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A high-precision measurement of the pion branching ratio Sullivan, Tristan 2017

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A high-precision measurement of thepi → eν branching ratiobyTristan SullivanBSc., University of Victoria, 2009A 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 2017© Tristan Sullivan 2017AbstractThe pion decay branching ratio, Rpi =Γ(pi+→e+νe+pi+→e+νeγ)Γ(pi+→µ+νµ+pi+→µ+νµγ) , is an impor-tant observable in the Standard Model of particle physics. The value of thepi → eν branching ratio has been calculated within the Standard Model tobe (1.2352 ± 0.0002) ×10−4[1] [2]. The PIENU experiment at TRIUMFaims to measure this quantity to a precision of < 0.1%. This tests thehypothesis that the leptons have identical weak couplings, known as leptonuniversality, at the 0.05% level. In addition, it provides stringent constraintson many other extensions to the Standard Model, such as R-parity violatingsupersymmetry, leptoquarks, and heavy neutrinos lighter than the pion. Incertain cases, these constraints can far exceed the reach of direct searchesat colliders. Most strikingly, a new pseudoscalar interaction whose energyscale were O(1000 TeV) would enhance the branching ratio by O(0.1%).The PIENU data set contains four years of data, taken between 2009 and2012. The analysis of a subset of the 2010 data was published in 2015 [3];the precision obtained for the branching ratio was approximately 0.25%.The 2012 data set is roughly five times larger than the 2010 data set, andits analysis is presented here. The statistical error using only 2012 datais 0.09%; incorporating the other data sets will reduce this to 0.07%. Thesystematic error in the 2012 analysis remains considerably larger than this,0.27%, and the prospects for reducing it to the level needed to reach theprecision goal of the experiment will be presented. The present work hasbeen done as a blind analysis, with an unknown factor masking the branchingratio result; the final value will be obtained when the analysis is completed.iiPrefaceThe PIENU collaboration consists of about twenty people from six differentcountries. Proposed in 2006, data-taking for the experiment began in thespring of 2009, and continued until December 2012. I joined the experimentas a Masters student in September 2009 and transferred to the PhD programthe following year. Throughout the data-taking period, from 2009 to 2012, Itook many shifts monitoring the data collection, and during 2011 and 2012was responsible for the trigger system, including modifying it as necessaryand documenting any changes.I was primarily responsible for measuring the response function of thePIENU calorimeter, which was necessary to obtain the largest correction tothe branching ratio, and also expected to be the largest source of systematicuncertainty in the experiment. Special data were taken to evaluate thiscorrection; this was first done at the end of 2009, but the uncertainty wastoo large, and it was done again in 2011. For this measurement I wasresponsible for devising a method to reduce the uncertainty and ensuringthe quality of the data was sufficient, as well as for its subsequent analysis.I also contributed to the main branching ratio analysis, which was pub-lished in 2015 using part of the data taken in 2010 [3]. I was primarilyresponsible for the analysis of the 2012 data set, which was approximatelyfive times larger than the 2010 data set. Although the desired level of pre-cision has not yet been achieved, the sources of systematic uncertainty thatstill need to be reduced, and ways of potentially achieving this, have beenidentified.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxxi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 The Standard Model . . . . . . . . . . . . . . . . . . . . . . 11.1.1 The weak interaction . . . . . . . . . . . . . . . . . . 21.2 Motivation for the PIENU experiment . . . . . . . . . . . . . 31.3 Brief history of pion branching ratio measurements . . . . . 41.4 Experimental technique . . . . . . . . . . . . . . . . . . . . . 51.4.1 PIENU experimental technique . . . . . . . . . . . . 91.5 Outline of the thesis . . . . . . . . . . . . . . . . . . . . . . . 112 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.1 Pions in the Standard Model . . . . . . . . . . . . . . . . . . 122.1.1 Pion decay modes . . . . . . . . . . . . . . . . . . . . 122.1.2 The weak interaction . . . . . . . . . . . . . . . . . . 122.1.3 Pion decay rate . . . . . . . . . . . . . . . . . . . . . 142.1.4 Corrections . . . . . . . . . . . . . . . . . . . . . . . . 162.2 Beyond the Standard Model . . . . . . . . . . . . . . . . . . 192.2.1 New pseudoscalar interactions . . . . . . . . . . . . . 20ivTable of Contents2.2.2 R-parity violating supersymmetry . . . . . . . . . . . 212.2.3 Charged Higgs . . . . . . . . . . . . . . . . . . . . . . 222.2.4 Leptoquarks . . . . . . . . . . . . . . . . . . . . . . . 242.2.5 Massive neutrinos . . . . . . . . . . . . . . . . . . . . 242.3 Lepton universality . . . . . . . . . . . . . . . . . . . . . . . 252.4 Physics Reach . . . . . . . . . . . . . . . . . . . . . . . . . . 283 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . 293.1 Beamline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.3 Trigger and DAQ . . . . . . . . . . . . . . . . . . . . . . . . 433.4 Event types . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.4.1 Beam-related background . . . . . . . . . . . . . . . . 533.5 Monte Carlo . . . . . . . . . . . . . . . . . . . . . . . . . . . 574 Data Taking and Processing . . . . . . . . . . . . . . . . . . . 584.1 Running periods . . . . . . . . . . . . . . . . . . . . . . . . . 584.1.1 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . 584.1.2 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1.3 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.1.4 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 594.2 Data processing . . . . . . . . . . . . . . . . . . . . . . . . . 594.2.1 Plastic scintillators . . . . . . . . . . . . . . . . . . . 594.2.2 Wire chambers . . . . . . . . . . . . . . . . . . . . . . 614.2.3 Silicon strips . . . . . . . . . . . . . . . . . . . . . . . 614.2.4 Crystal scintillators . . . . . . . . . . . . . . . . . . . 614.3 Blinding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 614.4 Energy calibration . . . . . . . . . . . . . . . . . . . . . . . . 624.5 Track reconstruction . . . . . . . . . . . . . . . . . . . . . . . 674.6 Data stability . . . . . . . . . . . . . . . . . . . . . . . . . . 724.7 Timing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745 Raw Branching Ratio Extraction . . . . . . . . . . . . . . . . 805.1 Event selection . . . . . . . . . . . . . . . . . . . . . . . . . . 805.1.1 Good run selection . . . . . . . . . . . . . . . . . . . 805.1.2 Time spectrum . . . . . . . . . . . . . . . . . . . . . . 805.1.3 Pion selection . . . . . . . . . . . . . . . . . . . . . . 825.1.4 Pileup rejection . . . . . . . . . . . . . . . . . . . . . 835.1.5 Acceptance cut . . . . . . . . . . . . . . . . . . . . . 915.1.6 Minor cuts . . . . . . . . . . . . . . . . . . . . . . . . 92vTable of Contents5.2 Fitting function . . . . . . . . . . . . . . . . . . . . . . . . . 945.2.1 Low energy time spectrum . . . . . . . . . . . . . . . 945.2.2 High energy time spectrum . . . . . . . . . . . . . . . 945.3 Fitting method . . . . . . . . . . . . . . . . . . . . . . . . . . 995.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.5 Systematic checks . . . . . . . . . . . . . . . . . . . . . . . . 1095.6 Summary of Chapter 5 . . . . . . . . . . . . . . . . . . . . . 1136 Tail Correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1156.2 Lower limit . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.3 Response function measurement . . . . . . . . . . . . . . . . 1206.3.1 Energy loss processes . . . . . . . . . . . . . . . . . . 1206.3.2 Detector setup . . . . . . . . . . . . . . . . . . . . . . 1226.4 Data-taking . . . . . . . . . . . . . . . . . . . . . . . . . . . 1246.4.1 Event selection cuts . . . . . . . . . . . . . . . . . . . 1256.4.2 Muon correction . . . . . . . . . . . . . . . . . . . . . 1266.5 Other systematics . . . . . . . . . . . . . . . . . . . . . . . . 1326.5.1 Background . . . . . . . . . . . . . . . . . . . . . . . 1336.5.2 Calibration and resolution . . . . . . . . . . . . . . . 1336.6 Positron beam simulation . . . . . . . . . . . . . . . . . . . . 1346.7 Determining the PIENU tail fraction . . . . . . . . . . . . . 1396.8 Tail as a function of R and Ecut . . . . . . . . . . . . . . . . 1456.9 Summary of Chapter 6 . . . . . . . . . . . . . . . . . . . . . 1487 Other Corrections . . . . . . . . . . . . . . . . . . . . . . . . . 1497.1 Acceptance correction . . . . . . . . . . . . . . . . . . . . . . 1497.1.1 Pion stopping position . . . . . . . . . . . . . . . . . 1507.1.2 Detector geometry . . . . . . . . . . . . . . . . . . . . 1537.1.3 Trigger thresholds . . . . . . . . . . . . . . . . . . . . 1537.2 Muon decay-in-flight correction . . . . . . . . . . . . . . . . . 1547.3 t0 correction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.4 Stability of the corrected branching ratio . . . . . . . . . . . 1557.5 Summary of Chapter 7 . . . . . . . . . . . . . . . . . . . . . 1588 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1608.1 Branching ratio result . . . . . . . . . . . . . . . . . . . . . . 1608.2 Future prospects . . . . . . . . . . . . . . . . . . . . . . . . . 1628.2.1 Systematic uncertainty . . . . . . . . . . . . . . . . . 1628.2.2 Limits on new physics . . . . . . . . . . . . . . . . . . 163viTable of ContentsBibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166AppendicesA Trigger Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . 172B Timing Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . 173C Event Selection For Positron Beam Data . . . . . . . . . . . 179D Positron Data Systematics . . . . . . . . . . . . . . . . . . . . 185viiList of Tables1.1 The particle content of the SM; e is the charge of the electron. 11.2 Coupling constants for SM interactions. . . . . . . . . . . . . 32.1 Pion decay modes and branching ratios. . . . . . . . . . . . . 122.2 Corrections to the leading-order value for Rpi. . . . . . . . . . 192.3 Experimental results on lepton universality tests from studiesof pion, kaon, tau, and W decays. Here B represents thebranching fraction of a particular decay mode. . . . . . . . . 273.1 Detector characteristics. The z position given is for the centreof the detector, except as noted for BINA. . . . . . . . . . . . 333.2 The number of events in one 2012 run caused by the physicstriggers, and the most important calibration triggers. . . . . . 463.3 Detector readout channels. . . . . . . . . . . . . . . . . . . . . 494.1 Running periods. . . . . . . . . . . . . . . . . . . . . . . . . . 585.1 The probability of fake hits after the real hit, in each T1 PMT. 895.2 Events removed by each cut, with every other cut applied. . . 935.3 Fit parameter list. . . . . . . . . . . . . . . . . . . . . . . . . 1005.4 Fit parameters for the 2010 and 2012 data sets. Note that the2012 branching ratios are still blinded by an unknown factoruniformly distributed between ±0.5%. . . . . . . . . . . . . . 1095.5 Systematic checks performed on the fit of the 2012 data. . . . 1106.1 Angles at which positron beam data were taken. . . . . . . . 1246.2 Properties of the muon energy spectra as a function of angle. 1306.3 ∆T at 0°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.4 ∆T at 11.8°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.5 ∆T at 20.9°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.6 ∆T at 30.8°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.7 ∆T at 41.6°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131viiiList of Tables6.8 ∆T at 6°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.9 ∆T at 16.5°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.10 ∆T at 24.4°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.11 ∆T at 36.2°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.12 ∆T at 47.7°. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1316.13 The tail fraction as a function of angle, with errors due tostatistics and the variation in the muon-corrected values. . . . 1326.14 Corrections and resulting pi+ → e+νe tail fractions as a func-tion of Ecut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1476.15 Tail fractions as a function of the maximum radius in whichevents are accepted. . . . . . . . . . . . . . . . . . . . . . . . 1476.16 Tail fractions as a function of the radius in which events areaccepted. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1487.1 The small corrections that must be made to the branching ra-tio. The values are multiplied by the tail-corrected branchingratio to give the final result. . . . . . . . . . . . . . . . . . . . 1498.1 Sources of error. The corrected branching ratio is given bythe product of the raw branching ratio and the corrections.The errors given for the corrections are the errors on the cor-rections themselves, not the resulting errors on the branchingratio. The stars indicate that the result is still blinded. . . . . 161D.1 The change in the tail fraction as beam parameters and detec-tor geometry were varied. The values given are the nominaltail fraction minus the new tail fraction (see Section 6.6 fora detailed description of what was changed). Note that thechange is given as a fraction of the total spectrum, not the tail.The upper part shows the results for variations that increasedthe tail, and the lower part shows the results for variationsthat decreased the tail. The errors are due to Monte Carlostatistics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185ixList of Figures1.1 The measured value of the branching ratio as a function oftime. The dashed line shows the SM prediction. The lastpoint indicates the expected level of precision that will ulti-mately be achieved by combining the results of the PIENUexperiment and the PEN experiment at PSI, as described inthe text. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.2 Simplified picture of the PIENU experimental technique. Pi-ons stop in the target and decay into either muons or positrons;muons also stop in the target, and decay into positrons. Positrons,and photons if any are produced, are detected by the crystalscintillator calorimeter. . . . . . . . . . . . . . . . . . . . . . . 71.3 GEANT4 [4] simulation of the time spectra of pi+ → e+νe(red) and pi → µ → e (blue) events. The distributions arenormalized to the same height. . . . . . . . . . . . . . . . . . 81.4 GEANT4 simulation of the starting energies of positrons dueto pi+ → e+νe (red) and pi → µ → e (blue) decays. Thedistributions are normalized to the same height. . . . . . . . . 81.5 The PIENU detector, with the pion beam incident from theleft. The region close to the target is shown in the blowup.Plastic scintillators (polyvinyltoluene) are shown in dark blue,wire chambers in green, silicon strip detectors in orange, andcrystal scintillators in light blue (NaI(Tl)) and red (CsI). . . . 102.1 Feynman diagram for the decay of the positively charged pion.W+ represents the positively charged gauge boson mediatingthe weak interaction, and u and d represent an up and ananti-down quark, respectively. . . . . . . . . . . . . . . . . . . 14xList of Figures2.2 The allowed directions of the spins and linear momenta ofthe pion decay products, in the pion rest frame. The require-ment, from angular momentum conservation, that the spinsof the positron and the neutrino must be in opposite direc-tions, leads to the suppression of the positron mode relativeto the muon (see text). . . . . . . . . . . . . . . . . . . . . . . 162.3 Feynman diagrams for the radiative corrections to pion decay,from real (a) and virtual (b) photons. l+ denotes an anti-lepton. 172.4 The constraint on the coefficients of R-parity violating inter-actions from a fit of electroweak observables, including Rpi.The blue curve shows the constraint using the PDG value forthe branching ratio, and the dashed red curve shows the pro-jected constraint from a 0.1% measurement of the branchingratio, with the same central value as the blue curve. Thegreen curve shows the expected limits with the results of theQweak experiment, measuring the weak charge of the proton,at Jefferson Lab [5]. . . . . . . . . . . . . . . . . . . . . . . . 232.5 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 [6], and thecircles and triangles are the limits from a subset of PIENUdata, published in 2011 [7]. The circles indicate a restrictedangular region was used when constructing the pi+ → e+νeenergy spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . 263.1 A schematic of the M13 beamline. . . . . . . . . . . . . . . . 303.2 The end of the M13 beamline, before (left) and after (right)the extension. Part of the detector was in place to measurethe particle content of the beam. . . . . . . . . . . . . . . . . 313.3 The transverse position distributions of pions, muons, andpositrons at F3; the particle type was identified using timeof flight. The black lines are Gaussian fits to the pion andpositron distributions. . . . . . . . . . . . . . . . . . . . . . . 323.4 One wire chamber plane and its preamplifier board; eachchamber consisted of three planes. . . . . . . . . . . . . . . . 343.5 Wire chambers 1 and 2 after installation on the beam pipe. . 343.6 The plastic scintillator readout scheme, for B1, B2, Tg, andT1. The light from the plastic scintillator (purple) was trans-mitted by four acrylic lightguides (light green) to PMTs (greycylinders). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35xiList of Figures3.7 The PIENU-I detector subsystem. . . . . . . . . . . . . . . . 353.8 The PIENU-II detector subsystem. . . . . . . . . . . . . . . . 363.9 The BINA detector on the test bench, with some of its PMTsin place. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363.10 BINA and the two CsI rings, with all the BINA PMTs in place. 373.11 A Solidworks drawing of PIENU-I and PIENU-II, along witha picture with a human for scale. During data-taking, PIENU-II was rolled forward around PIENU-I. . . . . . . . . . . . . . 383.12 Wires hit in each plane of the first wire chamber. . . . . . . . 393.13 Energy deposited in B1 vs. the time of the hit relative to thepeak of the cyclotron RF field. The cluster with the mostevents, labelled pi, is caused by pions; the cluster below that,with the same timing but less energy, is caused by pions thatdecayed in flight prior to reaching B1. The hit is caused bythe decay muon. The cluster labelled µ on the far left is dueto beam muons, and the low-energy cluster labelled e is dueto beam positrons. . . . . . . . . . . . . . . . . . . . . . . . . 403.14 The measured energy deposited in B1, B2, and Tg by pi-ons. The additional peak in the target spectrum is caused byevents in which the energy of the 4.1 MeV decay muon is alsoincluded. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 413.15 The simulated distance travelled by the 4.1 MeV pi → µνdecay muon in the target. . . . . . . . . . . . . . . . . . . . . 423.16 The simulated pion stopping position along the beam direction. 423.17 The simulated energy deposited for pi+ → e+νe decays inBINA (black) and BINA and CsI (blue), and for pi+ → µ+ →e+νeν¯µ decays in BINA and CsI (red). The distributions arenormalized to the maximum bin. The low-energy peak iscaused by the absorption of a single 511 keV photon from apositron annihilating at rest, in T2 or the front face of BINA. 443.18 Time spectra of events recorded by the Early (red) and Prescale(black) triggers. The Prescale trigger events are scaled bya factor of sixteen. The time spectrum is dominated bypi+ → µ+ → e+νeν¯µ events. Both the end of the early timewindow and the end of the trigger window can be seen. . . . 463.19 Measured energy deposited in the NaI(Tl) crystal, for eventsrecorded by the BinaHigh trigger. The sharp rise is causedby the threshold of the BinaHigh trigger, and the fall by theendpoint of the Michel distribution. The few events abovethe Michel edge are mostly due to pileup. . . . . . . . . . . . 47xiiList of Figures3.20 Time spectrum of hits in B1 (top) and T1 (bottom). . . . . . 503.21 The silicon strip readout. Each strip is connected to a capac-itor and there is one readout channel per four strips [8]. . . . 523.22 A muon decaying in the centre of the target and the decaypositron going past the edge of T1. The plastic scintillatorsand BINA are shown; distances are to scale. . . . . . . . . . . 553.23 Simulated x and y distributions at the first plane of WC3 forpi → µ → e events that deposit energy in BINA and not inT1 (left) and for pi → µ → e events that deposit energy inBINA (right). . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.24 The PIENU detector and beamline after the last bendingmagnet, showing the steel wall used for neutron shielding. . . 564.1 PMT waveform from a pion in B1. The points at the begin-ning and the end of the waveform are zero-suppressed; thedrop around -1380 ns is to the level of the pedestal. . . . . . . 604.2 The blinding technique. Events are removed at random inone of two regions of the spectrum of energy deposited inthe target counter, corresponding to either pi+ → e+νe orpi+ → µ+ → e+νeν¯µ events. . . . . . . . . . . . . . . . . . . . 624.3 The energy deposited in B1 before (upper panel) and after(lower panel) calibration. In the calibrated histogram the MCspectrum is shown in red, and a cut has been made to removeevents due to calibration triggers. The difference around 2.5MeV is due to the requirement of high energy deposit in B1for physics triggers, which was not included in the MC. . . . 644.4 Calibrated energy deposited in T1. The MC spectrum isshown in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . 654.5 Energy deposited in BINA + CsI for pi+ → e+νe events. Theblack histogram, fitted with the black curve, is data, and thered histogram, fitted with the red curve, is MC. The fits areGaussian; the fitting range is asymmetrical about the peakbecause the region to the left of the peak is not Gaussian,due to shower leakage. . . . . . . . . . . . . . . . . . . . . . . 664.6 Energy deposited in BINA + CsI for pi+ → µ+ → e+νeν¯µevents near the Michel edge. Data is shown in black and MCis shown in red. The lines are fits using Equation 4.1. . . . . 66xiiiList of Figures4.7 The calibrated BINA + CsI spectrum. The spectrum up to50 MeV is dominated by pi+ → µ+ → e+νeν¯µ events, and thepeak at approximately 65 MeV is due to pi+ → e+νe events.The high-energy tail is mainly due to pileup events, with asmall contribution from pi+ → e+νeγ events. . . . . . . . . . . 674.8 The beam spot reconstructed by the WC12 tracker. . . . . . 684.9 Ratios of the x and y momenta to the z momentum (tx andty), reconstructed by the WC12 tracker. . . . . . . . . . . . . 694.10 The beam spot reconstructed by the S12 tracker. . . . . . . . 704.11 Ratios of the x and y momenta to the z momentum (tx andty), reconstructed by the S12 tracker. The gaps are due tothe track reconstruction algorithm, which uses the centre ofthe plane hit as the position, leading to some values for txand ty never occurring. . . . . . . . . . . . . . . . . . . . . . 714.12 Decay positron position at the centre (along z) of WC3, re-constructed by the S3WC3 tracker. . . . . . . . . . . . . . . . 724.13 Ratios of the x and y momenta to the z momentum (tx andty) for decay positrons, reconstructed by the S3WC3 tracker. 734.14 The run by run variation in the pulse height of the beampositron peak in one BINA PMT. Similar variations were ob-served for the other PMTs. . . . . . . . . . . . . . . . . . . . 744.15 The run by run variation in the pion stopping position alongthe z axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 754.16 COPPER waveform fitted to a template. . . . . . . . . . . . . 764.17 Time difference between two T1 PMTs. The red line is aGaussian fit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 764.18 Time of the triggering pulse in the B1 counter. . . . . . . . . 774.19 Reduced χ2 distribution of the waveform fit in the B1 counter. 774.20 Time of the triggering pulse in the T1 counter. . . . . . . . . 784.21 Reduced χ2 distribution of the waveform fit in the T1 counter. 795.1 The time spectra for low energy (upper panel) and high en-ergy (lower panel) events. t = 0 is defined by the arrival timeof the pion. The repeating peaks are due to beam particlesand are separated by the cyclotron RF period (see text). . . . 815.2 Number of hits in B1 (left) and B2 (right). The average offour PMTs is taken. The peaks at whole numbers are due topileup events; the events with extra hits in only some of thetubes are due to noise (see text). . . . . . . . . . . . . . . . . 83xivList of Figures5.3 Energy deposited in B1 (left) and B2 (right). The three largepeaks in each spectrum are, from left to right, caused bypositrons, muons and pions. The smaller peaks are due toevents with two particles. The red lines indicate the cut values. 845.4 X (left) and Y (right) position at the centre of WC1. The redlines indicate the cut values. . . . . . . . . . . . . . . . . . . . 845.5 The time spectra after pion selection. The left-hand plotshows the low energy time spectrum and the right-hand plotshows the high energy time spectrum. . . . . . . . . . . . . . 855.6 High-energy time spectrum without (left) and with (right)the prepileup cut. . . . . . . . . . . . . . . . . . . . . . . . . . 865.7 The pulse height of the first hit in one of the T1 PMTs dividedby the pulse height of the second hit, if one was present. Thered line indicates the cut used to select events with fake hits,for plotting the time difference between the initial hit (thereal hit) and the fake hit (Figure 5.8). . . . . . . . . . . . . . 875.8 The time difference between the first and second hits in oneof the T1 PMTs, for events with a small second pulse. . . . . 875.9 The ratio of integrated charge in the T1 PMTs to the fittedpulse height as a function of the fitted pulse height. The redline indicates the cut used to separate real pileup from pileupdue to fake hits. . . . . . . . . . . . . . . . . . . . . . . . . . 885.10 Time spectra after prepileup and T1 pileup cuts. The left-hand plot shows the low energy time spectrum and the right-hand plot shows the high energy time spectrum. . . . . . . . 895.11 The time difference between the last hit in the T1 VT48 chan-nel and the first hit in the B1 VT48 channel versus the decaytime obtained from COPPER. . . . . . . . . . . . . . . . . . 915.12 The distance between the reconstructed positron track andthe centre of WC3 (R). The red line indicates the cut value. . 925.13 Time spectra following all cuts. The left-hand plot showsthe low energy time spectrum and the right-hand plot showsthe high energy time spectrum. The rise in the high energyspectrum near t = 0 at negative times is caused by the in-tegration window of the calorimeter; the closer in time thepileup positron is to the positron from the pion at t = 0, thegreater the probability that the measured energy in the eventwill be above Ecut. . . . . . . . . . . . . . . . . . . . . . . . . 93xvList of Figures5.14 The time difference between subsequent hits in each T1 PMT;leading times are fitted with an error function. The peakaround 30 ns is due to a fake hit at a characteristic time afterthe real hit. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965.15 The shape used in the fit for pileup events that pass the T1pileup cut due to the double-pulse resolution of the T1 counter. 975.16 The shape used in the fit for pileup events where only onepositron hit T1. . . . . . . . . . . . . . . . . . . . . . . . . . . 985.17 The shape used in the fit for pi → µνγ events. . . . . . . . . . 995.18 The T1res shape when the double-pulse resolution is set to100 ns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1015.19 The fitted T1res amplitude as the double-pulse resolution isincreased. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1025.20 The fit of the high-energy time spectrum with ∆T increasedto 100 ns. The red shape is the pi+ → e+νe signal, the blueis the pi+ → µ+ → e+νeν¯µ background, and the green is thesum of the other backgrounds. . . . . . . . . . . . . . . . . . 1035.21 The fitted time spectra from 2012. The left-hand panel showsthe low energy time spectrum, fitted with three components:the pi+ → µ+ → e+νeν¯µ signal shape, old muon decays, andpion decays-in-flight. The right-hand panel shows the highenergy time spectrum, fitted with six components: the pi+ →e+νe signal shape, pi+ → µ+ → e+νeν¯µ and pion decay-in-flight events promoted to the high energy time spectrum viatime independent mechanisms, two mechanisms of old muonpileup, pi → µνγ decays, and old muon decays. . . . . . . . . 1045.22 The residuals of the 2010 data set vs. the time of the event.Clockwise from top left, the panels show the residuals for thehigh energy t < 0 spectrum, the high energy t > 0 spectrum,the low energy t > 0 spectrum, and the low energy t < 0spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.23 The residuals of the 2011 data set vs. the time of the event.Clockwise from top left, the panels show the residuals for thehigh energy t < 0 spectrum, the high energy t > 0 spectrum,the low energy t > 0 spectrum, and the low energy t < 0spectrum. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106xviList of Figures5.24 The residuals of the 2012 data set vs. the time of the event,without the T2 pileup cut applied. Clockwise from top left,the panels show the residuals for the high energy t < 0 spec-trum, the high energy t > 0 spectrum, the low energy t > 0spectrum, and the low energy t < 0 spectrum. . . . . . . . . 1075.25 The residuals of the 2012 data set vs. the time of the event,with the T2 pileup cut applied. Clockwise from top left, thepanels show the residuals for the high energy t < 0 spec-trum, the high energy t > 0 spectrum, the low energy t > 0spectrum, and the low energy t < 0 spectrum. . . . . . . . . 1085.26 The variation of the branching ratio as more pileup eventsare allowed in the time spectrum, prior to applying the T2pileup cut. The x axis denotes the time prior to the pion stopin which events with hits in B1, B2, and Tg were rejected.The error bar on the point furthest to the left (with the leastpileup) is the error from the time spectrum fit, and the errorbars on the other points are the error on the change from theprevious point. The points are fitted to a parabola. . . . . . . 1115.27 The variation of the branching ratio as more pileup events areallowed in the time spectrum, after applying the T2 pileupcut. The x axis denotes the time prior to the pion stop inwhich events with hits in B1, B2, and Tg were rejected. Theerror bars represent the statistical variation from the pointfurthest to the left (with the least pileup). . . . . . . . . . . . 1125.28 The fitted branching ratio for groups of 1000 runs, fitted witha flat line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.1 The total energy deposited in B1, B2, S1, S2, and Tg by pi+ →e+νe (black), pi+ → µ+ → e+νeν¯µ (red), piDIF-µDAR (green),and piDAR-µDIF (blue). The distributions are normalized tothe same height. The solid red lines indicate the selected region.1176.2 The measured angle between the tracks reconstructed byWC12and S12 for pi+ → e+νe (black) and pi+ → µ+ → e+νeν¯µ (red)events after the time and energy loss cuts are applied. Thedouble peak in the pi+ → µ+ → e+νeν¯µ distribution is causedby pion decays-in-flight. The solid red line and arrow indicatethe selected region. . . . . . . . . . . . . . . . . . . . . . . . . 1186.3 The measured BINA+CsI spectra as the suppression cutsare applied. The legend indicates the fraction of low-energyevents (< 52 MeV) remaining after each cut. . . . . . . . . . 119xviiList of Figures6.4 Comparison between measured (filled circles with error bars)and simulated energy spectra. The simulation was performedwith (red) and without (blue) hadronic reaction contribu-tions. The histograms are normalized to the same area [9]. . . 1236.5 Beam particle energy measured by the crystals at 0°with nocuts applied. