UNIVERSIDAD NACIONAL AUTÓNOMA DE MÉXICO POSGRADO EN CIENCIAS FÍSICAS INSTITUTO DE CIENCIAS NUCLEARES IMPROVED HIGH PRECISION MEASUREMENT OF THE RARE PION DECAY π⁺→ e⁺νe TESIS QUE PARA OPTAR POR EL GRADO DE: DOCTOR EN CIENCIAS (FÍSICA) PRESENTA: CARLOS IVÁN ORTEGA HERNÁNDEZ TUTOR/A O TUTORES PRINCIPALES DR. ALEXIS AGUILAR AREVALO, INSTITUTO DE CIENCIAS NUCLEARES DR. SAUL CUEN ROCHIN, ESCUELA DE INGENIERÍA Y CIENCIAS. TECNOLÓGICO DE MONTERREY MIEMBROS DEL COMITÉ TUTOR DRA. MARÍA CATALINA ESPINOZA HERNÁNDEZ, INSTITUTO DE FÍSICA DR. ERIC VÁZQUEZ JÁUREGUI, INSTITUTO DE FÍSICA CIUDAD DE MÉXICO, MÉXICO, ABRIL DE 2025 UNAM – Dirección General de Bibliotecas Tesis Digitales Restricciones de uso DERECHOS RESERVADOS © PROHIBIDA SU REPRODUCCIÓN TOTAL O PARCIAL Todo el material contenido en esta tesis esta protegido por la Ley Federal del Derecho de Autor (LFDA) de los Estados Unidos Mexicanos (México). El uso de imágenes, fragmentos de videos, y demás material que sea objeto de protección de los derechos de autor, será exclusivamente para fines educativos e informativos y deberá citar la fuente donde la obtuvo mencionando el autor o autores. Cualquier uso distinto como el lucro, reproducción, edición o modificación, será perseguido y sancionado por el respectivo titular de los Derechos de Autor. Atentamente PROTESTA UNIVERSITARIA DE INTEGRIDAD Y HONESTIDAD ACADÉMICA Y PROFESIONAL (Graduación con trabajo escrito) (Nombre, firma y Número de cuenta de la persona alumna) De conformidad con lo dispuesto en los artículos 87, fracción V, del Estatuto General, 68, primer párrafo, del Reglamento General de Estudios Universitarios y 26, fracción I, y 35 del Reglamento General de Exámenes, me comprometo en todo tiempo a honrar a la Institución y a cumplir con los principios establecidos en el Código de Ética de la Universidad Nacional Autónoma de México, especialmente con los de integridad y honestidad académica. De acuerdo con lo anterior, manifiesto que el trabajo escrito titulado: que presenté para obtener el grado de - - - - D o c t o r a d o - - - - es original, de mi autoría y lo realicé con el rigor metodológico exigido por mi programa de posgrado, citando las fuentes de ideas, textos, imágenes, gráficos u otro tipo de obras empleadas para su desarrollo. En consecuencia, acepto que la falta de cumplimiento de las disposiciones reglamentarias y normativas de la Universidad, en particular las ya referidas en el Código de Ética, llevará a la nulidad de los actos de carácter académico administrativo del proceso de graduación. Carlos Iván Ortega Hernández 518026080 IMPROVED HIGH PRECISION MEASUREMENT OF THE RARE PION DECAY pi+ --> e+ nu_e Solicitud de Validación de Artículo Ciudad de México, 25 de Febrero de 2025 Comité Académico del Posgrado en Ciencias Físicas Universidad Autónoma de México P r e s e n t e Estimados miembros del Comité Académico, Por medio de la presente, yo Carlos Iván Ortega Hernández, alumno de Doctorado del posgrado en Ciencias Físicas, me permito solicitar formalmente la acreditación del artículo titulado Measurement of the response function of the PIENU calorimeter, publicado recientemente en la sección Nuclear Instrumentarion and Methods in Physics Research, A de la editorial Elsevier. A lo largo del proceso de desarrollo de esta investigación, mi participación fue crucial en diversas etapas, desde el análisis de datos hasta la elaboración y redacción del artículo. Mis aportes fueron especialmente relevantes en la depuración de datos para la selección de positrones, validación de la lista de integraciones electromagnéticas en la simulación Monte Carlo y el modelado geométrico y físico del calorimetro PIENU en dicha simulación. El impacto del artículo en el proyecto de tesis redice en la validación de la simulación Monte Carlo a través de este estudio. Esto permite la expansión de la simulación a la configuración PIENU para el calculo de las correcciones de primer orden del branching ratio del pion. Dicho parámetro es el principal resultado de mi tesis. Considero que mi contribución merece ser reconocida en el contexto académico que da sustento a este artículo. Por lo tanto, solicito que se me acredite adecuadamente en la autoría, conforme a las normativas de reconocimiento y autoría científica vigentes en el Posgrado en Ciencias Físicas. Los agradecimientos correspondientes a CONACHYT y al apoyo AANILD del Posgrado UNAM se encuentran en la sección Acknowledgments. Agradezco de antemano su consideración y quedo a su disposición para cualquier información adicional o documentación que necesiten para completar este proceso. Atentamente, _________________________________ Carlos Iván Ortega Hernández Instituto de Ciencias Nucleares 518026080 Vo. Bo. del comité tutor __________________________________ __________________________________ Dr. Alexis Armando Aguilar Arévalo Dr. Saúl Cuén Rochin Instituto de Ciencias Nucleares Escuela de Ingeniería y Ciencias Tecnológico de Monterrey __________________________________ __________________________________ Dra. María Catalina Espinoza Hernández Dr. Eric Vázquez Jáuregui Instituto de Física Instituto de Física Abstract The PIENU experiment at TRIUMF aims to measure the pion branching ratio Rπ = Γ(π+→e+νe+π+→e+νeγ) Γ(π+→µ+νe+π+→e+νeγ) to a precision level of O(0.1%). This is an important physical observable in the Standard Model (SM) of particle physics that provides stringent constraints to new physics scenarios. In par- ticular it is a sensitive test to the lepton universality hypothesis, which states the the weak couplings of leptons (e± ,µ± and τ±) to the W± bosons have the same strength (g = ge = gµ = gτ ). The current theoretical calculation of the pion branching ratio is RSM π = (1.2352±0.0002)×10−4, with a precision of 0.016%. The complete PIENU dataset consists of six data Runs taken be- tween 2009 and 2012 gathering around 5 million of π+ → e+νe events. The analysis of the Run IV dataset, with 0.4 million events, published in 2015, gave Rexp π = (1.2344±0.0023(stat.)±0.0019(syst.)), reaching a precision level of 0.24%. The agreement between this measurement and the SM translates to a 0.12% precision in the ratio of lepton couplings ge/gµ = 0.9996±0.0012. This thesis presents the blind analysis of the Run III dataset, which consists of 2 million events, comprising the second largest portion of data avail- able. Several improvements have been implemented giving a better handle on positron track reconstruction, the simulation-dependent corrections, and on the assessment of systematic uncertainties. One major contribution is the Monte Carlo validation with measurement of the PIENU calorimeter response function using a 70 MeV/c positron beam. The blinded value of Rπ from this dataset with a 0.12% precision, is at the same level of the com- bined blinded result from Runs IV-VI. The current work allows to project an expected precision of 0.11% on Rπ with the full dataset. The implica- tions of this ultimate precision for a variety of SM extensions are explored. In particular it would allow to test the lepton Universality hypothesis to a 0.05% precision level. i Acknowledgments My sincere thanks go to the PIENU collaboration for their contribution to my thesis work. Their intellectual discussions have provided a stimulating and collaborative environment, specially from those who I met personally: D.A. Bryman†, C. Malbrunot, R.E. Mischke⋆, T. Numao, A. Sher and T. Sullivan. To my advisors Alexis A. Aguilar-Arevalo and Saul Cuen-Rochin who invested uncountable effort in enriching my academic and professional de- velopment. To † for providing the resources and opportunities to participate in the PIENU experiment and the financial support that substantially contributed to the completion of my thesis. To ⋆ for the mentorship and the invaluable experience he transmitted me through effusive scientific discussions. To Universidad Nacional Autonoma de Mexico (UNAM), CONAHCYT doctoral fellowship (grant no. 894777) and the AANILD program for the financial support that made this reseach possible. ii Table of Contents Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . ii Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Review of the history of the Rπ measurement . . . . . . . . . 3 1.1.1 Lessons from the E248 experiment . . . . . . . . . . . 4 1.2 Overview of the PIENU technique . . . . . . . . . . . . . . . 6 1.3 Blind analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2 Theory background . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.1 The standard model . . . . . . . . . . . . . . . . . . . . . . . 12 2.1.1 Inter-generational mixing in the SM . . . . . . . . . . 13 2.1.2 Electroweak interaction . . . . . . . . . . . . . . . . . 15 2.1.3 Strong interaction . . . . . . . . . . . . . . . . . . . . 15 2.1.4 V-A theory . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 Pions in the Standard Model . . . . . . . . . . . . . . . . . . 17 2.2.1 Pion decay modes . . . . . . . . . . . . . . . . . . . . 17 2.2.2 Pion decay rate . . . . . . . . . . . . . . . . . . . . . 18 2.2.3 Helicity suppression . . . . . . . . . . . . . . . . . . . 20 2.2.4 Radiative corrections . . . . . . . . . . . . . . . . . . 21 2.3 Physics beyond the Standard model . . . . . . . . . . . . . . 24 2.3.1 Violation of the lepton universality . . . . . . . . . . 25 2.3.2 New pseudo-Scalar interactions . . . . . . . . . . . . 29 2.3.3 Massive neutrinos . . . . . . . . . . . . . . . . . . . . 33 2.3.4 Meson fields in dark-sector . . . . . . . . . . . . . . . 36 3 Description of the experiment . . . . . . . . . . . . . . . . . . 39 3.1 Beam-line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.1 TRIUMF cyclotron . . . . . . . . . . . . . . . . . . . 39 iii Table of Contents 3.1.2 M13 channel . . . . . . . . . . . . . . . . . . . . . . . 39 3.1.3 M13 extension . . . . . . . . . . . . . . . . . . . . . . 42 3.1.4 Beam-line calibration . . . . . . . . . . . . . . . . . . 43 3.2 Detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Scintillators . . . . . . . . . . . . . . . . . . . . . . . 45 3.2.2 Wire chambers . . . . . . . . . . . . . . . . . . . . . . 48 3.2.3 Silicon detectors . . . . . . . . . . . . . . . . . . . . . 50 3.2.4 Bina . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.2.5 CsI array . . . . . . . . . . . . . . . . . . . . . . . . . 52 3.3 Final detector assembly . . . . . . . . . . . . . . . . . . . . . 54 3.4 Tracking system . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.4.1 Track definition . . . . . . . . . . . . . . . . . . . . . 57 3.4.2 Track fitting . . . . . . . . . . . . . . . . . . . . . . . 58 3.4.3 Tracking quantities . . . . . . . . . . . . . . . . . . . 59 3.5 Data acquisition system and Triggers . . . . . . . . . . . . . 60 3.5.1 Reading boards . . . . . . . . . . . . . . . . . . . . . 60 3.5.2 Software . . . . . . . . . . . . . . . . . . . . . . . . . 62 3.5.3 Trigger . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.6 Data-Taking history . . . . . . . . . . . . . . . . . . . . . . . 66 3.6.1 2009 . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.6.2 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6.3 2011 . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 3.6.4 2012 . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.6.5 Full analysis . . . . . . . . . . . . . . . . . . . . . . . 68 4 Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.1 Variable extraction . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.1 From VT48 . . . . . . . . . . . . . . . . . . . . . . . . 71 4.1.2 From COPPER waveform . . . . . . . . . . . . . . . 71 4.1.3 From VF48 waveform . . . . . . . . . . . . . . . . . . 72 4.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 4.2.1 Gain stability . . . . . . . . . . . . . . . . . . . . . . 73 4.2.2 Energy calibration . . . . . . . . . . . . . . . . . . . . 77 4.3 The Run III dataset . . . . . . . . . . . . . . . . . . . . . . . 78 4.4 Event Selection . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.4.1 Pion selection cuts . . . . . . . . . . . . . . . . . . . . 80 4.4.2 Pileup rejection cuts . . . . . . . . . . . . . . . . . . . 82 4.4.3 Background suppression . . . . . . . . . . . . . . . . 88 4.4.4 Acceptance cut . . . . . . . . . . . . . . . . . . . . . 90 4.5 Summary of Event selection . . . . . . . . . . . . . . . . . . 91 iv Table of Contents 4.6 Data calibration to MC . . . . . . . . . . . . . . . . . . . . . 92 5 Raw branching ratio extraction . . . . . . . . . . . . . . . . . 96 5.1 Time spectrum . . . . . . . . . . . . . . . . . . . . . . . . . . 96 5.2 Signals and Backgrounds . . . . . . . . . . . . . . . . . . . . 97 5.2.1 Main signals . . . . . . . . . . . . . . . . . . . . . . . 97 5.2.2 Low energy background . . . . . . . . . . . . . . . . . 99 5.2.3 High energy background . . . . . . . . . . . . . . . . 100 5.3 The Fitting Function . . . . . . . . . . . . . . . . . . . . . . 105 5.3.1 Low-energy time spectrum . . . . . . . . . . . . . . . 107 5.3.2 High-energy time spectrum . . . . . . . . . . . . . . . 107 5.3.3 Fit parameters . . . . . . . . . . . . . . . . . . . . . . 108 5.3.4 Fit residuals . . . . . . . . . . . . . . . . . . . . . . . 109 5.4 Summary of Timefit . . . . . . . . . . . . . . . . . . . . . . . 117 6 Monte Carlo Corrections . . . . . . . . . . . . . . . . . . . . . 118 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 6.2 Energy loss mechanisms . . . . . . . . . . . . . . . . . . . . . 118 6.3 Response Function Measurement . . . . . . . . . . . . . . . . 120 6.3.1 Experimental setup . . . . . . . . . . . . . . . . . . . 120 6.3.2 Event selection . . . . . . . . . . . . . . . . . . . . . . 122 6.3.3 Low-momentum beam positrons . . . . . . . . . . . . 126 6.4 Lineshape simulation . . . . . . . . . . . . . . . . . . . . . . 126 6.4.1 Physics List . . . . . . . . . . . . . . . . . . . . . . . 126 6.4.2 Calorimeter tuning . . . . . . . . . . . . . . . . . . . 128 6.4.3 Positron beam tail fraction . . . . . . . . . . . . . . . 129 6.5 PIENU simulation . . . . . . . . . . . . . . . . . . . . . . . . 130 6.5.1 Beam Profile . . . . . . . . . . . . . . . . . . . . . . . 130 6.5.2 Tracking clustering . . . . . . . . . . . . . . . . . . . 139 6.5.3 WC3 position . . . . . . . . . . . . . . . . . . . . . . 141 6.6 Rπ corrections . . . . . . . . . . . . . . . . . . . . . . . . . . 144 6.6.1 Tail fraction . . . . . . . . . . . . . . . . . . . . . . . 144 6.6.2 Acceptance correction . . . . . . . . . . . . . . . . . . 146 6.6.3 Muon decay-in-flight . . . . . . . . . . . . . . . . . . 148 6.6.4 t0 correction . . . . . . . . . . . . . . . . . . . . . . . 152 6.7 Summary of Monte Carlo Corrections . . . . . . . . . . . . . 153 7 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 7.1 Systematic effects . . . . . . . . . . . . . . . . . . . . . . . . 154 7.1.1 Trigger inefficiencies . . . . . . . . . . . . . . . . . . . 155 v Table of Contents 7.1.2 Event selection . . . . . . . . . . . . . . . . . . . . . . 156 7.1.3 Timefit parameters . . . . . . . . . . . . . . . . . . . 159 7.1.4 Low Energy Tail (LET) correction . . . . . . . . . . . 162 7.1.5 Acceptance correction . . . . . . . . . . . . . . . . . . 164 7.1.6 µDIF correction . . . . . . . . . . . . . . . . . . . . . 165 7.2 Stability tests . . . . . . . . . . . . . . . . . . . . . . . . . . 165 7.3 Error Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 7.4 Combination of data blocks . . . . . . . . . . . . . . . . . . . 170 8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 8.1 Summary of results . . . . . . . . . . . . . . . . . . . . . . . 174 8.2 Limits on Physics beyond the Standard Model . . . . . . . . 176 8.2.1 Lepton universality . . . . . . . . . . . . . . . . . . . 177 8.2.2 New Pseudo-scalar interactions . . . . . . . . . . . . . 177 8.2.3 R-Parity violating SUSY . . . . . . . . . . . . . . . . 178 8.2.4 Charged Higgs Boson . . . . . . . . . . . . . . . . . . 178 8.2.5 Massive Neutrinos in the π+ → e+νe decay . . . . . . 178 8.3 Next generation of Rπ measurement . . . . . . . . . . . . . . 179 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Appendices A Trigger complete diagram . . . . . . . . . . . . . . . . . . . . 196 vi Chapter 1 Introduction The pion decay branching ratio, defined as Rπ = Γ(π+ → e+νe + π+ → e+νeγ) Γ(π+ → µ+νµ + π+ → µ+νµγ) , (1.1) is an important observable in the Standard Model of particle physics. The value of Rπ has been calculated within the Standard Model to be (1.2352 ± 0.0002)×10−4 [36] [37]. A precise measurement of Rπ provides stringent constraints on many extensions to the Standard Model (SM), such as R-parity violating super-symmetry [77], leptoquarks [72], and heavy neu- trinos lighter than the pion [85]. In certain cases, these constraints can far exceed the reach of direct searches at colliders. Most strikingly, a new pseu- doscalar interaction whose energy scale of O(1000 TeV) [76] would enhance the branching ratio by O(0.1%). The PIENU1 experiment at TRIUMF2 aims to measure Rπ to high preci- sion (O(0.1%)). To achieve this, PIENU has taken around 5 M of π+ → e+νe events that were divided in seven data runs along the data-taking history between 2009 and 2012. This thesis comprises the analysis of the Run III dataset, that consists of 2 M events from the 5 M available. The main outcome is a measurement of Rπ that will be later included in the global PIENU analysis, consisting of three runs (IV, V and VI), to reach its ex- pected precision goal. Run III dataset was taken in a period in which im- portant components of the PIENU detector were inactive or uncalibrated. These represent the main limitations for which Run III had not been consid- ered in the global analysis before. The measurement Rπ of Run III will be performed by selecting pion events via special cuts, then modeling the pion signal using a time spectrum fit function to extract a raw measurement of Rπ. Finally, important correction factors to Rπ will be extracted from the Monte Carlo simulation. 1Acronym of the π+ → e+νe decay. 2Canada’s particle accelerator centre. 1 Chapter 1. Introduction The key content of this thesis is providing a clear description of the special features of Run III dataset and detailing the corresponding compen- sations in various stages along the analysis to obtain a stable measurement of Rπ. The relevant areas in which the thesis made relevant contributions to the PIENU analysis are the implementation of new techniques to calibrate gain shifts in the calorimeter for the pion decay measurement, the devel- opment of an event selection criteria to clean the pion decay signal from other background sources, the implementation of a pion beam profile in the Monte Carlo simulation, the study of the response function of the PIENU calorimeter using a positron beam, the improvement of the positron track reconstruction and the update of the additive correction of muons decaying- in-flight for Rπ. At the end of the thesis, the results of Rπ of Run III dataset will be presented with a summary of the main contributions. The effect of incorporating such a results to the global analysis will be discussed as well as the prospects of finding physics beyond the Standard Model with the achieved precision. The outline of the thesis is detailed below. In Chapter 2, the calculation of Rπ under the SM framework is de- tailed. Possible deviations of this estimate due to potential effects Beyond the Standard Model (BSM) are also described with emphasis on the new theoretical constraints that could allow the violation of Lepton Flavor Uni- versality (LFU). In Chapter 3 a full description of the PIENU experiment is given. This includes the a detailed review of the PIENU detector; the calorimeter, the positron and pion tracking devices, the dedicated triggers and the data-taking software. In Chapter 4 the data analysis stage is de- tailed: key features of the experimental setup and the data-taking stage of the Run III dataset, the main subject of this thesis project, are described. In Chapter 5 the time analysis to measure a raw estimate of the branching ratio (Rraw π ) is reviewed. In Chapter, 6 the Monte Carlo simulation of the PIENU experiment is established; the validation of the geometry and physical inter- actions in the simulation is proven via the response function of the PIENU calorimeter. Then, the low-energy tail and other important corrections for the calculation of Rπ are presented. In Chapter 7, the stability studies of Rπ against important parameters in the analysis and the systematic uncertainty sources are summarized. Finally, in chapter 8 a quantification of including Run III dataset to the global PIENU analysis in the Rπ precision is given. At last, a brief description of the next generation of the Rπ measurement is discussed. 2 1.1. Review of the history of the Rπ measurement 1.1 Review of the history of the Rπ measurement The first precise measurement of the branching ratio was performed in 1960 by Anderson et al, using a magnetic spectrometer [38]. The measurement R1960 π = (1.21±0.07)×10−4 represents a precision level of 5%, and it was in complete agreement with the SM and the V-A structure of the weak interac- tion. The second milestone came in 1964, in which Di Capua measured the branching ratio using a NaI(Tℓ) calorimeter sensitive to positrons as well as photons from radiative decays. This led to R1964 π = (1.247± 0.028)× 10−4, achieving a precision of approximately 2% [24]. Di Capua’s measurement was later updated to R1975 π = (1.274 ± 0.024) × 10−4, owing to a more ac- curate determination of the pion lifetime and a 2σ agreement with the SM prediction. The next generation of experiments started in in 1983 at TRIUMF by Bryman et al [25] using a larger NaI(Tℓ) calorimeter, measuring R1983 π = (1.218±0.014)×10−4 using data from 0.032M events and achieving a preci- sion level of 1% and a 1σ agreement with the SM. In 1992 TRIUMF carried the experiment E248 [26] using a NaI(Tℓ) crystal as calorimeter. Right after in 1993, PSI carried its own experiment using a 4π steradian Bismuth germa- nium oxide (BGO) calorimeter [27]. The two experiments together gathered around 0.19 M events. After their respective analyses were concluded, both experiments reached comparable levels of statistical and systematic uncer- tainties yielding as a combined ratio of R1994 π = (1.231±0.005)×10−4. This result had a precision level of 0.5% and was again within 1σ agreement with SM. The current generation of Rπ measurement is being performed at TRI- UMF and PSI [23] with similar precision goals. In 2015, the PIENU experi- ment at TRIUMFmeasuredR2015 π = (1.2344±0.0023(stat.)±0.0019(syst.))× 10−4 [6] using only 0.4 M events from the 5 M available. This result rep- resents a 0.24% precision level and keeping the 1σ agreement with the SM. The current average reported by the Particle Data Group (PDG) is R2018 π = (1.2327 ± 0.0023) × 10−4 [44] to a precision level of 0.19% and includes all measurements from 1986 to 2015. Recently in 2019, a blind analysis was carried using 3 M events from the 5 M available from data- taking years 2011, 2012 and a portion of 2010. This analysis resulted in a stable measurement of Rπ to a 0.12% precision level. Once finalized the PIENU experiment, the Rπ measurement will have a precision level of 0.1% or better. A review of the experimental measurement of Rπ through the years is shown in Figure 1.1. 3 1.1. Review of the history of the Rπ measurement Figure 1.1: Measurement of Rπ branching ratio throughout the years. The red line represents the SM prediction [37] and the black dashed line is the current PDG experimental average [44]. 1.1.1 Lessons from the E248 experiment Prior to the current generation of experiments, the E248 experiment used a different approach to measure the decay positrons. Figure 1.2 shows a schematic of the E248 experimental setup [26]. The main detector was a cylindrical NaI(Tℓ) crystal whose axis was oriented at 90◦ with respect to the beam. This configuration was prompted to significantly reduce the contamination from background beam particles. Under this configuration, the solid angle was only 2% from 4π steradians. The experiment collected about 0.1 M π+ → e+νe events to measure a branching ratio R1992 π = (1.2265± 0.0034(stat.)± 0.0044(syst.))× 10−4 [26], [27]. The main systematic uncertainty came from the estimate of the π+ → e+νe low-energy tail buried under the π+ → µ+νµ → e+νeνµ spectrum. The approach to estimate the tail fraction consisted in suppressing the π+ → µ+νµ → e+νeνµ events. Figure 1.3 shows the resulting energy spectrum after the suppression analysis. There is a clear remainder of π+ → µ+νµ → e+νeνµ events, in most of them the pion decayed-in-flight (πDIF) before reaching the target and deposited a smaller amount of energy. For these 4 1.1. Review of the history of the Rπ measurement Figure 1.2: Schematic of the experimental setup of the E248 experiment carried at TRIUMF. Taken from [26]. Figure 1.3: Positron energy spectrum in the E248 calorimeter after sup- pressing π+ → µ+νµ → e+νeνµ events, the x-axis is energy in ADC units (channel 3400 corresponds to 56.4 MeV). Taken from [26]. 5 1.2. Overview of the PIENU technique events in the suppressed spectrum, the sum of the energy deposited in the target resembles to the energy deposited by π+ → e+νe events. Therefore, they could not be removed using a target energy cut, which was the only method to reject πDIF events since the detector was not equipped with upstream tracking capabilities. The π+ → e+νe tail fraction below the energy cut that separates these events from π+ → µ+νµ → e+νeνµ events was about 20%. This estimate was limited by the size of the data sample. The flaws from the E248 experiment were considered in the design of the PIENU experiment for a better control on the sources of systematic uncer- tainties. To increase the statistics, the calorimeter was placed downstream following the target scintillator, thereby increasing the angular acceptance of the isotropic decay positron tracks. Tracking devices were added before the target to reconstruct pion tracks and identify πDIF topologies and reduce the systematic uncertainty of the low-energy tail. Additionally, the PIENU detector was designed with the ability to rotate the calorimeter relative to the beam angle. This modified setup allows to measure the response func- tion of the calorimeter and helps to validate the detector geometry in the Monte Carlo simulation, constraining more the systematic uncertainty. 1.2 Overview of the PIENU technique The same fundamental technique has been used for every branching ratio measurement since Di Capua’s experiment in 1964 [24]. One stops a charged pion beam in a scintillator target, thick enough to allow the pion to decay within it to either the π+ → µ+νµ → e+νeνµ decay chain or the π+ → e+νe direct decay. A calorimeter located downstream the target captures the decay positron from either channels and measures its energy, as shown in Figure 1.4. The decay channels can be distinguished by measuring the decay time; the pion lifetime at τπ = 26.0 ns is two orders of magnitude smaller than the muon lifetime at τµ = 2197.0 ns. Energy-wise, the two decay channels are identified in a separate energy range. While the π+ → e+νe positrons give rise to a mono-energetic peak at 69.8 MeV, the positrons from the π+ → µ+νµ → e+νeνµ decay chain produce a broad energy distribution, known as Michel spectrum, between its rest mass 0.511 MeV and a sharp endpoint at half the muon’s mass of 52.8 MeV. The PIENU detector is fully described in Chapter 3 and Reference [8]. The PIENU apparatus is divided into an upstream and a downstream as- sembly for the measurement and tracking of the pion and decay positron, 6 1.2. Overview of the PIENU technique Figure 1.4: Schematic of the PIENU experimental technique. Pions stop in the target and decay into either muon or positron channels; muons also stop in the target, and decay into positrons. Positrons, and photons if any are produced, are detected by the calorimeter. respectively. A pion beam with momentum 75± 1 MeV/c [7] is produced at from the TRIUMF M13 beamline and injected into the detector. The pions were detected by the plastic scintillators B1 and B2, and tracked into the target ”B3”. The beam momentum is tuned so that the pions will stop in the middle of target B3. Muons from pions decaying at rest have a pene- tration range of 1 mm within B3. Therefore, there is sufficient material to contain the decay vertices from both π+ → µ+νµ → e+νeνµ and π+ → e+νe channels. Figure 1.5 shows the simulated energy spectra in the target for both decay channels. Although this quantity is not used in the analysis to distinguish the decay channels, it is useful to implement a blinding function, as described in Section 1.3. Positrons from both decay channels enter the PIENU calorimeter. The calorimeter consists of a 48 cm diameter × 48 cm long crystal made of Thallium-doped sodium iodide with 19 radiation-length long and an array of two CsI concentric rings made of 97 single crystals. This array was placed to intercept the energy leakage due to electromagnetic shower. Figure 1.6 (left) shows the simulated energy spectra in the calorimeter for both decay channels. A first estimate of the branching ratio is done by modeling the 7 1.2. Overview of the PIENU technique Figure 1.5: Simulated energy spectrum in the target B3 for π+ → e+νe (blue distribution) and π+ → µ+νµ → e+νeνµ decay channels. Figure 1.6: Simulated (Left) time spectra and (right) energy spectra in the PIENU calorimeter of π+ → e+νe and π + → µ+νµ → e+νeνµ channels. The spectra are normalized to the same amplitude. Using an energy cut above the π+ → µ+νµ → e+νeνµ energy spectrum edge, shown as a dashed line, the energy spectrum is separated into low-energy and high-energy regimes. The low energy tail from the π+ → e+νe energy spectrum is not visible due to scale. The π+ → e+νe time distribution peaks near t = 0 as a consequence of the short pion lifetime. 8 1.3. Blind analysis decay signals, shown in Figure 1.6 (right), and the various backgrounds that can not be suppressed by cuts in the data analysis stage. The time spectra of the events in the low and high-energy regions are fitted simultaneously with the functions that model the backgrounds in each energy regime. The result of this fit provides the raw branching ratio to which a number of corrections have to be applied. The time fit process is fully described in Chapter 5. The raw branching ratio is corrected by the π+ → e+νe low-energy tail, which is not included in the time fit functions. Other corrections related to the detector’s acceptance and the contribution of muon decay-in-flight events. These corrections are calculated using the Monte Carlo simulation of the PIENU detector. This simulation is validated via a modified setup of the PIENU experiment, known as Lineshape, in which the calorimeter is rotated with respect a positron beam coming from the M13 beamline to measure the detector response to be later compared with the simulation results. Chapter 6 describes the Lineshape validation and the calculation of the low-energy tail, acceptance and muon decay-in-flight corrections. By combining the raw estimate of the branching ratio and its corrections, a measurement of Rπ is provided. As a final part of the procedure, the measurement of Rπ is tested against several parameters involved in the analysis (e.g. the energy cut). Ideally these systematic checks should lead to no sensitivity in Rπ. Measuring the ratio of the decay rates or the energy distributions does not require knowledge of the total number of incoming pions, as positrons from both decay chains are measured regardless of the mode. Most efficiencies of the cuts and triggers cancel in the measured ratio of decay, thus reducing the systematic uncertainty. 1.3 Blind analysis A blind analysis is a research tool in whose main aim is to prevent any un- conscious or conscious influence from external knowledge, human conscious or unconscious bias or expectations. A clear example of a potential bias are the experimental results of the neutron lifetime measurements depicted in Figure 1.7, [44]; the good agreement between sets of measurements may be an indication of technique biases. The implementation of a blind analysis is performed to avoid bias, ensure objectivity, and maintain the integrity of the results. This method is especially useful in a high precision measure- ment that will then be compared with a precise theoretical prediction. The blindness can be implemented in different stages of the analysis of particle experiments such as data collection, data labeling blindness or calibration, 9 1.3. Blind analysis Figure 1.7: Measurement of the neutron lifetime throughout years [21S]. Figure 1.8: Blinding technique in the PIENU experiment: a smooth inef- ficiency function (unknown in the analysis) removes events based on their energy deposited in the target, lowering (case a) or raising (case b) the branching ratio. depending on the experiment design, as discussed in many papers [97] [98]. In any case, the blinding procedure should not artificially hide or create new systematic effects that would sabotage the analysis. The high-precision measurement of Rπ that the PIENU experiment aims to provide will be compared to the Standard Model theoretical prediction. 10 1.3. Blind analysis Therefore, a method for blinding the branching ratio at the stage of data processing was developed to mitigate potential biases in the analysis proce- dure. The ideal blinding procedure would randomly alter the quantity being measured without affecting the data in any other way. In the PIENU anal- ysis, a natural choice for a blinding quantity that is not used in the analysis but in which Rπ depends on is the energy deposited in the target counter [89]. In order to randomly shift the branching ratio, an inefficiency function was applied to the target energy. An unknown factor between 0 and 0.5% of events was excluded from the analysis in a region of the target energy spectrum containing mostly either π+ → e+νe or π + → µ+νµ → e+νeνµ de- cay events; the region in which events were excluded was chosen randomly. The blinded events will be included in the analysis as the blinding factor is removed once all the event selection, the time spectrum fit, the branching ratio corrections and the systematic uncertainties are finalized. Figure 1.8 shows the schematic of the blinding method in the PIENU experiment. A smooth rectangular function (red line in Figure 1.8) with hidden efficiency was used to remove π+ → e+νe or π+ → µ+νµ → e+νeνµ events indistinguishably. The blinding factor does not significantly affect the branching ratio stability tests, as the target energy is almost independent of other quantities used in the analysis, and the blinding factor is small enough that the dependence that may exist is negligible at the level of precision of the experiment. The same procedure was applied uniformly to all datasets so that they can be compared in systematic tests. 11 Chapter 2 Theory background 2.1 The standard model The standard model of elementary particles is the current theoretical frame- work that describes the most basic building blocks of matter and the inter- actions between them. Its early construction dates back to the mid-1960’s with Murray Gell-Man and Gerge Zweig proposing the idea of quarks as the fundamental pieces of mesons and baryons. This formulation was extended by David Plitzer and David Gross with the development and introduction of the color theory and the strong interaction. Later on, the measurements of SLAC and BNL concerning the J/ψ particle in 1974 [11][12] and the dis- covery of the tau lepton in 1976 [118] confirmed the existence of the quarks in bound states and the existence of a third lepton family. These discov- eries validated the formulations of quantum chromodynamics (QCD) and quantum electrodynamics (QED) as field theories and the foundations of a growing Standard Model. The standard model (SM) is a gauge theory based on the group SU(3)× SU(2)× U(1). (2.1) Each group corresponds to an interaction in the SM. The non-Abelian group SU(3) belongs to the strong interaction and the group SU(2)× U(1) is the unification of the electroweak interaction. Generation Charge I II III (q/|e|) Leptons e = 0.511 MeV/c2 µ = 105.658 MeV/c2 τ = 1.77686(12) GeV/c2 -1 νe < 2 eV/c2 νµ < 0.17 eV/c2 ντ < 18.2 eV/c2 0 Quarks u = 2.2(5) MeV/c2 c = 1.275(35) GeV/c2 t = 173.0(4) GeV/c2 +2/3 d = 4.7(5) MeV/c2 s = 95(9) MeV/c2 b=4.18(4) GeV/c2 -1/3 Table 2.1: Fermions (spin = 1/2). 12 2.1. The standard model Interaction Mediator Mass Coupling Range Strong gluon, g massless αs = 1.7 (at 100 MeV) 10−15 m Electromagnetic photon, γ < 1×10−8 eV α = 1/137 ∞ Weak W± 80.38(1) GeV/c2 GF ∼ g2/m2 W Z0 91.1876(21) GeV/c2 GF = 10−5 GeV−2 Higgs H = 125.11 GeV/c2 0 Table 2.2: Gauge bosons (spin = 1) and Scalars (spin = 0). The SM particles are classified into fermions, semi-integer spin particles that can be arranged in two groups of three families each, and gauge bosons, integer spin particles that mediates the interactions between fermions and/or with other bosons. Each particle is a point-like object that results from the excitation of a quantum field. Fermions are subdivided into leptons and quarks, as shown in table 2.1. Each particle has its own antiparticle with the opposite electric charge and the same mass. Gauge bosons are in charge to mediate interactions between particles. The strength and range of the interaction is given by the gauge coupling associated with each quantum field. Table 2.2 shows the properties of the gauge bosons included in the SM. Besides the fermions and bosons summarized in tables 2.2 and 2.1, there is another quantum field whose excitations produce a scalar particle called Higgs boson, with mass and charge shown on the bottom of Table 2.2. All massive particles in SM acquire their masses from the interactions of their respective fields with the Higgs field through a symmetry-breaking mecha- nism [13]. Its existence was confirmed by the ATLAS and CMS experiments at the Large Hadron Collider [14] [15]. 2.1.1 Inter-generational mixing in the SM The current understanding of the SM lies on symmetries of the mathemat- ical groups describing the SM. These symmetries lead to the conservation of a physical quantities, according to the Noether’s theorem. The symme- try of the group SU(3) under a the gauge transformation, containing the Gell-Mann matrices, allows the conservation of the color charge intrinsic of quarks and gluons. Analogously, the symmetry of the group U(1) under an inner complex rotation allow the conservation of the electric charge [117]. 