target consists of an oval tube orcassette measuring 18.8 mm by 11.3 mm, made from 0.25-mm-thick 316ELCstainless steel bent to shape, with 0.076-mm-thick 437 stainless steel win-dows welded at each end. The metal target is usually beryllium with a crosssection of 14.7 mm by 5.1 mm, and it is held in the center of the tube by awire frame. Cooling water enters the cassette near one end and leaves nearthe other end. The metal targets are completely immersed in the waterinflow path at the entrance and exit faces and the sides.(a) Full Frame (b) Zoom on production targetFigure D.1: T1 production target apparatus for M13 beam extension.Beam-line settings. The two principal beam-line configurations sim-ulated correspond to data run number #54880 and #81633. The mainparameters are listed in Table D.1. Such runs were chosen as they representdifferent positron data taking periods for different years. The approach wasto simulate their beam-line parameters and obtain the intrinsic positron tail184Appendix D. Beam-line SimulationTable D.1: Beam-line’s settings for positron runsParameter Run # 54880 Run # 81633Positron beam mean momentum 75 MeV/c 70 MeV/cMomentum spread σp 12 MeV/c 0.7 MeV/cBeam origin spot size (Gaussian) −→σ (1.67, 1.67, 1.67) mm (3.656, 3.133, 1.833) mmDipole field (B1,B2,B3) (0.2138,−0.2307, 0.2808) T (0.2077,−0.2077, 0.2630) TSlit width (F0,F1,F2) (102.4, 14.7, 30.0) mm (120.0, 15.5, 30.4) mmcontribution, and validate MC by comparison with data. The implementa-tion for run #54880 in the beryllium production target consisted of shootinga Gaussian positron beam from the origin ±1.67 mm in the x, y, and z co-ordinates (99.7 % of the origin of the beams will be within a sphere havinga radius of 10 mm ). The beams had a mean momentum of 75 MeV/cand RMS momentum width 12 MeV/c to give a wide range of momentumas input to the M13 beam-line. These Gaussian beams were directed tothe entrance of the beam-line from all angles within its acceptance cone.In Figure D.2(a), it can be seen how a Gaussian beam is directed to thecenter of the entrance of the beam-line, and in Figure D.2(b), how severalbeams are distributed within the acceptance cone. The implementation forrun #81633 was carried out similarly by increasing the 10-mm-radius modelthree times, modifying the mean momentum to 70 MeV/c width 0.7 MeV/c(meaning different gradients for the quads and magnets), and different slitwidths for focus points along the beams. Both implementations showed anegligible difference in the final beam intrinsic tail fraction.(a) Gaussian beam aligned to beam-linemain axis.(b) Beam-line’s acceptance coneFigure D.2: Beam input simulation185Appendix D. Beam-line SimulationIn Figure D.3(a), we can see the wide momentum range of the initialdistribution (red) at the beginning of the beam and how the momentumdistribution is affected after each beam component, and the final distribu-tion (black) at focus point F4 with the corresponding beam spot in FigureD.3(b). Position profiles at F4 (target B3) for the x and y axes for run 54880configured to a 75 MeV/c positron beam are shown in Figure D.4(c) andD.4(d) respectively, the F4 position profiles for x and y axis from G4beamlineoutput with the same settings from run #54880 are shown in Figure D.4(a)and D.4(b) respectively. Position profiles at F4 for for x and y axis for run#81633 configured to a 70 MeV/c positron beam are shown in Figure D.5(a)and D.5(b) respectively, each plot has data (blue) and MC (red) overlay-ed.There are additional systematic tests and data comparisons with previousstudies found in [6] and [20], including the number of events to determinethe statistical accuracy, beam-line steering for beam spot matching (throughadditional data runs with similar beam-line settings), simulated magneticfield vs. implementation of measured magnetic field at bending magnets,and beam rotation. All of them were taken into account to set an upperlimit to the intrinsic beam tail contribution.Tail origin. As there is an interest in where the tail comes from, in thissection, we show the results of tracing back the events in the tail of thepositron momentum distribution at the final focus point F4. It was foundthat there is indeed a tail contribution coming from the beam scattering atseveral points on the beam-line. A common tail event comes from focus pointF1 slit just after the first bending magnet B1. As shown in Figures D.6(a)and D.3(a)(green to light blue), the slit opening cleans low momenta fromthe positron