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.6 Energy measured by the crystals at 0° following event selec-tion cuts (see text). . . . . . . . . . . . . . . . . . . . . . . . . 1276.7 Energy measured by the crystals at 0° in the RF window cor-responding to muons. . . . . . . . . . . . . . . . . . . . . . . . 1276.8 Energy measured by the crystals at 0°. The red lines indicatethe region defined as the muon peak. . . . . . . . . . . . . . . 1286.9 Reconstructed track parameters for the positron beam at 0°.The top-left and top-right panels show the ratio of the x andy momenta to the z momentum, and the bottom-left andbottom-right panels show the reconstructed x and y positionsat z = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.10 Mean of the distribution of the x momentum, normalized tothe z momentum, as a function of angle. . . . . . . . . . . . . 1356.11 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. The green line shows the valueof Ecut. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.12 The energy spectrum from a 70 MeV positron beam at 11.8°(top left), 24.4° (top right), 36.2° (bottom left), and 47.7° (bot-tom right) to the crystal axis. Data is shown in black andsimulation is shown in red. . . . . . . . . . . . . . . . . . . . . 1376.13 The tail fraction as a function of angle in the positron beamdata. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1396.14 The difference between the tail fraction in the positron beamdata and the tail fraction from a simulated positron beam asa function of angle. . . . . . . . . . . . . . . . . . . . . . . . . 1406.15 The simulated BINA+CsI spectrum from pi+ → e+νe decayand pi+ → e+νeγ decay. . . . . . . . . . . . . . . . . . . . . . 1416.16 The simulated BINA+CsI spectrum from pi+ → e+νe de-cay and pi+ → e+νeγ decay, excluding events that underwentBhabha scattering in the target. . . . . . . . . . . . . . . . . 141xviiiList of Figures6.17 The tail fraction as a function of angle for the positron beamdata (left) and Monte Carlo (right), fitted to a fourth-degreepolynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1426.18 Simulated energy spectrum for pi+ → e+νe events emitted atsmall angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 1436.19 The tail fraction in the positron beam data minus the simu-lated tail fraction as a function of angle. The first 8 anglesare fitted to a straight line. . . . . . . . . . . . . . . . . . . . 1446.20 The probability distribution obtained by combining the upperand lower limits. . . . . . . . . . . . . . . . . . . . . . . . . . 1457.1 The ratio of pi+ → e+νe events to pi+ → µ+ → e+νeν¯µ eventswithin different radii of the centre of WC3, as reconstructedby the S3WC3 tracker. . . . . . . . . . . . . . . . . . . . . . . 1507.2 The z coordinate of the reconstructed pion stopping position.Data is shown in black and MC is shown in red. . . . . . . . 1517.3 A comparison of the actual and reconstructed pion stoppingpositions in MC. . . . . . . . . . . . . . . . . . . . . . . . . . 1517.4 Variation of the acceptance correction with the peak valueof the reconstructed pion stopping position. The peak wasvaried by ±0.2 mm; the largest variation in the correctionwas approximately ±0.05%, for R < 90 mm. . . . . . . . . . . 1527.5 Variation of the acceptance correction with the width of thereconstructed pion stopping position. . . . . . . . . . . . . . . 1527.6 Variation of the acceptance correction with the thresholds inthe T1 and T2 counters. . . . . . . . . . . . . . . . . . . . . . 1537.7 The decay time of muons in the target with non-zero kineticenergy at the time of the decay. . . . . . . . . . . . . . . . . . 1547.8 Simulated energy spectra measured by BINA+CsI for muondecays-at-rest (black) and decays-in-flight (red). . . . . . . . . 1557.9 Variation of the branching ratio as the radius in which eventsare accepted is varied. The left-hand panel shows the resultswith the T2 pileup cut, and the right-hand panel shows theresults without the T2 pileup cut. The red points show theraw branching ratio, and the black points show the branchingratio after all corrections. The error bars represent the erroron the change from the first point. . . . . . . . . . . . . . . . 156xixList of Figures7.10 The branching ratio for statistically independent rings in R,the distance between the reconstructed positron track andthe centre of WC3. The left-hand panel shows the resultswith the T2 pileup cut, and the right-hand panel shows theresults without the T2 pileup cut. The red points show theraw branching ratio, and the black points show the branchingratio after all corrections. The value along the x axis is thecentre of the ring under consideration; that is, the point at x= 35 mm is the branching ratio for events with R between 30and 40 mm. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1577.11 The corrected branching ratio for statistically independentrings in R, with (left) and without (right) the T2 pileup cut.This plot is identical to Figure 7.10, but zoomed in on thecorrected points. . . . . . . . . . . . . . . . . . . . . . . . . . 1587.12 The change in the branching ratio from the value at Ecut =52 MeV. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159A.1 PIENU Trigger Diagram . . . . . . . . . . . . . . . . . . . . . 172B.1 A simplified timing diagram for a PIMUE event in which themuon decayed 400 ns after the pion stop. . . . . . . . . . . . 174B.2 A simplified timing diagram for an event in which an old muondecays and 200 ns later a pion arrives. Due to the delaybetween the actual hits in T1 and T2 and the downstreamcoincidence signal, the event still triggers. . . . . . . . . . . . 175B.3 A simplified timing diagram for an event with two muons, inwhich the positron from the old muon decay completes thetrigger. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176B.4 A simplified timing diagram for an event with two muons,in which the positron from the muon from the decay of theprimary pion completes the trigger. . . . . . . . . . . . . . . . 177B.5 A simplified timing diagram for an event with two muons,in which both decay positrons enter the acceptance, but theevent passes the T1 pileup cut because the decays are tooclose together in time for the separate hits to be resolved. . . 178C.1 The beam spot in WC1 and WC2 for positron beam data. . . 179C.2 Energy measured by the crystals following the selection ofbeam particles. . . . . . . . . . . . . . . . . . . . . . . . . . . 180xxList of FiguresC.3 The time distribution of the first hit in the first plane of eachwire chamber. . . . . . . . . . . . . . . . . . . . . . . . . . . . 181C.4 Energy measured by the crystals at 0° following the removalof events with out-of-time hits. . . . . . . . . . . . . . . . . . 182C.5 Time of flight vs. T2 energy. . . . . . . . . . . . . . . . . . . 182C.6 Energy measured by the crystals at 0° for events with time-of-flight corresponding to positrons (left) and muons (right). . 183C.7 Energy measured by the crystals at 0° following the removalof events with high energy deposit in T2. . . . . . . . . . . . 184C.8 Simulated energy deposit in T2, with (red) and without (black)BINA in place. . . . . . . . . . . . . . . . . . . . . . . . . . . 184xxiGlossaryEach entry is followed by (G) if it is a term in general use or (E) if it isspecific to the experiment.Acceptance correction (E)A multiplicative factor that must be applied to the branching ratioobtained from the time spectrum fit to take into account the differencein geometrical acceptance between the two decay modes.ADC (G)Analog-to-digital converter.B1 (E)B1 and B2 are plastic scintillators in the PIENU detector.Bhabha scattering (G)Electron-positron scattering.BINA (E)The NaI(Tl) crystal at the centre of the PIENU calorimeter.BinaHigh (E)A trigger used in the PIENU experiment to record events with largeenergy deposit in BINA and CsI.Blinding(G) A procedure whereby the result of an analysis is changed by anunknown random factor, to avoid human bias.Branching ratio (G)The ratio of the rate of a specific decay mode to the full decay rate.Bremsstrahlung (G)A process by which charged particles emit photons when accelerating.xxiiGlossaryChPT (G)Chiral perturbation theory. A low-energy effective field theory forQCD.COPPER (E)A 500 MHz ADC used to digitize the waveforms from some elementsof the PIENU detector.CsI (G)Cesium iodide, a commonly-used type of crystal scintillator that formspart of the PIENU calorimeter.Cyclotron (G)A type of circular particle accelerator with constant magnetic field andvariable orbital radius.∆T (E)The time interval below which multiple hits cannot be resolved in T1.A parameter in the fitting function for the high energy time spectrum.Early (E)A trigger used in the PIENU experiment to record events shortly aftert0, the pion arrival time.Ecut (E)The threshold used to separate low energy and high energy events.Typically set at 52 MeV.Electroweak (G)Refers to the combined description of the electromagnetic and weakinteractions.Flavour (G)Particle species. In the Standard Model, there are three flavours ofquarks and three flavours of leptons; for example, the electron andelectron neutrino both have electron flavour.FPGA (G)Field-programmable gate array. A programmable chip that performslogic operations.GEANT4 (G)A software package common in high-energy physics for simulating par-ticles and their interactions with matter.xxiiiGlossaryGeneration (G)In the Standard Model, there are three generations of fermions, eachconsisting of two quarks, a charged lepton, and a neutrino. The parti-cles in the second and third generations differ from their counterpartsin the first generation only by mass. For the quarks and charged lep-tons, each generation is heavier than the one before; the mass orderingof the neutrinos is unknown.Kink (E)The angle between the track reconstructed using WC1 and WC2 andthe track reconstructed using S1 and S2. Used to identify pi-DIFevents.Lepton universality (G)The assumption in the Standard Model that the electroweak couplingsof the three lepton generations are the same.Lower limit (E)The lower limit on the tail correction; obtained from the energy spec-trum for pion decay events.M13 (E)A secondary beamline at TRIUMF, used to deliver particles to thePIENU experiment.Michel decay (G)The decay of a muon to an electron and two neutrinos.Michel edge (G)The endpoint of the energy spectrum of electrons produced via muondecay.MIDAS (E)A web-based DAQ interface developed at TRIUMF and PSI.Muon correction (E)A correction applied to the positron beam data to correct for thepresence of muons.Mu pie (E)A term in the fitting function for the high energy time spectrum. De-scribes events due to old muon decay, with energy added by a mecha-nism whose timing is independent of t0.xxivGlossaryMu pimu (E)A term in the fitting function for the low energy time spectrum. De-scribes events due to old muon decay.µ-DAR (E)Muon decay-at-rest.µ-DIF (E)Muon decay-in-flight.µ-DIF correction (E)A multiplicative factor that must be applied to the branching ratioobtained from the time spectrum fit to take into account muon decay-in-flight events, which have the same time dependence as pi → eνevents and can have measured energy above Ecut.NaI(Tl) (G)Thallium-doped sodium iodide, a commonly-used type of crystal scin-tillator.NMR (G)Nuclear magnetic resonance. A technique for measuring magneticfields based on excitation and relaxation of nuclear spins in a knownsample.oldmu both (E)A component in the fitting function for the high energy time spectrum.It describes events where two Michel positrons enter the crystal arraybut only passes through T1.Old muon (E)A muon present in Tg prior to the arrival of the primary pion.pi-DAR (E)Pion decay-at-rest.pi-DIF (E)Pion decay-in-flight.Pion data (E)Data taken in the usual mode, with a 75 MeV/c pion beam enteringthe detector.xxvGlossaryPositron beam data (E)Data taken with a positron beam, instead of the usual pion beam.Used to obtain the response function of the crystal array to 70 MeVpositrons.Prescale (E)One of the triggers used in the PIENU experiment. Also a genericterm for storing only one event out of a specified number that wouldnormally trigger, typically to reduce data size. In the PIENU experi-ment, the prescale factor is sixteen, meaning one event out of sixteenis stored.proot (E)Software written for the PIENU experiment to convert raw data intoROOT trees.PSI (G)Paul Scherrer Institute. A research facility in Villigen, Switzerland,with a cyclotron similar to the TRIUMF cyclotron.QCD (G)Quantum chromodynamics. The theory that describes the strong in-teraction.r (E)A parameter in the fitting functions for the time spectra. Describesthe proportion of low energy events promoted to the high energy timespectrum via mechanisms whose timing is independent of t0.R (E)The distance between the centre of WC3 and the reconstructed positrontrack.Rpi (E)The pion branching ratio.Reduced χ2The χ2 of a fit divided by the number of degrees of freedom of the fit.RF (G)Radio-frequency. In this context, refers to the accelerating electricfield used in the TRIUMF cyclotron.xxviGlossaryROOT (G)Object-oriented data analysis software commonly used in high-energyphysics.Scintillator (G)A material that emits photons of a characteristic wavelength whencharged particles pass through it. Common types include crystal, or-ganic, and liquid noble gas scintillators. The PIENU detector includescrystal and plastic scintillators.Silicon strip (G)A position-sensitive particle detector, consisting of segmented piecesof silicon with a bias voltage applied.S1 (E)S1, S2, and S3 are the three silicon strip detectors used in the PIENUapparatus.Standard Model (G)A theoretical description of fundamental particles and their interac-tions.t0 (E)The pion stop time.t0 correction (E)A multiplicative factor that must be applied to the branching ratioobtained from the time spectrum fit to take into account the depen-dence of the measured value of the decay time on the decay positronenergy.T1 (E)T1 and T2 are plastic scintillators in the PIENU detector.T1res (E)A component in the fitting function for the high energy time spectrum.It describes events where two Michel positrons pass through T1 intothe crystal array sufficiently close together in time that only a singlehit in T1 is recorded.Tail correction (E)A multiplicative factor that must be applied to the branching ratioobtained from the time spectrum fit to take into account pi → eνevents whose measured energy was less than Ecut.xxviiGlossaryTDC (G)Time-to-digital converter (records the time at which a signal wentabove a given threshold).Tg (E)A plastic scintillator in the PIENU detector that functions as the pionstopping target.TIGC (E)A VME module used in the BinaHigh trigger from 2010 onwards forsumming the pulse height of BINA and CsI, and issuing a trigger if itexceeds a defined threshold.TOF (E)Time of flight.Tpos (E)The time of the positron hit relative to t0. The quantity that is fittedto obtain the branching ratio.Trigger (G)A digital logic circuit that takes detector signals as inputs, and sendsa signal to the data acquistion system if the event should be stored.TRIUMF (G)Canada’s national laboratory for particle and nuclear physics. Thesite of the PIENU experiment.tx (E)tx and ty are parameters in the PIENU track reconstruction algorithm.They are the ratios of the x and y momenta to the z momentum.Upper limit (E)The upper limit on the tail correction; obtained from positron beamdata.VF48 (E)A 60 MHz ADC used to record waveforms from some elements of thePIENU detector.VT48 (E)A 1.6 GHz TDC used to record signals from some elements of thePIENU detector and trigger.xxviiiGlossaryWire chamber (G)A position-sensitive particle detector, consisting of wires in a gas-filledchamber with high voltage applied across it.WC1 (E)WC1, WC2, and WC3 are the three wire chambers used in the PIENUdetector.x0 (E)x0 and y0 are parameters in the PIENU track reconstruction algo-rithm. They are the x and y positions at z = 0.χ2 (G)A method for fitting data to a function based on the squared differencebetween the data and the function value, divided by the error on thedata; χ2 also refers to the goodness-of-fit parameter in this method.xxixAcknowledgementsI would like to thank the entire PIENU collaboration for their variouscontributions to the experimental effort, which are too many to list ex-haustively. Thank you to the current and former graduate students, Chloe´Malbrunot, Shintaro Ito, Saul Cuen-Rochin, and Dorothea vom Bruch forall the help, discussion, and shared confusion along the way. In particularI would like to thank Chloe´ Malbrunot for her analysis of the 2010 data,which formed the basis for the analysis of the 2011 data set, performed byShintaro Ito, and the 2012 data set, the main subject of this thesis. Thankyou to the postdocs, Aleksey Sher, Luca Doria, and Dima Vavilov, for theiressential contributions to every aspect of the experiment. Thank you toToshio Numao for his patience and willingness to explain the many thingsabout the experiment I found bewildering over the years. Thank you toDick Mischke for his thoroughness in checking not only the analysis itself,but also everything written about it. And my biggest thanks go to DougBryman, whom I was extremely fortunate to have as my supervisor.Lastly, I am extremely grateful to Abbra Latta, for all her invaluable helpand support.xxxTo my family.xxxiChapter 1Introduction1.1 The Standard ModelThe Standard Model (SM) of particle physics is a theory describing thecharacteristics and interactions of the most fundamental known constituentsof matter [10]. The particle content of the SM is divided into fermions, half-integer spin particles, and bosons, integer spin particles. Two forces arecontained in the SM, the strong force and the electroweak force. These forcesoccur through the exchange of spin one particles, called gauge bosons: gluonsin the case of the strong force, and photons, W±, and Z for the electroweakforce. Gluons and photons are massless, while W± and Z have masses of80.385 ± 0.0015 GeV and 91.1876 ± 0.0021 GeV, respectively [11]. Belowthe energy scale of the W± and Z masses the electroweak force appears toseparate into two forces, electromagnetism and the weak force.Fermions in the SM are further divided into two subcategories, quarksand leptons; the former feel the strong force in addition to the electroweakforce, while the latter do not. There are three generations of fermions,which have identical properties aside from mass, with particles from highergenerations being more massive than the corresponding particles from lowergenerations. The particle content of the SM is shown in Table 1.1; eachcolumn of fermions represents one generation, consisting of two quarks, acharged lepton, and an associated neutral lepton, called a neutrino.Table 1.1: The particle content of the SM; e is the charge of the electron.Fermions Charge (|e|) Bosons Charge (|e|)Quarksu c t +2/3 γ 0d s b -1/3 W± ±1Charged leptons e µ τ -1 Z 0Neutrinos νe νµ ντ 0 H 011.1. The Standard ModelIn the SM, these particles are viewed as quantized excitations of fields,in the mathematical sense of a function defined everywhere in spacetime.One more field exists, called the Higgs field; all massive particles in the SMacquire their masses through interactions with the Higgs field. Althoughthis mechanism was proposed as early as 1964 [12], the associated particlewas only discovered in 2012, by the ATLAS and CMS experiments at theLarge Hadron Collider [13] [14].1.1.1 The weak interactionThe weak interaction is unique in the SM in that it does not conserveparticle type, or flavour. Flavour refers to the generation to which a givenfermion belongs. In the strong and electromagnetic interactions, the numberof quarks and leptons of each flavour must be the same in the initial state asit is in the final state; in the weak interaction, this is not the case. Withoutthis property, many decays that are allowed in the SM would be forbidden,including that of the pion.The pion is the lightest particle made of quarks, consisting of a quarkanti-quark pair of the first generation. Pions decay either into electrons ormuons, which are the second-generation charged lepton, plus the associatedneutrinos. Pion decay in the SM proceeds via the weak interaction. Twotypes of weak interactions exist: charged-current, which occur via W± ex-change, and neutral-current, which occur via Z exchange; pion decay is ofthe former type.The strength of interactions in the SM is set by the coupling constantsα, for the electromagnetic interaction, αs, for the strong interaction, andg, for the weak interaction (see Appendix A of Reference [10]). The valuesof these coupling constants are shown in Table 1.2. In fact, these valuesdepend on the energy of the interacting particles; the quoted values applyat low energies, on the order of 100 MeV (the strong coupling, in particular,changes rapidly with energy) [10].It may seem surprising, given its name, that the coupling for the weakforce is so much greater than the coupling for the electromagnetic force. Thereason is the mass of the mediating particles, which reduces the coupling atlow energies. The “effective” coupling for the weak interaction, the Fermiconstant GF , is related to the “intrinsic” coupling g by GF = g24√2M2W.21.2. Motivation for the PIENU experimentTable 1.2: Coupling constants for SM interactions.Interaction Symbol Value (at ∼100 MeV)Strong αs 1.7Electromagnetic α 0.0073Weak g 0.652Another unique feature of the weak interaction is the violation of paritysymmetry, which is the symmetry of physical laws under a reflection of spa-tial coordinates. Fermions in the SM are represented by four-componentobjects called spinors; these can be either left-handed or right-handed, de-pending on their transformation properties. In technical terms, the twotypes are different representations of the Poincare´ algebra (see AppendixC of Reference [10]). Although the strong and electromagnetic interactionstreat left- and right-handed fermions identically, the weak interaction onlyaffects left-handed fermions and right-handed anti-fermions, which leads toparity violation. As will be explained in Chapter 2, this property is respon-sible for the fact that pions decay into muons overwhelmingly more oftenthan they decay into electrons.1.2 Motivation for the PIENU experimentThe PIENU experiment aims to make a precise measurement of the decayproperties of the pion. The pion branching ratio, Rpi, is defined as the rate atwhich pions decay into electrons and neutrinos divided by the rate at whichthey decay into muons and neutrinos. Also included are the associatedradiative modes, in which a photon is emitted as well:Rpi =Γ(pi+ → e+νe + pi+ → e+νeγ)Γ(pi+ → µ+νµ + pi+ → µ+νµγ) . (1.1)Rpi has been calculated within the SM to be (1.2352±0.0002)×10−4 [1], anunusual level of precision for a quantity involving quarks. Although stronginteraction effects influence the individual decay rates, to first order theycancel in the ratio, and also largely cancel in the higher-order corrections[15]. The small theoretical uncertainty, along with the ability to accuratelymeasure Rpi, makes pion decay a sensitive test of the structure of the weak31.3. Brief history of pion branching ratio measurementsinteraction and the manner in which it connects quarks and leptons, and apowerful tool in the search for new physics.Many new physics scenarios, such as lepton non-universality [2], R-parityviolating supersymmetry [16], leptoquarks [17], and heavy neutrinos [18],have an impact on Rpi. In general, new physics at the weak scale is expectedto impact Rpi on the level of 0.01-1% [1]. By reducing the experimentaluncertainty to the level of the theoretical uncertainty, stringent limits can beplaced on these theories. The branching ratio calculation, and the potentialphysics reach, will be discussed in more detail in Chapter 2.1.3 Brief history of pion branching ratiomeasurementsThe pion was discovered by Cecil Powell, Giuseppe Occhialini, Hugh Muir-head, and Cesare Lattes in 1947 [19]. For the first decade after their discov-ery, the only known decay mode of charged pions was to muons, pi → µνµ;it was unknown at this time what prevented the electron mode.In 1958, Sudarshan and Marshak [20] proposed that the weak interactionoperator has the form of a vector minus an axial vector (V-A), as comparedwith the pure vector operator of quantum electrodynamics (QED). The re-sulting theory explained almost all experimental observations of weak pro-cesses at the time, with one of the few exceptions being the non-observationof the electronic decay mode of the pion: pi → eνe [21]. This process waspredicted by the V-A theory to occur at a rate suppressed by a factor onthe order of 10−4 relative to the muon mode. At the time, the experimentalupper limit on the decay rate was on the level of 2 × 10−5; however, theexperiments setting these limits proved to be in error, and the decay wasfinally observed at CERN in 1958 [22]. Within two years, the branchingratioRpi =pi+ → e+νepi+ → µ+νµ (1.2)was measured in agreement with the predicted value at the 5% level [23],representing another success of the V-A theory. The branching ratio in-cluding radiative modes was measured in 1964 to a precision of approxi-mately 2% [24]. The result, Rpi = (1.247 ± 0.028) × 10−4, was within 0.5σof the theoretical prediction at the time; although it was later revised to41.4. Experimental techniqueRpi = (1.274 ± 0.024) × 10−4 due to a more accurate determination of thepion lifetime, it remained within 2σ [25]. The next measurement was doneat TRIUMF in 1983, giving Rpi = (1.218± 0.014)× 10−4 [26].The most recent experiments were done at TRIUMF [27] and PSI [28] inthe early 1990s. The Particle Data Group average of the last three experi-ments is Rpi = (1.230± 0.004)× 10−4, a precision of 0.33% [29]. The latestresult, Rpi = (1.2344±0.0023 (stat)±0.0019 (syst))×10−4, achieving 0.24%precision, was obtained from a subset of the PIENU data [3].Despite the precision obtained by the branching ratio measurements, thetheoretical uncertainty is smaller by a factor of 15. Nevertheless, the agree-ment between the theoretical and experimental values of Rpi provides thebest constraint on the ratio of the electroweak couplings of the electron andthe muon, ge/gµ (see Section 2.3). The PIENU experiment aims to mea-sure Rpi to a precision of < 0.1%, corresponding to a 0.05% measurement ofge/gµ. Figure 1.1 shows the uncertainty on the branching ratio achieved bythe previous measurements, along with the first PIENU result. Also shownis the projected final result from combining the results of PIENU and PEN[30], the latest PSI experiment. PEN is also attempting to make a < 0.1%measurement of Rpi.1.4 Experimental techniqueEvery branching ratio measurement since that by Di Capua in 1964 [24]has used the same fundamental technique: stopping a pion beam in a scintil-lator target and detecting the decay positrons, which result either from therare pi+ → e+νe process or the far more common pi → µ→ e decay chain, ina calorimeter. pi+ → e+νe decay is two-body, and produces a positron witha fixed energy of 69.8 MeV. Muon decay, µ+ → e+νeνµ, is three-body, andthe positron’s energy is distributed between 0.511 MeV (its rest energy) and52.8 MeV (half the muon mass). The energy distribution of positrons frommuon decay is referred to as the Michel spectrum, and the endpoint as theMichel edge. The lifetime of the muon, τµ = 2.197 µs, is much longer thanthe lifetime of the pion, τpi = 26.0 ns.The differing timing and energy distributions in the pi+ → e+νe case andthe pi → µ → e are used to distinguish the decay modes. The energyof the positrons, and photons if they are produced by either the decaying51.4. Experimental techniqueFigure 1.1: The measured value of the branching ratio as a function oftime. The dashed line shows the SM prediction. The last point indicatesthe expected level of precision that will ultimately be achieved by combiningthe results of the PIENU experiment and the PEN experiment at PSI, asdescribed in the text.61.4. Experimental techniquepion or muon, is measured using a crystal scintillator calorimeter. The twoprocesses, and the energies and lifetimes of the particles involved, are shownin Figure 1.2.Figure 1.2: Simplified picture of the PIENU experimental technique. Pionsstop in the target and decay into either muons or positrons; muons also stopin the target, and decay into positrons. Positrons, and photons if any areproduced, are detected by the crystal scintillator calorimeter.In the PIENU experiment, the branching ratio is obtained by separatingthe events into high- and low-energy regions, using a threshold just abovethe Michel edge, and fitting the time spectra in the two regions. These arereferred to as the high- and low-energy time spectra. The fitting functionused in the PIENU analysis is fully described in Chapter 5.The timing distribution of positrons arising from pi+ → e+νe decay issimply an exponential with the pion lifetime. The timing distribution of thepositrons from pi → µ→ e is determined by the decay time of the pion plusthe decay time of the muon, resulting in a more complicated distributionthat rises as the pions decay and then falls off with the muon lifetime, τµ= 2.197 µs [29]. Monte Carlo simulations of the time and energy spectra inthe two cases, excluding radiative modes, are shown in Figures 1.3 and 1.4(for ease of visualization, the decay modes are not shown in the correctproportion).71.4. Experimental techniqueFigure 1.3: GEANT4 [4] simulation of the time spectra of pi+ → e+νe (red)and pi → µ→ e (blue) events. The distributions are normalized to the sameheight.Figure 1.4: GEANT4 simulation of the starting energies of positrons dueto pi+ → e+νe (red) and pi → µ → e (blue) decays. The distributions arenormalized to the same height.81.4. Experimental techniqueEither pion decay mode, or the decay of the muon, can also result in aphoton being emitted. This does not affect the time spectra, but can affectthe measured energy. The branching ratio can be affected in two ways. If thepion decays to a muon, a photon emitted during the pi → µ→ e decay chaincan add sufficient energy for the event to be placed in the high-energy timespectrum. Depending on the timing of the event, it could be misidentifiedas a pi+ → e+νe decay. On the other hand, if the pion decays to a positronand a photon, the photon can carry away sufficient energy that the eventis placed in the low-energy time spectrum, if the photon is not absorbed inthe calorimeter. Both of these possibilities are taken into account via MonteCarlo simulation, as described in Chapters 5 and 6.Measuring the ratio of the decay rates, rather than simply the pi+ → e+νeyield, provides several benefits for reducing the systematic uncertainty. Thetotal number of incoming pions does not need to be known; since positronsare measured regardless of the decay mode of the pion, the efficiencies of thecuts and triggers cancel in the measured ratio of decay rates, to first order;and the geometrical acceptance enters only due to energy-dependent scat-tering, which results in a small correction to the measured branching ratio.Thus, many sources of systematic error1 are either reduced or eliminatedentirely.1.4.1 PIENU experimental techniqueThe PIENU detector (fully described in Chapter 3 and Reference [8]) isshown in Figure 1.5. Wire chambers and silicon strip detectors providedparticle tracking; thin plastic scintillators (including the pion stopping tar-get) provided timing information, particle ID, and were used in the trigger;and crystal scintillators provided the energy measurement.A pion beam with momentum 75 MeV/c ± 1 MeV/c [31] from the TRI-UMF M13 beamline was injected into the detector. The pions were detectedby the plastic scintillators B1 and B2, and tracked into the target (Tg) bya pair of wire chambers and a pair of double-sided silicon strip detectors.Decay positrons were detected by two more plastic scintillators, T1 andT2, and tracked by another silicon strip detector and wire chamber, beforeentering the crystal scintillators.1In this thesis, “error” and “uncertainty” are used interchangeably.91.4. Experimental techniqueFigure 1.5: The PIENU detector, with the pion beam incident from the left.The region close to the target is shown in the blowup. Plastic scintillators(polyvinyltoluene) are shown in dark