13 2.1. The standard model However, there are quantities that are not conserved in the SM such as the quark flavor. Indeed, the weak interaction is responsible for an inter- generational mixing in the quark sector. This is an important feature for the meson decays. This mixing is characterized by the Cabbibo-Kobayashi- Mashkawa (CKM) matrix, VCKM [17][18]. This is the common origin for the flavour mixing and the masses of the quarks in the SM. They VCKM elements arise from the Yukawa interactions with the Higgs field [19]. Both Kobayashi and Mashkawa were awarded with the Physics Nobel Prize in 2008 for this discovery. The CKM matrix is defined as follows:   d′ s′ b′   = VCKM   d s b   =   Vud Vus Vub Vcd Vcs Vcb Vtd Vts Vtb     d s b   , (2.2) Where (d′, s′, b′) are the weak eigenstates or the members of the weak isospin doublets and (d, s, b) their mass eigenstates. The formulation of the SM does not provide the components of VCKM, so they have to be determined experimentally. The latest results give the following modules of the CKM matrix elements [16]: VCKM =   0.97373± 0.00031 0.2243± 0.0008 (3.28± 0.020)× 10−3 0.221± 0.004 0.975± 0.006 (40.8± 1.4)× 10−3 (8.6± 0.2)× 10−3 (41.5± 0.9)× 10−3 1.014± 0.029   (2.3) These values are consistent with the unitarity of the CKM matrix. The VCKM entries are part of the free parameters in the SM such as mixing angles, masses and coupling that have to be determined or bounded ex- perimentally. Equation 2.3 show the amplitude of the CKM elements for simplicity. In reality these entries are complex numbers that can be written as Vij = |Vij | exp (iϕij), where ϕij are the relative phases between differ- ent elements that encode information about the strength of the quark flavor transitions. These phases are constrained to assure the unitarity of the CKM matrix. The CKM matrix introduces joint violation of two important symme- tries; the parity transformation (P) and charge conjugation (C), called CP violation, through complex phases, particularly in the transitions between different generations of quarks. A nonzero phase in any of the elements 14 2.1. The standard model of the matrix can lead to different behavior for particles and antiparticles, which is the essence of CP violation. 2.1.2 Electroweak interaction The electroweak sector is the unification of two of the fundamental forces in the SM; the electromagnetic interaction and weak interaction. The electro- magnetic force (EM) acts on fundamental particles or on hadrons carrying electric charge through a massless photon, γ. This interaction leaves the charge of the particle unaltered. The theory that describes the EM force is quantum electrodynamics (QED) whose foundations are based on the group U(1). The EM interaction explains the most common physical phenomena in everyday life such as the structure of atoms, the light scattering or chem- istry in general. QED gives exceptionally accurate predictions for quantities such as the magnetic moment of the electron, muon, proton or neutron [115] and the Lamb shift of the hydrogen energy level [20]. On the other hand, the weak interaction act on left-handed fermions and right-handed anti-fermions through the mediatorsW± for a charged current or Z0 for a neutral current decay channel. The fundamental framework of the weak force is Quantum flavor-dynamics (QFD) that provides a solid un- derstanding of the weak processes under the group SU(2). Besides the beta radioactive decay, n→ p+ e− + νe, the weak force is responsible of leptonic decays of the lightest hadron π±, whose measurement is the object of study of this thesis. Another important characteristic of the weak interaction is that due to its intrinsic preference to act on fermions of certain handedness, the parity P and charge-parity C symmetries are violated [21],[22]. One famous instance of parity violation is the so-called τ − θ puzzle. It consists in the finding that the Kaon K+ decays into two final states with opposite parity [116]. 2.1.3 Strong interaction The strong interactions act through a gauge boson called gluon on hadrons or any bound state that involves quarks such as protons (p), neutrons (n) or pions (π+, π−, π0). Quantum chromodynamics describe the strong interac- tion with the SU(3) Lie groups. Since SU(3) is a non-Abelian group, then the strong interaction is distinguished by the fact that the gauge bosons, gluons, are able to interact with themselves. The strong interaction is re- sponsible to the nuclear binding. This process is essential for the formation of a wide variety of nuclei, that have been possible in the core of young stars, 15 2.1. The standard model where the gravitational pressure is high enough to push quarks and gluons together. The strong interaction works in nuclear fission as well, which is the fundamental principle for the nuclear reactors to operate. Unlike the electroweak interaction, the strong coupling depends on the amount of transferred momentum in each vertex, becoming smaller with increasing energy in particle interactions. An important consequence of this is that at low energies quarks are confined in uncolored bound states (hadrons). In the low energy regime the QCD coupling constant, αs, cannot be considered small, meaning that a perturbative treatment is not possible. Non-perturbative methods such as Chiral Perturbation Theory (ChPT) and Lattice QCD have been used for strong interaction calculations in the low energy regime. 2.1.4 V-A theory The weak interaction plays a main role in light hadron decays to a lepton current, more specifically in the decay of mesons π±. In fact, an important feature of the weak sector is that the charged bosons W± only couple to left-handed species and this shapes the SM. To understand how, we recall Fermi’s assumption when considering the β-decay, n → p + e− + νe: At a single point in space-time, the wavefunction of the neutron becomes that of the proton and the wavefunction of the incoming neutrino is transformed into that of the electron. The description of this process is given by the amplitude M = GF ( ψpΓψn ) ( ψeΓψν ) , (2.4) where the factors Γ are responsible for particle transformation, and GF is the very well known Fermi coupling constant which determines the strength of the weak interaction. In 1956 Feynman and Gell-Mann proposed that the interaction factors Γ as a mixture of vector and axial-vector quantities, to account for parity violation in weak interactions. Considering also the fact that leptons and anti-leptons have opposite helicity states in weak decays that discard the S and T operators as well, the Γ operator can be reduced to: Γ = 1 2 γµ (CV + CAγ5) (2.5) 16 2.2. Pions in the Standard Model Where γ are the Dirac matrices and CV and CA are arbitrary coefficients. Many experiments at that time, in particular by the fact that neutrino is left-handed, meaning that its spin is antiparallel to its momentum, lead to CA = CV , leaving the universal weak operator: Γ = 1 2 γµ (1− γ5) . (2.6) This shows a maximal parity violation for the charge changing weak interactions. 2.2 Pions in the Standard Model 2.2.1 Pion decay modes Mesons are quark-antiquark pairs that are bounded together by the strong interaction. Pion are mesons made of quarks from the first generation; π+ is a bound state of an up (u) and an anti-down (d) quark; π− is a bound state of an anti-up (u) and a down (d) quark; and π0 is a combination of a u with an u or d with an d quark. Since pions are the lightest particles made of quarks, they can only decay via weak interactions. The π± has mass 139.57 MeV/c2 and a mean lifetime of 26.033 ns. It can only decay into lighter leptons, i.e., either a muon with a 105.658 MeV/c2 mass and a 2.2 µs mean lifetime or an electron with a 0.511 MeV/c2 mass. Decay mode Γπ l /Γ π Observations Γπ 1 π+ → µ+νµ 0.99987±0.00004 Γπ 2 π+ → µ+νµγ (2.00±0.25)×10−4 Eγ > 1 MeV Γπ 3 π+ → e+νe (1.230±0.004)×10−4 Γπ 4 π+ → e+νeγ (7.39±0.05)×10−7 Eγ > 10 MeV; θeγ > 40◦ Γπ 5 π+ → π0e+νe (1.036±0.006)×10−8 Γπ 6 π+ → e+νee +e− (3.2±0.5)×10−9 Table 2.3: Pion decay modes reported in [28]. The restrictions correspond to the cited experiment. Only the first four decay modes listed in the table 2.3, π+ → µ++ νµ(γ) and π+ → e++νe(γ), are relevant to the PIENU experiment. The remaining decay channels are of the order 10−8 or lower which is below the PIENU precision goal of 0.1%. 17 2.2. Pions in the Standard Model 2.2.2 Pion decay rate The π+ decay can be described with the Feynman diagram shown in figure 2.1. From a quark perspective it describes a scattering process u+d→ l+νl with l = e, µ. However, the pair ud can be effectively treated as a bound state, which couples with the W+ boson represented by the inner dashed line. The propagator of the W boson is igµν M2 W − q2 ∼ i gµν M2 W , (2.7) whereMW is the mass of theW+ boson, gµν is the Minkowski metric and the approximation made considers a small momentum transfer, compared to the mass of the W+ boson. The Lagrangian density describing the interaction mediated by the boson W+ is given by [29] LW+ = ig 2 √ 2 W+ µ (νmγ µ(1− γ5)qm + Vmnu ′ nγ µ(1− γ5)d′m), (2.8) where W+ µ are the generators of the group SU(2), associated with the elec- troweak sector, ν, q, u and d are the Dirac spinors, m is an index that runs over generations (q1 = e, q2 = µ and q3 = τ), Vmn are the CKM matrix elements, which link the interaction basis with the mass basis. Finally, γµ and γ5 are the Dirac matrices that satisfy the algebra: {γµ, γν} = 2gµνI4, γ5 = iγ0γ1γ2γ3. (2.9) The term (1−γ5) in equation 2.8 is responsible to the parity violation of the weak interaction, while the implicit sum over Vmn is responsible to the flavor change. A key piece to calculate the pion decay rate is the invariant amplitude M = ⟨l+νl|L|π+⟩, where l = e, µ and L is the lagrangian density from 2.8. Considering the approximation in 2.7, followed by expanding M in hadronic and leptonic parts we have: M = iGFVud√ 2 ⟨0|d(γµ(1− γ5)u|π⟩l(pl)γµ(1− γ5)ν(pν), (2.10) 18 2.2. Pions in the Standard Model Figure 2.1: Feynman diagram of the charged pion decay π+ → l+νl with l = e, µ. where GF = 1√ 2 ( g 2MW )2 is the Fermi constant, g is the coupling constant of the W boson to the lepton current and pl and pν are their respective four-momenta. The first part of the bra-ket in equation 2.10 links the pseudo-scalar pion with the scalar vacuum, hence the vector part of the weak interaction results in an odd-parity term that vanishes after the inte- gration over momenta is completed. The remaining term involves the strong interaction, however, since the pion decay rate is a Lorenz-invariant quan- tity, then the contraction of this element with itself must be Lorenz-invariant as well. Considering that the pion has zero spin, a plausible possibility is to associate this term to the transferred momentum to the W+ boson [30] ⟨0|dγµγ5u|π+⟩ = iFπq µ, (2.11) where Fπ parameterize the strong interaction and is known as the pion decay constant. According to the Fermi’s golden rule, the differential pion decay rate is [31] dΓ = 1 2mπ |M|2 1 ElEν d3pl (2π)3 d3pν (2π)3 (2π)4δ4(q − pl − pν), (2.12) where mπ is the pion mass, El and Eν are the lepton and neutrino energies, respectively. After performing the integration over the entire phase space and summing over the spin configurations of the lepton and the neutrino, the π+ decay rate at tree level is: 19 2.2. Pions in the Standard Model Γ0 π→l = 1 4π G2 F |Vud|2mπF 2 πm 2 l ( 1− m2 l m2 π )2 . (2.13) This decay rate is a function of the pion’s decay constant Fπ, that in- volves strong interaction calculations that are not very accurate. However, the ratio of decay rates to positron and muon channels is the first order branching ratio R0 π that does not depend on Fπ. R0 π = Γπ→e Γπ→µ = g2e g2µ m2 e m2 µ ( m2 π −m2 e m2 π −m2 µ )2 = (1.2833± 0.00002)× 10−4. (2.14) According to the SM, the coupling constant is the same (ge = gµ). This fact is known as lepton universality. The sources of the first-order branching ratio error are the uncertainties of the muon and positron masses. 2.2.3 Helicity suppression Since the positron has a mass nearly two hundred times smaller than the muon’s mass, the phase space of the decay through the positron channel is larger. This means that the integration of 2.12 should result in a larger pion decay rate through the positron channel, compared to the muon channel. The reason for which this does not affect the order of the branching ratio is the helicity suppression. The result in 2.14 contains the difference of the squared masses. This term is responsible for the small magnitude of the branching ratio and it comes from the term 1−γ5 in the Lagrangian density 2.8 that allows uniquely right-handed anti-leptons and left-handed neutrinos to emerge. For massless fermions, the right or left handedness is equivalent to the helicity, the spin direction relative to its momentum direction. Left-handed massless fermions have a negative helicity, while right-handed massless fermions have a positive helicity. For massive fermions both spin states are possible, but there is a sup- pression factor that depends on the energy. As the fermion starts exceeding its rest mass it starts behaving like a massless particle. In the pion at rest system, the neutrino and the anti-lepton emerge back to back; since the pion has spin zero, the neutrino and the anti-lepton are forced to have the same state of helicity. The allowed spin directions of the pion decay are shown in 20 2.2. Pions in the Standard Model Figure 2.2: The allowed directions of the spin and momenta in the pion decay products: pion (middle), anti-lepton (right) and neutrino (left). The conservation of the angular momentum requires the spin of the positron and the neutrino to be in opposite directions, which is indicated by the black arrows above the product particles. figure 2.2. Both for positron and the muon decay channel, there is a sup- pression factor due to the effect described above, this is 1− ( vl c )2 ∼ ( 2ml mπ )2 , where vl and ml are the velocity and the mass of the outgoing lepton [30]. This is the reason of the factor ( me mµ )2 in Eq. 2.14. 2.2.4 Radiative corrections The common denominator of the measurements of the branching ratio Rπ is that they are observed along with higher-order effects such as radiative processes. This includes decays in which the photon is actually a product of the point-like pion decay, which are commonly referred as Inner Bremm- srtahlung photons (IBγ) and pion decays in which the virtual photon is emitted and reabsorbed (ERγ). The possible configurations of these two radiative processes are shown in figure 2.3. The pion radiative decay is the first high-order corrections to the branching ratio Rπ, the contribution of the processes IBγ and ERγ was first calculated by Berman [32], and Kinoshita [33] in late 1950s assuming a point-like pion. The infrared and ultravio- let divergences ccoming from the phase space integration were handled by imposing energy cutoffs, whose effect on Rπ can still be estimated. The in- frared divergence terms from IBγ and ERγ cancel each other. On the other hand, the ultraviolet divergence cancels in branching ratio if the same cutoff is taken for both decay channels. The overall correction to Rπ was -3.929%. 21 2.2. Pions in the Standard Model This contribution was confirmed in 1976 by Marciano and Sirilin [34], who expanded the pion decay rate in powers of the lepton mass ml and its logarithm lnml, with l = e, µ. This process was held to consider the fact that the pion has an energy dependent structure made of a quark anti-quark pair, meaning that the photon, either real or virtual, can be emitted from the inner structure of the pion. In was found that the leading leading mass term is independent of strong interactions. This finding is consistent with the early calculation done by Kinoshita and Berman. A further assessment of the uncertainties from not well-known theoretical parameters involved in the calculation lead to result Rπ = (1.2352± 0.0005)× 10−4 [35]. The calculation of the Rπ corrections was later updated by Cirigliano and Rosell in 2007 using Chiral Perturbation Theory (ChPT) to improve the hadronic structure dependent effects [36]. Within ChPT, the invariant amplitudes of the radiative processes can be expanded in powers of the external masses and momenta and powers of the electromagnetic coupling. This expansion is preserved throughout the calculation of the branching ratio resulting in Rπ = R0 π [ 1 + ∆e2p2 +∆e2p4 +∆e2p6 + . . . ] [1 + ∆LL] , (2.15) where ∆e2pn are the addends in the expansion in powers of the pion mo- mentum and the electromagnetic coupling constant e and the ∆LL correction was introduced to incorporate the effects of the leading terms αn lnn(mµ/me). The leading term in the expansion is ∆e2p2 = α π { F ( me mµ ) − F ( mµ me )} , (2.16) which is consistent with the former calculation of the radiative correc- tions by Kinoshita and Berman, treating the pion as a point-like particle. The next term ∆e2p4 is the limit for the calculation and contains the pure structure-dependent radiative corrections with a prominent uncertainty in the prediction. The term ∆e2p6 is important since it comes from the emission of the photon from the decaying pion and evades the helicity suppression. Table 2.4 reports the size of each term within overall the correction [36] that leads to the branching ratio Rπ = (1.2352± 0.0001)× 10−4. The uncertainty from the two-loop diagrams contributions of O(α2) terms was updated in 2011 by Bryman et al., reporting an additional 0.01%. After summing up the uncertainties in quadrature, the branching ratio is [37] 22 2.2. Pions in the Standard Model b)a) Figure 2.3: Feynman diagrams for the radiative corrections to pion decay rate, of (a) Inner Bremmstrahlung (IBγ) and (b) Emission Re-absorption (REγ) processes. Rπ = (1.2352± 0.0002)× 10−4. (2.17) This result is in agreement with all previous predictions. This current calculation has a theoretical uncertainty of 0.016%. This level of precision is possible because the strong interaction effects vanish in the ratio of the pion decay rates and is only visible through the radiative corrections. Any 23 2.3. Physics beyond the Standard model Term Correction from [36] ∆e2p2 -3.929 ∆e2p4 0.053±0.011 ∆e2p6 0.073 ∆LL 0.054 Table 2.4: Summary of the radiative corrections to Rπ. effect of new physics at the weak (∼TeV) scale on Rπ is expected to be in the range of 1% to 0.01%. Therefore, it is mandatory to reduce the experimental uncertainty to observe phenomena beyond the standard model. The main candidates concerning deviations of Rπ within the precision of the SM are summarized in the next section. 2.3 Physics beyond the Standard model The SM has proven incredibly successful in describing phenomena in nature that has been confirmed by various experiments. The most famous example, is the agreement of the SM prediction and the experimental measurement of the electron magnetic dipole moment to a precision level of one over one hundred billion [39]. Another milestone of its success was the prediction of the W and Z boson’s mass 10 years before their observation in 1983 at the CERN pp collider. Since then, no experiment has reached the 5σ threshold necessary to claim a discovery outside the SM framework [41]. In spite of its successes, the SM has several flaws suggesting that it is a part of a more robust and complete description of nature. Oscillation of solar neutrinos observed in SNO [113] confirmed that these leptons do have mass. Cosmological observations give evidence of dark energy [42] and dark matter [41], both of them were attempted to be described within the confines of the SM without success. The asymmetry of observed antimatter compared to the matter caused by the CP violation is many times larger than the prediction of the SM [43]. The Higgs mechanism provides particles their mass but it does not predict the actual magnitude. Finally, the SM is unable to describe gravitational interaction between massive particles. Several extensions of the SM have been proposed to mend this and they are part of a framework beyond standard model (BSM). Due to the lack of any experimental evidence to indicate the nature of this theory, BSM remains as a compendium of proposals. The high precision measurement of 24 2.3. Physics beyond the Standard model Rπ can provide a natural way to test some of the BSM extensions. Examples include, lepton universality violation,and new pseudo-scalar interactions, in- cluding R-parity violating super-symmetry, leptoquarks, and charged Higgs (non-SM coupling). These will be summarized in the next sections. 2.3.1 Violation of the lepton universality An important feature of the SM is the existence of three fermion generations, the only difference between them is their mass. Because of this, the free Lagrangian of left-hand doublets and right-hand singlets in the electroweak sector can be written as LW = 3∑ j=1 iψj(x)γ µDµψj , (2.18) where Dµ is the covariant derivative and the index j sums over generations. The invariance of the global Lagrangian under gauge transformations of the group SU(2) × U(1) can only be assured if the global coupling of the left doublets to the gauge bosons Wµ is the same for the three generations (g = ge = gµ = gτ ). This result is known as the lepton flavor universality (LFU) and is a direct consequence of the global gauge invariance of the SM Lagrangian. This result allows to reduce R0 π to the mass ratio in equation 2.14. However, assuming that this is not true, at least to the precision level reached by the SM, the branching ratio expression becomes RSM π = ( gµ ge )2 Rexp π (2.19) Therefore, the measured Rexp π and the SM prediction RSM π provide a natural way to test the ratio of couplings gµ/ge for the electron-muon uni- versality. This indeed is the most precise measurement since the pion is the lightest meson and with least decay modes than other mesons. A higher precision level in the experimental measurement mean a higher constraint in this ratio. By reaching a precision level < 0.1%, PIENU would reach to test the LFU to a ∼0.05%. By measuring the branching ratio of kaons Γ[K+ → µ+νµ(γ)]/Γ[K + → e+νe(γ)], the experiments NA62 and KLOE [44] reported a precision of 0.18%. The constraints for different coupling ra- tios depend on different types of precision measurement experiments. Table 2.5 summarizes the most recent experimental values for the universality test using π, τ, B and W . 25 2.3. Physics beyond the Standard model Decay channel gµ/ge Bπ→µ/Bπ→e 1.0004 ± 0.0012 [6] Bτ→µ/Bτ→e 1.0018 ± 0.0014 [48] BK→µ/BK→e 0.996 ± 0.005 [49] BK→πµ/BK→πe 1.002 ± 0.002 [50] BW→µ/BW→e 0.997 ± 0.010 [50] gτ/gµ Bτ→eτµ/ττ 1.0011 ± 0.0015 [48] Bτ→π/Bπ→µ 0.9963 ± 0.0027 [48] Bτ→K/BK→µ 0.9858 ± 0.0071 [48] BW→τ/BW→µ 1.039 ± 0.013 [50] gτ/ge Bτ→µτµ/ττ 1.0029 ± 0.0015 [48] BW→τ/BW→e 1.036 ± 0.014 [50] Table 2.5: Experimental results of LFU tests performed on pion, kaon, tau y W decays. Here B represents the branching fraction of a particular decay mode. The LFU violation (LFUV) has been widely discussed in recent years in light of a pattern of anomalies, all pointing to LFUV. The observations indicate that electrons, muons, and tau leptons have a different behavior than the expected within the SM framework. This suggests that future tests of LFU will play a key role in elucidating new physics (NP). Some of the most prominent evidence pointing to the LFUV is summarized as follows [51]. b → sµ+µ− These are a set of process that are observed through scattering processes of bound states involving a quark interchange of the form b → sµ+µ−. The ratio of decay rates R(K(∗)) = Γ[B → K(∗)µ+µ−]/Γ[B → K(∗)e+e−] are of particular interest since they are expected to be approximately one in the SM [52], with small theoretical uncertainty. while the measured values lie significantly below unity. The most precise measurement by LHCb [53] reported a measurement significantly below the unit with a significance of 3.1σ. 26 2.3. Physics beyond the Standard model b → clν Analogously to R(K(∗)), the ratio R(D(∗)) = Γ[B → D(∗)τν]/Γ[B → D(∗)lν] show deviations from the SM predictions with a combined significance of about 3σ [54] provided by the Heavy Flavor Averaging Group. Here the transitions are not suppressed, therefore the BSM effects would have to be large, and the associated energy scale low for the LFUV to manifest. Cabbibo Angle Anomaly (CAA) and qq → e+e− The CAA has been recently prompted by the observation of fewer β-decays than expected in a survey of measurements with similar precision with a global significance of 3σ [55]. The relation between CAA and LFUV can be clarified as follows. The determination of Vud from β-decays is affected when the Wµν coupling is modified [57]. When this modification is not com- pensated in the coupling Weν , the ratios of decay rates Rπ = Γ(π→eν) Γ(π→µν) or Rτ = Γ(τ→eνν) Γ(τ→µνν) , which provide the best tests of LFU, would be affected. In contrast, the CMS experiment at CERN observed more high-energetic elec- trons in proton–proton collisions (qq → e+e−) compared to high-energetic muons than expected [56], pointing again to a different coupling magni- tude. The experimental bounds for LFUV can be interpreted in terms of constraints to physics BSM. Frameworks involving effective field theory and scenarios of new physics are summarized as follows. Modified Wlν couplings According to Loinaz et al. [61], the couplings gl (with l = e, µ, τ) can be parameterized as follows to quantify the current bounds gl → g ( 1− ϵll 2 ) . (2.20) where the double index in the deviations ε indicate the absence of flavor mixing. The linear combinations of ϵll constrained by W, τ , π and K decay measurements are given by gµ ge = 1+ ϵee − ϵµµ 2 , gτ gµ = 1+ ϵµµ − ϵττ 2 and gτ ge = 1+ ϵee − ϵττ 2 . (2.21) . Under this parametrization, the Fermi constant of the modifiedWlν cou- plings, GL F , can be written in terms of the SM Fermi constant and the devi- ations in 2.21 as 27 2.3. Physics beyond the Standard model Figure 2.4: Left: Global fit in the ϵττ − ϵµµ vs ϵee − ϵµµ plane, including kaon, pion, and tau decays, quantifying LFU in the charged current. Right: Global fit in the C llνν 23 − C llνν 12 vs C llνν 13 − C llνν 12 plane from leptonic tau and muon decays. Uncertainties are shown for one (dark red) and two (light red) standard deviations. The dashed lines are references for the zero points. GF = GL F (1 + ϵee + ϵµµ). (2.22) This substitution allows to quantify deviations from Vud caused by CAA, which is to a good approximation only sensitive to εµµ [62]. V CAA ud = Vud(1− ϵµµ). (2.23) The unitary relations in the CKM matrix are dominated by Vud leads to the good approximation εµµ ≈ 0.00098 ± 0.00027. This result can be used to parameterize the effects of physics BSM as direct measures of the LFUV. The experimental bounds in table 2.5 allow to perform a fit on the ϵττ − ϵµµ vs ϵee − ϵµµ plane, the results are shown in figure 2.4 (left). The hypothesis of LFU in the charged current is consistent with experimental data at 2σ and there is a slight preference for negative values in ϵee − ϵµµ. 4-lepton Operators 4-leptons operators are only allowed to act in leptonic decays. Since only left-handed vector operators with the same flavor structure as the SM have 28 2.3. Physics beyond the Standard model interference with the SM in these decay types, these are the only relevant processes in the context of LFUV and are described by [63] L4l = − g22 2m2 W C llνν fi lfγµ(1− γ5)liνiγ µ(1− γ5)νf , (2.24) where g2 is the coupling of the four-vertex, the indices i, f indicate the generation of the initial and final states and C llνν fi = 1+C llνν fi,NP. The effects of C llνν fi,NP are similar to those introduced by the modified Wlν couplings so the constraints can be done considering the three parameters C llνν 12,NP, C llνν 13,NP−C llνν 12,NP and C llνν 23,NP−C llνν 12,NP. Here C12 is determined from the CAA and has an impact on the global electroweak (EW) fit as it modifies the determination of the Fermi constant from muon decay [70] [71] C llνν 12,NP|CA ≈ 0.00098± 0.00027, C llνν 12,NP|EW ≈ −0.00067± 0.00033, (2.25) Using these approximations both C llνν 13,NP−C llνν 12,NP and C llνν 23,NP−C llνν 12,NP can be determined from the ratios of rates τ → µνν/τ → eνν, τ → µνν/µ→ eνν and τ → eνν/µ→ eνν. The fit is shown in figure 2.4 (right). Leptoquarks Extensions of the SM define leptoquarks as particles with both lepton and baryon quantum numbers, their further properties depend of the model. For instance, non-Chiral leptons couple to particles of either handedness. The constraints from the experimental results of Rπ are strong such that M2 LQ/gLgR > (100 TeV)2, with MQL the mass of the leptoquark and gL, gR are the couplings of the leptoquark to the left- and right-handed particles respectively [72]. Leptoquarks arise from SU(5) Grand Unified theories [73] and in the R-parity violating Minimal Super-symmetric Model. They were studied in the context of LFUV in kaon, tau and pion decays in [74],[75]. 2.3.2 New pseudo-Scalar interactions The pion is a particle with zero spin and odd parity, a pseudoscalar meson. Therefore, in the SM, it is allowed to decay only through a pseudoscalar operator. However, effective pseudoscalar interactions arising from scalar operators via electroweak renormalization effects can be induced by scalar interactions that are part of many proposed extensions of the SM. Some 29 2.3. Physics beyond the Standard model scenarios are Higgs multiplets, leptoquarks, and compositeness of quarks and leptons [76]. A key feature of these interactions is that it allows the decay into the positron channel without helicity suppression, meaning a larger measurable effect in Rπ. This in turn can constrain the pseudoscalar interactions and implies limits on the underlying fundamental scalar interactions. According to Ref [76], the transition amplitude through a pseudo scalar interaction is given by ⟨0|uγ5d⟩ = i √ 2 fπm 2 π mu +md = i √ 2f̃π, (2.26) where fπ = 93 MeV and f̃π = 1.8×105 MeV2. The pseudoscalar interaction with left-handed neutrinos in the final state is described by the Lagrangian: LP = −i ρ 2Λ2 [ l(1− γ5)νl ] [ uγ5d ] , (2.27) where ρ is the coupling constant for the new pseudoscalar Lagrangian and Λ its mass scale. This expression allows to calculate the invariant matrix element MP = ρ f̃π√ 2Λ2 [ l(1− γ5)νl ] . (2.28) The total matrix element will be then the sum of the pseudoscalar in- teraction and the matrix element of the SM in eq. 2.10. After squaring, summing over final states and integrating over the phase space and assum- ing the lepton universality in the coupling ρ, the branching ratio can be approximated by [76] RBSM π ≈ R0 π ( 1 + √ 2 f̃πRe(ρ) GFΛ2fπVudme +O ( 1 Λ4 )) , (2.29) with R0 π is the SM branching ratio to tree level in eq. 2.14. Considering a constant coupling with a similar strength of the weak coupling, a deviation of the SM to tree level via new pseudoscalar interactions is RBSM π R0 π − 1 ∼ ( 1TeV Λ )2 × 103. (2.30) This means that a precision measurement of Rπ is highly sensitive to the mass scale, Λ, of the pseudoscalar interaction. PIENU goal to reach a 30 2.3. Physics beyond the Standard model Figure 2.5: Tree level RPV contributions to Rπ [77]. ∼ 0.1% precision would allows potential access up to the mass scale Λ of 1000 TeV. However, the coupling would depend on the interaction itself and the actual mass scale could be smaller. The pseudoscalar interaction can be induced by a scalar interactions through three classes: scalar-dressed Z ex- change box diagrams, scalar-dressed W exhange box diagrams and radiative corrections to the quark vertex. In the following sections, some candidates are discussed [78]. R-Parity Violation SUSY Supersymmetry (SUSY) is one of the extensions of the SM, as it can be constrained by low-energy precision tests such effects of weak-scale. It has the potential to solve interactions of the Higgs boson that causes a large renormalization of the Higgs mass. In the Minimal Symmetric Standard Model (MSSM), deviations from the branching ratio, ∆RSUSY π , may arise from tree-level or one loop corrections [77]. The R-parity is defined by PR = (−1)3B+L+2S , where S is the spin, B is baryon number, and L is lepton number. All SM particles have R-parity of +1, while supersymmetric particles have R-parity of −1. Two scenarios can occur in the MSSM framework; the R-parity is conserved producing effects from SUSY loops leading to a deviation in Rπ at the level of 0.0005 ≤ ∣ ∣ ∣ ∣ ∆RRPC π Rπ ∣ ∣ ∣ ∣ ≤ 0.001, (2.31) where RPC stands for R-parity conservation. This hypothetical deviation is still far from the current experimental precision, and will be an achievable goal for the next generation of experiments [63]. On the other hand, in pres- ence of R-parity violation (RPV), three-level exchanges of sfermions, a hy- pothetical spin-0 super-partner particle (sparticle) of its associated fermion, shown in figure 2.5, allow the violation of LFU and non-vanishing effects on 31 2.3. Physics beyond the Standard model Figure 2.6: Current 95% confidence level constraints on RPV parameters ∆′ 11k and ∆′ 21k that fit Rπ. The blue curve corresponds to the fit using the PDG value, the dashed red contour corresponds to the fit using the future expected experimental precision of 0.1%. The light green curve indicates prospective impact of a future measurement of the proton weak charge at Jefferson Lab [79]. Rπ. The magnitude of their contribution is given by the sfermion masses and the coefficients λ′11k and λ′21k in the RPV interactions described by the general Lagrangian [77] LRPV,∆L=1 = λ′ijkLiQjd † k + . . . (2.32) Hence, the contributions to Rπ from RPV effects are given by ∆′ ijk(f) = |λijk|2 4 √ 2GFm2 f ≥ 0 ∣ ∣ ∣ ∣ ∆RRPV π Rπ ∣ ∣ ∣ ∣ = 2(∆′ 11k −∆′ 21k), (2.33) where mf is the mass of the exchange sfermion. The parameters ∆′ 11k and ∆′ 21k are constrained by the precision measurement of Rπ. Using the PDG 32 2.3. Physics beyond the Standard model value of the branching ratio, (1.230 ± 0.004)×10−4, the fit done to a 95% confidence level is shown in the blue curve of figure 8.2.3. The green curve indicates the possible implication of a future measurement of the proton weak charge planned at Jefferson Lab [79], assuming agreement with the Standard Model prediction for this quantity and the anticipated experimen- tal uncertainty. The dashed red curve shows the possible impact for the PIENU’s precision goal of ∼ 0.1% in Rπ. Charged Higgs The MSSM contains a neutral Higgs doublet, a neutral Higgs singlet, and a charged Higgs doublet [80]. The type-X two-Higgs-doublet model (2HDM) is part of this framework and it is constrained by loop effects in τ → µντνµ [81], [82] that are relavant at large values of tanβ. These charged Higgs could replace the W boson as the mediator of the pion decay [77]. Considering the coupling of the Higgs boson λud g 2 √ 2 to the pion pseudoscalar current and λlν g 2 √ 2 to the leptonic current, deviations from the SM branching ratio in 2.14 are 1− Rπ RSM π = 2m2 π me(mu +md) m2 W m2 H± λud ( λeν − me mµ λµν ) , (2.34) where mH is the mass of the charged Higgs boson. With the PIENU’s precision goal of 0.1% in the measurement of Rπ, this probes mH± ∼= 200 TeV× √ λud ( λeν − me mµ λµν ) . (2.35) In the case where λeν λµν = me mµ , as in the minimal two–Higgs doublet model, Rπ is not sensitive to a charged Higgs boson. If the coupling are λeν ∼= λµν ∼= λud ∼= α π under the conditions for eq. 2.35 then mH± ≈ 400 GeV [37]. 2.3.3 Massive neutrinos Neutrinos are chargeless particles with zero mass in the SM that interact solely through weak interaction. Due to their presumable lack of mass, they exist in a single helicity state, either νL or νR. However, this turned out to be incorrect when it was observed at SNO [114] that neutrinos oscillate between different generations, showing that, indeed, at least two neutrino flavors have mass. This intrinsic property implies the existence of a 3 × 3 33 2.3. Physics beyond the Standard model mixing matrix, UPMNS. This matrix must be unitary in case there are only three generations. The Neutrino Minimal Standard Model (νMSM) [83] is an extension of the SM that postulates the existence of three right-handed neutrinos, whose masses are below the electroweak scale (energy of electroweak symmetry breaking, 159.5±1.5 GeV). Right-handed neutrinos do not have mass and electric or strong charge but, unlike the SM left-handed neutrinos, they do not have weak charge or coupling to the Z0 boson, therefore, they are called sterile. This model was initially prompted to explain the observed small masses of the left-handed neutrinos and resolve the experimental anomalies observed at LSND and MiniBooNE [84]. For the νMSM to be consistent with experimental data, the Yukawa couplings of the right-handed neutrinos have to be very small, f2i ∼ O(MImν/v 2), where mν are the masses of the SM neutrinos, MI are the masses of the right-handed sterile neutrinos and v = 164 GeV is the vacuum expectation value of the Higgs field. This requirement leads to the small masses of the left-handed neutrinos, which is consistent with the gauge symmetries and the see-saw mechanism of the SM. The two more massive states of the sterile neutrinos are responsible for baryon asymmetry, and the lightest one can be a candidate for dark matter in the low energy range (keV/c2). Re-normalizable dark models [85] postulate a dark matter candidate that can couple to a sterile heavy neutrino via a new dark sector boson. This model requires that heavy neutrinos in the 100 MeV mass range. The model can be generalized for k sterile neutrinos, where the weak eigenstates νχk are related to the mass eigenstates νi through the matrix of change of basis Uli by νl = 3+k∑ i=1 Uliνi, (2.36) where l = e, µ, τ, χ1, ..., χk. Considering that the kinematic energy space of the decay π+ → e+νe is at most 140 MeV/c2, then, to measure effects caused by sterile neutrinos in the high precision measurement of Rπ their mass scale must be MeV/c2 to GeV/c2.The presence of any neutrino heavier than a few MeV/c2 will weaken the helicity-suppression mechanism and enhance the brancing ratio Rπ. Considering this, the PIENU experiment is sensitive to massive neutrinos above 55 MeV. 34 2.3. Physics beyond the Standard model Below 55 MeV/c2. The ratio of the decay width π+ → e+νi, with νi a massive neutrino, to the decay rate π+ → e+νe is [86] Γ(π+ → e+νi) Γ(π+ → e+νe) = |Uei|2ρe, (2.37) where Uei is the the element of the UPMNS matrix between νe an νi and ρe is a kinematic factor given by [86] ρe = √ 1 + δ2e + δ2i − 2(δi + δe + δiδe) δe(1− δe)2 δi + δe − (δi − δe) 2, (2.38) where δe = m2 e/m 2 π and δi = m2 νi/m 2 π, the massive neutrino mass is bounded to the two-body decay kinematics given by mνi = √ m2 π − 2mπEe +m2 e. (2.39) The presence of a massive neutrino would change the branching ratio by Rexp π = Γ(π → eν) + Γ(π → eνM ) Γ(π → µ→ e) = RSM π + Γ(π → eν)Γ(π → eνM ) Γ(π → νe)Γ(π → µ→ e) = RSM π (1 + |Uei|2ρe), (2.40) leading to |Uei|2 = r − 1 ρe , (2.41) where r = Rexp π /RSM π and νM is the massive neutrino. Hence, the limits of the mixing matrix UPMNS can be calculated as function of the neutrino mass mνi . Above 55 MeV/c2 Massive neutrino states with masses smaller than the pion mass mπ can be identified in the lepton energy spectrum. Since this is a two-body decay, the kinematics restrict the decay to have a fixed energy peak. For instance, in the PIENU experiment, the pion decays at rest and the lepton energy is fixed by the conservation of momentum-energy Ee+ = m2 π +m2 e −m2 ν 2mπ . (2.42) 35 2.3. Physics beyond the Standard model Figure 2.7: The 90% confidence level upper limit on the heavy neutrino mixing parameter |Uei|2, as a function mass. The dashed line shows the re- sult from the previous PIENU experiment [88], and the circles and triangles are the limits from a subset of PIENU data, published in 2011 [87]. The circles indicate a restricted angular region was used when constructing the π+ → e+νe energy spectrum. In the 55 MeV neighborhood and above, the positron peak has an en- ergy low enough to distinguish the peak of the sterile massive neutrino in the π+ → e+νe energy spectrum. Figure 2.7 shows an upper limit on |Uei|2, established a the search of extra peaks in the PIENU data from 2009, com- pared with the previous PIENU experiment [87][88]. 2.3.4 Meson fields in dark-sector The electron-like excess of events at a 4.8σ observed at the MiniBooNE experiment [90] [91] remains as a strong hint to the existence of physics BSM. The energy and angular constraints make new physics in the neutrino sector good candidates to map onto the observed spectra. However, it has been recently tested [92] that the production of long-lived particle (LLP) boson state from the three-body decay of charged mesons can potentially explain the electron excess. Particularly, a single long-lived vector boson that couples to quarks and enters the pion sector is a remarkable candidate. 