beam but generates scattering, which will eventually contributeto the intrinsic beam momenta at the final focus point F4. Figure D.6(b)shows the x-axis position distribution after F1 slit; the events near the maindistribution peak contribute to the tail.Pion beam. Similarly, a pion beam was successfully simulated in ac-cordance with pion data taking runs settings to evaluate and validate thebeam-line’s G4beamline MC implementation. Figures D.7(a) and D.7(b)show the main components of the beam simulation including the positronand pion beam settings, respectively. The implementation of the pion beamincluded additional components in the beam, a beam degrader after F1 toseparate particles and allow pion selection further down the beam, and acollimator just before the third bending magnet as described in Section 3.1.186Appendix D. Beam-line Simulation(a) Evolution of beam momentum(b) Beam spot for final beam momentum at F4.Figure D.3: Beam low momenta cleaning sequentially through differentbeam components and the final beam spot at F4.187Appendix D. Beam-line Simulation(a) X-axis position distribution in MC (b) Y-axis position distribution in MC(c) X-axis position distribution in Data (d) Y-axis position distribution in DataFigure D.4: Position profiles from MC and from positron run #54880 at F4.(a) X-axis position distribution (b) Y-axis position distributionFigure D.5: Position profiles from MC and from positron run #81633 at F4.188Appendix D. Beam-line Simulation(a) Beam momenta cleanse(b) Beam x-axis position distribution after F1’s slitFigure D.6: Focus point F1 slit simulation189Appendix D. Beam-line Simulation(a) Positron beam simulation.(b) Pion beam simulation. Pions in green, muons in blue, positrons in red.Figure D.7: Aerial view of beam-line simulation including all main compo-nents. Please refer to Figure 3.2 for blueprint.a) Right to Left: Starting from the T1 production target 75 MeV width ±12MeV positrons (red) are isotropically simulated and go trough the first twofocusing quadrupoles Q2. Only a small solid angle is displayed. Positronpasses horizontal slit (F0SL) and vertical jaws (F0JA) combo, then thefirst bending dipole steers the beam CW, then low momenta cleanse is donetrough F1SL/JA. Beam gets re-focused with three quads Q3, Q4, and Q5,enters another F2SL/JA to then get bended CC and further focused byquads Q6 and Q7. Positrons enter the beam-line extension and positronsbend CW trough dipole B3 and final focusing is done with Q8, Q9 andQ10.b) Same configuration but in this case only pions (green) are produced ini-tially. Muons (blue) and positrons are produced along each pion tree eventbut limited to one vertex. Additionally, a Lucite absorber is inserted af-ter F1SL to separate the beam composition to enable magnetic selection ofpions further downstream and finally a collimator at the beginning of thebeam-line extension to filter the pions.190Appendix ETrigger DiagramFigure E.1: Complete trigger diagram of the PIENU Experiment [19].191Appendix FTechnical DrawingsFigure F.1: Side view of the PIENU Detector. The pion beam comes fromthe right side.192Appendix F. Technical DrawingsFigure F.2: Cross section of the PIENU Detector.193AppendixF.TechnicalDrawingsACHICAGO MAGNET SHIELDING WALLASSEMBLY -TDE0601TDE0539 BEAMLINE ASSEMBLYTDE0548 OVERALL AREA LAYOUT31.5210.50DETAIL A SCALE 1 : 3TDE0589-BEAMLINE VACUUMTUBE ASSEMBLYTDE0637PIENU 1 OVERALL ASSEMBLYDIMENSIONS QUOTED ARE FINISHED DIMENSIONS, NO ALLOWANCE HAS BEEN MADE FOR MANUFACTURE.REMOVE ALL BURRS AND SHARP EDGESREV DATE ZONE REVISION DESCRIPTION APPROVEDD68311TDE0613DEC/081:32PIENU BEAMLINE END WITH PIENU1ASSEMBLYRoland Kokke0.010.5°0.005 125 .010.1h1gf2345hgfedcba1edc234ba5THIS DRAWING, SUBJECT MATTER AND INFORMATIONCONTAINED THEREIN, IS THE SOLE, EXCLUSIVE ANDCONFIDENTIAL PROPERTY OF  TRIUMF LABORATORY,AND AS SUCH, SHALL NOT BE DISCLOSED, COPIED,REPRODUCED  OR USED,  IN WHOLE  OR IN PART,WHITHOUT EXPRESSED WRITTEN PERMISSION OF THETRIUMF LABORATORY  OR  ITS  REPRESENTATIVES.DO NOT COPY, THIS DOCUMENTCONTAINS PROPRIETARY INFORMATIONCANADA'S NATIONAL LABORATORY FORPARTICLE AND NUCLEAR PHYSICSTRIUMFOFSHEETSIZENEXT ASSY:DRAWNREA #CHECKEDSCALEDATETOLERANCES UNLESS OTHERWISE SPECIFIEDALL DIMS IN INCHESDESIGNEDANGULARSURFACE FINISHDECIMALS.XXX.XX±±±µ inchDWG NO. REVVANCOUVER, BRITISH COLUMBIACANADA V6T-2A34004 WESBROOK MALLTHIRD-ANGLE PROJECTION.X ±Figure F.3: The PIENU detector mounted to TRIUMF’s M13 beam-line.194

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