blue, wire chambers in green, siliconstrip detectors in orange, and crystal scintillators in light blue (NaI(Tl)) andred (CsI).101.5. Outline of the thesisThe central element of the crystal calorimeter array was BINA, a singlecylindrical NaI(Tl) crystal borrowed from Brookhaven National Laboratory,48 cm long by 48 cm in diameter, which was read out by 19 photomultipliertubes (PMTs). BINA was surrounded by two concentric rings of pure CsIcrystals to capture shower leakage. There were a total of 97 individual CsIcrystals, which were pentagonal in cross section, 25 cm long, 9 cm wide, and8 cm high. Each crystal was read out by a single PMT.The precision of the branching ratio measurement ultimately rests on theextent to which pi+ → e+νe events can be separated from background events.Figures 1.3 and 1.4 show that backgrounds with large energy deposit at earlytime, relative to the stopping time of the pion, have the potential to bemisidentified as pi+ → e+νe events. The three main sources of backgroundin the PIENU experiment are beam-related background, pileup of muonsfrom previous pion decays, and pileup of neutral particles. These will bediscussed in section 3.4.1.5 Outline of the thesisIn Chapter 2, the theory of pion decay and the sensitivity of the branchingratio to various beyond the Standard Model scenarios are briefly presented.Chapter 3 gives an overview of the experimental apparatus, including theM13 beamline. Chapter 4 describes the data-taking conditions, and de-scribes the data processing. Chapter 5 describes the event selection andthe timing fit done to obtain the raw branching ratio. Chapters 6 and 7describe the corrections that must be applied to the raw branching ratio toobtain the final result, and conclusions are presented in Chapter 8.11Chapter 2Theory2.1 Pions in the Standard Model2.1.1 Pion decay modesTable 2.1 shows the decay modes of the charged pion listed by the ParticleData Group [11]. The branching ratios quoted are experimental values; thevalues for the radiative modes apply to a restricted kinematic range for thephoton, as indicated in the table.Table 2.1: Pion decay modes and branching ratios.Mode Branching ratio Notesµ+νµ 0.9998770± 0.0000004µ+νµγ (2.00± 0.25)× 10−4 Eγ > 1 MeVe+νe (1.230± 0.004)× 10−4e+νeγ (7.39± 0.05)× 10−7 Eγ > 10 MeV, θeγ > 40°e+νepi0 (1.036± 0.006)× 10−8e+νee+e− (3.2± 0.5)× 10−9e+νeνν < 5× 10−6Only the first four decay modes, pi+ → µ+νµ(γ) and pi+ → e+νe(γ), arerelevant to the PIENU experiment. The modes with the pi0 and the extrae+e− pair in the final state occur < 0.01% as often as the pi+ → e+νe mode,which is negligible at the level of precision of the PIENU experiment.2.1.2 The weak interactionPion decay to leptons in the SM proceeds via the weak interaction. Sincethe pion is the lightest hadronic particle, the strong and electromagnetic122.1. Pions in the Standard Modeldecay modes are forbidden, as these interactions conserve flavour. The La-grangian for the part of the interaction mediated by the W+ is given by[10].LW+ =ig2√2W+µ (νmγµ(1 + γ5)em + umγµ(1 + γ5)dm). (2.1)Here there is an implied sum over the index m, which goes from one to threeand indicates the generation of the particle involved; that is, e1 = e, e2 = µ,and e3 = τ . γµ, where µ goes from zero to three, are the Dirac gammamatrices; γ5 is given by iγ0γ1γ2γ3.This expression is valid in the “interaction” basis, that is, for particleswhich carry definite flavour; however, in the case of the quarks, this is notthe same as the mass basis, which is the physical one. The mass basis andthe interaction basis are connected via the Cabbibo-Kobayashi-Maskawa(CKM) matrix, in the following manner:d′s′b′ =Vud Vus VubVcd Vcs VcbVtd Vts Vtbdsb . (2.2)Here the primes indicate particles in the interaction basis, while the un-primed vector represents particles in the mass basis. The Lagrangian thenbecomesLW+ =ig2√2W+µ (νmγµ(1 + γ5)em + Vmnu′mγµ(1 + γ5)d′m) (2.3)The term 1+γ5 is responsible for the parity-violating nature of the weak in-teraction. This is most easily seen by writing γ5 and the spinor representinga fermion in the Weyl, or chiral, basis [32, p. 43-50]:γ5 =1 0 0 00 1 0 00 0 −1 00 0 0 −1 , ψ = [ψLψR]. (2.4)Here ψL and ψR are two-component objects, where the components representthe two available spin states of a spin 1/2 particle. In this basis it is clearthat 1+γ52 and1−γ52 , when multiplied by a spinor, project out the left- andright-handed components, respectively.132.1. Pions in the Standard Model2.1.3 Pion decay rateA representation of the lowest-order Feynman diagram for the decay ofthe pion into a lepton and neutrino is shown in Figure 2.1.Figure 2.1: Feynman diagram for the decay of the positively charged pion.W+ represents the positively charged gauge boson mediating the weak in-teraction, and u and d represent an up and an anti-down quark, respectively.To compute the rate of this process, it is necessary to calculate the matrixelement M = 〈l+νl|L|pi+〉, where l = e, µ and L is the Lagrangian given inEquation 2.3 [10]. Expanding this into a hadronic part and a leptonic partgivesM = iGFVud√2〈0|dγµγ5u|pi+〉 l(pl)γµ(1 + γ5)ν(pν), (2.5)where pl and pν are the momentum carried by the outgoing anti-lepton andneutrino, respectively. In the hadronic matrix element we have used the factthat only the axial-vector part of the weak interaction, γµγ5, contributes tothe transition amplitude. The pion itself is pseudoscalar, so the vector partof the weak interaction results in an expression with odd parity; when theintegral is done over all space, it therefore vanishes.The remaining hadronic matrix element is not simple to calculate dueto strong interaction effects; however, this term must be a Lorentz four-vector. Since the pion is spinless, the only four-vector that can contributein general is the momentum transfer qµ to the virtual W [33]. The most142.1. Pions in the Standard Modelgeneral expression for the hadronic part of Equation 2.5 can be written as[10] 〈0|dγµγ5u|pi+〉 = iFpiqµeiqx, (2.6)where Fpi is a constant parameterizing the strong interaction effects, usuallycalled the pion decay constant. Summing over spin states and integratingover outgoing particle energies leads to the following expression for the decayrate [10]:Γ0pi→l =G2FV2udmpiF2pim2l4pi(1− m2lm2pi)2. (2.7)Although this formula contains Fpi, the value of which must be taken fromexperiment, taking the ratio of the electron mode to the muon mode givesthe simple expressionR0pi =Γpi→eΓpi→µ=m2em2µ(m2pi −m2em2pi −m2µ)2= 1.283× 10−4. (2.8)This expression contains the ratio of the electron mass to the muon masssquared, which is responsible for the small value of the branching ratio. Thephysical origin of this factor is the 1+γ5 term in the electroweak Lagrangian(Equation 2.3), which allows only right-handed anti-leptons and left-handedneutrinos to emerge from this decay. For massless fermions, handedness isequivalent to helicity, the direction of a particle’s spin relative to its directionof motion. Left-handed massless fermions have negative helicity, and right-handed massless fermions have positive helicity.For massive fermions, both spin states are possible, but there is an energy-dependent suppression factor associated with the helicity state that wouldbe forbidden if the fermion were massless. That is, as the energy of afermion begins to exceed its rest mass, it begins to behave approximatelylike a massless particle. In the rest frame of the pion, the neutrino andthe anti-lepton must emerge back-to-back; since the pion is spin zero, theneutrino and the anti-lepton are forced into the same helicity state. Theallowed direction of the spins is shown in Figure 2.2.For both the positron and the muon decay modes, there is a suppressionfactor owing to this effect of 1 − (vlc )2 ≈ (2mlmpi )2, where vl and ml are the152.1. Pions in the Standard ModelFigure 2.2: The allowed directions of the spins and linear momenta of thepion decay products, in the pion rest frame. The requirement, from angularmomentum conservation, that the spins of the positron and the neutrinomust be in opposite directions, leads to the suppression of the positronmode relative to the muon (see text).velocity and mass of the outgoing lepton, mpi is the pion mass, and c is thespeed of light [33]. When the ratio of the rates of the two modes is taken,the factor of(memµ)2in Equation 2.8 is obtained.2.1.4 CorrectionsEquation 2.8 must be corrected for higher-order effects. Leading-order ra-diative corrections, still treating the pion as point-like, were first calculatedby Berman [34] and Kinoshita [35] in the late 1950s. They considered theinfluence of diagrams involving both real and virtual photons. Diagrams in-volving a real photon are referred to as Inner Bremsstrahlung (IB) processes,and diagrams involving a virtual photon are referred to as emission and re-absorbtion (ER). The diagrams that contribute are shown in Figure 2.3.Although the calculation of these diagrams requires both infrared andultraviolet cutoffs to be imposed, their effect on Rpi can still be rigorouslycomputed. The term involving the infrared cutoff cancels exactly for IBand ER processes, and the ultraviolet cutoff cancels in the branching ratio,although it affects the individual decay rates. Ultimately, a correction of-3.929% to Rpi was obtained.162.1. Pions in the Standard Modell+!"+!l+!"+!l+!"+!a)l+!"+!l+!"+!l+!"+!b)Figure 2.3: Feynman diagrams for the radiative corrections to pion decay,from real (a) and virtual (b) photons. l+ denotes an anti-lepton.172.1. Pions in the Standard ModelBy the 1970s, however, it was clear the pion was not a fundamental parti-cle, but rather a quark anti-quark pair, and the validity of this approach wasunclear. In 1976, Marciano and Sirlin expanded the decay rate in powers oflepton mass, ml, including the contribution of structure-dependent effects.These are diagrams where a photon, either real or virtual, is emitted fromthe internal structure of the pion. They showed that neither the leading-order ml term (the leading term in Rpi) nor the leading correction term, oforder α ln(mµ/me), depends on strong interaction effects. They obtained-3.7% for the leading-order correction, and argued that the higher-ordercorrections were likely to be significantly less than 1% [15]. They repeatedthe calculation in 1993 with a more rigorous assessment of the theoreticaluncertainty, and obtained Rpi = (1.2352± 0.0005)× 10−4 [36].The most recent calculation was done by Cirigliano and Rosell [1] incorpo-rating strong interaction effects using Chiral Perturbation Theory (ChPT),a low-energy effective field theory for quantum chromodynamics (QCD), thetheory which describes the strong interaction. ChPT allows for an expan-sion of the decay rates in powers of the pion mass (in this case) and theelectromagnetic coupling, through which the uncertainty on the ratio canbe tightly constrained. The calculation in Reference [1] is done to O(e2p4),where e is the electromagnetic coupling constant and p is proportional tothe pion mass. The branching ratio can be written asRpi = R0pi[1 + ∆e2p2 +∆e2p4 +∆e2p6 + ...][1 + ∆LL] . (2.9)Here ∆e2pn are the successive terms in the chiral expansion, and ∆LL rep-resents corrections of order αn lnn(mµ/me). Although the calculation isdone to O(e2p4), this expression contains a term to O(e2p6). This termarises from the emission of a photon by the decaying pion, which evadesthe helicity suppression, and so must be taken into account despite beinghigher-order. Photons emitted at any other part of the pion decay diagram,such as real bremsstrahlung from the decay lepton or a loop starting on theW line, do not affect the helicity suppression. Table 2.2 gives the size ofeach of these corrections [1].The result of this calculation isRpi = (1.2352± 0.0001)× 10−4. (2.10)182.2. Beyond the Standard ModelTable 2.2: Corrections to the leading-order value for Rpi.Term Value (%)∆e2p2 -3.929∆e2p4 0.053 ± 0.011∆e2p6 0.073∆LL 0.054However, it was shown in Reference [2] that O(α2) two-loop diagrams con-tribute an additional 0.0001 to the uncertainty, givingRpi = (1.2352± 0.0002)× 10−4. (2.11)The uncertainty on the SM prediction for Rpi is approximately an order ofmagnitude smaller than the experimental uncertainty. Thus, reducing theexperimental uncertainty closer to the level of the theoretical uncertaintywill improve the constraints on many extensions to the SM, which predictdeviations from the above value. Indeed, the effect of new physics at theweak (∼TeV) scale on Rpi is expected to be in the range of 1% to 0.01% [1];much of this parameter space will therefore be constrained by the combinedPIENU/PEN result.2.2 Beyond the Standard ModelThe SM is a very successful theory, having stood up to decades of exper-imental scrutiny. Although a number of intriguing experimental anomaliescurrently exist, such as the discrepancy between the experimental and the-oretical values of the anomalous magnetic moment of the muon [37] andthe proton radius puzzle [38], no experiment has reached the 5σ thresholdnecessary to claim a discovery (different techniques for measuring the pro-ton radius differ by 7σ, but no discrepancy of this size exists between anexperimental result and the SM prediction).Despite the many successes of the SM, it is known to be an incompletedescription of nature. Cosmological observations give evidence for the exis-tence of dark matter [39] and dark energy [40], neither of which is explainedby the SM. The asymmetry between matter and antimatter in the SM, CP192.2. Beyond the Standard Modelviolation, is very small, but the universe appears to be almost entirely com-posed of matter [41]. That neutrinos have mass is now proven by oscillationexperiments, but the mass generation mechanism is not included in the SM.If they obtain mass via the usual Higgs mechanism, there must exist as yetunobserved right-handed neutrino states. Finally, and most obviously, theSM does not describe the gravitational interaction between particles.In addition to these missing pieces, several aspects of the SM as it standsare considered theoretically unsatisfactory, either because of the necessity offine-tuning or the large number of parameters whose values have no explana-tion, and must be determined from experiment. These include the apparentlack of CP violation in the strong interaction (the “strong CP” problem)[42], the existence of three generations of particles that differ only by mass,and the lack of a mechanism to keep quantum corrections to the Higgs masssmall, in the presence of fermions (the “naturalness” or “hierarchy” prob-lem) [43].Given the seeming arbitrariness of several aspects of the SM, and itsknown shortcomings, it is generally believed that it will eventually be su-perseded by a more complete theory. However, given the lack of any exper-imental evidence to indicate the nature of this theory, many extensions tothe SM have been proposed; a selection of these with consequences for thepredicted value of Rpi will now be discussed.2.2.1 New pseudoscalar interactionsThe largest effect on Rpi would come from some new interaction that al-lowed the positron mode without helicity suppression. This could take theform of either a scalar or pseudoscalar operator, although the pion can-not decay directly through a scalar interaction. Since the pion is itself apseudoscalar particle, an operator of odd parity is required in order for thetransition amplitude between the pion and the vacuum to be non-zero.However, scalar interactions can induce pseudoscalar interactions throughelectroweak renormalization effects. This possibility is of particular interest,as new scalar interactions appear in several well-motivated extensions tothe SM, including extra Higgs multiplets, leptoquarks, and compositenessof quarks or leptons [44].202.2. Beyond the Standard ModelThe transition amplitude for pion decay through a pseudoscalar interac-tion is given by [44]〈0|uγ5d|pi〉 = i√2 fpim2pimu +md= i√2f˜pi. (2.12)Assuming only left-handed neutrinos are produced by the new pseudoscalar,the Lagrangian for this interaction is given byLP = −i ρ2Λ2[l(1− γ5)νl][uγ5d], (2.13)where ρ is the coupling constant for the new pseudoscalar and Λ is its massscale. This leads to a matrix elementMP = ρ f˜pi√2Λ2[l(1− γ5)νl]. (2.14)The total matrix element for pion decay then becomes the sum of the SMma-trix element andMP . After squaring the total matrix element and summingover final states, the branching ratio becomes, assuming lepton universalityholds for the new interaction [44],Rpi = RSMpi(1 +√2f˜piRe(ρ)GFΛ2fpiVudme+O(1Λ4)). (2.15)Assuming real coupling of approximately the same strength as the weakinteraction, this becomesRpiRSMpi− 1 ∼(1TeVΛ)2× 103. (2.16)This expression applies to a new interaction whose nature is fundamentallypseudoscalar; the effect on the branching ratio from an induced pseudoscalar,from a fundamentally scalar interaction, depends on the details of the newinteraction.2.2.2 R-parity violating supersymmetrySupersymmetry (SUSY) is widely considered to be one of the most well-motivated extensions to the SM, as it has the potential to solve the Higgsself-coupling problem and explain the nature of dark matter [43]. Versionsof SUSY that preserve R-parity are unlikely to have a measurable effect onthe branching ratio, at the level of precision of the PIENU experiment. R-parity is a quantum number defined as PR = −13B+L+2s [45], where B is212.2. Beyond the Standard Modelbaryon number, L is lepton number, and s is spin. R-parity is conserved inthe SM.In SUSY models that conserve R-parity, there are a variety of processesthat could contribute to Rpi, but the contribution would not be large exceptin specific, theoretically disfavoured regions of parameter space. However,if both R-parity and lepton number conservation are violated, tree-levelsfermion exchange (the supersymmetric partner to the electron or muon, inthis case) would contribute to Rpi at a potentially measurable level [16].The effect on Rpi is determined by both the sfermion masses, me and mµ,and the coefficients of the R-parity violating interactions in the Lagrangian,λ′11k for the decay to an electron and λ′21k for the decay to a muon. Defining∆ijk in terms of the coefficients λ′ijk as∆′ijk =|λ′ijk|24√2GFm2f, (2.17)where G is the Fermi coupling constant and mf is the sfermion mass, theeffect on Rpi is given by [16]∆RpiRSMpi= 2(∆′11k −∆′21k). (2.18)The allowed regions for ∆′11k and ∆′21k from precision measurements ofelectroweak parameters are shown in Figure 2.4, at 95% confidence level[16]. Shown are the constraints using the PDG value of the branching ratio((1.230±0.004)×10−4) in the fit and, in the dashed red curve, the projectedconstraint from a 0.1% measurement of Rpi, assuming the same central value.2.2.3 Charged HiggsThe Minimal Supersymmetric Standard Model (MSSM) contains a neutralHiggs doublet, a neutral Higgs singlet, and a charged Higgs doublet [46].A charged Higgs particle could replace the W in the pion decay diagram(Figure 2.1) [16]. If the coupling of the charged Higgs is g2√2λud to pionsand g2√2λlν to leptons, the effect on the branching ratio is [2]1− RpiRSMpi=2m2pime(mu +md)m2Wm2H±λud(λeν − memµλµν). (2.19)222.2. Beyond the Standard ModelFigure 2.4: The constraint on the coefficients of R-parity violating interac-tions from a fit of electroweak observables, including Rpi. The blue curveshows the constraint using the PDG value for the branching ratio, and thedashed red curve shows the projected constraint from a 0.1% measurementof the branching ratio, with the same central value as the blue curve. Thegreen curve shows the expected limits with the results of the Qweak exper-iment, measuring the weak charge of the proton, at Jefferson Lab [5].232.2. Beyond the Standard ModelIf, for example, the couplings λud, λeν , and λµν are all equal to αpi , whereα is the electromagnetic coupling, a 0.1% measurement of Rpi correspondsto mH± ≈ 400 GeV. If, however, λeν/λµν = me/mµ, as for the StandardModel Higgs, the term in the brackets cancels out and Rpi is insensitive tothe presence of charged Higgs bosons.2.2.4 LeptoquarksLeptoquarks are defined by having both baryon and lepton number, andpossess a range of other properties in different models. Chiral leptoquarkscouple only to left- or right-handed particles, whereas non-chiral leptoquarkscouple to particles of either handedness. The bounds from Rpi on the latterare very strong: M2LQ/(gLgR) > (100 TeV)2, where MLQ is the leptoquarkmass and gL and gR are the couplings. Rpi also constrains the M/g ratio forchiral leptoquarks that couple to left-handed particles; the exact constraintdepends on the leptoquarks under consideration, but for scalar leptoquarksare in the range of 1-10 TeV [17].2.2.5 Massive neutrinosAlthough the number of stable light neutrinos with SM couplings is re-stricted to three by the measurements of the Z peak done at LEP in the late1980s [47], no such restriction exists on heavy or weakly coupled neutrinos.Indeed, these appear in many extensions to the SM; heavy neutrinos are apromising dark matter candidate and could provide a natural way for the SMneutrinos to acquire their small masses [48]. Many experiments currentlyexist or are under construction to better measure neutrino properties. Someanomalies have been observed, for example by LSND and MiniBOONE, thatcould be resolved by heavy neutrinos [49].The PIENU experiment is sensitive to neutrinos in the mass range of 0-130 MeV, and particularly to those above 55 MeV. The presence of anyneutrino heavier than a few MeV would reduce the helicity suppression ofthe pi+ → e+νe mode, and thus enhance the branching ratio. The rate ofthe decay pi+ → e+νi, where νi is a heavy neutrino, relative to the rate ofthe pi+ → e+νe decay, is given by [18]Γ(pi+ → e+νi)Γ(pi+ → e+νe) = |Uei|2ρe. (2.20)242.3. Lepton universalityUei is the mixing parameter between νe and νi (analagous to a CKM matrixelement) and ρe is a function of δi = m2νi/m2pi and δe = m2e/m2pi, where mνi isthe heavy neutrino mass, me is the electron mass, and mpi is the pion mass:ρe =[1 + δ2e + δ2i − 2(δi + δe + δiδe)]1/2[δi + δe − (δi − δe)2]δe(1− δe)2 . (2.21)Using the value for Rpi based on the 2010 data, this translates to a limitof |Uei|2 < 1.32 × 10−6 for mνi = 50 MeV, at 90% confidence. The limitincreases as mνi goes to zero.Above 55 MeV, the positron energy would be sufficiently reduced that anextra peak would appear in the pi+ → e+νe energy spectrum. Figure 2.5shows the upper limit on |Uei|2 obtained through a search for extra peaksin PIENU data taken in 2009, compared with the limits from the previousPIENU experiment [7] [6]. The same analysis using the full PIENU data setis under way, which is expected to improve the limit by a factor of 3-5.2.3 Lepton universalityA key feature of the SM is the presence of three generations of fermions,which differ only by mass. The couplings to gauge bosons are assumed to beidentical; whether this is exactly true, or only an approximation, is a crucialtest of the structure of the SM. Referring to Equation 2.7, the rate of piondecay to either electrons or muons is proportional to g4, where g is the weakinteraction coupling constant. One factor of g2 arises from the quark vertex,and the other from the lepton vertex. If the couplings to the W of electronsand muons are not assumed to be equal, Rpi is then given byRpi =(gegµ)2RSMpi . (2.22)Constraints on the ratios of the coupling constants come from many differenttypes of measurement, as shown in Table 2.3.Rpi currently provides the most precise test of electron-muon universality,although the branching ratio of tau decays to muons and electrons is close.However, these tests are not exactly equivalent; since the pion is spin zerowhile the tau is spin 1/2, the mediating W boson in the former case must bein the spin zero state, whereas in the latter case all spin states contribute.252.3. Lepton universality5total energy cut. The largest energy-dependent effectwas in the vertex consistency requirement for pion andpositron tracks, which reduced the acceptance of lowenergy positrons by 60 %. The combined acceptancesfor 10 MeV positrons with respect to 70 MeV positronswere 45 % (35◦ data) and 42 % (no cut).RESULTSNo significant peaks above statistical fluctuations wereobserved. After correcting for the acceptance and thehelicity-suppression and phase-space terms, the ampli-tudes and associated errors were converted to 90 % C.L.upper limits on |Uei|2, assuming a Gaussian probabilitydistribution with a constraint that the physical regionof a peak area be positive. Figure 6 shows the combinedresults for the fits with the 35◦ angle cut (below 80MeV/c2 in neutrino mass), and without the angle cut(above 80 MeV/c2). The region below 60 MeV/c2(Ee+ > 57 MeV) was excluded in the plot because ofthe strong bias caused by the background subtractionprocedure. For comparison, the 90 % C.L. upper lim-its obtained in Ref. [7] are also plotted by a dashed curve.)2Neutrino mass (MeV/c50 60 70 80 90 100 110 120 1302 | ei|U-810-710FIG. 6: Combined 90 % C.L. upper limits obtained from the35◦ spectrum (circles) and no-cut spectrum (triangles) to-gether with the previous limits (dashed line) [7].CONCLUSIONSThe present experiment improved the upper limits onthe neutrino mixing matrix element |Uei|2 by a factor ofup to four in the mass region 68–129 MeV/c2.This work was supported by the Natural Science andEngineering Research Council and TRIUMF througha contribution from the National Research Council ofCanada, and by Research Fund for the Doctoral Pro-gram of Higher Education of China, and partially sup-ported by KAKENHI (21340059) in Japan, One of theauthors (M.B.) has been supported by US National Sci-ence Foundation Grant Phy-0553611. We are grateful toBrookhaven National Laboratory for the loan of the crys-tals, and to the TRIUMF detector, electronics and DAQgroups for their engineering and technical support.[1] A. Kusenko, Phys. Rep. 481, 1 (2009) and referencestherein.[2] A. Boyarsky, O. Ruchayskiy and M. Shaposhnikov, Ann.Rev. Nucl. Part. Sci. 59 (2009) 191-214.[3] T. Asaka, S. Eijima and H. Ishida, J. High Energy Phys.1104, 011 (2011).[4] D.I. Britton et al., Phys. Rev. Lett. 68, 3000 (1992); andD.I. Britton et al., Phys. Rev. D49, 28 (1994).[5] G. Czapek et al., Phys. Rev. Lett. 70, 17 (1993).[6] K. Nakamura et al. (Particle Data Group), J. Phys. G37,075021 (2010).[7] D.I. Britton et al., Phys. Rev. D46, R885 (1992).[8] PIENU experiment, TRIUMF Proposal S1072, (2005).[9] A. Aguilar-Arevalo et al., Nucl. Instrum. Method A609,102 (2009).[10] G. Blanpied et al., Phys. Rev. Lett. 76 (1996) 1023.[11] I-H. Chiang et al., IEEE NS-42 (1995) 394.[12] A. Aguilar-Arevalo et al., Nucl. Instrum. Method A621,188 (2010).Figure 2.5: 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 [6], and the circles and triangles are the limitsfrom a subset of PIENU data, published in 2011 [7]. The circles indicate arestricted angular region was used when constructing the pi+ → e+νe energyspectrum.262.3. Lepton universalityTable 2.3: Experimental results on lepton universality tests from studies ofpion, kaon, tau, and W decays. Here B represents the branching fraction ofa particular decay mode.Decay mode gµ/geBpi→µ/Bpi→e 1.0004± 0.0012[3]Bτ→µ/Bτ→e 1.0018± 0.0014 [50]BK→µ/BK→e 0.996± 0.005 [51]BK→piµ/BK→pie 1.002± 0.002 [52]BW→µ/BW→e 0.997± 0.010 [52]gτ/gµBτ→eτµ/ττ 1.0011± 0.0015 [50]Bτ→pi/Bpi→µ 0.9963± 0.0027 [50]Bτ→K/BK→µ 0.9858± 0.0071 [50]BW→τ/BW→µ 1.039± 0.013 [52]gτ/geBτ→µτµ/ττ 1.0029± 0.0015 [50]BW→τ/BW→e 1.036± 0.014 [52]The most precise tests of electron-tau and muon-tau universality come fromτ → e and τ → µ decay rates at B factories [50].Other interesting results, that have generated much attention, are themeasurements by LHCb of the flavour-changing neutral current processesB+ → K+l+l− [53], where l = e, µ, and the charged-current processesB0 → D∗+l−νl [54], where l = µ, τ . The first found an excess of 2.8σ in theelectron mode, and the second found an excess of 2.1σ in the τ mode. TheBaBar collaboration also reported a 2.7σ excess in this mode, and a 2.0σ inthe similar B0 → D+τ−ντ [55].The deviations from universality required to explain these measurementsare large compared to the uncertainties quoted in Table 2.3. In order to ex-plain these results in terms of new physics, while remaining consistent withother measurements, generally requires the new physics to couple preferen-tially to the third generation of particles [56].272.4. Physics ReachThis can be done by assuming the new physics couples only to the thirdgeneration in the interaction basis, with effects on the first two generationscoming from the mismatch between the mass basis and the interaction basis,or by assuming a mass-dependent coupling for the new physics. In eithercase, the effect on Rpi would be highly suppressed; the CKM matrix elementslinking the first and third generations of quarks, for example, are Vub =0.0035 and Vtd = 0.0087.2.4 Physics ReachAnalysis of the full PIENU data set will test electron-muon universalityat the 0.05% level, improve the limits on the couplings of heavy neutrinosin the mass range from 0-130 MeV, constrain new pseudoscalar interactionsup to O(1000 TeV), and provide improved constraints on various leptoquarkand R-parity violating SUSY models.28Chapter 3Experimental Setup3.1 BeamlineThe TRIUMF cyclotron accelerates H− ions to a maximum energy of520 MeV and a typical intensity of 300 µA, divided amongst four primarybeamlines. The accelerating gradient is provided by a 23.05 MHz 93 kVradiofrequency (RF) field; this corresponds to a bunch spacing of 43.4 ns,with a typical bunch width of 4 ns. Extraction from the cyclotron to thebeamlines is accomplished by stripping the electrons off the H− with a thinfoil, thereby reversing the direction of the magnetic steering. The PIENUproduction target, a 1cm thick piece of beryllium, was along beamline 1A,which sees a typical proton current of about 100 µA. At the productiontarget, pions, muons, and positrons were produced with a wide range ofenergies. The M13 secondary beamline selected positively charged particlesof momentum 75 MeV/c, with a roughly 1% spread. A schematic of theextended beamline is shown in Figure 3.1 [31]; the extension was added in2008 for the PIENU experiment. There were three dipole magnets, labelledB1-3, and ten quadrupoles, labelled Q1-10. The last dipole and the lastthree quadrupoles make up the extension. In addition, there were three setsof adjustable vertical and horizontal slits, an absorber, and a collimator.There were four focus points, labelled F1-4. The last was located at thecentre of the pion stopping target.The first bending magnet (B1) selected particles with a momentum ofroughly 78 MeV/c. At this point the beam contained significant numbers ofpions, muons, and positrons. While small numbers of muons and positronswere desirable for calibration purposes, the fact that the PIENU detectorwas in line with the beam meant that positron contamination, in particular,had to be reduced to the level of a few percent. Beam positrons traversedthe entire detector and were absorbed in the calorimeter, leaving a similaramount of energy to positrons from pi+ → e+νe decay. Thus, the design ofthe beamline was intended primarily to eliminate positrons. The end of thebeamline before and after the extension is shown in Figure 3.2.293.2. DetectorFigure 3.1: A schematic of the M13 beamline.Positron contamination was reduced through the use of an absorber andcollimator. The absorber (a thin piece of Lucite) was placed after the firstbending magnet; the three particle species lost different amounts of energyin the absorber, causing them to separate horizontally when they traversedthe second bending magnet. The second and third bending magnets selectedparticles with a momentum of about 75 MeV/c. A collimator was placednear the third focus, and positioned so as to block positrons and some muons.Because of variations in the energy loss in the absorber, many muons wentthrough the collimator as well, along with some positrons. During beamcommissioning tests, the particle content of the beam and the horizontalseparation of the three particle types were measured at F3; the results areshown in Figure 3.3. The final beam composition at F4 was approximately85% pion, 14% muon, and 1% positron [31].3.2 DetectorFollowing the last set of quadrupoles (Q8-10), the pion beam exited througha stainless steel vacuum window 76.2 microns thick and entered the PIENU303.2. DetectorFigure 3.2: The end of the M13 beamline, before (left) and after (right) theextension. Part of the detector was in place to measure the particle contentof the beam.313.2. DetectorFigure 3.3: The transverse position distributions of pions, muons, andpositrons at F3; the particle type was identified using time of flight. Theblack lines are Gaussian fits to the pion and positron distributions.detector, which has been briefly discussed in Section 1.4.1. The physicaldimensions of the detectors are shown in Table 3.1.Physically, the detector consisted of three modular subelements: the beamwire chambers, PIENU-I, and PIENU-II. The beam wire chambers, namedWC1 and WC2, were mounted on the beam pipe downstream of the vacuumwindow, and were the first detectors seen by the beam. One wire plane, alongwith its preamplifier, is shown in Figure 3.4; the chambers in place on thebeam pipe are shown in Figure 3.5.PIENU-I comprised the first four plastic scintillators, B1, B2, Tg, and T1,with their lightguides and PMTs, as well as the three silicon strip detectors,S1-3. PIENU-II comprised the last plastic scintillator, T2, a third wirechamber, WC3, and the NaI(Tl) and CsI crystals, along with their supportstructures. A schematic of the readout for the plastic scintillators in PIENU-I is shown in Figure 3.6; T2 was read out with optical fibres. PIENU-I andII are shown in Figures 3.7 and 3.8. BINA (the NaI(Tl) crystal) is shown inFigure 3.9, and the full crystal array is shown in Figure 3.10. A Solidworks323.2. DetectorTable 3.1: Detector characteristics. The z position given is for the centre ofthe detector, except as noted for BINA.Name Z (mm) Thickness (mm) Shape Dimensions (mm)Plastic ScintillatorsB1 -39.03 6.6 Square 100 x 100B2 -30.02 3.07 Square 45 x 45Tg 0.00 8.05 Square 70 x 70T1 19.92 3.04 Square 80 x 80T2 72.18 6.6 Circular 171.45 diameterName Z (mm) Diameter Wires per plane Wire spacing (mm)Wire ChambersWC1 -112.55 96.0 120 0.8WC2 -74.41 96.0 120 0.8WC3 55.86 230.4 96 2.4Name Z (mm) Dimensions (mm) Channels per plane Strip pitch (mm)Silicon StripsS1 -23.54 61 x 61 48 0.32S2 -11.76 61 x 61 48 0.32S3 10.50 61 x 61 48 0.32Name Z (front face, mm) Thickness (mm) Shape Dimensions (mm)Crystal scintillatorsBINA 84 480 Cylinder 240 radiusCsI - 250 Pentagon 90 x 80333.2. DetectorFigure 3.4: One wire chamberplane and its preamplifier board;each chamber consisted of threeplanes.Figure 3.5: Wire chambers 1 and2 after installation on the beampipe.