36 2.3. Physics beyond the Standard model Figure 2.8: Parameter space for the single mediator scenario where a massive vector V couples to the pion doublet via charged pion coupling gπ± and neutral pion coupling gπ0 as in Eq. 2.43. Exclusion zones and constrains by several experiments, including PIENU are detailed in text. The Lagrangian of a hadrophillic model that only couples to first generation quarks in the Chiral Perturbation Theory framework (χPT) is [93], Lint ⊃ igπ±Vµπ +∂µπ− + gπ0 e 16πfπ π0FµνH̃ µν , (2.43) where gπ± and gπ0 are the couplings of the vector boson field Vµ and H̃µν is defined as H̃µν = ϵµναβ(∂αVβ − ∂βVα). From this Lagrangian, the squared decay amplitude can be inferred by mapping the radiative charged pion decay π → eνγ to the three-body decay π → eνV . A detailed description of this process can be found in Ref [92]. This case can be sensitive to PIENU measurement of Rπ depending of the mass scale of the boson vector mass mV . Figure 2.8 shows the parameter phase space of the pion couplings gπ0 , gπ± to the vector boson for mV = 5 MeV/c2 (left) and mV = 20 MeV/c2 (right). The Fit of MiniBooNE data to 1σ and 2σ, shown in blue color, the constraints for the rare charged pion decay from PIENU [6], LSND [94] and KARMEN [95] experiments are shown in gray, light yellow and brown respectively. Also the exclusions set by CCM120 engineering [96] and the future sensitivity expected in CCM200 at a 95% C.L. are shown in black hatched and red, respectively. Constraints from π± decay width measurements can be directly applied to gπ± by taking the bounds from PIENU early results [6]. The constraints from PIENU also begin to relax while moving to larger vector boson masses, due to the weaker branching ratio of π → eνV with higher mass. By enhancing the precision of the pion 37 2.3. Physics beyond the Standard model branching ratio Rπ in the PIENU experiment, the hypothetical vector boson mass mV can be bounded further. 38 Chapter 3 Description of the experiment 3.1 Beam-line 3.1.1 TRIUMF cyclotron The TRIUMF cyclotron produces a 520 MeV proton beam (H− ions) with an average intensity of ∼ 100 µA that is divided among four primary beam- lines. The cyclotron has a diameter of 18 m and an accelerating gradient provided by a 23.05 MHz 93 kV radio-frequency (RF), it accelerates pro- tons to be delivered in bunches of 4-ns width with a 43-ns spacing between them. The production of the beam is accomplished by stripping electrons off the Hydrogen atoms using a thin foil, hence reversing the direction of the magnetic steering. The proton bunches are delivered through the primary beam-line BL1A with an intensity of 100 µA to impinge on a 1cm-thick Beryllium production target T13 and produces different types of particles such as photons, neutrons, positrons, muons or pions. The resulting charged particles are taken by the secondary beam-line M13 that has been config- ured [7] to select 75 MeV/c momentum particles with a 1% spread. Figure 3.1 shows the overall arrangement of the beam-line. The composition of the beam at the end of the secondary beam-line M13 depends on its configura- tion, but under a nominal setup for the PIENU detector it is composed by 85% pions, 14% muons and 1% positrons. 3.1.2 M13 channel M13 is a low-momentum achromatic channel rotated by 135◦ with respect to the primary beam-line BL1A axis from T1 and with a maximum angular acceptance of 29 mili-steradians. The schematic of the M13 channel is shown in Figure 3.2. The components of the M13 channel are two dipole magnets 3Note that components in bold refer to the M13 channel components, not to the PIENU detector components. 39 3.1. Beam-line Figure 3.1: Schematic of the TRIUMF cyclotron, the primary beam-lines, the meson hall and the secondary beam-lines. B1 and B2 rotated by -60◦ and +60◦ respectively. A quadrapole doublet Q1-2 is located right after T1. B1 is in charge of selecting pions by the par- ticle momenta. A focal point F1 to remove beam spread after the first beam selection is located downstream B1. F1 is followed by a quadrapole triplet Q3-5 before the focal point F2 that collimates the beam. A quadrupole doublet Q6-7 is located downstream of B2 to focus the beam after the pion filtering provided by this. The doublet is followed by another focal point F3. A pair of adjustable vertical and horizontal slits, an absorber, and a collimator SL1 and SL2 are located at F1 and F2 respectively. The beam particle contamination at F3 mainly consists of positrons. The pions and positrons energy loss difference is large enough to obtain a clear particle sep- aration at F3 as shown in figure 3.3. A collimator (5 cm-thick lead bricks with a 3 cm square hole) is placed at F3 to suppress the displaced positrons and redefine the pion image. Prior to the construction of the beam-line, Monte Carlo simulations were carried using the beam transport package called REVMOC [110], which cal- culates up to second order optics and Coulomb scattering effects, to de- termine the beam content and quality. This allowed to design the M13 channel extension that provides a controlled pion beam, although there is a 40 3.1. Beam-line Figure 3.2: M13 beam channel with the extension. Taken from [7]. Figure 3.3: Position distribution of π+, µ+ and e+ in the focus F3. The solid lines represent Gaussian fits. Taken from [7]. loss of intensity due to decay-in-flight. The MC simulation also helped to place a 1.45-mm-thick Lucite4 absorber near F1 that allowed to differenti- ate between particles in the beam by the momentum change due to different masses and deposited energies in F3 as shown in figure 3.3 (left). Originally the pion beam had a positron contamination of 25%, which severely increased trigger rates. After testing a suggested setup in the M13 4Polymethil methacrylate or PMMA. Lucite is one of the commercial names for this material. 41 3.1. Beam-line Figure 3.4: M13 channel before (left) and after (right) implementing the extension. The calorimeter of the detector was placed at the end of the extension to measure the composition of the beam. Taken from [2]. beamline by the REVMOC simulation in 2008, this contamination was low- ered to 2%. Another side result from this study was that a background consisting of neutrons and gamma rays in the beam-lime coming from the production target T1 was identified and suppressed. These particles boost the energy of the π+ → µ+νµ → e+νeνµ decay to the energy region of π+ → e+νe in the calorimeter [120]. All these considerations lead to the modification of the M13 channel, to achieve this an extension was added starting at the focal point F3. 3.1.3 M13 extension The M13 extension was designed to improve the purity of the pion beam. It starts 0.9 m downstream of Q7, right after F3. It consists of a dipole magnet B3 and quadrupole triplet Q8, Q9, Q10 that focuses the beam at the focal point F4 that has the same collimator specifications as F3. After implementing these upgrades, the desired rate of particles as function of the momentum, shown in figure 3.3 (right), was obtained [7]. By closing the 42 3.2. Detector horizontal SL1 slit to 1.5 cm, the momentum width of the beam is restricted to 1.5 % (FWHM). A large 20-cm thick steel wall isolates the location of the collimator from the detector, allowing better shielding from the γ rays emitted by the stopped positrons in the collimator. The M13 channel before and after including the extension is shown in Figure 3.4. 3.1.4 Beam-line calibration The dipole magnets (B1, B2 and B3) have a fringe field produced by nu- clear magnetic resonance (NMR) that comes with an intrinsic uncertainty. This makes hard to obtain a beam calibration with absolute precision. The calibration is carried considering two points that are well identified in en- ergy scale; the endpoint of the muon decay spectrum µ+ → e+νe and the peak of the π+ → e+νe decay. The positrons from the two chains come from the decays occurring in the detector target. An uncertainty of 1% in the beam-line calibration was estimated by using those measurements. The full width at the half maximum (FWHM) of the beam can be re- stricted by closing the SL1 slit, located downstream the dipole magnet B1. A suitable combination of slit openings that optimize the beam yield and minimize the momentum spread in the target a study was found by per- forming tests on SL1 and SL2. By tuning the quadrupoles upstream and downstream of B3, the steering and place the beam close to the center of our target detector was enhanced. For all the improvements the control and monitoring of the beam-line was done using Experimental Physics and Industrial Control System (EPICS) [144]. 3.2 Detector The PIENU detector is located downstream of the exit of the extended M13 channel. To reduce the beam background coming from the production target area and the M13 channel itself, a 24.8 cm-thick steel wall of 2.57×2.29 m2 and with a 26.67 cm-diameter hole for the beam pipe to go through was installed as shown in the figure 3.5. The PIENU apparatus, shown in figure 3.6, is divided into two assemblies PIENU-I and PIENU-II. PIENU-I was designed for the identification, detection and tracking capabilities of the incoming beam pion. From left to right in the schematic the components are; two wire chambers (WC1 and WC2) that provide the profile of the incoming beam; two plastic scintillator counters (B1 and B2) that degrade the beam and allows to identify pions; two pairs of Silicon-strip detectors 43 3.2. Detector Figure 3.5: PIENU detector and the steel wall used as shielding at the end of the M13 beam-line. Taken from [8]. (S1 and S2) to provide an intermediate profile of the beam, each detector is combined in pairs to cover the XY plane; the beam stops in a target plastic scintillator (B3, see Section 3.2.1) and decays producing an positron aura that sprays the remaining components of the detector; before entering the PIENU-II assembly, the positrons undergo through a silicon-trip detector (S3) and a plastic scintillator (T1). PIENU-II is a positron telescope with four main components; a wire chamber (WC3) made of 3 layers, together with S3 from PIENU-I, it pro- vides the tracking capabilities for the outgoing positron; a plastic scintillator (T2) that covers the front face of an inorganic NaI(Tℓ) calorimeter (Bina); Finally, to contain the energy leakage produced by electromagnetic shower, Bina is surrounded by 97 pure CsI crystals arranged in two rings. The detector is fully described in [8] and its features are summarized in table 3.1. 44 3.2. Detector Figure 3.6: Schematic diagram of the PIENU detector, see text. Taken from [8]. 3.2.1 Scintillators The plastic scintillator counters are made of Bicron BC-408 (polyvinyltolu- lene) scintillator and they were created with different dimensions according to the function they perform. The square scintillators B1 and B2 are located upstream of the target to degrade the beam and define the pion timing. As shown in figure 3.6, B2 is considerably smaller than B1 and B3 to restrict the beam acceptance. The energy deposited by the beam particles in these counters serves to identify pions. The target (B3) defines the center of the PIENU detector, it is thicker than B1 and B2 to stop the pion and to contain all the decay vertices. B3 is followed by the positron telescope counters T1 and T2. T1 defines the decay positron timing. Both B1 and T1 are rotated by +45◦ with respect to B2 and B3 around the beam axis, this helps to fit 45 3.2. Detector Plastic scintillator counters Trigger counters B1 B2 B3 T1 T2 Dimensions (XY in mm2) 100×100 45×45 70×70 80×80 171.45 (radius) Thickness (mm) 6.604 3.07 8.05 3.04 6.6 Z position (mm) -39.03 -30.02 0 19.92 72.18 Veto counters V1 V2 V3 Inner radius (mm) 40 107.95 177.8 Outer radius (mm) 52 150.65 241.3 Thickness (mm) 3.175 6.35 6.35 Tracking detectors Multi-wire proportional chambers WC1 WC2 WC3 Wire spacing (mm) 0.8 2.4 Number of wires/readout channels 120/40 96/48 Active area diameter (mm) 96.0 230.4 Silicon-strip detector pair S1 S2 S3 Active area 61×61 mm2 Silicon-strip pitch 80 µm Number of strips 189 Thickness 0.285 mm Separation between X and Y planes 12 mm Electromagnetic calorimeter Component NaI(Tℓ) CsI Number of crystals 1 97 Energy resolution (FWHM) at 70 MeV 2.2% 10% Thickness 480 mm 250 mm Outer radius 240 mm · · · Approximate width × height for pentagon-shaped crystals . . . 90×80 mm2 Table 3.1: Summary of the PIENU components [8]. the light guides in a compact arrangement. While T1 is a square counter, T2 has a round shape that covers entirely the front face of the NaI(Tℓ) calorime- ter. To improve the acceptance of the positron spraying in the calorimeter the detectors between the target (B3) and the calorimeter were designed to form a compact assembly, so the readout of T2 is performed by wavelength shifting fibers. There are also three ring-shaped veto counters (V1, V2, V3) covering the flanges of the WC1-2 assembly, WC3 and the NaI(Tℓ) crystal. The signals of all plastic scintillators, excluding those from T2 and the veto 46 3.2. Detector Figure 3.7: Left: Readout system of plastic scintillators B1, B2, B3 and T1. Light is collected by light-guides and read by four PMTs. Right: Schematic of the T2 readout system with wavelength-shifting fibers. Taken from [8]. Manufacturer Model Photo-cathode diameter Trigger counters B1 Hamamatsu H3178-51 34 mm B2 Burle 83112-511 22 mm B3 Photonis XP2262B 44 mm T1 Burle 83112-511 22 mm T2 Photonis H3165-10 10 mm Veto counters V1 Hamamatsu H3164-10 8 mm V2 Hamamatsu H3165-10 10 mm V3 Hamamatsu H3165-10 10 mm Crystal NaI(Tℓ) Hamamatsu R1911 76.2 mm CsI Hamamatsu R5543 76.2 mm Table 3.2: Readout scheme of the PIENU components PMTs. counters, are conducted by acrylic light-guides attached to each of the four sides and read by a photomultiplier tube (PMT). A schematic of the scin- tillators readout configuration is shown in figure 3.7. The specifications of the PMTs used in each scintillator are shown in table 3.2. 47 3.2. Detector Figure 3.8: Left: A beam wire chamber plane and its pre-amplifier board. Right: Wire Chambers 1 and 2 installed at the rear end of the beam pipe. Taken from [2]. 3.2.2 Wire chambers The multi-wire proportional chambers consist of three wire planes rotated by -120◦, 0◦ and +120◦, respectively, to form an X-U-V assembly. Figure 3.9 shows a technical drawing of the front face (up) and the cross section of a wire plane. Each sensitive wire is connected to a multi-hit TDC chan- nel, it records the hit time after the signal goes through pre-amplifiers and discriminators. The chambers has a gas mixture of 80% tetrafluoromethane (CF4) and 20% isobutane (C4H10) at atmospheric pressure. Beam Wire chambers (WC1 and WC2) The wire chambers WC1 and WC2 establish the first contact between the pion beam and the PIENU detector. Both wire chambers are mounted on the beam pipe, right next to the steel vacuum window foil. Each wire plane has 120 wires in an active diameter of 96 mm, the wires are bounded in groups of 3 wires that are connected to 40 readout channels. The effective pitch after the bounding is is 2.4 mm. The pair WC1 and WC2 is used to obtain information about the position and the angle of the incoming beam particles. Figure 3.8 (Left) shows a wire plane with the pre-amplifier board. 48 3.2. Detector Figure 3.9: Technical drawing of the front face (up) and the the cross section (down) of a wire plane. Taken from [1]. 49 3.2. Detector Decay positron wire chamber (WC3) The wire chamber WC3 is part of the tracking system for the decay positrons emerging from the target. It is composed by three wire planes that, unlike those in WC1-2, have 96 wires in an active diameter of 230.4 mm. The wires are attached by pairs to form 48 readout channels. This wire chamber is mounted on the flange of the NaI(Tℓ) crystal enclosure to define the ac- ceptance of the calorimeter. After bounding the wires, the effective readout pitch is of 4.8 mm. Using beam positrons, the efficiency of the wire planes were measured to be larger than 99%. Figure 3.9 shows an schematic of the front face and cross section of WC3. 3.2.3 Silicon detectors The silicon detectors (S1, S2, S3) share the same design. They are composed by two perpendicular planes for the X and Y coordinates with an active volume of 61 mm × 61 mm × 285 µm. Each plane is made of 189 strips that are bounded in groups of four (the last strip in each plane remains alone) and they are interconnected through capacitors to form 48 readout channels. Each plane has a single sided AC-coupled micro-strip device with the same design of those used in the ATLAS central tracker [99]. The network for capacitors shown in figure 3.10 forms a charge division line where the reconstruction of the ionization amplitude and position is made by proper weighting of the channels that fire when the charged particle passes through strips, figure 3.10 (up) shows a diagram of the charge-sharing readout system. After the readout reduction the strip pitch changed from 80 µm to 300 µm, as required by the PIENU experiment. S1 and S2 are placed upstream of the target and S3 is placed downstream of the target. Hence, S1-2 and S3 provide the most accurate information for the incoming pion and the outgoing decay positron, respectively. Figure 3.10 (down) shows the first two silicon strip detectors S1 and S2 mounted in the PIENU-I assembly. The signals were read out by VF48 60 MHz ADCs. In order to reduce the data size by suppressing channels without hits using predefined thresholds for pulse-signal waveforms. The thresholds in S1 and S2 were tuned for beam pions, and S3 thresholds were set lower to ensure an efficiency of 99% for the decay positrons. 50 3.2. Detector Figure 3.10: Up: Schematic of the capacitors line in the Silicon strip readout channels. Down: S1 and S2 assembly on their support structure [142] [105]. Taken from [8]. 51 3.2. Detector 3.2.4 Bina The inorganic calorimeter Bina is a single crystal of Thallium-doped Sodium Iodide (NaI(Tℓ)) that has been loaned from Brookhaven National Labora- tory after it was used by the LEGS collaboration [101] [102]. This detector is the largest ever grown of its type, the dimensions are shown in table 3.1. The crystal has been enclosed in a 3-mm-thick aluminum enclosure with 19 circular quartz windows at the rear end to mount a PMT in each window. In order to avoid extra spread for the incoming particles, an 0.5 mm-thick aluminum plate was installed in Bina’s front face. A thin layer of reflect- ing material was used to cover the crystal. Using the software Detect2000 [103] the dependence of the deposited energy in the crystal on the particle’s entrance point to Bina was studied. The results showed that the light emit- ted by the crystal is uniformly reflected [104], hence every PMT receives an equal amount of light independently of its position and the particle’s entrance point in the front face of Bina. This results were later confirmed with a 2% accuracy using a positron point-like source 22Na [121]. Figure 3.11 (left) shows the back face of Bina with the quartz windows for the PMTs. In each window a 3 inch-diameter PMT was mounted (see further specifications in table 3.2), they are surrounded by a 1-µm thin metal shield to reduce the effect of the cyclotron’s fringe field on the PMT gain. The bases of the PMTs were modified to allow the last two dynodes in each PMT to have a higher voltage (∼ ×0.21 and ×0.37 of the high voltage applied to the resistor chain). This modification was required to improve the PMT performance at high count rates for the PIENU experiment. 3.2.5 CsI array Bina has been covered with a reflecting material to prevent optical photons to reach the calorimeter. This does not prevent that a considerable amount of electromagnetic shower escapes from the crystal. In order to contain the energy leakage produced by this shower and reduce the systematic uncer- tainty of Rexp π , Bina is surrounded by an array of 97 pentagon-shaped CsI crystals with 25 cm in length (13.5 radiation length) and about 9 radiation length radially. A single CsI crystal is shown in figure 3.12. Figures 3.11 and 3.6 show the array of CsI crystals that were arranged in the form of two pairs of concentric rings. The rings are classified into upstream or downstream and inner or outer, according to the radial and beam-oriented position [123]. Each crystal is connected to a 3 inches-diameter Hamamatsu R5543 PMT 52 3.2. Detector Figure 3.11: Left: Back side of the NaI(Tℓ) crystal on the test bench. Right: The NaI(Tℓ) crystal and the 97 CsI crystals while the calorimeter was under construction [142]. Taken from [1]. Figure 3.12: Picture of one CsI crystal. Taken from [1]. [107] that has been designed to operate in high magnetic fields5. Each crystal has a built-in YAlO3:Ce 245 Am source [109] which produces light pulses (about 8 MeV) at a rate of ∼50 Hz with wavelength and pulse 5The components of the magnetic background in the M13 area are at maximum of 2 Gauss at the location of the detector. 53 3.3. Final detector assembly width similar to the CsI scintillation to monitor the crystal’s light output and the PMT’s gain. Each crystal is also connected via a quartz fiber to the output of a Xe lamp pulser, which flashes twice a second during data taking [122]. This Xe lamp monitoring system traces the changes in the CsI PMT’s gains only. Therefore, a comparison between the YAlO3:Ce 245 crystal and Xe-lamp monitoring gives access to the evolution of the crystal’s light collection efficiency. The stability of the Xe lamp was measured to be ∼1%. 3.3 Final detector assembly A technical drawing of the PIENU detector made in CAD is shown in figure 3.13. PIENU-I was mounted to the beam pipe and PIENU-II was mounted on a supporting structure on wheels which were guided by rails to ensure correct alignment to PIENU-I. In order to prevent the two sub-assemblies to collide when attaching them together, four safety rods were installed in PIENU-I. The rods were used to measure the position of WC3 along the beam axis relative to B3, it was concluded that there was a shift of nearly 0.4 mm downstream compared to the design sheet [108]. The flexible arrangement makes possible that PIENU-I could be removed and enables the inner rotation of PIENU-II with respect to the beam. This feature is specially useful to study the response of the calorimeter to a positron beam as a function of different entrance angles. This information is crucial to tune the Monte Carlo simulation and the estimate of low en- ergy tail in the energy spectrum due to energy leakage. Due to temperature variations in the experimental hall, there was variations in the PMTs gain. To suppress this effect, a temperature-controlled enclosure was constructed to house the detector and maintain temperatures at 20◦ C within ±0.5◦ C to keep the gain variations within acceptable limits. 3.4 Tracking system The PIENU tracking system provides position and direction information of the incoming pion and the outgoing positron at different stages. It is com- posed by three sub-systems. The first tracker (Trk1) is composed by the beam wire chambers WC1 and WC2, it provides six position measurements corresponding to the pair of three wire planes. The second tracker (Trk2) is made of the Silicon-strip detectors S1 and S2 and provides four position measurements corresponding to the pair of XY strips planes. The third 54 3.4. Tracking system Figure 3.13: CAD drawing of the PIENU detector and picture of the detector setup. Taken from [1]. tracker (Trk3) is composed by S3 and WC3, it provides five position mea- surements corresponding to two strip planes and three wire planes. Trk1 provides the profile of the beam pions coming out from the beam pipe, Trk2 gives a second view of the beam right before the pion enters the target. This tracking device is specially useful to identify different decay topologies, these are pion decay-at-rest (πDAR) and pion decay-in-flight (πDIF). Topologies such as muon decay-at-rest (µDAR), and muon decay-in-flight (µDIF) can not be identified by the trackers. In the case of the latter a correction is needed to quantify its effect on the branching ratio. Finally, Trk3 is used to identify the trajectories of decay positrons entering the calorimeter. Figure 3.14 shows the PIENU tracking devices and the different decay topologies. The decay topologies of the π+ → µ+νµ → e+νeνµ in which a decay-in-flight is involved are a problem since their energies are prompted due to a Lorentz boost effect, making them hard to distinguish from π+ → 55 3.4. Tracking system Figure 3.14: Schematic drawing of the tracking devices in PIENU and the different decay topologies. ”u” orientation of a WC plane corresponds to a rotation of +60◦ with respect to the ”x” plane while ”v” to -60◦. Taken from [8]. e+νe. A data quality check is verifying that the vertex (Zv) of the pion decay is within the target B3. A further classification of the decay topologies is listed as follows • π+ → e+νe: The pion stops in the target and decays directly to the positron channel. • πDAR→ µDIF: In the π+ → µ+νµ → e+νe decay, the muon decays in flight within the target. These events are problematic since they have the same timing distribution as the direct decay π+ → e+νe. There- fore, a correction obtained by a Monte Carlo simulation is needed. • πDIF upstream of target (”up”) → µDAR: The pion decays in flight before impinging the target, muon decay at rest within the target. 56 3.4. Tracking system Figure 3.15: Monte Carlo simulation of the kink angle Kθ in Geant4 for different decay topologies. Taken from [8]. These events are characterized by a kink angle Kθ between Trk1 and Trk2 as shown in figure 3.14. • πDAR→ µDAR: In the decay π+ → µ+νµ → e+νe, the pion and the muon decay happening at rest within the target. • πDIF inside the target (”it”)→ µDAR: Pion decay-in-flight in the target and muon decay-in-rest. Besides the topologies in which all the decays happen at rest within the target, the PIENU tracking system can identify only πDIF (up) - µDAR events by requiring a non-zero kink angle Kθ. Figure 3.15 shows MC simu- lation of Kθ for different decay topologies. The signals from πDIF decaying directly to positron and πDIF to µDIF events are prompt in time, therefore the will be neglected in a time analysis as we will see in Section 5.3.4. 3.4.1 Track definition The link between the actual position coordinates (x, y) and the measured wire/strips planes is done by the algorithm described in [124]. In reality, 57 3.4. Tracking system more than one single wire/strip can be activated, this case is considered in the data analysis before performing the transformation. This process, called clustering, groups activated strips/wires to determine a weighted position as a function of the charge in each readout channel. The clustering is performed separately for each Silicon detector and Wire chamber. Due to the absence of magnetic field within the detector, the trajectory of the particle can be described as a straight line as a function of time and parameterized by x = x0 + vxt, y = y0 + vyt, (3.1) z = z0 + vzt. The reference frame is chosen to be at the center of the target B3 de- fined as the projection point (x0, y0, z0), where the z-axis point in the beam direction, x is the horizontal coordinate and y is the vertical coordinate. In this notation the vector v⃗ =(vx, vy, vz) are the velocity of the particle. This description has six parameters from which only four are completely inde- pendent. If we choose z0 = 0 and set a normalization for the vector v⃗ with vz = 1, we have v⃗ = (tx, ty, 1), with tx = vx/vz and ty = vy/vz. Then z = t and x = x0 + txz, y = y0 + tyz. (3.2) This parametrization is convenient when the particle goes preferentially in the z direction, which has been chosen to be the beam direction. This means that the points x0 and y0 identifies the point in which the track intersects the plane z = 0, which was chosen to be the center of the target. The case in which the particle goes parallel to the xy is not relevant here. 3.4.2 Track fitting After performing the clustering process for each plane in the Silicon detectors and the wire chambers the track fitting algorithm can relate a coordinate system uv in the tracker’s plane with the z coordinate fixed and the u- axis is orthogonal to the wires/strips plane. Using a rotation matrix it is possible to transform the uv system to the xy system of the experimental hall, as the rotation angles are known. This transformation includes the velocity components of the particle in the hall system. The details can be 58 3.4. Tracking system found in [124], essentially the algorithm minimizes the squared deviation from fitted track to the measurements, or χ2 function, to select the most accurate parameterization. 3.4.3 Tracking quantities The following variables are defined to characterize the pion and decay positron by defining the acceptance of the detector and suppress background events in the clean spectrum: • Acceptance Radius R = |r⃗|z=zWC3 (3.3) is the acceptance of the calorimeter. Here r⃗ is the parameterized posi- tion in eq. 3.3 is fitted by tracker Trk3, zWC3 is the position of WC3 along the beam axis with respect to the center of B3. • Kink angle Kθ = arccos ( v⃗A · v⃗B |v⃗A||v⃗B| ) , (3.4) where v⃗ are the parameterized directions fitted by the trackers Trk1 for A and Trk2 for B. • Z-vertex: Zv = [ (r⃗A − r⃗B) · (v⃗A − v⃗B) |v⃗A − v⃗B|2 ] z=0 , (3.5) where the parameterized positions and velocities are fitted by Trk2 for A and Trk3 for B. R is used as a parameter to test the stability of the branching ratio independently from the chosen acceptance (discussed in Section 7.2). On the other hand Zv is used to track momenta differences in the beam along all the data taking time periods. In order to set the MC event initialization, it is crucial to properly set the momenta differences. We will discuss this topic in Section 6.5.1. 59 3.5. Data acquisition system and Triggers 3.5 Data acquisition system and Triggers 3.5.1 Reading boards COPPER The PMT signals of the scintillation counters (B1, B2, B3, T1, T2 and veto counters) are read by a 500-MHz waveform digitizer (Flash-ADC). The FADC system is built with the Common Pipelined Platform for Electron- ics Readout (COPPER) platform that is fully described in [111] [4]. The COPPER board was designed for the Belle experiment at the JPARC labo- ratory [45]. Each COPPER board has four front-end modules “FINESSE”. Each FINESSE module can receive two analog input signals. Therefore each COPPER board can receive a total of 8 signals to digitize. The PIENU ex- periment is equipped with 4 COPPER boards. On each FINESSE card, two 250-MHz FADC devices are driven in alternating phases in order to obtain 500-MHz sampling. If the inner 250-MHz clock in the two FINESSE channels were not syn- chronized, the pulse timing in each ADC was shifted, causing the degra- dation of time resolution in the FINESSE boards. To synchronize the FI- NESSE modules in the each COPPER board a Clock Distributor module was developed by the PIENU collaboration. Clock Distributor provided syn- chronized 250 MHz sampling clocks, latched-gate signals, and latched-reset signals to all FINESSE cards. Figure 3.16 shows the digitized waveform from a PMT obtained with COPPER. A General Purpose Input Output (GPIO) module developed by KEK is charged to provide reset signals for the Clock Distributor module. GPIO is the interface between the COPPER system and the trigger. The time window of the signals recorded by COPPER covers approximately 8µs (1.35 µs after and 7.75 µs before the trigger timing) to be able to detect pile- up particles in the pre and post-time regions. The signals below a given threshold were suppressed to reduce the amount of data, more specifically noise, except for a given region around detected peaks to be able to record pedestals. VF48 The VF48 board is a 60 MHz FACD single-width 6U-size VME module, originally designed by the University of Montréal in 2004 [112]. It has a resolution of 10 bits and a dynamic range of ±250 mV. PMTs from Bina 60 3.5. Data acquisition system and Triggers Figure 3.16: Typical waveform digitized by COPPER. Blue plots indicate sampling points and red curve shows the fitting template. Taken from [3]. and CsI array as well as the Silicon detector channels giving a total of 404 channels (Bina: 19, CsI array: 97 and SSD: 288) are read out by 10 VF80 modules. To achieve this each VF48 module has been configured to read 48 channels. All the VF48 modules are fed with the same 20-MHz synchronized signal from the TIGC module described below. The this clock signal is internally multiplied to reach a sampling of 60 MHz. Due to the large amount of reading channels, a data suppression threshold for the wave- forms was implemented in the modules. The threshold was set differently depending on the input channel, for the CsI channels it set at 2 MeV, while it was 0.2 MeV for S1 and S2, and 0.1 MeV for S3. The full waveform is recorded only with the following logic: if two subsequent samples have a pulse height difference higher than a given threshold. The suppression technique was different for Bina, it consisted in reducing the data sampling frequency to 30 MHz since the waveform length in Bina was long ∼1.3 µs. The number of samples recorded by the VF48 is 40 (666 ns) for the CsI array, 40 (1333 ns)6 for Bina and 70 (1162 ns) for the Silicon detectors. 6Beacuse the NaI(Tℓ) signal is slow the sampling in the VF48 is done at a rate of 30 Hz instead of 60 Hz for the other detectors. 61 3.5. Data acquisition system and Triggers TIGC The TIGRESS collector (TIGC) is a VME-based module developed by the University of Montréal and TRIUMF for the TIGRESS experiment at TRI- UMF [119]. This module can read in the digitized output data from the VF48 modules and perform the data process for decision in high rate. For this, the on-the-fly summing of VF48 signals is carried before the read-out. The highest samples of the waveforms in every 250 ns of all NaI and CsI channels were sent to the TIGC module. If the highest samples pass a given threshold, TIGC provided a signal used for the TIGC trigger. Although the threshold was set to be about 4 MeV lower than the Michel edge, there are slight variations in the threshold not greater than 2 MeV that depend on the data taking period and do not affect the analysis. The TIGC module also provided the clock to all the VF48 modules to synchronize to each other. VT48 The analog signals from three wire chambers (WC1, WC2 and WC3), and the major trigger logic are recorded by multi-hit TDC modules, VT48. This module was a single width VME 6-U module, and designed at TRIUMF in 2006 for the KOPIO experiment [46]. The AMT-2 module [47] developed for ATLAS was used as a TDC chip on the VT48 module. This device has a 25-MHz inner clock which was multiplied internally to achieve 0.625 ns resolution. All VT48 modules are fed with an external 25-MHz clock to ensure the synchronization of all modules. One board could read out 48 channels for up to 20 µs. In order to minimize the dead time between signals, only two channels were read out for 20 µs to detect long lifetime backgrounds while all other channels were read out 4.0 µs before and after the decay positron signal timing. A total of 11 VT48 modules were installed in the PIENU experiment. 3.5.2 Software The PIENU data acquisition system is made of three VME crates. Two of these are used for the VF48 and VT48 boards while the third is used for Slow Control modules and COPPER boards with a processor on each board. The Slow Control modules record quantities such as high voltage of PMTs, gas pressure in the Wire Chambers, hardware thresholds (e.g. threshold of TIGC module), to monitor the data taking conditions of the experiment. The collection of the data is done by MIDAS acquisition system [141], which incorporates an integrated Slow Control system with a fast on-line database 62 3.5. Data acquisition system and Triggers Figure 3.17: Web interface of MIDAS data acquisition system. All the VME modules were integrated and easily controlled via this interface. Taken from [3]. (ODB) and a history system. The MIDAS server computer can be controlled via a web interface. Figure 3.17 shows the MIDAS interface during a run data-taking. All the information and errors from each front-end are shown on the web page and several programs checking the quality of the data online are connected to MIDAS during data taking. An online record history is included to track down possible problems in the experiment that are not explicit in the variables. 3.5.3 Trigger The trigger logic of the PIENU experiment was build using NIM7 modules. Figure 3.18 shows a schematic of the simplified PIENU trigger logic, the full diagram is shown in Appendix A. The logic has been designed for two 7Acronym for Nuclear Instrumentation Module. It defines mechanical and electrical specifications for electronics modules used in experimental particle and nuclear physics. 63 3.5. Data acquisition system and Triggers main purposes; calibration data and physics data recording. The particle identification (pions for physics and positrons for the calibration of some detectors) starts by requiring a time coincidence between the counters B1 and B2 and the target scintillator B3. To chose exclusively pions, proper energy cuts in B1-B2 are set. This signal is called the pion signal. A time coincidence in T1-T2 is required on the decay positrons either from π+ → e+νe or π+ → µ+νµ → e+νeνµ coming out downstream the target. This coincidence defines the decay-positron-signal. A pion-decay-positron- signal coincidence within the time window of -300 ns to 540 ns with respect to the pion stop in target B3 is the basis of the trigger logic. We will refer to these events “PIE” events. On top on these logic, three main configurations are used for physics events data-taking. These are called “physics triggers” and are summarized as follows • Prescale: Since the π+ → µ+νµ → e+νeνµ events dominate over the π+ → e+νe events by four orders of magnitude, a Prescale unbiased trigger is set to select 1/16 of the PIE events that can be of the type π+ → e+νe as well. • Early: This trigger is fired if the pion signal is in coincidence with the decay-positron signal within 5-40 ns with respect to the pion stop time. Since the pion mean life time is 26 ns, this trigger is enhances the detection of π+ → e+νe events, 70% of them are within this time range. • TIGC or BinaHigh: This trigger selects events which deposit a high energy in the calorimeters (Bina and CsI). The logic of the TIGC flag is described in 3.5.1. The energy threshold is set bellow the Michel edge (around 46 MeV). Almost all the π+ → e+νe events (with the exclusion of the tail events which are buried under the Michel spectrum) are selected by this trigger. The there are three additional trigger configurations for data quality checks and calibration purposes, these are summarized as follows. • Cosmic: The cosmic-ray events, most of them energy cosmic muons, are selected by requiring a high energy deposit in the CsI outer layers (either upstream or downstream) or the coincidence of inner and outer layers. A prescaling factor of 16 is applied to reduce the rate of this trigger. These events are used for CsI calibration. 