(computer-aided design software) drawing of the full detector, along with apicture, is shown in Figure 3.11.343.2. DetectorFigure 3.6: The plastic scintillator readout scheme, for B1, B2, Tg, andT1. The light from the plastic scintillator (purple) was transmitted by fouracrylic lightguides (light green) to PMTs (grey cylinders).Figure 3.7: The PIENU-I detector subsystem.353.2. DetectorFigure 3.8: The PIENU-II detector subsystem.Figure 3.9: The BINA detector on the test bench, with some of its PMTsin place.363.2. DetectorFigure 3.10: BINA and the two CsI rings, with all the BINA PMTs in place.373.2. DetectorzyxFigure 3.11: A Solidworks drawing of PIENU-I and PIENU-II, along witha picture with a human for scale. During data-taking, PIENU-II was rolledforward around PIENU-I.383.2. DetectorThe beam wire chambers, WC1 and WC2, provided tracking for the in-coming pion beam. Each wire chamber contained three planes, inclined at0, 120, and 240 degrees [8]. Each plane contained 120 wires spaced at 0.8mm. Since each plane is sensitive to one coordinate, at fixed z, togetherWC1 and WC2 provided three points with which to reconstruct the particletrack. The distributions of wires hit are shown in Figure 3.12; the size of thebeam spot is slightly less than 1 cm (FWHM) in each of these dimensions.Events without a track in the beam spot were rejected in the analysis.Figure 3.12: Wires hit in each plane of the first wire chamber.Following the wire chambers, the beam passed through two plastic scin-tillators (B1 and B2) and two pairs of x-y silicon strip detectors (S1 andS2) [8], before reaching the target (Tg). X and y are defined in the PIENUcoordinate system as being the horizontal and vertical dimensions trans-verse to the beam direction, which points towards positive z. The particlecontent of the beam can be determined from the correlation of the energydeposited in B1 and the time of the hit in B1 relative to the cyclotron RF;the distribution is shown in Figure 3.13.393.2. DetectorFigure 3.13: Energy deposited in B1 vs. the time of the hit relative to thepeak of the cyclotron RF field. The cluster with the most events, labelledpi, is caused by pions; the cluster below that, with the same timing but lessenergy, is caused by pions that decayed in flight prior to reaching B1. Thehit is caused by the decay muon. The cluster labelled µ on the far left isdue to beam muons, and the low-energy cluster labelled e is due to beampositrons.403.2. DetectorThe measured energy loss distributions for pions in each of the first threeplastic scintillators are shown in Figure 3.14. The stopping distributionof the beam was centred in z in the target of thickness 8.05 mm. Thedistribution was approximately Gaussian, with σ = 0.8 mm. The range ofthe decay muon in plastic is less than 2 mm; thus, it was always containedwithin the target as well. A simulation of the distance travelled by the decaymuon is shown in Figure 3.15. The first peak in the right-hand panel ofFigure 3.14 corresponds to the full energy of the pion prior to reaching thetarget, while the second peak is caused by events where the muon emergedwithin the integration time of the pulse, leading to its energy being includedas well.Figure 3.14: The measured energy deposited in B1, B2, and Tg by pions.The additional peak in the target spectrum is caused by events in which theenergy of the 4.1 MeV decay muon is also included.Because of the narrow energy spread of the beam and predictable energyloss in the counters, the pions almost always stopped in the target unless theydecayed before reaching it. A Monte Carlo simulation of the pion stoppingposition along the z axis is shown in Figure 3.16; z = 0 is the centre ofthe target. The fraction of events outside the peak is approximately 0.02%,prior to the application of any cuts.The PIENU trigger selected events in which the decay positron enteredthe downstream part of the detector (in the direction of the beam, towards413.2. DetectorFigure 3.15: The simulated distance travelled by the 4.1 MeV pi → µν decaymuon in the target.Figure 3.16: The simulated pion stopping position along the beam direction.423.3. Trigger and DAQpositive z). Only events with coincident hits in the plastic scintillators down-stream of the target, T1 and T2, were recorded. This ensured the decaypositron would enter the crystal array. Typically, the full energy of thepositron was absorbed by the crystals, although there was some probabilityof leakage. Since the PIENU analysis hinges on distinguishing pi+ → e+νepositrons from pi+ → µ+ → e+νeν¯µ positrons based on their energy, detailedcharacterization of the calorimeter response function was required.The axis of BINA was aligned with the axis of the beam, correspondingto 19 radiation lengths of NaI(Tl) for a 70 MeV positron entering on-axis.The amount of material along the particle trajectory was much less forpositrons that entered at high angles relative to the crystal axis, increasingthe probability that part of the electromagnetic shower could escape. TheCsI crystals, which provided an additional 13.5 radiation lengths [8], wereplaced so as to capture much of the leakage from these high-angle events.Figure 3.17 shows a Monte Carlo simulation of the sum of the energy de-posited in BINA and CsI by pi+ → µ+ → e+νeν¯µ positrons and pi+ → e+νepositrons, and the energy deposited solely in BINA by pi+ → e+νe positrons.The overlap between the two spectra is much greater without the additionof the CsI energy. The analysis of the crystal response is discussed in detailin Chapter 6.3.3 Trigger and DAQThe signals from the plastic scintillator PMTs were used as inputs tothe trigger system, a digital logic circuit responsible for identifying eventsthat should be read out and stored. Several triggers were implemented,divided into two categories: physics triggers and calibration triggers. Physicstriggers were used to select events where a pion stopped in the target andthe decay positron entered the crystal array, while calibration triggers wereused to select other types of events to be used for calibration. If an eventpassed one of the triggers, the signals from all the detectors were digitizedand written to disk.The basis of the logic for physics triggers was a coincidence between B1,B2, and Tg, made by the pion, followed within 540 ns by a coincidencebetween T1 and T2, made by the decay positron. These were referredto as “upstream coincidence” and “downstream coincidence”, respectively.Events where the positron preceded the pion by up to 300 ns were also433.3. Trigger and DAQFigure 3.17: The simulated energy deposited for pi+ → e+νe decays in BINA(black) and BINA and CsI (blue), and for pi+ → µ+ → e+νeν¯µ decays inBINA and CsI (red). The distributions are normalized to the maximum bin.The low-energy peak is caused by the absorption of a single 511 keV photonfrom a positron annihilating at rest, in T2 or the front face of BINA.443.3. Trigger and DAQaccepted, in order to characterize backgrounds caused by the presence ofmuons in Tg from previous pion decays. A discriminator was used to selecta large pulse in B1, to suppress beam muon and positron events. Threedifferent triggers were used to record pion decay events; one of the triggershad only the requirements of upstream and downstream coincidence with alarge pulse in B1, and thus recorded almost entirely pi+ → µ+ → e+νeν¯µevents. The time window chosen reduced the pi+ → µ+ → e+νeν¯µ eventsby approximately a factor of five compared to pi+ → e+νe events, and thetrigger was prescaled by a factor of sixteen, meaning that only one event outof sixteen that triggered was actually recorded. This was done to reduce thedata size, since it was not neccessary to record every pi+ → µ+ → e+νeν¯µevent. This trigger was called the Prescale trigger.Two other, unprescaled, triggers were used to enhance the proportion ofpi+ → e+νe events: these were referred to as Early and BinaHigh. The Earlytrigger fired only if the downstream coincidence was within 5-40 ns of theupstream coincidence, and BinaHigh triggered only for large energy depositin the calorimeter. Initially this was accomplished by putting an analogsum of the BINA and CsI PMTs into a discriminator, but this trigger waslater modified to use a field-programmable gate array (FPGA), which allowslogic circuitry to be implemented in software. The energy threshold for thistrigger was set a few MeV below the Michel edge, to ensure 100% efficiencyfor pi+ → e+νe events. The measured time spectra of Prescale events, scaledup by a factor of sixteen, and Early events are shown in Figure 3.18, andthe energy spectrum of BinaHigh events is shown in Figure 3.19.Several calibration triggers were also used. The most important of thesewere the Positron trigger and the Cosmics trigger. The Positron triggerrequired a coincidence of B1, B2, and Tg with T1, and small energy depositin B1, thus recording beam positrons. The Cosmics trigger required energybe deposited in coincidence in the outer and inner layers of the CsI crystals,thereby recording cosmic ray muons. The Positron trigger was used to checkthe calibration of the scintillators and the NaI crystal, while the Cosmicstrigger was used to calibrate the CsI crystals. The full trigger diagram isshown in Appendix A. The proportions of each of these triggers within atypical 2012 run are shown in Table 3.2.Three different systems were used to digitize the signals from the variousdetectors. Two of these, VF48 [57] and COPPER (the COmmon Pipelined453.3. Trigger and DAQFigure 3.18: Time spectra of events recorded by the Early (red) and Prescale(black) triggers. The Prescale trigger events are scaled by a factor of sixteen.The time spectrum is dominated by pi+ → µ+ → e+νeν¯µ events. Both theend of the early time window and the end of the trigger window can be seen.Table 3.2: The number of events in one 2012 run caused by the physicstriggers, and the most important calibration triggers.Trigger Events FractionPrescale 84,097 29.5 %Early 52,457 18.4 %BinaHigh 157,784 55.3 %Positron 6,683 2.3 %Cosmics 4,094 1.4 %463.3. Trigger and DAQFigure 3.19: Measured energy deposited in the NaI(Tl) crystal, for eventsrecorded by the BinaHigh trigger. The sharp rise is caused by the thresholdof the BinaHigh trigger, and the fall by the endpoint of the Michel distribu-tion. The few events above the Michel edge are mostly due to pileup.473.3. Trigger and DAQPlatform for Electronics Readout) [58], were analog-to-digital converterswith the ability to record waveforms, while the third, VT48 [59], was atime-to-digital converter, which only recorded the time a signal went abovea certain threshold. The digitization frequencies were 60 MHz, 500 MHz,and 1.6 GHz, respectively. The VF48 dynamic range was ±250 mV with10-bit resolution, and the COPPER dynamic range was ±500 mV with 8-bitresolution. The signals recorded by each digitizer are shown in Table 3.3.The VT48 also recorded many other signals at various points along the trig-ger chain; these are shown on the trigger diagram. The VF48 was usedto record the signals from the crystals and the silicon detectors, which aremuch slower than the plastic scintillators. For the scintillators, each channelrepresents one PMT. COPPER was used mainly to record the plastic scin-tillators, although analog sums of the BINA and CsI PMTs were digitizedby COPPER as well.The readout window of COPPER was 8 µs wide, and defined by the piontime. Approximately 1.3 µs after and 6.6 µs before the pion time wererecorded. This gave three muon lifetimes of pileup coverage before the pion,reducing the background due to old muons (muons left in the target from aprevious incident pion) by a factor of about twenty. The VT48 and VF48readout windows were defined by the positron time, with readout windowsof about 1 and 8 µs, respectively. The times of the first hits in B1 and T1 areshown in Figure 3.20. The times plotted are the averages of the four PMTsreading out each counter. The time spectrum that is fitted to extract thebranching ratio is composed of the difference of these two times. Using twodetectors triggered by the same signal minimized the impact on the timespectrum of jitter in the timing system.The large peak in the T1 spectrum, the bottom panel of Figure 3.20,is due to beam particles, primarily muons, which passed through all thescintillators. For such events the T1 and B1 hits were almost coincident.The trigger window of 300 ns before the pion and 540 ns after can be clearlyseen in the T1 spectrum, and repeating peaks due to beam pileup can beseen in the B1 spectrum. These are spaced according to the cyclotron RFperiod of 43 ns, and are caused by events where the first hit in B1 did nottrigger. The width of the peak in B1 is set by the width and timing of thegate signal to COPPER. The plots are zoomed around the pion time, andthus do not show the whole readout window.483.3. Trigger and DAQTable 3.3: Detector readout channels.Detector Number of ChannelsVT48 (1.6 GHz TDC)WC1 120WC2 120WC3 120B1 4B2 4Tg 4T1 4T2 4V1 1V2 1V3 1VF48 (60 MHz ADC)S1 96S2 96S3 96BINA (30 MHz) 19CsI 97COPPER (500 MHz ADC)B1 4B2 4TG 4T1 4T2 4V1 1V2 1V3 1BINA 1CsI 4493.3. Trigger and DAQFigure 3.20: Time spectrum of hits in B1 (top) and T1 (bottom).503.3. Trigger and DAQThe primary purpose of the VF48s was to digitize the signals from thecrystals; each CsI crystal had its own PMT digitized at 60 MHz, whereasBINA had 19 PMTs digitized at 30 MHz, due to the slowness of its responsecompared to the pure CsI crystals. The CsI and BINA readout windowswere 60 and 40 counts long, respectively, corresponding to about 1 and 1.3µs. The positron signal was placed near the beginning of the BINA readoutwindow, in order to allow for full integration of the pulse; the positron signalfell roughly in the middle of the CsI readout window.The silicon strip detectors were also digitized by the VF48. The actualstrip pitch was 80 µm, but this was considerably smaller than the requiredresolution, so to reduce the number of readout channels four strips werecombined into one, leading to a strip pitch of 320 µm [8]. A further reductionwas achieved by connecting each strip to a capacitor and reading out everyfourth channel with an amplifier; the charge in neighbouring channels couldthen be used to determine the position of the hit. A schematic of the readoutscheme is shown in Figure 3.21.In addition to the detectors themselves, several experimental parametershad to be constantly monitored, such as the fields of the bending magnets,the temperature of the detector enclosure and electronics racks, the voltageacross the PMTs and wire chambers, and the pressure of the wire chambergas system. All of this information was aggregated and sent to MIDAS [60],a web-based DAQ interface developed at TRIUMF and PSI. Each subsystemran its own front end program, which connected MIDAS to the underlyinghardware and sent its data to MIDAS, meaning the individual front endscould crash and restart without affecting the entire DAQ. For example, eachCOPPER board ran its own front end, as did the VME crates housing theVF48 and VT48 modules.MIDAS then concatenated and compressed the digitized data from thedetectors themselves into so-called “raw data” files, and stored all the infor-mation from each front end in a text file. The raw data were then processedusing a program called “proot”, developed for the experiment, which con-verted the raw data into ROOT [61] trees.513.3. Trigger and DAQFigure 3.21: The silicon strip readout. Each strip is connected to a capacitorand there is one readout channel per four strips [8].523.4. Event types3.4 Event typesMany types of event could cause triggers other than single pi → µ → eor pi+ → e+νe decays. Some of these backgrounds could be removed withevent selection cuts, while others had to be included in the time spectrumfit. In general, background processes resulting in high energy deposit inthe crystal calorimeter were the most important to properly characterize, asthese had the potential to be misidentified as pi+ → e+νe events. The threemain types of background in the PIENU experiment were beam particles,multiple muons decaying in the same event, and neutral pileup.3.4.1 Beam-related backgroundThe beam delivered to the detector consisted of pions, muons, and positrons,as discussed in Section 3.1. Beam positrons traversed all the plastic scin-tillators and were absorbed by BINA, while beam muons stopped in eitherT2 or BINA. Triggers caused by muons or positrons could be removed inoffline analysis based on their timing, as the upstream and downstream co-incidences came at approximately the same time.Events where the upstream coincidence was caused by a pion and thetrigger was subsequently completed by a muon or, especially, a positron hadthe potential to bias the branching ratio, as beam positrons deposited asimilar amount of energy in the calorimeter as pi+ → e+νe events. Theseevents came at characteristic times relative to the pion, defined by the 43.4ns RF period of the cyclotron. These events were removed in the offlineanalysis by rejecting events with extra hits in B1, B2, or T1 in a widewindow around the pion. The window used was -6.6 µs to 1.4 µs, with t =0 defined by the upstream coincidence signal.Old muon pileupThe rate of particles reaching the PIENU detector was approximately oneevery fifteen µs, on average. Sometimes, of course, the separation betweentwo particles was considerably less than this. Due to the long lifetime of themuon, τµ = 2.197 µs, there was a significant probability that there wouldbe a muon already in the target when a pion arrived. These were referredto as “old muons”. The cut applied in the offline analysis removing eventswith extra hits in B1 and B2 mitigated this background, but did not remove533.4. Event typesit entirely, since approximately 5% of muons took longer than 6.6 µs (thepileup inspection window) to decay.The energy spectrum and the time spectrum for such events were bothaffected by the presence of the additional muon in the target. The timespectrum was affected because either muon could decay at any given time,resulting in an effectively shorter lifetime for this type of event. The energyspectrum was affected because both muons could decay within the inte-gration time of the calorimeter, and both decay positrons could enter thecrystals. This could result in enough energy being deposited to place theevent in the high-energy time spectrum.If both positrons went through T1 into BINA, the event would be removedby the pileup cut in T1, unless the two hits were too close together in time(∼15 ns) to resolve. A larger contribution came from events where onepositron went through T1 into BINA, and the other missed T1 while stillentering either BINA or CsI. More solid angle existed for this to occur inCsI, but the narrower integration window of CsI meant that the energy wasonly added to the event if the two muon decays were less than 80 ns apart.The solid angle for BINA was smaller than for CsI, but the integration timewas much longer. A diagram showing a positron emerging from the centreof the target, going past the edge of T1, going through T2, and enteringBINA is shown in Figure 3.22.The proportion of decay positrons that miss T1 and still enter BINA canbe determined from Monte Carlo simulation. Figure 3.23 shows the x and ydistributions in the first plane of WC3, for pi → µ → e events that depositenergy in BINA, and for pi → µ → e that deposit energy in BINA but notT1. The hole due to T1 in the latter plot is clearly visible. The ratio ofevents in which the decay positron misses T1 and deposits energy in BINAto the total number of events in which the decay positron deposits energyin BINA was approximately 10%.Since old muons could not be removed entirely, it was necessary to includetheir effects in the time spectrum fit. The trigger window was extended intonegative times, corresponding to events where the downstream coincidencepreceded the upstream coincidence. In the low-energy time spectrum, thisregion was populated almost entirely by events where an old muon decayedinto T1 and T2, and a pion subsequently completed the trigger. The time543.4. Event typesFigure 3.22: A muon decaying in the centre of the target and the decaypositron going past the edge of T1. The plastic scintillators and BINA areshown; distances are to scale.Figure 3.23: Simulated x and y distributions at the first plane of WC3 forpi → µ→ e events that deposit energy in BINA and not in T1 (left) and forpi → µ→ e events that deposit energy in BINA (right).553.4. Event typesspectrum for this component is a simple muon lifetime, which can then beextrapolated to positive times. For events in which both muons decayed, thetime spectrum was obtained by a Monte Carlo simulation which included thewaveforms and integration windows of BINA and CsI. This, and the othercomponents of the time spectrum fit, are described in detail in Chapter 5.Simplified timing diagrams for a regular pi+ → µ+ → e+νeν¯µ event, an eventwhere only the old muon decays, and events where two muons decay can befound in Appendix B.Neutral pileupThe neutral pileup background was mainly due to neutrons produced atthe production target, which then thermalized in the shielding around thebeamline before being captured in the calorimeter or its support structure,leading to gamma emission. This background was time-independent relativeto the pion decay time, since it took on the order of tens of microsecondsfor the neutrons to make their way from the primary target to the detector.To mitigate this background, a steel wall was placed between the detectorand the beamline, with a narrow opening for the beampipe, as shown inFigure 3.24.Figure 3.24: The PIENU detector and beamline after the last bending mag-net, showing the steel wall used for neutron shielding.563.5. Monte CarloDespite the extra shielding, some neutral pileup remained. This was in-cluded in the time spectrum fit by including a parameter representing theproportion of low-energy events promoted to the high-energy time spectrumvia time-independent means. Neutral pileup was the primary mechanismfor such events, though energy resolution could also contribute in principle.3.5 Monte CarloThe full PIENU detector geometry was implemented in GEANT4 [4],including all support structures and dead material. The parameters of thepion beam were taken from data. No waveform digitization was included inthe simulation. Gaussian energy resolution was added to each detector; thisresulted in good agreement for the plastic scintillators, but the response ofthe crystal scintillators was taken from data, as this was a crucial aspect ofthe analysis. The response function of BINA and CsI is described in detailin Chapter 6. Monte Carlo was used to determine the size of several othersystematic effects, as explained in Chapter 7.57Chapter 4Data Taking and Processing4.1 Running periodsData-taking began in April 2009 and concluded in December 2012. TheTRIUMF cyclotron is shut down at least three months of each year formaintenance, between January and March, so there is a separate data setfor each year. The data were divided into runs containing approximately300,000 events, which took about ten minutes at an incident pion rate of50-60 kHz. Table 4.1 shows the approximate number of runs taken in eachyear used thus far in the branching ratio analysis, as well as any specialruns. In the following sections, the conditions for each running period aredescribed.Table 4.1: Running periods.Year Runs used Special runs2009 0 One week positron beam data2010 2400 Eight hours per week muon beam data2011 3600 One month positron beam data2012 13000 One week beamline tests4.1.1 2009A discriminator was used to determine the pulse height of the sum of theNaI(Tl) and CsI PMTs for the BinaHigh trigger, rather than the digitalmodule that was used later. The signal used was an analog sum withoutgain correction, so the threshold of the trigger was not stable, leading to apotential loss of pi+ → e+νe events. Also, the trigger for recording cosmicrays, used to calibrate the CsI crystals, did not exist. The calibration wasinstead attempted using a xenon lamp with a dedicated trigger; however,this proved inadequate.584.2. Data processing4.1.2 2010The final trigger was in place in 2010; however, until November, the sig-nals from the CsI crystals were out of time with the trigger, and thus notrecorded. Consequently, the largest source of systematic error in the ex-periment, the estimated uncertainty in the low-energy tail of the measuredpi+ → e+νe energy spectrum, was larger by approximately a factor of two.The November 2010 data were analyzed in [3]; the uncertainty obtained onthe branching ratio was 0.24%, with about equal contributions from statis-tics and systematics.4.1.3 2011The cyclotron was shut down until September due to a failure in thevacuum system. In September and part of October, data were taken tomeasure the response function of the crystals; PIENU data were then takenuntil the end of the year.4.1.4 2012The largest easily useable data set was recorded in 2012. However, at thestart of this running period, the energy threshold of the BinaHigh triggerwas lowered, to ensure that no pi+ → e+νe decays were being missed. Thisresulted in more pi+ → µ+ → e+νeν¯µ events causing BinaHigh triggers.Since these events are not used in the analysis, the number of useable eventsper run is lower by approximately a factor of 1.5 in 2012 compared with2010 and 2011.4.2 Data processingThe raw data from each run were processed into a ROOT [61] tree con-taining information from each detector. From the point of view of dataprocessing, there were five separate types of detectors that must be consid-ered: the plastic scintillators, the wire chambers, the silicon strip detectors,BINA, and the CsI array. Each of these had a different characteristic re-sponse, and different algorithms were employed.4.2.1 Plastic scintillatorsEach plastic scintillator (B1, B2, Tg, T1, and T2) was read out by fourPMTs. The signal from each PMT was sent to VT48 and COPPER. De-594.2. Data processingspite the higher frequency of the VT48, 1.6 GHz compared to 500 MHz,the COPPER waveform gave not only energy information but also superiortiming, since it can be fitted. Thus, for the plastic scintillators, VT48 wasused mainly as a diagnostic check of COPPER information. Since the VT48is a TDC, the only information available to store in the output trees wasthe time at which the PMT voltage went above a given threshold.Since COPPER digitized the full PMT waveform, more sophisticated anal-ysis was required. COPPER digitized a time window 8 µs wide, with asample every 2 ns, so to keep the data size manageable it was necessaryto zero-suppress the waveform except near hits. A hit was defined by anincrease in the waveform from one sample to the next with the waveform ata level of at least three counts; the noise in the ADC was at the level of onecount. For each hit, the time and height of the peak was stored, in additionto integrals of the waveform in different time regions, from a narrow regionaround the peak to the full unsuppressed waveform. The level of the wave-form prior to the hit, referred to as the pedestal, was also stored; the zerosuppression was removed approximately 30 ns before the hit.A typical raw PMT waveform (in this case from a pion in B1) is shownin Figure 4.1. The red and black points display the values recorded by two250 MHz ADCs. The full digitization window is 8 µs wide, and the dynamicrange of the ADCs is 0 to 255. The noise in the ADCs was at the level of 1count, and a typical pion signal in B1 was 50-60 counts.Figure 4.1: PMT waveform from a pion in B1. The points at the beginningand the end of the waveform are zero-suppressed; the drop around -1380 nsis to the level of the pedestal.604.3. Blinding4.2.2 Wire chambersFor each wire chamber plane, the wires were bundled in groups of three,and each group was fed into one channel of the VT48; the times of the wireshit were stored in the output trees.4.2.3 Silicon stripsEach channel of S1, S2, and S3 was digitized by the VF48, a 60 MHz ADC;channels without hits were zero-suppressed. The position of the particle wasreconstructed based on a weighted average of the charge deposited in stripswith hits.4.2.4 Crystal scintillatorsThe scintillation light from BINA was collected by 19 PMTs, each of whichwas digitized by the VF48. The time and height of the peak sample werestored in the output trees, as well as integrals of both the full waveformand a narrow region around the peak. The former gives the best energyresolution, but is more affected by pileup.The CsI crystals were read out by the VF48 as well. Similarly to theNaI(Tl) crystal, timing, pulse height, and integrated charge informationwere stored in the trees.4.3 BlindingSince the PIENU experiment aims to make a high-precision measurementwhich will then be compared to a precise theoretical prediction, it is nec-essary to mitigate potential biases in the analysis procedure. Therefore, amethod for blinding the branching ratio at the stage of raw data process-ing was developed. The ideal blinding procedure would randomly alter thequantity being measured without affecting the data in any other way; inpractice, it is necessary to make use of some quantity that is not used in theanalysis but which depends on the quantity being measured. Knowing theexact nature of the dependence is not required.In the PIENU analysis, a natural choice for a quantity to use for theblinding is the energy deposited in the target counter [62]. The presence orabsence of the decay muon causes a significant difference between pi+ → e+νe614.4. Energy calibrationand pi+ → µ+ → e+νeν¯µ events, but the time and energy resolutions ofthe target are not sufficient to make use of this information at the level ofprecision required.In order to alter the branching ratio, an inefficiency function was appliedto the target energy. An unknown factor between 0 and 0.5% of events wasexcluded from the analysis in a region of the target energy spectrum con-taining mostly either pi+ → e+νe events or pi+ → µ+ → e+νeν¯µ events; theregion in which events were excluded was chosen randomly. The blindedevents will be included in the analysis once all the event selection cuts,shapes used in the time spectrum fit, and branching ratio corrections arefinalized, the blinded branching ratio is stable as parameters in the analysisare varied, and all systematic errors have been assigned. The blinding pro-cedure does not significantly affect the branching ratio stability tests, as thetarget energy is almost independent of other quantities used in the analy-sis, and the blinding factor is small enough that the dependence that mayexist is negligible at the level of precision of the experiment. A graphicalillustration of the technique is shown in Figure 4.2.Figure 4.2: The blinding technique. Events are removed at random in oneof two regions of the spectrum of energy deposited in the target counter,corresponding to either pi+ → e+νe or pi+ → µ+ → e+νeν¯µ events.4.4 Energy calibrationEnergy calibration in the PIENU experiment was done for each plasticscintillator as well as the crystals. The energy deposited in the plasticscintillators was obtained from the COPPER waveforms; each PMT was624.4. Energy calibrationdigitized by two 250 MHz ADCs. The offset of each ADC was obtainedfrom the average of the first three non-suppressed samples, just prior to thehit, and the gains were aligned using the peak in the energy deposited bybeam muons. The combined waveform from the two ADCs was then in-tegrated in a window around the peak to give the energy in ADC counts.This was converted to MeV using the Monte Carlo, with a Birks’ correctionincluded [63]. The consistency of the peaks of the different particle speciesin each counter provided a test of the accuracy of the calibration. As anexample, both the raw and calibrated spectra for the B1 counter are shownin Figure 4.3. In the calibrated histogram, only physics triggers have beenincluded, which removes the positrons visible in the raw histogram. Onerun of data is shown, along with simulations of beam muons and pions, nor-malized to the number of events in each peak. The tail on the left is largerin the simulated muon spectrum than the data because the trigger requiredhigh energy deposit in B1.The calibration for the other counters was done similarly, although in T1and T2 there were only muons and positrons, since pions stopped in thetarget. The T1 energy from pion decay events is shown in Figure 4.4, alongwith the simulated spectrum. The energy deposited in the target counterwas not used in the branching ratio analysis, to avoid biasing pi+ → e+νeor pi+ → µ+ → e+νeν¯µ events; the energy spectra of the two decay modesare significantly different, and the response of the target would have to bevery well understood to make use of this information without increasing thesystematic error to an unacceptable level. Furthermore, the presence of avariable with a strong dependence on the decay mode, that neither is usedin the analysis nor has a strong correlation with any variable that is usedin the analysis, provides a simple tool for blinding the branching ratio (seeSection 4.3).The calibration of BINA, the NaI(Tl) crystal, was done using the pi+ →e+νe peak and the endpoint of the pi+ → µ+ → e+νeν¯µ spectrum, with theenergy deposited by beam positrons providing an additional check. Becausethe analysis relies on separating the events into those that deposited less than52 MeV in the crystals from those that deposited more than 52 MeV in thecrystals, the calibration of BINA and CsI was by far the most importantof all the detectors. The position of the pi+ → e+νe peak was obtainedfrom MC, and the consistency of the Michel edge in data and simulationwas used to assign an uncertainty in the energy calibration. The position of634.4. Energy calibrationFigure 4.3: The energy deposited in B1 before (upper panel) and after (lowerpanel) calibration. In the calibrated histogram the MC spectrum is shown inred, and a cut has been made to remove events due to calibration triggers.The difference around 2.5 MeV is due to the requirement of high energydeposit in B1 for physics triggers, which was not included in the MC.644.4. Energy calibrationFigure 4.4: Calibrated energy deposited in T1. The MC spectrum is shownin red.the Michel edge was obtained by fitting the spectra with a convolution of aunit step function with a Gaussian representing the energy resolution. Theresult of this convolution is an error function:f(E) ∝ 12[1 + erf(E − aσ√2)]. (4.1)Here a is the value at which the unit step transitions from 0 to 1, and σ isthe standard deviation of the Gaussian resolution function. The mean of theresolution is assumed to be 0. Figure 4.5 shows the measured energy spec-trum in BINA + CsI zoomed in on the pi+ → e+νe peak for the 2012 data,as well as a simulation of pi+ → e+νe positrons in red. Both distributionsare fitted with Gaussians. Figure 4.6 shows the measured and simulatedenergy spectra for pi+ → µ+ → e+νeν¯µ positrons, fitted with Eqn 4.1.When this was done for the 2010 data, the energy calibration was found tobe accurate to 0.1 MeV [62]. The results for the edge of the Michel spectrumin data and MC were 48.50 MeV and 48.47 MeV; for the pi+ → e+νe peak,the results were 65.49 MeV and 65.38 MeV. Thus, the uncertainty on theenergy calibration between 48 and 65 MeV is again taken as 0.1 MeV. Thefull calibrated energy spectrum is shown in Figure 4.7.654.4. Energy calibrationFigure 4.5: Energy deposited in BINA + CsI for pi+ → e+νe events. Theblack histogram, fitted with the black curve, is data, and the red histogram,fitted with the red curve, is MC. The fits are Gaussian; the fitting range isasymmetrical about the peak because the region to the left of the peak isnot Gaussian, due to shower leakage.Figure 4.6: Energy deposited in BINA + CsI for pi+ → µ+ → e+νeν¯µ eventsnear the Michel edge. Data is shown in black and MC is shown in red. Thelines are fits using Equation 4.1.664.5. Track reconstructionFigure 4.7: The calibrated BINA + CsI spectrum. The spectrum up to50 MeV is dominated by pi+ → µ+ → e+νeν¯µ events, and the peak atapproximately 65 MeV is due to pi+ → e+νe events. The high-energy tailis mainly due to pileup events, with a small contribution from pi+ → e+νeγevents.4.5 Track reconstructionThe incoming pion track was reconstructed using the beam wire cham-bers, WC1 and WC2, and the silicon strips near the target, S1 and S2.Each pair was used independently to reconstruct the track, to reduce theimpact of multiple scattering in the intervening detectors. Each wire cham-ber contained 3 planes, each sensitive to a single coordinate; the siliconstrips each contained 2 orthogonal planes. The tracking algorithm dividedthe wire chambers and silicon strips into three groups, called trackers. Thefirst tracker, WC12, contained the beam wire chambers, WC1 and WC2.The second tracjer, S12, contained the two silicon strips before the target,S1 and S2. The third tracker, S3WC3, contained the silicon strip after thetarget and the wire chamber near the crystal face, S3 and WC3. S3WC3was used to reconstruct the decay positron track.For a given event in a given tracker, the locations of the hits in each planewere fitted to a straight line. The position of the hit was taken as the centreof the plane. The parameterization used is given in Equations 4.2 and 4.3;z is the coordinate in the direction of the beam, x is horizontal, and y is674.5. Track reconstructionvertical. tx and ty are the ratios of the x and y momenta to the z momentum,and x0 and y0 are the x and y coordinates at z = 0, the centre of the target.The tracks are parameterized according to the following equations:x = tx ∗ z + x0 (4.2)andy = ty ∗ z + y0. (4.3)The reconstructed y vs. x position at the centre of WC1 using the WC12tracker is shown in Figure 4.8, and the tx and ty distributions are shown inFigure 4.9. The same quantities reconstructed by the S12 tracker are shownin Figures 4.10 and 4.11. These distributions are used in the Monte Carlosimulation to provide realistic beam parameters for the incoming pions.Figure 4.8: The beam spot reconstructed by the WC12 tracker.684.5. Track reconstructionFigure 4.9: Ratios of the x and y momenta to the z momentum (tx and ty),reconstructed by the WC12 tracker.694.5. Track reconstructionFigure 4.10: The beam spot reconstructed by the S12 tracker.The decay positron track was reconstructed using S3 and WC3. The y vsx position in the middle of WC3 and the tx and ty distributions are shownin Figures 4.12 and 4.13, showing the positrons emerging isotropically ratherthan the focussed beam seen upstream of the target.704.5. Track reconstructionFigure 4.11: Ratios of the x and y momenta to the z momentum (tx andty), reconstructed by the S12 tracker. The gaps are due to the track recon-struction algorithm, which uses the centre of the plane hit as the position,leading to some values for tx and ty never occurring.714.6. Data stabilityFigure 4.12: Decay positron position at the centre (along z) of WC3, recon-structed by the S3WC3 tracker.4.6 Data stabilityDue to the length of time over which data were taken, compensating forthe variation of several experimental parameters was essential. Although thedetector enclosure and electronics racks were maintained at roughly constanttemperature and humidity, the PMT gains and thresholds, and the offsetsin the trigger logic, still varied considerably over the running period. Thecyclotron current was often unstable, leading to variations in the beam rate.The dipole magnets were monitored with nuclear magnetic resonance (NMR)probes, which were used to automatically adjust the current to maintain aconstant field, so the beam momentum was largely stable. However, thiswas not done for the quadrupole magnets, meaning that the exact locationof the beam was potentially variable.Although the branching ratio analysis is, in principle, insensitive to mostof these changes, the energy measured by the crystals is of crucial impor-tance. Therefore, automatic run-by-run gain correction was performed; inthe case of BINA, the beam positron peak in each tube was aligned with areference run prior to being included in the calibrated sum. This was done724.6. Data stabilityFigure 4.13: Ratios of the x and y momenta to the z momentum (tx and ty)for decay positrons, reconstructed by the S3WC3 tracker.734.7. Timingat the stage of converting the raw data into a tree. Similar procedures wereimplemented for the plastic scintillators as well, based on the peaks due toeither beam muons or positrons. Figure 4.14 shows the peak of the pulseheight distribution of beam positron events in the first BINA tube over the2012 running period. Significant variation was observed, particularly whenthe beam resumed after having been off. The gain changes were correctedfor on a run-by-run basis, for each PMT individually. Since the runs wereshort, no significant gain changes within a run were observed, so this levelof stability was acceptable for the PMTs.Figure 4.14: The run by run variation in the pulse height of the beampositron peak in one BINA PMT. Similar variations were observed for theother PMTs.Figure 4.15 shows the peak of the reconstructed pion stopping positiondistribution over the same period. The stopping position refers to the zcoordinate, and thus corresponds directly to the beam momentum. It isobtained from the point of closest approach of the tracks reconstructed byS12 and S3WC3. The gaps are due to regions of bad data.4.7 TimingThe primary method used to extract timing information was to fit thewaveforms produced by the plastic scintillator PMTs, digitized by COPPER.744.7. TimingFigure 4.15: The run by run variation in the pion stopping position alongthe z axis.Through this technique, sub-ns resolution was obtained. The pion time wastaken from the B1 waveform, and the positron time from the T1 waveform.The fitting function for each counter used was a template constructed fromthe average of many waveforms from that counter; a typical fitted pulse isshown in Figure 4.16. Figure 4.17 shows the time difference between fittedpulses in two of the T1 PMTs, fitted with a Gaussian; the resolution is givenby the width of the distribution, which is σ = 0.49 ns.The trigger signal going to COPPER comes from the counters upstreamof the target, and thus carries the pion time, meaning the B1, B2, and Tgpulses corresponding to the triggering particle are fixed within the readoutwindow. The time of the fitted pulse in B1, averaged over the four PMTs,is shown in Figure 4.18; the width of the distribution is due to the variablearrival time of the trigger signal relative to the gate signal. The reducedχ2 distribution (χ2 per degree of freedom) of the waveform fit is shown inFigure 4.19. The number of degrees of freedom varied depending on thenumber of unsuppressed samples in the waveform, but was typically around50.By contrast, the timing of the T1 and T2 pulses, caused by decay positrons,varied within the COPPER window. To account for the variation in the ar-rival time of the trigger signal, the B1 time was subtracted from the T1754.7. TimingFigure 4.16: COPPER waveform fitted to a template.Figure 4.17: Time difference between two T1 PMTs. The red line is aGaussian fit.764.7. TimingFigure 4.18: Time of the triggering pulse in the B1 counter.Figure 4.19: Reduced χ2 distribution of the waveform fit in the B1 counter.774.7. Timingtime. This quantity is referred to as Tpos, and is what is ultimately fittedto obtain the branching ratio. The T1 time and χ2 distributions are shownin Figures 4.20 and 4.21. The time spectrum shows the size of the triggerwindow, both before and after t = 0, the time of the pion stop. The largepeak at t = 0 is due to beam muons and positrons traversing the entiredetector.Figure 4.20: Time of the triggering pulse in the T1 counter.Prior to any event selection, many components exist in the Tpos spectrumother than the pi+ → e+νe and pi+ → µ+ → e+νeν¯µ signals. Some of thesecomponents can be removed or reduced with cuts, whereas some must bemodelled and included in the time spectrum fit; this is the subject of thenext chapter.784.7. TimingFigure 4.21: Reduced χ2 distribution of the waveform fit in the T1 counter.79Chapter 5Raw Branching RatioExtraction5.1 Event selection5.1.1 Good run selectionThe first step in the analysis is the removal of runs in which some aspect ofthe detector, trigger, or DAQ was malfunctioning, or which were not regulardata runs. This was done by eliminating the runs with errors based on thelogs kept by MIDAS (see Section 3.3), which indicated errors recorded byany of the front end programs monitoring the hardware. Furthermore, runswhose duration fell outside the normal range were excluded; this removedthe majority of test runs and runs from periods with no beam, during whichcosmic ray data were taken. Finally, the remaining runs taken under specialconditions, mainly for evaluating systematic effects, were removed. Theentire 2012 data set contains approximately 20,000 runs; 13,210 acceptabledata runs were included in the analysis.5.1.2 Time spectrumThe time of each event is defined as the time of the pulse in the T1 counterminus the time of the pulse in the B1 counter. The events were divided intohigh and low energy regions based upon the energy deposited in the crystals.The threshold is normally set at 52 MeV, and is referred to as Ecut. Bothtime spectra for the full 2012 data set are shown in Figure 5.1, excludingcalibration triggers but with no other cuts.Several important features in each spectrum can be clearly seen. The largepeak at t = 0, referred to as the “prompt peak”, is primarily due to beamparticles traversing the entire detector; muons, in particular, often fulfilledthe trigger requirement of pion-like energy deposit in B1. In the high energyspectrum the same peak is due to beam positrons, which tended to deposit805.1. Event selectionFigure 5.1: The time spectra for low energy (upper panel) and high energy(lower panel) events. t = 0 is defined by the arrival time of the pion. Therepeating peaks are due to beam particles and are separated by the cyclotronRF period (see text).815.1. Event selectionabout 70 MeV in BINA. The repeating peaks (obvious in the high energyspectrum but also present in the low energy spectrum) were caused by apion at t = 0, followed by a muon (low energy) or a positron (high energy)from a later beam spill. The peaks are separated by approximately 43 ns,which corresponds to the cyclotron frequency of 23 MHz.The trigger window was extended into negative times as well, correspond-ing to events where the coincidence in T1 and T2 came before the coincidencein B1, B2 and Tg. Peaks from beam particles can be seen in this region;these occurred either because the beam particle did not deposit enough en-ergy in B1 to trigger, or, more probably in the low energy time spectrum, didnot trigger due to the prescale factor of 16. In the high energy spectrum,there is no prescaling, but the beam particles in question were positrons,which did not usually deposit enough energy in B1 to trigger.5.1.3 Pion selectionTriggers due to beam particles other than pions can be removed fromthe time spectra with simple cuts. The prompt events can be removed al-together; i.e., when the actual time spectrum fit is performed, the regionaround t = 0 is excluded. Pileup events involving beam particles can beremoved by cuts on the number of hits in B1 and B2. This is done by re-quiring that at least one phototube (out of the four) for each scintillatorrecorded only one hit, in a window extending approximately 2.2 µs beforethe pion time. The efficiency of the PMTs for real particles hitting the scin-tillators was very high, > 99%, and due to frequent fake hits (see “T1Pileupcut” in Section 5.1.4) a tighter cut on the number of hits would remove anunacceptably large number of events. The distribution of hits in B1 and B2is shown in Figure 5.2. The four PMTs are averaged for each counter. Onehit in each PMT is the most probable case, but occurs only about half thetime. The presence of fake hits can be inferred from the many events thathave an extra hit in only one tube.Two other cuts were done to ensure the selection of only events in whichthe triggering particle was a pion: the selection of events inside the beamspot in wire chambers 1 and 2, and the selection of events with pion-likeenergy deposit in B1 and B2. Although the energy deposit requirement forB1 was present in the trigger, an offline version was implemented as well.This was based on the calibrated energy determined from the COPPERwaveform, whereas in the trigger it was done by placing a threshold on825.1. Event selectionFigure 5.2: Number of hits in B1 (left) and B2 (right). The average of fourPMTs is taken. The peaks at whole numbers are due to pileup events; theevents with extra hits in only some of the tubes are due to noise (see text).the raw output of the B1 PMTs, and thus included no gain correction.The distributions on which the cuts were applied are shown in Figures 5.3and 5.4, with red lines indicating the cut values. The time spectra followingall of these cuts are shown in Figure 5.5.No cuts have been applied to the distributions in Figures 5.1 to 5.4. Thethree largest peaks in Figure 5.3 are caused by pions, muons, and positrons,respectively. The high energy peak is caused by events in which two pionsarrived simultaneously. The requirement in the trigger of large pulse heightin B1 is responsible for the difference between the shape of the left-hand sideof the muon peak in the two counters (the positrons are present because ofa special trigger for selecting them, for calibration purposes).5.1.4 Pileup rejectionFollowing the B1B2 pileup, prompt, WC12, and B1B2 energy cuts, eventswhere the trigger involved beam muons or positrons are reduced to a negli-gible level. Only events with a triggering pion at t = 0 survive. However, ina significant fraction of events, a muon was left over from a previous event,due to its long lifetime. This could occur either in the target from a previouspion decay or from a beam muon, which would typically stop in either T2 orthe front face of BINA. Events with additional muons had a different time835.1. Event selectionFigure 5.3: Energy deposited in B1 (left) and B2 (right). The three largepeaks in each spectrum are, from left to right, caused by positrons, muonsand pions. The smaller peaks are due to events with two particles. The redlines indicate the cut values.Figure 5.4: X (left) and Y (right) position at the centre of WC1. The redlines indicate the cut values.845.1. Event selectionFigure 5.5: The time spectra after pion selection. The left-hand plot showsthe low energy time spectrum and the right-hand plot shows the high energytime spectrum.distribution than those without, since either the muon originating from theprimary pion or the pileup muon could potentially decay, causing a trigger;also, if both muons decayed, the event could be boosted from the low energytime spectrum to the high energy spectrum.Prepileup cutMost of these “old muons” were rejected by requiring no hits in any ofthe plastic scintillators in the time range from 6.6 µs to 2.2 µs before thepion time. 6.6 µs corresponds to three muon lifetimes, in which time 95%of muons will decay. The high energy time spectra with and without thiscut, referred to as the prepileup cut, are shown in Figure 5.6. The numberof events at negative times is much reduced, as expected.T1 pileup cutThe remaining old-muon background was mitigated by a further pileupcut in T1 similar to the pileup cuts for B1 and B2 described above; thatis, at least one PMT was required to have only one hit in a time windowextending 2.2 µs before the pion stop and 1.3 µs after. Since the decaypositron could emerge at any time up to 540 ns after the pion stop, thepileup rejection window after the trigger depended on the decay time. This855.1. Event selectionFigure 5.6: High-energy time spectrum without (left) and with (right) theprepileup cut.was potentially problematic due to the presence of fake hits after the realhit in the PMTs, which were observed to occur frequently. Generally, theycame soon after the initial hit (within 50-100 ns), but occurred sometimesbeyond 1 µs after the original pulse. The source of the early fake hits wasassumed to be in the amplifier or elsewhere in the electronics, but this couldnot explain fake hits on the time scale of µs.PMT afterpulsing is a well-known source of noise in these detectors [64];one mechanism that has been identified is the ionization of helium, whichdiffuses into the PMTs over time, by the photoelectrons. The ionization canthen be amplified and cause a measurable signal. This cut therefore had tobe refined, or it would have the potential to preferentially reject positronsfrom earlier decays. This could in turn bias the branching ratio, as virtuallyall pions decay within 100 ns, but most muons do not.To illustrate the issue, Figure 5.7 shows the ratio of the pulse heights ofthe first and second hits in one of the T1 PMTs. The ratio is generally closeto one for real pileup events, and often much greater than one for eventswith fake hits. Figure 5.8 shows the time difference between the two pulses,in the region to the right of the line in Figure 5.7, which is populated almostentirely by fake hits.865.1. Event selectionFigure 5.7: The pulse height of the first hit in one of the T1 PMTs divided bythe pulse height of the second hit, if one was present. The red line indicatesthe cut used to select events with fake hits, for plotting the time differencebetween the initial hit (the real hit) and the fake hit (Figure 5.8).Figure 5.8: The time difference between the first and second hits in one ofthe T1 PMTs, for events with a small second pulse.875.1. Event selectionFor the latest positrons accepted by the trigger, the post-pileup rejectionwindow is approximately 800 ns, as compared with approximately 1.3 µsfor the earliest. The presence of fake hits in this time range raises thepossibility that the T1 pileup cut will preferentially reject earlier decays,unless an additional requirement is imposed to ensure real pileup. This wasdone via the ratio of the full integrated charge to the fitted pulse height ofthe triggering hit. This ratio is shown in Figure 5.9, as a function of thepulse height, for events with multiple hits in every tube.Figure 5.9: 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 from pileup due to fake hits.The fake hits and real pileup separate clearly into two bands; by only re-jecting events where the ratio of integrated charge to pulse height is higherthan the red line, only real pileup will be removed, and events with fakepileup will be preserved. This prevents the probability of being rejected bythe pileup cut from depending on the positron decay time. The number ofevents below the red line in each PMT, divided by the number of eventswith at least one hit, gives the probability of fake hits for that PMT. Theprobabilities are shown in Table 5.1, as well as the probability of fake hits inevery PMT. This is much higher than the product of the individual proba-bilities would suggest, indicating that the probabilities are correlated. Thismay simply be due to the fact that large initial pulses are more likely to885.1. Event selectioncause fake hits. The time spectra after this cut are shown in Figure 5.10.Table 5.1: The probability of fake hits after the real hit, in each T1 PMT.Tube Probability of fake hitsT1 1 20.7%T1 2 15.5%T1 3 15.1%T1 4 17.5%All 2.0%Figure 5.10: Time spectra after prepileup and T1 pileup cuts. The left-handplot shows the low energy time spectrum and the right-hand plot shows thehigh energy time spectrum.T2 pileup cutA pileup cut can be performed in T2 as well; some solid angle existsfor a decay positron emerging from the target to miss T1 but go throughT2 and into BINA. Furthermore, beam muons typically stop in T2 or thefront face of BINA. The T2 waveforms are substantially noisier than the T1waveforms, due to the fibre readout of T2, and they are not fitted in theanalysis, so distinguishing real pileup from fake pileup is more difficult. Thedependence of the probability of fake hits in T2 on the energy deposited in895.1. Event selectionT2 also presents a more serious problem than in the case of T1; it will beshown in Section 6.4.1 that the energy spectrum of T2 is significantly affectedby energy leaking backwards out of BINA. Therefore, a possible distortionof the time spectrum is expected when this cut is applied; however, it iseffective in removing pileup. The analysis is done both with and withoutthis cut, and it will ultimately be shown to slightly reduce the systematicuncertainty. Currently, the cut removes any event where the time differencebetween the first and last hits in an analog sum of the T2 PMTs is morethan 100 ns.Muhit cutA special VT48 channel was connected to B1 extending approximately 25µs before the trigger time; this was done to provide extra protection againstbeam muon pileup. A discriminator was used to require muon-like pulseheight in B1. Events with hits in this channel up to 10 µs before the triggerwere rejected. Rejecting any event with a hit in the full window was foundto provide no benefit relative to the shortened version; its only noticeableeffect was to reduce the available statistics.Postpileup cutOne further minor cut is needed to protect against very late pileup. TheBINA integration window extends for approximately 1 µs after the positrontime, which, for late-decaying positrons, is slightly past the COPPER win-dow for pileup rejection. Thus, there is a small time window, overlappingwith the BINA integration window, in which a positron from old muon decaycould enter BINA through T1 and T2 and the event would not be rejectedby any of the previous pileup cuts. The VT48, however, covers 4 µs beforeand after the positron. In general, it was preferred to use the full wave-form information provided by COPPER, but in this case it was necessaryto resort to VT48 information.Figure 5.11 shows the time difference between the last hit in T1 andthe triggering hit in B1, recorded by the VT48, versus the T1 minus B1time plotted in the time spectrum, for high energy events. The diagonalstructure corresponds to events with no pileup, where the two times are thesame, within the resolution. The small cluster of events at the far right,slightly above 1 µs on the y-axis, is due to the mechanism just discussed;the cut removes these events. For earlier decay times, the pileup rejection905.1. Event selectioncovers the full BINA integration window, and thus events with late pileupare rejected. For very late decay times, this is no longer the case, and theextra energy deposited in BINA can result in the event moving to the highenergy time spectrum.Figure 5.11: The time difference between the last hit in the T1 VT48 channeland the first hit in the B1 VT48 channel versus the decay time obtained fromCOPPER.5.1.5 Acceptance cutThe most important cut still to be addressed restricts the radius fromthe centre of WC3 in which events are accepted, and thus also restricts theangle between the positron track and the crystal axis. The measured energyspectrum is highly dependent on the angle and position at which decaypositrons enter BINA (see Chapter 6). The distance between the centreof WC3 and the positron track reconstructed using the S3WC3 tracker, isshown in Figure 5.12. This distance is referred to as R. For events withmultiple tracks, the track with the minimum distance from the centre istaken. The usual value of the cut is shown by the red line. The placementof this cut is a tradeoff between the increasing systematic error as the lowenergy tail of the pi+ → e+νe energy spectrum increases and the decreasingstatistical error as more events are included in the analysis.915.1. Event selectionFigure 5.12: The distance between the reconstructed positron track and thecentre of WC3 (R). The red line indicates the cut value.5.1.6 Minor cutsFour other cuts are applied, which affect a very small proportion of events,but are necessary either to remove rare processes or to ensure data integrity.The first category consists of a cut to remove events triggered by protonsemitted when the stopping pion undergoes nuclear reactions in the target,and a cut to remove events in which the pion stopped upstream of the targetand the decay muon completed the B1 B2 Tg coincidence. This is referredto as the false trigger cut. The cuts in the second category ensure that thetrigger is caused by the first hit in each T1 PMT, and that the triggeringhits in T1 and T2 are coincident within 20 ns (the coincidence window inthe trigger itself was 100 ns). The proton events are removed based on theenergy deposited in T1, T2, S3, and BINA; the events where the pion didnot stop in the target are removed based on the energy deposited in thetarget and the time difference between the hits in B1 and Tg. The timespectra after all cuts are shown in Figure 5.13, and Table 5.2 shows thefraction of events removed by each cut, with all other cuts applied.925.1. Event selectionFigure 5.13: Time spectra following all cuts. The left-hand plot shows thelow energy time spectrum and the right-hand plot shows the high energytime spectrum. The rise in the high energy spectrum near t = 0 at negativetimes is caused by the integration window of the calorimeter; the closer intime the pileup positron is to the positron from the pion at t = 0, the greaterthe probability that the measured energy in the event will be above Ecut.Table 5.2: Events removed by each cut, with every other cut applied.Cut name Low energy High energyPrompt 4.2% 42.2%B1B2 pileup 9.5% 89.2%WC12 4.7% 4.7%B1B2 energy 3.1% 5.5%Prepileup 27.6% 49.2%T1 pileup 0.03% 7.0%T2 pileup 1.5% 11.4%Muhit 0.6% 2%Acceptance 43.1% 34.7%Proton 0% 1.9%False trigger 0.03% 0.03%T1 trigger 0.1% 0.7%T1T2 coincidence 0.0% 0.8%935.2. Fitting function5.2 Fitting function5.2.1 Low energy time spectrumFollowing all of these cuts, only two backgrounds remain in the low energytime spectrum at a non-negligible level: old muon decays and pion decays-in-flight (pi-DIF). The time dependence of the former is an exponential withthe muon lifetime, starting at the beginning of the trigger window, -300 ns.pi-DIF events are only included in the fit if the decay muon stops before T1,which is typically only the case if the pion decays in flight within the target.If the pion decays before the target, the muon will pass through and stop inS3 or T1. If it stops in T1, the event is prompt, and thus outside the fittingrange. The only pi-DIF events included in the fit are those for which thereis a muon in the target or S3 at t = 0; in either case, the time dependenceof these events is also an exponential with the muon lifetime, but startingat t = 0.The pi+ → µ+ → e+νeν¯µ signal shape is the convolution of exponentialswith the pion lifetime and the muon lifetime. The fitting function usedin the low energy time spectrum is the sum of these three shapes, shownin equation 5.1. H is the Heaviside function, t0 is the offset in the timespectrum (determined from through-going particles), τµ and τpi are the muonand pion lifetimes, A is the amplitude of the pi+ → µ+ → e+νeν¯µ shape, Bis the amplitude of the pion decay-in-flight shape, and C is the amplitudeof the old muon background. The low energy time distribution is given asfollows:f(t) = H(t)[A1τµ − τpi(et−t0τµ − e t−t0τpi)+B1τµet−t0τµ]+ C1τµet−t0τµ . (5.1)5.2.2 High energy time spectrumThe high energy time spectrum is significantly more complicated, as sev-eral mechanisms can result in extra energy being deposited in the calorimeter(see Section 3.4). These are old muon pileup, which has its own time depen-dence; neutrons from the cyclotron, which are time independent relative tothe decay; energy resolution effects, which are also decay time independent;and radiative pion decay, which has its own time dependence. Furthermore,there are two separate mechanisms by which old muon pileup events can beincluded in the time spectrum. If both decay positrons hit T1, the event945.2. Fitting functionis rejected by the T1 pileup cut, unless they are sufficiently close in time,within about 15 ns, to be recorded as a single hit in T1. If the decaysare separated in time, one of the positrons must miss T1 for the event topass the cuts. Simplified timing diagrams for the various types of old muonevents can be found in Appendix B. The time dependence of each of theseprocesses will now be evaluated in turn.Time-independent addition of energyTime-independent mechanisms by which energy is added to events resultin the components of the low energy time spectrum being present in the highenergy time spectrum. The term included in the fit is thus Equation 5.1multiplied by a free parameter, called r.Old-muon pileup IAlthough there is a pileup cut in T1, if two positrons pass through itsufficiently close together in time, the waveforms will overlap, and only asingle hit will be recorded. The time spectrum for this component dependson whether the trigger was caused by the positron from the old muon or thepositron from the primary pion, since the latter can only occur at positivetime but the former can occur at any time. Let ∆T be the minimum timedifference for which T1 can resolve hits. The shape for the old-muon triggercase is then given by the product of the amplitude of the old-muon shapeand the probability that the pi+ → µ+ → e+νeν¯µ positron will emerge within∆T. This component is called T1res; the expression isf(t) =0 t < −∆Texp(− tτµ)τµ∫ t+∆T0exp(− yτµ)−exp(− yτpi)τµ−τpi dy −∆T < t < 0exp(− tτµ)τµ∫ t+∆Ttexp(− yτµ)−exp(− yτpi)τµ−τpi dy t > 0.