64 3.5. Data acquisition system and Triggers Figure 3.18: Simplified diagram of the trigger logic for the three physics triggers in the PIENU experiment. Taken from [3]. • Xe-lamp: The Xe lamp provided flashes to all the CsI crystals for monitoring PMT gain variations. The lamp is triggered by a pulse generator twice per second. • Beam-Positron: This trigger is fired by beam positrons that are pre- scaled by a factor of 32. This trigger is used for the gain adjustment of plastic scintillators and Bina. The details of this procedure is in Section 4.2.1. Since the logic of these triggers is not mutually exclusive in most of the cases, an event can have more than one trigger label for physics triggers. This allows to assess the systematic uncertainties of not including a given trigger in the analysis. The rates of the triggers are shown in Table 3.3. The total pion stop rate in the target is ∼50 kHz and the total trigger rate is about 600 Hz. The trigger signal issued by any of the six triggers mentioned above is then latched by the positron (te+) and the pion (tπ+) timings. These latched signals trigger the acquisition of the data. te+ is used to trigger the 65 3.6. Data-Taking history data acquisition by the VME modules (VF48 and VT48) while tπ+ triggers the COPPER-boards data acquisition. Trigger Rate [Hz] Physics triggers Early 160 TIGC 170 Prescale 240 Calibration triggers Cosmics 15 Beam-positrons 5 Xe-lamp 2 Table 3.3: Rate of the different triggers logic in the PIENU experiment. 3.6 Data-Taking history The data for the PIENU experiment was taken from 2009 to 2012. The TRIUMF cyclotron is shut down each year from January to March to receive maintenance. This marks the separation between data-taking periods, or data-sets, as the data quality tests have to be performed separately. The most remarkable difference between data-sets is the difference in the pion beam momenta, this topic is discussed in Section 6.5.1. The data-sets were divided into runs containing approximately 300k events, which took about ten minutes at an incident pion rate of 50-60 kHz. The MIDAS files are about 1.8-GB in size. The proposal for the PIENU experiment was approved by TRIUMF committee in 2005, the PIENU detector was designed in 2006, an early design was tested with the M9 beam-line in 2007, and the full detector constructed and fully tested with M13 beam-line in 2008. In the following sections, the conditions for each running period are described and the overall data-taking history is summarized in Table 3.4. 3.6.1 2009 This data-set was taken under the first stable run conditions, data were separated as Run I and Run II, with about 1 M and 0.5 M π+ → e+νe events respectively. During this time period the module TIGC was still being tested but a discriminator for high energy was used. This former module summed 66 3.6. Data-Taking history the determined the pulse height of the sum of the Bina and CsI PMTs waveform. However, the analog sum did not consider any gain correction, leading to an unstable threshold and therefore to a potential loss of π+ → e+νe events. On top of this, the Cosmic trigger was not yet developed, leading to a non-calibrated energy spectrum. The CsI array calibration was instead attempted using a Xe-lamp dedicated trigger, however, this was inadequate. These constraints limited this data-set to exotic neutrino search in the π+ → e+νe energy spectrum. The learnings from an attempt to analyze this data-set will be of im- portance for the ongoing developing of the new-generation experiment PI- ONEER [63]. Finally, the preliminary results for massive neutrino analysis of 2009 data-set were deepened in the Ph.D. thesis of K. Yamada [4] and published in Ref. [87]. 3.6.2 2010 The final trigger configuration was in place at the beginning of the 2010 data-taking. The data-set is divided in Run III and Run IV, with about 2 M and 0.4 M of clean π+ → e+νe events respectively. During Run III the CsI PMTs signals were out of time with the trigger logic, and thus not recorded. Initially, the uncertainty in the low-energy tail of the measured π+ → e+νe energy spectrum, was estimated to be larger by a factor of two for Run III due to the lack of the CsI information. However, several upgrades in the MC simulation lowered the systematic uncertainty in the low energy tail correction making feasible to include this portion of data to the overall analysis. On the other hand, Run IV was the first high-quality data with the full triggers logic implemented and all the detector capabilities available. An initial analysis of Run IV is detailed in the Ph.D. thesis of C. Malbrunot [1]. Further investigation of Run IV resulted in the first publication of an improved branching ratio measurement [6]. The branching ratio uncertainty reached 0.24 % precision with similar contributions from statistics and sys- tematic sources. 3.6.3 2011 During 2011, the cyclotron was shut down until September due to a failure in the vacuum system. In September and early October, the PIENU-I sub assembly was removed to measure the response of the calorimeter as function 67 3.6. Data-Taking history of the angle of incidence with a special positron beam. Physics data-taking was resumed in mid October and named Run V with about 0.5 M clean π+ → e+νe events. Preliminary results of the analysis of the 2011 data-set were published in the Ph.D. thesis of S. Ito [3]. 3.6.4 2012 The largest and most easily usable high quality data-set was recorded in 2012, named Run VI, with around 2 M π+ → e+νe events. At the start of this running period, the energy threshold of the TIGC trigger was lowered to make sure that there was no potential loss of π+ → e+νe events. This resulted in more π+ → µ+νµ → e+νeνµ events firing the TIGC trigger. This decreased of the number of events per run is by a factor of 1.5 in 2012 compared to 2010 and 2011. Preliminary results of the analysis of the 2012 data-set were published in the Ph.D. thesis of T. Sullivan [2]. 3.6.5 Full analysis The integration of the analyses for Runs IV, V and VI, with a total of 3 M events, resulting in a blind measurement of the branching ratio was presented in the Ph.D. thesis of S. Cuen-Rochin [5]. This result represent the most precise result of the PIENU experiment until now. This thesis project concerns to the analysis of the Run III, with 2 M π+ → e+νe events, to be included later in the overall PIENU analysis. This thesis also aims to detail the upgrades in the analysis concerning the MC simulation since the version presented in [5]. 68 3.6. Data-Taking history Year Month Item 2005 Dec. PIENU proposal approved by TRIUMF 2006 - Detector design and test of concept 2007 - Detector construction and beam test on M9 beamline. 2008 May Oct. Oct.-Nov. Beam test in M13 channel M13 channel extension completed Test in the extended M13 channel with most of the detectors 2009 May May-Sep. Oct.-Dec Nov. 26 Nov. - PIENU detector completed Run I data-taking Run II data-taking Lineshape tests Lineshape measurements Beamline NIM publication [7] 2010 March Apr.-Sep. Oct.-Dec. - - Enclosure for the temperature control completed Run III data-taking Run IV data-taking Calorimeter NIM publication [9] Eight hours per week muon beam data 2011 Aug. Sep.-Oct. Nov. - Positron beam data-taking Lineshape measurements Run V data-taking Neutrino analysis for 2009’s data-set publication [84] 2012 Apr.-Dec. Dec. Run VI data-taking Special runs for systematics 2015 - - Detector NIM publication [8] Analysis of Run IV publication [6] 2025 January Detector’s energy response NIM publication [10] In development - Closing of the PIENU analysis and Final Rexp π publication. Table 3.4: Timeline of the PIENU experiment. 69 Chapter 4 Data Analysis Physics and calibration data are stored in raw format, organized in data- taking runs with an average duration of 10 minutes. Data have been clas- sified in six Runs. The size of each Run depends on the taking time period and data quality. Each Run was preceded by a scheduled maintenance in the TRIUMF cyclotron or by important changes in the detector. Table 4.1 shows a summary of Runs I-VI size and data-taking periods. Run Number of 10-min runs Data-taking period I 4,711 2009, May.-Sep. II 4,313 2009, Oct.-Dec. III 14,433 2010, Apr.-Sep. IV 2,041 2010, Oct.-Dec. V 3,688 2011, Sep.-Oct. VI 15,352 2012, Apr.-Dec. Table 4.1: Datasets along PIENU timeline. A decoding tool called PROOT was used to process raw data. It takes the digitized information from the ADC waveform (WF) and the TDC hits, and stores them in ROOT [145] trees. The trees are later processed with Clark; a tool to apply selection criteria on data. It was originally developed for the TWIST experiment [146]. Clark was later adapted to the PIENU experiment with specific analysis cuts to select pion-decay events and remove beam background and pileup events. The outputs of the analysis package are in the form of ROOT trees. In the first part of this chapter, the process of digitizing the analogue signal from different components of the PIENU experiment through the reading boards (see Section 3.5.1) is detailed. 70 4.1. Variable extraction 4.1 Variable extraction 4.1.1 From VT48 The VT48 board records hits in a time window from -3.6 µs to +4.4 µs around the prompt region, defined to be the hit coincidence in B1 and T1 counters. For the Wire Chambers, the hit provides spatial and time infor- mation that allows the tracking systems to reconstruct the particle track. The particle can activate several wires (or strips in the case of the Silicon detectors). If the activated wires are adjacent, the wires are clustered into a single hit. The wide time recording window allow VT48 to identify several clusters and for each reconstructed track, number of degrees of freedom, the χ2, the residuals and position information are stored. VT48 records two variables corresponding to the number of identified hits and the time of the i-th hit for each scintillation detector. 4.1.2 From COPPER waveform COPPER records all hits in the time window from -6.4 µs to 1.35 µs around the prompt time. Different hits are identified by a finding algorithm that determines the highest sample after a prolonged drop from the WF. Ten variables are recorded by COPPER, among which the base variables are the time (thit), the pulse height (PH) and the charge (Q) from the WF for all identified hits. The remaining variables are variations on the integration time and are described below. For Q, the charge is integrated in a time range of (thit - 20, thit + 20) ns. The charge was recorded over different time intervals to determine the sensitivity of the algorithm to pileup events, these are Qw in (thit - 20, thit + 80) ns and Qww (thit - 20, thit + 600) ns. An analogue version of these variables can be recorded taking the trigger time tf as a fixed reference instead of the peak time of the highest sample. For example Qfw in the time interval (tf - 20, tf + 80) ns. Additionally the charge over the full COPPER gate (7.75 µs) is integrated. All these variables are stored as arrays, whose length depends on the number of identified hits. Figure 4.1 illustrates the process of the charge integration for different time intervals. For B1, B2 and Tg counters, the WF is fitted using a fit template that is obtained from a spline interpolation of the average PMT pulse shape. The amplitude, time, χ2 of each pulse hit is recorded in the ROOT tree. The time interval from -6.4 µs to -2.15 µs before the prompt is defined as the ”Pre” region. The integrated charge (Q and Qw), PH and the time of each hit in this region are stored in the tree as well to study specific types of pileup events contaminating the pion signal. The number of samples read out by 71 4.1. Variable extraction Figure 4.1: Schematic drawing of the variables extraction from a waveform readout by COPPER [3]. each channel are stored in the tree as well to monitor the data integrity and track down possible errors in data-taking process. 4.1.3 From VF48 waveform The read out time window of VF48 is usually 1 µs around the hit that fired the positron trigger. Hits are identified by a hit finding algorithm with a different time identification interval since NaI crystal has a different sampling rate. Eleven variables are recorded considering wider time ranges for pileup tracking, they are described below. Unlike COPPER, the VF48 integration window is defined in terms of samples (Q: thit - 5 ≤ t ≤ thit + 5 samples). For extended studies, the charge is integrated before (Qpre: thit - 15 ≤ t ≤ thit - 5 samples) and after (Qpos: thit + 5 ≤ t ≤ thit + 15 samples) the peak time. Additionally, the charge in extended gates (Qw: thit - 10 ≤ t ≤ thit + 10 samples and Qww: thit - 10 ≤ t ≤ thit + 25 samples) are also recorded. These variables are particularly useful since NaI pulses are very broad. Analogously to COPPER, the integrated charge variables are also taken with the positron trigger time as a fixed reference. These are Qf (tf - 5 ≤ t ≤ tf + 5 samples), Qfw (tf - 10 ≤ t ≤ tf + 10 samples), Qfpos (tf + 5 ≤ t ≤ tf + 15 samples), Qfpre (tf - 15 ≤ t ≤ tf - 5 samples) and Qfww (tf - 10 ≤ t ≤ tf + 25 samples). For every event the fit is performed in each of 72 4.2. Calibration the 19 PMTs of Bina and each of the four quadrants in CsI array. For the Silicon detectors, the process of charge extraction is more com- plex due to the charge sharing mechanism between strips discussed in Section 3.2.3. As for Wire Chambers, hits in adjacent strips are clustered. For each cluster, the two strips in which the highest charge (Q: thit - 128 ≤ t ≤ thit + 128 ns) was deposited are tagged. The position of the hit is determined by weighting the charge of the two high-strips with a spatial resolution of ∼95 µm. By considering the charge sharing as ideal, i.e., that the charge is equally distributed in strips depending of the relative position to one of the high-strips, the mechanism has been implemented in MC. This have lead to the treatment of the analysis with one single track or multiple tracks sepa- rately and it will be discussed in Section 6.5.2. The time of the hit is the average of the time recorded in the high-strips weighted by their respective charge. The number and size (corresponding to the number of strips) of clusters are also recorded. Charge and pulse height variables recorded by VF48 are used to obtain the calibrated energy of the calorimeter and the scintillation counters. 4.2 Calibration 4.2.1 Gain stability Bina Since PMTs are highly sensitive to external factors such as temperature, exposure to light and, in general, to aging, their gain might change along time and be different relative to one another. Therefore, all counters read out by PMTs (excluding the CsI array for which the procedure is different and it is described in next section) have to be adjusted to a global gain factor. For the scintillation counters and the NaI crystal, the gain factor as a function of time has been calculated by determining the peak position of the calibration-positron mono-energetic energy spectrum as a function of the run number. The beam-positron trigger is set on to select calibration positrons. How- ever, due to the small rate of this trigger, there is a remaining portion of beam muons. To remove these beam muons, energy cuts are applied on the non-gain corrected energy spectrum of B1 and B2. Figure 4.2 shows the cuts applied on the B1/B2 energy spectra, their value are valid for all datasets. Since the goal is to estimate the peak position in the positron spectrum, 73 4.2. Calibration 0 0.5 1 1.5 2 2.5 3 B1 energy [MeV] 1 10 210 c o u n t / b in 0 0.5 1 1.5 2 2.5 3 B2 energy [MeV] 1 10 210 c o u n t / b in Figure 4.2: Non-gain corrected energy spectra of B1 (up) and B2 (down) using the calibration positron trigger. Blue solid lines represent the cuts that remove beam muons. resolution effects are negligible for the scintillation counters and Bina. To accomplish this, the variable Qwwf was chosen to improve the resolution on the pion peak. Due to its wider integration window, as described in Figure 4.1, it enables more pileup accumulation. For the PH variables, the calibration is done by requiring a further sum over a predefined number of hits for each event. The process consists in fitting the positron peak using a Gaussian function with an asymmetric range. Figure 4.3 shows the positron spectrum with the Gaussian fit over the restricted range for different PMTs and various run ranges. The process is performed for the 19 PMTs of Bina. By default this process is performed by PROOT on a run-by-run basis, however, to improve the fit quality the process was updated to use groups of 20 runs per fit. This grouping is small enough to identify gain shifts, the details of the process are in ref [130]. The gain factor is the ratio of the 74 4.2. Calibration Figure 4.3: Positron peak fitted with Gaussian for several Bina PMTs and various run groups. Integrated charge integrated in wide range (Qwwf) is used of a better peak resolution. positron peak mean for a given reference over the peak mean of the group in question. For convenience, the reference was chosen to be the run 11700 as the condition of the PMTs was optimal when the run was taken. Figure 4.4 (a) shows the gain factor of five Bina’s PMTs as function of run group for an early portion of Run IV data. The prominent shift is caused by the relative difference of gain between PMTs. Some PMTs show a sharper gain shifting as function of time. To ensure that these shifts are properly corrected, the gain factor is applied to their respective variables as function of run. Figure 4.4 (b) shows the gain shift after repeating the fitting process on the corrected positron spectra on each PMT. The gains fluctuate around the unit producing a normal distribution centered at one if they are projected on the y-axis. In average, the bell with of the normal distributions is smaller than 0.001 [130], which is good enough for the PIENU’s precision goal. 75 4.2. Calibration 0 10 20 30 40 50 run group 0.96 0.98 1 1.02 1.04 1.06 g a in PMT 1 PMT 18 PMT 5 PMT 6 (a) Gain factor before shift correction. 0 10 20 30 40 50 run group 0.998 0.999 1 1.001 1.002 1.003 g a in PMT 1 PMT 18 PMT 5 PMT 12 (b) Gain factor after shift correction. Figure 4.4: Gain of various PMTs from Bina using 10 runs per correction group. 76 4.2. Calibration CsI All CsI PMTs are connected to a Xe lamp using a quartz fiber. The pulse height of a given signal in the Xe lamp is compared run by run to a reference run to adjust any PMTs gain fluctuation in time. For consistency the signal is taken from cosmic events. The Xe reference run is replaced every 20 runs to be consistent with the energy calibration process. 4.2.2 Energy calibration Bina The energy calibration of Bina is based on the total deposited energy by π+ → e+νe events. This is achieved by properly aligning the energy spectra from data to MC. Figure 4.17 shows the comparison of the Bina’s energy spectra between MC and data. The match is achieved by locating the peak in both cases and applying a correction factor to Bina’s raw energy. The agreement is achieved for different openings of WC3 acceptance window, R in eq. 3.3. This process is better described in Section 6.5.3, as the match depends, among several factors, on the position of the WC3 chamber along the beam axis. The calibration has a 0.1 MeV precision level in the π+ → e+νe peak. Plastic scintillators The energy calibration is based on the deposited energy by beam positrons (a minimum ionizing particle) going through a known a Polyvinyltolulene- based scintillator with known dimensions. The deposited energy was cal- culated using important constants of the material from PDG [16]. The scintillator counters in the MC were tuned by setting the Birks’ constant [147] according to the Birks law that relates the quenching effect with the light saturation as dY dx = S dE/dx 1 + kB(dE/dx) , (4.1) where S is the scintillation efficiency and kB is the Birks constant. Data’s positron peak was later aligned to the PDG calculation and verified with MC spectrum. 77 4.3. The Run III dataset Silicon detectors The energy calibration of the Silicon detectors is based on the total deposited energy in the activated strips by a minimum-ionizing particle, i.e. a beam positron, traversing a known thickness of silicon wafer. CsI calibration The CsI crystal array was calibrated using the Cosmic trigger. This trigger enables to perform a calibration each 20 runs to collect enough amount of Cosmic events for a reliable calibration. The cosmic peak, made up of cosmic muons, in the raw variables is associated with the prediction given by MC using a cosmic shower simulation in the CRY [148] package. This package is used to generate correlated cosmic-ray particle showers at sea level and at the coordinates of the PIENU experiment. The scaling factor that multiplies the raw variables is α = CMC Cdata Xeref Xe , (4.2) where Cdata and CMC are the fitted peak parameters from data integrated charge and MC energy spectrum respectively and Xeref is the reference charge of Lamp trigger every 20 runs from section 4.2.1. Figure 4.5 shows the comparison between data and MC for the inner upstream set of crystals in the array. The peak position varies up to 20% in energy scale with the position of the crystal in the array, this difference is well emulated by the MC. 4.3 The Run III dataset The analyses of the datasets taken in Runs IV, V and VI corresponding to November-December 2010, 2011 and 2012 respectively are described in [1], [2, 3] and [5] respectively. These datasets were taken when the experiment reached a stable configurations, hence, their analyses are consistent with one another. We will refer to such as nominal analysis. When Run III data was taken, there were components in the experiment still under testing or were not stable in time, therefore, they were not included in the analysis. In order to correct its effect within the global analysis, the Run III dataset was divided into three blocks to be treated separately and merged after dealing with their particularities. Table 4.2 shows the division of Run III in blocks 78 4.3. The Run III dataset Figure 4.5: Cosmic signal in the inner upstream section of the CsI crystal array. Data is shown in black solid line and MC in hatched red line. The energy resolution was not implemented in MC for this figure. Taken from [5]. Block Number of runs Run range 1 7402 31872 - 40967 2 3530 40972 - 45151 3 3501 45838 - 49665 Table 4.2: Data blocks of Run III. of data. The shortcomings of the data blocks compared to Runs IV, V and VI are listed as follows • Downstream trigger was turned on until Block 2. • TIGC inefficiency was slightly worse for Block 1. 79 4.4. Event Selection • Upstream shift between sub-assemblies PIENU-I and PIENU-II that was corrected until Block 3. On top of these shortcomings, the CsI PMTs were out of time during the Run III data-taking period, leading to an unreliable calibration, hence the CsI information was not used in its analysis. This includes the analysis of raw data and the calculation of MC corrections, more details are described in Section 6.5.3. This chapter will describe how these features were treated to make the Run III analysis compatible with nominal analyses. 4.4 Event Selection To remove events that are not of interest for the branching ratio measure- ment, cuts are applied on data. These cuts can be classified in categories: i) the pion event selection cuts, ii) background event suppression cuts, and iii) pileup identification cuts. The PIENU-I sub-assembly is mostly in charge of the pion identification as the pion decays within the target B3, its most downstream component. Then, the positron tracks are measured in the sub-assembly PIENU-II where the components prior to the calorimeter are used to suppress background components out of the pion time and decay topologies that are not of interest. 4.4.1 Pion selection cuts The main function of scintillation counters B1 and B2 is to identify beam pions in their energy spectra. The position of the peak is obtained from the MC. Figure 4.6 shows the energy spectra of B1 (up) and B2 (bottom) respectively, the beam particle distribution going from left to right corre- sponds to positrons at 1.2 MeV, muons at 3.2 MeV, pions at 4 MeV and a pileup peak of two pions hitting B1 or B2 within the integration window at 8.5 MeV in B1 energy spectrum. The solid lines represent the cuts to remove beam particles other than pions. The branching ratio have shown to be stable against the B2 cut parameters within a reasonable range (more details in Section 7.1.2. On the other hand the branching ratio was found to have a non-negligible sensitivity to B1 upper limit. This sensitivity is re- lated with the existence of a beam background that was identified as a beam pion and a beam pion in the same RF bunch. The systematic uncertainty depends on the content of the beam according to the Ref. [128]. To suppress particles from the beam halo (such as positrons and muons), the beam profile was delimited by two acceptance cuts are in WC1 and WC2. 80 4.4. Event Selection 0 1 2 3 4 5 6 7 8 9 10 B1 energy [MeV] 410 5 10 6 10 710 C o u n ts / B in 0 1 2 3 4 5 B2 energy [MeV] 3 10 410 5 10 6 10 710 C o u n ts / B in Figure 4.6: Energy deposited in B1 (up) and B2 (bottom). The three main peaks in each spectrum are, from left to right, caused by positrons, muons, and pions in the beam. The blue lines indicate the cuts. This cut is dataset-dependent as the beam collimation depends on the M13 channel setup. For the entire Run III dataset, the cuts were set from -23.0 to 18.0 mm in the x-axis, and from -19.0 to 21.0 mm in the y-axis. Given that both wire chambers (WC1 and WC2) have the same dimensions and due to the proximity of one to the other, the same cuts were applied for both of them. 81 4.4. Event Selection 30− 20− 10− 0 10 20 30 WC1_X [mm] 30− 20− 10− 0 10 20 30 W C 1 _ Y [ m m ] 1 10 210 3 10 410 5 10 6 10 30− 20− 10− 0 10 20 30 WC1_X [mm] 30− 20− 10− 0 10 20 30 W C 1 _ Y [ m m ] 1 10 210 3 10 410 5 10 6 10 Figure 4.7: Distribution of XY positions at the center of WC1 (up) and WC2 (bottom). The red lines indicate the cuts. 4.4.2 Pileup rejection cuts Multiple pulses The waveform from B1 and B2 signals are fitted and digitized by COPPER boards. Information such as the timing and the integrated charge of the pulse is used to determine the presence of events arriving within the same trigger time window, these events are called pileup. These events are not useful for a physical measurement so they have to be rejected from the data. The first technique to do so is by requiring a single fitted pulse in at least 82 4.4. Event Selection 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 B1_Q/B1_Qww 1 10 210 3 10 410 5 10 6 10 710 8 10 9 10 C o u n t / B in 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 B2_Q/B2_Qww 10 210 3 10 410 5 10 6 10 710 8 10 C o u n t / B in Figure 4.8: Ratio of integrated charge in short gate to wide gate (Q/Qw) for the four PMTs of B1 (top) and B2 (bottom). Distributions with all cuts (excluding the one being discussed) and without cuts are shown in red and black colors, respectively. one of the four PMTs from the scintillators B1, B2 and T1. This condition has the following logical expression to be coded in Clark. { (NPMT1 B1 = 1) ∨ (NPMT2 B1 = 1) ∨ (NPMT3 B1 = 1) ∨ (NPMT4 B1 = 1) } ∧ { (NPMT1 B2 = 1) ∨ (NPMT2 B2 = 1) ∨ (NPMT3 B2 = 1) ∨ (NPMT4 B2 = 1) } ∧ { (NPMT1 T1 = 1) ∨ (NPMT2 T1 = 1) ∨ (NPMT3 T1 = 1) ∨ (NPMT4 T1 = 1) } . (4.3) 83 4.4. Event Selection This condition can not be required on B3 as the pion decay occurs within its volume and there is a different signal per particle. In T2 this requirement can not be applied as there is a position and energy dependence associated to the optical fiber detector topology. By itself the condition above is not enough to trim all pileup events within the same integration range that might be misidentified as single pulses. These events are usually caused by optical reflections within the counters and electronic noise. To remove them and enhance further the pileup rejection, the ratio of integrated charge in a short integration window ([-20,20] ns, Q) to a wider integration window ([-20,80] ns, Qw) is used. Figure 4.8 (up) shows the distribution of the ratio Q/Qw for B1 PMTs and figure 4.8 (bottom) shows the same ratio for B2 PMTs, the black distributions has no cuts while the red distributions consider the cuts described. The cuts were set to 0.78 ≤ Q/Qw ≤ 0.98 for B1 and 0.75 ≤ Q/Qw ≤ 0.96 for B2. For Run III data, there is a slight displacement in B2’s PMT4 that is caused by a gain shift corrected by a similar algorithm in PROOT as the one described in Section 4.2.1. The cuts for this PMT were set to 0.8≤ Q/Qw ≤1.01. Early times Another way to identify particles other than pions that can not be isolated by energy cuts is by their time of flight that is measured in the B1 first hit time compared to the pion stopping time within the target B3. This cut showed to be constant along Runs III - VI, showing the stability of the COPPER inner clock calibration. The time cuts were set to -1385 ≤ E.B1 j WF t[0] ≤ -1345 ns, where WF refers to a COPPER quantity, j=1,..,4 stands for the PMT number and the index 0 refers to the first fitted pulse. A similar condition is applied to the T1 counter to discriminate prompt events in the time coincidence of ±2 ns within the pion stopping time. This cut kills beam positrons, beam muons and protons produced from nuclear interactions happening within the target. In any case, the particle reaches T1 in prompt time. This cut is also useful to reject muon coming from previous pion decays (Old-Muon). This background will be reviewed in Section 5.2.2. Unlike B1, in T1 all the hits have to be examined and removing all of them in which the time difference between T1 and B1 is smaller than 2 ns. In normal conditions this region is not relevant in the Time spectrum analysis, but since the time variable considers the average of PMTs time some of these events can leak into the analysis. The cut is designed to cut prompt events from both π+ → µ+νµ → e+νeνµ and π+ → e+νe decay 84 4.4. Event Selection 8000− 7500− 7000− 6500− 6000− 5500− 5000− 4500− 4000− 3500− B1 Pre-Time [ns] 210 3 10 410 5 10 6 10 C o u n t / B in Figure 4.9: Pileup events distribution in the Pre-region (-7.7 to 3.5 µs) of B1. Distributions corresponding to first, second and third hits are shown in black, blue and red, respectively. channels. Pre-Pileup Another portion of pileup events is identified in a -7.7 µs to -3.5 µs sub-range from the integration window of scintillator counters, as shown in figure 4.1. The condition of not allowing events in the Pre-region is required for the scintillator counters B1, B2, B3 for the pion signal and in T1 and T2 for the positron signal. Figure 4.9 shows the distribution of the Pre-region in B1, a similar behavior belongs to counters B2, B3, T1 and T2. In principle these events are trimmed from data by requiring a condition similar to eq. 4.3 for the Pre-region. Since the COPPER digitization rate does not account for the pion beam rate, this condition is still not enough to suppress all the pileup events in the Pre-region. Therefore, a systematic uncertainty has to be assigned to this cut, as this is the only way to suppress this kind of pileup. To estimate such uncertainty, the upper limit in the Pre-Pileup distribution is relaxed to allow more events in the Time spectrum. This process will be described in Section 7.1.2. 85 4.4. Event Selection T1 Fake pileup The cuts on B1 and B2 remove most of the real pileup events caused by the beam pion rate and the existence of beam particles other than pions. However, there is another kind of pileup caused by inner reflections of noise fluctuations happening within scintillation counters read by PMTs, this is the case of T1. The condition 4.3 does not account for this kind of pileup. Its identification has been achieved by taking the ratio of the charge integrated in the full range to the amplitude of the fitted pulse of the triggering hit as function of the pulse height, any disproportion between these two quantities suggests the existence of overlapping signals. Figure 4.10 shows the fake hits and real pileup separated into two bands; events with a small charge ratio are identified as real pileup. They were separated from fake pileup (with large charge ratio). Additionally, this dis- tinction protects the event being rejected by the pileup cut from depending on the positron decay time. The change in the T1 energy spectrum because of this cut has an effect of < 1 × 10−8 in the branching ratio units [125], negligible for the PIENU’s precision goal. Given the topology of the two bands, the cut was designed to be a smooth function of the pulse height. It built by taking projections of slices along the pulse height and identifying local minimum points or inflection points. A quadratic model was fitted to this set of points, giving rise to the cut shown in figure 4.10 represented by the black line. This process was repeated for the four data blocks of Run III. The cut proved to be consistent for all data blocks. T1-T2 coincidence A further suppression of pileup in the downstream counters is done by re- quiring that in all PMTs of T1 only the closest pulses to the pion signal time are fitted. To reduce this component by an order a magnitude, a cut on the number of hits recorded by T1 counter is implemented: events which have more than one hit in all of T1 channels are removed. Additionally, a coincidence of ±20 ns is required between the T1 and T2 counters. Additional Pileup cut in T1 Only the closest pulse to the trigger time is fitted for every PMT in T1. This cut makes sure that the fitted pulse is also the first seen by the PMT. Events with additional pulses before the fitted one are discarded as the are off time. The closest pulse is identified by requiring the time difference of 86 4.4. Event Selection Figure 4.10: Ratio of the full integrated charge to the pulse amplitude vs the fitted pulse amplitude in T1. The black curve indicates the cut to separate the real pileup events (below) from false hits (above). Figure 4.11: Correlation between the minimum energy dE/dx in the coun- ters S3, T1 and T2 and the energy deposited in Bina, the red line indicates the cutoff that discriminates the events produced by protons. the positron time in T1 and the Downstream trigger within a time window that is beam-independent. This pulse is compared to the difference between 87 4.4. Event Selection 200− 100− 0 100 200 T2-T1 time [ns] 1 10 210 3 10 410 5 10 6 10 710 8 10 9 10 C o u n t / B in Figure 4.12: Time difference of T2 and T1 counters. In blue color the time distribution without cuts. In red color the time distributions with all cuts discussed so far. The vertical lines represent the coincidence cuts. T1 and the positron to Upstream trigger times for each PMT. Only the first hits in both categories pass this cut as they are the first detected by each PMT in T1. Figure 4.13 shows the difference of the positron time and the Downstream trigger to identify the first fitted pulse in T1. This cut was re-adapted for blocks 1 and 2 since the Downstream trigger inner clock was off-time during the data-taking period. The first fitted pulse was identified in the global positron time using a peak finding algorithm. The algorithm was validated using data from Run IV and Run V. 4.4.3 Background suppression Protons The interaction between the beam pions and the target nuclei releases pro- tons that deposit a large amount of energy in downstream counters and can be distinguished from minimum-ionizing positrons. On the other hand, the energy distribution in Bina is broad enough to span energies from few MeV to above 100 MeV. Yet, the largest accumulation of protons is located past the Michel shoulder at 48 MeV leading to a biasing in the branching 88 4.4. Event Selection 3800− 3790− 3780− 3770− 3760− 3750− 3740− 3730− (e_time_VT_t - Dwnstr_VT_t) [ns] 1 10 210 3 10 410 5 10 6 10 710 8 10 9 10 10 10 1110 c o u n t Figure 4.13: Time difference between the positron time and Downstream trigger. The peak with the smallest difference at -3760 ns corresponds to the first fitted pulse in T1. The distributions without cuts and with all cuts discussed so far are shown in black and red, respectively. The blue solid lines represent the cut values, all events outside the lines are rejected. ratio measurement if they were not removed. Since the two energies are anti-correlated, so the proton events are identified by a high energy band. Figure 4.11 show the correlation between the minimum energy loss per path length (dE/dx) in the downstream counters (S3, T1, T2) and energy deposit in Bina. Given the anti-correlation between these two energies, protons were identified in the upper band. Most of the protons are prompt in time, there- fore the early time cuts should remove a significant portion of them. The proton cut was modeled as a straight line and it works as a backup as it only removes 0.008% of events. There is a small energy dependence on this cut but it is negligible for the precision goal. Extra Muon signal A special channel in the VT48 TDC was set up with the purpose of moni- toring the muon signals in B1 and to provide additional protection against beam muon pileup. This channel had an extended time range of 20 µs before the trigger time, instead of the usual 8 µs. In order to prevent signals from particles other than beam muons, a discriminator was installed to require 89 4.4. Event Selection 0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000 Muon channel time [ns] 410 5 10 6 10 710 8 10 C o u n t / B in Figure 4.14: Distribution of events with hit in the B1 VT48 special channel for beam muon detection. Red solid lines indicate the cuts. Events within the arrow regions are rejected in the analysis. muon-like pulse height in B1. Rejecting any events in the full window was found to provide no benefit compared to a cut-based selection that loops over all hits in this channel. The specific location of the cuts are the result of studying the change in upstream and downstream energy distributions [139] leading to the cuts shown as red solid lines in figure 4.14. The only noticeable difference between this approach and removing all events is to reduce the available statistics. 4.4.4 Acceptance cut To restrict the the angle between the positron track and the beam axis, an acceptance cut is required on the radial position, with respect to the center of WC3, where the positron track crosses this detector. The energy spectrum is highly dependent on this cut as a larger opening allows more energy leakage produced by the electromagnetic showering. The radius from the center of WC3 wire chamber to the positron track is measured using the S3/WC3 tracker according to equation 3.3. This radius will be referred as R. This quantity is of extreme importance to estimate the branching ratio corrections such as the low energy tail (Section 6.6.1, and the acceptance 90 4.5. Summary of Event selection 0 10 20 30 40 50 60 70 80 90 100 R [mm] 0 20 40 60 80 100 120 140 6 10× C o u n t / B in Figure 4.15: Radius of the reconstructed positron track at WC3 position along the beam axis. The blue solid line represents the cut for the Run III dataset. All events above R > 40 mm are excluded from the analysis. correction (Section 6.6.2. If an event has multiple tracks, only the track with the smallest radius R is taken. Figure 4.15 shows the distribution of R radii of data block 3 in the Run III dataset. Nominally, this cut was taken at R < 60 mm in previous analyses [5] [2] [3]. For Run III, the nominal acceptance cut was chosen to be R < 40 mm as these data do not rely on CsI information, thus the systematic uncertainty assigned to the tail correction in the π+ → e+νe energy spectrum increases rapidly. Section 6.5.3 deals with the compatibility data-MC in the R distribution for the specific case of the Run III setup. 