(5.2)If the trigger is caused by the pi+ → µ+ → e+νeν¯µ positron, the shape isinstead given by the product of the pi+ → µ+ → e+νeν¯µ shape and theprobability that the old muon will decay within ∆T. This case is given inEquation 5.3:f(t) =0 t < 0exp(− tτµ )−exp(− tτpi )τµ−τpi∫ t+∆Ttexp(− yτµ)τµdy t > 0.(5.3)955.2. Fitting functionTo obtain ∆T, the time difference between subsequent T1 hits was plotted,and the edge fitted with an error function (see Equation 4.1). The fittedspectra for the T1 PMTs are shown in Figure 5.14; the average is ∆T =15.7 ns. The shape with this value of ∆T is shown in Figure 5.15.Figure 5.14: The time difference between subsequent hits in each T1 PMT;leading times are fitted with an error function. The peak around 30 ns isdue to a fake hit at a characteristic time after the real hit.Old-muon pileup IIOld-muon pileup events can also appear in the high energy time spectrumif one of the decay positrons misses T1, but still enters the crystal array;some solid angle exists for this to occur in either BINA or CsI (in BINAthis is largely possible because of the rotation of T1 by 45°about the beamaxis, which was necessary due to spatial constraints). The shape of thiscomponent is made more complicated by the BinaHigh trigger requirementin the high energy spectrum. A running sum of the BINA + CsI pulse heightwas used as an input for this trigger, which would only pass events above acertain threshold. The window in which this was done was 250 ns. However,the integration time used for the calibrated BINA energy was 1 µs, meaningthat if hits were sufficiently separated in time, the calibrated energy could be965.2. Fitting functionFigure 5.15: The shape used in the fit for pileup events that pass the T1pileup cut due to the double-pulse resolution of the T1 counter.above Ecut and a BinaHigh trigger would still not be present. Such eventsare excluded from the time spectrum, so this effect had to be taken intoaccount when determining the shape of this pileup mechanism.The shape was determined using Monte Carlo. Simulations were done ofevents caused by a pi+ → µ+ → e+νeν¯µ positron and of events caused by apositron from old muon decay. One event was then drawn at random fromeach simulation, to form one pileup event. If only one positron hit T1, theacceptance cut was passed, and the BinaHigh trigger requirement was met,the time of the event was included in the shape. The presence or absenceof the trigger was determined using BINA and CsI waveform templates inconjunction with the energy deposited in the simulation. This term is calledoldmu both; its shape is shown in Figure 5.16.Radiative decayIf the decay positron was produced in association with a photon via µ+ →e+νeνµγ the energy spectrum of the positron was altered, but the timedependence was not, and a separate shape is not required. If, however, the975.2. Fitting functionFigure 5.16: The shape used in the fit for pileup events where only onepositron hit T1.pion decayed radiatively to a muon, followed by µ+ → e+νeν¯µ decay, themeasured energy in the event could be above Ecut. The probability for thisto occur is dependent on the relative timing of the photon and the positron,again because of the BinaHigh trigger requirement. This shape was thereforealso taken from simulation; it is shown in Figure 5.17.The full high energy fitting consists of all of these shapes and the pi+ →e+νe signal shape, the amplitude of which is (A−A∗ r)∗Br, where A is thepi+ → µ+ → e+νeν¯µ amplitude, r is the proportion of the low energy timespectrum that is present in the high energy time spectrum, and Br is thebranching ratio. The fitting function is given in Equation 5.4. F1, F2, andF3 are the radiative decay shape and the two old muon pileup shapes, andL is the low energy fitting function, given in Equation 5.1. The function forthe high energy spectrum is given byf(t) = H(t)[(A−A ∗ r) ∗Br ∗ 1τpiet−t0τpi]+ r ∗ L+ F1 + F2 + F3. (5.4)985.3. Fitting methodFigure 5.17: The shape used in the fit for pi → µνγ events.5.3 Fitting methodIn all, the functions given in Equations 5.4 and 5.1 contain 12 parameters;their names, the symbols used in the fit, and a brief description of each, aregiven in Table 5.3. Of these parameters, five are typically fixed: t0, τµ, τpi,∆T (the T1 double-pulse resolution), and F2, the amplitude of the T1resshape.τµ, τpi, t0, and ∆T are all constants, and therefore fixing them in thefit is the natural approach, although the stability of the branching ratiowhen τµ and τpi are freed in the fit is an important systematic check. Theamplitude of T1res is fixed because of the similarity of its shape to F3, theoldmu both shape, which makes them difficult to fit simultaneously. It wasdecided to fix the first, rather than the second, because ∆T can be increasedarbitrarily by allowing events where successive hits in T1 are separated byless than some time interval. This causes the shape to change significantly,particularly at negative times, making it possible for the fit to determine thecorrect amplitude. As an example, the shape with ∆T = 100 ns is shown inFigure 5.18. The fit was done for several values of ∆T, and an extrapolation995.3. Fitting methodTable 5.3: Fit parameter list.ParameternameSymbol DescriptionA A The amplitude of the pi+ → µ+ → e+νeν¯µshape.r r The fraction of events promoted from thelow to high energy time spectrum via timeindependent mechanisms.t0 t0 The pion stop time.Mu Pimu B The amplitude of the old muon back-ground in the low-energy time spectrum.Mu Pie F4 The amplitude of the pure old-muon back-ground in the high energy time spectrum.piDIF C The amplitude of the pion decay-in-flightcomponent.BR Br The branching ratio: the pi+ → e+νeamplitude divided by the pi+ → µ+ →e+νeν¯µ amplitude.τµ τµ The lifetime of the muon.τpi τpi The lifetime of the pion.T1res F2 The amplitude of the old muon pileupbackground in the high energy time spec-trum, for the case where both decaypositrons hit T1 too close together in timeto resolve as separate hits.oldmu both F3 The amplitude of the old muon pileupbackground in the high energy time spec-trum, for the case where one of the decaypositrons missed T1.∆T ∆T The T1 double-pulse resolution.pimugamma F1 The amplitude of the shape for pi+ →µ+ → e+νeν¯µ events where the initial piondecay was radiative, and the photon en-ergy moved the measured energy in theevent above Ecut.1005.4. Resultswas performed to the true value, ∆T = 15.7 ns. The fitted amplitude as afunction of ∆T is shown in Figure 5.19, and the fit of the high energy timespectrum with ∆T = 100 ns is shown in Figure 5.20.Figure 5.18: The T1res shape when the double-pulse resolution is set to 100ns.One other modification had to be made in order to make the fit converge.In principle, the value of the parameter r, describing the fraction of eventswith low energy timing that are in the high energy time spectrum, shouldbe the same for negative and positive times. However, it was necessary tofree the old muon amplitude at negative times in the high energy spectrumto obtain a satisfactory fit; thus, another parameter F4 must be addedto Equation 5.4. The reasons for this are not currently understood. Theadditional parameter is referred to as Mu Pie.5.4 ResultsThe fitted 2012 time spectra are shown in Figure 5.21. The fit is performedfrom -290 to -20 ns, and 10 to 520 ns; the region -20 ns < t < 10 ns isexcluded.1015.4. ResultsFigure 5.19: The fitted T1res amplitude as the double-pulse resolution isincreased.The values of the fitting parameters and their errors are given in Table 5.4,along with the corresponding values for the 2010 data set. Values are givenboth with and without the T2 pileup cut (see Section 5.1.4). The total χ2 /d.o.f. of the fits is 1.39 with the T2 pileup cut and 1.23 without, in each caseconsiderably worse than the values for the 2010 and 2011 data sets. Theresiduals of the fits of the 2010 and 2011 data sets, and of the 2012 data setwith and without the T2 pileup cut, are shown in Figures 5.22, 5.23, 5.24,and 5.25.Examining the residuals, it is clear that the deterioration in the qualityof the fit is driven by the low energy t > 0 spectrum, where large, seeminglyperiodic, distortions can be seen. In principle, a small oscillating distortionof the time spectrum will not affect the branching ratio as long as events arenot added or removed, but only shifted. Since the shape of the spectrum isvery sensitive to both the pion and muon lifetimes, if these are fixed to thecorrect values and the integral of the residuals is consistent with zero, thebranching ratio should be correct. The result obtained for the integral of theresiduals was 551, 831 ± 223, 327, which is approximately 2.5 σ away fromzero. For reference, the number of events in the low energy t > 0 spectrum is3.269×109. So, although there may be evidence for a slight excess of eventsin the low energy time spectrum, it is on the level of 1.5× 10−4, which will1025.4. ResultsFigure 5.20: The fit of the high-energy time spectrum with ∆T increased to100 ns. The red shape is the pi+ → e+νe signal, the blue is the pi+ → µ+ →e+νeν¯µ background, and the green is the sum of the other backgrounds.1035.4. ResultsFigure 5.21: The fitted time spectra from 2012. The left-hand panel showsthe low energy time spectrum, fitted with three components: the pi+ →µ+ → e+νeν¯µ signal shape, old muon decays, and pion decays-in-flight.The right-hand panel shows the high energy time spectrum, fitted with sixcomponents: the pi+ → e+νe signal shape, pi+ → µ+ → e+νeν¯µ and piondecay-in-flight events promoted to the high energy time spectrum via timeindependent mechanisms, two mechanisms of old muon pileup, pi → µνγdecays, and old muon decays.1045.4. ResultsFigure 5.22: The residuals of the 2010 data set vs. the time of the event.Clockwise from top left, the panels show the residuals for the high energyt < 0 spectrum, the high energy t > 0 spectrum, the low energy t > 0spectrum, and the low energy t < 0 spectrum.1055.4. ResultsFigure 5.23: The residuals of the 2011 data set vs. the time of the event.Clockwise from top left, the panels show the residuals for the high energyt < 0 spectrum, the high energy t > 0 spectrum, the low energy t > 0spectrum, and the low energy t < 0 spectrum.1065.4. ResultsFigure 5.24: The residuals of the 2012 data set vs. the time of the event,without the T2 pileup cut applied. Clockwise from top left, the panelsshow the residuals for the high energy t < 0 spectrum, the high energy t> 0 spectrum, the low energy t > 0 spectrum, and the low energy t < 0spectrum.1075.4. ResultsFigure 5.25: The residuals of the 2012 data set vs. the time of the event,with the T2 pileup cut applied. Clockwise from top left, the panels showthe residuals for the high energy t < 0 spectrum, the high energy t >0 spectrum, the low energy t > 0 spectrum, and the low energy t < 0spectrum.1085.5. Systematic checksTable 5.4: Fit parameters for the 2010 and 2012 data sets. Note that the2012 branching ratios are still blinded by an unknown factor uniformly dis-tributed between ±0.5%.Parameter 2010 value 2012 value (T2 cut) 2012 value (no T2 pileup cut)A (109) 3.2903± 0.0010 16.227± 0.002 16.449± 0.002r (10−4) 2.458± 0.0047 2.94± 0.02 3.40± 0.02Mu Pimu (107) 1.533± 0.004 7.65± 0.01 8.03± 0.01Mu Pie (103) 7.18± 0.65 44.7± 1.7 38.8± 5.5piDIF(107) 3.821± 0.067 25.3± 0.2 27.6± 0.2BR (10−4) 1.1971± 0.0022 1.2005± 0.0011 1.2027± 0.0011Oldmu both (104) 2.93± 0.39 8.92± 0.4 18.1± 0.5not be negligible when the error on the branching ratio reaches the 0.1%level, but will not contribute greatly to the systematic error, since it woulddecrease the branching ratio by < 0.02%.5.5 Systematic checksSeveral different tests were performed to ensure the stability of the fit:the fitting range was changed, the bin size was changed from the usual valueof 2 ns, τµ and τpi were made free parameters, a flat component was added,and the amplitudes of the fixed shapes were changed. A summary of thesesystematic checks is given in Table 5.5. By far the largest variation ariseswhen the lifetimes are freed, if the T2 pileup cut is applied.Another important diagnostic, particularly with regard to the fitting func-tion, is the stability of the raw branching ratio when more pileup events areallowed into the time spectra. The prepileup cut normally rejects eventswith hits in any of the plastic scintillators in a window 6.6 µs to 2.2 µsbefore the pion stop (see Section 5.1.4). The fit was done for several differ-ent prepileup rejection windows in B1, B2, and Tg, from the full cut to noprepileup cut. The branching ratio as a function of the left-hand edge of theprepileup rejection window, before and after applying the T2 pileup cut, isshown in Figures 5.26 and 5.27, respectively.Finally, the stability of the branching ratio as a function of time wastested. The fit was performed for groups of 1000 runs, and the result fitted1095.5. Systematic checksTable 5.5: Systematic checks performed on the fit of the 2012 data.Test ∆BR (10−8) ∆BRerror (10−8) T2pileupFitting ranget < 420 0.1 0.0 Not < 420 2.0 0.1 Yes12.5 < t -3.7 0.5 No12.5 < t -8.4 0.5 Yest < 7.5 -2.6 -0.5 Not < 7.5 2.3 -0.4 Yest < −30 0.0 0.0 Not < −30 -0.8 0.0 Yes−250 < t 0.0 0.0 No−250 < t 0.2 0.0 YesBin sizeBin = 1 ns -2.4 0.3 NoBin = 1 ns -4.5 0.2 YesBin = 4 ns 3.0 0.0 NoBin = 4 ns 1.4 0.0 YesLifetimeτµ and τpi free -6.1 6.8 Noτµ and τpi free 43.9 6.8 YesAdditional backgroundFlat component 1.4 0.0 NoFlat component 2.6 0.0 YesFixed parameter±20% pimugamma ±2.6 0.0 No±20% pimugamma ±2.6 0.0 Yes1105.5. Systematic checksFigure 5.26: The variation of the branching ratio as more pileup events areallowed in the time spectrum, prior to applying the T2 pileup cut. The xaxis denotes the time prior to the pion stop in which events with hits inB1, B2, and Tg were rejected. The error bar on the point furthest to theleft (with the least pileup) is the error from the time spectrum fit, and theerror bars on the other points are the error on the change from the previouspoint. The points are fitted to a parabola.1115.5. Systematic checksFigure 5.27: The variation of the branching ratio as more pileup events areallowed in the time spectrum, after applying the T2 pileup cut. The x axisdenotes the time prior to the pion stop in which events with hits in B1, B2,and Tg were rejected. The error bars represent the statistical variation fromthe point furthest to the left (with the least pileup).1125.6. Summary of Chapter 5to a straight line, as shown in Figure 5.28. The p-value of the fit is 40.7%,meaning there is no evidence for time dependence.Figure 5.28: The fitted branching ratio for groups of 1000 runs, fitted witha flat line.5.6 Summary of Chapter 5In the absence of the T2 pileup cut, the branching ratio varies considerablyas pileup is added to the spectrum, indicating the presence of an incorrectshape or missing component in the time spectrum fit. However, the impacton the branching ratio becomes almost negligible as the cut approachesits nominal value. The fit shown in Figure 5.26 was extrapolated to itsminimum; the corresponding branching ratio was within 2 × 10−8 of thelowest point on the graph. Variations at this level are negligible for thepresent analysis; see Chapter 8.The other large variation in the branching ratio occurs when the T2 pileupcut is applied and τµ and τpi are made free parameters in the fit. It is possiblethat the value of the branching ratio when the lifetimes are freed is closer tothe correct value; the change in the lifetimes may compensate for whateverdistortion exists in the time spectrum. However, the lifetimes are highlycorrelated with the branching ratio, and, particularly in the presence ofdistortions, spurious minima may exist in the χ2 landscape. In the absenceof a reason to prefer one value of the branching ratio over the other, the1135.6. Summary of Chapter 5average is taken, and half the variation assigned as the 1σ error; in this caseσ = 21.5×10−8. This is to be compared with a statistical error of 11×10−8.Several corrections must be applied to the raw branching ratio in orderto obtain the final result. These are the subjects of the next two chapters;the final systematic error will be given in Chapter 8.114Chapter 6Tail Correction6.1 IntroductionBy far the largest correction to the branching ratio arises from the tailof the measured pi+ → e+νe energy spectrum below the cutoff value Ecutbetween the low and high energy time spectra. The standard condition forthe analysis is Ecut = 52 MeV. Let the measured pi+ → e+νe energy spectrumbe denoted by N(E). The tail fraction T is defined as the proportion of thisspectrum below the cutoff; that is,T =∫ Ecut0 N(E)dE∫∞0 N(E)dE. (6.1)The size of this tail will be shown to be approximately three percent. Theamplitude of the pi+ → e+νe component in the low energy time spectrum isequal to the amplitude of the pi+ → µ+ → e+νeν¯µ component, multipliedby T and the branching ratio. This is too small to fit with the PIENU dataset; neglecting other corrections, the raw branching ratio obtained from thefit is thus related to the true branching ratio by(1− T )BRtrue = BRraw. (6.2)Since this component cannot be fitted, the response of the crystal calorimeterto a 70 MeV positron beam had to be determined independently, and theraw branching ratio multiplied directly by 11−T . If ∆T is the uncertaintyon T, this results in an uncertainty on the branching ratio of ∆T1−T ∗ BR; inorder to meet the precision goal of the experiment, ∆BRBR < 0.1%, ∆T mustitself be substantially less than 0.1%.Two fundamentally different approaches were used to determine T . Onewas to inject a positron beam with almost the same energy as pi+ → e+νedecay positrons, about 70 MeV, into BINA at several angles and directlymeasure the proportion of the spectrum below Ecut. The other was to, as far1156.2. Lower limitas possible, suppress pi+ → µ+ → e+νeν¯µ events in the data, and attemptto deduce T from the energy spectrum itself, using the known shape of theMichel spectrum.The first of these techniques must be regarded as giving an upper limit onT , because of the potential for the positrons to scatter in the beamline, givinga low momentum tail to the beam itself. The second uses the approximationthat the pi+ → e+νe tail is zero at very low energies, which leads to aslight over-subtraction of pi+ → µ+ → e+νeν¯µ events from the measuredenergy spectrum. This results in an underestimation of T , so this techniqueis regarded as giving a lower limit. The upper and lower limits are thencombined to give the best estimate of T . For a detailed description of themethod for obtaining the lower limit, see [65].6.2 Lower limitThe first step in the lower limit procedure was to obtain an energy spec-trum with as many pi+ → e+νe events and as few pi+ → µ+ → e+νeν¯µevents as possible remaining, and other backgrounds suppressed as far aspossible without distorting the energy spectrum for pi+ → e+νe positrons.Five additional cuts were imposed to accomplish this: a timing cut, a cut onthe energy deposited in all detectors up to and including the target, a cuton the angle between the tracks reconstructed by WC12 and S12 (to removepion decays-in-flight), a cut on the energy deposited in S3, and a cut on theshape of the Tg waveform.The timing cut removed any event outside the time window 7-42 ns; thiswindow contains 80% of pi+ → e+νe events, and less than 2% of pi+ → µ+ →e+νeν¯µ events. The energy loss cut selected events depositing between 15.5and 16.5 MeV in total in B1, B2, S1, S2, and Tg. Simulations of the energydeposited by the four relevant processes are shown in Figure 6.1: pi+ → e+νedecay-at-rest (pi-DAR), pi+ → µ+ → e+νeν¯µ decay-at-rest (pi-DARµ-DAR),pi decay-at-rest followed by µ decay-in-flight (pi-DARµ-DIF), and pi decay-in-flight followed by µ decay-at-rest (pi-DIFµ-DAR).The energy loss cut was very effective in removing pi+ → µ+ → e+νeν¯µevents where both particles decayed at rest. However, contamination re-mained from events where either particle decayed in flight. To suppresspi-DIF events, a cut was done on the angle between the tracks reconstructed1166.2. Lower limit[MeV]totE12 14 16 18 20 22 24Normalized counts00.20.40.60.81 ei+eA+/DARµDAR-/DARµDIF-/DIFµFigure 6.1: The total energy deposited in B1, B2, S1, S2, and Tg by pi+ →e+νe (black), pi+ → µ+ → e+νeν¯µ (red), piDIF-µDAR (green), and piDAR-µDIF (blue). The distributions are normalized to the same height. Thesolid red lines indicate the selected region.1176.2. Lower limitby WC12 and S12. The distributions in the two cases are shown in Fig-ure 6.2.Figure 6.2: The measured angle between the tracks reconstructed by WC12and S12 for pi+ → e+νe (black) and pi+ → µ+ → e+νeν¯µ (red) eventsafter the time and energy loss cuts are applied. The double peak in thepi+ → µ+ → e+νeν¯µ distribution is caused by pion decays-in-flight. Thesolid red line and arrow indicate the selected region.pi-DIF events were further suppressed using the energy deposited in S3.About 30% of muons arising from pi-DIF stopped in S3 rather than thetarget, resulting in larger energy deposit than was typical for positrons.Finally, the target waveform was fitted with a template containing two peaksand a template containing three peaks, corresponding to pi+ → e+νe andpi+ → µ+ → e+νeν¯µ events, respectively. Events were removed if the χ2of the three-pulse fit was better than the two-pulse fit. This was done toensure as few as possible pi+ → µ+ → e+νeν¯µ events remained, but afterall the other suppression cuts the effect was almost negligible. The effectof each cut in turn upon the energy spectrum is shown in Figure 6.3; the1186.2. Lower limitspectrum after all cuts is referred to as the suppressed spectrum.[MeV]NaI+CsIE0 10 20 30 40 50 60 70 80 90 100Counts110210310410510Time 0.9916Energy loss 0.2947Kink 0.1542S3 0.1457Pulse fit 0.1441Figure 6.3: The measured BINA+CsI spectra as the suppression cuts areapplied. The legend indicates the fraction of low-energy events (< 52 MeV)remaining after each cut.There are expected to be three components below 52 MeV in Figure 6.3:the pi+ → e+νe tail, µ-DIF events, and pi-DIFµ-DAR events. The last haveessentially the same energy spectrum as normal pi+ → µ+ → e+νeν¯µ events,whereas µ-DIF events are somewhat higher energy. The energy spectrumfor µ-DIF events was obtained from Monte Carlo. The energy spectrum forpi-DIFµ-DAR events was taken from data, assuming the same distributionas pi+ → µ+ → e+νeν¯µ events. These spectra were then combined to givethe total background spectrum. The number of background events wasestimated by extrapolating from the number of events in the suppressedspectrum at low energy, where the pi+ → e+νe tail is very small. Thisestimate of the number of background events was then subtracted from thetotal number of events below 52 MeV in the suppressed spectrum, and theremainder was taken as the lower limit on the pi+ → e+νe tail.1196.3. Response function measurementA further correction had to be applied because of the cut on the totalenergy deposited in the detectors up to the target, which tended to removepi+ → e+νe events that underwent Bhabha (electron-positron) scattering.This correction was taken from Monte Carlo. The final result for the lowerlimit was TLL = 2.95% ± 0.07%(stat) ± 0.08%(syst) for the 2010 data andTLL = 3.22% ± 0.09%(stat) ± 0.05%(syst) for the 2011 data. However, the2011 lower limit analysis was done with a modified procedure where part ofthe pi+ → e+νe energy spectrum was taken from Monte Carlo and used in afit of the measured energy spectrum. In order to ensure the crystal responseis determined entirely from data, the 2010 value was used in combinationwith the upper limit for this analysis. Analysis of the 2012 suppressedspectrum is underway.6.3 Response function measurement6.3.1 Energy loss processesTo obtain the upper limit on the pi+ → e+νe tail, a positron beam was in-jected into the crystal array. The PIENU beamline was designed to be ableto deliver either a pion, muon, or positron beam, through the adjustment ofthe absorber and collimator configuration and the fields of the bending mag-nets. Special data were taken in 2009 and again in 2011 with the beamlinetuned for positrons at 70 MeV. The 2011 data were of higher quality, and areall that will be discussed here. There were two primary contributions to thelow energy tail in the measured pi+ → e+νe energy spectrum: electromag-netic shower leakage and energy loss upstream of the calorimeter. Anothersmall contribution arose from photonuclear interactions within BINA [9].Electromagnetic showersWhen a 70 MeV positron enters a material, in this case a NaI(Tl) crys-tal, it immediately begins to lose energy via bremsstrahlung and ionization.The bremsstrahlung photons in turn undergo pair production, resulting inelectron-positron pairs that themselves undergo bremsstrahlung, ionization,and, in the case of positrons, annihilation. This process, referred to as anelectromagnetic shower, continues until the initial positron runs out of en-ergy and annihilates. Eventually, the energy thus absorbed by the crystal isconverted into scintillation light, which can be read out by PMTs.1206.3. Response function measurementFor any detector of finite size, some of the shower will occasionally escape,resulting in less measured light. If enough energy escaped from BINA, themeasured energy could fall below Ecut, putting the event in the tail. Becausethe shower mainly proceeded via bremsstrahlung and pair production, itwas highly forward-peaked, so the amount of material along the path of theinitial positron essentially determined the probability that less than the fullpositron energy would be absorbed. Thus, it was essential to measure theresponse of the crystals as a function of the angle and entrance positionof the positron on the front face of BINA. Because the orientation of thebeamline was fixed, the crystal array was rotated relative to the beam.Upstream interactionsDuring normal PIENU data-taking, the pions stopped near the centre ofthe target, approximately 8 cm upstream from the front face of BINA. Ifthe pion underwent pi+ → e+νe decay, a positron with kinetic energy of69.3 MeV was emitted isotropically from the centre of the target. Positronsfurther than 6 cm away from the centre of WC3, 5.5 cm downstream ofthe centre of the target, were rejected in the analysis; this correspondedto an angle of slightly more than 45° relative to the axis of BINA. Prior toreaching BINA, the positrons traversed half of Tg, and all of S3, T1, andT2. For a positron travelling on-axis, this corresponded to about 1.4 cm ofplastic scintillator and 0.6 mm of silicon, or about 3 MeV of energy loss onaverage. However, sometimes significantly more energy was lost, raising theprobability that the measured energy would fall below Ecut.The effect of the material in the path of the pi+ → e+νe positrons couldnot be replicated using the positron beam. Even if the correct amount ofmaterial were placed in the path of the beam at each angle, which due tomechanical constraints was not straightforward, one significant differencewould still remain. pi+ → e+νe positrons emitted at high angles relativeto the axis of BINA would not normally trigger; however, a non-negligibleproportion of such positrons underwent Bhabha scattering while exiting thetarget. The scattered, generally low energy, electron could then trigger andenter the calorimeter, producing an event with low measured energy andpi+ → e+νe time dependence. If the same thing happened to a pi+ → µ+ →e+νeν¯µ event, it was included in the branching ratio, so pi+ → e+νe eventsthat underwent this process had to be included as well.1216.3. Response function measurementThe effect of high angle positrons scattering electrons into the detectorcould not be replicated using the positron beam. However, the interactionsof positrons in matter are well-understood, and can be precisely calculatedusing GEANT4. The electromagnetic shower inside BINA, on the otherhand, was sensitive to its real properties, such as defects or inhomogeneities.Thus, the decision was made to remove as much material as possible infront of BINA in order to obtain as accurate a measurement as possibleof the response function, and rely on MC for the contribution to the tailof energy loss upstream of BINA. This meant the value of Ecut for theresponse function measurement was effectively different from its value forthe pi+ → e+νe case, as the peak of the energy spectrum was shifted.Photonuclear effectsWhen the PIENU detector was initially being commissioned and char-acterized, the response of BINA to a positron beam parallel to the crystalaxis was measured. In addition to the expected peak around 70 MeV, theinitial positron energy, with a low energy tail, additional small peaks wereseen at 58 and 50.5 MeV. After ruling out explanations such as scattering inthe beamline, the experimental configuration was simulated using GEANT4,and peaks were seen at similar energies, but only if photonuclear interactionswere included in the simulation. By examining the events in these peaks,it was determined that they were caused by either one or two neutrons be-ing emitted from iodine and escaping BINA. Although GEANT4 did notprecisely reproduce the shape of these peaks, it confirmed their existence,and the mechanism of their creation. The energy spectrum obtained duringcommissioning, and simulations both including and not including hadronicinteractions, are shown in Figure 6.4 [9]. The fact that the second peakwas near the nominal value of Ecut, and that the MC did not accuratelyreproduce the size of the peak, was an additional reason to rely on data todetermine the true crystal response.6.3.2 Detector setupThe PIENU-I subassembly, comprising B1, B2, S1, S2, Tg, S3, and T1,was removed from the detector, leaving only the wire chambers, T2, and thecalorimeter. This reduced the momentum and position divergence of thebeam, allowing for a more accurate measurement of the crystal response.Finally, the PIENU-II subassembly (WC3, T2, and the crystals) was re-moved from the rails to which it was normally affixed so that it could be1226.3. Response function measurementFigure 6.4: Comparison between measured (filled circles with error bars)and simulated energy spectra. The simulation was performed with (red)and without (blue) hadronic reaction contributions. The histograms arenormalized to the same area [9].1236.4. Data-takingrotated with respect to the beam.The amount of material in the path of the beam depended not only onthe angle θ between the beam and the crystal axis, but also on the centreof rotation. In the pi+ → e+νe case, the centre of rotation of the positrontracks was the centre of the target. In order to mimic this configurationusing a straight beam and rotated detector, the centre of rotation had to bethe same distance from the front face of BINA as the centre of the target.To this end, a shaft was attached to the cart on which PIENU-II sat, at thesame distance in z from the front face of BINA as the centre of the target.6.4 Data-takingThe angles at which data were taken, and the number of good positron eventsrecorded at each angle, are shown in Table 6.1. To accurately determine theangle, markers were placed along the beamline and along a bar parallel tothe crystal axis. A theodolite was then used to measure the position ofthese markers each time the detector was rotated. The accuracy of thetheodolite was found to be 0.25 mm over the distance used; the targets were30 cm apart, resulting in an error in the angle measurement of less than 0.1°.This was the primary improvement made during 2011 data-taking relativeto 2009.Table 6.1: Angles at which positron beam data were taken.Angle (◦) # of Events (106)0.0 5.9736.0 11.6511.8 6.50916.5 7.25320.9 6.53224.4 6.27830.8 5.77936.2 5.86141.6 6.42947.7 9.0611246.4. Data-taking6.4.1 Event selection cutsSince T2 was the only plastic scintillator present for these data, it was theonly detector required in the trigger. To help reduce background, approx-imately half the 43 ns cyclotron RF period was vetoed in the trigger; thisregion contained almost all the pions in the beam, and very few positrons.Due to the looseness of this requirement, many types of event besides beampositrons could cause triggers. The energy spectrum in BINA + CsI priorto any cuts is shown in Figure 6.5. The data shown were taken with thebeam aligned with the crystal axis, that is, at 0°.Figure 6.5: Beam particle energy measured by the crystals at 0°with no cutsapplied.The spectrum contains many backgrounds that can be removed with sim-ple cuts. The low energy peak contains events without hits in the beam wirechambers in the region corresponding to the beam spot, suggesting they arenot due to beam particles; they are possibly due to muons decaying back-wards out of BINA. The peaks at 14 MeV and 18 MeV are due to beampions and muons, respectively. The slight ridge around 30 MeV appears tobe caused by pions decaying in flight; this was determined by Monte Carlo.The slight peak at 50 MeV and the clear shoulder near 60 MeV are the1256.4. Data-takingphotonuclear peaks. The main peak just below 70 MeV is due to beampositrons. The peak immediately to the right of the main peak is due to apositron and pion arriving simultaneously, and the shoulder to the right ofthat peak is due to a positron and muon arriving simultaneously. This wasdetermined by the correspondence of the peaks to the sum of the individualparticle peaks, and by the energy deposited in T2. The pion peak is largerthan the muon peak because pions did not generally trigger, as they arrivedin the region vetoed by the RF. Muons, however, overlapped with positronsin RF time. The peak around 130 MeV is due to events with two positrons.The beam wire chambers, WC1 and WC2, were used to remove mostevents not due to beam particles. The incoming positron track was recon-structed and events outside the beam spot were removed. The backgroundwas further reduced by eliminating events with out-of-time hits in the wirechambers, and selecting the RF window for positrons.Following these cuts, the spectrum contained events due to beam positronsand beam muons, which were reduced but not entirely eliminated by theRF cut. It was possible to remove them entirely with a cut on the energydeposited in T2; however, many positrons were removed as well. Any cutremoving positrons in a way that depends on the energy deposited in thecrystals could bias the response function measurement. In the absence ofleakage from the crystals, the initial energy of the positron is equal to theenergy deposited in T2 plus the energy deposited in BINA. In addition,GEANT4 simulation revealed that a significant fraction of events with highenergy deposit in T2 were due to shower leakage backwards out of BINA.More detail is provided in Appendix C. The energy spectrum following thewire chamber position cut, the wire chamber timing cut, and the RF cut isshown in Figure 6.6.6.4.2 Muon correctionAlthough the dependence of the response function on the T2 energy cutcould perhaps be taken into account by Monte Carlo, it would be safer notto use any information so closely related to the BINA response. Thus, analternative means to remove the muon contamination from the energy spec-trum was developed. The energy spectrum in the RF window correspondingto muons is shown in Figure 6.7.1266.4. Data-takingFigure 6.6: Energy measured by the crystals at 0° following event selectioncuts (see text).Figure 6.7: Energy measured by the crystals at 0° in the RF window corre-sponding to muons.1276.4. Data-takingThis spectrum is almost devoid of positrons, by the absence of any visiblepeak around 70 MeV. The large, almost flat distribution between 20 and70 MeV arises from the long integration time of the BINA pulse, which isabout 1 µs. There was a significant probability that the muon would decayin this time, resulting in all or part of the decay positron energy being addedto the muon energy. The cut-off of the energy spectrum corresponds to theenergy of a beam muon plus the highest energy Michel positron.The fraction of positrons below the muon peak is very small, on the orderof 10−5 (see Appendix C, Figures C.4 and C.7). The BINA + CsI spectrumobtained by selecting muons by RF time can be used to determine the ratioof muon events in the peak to the total number of muon events. Sincethere are essentially no positron events in the energy region containing themuon peak, the total number of muons in the spectrum can be accuratelydetermined from the size of the muon peak. This allows the muon componentto be subtracted from the energy spectrum, giving the tail due to positronsonly. Figure 6.8 shows the muon energy spectrum zoomed in on the peak,with red lines at 15 and 19 MeV defining the region considered as the peak.Due to energy loss in T2, these values change as a function of the anglebetween the beam and the axis of BINA.Figure 6.8: Energy measured by the crystals at 0°. The red lines indicatethe region defined as the muon peak.1286.4. Data-takingTo correct Equation 6.1, both the number of muons in the tail and thetotal number of muons in the spectrum must be known; these quantities arelinearly related to the number of events in the muon peak. From the BINA+ CsI spectrum containing only muons, the scaling factor can be obtained.Let NµRF (E) denote the energy spectrum in the RF window correspondingto muons, Nµpeak be the number of events in the peak (defined by the redlines in Figure 6.8), Nµtail be the number of events in the tail (below Ecut),and Nµtotal be the total number of muon events in the spectrum. Then,Nµpeak =∫ 19MeV15MeVNµRF (E)dE, (6.3)Nµtail =∫ Ecut0MeVNµRF (E)dE, (6.4)andNµtotal =∫ ∞0NµRF (E)dE. (6.5)Now considering the energy spectrum N(E) with the RF cut correspondingto positrons, Figure 6.6, let Npeak be the number of events in the muon peak,Ntail be the number of events in the tail, and Ntotal be the total number ofevents in the spectrum; i.e.,Npeak =∫ 19MeV15MeVN(E)dE, (6.6)Ntail =∫ Ecut0MeVN(E)dE, (6.7)andNtotal =∫ ∞0N(E)dE. (6.8)Let C1 denote the number of muons present in N(E) and C2 denote thenumber of muons present below Ecut in N(E); these are given byC1 = Npeak ×NµtotalNµpeak(6.9)andC2 = Npeak ×NµtailNµpeak. (6.10)1296.4. Data-takingThe formula for obtaining the tail fraction following the muon correction,Tcorr, is thenTcorr =Ntail − C2Ntotal − C1 . (6.11)This formula is valid so long as the shape of the muon energy spectrum inN(E) is identical to the shape of the muon energy spectrum in NµRF (E).The validity of this assumption can be determined by varying the RF win-dow used for N(E), from selecting as few muons as possible (Figure 6.6)to including the entire muon region in N(E). The stability of Tcorr as theRF window is varied can then be used to estimate the uncertainty of thiscorrection. Table 6.2 shows the ratios of Nµpeak to Nµtail and Nµpeak to Nµtotalfor each angle when the RF window corresponding to muons is selected.Table 6.2: Properties of the muon energy spectra as a function of angle.Angle (◦) µ RF window (ns) µ peak (MeV) Peak / tail Peak / total0.0 13-16 15-19 0.734 0.6886.0 12-15 15-19 0.731 0.68611.8 12-15 15-19 0.731 0.68616.5 11-14 15-19 0.730 0.68320.9 11-14 15-19 0.730 0.68824.4 11-14 15-19 0.728 0.68430.8 10-13 14-18 0.721 0.67336.2 13-16 14-18 0.710 0.67841.6 13-16 13-17 0.711 0.68647.7 13-16 12-16 0.670 0.660These numbers, along with the total number of muons in the peak, canthen be used in Equation 6.11 to calculate the tail fraction. Tables 6.3to 6.12 show the results for the tail fraction after the muon correction, foreach angle, as the RF window is broadened to include more muons.Considering the size of the correction, the tail fraction is quite stable as theRF window is broadened. The largest relative variation was at 36.2°, wherethe spread was slightly under 2% of the tail at that angle. At each angle,the tail fraction was taken to be the average of the highest and lowest valuesin the tables shown above, and the systematic error due to the correction1306.4. Data-takingTable 6.3: ∆T at 0°.RF window Tail fraction (%)5-12 0.5905-13 0.5815-14 0.5775-15 0.5845-16 0.590Table 6.4: ∆T at 11.8°.RF window Tail fraction (%)3-11 0.6303-12 0.6313-13 0.6323-14 0.6283-15 0.630Table 6.5: ∆T at 20.9°.RF window Tail fraction (%)3-10 0.7763-11 0.7733-12 0.7703-13 0.7763-14 0.776Table 6.6: ∆T at 30.8°.RF window Tail fraction (%)2-9 1.1892-10 1.1992-11 1.1892-12 1.1862-13 1.189Table 6.7: ∆T at 41.6°.RF window Tail fraction (%)6-12 1.9876-13 1.9846-14 1.9506-15 1.9676-16 1.987Table 6.8: ∆T at 6°.RF window Tail fraction (%)3-11 0.6023-12 0.6033-13 0.6013-14 0.6013-15 0.602Table 6.9: ∆T at 16.5°.RF window Tail fraction (%)1-10 0.6741-11 0.6801-12 0.6751-13 0.6731-14 0.674Table 6.10: ∆T at 24.4°.RF window Tail fraction (%)3-10 0.8743-11 0.8843-12 0.8823-13 0.8733-14 0.874Table 6.11: ∆T at 36.2°.RF window Tail fraction (%)6-12 1.3866-13 1.3786-14 1.3426-15 1.3656-16 1.386Table 6.12: ∆T at 47.7°.RF window Tail fraction (%)6-12 3.3166-13 3.3136-14 3.2586-15 3.2786-16 3.3161316.5. Other systematicswas assigned to be the variation from the average. The tail fractions withstatistical error and this source of systematic error are shown in Table 6.13.Note that the errors are given as percentages of the total number of events;i.e., they are the absolute errors on the tail fraction, not the relative errors.Table 6.13: The tail fraction as a function of angle, with errors due tostatistics and the variation in the muon-corrected values.Angle (◦) Tail fraction (%) Statistical error (%) Muon correction error (%)0.0 0.583 0.008 0.0066.0 0.603 0.004 0.00111.8 0.630 0.012 0.00216.5 0.677 0.012 0.00420.9 0.773 0.009 0.00324.4 0.879 0.006 0.00530.8 1.193 0.005 0.00636.2 1.364 0.005 0.02241.6 1.969 0.006 0.01847.7 3.287 0.006 0.0296.5 Other systematicsMany systematic effects besides the muon correction could potentiallycontribute to the error on the tail fraction. These come in two varieties:actual distortions in the energy spectra measured with the positron beam,and uncertainties in experimental parameters that must be known in orderto translate the tail fractions in the positron beam data into the predictedtail fraction for the pi+ → e+νe energy spectrum. The latter can be evaluatedonly through simulation, and will be addressed in Section 6.6.Since none of the cuts make any reference to what is measured by thecrystals, they should leave the energy spectrum for good positron eventsundistorted. Thus, the only errors on the measured tail fraction per se arisefrom the presence of backgrounds and uncertainty on the measured energyitself (i.e. the BINA and CsI resolution and calibration).1326.5. Other systematics6.5.1 BackgroundThe level of pileup background can be obtained from Figure 6.6. Theevents to the right of the main peak were caused by a normal positron eventhappening at the same time as a Michel positron entered BINA. This isclear from the fact that the endpoint of the spectrum is at approximately120 MeV, which is very close to the sum of the main peak and the highest-energy Michel positron. This could affect the tail fraction only in the casewhere an event that would have been in the <52 MeV region ended upabove 52 MeV due to the presence of the extra particle. Events where aMichel positron made the trigger were almost completely removed by thecut requiring hits consistent in time in all three wire chambers. The fractionof events to the right of the main peak is approximately 0.2% of the mainpeak itself; therefore, since the tail itself will be seen to be approximately3%, the impact on the tail will be less than 0.01%, which is negligible forthe purposes of this analysis.Other backgrounds could only be significant if they added events to thetail, since they would have to be present at a much higher level to be relevantin the rest of the spectrum. If muons are suppressed either by a T2 energycut or a tighter RF cut, the very low energy part of the zero-degree spectrumshows that such backgrounds, if present at all, must be present only at anegligible level. The fraction of events under 10 MeV at 0 degrees, forexample, is less than 10−5. Thus, a flat background would contribute at alevel of much less than 10−4, and indeed the only type of background thatcould be present would have a very similar shape to the positron energyspectrum, which is implausible.The possibility of scattering in the beamline leading to a low momentumtail in the beam momentum spread is taken into account by regarding thisanalysis as providing an upper limit on the tail due to the response functionof the calorimeter.6.5.2 Calibration and resolutionUsing the normal data, the BINA calibration has been established to beaccurate within 0.1 MeV (see Section 4.4). This uncertainty is taken intoaccount by varying the value of Ecut by ±0.1 MeV. The energy resolutioncould only affect the tail fraction insofar as it could move events belowEcut above it, and vice versa, so the actual error arises from the difference1336.6. Positron beam simulationbetween events moved into the tail region and those moved out of the tailregion. Fitting the 0° peak with a Gaussian gives σ = 0.6 MeV, whichincludes the beam momentum spread. Even if this were entirely due to theenergy resolution, its effect on the tail fraction could be neglected, sincethe energy range in which this can happen is only about 1 MeV. This wasverified by adding Gaussian resolution to the simulated BINA energy; nosignificant effect on the tail fraction was observed.6.6 Positron beam simulationSince there are contributions to the pi+ → e+νe tail that were not presentin the positron beam data, namely interactions upstream of the crystals andradiative decay, the ideal outcome of the response function analysis wouldbe the development of a simulation that could reproduce the data at eachangle, which could then be used to simulate pi+ → e+νe decay. Also, severalsources of systematic uncertainty, such as the characteristics of the beamand the detector geometry, could only be assessed via simulation. Thus, thenormal PIENU Monte Carlo was modified to match the conditions of thepositron data-taking. This required the removal of all detectors from thesimulation except for the three wire chambers, T2, and the crystals, and thereplacement of the pion beam with a positron beam.The beam parameters for the 0° data, reconstructed by the WC12 tracker,are shown in Figure 6.9; the quantities plotted are x, y, tx, and ty (seeSection 4.5). x and y are the positions at z = 0, and tx and ty are the ratiosof the x and y momenta to the z momentum. As is done for the normal pionbeam, the correlations between these quantities were determined, and theCholesky decomposition of the correlation matrix was multiplied by a vectorof these values sampled independently, thereby reproducing the correlations[66].The beam parameters changed considerably over the running period; themean of the tx distribution for the data set taken at each angle is shown inFigure 6.10. Although the ideal procedure would be to use a separate beamat each angle in the simulation, the tail fraction is insensitive to the beamparameters except at high angles. Thus, one beam was used for the firstseven angles, and a different beam was used for the 36.2°, 41.6°, and 47.7°angles.1346.6. Positron beam simulationFigure 6.9: Reconstructed track parameters for the positron beam at 0°.The top-left and top-right panels show the ratio of the x and y momentato the z momentum, and the bottom-left and bottom-right panels show thereconstructed x and y positions at z = 0.Figure 6.10: Mean of the distribution of the x momentum, normalized tothe z momentum, as a function of angle.1356.6. Positron beam simulationThe energy spectrum at 0° is shown in Figure 6.11, from both data andMonte Carlo. For this plot, events depositing more than 1.6 MeV in T2have been removed, so as to show clearly the shape of the energy spectrumdue to leakage, and the photonuclear peaks.Figure 6.11: 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. Thegreen line shows the value of Ecut.Several differences between data and simulation are apparent. There isno pileup in the simulation, resulting in the difference to the right of themain peak. There is also no energy resolution applied to the simulatedBINA energy, although the simulated CsI energies have a Gaussian resolu-tion applied; the main peak in the simulation is thus sharper than in thedata. Finally, the photonuclear peaks (see Section 6.3.1) are larger in thedata than in the simulation. Only the last disagreement has any signifi-cant impact on the tail, as pileup and crystal resolution are negligible. Theposition of the second photonuclear peak just below the standard value ofEcut means that its amplitude must be correct for the tail to be properlyreproduced by the simulation. It in fact contains the majority of the tailat 0°, although its importance diminishes at large angles. The energy spec-1366.6. Positron beam simulationtra at 11.8°, 24.4°, 36.2°, and 47.7° are shown in Figure 6.12. The same T2energy cut is applied, scaled by the path length difference through T2 as itis rotated. Qualitatively, the differences between data and simulation at 0°persist as the angle increases.Figure 6.12: The energy spectrum from a 70 MeV positron beam at 11.8°(top left), 24.4° (top right), 36.2° (bottom left), and 47.7° (bottom right) tothe crystal axis. Data is shown in black and simulation is shown in red.The effect of uncertainty on the angle between the crystal axis and thebeam, the centre of rotation of the crystal array, the momentum of the beam,and the x and y divergences of the beam were assessed via simulation. For1376.6. Positron beam simulationthe most part, these errors were negligible, although at the highest anglesthey contribute significantly. Appendix D shows the complete results.The error on the crystal-beam angle was taken as ±0.1°, determined froma test of the accuracy of the measurement system (see Section 6.4). Theposition of the centre of rotation was fixed by the shaft attached to thecart on which the crystals sat; this was positioned according to alignmentmarkings made by the TRIUMF beamlines group, so an error of ±1 mmwas assigned. The uncertainty in the divergence of the beam was takenfrom the spread in Figure 6.10. For the first seven angles, the mean of thetx distribution was varied by ±0.02, and for the last three, it was varied by±0.004.In principle, the beam momentum was fixed by the fields of the bend-ing magnets, which were monitored by NMR probes to less than 0.1% andmaintained at a constant value. Thus, the momentum should be known tohigh accuracy, and was nominally set to 70 MeV/c for this data. However,the ratio of the positions of the positron and muon peaks was inconsistentwith this value, and instead suggested a momentum of approximately 74MeV/c. This degree of deviation from the field determined by the NMRprobes was not plausible, but in light of this fact an error was assigned of1%, or 0.7 MeV/c. That the usual settings for pion data-taking correspondto 75 MeV/c was verified to within a few hundred keV by the pion stoppingposition (see Section 7.1.1). The uncertainty in the beam momentum is thelargest uncertainty in the positron beam data.The tail fraction as a function of angle is shown in Figure 6.13, and the dif-ference between the MC and data tail fractions is shown in Figure 6.14. Theuncertainties from the various sources of error have been added in quadra-ture for visualization purposes, but note that the errors at each angle dueto beam momentum, beam divergence, and the position of the centre of ro-tation are not independent; they will shift each point in the same direction.Some difference is expected due to the photonuclear peaks (see Figure 6.6);this difference should increase as a function of angle, since the tail itselfincreases, making it more likely that an event in the first photonuclear peakwill be in the tail. However, the size of the difference at the last two anglescannot be explained by this effect. The difference in counts at 0° in theregion containing both photonuclear peaks is approximately 0.25% of the1386.7. Determining the PIENU tail fractiontotal spectrum, and most of the first peak remains out of the tail even at47.7°.Figure 6.13: The tail fraction as a function of angle in the positron beamdata.Currently, the cause of the discrepancy at the two highest angles is un-known. Two possibilities must be considered: that it is due to a genuinefeature of the calorimeter response that is not included in the simulation,or that it is due to some unconsidered or underestimated systematic errorin the data. In the latter case, the tail fraction for pi+ → e+νe events canbe taken from simulation, and corrected based on the difference measuredfor the first eight angles, which is consistent with being entirely due to thephotonuclear effect. In the former case, the simulated pi+ → e+νe tail mustbe corrected for the difference measured at all angles. In order to ensurethat the uncertainty was properly covered, corrections for both cases weredetermined, the average used as the correction, and half the difference takenas the 1σ error.6.7 Determining the PIENU tail fractionThe simulated pi+ → e+νe energy spectrum is shown in Figure 6.15; thenormal acceptance cut was applied, requiring at least one reconstructedtrack to be within 60 mm of the centre of WC3, but no other cuts were1396.7. Determining the PIENU tail fractionFigure 6.14: The difference between the tail fraction in the positron beamdata and the tail fraction from a simulated positron beam as a function ofangle.applied besides the trigger condition (energy in T1 and T2) and that thepion decayed at rest within the target. The high energy tail is caused bypi+ → e+νeγ events. The proportion of the spectrum less than 52 MeV is2.99%, almost as high as the tail at the largest angle at which positron beamdata were taken. The reason for this, and the long, almost flat, part of thespectrum extending down to zero, is interactions in the scintillators upstreamof BINA, particularly the target. Thus, the tail fraction must be extractedfrom a combination of the pi+ → e+νe simulation and the positron beamdata; it cannot be obtained from the latter alone. The spectrum withoutevents undergoing Bhabha scattering in the target is shown in Figure 6.16;the tail fraction is 1.82%.To obtain a correction for the simulated pi+ → e+νe tail fraction, the tailfraction in the positron beam data must be averaged as a function of angle.Because the tail fraction is a smooth function of the angle, it can be fittedwith a polynomial. This is shown in Figure 6.17 for both data and MonteCarlo. For this plot, the correlated errors (from point to point) have beenomitted. The average of the fitted function can then be taken, weighted for1406.7. Determining the PIENU tail fractionFigure 6.15: The simulated BINA+CsI spectrum from pi+ → e+νe decayand pi+ → e+νeγ decay.Figure 6.16: The simulated BINA+CsI spectrum from pi+ → e+νe decayand pi+ → e+νeγ decay, excluding events that underwent Bhabha scatteringin the target.1416.7. Determining the PIENU tail fractionan isotropic distribution, according to the formulaTaverage =∫ θmax0 w(θ)f(θ)dθ∫ θmax0 w(θ)dθ. (6.12)Figure 6.17: The tail fraction as a function of angle for the positron beamdata (left) and Monte Carlo (right), fitted to a fourth-degree polynomial.Here w(θ) is simply sinθ. The results are T dataaverage = 1.42% ± 0.01% andTMCaverage = 1.26%± 0.01%; the uncertainties are those returned by the poly-nomial fit on the weighted average, due to the uncertainty in the parameters.In order to take the correlated errors into account, each point was shiftedby the 1-sigma error and the fit was redone, giving 1.45% ± 0.01% and1.39%± 0.01%. The correction is obtained by subtracting the MC tail fromthe data tail; this gives a correction of 0.16%± 0.03%.In this case Ecut = 53.7 MeV, which takes into account the difference inpeak position between the pi+ → e+νe case and the positron beam case. Thisdifference is primarily due to the extra material present in the pi+ → e+νecase, although the starting energy is slightly different as well. The peak forsimulated pi+ → e+νe events emitted at small angles is shown in Figure 6.18;the difference between this peak and the 0° peak is 1.7 MeV. Thus, an Ecut1426.7. Determining the PIENU tail fractionvalue of 52 MeV in the pi+ → e+νe case corresponds to 53.7 MeV in thepositron beam case.Figure 6.18: Simulated energy spectrum for pi+ → e+νe events emitted atsmall angles.The difference between data and Monte Carlo, with the first eight pointsfitted to a straight line, is shown in Figure 6.19. Averaging this, againusing w(θ) = sin θ, gives 0.11% ± 0.01%. Again, the correlated errors wereused to shift each point by the same amount; the resulting averages were0.14%± 0.01% and 0.08%± 0.01%.The final correction is obtained by averaging the highest and lowest cor-rections, and taking half the difference as the error. The largest correctioncomes from the method with the polynomial fit, and is 0.19%. The smallestcomes from the method with the straight line fit, and is 0.08%. The resultis therefore a final correction to the pi+ → e+νe tail of 0.14%± 0.06%. Thesimulated pi+ → e+νe tail fraction was 2.99%; thus, the corrected value isT = 3.13%± 0.06%.Since the correction takes into account the difference between the responsefunction obtained from data and the simulated response function, the only1436.7. Determining the PIENU tail fractionuncertainties that must be taken into account in the pi+ → e+νe simulationitself are those arising from the detector geometry and the pion stoppingposition. The uncertainties on the detector positions and thicknesses arevery small [8], and the simulated pion stopping distribution agrees very wellwith the data (see Section 7.1.1). The variation in the simulated tail fractionfrom these effects was found to be < 0.01%, which is negligible compared tothe error on the correction.The only further uncertainty entering into the upper limit on the tailfraction comes from the uncertainty on the energy calibration itself (seeSection 4.4) of 0.1 MeV between Ecut and the peak of the pi+ → e+νeenergy spectrum. This was obtained simply from the tail < 51.9 MeV andthe tail < 52.1 MeV, which varied by 0.04% from the tail < 52.0 MeV. Thisincreased the error on the upper limit from 0.06% to 0.07%.Figure 6.19: The tail fraction in the positron beam data minus the simulatedtail fraction as a function of angle. The first 8 angles are fitted to a straightline.This result for the upper limit on the tail correction is combined with thelower limit to give the best estimate of the pi+ → e+νe tail. The limits arecombined in the following way: a probability distribution is constructed byassuming the tail is equally likely to be any value above the lower limit andany value below the upper limit, and that the errors on each quantity areGaussian. The peak of the resulting probability distribution is then taken1446.8. Tail as a function of R and Ecutas the central value for the tail. This procedure is graphically illustrated inFigure 6.20, for the lower limit obtained from the 2010 data.Tail Fraction [%]2.6 2.8 3 3.2 3.4Probability00.20.40.60.81Lower LimitUpper BoundCombined regionAllowed Probability Regions for Upper and Lower LimitFigure 6.20: The probability distribution obtained by combining the upperand lower limits.6.8 Tail as a function of R and EcutThe stability of the branching ratio as the acceptance cut and Ecut arevaried is a crucial systematic check of the analysis. The tail correctiondepends strongly on both of these parameters. Obtaining corrections as afunction of Ecut is straightforward; the only modification is the value up towhich the spectra are integrated. Obtaining corrections for acceptance cutsless than 60 mm is also straightforward, as the positron beam tail fractionsmust simply be averaged up to a smaller angle. However, since the largestangle at which positron beam data were taken corresponds to R = 62 mm,obtaining corrections above this value is not straightforward.The obvious approach is to extrapolate the function obtained from fittingthe tail fractions and average that; however, when this approach was testedvia simulation, it was found to significantly underestimate the tail for anglesabove those that were included in the fit. When the function in the right-1456.8. Tail as a function of R and Ecuthand panel of Figure 6.17 was extrapolated to 52.0°, 54.0°, and 56.0°, the tailfractions obtained were 3.63%, 4.14%, and 4.72%. When these angles weresimulated directly, the tail fractions were 6.40%, 10.26%, and 16.2%. Forreference, R = 80 mm corresponds to 55.1°; to verify the analysis, extendingthe acceptance cut to 90 mm is desirable.To a reasonable approximation, the size of the correction to the tail fora given acceptance cut should be the same as the size of the correction fora value of Ecut for which the uncorrected size of the tail is the same. Thevalidity of this approximation can be tested using acceptance cuts < 60 mm(this is the reason for including results for the upper limit for Ecut < 50MeV).Consider the uncorrected tail fractions for R < 30 mm, 40 mm, and 50mm; these are approximately equal to the tail