4.5 Summary of Event selection After the event selection only ∼10% of all the events will be used to extract a branching ratio measurement. The most restrictive stages are the pion energy cuts, scintillator pileup cuts and the acceptance cut. Combined to- gether, they trim 88% from the available data. Because of their high impor- tance in the analysis, the parameters involved in these cuts will be assessed. 91 4.6. Data calibration to MC This will be done by measuring the systematic change of the branching ratio to variations in the cut parameters within reasonable ranges, see Section 7.1 for more details. Although the acceptance cut highly restrictive, there is a trade-off between the available statics and the systematic uncertainty from the tail correction that increases rapidly for large R radii, specially for Run III MC that does not deal with CsI information. A detailed decription can be found in Section 7.1.4. Cut Surviving events (%) Data integrity 99.15 Pion energy cut 74.28 WC12 acceptance 69.89 Scintillator pileup 49.74 T1 fake pileup 47.15 Pre-Pileup 31.18 Proton cut 30.44 B1 prompt 30.43 T1 prompt 29.59 Additional T1 PU 28.14 T1-T2 coincidence 28.13 Mu hit channel 27.61 Acceptance radius R < 60 mm 16.45 R < 40 mm 9.93 Table 4.3: Fraction of events in the cut flow of the event selection. The number of events before cuts is 1.0837×109 for the Run III dataset. 4.6 Data calibration to MC An important part of the analysis is the proper alignment between data and MC of the energy spectra for the scintillation counters and the calorimeter components. In contrast to previous analyses, in which a single MC was used for different datasets, this work considers the implementation of particular pion beam profile for each dataset. This is a major improvement from the nominal analysis. Such profile contains the beam momentum distribution and the spacial distributions, this process is described in detail in Section 6.5.1. This lead to a better agreement between MC and data for upstream 92 4.6. Data calibration to MC Figure 4.16: Alignment of B1 (up) and B2 (bottom) energy spectra. The distributions of data without cuts and MC simulation are shown in black and red color respectively. The peaks at 2.2 and 1.1 MeV in B1 and B2, respectively, are pion decay-in-flight events. counters energy spectra. Figure 4.16 shows the comparison of B1 (up) and B2 (down) particle energy distribution. The pion peak was matched to MC via an alignment factor that mainly depends on the beam momenta. On the other hand, the calibration of the calorimeter energy spectra is 93 4.6. Data calibration to MC E.Cal_eBina [MeV] 𝞹⁺→𝞵⁺𝞶𝞵 →e⁺𝞶e 𝞹⁺→e⁺𝞶e Ecut = 52 MeV Figure 4.17: Alignment of Bina energy spectra. Data’s distribution in blue color; the physics triggers are active. In red color the MC simulation of the π+ → µ+νµ → e+νeνµ (0 to 52.8 MeV) and the π+ → e+νe (peak at 68 MeV) decay channels. The black solid line represents the energy cut to distinguish the two decays to first order. done by aligning the π+ → e+νe peak to MC using scale factors. Unlike the scintillation counters calibration, the alignment factors does not depend on the beam momentum but it depends on the R cut opening as this cut allows less electromagnetic showering events to enter in the spectrum. However the R radius defines a different acceptance for blocks 1, 2 and 3 than it does for block 4. This is because the PIENU-II sub-assembly suffered a displacement of nearly 2 mm with respect to the nominal distance to PIENU- I, changing the relative distance between the target B3 and WC3. The estimated displacement between sub-assemblies via the MC simulation and a further description of the process is discussed in Section 6.5.3. Figure 4.17 shows the comparison between data and MC energy spectra. For data, the physics triggers are active to obtain the right amplitude of the components; the Michel spectrum from 0 to 48 MeV and the π+ → e+νe at 65 MeV. For MC there are two distributions corresponding to the muon decay channel and the positron decay channel, both of them are normalized to data amplitude. As a first approach the two decay channels can be distinguished in the energy spectrum by setting an energy cut-off. There is a remaining part of the π+ → e+νe tail buried under the Michel spectrum 94 4.6. Data calibration to MC that has to be corrected after validating the MC simulation. Under these conditions a raw estimate of the branching ratio can be done, this will be the main topic of Chapter 5. 95 Chapter 5 Raw branching ratio extraction 5.1 Time spectrum The decay times of the pion-like events passing the event selection criteria are used to build the time spectrum (TS). For this, the time difference between the decay positron and the beam pion is taken and averaged over the four PMTs of the B1 and T1 scintillation counters as follows, Tpos = 1 4 4∑ i=1 (tT1 i − tB1 i) , (5.1) where the mute suffix i stands for the PMT and times are taken from the first fitted pulses of B1 and T1. The pulse shape fitting is done using a spline interpolation, as detected in figure 3.16, so the achieved resolution of the time difference T1 to B1 is 270 ps. The events are separated into low-energy (LE) and high-energy (HE) regions, which correspond to π+ → µ+ → e+ and π+ → e+νe channels respectively, by an energy cut of Ecut = 52 MeV in Bina’s calibrated energy spectrum as depicted in figure 4.17. Besides the selection criteria described in Chapter 4, the histograms are filled when the physics triggers from section 3.5.3 are activated. Events in the LE region are required to trigger Early condition in the early time window and the remaining events are required to trigger the Prescale. A LE event can trigger one or the two triggers at the same time, but only those outside the early time region are unbiased, and are prescaled by a factor of 16 8 when added to the TS, the errors for each bar bin are scaled accordingly. Figure 5.1 shows the time spectra of LE (left) and HE (right) regions with and without applied cuts using Ecut = 52 MeV and R < 40 mm. The curve with no cuts depicts a periodic distribution of peaks in both LE and HE, which is caused by the cyclotron 42 ns RF in which proton bunches are delivered. 8Since π+ → µ+ → e+ events dominate over π+ → e+νe events by four orders of magnitude, a Prescale unbiased trigger selects only 1/16 of events. 96 5.2. Signals and Backgrounds The Tpos < 0 region is included in the analysis to measure the remaining amount of pileup and background events after the event selection in Tpos > 0 region. In order to extract a raw estimate of the pion branching ratio (Rraw π ) the background components are modeled in the time spectra for HE and LE regions separately. 5.2 Signals and Backgrounds 5.2.1 Main signals Throughout this section the components in TS are briefly described for both low and high energy regions, including the main signals of the π+ → µ+ → e+ and π+ → e+νe decay channels, respectively. There is essentially no leakage from the LE signal to the HE region. The leakage from the HE signal to the LE region will be described in the next chapter. The signals in LE and HE are described by the probability density functions of the pion undergoing to either channel by Eq. 5.2 and Eq. 5.3, respectively. επ→µ→e(t) = e−t/τµ − e−t/τπ τµ − τπ , (5.2) επ→e(t) = e−t/τπ τπ , (5.3) where τπ and τµ are the pion and muon mean lifetimes, respectively. Since the detector array T1-B1 has an intrinsic time resolution, its effect has to be included on the main signals. This is performed by convoluting the signals 5.2 and 5.3 with a Gaussian kernel G(T ) of a σ standard deviation that accounts for the time resolution, resulting in ε′π→µ→e(t, σ) = ρ(t, σ, τµ)− ρ(t, σ, τπ) 2 (τµ − τπ) , (5.4) ε′π→e(t, σ) = ρ(t, σ, τπ) 2τπ , (5.5) where ρ(t, σ, τ) is the shorthand notation for the composed function ρ(t, σ, τ) = exp ( σ2 − 2τt 2τ2 ) erfc ( σ2 − τt√ 2στ ) , (5.6) 97 5.2. Signals and Backgrounds Figure 5.1: Time spectra of LE (up) and HE (bottom) regions corresponding to π+ → µ+ → e+ and π+ → e+νe channels respectively. No cuts applied in black and event selection with nominal cuts in red color. In both cases Ecut = 52 MeV and an acceptance of R <40 mm. 98 5.2. Signals and Backgrounds and the error complementary function is defined by erfc(t) = ∫∞ t e−x2 dx. Introducing the time resolution effects via a Gaussian convolution was im- plemented in the previous version of the experiment with favorable results [40]. In order to clean the signals from the background and pileup events that can not be suppressed directly by nominal cuts in the event selection, these latter have to be included via a realistic model in the time analysis for both LE and HE spectra. The following section discusses the modeling process of these physical processes. 5.2.2 Low energy background Old-muon decays There is a remaining amount of muons from previous decays coming out the target or beam muons triggering B1 time. Both the positron from the actual decay and the positron from Old-muon hit T1 within the pion decay time. However these events can not be directly separated by setting energy cuts as this could bias the branching ratio. These events have the PDF from Eq. 5.7. εµ→e(t) = e−t/τµ τµ . (5.7) The procedure to introduce time resolution effects is similar to Eq. 5.5. Since part of this background comes from the beam, then it has to be replen- ished 43 ns periodically. However, the rejection of events with an additional hit in the Pre-time region extended to 6.4 µs prior to the prompt implies that no further supplies of beam particles can add to the background after this time. Thus, this background contributes as a flat component to the negative time spectrum. The probability of a decay from an old-muon from beam pion stopping within the target is of the order of 0.8%, which is con- sistent with the relative amplitude of positive and negative time regions in Figure 5.1 (left). Pion decay-in-flight Pion decay-in-flight events are relevant only when the decay muon stops before T1, otherwise the event will be prompt in time and not included in the Timefit. According to MC estimates, about 2 % of the π+ → µ+νµ → e+νe events decaying at rest (πDAR→ µDAR) undergo through the topology 99 5.2. Signals and Backgrounds 0 100 200 300 400 500 600 Tpos [ns] 0 10 20 30 40 50 60 70 80 90 N a I e n e rg y [ M e V ] 300− 200− 100− 0 100 200 300 400 500 600 Tpos [ns] 0 2000 4000 6000 8000 10000 12000 C o u n t / B in Figure 5.2: Simulation of the π+ → µ+νµγ decay followed by µ+ → e+νeνµ. Left: Positron time vs energy in the NaI(Tℓ) crystal. Right: Projection of the time spectrum for energies above 52 MeV (HE region). πDIF→ µDAR. This background is present only in positive times and decay with muon lifetime as in Eq. 5.7. 5.2.3 High energy background Pion radiative decay The are two mechanisms where a γ can be produced as part of the final state and potentially distorting positron energy and/or time distribution. If a γ is produced in the muon radiative decay µ+ → e+νeνµγ the positron energy spectrum will change without being boosted to the HE region because of energy conservation, while the time spectrum remains the same. Therefore, no additional shape needs to be included in time spectrum modeling. How- ever, if the pion undergoes the radiative decay π+ → µ+νµγ, followed by µ+ → e+νeνµ decay, both the energy and the time spectra are altered. Since the photon has the time of the pion decay and the positron has the time of the muon decay, the probability that the positron gets boosted to the HE region depends on the time difference between the photon and the positron entering the calorimeter. Similarly to the two Old-muon cases, the TIGC integration time plays an important role. If the γ from a pion radiative decay is recorded in the calorimeter, it will look like a pre-pileup event as it has the time of the pion decay instead of the 100 5.2. Signals and Backgrounds muon decay. These events are very likely to persist in the HE region after the pion decay time even if pre-pileup cuts are applied due to the long NaI pulse time. In contrast, the CsI crystals have better time resolution than the NaI crystal, thus they can reject events with smaller time difference. The time spectrum of the pion radiative decay has been generated using the Monte Carlo simulation. The probability of an event being boosted to the HE region as depends on the time difference between the photon and the positron entering the calorimeter. This probability was coded in the simulation. The TIGC trigger algorithm uses the waveforms from NaI. Figure 5.2 shows the pion radiative decay time spectrum. The amplitude of this component is a fixed parameter in the Timefit. Old-muon with no hit in T1 A mechanism of pileup in the HE region happens when a positron from a nominal decay or a positron from an old-muon decay impinges on the calorimeter without triggering the positron signal (missing T1). Such as a trajectory is geometrically possible, since the outer acceptance not covering T1 is dominated by the outer shell of Bina and the CsI rings. The rate of these events reduce significantly with tighter cuts on the acceptance R at WC3 (see Eq. 3.3). Since the positrons do not hit T1, the pileup cut in T1 do not remove these events. When a π+ → µ+νµ → e+νe event hits the calorimeter and trigger the positron signal within the old-muon time, then the total energy may get promoted to the HE region. Since these events are HE (the LE component is negligible), the event has to be fired by the TIGC trigger. For this the summed signal of Bina and the CsI PMTs is required to meet a certain energy threshold in a periodic 250 ns window. However, since the integration window for Bina is 1 µs, when these two events are separated by one trigger time, the energy would add up above Ecut and the TIGC flag would not be present and the event would not be recorded. The shape of this backgroud has been generated by the MC to be included in the HE time spectrum. Two cases were considered; either the positron from the π+ → µ+νµ → e+νe or from the old-muon fires the downstream trigger and the positron from old-muon or from π+ → µ+νµ → e+νe, respectively, enters the calorimeter without hitting T1. These two cases are combined to generate the Time spectrum. Additionally, the TIGC trigger mechanism was coded to emulate the events without the TIGC flag. For this, the waveform templates of the NaI and CsI detectors were used. Figure 5.3 depicts the resulting time spectrum for different acceptance cuts at WC3 in the HE region. 101 5.2. Signals and Backgrounds 300− 200− 100− 0 100 200 300 400 500 600 Tpos [ns] 5000 10000 15000 20000 25000 30000 35000 40000 45000 C o u n ts /b in Figure 5.3: Time spectra of Old-muons with no hit in T1 for the HE region. Acceptance cuts of R < 40, R < 50 and R < 60 were applied for black, blue and red color, respectively. The difference in the amplitude shows the suppression of the Old-muon component with smaller WC3 acceptances. T1 double pulse resolution Given that T1 has an integration window considerably larger than the decay positron rate, the case when two positrons hit T1 can not be removed by the pileup cuts described in section 4.4.2. This means that only one of them makes it to the downstream trigger to be recorded. Therefore, the combined signal has to be modeled to be included in Timefit. There are two cases contributing to the situation when the waveform of two positrons that are close in time overlap, with only one of them being recorded, due to the time resolution ∆ of T1; Case A, modeled by the functional form FA(t) in Eq 5.8, is when T1 is trigger by a positron from an old-muon decay in either positive or negative times. This means that the pion triggers the signal in B1 but the positron from old-muon hits T1 before the real decay positron. In this case the old-muon event can only be followed by the positron from pion decay in positive times. The amplitude of this case is the product of the old-muon probability (Eq. 5.7) times the integrated probability that the π+ → µ+νµ → e+νeνµ decay (Eq. 5.2) triggers T1 within ∆. 102 5.2. Signals and Backgrounds FA(t) =    0 t < −∆ Eµ→e(t) ∫ t+∆ 0 Eπ→µ→e(x)dx −∆ ≤ t ≤ 0 Eµ→e(t) ∫ t+∆ t Eπ→µ→e(x)dx t > 0. (5.8) Case B, modeled by the functional form FB(t) in Eq. 5.9, is the opposite case. Here the real pion triggers B1 and the decay positron hits T1 before the positron from the old-muon. This case is modeled as the product of the probability of π+ → µ+νµ → e+νeνµ signal times the integrated probability that the old-muon decaying within ∆ as follows: FB(t) = { 0 t < 0 Eπ→µ→e(t) ∫ t+∆ t Eµ→e(x)dx t ≥ 0. (5.9) The actual time resolution ∆ is the response of T1 signal readout mea- sured as the average difference between the first and second hit of the four PMTs. This difference is obtained from fitting the distribution of subsequent hits to step function convoluted by a Gaussian PDF as follows: H(t,∆, σ) = ∫ ∞ −∞ H(t′,∆)G(t− t′, σ)dt′ = 1 2 { 1 + erf ( t−∆√ 2σ )} , (5.10) where H is the Heaviside step function (H(t ≥ ∆) = 1, H(t < ∆) = 0), G(t − t′, σ) is the gaussian kernel and erf(t) is the Gauss error function. Figure 5.4 depicts the time distribution of hits in T1 PMTs along with the described fit. The resulting average from fours PMTs is ∆ =15.7 ns. Since the trigger timing can not distinguish between the two cases they should have a common amplitude in the time spectrum. The overall component is a sum of these cases, defined as F2(t) = FA(t) + FB(t). To estimate the amplitude f of F2(t) in the time spectrum, the T1 double pulse resolution (∆) was artificially increased up to 100 ns in steps of 20 ns. This is done by rejecting events after the first hit in a given window. Figure 5.5 depicts the effect of increasing such time difference in the time spectrum using nominal cuts. There is an accumulation of events that contaminates the t < 0 region in HE spectrum as the time difference increases, which is consistent with F2(t) modeling. To enhance the T1 double pulse resolution 103 5.2. Signals and Backgrounds Figure 5.4: The time difference between first and second hits in each of the T1 PMTs. Black solid line represents the fitting function modeled as a smooth step. component, only T1 pileup events were used to fill the time spectrum. This leaves three active components; Cases A, B and Old-muon not hitting T1. By performing a fit on these three components f can be obtained. Figure 5.6 (up) shows the time spectrum of T1 pileup events with the fitting compo- nents. The results of the fit are shown in figure 5.6 (down). The projection at ∆ = 0 ns has been fixed to a zero amplitude for consistency and the points were fitted to a quadratic model. The projection of the model to the actual value of T1 time resolution ∆ =15.7 ns resulted in 1.23×10−3. This amplitude was fixed in Timefit to be discussed later. This result is valid for 2010, 2011 and 2012 datasets. 104 5.3. The Fitting Function 300− 200− 100− 0 100 200 300 400 500 Tpos [ns] 10 2 10 3 10 4 10 C o u n ts /b in Figure 5.5: Time spectrum in the HE region. Events after the first hit in T1 were artificially rejected in different time windows; 40, 100 and 160 ns in black, blue and red color respectively. 5.3 The Fitting Function The signals and backgrounds described are fitted simultaneously to the time spectra of LE and HE regions in Figure 5.1 using the fitting package MINUIT [149]. This allows to extract a raw measurement of the branching ratio Rraw π . The fit is performed in the time rage from -270 to -20 ns and from 5 to 520 ns. Although the T1 prompt cut described in section 4.4.2 removes beam positrons, muons, and protons, the prompt region between -20 and 5 ns was excluded to avoid distortions in Timefit the could affect Rraw π . The real time t is measured with respect to the pion stopping time in B3 t0, or t = t′ − t0, where t′ is the measured time defined in 5.1. This parameter is fixed along the fitting process and was determined by fitting the rising time from the time spectrum from a special set of muon runs. This measurement was repeated consecutively in years 2010, 2011 and 2012. The result from 2012 dataset t0 = 2.5 ns [2] was used for all Run’s analyses including this since it is the most statistically significant result. 105 5.3. The Fitting Function Figure 5.6: Up: Time spectrum in the HE region of T1 pileup events which includes Old-muon positrons not hitting T1 in magenta, Case A and B with an artificial time difference after first hit ∆=100 ns in green and blue respec- tively (see text for details). The sum of the three components is represented by the red solid line. Down: T1 double pulse resolution amplitude f as a function of the difference time difference after first hit in which events are rejected ∆. Error bars comes from the fit quality, the red line represents the quadratic model and the red point is the projection of the model at the real T1 time resolution ∆=15.7 ns. 106 5.3. The Fitting Function 5.3.1 Low-energy time spectrum The LE region is dominated by the pion chain signal π+ → µ+νµ → e+νeνµ and a portion of πDIF and old-muon backgrounds. There is also contribution from π+ → e+νe events in the low-energy tail and µDIF that decay with the pion lifetime and are negligible in the time spectrum but have to be corrected via Monte Carlo. The fitting function of the low-energy region is shown in Eq. 5.11, where H is the Heaviside function, a is the amplitude of the π+ → µ+νµ → e+νeνµ signal, (1 − r) is a correction for the loss of LE events that are boosted to HE region through the described mechanisms, b is the amplitude of pion decay-in-flight and c is the amplitude of old-muon decays from muon at rest in the target Tg. ϕLE(t) = H(t) [ a(1− r)Eπ→µ→e(t) ︸ ︷︷ ︸ LE main signal + bEµ→e(t) ︸ ︷︷ ︸ LE πDIF ] + cEµ→e(t) ︸ ︷︷ ︸ LE old-muon from Tg . (5.11) 5.3.2 High-energy time spectrum The fitting function of the HE region contains the signal of the π+ → e+νe decay and the background terms described in section 5.2.3. The mathe- matical formulation of the HE fitting function is shown in Eq. 5.12. The fitting parameters are the raw branching ratio (Rraw π ), the fraction of boosted π+ → µ+νµ → e+νeνµ LE events to the HE region through the described mechanisms (a× r), the amplitude of the pion decay-in-flight events in the HE region (b′), the amplitude of the old-muon background (c′), the ampli- tude of radiative pion (d), the amplitude of the old-muon with no hit in T1 (e), the amplitude of T1 double pulse resolution. The HE fitting function is, ϕHE(t) = H(t) [ a {( Rraw π + CµDIF ) Eπ→e(t) ︸ ︷︷ ︸ HE main signal + dG(t) ︸ ︷︷ ︸ Radiative Pion + rEπ→µ→e(t) ︸ ︷︷ ︸ LE boosted signal } + b′Eµ→e(t) ︸ ︷︷ ︸ HE πDIF ] + c′Eµ→e(t) ︸ ︷︷ ︸ HE old-muon + eF1(t) ︸ ︷︷ ︸ Old-muon no hit in T1 + fF2(t) ︸ ︷︷ ︸ T1 resolution . (5.12) Parameters b′ and c′ were scaled to r from their respective components in the LE region to account for those events boosted to the HE region, i.e. b′ = rb. The term CµDIF is the correction for µDIF events in the target that 107 5.3. The Fitting Function undergoes with the pion lifetime. This is not a fit parameter, as it is not part of the Timefit but its effect on the branching ratio is quantified using Monte Carlo calculations. The procedure to calculate such a correction will be described in Section 6.6.3. 5.3.3 Fit parameters In the fit, there are parameters that are known to high precision so they are fixed for a better fit performance. These parameters are the pion lifetime (τπ), the muon lifetime (τµ), the pion stopping time (t0), T1 double pulse amplitude (f) and T1 time resolution (∆). An important test on Rπ is allowing these parameters to be free within reasonable intervals to verify the result is consistent within statistical errors, this will be discussed in Chapter 7. The results from the Timefit of the three data blocks in Run III are shown in Table 5.1. Left column corresponds to fit parameters and the three consequent columns correspond to results from the three blocks. Shown errors are statistical coming from MINUIT fit. The number of events show the difference in statistics between data blocks that can be inferred from table 4.2. Such a difference is the same for the LE (NLE) and HE (NHE) regions. The amplitude of the π+ → µ+νµ → e+νeνµ signal (a) is consistent with the data block size. The number of πDIF events does not follow the statistics ratio between blocks. This is a compensation performed by the fit to provide the smallest χ2 for all blocks and does not affect the branching ratio stability. Positrons coming from old-muon decay within the target events in LE (c) have the same amplitude between blocks, showing the consistency of the downstream trigger integration window although the trigger clock was not calibrated for block 1. The amplitude of old-muon with no hit in T1 follows the same conclusion given the ratio between parameters (e), this is possible since the T1 resolution related pileup (f) has been fixed for the three data blocks. The fraction of LE events boosted to HE region (r) is consistent within errors for data blocks 1 and 2 but slightly smaller for block 3. This is because r depends on different pileup mechanisms that in turn depend on the acceptance cut R at the front face of the calorimeter. The acceptance cut is slightly less restrictive for positron tracks in blocks 1 and 2 due to an extra 2.5 mm displacement between S3 and WC3, both components of the tracker that reconstruct positron tracks. This difference is corrected via Monte Carlo simulation and it will be discussed in Section 6.5.3. The degrees of freedom (DOF) are given by the number of independent 108 5.3. The Fitting Function data points minus the numbers of free parameters in the fit. The number of data points is given by the number of bins in the time spectrum. For the three data blocks in Run III dataset DOF = 1527. The total χ2 over the degrees of freedom in the fit (χ2/DOF) tends to be larger in t < 0 for both LE and HE regions. This is because such regions quantify the amount of pileup in the time analysis and are influenced by the low statistics. The distortions are negligible for positive times, and do not affect the branching ratio more than 10−8 units. The bottom row in Table 5.1 shows the χ2 probability P (χ2 ≥ X), i.e. the probability of obtaining a test statistic in the χ2 distribution at least as extreme as the observed value. 5.3.4 Fit residuals The fitted functions ϕLE(t) and ϕHE(t) on the LE and HE time spectra are shown in figures 5.7, 5.9 and 5.11 for data blocks 1, 2 and 3 respectively. The residuals per bin (data - fit function) for HE t < 0, HE t > 0, LE t > 0 and LE t < 0 are shown clockwise in figures 5.8, 5.10 and 5.12 for data blocks 1, 2 and 3 respectively. In order to improve the fit performance the range from 150 to 180 ns in LE t > 0 was blanked off. This region is not related with an unknown pileup component but introduced a small distortion in the overall fit. In the LE region the two background components; Old-muon from target and πDIF are represented by the pink dashed line and blue dashed line, respectively. The green solid line and red solid line are sum of backgrounds and the π+ → µ+νµ → e+νeνµ signal, respectively. In the HE region the red solid line, blue solid line and the green solid line represents the π+ → e+νe signal, the boosted π+ → µ+νµ → e+νeνµ events to HE and the sum of backgrounds. Here the backgrounds are T1 double pulse resolution pileup in dashed blue line, pion radiative decay π+ → µ+νµγ in dashed gray line, πDIF events boosted to HE in dashed light blue line, and old-muon decay with no hit in T1 in dashed light green line. 109 5 .3 . T h e F ittin g F u n ctio n Data Block Parameter Description Block 1 Block 2 Block 3 NLE [108] Number of events in Low-energy region 13.596 6.3134 5.1554 NHE [106] Number of events in High-energy region 9.8915 4.6783 3.8363 a [109] Amplitude of π+ → µ+νµ → e+νeνµ signal 6.3716±0.0013 2.9214±0.0009 2.3852±0.0008 b [108] πDIF events 1.1047±0.0085 0.7674±0.0059 0.7395±0.0055 c [107] Positron from old-muon decay from Tg in LE 3.7347±0.0069 1.7920±0.0048 1.3631±0.0042 r [10−4] Fraction of LE events boosted to HE region 1.0292±0.0091 1.0416±0.0142 0.922±0.0165 Rraw π [10−4] Raw estimate of branching ratio Rπ before corrections 1.1898±0.0015 1.1893±0.0023 1.1894±0.0025 e [104] Old-muon events with no hit in T1 10.343±0.1350 5.5765±0.0997 5.2142±0.0964 d [109] Pion radiative decay π+ → µ+νµγ 3.860 (fixed) — — f [10−4] T1 double pulse resolution pileup: Case A + Case B 1.230 (fixed) — — ∆ [ns] T1 time resolution 15.69 (fixed) — — t0 [ns] Pion stopping time 2.15 (fixed) — — CµDIF [10−7] Muon decay-in-flight correction 1.4525 (fixed) — — τπ [ns] Pion lifetime 26.033 (fixed) — — τµ [ns] Muon lifetime 2197.03 (fixed) — — χ2/DOF 1.112 1.053 1.079 HE, t < 0 1.144 0.933 1.308 HE, t > 0 1.039 1.027 1.068 LE, t < 0 1.444 1.036 1.025 LE, t > 0 1.056 1.187 1.050 P (χ2 ≥ X) 0.0014 0.0733 0.0162 Table 5.1: Fit parameters from the Timefit of the three data blocks of Run III. Results are presented for integrated charge. The errors are statistical, as obtained by MINUIT [149]. The acceptance radius was R < 40 mm, and the cut-off energy Ecut =52 MeV. The fitting range for both low- and high-energy regions is from -270 to 520 ns excluding the prompt region from -20 to 5 ns. The DOF are the same for the three data blocks (1527). 110 5 .3 . T h e F ittin g F u n ctio n Figure 5.7: Timefit of charge-based time spectrum from data block 1 of Run III 4.2. The left panel shows the low- energy time spectrum, dominated by the π+ → µ+νµ → e+νeνµ signal and two background components; old-muon decays inside the target, and pion decays-in-flight. The right panel shows the high-energy time spectrum, fitted with six components: the π+ → e+νe signal, π+ → µ+νµ → e+νeνµ and pion decay-in-flight events promoted from the low-energy region via time independent mechanisms, two mechanisms of old muon pileup, π+ → µ+νµγ decays, and old muon decays. 111 5 .3 . T h e F ittin g F u n ctio n 250− 200− 150− 100− 50− Time[ns] 30− 20− 10− 0 10 20 30 F u n c -N D a ta N /n=1.142χ residuals 100 200 300 400 500 Time[ns] 150− 100− 50− 0 50 100 150 F u n c -N D a ta N /n=1.042χ residuals 250− 200− 150− 100− 50− Time[ns] 3000− 2000− 1000− 0 1000 2000 3000 F u n c -N D a ta N /n=1.442χ residuals 100 200 300 400 500 Time[ns] 30000− 20000− 10000− 0 10000 20000 30000 F u n c -N D a ta N /n=1.062χ residuals Figure 5.8: The residuals from the Timefit performed on charge-based data block 1 4.2 of Run III. Clockwise from top left, the panels show 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 < 0 spectrum. In this fit DOF = 1527 and P (χ2 ≥ X) = 0.0014, see Table 5.1. 112 5 .3 . T h e F ittin g F u n ctio n Figure 5.9: Timefit of charge-based time spectrum from data block 2 of Run III 4.2. The left panel shows the low- energy time spectrum, dominated by the π+ → µ+νµ → e+νeνµ signal and two background components; old-muon decays inside the target, and pion decays-in-flight. The right panel shows the high-energy time spectrum, fitted with six components: the π+ → e+νe signal, π+ → µ+νµ → e+νeνµ and pion decay-in-flight events promoted from the low-energy region via time independent mechanisms, two mechanisms of old muon pileup, π+ → µ+νµγ decays, and old muon decays. 113 5 .3 . T h e F ittin g F u n ctio n 250− 200− 150− 100− 50− Time[ns] 20− 15− 10− 5− 0 5 10 15 20 F u n c -N D a ta N /n=0.932χ residuals 100 200 300 400 500 Time[ns] 100− 50− 0 50 100 F u n c -N D a ta N /n=1.032χ residuals 250− 200− 150− 100− 50− Time[ns] 2000− 1500− 1000− 500− 0 500 1000 1500 2000 F u n c -N D a ta N /n=1.042χ residuals 100 200 300 400 500 Time[ns] 20000− 15000− 10000− 5000− 0 5000 10000 15000 20000 F u n c -N D a ta N /n=1.192χ residuals Figure 5.10: The residuals from the Timefit performed on charge-based data block 2 4.2 of Run III. Clockwise from top left, the panels show 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 < 0 spectrum. In this fit DOF = 1527 and P (χ2 ≥ X) = 0.0733, see Table 5.1. 114 5 .3 . T h e F ittin g F u n ctio n Figure 5.11: Timefit of charge-based time spectrum from data block 3 of Run III 4.2. The left panel shows the low-energy time spectrum, dominated by the π+ → µ+νµ → e+νeνµ signal and two background components; old- muon decays inside the target, and pion decays-in-flight. The right panel shows the high-energy time spectrum, fitted with six components: the π+ → e+νe signal, π + → µ+νµ → e+νeνµ and pion decay-in-flight events promoted from the low-energy region via time independent mechanisms, two mechanisms of old muon pileup, π+ → µ+νµγ decays, and old muon decays. 115 5 .3 . T h e F ittin g F u n ctio n 250− 200− 150− 100− 50− Time[ns] 20− 15− 10− 5− 0 5 10 15 20 F u n c -N D a ta N /n=1.312χ residuals 100 200 300 400 500 Time[ns] 100− 50− 0 50 100 F u n c -N D a ta N /n=1.072χ residuals 250− 200− 150− 100− 50− Time[ns] 1500− 1000− 500− 0 500 1000 1500 F u n c -N D a ta N /n=1.032χ residuals 100 200 300 400 500 Time[ns] 20000− 15000− 10000− 5000− 0 5000 10000 15000 20000 F u n c -N D a ta N /n=1.052χ residuals Figure 5.12: The residuals from the Timefit performed on charge-based data block 3 4.2 of Run III. Clockwise from top left, the panels show 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 < 0 spectrum. In this fit DOF = 1527 and P (χ2 ≥ X) = 0.0162, see Table 5.1. 116 5.4. Summary of Timefit 5.4 Summary of Timefit The raw branching ratio Rraw π was calculated from the simultaneous fitting in HE and LE time spectra with statistical errors of 15×10−8, 23×10−8 and 25×10−8 for data blocks 1, 2 and 3 respectively. Results are shown in Ta- ble 5.1, fit parameters showed consistency on the data size ratio between blocks. Different event selection criteria were developed and used for the three blocks of Run III to compensate possible effects due to non-calibrated downstream clock in data blocks 1 and an extra displacement of 2.5 mm be- tween sub-assemblies PIENU-I and PIENU-II in blocks 1 and 2, see Section 4.3. While the first has a negligible effect on Rraw π (< 10−8), the second has a direct effect on the low-energy tail of π+ → e+νe energy spectrum. The Monte Carlo adaptation to data blocks conditions will be a main subject in Chapter 6. This is important to reproduce key data features, including energy spectrum, and calculating first (low-energy tail) and second order corrections to Rraw π (acceptance ratio and CµDIF). 117 Chapter 6 Monte Carlo Corrections 6.1 Introduction The raw measurement of Rπ needs corrections that cannot be calculated using data and rely on a Monte Carlo (MC) simulation. All branching ratio measurements using the technique in which a pion stops within a target, including PIENU, share the same prominent correction. This is the tail of π+ → e+νe events measured in the calorimeter (NaI cystal and CsI array) that are below the energy cut. The size of this tail will be shown to be approximately three percent, small enough to be included in the Time fit and it is in turn the largest source of systematic uncertainty. Let λ(E) be the energy spectrum of π+ → e+νe measured in the calorimeter. The tail fraction T , defined as T = ∫ Ecut 0 λ(E)dE ∫∞ 0 λ(E)dE , (6.1) is highly influenced by the response of the calorimeter as a function of the incident angle. Therefore, the MC has to be adapted to reproduce the response function of the calorimeter. This subject will play an essential role in this chapter. Among other important features of the MC, the pion beam simulation will be described to prepare the necessary tools that require the two second-order corrections; the acceptance ratio between decays π+ → µ+νµ → e+νeνµ and π+ → e+νe; and the muon decay-in-flight correction (µDIF). These elements will allow us to report a corrected measurement of Rπ. 6.2 Energy loss mechanisms When the pion decays within the target, the outgoing positron impinges on the calorimeter at different angles sweeping a maximum acceptance of R = 90 mm that correspond to an incidence angle of 66◦ with respect to the 118 6.2. Energy loss mechanisms beam axis. The tail of π+ → e+νe events is produced by several energy loss mechanisms described in the following sections. Electromagnetic showers When a decay positron enters the calorimeter, it begins loosing energy via Bremsstrahlung radiation and ionization. The Bremsstrahlung photons in turn undergo electron-positron pair production, that themselves undergo Bremsstrahlung, ionization, and annihilation in the case of positrons. This process, known as electromagnetic shower, continues until all the positrons offspring of the initial one run out of energy and annihilate. When this happens, the absorbed energy by the crystal is converted into scintillation light, that is read by the PMTs at the rear end of NaI(Tℓ) and CsI(Tℓ). Because of the finite size of the detector, some of the scintillation light shower will escape. Occasionally, the amount of not measured light is enough to move the positron energy below the energy cut in the tail. The probability that this happens depend on the amount of material in the positron path. Bhabha scattering The beam pion decays in the target approximately 80 mm upstream the front face of Bina. If the decay positron undergos Bhabha scattering within the target, the low energy electron could make the downstream trigger and enter the calorimeter. This would produce an event with low energy and with π+ → e+νe time. When this happens in the π+ → µ+νµ → e+νeνµ decay, the energy falls in the low energy range. Therefore, it is included in the time fit. The Bhabha scattering undergoes through two channels; the t and s channels. In the former, the positron and the electron scatter by exchanging a virtual photon. The s-channel happens when the positron and the electron annihilates and produces a virtual photon which subsequently decays in a positron-electron pair. In both cases the probability of scattering depends on the scattering angle θs = cos−1Ω·Ω′ [31], where Ω and Ω′ are the normal momenta vectors of the positron and electron, and on the positron energy. The t-channel favors small scattering angles relative to the initial direction of the positron, while the s-channel allows higher scattering angles (up to π radians). A portion of low-energy tail events is caused by Bhabha scattering. 119 6.3. Response Function Measurement Photonuclear peaks When the incident positron produces an electromagnetic shower contain- ing several high-energy photons, they can interact with nuclei and induce neutron emission. This process typically occurs in nuclear reactions when photons with energies close to the giant dipole resonance (GDR) are ab- sorbed by the nucleus. The photon energy excites the nucleus, pushing it to a higher energy. The nucleus may then undergo a reaction in which it emits one or two neutrons to reach a stable state. Since neutrons are charge-less, they escape from the calorimeter leading to a specific peak in the spectrum. If the neutron energy is enough to leave the peak below the energy cut the photonuclear peak will contribute significantly to the tail. As we will see, the two photonuclear peaks of NaI(Tℓ) are located at 50 and 58 MeV, respectively [9] for a 70 MeV incident positron. 6.3 Response Function Measurement The net effect of the energy loss mechanisms in a calorimeter is referred to as the response function. Said mechanisms in turn depend on the geometry of the calorimeter itself and the amount of material in the positron path. To measure the response function, it is important to have control on the positron incidence angle, that in turn determines the amount of material in the positron path. For this, a 70 MeV/c positron beam is used to impinge on the PIENU calorimeter at ten different angles with respect to the beam axis. For each angle the tail fraction is used as a means to quantify the response function of the detector. 6.3.1 Experimental setup The magnetic bends in the M13 secondary beam-line were tuned to produce a 70±1 MeV/c positron beam. The beam composition is approximately 63% pions, 11% muons and 26% positrons. To reduce energy loss due to interactions in the counters upstream the calorimeter and allow the direct exposure of the calorimeter to the positron beam,the PIENU-I sub-assembly, comprising counters B1, B2, S1, S2, B3, S3 and T1, was removed. Wire chamber WC1 and WC2 were left in place to measure the beam profile. The sub-assembly PIENU-II, made of WC3, T2, Bina and the CsI array, was detached from the fixed rail in which it sits and is rotated by an angle θ with respect to the beam axis. Figure 6.1 shows an schematic of the experimental setup that will be referred as the “Lineshape setup” from now 120 6.3. Response Function Measurement Figure 6.1: Schematic drawing of the Lineshape setup for the response func- tion measurement using a positron beam, showing the angle of rotation θ between the beam and calorimeter. on. The amount of material in the positron path also depends on the centre of rotation, which is located at the same distance between the front face of Bina and the centre of the target (B3). This was done to mimic the decay positron track angle from the π+ → e+νe decay. To achieve this, a shaft was attached to the cart on which PIENU-II sat, at the mentioned distance in along the beam axis (z). The angle θ was measured using a theodolite by placing targets on the beampipe and on the PIENU-II cart. The precise angle was determined by measuring the relative position between the theodolite and the targets with high accuracy resulting in an angle uncertainty better than 0.1◦. 121 6.3. Response Function Measurement Angle (θ) Events (106) Tail (%) Systematic (%) 0.0 5.97 0.54 0.02 6.0 11.65 0.55 0.02 11.8 6.51 0.58 0.02 16.5 7.25 0.63 0.02 20.9 6.53 0.71 0.02 24.4 6.28 0.80 0.03 30.8 5.78 1.14 0.04 36.2 5.86 1.45 0.04 41.6 6.43 2.12 0.06 47.7 9.06 3.61 0.10 Table 6.1: The number of events for each data-taking angle θ followed by the tail fractions for data and systematic uncertainties. The statistical un- certainties were negligible. The downstream trigger of the PIENU setup was modified to require at least one hit in T2 since this was the only scintillator counter in the setup. Additionally, to suppress the background from beam pions, approximately half of the 43 ns radio-frequency (RF) from the cyclotron was vetoed in the trigger configuration. Table 6.1 shows the ten θ angles at which data were taken and the number of events captured in each data block. Given that the T2-hit trigger requirement is not rigorous for the exclusive selection of positrons, data have to be selected using specific cuts. 6.3.2 Event selection For a further suppression of background and beam particles three cut stages were applied to data. The first stage requires that at least one wire plane gets activated in wire chambers WC1 and WC2. Also, a timing cut was applied in these wire chambers to reject events outside a 100 ns window around the trigger time. To remove particles from the beam halo, acceptance cuts were applied in WC1 and WC2 similar to those in the event selection of the nominal analysis of section 4.4.2. The second selection stage removes a portion of beam muons. A cut on T2 energy can reject a significant potion of beam muons, but it also removes positrons with energy deposit in T2 due to the Landau distribution, and positrons from muon decay going backwards out of Bina. To mitigate these effects, a two-dimensional cut in T2 versus calorimeter energy was employed. The cut removed events with energies 122 6.3. Response Function Measurement 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 10 2 10 3 10 4 10 5 10 C o u n ts /b in Figure 6.2: Energy spectrum in PIENU calorimeter for the 70 MeV positron beam with a crystal-beam angle θ = 0◦. Acceptance and timing cuts were applied in WC1/2 and a two-dimensional cut was applied in calorimeter energy vs T2 energy for beam halo particles and beam muon suppression, respectively. in T2 higher than 2 MeV and calorimeter energy less than 35 MeV. These boundaries were set to remove most beam muons without compromising the tail fraction [10]. The resulting energy spectrum for θ=0◦ is shown in Figure 6.2. From left to right the features are: a small peak from beam pions around 13 MeV, a beam muon peak at 17 MeV, a slight bump at about 30 MeV caused by pion decays in flight, photonuclear peaks at 50 and 58 MeV, the positron beam at 70 MeV, and a high energy tail above 70 MeV due to pileup from beam muons decaying in Bina. The third selection stage removes the remaining beam muon in the positron energy spectrum. For this beam muons were selected by time of flight (TOF). Figure 6.3 depicts the correlation between the calorime- ter energy and TOF, the energy spots are consistent with features in Figure 6.2. By selecting bands of 4-11 ns and 12-15 ns in TOF, the positron and muon energy spectra are obtained, these spectra are depicted in figures 6.4 123 6.3. Response Function Measurement 0 10 20 30 40 50 60 70 80 Calorimeter energy [MeV] 0 2 4 6 8 10 12 14 16 18 20 R F T O F [ n s ] 10 2 10 3 10 4 10 Figure 6.3: The calorimeter energy versus time of flight (TOF). The upper band (12-15 ns) corresponds to muons and the lower band (4-11 ns) corre- sponds to positrons. and 6.5, respectively. The muon energy spectrum has virtually no positron events remaining, while the positron spectrum has a remaining portion of beam muons. By normalizing the muon peak in the muon spectrum to the remaining muon peak at 17 MeV in the positron spectrum, the muon contribution was subtracted. The resulting energy spectrum is shown in Figure 6.6. There is a remaining small portion of muons remaining in the 17 MeV peak and a small number of pion decaying-in-flight events at 30 MeV, confirmed by MC simulation. Both had negligible contributions to the tail fraction (<0.01%). The results of tail fraction for the ten angles θ are shown in table 6.1. The systematic uncertainty due to the muon subtrac- tion is evaluated by varying the TOF cut applied to the positron spectrum. For example, the cut at θ = 0◦ is varied from the default at 4-11 ns to 4-16 ns in 1 ns steps, where essentially all the muons have been allowed into the spectrum. By averaging the resulting change in the tail fraction, the systematic uncertainty was assessed. 124 6.3. Response Function Measurement 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n ts /b in Figure 6.4: Positron energy spectrum in the calorimeter identified by its time of flight (4 - 11 ns). 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 C o u n ts /b in Figure 6.5: Muon energy spectrum in the calorimeter identified by its time of flight (4 - 11 ns). 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n ts /b in Figure 6.6: Positron energy spectrum in the calorimeter after the subtraction of the remaining muon component, see text. 125 6.4. Lineshape simulation 6.3.3 Low-momentum beam positrons Since positrons have the potential to scatter in the beamline, a small por- tion of positrons could lose energy along and after the rear end of the beam line and fall below the energy cut. This means that the tail distribution could be contaminated by intrinsic low-momentum positrons coming from the M13 beamline. This possible low-energy component could not be di- rectly identified in data. To quantify the effect of low-momentum positrons in tail fraction, the transport of positrons through the M13 channel was simulated using G4beamline [151]. This resulted in an upper limit to the low-momentum tail contribution of (2.8±0.5)×10−4 [5], thus negligible. 6.4 Lineshape simulation The baseline of the MC simulation for the Lineshape setup consists of the modeling of the PIENU calorimeter geometry and the positron beam. The simulation was developed using the simulation toolkit Geant4 version 10.06 [150]. The geometrical dimensions and material specifications of the calorimeter were taken from the BICRON manual [9]. The positron beam is generated by following the beam profile distributions measured in WC1 and WC2. This process will be described in detail in Section 6.5.1. Due to changes in the beam conditions, two beam profiles corresponding to θ = 0◦ and θ = 36.2◦ were used for the first 7 angles and the last 3 angles respec- tively. The shower leakage due to electromagnetic cascade is very important for large angles. Therefore, matching the angular distribution along the X axis in the beam profile to data is specially important, see section 6.5.1. The beam momentum distribution was modeled as a Gaussian PDF with a central value of 70 MeV/c and a sigma of 0.64 MeV/c. As proof of the proper beam tuning, Figure 6.7 illustrates the good agreement between data in Monte Carlo in the activated wire distribution on the first plane of WC3 for θ = 47.7◦. 6.4.1 Physics List To reproduce the tail fraction in data, the positron energy spectra should ideally reproduce every feature, as depicted in Fig. 6.6 for θ = 0◦. The electromagnetic interactions are by far the most important, hence a suitable list of EM physics comprising the energy range of the positron beam and the calorimeter nuclear structure was needed. Among several options, Geant4’s 126 6.4. Lineshape simulation 0 5 10 15 20 25 30 35 40 45 WC3_1 wire 210 3 10 410 5 10 6 10 C o u n ts Figure 6.7: Distribution of activated wires in the first plane of WC3 for a positron beam wit θ = 47.7◦. Data are shown in black and MC in red. PENELOPE (Penetration and Energy Loss of Positrons and Electrons) code system [152] was chosen. The relevant characteristics of PENELOPE for the Lineshape analysis are: 1. It generates electron-photon showers produced by primary positrons, electrons or photons. 2. Contains improved positron interaction models and more elaborate tracking algorithms for simulation of intermediate-energy positron trans- port involving complex geometries. 3. Energy range of operation from 100 eV up to 1 GeV. 4. Simulations verified with high accuracy for experimental data involv- ing detectors with atomic nuclear number Z = 1− 92 [153]. On the other hand, the hadronic physics list was chosen to obtain a good agreement in the photonuclear peaks at 58 and 50 MeV corresponding to the emission and escape of one and two neutrons with an approximated energy of 8 MeV, respectively. The photonuclear cross-section giving rise to these 127 6.4. Lineshape simulation peaks is implemented in Geant4 for relevant nuclei target including iodine (prominent in the NaI(Tℓ) crystal). This experimental data is coded in the list of hadronic interactions [154]. The neutron emission yield depends on the neutron escape probability, that in turn depends on the target nucleus de-excitation model, as well as on the interaction cross-section of the emitted neutron with the crystal nuclei. Given that G4NDL library for high precision neutron physics [155] did not give a good agreement, several libraries were tested. The optimal choice was found to be FTFP BERT PEN and the photonuclear cross-sections was scaled by a factor 1.5 from the encoded values in Geant4. This allows to reproduce the amplitude of the second photonuclear peak; the peak below the energy cut and the relevant peak for the tail fraction. 6.4.2 Calorimeter tuning Two key features in the energy spectrum for the tail fraction are the shape and position of the positron peak. Besides the beam momentum, these parameters involve several properties of the calorimeter. On top of the baseline of the simulation and the modifications discussed, the calorimeter geometry was tuned to achieve agreement between data and MC. Several models were tested as fitting functions to extract the peak posi- tion including a restricted Gauss PDF and Crystal ball function. It was con- cluded that modeling the positron peak as a Gaussian function convoluted with a falling exponential to mimic the energy loss due to electromagnetic shower gives the best performance. The mathematic expression of the model is F(E) = N ∫ 0 −∞ exp (t/ϵ)G(E − t, µ, σ)dt, (6.2) where N is the amplitude of the peak, ϵ is the decay rate of the expo- nential and G is the shifted Gauss PDF centered at µ with σ width. Figure 6.8 shows the fit applied on the positron energy spectrum for θ = 0◦. This formulation allows to align the uncalibrated energy spectrum from data to the MC at each angle. An important tuning came from an expected amount of material tra- versed by positrons entering the calorimeter. It was concluded that there was less material in the Al2O3 layer at the front face of Bina than given in the manufacturer’s specifications. By scaling its nominal density of 3.99 128 6.4. Lineshape simulation Calorimeter energy [MeV] 62 63 64 65 66 67 68 69 70 C o u n ts /b in 0 50 100 150 200 250 3 10× /ndf = 12.33/122χ = 67.36 MeVmaxE Figure 6.8: Fit for the positron peak of θ = 0◦ data. The fitting function consisted of a Gaussian convolved with a falling exponential (see text). Data are shown in black, the red solid line is the fit, the dashed blue line represents the falling exponential, and the dashed red line represents the Gaussian before convolution. g/cm3 to 50% of this value, a ∼10% correction for the amount of energy loss for positrons incident on the calorimeter was obtained. At large θ angles shower leakages become important and the contribution from the energy deposited in the CsI to the total energy deposited is large. The simulated tail fraction resulted to be slightly smaller than data’s for the largest three angles. A plausible cause of the mismatch between MC and data is missing material between Bina and CsI in the form of irregular or unspecified shims and spacers. By introducing an attenuation factor or M = 0.92 to the CsI contribution in the MC, a good agreement between MC and data at all angles in the energy spectra was achieved. 6.4.3 Positron beam tail fraction Figures 6.9 and 6.18 show the good agreement between MC and data af- ter implementing the modifications discussed in this section for θ = 0◦ and θ = 47.7◦, respectively. For θ = 0◦, the little disagreement at the first pho- 129 6.5. PIENU simulation tonuclear peak (58 MeV) is due to the scaling factor introduced to achieve a good match at the second photonuclear peak (50 MeV). There is a remain- ing portion of beam muons at 18 MeV that is negligible for the tail fraction (< 0.01%). Given the lack of the upstream counters in the Lineshape setup, the energy cut was set at 53.7 MeV to account for the non-deposited energy in the PIENU-I assembly’s detectors. The precision of the energy calibra- tion described above is bounded by 0.1 MeV, which was used as systematic uncertainty. A second source of uncertainty came from the photonuclear scaling. For this a 0.1 variation to the default value 1.5 was considered to produce an agreement envelope in the first photonuclear peak. This was the dominant systematic uncertainty for the MC tail fraction when added in quadrature with the two othe systematic sources corresponding to the attenuation factor M and the Al2O3 density. This process was repeated for all angles. Figure 6.19 shows the 1σ MC-data agreement in the positron tail fraction for the ten crystal-beam angles. The level of agreement obtained supports the use of MC and allows to expand the simulation to the PIENU setup for the simulation of the π+ → e+νe and π+ → µ+νµ → e+νeνµ decays. 6.5 PIENU simulation For the simulation of the π+ → e+νe decay in the PIENU setup, the upstream detectors of the PIENU-I assembly were included as shown in schematic 3.6. Their geometrical dimensions were adjusted according to manufacturer’s specifications. The positron beam was substituted by a pion beam that is required to stop within the target an decay in either π+ → e+νe or π+ → µ+νµ → e+νeνµ channel. They are simulated separately, which allows to calculate the Rπ corrections described in the following sections. 6.5.1 Beam Profile The beam profile is important for the pion tracking and to reproduce the decay vertices in the target B3. The pion stopping position distribution is a key feature to estimate the acceptance ratio between π+ → e+νe and π+ → µ+νµ → e+νeνµ channel. This requires that the beam profile features in MC are representative of data. The pion stopping position (Zv) from Eq. 3.5 is shown in Figure 6.20 for data block 2. Although the shape of the distribution is not entirely symmetric, it can be modeled as a Gaussian PDF in the central region to 130 6.5. PIENU simulation 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n t / 0 .1 M e V Figure 6.9: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 0.0◦. Data are shown in black and MC in red. 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 6 10 C o u n t / 0 .1 M e V Figure 6.10: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 6.0◦. Data are shown in black and MC in red. 131 6.5. PIENU simulation 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n t / 0 .1 M e V Figure 6.11: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 11.8◦. Data are shown in black and MC in red. 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n t / 0 .1 M e V Figure 6.12: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 16.5◦. Data are shown in black and MC in red. 132 6.5. PIENU simulation 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n t / 0 .1 M e V Figure 6.13: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 20.9◦. Data are shown in black and MC in red. 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n t / 0 .1 M e V Figure 6.14: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 24.4◦. Data are shown in black and MC in red. 133 6.5. PIENU simulation 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n t / 0 .1 M e V Figure 6.15: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 30.8◦. Data are shown in black and MC in red. 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n t / 0 .1 M e V Figure 6.16: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 36.2◦. Data are shown in black and MC in red. 134 6.5. PIENU simulation 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n t / 0 .1 M e V Figure 6.17: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 41.6◦. Data are shown in black and MC in red. 0 10 20 30 40 50 60 70 80 90 100 Calorimeter energy [MeV] 1 10 210 3 10 410 5 10 C o u n t / 0 .1 M e V Figure 6.18: Energy spectrum of positron beam in the calorimeter for a crystal-beam angle rotation of θ = 47.7◦. Data are shown in black and MC in red. 135 6.5. PIENU simulation 0 10 20 30 40 50 Angle (degree) 0.5 1 1.5 2 2.5 3 3.5 T a il F ra c ti o n ( % ) Figure 6.19: Tail fraction of the 70 MeV/c positron beam in the PIENU calorimeter as a function of the crystal-beam angle. Statistical uncertain- ties are too small to be visible. Data are shown in blue and MC in red. Systematic uncertainties are displayed for data (muon subtraction) and MC (sum in quadrature, see text). MC points were virtually displaced by 0.7◦ to avoid overlap with data. estimate the mean value. As discussed in section 4.3, a criterion to split Run III dataset in blocks was Zv. Table 6.2 shows the mean values of Zv for the three data blocks. A beam profile was developed for each data block to account for the different Zv means and for the specific features described in section 4.3. The beam profile contains the momentum (P ), the position (x, y) and angular information (tx, ty) of the incoming pion. These latter are mea- sured in the XY plane (perpendicular to the beam axis) of wire chambers WC1 and WC2. In order to estimate the beam momentum, it was assumed that Zv can be used to make to match the beam momentum to data is an independent variable. However, the momentum of each track can not be measured directly, but presumably can be estimated from the visible pion energy. Therefore, an estimate of the beam momentum is obtained by scal- 136 6.5. PIENU simulation / ndf 2χ 199 / 147 Constant 4.162e+01± 1.122e+05 Mean 0.00140± 0.07931 Sigma 0.004± 2.135 5− 4− 3− 2− 1− 0 1 2 3 4 5 Zv [mm] 20 40 60 80 100 3 10× c o u n t/ b in Figure 6.20: Pion stopping position within the target fitted with a Gaussian from -1.5 to 1.5 mm. Data block Zv(data) [mm] Zv(MC) [mm] P0 [MeV/c] 1 1.13 1.134 75.694 2 1.04 1.05 75.58 3 0.114 0.11 74.29 Table 6.2: Mean value of the pion stopping position Zv of Run III divided in data blocks with their corresponding MC values after tuning the beam momentum. ing the sum of the pion energies in B1, B2, S1, S2 and B3 to a nominal value 75 MeV/c [129]. This scaling factor is arbitrary and only serves to obtain a momentum distribution that keeps the correlation with position and angular variables. A subset of data events encode the beam phase space to give the proper distributions for the mentioned variables and their correlations. A total of 250k events were sampled uniformly from each data block. This guarantees that the sample dataset is representative of the entire block. Figure 6.21 shows the beam profile variables sampled from a data block in Run III. The correlations of these variables are encoded in the covariance matrix V of the set of the variables {ηi} = {P , x, y, tx, ty} [156]. Since V is a positive defined matrix, it can be factorized into a lower triangular matrix L using 137 6.5. PIENU simulation Figure 6.21: Beam profile of a sample of 250k evenly distributed events from Run III. The profile consist of the beam momentum (upper left panel), posi- tion (panels top center and right) and angular information (bottom panels) measured in WC1 and WC2 in unitary units, according to Eqs. 3.2. the Cholesky decomposition [157] according to: Ljj = ( Vjj − j−1 ∑ k=1 L2 jk )1/2 Lij = 1 Ljj ( Vij − j−1 ∑ k=1 LikLjk ) , j = 1, ..., n; i = j + 1, ..., n, (6.3) where n is the total number of events. This factorization is specially useful to generate a set of random variables {xi} that are correlated with each other from a set of completely independent random variables {ηi} by xi = ηi + ∑ j Lijηj , where ηi is the mean value of the variable ηi. The variables ηi are randomly generated following their respective PDF from the beam profile shown in Figure 6.21 excluding the beam momentum. Although the scaled pion energy serves to obtain the correlation of mo- mentum to position and angle variables, it does not model the actual pion beam momentum as it can not reproduce Zv in Figure 6.20. Instead, the 138 6.5. PIENU simulation 5− 4− 3− 2− 1− 0 1 2 3 4 5 Zv [mm] 20 40 60 80 100 3 10× c o u n t/ b in Figure 6.22: Comparison of the pion stopping position within the target B3 after tuning the beam momentum in MC. Data from block 2 in Run III dataset in black and MC with its respective beam profile tuning in red. beam momentum was modeled as a Gaussian PDF with a mean value P0 as a free parameter and a nominal width of 0.64 MeV/c. The optimization of P0 allows to match the Zv peak. Figure 7.1.5 shows the comparison of the Zv distribution between MC and data after tuning P0 and table 6.2 re- ports the results of the agreement for the three data blocks. The agreement is remarkably good and validates the beam profile to be used in the event generator of MC. 6.5.2 Tracking clustering An important feature of the downstream assembly PIENU-II is the capa- bility to reconstruct the positron track after the pion decay. A good recon- struction of tracks allows to calculate the acceptance radius at WC3 (R) as defined in Eq. 3.3. This parameter is necessary to calculate the raw branch- ing ratio and its corrections and allows to achieve a 0.1 MeV precision in Bina calibration. An essential test to the analysis is verifying that Rπ is independent from the acceptance R cut. Additionally, Rπ should be insen- sitive to the number of identified tracks, which is Ntrk = 1 or Ntrk ≥ 1. Both tests will be discussed in detail in Section 7.2. According to equation 3.2, measuring the x and y positions allows the tracking system of tracker 3 (Trk3, see figure 3.14) to fit a positron trajec- 139 6.5. PIENU simulation tory and assess the fit parameters x0, y0, tx, ty, which are essential for the calculation of R and other important tracking quantities. Also, the num- ber of identified hits in the downstream detectors is related with physical processes like Bhabha-scattering or delta rays which can blur the positron trajectory. To avoid this the track with the χ2 closest to 1 is chosen and at least one hit is required in WC3 and S3. The hit identification and position measurement in these detectors is done by determining the number of ac- tivated clusters. A cluster is a set of adjoining activated wires or strips in WC3 or S3 respectively. The clustering is performed by the tracking system called TrReco. This process determines the number of hits in a tracking detector. WC3 clustering In the case of WC3 the clustering process is simpler since the 96 wires are connected by pairs on the same pad which reduces the number of channels to 48 with an effective read-out pitch of 4.8 mm. The clustering is done by simply grouping pairs of activated wires and assigning an intermediate position depending on the charge ratio between pairs. This is done for the three wire planes and by interpolating the three inner positions a XY coordinate is fitted. This process is fully described in Ref. [124] and was implemented in MC since the earliest PIENU analysis [1]. S3 clustering The clustering process in S3 is more complicated since the silicon detectors are made of two planes. The front and rear planes determine the X and Y position, respectively. Each plane consists of 189 silicon strips that are grouped by four to form 48 readout channels through an amplifier. The grouped strips are sequentially connected by capacitors as shown in the schematic of figure 3.10. This implies a charge sharing system between readout channels. The clustering process is coded in proot, its pipeline routine goes as follows: 1. Sorts activated amplifiers according to their channel number. 2. It identifies and groups together adjacent activated amplifiers. Each group of amplifiers is a cluster and they are separated by at least one empty channel between them. 140 6.5. PIENU simulation 3. For every cluster of a given event, the amplifiers with the highest and second highest charges are identified. 4. The position of the hit is calculated by assigning a weighted position between amplifiers. The assignment depends upon the ratio of the charge in the highest amplifier to the sum of charges of the two highest adjacent amplifiers. The implementation of this routine in MC is fully described in Ref. [131] and it is an important upgrade developed by the author in the simulation from the latest reported analysis [5]. Prior to this implementation, the four- strip arrangement per amplifier was simulated as a solid sensitive volume that fills the S3 plane. This approach was unrealistic as several strips can be activated by the same particle and the number of hits was overestimated as the sum of activated strips. This was particularly wrong for large positron angles as their probability to activate more than one strip is greater. Figures 6.23 and 6.24 show the match of data to MC in the number of clusters and position in the X plane of S3, Y plane has the same level of agreement. In particular, the position distribution reproduce the periodic quantized shape that reflects the amplifier arrangement in the silicon planes. The quality of these results is inherited by the number of tracks, which is shown in Figure 6.25 and was the main reason to implement the clustering in S3. 6.5.3 WC3 position The track radius at WC3 (R) is highly sensitive to WC3 position along the beam axis (zWC3). This position is well determined for Runs IV, V and VI, corresponding to years 2010 (Nov.-Dic.), 2011 and 2012, respectively. In order to determine the distance between PIENU-I and PIENU-II [108] and in turn the relative position of WC3 to B3, a survey was performed using the safety rods that supports the coupling between both sub-assemblies and prevents they collide. It was concluded PIENU-I was bolted to PIENU- II, fixing the WC3 position at the beginning of Run IV with a ±0.4 mm uncertainty. According to the MIDAS log, the TRIUMF cyclotron was turned off for maintenance between block 2 and block 3. During this time period, PIENU- I was removed to test the response time of WC2. Therefore, there was a possibility that the PIENU-I assembly was slightly displaced upstream. The magnitude of the displacement was estimated using two criteria by 141 6.5. PIENU simulation 0 1 2 3 4 5 6 7 8 S3_X clusters 1 10 210 3 10 410 5 10 6 10 710 8 10 C o u n t Figure 6.23: Number of sorted clusters in the X plane of S3. Data from block 3 in Run III dataset is shown in blue and MC in red. 30− 20− 10− 0 10 20 30 S3_X [mm] 410 5 10 6 10 C o u n t / b in Figure 6.24: Positron track position in the X plane of S3. Data from block 3 in Run III dataset is shown in blue and MC in red. The spike-like shape is caused by amplifier arrange in S3 silicon planes. assuming and coding a virtual displacement δ between B3 and WC3 in MC. The position of S3 is fixed relative to B3 since they are part of the same subassembly. Considering that the position measurements of S3 and WC3 in both cases is the same, the displacement has effect on the reconstructed 142 6.5. PIENU simulation Figure 6.25: Number of reconstructed positron tracks by S3-WC3 tracker. Data is shown in blue and MC outcome after the S3 clustering implementa- tion is shown in red. positron track as shown in the schematic of Figure 6.26. More specifically, the projections of position and angle in B3 is changed. The description of tuning δ in MC to reproduce data’s B3 projections is detailed in Refs. [137] and [138]. It was concluded that an upstream shift of 2.5 mm was consistent along the taking periods of block 1 and block 2. A second criterion to estimate δ is given by the energy calibration of Bina as a function of the positron angle. As described along this chapter, the R cut determines the response of Bina as a larger cut allows more positron tracks with different amount of material in their path. Having consistency between data and MC in the position of π+ → e+νe peak for differential cuts in R verifies the δ = 2.5 mm displacement between PIENU-I and PIENU-II. The process has the following sequence: 1. Apply slice cuts 0 < R < 10, 10 < R < 20, ... , 70 < R < 80 mm on the Bina energy spectra for data and MC. In case of MC, cuts are applied on the π+ → e+νe energy spectrum. 2. Fit the π+ → e+νe peak to the falling Gaussian model in Eq. 6.2 for both data and MC. 3. Get the difference MC - data in the peak parameter as a function of 143 6.6. Rπ corrections differential R. Figure 6.27 depicts the peak difference smaller than 0.1 MeV, which is the Bina calibration precision. The results block 3 are presented together with blocks 1 and 2 to highlight the stability of the R cut along the Run III dataset after implementing the changes described along this section. 6.6 Rπ corrections 6.6.1 Tail fraction The simulated π+ → e+νe energy spectrum is depicted in Figure 6.28. To have consistency between data and MC, CsI energy was not added. Energy cuts in B1 and B2, acceptance cuts in WC1/WC2 are required, the bounds are the same than those used for data. The acceptance radial cut was applied at R <40 mm, requiring at least one reconstructed track. The black solid line represents the energy cut that divides the LE and the HE region. The spike in the LE region is due to electromagnetic showers not captured by the CsI detector, and the small bumps at 58 and 50 MeV are the first and second photonuclear peaks, respectively. These are visible since cut on R is highly restrictive on the positron track. The tail correction CT is applied as a multiplicative factor to Rraw π as Rπ = ( 1 1− T ) Rraw π , (6.4) where T is the π+ → e+νe tail fraction as defined in Eq. 6.1. The resulting tail correction is 4.33% for Ecut = 52 MeV and R <40 mm. In contrast to previous analyses, in which the acceptance radial cut was R <60 mm [5] [2], R <40 mm was chosen to decrease the tail component caused by non- captured electromagnetic shower and the systematic uncertainty associated to it, other sources of uncertainty will be discussed in Chapter 7. To depict the effect of WC3 position along the beam axis on T , Figure 6.29 shows the tail fraction as a function of the radial acceptance cut for blocks 1, 2 and 3. For each data block, 60 M events were simulated, statistical errors are too small to be visible. The tail fraction of blocks 1 and 2 for is consistent along different choices of R cut. Given that WC3 is approximately 2.5 mm closer to B3 in block 3, its tail fraction has a sharper dependence on the acceptance cut. As will be discussed in Chapter 7, this effect is compensated 144 6.6. Rπ corrections Figure 6.26: Schematic of the virtual positron track produced by an extra displacement δ between PIENU-I and PIENU-II. Figure 6.27: Difference of the π+ → e+νe peak MC- data as a function of differential radial acceptance cuts at WC3, see text. 145 6.6. Rπ corrections 0 10 20 30 40 50 60 70 80 90 100 Bina energy [MeV] 1 10 210 3 10 410 5 10 c o u n t/ b in Figure 6.28: π+ → e+νe simulated energy spectrum using beam profile of data block 2 with acceptance radial cut R <40 mm, the black solid line represents the energy cut Ecut = 52 MeV. after combining Rraw π with MC corrections to achieve consistency between data blocks. 6.6.2 Acceptance correction Although the π+ → e+νe and π+ → µ+νµ → e+νeνµ are measured within the same time interval in the calorimeter, two physical effects could po- tentially change the ratio of channel events within the geometrical accep- tance. These are the extra spread in the starting position of the decay positron caused due to the 4.1 MeV muon before decaying and energy de- pendent interactions upstream of Bina. Figure 6.30 show the simulated three-dimension distance traveled by muon before decay, two versions are depicted here: EM0 and PENELOPE code systems for EM interactions in blue and red, respectively. The narrower distribution of PENELOPE de- picts its accuracy in the low-energy regime. The acceptance correction is a multiplicative factor to Rraw π defined as Cacc(RA) = λ(π+ → µ+νµ → e+νeνµ, RA) λ(π+ → e+νe, RA) , (6.5) 146 6.6. Rπ corrections 30 40 50 60 70 80 R cut [mm] 4 5 6 7 8 9 10 11 12 T a il F ra c ti o n ( % ) Figure 6.29: Tail fraction of the π+ → e+νe decay below the energy cut at 52 MeV as a function of the radial acceptance cut. Results from data blocks 1, 2 and 3 are shown in black, red and blue, respectively. Points from blocks 2 and 3 were virtually shifted for display purposes. where RA is the radial cut imposed on R and λ is the energy spectrum of the specified decay channel measured in Bina. On top of the two men- tioned physical effects, the acceptance correction is highly sensitive to the pion stopping position Zv, as the positrons tracks emerge from this vertex in the case of the π+ → e+νe decay and about ∼1 mm downstream further in the case of the π+ → µ+νµ → e+νeνµ decay. The match between MC and data in Zv that comes with the tuning of the beam profile is important for the calculation of Cacc(RA), as described in Section 6.5.1. Hence, each dataset requires its acceptance correction to be calculated separately. An- other important factor that have the potential to affect Cacc is the lack of a material upstream of Bina, as for Run V (year 2011) dataset that had an extra piece of material left near target B3 for a special set of runs for direct muon capture [100]. Figure 6.31 depicts the acceptance corrections as a function of RA for the data blocks 2 and 3 in Run III. Block 2 has a systematic downwards shift caused by the additional 2.5 mm displacement between B3 and WC3. The 147 6.6. Rπ corrections 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 xMuon [mm] 1 10 210 3 10 410 5 10 c o u n t/ b in Figure 6.30: Distance traveled by the muon from the π+ → µ+νµ → e+νeνµ decay within the target B3. MC simulation using EM0 and PENELOPE Geant4 physics lists for EM interactions are shown in red and blue, respec- tively. ratio defined in Eq. 6.5 requires high statistics for an accurate estimate. For each data block, 300 M events were simulated both for π+ → e+νe and π+ → µ+νµ → e+νeνµ decay channels, statistical errors are displayed. Geometrical features concerning the position of the downstream tracking detectors (S3 and WC3) are considered as sources of systematic uncertainty as long with other important parameters in Cacc calculation. This will be a main topic in the next chapter. 6.6.3 Muon decay-in-flight Muons emerging from π+ → µ+νµ vertex can decay in flight within the target (µDIF). Although this topology can be identified by the tracking system, as shown in Fig. 3.14, it has the same time distribution as the π+ → e+νe decay. The energy cut Ecut can not suppress entirely these events since the µDIF kinetic energy can boost the LE positrons to the HE region. Given that these events can not be separated entirely using time or energy cuts, a correction to Rπ is needed. Figure 6.32 shows the simulated decay time of muons in flight; the stopping time is approximately 19 ps. 148 6.6. Rπ corrections 30 40 50 60 70 80 [mm]A R 0.996 0.997 0.998 0.999 1 1.001 1.002 A c c Figure 6.31: Acceptance correction from the ratio of the π+ → µ+νµ → e+νeνµ decay to the π+ → e+νe decay as a function of the radial cut RA. Correction of blocks 2 and 3 from Run III dataset are shown in red and blue, respectively. Points from block 3 were virtually shifted for display purposes. Hence, the probability of a muon decay in flight can be estimated as p ∼ 1− exp ( −τµDIF /γ τµ ) = 8.3× 10−6, (6.6) where γ = 1/ √ 1− v2/c2 = 1.039 for the muon kinetic energy Tµ = 4.1 MeV, τµDIF = 19 ps, and τµ is the muon lifetime. The muon kinetic energy Tµ is deposited in the target before decaying, with was confirmed by the MC simulation. The additive correction (CµDIF ) is calculated as the fraction of events above Ecut of the µDIF energy spectrum multiplied by the probability of the muon decay in flight in Eq. 6.6. The previous unblinded analysis of PIENU [6] used CµDIF = 2.406×10−7, that was the result of two consecutive internal studies [134] [135]. However, the PIENU MC simulation received several upgrades since then, and some of these upgrades could potentially change the estimate of CµDIF . By using a reverse engineering method, the relevant changes in MC for µDIF were identified. The complete description of this process and the estimate of the effect on CµDIF is documented in 149 6.6. Rπ corrections Figure 6.32: The decay time of muons in the target with non-zero kinetic energy at the time of the decay. Ref. [132]. For a clearer comparison MC simulation of Run V (2011 year) was used along the process, as the features of this dataset were considered for the former calculation of CµDIF . The MC modifications are summarized as below. • S3 clustering: Implementation of silicon strip clustering in S3. • Muon radiative decay: Update of the muon radiative decay branch- ing ratio from 0.14 to 0.09. • Birks constant: Joint modification that includes update in Bina energy resolution from 0.05 MeV to 0.035 MeV and the implementation of the Birks constant 0.024 mm/MeV. The prominent modification was the Birks constant giving a change of order ∼20%. • CsI geometry: The modeling of CsI rings geometry was updated from a solid volume surrounding Bina to an array of four quadrants of upstream, downstream, inner and outer volumes. This new approach allows to apply a 2 MeV threshold in each quadrant. In either case an energy resolution of 0.25 MeV was implemented. Additionally, an attenuation factor M=0.82 was applied on the sum of CsI quadrants to account for an excess of electromagnetic showering, see section 6.4.2. 