fractions at 48 MeV, at 49.5MeV, and halfway between 50.5 MeV and 51 MeV. The corrections for thesevalues of R are 0.09%±0.02%, 0.10%±0.03%, and 0.12%±0.04%; for thesevalues of Ecut, they are 0.05%± 0.05%, 0.07%± 0.05%, and 0.10%± 0.06%(averaging 50.5 MeV and 51 MeV). These are consistent within errors. Theuncorrected tail fractions for R < 70 mm, 80 mm, and 90 mm are 3.46%,3.99%, and 4.63%. These are approximately equal to the uncorrected tailfractions for 53 MeV (3.40%), 54 MeV (3.92%), and 55 MeV (4.58%); thesame corrections are therefore applied.It is also desirable to check the stability of the branching ratio in statis-tically independent regions of R; to this end, the tail correction in 10 mmslices is given in Table 6.16. For large R the tail grows quickly, and thecorrection cannot be obtained by using a value of Ecut for which the tail iscomparable. Instead, the tails for large R slices were estimated using thebehaviour of the tail as larger R values are included. For example, the tailin the 60-70 mm slice is equal to the average of the 0-60 mm tail and the60-70 mm tail, weighted by the number of events in each slice. The relativeerror of the correction is taken to be the same as the 50-60 mm slice.1466.8. Tail as a function of R and EcutTable 6.14: Corrections and resulting pi+ → e+νe tail fractions as a functionof Ecut.Ecut (MeV) Correction (%) Upper limit (%) Combined tail (%)48.0 0.05± 0.05 2.02± 0.0548.5 0.05± 0.05 2.11± 0.0549.0 0.06± 0.05 2.22± 0.0549.5 0.07± 0.05 2.34± 0.0550.0 0.08± 0.05 2.46± 0.06 2.34± 0.1050.5 0.09± 0.05 2.60± 0.06 2.49± 0.1051.0 0.11± 0.06 2.76± 0.07 2.65± 0.1051.5 0.12± 0.06 2.93± 0.07 2.83± 0.1052.0 0.14± 0.06 3.13± 0.07 3.03± 0.1052.5 0.16± 0.07 3.34± 0.08 3.25± 0.1053.0 0.18± 0.07 3.58± 0.08 3.50± 0.1053.5 0.21± 0.08 3.86± 0.09 3.78± 0.1154.0 0.24± 0.08 4.16± 0.10 4.09± 0.1254.5 0.28± 0.09 4.51± 0.11 4.45± 0.1355.0 0.33± 0.09 4.91± 0.12 4.85± 0.13Table 6.15: Tail fractions as a function of the maximum radius in whichevents are accepted.Max R (mm) Correction (%) Upper limit (%) Combined tail (%)20 0.07± 0.02 1.79± 0.0330 0.09± 0.02 1.99± 0.03 1.91± 0.0840 0.10± 0.03 2.31± 0.04 2.21± 0.0950 0.12± 0.04 2.69± 0.05 2.58± 0.1060 0.14± 0.06 3.13± 0.07 3.03± 0.1070 0.18± 0.07 3.64± 0.08 3.55± 0.1080 0.24± 0.08 4.23± 0.10 4.12± 0.1590 0.33± 0.09 4.96± 0.11 4.84± 0.171476.9. Summary of Chapter 6Table 6.16: Tail fractions as a function of the radius in which events areaccepted.R range (mm) Correction (%) Upper limit (%) Combined tail (%)20-30 0.10± 0.03 2.18± 0.04 2.03± 0.1330-40 0.12± 0.04 2.84± 0.05 2.63± 0.1840-50 0.16± 0.08 3.70± 0.09 3.60± 0.1250-60 0.23± 0.14 4.88± 0.15 4.77± 0.1860-70 0.42± 0.26 6.48± 0.27 6.52± 0.2670-80 0.64± 0.39 8.76± 0.41 8.99± 0.3680-90 1.2± 0.73 12.3± 0.75 12.5± 0.606.9 Summary of Chapter 6The combined tail fraction is used to correct the branching ratio. Thecombined tail fraction is defined as the peak of the probability distributionobtained by combining the upper and lower limits, as shown in Figure 6.20.The formula for obtaining the corrected branching ratio isBRtailcorrected =11− T BRraw. (6.13)For the nominal values of R = 60 mm and Ecut = 52 MeV, the combinedtail fraction is T = 3.03%±0.10%, resulting in a multiplicative correction of1.0312 ± 0.0011. The raw branching ratio with the T2 pileup cut applied,given in Table 5.4, is BRraw = (1.2005± 0.0011)× 10−4. Applying the tailcorrection givesBRtailcorrected = (1.2380± 0.0011(stat)± 0.0013(tail))× 10−4. (6.14)This result is still blinded by an unknown factor between ±0.5%, so itcannot be directly compared to the theoretical prediction. The correctionsthat must be applied to the tail-corrected branching ratio are described inthe next chapter.148Chapter 7Other CorrectionsThree other, much smaller, corrections must be made to the raw branchingratio. These are referred to as the acceptance correction, the muon decay-in-flight correction, and the t0 correction. The first corrects for the differencein acceptance between pi+ → e+νe and pi+ → µ+ → e+νeν¯µ events, whicharises primarily from the energy dependence of multiple scattering. Thesecond corrects for the presence of muon decay-in-flight events in the highenergy time spectrum, which are indistinguishable from pi+ → e+νe eventsbased on their timing. The last corrects for the slight difference in themeasured time of a very low energy positron and a higher energy positron.Their values are shown in Table 7.1.Table 7.1: The small corrections that must be made to the branching ratio.The values are multiplied by the tail-corrected branching ratio to give thefinal result.Correction Value ErrorAcceptance 0.9991 0.0003µ-DIF 0.9983 < 0.0001t0 1.0006 0.00037.1 Acceptance correctionTwo effects could potentially change the ratio of pi+ → e+νe events topi+ → µ+ → e+νeν¯µ events within the geometrical acceptance: the extraspread in the starting position distribution of the decay positron caused bythe distance travelled by the 4.1 MeV muon, and energy dependent interac-tions upstream of BINA. Both of these can be assessed with Monte Carlo,since they depend on well-understood electromagnetic physics.Each of the 2010, 2011, and 2012 data sets required its own correction,since the beam momentum (and thus the pion stopping position) and de-1497.1. Acceptance correctiontector geometry were slightly different. For each case, 1 billion of eachdecay were simulated, and the ratio of events within different acceptanceregions was calculated. The ratio of pi+ → e+νe to pi+ → µ+ → e+νeν¯µevents within different radii of the centre of WC3 is shown in Figure 7.1.The errors are due to Monte Carlo statistics. The systematic error on thecorrection was obtained by varying several parameters in the simulation:the position and width of the pion stopping distribution, the positions andthicknesses of various detectors, and the trigger thresholds in T1 and T2.Figure 7.1: The ratio of pi+ → e+νe events to pi+ → µ+ → e+νeν¯µ eventswithin different radii of the centre of WC3, as reconstructed by the S3WC3tracker.7.1.1 Pion stopping positionThe pion stopping position was calculated from the point of closest ap-proach in the tracks reconstructed by the S12 and S3WC3 trackers. This wasdone in both data and Monte Carlo; a comparison of the two distributionsfor the 2012 data set is shown in Figure 7.2. Figure 7.3 shows the differencebetween the reconstructed stopping position and the actual stopping posi-tion in the simulation, Figure 7.4 shows the variation in the correction withthe mean of the reconstructed stopping position, and Figure 7.5 shows thevariation in the acceptance correction with the width of the reconstructedstopping position.1507.1. Acceptance correctionZ stoppping position (mm)-10 -5 0 5 10Normalized counts0102030405060708090610×DataMCFigure 7.2: The z coordinate of the reconstructed pion stopping position.Data is shown in black and MC is shown in red.Z stopping position (mm)-10 -5 0 5 10Normalized counts00.10.20.30.40.50.60.70.8ReconstructedActual positionFigure 7.3: A comparison of the actual and reconstructed pion stoppingpositions in MC.1517.1. Acceptance correctionRadius at WC3 (mm)20 30 40 50 60 70 80 90Acceptance ratio0.9940.9950.9960.9970.9980.99911.0011.002-0.2mm-0.1mmnominal+0.1mm+0.2mmFigure 7.4: Variation of the acceptance correction with the peak value ofthe reconstructed pion stopping position. The peak was varied by ±0.2 mm;the largest variation in the correction was approximately ±0.05%, for R <90 mm.Radius at WC3 (mm)20 30 40 50 60 70 80 90Acceptance ratio0.9940.9950.9960.9970.9980.99911.0011.002- 20% nominal+ 20%Figure 7.5: Variation of the acceptance correction with the width of thereconstructed pion stopping position.1527.1. Acceptance correction7.1.2 Detector geometryThe relevant detector parameters for the pion stopping position are thethicknesses of Tg, S3, and T1 and the positions of Tg, S1, S2, S3, and WC3;these are known to the level of microns and tens of microns, respectively.The effect on the correction of varying each of these parameters within theiruncertainties was found to be much less than 0.0001.7.1.3 Trigger thresholdsThe energy thresholds in T1 and T2 were each approximately 0.1 MeV.Since the acceptance correction could be sensitive to the exact value ofthe thresholds, they were varied by ±25 keV. The effect on the acceptancecorrection is shown in Figure 7.6. At R = 60, it was found to be 0.0001.These uncertainties are all negligible in the present analysis.Radius at WC3 (mm)20 30 40 50 60 70 80 90Acceptance ratio0.9940.9950.9960.9970.9980.99911.0011.00275 KeV100 KeV (nominal)125 KeVFigure 7.6: Variation of the acceptance correction with the thresholds in theT1 and T2 counters.All the uncertainties on this correction, both statistical and systematic,are on the level of a few parts in 10−8, which is negligible for the purposesof this analysis (see Chapter 8). The variations are at the same level as the2010 analysis, for which the error was 0.0003 at the nominal value of R.1537.2. Muon decay-in-flight correction7.2 Muon decay-in-flight correctionDespite the very low energy of muons arising from pi → µν decay, theinfluence of decays-in-flight upon the branching ratio is not negligible. Fig-ure 7.7 shows a simulation of the decay time of muons that were not at restwhen they decayed; the stopping time is approximately 19 ps. The proba-bility of this occurring can be approximated by the proportion of the muondecay curve between 0 and 19 ps; i.e.1− e−0.0192197 = 8.3× 10−6. (7.1)Figure 7.7: The decay time of muons in the target with non-zero kineticenergy at the time of the decay.Since the time of these events is distributed according to a pion life-time, any event with measured energy above Ecut will be misidentified as api+ → e+νe event by the time spectrum fit. Figure 7.8 shows simulationsof the measured energy of both muon decay-in-flight and muon decay-at-rest events. The proportion of the decay-in-flight spectrum above 52 MeVwas found to be 2.37%, giving a total correction of 0.0237 × 8.3 × 10−6 =1.97 × 10−7. Given the level of agreement in the measured energy spectrabetween Monte Carlo and data for both pi+ → µ+ → e+νeν¯µ events and thepositron beam, the relative error on the proportion of the spectrum above1547.3. t0 correction52 MeV is on the order of a few percent, which results in an uncertainty onthe correction of less than 10−8, which is negligible for this analysis.Energy in NaI+CsI[MeV]0 10 20 30 40 50 60Normalized Counts00.0010.0020.0030.0040.0050.0060.0070.0080.009DARµDIFµFigure 7.8: Simulated energy spectra measured by BINA+CsI for muondecays-at-rest (black) and decays-in-flight (red).7.3 t0 correctionIf the shape of the T1 waveform were to depend on the positron energy,there could also be a dependence of the fitted time on the positron energy.This would result in an effectively different value of t0 for pi+ → e+νe andpi+ → µ+ → e+νeν¯µ events. To determine if this effect exists, the timespectra for different energy regions were plotted and t0 was obtained by fit-ting the edge with a step function with Gaussian resolution. The correctionobtained using 2010 data was 1.0004 ± 0.0005 [3]. It was done again using2011 data, obtaining 1.0006 ± 0.0003 [65]. The error is due to statistics; itcould be reduced still further using 2012 data. For this analysis, the 2011value was used.7.4 Stability of the corrected branching ratioTwo crucial checks on the analysis are the stability of the branching ratiowhen the radius in which events are accepted is varied, and when the value1557.4. Stability of the corrected branching ratioof Ecut is varied. The fitting functions and the corrections both change withthese parameters, so any variation is an indication of a flaw in the analysis.Figure 7.9 shows the raw and corrected branching ratios for different accep-tance cuts; in each case the results are plotted both with and without the T2pileup cut applied. The error bars represent the error on the change fromthe first point; for the statistical error, this is done by taking the square rootof the squared sum of the difference of the statistical error at each point. Forthe tail correction, the difference in the error from the first point is takenas the error on the change: if the tail is above its central value at the firstpoint, it will also be above its central value at each subsequent point.Figure 7.9: Variation of the branching ratio as the radius in which events areaccepted is varied. The left-hand panel shows the results with the T2 pileupcut, and the right-hand panel shows the results without the T2 pileup cut.The red points show the raw branching ratio, and the black points show thebranching ratio after all corrections. The error bars represent the error onthe change from the first point.In both cases a trend is apparent from R > 60 mm onwards, althoughit is reduced when the T2 pileup cut is applied. The fit was also done forevents in 10 mm rings from the centre of WC3; these have the advantageof being statistically independent, although the errors are large. The erroron the tail is again taken as the difference in the error from the first point;it is then added in quadrature with the statistical error. The results are1567.4. Stability of the corrected branching ratioshown in Figure 7.10. Figure 7.11 is the same plot, zoomed in on the regioncontaining the corrected points. The branching ratio appears to take on onevalue for R < 50 mm, and another, lower, value after, thus causing the trendin Figure 7.9.Figure 7.10: The branching ratio for statistically independent rings in R, thedistance between the reconstructed positron track and the centre of WC3.The left-hand panel shows the results with the T2 pileup cut, and the right-hand panel shows the results without the T2 pileup cut. The red points showthe raw branching ratio, and the black points show the branching ratio afterall corrections. The value along the x axis is the centre of the ring underconsideration; that is, the point at x = 35 mm is the branching ratio forevents with R between 30 and 40 mm.Since the downward trend in the branching ratio only becomes statisti-cally significant after R > 60 mm, which is beyond the usual acceptancecut, it could perhaps be argued that the behaviour of the branching ratioin this region does not affect the systematic error on the result for R < 60mm. This would be the case if, for example, the tail correction at large Rwere underestimated. However, if instead there were a systematic effect inthe track reconstruction by which pi+ → e+νe events were shifted to lowerradii, the branching ratio would be overestimated for small R, and underes-timated for large R. Without knowing the source of the R dependence, thetrue branching ratio could be anywhere within the observed range, so the1577.5. Summary of Chapter 7Figure 7.11: The corrected branching ratio for statistically independent ringsin R, with (left) and without (right) the T2 pileup cut. This plot is identicalto Figure 7.10, but zoomed in on the corrected points.variation is included in the systematic error.The change in the branching ratio as Ecut is varied is shown in Figure 7.12.Aside from the point at 50 MeV, no significant variation is observed. Thesignal to background ratio in the high energy spectrum decreases rapidlystarting near 50 MeV, and the quality of the fit at this point is poor. Inlight of the stability above 50 MeV, no systematic error is assigned for theselection of Ecut.7.5 Summary of Chapter 7The systematic uncertainties on the three small corrections are all lessthan 0.05% (see Table 7.1). The dependence of the corrected branchingratio on the radius at WC3 in which events are accepted, however, is muchlarger, and contributes significantly to the total uncertainty on the branchingratio (see Figure 7.9). The size of each source of error, and the total error,are given in the next chapter.1587.5. Summary of Chapter 7Figure 7.12: The change in the branching ratio from the value at Ecut = 52MeV.159Chapter 8Conclusion8.1 Branching ratio resultThe major sources of error on the branching ratio obtained from the 2012data are as follows:ˆ Statistics: 11× 10−8 (see Section 5.4)ˆ Low-energy tail: 13× 10−8 (see Section 6.8)ˆ R dependence: 48× 10−8 (see Section 7.4)ˆ R dependence: 29× 10−8 (T2 pileup cut applied) (see Section 7.4)ˆ τµ, τpi dependence: 22×10−8 (T2 pileup cut applied) (see Section 5.5)Since this analysis represents a work in progress, a conservative approachto the final systematic error is taken; namely, the full variations due tothe R dependence and the lifetime dependence are taken as the 1σ rangeof the branching ratio. For example, with the T2 pileup cut applied, thevariation as a function of R is 58 × 10−8; thus, a systematic uncertaintyof ±29 × 10−8 is assigned. The impacts of the small (< 0.05%) sources ofsystematic uncertainty on the raw branching ratio are calculated in the sameway (see Table 5.5). They do not affect the final error.Adding the systematic uncertainties in quadrature givesRpi = (1.2∗∗∗ ± 0.0011(stat)± 0.0040(syst))× 10−4. (8.1)Adding the statistical and systematic uncertainties in quadrature gives 41×10−8, or 0.33% of the theoretical value. For reference, the blinded values areRrawpi = 1.2005× 10−4 and Rcorrpi = 1.2355× 10−4.1608.1. Branching ratio resultTable 8.1: Sources of error. The corrected branching ratio is given by theproduct of the raw branching ratio and the corrections. The errors givenfor the corrections are the errors on the corrections themselves, not theresulting errors on the branching ratio. The stars indicate that the result isstill blinded.Values UncertaintiesStat SystRrawpi (10−4) § 5.4 1.2*** 0.0011pi, µ lifetimes §5.5 0.0022Fitting range §5.5 0.0004Bin size §5.5 0.0003Fixed parameters §5.5 0.0001Additional components §5.5 0.0001CorrectionsLow energy tail §6.8 1.0312 0.0011t0 §7.3 1.0006 0.0003Acceptance §7.1 0.9991 0.0003µ-DIF §7.2 0.9983 < 0.0001R dependence §7.4 0.0029Rcorrpi (10−4) 1.2*** 0.0011 0.00401618.2. Future prospects8.2 Future prospects8.2.1 Systematic uncertaintyClearly, the largest issue with the analysis is the R dependence. The factthat the dependence is reduced when the T2 pileup cut is applied indicatesthat some pileup mechanism not included in the fit could be responsible. Ifthis is the case, identifying and characterizing the component, and includingit in the fit, would remove the dependence. Alternatively, if the T2 pileupcut could be improved in such a way as to not distort the time spectrum, itcould perhaps be tightened, which might also remove the dependence.The time spectrum distortion caused by the cut is likely due to the re-moval of events with fake hits; this effect could be reduced or removed ifthe cut were made more sophisticated, by comparing the pulse height of thetriggering hit to the pileup hit, for example. It is worth mentioning thatthe R dependence is not present if the T2 pileup cut is applied and the life-times are freed in the fit; however, at large R the fitted lifetimes are severalstandard deviations away from their accepted values.Another possibility to reduce the influence of pileup is to evaluate theBINA energy based on the pulse height of the waveform, rather than theintegrated charge in a wide gate. Although the energy resolution may sufferslightly, this will not appreciably affect the result at least as long as the res-olution remains Gaussian. This was done for the 2011 data, but is currentlybeing implemented for the 2012 data.Once the systematics due to the R dependence and the lifetime depen-dence are removed or reduced to an acceptable level, the largest remainingsystematic error will be due to the low energy tail; this was the largest sourceof systematic error in the 2010 and 2011 analyses. The error coming fromthe tail correction could be reduced if the influence of beamline scatteringin the response function measurement could be either characterized or es-tablished as negligible; then what is now considered the upper limit on thetail correction could be taken as the true tail correction. This would reducethe error from 13× 10−8 to 8× 10−8, even if no other improvements in thetail analysis were made. An effort to simulate the positron transport downthe beamline, and evaluate the low momentum tail of the beam entering thedetector, is ongoing, but has not yet produced conclusive results.1628.2. Future prospectsUltimately, the analysis of at least the 2010, 2011, and 2012 data will becombined to obtain a single result for the branching ratio. The statisticalerror using the current cuts will be 8× 10−8. If the systematic error can bereduced to 9 × 10−8 or less, the goal of 0.1% uncertainty on the branchingratio will have been achieved. If the systematic error can be reduced belowthis level, the data from 2009 and the beginning of 2010 could be incorpo-rated as well, reducing the statistical error further. The different data-takingconditions for these running periods would make this a challenge, but if theresult from the easily-useable data turns out to be statistics-limited, it couldbe attempted.8.2.2 Limits on new physicsIf there were a difference in the couplings of the W to the electron andmuon, ge and gµ, the branching ratio would be related to the StandardModel prediction by Rpi = (ge/gµ)2RSMpi (see Section 2.3). Thus, achievinga precision of 0.1% on the branching ratio would correspond to a 0.05% leveltest of lepton universality.This would make pion decay once again the most sensitive test of lep-ton universality, and improve the already stringent constraints on modelsattempting to explain the hints of possible lepton nonuniversality seen bythe LHCb [53] [54] and BaBar [55] experiments. Essentially, the modelsmust include the property that the mechanism that couples differently tothe different generations be greatly enhanced for the third generation [56].The limits placed on specific processes are calculated using formulae foundin Section 2.2. The unblinded result of the 2010 analysis wasR2010pi = (1.2344± 0.0030)× 10−4. (8.2)To translate this into an upper limit on the branching ratio, the Feldman-Cousins technique [67] will be used. This provides a frequentist confidenceinterval based on an ordering of likelihood ratios, avoiding such problems asnon-physical confidence regions. Notably, it also provides consistent treat-ment when the data itself is used to make the decision to assign an upperlimit or a two-sided interval. Consulting Table X in Reference [67], the upperlimit on the branching ratio at 95% confidence is 1.67 standard deviationsabove the SM prediction, orRULpi = 1.2402× 10−4. (8.3)1638.2. Future prospectsNew pseudoscalar interactionsSubstituting the SM prediction and the value from Equation 8.3 intoEquation 2.16 gives1.24021.2352− 1 =(1 TeVΛ)2× 103, (8.4)givingΛ = 497 TeV. (8.5)Thus, the mass scale of a new fundamental pseudoscalar, with the samecoupling strength to quarks and leptons as the weak interaction, must be> 500 TeV at 95% C.L. A 0.1% measurement, with the same central value,would give Λ > 880 TeV.R-parity violating SUSYThe relationship between Rpi and the R-parity violating parameters ∆′11kand ∆′21k (see Section 2.2.2) is∆RpiRSMpi= 2(∆′11k −∆′21k). (8.6)Rpi itself does not provide any constrant on the size of ∆′11k and ∆′21k in thecase where they are equal in value. In the case where ∆′11k = 0, ∆′21k <0.0020 at 95% C.L.; in the case where ∆′21k = 0, ∆′11k < 0.0028 at 95% C.L.Charged Higgs bosonsAs discussed in Section 2.2.3, 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, if thecoupling is independent of the lepton mass, this is no longer the case. For acoupling of α/pi, the limit at 95% C.L. isMH+ > 144 GeV. (8.7)Massive neutrinosThe limits obtained for massive neutrino mixing, for neutrino mass in therange 55-130 MeV, using a search for extra peaks in the pi+ → e+νe energy1648.2. Future prospectsspectrum with PIENU data taken in 2009, were shown in Figure 2.5. Asimilar analysis using the 2010, 2011, and 2012 data is underway, which isexpected to improve the limits by up to a factor of 5. 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D, 57:3873–3889, 1998. doi:10.1103/PhysRevD.57.3873.171Appendix ATrigger DiagramFigure A.1: PIENU Trigger Diagram172Appendix BTiming DiagramsThis appendix contains simplified timing diagrams for various types ofevent, depicting the pion stop time, the upstream coincidence signal, themuon decay time, the downstream coincidence signal, and the trigger signal.The events shown are a regular PIMUE event, an event where an old muondecays and the muon from the primary pion decay does not, an event whereboth muons decay and the positron from the old muon decay completes thetrigger, an event where both muons decay and the positron from the decayof the muon from the decay of the primary pion completes the trigger, andan event where both muons decay sufficiently close together in time thatonly one hit is recorded by T1. The only timing offset shown is the 300 nsdelay between the upstream coincidence signal and the pion stop time.173Appendix B. Timing Diagrams  Pion stopUpstream coincidenceMuon decayDownstream coincidence-300 ns 500 ns400 ns400 nsTrigger0 nsFigure B.1: A simplified timing diagram for a PIMUE event in which themuon decayed 400 ns after the pion stop.174Appendix B. Timing Diagrams  Pion stopUpstream coincidenceOld muon decayDownstream coincidenceTrigger-300 ns 500 ns-200 ns-200 ns0 nsFigure B.2: A simplified timing diagram for an event in which an old muondecays and 200 ns later a pion arrives. Due to the delay between the actualhits in T1 and T2 and the downstream coincidence signal, the event stilltriggers.175Appendix B. Timing Diagrams  Pion stopUpstream coincidenceMuon decay (into acceptance)-300 ns 500 ns400 nsOld muon decay (out of acceptance) 50 ns0 nsDownstream coincidence400 nsTriggerFigure B.3: A simplified timing diagram for an event with two muons, inwhich the positron from the old muon decay completes the trigger.176Appendix B. Timing Diagrams  Pion stopMuon decay (out of acceptance)Downstream coincidence400 ns50 nsTriggerOld muon decay (into acceptance) 50 nsUpstream coincidence-300 ns 500 ns0 nsFigure B.4: A simplified timing diagram for an event with two muons, inwhich the positron from the muon from the decay of the primary pion com-pletes the trigger.177Appendix B. Timing Diagrams  Pion stopUpstream coincidenceMuon decay (into acceptance)-300 ns 500 ns60 nsOld muon decay (into acceptance) 50 ns0 nsDownstream coincidence50 nsTriggerFigure B.5: A simplified timing diagram for an event with two muons, inwhich both decay positrons enter the acceptance, but the event passes theT1 pileup cut because the decays are too close together in time for theseparate hits to be resolved.178Appendix CEvent Selection For PositronBeam DataThis Appendix contains the details of the cuts used to produce the energyspectrum used for the upper limit analysis (see Chapter 6). The purpose ofthe analysis was to determine the response of the crystal calorimeter arrayto a 70 MeV positron beam, as a function of angle. The distributions shownwere taken with the beam axis aligned with the crystal axis. The recon-structed x and y distributions in the WC12 tracker are shown in Figure C.1;events outside the beam spot were removed. The energy spectrum followingthis cut is shown in Figure C.2.Figure C.1: The beam spot in WC1 and WC2 for positron beam data.179Appendix C. Event Selection For Positron Beam DataFigure C.2: Energy measured by the crystals following the selection of beamparticles.The timing distributions in the first plane of WC1, WC2, and WC3 isshown in Figure C.3. Event with times outside the peak in any of thedistributions were removed. The energy spectrum following this cut is shownin Figure C.4.The two main features remaining in the energy spectrum are the peaksdue to beam positrons and beam muons. A time-of-flight cut was used tomitigate the muon background. A plot of the time of the hit in T2 relativeto the RF versus the energy deposited in T2 is shown in Figure C.5. They-axis is the time difference between the peak of the 23 MHz cyclotron RFand the hit in T2; because the particles travelled the entire length of thebeamline before reaching the detector, the three particle species separatedin time. The x-axis is the integral of the T2 waveform in a region aroundthe peak, which is proportional to the energy deposited. The two dark blobsare positrons and muons; the RF region vetoed in the trigger can be seen.The energy spectra in the region from 3-12 ns and 12-16 ns, correspondingto positrons and muons, respectively, are shown in Figure C.6.180Appendix C. Event Selection For Positron Beam DataFigure C.3: The time distribution of the first hit in the first plane of eachwire chamber.181Appendix C. Event Selection For Positron Beam DataFigure C.4: Energy measured by the crystals at 0° following the removal ofevents with out-of-time hits.Figure C.5: Time of flight vs. T2 energy.182Appendix C. Event Selection For Positron Beam DataFigure C.6: Energy measured by the crystals at 0° for events with time-of-flight corresponding to positrons (left) and muons (right).Although selecting the RF window corresponding to positrons substan-tially reduced the muon background, it did not eliminate it entirely. Thiscould be accomplished with a cut in the T2 energy, as shown in Figure C.7.In this plot, any event with a total T2 energy greater than 400 ADC countswas rejected (see Figure C.5).This cut is effective in removing muons; no trace of the peak around 18MeV remains. However, inspection of Figure C.5 shows that positrons whichdeposited an unusually large amount of energy in T2 were removed as well.This could bias the response function measurement, since it alters the energydistribution of the positrons entering BINA. Furthermore, shower leakagebackwards out of BINA affected the T2 energy spectrum, meaning a cutin the T2 energy would preferentially remove events with shower leakage,worsening the potential for bias. Figure C.8 shows simulated T2 energyspectra with and without BINA in place. Ultimately, no cut was done onthe energy deposited in T2.183Appendix C. Event Selection For Positron Beam DataFigure C.7: Energy measured by the crystals at 0° following the removal ofevents with high energy deposit in T2.Figure C.8: Simulated energy deposit in T2, with (red) and without (black)BINA in place.184Appendix DPositron Data SystematicsTable D.1: The change in the tail fraction as beam parameters and detectorgeometry were varied. The values given are the nominal tail fraction minusthe new tail fraction (see Section 6.6 for a detailed description of what waschanged). Note that the change is given as a fraction of the total spectrum,not the tail. The upper part shows the results for variations that increasedthe tail, and the lower part shows the results for variations that decreasedthe tail. The errors are due to Monte Carlo statistics.Crystal-beam angle Centre of rotation Beam momentum Beam divergence0° (−5.2± 3.1)× 10−5 (−3.8± 4.1)× 10−5 (−11± 4)× 10−5 (−3.1± 4.1)× 10−56° (−0.7± 3.2)× 10−5 (−2.2± 4.1)× 10−5 (−13± 4)× 10−5 (5.8± 4.1)× 10−511.8° (−1.7± 3.3)× 10−5 (−7.4± 4.2)× 10−5 (−13± 4.0)× 10−5 (−3.6± 4.2)× 10−516.5° (3.4± 3.4)× 10−5 (7.9± 4.3)× 10−5 (−2.0± 4.4)× 10−5 (1.5± 4.4)× 10−520.9° (−12± 4)× 10−5 (−12± 5)× 10−5 (−12± 4)× 10−5 (−12± 5)× 10−524.4° (−4.5± 3.9)× 10−5 (−12± 5)× 10−5 (−18± 5.0)× 10−5 (−11± 5)× 10−530.8° (−11± 5)× 10−5 (−21± 6)× 10−5 (−13± 6)× 10−5 (−19± 6)× 10−536.2° (−3.9± 6.4)× 10−5 (−3.3± 6.5)× 10−5 (−11± 6)× 10−5 (−3.9± 6.4)× 10−541.6° (−8.1± 7.7)× 10−5 (−12± 8)× 10−5 (−27± 8)× 10−5 (−4.8± 7.7)× 10−547.7° (−36± 10)× 10−5 (−37± 10)× 10−5 (−49± 10)× 10−5 (−37± 10)× 10−50° (3.1± 3.1)× 10−5 (2.9± 4.0)× 10−5 (7.2± 3.8)× 10−5 (−0.6± 4.1)× 10−56° (1.3± 3.1)× 10−5 (−1.5± 4.1)× 10−5 (13± 4)× 10−5 (−3.8± 4.1)× 10−511.8° (0.1± 3.3)× 10−5 (−2.4± 4.2)× 10−5 (6.3± 4.0)× 10−5 (2.1± 4.2)× 10−516.5° (5.8± 3.4)× 10−5 (10± 4)× 10−5 (13± 4)× 10−5 (−1.0± 4.4)× 10−520.9° (−4.0± 3.6)× 10−5 (−8.1± 4.6)× 10−5 (4.9± 4.4)× 10−5 (3.9± 4.6)× 10−524.4° (−5.7± 3.9)× 10−5 (1.5± 4.9)× 10−5 (10± 5)× 10−5 (1.5± 4.9)× 10−530.8° (−1.5± 4.5)× 10−5 (−1.8± 5.7)× 10−5 (10± 6)× 10−5 (10± 6)× 10−536.2° (6.1± 6.4)× 10−5 (6.0± 6.5)× 10−5 (22± 6)× 10−5 (7.2± 6.4)× 10−541.6° (17± 6)× 10−5 (20± 8)× 10−5 (12± 8)× 10−5 (4.4± 7.7)× 10−547.7° (14± 10)× 10−5 (44± 10)× 10−5 (56± 10)× 10−5 (31± 10)× 10−5185

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