150 6.6. Rπ corrections 0 10 20 30 40 50 60 70 Ecal [MeV] 1 10 210 Figure 6.33: Energy spectra of µDIF for R <60 mm. The spectrum of the former version of µDIF and the spectrum of the current version of MC are shown in black and cyan, respectively. The red solid line represents the energy cut. 30 40 50 60 70 80 90 R [mm] 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.22 0.24 0.26 6− 10× D IF µ C Figure 6.34: Muon decay-in-flight correction as a function of the radial acceptance cut. Results from the former version of µDIF and the current version of MC are shown in black and cyan, respectively. 151 6.6. Rπ corrections • Beam profile: The Beam profile development described in section 6.5.1 was implemented in MC. The previous version did not used Zv as a means to validate the intensity and shape of the beam momentum. Instead a nominal momentum of 75 MeV/c was used. • EM physics list: Migration from the usage of the EM0 physics list to the PENELOPE implementation of EM interactions in Geant4. Two additional changes were needed to achieve consistency in the Lineshape analysis: photonuclear cross-section scaling update from 1.1 to 1.5, and the modification of CsI attenuation factor to M=0.92. MC upgrade ∆CµDIF [10−8] S3 clustering +2.0±0.5 Muon radiative decay +0.5±0.2 Birks constant -4.5±0.5 CsI geometry -1.5±0.5 Beam Profile -4.5±0.5 EM physics list -2.0±0.5 Total 10.0±1.0 Table 6.3: Change in CµDIF after implementing upgrades in MC relevant for the µDIF decay. Figure 6.33 show the energy spectra of µDIF before and after the MC modifications listed above in black and cyan, respectively. There is a clear decrease of positrons being boosted to the HE region, which is reflected on CµDIF . Table 6.3 show the effect on CµDIF of all MC modifications listed above considering Ecut = 52 MeV and R <60 mm. Figure 6.34 depicts CµDIF as a function of the acceptance cut on R for the former and current version of MC in black and cyan, respectively. The -10×10−8 difference have a direct effect on Rπ according to Eq. 5.12. This difference is reflected as a 0.08% effect in the measurement of the branching ratio and will be incorporated to the global analysis comprising data from Runs III-VI. 6.6.4 t0 correction The reference time of the time spectrum is t0. The timing of the positron signal is extracted by fitting the waveforms from T1 signal. If the shape 152 6.7. Summary of Monte Carlo Corrections of the waveform depends on the positron energy, the extracted time can be energy-dependent, and biasing the branching ratio. To investigate this effect, special muon runs at 62 MeV/c were taken during 2011. The muons stopped at the center of target B3 and the time spectra for different energy regions were constructed, giving and t0 as main outcome by fitting the edge with a step function with Gaussian resolution. The multiplicative correction associated was Ct0 = 1.0006±0.0003 [3]. 6.7 Summary of Monte Carlo Corrections The development of the MC simulation and the important upgrades for the analysis were detailed along this chapter. A cornerstone of the MC validation was the Lineshape analysis development described in section 6.3 and in Ref. [10], in which a 70 MeV positron beam was used to measure the response function of the PIENU calorimeter. By setting the MC simulation with suitable physics lists and tuning the calorimeter non-specified features, a 1σ agreement to data was obtained. From the PIENU MC, the tail fraction of the π+ → e+νe below Ecut = 52 MeV was calculated, resulting in a first order correction of 4.33%. Two second-order corrections were calculated, corresponding to the detector’s acceptance ratio of π+ → µ+νµ → e+νeνµ decay to π+ → e+νe decay and the component of misidentified muon decay- in-flight boosted to HE events. Finally a third-order correction for energy- dependent effects on T1 waveform was reported. The raw measurement of the branching ratio Rraw π and the MC corrections are combined to obtain a blinded measurement of Rπ according to Rπ = (Rraw π − CµDIF )× CT × Cacc × Ct0 . (6.7) In the next chapter, the systematic uncertainties of the parameter related with each component in Eq. 6.7 will be assessed and the measurements of the three data blocks will be combined to report a Rπ measurement, with statistical and systematic errors included, of the Run III dataset. 153 Chapter 7 Results The results of the branching ratio (Rπ) are presented in this chapter. The time spectra is separated in LE and HE via an energy cut, nominally 52 MeV, to perform a fit to the time distribution of the events (the Timefit). This resulted a raw estimate of the branching ratio (Rraw π ) as a fit parameter. The usage of an energy cut left a remaining portion of π+ → e+νe events buried under the π+ → µ+ → e+νeνµ energy spectrum known as low energy tail (LET). This remainder is corrected via Monte Carlo simulation that allows to obtain the π+ → e+νe energy spectrum separately. Two second- order corrections are calculated using the MC simulation; the acceptance ratio of π+ → µ+ → e+νeνµ to π+ → e+νe decay, and the muon decay-in- flight contamination in the HE region. The blinded measurement of Rπ is the result of combining Rraw π from Chapter 5 and the MC corrections from Chapter 6. The results are reported for the three data blocks in Run III. Throughout the chapter the sensitivity of the branching ratio against parameters involved in Timefit and MC modeling are tested and their corre- sponding systematic uncertainties are assessed. Finally, the separate results of Rπ of the three blocks that include statistical and systematic uncertainties are merged to report a measurement of Rπ for the Run III dataset, ready to be included in the complete analysis comprising Runs IV-VI. 7.1 Systematic effects The statistical uncertainties of Rπ are the result of limited statistics and cause fluctuations in unpredictable directions from one measurement to the next. On the other hand, systematic uncertainties affect the measured value of Rπ in a particular direction. Systematic uncertainties are inconsistent or biased measurement techniques, or assumptions made during the analysis that could lead to a systematic shift [158]. The sensitivity of the branching ratio to various parameters along the analysis were calculated following standard methodology [159]. The system- atic uncertainties were assessed by varying the analysis parameters within 154 7.1. Systematic effects reasonable ranges and comparing the difference to a given anchor point. The anchor point are the set of parameters at which the result will be reported. In the case of Rπ, the anchor point is R < 40 mm and Ecut = 52 MeV. The branching ratio difference (∆Rπ) and the uncorrelated statistical error (∆e) between two different branching ratio calculations is defined as, ∆Rπ ±∆e = (Rπ −R′ π)± √ |e2st − e′2st|, (7.1) where Rπ and R′ π are the branching ratios from the anchor and test points respectively, and e′st, est are their respective raw statistical errors. The uncorrelated statistical error ∆e measures the contribution of errors in two of measurements of Rπ are statistically independent of each other. Normally the errors are summed in quadrature, i.e., √ e2st + e′2st, but we are dealing with two different Rπ measurements. For simplicity results of ∆Rπ will be reported in units of 10−8, unless specified otherwise. The sources of systematic uncertainty are classified in five categories and will be discussed along the following sections. 7.1.1 Trigger inefficiencies T1 inefficiency Inefficiencies of the downstream trigger counters could distort the time spec- trum if they were time dependent. The inefficiency of the T1 counter is mea- sured using special muon runs during which this counter is excluded from the trigger and instead T2 is adapted to measure the positron time. From this study, it was concluded that COPPER times are not affected when in- troducing a cut in prompt times of T1 (see Section 4.4.2) [106]. The overall effect on Rπ is 1× 10−8. Muon leakage The muon from a π+ → µ+νµ decay travels ∼1.4 mm inside the plastic scintillator B3 berofe decaying. When its trajectory is contained in the target counter, the muons are expected to stop within a spherical shell of radius ∼1.4 mm. However, if a muon leaks out from the target via the upstream or downstream side, it most likely stops in either S2 or S3, and the solid angle for positron detection is different by 20% from the π+ → e+νe decay. MC simulation indicated that the effect of muon leakage from the target can be corrected by the acceptance correction and the effect on the 155 7.1. Systematic effects branching ratio is less than 1 × 10−8, smaller than PIENU precision goal. Therefore, its effect can be ignored. 7.1.2 Event selection Pion selection An important test to verify the robustness of the branching ratio consists of determining that the sensitivity of Rπ to the energy cuts in B1 to select beam pions, described in Section 4.4.1 and shown in Figure 4.6, is negligible to a 10−8 limit. The lower cut in B1 is set at the point where π+ → µ+νµ → e+νeνµ decays rise above the πDIF background. By setting this cut lower more πDIF background is allowed in Time spectrum, which is taken into account by Timefit. On the B1 upper cut side, there is contamination of beam pions and beam muons within the same RF bunch. Timefit is not sufficiently robust to account for this kind of events. In this case the upper cut (B1h) is varied up to 6.4 MeV from its nominal value of 5.2 MeV in data analysis and MC corrections consistently, resulting in a maximum ∆Rπ value of ±2 about the anchor point, as depicted in figure 7.1. The systematic uncertainty can be taken conservatively as ±1 [133]. A second set of parameters involved in the pion selection is the windows to delimit the WC1/2 acceptance to suppress beam halo particles. By in- creasing the broadening of the cut in the X and Y axes for both WC1 and WC2, it was determined that Rπ is sensitive below 1 × 10−8 [140]. The change was done in data and MC consistently. Thus, no systematic uncer- tainty was assigned. Pileup in scintillation counters As discussed in Section 4.4.2 and shown in Figure 4.8, pileup events are suppressed by requiring a single hit in COPPER time integration window for scintillation counters. However, multiple hits can sometimes be misidentified as only one hit by the waveform identifying algorithm. To avoid this, a cut criterion was implemented on the charge ratio Q/Qw for all PMTs of B1 and B2 as shown in figure 4.8. There is a negligible effect on shifting the upper cut as the number of events in this region is down by five orders of magnitude. On the left side, the cut was shifted by 0.1 units in Q/Qw to allow the immediate left pileup peak in the Time spectrum. Time does not takes care for this amount of pileup and a ∆Rπ = 2 change was observed [140]. A conservative systematic uncertainty of ±1 to Rπ was assigned. 156 7.1. Systematic effects ΔR𝞹 B1 upper cut [MeV] Figure 7.1: ∆Rπ vs. B1 upper energy cut data block 3 in Run III dataset: The x-axis is in MeV units. The y-axis is presented in 10−8 units, with zero change representing the nominal cut at 5.2 MeV. Uncorrelated statistical error are included considering 1-ns binning as the anchor point. Data blocks 1 and 2 behave similarly. In the time window before the pion arrival (Pre-region), no hits are allowed in B1, B2, Tg and T1 by the PrePileup suppression requirement in Eq. 4.3. This cut normally rejects events in a -6.4 µs to -2.2 µs window before the arrival of the pion (-7.7 µs to -3.5 µs window with respect to trigger time as shown in Figure 4.1. Any remainder of PrePileup events in the time spectrum should be compensated by the robust PIENU spectrum analysis by identifying and modeling pileup correctly. The PrePileup condition was removed from the event selection and the upper rejection window of B1, B2, Tg and T1 was relaxed starting from -7.7 µs to allow more PrePileup events in the time spectrum, Figure 4.9 shows the distribution of pileup events in the Pre-region. Figure 7.2 shows ∆Rπ as a function of the upper window limit in the Pre-region for the three data blocks. Uncorrelated statistical errors are displayed. Here the anchor point is the nominal analysis, in which the PrePileup condition was imposed. The time spectrum analysis is stable within errors up to a window broadening of 2.6 µs past the lower bound at -7.7 µs. This is equivalent to allow ∼60% of pileup events in the Pre-region to be included in the time spectrum. This range was enough to assure the 157 7.1. Systematic effects ΔR𝞹 Upper limit in Pre-time [ns] Figure 7.2: ∆Rπ vs. Upper window limit of Pre-Pileup events in Scintilla- tion counters for data blocks 1,2 and 3 of Run III in black, red and blue, respectively: The x-axis is in ns units. The y-axis is presented in 10−8 units, with zero change representing the condition of no-hits in the Pre-region. Un- correlated statistical error are displayed. stability of Rπ against a deficit pileup events that survive the PrePileup condition, which was estimated to be about ∼12% from the total pileup events in the Pre-region. Thus, no systematic uncertainty was assigned. T1 prompt events Usually prompt events are rejected in the Timefit by excluding the -20 to 5 ns region of the time spectrum. The time variable Tpos in Eq. 5.1 is the average of the times of the first hits in each PMTs. In principle a given hit combination between T1 and B1 that falls in the prompt region could sneak into the analysis and distort the fitted branching ratio. A cut in T1 prompt times removes remaining potential events of this kind, see Section 4.4.2. The sensitivity of Rπ to this cut was tested by making the prompt window broader up to 3 ns. It was found a statistically significant difference of ∆Rπ = ±4, consistent with the Run VI analysis [126]. This number was set as systematic uncertainty. 158 7.1. Systematic effects Extra Muon pileup The cut that removes muon pileup events in the B1’s special muon chan- nel, consisting of 20 µs before the pion trigger, is detailed in Section 4.4.3. This cut was designed by studying the change in upstream and downstream energy distributions of the beam pion and decay positron energy, respec- tively. Resulting boundaries that are shown in Figure 4.14. The boundaries between 16 and 18 µs are attached to features in the distribution that cor- respond to the beam muon signal, so the only semi-free parameter is the boundary at 8 µs. It was found that varying this parameter from 4 µs to 10 µs produce a statistically significant change of ∆Rπ = 4. Since the lower boundary was restricted between 5 to 7 µs, a systematic uncertainty of ±2 was assigned. T1-T2 coincidence The cut that suppresses the pileup in downstream counters requires a coin- cidence of ±20 ns between T2 and T1. This condition is independent from the downstream trigger time coincidence that is embedded to the VT48 hardware. Since the single hot requirement in T1 and T2 does not takes time difference into account, the window opening is a free parameter in the analysis. By changing the opening of the window up to ±50 ns, a change of 4 units in Rπ was observed. Since the window maximum opening was ± 25 ns in early analyses, a systematic uncertainty of ∆Rπ = 2 was assigned to T1-T2 time coincidence. 7.1.3 Timefit parameters The time analysis returns the background components such as Old-Muon with no hit in T1 and pion radiative decay whose shape is obtained from MC. To generate these backgrounds, assumptions on the trigger logic were made. This opened the possibly of potential systematic uncertainties in Rπ. Other examples of potential sources are the conditions under which the Timefit is performed such as the binning of time spectrum, fitting range or time resolution. Binning The time spectrum has 1 ns per bin in the nominal analysis. Given that the fit function is smooth along the time range, the branching ratio should be independent of the bin size. However, Run III data shown to be more 159 7.1. Systematic effects sensitive to bin size change, mainly due to statistical fluctuations caused by small statistics in data blocks and a more restrictive acceptance cut at R < 40 mm. Also, there was a large difference in sampling rates from different DAQ hardware components. By changing the binning size from 1 ns to 4 ns per bin, a systematic change in Rπ of 4 units was measured. The bin size does not exceed 4 ns in normal circumstances. In this range the ∆Rπ variations are statistically significant, therefore a conservative systematic uncertainty of ∆Rπ=2 was applied. Fitting range The nominal range for our fitting function for both LE and HE time spectra is from -270 to 520 ns, excluding prompt events from -20 to 5 ns. In order to test the stability of Rπ vs the fit boundaries, these values were shifted. the bottom limit goes from -290 to -250 ns, the lower prompt limit from -20 to -30 ns, the upper prompt limit from 5 to 8 ns, and the top limit from 520 to 490 ns. In all the cases the Rπ changes were 1σ deviations or ∆Rπ < 1, which are not significant for the PIENU precision goal. Lifetimes The pion and muon lifetimes (LT), τπ and τµ respectively, are fixed in the fitting function to the PDG central values [16]. Ideally, the fitting function should return LT values consistent with PDG when they are set as free parameters in the fit. Timefit provides LT consistent with PDG within 1σ for all data blocks. However allowing these LTs to float can cause Rπ to change significantly. To assess any potential systematic uncertainty to LTs, τπ and τµ were declared as the central values of PGD plus a deviation of ±5σ in 1σ steps. Only one LT had the deviation at a time, the other LT was set as the central value of PDG. Figure 7.3 shows the change of Rπ as a function of PDG’s σ displacement in pion and muon LTs. In the case of the muon LT, to measure a significant Rπ change, the range was extended up to ±200σ. The systematic uncertainty was assessed by dividing the total change between the extreme point by the number of σ steps. Both for pion and muon LTs ∆Rπ < 1, hence no systematic uncertainty was added to the error budget. Time resolution The time analysis nominally uses the probability density functions επ→µ→e(t) (Eq. 5.2) and επ→e(t) (Eq. 5.3) for the π + → µ+ → e+νeνµ and π+ → e+νe 160 7.1. Systematic effects Figure 7.3: ∆Rπ vs. σ deviations from PDG central value of τπ (left) and τµ (right) for data block 3 in the Run III dataset. The y-axis is presented in units of 10−8, with zero change representing the PDG fixed values. Un- correlated statistical error are negligible in this scale. Figure 7.4: Change in the branching ratio ∆Rπ vs the time resolution from T1-B1 for data block 3 in Run III data set. The point at zero ∆Rπ corre- sponds to not including any time resolution effects. The red solid line rep- resents the actual time resolution from the scintillators T1 and B1, which was measured at σ = 0.3± 0.1 ns. Other data blocks behave similarly. 161 7.1. Systematic effects signals, respectively. In order to test the sensitivity of the time analysis to time resolution effects, the resolution of the time difference between T1 and B1 is modified to non-zero values in the fitting function. This implies that the signals were replaced by ε′π→µ→e(t, σ) (Eq. 5.4) and ε ′ π→e(t, σ) (Eq. 5.5). Figure 7.4 depicts ∆Rπ in nominal scale vs. time resolution σ for data block 3, the anchor point corresponds to σ = 0 ns, uncorrelated statistical errors are negligible. Other data blocks behave similarly. Resolution effects are relevant in Rπ for values of σ greater than 1.2 ns. Since time difference between T1 and B1 was measured to have a resolution of σ = 0.3± 0.1 ns, which is indicated in the figure by a red solid line, thus resolution effects are far to cause a systematic deviation in Rπ. Therefore, no systematic uncertainty was included in the error budget. Fixed Parameters The fitting function contains background terms that are either simulated or modeled and whose amplitude is fixed to a given value. This is the case of the π+ → µ+νµγ time spectrum amplitude, the T1 time resolution and the background from the Old-muon with no hit in T1. In order to test the sensitivity of Rπ to the π+ → µ+νµγ background the amplitude is scaled and fixed to ±50%, five times the expected difference of the parameter. Scaling the change allow us to measure the sensitivity of Rπ, otherwise the effect would be to small to quantify. The change in the branching ratio between the two extreme points is divided by 5 to set a systematic uncertainty. This resulted in an assigned systematic of ∆Rπ = ±3. The process of determining the amplitude of T1 double pulse resolu- tion background is done to high precision following the procedure depicted in Figure 5.6. The only possible of uncertainty comes from the quadratic model that relates the resolution between hits (∆) and the T1 double pulse resolution. To account for a possible uncertainty, the amplitude was scaled to a factor of ±10%. This scaling was chosen to be conservative because of the high precision to determine the amplitude of this background, see Section 5.2.3. A change in Rπ of ±1.5 was observed, this number was added to the error budget. 7.1.4 Low Energy Tail (LET) correction The LET is the main correction of Rπ and it has been historically the main source of systematic uncertainty. The efforts made to reduce the LET con- tribution to the error budget are described in Sections 6.3, 6.4 and 6.5. In 162 7.1. Systematic effects particular, the Lineshape analysis [10] is an enhanced version from previ- ous analyses [5] [2]. Its development had as its main objective to achieve a better agreement data-MC and to constraint the systematic sources. Unlike the Lineshape tail fraction, the π+ → e+νe tail fraction has three sources of systematic uncertainty. Only two of them were addressed; the energy cal- ibration and the photonuclear cross-section scaling factor (PN). The other belongs to WC3 position along the beam axis. WC3 position As described in Section 6.5.3, the track reconstruction of decay positrons highly depends on the WC3 position along the beam axis. In particular the radius at WC3 R is sensitive to this parameter. To estimate the distance between PIENU-I and PIENU-II, that in turn determines the WC3 position, a survey was performed [108]. It was concluded that an uncertainty of 0.4 mm has to be considered as a LET systematic source. This value is valid for data block 3, since sub-assemblies PIENU-I and PIENU-II were bolted together prior to this data period. In contrast, the process of determining the position of WC3 for data blocks 1 and 2 is attached to achieve a 0.1 MeV agreement in the π+ → e+νe peak between data and MC. Since no constraint is indicated here, the uncertainty in WC3 position is assessed to fulfill the 0.1 MeV agreement. It was determined that a conservative uncertainty of 1 mm is required for data blocks 1 and 2, see more details in Section 6.5.3. The effect of these uncertainties has no significant influence as the statistics are the same, the only change is done in MC. The systematic uncertainties in Rπ caused by changes in the LET correction were found to be ±3 and ±2 for data blocks 1 and 2, and 3, respectively using a radial acceptance cut of R < 40 mm and an energy cut of 52 MeV. Energy calibration The LET correction is highly sensitive to the energy cut. To assess possible systematic deviations to Rπ due to uncertainty in the energy calibration the energy cut was modified by ±0.1 MeV and the change in LET from the nominal Ecut = 52 MeV was measured. A systematic change of ∆Rπ = ±6 was obtained for R < 40 mm. Since all data blocks have the same 0.1 MeV agreement in the π+ → e+νe peak, this number was set as a common systematic uncertainty. 163 7.1. Systematic effects Photonuclear cross-section scaling The choice of the photonuclear scaling is done by considering the photonu- clear peaks at 58 and 50 MeV in the Lineshape energy spectrum shown in Figure 6.9, which are caused by one and two neutron emission respectively. These interactions are important and are challenging to reproduce. Geant4 libraries contain measured photonuclear cross-sections for the relevant tar- get nuclei [154]. However, the neutron escape probability depends on the target nucleus de-excitation model. The optimal choice was found to be FTFP BERT PEN and scaling the photonuclear cross-sections by a factor of 1.5 with an uncertainty of ±0.1. The change in Rπ via the LET correction was found to be ∆Rπ = ±5 for Ecut = 52 MeV and R <40 mm. In the next section, the combined systematic uncertainty of these three sources will be displayed for different integral cuts on R. 7.1.5 Acceptance correction The acceptance ratio in Eq. 6.5 is the second-order correction of Rπ. Al- though energy-related effects are canceled, other parameters such as the pion stopping position and the positron track reconstruction are relevant for the multiplicative correction CAcc and have direct impact on Rπ. Beam momentum The mean momentum in the beam profile is set to reproduce the pion stop- ping position within the target (Zv). The distance traveled within upstream detectors by positrons from either π+ → µ+νµ → e+νeνµ or π+ → e+νe decays depend on Zv. Since the decay channels are susceptible to energy- dependent scattering, the acceptance ratio is sensitive to Zv and thus to beam momentum. The process of beam profile tuning is done to high preci- sion as depicted in Figure . To test the sensitivity of Rπ, to beam mean was shifted by 5 times ±0.5%, which the uncertainty of the measured beam in the M13 beamline using nuclear magnetic resonance (NMR). Just as before, the change was scaled to quantify the change in Rπ. The measured effect on the branching ratio was ∆Rπ < 1, so no systematic uncertainty is needed. WC3 geometrical constraints WC3 position is also important for the acceptance ratio. In this case the central position along the XY axes were shifted by 1 mm in both directions. This change was suggested to match the uncertainty in WC3 position along 164 7.2. Stability tests the z-axis, since no estimate of the actual uncertainty was reported before. No changes were observed in Rπ. Since the CAcc depend on the radial acceptance cut, it is expected to be sensitive to WC3 position along the beam axis. However, any change ∆Rπ is entirely correlated to the systematic effect included in the LET correction budget. S3 position The positron radius R also depends on the S3 position along the beam axis. Due to geometrical constraints the uncertainty in S3 position is two times smaller than WC3 [1] (±0.2 mm). This shift was tested on the MC of data blocks 1, 2 and 3 resulting in ∆Rπ = ±4, ±0.5 for blocks 1 and 2, and 3, respectively. The extra 2.5 mm displacement between PIENU-I and PIENU- II makes the tracker 3 more sensitive to changes in the position of either S3 or WC3. Only the systematic error for blocks 1 and 2 was included in the error budget. 7.1.6 µDIF correction There are three potential sources of systematic uncertainty in the µDIF cor- rection (CµDIF ); beam momentum, Birks constant and muon lifetime (τµ). By following the same procedure described for the acceptance correction, it was estimated that only changes greater than 0.2 MeV in the beam mo- mentum could produce measurable effects in the branching ratio (∆Rπ ≥ 1) [132]. The current precision to match Zv via the beam momentum is smaller than 0.05 MeV. Although the µDIF correction is highly sensitive to the Birks constant, the value 0.024 mm/MeV was set consistently in the MC simu- lation and in data. With inner consistency in the analysis, a systematic uncertainty can not be assigned. Finally, to run the µDIF simulation τµ was shortened to 2 ns, otherwise the simulation would take large amount of time and computing resources. By running the simulation with τµ = 5 and 10 ns, no statistically significant difference in Rπ was observed. 7.2 Stability tests A crucial test to Rπ is verifying it independence from the radial acceptance cut (RA) and the energy cut (Ecut), as long as these parameters are set consistently in Timefit and MC. Figure 7.5 show ∆Rπ vs RA for the three data blocks in Run III. The anchor point in each sub-plot corresponds to R < 40 mm from each data block, uncorrelated statistical errors (∆e) are 165 7.2. Stability tests displayed. The dashed lines represent the systematic envelope dominated by the error sources of the LET correction. The total systematic error increases consistently with RA making R < 40 mm an optimal point to report Rπ. Excluding R < 30 mm, all points are consistent within error bars and with the systematic envelope. The anomaly at R < 30 mm due to poor statistics, which can be visualized in the magnitude of the error bars. Figure 7.6 shows ∆Rπ vs Ecut for the three data blocks in Run III. The anchor point in this case corresponds to Ecut = 52 MeV from each data block, uncorrelated statistical errors (∆e) are displayed. Here ∆e is smaller since there is no change in statistics. Since the acceptance correction is independent from the energy cut, the systematic envelope only accounts LET error sources. There is a clear systematic effect that makes Rπ to decrease consistently as Ecut increases. This is an expected behavior as the π+ → e+νe events are more suppressed with higher energy cuts. The systematic envelope takes care of this systematic effect. This description is valid for the three data blocks. Achieving this consistency has been specially challenging since there were energy shifts related with gain-shift effects, as described in Section 4.2.1, that lead to different energy calibration factors calculated via MC along the Run III dataset. Although there are various disparities in data blocks described in Section 4.3, these differences were compensated in the development of their respective event selection stages. Ideally they should not affect inner consistency of the branching ratio along the Run III dataset. Figures 7.7 (up) and (bottom) show the blinded re- sults of Rπ as a function of RA and Ecut respectively for the three data blocks in Run III. Here, absolute statistical error are displayed for a clearer comparison between blocks. An important outcome of this thesis is the validation of the Run III analysis development by reproducing results from other data sets that were parallel studied in the PIENU experiment. In this case, figure 7.7 (up) shows Rπ as a function of the R cut for the Run VI (2012) dataset in green points. These results were compared with the current version of the PIENU analysis, resulting in an agreement of the order of ∼ 5×10−9. The agreement was verified at all the stages of the analysis; Event selection, Timefit and MC corrections. Additionally, the results of Run VI analysis are consistent with those obtained for Run III within statistical errors for all choices of the R-cut as shown in figure 7.7 (up). A secondary consequence of this is that the weighted statistical errors will not increase more when averaging the result for datasets Run III-VI. The formulae of Section 7.4 will clarify this. 166 7.2. Stability tests ΔR𝞹 RA [mm] ΔR𝞹 RA [mm] ΔR𝞹 RA [mm] Figure 7.5: ∆Rπ±∆e (Eq. 7.1) vs RA for data blocks 1 (up), 2 (middle) and 3 (bottom). The error bars (∆e) on each point represent the uncorrelated statistical error between the point in question and the anchor point. The x-axis is in mm units. In the y-axis is in 10−8 units and the anchor point in each sub-plot corresponds to R < 40 mm from each data block. The dashed lines represent the envelope of systematic uncertainty dominated by low energy tail error sources that increases consistently with the radial acceptance. 167 7.2. Stability tests Ecut [MeV] ΔR𝞹 Ecut [MeV] ΔR𝞹 ΔR𝞹 Ecut [MeV] Figure 7.6: ∆Rπ±∆e (Eq. 7.1) vs Ecut for data blocks 1 (up), 2 (middle) and 3 (bottom). The error bars (∆e) on each point represent the uncorrelated statistical error between the point in question and the anchor point. The x-axis is in MeV units. In the y-axis is in 10−8 units and the anchor point in each sub-plot corresponds to Ecut = 52 MeV from each data block. The dashed lines represent the envelope of systematic uncertainty dominated by low energy tail error sources. 168 7.2. Stability tests R𝞹 RA [mm] R𝞹 Ecut [MeV] Figure 7.7: The blinded branching ratio Rπ as a function of RA (top) and Ecut (bottom) for different data blocks: 1 (black), 2 (red), and 3 (blue) and the Run VI dataset (green). The error bars are absolute statistical uncertainties displayed to compare Rπ results between data blocks along different acceptance and energy cuts. 169 7.3. Error Budget 7.3 Error Budget Table 7.1 presents the systematic and statistical errors for Run III data blocks. The rows with bold titles correspond to the categories described in Section 7.1. These rows show the sum in quadrature of their sources listed below. The error categories can be classified into common or errors that de- pend on the data-taking period. Trigger inefficiencies, Event selection and Timefit associated errors are considered common as the three data blocks share similar analyses. The symbols to denote the sum in quadrature for common error sources will be clarified in the next section. The same tests (7.1.1, 7.1.2, 7.1.3) were performed on the three data blocks. It was con- cluded that the difference in these error sources were negligible (∆Rπ < 0.5). The error categories that depend on the data-taking period are those comprising specific modifications in MC simulation. Specifically, the LET correction is highly sensitive to WC3 position along the beam axis relative to the target B3. As discussed in Section 7.1.4, the 2.5 mm displacement of WC3 in data blocks 1 and 2 causes LET error sources to vary with the data-taking period. This classification prevents error overestimate when combining data blocks to derive the Run III measurement in the next section. The sum in quadrature in quadrature of these dataset-dependent sources is denoted by √ Σ2 α. 7.4 Combination of data blocks Data blocks have clear differences along the analysis besides the statistics collected. The WC3 position difference prevents a global fit of all Run III data. Therefore, the three branching ratios have to be combined after the separate fits to the timing spectra and their respective MC corrections. The procedure to combine several measurements of Rπ with common and inde- pendent systematic errors is outlined in Ref [136]. Let the raw branching ratios (Rraw π ) for each data block with their respective statistical and sys- tematic errors be Yi±δY st i ±δY sy i , where the index i corresponds to the data block number. Here the systematic error δY sy i corresponds to the Timefit category in Table 7.1. Since MINUIT [149] finds a balance between fit pa- rameters to give the least χ2, the systematic error associated to parameters in Timefit category are correlated. The squared error is calculated as 170 7.4. Combination of data blocks Error source Block 1 Block 2 Block 3 Statistical, R < 40 mm (δY st i ) 15.6 22.9 21.8 T1 inefficiency 7.1.1 1.0 Event Selection ( √ Σ2 β) 7.1.2 5.1 B1, B2 Pileup 1.0 B1, B2, T1 PrePU 1.0 T1 prompt 4.0 Extra Muon PU 2.0 T1-T2 coincidence 2.0 Timefit (δY sy i ) 7.1.3 3.5 Binning 1.0 T1 resolution 1.5 π+ → µ+νµγ amplitude 3.0 LET correction ( √ Σ2 α) 7.1.4 8.4 8.4 8.0 WC3 position 3.0 3.0 2.0 Energy calibration 6.0 6.0 6.0 Photonuclear scaling 5.0 5.0 5.0 Acceptance correction ( √ Σ2 α) 7.1.5 0.0 0.0 0.0 Beam momentum < 1.0 < 1.0 < 1.0 Geometrical constraints < 1.0 < 1.0 < 1.0 Table 7.1: Error budget in [10−8] branching ratio units. Data blocks share the same systematic uncertainty, unless the columns are split. (δY sy)2 = ∑ l ∑ m (δY sy l )(δY sy m )ρlm, (7.2) where δY sy l is the sensitivity of Y caused to the parameter l and ρlm the correlation between parameters l and m. MINUIT provides the covariance matrix of the fit parameters after the least squares minimization of the Timefit is executed. For the relevant parameters in Table 7.1, the correlation entries are O(ρlm) ∼ 0.001 and have negligible effect so the usual sum of squares was adopted. Let also the dataset-dependent MC corrections with their uncertain- ties be Cij ± δCst ij ± δCsy ij , where the index j corresponds to multiplica- tive independent corrections j = 1, .., J , which is the case of CT and Cacc. Analogously, the common corrections with their respective uncertainties are 171 7.4. Combination of data blocks Ck ± δCst k ± δCsy k that is the case only for Ct0 . The common global sys- tematics contribution is simply ±δS = √ Σ2 β , where the index β runs over the error sources in the Trigger inefficiencies and Event selection categories, see Table 7.1. Let the partially-corrected branching ratio for each dataset- dependent correction with uncertainties be Ri ± δRst i ± δRsy i with the index i running over block number. Ri and its uncertainties are Ri = YiΠjCij , (7.3) δRi = √ (R2 i ) [(δYi/Yi) 2 +Σj(δCij/Cij)2]. (7.4) Let branching ratio weighted average with uncertainties before common corrections be Rs ± δRst s ± δRsy s defined by Rs = ΣiRiwi/(Σiwi), (7.5) δRs = √ Σi(δRiwi/Σiwi)2, (7.6) where wi is the statistical plus systematic weight for each data block, wi = { (δY st i )2 + (Ri) 2 [ (δY sy i /Yi) 2 +Σj(δC st ij/Cij) 2 +Σj(δC sy ij /Cij) 2 ]}−1 . (7.7) The final combined and weighted branching ratio, including global and common systematic uncertainties is Rf ± δRst f ± δRsy f , defined by Rf = RsΠkCk ± δRst f ± δRsy f , (7.8) δRst f = RsΠkCk √ (δRst s /Rs)2 +Σk(δC st k /Ck)2, (7.9) δRsy f = √ (RsΠkCk)2[(δR sy s /Rs)2 +Σk(δC sy k /Ck)2] + δS2. (7.10) The Rπ weighted average of the Run III dataset is summarized in Table 7.2. The dominant error is statistical at 12×10−8, the main reason for this is the restrictive cut at R < 40 mm. The statistical uncertainty achieved has a similar magnitude to that from the most recent analysis [5], in which datasets Run IV-VI were combined. Although that analysis carried about 3 M events, the central values of Rπ were apart by approximately 30×10−8 units, which made the statistical errors to be inflated when averaging. Although the lack of the CsI information in the Run III analysis, the systematic error was reduced from 9 to 7.6 10−8 units with respect to the previous analysis. This is because of the several improvements to the MC simulation summarized in Section 8.1. 172 7.4. Combination of data blocks Value Stat. Error Syst. Error Rraw π Yi δY st i δY sy i Block 1 1.2∗∗∗ 0.0016 0.0004 Block 2 1.2∗∗∗ 0.0023 0.0004 Block 3 1.2∗∗∗ 0.0022 0.0004 Common systematics δS √ Σ2 β 0.0005 Dataset-dependent corrections LET ( √ Σ2 α) CT δCst T δCsy T Block 1 1.0454 < 0.0001 0.00073 Block 2 1.0454 < 0.0001 0.00073 Block 3 1.0453 < 0.0001 0.00069 Acceptance ( √ Σ2 α) Cacc δCst acc δCsy acc Block 1 0.99723 0.00032 - Block 2 0.99723 0.00032 - Block 3 0.99737 0.00044 - Common corrections Ct0 δCst t0 δCst t0 t0 1.0006 0.0003 - Corrected measurement Ri δRst i δRsy i Block 1 1.2∗∗∗ 0.0016 0.00096 Block 2 1.2∗∗∗ 0.0026 0.00096 Block 3 1.2∗∗∗ 0.0023 0.00092 Weighted average Rs δRst s δRsy s 1.2∗∗∗ 0.0012 0.00058 Final average Rf δRst f δRsy f 1.2∗∗∗ 0.00124 0.00076 Table 7.2: Combination of the three data blocks in Run III for R < 40 mm. The branching ratios for all datasets are still blinded. Nomenclature is explained in text. 173 Chapter 8 Conclusions 8.1 Summary of results A weighted average of Rπ was measured for the Run III dataset. Run III contains 2 M events that will be incorporated to the global analysis that includes Runs IV-VI datasets consisting of 2.9 M events. Performing a measurement of Rπ required specific developments in the various stages of the analysis to compensate special features in the data-taking history of Run III. This thesis presents the implementation and testing of the cuts from the nominal analyses developed so far ([1], [2], [3] and [5]) to the Run III dataset. The development of adjustments to Data selection stage is reported in Chapter 4. Additionally, a list of specific implementations to the Timefit of Run III for the extraction of the raw branching ratio is summarized below. • Implementation of an improved method for the gain-shift correction of the NaI(Tℓ) PMTs. • Given the lack of time calibration in the downstream trigger clock, the cut in T1 that matches the fitted pulse with the first hit seen by the PMT has to be adapted. • Updating the amplitude of the T1 double pulse resolution background in Timefit with a higher accuracy. • Estimation of a non-documented∼2.5 mm displacement between PIENU- I and PIENU-II sub-assemblies, valid for 70% of the data in the Run III dataset. Compensations were done consistently for Event selection, Timefit and MC corrections. Once extracted the branching ratio, a MC simulation was needed to calculate three order corrections that are not possible to estimate using data. In order of importance these were; Tail fraction, Acceptance correction and µ decay-in-flight, all of them described in chapter 6. A Due to the differences in the beam momenta, a MC was developed for each block. The development 174 8.1. Summary of results on MC required involved improvements that have been incorporated to the global analysis, these are summarized below. • Development of the Beam Profile to initialize events in the MC simu- lation by matching the pion stopping position Zv. • Implementation of Strip clustering in the MC simulation of the Silicon detectors. This resulted into a more realistic positron track recon- struction. • Muon decay-in-flight correction update from the previous analysis that resulted in a −10 × 108 additive factor to Rπ not considered in the previous PIENU measurement [6]. A important cornerstone in the analysis is the conclusion of the Line- shape analysis [10], where the response function of the PIENU calorimeter was measured using a 70 MeV positron beam. This study resulted in a better understanding of the physical processes within the calorimeter and a significant reduction of the low-energy tail systematic sources. A 1σ match between data and MC was achieved. By combining the raw branching ratio and the MC corrections, a cor- rected measurement of Rπ was obtained for each data block in Run III dataset. The results were tested against two parameters R and Ecut to ver- ify the stability of the measurement. The systematic sources related to the various stages of the analysis were assessed and summarized in the error budget. Finally, the three data blocks were combined to obtain a weighted average of Rπ with statistical and systematic errors yielding Rπ = (1.2∗∗∗ ± 0.0012(stat.)± 0.0008(syst.))× 10−4. (8.1) The statistical error of the weighted average is similar to the previous measurement [5], where data Runs IV-VI were combined to gather a total of 3 M events. In contrast Run III consists of 2 M events. Although the lack of CsI limited the available statistics, the magnitude of the statistical error is a side result of the consistency between central values of the three data blocks. Given the various improvements in the MC analysis, the systematic uncertainty in the LET correction was reduced from 9 to 8 in 10−8 units. This was an important improvement to the analysis considering the fact that CsI was included neither in the MC simulation. Finally, it was verified that results from the Run III were consistent with the prominent and most controlled dataset in the PIENU analysis (Run VI). By combining the 2 175 8.2. Limits on Physics beyond the Standard Model M events of Run III with the 3 M events of Runs IV-VI from the nominal analysis, the statistical uncertainty could be enhanced from 13.0 to 10.3 10−8 units considering the formulation described along the Section 7.4 and the consistency between blinded branching ratios shown in Figure 7.7. At the moment of writing this thesis, the global analysis is almost com- pleted. Three sources of systematic uncertainty in the acceptance correction are being studied. Under a conservative estimate and considering the im- provements made in the analysis and in the MC simulation mentioned above, the systematic uncertainty for Runs III-VI would add to 9×10−8 units for R < 40 mm. Under this approach, the PIENU Rπ measurement would reach a 0.11% precision level. 8.2 Limits on Physics beyond the Standard Model The blinded measurement of Rπ using Run III dataset with 2 million π+ → e+νe events is shown in Eq. 8.1. Under a conservative estimation, the precision level of the PIENU measurement of Rπ could be reduced to 0.11%, a factor of 2.2 from the the analysis published in 2015 in which a subset of 0.4 million π+ → e+νe events was used yielding R2015 π = (1.2344± 0.0023(stat.)± 0.0019(syst.))× 10−4. (8.2) Stringent constraints can be set to new physics frameworks by estimat- ing an upper limit to the branching ratio measurement RUL π . This can be achieved using Feldman-Cousins frequentist method described in Ref. [160] by comparing the experimental result to the SM prediction in σ units of experimental error (squared sum of statistical and systematic). Considering that Rπ is quantity physically bounded to be non-negative and according to Table X of Reference [160]. Using the published result from 2015 in Eq. 8.2 with an error of σ = 0.003 × 10−4, the upper limit at 95% C.L. is 1.67 standard deviations above the SM prediction or RUL π = 1.2402× 10−4. (8.3) In contrast, using the conservative estimation of the PIENU Rπ mea- surement once the datasets are combined with the result of Eq. 8.1 yielding an combined error σ = 0.014×10−4, and the central value of R2015 π the upper limit to the experimental value of Rπ would decrease to RUL π = 1.2371×10−4. 176 8.2. Limits on Physics beyond the Standard Model 8.2.1 Lepton universality Lepton universality is a consequence of requiring the SM Lagrangian to be invariant under gauge transformations, this translated in the a coupling strength between each lepton generation and the W boson, i.e., ge = gµ = gτ = g. According to Section 2.3.1, any difference between the coupling constants can be estimated by the deviation between the SM prediction (RSM π ) and the experimental measurement (Rexp π ) via Rexp π /RSM π = (ge/gµ) 2. Using the PIENU result from 2015 (R2015 π ) with a 0.24% precision level, the coupling constants ratio is ge gµ = 0.9996± 0.0012, (8.4) which translates into a 0.12% precision in the lepton universality test. In contrast, using the conservative projection for the final PIENU Rπ measure- ment (0.11% precision level) and the central value of R2015 π the errors of the ge/gµ ratio would improve to ±0.0005, reaching therefore a 0.05% precision in the lepton universality test. This would consolidate the pion decay as the most sensitive test of lepton universality, and improve the already existing constraints on BSM models attempting to explain lepton non-universality seen by the LHCb [58] [59] and BaBar [60] experiments. 8.2.2 New Pseudo-scalar interactions As discussed in Section 2.3.2, the pion decay is a reliable test for effec- tive pseudo-scalar interactions arising from scalar operators via electroweak renormalization effects. The energy scale of these interactions can be es- timated by substituting the SM prediction and the upper limit of the Rπ measurement from Eq. 8.3 in Eq. 2.30, giving RUL π RSM π − 1 = 1.2402 1.2352 − 1 ∼ ( 1TeV Λ )2 × 103. (8.5) resulting in Λ ∼ 497 TeV. Therefore, the mass scale of a new fundamental pseudo-scalar mediators having the same coupling strength to quarks and leptons as the W bosons, must be above 500 TeV at 95% C.L. If the upper limit on Rπ for the conservative improvement of PIENU is used, the mass scale of the hypothetical pseudo-scalar interaction would increase to Λ ∼ 806 TeV. 177 8.2. Limits on Physics beyond the Standard Model 8.2.3 R-Parity violating SUSY Any deviation from the SM caused by the violation of the R-parity can be explained by MSSM framework (see Section 2.3.2) in terms of the parameters ∆′ 11k and ∆′ 21k [77] according to Eq. 2.33 ∣ ∣ ∣ ∣ ∆RUL π RSM π ∣ ∣ ∣ ∣ = 2(∆′ 11k −∆′ 21k) (8.6) Further constraints on the parameters ∆′ 11k and ∆′ 21k can not be set only through ∆RUL π , only considering special cases. According to Figure , with a experimental precision level 0.1% in the branching ratio measurement and in the extreme case of ∆′ 11k = 0 then ∆′ 21k could be restricted to 0.002±0.001, at 95% C.L. 8.2.4 Charged Higgs Boson The MSSM contains a doublet of two charged Higgs (2HDM) that could replace the W boson as a mediator of the pion decay (see Section 2.3.2). Only if the coupling between this hypothetical Higgs to leptons is independent of the lepton masses, Rπ would be directly affected, otherwise no deviation from RSM π could be observed. Assuming similar order couplings λeν ∼ λµν ∼ λud ∼ α/π and substituting in Eq. 2.34 we have m± H ∼ mπmWα π √ 2 me(mu +md) ( 1− me mµ ) RSM π RSM π −Rexp π . (8.7) Using the upper limit of Rexp π estimated with 95% C.L. at Eq. 8.3 a lower limit to m± H can be identified as m± H ≥ 182 GeV. (8.8) This lower limit can increase up to m± H ≥ 295 GeV when using the conservative projection of the PIENU final precision. 8.2.5 Massive Neutrinos in the π+ → e+νe decay Massive neutrinos described in Section 2.3.3 below 55 MeV/c2 can be bounded with the upper limit to Rexp π using the Feldman-Cousins approach. Specif- ically, the element |Uei|2, from the mixing matrix UPMNS , can be assessed via Eq. 2.41 according to 178 8.3. Next generation of Rπ measurement 5 10 15 20 25 30 35 40 45 50 ] 2 Neutrino Mass [MeV/c 6− 10 5− 10 4− 10 3− 10 2− 102 | e i | U Figure 8.1: Upper limit of the |Uei|2 mixing parameter as a function of the heavy neutrino mass to a 95% C.L. The blue line shows the result from the estimated upper limit to the pion branching ratio using a subset of data (Run IV) published in 2015 [6] with the Feldman-Cousins approach. The red line considers the upper limit to Rπ using the conservative projection of the PIENU final precision. |Uei|2 = RUL π /RSM π − 1 ρe , (8.9) where ρe is a function of the neutrino mass mνi , as shown in Eq. 2.38. Figure 8.1 shows the upper limit of |Uei|2 to a 95% C.L. as a function of the neutrino mass using Eq. 8.3 from the publication in 2015 using a subset of 0.4 M events (Run IV) [6] in blue, and the conservative projection of the final PIENU result that compiles the result of this thesis in red. A heavy neutrino mass of 50 MeV/c2 has a limit of between 10−7 and 10−6 in the mixing parameter |Uei|2 and the limit increases as the mass goes to zero. 8.3 Next generation of Rπ measurement The next generation experiment for a high precision measurement of the pion branching ratio will be leaded by PIONEER [64] at PSI. Its current 179 8.3. Next generation of Rπ measurement design is strongly influenced by the experience gather from PIENU [8] and PEN [23], aiming to suppress sources of systematic uncertainties and han- dling increased rates. The features of the PIONEER detector are shown in the simplified schematic of Figure 8.2. PIONEER follows the experimental technique to measure Rπ used across many experiment generations, in which an pion beam is stopped within a target (ATAR) and the decay positron is measured by a calorimeter (CALO). The PIONEER detector will feature time and energy resolutions, high-speed detector and electronic response, large solid angle coverage, and a complete event reconstruction. It is ex- pected that the net effect of these improvements compared, to the PIENU analysis, will yield a Rπ measurement with precision of O(0.01%). For this, two conditions need to be fulfilled; the uncertainty budget is equally allo- cated to statistics and systematics and about 2 × 108 π+ → e+νe events to be gathered. The main features of PIONEER to achieve such a level of precision are summarized below. High-rate pion beam The basic beam requirements for PIONEER comprise a beam intensity of 55 to 70 MeV/c with a reduced spread (∆p/p) of 2%; a maximum width ∆Zv = 1 mm in the stopping position of the pion; a transverse beam size (FWHM) at the target location (∆X ×∆Y ) of 10× 10 mm2; a maximum transverse angular deflection (∆X ′,∆Y ′) of ±10◦; and a pion rate of 3× 105/s. Active target (ATAR) The key feature of PIONEER is a highly segmented active target (ATAR) [67] that will define the fiducial pion stop region and provide high-resolution timing information of the pion decay. For the measurement of the calorime- ter response to π+ → e+νe decays (similar to the Lineshape measurement from Section 6.4), ATAR will be able to suppress π+ → µ+νµ → e+νeνµ. Furthermore, ATAR will suppress accidental muon stops that precede the trigger signal; these were a significant source of pile-up background in the PIENU experiment. The decay muons must be fully contained in ATAR. This drives the requirement for small longitudinal and lateral size of the π stopping distribution. Cylindrical target A dual layer cylindrical silicon strip tracker will be situated between the ATAR and the calorimeter to measure the positron position in two dimen- 180 8.3. Next generation of Rπ measurement Figure 8.2: Schematic of the PIONEER detector. The intense pion beam enters from the left and is brought to rest in a highly segmented active target (ATAR). Decay positron trajectories are measured from the ATAR to an outer electromagnetic calorimeter (CALO) through a tracker. The CALO records the positron energy, time and location. sions and its time with respect to the pion stop. Two layers of strips with a small stereo angle between them will provide O(mm) z-resolution and 300 µm resolution in the perpendicular direction to the strips. LXe Calorimeter The prominent candidate to be used in the PIONEER calorimeter is liquid Xenon (LXe) due to its fast timing properties, high light yield, excellent energy resolution and highly uniform response. The calorimeter (CALO) will be read out by UV sensitive phototubes. The suggestion of using LXe was raised by the MEG [65] and MEG-II [66] experiments, in which a high- rate LXe detector is used to search for the lepton flavor violating muon decay, µ+ → e+γ. The PIONEER LXe calorimeter is expected to be a 25 radiation length, 3π sr sphere surrounding ATAR. The homogeneity of the LXe detector is an advantage in achieving the high energy resolution 181 8.3. Next generation of Rπ measurement which, in turn, is important to accurately determine the LET fraction of the π+ → e+νe energy spectrum and to reduce the associated systematic uncertainty. Trigger The main improvement in the trigger design is that the logic does not intro- duce any bias between the π+ → e+νe and the π+ → µ+νµ → e+νeνµ sig- nals. This could be achieved by selecting prompt events with a TRACKER hit in time range [2,32] ns relative to the pion signal. This prompt trigger is critical to measure π+ → e+νe events which are lost because the CALO response and fall below the energy cut that sets apart the low-energy and high-energy regions. The uncertainty due to those losses is part of the sys- tematic associated to the LET correction, which leads the systematic sources in the last generation experiments. The design of the trigger takes advan- tage of the APOLLO [68] platform designed for the trigger track finder and pixel readout in the CMS experiment at the LHC. PIONEER with two orders of magnitude more statistics has the potential to improve the existing precision level by at least an order of magnitude. In addition, PIONEER aims to bring significant additional advantages in lowering the limits and in reducing the systematic errors. For example, the π+ → e+νe represents the main background for the π+ → e+νH , π+ → e+νeX, and π+ → e+νeνν searches. A knowledge in-depth of the tail and its further reduction will significantly improve the upper limits beyond the statistics. The search for rare and exotic decays involving muons like π+ → µ+νH , π+ → µ+νH , and π+ → µ+νµνν can benefit from an improved stopping target and faster electronics, which will allow better separation of muons from pions. 182 Bibliography [1] C. Malbrunot. Study of π+ → e+νe decay. PhD thesis, The University of British Columbia, 2012. [2] T. Sullivan. A high-precision measurement of the π → eν branching ratio. PhD thesis, The University of British Columbia, 2017. [3] S. Ito. Measurement of the π+ → e+νe Branching Ratio. PhD thesis, Osaka University, 2016. [4] K. Yamada. Search for Massive Neutrinos in π+ → e+νe Decay. PhD thesis, Osaka University, 2010. [5] S. Cuen-Rochin. Precise Measurement of Rare Pion Decay. PhD thesis, The University of British Columbia, 2019. [6] A. Aguilar-Arevalo et al. (PIENU Collaboration). Improved measure- ment of the π → eν branching ratio. Phys. Rev. Lett., 115:071801, 2015. doi:10.1103/PhysRevLett.115.071801. [7] A. Aguilar-Arevalo et al. (PIENU collaboration). High Purity Pion Beam at TRIUMF. Nucl. Instrum. Methods Phys. Res., Sect. A, 609:102–105, 2009. doi:10.48550/arXiv.1001.3121. [8] A. Aguilar-Arevalo et al. (PIENU collaboration). Detector for measuring the π+ → branching fraction. Nucl. Instrum. Methods Phys. Res., Sect. A, 791:38–46, 2015. [9] A. Aguilar-Arevalo et al. (PIENU collaboration). Study of a large NaI(Tl) crystal. Nucl. Instrum. Methods Phys. Res., Sect. A, 621:188- 191, 2010. [10] A. Aguilar-Arevalo et al. (PIENU collaboration). Measurement of the response function of the PIENU calorimeter. Nucl. Instrum. Methods Phys. Res., Sect. A, 621:188-191, 2010. 183 Bibliography [11] E. D. Bloom, et al. Hartmann, and H. W. Kendall. High-energy inelastic e−p scattering at 6◦ and 10◦. Phys. Rev. Lett., 23(16):930–934, Oct 1969. doi:10.1103/PhysRevLett.23.930. [12] M. Breidenbach, et al. Observed behavior of highly inelastic electron-proton scattering. Phys. Rev. Lett., 23(16):935–939, Oct 1969. doi:10.1103/PhysRevLett.23.935. [13] Peter W. Higgs. Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett., 13:508–509, 1964. [14] G. Aad et al. Observation of a new particle in the search for the Stan- dard Model Higgs boson with the ATLAS detector at the LHC. Physics Letters B, 716:1–29, 2012. doi:10.1016/j.physletb.2012.08.020. [15] W. J. Marciano and A. Sirlin. Theorem on πl2 decays and electron- muon universality. Phys. Rev. Lett., 36(24):1425–1428, Jun 1976. doi:10.1103/PhysRevLett.36.1425. [16] R. L. Workman et. al. (Particle Data Group). Review of parti- cle physics. Prog. Theor. Exp. Phys., 083C01 (2022), Mar. 2022. https://pdg.lbl.gov/2022/reviews/contents sports.html. [17] Makoto Kobayashi and Toshihide Maskawa. cp-violation in the renor- malizable theory of weak interaction. Progress of Theoretical Physics, 49(2):652–657, 1973. http://dx.doi.org/10.1143/PTP.49.652. [18] Nicola Cabibbo. Unitary symmetry and leptonic decays. Phys. Rev. Lett., 10(12):531–533, Jun 1963. doi:10.1103/PhysRevLett.10.531. [19] Scott Willenbrock. Symmetries of the Standard Model. Phys. Rev. Lett.. arXiv:hep-ph/0410370. [20] R. P. Feynman. QED: The Strange Theory of Light and Matter. Prince- ton University Press, new ed edition, 1988. [21] C. Burgess and G. Moore. The Standard Model: A Primer. Cambridge University Press, 2006. [22] G. D. Coughlan, J. E. Dodd, and B. M. Gripaios. The Ideas of Particle Physics: An Introduction for Scientists. Cambridge University Press, 3 edition, 2006. [23] The PEN Collaboration. Precise measurement of the π+ → e+νe branching ratio. psi proposal r-05-01. 2006. 184 Bibliography [24] E. Di Capua, R. Garland, L. Pondrom, and A. Strelzoff. Study of the decay π → e + ν. Phys. Rev., 133(5B):B1333–B1340, Mar 1964. doi:10.1103/PhysRev.133.B1333. [25] D. A. Bryman, R. Dubois, T. Numao, B. Olaniyi, A. Olin, M. S. Dixit, D. Berghofer, J. M. Poutissou, J. A. Macdonald, and B. C. Robertson. New measurement of the π → eν branching ratio. Phys. Rev. Lett., 50:7- 10, 1983. [26] D. I. Britton, S. Ahmad, D. A. Bryman, R. A. Burnham, E. T. H. Clifford, P. Kitching, Y. Kuno, J. A. Macdonald, T. Numao, A. Olin, J-M. Poutissou, and M. S. Dixit. Measurement of the π → eν branching ratio. Phys. Rev. D, 49:28–39, 1994. [27] G. Czapek, A. Federspiel, A. Flükiger, D. Frei, B. Hahn, C. Hug, E. Hugentobler, W. Krebs, U. Moser, D. Muster, E. Ramseyer, H. Scheidi- ger, P. Schlatter, G. Stucki, R. Abela, D. Renker, and E. Steiner. Branch- ing ratio for the rare pion decay into positron and neutrino. Phys. Rev. Lett., 70:17–20, 1993. [28] C. Patrignani et al. Particle data group. Chin. Phys. C, 40, 2016. [29] Cliff Burgess and Guy Moore. The Standard Model: A Primer. Cam- bridge University Press, 2007. [30] James D. Bjorken and Sidney D. Drell. Relativistic Quantum Mechan- ics. Mcgraw-Hill Book Company, 1964. [31] Micheal E. Peskin and Daniel V. Schroeder. An Introduction to Quan- tum Field Theory. Westview Press, 1995. [32] S. M. Berman. Radiative corrections to pion beta decay. Phys. Rev. Lett., 1(12):468–469, Dec 1958. doi:10.1103/PhysRevLett.1.468. [33] Toichiro Kinoshita. Radiative corrections to π − e decay. Phys. Rev. Lett., 2(11):477–480, Jun 1959. doi:10.1103/PhysRevLett.2.477. [34] W. J. Marciano and A. Sirlin. Theorem on πl2 decays and electron- muon universality. Phys. Rev. Lett., 36(24):1425–1428, Jun 1976. doi:10.1103/PhysRevLett.36.1425. [35] William J. Marciano and A. Sirlin. Radiative corrections to πl2 decays. Phys. Rev. Lett., 71(22):3629–3632, Nov 1993. doi:10.1103/PhysRevLett.71.3629. 185 Bibliography [36] V. Cirigliano and I. Rosell. π/K → eν branching ratios to O(e2p4) in Chiral Perturbation Theory. JHEP, 10:005, 2007. arXiv:0707.4464, doi:10.1088/1126-6708/2007/10/005. [37] D. A. Bryman, W. J. Marciano, R. Tschirhart and T. Yamanaka. Rare kaon and pion decays: Incisive probes for new physics beyond the stan- dard model. Annual Review of Nuclear and Particle Science, 61:331–354, 2011. doi:10.1146/annurev-nucl-102010-130431. [38] H.L. Anderson et al. Branching Ratio of the Electronic Mode of Positive Pion Decay. Phys. Rev., 119:2050–2067, 1960. [39] Adam West. Lepton Dipole Moments: Physics in Collition, 2015 arXiv:1607.00925v1 [40] D. I. Britton, S. Ahmad, D. A. Bryman, R. A. Burnham, E. T. H. Clifford, P. Kitching, Y. Kuno, J. A. Macdonald, T. Numao, A. Olin, J- M. Poutissou, and M. S. Dixit. Measurement of the π+ → e+ν branching ratio. Phys. Rev. Lett., 68:3000–3003, May 1992. [41] Guido Altarelli and Martin W. Grunewald. Precision electroweak tests of the standard model. Phys. Rept., 403-404:189–201, 2004. arXiv:hep-ph/0404165, doi:10.1016/j.physrep.2004.08.013. [42] Joshua A. Frieman, Micheal S. Turner, and Dragan Huterer. Dark en- ergy and the accelerating universe. Annual Review of Astronomy and Astrophysics, 46:1–572, 2008. [43] Edward W. Kolb and Michael S. Turner. Grand unified theories and the origin of the baryon asymmetry. Annual Review of Nuclear and Particle Science, 33:645–696, 1983. [44] P. A. Zyla et al. (Particle Data Group). PTEP 2020, 083C01 (2020). [45] Bevan, A. J. et al. Physics of the B Factories. The European Physical Journal C. 74 (11): 3026. arXiv:1406.6311. [46] M. González-Alonso, J. Martin Camalich, and K. Mimouni. Renormalization-group evolution of new physics contributions to (semi)leptonic meson decays. Physics Letters B, 772:777–785, 2017. [47] Y. Arai. AMT-3 -ATLAS Muon TDC version 3 & AMT-2- User’s Man- ual, 2005. 186 Bibliography [48] Lusiani, Alberto. Lepton universality and lepton flavour violation tests at the b-factories. EPJ Web of Conferences, 118:01018, 2016. doi:10.1051/epjconf/201611801018. [49] C. Lazzeroni et al. Test of lepton flavour universality in K+ → l+ν decays. Physics Letters B, 698(2):105 – 114, 2011. doi:10.1016/j.physletb.2011.02.064. [50] A. Pich. Tau Physics: Theory Overview. Nucl. Phys. Proc. Suppl., 181- 182:300–305, 2008. arXiv:0806.2793. [51] Andreas Crivellin and Martin Hoferichter, Mounting Evidence for the Violation of Lepton Flavor Universality. Nucl. Phys. Proc. Suppl. arXiv:2111.12739v1. [52] C. Bobeth, G. Hiller, G. Piranishvili, Angular Distributions of B → Kll Decays. JHEP 12, 040 (2007). arXiv:0709.4174v2. [53] R. Aaij, et al. Test of lepton universality in beauty-quark decays. arXiv:2103.11769v2. [54] Y. S. Amhis, et al. Averages of b-hadron, c-hadron, and τ -lepton prop- erties as of 2018, Eur. Phys. J. C 81, 226 (2021). arXiv:1909.12524v3. [55] J. C. Hardy, I. S. Towner, Superallowed 0+ → 0+ nuclear β decays: 2020 critical survey, with implications for Vud and CKM unitarity Phys. Rev. https://doi.org/10.1103/PhysRevC.102.045501. [56] A. M. Sirunyan, et al, Search for resonant and non-resonant new phe- nomena in high-mass dilepton final states at √ s = 13 TeV. JHEP 07, 208 (2021). arXiv:2103.02708v2. [57] A. Crivellin and M. Hoferichter. β Decays as Sensitive Probes of Lepton Flavor Universality. Phys. Rev. Lett. 125, 111801 (2020). https://doi.org/10.1103/PhysRevLett.125.111801. [58] R. Aaij, et al (The LHCb Collaboration). Test of lepton universality using B+ → K+l+l+ decays. Phys. Rev. Lett., 113(15):151601 (2014). [59] R. Aaij, et al (The LHCb Collaboration). Measurement of the ratio of branching fractions B(B0 → D∗+τ−ντ )/B(B0 → D∗+µ−νµ) Phys. Rev. Lett., 115(11):111803 (2015). 187 Bibliography [60] J.P. Lees, et al (BaBar Collaboration). Evidence for an excess of B → D(∗)τ−ντ decays. Phys. Rev. Lett., 109:101802 (2012). [61] W. Loinaz et al., Phys Rev D70 113004 2004. http://dx.doi.org/10.1103/PhysRevD.70.113004 [62] V. Tishchenko et al. Detailed Report of the MuLan Measurement of the Positive Muon Lifetime and Determination of the Fermi Constant, Phys. Rev. D 87, 052003 (2013), arXiv:1211.0960. [63] D. A. Bryman, et al (PIONEER collaboration). Testing Lep- ton Flavor Universality with Pion, Kaon, Tau, and Beta Decays. arXiv:2111.05338, https://doi.org/10.48550/arXiv.2111.05338. [64] W. Altmannshofer et al (PIONEER collaboration). PIO- NEER: Studies of Rare Pion Decays. arXiv:2203.01981, https://doi.org/10.48550/arXiv.2203.01981. [65] A. M. Baldini, et al (MEG collaboration). Search for the Lepton Flavour Violating Decay µ+ → e+γ with the Full Dataset of the MEG Experiment. arXiv:1605.05081, https://doi.org/10.48550/arXiv.1801.04688. [66] A. M. Baldini, et al (MEG collaboration). The de- sign of the MEG II experiment. arXiv:1801.04688, https://doi.org/10.48550/arXiv.1801.04688. [67] S. M. Mazza, et al (PIONEER collaboration). An LGAD-based full active target for the PIONEER experiment. arXiv:2111.05375, https://doi.org/10.48550/arXiv.2111.05375. [68] R. Zou, The Apollo ATCA design for CMS Track Finder and Pixel Readout at the HL-LHC, indico.cern.ch/event/1019078/contributions/4444387/ (2021). [69] O. Fischer, et al, Unveiling Hidden Physics at the LHC. arXiv:2109.06065, https://doi.org/10.48550/arXiv.2109.06065. [70] W. J. Marciano. Fermi Constants and ”New Physics”. Phys. Rev. D 60, 093006 (1999), arXiv:hep-ph/9903451. [71] A. Crivellin, M. Hoferichter, and C. A. Manzari. The Fermi constant from muon decay versus electroweak fits and CKM unitarity. Phys. Rev. Lett. 127, 071801 (2021). arXiv:2102.02825 [hep-ph] 188 Bibliography [72] Miriam Leurer. Comprehensive study of leptoquark bounds. Phys. Rev. D, 49:333–342, 1994. [73] H. Georgi and S. L. Glashow. Unity of All Elementary- Particle Forces. Phys. Rev. Lett. 32, 438 (1974). https://doi.org/10.1103/PhysRevLett.32.438. [74] S. Davidson, et al.Model independent constraints on leptoquarks from rare processes. Z. Phys. C 61, 613 (1994), arXiv:hep-ph/9309310. [75] B. A. Campbell and D. W. Maybury. Constraints on Scalar Couplings from π± → l±νl. Nucl. Phys. B 709, 419 (2005). arXiv:hep-ph/0303046. [76] Joshua A. Frieman, Micheal S. Turner, and Dragan Huterer. Dark en- ergy and the accelerating universe. Annual Review of Astronomy and Astrophysics, 46:1–572, 2008. [77] M. J. Ramsey-Musolf, S. Su, and S. Tulin. Pion Leptonic Decays and Supersymmetry. Phys. Rev., D76:095017, 2007. [78] O. Shanker. Nuclear Physics B, 204(3):375-386. 1982 http://dx.doi.org/10.1016/0550-3213(82)90196-1 [79] R. Carlini, J.M. Finn, S.Kowalski, and S. Page, spokespersons. JLab Experiment E-02-020. [80] Ian Hinchliffe. Supersymmetric models of particle physics and their phe- nomenology. Annual Review of Nuclear and Particle Science, 36:505-543, 1986. [81] M. Krawczyk and D. Temes. Large 2HDM(II) one-loop correc- tions in leptonic tau decays. Eur. Phys. J. C 44, 435 (2005). arXiv:hep-ph/0410248. [82] A. Broggio, et al. Limiting two-Higgs-doublet models. JHEP 11, 058 (2014). arXiv:1409.3199 [hep-ph]. [83] T. Asaka and M. Shaposhnikov. The nuMSM, dark matter and baryon asymmetry of the universe. Phys. Lett., B620:17–26, 2005. [84] A. Aguilar-Arevalo et al. (MiniBooNE Collaboration). Event Excess in the MiniBooNE Search for νµ → νe Oscillations. Phys. Rev. Lett., 105:181801, 2010. 189 Bibliography [85] B. Bertoni, S. Ipek, D. McKeen, and A. E. Nelson. Constraints and consequences of reducing small scale structure via large dark matter- neutrino interactions. Journal of High Energy Physics, 2015(4):170, 2015. [86] R. E. Shrock. General theory of weak processes involving neutrinos. ii. pure leptonic decays. Phys. Rev. D, 24:1275–1309, 1981 [87] M. Aoki et al. (PIENU collaboration). Search for massive neutrinos in the decay π → eν. Phys. Rev. D, 84:052002, 2011. [88] D.I. Britton et al. Improved search for massive neutrinos in π+ → e+ν decay. Phys. Rev. D, 46:R885–R887, 1992. [89] Tatsu Takeuchi. Future constraints on and from lepton universality. Journal of Physics: Conference Series, 136(4):042045, 2008. [90] A. Aguilar-Arevalo et al, (MiniBooNE Collaboration). Unexplained Ex- cess of Electronlike Events from a 1-GeV Neutrino Beam. Phys Rev. Lett. 102, 101802 (2009). [91] A. Aguilar-Arevalo et al, (MiniBooNE Collaboration). Significant Ex- cess of Electronlike Events in the MiniBooNE Short-Baseline Neutrino Experiment. Phys Rev. Lett. 102, 221801 (2018). [92] A. Aguilar-Arevalo et al, (MiniBooNE Collaboration). Testing meson portal dark sector solutions to the MiniBooNE anomaly at the Coherent CAPTAIN Mills experiment. Phys Rev. Lett. 109, 095017 (2024). [93] D. Berger, A. Rajaraman, and J. Kumar. Dark matter through the quark vector current portal. Pramana 94, 133 (2020) [94] L. B. Auerbach et al. (LSND Collaboration). Measurement of electron- neutrino electron elastic scattering. Phys. Rev. D 63, 112001 (2001). [95] B. Zeitnitz (KARMEN Collaboration). Limits on neutrino oscillations in the appearance channels νµ → νe and νµ → νe Prog. Part. Nucl. Phys. 32, 351 (1994). [96] A. A. Aguilar-Arevalo et al. (CCM Collaboration). First dark matter search results from Coherent CAPTAIN-Mills. Phys. Rev. D 106, 012001 (2022). 190 Bibliography [97] J. R. Klein and A. Roodman. Blind analysis in nuclear and particle physics. Annual Review of Nuclear and Particle Science, 55(1):141-163, 2005. [98] P. F. Harrison. Blind analysis. Journal of Physics G: Nuclear and Par- ticle Physics, 28(10):2679, 2002. [99] Murray Gell-Mann. A Schematic Model of Baryons and Mesons. Phys. Lett., 8:214–215, 1964. doi:10.1016/S0031-9163(64)92001-3. [100] D. vom Bruch. Studies for the PIENU Experiment and on the Direct Radiative Capture of Muons in Zirconium. M.Sc. thesis, The University of British Columbia, 2013. [101] G. Blanpied, et al. N→ ∆Transition from Simultaneous Measurements of p(γ⃗, π) and p(γ⃗, γ). Phys. Rev. Lett., 79(22):4337–4340, Dec 1997. doi:10.1103/PhysRevLett.79.4337. [102] V. Bellini, et al. Polarized Compton scattering from 4 He in the ∆ region. Phys. Rev. C, 68(5):054607, Nov 2003. doi:10.1103/PhysRevC.68.054607. [103] Cayouette F., et al. DETECT2000: An Improved Monte-Carlo Simu- lator for the Computer Aided Design of Photon Sensing Devices. IEEE Transactions on Nuclear Science, 49:624–628, 2002. [104] Chloé Malbrunot. Presentation: Bina optical simulation, April 2008. Available from: https://pienu.triumf.ca/InternalDocuments/. [105] Y. Unno. ATLAS silicon microstrip detector system (SCT). Nucl. In- strum. Meth., A511:58–63, 2003. [106] D. Vavilov. T1 inefficiency with special runs of Aug 2011. Available from: https://pienu.triumf.ca/InternalDocuments/. [107] T. K. Komatsubara et al. Performance of fine mesh photomultiplier tubes designed for an undoped CsI endcap photon detector. Nucl. In- strum. Meth., A404:315–326, 1998. [108] Aleksey Sher. Presentation: WC3 position, April 2019. Available from: https://pienu.triumf.ca/InternalDocuments/. 191 Bibliography [109] Masaaki Kobayashi, et al. YAlO3: Ce-Am light pulsers as a gain monitor for undoped CsI detectors in a magnetic field. Nucl. In- strum. and Meth. in Physics Research Sec. A: Accelerators, Spec- trometers, Detectors and Associated Equipment, 337(2-3):355–361, 1 1994. http://www.sciencedirect.com/science/article/B6TJM- 473FRB0-2N4/2/21b282cb90c7601d8eb982dd9abcb54c. [110] C.Kost, P.Reeve. REVMOC: TR-DN-82-28, 1982. [111] Y. Igarashi, et al. A common data acquisition system for high-intensity beam experiments. IEEE Transactions on Nuclear Science, 52(6):2866 –2871, 2005. [112] J.-P. Martin and P.-A. Amaudruz. A 48 channel pulse shape digitizer with dsp. IEEE Transactions on Nuclear Science, 53(3):715 – 719, 2006. [113] Ahmad, Q.R., et al. (SNO Collaboration). Measurement of the rate of νe + d → p + p + e− interactions produced by 8B Solar neu- trinos at the Sudbury Neutrino Observatory. Phys. Review Letters. arXiv:nucl-ex/0106015 [114] Arthur B. McDonald. The Sudbury Neutrino Observa- tory: Observation of flavor change for solar neutrinos. https://doi.org/10.1002/andp.201600031. [115] Combe, J. C., Magnetic moment experiments, Nuclear Physics Divi- sion, CERN, 1967 10.5170/CERN-1964-013-V-4.97 [116] R.H. Dalitz. The τ − θ puzzle, Oxford U. Jun, 1994. doi:10.1063/1.45424. [117] Scott Willenbrock. Symmetries of the Standard Model. February 2, 2008. http://arxiv.org/abs/hep-ph/0410370v2 [118] Martin L. Perl. The Discovery of the Tau Lepton. Stanford Linear Accelerator Center, SLAC-PUB-5937 September 1992. [119] J.-P. Martin, C. Mercier, N. Starinski, C.J. Pearson, and P.-A. Amau- druz. The TIGRESS DAQ/Trigger System. IEEE Transactions on Nu- clear Science, 55(1):84 –90, 2008. [120] T. Numao. Pienu technote 07: Effects of positrons in the beam, PIENU Technical Report, 2008. 192 Bibliography [121] Alexis A. Aguilar-Arevalo, Ahmed Hussein and Chloé Malbrunot. TN08 : Testing and Setting Up BINA Using Gamma Ray Calibration Sources. Technical report, PIENU internal documents 2008. [122] Alexis A. Aguilar-Arevalo. TN10 : Stand-Alone Setup and Testing of the Xe-Flash Tube Monitoring System. Technical report, PIENU internal documents 2008. [123] L. Doria. TN11 : Installation of the csi calorimeter for the pienu ex- periment. Technical report, PIENU internal documents2009. [124] L. Doria. A tracking algorithm for the PIENU experiment. Technical report, PIENU internal documents, 2009. [125] S. Cuen-Rochin. TN37: T1 single-multi pulse correction. Technical report, TRIUMF-PIENU’s internal documents, 2014. [126] S. Cuen-Rochin. TN41: Master Technote for 2010 Analysis. Technical report, TRIUMF-PIENU’s internal documents, 2015. [127] S. Ito, T. Numao, TN44: Muon Leakage. Technical report, PIENU internal documents, 2015. [128] R. Mischke, TN58: Pion cut studies and systematic uncertainty. Tech- nical report, PIENU internal documents, 2018. [129] R. Mischke. TN60: Beam Vector Files. Technical report, PIENU in- ternal documents, 2021. [130] I. Hernandez. TN63: Improved gain-shift correction for the calibration of the BINA’s PMT array. Technical report, PIENU internal documents, 2021. [131] I. Hernandez. TN66: Silicon Strip Clustering. Technical report, PIENU internal documents, 2023. [132] I. Hernandez. TN67: µDIF correction . Technical report, PIENU in- ternal documents, 2023. [133] R. Mischke. TN68: Pion cut studies and systematic uncertainty . Tech- nical report, PIENU internal documents, 2024. [134] S. Ito. The Table of Muon Decay In Flight Correction. Technical re- port, PIENU internal documents, September, 2014. 193 Bibliography [135] S. Ito. The Latest Table of Muon Decay In Flight Correction. Technical report, PIENU internal documents, January, 2015. [136] R. Mischke. Formulae to combine years. Technical report, PIENU in- ternal documents, 2018. [137] I. Hernandez. Presentation: Parts 22-25: 2010preNov, July 2023. Available from: https://pienu.triumf.ca/InternalDocuments/. [138] I. Hernandez. Presentation: Systematic Checks up- date: 2010preNov, September 2023. Available from: https://pienu.triumf.ca/InternalDocuments/. [139] L. Doria. Presentation: MuHit cut study, July 2014. Available from: https://pienu.triumf.ca/InternalDocuments/. [140] I. Hernandez. Presentation: Error budget (pt. 2), May 2024. Available from: https://pienu.triumf.ca/InternalDocuments/. [141] S. Ritt and P. A. Amaudruz. The midas data acquisition system. [142] PIENU webpage. https://pienu.triumf.ca/. Accessed: 2018. [143] C.J. Oram, J.B. Warren, G.M. Marshall, and J. Doornbos. Commis- sioning of a new low energy π−µ at TRIUMF. Nuclear Instruments and Methods, 179(1):95-103, 1981. doi:10.1016/0029-554X(81)91166-6. [144] Martin R. Kraimer. EPICS Application Developers Guide, 1998. http://www.aps.anl.gov/epics/base/R3-13.php. [145] R. Brun and F. Rademakers. ROOT - An Object Oriented Data Anal- ysis Framework. Nucl. Inst. & Meth. in Phys. Res. A, 389:81–86, 1997. [146] TWIST experiment at TRIUMF. Available from: http://twist.triumf.ca/ e614/experiment.html. [147] J. Birks. Theory and Practice of Scintillation Counting. Proc. Phys. Soc., A64:874, 1951. [148] NADS. Cry simulation package for cosmics, May 2010. Available from: http://nuclear.llnl.gov/simulation/. [149] F. James. Minuit function minimization and error analysis. Technical report, CERN Program Library Entry D 506. 194 [150] Geant4 Simulation toolkit. Geant4 10.6 Release Notes, https://geant4.web.cern.ch/download/release-notes /notes-v10.6.0.html [151] T. J. Roberts and D. M. Kaplan. G4beamline simulation program- for matter-dominated beamlines. IEEE Particle Accelerator Conference (PAC), pages 3468–3470, 2007. [152] Makoto Asai, et al. The PENELOPE Physics Models and Trans- port Mechanics. Implementation into Geant4. Frontiers in Physics, 2021. DOI: https://doi.org/10.3389/fphy.2021.738735. [153] J. Sempau, et al. Experimental benchmarks of the Monte Carlo code PENELOPE. Nucl. Instrum. Methods Phys. Res., Sect. B, doi:10.1016/S0168-583X(03)00453-1, 2003. [154] INTERNATIONAL ATOMIC ENERGY AGENCY, Handbook on Photonuclear Data for Applications Cross-sections and Spectra, IAEA- TECDOC-1178, IAEA, Vienna (2000). [155] Geant4 Collaboration, Geant4: Guide for Physics List, Re- lease 10.6, https://geant4-userdoc.web.cern.ch/UsersGuides/ PhysicsListGuide/BackupVersions/V10.6/html/index.html [156] Richard A. Johnson and Dean W. Wichern. Applied Multivariate Sta- tistical Analysis. Pearson, 2007 [157] Burgess Nicholas. Correlated Monte Carlo Simulation using Cholesky Decomposition. Said Business School, Oxford University. https://ssrn.com/abstract=4066115. [158] L. Lyons. Practical Statistics. Imperial College, London, UK, 2017. https://doi.org/10.48550/arXiv.1708.01007. [159] R. Barlow. Systematic errors: Facts and fictions. In Advanced Statis- tical Techniques in Particle Physics. Proceedings, Conference, Durham, UK, March 18-22, 2002, pages 134–144, 2002. [160] G. J. Feldman and R. D. Cousins, Unified approach to the classical statistical analysis of small signals. Phys. Rev. D, 57:3873-3889, 1998. Appendix A Trigger complete diagram Figure A.1: Complete diagram of the physics trigger in the PIENU experi- ment. Taken from [2]. 196